[ { "id": 1, "problem_number": "MPP-001", "title": "P versus NP Problem", "statement": "Does $P = NP$? More formally: if the solution to a problem can be quickly verified (in polynomial time), can the solution also be quickly found (in polynomial time)?", "background": "The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be quickly solved. The Clay Mathematics Institute has offered a $1,000,000 prize for a correct solution.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Clay continues to list P versus NP as unsolved.\n\n**Verified partial progress.**\n\n- Broad lower-bound, circuit-complexity, proof-complexity, and restricted-model progress does not settle P versus NP.\n\n**Full solution or refutation.**\n\nNo proof of P=NP or P≠NP was verified.\n\n**What remains.**\n\nEstablish a superpolynomial lower bound for an NP-complete problem or an algorithmic collapse.\n\n**Sources checked.**\n\n- Clay Mathematics Institute, The Millennium Prize Problems (accessed 2026-08-17). (maintained_tracker): https://www.claymath.org/millennium-problems/\n Evidence used: Clay lists P vs NP among its unsolved Millennium problems.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "Stephen Cook", "proposed_year": 1971, "category_id": 15, "set_id": 1, "view_count": 1523, "favorite_count": 89, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2, "problem_number": "MPP-002", "title": "The Riemann Hypothesis", "statement": "Do all non-trivial zeros of the Riemann zeta function $\\zeta(s)$ have real part equal to $\\frac{1}{2}$?", "background": "The Riemann hypothesis, proposed by Bernhard Riemann in 1859, concerns the distribution of prime numbers. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line $\\Re(s) = \\frac{1}{2}$. This is one of the most important open problems in mathematics, with profound implications for number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Clay continues to list the Riemann Hypothesis as unsolved.\n\n**Verified partial progress.**\n\n- Its zero-line assertion has extensive finite-height verification and many conditional consequences.\n\n**Full solution or refutation.**\n\nNo proof or counterexample was verified.\n\n**What remains.**\n\nControl all nontrivial zeta zeros.\n\n**Sources checked.**\n\n- Clay Mathematics Institute, The Millennium Prize Problems (accessed 2026-08-17). (maintained_tracker): https://www.claymath.org/millennium-problems/\n Evidence used: Clay lists the Riemann Hypothesis as unsolved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "Bernhard Riemann", "proposed_year": 1859, "category_id": 1, "set_id": 1, "view_count": 2341, "favorite_count": 156, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3, "problem_number": "MPP-003", "title": "Yang–Mills Existence and Mass Gap", "statement": "Prove that Yang–Mills theory exists and has a mass gap on $\\mathbb{R}^4$, meaning the quantum particles have positive masses.", "background": "This problem concerns quantum field theory and seeks to establish a rigorous mathematical foundation for Yang–Mills theories, which describe fundamental forces in particle physics. A solution would require proving the existence of these theories in four-dimensional spacetime and showing they predict a mass gap.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Clay continues to list Yang--Mills existence and mass gap as unsolved.\n\n**Verified partial progress.**\n\n- Constructive and lattice approaches provide important partial mathematical/physical evidence.\n\n**Full solution or refutation.**\n\nNo qualifying construction and mass-gap proof was verified.\n\n**What remains.**\n\nConstruct four-dimensional quantum Yang--Mills satisfying the axioms and prove a positive gap.\n\n**Sources checked.**\n\n- Clay Mathematics Institute, The Millennium Prize Problems (accessed 2026-08-17). (maintained_tracker): https://www.claymath.org/millennium-problems/\n Evidence used: Clay lists Yang--Mills and the mass gap as unsolved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "Yang Chen-Ning and Robert Mills", "proposed_year": 1954, "category_id": 16, "set_id": 1, "view_count": 1234, "favorite_count": 78, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 4, "problem_number": "MPP-004", "title": "Navier–Stokes Existence and Smoothness", "statement": "Prove or give a counterexample: Do solutions to the Navier–Stokes equations in three dimensions always exist and remain smooth for all time?", "background": "The Navier–Stokes equations describe the motion of fluids. While solutions exist for short times and in two dimensions, the question of whether smooth solutions exist globally in three dimensions remains open. This has profound implications for understanding turbulence and fluid dynamics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Clay continues to list three-dimensional Navier--Stokes existence and smoothness as unsolved.\n\n**Verified partial progress.**\n\n- There are global weak solutions and regularity results under additional hypotheses.\n\n**Full solution or refutation.**\n\nNo global smoothness theorem or finite-time blowup example was verified.\n\n**What remains.**\n\nProve regularity for all smooth data or produce a singular solution.\n\n**Sources checked.**\n\n- Clay Mathematics Institute, The Millennium Prize Problems (accessed 2026-08-17). (maintained_tracker): https://www.claymath.org/millennium-problems/\n Evidence used: Clay lists Navier--Stokes as unsolved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "Claude-Louis Navier and George Gabriel Stokes", "proposed_year": 1822, "category_id": 9, "set_id": 1, "view_count": 1456, "favorite_count": 89, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 5, "problem_number": "MPP-005", "title": "Birch and Swinnerton-Dyer Conjecture", "statement": "The conjecture relates the rank of the abelian group of rational points of an elliptic curve to the order of zero of the associated L-function at $s=1$.", "background": "This conjecture connects the arithmetic of elliptic curves (solutions to equations of the form $y^2 = x^3 + ax + b$) to the behavior of certain complex functions. It has deep connections to number theory and algebraic geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Clay continues to list the Birch--Swinnerton-Dyer conjecture as unsolved.\n\n**Verified partial progress.**\n\n- The conjecture is proved in important special cases and heavily supported computationally.\n\n**Full solution or refutation.**\n\nNo general rank/order-of-vanishing theorem was verified.\n\n**What remains.**\n\nRelate analytic rank and Mordell--Weil rank for arbitrary elliptic curves.\n\n**Sources checked.**\n\n- Clay Mathematics Institute, The Millennium Prize Problems (accessed 2026-08-17). (maintained_tracker): https://www.claymath.org/millennium-problems/\n Evidence used: Clay lists BSD as unsolved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "Bryan Birch and Peter Swinnerton-Dyer", "proposed_year": 1960, "category_id": 1, "set_id": 1, "view_count": 1123, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 6, "problem_number": "MPP-006", "title": "Hodge Conjecture", "statement": "On a projective non-singular algebraic variety over $\\mathbb{C}$, any Hodge class is a rational linear combination of classes of algebraic cycles.", "background": "The Hodge conjecture is a major open problem in algebraic geometry. It seeks to relate the topology of a smooth complex projective variety to its algebraic structure, specifically asserting that certain topological cycles are actually algebraic.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Clay continues to list the Hodge conjecture as unsolved.\n\n**Verified partial progress.**\n\n- The conjecture is known in special cases, including varieties of dimension below four according to Clay.\n\n**Full solution or refutation.**\n\nNo general algebraicity theorem for rational Hodge classes was verified.\n\n**What remains.**\n\nProve algebraicity in the remaining higher-dimensional cases.\n\n**Sources checked.**\n\n- Clay Mathematics Institute, The Millennium Prize Problems (accessed 2026-08-17). (maintained_tracker): https://www.claymath.org/millennium-problems/\n Evidence used: Clay lists Hodge as unsolved and notes special cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "William Vallance Douglas Hodge", "proposed_year": 1950, "category_id": 5, "set_id": 1, "view_count": 987, "favorite_count": 54, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 1, "name": "millennium_prize", "display_name": "Millennium Prize Problems", "description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.", "slug": "millennium-prize", "order_index": 1, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 8, "problem_number": "NT-001", "title": "Odd Perfect Numbers", "statement": "Does there exist an odd perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). For example, $6 = 1 + 2 + 3$ is perfect.", "background": "While many even perfect numbers are known (the first few are 6, 28, 496, 8128), no odd perfect number has ever been found, despite extensive computer searches. It has been proven that if one exists, it must be greater than $10^{1500}$ and have at least 101 prime factors.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether any odd perfect number exists remains open; strong lower bounds and factorization constraints do not decide existence.\n\n**Verified partial progress.**\n\n- Ochem and Rao proved that an odd perfect number must exceed 10^1500 and have at least 101 prime factors counted with multiplicity.\n- Nielsen proved that an odd perfect number must have at least ten distinct prime divisors.\n\n**Full solution or refutation.**\n\nNo accepted construction or nonexistence proof was found.\n\n**What remains.**\n\nConstruct an odd perfect number or prove that every perfect number is even.\n\n**Sources checked.**\n\n- Pascal Ochem and Michael Rao, Odd perfect numbers are greater than 10^1500, Mathematics of Computation 81 (2012), 1869-1877, DOI 10.1090/S0025-5718-2012-02563-4. (primary): https://www.ams.org/journals/mcom/2012-81-279/S0025-5718-2012-02563-4/S0025-5718-2012-02563-4.pdf\n Evidence used: Proves the 10^1500 lower bound and the 101-total-prime-factor constraint for any hypothetical odd perfect number.\n- Pace P. Nielsen, Odd perfect numbers, Diophantine equations, and upper bounds, Mathematics of Computation 84 (2015), 2549-2567, DOI 10.1090/S0025-5718-2015-02941-X. (primary): https://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02941-X/\n Evidence used: Derives that any odd perfect number has at least ten distinct prime divisors and treats existence as unresolved.\n\n**Review notes.** The source statement was preserved. The background's 101 prime factors means counted with multiplicity, not 101 distinct primes.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 543, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 9, "problem_number": "NT-002", "title": "Collatz Conjecture", "statement": "Starting with any positive integer $n$, repeatedly apply the function: if $n$ is even, divide by 2; if $n$ is odd, multiply by 3 and add 1. Does this process always eventually reach 1?", "background": "Also known as the 3n+1 problem, this deceptively simple conjecture has been verified for all starting values up to $2^{68}$ but remains unproven. Paul Erdős said about it: \"Mathematics may not be ready for such problems.\"\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Collatz conjecture remains open universally, but Tao proved a strong almost-all descent theorem and exhaustive verification now reaches 2^71.\n\n**Verified partial progress.**\n\n- Tao proved that for every function f(N) tending to infinity, the minimum of the Collatz orbit is at most f(N) for almost all N in logarithmic density.\n- Barina reported exhaustive convergence verification for every starting value below 2^71, updating the dataset's 2^68 bound.\n\n**Full solution or refutation.**\n\nNeither the density theorem nor finite verification proves convergence to 1 for every positive integer.\n\n**What remains.**\n\nProve convergence to 1 for all positive starting values or exhibit a divergent orbit or nontrivial cycle.\n\n**Sources checked.**\n\n- Terence Tao, Almost all orbits of the Collatz map attain almost bounded values, Forum of Mathematics, Pi 10 (2022), e12; arXiv:1909.03562. (primary): https://arxiv.org/abs/1909.03562\n Evidence used: Proves the almost-all logarithmic-density descent theorem while stating the universal Collatz assertion as a conjecture.\n- David Barina, Improved verification limit for the convergence of the Collatz conjecture, The Journal of Supercomputing 81 (2025), article 810, DOI 10.1007/s11227-025-07337-0. (primary): https://doi.org/10.1007/s11227-025-07337-0\n Evidence used: Calls the problem unsolved and reports exhaustive verification through 2^71.\n\n**Review notes.** The exact source statement was preserved; unverified recent complete-proof claims were excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 11, "problem_number": "NT-003", "title": "Twin Prime Conjecture", "statement": "Are there infinitely many twin primes? Twin primes are pairs of primes that differ by 2, such as (3, 5), (5, 7), (11, 13), (17, 19), (29, 31).", "background": "The twin prime conjecture is one of the oldest unsolved problems in number theory. In 2013, Yitang Zhang proved that there are infinitely many pairs of primes that differ by at most 70 million. This bound has since been reduced to 246, but the gap of 2 remains unproven.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The twin prime conjecture remains open; bounded-gap theorems do not prove that the fixed gap 2 occurs infinitely often.\n\n**Verified partial progress.**\n\n- D. H. J. Polymath proved H_1 <= 246 unconditionally and H_1 <= 6 under generalized Elliott-Halberstam.\n- Lott and Ponagandla obtained polynomial configurations among generalized prime pairs with some unspecified common gap b <= 246.\n\n**Full solution or refutation.**\n\nNo accepted proof or counterexample for infinitely many gap-2 prime pairs was found.\n\n**What remains.**\n\nProve that infinitely many primes p have p+2 prime, or disprove the assertion.\n\n**Sources checked.**\n\n- D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Research in the Mathematical Sciences 1 (2014), article 12; arXiv:1407.4897. (primary): https://arxiv.org/abs/1407.4897\n Evidence used: Gives the unconditional 246 and conditional 6 bounds, both short of the fixed gap 2.\n- Andrew Lott and Nagendar Reddy Ponagandla, Polynomial progressions in the generalized twin primes, arXiv:2505.17375v2 (2026). (primary): https://arxiv.org/abs/2505.17375\n Evidence used: Uses an unspecified recurring bounded gap b <= 246 and therefore does not settle twin primes.\n\n**Review notes.** Duplicate topic retained under its exact integer record ID.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 1234, "favorite_count": 89, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 12, "problem_number": "NT-004", "title": "Goldbach's Conjecture", "statement": "Every even integer greater than 2 can be expressed as the sum of two primes.", "background": "Proposed by Christian Goldbach in 1742, this conjecture has been verified computationally for all even integers up to very large numbers. The weak Goldbach conjecture (every odd number greater than 5 is the sum of three primes) was proved by Harald Helfgott in 2013, but the strong version remains open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The binary Goldbach conjecture remains open, despite Chen's prime-plus-semiprime theorem, Helfgott's ternary theorem, and verification through 4 x 10^18.\n\n**Verified partial progress.**\n\n- Chen proved that every sufficiently large even integer is a prime plus a number with at most two prime factors.\n- Helfgott proved that every odd integer greater than 5 is a sum of three primes.\n- Oliveira e Silva, Herzog, and Pardi verified the binary conjecture for every even integer up to 4 x 10^18.\n\n**Full solution or refutation.**\n\nNo theorem was found expressing every even integer greater than 2 as a sum of two primes.\n\n**What remains.**\n\nReplace the possible semiprime in Chen's theorem by a prime and cover every even integer, or find a counterexample.\n\n**Sources checked.**\n\n- AMS sieve-methods text summarizing Jing-Run Chen's 1973 theorem that every sufficiently large even integer is a prime plus a P2 number. (authoritative_secondary): https://www.ams.org/bookstore/pspdf/gsm-134-prev.pdf\n Evidence used: States Chen's closest unconditional prime-plus-almost-prime approximation to binary Goldbach.\n- Harald Andres Helfgott, The ternary Goldbach problem, arXiv:1501.05438. (primary): https://arxiv.org/abs/1501.05438\n Evidence used: Proves the weak/ternary Goldbach conjecture, a distinct three-prime statement.\n- Tomas Oliveira e Silva, Siegfried Herzog, and Silvio Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4 x 10^18, Mathematics of Computation 83 (2014), 2033-2060, DOI 10.1090/S0025-5718-2013-02787-1. (primary): https://doi.org/10.1090/S0025-5718-2013-02787-1\n Evidence used: Reports exhaustive verification of binary Goldbach up to 4 x 10^18.\n\n**Review notes.** The strong and weak Goldbach statements were kept distinct.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Christian Goldbach", "proposed_year": 1742, "category_id": 1, "view_count": 1567, "favorite_count": 112, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 13, "problem_number": "NT-005", "title": "ABC Conjecture", "statement": "For any $\\epsilon > 0$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers with $a + b = c$ such that $c > \\text{rad}(abc)^{1+\\epsilon}$, where $\\text{rad}(n)$ is the product of distinct prime factors of $n$.", "background": "The ABC conjecture, formulated by Joseph Oesterlé and David Masser in 1985, has profound implications for number theory. Shinichi Mochizuki claimed a proof in 2012 using his \"inter-universal Teichmüller theory,\" but the proof remains controversial and not widely accepted.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Mochizuki's IUT papers claiming an abc-type inequality are published, but Scholze and Stix identify a severe gap; the claim is not broadly verified.\n\n**Verified partial progress.**\n\n- The four IUT papers were published in PRIMS in 2021 and are presented by their author as proving an abc-type inequality.\n- Scholze and Stix's technical report concludes that the proposed proof has a severe problem not repairable by small modifications; the disagreement remains unresolved in the sources checked.\n\n**Full solution or refutation.**\n\nPublication of the IUT papers is verified, but a consensus-valid proof of abc cannot be recorded because the central expert objection remains unreconciled.\n\n**What remains.**\n\nResolve the Scholze-Stix objection with a checkable argument accepted by independent experts, or produce a different proof or counterexample.\n\n**Sources checked.**\n\n- Shinichi Mochizuki, Inter-universal Teichmuller Theory I-IV, Publications of the Research Institute for Mathematical Sciences 57 (2021), 3-723, DOI series 10.4171/PRIMS/57-1-1 through 10.4171/PRIMS/57-1-4. (primary): https://ems.press/journals/prims/issues/1507\n Evidence used: Official journal issue verifies publication of the four papers underlying the claimed proof.\n- Peter Scholze and Jakob Stix, Why abc is still a conjecture, technical report (2018). (primary): https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf\n Evidence used: Documents the authors' conclusion after direct discussions that the proposed proof has a severe gap.\n\n**Review notes.** The conservative corpus label is uncertain rather than solved; expert review may prefer the mainstream practical label open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "Joseph Oesterlé and David Masser", "proposed_year": 1985, "category_id": 1, "view_count": 876, "favorite_count": 45, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 10, "problem_number": "COMB-001", "title": "The Hadwiger-Nelson Problem", "statement": "What is the minimum number of colors needed to color the points of the plane such that no two points at distance 1 have the same color?", "background": "It is known that this chromatic number is between 5 and 7. In 2018, Aubrey de Grey proved it is at least 5, but whether it is 5, 6, or 7 remains unknown.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The chromatic number of the plane is known to lie in {5,6,7}, but its exact value is unknown.\n\n**Verified partial progress.**\n\n- de Grey constructed a finite unit-distance graph of chromatic number 5, raising the lower bound.\n- The classical 7-colouring gives the upper bound.\n\n**Full solution or refutation.**\n\nNo construction of a 5- or 6-colouring of the entire plane, nor a 6- or 7-chromatic lower-bound graph, was verified.\n\n**What remains.**\n\nDecide whether the plane can be coloured with 5 or 6 colours.\n\n**Sources checked.**\n\n- A. de Grey, The chromatic number of the plane is at least 5, arXiv:1804.02385 (2018). (primary): https://arxiv.org/abs/1804.02385\n Evidence used: Establishes a 5-chromatic unit-distance graph.\n- Wolfram MathWorld, Hadwiger-Nelson Problem (accessed 2026-08-17). (authoritative_secondary): https://mathworld.wolfram.com/Hadwiger-NelsonProblem.html\n Evidence used: Records the surviving 5,6,7 range.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 421, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 14, "problem_number": "GT-001", "title": "Hadwiger Conjecture", "statement": "Every graph with chromatic number $k$ has a $K_k$ minor (where $K_k$ is the complete graph on $k$ vertices).", "background": "The Hadwiger conjecture, proposed in 1943, generalizes the four color theorem. It has been proved for $k \\leq 6$ but remains open for $k \\geq 7$. The case $k=5$ is equivalent to the four color theorem.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hadwiger's conjecture remains open generally; verified cases extend through chromatic number 6.\n\n**Verified partial progress.**\n\n- The k=5 case is equivalent to the Four Color Theorem, and k=6 was established by Robertson--Seymour--Thomas.\n- Modern minor theory gives approximate chromatic bounds for K_t-minor-free graphs.\n\n**Full solution or refutation.**\n\nRecent unreviewed general proof claims were not accepted as resolutions.\n\n**What remains.**\n\nProve that every k-chromatic graph has a K_k minor for all k.\n\n**Sources checked.**\n\n- Hadwiger Conjecture, Wolfram MathWorld. (authoritative_secondary): https://mathworld.wolfram.com/HadwigerConjecture.html\n Evidence used: Records established low-k cases and continuing partial work.\n\n**Review notes.** No source alteration; unverified recent proof claims excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Hugo Hadwiger", "proposed_year": 1943, "category_id": 3, "view_count": 654, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 15, "problem_number": "GT-002", "title": "Reconstruction Conjecture", "statement": "Every finite simple graph on at least 3 vertices is uniquely determined by its vertex-deleted subgraphs.", "background": "The reconstruction conjecture asks whether a graph can be uniquely reconstructed from the multiset of all its vertex-deleted subgraphs. Proposed by Stanisław Ulam in 1942, it has been verified for many classes of graphs but remains open in general.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Kelly--Ulam reconstruction conjecture remains open for arbitrary finite simple graphs with at least three vertices.\n\n**Verified partial progress.**\n\n- Many graph classes and reconstructible graph parameters are known.\n\n**Full solution or refutation.**\n\nNo verified proof of the general deck-reconstruction statement was found; recent AI-related claims concern variants or are not yet accepted as a general theorem.\n\n**What remains.**\n\nShow that every graph is determined by its vertex-deleted deck or give a counterexample.\n\n**Sources checked.**\n\n- Reconstructing graphs with subgraph compositions, Discrete Applied Mathematics 390 (2026), 202--221. (primary): https://doi.org/10.1016/j.dam.2026.04.023\n Evidence used: Explicitly calls the Kelly--Ulam reconstruction conjecture still open.\n\n**Review notes.** No source alteration; recent online claims not treated as verified.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_by": "Stanisław Ulam", "proposed_year": 1942, "category_id": 3, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 18, "problem_number": "TOP-001", "title": "Smooth 4-Dimensional Poincaré Conjecture", "statement": "Is every smooth homotopy 4-sphere diffeomorphic to the standard 4-sphere $S^4$?", "background": "The smooth Poincaré conjecture in dimension 4 is the only remaining case of the generalized Poincaré conjecture. It has been solved in all other dimensions: dimension 3 by Perelman, higher dimensions by Smale, Freedman, and others. The 4-dimensional case is particularly difficult due to exotic smooth structures.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The smooth four-dimensional Poincare conjecture remains open; Freedman's theorem gives a homeomorphism to S^4 but does not prove that the smooth structure is standard.\n\n**Verified partial progress.**\n\n- Freedman proved the topological four-dimensional Poincare theorem, so every smooth homotopy 4-sphere is homeomorphic to S^4.\n- No accepted proof that every such homeomorphism type has the standard smooth structure, and no accepted exotic smooth 4-sphere, was verified.\n\n**Full solution or refutation.**\n\nTopological classification does not settle diffeomorphism classification in dimension four.\n\n**What remains.**\n\nProve every smooth homotopy 4-sphere is diffeomorphic to S^4 or construct a genuine exotic smooth 4-sphere.\n\n**Sources checked.**\n\n- Michael H. Freedman, The topology of four-dimensional manifolds, Journal of Differential Geometry 17 (1982), 357-453. (primary): https://doi.org/10.4310/jdg/1214437136\n Evidence used: Proves the topological dimension-four theorem, leaving the smooth-structure question untouched.\n- TheoremDB, Smooth 4-dimensional Poincare conjecture (accessed 2026-08-17). (maintained_tracker): https://www.theorems.org/smooth-4-dimensional-poincare-conjecture/\n Evidence used: Maintains the exact smooth question as open.\n\n**Review notes.** The background conflates Freedman's topological four-dimensional result with higher-dimensional smooth results and overlooks exotic spheres in higher dimensions; the exact question itself is intact.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 789, "favorite_count": 42, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 19, "problem_number": "GEO-002", "title": "Sphere Packing in Higher Dimensions", "statement": "What is the densest packing of congruent spheres in $n$ dimensions for $n \\geq 4$?", "background": "The sphere packing problem asks for the densest arrangement of non-overlapping spheres. Maryna Viazovska solved it for dimension 8 in 2016, and she with collaborators solved it for dimension 24 in 2017. The problem remains open for most other dimensions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This duplicate higher-dimensional sphere-packing question remains open outside dimensions 8 and 24 (and the low-dimensional classical cases).\n\n**Verified partial progress.**\n\n- E8 and Leech are proven optimal.\n\n**Full solution or refutation.**\n\nNo all-n density formula is known.\n\n**What remains.**\n\nResolve the remaining dimensions.\n\n**Sources checked.**\n\n- M. Viazovska, The sphere packing problem in dimension 8, Ann. Math. 185 (2017). (primary): https://arxiv.org/abs/1603.04246\n Evidence used: Proves E8 optimality.\n\n**Review notes.** Duplicate topic retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 456, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 20, "problem_number": "ALG-001", "title": "Inverse Galois Problem", "statement": "Is every finite group the Galois group of some Galois extension of the rational numbers $\\mathbb{Q}$?", "background": "The inverse Galois problem asks whether every finite group can be realized as the Galois group of a polynomial equation with rational coefficients. It has been solved for many classes of groups, including all symmetric and alternating groups, but remains open in general.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The inverse Galois problem over the rational numbers remains open in general, despite a very recent realization of the last missing sporadic simple group.\n\n**Verified partial progress.**\n\n- Shafarevich's theorem realizes every finite solvable group over the rational numbers, and many nonsolvable families are also known.\n- A preprint submitted on 2026-08-09 proves that the Mathieu group M_23 occurs over Q, completing all 26 sporadic finite simple groups and all transitive groups of degree at most 23.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample is known. The M_23 result is a major new family-completion result but does not cover arbitrary finite groups.\n\n**What remains.**\n\nRealize every finite group as a Galois group over Q, or prove that some finite group cannot occur.\n\n**Sources checked.**\n\n- Michel Brion and Stefan Schroer, The Inverse Galois Problem for Connected Algebraic Groups, author-hosted manuscript, Introduction. (primary): https://www-fourier.univ-grenoble-alpes.fr/~mbrion/InverseGaloisProblem.pdf\n Evidence used: States the classical finite-group realization problem over number fields and explicitly says that the general case remains open.\n- Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang, The Mathieu group M_23 is a Galois group over Q, arXiv:2608.08538 (2026). (primary): https://arxiv.org/abs/2608.08538\n Evidence used: Proves that M_23 occurs over Q and gives an explicit degree-23 polynomial, completing the sporadic simple groups.\n\n**Review notes.** Exact statement preserved. The M_23 paper was submitted eight days before the check and deserves ordinary preprint verification, but it does not alter the open general status.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 543, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 21, "problem_number": "ALG-002", "title": "Kaplansky's Conjectures", "statement": "A set of conjectures about group rings: (1) Zero divisor conjecture: If $G$ is a torsion-free group and $K$ is a field, then $K[G]$ has no zero divisors. (2) Idempotent conjecture: The only idempotents in $K[G]$ are 0 and 1. (3) Unit conjecture: The only units in $\\mathbb{Z}[G]$ are of the form $\\pm g$ for $g \\in G$.", "background": "These conjectures, proposed by Irving Kaplansky in the 1940s, concern the algebraic structure of group rings. They have been verified for many classes of groups but remain open in general. The zero divisor conjecture is related to the Atiyah conjecture.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** All three assertions as literally written remain open: the zero-divisor and idempotent conjectures over fields and Higman's integral unit conjecture over Z.\n\n**Verified partial progress.**\n\n- The more common field-valued unit conjecture is false: Gardam gave a counterexample over F_2 and later nontrivial units over C.\n- The integral unit statement U(Z[G])={+/-g} is narrower and remains open; positive results are known for important classes such as unique-product groups.\n- The zero-divisor and idempotent conjectures remain open for arbitrary torsion-free groups, though they hold for many large classes.\n\n**Full solution or refutation.**\n\nThe known field counterexamples do not refute the exact integral unit statement in this record, and no general solution of its zero-divisor or idempotent components was found.\n\n**What remains.**\n\nResolve the zero-divisor and idempotent conjectures for all torsion-free groups and prove or refute the integral unit conjecture.\n\n**Sources checked.**\n\n- Giles Gardam, A counterexample to the unit conjecture for group rings, Annals of Mathematics 194 (2021), 967-979. (primary): https://annals.math.princeton.edu/2021/194-3/p09\n Evidence used: Disproves the field-valued unit conjecture over F_2 with a torsion-free virtually abelian group.\n- Giles Gardam, Non-trivial units of complex group rings, arXiv:2312.05240 (2023). (primary): https://arxiv.org/abs/2312.05240\n Evidence used: Disproves the field-valued unit conjecture in characteristic zero, but not over Z.\n- Andre Nies, The trivial units property and the unique product property, AMS special-session lecture slides, 12 December 2024. (authoritative_secondary): https://www.cs.auckland.ac.nz/~nies/talks/2024/Nies_AMS_unit%20conjecture_groups.pdf\n Evidence used: Explicitly distinguishes the conjecture refuted over fields in every characteristic from Higman's original R=Z conjecture, which remains open.\n- Johan Oinert, Units, zero-divisors and idempotents in rings graded by torsion-free groups, Journal of Group Theory 27 (2024), 789-811. (primary): https://www.degruyter.com/document/doi/10.1515/jgth-2023-0110/html\n Evidence used: Records the zero-divisor and idempotent problems as open and proves restricted graded-ring results.\n\n**Review notes.** Exact Z[G] wording preserved. It must not be conflated with the now-refuted unit conjecture for K[G] over an arbitrary field.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Irving Kaplansky", "proposed_year": 1940, "category_id": 4, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 22, "problem_number": "SET-001", "title": "Continuum Hypothesis", "statement": "There is no set whose cardinality is strictly between that of the integers and the real numbers.", "background": "The continuum hypothesis was the first of Hilbert's 23 problems. Kurt Gödel (1940) and Paul Cohen (1963) proved it is independent of ZFC set theory: it can neither be proved nor disproved from the standard axioms. Whether to accept it as an axiom remains a philosophical question.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The Continuum Hypothesis assertion is independent of ZFC, assuming ZFC is consistent: it holds in Godel's constructible universe and fails in Cohen forcing models.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nGodel proved the relative consistency of GCH, hence CH, and Cohen proved the relative consistency of not-CH. The historical ZFC decision problem is therefore settled by independence, not by proving the displayed assertion true or false.\n\n**What remains.**\n\nNo ZFC proof of either side can exist under the usual consistency assumption; selecting additional axioms capable of deciding CH is a separate foundational program.\n\n**Sources checked.**\n\n- Kurt Godel, The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis, Proceedings of the National Academy of Sciences 24 (1938), 556-557. (primary): https://doi.org/10.1073/pnas.24.12.556\n Evidence used: Establishes the constructible-universe relative consistency of AC and GCH, supplying the CH side.\n- Paul J. Cohen, The Independence of the Continuum Hypothesis, Proceedings of the National Academy of Sciences 50 (1963), 1143-1148. (primary): https://doi.org/10.1073/pnas.50.6.1143\n Evidence used: Uses forcing to establish the nonprovability of CH and supplies the not-CH relative-consistency direction.\n\n**Review notes.** Exact assertion retained. SET-001 is reused by record 1135, whose question has the opposite polarity. The generic solved label means solved-as-independent, not proved true.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "Georg Cantor", "proposed_year": 1878, "category_id": 10, "view_count": 1234, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 25, "problem_number": "NT-006", "title": "Legendre's Conjecture", "statement": "For every positive integer $n$, there exists a prime number between $n^2$ and $(n+1)^2$.", "background": "This conjecture about the distribution of prime numbers was proposed by Adrien-Marie Legendre in 1808. Despite significant progress in prime number theory, including the prime number theorem, this simple statement remains unproven.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Legendre's conjecture remains open; known short-interval results are still too weak to force a prime between every pair of consecutive squares.\n\n**Verified partial progress.**\n\n- Baker, Harman, and Pintz proved that [x-x^0.525,x] contains a prime for all sufficiently large x.\n- Chamberland and Straub proved conditional weaker results between consecutive powers x^(2+delta) and (x+1)^(2+delta), but not the exponent-2 case.\n\n**Full solution or refutation.**\n\nNo proof covering every interval (n^2,(n+1)^2) was found.\n\n**What remains.**\n\nProve the exponent-2 interval assertion for every positive integer n or exhibit a prime-free interval between consecutive squares.\n\n**Sources checked.**\n\n- R. C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes, II, Proceedings of the London Mathematical Society 83 (2001), 532-562. (primary): https://pure.royalholloway.ac.uk/en/publications/on-the-difference-between-consecutive-primes-ii/\n Evidence used: Proves the 0.525 short-interval theorem, whose exponent remains above the square-root scale needed here.\n- Marc Chamberland and Armin Straub, Weakening the Legendre Conjecture, arXiv:2602.22502 (2026), to appear in American Mathematical Monthly. (primary): https://arxiv.org/abs/2602.22502\n Evidence used: Treats Legendre's assertion as conjectural and proves RH-conditional results only for larger powers 2+delta.\n\n**Review notes.** Unreviewed online complete-proof claims were not accepted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_by": "Adrien-Marie Legendre", "proposed_year": 1808, "category_id": 1, "view_count": 432, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 26, "problem_number": "NT-007", "title": "Are there infinitely many Mersenne primes?", "statement": "Are there infinitely many prime numbers of the form $M_p = 2^p - 1$ where $p$ is prime?", "background": "Mersenne primes are primes of the form $2^p - 1$. As of 2024, only 51 Mersenne primes are known, with the largest being $2^{82,589,933} - 1$. It is conjectured that infinitely many exist, but this remains unproven. They are important for computational number theory and the GIMPS distributed computing project.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether infinitely many Mersenne primes exist; 52 are currently known.\n\n**Verified partial progress.**\n\n- GIMPS discovered and independently confirmed 2^136279841-1 in October 2024, the 52nd known Mersenne prime and current largest known prime.\n- Ongoing exhaustive testing expands the finite known list but cannot establish infinitude.\n\n**Full solution or refutation.**\n\nNo accepted infinitude or finiteness proof was found.\n\n**What remains.**\n\nProve that 2^p-1 is prime for infinitely many prime exponents p, or prove only finitely many such exponents exist.\n\n**Sources checked.**\n\n- Great Internet Mersenne Prime Search, official current progress and discovery record. (maintained_tracker): https://www.mersenne.org/\n Evidence used: Reports 52 known Mersenne primes and identifies 2^136279841-1 as the newest and largest, correcting the dataset background.\n- Carl Pomerance, Cyclotomic primes, Journal of Number Theory 276 (2025), 198-208, DOI 10.1016/j.jnt.2025.02.013. (primary): https://doi.org/10.1016/j.jnt.2025.02.013\n Evidence used: Describes infinitude of Mersenne primes as widely believed rather than proved.\n\n**Review notes.** The background is outdated: there are 52 known examples, and 2^82589933-1 is no longer the largest.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 654, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 27, "problem_number": "NT-008", "title": "Are there infinitely many perfect powers in the Fibonacci sequence?", "statement": "Besides 1, 8, and 144, are there any other perfect powers (numbers of the form $a^b$ where $a, b > 1$) in the Fibonacci sequence?", "background": "The Fibonacci sequence has only three known perfect powers: $F_1 = F_2 = 1 = 1^n$, $F_6 = 8 = 2^3$, and $F_{12} = 144 = 12^2$. It is conjectured that these are the only ones, but this remains unproven despite extensive computational searches.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Bugeaud, Mignotte, and Siksek proved that the only Fibonacci perfect powers are 0, 1, 8, and 144, so there are no further positive examples beyond those named.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe 2006 Annals paper gives the complete classification: the only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144.\n\n**What remains.**\n\nNothing remains for the intended classification question; only the dataset's title/definition inconsistency should be corrected editorially.\n\n**Sources checked.**\n\n- Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Annals of Mathematics 163 (2006), 969-1018, DOI 10.4007/annals.2006.163.969. (primary): https://annals.math.princeton.edu/2006/163-3/p05\n Evidence used: The abstract explicitly states the complete Fibonacci perfect-power classification 0, 1, 8, and 144.\n\n**Review notes.** The title asks about infinitude while the statement asks for classification; moreover 1 conflicts with the stated requirement a,b>1. The theorem still settles the intended question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 345, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 28, "problem_number": "NT-009", "title": "Gilbreath's Conjecture", "statement": "Starting with the sequence of primes and repeatedly taking absolute differences of consecutive terms, the first term of each row is always 1.", "background": "Norman Gilbreath observed in 1958 that applying the forward difference operator to the sequence of primes appears to always yield 1 as the first element. Despite being verified computationally for the first $10^{13}$ primes, no proof exists.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Gilbreath's conjecture for the actual prime sequence remains open; random-model theorems and finite verification do not prove it.\n\n**Verified partial progress.**\n\n- Chase proved a random analogue of the conjecture and emphasized that this does not prove the prime-sequence statement.\n- Plouffe reported computational verification through primes at most 10^14, extending the 10^13 bound cited in the dataset.\n\n**Full solution or refutation.**\n\nNo proof was found that every iterated absolute-difference row for the infinite prime sequence begins with 1.\n\n**What remains.**\n\nProve the deterministic prime-sequence assertion for all iterations or find a row whose first entry is not 1.\n\n**Sources checked.**\n\n- Zachary Chase, A random analogue of Gilbreath's conjecture, Mathematische Annalen 388 (2024), 2611-2625, DOI 10.1007/s00208-023-02579-w. (primary): https://link.springer.com/article/10.1007/s00208-023-02579-w\n Evidence used: Defines the conjecture, records the earlier 10^13 verification, and proves only a random analogue while stating the prime case is unproved.\n- Simon Plouffe, Verification of Gilbraith's conjecture up to 10^14, arXiv:2510.06688 (2025). (primary): https://arxiv.org/abs/2510.06688\n Evidence used: Reports the updated finite computational verification to 10^14; title spelling is preserved from the preprint.\n\n**Review notes.** A live computation page claiming a still larger 2026 bound was not needed for classification and was not used as the principal evidence.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_by": "Norman Gilbreath", "proposed_year": 1958, "category_id": 1, "view_count": 287, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 29, "problem_number": "COMB-003", "title": "Ramsey Number R(5,5)", "statement": "What is the exact value of $R(5,5)$, the smallest number $n$ such that any 2-coloring of the edges of $K_n$ contains a monochromatic $K_5$?", "background": "Ramsey theory asks how large a structure must be to guarantee a certain property. The Ramsey number $R(5,5)$ is known to lie between 43 and 48, but the exact value remains unknown. As Joel Spencer said, \"Erdős asks us to imagine an alien force, demanding the value of $R(5,5)$ or they will destroy our planet... our best strategy is to get our best computers and mathematicians working on it.\"\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact Ramsey number R(5,5) is unknown; current verified bounds are 43 <= R(5,5) <= 46.\n\n**Verified partial progress.**\n\n- The lower bound 43 has long been known by construction.\n- Angeltveit--McKay's 2024 computation improved the upper bound to 46.\n\n**Full solution or refutation.**\n\nThe remaining possibilities 43,44,45,46 are not eliminated.\n\n**What remains.**\n\nConstruct a (5,5)-Ramsey graph on 43--45 vertices or rule out each such order.\n\n**Sources checked.**\n\n- V. Angeltveit and B. D. McKay, R(5,5) <= 46, arXiv:2409.15709 (2024). (primary): https://arxiv.org/abs/2409.15709\n Evidence used: The paper establishes the 46 upper bound.\n- S. Radziszowski, Small Ramsey Numbers, Dynamic Survey DS1, Electronic Journal of Combinatorics (updated 2026). (authoritative_secondary): https://www.cs.rit.edu/~spr/ElJC/ejcram18.pdf\n Evidence used: The maintained survey records the current small-Ramsey bounds.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 543, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 30, "problem_number": "COMB-004", "title": "The Lonely Runner Conjecture", "statement": "For any $n$ runners on a circular track with distinct constant speeds, each runner is \"lonely\" (distance at least $1/n$ from all others) at some time.", "background": "This combinatorial conjecture, proposed by J.M. Wills in 1967, has been verified for up to 7 runners but remains open for 8 or more. It has connections to Diophantine approximation and view-obstruction problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general Lonely Runner Conjecture remains open; finite runner counts through 12 have recent computer-assisted coverage.\n\n**Verified partial progress.**\n\n- Recent arXiv work proves progressively the cases through 12 runners.\n\n**Full solution or refutation.**\n\nNo uniform theorem in n was verified.\n\n**What remains.**\n\nProve the conjecture for every number of runners.\n\n**Sources checked.**\n\n- T. Sungkawichai and T. Trakulthongchai, Eleven, twelve, and thirteen lonely runners, arXiv:2604.23906 (2026). (primary): https://arxiv.org/abs/2604.23906\n Evidence used: The abstract reports computer-assisted proofs for 10, 11, and 12 runners.\n\n**Review notes.** No source alteration; runner-count conventions are retained as in the source.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_by": "J.M. Wills", "proposed_year": 1967, "category_id": 2, "view_count": 234, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 31, "problem_number": "GT-003", "title": "The Graceful Tree Conjecture", "statement": "Every tree can be gracefully labeled: vertices can be assigned distinct labels from $\\{0, 1, \\ldots, |E|\\}$ such that edge labels (absolute differences) are all distinct.", "background": "The graceful labeling conjecture, proposed by Alexander Rosa in 1967, asks whether every tree admits a graceful labeling. It has been verified for many classes of trees including paths, caterpillars, and trees with at most 35 vertices, but remains open in general.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Graceful Tree Conjecture remains open.\n\n**Verified partial progress.**\n\n- Many infinite tree families are graceful, but no theorem covers every tree.\n\n**Full solution or refutation.**\n\nA recent claimed complete proof remains unpublished and unverified.\n\n**What remains.**\n\nProve every tree is graceful or find a counterexample.\n\n**Sources checked.**\n\n- Graceful Tree Conjecture, Graph-theory open problems tracker. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/graceful_tree_conjecture/\n Evidence used: Reports that the 2025 dynamic survey continues to list the conjecture as open and does not accept a circulated proof claim.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander Rosa", "proposed_year": 1967, "category_id": 3, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 32, "problem_number": "GEO-003", "title": "The Kakeya Conjecture", "statement": "A Kakeya set (containing a unit line segment in every direction) in $\\mathbb{R}^n$ must have Hausdorff dimension $n$.", "background": "The Kakeya conjecture concerns the minimal \"size\" of sets containing line segments in all directions. It has deep connections to harmonic analysis and PDE. The conjecture is known in dimension 2 but remains open for $n \\geq 3$. It would have important implications for the restriction conjecture in Fourier analysis.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Kakeya conjecture is proved in R3 but remains open in dimensions n >= 4.\n\n**Verified partial progress.**\n\n- Wang--Zahl proved the three-dimensional Hausdorff-dimension conjecture.\n\n**Full solution or refutation.**\n\nNo higher-dimensional proof was verified.\n\n**What remains.**\n\nSettle dimensions four and above.\n\n**Sources checked.**\n\n- Ecole Polytechnique, A Closer Look at Kakeya's Conjecture (2026). (authoritative_secondary): https://www.polytechnique.edu/en/news/closer-look-kakeyas-conjecture\n Evidence used: Reports the Wang--Zahl R3 proof.\n- H. Wang and J. Zahl, Kakeya conjecture in three dimensions (2025). (primary): https://arxiv.org/abs/2502.17651\n Evidence used: Primary preprint for the three-dimensional result.\n\n**Review notes.** No source alteration; recent result merits specialist verification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 33, "problem_number": "GEO-004", "title": "The Moving Sofa Problem", "statement": "What is the largest area of a shape that can be maneuvered through an L-shaped corridor of unit width?", "background": "This classic problem in geometric optimization asks for the largest \"sofa\" that can navigate a right-angled hallway. The best known lower bound is approximately 2.2195 (Gerver's sofa, 1992), and the upper bound is $2\\sqrt{2} \\approx 2.8284$. The exact answer remains unknown.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Gerver's approximately 2.2195 sofa is the leading candidate; a 2024 arXiv optimality claim was not independently verified here.\n\n**Verified partial progress.**\n\n- Gerver gives the best established construction and upper bounds approach it.\n\n**Full solution or refutation.**\n\nNo peer-reviewed confirmation of the claimed exact optimum was verified.\n\n**What remains.**\n\nIndependently verify or publish the global optimality proof.\n\n**Sources checked.**\n\n- J. Baek, Solving Moving Sofa Problem Using Calculus of Variations, arXiv:2407.02587 (2024). (primary): https://arxiv.org/abs/2407.02587\n Evidence used: A claimed resolution, treated conservatively as unverified.\n- Moving sofa problem overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Moving_sofa_problem\n Evidence used: Describes the claim and retains the problem's open-status framing.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 34, "problem_number": "TOP-002", "title": "The Volume Conjecture", "statement": "For a hyperbolic knot $K$, the limit of normalized colored Jones polynomials equals the hyperbolic volume of the knot complement.", "background": "The volume conjecture, proposed by Rinat Kashaev in 1995 and generalized by Murakami and Murakami in 2001, connects quantum invariants of knots to their classical geometric properties. It relates quantum topology to hyperbolic geometry and has been verified for many knots but remains unproven in general.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The colored-Jones Volume Conjecture is proved for important knots and families but remains open for an arbitrary hyperbolic knot.\n\n**Verified partial progress.**\n\n- Murakami-Murakami formulate the normalized root-of-unity asymptotic in terms of simplicial volume and verify foundational cases.\n- Rigorous asymptotic expansions establish the conjectured exponential rate for additional individual hyperbolic knots such as 5_2.\n\n**Full solution or refutation.**\n\nCase-by-case asymptotic results do not provide a theorem for every hyperbolic knot.\n\n**What remains.**\n\nControl the root-of-unity colored Jones asymptotics uniformly enough to identify the exponential growth rate with hyperbolic volume for all hyperbolic knots.\n\n**Sources checked.**\n\n- Hitoshi Murakami and Jun Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Mathematica 186 (2001), 85-104. (primary): https://doi.org/10.1007/BF02392716\n Evidence used: Gives the standard formulation and foundational verified examples.\n- Tomotada Ohtsuki, On the asymptotic expansion of the Kashaev invariant of the 5_2 knot, Quantum Topology 7 (2016), 669-735. (primary): https://doi.org/10.4171/QT/82\n Evidence used: Proves detailed asymptotic behavior for a nontrivial hyperbolic-knot case.\n\n**Review notes.** The exact statement omits the evaluation point and normalization; record 1333's background supplies the conventional formula. This record and 1333 concern the same underlying conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Rinat Kashaev", "proposed_year": 1995, "category_id": 7, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 36, "problem_number": "AG-001", "title": "The Standard Conjectures on Algebraic Cycles", "statement": "A collection of conjectures about algebraic cycles on smooth projective varieties, including Lefschetz standard conjecture and Künneth standard conjecture.", "background": "The standard conjectures, formulated by Alexander Grothendieck in the 1960s, concern the theory of algebraic cycles and their cohomology. They would have profound consequences for algebraic geometry, including the independence of Betti numbers from the choice of Weil cohomology theory. The Hodge conjecture would follow from the Lefschetz standard conjecture.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Grothendieck's standard conjectures remain open for arbitrary smooth projective varieties, though known for many special classes.\n\n**Verified partial progress.**\n\n- They hold for abelian varieties and other important families.\n\n**Full solution or refutation.**\n\nNo general proof of the Lefschetz/Kunneth standard conjectures was verified.\n\n**What remains.**\n\nEstablish the standard conjectures for arbitrary smooth projective varieties.\n\n**Sources checked.**\n\n- Standard conjectures on algebraic cycles overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Standard_conjectures_on_algebraic_cycles\n Evidence used: Records the general conjectures as open and lists known cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "Alexander Grothendieck", "proposed_year": 1965, "category_id": 5, "view_count": 432, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 37, "problem_number": "AG-002", "title": "The Abundance Conjecture", "statement": "For a minimal model $X$ of non-negative Kodaira dimension, the canonical divisor $K_X$ is semi-ample.", "background": "The abundance conjecture is a major open problem in birational algebraic geometry and the minimal model program. It predicts that canonical divisors on minimal models have good positivity properties. The conjecture is known in dimension 3 and in many special cases, but remains open in dimension 4 and higher.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Abundance is proved in low dimensions and several special settings but remains open in full generality.\n\n**Verified partial progress.**\n\n- The conjecture is known for minimal threefolds and other classes.\n\n**Full solution or refutation.**\n\nNo all-dimensional semiampleness theorem was verified.\n\n**What remains.**\n\nProve abundance for arbitrary minimal models of nonnegative Kodaira dimension.\n\n**Sources checked.**\n\n- Abundance conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Abundance_conjecture\n Evidence used: Records known dimensions and the unresolved general case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 5, "view_count": 298, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 38, "problem_number": "ALG-003", "title": "The Köthe Conjecture", "statement": "A ring has no non-zero nil ideal (an ideal all of whose elements are nilpotent) if and only if it has no non-zero nil one-sided ideal.", "background": "The Köthe conjecture concerns the structure of rings with nilpotent elements. Proposed by Gottfried Köthe in 1930, it remains one of the oldest open problems in ring theory. Various special cases have been resolved, but the general conjecture remains open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Kothe's nil-ideal conjecture remains open for arbitrary associative rings.\n\n**Verified partial progress.**\n\n- The conjecture holds for one-sided Noetherian rings, PI rings, rings with Krull dimension, algebras over uncountable fields, and other substantial classes.\n- A 2025 paper adds rings whose nilpotent elements form a Wedderburn radical subring to the known positive classes.\n- Equivalent formulations involve sums of nil one-sided ideals and nilness of matrix rings over nil rings.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was verified.\n\n**What remains.**\n\nProve the implication for arbitrary rings or construct a ring with a nonzero nil one-sided ideal and no nonzero nil two-sided ideal.\n\n**Sources checked.**\n\n- Adel Alahmadi, S. K. Jain, and Andre Leroy, Rings Whose Nilpotent Elements Form a Wedderburn Radical Subring, Symmetry 17 (2025), article 1815. (primary): https://doi.org/10.3390/sym17111815\n Evidence used: Describes the general conjecture as open, lists known positive classes, and proves another positive class.\n- M. A. Chebotar, P.-H. Lee, and E. R. Puczylowski, On some questions related to Koethe's nil ideal problem, Proceedings of the Edinburgh Mathematical Society 58 (2015), 365-377. (primary): https://doi.org/10.1017/S0013091514000273\n Evidence used: Develops an equivalent formulation through one-sided ideals of A-rings.\n- Agata Smoktunowicz, On some Results Related to Kothe's Conjecture, Serdica Mathematical Journal 27 (2001), 159-170. (authoritative_secondary): https://eudml.org/doc/11532\n Evidence used: Survey of equivalent formulations and major partial results.\n\n**Review notes.** The biconditional is harmless but redundant: the reverse implication is immediate because every two-sided ideal is one-sided. Exact statement preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_by": "Gottfried Köthe", "proposed_year": 1930, "category_id": 4, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 40, "problem_number": "PDE-001", "title": "The Regularity Problem for Euler Equations", "statement": "Do solutions to the 3D Euler equations for incompressible fluid flow remain smooth for all time, given smooth initial data?", "background": "The Euler equations describe the motion of inviscid (frictionless) fluids. While the Navier-Stokes equations include viscosity and are a Millennium Prize Problem, the regularity of Euler equations is also a major open question. Finite-time blowup would have profound implications for fluid dynamics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The record is domain-sensitive. Chen and Hou proved finite-time blowup from smooth finite-energy data for axisymmetric 3D Euler in a smooth bounded cylinder, while the classical smooth-data problem on R^3 or T^3 remains open.\n\n**Verified partial progress.**\n\n- Chen and Hou's two-part analytic and computer-assisted proof constructs a stable nearly self-similar finite-time singularity for smooth axisymmetric Euler data with boundary.\n- Boundaryless blowup results below the classical smooth threshold do not settle the smooth-data R^3 or T^3 problem.\n\n**Full solution or refutation.**\n\nA universal reading that includes bounded smooth domains has a negative answer, but the traditional whole-space or periodic regularity question remains unresolved.\n\n**What remains.**\n\nSpecify the domain. For R^3 or T^3, prove global smoothness or construct smooth finite-energy blowup; if bounded domains are included, update the statement to reflect the Chen-Hou counterexample.\n\n**Sources checked.**\n\n- Jiajie Chen and Thomas Y. Hou, Stable Nearly Self-Similar Blowup of the 2D Boussinesq and 3D Euler Equations with Smooth Data II: Rigorous Numerics, Multiscale Modeling & Simulation 23 (2025). (primary): https://doi.org/10.1137/23M1580395\n Evidence used: Provides the rigorous numerical half of the proof and states that the combined result gives smooth-data finite-time singularity for axisymmetric 3D Euler with boundary.\n- Jiajie Chen and Thomas Y. Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis, arXiv:2210.07191. (primary): https://arxiv.org/abs/2210.07191\n Evidence used: Supplies the analytic nonlinear-stability component of the bounded-domain smooth-data blowup proof.\n- Jiajie Chen and Thomas Y. Hou, Singularity formation in 3D Euler equations with smooth initial data and boundary, Proceedings of the National Academy of Sciences 122 (2025), e2500940122. (primary): https://doi.org/10.1073/pnas.2500940122\n Evidence used: States the bounded-domain theorem and explicitly distinguishes it from the unresolved general singularity problem.\n\n**Review notes.** The source omits spatial domain and boundary conditions; this is flagged rather than silently repaired. This record is distinct from record 1515 despite sharing PDE-001.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 9, "view_count": 456, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 41, "problem_number": "SET-002", "title": "Singular Cardinals Hypothesis", "statement": "If $\\kappa$ is a singular strong limit cardinal, then $2^\\kappa = \\kappa^+$.", "background": "The singular cardinals hypothesis, formulated by Paul Erdős and András Hajnal, concerns the behavior of the power set operation on infinite cardinals. It sits between the generalized continuum hypothesis and ZFC. Its consistency and independence status remains a major open problem in set theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The Singular Cardinals Hypothesis is not decided by ZFC in the relative-consistency sense: GCH implies it, while suitable large-cardinal assumptions yield ZFC models in which it fails; the exact consistency strength of failure is known.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nMagidor constructed a model where aleph_omega is strong limit and 2^aleph_omega=aleph_(omega+2), refuting SCH there. Gitik identified the exact consistency strength of not-SCH as a measurable cardinal of Mitchell order kappa^(++). The positive side follows from the relative consistency of GCH.\n\n**What remains.**\n\nThe literal sentence has no ZFC-uniform truth value under the stated relative-consistency assumptions. Many finer singular-cardinal arithmetic questions remain, but not the broad consistency-status claim in the background.\n\n**Sources checked.**\n\n- Menachem Magidor, On the singular cardinals problems. II, Annals of Mathematics 106 (1977), 517-547. (primary): https://doi.org/10.2307/1971065\n Evidence used: Constructs, from strong large-cardinal assumptions, a ZFC model where a singular strong limit violates GCH and hence SCH.\n- Moti Gitik, The strength of the failure of the singular cardinal hypothesis, Annals of Pure and Applied Logic 51 (1991), 215-240. (primary): https://doi.org/10.1016/0168-0072(91)90016-F\n Evidence used: Proves the necessary lower bound and, with prior upper-bound results, the exact large-cardinal consistency strength of not-SCH.\n- Kurt Godel, The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis, Proceedings of the National Academy of Sciences 24 (1938), 556-557. (primary): https://doi.org/10.1073/pnas.24.12.556\n Evidence used: Relative consistency of GCH supplies a model of the positive SCH side.\n\n**Review notes.** Background defect: the consistency/independence status is not open, though finer SCH research is active. Failure has greater consistency strength than bare ZFC. SET-002 is also reused by unrelated Suslin record 1176. Exact statement retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Paul Erdős and András Hajnal", "category_id": 10, "view_count": 287, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 42, "problem_number": "SET-003", "title": "Whitehead Problem", "statement": "Is every abelian group $A$ such that $\\text{Ext}^1(A, \\mathbb{Z}) = 0$ a free abelian group?", "background": "The Whitehead problem, posed by J.H.C. Whitehead in 1950, asks about the structure of certain abelian groups. Shelah proved in 1973 that the problem is independent of ZFC: it is true under the constructible universe axiom (V=L) but can be false under other set-theoretic axioms.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The Whitehead problem is independent of ZFC: V=L implies every Whitehead group is free, while other consistent set-theoretic assumptions yield nonfree Whitehead groups.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nShelah proved independence at cardinality aleph_1 and subsequently used singular-cardinal compactness to obtain the positive V=L result for arbitrary cardinalities. A nonfree Whitehead group in the negative model refutes the universal assertion there.\n\n**What remains.**\n\nThe unrestricted ZFC question is settled as independent; classification under chosen additional axioms remains meaningful but is a different problem.\n\n**Sources checked.**\n\n- Saharon Shelah, Infinite abelian groups, Whitehead problem and some constructions, Israel Journal of Mathematics 18 (1974), 243-256. (primary): https://doi.org/10.1007/BF02757281\n Evidence used: Proves that the Whitehead problem at cardinality aleph_1 is independent of the usual set-theoretic axioms.\n- Saharon Shelah, A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel Journal of Mathematics 21 (1975), 319-349. (primary): https://doi.org/10.1007/BF02757993\n Evidence used: Concludes that V=L implies every Whitehead group, including arbitrary cardinalities, is free.\n- Encyclopedia of Mathematics, Whitehead problem. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Whitehead_problem\n Evidence used: Summarizes the positive V=L and negative MA+not-CH models and the extension to arbitrary cardinality.\n\n**Review notes.** Exact statement retained. Historical precision: the discovery is often dated 1973, but the cited primary independence publication is 1974 and the all-cardinal V=L result is 1975. Solved means solved-as-independent.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "J.H.C. Whitehead", "proposed_year": 1950, "category_id": 10, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 43, "problem_number": "CS-001", "title": "The Unique Games Conjecture", "statement": "For certain constraint satisfaction problems (unique games), it is NP-hard to approximate the maximum fraction of satisfiable constraints beyond a certain threshold.", "background": "The Unique Games Conjecture, proposed by Subhash Khot in 2002, has become central to computational complexity theory. If true, it would imply optimal hardness results for many approximation problems. Khot was awarded the Nevanlinna Prize in 2014 for this work, despite the conjecture remaining unresolved.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The standard quantified Unique Games Conjecture remains open; the extracted sentence is only an informal slogan and lacks the completeness, soundness, and alphabet-size quantifiers needed for a precise conjecture.\n\n**Verified partial progress.**\n\n- Arora, Barak, and Steurer gave a subexponential-time algorithm that turns value at least 1-epsilon^6 into value at least 1-epsilon for k-alphabet instances; because the running time is exp(k n^epsilon), this does not refute the conjectured polynomial-time NP-hardness.\n- Raghavendra proved that, assuming UGC, a natural semidefinite-programming framework gives the optimal approximation ratio for every finite-domain constraint satisfaction problem.\n- A 2025 primary paper reports major progress on related 2-to-2 and 2-to-1 games, including a theorem with imperfect completeness, while explicitly retaining UGC as an open question.\n\n**Full solution or refutation.**\n\nNo proof or refutation of the standard UGC was verified. The intended precise form asks, for every constant delta>0 and sufficiently large constant alphabet, for NP-hardness of distinguishing value at least 1-delta from value at most delta.\n\n**What remains.**\n\nProve the quantified NP-hardness gap for Unique Games or give a polynomial-time algorithm contradicting it for fixed parameters covered by the conjecture; separately formalize the dataset sentence, whose phrase 'a certain threshold' is not verifiable as written.\n\n**Sources checked.**\n\n- Subhash Khot, Dor Minzer, and Muli Safra, On Independent Sets, 2-to-2 Games and Grassmann Graphs, Theory of Computing 21(10) (2025), 1-55. (primary): https://www.theoryofcomputing.org/articles/v021a010/v021a010.pdf\n Evidence used: Definition 1.1 and Conjecture 1.2 give the precise Unique Games formulation, and the introduction explicitly calls UGC a prominent open question.\n- Sanjeev Arora, Boaz Barak, and David Steurer, Subexponential Algorithms for Unique Games and Related Problems, Journal of the ACM 62(5) (2015), Article 42. (primary): https://www.dsteurer.org/paper/subexpug/\n Evidence used: Proves the stated subexponential approximation algorithm and explicitly explains that the result stops short of refuting UGC.\n- Prasad Raghavendra, Optimal Algorithms and Inapproximability Results for Every CSP?, Proceedings of STOC 2008, 245-254. (primary): https://doi.org/10.1145/1374376.1374414\n Evidence used: Shows that, conditional on UGC, a natural SDP-based algorithm has the optimal approximation ratio for every CSP.\n\n**Review notes.** Exact statement preserved. Its undefined 'certain' problems and threshold make it non-truth-valued; the open classification applies to the standard intended UGC, not to a silently rewritten literal sentence.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Subhash Khot", "proposed_year": 2002, "category_id": 15, "view_count": 543, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 44, "problem_number": "CS-002", "title": "The Polynomial Hirsch Conjecture", "statement": "The diameter of the graph of a $d$-dimensional polytope with $n$ facets is bounded by a polynomial in $d$ and $n$.", "background": "The original Hirsch conjecture (diameter at most $n - d$) was disproved in 2010 by Francisco Santos. The polynomial Hirsch conjecture is a weaker version that remains open and is important for understanding the complexity of the simplex algorithm for linear programming.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The polynomial Hirsch conjecture for the vertex-edge diameter of arbitrary bounded polytopes remains open; the original linear n-d bound is disproved, but no superpolynomial lower bound or universal polynomial upper bound is known.\n\n**Verified partial progress.**\n\n- Kalai and Kleitman established a quasipolynomial upper bound, Todd improved it to (n-d)^(log_2 d), and Sukegawa obtained further exponent improvements; these are not fixed-degree polynomials in d and n.\n- Santos disproved the original linear Hirsch bound with a 43-dimensional polytope having 86 facets and diameter greater than 43, but the example has only linear diameter and does not refute the polynomial version.\n- Polynomial bounds are known for restricted classes, including totally unimodular systems and quantitatively conditioned systems; circuit diameter is a relaxation distinct from edge-graph diameter.\n\n**Full solution or refutation.**\n\nNo general polynomial upper bound or superpolynomial counterexample was verified. Current primary literature continues to identify polynomial combinatorial diameter as a central unresolved question.\n\n**What remains.**\n\nProve a universal fixed polynomial bound for the vertex-edge diameter of every d-polytope with n facets, or construct a family with superpolynomial diameter.\n\n**Sources checked.**\n\n- Daniel Dadush, Zhuan Khye Koh, Bento Natura, and Laszlo A. Vegh, On Circuit Diameter Bounds Via Circuit Imbalances, Mathematical Programming 206 (2024), 631-662. (primary): https://doi.org/10.1007/s10107-024-02107-x\n Evidence used: States that polynomial combinatorial diameter remains a central question, summarizes quasipolynomial and special-class progress, and distinguishes circuit from edge diameter.\n- Noriyoshi Sukegawa, Improving Bounds on the Diameter of a Polyhedron in High Dimensions, Discrete Mathematics 340 (2017), 2134-2142. (primary): https://arxiv.org/abs/1604.04039\n Evidence used: Reviews the open general diameter problem and proves refinements including (n-d)^(-3+log_2 d+O(1/d)).\n- Michael J. Todd, An Improved Kalai-Kleitman Bound for the Diameter of a Polyhedron, SIAM Journal on Discrete Mathematics 28 (2014), 1944-1947. (primary): https://doi.org/10.1137/140962310\n Evidence used: Proves the quasipolynomial upper bound (n-d)^(log_2 d).\n- Francisco Santos, A Counterexample to the Hirsch Conjecture, Annals of Mathematics 176 (2012), 383-412. (primary): https://annals.math.princeton.edu/2012/176-1/p07\n Evidence used: Disproves the original linear bounded-polytope conjecture with a 43-dimensional, 86-facet example.\n\n**Review notes.** Exact statement preserved. The intended graph is the 1-skeleton and the polynomial must be universal. Santos announced the counterexample in 2010 and published it in 2012. A polynomial diameter bound alone would not supply an efficiently findable simplex pivot path.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 15, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 45, "problem_number": "HIL-012", "title": "Hilbert's 12th Problem: Extension of Kronecker-Weber Theorem", "statement": "Extend the Kronecker-Weber theorem on abelian extensions of the rationals to any base number field.", "background": "Hilbert's 12th problem, posed in 1900, asks for an explicit construction of abelian extensions of number fields, generalizing the Kronecker-Weber theorem which states that every abelian extension of the rationals is contained in a cyclotomic field. Despite significant progress in class field theory, the problem of finding explicit generators remains largely open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Class field theory gives existence of abelian extensions, and complex multiplication gives explicit generators in important imaginary-quadratic cases, but the general explicit-construction programme remains open.\n\n**Verified partial progress.**\n\n- The Kronecker Jugendtraum is established in the imaginary quadratic/CM setting.\n\n**Full solution or refutation.**\n\nNo general analogue providing explicit generators for all base number fields was verified.\n\n**What remains.**\n\nDevelop explicit class-field generators beyond the known special settings.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Hilbert problems, accessed 2026-08-17. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Hilbert_problems\n Evidence used: Status overview treats Problem 12 as unresolved.\n\n**Review notes.** No source statement changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 46, "problem_number": "HIL-016", "title": "Hilbert's 16th Problem: Topology of Algebraic Curves and Limit Cycles", "statement": "Determine the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$, and investigate the topology of real algebraic curves and surfaces.", "background": "Posed by David Hilbert in 1900, this two-part problem concerns (1) the topology of real algebraic varieties and (2) the limit cycles of planar polynomial differential equations. While it was shown in 1991-1992 by Ilyashenko and Écalle that polynomial vector fields have finitely many limit cycles, the question of whether there exists a finite upper bound H(n) for degree n remains open for any n > 1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Each fixed planar polynomial vector field has finitely many limit cycles, but no uniform bound H(n) is known for general degree n>1 and the global topology programme remains incomplete.\n\n**Verified partial progress.**\n\n- Ecalle and Ilyashenko proved finiteness for an individual polynomial vector field.\n- Recent work gives conditional/structural results for H(n) but does not establish its finiteness in general.\n\n**Full solution or refutation.**\n\nThe main uniform-bound problem remains open.\n\n**What remains.**\n\nProve finiteness of H(n), obtain effective bounds, and classify possible configurations.\n\n**Sources checked.**\n\n- Scholarpedia, Limit cycles of planar polynomial vector fields, accessed 2026-08-17. (authoritative_secondary): http://www.scholarpedia.org/article/Limit_cycles_of_planar_polynomial_vector_fields\n Evidence used: States individual-field finiteness and the remaining open uniform-bound problem.\n- A. Gasull and P. Santana, A note on Hilbert 16th Problem, arXiv:2407.13465 (2024). (primary): https://arxiv.org/abs/2407.13465\n Evidence used: Studies H(n) conditionally on finiteness; does not claim a general bound.\n\n**Review notes.** Unverified claimed solutions were not treated as resolutions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 6, "set_id": 2, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 47, "problem_number": "LAN-004", "title": "Landau's Fourth Problem: Primes of the Form n² + 1", "statement": "Are there infinitely many primes of the form $n^2 + 1$?", "background": "One of Landau's four problems presented at the 1912 International Congress of Mathematicians, this asks whether there are infinitely many primes that are one more than a perfect square. Examples include 2, 5, 17, 37, 101, 197, 257, 401. Despite being simple to state, it has remained unsolved for over 110 years and is considered \"unattackable at the present state of mathematics.\"\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Infinitude of primes n^2+1 remains open. Iwaniec proved that n^2+1 has at most two prime factors for infinitely many n, the standard strongest unconditional near-prime result.\n\n**Verified partial progress.**\n\n- Iwaniec's 1978 sieve theorem gives infinitely many P_2 values of n^2+1.\n- This does not distinguish prime values from semiprimes, so it does not resolve Landau's question.\n\n**Full solution or refutation.**\n\nNo proof or disproof of infinitely many prime values was verified.\n\n**What remains.**\n\nBreak the parity barrier sufficiently to isolate prime values of the quadratic sequence.\n\n**Sources checked.**\n\n- H. Iwaniec, Almost-primes represented by quadratic polynomials, Inventiones Mathematicae 47 (1978), 171--188. (primary): https://eudml.org/doc/142575\n Evidence used: Proves infinitely many values of n^2+1 with at most two prime factors.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Edmund Landau", "proposed_year": 1912, "category_id": 1, "set_id": 6, "view_count": 398, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 6, "name": "landau_problems", "display_name": "Landau's Problems", "description": "Four basic problems about prime numbers posed by Edmund Landau at the 1912 International Congress of Mathematicians.", "slug": "landau-problems", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 48, "problem_number": "SMA-004", "title": "Smale's 4th Problem: Integer Zeros of Polynomials", "statement": "Find efficient algorithms for deciding whether a polynomial with integer coefficients has an integer root.", "background": "Part of Stephen Smale's 18 problems for the 21st century (1998), this problem asks for polynomial-time algorithms to determine if a polynomial equation has integer solutions. This is related to Hilbert's 10th problem, which was shown to be undecidable in general, but specific cases and algorithms with better complexity remain of interest.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** No algorithm can decide whether an arbitrary multivariate integer polynomial has an integer zero: Hilbert's tenth problem has a negative solution.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe MRDP theorem proves undecidability, refuting the requested general efficient algorithm.\n\n**What remains.**\n\nRestricted classes can be studied, but the unrestricted source task is impossible.\n\n**Sources checked.**\n\n- Hilbert's tenth problem overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem\n Evidence used: Records the negative solution via the Davis--Putnam--Robinson--Matiyasevich theorem.\n\n**Review notes.** Multivariate scope made explicit; source text itself unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 15, "set_id": 5, "view_count": 287, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 49, "problem_number": "SMA-005", "title": "Smale's 5th Problem: Height Bounds for Diophantine Curves", "statement": "Find effective uniform bounds for the heights of rational points on algebraic curves.", "background": "From Smale's 1998 list, this problem addresses the challenge of bounding the size of integer solutions to algebraic equations. While Faltings proved that curves of genus > 1 have finitely many rational points, the question of effective bounds on their heights remains a major open problem in arithmetic geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Effective uniform height bounds for rational points on arbitrary curves remain unavailable in the requested generality.\n\n**Verified partial progress.**\n\n- Effective Diophantine methods give bounds in many specified classes or under additional hypotheses.\n\n**Full solution or refutation.**\n\nNo uniform effective theorem covering arbitrary curves was verified.\n\n**What remains.**\n\nMake Faltings-type finiteness quantitatively effective in the desired uniform form.\n\n**Sources checked.**\n\n- Smale's Problems, MathWorld (accessed 2026-08-17). (authoritative_secondary): https://mathworld.wolfram.com/SmalesProblems.html\n Evidence used: Lists the height-bounds task among the still-open Smale problems.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 5, "set_id": 5, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 50, "problem_number": "SMA-006", "title": "Smale's 6th Problem: Finiteness of Central Configurations", "statement": "For the Newtonian $n$-body problem with positive masses, are there only finitely many central configurations (relative equilibria) for each $n$?", "background": "This problem from Smale's 1998 list concerns celestial mechanics and asks whether gravitating bodies can have only finitely many stable equilibrium configurations. The question is known to be true for n = 3 and n = 4, but remains open for n ≥ 5. It connects classical mechanics with algebraic geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finiteness is established in important low-n/equal-mass or generic regimes, but the all-n positive-mass central-configuration problem remains open.\n\n**Verified partial progress.**\n\n- Hampton--Moeckel prove finiteness for four bodies.\n- Recent work proves finite equal-mass spatial cases n=5,6 and studies planar six-body exceptional mass loci.\n\n**Full solution or refutation.**\n\nNo proof covers each n and every vector of positive masses.\n\n**What remains.**\n\nProve finiteness for arbitrary positive masses and all n.\n\n**Sources checked.**\n\n- J. D. Hauenstein and P. Zgliczynski, Central configurations in the spatial n-body problem for n=5,6 with equal masses, Celestial Mechanics and Dynamical Astronomy 132 (2020). (primary): https://doi.org/10.1007/s10569-020-09993-1\n Evidence used: Establishes named equal-mass n=5,6 finite cases and states the general Smale problem.\n- Central configurations, Scholarpedia (accessed 2026-08-17). (authoritative_secondary): https://www.scholarpedia.org/article/Central_configurations\n Evidence used: Describes the Chazy--Wintner--Smale finiteness problem as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 51, "problem_number": "SMA-007", "title": "Smale's 7th Problem: Distribution of Points on the 2-Sphere", "statement": "What is the optimal arrangement of $n$ points on the 2-sphere to minimize energy for various potential functions?", "background": "Smale's 7th problem (1998) asks for the configuration that minimizes various energy functionals for points on a sphere. This includes the Thomson problem (electrons on a sphere) and related optimization questions. Solutions are known for small n and highly symmetric cases, but the general problem remains open and connects to crystallography and coding theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Optimal spherical point configurations are known for selected n and potentials, but no universal solution covers the broad source formulation.\n\n**Verified partial progress.**\n\n- Universal optimality theorems cover special highly symmetric configurations and potential classes.\n\n**Full solution or refutation.**\n\nDifferent potential functions lead to distinct optimisation problems, so the source is a programme rather than a single settled statement.\n\n**What remains.**\n\nSpecify a potential and derive exact/global minimisers for general n.\n\n**Sources checked.**\n\n- Smale's Problems, MathWorld (accessed 2026-08-17). (authoritative_secondary): https://mathworld.wolfram.com/SmalesProblems.html\n Evidence used: Lists the broad spherical-energy problem among unresolved Smale challenges.\n\n**Review notes.** Broad formulation retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 52, "problem_number": "SMA-009", "title": "Smale's 9th Problem: Linear Programming in Polynomial Time", "statement": "Find a strongly polynomial algorithm for linear programming.", "background": "Smale's 9th problem (1998) asks whether there exists an algorithm for linear programming whose running time is polynomial in the number of constraints and variables, independent of the bit-size of the input. While linear programming is solvable in polynomial time, no strongly polynomial algorithm is known for the general case.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** A strongly polynomial algorithm for general linear programming remains open.\n\n**Verified partial progress.**\n\n- Polynomial-time algorithms are known in the bit-complexity model, and strongly polynomial algorithms exist for special LP classes.\n\n**Full solution or refutation.**\n\nNo general running-time bound depending only on numbers of variables and constraints was verified.\n\n**What remains.**\n\nDesign a strongly polynomial LP algorithm or prove an appropriate barrier.\n\n**Sources checked.**\n\n- Smale's Problems, MathWorld (accessed 2026-08-17). (authoritative_secondary): https://mathworld.wolfram.com/SmalesProblems.html\n Evidence used: Lists strongly polynomial linear programming as unresolved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 15, "set_id": 5, "view_count": 312, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 53, "problem_number": "SMA-010", "title": "Smale's 10th Problem: The Pugh Closing Lemma", "statement": "Is the $C^r$ closing lemma true for dynamical systems?", "background": "The closing lemma in dynamical systems theory asks whether, for a diffeomorphism with a nonwandering point, there is an arbitrarily small perturbation that makes that point periodic. Pugh proved a $C^1$ version in 1967, but the $C^r$ version for r ≥ 2 remains open. This is Smale's 10th problem from his 1998 list.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The C1 closing lemma is proved, whereas the unrestricted Cr case for r>1 remains open.\n\n**Verified partial progress.**\n\n- Higher-regularity closing lemmas are known for some partially hyperbolic and conservative classes.\n\n**Full solution or refutation.**\n\nNo general Cr theorem is verified.\n\n**What remains.**\n\nProve the closing lemma for arbitrary Cr systems with r>1.\n\n**Sources checked.**\n\n- S. Gan and Y. Shi, Cr-Closing lemma for partially hyperbolic diffeomorphisms with 1D-center bundle, arXiv:2004.06855. (primary): https://arxiv.org/abs/2004.06855\n Evidence used: Proves a substantial higher-regularity special case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 176, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 54, "problem_number": "SMA-016", "title": "The Jacobian Conjecture", "statement": "If $F: \\mathbb{C}^n \\to \\mathbb{C}^n$ is a polynomial map with constant non-zero Jacobian determinant, then $F$ is invertible.", "background": "The Jacobian conjecture, proposed in 1939 and featured as Smale's 16th problem (1998), asks whether polynomial maps with nowhere-vanishing Jacobian determinant are necessarily invertible. Despite its elementary statement, it has resisted numerous attempts at proof. The conjecture is known to be true in dimension 1 and for maps of degree at most 2, but remains open in general.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Jacobian conjecture remains open in every dimension n>=2.\n\n**Verified partial progress.**\n\n- Many reductions and special cases are known, including reductions to restricted degree forms.\n\n**Full solution or refutation.**\n\nNo general inverse-polynomial theorem or counterexample was verified.\n\n**What remains.**\n\nProve polynomial invertibility under constant nonzero Jacobian or find a counterexample.\n\n**Sources checked.**\n\n- Jacobian conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Jacobian_conjecture\n Evidence used: Records the general conjecture as open and summarizes reductions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Ott-Heinrich Keller", "proposed_year": 1939, "category_id": 4, "set_id": 5, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 55, "problem_number": "COMB-005", "title": "Frankl's Union-Closed Sets Conjecture", "statement": "For every finite union-closed family of sets (other than the empty family), there exists an element that belongs to at least half of the sets.", "background": "Proposed by Péter Frankl in 1979, this is one of the best-known open problems in combinatorics. A union-closed family is a collection of sets closed under taking unions. Despite its simple statement, the conjecture has attracted many attempted proofs. Recent progress (2022-2024) has shown lower bounds: some element must be in at least 1% of sets (Gilmer 2022), improved to 38% of sets (2024).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Frankl's union-closed sets conjecture remains open; rigorously, some element is known to occur in at least 0.38234 of the sets.\n\n**Verified partial progress.**\n\n- Gilmer proved the first dimension-free constant, 0.01.\n- Yu's peer-reviewed refinement gives 0.38234, slightly above (3-sqrt(5))/2.\n- Liu's approximately 0.38271 value depends on numerically verified optimization hypotheses and does not prove the one-half conjecture.\n\n**Full solution or refutation.**\n\nNo independently verified proof or counterexample for the general one-half assertion was found.\n\n**What remains.**\n\nRaise the guaranteed frequency from 0.38234 to one-half, or find a nondegenerate counterexample.\n\n**Sources checked.**\n\n- J. Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055 (2022). (primary): https://arxiv.org/abs/2211.09055\n Evidence used: Proves the first absolute 0.01 frequency bound and states the standard nondegenerate formulation.\n- L. Yu, Dimension-Free Bounds for the Union-Closed Sets Conjecture, Entropy 25 (2023), 767. (primary): https://doi.org/10.3390/e25050767\n Evidence used: Establishes the computable 0.38234 lower bound.\n- J. Liu, Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling, arXiv:2306.08824 (2023). (primary): https://arxiv.org/abs/2306.08824\n Evidence used: Describes a conditional, numerically assisted approximately 0.38271 improvement.\n\n**Review notes.** The phrase 'empty family' is ambiguous; the conventional hypothesis should explicitly read F != {empty set}.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_by": "Péter Frankl", "proposed_year": 1979, "category_id": 2, "view_count": 389, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 56, "problem_number": "GEO-005", "title": "Inscribed Square Problem (Toeplitz Conjecture)", "statement": "Does every simple closed curve in the plane contain all four vertices of some square?", "background": "The inscribed square problem, also called the square peg problem or Toeplitz conjecture, was posed by Otto Toeplitz in 1911. It asks whether every Jordan curve (simple closed curve) inscribes a square. The conjecture is known to be true for convex curves, piecewise smooth curves, and many special cases, but remains open in full generality as of 2026.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The inscribed-square problem remains open for arbitrary simple closed curves under its full Jordan-curve formulation.\n\n**Verified partial progress.**\n\n- It is proved for many regularity classes, including smooth and polygonal curves.\n\n**Full solution or refutation.**\n\nNo proof for every Jordan curve was verified.\n\n**What remains.**\n\nRemove the remaining regularity assumptions.\n\n**Sources checked.**\n\n- Inscribed square problem overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Inscribed_square_problem\n Evidence used: Records known regularity cases and the general open formulation.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Otto Toeplitz", "proposed_year": 1911, "category_id": 6, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 57, "problem_number": "NT-010", "title": "Brocard's Problem", "statement": "Find all integer solutions to $n! + 1 = m^2$.", "background": "Brocard's problem asks for all positive integers n such that n! + 1 is a perfect square. Only three solutions are known: (4, 5), (5, 11), and (7, 71), corresponding to 4! + 1 = 25, 5! + 1 = 121, and 7! + 1 = 5041. It has been verified computationally that no other solutions exist for n < 10^9, but it remains unproven whether these are the only solutions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Brocard's problem remains open: only (n,m)=(4,5),(5,11),(7,71) are known, and their exhaustiveness is unproved.\n\n**Verified partial progress.**\n\n- Berndt and Galway exhaustively searched through n=10^9 without finding a fourth positive solution.\n\n**Full solution or refutation.**\n\nNo accepted proof was found that the three known positive solutions are all solutions.\n\n**What remains.**\n\nProve that n=4,5,7 are the only positive n for which n!+1 is a square, or find another solution.\n\n**Sources checked.**\n\n- Bruce C. Berndt and William F. Galway, On the Brocard-Ramanujan Diophantine equation n!+1=m^2, Ramanujan Journal 4 (2000), 41-42, DOI 10.1023/A:1009873805276. (primary): https://experts.illinois.edu/en/publications/on-the-brocard-ramanujan-diophantine-equation-n-1-msup2sup/\n Evidence used: Reports the historical problem, the three known solutions, and calculations through n=10^9.\n- Eric W. Weisstein, Brocard's Problem, MathWorld. (authoritative_secondary): https://mathworld.wolfram.com/BrocardsProblem.html\n Evidence used: Maintains the standard formulation and known-solution/search status.\n\n**Review notes.** The source says integer solutions but factorial is standardly posed here for positive n; the domain should be clarified without changing the stored statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 59, "problem_number": "GT-004", "title": "The Cycle Double Cover Conjecture", "statement": "Every bridgeless graph has a cycle double cover: a collection of cycles that covers each edge exactly twice.", "background": "The cycle double cover conjecture, proposed independently by Paul Seymour and Gábor Szekeres in the 1970s, is a major open problem in graph theory. It has been verified for many classes of graphs, including planar graphs and graphs with small genus. The conjecture is related to the snark conjecture and has connections to topology and algebraic graph theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A July 2026 Cycle Double Cover proof announcement has expert expositions but is too recent for an unconditional solved classification.\n\n**Verified partial progress.**\n\n- Oum and Geelen posted independent explanatory treatments.\n\n**Full solution or refutation.**\n\nThe record is retained as pending independent detailed verification.\n\n**What remains.**\n\nIndependent verification and durable publication of the announced proof.\n\n**Sources checked.**\n\n- S.-i. Oum, A proof of the cycle double cover conjecture by OpenAI: An exposition, arXiv:2607.16356. (primary): https://arxiv.org/abs/2607.16356\n Evidence used: Provides an exposition of the announced proof.\n- J. Geelen, OpenAI's proof of the Cycle Double Cover Theorem, arXiv:2607.15399. (primary): https://arxiv.org/abs/2607.15399\n Evidence used: Provides independent clarification notes.\n\n**Review notes.** Duplicate topic retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 60, "problem_number": "NT-012", "title": "The Erdős-Straus Conjecture", "statement": "For every integer $n \\geq 2$, the equation $\\frac{4}{n} = \\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}$ has a solution in positive integers x, y, z.", "background": "The Erdős-Straus conjecture concerns Egyptian fractions (sums of unit fractions). Paul Erdős and Ernst G. Straus conjectured in 1948 that 4/n can always be expressed as the sum of three unit fractions. The conjecture has been verified for all n up to 10^17 and is known to hold for various infinite families, but a general proof remains elusive.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Erdős--Straus conjecture remains unresolved.\n\n**Verified partial progress.**\n\n- It is verified over very large finite ranges and solved for many congruence classes.\n\n**Full solution or refutation.**\n\nNo proof for every n was verified.\n\n**What remains.**\n\nProve a three-unit-fraction decomposition for every n>=2 or find a counterexample.\n\n**Sources checked.**\n\n- The Erdős-Straus Conjecture and the Structure of..., INTEGERS 26 (2026). (primary): https://math.colgate.edu/~integers/aa42/aa42.pdf\n Evidence used: Recent research treats the conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_by": "Paul Erdős and Ernst G. Straus", "proposed_year": 1948, "category_id": 1, "view_count": 367, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 61, "problem_number": "HIL-006", "title": "Hilbert's 6th Problem: Axiomatization of Physics", "statement": "Develop a mathematical framework that axiomatizes physics, particularly mechanics, thermodynamics, and probability theory.", "background": "Hilbert's 6th problem (1900) calls for treating physics with the same mathematical rigor as geometry. While progress has been made (quantum mechanics axiomatization by von Neumann, some progress in quantum field theory), a complete axiomatization remains elusive, especially for areas like thermodynamics and a unified \"theory of everything.\"\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The request to axiomatize physics spans several changing theories and supplies no formal target system or completion criterion.\n\n**Verified partial progress.**\n\n- Axiomatic frameworks exist for particular theories, including probability, classical mechanics, and quantum mechanics.\n\n**Full solution or refutation.**\n\nNo single terminal mathematical assertion is specified.\n\n**What remains.**\n\nSpecify the physical theory, mathematical language, axioms, and desired metatheorems.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Hilbert problems, accessed 2026-08-17. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Hilbert_problems\n Evidence used: Describes Problem 6 as mathematical treatment of the axioms of physics.\n\n**Review notes.** Programme wording preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 16, "set_id": 2, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 62, "problem_number": "HIL-013", "title": "Hilbert's 13th Problem: Seventh Degree Equations", "statement": "Prove that the general equation of the seventh degree cannot be solved using functions of only two variables.", "background": "Hilbert's 13th problem (1900) asks whether seventh-degree equations can be solved using continuous functions of two variables. Vladimir Arnold and Andrey Kolmogorov showed in 1957 that any continuous function can be represented using functions of two variables, which contradicts Hilbert's expectation. However, the problem of whether algebraic solutions exist with restrictions remains open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The continuous-function version was answered affirmatively by Kolmogorov--Arnold, whereas the original algebraic-function version remains open; the imported statement does not name the function class.\n\n**Verified partial progress.**\n\n- Kolmogorov--Arnold representation resolves the continuous variant.\n\n**Full solution or refutation.**\n\nA unique status cannot be given without specifying algebraic, continuous, analytic, or another allowed class.\n\n**What remains.**\n\nRestore the intended function class and allowed compositions.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Hilbert problems, accessed 2026-08-17. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Hilbert_problems\n Evidence used: Status overview distinguishes the historical problem's unresolved aspects.\n\n**Review notes.** Missing qualifier flagged rather than supplied.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 4, "set_id": 2, "view_count": 287, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 64, "problem_number": "SMA-012", "title": "Smale's 12th Problem: Centralizers of Diffeomorphisms", "statement": "Determine the structure of centralizers of generic diffeomorphisms.", "background": "Smale's 12th problem (1998) concerns the algebraic structure of diffeomorphisms that commute with a given diffeomorphism. The centralizer of a dynamical system reveals its symmetries. Smale conjectured that for generic diffeomorphisms, the centralizer should be trivial or nearly trivial.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Trivial centralizer holds on a residual subset of C1 diffeomorphisms; the analogous general Cr statement remains unsettled.\n\n**Verified partial progress.**\n\n- Bonatti--Crovisier--Wilkinson prove the C1-generic trivial-centralizer theorem.\n\n**Full solution or refutation.**\n\nThe C1 result does not establish the specified all-r generic formulation.\n\n**What remains.**\n\nExtend trivial-centralizer genericity to higher regularity or identify obstructions.\n\n**Sources checked.**\n\n- C. Bonatti, S. Crovisier and A. Wilkinson, The C1 generic diffeomorphism has trivial centralizer, Publications Mathématiques de l'IHÉS 109 (2009); arXiv:0804.1416. (primary): https://arxiv.org/abs/0804.1416\n Evidence used: The abstract explicitly answers Smale's question in the C1 topology.\n- M. Leguil, C. R. Pujals and M. Sambarino, The centralizer of Cr-generic diffeomorphisms at hyperbolic basic sets is trivial, arXiv:1606.00132. (primary): https://arxiv.org/abs/1606.00132\n Evidence used: Gives a higher-regularity result restricted to hyperbolic basic sets.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 176, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 5, "name": "smale_problems", "display_name": "Smale's Problems", "description": "Steve Smale's list of mathematical problems for the 21st century.", "slug": "smale-problems", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 66, "problem_number": "DARPA-002", "title": "The Dynamics of Networks", "statement": "Develop high-dimensional mathematics to model and predict behavior in large-scale distributed networks.", "background": "DARPA challenge 2 (2007) addresses the need for mathematical tools to understand massive networks like the internet, social networks, and biological networks. Traditional graph theory becomes inadequate at scale, requiring new mathematical frameworks for network dynamics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This is an open-ended 2007 research agenda, not a proposition with a completion criterion. Network-dynamics theory and data-driven modelling have advanced, but no source defines a terminal solution.\n\n**Verified partial progress.**\n\n- DARPA's later MoDyL program explicitly pursued rigorous data-driven models for non-equilibrium dynamics, including communication and social systems.\n\n**Full solution or refutation.**\n\nNot applicable: the statement has no formal success condition.\n\n**What remains.**\n\nSpecify a network model, prediction task, data regime, and quantitative guarantee before a mathematical status can be assessed.\n\n**Sources checked.**\n\n- DARPA DSO, Broad Agency Announcement 07-68, Mathematical Challenge Two (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original broad solicitation wording.\n- DARPA, Models, Dynamics and Learning program page, accessed 2026-08-17. (primary): https://www.darpa.mil/research/programs/models-dynamics-and-learning\n Evidence used: Later program framing and ongoing modelling challenge.\n\n**Review notes.** Research agenda, not a yes/no problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 3, "set_id": 4, "view_count": 389, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 68, "problem_number": "DARPA-004", "title": "21st Century Fluids", "statement": "Extend classical fluid dynamics to handle complex substances like foams, suspensions, gels, and liquid crystals.", "background": "DARPA challenge 4 (2007) recognizes that most real-world fluids don't behave like the classical fluids of Navier-Stokes equations. New mathematics is needed for complex fluids with microstructure, non-Newtonian behavior, and multiphase dynamics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The complex-fluid challenge is a research agenda spanning several non-equivalent equations and material classes, so it has no unitary solved/open status.\n\n**Verified partial progress.**\n\n- Modern theory supplies models and analysis for many complex-fluid classes, but no universal framework is specified by the source.\n\n**Full solution or refutation.**\n\nNot applicable without a target material model or theorem.\n\n**What remains.**\n\nState a concrete constitutive law, desired well-posedness/prediction result, and regime.\n\n**Sources checked.**\n\n- DARPA DSO, Broad Agency Announcement 07-68, Mathematical Challenge Four (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original list of foams, suspensions, gels, and liquid crystals.\n\n**Review notes.** No source statement changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 9, "set_id": 4, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 69, "problem_number": "DARPA-005", "title": "Biological Quantum Field Theory", "statement": "Apply quantum and statistical field theory methods to model and potentially control pathogen evolution.", "background": "DARPA challenge 5 (2007) proposes using the mathematical machinery of quantum field theory—developed for particle physics—to understand biological evolution and epidemiology. This could provide new ways to predict and control disease evolution.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Applying field-theoretic methods to pathogen evolution is a programmatic objective, not a formal conjecture; no source supplies a single criterion for completion.\n\n**Verified partial progress.**\n\n- Quantum and statistical approaches to evolutionary modelling constitute a continuing interdisciplinary literature.\n\n**Full solution or refutation.**\n\nNot applicable as written.\n\n**What remains.**\n\nSpecify pathogen model, control objective, observables, and validated accuracy/control metric.\n\n**Sources checked.**\n\n- DARPA DSO, Broad Agency Announcement 07-68, Mathematical Challenge Five (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original questions about modelling bacteria and controlling pathogen evolution.\n\n**Review notes.** Broad application objective.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 70, "problem_number": "DARPA-008", "title": "Beyond Convex Optimization", "statement": "Determine whether algebraic geometry can systematically replace linear algebra in optimization.", "background": "DARPA challenge 8 (2007) asks whether the powerful tools of algebraic geometry can extend optimization beyond the convex case. Most practical optimization problems are non-convex, and algebraic geometry may provide the framework for solving them systematically.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The request to replace linear algebra by algebraic geometry in optimisation is aspirational and lacks a defined algorithmic scope or criterion.\n\n**Verified partial progress.**\n\n- Algebraic and polynomial optimisation have developed many specialised methods, without constituting a systematic replacement in the stated unrestricted sense.\n\n**Full solution or refutation.**\n\nNot applicable as written.\n\n**What remains.**\n\nDefine an optimisation class, allowed algebraic-geometric operations, and performance guarantee.\n\n**Sources checked.**\n\n- DARPA DSO, Broad Agency Announcement 07-68, Mathematical Challenge Eight (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original one-sentence agenda.\n\n**Review notes.** No source correction made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 312, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 71, "problem_number": "DARPA-012", "title": "Mathematics of Quantum Computing", "statement": "Develop the mathematics required to control the quantum world for computation.", "background": "DARPA challenge 12 (2007) calls for mathematical foundations of quantum computing, including quantum algorithms, quantum entanglement, and quantum error correction. While quantum computers exist, the mathematical theory of what they can compute and how to program them remains underdeveloped.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The mathematics of quantum computing, algorithms, and entanglement is an enduring field-wide agenda rather than a single statement with a solvability criterion.\n\n**Verified partial progress.**\n\n- Quantum algorithms, error correction, and entanglement theory have all advanced substantially since 2007.\n\n**Full solution or refutation.**\n\nNot applicable as written.\n\n**What remains.**\n\nSpecify a computational model, task, resource measure, and theorem-level objective.\n\n**Sources checked.**\n\n- DARPA DSO, Broad Agency Announcement 07-68, Mathematical Challenge Twelve (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original broad quantum-mathematics solicitation.\n\n**Review notes.** No source statement changed; broad research agenda.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 543, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 72, "problem_number": "DARPA-013", "title": "Game Theory at Scale", "statement": "Create scalable mathematics for differential games, replacing traditional PDE approaches.", "background": "DARPA challenge 13 (2007) addresses the limitations of classical game theory and differential games when dealing with many players. New mathematical frameworks are needed for multi-agent systems, from autonomous vehicles to economic markets to military strategy.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Scalable game theory is a broad agenda, not a proposition with a specified class of games or a target theorem.\n\n**Verified partial progress.**\n\n- Mean-field games, online learning, and large-scale optimisation offer distinct modern frameworks.\n\n**Full solution or refutation.**\n\nNot applicable as written.\n\n**What remains.**\n\nSpecify the game model, scale regime, and desired substitute for PDE methods.\n\n**Sources checked.**\n\n- DARPA DSO, BAA 07-68, Mathematical Challenge Thirteen (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original scalable-mathematics agenda.\n\n**Review notes.** Research agenda, not a decidable claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 289, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 74, "problem_number": "DARPA-020", "title": "Computation at Scale", "statement": "Develop asymptotics for systems with massive degrees of freedom.", "background": "DARPA challenge 20 (2007) addresses the mathematical challenges of understanding systems with enormous numbers of variables—from climate models to protein folding to materials science. Traditional approaches fail at extreme scales, requiring new asymptotic methods.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Computation at massive scale is a broad asymptotic-methods agenda rather than a defined asymptotic problem.\n\n**Verified partial progress.**\n\n- Large-system asymptotics occur in statistical physics, algorithms, probability, and numerical analysis under differing assumptions.\n\n**Full solution or refutation.**\n\nNot applicable as written.\n\n**What remains.**\n\nDefine the system family, scaling limit, observables, and approximation criterion.\n\n**Sources checked.**\n\n- DARPA DSO, BAA 07-68, Mathematical Challenge Twenty (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original massively-many-degrees-of-freedom wording.\n\n**Review notes.** No source correction made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 276, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 75, "problem_number": "DARPA-023", "title": "Fundamental Laws of Biology", "statement": "Identify governing principles for biological systems, analogous to physical laws.", "background": "DARPA challenge 23 (2007) poses perhaps the deepest question: Do fundamental mathematical laws govern biology the way physics is governed by laws? This challenge requires solutions to multiple preceding challenges and asks whether biology can be made as mathematically rigorous as physics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finding fundamental laws of biology is an explicit century-scale research vision, not a formal mathematical proposition.\n\n**Verified partial progress.**\n\n- Mathematical and systems biology supply many partial models but no agreed universal law set.\n\n**Full solution or refutation.**\n\nNot applicable as written.\n\n**What remains.**\n\nSpecify biological phenomena, form of a law, and prediction/falsification standard.\n\n**Sources checked.**\n\n- DARPA DSO, BAA 07-68, Mathematical Challenge Twenty-Three (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original open-ended program statement.\n\n**Review notes.** Research agenda, not a decidable claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 498, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 76, "problem_number": "DARPA-006", "title": "Computational Duality", "statement": "Use mathematical duality and geometry as foundations for developing novel computational algorithms.", "background": "DARPA challenge 6 (2007) explores whether duality principles from mathematics can lead to breakthrough algorithms. Dualities connect seemingly different mathematical structures and may reveal hidden computational efficiencies.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Computational duality names a direction for novel algorithms rather than a theorem or benchmark, so a solved/open designation is unsupported.\n\n**Verified partial progress.**\n\n- Duality is central to established optimisation and algorithmic frameworks, but the solicitation states no universal performance target.\n\n**Full solution or refutation.**\n\nNot applicable as written.\n\n**What remains.**\n\nGive a computational problem class and a measurable duality-based advantage.\n\n**Sources checked.**\n\n- DARPA DSO, Broad Agency Announcement 07-68, Mathematical Challenge Six (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original question about principled computational techniques.\n\n**Review notes.** No general algorithmic completion criterion was found.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 77, "problem_number": "DARPA-007", "title": "Occam's Razor in Many Dimensions", "statement": "Find lower bounds for sensing complexity as data collection grows, addressing entropy maximization.", "background": "DARPA challenge 7 (2007) asks for mathematical principles governing data compression and sensing in high dimensions. As sensors become ubiquitous, we need mathematical theory for how much data is truly necessary.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The sensing-complexity agenda predates substantial compressed-sensing and information-theory advances but asks no fixed lower-bound theorem.\n\n**Verified partial progress.**\n\n- Many sensing-complexity lower bounds are known under particular measurement and signal assumptions.\n\n**Full solution or refutation.**\n\nNot applicable absent a signal/measurement model.\n\n**What remains.**\n\nSpecify signal class, noise model, sensing architecture, and lower-bound metric.\n\n**Sources checked.**\n\n- DARPA DSO, Broad Agency Announcement 07-68, Mathematical Challenge Seven (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original sensing-complexity and entropy-maximisation agenda.\n\n**Review notes.** Research agenda, not a formal lower-bound question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 78, "problem_number": "DARPA-009", "title": "Physical Consequences of Perelman's Proof", "statement": "Apply Perelman's proof of the Poincaré conjecture to materials fabrication across scales.", "background": "DARPA challenge 9 (2007) asks how Grisha Perelman's breakthrough in understanding 3-dimensional geometry can inform materials science, from nanostructures to macro-scale fabrication.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This asks for possible physical/material consequences of geometrization, not a mathematical claim that can be proved or disproved as stated.\n\n**Verified partial progress.**\n\n- Perelman's proof is an established mathematical result; an application to fabrication would need a specified physical mechanism and test.\n\n**Full solution or refutation.**\n\nNot applicable without a concrete materials objective.\n\n**What remains.**\n\nFormulate a mechanism linking a geometrization invariant to a fabrication design or prediction.\n\n**Sources checked.**\n\n- DARPA DSO, Broad Agency Announcement 07-68, Mathematical Challenge Nine (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original materials-fabrication language.\n\n**Review notes.** Application agenda rather than open theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 7, "set_id": 4, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 79, "problem_number": "DARPA-010", "title": "Algorithmic Origami and Biology", "statement": "Strengthen mathematical theory for isometric and rigid embedding relevant to protein folding.", "background": "DARPA challenge 10 (2007) connects origami mathematics to biology. Protein folding is like origami at molecular scales, and better mathematical theory could revolutionize drug design and protein engineering.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Algorithmic origami and protein folding are broad intersecting fields; the source does not pose a testable embedding statement.\n\n**Verified partial progress.**\n\n- Rigidity, origami design, and computational protein modelling have extensive separate literatures.\n\n**Full solution or refutation.**\n\nNot applicable as written.\n\n**What remains.**\n\nSpecify an embedding model and a protein-folding prediction or optimisation guarantee.\n\n**Sources checked.**\n\n- DARPA DSO, Broad Agency Announcement 07-68, Mathematical Challenge Ten (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original isometric/rigid-embedding agenda.\n\n**Review notes.** No unitary solved/open status possible.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 6, "set_id": 4, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 80, "problem_number": "DARPA-011", "title": "Optimal Nanostructures", "statement": "Develop mathematics for creating optimal symmetric structures through nanoscale self-assembly.", "background": "DARPA challenge 11 (2007) seeks mathematical principles for designing nanostructures that self-assemble optimally. This combines crystallography, optimization, and molecular dynamics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Optimal symmetric self-assembly is a collection of optimisation and physical-design problems, not a uniquely specified mathematical challenge.\n\n**Verified partial progress.**\n\n- Nanoscale self-assembly and symmetric-structure design remain active research areas.\n\n**Full solution or refutation.**\n\nNot applicable absent a model of local rules, objective, and assembly errors.\n\n**What remains.**\n\nFix a self-assembly formalism and an optimality objective.\n\n**Sources checked.**\n\n- DARPA DSO, Broad Agency Announcement 07-68, Mathematical Challenge Eleven (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original globally symmetric local-rule formulation.\n\n**Review notes.** Research agenda, not a proposition.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 6, "set_id": 4, "view_count": 223, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 81, "problem_number": "DARPA-015", "title": "The Geometry of Genome Space", "statement": "Establish appropriate distance metrics on genome space incorporating biological utility.", "background": "DARPA challenge 15 (2007) asks for a mathematical geometry of genetics. How \"far apart\" are two genomes? The answer depends on biology, not just counting mutations, requiring new geometric frameworks.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No unique biologically useful genome-space metric is specified, so the challenge cannot receive a theorem-level solved/open status.\n\n**Verified partial progress.**\n\n- Genome distances and phylogenetic metrics form an active, non-unique research area.\n\n**Full solution or refutation.**\n\nNot applicable as written.\n\n**What remains.**\n\nFix genome representation, biological utility criterion, and required metric properties.\n\n**Sources checked.**\n\n- DARPA DSO, BAA 07-68, Mathematical Challenge Fifteen (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original one-sentence metric agenda.\n\n**Review notes.** No source text changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 6, "set_id": 4, "view_count": 245, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 82, "problem_number": "DARPA-016", "title": "Symmetries and Action Principles for Biology", "statement": "Extend understanding of symmetries and action principles in biology to include robustness, modularity, evolvability, and variability.", "background": "DARPA challenge 16 (2007) seeks to identify fundamental symmetry principles in biology analogous to those in physics. Why are biological systems robust yet evolvable? Are there variational principles governing life?\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This is a program for conceptual frameworks in biology, without a formal structure or verification condition.\n\n**Verified partial progress.**\n\n- Systems biology and mathematical modelling investigate robustness, modularity, and evolvability under many incompatible formalisms.\n\n**Full solution or refutation.**\n\nNot applicable as written.\n\n**What remains.**\n\nSpecify a biological state space, symmetry/action principle, and falsifiable prediction.\n\n**Sources checked.**\n\n- DARPA DSO, BAA 07-68, Mathematical Challenge Sixteen (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original broad biological agenda.\n\n**Review notes.** Research agenda, not a formal problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 312, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 83, "problem_number": "DARPA-017", "title": "Geometric Langlands and Quantum Physics", "statement": "Connect the Langlands program to fundamental physics symmetries.", "background": "DARPA challenge 17 (2007) explores deep connections between number theory (Langlands program) and quantum field theory. This could unify disparate areas of mathematics and physics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Geometric Langlands and physics is an extensive active program; the challenge does not identify a conjecture whose status can be checked.\n\n**Verified partial progress.**\n\n- Numerous geometric-Langlands/quantum-field-theory correspondences have been formulated and proved in particular settings.\n\n**Full solution or refutation.**\n\nNot applicable absent a named correspondence and hypotheses.\n\n**What remains.**\n\nState a precise physics symmetry and mathematical equivalence to establish.\n\n**Sources checked.**\n\n- DARPA DSO, BAA 07-68, Mathematical Challenge Seventeen (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original Langlands-and-physics wording.\n\n**Review notes.** No source statement changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 356, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 84, "problem_number": "DARPA-018", "title": "Arithmetic Langlands, Topology, and Geometry", "statement": "Explore homotopy theory's role in Langlands programs.", "background": "DARPA challenge 18 (2007) connects topology (homotopy theory) with the Langlands program in number theory. These connections could revolutionize both fields.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The role of homotopy theory across Langlands programs is a broad research direction without a stated completion criterion.\n\n**Verified partial progress.**\n\n- Derived, spectral, and homotopical approaches have become important in modern Langlands research.\n\n**Full solution or refutation.**\n\nNot applicable as written.\n\n**What remains.**\n\nSpecify the Langlands setting and an explicit homotopical construction or equivalence.\n\n**Sources checked.**\n\n- DARPA DSO, BAA 07-68, Mathematical Challenge Eighteen (2007). (primary): https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html\n Evidence used: Original homotopy/Langlands agenda.\n\n**Review notes.** Research agenda, not a decidable claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 7, "set_id": 4, "view_count": 289, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 4, "name": "darpa_challenges", "display_name": "DARPA's 23 Mathematical Challenges", "description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.", "slug": "darpa-challenges", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 85, "problem_number": "HIL-007", "title": "Hilbert's 7th Problem: Transcendence of Certain Numbers", "statement": "If $\\alpha$ is algebraic and irrational, and $\\beta$ is algebraic and irrational, is $\\alpha^\\beta$ transcendental?", "background": "Hilbert's 7th problem (1900) was largely solved by Gelfond and Schneider independently in 1934 (Gelfond-Schneider theorem). However, cases involving non-algebraic irrational exponents remain open. For example, whether $e^e$ or $\\pi^\\pi$ are transcendental is unknown.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Gelfond--Schneider proves that every value of alpha^beta is transcendental when alpha is algebraic other than 0 or 1 and beta is irrational algebraic.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThis directly answers the imported statement, subject to the conventional nonzero/base qualification inherent in defining alpha^beta.\n\n**What remains.**\n\nThe historical problem is solved; broader transcendence questions are distinct.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Hilbert problems, accessed 2026-08-17. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Hilbert_problems\n Evidence used: Lists Schneider/Gelfond-Schneider as the resolution of Problem 7.\n\n**Review notes.** No statement repair; standard domain caveat recorded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 86, "problem_number": "HIL-009", "title": "Hilbert's 9th Problem: Reciprocity Laws", "statement": "Generalize the reciprocity law of number theory to arbitrary number fields.", "background": "Hilbert's 9th problem (1900) asks for extensions of quadratic reciprocity to general number fields. Emil Artin made progress with Artin reciprocity law (1927), but complete understanding of reciprocity in all cases remains an active research area.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Artin reciprocity gives the standard general reciprocity law for abelian extensions of global fields, resolving the intended class-field-theoretic generalization.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nModern class field theory supplies the requested reciprocity framework.\n\n**What remains.**\n\nFurther nonabelian reciprocity programmes are outside this historical statement.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Hilbert problems, accessed 2026-08-17. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Hilbert_problems\n Evidence used: Status overview identifies the classical resolution context for Hilbert's reciprocity problem.\n\n**Review notes.** The broad word arbitrary is read in the standard abelian class-field-theory sense.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 87, "problem_number": "HIL-011", "title": "Hilbert's 11th Problem: Quadratic Forms over Algebraic Number Fields", "statement": "Extend the theory of quadratic forms with algebraic numerical coefficients.", "background": "Hilbert's 11th problem (1900) concerns arithmetic of quadratic forms over number fields. Partial progress has been made through class field theory and the Hasse-Minkowski theorem, but general questions about representations remain open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The classical arithmetic theory of quadratic forms over algebraic number fields, including local--global classification tools, is conventionally treated as a resolution of Hilbert's eleventh problem.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe imported programme statement has a standard mature theory; later representation questions do not re-open the original programme.\n\n**What remains.**\n\nNo single residual claim is specified in this import.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Hilbert problems, accessed 2026-08-17. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Hilbert_problems\n Evidence used: Its status overview lists Problem 11 among consensus resolutions.\n\n**Review notes.** Broad formulation has no particular representation problem attached.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 88, "problem_number": "HIL-014", "title": "Hilbert's 14th Problem: Finite Generation of Rings", "statement": "Is the ring of invariants of a linear algebraic group acting on a polynomial ring always finitely generated?", "background": "Hilbert's 14th problem (1900) was answered negatively by Nagata in 1958, who found counterexamples. However, the problem remains interesting for special cases, and understanding when finite generation holds is an active area.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Nagata constructed counterexamples, so invariant rings for linear algebraic-group actions are not always finitely generated.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThis is a negative resolution of the universal assertion.\n\n**What remains.**\n\nClassify actions for which finite generation holds.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Hilbert problems, accessed 2026-08-17. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Hilbert_problems\n Evidence used: Status overview lists Problem 14 as resolved.\n\n**Review notes.** No statement change.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 4, "set_id": 2, "view_count": 176, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 89, "problem_number": "HIL-015", "title": "Hilbert's 15th Problem: Schubert's Enumerative Calculus", "statement": "Rigorously justify Schubert's enumerative geometry.", "background": "Hilbert's 15th problem (1900) calls for making Schubert's 19th century enumerative geometry rigorous. While intersection theory and Schubert calculus have been developed (Chow rings, Gromov-Witten theory), some classical problems remain open and new questions arise.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Intersection theory, Chow rings, and modern Schubert calculus give rigorous foundations for the classical enumerative calculus, while the programme has expanded into new enumerative questions.\n\n**Verified partial progress.**\n\n- Modern Schubert calculus rigorously computes many classical enumerative numbers via intersection theory.\n\n**Full solution or refutation.**\n\nThe foundational thrust is achieved, but the import is programme-like rather than a single criterion with an agreed terminal condition.\n\n**What remains.**\n\nFor a sharper label, identify a specific historical Schubert rule or enumerative assertion.\n\n**Sources checked.**\n\n- American Mathematical Society, The Calculus of Enumerative Geometry, book description, accessed 2026-08-17. (authoritative_secondary): https://bookstore.ams.org/CTM/11\n Evidence used: Identifies Hilbert's fifteenth problem as rigorous justification of Schubert calculus and presents the modern framework.\n\n**Review notes.** Programme wording retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 5, "set_id": 2, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 90, "problem_number": "HIL-017", "title": "Hilbert's 17th Problem: Expression of Definite Forms", "statement": "Can every non-negative rational function be expressed as a sum of squares of rational functions?", "background": "Hilbert's 17th problem (1900) was solved affirmatively by Artin in 1927: every non-negative polynomial can be written as a sum of squares of rational functions. However, questions about minimal representations and related problems in real algebraic geometry remain active.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Artin proved that every nonnegative polynomial is a sum of squares of rational functions, answering the imported question affirmatively.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThis is the accepted affirmative solution to Hilbert's seventeenth problem.\n\n**What remains.**\n\nQuantitative and denominator-complexity refinements are separate questions.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Hilbert problems, accessed 2026-08-17. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Hilbert_problems\n Evidence used: Status overview lists Problem 17 among consensus resolutions.\n\n**Review notes.** No source statement altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 4, "set_id": 2, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 91, "problem_number": "HIL-018", "title": "Hilbert's 18th Problem: Polyhedra and Space-Filling", "statement": "Are there only finitely many essentially different space-filling convex polyhedra? Is there a polyhedron which tiles space but not in a lattice arrangement?", "background": "Hilbert's 18th problem (1900) has multiple parts. Non-lattice tilings (aperiodic tilings) were discovered by Heesch and others. The Kepler conjecture about sphere packing was proved by Hales. However, classification questions about space-filling polyhedra remain open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The standard anisohedral/fundamental-domain question has an affirmative three-dimensional answer (Reinhardt, 1928), while the imported broad classification question has no formal equivalence relation or tiling convention.\n\n**Verified partial progress.**\n\n- Reinhardt settled Hilbert's second tiling question by an anisohedral three-dimensional tiling.\n- The associated Kepler sphere-packing component was later resolved by Hales.\n\n**Full solution or refutation.**\n\nThe explicit existence subquestion is solved; the unqualified finiteness/classification wording cannot be assigned a single exact status.\n\n**What remains.**\n\nSpecify convexity, face-to-face and symmetry conventions, and the equivalence relation for a precise classification problem.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Hilbert problems, accessed 2026-08-17. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Hilbert_problems\n Evidence used: Lists the original subproblem concerning a tile that is not a fundamental domain.\n- Wolfram MathWorld, Hilbert's Problems, accessed 2026-08-17. (authoritative_secondary): https://mathworld.wolfram.com/HilbertsProblems.html\n Evidence used: Current status overview attributes resolution of intended Problem 18 questions to Bieberbach, Reinhardt and Hales.\n\n**Review notes.** The broad first clause is flagged, not rewritten.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 6, "set_id": 2, "view_count": 289, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 2, "name": "hilbert_problems", "display_name": "Hilbert's 23 Problems", "description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.", "slug": "hilbert-problems", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 92, "problem_number": "GREEN-001", "title": "Large Sum-Free Sets", "statement": "Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $n/3 + \\Omega(n)$, where $\\Omega(n) \\to \\infty$ as $n \\to \\infty$?", "background": "This is a pretty old and increasingly notorious problem, first mentioned over 50 years ago. The best known bounds are in Bourgain's paper, where he shows that there is necessarily a sum-free set of size at least $(n+2)/3$. In fact, Eberhard, Manners and Green (unpublished) worked out a proof that Problem 1 has a positive solution under certain structural assumptions. It is known that there do exist sets with no sum-free set of size larger than $(1/3 + o(1))n$. However, the $o(1)$ term in these results is more-or-less ineffective; it would be interesting to get a reasonable bound. [1] P. Erdős. Extremal problems in number theory, In Proc. Sympos. Pure Math., Vol. VIII, pages 181–189. Amer. Math. Soc., Providence, R.I., 1965. [2] J. Bourgain, Estimates related to sumfree subsets of sets of integers, Israel J. Math. 97 (1997), 71–92. [3] S. Eberhard, Følner sequences and sum-free sets, Bull. Lond. Math. Soc. 47 (2015), no. 1, 21–28. [4] S. Eberhard, B. J. Green and F. Manners, Sets of integers with no large sum-free subset, Ann. of Math. (2) 180 (2014), no. 2, 621–652.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Bedert proves every n-element set of integers has a sum-free subset of size at least n/3+c log log n, giving the requested divergent additive improvement.\n\n**Verified partial progress.**\n\n- Bourgain previously obtained (n+2)/3.\n- Eberhard--Green--Manners give near-matching upper examples of size (1/3+o(1))n.\n\n**Full solution or refutation.**\n\nThe displayed existence question is answered affirmatively by the 2025 preprint.\n\n**What remains.**\n\nDetermine the optimal secondary term, which is beyond the stated question.\n\n**Sources checked.**\n\n- B. Bedert, Large sum-free subsets of sets of integers via L1-estimates for trigonometric series, arXiv:2502.08624 (2025). (primary): https://arxiv.org/abs/2502.08624\n Evidence used: Abstract states n/3+c log log n and that it answers the longstanding Erdős problem.\n- B. Green, 100 Open Problems, updated December 2025. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Problem marked solved in the maintained author notes.\n\n**Review notes.** A preprint is a primary source; no claim of peer review is made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_by": "Erdős and Cameron", "category_id": 2, "set_id": 3, "view_count": 145, "favorite_count": 8, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 93, "problem_number": "GREEN-002", "title": "Restricted Sumset Problem", "statement": "Let $A \\subset \\mathbb{Z}$ be a set of $n$ integers. Is there a subset $S \\subset A$ of size $(\\log n)^{100}$ such that $S \\hat{+} S$ is disjoint from $A$?", "background": "Here $S \\hat{+} S$ denotes the restricted sumset $\\{s_1 + s_2 : s_1, s_2 \\in S, s_1 \\neq s_2\\}$. Problems of this type are also at least 50 years old, being once again mentioned (and attributed to joint discussions of Erdős and Moser). It is known from very recent work of Sanders that there is always such an $S$ with $|S| \\geq (\\log n)^{1+c}$. By contrast the best-known upper bound is due to Ruzsa, showing that one cannot in general hope to take $|S|$ bigger than $e^{C\\sqrt{\\log n}}$. [1] P. Erdős. Extremal problems in number theory, In Proc. Sympos. Pure Math., Vol. VIII, pages 181–189. Amer. Math. Soc., Providence, R.I., 1965. [2] T. Sanders, The Erdős-Moser sum-free set problem, Canad. J. Math. 73 (2021), no. 1, 63–107. [3] I. Z. Ruzsa, Sum-avoiding subsets. Ramanujan J., 9 (2005) (1-2):77–82.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Polylogarithmic-size disjoint restricted-sumset subsets are known, but the requested log-power 100 scale is not established.\n\n**Verified partial progress.**\n\n- The source attributes a lower bound of log^(1+c) n to Sanders.\n- Ruzsa's construction gives an upper obstruction of exp(C sqrt(log n)).\n\n**Full solution or refutation.**\n\nThe large gap remains open.\n\n**What remains.**\n\nImprove the guaranteed subset size toward a fixed high log power.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, updated December 2025. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Problem statement and cited Sanders/Ruzsa bounds.\n\n**Review notes.** The exact bibliographic identity of the Sanders result should be checked before a stronger claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_by": "Erdős and Moser", "category_id": 2, "set_id": 3, "view_count": 123, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 95, "problem_number": "GREEN-005", "title": "Product-Free Sets in Finite Groups", "statement": "Which finite groups have the smallest largest product-free sets?", "background": "Kedlaya (2003) showed that every finite group $G$ of order $n$ has a product-free subset of size $\\gg n^{11/14}$, using the classification of finite simple groups. Understanding which groups achieve the minimum and improving bounds remains an open question connecting group theory and combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Kedlaya's n^(11/14) universal product-free lower bound gives nontrivial progress, but the extremal finite groups are not classified.\n\n**Verified partial progress.**\n\n- Every finite group of order n has a product-free subset of order at least n^(11/14).\n\n**Full solution or refutation.**\n\nNo minimising family/classification was located.\n\n**What remains.**\n\nIdentify extremal groups and the true minimum order of the largest product-free subset.\n\n**Sources checked.**\n\n- Ben Green problem collection, GREEN-005, checked 2026-08-17. (authoritative_secondary): https://www.unsolvedmath.com/problems/GREEN-005\n Evidence used: Kedlaya bound and current open framing.\n\n**Review notes.** No source statement changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_by": "Kedlaya", "proposed_year": 2003, "category_id": 4, "set_id": 3, "view_count": 134, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 96, "problem_number": "GREEN-007", "title": "Ulam's Sequence", "statement": "Define Ulam's sequence $1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, \\ldots$ where $u_1 = 1, u_2 = 2$, and $u_{n+1}$ is the smallest number uniquely expressible as $u_i + u_j$ for $i < j \\leq n$. Does this sequence have positive density? Can one explain its curious Fourier properties?", "background": "Ulam's sequence exhibits mysterious quasi-periodic behavior in its Fourier transform. While it appears to have density around $0.07$, proving it has positive density remains open. The sequence's additive structure and apparent regularity in numerical experiments are not well understood theoretically.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Positive density and an explanation of the observed Fourier phenomena for U(1,2) remain unproved.\n\n**Verified partial progress.**\n\n- Numerical work observes density near 0.07 and prominent Fourier structure.\n\n**Full solution or refutation.**\n\nNo rigorous positive-density theorem was found.\n\n**What remains.**\n\nProve positive density or give a rigorous structural description.\n\n**Sources checked.**\n\n- Ben Green problem collection, GREEN-007, checked 2026-08-17. (authoritative_secondary): https://www.unsolvedmath.com/problems/GREEN-007\n Evidence used: Current open framing and numerical context.\n\n**Review notes.** Recent speculative/AI notes are not used as a result.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "proposed_by": "Stanisław Ulam", "category_id": 1, "set_id": 3, "view_count": 187, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 97, "problem_number": "GREEN-008", "title": "Almost Sum-Free Sets", "statement": "Suppose that $A \\subset [N]$ has no more than $\\varepsilon N^2$ solutions to $x + y = z$. Can one remove $\\varepsilon' N$ elements to leave a sum-free set, where $\\varepsilon' \\to 0$ as $\\varepsilon \\to 0$, with a reasonable bound?", "background": "It is known that one can remove $\\varepsilon' N$ elements to obtain a sum-free set, but the quantitative dependence of $\\varepsilon'$ on $\\varepsilon$ is very poor. Finding explicit reasonable bounds would significantly improve our understanding of the structure of almost sum-free sets.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A qualitative removal theorem is known, but the requested effective/reasonable dependence remains an open quantitative problem.\n\n**Verified partial progress.**\n\n- One can delete epsilon-prime N elements with epsilon-prime tending to zero as epsilon tends to zero.\n\n**Full solution or refutation.**\n\nNo verified sharp quantitative dependence was located.\n\n**What remains.**\n\nGive explicit effective, preferably polynomial, removal bounds.\n\n**Sources checked.**\n\n- Ben Green problem collection, GREEN-008, checked 2026-08-17. (authoritative_secondary): https://www.unsolvedmath.com/problems/GREEN-008\n Evidence used: Qualitative theorem and quantitative gap.\n\n**Review notes.** The word reasonable has no formal threshold; status concerns the quantitative agenda.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 109, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 98, "problem_number": "GREEN-006", "title": "Sum-Free Subsets of [N]^d", "statement": "Fix an integer $d$. What is the largest sum-free subset of $[N]^d$?", "background": "This multi-dimensional generalization asks for the maximum size of a set in the $d$-dimensional grid with no solutions to $x + y = z$. Lepsveridze and Sun (2023) determined the constants $c_3, c_4, c_5$ and confirmed that the \"slice example\" is asymptotically optimal in these cases.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The asymptotic density is determined for d=3 and d=4, with the original preprint also reporting d=5; arbitrary fixed d remains open.\n\n**Verified partial progress.**\n\n- Lepsveridze--Sun solve the Cameron--Aydinian conjecture in dimensions 3 and 4.\n\n**Full solution or refutation.**\n\nThe all-fixed-d question is not resolved by the low-dimensional theorem.\n\n**What remains.**\n\nDetermine the maximum density for every fixed d.\n\n**Sources checked.**\n\n- S. Lepsveridze and Y. Sun, Size of the largest sum-free subset of [n]^3, [n]^4, and [n]^5, arXiv:2311.18289 (2023). (primary): https://arxiv.org/abs/2311.18289\n Evidence used: Abstract reports d=3,4,5.\n- S. Lepsveridze and Y. Sun, Size of the Largest Sum-Free Subset of [n]^3 and [n]^4, IMRN (2026). (primary): https://doi.org/10.1093/imrn/rnag081\n Evidence used: Published d=3,4 result.\n\n**Review notes.** The preprint/journal dimension discrepancy is made explicit.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 118, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 99, "problem_number": "GREEN-009", "title": "Progressions in Subsets of Z/NZ", "statement": "Is $r_5(N) \\ll N(\\log N)^{-c}$? Is $r_4(\\mathbb{F}_5^n) \\ll N^{1-c}$ where $N = 5^n$?", "background": "Here $r_k(N)$ denotes the maximum size of a subset of $\\{1, \\ldots, N\\}$ with no $k$-term arithmetic progression. Kelley-Meka (2024) resolved the $k=3$ case. For $k \\geq 5$, Leng-Sah-Sawhney (2024) proved bounds of shape $r_k(N) \\ll Ne^{-(\\log \\log N)^{c_k}}$. Finding polynomial savings remains a central challenge in additive combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Recent work gives stretched-double-log savings for k>=5, not the requested log-power saving; the finite-field 4-AP power saving also remains open.\n\n**Verified partial progress.**\n\n- Leng--Sah--Sawhney prove r_k(N) at most N exp(-(log log N)^c_k) for k>=5.\n- The k=3 problem has been dramatically advanced but is not the displayed k=5/finite-field k=4 question.\n\n**Full solution or refutation.**\n\nNeither displayed estimate is verified.\n\n**What remains.**\n\nObtain polynomial logarithmic/exponential savings in the stated settings.\n\n**Sources checked.**\n\n- Ben Green problem collection, GREEN-009, checked 2026-08-17. (authoritative_secondary): https://www.unsolvedmath.com/problems/GREEN-009\n Evidence used: Current stated advances and residual targets.\n\n**Review notes.** Do not conflate Kelley--Meka's 3-AP work with this question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 142, "favorite_count": 8, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 100, "problem_number": "GREEN-010", "title": "Roth's Theorem with Random Common Differences", "statement": "Let $S \\subset \\mathbb{N}$ be random. Under what conditions is Roth's theorem for progressions of length 3 true with common differences in $S$?", "background": "This asks when Roth's theorem holds if we restrict common differences to a random set. Briët and Castro-Silva (2023) advanced bounds for odd $k$. The problem explores how randomness interacts with additive structure.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Briët--Castro-Silva establish upper threshold bounds for random differences, with polynomial improvements for odd progression lengths; the broad classification question remains open.\n\n**Verified partial progress.**\n\n- For length k, their upper critical-size bound is N^(1-2/k+o(1)).\n\n**Full solution or refutation.**\n\nThis does not provide a complete necessary-and-sufficient condition for the 3-AP formulation.\n\n**What remains.**\n\nDetermine the sharp threshold/conditions, especially for Roth's 3-AP case.\n\n**Sources checked.**\n\n- J. Briët and D. Castro-Silva, On the threshold for Szemerédi's theorem with random differences, Electron. J. Combin. 31 (2024), P4.8. (primary): https://doi.org/10.37236/12415\n Evidence used: Published threshold upper bound.\n\n**Review notes.** The source asks an intentionally broad conditions question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 126, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 102, "problem_number": "GREEN-012", "title": "Tuples in Dense Sets", "statement": "Let $G$ be an abelian group of size $N$, and suppose that $A \\subset G$ has density $\\alpha$. Are there at least $\\alpha^{15}N^{10}$ tuples $(x_1, \\ldots, x_5, y_1, \\ldots, y_5) \\in G^{10}$ such that $x_i + y_j \\in A$ whenever $j \\in \\{i, i+1, i+2\\}$?", "background": "This problem asks about higher-order additive structures in dense sets. Deng-Tidor-Zhao (2023) considered this problem and conjectured a negative answer, suggesting the exponent might not be optimal.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Deng--Tidor--Zhao considered the configuration and conjectured a negative answer, but no verified theorem resolving the exact alpha^15 lower bound was located.\n\n**Verified partial progress.**\n\n- The problem is connected to recent work on uniform sets with few patterns.\n\n**Full solution or refutation.**\n\nA conjectural negative answer is not classified as a refutation.\n\n**What remains.**\n\nProduce a counterexample or prove the stated lower bound.\n\n**Sources checked.**\n\n- M. Deng, J. Tidor, Y. Zhao, Uniform sets with few progressions via colourings, Math. Proc. Camb. Phil. Soc. 179 (2025), 79--103. (primary): https://doi.org/10.1017/S0305004125000106\n Evidence used: Related modern pattern work; not asserted to resolve this exact tuple inequality.\n\n**Review notes.** The claimed negative conjecture is intentionally not upgraded to proof.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 108, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 103, "problem_number": "GREEN-013", "title": "4-term APs in Fourier Uniform Sets", "statement": "Suppose that $A \\subset \\mathbb{Z}/N\\mathbb{Z}$ has density $\\alpha$ and is Fourier uniform (all Fourier coefficients of $1_A - \\alpha$ are $o(N)$). Does $A$ contain at least $\\gg \\alpha^{100}N^2$ 4-term arithmetic progressions?", "background": "Fourier uniformity means the set \"looks random\" from a Fourier perspective. The question asks if this forces many 4-APs. Deng-Tidor-Zhao (2023) conjectured a negative answer, suggesting Fourier uniformity alone may not suffice.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Fourier uniformity alone does not control 4-AP counts at the random scale, and recent work studies very few 4-APs; no source was found proving or refuting the exact alpha^100 lower bound.\n\n**Verified partial progress.**\n\n- Deng--Tidor--Zhao reduce a broader few-4-AP question to a colouring problem and establish related even-length results.\n\n**Full solution or refutation.**\n\nThe exact displayed lower bound remains unverified.\n\n**What remains.**\n\nConstruct Fourier-uniform counterexamples below alpha^100, or prove the bound.\n\n**Sources checked.**\n\n- M. Deng, J. Tidor, Y. Zhao, Uniform sets with few progressions via colourings, Math. Proc. Camb. Phil. Soc. 179 (2025), 79--103. (primary): https://doi.org/10.1017/S0305004125000106\n Evidence used: Primary discussion of Fourier-uniform sets with few 4-APs.\n\n**Review notes.** Fourier uniformity versus random-count behaviour is kept distinct from the requested alpha^100 threshold.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 115, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 104, "problem_number": "GREEN-015", "title": "Lipschitz AP-Free Graphs", "statement": "Does there exist a Lipschitz function $f : \\mathbb{N} \\to \\mathbb{Z}$ whose graph $\\Gamma = \\{(n, f(n)) : n \\in \\mathbb{Z}\\} \\subset \\mathbb{Z}^2$ is free of 3-term progressions?", "background": "This asks whether a \"smooth\" (Lipschitz) function can have a graph avoiding arithmetic progressions. The Lipschitz condition prevents wildly oscillating behavior, making AP-avoidance more constrained.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The 3-term Lipschitz-graph problem remains open in Green's January 2026 list; a 4-term analogue is known.\n\n**Verified partial progress.**\n\n- Brown, Jungić, and Poelstra develop the equivalent bounded-gap/double-3-term-progression formulation and retain it as an open problem.\n- Cassaigne, Currie, Schaeffer, and Shallit construct an infinite word avoiding three consecutive equal-length, equal-sum blocks, yielding the positive 4-term analogue rather than the requested 3-term result.\n\n**Full solution or refutation.**\n\nNo construction or impossibility theorem for the exact 3-term problem was found.\n\n**What remains.**\n\nResolve the 3-term problem after fixing whether the intended domain is N or Z.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 15, January 2026 revision, checked 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Explicitly says the additive-word formulation of Problem 15 is unsolved and records the 4-term construction.\n- Tom Brown, Veselin Jungić, and Andrew Poelstra, On Double 3-Term Arithmetic Progressions, arXiv:1304.1829 (2013/2014). (primary): https://arxiv.org/abs/1304.1829\n Evidence used: Studies the equivalent bounded-gap sequence problem and identifies the double 3-term progression question as open.\n- Julien Cassaigne, James Currie, Luke Schaeffer, and Jeffrey Shallit, Avoiding Three Consecutive Blocks of the Same Size and Same Sum, arXiv:1106.5204. (primary): https://arxiv.org/abs/1106.5204\n Evidence used: Proves the three-block additive-cube avoidance result underlying the 4-term analogue.\n\n**Review notes.** The preserved statement says f:N->Z but defines the graph using every n in Z; the same inconsistency is present in Green's PDF.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 121, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 105, "problem_number": "GREEN-016", "title": "Linear Equation x + 3y = 2z + 2w", "statement": "What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?", "background": "This asks about sets avoiding a specific linear configuration. Understanding which linear equations are easier or harder to avoid is a fundamental question in additive combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The extremal order remains unknown between a square-root lower bound and a subdensity upper bound.\n\n**Verified partial progress.**\n\n- Writing f(N) for the maximum, Ruzsa's construction gives N^(1/2) << f(N).\n- Schoen and Sisask give f(N) << N exp(-c(log N)^(1/7)); Green points to Section 9 of their paper.\n\n**Full solution or refutation.**\n\nThe known bounds leave a large gap and do not determine the largest solution-free subset.\n\n**What remains.**\n\nDetermine the asymptotic order of f(N) or substantially narrow the gap.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 16, January 2026 revision, checked 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Retains the problem and records N^(1/2) << f(N) << N exp(-c(log N)^(1/7)), with attributions.\n- Tomasz Schoen and Olof Sisask, Roth's theorem for four variables and additive structures in sums of sparse sets, arXiv:1408.2568; Forum Math. Sigma 4 (2016), e5. (primary): https://arxiv.org/abs/1408.2568\n Evidence used: Contains the four-variable Roth/sparse-sumset machinery cited by Green for the upper bound.\n\n**Review notes.** No later improvement specific to this equation was verified.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 98, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 106, "problem_number": "GREEN-017", "title": "Progressions in F_3^n with Boolean Common Differences", "statement": "Suppose that $A \\subset \\mathbb{F}_3^n$ is a set of density $\\alpha$. Under what conditions on $\\alpha$ is $A$ guaranteed to contain a 3-term progression with nonzero common difference in $\\{0, 1\\}^n$?", "background": "This constrains the progression to have Boolean-like common differences. Bhangale-Khot-Minzer (2023) showed sets avoiding such progressions have density $\\ll_p (\\log \\log \\log n)^{-c_p}$, using extraordinarily difficult techniques.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A 2025 theorem gives a finite-iterated-log sufficient density for the exact {0,1}^n-difference problem, but the optimal threshold is unknown.\n\n**Verified partial progress.**\n\n- Bhangale, Khot, and Minzer proved density O_p((log log log n)^(-c_p)) for avoiding progressions with differences in the larger set {0,1,2}^n.\n- Bhangale, Khot, Liu, and Minzer proved that density at least C(log log log n)^(-c) suffices for the original {0,1}^n-difference problem.\n- A 2026 slice-rank note obtains exponential savings when the allowed one-coordinate difference set has size greater than (q+1)/2, but explicitly leaves the two-element case open.\n\n**Full solution or refutation.**\n\nThe first reasonable threshold is known, but it is not believed to be optimal and does not decide the possible exponential-scale threshold.\n\n**What remains.**\n\nDetermine the true threshold, including whether a bound of the form (1-c)^n suffices.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 17 and 2023/2025 updates, January 2026 revision, checked 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Distinguishes the {0,1,2}^n result from the later theorem for the original {0,1}^n question and retains the optimum as open.\n- Amey Bhangale, Subhash Khot, and Dor Minzer, Effective Bounds for Restricted 3-Arithmetic Progressions in F_p^n, arXiv:2308.06600; Discrete Analysis 2024:16. (primary): https://arxiv.org/abs/2308.06600\n Evidence used: Proves the triple-logarithmic density bound for common differences in {0,1,2}^n.\n- Amey Bhangale, Subhash Khot, Yang P. Liu, and Dor Minzer, On Approximability of Satisfiable k-CSPs: VI, arXiv:2411.15133 (2024; FOCS 2025 version titled On inverse theorems and combinatorial lines). (primary): https://arxiv.org/abs/2411.15133\n Evidence used: Its additive-combinatorics application gives the first reasonable bound for the original restricted 3-AP problem, including S={0,1}.\n\n**Review notes.** The dataset background conflates the relaxed {0,1,2}^n theorem with the later exact {0,1}^n result. Green's separate supersaturation question is omitted from the dataset statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 104, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 107, "problem_number": "GREEN-018", "title": "Corner Problem in Product Sets", "statement": "Suppose $G$ is a finite group, and let $A \\subset G \\times G$ be a subset of density $\\alpha$. Are there $\\gg_\\alpha |G|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$ all lie in $A$?", "background": "This is a \"corner-type\" problem in the group product setting. Dense sets should contain many axis-aligned corners. The problem connects additive combinatorics with group theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Strong special-case and corner-free-set bounds are known, but the uniform all-finite-groups lower bound of c(alpha)|G|^3 naïve corners remains open.\n\n**Verified partial progress.**\n\n- Austin proves strong counting statements for sufficiently quasirandom groups.\n- Recent corner theorems give strong density upper bounds for corner-free sets in abelian groups and, through additional arguments, existence bounds for broad nonabelian classes.\n- The maintained formal-conjecture entry still marks the uniform supersaturation assertion as open.\n\n**Full solution or refutation.**\n\nExistence and quasirandom-group results do not establish the requested cubic count uniformly over all finite groups.\n\n**What remains.**\n\nProve or refute a uniform c(alpha)|G|^3 lower bound for nontrivial naïve corners in every sufficiently large finite group.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 18, January 2026 revision, checked 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the all-groups counting problem and records Austin's quasirandom-group partial theorem.\n- Tim Austin, Ajtai-Szemerédi Theorems over quasirandom groups, arXiv:1503.08746; Recent Trends in Combinatorics (2016), 453-484. (primary): https://arxiv.org/abs/1503.08746\n Evidence used: Proves strong forms of the relevant corner theorems under quasirandomness hypotheses.\n- Michael Jaber, Yang P. Liu, Shachar Lovett, Anthony Ostuni, and Mehtaab Sawhney, Quasipolynomial bounds for the corners theorem, arXiv:2504.07006. (primary): https://arxiv.org/abs/2504.07006\n Evidence used: Corollary 1.8 gives a quasipolynomial density bound for sets with no nontrivial naïve corner in an arbitrary finite group; it is an existence/density result rather than the requested cubic supersaturation count.\n- Formal Conjectures, Ben Green Open Problem 18, checked 2026-08-17. (authoritative_secondary): https://firsching.ch/formal-conjectures/src/FormalConjectures/GreensOpenProblems/%C2%AB18%C2%BB/\n Evidence used: Formalizes the nontrivial g!=e count and labels the c(alpha)|G|^3 statement research-open.\n\n**Review notes.** The shortened Green/dataset statement omits the original condition g!=e. Identity-difference triples contribute only O(|G|^2), so the omission does not trivially imply the cubic conclusion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 110, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 108, "problem_number": "GREEN-020", "title": "Multidimensional Szemerédi Theorem Bounds", "statement": "Find reasonable bounds for instances of the multidimensional Szemerédi theorem.", "background": "Szemerédi's theorem extends to multiple dimensions (finding combinatorial lines in dense sets). Pohoata-Zakharov (2024) improved bounds for skew corners to $N^{5/4}$. Quantitative bounds remain a major challenge.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Reasonable quantitative bounds are now known for several multidimensional configurations, but the umbrella programme remains open for basic cases such as squares and 3-dimensional corners.\n\n**Verified partial progress.**\n\n- Jaber, Liu, Lovett, Ostuni, and Sawhney proved quasipolynomial bounds for ordinary corners in finite abelian groups.\n- Peluse proved finite-iterated-log bounds for an L-shaped four-point configuration.\n- Guo, Miao, and Zhan (May 2026) proved logarithmic-density bounds for multidimensional configurations with distinct polynomial coordinate directions.\n- For skew corners, the dataset's N^(5/4) lower bound was quickly superseded by a nearly quadratic construction; 2025 work also improved upper bounds.\n\n**Full solution or refutation.**\n\nThis is a programme rather than one theorem; meaningful subclasses are solved quantitatively, while central configurations remain unresolved.\n\n**What remains.**\n\nObtain reasonable bounds for axis-parallel squares, 3-dimensional corners, and other unresolved multidimensional patterns.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 20 and updates through 2025, January 2026 revision, checked 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Defines reasonable bounds, inventories solved configurations, and identifies squares and 3-dimensional corners as open.\n- Michael Jaber, Yang P. Liu, Shachar Lovett, Anthony Ostuni, and Mehtaab Sawhney, Quasipolynomial bounds for the corners theorem, arXiv:2504.07006. (primary): https://arxiv.org/abs/2504.07006\n Evidence used: Proves a corner-free density bound exp(-(log |G|)^Omega(1)) for finite abelian groups.\n- Jingwei Guo, Changxing Miao, and Guoqing Zhan, A multidimensional Szemerédi theorem in integers, arXiv:2605.06360 (2026). (primary): https://arxiv.org/abs/2605.06360\n Evidence used: Proves a (log N)^(-c) density threshold for a new family of multidimensional polynomial-direction configurations.\n\n**Review notes.** The record is deliberately broad. Its Pohoata-Zakharov N^(5/4) background is now superseded and cannot summarize the full problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 127, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 109, "problem_number": "GREEN-021", "title": "Large Sieve and Quadratic Sets", "statement": "Suppose that a large sieve process leaves a set of quadratic size. Is that set quadratic?", "background": "Sieve methods remove arithmetic structure from sets. This problem asks whether a set that \"survives\" a large sieve and has size $\\sim N^2$ must actually be a quadratic sequence or similar structured set. Understanding the structure of sieved sets is fundamental in analytic number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The quadratic inverse large-sieve conjecture remains open, but a July 2026 paper proves the strongest verified direct intersection-with-a-quadratic result.\n\n**Verified partial progress.**\n\n- Green and Harper prove several inverse results and explicitly state that they cannot solve the simplest quadratic inverse conjecture.\n- Croot and Yip improve Hanson's logarithmic intersection bound: a square-root-sized ill-distributed set contains exp(c sqrt(log N)/log log N) elements in the image of one quadratic.\n\n**Full solution or refutation.**\n\nA large structured intersection is now forced, but global containment in a single quadratic image—the conjectured dichotomy—is not known.\n\n**What remains.**\n\nUpgrade partial concentration on a quadratic image to the proposed containment/density-loss dichotomy, or find a counterexample.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 47, January 2026 revision, checked 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Gives the precise residue-class hypothesis, square-root scale, and quadratic-image dichotomy.\n- Ben Green and Adam Harper, Inverse questions for the large sieve, arXiv:1311.6176; GAFA 24 (2014), 1167-1203. (primary): https://arxiv.org/abs/1311.6176\n Evidence used: Formulates the inverse problem, proves partial results, and explicitly says the problem itself is not solved.\n- Ernie Croot and Chi Hoi Yip, A weighted entropy approach for the quadratic inverse large sieve conjecture, arXiv:2607.15311 (2026). (primary): https://arxiv.org/abs/2607.15311\n Evidence used: Proves an exp(c sqrt(log N)/log log N)-sized intersection with a single quadratic image under the finite quadratic-inverse-sieve hypotheses.\n\n**Review notes.** The dataset background's size ~N^2 gloss is wrong: the relevant cardinality scale up to X is X^(1/2), as for values of a quadratic polynomial.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 87, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 110, "problem_number": "GREEN-022", "title": "Small Sieve Maximal Sets", "statement": "Suppose that a small sieve process leaves a set of maximal size. What is the structure of that set?", "background": "When a small sieve (sieving by small primes) leaves the maximum possible density of survivors, what structure must the original set have? This connects sieve theory with the structural theory of sets in number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The inverse small-sieve programme remains open, and Green explicitly says that a satisfactory precise formulation is itself unclear.\n\n**Verified partial progress.**\n\n- In a model removing one residue class modulo each prime p<=sqrt(X), Selberg/Rosser-Iwaniec upper-bound sieves leave at most (2+o(1))X/log X survivors.\n- The factor-2 extremal phenomenon can be tied to a Siegel-zero obstruction; Green proposes characterizing examples arising from that mechanism.\n- For sieve dimension kappa=1/2 the optimal size is known, suggesting a more tractable inverse problem of classifying extremizers related to sums of two squares.\n\n**Full solution or refutation.**\n\nThere is no single settled theorem because the source heading intentionally describes a family of possible inverse problems.\n\n**What remains.**\n\nChoose a precise sieve model and structural equivalence, then characterize all asymptotically extremal survivor sets.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 48, January 2026 revision, checked 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States that a good formulation is unclear and lays out the Siegel-zero and sieve-dimension candidate versions.\n\n**Review notes.** The dataset does not define small sieve process, maximality, or the intended structural equivalence; it is a programme, not a determinate yes/no proposition.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 82, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 111, "problem_number": "GREEN-023", "title": "Large Cosets in Iterated Sumsets", "statement": "Suppose that $A \\subset \\mathbb{F}_2^n$ has density $\\alpha$. Does $10A$ contain a coset of some subspace of dimension at least $n - O(\\log(1/\\alpha))$?", "background": "This asks how many times we must add a set to itself before it contains a large subspace coset. Kosciuszko (2024), building on Konyagin, showed that $mA - mA$ contains a subspace of dimension $\\geq n - O(\\log^{3+\\eta}(1/\\alpha))$ for suitable $m$. The problem asks if fewer iterations suffice.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Polynomial Bogolyubov conjecture remains open and is not implied by the 2023 Polynomial Freiman-Ruzsa theorem.\n\n**Verified partial progress.**\n\n- Sanders obtains a polylogarithmic-codimension result, stated by Green as n-O(log^(4+o(1))(1/alpha)), in a small iterated sumset.\n- Kościuszko, building on Konyagin, proves that for every eta>0 there is an m for which mA-mA contains a subspace of codimension O(log^(3+eta)(1/alpha)).\n- Green explicitly notes that the Polynomial Freiman-Ruzsa solution does not address this stronger containment problem.\n\n**Full solution or refutation.**\n\nKnown theorems miss both the desired linear logarithmic codimension and/or the fixed 10A target.\n\n**What remains.**\n\nShow that a fixed bounded iterated sumset, specifically 10A, contains a coset of codimension O(log(1/alpha)).\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 50 and 2024 update, January 2026 revision, checked 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Labels the problem Polynomial Bogolyubov, distinguishes it from solved PFR, and records the Sanders and Kościuszko bounds.\n- Tom Sanders, On the Bogolyubov-Ruzsa lemma, arXiv:1011.0107; Analysis & PDE 5 (2012), 627-655. (primary): https://arxiv.org/abs/1011.0107\n Evidence used: Proves the quasipolynomial Bogolyubov-Ruzsa-type structural bound underlying the maintained best estimate.\n- Tomasz Kościuszko, Counting solutions to invariant equations in dense sets, arXiv:2306.08567, Theorem 9. (primary): https://arxiv.org/abs/2306.08567\n Evidence used: Contains the improved iterated-sumset/subspace bound cited in Green's 2024 update.\n\n**Review notes.** The dataset background is broadly consistent with Green's update; the exact quantifier is that m may depend on eta but not on A or n.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 93, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 112, "problem_number": "GREEN-024", "title": "Largest Coset in 2A", "statement": "Suppose that $A \\subset \\mathbb{F}_2^n$ has density $\\alpha$. What is the largest size of coset guaranteed to be contained in $2A$?", "background": "This asks for the largest affine subspace (coset) contained in the doubling $2A = A + A$. Unlike the previous problem about many iterations, this focuses on just $2A$. Determining the optimal bound is a fundamental question in additive combinatorics over $\\mathbb{F}_2^n$.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The optimal affine-subspace size guaranteed inside 2A is unknown, despite linear-dimension lower bounds and n-sqrt(n)-scale obstructions.\n\n**Verified partial progress.**\n\n- For fixed alpha>0, 2A contains a coset of dimension at least c(alpha)n; Sanders's refinement gives dimension on the order of alpha n.\n- There are constant-density examples for which 2A contains no coset of dimension n-sqrt(n).\n- Near density 1/2, Sanders proves codimension-one and o(n)-codimension theorems in specified regimes and asks for an O_K(1) transition result.\n\n**Full solution or refutation.**\n\nThe general extremal function of alpha and n remains undetermined.\n\n**What remains.**\n\nDetermine the largest guaranteed dimension/cardinality, including its dependence on alpha and the transition near alpha=1/2.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 51, January 2026 revision, checked 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Records the linear-dimension lower bound, n-sqrt(n) obstruction, and near-half-density subproblem.\n- Tom Sanders, Green's sumset problem at density one half, arXiv:1003.5649; Acta Arith. 146 (2011), 91-101. (primary): https://arxiv.org/abs/1003.5649\n Evidence used: Proves codimension-one and o(n)-codimension results near density one half and records the relevant obstruction.\n- Formal Conjectures, Ben Green Open Problem 51, checked 2026-08-17. (authoritative_secondary): https://firsching.ch/formal-conjectures/src/FormalConjectures/GreensOpenProblems/%C2%AB51%C2%BB/\n Evidence used: Keeps the extremal function and near-half-density assertion research-open while separating known lower and upper results.\n\n**Review notes.** The statement asks for a function of both alpha and n, not merely a yes/no answer.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 88, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 113, "problem_number": "GREEN-025", "title": "Additive Complements and Cosets", "statement": "Suppose that $A \\subset \\mathbb{F}_2^n$ has an additive complement of size $K$. Does $2A$ contain a coset of codimension $O_K(1)$?", "background": "If $A$ has a small additive complement (a set $B$ with $A + B = \\mathbb{F}_2^n$), does this force $2A$ to contain a large coset? This problem explores the relationship between additive complements and the structure of sumsets.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact O_K(1) codimension question remains open, while the stronger O(log K) variant was refuted in a 2025 update.\n\n**Verified partial progress.**\n\n- Green reports independent Hamming-ball observations by Kaave Hosseini and Ryan Alweiss: a ball of radius n/2-sqrt(n) has an additive complement of size O(n), but every subspace in 2A has codimension Omega(sqrt(n)).\n- This disproves O(log K), since K=O(n), but does not disprove a bound depending arbitrarily on K.\n- The stronger logarithmic statement would have implied Polynomial Bogolyubov, clarifying why the obstruction is significant.\n\n**Full solution or refutation.**\n\nOnly the proposed logarithmic quantitative strengthening is refuted; the preserved qualitative O_K(1) assertion is still open.\n\n**What remains.**\n\nProve or disprove the existence of some codimension bound depending only on K; any valid bound must exceed logarithmic growth along the Hamming-ball family.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 52 and 2025 update, January 2026 revision, checked 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Retains the O_K(1) question and gives the Hamming-ball counterexample to the separate O(log K) strengthening.\n\n**Review notes.** Do not classify the dataset's exact question as disproved: the counterexample has K growing with n and only rules out the specific logarithmic dependence.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 91, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 114, "problem_number": "GREEN-026", "title": "Partitions and Large Cosets", "statement": "Suppose that $\\mathbb{F}_2^n$ is partitioned into sets $A_1, \\dots, A_K$. Does $2A_i$ contain a coset of codimension $O_K(1)$ for some $i$?", "background": "When partitioning a vector space into $K$ parts, at least one part must have substantial additive structure. This problem asks if one piece must have a doubling containing a large coset. It's a partitioning variant of the previous coset problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The partition-coset assertion remains open; Green's maintained collection says it is not known even for K=3.\n\n**Verified partial progress.**\n\n- The Hamming-ball example shows that a single candidate part can have 2A containing no bounded-codimension coset, reducing one concrete subcase to ruling out a cover by three such balls in different bases.\n- An O(log K) codimension version would imply polynomial Bogolyubov and polynomial Freiman--Ruzsa consequences, but no such theorem is claimed.\n\n**Full solution or refutation.**\n\nNo verified proof or counterexample to the stated partition theorem was located.\n\n**What remains.**\n\nProve that some part has the required coset, already for K=3, or construct a partition disproving it.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, Problem 53, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States that the problem is not known even for K=3, records the Hamming-ball obstruction, and explains the polynomial Freiman--Ruzsa connection.\n\n**Review notes.** The local label GREEN-026 matches Problem 53 in Green's current PDF by exact statement; the local ID and statement were preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 86, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 116, "problem_number": "GREEN-028", "title": "Gowers Box Norms over Finite Fields", "statement": "Let $p$ be an odd prime and suppose $f : \\mathbb{F}_p^n \\times \\mathbb{F}_p^n \\to \\mathbb{C}$ is bounded pointwise by 1. Suppose $\\mathbb{E}_h \\|\\Delta_{(h,h)}f\\|_\\square^4 \\geq \\delta$. Does $f$ correlate with a function of the form $a(x)b(y)c(x+y)(-1)^{q(x,y)}$?", "background": "This asks for an inverse theorem for a particular Gowers norm in product spaces over finite fields. Understanding which structured functions correlate with high Gowers norm is central to higher-order Fourier analysis.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The intended box-norm inverse theorem remains open in Green's maintained collection, which describes it as the simplest Gowers-type norm for which an inverse theorem is unknown.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo verified inverse theorem of the displayed form was located.\n\n**What remains.**\n\nClarify the class and codomain of q and the intended phase, then prove the stated structural correlation or find a counterexample.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, Problem 55, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Gives the same formula, defines the diagonal derivative and box configuration, and says the inverse theorem is unknown.\n\n**Review notes.** The local label GREEN-028 matches Problem 55 by statement. Both the imported record and Green PDF write (-1)^{q(x,y)} for odd p without defining q; this potentially non-intrinsic phase notation was flagged, not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 84, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 117, "problem_number": "GREEN-029", "title": "Inverse Theorem for Gowers Norms", "statement": "Determine bounds for the inverse theorem for Gowers norms.", "background": "The inverse theorem characterizes functions with large Gowers $U^{s+1}$ norm. Leng-Sah-Sawhney (2024) established a quasi-polynomial inverse theorem for $\\|\\cdot\\|_{U^{s+1}[N]}$ norms for all $s \\geq 3$. Improving to polynomial bounds remains a major challenge in additive combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Major quantitative advances give quasipolynomial inverse theorems over the integers and for finite-field U^4, plus broader U^4 theory and nearly-polynomial bounds for restricted polynomial inputs, but a general polynomial inverse theorem remains open.\n\n**Verified partial progress.**\n\n- Leng--Sah--Sawhney prove quasipolynomial bounds for the U^{s+1}[N] inverse theorem for all s at least 3.\n- Milićević proves a quasipolynomial U^4 inverse theorem in finite vector spaces and, in 2026, quantitative quasipolynomial U^4 inverse theorems for general finite abelian groups.\n- Milo--Moshkovitz prove a nearly-polynomial theorem over finite fields for polynomial inputs of degree d+1 and homogeneous inputs below degree 2d; this is not the general bounded-function conjecture.\n\n**Full solution or refutation.**\n\nThe broad bounds agenda has substantial partial solutions, but the polynomial dependence sought in the canonical finite-field formulation is not known in general and the integer version has not even been fully formulated at that strength.\n\n**What remains.**\n\nFormulate and prove a truly polynomial inverse theorem in the unresolved general settings, especially higher-order finite-field bounded functions and the integer nilsequence formulation.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, Problem 56, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Sets out the polynomial Gowers inverse conjecture and records the 2023--2025 quantitative advances and remaining polynomial gap.\n- J. Leng, A. Sah, M. Sawhney, Quasipolynomial bounds on the inverse theorem for the Gowers U^{s+1}[N]-norm, arXiv:2402.17994 (2024). (primary): https://arxiv.org/abs/2402.17994\n Evidence used: Abstract states a quasipolynomial inverse theorem for U^{s+1}[N].\n- L. Milićević, Quasipolynomial inverse theorem for the U^4(F_p^n) norm, arXiv:2410.08966 (2024). (primary): https://arxiv.org/abs/2410.08966\n Evidence used: Abstract states a quasipolynomial U^4 inverse theorem in finite vector spaces.\n- L. Milićević, General inverse theory for the U^4 norm, arXiv:2601.01682 (2026). (primary): https://arxiv.org/abs/2601.01682\n Evidence used: Abstract gives quantitative quasipolynomial U^4 inverse theorems in general finite abelian groups.\n- T. Milo, G. Moshkovitz, Nearly-polynomial inverse theorem for the U^d norm in degree d+1, arXiv:2603.16836v2 (2026). (primary): https://arxiv.org/abs/2603.16836\n Evidence used: Abstract explicitly limits the nearly-polynomial theorem to polynomial inputs of degree d+1 and related homogeneous inputs.\n\n**Review notes.** The local label GREEN-029 matches Problem 56 by statement. The imported wording is an agenda rather than one fully quantified theorem, so partial status reflects bounds progress without implying a complete resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 95, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 118, "problem_number": "GREEN-030", "title": "Φ(G) and Φ'(G) Coincidence", "statement": "Do $\\Phi(G)$ and $\\Phi'(G)$ coincide?", "background": "This asks whether two different notions of the Frattini-like subgroup of $G$ are equal. The Frattini subgroup consists of non-generators; different definitions can arise in different contexts. Determining their equivalence has implications for group theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact imported statement is not independently meaningful because Phi(G) and Phi'(G) are undefined, and its background incorrectly identifies them as Frattini-like subgroups; Green's intended additive-combinatorics Problem 57 remains open.\n\n**Verified partial progress.**\n\n- In the source, Phi(G) is a convex hull of generalized convolutions on an abelian group, while Phi'(G) imposes that one factor depend only on x1+x2.\n- Green conjecturally expects the two source-defined function spaces not to coincide, but gives no resolution.\n\n**Full solution or refutation.**\n\nNo status can be assigned to the literal undefined statement without importing definitions that the record omits; the identifiable intended source problem is still open.\n\n**What remains.**\n\nCorrect the record by supplying the exact source definitions and removing the erroneous Frattini interpretation, then triage the resulting well-posed function-space equality.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, Problem 57, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Defines both spaces as classes of functions on an abelian group, links them to generalized convolution algebras and UAP^2 functions, and guesses the answer is no.\n\n**Review notes.** The local label GREEN-030 matches Problem 57 only after consulting the source. The imported category algebra and Frattini-subgroup background are substantively wrong. No silent repair was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 4, "view_count": 73, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 119, "problem_number": "GREEN-031", "title": "Sumsets Containing Composites", "statement": "Suppose $A, B \\subset \\{1, \\dots, N\\}$ both have size $N^{0.49}$. Does $A + B$ contain a composite number?", "background": "This asks whether sumsets of moderately large sets must contain composite numbers. Since primes have density $1/\\log N$, sets of size $N^{0.49}$ are much denser, suggesting their sumset should hit composites. However, proving this rigorously requires understanding the additive structure of primes.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The N^0.49 sumset-composite question remains open; the maintained collection records a conditional route through the inverse large sieve, not an unconditional theorem.\n\n**Verified partial progress.**\n\n- Green--Harper show that suitable control of the inverse large sieve would imply a positive answer.\n- A positive answer would rule out an eventual additive decomposition of the primes via earlier work of Elsholtz.\n\n**Full solution or refutation.**\n\nNo unconditional proof that A+B contains a composite, and no counterexample, was located.\n\n**What remains.**\n\nProve the assertion below the square-root threshold, obtain the needed inverse-large-sieve theorem, or construct counterexamples.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, Problem 58, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Retains the question as open and states the Green--Harper inverse-large-sieve implication and inverse-Goldbach connection.\n\n**Review notes.** The local label GREEN-031 matches Problem 58 by exact statement; no wording was changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 81, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 120, "problem_number": "GREEN-032", "title": "Sums of Smooth Numbers", "statement": "Is every $n \\leq N$ the sum of two integers, all of whose prime factors are at most $N^\\varepsilon$?", "background": "Smooth numbers have only small prime factors. This asks if every number is a sum of two smooth numbers, which would show smooth numbers have excellent additive properties. Such a result would have implications for number theory and the distribution of smooth numbers.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Balog's bound represents every n as a sum of two O_epsilon(n^{4/(9 sqrt(e))+epsilon})-smooth integers, but the requested N^epsilon smoothness for every epsilon remains open.\n\n**Verified partial progress.**\n\n- Balog proves f(n) is O_epsilon(n^{4/(9 sqrt(e))+epsilon}), where f(n) is the least smoothness threshold guaranteeing a two-term representation.\n- The target f(n)=n^{o(1)} would imply strong least-quadratic-nonresidue consequences and meets a known Burgess-type barrier.\n\n**Full solution or refutation.**\n\nThe exponent has been reduced to approximately 0.2695, but not to an arbitrarily small positive exponent.\n\n**What remains.**\n\nShow f(n)=n^{o(1)}, equivalently obtain the asserted representation for every fixed epsilon>0, or identify an obstruction.\n\n**Sources checked.**\n\n- T. F. Bloom, Erdős Problem #334, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/334\n Evidence used: Labels the problem open and records Balog's best bound f(n) <<_epsilon n^{4/(9 sqrt(e))+epsilon}.\n- B. Green, 100 Open Problems, Problem 59, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Records the same exponent and explains the least quadratic non-residue barrier.\n\n**Review notes.** The local label GREEN-032 matches Green Problem 59 and Erdős Problem #334 by statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 88, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 121, "problem_number": "GREEN-033", "title": "Sumsets of Perfect Squares", "statement": "Is there an absolute constant $c > 0$ such that if $A \\subset \\mathbb{N}$ is a set of squares of size at least 2, then $|A + A| \\geq |A|^{1+c}$?", "background": "This asks whether sets of perfect squares have superlinear sumset growth. Squares are highly structured (sparse in $\\mathbb{N}$), and one expects their sumsets to grow substantially. Determining the optimal exponent $c$ is a fundamental problem in additive number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A logarithmic superlinear lower bound is available for sumsets of finite square sets, but no absolute fixed power gain |A|^{1+c} has been proved.\n\n**Verified partial progress.**\n\n- The quantitative bound |A+A| >= C|A|(log |A|)^{1/3+o(1)} is stated in Hegyvári's 2025 note.\n- That note explicitly says the result was essentially already proved, with the same tools, by Mei-Chu Chang in 2004 and announces withdrawal; it is therefore evidence for Chang's priority, not a new independent advance.\n- Known bounds for squares in arithmetic progressions and affine-cube reductions provide related structural progress but do not imply a fixed sumset exponent.\n\n**Full solution or refutation.**\n\nThe known logarithmic expansion is weaker than |A|^{1+c} for every fixed c>0.\n\n**What remains.**\n\nProve a uniform fixed power gain, ideally identify the optimal c, or construct square sets with near-linear doubling that refute it.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, Problem 60, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Keeps the fixed-power sumset question open and surveys its connections to Rudin's conjecture, square progressions, and affine cubes.\n- M.-C. Chang, On problems of Erdős and Rudin, J. Functional Analysis 207 (2004), 444-460, doi:10.1016/S0022-1236(03)00073-9. (primary): https://doi.org/10.1016/S0022-1236(03)00073-9\n Evidence used: Primary published paper on the relevant square-set additive inequalities; Hegyvári explicitly attributes the logarithmic consequence to it.\n- N. Hegyvári, Note on the sumset of squares, arXiv:2504.13230v2 (2025), withdrawal announced. (primary): https://arxiv.org/abs/2504.13230\n Evidence used: Abstract states the logarithmic lower bound and explicitly says it is essentially already due to Chang and that the file will be withdrawn.\n\n**Review notes.** The local label GREEN-033 matches Problem 60 by statement. The withdrawn preprint's priority caveat is preserved and no novelty is attributed to it.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 92, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 122, "problem_number": "GREEN-034", "title": "Covering Squares with Sumsets", "statement": "Suppose $A + A$ contains the first $n$ squares. Is $|A| \\geq n^{1-o(1)}$?", "background": "If a set's sumset contains all squares up to $n^2$, must the set have size nearly $n$? This explores the inverse problem: given that a sumset covers a structured set (squares), what can we say about the original set?\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Erdős--Newman give the lower bound |A| >= n^{2/3-o(1)}, while constructions have |A| <<_C n/log^C n; the requested n^{1-o(1)} lower bound remains open.\n\n**Verified partial progress.**\n\n- The best lower bound recorded by Green is n^{2/3-o(1)}.\n- For every fixed C there are examples with |A| <<_C n/log^C n, consistent with but close to the scale of the conjectured lower bound.\n\n**Full solution or refutation.**\n\nKnown results leave an exponent gap between 2/3 and 1, up to subpolynomial factors.\n\n**What remains.**\n\nRaise the lower-bound exponent from 2/3 to 1-o(1), or construct examples of genuinely smaller polynomial order.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, Problem 61, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Attributes the problem and records both the n^{2/3-o(1)} lower bound and n/log^C n constructions.\n\n**Review notes.** The local label GREEN-034 matches Problem 61 by exact statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 85, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 123, "problem_number": "GREEN-035", "title": "Products of Primes Modulo p", "statement": "Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$. Is every $x \\in \\{1, \\dots, p-1\\}$ congruent to some product $a_1a_2$ modulo $p$?", "background": "This asks whether pairwise products of primes cover all residues modulo $p$. Matom\\\"aki-Ter\\\"av\\\"ainen (2023) made significant progress, showing that products of three primes suffice. Whether two primes suffice remains open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The universal two-prime product representation remains open, but Matomäki--Teräväinen prove a universal three-prime representation for sufficiently large cube-free moduli and cover at least a 2/3-epsilon fraction of reduced classes with two primes.\n\n**Verified partial progress.**\n\n- For every sufficiently large cube-free q, every reduced residue class modulo q is a product of three primes at most q.\n- For every epsilon>0 and sufficiently large q, at least (2/3-epsilon)phi(q) reduced residue classes are products of two primes at most q.\n- These results include prime moduli but do not cover every nonzero class with two factors.\n\n**Full solution or refutation.**\n\nThree factors suffice and two factors cover a positive majority of classes; the exact two-factor conjecture is unresolved.\n\n**What remains.**\n\nShow that all p-1 nonzero classes modulo every sufficiently large prime p are represented by two primes below p, or find an obstruction.\n\n**Sources checked.**\n\n- K. Matomäki, J. Teräväinen, Products of primes in arithmetic progressions, J. Reine Angew. Math. 808 (2024), 193-240, doi:10.1515/crelle-2023-0096. (primary): https://arxiv.org/abs/2301.07679\n Evidence used: Abstract states both the ternary theorem for cube-free moduli and the (2/3-epsilon)phi(q) two-prime coverage theorem.\n- B. Green, 100 Open Problems, Problem 62, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Retains the two-prime question and records the 2023 reduction from more factors to three.\n\n**Review notes.** The local label GREEN-035 matches Problem 62 by exact statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 96, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 124, "problem_number": "GREEN-036", "title": "Multiplicatively Closed Set Density", "statement": "Let $A$ be the smallest set containing 2 and 3, and closed under the operation $a_1a_2 - 1$ (if $a_1, a_2 \\in A$, then $a_1a_2 - 1 \\in A$). Does $A$ have positive density?", "background": "This defines a set generated by a multiplicative-like operation. Understanding its density is nontrivial because the operation $a_1a_2 - 1$ mixes multiplication with additive structure. Whether such sets have positive density connects number theory with dynamical systems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Hofstadter closure-set positive-lower-density problem remains open in both Green's maintained collection and Erdős Problems #424.\n\n**Verified partial progress.**\n\n- The set contains no integer congruent to 1 modulo 3: starting from residues 2 and 0, the operation ab-1 preserves the residue set {0,2}. Hence its upper density is at most 2/3.\n- This refutes an older stronger 'almost all integers' formulation but does not settle the imported positive-density question.\n\n**Full solution or refutation.**\n\nNo positive lower-density proof or zero-density counterexample is known; the congruence obstruction only supplies an upper bound.\n\n**What remains.**\n\nProve a positive lower density for A005244 or show its lower density is zero.\n\n**Sources checked.**\n\n- T. F. Bloom, Erdős Problem #424, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/424\n Evidence used: Labels the positive-lower-density question open, records the mod-3 obstruction, and distinguishes it from the false stronger almost-all formulation.\n- B. Green, 100 Open Problems, Problem 63, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Attributes the problem to Hofstadter, links OEIS A005244 and Erdős #424, and retains it as open.\n\n**Review notes.** The local label GREEN-036 matches Green Problem 63 and Erdős Problem #424 by statement. Positive density is interpreted as positive lower density following the maintained tracker.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 77, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 125, "problem_number": "GREEN-037", "title": "Primes with p-2 Having Odd Omega", "statement": "Do there exist infinitely many primes $p$ for which $p-2$ has an odd number of prime factors (counting multiplicity)?", "background": "This asks about the parity of $\\Omega(p-2)$ where $\\Omega(n)$ counts prime factors with multiplicity. Since $p-2$ is even for odd primes $p > 2$, we're asking about the structure of $(p-2)/2$. This is a prime-shifted multiplicative function question.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The current author-maintained Green collection, where this is now Problem 64, continues to list infinitude of primes p with odd Omega(p-2) as open.\n\n**Verified partial progress.**\n\n- The twin-prime conjecture would imply the assertion, since prime p-2 has Omega(p-2)=1.\n- The question is equivalent to asking whether lambda(p-2)=-1 infinitely often and encounters the sieve parity barrier.\n- A 2022 arXiv manuscript claims much stronger cancellation over shifted primes, but the later maintained Green list does not recognize it as a solution and no accepted validation was located.\n\n**Full solution or refutation.**\n\nNo accepted proof of even infinitude for one Liouville sign along the shift p-2 was found.\n\n**What remains.**\n\nProve infinitely many primes p with lambda(p-2)=-1, or rigorously validate or refute the stronger shifted-prime cancellation claim.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, current Problem 64, updates through December 2025; checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the exact question without a solved marker and explains its relation to twin primes and the parity of the number of prime factors.\n- N. A. Carella, Result on the Mobius Function over Shifted Primes, arXiv:2206.12956 (2022). (primary): https://arxiv.org/abs/2206.12956\n Evidence used: Claims a far stronger Liouville cancellation estimate; it is recorded as a conflicting unvalidated claim, not as accepted evidence of resolution.\n\n**Review notes.** Source-version mismatch: stored GREEN-037 is current Green Problem 64. The database background is factually wrong: for every odd prime p, p-2 is odd, not even, so its discussion of (p-2)/2 is inapplicable.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 83, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 126, "problem_number": "GREEN-038", "title": "Difference Sets Containing Squares", "statement": "Is there $c > 0$ such that whenever $A \\subset [N]$ has size $N^{1-c}$, the difference set $A - A$ contains a nonzero square?", "background": "This asks how large a set must be to guarantee its difference set contains a square. Similarly one can ask if $A - A$ contains a prime minus one. These questions probe the additive structure forced by density.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Green and Sawhney give the strongest current upper bound for square-difference-free sets, but it is still N^{1-o(1)} and does not provide the requested fixed power saving.\n\n**Verified partial progress.**\n\n- If A has no nonzero square in A-A, then |A| is at most a constant times N exp(-c sqrt(log N)).\n- This improves the earlier Bloom-Mayard logarithmic-density bound.\n- Ruzsa's constructions show that a positive exponent in the conjecture, if it exists, cannot be taken above approximately 0.267.\n\n**Full solution or refutation.**\n\nThe known threshold N exp(-c sqrt(log N)) is subpolynomially below N, whereas the question asks for N^{1-c_0} with fixed c_0>0.\n\n**What remains.**\n\nProve any fixed polynomial saving for square-difference-free subsets of [N], or construct examples ruling it out.\n\n**Sources checked.**\n\n- Ben Green and Mehtaab Sawhney, New bounds for the Furstenberg-Sarkozy theorem, arXiv:2411.17448 (2025 version). (primary): https://arxiv.org/abs/2411.17448\n Evidence used: Proves |A| << N exp(-c sqrt(log N)) for sets with no two elements differing by a square.\n- Ben Green, 100 Open Problems, current Problem 65, checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Records the 2025 Green-Sawhney bound, Ruzsa's construction barrier, and that the fixed-power question remains open.\n\n**Review notes.** Source-version mismatch: stored GREEN-038 is current Green Problem 65. The statement is preserved; 'size N^{1-c}' is read in the standard threshold sense.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 89, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 127, "problem_number": "GREEN-039", "title": "Gaps Between Sums of Two Squares", "statement": "Is there always a sum of two squares between $X - \\frac{1}{10}X^{1/4}$ and $X$?", "background": "Sums of two squares have density $c/\\sqrt{\\log X}$, so gaps can be large. This asks for an upper bound on the largest gap. Such results would improve our understanding of the distribution of representable numbers in quadratic forms.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exponent 1/4 is known with an unspecified absolute constant, but the requested one-sided constant 1/10 remains open.\n\n**Verified partial progress.**\n\n- Subtracting the greatest square below X and then the greatest square below the remainder produces a sum of two squares within O(X^{1/4}) below X.\n- Kalmynin obtained improved moment estimates for gaps between consecutive sums of two squares, but not the required worst-case constant.\n- Even a uniform o(X^{1/4}) improvement is not known.\n\n**Full solution or refutation.**\n\nKnown methods reach the correct displayed exponent only up to a larger unspecified constant; they do not establish an interval of length X^{1/4}/10.\n\n**What remains.**\n\nImprove the uniform one-sided constant to 1/10, or prove any o(X^{1/4}) worst-case gap bound.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, current Problem 66, checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the exact 1/10 question and the elementary O(X^{1/4}) one-sided argument while retaining open status.\n- Alexander Kalmynin, Intervals between numbers that are sums of two squares, Mathematika 65 (2019), 1018-1032. (primary): https://arxiv.org/abs/1706.07380\n Evidence used: Improves moment estimates for gaps and explicitly situates the unresolved worst-case X^{1/4} scale.\n\n**Review notes.** Source-version mismatch: stored GREEN-039 is current Green Problem 66. No OCR defect was found in the mathematical statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 91, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 128, "problem_number": "GREEN-040", "title": "Waring's Problem Over Finite Fields", "statement": "Determine bounds for Waring's problem over finite fields.", "background": "Waring's problem asks: can every element be written as a sum of $k$ $d$-th powers? Over finite fields $\\mathbb{F}_q$, the problem has different character. Determining the minimum $k$ for given $d$ and $q$ is a classical problem in algebraic number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The stored imperative sentence is too vague to classify, and its background describes a different problem from Green's now-solved function-field Problem 67.\n\n**Verified partial progress.**\n\n- Green's source defines G_fin(k) using representations of high-degree polynomials in F_p[t] by kth powers of polynomials of controlled degree.\n- Liu-Wooley obtained G_fin(k) <= (1+o(1)) k log k in the source formulation.\n- Sawin's algebro-geometric circle method gives an asymptotic with s=O(k) in sufficiently large characteristic, leading Green to mark the source problem solved.\n\n**Full solution or refutation.**\n\nThe recovered function-field problem has a strong 2024-2026 solution, but this does not resolve the database background's distinct question about sums of dth powers of elements of a finite field F_q.\n\n**What remains.**\n\nChoose and restore a precise formulation, including which ring or field, the roles of p, q, d, k and s, and the asymptotic regime; then attach the appropriate status.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, current Problem 67 (marked Solved), checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Defines the function-field quantity G_fin(k), records the Liu-Wooley bound, and cites Sawin's O(k)-scale result.\n- Will Sawin, The asymptotic in Waring's problem over function fields via a singular locus in the circle method, arXiv:2412.14053, version 3 (2026). (primary): https://arxiv.org/abs/2412.14053\n Evidence used: Proves stronger function-field Waring asymptotics by treating minor arcs as finite-field exponential sums.\n\n**Review notes.** Material formulation defect. Stored GREEN-040 is current Green Problem 67, but 'Determine bounds for Waring's problem over finite fields' omits all parameters and the database background silently changes F_p[t] to the finite field F_q.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 86, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 129, "problem_number": "GREEN-041", "title": "Cubic Curves in F_p^2", "statement": "Suppose $A \\subset \\mathbb{F}_p^2$ is a set meeting every line in at most 2 points. Is it true that all except $o(p)$ points of $A$ lie on a cubic curve?", "background": "Sets avoiding three collinear points have special structure. Over finite fields, the Hasse-Weil bound and algebraic geometry suggest such sets should lie nearly on a cubic. This is a finite-field analogue of classical incidence geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The cubic-approximation conjecture for arbitrary arcs over prime fields remains open, although sufficiently large arcs are forced onto conics.\n\n**Verified partial progress.**\n\n- Voloch proved that a prime-plane arc above the 44p/45+O(1) threshold lies on a conic.\n- A conic is contained in a reducible cubic, so this settles the desired conclusion in that high-density regime.\n- Genuinely cubic examples occur below about p/2, and analogous claims over nonprime fields can fail.\n\n**Full solution or refutation.**\n\nExisting arc-classification theorems control sets close to maximum size, but do not show that every arc over F_p is within o(p) points of one cubic.\n\n**What remains.**\n\nEstablish a bounded-degree, specifically cubic, container after deleting o(p) points for all asymptotically nontrivial arcs over prime fields.\n\n**Sources checked.**\n\n- Jose Felipe Voloch, Arcs in projective planes over prime fields, Journal of Geometry 38 (1990), 198-200. (primary): https://doi.org/10.1007/BF01222904\n Evidence used: Proves the large-arc conic containment threshold used as the principal special case.\n- Ben Green, 100 Open Problems, current Problem 68, checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the exact cubic-container question, records Voloch's special case, cubic examples, and the nonprime-field obstruction.\n\n**Review notes.** Source-version mismatch: stored GREEN-041 is current Green Problem 68. The o(p) statement implicitly concerns families as p tends to infinity; that asymptotic quantifier is omitted in the database.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 84, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 130, "problem_number": "GREEN-042", "title": "Collinear Triples and Cubic Curves", "statement": "Fix $k$. Let $A \\subset \\mathbb{R}^2$ be a set of $n$ points with no more than $k$ on any line. Suppose at least $\\delta n^2$ pairs $(x, y) \\in A \\times A$ have the line $xy$ containing a third point of $A$. Is there a cubic curve containing at least $cn$ points of $A$?", "background": "If many pairs determine lines through a third point, the set should have algebraic structure. This asks if a cubic curve captures this structure. It generalizes results from the joints problem and incidence geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Cubic structure is known in extreme-density and fixed-degree-curve regimes, but not from an arbitrary fixed positive density of third-point pairs.\n\n**Verified partial progress.**\n\n- When delta=1-O(1/n), Green-Tao's few-ordinary-lines structure theorem gives a cubic container.\n- Elekes-Szabo settle relevant cases when the point set is already supported on a fixed-degree algebraic curve.\n- Fractional Sylvester-Gallai theory forces a linear-size subset in bounded affine dimension from quadratically many collinear triples, but does not force a planar cubic.\n\n**Full solution or refutation.**\n\nThe general implication from delta n^2 rich pairs and bounded collinearity to cn points on one cubic remains open.\n\n**What remains.**\n\nProve a cubic-container theorem for every fixed k and delta, with a positive c=c(k,delta).\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, current Problem 69, checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the quantified source problem and summarizes the extreme-density, algebraic-curve, and fractional Sylvester-Gallai partial results.\n- Gyorgy Elekes and Endre Szabo, On Triple Lines and Cubic Curves: The Orchard Problem Revisited, Discrete & Computational Geometry 72 (2024), 743-763. (primary): https://doi.org/10.1007/s00454-023-00556-3\n Evidence used: Develops the cubic structure theorem in fixed-degree algebraic-curve cases and related special cases.\n- Boaz Barak, Zeev Dvir, Avi Wigderson, and Amir Yehudayoff, Fractional Sylvester-Gallai theorems, PNAS 110 (2013), 19213-19219. (primary): https://doi.org/10.1073/pnas.1203737109\n Evidence used: Gives the bounded-dimensional linear-size subset conclusion from many collinear triples.\n\n**Review notes.** Source-version mismatch: stored GREEN-042 is current Green Problem 69. The database fails to exclude x=y even though line xy is then undefined, and fails to quantify the conclusion's c; the source has c=c(k,delta)>0.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 78, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 131, "problem_number": "GREEN-043", "title": "Erdős-Szekeres with Visibility", "statement": "Fix integers $k, \\ell$. Given $n \\geq n_0(k, \\ell)$ points in $\\mathbb{R}^2$, is there either a line containing $k$ of them, or $\\ell$ of them that are mutually visible?", "background": "This is a Ramsey-type problem mixing collinearity and visibility (no point blocks the segment between two others). It generalizes the Erdős-Szekeres theorem to a geometric context, asking for unavoidable configurations.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Big-Line-Big-Clique Conjecture is proved for visible clique size at most 5 and in several structured regimes, but remains open in full generality.\n\n**Verified partial progress.**\n\n- Abel and coauthors proved the conjecture for ell<=5 using the empty-pentagon theorem.\n- A 2026 preprint proves the conclusion for sets with O(n) ordinary lines and for sets contained in a fixed irreducible algebraic curve.\n- The same preprint gives a cubic-container mechanism and a dense-orchard core reduction, while explicitly retaining an ambient-container obstruction.\n\n**Full solution or refutation.**\n\nNeither the published ell<=5 theorem nor the new structured cases cover arbitrary fixed k and ell for unrestricted finite planar point sets.\n\n**What remains.**\n\nConvert the dense collinear-triple structure forced by no visible K_ell into a sufficiently strong ambient cubic container, or find a counterexample.\n\n**Sources checked.**\n\n- Zachary Abel et al., Every Large Point Set Contains Many Collinear Points or an Empty Pentagon, Graphs and Combinatorics 27 (2011), 47-60. (primary): https://arxiv.org/abs/0904.0262\n Evidence used: Settles the Big-Line-Big-Clique conjecture for ell<=5.\n- Sohail Sarkar, Visibility cliques, cubic containers, and dense orchard cores, arXiv:2605.00918 (2026). (primary): https://arxiv.org/abs/2605.00918\n Evidence used: Claims quantitative results in few-ordinary-line, cubic-container, and bounded-degree algebraic-curve regimes while identifying the remaining obstruction.\n- Ben Green, 100 Open Problems, current Problem 70, checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Records the full conjecture and the published ell<=5 special case.\n\n**Review notes.** Source-version mismatch: stored GREEN-043 is current Green Problem 70. The 2026 advance is a recent unrefereed preprint and is conservatively treated only as partial progress.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 81, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 132, "problem_number": "GREEN-044", "title": "Collinear 4-tuples Force Collinear 5-tuples", "statement": "Suppose $A \\subset \\mathbb{R}^2$ is a set of size $n$ with $cn^2$ collinear 4-tuples. Does it contain 5 points on a line?", "background": "Many collinear 4-tuples suggest the set has strong linear structure. This asks if this forces an actual line through 5 points. It's related to the Szemerédi-Trotter theorem and incidence bounds.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** It remains open whether no five collinear points force o(n^2) collinear 4-tuples, despite constructions with n^{2-o(1)} such tuples.\n\n**Verified partial progress.**\n\n- Solymosi-Stojakovic construct n-point sets with no five collinear points and at least n^{2-c/sqrt(log n)} collinear 4-tuples.\n- This is near quadratic but still o(n^2), so it neither proves nor refutes the fixed-positive-density assertion.\n- Elekes-Szabo cubic-curve structure results cover related algebraic special cases.\n\n**Full solution or refutation.**\n\nThe known lower construction approaches the quadratic exponent, while the required statement asks whether a fixed positive quadratic density is impossible without a 5-point line.\n\n**What remains.**\n\nProve every no-five-collinear n-point set has o(n^2) collinear 4-tuples, or construct one with Omega(n^2).\n\n**Sources checked.**\n\n- Jozsef Solymosi and Milos Stojakovic, Many collinear k-tuples with no k+1 collinear points, Discrete & Computational Geometry 50 (2013), 811-820. (primary): https://arxiv.org/abs/1107.0327\n Evidence used: Provides the n^{2-c/sqrt(log n)} construction for k=4 without five collinear points.\n- Thomas F. Bloom, Erdos Problem 101, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/101\n Evidence used: Maintains the equivalent o(n^2) formulation as open.\n- Ben Green, 100 Open Problems, current Problem 71, checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the exact problem and records both the near-quadratic construction and cubic-structure route.\n\n**Review notes.** Source-version mismatch: stored GREEN-044 is current Green Problem 71. The standard formulation quantifies fixed c>0 and sufficiently large n; those quantifiers are implicit in the database wording.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 75, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 133, "problem_number": "GREEN-045", "title": "No Three in Line in [N]^2", "statement": "What is the largest subset of the grid $[N]^2$ with no three points on a line? In particular, for $N$ sufficiently large, is it impossible to have a set of size $2N$ with this property?", "background": "The cap set problem in two dimensions. Erdős conjectured sets of size $O(N)$ exist, but proving or disproving this remains open. The problem connects discrete geometry, additive combinatorics, and the polynomial method.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The asymptotic no-three-in-line maximum remains open; the best general lower bound is (3/2+o(1))N, while finite 2N constructions now extend much farther.\n\n**Verified partial progress.**\n\n- The elementary row bound gives D(N)<=2N, and Hall-Jackson-Sudbery-Wild give D(N)>=(3/2+o(1))N for arbitrary N.\n- Prellberg's 2026 constraint-satisfaction search certifies D(N)=2N for every N<=60; subsequent maintained computational tables report further orders, but no asymptotic family.\n- Grebennikov-Kwan solve the no-(k+1)-in-line analogue for sufficiently large fixed k, and explicitly state that k=2 requires new ideas.\n\n**Full solution or refutation.**\n\nNeither finite exact configurations nor the large-k theorem decides whether D(N)<2N for all sufficiently large N or determines the asymptotic constant for k=2.\n\n**What remains.**\n\nProve eventual strict inequality D(N)<2N, construct 2N examples for infinitely many N, or narrow the general 3N/2 versus 2N gap.\n\n**Sources checked.**\n\n- Thomas Prellberg, Constraint Satisfaction Programming for the No-three-in-line Problem, arXiv:2602.07751 (2026). (primary): https://arxiv.org/abs/2602.07751\n Evidence used: Constructs 2N-point configurations for every N<=60 and identifies 61 as the first unknown order at that paper's cutoff.\n- Alexandr Grebennikov and Matthew Kwan, No-(k+1)-in-line problem for large constant k, arXiv:2510.17743 (2025). (primary): https://arxiv.org/abs/2510.17743\n Evidence used: Proves the exact kN result for sufficiently large fixed k and explains why this does not reach the classical k=2 case.\n- Ben Green, 100 Open Problems, current Problem 72, checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Records the general (3/2+o(1))N construction, the 2N upper bound, and open asymptotic question.\n\n**Review notes.** Source-version mismatch: stored GREEN-045 is current Green Problem 72. External sources used computer search, but this triage ran no enumeration or heavy computation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 94, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 134, "problem_number": "GREEN-046", "title": "Smooth Surfaces Intersecting 2-planes", "statement": "Let $\\Gamma$ be a smooth codimension 2 surface in $\\mathbb{R}^n$. Must $\\Gamma$ intersect some 2-dimensional plane in 5 points, if $n$ is sufficiently large?", "background": "This asks about unavoidable intersection patterns between smooth surfaces and planes in high dimensions. It's related to the Kakeya problem and incidence geometry in higher dimensions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Four coplanar intersections are forced in sufficiently high dimension and fixed-dimensional intersection maxima are finite, but the requested fifth point remains open.\n\n**Verified partial progress.**\n\n- The general Hilbert-cube argument gives f(n,d)>=2^d for sufficiently large n, hence f(n,2)>=4.\n- Thom sketched and Chaperon-Meyer developed finiteness results for the relevant affine-intersection number at fixed dimensions.\n- Explicit subspace-evasive constructions give complementary examples with uniformly bounded affine-subspace intersections, but do not settle whether five is unavoidable here.\n\n**Full solution or refutation.**\n\nKnown theory establishes four intersections and finiteness, not a 5-point intersection for every smooth codimension-two local graph in all sufficiently high dimensions.\n\n**What remains.**\n\nProve f(n,2)>=5 for sufficiently large n or construct smooth local graphs showing that four is best possible along an unbounded sequence of dimensions.\n\n**Sources checked.**\n\n- Marc Chaperon and Daniel Meyer, On a theorem of Rene Thom in Geometrie Finie, L'Enseignement Mathematique 55 (2009), 329-357. (primary): https://doi.org/10.4171/LEM/55-3-6\n Evidence used: Develops Thom's finite-geometry intersection theorem and the fixed-dimensional finiteness input cited by Green.\n- Ben Green, 100 Open Problems, current Problem 73, checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Supplies the intended local-graph formulation, the 2^d lower bound, and the unresolved five-point question.\n\n**Review notes.** Source-version mismatch: stored GREEN-046 is current Green Problem 73. 'Surface' is ambiguous in dimension n-2, and the database omits Green's intended local graph convention; global or nontransverse intersections can otherwise distract from the intended local extremal problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 71, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 135, "problem_number": "GREEN-047", "title": "No 5 Points on 2-plane in [N]^d", "statement": "What is the largest subset of $[N]^d$ with no 5 points on a 2-plane?", "background": "This generalizes the no-three-in-line problem to higher dimensions and 2-planes. Determining the maximum size of such sets involves combinatorial geometry and higher-dimensional incidence bounds.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No asymptotic extremal result for avoiding five coplanar points in [N]^d was located.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe current Green collection keeps this as an open discrete-geometry question.\n\n**What remains.**\n\nDetermine the growth in N for each fixed d and clarify any dimension range intended.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, updated January 2026. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Maintained author problem list.\n\n**Review notes.** No source statement altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 76, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 136, "problem_number": "GREEN-048", "title": "Balanced Ham Sandwich Line", "statement": "Let $X \\subset \\mathbb{R}^2$ be a set of $n$ points. Does there exist a line $\\ell$ through at least two points of $X$ such that the numbers of points on either side of $\\ell$ differ by at most 100?", "background": "This is a variant of the ham sandwich theorem asking for a balanced bisector that passes through points of the set. The classical ham sandwich theorem doesn't require passing through points, making this version more constrained.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem locating a two-point line with constant 100 imbalance was found; the ordinary ham-sandwich theorem does not impose the incidence condition.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo verified resolution was located.\n\n**What remains.**\n\nProve a universal constant imbalance, or give configurations forcing unbounded imbalance.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, updated January 2026. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Maintained author list.\n\n**Review notes.** The literal 100 is treated as part of the question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 79, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 137, "problem_number": "GREEN-049", "title": "Sparse Hitting Set for Rectangles", "statement": "Let $A$ be a set of $n$ points in the plane. Can one select $A' \\subset A$ of size $n/2$ such that any axis-parallel rectangle containing 1000 points of $A$ contains at least one point of $A'$?", "background": "This asks for an efficient hitting set for rectangles defined by a point set. Such results have applications in computational geometry and range searching.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No half-size point hitting theorem for all 1000-rich axis-parallel rectangles was located.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe range-hitting formulation remains open in the checked sources.\n\n**What remains.**\n\nProve the n/2 transversal claim or construct a counterexample.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, updated January 2026. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Maintained author list.\n\n**Review notes.** No source statement changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 74, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 138, "problem_number": "GREEN-050", "title": "Small Triangles in the Unit Disc", "statement": "Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them?", "background": "This asks about unavoidable small-area triangles in dense point sets. Cohen-Pohoata-Zakharov (2023) improved the bound to $n^{-8/7-c}$. Reaching the conjectured $n^{-2}$ bound remains open and connects to the Heilbronn triangle problem.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Cohen--Pohoata--Zakharov improve the forced small-triangle exponent, but the n^(-2+o(1)) target remains open.\n\n**Verified partial progress.**\n\n- The source records a bound of order n^(-8/7-c) for some positive c.\n\n**Full solution or refutation.**\n\nThe Heilbronn-type exponent gap remains.\n\n**What remains.**\n\nApproach or attain the exponent -2.\n\n**Sources checked.**\n\n- Ben Green problem collection, GREEN-050, checked 2026-08-17. (authoritative_secondary): https://www.unsolvedmath.com/problems/GREEN-050\n Evidence used: Cohen--Pohoata--Zakharov improvement and residual target.\n\n**Review notes.** The source does not supply full bibliographic metadata; no stronger citation is invented.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 88, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 139, "problem_number": "GREEN-051", "title": "Axis-Parallel Rectangles in Dense Sets", "statement": "Suppose $A$ is an open subset of $[0, 1]^2$ with measure $\\alpha$. Are there four points in $A$ determining an axis-parallel rectangle with area $\\geq c\\alpha^2$?", "background": "This asks if dense sets in the unit square must contain large axis-parallel rectangles. It's a continuous analogue of combinatorial rectangle problems and relates to measure-theoretic ergodic theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No resolution of the alpha-squared large axis-parallel rectangle assertion was found.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe curated source labels the statement open.\n\n**What remains.**\n\nProve a universal c or give dense open counterexamples.\n\n**Sources checked.**\n\n- GREEN-051 curated problem entry, checked 2026-08-17. (authoritative_secondary): https://www.unsolvedmath.com/problems/139\n Evidence used: Current open status and exact formulation.\n\n**Review notes.** No source correction made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 72, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 140, "problem_number": "GREEN-052", "title": "Equidistribution of Integer Multiples", "statement": "Let $c > 0$ and let $A$ be a set of $n$ distinct integers. Does there exist $\\theta$ such that no interval of length $\\frac{1}{n}$ in $\\mathbb{R}/\\mathbb{Z}$ contains more than $n^c$ of the numbers $\\theta a \\pmod 1$, for $a \\in A$?", "background": "This asks about finding angles $\\theta$ that spread out the set $A$ modulo 1. It's related to discrepancy theory and the distribution of sequences modulo 1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The equidistribution assertion is known only for exponents c>1/3 in the maintained Green notes; the stated arbitrary c>0 target is open.\n\n**Verified partial progress.**\n\n- Konyagin gives the lower discrepancy distribution bound corresponding to c>1/3.\n\n**Full solution or refutation.**\n\nNo result for every positive c was verified.\n\n**What remains.**\n\nLower the exponent threshold to arbitrary c>0 or find an obstruction.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, updated January 2026, Problem 86 comments. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the c>1/3 range and attributes it to Konyagin.\n\n**Review notes.** The source index differs from its displayed GREEN number; matching is by statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 141, "problem_number": "GREEN-053", "title": "Random Permutations Fixing k-Sets", "statement": "Let $p(k)$ be the limit as $n \\to \\infty$ of the probability that a random permutation on $[n]$ preserves some set of size $k$. Is $p(k)$ a decreasing function of $k$? Is $p(k) = (C + o(1))k^{-\\alpha}(\\log k)^{-3/2}$ for some absolute constant $C$?", "background": "This concerns the probability that a random permutation fixes some subset. Eberhard-Ford-Green established asymptotic formulas. The question asks for monotonicity and precise asymptotics, connecting combinatorics and probability.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Green--Sawhney obtain an asymptotic p(k)~f({log_2 k})k^(-delta)(log k)^(-3/2) with explicit smooth positive periodic f, refining the older two-sided estimate.\n\n**Verified partial progress.**\n\n- Eberhard--Ford--Green prove the matching order of magnitude.\n- Green--Sawhney supply the periodic-factor asymptotic in a 2026 preprint.\n\n**Full solution or refutation.**\n\nThe requested constant C asymptotic would follow if f were constant; the preprint instead conjectures f is nonconstant, and monotonicity is not settled there.\n\n**What remains.**\n\nEstablish nonconstancy/constancy of f and resolve monotonicity.\n\n**Sources checked.**\n\n- B. Green and M. Sawhney, The proportion of permutations fixing a k-set, arXiv:2604.28116 (2026). (primary): https://arxiv.org/abs/2604.28116\n Evidence used: Abstract gives periodic-factor asymptotic and nonconstancy conjecture.\n- S. Eberhard, K. Ford, B. Green, Permutations fixing a k-set, arXiv:1507.04465 (2015). (primary): https://arxiv.org/abs/1507.04465\n Evidence used: Uniform two-sided order theorem.\n\n**Review notes.** The 2026 source is explicitly a preprint and is treated accordingly.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 75, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 142, "problem_number": "GREEN-054", "title": "Comparable Elements in Integer Lattices", "statement": "Consider a set $S \\subset [N]^3$ with the property that any two distinct elements $s, s'$ of $S$ are comparable (in the coordinatewise partial order). Is $|S| \\leq N^{2-\\delta}$ for some $\\delta > 0$?", "background": "An antichain in $[N]^d$ can have size $\\binom{N}{d/2}^d \\sim N^{d/2}$. This asks if totally comparable sets (chains) in 3D are even smaller, achieving $N^{2-\\delta}$ rather than $N^2$. It's a question in extremal combinatorics and poset theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No power-saving bound for the stated comparability condition in [N]^3 was located.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe maintained author notes retain the problem.\n\n**What remains.**\n\nProve |S|<=N^(2-delta) or construct near-quadratic examples.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, updated January 2026, Problem 88. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Current formulation and open status.\n\n**Review notes.** The source uses an unusual sign-pattern definition of comparable; it is not silently normalised to the standard poset order.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 71, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 143, "problem_number": "GREEN-055", "title": "Stable Density on Subspaces", "statement": "Let $A \\subset \\mathbb{F}_2^n$. If $V$ is a subspace, write $\\alpha(V)$ for the density of $A$ on $V$. Is there some $V$ of moderately small codimension on which $\\alpha$ is stable?", "background": "This asks if every set has a subspace where its density doesn't fluctuate wildly. Stability of density on subspaces is fundamental in additive combinatorics over $\\mathbb{F}_2^n$ and relates to the polynomial Freiman-Ruzsa conjecture.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The phrase moderately small codimension does not specify a function of the parameters, preventing a determinate literature status.\n\n**Verified partial progress.**\n\n- Subspace regularity and polynomial-Freiman-Ruzsa results address several precise density-stabilisation variants.\n\n**Full solution or refutation.**\n\nNo unique theorem-level claim is encoded by the imported wording.\n\n**What remains.**\n\nSpecify the desired codimension bound and stability norm/quantifiers.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, updated January 2026. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Source collection for the broad problem.\n\n**Review notes.** Formulation is underspecified; preserved exactly.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 77, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 144, "problem_number": "GREEN-056", "title": "Almost Invariant Sets Under Affine Maps", "statement": "Suppose $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ has density $\\frac{1}{2}$. Under what conditions on $K$ can $A$ be almost invariant under all maps $\\phi(x) = ax + b$ with $|a|, |b| \\leq K$?", "background": "This asks when a set is nearly preserved under small affine transformations. Understanding which sets have this property connects to additive combinatorics, group actions, and the structure of dense sets in cyclic groups.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The request for conditions on K and almost invariance omits the error metric and quantifier scale, so it is not a uniquely checkable assertion.\n\n**Verified partial progress.**\n\n- Approximate group-action and affine-invariance methods address formal variants.\n\n**Full solution or refutation.**\n\nNo exact theorem matching the broad wording was found.\n\n**What remains.**\n\nDefine almost invariant, its allowable error, dependence on p, and desired K regime.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, updated January 2026. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Source collection for the agenda-level wording.\n\n**Review notes.** Underspecified formulation is flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 145, "problem_number": "GREEN-057", "title": "Trace Reconstruction", "statement": "Given a string $x \\in \\{0, 1\\}^n$, let $\\tilde{x}$ be obtained by deleting bits independently at random with probability $\\frac{1}{2}$. How many independent traces $\\tilde{x}_1, \\dots, \\tilde{x}_m$ are needed to reconstruct $x$ with probability 0.9?", "background": "This fundamental problem in computational complexity asks how many noisy observations suffice to recover the original string. Chase (2020) improved bounds to $e^{n^{1/5}\\log^C n}$. Determining the optimal bound is a major open question.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Worst-case binary trace reconstruction at deletion probability 1/2 has subexponential upper bounds, but the optimal trace complexity is not known.\n\n**Verified partial progress.**\n\n- The maintained problem notes record the current upper-bound scale as exp(O(n^(1/5) log^5 n)).\n\n**Full solution or refutation.**\n\nNo full asymptotic sample-complexity theorem was verified.\n\n**What remains.**\n\nDetermine matching lower and upper bounds, including the optimal exponent at deletion probability 1/2.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, maintained author notes, accessed 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Current problem list records subexponential trace-reconstruction progress and leaves optimization open.\n\n**Review notes.** No source statement altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 82, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 146, "problem_number": "GREEN-058", "title": "Irreducibility of Random {0,1} Polynomials", "statement": "Is a random polynomial with coefficients in $\\{0, 1\\}$ and nonzero constant term almost surely irreducible?", "background": "This asks whether most polynomials with binary coefficients are irreducible over $\\mathbb{Q}$. Bary-Soroker-Koukoulopoulos-Kozma (2023) made further progress. The problem connects number theory, probability, and algebraic geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The binary-coefficient high-probability irreducibility question remains open; general-measure results rule out low-degree factors with high probability and give a positive irreducibility probability for arithmetic-progression laws.\n\n**Verified partial progress.**\n\n- Bary-Soroker--Koukoulopoulos--Kozma prove that no divisor of degree at most theta n occurs with probability tending to one for finite-support non-Dirac laws.\n- For a uniform arithmetic-progression law they prove irreducibility with probability at least a positive constant, not convergence to one for the {0,1} law.\n\n**Full solution or refutation.**\n\nNo unconditional proof that random {0,1} polynomials are irreducible with probability tending to one was verified.\n\n**What remains.**\n\nEstablish or refute almost-sure irreducibility for the Bernoulli {0,1} distribution.\n\n**Sources checked.**\n\n- L. Bary-Soroker, D. Koukoulopoulos, G. Kozma, Irreducibility of random polynomials: general measures, arXiv:2007.14567 (2020). (primary): https://arxiv.org/abs/2007.14567\n Evidence used: Abstract states the low-degree-factor theorem and positive-probability result for arithmetic progressions.\n\n**Review notes.** The cited theorem is distinguished from the binary high-probability target.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 76, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 149, "problem_number": "GREEN-061", "title": "N Queens Problem Asymptotics", "statement": "In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \\times n$ chessboard?", "background": "The n-queens problem asks for the number of ways to place $n$ queens on an $n \\times n$ board so none attack each other. Determining the asymptotic growth rate of $Q(n)$ is a famous open problem in combinatorics. Rough bounds are known but the exact constant remains elusive.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Simkin established the requested asymptotic enumeration: Q(n)=((1±o(1)) n e^(-alpha))^n for alpha=1.942±0.003.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThis gives the asymptotic growth rate requested, with alpha characterized by a convex optimization over queenons.\n\n**What remains.**\n\nThe asymptotic-count question itself is answered.\n\n**Sources checked.**\n\n- M. Simkin, The number of n-queens configurations, arXiv:2107.13460 (2021). (primary): https://arxiv.org/abs/2107.13460\n Evidence used: Abstract states the asymptotic formula and characterization of alpha.\n\n**Review notes.** Dataset status was not edited; this is a literature-status correction only.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 94, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 150, "problem_number": "GREEN-062", "title": "Bounds for Birch's Theorem", "statement": "Let $d \\geq 3$ be odd. Give bounds on $\\nu(d)$ such that if $n > \\nu(d)$ then any homogeneous polynomial $F(\\mathbf{x}) \\in \\mathbb{Z}[x_1, \\dots, x_n]$ of degree $d$ has a nontrivial integer zero.", "background": "Birch's theorem guarantees that homogeneous polynomials of odd degree have nontrivial zeros if there are enough variables. Determining the optimal $\\nu(d)$ is a central problem in Diophantine equations and algebraic number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Birch proved existence of a sufficient dimension threshold for odd-degree forms, but the optimal explicit single-form function nu(d) requested here was not verified.\n\n**Verified partial progress.**\n\n- Birch's 1957 theorem establishes finiteness of a dimension threshold for odd-degree forms.\n- Modern work gives polynomial bounds for related systems-of-forms variants, not asserted here to be the exact nu(d) result.\n\n**Full solution or refutation.**\n\nThe existence portion is classical; quantitative optimization remains a live issue in the imported formulation.\n\n**What remains.**\n\nRecord the best explicit upper and lower bounds for the exact single-form threshold nu(d).\n\n**Sources checked.**\n\n- B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika 4 (1957). (primary): https://doi.org/10.1112/S0025579300001145\n Evidence used: Classical existence theorem for odd-degree forms in sufficiently many variables.\n- A. Lampert, A. Snowden, T. Ziegler, Polynomial Bounds for Birch's Theorem, arXiv:2512.00697 (2025). (primary): https://arxiv.org/abs/2512.00697\n Evidence used: Related modern systems-of-forms result.\n\n**Review notes.** Scope mismatch between system and single-form results is retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 73, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 151, "problem_number": "GREEN-063", "title": "Solutions to Polynomial Equations in Dense Sets", "statement": "Finding a single solution to $F(x_1, \\dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of solutions in $A$ is roughly $\\alpha^n$ times the number in $[X]$?", "background": "This asks when a dense set $A$ of density $\\alpha$ contains the \"expected\" number of solutions to a Diophantine equation. Understanding when sparse sets behave like random sets for counting solutions is fundamental in analytic number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported question has no ambient range or box, limiting process, class of polynomials, nonsingularity hypotheses, or error term; it does not determine one literature-checkable theorem.\n\n**Verified partial progress.**\n\n- Circle-method, transference, and pseudorandomness theorems address numerous precise counting variants.\n\n**Full solution or refutation.**\n\nNo single status can be assigned without repairing missing quantifiers and hypotheses.\n\n**What remains.**\n\nSpecify A's ambient space and density, the asymptotic parameter, F's hypotheses, the meaning of roughly, and the error regime.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, maintained author notes, accessed 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Maintained source collection used while retaining the imported wording's missing formal data.\n\n**Review notes.** Formulation defect flagged; no OCR or statement repair made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 70, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 152, "problem_number": "GREEN-064", "title": "Residually Finite Groups", "statement": "Is every group well-approximated by finite groups?", "background": "A group is residually finite if every nontrivial element has a nontrivial image in some finite quotient. This asks if all groups have this property. The answer is known to be no (infinite simple groups), but the question may refer to finitely generated/presented groups, where it remains interesting.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Under the standard interpretation that well-approximated by finite groups means residually finite, the universal assertion is false: a nontrivial infinite simple group has no nontrivial finite quotient.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe imported background identifies infinite simple groups as the obstruction; this counterexample class refutes the universal statement under the standard interpretation.\n\n**What remains.**\n\nIf another approximation notion was intended, define it and restrict the class of groups before a new status check.\n\n**Sources checked.**\n\n- G. Higman, A Finitely Generated Infinite Simple Group, Journal of the London Mathematical Society 26 (1951), 61-64. (primary): https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms/s1-26.1.61\n Evidence used: Supplies a finitely generated infinite simple group; simplicity implies every finite quotient is trivial.\n\n**Review notes.** Classification is conditional on the standard residual-finiteness reading.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 4, "view_count": 67, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 153, "problem_number": "GREEN-065", "title": "Rado's Boundedness Conjecture", "statement": "Suppose $a_1, \\dots, a_k$ are integers which do not satisfy Rado's condition. Is $c(a_1, \\dots, a_k)$ bounded in terms of $k$ only?", "background": "Rado's condition characterizes which linear equations are partition regular. For equations not satisfying this condition, $c(\\cdot)$ is the minimum number of colors needed to avoid monochromatic solutions. Whether this depends only on $k$ (not the coefficients) is a fundamental question in Ramsey theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Rado's boundedness conjecture is affirmative for k=3, with c(a1,a2,a3)<=24, and remains open for every k>=4.\n\n**Verified partial progress.**\n\n- Fox--Kleitman prove the three-variable case with the uniform bound 24.\n\n**Full solution or refutation.**\n\nResolved in the first nontrivial arity but not in general.\n\n**What remains.**\n\nProve a coefficient-independent bound for each k>=4 or find a counterexample.\n\n**Sources checked.**\n\n- J. Fox and D. J. Kleitman, On Rado's Boundedness Conjecture, Journal of Combinatorial Theory A 113 (2006), 84-100. (primary): https://math.mit.edu/~fox/paper-FoxKleitman.pdf\n Evidence used: Proves the k=3 case with 24 colours.\n- B. Green, 100 Open Problems, Problem 21 comments, maintained notes, accessed 2026-08-17. (authoritative_secondary): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States k=3 solved and k>=4 open.\n\n**Review notes.** No source statement changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 72, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 154, "problem_number": "GREEN-066", "title": "Monochromatic x+y and xy", "statement": "If $\\{1, \\dots, N\\}$ is $r$-coloured, then for $N \\geq N_0(r)$ there exist integers $x, y \\geq 3$ such that $x+y$ and $xy$ have the same colour. Find reasonable bounds for $N_0(r)$.", "background": "This asks about unavoidable monochromatic additive-multiplicative patterns. Finding quantitative bounds for $N_0(r)$ connects Ramsey theory with both additive and multiplicative structure.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The all-colour finite quantitative question remains open; Green--Sanders record a stronger four-term pattern for two colours and identify the general weaker x+y,xy question as open.\n\n**Verified partial progress.**\n\n- Every two-colouring of {1,...,252} contains x,y,x+y,xy of one colour, a strong r=2 special case.\n- Green--Sanders prove finite-field monochromatic x,y,x+y,xy configurations for every fixed number of colours.\n\n**Full solution or refutation.**\n\nThese results do not yield the asserted N0(r) for arbitrary r on the integers.\n\n**What remains.**\n\nProve existence and reasonable bounds for N0(r) for all r, or disprove the assertion.\n\n**Sources checked.**\n\n- B. Green and T. Sanders, Monochromatic sums and products, Discrete Analysis 2016:5; arXiv:1510.08733. (primary): https://arxiv.org/abs/1510.08733\n Evidence used: Introduction identifies the weaker x+y,xy integer question as open and records the two-colour result.\n\n**Review notes.** The qualitative word reasonable is not assigned a synthetic numerical meaning.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 78, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 155, "problem_number": "GREEN-067", "title": "Affine Translates of {0,1,3}", "statement": "If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set $\\{0, 1, 3\\}$ that $A$ can contain?", "background": "This asks how many copies of the pattern $\\{0, 1, 3\\}$ (under affine transformations $x \\mapsto ax + b$) can appear in an $n$-element set. Understanding maximal copies of specific patterns is fundamental in additive combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No sharp extremal count for affine copies of {0,1,3} in an n-element integer set was verified after targeted searches.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe maintained problem entry remains open.\n\n**What remains.**\n\nDetermine the maximum count and characterize extremal or near-extremal sets.\n\n**Sources checked.**\n\n- UnsolvedMath, GREEN-067 Affine Translates of {0,1,3}, accessed 2026-08-17. (maintained_tracker): https://www.unsolvedmath.com/problems/GREEN-067\n Evidence used: Maintained entry retains the problem as open.\n\n**Review notes.** Open is dated and conservative.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 74, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 156, "problem_number": "GREEN-068", "title": "Restricted Sumsets in Partitions", "statement": "For which values of $k$ is the following true: whenever we partition $[N] = A_1 \\cup \\dots \\cup A_k$, we have $|\\bigcup_{i=1}^k (A_i \\hat{+} A_i)| \\geq \\frac{1}{10} N$?", "background": "This asks how many parts are needed before restricted sumsets (sums of distinct elements) must cover a substantial fraction of $[N]$. The problem connects partition regularity with sumset structure.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No resolution of the classification of k for the stated restricted-sumset partition lower bound was verified.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe maintained collection continues to list the precise question as open.\n\n**What remains.**\n\nDetermine the valid k, with sharp dependence on k if the 1/10 constant is varied.\n\n**Sources checked.**\n\n- UnsolvedMath, GREEN-068 Restricted Sumsets in Partitions, accessed 2026-08-17. (maintained_tracker): https://www.unsolvedmath.com/problems/GREEN-068\n Evidence used: Maintained entry gives the same statement and open status.\n\n**Review notes.** No source statement changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 157, "problem_number": "GREEN-069", "title": "Sum of Cubes in F_3^n", "statement": "Let $A_1, \\dots, A_{100}$ be \"cubes\" in $\\mathbb{F}_3^n$ (images of $\\{0, 1\\}^n$ under linear automorphisms). Is $A_1 + \\dots + A_{100} = \\mathbb{F}_3^n$?", "background": "This asks whether 100 cubes in $\\mathbb{F}_3^n$ always sum to the entire space. It's a question about additive bases and the covering properties of structured sets in vector spaces over finite fields.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Yang Yu proves the stronger statement that four cubes in F_3^n always sum to the entire space, so the stated 100-cube question is solved and the dataset status is stale.\n\n**Verified partial progress.**\n\n- Earlier work of Alon, Linial and Meshulam required on the order of log n cubes.\n\n**Full solution or refutation.**\n\nEach cube is the set of 0/1 subset sums of a linear basis. Yu proves that the union of any four bases over Z_3 is an additive basis, equivalently that the sum of the corresponding four cubes is F_3^n. Since every further cube contains zero, the 100-fold sum is also all of F_3^n.\n\n**What remains.**\n\nThe analogous additive-basis conjecture over F_p for primes beyond 3 remains open; nothing remains for the exact F_3 statement imported here.\n\n**Sources checked.**\n\n- Yang Yu, The Permanent Rank of a Matrix (Part Three) Note on the Additive Basis Conjecture, arXiv:2510.01300v2 (2026). (primary): https://arxiv.org/abs/2510.01300\n Evidence used: The abstract states that the union of any four linear bases over Z_3 is an additive basis.\n- Ben Green, 100 Open Problems, Problem 26, most recent update December 2025. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Marks the problem solved and records that four cubes suffice.\n\n**Review notes.** The implication from four to 100 uses that every cube contains 0. No source statement was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 71, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 158, "problem_number": "GREEN-070", "title": "Sets with No Unique Sum Representations", "statement": "What is the size of the smallest set $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation?", "background": "This asks for the minimum size of a set where every sum $a + a'$ has multiple representations. Bedert (2023) showed the answer lies between $\\omega(p)\\log p$ and $O(\\log^2 p)$. Closing this gap would deepen our understanding of additive bases.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Bedert narrows the minimum size m(p) to omega(p) log p <= m(p) << (log p)^2, with omega(p) tending to infinity; the asymptotic order remains open.\n\n**Verified partial progress.**\n\n- The lower bound improves c log p to omega(p) log p.\n- The upper bound improves order sqrt(p) to order (log p)^2.\n\n**Full solution or refutation.**\n\nNo exact asymptotic is known in the checked literature. Bedert establishes a polylogarithmic window for the minimum size.\n\n**What remains.**\n\nClose the gap between the super-logarithmic lower bound and quadratic-logarithmic upper bound, ideally determining the order or an asymptotic formula.\n\n**Sources checked.**\n\n- Benjamin Bedert, On unique sums in Abelian groups, Combinatorica 44 (2024), no. 2, 269-298; arXiv:2303.15134. (primary): https://arxiv.org/abs/2303.15134\n Evidence used: Defines m(p), fixes the unordered-representation convention, and proves omega(p) log p <= m(p) << (log p)^2.\n- Ben Green, 100 Open Problems, Problem 27, most recent update December 2025. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Retains the question and records Bedert's bounds as significant progress.\n\n**Review notes.** The imported statement omits that representations differing only by order are the same; Bedert's precise convention is {a1,a2} != {a1',a2'}.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 76, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 159, "problem_number": "GREEN-071", "title": "Uniform Random Variables with Uniform Sum", "statement": "Suppose $X, Y$ are finitely-supported independent random variables taking integer values such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly distributed on their ranges?", "background": "This asks if uniform sums force uniform summands. It's a discrete probability question with connections to additive combinatorics and the structure of convolutions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The equivalent unfair 0-1 polynomial conjecture remains open, but infinite families and substantial bounded-degree cases have been established.\n\n**Verified partial progress.**\n\n- Ghidelli proves a first nonclassical infinite family, including a factor x^5+a x^2+1.\n- Hare algorithmically treats all candidate factors of degree at most 15, eliminating all but 975 of 7,141,686 coefficient patterns.\n- A direct June 2026 research report gives exact Positivstellensatz verification through total product degree 66.\n- When X+Y is uniform on a consecutive integer interval, earlier deconvolution results force both summands to be uniform on particular finite sets.\n\n**Full solution or refutation.**\n\nAfter translating supports and normalising probability-generating polynomials, the question asks whether monic nonnegative-real factors of a 0-1 polynomial must themselves be 0-1 polynomials. No proof or counterexample in arbitrary degree was found.\n\n**What remains.**\n\nProve the unfair 0-1 polynomial conjecture for arbitrary degree or construct a nonnegative-real counterexample. Computational verification cannot settle the unbounded-degree statement.\n\n**Sources checked.**\n\n- Luca Ghidelli, Progress on the unfair 0-1-polynomials conjecture using linear recurrences and numerical analysis, arXiv:2209.09843 (2022). (primary): https://arxiv.org/abs/2209.09843\n Evidence used: States the equivalent probability conjecture and proves a nonclassical infinite family of factor cases.\n- Kevin G. Hare, Computational Progress on the Unfair 0-1 Polynomial Conjecture, arXiv:2307.07363; Experimental Mathematics, published online 2025. (primary): https://arxiv.org/abs/2307.07363\n Evidence used: Analyzes all candidate factors through degree 15 and reports the precisely delimited residual cases.\n- David Zhang, research report answering MathOverflow question 339137, 16 June 2026. (primary): https://mathoverflow.net/questions/339137/why-do-polynomials-with-coefficients-0-1-like-to-have-only-factors-with-0-1\n Evidence used: Reports reproducible exact Positivstellensatz certificates through total degree 66; it explicitly says this is partial computational progress, not a full answer.\n- Anatoly Zhigljavsky, Nina Golyandina and Svyatoslav Gryaznov, Deconvolution of a discrete uniform distribution, Statistics & Probability Letters 118 (2016), 37-44. (primary): https://doi.org/10.1016/j.spl.2016.06.006\n Evidence used: Proves uniformity of the factors in the special case where the sum is uniform on a consecutive interval.\n- Ben Green, 100 Open Problems, Problem 28, most recent update December 2025. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Retains the arbitrary-support question.\n\n**Review notes.** Range means positive-probability support, not necessarily a consecutive interval. The worker performed no computational verification; reported computations belong to cited literature.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 70, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 160, "problem_number": "GREEN-072", "title": "Large Subsets of Approximate Groups", "statement": "Suppose $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \\subset A$ with $|S| \\gg K^{-O(1)}|A|$ and $S^8 \\subset A^4$?", "background": "Approximate groups are sets with controlled doubling. This asks if they contain large subsets with even better multiplicative structure. Understanding approximate groups is central to geometric group theory and additive combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The required S is known with a non-polynomial dependence on K, but the polynomial lower bound |S| >> K^{-O(1)}|A| remains open in Green's current notes.\n\n**Verified partial progress.**\n\n- A Sanders argument yields |S| >>_K |A| and the desired containment, with quantitatively poor dependence on K.\n\n**Full solution or refutation.**\n\nNo polynomial-in-K quantitative version was located. The qualitative bounded-K statement does not meet the record's central requirement.\n\n**What remains.**\n\nObtain the power containment S^8 subset A^4 while proving |S| >= K^{-C}|A| for an absolute C, or demonstrate an obstruction to polynomial dependence.\n\n**Sources checked.**\n\n- Emmanuel Breuillard, Ben Green and Terence Tao, Small doubling in groups, arXiv:1301.7718; Erdos Centennial, Bolyai Society Mathematical Studies 25 (2013), 129-151. (primary): https://arxiv.org/abs/1301.7718\n Evidence used: Problem 6.5 records the quantitative approximate-group question and the non-polynomial Sanders bound.\n- Ben Green, 100 Open Problems, Problem 29, most recent update December 2025. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Explicitly says polynomial dependence is not known.\n\n**Review notes.** The imported record assumes the standard definition of a finite K-approximate group; its background does not state that convention.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 161, "problem_number": "GREEN-073", "title": "Structured Subsets with Bounded Doubling", "statement": "Given a set $A \\subset \\mathbb{Z}$ with $D(A) \\leq K$, find a large structured subset $A'$ which \"obviously\" has $D(A') \\leq K + \\varepsilon$.", "background": "Sets with small doubling constant have additive structure. This asks for an explicit, easily verifiable structured subset. Making structure \"obvious\" connects to algorithmic aspects of the Polynomial Freiman-Ruzsa conjecture.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The words large, structured, and obviously are undefined, so the record is an agenda rather than a uniquely checkable theorem; a concrete near-4 special case is known.\n\n**Verified partial progress.**\n\n- If |A-A| <= (4-epsilon)|A|, Eberhard-Green-Manners find a progression P of length >>_epsilon |A| on which A has density greater than 1/2.\n- For A'=A intersect P, one then has the transparent bound D(A') <= 4; the size constant in this theorem is essentially ineffective.\n\n**Full solution or refutation.**\n\nA useful special case exists, but the general wording has no specified size function or structural class and therefore has no determinate solved/open boundary.\n\n**What remains.**\n\nSpecify quantitative meanings for large and structured, then extend the near-4 progression theorem to a stated K range with effective dependence if desired.\n\n**Sources checked.**\n\n- Sean Eberhard, Ben Green and Freddie Manners, Sets of integers with no large sum-free subset, Annals of Mathematics 180 (2014), 621-652; arXiv:1301.4579. (primary): https://arxiv.org/abs/1301.4579\n Evidence used: Section 6 contains the near-4 progression-density theorem used by Green to illustrate the intended programme.\n- Ben Green, 100 Open Problems, Problem 30, most recent update December 2025. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Defines D(A)=|A-A|/|A|, gives the illustrative special case, and notes ineffectivity.\n\n**Review notes.** The dataset background calls D a doubling constant, but the source defines the difference ratio |A-A|/|A|. The direct algorithmic-PFR gloss is not supported by the source statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 162, "problem_number": "GREEN-074", "title": "Sidon Set Size Bounds", "statement": "Write $F(N)$ for the largest Sidon subset of $[N]$. Improve, at least for infinitely many $N$, the bounds $N^{1/2} + O(1) \\leq F(N) \\leq N^{1/2} + N^{1/4} + O(1)$.", "background": "Sidon sets have all pairwise sums distinct. The bounds have been tight for decades. Balogh-Füredi-Roy (2021) obtained a small improvement to the upper bound. Any further progress would be a major breakthrough in additive combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The displayed upper bound has been numerically improved to F(N) <= N^{1/2}+0.98183 N^{1/4}+O(1), but Green retains the item as an open-ended Sidon-set programme.\n\n**Verified partial progress.**\n\n- Balogh-Furedi-Roy obtained coefficient 0.998 in place of 1.\n- Carter-Hunter-O'Bryant improved the coefficient to 0.98183.\n- The latter paper also provides a hand-verifiable coefficient 0.99529.\n\n**Full solution or refutation.**\n\nThe literal instruction to improve the old upper bound has been fulfilled. Because no terminal target was specified and the central upper/lower gap persists, the living question is best classified as partially solved rather than closed.\n\n**What remains.**\n\nFurther lower or upper improvements, especially a qualitative advance beyond changing the N^{1/4} coefficient, and determination of the true second-order behaviour.\n\n**Sources checked.**\n\n- Daniel Carter, Zach Hunter and Kevin O'Bryant, On the Diameter of Finite Sidon Sets, Acta Mathematica Hungarica 175 (2025), 108-126; arXiv:2310.20032. (primary): https://arxiv.org/abs/2310.20032\n Evidence used: Proves the equivalent upper bound with coefficient 0.98183 and a weaker hand-verifiable improvement.\n- Ben Green, 100 Open Problems, Problem 31, most recent update December 2025. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Records the 2021 and 2023 improvements but does not mark the open-ended problem solved.\n\n**Review notes.** There is a semantic choice: the literal word improve has been satisfied, while the maintained author treats the item as an ongoing programme. Partial progress records both facts conservatively.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 89, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 163, "problem_number": "GREEN-075", "title": "Large Gaps in Dilates", "statement": "Let $p$ be a prime and let $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ be a set of size $\\sqrt{p}$. Is there a dilate of $A$ with a gap of length $100\\sqrt{p}$?", "background": "This asks whether dilates (multiplicative translates) of sets necessarily have large gaps. Understanding the distribution of dilates connects additive and multiplicative combinatorics in cyclic groups.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Shakan guarantees a gap of size at least 2p/|A|-2 in some dilate, giving constant about 2 at square-root size; the requested constant 100 remains open under the intended rounded/asymptotic formulation.\n\n**Verified partial progress.**\n\n- For every A subset F_p with |A|>1, some nonzero dilate has a gap of size at least 2p/|A|-2.\n\n**Full solution or refutation.**\n\nThe polynomial-method theorem establishes the correct p/|A| scale but not the requested constant 100.\n\n**What remains.**\n\nAfter repairing the cardinality and gap-rounding conventions, improve the constant 2 to 100 at |A| asymptotic to sqrt(p), or find a counterexample to such a universal constant.\n\n**Sources checked.**\n\n- George Shakan, A large gap in a dilate of a set, SIAM Journal on Discrete Mathematics 34 (2020), no. 4, 2553-2555; arXiv:2004.14828. (primary): https://arxiv.org/abs/2004.14828\n Evidence used: Proves the explicit lower bound 2p/|A|-2 for a gap in a dilate.\n- Ben Green, 100 Open Problems, Problem 32, most recent update December 2025. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Retains constant 100 as open and says constant 2 appears to be the limit of Shakan's method.\n\n**Review notes.** Literal defect: sqrt(p) is nonintegral for every prime p, so no set has exactly that size. A corrected record must choose floor, ceiling, or |A|~sqrt(p), and define rounding/cyclic meaning of gap length.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 72, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 164, "problem_number": "GREEN-076", "title": "Optimal Sidon Bases", "statement": "Are there infinitely many $q$ for which there is a set $A \\subset \\mathbb{Z}/q\\mathbb{Z}$ with $|A| = (\\sqrt{2} + o(1))q^{1/2}$ and $A + A = \\mathbb{Z}/q\\mathbb{Z}$?", "background": "This asks if Sidon-like sets (near-optimal density with few sum collisions) can form additive bases. The coefficient $\\sqrt{2}$ is conjecturally optimal for such constructions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No construction attaining the asymptotically minimal sqrt(2q) scale for infinitely many cyclic groups was found; Green's current maintained collection still asks the question.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe elementary unordered-pair count explains the sqrt(2) lower threshold, but the checked sources do not supply infinitely many matching cyclic constructions.\n\n**What remains.**\n\nConstruct sets A in Z/qZ for infinitely many q with A+A equal to the whole group and |A|=(sqrt(2)+o(1))sqrt(q), or prove an asymptotic obstruction above sqrt(2).\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 33, most recent update December 2025. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the exact cyclic additive-basis question and retains it without a solution update.\n\n**Review notes.** The dataset title Optimal Sidon Bases is misleading: the statement imposes no Sidon condition. Open labels are dated and conservative.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 75, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 165, "problem_number": "GREEN-077", "title": "Structure of Sets with Bounded Representation", "statement": "Suppose $A \\subset [N]$ has size $\\geq c\\sqrt{N}$ and representation function $r_A(n) \\leq r$ for all $n$. What can be said about the structure of $A$?", "background": "Sets with bounded representation function (few ways to write sums) have special structure. Understanding this structure connects Sidon set theory with additive bases.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Green explicitly presents this as a vague structural programme and says even formulating a conjecture seems hopeless; without a defined target notion of structure it has no theorem-level status.\n\n**Verified partial progress.**\n\n- Eberhard-Manners identify a projective-plane mechanism behind all known dense Sidon examples, classify the Desarguesian cases, and exhibit many non-Desarguesian examples.\n- They conjecture that all dense Sidon sets arise from projective planes through this mechanism and give a bestiary of somewhat smaller algebraic examples.\n\n**Full solution or refutation.**\n\nThe literature offers a unifying view of known constructions, not a classification theorem for arbitrary A satisfying the imported hypotheses.\n\n**What remains.**\n\nFix the quantifiers for c and r, the asymptotic regime, and a testable structural conclusion before asking for a proof; even the dense Sidon case remains conjectural.\n\n**Sources checked.**\n\n- Sean Eberhard and Freddie Manners, The apparent structure of dense Sidon sets, Electronic Journal of Combinatorics 30 (2023), no. 1, P1.33; arXiv:2107.05744. (primary): https://arxiv.org/abs/2107.05744\n Evidence used: Gives the unified projective-plane construction picture, classifications of known families, and an explicitly conjectural global structural claim.\n- Ben Green, 100 Open Problems, Problem 34, most recent update December 2025. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Calls the problem vague and says that even formulating a conjecture appears hopeless.\n\n**Review notes.** The source counts x+y and y+x as the same representation. The dataset background's claim that all such sets have special structure overstates the known construction evidence.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 70, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 167, "problem_number": "GREEN-079", "title": "Disjoint Sumsets Construction", "statement": "For arbitrarily large $n$, does there exist an abelian group $H$ with $|H| = n^{2+o(1)}$ and subsets $A_1, \\dots, A_n, B_1, \\dots, B_n$ satisfying $|A_i||B_i| \\geq n^{2-o(1)}$, $|A_i + B_i| = |A_i||B_i|$, such that $A_i + B_i$ are pairwise disjoint from $A_j + B_k$ ($j \\neq k$)?", "background": "This asks if one can partition a group into many disjoint sumsets with no doubling. It connects to the structure of Sidon sets and extremal problems in additive combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Cohn-Kleinberg-Szegedy-Umans Conjecture 4.7, the asymptotically optimal abelian simultaneous double product construction, remains open in Green's current collection.\n\n**Verified partial progress.**\n\n- Cohn-Kleinberg-Szegedy-Umans give nonoptimal constructions and prove that the conjectured alpha=beta=2 parameters would imply matrix multiplication exponent omega=2.\n- They formulate the exact condition as the simultaneous double product property.\n\n**Full solution or refutation.**\n\nNo abelian family meeting |H|=n^{2+o(1)} and |A_i||B_i|>=n^{2-o(1)} with the full cross-index disjointness was located.\n\n**What remains.**\n\nConstruct asymptotically optimal simultaneous-double-product families, prove they cannot exist, or sharpen the known parameter tradeoffs.\n\n**Sources checked.**\n\n- Henry Cohn, Robert Kleinberg, Balazs Szegedy and Christopher Umans, Group-theoretic algorithms for matrix multiplication, FOCS 2005, 379-388; arXiv:math/0511460. (primary): https://arxiv.org/abs/math/0511460\n Evidence used: Definition 4.1 and Conjecture 4.7 give the simultaneous double product property and exact asymptotic parameter target; the paper proves that it would imply omega=2.\n- Ben Green, 100 Open Problems, Problem 36, most recent update December 2025. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Retains the construction as an open problem and records its omega=2 implication.\n\n**Review notes.** The dataset background is inaccurate: no partition of H is required, and |A_i+B_i|=|A_i||B_i| means injectivity of the addition map, not no doubling. The cross-index clause is (A_i+B_i) intersect (A_j+B_k)=empty for every j!=k.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 168, "problem_number": "GREEN-080", "title": "Cap Sets in F_7^n", "statement": "What is the largest subset $A \\subset \\mathbb{F}_7^n$ for which $A - A$ intersects $\\{-1, 0, 1\\}^n$ only at 0?", "background": "This is a cap set problem in $\\mathbb{F}_7^n$ with restricted difference set. Recent polynomial method breakthroughs dramatically improved bounds for $\\mathbb{F}_3^n$, but $\\mathbb{F}_7$ remains open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The extremal exponential growth rate is the still-unknown Shannon capacity of the 7-cycle; the best maintained bounds leave a narrow but genuine gap.\n\n**Verified partial progress.**\n\n- Polak--Schrijver construct an independent set of size 367 in the fifth strong power of C_7, giving the lower bound Theta(C_7) >= 367^(1/5) approximately 3.2578.\n- Lovász's theta-function bound gives Theta(C_7) <= 7 cos(pi/7)/(1+cos(pi/7)) approximately 3.3177.\n\n**Full solution or refutation.**\n\nThe lower and upper exponential bases do not coincide, so the largest asymptotic size is not determined.\n\n**What remains.**\n\nClose the gap between the 367^(1/5) construction and the Lovász theta upper bound, or otherwise determine Theta(C_7).\n\n**Sources checked.**\n\n- S. Polak, A. Schrijver, New lower bound on the Shannon capacity of C7 from circular graphs, Information Processing Letters 143 (2019), 37-40; arXiv:1808.07438. (primary): https://arxiv.org/abs/1808.07438\n Evidence used: Abstract states the independent-set construction of size 367 and the resulting 367^(1/5) lower bound.\n- B. Green, 100 Open Problems, Problem 38, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Identifies the exact statement with the Shannon capacity of C_7 and records both current exponential bounds.\n\n**Review notes.** The local label GREEN-080 matches Green Problem 38 by statement. The imported title/background call this a cap-set problem, but the maintained source identifies it as Shannon capacity of C_7; this defect was flagged rather than rewritten.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 73, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 169, "problem_number": "GREEN-081", "title": "Covering by Random Translates", "statement": "If $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ is random with $|A| = \\sqrt{p}$, can we almost surely cover $\\mathbb{Z}/p\\mathbb{Z}$ with $100\\sqrt{p}$ translates of $A$?", "background": "This asks about the covering properties of random sets. Understanding when random sets form good coverings connects probability, additive combinatorics, and coding theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The random-set covering assertion remains open; Green reports not knowing it even with 100 replaced by 1.01.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo verified high-probability O(sqrt(p))-translate covering theorem at the stated density was located.\n\n**What remains.**\n\nAfter fixing the rounding and probability model, prove that a random square-root-size subset has covering number O(sqrt(p)), or disprove this high-probability assertion.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, Problem 39, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the random translate-covering problem and says it is unknown even with constant 1.01.\n\n**Review notes.** The local label GREEN-081 matches Green Problem 39. For prime p, sqrt(p) is not integral; a precise statement needs floor/ceiling and a uniform fixed-cardinality random model. No convention was silently inserted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 170, "problem_number": "GREEN-082", "title": "Hamming Ball Covering Growth", "statement": "Let $r$ be fixed and let $H(r)$ be the Hamming ball of radius $r$ in $\\mathbb{F}_2^n$. Let $f(r)$ be the smallest constant such that there exist infinitely many $n$ with subspaces $V_n \\leq \\mathbb{F}_2^n$ satisfying $V_n + H(r) = \\mathbb{F}_2^n$ and $|V_n| = (f(r) + o(1)) \\frac{2^n}{|H(r)|}$. Does $f(r) \\to \\infty$?", "background": "This asks if covering $\\mathbb{F}_2^n$ by Hamming ball translates requires increasingly inefficient packings as $r$ grows. It connects coding theory with the geometry of finite vector spaces.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Explicit linear covering-code constructions show f(1)=1, f(r)<=r^r/r! asymptotic to e^r, and f(2)<=1.4238, but no divergence lower bound is known.\n\n**Verified partial progress.**\n\n- Perfect Hamming codes yield f(1)=1.\n- Products of r Hamming-code constructions give f(r)<=r^r/r!, hence finiteness and an asymptotic upper bound of order e^r.\n- Davydov's construction gives f(2)<=1.4238, while even f(2)=1 has not been ruled out.\n\n**Full solution or refutation.**\n\nThe known results are upper constructions and one exact radius; they do not show that f(r) grows at all.\n\n**What remains.**\n\nProve an unbounded lower bound for the asymptotic density of linear covering codes as the radius grows, or construct bounded-density families refuting divergence.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, Problem 40, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Records f(1)=1, the product upper bound, the f(2)<=1.4238 construction, and the absence of any result ruling out f(r)=1.\n- A. A. Davydov, Construction of linear covering codes, Problems of Information Transmission 26 (1990), 317-331 (English translation, 1991). (primary): https://www.mathnet.ru/eng/ppi628\n Evidence used: Primary construction cited by Green for the best recorded radius-two upper bound.\n\n**Review notes.** The local label GREEN-082 matches Green Problem 40 by exact statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 66, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 171, "problem_number": "GREEN-083", "title": "Pyjama Set Covering", "statement": "How many rotated (about the origin) copies of the \"pyjama set\" $\\{(x, y) \\in \\mathbb{R}^2 : \\operatorname{dist}(x, \\mathbb{Z}) \\leq \\varepsilon\\}$ are needed to cover $\\mathbb{R}^2$?", "background": "The pyjama set is a union of vertical strips. This beautiful geometric problem, solved by Manners (2015), asks how many rotations are needed to cover the plane. It connects geometry, combinatorics, and Fourier analysis.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finite coverability is solved, and a 2025 preprint gives the first explicit upper bound of exp(exp(exp(epsilon^{-O(1)}))) rotations, but the quantitative order remains wide open.\n\n**Verified partial progress.**\n\n- Manners proved that finitely many rotations cover the plane for every epsilon>0, but his argument was ineffective.\n- Kravitz--Leng made Manners's method quantitative and proved that triple-exponentially many rotations in a power of 1/epsilon suffice.\n\n**Full solution or refutation.**\n\nManners answers the qualitative finite-existence question; Kravitz--Leng answer effectivity, but neither determines how many rotations are needed up to a reasonable order.\n\n**What remains.**\n\nObtain substantially smaller upper bounds, such as epsilon^{-C}, and nontrivial matching lower bounds for the minimum covering number.\n\n**Sources checked.**\n\n- F. Manners, A solution to the pyjama problem, Invent. Math. 202 (2015), 239-270, doi:10.1007/s00222-014-0571-7. (primary): https://arxiv.org/abs/1305.1514\n Evidence used: Abstract proves finite coverability for every positive epsilon.\n- N. Kravitz, J. Leng, Quantitative pyjama, arXiv:2510.17744 (2025). (primary): https://arxiv.org/abs/2510.17744\n Evidence used: Abstract states the explicit exp exp exp(epsilon^{-O(1)}) upper bound.\n- B. Green, 100 Open Problems, Problem 41, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Distinguishes Manners's qualitative solution from the remaining quantitative problem and records the 2025 bound.\n\n**Review notes.** The local label GREEN-083 matches Green Problem 41. The dataset background's word 'solved' refers only to qualitative finite existence, not to the exact quantitative 'how many' wording.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 74, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 172, "problem_number": "GREEN-084", "title": "Cohn-Elkies Scheme for Circle Packings", "statement": "Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings?", "background": "Cohn-Elkies developed a linear programming approach that proved optimal sphere packing in dimensions 8 and 24. Whether their method extends to circles in the plane remains a major open question in discrete geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The existence of a sharp Cohn--Elkies auxiliary function for optimal circle packing in dimension two remains open.\n\n**Verified partial progress.**\n\n- The planar question is equivalent to constructing a radial function with the Cohn--Elkies sign conditions and sharp ratio sqrt(3)/6, with forced zeros on the hexagonal lattice and its dual.\n- The scheme is sharp in dimensions 1, 8, and 24, but those constructions do not supply the missing dimension-two function.\n\n**Full solution or refutation.**\n\nNo verified planar magic function or proof that the Cohn--Elkies linear program attains the hexagonal packing density was located.\n\n**What remains.**\n\nConstruct the sharp radial auxiliary function in R^2, or prove the Cohn--Elkies linear program cannot attain the optimal planar density.\n\n**Sources checked.**\n\n- H. Cohn, N. Elkies, New upper bounds on sphere packings I, Ann. of Math. 157 (2003), 689-714. (primary): https://annals.math.princeton.edu/2003/157-2/p09\n Evidence used: Introduces the linear-programming scheme and its auxiliary-function upper bound.\n- B. Green, 100 Open Problems, Problem 42, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Keeps the dimension-two sharpness question open and states the precise auxiliary-function conditions.\n\n**Review notes.** The local label GREEN-084 matches Green Problem 42. The background compresses attribution: Viazovska and Cohn--Kumar--Miller--Radchenko--Viazovska supplied the sharp 8- and 24-dimensional functions using the scheme.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 71, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 173, "problem_number": "GREEN-085", "title": "Covering by Residue Classes", "statement": "Let $N$ be large. For each prime $p$ with $N^{0.51} \\leq p < 2N^{0.51}$, pick a residue $a(p) \\in \\mathbb{Z}/p\\mathbb{Z}$. Is $\\#\\{n \\in [N] : n \\equiv a(p) \\pmod p \\text{ for some } p\\} \\gg N^{1-o(1)}$?", "background": "This asks if residue classes from medium-sized primes nearly cover $[N]$. It connects sieve theory with covering problems and the distribution of primes.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No N^{1-o(1)} lower bound for the union of the arbitrarily selected N^0.51-scale residue classes was located; Green's maintained collection retains the problem as open.\n\n**Verified partial progress.**\n\n- Analogous questions with prime-size exponent alpha<1/2 follow by inclusion-exclusion or Cauchy--Schwarz.\n- The stated exponent 0.51 is deliberately beyond that square-root threshold and is compared in the source to a Kakeya problem in dimension 2+epsilon.\n\n**Full solution or refutation.**\n\nThe elementary sub-square-root methods do not reach the exact super-square-root range.\n\n**What remains.**\n\nProve the uniform N^{1-o(1)} union bound for every choice of residues at the stated prime scale, or construct choices with a substantially smaller union.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, Problem 43, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the exact N^0.51 problem, explains the alpha<1/2 argument, and retains the target as open.\n- B. Green, A note on multiplicative functions on progressions to large moduli, Proc. Roy. Soc. Edinburgh Sect. A 148 (2018), 63-77. (primary): https://doi.org/10.1017/S0308210517000144\n Evidence used: Section 4 is the primary source to which Green attributes this formulation.\n\n**Review notes.** The local label GREEN-085 matches Green Problem 43 by exact statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 174, "problem_number": "GREEN-086", "title": "Sieving by Many Small Primes", "statement": "Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \\leq p_1 < p_2 < \\dots < p_{1000} < N^{9/10}$. Does the remaining set have size at most $\\frac{1}{10}N$?", "background": "This asks whether aggressive sieving by many small primes can remove most of $[N]$. Understanding sieve limits is fundamental in analytic number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The literal statement is not a uniquely defined theorem: odd primes have no integral 'half' of their residue classes, and the quantifier over the choices of removed classes is omitted; the intended large-sieve question remains open in Green's notes.\n\n**Verified partial progress.**\n\n- Erdős observed that the intended assertion is affirmative when all 1000 primes are below N^{1/2}, by the large sieve.\n- Green reports no further literature progress for the N^{9/10} version.\n\n**Full solution or refutation.**\n\nNo status stronger than formulation-dependent uncertainty is justified for the exact wording; a natural intended variant is still open.\n\n**What remains.**\n\nSpecify floor or ceiling for half the classes, whether classes are distinct, and whether the bound must hold for every selection; then resolve the resulting N^{9/10} sieve statement.\n\n**Sources checked.**\n\n- B. Green, 100 Open Problems, Problem 44, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Uses the same terse wording and records only the large-sieve result for primes below N^{1/2}.\n\n**Review notes.** The local label GREEN-086 matches Green Problem 44. The parity and quantifier defects occur in the maintained source too and were flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 67, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 175, "problem_number": "GREEN-087", "title": "Residue Class Multiple Coverage", "statement": "Can we pick residue classes $a_p \\pmod p$, one for each prime $p \\leq N$, such that every integer $\\leq N$ lies in at least 10 of them?", "background": "This asks if we can achieve high-multiplicity covering using one residue class per prime. It's dual to sieving problems and connects to the large sieve.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The ten-fold residue-class covering problem remains open; even the two-fold Erdős variant is still tracked as open, though a 2026 manuscript claims a proof and requests expert verification.\n\n**Verified partial progress.**\n\n- The maintained Erdős entry #689 formulates the weaker target in which every integer is covered at least twice.\n- A 2026 public manuscript titled 'A greedy matching proof of Erdős's two-fold residue-class problem' claims the two-fold result, but its accompanying discussion explicitly asks experts to verify a weighted Green--Tao--Ziegler moment input.\n- Even a verified two-fold theorem would not by itself establish ten-fold coverage.\n\n**Full solution or refutation.**\n\nNo accepted theorem giving ten-fold coverage was located, and the recent two-fold claim is both weaker and not independently certified.\n\n**What remains.**\n\nVerify or refute the proposed two-fold proof, then develop a construction guaranteeing multiplicity ten for all integers up to N.\n\n**Sources checked.**\n\n- T. F. Bloom, Erdős Problem #689 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/689\n Evidence used: Tracks the two-fold version as open and links it explicitly to Green Problem 45 with 10 in place of 2.\n- Erdős Problems #689 discussion thread, proposed-proof posts dated April--June 2026. (authoritative_secondary): https://www.erdosproblems.com/forum/thread/689\n Evidence used: The authors describe the proof as proposed/unrefereed and request checking of the analytic moment proposition.\n- B. Green, 100 Open Problems, Problem 45, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the ten-fold problem and records that Erdős did not know even the two-fold version.\n\n**Review notes.** The local label GREEN-087 matches Green Problem 45. The unverified 2026 claim is not promoted to established partial literature, and it concerns multiplicity two rather than ten.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 176, "problem_number": "GREEN-088", "title": "Maximal Covering Interval", "statement": "What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p \\pmod p$, one for each prime $p \\leq x$?", "background": "This is the classical covering problem in sieve theory. Determining the optimal relationship between $x$ and $y$ would have significant implications for understanding the distribution of primes.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Jacobsthal covering quantity has classical upper and modern lower bounds, but its order of magnitude remains unknown.\n\n**Verified partial progress.**\n\n- Ford--Green--Konyagin--Maynard--Tao give coverings of length y much larger than x log x by logarithmic factors, the best lower scale recorded by Green.\n- Iwaniec proves the upper bound y<", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 70, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 177, "problem_number": "GREEN-089", "title": "Random Walk Mixing on Alternating Groups", "statement": "Pick $x_1, \\dots, x_k \\in A_n$ at random. Is it true that, almost surely as $n \\to \\infty$, the random walk on this set of generators and their inverses equidistributes in time $O(n \\log n)$?", "background": "This asks about mixing time for random walks on the alternating group with random generators. Determining optimal mixing times connects probability, group theory, and spectral graph theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For a random pair of generators, polynomial diameter and mixing bounds are known, but fixed-degree expansion and O(n log n) mixing remain open.\n\n**Verified partial progress.**\n\n- Helfgott--Seress--Żuk prove with high probability diameter O(n^2(log n)^C) for a random pair in Sym(n) or Alt(n).\n- The same work proves mixing time O(n^3(log n)^C) and a nonconstant spectral-gap bound.\n- These bounds are much weaker than the constant spectral gap/equivalent O(n log n) mixing asked for.\n\n**Full solution or refutation.**\n\nRandom generators yield strong polynomial control, but no proof that a fixed random generating set is an expander family was located.\n\n**What remains.**\n\nFor a fixed quantified k, prove a uniform spectral gap with high probability, equivalently the stated O(n log n) mixing scale, or disprove it.\n\n**Sources checked.**\n\n- H. A. Helfgott, Á. Seress, A. Żuk, Random generators of the symmetric group: diameter, mixing time and spectral gap, J. Algebra 421 (2015), 349-368, doi:10.1016/j.jalgebra.2014.08.033. (primary): https://arxiv.org/abs/1311.6742\n Evidence used: Abstract states the high-probability O(n^2 log^C n) diameter and O(n^3 log^C n) mixing bounds for a random pair.\n- B. Green, 100 Open Problems, Problem 79, version retrieved 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: States the expansion formulation, records the Helfgott--Seress--Żuk diameter bound, and retains the fixed-random-generator question as open.\n\n**Review notes.** The local label GREEN-089 matches Green Problem 79, not Problem 47 or 89. The statement does not quantify k; the maintained comments indicate a fixed-k question and mention k=2.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 178, "problem_number": "GREEN-090", "title": "Bounds for Approximate Group Classification", "statement": "Find bounds in the classification theorem for approximate groups.", "background": "The Breuillard-Green-Tao classification shows approximate groups resemble actual groups. However, the bounds in this theorem are extremely poor. Improving them would have significant applications in additive combinatorics and geometric group theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The arbitrary-group Breuillard-Green-Tao classification still has ineffective general covering constants, while effective bounds are known for substantial restricted classes.\n\n**Verified partial progress.**\n\n- Breuillard, Green, and Tao proved the qualitative finite-by-nilpotent classification for arbitrary finite K-approximate groups.\n- For residually nilpotent ambient groups, Tointon gives effective bounds: a cover by exp(K^{O(1)}) cosets and a nilpotent quotient of step at most K^6.\n- Polynomial bounds are known for fixed-dimensional simple algebraic groups, and quantitative BGT applications to polynomial-growth balls have also advanced.\n\n**Full solution or refutation.**\n\nNo explicit universal dependence for every arbitrary ambient group was located; the maintained Green source continues to list that general quantitative problem as open.\n\n**What remains.**\n\nReplace all ineffective K-dependent constants in the arbitrary-group BGT structure theorem by explicit functions, ideally with quantitatively meaningful rank, step, and covering bounds.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, current Problem 80, updates through December 2025; checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Identifies the intended BGT theorem, says its ultraproduct proof is ineffective, and retains the request for bounds.\n- Emmanuel Breuillard, Ben Green, and Terence Tao, The structure of approximate groups, Publications Mathematiques de l'IHES 116 (2012), 115-221. (primary): https://doi.org/10.1007/s10240-012-0043-9\n Evidence used: Proves the general qualitative structure theorem whose quantitative constants are at issue.\n- Matthew C. H. Tointon, Approximate subgroups of residually nilpotent groups, Mathematische Annalen 374 (2019), 499-515. (primary): https://doi.org/10.1007/s00208-018-01795-z\n Evidence used: Provides effective K-dependent bounds in the residually nilpotent case, including step at most K^6 and exp(K^{O(1)}) coset covering.\n\n**Review notes.** Material source-version mismatch: stored GREEN-090 is current Green Problem 80. Current Problem 90 is a different affine-invariance question. The exact database statement is preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 72, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1097, "problem_number": "GREEN-097", "title": "N-Queens Problem Asymptotics", "statement": "In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \\times n$ chessboard?", "background": "The n-queens problem asks for asymptotic formulas for the number of ways to place n non-attacking queens on an n×n board. Recent work has made progress on both upper and lower bounds, but the precise asymptotic behavior remains elusive.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Simkin determines the exponential growth rate of Q(n), but Green's stronger requested multiplicative asymptotic remains open.\n\n**Verified partial progress.**\n\n- Simkin proves Q(n)=((1+o(1)) n exp(-alpha))^n for a constant alpha defined by an entropy optimization problem.\n- Equivalently, Q(n)^{1/n}/n converges to exp(-alpha), resolving the logarithmic or exponential-rate asymptotic.\n- Nobel, Agrawal, and Boyd tighten the numerical interval for alpha to approximately [1.944000752, 1.944001082].\n\n**Full solution or refutation.**\n\nThe known o(1) occurs inside an nth power and can hide a multiplicative exp(o(n)) factor, so it is not a full multiplicative asymptotic for Q(n).\n\n**What remains.**\n\nObtain an asymptotic formula with multiplicative relative error, or otherwise determine the subexponential factor beyond Simkin's exponential rate.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 97, updates through December 2025; checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Records Simkin's formula and explicitly states that an asymptotic rather than an asymptotic for the logarithm remains open.\n- Michael Simkin, The number of n-queens configurations, Advances in Mathematics 427 (2023), 109127. (primary): https://doi.org/10.1016/j.aim.2023.109127\n Evidence used: Proves existence and the variational characterization of the limiting n-queens constant and the resulting exponential-rate formula.\n- Parth Nobel, Akshay Agrawal, and Stephen Boyd, Computing tighter bounds on the n-queens constant via Newton's method, Optimization Letters 17 (2023), 1229-1240. (primary): https://doi.org/10.1007/s11590-022-01933-2\n Evidence used: Gives the tight numerical enclosure for the constant in Simkin's exponent.\n\n**Review notes.** The stored statement is exact but the word asymptotically is ambiguous. Simkin solves the exponential-rate reading; Green's maintained comments require a stronger multiplicative asymptotic.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 145, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" } }, { "id": 1098, "problem_number": "GREEN-098", "title": "Bounds for Homogeneous Polynomial Zeros", "statement": "Let $d \\geq 3$ be an odd integer. Give bounds on $\\nu(d)$ such that if $n > \\nu(d)$ the following is true: given any homogeneous polynomial $F(\\mathbf{x}) \\in \\mathbb{Z}[x_1, \\dots, x_n]$ of degree $d$, there is some $\\mathbf{x} \\in \\mathbb{Z}^n \\setminus \\{\\mathbf{0}\\}$ such that $F(\\mathbf{x}) = 0$.", "background": "This asks for explicit bounds on how many variables are needed to guarantee integer zeros of homogeneous polynomials. Classical results give existence but quantitative bounds remain challenging, especially for higher degrees.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Existence and explicit general bounds for ν(d) are known, including ν(3) at most 13, but the general bounds remain tower-sized and far from the conjectural d^2 scale.\n\n**Verified partial progress.**\n\n- Birch proved that a finite threshold exists for every odd degree.\n- Wooley obtained the first fully explicit general bounds, with iterated-exponential or tower-type growth as the degree varies.\n- Heath-Brown proved that every cubic form in at least 14 variables has a nontrivial integer zero, giving ν(3) at most 13.\n- Lampert, Snowden, and Ziegler prove polynomial dependence on the number of forms when the degree is fixed, a substantial multi-form variant.\n\n**Full solution or refutation.**\n\nThe imperative request to give bounds has been met in a weak literal sense, but the maintained problem asks for dramatically improved degree dependence and continues to identify even the analogous local scale as open.\n\n**What remains.**\n\nReduce the dependence of ν(d) on odd d, ideally toward the proposed ν(d)=d^2 scale, and understand the corresponding p-adic thresholds.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 98, updates through December 2025; checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Records Birch's existence theorem, Wooley's tower bounds, ν(3) at most 13, the possible d^2 target, and the 2025 multiple-form update.\n- Trevor D. Wooley, An explicit version of Birch's theorem, Acta Arithmetica 85 (1998), 79-96. (primary): https://www.math.purdue.edu/~twooley/publ/1998%20birch.pdf\n Evidence used: Gives the first entirely explicit general recursive bounds for odd-degree forms.\n- D. R. Heath-Brown, Cubic forms in 14 variables, Inventiones Mathematicae 170 (2007), 199-230. (primary): https://doi.org/10.1007/s00222-007-0062-1\n Evidence used: Proves nontrivial integral solubility for every cubic form in at least 14 variables.\n- Amichai Lampert, Andrew Snowden, and Tamar Ziegler, Polynomial Bounds for Birch's Theorem, arXiv:2512.00697 (2025). (primary): https://arxiv.org/abs/2512.00697\n Evidence used: Proves polynomial variable dependence on the number of equations for fixed odd degree.\n\n**Review notes.** The database background is stale in suggesting that only existence is classical: explicit bounds have been known since Wooley, though they are extraordinarily weak. The exact stored statement is preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 78, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1099, "problem_number": "GREEN-099", "title": "Polynomial Solutions in Dense Sets", "statement": "Finding a single solution to a polynomial equation $F(x_1, \\dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of such solutions in $A$ is roughly $\\alpha^n$ times the number of solutions in $[X]$?", "background": "This problem asks when dense sets contain the \"expected\" number of polynomial solutions. Understanding density conditions that guarantee proportional solution counts connects number theory with additive combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Expected-count criteria are known for linear systems and several nonlinear or high-variable regimes, but Green's specific low-variable quadratic, cubic, and four-square targets remain open.\n\n**Verified partial progress.**\n\n- For linear equations with at least three variables, ordinary Fourier uniformity controls the expected solution count.\n- Cook and Magyar establish high-variable local-to-global counting for polynomial systems with prime coordinates.\n- Green proves an asymptotic for nondegenerate quadratic forms in eight prime variables; his maintained notes say the methods should give an eight-variable generic dense-set result under non-correlation with progressions.\n- Browning and Prendiville prove dense-set transference for translation-invariant diagonal quadrics in at least five variables within the squares.\n\n**Full solution or refutation.**\n\nThe stored row is an under-specified research program rather than a binary proposition. No general criterion or resolution of the three specific source subquestions was located.\n\n**What remains.**\n\nResolve generic quadratic forms in seven variables, determine a useful least variable count for generic cubics, and handle the four-square equation under sharp pseudorandomness conditions on A.\n\n**Sources checked.**\n\n- Ben Green, 100 Open Problems, Problem 99, updates through December 2025; checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Supplies the omitted quantifiers and three concrete subquestions, summarizes the linear case and nearby nonlinear results, and retains all targets as open.\n- Ben Green, Quadratic forms in 8 prime variables, Geometric and Functional Analysis 35 (2025), 1587-1637. (primary): https://doi.org/10.1007/s00039-025-00727-9\n Evidence used: Proves an asymptotic count for prime solutions to a mildly nondegenerate quadratic form in eight variables, the method cited for the nearby generic quadratic regime.\n- Brian Cook and Akos Magyar, Diophantine equations in the primes, Inventiones Mathematicae 198 (2014), 701-737. (primary): https://doi.org/10.1007/s00222-014-0508-1\n Evidence used: Provides general high-variable local-to-global polynomial counting machinery, illustrating a broad solved regime away from the desired thresholds.\n- Tim Browning and Sean Prendiville, A transference approach to a Roth-type theorem in the squares, International Mathematics Research Notices 2017, 2219-2248. (primary): https://doi.org/10.1093/imrn/rnw096\n Evidence used: Proves a concrete nonlinear dense-subset theorem for translation-invariant diagonal quadrics in at least five variables.\n\n**Review notes.** Material formulation omissions: the stored statement leaves out A subset [X], alpha=|A|/X, fixed density and the X-to-infinity limit, hypotheses on F and C, and the source's three concrete subquestions. No repair was silently inserted into the preserved statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 71, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1100, "problem_number": "GREEN-100", "title": "Sofic Groups", "statement": "Is every group well-approximated by finite groups?", "background": "A group is sofic if it can be approximated by finite symmetric groups in a precise sense. Whether all groups are sofic is a major open question in group theory with connections to dynamics, graph theory, and combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The universal soficity assertion in the database background was refuted in August 2026 by an explicit nonsofic group, with an independent expert preprint already deriving a torsion-free example from the same key criterion.\n\n**Verified partial progress.**\n\n- The released manuscript proves that the unit group of the binary Leavitt algebra over F_2 is not sofic.\n- Its key criterion combines property (T), nested conjugation, and a commuting non-LEF subgroup, and is applied using Thompson's group V.\n- Fournier-Facio independently explains the criterion and constructs a finitely presented torsion-free nonsofic group by small-cancellation methods.\n- The new example is not known to be nonhyperlinear; the broader hyperlinear interpretation remains open.\n\n**Full solution or refutation.**\n\nThere exists a group that cannot be approximated by finite symmetric groups in the sofic sense, so the database row's sofic interpretation has a negative answer.\n\n**What remains.**\n\nObtain ordinary peer-reviewed verification of the extremely recent proofs, clarify the maintained collection's wording, and separately determine whether every group is hyperlinear.\n\n**Sources checked.**\n\n- OpenAI, Nonsofic groups exist, Chapter 3 of Ten Advances in Mathematics and Theoretical Computer Science, updated 2026-08-06. (primary): https://cdn.openai.com/pdf/ten-proofs-oai.pdf\n Evidence used: States and proves that the unit group L_{F_2}(1,2)^x is not sofic, giving an unconditional counterexample to universal soficity.\n- Francesco Fournier-Facio, A torsion-free non-sofic group, arXiv:2608.02025 (2026-08-03). (primary): https://arxiv.org/abs/2608.02025\n Evidence used: Accepts and analyzes the key new nonsoficity criterion, then independently applies it to construct a finitely presented torsion-free nonsofic group.\n- Ben Green, 100 Open Problems, Problem 100, incorporated updates through December 2025; checked 2026-08-17. (maintained_tracker): https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf\n Evidence used: Shows that the source headline encompasses distinct sofic and hyperlinear questions; it predates the August 2026 nonsofic construction.\n\n**Review notes.** The classification applies to the sofic reading explicitly given in the database background. Green's broader wording also mentions hyperlinearity, which is not refuted by these sources. Confidence is medium because both proofs are only about two weeks old and not yet journal-refereed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 92, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "set": { "id": 3, "name": "green_problems", "display_name": "Ben Green's 100 Open Problems", "description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.", "slug": "green-problems", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1102, "problem_number": "ALG-002", "title": "Hadamard Conjecture", "statement": "For every positive integer $k$, does there exist a Hadamard matrix of order $4k$?", "background": "A Hadamard matrix is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. The Hadamard conjecture, dating back to 1893, states that such matrices exist for all orders that are multiples of 4. These matrices have important applications in coding theory, signal processing, and quantum information theory. While Hadamard matrices have been constructed for many values of $k$, the smallest order for which existence is unknown is 668. The conjecture has deep connections to combinatorial design theory and remains one of the central problems in discrete mathematics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The universal Hadamard conjecture remains open; no construction for every order divisible by four and no impossible order is known.\n\n**Verified partial progress.**\n\n- At the 2025 peer-reviewed baseline, orders 668, 716, and 892 were the unresolved cases at most 1000.\n- Eliahou constructed a 64-modular Hadamard matrix of order 668, improving the earlier 32-modular approximation but not producing a genuine Hadamard matrix.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was verified. A days-old nonarchival claim of explicit matrices for formerly unknown orders was not used as evidence and in any event would not prove the universal statement.\n\n**What remains.**\n\nConstruct a Hadamard matrix of order 4k for every positive integer k, or prove nonexistence for some admissible order.\n\n**Sources checked.**\n\n- Shalom Eliahou, A 64-modular Hadamard matrix of order 668, Australasian Journal of Combinatorics 93(2) (2025), 422-427. (primary): https://ajc.maths.uq.edu.au/pdf/93/ajc_v93_p422.pdf\n Evidence used: States the universal conjecture, documents the 2025 unresolved finite frontier, and gives a modular near-construction at order 668.\n- The Encyclopaedia of Design Theory, Hadamard matrices. (authoritative_secondary): https://maths.qmul.ac.uk/~lsoicher/designtheory.org/library/encyc/topics/had.pdf\n Evidence used: Explains the conjecture, necessary divisibility condition, and standard construction background.\n\n**Review notes.** The background's smallest-unknown-order claim may have become stale days before this check, but no archival primary source or authoritative updated tracker was found. This does not affect the open general classification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 387, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1103, "problem_number": "ALG-003", "title": "Köthe Conjecture", "statement": "If a ring has no nil ideal other than $\\{0\\}$, does it follow that it has no nil one-sided ideal other than $\\{0\\}$?", "background": "The Köthe conjecture, proposed by Gottfried Köthe in 1930, is a fundamental problem in ring theory concerning the structure of nil ideals. A nil ideal is one in which every element is nilpotent. The conjecture asks whether the absence of two-sided nil ideals implies the absence of one-sided nil ideals. Despite being studied for over 90 years, the problem remains open even for Noetherian rings. The conjecture is related to the Jacobson conjecture and has implications for understanding the structure of general rings. Counterexamples would reveal unexpected asymmetry in the behavior of left and right ideals.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Kothe's nil-ideal conjecture remains open for arbitrary associative rings.\n\n**Verified partial progress.**\n\n- The conjecture holds for one-sided Noetherian rings, PI rings, rings with Krull dimension, algebras over uncountable fields, and other substantial classes.\n- A 2025 paper adds rings whose nilpotent elements form a Wedderburn radical subring to the known positive classes.\n- Equivalent formulations involve sums of nil one-sided ideals and nilness of matrix rings over nil rings.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was verified.\n\n**What remains.**\n\nProve the implication for arbitrary rings or construct a ring with a nonzero nil one-sided ideal and no nonzero nil two-sided ideal.\n\n**Sources checked.**\n\n- Adel Alahmadi, S. K. Jain, and Andre Leroy, Rings Whose Nilpotent Elements Form a Wedderburn Radical Subring, Symmetry 17 (2025), article 1815. (primary): https://doi.org/10.3390/sym17111815\n Evidence used: Describes the general conjecture as open, lists known positive classes, and proves another positive class.\n- M. A. Chebotar, P.-H. Lee, and E. R. Puczylowski, On some questions related to Koethe's nil ideal problem, Proceedings of the Edinburgh Mathematical Society 58 (2015), 365-377. (primary): https://doi.org/10.1017/S0013091514000273\n Evidence used: Develops an equivalent formulation through one-sided ideals of A-rings.\n- Agata Smoktunowicz, On some Results Related to Kothe's Conjecture, Serdica Mathematical Journal 27 (2001), 159-170. (authoritative_secondary): https://eudml.org/doc/11532\n Evidence used: Survey of equivalent formulations and major partial results.\n\n**Review notes.** The background's claim that the conjecture remains open even for Noetherian rings is false or seriously misleading: Levitzki's theorem settles one-sided Noetherian rings. Exact statement preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1104, "problem_number": "ALG-004", "title": "Connes Embedding Problem", "statement": "Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?", "background": "The Connes embedding problem, formulated by Alain Connes in 1976, is a central question in the theory of von Neumann algebras. In 2020, Ji, Natarajan, Vidick, Wright, and Yuen published a paper claiming to have shown the problem has a negative answer, based on connections to quantum complexity theory and the equivalence with Tsirelson's problem in quantum information. However, the problem's status remains subject to verification of their approach. The problem has deep connections to free probability, quantum groups, and mathematical physics, making it one of the most important questions in operator algebra theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Connes' embedding conjecture has a negative answer as a consequence of MIP*=RE.\n\n**Verified partial progress.**\n\n- MIP*=RE implies a strict separation between tensor-product and commuting-operator quantum correlation sets.\n- Established equivalences through Tsirelson's problem transfer that separation to a non-embeddable finite tracial von Neumann algebra.\n\n**Full solution or refutation.**\n\nJi, Natarajan, Vidick, Wright, and Yuen refute the standard separable Connes embedding problem; the record background's language of a merely claimed solution is stale.\n\n**What remains.**\n\nStudy explicit and structurally natural non-embeddable factors, quantitative witnesses, and restricted classes where embeddability may still hold.\n\n**Sources checked.**\n\n- Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen, MIP*=RE, arXiv:2001.04383 (2020; revised 2021). (primary): https://arxiv.org/abs/2001.04383\n Evidence used: The paper explicitly states that its correlation-set separation refutes Connes' embedding conjecture.\n- Isaac Goldbring, The Connes Embedding Problem: A Guided Tour, arXiv:2109.12682 (2021). (authoritative_secondary): https://arxiv.org/abs/2109.12682\n Evidence used: Explains the negative solution and gives two routes from MIP*=RE to the operator-algebraic conclusion.\n\n**Review notes.** The classical formulation normally includes separability. The background is stale in treating the 2020 result as only a claim subject to verification. Exact statement preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1105, "problem_number": "ALG-005", "title": "Jacobson's Conjecture", "statement": "For a left-and-right Noetherian ring $R$, is the intersection of all powers of the Jacobson radical $J(R)$ equal to zero?", "background": "Jacobson's conjecture addresses a fundamental question about the structure of Noetherian rings. The Jacobson radical of a ring consists of elements that annihilate all simple modules, and understanding its intersection over all powers relates to the ring's nilpotent elements and its representation theory. While the conjecture holds for many important classes of rings (including commutative Noetherian rings), the general case remains open. The problem is closely related to other structural conjectures in non-commutative ring theory, including the Köthe conjecture.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The intersection-of-radical-powers conjecture remains open for arbitrary rings that are Noetherian on both sides. One-sided variants have counterexamples, while many two-sided classes satisfy the conjecture.\n\n**Verified partial progress.**\n\n- Jategaonkar proved the conjecture for fully bounded Noetherian rings.\n- Positive results also cover Noetherian rings of Krull dimension one and rings satisfying the second-layer condition.\n- Reyes proves that bijective skew PBW extensions over domains have zero Jacobson radical and hence satisfy the conjecture.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for every left-and-right Noetherian ring was verified.\n\n**What remains.**\n\nEither extend the known module-theoretic/FBN arguments to all two-sided Noetherian rings or construct a genuinely two-sided Noetherian counterexample; one-sided examples do not suffice.\n\n**Sources checked.**\n\n- Arun Vinayak Jategaonkar, Jacobson's conjecture and modules over fully bounded Noetherian rings, Journal of Algebra 30 (1974), 103-121. (primary): https://doi.org/10.1016/0021-8693(74)90195-1\n Evidence used: Proves Jacobson's conjecture for fully bounded Noetherian rings.\n- Armando Reyes, Jacobson's conjecture and skew PBW extensions, Revista Integración 32 (2014), 139-152. (primary): https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/4389\n Evidence used: Describes the two-sided question and known classes, distinguishes one-sided counterexamples, and proves the property for bijective skew PBW extensions over domains.\n\n**Review notes.** The left-and-right hypothesis is essential. Counterexamples to Jacobson's original one-sided Noetherian question do not refute this exact record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 198, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1106, "problem_number": "ALG-006", "title": "Zauner's Conjecture", "statement": "Do SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite dimensions?", "background": "Zauner's conjecture, proposed in 1999, concerns the existence of a special type of quantum measurement in Hilbert spaces of all finite dimensions. A SIC-POVM consists of d² unit vectors in a d-dimensional complex Hilbert space that are equiangular - the absolute inner product of any two distinct vectors is constant. These structures have applications in quantum information theory, quantum state tomography, and quantum cryptography. While SIC-POVMs have been found numerically for all dimensions up to 151 and proven to exist analytically in some special cases, the general existence question remains open. The conjecture has surprising connections to number theory and algebraic geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Universal SIC existence was open in peer-reviewed literature through 2024; a 2025 construction is conditional on two number-theoretic conjectures, and a January 2026 arXiv manuscript claims an unconditional proof. No independent or peer-reviewed confirmation of that claim was located.\n\n**Verified partial progress.**\n\n- Appleby--Flammia--Kopp give a construction in all dimensions d>3 conditional on the abelian Stark conjecture and a Shintani--Faddeev special-value identity.\n- Exact constructions are known in many finite dimensions, and author reports give numerical solutions in every dimension through at least 196 plus additional higher examples.\n- Joka's arXiv:2601.13475 claims SIC-POVM existence in every finite dimension.\n\n**Full solution or refutation.**\n\nA complete proof is claimed but is not yet verified strongly enough for a solved classification.\n\n**What remains.**\n\nObtain expert audit or peer-reviewed acceptance of the 2026 unconditional proof claim, or discharge the two conjectural inputs in the 2025 Stark-based construction.\n\n**Sources checked.**\n\n- Danylo Yakymenko, SICs and the Triangle Group (3,3,3), SIGMA 20 (2024), 044. (primary): https://doi.org/10.3842/SIGMA.2024.044\n Evidence used: Peer-reviewed paper explicitly states that existence in every dimension remains open and proves structural results about order-three symmetries.\n- Marcus Appleby, Steven T. Flammia, and Gene S. Kopp, A Constructive Approach to Zauner's Conjecture via the Stark Conjectures, arXiv:2501.03970. (primary): https://arxiv.org/abs/2501.03970\n Evidence used: Proves an all-dimension SIC construction conditional on two explicitly named conjectures and validates it against known solutions.\n- Stefan Joka, Symmetric Informationally Complete Positive Operator Valued Measure and Zauner conjecture, arXiv:2601.13475. (primary): https://arxiv.org/abs/2601.13475\n Evidence used: Recent standalone manuscript claiming unconditional existence in every finite dimension; no independent validation was found.\n- Markus Grassl, Exact SIC-POVMs from Permutation Symmetries, Hadamard 2025 slides. (primary): https://us.ticmeet.com/assets/archivos/d6f1d9b8-d39f-4888-925a-7eb81c4905cc/Grassl.pdf\n Evidence used: Author report of current exact and numerical construction ranges, including numerical solutions through dimension 196.\n\n**Review notes.** The displayed question asks only universal SIC existence. Stronger versions additionally require Weyl--Heisenberg covariance or Zauner order-three symmetry; those were not silently substituted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 176, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1107, "problem_number": "ALG-007", "title": "Casas-Alvero Conjecture", "statement": "If a univariate polynomial $f$ of degree $d$ over a field of characteristic 0 shares a common factor with each of its first $d-1$ derivatives, must $f$ be a power of a linear polynomial?", "background": "The Casas-Alvero conjecture, proposed in 2001, connects the factorization of a polynomial with the factorization of its derivatives. If $f(x)$ has degree $d$ and for each $k = 1, 2, \\ldots, d-1$, the polynomial $f(x)$ shares a root with its $k$-th derivative $f^{(k)}(x)$, the conjecture states that $f$ must be of the form $f(x) = (x - a)^d$ for some constant $a$. While proven for various special cases and low degrees, the general conjecture remains open. It has connections to algebraic geometry and the theory of polynomial equations.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ghosh's 2025 arXiv manuscript claims a proof in all characteristic-zero degrees, but a peer-reviewed article published in May 2026 still says the conjecture remains open with degree 24 the least unsettled case. No authoritative reconciliation was found.\n\n**Verified partial progress.**\n\n- The conjecture is established for many infinite families of degrees and in all degrees through 12.\n- Ghosh's 2024 work proves finiteness, up to affine transformations, of possible counterexamples in every fixed degree via arithmetic Casas--Alvero schemes.\n- Ghosh's 2025 manuscript claims a complete proof using downward induction and Koszul regularity.\n\n**Full solution or refutation.**\n\nA full solution is claimed, but current peer-reviewed literature still treats the problem as open; expert proof audit is required.\n\n**What remains.**\n\nResolve the discrepancy by checking the downward-induction/Koszul argument in arXiv:2501.09272 and obtaining independent or journal verification, or identify a precise gap.\n\n**Sources checked.**\n\n- Soham Ghosh, Proof of the Casas-Alvero conjecture, arXiv:2501.09272. (primary): https://arxiv.org/abs/2501.09272\n Evidence used: States Theorem A proving the exact displayed characteristic-zero conjecture in every degree.\n- Armengol Gasull, A Primer on Resultants and Their Applications, Matemática Contemporânea (2026). (authoritative_secondary): https://doi.org/10.1007/s44425-026-00047-6\n Evidence used: Peer-reviewed survey published after the proof claim says the conjecture remains open, identifies degree 24 as least open, and gives direct proofs in degrees four and five.\n- Soham Ghosh, A finiteness result towards the Casas-Alvero Conjecture, arXiv:2402.18717. (primary): https://arxiv.org/abs/2402.18717\n Evidence used: Proves finiteness and rigidity results for the arithmetic Casas--Alvero schemes in each degree.\n\n**Review notes.** The source statement is well formed. The common factor/root may depend on derivative order; no stronger same-root condition was substituted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 154, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1108, "problem_number": "ALG-008", "title": "Andrews-Curtis Conjecture", "statement": "Can every balanced presentation of the trivial group be transformed into a trivial presentation by a sequence of Nielsen transformations and conjugations of relators?", "background": "The Andrews-Curtis conjecture, proposed in 1965, is a central problem in combinatorial group theory. A balanced presentation has an equal number of generators and relators. The conjecture asks whether such presentations of the trivial group can always be simplified to the form $\\langle x_1, \\ldots, x_n \\mid x_1, \\ldots, x_n \\rangle$ using only Andrews-Curtis moves (Nielsen transformations on relators and conjugations). Despite extensive computational searches and partial results, no counterexample has been found, yet no proof exists. The conjecture has important implications for 3-manifold theory and the classification of homotopy types.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The original unstable Andrews--Curtis conjecture remains unsolved, with no accepted counterexample. Recent work proves new restricted classes and certifies very long simplifications of difficult individual presentations.\n\n**Verified partial progress.**\n\n- A Miller--Schupp presentation has a verified 8,634-step Andrews--Curtis simplification.\n- Current automated-search work resolves additional automorphic equivalences but leaves several standard candidate families open.\n- Lackenby proves the unstable conjecture for all thickenable balanced presentations and gives an explicit stable-move bound for that class.\n\n**Full solution or refutation.**\n\nNeither a proof for every balanced trivial-group presentation nor a counterexample is known.\n\n**What remains.**\n\nHandle non-thickenable balanced presentations or find a move-invariant obstruction producing a genuine counterexample; stable and unstable formulations must remain distinct.\n\n**Sources checked.**\n\n- Michael Fairbank, Alexei Lisitsa, and Alexei Vernitski, Probabilistic Automaton Classifier Applied to Examples Related to the Andrews--Curtis Conjecture, Journal of Automated Reasoning 70 (2026), article 11. (primary): https://doi.org/10.1007/s10817-026-09759-8\n Evidence used: Peer-reviewed July 2026 paper explicitly calls the conjecture unsolved, states no counterexamples are known, and documents solved and unresolved computational instances.\n- Marc Lackenby, The stable Andrews--Curtis conjecture and thickenable presentations of the trivial group, arXiv:2606.06122. (primary): https://arxiv.org/abs/2606.06122\n Evidence used: Proves the unstable conjecture for thickenable balanced presentations and a bounded stable result for the same class.\n- Alexei Lisitsa, Automated theorem proving reveals a lengthy Andrews--Curtis trivialization for a Miller--Schupp trivial group presentation, Examples and Counterexamples 8 (2025), 100201. (primary): https://doi.org/10.1016/j.exco.2025.100201\n Evidence used: Provides an explicit 8,634-move simplification certificate for a formerly unsettled Miller--Schupp instance.\n\n**Review notes.** The wording is broadly standard but should formally specify the elementary AC moves. Adding or deleting generator-relator pairs would change the problem to the stable conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 212, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1109, "problem_number": "ALG-009", "title": "Bounded Burnside Problem", "statement": "For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2,5)$ finite?", "background": "The Bounded Burnside problem asks which free Burnside groups are finite. A free Burnside group $B(m,n)$ is the largest group with $m$ generators in which every element has order dividing $n$. It is known that $B(m,n)$ is finite for $n \\in \\{2, 3, 4, 6\\}$ and for certain other special values, and infinite for most large odd exponents. The case $B(2,5)$ has been the subject of extensive computational investigation but remains open. Solving this problem would significantly advance our understanding of periodic groups and torsion in group theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many Burnside groups are known finite or infinite, but B(2,5) remains an open special case.\n\n**Verified partial progress.**\n\n- Large odd and sufficiently large even exponents yield infinite free Burnside groups.\n\n**Full solution or refutation.**\n\nThe source explicitly highlights an unresolved exponent-five case.\n\n**What remains.**\n\nDecide finiteness of B(2,5) and extend the classification.\n\n**Sources checked.**\n\n- Burnside problem history, MacTutor (accessed 2026-08-17). (authoritative_secondary): https://mathshistory.st-andrews.ac.uk/HistTopics/Burnside_problem/\n Evidence used: Records B(2,5) as open and summarizes known finite/infinite regimes.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1110, "problem_number": "ALG-010", "title": "Herzog-Schönheim Conjecture", "statement": "If a finite system of left cosets of subgroups of a group $G$ partitions $G$, then must at least two of the subgroups have the same index in $G$?", "background": "The Herzog-Schönheim conjecture, proposed in 1974, concerns coset decompositions of groups. If $G$ is partitioned by cosets $g_1H_1, g_2H_2, \\ldots, g_kH_k$ where each $H_i$ is a subgroup of $G$ and the cosets are pairwise disjoint, the conjecture states that at least two of the indices $[G:H_i]$ must be equal. While proven for abelian groups and various other special cases, the general conjecture remains open. It has connections to combinatorial number theory and the structure theory of groups.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Herzog--Schonheim index-repetition conjecture remains open generally, with many special groups and partition types settled.\n\n**Verified partial progress.**\n\n- Numerous restrictions on possible distinct-index coset partitions are known.\n\n**Full solution or refutation.**\n\nNo general proof that two indices must coincide was verified.\n\n**What remains.**\n\nProve or disprove the conjecture for arbitrary group coset partitions.\n\n**Sources checked.**\n\n- Herzog--Schonheim conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Herzog%E2%80%93Sch%C3%B6nheim_conjecture\n Evidence used: Records the general problem and partial results.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 142, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1112, "problem_number": "ALG-012", "title": "Existence of Perfect Cuboids", "statement": "Does there exist a rectangular cuboid where all edges, face diagonals, and space diagonals have integer lengths?", "background": "A perfect cuboid (also called a perfect box or Euler brick with space diagonal) would be a rectangular parallelepiped with integer edge lengths $a$, $b$, $c$ such that the face diagonals $\\sqrt{a^2+b^2}$, $\\sqrt{b^2+c^2}$, $\\sqrt{a^2+c^2}$ and the space diagonal $\\sqrt{a^2+b^2+c^2}$ are all integers. Despite extensive computational searches up to very large bounds and numerous partial results, no perfect cuboid has been found, nor has impossibility been proven. The problem has connections to Diophantine equations and elliptic curves, and has fascinated both amateur and professional mathematicians for centuries.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** This duplicates the perfect-cuboid existence question and remains open.\n\n**Verified partial progress.**\n\n- Same computational and parametrisation evidence as the distinct perfect-cuboid record.\n\n**Full solution or refutation.**\n\nNo perfect cuboid or impossibility proof is verified.\n\n**What remains.**\n\nResolve existence.\n\n**Sources checked.**\n\n- Perfect cuboid overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Perfect_cuboid\n Evidence used: Records the problem as open.\n\n**Review notes.** Duplicate source statement retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 234, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1114, "problem_number": "ALG-014", "title": "McKay Conjecture", "statement": "For a finite group $G$ and prime $p$, is the number of irreducible complex characters of $G$ whose degree is not divisible by $p$ equal to the corresponding number for the normalizer of a Sylow $p$-subgroup?", "background": "The McKay conjecture, proposed in the 1970s, is a central problem in the representation theory of finite groups. It predicts a surprising relationship between the character degrees of a group and those of a much smaller subgroup (the normalizer of a Sylow $p$-subgroup). The conjecture has been verified for many important classes of groups and has led to deep insights about the structure of character tables. A proof was announced in 2007 by Isaacs, Malle, and Navarro assuming the classification of finite simple groups, though subtle gaps in the argument have led to ongoing refinement of the proof.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The McKay conjecture has a major reduction to finite simple groups but is not a general theorem.\n\n**Verified partial progress.**\n\n- Isaacs--Malle--Navarro reduce the conjecture to finite simple groups.\n\n**Full solution or refutation.**\n\nNo full proof for every finite group was verified.\n\n**What remains.**\n\nComplete the remaining simple-group cases.\n\n**Sources checked.**\n\n- McKay conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/McKay_conjecture\n Evidence used: Records reduction results and open general status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1115, "problem_number": "ALG-015", "title": "Are All Groups Surjunctive?", "statement": "Is every group surjunctive? That is, for any group $G$, if $\\phi: A^G \\to A^G$ is a cellular automaton that is injective, must it also be surjective?", "background": "A group $G$ is called surjunctive if every injective cellular automaton on $G$ is automatically surjective. Equivalently, this asks whether the dynamical system defined by a cellular automaton on the group can be injective without being bijective. Gromov and Weiss proved that all sofic groups are surjunctive, and all known groups are sofic, but it remains unknown whether all groups are surjunctive. The question has deep connections to symbolic dynamics, geometric group theory, and the Garden of Eden theorem from cellular automaton theory. A negative answer would be quite surprising and would reveal fundamental limitations in our understanding of group actions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Gottschalk's surjunctivity conjecture remains open for arbitrary groups.\n\n**Verified partial progress.**\n\n- It is proved for sofic groups, including amenable and residually finite groups.\n\n**Full solution or refutation.**\n\nNo theorem covers every group.\n\n**What remains.**\n\nProve all groups surjunctive or find a nonsurjunctive group.\n\n**Sources checked.**\n\n- Surjunctive group overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Surjunctive_group\n Evidence used: Records the all-groups question as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 143, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1119, "problem_number": "NT-016", "title": "Catalan-Mersenne Conjecture", "statement": "Are all Catalan-Mersenne numbers $C_n$ composite for $n > 4$? Here $C_0 = 2$ and $C_{n+1} = 2^{C_n} - 1$.", "background": "The Catalan-Mersenne conjecture concerns a doubly exponential sequence where each term is a Mersenne number with exponent equal to the previous term. The sequence grows extraordinarily rapidly: $C_0 = 2$, $C_1 = 3$, $C_2 = 7$, $C_3 = 127$, $C_4 = 170141183460469231731687303715884105727$ (a 39-digit number). The first four terms are prime, but $C_5$ has over $10^{38}$ digits, making it far beyond reach of current computational methods. The conjecture predicts that all subsequent terms are composite. This problem connects to deep questions about the distribution of Mersenne primes and the limitations of our ability to determine primality for extremely large numbers.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Catalan--Mersenne compositeness assertion after C4 remains open.\n\n**Verified partial progress.**\n\n- The recurrence makes each term a Mersenne number; known early terms motivate the conjecture.\n\n**Full solution or refutation.**\n\nNo all-n compositeness proof was verified.\n\n**What remains.**\n\nProve each C_n for n>4 composite or find a later prime term.\n\n**Sources checked.**\n\n- Double Mersenne number overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Double_Mersenne_number\n Evidence used: Records the Catalan--Mersenne recurrence and associated conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 287, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1120, "problem_number": "NT-017", "title": "Are There Infinitely Many Mersenne Primes?", "statement": "Are there infinitely many prime numbers of the form $2^p - 1$ where $p$ is prime?", "background": "Mersenne primes are primes of the form $M_p = 2^p - 1$ where $p$ is itself prime. As of 2024, only 51 Mersenne primes are known, the largest being $2^{82589933} - 1$ discovered in 2018. Despite their rarity, it is conjectured that infinitely many exist. Mersenne primes are intimately connected to perfect numbers through the Euclid-Euler theorem: an even number is perfect if and only if it has the form $2^{p-1}(2^p-1)$ where $2^p-1$ is a Mersenne prime. The question of whether infinitely many Mersenne primes exist is closely related to our understanding of the distribution of primes and has implications for both pure and applied mathematics, as Mersenne primes are used in pseudorandom number generation and cryptography.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Infinitude of Mersenne primes remains open.\n\n**Verified partial progress.**\n\n- GIMPS has verified 52 Mersenne primes and continues expanding the tested range.\n\n**Full solution or refutation.**\n\nNo infinitude theorem was verified.\n\n**What remains.**\n\nProve infinitely many prime exponents p yield 2^p-1 prime.\n\n**Sources checked.**\n\n- GIMPS list of known Mersenne primes. (maintained_tracker): https://www.mersenne.org/primes/\n Evidence used: Provides the current verified list and search status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 49, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1121, "problem_number": "GEO-001", "title": "Sphere Packing Problem in Higher Dimensions", "statement": "What is the densest packing of spheres in dimensions 4 through 23? More generally, what is the optimal sphere packing density in dimension $n$?", "background": "The sphere packing problem asks for the densest arrangement of non-overlapping spheres in $n$-dimensional space. In dimension 3, Kepler's conjecture (proved by Hales in 1998) shows the densest packing has density $\\pi/\\sqrt{18} \\approx 0.7405$. In 2016, Maryna Viazovska proved that the E₈ lattice gives the densest packing in dimension 8, and shortly after, Cohn, Kumar, Miller, Radchenko, and Viazovska proved the Leech lattice is optimal in dimension 24. However, dimensions 4-7 and 9-23 remain open, as do almost all higher dimensions. The problem has deep connections to coding theory, number theory, and optimization.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Optimal unrestricted sphere packing is known only in dimensions 1--3, 8, and 24.\n\n**Verified partial progress.**\n\n- Viazovska proved E8 optimal in dimension 8; Cohn--Kumar--Miller--Radchenko--Viazovska proved Leech optimal in 24.\n\n**Full solution or refutation.**\n\nThe requested dimensions 4--23 remain mostly open.\n\n**What remains.**\n\nDetermine optimal densities outside the solved dimensions.\n\n**Sources checked.**\n\n- M. Viazovska, The sphere packing problem in dimension 8, Ann. Math. 185 (2017). (primary): https://arxiv.org/abs/1603.04246\n Evidence used: Proves dimension-8 optimality.\n- Sphere packing overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Sphere_packing\n Evidence used: Records unrestricted optima only in 1--3, 8, and 24.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 398, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1122, "problem_number": "GEO-002", "title": "Mahler's Conjecture", "statement": "Among all centrally symmetric convex bodies in $\\mathbb{R}^n$, does the cube (or cross-polytope) minimize the product of the body's volume and the volume of its polar dual?", "background": "Mahler's conjecture, proposed in 1939, concerns a fundamental geometric quantity called the Mahler volume, defined as the product of a convex body's volume with the volume of its polar dual. Kurt Mahler conjectured that among all centrally symmetric convex bodies in $\\mathbb{R}^n$, this product is minimized by the cube and the cross-polytope (which are dual to each other). The conjecture has been proved in dimension 2 by Mahler himself, and partial results exist for special classes of bodies, but the general case remains open. The problem connects convex geometry, functional analysis, and the theory of Banach spaces.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The symmetric Mahler conjecture is open in general dimensions at least four.\n\n**Verified partial progress.**\n\n- The symmetric problem is proved in dimensions two and three; strong general lower bounds are known.\n\n**Full solution or refutation.**\n\nNo general cube/cross-polytope minimization theorem was verified.\n\n**What remains.**\n\nProve Mahler's sharp bound in all dimensions.\n\n**Sources checked.**\n\n- Mahler volume overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Mahler_volume\n Evidence used: States that the symmetric conjecture remains unsolved for n >= 4.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 245, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1123, "problem_number": "GEO-003", "title": "The Illumination Conjecture", "statement": "Can every convex body in $n$-dimensional space be illuminated by at most $2^n$ point light sources?", "background": "The illumination conjecture, also known as Hadwiger's covering conjecture in one of its forms, asks whether every convex body in $\\mathbb{R}^n$ can be illuminated by at most $2^n$ point light sources placed outside the body. A point on the surface is considered illuminated if the ray from the light source to that point does not pass through the interior of the body. The conjecture has been proven for $n = 2$ and $n = 3$, but remains open for higher dimensions. The problem is closely related to covering problems and has connections to discrete geometry and combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The illumination conjecture remains open generally.\n\n**Verified partial progress.**\n\n- It is proved for several special body classes and low-dimensional cases.\n\n**Full solution or refutation.**\n\nNo general 2^n illumination theorem was verified.\n\n**What remains.**\n\nProve the sharp universal illumination bound.\n\n**Sources checked.**\n\n- Illumination problem overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Illumination_problem\n Evidence used: Records the general conjecture and special cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 187, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1124, "problem_number": "GEO-004", "title": "Kakeya Needle Problem", "statement": "What is the minimum area of a region in the plane in which a unit line segment can be continuously rotated through 360 degrees?", "background": "The Kakeya needle problem asks for the smallest area set in the plane within which a unit line segment can be rotated continuously through 360 degrees, returning to its initial position. While Besicovitch showed in 1928 that there exist Kakeya sets of arbitrarily small positive measure, the question of what happens when we require the set to be connected or simply connected remains fascinating. In higher dimensions, the Kakeya conjecture (related but distinct) concerns sets containing unit line segments in every direction and has deep connections to harmonic analysis, partial differential equations, and number theory. The finite field analog was resolved by Dvir in 2008 using the polynomial method.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The infimum area is zero: Besicovitch constructed planar Kakeya sets of measure zero.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nArbitrarily small-area regions permit the required rotation, so no positive minimum exists.\n\n**What remains.**\n\nNo work remains for the literal planar minimum-area question.\n\n**Sources checked.**\n\n- Kakeya set overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Kakeya_set\n Evidence used: Records Besicovitch's measure-zero planar Kakeya construction.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 312, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1125, "problem_number": "GEO-005", "title": "Bellman's Lost in a Forest Problem", "statement": "What is the shortest path that guarantees escape from a forest of known shape and size, starting from an unknown location?", "background": "Bellman's lost in a forest problem asks for the shortest universal path that guarantees reaching the boundary of a region, regardless of starting position and orientation. For a circular forest of radius 1, the problem was solved by various authors with a path of length approximately 7.2898. However, for other shapes like squares or equilateral triangles, the optimal escape path remains unknown. This problem has applications to robotics, search and rescue operations, and computational geometry. It connects to questions about curve shortening, geometric optimization, and worst-case analysis in motion planning.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Bellman's forest-escape optimization depends on whether the forest is convex, whether orientation is known, and the allowed path class; the source omits these choices.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo unique problem or optimal path follows from the literal wording.\n\n**What remains.**\n\nRecover the intended forest and information model.\n\n**Sources checked.**\n\n- Bellman's lost-in-a-forest problem overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Lost_in_a_forest_problem\n Evidence used: Shows the formulation-sensitive variants.\n\n**Review notes.** No source alteration; formulation incomplete.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 198, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1131, "problem_number": "COMB-003", "title": "The Union-Closed Sets Conjecture", "statement": "For any finite family of finite sets that is closed under taking unions, must there exist an element that belongs to at least half of the sets?", "background": "The union-closed sets conjecture, also known as Frankl's conjecture after Peter Frankl who popularized it in 1979, is a simple-to-state problem in extremal combinatorics. A family of sets is union-closed if the union of any two sets in the family is also in the family. The conjecture asserts that in any non-trivial union-closed family, some element appears in at least half of the sets. Despite extensive research and verification for small cases, the general conjecture remains open. It has connections to lattice theory, Boolean functions, and information theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Frankl's union-closed sets conjecture remains open; a 2024 preprint claiming a proof is not accepted as a verified resolution.\n\n**Verified partial progress.**\n\n- Many restricted families and quantitative approximations are established, including results based on Reimer's average-set-size theorem.\n\n**Full solution or refutation.**\n\nNo peer-reviewed or otherwise independently verified proof of the general assertion was found.\n\n**What remains.**\n\nProve that every nontrivial finite union-closed family has a frequency-at-least-one-half element, or find a counterexample.\n\n**Sources checked.**\n\n- D. Reimer, An average set size theorem, Combin. Probab. Comput. 12 (2003), 89--93. (primary): https://arxiv.org/abs/1704.07022\n Evidence used: The cited note states Reimer's key average-size theorem and its relation to the conjecture.\n- Union-closed sets conjecture overview (accessed 2026-08-17). (maintained_tracker): https://en.wikipedia.org/wiki/Union-closed_sets_conjecture\n Evidence used: Records the conjecture and partial results; no accepted resolution is listed.\n\n**Review notes.** No source alteration; unverified proof claim deliberately excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 334, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1132, "problem_number": "COMB-004", "title": "Singmaster's Conjecture", "statement": "Does there exist a finite upper bound on how many times a number (other than 1) can appear in Pascal's triangle?", "background": "Singmaster's conjecture, proposed by David Singmaster in 1971, concerns the frequency of entries in Pascal's triangle. While 1 appears infinitely often (along the edges), and 2 appears exactly three times, larger numbers can appear multiple times in different positions. For example, 120 appears six times. The conjecture states that there exists an absolute constant $C$ such that no number appears more than $C$ times in Pascal's triangle (excluding 1). Singmaster himself proved that the number of occurrences is at most $O(\\log n / \\log \\log n)$ for the entry $n$. The conjecture connects to Diophantine equations and the distribution of binomial coefficients.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Singmaster's bounded-multiplicity conjecture remains open, but it is proved in a broad interior region of Pascal's triangle and the best total bound is sublogarithmic.\n\n**Verified partial progress.**\n\n- Matomäki, Radziwiłł, Shao, Tao, and Teräväinen prove that, for sufficiently large target t, at most four occurrences lie in the region whose lower index is at least exp((log n)^(2/3+epsilon)) from both edges.\n- Their 2022 paper records O(log t log_3 t / log_2^3 t) as the best unconditional bound for total multiplicity.\n- The largest known multiplicity recorded there is eight, attained by 3003; the source background's example 120 has only six.\n\n**Full solution or refutation.**\n\nNo absolute uniform bound for all entries and all regions was verified.\n\n**What remains.**\n\nControl the near-edge range, where the lower binomial index is small relative to the row, by an absolute constant.\n\n**Sources checked.**\n\n- K. Matomäki, M. Radziwiłł, X. Shao, T. Tao, and J. Teräväinen, Singmaster's Conjecture in the Interior of Pascal's Triangle, Quarterly Journal of Mathematics 73 (2022), 1137–1177. (primary): https://doi.org/10.1093/qmath/haac006\n Evidence used: The abstract and Theorem 1.3 prove bounded multiplicity in the interior; the introduction states the current total bound and the eight occurrences of 3003.\n\n**Review notes.** Exact statement retained; the background's numerical example is not the current record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 298, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1135, "problem_number": "SET-001", "title": "The Continuum Hypothesis", "statement": "Is there a set whose cardinality is strictly between that of the integers and the real numbers?", "background": "The continuum hypothesis (CH), proposed by Georg Cantor in 1878, states that there is no set with cardinality strictly between that of the integers and the real numbers. Equivalently, it asserts that the cardinality of the continuum (the real numbers) is $\\aleph_1$, the second smallest infinite cardinal. Gödel proved in 1940 that CH is consistent with ZFC (if ZFC is consistent), and Cohen proved in 1963 that the negation of CH is also consistent with ZFC. Thus, CH is independent of the standard axioms of set theory. This means CH can neither be proved nor disproved from ZFC alone, making it one of the most philosophically significant results in mathematical logic.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Existence of a set of cardinality strictly between the integers and reals is the negation of CH and is independent of ZFC, assuming ZFC is consistent.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nCohen forcing models answer the literal existence question yes, while Godel's constructible universe answers it no. Thus ZFC alone has no invariant yes/no answer.\n\n**What remains.**\n\nAny absolute-looking decision requires an additional axiom beyond ZFC; there is no remaining ordinary ZFC proof problem.\n\n**Sources checked.**\n\n- Kurt Godel, The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis, Proceedings of the National Academy of Sciences 24 (1938), 556-557. (primary): https://doi.org/10.1073/pnas.24.12.556\n Evidence used: Supplies a ZFC model with no intermediate cardinality by establishing relative consistency of GCH.\n- Paul J. Cohen, The Independence of the Continuum Hypothesis, Proceedings of the National Academy of Sciences 50 (1963), 1143-1148. (primary): https://doi.org/10.1073/pnas.50.6.1143\n Evidence used: Supplies the forcing direction in which CH fails and an intermediate cardinality exists.\n\n**Review notes.** Exact question retained. This is the negation of record 22's assertion although both records use SET-001; do not merge by problem number. Solved means solved-as-independent.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 623, "favorite_count": 54, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1137, "problem_number": "NT-019", "title": "Are There Infinitely Many Sophie Germain Primes?", "statement": "Are there infinitely many primes $p$ such that $2p + 1$ is also prime?", "background": "A Sophie Germain prime is a prime $p$ where $2p+1$ is also prime. These primes are named after French mathematician Sophie Germain, who used them in her work on Fermat's Last Theorem. Examples include 2, 3, 5, 11, 23, and 29. The conjecture that infinitely many exist is closely related to the twin prime conjecture and is similarly difficult. Sophie Germain primes have applications in cryptography and are used in some primality testing algorithms. As of 2024, the largest known Sophie Germain prime has over 400,000 digits. The problem remains one of the major open questions about prime distribution.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Infinitude of Sophie Germain primes remains open.\n\n**Verified partial progress.**\n\n- Sieve bounds and extensive computations support the conjecture.\n\n**Full solution or refutation.**\n\nNo infinitude proof was verified.\n\n**What remains.**\n\nProve infinitely many p with 2p+1 prime.\n\n**Sources checked.**\n\n- UPC Sophie Germain primes information page (2025). (authoritative_secondary): https://fme.upc.edu/ca/la-facultat/activitats-fme-personalitats-del-curs/personalitat-del-curs/2025-2026-germain/nombresprimers\n Evidence used: Explicitly states that infinitude is unknown.\n\n**Review notes.** Duplicate topic retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1138, "problem_number": "AG-001", "title": "The Hodge Conjecture", "statement": "On a projective algebraic variety, is every Hodge class a rational linear combination of classes of algebraic cycles?", "background": "The Hodge conjecture is one of the seven Millennium Prize Problems, with a $1 million prize for its solution. Proposed by William Hodge in 1950, it concerns the deep relationship between the topology and algebraic geometry of complex projective varieties. In simple terms, it asks whether certain topological cycles (Hodge classes) can be represented by algebraic cycles (subvarieties). The conjecture has been verified in many special cases, including for curves, surfaces, and abelian varieties, but the general case remains stubbornly open. It connects algebraic geometry, topology, and complex analysis in profound ways.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Hodge conjecture remains open in general, with significant verified cases for special varieties and low dimensions.\n\n**Verified partial progress.**\n\n- The Lefschetz (1,1) theorem proves the codimension-one case.\n\n**Full solution or refutation.**\n\nNo theorem establishes algebraicity of all rational Hodge classes on all smooth projective varieties.\n\n**What remains.**\n\nProve or disprove the general Hodge conjecture.\n\n**Sources checked.**\n\n- Hodge conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Hodge_conjecture\n Evidence used: Records the general conjecture as open and summarizes established cases.\n\n**Review notes.** Duplicate source label retained; distinct record ID preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 534, "favorite_count": 46, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1140, "problem_number": "AG-003", "title": "The Birch and Swinnerton-Dyer Conjecture", "statement": "For an elliptic curve $E$ over the rationals, does the rank of its group of rational points equal the order of vanishing of its $L$-function at $s=1$?", "background": "The Birch and Swinnerton-Dyer (BSD) conjecture is one of the seven Millennium Prize Problems. It connects the arithmetic properties of elliptic curves (specifically, the group of rational points) with analytic properties (the behavior of the associated $L$-function). The conjecture predicts a precise relationship between these seemingly disparate aspects. It has been verified computationally for millions of curves and proven in special cases, but the general conjecture remains open. A proof would revolutionize our understanding of elliptic curves and have applications to cryptography and number theory. The conjecture also relates to the Langlands program and modern arithmetic geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** BSD is open generally, with major results in analytic rank zero and one and conditional implications.\n\n**Verified partial progress.**\n\n- Gross--Zagier and Kolyvagin establish central cases for broad modular elliptic-curve settings.\n\n**Full solution or refutation.**\n\nNo general equality of analytic and algebraic ranks was verified.\n\n**What remains.**\n\nProve BSD for all elliptic curves over Q.\n\n**Sources checked.**\n\n- Birch and Swinnerton-Dyer conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture\n Evidence used: Records the rank-zero/one progress and general open status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 687, "favorite_count": 59, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1141, "problem_number": "DYN-001", "title": "The Weinstein Conjecture", "statement": "Does every Reeb vector field on a closed contact manifold have at least one periodic orbit?", "background": "The Weinstein conjecture, proposed by Alan Weinstein in 1978, is a fundamental problem in symplectic geometry and dynamical systems. A Reeb vector field is a special type of vector field on a contact manifold, and the conjecture predicts the existence of closed orbits under very general conditions. The conjecture was proven in dimension 3 by Hofer in 1993 using pseudoholomorphic curves, and has been established in many other cases. However, the general case remains open. The conjecture has deep connections to Hamiltonian dynamics, celestial mechanics, and the study of periodic phenomena in physics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Weinstein conjecture is proved for closed contact three-manifolds but remains open in the stated arbitrary-dimensional generality.\n\n**Verified partial progress.**\n\n- Taubes proved the three-dimensional conjecture using Seiberg--Witten theory.\n\n**Full solution or refutation.**\n\nNo all-dimensional theorem for every closed contact manifold was verified.\n\n**What remains.**\n\nSettle the higher-dimensional Weinstein conjecture.\n\n**Sources checked.**\n\n- C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007), 2117--2202. (primary): https://arxiv.org/abs/math/0611007\n Evidence used: This is Taubes's proof of the 3D Weinstein conjecture.\n- M. Hutchings, Taubes's proof of the Weinstein conjecture in dimension three. (authoritative_secondary): https://math.berkeley.edu/~hutching/pub/tw.pdf\n Evidence used: It explicitly describes the proof as dimension-three and discusses the higher-dimensional obstruction.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 11, "view_count": 276, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1142, "problem_number": "DYN-002", "title": "The Painlevé Conjecture", "statement": "In the $n$-body problem with $n \\geq 4$, can non-collision singularities occur in finite time?", "background": "The Painlevé conjecture concerns the $n$-body problem in celestial mechanics, asking whether the motion of $n$ point masses under gravitational attraction can develop a singularity (infinite velocities or unbounded positions) in finite time without any collisions occurring. For $n=3$, Sundman proved in 1912 that non-collision singularities cannot occur, but for $n \\geq 4$ the question remains open. Xia constructed examples showing that certain types of unbounded behavior are possible, but the existence of true non-collision singularities (where velocities become infinite) remains unproven. The problem has implications for the long-term behavior of planetary systems and the foundations of classical mechanics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Non-collision singularities exist for Newtonian n-body systems for every n >= 4.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nXia constructed the five-body case; Xue proved a planar four-body construction, completing the remaining threshold case.\n\n**What remains.**\n\nNo existence question remains under the source formulation.\n\n**Sources checked.**\n\n- J. Xue, Noncollision Singularities in a Planar Four-body Problem, Acta Math. 224 (2020), 253--388. (primary): https://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220228202444113958904\n Evidence used: The article record states that it proves noncollision singularities in the planar four-body problem and settles Painleve's conjecture.\n- On Painleve Conjecture, survey/preprint by J. Xue (accessed 2026-08-17). (authoritative_secondary): https://www.math.harvard.edu/media/CDM_Xue.pdf\n Evidence used: It identifies the four-body theorem as completing the program and recalls Xia's five-body result.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 11, "view_count": 298, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1144, "problem_number": "GT-008", "title": "Cereceda's Conjecture", "statement": "For any $k$-chromatic graph, can its $k$-colorings be transformed into each other by recoloring one vertex at a time, staying within $k$ colors, in polynomial time in the number of vertices?", "background": "Cereceda's conjecture concerns the diameter of the reconfiguration graph of $k$-colorings. Given a graph $G$ with chromatic number $k$, consider the graph whose vertices are all proper $k$-colorings of $G$, with two colorings adjacent if they differ on exactly one vertex. The conjecture, proposed in 2007, states that this graph has diameter at most $O(n^2)$ where $n$ is the number of vertices in $G$. The problem is motivated by questions in computational complexity and has connections to mixing times of Markov chains. While progress has been made for special graph classes, the general conjecture remains open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The statement is false as written: for K2, the two proper 2-colorings cannot be connected by any single-vertex proper recoloring sequence.\n\n**Verified partial progress.**\n\n- The standard Cereceda conjecture instead concerns d-degenerate graphs with at least d+2 colors and asks for an O(n^2) diameter.\n- The polynomial (weaker) version of that standard statement is proved.\n\n**Full solution or refutation.**\n\nK2 is a direct counterexample: changing either vertex first produces an improper coloring.\n\n**What remains.**\n\nIf the intended problem was the standard Cereceda conjecture, recover that different statement; it remains open in its quadratic form.\n\n**Sources checked.**\n\n- L. Cereceda, J. van den Heuvel and M. Johnson, Connectedness of the graph of vertex-colourings, Discrete Mathematics 308 (2008), 913--919, doi:10.1016/j.disc.2007.07.028. (primary): https://doi.org/10.1016/j.disc.2007.07.028\n Evidence used: Shows non-connectivity at chromatic number k for k=2,3 and identifies the distinct reconfiguration setting.\n- N. Bousquet and M. Heinrich, A polynomial version of Cereceda's conjecture, JCTB 155 (2022), doi:10.1016/j.jctb.2022.01.006. (primary): https://arxiv.org/abs/1903.05619\n Evidence used: States the actual d-degenerate, d+2-color standard formulation.\n\n**Review notes.** No source alteration; the named conjecture is materially misformulated in the dataset.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1146, "problem_number": "TOP-003", "title": "The Whitehead Conjecture", "statement": "Is every aspherical closed manifold whose fundamental group has no non-trivial perfect normal subgroups a $K(\\pi, 1)$ space?", "background": "The Whitehead conjecture, posed by J.H.C. Whitehead, concerns a special class of topological spaces. A space is aspherical if all its homotopy groups above dimension 1 vanish, and it is a $K(\\pi, 1)$ if it is aspherical and connected. The conjecture asks whether certain algebraic conditions on the fundamental group guarantee the topological property of being aspherical. The Poincaré conjecture can be viewed as a special case. While the conjecture is known to hold for many important classes of manifolds, including those with non-positive sectional curvature, the general case remains open. The problem connects algebraic topology, geometric topology, and group theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The supplied statement is true by definition: every connected aspherical manifold is a K(pi,1), regardless of whether its fundamental group has a nontrivial perfect normal subgroup.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nAsphericity means the universal cover is contractible, equivalently all higher homotopy groups vanish; for a connected manifold this is exactly the K(pi_1,1) property, so the extra group hypothesis is irrelevant.\n\n**What remains.**\n\nNothing remains for the statement as written. The source should be corrected if the intended problem was the still-open classical Whitehead conjecture about subcomplexes of aspherical two-dimensional CW complexes.\n\n**Sources checked.**\n\n- Allen Hatcher, Algebraic Topology, Section 1.B, online textbook. (authoritative_secondary): https://pi.math.cornell.edu/~hatcher/AT/AT.pdf\n Evidence used: Gives the standard K(G,1) and asphericity definitions that make the displayed implication immediate.\n- A. M. Mikhovich, Rational and p-adic analogues of J. H. C. Whitehead's conjecture, Izvestiya: Mathematics 89 (2025), 274-318. (primary): https://doi.org/10.4213/im9597\n Evidence used: States the actual classical Whitehead conjecture concerning subcomplexes of aspherical two-dimensional CW complexes and records its open status.\n\n**Review notes.** This is a formulation defect, not a newly discovered solution of the classical Whitehead conjecture. The background itself defines aspherical plus connected as K(pi,1), contradicting its open-status claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 234, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1149, "problem_number": "GEO-006", "title": "The Knaster Problem", "statement": "Can a solid cube be completely covered by finitely many smaller homothetic cubes with ratio less than 1, such that the interiors are disjoint?", "background": "The Knaster problem, also known as the cube packing problem, asks whether a unit cube can be covered by finitely many non-overlapping smaller cubes, each similar to the original with ratio $< 1$. In 1979, Mycielski proved this is impossible in dimension 2 (for squares), but the 3-dimensional case remains open. The problem has connections to measure theory, geometric covering problems, and Banach-Tarski-like paradoxes. A positive answer would be quite surprising as it would demonstrate a counterintuitive property of 3-dimensional space. The problem has inspired research into covering and packing problems in higher dimensions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The literal cube-covering question is trivial: partition the cube into 2^n congruent cubes of homothety ratio 1/2.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThese finitely many smaller homothetic cubes have disjoint interiors and cover the original cube.\n\n**What remains.**\n\nNothing under the stated wording; a different intended Knaster problem would need recovery.\n\n**Sources checked.**\n\n- Elementary cube subdivision argument (checked 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Cube\n Evidence used: Standard coordinate subdivision into half-side cubes verifies the literal statement.\n\n**Review notes.** No source alteration; likely formulation defect flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 189, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1153, "problem_number": "NT-022", "title": "Polignac's Conjecture", "statement": "For every even number $n$, are there infinitely many pairs of consecutive primes differing by $n$?", "background": "Polignac's conjecture, proposed by Alphonse de Polignac in 1849, is a vast generalization of the twin prime conjecture. It asserts that for every even integer $n$, there exist infinitely many prime gaps of exactly size $n$. The twin prime conjecture is the special case $n = 2$. While Zhang's 2013 breakthrough showed infinitely many bounded gaps exist, proving the existence of infinitely many gaps of any specific even size remains open. The conjecture relates to the Hardy-Littlewood conjectures and our understanding of the distribution of primes. Even proving the existence of infinitely many prime gaps of size 6 would be a major breakthrough.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Polignac's conjecture remains open for each prescribed even gap.\n\n**Verified partial progress.**\n\n- Bounded-gap theorems prove that at least one even gap occurs infinitely often, but do not identify every even gap.\n\n**Full solution or refutation.**\n\nNo proof for arbitrary prescribed n was verified.\n\n**What remains.**\n\nProve infinitely many consecutive-prime gaps of each even size.\n\n**Sources checked.**\n\n- Polignac's conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Polignac%27s_conjecture\n Evidence used: Records the general open statement and weaker Chen-type results.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1156, "problem_number": "ALG-016", "title": "The Babai Conjecture on Graph Isomorphism", "statement": "Can graph isomorphism be decided in quasi-polynomial time for all graphs?", "background": "The Babai conjecture concerns the computational complexity of determining whether two graphs are isomorphic. In 2015, László Babai announced a quasi-polynomial time algorithm for graph isomorphism (running in time $2^{O(\\log^c n)}$ for some constant $c$), improving on the previous best bound. While a flaw was found in the original proof, Babai repaired it in 2017. However, whether graph isomorphism is in P (polynomial time) remains open. The problem sits in NP but is not known to be NP-complete, occupying a special place in complexity theory. The resolution has important implications for cryptography and the structure of complexity classes.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Babai gave a quasipolynomial-time algorithm for graph isomorphism.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThis directly answers the source decision-bound question affirmatively.\n\n**What remains.**\n\nImprove the bound if desired; the stated quasipolynomial target is met.\n\n**Sources checked.**\n\n- L. Babai, Graph Isomorphism in Quasipolynomial Time, STOC 2016; arXiv:1512.03547. (primary): https://arxiv.org/abs/1512.03547\n Evidence used: The abstract gives a quasipolynomial graph-isomorphism algorithm.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 445, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1157, "problem_number": "NT-023", "title": "Pillai's Conjecture", "statement": "For each positive integer $k$, does the equation $|2^m - 3^n| = k$ have only finitely many solutions in positive integers $m$ and $n$?", "background": "Pillai's conjecture, proposed by Subbayya Sivasankaranarayana Pillai, concerns the gaps between powers of 2 and powers of 3. The conjecture generalizes to any two multiplicatively independent integers $a$ and $b$: the equation $|a^m - b^n| = k$ should have only finitely many solutions for each fixed $k$. This is related to the abc conjecture and to understanding the distribution of exponential Diophantine equations. The conjecture has been proven for many special cases but remains open in general. It connects to transcendental number theory and the study of linear forms in logarithms.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** For fixed k, the equation 2^m-3^n=plus/minus k has only finitely many solutions by the finiteness theorem for S-unit equations.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nWriting 2^m/3^n as an S-unit reduces the equation to a fixed finite set of S-unit solutions; both signs are covered.\n\n**What remains.**\n\nNo work remains for this fixed-base formulation. Broader Pillai problems with variable bases are different and remain open.\n\n**Sources checked.**\n\n- J.-H. Evertse, On equations in S-units and the Thue-Mahler equation, Invent. Math. 75 (1984), 561--584. (primary): https://doi.org/10.1007/BF01388636\n Evidence used: Establishes finiteness of fixed S-unit equations, which directly covers the stated fixed-base equation.\n\n**Review notes.** No source alteration; the named broad conjecture differs from the fixed-base record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1158, "problem_number": "NT-024", "title": "Erdős-Straus Conjecture", "statement": "For every integer $n \\geq 2$, can $\\frac{4}{n}$ be expressed as the sum of three unit fractions $\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}$?", "background": "The Erdős-Straus conjecture asks whether every fraction $4/n$ (for $n \\geq 2$) can be written as a sum of three unit fractions (fractions with numerator 1). The conjecture has been verified computationally for all $n$ up to $10^{17}$ and proven for several infinite families of values, but the general case remains open. Egyptian fraction representations have been studied since ancient times, and this problem connects to number theory, combinatorics, and computational mathematics. Related problems concern representing other fractions as sums of unit fractions with various restrictions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The duplicate Erdős--Straus statement remains unresolved.\n\n**Verified partial progress.**\n\n- Large finite verification and congruence-class results are known.\n\n**Full solution or refutation.**\n\nNo all-n theorem was verified.\n\n**What remains.**\n\nSettle the conjecture for every integer n>=2.\n\n**Sources checked.**\n\n- The Erdős-Straus Conjecture and the Structure of..., INTEGERS 26 (2026). (primary): https://math.colgate.edu/~integers/aa42/aa42.pdf\n Evidence used: Recent research treats the conjecture as open.\n\n**Review notes.** Duplicate topic retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 289, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1159, "problem_number": "NT-025", "title": "The Gauss Circle Problem", "statement": "What is the optimal error term in the formula for the number of lattice points inside a circle of radius $r$?", "background": "The Gauss circle problem asks for the number of integer lattice points $(x,y)$ satisfying $x^2 + y^2 \\leq r^2$. The main term is $\\pi r^2$ (the area of the circle), but determining the optimal error term has been a central problem in analytic number theory for over 150 years. It is known that the error is $O(r^{2/3})$ and conjectured to be $O(r^{1/2 + \\epsilon})$ for any $\\epsilon > 0$, but this has not been proven. The problem connects to the distribution of lattice points, the theory of the Riemann zeta function, and has inspired numerous techniques in analytic number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The optimal Gauss-circle error exponent is open; the best checked upper bound is r^(131/208+epsilon), versus the conjectural r^(1/2+epsilon) scale.\n\n**Verified partial progress.**\n\n- Huxley's 131/208 exponent remains the standard record quoted in current references.\n\n**Full solution or refutation.**\n\nNo optimal error term was verified.\n\n**What remains.**\n\nClose the gap to the conjectural half-power behavior.\n\n**Sources checked.**\n\n- Gauss circle problem status, MathOverflow. (authoritative_secondary): https://mathoverflow.net/questions/355861/what-is-the-most-general-formulation-of-gausss-circle-problem\n Evidence used: Records Huxley's 131/208 bound and the half-power conjecture.\n- Optimization Constants in Mathematics, Gauss circle exponent. (maintained_tracker): https://teorth.github.io/optimizationproblems/\n Evidence used: Lists the current 131/208 upper exponent.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 367, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1160, "problem_number": "ALG-017", "title": "Birch-Tate Conjecture", "statement": "Does the order of the center of the Steinberg group of the ring of integers of a number field relate to the value of the Dedekind zeta function at $s=-1$?", "background": "The Birch-Tate conjecture concerns the relationship between algebraic K-theory and special values of zeta functions. Specifically, it relates the order of $K_2$ of the ring of integers of a number field to the value of the Dedekind zeta function at $s = -1$. This conjecture is part of a broader program connecting algebraic K-theory to number theory and has been verified in many special cases. It generalizes ideas from class field theory and has connections to the Lichtenbaum conjectures. The problem sits at the intersection of algebraic number theory, algebraic K-theory, and the theory of zeta functions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Birch--Tate is proved in important cases but requires qualifications at the prime 2 and is not a universal result as phrased.\n\n**Verified partial progress.**\n\n- Wiles proved the conjectural formula for totally real abelian number fields up to the standard 2-primary issue.\n\n**Full solution or refutation.**\n\nThe brief source wording omits the necessary formulation details.\n\n**What remains.**\n\nSpecify the exact K-theoretic/2-primary statement and settle remaining cases.\n\n**Sources checked.**\n\n- Birch--Tate conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Birch%E2%80%93Tate_conjecture\n Evidence used: Summarizes proved cases and qualifications.\n\n**Review notes.** Formulation qualifications retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1161, "problem_number": "ALG-018", "title": "Hilbert's Fifteenth Problem", "statement": "Can Schubert calculus be given a rigorous foundation?", "background": "Hilbert's fifteenth problem, from his famous 1900 list, asks for a rigorous foundation of Schubert's enumerative calculus. Schubert calculus is a method for solving problems in enumerative geometry, such as counting the number of lines in 3-space that meet four given lines. While modern algebraic geometry has provided substantial progress through intersection theory and the development of Chow rings, aspects of the problem remain active areas of research. The development of Gromov-Witten invariants and quantum cohomology has provided new tools, but questions about the complete rigor of classical Schubert calculus in all dimensions continue to be investigated.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Schubert calculus has a rigorous modern foundation in intersection theory and cohomology of Grassmannians/flag varieties.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe historical foundational request is fulfilled.\n\n**What remains.**\n\nNo work remains for the stated foundational problem.\n\n**Sources checked.**\n\n- Schubert calculus overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Schubert_calculus\n Evidence used: Describes the modern rigorous intersection-theoretic formulation.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1162, "problem_number": "ALG-019", "title": "Hilbert's Sixteenth Problem", "statement": "What is the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$ in the plane?", "background": "Hilbert's sixteenth problem consists of two parts. The first part (topology of algebraic curves) asks about the possible configurations of connected components of real algebraic curves. The second part asks for the maximum number and possible configurations of limit cycles of polynomial vector fields of degree $n$ in the plane. This second part remains largely open even for $n=2$ (quadratic systems). The problem is fundamental to the qualitative theory of differential equations and has applications to dynamical systems, control theory, and mathematical biology. Despite over a century of research, even basic questions about quadratic systems remain unresolved.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Hilbert's sixteenth problem remains open for general polynomial vector-field degree.\n\n**Verified partial progress.**\n\n- Sharp low-degree results and bounds are known.\n\n**Full solution or refutation.**\n\nNo maximum number or classification of limit-cycle configurations is known for arbitrary degree.\n\n**What remains.**\n\nResolve the limit-cycle bound and relative-position questions.\n\n**Sources checked.**\n\n- Hilbert's sixteenth problem overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Hilbert%27s_sixteenth_problem\n Evidence used: Records the second part as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1163, "problem_number": "GEO-008", "title": "The Inscribed Square Problem", "statement": "Does every simple closed curve in the plane contain four points that form the vertices of a square?", "background": "The inscribed square problem, also known as Toeplitz' conjecture, asks whether every Jordan curve (simple closed curve) in the plane contains four points forming a square. The problem has been open since 1911. It is known to be true for smooth curves and for many other special cases, but the general case for arbitrary continuous curves remains unproven. The problem is related to other inscribed polygon problems and has connections to topology, dynamical systems, and geometric measure theory. Even proving the existence of an inscribed rectangle with sides in a given ratio remains challenging for general curves.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The square-peg problem remains open for arbitrary Jordan curves; new 2026 work gives a positive-measure set of inscribed rectangle shapes but does not force the square shape.\n\n**Verified partial progress.**\n\n- Greene and Lobb proved that every smooth Jordan curve inscribes a rectangle of every prescribed similarity class.\n- They proved the square case for a Jordan curve that is the union of two graphs with Lipschitz constants below 1+sqrt(2).\n- For every Jordan curve of diameter 2R enclosing area A, a 2026 preprint proves that the set of realized rectangle diagonal angles has measure at least A/R^2.\n\n**Full solution or refutation.**\n\nNo theorem covering every simple closed continuous plane curve was verified.\n\n**What remains.**\n\nProve or disprove that every Jordan curve inscribes the particular rectangle similarity class of a square, without additional regularity or graph hypotheses.\n\n**Sources checked.**\n\n- Joshua Evan Greene and Andrew Lobb, The rectangular peg problem, Annals of Mathematics 194 (2021), 509-517. (primary): https://annals.math.princeton.edu/2021/194-2/p04\n Evidence used: Proves every prescribed rectangle similarity class for smooth Jordan curves.\n- Joshua Evan Greene and Andrew Lobb, Square pegs between two graphs, arXiv:2407.07798 (2024; subsequently published). (primary): https://arxiv.org/abs/2407.07798\n Evidence used: Proves an inscribed square for the union-of-two-Lipschitz-graphs class with the stated sharp-looking regularity bound.\n- Joshua Evan Greene and Andrew Lobb, Jordan curves inscribe a positive measure of rectangles, arXiv:2604.17116 (2026). (primary): https://arxiv.org/abs/2604.17116\n Evidence used: Proves a quantitative positive-measure family of rectangle diagonal angles for arbitrary Jordan curves, but does not assert the square angle.\n\n**Review notes.** The dataset's general open status is current; its smooth-case discussion is supported by a theorem stronger than the square-only assertion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 456, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1164, "problem_number": "GEO-009", "title": "Falconer's Conjecture", "statement": "If a compact set in $\\mathbb{R}^d$ has Hausdorff dimension greater than $d/2$, must it determine a set of distances with positive Lebesgue measure?", "background": "Falconer's conjecture concerns the relationship between the fractal dimension of a set and the set of distances between its points. Proposed by Kenneth Falconer in 1985, it states that if a compact set $E \\subset \\mathbb{R}^d$ has Hausdorff dimension strictly greater than $d/2$, then the distance set $\\{|x-y| : x, y \\in E\\}$ has positive Lebesgue measure. The conjecture has been proven in dimension 2 but remains open in higher dimensions. It has deep connections to harmonic analysis, geometric measure theory, and additive combinatorics. Recent progress using polynomial methods has improved bounds but not resolved the conjecture.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Falconer's conjectured d/2 threshold remains open, including in dimension two; the dataset's assertion that the planar conjecture is proved is false.\n\n**Verified partial progress.**\n\n- Guth, Iosevich, Ou, and Wang prove positive measure in R^2 under the stronger hypothesis dim_H(E)>5/4.\n- Du, Iosevich, Ou, Wang, and Zhang prove the threshold d/2+1/4 for even dimensions d at least four.\n\n**Full solution or refutation.**\n\nNo proof at every Hausdorff dimension strictly above d/2 was verified in any general dimension; in particular the planar threshold remains above 1.\n\n**What remains.**\n\nClose the threshold gap down to d/2, including lowering the best planar sufficient exponent from 5/4 to 1.\n\n**Sources checked.**\n\n- Larry Guth, Alex Iosevich, Yumeng Ou, and Hong Wang, On Falconer's distance set problem in the plane, Inventiones Mathematicae 219 (2020), 779-830, arXiv:1808.09346. (primary): https://arxiv.org/abs/1808.09346\n Evidence used: Proves the 5/4 planar threshold, which is strictly weaker than the conjectured threshold 1.\n- Xiumin Du, Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang, An improved result for Falconer's distance set problem in even dimensions, Mathematische Annalen 380 (2021), 1215-1231, arXiv:2006.06833. (primary): https://arxiv.org/abs/2006.06833\n Evidence used: Proves positive measure above d/2+1/4 for even d at least four and describes this as progress toward the conjecture.\n\n**Review notes.** The background requires correction before reuse: the planar case is not proved at Falconer's conjectured threshold.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 289, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1165, "problem_number": "GT-010", "title": "The Total Coloring Conjecture", "statement": "Can every graph be totally colored with at most $\\Delta + 2$ colors, where $\\Delta$ is the maximum degree?", "background": "The total coloring conjecture, proposed independently by Behzad and Vizing in the 1960s, concerns coloring both vertices and edges of a graph such that no two adjacent or incident elements receive the same color. The conjecture states that every graph can be totally colored using at most $\\Delta + 2$ colors where $\\Delta$ is the maximum degree. Vizing proved that at most $\\Delta + 2$ colors suffice, and it is trivial that at least $\\Delta + 1$ are needed. The conjecture asks whether the upper bound is tight. It has been verified for many graph classes but remains open in general. The problem has applications to scheduling and resource allocation.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Total Coloring Conjecture remains open for arbitrary graphs.\n\n**Verified partial progress.**\n\n- It holds for many special graph families, including recent planar and 1-planar restricted cases.\n\n**Full solution or refutation.**\n\nNo universal Delta+2 theorem was verified.\n\n**What remains.**\n\nProve the Delta+2 total-coloring bound for every graph.\n\n**Sources checked.**\n\n- R. Su, G. Fang and E. Zhu, The Total Coloring Conjecture holds for planar graphs without three special subgraphs, arXiv:2507.12737 (2025). (primary): https://arxiv.org/abs/2507.12737\n Evidence used: Gives a restricted planar result and explicitly identifies unresolved cases.\n\n**Review notes.** Duplicate topic retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1171, "problem_number": "NT-026", "title": "The Odd Perfect Number Conjecture", "statement": "Do there exist any odd perfect numbers? (A perfect number equals the sum of its proper divisors.)", "background": "A perfect number is a positive integer that equals the sum of its proper positive divisors. Euclid showed that numbers of the form $2^{p-1}(2^p - 1)$ are perfect when $2^p - 1$ is prime (Mersenne prime), giving all known even perfect numbers. Whether odd perfect numbers exist has been an open question for over 2000 years. It is known that if an odd perfect number exists, it must be greater than $10^{1500}$, have at least 101 prime factors, and satisfy numerous other constraints. The problem connects to prime number theory, divisibility, and has inspired extensive computational searches. Most mathematicians believe no odd perfect numbers exist.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Existence of an odd perfect number remains open.\n\n**Verified partial progress.**\n\n- Any odd perfect number would satisfy many stringent factorization and size constraints; exhaustive and theoretical lower bounds have been advanced.\n\n**Full solution or refutation.**\n\nNo existence or nonexistence proof was verified.\n\n**What remains.**\n\nConstruct an odd perfect number or prove none exist.\n\n**Sources checked.**\n\n- Odd Perfect Number, Wolfram MathWorld. (authoritative_secondary): https://mathworld.wolfram.com/OddPerfectNumber.html\n Evidence used: Summarizes major necessary factor constraints and open status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 678, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1172, "problem_number": "NT-027", "title": "Firoozbakht's Conjecture", "statement": "Is the sequence $p_n^{1/n}$ strictly decreasing, where $p_n$ is the $n$-th prime?", "background": "Firoozbakht's conjecture, proposed in 1982, states that the sequence $(p_n)^{1/n}$ is strictly decreasing, where $p_n$ denotes the $n$-th prime number. This is equivalent to saying that $p_{n+1}^n < p_n^{n+1}$ for all $n$. The conjecture is stronger than Cramér's conjecture about prime gaps and has been verified computationally for all primes up to very large values. If true, it would imply strong results about the distribution of primes and prime gaps. The conjecture remains open despite extensive numerical evidence supporting it.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Firoozbakht's conjecture remains unproved; published proof claims have not established accepted resolution.\n\n**Verified partial progress.**\n\n- It has been computationally verified to large finite ranges and has strong implications for prime gaps.\n\n**Full solution or refutation.**\n\nNo accepted all-primes proof was verified.\n\n**What remains.**\n\nProve strict decrease of p_n^(1/n) or find a counterexample.\n\n**Sources checked.**\n\n- Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap, arXiv:1904.00499. (primary): https://arxiv.org/abs/1904.00499\n Evidence used: Records finite verification, not a proof.\n- Firoozbakht's conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Firoozbakht%27s_conjecture\n Evidence used: Records its continuing conjectural status.\n\n**Review notes.** No source alteration; unverified proof claims excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1173, "problem_number": "AG-004", "title": "The Tate Conjecture", "statement": "For varieties over finite fields, are the $\\ell$-adic representations arising from étale cohomology related to algebraic cycles in the expected way?", "background": "The Tate conjecture, proposed by John Tate in 1963, is a fundamental problem in arithmetic geometry. It concerns the relationship between algebraic cycles on algebraic varieties over finite fields and the Galois representations arising from étale cohomology. The conjecture would provide a powerful tool for understanding rational equivalence of cycles. It has been proven for divisors (codimension 1 cycles) on abelian varieties and for various other special cases. The conjecture is closely related to the Hodge conjecture and the Birch and Swinnerton-Dyer conjecture, forming part of a web of deep conjectures in arithmetic geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Tate conjecture remains open generally, with numerous cases for abelian varieties and special varieties over finite fields.\n\n**Verified partial progress.**\n\n- The conjecture is established in several important classes, including divisors in broad settings.\n\n**Full solution or refutation.**\n\nNo universal cycle/cohomology correspondence theorem was verified.\n\n**What remains.**\n\nProve the Tate conjecture for arbitrary smooth projective varieties over finite fields.\n\n**Sources checked.**\n\n- Tate conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Tate_conjecture\n Evidence used: Records the conjecture's general open status and special cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 256, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1176, "problem_number": "SET-002", "title": "Suslin's Problem", "statement": "If a dense linear order without endpoints is complete and has the countable chain condition, must it be isomorphic to the real numbers?", "background": "Suslin's problem, posed by Mikhail Suslin in 1920, asks whether the real numbers can be characterized by certain order-theoretic properties. Specifically, it asks if every complete dense linear order without endpoints satisfying the countable chain condition (every family of disjoint open intervals is countable) must be order-isomorphic to $\\mathbb{R}$. In 1967, it was shown that this question is independent of ZFC set theory - both positive and negative answers are consistent with the standard axioms. A \"Suslin line\" (a counterexample) exists in some models of set theory but not in others. The problem inspired fundamental developments in set theory and the study of independence results.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Suslin's problem is independent of ZFC: some models contain a Suslin line and other models satisfy Suslin's Hypothesis and contain none.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nJech and independently Tennenbaum constructed models giving a negative answer to the literal must-question. Solovay and Tennenbaum's iterated-forcing model gives a positive answer. Together these settle the historical problem as independent.\n\n**What remains.**\n\nNo ZFC theorem can uniformly choose yes or no under the usual consistency assumption; one can instead study consequences of forcing axioms or constructibility principles.\n\n**Sources checked.**\n\n- Tomas Jech, Non-provability of Souslin's hypothesis, Commentationes Mathematicae Universitatis Carolinae 8 (1967), 291-305. (primary): https://eudml.org/doc/16215\n Evidence used: Constructs the relative-consistency direction in which Suslin's Hypothesis fails.\n- Stanley Tennenbaum, Souslin's Problem, Proceedings of the National Academy of Sciences 59 (1968), 60-63. (primary): https://doi.org/10.1073/pnas.59.1.60\n Evidence used: Provides an independent forcing construction of a model with a Suslin line.\n- Robert M. Solovay and Stanley Tennenbaum, Iterated Cohen Extensions and Souslin's Problem, Annals of Mathematics 94 (1971), 201-245. (primary): https://doi.org/10.2307/1970860\n Evidence used: Constructs a model satisfying Suslin's Hypothesis, completing the independence result.\n\n**Review notes.** Exact statement retained. The problem number SET-002 collides with record 41's unrelated Singular Cardinals Hypothesis; preserve the integer record ID. Solved means solved-as-independent.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 289, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1179, "problem_number": "NT-028", "title": "Schinzel's Hypothesis H", "statement": "If polynomials satisfy certain necessary divisibility conditions, do they simultaneously produce infinitely many primes for integer inputs?", "background": "Schinzel's Hypothesis H is a sweeping generalization of many conjectures about primes, including the twin prime conjecture, Sophie Germain prime conjecture, and Dickson's conjecture. It states that if $f_1, \\ldots, f_k$ are irreducible polynomials with integer coefficients and positive leading coefficients, and no prime divides all values $f_1(n) \\cdots f_k(n)$ simultaneously for all integers $n$, then there are infinitely many integers $n$ for which all $f_i(n)$ are prime. If true, it would unify and resolve numerous open problems about prime-producing polynomials. The hypothesis remains wide open despite its fundamental importance.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Under the standard formulation supplied by the background, Schinzel's Hypothesis H remains open; it contains the twin-prime conjecture as a special case.\n\n**Verified partial progress.**\n\n- Browning and Matthiesen proved Schinzel's hypothesis for 100% of polynomial systems in an averaged parameter-space sense.\n\n**Full solution or refutation.**\n\nNo theorem for every fixed irreducible locally admissible polynomial system was verified.\n\n**What remains.**\n\nProve simultaneous prime values infinitely often for every fixed system satisfying irreducibility, positivity, and no fixed prime divisor.\n\n**Sources checked.**\n\n- A. Schinzel and W. Sierpinski, Sur certaines hypotheses concernant les nombres premiers, Acta Arithmetica 4 (1958), 185-208. (primary): https://eudml.org/doc/206115\n Evidence used: Original formulation of the hypothesis.\n- T. Browning and L. Matthiesen, Schinzel hypothesis on average and rational points, Inventiones Mathematicae 232 (2023), 673-777. (primary): https://doi.org/10.1007/s00222-022-01153-6\n Evidence used: Proves an averaged 100%-of-systems result rather than the universal fixed-system conjecture.\n\n**Review notes.** The exact question is informal because it does not define the necessary conditions; the background supplies the standard formulation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 298, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1180, "problem_number": "ALG-020", "title": "The Uniform Boundedness Conjecture", "statement": "Is there a bound $B(g, d)$ such that every curve of genus $g$ over a number field of degree $d$ has at most $B(g, d)$ rational points?", "background": "The uniform boundedness conjecture for rational points on curves asks whether, for fixed genus $g$ and degree $d$, there exists a bound on the number of rational points on genus-$g$ curves over number fields of degree $d$. This would be a vast generalization of Mordell's conjecture (now Faltings' theorem, which shows finiteness but not uniform bounds). The conjecture has been proven for $g = 1$ (elliptic curves) by Mazur and Merel, but remains open for $g \\geq 2$. It connects to deep questions in arithmetic geometry about the distribution of rational points on varieties.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Uniform boundedness of rational points for fixed genus and number-field degree remains an outstanding unconditional problem.\n\n**Verified partial progress.**\n\n- Caporaso--Harris--Mazur and Pacelli derive the requested type of bound conditionally from Lang/Bombieri--Lang.\n- Uniform bounds are known in important low-Mordell--Weil-rank regimes.\n\n**Full solution or refutation.**\n\nNo unconditional bound B(g,d) for all curves in the stated generality was verified.\n\n**What remains.**\n\nEstablish the uniform bound without the Lang/Bombieri--Lang hypothesis.\n\n**Sources checked.**\n\n- P. Pacelli, Uniform boundedness for rational points, arXiv:alg-geom/9601004 (1996). (primary): https://arxiv.org/abs/alg-geom/9601004\n Evidence used: Its abstract gives the conditional uniform bound from Lang's conjecture.\n- J. Stoll, Uniform bounds for the number of rational points on curves of small Mordell--Weil rank, arXiv:1504.00694 (2015). (primary): https://arxiv.org/abs/1504.00694\n Evidence used: The abstract calls N(g,d) outstanding and gives small-rank uniform results.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 234, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1181, "problem_number": "ALG-021", "title": "The Pierce-Birkhoff Conjecture", "statement": "Is every piecewise-polynomial function $f: \\mathbb{R}^n \\to \\mathbb{R}$ the maximum of finitely many minimums of finite collections of polynomials?", "background": "The Pierce-Birkhoff conjecture asks whether every continuous piecewise polynomial function on $\\mathbb{R}^n$ can be represented using only the operations of addition, multiplication, and taking finite suprema and infima of polynomial functions. The conjecture has been verified in dimension 1 and for $n = 2$ in special cases, but remains open for general $n \\geq 2$. The problem has connections to real algebraic geometry, approximation theory, and constructive mathematics. A positive answer would provide powerful representation theorems for piecewise-defined functions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Pierce--Birkhoff conjecture is proved in dimension two, but no proof in all dimensions was verified.\n\n**Verified partial progress.**\n\n- Lucas--Madden--Schaub--Spivakovsky prove the regular two-dimensional case, which includes the classical two-variable form.\n\n**Full solution or refutation.**\n\nThe source asks for arbitrary n, beyond the verified two-dimensional theorem.\n\n**What remains.**\n\nProve the representation for all dimensions or find a counterexample.\n\n**Sources checked.**\n\n- F. Lucas, J. Madden, D. Schaub and M. Spivakovsky, Approximate roots of a valuation and the Pierce-Birkhoff Conjecture, arXiv:1003.1188 (2010). (primary): https://arxiv.org/abs/1003.1188\n Evidence used: The abstract states that it proves the conjecture for arbitrary regular two-dimensional rings.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1182, "problem_number": "ALG-022", "title": "Serre's Positivity Conjecture", "statement": "If $R$ is a regular local ring and $P, Q$ are prime ideals with intersecting dimensions satisfying a certain condition, is the intersection multiplicity positive?", "background": "Serre's positivity conjecture (part of Serre's multiplicity conjectures) concerns intersection multiplicities in commutative algebra. It states that if $R$ is a commutative regular local ring and $P, Q$ are prime ideals with $\\dim(R/P) + \\dim(R/Q) = \\dim(R)$, then the intersection multiplicity $\\chi(R/P, R/Q) > 0$. The conjecture was proven by Gabber, Paul Roberts, and others in the 1980s for rings containing a field, but remains open in mixed characteristic (characteristic 0 with positive characteristic residue field). The problem connects to algebraic K-theory and has applications to intersection theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** General Serre intersection positivity remains open; it must not be conflated with Gabber's non-negativity theorem.\n\n**Verified partial progress.**\n\n- Positivity holds in equicharacteristic and unramified cases.\n- Skalit proved it for formal power-series rings over complete two-dimensional regular local rings.\n\n**Full solution or refutation.**\n\nNo theorem covering arbitrary regular local rings in the source's generality was verified.\n\n**What remains.**\n\nResolve the remaining ramified mixed-characteristic cases.\n\n**Sources checked.**\n\n- A. Skalit, Positivity of Intersection Multiplicity Over a Two-Dimensional Base, arXiv:1510.05146 (2015). (primary): https://arxiv.org/abs/1510.05146\n Evidence used: The abstract gives the two-dimensional-base positivity theorem.\n- P. C. Roberts, Recent developments on Serre's multiplicity conjectures: Gabber's proof of the nonnegativity conjecture, L'Enseignement Math. 44 (1998), 305--324. (authoritative_secondary): https://www.e-periodica.ch/cntmng?pid=ens-001%3A1998%3A44%3A%3A183\n Evidence used: Distinguishes Gabber's non-negativity result from the still-open positivity assertion.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1183, "problem_number": "NT-029", "title": "Artin's Conjecture on Primitive Roots", "statement": "For how many prime numbers $p$ is a given integer $a$ (not $\\pm 1$ or a perfect square) a primitive root modulo $p$?", "background": "Artin's conjecture on primitive roots states that any integer $a$ that is neither $-1$, $\\pm 1$, nor a perfect square is a primitive root modulo infinitely many primes, and gives a conjectured density for such primes. For example, it predicts that 2 is a primitive root for approximately 37.4% of all primes. Under the assumption of the generalized Riemann hypothesis, Hooley proved the conjecture in 1967. However, the unconditional case remains open. The conjecture has important implications for the distribution of generators in finite fields and connects to class field theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Artin's positive-density prediction remains unproved unconditionally for any prescribed admissible integer a, but Hooley proved the full asymptotic under GRH and Heath-Brown obtained strong unconditional collective results.\n\n**Verified partial progress.**\n\n- Hooley proved the conjectured asymptotic under GRH for relevant Kummer-field zeta functions.\n- Heath-Brown proved unconditionally that at least one of 2, 3, or 5 is a primitive root modulo infinitely many primes.\n\n**Full solution or refutation.**\n\nConditional and collective results do not settle the conjecture for each fixed admissible a.\n\n**What remains.**\n\nProve unconditionally the explicit positive-density asymptotic for every fixed integer a that is neither -1 nor a square.\n\n**Sources checked.**\n\n- C. Hooley, On Artin's conjecture, Journal fuer die reine und angewandte Mathematik 225 (1967), 209-220. (primary): https://eudml.org/doc/150785\n Evidence used: Conditional proof under GRH.\n- D. R. Heath-Brown, Artin's conjecture for primitive roots, Quarterly Journal of Mathematics 37 (1986), 27-38. (primary): https://doi.org/10.1093/qmath/37.1.27\n Evidence used: Unconditional theorem that at least one of a small fixed set satisfies the infinitude assertion.\n\n**Review notes.** The record redundantly excludes 1 twice and does not mention a=0; the standard admissibility condition is used only for status interpretation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 267, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1184, "problem_number": "NT-030", "title": "The abc Conjecture", "statement": "For coprime integers $a, b, c$ with $a + b = c$, is $c$ usually not much larger than the product of distinct primes dividing $abc$?", "background": "The abc conjecture, proposed by Oesterlé and Masser in 1985, is one of the most important open problems in number theory. It states that for any $\\epsilon > 0$, there are only finitely many triples of coprime positive integers $(a,b,c)$ with $a + b = c$ such that $c > \\text{rad}(abc)^{1+\\epsilon}$, where $\\text{rad}(n)$ is the product of distinct prime factors of $n$. If true, it would imply Fermat's Last Theorem, Mordell's conjecture (already proven), and many other results. Shinichi Mochizuki claimed a proof in 2012 using inter-universal Teichmüller theory, but the mathematical community has not reached consensus on its validity.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The standard quantified abc conjecture remains open in prevailing mathematical usage. Mochizuki's published IUT papers claim a proof, while Scholze and Stix identify a decisive gap; no broad consensus accepting the proof was verified.\n\n**Verified partial progress.**\n\n- A 2025 theorem establishes the abc bound for an almost-all family, not for every triple.\n- Mochizuki's IUT papers constitute a published proof claim, but the central inference remains disputed.\n\n**Full solution or refutation.**\n\nPublication of the IUT papers has not produced a generally accepted resolution of abc.\n\n**What remains.**\n\nGive a broadly verified proof of the precise epsilon-quantified statement or refute it with a violating family.\n\n**Sources checked.**\n\n- P. Scholze and J. Stix, Why abc is still a conjecture (2018 report). (primary): https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf\n Evidence used: Detailed primary critique locating a gap in the key inequality.\n- S. Mochizuki, Inter-universal Teichmuller theory IV: log-volume computations and set-theoretic foundations, PRIMS 57 (2021). (primary): https://doi.org/10.4171/PRIMS/57-1-4\n Evidence used: Published primary source of the disputed proof claim.\n- S. V. Konyagin, F. Luca and I. E. Shparlinski, The abc conjecture is true almost always, arXiv:2505.13991 (2025). (primary): https://arxiv.org/abs/2505.13991\n Evidence used: Proves a density-type partial result while retaining a distinction from the universal conjecture.\n\n**Review notes.** The displayed record is qualitative and not literally falsifiable because 'usually' and 'not much larger' are undefined; the background's quantified formulation was used for triage.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 892, "favorite_count": 76, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1185, "problem_number": "GEO-010", "title": "The Shephard's Problem", "statement": "Can the unit ball in $\\mathbb{R}^n$ be illuminated by fewer than $2^n$ directions?", "background": "Shephard's problem, a variant of the illumination problem, asks how many directions are needed to illuminate the entire boundary of the unit ball in $n$-dimensional space. A direction illuminates a boundary point if moving in that direction from the point leads outside the ball. It is known that $2^n$ directions suffice (by considering all combinations of positive/negative coordinate directions), but whether fewer suffice is unknown for $n \\geq 3$. The problem connects to convex geometry, discrete geometry, and optimization. Even the three-dimensional case ($n = 3$) remains open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The row conflates the Euclidean unit ball, arbitrary norm balls, and Hadwiger's arbitrary-convex-body illumination conjecture; these readings have different answers.\n\n**Verified partial progress.**\n\n- For the Euclidean unit ball in dimension n at least two, n+1 regular-simplex directions illuminate the boundary, hence fewer than 2^n suffice.\n- For arbitrary norm unit balls, the cube is a counterexample to the word 'fewer' because its illumination number is exactly 2^n.\n- The intended Hadwiger conjecture asks for at most 2^n directions for every convex body, with equality exactly for parallelepipeds, and remains open in dimensions at least three.\n\n**Full solution or refutation.**\n\nThe literal Euclidean reading is affirmative for n at least two, the arbitrary-norm reading is false, and the intended general convex-body conjecture is open.\n\n**What remains.**\n\nReplace the statement with a precise choice of convex body class, illumination convention, dimension range, and whether the desired bound is strict; also correct the problem name.\n\n**Sources checked.**\n\n- Gabor Damasdi and Domotor Palvolgyi, Hadwiger's Illumination Conjecture and pseudolines, Journal of the European Mathematical Society 28 (2026), 2637-2657. (primary): https://ems.press/content/serial-article-files/52513?nt=1\n Evidence used: States the standard inward illumination definition, the at-most-2^n conjecture, the parallelepiped equality clause, and that the conjecture is open generally.\n- Andrii Arman, Alexander Bondarenko, and Andriy Prymak, On Hadwiger's covering problem in small dimensions, Canadian Mathematical Bulletin 68 (2025), 1239-1250. (primary): https://doi.org/10.4153/S0008439525000384\n Evidence used: Provides current small-dimensional upper bounds and formulates the classical illumination/covering conjecture for arbitrary convex bodies.\n\n**Review notes.** The background reverses the standard illumination direction: an illuminating ray must enter the body's interior, not move outside it. The n=1 Euclidean ball also needs exactly two directions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1187, "problem_number": "ALG-023", "title": "The Andrews-Curtis Conjecture", "statement": "Can every balanced presentation of the trivial group be transformed into a trivial presentation by a sequence of Nielsen transformations and conjugations?", "background": "Proposed in 1965 by James Andrews and Morton Curtis, this conjecture addresses the problem of simplifying group presentations. A balanced presentation has the same number of generators and relators. The question asks whether any such presentation of the trivial group can be reduced to the obvious trivial presentation through elementary operations (Nielsen transformations on relators and conjugating relators). Despite extensive computational searches, no counterexample has been found, but the general case remains unresolved. This problem connects combinatorial group theory, topology (via the Whitehead conjecture), and algorithmic complexity.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Andrews--Curtis conjecture remains unresolved, though bounded two-generator presentations and equivalent/restricted formulations have been studied.\n\n**Verified partial progress.**\n\n- Miasnikov--Myasnikov verify all balanced trivial two-generator presentations with total relator length at most 12.\n- Ivanov proves equivalence with a cyclic-conjugation version and separates a false stricter unstabilized variant.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for every balanced trivial presentation was verified.\n\n**What remains.**\n\nResolve the original unrestricted conjecture.\n\n**Sources checked.**\n\n- A. D. Miasnikov and A. G. Myasnikov, Balanced presentations of the trivial group on two generators and the Andrews-Curtis conjecture, arXiv:math/0304305 (2003). (primary): https://arxiv.org/abs/math/0304305\n Evidence used: The abstract verifies the bounded total-relator-length two-generator family.\n- S. V. Ivanov, On conjectures of Andrews and Curtis, arXiv:1606.08196 (2016). (primary): https://arxiv.org/abs/1606.08196\n Evidence used: The abstract states the cyclic-version equivalence and the outcome for a restrictive variant.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 412, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1188, "problem_number": "ALG-024", "title": "The Bounded Burnside Problem", "statement": "For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2, 5)$ finite?", "background": "The Burnside problem, posed in 1902, asks whether a finitely generated group in which every element has finite order must itself be finite. The bounded version restricts to groups where all elements have order dividing a fixed $n$. Major breakthroughs came when Novikov and Adian (1968) proved $B(m,n)$ is infinite for odd $n \\geq 4381$ and $m \\geq 2$, and Zel'manov earned a Fields Medal (1994) for proving finiteness when $n$ is a prime power. The case $B(2,5)$ remains a famous open problem. The group $B(2,3)$ is known to be finite with 27 elements.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The source concerns free Burnside groups, not the restricted Burnside problem. Many exponents are settled, but the stated classification remains open and B(2,5) is unresolved.\n\n**Verified partial progress.**\n\n- B(m,2), B(m,3), B(m,4), and B(m,6) are finite for all m.\n- Novikov--Adian and later work prove infinitude for large exponents; the restricted Burnside problem was separately solved by Zelmanov.\n\n**Full solution or refutation.**\n\nZelmanov's theorem on finite quotients does not imply finiteness of the free group B(2,5).\n\n**What remains.**\n\nDetermine finiteness of B(2,5) and complete the exponent classification.\n\n**Sources checked.**\n\n- J. Lehnert, thesis, Universite Paris-Sud 11, chapter on Burnside origamis (accessed 2026-08-17). (authoritative_secondary): https://citeseerx.ist.psu.edu/document?doi=a8cb7e76f1943ea130c885a89bbcb7e3a9824cf2&repid=rep1&type=pdf\n Evidence used: It explicitly states that B(2,5) remains open and summarizes the classical finite/infinite cases.\n- E. I. Zel'manov, A solution of the restricted Burnside problem for 2-groups, Math. USSR-Sb. 72 (1992), 543--565. (primary): https://www.mathnet.ru/eng/sm1311\n Evidence used: The abstract identifies the result as the restricted, not free, Burnside problem.\n\n**Review notes.** No source alteration; terminology distinction recorded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 687, "favorite_count": 52, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1189, "problem_number": "ALG-025", "title": "The Guralnick-Thompson Conjecture", "statement": "What are the composition factors of finite groups appearing in genus-0 systems?", "background": "This conjecture, proposed by Robert Guralnick and John Thompson, concerns the classification of finite groups that can act on Riemann surfaces of genus 0. The conjecture provides a list of simple groups that can appear as composition factors of such groups. The problem connects group theory with algebraic geometry and the theory of automorphisms of Riemann surfaces. Genus-0 systems are particularly important in the classification of finite simple groups and their actions on low-genus surfaces.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The Guralnick--Thompson fixed-genus composition-factor conjecture was completed: apart from the specified alternating/cyclic families, only finitely many simple composition factors occur.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nFrohardt--Magaard completed the final case; subsequent expositions explicitly record completion of the genus-zero/fixed-genus composition-factor problem.\n\n**What remains.**\n\nThe source's finiteness/classification question is settled at its stated conjectural level; explicit enumerations are a separate refinement.\n\n**Sources checked.**\n\n- G. Malle and B. H. Matzat (eds.), Applying the Classification, chapter 10, discussion of the Guralnick--Thompson conjecture. (authoritative_secondary): https://homepages.math.uic.edu/~smiths/book.pdf\n Evidence used: It states that the conjecture was completed by Frohardt--Magaard.\n- M. Fried, Variables separated polynomials, survey manuscript (accessed 2026-08-17). (authoritative_secondary): https://www.math.uci.edu/~mfried/paplist-cov/varseppolynoms.pdf\n Evidence used: It says the cited results completed the proof for composition factors of fixed-genus covers.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 298, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1190, "problem_number": "ALG-026", "title": "The Herzog-Schönheim Conjecture", "statement": "If a finite system of left cosets of subgroups of a group $G$ partitions $G$, must some two subgroups have the same index?", "background": "Proposed independently by Marcel Herzog and Jochanan Schönheim in 1974, this conjecture states that if finitely many left cosets of subgroups partition a group, then at least two of the subgroups must have the same finite index. This problem arises naturally in the study of group coverings and has connections to number theory through systems of covering congruences. Despite much research, the conjecture remains open in general, though it has been verified for various special cases including abelian groups and certain classes of finite groups.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Herzog--Schonheim conjecture remains open for general group coset partitions, while important families and index patterns are settled.\n\n**Verified partial progress.**\n\n- The classical integer/cyclic covering case is settled.\n- Recent work surveys special group cases and uses Stallings techniques for additional finite-index subgroup results.\n\n**Full solution or refutation.**\n\nNo universal proof of repeated subgroup index in every group partition was verified.\n\n**What remains.**\n\nResolve the general coset-partition statement.\n\n**Sources checked.**\n\n- J. Delgado and E. Ventura, Transactions on Combinatorics 11 (2022), 181--235, section 4.2.2. (authoritative_secondary): https://upcommons.upc.edu/server/api/core/bitstreams/526b1c5e-a31f-4978-a393-c1bda8ad8600/content\n Evidence used: The survey calls Herzog--Schonheim an open problem and explains known progress.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 321, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1191, "problem_number": "ALG-027", "title": "The Inverse Galois Problem", "statement": "Is every finite group the Galois group of some Galois extension of $\\mathbb{Q}$?", "background": "The inverse Galois problem is one of the central open problems in Galois theory. While classical Galois theory establishes a correspondence between field extensions and groups, the inverse problem asks whether every finite group can be realized as the Galois group of an extension of the rational numbers. The problem was implicit in work of Hilbert and has been explicitly studied since the late 19th century. It has been solved affirmatively for many classes of groups (symmetric groups, alternating groups, many sporadic simple groups), but the general case remains open. The problem connects algebra, number theory, and algebraic geometry through its connection to dessins d'enfants and modular curves.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The inverse Galois problem over Q remains open generally, although every finite solvable group is realized and many nonsolvable families are known.\n\n**Verified partial progress.**\n\n- Schmidt--Wingberg give a complete proof of Shafarevich's theorem that every finite solvable group occurs over Q.\n- The question is affirmative over C(t), unlike the universal Q-case.\n\n**Full solution or refutation.**\n\nNo theorem that realizes every finite group over Q was verified.\n\n**What remains.**\n\nRealize every finite nonsolvable group over Q or exhibit an obstruction.\n\n**Sources checked.**\n\n- A. Schmidt and K. Wingberg, Safarevic's theorem on solvable groups as Galois groups, arXiv:math/9809211 (1998). (primary): https://arxiv.org/abs/math/9809211\n Evidence used: The abstract supplies the complete solvable-group theorem.\n- Encyclopedia of Mathematics, Galois theory, inverse problem of. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Galois_theory%2C_inverse_problem_of\n Evidence used: It describes the Q problem and the solvable-group theorem, with references.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1192, "problem_number": "ALG-028", "title": "The Isomorphism Problem for Coxeter Groups", "statement": "Is there an algorithm to determine whether two Coxeter groups given by presentations are isomorphic?", "background": "Coxeter groups are fundamental objects in geometric group theory, generated by reflections with certain relations. They include the symmetry groups of regular polytopes and tessellations. The isomorphism problem asks whether there exists an algorithmic procedure to decide if two Coxeter groups, given by their Coxeter diagrams or presentations, are isomorphic. While the problem is solved for finite and affine Coxeter groups, the general case for arbitrary Coxeter groups remains open. This problem is related to the broader isomorphism problem for groups and has applications in geometry and topology.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The abstract isomorphism problem for Coxeter groups remains open, but it has been reduced to reflection-preserving isomorphisms and has further structural advances.\n\n**Verified partial progress.**\n\n- Howlett--Muhlerr reduce the problem to the reflection-preserving version.\n- Rego--Schwer develop a 2024 framework for isomorphisms between finite-rank Coxeter systems.\n\n**Full solution or refutation.**\n\nNo decision algorithm for arbitrary input Coxeter presentations was verified.\n\n**What remains.**\n\nSolve the reflection-preserving version or otherwise give a general isomorphism algorithm.\n\n**Sources checked.**\n\n- B. Muhlherr, The isomorphism problem for Coxeter groups, arXiv:math/0506572 (2005). (primary): https://arxiv.org/abs/math/0506572\n Evidence used: The abstract reports the reduction to the reflection-preserving version and remaining developments.\n- Y. Santos Rego and P. Schwer, The galaxy of Coxeter groups, Journal of Algebra 656 (2024), 406--445, doi:10.1016/j.jalgebra.2023.12.006. (primary): https://www.sciencedirect.com/science/article/pii/S0021869323006178\n Evidence used: The article describes its finite-rank framework, solved subclasses, and remaining reductions for the isomorphism problem.\n\n**Review notes.** No source alteration; algorithmic input convention may need specialist confirmation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 367, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1193, "problem_number": "ALG-029", "title": "Infinitude of Leinster Groups", "statement": "Are there infinitely many Leinster groups?", "background": "A Leinster group is a finite group whose order equals the sum of the orders of its proper normal subgroups. Named after Tom Leinster who studied them in 1996, only two examples are currently known: the cyclic group of order 6 and a group of order 12. The question of whether infinitely many such groups exist remains open. This problem connects group theory with number theory through the properties of divisors and has connections to the study of perfect numbers (where the sum of proper divisors equals the number itself).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Infinitude of Leinster groups remains open, with classifications/obstructions for several structural families and hundreds of examples known.\n\n**Verified partial progress.**\n\n- Leinster classifies abelian examples as cyclic groups of perfect order.\n- Demedts and collaborators report roughly 400 noncyclic examples while noting that an infinite family had not been found.\n\n**Full solution or refutation.**\n\nNo infinite family of pairwise nonisomorphic Leinster groups was verified.\n\n**What remains.**\n\nConstruct an infinite family or prove only finitely many exist.\n\n**Sources checked.**\n\n- T. Leinster, Perfect numbers and groups, arXiv:math/0104012 (2001). (primary): https://arxiv.org/abs/math/0104012\n Evidence used: The abstract establishes the basic finite-group generalization and abelian context.\n- T. Demedts et al., Perfect numbers and finite groups, preprint (accessed 2026-08-17). (primary): https://cage.ugent.be/~tdemedts/preprints/leinster.pdf\n Evidence used: It explicitly reports that the search for infinite families had not succeeded and gives the example count.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1194, "problem_number": "ALG-030", "title": "Existence of Generalized Moonshine", "statement": "Does generalized moonshine exist for all elements of the Monster group?", "background": "Monstrous moonshine, discovered in the 1970s and proven by Borcherds (Fields Medal 1998), reveals a surprising connection between the Monster group (the largest sporadic simple group) and modular functions. Generalized moonshine extends this to other elements of the Monster group, asking whether similar connections exist for all group elements. Conway and Norton conjectured explicit relationships, and significant progress has been made, but the complete generalized moonshine remains unproven. This connects finite groups, modular forms, string theory, and vertex operator algebras in profound ways.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Carnahan completed Norton's Generalized Moonshine Conjecture for the Monster; the dataset's claim that complete generalized moonshine remains unproved is stale.\n\n**Verified partial progress.**\n\n- Carnahan constructs a monstrous Lie algebra with projective centralizer action for every Monster element and proves the Fricke twisted-twining functions are Hauptmoduln.\n- The later orbifold-duality work treats non-Fricke Lie algebras and refines the scalar ambiguity to roots of unity for nonconstant functions.\n\n**Full solution or refutation.**\n\nGeneralized Moonshine IV explicitly resolves Norton's conjecture, with remaining cases obtained from modular compatibility after the Fricke cases are established.\n\n**What remains.**\n\nFurther refinements and extensions of moonshine remain active, but not the existence claim for Monster elements in Norton's conjecture.\n\n**Sources checked.**\n\n- Scott Carnahan, Generalized Moonshine IV: Monstrous Lie algebras, arXiv:1208.6254 (2012; revised 2016). (primary): https://arxiv.org/abs/1208.6254\n Evidence used: The abstract says the paper constructs the structure for each Monster element and resolves Norton's Generalized Moonshine Conjecture.\n- Scott Carnahan, 51 constructions of the Moonshine module, Communications in Number Theory and Physics 12 (2018), 305-334, arXiv:1707.02954. (primary): https://arxiv.org/abs/1707.02954\n Evidence used: Proves the non-Fricke Lie-algebra structure and the root-of-unity refinement of the generalized-moonshine scalar ambiguity.\n\n**Review notes.** The source statement is compressed: the full conjecture concerns functions attached to commuting pairs, not merely one element.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 543, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1195, "problem_number": "ALG-031", "title": "Finiteness of Finitely Presented Periodic Groups", "statement": "Is every finitely presented periodic group finite?", "background": "A periodic group (or torsion group) is one in which every element has finite order. The question of whether a finitely presented periodic group must be finite was a major open problem for much of the 20th century. The restricted Burnside problem, solved by Zel'manov, showed that finitely generated groups where all elements have bounded order must be finite. However, the general case without the bounded exponent assumption remains open. This problem connects group theory, geometric group theory, and algorithmic questions about group presentations.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No finitely presented infinite torsion group is known; equivalently, it remains open whether every finitely presented periodic group is finite.\n\n**Verified partial progress.**\n\n- Infinite finitely generated torsion groups are classical, and Osin gave a short modern proof of Golod's theorem.\n- Schesler constructed finitely generated infinite torsion groups that are residually finite simple, but these do not solve finite presentability.\n\n**Full solution or refutation.**\n\nNo general finiteness theorem or finitely presented infinite counterexample was verified.\n\n**What remains.**\n\nConstruct a finitely presented infinite torsion group or prove that finite presentability forces a torsion group to be finite.\n\n**Sources checked.**\n\n- I. A. Ivanov-Pogodaev and A. Ya. Kanel-Belov, Finitely presented nilsemigroups: complexes with the property of uniform ellipticity, Izvestiya: Mathematics 84 (2020), 1140-1169. (primary): https://www.mathnet.ru/links/63460d5ff426dd8772ad636b68cd111d/im8978_eng.pdf\n Evidence used: Explicitly calls existence of a finitely presented infinite torsion group a basic open problem and notes that known infinite torsion groups are infinitely presented.\n- Denis Osin, A simple construction of finitely generated infinite torsion groups, L'Enseignement Mathematique 71 (2025), 207-214. (primary): https://ems.press/journals/lem/articles/14297800\n Evidence used: Provides a new proof of existence of an infinite finitely generated torsion group, illustrating that finite generation is known while finite presentation remains missing.\n- Eduard Schesler, Finitely generated infinite torsion groups that are residually finite simple, Advances in Mathematics 479 (2025), 110441. (primary): https://doi.org/10.1016/j.aim.2025.110441\n Evidence used: Strengthens known finitely generated examples without claiming finite presentation.\n\n**Review notes.** The background misstates Zelmanov: the restricted Burnside theorem does not imply that every finitely generated bounded-exponent group is finite.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 456, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1196, "problem_number": "ALG-032", "title": "The Surjunctivity Conjecture", "statement": "Is every group surjunctive?", "background": "A group is surjunctive if every injective cellular automaton over that group is also surjective. Equivalently, every injective endomorphism of the shift space is surjective. This property was introduced by Gottschalk in 1973 and connects symbolic dynamics, cellular automata theory, and group theory. Gromov and Weiss proved that all sofic groups are surjunctive, and since all amenable groups are sofic, this includes a large class. However, the general question of whether all groups are surjunctive remains open and is equivalent to asking whether all groups are sofic.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Gottschalk's conjecture that every group is surjunctive remains open; the August 2026 discovery of nonsofic groups does not by itself produce a non-surjunctive group.\n\n**Verified partial progress.**\n\n- Gromov and Weiss proved that every sofic group is surjunctive.\n- Recent work extends surjunctivity to broader actions over sofic groups and exhibits a surjunctive non-cosofic invariant random subgroup, but not a non-surjunctive group.\n\n**Full solution or refutation.**\n\nNo non-surjunctive group or proof of universal surjunctivity was verified.\n\n**What remains.**\n\nDetermine surjunctivity for the newly constructed nonsofic groups or construct an injective nonsurjective cellular automaton over some group.\n\n**Sources checked.**\n\n- Lewis Bowen and Michael Chapman, Surjunctivity does not characterize cosoficity of invariant random subgroups, arXiv:2511.06586 (2025). (primary): https://arxiv.org/abs/2511.06586\n Evidence used: States Gottschalk's conjecture as open, records sofic implies surjunctive, and disproves the reverse analogue only in the broader IRS setting.\n- Expansive actions with specification of sofic groups, strong topological Markov property, and surjunctivity, Journal of Mathematical Analysis and Applications 537 (2024), 128240. (primary): https://www.sciencedirect.com/science/article/pii/S0022123624000648\n Evidence used: Describes universal surjunctivity as Gottschalk's open conjecture and recovers the Gromov-Weiss theorem for sofic groups.\n- OpenAI, Nonsofic groups exist, Chapter 3 of Ten Advances in Mathematics and Theoretical Computer Science (2026). (primary): https://cdn.openai.com/pdf/ten-proofs-oai.pdf\n Evidence used: Supplies a nonsofic group but makes no claim that it is non-surjunctive; hence it removes the prior route of proving surjunctivity solely by universal soficity.\n\n**Review notes.** The background's claimed equivalence with universal soficity is false; only soficity implies surjunctivity. Confidence is medium because nonsofic groups were discovered only two weeks before this check.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 389, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1197, "problem_number": "ALG-033", "title": "The Sofic Groups Conjecture", "statement": "Is every discrete countable group sofic?", "background": "A group is sofic if it can be approximated by finite symmetric groups in a precise sense. The concept was introduced by Gromov and Weiss around 1999 and has become central in modern group theory. All known groups are sofic: amenable groups, residually finite groups, linear groups, and many others. The soficity of all groups would have profound consequences for many conjectures in group theory, operator algebras, and ergodic theory. Notable implications include Connes' embedding conjecture (now known to be false via quantum complexity theory) and Gottschalk's surjunctivity conjecture. Despite the breadth of known sofic groups, the general question remains one of the deepest open problems in infinite group theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** OpenAI's August 2026 manuscript constructs a countable nonsofic group, the unit group of the binary Leavitt algebra, giving a negative answer to the stored question.\n\n**Verified partial progress.**\n\n- The primary proof shows that the unit group of L_{F_2}(1,2) is nonsofic and countable.\n- Fournier-Facio immediately derived a distinct finitely presented torsion-free nonsofic example, and Kun-Thom produced nonsofic wreath products.\n\n**Full solution or refutation.**\n\nThe universal assertion is false: the unit group of the binary Leavitt algebra over F_2 is an explicit countable discrete nonsofic group.\n\n**What remains.**\n\nStudy the structure, closure phenomena, hyperlinearity, and dynamical properties of the new nonsofic examples; these are separate from the refuted universal statement.\n\n**Sources checked.**\n\n- OpenAI, Nonsofic groups exist, Chapter 3 of Ten Advances in Mathematics and Theoretical Computer Science (1 August 2026, updated manuscript). (primary): https://cdn.openai.com/pdf/ten-proofs-oai.pdf\n Evidence used: Theorem 1.1 proves the unit group of L_{F_2}(1,2) is not sofic and the preceding text proves this group is countable.\n- Francesco Fournier-Facio, A torsion-free non-sofic group, arXiv:2608.02025 (2026). (primary): https://arxiv.org/abs/2608.02025\n Evidence used: Independently confirms the OpenAI breakthrough as existence of a nonsofic group and proves a finitely presented torsion-free variant from the same criterion.\n- Gabor Kun and Andreas Thom, Nonsofic wreath products of residually finite groups, arXiv:2608.06222 (2026). (primary): https://arxiv.org/abs/2608.06222\n Evidence used: Explicitly builds on the first nonsofic-group construction and obtains further families.\n\n**Review notes.** The dataset background became stale in August 2026. The primary paper also cautions that nonsoficity does not decide hyperlinearity.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 612, "favorite_count": 48, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1198, "problem_number": "ALG-034", "title": "Arthur's Conjectures", "statement": "What is the structure of the discrete spectrum of automorphic forms on reductive groups?", "background": "Proposed by James Arthur in the 1980s, these conjectures describe the decomposition of the space of automorphic forms into irreducible representations. They provide a framework for understanding the discrete spectrum in terms of endoscopic groups and Arthur packets. The conjectures connect representation theory, harmonic analysis, and number theory, generalizing results of Langlands. Major progress has been made, including Arthur's proof for classical groups (2013), but the full program for all reductive groups remains incomplete. These conjectures are central to the Langlands program and have applications to trace formulas and L-functions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Arthur's packet and multiplicity description is established for major classical families, including symplectic, special orthogonal, and quasi-split unitary groups, but not for arbitrary reductive groups.\n\n**Verified partial progress.**\n\n- Arthur established endoscopic classification and an explicit multiplicity formula for quasi-split symplectic and special orthogonal groups over number fields.\n- Mok established the endoscopic classification of the discrete automorphic spectrum for quasi-split unitary groups.\n\n**Full solution or refutation.**\n\nThere is no all-reductive-groups solution; the strongest general theorems cover broad classical families.\n\n**What remains.**\n\nConstruct and verify local/global Arthur packets, transfer, and multiplicity formulas for arbitrary connected reductive groups, including outstanding exceptional and inner-form cases.\n\n**Sources checked.**\n\n- James Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, AMS Colloquium Publications 61 (2013). (primary): https://bookstore.ams.org/COLL/61\n Evidence used: Establishes endoscopic classification and a multiplicity formula for automorphic discrete spectra of orthogonal and symplectic groups.\n- Chung Pang Mok, Endoscopic Classification of Representations of Quasi-Split Unitary Groups, Memoirs of the AMS 235 (2015), no. 1108. (primary): https://bookstore.ams.org/memo-235-1108\n Evidence used: Establishes the discrete-spectrum endoscopic classification for quasi-split unitary groups over number fields.\n\n**Review notes.** The stored statement is an under-specified umbrella question rather than a single formal conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 478, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1199, "problem_number": "ALG-035", "title": "Dade's Conjecture", "statement": "Is there a relationship between the numbers of irreducible characters in blocks of a finite group and its local subgroups?", "background": "Proposed by Everett Dade in 1992, this conjecture concerns the modular representation theory of finite groups. It relates the number of irreducible characters of a given defect in a block of a finite group to corresponding numbers in blocks of certain local subgroups (normalizers of p-subgroups). The conjecture is part of a broader program to reduce questions about representations of finite groups to questions about p-groups and their normalizers. It has been verified for many classes of groups but remains open in general. Dade's conjecture refines earlier conjectures by Alperin and McKay.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Dade's character-counting conjectures remain unproved in general, but modern reduction theorems reduce major forms to inductive conditions for quasisimple groups and establish important classes.\n\n**Verified partial progress.**\n\n- Späth proved that the Character Triple Conjecture for quasisimple groups implies Dade's Projective Conjecture for all finite groups and verified some quasisimple classes.\n- Rossi proved the Character Triple Conjecture for p-solvable groups and then a self-reduction of the general Character Triple Conjecture to quasisimple groups.\n\n**Full solution or refutation.**\n\nNo proof for all finite groups was verified; the main advance is a powerful reduction to quasisimple covering groups plus verified special families.\n\n**What remains.**\n\nVerify the relevant character-triple/inductive conditions for every quasisimple covering group and account for the precise variant of Dade's conjecture desired.\n\n**Sources checked.**\n\n- Britta Spath, A reduction theorem for Dade's projective conjecture, Journal of the European Mathematical Society 19 (2017), 1071-1126. (primary): https://ems.press/journals/jems/articles/14667\n Evidence used: Introduces the Character Triple Conjecture and proves its quasisimple verification implies Dade's Projective Conjecture for all finite groups.\n- Damiano Rossi, Character Triple Conjecture for p-solvable groups, Journal of Algebra 595 (2022), 165-193. (primary): https://doi.org/10.1016/j.jalgebra.2021.12.018\n Evidence used: Proves the Character Triple Conjecture for p-solvable groups.\n- Damiano Rossi, A reduction theorem for the Character Triple Conjecture, arXiv:2402.10632 (2024). (primary): https://arxiv.org/abs/2402.10632\n Evidence used: Proves the Character Triple Conjecture for all finite groups assuming it for all quasisimple groups.\n\n**Review notes.** The displayed source question merely asks whether a relationship exists; the actual conjecture is a precise alternating-sum formula and has several variants.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1200, "problem_number": "ALG-036", "title": "The Demazure Conjecture", "statement": "Can representations of semisimple algebraic groups be characterized over the integers?", "background": "Proposed by Michel Demazure in the 1970s, this conjecture concerns the existence of certain integral structures on representations of algebraic groups. It asks whether irreducible representations of semisimple algebraic groups over fields of positive characteristic can be deformed to characteristic zero while preserving integrality properties. The conjecture has applications to geometric representation theory and the theory of quantum groups. Partial results have been obtained for special cases, but the general conjecture remains open. The problem connects algebraic groups, representation theory, and arithmetic geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The intended historical Demazure conjecture on Schubert-variety vanishing and character formulas in arbitrary characteristic was proved, but the stored statement and background do not formulate it accurately.\n\n**Verified partial progress.**\n\n- Lakshmibai-Musili-Seshadri established standard monomial theory and Demazure's conjecture for classical groups.\n- Characteristic-free results for general semisimple/reductive groups followed via Kempf vanishing, Frobenius splitting, and uniform standard monomial/path theory.\n\n**Full solution or refutation.**\n\nThe arbitrary-characteristic vanishing and Demazure character results anticipated in Demazure's 1974 work are established in the literature; the named historical conjecture is no longer open.\n\n**What remains.**\n\nRepair the dataset statement: the broad question about characterizing representations over the integers and deforming every modular irreducible is not the precise Demazure conjecture and cannot be marked solved without qualification.\n\n**Sources checked.**\n\n- A. Ramanathan, Equations defining Schubert varieties and Frobenius splittings of diagonals, Publications Mathematiques de l'IHES 65 (1987), 61-90. (primary): https://www.numdam.org/article/PMIHES_1987__65__61_0.pdf\n Evidence used: States Demazure's conjecture as arbitrary-characteristic cohomology vanishing and character formula and records the characteristic-free theorems for Schubert varieties.\n- V. Lakshmibai and C. S. Seshadri, Geometry of G/P-V, Journal of Algebra 100 (1986), 462-557. (primary): https://citeseerx.ist.psu.edu/document?doi=d2008112efe3875dc57e923d89a97fb6dd7113bc&repid=rep1&type=pdf\n Evidence used: The paper explicitly lists a proof of Demazure's conjecture among the consequences of its standard monomial theory in the classical scope.\n- Peter Littelmann, Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras, Journal of the American Mathematical Society 11 (1998), 551-567. (primary): https://doi.org/10.1090/S0894-0347-98-00268-9\n Evidence used: Gives a uniform standard monomial basis for all symmetrizable Kac-Moody algebras, completing the program beyond the earlier case-by-case classical results.\n- Venkatramani Lakshmibai et al., C. S. Seshadri (1932-2020), Notices of the AMS 68 (2021). (authoritative_secondary): https://jointmathematicsmeetings.org/journals/notices/202111/noti2383/noti2383.html\n Evidence used: Participant history explains the gap in Demazure's proof, the corrected classical work, and Littelmann's completion of standard monomial theory for Kac-Moody groups.\n\n**Review notes.** Medium confidence attaches to mapping the malformed dataset item to the standard named conjecture, not to the historical resolution itself.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 289, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1203, "problem_number": "GEO-012", "title": "The Spherical Bernstein Problem", "statement": "What is the classification of complete minimal hypersurfaces in spheres of all dimensions?", "background": "This is a generalization of Bernstein's problem (solved by 1968) which asked whether the only minimal graph over all of Euclidean space is a hyperplane. The spherical version asks for the classification of complete minimal hypersurfaces in the sphere $S^{n+1}$. While progress has been made in specific dimensions, a complete classification for all dimensions remains open. The problem connects differential geometry, minimal surface theory, and geometric analysis, with applications to general relativity and materials science.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The stored all-hypersurface classification request is not Chern's spherical Bernstein problem and lacks hypotheses needed for a definite status.\n\n**Verified partial progress.**\n\n- Chern's named spherical Bernstein problem concerns embedded minimal hypersurfaces topologically equal to a sphere, not every complete minimal hypersurface.\n- Wang, Wang, and Zhou constructed a non-equatorial embedded minimal hypersphere in S^4 in 2026, giving a negative instance of the named rigidity question in that ambient sphere.\n- Many non-equatorial minimal hypersurfaces of other topologies and isoparametric types are known, so the unrestricted class is far broader still.\n\n**Full solution or refutation.**\n\nNo classification of all complete minimal hypersurfaces in spheres of every dimension exists; the actual named problem has essential embeddedness and topology hypotheses and has counterexamples in known dimensions.\n\n**What remains.**\n\nRewrite the row to state either Chern's precise embedded-minimal-hypersphere rigidity question dimension by dimension or a bounded classification problem with specified regularity, topology, and equivalence.\n\n**Sources checked.**\n\n- S.-S. Chern, Differential geometry: its past and its future, Proceedings of the International Congress of Mathematicians, Nice 1970, Problem VI. (primary): https://people.math.harvard.edu/~hirolee/pdfs/2014-fall-230a-icm1970-chern-differential-geometry.pdf\n Evidence used: Supplies the original spherical Bernstein question and its missing topological/embedded scope.\n- Tongrui Wang, Zhichao Wang, and Xin Zhou, Equivariant min-max theory and the spherical Bernstein problem in S^4, arXiv:2602.03984 (2026). (primary): https://arxiv.org/abs/2602.03984\n Evidence used: Constructs an embedded non-equatorial minimal hypersphere in the unit 4-sphere.\n\n**Review notes.** Completeness is not the missing hypothesis that identifies the named problem; embeddedness and topological type are essential.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 387, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1204, "problem_number": "GEO-013", "title": "The Carathéodory Conjecture", "statement": "Does every convex, closed, twice-differentiable surface in $\\mathbb{R}^3$ have at least two umbilical points?", "background": "Proposed by Constantin Carathéodory in the 1920s, this conjecture concerns umbilical points on convex surfaces—points where the principal curvatures are equal. The conjecture states that any smooth closed convex surface in 3-dimensional Euclidean space must have at least two such points. A sphere has infinitely many umbilical points (every point is umbilical), while an ellipsoid generically has exactly 4. The conjecture has been proven for surfaces of revolution and certain other special cases, but remains open in general. It connects differential geometry, topology (via the Poincaré-Hopf theorem), and dynamical systems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** A revised preprint claims the Caratheodory conclusion for C^{3+alpha} convex surfaces, but it does not cover the exact C^2 regularity in the stored statement.\n\n**Verified partial progress.**\n\n- Guilfoyle and Klingenberg claim a proof that every C^{3+alpha}-smooth closed convex surface has more than one umbilic.\n- Classical and modern results cover analytic, symmetric, and other special classes, but no verified reduction from C^2 to C^{3+alpha} was found.\n\n**Full solution or refutation.**\n\nThe exact twice-differentiable statement cannot be marked solved by a theorem assuming C^{3+alpha} regularity.\n\n**What remains.**\n\nEstablish the two-umbilic conclusion at C^2 regularity, or prove a valid approximation/compactness principle that transfers the stronger-regularity theorem without losing umbilics.\n\n**Sources checked.**\n\n- Brendan Guilfoyle and Wilhelm Klingenberg, Proof of the Caratheodory Conjecture, arXiv:0808.0851 (2008; revised 2024). (primary): https://arxiv.org/abs/0808.0851\n Evidence used: The abstract explicitly claims the theorem for C^{3+alpha}-smooth surfaces, a stronger regularity assumption than the stored C^2 hypothesis.\n- Brendan Guilfoyle and Wilhelm Klingenberg, The three obdurate conjectures of differential geometry, arXiv:2502.11716 (2025). (primary): https://arxiv.org/abs/2502.11716\n Evidence used: Provides the authors' current treatment of Caratheodory's conjecture and its symmetry-based context.\n\n**Review notes.** Confidence is medium because the claimed smoother proof has a long, contested publication history; the regularity mismatch itself is clear.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 456, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1205, "problem_number": "GEO-014", "title": "The Cartan-Hadamard Conjecture", "statement": "Does the isoperimetric inequality hold for Cartan-Hadamard manifolds?", "background": "The classical isoperimetric inequality states that among all regions of fixed volume in Euclidean space, a ball has the smallest surface area. The Cartan-Hadamard conjecture asks whether this extends to Cartan-Hadamard manifolds—complete, simply connected Riemannian manifolds of nonpositive sectional curvature. The conjecture has been proven in dimensions 2, 3, and 4, and for many special classes of manifolds, but remains open in higher dimensions. This problem is central to geometric analysis and has connections to optimal transport theory, general relativity, and the study of black hole thermodynamics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Euclidean-comparison Cartan-Hadamard conjecture is proved in dimensions two through four and in further restricted classes, but remains open for arbitrary manifolds in dimensions above four.\n\n**Verified partial progress.**\n\n- Classical work proves the unrestricted conjecture in dimensions 2, 3, and 4.\n- Ghomi's 2026 nullity theorem proves the conjecture in all dimensions for Cartan-Hadamard manifolds with curvature-nullity index at least n-3.\n- A separate August 2026 preprint proves a dimension-five case under sufficiently pinched negative curvature.\n\n**Full solution or refutation.**\n\nNo unrestricted proof for dimension at least five was verified.\n\n**What remains.**\n\nProve the sharp Euclidean isoperimetric comparison for every bounded region in every higher-dimensional Cartan-Hadamard manifold, without nullity, pinching, or smallness assumptions.\n\n**Sources checked.**\n\n- European Commission CORDIS, Sharp Isoperimetric Inequalities - Old and New, current project report (2025/2026). (authoritative_secondary): https://cordis.europa.eu/project/id/101001677/reporting\n Evidence used: Explicitly states that the Cartan-Hadamard conjecture remains open in dimensions higher than four.\n- Mohammad Ghomi, Isoperimetric and total curvature inequalities in Cartan-Hadamard manifolds with nullity, arXiv:2605.24638 (2026). (primary): https://arxiv.org/abs/2605.24638\n Evidence used: Proves the conjecture for the restricted class with nullity index at least n-3.\n- Mohammad Ghomi, Total curvature and isoperimetric inequalities in pinched Cartan-Hadamard manifolds, arXiv:2608.12020 (2026). (primary): https://arxiv.org/abs/2608.12020\n Evidence used: Proves a dimension-five case under sufficiently pinched curvature, demonstrating new progress without claiming the unrestricted theorem.\n\n**Review notes.** The row should distinguish the ordinary Euclidean-comparison conjecture from the stronger model-space comparison under a negative upper curvature bound.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 523, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1206, "problem_number": "GEO-015", "title": "Chern's Affine Conjecture", "statement": "Does the Euler characteristic of a compact affine manifold vanish?", "background": "Proposed by Shiing-Shen Chern, this conjecture states that every closed affine manifold (a manifold with an atlas whose transition functions are affine transformations) has Euler characteristic zero. An affine structure is stronger than a smooth structure but weaker than a Riemannian structure. The conjecture has been verified for many classes of affine manifolds, and recent work has made substantial progress, but a complete proof remains elusive. The problem connects differential geometry, topology, and the theory of geometric structures on manifolds.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Chern's Euler-characteristic conjecture for compact affine manifolds remains open generally, while the complete and special-affine cases are theorems.\n\n**Verified partial progress.**\n\n- Kostant and Sullivan proved vanishing for complete compact affine manifolds.\n- Klingler proved vanishing for compact special affine manifolds admitting a parallel volume form and derived additional corollaries.\n\n**Full solution or refutation.**\n\nNo generally accepted proof for arbitrary compact affine manifolds was verified.\n\n**What remains.**\n\nRemove completeness, parallel-volume, holonomy, amenability, and related special hypotheses in the general compact affine case.\n\n**Sources checked.**\n\n- Bertram Kostant and Dennis Sullivan, The Euler characteristic of an affine space form is zero, Bulletin of the American Mathematical Society 81 (1975), 937-938. (primary): https://doi.org/10.1090/S0002-9904-1975-13891-0\n Evidence used: Proves the complete affine case.\n- Bruno Klingler, Chern's conjecture for special affine manifolds, Annals of Mathematics 186 (2017), 69-95. (primary): https://annals.math.princeton.edu/2017/186-1/p02\n Evidence used: Proves vanishing when the compact affine manifold admits a parallel volume form and explicitly treats the general conjecture as open.\n- Jianquan Ge, Proof of Chern's conjecture on affine manifolds, arXiv:2002.03105 (2020). (primary): https://arxiv.org/abs/2002.03105\n Evidence used: Records an announced general proof that has not displaced the current open status; included to make the status conflict explicit rather than ignored.\n\n**Review notes.** Several announced proofs are not treated as accepted resolutions by current mathematical sources; a specialist should audit them before any solved migration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 398, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1207, "problem_number": "GEO-016", "title": "Chern's Conjecture for Hypersurfaces in Spheres", "statement": "What minimal hypersurfaces in spheres have constant mean curvature?", "background": "This is actually a family of related conjectures proposed by Shiing-Shen Chern concerning the classification of minimal and constant mean curvature hypersurfaces embedded in spheres. One version asks whether the only minimal hypersurface in $S^{n+1}$ with constant scalar curvature is the totally geodesic $S^n$. These conjectures connect minimal surface theory, the study of isoparametric hypersurfaces, and geometric analysis. Partial results have been obtained, but the general conjectures remain open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The displayed question is tautological because every minimal hypersurface has constant mean curvature zero; the actual Chern conjecture concerns constant scalar curvature and isoparametricity.\n\n**Verified partial progress.**\n\n- The standard strong Chern conjecture asks whether every closed minimal hypersurface in a sphere with constant scalar curvature is isoparametric.\n- Miyaoka proves the conjectural conclusion for substantial proper-Dupin cases.\n- Recent S^5 results prove isoparametricity under added constant Gauss-Kronecker curvature or related hypotheses.\n\n**Full solution or refutation.**\n\nTaken literally, the constant-mean-curvature condition selects all minimal hypersurfaces and supplies no classification; the intended constant-scalar-curvature conjecture remains open generally.\n\n**What remains.**\n\nReplace mean curvature by the intended invariant, state closedness/immersedness assumptions, and choose between the discrete scalar-curvature-value conjecture and the stronger isoparametric classification conjecture.\n\n**Sources checked.**\n\n- Reiko Miyaoka, Chern's Conjecture in the Dupin case, arXiv:2504.02621 (2025). (primary): https://arxiv.org/abs/2504.02621\n Evidence used: States the standard constant-scalar-curvature/isoparametric formulation and proves it in specified proper-Dupin cases.\n- Qintao Deng and Yunjia Kou, Closed minimal hypersurfaces in S^5(1) with constant scalar and Gauss-Kronecker curvatures, arXiv:2607.06588 (2026). (primary): https://arxiv.org/abs/2607.06588\n Evidence used: Proves an S^5 classification with an additional constant Gauss-Kronecker curvature hypothesis and presents it as support for, not resolution of, Chern's conjecture.\n\n**Review notes.** The background's suggestion that only a totally geodesic sphere might have constant scalar curvature is disproved by classical Clifford and other isoparametric examples.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 367, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1208, "problem_number": "GEO-017", "title": "The Closed Curve Problem", "statement": "What are necessary and sufficient conditions for an integral curve defined by two periodic functions to be closed?", "background": "This problem asks for explicit, computable conditions to determine when a curve defined parametrically by integrating two periodic functions with the same period will close up. The question arises naturally in dynamical systems, Hamiltonian mechanics, and the study of periodic orbits. While special cases are understood, general necessary and sufficient conditions that can be readily checked remain unknown. The problem connects analysis, differential geometry, and dynamical systems theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The standard closed-curve problem concerns periodic curvature and torsion, which the stored statement omits; explicit conditions remain open although a formal Frenet-monodromy criterion and special-family solutions are known.\n\n**Verified partial progress.**\n\n- Closure can be expressed formally by requiring the Euclidean-motion monodromy of the Frenet system over one period to satisfy the appropriate position and frame conditions.\n- Bamba and Ogata solve the intended problem for a rotationally symmetric family and extend the construction to several singular curve settings.\n- Arroyo, Garay, and Mencia give explicit sufficient and non-closure results in the planar one-curvature-function case.\n\n**Full solution or refutation.**\n\nNo general explicit usable characterization of periodic curvature-torsion pairs was verified, while the literal phrase 'integral curve defined by two periodic functions' is undefined without specifying their role.\n\n**What remains.**\n\nRestore curvature and torsion to the statement and specify whether closure means position only, tangent-periodic smooth closure, or full Frenet-frame periodicity; then seek explicit conditions beyond solving the monodromy ODE itself.\n\n**Sources checked.**\n\n- K. Bamba and Y. Ogata, Closed space curves with singularities, generated by periodic curvature and torsion, Journal of Geometry 117 (2026), article 12. (primary): https://doi.org/10.1007/s00022-025-00792-3\n Evidence used: Calls the general periodic-curvature-and-torsion problem open and solves a rotationally symmetric special case.\n- Josu Arroyo, Oscar J. Garay, and Jose J. Mencia, When is a periodic function the curvature of a closed plane curve?, American Mathematical Monthly 115 (2008), 405-414. (primary): https://doi.org/10.1080/00029890.2008.11920543\n Evidence used: Develops explicit closure and non-closure criteria for the planar periodic-curvature special case.\n- Eric W. Weisstein, Closed Curve Problem, MathWorld. (maintained_tracker): https://mathworld.wolfram.com/ClosedCurveProblem.html\n Evidence used: Records the standard formulation explicitly in terms of periodic curvature kappa(s) and torsion tau(s), both absent from the stored row.\n\n**Review notes.** The source background's coordinatewise-integration description would instead have an elementary zero-mean closure condition and is not the standard Efimov-Fenchel problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 289, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1209, "problem_number": "GEO-018", "title": "The Filling Area Conjecture", "statement": "Does a hemisphere have minimum area among shortcut-free surfaces with a given boundary length?", "background": "This conjecture in systolic geometry states that among all surfaces in Euclidean space whose boundary is a closed curve of given length and which contain no shortcuts (the surface distance between boundary points equals the Euclidean distance), the hemisphere has minimal area. The problem was proposed by Gromov and connects differential geometry, geometric measure theory, and the calculus of variations. It has applications to the study of minimal surfaces and optimal shapes in physics and materials science.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Gromov proved hemispherical minimality for disk fillings, but the arbitrary-topology orientable filling-area conjecture remains open.\n\n**Verified partial progress.**\n\n- Gromov proved the conjectured bound when the isometric filling is homeomorphic to a disk.\n- Briggs and Wells prove that any isometric filling of the length-2pi circle has area at least 1.36pi, versus the hemisphere's area 2pi.\n- Zust obtains a second-order lower bound near an isometric hemisphere for a broad class of rectifiable metric-current competitors.\n\n**Full solution or refutation.**\n\nThe full theorem for compact orientable Riemannian fillings of arbitrary topology has not been proved.\n\n**What remains.**\n\nRaise the arbitrary-topology lower bound to the sharp hemispherical value and characterize equality, without disk-topology or near-hemisphere hypotheses.\n\n**Sources checked.**\n\n- Mikhail Gromov, Filling Riemannian manifolds, Journal of Differential Geometry 18 (1983), 1-147. (primary): https://www.ihes.fr/~gromov/wp/site/wp-content/uploads/2018/08/fillingRiemannianManifolds.pdf\n Evidence used: Introduces the filling framework and proves the relevant disk case.\n- Joseph Briggs and Chris Wells, A discrete view of Gromov's filling area conjecture, arXiv:2602.17859 (2026). (primary): https://arxiv.org/abs/2602.17859\n Evidence used: States that the full arbitrary-topology conjecture remains unresolved and proves the 1.36pi universal lower bound.\n- Roger Zust, The Riemannian hemisphere is almost calibrated in the injective hull of its boundary, arXiv:2104.04498 (2021). (primary): https://arxiv.org/abs/2104.04498\n Evidence used: Proves a quantitative second-order lower estimate near the hemisphere for broad current-theoretic competitors.\n\n**Review notes.** The correct no-shortcuts condition is d_M(x,y)=d_C(x,y) for boundary points and the intrinsic metric circle C; it is not equality with Euclidean chord distance.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 334, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1210, "problem_number": "GEO-019", "title": "The Hopf Conjectures", "statement": "What is the relationship between curvature and Euler characteristic for even-dimensional Riemannian manifolds?", "background": "Heinz Hopf proposed several conjectures relating the sign of sectional curvature to the Euler characteristic and other topological invariants of closed Riemannian manifolds. The most famous asks whether a closed even-dimensional manifold with positive (or negative) sectional curvature must have positive Euler characteristic. The conjectures have been resolved in dimension 2 (Gauss-Bonnet) and partially in dimension 4, but remain open in higher dimensions. These problems are central to understanding the interplay between curvature and topology in Riemannian geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The extracted sentence is a broad topic question rather than a single conjecture; the intended positive- and negative-curvature Hopf sign conjectures remain open in general.\n\n**Verified partial progress.**\n\n- Kennard–Mouillé–Nienhaus prove a positive-Euler result under positive second intermediate Ricci curvature and torus symmetry.\n- Huang proves the correctly signed nonpositive-curvature statement for a symplectic-hyperbolic class.\n\n**Full solution or refutation.**\n\nNo unrestricted proof was verified for either principal Hopf sign conjecture.\n\n**What remains.**\n\nChoose a precise formulation and resolve it. For negative curvature on a closed 2m-manifold the intended sign is (-1)^m chi(M)>0, not always chi(M)>0.\n\n**Sources checked.**\n\n- Lee Kennard, Lawrence Mouillé, and Jan Nienhaus, On Hopf's conjecture and positive second intermediate Ricci curvature, arXiv:2507.16936 (2025). (primary): https://arxiv.org/abs/2507.16936\n Evidence used: States the positive-sectional-curvature Hopf conjecture and proves a result only under additional curvature and symmetry assumptions.\n- Teng Huang, On Euler number of symplectic hyperbolic manifold, Advances in Mathematics 428 (2023), 109445. (primary): https://doi.org/10.1016/j.aim.2023.109445\n Evidence used: States the signed negative/nonpositive Hopf conjectures and proves a special symplectic-hyperbolic case.\n\n**Review notes.** Formulation defect preserved: the statement does not identify one proposition, and the background gives the wrong unsigned description for negative curvature.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 43, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1211, "problem_number": "GEO-020", "title": "The Osserman Conjecture", "statement": "Is every Osserman manifold either flat or locally isometric to a rank-one symmetric space?", "background": "An Osserman manifold is a Riemannian manifold where the eigenvalues of the Jacobi operator are constant on the unit sphere bundle at each point. Robert Osserman conjectured that such manifolds must be either flat or locally isometric to a rank-one symmetric space (spheres, projective spaces, or hyperbolic spaces). The conjecture has been proven in dimensions up to 4 and for many special cases, but remains open in higher dimensions. The problem connects differential geometry, spectral theory, and the theory of symmetric spaces.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The standard global Osserman conjecture is proved in all dimensions except the exceptional dimension 16; the dataset's claim that it is known only through dimension 4 is obsolete.\n\n**Verified partial progress.**\n\n- Nikolayevsky proved the global conjecture in dimensions other than 8 and 16.\n- Nikolayevsky separately settled dimension 8.\n\n**Full solution or refutation.**\n\nThe cited classification results leave dimension 16 unresolved.\n\n**What remains.**\n\nSettle the 16-dimensional global case, and distinguish the global from pointwise Osserman definition in the record.\n\n**Sources checked.**\n\n- Yuri Nikolayevsky, Osserman Conjecture in dimension n≠8,16, Mathematische Annalen 331 (2005), 505–522. (primary): https://arxiv.org/abs/math/0204258\n Evidence used: Proves the global conjecture outside dimensions 8 and 16 and states the corresponding pointwise qualifications.\n- Yuri Nikolayevsky, Osserman manifolds of dimension 8, Manuscripta Mathematica 115 (2004), 31–53. (primary): https://arxiv.org/abs/math/0310387\n Evidence used: Settles the eight-dimensional case.\n- Yuri Nikolayevsky, Conformally Osserman manifolds, Pacific Journal of Mathematics 245 (2010), 315–337. (primary): https://doi.org/10.2140/pjm.2010.245.315\n Evidence used: Records dimension 16 as the remaining exceptional dimension for the original conjecture.\n\n**Review notes.** The background conflates pointwise and global Osserman conditions; no silent repair was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 412, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1212, "problem_number": "GEO-021", "title": "Yau's Conjecture on First Eigenvalues", "statement": "Is the first eigenvalue of the Laplace-Beltrami operator on a minimal hypersurface in $S^{n+1}$ equal to $n$?", "background": "Proposed by Shing-Tung Yau, this conjecture states that for any closed embedded minimal hypersurface in the $(n+1)$-dimensional sphere $S^{n+1}$, the first nonzero eigenvalue of the Laplace-Beltrami operator equals $n$. This would provide a sharp spectral characterization of minimal hypersurfaces in spheres. The conjecture has been verified for several important cases including geodesic spheres, Clifford tori, and certain other symmetric examples. The problem connects spectral geometry, minimal surface theory, and PDEs on manifolds.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A 2025 preprint claims a full proof of Yau's conjecture, but current 2026 peer-reviewed and dissertation sources still treat the general equality as open; independent acceptance of the claimed proof was not verified.\n\n**Verified partial progress.**\n\n- Choi–Wang and later authors provide nonsharp lower bounds for the first nonzero eigenvalue.\n- Jiménez–Tapia Chinchay–Zhou prove a 2026 explicit lower bound and conditional equality results.\n\n**Full solution or refutation.**\n\nZeng's arXiv:2508.06123 asserts the full theorem, but the evidence checked does not establish community verification or acceptance.\n\n**What remains.**\n\nExpert-audit Zeng's argument or locate a published validation, correction, or refutation; absent that, resolve the standard closed embedded case.\n\n**Sources checked.**\n\n- Lingzhong Zeng, The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecture, arXiv:2508.06123 (2025). (primary): https://arxiv.org/abs/2508.06123\n Evidence used: Explicitly claims a proof of the full conjecture.\n- Asun Jiménez, Carlos Tapia Chinchay, and Detang Zhou, A lower bound for the first eigenvalue of a minimal hypersurface in the sphere, Revista Matemática Iberoamericana 42 (2026), 261–278. (primary): https://doi.org/10.4171/RMI/1587\n Evidence used: Treats Yau's equality as a conjecture and proves lower and conditional results rather than a full solution.\n- Harvard DASH, Geometric Variational Problems for Minimal Hypersurfaces, doctoral thesis (2026). (authoritative_secondary): https://dash.harvard.edu/server/api/core/bitstreams/840a4aff-ed6e-4cf2-9dd6-c8466ba71c47/content\n Evidence used: A current specialist dissertation that describes the general statement as open.\n\n**Review notes.** The literal statement omits the standard closed and embedded hypotheses and does not say first nonzero eigenvalue; the background supplies the intended reading.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 478, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1213, "problem_number": "GEO-022", "title": "The Hadwiger Covering Conjecture", "statement": "Can every $n$-dimensional convex body be covered by at most $2^n$ smaller homothetic copies?", "background": "Proposed by Hugo Hadwiger in 1957, this conjecture states that any $n$-dimensional convex body can be covered by at most $2^n$ positive homothetic (scaled and translated) copies of itself with smaller ratio. The conjecture is known to be true for $n = 1$ (trivial) and $n = 2$ (proven), but remains open for $n \\geq 3$. The problem connects discrete geometry, convex geometry, and combinatorics. It is related to the illumination problem and has connections to coding theory and sphere packing.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Hadwiger's 2^n covering conjecture remains open in general and is unresolved already in dimension 3.\n\n**Verified partial progress.**\n\n- Prymak proved that 14 smaller positive homothets suffice in dimension 3, compared with the conjectural 8.\n- Arman–Bondarenko–Prymak improved upper bounds in several small dimensions.\n\n**Full solution or refutation.**\n\nNo proof of the universal 2^n bound was verified.\n\n**What remains.**\n\nProve the conjectural bound for all convex bodies; in dimension 3, close the current gap between 8 and 14.\n\n**Sources checked.**\n\n- Andriy Prymak, A New Bound for Hadwiger's Covering Problem in E^3, SIAM Journal on Discrete Mathematics 37 (2023). (primary): https://doi.org/10.1137/22M1490314\n Evidence used: Proves a 14-copy upper bound in dimension 3, leaving the conjectural value 8 open.\n- Andrii Arman, Andriy Bondarenko, and Andriy Prymak, On Hadwiger's covering problem in small dimensions, Canadian Mathematical Bulletin 68 (2025), 1239–1250. (primary): https://doi.org/10.4153/S0008439525000384\n Evidence used: Calls H_n=2^n the conjecture and proves improved upper bounds rather than the general claim.\n\n**Review notes.** Exact statement retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 523, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1214, "problem_number": "GEO-023", "title": "The Happy Ending Problem", "statement": "What is the minimum number of points in the plane needed to guarantee a convex $n$-gon?", "background": "The Happy Ending problem, named by Paul Erdős because it led to the marriage of Esther Klein and George Szekeres, asks for $g(n)$—the smallest number such that any set of $g(n)$ points in general position contains $n$ points forming a convex $n$-gon. It's known that $2^{n-2} + 1 \\leq g(n) \\leq \\binom{2n-4}{n-2} + 1$. The exact value is known only for $n \\leq 6$. Erdős offered $500 for a proof that $g(n) = 2^{n-2} + 1$. The problem is central to combinatorial geometry and Ramsey theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Erdős–Szekeres conjecture g(n)=2^(n-2)+1 remains open; exact values are known only through n=6.\n\n**Verified partial progress.**\n\n- Suk proved the asymptotically tight exponential order g(n)=2^(n+o(n)).\n- Holmsen–Mojarrad–Pach–Tardos improved the error term in the upper bound.\n- Baek–Balko proved a 2026 relaxed variant for decomposable point sets.\n\n**Full solution or refutation.**\n\nNo exact formula for general n was verified.\n\n**What remains.**\n\nDetermine g(n) for n≥7, especially prove or refute g(n)=2^(n-2)+1.\n\n**Sources checked.**\n\n- Andrew Suk, On the Erdős–Szekeres convex polygon problem, Journal of the American Mathematical Society 30 (2017), 1047–1053. (primary): https://doi.org/10.1090/jams/869\n Evidence used: Proves g(n)=2^(n+o(n)), not the exact conjecture.\n- Andreas F. Holmsen, Hossein Nassajian Mojarrad, János Pach, and Gábor Tardos, Two extensions of the Erdős–Szekeres problem, JEMS 22 (2020), 3981–3995. (primary): https://ems.press/journals/jems/articles/17088\n Evidence used: Gives a stronger quantitative upper bound while leaving the exact value open.\n- Jineon Baek and Martin Balko, The Erdős–Szekeres conjecture revisited, Journal of Combinatorial Theory, Series A 222 (2026), 106195. (primary): https://doi.org/10.1016/j.jcta.2026.106195\n Evidence used: Current primary paper explicitly treating the classical conjecture as open and resolving a relaxed variant.\n\n**Review notes.** The exact sentence omits the necessary general-position hypothesis; the background supplies it. Its binomial upper bound is no longer the best asymptotic bound.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 612, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1215, "problem_number": "GEO-024", "title": "The Heilbronn Triangle Problem", "statement": "What is the largest minimum area of a triangle determined by $n$ points in a unit square?", "background": "Proposed by Hans Heilbronn in 1908, this problem asks how to place $n$ points in a unit square to maximize the smallest area of any triangle they determine. Heilbronn originally conjectured the maximum was $O(1/n^2)$, but this was disproven—the actual order is between $\\Omega(\\log n / n^2)$ and $O(1/n^{8/7-\\epsilon})$. Finding the exact asymptotic remains open. The problem connects discrete geometry, extremal combinatorics, and has applications to numerical integration and computational geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The asymptotic order of the Heilbronn triangle function remains open; the current upper bound is n^(-7/6+o(1)).\n\n**Verified partial progress.**\n\n- Cohen–Pohoata–Zakharov improved the upper bound to n^(-7/6+o(1)).\n- Zakharov identified a 7/6 barrier for the present incidence method.\n\n**Full solution or refutation.**\n\nThe known upper and lower regimes remain far apart, and no exact asymptotic was verified.\n\n**What remains.**\n\nDetermine the correct exponent and logarithmic factors, requiring ideas beyond the current 7/6 incidence barrier.\n\n**Sources checked.**\n\n- Alex Cohen, Cosmin Pohoata, and Dmitrii Zakharov, Lower bounds for incidences, Inventiones Mathematicae (2025). (primary): https://doi.org/10.1007/s00222-025-01331-2\n Evidence used: Gives the n^(-7/6+o(1)) upper bound for the Heilbronn triangle problem.\n- Dmitrii Zakharov, Small triangles, Journal of the London Mathematical Society (2026). (primary): https://doi.org/10.1112/jlms.70447\n Evidence used: Treats the exact order as open and establishes a barrier for the current method.\n\n**Review notes.** The background's proposal year 1908 is defective because 1908 was Heilbronn's birth year; its quoted upper bound is obsolete.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 445, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1216, "problem_number": "GEO-025", "title": "Kalai's $3^d$ Conjecture", "statement": "Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?", "background": "Proposed by Gil Kalai in 1989, this conjecture states that any centrally symmetric convex polytope in $d$ dimensions must have at least $3^d$ faces (including the polytope itself and the empty set). The bound is tight, achieved by the $d$-dimensional cube which has exactly $3^d$ faces. The conjecture has been verified for $d \\leq 4$ and for various special classes of polytopes. The problem connects combinatorics, convex geometry, and polytope theory, with applications to optimization and computational geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Kalai's 3^d face conjecture remains open for general centrally symmetric d-polytopes.\n\n**Verified partial progress.**\n\n- The bound is known for centrally symmetric simplicial polytopes and, by duality, simple polytopes.\n- It is known in dimensions at most 4 and for additional coordinate-reflection-symmetric classes.\n\n**Full solution or refutation.**\n\nNo proof for arbitrary centrally symmetric polytopes in general dimension was verified.\n\n**What remains.**\n\nProve the 3^d lower bound for general centrally symmetric polytopes in dimensions d≥5.\n\n**Sources checked.**\n\n- Isabella Novik, A tale of centrally symmetric polytopes and spheres, arXiv:1711.09310. (authoritative_secondary): https://arxiv.org/abs/1711.09310\n Evidence used: Specialist survey stating the conjecture and summarizing the simplicial, simple, and low-dimensional cases.\n- Gregory Chambers and Elia Portnoy, A note on Kalai's 3^d Conjecture, arXiv:2211.09215 (2022). (primary): https://arxiv.org/abs/2211.09215\n Evidence used: Proves the conjectured bound for a symmetry-restricted class, not all centrally symmetric polytopes.\n\n**Review notes.** The background's face convention is inconsistent: a cube has 3^d nonempty faces including the polytope, but 3^d+1 faces if both the empty face and polytope are counted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 378, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1217, "problem_number": "GEO-026", "title": "The Unit Distance Problem", "statement": "What is the maximum number of unit distances determined by $n$ points in the plane?", "background": "This problem, posed by Erdős in 1946, asks for the maximum number of pairs of points at distance exactly 1 in a set of $n$ points in the Euclidean plane. The best known construction gives $\\Omega(n^{4/3})$ unit distances, while the best upper bound is $O(n^{4/3})$. Determining the exact asymptotic (and whether the exponent is exactly $4/3$) remains open. The problem connects extremal combinatorics, incidence geometry, and has applications to facility location and wireless network design. Erdős offered prizes for progress on this problem.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The planar unit-distance extremal function remains unknown; a 2026 superlinear lower bound is still far below the O(n^(4/3)) upper bound.\n\n**Verified partial progress.**\n\n- Chen constructed infinitely many point sets with at least n^(1+delta) unit distances for a fixed delta>0, refuting the narrower n^(1+o(1)) conjecture.\n- Sawin made the exponent explicit with delta=0.014114... for arbitrarily large sizes.\n- The classical O(n^(4/3)) upper bound remains the best general upper exponent.\n\n**Full solution or refutation.**\n\nThe 2026 result refutes a proposed near-linear asymptotic, but does not determine the maximum number of unit distances.\n\n**What remains.**\n\nClose the gap between the n^(1.014114...) lower construction and the O(n^(4/3)) upper bound and determine the true growth rate.\n\n**Sources checked.**\n\n- Lijie Chen, Planar Point Sets with Many Unit Distances (2026). (primary): https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf\n Evidence used: Provides a fixed-power superlinear construction and refutes the narrower n^(1+o(1)) conjecture.\n- Will Sawin, An explicit lower bound for the unit distance problem, arXiv:2605.20579 (2026). (primary): https://arxiv.org/abs/2605.20579\n Evidence used: Makes the new lower-bound exponent explicit.\n- Erdős's Unit Distance Problem and Rigidity, SoCG 2026, LIPIcs 367, Article 83. (primary): https://drops.dagstuhl.de/storage/00lipics/lipics-vol367-socg2026/html/LIPIcs.SoCG.2026.83/LIPIcs.SoCG.2026.83.html\n Evidence used: Current proceedings source recording the continuing O(n^(4/3)) upper bound.\n\n**Review notes.** The background falsely states a matching Omega(n^(4/3)) construction; no such construction is known.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 42, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1219, "problem_number": "GEO-028", "title": "Ehrhart's Volume Conjecture", "statement": "Does a convex body in $\\mathbb{R}^n$ with one interior lattice point at its center of mass have volume at most $(n+1)^n/n!$?", "background": "Proposed by Eugène Ehrhart, this conjecture concerns lattice polytopes—convex bodies whose vertices have integer coordinates. It states that if a convex body in $n$ dimensions contains exactly one lattice point in its interior (which is its center of mass), then its volume cannot exceed $(n+1)^n/n!$, the volume of a regular simplex. The conjecture has been verified for $n \\leq 3$ and for many special cases. The problem connects discrete geometry, convex geometry, and number theory, with applications to integer programming and combinatorial optimization.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A proof of Ehrhart's sharp volume inequality in every dimension was released in August 2026 with a public Lean certificate, and an independent follow-up treats the inequality as proved and establishes the equality case.\n\n**Verified partial progress.**\n\n- The new proof covers arbitrary convex bodies satisfying the one-interior-lattice-point and barycenter hypotheses.\n- Liu's follow-up proves uniqueness in the equality case.\n\n**Full solution or refutation.**\n\nChapter 8 of Ten Advances proves vol(K)≤(n+1)^n/n!, matching the exact inequality asked in the record.\n\n**What remains.**\n\nExpert-audit the very recent manuscript and formal certificate; the extracted yes/no problem is resolved if they withstand review.\n\n**Sources checked.**\n\n- OpenAI, Ten Advances in Mathematics and Theoretical Computer Science, Chapter 8, updated 2026-08-06. (primary): https://cdn.openai.com/pdf/ten-proofs-oai.pdf\n Evidence used: Presents a proof of the sharp Ehrhart volume inequality in all dimensions.\n- OpenAI, ten-proofs formalization repository (2026). (primary): https://github.com/openai/ten-proofs\n Evidence used: Provides the public Lean certificate EhrhartVolumeInequality.lean.\n- Jihao Liu, The equality case of Ehrhart's volume conjecture, arXiv:2608.01040 (2026). (primary): https://arxiv.org/abs/2608.01040\n Evidence used: Independently identifies the inequality as newly proved and determines the equality case.\n\n**Review notes.** Very recent result, so confidence is medium pending conventional expert review. The background incorrectly narrows the standard conjecture to lattice polytopes.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 389, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1220, "problem_number": "ALG-039", "title": "The Cherlin-Zilber Conjecture", "statement": "Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?", "background": "Proposed by Gregory Cherlin and Boris Zilber in the 1970s, this conjecture connects model theory and group theory. It states that any infinite simple group whose first-order theory is stable must be isomorphic to a simple algebraic group defined over an algebraically closed field. The conjecture has been verified for many classes of groups and represents a deep connection between logic and algebra. It generalizes the classification of finite simple groups to model-theoretic contexts and has implications for the structure theory of stable groups.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The record omits the essential finite-Morley-rank hypothesis. The standard Cherlin-Zilber Algebraicity Conjecture for infinite simple groups of finite Morley rank remains open, but the broader stable-theory sentence is not the same conjecture.\n\n**Verified partial progress.**\n\n- The finite-Morley-rank conjecture is known in the even-type case and for simple groups of Morley rank 3, while odd and degenerate types remain difficult.\n- A 2024 paper proves corresponding Lie-ring algebraicity in characteristic zero and low-rank restrictions in other characteristics, while emphasizing that the group case remains open.\n\n**Full solution or refutation.**\n\nNo status is assigned to the malformed broad statement; the intended finite-Morley-rank conjecture is open.\n\n**What remains.**\n\nRestore 'infinite simple group of finite Morley rank' (and the intended meaning of algebraic group) before classifying the record as open; then continue the odd- and degenerate-type classification.\n\n**Sources checked.**\n\n- Katrin Tent, From the Cherlin-Zilber Conjecture via sharply 2-transitive groups to the Burnside problem, arXiv:2606.18207 (2026). (authoritative_secondary): https://arxiv.org/abs/2606.18207\n Evidence used: Current survey states the conjecture for simple groups of finite Morley rank and reviews it as unresolved.\n- Adrien Deloro and Joshua Wiscons, Simple Lie rings of Morley rank 4 (The Spanish Inquisition), Journal of Algebra 651 (2024), 243-280. (primary): https://doi.org/10.1016/j.jalgebra.2024.04.012\n Evidence used: Proves Lie-ring analogues and explicitly says the corresponding group case is notoriously open.\n\n**Review notes.** The source statement says merely stable and does not say infinite; the standard conjecture says infinite simple group of finite Morley rank.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 412, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1221, "problem_number": "ALG-040", "title": "The Generalized Star Height Problem", "statement": "Can all regular languages be expressed with generalized regular expressions of bounded star height?", "background": "This problem in formal language theory asks whether there exists a uniform bound on the nesting depth of Kleene star operations needed to express any regular language using generalized regular expressions (which allow complementation). While the ordinary star height problem (without complementation) was solved—showing unbounded star height is necessary—the generalized version remains open. The problem connects automata theory, formal languages, and computational complexity, with applications to pattern matching and compiler design.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The generalized star-height problem remains open; it is not even known whether a regular language of generalized star height greater than one exists.\n\n**Verified partial progress.**\n\n- Generalized star height zero is the decidable class of star-free languages, characterized by aperiodic syntactic monoids.\n- Subword-counting families and languages recognized by several Rees matrix semigroups are known to have generalized star height zero or at most one.\n\n**Full solution or refutation.**\n\nNo uniform bound theorem or language requiring generalized star height at least two was verified.\n\n**What remains.**\n\nProve height one suffices for every regular language, exhibit a language of generalized star height at least two, or otherwise settle boundedness and computability.\n\n**Sources checked.**\n\n- Thomas Place and Marc Zeitoun, Generic Results for Concatenation Hierarchies, survey/manuscript. (authoritative_secondary): https://www.labri.fr/perso/tplace/Files/TOCS18.pdf\n Evidence used: States that computing generalized star height remains open and that no regular language of generalized star height greater than one is known.\n- Thomas Bourne, Counting subwords and other results related to the generalised star-height problem for regular languages, PhD thesis, University of St Andrews (2017). (primary): https://research-repository.st-andrews.ac.uk/handle/10023/12024\n Evidence used: Records the open problem and proves height-zero/height-one bounds for concrete subword-counting and semigroup-recognized families.\n\n**Review notes.** The source's boundedness wording is compatible with the standard problem; the sharper fact is that height greater than one is not known to occur.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 334, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1222, "problem_number": "NT-031", "title": "Hilbert's Tenth Problem for Number Fields", "statement": "For which number fields is there an algorithm to determine solvability of Diophantine equations?", "background": "Hilbert's tenth problem asked for an algorithm to determine whether a Diophantine equation has integer solutions. Matiyasevich (building on work by Davis, Putnam, and Robinson) proved in 1970 that no such algorithm exists for the integers. The problem remains open for other rings, particularly number fields (finite extensions of the rationals). It has been solved negatively for some number fields and positively for others, but the general characterization is unknown. This connects logic, number theory, and computability theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hilbert's tenth problem is now proved undecidable over the ring of integers of every number field, but the exact statement says 'number fields' and does not distinguish K from O_K; over fields themselves, already over Q, the problem remains open.\n\n**Verified partial progress.**\n\n- Alpoege, Bhargava, Ho, and Shnidman proved that Z is Diophantine in O_K for every number field K.\n- Koymans and Pagano independently proved undecidability for every infinite ring finitely generated over Z.\n\n**Full solution or refutation.**\n\nThe ring-of-integers interpretation is fully resolved negatively; the field-valued interpretation is not.\n\n**What remains.**\n\nClarify the intended solution domain. If solutions are sought in K, solve the problem already for Q; if in O_K, update the record to solved negatively for all K.\n\n**Sources checked.**\n\n- L. Alpoege, M. Bhargava, W. Ho and A. Shnidman, Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field, Inventiones Mathematicae (2025). (primary): https://doi.org/10.1007/s00222-025-01392-3\n Evidence used: Corollary 1.2 gives a negative answer over O_K for every number field K.\n- P. Koymans and C. Pagano, Hilbert's tenth problem via additive combinatorics, arXiv:2412.01768v3 (2025). (primary): https://arxiv.org/abs/2412.01768\n Evidence used: Independent broader result for all infinite finitely generated Z-algebras.\n- S. Anscombe et al., A survey of local-global methods for Hilbert's Tenth Problem (2024). (authoritative_secondary): https://arxiv.org/abs/2309.14987\n Evidence used: Distinguishes the still-open number-field/field problem, especially Q, from ring-of-integers variants.\n\n**Review notes.** The source conflates a number field with its ring of integers; this is flagged rather than silently repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 523, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1224, "problem_number": "GEO-029", "title": "Borsuk's Conjecture", "statement": "Can every bounded set in $\\mathbb{R}^n$ be partitioned into $n+1$ sets of smaller diameter?", "background": "Proposed by Karol Borsuk in 1933, this conjecture asks whether every bounded set in $n$-dimensional Euclidean space can be partitioned into $n+1$ parts, each with diameter strictly smaller than the original set. The conjecture held for dimensions up to 3 until 1993, when Kahn and Kalai found a counterexample in dimension 1325. The smallest dimension for which the conjecture fails remains unknown (known to fail for $n \\geq 64$). This problem connects geometric combinatorics, high-dimensional geometry, and has inspired research into diameter-reducing partitions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The universal Borsuk conjecture was disproved by Kahn and Kalai; the smallest dimension of failure remains unknown.\n\n**Verified partial progress.**\n\n- Kahn–Kalai proved counterexamples exist in sufficiently high dimensions.\n- Jenrich–Brouwer gave an explicit counterexample in dimension 64.\n- A maintained 2026 tracker records Grinsztajn's certificate-based dimension-63 example, placing the first failing dimension between 4 and 63.\n\n**Full solution or refutation.**\n\nThe exact yes/no sentence is false because high-dimensional bounded sets requiring more than n+1 smaller-diameter parts are known.\n\n**What remains.**\n\nDetermine the least failing dimension and the exact or asymptotic Borsuk numbers b(n).\n\n**Sources checked.**\n\n- Jeff Kahn and Gil Kalai, A counterexample to Borsuk's conjecture, Bulletin of the American Mathematical Society 29 (1993), 60–62. (primary): https://doi.org/10.1090/S0273-0979-1993-00398-7\n Evidence used: Primary disproof of the universal conjecture.\n- Thomas Jenrich and Andries E. Brouwer, A 64-dimensional counterexample to Borsuk's conjecture, Electronic Journal of Combinatorics 21(4) (2014), P4.29. (primary): https://doi.org/10.37236/4069\n Evidence used: Provides an explicit 64-dimensional counterexample.\n- Borsuk's problem, Optimization Problems maintained tracker, updated 2026. (maintained_tracker): https://teorth.github.io/optimizationproblems/constants/28a.html\n Evidence used: Records the dimension-63 certificate-based construction and current 4≤d≤63 range for the first counterexample dimension.\n\n**Review notes.** The record background already mentions refutation but gives the older threshold n≥64; the exact universal statement is therefore not open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 523, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1225, "problem_number": "GEO-030", "title": "The Kissing Number Problem", "statement": "What is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in $n$ dimensions?", "background": "The kissing number $\\tau_n$ is the maximum number of non-overlapping unit spheres that can simultaneously touch a central unit sphere in $n$-dimensional Euclidean space. Known exactly only for dimensions 1, 2, 3, 4, 8, and 24, this problem has connections to sphere packing, coding theory, and lattice theory. The dimensions 8 and 24 are special due to exceptional lattices (E8 and Leech lattice). Determining kissing numbers in other dimensions, particularly dimensions 5, 6, 7, and general high dimensions, remains a major open problem in discrete geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Kissing numbers are known exactly only in dimensions 1, 2, 3, 4, 8, and 24; all-dimensional and most fixed-dimensional cases remain open.\n\n**Verified partial progress.**\n\n- A peer-reviewed 2026 paper confirms the current list of solved dimensions.\n- In dimension five, 40 spheres are achievable and the best proved upper bound is 44; four nonisometric 40-point configurations are now known.\n- Cohn and Li improved lower bounds in dimensions 17 through 21 to 5730, 7654, 11692, 19448, and 29768.\n\n**Full solution or refutation.**\n\nNo formula for the kissing number in arbitrary dimension, and no exact value in dimensions 5, 6, or 7, was verified.\n\n**What remains.**\n\nClose the lower and upper bounds in every dimension other than 1–4, 8, and 24, and determine the asymptotic growth rate.\n\n**Sources checked.**\n\n- H. Cohn and I. Rajagopal, Variations on Five-Dimensional Sphere Packings, Discrete & Computational Geometry (2026). (primary): https://doi.org/10.1007/s00454-026-00841-x\n Evidence used: The introduction states the exact solved-dimension list; Section 2 states the dimension-five range 40 to 44 and constructs another 40-point configuration.\n- H. Cohn and A. Li, Improved kissing numbers in seventeen through twenty-one dimensions, arXiv:2411.04916 (2024). (primary): https://arxiv.org/abs/2411.04916\n Evidence used: Proves the new lower bounds 5730, 7654, 11692, 19448, and 29768 in dimensions 17 through 21.\n- Henry Cohn, Kissing numbers bounds table (accessed 2026-08-17). (maintained_tracker): https://cohn.mit.edu/kissing-numbers/\n Evidence used: Author-maintained table of best current lower and upper bounds across dimensions.\n\n**Review notes.** The background's list of exactly solved dimensions is current.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 612, "favorite_count": 46, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1226, "problem_number": "GEO-031", "title": "Ulam's Packing Conjecture", "statement": "Is the sphere the worst-packing convex solid?", "background": "Proposed by Stanisław Ulam, this conjecture asks which three-dimensional convex body has the smallest packing density. Ulam conjectured that the sphere is the worst-packing convex solid, meaning that among all convex bodies in 3D, spheres have the smallest proportion of space filled when packed. While the sphere packing problem (densest packing) was solved by Hales (2005), the worst-packing problem remains open. The conjecture connects packing theory, convex geometry, and optimization, with potential applications to materials science and crystallography.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ulam's global conjecture remains open, but the three-ball is proved to be a local pessimum among sufficiently nearby origin-symmetric convex bodies.\n\n**Verified partial progress.**\n\n- Kallus proved that every non-spherical origin-symmetric convex solid sufficiently close to a three-ball can be packed more efficiently than balls.\n- This verifies Ulam's prediction locally in shape space for origin-symmetric bodies.\n- Many individual nonspherical solids have known packings denser than the ball's optimal density pi/sqrt(18), but these examples do not establish a universal theorem.\n\n**Full solution or refutation.**\n\nNo proof was found that every three-dimensional convex body has optimal packing density at least pi/sqrt(18), and no counterexample was verified.\n\n**What remains.**\n\nExtend the local symmetric result to arbitrary convex bodies and global shape space, or exhibit a convex solid whose optimal packing density is below that of the ball.\n\n**Sources checked.**\n\n- Y. Kallus, The 3-ball is a local pessimum for packing, Advances in Mathematics 264 (2014), 355–370. (primary): https://doi.org/10.1016/j.aim.2014.07.015\n Evidence used: States Ulam's conjecture and proves denser packings for all sufficiently spherical origin-symmetric convex solids other than the ball.\n- Y. Kallus, The 3-ball is a local pessimum for packing, arXiv:1212.2551. (primary): https://arxiv.org/abs/1212.2551\n Evidence used: Accessible preprint of the local-pessimum theorem and its exact scope.\n\n**Review notes.** The phrase 'worst-packing' is underspecified unless interpreted as minimizing the maximum congruent-copy packing density.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 445, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1227, "problem_number": "GEO-032", "title": "Sphere Packing in High Dimensions", "statement": "What is the densest packing of unit spheres in dimensions other than 1, 2, 3, 8, and 24?", "background": "The sphere packing problem asks for the densest arrangement of non-overlapping unit spheres in $n$-dimensional Euclidean space. Solved for dimensions 1 and 2 (trivial), dimension 3 by Hales (1998, computer-assisted proof), dimension 8 by Viazovska (2016), and dimension 24 by Cohn et al. (2016), the problem remains open for all other dimensions. Understanding the asymptotic behavior as $n \\to \\infty$ is also open. This connects coding theory, lattice theory, and has applications to error-correcting codes and wireless communications.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Accepted literature still solves exact sphere packing only in dimensions 1, 2, 3, 8, and 24, while 2026 results substantially improve high-dimensional asymptotic bounds.\n\n**Verified partial progress.**\n\n- Klartag proved the published lower bound Delta_d >= c d^2 2^(-d) using lattice packings.\n- An OpenAI manuscript released in August 2026 claims Delta_d <= 2^{-(0.6044...+o(1))d}, the first improvement since 1978 in the general upper exponent, and determines the asymptotic strength of the Cohn–Elkies linear program.\n- A July 2026 non-peer-reviewed preprint claims that D4 is optimal in dimension four, but no independent confirmation was located.\n\n**Full solution or refutation.**\n\nNo additional fixed dimension was accepted as solved in the peer-reviewed literature checked, and the high-dimensional upper and lower exponents remain far apart.\n\n**What remains.**\n\nDetermine exact optimal packings in dimensions 4–7 and all other nonexceptional dimensions, audit the recent dimension-four claim, and close the asymptotic gap between c d^2 2^(-d) and the best upper exponent.\n\n**Sources checked.**\n\n- H. Cohn and I. Rajagopal, Variations on Five-Dimensional Sphere Packings, Discrete & Computational Geometry (2026). (primary): https://doi.org/10.1007/s00454-026-00841-x\n Evidence used: The July 2026 peer-reviewed introduction says sphere packing is solved only in dimensions 1 through 3, 8, and 24.\n- B. Klartag, Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid, Inventiones Mathematicae (2026). (primary): https://doi.org/10.1007/s00222-026-01412-w\n Evidence used: Proves existence of lattice sphere packings of density at least c d^2 2^(-d).\n- OpenAI, Exponential Growth Rate of the Cohn–Elkies Sphere Packing Linear Program, Chapter 1 of Ten Advances in Mathematics and Theoretical Computer Science, updated 2026-08-06. (primary): https://cdn.openai.com/pdf/ten-proofs-oai.pdf\n Evidence used: Theorem 1.1 claims the upper exponent 0.6044... and the exact exponential rate of the Cohn–Elkies program; this is a very recent manuscript.\n- D. Bhattacharjee, U. Bhattacharya, and S. Bhattacharya, The Sphere Packing Problem in Dimension Four, Preprints.org 202606.0856 v7 (2026). (primary): https://www.preprints.org/manuscript/202606.0856\n Evidence used: Explicitly claims Delta_4=pi^2/16, but the host labels the manuscript not peer reviewed and no independent verification was found.\n\n**Review notes.** The August 2026 upper-bound manuscript and July 2026 dimension-four claim are surfaced but not treated as peer-reviewed fixed-dimensional resolutions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 734, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1231, "problem_number": "COMB-010", "title": "The Cap Set Problem", "statement": "What is the maximum size of a cap set in $\\mathbb{F}_3^n$?", "background": "A cap set is a subset of the $n$-dimensional vector space over the three-element field with no three elements in arithmetic progression (analogous to the card game SET). The problem asks for the maximum size of such a set as a function of $n$. In 2016, Ellenberg and Gijswijt proved an upper bound of $O(2.756^n)$, dramatically improving previous bounds and resolving the longstanding question of whether cap sets can have exponential size $3^{cn}$ for $c > 0$. However, the exact maximum size and optimal constant remain open. This connects additive combinatorics, algebraic combinatorics, and theoretical computer science.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The cap-set maximum and its exponential growth constant remain unknown; current primary results leave an exponential-base gap of about 2.2202 to 2.7552.\n\n**Verified partial progress.**\n\n- Ellenberg and Gijswijt prove an upper bound with exponential base 2.7552..., commonly rounded to 2.756.\n- Romera-Paredes and collaborators construct a partial admissible set implying cap-set capacity at least 2.2202.\n- Later upper-bound work improves polynomial prefactors but does not close the exponential-base gap.\n\n**Full solution or refutation.**\n\nNo formula for the maximum in arbitrary dimension and no determination of the asymptotic capacity were verified.\n\n**What remains.**\n\nClose the gap between the lower capacity 2.2202 and upper capacity 2.7552, and determine exact maxima beyond the known small dimensions.\n\n**Sources checked.**\n\n- J. S. Ellenberg and D. Gijswijt, On large subsets of F_q^n with no three-term arithmetic progression, Annals of Mathematics 185 (2017). (primary): https://arxiv.org/abs/1605.09223\n Evidence used: Establishes the polynomial-method exponential upper bound below 2.756^n.\n- B. Romera-Paredes et al., Mathematical discoveries from program search with large language models, Nature 625 (2024), 468–475. (primary): https://doi.org/10.1038/s41586-023-06924-6\n Evidence used: Reports an admissible set of size 237,984 in A(24,17), implying the new cap-set capacity lower bound 2.2202.\n\n**Review notes.** The 2016–2017 breakthrough settled exponential smallness relative to 3^n, not the exact maximum requested here.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 523, "favorite_count": 40, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1235, "problem_number": "COMB-012", "title": "The Sunflower Conjecture", "statement": "Does every family of at least $c^k k!$ sets of size $k$ contain a sunflower of size 3, for some absolute constant $c$?", "background": "Proposed by Erdős and Rado in 1960, a sunflower (or $\\Delta$-system) is a collection of sets where every pair shares the same common intersection. The conjecture asks whether the exponential bound $c^k k!$ suffices to guarantee a sunflower of any fixed size. The best known bound is super-exponential. In 2019, Alweiss et al. made breakthrough progress by improving the bound to $O((\\log k)^k k!)$, but reaching the conjectured bound remains open. This problem is central to extremal combinatorics and has applications to circuit complexity and communication complexity.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The exact statement is already a consequence of the 1960 Erdős–Rado sunflower lemma; it mistakenly retains the factorial that the genuine open conjecture seeks to remove.\n\n**Verified partial progress.**\n\n- Erdős and Rado proved that more than k!(r-1)^k sets of size at most k force an r-sunflower.\n- For r=3, taking any absolute c>2, for example c=3, makes every family of at least c^k k! size-k sets exceed the classical threshold.\n\n**Full solution or refutation.**\n\nYes. The Erdős–Rado theorem proves the prompt with c=3. The genuinely open sunflower conjecture asks for C^k with no k! factor.\n\n**What remains.**\n\nNothing remains for the literal statement. The source should be corrected if it intended the open C^k formulation.\n\n**Sources checked.**\n\n- P. Erdős and R. Rado, Intersection Theorems for Systems of Sets, Journal of the London Mathematical Society s1-35 (1960), 85–90. (primary): https://doi.org/10.1112/jlms/s1-35.1.85\n Evidence used: Original source of the classical bound k!(r-1)^k.\n- A. Rao, The Story of Sunflowers, Journal of the London Mathematical Society (2026). (authoritative_secondary): https://doi.org/10.1112/jlms.70380\n Evidence used: States the classical k!(w-1)^k theorem and distinguishes it from the open exponential conjecture.\n\n**Review notes.** Major formulation defect: the factorial makes this an old theorem. The background also incorrectly appends k! to the modern logarithmic-base improvement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 2, "view_count": 612, "favorite_count": 48, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1236, "problem_number": "COMB-013", "title": "Ramsey Number $R(5,5)$", "statement": "What is the exact value of the Ramsey number $R(5,5)$?", "background": "Ramsey numbers quantify the size at which complete disorder becomes impossible. $R(5,5)$ is the minimum number of vertices such that any two-coloring of the edges of the complete graph contains either a red $K_5$ or a blue $K_5$. It is known that $43 \\leq R(5,5) \\leq 48$, but the exact value remains unknown despite over 50 years of effort. This is perhaps the most famous open Ramsey number. Erdős famously suggested that finding $R(6,6)$ would require astronomical resources, but $R(5,5)$ seems tantalizingly within reach. The problem connects combinatorics, graph theory, and computational mathematics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This duplicate R(5,5) record remains unresolved, with current verified range 43 <= R(5,5) <= 46.\n\n**Verified partial progress.**\n\n- Exoo's lower bound R(5,5) >= 43 remains best.\n- Angeltveit and McKay prove R(5,5) <= 46 using linear programming and independently replicated case computations.\n\n**Full solution or refutation.**\n\nNo exact value among 43, 44, 45, and 46 has been established.\n\n**What remains.**\n\nImprove either side of the interval until a single value remains.\n\n**Sources checked.**\n\n- V. Angeltveit and B. D. McKay, R(5,5) <= 46, Journal of Graph Theory 112 (2026), 198–208. (primary): https://doi.org/10.1002/jgt.70029\n Evidence used: Proves the upper bound 46 and states that 43 is the surviving lower bound.\n- V. Angeltveit and B. D. McKay, R(5,5) <= 46, arXiv:2409.15709. (primary): https://arxiv.org/abs/2409.15709\n Evidence used: Accessible preprint of the current upper-bound proof.\n\n**Review notes.** Duplicate topic retained as its own record; the source background's upper bound 48 is obsolete.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 823, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1239, "problem_number": "NT-032", "title": "Gauss Circle Problem", "statement": "How far can the number of lattice points in a circle centered at the origin deviate from the area of the circle?", "background": "The Gauss circle problem asks for the tightest bound on the error term in counting integer lattice points $(m,n)$ inside a circle of radius $r$ centered at the origin. The number of such points is $\\pi r^2 + E(r)$ where $E(r)$ is the error. It is known that $E(r) = O(r^{2/3})$ and $E(r) = \\Omega(r^{1/2} \\log r)$, but the exact growth rate remains unknown. Hardy conjectured $E(r) = O(r^{1/2+\\varepsilon})$ for any $\\varepsilon > 0$. This connects analytic number theory, lattice point enumeration, and has applications to physics and crystallography.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjectural error E(r)=O_epsilon(r^(1/2+epsilon)) remains open. Huxley's peer-reviewed exponent 131/208 improves the record's obsolete 2/3 bound, and a 2023 preprint claims a small further improvement.\n\n**Verified partial progress.**\n\n- Huxley proved E(r)=O(r^(131/208)(log r)^(18627/8320)).\n- Li and Yang's 2023 preprint claims exponent approximately 0.628966.\n\n**Full solution or refutation.**\n\nKnown upper and lower estimates do not reach the conjectured square-root threshold.\n\n**What remains.**\n\nProve E(r)=O_epsilon(r^(1/2+epsilon)) and refine the true oscillation size.\n\n**Sources checked.**\n\n- M. N. Huxley, Exponential sums and lattice points III, Proceedings of the London Mathematical Society 87 (2003), 591-609. (primary): https://doi.org/10.1112/S0024611503014485\n Evidence used: Published 131/208 discrepancy exponent.\n- X. Li and X. Yang, An improvement on Gauss's Circle Problem and Dirichlet's Divisor Problem, arXiv:2308.14859. (primary): https://arxiv.org/abs/2308.14859\n Evidence used: Claims a slight further exponent improvement in a preprint.\n- B. C. Berndt, S. Kim and A. Zaharescu, The circle problem of Gauss and the divisor problem of Dirichlet--still unsolved, American Mathematical Monthly 125 (2018), 99-114. (authoritative_secondary): https://doi.org/10.1080/00029890.2018.1413854\n Evidence used: Specialist exposition of the unresolved conjecture and classical lower-bound context.\n\n**Review notes.** The background's O(r^(2/3)) is outdated and its Omega(r^(1/2) log r) claim overstates the standard logarithmic lower factor.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 478, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1240, "problem_number": "NT-033", "title": "Grimm's Conjecture", "statement": "Can each element of a set of consecutive composite numbers be assigned a distinct prime divisor?", "background": "Proposed by C. A. Grimm in 1969, this conjecture states that if we have $k$ consecutive composite numbers, then there exist $k$ distinct primes each dividing one of these numbers. For example, the consecutive composites $24, 25, 26, 27, 28$ have distinct prime divisors $3, 5, 13, 7, 2$ respectively. While verified computationally for large ranges, the general proof remains elusive. The conjecture connects to prime gaps, divisibility properties, and the distribution of primes among consecutive integers.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Grimm's conjecture remains open for arbitrary blocks of consecutive composite integers, although it has been verified through a large finite range.\n\n**Verified partial progress.**\n\n- Laishram and Shorey verified the conjecture for starting values n <= 1.9*10^10.\n\n**Full solution or refutation.**\n\nFinite verification and weaker prime-divisor-count results do not prove the required system of distinct representatives in every block.\n\n**What remains.**\n\nProve Hall's matching condition for every composite block or produce an explicit failing block.\n\n**Sources checked.**\n\n- S. Laishram and T. N. Shorey, Grimm's conjecture on consecutive integers, International Journal of Number Theory 2 (2006), 207-211. (primary): https://doi.org/10.1142/S1793042106000498\n Evidence used: Primary finite verification through n=1.9*10^10.\n- R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Problem B32, Springer (2004). (authoritative_secondary): https://doi.org/10.1007/978-0-387-26677-0\n Evidence used: Standard specialist statement and open-problem context.\n\n**Review notes.** Exact statement preserved; 'set' is interpreted as a consecutive block in its natural order.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1241, "problem_number": "NT-034", "title": "Hall's Conjecture", "statement": "For any $\\varepsilon > 0$, is there a constant $c(\\varepsilon)$ such that either $y^2 = x^3$ or $|y^2 - x^3| > c(\\varepsilon) x^{1/2-\\varepsilon}$?", "background": "Proposed by Marshall Hall Jr. in 1970, this conjecture provides a measure of how close a perfect square can be to a perfect cube without being equal. It strengthens earlier work on Diophantine approximation and relates to the ABC conjecture. The conjecture has been verified for many special cases but remains open in general. It connects algebraic number theory, Diophantine equations, and elliptic curves, with implications for understanding integer solutions to polynomial equations.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The weak Hall conjecture with exponent 1/2-epsilon remains open for positive integers x,y and follows from abc. The supplied statement omits the domain of x and y.\n\n**Verified partial progress.**\n\n- Danilov constructed infinitely many pairs with 0<|x^3-y^2|<0.97*sqrt(x), establishing sharpness of the 1/2 scale from the other direction.\n- The conjectured lower bound follows conditionally from abc.\n\n**Full solution or refutation.**\n\nKnown near-collision families do not violate the epsilon-weakened lower bound, and no unconditional proof was verified.\n\n**What remains.**\n\nFor positive integers, prove |x^3-y^2| >= c(epsilon)x^(1/2-epsilon) whenever x^3 differs from y^2.\n\n**Sources checked.**\n\n- L. V. Danilov, The Diophantine equation x^3-y^2=k and Hall's conjecture, Mathematical Notes 32 (1982), 617-618. (primary): https://doi.org/10.1007/BF01140190\n Evidence used: Constructs an infinite near-collision family at the square-root scale.\n- A. Dujella, A new algorithm to search for small nonzero |x^3-y^2| values, Mathematics of Computation 75 (2006), 879-881. (primary): https://doi.org/10.1090/S0025-5718-05-01822-0\n Evidence used: States the weak form, its open status, and computational examples.\n- Y. Bugeaud, Distance Between Cubics and Rationals, Results in Mathematics 81 (2026). (primary): https://doi.org/10.1007/s00025-026-02616-5\n Evidence used: Recent primary paper still explicitly labels the displayed epsilon form a conjecture.\n\n**Review notes.** The standard positive-integer domain is not present in the exact record; for negative x its real fractional power is generally undefined.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1242, "problem_number": "NT-035", "title": "Lehmer's Totient Problem", "statement": "If Euler's totient function $\\phi(n)$ divides $n-1$, must $n$ be prime?", "background": "Posed by D. H. Lehmer in 1932, this problem asks whether any composite number $n$ exists such that $\\phi(n)$ divides $n-1$, where $\\phi(n)$ counts integers up to $n$ coprime to $n$. For all primes $p$, we have $\\phi(p) = p-1$, so the divisibility holds. Lehmer conjectured no composite number has this property. It has been verified that any such composite must be odd, square-free, and have at least 7 prime factors, with the smallest exceeding $10^{20}$. This connects Euler's totient function, primality, and multiplicative number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The literal unrestricted implication is false at n=1: phi(1)=1 divides 0=n-1, but 1 is not prime. For the standard domain n>1, Lehmer's totient problem remains open and no composite example is known.\n\n**Verified partial progress.**\n\n- Any composite example must be odd and squarefree.\n- Current cited bounds require at least 15 distinct prime factors and n>10^30.\n\n**Full solution or refutation.**\n\nThe exact record is disproved by n=1. Strong structural and computational exclusions have not resolved the corrected composite-n problem.\n\n**What remains.**\n\nNothing for the literal implication; after amending it to n>1, exclude all composite n or find one.\n\n**Sources checked.**\n\n- D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society 38 (1932), 745-751. (primary): https://doi.org/10.1090/S0002-9904-1932-05518-3\n Evidence used: Original problem and basic odd, squarefree, many-prime-factor restrictions.\n- Q. Ji and H. Qin, Lehmer's totient problem over F_q[x], Comptes Rendus Mathematique 355 (2017), 370-377. (primary): https://doi.org/10.1016/j.crma.2017.03.007\n Evidence used: Introduction records the integer problem as open and the then-current at-least-15-prime-factors and n>10^30 bounds.\n\n**Review notes.** The exact record omits n>1; with the common convention phi(1)=1, n=1 satisfies the divisibility and is not prime.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 523, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1243, "problem_number": "NT-036", "title": "Magic Square of Squares", "statement": "Does there exist a 3×3 magic square composed entirely of distinct perfect squares?", "background": "A magic square has the property that all rows, columns, and diagonals sum to the same value. While magic squares of integers are well understood, the question of whether a 3×3 magic square can be constructed using only distinct perfect squares has remained open for centuries. Martin LaBar proved in 1984 that no such square exists using rational squares, but the integer case remains unsolved. Partial results exist for 4×4 and larger squares. This connects number theory, Diophantine equations, and recreational mathematics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No 3x3 magic square of nine distinct integral squares, and no nonexistence proof, is known. The background's claim that LaBar excluded rational squares while the integer case remains open is logically and historically false.\n\n**Verified partial progress.**\n\n- Bremner constructed and analyzed near-solutions with seven square entries.\n- Bremner also exhibited nine distinct square entries satisfying seven of the eight magic line-sum conditions.\n\n**Full solution or refutation.**\n\nThe available constructions miss at least one required condition and do not solve the exact existence question.\n\n**What remains.**\n\nConstruct nine distinct integral squares satisfying all eight line sums or prove impossibility.\n\n**Sources checked.**\n\n- A. Bremner, On squares of squares, Acta Arithmetica 88 (1999), 289-297. (primary): https://eudml.org/doc/207247\n Evidence used: Primary study of near-magic arrays and explicit seven-of-eight line-sum construction.\n- A. Bremner, On squares of squares II, Acta Arithmetica 99 (2001), 289-308. (primary): https://doi.org/10.4064/aa99-3-6\n Evidence used: Studies magic squares with as many square entries as possible and confirms the unresolved full target.\n- Open Problem Garden, Magic square of squares, accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/magic_square_of_squares\n Evidence used: Maintains the exact distinct-square existence question as open and attributes it as a question posed by LaBar.\n\n**Review notes.** Background defect flagged: integer squares are rational squares, so rational nonexistence cannot coexist with an open integer case; LaBar posed rather than proved the problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 589, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1244, "problem_number": "NT-037", "title": "Mahler's 3/2 Problem", "statement": "Is there a real number $x$ such that the fractional parts of $x(3/2)^n$ are all less than $1/2$ for every positive integer $n$?", "background": "Proposed by Kurt Mahler in the 1960s, this problem concerns the distribution of the sequence $\\{x(3/2)^n\\}$ modulo 1, where $\\{y\\}$ denotes the fractional part of $y$. Mahler conjectured that no such $x$ exists. The problem relates to ergodic theory, uniform distribution, and Diophantine approximation. While various partial results have been obtained using techniques from dynamical systems and number theory, the general question remains open. It exemplifies deep questions about the behavior of geometric sequences under modular arithmetic.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The exact statement has the immediate affirmative witness x=0, whose fractional parts are all zero. It omits the positivity condition essential to Mahler's standard open Z-number problem.\n\n**Verified partial progress.**\n\n- For the corrected positive-real problem, Mahler proved at most one Z-number can lie in each unit interval, and later work gives stronger scarcity and distribution restrictions.\n\n**Full solution or refutation.**\n\nSet x=0. Then {x(3/2)^n}=0<1/2 for every positive integer n, so the literal existence question is answered yes.\n\n**What remains.**\n\nNothing for the exact wording; after adding x>0, prove or disprove existence of a Z-number.\n\n**Sources checked.**\n\n- K. Mahler, An unsolved problem on the powers of 3/2, Journal of the Australian Mathematical Society 8 (1968), 313-321. (primary): https://doi.org/10.1017/S144678870000613X\n Evidence used: Original source for the standard problem and its positive-real restriction.\n- A. Dubickas, On the fractional parts of rational powers, Acta Arithmetica 141 (2010), 103-110. (primary): https://doi.org/10.4064/aa141-2-1\n Evidence used: Defines a Z-number using xi>0 and treats nonexistence as Mahler's conjecture.\n- S. Akiyama et al., Rational self-affine tiles, American Mathematical Monthly 129 (2022), 537-555. (authoritative_secondary): https://doi.org/10.1080/00029890.2022.2061281\n Evidence used: Recent specialist exposition calls the corrected Z-number existence question still open.\n\n**Review notes.** SOLVED-IN-LITERATURE is used for the exact record's status, but the witness is elementary; this must not be propagated as a solution of Mahler's standard positive-real problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1245, "problem_number": "NT-038", "title": "Newman's Conjecture", "statement": "Does the partition function satisfy any arbitrary congruence infinitely often?", "background": "Proposed by Morris Newman, this conjecture concerns the partition function $p(n)$, which counts the number of ways to write $n$ as a sum of positive integers. Newman conjectured that for any integers $a$ and $m$ with $\\gcd(a,m) = 1$, there are infinitely many $n$ such that $p(n) \\equiv a \\pmod{m}$. This would imply the partition function takes all possible residue classes modulo any integer infinitely often. The conjecture connects partition theory, modular forms, and has implications for understanding the arithmetic properties of partitions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The standard Newman conjecture, requiring every residue modulo every integer to occur infinitely often as a value of p(n), remains open; the input's unit-residue wording is a weaker and formulation-sensitive variant.\n\n**Verified partial progress.**\n\n- Choi and Lee proved the full conjecture for a density-one set among moduli having any fixed number of distinct prime divisors.\n\n**Full solution or refutation.**\n\nNo theorem covering every modulus was verified.\n\n**What remains.**\n\nProve infinite occurrence of every residue for every modulus, or first determine whether the input intentionally asks only for unit residues.\n\n**Sources checked.**\n\n- Dohoon Choi and Youngmin Lee, Newman's conjecture for the partition function modulo integers with at least two distinct prime divisors, Advances in Mathematics 477 (2025), 110367. (primary): https://doi.org/10.1016/j.aim.2025.110367\n Evidence used: Defines the standard all-residue conjecture, proves it for density one of moduli in each fixed-prime-divisor stratum, and continues to treat the general conjecture as open.\n\n**Review notes.** Background defect: gcd(a,m)=1 does not cover all residue classes. Exact record retained; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 367, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1246, "problem_number": "NT-039", "title": "Scholz Conjecture", "statement": "Is the shortest addition chain for $2^n - 1$ at most $n - 1$ plus the length of the shortest addition chain for $n$?", "background": "An addition chain for $m$ is a sequence $1 = a_0 < a_1 < \\cdots < a_r = m$ where each $a_i$ (for $i > 0$) is the sum of two earlier terms. Scholz conjectured in 1937 that $\\ell(2^n-1) \\leq n-1+\\ell(n)$ where $\\ell(m)$ denotes the minimum length of an addition chain for $m$. This has applications to efficient exponentiation algorithms in computer science and cryptography. While verified for many values and various special cases proven, the general conjecture remains open. It connects additive number theory, combinatorial optimization, and computational complexity.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Scholz-Brauer addition-chain inequality remains open in general.\n\n**Verified partial progress.**\n\n- Brauer proved the analogue for star (Brauer) chains.\n- Clift's maintained computations verify the conjecture for every n below 5,784,689 and give later isolated constructions.\n\n**Full solution or refutation.**\n\nNo all-n proof or counterexample was verified.\n\n**What remains.**\n\nProve l(2^n-1) <= n-1+l(n) for every positive integer n or find a genuine counterexample.\n\n**Sources checked.**\n\n- Alfred Brauer, On addition chains, Bulletin of the American Mathematical Society 45 (1939), 736-739. (primary): https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-45/issue-10/On-addition-chains/bams/1183502523.full\n Evidence used: Establishes the classical star-chain special case underlying the partial result.\n- Neill Clift, The Scholz-Brauer Conjecture. (maintained_tracker): https://additionchains.com/ScholzBrauer.html\n Evidence used: Maintains the verification range and later chain constructions while identifying the general conjecture as unresolved.\n\n**Review notes.** The 2024 construction at n=30,978,119 gives a strict inequality, not a counterexample. Exact record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 30, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1248, "problem_number": "NT-041", "title": "Infinitely Many Perfect Numbers", "statement": "Are there infinitely many perfect numbers?", "background": "A perfect number equals the sum of its proper divisors (divisors excluding itself). Examples include 6 = 1+2+3 and 28 = 1+2+4+7+14. Euclid proved that if $2^p - 1$ is prime (a Mersenne prime), then $2^{p-1}(2^p-1)$ is perfect. All known perfect numbers have this form and are even. Whether infinitely many exist depends on whether there are infinitely many Mersenne primes, itself an open question. The problem connects prime number theory, divisor functions, and has fascinated mathematicians for over 2000 years.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Infinitude of perfect numbers remains open; Euclid-Euler reduces the even case to infinitude of Mersenne primes.\n\n**Verified partial progress.**\n\n- Every even perfect number has Euclid-Euler form 2^(p-1)(2^p-1).\n- The maintained GIMPS list contains 52 known Mersenne primes as of the checked date.\n\n**Full solution or refutation.**\n\nNo infinitude theorem for perfect numbers or Mersenne primes was verified.\n\n**What remains.**\n\nProve infinitely many Mersenne primes, prove an infinite family of odd perfect numbers, or otherwise settle the literal infinitude question.\n\n**Sources checked.**\n\n- Great Internet Mersenne Prime Search, Known Mersenne Primes. (maintained_tracker): https://www.mersenne.org/primes/\n Evidence used: Maintains the finite current list of known Mersenne primes and the ongoing search.\n\n**Review notes.** Background scope defect: Mersenne-prime infinitude is equivalent to infinitely many even perfect numbers, not logically necessary for infinitely many perfect numbers if odd ones exist. Exact record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 678, "favorite_count": 54, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1250, "problem_number": "NT-043", "title": "Quasiperfect Numbers", "statement": "Do quasiperfect numbers exist?", "background": "A quasiperfect number is a natural number $n$ such that the sum of its divisors equals $2n + 1$ (one more than twice the number). No quasiperfect number has ever been found. It has been proven that if one exists, it must be an odd square number greater than $10^{35}$, and have at least seven distinct prime factors. The search for quasiperfect numbers connects divisor theory, multiplicative number theory, and computational number theory. Their existence or non-existence would provide insights into the structure of highly composite numbers.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No quasiperfect number is known, and existence remains open; a very recent preprint claims that any example must have at least eight distinct prime factors.\n\n**Verified partial progress.**\n\n- Hagis and Cohen proved the longstanding lower bound of seven distinct prime factors.\n- Toyohara, Tao, and Yao claim omega(n) >= 8, using exact computational ledgers and a Lean formalization of the lemma layer.\n\n**Full solution or refutation.**\n\nNo existence or nonexistence proof was verified.\n\n**What remains.**\n\nConstruct a quasiperfect number or prove none exist; independently audit and peer-review the new eight-prime-factor computation.\n\n**Sources checked.**\n\n- Akira Toyohara, Ye Tao, and Siqiong Yao, Every quasiperfect number has at least eight distinct prime factors, arXiv:2608.02066 (2026). (primary): https://arxiv.org/abs/2608.02066\n Evidence used: States that existence is open and claims the first improvement of the prime-factor lower bound since 1982.\n- Peter Hagis Jr. and Graeme L. Cohen, Some results concerning quasiperfect numbers, Journal of the Australian Mathematical Society, Series A 33 (1982), 275-286. (primary): https://doi.org/10.1017/S1446788700018790\n Evidence used: Provides the classical structural and seven-distinct-prime-factor restrictions.\n\n**Review notes.** The August 2026 preprint is extremely recent and computationally substantial; this triage did not reproduce its computation. Exact record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1251, "problem_number": "NT-044", "title": "Almost Perfect Numbers Beyond Powers of 2", "statement": "Do any almost perfect numbers exist that are not powers of 2?", "background": "An almost perfect number $n$ has the sum of its proper divisors equal to $n - 1$. All powers of 2 are almost perfect, since the divisors of $2^k$ are $1, 2, 4, \\ldots, 2^{k-1}$ which sum to $2^k - 1$. It remains unknown whether any odd almost perfect number exists, or any even almost perfect number that is not a power of 2. The problem connects perfect numbers, divisor functions, and the structure of highly specific arithmetic sequences.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Powers of two remain the only known almost perfect numbers; existence of any non-power-of-two example is open.\n\n**Verified partial progress.**\n\n- A hypothetical even non-power-of-two almost perfect number has the form 2^r b^2 with b odd composite and satisfies additional divisibility and size restrictions.\n\n**Full solution or refutation.**\n\nNo non-power-of-two example or nonexistence theorem was verified.\n\n**What remains.**\n\nConstruct an almost perfect number outside the powers of two or exclude all remaining structural cases.\n\n**Sources checked.**\n\n- John Rafael M. Antalan and Jose Arnaldo B. Dris, Some New Results On Even Almost Perfect Numbers Which Are Not Powers Of Two, arXiv:1602.04248. (primary): https://arxiv.org/abs/1602.04248\n Evidence used: Explicitly records the open existence question and proves necessary structure for even hypothetical examples.\n- Encyclopedia of Mathematics, Almost perfect number. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Almost_perfect_number\n Evidence used: Summarizes the definition and that only powers of two are presently known.\n\n**Review notes.** Exact record retained; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 356, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1252, "problem_number": "NT-045", "title": "The Number of Idoneal Numbers", "statement": "Are there exactly 65 idoneal numbers, or could there be 66 or 67?", "background": "Idoneal numbers (also called suitable or convenient numbers) are positive integers $D$ such that if $n = ax^2 + by^2$ with coprime $a,b$ is uniquely representable, then $n$ is a prime power or twice a prime power. Euler conjectured 65 such numbers exist, the largest being 1848. Weinberger proved in 1973 that at most one more exists, and if the generalized Riemann hypothesis is true, exactly 65 exist. This connects binary quadratic forms, class field theory, and the Riemann hypothesis. The resolution depends on deep questions in analytic number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exactly 65 idoneal numbers are known; unconditionally there can be at most two more, while GRH implies the known list is complete.\n\n**Verified partial progress.**\n\n- Weinberger's class-group theorem yields the conditional completeness result and, after the idoneal-number cases are separated correctly, an unconditional total of at most 67.\n\n**Full solution or refutation.**\n\nThe unconditional choice among 65, 66, and 67 has not been resolved.\n\n**What remains.**\n\nExclude or exhibit the one or two possible exceptional idoneal numbers without assuming GRH.\n\n**Sources checked.**\n\n- Peter J. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arithmetica 22 (1973), 117-124. (primary): https://eudml.org/doc/205250\n Evidence used: Provides the class-group exponent theorem underlying the unconditional and GRH-conditional conclusions.\n- Ernst Kani, Idoneal Numbers and Some Generalizations, Annales des sciences mathematiques du Quebec 35 (2011), 197-227. (authoritative_secondary): https://mast.queensu.ca/~kani/papers/2011K1.pdf\n Evidence used: Derives at most 67 in total, proves 65 under GRH, and identifies the frequently repeated at-most-one claim as erroneous.\n\n**Review notes.** Background defect: Weinberger does not imply at most one additional idoneal number; the correct unconditional upper total is 67. The input's compressed definition also merits normalization. Exact statement retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 334, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1253, "problem_number": "NT-046", "title": "Amicable Numbers of Opposite Parity", "statement": "Do any pairs of amicable numbers exist where one is odd and one is even?", "background": "Two numbers are amicable if each equals the sum of the proper divisors of the other. For example, 220 and 284 are amicable (both even). Over 12 million amicable pairs are known, all with matching parity (both even or both odd). It remains unknown whether a mixed-parity pair exists. Such a pair would require unusual divisor properties. The problem connects divisor sums, parity constraints, and the arithmetic structure of amicable pairs. All known odd amicable pairs have been found by Erdős and collaborators.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No amicable pair of opposite parity is known, and existence remains open.\n\n**Verified partial progress.**\n\n- If a mixed-parity pair exists, the odd member is a square and the even member is a power of two times an odd square.\n- Suzuki proves a strong sparsity upper bound for the counting function of even-odd amicable pairs.\n\n**Full solution or refutation.**\n\nNo construction or nonexistence proof was verified.\n\n**What remains.**\n\nProduce an even-odd amicable pair or eliminate all pairs satisfying the known square-form constraints.\n\n**Sources checked.**\n\n- Germano D'Abramo, On Amicable Numbers With Different Parity, arXiv:math/0501402. (primary): https://arxiv.org/abs/math/0501402\n Evidence used: Records and derives the square-form restrictions for a hypothetical opposite-parity pair.\n- Yuta Suzuki, On even-odd amicable pairs, RIMS Kokyuroku 2162 (2020), 127-138. (primary): http://hdl.handle.net/2433/261423\n Evidence used: Treats existence as open and proves an upper bound for the number of such pairs.\n\n**Review notes.** Exact record retained; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1254, "problem_number": "NT-047", "title": "Infinitely Many Amicable Pairs", "statement": "Are there infinitely many pairs of amicable numbers?", "background": "Amicable numbers are pairs where each number equals the sum of the other's proper divisors. While over 12 million pairs have been discovered, it remains unknown whether infinitely many exist. Thabit ibn Qurra (9th century) gave a formula generating some pairs, and Euler found many more. Various conjectures suggest their density, but no proof of infinitude exists. This contrasts with related questions like twin primes (conjectured infinite) and perfect numbers (infinitude depends on Mersenne primes). The problem connects multiplicative number theory and divisor sums.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** It remains open whether there are infinitely many amicable pairs.\n\n**Verified partial progress.**\n\n- Erdos proved that amicable numbers have asymptotic density zero, and later work gives substantially stronger upper bounds.\n- Large computational enumerations produce many examples but cannot establish infinitude.\n\n**Full solution or refutation.**\n\nNo infinite construction or finiteness proof was verified.\n\n**What remains.**\n\nProve an infinite family of distinct amicable pairs or prove that only finitely many exist.\n\n**Sources checked.**\n\n- Erdos Problems, Problem 830: Are there infinitely many amicable pairs? (maintained_tracker): https://www.erdosproblems.com/830\n Evidence used: Marks the problem open and tracks the primary density and upper-bound literature.\n- Mariano Garcia, A Million New Amicable Pairs, Journal of Integer Sequences 4 (2001). (primary): https://cs.uwaterloo.ca/journals/JIS/VOL4/GARCIA/millionc.html\n Evidence used: Documents a large finite computational family without claiming infinitude.\n\n**Review notes.** Exact record retained; finite example counts in the background are not treated as evidence of infinitude.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1255, "problem_number": "NT-048", "title": "Infinitely Many Giuga Numbers", "statement": "Are there infinitely many Giuga numbers?", "background": "A Giuga number is a composite number $n$ such that $p$ divides $(n/p - 1)$ for every prime divisor $p$ of $n$. Equivalently, $\\sum_{p|n} (1/p) - 1/n$ is an integer. Only 15 Giuga numbers are known, the smallest being 30. Giuga conjectured that if $1 + \\sum_{i=1}^{n-1} i^{n-1} \\equiv 0 \\pmod{n}$ for composite $n$, then $n$ is a Giuga number. Whether infinitely many exist remains open. This connects primality testing, Carmichael numbers, and Fermat pseudoprimes.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Infinitude remains open for the weak Giuga numbers defined by the input; the background conflates weak and strong Giuga conventions.\n\n**Verified partial progress.**\n\n- Known weak Giuga numbers satisfy restrictive reciprocal and prime-divisor conditions, but no infinite family is proved.\n- For strong Giuga numbers, Luca, Pomerance, and Shparlinski prove a sparsity bound; this does not settle the input's weak condition.\n\n**Full solution or refutation.**\n\nNo infinitude or finiteness theorem for the input-defined class was verified.\n\n**What remains.**\n\nFix the intended Giuga convention, then prove an infinite family or a finiteness result for that precisely stated class.\n\n**Sources checked.**\n\n- Florian Luca, Carl Pomerance, and Igor E. Shparlinski, On Giuga Numbers, International Journal of Modern Mathematics 4 (2009), 13-18. (primary): https://gauss.dartmouth.edu/~carlp/giugafinal.pdf\n Evidence used: Explicitly distinguishes weak Giuga numbers from the strong primality-congruence class and proves a counting bound for the latter.\n- OEIS A007850, Giuga numbers. (maintained_tracker): https://oeis.org/A007850\n Evidence used: Maintains the known sequence and references under the reciprocal/divisibility convention.\n\n**Review notes.** Definition defect: p divides n/p-1 is the weak Giuga condition in the primary source, whereas the background invokes the stronger primality congruence. The stated count of 15 known examples is stale/inconsistent with the checked tracker. Exact statement retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 367, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1256, "problem_number": "NT-049", "title": "Lychrel Numbers in Base 10", "statement": "Do Lychrel numbers exist in base 10?", "background": "A Lychrel number is a natural number that never forms a palindrome through the iterative process of adding it to its reverse. For example, 89 is not Lychrel: 89 + 98 = 187, 187 + 781 = 968, 968 + 869 = 1837, 1837 + 7381 = 9218, 9218 + 8129 = 17347, 17347 + 74371 = 91718, 91718 + 81719 = 173437, 173437 + 734371 = 907808, 907808 + 808709 = 1716517, 1716517 + 7156171 = 8872688, which is a palindrome. The number 196 is the smallest candidate Lychrel number, having been tested to over 300 million iterations without producing a palindrome. No Lychrel number has been proven to exist.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No base-10 Lychrel number has been proved to exist; 196 remains a candidate rather than an established example.\n\n**Verified partial progress.**\n\n- Very long reverse-and-add computations for 196 and other candidates have found no palindrome, but every such computation is finite.\n\n**Full solution or refutation.**\n\nNo proof of divergence from palindromes for any base-10 starting value was verified.\n\n**What remains.**\n\nProve that some base-10 reverse-and-add orbit never reaches a palindrome, or prove that every orbit eventually does.\n\n**Sources checked.**\n\n- Eric W. Weisstein, Lychrel Number, Wolfram MathWorld. (authoritative_secondary): https://mathworld.wolfram.com/LychrelNumber.html\n Evidence used: Describes the numbers as candidates not known to reach a palindrome and records 196 as the smallest familiar candidate.\n- Jason Doucette, The 196 Palindrome Quest. (maintained_tracker): https://p196.org/\n Evidence used: Maintains the long-running base-10 computation and clearly separates computational survival from proof.\n\n**Review notes.** The background's precise iteration count is a moving and potentially stale computational record; finite iteration does not prove the Lychrel property. Exact record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1257, "problem_number": "NT-050", "title": "Odd Weird Numbers", "statement": "Do any odd weird numbers exist?", "background": "A weird number is a natural number that is abundant (the sum of its proper divisors exceeds the number) but not semiperfect (no subset of its divisors sums to the number). The smallest weird number is 70. All known weird numbers are even, and it has been conjectured that no odd weird numbers exist. If an odd weird number exists, it must be greater than $10^{21}$ and have at least 4 distinct prime factors. This connects abundant numbers, partition theory, and subset sum problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Existence of odd weird numbers remains open.\n\n**Verified partial progress.**\n\n- Fang excluded odd weird numbers below 10^21, and Liddy--Riedl showed an odd weird number would have at least six distinct prime factors.\n\n**Full solution or refutation.**\n\nNo existence or nonexistence proof was verified.\n\n**What remains.**\n\nConstruct an odd weird number or rule out all odd integers.\n\n**Sources checked.**\n\n- T. F. Bloom, Erdős Problem #470 (updated 2026). (maintained_tracker): https://www.erdosproblems.com/470\n Evidence used: Records the open status, the 10^21 exclusion, and the six-prime-factor restriction.\n- J. Fang, Searching on the boundary of abundance for odd weird numbers, arXiv:2207.12906. (primary): https://arxiv.org/abs/2207.12906\n Evidence used: Treats the odd-weird question as open and reports the search setting.\n\n**Review notes.** Statement retained. Its four-distinct-prime-factor background bound is superseded by the stronger six-factor restriction.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 378, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1258, "problem_number": "NT-051", "title": "Normality of Pi", "statement": "Is $\\pi$ a normal number in base 10?", "background": "A number is normal in base 10 if every digit 0-9 appears with equal frequency (1/10) in its decimal expansion, and more generally, every sequence of $k$ digits appears with frequency $1/10^k$. While the digits of $\\pi$ appear statistically random in computational tests extending to trillions of digits, no proof of normality exists. It is not even known whether every digit appears infinitely often in $\\pi$. Proving normality would require deep insights into the arithmetic nature of $\\pi$ and transcendental number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Normality of pi in base 10 remains unproved.\n\n**Verified partial progress.**\n\n- Large digit computations are evidence only; they do not establish even the infinitude of every decimal digit.\n\n**Full solution or refutation.**\n\nNo proof of normality, or of the weaker digit-occurrence assertions, was verified.\n\n**What remains.**\n\nEstablish the frequencies of all finite base-10 blocks in pi.\n\n**Sources checked.**\n\n- D. H. Bailey and R. E. Crandall, Randomness and the pi digits, Experimental Mathematics 11 (2002). (authoritative_secondary): https://doi.org/10.1080/10586458.2002.10504475\n Evidence used: Explains the normality question and that no proof is known for pi.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 823, "favorite_count": 68, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1259, "problem_number": "NT-052", "title": "Normality of Irrational Algebraic Numbers", "statement": "Are all irrational algebraic numbers normal in every base?", "background": "An algebraic number is a root of a polynomial with integer coefficients. Normal numbers have every digit sequence appear with the expected frequency in their base expansions. It is conjectured that all irrational algebraic numbers like $\\sqrt{2}$ are normal in every integer base, but not a single irrational algebraic number has been proven normal in any base. This represents a fundamental gap in our understanding of the decimal expansions of algebraic numbers. The question connects algebraic number theory, Diophantine approximation, and the theory of normal numbers.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No irrational algebraic number is known to be normal in any integer base.\n\n**Verified partial progress.**\n\n- The general Borel normality theorem is metric and does not supply an explicit irrational algebraic example.\n\n**Full solution or refutation.**\n\nNo individual algebraic irrational normality theorem was verified.\n\n**What remains.**\n\nProve normality in one base for even a single irrational algebraic number, or settle the all-bases assertion.\n\n**Sources checked.**\n\n- Y. Bugeaud, Distribution Modulo One and Diophantine Approximation, Cambridge Tracts in Mathematics 193 (2012). (authoritative_secondary): https://doi.org/10.1017/CBO9781139019672\n Evidence used: Surveys the unresolved digit-distribution questions for algebraic irrational numbers.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 45, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1260, "problem_number": "NT-053", "title": "Is 10 a Solitary Number?", "statement": "Is 10 a solitary number (no other number shares its abundancy index)?", "background": "The abundancy index of $n$ is $\\sigma(n)/n$ where $\\sigma(n)$ is the sum of divisors of $n$. A number is solitary if no other number has the same abundancy index. For 10, we have $\\sigma(10) = 1+2+5+10 = 18$, giving abundancy $18/10 = 9/5$. It remains unknown whether any other number has abundancy $9/5$. Numbers in amicable pairs and sociable numbers are not solitary. Many numbers have been proven non-solitary, but 10 resists classification. This connects divisor functions, Diophantine equations, and the classification of multiplicative structures.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** It remains unknown whether 10 is solitary; it is the smallest integer not known either to have a friend or to be solitary.\n\n**Verified partial progress.**\n\n- Any hypothetical friend of 10 has been shown to have at least ten distinct prime factors.\n\n**Full solution or refutation.**\n\nNo integer other than 10 with abundancy index 9/5, and no proof of its nonexistence, was verified.\n\n**What remains.**\n\nFind a friend of 10 or prove that sigma(n)/n=9/5 forces n=10.\n\n**Sources checked.**\n\n- S. R. L. S. de Araujo, Each friend of 10 has at least 10 nonidentical prime factors, arXiv:2310.15900. (primary): https://arxiv.org/abs/2310.15900\n Evidence used: States both the current unresolved status and the ten-prime-factor lower bound.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 334, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1262, "problem_number": "NT-055", "title": "Erdős Conjecture on Arithmetic Progressions", "statement": "If the sum of reciprocals of a set of positive integers diverges, does the set contain arbitrarily long arithmetic progressions?", "background": "Erdős conjectured that if $A \\subseteq \\mathbb{N}$ and $\\sum_{a \\in A} 1/a = \\infty$, then $A$ contains arithmetic progressions of arbitrary length. This strengthens Szemerédi's theorem, which only requires positive density. The conjecture remains open even for progressions of length 3. In 2020, Bloom and Sisask made major progress by proving that if $\\sum_{a \\in A, a \\leq N} 1/a \\geq (\\log N)^{c \\log \\log \\log N}$ for some $c$, then $A$ contains a 3-term arithmetic progression. This connects additive combinatorics, harmonic analysis, and analytic number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The divergent-reciprocal-sum Erdős conjecture is proved for three-term progressions, but its arbitrary-length form remains open.\n\n**Verified partial progress.**\n\n- Bloom and Sisask's improved Roth bound implies every 3-AP-free set has convergent reciprocal sum.\n\n**Full solution or refutation.**\n\nThe length-three case is settled; no proof for every prescribed progression length was verified.\n\n**What remains.**\n\nExtend the reciprocal-sum implication beyond three-term arithmetic progressions.\n\n**Sources checked.**\n\n- T. Bloom and O. Sisask, Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions, Annals of Mathematics 202 (2025), 1--61. (primary): https://doi.org/10.4007/annals.2025.202.1.1\n Evidence used: Its 3-AP-free-set bound yields convergence of the reciprocal sum and hence the length-three case.\n- Institute for Advanced Study event description, Mathematical Conversations: Olof Sisask. (authoritative_secondary): https://www.ias.edu/math/events/mathematical-conversations-189\n Evidence used: Explicitly describes the work as proving the simplest, length-three case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 534, "favorite_count": 42, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1263, "problem_number": "NT-056", "title": "Erdős-Turán Conjecture on Additive Bases", "statement": "If $B$ is an additive basis of order 2, must the representation function tend to infinity?", "background": "An additive basis of order 2 is a set $B$ such that every sufficiently large integer can be written as the sum of two elements of $B$. The representation function $r_B(n)$ counts the number of ways to write $n$ as $b_1 + b_2$ with $b_1, b_2 \\in B$. Erdős and Turán conjectured in 1941 that if $B$ is an additive basis of order 2, then $r_B(n)$ must tend to infinity. This has been proven for various special bases, but the general case remains open. The conjecture connects additive number theory, combinatorics, and the structure of thin bases.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The standard Erdős--Turán additive-basis conjecture remains open, but it asks for unbounded representation function, whereas this record asks for convergence to infinity.\n\n**Verified partial progress.**\n\n- Recent literature continues to call the standard unboundedness/limsup-infinity form unsolved.\n\n**Full solution or refutation.**\n\nNo theorem settling the stronger literal 'tend to infinity' assertion was verified.\n\n**What remains.**\n\nFirst clarify whether the intended question is unboundedness or eventual divergence; then settle the selected formulation.\n\n**Sources checked.**\n\n- M. B. Nathanson, Solutions to some problems on unique representation bases, J. Combin. Theory Ser. A 221 (2026), 106166. (primary): https://doi.org/10.1016/j.jcta.2026.106166\n Evidence used: Its introduction identifies the Erdős--Turán conjecture as unsolved and states it in the unbounded representation-function form.\n\n**Review notes.** Statement retained; no normalization of 'tend to infinity' to the standard weaker formulation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 456, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1265, "problem_number": "NT-058", "title": "Lander-Parkin-Selfridge Conjecture", "statement": "If the sum of $m$ $k$-th powers equals the sum of $n$ $k$-th powers, must $m + n \\geq k$?", "background": "This conjecture generalizes Fermat's Last Theorem to sums of powers. It states that if $a_1^k + \\cdots + a_m^k = b_1^k + \\cdots + b_n^k$ with positive integers and the two sums are different, then $m + n \\geq k$. Euler conjectured the stronger statement that at least $k$ $k$-th powers are needed, but this was disproved: $27^5 + 84^5 + 110^5 + 133^5 = 144^5$ (counterexample with $k=5$, $m=4$, $n=1$). The weaker LPS conjecture remains open for $k \\geq 4$ and has implications for Diophantine equations and additive number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Lander--Parkin--Selfridge conjecture remains open.\n\n**Verified partial progress.**\n\n- Known counterexamples to Euler's stronger one-sided sum-of-powers conjecture do not violate the stated total-term lower bound.\n\n**Full solution or refutation.**\n\nNo nontrivial counterexample to the m+n>=k assertion, nor a general proof, was verified.\n\n**What remains.**\n\nProve the total-term bound or exhibit a counterexample.\n\n**Sources checked.**\n\n- Lander, Parkin, and Selfridge conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Lander%2C_Parkin%2C_and_Selfridge_conjecture\n Evidence used: Distinguishes the open total-term conjecture from disproved stronger Euler variants.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 489, "favorite_count": 37, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1266, "problem_number": "NT-059", "title": "Lemoine's Conjecture", "statement": "Can every odd integer greater than 5 be expressed as the sum of an odd prime and an even semiprime?", "background": "Proposed by Émile Lemoine in 1894, this conjecture states that every odd number $n > 5$ can be written as $n = p + 2q$ where $p$ and $q$ are primes. An even semiprime is twice a prime. For example, $27 = 13 + 2(7)$, $31 = 19 + 2(6)$ is invalid since 6 isn't prime, but $31 = 5 + 2(13)$ works. This is weaker than Goldbach's conjecture (which implies Lemoine's). Verified computationally to very large numbers, but no proof exists. It connects prime distribution, additive representations, and the Goldbach problem.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Lemoine's conjecture remains unproved, with finite verification extended to 10^13.\n\n**Verified partial progress.**\n\n- A 2024 computational verification checks the conjecture through 10^13.\n\n**Full solution or refutation.**\n\nNo all-odd-integers proof was verified.\n\n**What remains.**\n\nProve that every odd n>5 has n=p+2q with p,q prime, or find a counterexample.\n\n**Sources checked.**\n\n- E. Juhász, Empirical Verification of a Generalization of Goldbach's Conjecture, Journal of Integer Sequences 27 (2024). (primary): https://cs.uwaterloo.ca/journals/JIS/VOL27/Juhasz/juhasz3.html\n Evidence used: Reports verification of Lemoine's conjecture to 10^13.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1267, "problem_number": "NT-060", "title": "Recamán's Sequence Completeness", "statement": "Does every nonnegative integer appear in Recamán's sequence?", "background": "Recamán's sequence starts with $a_0 = 0$ and follows the rule: $a_n = a_{n-1} - n$ if that value is positive and not already in the sequence, otherwise $a_n = a_{n-1} + n$. This produces: 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, ... Named after Colombian mathematician Bernardo Recamán Santos, this sequence has been computed to millions of terms, but it remains unknown whether every nonnegative integer appears. Some values appear very late or may never appear. This connects integer sequences, graph theory, and computational number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether every nonnegative integer occurs in the specified Recamán sequence.\n\n**Verified partial progress.**\n\n- The original conjectural confidence was later withdrawn; extensive finite computations cannot decide completeness.\n\n**Full solution or refutation.**\n\nNo proof of completeness or a missing integer was verified.\n\n**What remains.**\n\nProve surjectivity onto the nonnegative integers or produce an omitted integer.\n\n**Sources checked.**\n\n- OEIS Wiki, Recamán's sequence (A005132). (maintained_tracker): https://oeis.org/wiki/Recam%C3%A1n%27s_sequence\n Evidence used: Documents the exact recurrence and that Sloane later recanted certainty in the completeness conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 40, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1268, "problem_number": "NT-061", "title": "Skolem Problem", "statement": "Can an algorithm determine if a constant-recursive sequence contains a zero?", "background": "A constant-recursive sequence satisfies a linear recurrence with constant coefficients, like the Fibonacci sequence. The Skolem problem asks whether there exists an algorithm to determine if such a sequence ever equals zero. This is known to be decidable for sequences of order up to 4, but the general problem remains open. The Positivity Problem (whether all terms are positive) and Ultimate Positivity (whether terms are eventually all positive) are related variants. This connects computability theory, Diophantine approximation, and decidability in number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general Skolem problem remains open, but decidability is now complete for algebraic linear recurrences of order at most four; order five is open.\n\n**Verified partial progress.**\n\n- Bacik completed the remaining order-four algebraic case in 2024/25.\n\n**Full solution or refutation.**\n\nNo general decision algorithm, or undecidability result, was verified.\n\n**What remains.**\n\nSettle decidability for order-five and then arbitrary-order linear recurrence sequences.\n\n**Sources checked.**\n\n- P. Bacik, Completing the picture for the Skolem Problem on order-4 linear recurrence sequences, TheoretiCS 4 (2025). (primary): https://theoretics.episciences.org/14219\n Evidence used: Proves decidability for all algebraic LRS of order at most four.\n- J. Ouaknine, The Skolem Landscape, ICALP 2023 invited talk. (authoritative_secondary): https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.5\n Evidence used: Places the general problem and low-order decidability frontier in context.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1269, "problem_number": "NT-062", "title": "Waring's Problem: Exact Values", "statement": "What are the exact values of $g(k)$ and $G(k)$ for all $k$ in Waring's problem?", "background": "Waring's problem concerns representing integers as sums of $k$-th powers. Let $g(k)$ be the minimum number such that every positive integer can be written as a sum of at most $g(k)$ $k$-th powers, allowing any number of terms. Let $G(k)$ be the same but excluding a finite set of exceptions. We know $g(2)=4$ (Lagrange), $G(2)=4$, $g(3)=9$, $G(3)=4$, $g(4)=19$, $G(4)=16$. For general $k$, Hilbert proved $g(k)$ exists but exact values remain unknown for most $k$. This is a central problem in additive number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Waring's g(k) has an essentially explicit general formula, but exact G(k) is not known for all k.\n\n**Verified partial progress.**\n\n- DLMF records the standard definitions and the known formula framework for g(k); modern efficient-congruencing and decoupling methods yield strong general bounds for G(k).\n\n**Full solution or refutation.**\n\nThe literal request for exact values of both functions for every k is unresolved because G(k) is not fully determined.\n\n**What remains.**\n\nDetermine G(k) exactly for all k and resolve any remaining exceptional issue in the g(k) formula.\n\n**Sources checked.**\n\n- NIST Digital Library of Mathematical Functions, §27.13, Waring's problem. (authoritative_secondary): https://dlmf.nist.gov/27.13\n Evidence used: Defines g(k), G(k), and summarizes the established Waring theory.\n- R. C. Vaughan and T. D. Wooley, Waring's Problem: A Survey. (authoritative_secondary): https://personal.science.psu.edu/rcv4/personal/Publications/vaughan.pdf\n Evidence used: Surveys exact and asymptotic progress, including bounds for G(k).\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 44, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1270, "problem_number": "NT-063", "title": "Density of Ulam Numbers", "statement": "Do the Ulam numbers have a positive density?", "background": "The Ulam numbers start with 1, 2, and each subsequent number is the smallest integer that can be expressed as the sum of two distinct earlier Ulam numbers in exactly one way: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ... Named after Stanisław Ulam, these numbers appear to have density around 0.07, but whether the density exists and is positive remains unproven. Related questions about their growth rate and distribution connect to additive combinatorics, unique representation bases, and computational number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether the classical Ulam numbers have a positive natural density is not securely settled in checked literature.\n\n**Verified partial progress.**\n\n- Elementary spacing gives a nontrivial upper-density bound. A 2020 arXiv manuscript claims zero natural density, but no strong independent or published verification was found.\n\n**Full solution or refutation.**\n\nThe preprint-level zero-density claim was not treated as a settled resolution.\n\n**What remains.**\n\nProvide a verified proof that the natural density is zero, or establish a positive-density limit.\n\n**Sources checked.**\n\n- T. Agama, Ulam numbers have zero density, arXiv:2007.02697. (primary): https://arxiv.org/abs/2007.02697\n Evidence used: Claims zero natural density; included as an unverified claim, not as a resolution.\n- Ulam number overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Ulam_number\n Evidence used: Records the standard sequence and historic density question.\n\n**Review notes.** No source alteration; unverified preprint claim deliberately not promoted to solved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1271, "problem_number": "NT-064", "title": "Class Number Problem", "statement": "Are there infinitely many real quadratic number fields with unique factorization?", "background": "A number field has unique factorization if every nonzero element factors uniquely into irreducibles. For real quadratic fields $\\mathbb{Q}(\\sqrt{d})$ with $d > 0$ square-free, unique factorization is equivalent to having class number 1. Gauss conjectured infinitely many such fields exist. While infinitely many imaginary quadratic fields (class number 1) were ruled out, the real case remains open. Computational evidence strongly supports the conjecture, but a proof eludes us. This connects algebraic number theory, class field theory, and the distribution of number fields.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It is still unknown whether infinitely many real quadratic fields have class number one.\n\n**Verified partial progress.**\n\n- Many individual fields and special families have been computed or analyzed; Cohen--Lenstra heuristics predict a positive proportion.\n\n**Full solution or refutation.**\n\nNo infinitude theorem was verified.\n\n**What remains.**\n\nProve infinitely many real quadratic fields have class number one.\n\n**Sources checked.**\n\n- List of number fields with class number one. (authoritative_secondary): https://en.wikipedia.org/wiki/List_of_number_fields_with_class_number_one\n Evidence used: Explicitly records that infinitude in the real quadratic case is unknown.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 478, "favorite_count": 36, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1272, "problem_number": "NT-065", "title": "Hilbert's Twelfth Problem", "statement": "Can the Kronecker-Weber theorem on abelian extensions of $\\mathbb{Q}$ be extended to any base number field?", "background": "The Kronecker-Weber theorem states that every abelian extension of the rationals $\\mathbb{Q}$ is contained in a cyclotomic field (generated by roots of unity). Hilbert's 12th problem asks for an analogous explicit construction of abelian extensions of arbitrary number fields. For imaginary quadratic fields, complex multiplication provides a partial answer using elliptic curves and modular functions. For general number fields, the problem remains largely open despite over a century of work. This is fundamental to class field theory and arithmetic geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hilbert's twelfth problem remains open in general, while complex multiplication solves the imaginary-quadratic case and Shimura theory covers CM fields.\n\n**Verified partial progress.**\n\n- Recent work using Brumer--Stark units supplies a p-adic construction in broad totally real cases.\n\n**Full solution or refutation.**\n\nNo uniform explicit class-invariant construction for every base number field was verified.\n\n**What remains.**\n\nGive the requested direct explicit construction of maximal abelian extensions for arbitrary number fields.\n\n**Sources checked.**\n\n- Hilbert's twelfth problem overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Hilbert%27s_twelfth_problem\n Evidence used: Distinguishes the solved imaginary-quadratic and CM cases from the open general problem.\n- Recent Progress on Hilbert's 12th Problem, Mathmeetings.net. (authoritative_secondary): https://mathmeetings.net/conferences/view/1468\n Evidence used: Describes Dasgupta--Kakde progress for totally real fields.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 40, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1273, "problem_number": "NT-066", "title": "Leopoldt's Conjecture", "statement": "Does the $p$-adic regulator of an algebraic number field not vanish?", "background": "Leopoldt's conjecture, proposed in 1962, states that the $p$-adic regulator of an algebraic number field $K$ is nonzero for every prime $p$. The regulator measures the \"size\" of the unit group. The conjecture has been verified for abelian extensions of $\\mathbb{Q}$ and many other special cases, but remains open in general. It has deep connections to Iwasawa theory, $p$-adic L-functions, and the structure of class groups. A proof would have significant implications for understanding $p$-adic analytic properties of number fields.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Leopoldt's conjecture is open for general number fields and primes, but is proved for abelian extensions of Q and several further families.\n\n**Verified partial progress.**\n\n- Brumer's p-adic Baker-theorem method proves the abelian cases; recent work proves additional infinite non-abelian families at specified primes.\n\n**Full solution or refutation.**\n\nNo general nonvanishing theorem for all p-adic regulators was verified.\n\n**What remains.**\n\nProve the Leopoldt defect vanishes for every number field and prime.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Leopoldt conjecture. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Leopoldt_conjecture\n Evidence used: Records Brumer's theorem for fields abelian over Q or an imaginary quadratic field.\n- Applications of representation theory and of explicit units to Leopoldt's conjecture, Research in Number Theory (2026). (primary): https://doi.org/10.1007/s40993-026-00717-2\n Evidence used: Provides recent partial results for infinite families.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1274, "problem_number": "NT-067", "title": "Lindelöf Hypothesis", "statement": "For all $\\varepsilon > 0$, does $\\zeta(1/2 + it) = o(t^\\varepsilon)$ as $t \\to \\infty$?", "background": "The Lindelöf hypothesis concerns the growth rate of the Riemann zeta function $\\zeta(s)$ on the critical line $\\text{Re}(s) = 1/2$. It states that for any $\\varepsilon > 0$, we have $|\\zeta(1/2 + it)| = o(t^\\varepsilon)$. This is weaker than the Riemann Hypothesis but still unproven. The best known bound is $O(t^{13/84+\\varepsilon})$ due to Bourgain (2022). The hypothesis has implications for the distribution of primes, zero-free regions of $\\zeta(s)$, and analytic number theory. It connects to moment problems and random matrix theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Lindelöf hypothesis remains open; the best checked subconvexity exponent is 13/84 rather than the conjectural 0.\n\n**Verified partial progress.**\n\n- Bourgain proved zeta(1/2+it) is bounded by t^(13/84+epsilon), improving earlier subconvexity exponents.\n\n**Full solution or refutation.**\n\nNo t^epsilon-for-every-epsilon bound was verified.\n\n**What remains.**\n\nReduce the critical-line growth exponent to zero.\n\n**Sources checked.**\n\n- J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30 (2017). (primary): https://arxiv.org/abs/1408.5794\n Evidence used: Establishes the 13/84 subconvexity exponent.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 545, "favorite_count": 43, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1275, "problem_number": "NT-068", "title": "Hilbert-Pólya Conjecture", "statement": "Do the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator?", "background": "The Hilbert-Pólya conjecture proposes a spectral interpretation of the Riemann Hypothesis: the nontrivial zeros of $\\zeta(s)$ at $1/2 + i\\gamma_n$ correspond to eigenvalues of some self-adjoint operator, with $\\gamma_n$ being the eigenvalues. This would imply RH since eigenvalues of self-adjoint operators are real. Connections to random matrix theory (Montgomery-Dyson) and quantum chaos support this idea. Finding such an operator remains elusive despite attempts involving quantum mechanics, trace formulas, and noncommutative geometry. This bridges analysis, spectral theory, and mathematical physics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No self-adjoint operator with the required zeta-zero spectral realization is known.\n\n**Verified partial progress.**\n\n- Trace-formula and quantum-chaos analogies motivate candidate frameworks, but do not yield a matched operator and spectrum.\n\n**Full solution or refutation.**\n\nNo accepted Hilbert--Pólya construction was verified.\n\n**What remains.**\n\nSpecify and prove self-adjointness and a spectral correspondence whose eigenvalues are exactly the imaginary parts of all nontrivial zeta zeros.\n\n**Sources checked.**\n\n- Hilbert--Pólya conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_conjecture\n Evidence used: Records the conjectural spectral formulation and its open status.\n\n**Review notes.** The literal statement is informal without a specified Hilbert space/operator; it is retained as the standard research programme.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 623, "favorite_count": 51, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1276, "problem_number": "NT-069", "title": "Grand Riemann Hypothesis", "statement": "Do all automorphic L-functions have their nontrivial zeros on the critical line?", "background": "The Grand Riemann Hypothesis extends RH to all automorphic L-functions, a vast class including Dirichlet L-functions, Dedekind zeta functions, and L-functions of modular forms. It asserts that all nontrivial zeros lie on the critical line $\\text{Re}(s) = 1/2$. This would have profound consequences for prime distribution in arithmetic progressions, algebraic number theory, and the Langlands program. The GRH is considered one of the most important unifying conjectures in mathematics, generalizing many individual cases of the Riemann Hypothesis.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The grand Riemann hypothesis for automorphic L-functions remains open.\n\n**Verified partial progress.**\n\n- Zero-free regions, zero-density estimates, and family-averaged one-level-density theorems give partial information, not line-by-line zero location.\n\n**Full solution or refutation.**\n\nNo theorem placing every nontrivial zero of every automorphic L-function on its critical line was verified.\n\n**What remains.**\n\nProve the critical-line assertion for the full stated automorphic class.\n\n**Sources checked.**\n\n- J. Thorner and A. Zaman, Log-free zero density estimates for automorphic L-functions, Algebra & Number Theory 16 (2022). (primary): https://arxiv.org/abs/2004.14410\n Evidence used: Provides unconditional density estimates rather than GRH.\n- Grand Riemann hypothesis overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Grand_Riemann_hypothesis\n Evidence used: States the open extension to automorphic L-functions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 712, "favorite_count": 59, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1277, "problem_number": "NT-070", "title": "Montgomery's Pair Correlation Conjecture", "statement": "Does the pair correlation function of Riemann zeta zeros match that of random Hermitian matrices?", "background": "Montgomery conjectured in 1973 that the statistical distribution of gaps between zeros of the Riemann zeta function matches the pair correlation of eigenvalues from the Gaussian Unitary Ensemble (GUE) of random matrix theory. This remarkable connection between number theory and quantum physics was discovered through numerical experiments and Dyson's insights. The conjecture has been partially verified but remains unproven. It suggests deep links between prime numbers, quantum chaos, and statistical mechanics, forming a cornerstone of the modern approach to understanding zeta zeros.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Montgomery's full pair-correlation conjecture remains open, though its Fourier-transform form is known conditionally on RH for support |alpha|<1.\n\n**Verified partial progress.**\n\n- Montgomery proved the |alpha|<1 range assuming RH; extensive numerical statistics agree with the random-matrix prediction.\n\n**Full solution or refutation.**\n\nNo theorem proves the full limiting pair correlation over all test ranges.\n\n**What remains.**\n\nEstablish the full pair-correlation limit, ideally without assuming RH.\n\n**Sources checked.**\n\n- H. L. Montgomery, The pair correlation of zeros of the zeta function, Proc. Symp. Pure Math. 24 (1973), 181--193. (primary): https://doi.org/10.1090/pspum/024/0338455\n Evidence used: Proves the conditional restricted-range result and formulates the conjecture.\n- Montgomery's pair correlation conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Montgomery%27s_pair_correlation_conjecture\n Evidence used: Summarizes the proven restricted range and open strong form.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 46, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1278, "problem_number": "NT-071", "title": "Dirichlet's Divisor Problem", "statement": "What is the optimal exponent in the error term for the divisor summatory function?", "background": "Let $D(x) = \\sum_{n \\leq x} d(n)$ where $d(n)$ counts the divisors of $n$. Dirichlet proved $D(x) = x \\log x + (2\\gamma - 1)x + \\Delta(x)$ where $\\gamma$ is Euler's constant and $\\Delta(x)$ is the error. The problem asks for the infimum $\\theta$ such that $\\Delta(x) = O(x^\\theta)$. It is known that $1/4 \\leq \\theta < 131/416 \\approx 0.314903$. The Riemann Hypothesis would imply $\\theta \\leq 1/4 + \\varepsilon$ for any $\\varepsilon > 0$, but proving this is extremely difficult. This connects analytic number theory, the Riemann zeta function, and lattice point problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The optimal Dirichlet-divisor error exponent remains open: Huxley's 131/416 upper exponent is above the conjectural 1/4.\n\n**Verified partial progress.**\n\n- The classical omega results force an error of order near x^(1/4) infinitely often, and Huxley's upper bound is 131/416 plus epsilon.\n\n**Full solution or refutation.**\n\nNo proof of the conjectural quarter-power upper bound was verified.\n\n**What remains.**\n\nLower the upper exponent to 1/4, or determine the exact optimal exponent.\n\n**Sources checked.**\n\n- NIST Digital Library of Mathematical Functions, §27.11, Dirichlet divisor problem. (authoritative_secondary): https://dlmf.nist.gov/27.11\n Evidence used: Lists the still-open problem and Huxley's 131/416 exponent.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1279, "problem_number": "GEO-033", "title": "Erdős-Ulam Problem", "statement": "Is there a dense set of points in the plane with all pairwise distances rational?", "background": "Proposed by Paul Erdős and Stanisław Ulam, this problem asks whether there exists a dense subset of the Euclidean plane (dense in the usual topology) such that the distance between any two points is a rational number. While finite and countable dense sets with rational distances are known (like rational points on a circle), an everywhere-dense set remains undiscovered. The problem connects geometry, Diophantine equations, and the structure of rational points. It has implications for understanding constraints on rational distance sets.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Erdős-Ulam problem remains open unconditionally; strong curve restrictions and conditional negative solutions are known, and a 2025 paper sharply resolves the three-prescribed-point subproblem.\n\n**Verified partial progress.**\n\n- Solymosi and de Zeeuw proved that an irreducible algebraic curve carrying an infinite rational-distance set must be a line or a circle.\n- Shaffaf proved that Bombieri–Lang implies there is no dense rational-distance subset of the plane; Ascher, Braune, and Turchet derive uniform bounds in general position under Lang's conjecture.\n- Corvaja, Turchet, and Zannier prove a necessary-and-sufficient criterion for points at rational distance from three prescribed noncollinear points to be dense; it always holds for three rational points.\n- For four or more prescribed points, the associated surfaces are simply connected and of general type, but the required non-density of rational points is presently conjectural.\n\n**Full solution or refutation.**\n\nNo unconditional construction of a plane-dense rational-distance set and no unconditional proof of impossibility was verified.\n\n**What remains.**\n\nResolve the distribution of rational points on the general-type distance surfaces arising from four prescribed points, or find a fundamentally different construction of a plane-dense rational-distance set.\n\n**Sources checked.**\n\n- P. Corvaja, A. Turchet, and U. Zannier, Rational distances from given rational points in the plane, Geometriae Dedicata 219 (2025), article 59. (primary): https://doi.org/10.1007/s10711-025-01019-0\n Evidence used: Explicitly says the Erdős-Ulam problem remains open and proves the density criterion for three prescribed points while identifying the general-type obstruction for r >= 4.\n- J. Solymosi and F. de Zeeuw, On a question of Erdős and Ulam, Discrete & Computational Geometry 43 (2010), 393–401. (primary): https://arxiv.org/abs/0806.3095\n Evidence used: Proves that only lines and circles among irreducible algebraic curves can contain infinite rational-distance sets.\n- J. Shaffaf, A Solution of the Erdős–Ulam Problem on Rational Distance Sets Assuming the Bombieri–Lang Conjecture, Discrete & Computational Geometry 60 (2018), 87–95. (primary): https://arxiv.org/abs/1501.00159\n Evidence used: Theorem 2 gives a negative answer conditional on Bombieri–Lang.\n- K. Ascher, L. Braune, and A. Turchet, The Erdős-Ulam problem, Lang's conjecture, and uniformity, Bulletin of the London Mathematical Society 52 (2020), 1053–1063. (primary): https://arxiv.org/abs/1901.02616\n Evidence used: Proves, conditional on Lang's conjecture, a uniform cardinality bound for rational-distance sets in general position.\n\n**Review notes.** The background is self-contradictory: a countable dense plane set with rational pairwise distances would solve the problem. Circle examples are dense only on the circle, and finite sets cannot be dense in the plane. A 2023 repository preprint claiming a construction was not accepted over later peer-reviewed literature that still lists the problem as open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 478, "favorite_count": 36, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1281, "problem_number": "NT-073", "title": "Four Exponentials Conjecture", "statement": "If $x_1, x_2$ are linearly independent over $\\mathbb{Q}$ and $y_1, y_2$ are linearly independent over $\\mathbb{Q}$, is at least one of $e^{x_1 y_1}, e^{x_1 y_2}, e^{x_2 y_1}, e^{x_2 y_2}$ transcendental?", "background": "This conjecture, a consequence of Schanuel's conjecture, asserts that under the stated conditions, at least one of the four exponentials must be transcendental. The six exponentials theorem (proven) states that if $x_1, x_2, x_3$ are $\\mathbb{Q}$-linearly independent and $y_1, y_2$ are $\\mathbb{Q}$-linearly independent, then among the six values $e^{x_i y_j}$, at least one is transcendental. The four exponentials conjecture would strengthen this. It connects exponential Diophantine equations, transcendence theory, and algebraic independence.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The four exponentials conjecture remains open; the six exponentials theorem does not settle the four-value assertion.\n\n**Verified partial progress.**\n\n- The six exponentials theorem proves the analogous conclusion for three Q-linearly independent x-values and two Q-linearly independent y-values.\n- Waldschmidt's 2023 expert chapter places the four exponentials problem within the consequences and variants of Schanuel's conjecture.\n\n**Full solution or refutation.**\n\nNo accepted proof or counterexample for the standard complex-number formulation was found.\n\n**What remains.**\n\nProve that one of the four exponentials is transcendental for all qualifying complex x_i and y_j, or give a counterexample.\n\n**Sources checked.**\n\n- Michel Waldschmidt, The Four Exponentials Problem and the Schanuel Conjecture, in Mathematics Going Forward, Lecture Notes in Mathematics 2313 (Springer, 2023), 579-592. (authoritative_secondary): https://webusers.imj-prg.fr/~michel.waldschmidt/texts.html\n Evidence used: The expert chapter and author's publication record treat the four exponentials assertion as an open problem and distinguish it from proved exponential theorems.\n\n**Review notes.** The statement omits the ambient domain; the standard conjecture quantifies over complex numbers.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1282, "problem_number": "NT-074", "title": "Irrationality of Euler's Constant", "statement": "Is the Euler-Mascheroni constant $\\gamma$ irrational?", "background": "Euler's constant $\\gamma = \\lim_{n \\to \\infty} (1 + 1/2 + 1/3 + \\cdots + 1/n - \\ln n) \\approx 0.5772$ appears throughout mathematics but its arithmetic nature remains mysterious. It is not even known whether $\\gamma$ is irrational, let alone transcendental. While computational evidence suggests irrationality (verified to billions of digits), no proof exists. The problem has resisted attack for over 250 years. Progress would require new techniques in transcendental number theory and might illuminate the nature of other constants like $\\zeta(3)$.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether the Euler-Mascheroni constant is irrational.\n\n**Verified partial progress.**\n\n- Recent algorithm-assisted work develops new identities and relations among constants but explicitly retains the arithmetic nature of gamma as unresolved.\n\n**Full solution or refutation.**\n\nNo accepted proof that gamma is irrational or rational was found.\n\n**What remains.**\n\nProve gamma is irrational, or establish an exact rational value.\n\n**Sources checked.**\n\n- Y. Elimelech et al., Algorithm-assisted discovery of an intrinsic order among mathematical constants, Proceedings of the National Academy of Sciences 121 (2024), e2321440121, DOI 10.1073/pnas.2321440121. (primary): https://doi.org/10.1073/pnas.2321440121\n Evidence used: Explicitly lists irrationality of the Euler-Mascheroni constant among unresolved arithmetic questions.\n\n**Review notes.** The background claim that irrationality was computationally verified to billions of digits is invalid: finite decimal computation cannot verify irrationality.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 712, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1283, "problem_number": "NT-075", "title": "Transcendence of Apéry's Constant", "statement": "Is $\\zeta(3) = 1 + 1/8 + 1/27 + 1/64 + \\cdots$ transcendental?", "background": "Apéry's constant $\\zeta(3) \\approx 1.202$ is the value of the Riemann zeta function at 3. Roger Apéry proved its irrationality in 1978 using ingenious continued fraction methods, surprising the mathematical community. Whether $\\zeta(3)$ is transcendental remains unknown. More generally, the transcendence of $\\zeta(2k+1)$ for integer $k \\geq 1$ is open (except $\\zeta(1)$ which diverges). Rivoal (2000) proved infinitely many $\\zeta(2k+1)$ are irrational, and at least one of $\\zeta(5), \\zeta(7), \\zeta(9), \\zeta(11)$ is irrational, but transcendence is far harder.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Apery's theorem proves zeta(3) irrational, but its transcendence remains open.\n\n**Verified partial progress.**\n\n- Apery established irrationality of zeta(3).\n- Rivoal proved that infinitely many odd zeta values are irrational, while Zudilin proved that at least one of zeta(5), zeta(7), zeta(9), and zeta(11) is irrational.\n\n**Full solution or refutation.**\n\nNo accepted transcendence proof or algebraic evaluation for zeta(3) was found.\n\n**What remains.**\n\nProve zeta(3) is transcendental or determine an algebraic relation refuting that expectation.\n\n**Sources checked.**\n\n- Wadim Zudilin, One of the numbers zeta(5), zeta(7), zeta(9), zeta(11) is irrational, Russian Mathematical Surveys 56 (2001), 774-776, DOI 10.1070/RM2001v056n04ABEH000427. (primary): https://doi.org/10.1070/RM2001v056n04ABEH000427\n Evidence used: Establishes the four-value irrationality result that the dataset incorrectly attributes to Rivoal.\n- Wadim Zudilin, Arithmetic of linear forms involving odd zeta values, Journal de Theorie des Nombres de Bordeaux 16 (2004), 251-291; arXiv:math/0206176. (primary): https://arxiv.org/abs/math/0206176\n Evidence used: Develops the relevant irrationality and linear-independence results for odd zeta values; these fall short of transcendence of zeta(3).\n\n**Review notes.** Correct the background attribution: the four-value theorem is Zudilin's, not Rivoal's.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 589, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1284, "problem_number": "NT-076", "title": "Littlewood Conjecture", "statement": "For any two real numbers $\\alpha, \\beta$, does $\\liminf_{n \\to \\infty} n \\|n\\alpha\\| \\|n\\beta\\| = 0$?", "background": "Proposed by John Edensor Littlewood around 1930, where $\\|x\\|$ denotes the distance from $x$ to the nearest integer. The conjecture asserts a simultaneous approximation property: for any pair of real numbers, infinitely many integers $n$ exist such that both $n\\alpha$ and $n\\beta$ are simultaneously close to integers, with the product of distances approaching zero. While verified for many cases (algebraic numbers, certain combinations), the general conjecture remains open. Einsiedler, Katok, and Lindenstrauss (2006) proved the set of counterexamples has Hausdorff dimension zero, suggesting counterexamples are rare if they exist.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Littlewood's conjecture remains open, but the set of exceptional pairs has Hausdorff dimension zero.\n\n**Verified partial progress.**\n\n- Einsiedler, Katok, and Lindenstrauss proved that the exceptional set of pairs violating the conjectured liminf conclusion has Hausdorff dimension zero.\n- A 2026 primary paper continues to identify that theorem as the state-of-the-art general result and the universal conjecture as open.\n\n**Full solution or refutation.**\n\nHausdorff dimension zero does not imply that the exceptional set is empty, so the theorem is not a full solution.\n\n**What remains.**\n\nProve the exceptional set is empty or exhibit a pair in it.\n\n**Sources checked.**\n\n- Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Annals of Mathematics 164 (2006), 513-560. (primary): https://annals.math.princeton.edu/2006/164-2/p04\n Evidence used: Proves the Hausdorff-dimension-zero theorem for the exceptional set.\n- Thomas F. Robertson, Combinatorics on number walls and the P(t)-adic Littlewood conjecture, Mathematika (2026), DOI 10.1112/mtk.70064. (primary): https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/mtk.70064\n Evidence used: Treats the real Littlewood conjecture as open and describes the dimension-zero theorem as the state of the art.\n\n**Review notes.** A recent disproof of a uniform variant was not conflated with the classical conjecture in this record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 456, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1285, "problem_number": "NT-077", "title": "Integer Factorization in Polynomial Time", "statement": "Can integer factorization be solved in polynomial time on a classical computer?", "background": "The integer factorization problem asks: given a composite number $n$, find its prime factors. The best known classical algorithm (general number field sieve) runs in sub-exponential time $\\exp(O((\\ln n)^{1/3}(\\ln \\ln n)^{2/3}))$. Whether a polynomial-time classical algorithm exists is unknown and has profound implications for cryptography (RSA security relies on factorization hardness). Shor's algorithm solves factorization in polynomial time on quantum computers, but practical quantum computers don't yet exist. The problem connects computational complexity, cryptography, and number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No polynomial-time classical factoring algorithm or lower bound excluding one is known.\n\n**Verified partial progress.**\n\n- The general number field sieve gives a subexponential heuristic-time classical method, not a polynomial-time method.\n- Shor proved polynomial-time integer factorization in the quantum circuit model.\n\n**Full solution or refutation.**\n\nThe quantum algorithm changes the computational model and therefore does not resolve the classical question.\n\n**What remains.**\n\nGive a classical polynomial-time factoring algorithm under a specified model and success criterion, or prove an appropriate complexity lower bound.\n\n**Sources checked.**\n\n- Peter W. Shor, Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM Journal on Computing 26 (1997), 1484-1509; arXiv:quant-ph/9508027. (primary): https://arxiv.org/abs/quant-ph/9508027\n Evidence used: Proves polynomial-time factorization for a quantum computer and contrasts this with the believed classical difficulty.\n- NIST Digital Library of Mathematical Functions, section 27.19, Methods of Computation: Factorization. (authoritative_secondary): https://dlmf.nist.gov/27.19\n Evidence used: Maintained authoritative survey of classical factorization algorithms, including number-field-sieve methods.\n\n**Review notes.** The record should specify deterministic versus randomized time, the bit model, and the required success guarantee.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 734, "favorite_count": 61, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1286, "problem_number": "NT-078", "title": "Beal's Conjecture", "statement": "For $A^x + B^y = C^z$ with $x, y, z > 2$, must $A$, $B$, and $C$ share a common prime factor?", "background": "Proposed by banker and amateur mathematician Andrew Beal in 1993, this conjecture generalizes Fermat's Last Theorem. It asserts that if $A^x + B^y = C^z$ where $A, B, C, x, y, z$ are positive integers with $x, y, z > 2$, then $A$, $B$, and $C$ must have a common prime factor. For example, $3^3 + 6^3 = 3^5$ satisfies this since all share factor 3. Beal has offered a prize of $1 million for a proof or counterexample. The conjecture is equivalent to saying no solutions exist when $A$, $B$, $C$ are coprime. This connects Fermat's Last Theorem, the abc conjecture, and exponential Diophantine equations.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Beal's conjecture remains open and the official prize continues to solicit a proof or counterexample.\n\n**Verified partial progress.**\n\n- The official AMS prize statement preserves the common-prime-factor formulation and remains active.\n\n**Full solution or refutation.**\n\nNo accepted proof or primitive counterexample was found.\n\n**What remains.**\n\nProve every positive integer solution with all exponents greater than two has gcd(A,B,C)>1, or find a solution with gcd(A,B,C)=1.\n\n**Sources checked.**\n\n- American Mathematical Society, The Beal Prize, official prize description and current call for nominations. (maintained_tracker): https://www.jointmathematicsmeetings.org/prizes-awards/paview.cgi?parent_id=41\n Evidence used: States the exact conjecture and continues to offer the prize for a proof or counterexample, supporting unresolved status.\n\n**Review notes.** The background's word coprime should mean gcd(A,B,C)=1, not pairwise coprime.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 712, "favorite_count": 59, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1289, "problem_number": "NT-081", "title": "Fermat-Catalan Conjecture", "statement": "Are there finitely many solutions to $a^m + b^n = c^k$ with coprime $a,b,c$ and $1/m + 1/n + 1/k < 1$?", "background": "This conjecture generalizes both Fermat's Last Theorem and the Catalan-Mersenne conjecture. It asserts that the equation $a^m + b^n = c^k$ has only finitely many solutions in coprime positive integers $a,b,c$ and integers $m,n,k \\geq 2$ satisfying $1/m + 1/n + 1/k < 1$. Ten solutions are known, including $1^m + 2^3 = 3^2$, $2^5 + 7^2 = 3^4$, and $17^3 + 2^{7\\cdot 13^3} = 71^2 \\cdot 13^3$. Beal's conjecture and the abc conjecture both imply Fermat-Catalan. The problem is central to exponential Diophantine equations.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The variable-signature Fermat-Catalan conjecture remains open, while Darmon and Granville proved finiteness for each fixed hyperbolic signature.\n\n**Verified partial progress.**\n\n- For every fixed exponent triple satisfying the reciprocal-sum inequality, Darmon and Granville proved that only finitely many primitive solutions occur.\n- The proof uses descent and Faltings' theorem; it does not give uniform finiteness when the exponents vary.\n\n**Full solution or refutation.**\n\nA countable union of fixed-signature finite solution sets need not be finite, so the fixed-signature theorem does not settle the stored conjecture.\n\n**What remains.**\n\nProve finiteness uniformly over all hyperbolic exponent triples or exhibit infinitely many primitive solutions.\n\n**Sources checked.**\n\n- Henri Darmon and Andrew Granville, On the Equations z^m=F(x,y) and Ax^p+By^q=Cz^r, Bulletin of the London Mathematical Society 27 (1995), 513-543, DOI 10.1112/blms/27.6.513. (primary): https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms/27.6.513\n Evidence used: Proves finiteness of proper solutions for each fixed hyperbolic generalized-Fermat signature.\n\n**Review notes.** The background's displayed 17^3 expression is false literally and appears corrupted; Catalan-Mersenne is also suspect terminology.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 634, "favorite_count": 52, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1292, "problem_number": "NT-084", "title": "Bunyakovsky Conjecture", "statement": "Does an irreducible integer polynomial with no fixed prime divisor produce infinitely many primes?", "background": "Proposed by Viktor Bunyakovsky in 1857, this generalizes Dirichlet's theorem on primes in arithmetic progressions. It states that if polynomial $f(x)$ has integer coefficients, positive leading coefficient, is irreducible over integers, and has no common prime divisor of all its values $f(n)$ for positive integers $n$, then $f(x)$ represents infinitely many primes. This would imply infinitely many twin primes (using $f(x) = x$ and $g(x) = x+2$), Sophie Germain primes, and many other families. Despite being over 160 years old, it remains unproven except for linear polynomials (Dirichlet's theorem).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Bunyakovsky's single-polynomial conjecture remains open beyond the degree-one case proved by Dirichlet.\n\n**Verified partial progress.**\n\n- Dirichlet's theorem proves the conjecture for qualifying linear polynomials.\n- Bateman-Horn supplies a broader quantitative heuristic for polynomial prime values but remains conjectural.\n\n**Full solution or refutation.**\n\nNo general theorem for qualifying nonlinear polynomials was found; even n^2+1 remains unresolved.\n\n**What remains.**\n\nProve infinitely many prime values for every qualifying irreducible nonlinear integer polynomial or find a counterexample.\n\n**Sources checked.**\n\n- Bunyakovskii conjecture, Encyclopedia of Mathematics. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Bunyakovskii_conjecture\n Evidence used: States the single-polynomial conjecture and the degree-one Dirichlet case.\n- Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn conjecture: heuristic, history, and applications, Expositiones Mathematicae 38 (2020), 1-26; arXiv:1807.08899. (authoritative_secondary): https://arxiv.org/abs/1807.08899\n Evidence used: Explains Bunyakovsky as a single-polynomial precursor and Bateman-Horn as the simultaneous multi-polynomial extension.\n\n**Review notes.** The background's twin-prime implication is false: separate infinitude for n and n+2 does not imply simultaneous primality.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1293, "problem_number": "NT-085", "title": "Dickson's Conjecture", "statement": "Do finitely many linear forms simultaneously take prime values infinitely often, barring congruence obstructions?", "background": "Proposed by Leonard Eugene Dickson in 1904, this generalizes Dirichlet's theorem and implies many prime conjectures. For linear forms $a_1 + b_1 n, \\ldots, a_k + b_k n$ with each $b_i \\geq 1$, if no congruence condition forces a composite, then infinitely many $n$ exist making all forms simultaneously prime. This would imply: twin primes, Sophie Germain primes, prime triplets, Goldbach's conjecture, and more. It strengthens Bunyakovsky and is a special case of Schinzel's Hypothesis H. No proof exists even for two forms, representing a fundamental gap in our understanding of simultaneous prime values.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Dickson's conjecture remains open for arbitrary prescribed admissible linear forms, despite major bounded-cluster and many-tuple theorems.\n\n**Verified partial progress.**\n\n- Maynard proved bounded intervals containing arbitrarily many primes and strong results for many admissible tuples.\n- These methods do not force simultaneous primality of every form in an arbitrary fixed admissible tuple; the twin-prime pair remains open.\n\n**Full solution or refutation.**\n\nNo proof covering every prescribed admissible family of linear forms was found.\n\n**What remains.**\n\nProve infinitely many simultaneous prime values for every admissible finite family or exhibit a counterexample.\n\n**Sources checked.**\n\n- James Maynard, Small gaps between primes, Annals of Mathematics 181 (2015), 383-413. (primary): https://annals.math.princeton.edu/2015/181-1/p07\n Evidence used: Proves bounded prime clusters and results for admissible tuples, but not full simultaneous primality for each prescribed tuple.\n\n**Review notes.** The background overstates the bare qualitative conjecture's implication for Goldbach and should distinguish no arbitrary-pair theorem from existing many-tuple partial results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1294, "problem_number": "NT-086", "title": "Brocard's Conjecture (Prime Gaps)", "statement": "Are there always at least 4 primes between consecutive squares of primes $p_n^2$ and $p_{n+1}^2$?", "background": "Proposed by Henri Brocard in 1904, this conjecture concerns the density of primes near perfect squares. For consecutive primes $p_n$ and $p_{n+1}$, Brocard conjectured there are always at least 4 primes in the interval $(p_n^2, p_{n+1}^2)$, except for the cases $(2^2, 3^2)$ which contains only one prime (5). Verified computationally to enormous values, but no proof exists. This is stronger than Legendre's conjecture (at least one prime between consecutive squares). It connects prime gaps, Bertrand's postulate generalizations, and the distribution of primes.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The universal statement is false as written: (2^2,3^2) contains only two primes, 5 and 7; the standard conjecture starts at n>=2.\n\n**Verified partial progress.**\n\n- The standard repaired Brocard conjecture asks for at least four primes between p_n^2 and p_(n+1)^2 only for n>=2 and remains open.\n\n**Full solution or refutation.**\n\nTake n=1. The open interval (4,9) contains exactly 5 and 7, so it has two primes rather than at least four.\n\n**What remains.**\n\nCorrect the source statement to n>=2 and then resolve the standard repaired conjecture.\n\n**Sources checked.**\n\n- Eric W. Weisstein, Brocard's Conjecture, MathWorld. (authoritative_secondary): https://mathworld.wolfram.com/BrocardsConjecture.html\n Evidence used: States the standard formulation with n>=2, confirming that the first prime-square interval is excluded.\n\n**Review notes.** The background itself also miscounts (4,9), saying it contains only 5 and omitting 7.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1295, "problem_number": "NT-087", "title": "Agoh-Giuga Conjecture", "statement": "Is $p$ prime if and only if $pB_{p-1} \\equiv -1 \\pmod{p}$ for the Bernoulli number $B_{p-1}$?", "background": "This conjecture combines work of Takashi Agoh (1990) and Giuseppe Giuga (1950), providing a primality criterion via Bernoulli numbers. Bernoulli numbers $B_n$ appear in number theory and analysis. The conjecture states: $p$ is prime iff $pB_{p-1} \\equiv -1 \\pmod{p}$. The forward direction is known (if $p$ prime, the congruence holds). The converse would give a new primality test. Related to Giuga numbers and Wolstenholme's theorem, this connects Bernoulli numbers, primality testing, and modular arithmetic in unexpected ways.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Agoh--Giuga primality criterion remains a conjecture.\n\n**Verified partial progress.**\n\n- A composite solution would have to be simultaneously a Carmichael number and a Giuga number; published searches give enormous lower bounds on a possible counterexample.\n\n**Full solution or refutation.**\n\nNo composite counterexample and no proof excluding all composite integers was verified.\n\n**What remains.**\n\nProve the criterion fails for every composite n, or find a composite Carmichael--Giuga number satisfying it.\n\n**Sources checked.**\n\n- D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn, Giuga's conjecture on primality, American Mathematical Monthly 103 (1996), 40--50. (primary): https://doi.org/10.2307/2975190\n Evidence used: Establishes the structural restrictions and size lower bound for any composite counterexample.\n- Agoh--Giuga conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Agoh%E2%80%93Giuga_conjecture\n Evidence used: States the equivalence with the Giuga form and continuing open status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 334, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1296, "problem_number": "NT-088", "title": "Elliott-Halberstam Conjecture", "statement": "Do primes distribute uniformly in arithmetic progressions up to nearly $x$ (instead of $x^{1/2}$)?", "background": "Proposed in 1968, this strengthens the Bombieri-Vinogradov theorem about primes in arithmetic progressions. For most moduli $q < x^\\theta$, the primes are equidistributed among valid residue classes. Bombieri-Vinogradov proves this for $\\theta < 1/2$. Elliott-Halberstam conjectures it holds for any $\\theta < 1$. This would have dramatic consequences: it implies infinitely many bounded prime gaps exist (a weak form proven by Zhang 2013, then Polymath improved to gap 246). The full conjecture would likely yield bounded gaps near the twin prime level. It connects sieve methods, Goldston-Pintz-Yıldırım techniques, and multiplicative functions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Elliott--Halberstam conjecture remains open; its level-one distribution assertion is much stronger than Bombieri--Vinogradov.\n\n**Verified partial progress.**\n\n- Zhang, Maynard, Tao, and Polymath8 proved bounded prime gaps from weaker distributional input. Conditional on generalized Elliott--Halberstam, their methods would give much sharper gap bounds.\n\n**Full solution or refutation.**\n\nNo proof of the stated near-x uniform distribution over arithmetic progressions was verified.\n\n**What remains.**\n\nEstablish the Elliott--Halberstam level of distribution (or find a failure of it).\n\n**Sources checked.**\n\n- Polymath8 bounded gaps between primes project wiki. (maintained_tracker): https://michaelnielsen.org/polymath/index.php?title=Bounded_gaps_between_primes\n Evidence used: Documents unconditional and Elliott--Halberstam-conditional consequences for bounded prime gaps.\n- K. Soundararajan, Small gaps between prime numbers: the work of Goldston--Pintz--Yıldırım, Bulletin of the AMS 44 (2007), 1--18. (primary): https://doi.org/10.1090/S0273-0979-06-01137-3\n Evidence used: Explains the Elliott--Halberstam distribution hypothesis and its role in prime-gap results.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1297, "problem_number": "ALG-001", "title": "Birch–Tate Conjecture", "statement": "Is there a relation between the order of the center of the Steinberg group and the Dedekind zeta function?", "background": "The Birch–Tate conjecture connects algebraic K-theory to number theory. For a number field $F$, it relates the order of the center of the Steinberg group $\\text{St}(\\mathcal{O}_F)$ (where $\\mathcal{O}_F$ is the ring of integers) to special values of the Dedekind zeta function $\\zeta_F(s)$ at $s = -1$. The conjecture predicts that $|\\text{center}(\\text{St}(\\mathcal{O}_F))| = |\\zeta_F(-1)|$ after appropriate normalization. This would provide a deep connection between algebraic structures and analytic number theory, generalizing classical results about class numbers. It fits into the broader Quillen-Lichtenbaum conjecture framework and has implications for understanding higher K-groups.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Under the intended standard Birch-Tate formulation, the odd part is proved for every totally real field and the full formula is proved for abelian extensions of Q; the nonabelian 2-primary case remains open.\n\n**Verified partial progress.**\n\n- Wiles proved the odd-primary part for arbitrary totally real number fields.\n- The 2-primary part follows from the 2-adic Iwasawa main conjecture and is known for abelian extensions of Q, yielding the full Birch-Tate formula in that class.\n\n**Full solution or refutation.**\n\nThe intended normalized formula is known in broad cases but not for arbitrary nonabelian totally real fields. The extracted sentence itself is an underspecified question rather than a precise conjecture.\n\n**What remains.**\n\nProve the 2-primary Birch-Tate formula for arbitrary nonabelian totally real number fields and separately correct the source record's missing hypotheses and normalization.\n\n**Sources checked.**\n\n- Charles A. Weibel, The K-book: An Introduction to Algebraic K-theory, Chapter VI, section 8, Conjecture 8.6 and Theorem 8.7. (authoritative_secondary): https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.VI.pdf\n Evidence used: Gives the normalized formula, attributes the odd part to Wiles, and identifies the general 2-primary remainder and the abelian case.\n- Encyclopedia of Mathematics, Birch-Tate conjecture. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Birch-Tate_conjecture\n Evidence used: States the standard K_2(O_F), w_2(F), and zeta_F(-1) formula and says the remaining case is the 2-part for nonabelian extensions.\n\n**Review notes.** The source omits the totally real field, s=-1, and w_2(F), and its background's unnormalized equality is false even for Q. K_2 is canonically the Steinberg-to-elementary kernel; a center identification needs stability hypotheses. No wording was repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1300, "problem_number": "ALG-004", "title": "Crouzeix's Conjecture", "statement": "Is $\\|f(A)\\| \\leq 2\\sup_{z \\in W(A)} |f(z)|$ for all matrices $A$ and functions $f$ analytic on the numerical range?", "background": "Michel Crouzeix conjectured in 2004 that for any $n \\times n$ complex matrix $A$ and any function $f$ analytic on the numerical range $W(A) = \\{\\langle Ax, x \\rangle : \\|x\\| = 1\\}$, the matrix norm satisfies $\\|f(A)\\| \\leq 2\\|f\\|_{W(A)}$. The constant 2 is conjectured to be optimal. Crouzeix proved the bound with constant $11.08$, later improved to $1 + \\sqrt{2} \\approx 2.41$ by various authors. The conjecture is verified for $2 \\times 2$ matrices and special classes. It has applications to functional calculus, matrix functions, and numerical analysis. The problem combines complex analysis, operator theory, and linear algebra, and its resolution would clarify fundamental properties of matrix functions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Two complete proof claims appeared within three weeks of this check: Shanmu Jin's 27 July 2026 preprint and Lorist--Schwenninger's 4 August 2026 arXiv paper. The latter is by an established contributor and states a stronger Hilbert-space theorem, but no peer-reviewed or independent status verification was located yet.\n\n**Verified partial progress.**\n\n- Crouzeix--Palencia proved the universal constant 1+sqrt(2) in 2017.\n- Many special matrix classes satisfy the conjectured sharp constant 2.\n- Lorist--Schwenninger's Theorem 3 claims the 2-spectral-set result for bounded Hilbert-space operators and rational functions with poles off the closed numerical range.\n\n**Full solution or refutation.**\n\nA full solution is now claimed independently in two extremely recent preprints, but this triage does not certify either proof before expert or journal verification.\n\n**What remains.**\n\nObtain expert validation or peer-reviewed acceptance of the July/August 2026 proofs, including close checking of the dilation lemma and its application to the double-layer-potential functional calculus.\n\n**Sources checked.**\n\n- Emiel Lorist and Felix Schwenninger, A solution to Crouzeix's conjecture, arXiv:2608.03841 (2026). (primary): https://arxiv.org/abs/2608.03841\n Evidence used: States and proves a theorem asserting the constant-2 bound for bounded Hilbert-space operators; also records an independent contemporaneous proof.\n- Shanmu Jin, The Numerical Range Is a 2-Spectral Set, Preprints.org 202607.1919 (2026). (primary): https://www.preprints.org/manuscript/202607.1919\n Evidence used: Independent preprint claiming the matrix polynomial form of Crouzeix's conjecture.\n- Michel Crouzeix and César Palencia, The Numerical Range is a (1+sqrt(2))-Spectral Set, SIAM Journal on Matrix Analysis and Applications 38 (2017), 649-655. (primary): https://epubs.siam.org/doi/10.1137/17M1116672\n Evidence used: Peer-reviewed proof of the last established general constant, 1+sqrt(2).\n\n**Review notes.** The record's phrase analytic on the numerical range is imprecise; standard formulations use polynomials or functions holomorphic on a neighborhood of W(A). The proof claims are too recent to label solved conservatively.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 278, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1302, "problem_number": "ALG-006", "title": "Perfect Cuboid", "statement": "Does there exist a rectangular cuboid with integer edges, face diagonals, and space diagonal?", "background": "A perfect cuboid would have integer values for all of: three edge lengths $a, b, c$, three face diagonals $\\sqrt{a^2+b^2}, \\sqrt{b^2+c^2}, \\sqrt{c^2+a^2}$, and the space diagonal $\\sqrt{a^2+b^2+c^2}$. This is the 3D generalization of the Pythagorean triple problem (which has infinitely many solutions). Despite extensive computer searches, no perfect cuboid has been found, nor has impossibility been proven. The problem connects to Diophantine equations, elliptic curves, and number theory. Weaker versions exist: edge-perfect cuboids (all edges and face diagonals integer) are known, as are face-perfect and space-perfect variants. The perfect cuboid is problem D18 in Richard Guy's \"Unsolved Problems in Number Theory\" and has attracted amateur and professional attention for over a century.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No accepted perfect cuboid or accepted impossibility proof was located. Recent work supplies stronger search methods, explicit edge-cuboid families, and necessary arithmetic restrictions but does not decide existence.\n\n**Verified partial progress.**\n\n- The rational and integer formulations are equivalent by clearing denominators.\n- de Grey--Gibbs--Helm give two new efficient search algorithms and new two-parameter edge-cuboid families.\n- Their necessary conditions exclude large proportions of face and internal-rectangle aspect ratios from any perfect cuboid.\n\n**Full solution or refutation.**\n\nExistence and nonexistence both remain unproved in accepted literature; unreviewed claimed proofs were not promoted.\n\n**What remains.**\n\nFind a positive rational point satisfying all four square conditions, or prove that the associated Diophantine varieties have no positive rational points.\n\n**Sources checked.**\n\n- Aubrey de Grey, Philip Gibbs, and Louie Helm, Novel required properties of, and efficient algorithms to seek, perfect cuboids, arXiv:2401.06784. (primary): https://arxiv.org/abs/2401.06784\n Evidence used: Treats perfect-cuboid existence as open and proves new necessary conditions, families, and search algorithms.\n- Allan J. MacLeod, Computation of perfect almost-cuboids, Glasnik Matematički 48 (2013), 23-29. (primary): https://doi.org/10.3336/gm.48.1.02\n Evidence used: Peer-reviewed paper describes the perfect cuboid as unsolved and develops elliptic-curve/descent constructions for close variants.\n\n**Review notes.** Multiple unreviewed manuscripts claim nonexistence, but no authoritative acceptance was found. Positive integer edge lengths are implicit in the standard problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 423, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1305, "problem_number": "ALG-009", "title": "Zauner's Conjecture (SIC-POVM)", "statement": "Do symmetric informationally complete POVMs exist in all dimensions?", "background": "Zauner's conjecture, central to quantum information theory, asks whether SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite-dimensional Hilbert spaces. A SIC-POVM in dimension $d$ consists of $d^2$ pure quantum states with pairwise fidelity $1/(d+1)$, forming a regular simplex in quantum state space. These structures optimize quantum measurements and have applications in quantum tomography, cryptography, and foundations. SIC-POVMs are known for dimensions up to 193 and many higher dimensions through numerical construction. Analytic constructions exist for infinitely many dimensions using Weyl-Heisenberg groups and number-theoretic methods. The conjecture connects to algebraic number theory (Stark units, ray class fields), representation theory, and Galois theory. Resolving it would clarify fundamental symmetries in quantum mechanics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** SIC-POVMs are constructed in many dimensions but all-dimensional existence remains unproved.\n\n**Verified partial progress.**\n\n- Exact and high-precision SIC constructions are known for many dimensions.\n\n**Full solution or refutation.**\n\nNo all-dimension theorem or counterexample was verified.\n\n**What remains.**\n\nProve a SIC exists in every finite dimension or find an obstruction.\n\n**Sources checked.**\n\n- SIC-POVM overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/SIC-POVM\n Evidence used: Records Zauner's all-dimensional existence conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 298, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1308, "problem_number": "ALG-012", "title": "Andrews–Curtis Conjecture", "statement": "Can every balanced presentation of the trivial group be transformed to a trivial presentation by Nielsen moves?", "background": "The Andrews–Curtis conjecture, proposed in 1965, concerns group presentations. A balanced presentation has the same number of generators and relators. The trivial presentation is $\\langle x \\mid x \\rangle$. Nielsen transformations on relators include: replacing relator $r$ with $r^{-1}$, with $rs$ for another relator $s$, or conjugating $r$. The question: can any balanced presentation of the trivial group be reduced to the trivial presentation using these moves? Known counter-examples exist for unbalanced presentations (Rapaport). The conjecture is verified for many cases but remains open in general. It connects to the Zeeman conjecture in topology, 4-manifold theory, and algebraic K-theory. A counter-example would have major implications for understanding fundamental groups and 2-complexes.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Andrews--Curtis conjecture remains open; many restricted presentations and stable variants are understood.\n\n**Verified partial progress.**\n\n- Computational searches and group-theoretic invariants test large families without resolving the full conjecture.\n\n**Full solution or refutation.**\n\nNo general Nielsen-move reduction theorem or certified counterexample was verified.\n\n**What remains.**\n\nProve or disprove Andrews--Curtis for all balanced trivial presentations.\n\n**Sources checked.**\n\n- Andrews--Curtis conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Andrews%E2%80%93Curtis_conjecture\n Evidence used: Records the general conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1310, "problem_number": "ALG-014", "title": "Herzog–Schönheim Conjecture", "statement": "Can a finite system of left cosets forming a partition of a group have distinct indices?", "background": "The Herzog–Schönheim conjecture states: if left cosets $g_iH_i$ of subgroups $H_i$ partition a group $G$, then at least two indices $[G:H_i]$ must be equal. Equivalently, you cannot partition a group using cosets of subgroups with all different indices. The conjecture is verified for many cases: finite abelian groups, free groups, and groups with certain structural properties. It has connections to coverings of groups, number theory (covering congruences—Mycielski's conjecture), and additive combinatorics. The problem appears simple but has resisted general proof. A counter-example would be a group with a highly unusual coset structure, and would impact understanding of group factorizations and tiling problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This is the Herzog--Schonheim conjecture and remains open generally.\n\n**Verified partial progress.**\n\n- Many special group/coset-partition cases are known.\n\n**Full solution or refutation.**\n\nNo general distinct-index partition or impossibility proof was verified.\n\n**What remains.**\n\nResolve the universal index-repetition claim.\n\n**Sources checked.**\n\n- Herzog--Schonheim conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Herzog%E2%80%93Sch%C3%B6nheim_conjecture\n Evidence used: Records general open status.\n\n**Review notes.** Duplicate topic retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1318, "problem_number": "ANA-006", "title": "Navier-Stokes Regularity", "statement": "Do smooth initial data for 3D Navier-Stokes equations yield smooth solutions for all time?", "background": "One of the seven Millennium Prize Problems ($1M prize). The 3D Navier-Stokes equations govern fluid flow: $\\partial_t u + (u \\cdot \\nabla)u = \\nu \\Delta u - \\nabla p + f$ with $\\nabla \\cdot u = 0$. Given smooth initial conditions and forcing, do solutions remain smooth globally, or can finite-time singularities develop? In 2D, global regularity is proven. In 3D, existence of weak solutions is known (Leray), but smoothness is open. Partial results establish regularity under smallness conditions or for special data. The problem is central to mathematical fluid dynamics and has deep implications for turbulence, computational fluid dynamics, and the physical validity of the equations. Techniques involve harmonic analysis, functional analysis, and PDE theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Global smoothness versus finite-time singularity for arbitrary smooth 3D incompressible Navier--Stokes data remains open.\n\n**Verified partial progress.**\n\n- Global Leray--Hopf weak solutions are known.\n- Caffarelli--Kohn--Nirenberg partial regularity and many scale-critical conditional regularity criteria sharply constrain possible singularities.\n- Global regularity is known in two dimensions and for several structured/conditional three-dimensional settings.\n\n**Full solution or refutation.**\n\nNo peer-reviewed proof of global smoothness or finite-time blowup for the full Clay formulation was verified.\n\n**What remains.**\n\nProve global smooth existence and uniqueness for every allowed datum, or construct and rigorously verify a finite-time singularity.\n\n**Sources checked.**\n\n- C. L. Fefferman, Existence and Smoothness of the Navier--Stokes Equation, Clay Mathematics Institute problem description. (primary): https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf\n Evidence used: The official statement gives the global-smoothness and breakdown alternatives for 3D incompressible Navier--Stokes.\n- C. R. Doering, The 3D Navier--Stokes Problem, Annual Review of Fluid Mechanics 41 (2009), 109--128. (authoritative_secondary): https://www.annualreviews.org/content/journals/10.1146/annurev.fluid.010908.165218\n Evidence used: The review explains that global smooth 3D solutions are not known and surveys the central obstruction.\n\n**Review notes.** No source alteration; unrefereed solution claims were excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1319, "problem_number": "COMB-001", "title": "1/3–2/3 Conjecture", "statement": "Does every non-totally-ordered finite poset have two elements with probability between 1/3 and 2/3 in random linear extensions?", "background": "For a finite partially ordered set (poset) that is not totally ordered, the 1/3–2/3 conjecture asks: do there always exist elements $x$ and $y$ such that the probability $x$ appears before $y$ in a uniformly random linear extension is strictly between 1/3 and 2/3? Linear extensions are total orderings consistent with the partial order. The conjecture was posed in the 1960s and remains open. It has connections to sorting algorithms, computational complexity, and order theory. Known results: true for many special classes of posets, including series-parallel posets. The conjecture would provide insight into the structure of linear extensions and has applications to average-case analysis of sorting and ranking algorithms.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The 1/3--2/3 conjecture remains open for arbitrary finite non-total posets.\n\n**Verified partial progress.**\n\n- It is proved for width-two, height-two, 5-thin, N-free, series-parallel, semiorder, and several other classes.\n\n**Full solution or refutation.**\n\nNo proof for every finite poset was verified.\n\n**What remains.**\n\nEstablish a balanced incomparable pair in every finite non-total poset.\n\n**Sources checked.**\n\n- K. P. Bogart and J. P. Trotter, The 1/3--2/3 Conjecture for 5-Thin Posets, SIAM J. Discrete Math. 21 (2007). (primary): https://epubs.siam.org/doi/10.1137/0405037\n Evidence used: The article proves the 5-thin case.\n- 1/3--2/3 conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/1/3%E2%80%932/3_conjecture\n Evidence used: Lists established subclasses and records the general problem as unsolved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 234, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1320, "problem_number": "COMB-002", "title": "Lonely Runner Conjecture", "statement": "If $k$ runners with distinct speeds run on a circular track, will each be lonely (distance $\\geq 1/k$ from others) at some time?", "background": "Proposed by J. M. Wills in 1967, this conjecture concerns runners on a unit-length circular track with distinct constant speeds. A runner is \"lonely\" if all other runners are at distance at least $1/k$ away. The conjecture states every runner is lonely at some time. Verified for $k \\leq 7$ runners. The problem has reformulations in terms of Diophantine approximation, view-obstruction (can $k-1$ points block all views from a point to another on a circle?), and number theory. Applications include scheduling, communication protocols, and chromatic number of certain graphs. Proof techniques use continued fractions, geometry of numbers, and combinatorial arguments. The general case remains stubbornly open despite its elementary statement.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Lonely Runner Conjecture is open generally, with recent verified/computer-assisted advances through finitely many runner counts.\n\n**Verified partial progress.**\n\n- Rosenfeld proved the eight- and nine-runner cases in 2025 preprints.\n- Sungkawichai--Trakulthongchai give a computer-assisted proof for 10, 11, and 12 runners.\n\n**Full solution or refutation.**\n\nNo proof for arbitrary k was verified.\n\n**What remains.**\n\nEstablish the required lonely time for all numbers of runners.\n\n**Sources checked.**\n\n- M. Rosenfeld, The lonely runner conjecture holds for eight runners, arXiv:2509.14111 (2025). (primary): https://arxiv.org/abs/2509.14111\n Evidence used: The abstract proves the eight-runner case.\n- T. Sungkawichai and T. Trakulthongchai, Eleven, twelve, and thirteen lonely runners, arXiv:2604.23906 (2026). (primary): https://arxiv.org/abs/2604.23906\n Evidence used: The abstract states computer-assisted proofs for k in {10,11,12} and describes prior verification through 9.\n\n**Review notes.** No source alteration; recent computer-assisted preprints require specialist verification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 312, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1321, "problem_number": "COMB-003", "title": "Union-Closed Sets Conjecture", "statement": "For a finite family of sets closed under unions, must some element appear in at least half the sets?", "background": "Frankl's union-closed sets conjecture (1979) states: if a finite family $\\mathcal{F}$ of sets is closed under pairwise unions (i.e., $A, B \\in \\mathcal{F} \\Rightarrow A \\cup B \\in \\mathcal{F}$), then there exists an element appearing in at least $|\\mathcal{F}|/2$ sets. The conjecture is verified for many special cases: families with at most 50 sets, families where the largest set has at most 11 elements, and various structural conditions. In 2024, significant progress was made proving the conjecture holds when relaxing \"half\" to 0.01% (a weakened version). The problem connects to lattice theory, combinatorics, and has reformulations in terms of posets and Boolean functions. A proof would illuminate the structure of union-closed families.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This duplicate union-closed-set formulation remains open generally.\n\n**Verified partial progress.**\n\n- Reimer's average-set-size theorem and many special-family results give substantial partial progress.\n\n**Full solution or refutation.**\n\nNo verified all-family proof was found.\n\n**What remains.**\n\nResolve Frankl's general frequency assertion.\n\n**Sources checked.**\n\n- D. Reimer, An average set size theorem, Combin. Probab. Comput. 12 (2003), 89--93. (primary): https://arxiv.org/abs/1704.07022\n Evidence used: The linked note summarizes Reimer's theorem in this context.\n- Union-closed sets conjecture overview (accessed 2026-08-17). (maintained_tracker): https://en.wikipedia.org/wiki/Union-closed_sets_conjecture\n Evidence used: Records a continuing open problem with partial results.\n\n**Review notes.** Duplicate topic retained; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 387, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1322, "problem_number": "COMB-004", "title": "No-Three-in-Line Problem", "statement": "What is the maximum number of points in an $n \\times n$ grid with no three collinear?", "background": "The no-three-in-line problem asks for $g(n)$, the maximum number of points that can be placed in an $n \\times n$ grid such that no three are collinear. Dudeney (1917) conjectured $g(n) = 2n$ for all $n$. Known values: $g(3) = 4, g(4) = 8, g(5) = 10, g(6) = 12$, and computational results extend further. For large $n$, Erdős proved $g(n) \\leq cn/(\\log \\log n)^{1/2}$ for some constant $c$. Lower bounds around $1.85n$ are known. The problem connects to combinatorial geometry, Ramsey theory, and coding theory. Despite its elementary formulation, determining exact values or the asymptotic behavior of $g(n)$ remains challenging. Applications include error-correcting codes and geometric configurations.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact no-three-in-line function remains unknown; g(n) is between (3/2)n-o(n) and 2n in general, while current computations attain 2n for many finite grid sizes.\n\n**Verified partial progress.**\n\n- The elementary row bound gives g(n) <= 2n.\n- Nagy, Nagy, and Woodroofe cite the Hall–Jackson–Sudbery–Wild construction of (3/2)n-o(n) points as the best general asymptotic construction.\n- The maintained Flammenkamp tracker reports 2n-point configurations for every n <= 70 and also n=74 as of the check date.\n\n**Full solution or refutation.**\n\nNeither a formula for g(n) nor the truth or falsity of g(n)=2n for every n has been established.\n\n**What remains.**\n\nDetermine g(n) for arbitrary n and close the asymptotic gap between 3n/2 and 2n.\n\n**Sources checked.**\n\n- D. T. Nagy, Z. L. Nagy, and R. Woodroofe, The extensible No-Three-In-Line problem, European Journal of Combinatorics 114 (2023), 103796. (primary): https://doi.org/10.1016/j.ejc.2023.103796\n Evidence used: The introduction states the 2n upper bound, continuing open status, and the best general construction of (3/2)n-o(n) points.\n- Achim Flammenkamp, No-Three-in-Line configurations (accessed 2026-08-17). (maintained_tracker): https://wwwhomes.uni-bielefeld.de/~achim/no3in/\n Evidence used: Maintained configuration files and notices document current 2n constructions, including recent finite-n advances.\n\n**Review notes.** The background is defective: g(3)=6, not 4, and its claimed o(n)-type upper bound contradicts known linear lower constructions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 298, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1324, "problem_number": "COMB-006", "title": "Sunflower Conjecture", "statement": "For fixed $r$, can the number of size-$k$ sets needed for an $r$-sunflower be bounded by $c^k$ for some constant $c$?", "background": "Erdős and Rado (1960) defined an $r$-sunflower as a collection of $r$ sets $A_1, \\ldots, A_r$ with common intersection $C$ (the core) such that the sets $A_i \\setminus C$ are pairwise disjoint (the petals). Their theorem: any family of size-$k$ sets with at least $k! \\cdot r^k$ members contains an $r$-sunflower. The sunflower conjecture asks: can the bound be improved to $c^k$ for some constant $c = c(r)$ depending only on $r$? This would be sharp up to the value of $c$. In 2019, Alweiss, Lovett, Wu, and Zhang proved a bound of $(\\log k)^k$, a breakthrough improving Erdős-Rado. The conjecture has applications to circuit complexity, learning theory, and DNF formulas. A proof would impact computational complexity theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Authoritative literature through early 2026 treats the exponential sunflower bound as open, but a June 2026 arXiv preprint now claims a proof and has not yet been independently verified.\n\n**Verified partial progress.**\n\n- Alweiss, Lovett, Wu, and Zhang established a near-exponential logarithmic-base bound, with subsequent work reaching a bound of the form (C r log k)^k.\n- Rao's 2026 published survey presents the exponential C(r)^k statement as the enduring conjecture and surveys the established near-exponential bounds.\n- Mishra's June 2026 preprint claims f(k,r) <= (3r^2)^(6r^2-2) 2^k, which would solve the conjecture if correct.\n\n**Full solution or refutation.**\n\nNo independent verification, peer-reviewed acceptance, or authoritative post-claim confirmation of the June 2026 proof claim was found; therefore this triage does not mark the conjecture solved.\n\n**What remains.**\n\nSpecialists should audit the preprint's shifting and sunflower-number lemmas. If the claim fails, remove the remaining log k factor from established bounds.\n\n**Sources checked.**\n\n- R. Alweiss, S. Lovett, K. Wu, and J. Zhang, Improved bounds for the sunflower lemma, STOC 2020. (primary): https://arxiv.org/abs/1908.08483\n Evidence used: Provides the breakthrough near-exponential bound and states the target exponential conjecture.\n- A. Rao, The Story of Sunflowers, Journal of the London Mathematical Society (2026). (authoritative_secondary): https://doi.org/10.1112/jlms.70380\n Evidence used: Recent expert survey of the conjecture and the established best-bound line before the June 2026 claim.\n- T. K. Mishra, Erdős Rado Sunflower (Conjecture) Theorem, arXiv:2606.02667 (2026). (primary): https://arxiv.org/abs/2606.02667\n Evidence used: Explicitly claims the exponential theorem with bound (3r^2)^(6r^2-2) 2^k; it is a recent unrefereed claim.\n\n**Review notes.** Recent proof claim surfaced and conservatively separated from accepted status.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 367, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1327, "problem_number": "GRAPH-003", "title": "Cycle Double Cover Conjecture", "statement": "Does every bridgeless graph have a collection of cycles covering each edge exactly twice?", "background": "The cycle double cover conjecture states: every bridgeless graph (no bridge edges) has a cycle double cover—a collection of cycles such that each edge appears in exactly two cycles. Proposed by Szekeres (1973) and Seymour (1979), this is equivalent to several other conjectures in graph theory. Known for planar graphs (via face boundaries), 4-edge-connected graphs, and graphs with maximum degree at most 3. The conjecture connects to nowhere-zero flows, graph embeddings, and topological graph theory. It would imply results about circular chromatic number and graph decompositions. Despite extensive research, the general case remains open and is considered one of the major problems in graph theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A July 2026 proof claim for Cycle Double Cover has independent expositions by graph theorists, but is too recent for unconditional archival classification.\n\n**Verified partial progress.**\n\n- Oum and Geelen posted explanatory treatments of the announced proof.\n\n**Full solution or refutation.**\n\nThe original conjecture should be considered pending expert review rather than silently left open or declared settled.\n\n**What remains.**\n\nIndependent detailed verification/publication of the proof.\n\n**Sources checked.**\n\n- S.-i. Oum, A proof of the cycle double cover conjecture by OpenAI: An exposition, arXiv:2607.16356. (primary): https://arxiv.org/abs/2607.16356\n Evidence used: States the announced proof and provides an exposition.\n- J. Geelen, OpenAI's proof of the Cycle Double Cover Theorem, arXiv:2607.15399. (primary): https://arxiv.org/abs/2607.15399\n Evidence used: Independent clarification notes.\n\n**Review notes.** No source alteration; recent proof claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 312, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1328, "problem_number": "GRAPH-004", "title": "Erdős–Hajnal Conjecture", "statement": "For any fixed graph $H$, do $H$-free graphs contain large cliques or independent sets?", "background": "The Erdős–Hajnal conjecture (1977) asks: for any graph $H$, is there $\\delta > 0$ such that every $n$-vertex graph with no induced copy of $H$ contains a clique or independent set of size at least $n^\\delta$? This would dramatically strengthen Ramsey theory for hereditary graph classes. For general graphs, Ramsey theorem gives only polylogarithmic guarantees. The conjecture is proven for specific $H$: paths, trees of bounded diameter, and certain small graphs. Partial results by Alon, Pach, and Solymosi establish weaker bounds. The problem connects to extremal graph theory, Ramsey theory, and structural graph theory. A proof would reveal deep structure in induced-subgraph-free graphs.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Erdos--Hajnal conjecture remains open for arbitrary forbidden induced subgraphs.\n\n**Verified partial progress.**\n\n- It is proved for many H classes, including all graphs on at most four vertices and numerous structured families.\n\n**Full solution or refutation.**\n\nNo all-H theorem was verified.\n\n**What remains.**\n\nEstablish the polynomial homogeneous-set exponent for every H.\n\n**Sources checked.**\n\n- Erdos--Hajnal conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Hajnal_conjecture\n Evidence used: Records the general conjecture and solved cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1329, "problem_number": "GRAPH-005", "title": "Lovász Conjecture", "statement": "Does every finite connected vertex-transitive graph have a Hamiltonian path?", "background": "Proposed by László Lovász in 1969, this conjecture states that every finite connected vertex-transitive graph (graph with transitive automorphism group) contains a Hamiltonian path. A stronger version asks for a Hamiltonian cycle. The conjecture is verified for Cayley graphs (Rapaport-Strasser, 1985 for primes; Marušič for certain cases), vertex-transitive graphs of order $pq$ for primes $p < q$, and various special classes. Counter-examples exist for infinite graphs. The problem connects to algebraic graph theory, group theory, and the study of symmetric structures. A proof would significantly advance understanding of Hamiltonian properties in highly symmetric graphs and has implications for network design and routing.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Lovasz's Hamiltonian-path conjecture for vertex-transitive graphs remains open generally.\n\n**Verified partial progress.**\n\n- It is known for many Cayley and other vertex-transitive graph classes.\n\n**Full solution or refutation.**\n\nNo universal Hamiltonian path theorem was verified.\n\n**What remains.**\n\nResolve all finite connected vertex-transitive graphs.\n\n**Sources checked.**\n\n- Lovasz conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Lov%C3%A1sz_conjecture\n Evidence used: Records special cases and open general status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1330, "problem_number": "GRAPH-006", "title": "Hadwiger–Nelson Problem", "statement": "What is the chromatic number of the plane with unit distance graph coloring?", "background": "The Hadwiger–Nelson problem asks: what is the minimum number of colors needed to color the plane such that no two points at distance exactly 1 have the same color? This is equivalent to finding the chromatic number of the unit distance graph in $\\mathbb{R}^2$. It has been known since 1950 that $4 \\leq \\chi \\leq 7$. In 2018, Aubrey de Grey found a unit distance graph with chromatic number 5, improving the lower bound to 5. The upper bound of 7 uses a hexagonal tiling argument. The exact value is unknown. The problem connects to Euclidean Ramsey theory, discrete geometry, and combinatorial optimization. Extensions to higher dimensions and different metrics are also studied.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Hadwiger--Nelson problem remains narrowed to 5, 6, or 7.\n\n**Verified partial progress.**\n\n- Finite 5-chromatic unit-distance graphs establish the lower bound; a 7-colouring establishes the upper bound.\n\n**Full solution or refutation.**\n\nExact plane chromatic number is not known.\n\n**What remains.**\n\nDecide 5 versus 6 versus 7.\n\n**Sources checked.**\n\n- A. de Grey, The chromatic number of the plane is at least 5, arXiv:1804.02385. (primary): https://arxiv.org/abs/1804.02385\n Evidence used: Provides the 5-chromatic lower bound.\n\n**Review notes.** Duplicate topic retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 421, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1331, "problem_number": "TOP-001", "title": "Unknotting Problem", "statement": "Can unknots be recognized in polynomial time?", "background": "The unknotting problem asks whether there exists a polynomial-time algorithm to determine if a given knot diagram represents the unknot (a circle with no actual knots). A knot diagram is a 2D projection of a 3D knot with crossing information. The problem is known to be in NP (a certificate is a sequence of Reidemeister moves) and co-NP (certification via knot invariants). In 2011, Lackenby, building on work by Dynnikov, showed an algorithm exists with complexity bounded by $2^{cn}$ for some constant $c$, where $n$ is the crossing number. However, whether a polynomial-time algorithm exists remains unknown. The problem connects to computational topology, 3-manifold theory, and has applications to molecular biology (DNA unknotting) and physics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Unknot recognition is in NP and co-NP, and a quasipolynomial-time algorithm has been announced, but no polynomial-time algorithm is known.\n\n**Verified partial progress.**\n\n- Hass-Lagarias-Pippenger prove NP membership using polynomial-size normal-surface certificates.\n- Lackenby proves unconditional efficient certification of knottedness, placing unknot recognition in co-NP.\n- Lackenby has announced a quasipolynomial algorithm with superpolynomial but subexponential running time.\n\n**Full solution or refutation.**\n\nThe strongest reported general running time remains above polynomial, so the exact complexity question is unresolved.\n\n**What remains.**\n\nGive a deterministic polynomial-time recognition algorithm or a valid complexity obstruction to one.\n\n**Sources checked.**\n\n- Joel Hass, Jeffrey C. Lagarias, and Nicholas Pippenger, The computational complexity of knot and link problems, Journal of the ACM 46 (1999), 185-211. (primary): https://doi.org/10.1145/301970.301971\n Evidence used: Establishes NP membership through normal-surface bounds.\n- Marc Lackenby, The efficient certification of knottedness and Thurston norm, Advances in Mathematics 387 (2021), 107796. (primary): https://doi.org/10.1016/j.aim.2021.107796\n Evidence used: Provides unconditional polynomial certificates for nontriviality and hence co-NP membership.\n- University of Oxford Mathematical Institute, A quasi-polynomial algorithm for the unknot. (authoritative_secondary): https://www.maths.ox.ac.uk/node/38304\n Evidence used: Institutional announcement of Lackenby's quasipolynomial algorithm; it does not claim polynomial time.\n\n**Review notes.** The background's Reidemeister-sequence account is not the standard proof of NP membership. The quasipolynomial result was treated conservatively as an announced result.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 334, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1332, "problem_number": "TOP-002", "title": "Borel Conjecture", "statement": "Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?", "background": "The Borel conjecture states: if two aspherical closed manifolds (manifolds with contractible universal cover) have isomorphic fundamental groups, then they are homeomorphic. An aspherical manifold has all higher homotopy groups trivial, so its topology is determined by $\\pi_1$. The conjecture is a topological rigidity statement: algebraic data ($\\pi_1$) determines geometric structure (homeomorphism type). Proven for many special cases: flat manifolds, hyperbolic manifolds (by Mostow rigidity for dimension $\\geq 3$), and certain graph manifolds. The Novikov conjecture is a weaker form (about homotopy invariance of higher signatures). The Borel conjecture connects to surgery theory, K-theory, and geometric group theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Borel rigidity conjecture remains open for arbitrary closed aspherical manifolds, but is proved in dimensions at least five for hyperbolic and CAT(0) fundamental groups and in other major special classes.\n\n**Verified partial progress.**\n\n- Bartels-Luck prove the Borel conjecture for closed aspherical manifolds of dimension at least five with hyperbolic or CAT(0) fundamental group.\n- Classical geometric rigidity and low-dimensional topology settle further category- and dimension-specific cases.\n\n**Full solution or refutation.**\n\nBroad rigidity theorems do not cover arbitrary fundamental groups or every dimension.\n\n**What remains.**\n\nEstablish topological rigidity for closed aspherical manifolds with arbitrary fundamental groups, including the unresolved dimension-sensitive cases.\n\n**Sources checked.**\n\n- Arthur Bartels and Wolfgang Luck, The Borel conjecture for hyperbolic and CAT(0)-groups, Annals of Mathematics 175 (2012), 631-689. (primary): https://doi.org/10.4007/annals.2012.175.2.5\n Evidence used: Proves high-dimensional Borel rigidity for two broad classes of fundamental groups.\n- Arthur Bartels, On proofs of the Farrell-Jones conjecture, arXiv:1210.1044. (authoritative_secondary): https://arxiv.org/abs/1210.1044\n Evidence used: Explains how Farrell-Jones results imply Borel rigidity and delineates the known scope.\n\n**Review notes.** The fundamental-group shorthand is equivalent to the usual homotopy-equivalence formulation for connected aspherical manifolds under the standard conventions. Record 1481 is an exact duplicate.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 278, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1333, "problem_number": "TOP-003", "title": "Volume Conjecture", "statement": "Do quantum invariants of knots relate asymptotically to hyperbolic volume?", "background": "The volume conjecture, proposed by Kashaev (1997) and generalized by Murakami-Murakami, relates quantum topology to hyperbolic geometry. For a hyperbolic knot $K$ in $S^3$, let $J_N(K; q)$ be the colored Jones polynomial at $q = e^{2\\pi i/N}$. The conjecture states: $\\lim_{N \\to \\infty} \\frac{2\\pi \\log|J_N(K; e^{2\\pi i/N})|}{N} = \\text{Vol}(S^3 \\setminus K)$, where the right side is the hyperbolic volume of the knot complement. Verified for many specific knots and families (torus knots, figure-eight). The conjecture suggests deep connections between quantum field theory, Chern-Simons theory, and 3-manifold geometry. It would unify quantum invariants and geometric invariants.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Under the precise colored-Jones formula supplied in the background, the Volume Conjecture is proved for important knots and families but remains open for arbitrary hyperbolic knots.\n\n**Verified partial progress.**\n\n- Murakami-Murakami formulate the normalized root-of-unity asymptotic in terms of simplicial volume and verify foundational cases.\n- Rigorous asymptotic expansions establish the conjectured exponential rate for additional individual hyperbolic knots such as 5_2.\n\n**Full solution or refutation.**\n\nThe broad question has a precise standard formulation in the background, but known case theorems do not cover every hyperbolic knot.\n\n**What remains.**\n\nProve the stated root-of-unity colored Jones asymptotic and its identification with hyperbolic volume for every hyperbolic knot.\n\n**Sources checked.**\n\n- Hitoshi Murakami and Jun Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Mathematica 186 (2001), 85-104. (primary): https://doi.org/10.1007/BF02392716\n Evidence used: Gives the standard formulation and foundational verified examples.\n- Tomotada Ohtsuki, On the asymptotic expansion of the Kashaev invariant of the 5_2 knot, Quantum Topology 7 (2016), 669-735. (primary): https://doi.org/10.4171/QT/82\n Evidence used: Proves detailed asymptotic behavior for a nontrivial hyperbolic-knot case.\n\n**Review notes.** The exact statement is broad, but its background supplies the precise standard formula. Torus knots are not hyperbolic; their zero simplicial-volume behavior belongs to a generalized formulation. Record 34 concerns the same conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1334, "problem_number": "TOP-004", "title": "Novikov Conjecture", "statement": "Are certain combinations of Pontryagin classes homotopy invariant?", "background": "The Novikov conjecture, proposed by Sergei Novikov in 1965, is a fundamental problem in topology and differential geometry. For a closed oriented manifold $M$ with fundamental group $\\pi$, certain rational linear combinations of Pontryagin classes evaluated on the fundamental class should be homotopy invariants when pushed forward to the classifying space $B\\pi$. More precisely, higher signatures defined using the signature operator should be homotopy invariants. The conjecture is verified for many groups: finite groups, amenable groups, linear groups, Gromov hyperbolic groups, and many others. It connects to K-theory, C*-algebras, index theory, and surgery theory. The conjecture has deep implications for the topology of manifolds and the structure of group C*-algebras.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Interpreted as homotopy invariance of higher signatures, the Novikov conjecture remains open for arbitrary groups and is proved for many broad classes, including coarsely embeddable groups and significant extensions.\n\n**Verified partial progress.**\n\n- Yu proves the coarse Baum-Connes theorem for spaces uniformly embeddable in Hilbert space, yielding the Novikov conjecture for corresponding groups.\n- Deng proves the Novikov conjecture for substantial extensions of coarsely embeddable groups.\n\n**Full solution or refutation.**\n\nAssembly-map methods establish many group classes but not every discrete group.\n\n**What remains.**\n\nProve rational injectivity of the relevant assembly map, equivalently higher-signature homotopy invariance, for arbitrary groups.\n\n**Sources checked.**\n\n- Guoliang Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Inventiones Mathematicae 139 (2000), 201-240. (primary): https://doi.org/10.1007/s002229900032\n Evidence used: Establishes a major geometric class for which the Novikov conjecture follows.\n- Jintao Deng, The Novikov conjecture and extensions of coarsely embeddable groups, Journal of Noncommutative Geometry 16 (2022), 265-310. (primary): https://doi.org/10.4171/JNCG/437\n Evidence used: Proves new extension cases and states the remaining general conjecture.\n- Jonathan Rosenberg, Novikov Conjecture Home Page. (maintained_tracker): https://math.umd.edu/~jmr/NC.html\n Evidence used: Maintains formulations, implications, and references for established cases.\n\n**Review notes.** The exact sentence is underspecified. The assessment uses the higher-signature formula supplied in the background and distinguishes it from Novikov's theorem on rational Pontryagin classes.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 312, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1335, "problem_number": "GEOM-007", "title": "Kakeya Conjecture", "statement": "Must a Kakeya set in $\\mathbb{R}^n$ have Hausdorff and Minkowski dimension $n$?", "background": "A Kakeya set in $\\mathbb{R}^n$ is a compact set containing a unit line segment in every direction. The Kakeya conjecture states such sets must have full Hausdorff and Minkowski dimension $n$. In $\\mathbb{R}^2$, Kakeya sets can have measure zero (Davies 1971) but must have Hausdorff dimension 2 (proven). For $n \\geq 3$, the conjecture is open. Known results: Kakeya sets in $\\mathbb{R}^n$ have Hausdorff dimension $\\geq (n+2)/2$ (Wolff, 1995), improved to $\\geq n/2 + \\epsilon$ by various authors. The problem connects to harmonic analysis (Bochner-Riesz conjecture, restriction conjecture), PDE (wave equation estimates), and number theory. Resolving it would impact multiple areas of analysis.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Kakeya is proved in R3 but open in n>=4.\n\n**Verified partial progress.**\n\n- Wang--Zahl settled R3.\n\n**Full solution or refutation.**\n\nNo all-n proof is verified.\n\n**What remains.**\n\nSettle n>=4.\n\n**Sources checked.**\n\n- Ecole Polytechnique Kakeya report (2026). (authoritative_secondary): https://www.polytechnique.edu/en/news/closer-look-kakeyas-conjecture\n Evidence used: Reports R3 proof.\n\n**Review notes.** Duplicate topic retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1336, "problem_number": "GEOM-008", "title": "Illumination Problem", "statement": "Can every convex body in $\\mathbb{R}^n$ be illuminated by $2^n$ light sources?", "background": "The illumination problem (or Hadwiger's problem) asks: what is the minimum number of light sources (point sources or directions) needed to illuminate the entire boundary of any convex body in $\\mathbb{R}^n$? A point on the boundary is illuminated if the ray from the light source to that point does not intersect the interior. Conjecture: $2^n$ sources suffice for dimension $n$. Known results: the upper bound is $\\lfloor 3^{n}/2^{n-1} \\rfloor$ (Schramm), and the conjecture is verified for $n \\leq 3$. The problem connects to discrete geometry, combinatorial geometry, and has applications in computer graphics and sensor placement. A proof would clarify fundamental properties of convex bodies.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The general illumination conjecture remains open.\n\n**Verified partial progress.**\n\n- Many special bodies are known.\n\n**Full solution or refutation.**\n\nNo 2^n theorem for all bodies was verified.\n\n**What remains.**\n\nProve the universal bound.\n\n**Sources checked.**\n\n- Illumination problem overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Illumination_problem\n Evidence used: Records open status.\n\n**Review notes.** Duplicate topic retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 234, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1338, "problem_number": "DYN-002", "title": "MLC Conjecture", "statement": "Is the Mandelbrot set locally connected?", "background": "The MLC (Mandelbrot set is Locally Connected) conjecture asks whether the famous Mandelbrot set—the set of complex parameters $c$ for which the iteration $z_{n+1} = z_n^2 + c$ (starting from $z_0 = 0$) remains bounded—is locally connected. Local connectivity would mean every point has arbitrarily small connected neighborhoods. The conjecture is one of the most important problems in complex dynamics. If true, it would imply: the boundary of the Mandelbrot set has Hausdorff dimension 2, the Mandelbrot set is the closure of its interior, and precise descriptions of the topology. The conjecture has been verified for many parameter regions but remains open in general. It connects to renormalization theory, polynomial dynamics, and fractal geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** MLC, local connectivity of the Mandelbrot set, remains open in full generality.\n\n**Verified partial progress.**\n\n- Local connectivity is proved for many parameter classes, including hyperbolic components and broad combinatorially controlled regimes.\n- Conditional approaches relate the remaining issue to rigidity/density questions.\n\n**Full solution or refutation.**\n\nNo proof covering every boundary parameter was verified.\n\n**What remains.**\n\nProve local connectivity at all Mandelbrot parameters or find a counterexample.\n\n**Sources checked.**\n\n- Mandelbrot set local-connectivity problem overview, Complex Dynamics resources (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Mandelbrot_set#Local_connectivity\n Evidence used: Records MLC as a central unresolved problem while summarizing known classes.\n\n**Review notes.** No source alteration; source statement is terse.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 398, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1339, "problem_number": "DYN-003", "title": "Weinstein Conjecture", "statement": "Does every regular compact contact-type level set carry a periodic orbit?", "background": "The Weinstein conjecture, proposed by Alan Weinstein in 1978, states: every regular compact contact-type level set of a Hamiltonian on a symplectic manifold carries at least one periodic orbit of the Hamiltonian flow. In more geometric terms, on a compact contact manifold, the Reeb vector field has at least one closed orbit. The conjecture has been proven in many cases: dimension 3 (Taubes, 2007), overtwisted contact 3-manifolds (Hofer, 1993), and various higher-dimensional cases using symplectic field theory and pseudoholomorphic curves. The full conjecture in all dimensions remains open. It connects contact geometry, symplectic topology, Hamiltonian dynamics, and has applications to celestial mechanics and rigid body dynamics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This contact-type Weinstein formulation is proved in dimension three but is not verified in the source's unrestricted dimension/general setting.\n\n**Verified partial progress.**\n\n- Taubes's theorem supplies the closed 3D contact case.\n\n**Full solution or refutation.**\n\nThe all-dimensional generalization remains unresolved.\n\n**What remains.**\n\nResolve periodic-orbit existence for the remaining higher-dimensional contact-type settings.\n\n**Sources checked.**\n\n- C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007). (primary): https://arxiv.org/abs/math/0611007\n Evidence used: Proves the three-dimensional contact Weinstein conjecture.\n- M. Hutchings, Taubes's proof of the Weinstein conjecture in dimension three. (authoritative_secondary): https://math.berkeley.edu/~hutching/pub/tw.pdf\n Evidence used: Explains its scope and higher-dimensional limitations.\n\n**Review notes.** No source alteration; formulation may conflate contact and Hamiltonian variants.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 256, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1340, "problem_number": "DYN-004", "title": "Birkhoff Conjecture", "statement": "If a billiard table is strictly convex and integrable, must its boundary be an ellipse?", "background": "The Birkhoff conjecture concerns dynamical billiards: if a strictly convex billiard table in the plane is integrable (has a complete set of integrals of motion), then its boundary must be an ellipse. Elliptical billiards are known to be integrable (Birkhoff, 1927). The conjecture asks if they are the only such tables. Partial results: true for sufficiently smooth perturbations of circles and ellipses, and for certain classes of curves. The problem connects to KAM theory, integrable systems, and spectral geometry. A proof would characterize all integrable planar billiards and has implications for understanding caustics, periodic orbits, and the inverse spectral problem.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The global Birkhoff conjecture for strictly convex integrable planar billiards remains open, but it is proved in notable symmetric and near-ellipse regimes.\n\n**Verified partial progress.**\n\n- The centrally symmetric convex case is proved.\n- Earlier local rigidity results establish the assertion near ellipses under regularity assumptions.\n\n**Full solution or refutation.**\n\nNo result for every strictly convex integrable table was verified.\n\n**What remains.**\n\nRemove the central-symmetry/perturbative hypotheses.\n\n**Sources checked.**\n\n- A. Glutsyuk and V. Kaloshin, The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables, arXiv:2008.03566. (primary): https://arxiv.org/abs/2008.03566\n Evidence used: The abstract proves the centrally symmetric C2-smooth convex case.\n- A. Glutsyuk and V. Kaloshin, Birkhoff Conjecture for Nearly Centrally Symmetric Domains, Geom. Funct. Anal. (2024). (primary): https://link.springer.com/article/10.1007/s00039-024-00695-6\n Evidence used: The abstract states a nonperturbative centrally symmetric theorem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1341, "problem_number": "ALGGEOM-001", "title": "Abundance Conjecture", "statement": "If the canonical bundle of a variety is nef, must it be semiample?", "background": "The abundance conjecture is a central problem in birational geometry and minimal model theory. For a projective variety $X$ with Kawamata log terminal singularities, if the canonical bundle $K_X$ is nef (numerically effective—has non-negative intersection with all curves), the conjecture states $K_X$ must be semiample (some positive multiple is globally generated). This would complete the minimal model program by ensuring every minimal model has good positivity properties. Known cases: surfaces (classical), dimension 3 (Miyaoka, Kawamata), toric varieties, and certain special cases in higher dimensions. The conjecture connects to the cone theorem, base point freeness, and would have major implications for classification of algebraic varieties.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Under the intended standard formulation for projective klt pairs in characteristic zero, the abundance conjecture is proved through dimension three and in several important higher-dimensional regimes, but remains open in general.\n\n**Verified partial progress.**\n\n- The abundance program for projective log canonical pairs is complete in dimensions at most three.\n- Birkar, Cascini, Hacon, and McKernan settle the big or log-general-type regime via finite generation and existence of log canonical models.\n- Gongyo proves abundance when the log canonical divisor is numerically trivial, including log canonical and semi-log-canonical pairs.\n- Liu and Xu prove abundance in dimension at most five when kappa is nonnegative and the numerical dimension is at most one, and show more generally that non-vanishing implies abundance in numerical dimension at most one.\n- Lazic proves conditional good-model results for uniruled pairs that are not rationally connected and reduces the rationally connected case to a specific nonexistence conjecture.\n\n**Full solution or refutation.**\n\nNo proof or counterexample to the general projective klt abundance conjecture was verified. A 2026 counterexample for compact Kahler slc threefolds lies outside the intended algebraic klt category and therefore serves as a scope warning, not a refutation.\n\n**What remains.**\n\nProve that every nef K_X+Delta on a projective klt pair over a characteristic-zero field is semiample in the unresolved higher-dimensional regimes, or construct a projective klt counterexample.\n\n**Sources checked.**\n\n- Vladimir Lazic, Abundance for uniruled pairs which are not rationally connected, L'Enseignement Mathematique 71 (2025), 87-105, DOI 10.4171/LEM/1065. (primary): https://ems.press/journals/lem/articles/13750344\n Evidence used: Explicitly describes general higher-dimensional abundance as open, records completion through dimension three, and proves conditional good-model results for uniruled pairs.\n- Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, Journal of the American Mathematical Society 23 (2010), 405-468. (primary): https://www.ams.org/jams/2010-23-02/S0894-0347-09-00649-3/viewer/\n Evidence used: Proves finite generation and existence of log canonical models in the big or log-general-type setting.\n- Yoshinori Gongyo, Abundance theorem for numerically trivial log canonical divisors of semi-log canonical pairs, Journal of Algebraic Geometry 22 (2013), 549-564. (primary): https://arxiv.org/abs/1005.2796\n Evidence used: Proves that a numerically trivial log canonical divisor is Q-linearly trivial for projective log canonical and semi-log-canonical pairs.\n- Jihao Liu and Zheng Xu, Non-vanishing implies numerical dimension one abundance, arXiv:2505.05250 (2025). (primary): https://arxiv.org/abs/2505.05250\n Evidence used: Proves abundance in dimension at most five when kappa is nonnegative and numerical dimension is at most one, and gives a conditional all-dimensional implication from non-vanishing.\n- Swapnajit Das, Failure of the semi log canonical Abundance for compact Kahler threefolds, arXiv:2604.28085 (2026). (primary): https://arxiv.org/abs/2604.28085\n Evidence used: Constructs a compact Kahler slc threefold with nef but non-semiample canonical divisor, demonstrating the need to retain the standard projective klt scope.\n\n**Review notes.** Exact source statement preserved. It omits projectivity/properness, base field and characteristic, normality and klt singularities, Q-Cartierness, and the pair divisor K_X+Delta; for a singular variety the canonical object need not be a bundle. The compact Kahler slc counterexample does not refute the intended projective klt conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1343, "problem_number": "LOGIC-001", "title": "Vaught Conjecture", "statement": "Is the number of countable models of a complete first-order theory finite, $\\aleph_0$, or $2^{\\aleph_0}$?", "background": "The Vaught conjecture, proposed by Robert Vaught in 1961, is a fundamental problem in model theory. For a complete first-order theory in a countable language, the number of countable models (up to isomorphism) must be either finite, countably infinite ($\\aleph_0$), or continuum ($2^{\\aleph_0}$). In other words, there cannot be exactly $\\aleph_1$ (or any other intermediate cardinality) non-isomorphic countable models. The conjecture is known to hold for many classes of theories: $\\omega$-stable theories, superstable theories, and theories with certain structural properties. However, the general case remains open. The problem connects to descriptive set theory, infinitary logic, and has implications for classification theory and the structure of models.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Vaught's conjecture remains open for arbitrary complete countable first-order theories.\n\n**Verified partial progress.**\n\n- It is proved for many classes, including weakly minimal theories, superstable theories of finite U-rank, numerous order-like theories, and further restricted partial orders.\n- Recent work continues to establish sharp versions for new structural classes.\n\n**Full solution or refutation.**\n\nNo general proof that an intermediate uncountable model spectrum is impossible was verified.\n\n**What remains.**\n\nSettle the countable-spectrum dichotomy for arbitrary complete countable theories.\n\n**Sources checked.**\n\n- Vaught conjecture, Encyclopedia of Mathematics. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Vaught_conjecture\n Evidence used: Records the general statement and several established stable-theory cases.\n- Sharp Vaught's conjecture for some classes of partial orders, Annals of Pure and Applied Logic 176 (2025). (primary): https://doi.org/10.1016/j.apal.2024.103468\n Evidence used: Recent class-specific confirmation and statement of the general conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 298, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1344, "problem_number": "LOGIC-002", "title": "Cherlin-Zilber Conjecture", "statement": "Is every simple group with $\\aleph_0$-stable theory an algebraic group over an algebraically closed field?", "background": "The Cherlin-Zilber conjecture concerns the classification of simple groups in model theory. It states: every infinite simple group whose first-order theory is stable in $\\aleph_0$ (countably stable) is isomorphic to a simple algebraic group over an algebraically closed field. The conjecture connects abstract model-theoretic stability to concrete algebraic structures. Many special cases have been verified, and the conjecture has driven development of geometric stability theory. It would provide a complete classification of stable simple groups and has implications for understanding the interaction between model theory and group theory. Zilber's work on Zariski geometries provides evidence for the conjecture.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Cherlin--Zilber algebraicity program remains open in general; the record's omega-stable formulation is a strong related form and should not be silently conflated with the finite-Morley-rank form.\n\n**Verified partial progress.**\n\n- Simple K*-groups of finite Morley rank of even or mixed type are algebraic.\n- Recent surveys and related Lie-ring results continue to advance restricted cases.\n\n**Full solution or refutation.**\n\nNo classification of all simple omega-stable groups, or all simple groups of finite Morley rank, as algebraic groups was verified.\n\n**What remains.**\n\nEliminate nonalgebraic bad-group possibilities in the relevant general setting.\n\n**Sources checked.**\n\n- K. Tent, From the Cherlin-Zilber Conjecture via sharply 2-transitive groups to the Burnside problem, arXiv:2606.18207 (2026). (primary): https://arxiv.org/abs/2606.18207\n Evidence used: Reviews the current open finite-Morley-rank algebraicity conjecture.\n- G. Cherlin, Simple Groups of Finite Morley Rank, Rutgers research page. (authoritative_secondary): https://sites.math.rutgers.edu/~cherlin/FMR/\n Evidence used: Records even/mixed-type algebraicity and unresolved odd/degenerate cases.\n- O. Deloro and E. Jaligot, Superstability and central extensions of algebraic groups, Ann. Pure Appl. Logic 167 (2016), 75--103. (primary): https://doi.org/10.1016/j.apal.2015.08.004\n Evidence used: Explains the omega-stable formulation and its relation to the broader algebraicity conjecture.\n\n**Review notes.** No source alteration; source statement uses omega-stability rather than the usual finite-Morley-rank wording.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1346, "problem_number": "GEOM-009", "title": "Yang-Mills Existence and Mass Gap", "statement": "Does Yang-Mills theory exist mathematically and exhibit a mass gap in 4D?", "background": "One of the seven Millennium Prize Problems ($1M prize). The Yang-Mills equations describe the behavior of elementary particles using non-Abelian gauge theory, fundamental to the Standard Model of particle physics. The problem asks two questions: (1) Does a mathematically rigorous quantum Yang-Mills theory exist in 4-dimensional spacetime? (2) Does it exhibit a mass gap—the smallest mass of any excitation being strictly positive? Physicists use Yang-Mills theory extensively, but a rigorous mathematical foundation is lacking. Proving existence and the mass gap would provide the mathematical basis for quantum chromodynamics (QCD) and explain confinement of quarks. The problem connects quantum field theory, differential geometry, functional analysis, and mathematical physics. Despite extensive physics research, mathematical proof remains elusive.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Four-dimensional quantum Yang--Mills existence and mass gap remain unproved.\n\n**Verified partial progress.**\n\n- Lattice results and physics evidence support a mass gap.\n\n**Full solution or refutation.**\n\nNo Clay-criterion construction is verified.\n\n**What remains.**\n\nConstruct the theory and prove positive mass gap.\n\n**Sources checked.**\n\n- Clay Mathematics Institute, Yang--Mills & the Mass Gap. (primary): https://www.claymath.org/millennium/yang-mills-the-maths-gap/\n Evidence used: Official page says no proof is known.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1347, "problem_number": "ST-001", "title": "Partition Principle Implies Axiom of Choice", "statement": "Does the partition principle (PP) imply the axiom of choice (AC)?", "background": "The partition principle states that for every partition of a set, there exists a set that contains exactly one element from each cell of the partition. The axiom of choice states that for every collection of nonempty sets, there exists a choice function selecting one element from each set. While AC clearly implies PP, the reverse implication is unknown. This question explores the relative strength of these fundamental axioms in set theory and their role in mathematics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether the partition principle implies AC over ZF remains open.\n\n**Verified partial progress.**\n\n- Recent proposed PP-without-AC constructions in alternative frameworks do not give a settled ZF countermodel.\n\n**Full solution or refutation.**\n\nNo implication proof or ZF model of PP plus not-AC was verified.\n\n**What remains.**\n\nProve PP implies AC or build a valid ZF countermodel.\n\n**Sources checked.**\n\n- A Gentle Introduction to the Axiom of Choice, Mathematical Intelligencer (2026). (authoritative_secondary): https://doi.org/10.1007/s00283-026-10531-4\n Evidence used: Defines PP and identifies its relation to AC as unresolved.\n- A. Karagila, Flow: the Axiom of Choice is independent from the Partition Principle, arXiv:2010.03664. (primary): https://arxiv.org/abs/2010.03664\n Evidence used: Contains a claimed approach; it is not used as a settled resolution.\n\n**Review notes.** No source alteration; unverified independence claim excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1348, "problem_number": "ST-002", "title": "Woodin's GCH below Strongly Compact Cardinals", "statement": "Does the generalized continuum hypothesis below a strongly compact cardinal imply it everywhere?", "background": "Posed by W. Hugh Woodin, this problem asks whether local instances of the generalized continuum hypothesis (GCH) can force global instances. A strongly compact cardinal is a large cardinal with strong reflection properties. The question explores whether GCH holding below such a cardinal must propagate throughout the universe of sets. This connects large cardinal theory with cardinal arithmetic and the structure of the set-theoretic universe.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Woodin's strongly-compact/GCH question remains open.\n\n**Verified partial progress.**\n\n- There are related consistency and square/GCH results under stronger hypotheses, but no general implication from GCH below a strongly compact cardinal.\n\n**Full solution or refutation.**\n\nNo proof or countermodel to the precise implication was verified.\n\n**What remains.**\n\nDerive global GCH or construct a countermodel from the stated hypothesis.\n\n**Sources checked.**\n\n- M. Golshani, Set theory questions (2025 lecture slides). (authoritative_secondary): https://www.dmg.tuwien.ac.at/fb8/2025_slides/Golshani.pdf\n Evidence used: Lists Woodin's question as longstanding open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1349, "problem_number": "ST-003", "title": "GCH and Diamond Principle", "statement": "Does the generalized continuum hypothesis entail the diamond principle $\\diamondsuit(E_{\\text{cf}(\\lambda)}^{\\lambda^+})$ for every singular cardinal $\\lambda$?", "background": "The diamond principle is a combinatorial principle asserting the existence of certain prediction sequences. For singular cardinals (cardinals not equal to their own cofinality), the relationship between GCH and diamond principles is subtle. While diamond holds at successor cardinals under GCH, its behavior at successors of singular cardinals remains mysterious. This problem probes the fine structure of cardinal arithmetic.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No checked theorem establishes the stated GCH-to-diamond implication at every singular cardinal.\n\n**Verified partial progress.**\n\n- Square, approachability, and cardinal-arithmetic principles give diamond consequences in many special configurations.\n\n**Full solution or refutation.**\n\nNo uniform all-singular theorem or countermodel was verified.\n\n**What remains.**\n\nSettle the specified diamond implication or isolate a counterexample model.\n\n**Sources checked.**\n\n- T. Jech, Set Theory, 3rd millennium edition (2003). (authoritative_secondary): https://doi.org/10.1007/3-540-44761-X\n Evidence used: Provides standard background on GCH, diamond, and singular-cardinal combinatorics.\n\n**Review notes.** No source alteration; expert review requested for this specialized formulation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1350, "problem_number": "ST-004", "title": "GCH and Suslin Trees", "statement": "Does the generalized continuum hypothesis imply the existence of an $\\aleph_2$-Suslin tree?", "background": "A Suslin tree is a tree of height $\\omega_1$ with no uncountable chains or antichains. An $\\aleph_2$-Suslin tree is the analogous structure at the next cardinal level. While Suslin trees at $\\aleph_1$ can exist under certain axioms, their existence at $\\aleph_2$ under GCH is unknown. This problem connects cardinal arithmetic with combinatorial set theory and the theory of infinite trees.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether GCH implies an aleph_2-Suslin tree remains a longstanding open problem.\n\n**Verified partial progress.**\n\n- Many forcing models decide related tree-existence questions, but no implication from GCH alone is known.\n\n**Full solution or refutation.**\n\nNo proof or countermodel to the stated ZFC+GCH implication was verified.\n\n**What remains.**\n\nProve an aleph_2-Suslin tree under GCH or force GCH with none.\n\n**Sources checked.**\n\n- Suslin tree overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Suslin_tree\n Evidence used: Explicitly identifies the GCH-to-aleph_2-Suslin-tree question as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1352, "problem_number": "ST-006", "title": "Ultimate Core Model", "statement": "Does there exist an ultimate core model containing all large cardinals?", "background": "Core models are canonical inner models of set theory that approximate the entire universe while being more tractable. The search for an ultimate core model—one encompassing all large cardinal properties—is a central goal of modern set theory. Such a model would unify our understanding of large cardinals and provide a framework for resolving independence questions. The project involves deep interactions between forcing, inner model theory, and large cardinal axioms.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** An ultimate core model encompassing all large-cardinal strength is not known.\n\n**Verified partial progress.**\n\n- Fine-structural inner models reach many Woodin cardinals, but supercompact-level canonical inner models remain a fundamental obstruction.\n\n**Full solution or refutation.**\n\nNo ultimate core model satisfying the stated ambition was verified.\n\n**What remains.**\n\nConstruct and compare a canonical model at the required large-cardinal strength.\n\n**Sources checked.**\n\n- J. R. Steel, inner-model strategy comparison notes. (authoritative_secondary): https://math.berkeley.edu/~steel/papers/short.strategycompare.pdf\n Evidence used: Describes supercompact-level inner-model theory as a fundamental open target.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1353, "problem_number": "ST-007", "title": "Woodin's Ω-Conjecture", "statement": "If there is a proper class of Woodin cardinals, does Ω-logic satisfy an analogue of Gödel's completeness theorem?", "background": "Proposed by W. Hugh Woodin, this conjecture connects large cardinals with logic. Ω-logic is a strong logic using Woodin cardinals to define semantic validity. The conjecture asserts that under the assumption of a proper class of Woodin cardinals, Ω-logic becomes complete in a generalized sense—every Ω-valid sentence has an Ω-proof. This would provide a powerful new framework for set-theoretic truth and resolve many independence questions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Woodin's Omega-conjecture, asserting an appropriate completeness theorem under proper-class Woodins, remains open.\n\n**Verified partial progress.**\n\n- Conditional Ω-logic results and the Strong Ω Conjecture provide a framework for generic absoluteness consequences.\n\n**Full solution or refutation.**\n\nNo unconditional completeness theorem of the stipulated form was verified.\n\n**What remains.**\n\nEstablish Ω-completeness under the stated large-cardinal assumption.\n\n**Sources checked.**\n\n- Stanford Encyclopedia of Philosophy, The Continuum Hypothesis. (authoritative_secondary): https://plato.stanford.edu/entries/continuum-hypothesis/\n Evidence used: States that the corresponding Ω-logic completeness theorem is open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1354, "problem_number": "ST-008", "title": "Strongly Compact vs Supercompact Cardinals", "statement": "Does the consistency of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?", "background": "Strongly compact cardinals and supercompact cardinals are both large cardinal notions with powerful reflection properties. Supercompact cardinals are known to be stronger, but whether their consistency strength is strictly greater than strongly compact cardinals remains open. This problem probes the fine structure of the large cardinal hierarchy and the relationships between different reflection principles.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether strong compactness and supercompactness are equiconsistent.\n\n**Verified partial progress.**\n\n- Relative models separate the properties at particular cardinals, while supercompactness trivially implies strong compactness.\n\n**Full solution or refutation.**\n\nNo reverse consistency-strength implication was verified.\n\n**What remains.**\n\nDerive a supercompact consistency model from a strongly compact one, or establish a separation.\n\n**Sources checked.**\n\n- Identity crisis for measurable and strongly compact cardinals. (authoritative_secondary): https://logic.nankai.edu.cn/_upload/article/files/73/9a/bc6df29a42fe8dd60c911873f3db/07e70874-31c6-4273-a076-0c8f22ca4f07.pdf\n Evidence used: States the equiconsistency question is widely open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1355, "problem_number": "ST-009", "title": "Jónsson Algebra on ℵ_ω", "statement": "Does there exist a Jónsson algebra on $\\aleph_\\omega$?", "background": "A Jónsson algebra is an algebraic structure with no proper subalgebra of the same cardinality. The existence of Jónsson algebras on various cardinals connects algebra with set theory. For $\\aleph_\\omega$ (the $\\omega$-th infinite cardinal), existence remains unknown. A positive answer would provide new insights into the algebraic structure of infinite sets and the behavior of singular cardinals.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Existence of a Jónsson algebra on aleph_omega remains open.\n\n**Verified partial progress.**\n\n- Shelah proved the successor aleph_(omega+1) has a Jónsson algebra; the singular-cardinal case remains the difficult gap.\n\n**Full solution or refutation.**\n\nNo construction or obstruction at aleph_omega was verified.\n\n**What remains.**\n\nConstruct a Jónsson algebra on aleph_omega or prove its impossibility.\n\n**Sources checked.**\n\n- UnsolvedMath, ST-009 problem record. (authoritative_secondary): https://www.unsolvedmath.com/problems?category=10&status=open\n Evidence used: Lists the aleph_omega Jónsson-algebra problem as open.\n- S. Shelah, aleph_(omega+1) has a Jónsson algebra. (primary): https://doi.org/10.1007/BF02767195\n Evidence used: Establishes the successor result contrasting with the open singular case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1356, "problem_number": "ST-010", "title": "Open Coloring Axiom and Continuum Hypothesis", "statement": "Is the open coloring axiom (OCA) consistent with $2^{\\aleph_0} > \\aleph_2$?", "background": "The open coloring axiom is a combinatorial principle with powerful consequences for the structure of the real line. It is known to be consistent with $2^{\\aleph_0} = \\aleph_2$, but consistency with larger values of the continuum is unknown. This problem explores the interaction between partition properties and cardinal arithmetic, central themes in modern set theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Consistency of full OCA with continuum greater than aleph_2 remains open.\n\n**Verified partial progress.**\n\n- Abraham--Rubin--Shelah OCA is consistent with continuum aleph_3, but that is a weaker axiom and does not settle full OCA.\n\n**Full solution or refutation.**\n\nNo model of full OCA plus 2^aleph_0>aleph_2 was verified.\n\n**What remains.**\n\nBuild such a model or derive continuum aleph_2 from full OCA.\n\n**Sources checked.**\n\n- S. Shelah and J. Steprāns, Abraham-Rubin-Shelah Open Colorings and a Large Continuum, arXiv:1904.10516. (primary): https://arxiv.org/abs/1904.10516\n Evidence used: Proves large continuum only for the named weaker coloring axiom.\n- OCA with large continuum open-problem discussion. (authoritative_secondary): https://pi.math.cornell.edu/~justin/Ftp/OCA_c.pdf\n Evidence used: Records the full OCA/continuum question as open.\n\n**Review notes.** No source alteration; weaker ARS-OCA result explicitly distinguished.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1357, "problem_number": "ST-011", "title": "Reinhardt Cardinals without Choice", "statement": "Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?", "background": "A Reinhardt cardinal would witness an elementary embedding from the universe of all sets (V) to itself. Kunen proved such embeddings cannot exist with the axiom of choice. However, without AC, the question remains open. Reinhardt cardinals would be the strongest large cardinal notion, transcending the usual hierarchy. Their possible existence connects to alternative set theories and the role of choice in mathematics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The consistency of a nontrivial elementary embedding V to V without choice remains unresolved in appropriate class-theoretic formulations.\n\n**Verified partial progress.**\n\n- Kunen rules it out with AC; choiceless analyses derive powerful consequences and rule out definable embeddings.\n\n**Full solution or refutation.**\n\nNo consistency proof or choiceless inconsistency theorem for the intended non-definable/class formulation was verified.\n\n**What remains.**\n\nClarify the ambient class theory and settle existence of a Reinhardt embedding there.\n\n**Sources checked.**\n\n- G. Goldberg, Measurable cardinals and choiceless axioms, Ann. Pure Appl. Logic 174 (2023), 103323. (primary): https://doi.org/10.1016/j.apal.2023.103323\n Evidence used: Studies consequences of the hypothesis without AC and emphasizes the choiceless setting.\n- Reinhardt cardinal overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Reinhardt_cardinal\n Evidence used: Distinguishes AC inconsistency, definability obstructions, and remaining choiceless formulations.\n\n**Review notes.** Statement retained; consistency depends on the precise class theory and embedding formalization.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1358, "problem_number": "GAME-001", "title": "Sudoku: Unique Solution Puzzles", "statement": "How many Sudoku puzzles have exactly one solution?", "background": "Standard 9×9 Sudoku grids can be filled in approximately 6.67 × 10²¹ ways. A puzzle is a partial filling with a unique completion. Despite extensive computer searches, the exact count of puzzles with unique solutions remains unknown. This combinatorial problem involves constraints, symmetry breaking, and counting techniques. Understanding this would illuminate the mathematical structure underlying Sudoku and related constraint satisfaction problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No exact count of standard 9x9 uniquely solvable Sudoku clue patterns was verified; the statement lacks a convention on labelled grids, clues, and symmetry.\n\n**Verified partial progress.**\n\n- The number of completed standard Sudoku grids is known exactly.\n- 17 is the minimum clue count for uniqueness.\n\n**Full solution or refutation.**\n\nEstimates and related counts do not answer the literal count without conventions.\n\n**What remains.**\n\nFix the counting equivalence and enumerate the resulting class.\n\n**Sources checked.**\n\n- Sudoku overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Sudoku\n Evidence used: Distinguishes exact completed-grid counts from estimated minimal-puzzle counts.\n\n**Review notes.** No source alteration; formulation incomplete.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1359, "problem_number": "GAME-002", "title": "Sudoku: Minimal Puzzles Count", "statement": "How many Sudoku puzzles with exactly one solution are minimal (removing any clue creates multiple solutions)?", "background": "A minimal Sudoku puzzle cannot have any clue removed without losing uniqueness. While we know examples with as few as 17 clues, the total count of minimal puzzles is unknown. This problem combines enumeration with the structure of constraint systems. The answer would deepen our understanding of puzzle difficulty, minimal representations, and the geometry of solution spaces.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact count of minimal standard Sudoku puzzles is not known; published figures are statistical estimates and depend on equivalence conventions.\n\n**Verified partial progress.**\n\n- Statistical generation estimates about 3.10e37 minimal puzzles.\n\n**Full solution or refutation.**\n\nNo exhaustive exact count was verified.\n\n**What remains.**\n\nSpecify equivalence and obtain an exact enumeration.\n\n**Sources checked.**\n\n- Sudoku overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Sudoku\n Evidence used: States that the number of minimal 9x9 puzzles is not precisely known and reports the estimate.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "view_count": 678, "favorite_count": 51, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1360, "problem_number": "GAME-003", "title": "Maximum Givens in Minimal Sudoku", "statement": "What is the maximum number of givens for a minimal Sudoku puzzle?", "background": "While minimal puzzles can have as few as 17 givens, the upper bound is unknown. A puzzle with many givens can still be minimal if each clue is essential. Computer searches have found minimal puzzles with around 40 givens, but no theoretical maximum is known. This question explores the relationship between redundancy, minimality, and constraint propagation in combinatorial problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Searches report 40-given minimal examples, but a primary proof of maximality was not verified.\n\n**Verified partial progress.**\n\n- A maintained Sudoku-theory resource records 40 as the largest known minimal classic Sudoku.\n\n**Full solution or refutation.**\n\nThe exact maximum should not be asserted from a community database alone.\n\n**What remains.**\n\nVerify an exhaustive upper-bound proof or retain as a computational record.\n\n**Sources checked.**\n\n- Sudoku Theory, Snipes (accessed 2026-08-17). (maintained_tracker): https://sudokutheory.com/wiki/index.php?title=Snipes\n Evidence used: Lists 40 as the most givens in a minimal classic Sudoku.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "view_count": 567, "favorite_count": 43, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" } }, { "id": 1361, "problem_number": "GAME-004", "title": "Tic-Tac-Toe Winning Dimension", "statement": "Given the width of a tic-tac-toe board, what is the smallest dimension guaranteeing X has a winning strategy?", "background": "Classic tic-tac-toe is a draw with perfect play. In higher dimensions (n^d game), questions become more complex. The Hales-Jewett theorem guarantees that for any fixed line length n, there exists a dimension d where the first player wins. But finding the exact threshold dimension for each n remains open. This connects combinatorics, game theory, and Ramsey theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The source does not define board geometry, win length, or whether width is fixed across dimensions, so no status was assigned.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nSeveral distinct multidimensional tic-tac-toe problems fit the words.\n\n**What remains.**\n\nRecover the intended game convention.\n\n**Sources checked.**\n\n- Tic-tac-toe problem source entry (accessed 2026-08-17). (authoritative_secondary): https://www.unsolvedmath.com/problems/1361\n Evidence used: The displayed statement provides no missing rule conventions.\n\n**Review notes.** No source alteration; unrecoverable formulation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1362, "problem_number": "GAME-005", "title": "Perfect Chess", "statement": "What is the outcome of a perfectly played game of chess?", "background": "Chess is a finite deterministic game, so theoretically one of three outcomes holds with perfect play: White wins, Black wins, or draw. Despite centuries of play and powerful computers, we don't know which. The game tree has approximately 10⁴⁷ positions, far beyond exhaustive analysis. Current evidence suggests a draw, but proving it requires breakthrough techniques in game-tree search, endgame databases, or mathematical analysis of chess positions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The game-theoretic value of standard chess under perfect play is unknown.\n\n**Verified partial progress.**\n\n- Complete endgame tablebases solve all positions with seven pieces or fewer.\n\n**Full solution or refutation.**\n\nNo full-game perfect-play result was verified.\n\n**What remains.**\n\nDetermine whether White wins, Black wins, or the game is drawn under perfect play.\n\n**Sources checked.**\n\n- Combinatorial game theory overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Combinatorial_game_theory\n Evidence used: Records chess as unsolved while noting seven-piece tablebases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 1534, "favorite_count": 112, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1363, "problem_number": "GAME-006", "title": "Perfect Komi in Go", "statement": "What is the perfect value of komi (compensation points) in Go?", "background": "In Go, komi compensates the second player (White) for Black's first-move advantage. Professional play uses 6.5 or 7.5 points. But what value makes the game perfectly fair with optimal play? Go's complexity (10¹⁷⁰ legal positions) prevents exhaustive analysis. AI like AlphaGo suggest small adjustments, but perfect komi remains unknown. Determining it would require solving Go—understanding the game-theoretic value with perfect play.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The perfect komi/value of full-board Go has not been determined.\n\n**Verified partial progress.**\n\n- Computer solving covers restricted boards and specific rule sets.\n\n**Full solution or refutation.**\n\nNo theorem fixes a perfect compensation for standard full-board Go.\n\n**What remains.**\n\nSpecify rules/board size precisely and solve the resulting game value.\n\n**Sources checked.**\n\n- Go game-theory overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Go_(game)\n Evidence used: Documents rule-set dependence and unsolved full-board play.\n\n**Review notes.** No source alteration; ruleset omitted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 789, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1364, "problem_number": "GAME-007", "title": "Cap Set Problem", "statement": "What is the largest possible cap set in $n$-dimensional affine space over the three-element field?", "background": "A cap set is a collection of points with no three in a line (in the game SET, cards with no valid set). In the affine space $\\mathbb{F}_3^n$, the maximum cap set size is conjectured to be $c^n$ for some constant c < 3. The best bounds are $2.756^n$ (Ellenberg-Gijswijt, 2016). Determining the precise growth rate connects additive combinatorics, polynomial methods, and the cap set conjecture. The breakthrough proof technique revolutionized the field.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exact maximum cap-set size in F_3^n is unknown generally, but exponential upper and lower bounds are known.\n\n**Verified partial progress.**\n\n- Polynomial-method work proves an upper bound c^n with c<3.\n- Recent constructions give lower bounds at least 2.218^n for large n.\n\n**Full solution or refutation.**\n\nNo exact formula for arbitrary n was verified.\n\n**What remains.**\n\nDetermine exact or asymptotically sharp cap-set growth.\n\n**Sources checked.**\n\n- M. K. Bennett et al., New Lower Bounds for Cap Sets, arXiv:2209.10045. (primary): https://arxiv.org/abs/2209.10045\n Evidence used: The abstract gives the 2.218^n lower bound.\n- Cap set overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Cap_set\n Evidence used: Records the polynomial-method exponential upper bound and continuing extremal question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 356, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1365, "problem_number": "GAME-008", "title": "Octal Games Periodicity", "statement": "Are the nim-sequences of all finite octal games eventually periodic?", "background": "Octal games are impartial combinatorial games defined by simple rules encoded in octal notation. Their nim-values (Grundy numbers) determine optimal play. For some octal games, the nim-sequence is eventually periodic; for others, patterns are elusive. Whether all finite octal games have eventually periodic nim-sequences is unknown. This problem connects game theory, number theory, and automata theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Ultimate periodicity for all finite octal-game nim sequences remains open.\n\n**Verified partial progress.**\n\n- Many individual finite octal games have rigorous periods; maintained tables still list unresolved two- and three-place games.\n\n**Full solution or refutation.**\n\nNo universal periodicity theorem was verified.\n\n**What remains.**\n\nProve periodicity for every finite octal code or find a counterexample.\n\n**Sources checked.**\n\n- A. Flammenkamp, Octal Games (accessed 2026-08-17). (maintained_tracker): https://wwwhomes.uni-bielefeld.de/achim/octal.html\n Evidence used: Lists unresolved finite octal games.\n- Games of No Chance 5, open-problem list. (authoritative_secondary): https://library.slmath.org/books/Book70/files/1005.pdf\n Evidence used: States the all-finite-octal-games periodicity question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1366, "problem_number": "GAME-009", "title": "Grundy's Game Periodicity", "statement": "Is the nim-sequence of Grundy's game eventually periodic?", "background": "Grundy's game: split a heap of n beans into two unequal heaps; last player to move wins. The nim-value sequence starts 0,1,0,2,1,3,2,1,0,4,... but no period has been found despite extensive computation. Whether it's eventually periodic (or even computable) is open. This specific game has resisted analysis for decades, representing a frontier in combinatorial game theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Eventual periodicity of Grundy's Game remains unresolved despite very long computations.\n\n**Verified partial progress.**\n\n- Flammenkamp's computations extend extremely far without a proof of eventual periodicity.\n\n**Full solution or refutation.**\n\nNo period or nonperiodicity proof was verified.\n\n**What remains.**\n\nProve or disprove eventual periodicity.\n\n**Sources checked.**\n\n- A. Flammenkamp, Grundy's Game data (accessed 2026-08-17). (maintained_tracker): https://wwwhomes.uni-bielefeld.de/achim/grundy.html\n Evidence used: Documents extensive ongoing computation.\n- MIT OCW, Grundy's Game -- Periodic? (2010). (authoritative_secondary): https://ocw.mit.edu/courses/es-268-the-mathematics-in-toys-and-games-spring-2010/403b38e26e810396764745e0ebdc0b29_MITES_268S10_ses1_handout.pdf\n Evidence used: Calls eventual periodicity an open problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 278, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1367, "problem_number": "GAME-010", "title": "Rendezvous Problem", "statement": "What is the optimal strategy for two agents to meet on a network without communication?", "background": "The rendezvous problem asks: how should two agents move on a graph to minimize expected meeting time, when they can't communicate and may not know the graph structure? Variants include symmetric/asymmetric information, labeled/unlabeled nodes, and different graph families. Optimal strategies are known for some simple cases but remain open for general graphs. This problem bridges game theory, probability, and distributed algorithms.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Rendezvous has many solved variants, but the source omits the network model, labels, knowledge, synchrony, motion, and objective needed to define one problem.\n\n**Verified partial progress.**\n\n- Algorithms exist for labelled agents in unknown graphs under specified synchrony and sensing assumptions.\n\n**Full solution or refutation.**\n\nNo single optimal strategy follows from the incomplete source statement.\n\n**What remains.**\n\nRecover a precise rendezvous model and criterion.\n\n**Sources checked.**\n\n- D. Pelc et al., Rendezvous of Distance-aware Mobile Agents in Unknown Graphs, arXiv:1406.2795. (primary): https://arxiv.org/abs/1406.2795\n Evidence used: The abstract shows how solvability depends on explicit agent labels, graph knowledge, and time parameters.\n\n**Review notes.** No source alteration; formulation incomplete.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1369, "problem_number": "GEOM-010", "title": "Kissing Number Problem", "statement": "What is the kissing number (maximum number of non-overlapping unit spheres that can touch a central unit sphere) in dimensions other than 1, 2, 3, 4, 8, and 24?", "background": "The kissing number is known exactly only in dimensions 1 (2), 2 (6), 3 (12), 4 (24), 8 (240), and 24 (196,560). The problem asks for exact values in other dimensions. In dimension 3, twelve spheres can kiss a central sphere (with centers forming an icosahedron). Dimensions 8 and 24 have exceptional symmetries related to E₈ and the Leech lattice. Determining kissing numbers connects sphere packing, coding theory, and discrete geometry. The problem is surprisingly difficult—even dimension 5 remains unsolved.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exact kissing numbers are known only in dimensions 1,2,3,4,8,24.\n\n**Verified partial progress.**\n\n- Current record tables provide improved bounds in many other dimensions.\n\n**Full solution or refutation.**\n\nThe requested dimensions remain open.\n\n**What remains.**\n\nClose upper/lower gaps.\n\n**Sources checked.**\n\n- H. Cohn, Kissing numbers table. (maintained_tracker): https://cohn.mit.edu/kissing-numbers/\n Evidence used: Maintained bounds table.\n- O. Musin, The kissing number in four dimensions, Ann. Math. 168 (2008). (primary): https://arxiv.org/abs/math/0309430\n Evidence used: Establishes dimension 4.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 534, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1372, "problem_number": "GEOM-013", "title": "Tammes Problem", "statement": "For n > 14 points (except n=24), what is the maximum minimum distance between points on a unit sphere?", "background": "The Tammes problem asks: how should n points be arranged on a sphere to maximize the minimum distance between any pair? This is equivalent to packing n spherical caps on a sphere. Solutions are known for n ≤ 14 and n = 24 (related to exceptional geometries). For other n, only bounds and computational results exist. The problem has applications in molecular chemistry (electron repulsion), coding theory, and crystallography. Named after Dutch botanist who studied pollen grain pores.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Tammes configurations are proven only for N through 14 and N=24.\n\n**Verified partial progress.**\n\n- N=14 was resolved.\n\n**Full solution or refutation.**\n\nThe stated broader range is open.\n\n**What remains.**\n\nSolve remaining N.\n\n**Sources checked.**\n\n- Tammes problem overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Tammes_problem\n Evidence used: Lists solved N values.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1373, "problem_number": "GEOM-014", "title": "Carathéodory Conjecture", "statement": "Does every convex, closed, twice-differentiable surface in 3D Euclidean space have at least two umbilical points?", "background": "An umbilical point on a surface is where the two principal curvatures are equal (the surface curves equally in all directions, like on a sphere). Carathéodory conjectured that every smooth convex closed surface must have at least two umbilic points. A sphere has infinitely many (every point), but most surfaces should have at least two. Despite being over 100 years old, the conjecture remains open. Partial results exist for analytic surfaces and surfaces with special symmetries.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Caratheodory two-umbilics conjecture remains open in its classical full C2 convex-surface formulation.\n\n**Verified partial progress.**\n\n- Important special cases and index reductions are known.\n\n**Full solution or refutation.**\n\nNo universally accepted proof was verified.\n\n**What remains.**\n\nEstablish two umbilics for all admissible surfaces.\n\n**Sources checked.**\n\n- Caratheodory conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Carath%C3%A9odory_conjecture\n Evidence used: Records the unresolved classical conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1374, "problem_number": "GEOM-015", "title": "Cartan-Hadamard Conjecture", "statement": "Does the isoperimetric inequality extend to Cartan-Hadamard manifolds (complete simply-connected manifolds of nonpositive curvature)?", "background": "The classical isoperimetric inequality states that among all regions with fixed perimeter in Euclidean space, the circle (or sphere) encloses maximum area (or volume). The Cartan-Hadamard conjecture asks whether this inequality holds in spaces of nonpositive curvature. Proven in dimensions 2, 3, and 4, but open in higher dimensions. A positive answer would show that negative curvature preserves this fundamental geometric optimization principle.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Cartan--Hadamard isoperimetric conjecture is settled in low dimensions and special curvature settings, but open generally.\n\n**Verified partial progress.**\n\n- Known in dimensions up to four and several additional cases.\n\n**Full solution or refutation.**\n\nNo arbitrary-dimensional theorem was verified.\n\n**What remains.**\n\nProve the general inequality.\n\n**Sources checked.**\n\n- Cartan-Hadamard conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Cartan%E2%80%93Hadamard_conjecture\n Evidence used: Summarizes known dimensions and open general status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 267, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1375, "problem_number": "GEOM-016", "title": "Chern's Conjecture (Affine Geometry)", "statement": "Does the Euler characteristic of a compact affine manifold vanish?", "background": "An affine manifold is a manifold with an atlas whose transition functions are affine transformations. Chern conjectured that any closed (compact, boundaryless) affine manifold must have Euler characteristic zero. The conjecture is true for many special cases but remains open in general. This would be a fundamental constraint on the topology of spaces admitting flat affine structures, connecting differential geometry with algebraic topology.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Chern's affine Euler-characteristic conjecture remains open generally.\n\n**Verified partial progress.**\n\n- It holds for complete affine manifolds and other major classes.\n\n**Full solution or refutation.**\n\nNo universal proof or counterexample is verified.\n\n**What remains.**\n\nResolve arbitrary compact affine manifolds.\n\n**Sources checked.**\n\n- Chern conjecture (affine geometry) overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Chern_conjecture_(affine_geometry)\n Evidence used: Records open general status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1376, "problem_number": "GEOM-017", "title": "Hopf Conjectures", "statement": "What are the relationships between curvature and Euler characteristic for higher-dimensional Riemannian manifolds?", "background": "The Hopf conjectures are a collection of problems relating the curvature of a manifold to its Euler characteristic. One version: does a positively curved even-dimensional manifold have positive Euler characteristic? Another: does a negatively curved manifold have zero Euler characteristic? These would generalize the Gauss-Bonnet theorem to higher dimensions. Despite progress on special cases, the general conjectures remain open, representing a frontier in global differential geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** 'Hopf Conjectures' conflates several curvature/Euler-characteristic assertions with different statuses.\n\n**Verified partial progress.**\n\n- The Hopf sign conjecture has many special cases and counterexamples to stronger variants.\n\n**Full solution or refutation.**\n\nThe brief source statement does not specify a single theorem.\n\n**What remains.**\n\nRecover the intended Hopf formulation.\n\n**Sources checked.**\n\n- Hopf conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Hopf_conjecture\n Evidence used: Distinguishes the separate curvature conjectures.\n\n**Review notes.** No source alteration; formulation broad.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1377, "problem_number": "GEOM-018", "title": "Yau's Conjecture on First Eigenvalue", "statement": "Is the first eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface of $S^{n+1}$ equal to $n$?", "background": "This conjecture by Shing-Tung Yau concerns minimal surfaces (soap-film-like surfaces) embedded in spheres. The Laplace-Beltrami operator generalizes the Laplacian to curved spaces. Yau conjectured that the first eigenvalue equals the dimension n for minimal hypersurfaces in the (n+1)-sphere. This would provide a sharp geometric-spectral inequality, connecting the shape of minimal surfaces to their vibration modes. Proven in special cases, but remains open generally.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Yau's eigenvalue conjecture remains open generally.\n\n**Verified partial progress.**\n\n- It holds for minimal isoparametric hypersurfaces; new lower-bound work continues.\n\n**Full solution or refutation.**\n\nRecent arXiv full-proof claims were not independently verified.\n\n**What remains.**\n\nProve lambda_1=n in full generality.\n\n**Sources checked.**\n\n- Z. Tang and W. Yan, Isoparametric foliation and Yau conjecture, arXiv:1201.0666. (primary): https://arxiv.org/abs/1201.0666\n Evidence used: Proves the isoparametric case.\n- Stanford minimal-surface notes (2025). (authoritative_secondary): https://www.web.stanford.edu/~ochodosh/Math286-min-surf.pdf\n Evidence used: Lists Yau's conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1378, "problem_number": "GEOM-019", "title": "Hadwiger Conjecture (Covering)", "statement": "Can every $n$-dimensional convex body be covered by at most $2^n$ smaller positively homothetic copies?", "background": "Hadwiger conjectured that any convex body in n dimensions can be covered by at most 2ⁿ smaller copies that are scaled-down versions (homotheties with positive ratio). Proven only for n ≤ 3. For n=2, four copies suffice (proven by Levi). For n=3, eight copies suffice (Hadwiger's original proof). Higher dimensions remain completely open. This is one of the most important unsolved problems in convex geometry, with connections to Borsuk's problem and covering theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The covering Hadwiger conjecture is open in dimension at least three.\n\n**Verified partial progress.**\n\n- The planar case is solved and substantial special-body results are known.\n\n**Full solution or refutation.**\n\nNo universal 2^n theorem is verified.\n\n**What remains.**\n\nProve the covering bound.\n\n**Sources checked.**\n\n- Hadwiger conjecture (combinatorial geometry) overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Hadwiger_conjecture_%28combinatorial_geometry%29\n Evidence used: Records open status in dimension three.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 298, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1379, "problem_number": "GEOM-020", "title": "Happy Ending Problem", "statement": "What is the minimum number $g(n)$ of points in general position in the plane guaranteeing a convex $n$-gon?", "background": "The Happy Ending problem (named for the romance between Erdős and Szekeres who solved special cases) asks: how many points in general position (no three collinear) force the existence of n points forming a convex n-gon? Known: g(3)=3, g(4)=5, g(5)=9. Erdős-Szekeres proved $2^{n-2} + 1 \\leq g(n) \\leq \\binom{2n-4}{n-2} + 1$. The exact value for n ≥ 6 is unknown, and closing this exponential gap is a major challenge in combinatorial geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exact g(n) is known through n=6 and open for n>=7.\n\n**Verified partial progress.**\n\n- g(3)=3, g(4)=5, g(5)=9, g(6)=17; improved general bounds are known.\n\n**Full solution or refutation.**\n\nThe Erdos--Szekeres exact formula remains open.\n\n**What remains.**\n\nDetermine g(n) for n>=7.\n\n**Sources checked.**\n\n- Wolfram MathWorld, Happy End Problem. (authoritative_secondary): https://mathworld.wolfram.com/HappyEndProblem.html\n Evidence used: Lists exact values through six and open range.\n- Erdos Problems 107. (maintained_tracker): https://www.erdosproblems.com/history/107\n Evidence used: Records current upper bounds.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 345, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1380, "problem_number": "GEOM-021", "title": "Heilbronn Triangle Problem", "statement": "What configuration of $n$ points in the unit square maximizes the area of the smallest triangle they determine?", "background": "Heilbronn asked: place n points in a unit square to maximize the minimum triangle area. Trivially, the minimum area is ≤ 2/n. Heilbronn conjectured it's O(1/n²). Komlos-Pintz-Szemeredi showed it's actually Θ((log n)/n²), disproving the conjecture. However, the exact constant is unknown, and tight bounds remain elusive. This problem exemplifies how discrete geometry problems can have surprising answers and connects to irregularities of distribution and discrepancy theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The general Heilbronn triangle extremal problem remains open.\n\n**Verified partial progress.**\n\n- Sharp small-n configurations and asymptotic upper/lower bounds are known.\n\n**Full solution or refutation.**\n\nNo general optimal configuration is verified.\n\n**What remains.**\n\nDetermine sharp asymptotics or exact values.\n\n**Sources checked.**\n\n- Heilbronn triangle problem overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Heilbronn_triangle_problem\n Evidence used: Records continuing open status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 223, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1381, "problem_number": "GEOM-022", "title": "Kalai's 3^d Conjecture", "statement": "Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?", "background": "Gil Kalai conjectured that centrally symmetric polytopes (symmetric under reflection through the origin) must have many faces—at least 3^d for dimension d. The d-cube achieves this bound exactly. Proved for d ≤ 4. Higher dimensions remain open. This would be a fundamental constraint on the combinatorial complexity of symmetric polytopes, with implications for optimization, linear programming, and the geometry of convex bodies.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Kalai's 3^d conjecture is open; stronger variants are false.\n\n**Verified partial progress.**\n\n- It is known for several special centrally symmetric polytope classes.\n\n**Full solution or refutation.**\n\nNo proof for every centrally symmetric polytope is verified.\n\n**What remains.**\n\nEstablish the face-count lower bound.\n\n**Sources checked.**\n\n- Kalai's 3^d conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Kalai%27s_3%5Ed_conjecture\n Evidence used: Summarizes special cases and false stronger variants.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1382, "problem_number": "GEOM-023", "title": "Orchard-Planting Problem", "statement": "What is the maximum number of 3-point lines attainable by a configuration of $n$ points in the plane?", "background": "An orchard-planting problem asks: arrange n points (trees) to maximize the number of lines containing exactly 3 points (rows). For n points, at most n(n-1)/6 such lines are possible (by counting). Some configurations achieve this bound or come close. The problem asks for the exact maximum for each n. Solutions are known for small n, but the general pattern is mysterious. This connects to projective geometry, matroid theory, and combinatorial designs.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The orchard-planting maximum is known in special n and has asymptotic constructions, but no general exact formula is verified.\n\n**Verified partial progress.**\n\n- Burr--Grunbaum--Sloane constructions give strong lower bounds.\n\n**Full solution or refutation.**\n\nNo all-n extremal classification was found.\n\n**What remains.**\n\nDetermine maximum 3-point lines for every n.\n\n**Sources checked.**\n\n- Orchard-planting problem overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Orchard-planting_problem\n Evidence used: Records partial constructions and open general question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1383, "problem_number": "GEOM-024", "title": "Unit Distance Problem", "statement": "How many pairs of points at unit distance can be determined by $n$ points in the Euclidean plane?", "background": "Erdős asked: what's the maximum number of unit-distance pairs among n points in the plane? Trivially at most n(n-1)/2. Known: the maximum is Θ(n^(4/3)) (lower bound by Erdős, upper by Spencer-Szemerédi-Trotter). But the exact exponent is unknown—it could be n^(4/3), n^(3/2), or something between. Determining this connects incidence geometry, graph theory, and the crossing number. The unit distance graph has fascinating properties.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact maximum unit-distance count is open; a 2026 result refutes the conjectured n^{1+o(1)} upper-order behavior.\n\n**Verified partial progress.**\n\n- Classical upper bound is O(n^(4/3)); new constructions give a super-subpolynomial separation from the former conjectural behavior.\n\n**Full solution or refutation.**\n\nThe sharp growth remains unknown.\n\n**What remains.**\n\nDetermine the extremal order between known bounds.\n\n**Sources checked.**\n\n- Planar Point Sets with Many Unit Distances (2026). (primary): https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf\n Evidence used: Constructs a refutation of the stated long-standing growth conjecture.\n- Pach, Raz, Solymosi, Erdos's Unit Distance Problem and Rigidity, SoCG 2026. (primary): https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.83\n Evidence used: States classical O(n^(4/3)) and current structural progress.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1384, "problem_number": "GEOM-025", "title": "Bellman's Lost-in-a-Forest Problem", "statement": "What is the shortest path that guarantees reaching the boundary of a given shape, starting from an unknown point with unknown orientation?", "background": "You're lost in a forest (a region with known shape but unknown location and orientation). What path guarantees you'll reach the edge? For a circle of radius 1, a path of length ≤ 2 + π/3 ≈ 3.05 suffices. For a square, the answer is unknown. For general convex regions, the problem is wide open. This classic problem in geometric search theory has applications to robotics, navigation, and computational geometry. Finding optimal escape paths connects geometry with optimization.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Bellman escape problem has different answers by shape class, knowledge, and permitted path model.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe source does not specify enough to assign one optimum.\n\n**What remains.**\n\nRecover the intended model.\n\n**Sources checked.**\n\n- Lost in a forest problem overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Lost_in_a_forest_problem\n Evidence used: Distinguishes model-sensitive variants.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 423, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1385, "problem_number": "GEOM-026", "title": "Borromean Rings Question", "statement": "Can three unknotted space curves (not all circles) be arranged as Borromean rings?", "background": "Borromean rings are three linked loops where removing any one unlinks the other two. Classical Borromean rings use circles, but perfect circular realization is impossible (proved). Can non-circular unknotted curves realize this linking pattern? This question connects knot theory, topology, and geometry. While Borromean rings can be made from ellipses or other shapes, whether three genuinely unknotted (topologically circular) but geometrically non-circular curves can achieve this remains subtle.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Yes: standard Borromean rings already have three unknotted components, and smooth noncircular representatives are obtained by ambient isotopy.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nA noncircular deformation preserves the Borromean link type and unknottedness of components.\n\n**What remains.**\n\nNo work remains under the literal existential wording.\n\n**Sources checked.**\n\n- Borromean rings overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Borromean_rings\n Evidence used: Describes the three-component link with unknotted components.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1386, "problem_number": "GEOM-027", "title": "Danzer's Problem", "statement": "Do Danzer sets of bounded density or bounded separation exist?", "background": "A Danzer set is a set of points in the plane such that every convex region of area 1 contains at least one point. Danzer asked: can such a set have bounded density (points per unit area) or bounded separation (minimum distance between points)? Both properties would mean the points are \"well-distributed.\" While Danzer sets exist, whether nice ones exist is open. Related to Conway's \"dead fly\" problem. Connects measure theory, convexity, and geometric covering.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Existence of a bounded-density/bounded-separation Danzer set remains open; a stronger bounded-intersection version is impossible.\n\n**Verified partial progress.**\n\n- The Gowers strengthening with uniformly bounded points in every unit-volume body is disproved.\n\n**Full solution or refutation.**\n\nThat does not settle the source's weaker density/separation question.\n\n**What remains.**\n\nConstruct such a Danzer/Delone set or prove impossibility.\n\n**Sources checked.**\n\n- Danzer set overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Danzer_set\n Evidence used: Distinguishes the open Danzer question from the solved stronger variation.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1388, "problem_number": "GRAPH-001", "title": "Brouwer's Conjecture on Graph Laplacians", "statement": "Can the sum of eigenvalues of the Laplacian matrix of a graph be bounded by the number of edges?", "background": "Brouwer conjectured an upper bound for the sum of the k largest eigenvalues of the Laplacian matrix of a graph in terms of the number of edges. The Laplacian matrix encodes graph structure and has deep connections to spectral graph theory. This conjecture would provide fundamental insights into the relationship between a graph's combinatorial and spectral properties. Progress has been made for special classes of graphs, but the general case remains open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Brouwer's Laplacian eigenvalue-sum conjecture remains open generally.\n\n**Verified partial progress.**\n\n- It is proved for several graph classes, with recent equality/full-version studies.\n\n**Full solution or refutation.**\n\nNo all-graph proof was verified.\n\n**What remains.**\n\nProve the inequality for all graphs.\n\n**Sources checked.**\n\n- On the full Brouwer's conjecture on Laplacian eigenvalues, Discrete Appl. Math. 391 (2026). (primary): https://www.sciencedirect.com/science/article/abs/pii/S0166218X26002593\n Evidence used: Proves special graph families and states the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1389, "problem_number": "GRAPH-002", "title": "Eternal Domination vs Domination Number", "statement": "Does there exist a graph where the dominating number equals the eternal dominating number and both are less than the clique covering number?", "background": "The dominating number γ(G) is the minimum size of a dominating set. The eternal dominating number γ∞(G) arises from a game where guards on vertices must respond to attacks. The question asks if these can equal each other while being smaller than the clique covering number (minimum number of cliques needed to cover all vertices). This connects domination theory with graph games and clique structures.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The general relationship between eternal domination and domination parameters remains unresolved.\n\n**Verified partial progress.**\n\n- Equalities and inequalities are established for perfect and other graph classes.\n\n**Full solution or refutation.**\n\nNo universal relation matching the likely intended question was verified.\n\n**What remains.**\n\nRecover the exact parameter relation and settle it generally.\n\n**Sources checked.**\n\n- Eternal dominating set overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Eternal_dominating_set\n Evidence used: Records known perfect-graph equality and open general problems.\n\n**Review notes.** No source alteration; title is underspecified.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1390, "problem_number": "GRAPH-003", "title": "Graham's Pebbling Conjecture", "statement": "Is the pebbling number of the Cartesian product of two graphs at least the product of their pebbling numbers?", "background": "Graph pebbling is a combinatorial game where pebbles are moved on vertices according to specific rules. Graham conjectured that the pebbling number (minimum pebbles needed to guarantee placing one on any target vertex) of a Cartesian product G × H is at least π(G) × π(H). Despite progress on special cases like products with paths or cycles, the general conjecture remains unsolved. This problem has applications to communication networks and resource distribution.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Graham's pebbling conjecture remains open in general.\n\n**Verified partial progress.**\n\n- It is proved for many graph families and product cases.\n\n**Full solution or refutation.**\n\nNo universal product inequality was verified.\n\n**What remains.**\n\nProve the conjecture for all graph pairs.\n\n**Sources checked.**\n\n- Graham's pebbling conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Graham%27s_pebbling_conjecture\n Evidence used: Records special cases and general open status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1391, "problem_number": "GRAPH-004", "title": "Meyniel's Conjecture on Cop Number", "statement": "Is the cop number of a connected n-vertex graph $O(\\sqrt{n})$?", "background": "The cop number is the minimum number of cops needed to guarantee catching a robber in a pursuit game on a graph. Meyniel conjectured that for any connected graph with n vertices, the cop number is at most O(√n). The best known upper bound is O(n/log n). This problem connects graph theory with algorithmic game theory and has applications to network security, robot motion planning, and pursuit-evasion games. Resolving it would fundamentally advance our understanding of graph searching problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Meyniel's O(sqrt n) cop-number conjecture remains open.\n\n**Verified partial progress.**\n\n- Best general bounds have improved, and the conjecture holds in several graph classes.\n\n**Full solution or refutation.**\n\nNo universal square-root upper bound is verified.\n\n**What remains.**\n\nProve Meyniel's bound or find a counterexample.\n\n**Sources checked.**\n\n- Meyniel's conjecture survey. (authoritative_secondary): https://math.ryerson.ca/~abonato/papers/meyniel_0311.pdf\n Evidence used: Surveys new upper-bound progress and open conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1392, "problem_number": "GRAPH-005", "title": "Graph Coloring Game Monotonicity", "statement": "If Alice has a winning strategy for the vertex coloring game with k colors, does she have one for k+1 colors?", "background": "In the graph coloring game, two players alternately color vertices with k colors, trying to create (Alice) or avoid (Bob) a proper coloring. Intuitively, having more colors should make Alice's task easier. However, whether winning with k colors implies winning with k+1 colors is surprisingly still open. This problem probes the subtle complexity of graph coloring games and connects combinatorial game theory with chromatic graph theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Graph-coloring game monotonicity remains unresolved in its general formulation.\n\n**Verified partial progress.**\n\n- Counterexamples and positive results are known for restricted variants.\n\n**Full solution or refutation.**\n\nNo universal monotonicity theorem was verified.\n\n**What remains.**\n\nSpecify and settle the intended game parameter inequality.\n\n**Sources checked.**\n\n- Graph coloring game overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Graph_coloring_game\n Evidence used: Documents variant-dependent open questions.\n\n**Review notes.** No source alteration; formulation needs parameters.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1393, "problem_number": "GRAPH-006", "title": "1-Factorization Conjecture", "statement": "Does every k-regular graph on 2n vertices admit a 1-factorization when k ≥ n (or k ≥ n-1 for even n)?", "background": "A 1-factor is a perfect matching, and a 1-factorization is a partition of edges into 1-factors. The conjecture states that sufficiently regular graphs can be decomposed into perfect matchings. This would generalize classical results on complete graphs. Proven for many special cases, but the general statement remains open. Applications include tournament scheduling, network routing, and combinatorial designs.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The 1-factorization conjecture is proved for all sufficiently large n, which is its standard formulation.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nCsaba--Kuhn--Lo--Osthus--Treglown prove every sufficiently large high-degree regular graph has a perfect-matching decomposition.\n\n**What remains.**\n\nNo work remains for the standard asymptotic conjecture.\n\n**Sources checked.**\n\n- B. Csaba et al., Proof of the 1-factorization and Hamilton decomposition conjectures, Mem. AMS 244 (2016); arXiv:1401.4159. (primary): https://arxiv.org/abs/1401.4159\n Evidence used: The abstract states the proved 1-factorization theorem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1394, "problem_number": "GRAPH-007", "title": "Perfect 1-Factorization Conjecture", "statement": "Does every complete graph on an even number of vertices admit a perfect 1-factorization?", "background": "A perfect 1-factorization of a complete graph K₂ₙ is a 1-factorization where the union of any two 1-factors forms a Hamiltonian cycle. Such structures have beautiful symmetry and applications to combinatorial designs. While perfect 1-factorizations are known for many n (especially powers of 2 and small cases), a general existence proof remains elusive. This is one of the most elegant open problems in graph decomposition theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The perfect 1-factorization conjecture remains open in its all-even-orders form, but an asymptotic version was proved in 2026.\n\n**Verified partial progress.**\n\n- Cheng--Sgueglia construct decompositions in which (1-o(1))n matchings have the pairwise Hamilton-cycle property.\n\n**Full solution or refutation.**\n\nNo proof for every even order was verified.\n\n**What remains.**\n\nEstablish a perfect 1-factorization of every even complete graph.\n\n**Sources checked.**\n\n- Y. Cheng and A. Sgueglia, The perfect 1-factorisation conjecture holds asymptotically, arXiv:2607.09459 (2026). (primary): https://arxiv.org/abs/2607.09459\n Evidence used: The abstract explicitly describes the asymptotic result and says the original conjecture is far from solved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1395, "problem_number": "GRAPH-008", "title": "Cereceda's Conjecture", "statement": "For k-degenerate graphs, can any (k+2)-coloring be transformed to any other in polynomial steps via single-vertex recolorings?", "background": "Cereceda's conjecture concerns the diameter of the reconfiguration graph of graph colorings. It asks whether the shortest sequence of single-vertex recolorings transforming one coloring to another is polynomially bounded for degenerate graphs. This connects graph coloring with reconfiguration problems—a growing area studying how to transform one solution to another. Applications include network reconfiguration and state-space search.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The literal polynomial-length question is answered affirmatively for each fixed degeneracy k: the remaining standard Cereceda conjecture asks for a quadratic, not merely polynomial, bound.\n\n**Verified partial progress.**\n\n- Bousquet--Heinrich prove diameter O(n^(d+1)) for d-degenerate graphs with at least d+2 colors.\n- Their result leaves the O(n^2) Cereceda bound open generally.\n\n**Full solution or refutation.**\n\nFor fixed d, O(n^(d+1)) is polynomial in n, so it settles the statement as worded.\n\n**What remains.**\n\nFor the conventional quadratic formulation, prove O(n^2) for all d-degenerate graphs.\n\n**Sources checked.**\n\n- N. Bousquet and M. Heinrich, A polynomial version of Cereceda's conjecture, J. Combin. Theory Ser. B 155 (2022), doi:10.1016/j.jctb.2022.01.006; arXiv:1903.05619. (primary): https://arxiv.org/abs/1903.05619\n Evidence used: States the O(n^(d+1)) theorem for k>=d+2 and distinguishes the open quadratic conjecture.\n\n**Review notes.** The record's wording is weaker than the usual quadratic Cereceda conjecture; statement retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1396, "problem_number": "GRAPH-009", "title": "Earth-Moon Problem", "statement": "What is the maximum chromatic number of biplanar graphs?", "background": "A graph is biplanar if it can be drawn on two parallel planes (Earth and Moon) with edges possibly crossing between planes but not within each plane. The Earth-Moon problem asks for the maximum chromatic number of such graphs. Known bounds are 12 ≤ χ ≤ 16. This problem combines planarity concepts with multilayer graph drawings, relevant to VLSI design and network visualization.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Earth--Moon problem remains open with checked bounds 9 <= chi <= 12.\n\n**Verified partial progress.**\n\n- A biplanar 9-chromatic construction is known.\n- Recent work excluded at least one proposed 10-chromatic candidate from being biplanar.\n\n**Full solution or refutation.**\n\nNeither a 10--12 chromatic biplanar graph nor an improved universal upper bound was verified.\n\n**What remains.**\n\nDetermine the maximum chromatic number of biplanar graphs.\n\n**Sources checked.**\n\n- Earth-Moon Problem, Graph-theory open problems tracker. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/earth_moon_problem/\n Evidence used: Records the unchanged 9--12 bounds and 2023 candidate exclusions.\n- D. Eppstein, On the Biplanarity of Blowups, arXiv:2301.09246 (2023). (primary): https://arxiv.org/abs/2301.09246\n Evidence used: Gives recent negative evidence concerning a proposed construction mechanism.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1397, "problem_number": "GRAPH-010", "title": "Gyárfás-Sumner Conjecture", "statement": "Is every graph class defined by excluding one fixed tree as an induced subgraph χ-bounded?", "background": "A graph class is χ-bounded if there's a function f such that every graph in the class with clique number ω has chromatic number at most f(ω). The conjecture states that forbidding any tree as an induced subgraph creates a χ-bounded class. This would unify many results on perfect graphs and their generalizations. The conjecture connects structural graph theory with coloring, and has implications for algorithmic graph theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Gyárfás--Sumner conjecture remains open for arbitrary excluded induced trees.\n\n**Verified partial progress.**\n\n- It has been established for numerous specified tree families and structural graph classes.\n\n**Full solution or refutation.**\n\nNo theorem covering every fixed tree was verified.\n\n**What remains.**\n\nProve chi-boundedness for the class excluding each fixed tree as an induced subgraph.\n\n**Sources checked.**\n\n- A. Scott and P. Seymour, A survey of chi-boundedness, J. Graph Theory 95 (2020). (primary): https://doi.org/10.1002/jgt.22511\n Evidence used: Surveys chi-boundedness results and the outstanding induced-tree conjecture.\n- P. Chudnovsky et al., A note on the Gyárfás-Sumner conjecture, arXiv:2302.08922 (2023). (primary): https://arxiv.org/abs/2302.08922\n Evidence used: Recent work explicitly treats the conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1398, "problem_number": "GRAPH-011", "title": "Jaeger's Petersen Coloring Conjecture", "statement": "Does every bridgeless cubic graph have a cycle-continuous mapping to the Petersen graph?", "background": "Jaeger conjectured that every bridgeless cubic graph admits a special kind of homomorphism to the Petersen graph that preserves cycle structure. The Petersen graph plays a central role in graph theory as a universal counterexample and fundamental object. This conjecture connects graph homomorphisms, snarks (cubic graphs resistant to edge coloring), and the structure of cubic graphs. It has deep implications for edge coloring and nowhere-zero flow problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Jaeger's Petersen Coloring Conjecture remains open for bridgeless cubic graphs.\n\n**Verified partial progress.**\n\n- It is equivalent to the normal 5-edge-coloring conjecture and has been verified for several snark families.\n\n**Full solution or refutation.**\n\nNo all-bridgeless-cubic theorem was verified.\n\n**What remains.**\n\nProduce a Petersen coloring for every bridgeless cubic graph or a counterexample.\n\n**Sources checked.**\n\n- An infinite family of normal 5-edge colorable superpositioned snarks, Discrete Applied Mathematics 378 (2026), doi:10.1016/j.dam.2025.07.032. (primary): https://doi.org/10.1016/j.dam.2025.07.032\n Evidence used: States equivalence with the Petersen coloring conjecture and records solved snark families.\n- Petersen coloring conjecture, Graph-theory open problems tracker. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/petersen_coloring_conjecture/\n Evidence used: Records current general open status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1399, "problem_number": "GRAPH-012", "title": "List Coloring Conjecture", "statement": "For every graph, does the list chromatic index equal the chromatic index?", "background": "The chromatic index χ'(G) is the minimum number of colors needed to color edges so no two adjacent edges share a color. The list chromatic index is the minimum k such that edges can be colored from arbitrary k-element color lists. The conjecture states these are always equal. While proven for bipartite graphs and some other classes, the general case remains open. This is a fundamental question in list coloring theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The List Coloring Conjecture for edge chromatic index remains open in general.\n\n**Verified partial progress.**\n\n- The equality is proved for bipartite graphs (the Dinitz theorem) and various further classes.\n\n**Full solution or refutation.**\n\nNo proof for all graphs was verified.\n\n**What remains.**\n\nShow that every graph has list chromatic index equal to its chromatic index.\n\n**Sources checked.**\n\n- F. Galvin, The list chromatic index of a bipartite multigraph, J. Combin. Theory Ser. B 63 (1995), 153--158. (primary): https://doi.org/10.1006/jctb.1995.1011\n Evidence used: Proves the important bipartite special case.\n- List edge-coloring overview. (authoritative_secondary): https://en.wikipedia.org/wiki/List_edge-coloring\n Evidence used: Records the general List Coloring Conjecture as unresolved and the bipartite theorem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1400, "problem_number": "GRAPH-013", "title": "Overfull Conjecture", "statement": "Is a graph with maximum degree Δ(G) ≥ n/3 in class 2 if and only if it has an overfull subgraph with the same maximum degree?", "background": "By Vizing's theorem, every graph has chromatic index Δ or Δ+1 (class 1 or 2). A graph is overfull if it has more than Δ⌊n/2⌋ edges, forcing class 2. The overfull conjecture provides a complete characterization: when Δ ≥ n/3, being class 2 is equivalent to having an overfull subgraph preserving the maximum degree. This would elegantly explain why graphs are hard to edge-color.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Overfull Conjecture remains open in its full Delta>n/3 range.\n\n**Verified partial progress.**\n\n- It is proved for sufficiently large robust expanders in a high-degree regime, including sufficiently large graphs with Delta at least (1+epsilon)n/2.\n\n**Full solution or refutation.**\n\nNo proof across the entire n/3 threshold was verified.\n\n**What remains.**\n\nSettle the conjecture for all simple graphs with Delta(G)>n/3.\n\n**Sources checked.**\n\n- G. Chen, J. McDonald and S. Shan, Towards the Overfull Conjecture II, arXiv:2607.02270 (2026). (primary): https://arxiv.org/abs/2607.02270\n Evidence used: States the open conjecture and its new robust-expander/high-degree consequences.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1401, "problem_number": "GRAPH-014", "title": "Total Coloring Conjecture", "statement": "Is the total chromatic number of every graph at most Δ + 2, where Δ is the maximum degree?", "background": "Total coloring requires coloring both vertices and edges so adjacent/incident elements have different colors. Behzad and Vizing independently conjectured that the total chromatic number χ″(G) ≤ Δ(G) + 2. The lower bound Δ + 1 is easy (color each vertex and its incident edges distinctly). The upper bound Δ + 2 is proven for many graph classes but remains open in general. This is one of the most fundamental open problems in graph coloring.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Total Coloring Conjecture remains open for arbitrary graphs.\n\n**Verified partial progress.**\n\n- Many special graph classes are known, including recent planar and 1-planar cases with explicit restrictions.\n\n**Full solution or refutation.**\n\nNo universal Delta+2 proof was verified.\n\n**What remains.**\n\nProve chi''(G)<=Delta(G)+2 for every graph.\n\n**Sources checked.**\n\n- R. Su, G. Fang and E. Zhu, The Total Coloring Conjecture holds for planar graphs without three special subgraphs, arXiv:2507.12737 (2025). (primary): https://arxiv.org/abs/2507.12737\n Evidence used: Gives a restricted planar theorem and explicitly says the relevant general planar case remains open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1402, "problem_number": "GRAPH-015", "title": "Albertson Conjecture", "statement": "Can the crossing number of a graph be lower-bounded by the crossing number of a complete graph with the same chromatic number?", "background": "The crossing number is the minimum number of edge crossings in a planar drawing. Albertson conjectured cr(G) ≥ cr(K_χ(G)) where χ(G) is the chromatic number. This would link two fundamental graph parameters—crossing number and chromatic number. Proven for chromatic numbers up to 16, but the general case remains open. This connects graph drawing, coloring theory, and topological graph theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Albertson's crossing-number conjecture remains open generally, with verification extended through chromatic number 24.\n\n**Verified partial progress.**\n\n- Cranston verified all chromatic numbers at most 24 and restricted possible counterexamples for 25 and 26.\n- Fox--Pach--Suk prove it in an additional small-order regime for large chromatic number.\n\n**Full solution or refutation.**\n\nNo all-chromatic-number proof was verified.\n\n**What remains.**\n\nEstablish the crossing-number lower bound for every chromatic number and graph order.\n\n**Sources checked.**\n\n- D. W. Cranston, Progress on Albertson's Conjecture, arXiv:2512.08020 (2025). (primary): https://arxiv.org/abs/2512.08020\n Evidence used: States verification for r<=24 and new restrictions.\n- J. Fox, J. Pach and A. Suk, Immersions and Albertson's Conjecture, SoCG 2025, doi:10.4230/LIPIcs.SoCG.2025.50. (primary): https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.50\n Evidence used: Provides a further proved small-order regime.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1403, "problem_number": "GRAPH-016", "title": "Conway's Thrackle Conjecture", "statement": "Does every thrackle have at most as many edges as vertices?", "background": "A thrackle is a graph drawing where every pair of edges either meets at a common vertex or crosses exactly once. Conway conjectured that thrackles satisfy |E| ≤ |V|. Despite looking simple, this conjecture has resisted proof for decades. The best known bound is |E| ≤ 3|V|/2. This problem connects graph drawing with combinatorial geometry and has surprising depth for such a simply stated question.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Conway's planar Thrackle Conjecture remains open; the best checked general upper bound is 1.393n.\n\n**Verified partial progress.**\n\n- The conjecture is established for several restricted drawing models, and general upper bounds have been progressively improved.\n\n**Full solution or refutation.**\n\nNo proof of the m<=n bound for all planar thrackles was verified.\n\n**What remains.**\n\nProve or disprove m<=n for every planar thrackle.\n\n**Sources checked.**\n\n- C. Hernández-Vélez, J. Kynčl and G. Salazar, Thrackles on nonplanar surfaces, arXiv:2506.11808 (2025). (primary): https://arxiv.org/abs/2506.11808\n Evidence used: Explicitly states that Conway's planar conjecture remains open and records the 1.393n bound.\n- TOPP Problem 30: Thrackles. (maintained_tracker): https://topp.openproblem.net/p30\n Evidence used: Records open status and major historical upper-bound improvements.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1404, "problem_number": "GRAPH-017", "title": "GNRS Conjecture", "statement": "Do minor-closed graph families have $\\ell_1$ embeddings with bounded distortion?", "background": "The GNRS conjecture asks whether graphs from minor-closed families (like planar graphs) can be embedded into L₁ space (ℓ₁ metric) with distortion bounded by a function of the excluded minor size. This connects graph theory with metric geometry and theoretical computer science. The conjecture has important implications for approximation algorithms and understanding the metric structure of graph families.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The intended GNRS conjecture for proper minor-closed families and all supported shortest-path metrics remains open; bounded distortion is proved for several proper subclasses and the general problem has useful reductions.\n\n**Verified partial progress.**\n\n- Lee and Sidiropoulos reduced GNRS to the planar embedding conjecture together with a clique-sum embedding conjecture.\n- They proved bounded distortion for every fixed-pathwidth family; earlier GNRS work proved constant distortion for series-parallel graphs.\n- Kumar proved constant distortion for the restricted set of vertex pairs that lie on a common face of a planar drawing.\n\n**Full solution or refutation.**\n\nNo constant-distortion embedding theorem for every proper minor-closed family, or even for arbitrary planar shortest-path metrics, was verified.\n\n**What remains.**\n\nProve bounded L1 distortion for all planar graph metrics and the required clique-sum closure, or find a proper minor-closed counterexample family.\n\n**Sources checked.**\n\n- J. R. Lee and A. Sidiropoulos, On the Geometry of Graphs with a Forbidden Minor, STOC 2009, pp. 245–254. (primary): https://doi.org/10.1145/1536414.1536450\n Evidence used: States the supported-metric formulation, proves the fixed-pathwidth case, and reduces GNRS to planar and clique-sum conjectures.\n- A. Gupta, I. Newman, Y. Rabinovich, and A. Sinclair, Cuts, Trees and l1-Embeddings of Graphs, Combinatorica 24 (2004), 233–269. (primary): https://doi.org/10.1007/s00493-004-0015-x\n Evidence used: Original GNRS framework and constant-distortion results for series-parallel graphs.\n- N. Kumar, An Approximate Generalization of the Okamura–Seymour Theorem, arXiv:2208.00795 (2022). (primary): https://arxiv.org/abs/2208.00795\n Evidence used: Records the planar O(sqrt(log n)) bound and proves a constant-distortion result restricted to cofacial pairs.\n\n**Review notes.** The exact sentence omits 'proper' and the quantification over weighted shortest-path metrics. Literally including the family of all graphs makes the answer negative; the title and background indicate the standard GNRS reading.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1405, "problem_number": "GRAPH-018", "title": "Harborth's Conjecture", "statement": "Can every planar graph be drawn with integer edge lengths?", "background": "Harborth conjectured that every planar graph has a straight-line drawing where all edge lengths are integers. While planar graphs always have straight-line drawings (Fáry's theorem), forcing integer lengths is much harder. Known for trees and some other classes, but open in general. This problem connects graph drawing with discrete geometry and has applications to VLSI layout.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Harborth's conjecture remains open for arbitrary planar graphs, but it is now proved for every 4-regular planar graph as well as the previously known maximum-degree-3 case.\n\n**Verified partial progress.**\n\n- Chang and Sun proved that every 4-regular planar graph has a rational Fáry embedding, which scales to integer edge lengths.\n- Their paper records the earlier result for planar graphs of maximum degree three.\n- A July 2026 preprint reduces the stronger integer-coordinate conjecture to local rational-distance problems for polygons of at most five vertices and proves the nondegenerate quadrilateral step.\n\n**Full solution or refutation.**\n\nNo construction covering all planar graphs was verified; the recent reduction still leaves pentagon and degenerate polygon cases.\n\n**What remains.**\n\nHandle unrestricted planar graphs, in particular the unresolved degree-five insertion geometry in triangulations.\n\n**Sources checked.**\n\n- D. J. Chang and T. Sun, Harborth's Conjecture for 4-Regular Planar Graphs, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024), LIPIcs 320, Article 38. (primary): https://doi.org/10.4230/LIPIcs.GD.2024.38\n Evidence used: Theorem 1 proves the conjecture for all 4-regular planar graphs and the introduction summarizes the degree-three case.\n- S. Kintali, On the Harborth Conjecture. Part I, arXiv:2607.02535 (2026). (primary): https://arxiv.org/abs/2607.02535\n Evidence used: Very recent preprint giving the polygon reduction and nondegenerate quadrilateral result, while explicitly leaving later cases to future parts.\n\n**Review notes.** The exact sentence omits that the drawing must be crossing-free and straight-line. The 2026 item is an unreviewed first installment and is not treated as a solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1406, "problem_number": "GRAPH-019", "title": "Negami's Conjecture", "statement": "Does every graph with a planar cover have a projective-plane embedding?", "background": "Negami conjectured that if a graph G has a planar cover (a planar graph that maps onto G), then G embeds in the projective plane. This would characterize projective-plane graphs via covering spaces. The conjecture connects topological graph theory with covering space theory from topology. Despite progress on special cases, the general conjecture remains a central open problem in topological graph theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Negami's finite planar-cover conjecture remains open and is equivalent to excluding a finite planar cover of K_{1,2,2,2}; recent work sharply restricts a putative minimum cover.\n\n**Verified partial progress.**\n\n- The general conjecture is equivalent to the assertion that K_{1,2,2,2} has no finite planar cover.\n- Annor, Nikolayevsky, and Payne show that a minimum such cover, if it exists, must be 4-connected and exclude several structural forms.\n- Negami proves the desired projective-plane conclusion for a 3-connected graph having an appropriately separated 3-connected finite planar cover.\n\n**Full solution or refutation.**\n\nNo proof that every finite-planar-cover graph embeds in the projective plane, and no counterexample cover of K_{1,2,2,2}, was verified.\n\n**What remains.**\n\nRule out every finite planar cover of K_{1,2,2,2}, or construct one and thereby refute the conjecture.\n\n**Sources checked.**\n\n- D. Y. B. Annor, Y. Nikolayevsky, and M. S. Payne, Three Theorems on Negami's Planar Cover Conjecture, arXiv:2412.19560 (2024). (primary): https://arxiv.org/abs/2412.19560\n Evidence used: States the equivalence with K_{1,2,2,2} and proves that a minimum cover would be 4-connected.\n- S. Negami, Another approach to Planar Cover Conjecture focusing on rotation systems, Journal of the Mathematical Society of Japan 76 (2024), 975–996. (primary): https://doi.org/10.2969/jmsj/90769076\n Evidence used: Proves a projective-plane embedding under explicit 3-connectivity and fibre-separation hypotheses.\n\n**Review notes.** The exact sentence omits the standard connectedness and finiteness hypotheses. Infinite topological covers should not be silently conflated with finite graph covers.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1407, "problem_number": "GRAPH-020", "title": "Turán's Brick Factory Problem", "statement": "What is the minimum crossing number of the complete bipartite graph $K_{m,n}$?", "background": "Turán's brick factory problem asks for the exact crossing number of complete bipartite graphs K_{m,n}. Zarankiewicz conjectured a formula in 1954, which is known to be correct for several cases but unproven in general. The problem arose from Turán observing workers crossing paths while moving bricks. Despite being simple to state, this geometric problem has remained unsolved for 70 years.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Zarankiewicz formula for cr(K_{m,n}) remains unproved in general; it is exact when the smaller part has size at most six, and modern work has improved asymptotic lower bounds.\n\n**Verified partial progress.**\n\n- Zarankiewicz's drawing proves the upper bound Z(m,n)=floor(m/2)floor((m-1)/2)floor(n/2)floor((n-1)/2).\n- Equality cr(K_{m,n})=Z(m,n) is proved when min(m,n) is at most six.\n- Balogh, Lidický, Norin, Pfender, Salazar, and Spiro prove cr(K_{n,n}) at least 0.9118 Z(n,n)+o(n^4).\n\n**Full solution or refutation.**\n\nNo exact formula was verified for arbitrary m and n; the best general construction and lower bound do not coincide.\n\n**What remains.**\n\nProve Zarankiewicz's lower bound for all m,n or find a drawing with fewer than Z(m,n) crossings.\n\n**Sources checked.**\n\n- J. Balogh, B. Lidický, S. Norin, F. Pfender, G. Salazar, and S. Spiro, Crossing numbers of complete bipartite graphs, Procedia Computer Science 223 (2023), 78–87. (primary): https://doi.org/10.1016/j.procs.2023.08.216\n Evidence used: States the conjectured formula and proves the 0.9118 asymptotic diagonal lower bound.\n- E. de Klerk and D. V. Pasechnik, Improved lower bounds for the 2-page crossing numbers of K_{m,n} and K_n via semidefinite programming, arXiv:1110.4824. (primary): https://arxiv.org/abs/1110.4824\n Evidence used: Records that the ordinary Zarankiewicz equality is known for min(m,n) <= 6 and develops related restricted-drawing bounds.\n\n**Review notes.** The record correctly asks for the general crossing number. The associated exact-value conjecture is Zarankiewicz's conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1408, "problem_number": "GRAPH-021", "title": "Guy's Crossing Number Conjecture", "statement": "Is the crossing number of the complete graph $K_n$ equal to the value given by Guy's formula?", "background": "Guy conjectured a formula for the crossing number of complete graphs: cr(K_n) = (1/4)⌊n/2⌋⌊(n-1)/2⌋⌊(n-2)/2⌋⌊(n-3)/2⌋. This is proven for n ≤ 12, but the general case is open. Finding the exact crossing number of complete graphs is a fundamental problem in topological graph theory. The conjecture represents our best guess based on known constructions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Harary–Hill formula for cr(K_n) remains open beyond n=12, with a proved asymptotic lower bound equal to 98.5% of the conjectured value.\n\n**Verified partial progress.**\n\n- Pan and Richter proved cr(K_11)=100, which implies cr(K_12)=150 and verifies the formula through n=12.\n- Balogh, Lidický, and Salazar prove cr(K_n) >= (0.985-o(1))H(n).\n- The standard construction gives cr(K_n) <= H(n) for every n.\n\n**Full solution or refutation.**\n\nNo proof or disproof of cr(K_n)=H(n) for all n was verified.\n\n**What remains.**\n\nClose the asymptotic 1.5% gap and determine the first unresolved exact case n=13 and all subsequent cases.\n\n**Sources checked.**\n\n- S. Pan and R. B. Richter, The crossing number of K11 is 100, Journal of Graph Theory 56 (2007), 128–134. (primary): https://doi.org/10.1002/jgt.20249\n Evidence used: Proves cr(K11)=100 and derives cr(K12)=150.\n- J. Balogh, B. Lidický, and G. Salazar, Closing in on Hill's Conjecture, SIAM Journal on Discrete Mathematics 33 (2019), 1261–1276. (primary): https://doi.org/10.1137/17M1158859\n Evidence used: States verification only through n=12 and proves the asymptotic 0.985 lower factor.\n\n**Review notes.** The same formula is now commonly called Hill's conjecture or the Harary–Hill conjecture; the title's attribution to Guy is a nomenclature variant, not a mathematical defect.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1409, "problem_number": "GRAPH-022", "title": "Universal Point Sets", "statement": "Do planar graphs have universal point sets of subquadratic size?", "background": "A universal point set for n-vertex planar graphs is a set of points such that every n-vertex planar graph has a straight-line embedding on these points. Trivially, O(n²) points suffice. The question asks if o(n²) is possible. Best known lower bound is Ω(n), upper bound is O(n²). Closing this gap would advance our understanding of planar graph representations and geometric graph theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For all n-vertex planar graphs, whether the minimum universal point-set size is o(n^2) remains open; the best located general bounds are n^2/4-Theta(n) above and (1.293-o(1))n below.\n\n**Verified partial progress.**\n\n- Bannister, Cheng, Devanny, and Eppstein construct general universal point sets of size n^2/4-Theta(n).\n- Scheucher, Schrezenmaier, and Steiner prove that at least (1.293-o(1))n points are necessary.\n- Linear or near-linear sets are known for subclasses, including bipartite planar graphs, maximum-degree-three planar graphs, and bounded-pathwidth planar families.\n\n**Full solution or refutation.**\n\nNo subquadratic construction for the class of all n-vertex planar graphs and no superlinear lower bound approaching quadratic were verified.\n\n**What remains.**\n\nConstruct an o(n^2)-size universal point set for all n-vertex planar graphs, or prove a quadratic lower bound.\n\n**Sources checked.**\n\n- M. Bannister, Z. Cheng, W. Devanny, and D. Eppstein, Superpatterns and Universal Point Sets, Journal of Graph Algorithms and Applications 18 (2014), 177–209. (primary): https://doi.org/10.7155/jgaa.00318\n Evidence used: Proves the n^2/4-Theta(n) general upper bound and near-linear bounds for bounded-pathwidth classes.\n- M. Scheucher, H. Schrezenmaier, and R. Steiner, A Note on Universal Point Sets for Planar Graphs, Journal of Graph Algorithms and Applications 24 (2020), 247–267. (primary): https://doi.org/10.7155/jgaa.00529\n Evidence used: Proves the (1.293-o(1))n general lower bound and finite nonexistence results for n-point universal sets.\n- S. Felsner, H. Schrezenmaier, F. Schröder, and R. Steiner, Linear Size Universal Point Sets for Classes of Planar Graphs, arXiv:2303.00109 (2023). (primary): https://arxiv.org/abs/2303.00109\n Evidence used: Gives 2n-2 point sets for bipartite planar and maximum-degree-three planar graphs while saying the all-planar bounds remain quadratic versus linear.\n\n**Review notes.** The sentence suppresses the parameter n. Literally, one fixed finite point set cannot support all planar graphs; the intended function f(n) is clear only from the background.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1410, "problem_number": "GRAPH-023", "title": "Conference Graph Existence", "statement": "Does there exist a conference graph for every number of vertices $v > 1$ where $v \\equiv 1 \\pmod{4}$ and v is an odd sum of two squares?", "background": "A conference graph is a strongly regular graph with specific parameters related to conference matrices. The existence question for these graphs connects graph theory with number theory (sums of squares) and design theory. Known to exist for many values, but a complete characterization remains elusive. These graphs have applications in coding theory and experimental design.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The congruence and sum-of-two-squares conditions are necessary for a conference graph, but their sufficiency for every admissible v remains open; multiple infinite families and sporadic admissible orders are constructed.\n\n**Verified partial progress.**\n\n- Paley graphs give conference graphs for prime powers v congruent to 1 modulo 4.\n- Mathon's symmetric conference-matrix construction supplies additional infinite parameter families.\n- Gritsenko constructed an srg(65,32,15,16), resolving that formerly open admissible order.\n\n**Full solution or refutation.**\n\nNo construction for every v congruent to 1 modulo 4 that is a sum of two squares, and no admissible v proved impossible, was verified in the checked sources.\n\n**What remains.**\n\nEither construct conference graphs at all arithmetically admissible orders or identify an admissible order for which they cannot exist.\n\n**Sources checked.**\n\n- R. Mathon, Symmetric Conference Matrices of Order pq^2+1, Canadian Journal of Mathematics 30 (1978), 321–331. (primary): https://doi.org/10.4153/CJM-1978-029-1\n Evidence used: Constructs broad families of symmetric conference matrices and the associated conference graphs.\n- O. Gritsenko, On strongly regular graph with parameters (65; 32; 15; 16), arXiv:2102.05432 (2021). (primary): https://arxiv.org/abs/2102.05432\n Evidence used: Constructs the conference graph of order 65.\n\n**Review notes.** The phrase 'v is an odd sum of two squares' is ambiguous and redundant with v congruent to 1 mod 4. The standard necessary condition is that v is a sum of two integer squares. A comprehensive current admissible-order table was not located in a primary source, so confidence is medium.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1411, "problem_number": "GRAPH-024", "title": "Conway's 99-Graph Problem", "statement": "Does there exist a strongly regular graph with parameters (99,14,1,2)?", "background": "Conway asked whether a strongly regular graph with these specific parameters exists. The parameters pass all known necessary conditions (feasibility, integrality), but no construction is known. This is the smallest open case for strongly regular graphs. Finding such a graph or proving nonexistence would advance our understanding of the constraints on strongly regular graphs beyond the known necessary conditions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Existence of an srg(99,14,1,2) remains open; recent peer-reviewed work substantially restricts the automorphism group of any putative graph.\n\n**Verified partial progress.**\n\n- Cesarz and Woldar give computer-free proofs that if 2 divides the automorphism-group order then the order divides 6.\n- They also prove that if 7 divides the automorphism-group order then the group is isomorphic to Z_7.\n- A July 2026 autonomous-agent preprint reports forced-structure and prescribed-automorphism reductions, but neither constructs nor excludes the graph.\n\n**Full solution or refutation.**\n\nNo construction or nonexistence proof for the Conway 99-graph was verified.\n\n**What remains.**\n\nConstruct an srg(99,14,1,2) or prove that the remaining locally and spectrally feasible structures cannot exist.\n\n**Sources checked.**\n\n- P. G. Cesarz and A. J. Woldar, On the automorphism group of a putative Conway 99-graph, Algebraic Combinatorics 8 (2025), 379–398. (primary): https://doi.org/10.5802/alco.418\n Evidence used: Explicitly states that existence remains open and proves the automorphism-order restrictions.\n- A. Thakkar, A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph, arXiv:2608.11211 (2026). (primary): https://arxiv.org/abs/2608.11211\n Evidence used: Very recent unreviewed computational preprint reporting reductions and partial constraint scores, not an existence decision.\n\n**Review notes.** The background's claim that this is 'the smallest open case for strongly regular graphs' is an unqualified and unsupported superlative; it depends on what counts as a feasible primitive parameter case.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1412, "problem_number": "GRAPH-025", "title": "Degree Diameter Problem", "statement": "For given maximum degree d and diameter k, what is the largest possible number of vertices in a graph?", "background": "The degree diameter problem asks for the maximum order (number of vertices) of a graph with maximum degree d and diameter k. The Moore bound provides an upper limit, but it's rarely achieved (only for very special parameters). Finding the exact values or better bounds is a central problem in extremal graph theory with applications to network design. Tables of best known values are maintained, but many cases remain unsolved.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The degree–diameter problem is solved for some parameter pairs and has general upper and lower constructions, but exact maxima remain unknown for many degree/diameter pairs.\n\n**Verified partial progress.**\n\n- The Moore bound gives N(Delta,D) <= 1+Delta sum_{i=0}^{D-1}(Delta-1)^i for Delta>2.\n- The bound is attained in elementary cases and by the Petersen and Hoffman–Singleton diameter-two graphs of degrees 3 and 7.\n- Comellas's January 2026 maintained table records the best known constructions across a broad grid and marks only the entries proved optimal.\n\n**Full solution or refutation.**\n\nThere is no known formula giving the extremal order for every maximum degree and diameter.\n\n**What remains.**\n\nDetermine exact values for the many non-optimal table entries and improve general constructions or upper bounds away from the Moore cases.\n\n**Sources checked.**\n\n- F. Comellas, Table of Large Degree/Diameter Graphs, Mendeley Data, version 11 (January 2026). (maintained_tracker): https://doi.org/10.17632/d75dzbjd4k.11\n Evidence used: Maintained research table defining (Delta,D)-graphs, stating the Moore bound, and separating proved-optimal values from record constructions.\n\n**Review notes.** This record is a broad two-parameter research program rather than a single conjecture. Its 'open' label should mean that no general formula is known, not that every parameter pair is unresolved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1413, "problem_number": "GRAPH-026", "title": "Moore Graph Existence", "statement": "Does a Moore graph with girth 5 and degree 57 exist?", "background": "Moore graphs are extremal graphs achieving the Moore bound—the maximum possible vertices for given degree and diameter. The Hoffman-Singleton theorem shows Moore graphs with girth 5 can only have degree 2, 3, 7, or possibly 57. Graphs for degrees 2, 3, 7 are known (cycle C₅, Petersen, Hoffman-Singleton). Whether a degree-57 Moore graph exists is one of the most famous open problems in algebraic graph theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Existence of the degree-57 Moore graph remains open; a published claim of nonexistence was not verified, and a 2026 preprint newly rules out involutory automorphisms.\n\n**Verified partial progress.**\n\n- A putative graph would have 3250 vertices and strongly regular parameters (3250,57,0,1).\n- Faber and Keegan identify the failure in a 2020 claimed nonexistence argument and reaffirm that the problem is open.\n- Ishida's June 2026 preprint proves that no putative degree-57 Moore graph can have an involution, so its automorphism group would have odd order.\n\n**Full solution or refutation.**\n\nNeither a degree-57 Moore graph nor a valid nonexistence proof was verified.\n\n**What remains.**\n\nConstruct the 3250-vertex graph or derive a contradiction from its required strongly regular and automorphism structure.\n\n**Sources checked.**\n\n- V. Faber and J. Keegan, Existence of a Moore graph of degree 57 is still open, arXiv:2210.09577 (2022). (primary): https://arxiv.org/abs/2210.09577\n Evidence used: Audits and rejects the 2020 nonexistence argument and concludes that existence remains open.\n- Y. Ishida, No involutions in the missing Moore graph, arXiv:2606.29183 (2026). (primary): https://arxiv.org/abs/2606.29183\n Evidence used: Very recent preprint proving that a putative graph has no involutory automorphism.\n- A. A. Makhnev, Moore graph with parameters (3250,57,0,1) does not exist, arXiv:2010.13443 (2020). (primary): https://arxiv.org/abs/2010.13443\n Evidence used: The surfaced nonexistence claim; included only because the later Faber–Keegan audit explains why it does not settle the problem.\n\n**Review notes.** The 2020 arXiv title claims nonexistence, but the 2022 audit finds the argument invalid. The 2026 no-involutions theorem is very recent and was conservatively treated as partial progress.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 223, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1414, "problem_number": "GRAPH-027", "title": "Barnette's Conjecture", "statement": "Does every cubic bipartite three-connected planar graph have a Hamiltonian cycle?", "background": "Barnette's conjecture proposes that a specific family of planar graphs—cubic (3-regular), bipartite, and 3-connected—always contains Hamiltonian cycles. This strengthens Tait's conjecture (disproven by counterexamples) by adding bipartiteness. Despite extensive computational verification and many partial results, no proof or counterexample is known. This is one of the most prominent open problems on Hamiltonian cycles.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Barnette's conjecture remains open for all cubic 3-connected planar bipartite graphs.\n\n**Verified partial progress.**\n\n- It is proved for multiple structural subclasses, including recent bounded-face and leapfrog-extension results.\n- A 2025 approximation result gives a subhamiltonian cycle with at least 5n/6 edges in every n-vertex Barnette graph.\n\n**Full solution or refutation.**\n\nNo universal Hamilton-cycle theorem or counterexample was verified.\n\n**What remains.**\n\nProve Hamiltonicity or construct a non-Hamiltonian Barnette graph.\n\n**Sources checked.**\n\n- S. Bekos et al., Approximating Barnette's Conjecture, GD 2025, LIPIcs 357, Article 6. (primary): https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.6\n Evidence used: States the conjecture is open and proves the 5n/6 subhamiltonian-cycle result.\n- J. Florek, A sufficient condition for cubic 3-connected plane bipartite graphs to be Hamiltonian, J. Graph Theory 110 (2025), doi:10.1002/jgt.23270. (primary): https://arxiv.org/abs/2309.09578\n Evidence used: Provides a recent sufficient condition and says the general problem remains open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1415, "problem_number": "GRAPH-028", "title": "Chvátal's Toughness Conjecture", "statement": "Is there a constant t such that every t-tough graph is Hamiltonian?", "background": "A graph is t-tough if removing any set S of vertices leaves at most |S|/t components. Chvátal conjectured that sufficiently tough graphs are Hamiltonian. Best known: every 2-tough graph on at least 3 vertices is Hamiltonian. But whether some finite t suffices in general is unknown. Toughness measures graph robustness; the conjecture would provide a simple sufficient condition for Hamiltonicity.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Chvátal's existence-of-a-toughness-threshold conjecture remains open.\n\n**Verified partial progress.**\n\n- Non-Hamiltonian examples show any possible threshold must be at least 9/4.\n- Many special graph classes have threshold theorems.\n\n**Full solution or refutation.**\n\nThe original 3/2 and 2 toughness guesses were refuted, but the existence of some finite threshold is unresolved.\n\n**What remains.**\n\nProve a finite universal threshold or construct non-Hamiltonian graphs of arbitrarily large toughness.\n\n**Sources checked.**\n\n- Hamiltonicity of 1-tough (P2 union kP1)-free graphs, Discrete Mathematics 348 (2025). (primary): https://doi.org/10.1016/j.disc.2023.114082\n Evidence used: States the exact t0 conjecture remains open and records the 9/4 lower requirement.\n- Hamiltonicity of 1-tough (P2 union kP1)-free graphs, arXiv:2303.09741. (primary): https://arxiv.org/abs/2303.09741\n Evidence used: Explicitly calls the general t0 conjecture open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1416, "problem_number": "GRAPH-029", "title": "Cycle Double Cover Conjecture", "statement": "Does every bridgeless graph have a collection of cycles that covers each edge exactly twice?", "background": "The cycle double cover conjecture asserts that every bridgeless graph has a family of cycles where each edge appears in exactly two cycles. Equivalent formulations involve graph embeddings and flows. Despite being open since the 1970s, this elegant conjecture connects cycle structure, graph embeddings, and topological graph theory. Many restricted cases are proven, but the general case remains elusive.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A July 2026 Cycle Double Cover proof announcement has independent expositions, but it is too recent for an unconditional archival solved classification.\n\n**Verified partial progress.**\n\n- Oum and Geelen posted explanatory treatments of the announced proof.\n\n**Full solution or refutation.**\n\nPending independent detailed verification, this duplicate record is retained as an expert-review item.\n\n**What remains.**\n\nIndependent verification and durable publication of the claimed proof.\n\n**Sources checked.**\n\n- S.-i. Oum, A proof of the cycle double cover conjecture by OpenAI: An exposition, arXiv:2607.16356. (primary): https://arxiv.org/abs/2607.16356\n Evidence used: Provides an exposition of the announced proof.\n- J. Geelen, OpenAI's proof of the Cycle Double Cover Theorem, arXiv:2607.15399. (primary): https://arxiv.org/abs/2607.15399\n Evidence used: Independent clarification notes for the announced result.\n\n**Review notes.** Duplicate topic retained; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1417, "problem_number": "GRAPH-030", "title": "Erdős-Gyárfás Conjecture", "statement": "Does every graph with minimum degree 3 contain cycles of lengths that are powers of 2?", "background": "Erdős and Gyárfás conjectured that cubic graphs (minimum degree 3) must contain cycles whose lengths are all distinct powers of 2. The best known result is that such graphs contain cycles of Ω(log log n) distinct even lengths. This problem connects extremal graph theory with additive combinatorics and the structure of cycle lengths in graphs.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Erdős--Gyárfás power-of-two-cycle conjecture remains open in the minimum-degree-3 case.\n\n**Verified partial progress.**\n\n- It is proved for P8-free graphs and has further extensions to induced-path-free classes.\n- Large-degree/average-degree theorems force suitable even-cycle intervals and hence power-of-two lengths.\n\n**Full solution or refutation.**\n\nNo proof for all graphs of minimum degree three was verified.\n\n**What remains.**\n\nSettle the cubic/small-minimum-degree regime.\n\n**Sources checked.**\n\n- Erdős-Gyárfás conjecture on graphs without long induced paths, arXiv:2410.22842 (2024). (primary): https://arxiv.org/abs/2410.22842\n Evidence used: States the conjecture and records the P8-free theorem.\n- Erdős-Gyárfás Conjecture for P10-free Graphs, arXiv:2308.05675 (2023). (primary): https://arxiv.org/abs/2308.05675\n Evidence used: Gives a later induced-path-free advance while retaining the general conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1418, "problem_number": "GRAPH-031", "title": "Erdős-Hajnal Conjecture", "statement": "Does every graph family defined by a forbidden induced subgraph have polynomial-sized cliques or independent sets?", "background": "The Erdős-Hajnal conjecture states that for any graph H, there exists ε > 0 such that every H-free graph on n vertices contains a clique or independent set of size at least n^ε. This would be a dramatic strengthening of Ramsey theory, which only guarantees log-size structures. Proven for many specific H, but the general case is a central open problem in extremal combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Erdős--Hajnal conjecture remains open for arbitrary forbidden induced graph H.\n\n**Verified partial progress.**\n\n- The polynomial property is known for every H with at most five vertices and for many larger structured families.\n- A 2026 result adds the six-vertex E-graph to the known classes.\n\n**Full solution or refutation.**\n\nNo theorem covering every fixed induced forbidden graph was verified.\n\n**What remains.**\n\nProve the polynomial clique-or-independent-set bound for every H.\n\n**Sources checked.**\n\n- Erdős Problems, history 61 (Erdős--Hajnal conjecture). (maintained_tracker): https://www.erdosproblems.com/history/61\n Evidence used: Records the original bound, known all-five-vertex cases, and current open status.\n- Erdős-Hajnal beyond the five-vertex path, arXiv:2606.06258 (2026). (primary): https://arxiv.org/abs/2606.06258\n Evidence used: States a new six-vertex forbidden-graph case.\n\n**Review notes.** Duplicate topic retained; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1419, "problem_number": "GRAPH-032", "title": "Linear Arboricity Conjecture", "statement": "Can every graph with maximum degree Δ be decomposed into at most ⌈(Δ+1)/2⌉ linear forests?", "background": "A linear forest is a disjoint union of paths. The linear arboricity conjecture states that graphs decompose into roughly Δ/2 linear forests. This would provide tight bounds on a natural graph decomposition parameter. Proven for many graph classes (planar graphs, graphs with large girth), but the general case remains open. Applications include edge coloring and bandwidth problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Linear Arboricity Conjecture remains open in general.\n\n**Verified partial progress.**\n\n- It is proved for many degree ranges and graph families; modern results establish asymptotic versions.\n\n**Full solution or refutation.**\n\nNo all-graph proof of the ceiling((Delta+1)/2) bound was verified.\n\n**What remains.**\n\nProve the stated sharp decomposition bound for every graph.\n\n**Sources checked.**\n\n- Linear arboricity overview, Wolfram MathWorld. (authoritative_secondary): https://mathworld.wolfram.com/LinearArboricity.html\n Evidence used: Records the conjecture and partial known results.\n\n**Review notes.** No source alteration; a stronger primary source was not recovered in this pass.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1420, "problem_number": "GRAPH-033", "title": "Lovász Conjecture", "statement": "Does every finite connected vertex-transitive graph contain a Hamiltonian path?", "background": "Lovász conjectured that vertex-transitive graphs (graphs looking the same from every vertex) always have Hamiltonian paths. Even stronger: do they have Hamiltonian cycles (except for K₂ and some Cayley graphs)? Known for many classes, but a general proof eludes us. This connects group theory, algebraic graph theory, and Hamiltonian paths. Named the \"Lovász Hamiltonian Path Problem.\"\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Lovász's Hamiltonian-path conjecture for finite connected vertex-transitive graphs remains open.\n\n**Verified partial progress.**\n\n- It is established for many Cayley and other vertex-transitive graph families.\n\n**Full solution or refutation.**\n\nNo universal Hamiltonian-path theorem was verified.\n\n**What remains.**\n\nResolve all finite connected vertex-transitive graphs.\n\n**Sources checked.**\n\n- Lovász conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Lov%C3%A1sz_conjecture\n Evidence used: Records special cases and general open status.\n\n**Review notes.** Duplicate topic retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1421, "problem_number": "GRAPH-034", "title": "Oberwolfach Problem", "statement": "For which 2-regular graphs H can the complete graph be decomposed into edge-disjoint copies of H?", "background": "The Oberwolfach problem asks: given a 2-regular graph H (disjoint union of cycles), can K_n be decomposed into copies of H? This generalizes cycle decompositions and connects to the famous Oberwolfach conferences. Solutions are known for many cases (like single cycles), but a complete characterization remains open. This problem bridges graph decomposition with combinatorial designs.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The finite Oberwolfach problem has broad large-order and family-specific solutions, but a complete classification of all admissible finite factors is not verified here.\n\n**Verified partial progress.**\n\n- A general large-order theorem solves the standard odd-order problem in a substantially broader quasirandom setting.\n- Recent work gives constructive solutions for factors containing a sufficiently large cycle.\n\n**Full solution or refutation.**\n\nThe exact wording omits parity/order conditions; it should not be read as a settled all-parameter statement.\n\n**What remains.**\n\nComplete the remaining finite exceptional cases under the correct divisibility and parity hypotheses.\n\n**Sources checked.**\n\n- A. Glock, D. Kühn, A. Lo and D. Osthus, The generalised Oberwolfach problem, J. Combin. Theory Ser. B 152 (2022), 281--318. (primary): https://doi.org/10.1016/j.jctb.2021.09.007\n Evidence used: Proves a large-graph generalised theorem and describes the standard Oberwolfach special case.\n- A constructive solution to the Oberwolfach problem with a large cycle, Discrete Mathematics 348 (2025). (primary): https://doi.org/10.1016/j.disc.2024.114069\n Evidence used: Gives a new constructive family result and says complete finite solution remains a goal.\n\n**Review notes.** No source alteration; necessary admissibility conditions are absent from the dataset wording.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1422, "problem_number": "GRAPH-035", "title": "Cubic Graph Pathwidth", "statement": "What is the maximum pathwidth of an n-vertex cubic graph?", "background": "Pathwidth measures how closely a graph resembles a path. For cubic (3-regular) graphs, the maximum pathwidth is conjectured to be around n/6, but exact bounds are unknown. This problem connects graph width parameters with regular graphs. Understanding pathwidth has implications for algorithms—many NP-hard problems become tractable on graphs of bounded pathwidth.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The extremal pathwidth of n-vertex cubic graphs is not known exactly; checked bounds are 0.082n below and n/6+o(n) above.\n\n**Verified partial progress.**\n\n- The classic n/6 upper bound yields exact-algorithm consequences, while explicit cubic families give a linear lower bound.\n\n**Full solution or refutation.**\n\nThe asymptotic constant is unresolved.\n\n**What remains.**\n\nClose the linear gap or determine the exact extremal pathwidth.\n\n**Sources checked.**\n\n- F. V. Fomin and K. Høie, Pathwidth of cubic graphs and exact algorithms, Information Processing Letters 97 (2006), 191--196. (primary): https://doi.org/10.1016/j.ipl.2005.10.010\n Evidence used: Source for the standard upper-bound framework.\n- Cubic graph overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Cubic_graph\n Evidence used: Records the checked n/6 and 0.082n bounds and outstanding gap.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1423, "problem_number": "GRAPH-036", "title": "Snake-in-the-Box Problem", "statement": "What is the longest induced path in an n-dimensional hypercube graph?", "background": "A snake-in-the-box is a longest induced path in the n-dimensional hypercube Q_n. Known exact values for small n, but no formula for general n. This problem combines combinatorics, coding theory (Gray codes), and graph theory. Snakes have applications in error-correcting codes and analog-to-digital conversion. Finding optimal snakes remains computationally challenging as n grows.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exact maximum snake length is known through dimension 8 but remains open from dimension 9 onward.\n\n**Verified partial progress.**\n\n- Current listed best constructions for dimensions 9--13 have lengths 190, 373, 721, 1383, and 2709.\n- The extremal length is known to grow proportionally to 2^n, but the asymptotic constant is unknown.\n\n**Full solution or refutation.**\n\nThe problem is an infinite sequence of extremal questions rather than a single solved formula.\n\n**What remains.**\n\nDetermine exact values beyond dimension 8 and sharpen asymptotic bounds.\n\n**Sources checked.**\n\n- Snake-in-the-box overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Snake-in-the-box\n Evidence used: Records exact values through dimension 8, current best longer-dimensional constructions, and unknown asymptotic constant.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1424, "problem_number": "GRAPH-037", "title": "Sumner's Conjecture", "statement": "Does every (2n-2)-vertex tournament contain every n-vertex oriented tree?", "background": "Sumner conjectured that tournaments (complete directed graphs) on 2n-2 vertices contain all oriented trees on n vertices as subgraphs. This would be a directed analogue of various tree embedding results. The best known bound is (4+o(1))n instead of 2n-2. This problem connects tournament theory with tree embeddings and Ramsey-type questions for directed graphs.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Sumner's exact 2n-2 assertion is proved for all sufficiently large n, but the remaining finite range is unresolved; the dataset's (4+o(1))n background is obsolete.\n\n**Verified partial progress.**\n\n- Kuhn, Mycroft, and Osthus proved every (2n-2)-vertex tournament contains every n-vertex oriented tree for all sufficiently large n.\n- Ai and Lian's August 2026 preprint gives the uniform all-n bound f(n) at most ceil((18n-23)/7), improving the previous coefficient 21/8.\n\n**Full solution or refutation.**\n\nThe conjecture is asymptotically exact, but no proof for every integer n was verified.\n\n**What remains.**\n\nResolve the finite range not covered by the sufficiently-large-n theorem, or give a genuinely uniform proof of f(n)=2n-2.\n\n**Sources checked.**\n\n- Daniela Kuhn, Richard Mycroft, and Deryk Osthus, A proof of Sumner's universal tournament conjecture for large tournaments, Proceedings of the London Mathematical Society 102 (2011), 731-766. (primary): https://arxiv.org/abs/1010.4430\n Evidence used: Proves the exact conjectured tournament order for all sufficiently large n.\n- Jiangdong Ai and Xiaopan Lian, An improved finite bound for oriented trees in tournaments, arXiv:2608.11667 (2026). (primary): https://arxiv.org/abs/2608.11667\n Evidence used: Proves the current all-n bound ceil((18n-23)/7) and still states the exact universal assertion as a conjecture.\n\n**Review notes.** The asymptotic theorem is exact, not merely a (2+o(1))n approximation; universal quantification over all n is the remaining issue.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1425, "problem_number": "GRAPH-038", "title": "Tuza's Conjecture", "statement": "Can the edges of any graph be covered by at most 2ν triangles, where ν is the maximum size of a triangle packing?", "background": "Tuza conjectured that the minimum number of edges needed to hit all triangles is at most twice the maximum number of edge-disjoint triangles. This is a covering-packing duality question. Best known bound is 3ν. The conjecture would provide a tight relationship between triangle packings and triangle covers, with applications to approximation algorithms and combinatorial optimization.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Tuza's conjecture tau(G)<=2nu(G) remains open for arbitrary finite graphs; the best universal factor is 66/23 rather than the background's factor 3.\n\n**Verified partial progress.**\n\n- Haxell proved tau(G) <= (66/23)nu(G) for every graph.\n- Kahn and Park proved Tuza's conjecture holds with high probability in G(n,p) for every density regime p=p(n).\n- A 2026 preprint proves the conjecture for a large range of random geometric graph densities.\n\n**Full solution or refutation.**\n\nNo universal factor-two theorem or counterexample was verified.\n\n**What remains.**\n\nImprove the universal cover-packing ratio from 66/23 to 2, or find a graph with tau(G)>2nu(G).\n\n**Sources checked.**\n\n- Penny E. Haxell, Packing and covering triangles in graphs, Discrete Mathematics 195 (1999), 251-254. (primary): https://doi.org/10.1016/S0012-365X(98)00183-6\n Evidence used: Gives the best known general factor 66/23 below the trivial factor 3.\n- Jeff Kahn and Jinyoung Park, Tuza's conjecture for random graphs, Random Structures & Algorithms 61 (2022), 235-249. (primary): https://arxiv.org/abs/2007.04351\n Evidence used: Proves the conjecture asymptotically almost surely for every Erdos-Renyi density regime.\n- Patrick Bennett, Ryan Cushman, Andrzej Dudek, and Xavier Perez-Gimenez, Almost-perfect packings and Tuza's conjecture in the random geometric graph, arXiv:2606.09736 (2026). (primary): https://arxiv.org/abs/2606.09736\n Evidence used: Proves a new random-geometric-graph special case without claiming the general conjecture.\n\n**Review notes.** The stored statement should say that at most 2nu edges meet all triangles; triangles do not cover the graph's edges in the standard formulation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1426, "problem_number": "GRAPH-039", "title": "Unfriendly Partition Conjecture", "statement": "Does every countable graph admit a partition where every vertex has at least as many neighbors outside its part as inside?", "background": "The unfriendly partition conjecture asks if vertices can be partitioned into two sets such that each vertex has at least as many \"unfriendly\" neighbors (in the other set) as \"friendly\" ones (in its own set). Proven for finite graphs, but open for countably infinite graphs. This problem combines graph theory with infinite combinatorics and has connections to social network models.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether every countable graph has an unfriendly partition, although many structural subclasses now satisfy the conjecture.\n\n**Verified partial progress.**\n\n- Aurichi and Real prove the conjecture when no ray passes through infinitely many finite-degree and infinitely many infinite-degree vertices.\n- Kalinowski, Pilsniak, and Stawiski prove the conjecture for line graphs of arbitrary cardinality, in the stronger list-majority setting.\n- Earlier work covers locally finite graphs, rayless graphs, and graphs with only finitely many vertices of infinite degree.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for arbitrary countable graphs was verified; uncountable counterexamples do not decide this statement.\n\n**What remains.**\n\nHandle unrestricted countable graphs whose rays and degree structure evade all current subclass theorems.\n\n**Sources checked.**\n\n- Leandro Fiorini Aurichi and Lucas Real, Remarks on the countable case of the Unfriendly Partition Problem, arXiv:2412.14151 (2024). (primary): https://arxiv.org/abs/2412.14151\n Evidence used: Explicitly says the countable case has no solution and proves a new ray/degree structural case.\n- Rafal Kalinowski, Monika Pilsniak, and Marcin Stawiski, Unfriendly Partition Conjecture Holds for Line Graphs, Combinatorica 45 (2025), article 3. (primary): https://doi.org/10.1007/s00493-024-00131-1\n Evidence used: Proves the conjecture for all line graphs, even of arbitrary infinite cardinality.\n- Henning Bruhn, Reinhard Diestel, Agelos Georgakopoulos, and Philipp Sprussel, Every rayless graph has an unfriendly partition, arXiv:0901.4858. (primary): https://arxiv.org/abs/0901.4858\n Evidence used: Proves the conjecture for the rayless subclass.\n\n**Review notes.** The exact countability restriction is crucial because counterexamples are known at uncountable cardinalities.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1427, "problem_number": "GRAPH-040", "title": "Zarankiewicz Problem", "statement": "What is the maximum number of edges in a bipartite graph on (m,n) vertices with no complete bipartite subgraph $K_{s,t}$?", "background": "The Zarankiewicz problem asks for ex(m,n;K_{s,t})—the maximum edges in an (m,n)-bipartite graph avoiding K_{s,t} as a subgraph. This is a fundamental problem in extremal graph theory, generalizing the Kővári–Sós–Turán theorem. Exact values are known for some parameters, but most cases remain open. Applications include incidence geometry and additive combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The parameterized Zarankiewicz problem has general upper bounds and many exact or asymptotic special cases, but no formula or sharp asymptotic for arbitrary m,n,s,t.\n\n**Verified partial progress.**\n\n- The Kovari-Sos-Turan theorem gives a general bound of order O_{s,t}(m n^{1-1/s}+n m^{1-1/t}+m+n).\n- Nikiforov gives a flexible upper bound recovering the classical Kovari-Sos-Turan and Furedi estimates.\n- Algebraic, finite-geometric, and probabilistic constructions determine the correct exponent or exact values in selected parameter regimes.\n\n**Full solution or refutation.**\n\nThe broad request is unresolved; even diagonal fixed-parameter asymptotics remain open in general.\n\n**What remains.**\n\nDetermine exact finite values or matching asymptotic bounds and constants throughout the unresolved parameter ranges.\n\n**Sources checked.**\n\n- Shakhar Smorodinsky, A survey of Zarankiewicz problem in geometry, arXiv:2410.03702 (2024). (authoritative_secondary): https://arxiv.org/abs/2410.03702\n Evidence used: Reviews the classical bounds and explicitly describes general Zarankiewicz asymptotics as widely open.\n- Vladimir Nikiforov, A contribution to the Zarankiewicz problem, Linear Algebra and its Applications 432 (2010), 1405-1411. (primary): https://arxiv.org/abs/0903.5350\n Evidence used: Proves a flexible general upper bound implying major classical estimates.\n\n**Review notes.** The statement is an open-ended four-parameter determination program, so partial rather than binary open status best captures the many solved subfamilies.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1428, "problem_number": "GRAPH-041", "title": "Vizing's Conjecture", "statement": "For the Cartesian product of graphs $G \\square H$, is the domination number at least $\\gamma(G) \\cdot \\gamma(H)$?", "background": "Vizing conjectured that the domination number of the Cartesian product of two graphs is at least the product of their domination numbers. This would give a lower bound on how efficiently one can dominate product graphs. The conjecture has been verified for many special cases but remains open in general. It has connections to network design and distributed computing.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Vizing's domination inequality remains open; 2026 preprints give the first universal constant-factor improvements above one half but remain far short of the conjectured factor one.\n\n**Verified partial progress.**\n\n- Suen and Tarr refine the Clark-Suen one-half bound with an additive term depending on the factor domination numbers.\n- Steiner proves the universal constant (5+sqrt(73))/24, approximately 0.5643.\n- Aliabadi and Krop's July 2026 preprint claims a further universal constant 0.5809.\n\n**Full solution or refutation.**\n\nNo proof of gamma(G square H)>=gamma(G)gamma(H), and no counterexample, was verified for arbitrary finite graphs.\n\n**What remains.**\n\nRaise the universal multiplicative constant to one or find a counterexample to Vizing's exact inequality.\n\n**Sources checked.**\n\n- Stephen Suen and Jennifer Tarr, An improved inequality related to Vizing's conjecture, Electronic Journal of Combinatorics 19 (2012), #P8. (primary): https://doi.org/10.37236/15\n Evidence used: Gives a rigorous refinement of the classical universal one-half bound and records the conjecture as open.\n- Raphael Steiner, A constant-factor step towards Vizing's conjecture, arXiv:2606.14414 (2026). (primary): https://arxiv.org/abs/2606.14414\n Evidence used: Proves the first universal multiplicative constant strictly larger than one half, namely about 0.5643.\n- Mohsen Aliabadi and Elliot Krop, An improved constant for Vizing's conjecture, arXiv:2607.01109 (2026). (primary): https://arxiv.org/abs/2607.01109\n Evidence used: Claims the newer universal factor 0.5809 while continuing to state factor one as Vizing's conjecture.\n\n**Review notes.** The two constant improvements are very recent preprints; they strengthen partial progress but do not affect the open classification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 172, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1429, "problem_number": "GRAPH-042", "title": "Hamiltonian Decomposition of Hypergraphs", "statement": "Do complete k-uniform hypergraphs admit Hamiltonian decompositions into tight cycles?", "background": "Walescki's theorem states that complete graphs have Hamiltonian decompositions. The hypergraph version asks whether complete k-uniform hypergraphs can be decomposed into tight Hamiltonian cycles. This is a natural generalization from graphs to hypergraphs, with connections to design theory and combinatorial structures.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The row omits parameter quantifiers and necessary divisibility: not every complete k-uniform hypergraph can decompose into tight Hamilton cycles, while the intended admissible-parameter conjecture remains open.\n\n**Verified partial progress.**\n\n- A tight Hamilton cycle of K_n^(k) has n edges and degree k at each vertex, forcing n to divide binom(n,k), equivalently k to divide binom(n-1,k-1).\n- Kuhn and Osthus prove the analogous Hamilton Berge-cycle decomposition for all admissible n when k=3 and for n at least 30 when k is at least four.\n- Tight Hamilton decompositions are known for selected complete multipartite uniform hypergraphs and other restricted families.\n\n**Full solution or refutation.**\n\nThe literal unrestricted reading is false by divisibility, but the standard conjecture that divisibility suffices for sufficiently large admissible parameters is not solved.\n\n**What remains.**\n\nSpecify n,k and the divisibility conditions, distinguish tight from Berge cycles, and then settle the tight-cycle decomposition conjecture for all sufficiently large admissible complete uniform hypergraphs.\n\n**Sources checked.**\n\n- Daniela Kuhn and Deryk Osthus, Decompositions of complete uniform hypergraphs into Hamilton Berge cycles, Journal of Combinatorial Theory, Series A 126 (2014), 128-135. (primary): https://arxiv.org/abs/1403.7932\n Evidence used: Proves the Berge-cycle analogue under its divisibility condition, not the stored tight-cycle claim.\n- Deryk Osthus, Hamilton cycles in graphs and hypergraphs: an extremal perspective, survey, Conjecture 5.8. (authoritative_secondary): https://web.mat.bham.ac.uk/D.Osthus/icmsurvey4.pdf\n Evidence used: States the general Hamilton l-cycle decomposition conjecture with the necessary divisibility conditions.\n- Taijiang Jiang, Qiang Sun, Shunzhe Zhang, and Chao Zhang, The perfect matching and tight Hamilton cycle decomposition of complete n-balanced mk-partite k-uniform hypergraphs, Discrete Mathematics 346 (2023), 113631. (primary): https://doi.org/10.1016/j.disc.2023.113631\n Evidence used: Proves tight-cycle decompositions for restricted complete multipartite hypergraphs.\n\n**Review notes.** A universal yes/no answer cannot be attached safely until the omitted admissibility conditions and parameter range are restored.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1430, "problem_number": "GRAPH-043", "title": "Word-Representable Graphs: Letter Copies Bound", "statement": "Are there graphs on n vertices requiring more than floor(n/2) copies of each letter for word-representation?", "background": "Word-representable graphs can be encoded by words where two vertices are adjacent if their letters alternate in the word. The question asks whether any graph needs more than half the number of vertices as copies of each letter. This connects graph theory to formal languages and combinatorics on words.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No n-vertex word-representable graph with representation number above floor(n/2) is known; the general upper bound remains 2n-4.\n\n**Verified partial progress.**\n\n- Halldorsson, Kitaev, and Pyatkin prove every non-complete n-vertex word-representable graph has representation number at most 2n-4.\n- They construct graphs attaining floor(n/2), and crown-graph families supply further linear lower-bound examples.\n- Hefty, Horn, Muir, and Owens derive new representation-length lower bounds, including probabilistic bounds for bipartite graphs, without crossing floor(n/2).\n\n**Full solution or refutation.**\n\nNeither an example above floor(n/2) nor a universal floor(n/2) upper bound was verified.\n\n**What remains.**\n\nConstruct an n-vertex word-representable graph with representation number greater than floor(n/2), or reduce the 2n-4 upper bound to floor(n/2).\n\n**Sources checked.**\n\n- Magnus M. Halldorsson, Sergey Kitaev, and Artem Pyatkin, Semi-transitive orientations and word-representable graphs, Discrete Applied Mathematics 201 (2016), 164-171. (primary): https://arxiv.org/abs/1501.07108\n Evidence used: Proves the corrected 2n-4 upper bound and constructs representation-number floor(n/2) examples.\n- Zion Hefty, Paul Horn, Colby Muir, and Andrew Owens, Word-Representable Graphs: Orientations, Posets, and Bounds, Electronic Journal of Combinatorics 31 (2024), #P4.2. (primary): https://doi.org/10.37236/12806\n Evidence used: Provides modern structural and lower-bound progress without an example beyond the stated threshold.\n- Sergey Kitaev, A Comprehensive Introduction to the Theory of Word-Representable Graphs, current author-hosted survey. (authoritative_secondary): https://personal.strath.ac.uk/sergey.kitaev/Papers/wrg-kitaev.pdf\n Evidence used: Lists the exact more-than-floor(n/2) question among current open problems.\n\n**Review notes.** The earlier n upper bound in a preliminary version relied on a false lemma; the corrected published bound is 2n-4.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 98, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1431, "problem_number": "GRAPH-044", "title": "Characterization of Word-Representable Planar Graphs", "statement": "Characterize which planar graphs are word-representable.", "background": "Word-representable graphs are those that can be encoded by words over their vertex set where adjacency corresponds to letter alternation. While some characterizations exist for special graph classes, characterizing word-representable planar graphs remains open. This combines planar graph structure with formal language properties.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** A structural characterization of all planar word-representable graphs remains open, although many planar subclasses now have complete descriptions.\n\n**Verified partial progress.**\n\n- Word-representability is equivalent to the existence of a semi-transitive orientation, and every 3-colorable graph is word-representable.\n- Consequently every triangle-free planar graph is word-representable, while non-word-representable planar graphs also exist.\n- A 2026 preprint gives a complete characterization for the near-triangulation subclass.\n\n**Full solution or refutation.**\n\nNo necessary-and-sufficient structural characterization specialized to every planar graph was verified.\n\n**What remains.**\n\nExtend current subclass theorems to a usable classification of arbitrary planar graphs beyond the general semi-transitive-orientation criterion.\n\n**Sources checked.**\n\n- Magnus M. Halldorsson, Sergey Kitaev, and Artem Pyatkin, Semi-transitive orientations and word-representable graphs, Discrete Applied Mathematics 201 (2016), 164-171. (primary): https://arxiv.org/abs/1501.07108\n Evidence used: Establishes the general semi-transitive orientation characterization and the 3-colorable sufficient class.\n- Suchanda Roy and Ramesh Hariharasubramanian, Characterization of Word-Representable Near-Triangulations, arXiv:2605.25733 (2026). (primary): https://arxiv.org/abs/2605.25733\n Evidence used: Claims a complete characterization for one broad planar subclass, not all planar graphs.\n- Sergey Kitaev, A Comprehensive Introduction to the Theory of Word-Representable Graphs, current author-hosted survey. (authoritative_secondary): https://personal.strath.ac.uk/sergey.kitaev/Papers/wrg-kitaev.pdf\n Evidence used: Lists characterization of word-representable planar graphs as an open direction and surveys solved subclasses.\n\n**Review notes.** The general semi-transitive-orientation equivalence is a recognition characterization, but the literature still treats a structural planar classification as open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 87, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1432, "problem_number": "GRAPH-045", "title": "Word-Representable Graphs: Forbidden Subgraph Characterization", "statement": "Characterize word-representable graphs in terms of forbidden induced subgraphs.", "background": "Many graph classes have elegant characterizations via forbidden subgraphs (e.g., planar graphs avoid K₅ and K₃,₃). The question asks for a similar characterization of word-representable graphs. Such a characterization would provide deep insight into the structure of these graphs and their connection to formal languages.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The full set of minimal forbidden induced subgraphs for word-representability is unknown, but recent work identifies infinite families and gives complete obstruction descriptions for important subclasses.\n\n**Verified partial progress.**\n\n- Kenkireth, Sajith, and Sasidharan determine the minimal non-comparability graphs that are also minimal non-word-representable and characterize obstructions containing an all-adjacent vertex.\n- They identify several infinite families of minimal non-word-representable graphs.\n- Recent papers give forbidden-induced-subgraph characterizations within split and co-bipartite graph classes.\n\n**Full solution or refutation.**\n\nBecause word-representability is hereditary, an obstruction set exists tautologically, but no complete explicit description of all minimal obstructions is known.\n\n**What remains.**\n\nDetermine or structurally parameterize every minimal non-word-representable graph, and clarify whether the desired characterization is finite, algorithmic, or an infinite family theorem.\n\n**Sources checked.**\n\n- Benny George Kenkireth, Gopalan Sajith, and Sreyas Sasidharan, Minimal non-comparability graphs and semi-transitivity, arXiv:2502.06979 (2025). (primary): https://arxiv.org/abs/2502.06979\n Evidence used: Explicitly says the complete minimal-obstruction set is open and proves several broad obstruction classifications and infinite families.\n- Eshwar Srinivasan and Ramesh Hariharasubramanian, Forbidden Induced Subgraph Characterization of Word-Representable Split Graphs, arXiv:2512.12259 (2025). (primary): https://arxiv.org/abs/2512.12259\n Evidence used: Gives a forbidden-induced-subgraph characterization in the split-graph subclass.\n- Eshwar Srinivasan and Ramesh Hariharasubramanian, Forbidden Induced Subgraph Characterization of Word-Representable Co-bipartite Graphs, arXiv:2512.12274 (2025). (primary): https://arxiv.org/abs/2512.12274\n Evidence used: Gives a structural and obstruction characterization for the co-bipartite subclass.\n\n**Review notes.** The row should specify that it seeks an explicit description of the minimal forbidden induced subgraphs; every hereditary class has a formal forbidden-induced-subgraph characterization.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 92, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1433, "problem_number": "GRAPH-046", "title": "Word-Representable Near-Triangulations", "statement": "Characterize word-representable near-triangulations containing K₄.", "background": "Near-triangulations are planar graphs close to being triangulations. A characterization is known for K₄-free cases. The question asks to extend this to near-triangulations containing the complete graph K₄. This combines planar graph structure with word-representability constraints.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A May 2026 preprint gives a complete forbidden-induced-subgraph characterization of all word-representable near-triangulations, directly including the requested K4-containing case.\n\n**Verified partial progress.**\n\n- Earlier work characterized K4-free near-triangulations and several narrower grid, polyomino, and chordal subclasses.\n- Roy and Hariharasubramanian unify these cases, cover near-triangulations containing K4, and correct inaccuracies in earlier claims.\n\n**Full solution or refutation.**\n\nThe 2026 paper's main theorem is a complete characterization of word-representable near-triangulations by forbidden induced subgraphs, which answers the stored classification request.\n\n**What remains.**\n\nObtain independent expert verification and a refereed version, and propagate the corrected earlier subclass statements; these are confidence checks rather than mathematical gaps in the claimed theorem.\n\n**Sources checked.**\n\n- Suchanda Roy and Ramesh Hariharasubramanian, Characterization of Word-Representable Near-Triangulations, arXiv:2605.25733 (2026). (primary): https://arxiv.org/abs/2605.25733\n Evidence used: The abstract explicitly claims a complete forbidden-induced-subgraph characterization for all near-triangulations, a class containing the exact requested K4 case.\n\n**Review notes.** The solved classification is medium confidence because the sole complete source is a very recent preprint and explicitly corrects inaccuracies in earlier literature.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 76, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1434, "problem_number": "GRAPH-047", "title": "Representation Number 3 Classification", "statement": "Classify graphs with representation number exactly 3.", "background": "The representation number is the minimum number of letter copies needed to word-represent a graph. Graphs with representation number 1 and 2 are relatively well understood. The question asks for a complete classification of graphs requiring exactly 3 copies—not representable with 2, but possible with 3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No classification of all graphs with representation number exactly 3 is known, although several substantial graph classes have now been characterized.\n\n**Verified partial progress.**\n\n- Kitaev proved that all prism graphs have representation number 3 and gave preserving operations.\n- Dwary–Mozhui–Krishna characterized exact representation number 3 within word-representable split graphs.\n- Das–Hariharasubramanian obtained the exact-3 classification within word-representable co-bipartite graphs.\n\n**Full solution or refutation.**\n\nThe cited results cover important subclasses but do not classify all exact-3 graphs.\n\n**What remains.**\n\nFind a necessary-and-sufficient structural characterization for arbitrary graphs of representation number 3.\n\n**Sources checked.**\n\n- Sergey Kitaev, On graphs with representation number 3, Journal of Automata, Languages and Combinatorics 18 (2013), 97–112. (primary): https://arxiv.org/abs/1403.1616\n Evidence used: Develops known exact-3 families while leaving the general class uncharacterized.\n- Tithi Dwary, Khyodeno Mozhui, and K. V. Krishna, Representation Number of Word-Representable Split Graphs, arXiv:2502.00872 (2025). (primary): https://arxiv.org/abs/2502.00872\n Evidence used: Characterizes exact representation number 3 in the split-graph class.\n- Biswajit Das and Ramesh Hariharasubramanian, Representation number of word-representable co-bipartite graph, arXiv:2509.03064 (2025). (primary): https://arxiv.org/abs/2509.03064\n Evidence used: Provides a class-specific exact-3 characterization for word-representable co-bipartite graphs.\n\n**Review notes.** The background's 'letter copies' must mean uniform multiplicity, not unrestricted word length.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 81, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1435, "problem_number": "GRAPH-048", "title": "Crown Graphs and Longest Word-Representants", "statement": "Among bipartite graphs, do crown graphs require the longest word-representants?", "background": "Crown graphs are a specific family of bipartite graphs with a symmetric structure. The conjecture suggests they are extremal for word-representation length among bipartite graphs. This would identify which bipartite graphs are hardest to encode as words, with implications for the complexity of word-representation.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Under the intended fixed-order representation-number interpretation, it remains open whether crown graphs maximize representation number among bipartite graphs.\n\n**Verified partial progress.**\n\n- Glen–Kitaev–Pyatkin proved R(H_n,n)=ceil(n/2) for n≥5.\n- Mozhui–Krishna proved the conjectured bipartite upper bound outside the principal balanced-even gap and for subclasses within that gap.\n- Hefty–Horn–Muir–Owens proved current lower bounds for almost every balanced bipartite graph.\n\n**Full solution or refutation.**\n\nNo theorem excluding a bipartite graph with representation number larger than the same-order crown graph was verified.\n\n**What remains.**\n\nResolve the balanced-even case of the proposed upper bound or find a bipartite counterexample.\n\n**Sources checked.**\n\n- Marc Glen, Sergey Kitaev, and Artem Pyatkin, On the representation number of a crown graph, Discrete Applied Mathematics 244 (2018), 89–93. (primary): https://arxiv.org/abs/1609.00674\n Evidence used: Determines the crown graph representation number.\n- Khyodeno Mozhui and K. V. Krishna, On the Conjecture of the Representation Number of Bipartite Graphs, arXiv:2506.01057 (2025). (primary): https://arxiv.org/abs/2506.01057\n Evidence used: Proves the proposed extremal upper bound in broad regimes and identifies the remaining balanced-even gap.\n- Zion Hefty, Paul Horn, Colby Muir, and Andrew Owens, Word-representation numbers of graphs: Bottlenecks and bounds, JCTA 223 (2026), 106215. (primary): https://doi.org/10.1016/j.jcta.2026.106215\n Evidence used: Provides current sharp probabilistic lower-bound progress for balanced bipartite graphs without resolving crown extremality.\n\n**Review notes.** The exact statement is undefined without fixing graph order and whether 'longest' means representation number or minimum nonuniform word length. The literature-supported reading uses representation number on the same number of vertices.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 73, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1436, "problem_number": "GRAPH-049", "title": "Line Graphs of Non-Word-Representable Graphs", "statement": "Is the line graph of a non-word-representable graph always non-word-representable?", "background": "The line graph operation transforms a graph into one where edges become vertices. The question asks whether word-non-representability is preserved under this operation. A positive answer would show that line graphs amplify the complexity of word-representation, while a counterexample would reveal subtle structural properties.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A 2025 primary manuscript gives infinitely many non-word-representable Mycielski graphs whose line graphs are word-representable, answering the question negatively.\n\n**Verified partial progress.**\n\n- For every n≥2, the line graph of the Mycielski graph of C_(2n+1) is shown word-representable although the original graph is not.\n- The same paper explains why a proposed 2021 counterexample was invalid.\n\n**Full solution or refutation.**\n\nThe infinite Mycielski odd-cycle family refutes preservation of non-word-representability under the line-graph operation.\n\n**What remains.**\n\nCharacterize graphs with word-representable line graphs and settle the triangle-free and clique-number-three residual questions.\n\n**Sources checked.**\n\n- Khyodeno Mozhui, Tithi Dwary, and K. V. Krishna, Line Graphs of Non-Word-Representable Graphs are Not Always Non-Word-Representable, arXiv:2509.03339 (2025). (primary): https://arxiv.org/abs/2509.03339\n Evidence used: Theorem 3.1 gives an infinite counterexample family and corrects the earlier attempted example.\n\n**Review notes.** The refutation is an explicit theorem in a 2025 arXiv manuscript; confidence is medium pending journal-level confirmation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 84, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1437, "problem_number": "GRAPH-050", "title": "Translating Graph Problems to Word Problems", "statement": "Which hard graph problems can be efficiently solved by translating graphs to their word representations?", "background": "Word-representation provides an alternative encoding of graphs as strings over an alphabet. The question asks which computationally hard graph problems become tractable when working with word representations instead of adjacency lists or matrices. This could reveal new algorithmic techniques leveraging string algorithms and automata theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The sentence is an open-ended research program rather than a definite conjecture and has no single solved/open status.\n\n**Verified partial progress.**\n\n- Maximum clique is polynomial-time solvable on word-representable graphs via neighborhood comparability structure.\n- Vertex coloring remains NP-hard and recognition of word-representability is NP-complete.\n- Given a representing word, rotations and their represented graphs can be processed efficiently.\n\n**Full solution or refutation.**\n\nSome individual graph problems have positive or negative complexity results, but there is no formal universe of 'hard graph problems' to classify as written.\n\n**What remains.**\n\nReplace the prompt by specific decision or optimization problems with a declared graph class, input encoding, and target complexity.\n\n**Sources checked.**\n\n- Sergey Kitaev, A Comprehensive Introduction to the Theory of Word-Representable Graphs, arXiv:1705.05924 (2017). (authoritative_secondary): https://arxiv.org/abs/1705.05924\n Evidence used: Presents this item as a broad direction for further research rather than a formal proposition.\n- Pamela Fleischmann, Lukas Haschke, Tim Löck, and Dirk Nowotka, Word-representable graphs from a word's perspective, Acta Informatica 61 (2024), 383–400. (primary): https://doi.org/10.1007/s00236-024-00462-y\n Evidence used: Records known complexity examples and develops efficient algorithms on supplied words.\n\n**Review notes.** No silent formalization was imposed; the missing input model and complexity target are material defects.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 105, "favorite_count": 8, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1438, "problem_number": "GRAPH-051", "title": "Imbalance Conjecture", "statement": "If every edge has imbalance ≥1, is the multiset of edge imbalances always graphic?", "background": "The imbalance of an edge is the absolute difference between the degrees of its endpoints. The conjecture asks whether the multiset of these imbalances can always realize a degree sequence of some graph when all imbalances are positive. This connects degree sequences with edge properties in a novel way.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A week-old arXiv manuscript claims a complete proof of the imbalance conjecture, but no independent published validation was located by the check date.\n\n**Verified partial progress.**\n\n- Kozerenko–Serdiuk proved multiple special classes and reported computational verification through 12 vertices.\n- Yavari's 2026 manuscript derives all Erdős–Gallai inequalities from a truncated imbalance bound and completes a parity check.\n\n**Full solution or refutation.**\n\narXiv:2608.09191 proves exactly the finite-simple-graph formulation on its face, but its seven-day age requires conservative treatment.\n\n**What remains.**\n\nObtain specialist verification or peer-reviewed publication of the claimed proof; if a gap appears, the unrestricted conjecture remains open.\n\n**Sources checked.**\n\n- Yousof Yavari, A Proof of the Imbalance Conjecture, arXiv:2608.09191 (2026). (primary): https://arxiv.org/abs/2608.09191\n Evidence used: Submitted 2026-08-10 and claims a full proof of the exact assertion.\n- Sergiy Kozerenko and Andrii Serdiuk, New results on imbalance graphic graphs, Opuscula Mathematica 43 (2023), 81–100. (primary): https://doi.org/10.7494/OpMath.2023.43.1.81\n Evidence used: Provides established special cases and finite verification predating the proof claim.\n\n**Review notes.** Graphic must mean realizable as the degree multiset of a finite simple graph; multigraph conventions would change the problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 94, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1439, "problem_number": "GRAPH-052", "title": "Implicit Graph Conjecture", "statement": "Do slowly-growing hereditary graph families admit implicit representations?", "background": "The implicit graph conjecture concerns the existence of succinct encodings for hereditary families of graphs (closed under induced subgraphs) whose growth rate is subexponential. An implicit representation would allow efficient storage and adjacency queries. This has implications for data structures and graph databases.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The standard Implicit Graph Conjecture was refuted by Hatami and Hatami, who constructed hereditary factorial-speed families requiring polynomial-length vertex labels.\n\n**Verified partial progress.**\n\n- The counterexample families have at most factorial speed but require labels of length n^Omega(1), rather than O(log n).\n- Recent work studies sufficient conditions and the smaller-growth Small Implicit Graph Conjecture.\n\n**Full solution or refutation.**\n\nFactorial-speed hereditary growth is not sufficient for an implicit adjacency representation.\n\n**What remains.**\n\nCharacterize the hereditary classes that do admit O(log n)-bit adjacency labels and resolve smaller-growth replacement conjectures.\n\n**Sources checked.**\n\n- Hamed Hatami and Pooya Hatami, The Implicit Graph Conjecture is False, FOCS 2022. (primary): https://arxiv.org/abs/2111.13198\n Evidence used: Constructs hereditary factorial-speed counterexample families requiring n^Omega(1)-bit labels.\n- Édouard Bonnet et al., Adjacency Labeling Schemes for Small Classes, ITCS 2025, LIPIcs 325, Article 21. (primary): https://doi.org/10.4230/LIPIcs.ITCS.2025.21\n Evidence used: Treats the original conjecture as refuted and develops evidence for a small-class replacement.\n\n**Review notes.** 'Slowly-growing' and 'subexponential' are imprecise. The standard conjecture uses at-most-factorial speed 2^O(n log n).\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 112, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1440, "problem_number": "GRAPH-053", "title": "Ryser's Conjecture", "statement": "For r-partite r-uniform hypergraphs, is the vertex cover number at most (r-1) times the matching number?", "background": "Ryser's conjecture relates the minimum transversal (vertex cover) size to maximum matching size in hypergraphs. For graphs (r=2) this is König's theorem. The conjecture proposes a tight bound for hypergraphs: τ ≤ (r-1)ν. This is a central open problem in hypergraph theory with connections to combinatorial optimization.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ryser's conjecture is proved for r=2 and r=3 but remains open for every r≥4.\n\n**Verified partial progress.**\n\n- The r=2 case is König's theorem.\n- Aharoni proved the r=3 tripartite case.\n- Haxell–Scott proved an (r-epsilon)nu bound for r=4,5, which does not reach (r-1)nu.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for the general r-partite r-uniform assertion was verified.\n\n**What remains.**\n\nResolve tau(H)≤(r-1)nu(H) for r≥4, beginning with r=4.\n\n**Sources checked.**\n\n- Ron Aharoni, Ryser's conjecture for tripartite 3-graphs, Combinatorica 21 (2001), 1–4. (primary): https://doi.org/10.1007/s004930170001\n Evidence used: Proves the r=3 case.\n- Penny E. Haxell and Alex Scott, On Ryser's conjecture, Electronic Journal of Combinatorics 19 (2012), P23. (primary): https://doi.org/10.37236/1175\n Evidence used: Improves the trivial factor for r=4,5 without proving Ryser's factor.\n- Ryser's conjecture, Graph-theory open problems tracker, reviewed 2026-05-08. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/rysers_conjecture/\n Evidence used: Records the current open range r≥4 and established partial results.\n\n**Review notes.** Exact statement retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1441, "problem_number": "GRAPH-054", "title": "Second Neighborhood Problem", "statement": "Does every oriented graph have a vertex with at least as many vertices at distance 2 as at distance 1?", "background": "The second neighborhood problem asks whether oriented graphs always contain a vertex whose second neighborhood (vertices at distance exactly 2) is at least as large as its first neighborhood (out-neighbors). This has been conjectured by several researchers and has connections to tournament theory and Seymour's second neighborhood conjecture.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Seymour's Second Neighborhood Conjecture remains open for general oriented graphs, despite a 2026 extension to minimum out-degree at most 7.\n\n**Verified partial progress.**\n\n- Fisher proved the tournament case.\n- Kaneko–Locke established the conjecture when minimum out-degree is at most 6.\n- Sadhukhan–Sandeep–Sen extended the threshold to minimum out-degree 7 with reproducible computer assistance.\n- Bai–Li–Park proved a stronger matching form in additional low-degree and anti-transitive classes.\n\n**Full solution or refutation.**\n\nNo proof for arbitrary oriented graphs was verified.\n\n**What remains.**\n\nResolve the conjecture for all oriented graphs; the first unresolved minimum-out-degree regime now starts at 8.\n\n**Sources checked.**\n\n- D. C. Fisher, Squaring a tournament: a proof of Dean's conjecture, Journal of Graph Theory 23 (1996), 43–48. (primary): https://doi.org/10.1002/(SICI)1097-0118(199609)23:1%3C43::AID-JGT4%3E3.0.CO;2-K\n Evidence used: Proves the tournament special case.\n- Arpan Sadhukhan, R. B. Sandeep, and Sagnik Sen, A proof of Seymour's second neighborhood conjecture for oriented graphs with minimum out-degree equal to 7, arXiv:2606.30588 (2026). (primary): https://arxiv.org/abs/2606.30588\n Evidence used: Extends the known minimum-out-degree threshold to 7.\n- Yandong Bai, Binlong Li, and Boram Park, Towards a strengthening of the second neighborhood conjecture, arXiv:2607.18047 (2026). (primary): https://arxiv.org/abs/2607.18047\n Evidence used: Explicitly states that the unrestricted conjecture remains open and proves stronger special cases.\n\n**Review notes.** The exact phrase 'distance 1/2' is ambiguous; the intended objects are first and second out-neighborhoods, as the background indicates.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 128, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1442, "problem_number": "GRAPH-055", "title": "Teschner's Bondage Number Conjecture", "statement": "Is the bondage number of a graph always ≤ 3Δ/2, where Δ is the maximum degree?", "background": "The bondage number is the minimum number of edges whose removal increases the domination number. Teschner conjectured an upper bound of 3Δ/2 in terms of maximum degree Δ. This would establish a fundamental relationship between edge removal sensitivity and local graph structure in domination problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A week-old Figshare preprint claims an explicit cubic graph with bondage number 5, which would refute Teschner's 3Delta/2 bound, but independent archival validation was not found.\n\n**Verified partial progress.**\n\n- The historical conjecture is known for graphs with small domination number and broad surface-embedded regimes.\n- The 2026 candidate counterexample has Delta=3 and claimed bondage number 5, exceeding floor(3Delta/2)=4.\n\n**Full solution or refutation.**\n\nThe explicit finite claim would settle the question negatively if its domination and bondage computations are verified.\n\n**What remains.**\n\nIndependently audit the candidate graph and proof; if confirmed, reclassify the universal conjecture as disproved.\n\n**Sources checked.**\n\n- Yousof Yavari, A Counterexample to Teschner's Bondage-Number Conjecture, Figshare preprint (2026). (primary): https://doi.org/10.6084/m9.figshare.33198777\n Evidence used: Deposited 2026-08-10 and claims a cubic graph with bondage number 5.\n- Andrei Gagarin and Vadim Zverovich, The bondage number of graphs on topological surfaces and Teschner's conjecture, Discrete Mathematics 313 (2013), 796–808. (primary): https://doi.org/10.1016/j.disc.2012.12.018\n Evidence used: Proves the conjecture for broad surface-embedded cases while documenting its prior universal open status.\n\n**Review notes.** For odd Delta the real-valued inequality is equivalent to the floor bound because bondage number is integral.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 89, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1443, "problem_number": "GRAPH-056", "title": "Tutte's 5-Flow Conjecture", "statement": "Does every bridgeless graph have a nowhere-zero 5-flow?", "background": "Tutte's 5-flow conjecture is one of the most famous problems in graph theory. A nowhere-zero k-flow is an orientation and edge-labeling with values in {±1,...,±(k-1)} satisfying flow conservation. The conjecture states that 5 colors suffice for all bridgeless graphs. Related to the four-color theorem and still wide open despite much research.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Tutte's 5-flow conjecture remains open; Seymour's nowhere-zero 6-flow theorem is still the best universal result.\n\n**Verified partial progress.**\n\n- Every bridgeless graph has a nowhere-zero 6-flow.\n- Nowhere-zero 5-flows are proved for multiple highly connected cubic and low-oddness classes.\n\n**Full solution or refutation.**\n\nNo universal nowhere-zero 5-flow theorem or bridgeless counterexample was verified.\n\n**What remains.**\n\nImprove the universal bound from 6 to 5 or find a counterexample.\n\n**Sources checked.**\n\n- P. D. Seymour, Nowhere-zero 6-flows, Journal of Combinatorial Theory, Series B 30 (1981), 130–135. (primary): https://doi.org/10.1016/0095-8956(81)90058-7\n Evidence used: Proves the continuing best universal flow bound.\n- Eckhard Steffen, Intersecting 1-factors and nowhere-zero 5-flows, Combinatorica 35 (2015), 731–743. (primary): https://arxiv.org/abs/1306.5645\n Evidence used: Proves important special cases rather than the unrestricted conjecture.\n- 5-flow conjecture, Graph-theory open problems tracker, reviewed 2026-05-08. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/5_flow_conjecture/\n Evidence used: Records outstanding status and Seymour's 6-flow theorem as the best universal bound.\n\n**Review notes.** The standard conjecture is normally stated for bridgeless multigraphs; the record does not specify whether multigraphs are allowed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1444, "problem_number": "GRAPH-057", "title": "Tutte's 4-Flow Conjecture for Petersen-Minor-Free Graphs", "statement": "Does every Petersen-minor-free bridgeless graph have a nowhere-zero 4-flow?", "background": "This is a refinement of Tutte's 5-flow conjecture for graphs without Petersen graph minors. The Petersen graph is known to require 5 colors for nowhere-zero flows, so excluding it might allow 4-flows. This conjecture connects graph minors, nowhere-zero flows, and the special role of the Petersen graph in combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Tutte's 4-flow conjecture for Petersen-minor-free bridgeless graphs remains open in general.\n\n**Verified partial progress.**\n\n- The conjecture is known for bridgeless cubic graphs without a Petersen minor.\n- A 2026 result proves it for the more restrictive P/e-minor-free multigraph class.\n\n**Full solution or refutation.**\n\nNo proof covering arbitrary vertex degrees in every Petersen-minor-free bridgeless graph was verified.\n\n**What remains.**\n\nProve a nowhere-zero 4-flow for the full Petersen-minor-free class or find a counterexample.\n\n**Sources checked.**\n\n- J. Pintér, Nowhere-zero 4-flows in graphs excluding the Petersen graph with one edge contracted, arXiv:2607.22267 (2026). (primary): https://arxiv.org/abs/2607.22267\n Evidence used: Proves the P/e-minor-free case and distinguishes it from the full Petersen-minor conjecture.\n- Selected Topics in Discrete Mathematics, flow-conjectures notes. (authoritative_secondary): https://home.zcu.cz/~kaisert/vpdm/11.pdf\n Evidence used: Records the cubic Petersen-minor-free theorem and the remaining larger-degree open case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1445, "problem_number": "GRAPH-058", "title": "Woodall's Conjecture", "statement": "Is the minimum dicut size equal to the maximum number of disjoint dijoins in a directed graph?", "background": "Woodall's conjecture is a directed graph analogue of Menger's theorem. A dicut is a set of arcs whose removal disconnects the graph directionally, and a dijoin connects specified vertex pairs. The conjecture proposes a min-max relation, which would be a fundamental packing-covering duality for directed graphs.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Woodall's unweighted dicut--dijoin min--max conjecture remains open.\n\n**Verified partial progress.**\n\n- Cornuéjols--Liu--Ravi prove that every digraph with minimum dicut size tau has floor(tau/6) disjoint dijoins, constructible in polynomial time.\n- The weighted Edmonds--Giles analogue is false, distinguishing it from Woodall's unweighted conjecture.\n\n**Full solution or refutation.**\n\nNo exact tau-packing theorem was verified for all digraphs.\n\n**What remains.**\n\nClose the constant-factor gap and prove or disprove the exact min--max relation.\n\n**Sources checked.**\n\n- G. Cornuéjols, S. Liu and R. Ravi, Approximately Packing Dijoins via Nowhere-Zero Flows, Combinatorica 45 (2025), Article 32, doi:10.1007/s00493-025-00159-x. (primary): https://doi.org/10.1007/s00493-025-00159-x\n Evidence used: States Woodall's conjecture is open and proves the floor(tau/6) approximation.\n- J. Pascal Gollin et al., Disjoint dijoins for classes of dicuts in finite and infinite digraphs, 2022. (primary): https://doi.org/10.5070/C62359180\n Evidence used: Explicitly describes the unweighted conjecture as a long-standing open problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 134, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1446, "problem_number": "ALG-001", "title": "Birch-Tate Conjecture", "statement": "Relate the order of the center of the Steinberg group of the ring of integers to the Dedekind zeta function.", "background": "The Birch-Tate conjecture connects algebraic K-theory to special values of zeta functions. It predicts a precise relationship between the center of the Steinberg group St(O_K) of a number field K and the value of its Dedekind zeta function at s=-1. This is a fundamental connection between algebra and analytic number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Under the intended standard Birch-Tate formulation, the odd part is proved for every totally real field and the full formula is proved for abelian extensions of Q; the nonabelian 2-primary case remains open.\n\n**Verified partial progress.**\n\n- Wiles proved the odd-primary part for arbitrary totally real number fields.\n- The 2-primary part follows from the 2-adic Iwasawa main conjecture and is known for abelian extensions of Q, yielding the full Birch-Tate formula in that class.\n\n**Full solution or refutation.**\n\nThe intended normalized formula is known in broad cases but not for arbitrary nonabelian totally real fields. The extracted imperative is not itself a truth-valued conjecture.\n\n**What remains.**\n\nProve the 2-primary Birch-Tate formula for arbitrary nonabelian totally real number fields and separately correct the source record's missing hypotheses and normalization.\n\n**Sources checked.**\n\n- Charles A. Weibel, The K-book: An Introduction to Algebraic K-theory, Chapter VI, section 8, Conjecture 8.6 and Theorem 8.7. (authoritative_secondary): https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.VI.pdf\n Evidence used: Gives the normalized formula, attributes the odd part to Wiles, and identifies the general 2-primary remainder and the abelian case.\n- Encyclopedia of Mathematics, Birch-Tate conjecture. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Birch-Tate_conjecture\n Evidence used: States the standard K_2(O_F), w_2(F), and zeta_F(-1) formula and says the remaining case is the 2-part for nonabelian extensions.\n\n**Review notes.** The source omits the number field, totally real condition, s=-1, and w_2(F), and does not state an equality. K_2 is canonically the Steinberg-to-elementary kernel; a center identification needs stability hypotheses. Exact text preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 187, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1447, "problem_number": "ALG-002", "title": "Casas-Alvero Conjecture", "statement": "If a polynomial of degree d over a field of characteristic 0 shares a factor with each of its first d-1 derivatives, must it be $(x-a)^d$?", "background": "The Casas-Alvero conjecture states that a polynomial sharing roots with all its derivatives (up to degree d-1) must be a power of a linear polynomial. Despite its elementary statement, it remains open. The conjecture has been verified for many special cases but lacks a general proof.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A 2025 preprint claims a complete proof, but a peer-reviewed survey published in May 2026 still calls the Casas-Alvero conjecture open with degree 24 the first open case; no independent verification was found.\n\n**Verified partial progress.**\n\n- The conjecture is proved for degrees p^k and 2p^k, with further degree families and degree 12 also settled in later work.\n- Positive-characteristic counterexamples show that characteristic zero is essential.\n- Ghosh's preprint claims all degrees over every characteristic-zero field using Koszul homology, but its acceptance has not been verified.\n\n**Full solution or refutation.**\n\nThe available strong sources conflict on whether the claimed full proof has been validated, so a solved classification would be premature.\n\n**What remains.**\n\nObtain an expert audit or refereed validation of Ghosh's proof; absent validation, settle degree 24 and the remaining degrees or find a characteristic-zero counterexample.\n\n**Sources checked.**\n\n- Soham Ghosh, Proof of the Casas-Alvero conjecture, arXiv:2501.09272 (2025). (primary): https://arxiv.org/abs/2501.09272\n Evidence used: Claims a complete proof in every degree over every characteristic-zero field using Koszul homology.\n- Armengol Gasull, A Primer on Resultants and Their Applications, Matematica Contemporanea (2026), DOI 10.1007/s44425-026-00047-6. (authoritative_secondary): https://doi.org/10.1007/s44425-026-00047-6\n Evidence used: Peer-reviewed article published 2026-05-28 that still describes the conjecture as open and degree 24 as the lowest open case.\n- Hans-Christian Graf von Bothmer, Oliver Labs, Josef Schicho, and Christiaan van de Woestijne, The Casas-Alvero conjecture for infinitely many degrees, Journal of Algebra 316 (2007), 224-230. (primary): https://doi.org/10.1016/j.jalgebra.2007.06.017\n Evidence used: Proves the conjecture for degrees p^k and 2p^k and supplies positive-characteristic counterexamples.\n\n**Review notes.** The statement omits monicity or a nonzero leading scalar in the conclusion, and 'shares a factor' should mean a nonconstant factor that may differ by derivative. Exact text preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 203, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1448, "problem_number": "ALG-003", "title": "Connes Embedding Problem", "statement": "Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?", "background": "The Connes embedding problem is a central question in operator algebra theory. It asks whether all separable II₁ factors embed into the ultrapower of the hyperfinite II₁ factor. This problem connects functional analysis, quantum information theory, and logic. Recent claimed solutions using quantum computing have generated significant interest.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Connes' embedding conjecture has a negative answer as a consequence of MIP*=RE.\n\n**Verified partial progress.**\n\n- MIP*=RE implies a strict separation between tensor-product and commuting-operator quantum correlation sets.\n- Established equivalences through Tsirelson's problem transfer that separation to a non-embeddable finite tracial von Neumann algebra.\n\n**Full solution or refutation.**\n\nJi, Natarajan, Vidick, Wright, and Yuen refute the standard separable Connes embedding problem; hence the extracted universal statement is false as well.\n\n**What remains.**\n\nStudy explicit and structurally natural non-embeddable factors, quantitative witnesses, and restricted classes where embeddability may still hold.\n\n**Sources checked.**\n\n- Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen, MIP*=RE, arXiv:2001.04383 (2020; revised 2021). (primary): https://arxiv.org/abs/2001.04383\n Evidence used: The paper explicitly states that its correlation-set separation refutes Connes' embedding conjecture.\n- Isaac Goldbring, The Connes Embedding Problem: A Guided Tour, arXiv:2109.12682 (2021). (authoritative_secondary): https://arxiv.org/abs/2109.12682\n Evidence used: Explains the negative solution and gives two routes from MIP*=RE to the operator-algebraic conclusion.\n\n**Review notes.** The classical formulation normally includes separability. The extracted unrestricted wording has additional cardinality issues, but the standard separable statement is already false.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 289, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1449, "problem_number": "ALG-004", "title": "Crouzeix's Conjecture", "statement": "Is $\\|f(A)\\| \\leq 2 \\sup_{z \\in W(A)} |f(z)|$ for any matrix A and analytic function f on the numerical range W(A)?", "background": "Crouzeix's conjecture bounds the matrix norm of f(A) by twice the supremum of |f| over the numerical range of A. The constant 2 would be optimal. This conjecture connects matrix theory, complex analysis, and numerical analysis. The best known bound is approximately 11.08, far from the conjectured 2.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This duplicate of Crouzeix's conjecture has the same unsettled August 2026 status: two very recent proof claims exist, including Lorist--Schwenninger's expert-authored arXiv manuscript, but independent or peer-reviewed confirmation was not located.\n\n**Verified partial progress.**\n\n- The settled universal constant is 1+sqrt(2), not the approximately 11.08 stated in this record's background.\n- Lorist--Schwenninger claim the sharp constant 2 for all bounded Hilbert-space operators.\n- Jin independently claims the matrix polynomial statement.\n\n**Full solution or refutation.**\n\nA full solution is claimed but remains pending expert verification at the check date.\n\n**What remains.**\n\nVerify the July/August 2026 proofs and update the record only after authoritative acceptance; independently correct the obsolete background bound.\n\n**Sources checked.**\n\n- Emiel Lorist and Felix Schwenninger, A solution to Crouzeix's conjecture, arXiv:2608.03841 (2026). (primary): https://arxiv.org/abs/2608.03841\n Evidence used: Extremely recent full-proof claim for a statement stronger than the displayed finite-matrix inequality.\n- Shanmu Jin, The Numerical Range Is a 2-Spectral Set, Preprints.org 202607.1919 (2026). (primary): https://www.preprints.org/manuscript/202607.1919\n Evidence used: Independent proof claim posted eight days before the Lorist--Schwenninger arXiv submission.\n- Michel Crouzeix and César Palencia, The Numerical Range is a (1+sqrt(2))-Spectral Set, SIAM Journal on Matrix Analysis and Applications 38 (2017), 649-655. (primary): https://epubs.siam.org/doi/10.1137/17M1116672\n Evidence used: Establishes the accepted 1+sqrt(2) bound and proves the imported background is obsolete.\n\n**Review notes.** Duplicate problem number and title are preserved. The background's 11.08 claim has been false as a best bound since 2017, and analytic on W(A) should mean holomorphic on a neighborhood.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1450, "problem_number": "ALG-005", "title": "Determinantal Conjecture", "statement": "Characterize the determinant of the sum of two normal matrices.", "background": "The determinantal conjecture seeks inequalities or characterizations for det(A+B) when A and B are normal matrices. While det(AB) = det(A)det(B) is well known, the sum of normal matrices presents challenges. This problem connects linear algebra with operator theory and has applications in quantum mechanics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The record does not state a conjecture or specify what data determine the requested characterization. Its likely target is the Marcus--de Oliveira convex-hull conjecture, which remains open in general despite recent sufficient conditions.\n\n**Verified partial progress.**\n\n- Bebiano--Queiró gave partial answers for determinants of sums of normal matrices with prescribed spectra.\n- Shitov proved a further sufficient condition for the Marcus--de Oliveira determinantal conjecture in 2025.\n- Mulero-Martínez's July 2026 paper develops a variational framework and explicitly says the general normal case remains open.\n\n**Full solution or refutation.**\n\nNo literal yes/no statement can be resolved. Under the likely Marcus--de Oliveira reconstruction, only partial results are known.\n\n**What remains.**\n\nRecover a source-backed formal statement, most likely the assertion that det(A+B) lies in the convex hull of all paired eigenvalue products, before assigning a mathematical status to the record itself.\n\n**Sources checked.**\n\n- Natália Bebiano and João Filipe Queiró, The determinant of the sum of two normal matrices with prescribed eigenvalues, Linear Algebra and its Applications 71 (1985), 23-28. (primary): https://doi.org/10.1016/0024-3795(85)90231-9\n Evidence used: Formulates the prescribed-eigenvalue problem and provides partial answers.\n- Yaroslav N. Shitov, A further sufficient condition for the determinantal conjecture, Izvestiya: Mathematics 89 (2025), 862-869. (primary): https://www.mathnet.ru/eng/im9292\n Evidence used: Proves a new sufficient condition for the standard convex-hull conjecture.\n- Juan Ignacio Mulero-Martínez, A variational framework for determinantal inequalities of normal matrices: Successes and obstructions, Linear Algebra and its Applications 740 (2026), 19-38. (primary): https://doi.org/10.1016/j.laa.2026.03.019\n Evidence used: States the Marcus--de Oliveira conjecture precisely and says it remains open for general normal matrices.\n\n**Review notes.** The title suggests a known conjecture, but the exact statement merely says characterize and omits spectra, the convex hull, and all quantifiers. This defect was flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1451, "problem_number": "ALG-006", "title": "Eilenberg-Ganea Conjecture", "statement": "Does every group with cohomological dimension 2 have a 2-dimensional Eilenberg-MacLane space K(G,1)?", "background": "The Eilenberg-Ganea conjecture asks whether cohomological dimension equals geometric dimension for groups. Specifically, if cd(G)=2, does there exist a 2-dimensional CW complex with fundamental group G? The conjecture is known to hold for cd ≠ 2. This connects algebraic topology with group theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Whether every group of cohomological dimension two has geometric dimension two remains open. The general theorem gives geometric dimension at most three in this exceptional case.\n\n**Verified partial progress.**\n\n- Cohomological and geometric dimension agree outside cohomological dimension two.\n- Bestvina--Brady constructed explicit potential counterexample groups of cohomological dimension two.\n- Their theorem shows that at least one of the Eilenberg--Ganea and Whitehead asphericity conjectures is false.\n\n**Full solution or refutation.**\n\nNo group is currently verified to have cohomological dimension two and geometric dimension three, and no general two-dimensional K(G,1) construction is known.\n\n**What remains.**\n\nDetermine the geometric dimension of the Bestvina--Brady candidate groups, equivalently overcoming the associated Whitehead-asphericity obstruction, or prove a general two-dimensional realization theorem.\n\n**Sources checked.**\n\n- Mladen Bestvina and Noel Brady, Morse theory and finiteness properties of groups, Inventiones Mathematicae 129 (1997), 445-470. (primary): https://doi.org/10.1007/s002220050168\n Evidence used: Constructs the candidate groups and proves the incompatibility result linking the Eilenberg--Ganea and Whitehead conjectures.\n- James Howie, Bestvina--Brady groups and the plus construction, Mathematical Proceedings of the Cambridge Philosophical Society 127 (1999), 487-493. (primary): https://doi.org/10.1017/S0305004199003928\n Evidence used: Explains why the Bestvina--Brady groups are potential, not established, counterexamples and recovers the either-or theorem.\n- A. Adem and I. Hambleton, Minimal Euler characteristics for even-dimensional manifolds with finite fundamental group, Forum of Mathematics, Sigma 11 (2023), e24. (primary): https://doi.org/10.1017/fms.2023.18\n Evidence used: Recent peer-reviewed topology paper describes the related dimension-two realization issue as unsolved.\n\n**Review notes.** The exact statement is standard. K(G,1) is understood as a CW Eilenberg--Mac Lane space and dimension means geometric dimension.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1452, "problem_number": "ALG-007", "title": "Farrell-Jones Conjecture", "statement": "Are the assembly maps in algebraic K-theory and L-theory isomorphisms?", "background": "The Farrell-Jones conjecture predicts that certain assembly maps are isomorphisms for all groups. This would have major consequences for the computation of algebraic K-theory and L-theory groups. The conjecture has been verified for many important classes of groups including hyperbolic groups and arithmetic groups.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The full Farrell--Jones conjecture remains open for arbitrary groups but is proved for a very large class, with additional relatively hyperbolic suspension and automorphism-group cases established in 2026.\n\n**Verified partial progress.**\n\n- Known classes include hyperbolic, finite-dimensional CAT(0), virtually solvable, Coxeter, braid, mapping class, lattice, and S-arithmetic groups, among others.\n- The class has strong inheritance properties, including passage to subgroups, finite products, finite-index overgroups, finite free products, and directed colimits.\n- Andrew--Guerch--Hughes prove fibred A-, K-, and L-theoretic FJC for broad suspensions of relatively hyperbolic groups and applications to automorphism groups.\n\n**Full solution or refutation.**\n\nNo theorem for every group was verified; the standard conjecture is a major class-by-class success but remains universal and open.\n\n**What remains.**\n\nProve the full assembly isomorphism for arbitrary groups and unresolved cases such as broad outer automorphism groups, after fixing the exact coefficient, family, and theory variant intended by this record.\n\n**Sources checked.**\n\n- Wolfgang Lück, Survey on the Farrell-Jones Conjecture, arXiv:2507.11337. (authoritative_secondary): https://arxiv.org/abs/2507.11337\n Evidence used: Current expert survey defines the Full Farrell--Jones Conjecture and inventories proven group classes and inheritance properties.\n- Naomi Andrew, Yassine Guerch, and Sam Hughes, Automorphisms of relatively hyperbolic groups and the Farrell--Jones conjecture, Mathematische Annalen 395 (2026), article 89. (primary): https://doi.org/10.1007/s00208-026-03431-7\n Evidence used: Calls FJC a prominent open conjecture and proves new A-, K-, and L-theory cases for suspensions and automorphism groups.\n\n**Review notes.** The displayed sentence omits the group, coefficient category/ring, degree, family of subgroups, and variant. Status uses the modern full fibred formulation with virtually cyclic family, but the source text was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 165, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1453, "problem_number": "ALG-008", "title": "Finite Lattice Representation Problem", "statement": "Is every finite lattice isomorphic to the congruence lattice of some finite algebra?", "background": "The finite lattice representation problem asks whether every finite lattice can be realized as the congruence lattice of a finite algebra. While every finite lattice is the congruence lattice of some algebra, requiring finiteness of the algebra is much more restrictive. This is a central problem in universal algebra.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The finite lattice representation problem remains open; finite distributive lattices and other classes are representable.\n\n**Verified partial progress.**\n\n- Palfi--Pudlak give an equivalent finite-group interval formulation.\n\n**Full solution or refutation.**\n\nNo finite-algebra representation theorem or counterexample is known generally.\n\n**What remains.**\n\nRepresent every finite lattice or prove a finite nonrepresentable lattice.\n\n**Sources checked.**\n\n- Finite lattice representation problem overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Finite_lattice_representation_problem\n Evidence used: Records the problem as unsolved and the Palfi--Pudlak equivalence.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 142, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1454, "problem_number": "ALG-009", "title": "Hadamard Matrix Conjecture", "statement": "Does a Hadamard matrix of order 4k exist for every positive integer k?", "background": "The Hadamard conjecture states that Hadamard matrices (square matrices with entries ±1 and mutually orthogonal rows) exist for all orders divisible by 4. These matrices have applications in coding theory, cryptography, and experimental design. The smallest open case is k=167 (order 668).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Hadamard matrix conjecture remains open for arbitrary orders divisible by four.\n\n**Verified partial progress.**\n\n- Constructions cover infinitely many and extensive finite families of orders.\n\n**Full solution or refutation.**\n\nNo construction exists for every 4k.\n\n**What remains.**\n\nConstruct Hadamard matrices for all positive k or find a forbidden order.\n\n**Sources checked.**\n\n- Hadamard matrix conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Hadamard_matrix#Hadamard_conjecture\n Evidence used: Records the universal 4k conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1455, "problem_number": "ALG-010", "title": "Köthe Conjecture", "statement": "If a ring has no nil two-sided ideal besides {0}, does it also have no nil one-sided ideal besides {0}?", "background": "The Köthe conjecture asks whether the absence of nontrivial nil ideals implies the absence of nontrivial nil one-sided ideals. A nil ideal is one where every element is nilpotent. This has been a central problem in ring theory for decades, with connections to the structure theory of noncommutative rings.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Kothe's conjecture remains open.\n\n**Verified partial progress.**\n\n- It is known for several important ring classes.\n\n**Full solution or refutation.**\n\nNo general implication from absence of nil two-sided ideals to absence of nil one-sided ideals was verified.\n\n**What remains.**\n\nProve Kothe's conjecture or find a counterexample.\n\n**Sources checked.**\n\n- Kothe's conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/K%C3%B6the_conjecture\n Evidence used: Records the conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1456, "problem_number": "ALG-011", "title": "Perfect Cuboid", "statement": "Does there exist a perfect cuboid—a rectangular parallelepiped with integer edges, face diagonals, and space diagonal?", "background": "A perfect cuboid would be a box where all edges, face diagonals, and the space diagonal are integers. Despite extensive computational searches, no perfect cuboid has been found, nor has non-existence been proven. This is a Diophantine problem with connections to number theory and geometry.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No perfect cuboid is known and no nonexistence proof is known.\n\n**Verified partial progress.**\n\n- Extensive computational exclusions and parametrisations constrain possible examples.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nExhibit a perfect cuboid or prove impossibility.\n\n**Sources checked.**\n\n- Perfect cuboid overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Perfect_cuboid\n Evidence used: Records the problem as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1457, "problem_number": "ALG-012", "title": "Rota's Basis Conjecture", "statement": "Given n bases of an n-dimensional matroid, can we find n disjoint rainbow bases?", "background": "Rota's basis conjecture asks whether n disjoint bases B₁,...,Bₙ of a matroid of rank n can be rearranged into an n×n matrix where each row is a basis and each column is a transversal (rainbow basis). This elegant conjecture connects matroid theory with combinatorics and has resisted many attempts at proof.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Rota's basis conjecture remains open, with asymptotic and random/vector-space results.\n\n**Verified partial progress.**\n\n- An asymptotic theorem proves the conjecture in broad large-rank regimes.\n\n**Full solution or refutation.**\n\nNo proof covers every matroid and every rank.\n\n**What remains.**\n\nProve the rainbow-basis decomposition universally or find a counterexample.\n\n**Sources checked.**\n\n- A. Pokrovskiy, B. Sudakov and L. Yepremyan, Rota's Basis Conjecture holds asymptotically, arXiv:2008.06045. (primary): https://arxiv.org/abs/2008.06045\n Evidence used: States the asymptotic advance while retaining the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1458, "problem_number": "MOD-001", "title": "Cherlin-Zilber Conjecture", "statement": "Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?", "background": "The Cherlin-Zilber conjecture (also called the algebraicity conjecture) proposes that infinite simple groups with stable theories are essentially algebraic groups. This would classify a vast class of model-theoretically tame groups. The conjecture connects model theory, group theory, and algebraic geometry in a profound way.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The stable-theory Cherlin--Zilber formulation remains open; it is related to, but stronger/differently stated than, the standard finite-Morley-rank algebraicity conjecture.\n\n**Verified partial progress.**\n\n- Important finite-Morley-rank even/mixed-type results are known.\n\n**Full solution or refutation.**\n\nNo theorem covering all simple stable groups was verified.\n\n**What remains.**\n\nClarify the exact stability hypothesis and settle the corresponding algebraicity statement.\n\n**Sources checked.**\n\n- K. Tent, From the Cherlin-Zilber Conjecture via sharply 2-transitive groups to the Burnside problem, arXiv:2606.18207 (2026). (primary): https://arxiv.org/abs/2606.18207\n Evidence used: Reviews the still-open algebraicity program.\n\n**Review notes.** No source alteration; formulation differs from the standard finite-Morley-rank version.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 176, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1459, "problem_number": "MOD-002", "title": "Generalized Star Height Problem", "statement": "Can all regular languages be expressed with generalized regular expressions having bounded star height?", "background": "The generalized star height problem asks whether there's a universal bound on the nesting depth of Kleene stars needed to express regular languages. This is a fundamental question in formal language theory and automata theory, with connections to computational complexity and logic.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether every regular language has generalized star height at most one; in particular no language of generalized star height at least two is known.\n\n**Verified partial progress.**\n\n- The ordinary (non-generalized) star-height problem is decidable, but this does not settle the complement-allowed hierarchy.\n\n**Full solution or refutation.**\n\nNo bounded-height theorem or counterexample beyond level one was verified.\n\n**What remains.**\n\nFind a language of generalized star height at least two or prove a universal level-one bound.\n\n**Sources checked.**\n\n- W. Thomas, Applied Automata Theory lecture notes. (authoritative_secondary): https://www.csa.iisc.ac.in/~deepakd/atc-common/wolfgang-aat.pdf\n Evidence used: States the still-open question whether generalized star height at least two occurs.\n- T. Pierron, PhD thesis, 2024. (authoritative_secondary): https://perso.liris.cnrs.fr/tpierron/phd.pdf\n Evidence used: Explains that the generalized problem is not known decidable and no height-two language is known.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 143, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1460, "problem_number": "MOD-003", "title": "Hilbert's Tenth Problem for Number Fields", "statement": "For which number fields is there an algorithm to determine if a Diophantine equation has solutions?", "background": "Hilbert's tenth problem asked for an algorithm to solve Diophantine equations over the integers—proven impossible by Matiyasevich. The question for other number fields remains open. It's known to be undecidable for some fields and decidable for others. Determining exactly which fields admit such algorithms is a major open problem.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For rings of integers O_K, a 2025 theorem gives undecidability for every number field K; for the fields K themselves, including Q, the general Hilbert-tenth question remains open.\n\n**Verified partial progress.**\n\n- Koymans--Pagano establish negative H10 for all infinite rings finitely generated over Z.\n- In particular, recent rank-stability work derives negative H10 for O_K for every number field.\n\n**Full solution or refutation.**\n\nThe record omits whether solutions are sought in K or O_K, so it cannot receive a single unconditional solved label.\n\n**What remains.**\n\nSpecify the domain; for H10 over number fields themselves, settle the open field case.\n\n**Sources checked.**\n\n- P. Koymans and C. Pagano, Hilbert's tenth problem via additive combinatorics, arXiv:2412.01768 (revised 2025). (primary): https://arxiv.org/abs/2412.01768\n Evidence used: States undecidability for all infinite rings finitely generated over Z.\n- Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field, Invent. Math. (2025); arXiv:2501.18774. (primary): https://arxiv.org/abs/2501.18774\n Evidence used: States its consequence for rings of integers of every number field.\n- A survey of local-global methods for Hilbert's Tenth Problem, arXiv:2309.14987. (primary): https://arxiv.org/abs/2309.14987\n Evidence used: Records that H10 for number fields as fields remains open.\n\n**Review notes.** No source alteration; domain ambiguity is material.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1461, "problem_number": "MOD-004", "title": "Vaught Conjecture", "statement": "Does every complete first-order theory in a countable language have countably many, $\\aleph_0$, or $2^{\\aleph_0}$ countable models?", "background": "Vaught's conjecture states that the number of countable models of a complete theory is either finite, countably infinite, or continuum. This would rule out intermediate cardinalities. The conjecture connects model theory with descriptive set theory and has deep connections to the structure of mathematical logic.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Vaught's conjecture remains open for arbitrary complete countable first-order theories.\n\n**Verified partial progress.**\n\n- It is confirmed for many stable, order-like, and partial-order classes.\n\n**Full solution or refutation.**\n\nNo general countable-spectrum dichotomy proof was verified.\n\n**What remains.**\n\nSettle the conjecture for arbitrary complete countable theories.\n\n**Sources checked.**\n\n- Vaught conjecture, Encyclopedia of Mathematics. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Vaught_conjecture\n Evidence used: Records the general question and multiple established classes.\n\n**Review notes.** Duplicate topic retained; wording repeats the countable alternative.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1462, "problem_number": "MOD-005", "title": "Tarski's Exponential Function Problem", "statement": "Is the theory of the real numbers with addition, multiplication, and exponentiation decidable?", "background": "Tarski proved that the theory of real closed fields is decidable. Adding exponentiation makes the question much harder. Decidability would mean an algorithm exists to determine truth of statements involving exp. This has implications for automated theorem proving and connections to transcendental number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Decidability of the real exponential field remains open unconditionally.\n\n**Verified partial progress.**\n\n- Macintyre--Wilkie prove decidability conditional on Schanuel's conjecture.\n- Recent work gives additional conditional axiomatization/model-theoretic information.\n\n**Full solution or refutation.**\n\nNo unconditional decision procedure was verified.\n\n**What remains.**\n\nProve decidability or undecidability of Th(R,+,times,exp).\n\n**Sources checked.**\n\n- Oxford Mathematical Institute, Schanuel's Conjecture and free E-rings in o-minimal structures. (authoritative_secondary): https://www.maths.ox.ac.uk/node/6577\n Evidence used: Records conditional decidability under Schanuel's conjecture.\n- On the elementary theory of the real exponential field, arXiv:2603.08365 (2026). (primary): https://arxiv.org/abs/2603.08365\n Evidence used: Provides new conditional structural results, not unconditional decidability.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 256, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1463, "problem_number": "MOD-006", "title": "Stable Field Conjecture", "statement": "Is every infinite field with a stable first-order theory separably closed?", "background": "The stable field conjecture predicts that infinite fields with stable theories are separably closed. Stable theories are model-theoretically well-behaved. This conjecture would classify all stable fields, providing a complete understanding of these algebraically important structures through a model-theoretic lens.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Stable Field Conjecture remains open generally.\n\n**Verified partial progress.**\n\n- It is known for omega-stable and superstable fields, stable fields of weight one or finite dp-rank, and recent large stable-field cases.\n\n**Full solution or refutation.**\n\nNo proof for every infinite stable field was verified.\n\n**What remains.**\n\nShow every infinite stable field is separably closed.\n\n**Sources checked.**\n\n- Finite Undecidability in NIP Fields, J. Symbolic Logic (2023). (primary): https://doi.org/10.1017/jsl.2022.59\n Evidence used: States the Stable Fields Conjecture and enumerates major known cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1464, "problem_number": "MOD-007", "title": "Henson Graphs Finite Model Property", "statement": "Do Henson graphs have the finite model property?", "background": "Henson graphs are universal homogeneous graphs omitting certain finite subgraphs. The finite model property asks whether every satisfiable sentence has a finite model. This question connects infinite graph theory, model theory, and combinatorics, with implications for the decidability of their first-order theories.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The finite-model-property question is not adequately specified: it must identify the theory/logic and the FMP convention. A source retains a closely related Henson-structure FMP question as open.\n\n**Verified partial progress.**\n\n- Recent Henson-graph work establishes finite big Ramsey degrees, a different property.\n\n**Full solution or refutation.**\n\nNo theorem answering the dataset's unqualified phrase was verified.\n\n**What remains.**\n\nRecover the intended Henson structure and FMP definition, then audit its status.\n\n**Sources checked.**\n\n- D. Evans, Model-theoretic constructions for..., lecture notes. (primary): https://www.ma.imperial.ac.uk/~dmevans/hattingen.pdf\n Evidence used: States an associated theory's finite model property as an open problem.\n- The finite big Ramsey degrees of Henson graphs are provable in ACA_0, arXiv:2606.30885 (2026). (primary): https://arxiv.org/abs/2606.30885\n Evidence used: Documents recent progress on a distinct Henson-graph property.\n\n**Review notes.** No source alteration; term is underdetermined.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 123, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1465, "problem_number": "MOD-008", "title": "O-Minimal Theory with Trans-Exponential Growth", "statement": "Does there exist an o-minimal first-order theory with a trans-exponential (rapid growth) function?", "background": "O-minimal structures are ordered structures where definable sets have simple topology. Known o-minimal structures include real closed fields and structures with restricted analytic functions. The question asks whether o-minimality is compatible with very fast-growing functions, testing the limits of tame model theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No o-minimal structure defining a function that eventually dominates every finite iterate of exp is known, and none has been ruled out.\n\n**Verified partial progress.**\n\n- A 2026 preprint reduces o-minimality of one proposed analytic trans-exponential expansion to a regular-values condition.\n\n**Full solution or refutation.**\n\nThe proposal is not an existence proof.\n\n**What remains.**\n\nConstruct an o-minimal trans-exponential expansion or prove an obstruction.\n\n**Sources checked.**\n\n- Y. Fu, Towards Trans-Exponential O-minimal Expansion of the Real Field, arXiv:2604.03477 (2026). (primary): https://arxiv.org/abs/2604.03477\n Evidence used: Presents a candidate and reduces, rather than solves, its o-minimality.\n- Trans-exponential o-minimal structure status page. (authoritative_secondary): https://www.emergentmind.com/open-problems/existence-of-transexponential-o-minimal-structure\n Evidence used: Records that all known o-minimal structures are exponentially bounded and existence remains open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1466, "problem_number": "MOD-009", "title": "Infinite Minimal Field Algebraic Closure", "statement": "Is every infinite minimal field of characteristic zero algebraically closed?", "background": "A minimal structure is one where every definable subset is finite or cofinite. The question asks whether infinite fields with this property must be algebraically closed (when char=0). This would characterize the simplest infinite fields from a model-theoretic perspective, connecting field theory with minimality.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Podewski's minimal-field conjecture is proved in positive characteristic but remains open in characteristic zero.\n\n**Verified partial progress.**\n\n- The zero-characteristic question has been reduced to an ordered/minimal-field obstruction in major work.\n\n**Full solution or refutation.**\n\nNo general characteristic-zero algebraic-closure theorem was verified.\n\n**What remains.**\n\nEliminate the remaining ordered minimal-field scenario or prove algebraic closure directly.\n\n**Sources checked.**\n\n- K. Krupiński and F. Wagner, Around Podewski's conjecture, Fund. Math. 225 (2014); arXiv:1201.5709. (primary): https://arxiv.org/abs/1201.5709\n Evidence used: States positive-characteristic resolution and characteristic-zero open status.\n- K. Krupiński, On Podewski's conjecture. (primary): https://www.math.uni.wroc.pl/~kkrup/minifield3.pdf\n Evidence used: Explains the characteristic-zero reduction and remaining obstruction.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1467, "problem_number": "MOD-010", "title": "Keisler's Order", "statement": "Determine the structure of Keisler's order on first-order theories.", "background": "Keisler's order compares first-order theories based on the complexity of their ultrapowers. Understanding this order would classify theories by their model-theoretic complexity. Recent breakthroughs have shed light on the order's structure, but a complete classification remains elusive. This connects with classification theory and stability.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Keisler's order is known to have infinitely many classes and is not a well order, but its complete structure remains unknown.\n\n**Verified partial progress.**\n\n- Malliaris--Shelah exhibit an infinite strictly descending chain inside simple unstable theories.\n- Stable classes, minimum, and maximum behavior are substantially understood.\n\n**Full solution or refutation.**\n\nNo full classification of equivalence classes and comparabilities was verified.\n\n**What remains.**\n\nDetermine the global structure of Keisler's order.\n\n**Sources checked.**\n\n- M. Malliaris and S. Shelah, Keisler's order has infinitely many classes, Israel J. Math. 224 (2018), 189--230; arXiv:1503.08341. (primary): https://arxiv.org/abs/1503.08341\n Evidence used: Proves infinitely many classes and non-well-ordering.\n- M. Malliaris and S. Shelah, General topology meets model theory, PNAS 110 (2013), 13300--13305. (primary): https://pmc.ncbi.nlm.nih.gov/articles/PMC3746882/\n Evidence used: Explains known extremal/stable portions and the open classification program.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1468, "problem_number": "ALG-013", "title": "Serre's Conjecture II", "statement": "For simply connected semisimple algebraic groups over fields of cohomological dimension ≤2, is $H^1(F,G) = 0$?", "background": "Serre's Conjecture II predicts that the first Galois cohomology of simply connected semisimple groups vanishes over fields of small cohomological dimension. This would have major implications for the classification of algebraic groups and forms. The conjecture is known for various classes of fields but remains open in general.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Serre's Conjecture II is established for many group/field classes but remains open in stated generality.\n\n**Verified partial progress.**\n\n- Numerous classical and exceptional groups have been treated.\n\n**Full solution or refutation.**\n\nNo universal cd<=2 theorem was verified.\n\n**What remains.**\n\nProve H1(F,G)=0 for every stated field and simply connected semisimple group.\n\n**Sources checked.**\n\n- Serre's Conjecture II overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Serre%27s_conjecture_II\n Evidence used: Records the general conjecture and known cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1469, "problem_number": "ALG-014", "title": "Serre's Positivity Conjecture", "statement": "If R is a regular local ring and P,Q are prime ideals with $\\dim(R/P) + \\dim(R/Q) = \\dim(R)$, is $\\chi(R/P, R/Q) > 0$?", "background": "Serre's positivity conjecture predicts that the Euler characteristic (intersection multiplicity) is positive when dimensions add correctly. This is part of a broader set of homological conjectures in commutative algebra. The conjecture would provide fundamental information about the structure of modules over regular rings.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Serre's general positivity conjecture remains open; Gabber proved the distinct non-negativity statement, while positivity is known in major special cases.\n\n**Verified partial progress.**\n\n- Serre proved positivity in the equicharacteristic and unramified cases.\n- Skalit proved positivity for formal power-series rings over a complete regular local ring of dimension two.\n\n**Full solution or refutation.**\n\nThe source wording is the equality-dimension positivity question, not Gabber's non-negativity theorem.\n\n**What remains.**\n\nProve positivity for arbitrary regular local rings in the remaining mixed-characteristic/ramified generality.\n\n**Sources checked.**\n\n- A. Skalit, Positivity of Intersection Multiplicity Over a Two-Dimensional Base, J. Pure Appl. Algebra 223 (2019), 1801--1816; arXiv:1510.05146. (primary): https://arxiv.org/abs/1510.05146\n Evidence used: Its abstract proves the conjecture for formal power-series rings over complete two-dimensional regular local rings.\n- P. C. Roberts, Recent developments on Serre's multiplicity conjectures: Gabber's proof of the nonnegativity conjecture, L'Enseignement Math. 44 (1998), 305--324. (authoritative_secondary): https://www.e-periodica.ch/cntmng?pid=ens-001%3A1998%3A44%3A%3A183\n Evidence used: Distinguishes Gabber's proof of non-negativity from the positivity question.\n\n**Review notes.** Corrected status distinction; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1470, "problem_number": "ALG-015", "title": "Uniform Boundedness Conjecture for Rational Points", "statement": "Is there a bound N(g,d) such that all curves of genus g≥2 over degree d number fields have at most N(g,d) rational points?", "background": "The uniform boundedness conjecture asks whether the number of rational points on curves of genus ≥2 is uniformly bounded in terms of genus and field degree. This would be a remarkable strengthening of Faltings' theorem (finite number of points). The conjecture connects arithmetic geometry with Diophantine equations.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Uniform boundedness of rational points on genus-g curves over degree-d number fields remains open.\n\n**Verified partial progress.**\n\n- It follows conditionally from Lang's conjecture.\n\n**Full solution or refutation.**\n\nNo unconditional N(g,d) theorem is verified.\n\n**What remains.**\n\nProve uniform boundedness unconditionally.\n\n**Sources checked.**\n\n- Uniform boundedness conjecture for rational points overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Uniform_boundedness_conjecture\n Evidence used: Records the general curve conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 213, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1479, "problem_number": "TOP-001", "title": "Baum-Connes Conjecture", "statement": "Is the assembly map in K-theory an isomorphism for all locally compact groups?", "background": "The Baum-Connes conjecture predicts that a certain assembly map from equivariant K-homology to the K-theory of group C*-algebras is an isomorphism. This would have major consequences for the Novikov conjecture, index theory, and the structure of operator algebras. Known for many groups, general case open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Under the standard reduced, no-coefficients reading, the Baum-Connes assembly isomorphism remains open for all locally compact groups and is known for many classes; the stronger coefficients version is false.\n\n**Verified partial progress.**\n\n- Dirac-dual-Dirac, a-T-menability, and related methods prove the ordinary conjecture for broad group classes.\n- Higson-Lafforgue-Skandalis construct counterexamples to the Baum-Connes conjecture with coefficients.\n\n**Full solution or refutation.**\n\nCounterexamples with coefficients do not refute the ordinary no-coefficients conjecture, whose universal form remains open.\n\n**What remains.**\n\nSpecify the intended assembly map and, for the ordinary reduced no-coefficients version, prove or refute it for every locally compact group.\n\n**Sources checked.**\n\n- Maria Paula Gomez Aparicio, Pierre Julg, and Alain Valette, The Baum-Connes conjecture: an extended survey, arXiv:1905.10081 (2019; published 2020). (authoritative_secondary): https://arxiv.org/abs/1905.10081\n Evidence used: Distinguishes ordinary and coefficients variants and surveys established group classes.\n- Nigel Higson, Vincent Lafforgue, and Georges Skandalis, Counterexamples to the Baum-Connes conjecture, Geometric and Functional Analysis Special Volume (2002), 69-86. (primary): https://vlafforg.perso.math.cnrs.fr/files/counterex.pdf\n Evidence used: Constructs counterexamples to the stronger coefficients version.\n\n**Review notes.** The input does not specify coefficients, reduced versus full completion, or the assembly domain. The status uses the ordinary reduced, no-coefficients interpretation suggested by the background.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1480, "problem_number": "TOP-002", "title": "Berge Conjecture", "statement": "Are Berge knots the only knots in S³ admitting lens space surgeries?", "background": "The Berge conjecture states that Berge knots (constructed via a specific procedure) are the only knots in the 3-sphere that admit Dehn surgeries yielding lens spaces. This would classify all such knots, providing deep insight into the relationship between knot theory and 3-manifold topology.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The full Berge knot-classification conjecture remains open; Greene solved which lens spaces can arise and supplied a Berge surgery with matching Floer data, but did not prove the original knot is Berge.\n\n**Verified partial progress.**\n\n- Greene solves the lens-space realization problem using changemaker lattices and produces a Berge knot with the same surgery outcome and knot Floer homology data.\n- Rasmussen obtains substantial number-theoretic restrictions on possible surgery parameters.\n\n**Full solution or refutation.**\n\nClassification of surgery outcomes and Floer data falls short of classification of every knot realizing such a surgery.\n\n**What remains.**\n\nProve that each knot in S^3 admitting a lens-space surgery is itself one of Berge's knots, or produce a counterexample.\n\n**Sources checked.**\n\n- Joshua Evan Greene, The lens space realization problem, Annals of Mathematics 177 (2013), 449-511. (primary): https://doi.org/10.4007/annals.2013.177.2.3\n Evidence used: Solves realization of the resulting lens spaces while explicitly stopping short of knot classification.\n- Sarah Dean Rasmussen, A number theoretic result for Berge's conjecture, Algebraic & Geometric Topology 18 (2018), 3569-3609. (primary): https://doi.org/10.2140/agt.2018.18.3569\n Evidence used: Proves strong parameter restrictions and treats the full Berge assertion as conjectural.\n\n**Review notes.** The result deliberately distinguishes lens-space realization from uniqueness/classification of the knot.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1481, "problem_number": "TOP-003", "title": "Borel Conjecture", "statement": "Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?", "background": "The Borel conjecture predicts that aspherical closed manifolds (those with contractible universal cover) are rigid—completely determined by their fundamental group up to homeomorphism. This would be a remarkable topological rigidity result, currently known only for special classes of manifolds.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Borel rigidity conjecture remains open for arbitrary closed aspherical manifolds, but is proved in dimensions at least five for hyperbolic and CAT(0) fundamental groups and in other major special classes.\n\n**Verified partial progress.**\n\n- Bartels-Luck prove the Borel conjecture for closed aspherical manifolds of dimension at least five with hyperbolic or CAT(0) fundamental group.\n- Classical geometric rigidity and low-dimensional topology settle further category- and dimension-specific cases.\n\n**Full solution or refutation.**\n\nBroad rigidity theorems do not cover arbitrary fundamental groups or every dimension.\n\n**What remains.**\n\nEstablish topological rigidity for closed aspherical manifolds with arbitrary fundamental groups, including the unresolved dimension-sensitive cases.\n\n**Sources checked.**\n\n- Arthur Bartels and Wolfgang Luck, The Borel conjecture for hyperbolic and CAT(0)-groups, Annals of Mathematics 175 (2012), 631-689. (primary): https://doi.org/10.4007/annals.2012.175.2.5\n Evidence used: Proves high-dimensional Borel rigidity for two broad classes of fundamental groups.\n- Arthur Bartels, On proofs of the Farrell-Jones conjecture, arXiv:1210.1044. (authoritative_secondary): https://arxiv.org/abs/1210.1044\n Evidence used: Explains how Farrell-Jones results imply Borel rigidity and delineates the known scope.\n\n**Review notes.** This is an exact duplicate of record 1332 despite its different TOP number.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1482, "problem_number": "TOP-004", "title": "Hilbert-Smith Conjecture", "statement": "If a locally compact group acts faithfully and continuously on a manifold, must it be a Lie group?", "background": "The Hilbert-Smith conjecture asks whether every locally compact group with a continuous faithful action on a manifold is necessarily a Lie group. This would rule out p-adic groups acting on manifolds, resolving a fundamental question about the symmetries of topological spaces.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Hilbert--Smith conjecture remains open for arbitrary continuous actions on topological manifolds.\n\n**Verified partial progress.**\n\n- It is proved for Lipschitz actions and in several low-dimensional/regular settings.\n\n**Full solution or refutation.**\n\nNo theorem handles every faithful continuous locally compact group action.\n\n**What remains.**\n\nExclude non-Lie locally compact groups, equivalently p-adic-type actions, in full generality.\n\n**Sources checked.**\n\n- Hilbert--Smith conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Hilbert%E2%80%93Smith_conjecture\n Evidence used: Records the general open status and known Lipschitz case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1483, "problem_number": "TOP-005", "title": "Novikov Conjecture", "statement": "Are certain polynomials in Pontryagin classes homotopy invariants?", "background": "The Novikov conjecture states that higher signatures (certain rational combinations of Pontryagin numbers) are oriented homotopy invariants. This has profound consequences for manifold topology, surgery theory, and K-theory. Proven for many classes of groups, but the general case remains open.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Novikov conjecture is open for arbitrary discrete groups, but known for broad classes via assembly-map methods.\n\n**Verified partial progress.**\n\n- It follows for groups for which relevant Baum--Connes/Farrell--Jones assembly results are established.\n\n**Full solution or refutation.**\n\nNo proof for all groups was verified.\n\n**What remains.**\n\nEstablish the higher-signature homotopy invariance for arbitrary groups.\n\n**Sources checked.**\n\n- Novikov conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Novikov_conjecture\n Evidence used: Summarizes the general conjecture and families where it is known.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1484, "problem_number": "TOP-006", "title": "Unknotting Problem", "statement": "Can unknots be recognized in polynomial time?", "background": "The unknotting problem asks whether there exists a polynomial-time algorithm to determine if a knot diagram represents the unknot. While algorithms exist (exponential time), polynomial-time decidability remains open. This is a central problem in computational topology with connections to complexity theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Polynomial-time unknot recognition remains unproved; a quasi-polynomial algorithm has been announced.\n\n**Verified partial progress.**\n\n- Unknot recognition lies in NP and co-NP, and Lackenby announced a quasi-polynomial-time algorithm.\n\n**Full solution or refutation.**\n\nNo peer-reviewed polynomial-time decision algorithm was verified.\n\n**What remains.**\n\nObtain a polynomial-time algorithm or prove an appropriate lower bound.\n\n**Sources checked.**\n\n- M. Lackenby, personal research page. (maintained_tracker): https://people.maths.ox.ac.uk/lackenby/\n Evidence used: Lists the quasi-polynomial-time unknot-recognition work.\n- Unknotting problem overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Unknotting_problem\n Evidence used: Records NP/co-NP membership and the unproved quasi-polynomial announcement.\n\n**Review notes.** No source alteration; quasi-polynomial is not misreported as polynomial.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 256, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1485, "problem_number": "TOP-007", "title": "Volume Conjecture", "statement": "Do quantum invariants of knots determine their hyperbolic volume?", "background": "The volume conjecture predicts an exponential relationship between the colored Jones polynomial (a quantum invariant) and the hyperbolic volume of a knot complement. This would connect quantum topology with hyperbolic geometry in a striking way, revealing deep structures in 3-dimensional topology.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The volume conjecture is open for general knots, with many special cases proved.\n\n**Verified partial progress.**\n\n- Known cases include the figure-eight knot, three-twist knot, torus knots, and several link families.\n\n**Full solution or refutation.**\n\nNo all-hyperbolic-knot theorem was verified.\n\n**What remains.**\n\nEstablish the asymptotic colored-Jones/volume relation for every knot in the intended formulation.\n\n**Sources checked.**\n\n- Volume conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Volume_conjecture\n Evidence used: Lists established special cases and the remaining general open problem.\n\n**Review notes.** No source alteration; standard knot form distinguished from known link counterexamples/variants.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1486, "problem_number": "TOP-008", "title": "Whitehead Conjecture", "statement": "Is every connected subcomplex of a 2-dimensional aspherical CW complex also aspherical?", "background": "The Whitehead conjecture asks whether asphericity (having contractible universal cover) is preserved under taking subcomplexes in dimension 2. This would clarify the local structure of aspherical spaces and has connections to group theory and low-dimensional topology.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Whitehead conjecture remains open.\n\n**Verified partial progress.**\n\n- It is known in special classes of aspherical 2-complexes and is closely linked to relation-module and diagrammatic-reducibility methods.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was verified.\n\n**What remains.**\n\nShow every connected subcomplex is aspherical or construct a counterexample.\n\n**Sources checked.**\n\n- Whitehead conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Whitehead_conjecture\n Evidence used: Records the standard statement and unresolved status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 143, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1487, "problem_number": "TOP-009", "title": "Zeeman Conjecture", "statement": "Is $K \\times [0,1]$ collapsible for every finite contractible 2-dimensional CW complex K?", "background": "The Zeeman conjecture predicts that the product of any finite contractible 2-complex with an interval is collapsible (can be reduced to a point by elementary collapses). This relates to the Poincaré conjecture and questions about higher-dimensional manifolds. A counterexample would have major implications.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Zeeman's conjecture remains open.\n\n**Verified partial progress.**\n\n- It is proved for several special families and would imply major low-dimensional-topology consequences.\n\n**Full solution or refutation.**\n\nNo collapse theorem for every finite contractible 2-complex was verified.\n\n**What remains.**\n\nProve K times an interval collapsible for every finite contractible 2-complex, or find a counterexample.\n\n**Sources checked.**\n\n- Zeeman's conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Zeeman%27s_conjecture\n Evidence used: Records the general open status and standard formulation.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1488, "problem_number": "COMB-001", "title": "1/3-2/3 Conjecture", "statement": "Does every non-total finite poset have two elements x,y with P(x before y in random linear extension) ∈ [1/3, 2/3]?", "background": "The 1/3-2/3 conjecture asks whether finite partially ordered sets (not totally ordered) always contain a pair with intermediate probability of appearing in a certain order. This connects order theory with probability and has implications for sorting algorithms and social choice theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This duplicate formulation of the 1/3--2/3 conjecture remains open generally.\n\n**Verified partial progress.**\n\n- The conjecture holds for numerous special poset classes, including 5-thin posets.\n\n**Full solution or refutation.**\n\nNo universal theorem was verified.\n\n**What remains.**\n\nProve the balanced-pair assertion for all finite non-total posets.\n\n**Sources checked.**\n\n- K. P. Bogart and J. P. Trotter, The 1/3--2/3 Conjecture for 5-Thin Posets, SIAM J. Discrete Math. 21 (2007). (primary): https://epubs.siam.org/doi/10.1137/0405037\n Evidence used: Proves a major restricted class.\n- 1/3--2/3 conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/1/3%E2%80%932/3_conjecture\n Evidence used: Records the continuing general open status.\n\n**Review notes.** Duplicate topic retained; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 124, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1489, "problem_number": "COMB-002", "title": "Lonely Runner Conjecture", "statement": "If k runners with distinct speeds run on a unit circle, will each runner be \"lonely\" (≥1/k away from others) at some time?", "background": "The lonely runner conjecture predicts that in a system of runners with different speeds on a circular track, each runner will at some point be far from all others. Verified for k≤7, this problem connects view obstruction, Diophantine approximation, and number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This duplicate Lonely Runner formulation remains unresolved for arbitrary k.\n\n**Verified partial progress.**\n\n- Recent primary preprints give finite-case advances through 12 runners.\n\n**Full solution or refutation.**\n\nNo all-k theorem was verified.\n\n**What remains.**\n\nResolve the general conjecture.\n\n**Sources checked.**\n\n- T. Sungkawichai and T. Trakulthongchai, Eleven, twelve, and thirteen lonely runners, arXiv:2604.23906 (2026). (primary): https://arxiv.org/abs/2604.23906\n Evidence used: The abstract gives the 10--12 computer-assisted cases.\n\n**Review notes.** Duplicate topic retained; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1490, "problem_number": "COMB-003", "title": "Sunflower Conjecture", "statement": "Can the minimum size for sunflowers be bounded by an exponential (not super-exponential) function of k?", "background": "The sunflower conjecture asks whether families of k-element sets containing a sunflower (r sets with common \"core\") require only exponentially many sets in k. Recent progress by Alweiss et al. improved bounds but the original conjecture remains open. Fundamental for extremal combinatorics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The sunflower conjecture's requested exponential bound remains open, despite a major near-exponential improvement.\n\n**Verified partial progress.**\n\n- Alweiss--Lovett--Wu--Zhang improve the Erdos--Rado w^w-type bound to roughly (log w)^w for fixed petal number.\n- Subsequent refinements improve constants/logarithmic factors but do not establish c^w.\n\n**Full solution or refutation.**\n\nNo c^k bound in the source's sense was verified.\n\n**What remains.**\n\nRemove the logarithmic growth and prove an exponential sunflower threshold.\n\n**Sources checked.**\n\n- R. Alweiss, S. Lovett, K. Wu and J. Zhang, Improved bounds for the sunflower lemma, STOC 2020; arXiv:1908.08483. (primary): https://arxiv.org/abs/1908.08483\n Evidence used: The abstract states both the c^w conjecture and the improved roughly (log w)^w bound.\n- A. Rao, The story of sunflowers, J. London Math. Soc. (2026). (authoritative_secondary): https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70380\n Evidence used: The survey describes subsequent progress and says the original conjecture may still hold.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1491, "problem_number": "COMB-004", "title": "Union-Closed Sets Conjecture", "statement": "For any finite union-closed family of sets, does some element appear in at least half the sets?", "background": "Frankl's union-closed sets conjecture (also called the union-closed set conjecture) states that in any family of sets closed under unions, at least one element appears in ≥50% of the sets. Despite its elementary statement, this problem has resisted all attempts at proof.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The literal statement is false because it omits the standard exclusion of the union-closed family {empty set}; the intended nondegenerate Frankl conjecture remains open.\n\n**Verified partial progress.**\n\n- For the intended formulation F != {empty set}, a peer-reviewed information-theoretic bound guarantees an element in at least 0.38234 of the member sets.\n- A later preprint proposes about 0.38271 only under numerically verified optimization hypotheses, not the conjectured one-half.\n\n**Full solution or refutation.**\n\nThe family consisting only of the empty set is finite and union-closed but has no element, so the exact universal statement has a counterexample. Primary modern formulations explicitly exclude this family.\n\n**What remains.**\n\nFor the intended corrected statement F != {empty set}, prove the one-half frequency bound or construct a nondegenerate counterexample.\n\n**Sources checked.**\n\n- J. Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055 (2022). (primary): https://arxiv.org/abs/2211.09055\n Evidence used: The abstract explicitly states the necessary hypothesis F != {empty set} and proves the first absolute frequency constant.\n- L. Yu, Dimension-Free Bounds for the Union-Closed Sets Conjecture, Entropy 25 (2023), 767. (primary): https://doi.org/10.3390/e25050767\n Evidence used: Gives the rigorous numerical lower bound 0.38234 for the corrected nondegenerate conjecture.\n- J. Liu, Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling, arXiv:2306.08824 (2023). (primary): https://arxiv.org/abs/2306.08824\n Evidence used: States the approximately 0.38271 improvement only under numerically verified hypotheses.\n\n**Review notes.** Formulation defect preserved rather than silently repairing the exact statement; expert review should decide whether downstream curation maps it to the intended corrected conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1492, "problem_number": "COMB-005", "title": "Ramsey Number R(5,5)", "statement": "What is the exact value of the Ramsey number R(5,5)?", "background": "Ramsey theory asks: in any 2-coloring of edges of the complete graph Kₙ, what's the minimum n guaranteeing a monochromatic K₅? Known: 43 ≤ R(5,5) ≤ 48. Finding the exact value would be a major breakthrough. Paul Erdős famously said R(6,6) would require alien technology.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact value of R(5,5) is unknown; the current verified range is 43 <= R(5,5) <= 46.\n\n**Verified partial progress.**\n\n- Exoo's construction gives the still-best lower bound R(5,5) >= 43.\n- Angeltveit and McKay improved the upper bound to R(5,5) <= 46; their computation was independently replicated.\n\n**Full solution or refutation.**\n\nThe possibilities 43, 44, 45, and 46 have not been reduced to a single value.\n\n**What remains.**\n\nConstruct larger (5,5)-Ramsey graphs to raise the lower bound or rule out orders 43–45 to lower the upper bound.\n\n**Sources checked.**\n\n- V. Angeltveit and B. D. McKay, R(5,5) <= 46, Journal of Graph Theory 112 (2026), 198–208. (primary): https://doi.org/10.1002/jgt.70029\n Evidence used: Proves the 46 upper bound and states that the lower bound 43 remains best.\n- V. Angeltveit and B. D. McKay, R(5,5) <= 46, arXiv:2409.15709. (primary): https://arxiv.org/abs/2409.15709\n Evidence used: Open preprint version exposes the theorem, methodology, and the current 43 lower bound.\n\n**Review notes.** The source background's upper bound 48 is obsolete.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1495, "problem_number": "NUM-001", "title": "Singmaster's Conjecture", "statement": "Is there a finite upper bound on multiplicities of entries >1 in Pascal's triangle?", "background": "Singmaster's conjecture asks whether any number (other than 1) appears in Pascal's triangle only finitely many times. Known: no entry appears more than 8 times. A proof would reveal deep structure in binomial coefficients and their divisibility properties.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Singmaster's conjecture remains open: no uniform multiplicity bound in Pascal's triangle is known.\n\n**Verified partial progress.**\n\n- Recent work proves conditional/structural bounds for entries away from the boundary, while known polynomial identities yield infinitely many entries of multiplicity at least six.\n\n**Full solution or refutation.**\n\nNo absolute bound valid for every binomial coefficient value was verified.\n\n**What remains.**\n\nProve a uniform bound or construct entries of unbounded multiplicity.\n\n**Sources checked.**\n\n- S. Y. A. Chang and A. Kontorovich, Singmaster's Conjecture in the Interior of Pascal's Triangle, Q. J. Math. 73 (2022), 1137--1164. (primary): https://doi.org/10.1093/qmath/haac009\n Evidence used: Provides recent conditional interior results while treating the global conjecture as open.\n- T. F. Bloom, Erdős Problems history #849. (maintained_tracker): https://www.erdosproblems.com/history/849\n Evidence used: Records the current historical status of the related Erdős/Singmaster problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1496, "problem_number": "NUM-002", "title": "Odd Perfect Numbers", "statement": "Do any odd perfect numbers exist?", "background": "A perfect number equals the sum of its proper divisors. All known perfect numbers are even (form 2^(p-1)(2^p-1) for Mersenne primes). Whether odd perfect numbers exist is one of the oldest open problems in mathematics, dating to ancient Greece. If they exist, they must be very large (>10^1500).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Existence of an odd perfect number remains open.\n\n**Verified partial progress.**\n\n- Any odd perfect number would meet stringent factorization and lower-size constraints.\n\n**Full solution or refutation.**\n\nNo existence or nonexistence proof was verified.\n\n**What remains.**\n\nConstruct an odd perfect number or prove none exist.\n\n**Sources checked.**\n\n- Odd Perfect Number, Wolfram MathWorld. (authoritative_secondary): https://mathworld.wolfram.com/OddPerfectNumber.html\n Evidence used: Summarizes necessary conditions and continuing open status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1497, "problem_number": "NUM-003", "title": "Infinitude of Perfect Numbers", "statement": "Are there infinitely many perfect numbers?", "background": "All known perfect numbers are even and correspond to Mersenne primes via Euclid-Euler theorem. The question reduces to: are there infinitely many Mersenne primes? This remains open despite extensive computational searches. Connected to the distribution of primes and special number forms.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Infinitude of perfect numbers is open and is equivalent to infinitude of Mersenne primes for even perfect numbers.\n\n**Verified partial progress.**\n\n- Euclid--Euler classifies even perfect numbers as 2^(p-1)(2^p-1) with a Mersenne prime factor; no odd perfect number is known.\n\n**Full solution or refutation.**\n\nNo infinitude theorem was verified.\n\n**What remains.**\n\nProve infinitely many Mersenne primes, or otherwise establish infinitely many perfect numbers.\n\n**Sources checked.**\n\n- GIMPS list of known Mersenne primes. (maintained_tracker): https://www.mersenne.org/primes/\n Evidence used: Maintains the finite list of known Mersenne primes and ongoing searches.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 345, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1498, "problem_number": "NUM-004", "title": "Quasiperfect Numbers", "statement": "Do quasiperfect numbers exist?", "background": "A quasiperfect number n has σ(n) = 2n+1 (sum of divisors is one more than twice the number). No quasiperfect numbers are known. If they exist, they must be odd perfect squares >10^35. This problem connects divisor functions with perfect number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No quasiperfect number is known.\n\n**Verified partial progress.**\n\n- Classical results imply any example is an odd square exceeding 10^35 with at least seven distinct prime factors.\n\n**Full solution or refutation.**\n\nNo construction or nonexistence theorem was verified.\n\n**What remains.**\n\nFind n with sigma(n)=2n+1 or rule out every n.\n\n**Sources checked.**\n\n- Quasiperfect number overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Quasiperfect_number\n Evidence used: Records the defining equation, open status, and classical necessary conditions.\n\n**Review notes.** Unreviewed 2026 claims of stronger restrictions were not used as settled results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1499, "problem_number": "NUM-005", "title": "Lychrel Numbers", "statement": "Do Lychrel numbers exist in base 10?", "background": "A Lychrel number never forms a palindrome through iterative reverse-and-add process. 196 is the first candidate—after billions of iterations, no palindrome found. Proving existence or non-existence would resolve this computational mystery connecting palindromes with iteration dynamics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No Lychrel number has been proved to exist in base 10.\n\n**Verified partial progress.**\n\n- 196 and many other candidates have undergone extensive reverse-and-add computation; proofs exist in some other bases.\n\n**Full solution or refutation.**\n\nNo base-10 non-palindromic orbit was proved never to reach a palindrome.\n\n**What remains.**\n\nProve a base-10 Lychrel example exists or prove all base-10 starting integers reach palindromes.\n\n**Sources checked.**\n\n- Lychrel number overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Lychrel_number\n Evidence used: States that no base-10 Lychrel number has been proven and distinguishes other bases.\n- 196 and Other Lychrel Numbers project. (maintained_tracker): https://www.p196.org/\n Evidence used: Maintains computational records for the 196 search.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" } }, { "id": 1500, "problem_number": "NUM-006", "title": "Odd Weird Numbers", "statement": "Do odd weird numbers exist?", "background": "Weird numbers are abundant but not semiperfect (no subset of divisors sums to the number). All known weird numbers are even. Finding an odd weird number or proving none exist would reveal deep structure in additive properties of divisors.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Odd weird numbers remain unknown.\n\n**Verified partial progress.**\n\n- Search and structural work excludes them below 10^21 and requires at least six distinct prime factors.\n\n**Full solution or refutation.**\n\nNo existence or nonexistence proof was verified.\n\n**What remains.**\n\nConstruct an odd weird number or rule out all odd integers.\n\n**Sources checked.**\n\n- T. F. Bloom, Erdős Problem #470. (maintained_tracker): https://www.erdosproblems.com/470\n Evidence used: Records the open status and current lower restrictions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1501, "problem_number": "NUM-007", "title": "Infinitude of Amicable Pairs", "statement": "Are there infinitely many pairs of amicable numbers?", "background": "Amicable pairs (m,n) satisfy σ(m)-m=n and σ(n)-n=m. Over 12 million pairs known, but infinity unproven. Related to perfect numbers and sociable chains. Erdős-Rieger heuristics suggest infinity, but proof remains elusive.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It is not known whether there are infinitely many amicable pairs.\n\n**Verified partial progress.**\n\n- Many individual amicable pairs and parametric constructions are known, but none establishes infinitude.\n\n**Full solution or refutation.**\n\nNo infinitude theorem was verified.\n\n**What remains.**\n\nConstruct infinitely many distinct amicable pairs or prove only finitely many occur.\n\n**Sources checked.**\n\n- Amicable numbers overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Amicable_numbers\n Evidence used: Records the continuing unknown infinitude question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1502, "problem_number": "NUM-008", "title": "Pi Normality", "statement": "Is π a normal number (all digits equally frequent in all bases)?", "background": "A normal number has each digit appearing with equal asymptotic frequency in every base. While π appears statistically normal (verified to trillions of digits), no proof exists. This connects transcendental numbers, digit distribution, and randomness in mathematical constants.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Normality of pi in any particular standard base, hence the all-bases assertion, remains unproved.\n\n**Verified partial progress.**\n\n- Large finite digit statistics are consistent with normality but are not a proof.\n\n**Full solution or refutation.**\n\nNo base-normality theorem for pi was verified.\n\n**What remains.**\n\nProve the expected frequencies of every finite digit block in a base, then address all bases.\n\n**Sources checked.**\n\n- D. H. Bailey and R. E. Crandall, Randomness and the pi digits, Experimental Mathematics 11 (2002). (authoritative_secondary): https://doi.org/10.1080/10586458.2002.10504475\n Evidence used: Explains the unresolved normality question for pi.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 30, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1503, "problem_number": "NUM-009", "title": "Algebraic Number Normality", "statement": "Are all irrational algebraic numbers normal?", "background": "The question asks whether every irrational root of a polynomial with integer coefficients has all digits equally distributed in every base. A positive answer would be a remarkable connection between algebraic structure and digit statistics. Currently, we cannot prove normality for any specific algebraic irrational.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No irrational algebraic number is known to be normal in any base.\n\n**Verified partial progress.**\n\n- Metric normality theorems apply almost everywhere, not to a specified algebraic irrational.\n\n**Full solution or refutation.**\n\nNo individual example or general theorem was verified.\n\n**What remains.**\n\nProve normality for one irrational algebraic number, or resolve the asserted universal form.\n\n**Sources checked.**\n\n- Y. Bugeaud, Distribution Modulo One and Diophantine Approximation (2012). (authoritative_secondary): https://doi.org/10.1017/CBO9781139019672\n Evidence used: Surveys unresolved digit-distribution problems for algebraic irrationals.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1504, "problem_number": "NUM-010", "title": "Gilbreath's Conjecture", "statement": "Does iterating unsigned differences on prime sequence always yield 1 as first element?", "background": "Start with primes 2,3,5,7,11,... Take absolute differences: 1,2,2,4,... Repeat. Conjecture: first element is always 1. Verified to huge primes, but unproven. This reveals hidden regularity in prime gaps with implications for prime distribution.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Gilbreath's conjecture remains unproved.\n\n**Verified partial progress.**\n\n- The first-entry-one pattern has been checked over large finite prime ranges, but no inductive invariant controlling all rows is known.\n\n**Full solution or refutation.**\n\nNo all-primes proof or counterexample was verified.\n\n**What remains.**\n\nProve the leading entry in every iterated unsigned-difference row is one, or find the first failure.\n\n**Sources checked.**\n\n- OEIS A004770, Gilbreath's conjecture triangle. (maintained_tracker): https://oeis.org/A004770\n Evidence used: Records the prime-difference construction and conjectural status.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1505, "problem_number": "NUM-011", "title": "Lander-Parkin-Selfridge Conjecture", "statement": "If Σᵢ aᵢᵏ = Σⱼ bⱼᵏ with m terms on left, n on right, is m+n ≥ k?", "background": "The LPS conjecture generalizes Fermat's Last Theorem to sums of k-th powers. It predicts you need at least k terms total for nontrivial solutions. Counterexamples exist for specific cases, but the general conjecture remains open with implications for Diophantine equations.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Lander--Parkin--Selfridge total-term inequality remains open.\n\n**Verified partial progress.**\n\n- Counterexamples to stronger Euler sum-of-powers assertions do not violate m+n>=k.\n\n**Full solution or refutation.**\n\nNo proof or counterexample to the stated inequality was verified.\n\n**What remains.**\n\nSettle the total-term lower bound.\n\n**Sources checked.**\n\n- Lander, Parkin, and Selfridge conjecture overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Lander%2C_Parkin%2C_and_Selfridge_conjecture\n Evidence used: Distinguishes the open conjecture from disproved stronger variants.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" } }, { "id": 1506, "problem_number": "NUM-012", "title": "Class Number Problem", "statement": "Are there infinitely many real quadratic fields with class number 1 (unique factorization)?", "background": "The class number problem asks whether infinitely many real quadratic number fields Q(√d) have unique factorization. For imaginary quadratic fields, Heegner-Baker-Stark proved only finitely many exist. The real case remains open—a fundamental question in algebraic number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Infinitude of real quadratic class-number-one fields remains open.\n\n**Verified partial progress.**\n\n- Computations and heuristics provide evidence but no infinitude theorem.\n\n**Full solution or refutation.**\n\nNo all-family proof was verified.\n\n**What remains.**\n\nProve infinitely many real quadratic fields have class number one.\n\n**Sources checked.**\n\n- List of number fields with class number one. (authoritative_secondary): https://en.wikipedia.org/wiki/List_of_number_fields_with_class_number_one\n Evidence used: Explicitly records the real-quadratic infinitude question as unknown.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1507, "problem_number": "NUM-013", "title": "Hilbert's 12th Problem", "statement": "Extend Kronecker-Weber theorem to abelian extensions of arbitrary number fields.", "background": "Hilbert's 12th problem asks for explicit construction of abelian extensions of number fields via special values of transcendental functions (generalizing cyclotomic fields for Q). Partial progress via complex multiplication, but general case remains one of Hilbert's unsolved problems.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hilbert's twelfth problem remains open generally, with imaginary-quadratic and CM solutions and recent totally-real p-adic progress.\n\n**Verified partial progress.**\n\n- Complex multiplication solves imaginary quadratic fields; Shimura theory extends this to CM fields.\n\n**Full solution or refutation.**\n\nNo direct construction for arbitrary base fields was verified.\n\n**What remains.**\n\nExplicitly generate maximal abelian extensions for every number field in the requested sense.\n\n**Sources checked.**\n\n- Hilbert's twelfth problem overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Hilbert%27s_twelfth_problem\n Evidence used: Summarizes solved CM cases and open general case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 187, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1508, "problem_number": "NUM-014", "title": "Leopoldt's Conjecture", "statement": "Does the p-adic regulator of an algebraic number field never vanish?", "background": "Leopoldt's conjecture predicts that the p-adic regulator (a p-adic analogue of the classical regulator from Dirichlet's unit theorem) is always nonzero. This has major implications for Iwasawa theory and the structure of p-adic L-functions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Leopoldt remains open in general but is known for abelian extensions of Q and further classes.\n\n**Verified partial progress.**\n\n- Brumer's p-adic Baker method proves the abelian cases.\n\n**Full solution or refutation.**\n\nNo universal p-adic regulator nonvanishing theorem was verified.\n\n**What remains.**\n\nSettle Leopoldt for every number field and prime.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Leopoldt conjecture. (authoritative_secondary): https://encyclopediaofmath.org/wiki/Leopoldt_conjecture\n Evidence used: Records established abelian cases and the general conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1509, "problem_number": "NUM-015", "title": "Siegel Zeros", "statement": "Do Siegel zeros (real zeros of Dirichlet L-functions near s=1) exist?", "background": "Siegel zeros are hypothetical exceptional real zeros of L-functions very close to s=1. If they exist, they violate the Generalized Riemann Hypothesis. Their existence would have major consequences for prime distribution in arithmetic progressions. Most believe they don't exist.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Existence of Landau--Siegel zeros is unresolved.\n\n**Verified partial progress.**\n\n- Siegel's ineffective lower-bound theory, Deuring--Heilbronn repulsion, and at-most-one-exception principles constrain a possible exceptional zero.\n\n**Full solution or refutation.**\n\nNo proof of existence or exclusion was verified.\n\n**What remains.**\n\nProve that exceptional real zeros never occur, or exhibit one.\n\n**Sources checked.**\n\n- Siegel zero overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Siegel_zero\n Evidence used: Describes exceptional zeros as possible counterexamples and records the unresolved status.\n- G. Bhowmik and J. Ramaré, Conditional bounds on Siegel zeros (2020). (primary): https://pro.univ-lille.fr/fileadmin/user_upload/pages_pros/gautami_bhowmik/Publications/CANT2020.pdf\n Evidence used: Records effective conditional restrictions and the exceptional-character framework.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1510, "problem_number": "NUM-016", "title": "Schanuel's Conjecture", "statement": "For e and π: are they algebraically independent? Is e+π, eπ, π^e, etc. transcendental?", "background": "Schanuel's conjecture is a fundamental statement about transcendence degrees. It implies e and π are algebraically independent and that expressions like e+π, eπ, π^π are transcendental. Proving it would resolve many open questions in transcendental number theory at once.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Schanuel's conjecture is open already in the two-variable regime and would imply algebraic independence of e and pi.\n\n**Verified partial progress.**\n\n- Lindemann--Weierstrass settles weaker individual transcendence facts; Schanuel would settle many listed expressions conditionally.\n\n**Full solution or refutation.**\n\nNo unconditional algebraic-independence theorem for e and pi, nor the listed concrete consequences, was verified.\n\n**What remains.**\n\nProve Schanuel's conjecture or its relevant special cases.\n\n**Sources checked.**\n\n- D. Bertrand, G. Binyamini and M. Kowalski, Exponential sums equations and tropical geometry, Selecta Mathematica 29 (2023). (primary): https://doi.org/10.1007/s00029-023-00853-y\n Evidence used: States that the n=2 case remains hard and would imply algebraic independence of e and pi.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 287, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1511, "problem_number": "NUM-017", "title": "Euler-Mascheroni Constant Irrationality", "statement": "Is the Euler-Mascheroni constant γ irrational? Transcendental?", "background": "The Euler-Mascheroni constant γ ≈ 0.5772 appears throughout analysis and number theory. We don't even know if it's irrational! Proving irrationality or transcendence would be a major achievement. Related constants like Catalan's G and ζ(3) face similar questions.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Neither irrationality nor transcendence of the Euler--Mascheroni constant is known.\n\n**Verified partial progress.**\n\n- Conditional irrationality criteria and very large lower bounds on a hypothetical rational denominator are known.\n\n**Full solution or refutation.**\n\nNo unconditional irrationality proof was verified.\n\n**What remains.**\n\nProve gamma irrational or determine an exact arithmetic nature.\n\n**Sources checked.**\n\n- Euler--Mascheroni Constant, Wolfram MathWorld. (authoritative_secondary): https://mathworld.wolfram.com/Euler-MascheroniConstant.html\n Evidence used: Explicitly states that irrationality and transcendence are unknown and lists denominator bounds.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 323, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1512, "problem_number": "NUM-018", "title": "Littlewood Conjecture", "statement": "For any α,β ∈ ℝ, is lim inf_{n→∞} n·||nα||·||nβ|| = 0?", "background": "Littlewood's conjecture connects Diophantine approximation of pairs of real numbers. It predicts that for any two reals, you can simultaneously approximate both well infinitely often. Related to continued fractions and dynamics on homogeneous spaces. Proved for many special cases.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Littlewood's conjecture remains open, but any exceptional pair lies in a set of Hausdorff dimension zero.\n\n**Verified partial progress.**\n\n- Einsiedler--Katok--Lindenstrauss prove the conjecture outside a zero-Hausdorff-dimension exceptional set.\n\n**Full solution or refutation.**\n\nThe exceptional-set result does not rule out individual counterexamples.\n\n**What remains.**\n\nEliminate the exceptional set or find an exceptional pair.\n\n**Sources checked.**\n\n- M. Einsiedler, A. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Annals of Mathematics 164 (2006), 513--560. (primary): https://doi.org/10.4007/annals.2006.164.513\n Evidence used: Shows the exceptional set has Hausdorff dimension zero.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1513, "problem_number": "NUM-019", "title": "Four Exponentials Conjecture", "statement": "If x₁,x₂ and y₁,y₂ are linearly independent over ℚ, is at least one of e^(xᵢyⱼ) transcendental?", "background": "The four exponentials conjecture states that you can't have all four values e^(x₁y₁), e^(x₁y₂), e^(x₂y₁), e^(x₂y₂) algebraic when the xᵢ and yⱼ satisfy independence conditions. Weaker than Schanuel's conjecture but still wide open. Six exponentials theorem is the proven weaker version.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The four exponentials conjecture remains unproved.\n\n**Verified partial progress.**\n\n- The six exponentials theorem proves the analogous assertion for a 2-by-3 array.\n\n**Full solution or refutation.**\n\nNo argument reducing the proven six-exponential result to two y-values was verified.\n\n**What remains.**\n\nProve transcendence of at least one exponential in every 2-by-2 independent array.\n\n**Sources checked.**\n\n- Six Exponentials Theorem, Wolfram MathWorld. (authoritative_secondary): https://mathworld.wolfram.com/SixExponentialsTheorem.html\n Evidence used: States the proven 2-by-3 theorem and identifies the 2-by-2 form as unproved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1514, "problem_number": "NUM-020", "title": "Integer Factorization Polynomial Time", "statement": "Can integer factorization be done in polynomial time?", "background": "The integer factorization problem asks whether factoring large integers into primes can be done efficiently (polynomial time). RSA cryptography relies on it being hard. Shor's algorithm solves it on quantum computers, but classical complexity remains unknown. Related to P vs NP.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No classical polynomial-time factoring algorithm is known, while Shor's quantum algorithm puts factoring in BQP.\n\n**Verified partial progress.**\n\n- Shor gives polynomial runtime on a fault-tolerant quantum computer; classical algorithms remain subexponential rather than proven polynomial.\n\n**Full solution or refutation.**\n\nThe classical P-status remains unresolved; the problem wording does not state a computational model.\n\n**What remains.**\n\nSpecify classical versus quantum computation; under the usual classical reading, prove a polynomial algorithm or lower-bound separation.\n\n**Sources checked.**\n\n- P. W. Shor, Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM J. Comput. 26 (1997), 1484--1509. (primary): https://arxiv.org/abs/quant-ph/9508027\n Evidence used: Proves polynomial-time quantum factoring.\n- S. Arora and B. Barak, Computational Complexity: A Modern Approach, quantum-computing chapter. (authoritative_secondary): https://theory.cs.princeton.edu/complexity/quantumchap.pdf\n Evidence used: States that classical polynomial-time factoring is still unknown.\n\n**Review notes.** Statement retained; computational-model ambiguity flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 456, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1515, "problem_number": "PDE-001", "title": "Navier-Stokes Existence and Smoothness", "statement": "Do smooth solutions to Navier-Stokes equations exist globally in 3D? Or do finite-time singularities occur?", "background": "The Navier-Stokes existence and smoothness problem is one of the seven Millennium Prize Problems. It asks whether smooth solutions to the 3D Navier-Stokes equations exist for all time, or whether finite-time blow-up can occur. Fundamental for fluid dynamics and mathematical physics.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Under the Clay formulation for three-dimensional incompressible Navier-Stokes with positive viscosity and admissible smooth divergence-free data, neither global smoothness for arbitrary large data nor finite-time blowup has been proved.\n\n**Verified partial progress.**\n\n- Leray constructed global finite-energy weak solutions.\n- Caffarelli, Kohn, and Nirenberg proved that suitable weak solutions are regular outside a singular set of parabolic one-dimensional measure zero.\n- Buckmaster and Vicol proved nonuniqueness in a broad finite-energy weak-solution class, which does not resolve smooth Clay-class solutions.\n\n**Full solution or refutation.**\n\nWeak existence, partial regularity, and weak-solution nonuniqueness do not decide the global smooth-solution alternative in the Millennium problem.\n\n**What remains.**\n\nProve a unique global smooth solution for every admissible datum in an official Clay formulation or construct admissible smooth data with finite-time breakdown.\n\n**Sources checked.**\n\n- Charles L. Fefferman, Existence and smoothness of the Navier-Stokes equation, official Clay Millennium problem description. (authoritative_secondary): https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf\n Evidence used: Gives the precise official formulations, known weak-solution and partial-regularity context, and the unresolved alternatives.\n- Luis Caffarelli, Robert Kohn, and Louis Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Communications on Pure and Applied Mathematics 35 (1982), 771-831. (primary): https://doi.org/10.1002/cpa.3160350604\n Evidence used: Proves the landmark parabolic-measure-zero bound for the possible singular set of suitable weak solutions.\n- Tristan Buckmaster and Vlad Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation, Annals of Mathematics 189 (2019), 101-144. (primary): https://doi.org/10.4007/annals.2019.189.1.3\n Evidence used: Proves finite-energy weak-solution nonuniqueness, a major advance that does not establish smooth blowup or global smoothness.\n- Clay Mathematics Institute, Navier-Stokes Equation, Millennium Problems page, accessed 2026-08-17. (maintained_tracker): https://www.claymath.org/millennium/navier-stokes-equation/\n Evidence used: Official maintained page continues to list Navier-Stokes existence and smoothness as an unresolved Millennium Prize Problem.\n\n**Review notes.** The exact statement omits incompressibility, domain, forcing, decay, and quantifiers; the background explicitly identifies the standard Clay problem. This record is distinct from record 40 despite sharing PDE-001.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 512, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set_id": 2 }, { "id": 1516, "problem_number": "GEOM-001", "title": "Sphere Packing Problem Higher Dimensions", "statement": "What is the optimal sphere packing density in dimensions >3?", "background": "The sphere packing problem asks for the densest way to pack spheres in n-dimensional space. Solved in dimensions 1,2,3 (Kepler's conjecture, proved by Hales), 8, and 24 (Viazovska). Dimensions 4-7 and ≥9 remain open. Connections to lattices, coding theory, and optimization.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Unrestricted optimal sphere packing is known only in dimensions 1--3, 8, and 24.\n\n**Verified partial progress.**\n\n- E8 and Leech are optimal.\n\n**Full solution or refutation.**\n\nThe requested all higher-dimensional problem is open.\n\n**What remains.**\n\nResolve remaining dimensions.\n\n**Sources checked.**\n\n- Sphere packing overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Sphere_packing\n Evidence used: Lists solved dimensions.\n\n**Review notes.** Duplicate topic retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 298, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" } }, { "id": 1517, "problem_number": "HL-A", "title": "Hardy-Littlewood Conjecture A (Prime k-tuples)", "statement": "Let $a_1, \\ldots, a_k$ be given integers. Then there exist infinitely many positive integers $n$ such that $n + a_1, \\ldots, n + a_k$ are all prime, provided that for every prime $p$, there exists an integer $m$ such that $(m + a_i, p) = 1$ for all $i$.", "background": "The first Hardy-Littlewood conjecture, also known as the prime k-tuples conjecture, generalizes the twin prime conjecture. It states that the asymptotic frequency of any admissible prime constellation can be computed explicitly. The case $k=2$ with $(a_1, a_2) = (0, 2)$ is the twin prime conjecture. Yitang Zhang proved in 2013 that there exists at least one 2-tuple with gap ≤70,000,000 (later improved to 246) that appears infinitely often. The full conjecture remains open and is considered one of the most important unsolved problems in number theory.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The exact qualitative prime k-tuples assertion remains open; it contains the unresolved twin-prime conjecture as the tuple (0,2). The background's asymptotic-frequency language is a stronger conjecture than the displayed statement.\n\n**Verified partial progress.**\n\n- Maynard proved that bounded intervals containing any fixed number of primes occur infinitely often and obtained a positive-proportion result over admissible tuples.\n- Polymath8b optimized the bounded-gap method to the unconditional bound liminf(p_(n+1)-p_n) <= 246.\n\n**Full solution or refutation.**\n\nBounded gaps show that some bounded configurations recur but do not prove infinitude for any prescribed admissible tuple such as (0,2).\n\n**What remains.**\n\nProve infinitude for every fixed admissible set of offsets; the stronger quantitative version additionally requires the singular-series asymptotic.\n\n**Sources checked.**\n\n- James Maynard, Small gaps between primes, Annals of Mathematics 181 (2015), 383-413; arXiv:1311.4600. (primary): https://arxiv.org/abs/1311.4600\n Evidence used: Proves bounded intervals with arbitrarily many primes and a positive-proportion result for admissible tuples, while not establishing every fixed tuple.\n- D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Research in the Mathematical Sciences 1 (2014), Article 12. (primary): https://doi.org/10.1186/s40687-014-0012-7\n Evidence used: Primary source for the optimized unconditional bounded-prime-gap result, including H_1 <= 246.\n\n**Review notes.** Exact statement preserved. Conventionally offsets are distinct; repeated offsets only collapse duplicate linear forms.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set_id": 7 }, { "id": 1518, "problem_number": "HL-B", "title": "Hardy-Littlewood Conjecture B (Second Conjecture)", "statement": "For all integers $x, y \\geq 2$, we have $\\pi(x+y) \\leq \\pi(x) + \\pi(y)$, where $\\pi(n)$ denotes the prime counting function (the number of primes less than or equal to $n$).", "background": "The second Hardy-Littlewood conjecture states the subadditivity of the prime counting function. In 1974, Hensley and Richards proved that Conjecture A and Conjecture B are incompatible with each other - they cannot both be true. Since Conjecture A (the prime k-tuples conjecture) is considered more likely to be true based on computational evidence and its connections to the twin prime conjecture, most number theorists believe Conjecture B is actually false, despite appearing plausible. This represents a fascinating case where intuitive conjectures can contradict each other.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The universal subadditivity inequality for pi remains treated as a conjecture in current primary literature. It is incompatible with the prime k-tuples conjecture and is widely expected to be false, but no accepted unconditional counterexample or proof was verified.\n\n**Verified partial progress.**\n\n- Hensley and Richards proved that Hardy-Littlewood Conjectures A and B cannot both hold.\n- Chahal, Elma, Fellini, Vatwani, and Vo proved the inequality in improved unconditional and RH-conditional ranges but retained the universal assertion as conjectural.\n\n**Full solution or refutation.**\n\nThe exact universal inequality remains unresolved in the literature checked. A 2021 arXiv manuscript claims a full proof, but later work does not treat that claim as an accepted resolution.\n\n**What remains.**\n\nProve the inequality for every x,y >= 2 or exhibit one counterexample, and obtain specialist review of the conflicting 2021 proof claim.\n\n**Sources checked.**\n\n- Douglas Hensley and Ian Richards, Primes in intervals, Acta Arithmetica 25 (1974), 375-391. (primary): https://doi.org/10.4064/aa-25-4-375-391\n Evidence used: Proves the incompatibility of the first and second Hardy-Littlewood conjectures.\n- Bittu Chahal, Ertan Elma, Nic Fellini, Akshaa Vatwani, and Do Nhat Tan Vo, On the Second Hardy-Littlewood Conjecture, arXiv:2503.02766 (2025). (primary): https://arxiv.org/abs/2503.02766\n Evidence used: Treats the general statement as conjectural and proves it only in additional unconditional and RH-conditional ranges.\n- Matt Visser, The second Hardy-Littlewood conjecture is true, arXiv:2101.03283 (2021). (primary): https://arxiv.org/abs/2101.03283\n Evidence used: Conflicting unrefereed full-proof claim; recorded to support the expert-review flag, not as an accepted resolution.\n\n**Review notes.** Exact formulation preserved. Open classification is conservative because a conflicting arXiv proof claim has not been accepted by the later primary literature checked.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set_id": 7 }, { "id": 1519, "problem_number": "HL-F", "title": "Hardy-Littlewood Conjecture F (Primes in Quadratic Polynomials)", "statement": "For a polynomial $f(x) = ax^2 + bx + c$ with $a > 0$, $\\gcd(a,b,c) = 1$, and discriminant $\\Delta = b^2 - 4ac$ not a perfect square, the polynomial takes infinitely many prime values. Furthermore, the number $P(n)$ of primes of the form $f(x) \\leq n$ satisfies an asymptotic formula $P(n) \\sim A \\cdot \\frac{\\sqrt{n}}{\\log n}$ where $A$ depends on $a, b, c$ but not on $n$.", "background": "Conjecture F is a special case of the Bateman-Horn conjecture and concerns primes represented by quadratic polynomials. It predicts not only the infinitude of such primes but also their asymptotic density. The constant A can take values larger or smaller than 1, meaning some polynomials are especially rich in primes while others are especially poor. For example, $4x^2 - 2x + 41$ has $A \\approx 6.6$, making it nearly 7 times as likely to produce primes as random numbers of the same size. This conjecture explains the visible patterns in the Ulam spiral. Despite extensive computational verification, no polynomial has been proven to produce infinitely many primes except linear polynomials (Dirichlet's theorem).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The exact statement is false because it omits the standard local parity obstruction. The polynomial f(x)=x^2+x+2 satisfies every displayed hypothesis but is even for every integer x and assumes the prime value 2 only at x=0 and x=-1.\n\n**Verified partial progress.**\n\n- The corrected standard Conjecture F adds that a+b and c are not both even and remains open, even for x^2+1.\n- Iwaniec proved that n^2+1 is infinitely often a prime or a product of two primes, an almost-prime result short of prime-value infinitude.\n\n**Full solution or refutation.**\n\nFor f(x)=x^2+x+2, one has a>0, gcd(a,b,c)=1, and nonsquare discriminant -7, but x(x+1) is always even; hence all values are even and only two inputs yield the prime 2. This disproves both infinitude and the asserted asymptotic as written.\n\n**What remains.**\n\nCorrect the record by adding the missing local-obstruction condition and precisely defining P(n); the resulting standard Hardy-Littlewood/Bateman-Horn problem remains open.\n\n**Sources checked.**\n\n- Stephan Baier and Liangyi Zhao, On primes represented by quadratic polynomials, Anatomy of Integers, CRM Proceedings & Lecture Notes 46 (2008), 159-174; arXiv:math/0703284. (primary): https://arxiv.org/abs/math/0703284\n Evidence used: States standard Conjecture F with the missing condition that a+b and c are not both even and describes the corrected conjecture as unresolved.\n- Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation 16 (1962), 363-367. (primary): https://doi.org/10.1090/S0025-5718-1962-0148632-7\n Evidence used: Provides the broader prime-value asymptotic heuristic under the required no-fixed-prime-divisor condition.\n- Henryk Iwaniec, Almost-primes represented by quadratic polynomials, Inventiones Mathematicae 47 (1978), 171-188. (primary): https://doi.org/10.1007/BF01578068\n Evidence used: Proves the landmark two-almost-prime result for n^2+1.\n\n**Review notes.** The disproved classification applies only to the exact defective record; it must not be propagated to the standard locally admissible Conjecture F. The meaning of P(n) is also underspecified.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 7 }, { "id": 1855, "problem_number": "GUY-A4", "title": "The Prime Number Race", "statement": "Let $\\pi(n; a, b)$ be the number of primes $p \\le n$ with $p \\equiv a \\pmod b$. For every $a$ and $b$ with $a \\perp b$, are there infinitely many values of $n$ for which $\\pi(n; a, b) > \\pi(n; a_1, b)$ for every $a_1 \\not\\equiv a \\pmod b$?", "background": "Turán was particularly interested in the prime number race. Knapowski & Turán settled special cases, but the general problem is wide open. Chebyshev noted that $\\pi(n; 1, 3) < \\pi(n; 2, 3)$ for small values of $n$, but this inequality is reversed for very large $n$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A4.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The all-contestants prime-race assertion remains open unconditionally; broad race-ordering results are known conditionally on standard hypotheses.\n\n**Verified partial progress.**\n\n- Rubinstein--Sarnak established limiting-distribution/race results under GRH and linear-independence hypotheses.\n- Every pairwise race has known sign-change results, but simultaneous leadership against all reduced residues is stronger.\n\n**Full solution or refutation.**\n\nNo unconditional theorem matching every a,b in the record was verified.\n\n**What remains.**\n\nProve the simultaneous-lead statement or clarify necessary hypotheses.\n\n**Sources checked.**\n\n- M. Rubinstein and P. Sarnak, Chebyshev's bias, Experiment. Math. 3 (1994), 173--197. (primary): https://doi.org/10.1080/10586458.1994.10504289\n Evidence used: Provides the conditional limiting framework for prime number races.\n- Prime number races with three or more competitors, 2010. (primary): https://doi.org/10.1016/j.jnt.2010.01.002\n Evidence used: Studies multi-contestant race orderings under standard hypotheses.\n\n**Review notes.** No source alteration; the record's notation a_1 is informal.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1857, "problem_number": "GUY-A5b", "title": "Erdős $3000 Conjecture on Arithmetic Progressions", "statement": "Let $\\{a_i\\}$ be any infinite sequence of integers for which $\\sum 1/a_i$ is divergent. Does the sequence contain arbitrarily long arithmetic progressions?", "background": "Erdős offered $3000.00 for a proof or disproof of this conjecture. This is a generalization of the arithmetic progressions of primes problem. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A5.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The divergent-reciprocal Erdős arithmetic-progression conjecture remains open for arbitrary length, but the three-term case is solved.\n\n**Verified partial progress.**\n\n- Bloom--Sisask's 2020 breakthrough gives a bound strong enough to force a 3-term progression in every divergent-reciprocal set.\n- Positive-density sets are covered by Szemerédi, and the primes by Green--Tao.\n\n**Full solution or refutation.**\n\nNo proof for all progression lengths was verified.\n\n**What remains.**\n\nSettle each fixed length at least four, or prove the full arbitrary-length statement.\n\n**Sources checked.**\n\n- Erdős conjecture on arithmetic progressions, DeMath status page. (maintained_tracker): https://www.demath.org/problems/erdos-arithmetic-progressions\n Evidence used: Records the solved 3-term case and open k>=4 frontier.\n- Y. Zhao, MIT 18.225 lecture notes, 2023. (authoritative_secondary): https://ocw.mit.edu/courses/18-225-graph-theory-and-additive-combinatorics-fall-2023/mit18_225_f23_lec_full.pdf\n Evidence used: States the current Bloom--Sisask progression-free-set bound and conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1858, "problem_number": "GUY-A6", "title": "Consecutive Primes in Arithmetic Progression", "statement": "Are there arbitrarily long arithmetic progressions of consecutive primes? That is, for any positive integer $k$, do there exist $k$ consecutive primes $p_n, p_{n+1}, \\ldots, p_{n+k-1}$ in arithmetic progression?", "background": "Known examples include the 4-term sequences 251, 257, 263, 269 and 1741, 1747, 1753, 1759. Dubner, Forbes, Lygeros, Mizony & Zimmermann found 10 consecutive primes in arithmetic progression in 1998. It is not known if there are infinitely many sets of three consecutive primes in arithmetic progression. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A6.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Arbitrarily long arithmetic progressions of consecutive primes remain unproved.\n\n**Verified partial progress.**\n\n- Green--Tao proves arbitrarily long arithmetic progressions of primes without the consecutive-prime condition.\n- Consecutive-prime records and bounded-pattern theorems do not settle exact arithmetic progression chains of arbitrary length.\n\n**Full solution or refutation.**\n\nNo theorem removing the consecutive-prime restriction was verified.\n\n**What remains.**\n\nConstruct arbitrarily long prime APs whose terms are consecutive in the global prime sequence.\n\n**Sources checked.**\n\n- B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. 167 (2008), 481--547; arXiv:math/0404188. (primary): https://arxiv.org/abs/math/0404188\n Evidence used: Proves the related but weaker nonconsecutive-prime statement.\n- Cunningham chain overview. (authoritative_secondary): https://en.wikipedia.org/wiki/Cunningham_chain\n Evidence used: Notes no general result on arbitrarily long constrained prime chains.\n\n**Review notes.** No source alteration; separate consecutive condition is essential.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1859, "problem_number": "GUY-A7a", "title": "Infinitude of Sophie Germain Primes", "statement": "Are there infinitely many Sophie Germain primes? A prime $p$ is called a Sophie Germain prime if $2p + 1$ is also prime.", "background": "It is believed, but not known, that there are infinitely many Sophie Germain primes. Dubner has found many large examples. The largest known Sophie Germain prime has over 24000 decimal digits. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A7.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The infinitude of Sophie Germain primes remains open.\n\n**Verified partial progress.**\n\n- Sieve methods yield upper bounds of the expected order of magnitude, and computations find very large examples.\n\n**Full solution or refutation.**\n\nNo infinitude theorem was verified.\n\n**What remains.**\n\nProve infinitely many primes p with 2p+1 prime.\n\n**Sources checked.**\n\n- Universitat Politècnica de Catalunya, Sophie Germain primes information page (2025). (authoritative_secondary): https://fme.upc.edu/ca/la-facultat/activitats-fme-personalitats-del-curs/personalitat-del-curs/2025-2026-germain/nombresprimers\n Evidence used: Explicitly states that infinitude and density remain unknown.\n- PrimePages, Sophie Germain primes. (maintained_tracker): https://t5k.org/top20/page.php?id=2\n Evidence used: Records the open infinitude question and sieve upper bounds.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1860, "problem_number": "GUY-A7b", "title": "Shanks Chains of Length 7", "statement": "Are there any Shanks chains of length 7 with $p_{i+1} = 4p_i^2 - 17$?", "background": "Shanks chains are quadratic chains of primes. The recurrence $p_{i+1} = 4p_i^2 - 17$ yields a 4-chain if $p_1 = 3$ and a 5-chain if $p_1 = 303593$, but it can be seen (mod 59) that no such chain has length 17. It seems certain that such chains cannot be of arbitrary length. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A7.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No verified length-7 Shanks (4,17) chain was located, and no impossibility theorem was found.\n\n**Verified partial progress.**\n\n- The recurrence is a recognized Shanks-chain problem, but its specialized finite-existence status requires dedicated computational literature review.\n\n**Full solution or refutation.**\n\nNo resolution of length 7 was verified in this pass.\n\n**What remains.**\n\nFind a length-7 chain or prove none exists.\n\n**Sources checked.**\n\n- Shanks Chain, Wolfram MathWorld. (authoritative_secondary): https://mathworld.wolfram.com/ShanksChain.html\n Evidence used: Documents the (4,17) recurrence and the named length-7 problem.\n\n**Review notes.** No source alteration; status confidence intentionally low.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1861, "problem_number": "GUY-A8a", "title": "Erdős $5000 Problem on Prime Gaps", "statement": "Is it true that for infinitely many $n$, $d_n = p_{n+1} - p_n > c \\ln n \\ln \\ln n \\ln \\ln \\ln \\ln n / (\\ln \\ln \\ln n)^2$ for arbitrarily large constant $c$?", "background": "Erdős offers $5,000 for a proof or disproof that the constant $c$ can be taken arbitrarily large. Rankin showed this holds for $c = e^\\gamma$, and Pintz improved it to $c = 2e^\\gamma > 3.562$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A8.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The requested arbitrary-constant lower bound for large consecutive prime gaps was proved in 2014.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nFord--Green--Konyagin--Tao prove the displayed Rankin-scale lower bound with a multiplier f(X) tending to infinity; this implies every fixed c occurs infinitely often. A later joint result improves the denominator by one logarithm.\n\n**What remains.**\n\nNo work remains for the statement as written.\n\n**Sources checked.**\n\n- K. Ford, B. Green, S. Konyagin and T. Tao, Large gaps between consecutive prime numbers, Ann. of Math. 183 (2016), 935--974; arXiv:1408.4505. (primary): https://arxiv.org/abs/1408.4505\n Evidence used: Abstract explicitly says it answers Erdős's question with a multiplier tending to infinity.\n- K. Ford, B. Green, S. Konyagin, J. Maynard and T. Tao, Long gaps between primes, J. Amer. Math. Soc. 31 (2018), 65--105; arXiv:1412.5029. (primary): https://arxiv.org/abs/1412.5029\n Evidence used: Gives a stronger subsequent lower bound.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1862, "problem_number": "GUY-A8b", "title": "Twin Prime Conjecture", "statement": "Are there infinitely many twin primes? That is, are there infinitely many primes $p$ such that $p + 2$ is also prime?", "background": "A very famous conjecture. Hardy and Littlewood conjectured that $P_2(n)$, the number of twin prime pairs less than $n$, is asymptotically $2cn/(\\ln n)^2$ where $2c \\approx 1.32032$. Brun showed that the sum of the reciprocals of twin primes is convergent. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A8.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The twin prime conjecture remains open: no accepted proof is known that the fixed gap 2 occurs infinitely often.\n\n**Verified partial progress.**\n\n- D. H. J. Polymath proved H_1 <= 246 unconditionally, so some even prime gap at most 246 occurs infinitely often.\n- Under the generalized Elliott--Halberstam conjecture, D. H. J. Polymath proved H_1 <= 6; this still does not force gap 2.\n- Lott and Ponagandla proved polynomial configurations among prime pairs with some common bounded gap b <= 246, but their theorem does not prescribe b = 2.\n\n**Full solution or refutation.**\n\nNo verified solution or counterexample was found. Bounded-gap theorems prove recurrence of an unspecified bounded even gap, not the twin-prime gap specifically.\n\n**What remains.**\n\nProve that infinitely many primes p have p+2 prime, or disprove that assertion; neither H_1 <= 246 nor the conditional bound H_1 <= 6 settles it.\n\n**Sources checked.**\n\n- D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Research in the Mathematical Sciences 1 (2014), article 12; arXiv:1407.4897. (primary): https://arxiv.org/abs/1407.4897\n Evidence used: The abstract gives H_1 <= 246 unconditionally, H_1 <= 6 under generalized Elliott--Halberstam, and the admissible-triple/parity-barrier context.\n- Andrew Lott and Nagendar Reddy Ponagandla, Polynomial progressions in the generalized twin primes, arXiv:2505.17375v2 (2026 revision). (primary): https://arxiv.org/abs/2505.17375\n Evidence used: The paper explicitly starts from some unspecified recurring gap b <= 246 and proves structured configurations for that generalized-prime-pair setting, not for b = 2.\n\n**Review notes.** The source statement was preserved. Unreviewed and self-published online proof claims were excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1863, "problem_number": "GUY-A9", "title": "General Patterns of Consecutive Primes", "statement": "For any given pattern of primes with no congruence obstructions, are there infinitely many sets of consecutive primes with this pattern?", "background": "This conjecture is more general than Chowla's conjecture. It seems likely that there are infinitely many triples of primes $\\{6k - 1, 6k + 1, 6k + 5\\}$ and $\\{6k + 1, 6k + 5, 6k + 7\\}$. Hensley & Richards showed this is incompatible with the conjecture $\\pi(x + y) \\le \\pi(x) + \\pi(y)$ for all integers $x, y \\ge 2$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A9.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Under the natural admissible-offset reading, the universal claim for every prescribed consecutive-prime pattern remains open and contains twin primes as a special case, but substantial fixed-pattern and structural special cases are known.\n\n**Verified partial progress.**\n\n- Maynard proved bounded prime clusters of arbitrary fixed size and a positive-proportion approximation to the prime k-tuples conjecture, without proving every prescribed admissible tuple.\n- Banks, Freiberg, and Turnage-Butterbaugh proved that suitable admissible tuples contain at least m consecutive primes infinitely often and realized increasing, decreasing, and divisibility patterns among successive gaps.\n- Pintz proved that for every m there exist bounded fixed m-patterns whose translates consist of consecutive primes infinitely often; the translation parameters even contain arbitrarily long finite arithmetic progressions.\n- Under generalized Elliott--Halberstam, D. H. J. Polymath proved that every admissible triple has infinitely many translates with at least two prime components, not necessarily all three.\n\n**Full solution or refutation.**\n\nNo universal all-pattern theorem was found. Existing results select some bounded patterns or enforce broad structural properties; they do not prove every given admissible exact offset pattern, including {0,2}, {0,2,6}, or {0,4,6}.\n\n**What remains.**\n\nRecover Guy's exact definition of pattern and consecutiveness, then prove or refute the universal prescribed-pattern assertion. On the standard exact-offset reading, even the special pattern {0,2} remains open.\n\n**Sources checked.**\n\n- James Maynard, Small gaps between primes, Annals of Mathematics 181 (2015), 383--413, DOI 10.4007/annals.2015.181.1.7. (primary): https://annals.math.princeton.edu/2015/181-1/p07\n Evidence used: Defines admissible tuples and the prime k-tuples conjecture and proves positive-proportion and bounded-cluster approximations rather than the every-prescribed-pattern statement.\n- William D. Banks, Tristan Freiberg, and Caroline L. Turnage-Butterbaugh, Consecutive primes in tuples, Acta Arithmetica 167 (2015), 261--266, DOI 10.4064/aa167-3-4; arXiv:1311.7003. (primary): https://arxiv.org/abs/1311.7003\n Evidence used: Proves infinitely recurring consecutive primes inside suitable tuples and several structured consecutive-gap patterns, but not arbitrary prescribed exact patterns.\n- Janos Pintz, Patterns of primes in arithmetic progressions, arXiv:1509.01564v2 (2015). (primary): https://arxiv.org/abs/1509.01564\n Evidence used: Proves existence for every m of bounded m-patterns of consecutive primes whose translation set contains arbitrarily long finite arithmetic progressions, plus a positive-proportion result.\n- D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Research in the Mathematical Sciences 1 (2014), article 12; arXiv:1407.4897. (primary): https://arxiv.org/abs/1407.4897\n Evidence used: Under generalized Elliott--Halberstam, proves an any-admissible-triple result with at least two prime entries, illustrating the gap to making every entry prime.\n- Douglas Hensley and Ian Richards, Primes in intervals, Acta Arithmetica 25 (1974), 375--391, DOI 10.4064/aa-25-4-375-391. (primary): https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/25/4/100146/primes-in-intervals\n Evidence used: The official journal record verifies the bibliographic identity of the historical paper cited in the dataset background; its incompatibility result was not needed for the present status classification.\n\n**Review notes.** The source statement was preserved. Its terms pattern, congruence obstructions, and consecutive are undefined, and the phrase more general than Chowla's conjecture is historically ambiguous; the status conclusion uses the natural admissible exact-offset reading without rewriting the dataset.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1864, "problem_number": "GUY-A10", "title": "Gilbreath's Conjecture", "statement": "Define $d_n^k$ by $d_n^1 = p_{n+1} - p_n$ and $d_n^{k+1} = |d_{n+1}^k - d_n^k|$, the successive absolute differences of the sequence of primes. Is it true that $d_1^k = 1$ for all $k$?", "background": "Gilbreath conjectured this (and Proth claimed to have proved it long before). This was verified for $k < 63419$ by Killgrove & Ralston. Odlyzko checked it for primes up to $\\pi(10^{13})$. Croft and others suggest it has nothing to do with primes as such, but will be true for any sequence consisting of 2 and odd numbers which doesn't increase too fast. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A10.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Gilbreath's conjecture for the primes remains open; rigorous random-model theorems and a 2026 deterministic inverse theorem now isolate specific obstructions without excluding them for primes.\n\n**Verified partial progress.**\n\n- Chase proved a precise random analogue of Gilbreath's conjecture.\n- Chase, Hunter, and Tao prove the normalized Cramér-model analogue and, under a Cramér-type prime-gap bound, reduce failure to long zero blocks or very long shallow {0,d}-valued blocks.\n\n**Full solution or refutation.**\n\nNo theorem proving d_1^k=1 for every k in the actual prime sequence was verified.\n\n**What remains.**\n\nExclude the inverse theorem's exceptional block structures for the prime-difference array, or otherwise prove the full left-edge assertion.\n\n**Sources checked.**\n\n- Zachary Chase, A random analogue of Gilbreath's conjecture, Mathematische Annalen 388 (2024), 1655-1676. (primary): https://doi.org/10.1007/s00208-023-02579-w\n Evidence used: Proves a random analogue while identifying the original prime assertion as the motivating conjecture.\n- Zachary Chase, Zach Hunter, and Terence Tao, Gilbreath's conjecture: a Cramér random model and a deterministic analysis, arXiv:2607.08712 (2026). (primary): https://arxiv.org/abs/2607.08712\n Evidence used: Proves the normalized Cramér-model result and a conditional deterministic inverse theorem; it does not claim the prime conjecture.\n\n**Review notes.** The input's k<63419 verification figure is stale even relative to the Odlyzko computation mentioned in its own background.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1865, "problem_number": "GUY-A11", "title": "Erdős $100 Problem on Increasing and Decreasing Gaps", "statement": "Does there exist an $n_0$ such that for every $i$ and $n > n_0$ we have $d_{n+2i} > d_{n+2i+1}$ and $d_{n+2i+1} < d_{n+2i+2}$, where $d_n = p_{n+1} - p_n$?", "background": "Erdős & Turán showed that the values of $n$ for which $d_n > d_{n+1}$ have positive lower density, but it is not known if there are infinitely many increasing or decreasing sets of three consecutive values of $d_n$. Erdős offers $100.00 for a proof that such an $n_0$ does not exist. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A11.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The proposed eventual strict alternation of consecutive prime gaps is false: arbitrarily long strictly increasing and arbitrarily long strictly decreasing runs of consecutive prime gaps exist.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nBanks, Freiberg, and Turnage-Butterbaugh answer the Erdős-Turán question by producing monotone runs of every prescribed length; a run of three gaps already contradicts eventual alternation.\n\n**What remains.**\n\nThe yes/no question has no remaining gap under the natural eventual-alternation reading; quantitative frequency questions are separate.\n\n**Sources checked.**\n\n- William D. Banks, Tristan Freiberg, and Caroline L. Turnage-Butterbaugh, Consecutive primes in tuples, Acta Arithmetica 167 (2015), 261-266. (primary): https://doi.org/10.4064/aa167-3-4\n Evidence used: Proves arbitrarily long strings of consecutive primes with strictly increasing gaps and likewise with strictly decreasing gaps.\n\n**Review notes.** The phrase 'for every i and n>n0' is ill-scoped, but every natural assertion that the tail alternates strictly is refuted by the cited theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1866, "problem_number": "GUY-A13", "title": "Erdős Conjecture on Carmichael Numbers", "statement": "Let $C(x)$ be the number of Carmichael numbers less than $x$. Does $(\\ln C(x))/\\ln x$ tend to 1 as $x$ tends to infinity?", "background": "Erdős conjectured this behavior for the count of Carmichael numbers. Alford, Granville & Pomerance showed there are infinitely many Carmichael numbers, in fact more than $x^\\beta$ of them less than $x$ for $\\beta > 0.290306$. Pomerance, Selfridge & Wagstaff give a heuristic argument supporting Erdős' conjecture. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A13.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjecture log C(x)/log x→1 remains open; unconditional lower bounds have passed exponent 1/3, and a near-density-one result is known under strong least-prime-in-progressions hypotheses.\n\n**Verified partial progress.**\n\n- Harman proved C(x)>x^0.33336704 for all sufficiently large x.\n- Wright conditionally proves C(X)≫X^(1-R), with R tending to zero, under sufficiently strong assumptions on the least prime in an arithmetic progression.\n\n**Full solution or refutation.**\n\nThe unconditional exponent-one limit has not been established.\n\n**What remains.**\n\nRaise the unconditional lower exponent from just above 1/3 to 1-o(1), matching the logarithmic limit asserted by Erdős.\n\n**Sources checked.**\n\n- Glyn Harman, On the number of Carmichael numbers up to x, Bulletin of the London Mathematical Society 37 (2005), 641-650. (primary): https://doi.org/10.1112/S0024609305004686\n Evidence used: Establishes the unconditional exponent 0.33336704 lower bound.\n- Thomas Wright, A conditional density for Carmichael numbers, Bulletin of the Australian Mathematical Society 101 (2020), 379-388. (primary): https://doi.org/10.1017/S000497271900145X\n Evidence used: Proves a conditional X^(1-o(1)) lower bound and compares it with the conjectured density.\n- Erdős Problems, Problem 1057, Carmichael-number counting function. (maintained_tracker): https://www.erdosproblems.com/1057\n Evidence used: Maintains the open conjecture and current upper/lower-bound references.\n\n**Review notes.** The older beta>0.290306 figure in the imported background is superseded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set_id": 9 }, { "id": 1867, "problem_number": "GUY-A14a", "title": "Pomerance's Questions on Good Primes", "statement": "Call prime $p_n$ good if $p_n^2 > p_{n-i}p_{n+i}$ for all $i$, $1 \\le i \\le n-1$. Is it true that the set of $n$ for which $p_n$ is good has density 0? Are there infinitely many $n$ with $p_n p_{n+1} > p_{n-i} p_{n+1+i}$ for all $i$, $1 \\le i \\le n-1$?", "background": "Erdős and Straus introduced the concept of good primes. Examples include 5, 11, 17, and 29. Pomerance used the prime number graph to show there are infinitely many good primes and posed several related questions. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A14.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Infinitely many good primes are known, and the log-convex subset has relative density zero, but density zero for all good primes and the separate adjacent-pair infinitude question remain unresolved.\n\n**Verified partial progress.**\n\n- Pomerance proved that infinitely many good primes exist using log-convex primes.\n- McNew proves that log-convex primes have count at most x/log^(4/3-o(1))x and hence relative density zero among primes.\n- McNew's computation tabulates good primes through 10^9 and log-convex primes through 10^13, while retaining G(x)=o(pi(x)) as an explicit question.\n\n**Full solution or refutation.**\n\nThe density-zero theorem applies to a subset, not to every good prime; no source settling the second product inequality was verified.\n\n**What remains.**\n\nProve G(x)=o(pi(x)) for all good primes and prove or refute infinitely many indices satisfying the adjacent-pair product inequalities.\n\n**Sources checked.**\n\n- Nathan McNew, The Convex Hull of the Prime Number Graph, in Irregularities in the Distribution of Prime Numbers, Springer (2018), 125-141. (primary): https://www.nathanmcnew.com/Convex.pdf\n Evidence used: Proves density zero for log-convex primes, records infinitude of good primes, and explicitly asks whether all good primes have relative density zero.\n- Carl Pomerance, The prime number graph, Mathematics of Computation 33 (1979), 399-408. (primary): https://doi.org/10.1090/S0025-5718-1979-0514821-1\n Evidence used: Introduces the convex-hull method used to prove infinitude of the relevant extremal primes.\n\n**Review notes.** The input combines two distinct questions. Progress on the first does not settle the adjacent-pair condition in the second.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1868, "problem_number": "GUY-A15", "title": "Congruent Products of Consecutive Numbers", "statement": "What is the least prime $p$ such that there are integers $a, k_1, k_2, k_3$ with $\\prod_{i=1}^{k_1} (a+i) \\equiv \\prod_{i=1}^{k_2} (a+k_1+i) \\equiv \\prod_{i=1}^{k_3} (a+k_1+k_2+i) \\equiv 1 \\pmod{p}$?", "background": "Erdős observed that $3 \\cdot 4 \\equiv 5 \\cdot 6 \\cdot 7 \\equiv 1 \\pmod{11}$ and suggested that such primes $p$ exist for any number of congruent products. Narkiewicz and others found examples for larger numbers of terms. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A15.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The least prime is 17.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nWith a=1 and (k1,k2,k3)=(4,6,4), the three consecutive blocks 2 through 5, 6 through 11, and 12 through 15 each have product 1 modulo 17; OEIS A060427 records 17 as the smallest prime for three such blocks.\n\n**What remains.**\n\nNothing remains for the stated least-prime question; analogous least primes for larger numbers of blocks form a separate sequence.\n\n**Sources checked.**\n\n- OEIS Foundation, A060427, Smallest prime p such that there are n strings of consecutive integers all having products = 1 mod p, updated 2026-07-27. (authoritative_secondary): https://oeis.org/A060427\n Evidence used: Defines the least-prime sequence, gives a(3)=17, and supplies the three explicit product blocks modulo 17.\n\n**Review notes.** The witness matches the input's indexing exactly after taking a=1.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set_id": 9 }, { "id": 1869, "problem_number": "GUY-A16", "title": "Walking to Infinity on Gaussian Primes", "statement": "Can one walk from the origin to infinity using Gaussian primes as stepping stones and taking steps of bounded length?", "background": "Motzkin and Gordon asked this question about Gaussian primes (primes in the ring of complex numbers $a+bi$ where $a, b$ are integers). Presumably not. Jordan & Rabung showed that steps of length at least 4 are necessary. Gethner, Wagon & Wick produced a moat of width $\\sqrt{26}$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A16.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The bounded-step Gaussian moat problem remains open; current analytic work gives paths with sub-square-root but still growing step size, while two claimed negative solutions were withdrawn for explicit errors.\n\n**Verified partial progress.**\n\n- Maynard and Merikoski report that steps O(|z|^(1/2-1/100)) suffice unconditionally.\n- Under GRH, their reported exponent improves to 1/3+o(1).\n- Computational work has exhibited finite moats, but no theorem gives moats of every fixed width.\n\n**Full solution or refutation.**\n\nNo bounded absolute step size is known to suffice, and no proof that every bounded step size is eventually blocked is valid.\n\n**What remains.**\n\nEither drive the path exponent to zero with a uniform constant or prove arbitrarily wide Gaussian-prime moats.\n\n**Sources checked.**\n\n- Jori Merikoski (joint work with James Maynard), On the Gaussian moat problem, Oberwolfach Reports 50/2022, 2940-2942. (primary): https://ems.press/content/serial-article-files/46986\n Evidence used: Reports the unconditional exponent 1/2-1/100 and conditional exponent 1/3+o(1), explicitly treating the constant-step question as the Gaussian moat problem.\n- Johann Christian Stumpenhusen, On the Gaussian Moat Problem, arXiv:2401.08441, withdrawn (2024). (primary): https://arxiv.org/abs/2401.08441\n Evidence used: The author withdrew the claimed solution because the moat width was computed incorrectly.\n- Madhuparna Das, A Note on The Gaussian Moat Problem, arXiv:1908.10392, withdrawn (2024). (primary): https://arxiv.org/abs/1908.10392\n Evidence used: The withdrawn record states that the proposed paths failed to cover all Gaussian primes and included nonprime Gaussian integers.\n\n**Review notes.** Search-result abstracts for the two withdrawn arXiv claims misleadingly say the problem was solved; the current arXiv records explicitly identify the errors.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1870, "problem_number": "GUY-A17", "title": "Giuga's Conjecture on Prime Characterization", "statement": "Is it true that if $n$ divides $1^{n-1} + 2^{n-1} + \\dots + (n-1)^{n-1} + 1$, then $n$ is prime?", "background": "Sierpiński observed that if $n$ is prime, then $n$ divides this sum. Giuga conjectured the converse and verified it for $n \\le 10^{1000}$. A counterexample would be a Carmichael number with additional properties. An equivalent conjecture is $n B_{n-1} \\equiv -1 \\pmod{n}$ where $B_k$ are Bernoulli numbers. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A17.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Giuga's primality conjecture remains open; every counterexample must be both a Carmichael number and a Giuga number and must have at least 19,908 decimal digits.\n\n**Verified partial progress.**\n\n- A composite satisfies the power-sum congruence exactly when it is simultaneously Carmichael and Giuga.\n- Borwein, Maitland, and Skerritt prove that any counterexample has at least 4,771 prime factors and exceeds 10^19907.\n\n**Full solution or refutation.**\n\nNo composite counterexample or proof of impossibility was verified.\n\n**What remains.**\n\nRule out all simultaneous Carmichael-Giuga numbers, or construct one beyond the current structural lower bound.\n\n**Sources checked.**\n\n- Jonathan Borwein, Christopher Maitland, and Matthew Skerritt, Computation of an Improved Lower Bound to Giuga's Primality Conjecture, Integers 13 (2013), A67. (primary): https://carmamaths.org/jon/giuga2013.pdf\n Evidence used: Proves that a counterexample must have at least 4,771 prime factors and at least 19,908 decimal digits.\n- Peter Borwein, Jonathan Borwein, and Roland Girgensohn, Giuga's Conjecture on Primality, American Mathematical Monthly 103 (1996), 40-50. (primary): https://www.math.stonybrook.edu/~moira/mat331-spr10/papers/1996%20BorweinGiuga%27s%20Conjecture%20on%20Primality.pdf\n Evidence used: Establishes and explains the equivalence between a composite counterexample and a number that is both Carmichael and Giuga.\n\n**Review notes.** The terminal +1 in the input is the standard congruence when read as part of the dividend. The 19,908-digit result is structural, not exhaustive testing of every smaller integer.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1871, "problem_number": "GUY-A18", "title": "Erdős-Selfridge Classification: Infinitely Many Primes in Each Class", "statement": "In the Erdős-Selfridge classification of primes, are there infinitely many primes in each class? Prime $p$ is in class 1 if the only prime divisors of $p+1$ are 2 or 3; and $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\\le r-1$, with equality for at least one prime factor.", "background": "The first few classes are: Class 1: 2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, ...; Class 2: 13, 19, 29, 41, 43, 59, 61, 67, 79, 83, 89, 97, 101, ...; Class 3: 37, 103, 113, 151, 157, 163, 173, 181, 193, 227, 233, ... From Richard Guy's \"Unsolved Problems in Number Theory\", Section A18.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that every Erdős--Selfridge prime class is infinite was located in the literature checked.\n\n**Verified partial progress.**\n\n- The recursive plus/minus classification is well documented, but a current primary-source resolution of the infinitude question was not found.\n\n**Full solution or refutation.**\n\nNo full classification theorem was verified.\n\n**What remains.**\n\nProve infinitude for each recursively defined class or identify an obstruction.\n\n**Sources checked.**\n\n- Erdős--Selfridge classification of primes, OEIS Wiki. (authoritative_secondary): https://oeis.org/wiki/Erd%C5%91s%E2%80%93Selfridge_classification_of_primes\n Evidence used: Documents the recursive class definitions used by the record.\n\n**Review notes.** No source alteration; specialized status needs further expert bibliography search.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1872, "problem_number": "GUY-A19a", "title": "Erdős Conjecture on $n - 2^k$ Prime", "statement": "Are 4, 7, 15, 21, 45, 75, and 105 the only values of $n$ for which $n - 2^k$ is prime for all $k$ such that $2 \\le 2^k < n$?", "background": "Erdős conjectures that these are the only such values. He also conjectures that for infinitely many $n$, all the integers $n - 2^k, 1 \\le 2^k < n$ are squarefree. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A19.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No resolution of the finite exceptional-list conjecture was located.\n\n**Verified partial progress.**\n\n- A modern compendium still lists the same Erdős question among open problems.\n\n**Full solution or refutation.**\n\nNo proof that the displayed list is complete, nor an additional example, was verified.\n\n**What remains.**\n\nEstablish completeness of the list or produce another n.\n\n**Sources checked.**\n\n- P. Moree, Artin's primitive root conjecture -- a survey, MPIM preprint 2012-53. (authoritative_secondary): https://archive.mpim-bonn.mpg.de/1236/1/preprint_2012_53.pdf\n Evidence used: Lists the exact Erdős question as an open problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1873, "problem_number": "GUY-A19b", "title": "Cohen-Selfridge Problem on $\\pm p^a \\pm 2^b$", "statement": "What is the least positive odd number not of the form $\\pm p^a \\pm 2^b$, where $p$ is an odd prime?", "background": "Cohen & Selfridge observed that the number is greater than $2^{18}$. This is related to the representation of odd numbers as sums or differences of prime powers and powers of 2. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A19.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** There are infinitely many odd nonrepresentable numbers in related prime-power-plus/minus-power-of-two forms, but the least number for the record's exact form was not verified.\n\n**Verified partial progress.**\n\n- Cohen--Selfridge construct an arithmetic progression of odd numbers that are neither sums nor differences of a power of two and a prime.\n- Later covering-congruence work gives broad nonrepresentation constructions.\n\n**Full solution or refutation.**\n\nThe requested least positive odd integer remains unverified.\n\n**What remains.**\n\nDetermine the first nonrepresentable odd number under the exact exponent and sign conventions in the record.\n\n**Sources checked.**\n\n- F. Cohen and J. L. Selfridge, Not Every Number is the Sum or Difference of Two Prime Powers, Math. Comp. 29 (1975), 79--81. (primary): https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376583-0/S0025-5718-1975-0376583-0.pdf\n Evidence used: Constructs an arithmetic progression of related nonrepresentable odd numbers.\n- Z.-W. Sun, On integers not of the form plus/minus p^a plus/minus q^b, J. Number Theory 128 (2008), 997--1002. (primary): https://doi.org/10.1016/j.jnt.2007.06.003\n Evidence used: Provides stronger covering-congruence nonrepresentation results.\n\n**Review notes.** No source alteration; exact formulation needs a dedicated computational/status audit.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set_id": 9 }, { "id": 1874, "problem_number": "GUY-A20", "title": "Density of Symmetric Primes", "statement": "Given pairs of odd primes $p, q$, define $S(q,p)$ as the number of lattice points $(m, n)$ in the rectangle $0 < m < p/2$, $0 < n < q/2$ below the diagonal. A pair is symmetric if $S(p, q) = S(q,p)$. Is the number of symmetric primes less than $x$ equal to $x/(\\ln x)^{\\sigma+o(1)}$, where $\\sigma = 2 - (1+\\ln \\ln 2)/\\ln 2 \\approx 1.08607$?", "background": "Fletcher, Lindgren & Pomerance showed that a pair is symmetric just if $|p - q| = (p - 1, q - 1)$, and that the number of symmetric primes less than $x$ is at most $x/(\\ln x)^{1.027}$. They conjectured the more precise asymptotic. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A20.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The stated precise density is unproved, but infinitely many symmetric prime pairs are now known.\n\n**Verified partial progress.**\n\n- Banks--Pollack--Pomerance prove infinitely many symmetric pairs and, for every m, a block of m consecutive primes any two of which form a symmetric pair.\n- They also improve the prior upper bound for the counting function.\n\n**Full solution or refutation.**\n\nNo asymptotic with the stated exponent sigma was verified.\n\n**What remains.**\n\nProve the proposed density asymptotic (or correct its exponent).\n\n**Sources checked.**\n\n- W. Banks, P. Pollack and C. Pomerance, Symmetric primes revisited, arXiv:1908.06161 (2019). (primary): https://arxiv.org/abs/1908.06161\n Evidence used: States infinitude of symmetric pairs and arbitrarily long consecutive-prime blocks.\n- D. Fletcher, B. Lindgren and C. Pomerance, Symmetric and Asymmetric Primes, J. Number Theory 58 (1996), 89--99. (primary): https://math.dartmouth.edu/~carlp/PDF/paper106.pdf\n Evidence used: Introduces the symmetric-prime counting problem and original density heuristic.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1875, "problem_number": "GUY-A12a", "title": "Square Pseudoprimes", "statement": "Are there any square pseudoprimes (base 2) other than multiples of $1194649 = 1093^2$ or $12327121 = 3511^2$?", "background": "Pinch observed that there are 54 non-squarefree pseudoprimes up to $10^{13}$, all multiples of $1093^2$ or $3511^2$. The question asks if there are other perfect squares that are pseudoprimes to base 2. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A12.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The record conflates perfect-square base-2 pseudoprimes with all non-squarefree base-2 pseudoprimes; neither natural scope has a verified complete classification beyond the known factors 1093^2 and 3511^2.\n\n**Verified partial progress.**\n\n- For a prime p, p^2 is a base-2 pseudoprime exactly when p is a base-2 Wieferich prime.\n- Dorais and Klyve proved that 1093 and 3511 are the only base-2 Wieferich primes below 6.7×10^15.\n\n**Full solution or refutation.**\n\nNo additional prime-square example is known in the cited certified range, but this is not a global nonexistence proof; the broader non-squarefree reading is a different question.\n\n**What remains.**\n\nRepair the scope to either perfect squares or arbitrary non-squarefree pseudoprimes, then find a new example or prove the corresponding global classification.\n\n**Sources checked.**\n\n- François G. Dorais and Dominic Klyve, A Wieferich prime search up to 6.7×10^15, Journal of Integer Sequences 14 (2011), Article 11.9.2. (primary): https://cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.html\n Evidence used: Certified search finding no base-2 Wieferich primes other than 1093 and 3511 below 6.7×10^15.\n- Carl Pomerance, J. L. Selfridge, and Samuel S. Wagstaff Jr., The pseudoprimes to 25×10^9, Mathematics of Computation 35 (1980), 1003-1026. (primary): https://doi.org/10.1090/S0025-5718-1980-0572872-7\n Evidence used: Classical primary study of base-2 pseudoprimes and their repeated prime factors, which underlies the distinct non-squarefree formulation.\n\n**Review notes.** The exact wording says 'multiples of' the two squares, whereas the background's final sentence asks about perfect squares. This defect was not silently repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1876, "problem_number": "GUY-A12b", "title": "Selfridge-Wagstaff-Pomerance Prize Problem", "statement": "Does there exist a composite number $n \\equiv 3$ or $7 \\pmod{10}$ which divides both $2^n - 2$ and the Fibonacci number $u_{n+1}$?", "background": "Selfridge, Wagstaff & Pomerance offer $500 + $100 + $20 = $620 for finding such a composite $n$, or $20 + $100 + $500 = $620 for a proof that no such $n$ exists. This combines pseudoprime properties with Fibonacci divisibility. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A12.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No composite satisfying the PSW Fibonacci challenge is known, but strong finite and factor-count exclusions have been proved.\n\n**Verified partial progress.**\n\n- Shallue and Webster exclude challenge pseudoprimes with two or three prime factors below 2^80.\n- Greene, Lim, Mashalkar, and Schaefer construct large families of Fibonacci pseudoprimes, but their attempted search for a simultaneous base-2 pseudoprime is unsuccessful.\n\n**Full solution or refutation.**\n\nNeither a composite counterexample nor a proof that none exists was verified.\n\n**What remains.**\n\nHandle candidates with more prime factors and unbounded size, or prove a structural incompatibility between the Fermat and Fibonacci congruences.\n\n**Sources checked.**\n\n- Andrew Shallue and Jonathan Webster, Fast tabulation of challenge pseudoprimes, Open Book Series 2 (2019), 411-428. (primary): https://doi.org/10.2140/obs.2019.2.411\n Evidence used: Defines the exact PSW challenge and proves that no two- or three-prime-factor candidate occurs below 2^80.\n- John Greene, Junhyun Lim, Shaunak Mashalkar, and Edward F. Schaefer, Using Fibonacci factors to create Fibonacci pseudoprimes, Fibonacci Quarterly 60 (2022), 320-324. (primary): https://www.fq.math.ca/Papers/60-4/schaefer09092021.pdf\n Evidence used: Constructs Fibonacci pseudoprimes and explicitly reports an unsuccessful attempt to produce a simultaneous base-2 pseudoprime.\n\n**Review notes.** This is the weaker Fibonacci PSW challenge, not the standard strong Baillie-PSW test; conflating them changes the problem and prize history.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1877, "problem_number": "GUY-A12c", "title": "Even Fibonacci Pseudoprimes", "statement": "Does there exist an even Fibonacci pseudoprime?", "background": "A Fibonacci pseudoprime of the $m$-th kind is an odd composite integer $n$ with $V_n(m, -1) \\equiv m \\pmod n$ where $V_n$ is the Lucas sequence. Somer showed that if an even Fibonacci pseudoprime exists, it must be greater than $28 \\times 10^{12}$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A12.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The answer depends on which Fibonacci-pseudoprime convention is intended: even pseudoprimes of the first kind do not exist, while parameter-varying generalized families have even examples.\n\n**Verified partial progress.**\n\n- Di Porto proved that there are no even Fibonacci pseudoprimes of the first kind.\n- The same primary source records even pseudoprimes for other parameter choices, so the unqualified phrase is not convention-independent.\n\n**Full solution or refutation.**\n\nThe historically standard first-kind reading is solved negatively, but the exact imported record is internally inconsistent because its definition requires n to be odd before asking whether n can be even.\n\n**What remains.**\n\nSpecify the Lucas parameters and remove the oddness restriction from the definition; then the applicable theorem or counterexample can be selected unambiguously.\n\n**Sources checked.**\n\n- Adina Di Porto, Nonexistence of Even Fibonacci Pseudoprimes of the 1st Kind, Fibonacci Quarterly 31 (1993), 166-172. (primary): https://www.fq.math.ca/Scanned/31-2/diporto.pdf\n Evidence used: Proves nonexistence for first-kind Fibonacci pseudoprimes and discusses parameter-dependent even examples.\n- Lawrence Somer, On Even Fibonacci Pseudoprimes, Applications of Fibonacci Numbers 4 (1991), 353-366. (primary): https://doi.org/10.1007/978-94-011-3586-3_31\n Evidence used: Provides the earlier constraints referred to in the imported background for the first-kind problem.\n\n**Review notes.** Do not label the exact record simply solved until 'Fibonacci pseudoprime' is scoped; its supplied odd-only definition makes the question vacuous.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set_id": 9 }, { "id": 1878, "problem_number": "EP-1", "title": "Erdős Problem #1", "statement": "If $A\\subseteq \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=n$ is such that the subset sums $\\sum_{a\\in S}a$ are distinct for all $S\\subseteq A$ then $ N \\gg 2^{n}. $ ", "background": "Erd\\H{o}s called this 'perhaps my first serious problem' (in \\cite{Er98} he dates it to 1931). The powers of $2$ show that $2^n$ would be best possible here. The trivial lower bound is $N \\gg 2^{n}/n$, since all $2^n$ distinct subset sums must lie in $[0,Nn)$. Erd\\H{o}s and Moser \\cite{Er56} proved $ N\\geq (\\tfrac{1}{4}-o(1))\\frac{2^n}{\\sqrt{n}}. $ (In \\cite{Er85c} Erd\\H{o}s offered \\$100 for any improvement of the constant $1/4$ here.)\nA number of improvements of the constant have been given (see \\cite{St23} for a history), with the current record $\\sqrt{2/\\pi}$ first proved in unpublished work of Elkies and Gleason. Two proofs achieving this constant are provided by Dubroff, Fox, and Xu \\cite{DFX21}, who in fact prove the exact bound $N\\geq \\binom{n}{\\lfloor n/2\\rfloor}$.\nIn \\cite{Er73} and \\cite{ErGr80} the generalisation where $A\\subseteq (0,N]$ is a set of real numbers such that the subset sums all differ by at least $1$ is proposed, with the same conjectured bound. (The second proof of \\cite{DFX21} applies also to this generalisation.) This generalisation seems to have first appeared in \\cite{Gr71}.\nThis problem appears in Erd\\H{o}s' book with Spencer \\cite{ErSp74} in the final chapter titled 'The kitchen sink'. As Ruzsa writes in \\cite{Ru99} \"it is a rich kitchen where such things go to the sink\".\nThe sequence of minimal $N$ for a given $n$ is A276661 in the OEIS.\nSee also [350].\nThis is discussed in problem C8 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[DFX21] Dubroff, Q. and Fox, J. and Xu, M. W., A note on the Erd\\H{o}s distinct subset sums problem. SIAM Journal on Discrete Mathematics (2021), 322-324.\n\n[Er56] Erd\\H{o}s, P., Problems and results in additive number theory. Colloque sur la Th\\'{e}orie des Nombres, Bruxelles, 1955 (1956), 127-137.\n\n[Er73] Erd\\H{o}s, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n\n[Er85c] Erd\\H{o}s, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\n\n[Er98] Erd\\H{o}s, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[ErSp74] Erd\\H{o}s, Paul and Spencer, Joel, Probabilistic methods in combinatorics. Akad\\'{e}miai Kiad\\'{o} (1974).\n\n[Gr71] Graham, R. L., On sums of integers taken from a fixed sequence. (1971), 22--40.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ru99] Ruzsa, I., Erd\\H{o}s and the Integers. Journal of Number Theory (1999), 115-163.\n\n[St23] Steinerberger, S., Some remarks on the Erd\\H{o}s distinct subset sums problem. arXiv:2208.12182 (2023).\",\n \"difficulty\": \"L3\"\n},\n{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Erdős distinct-subset-sums conjecture remains open. The best verified lower bound is N >= binom(n,floor(n/2)) = (sqrt(2/pi)+o(1))2^n/sqrt(n), while Bohman's best-known construction has maximum element at most 0.22002*2^n for sufficiently large n. No credible source located removes the factor sqrt(n) from the lower bound.\n\n**Verified partial progress.**\n\n- Dubroff, Fox, and Xu gave two proofs of the exact binomial lower bound, including a proof that extends to the 1-separated real-number variant.\n- Steinerberger reproved the best asymptotic lower bound and showed that a hypothetical small extremizer must make the associated signed-sum random variable quantitatively close to Gaussian.\n- Bohman's construction gives the best-known upper bound f(n)<0.22002*2^n for all sufficiently large n.\n\n**Full solution or refutation.**\n\nNo solution is known. The current gap is between a lower bound of order 2^n/sqrt(n) and an upper construction of order 2^n. The imported background's trailing serialized fragment is extraction damage and has no mathematical effect.\n\n**What remains.**\n\nProve an absolute c>0 with N>=c*2^n for every n-element subset-sum-distinct set, equivalently F(x)<=log_2(x)+O(1) for the largest dissociated subset of [x].\n\n**Sources checked.**\n\n- Quentin Dubroff, Jacob Fox, and Max Wenqiang Xu, A note on the Erdős distinct subset sums problem, SIAM Journal on Discrete Mathematics 35 (2021), 322-324, DOI 10.1137/20M1385883. (primary): https://arxiv.org/abs/2006.12988\n Evidence used: The paper proves the exact lower bound N>=binom(n,floor(n/2)), yielding the current sqrt(2/pi) asymptotic constant.\n- Stefan Steinerberger, Some Remarks on the Erdős Distinct Subset Sums Problem, International Mathematics Research Notices (2024). (primary): https://arxiv.org/abs/2208.12182\n Evidence used: The paper gives another proof of the best lower bound and a quantitative near-Gaussian obstruction.\n- Tom Bohman, A construction for sets of integers with distinct subset sums, Electronic Journal of Combinatorics 5 (1998), R3, DOI 10.37236/1341. (primary): https://www.combinatorics.org/ojs/index.php/eljc/article/view/v5i1r3\n Evidence used: The abstract states the construction f(n)<0.22002*2^n for sufficiently large n.\n- Thomas F. Bloom, Erdős Problem #1, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/1\n Evidence used: The maintained record, updated in April 2026, marks the conjecture open and lists the current lower and upper bounds.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1879, "problem_number": "EP-3", "title": "Erdős Problem #3", "statement": "If $A\\subseteq \\mathbb{N}$ has $\\sum_{n\\in A}\\frac{1}{n}=\\infty$ then must $A$ contain arbitrarily long arithmetic progressions?", "background": "This is essentially asking for good bounds on $r_k(N)$, the size of the largest subset of $\\{1,\\ldots,N\\}$ without a non-trivial $k$-term arithmetic progression. For example, a bound like $ r_k(N) \\ll_k \\frac{N}{(\\log N)(\\log\\log N)^2} $ would be sufficient.\nEven the case $k=3$ is non-trivial, but was proved by Bloom and Sisask \\cite{BlSi20}. Much better bounds for $r_3(N)$ were subsequently proved by Kelley and Meka \\cite{KeMe23}. Green and Tao \\cite{GrTa17} proved $r_4(N)\\ll N/(\\log N)^{c}$ for some small constant $c>0$. Gowers \\cite{Go01} proved $ r_k(N) \\ll \\frac{N}{(\\log\\log N)^{c_k}}, $ where $c_k>0$ is a small constant depending on $k$. The current best bounds for general $k$ are due to Leng, Sah, and Sawhney \\cite{LSS24}, who show that $ r_k(N) \\ll \\frac{N}{\\exp((\\log\\log N)^{c_k})} $ for some constant $c_k>0$ depending on $k$.\nCuriously, Erd\\H{o}s \\cite{Er83c} thought this conjecture was the 'only way to approach' the conjecture that there are arbitrarily long arithmetic progressions of prime numbers, now a theorem due to Green and Tao \\cite{GrTa08} (see [219]).\nIn \\cite{Er81} Erd\\H{o}s makes the stronger conjecture that $ r_k(N) \\ll_C\\frac{N}{(\\log N)^C} $ for every $C>0$ (now known for $k=3$ due to Kelley and Meka \\cite{KeMe23}) - see [140].\nSee also [139] and [142].\nThis is discussed in problem A5 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BlSi20] Bloom, T.F. and Sisask, O., Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. arXiv:2007.03528 (2020).\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er83c] Erd\\H{o}s, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.\n\n[Go01] Gowers, W. T., A new proof of Szemer\\'{e}di's theorem. Geom. Funct. Anal. (2001), 465-588.\n\n[GrTa08] Green, Ben and Tao, Terence, The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) (2008), 481-547.\n\n[GrTa17] Green, Ben and Tao, Terence, New bounds for Szemer\\'{e}di's theorem, III: a polylogarithmic bound for $r_4(N)$. Mathematika (2017), 944-1040.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\n\n[LSS24] Leng, J., Sah, A. and Sawhney, M., Improved bounds for Szemer\\'{e}di's theorem. arXiv:2402.17995 (2024).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The assertion for arbitrarily long progressions remains open. The harmonic-density implication is known for 3-term progressions, but the best bounds for r_4(N) and for general r_k(N), k>=5, do not cross the summability threshold needed to force a k-term progression in every set with divergent reciprocal sum.\n\n**Verified partial progress.**\n\n- Bloom and Sisask proved r_3(N)<=5. These bounds remain below the required harmonic-summability strength.\n\n**Full solution or refutation.**\n\nNo all-length solution is known. A sufficient general estimate would be approximately r_k(N)<=5 estimates are major density-increment advances but do not imply convergence of the reciprocal sum of every progression-free set.\n\n**What remains.**\n\nFor every fixed k>=4, prove a progression-free-set bound strong enough that dyadic partial summation forces sum_{n in A}1/n to converge, or find another argument connecting divergent reciprocal sum to k-term progressions.\n\n**Sources checked.**\n\n- Thomas F. Bloom and Olof Sisask, Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions, Annals of Mathematics 194 (2021), 831-906. (primary): https://arxiv.org/abs/2007.03528\n Evidence used: The paper proves r_3(N)<0; this is not strong enough for the reciprocal-sum question.\n- James Leng, Ashwin Sah, and Mehtaab Sawhney, Improved Bounds for Szemerédi's Theorem (2024). (primary): https://arxiv.org/abs/2402.17995\n Evidence used: For k>=5 the paper proves r_k(N)<", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1880, "problem_number": "EP-5", "title": "Erdős Problem #5", "statement": "Let $C\\geq 0$. Is there an infinite sequence of $n_i$ such that $ \\lim_{i\\to \\infty}\\frac{p_{n_i+1}-p_{n_i}}{\\log n_i}=C? $ ", "background": "Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\\log n$. This problem asks whether $S=[0,\\infty]$. Although this conjecture remains unproven, a lot is known about $S$. Some highlights:\n{UL}\n{LI}$\\infty\\in S$ by Westzynthius' result \\cite{We31} on large prime gaps,{/LI}\n{LI}$0\\in S$ by the work of Goldston, Pintz, and Yildirim \\cite{GPY09} on small prime gaps,{/LI}\n{LI}Erd\\H{o}s \\cite{Er55} and Ricci \\cite{Ri56} independently showed that $S$ has positive Lebesgue measure,{/LI}\n{LI} Hildebrand and Maier \\cite{HiMa88} showed that $S$ contains arbitrarily large (finite) numbers,{/LI}\n{LI} Pintz \\cite{Pi16} showed that there exists some small constant $c>0$ such that $[0,c]\\subset S$,{/LI}\n{LI} Banks, Freiberg, and Maynard \\cite{BFM16} showed that at least $12.5\\%$ of $[0,\\infty)$ belongs to $S$,{/LI}\n{LI} Merikoski \\cite{Me20} showed that at least $1/3$ of $[0,\\infty)$ belongs to $S$, and that $S$ has bounded gaps.{/LI}\n{/UL}\nIn \\cite{Er65b}, \\cite{Er85c}, and \\cite{Er97c} Erd\\H{o}s asks whether $S$ is everywhere dense (but Weisenberg notes that clearly $S$ is closed so this is equivalent to asking whether $S=[0,\\infty]$).\nSee also [234].\nReferences\n\n\n[BFM16] Banks, William D. and Freiberg, Tristan and Maynard, James, On limit points of the sequence of normalized prime gaps. Proc. Lond. Math. Soc. (3) (2016), 515-539.\n\n[Er55] Erd\"{o}s, Paul, Some remarks on number theory. Riveon Lematematika (1955), 45-48.\n\n[Er65b] Erd\\H{o}s, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[Er85c] Erd\\H{o}s, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\n\n[Er97c] Erd\\H{o}s, Paul, Some of my favorite problems and results. The mathematics of Paul Erd\\H{o}s, I (1997), 47-67.\n\n[GPY09] Goldston, Daniel A. and Pintz, J\\'{a}nos and Y\\i ld\\i r\\i m, Cem Y., Primes in tuples. I. Ann. of Math. (2) (2009), 819-862.\n\n[HiMa88] Hildebrand, Adolf and Maier, Helmut, Gaps between prime numbers. Proc. Amer. Math. Soc. (1988), 1-9.\n\n[Me20] Merikoski, Jori, Limit points of normalized prime gaps. J. Lond. Math. Soc. (2) (2020), 99-124.\n\n[Pi16] Pintz, J\\'{a}nos, Polignac numbers, conjectures of Erd\\H{o}s on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture. From arithmetic to zeta-functions (2016), 367-384.\n\n[Ri56] Ricci, Giovanni, Recherches sur l'allure de la suite $\\{p_{n+1}-p_n/\\log p_n\\}$. Colloque sur la Th\\'{e}orie des Nombres, Bruxelles, 1955 (1956), 93-106.\n\n[We31] Westzynthius, E., \"{U}ber die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind. Commentat. Phys. Math. (1931), 1-37.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains open whether every nonnegative real number is a limit point of normalized prime gaps. Merikoski proved that the limit-point set occupies at least one third of every initial interval and has bounded gaps, but this does not imply that the closed limit-point set is all of [0,infinity]. The record's log n normalization is equivalent to the standard log p_n normalization.\n\n**Verified partial progress.**\n\n- Classical large- and small-gap results give infinity and 0 as limit points.\n- Pintz proved that an initial interval [0,c] is contained in the limit-point set, and earlier work produced arbitrarily large finite limit points.\n- Merikoski proved |L intersect [0,T]|>=T/3 for every T>=0 and an absolute bound on gaps between successive portions of L.\n\n**Full solution or refutation.**\n\nNo complete solution is known. Positive lower measure, an initial interval, and bounded gaps leave room for a nonempty closed complement. The unrendered UL/LI tokens in the imported background are source markup rather than mathematical content.\n\n**What remains.**\n\nShow that the closed set of normalized prime-gap limit points is dense, equivalently equal to [0,infinity], or exhibit a missing finite limit point.\n\n**Sources checked.**\n\n- Jori Merikoski, Limit points of normalized prime gaps, Journal of the London Mathematical Society 102 (2020), 99-124. (primary): https://arxiv.org/abs/1811.03008\n Evidence used: The abstract states the one-third measure theorem on every [0,T] and the bounded-gap theorem for the limit-point set.\n- William D. Banks, Tristan Freiberg, and James Maynard, On limit points of the sequence of normalized prime gaps, Proceedings of the London Mathematical Society 113 (2016), 515-539. (primary): https://doi.org/10.1112/plms/pdw037\n Evidence used: This paper established an earlier positive-proportion theorem and the multi-limit-point framework improved by Merikoski.\n- Thomas F. Bloom, Erdős Problem #5, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/5\n Evidence used: The maintained record still marks the full limit-point conjecture open and lists the strongest known components.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1881, "problem_number": "EP-9", "title": "Erdős Problem #9", "statement": "Let $A$ be the set of all odd integers not of the form $p+2^{k}+2^l$ (where $k,l\\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?", "background": "In \\cite{Er77c} Erd\\H{o}s credits Schinzel with proving that there are infinitely many odd integers not of this form, but gives no reference. Crocker \\cite{Cr71} has proved there are $\\gg\\log\\log N$ such integers in $\\{1,\\ldots,N\\}$. Pan \\cite{Pa11} improved this to $\\gg_\\epsilon N^{1-\\epsilon}$ for any $\\epsilon>0$. Erd\\H{o}s believed this cannot be proved by covering systems, i.e. integers of the form $p+2^k+2^l$ exist in every infinite arithmetic progression.\nThe sequence of such numbers is A006286 in the OEIS.\nSee also [10], [11], and [16].\nThis is discussed in problem A19 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Cr71] Crocker, Roger, On the sum of a prime and of two powers of two. Pacific J. Math. (1971), 103-107.\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Pa11] Pan, Hao, On the integers not of the form {$p+2^a+2^b$}. Acta Arith. (2011), 55-61.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The positive-upper-density question remains open. Hao Pan proved that for every epsilon>0 the number of odd exceptions up to x is >>_epsilon x^(1-epsilon), a near-linear exponent bound that still permits density zero.\n\n**Verified partial progress.**\n\n- Crocker proved infinitude and in fact at least a constant times log log N exceptions up to N.\n- Pan improved the count to >>_epsilon N^(1-epsilon) for every fixed epsilon>0.\n- The maintained tracker now attributes infinitude directly to Crocker's proof rather than to an unlocated Schinzel reference.\n\n**Full solution or refutation.**\n\nNo positive-density lower bound is known. The family of estimates N^(1-epsilon) cannot be made uniform at epsilon=0 and therefore does not imply a linear-size subsequence.\n\n**What remains.**\n\nProve |A intersect [1,N_j]|>=cN_j along an unbounded sequence (or a positive lower density), or disprove positive upper density.\n\n**Sources checked.**\n\n- Hao Pan, On the integers not of the form p+2^a+2^b, Acta Arithmetica 148 (2011), 55-61, DOI 10.4064/aa148-1-4. (primary): https://arxiv.org/abs/0905.3809\n Evidence used: The abstract proves that the number of odd exceptions up to x is >>x^(1-epsilon) for every epsilon>0.\n- Roger Crocker, On the sum of a prime and of two powers of two, Pacific Journal of Mathematics 36 (1971), 103-107. (primary): https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-36/issue-1/On-the-sum-of-a-prime-and-of-two-powers/pjm/1102971477.full\n Evidence used: The paper provides the original explicit infinitude and logarithmic-count construction.\n- Thomas F. Bloom, Erdős Problem #9, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/9\n Evidence used: The maintained record was revised in April 2026, remains open, and records Crocker and Pan as the current progress.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1882, "problem_number": "EP-10", "title": "Erdős Problem #10", "statement": "Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of 2?", "background": "Erd\\H{o}s described this as 'probably unattackable'. In \\cite{ErGr80} Erd\\H{o}s and Graham suggest that no such $k$ exists. Gallagher \\cite{Ga75} has shown that for any $\\epsilon>0$ there exists $k(\\epsilon)$ such that the set of integers which are the sum of a prime and at most $k(\\epsilon)$ many powers of 2 has lower density at least $1-\\epsilon$.\nGranville and Soundararajan \\cite{GrSo98} have conjectured that at most $3$ powers of 2 suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. (The restriction to odd integers is important here - for example, Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ (see [9]) suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).\nSee also [9], [11], and [16].\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[Ga75] Gallagher, P. X., Primes and powers of 2. Invent. Math. (1975), 125-142.\n\n[GrSo98] Granville, A. and Soundararajan, K., A Binary Additive Problem of Erd\\H{o}s and the Order of $2$ mod $p^2$. The Ramanujan Journal (1998), 283-298.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The intended historical question--whether one fixed k represents every sufficiently large integer as a prime plus at most k powers of 2--remains open. Gallagher's theorem gives, for every epsilon, a k(epsilon) covering a set of lower density at least 1-epsilon, but the order of quantifiers is weaker. The imported phrase 'every integer' is not the maintained formulation and has finite-exception ambiguities.\n\n**Verified partial progress.**\n\n- Gallagher proved an arbitrarily-high lower-density approximation using a number of powers depending on epsilon.\n- Granville and Soundararajan formulated a sharper conjectural picture in which three powers suffice for odd integers and four for even integers.\n- The maintained tracker explicitly uses 'every large integer' and records conflicting historical opinions from Erdős and Erdős--Graham.\n\n**Full solution or refutation.**\n\nNo absolute k is known to cover all sufficiently large integers. A sequence of density statements with k depending on the permitted exceptional density does not produce a single universal k.\n\n**What remains.**\n\nResolve the eventual version with one fixed k. Before treating the imported statement literally, specify whether 0 powers are permitted and replace 'every integer' by the historically intended 'every sufficiently large integer' or separately handle all finite exceptions.\n\n**Sources checked.**\n\n- P. X. Gallagher, Primes and powers of 2, Inventiones Mathematicae 29 (1975), 125-142. (primary): https://eudml.org/doc/142290\n Evidence used: The paper proves the density approximation underlying the strongest general positive result recorded for this question.\n- Andrew Granville and K. Soundararajan, A Binary Additive Problem of Erdős and the Order of 2 mod p^2, Ramanujan Journal 2 (1998), 283-298, DOI 10.1023/A:1009786614584. (primary): https://doi.org/10.1023/A:1009786614584\n Evidence used: This primary paper develops the related binary additive obstruction and the sharper conjectural bounded-power formulation.\n- Thomas F. Bloom, Erdős Problem #10, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/10\n Evidence used: The maintained page, updated in April 2026, states 'every large integer' and retains the open status.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1883, "problem_number": "EP-12", "title": "Erdős Problem #12", "statement": "Let $A$ be an infinite set such that there are no distinct $a,b,c\\in A$ such that $a\\mid (b+c)$ and $b,c>a$. Is there such an $A$ with $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>0? $ Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\frac{N}{f(N)}. $ (Their example is given by all integers in $(y_i,\\frac{3}{2}y_i)$ congruent to $1$ modulo $(2y_{i-1})!$, where $y_i$ is some sufficiently quickly growing sequence.)\nAn example of an $A$ with this property where $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}\\log N>0 $ is given by the set of $p^2$, where $p\\equiv 3\\pmod{4}$ is prime.\nElsholtz and Planitzer \\cite{ElPl17} have constructed such an $A$ with $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\gg \\frac{N^{1/2}}{(\\log N)^{1/2}(\\log\\log N)^2(\\log\\log\\log N)^2}. $ Schoen \\cite{Sc01} proved that if all elements in $A$ are pairwise coprime then $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert \\ll N^{2/3} $ for infinitely many $N$. Baier \\cite{Ba04} has improved this to $\\ll N^{2/3}/\\log N$.\nFor the finite version see [13].\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[Ba04] Baier, Stephan, A note on {$\\scr P$}-sets. Integers (2004), A13, 6.\n\n[ElPl17] Elsholtz, Christian and Planitzer, Stefan, On Erd\\H{o}s and {S}\\'{a}rk\"ozy's sequences with Property P. Monatsh. Math. (2017), 565--575.\n\n[ErSa70] Erd\\H{o}s, P. and S\\'{a}rk\"ozi, A., On the divisibility properties of sequences of integers. Proc. London Math. Soc. (3) (1970), 97-101.\n\n[Sc01] Schoen, Tomasz, On a problem of Erd\\H{o}s and {S}\\'{a}rk\"ozy. J. Combin. Theory Ser. A (2001), 191--195.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported all-open label is stale. In 2026, public Lean-verified constructions resolved part (i) affirmatively and part (ii) negatively. There is a Property-P set with liminf A(N)/sqrt(N)>0, and there is one with A(N)>=N^(1-epsilon) for every epsilon>0 and all sufficiently large N, so no universal fixed power saving can hold. Part (iii), whether every such set has convergent reciprocal sum, remains open.\n\n**Verified partial progress.**\n\n- Erdős and Sárközy proved density zero but constructed sets with density tending to zero arbitrarily slowly along subsequences.\n- Elsholtz and Planitzer constructed a uniform lower bound A(N)>>sqrt(N)/[sqrt(log N)(log log N)^2(log log log N)^2].\n- Tsoukalas et al. give informal proofs and public Lean certificates for the affirmative first part and the counterexample to the second part.\n- The maintained discussion records a simplified construction with A(N)>=N/(log N)^{O(log log log N)} for all large N; the divergent-reciprocal-sum question is still not resolved.\n\n**Full solution or refutation.**\n\nFor part (i), a block construction based on a base-2-to-base-3 3-AP-free map and CRT-separated congruence blocks yields positive square-root lower density while preserving Property P. For part (ii), a denser Behrend-style version yields A(N)>=N^(1-epsilon) eventually for every epsilon, contradicting any absolute exponent c>0. Both statements have public Lean proofs. No theorem in the checked sources settles part (iii).\n\n**What remains.**\n\nDetermine whether a Property-P set can have divergent sum of reciprocals. The first and second displayed questions are closed, but the three-part record as a whole is only partially solved.\n\n**Sources checked.**\n\n- George Tsoukalas et al., Advancing Mathematics Research with AI-Driven Formal Proof Search, arXiv:2605.22763v2 (2026). (primary): https://arxiv.org/abs/2605.22763\n Evidence used: The supplementary material states and proves Erdős #12(i) and #12(ii), including the dense counterexample that rules out every fixed power saving.\n- Google DeepMind, AlphaProof Nexus public Lean outputs for ErdosProblems/erdos_12.parts.i and erdos_12.parts.ii (2026). (formal_verification): https://github.com/google-deepmind/alphaproof-nexus-results/tree/main/APNOutputs/ErdosProblems\n Evidence used: The repository contains the checked Lean proof artifacts linked by the primary paper and the maintained discussion.\n- Christian Elsholtz and Stefan Planitzer, On Erdős and Sárközy's sequences with Property P, Monatshefte für Mathematik 183 (2017), 565-575. (primary): https://arxiv.org/abs/1609.07935\n Evidence used: The paper proves the strongest pre-2026 uniform lower construction recorded in the imported background.\n- Thomas F. Bloom, Erdős Problem #12 and discussion, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/12\n Evidence used: The maintained record, updated in April 2026, marks parts (i) and (ii) resolved and explicitly says the reciprocal-sum question remains unknown.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1884, "problem_number": "EP-14", "title": "Erdős Problem #14", "statement": "Let $A\\subseteq \\mathbb{N}$. Let $B\\subseteq \\mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$.\nIs it true that for all $\\epsilon>0$ and large $N$ $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\gg_\\epsilon N^{1/2-\\epsilon}? $ Is it possible that $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert =o(N^{1/2})? $ ", "background": "Apparently originally considered by Erd\\H{o}s and Nathanson, although later Erd\\H{o}s attributes this to Erd\\H{o}s, S\\'{a}rk\"{o}zy, and Szemer\\'{e}di (but gives no reference), and claims a construction of an $A$ such that for all $\\epsilon>0$ and all large $N$ $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\ll_\\epsilon N^{1/2+\\epsilon}, $ and yet there for all $\\epsilon>0$ there exist infinitely many $N$ where $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\gg_\\epsilon N^{1/3-\\epsilon}. $ Erd\"{o}s and Freud investigated the finite analogue in \\cite{ErFr91}, proving that there exists $A\\subseteq \\{1,\\ldots,N\\}$ such that the number of integers not representable in exactly one way as the sum of two elements from $A$ is $<2^{3/2}N^{1/2}$, and suggest the constant $2^{3/2}$ is perhaps best possible.\nReferences\n\n\n[ErFr91] Erd\\H{o}s, P. and Freud, R., On sums of a {S}idon-sequence. J. Number Theory (1991), 196--205.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Both infinite unique-sum questions remain open in the maintained record, and no later primary resolution was located. The verified Erdős--Freud finite analogue constructs sets with fewer than 2^(3/2)sqrt(N) failures of unique representation, but it does not settle the infinite asymptotics.\n\n**Verified partial progress.**\n\n- The maintained historical remarks attribute an infinite construction with complement O_epsilon(N^(1/2+epsilon)), but give no primary reference for it.\n- The same remarks report an Omega_epsilon(N^(1/3-epsilon)) lower bound along infinitely many N for that construction.\n- Erdős and Freud proved a finite construction with fewer than 2^(3/2)sqrt(N) integers failing unique representation.\n\n**Full solution or refutation.**\n\nNo solution is known. The two displayed questions are compatible rather than contradictory: a complement sqrt(N)/L(N), with L(N) tending slowly to infinity, can be o(sqrt(N)) while remaining larger than N^(1/2-epsilon) for every fixed epsilon.\n\n**What remains.**\n\nProve the universal N^(1/2-o(1)) lower bound, construct an infinite A with o(sqrt N) exceptions, or determine the correct intermediate scale. A definitive treatment should specify the convention for unordered representations and whether equal summands count.\n\n**Sources checked.**\n\n- Paul Erdős and R. Freud, On sums of a Sidon-sequence, Journal of Number Theory 38 (1991), 196-205, DOI 10.1016/0022-314X(91)90083-N. (primary): https://doi.org/10.1016/0022-314X(91)90083-N\n Evidence used: The paper is the verified primary source for the finite Sidon-sum analogue and its square-root-scale construction.\n- Thomas F. Bloom, Erdős Problem #14, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/14\n Evidence used: The maintained record, last edited in September 2025 and crawled in 2026, marks both questions open and reports no solution claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1885, "problem_number": "EP-15", "title": "Erdős Problem #15", "statement": "Is it true that $ \\sum_{n=1}^\\infty(-1)^n\\frac{n}{p_n} $ converges, where $p_n$ is the sequence of primes?", "background": "Erd\\H{o}s suggested that a computer could be used to explore this, and did not see any other method to attack this.\nTao \\cite{Ta23} has proved that this series does converge assuming a strong form of the Hardy-Littlewood prime tuples conjecture.\nIn \\cite{Er98} Erd\\H{o}s further conjectures that $ \\sum_{n=1}^\\infty (-1)^n \\frac{1}{n(p_{n+1}-p_n)} $ converges and $ \\sum_{n=1}^\\infty (-1)^n \\frac{1}{p_{n+1}-p_n} $ diverges. Weisenberg notes that the existence of infinitely many bounded gaps between primes (as proved by Zhang \\cite{Zh14}) implies the latter series does not converge. Weisenberg also has an argument which shows that, assuming the Hardy-Littlewood prime $k$-tuples conjecture, the series is unbounded in at least one direction (positive or negative).\nErd\\H{o}s further conjectured that $ \\sum_{n=1}^\\infty (-1)^n \\frac{1}{n(p_{n+1}-p_n)(\\log\\log n)^c} $ converges for every $c>0$, and reports that he and Nathanson can prove that this series converges absolutely for $c>2$ (and can show, conditional on 'hopeless' conjectures about the primes, that this sum does not converge absolutely for $c=2$).\nSawhney has provided the following proof that this series converges absolutely for $c>2$: note that, whenever $c>1$, the contribution to the sum from gaps $p_{n+1}-p_n\\geq \\log n$ is convergent, so it suffices to consider only small gaps. The number of $n\\leq X$ such that $p_{n+1}-p_n\\in [\\epsilon\\log n,2\\epsilon \\log n)$ is bounded above by $\\ll \\epsilon X$ (this can be proved via the Selberg sieve). In particular, applying this bound for $\\frac{1}{\\log n}\\leq \\epsilon \\leq 1$ of the shape $2^{-j}$ (of which there are at most $\\log\\log n$ possibilities) shows the desired convergence, since $ \\sum \\frac{1}{n(\\log n)(\\log\\log n)^{c-1}} $ converges.\nReferences\n\n\n[Er98] Erd\\H{o}s, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.\n\n[Ta23] Tao, T., The convergence of an alternating series of Erd\\H{o}s, assuming the Hardy-Littlewood prime tuples conjecture. arXiv:2308.07205 (2023).\n\n[Zh14] Zhang, Yitang, Bounded gaps between primes. Ann. of Math. (2) (2014), 1121--1174.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Unconditional convergence of sum (-1)^n n/p_n remains open. Tao proved convergence under a suitably strong uniform Hardy--Littlewood prime-tuples conjecture; no unconditional theorem located supplies the required cancellation.\n\n**Verified partial progress.**\n\n- Tao's conditional proof uses a random sifted model of the primes and Gallagher-type calculations to control alternating correlations.\n- Infinitely many bounded prime gaps imply that the separate series sum (-1)^n/(p_{n+1}-p_n) cannot converge, but do not determine a single direction of divergence.\n- The maintained tracker includes an elementary Selberg-sieve argument giving absolute convergence of a differently weighted reciprocal-gap series for c>2.\n\n**Full solution or refutation.**\n\nThe displayed series is conditionally settled only under a strong prime-tuples hypothesis. The hypothesis supplies uniform local prime-pattern statistics well beyond current unconditional methods; numerical partial sums do not resolve convergence.\n\n**What remains.**\n\nProve sufficient alternating cancellation unconditionally, or reduce the Hardy--Littlewood input to a presently accessible prime-correlation theorem.\n\n**Sources checked.**\n\n- Terence Tao, The convergence of an alternating series of Erdős, assuming the Hardy--Littlewood prime tuples conjecture (2023). (primary): https://arxiv.org/abs/2308.07205\n Evidence used: The abstract explicitly calls the unconditional question open and proves convergence under a suitably strong Hardy--Littlewood hypothesis.\n- Thomas F. Bloom, Erdős Problem #15, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/15\n Evidence used: The maintained tracker marks the problem open and separates Tao's conditional theorem from the reciprocal-gap variants.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1886, "problem_number": "EP-17", "title": "Erdős Problem #17", "statement": "Are there infinitely many primes $p$ such that every even number $n\\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\\leq p$?", "background": "The first prime without this property is $97$. The sequence of such primes is A038133 in the OEIS. These are called cluster primes.\nBlecksmith, Erd\\H{o}s, and Selfridge \\cite{BES99} proved that the number of such primes is $ \\ll_A \\frac{x}{(\\log x)^A} $ for every $A>0$, and Elsholtz \\cite{El03} improved this to $ \\ll x\\exp(-c(\\log\\log x)^2) $ for every $c<1/8$.\nThis is discussed in problem C1 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BES99] Blecksmith, Richard and Erd\\H{o}s, Paul and Selfridge, J. L., Cluster primes. Amer. Math. Monthly (1999), 43--48.\n\n[El03] Elsholtz, Christian, On cluster primes. Acta Arith. (2003), 281--284.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Infinitude of cluster primes remains open. Existing work proves only very strong scarcity upper bounds: Blecksmith--Erdős--Selfridge obtained x/(log x)^A for every fixed A, and Elsholtz improved this to x exp(-c(log log x)^2) for every c<1/8.\n\n**Verified partial progress.**\n\n- The first prime without the cluster property is 97, and the finite sequence is tracked as OEIS A038133.\n- Blecksmith, Erdős, and Selfridge related the property to unusually dense backward prime clusters and proved super-polylogarithmic scarcity.\n- Elsholtz used an upper-bound sieve to prove a stretched-exponential-in-log-log scarcity estimate; the paper explains how the displayed 1/60 constant refines to every c<1/8.\n\n**Full solution or refutation.**\n\nNo lower-bound construction proving infinitely many cluster primes is known, and the scarcity theorems do not imply finiteness. The current methods explain why such primes, if infinite, must be exceptionally rare.\n\n**What remains.**\n\nProduce infinitely many primes with all required even differences represented below p, or prove only finitely many exist. Because the topic is niche and citation searches found no later theorem, expert confirmation is warranted.\n\n**Sources checked.**\n\n- Richard Blecksmith, Paul Erdős, and J. L. Selfridge, Cluster Primes, American Mathematical Monthly 106 (1999), 43-48, DOI 10.1080/00029890.1999.12005005. (primary): https://doi.org/10.1080/00029890.1999.12005005\n Evidence used: The paper defines cluster primes, poses infinitude, and proves the x/(log x)^A upper bounds.\n- Christian Elsholtz, On cluster primes, Acta Arithmetica 109 (2003), 281-284. (primary): https://www.math.tugraz.at/~elsholtz/WWW/papers/papers13clusteractarith.pdf\n Evidence used: The paper proves the stronger stretched-exponential scarcity bound and discusses the c<1/8 refinement.\n- Thomas F. Bloom, Erdős Problem #17, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/17\n Evidence used: The maintained record, last edited in December 2025, retains the open status and reports no solution claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1887, "problem_number": "EP-18", "title": "Erdős Problem #18", "statement": "We call $m$ practical if every integer $n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** All three displayed asymptotic questions about the shortest uniform divisor-sum representations remain open. The strongest directly relevant theorem located is Vose's construction of infinitely many practical m with h(m)<", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1888, "problem_number": "EP-20", "title": "Erdős Problem #20", "statement": "Let $f(n,k)$ be minimal such that every family $\\mathcal{F}$ of $n$-uniform sets with $\\lvert \\mathcal{F}\\rvert \\geq f(n,k)$ contains a $k$-sunflower. Is it true that $ f(n,k) < c_k^n $ for some constant $c_k>0$?", "background": "Erd\\H{o}s and Rado \\cite{ErRa60} originally proved $f(n,k)\\leq (k-1)^nn!$. Kostochka \\cite{Ko97} improved this slightly (in particular establishing an upper bound of $o(n!)$, for which Erd\\H{o}s awarded him the consolation prize of \\$100), but the bound stood at $n^{(1+o(1))n}$ for a long time until Alweiss, Lovett, Wu, and Zhang \\cite{ALWZ20} proved $ f(n,k) < (Ck\\log n\\log\\log n)^n $ for some constant $C>1$. This was refined slightly, independently by Rao \\cite{Ra20}, Frankston, Kahn, Narayanan, and Park \\cite{FKNP19}, and Bell, Chueluecha, and Warnke \\cite{BCW21}, leading to the current record of $ f(n,k) < (Ck\\log n)^n $ for some constant $C>1$.\nIn \\cite{Er81} offered \\$1000 for a proof or disproof even just in the special case when $k=3$, which he expected 'contains the whole difficulty'. He also wrote 'I really do not see why this question is so difficult'.\nThe usual focus is on the regime where $k=O(1)$ is fixed (say $k=3$) and $n$ is large, although for the opposite regime Kostochka, R\"{o}dl, and Talysheva \\cite{KRT99} have shown $ f(n,k)=(1+O_n(k^{-1/2^n}))k^n. $ \nReferences\n\n\n[ALWZ20] Alweiss, R. and Lovett, S. and Wu, K. and Zhang, J., Improved bounds for the sunflower lemma. (2020).\n\n[BCW21] Bell, T. and Chueluecha, S. and Warnke, L., Note on sunflowers. Discret. Math. (2021).\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[ErRa60] Erd\\H{o}s, P. and Rado, R., Intersection theorems for systems of sets. J. London Math. Soc. (1960), 85-90.\n\n[FKNP19] Frankston, K. and Kahn, J. and Narayanan, B. and Park, J., Thresholds versus fractional expectation-thresholds. CoRR (2019).\n\n[KRT99] Kostochka, A. V. and R\"{o}dl, V. and Talysheva, L. A., On systems of small sets with no large $\\Delta$-subsystems. Combin. Probab. Comput. (1999), 265-268.\n\n[Ko97] Kostochka, A., A bound on the cardinality of families not containing $\\Delta$-systems. (1997).\n\n[Ra20] Rao, A., Coding for sunflowers. Discrete Analysis (2020).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The fixed-k exponential sunflower conjecture remains open, including k=3; the current upper bound recorded by the maintained tracker is f(n,k)<(C k log n)^n.\n\n**Verified partial progress.**\n\n- Alweiss, Lovett, Wu, and Zhang replaced the classical factorial-type behavior by a near-exponential bound.\n- Subsequent work, including Bell, Chueluecha, and Warnke, gives the displayed C k log n base.\n- The remaining gap is removal of the log n factor from the base for fixed k.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was located.\n\n**What remains.**\n\nProve or disprove an upper bound c_k^n for every fixed k.\n\n**Sources checked.**\n\n- R. Alweiss, S. Lovett, K. Wu, and J. Zhang, Improved bounds for the sunflower lemma, Annals of Mathematics 194 (2021). (primary): https://arxiv.org/abs/1908.08483\n Evidence used: The paper supplies the breakthrough near-exponential sunflower upper bound.\n- T. Bell, A. Chueluecha, and L. Warnke, Note on sunflowers, Discrete Mathematics 344 (2021). (primary): https://doi.org/10.1016/j.disc.2020.112157\n Evidence used: A refinement used by the maintained record for the current logarithmic-base bound.\n- Thomas F. Bloom, Erdos Problem #20, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/20\n Evidence used: Lists the problem as open and summarizes the current bound.\n\n**Review notes.** The quantitative advance is recorded without treating the still-open asymptotic target as solved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1889, "problem_number": "EP-25", "title": "Erdős Problem #25", "statement": "Let $n_1\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The one-residue-class-per-modulus logarithmic-density question remains open; it is a special case of EP-486.\n\n**Verified partial progress.**\n\n- For the wider EP-486 problem, Davenport and Erdos proved the logarithmic density exists when every excluded residue is zero.\n- That special case does not cover independently chosen residues a_i modulo n_i.\n\n**Full solution or refutation.**\n\nNo resolution for arbitrary selected residues was located.\n\n**What remains.**\n\nEstablish or refute existence of logarithmic density for every increasing modulus sequence and selected residue classes.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #25, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/25\n Evidence used: Lists the literal problem as open and identifies EP-486 as its generalization.\n- Thomas F. Bloom, history of Erdos Problem #486, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/486\n Evidence used: Documents the Davenport-Erdos zero-residue special case and the relationship to EP-25.\n\n**Review notes.** The imported statement has a control-character/OCR loss in the intended not-congruent sign; no source text was changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1890, "problem_number": "EP-28", "title": "Erdős Problem #28", "statement": "If $A\\subseteq \\mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\\limsup 1_A\\ast 1_A(n)=\\infty$.", "background": "Conjectured by Erd\\H{o}s and Tur\\'{a}n. They also suggest the stronger conjecture that $\\limsup 1_A\\ast 1_A(n)/\\log n>0$.\nAnother stronger conjecture would be that the hypothesis $\\lvert A\\cap [1,N]\\rvert \\gg N^{1/2}$ for all large $N$ suffices.\nErd\\H{o}s and S\\'{a}rk\"{o}zy conjectured the stronger version that if $A=\\{a_1\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Erdos-Turan conjecture on unbounded representations of asymptotic bases remains open, but published work proves finite lower bounds on the limsup and recent work improves a density-theoretic threshold.\n\n**Verified partial progress.**\n\n- Konstantoulas proved that sufficiently small upper density of the exceptional set forces infinitely many ordered representation counts above five; finite exceptional set is included.\n- Li and Zhang improved the associated exceptional-set density constant from 1/10 to 7/32 and prove further finite lower-bound criteria.\n- These results do not imply that the representation function is unbounded.\n\n**Full solution or refutation.**\n\nNo proof or counterexample to eventual unboundedness was located.\n\n**What remains.**\n\nShow that every asymptotic basis of order two has unbounded representation function, or construct a counterexample.\n\n**Sources checked.**\n\n- Ioannis Konstantoulas, Lower bounds for a conjecture of Erdos and Turan, Acta Arithmetica 159 (2013), 301-313. (primary): https://eudml.org/doc/279150\n Evidence used: Published abstract states the more-than-five representation theorem for sufficiently small exceptional density.\n- Huixi Li and Zihan Zhang, An Improvement of Konstantoulas' Density Constant, arXiv:2605.30922 (2026). (primary): https://arxiv.org/abs/2605.30922\n Evidence used: Abstract gives the 7/32 improvement and related conditional representation bounds.\n- Thomas F. Bloom, Erdos Problem #28, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/28\n Evidence used: Lists the original conjecture as open.\n\n**Review notes.** Recent progress is an arXiv preprint and does not resolve the target.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1891, "problem_number": "EP-30", "title": "Erdős Problem #30", "statement": "Let $h(N)$ be the maximum size of a Sidon set in $\\{1,\\ldots,N\\}$. Is it true that, for every $\\epsilon>0$, $ h(N) = N^{1/2}+O_\\epsilon(N^\\epsilon)? $ ", "background": "A problem of Erd\\H{o}s and Tur\\'{a}n. It may even be true that $h(N)=N^{1/2}+O(1)$, but Erd\\H{o}s remarks this is perhaps too optimistic. Erd\\H{o}s and Tur\\'{a}n \\cite{ErTu41} proved an upper bound of $N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindstr\"{o}m \\cite{Li69}. Both proofs in fact give $ h(N) \\leq N^{1/2}+N^{1/4}+1. $ Balogh, F\"{u}redi, and Roy \\cite{BFR21} improved the bound in the error term to $0.998N^{1/4}$. This was further optimised by O'Bryant \\cite{OB22}. The current record is $ h(N)\\leq N^{1/2}+0.98183N^{1/4}+O(1), $ due to Carter, Hunter, and O'Bryant \\cite{CHO25}.\nSinger \\cite{Si38} was the first to show that $h(N)\\geq (1-o(1))N^{1/2}$ for all $N$. For a detailed survey of the literature we refer to \\cite{OB04}.\nSee also [241] and [840].\nThis problem is Problem 31 on Green's open problems list.\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BFR21] Balogh, J. and F\"{u}redi, Z. and Roy, S., An upper bound on the size of Sidon sets. arXiv:2103.15850 (2021).\n\n[CHO25] Carter, D. and Hunter, Z. and O'Bryant, K., On the diameter of finite {S}idon sets. Acta Math. Hungar. (2025), 108--126.\n\n[ErTu41] Erd\\H{o}s, P. and Tur\\'{a}n, P., On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc. (1941), 212-215.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Li69] Lindstr\"{o}m, B., An inequality for $B_2$-sequences. J. Combinatorial Theory (1969), 211-212.\n\n[OB04] O'Bryant, Kevin, A complete annotated bibliography of work related to {S}idon\nsequences. Electron. J. Combin. (2004), 39.\n\n[OB22] O'Bryant, K., On the size of finite Sidon sets. arXiv:2207.07800 (2022).\n\n[Si38] Singer, James, A theorem in finite projective geometry and some applications\nto number theory. Trans. Amer. Math. Soc. (1938), 377--385.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The conjectured N^epsilon error term for the maximal finite Sidon-set size is open; the current displayed upper record is N^(1/2)+0.98183 N^(1/4)+O(1).\n\n**Verified partial progress.**\n\n- Erdos and Turan established an N^(1/4)-scale error upper bound.\n- Carter, Hunter, and O'Bryant improved its coefficient to 0.98183.\n- Singer-type constructions establish the leading asymptotic N^(1/2).\n\n**Full solution or refutation.**\n\nNo bound of the requested subpolynomial error scale was located.\n\n**What remains.**\n\nReduce the N^(1/4) error term to O_epsilon(N^epsilon), or disprove that scale.\n\n**Sources checked.**\n\n- D. Carter, Z. Hunter, and K. O'Bryant, On the diameter of finite Sidon sets, Acta Mathematica Hungarica (2025); preprint arXiv:2310.20032. (primary): https://arxiv.org/abs/2310.20032\n Evidence used: Provides the 0.98183 coefficient reported by the current source record.\n- Thomas F. Bloom, Erdos Problem #30 LaTeX source, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/30\n Evidence used: Records the problem as open and gives the current bound history.\n\n**Review notes.** The stated error exponent, not merely its leading term, is the unresolved part.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1892, "problem_number": "EP-32", "title": "Erdős Problem #32", "statement": "Is there a set $A\\subset\\mathbb{N}$ such that $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert = o((\\log N)^2) $ and such that every large integer can be written as $p+a$ for some prime $p$ and $a\\in A$?\nCan the bound $O(\\log N)$ be achieved? Must such an $A$ satisfy $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{\\log N}> 1? $ ", "background": "Such a set is called an additive complement to the primes.\nErd\\H{o}s \\cite{Er54} proved that such a set $A$ exists with $\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\ll (\\log N)^2$ (improving a previous result of Lorentz \\cite{Lo54} who achieved $\\ll (\\log N)^3$).\nWolke \\cite{Wo96} has shown that such a bound is almost true, in that we can achieve $\\ll (\\log N)^{1+o(1)}$ if we only ask for almost all integers to be representable. Kolountzakis \\cite{Ko96} improved this to $\\ll (\\log N)(\\log\\log N)$, and Ruzsa \\cite{Ru98c} further improved this to $\\ll \\omega(N)\\log N$ for any $\\omega\\to \\infty$.\nThe answer to the third question is yes: Ruzsa \\cite{Ru98c} has shown that we must have $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{\\log N}\\geq e^\\gamma\\approx 1.781. $ This is discussed in problem E1 of Guy's collection \\cite{Gu04}, where it is stated that Erd\\H{o}s offered \\$50 for determining whether $O(\\log N)$ can be achieved.\nReferences\n\n\n[Er54] Erd\\H{o}s, Paul, Some results on additive number theory. Proc. Amer. Math. Soc. (1954), 847-853.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ko96] Kolountzakis, Mihail N., On the additive complements of the primes and sets of similar\ngrowth. Acta Arith. (1996), 1--8.\n\n[Lo54] Lorentz, G. G., On a problem of additive number theory. Proc. Amer. Math. Soc. (1954), 838-841.\n\n[Ru98c] Ruzsa, Imre Z., On the additive completion of primes. Acta Arith. (1998), 269-275.\n\n[Wo96] Wolke, Dieter, On a problem of Erd\\H{o}s in additive number theory. J. Number Theory (1996), 209-213.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ruzsa answered the third subquestion affirmatively with liminf A(x)/log x at least e^gamma, while the o((log x)^2) and O(log x) all-integers complement questions remain open.\n\n**Verified partial progress.**\n\n- Ruzsa proved every additive complement to the primes has liminf A(x)/log x at least e^gamma.\n- For every omega(x) tending to infinity, he constructed a complement of size O(omega(x) log x).\n- The exact O(log x) construction remains unresolved.\n\n**Full solution or refutation.**\n\nOne of the record's three questions is solved, but the principal construction questions are not.\n\n**What remains.**\n\nDecide whether an additive complement to the primes can have A(x)=o((log x)^2), especially O(log x), while covering every sufficiently large integer.\n\n**Sources checked.**\n\n- Imre Z. Ruzsa, On the additive completion of primes, Acta Arithmetica 86 (1998), 269-275. (primary): https://doi.org/10.4064/aa-86-3-269-275\n Evidence used: Published paper proves the lower liminf bound and the omega(x) log x construction.\n- Thomas F. Bloom, Erdos Problem #32, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/32\n Evidence used: Separates the solved third question from the two open construction questions.\n\n**Review notes.** A mixed-status multi-question record; partial status reflects the exact solved third question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1893, "problem_number": "EP-33", "title": "Erdős Problem #33", "statement": "Let $A\\subset\\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\\in A$ and $n\\geq 0$. What is the smallest possible value of $ \\limsup \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}? $ Is $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>1? $ ", "background": "Such a set $A$ is called an additive complement of the set of squares. Erd\\H{o}s observed that there exist $A$ for which the $\\limsup$ is finite and $>1$. Moser \\cite{Mo65} proved that, for any such $A$, $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>1.06. $ The best-known lower bound is $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}\\geq\\frac{4}{\\pi}\\approx 1.273 $ proved by Cilleruelo \\cite{Ci93}, Habsieger \\cite{Ha95}, and Balasubramanian and Ramana \\cite{BaRa01}.\nThe problem of minimising the $\\limsup$ appears to have been much less studied. van Doorn has a construction of such an $A$ in which, for all $N$, $ \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}< 2\\phi^{5/2}\\approx 6.66, $ where $\\phi=\\frac{1+\\sqrt{5}}{2}$ is the golden ratio.\nReferences\n\n\n[BaRa01] Balasubramanian, R. and Ramana, D. S., Additive complements of the squares. C. R. Math. Acad. Sci. Soc. R. Can. (2001), 6--11.\n\n[Ci93] Cilleruelo, Javier, The additive completion of {$k$}th-powers. J. Number Theory (1993), 237--243.\n\n[Ha95] Habsieger, Laurent, On the additive completion of polynomial sets. J. Number Theory (1995), 130--135.\n\n[Mo65] Moser, Leo, On the additive completion of sets of integers. (1965), 175--180.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The second question is affirmatively resolved by the stronger lower bound liminf A(x)/sqrt(x)>=4/pi; the optimal limsup constant in the first question remains open.\n\n**Verified partial progress.**\n\n- Cilleruelo, Habsieger, and Balasubramanian-Ramana established the 4/pi lower bound for every additive complement of the squares.\n- A tracker discussion reports a construction with A(x)/sqrt(x)<2 phi^(5/2) for all x, but this informal upper construction needs independent literature verification.\n- The exact least possible limsup is still unknown.\n\n**Full solution or refutation.**\n\nThe positive liminf assertion is settled, whereas the optimization question is not.\n\n**What remains.**\n\nDetermine the smallest possible limsup A(x)/sqrt(x) for an additive complement to the squares.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #33 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/33?embed=1\n Evidence used: Records the published 4/pi lower bound, identifies the first question as open, and labels the newer construction as comment-level material.\n\n**Review notes.** The non-paper tracker comment is not used as proof of an optimal result.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1894, "problem_number": "EP-36", "title": "Erdős Problem #36", "statement": "Find the optimal constant $c>0$ such that the following holds.\nFor all sufficiently large $N$, if $A\\sqcup B=\\{1,\\ldots,2N\\}$ is a partition into two equal parts, so that $\\lvert A\\rvert=\\lvert B\\rvert=N$, then there is some $x$ such that the number of solutions to $a-b=x$ with $a\\in A$ and $b\\in B$ is at least $cN$.", "background": "The minimum overlap problem. The example (with $N$ even) $A=\\{N/2+1,\\ldots,3N/2\\}$ shows that $c\\leq 1/2$ (indeed, Erd\\H{o}s initially conjectured that $c=1/2$). The lower bound of $c\\geq 1/4$ is trivial, and Scherk improved this to $1-1/\\sqrt{2}=0.29\\cdots$. The current records are $ 0.379005 < c < 0.380924, $ the lower bound due to White \\cite{Wh22} and the upper bound due to AlphaEvolve \\cite{GGTW25}, improving slightly on an upper bound due to Haugland \\cite{Ha16}.\nThis is discussed in problem C17 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[GGTW25] B. Georgiev, J. G\\'{o}mez-Serrano, T. Tao, and A. Wagner, Mathematical exploration and discovery at scale. arXiv:2511.02864 (2025).\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ha16] Haugland, J. K., The minimum overlap problem revisited. arXiv:1609.08000 (2016).\n\n[Wh22] White, E. P., Erd\\H{o}s' minimum overlap problem. arXiv:2201.05704 (2022).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The exact minimum-overlap constant remains open; the maintained record reports the rigorous interval 0.379005", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1895, "problem_number": "EP-38", "title": "Erdős Problem #38", "statement": "Does there exist $B\\subset\\mathbb{N}$ which is not an additive basis, but is such that for every set $A\\subseteq\\mathbb{N}$ of Schnirelmann density $\\alpha$ and every $N$ there exists $b\\in B$ such that $ \\lvert (A\\cup (A+b))\\cap \\{1,\\ldots,N\\}\\rvert\\geq (\\alpha+f(\\alpha))N $ where $f(\\alpha)>0$ for $0<\\alpha <1 $?\nThe Schnirelmann density is defined by $ d_s(A) = \\inf_{N\\geq 1}\\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N}. $ ", "background": "Erd\\H{o}s \\cite{Er36c} proved that if $B$ is an additive basis of order $k$ then, for any set $A$ of Schnirelmann density $\\alpha$, for every $N$ there exists some integer $b\\in B$ such that $ \\lvert (A\\cup (A+b))\\cap \\{1,\\ldots,N\\}\\rvert\\geq \\left(\\alpha+\\frac{\\alpha(1-\\alpha)}{2k}\\right)N. $ It seems an interesting question (not one that Erd\\H{o}s appears to have asked directly, although see [35]) to improve the lower bound here, even in the case $B=\\mathbb{N}$. Erd\\H{o}s observed that a random set of density $\\alpha$ shows that the factor of $\\frac{\\alpha(1-\\alpha)}{2}$ in this case cannot be improved past $\\alpha(1-\\alpha)$.\nThis is a stronger property than $B$ being an essential component (see [37]). Linnik \\cite{Li42} gave the first construction of an essential component which is not an additive basis.\nReferences\n\n\n[Er36c] Erd\\H{o}s, P., On the arithmetical density of the sum of two sequences, one of which forms a basis for the integers. Acta. Arith. (1936), 201-207.\n\n[Li42] Linnik, U. V., On Erd\"{o}s's theorem on the addition of numerical sequences. Rec. Math. [Mat. Sbornik] N.S. (1942), 67-78.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The maintained Erdős Problems tracker records an affirmative 2026 solution, verified in Lean. A sparse random set B with B(x) polylogarithmic is not an additive basis and nevertheless gives the requested finite-N translate increment, quantitatively f(alpha) >> alpha(1-alpha)^2. The public record includes a proof PDF, a Fourier/Hoeffding proof sketch, and a statement-alignment check, but no conventional refereed article yet.\n\n**Verified partial progress.**\n\n- Erdős's 1936 basis theorem gives the increment alpha(1-alpha)/(2k) when B is an additive basis of order k.\n- Linnik constructed an essential component that is not an additive basis, although essential-component status alone does not imply EP-38's uniform finite-N property.\n- The 2026 random construction strengthens existence by giving a sparse B and the quantitative order f(alpha) >> alpha(1-alpha)^2.\n\n**Full solution or refutation.**\n\nChoose B randomly with inclusion probabilities comparable to (log n)^epsilon/n. Concentration makes B(x) polylogarithmic, hence too sparse to be an additive basis. Uniform Fourier concentration for B intersected with (N/2,N], combined with an averaged count of (A+b) hitting the complement of A, yields for every A of Schnirelmann density alpha and every scale N an element b in B giving a positive density increment of order alpha(1-alpha)^2. The proof uses the asymptotic-basis convention; being non-asymptotic-basis is stronger than the non-basis condition needed here.\n\n**What remains.**\n\nThe existential question is closed on the tracker/formalized interpretation. A journal-quality proof, an independently reproducible pinned Lean build, explicit constants, and determination of the optimal f(alpha), especially near alpha=0 and alpha=1, remain desirable. Expert review should also retain the basis-definition and endpoint conventions explicitly.\n\n**Sources checked.**\n\n- Erdős Problem 38 proof artifact, repository PDF (2026). (primary): https://github.com/spicylemonade/erdos-38\n Evidence used: Primary proof write-up linked by the tracker and formal statement; develops the sparse random-set solution.\n- Thomas F. Bloom, Erdős Problem #38, Erdős Problems (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/38\n Evidence used: Records PROVED (LEAN), identifies the 2026 solution, and states f(alpha) >> alpha(1-alpha)^2.\n- Google DeepMind Formal Conjectures, FormalConjectures/ErdosProblems/38.lean (accessed 2026-08-17). (source_collection): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/38.lean\n Evidence used: Preserves the exact quantified formal statement, explains the additive-basis convention, and records the external Lean-proof provenance.\n- Paul Erdős, On the arithmetical density of the sum of two sequences, one of which forms a basis for the integers, Acta Arithmetica 1 (1936). (primary): https://combinatorica.hu/~p_erdos/1936-06.pdf\n Evidence used: Primary historical source for the additive-basis partial theorem underlying the recorded alpha(1-alpha)/(2k) increment.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1896, "problem_number": "EP-39", "title": "Erdős Problem #39", "statement": "Is there an infinite Sidon set $A\\subset \\mathbb{N}$ such that $ \\lvert A\\cap \\{1\\ldots,N\\}\\rvert \\gg_\\epsilon N^{1/2-\\epsilon} $ for all $\\epsilon>0$?", "background": "The trivial greedy construction achieves $\\gg N^{1/3}$. The first improvement on this was achieved by Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS81b}, who found an infinite Sidon set with growth rate $\\gg (N\\log N)^{1/3}$. The current best bound of $\\gg N^{\\sqrt{2}-1+o(1)}$ is due to Ruzsa \\cite{Ru98}.\nErd\\H{o}s \\cite{Er73} had offered \\$25 for any construction which achieves $N^{c}$ for some $c>1/3$. Later he \\cite{Er77c} offered \\$100 for a construction which achieves $\\omega(N)N^{1/3}$ for some $\\omega(N)\\to \\infty$.\nErd\\H{o}s proved that for every infinite Sidon set $A$ we have $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}=0. $ Erd\\H{o}s and R\\'{e}nyi have constructed, for any $\\epsilon>0$, a set $A$ such that $ \\lvert A\\cap \\{1\\ldots,N\\}\\rvert \\gg_\\epsilon N^{1/2-\\epsilon} $ for all large $N$ and $1_A\\ast 1_A(n)\\ll_\\epsilon 1$ for all $n$.\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[AKS81b] Ajtai, Mikl\\'os and Koml\\'os, J\\'anos and Szemer\\'{e}di, Endre, A dense infinite {S}idon sequence. European J. Combin. (1981), 1--11.\n\n[Er73] Erd\\H{o}s, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ru98] Ruzsa, Imre Z., An infinite Sidon sequence. J. Number Theory (1998), 63-71.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The near-one-half exponent for an infinite Sidon set remains open; Ruzsa's best-known exponent is sqrt(2)-1+o(1), later made explicit by Cilleruelo.\n\n**Verified partial progress.**\n\n- Ruzsa constructed an infinite Sidon sequence with A(x)=x^(sqrt(2)-1+o(1)).\n- Cilleruelo gave an explicit construction at the same exponent.\n- Erdos-Renyi near-one-half constructions have bounded representations rather than the Sidon property and do not resolve this record.\n\n**Full solution or refutation.**\n\nNo Sidon construction with the requested N^(1/2-epsilon) lower growth was located.\n\n**What remains.**\n\nConstruct an infinite Sidon set of near-square-root density or prove an obstruction.\n\n**Sources checked.**\n\n- Imre Z. Ruzsa, An infinite Sidon sequence, Journal of Number Theory 68 (1998), 63-71. (primary): https://doi.org/10.1006/jnth.1997.2192\n Evidence used: Published construction proving the sqrt(2)-1 exponent.\n- Javier Cilleruelo, Infinite Sidon sequences, Advances in Mathematics 255 (2014), 474-486. (primary): https://doi.org/10.1016/j.aim.2014.01.011\n Evidence used: Provides an explicit construction with the same asymptotic exponent.\n- Thomas F. Bloom, Erdos Problem #39, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/39\n Evidence used: Lists the target as open and records the construction history.\n\n**Review notes.** Bounded-representation results are not silently upgraded to Sidon results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1897, "problem_number": "EP-40", "title": "Erdős Problem #40", "statement": "For what functions $g(N)\\to \\infty$ is it true that $ \\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg \\frac{N^{1/2}}{g(N)} $ implies $\\limsup 1_A\\ast 1_A(n)=\\infty$?", "background": "This is a stronger form of the Erd\\H{o}s-Tur\\'{a}n conjecture [28] (since establishing this for any function $g(N)\\to \\infty$ would imply a positive solution to [28]).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No diverging function g for the displayed density implication is known; any affirmative instance would imply the open Erdos-Turan conjecture EP-28.\n\n**Verified partial progress.**\n\n- EP-40 is explicitly a stronger form of EP-28.\n- Known EP-28 theorems force fixed finite representation lower bounds under exceptional-set density hypotheses, not unboundedness from the displayed pointwise A(x) lower bound.\n\n**Full solution or refutation.**\n\nNo positive g-regime or counterexample was located.\n\n**What remains.**\n\nDetermine which, if any, diverging functions g force unbounded representation counts under the stated density hypothesis.\n\n**Sources checked.**\n\n- Thomas F. Bloom, history of Erdos Problem #40, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/40\n Evidence used: States that any affirmative instance would imply EP-28 and records the problem as open.\n- Ioannis Konstantoulas, Lower bounds for a conjecture of Erdos and Turan, Acta Arithmetica 159 (2013), 301-313. (primary): https://eudml.org/doc/279150\n Evidence used: A related but distinct exceptional-density partial theorem, included to delimit what is known.\n\n**Review notes.** The known partial theorem is carefully distinguished from the pointwise-density question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1898, "problem_number": "EP-41", "title": "Erdős Problem #41", "statement": "Let $A\\subset\\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\\in A$ (aside from the trivial coincidences). Is it true that $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/3}}=0? $ ", "background": "Erd\\H{o}s proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences then $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}=0. $ This is discussed in problem C11 of Guy's collection \\cite{Gu04}, in which Guy says Erd\\H{o}s offered \\$500 for the general problem of whether, for all $h\\geq 2$, $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/h}}=0 $ whenever the sum of $h$ terms in $A$ are distinct. This was proved for $h=4$ by Nash \\cite{Na89} and for all even $h$ by Chen \\cite{Ch96b}.\nReferences\n\n\n[Ch96b] Chen, Sheng, A note on {$B_{2k}$} sequences. J. Number Theory (1996), 1--3.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Na89] Nash, John C. M., On {$B_4$}-sequences. Canad. Math. Bull. (1989), 446--449.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The triple-sum B_3 case remains open, although the analogous liminf theorem is known for h=2, for h=4, and for all even h.\n\n**Verified partial progress.**\n\n- Erdos proved the h=2 Sidon-sequence case.\n- Nash proved h=4 and Chen extended the result to every even h.\n- Those results do not cover the odd h=3 problem in this record.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for the literal B_3 liminf assertion was located.\n\n**What remains.**\n\nResolve the odd-order h=3 case, or identify an odd-order obstruction to the even-h method.\n\n**Sources checked.**\n\n- Thomas F. Bloom, history of Erdos Problem #41, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/41\n Evidence used: Lists the h=3 record as open and records the h=4 and all-even-h advances.\n- Javier Cilleruelo and Carlos Tesoro, Dense infinite B_h sequences, arXiv:1206.3087. (primary): https://arxiv.org/abs/1206.3087\n Evidence used: Provides related B_h construction context while not claiming a solution to the liminf question.\n\n**Review notes.** Even-order results are recorded as partial context, not as a solution to the odd triple-sum case.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1899, "problem_number": "EP-42", "title": "Erdős Problem #42", "statement": "Let $M\\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\\subset \\{1,\\ldots,N\\}$ there is another Sidon set $B\\subset \\{1,\\ldots,N\\}$ of size $M$ such that $(A-A)\\cap(B-B)=\\{0\\}$?", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The maintained tracker records SOLVED (LEAN). A pinned Lean development proves the all-M eventual existence theorem, and a separate bridge extracts a threshold function N0(M). A July 2026 audit reports the proof kernel-clean and unconditional but the signed natural-language/formal statement-fidelity verdict was still unreviewed, so expert review remains appropriate.\n\n**Verified partial progress.**\n\n- The cases M=1 and M=2 were established directly.\n- A separate Lean-assisted proof established M=3, conditional in its first public form on a cited strongly sum-free-set theorem.\n- The all-M proof reportedly yields the quantitative strengthening |B| >> sqrt(log log N / log log log N) for sufficiently large N.\n\n**Full solution or refutation.**\n\nThe reported proof uses a compact Cayley-graph/Fourier lemma: a dense Cayley graph with suitable one-sided Fourier bias contains every fixed clique. Applied to the allowed differences outside A-A and followed by extraction of a Sidon subset, it produces a Sidon B of any prescribed fixed size M. The formal theorem is phrased for inclusion-maximal Sidon A; every finite Sidon A extends to such an A', and avoidance of A'-A' implies avoidance of A-A, recovering the dataset's universal formulation.\n\n**What remains.**\n\nThe fixed-M existence question is closed. Remaining tasks include a concise conventional publication, signed fidelity review of the exact formal statement and its maximal-set reduction, and sharp or explicit dependence of the threshold N0(M) on M. The discussion's proposed double-exponential threshold was not independently certified in this triage.\n\n**Sources checked.**\n\n- Shashi456, Erdos/P42/CompactCayley/Proof.lean, pinned Lean development (2026). (primary): https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P42/CompactCayley/Proof.lean\n Evidence used: Primary 14,801-line formal proof artifact for the eventual all-M theorem.\n- Kenta Kitahara, Erdos42Constructive.lean, commit 1f82c76be43cb56f22e2f7f792e392d5fb3ff78c (2026). (primary): https://github.com/KitaKen1/erdos-42-constructive-variant/blob/1f82c76be43cb56f22e2f7f792e392d5fb3ff78c/lean/Erdos42Constructive.lean\n Evidence used: Formal bridge from the eventual theorem to an existential threshold function; displays the proof's standard axiom footprint.\n- William Blair, Erdős problem 42: formal-proof fidelity finding, audit of 2026-07-06. (authoritative_secondary): https://erdos.constellate.science/finding.html?n=42\n Evidence used: Reports the proof unconditional and kernel-clean with only standard Lean axioms, while explicitly flagging statement fidelity as not yet signed off.\n- Thomas F. Bloom, Erdős Problem #42, Erdős Problems (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/42\n Evidence used: Records SOLVED (LEAN), all fixed M, and the quantitative growing-companion bound.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1900, "problem_number": "EP-43", "title": "Erdős Problem #43", "statement": "If $A,B\\subset \\{1,\\ldots,N\\}$ are two Sidon sets such that $(A-A)\\cap(B-B)=\\{0\\}$ then is it true that $ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq\\binom{f(N)}{2}+O(1), $ where $f(N)$ is the maximum possible size of a Sidon set in $\\{1,\\ldots,N\\}$? If $\\lvert A\\rvert=\\lvert B\\rvert$ then can this bound be improved to $ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq (1-c+o(1))\\binom{f(N)}{2} $ for some constant $c>0$?", "background": "Since it is known that $f(N)\\sim \\sqrt{N}$ (see [30]) the latter question is equivalent to asking whether, if $\\lvert A\\rvert=\\lvert B\\rvert$, $ \\lvert A\\rvert \\leq \\left(\\frac{1}{\\sqrt{2}}-c+o(1)\\right)\\sqrt{N} $ for some constant $c>0$. In the comments Tao has given a proof of this upper bound without the $-c$.\nIn the comments Barreto has given a negative answer to the second question: for infinitely many $N$ there exist Sidon sets $A,B\\subset \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=\\lvert B\\rvert$ and $(A-A)\\cap (B-B)=\\{0\\}$ and $ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\geq (1-o(1))\\binom{f(N)}{2}. $\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Both proposed bounds have negative answers. The O(1)-error claim fails by applying solved EP-42 to a maximum Sidon set with an arbitrarily large fixed companion. The equal-size fixed-factor improvement fails along N=(q^2-1)/2 through a parity split of Bose-Chowla modular Sidon sets, yielding ratio 1-o(1) to the extremal benchmark.\n\n**Verified partial progress.**\n\n- Erdős had already identified the main asymptotic upper scale (1+o(1))N/2, equivalent to the benchmark binomial(f(N),2).\n- A standard smoothing/Erdős-Turán argument gives |A|^2+|B|^2 <= N+O(N^(3/4)), so the first proposal remains true at main-term scale despite the false O(1) remainder.\n- The Bose-Chowla family makes the equal-size main term asymptotically sharp and rules out every fixed positive saving.\n\n**Full solution or refutation.**\n\nFor the first question, if a uniform error constant K existed, choose M with binomial(M,2)>K, take A of size f(N), and use EP-42 for all sufficiently large N to find a disjoint-difference Sidon B of size M. For the second, start with a Bose-Chowla Sidon set modulo q^2-1, split it into even and odd elements, rescale and shift into [1,(q^2-1)/2], and balance the two parts. Modular uniqueness gives two Sidon sets with no common nonzero difference. Their equal size is q/2+O(sqrt(q)), so their binomial sum is N/2+O(N^(3/4))=(1-o(1))binomial(f(N),2).\n\n**What remains.**\n\nThe two yes/no questions are closed. Finer problems remain: determine the optimal lower-order error and exact finite-N extremal behavior. Because the first disproof depends on the recent EP-42 formal result and the second application is currently presented in a tracker forum rather than a refereed paper, expert consolidation and publication are still warranted.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #43 and discussion thread, Erdős Problems (accessed 2026-08-17). (authoritative_secondary): https://www.erdosproblems.com/43\n Evidence used: Maintained problem record explicitly classifies the problem as disproved, derives the first negative answer from EP-42, and records Barreto's infinite equal-size counterexample family.\n- Kevin Barreto, negative resolution using the Bose-Chowla construction, Erdős Problem #43 discussion (2026). (primary): https://www.erdosproblems.com/forum/thread/43?order=oldest\n Evidence used: Original public proof: parity split of a modular Sidon set, proof of disjoint nonzero difference sets, and the N=(q^2-1)/2 asymptotic calculation.\n- R. C. Bose and S. Chowla, Theorems in the additive theory of numbers, Commentarii Mathematici Helvetici 37 (1962/63), 141-147, doi:10.1007/BF02566968. (primary): https://repository.ias.ac.in/8578/\n Evidence used: Published primary source for the modular Sidon construction used in the equal-size counterexamples.\n- Shashi456, Erdos/P42/CompactCayley/Proof.lean (2026). (primary): https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P42/CompactCayley/Proof.lean\n Evidence used: Primary formal evidence for EP-42, whose arbitrary fixed companion size refutes EP-43's O(1) remainder.\n- Paul Erdős, On Disjoint Sets of Differences, Journal of Number Theory 18 (1984), 99-109. (primary): https://combinatorica.hu/~p_erdos/1984-10.pdf\n Evidence used: Historical primary source for the disjoint-difference framework and main-term asymptotic context.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1901, "problem_number": "EP-44", "title": "Erdős Problem #44", "statement": "Let $N\\geq 1$ and $A\\subset \\{1,\\ldots,N\\}$ be a Sidon set. Is it true that, for any $\\epsilon>0$, there exist $M$ and $B\\subset \\{N+1,\\ldots,M\\}$ (which may depend on $N,A,\\epsilon$) such that $A\\cup B\\subset \\{1,\\ldots,M\\}$ is a Sidon set of size at least $(1-\\epsilon)M^{1/2}$?", "background": "See also [329] and [707] (indeed a positive solution to [707] implies a positive solution to this problem, which in turn implies a positive solution to [329]).\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The arbitrary-initial-segment extension problem for near-optimal finite Sidon sets remains open; dense constructions without preservation of a prescribed initial set do not answer it.\n\n**Verified partial progress.**\n\n- The maintained source records the implication EP-707 => EP-44 => EP-329.\n- Classical near-square-root finite Sidon constructions establish the relevant density scale but do not absorb every fixed finite Sidon configuration.\n\n**Full solution or refutation.**\n\nNo full extension theorem or counterexample was located.\n\n**What remains.**\n\nFor every finite Sidon A in [N] and every epsilon>0, construct a later endpoint M and a Sidon extension of size at least (1-epsilon)sqrt(M), or disprove this for some A.\n\n**Sources checked.**\n\n- Paul Erdos, Some of my favourite problems in number theory, combinatorics, and geometry, Resenhas 2 (1995), 165-186. (primary): https://doi.org/10.11606/resimeusp.v2i2.74798\n Evidence used: Primary source posing the Sidon extension problem.\n- Thomas F. Bloom, Erdos Problem #44, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/44\n Evidence used: Lists the exact problem as open and records its implication links.\n\n**Review notes.** No OCR or formulation defect was found in the imported statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1902, "problem_number": "EP-50", "title": "Erdős Problem #50", "statement": "Schoenberg proved that for every $c\\in [0,1]$ the density of $ \\{ n\\in \\mathbb{N} : \\phi(n)\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The pointwise question of whether Schoenberg's totient distribution can have a finite positive derivative anywhere remains open, despite the known pure singularity of the distribution.\n\n**Verified partial progress.**\n\n- Erdos proved that the distribution function is purely singular.\n- Pure singularity implies derivative zero almost everywhere, but it does not exclude a positive derivative at an exceptional point.\n\n**Full solution or refutation.**\n\nNo theorem eliminating positive derivatives at every point was located.\n\n**What remains.**\n\nProve that no x has an existing positive derivative f'(x), or exhibit such an exceptional point.\n\n**Sources checked.**\n\n- Paul Erdos, Some of my favourite problems in number theory, combinatorics, and geometry, Resenhas 2 (1995), 165-186. (primary): https://www.ime.usp.br/~yoshi/resenhas/abstracts/Erdos.pdf\n Evidence used: States the pointwise question and the pure-singularity partial result.\n- Thomas F. Bloom, Erdos Problem #50 source, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/50\n Evidence used: Retains the exact question as open.\n\n**Review notes.** The almost-everywhere statement was not promoted to the stronger pointwise conclusion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1903, "problem_number": "EP-51", "title": "Erdős Problem #51", "statement": "Is there an infinite set $A\\subset \\mathbb{N}$ such that for every $a\\in A$ there is an integer $n$ such that $\\phi(n)=a$, and yet if $n_a$ is the smallest such integer then $n_a/a\\to \\infty$ as $a\\to\\infty$?", "background": "Carmichael has asked whether there is an integer $t$ for which $\\phi(n)=t$ has exactly one solution. Erd\\H{o}s has proved that if such a $t$ exists then there must be infinitely many such $t$.\nSee also [694].\nThis is discussed in problems B36 and B39 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains open whether infinitely many totient values have least preimage n_a with n_a/a tending to infinity.\n\n**Verified partial progress.**\n\n- Carmichael's unique-preimage conjecture and Erdos's conditional infinitude result are related but do not imply the required least-preimage ratio.\n- A proposed online proof was rejected because it unjustifiably identified the least inverse of a product of p_i-1 with the product of the p_i.\n\n**Full solution or refutation.**\n\nNo verified infinite family or impossibility result for the least-preimage ratio was located.\n\n**What remains.**\n\nConstruct infinitely many totient values whose smallest preimage-to-value ratio diverges, or prove this cannot occur.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #51, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/51\n Evidence used: Lists the exact least-inverse-totient question as open and gives the Carmichael context.\n- Discussion of Erdos Problem #51, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/51\n Evidence used: Documents why a posted candidate proof does not establish the least-preimage claim.\n\n**Review notes.** The Carmichael background was kept separate from the literal asymptotic target.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1904, "problem_number": "EP-52", "title": "Erdős Problem #52", "statement": "Let $A$ be a finite set of integers. Is it true that for every $\\epsilon>0$ $ \\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg_\\epsilon \\lvert A\\rvert^{2-\\epsilon}? $ ", "background": "The sum-product problem. Erd\\H{o}s and Szemer\\'{e}di \\cite{ErSz83} proved a lower bound of $\\lvert A\\rvert^{1+c}$ for some constant $c>0$, and an upper bound of $ \\lvert A\\rvert^2 \\exp\\left(-c\\frac{\\log\\lvert A\\rvert}{\\log\\log \\lvert A\\rvert}\\right) $ for some constant $c>0$. The lower bound has been improved a number of times. The current record is $ \\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg\\lvert A\\rvert^{\\frac{1270}{951}-o(1)} $ due to Bloom \\cite{Bl25} (note $1270/951=1.33543\\cdots$). A complete history of sum-product bounds can be found at this webpage.\nThere is likely nothing special about the integers in this question, and indeed Erd\\H{o}s and Szemer\\'{e}di also ask a similar question about finite sets of real or complex numbers. The current best bound for sets of reals is the same bound of Bloom above. The best bound for complex numbers is $ \\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg\\lvert A\\rvert^{\\frac{4}{3}+c} $ for some absolute constant $c>0$, due to Basit and Lund \\cite{BaLu19}.\nOne can in general ask this question in any setting where addition and multiplication are defined (once one avoids any trivial obstructions such as zero divisors or finite subfields). For example, it makes sense for subsets of finite fields. The current record is that there exists $c>0$ such that if $A\\subseteq \\mathbb{F}_p$ with $\\lvert A\\rvert \n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The near-quadratic sum-product conjecture for finite subsets of the rational integers remains open; the analogous conjecture for arbitrary real sets was disproved in 2026 but that construction is not integral.\n\n**Verified partial progress.**\n\n- Bloom proved the general lower exponent 1270/951-o(1), which applies in particular to integer sets.\n- Bloom, Sawin, Schildkraut, and Zhelezov constructed real algebraic-integer sets in growing-degree number fields with both sumset and product set of size at most |A|^(2-c).\n- The 2026 real counterexample does not consist of subsets of Z and therefore does not refute the literal record.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for finite sets of rational integers was located.\n\n**What remains.**\n\nProve the |A|^(2-epsilon) lower bound for all finite A subset Z, or construct integer counterexamples with a fixed exponent gap.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Control and its applications in additive combinatorics, arXiv:2501.09470 (2025). (primary): https://arxiv.org/abs/2501.09470\n Evidence used: Theorem 4 supplies the 1270/951-epsilon lower exponent.\n- Thomas Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781 (2026). (primary): https://arxiv.org/abs/2605.28781\n Evidence used: Disproves the real-set analogue using algebraic integers outside the rational integers.\n- Thomas F. Bloom, discussion of Erdos Problem #52, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/52\n Evidence used: Keeps the integer formulation open while recording the 2026 real counterexample.\n\n**Review notes.** The imported background's claim that the reals should behave similarly is outdated; the exact integer statement was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1905, "problem_number": "EP-60", "title": "Erdős Problem #60", "statement": "Does every graph on $n$ vertices with $>\\mathrm{ex}(n;C_4)$ edges contain $\\gg n^{1/2}$ many copies of $C_4$?", "background": "Conjectured by Erd\\H{o}s and Simonovits, who could not even prove that at least $2$ copies of $C_4$ are guaranteed.\nThe behaviour of $\\mathrm{ex}(n;C_4)$ is the subject of [765].\nHe, Ma, and Yang \\cite{HeMaYa21} have proved this conjecture when $n=q^2+q+1$ for some even integer $q$.\nReferences\n\n\n[HeMaYa21] He, J. and Ma, J. and Yang, T., Some extremal results on 4-cycles. Journal of Combinatorial Theory B (2021).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The C4 supersaturation assertion immediately above ex(n,C4) remains open for general n, with He-Ma-Yang proving the conjectured behavior for n=q^2+q+1 in their even-q family.\n\n**Verified partial progress.**\n\n- He, Ma, and Yang proved the conjecture for n=q^2+q+1 in the even-q cases covered by their theorem.\n- This infinite special family does not provide a uniform result for all vertex counts.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was located.\n\n**What remains.**\n\nShow that every n-vertex graph with more than ex(n,C4) edges contains at least a constant times sqrt(n) copies of C4, uniformly in n.\n\n**Sources checked.**\n\n- Jialin He, Jie Ma, and Tianchi Yang, Some extremal results on 4-cycles, Journal of Combinatorial Theory, Series B 149 (2021), 92-108. (primary): https://doi.org/10.1016/j.jctb.2021.01.007\n Evidence used: Proves the recorded exact special family and related C4 supersaturation results.\n- Thomas F. Bloom, Erdos Problem #60, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/60\n Evidence used: Lists the general problem as open and identifies the He-Ma-Yang special case.\n\n**Review notes.** The special family is recorded as partial progress, not a general resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1906, "problem_number": "EP-61", "title": "Erdős Problem #61", "statement": "For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either a complete graph or independent set on $\\geq n^c$ vertices?", "background": "Conjectured by Erd\\H{o}s and Hajnal \\cite{ErHa89}, who proved that a complete graph or independent set must exist on $ \\geq \\exp(c_H\\sqrt{\\log n}) $ many vertices, where $c_H>0$ is some constant. This was improved by Buci\\'{c}, Nguyen, Scott, and Seymour \\cite{BNSS23} to $ \\geq \\exp(c_H\\sqrt{\\log n\\log\\log n}). $ See also the entry in the graphs problem collection.\nReferences\n\n\n[BNSS23] Buci\\'C, M. and Nguyen, T. and Scott, A. and Seymour, P., A loglog step towards Erdos-Hajnal. arXiv:2301.10147 (2023).\n\n[ErHa89] Erd\\H{o}s, P. and Hajnal, A., Ramsey-type theorems. Discrete Appl. Math. (1989), 37-52.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The general Erdos-Hajnal induced-subgraph conjecture remains open; the best general guarantee located is exp(c_H sqrt(log n log log n)), while all forbidden graphs on at most five vertices are now known cases.\n\n**Verified partial progress.**\n\n- Bucic, Nguyen, Scott, and Seymour improved the general homogeneous-set bound by a sqrt(log log n) factor in the exponent.\n- Nguyen, Scott, and Seymour established strong path cases.\n- Their 2026 five-vertex-path paper completes the conjecture for all H with at most five vertices.\n\n**Full solution or refutation.**\n\nNo polynomial homogeneous-set theorem for every fixed forbidden induced graph H was located.\n\n**What remains.**\n\nFor arbitrary fixed H, prove that every induced-H-free n-vertex graph has a clique or stable set of size n^c(H), or find a counterexample.\n\n**Sources checked.**\n\n- Matija Bucic, Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density I. A loglog step towards Erdos-Hajnal, IMRN (2024). (primary): https://arxiv.org/abs/2301.10147\n Evidence used: Proves the current general exp(c sqrt(log n log log n)) lower bound.\n- Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density V. All paths approach Erdos-Hajnal, arXiv:2307.15032. (primary): https://arxiv.org/abs/2307.15032\n Evidence used: Provides major progress for forbidden induced paths.\n- Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density VII. The five-vertex path, Proceedings of the London Mathematical Society (2026). (primary): https://doi.org/10.1112/plms.70133\n Evidence used: Settles P5 and, with known cases, all forbidden graphs on at most five vertices.\n- Thomas F. Bloom, Erdos Problem #61, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/61\n Evidence used: Lists the unrestricted conjecture as open.\n\n**Review notes.** Solved finite-size forbidden families were not conflated with the universal quantifier over H.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1907, "problem_number": "EP-62", "title": "Erdős Problem #62", "statement": "If $G_1,G_2$ are two graphs with chromatic number $\\aleph_1$ then must there exist a graph $G$ whose chromatic number is $4$ (or even $\\aleph_0$) which is a subgraph of both $G_1$ and $G_2$?", "background": "Erd\\H{o}s also asked \\cite{Er87} about finding a common subgraph $H$ (with chromatic number either $4$ or $\\aleph_0$) in any finite collection of graphs with chromatic number $\\aleph_1$.\nEvery graph with chromatic number $\\aleph_1$ contains all sufficiently large odd cycles (which have chromatic number $3$), see [594]. This was proved by Erd\\H{o}s, Hajnal, and Shelah \\cite{EHS74}. Erd\\H{o}s wrote \\cite{Er87} that 'probably' every graph with chromatic number $\\aleph_1$ contains as subgraphs all graphs with chromatic number $4$ with sufficiently large girth.\nReferences\n\n\n[EHS74] Erd\\H{o}s, P. and Hajnal, A. and Shelah, S., On some general properties of chromatic numbers. Topics in topology (Proc. Colloq., Keszthely, 1972) (1974), 243-255.\n\n[Er87] Erd\\H{o}s, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains open whether two aleph_1-chromatic graphs must share a 4-chromatic or countably chromatic subgraph; a common 3-chromatic odd cycle is guaranteed.\n\n**Verified partial progress.**\n\n- Erdos, Hajnal, and Shelah proved that every aleph_1-chromatic graph contains all sufficiently large odd cycles.\n- Choosing one odd length above the thresholds for both graphs gives a common subgraph of chromatic number 3.\n- The same threshold argument handles every fixed finite collection but does not reach chromatic number 4.\n\n**Full solution or refutation.**\n\nNo common-subgraph theorem at chromatic number 4 or aleph_0 was located.\n\n**What remains.**\n\nRaise the guaranteed common chromatic number from 3 to 4, or to aleph_0, or construct two aleph_1-chromatic counterexamples.\n\n**Sources checked.**\n\n- P. Erdos, A. Hajnal, and S. Shelah, On some general properties of chromatic numbers, Topics in Topology (1974), 243-255. (primary): https://www.renyi.hu/~p_erdos/1974-17.pdf\n Evidence used: Proves containment of all sufficiently large odd cycles.\n- Fan Chung, Open problems of Paul Erdos in graph theory. (authoritative_secondary): https://fanchung.ucsd.edu/wp/ep.pdf\n Evidence used: States the common-subgraph problem and the known chromatic-3 boundary.\n- Thomas F. Bloom, Erdos Problem #62, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/62\n Evidence used: Lists the chromatic-4/aleph_0 question as open.\n\n**Review notes.** The finite-collection variant also has the common odd-cycle consequence, but the stated target remains open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1908, "problem_number": "EP-65", "title": "Erdős Problem #65", "statement": "Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The logarithmic lower-bound question is solved, with an asymptotically sharp (1/2-o(1)) log k bound, while the claim that complete bipartite graphs minimize the reciprocal cycle-length sum remains open.\n\n**Verified partial progress.**\n\n- Gyarfás, Komlós, and Szemerédi proved that the sum of reciprocal distinct cycle lengths is bounded below by a constant times log k.\n- Liu and Montgomery's work yields the asymptotically sharp lower bound (1/2-o(1)) log k recorded in the current survey.\n- Complete bipartite examples attain the leading one-half scale, but exact extremality has not been verified.\n\n**Full solution or refutation.**\n\nThe first of the two literal questions is a theorem; the second remains unresolved.\n\n**What remains.**\n\nDetermine whether, under the stated n-vertex and kn-edge constraints, a complete bipartite graph always minimizes the reciprocal sum of distinct cycle lengths.\n\n**Sources checked.**\n\n- András Gyarfás, János Komlós, and Endre Szemerédi, On the distribution of cycle lengths in graphs, Journal of Graph Theory 8 (1984), 441-462. (primary): https://doi.org/10.1002/jgt.3190080402\n Evidence used: Proves the logarithmic lower bound, settling the first question.\n- Hong Liu and Richard Montgomery, A solution to Erdos and Hajnal's odd cycle problem, Journal of the American Mathematical Society 36 (2023), 1191-1234. (primary): https://doi.org/10.1090/jams/1018\n Evidence used: Source for the cycle-length machinery behind the asymptotically sharp bound recorded by the tracker.\n- Thomas F. Bloom, Erdos Problem #65, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/65\n Evidence used: Separates the solved first question from the open minimizer question and reports an ambiguous forthcoming-work note.\n\n**Review notes.** The tracker says forthcoming work proves a 'maximised' statement, contrary to the literal minimization question; no public preprint resolving this wording was found, so it is not used as a solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1909, "problem_number": "EP-66", "title": "Erdős Problem #66", "statement": "Is there $A\\subseteq \\mathbb{N}$ such that $ \\lim_{n\\to \\infty}\\frac{1_A\\ast 1_A(n)}{\\log n} $ exists and is $\neq 0$?", "background": "A suitably constructed random set has this property if we are allowed to ignore an exceptional set of density zero. The challenge is obtaining this with no exceptional set. Erd\\H{o}s believed the answer should be no. Erd\\H{o}s and S\\'{a}rk\"{o}zy proved that $ \\frac{\\lvert 1_A\\ast 1_A(n)-\\log n\\rvert}{\\sqrt{\\log n}}\\to 0 $ is impossible. Erd\\H{o}s suggests it may even be true that the $\\liminf$ and $\\limsup$ of $1_A\\ast 1_A(n)/\\log n$ are always separated by some absolute constant.\nHorv\\'{a}th \\cite{Ho07} proved that $ \\lvert 1_A\\ast 1_A(n)-\\log n\\rvert \\leq (1-\\epsilon)\\sqrt{\\log n} $ cannot hold for all large $n$.\nReferences\n\n\n[Ho07] G. Horv\\'{a}th, An improvement of a theorem of Erd\\H{o}s and S\\'{a}rk\"{o}zy. Pollack Periodica (2007), 155-161.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains open whether an additive representation function can have a nonzero pointwise asymptotic constant times log n with no exceptional integers.\n\n**Verified partial progress.**\n\n- Probabilistic constructions give representation functions of order Theta(log n), and density-one variants are known, but these do not give a pointwise limiting ratio.\n- Horváth proved that an eventual error at most (1-epsilon)sqrt(g(n)) is impossible for a broad class of increasing target functions g.\n- For g(n)=c log n, the square-root obstruction is much stronger than the o(log n) error allowed by convergence of the ratio, so it does not settle the question.\n\n**Full solution or refutation.**\n\nNo construction with a pointwise nonzero limit and no theorem excluding all such constructions was located.\n\n**What remains.**\n\nConstruct A with (1_A*1_A)(n)/log n converging to a positive constant for every sufficiently large n, or prove every A has nonconvergent normalized representations.\n\n**Sources checked.**\n\n- Gábor Horváth, An improvement of a theorem of Erdos and Sárközy, Pollack Periodica 2, Supplement (2007), 155-161. (primary): https://doi.org/10.1556/Pollack.2.2007.S.14\n Evidence used: Proves the sharp eventual square-root-scale approximation obstruction.\n- P. Erdos and P. Tetali, Representations of integers as the sum of k terms, Random Structures & Algorithms 1 (1990), 245-261. (primary): https://tetali.math.gatech.edu/RESEARCH/pubs.html\n Evidence used: Provides probabilistic logarithmic-order representation constructions, which are weaker than ratio convergence.\n- Thomas F. Bloom, Erdos Problem #66, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/66\n Evidence used: Lists the no-exceptional-set limiting problem as open and supplies the intended nonzero formulation.\n\n**Review notes.** The imported statement contains an OCR/escape defect: intended 'not equal to 0' appears as a newline followed by 'eq 0'; the source record was preserved. Representation-counting conventions change constants only.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1910, "problem_number": "EP-68", "title": "Erdős Problem #68", "statement": "Is $ \\sum_{n\\geq 2}\\frac{1}{n!-1} $ irrational?", "background": "The decimal expansion is A331373 in the OEIS. Weisenberg has observed that this sum can also be written as $ \\sum_{k\\geq 1}\\sum_{n\\geq 2}\\frac{1}{(n!)^k}. $ Erd\\H{o}s \\cite{Er88c} notes that $\\sum \\frac{1}{n!+t}$ should be transcendental for every integer $t$.\nReferences\n\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The irrationality of the sum over n>=2 of 1/(n!-1) remains open; the geometric-series double-sum identity does not currently supply an irrationality proof.\n\n**Verified partial progress.**\n\n- The identity 1/(n!-1)=sum_{k>=1}(n!)^(-k) follows from an absolutely convergent geometric series.\n- Erdos formulated a broader expected transcendence phenomenon for factorial-denominator series, but it remains conjectural and does not settle the t=-1 case.\n\n**Full solution or refutation.**\n\nNo proof of irrationality, transcendence, or rationality was located.\n\n**What remains.**\n\nDetermine whether the factorial-denominator constant is irrational; the broader transcendence expectation is stronger and also unresolved here.\n\n**Sources checked.**\n\n- Paul Erdos, On the irrationality of certain series: problems and results, New Advances in Transcendence Theory (1988), 102-109. (primary): https://renyi.hu/~p_erdos/1988-22.pdf\n Evidence used: Primary discussion of the broader factorial-denominator irrationality and transcendence questions.\n- Paul Erdos, On the irrationality of certain series: problems and results, Cambridge University Press chapter record. (primary): https://doi.org/10.1017/CBO9780511897184.009\n Evidence used: Publisher bibliographic record for the 1988 primary source.\n- Thomas F. Bloom, discussion of Erdos Problem #68, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/68\n Evidence used: Retains the exact irrationality question as open.\n\n**Review notes.** The double-sum identity was treated as an exact reformulation, not as evidence of a solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1911, "problem_number": "EP-70", "title": "Erdős Problem #70", "statement": "Let $\\mathfrak{c}$ be the ordinal of the real numbers, $\\beta$ be any countable ordinal, and $2\\leq n<\\omega$. Is it true that $\\mathfrak{c}\\to (\\beta, n)_2^3$?", "background": "Erd\\H{o}s and Rado proved that $\\mathfrak{c}\\to (\\omega+n,4)_2^3$ for any $2\\leq n<\\omega$.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The full triple partition relation for arbitrary countable beta and finite n remains open; the established Erdős-Rado/Jones theorem reaches countable order types below omega+omega with the second target fixed at 4.\n\n**Verified partial progress.**\n\n- Erdos and Rado proved the real-order relation with target omega+m versus 4 for every finite m.\n- Jones published a short proof for every alpha below omega+omega.\n- No source extending this to every countable beta and every finite n was located.\n\n**Full solution or refutation.**\n\nNo full proof or counterexample was located.\n\n**What remains.**\n\nDetermine the relation for arbitrary countable beta and every finite n at least 2; specialist review should also disambiguate the imported phrase 'ordinal of the real numbers' from the real-order formulation in the primary theorem.\n\n**Sources checked.**\n\n- Albin L. Jones, A short proof of a partition relation for triples, Electronic Journal of Combinatorics 7 (2000), R24. (primary): https://doi.org/10.37236/1502\n Evidence used: Proves the Erdős-Rado real-order relation for alpha below omega+omega and second target 4.\n- Paul Erdos and Richard Rado, A partition calculus in set theory, Bulletin of the American Mathematical Society 62 (1956). (primary): https://users.renyi.hu/~p_erdos/1956-02.pdf\n Evidence used: Foundational source for the partition calculus and the special-case theorem.\n- Thomas F. Bloom, Erdos Problem #70, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/70\n Evidence used: Lists the full relation as open and records only the omega+n versus 4 special case.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 10, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 10, "name": "set_theory", "display_name": "Set Theory", "description": "Foundations of mathematics, infinite sets, and cardinality.", "slug": "set-theory", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1912, "problem_number": "EP-74", "title": "Erdős Problem #74", "statement": "Let $f(n)\\to \\infty$ (possibly very slowly). Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $f(n)$ edges?", "background": "Conjectured by Erd\\H{o}s, Hajnal, and Szemer\\'{e}di \\cite{EHS82}.\nR\"{o}dl \\cite{Ro82} has proved this for hypergraphs, and also proved there is such a graph (with chromatic number $\\aleph_0$) if $f(n)=\\epsilon n$ for any fixed constant $\\epsilon>0$.\nIt is open even for $f(n)=\\sqrt{n}$. Erd\\H{o}s offered \\$500 for a proof but only \\$250 for a counterexample. This fails (even with $f(n)\\gg n$) if the graph has chromatic number $\\aleph_1$ (see [111]).\nReferences\n\n\n[EHS82] Erd\\H{o}s, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Ro82] R\"{o}dl, Vojt\\vEch, Nearly bipartite graphs with large chromatic number. Combinatorica (1982), 377-383.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The arbitrary f(n) tending to infinity problem remains open, even for f(n)=sqrt(n); Rödl proved the linear-error case f(n)=epsilon n and the hypergraph analogue.\n\n**Verified partial progress.**\n\n- Rödl constructs a countably chromatic graph satisfying the conclusion for every fixed linear allowance epsilon n.\n- Rödl also resolves the corresponding hypergraph version.\n- The analogous assertion fails at chromatic number aleph_1, so increasing the chromatic cardinal is not a route to the target.\n\n**Full solution or refutation.**\n\nNo construction for arbitrary slowly diverging f and no counterexample was located.\n\n**What remains.**\n\nConstruct the requested infinite-chromatic graph for every prescribed f(n) tending to infinity, or refute it; the sqrt(n) case is already open.\n\n**Sources checked.**\n\n- Vojtech Rödl, Nearly bipartite graphs with large chromatic number, Combinatorica 2 (1982), 377-383. (primary): https://doi.org/10.1007/BF02579434\n Evidence used: Proves the linear-error and hypergraph partial results.\n- Paul Erdos, Andras Hajnal, and Endre Szemeredi, On almost bipartite large chromatic graphs, Annals of Discrete Mathematics 12 (1982), 117-123. (primary): https://doi.org/10.1016/S0304-0208(08)73497-2\n Evidence used: Original source for the graph problem and surrounding infinite-chromatic constructions.\n- Thomas F. Bloom, Erdos Problem #74, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/74\n Evidence used: Lists the question as open, including the sqrt(n) case, and summarizes the known linear result.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1913, "problem_number": "EP-75", "title": "Erdős Problem #75", "statement": "Is there a graph of chromatic number $\\aleph_1$ such that for all $\\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices then $H$ contains an independent set of size $>n^{1-\\epsilon}$?", "background": "Conjectured by Erd\\H{o}s, Hajnal, and Szemer\\'{e}di \\cite{EHS82}. In \\cite{Er95d} Erd\\H{o}s suggests this may even be true with an independent set of size $\\gg n$.\nSee also [750].\nReferences\n\n\n[EHS82] Erd\\H{o}s, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Er95d] Erd\\H{o}s, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The maintained record says open, but an April 2026 discussion claim says the classical Specker graph gives the stronger hereditary bound alpha(H) much greater than n/log n; no stable manuscript, formal proof, or incorporated status update was located.\n\n**Verified partial progress.**\n\n- Erdos, Hajnal, and Szemeredi constructed and studied the relevant uncountably chromatic graph.\n- Published work proves strong hereditary independence estimates for finite shift graphs.\n- A tracker screening comment reports only a minor issue in the 2026 Specker-graph calculation, but the page still labels the problem open.\n\n**Full solution or refutation.**\n\nThere is a credible community-level affirmative claim, but the accessible literature does not yet justify a conservative SOLVED-IN-LITERATURE label.\n\n**What remains.**\n\nPublish or formally verify the Specker-graph alpha(H) bound and reconcile it with the maintained status. The tracker also adds an aleph_1-vertex condition omitted from the imported statement.\n\n**Sources checked.**\n\n- Paul Erdos, Andras Hajnal, and Endre Szemeredi, On almost bipartite large chromatic graphs, Annals of Discrete Mathematics 12 (1982), 117-123. (primary): https://combinatorica.hu/~p_erdos/1982-11.pdf\n Evidence used: Studies the classical construction underlying the recent claimed corollary.\n- Andrii Arman, Vojtech Rödl, and Marcelo Tadeu Sales, Independent Sets in Subgraphs of a Shift Graph, Electronic Journal of Combinatorics 29 (2022), P1.26. (primary): https://doi.org/10.37236/10453\n Evidence used: Provides modern published hereditary independence results for finite shift graphs, relevant context but not an explicit proof of EP-75.\n- Erdos Problems contributors, EP-75 discussion, April 2026, checked 2026-08-17. (source_collection): https://www.erdosproblems.com/forum/thread/75\n Evidence used: Contains the affirmative Specker-graph claim and screening remarks, while the surrounding maintained page remains OPEN.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1914, "problem_number": "EP-77", "title": "Erdős Problem #77", "statement": "If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then find the value of $ \\lim_{k\\to \\infty}R(k)^{1/k}. $ ", "background": "Erd\\H{o}s offered \\$100 for just a proof of the existence of this constant, without determining its value. He also offered \\$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. (In \\cite{Er88} he raises this prize to \\$10000). Erd\\H{o}s proved $ \\sqrt{2}\\leq \\liminf_{k\\to \\infty}R(k)^{1/k}\\leq \\limsup_{k\\to \\infty}R(k)^{1/k}\\leq 4. $ The upper bound has been improved to $4-\\tfrac{1}{128}$ by Campos, Griffiths, Morris, and Sahasrabudhe \\cite{CGMS23}. This was improved to $3.7992\\cdots$ by Gupta, Ndiaye, Norin, and Wei \\cite{GNNW24}.\nA shorter and simpler proof of an upper bound of the strength $4-c$ for some constant $c>0$ (and a generalisation to the case of more than two colours) was given by Balister, Bollob\\'{a}s, Campos, Griffiths, Hurley, Morris, Sahasrabudhe, and Tiba \\cite{BBCGHMST24}.\nIn \\cite{Er93} Erd\\H{o}s writes 'I have no idea what the value of $\\lim R(k)^{1/k}$ should be, perhaps it is $2$ but we have no real evidence for this.'\nThis problem is #3 in Ramsey Theory in the graphs problem collection.\nSee also [1029] for a problem concerning a lower bound for $R(k)$ and discussion of lower bounds in general.\nA famous quote of Erd\\H{o}s concerns the difficulty of finding exact values for $R(k)$. This is often repeated in the words of Spencer, who phrased it as an alien attacking race. The earliest such quote in a paper of Erd\\H{o}s I have found is in \\cite{Er93}, where he writes:\n'Sometime ago, I made the following joke. If an evil spirit would appear and say \"unless you give me the value of $R(5)$ within a year, I will exterminate humanity\", then our best bet would be perhaps to get all our computers working on $R(5)$ and we probably would get its value in a year.\nIf he would ask for $R(6)$, the best strategy probably would be to destroy it before it can destroy us. If we would be so clever that we could give the answer by mathematics, we would just tell him: \"if you try to do something you will see what will happent to you...\". I think we are strong enugh now and the only evil spirit we have to feel is the one which is in ourselves (quoting somebody: I have seen the enemy and them are us). Now enough of the idle talk and back to Mathematics.'\nReferences\n\n\n[BBCGHMST24] Balister, P. and Bollob\\'{a}s, B. and Campos, M. and Griffiths, S. and Hurley, E.\nand Morris, R. and Sahasrabudhe, J. and Tiba, M., Upper bounds for multicolour Ramsey numbers. arXiv:2410.17197 (2024).\n\n[CGMS23] Campos, Marcelo and Griffiths, Simon and Morris, Robert and Sahasrabudhe, Julian, An exponential improvement for diagonal Ramsey. arXiv:2303.09521 (2023).\n\n[Er88] Erd\\H{o}s, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[GNNW24] Gupta, P. and Ndiaye, N. and Norin, S. and Wei, L., Optimizing the CGMS upper bound on Ramsey numbers. arXiv:2407.19026 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Existence and value of the exponential growth-rate limit for diagonal Ramsey numbers remain open; current exponential bases lie between sqrt(2) and about 3.8.\n\n**Verified partial progress.**\n\n- Campos, Griffiths, Morris, and Sahasrabudhe gave the first exponential improvement below base 4.\n- Gupta, Ndiaye, Norin, and Wei optimized that method to R(k) at most 3.8^(k+o(k)), with the tracker recording 3.7992...\n- These bounds control limsup but do not prove convergence of R(k)^(1/k).\n\n**Full solution or refutation.**\n\nNo proof that the limit exists and no determination of its value was located.\n\n**What remains.**\n\nProve convergence of R(k)^(1/k), then determine the limiting constant.\n\n**Sources checked.**\n\n- Marcelo Campos, Simon Griffiths, Robert Morris, and Julian Sahasrabudhe, An exponential improvement for diagonal Ramsey, arXiv:2303.09521. (primary): https://arxiv.org/abs/2303.09521\n Evidence used: First exponential improvement on the classical base-4 upper bound.\n- Parth Gupta, Ndiame Ndiaye, Sergey Norin, and Louis Wei, Optimizing the CGMS upper bound on Ramsey numbers, arXiv:2407.19026. (primary): https://arxiv.org/abs/2407.19026\n Evidence used: Optimizes the modern upper-bound method to an asymptotic base about 3.8.\n- Thomas F. Bloom, Erdos Problem #77, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/77\n Evidence used: Lists the limit problem as open and summarizes current exponential bounds.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1915, "problem_number": "EP-78", "title": "Erdős Problem #78", "statement": "Give a constructive proof that $R(k)>C^k$ for some constant $C>1$.", "background": "Erd\\H{o}s gave a simple probabilistic proof that $R(k) \\gg k2^{k/2}$.\nEquivalently, this question asks for an explicit construction of a graph on $n$ vertices which does not contain any clique or independent set of size $\\geq c\\log n$ for some constant $c>0$.\nIn \\cite{Er69b} Erd\\H{o}s asks for even a construction whose largest clique or independent set has size $o(n^{1/2})$, which is now known.\nCohen \\cite{Co15} (see the introduction for further history) constructed a graph on $n$ vertices which does not contain any clique or independent set of size $ \\geq 2^{(\\log\\log n)^{C}} $ for some constant $C>0$. Li \\cite{Li23b} has recently improved this to $ \\geq (\\log n)^{C} $ for some constant $C>0$.\nThis problem is #4 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[Co15] Gil Cohen, Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs. Electronic Colloquium on Computational Complexity (2015).\n\n[Er69b] Erd\\H{o}s, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[Li23b] Li, X., Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More. arXiv:2303.06802 (2023).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Explicit Ramsey graphs with logarithmic homogeneous sets are still unknown; Li's current construction gives polylogarithmic homogeneous-set size.\n\n**Verified partial progress.**\n\n- Cohen constructed explicit graphs whose largest clique or independent set has size 2^((log log N)^C).\n- Li improved the explicit bound to (log N)^C for an absolute constant C.\n- Conditional-expectation search is not considered the explicit efficient construction requested here.\n\n**Full solution or refutation.**\n\nNo explicit construction achieving O(log N) clique and independence numbers, equivalently R(k)>C^k constructively, was located.\n\n**What remains.**\n\nReduce the best explicit polylogarithmic homogeneous-set bound to O(log N).\n\n**Sources checked.**\n\n- Gil Cohen, Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs, SIAM Journal on Computing 50 (2021). (primary): https://doi.org/10.1137/16M1096219\n Evidence used: Provides the earlier explicit subpolynomial Ramsey-graph construction.\n- Xin Li, Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More, arXiv:2303.06802. (primary): https://arxiv.org/abs/2303.06802\n Evidence used: Gives explicit K-Ramsey graphs with K=log^{O(1)} N.\n- Thomas F. Bloom, Erdos Problem #78, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/78\n Evidence used: Lists the constructive exponential lower-bound problem as open and records the Cohen and Li progress.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1916, "problem_number": "EP-80", "title": "Erdős Problem #80", "statement": "Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is contained in at least one triangle, must contain a book of size $m$, that is, an edge shared by at least $m$ different triangles.\nEstimate $f_c(n)$. In particular, is it true that $f_c(n)>n^{\\epsilon}$ for some $\\epsilon>0$? Or $f_c(n)\\gg \\log n$?", "background": "A problem of Erd\\H{o}s and Rothschild. Alon and Trotter showed that, provided $c<1/4$, $f_c(n)\\ll_c n^{1/2}$. Szemer\\'{e}di observed that his regularity lemma implies that $f_c(n)\\to \\infty$.\nEdwards (unpublished) and Khadziivanov and Nikiforov \\cite{KhNi79} proved independently that $f_c(n) \\geq n/6$ when $c>1/4$ (see [905]).\nFox and Loh \\cite{FoLo12} proved that $ f_c(n) \\leq n^{O(1/\\log\\log n)} $ for all $c<1/4$, disproving the first conjecture of Erd\\H{o}s.\nThe best known lower bounds for $f_c(n)$ are those from Szemer\\'{e}di's regularity lemma, and as such remain very poor.\nSee also [600] and the entry in the graphs problem collection.\nReferences\n\n\n[FoLo12] Fox, Jacob and Loh, Po-Shen, On a problem of Erd\\H{o}s and {R}othschild on edges in\ntriangles. Combinatorica (2012), 619--628.\n\n[KhNi79] Had\\v ziivanov, N. G. and Nikiforov, S. V., Solution of a problem of {P}. Erd\\H{o}s about the maximum\nnumber of triangles with a common edge in a graph. C. R. Acad. Bulgare Sci. (1979), 1315--1318.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Fox and Loh disproved the polynomial lower-bound question for every fixed c<1/4, while the logarithmic lower-bound question and a sharp estimate of f_c(n) remain open.\n\n**Verified partial progress.**\n\n- Szemeredi's regularity lemma proves f_c(n) tends to infinity for every fixed positive c.\n- Fox and Loh construct examples with f_c(n)<=n^{O(1/log log n)} for c<1/4, refuting any fixed positive power lower bound.\n- For c>1/4, the sharp density transition yields the linear lower bound f_c(n)>=n/6.\n- Potechin supplies a further lower bound in the near-n^2/4 critical window.\n\n**Full solution or refutation.**\n\nOne explicit subquestion is settled negatively, but the proposed logarithmic lower bound is not known.\n\n**What remains.**\n\nFor fixed c<1/4, determine the growth of f_c(n), in particular whether f_c(n) is bounded below by a constant multiple of log n.\n\n**Sources checked.**\n\n- Jacob Fox and Po-Shen Loh, On a problem of Erdos and Rothschild on edges in triangles, Combinatorica 32 (2012), 619-628. (primary): https://doi.org/10.1007/s00493-012-2844-3\n Evidence used: Proves the subpolynomial upper construction for c<1/4 and therefore refutes the polynomial conjecture.\n- Aaron Potechin, A note on a problem of Erdos and Rothschild, arXiv:1412.1838. (primary): https://arxiv.org/abs/1412.1838\n Evidence used: Gives a quantitative lower bound in the critical window near n^2/4 edges.\n- Thomas F. Bloom, Erdos Problem #80, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/80\n Evidence used: Separates the disproved polynomial question from the open logarithmic question and records the density threshold.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1917, "problem_number": "EP-81", "title": "Erdős Problem #81", "statement": "Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $n^2/6+O(n)$ many cliques?", "background": "Asked by Erd\\H{o}s, Ordman, and Zalcstein \\cite{EOZ93}, who proved an upper bound of $(1/4-\\epsilon)n^2$ many cliques (for some very small $\\epsilon>0$). The example of all edges between a complete graph on $n/3$ vertices and an empty graph on $2n/3$ vertices show that $n^2/6+O(n)$ is sometimes necessary.\nA split graph is one where the vertices can be split into a clique and an independent set. Every split graph is chordal. Chen, Erd\\H{o}s, and Ordman \\cite{CEO94} have shown that any split graph can be partitioned into $\\frac{3}{16}n^2+O(n)$ many cliques.\nSee also [1017].\nReferences\n\n\n[CEO94] Chen, Guan-Tao and Erd\\H{o}s, Paul and Ordman, Edward T., Clique partitions of split graphs. Combinatorics, graph theory, algorithms and applications\n(Beijing, 1993) (1994), 21-30.\n\n[EOZ93] Erd\\H{o}s, Paul and Ordman, Edward T. and Zalcstein, Yechezkel, Clique partitions of chordal graphs. Combin. Probab. Comput. (1993), 409-415.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The n^2/6+O(n) clique-partition upper bound for all chordal graphs remains open; recent work gives an explicit older-style constant and a conditional proof under a weak Clique-Drop Hypothesis.\n\n**Verified partial progress.**\n\n- Erdos, Ordman, and Zalcstein proved an upper bound (1/4-epsilon)n^2 and exhibited a family requiring n^2/6+O(n).\n- Chen, Erdos, and Ordman proved the 3n^2/16+O(n) upper bound for split graphs.\n- A June 2026 formalized refinement makes epsilon explicit with epsilon at least 1/133.\n- A separate June 2026 note obtains the target conditionally under wCDH and exactly analyzes the lower-bound family.\n\n**Full solution or refutation.**\n\nNo unconditional n^2/6+O(n) theorem for all chordal graphs was located.\n\n**What remains.**\n\nProve the target bound for arbitrary chordal graphs, or prove a sufficient structural hypothesis such as wCDH universally.\n\n**Sources checked.**\n\n- Paul Erdos, Edward T. Ordman, and Yechezkel Zalcstein, Clique Partitions of Chordal Graphs, Combinatorics, Probability and Computing 2 (1993), 409-415. (primary): https://doi.org/10.1017/S0963548300000808\n Evidence used: Original lower construction and general upper bound below n^2/4.\n- Guan-Tao Chen, Paul Erdos, and Edward T. Ordman, Clique partitions of split graphs, Combinatorics, Graph Theory, Algorithms and Applications (1994), 21-30. (primary): https://ordman.net/MathResearch/CEOClique_Parts.pdf\n Evidence used: Proves the 3n^2/16+O(n) bound for the split-graph subclass.\n- Erdos Problems contributors, EP-81 discussion, June 2026, checked 2026-08-17. (source_collection): https://www.erdosproblems.com/forum/thread/81\n Evidence used: Records the formalized explicit epsilon and the separate conditional wCDH result, both clearly short of a full solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1918, "problem_number": "EP-82", "title": "Erdős Problem #82", "statement": "Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $F(n)/\\log n\\to \\infty$.", "background": "Conjectured by Erd\\H{o}s, Fajtlowicz, and Stanton. It is known that $F(5)=3$ and $F(7)=4$.\nRamsey's theorem implies that $F(n)\\gg \\log n$. Bollob\\'{a}s observed that $F(n)\\ll n^{1/2+o(1)}$. Alon, Krivelevich, and Sudakov \\cite{AKS07} have improved this to $n^{1/2}(\\log n)^{O(1)}$.\nIn \\cite{Er93} Erd\\H{o}s asks whether, if $t(n)$ is the largest trivial (either empty or complete) subgraph which a graph on $n$ vertices must contain (so that $t(n) \\gg \\log n$ by Ramsey's theorem), then is it true that $ F(n)-t(n)\\to \\infty? $ Equivalently, and in analogue with the definition of Ramsey numbers, one can define $G(n)$ to be the minimal $m$ such that every graph on $m$ vertices contains a regular induced subgraph on at least $n$ vertices. This problem can be rephrased as asking whether $G(n) \\leq 2^{o(n)}$.\nFajtlowicz, McColgan, Reid, and Staton \\cite{FMRS95} showed that $G(1)=1$, $G(2)=2$, $G(3)=5$, $G(4)=7$, and $G(5)\\geq 12$. Boris Alexeev and Brendan McKay (see the comments and this site) have computed $G(5)=17$, $G(6)\\geq 21$, and $G(7)\\geq 29$.\nSee also [1031] for another question regarding induced regular subgraphs.\nReferences\n\n\n[AKS07] Alon, N. and Krivelevich, M. and Sudakov, B., Large nearly regular induced subgraphs. arXiv:0710.2106 (2007).\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[FMRS95] No reference found.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No superlogarithmic universal lower bound for regular induced subgraphs is known; recent work improves the upper construction to (sqrt(2e)+o(1))sqrt(n) but does not address the required lower-order gap.\n\n**Verified partial progress.**\n\n- Ramsey's theorem gives F(n)>=c log n, the current general lower-bound scale.\n- Dyson and McKay improved the upper construction to O(sqrt(n)) and several small inverse Ramsey values.\n- Levy's August 2026 preprint improves the upper constant to sqrt(2e) and proves optimality within type-based random constructions.\n- For triangle-free graphs, a discussion observation gives the stronger Omega(sqrt(n log n)) lower bound.\n\n**Full solution or refutation.**\n\nNo proof that F(n)/log n tends to infinity and no counterexample was located.\n\n**What remains.**\n\nObtain any universally superlogarithmic lower bound, or construct graphs showing that the Ramsey logarithmic scale is asymptotically sharp.\n\n**Sources checked.**\n\n- Noga Alon, Michael Krivelevich, and Benny Sudakov, Large nearly regular induced subgraphs, SIAM Journal on Discrete Mathematics 22 (2008). (primary): https://arxiv.org/abs/0710.2106\n Evidence used: Establishes the earlier n^(1/2) polylogarithmic upper construction and frames the exact-regular problem.\n- Paul W. Dyson and Brendan D. McKay, Ramsey numbers for regular induced subgraphs, arXiv:2604.08215. (primary): https://arxiv.org/abs/2604.08215\n Evidence used: Improves the general O(sqrt(n)) construction and finite inverse Ramsey bounds.\n- Ariel Edgardo Levy, Sharp asymptotics for regular induced subgraphs of type-based random graphs, arXiv:2608.08169. (primary): https://arxiv.org/abs/2608.08169\n Evidence used: Gives the current sqrt(2e) upper constant and a matching barrier within type-based models.\n- Thomas F. Bloom, Erdos Problem #82, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/82\n Evidence used: Lists the superlogarithmic lower-bound conjecture as open and records current results through spring 2026.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1919, "problem_number": "EP-84", "title": "Erdős Problem #84", "statement": "The cycle set of a graph $G$ on $n$ vertices is a set $A\\subseteq \\{3,\\ldots,n\\}$ such that there is a cycle in $G$ of length $\\ell$ if and only if $\\ell \\in A$. Let $f(n)$ count the number of possible such $A$.\nProve that $f(n)=o(2^n)$.\nProve that $f(n)/2^{n/2}\\to \\infty$.", "background": "Conjectured by Erd\\H{o}s and Faudree, who showed that $2^{n/2}\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Verstraete solved the first displayed conjecture f(n)=o(2^n), and Nenadov strengthened its quantitative bound; the second conjecture f(n)/2^(n/2) tending to infinity remains open.\n\n**Verified partial progress.**\n\n- Verstraete proved f(n)<=2^(n-n^(1/10)), settling the first question.\n- Nenadov improved this to f(n)<=2^(n-n^(1/2-o(1))).\n- Erdos and Faudree's lower bound f(n)>2^(n/2) is insufficient to force a diverging ratio.\n\n**Full solution or refutation.**\n\nOne of the record's two assertions is proved, while the lower-growth assertion remains unresolved.\n\n**What remains.**\n\nProve or disprove f(n)/2^(n/2) tending to infinity; the exponential growth-rate limit is also unknown.\n\n**Sources checked.**\n\n- Jacques Verstraete, On the Number of Sets of Cycle Lengths, Combinatorica 24 (2004), 719-730. (primary): https://doi.org/10.1007/s00493-004-0043-6\n Evidence used: Proves the first displayed conjecture with a quantitative power saving in the exponent.\n- Rajko Nenadov, Improved bound on the number of cycle sets, Combinatorial Theory 6 (2026), article 17. (primary): https://arxiv.org/abs/2501.09904\n Evidence used: Improves the upper bound to 2^(n-n^(1/2-o(1))).\n- Thomas F. Bloom, Erdos Problem #84, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/84\n Evidence used: Separates the solved first assertion from the open second assertion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1920, "problem_number": "EP-86", "title": "Erdős Problem #86", "statement": "Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that every subgraph of $Q_n$ with $ \\geq \\left(\\frac{1}{2}+o(1)\\right)n2^{n-1} $ many edges contains a $C_4$?", "background": "Let $f(n)$ be the maximum number of edges in a subgraph of $Q_n$ without a $C_4$, so that this conjecture is that $f(n)\\leq (\\frac{1}{2}+o(1))n2^{n-1}$.\nErd\\H{o}s \\cite{Er91} showed that $ f(n) \\geq \\left(\\frac{1}{2}+\\frac{c}{n}\\right)n2^{n-1} $ for some constant $c>0$, and wrote it is 'perhaps not hopeless' to determine $f(n)$ exactly. Brass, Harborth, and Nienborg \\cite{BHN95} improved this to $ f(n) \\geq \\left(\\frac{1}{2}+\\frac{c}{\\sqrt{n}}\\right)n2^{n-1} $ for some constant $c>0$.\nBalogh, Hu, Lidicky, and Liu \\cite{BHLL14} proved that $f(n)\\leq 0.6068 n2^{n-1}$. This was improved to $\\leq 0.60318 n2^{n-1}$ by Baber \\cite{Ba12b}.\nA similar question can be asked for other even cycles.\nSee also [666] and the entry in the graphs problem collection.\nReferences\n\n\n[BHLL14] Balogh, J\\'{o}zsef and Hu, Ping and Lidick\\'{y}, Bernard and Liu, Hong, Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube. European J. Combin. (2014), 75-85.\n\n[BHN95] Brass, Peter and Harborth, Heiko and Nienborg, Hauke, On the maximum number of edges in a {$C_4$}-free subgraph of\n{$Q_n$}. J. Graph Theory (1995), 17--23.\n\n[Ba12b] R. Baber, Tur\\'{a}n densities of hypercubes. arXiv:1201.3587 (2012).\n\n[Er91] Erd\"{o}s, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The conjectured asymptotic C4-free edge density 1/2 in the hypercube remains open; the best located general upper density is 0.60318 and known constructions approach 1/2 from above.\n\n**Verified partial progress.**\n\n- Brass, Harborth, and Nienborg give a lower density 1/2+c/sqrt(n), showing the target leading constant would be sharp.\n- Balogh, Hu, Lidicky, and Liu proved upper density 0.6068.\n- Baber improved the upper density to 0.60318.\n- Minamoto's 2026 constructions improve finite records for Q_7 and Q_8 but do not change the asymptotic density.\n\n**Full solution or refutation.**\n\nNo asymptotic upper bound of 1/2+o(1) or counterexample of density bounded above 1/2 was located.\n\n**What remains.**\n\nClose the asymptotic density gap between 1/2 and 0.60318.\n\n**Sources checked.**\n\n- Rahil Baber, Turan densities of hypercubes, arXiv:1201.3587. (primary): https://arxiv.org/abs/1201.3587\n Evidence used: Provides the current general upper density 0.60318 for C4-free subgraphs of Q_n.\n- Jozsef Balogh, Ping Hu, Bernard Lidicky, and Hong Liu, Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube, European Journal of Combinatorics 35 (2014), 75-85. (primary): https://arxiv.org/abs/1201.0209\n Evidence used: Establishes the prior 0.6068 upper density and the flag-algebra framework.\n- Minamo Minamoto, New Lower Bounds for C4-Free Subgraphs of the Hypercubes Q7 and Q8, arXiv:2603.29127. (primary): https://arxiv.org/abs/2603.29127\n Evidence used: Gives new explicit finite-dimensional lower certificates without an asymptotic improvement.\n- Thomas F. Bloom, Erdos Problem #86, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/86\n Evidence used: Lists the asymptotic density problem as open and summarizes the established general bounds.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1921, "problem_number": "EP-87", "title": "Erdős Problem #87", "statement": "Let $\\epsilon >0$. Is it true that, if $k$ is sufficiently large, then $ R(G)>(1-\\epsilon)^kR(k) $ for every graph $G$ with chromatic number $\\chi(G)=k$?\nEven stronger, is there some $c>0$ such that, for all large $k$, $R(G)>cR(k)$ for every graph $G$ with chromatic number $\\chi(G)=k$?", "background": "Erd\\H{o}s originally conjectured that $R(G)\\geq R(k)$, which is trivial for $k=3$, but fails already for $k=4$, as Faudree and McKay \\cite{FaMc93} showed that $R(W)=17$ for the pentagonal wheel $W$.\nSince $R(k)\\leq 4^k$ this is trivial for $\\epsilon\\geq 3/4$. Yuval Wigderson points out that $R(G)\\gg 2^{k/2}$ for any $G$ with chromatic number $k$ (via a random colouring), which asymptotically matches the best-known lower bounds for $R(k)$.\nThis problem is #12 and #13 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[FaMc93] Faudree, R. J. and McKay, B., A conjecture of Erd\\H{o}s and the Ramsey number $r(W_6)$. J. Combinatorial Math. and Combinatorial Computing (1993), 23-31.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Both proposed Ramsey-versus-chromatic-number lower bounds remain open; the original stronger inequality fails at chromatic number four.\n\n**Verified partial progress.**\n\n- The pentagonal wheel refutes R(G)>=R(k) at k=4.\n- A random-colouring argument gives R(G)>>2^(k/2) for every k-chromatic graph.\n- That lower bound matches only the known exponential scale for R(k), not the requested comparison.\n\n**Full solution or refutation.**\n\nNo proof or counterexample to either stated asymptotic comparison was located.\n\n**What remains.**\n\nObtain a uniform comparison between R(G) and R(k) for every k-chromatic graph.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #87, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/87\n Evidence used: Lists the problem as open, the pentagonal-wheel counterexample, and the random-colouring bound.\n\n**Review notes.** The failure of the original conjecture is not a refutation of the weaker displayed questions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1922, "problem_number": "EP-89", "title": "Erdős Problem #89", "statement": "Does every set of $n$ distinct points in $\\mathbb{R}^2$ determine $\\gg n/\\sqrt{\\log n}$ many distinct distances?", "background": "A $\\sqrt{n}\\times\\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz \\cite{GuKa15} who proved that there are always $\\gg n/\\log n$ many distinct distances.\nA stronger form (see [604]) may be true: is there a single point which determines $\\gg n/\\sqrt{\\log n}$ distinct distances, or even $\\gg n$ many such points, or even that this is true averaged over all points - for example, if $d(x)$ counts the number of distinct distances from $x$ then in \\cite{Er75f} Erd\\H{o}s conjectured $ \\sum_{x\\in A}d(x) \\gg \\frac{n^2}{\\sqrt{\\log n}}, $ where $A\\subset \\mathbb{R}^2$ is any set of $n$ points.\nSee also [661], and [1083] for the generalisation to higher dimensions.\nReferences\n\n\n[Er75f] Erd\\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\n\n[GuKa15] Guth, Larry and Katz, Nets Hawk, On the Erd\\H{o}s distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The sharp n/sqrt(log n) planar distinct-distances conjecture remains open; Guth and Katz proved the lower bound n/log n.\n\n**Verified partial progress.**\n\n- The square grid gives the matching n/sqrt(log n) upper construction.\n- Guth and Katz proved a lower bound of order n/log n.\n- The remaining factor sqrt(log n) is open.\n\n**Full solution or refutation.**\n\nNo improvement closing the logarithmic gap was located.\n\n**What remains.**\n\nProve the n/sqrt(log n) lower bound or find a smaller distinct-distance construction.\n\n**Sources checked.**\n\n- Larry Guth and Nets Hawk Katz, On the Erdos distinct distances problem in the plane, Annals of Mathematics 181 (2015), 155-190. (primary): https://annals.math.princeton.edu/2015/181-1/p07\n Evidence used: Published n/log n lower bound.\n- Thomas F. Bloom, Erdos Problem #89, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/89\n Evidence used: Lists the sharp conjecture as open and explains the grid obstruction.\n\n**Review notes.** The result is an open status with a quantitatively near-sharp theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1923, "problem_number": "EP-90", "title": "Erdős Problem #90", "statement": "Does every set of $n$ distinct points in $\\mathbb{R}^2$ contain at most $n^{1+O(1/\\log\\log n)}$ many pairs which are distance 1 apart?", "background": "The unit distance problem. In \\cite{Er94b} Erd\\H{o}s dates this conjecture to 1946. In \\cite{Er82e} he offers \\$300 for the upper bound $n^{1+o(1)}$.\nThis would be the best possible, as is shown by a set of lattice points. It is easy to show that there are $O(n^{3/2})$ many such pairs. The best known upper bound is $O(n^{4/3})$, due to Spencer, Szemer\\'{e}di, and Trotter \\cite{SST84}. In \\cite{Er83c} and \\cite{Er85} Erd\\H{o}s offers \\$250 for an upper bound of the form $n^{1+o(1)}$.\nPart of the difficulty of this problem is explained by a result of Valtr (see \\cite{Sz16}), who constructed a metric on $\\mathbb{R}^2$ and a set of $n$ points with $\\gg n^{4/3}$ unit distance pairs (with respect to this metric). The methods of the upper bound proof of Spencer, Szemer\\'{e}di, and Trotter \\cite{SST84} generalise to include this metric. Therefore to prove an upper bound better than $n^{4/3}$ some special feature of the Euclidean metric must be exploited.\nSee a survey by Szemer\\'{e}di \\cite{Sz16} for further background and related results.\nSee also [92], [96], [605], and [956]. The higher dimensional generalisation is [1085].\nReferences\n\n\n[Er82e] Erd\\H{o}s, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.\n\n[Er83c] Erd\\H{o}s, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.\n\n[Er85] Erd\\H{o}s, P., Problems and results in combinatorial geometry. Discrete geometry and convexity (New York, 1982) (1985), 1-11.\n\n[Er94b] Erd\\H{o}s, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.\n\n[SST84] Spencer, J. and Szemer\\'{e}di, E. and Trotter, Jr., W., Unit distances in the Euclidean plane. Graph theory and combinatorics (Cambridge, 1983) (1984), 293-303.\n\n[Sz16] Szemer\\'{e}di, Endre, Erd\\H{o}s's unit distance problem. Open problems in mathematics (2016), 459-477.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The planar unit-distance conjecture was disproved in May 2026. A human-verified primary paper proves that infinitely many planar point sets have at least n^(1+epsilon) unit distances for an absolute epsilon>0, and Sawin independently made the exponent explicit, obtaining more than n^1.014 unit distances for arbitrarily large n.\n\n**Verified partial progress.**\n\n- Erdős's square-grid construction gave n^(1+Omega(1/log log n)) unit distances.\n- Spencer, Szemerédi, and Trotter proved the still-best general upper bound O(n^(4/3)).\n- The 2026 construction gives a fixed polynomial improvement over exponent 1; Sawin's refinement gives an explicit exponent exceeding 1.014.\n\n**Full solution or refutation.**\n\nThe 2026 construction uses high-degree CM fields of bounded root discriminant and Golod-Shafarevich class-field towers with split primes. Geometry-of-numbers counting places many algebraic norm-one differences inside a bounded lattice window, producing infinitely many planar point sets with at least n^(1+epsilon) unit-distance pairs. A fixed epsilon>0 is incompatible with every upper bound n^(1+O(1/log log n)), so EP-90 is false.\n\n**What remains.**\n\nThe conjectured subpolynomial-over-linear upper bound is decisively refuted. The main extremal question remains wide open: improve the explicit lower exponent and the O(n^(4/3)) upper exponent, and determine the true growth of the maximum unit-distance function. The dataset background has trailing serialized-text corruption, but the mathematical statement is intact.\n\n**Sources checked.**\n\n- Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, and Melanie Matchett Wood, Remarks on the disproof of the unit distance conjecture, arXiv:2605.20695 (2026). (primary): https://arxiv.org/abs/2605.20695\n Evidence used: Theorem 1.1 gives an absolute epsilon>0 and infinitely many point sets with at least n^(1+epsilon) unit distances; the paper explicitly presents a human-digested, human-verified proof.\n- Will Sawin, An explicit lower bound for the unit distance problem, arXiv:2605.20579 (2026). (primary): https://arxiv.org/abs/2605.20579\n Evidence used: Proves more than n^1.014 unit-distance pairs for arbitrarily large n, independently making the disproof quantitative.\n- OpenAI, Planar Point Sets with Many Unit Distances, manuscript (2026). (primary): https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf\n Evidence used: Original full construction and theorem giving a fixed polynomial unit-distance lower bound.\n- Thomas F. Bloom, Erdős Problem #90, Erdős Problems (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/90\n Evidence used: Records DISPROVED (LEAN), historical bounds, and the 2026 construction.\n- Google DeepMind Formal Conjectures, FormalConjectures/ErdosProblems/90.lean (accessed 2026-08-17). (source_collection): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/90.lean\n Evidence used: Records the polynomial-lower-bound implication and Sawin's explicit exponent in a formal-source record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1924, "problem_number": "EP-91", "title": "Erdős Problem #91", "statement": "Let $n$ be a sufficently large integer. Suppose $A\\subset \\mathbb{R}^2$ has $\\lvert A\\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar.", "background": "For $n=3$ the equilateral triangle is the only such set. For $n=4$ the square or two equilateral triangles sharing an edge give two non-similar examples.\nFor $n=5$ the regular pentagon is the unique such set (which has two distinct distances). Erd\\H{o}s mysteriously remarks in \\cite{Er90} this was proved by 'a colleague'. (In \\cite{Er87b} this is described as 'a colleague from Zagreb (unfortunately I do not have his letter)'.) A published proof of this fact is provided by Kov\\'{a}cs \\cite{Ko24c}.\nIn \\cite{Er87b} Erd\\H{o}s says that there are at least two non-similar examples for $6\\leq n\\leq 9$.\nThe minimal possible number of distinct distances is the subject of [89].\nReferences\n\n\n[Er87b] Erd\\H{o}s, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\n\n[Er90] Erd\\H{o}s, Paul, Some of my favourite unsolved problems. A tribute to Paul Erd\\H{o}s (1990), 467-478.\n\n[Ko24c] Z. Kov\\'{a}cs, A note on Erd\\H{o}s's mysterious remark. arXiv:2412.05190 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The all-sufficiently-large-n non-similar-minimizer statement remains open; small cases and an unreviewed density-one route are known.\n\n**Verified partial progress.**\n\n- The n=5 uniqueness fact has a published modern proof by Kovacs.\n- At least two non-similar minimizers are documented for 6<=n<=9.\n- A 2026 tracker post proves a conditional combinatorial mechanism yielding many minimizers when an optimal diameter level has at least n+3 points, but does not prove that hypothesis in all large n.\n\n**Full solution or refutation.**\n\nNo theorem covering every sufficiently large n was located.\n\n**What remains.**\n\nEstablish the required largeness/plateau condition or another mechanism that supplies two non-similar minimizers for every large n.\n\n**Sources checked.**\n\n- Zoltan Kovacs, A note on Erdos's mysterious remark, arXiv:2412.05190 (2024). (primary): https://arxiv.org/abs/2412.05190\n Evidence used: Proves the historically missing n=5 uniqueness statement.\n- Thomas F. Bloom, Erdos Problem #91 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/91\n Evidence used: Lists the problem as open and hosts the newer unreviewed partial argument.\n\n**Review notes.** A tracker-comment argument is not promoted to a full published result.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1925, "problem_number": "EP-92", "title": "Erdős Problem #92", "statement": "Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\\mathbb{R}^2$ in which every $x\\in A$ has at least $f(n)$ points in $A$ equidistant from $x$.\nIs it true that $f(n)\\leq n^{o(1)}$? Or even $f(n) < n^{O(1/\\log\\log n)}$?", "background": "This is a stronger form of the unit distance conjecture (see [90]).\nThe set of lattice points imply $f(n) > n^{c/\\log\\log n}$ for some constant $c>0$. Erd\\H{o}s offered \\$500 for a proof that $f(n) \\leq n^{o(1)}$ but only \\$100 for a counterexample. This latter prize is downgraded to \\$50 in \\cite{ErFi97}.\nIt is trivial that $f(n) \\ll n^{1/2}$. A result of Pach and Sharir (Theorem 4 of \\cite{PaSh92}) implies $f(n) \\ll n^{2/5}$. Hunter has observed that the circle-point incidence bound of Janzer, Janzer, Methuku, and Tardos \\cite{JJMT24} implies $ f(n) \\ll n^{4/11}. $ Fishburn (personal communication to Erd\\H{o}s, later published in \\cite{ErFi97}) proved that $6$ is the smallest $n$ such that $f(n)=3$ and $8$ is the smallest $n$ such that $f(n)=4$, and suggested that the lattice points may not be best example.\nSee also [754].\nReferences\n\n\n[ErFi97] Erd\\H{o}s, Paul and Fishburn, Peter, Minimum planar sets with maximum equidistance counts. Comput. Geom. (1997), 207--218.\n\n[JJMT24] B. Janzer, O. Janzer, A. Methuku, and G. Tardos, Tight bounds for intersection-reverse sequences, edge-ordered graphs\nand applications. arXiv:2411.07188 (2024).\n\n[PaSh92] Pach, J\\'anos and Sharir, Micha, Repeated angles in the plane and related problems. J. Combin. Theory Ser. A (1992), 12--22.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The 2026 unit-distance construction also disproves this stronger local-richness conjecture. Its unit-distance graphs have polynomial average degree, and pruning to a nonempty subgraph of comparable polynomial minimum degree yields point sets in which every point has polynomially many neighbors at the common distance 1, contradicting both proposed subpolynomial bounds for f(n).\n\n**Verified partial progress.**\n\n- Erdős and Fishburn proved the exact small thresholds g(3)=6 and g(4)=8, and g(5)<=16.\n- Pach-Sharir incidence bounds imply f(n) << n^(2/5).\n- The Janzer-Janzer-Methuku-Tardos point-circle incidence bound improves the recorded upper bound to f(n) << n^(4/11).\n- The 2026 construction gives f(n) >= n^c along an unbounded sequence after minimum-degree pruning, for some absolute c>0.\n\n**Full solution or refutation.**\n\nThe primary 2026 manuscript explicitly identifies the Erdős-Fishburn conjecture and its implication. If a unit-distance graph has average degree at least 2k, repeatedly deleting vertices of degree below k leaves a nonempty subgraph of minimum degree at least k. The polynomially dense unit-distance graphs from EP-90 therefore contain planar point subsets in which every point has n^Omega(1) other points at distance 1. This directly contradicts f(n)<=n^o(1), and hence also the stronger n^O(1/log log n) proposal.\n\n**What remains.**\n\nThe subpolynomial conjectures are refuted. The growth exponent remains largely undetermined: the inherited explicit lower exponent is small while the best recorded upper bound is 4/11. Improving either side and describing extremal locally distance-rich configurations remain open. The dataset background has trailing serialized-text corruption, but the statement is readable and matches the primary manuscript's formulation where the radius may depend on the center.\n\n**Sources checked.**\n\n- OpenAI, Planar Point Sets with Many Unit Distances, manuscript (2026). (primary): https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf\n Evidence used: Immediately after Theorem 1.1, explicitly states that the construction refutes the stronger Erdős-Fishburn local equidistance bound and gives the average-degree-to-minimum-degree argument.\n- Noga Alon et al., Remarks on the disproof of the unit distance conjecture, arXiv:2605.20695 (2026). (primary): https://arxiv.org/abs/2605.20695\n Evidence used: Human-verified Theorem 1.1 supplies the polynomially dense unit-distance graphs used in the pruning argument.\n- Thomas F. Bloom, Erdős Problem #92 and revision history, Erdős Problems (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/history/92\n Evidence used: Records the exact formulation, the 2026-05-21 change to disproved, and the prior n^(2/5) and n^(4/11) upper bounds.\n- Paul Erdős and Peter Fishburn, Minimum planar sets with maximum equidistance counts, Computational Geometry 7 (1997), 207-218, doi:10.1016/0925-7721(95)00050-X. (primary): https://www.sciencedirect.com/science/article/pii/092577219500050X\n Evidence used: Published source for g(3)=6, g(4)=8, g(5)<=16 and the local equidistance problem's original context.\n- János Pach and Micha Sharir, Repeated angles in the plane and related problems, Journal of Combinatorial Theory Series A 59 (1992), 12-22, doi:10.1016/0097-3165(92)90094-B. (primary): https://doi.org/10.1016/0097-3165(92)90094-B\n Evidence used: Primary incidence result from which the tracker records f(n) << n^(2/5).\n- Barnabás Janzer, Oliver Janzer, Abhishek Methuku, and Gábor Tardos, Tight bounds for intersection-reverse sequences, edge-ordered graphs and applications, arXiv:2411.07188. (primary): https://arxiv.org/abs/2411.07188\n Evidence used: Primary point-circle incidence bound from which the maintained tracker derives f(n) << n^(4/11).\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1926, "problem_number": "EP-96", "title": "Erdős Problem #96", "statement": "If $n$ points in $\\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.", "background": "Conjectured by Erd\\H{o}s and Moser. In \\cite{Er92e} Erd\\H{o}s credits the conjecture that the true upper bound is $2n$ to himself and Fishburn. F\"{u}redi \\cite{Fu90} proved an upper bound of $O(n\\log n)$. A short proof of this bound was given by Brass and Pach \\cite{BrPa01}. The best known upper bound is $ \\leq n\\log_2n+4n, $ due to Aggarwal \\cite{Ag15}.\nEdelsbrunner and Hajnal \\cite{EdHa91} have constructed $n$ such points with $2n-7$ pairs distance $1$ apart. (This disproved an early stronger conjecture of Erd\\H{o}s and Moser, that the true answer was $\\frac{5}{3}n+O(1)$.)\nA positive answer would follow from [97]. See also [90].\nIn \\cite{Er92e} Erd\\H{o}s makes the stronger conjecture that, if $g(x)$ counts the largest number of points equidistant from $x$ in $A$, then $ \\sum_{x\\in A}g(x)< 4n. $ He notes that the example of Edelsbrunner and Hajnal shows that $\\sum_{x\\in A}g(x)>4n-O(1)$ is possible.\nReferences\n\n\n[Ag15] Aggarwal, Amol, On unit distances in a convex polygon. Discrete Math. (2015), 88-92.\n\n[BrPa01] Brass , Peter and Pach, J\\'{a}nos, The maximum number of times the same distance can occur among\nthe vertices of a convex {$n$}-gon is {$O(n\\log n)$}. J. Combin. Theory Ser. A (2001), 178-179.\n\n[EdHa91] Edelsbrunner, Herbert and Hajnal, P\\'{e}ter, A lower bound on the number of unit distances between the\nvertices of a convex polygon. J. Combin. Theory Ser. A (1991), 312-316.\n\n[Er92e] Erd\\H{o}s, P\\'{a}l, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.\n\n[Fu90] F\"{u}redi, Zolt\\'{a}n, The maximum number of unit distances in a convex {$n$}-gon. J. Combin. Theory Ser. A (1990), 316-320.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The O(n) unit-distance conjecture for convex polygons is open; the best displayed upper bound is n log_2 n+4n.\n\n**Verified partial progress.**\n\n- Furedi and later Brass-Pach gave O(n log n) bounds.\n- Aggarwal improved the explicit bound to n log_2 n+4n.\n- Edelsbrunner-Hajnal constructed 2n-7 unit-distance pairs, refuting only an older 5n/3+O(1) strengthening.\n\n**Full solution or refutation.**\n\nNo linear upper bound was located.\n\n**What remains.**\n\nProve O(n) or construct a superlinear family.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #96 LaTeX source, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/96\n Evidence used: Lists the problem as open and records the current upper and lower constructions.\n\n**Review notes.** The known lower construction does not contradict the O(n) conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1927, "problem_number": "EP-98", "title": "Erdős Problem #98", "statement": "Let $h(n)$ be such that any $n$ points in $\\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\\to \\infty$?", "background": "Erd\\H{o}s could not even prove $h(n)\\geq n$. Pach has shown $h(n)0$.\nReferences\n\n\n[EFPR93] Erd\\H{o}s, Paul and F\"{u}redi, Zolt\\'{a}n and Pach, J\\'{a}nos and\nRuzsa, Imre Z., The grid revisited. Discrete Math. (1993), 189--196.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether h(n)/n tends to infinity under the stated general-position restrictions.\n\n**Verified partial progress.**\n\n- Erdos-Furedi-Pach-Ruzsa constructed h(n)=n was not known in the tracker audit.\n- The upper construction leaves a broad gap to the desired superlinear lower bound.\n\n**Full solution or refutation.**\n\nNo resolution or decisive lower-bound advance was located.\n\n**What remains.**\n\nProve superlinear growth of h(n), or find a near-linear general-position construction.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #98, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/98\n Evidence used: Lists the literal question as open and records the known upper construction.\n\n**Review notes.** No three collinear/no four concyclic is a material restriction and was retained exactly.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1928, "problem_number": "EP-99", "title": "Erdős Problem #99", "statement": "Let $A\\subseteq\\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently large then must there be three points in $A$ which form an equilateral triangle of size 1?", "background": "Thue proved that the minimal such diameter is achieved (asymptotically) by the points in a triangular lattice intersected with a circle. In general Erd\\H{o}s believed such a set must have very large intersection with the triangular lattice (perhaps as many as $(1-o(1))n$).\nErd\\H{o}s \\cite{Er94b} wrote 'I could not prove it but felt that it should not be hard. To my great surprise both B. H. Sendov and M. Simonovits doubted the truth of this conjecture.' In \\cite{Er94b} he offers \\$100 for a counterexample but only \\$50 for a proof.\nThe stated problem is false for $n=4$, for example taking the points to be vertices of a square. The behaviour of such sets for small $n$ is explored by Bezdek and Fodor \\cite{BeFo99}.\nSee also [103].\nReferences\n\n\n[BeFo99] Bezdek, Andr\\'{a}s and Fodor, Ferenc, Minimal diameter of certain sets in the plane. J. Combin. Theory Ser. A (1999), 105-111.\n\n[Er94b] Erd\\H{o}s, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The eventual-unit-equilateral-triangle assertion for diameter-minimizing planar packings remains open.\n\n**Verified partial progress.**\n\n- Triangular-lattice disk sections are asymptotically optimal for the diameter problem.\n- A square is a counterexample at n=4.\n- No argument found forces an exact equilateral unit triangle in every large optimizer.\n\n**Full solution or refutation.**\n\nNo asymptotic proof or counterexample was located.\n\n**What remains.**\n\nDecide whether every sufficiently large diameter minimizer contains a unit equilateral triangle.\n\n**Sources checked.**\n\n- Thomas F. Bloom, history of Erdos Problem #99, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/99\n Evidence used: Lists the problem as open and records the asymptotic triangular-lattice context and n=4 exception.\n\n**Review notes.** A finite small-n counterexample does not refute the eventual claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1929, "problem_number": "EP-100", "title": "Erdős Problem #100", "statement": "Let $A$ be a set of $n$ points in $\\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\\gg n$?", "background": "Perhaps the diameter is even $\\geq n-1$ for sufficiently large $n$. Piepmeyer has an example of $9$ such points with diameter $<5$. Kanold proved the diameter is $\\geq n^{3/4}$. The bounds on the distinct distance problem [89] proved by Guth and Katz \\cite{GuKa15} imply a lower bound of $\\gg n/\\log n$.\nReferences\n\n\n[GuKa15] Guth, Larry and Katz, Nets Hawk, On the Erd\\H{o}s distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The linear diameter conclusion remains open; Guth-Katz distinct-distance bounds imply only a lower bound of order n/log n.\n\n**Verified partial progress.**\n\n- Kanold proved a n^(3/4) lower bound.\n- Guth-Katz technology improves this consequence to n/log n.\n- The desired linear scale is still separated by a logarithm.\n\n**Full solution or refutation.**\n\nNo linear lower bound or counterexample was located.\n\n**What remains.**\n\nProve a linear diameter lower bound from separated distances, or construct a sublinear-diameter family.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #100 LaTeX source, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/100\n Evidence used: Lists the problem as open and records the n/log n consequence.\n\n**Review notes.** The record assumes both minimum separation and unit separation between distinct distance values.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1930, "problem_number": "EP-101", "title": "Erdős Problem #101", "statement": "Given $n$ points in $\\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$.", "background": "There are examples of sets of $n$ points with $\\sim n^2/6$ many collinear triples and no four points on a line. Such constructions are given by Burr, Gr\"{u}nbaum, and Sloane \\cite{BGS74} and F\"{u}redi and Pal\\'{a}sti \\cite{FuPa84}.\nGr\"{u}nbaum \\cite{Gr76} constructed an example with $\\gg n^{3/2}$ such lines. Erd\\H{o}s speculated this may be the correct order of magnitude. This is false: Solymosi and Stojakovi\\'{c} \\cite{SoSt13} have constructed a set with no five on a line and at least $ n^{2-O(1/\\sqrt{\\log n})} $ many lines containing exactly four points.\nSee also [102] and [669]. A generalisation of this problem is asked in [588].\nThis problem is Problem 71 on Green's open problems list.\nReferences\n\n\n[BGS74] Burr, Stefan A. and Gr\"{u}nbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.\n\n[FuPa84] F\"{u}redi, Z. and Pal\\'{a}sti, I., Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. (1984), 561-566.\n\n[Gr76] Gr\"{u}nbaum, Branko, New views on some old questions of combinatorial geometry. Colloquio Internazionale sulle Teorie Combinatorie\n(Roma, 1973), Tomo I (1976), 451-468.\n\n[SoSt13] Solymosi, J\\'{o}zsef and Stojakovi\\'C, Milo\\vS, Many collinear {$k$}-tuples with no {$k+1$} collinear points. Discrete Comput. Geom. (2013), 811-820.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The o(n^2) upper-bound conjecture remains open despite a near-quadratic lower construction with no five collinear points.\n\n**Verified partial progress.**\n\n- Solymosi and Stojakovic constructed n^(2-O(1/sqrt(log n))) four-point lines with no five collinear points.\n- This refutes Erdos's earlier n^(3/2)-scale speculation.\n- The exponent is still strictly below two, so it does not refute o(n^2).\n\n**Full solution or refutation.**\n\nNo proof of o(n^2) and no quadratic-order counterexample was located.\n\n**What remains.**\n\nClose the subexponential exponent gap between the near-quadratic construction and o(n^2).\n\n**Sources checked.**\n\n- Jozsef Solymosi and Milos Stojakovic, Many collinear k-tuples with no k+1 collinear points, arXiv:1107.0327 (2011). (primary): https://arxiv.org/abs/1107.0327\n Evidence used: Abstract gives the n^(2-c/sqrt(log n)) construction.\n- Thomas F. Bloom, Erdos Problem #101, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/101\n Evidence used: Lists the problem as open and explains the distinction from the older speculation.\n\n**Review notes.** Near-quadratic lower growth is not conflated with a disproof of little-o behavior.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1931, "problem_number": "EP-102", "title": "Erdős Problem #102", "statement": "Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\\mathbb{R}^2$ such that there are $\\geq cn^2$ lines each containing more than three points, there must be some line containing $h_c(n)$ many points. Estimate $h_c(n)$. Is it true that, for fixed $c>0$, we have $h_c(n)\\to \\infty$?", "background": "A problem of Erd\\H{o}s and Purdy. It is not even known if $h_c(n)\\geq 5$ (see [101]).\nIt is easy to see that $h_c(n) \\ll_c n^{1/2}$, and Erd\\H{o}s at one point \\cite{Er95} suggested that perhaps a similar lower bound $h_c(n)\\gg_c n^{1/2}$ holds. Zach Hunter has pointed out that this is false, even replacing $>3$ points on each line with $>k$ points: consider the set of points in $\\{1,\\ldots,m\\}^d$ where $n\\approx m^d$. These intersect any line in $\\ll_d n^{1/d}$ points, and have $\\gg_d n^2$ many pairs of points each of which determine a line with at least $k$ points. This is a construction in $\\mathbb{R}^d$, but a random projection into $\\mathbb{R}^2$ preserves the relevant properties.\nThis construction shows that $h_c(n) \\ll n^{1/\\log(1/c)}$.\nReferences\n\n\n[Er95] Erd\\H{o}s, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It is unknown even whether h_c(n)>=5 for every fixed c>0; a grid-product construction disproves the earlier suggested square-root lower growth.\n\n**Verified partial progress.**\n\n- The elementary upper bound h_c(n)<<_c n^(1/2) is known.\n- Hunter's grid-product construction improves the obstruction to h_c(n)<", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1932, "problem_number": "EP-103", "title": "Erdős Problem #103", "statement": "Let $h(n)$ count the number of incongruent sets of $n$ points in $\\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\\geq 1$ for all points $x\neq y$. Is it true that $h(n)\\to \\infty$?", "background": "It is not even known whether $h(n)\\geq 2$ for all large $n$.\nSee also [99].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The number of incongruent diameter-minimizing unit-separated planar configurations is not known to diverge; even eventual nonuniqueness is open.\n\n**Verified partial progress.**\n\n- The problem is explicitly tied to EP-99.\n- No lower bound h(n)>=2 for all large n is known in the maintained record.\n\n**Full solution or refutation.**\n\nNo resolution was located.\n\n**What remains.**\n\nShow eventual nonuniqueness and then obtain a growing lower bound for h(n), or establish asymptotic uniqueness behavior.\n\n**Sources checked.**\n\n- Thomas F. Bloom, history of Erdos Problem #103, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/103\n Evidence used: Lists the problem as open and explicitly notes the lack of an eventual-two-minimizer result.\n\n**Review notes.** The source statement has OCR loss in x not-equal y; it was not changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1933, "problem_number": "EP-104", "title": "Erdős Problem #104", "statement": "Given $n$ points in $\\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.", "background": "In \\cite{Er81d} Erd\\H{o}s proved that $\\gg n$ many circles is possible, and that there cannot be more than $O(n^2)$ many circles. The argument is very simple: every pair of points determines at most $2$ unit circles, and the claimed bound follows from double counting. Erd\\H{o}s claims in a number of places this produces the upper bound $n(n-1)$, but Harborth and Mengerson \\cite{HaMe86} note that in fact this delivers an upper bound of $\\frac{n(n-1)}{3}$.\nElekes \\cite{El84} has a simple construction of a set with $\\gg n^{3/2}$ such circles. This may be the correct order of magnitude.\nIn \\cite{Er75h} and \\cite{Er92e} Erd\\H{o}s also asks how many such unit circles there must be if the points are in general position.\nIn \\cite{Er92e} Erd\\H{o}s offered £100 for a proof or disproof that the answer is $O(n^{3/2})$.\nThe maximal number of unit circles achieved by $n$ points is A003829 in the OEIS.\nSee also [506] and [831].\nReferences\n\n\n[El84] Elekes, G., {$n$} points in the plane can determine $n^{3/2}$ unit\ncircles. Combinatorica (1984), 131.\n\n[Er75h] Erd\\H{o}s, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.\n\n[Er81d] Erd\\H{o}s, P., Some applications of graph theory and combinatorial methods to number theory and geometry. Algebraic methods in graph theory, Vol. I, II (Szeged, 1978) (1981), 137-148.\n\n[Er92e] Erd\\H{o}s, P\\'{a}l, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.\n\n[HaMe86] Harborth, Heiko and Mengersen, Ingrid, Point sets with many unit circles. Discrete Math. (1986), 193--197.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The unit-circle incidence problem remains open. The current maintained record retains the elementary upper bound n(n-1)/3 and Elekes's Omega(n^(3/2)) construction; no verified source located proves the requested o(n^2) bound or the stronger conjectured O(n^(3/2)) bound.\n\n**Verified partial progress.**\n\n- Erdős's double-counting argument gives O(n^2); Harborth and Mengersen record the sharper explicit value n(n-1)/3 from the same argument.\n- Elekes constructed n-point configurations incident with Omega(n^(3/2)) distinct unit circles containing at least three points.\n- The maintained page distinguishes the o(n^2) prize question from the stronger O(n^(3/2)) question.\n\n**Full solution or refutation.**\n\nNo solution is known. Recent breakthroughs on the planar unit-distance problem concern pairs at distance one and do not resolve this different problem about three-rich unit circles. The trailing serialized difficulty fragment in the imported background is extraction damage only.\n\n**What remains.**\n\nProve any subquadratic upper bound for the number of distinct unit circles containing at least three of n points, or refute it; the sharper target is O(n^(3/2)).\n\n**Sources checked.**\n\n- Heiko Harborth and Ingrid Mengersen, Point sets with many unit circles, Discrete Mathematics 60 (1986), 193-197, DOI 10.1016/0012-365X(86)90011-7. (primary): https://doi.org/10.1016/0012-365X(86)90011-7\n Evidence used: The paper studies the extremal function, supplies exact small values, and is the cited source for the corrected quadratic double-counting constant.\n- György Elekes, n points in the plane can determine n^(3/2) unit circles, Combinatorica 4 (1984), 131. (primary): https://www.erdosproblems.com/latex/104\n Evidence used: The maintained bibliography and linked source record identify Elekes's Omega(n^(3/2)) lower construction.\n- Thomas F. Bloom, Erdős Problem #104, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/104\n Evidence used: The maintained record, crawled in June 2026, marks the problem open and records no improvement beyond the classical bounds.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1934, "problem_number": "EP-108", "title": "Erdős Problem #108", "statement": "For every $r\\geq 4$ and $k\\geq 2$ is there some finite $f(k,r)$ such that every graph of chromatic number $\\geq f(k,r)$ contains a subgraph of girth $\\geq r$ and chromatic number $\\geq k$?", "background": "Conjectured by Erd\\H{o}s and Hajnal. R\"{o}dl \\cite{Ro77} has proved the $r=4$ case (see [923]). The infinite version (whether every graph of infinite chromatic number contains a subgraph of infinite chromatic number whose girth is $>k$) is also open.\nIn \\cite{Er79b} Erd\\H{o}s also asks whether $ \\lim_{k\\to \\infty}\\frac{f(k,r+1)}{f(k,r)}=\\infty. $ See also the entry in the graphs problem collection and [740] for the infinitary version.\nReferences\n\n\n[Er79b] Erd\\H{o}s, Paul, Problems and results in graph theory and combinatorial analysis. Graph theory and related topics (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977) (1979), 153-163.\n\n[Ro77] R\"{o}dl, V., On the chromatic number of subgraphs of a given graph. Proc. Amer. Math. Soc. (1977), 370-371.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The general high-girth/high-chromatic-subgraph question remains open. Rödl proved the r=4 case, but the maintained January 2026 record explicitly keeps the cases r>=5, the infinite analogue, and the ratio question open.\n\n**Verified partial progress.**\n\n- Rödl proved that sufficiently high chromatic number forces a triangle-free subgraph with any prescribed finite chromatic number, settling r=4.\n- The maintained record reports no solution claims for the higher-girth cases.\n- The associated asymptotic question about f(k,r+1)/f(k,r) is also unresolved.\n\n**Full solution or refutation.**\n\nNo general solution is known. Rödl's theorem excludes triangles but does not exclude all cycles shorter than an arbitrary r.\n\n**What remains.**\n\nProve finiteness of f(k,r) for every r>=5 and k>=2 or construct a counterexample; quantitative estimates and the infinite version remain separate targets.\n\n**Sources checked.**\n\n- Vojtěch Rödl, On the chromatic number of subgraphs of a given graph, Proceedings of the American Mathematical Society 64 (1977), 370-371, DOI 10.1090/S0002-9939-1977-0469806-4. (primary): https://doi.org/10.1090/S0002-9939-1977-0469806-4\n Evidence used: The theorem gives the triangle-free prescribed-chromatic-number result corresponding to r=4.\n- Thomas F. Bloom, Erdős Problem #108, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/108\n Evidence used: The page was last edited in January 2026, marks the general problem open, and reports no claimed solutions in the comments.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1935, "problem_number": "EP-111", "title": "Erdős Problem #111", "statement": "If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G(n)$ edges.\nWhat is the behaviour of $h_G(n)$? Is it true that $h_G(n)/n\\to \\infty$ for every graph $G$ with chromatic number $\\aleph_1$?", "background": "A problem of Erd\\H{o}s, Hajnal, and Szemer\\'{e}di \\cite{EHS82}. Every $G$ with chromatic number $\\aleph_1$ must have $h_G(n)\\gg n$ since $G$ must contain, for some $r$, $\\aleph_1$ many vertex disjoint odd cycles of length $2r+1$.\nOn the other hand, Erd\\H{o}s, Hajnal, and Szemer\\'{e}di proved that there is a $G$ with chromatic number $\\aleph_1$ such that $h_G(n)\\ll n^{3/2}$. In \\cite{Er81} Erd\\H{o}s conjectured that this can be improved to $\\ll n^{1+\\epsilon}$ for every $\\epsilon>0$.\nSee also [74].\nReferences\n\n\n[EHS82] Erd\\H{o}s, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The question whether h_G(n)/n tends to infinity for every graph of chromatic number aleph_1 remains open. The maintained May 2026 record retains the Erdős-Hajnal-Szemerédi linear lower bound and their example with h_G(n)<>n because it contains uncountably many vertex-disjoint odd cycles of one fixed length.\n- Erdős, Hajnal, and Szemerédi constructed an aleph_1-chromatic graph satisfying h_G(n)<0.\n\n**Full solution or refutation.**\n\nNo complete solution is known. Results for finite chromatic number or related hypergraph deletion functions do not decide the universal statement over all aleph_1-chromatic graphs.\n\n**What remains.**\n\nProve universal superlinearity of h_G(n), or construct an aleph_1-chromatic graph with h_G(n)=O(n) along an unbounded sequence; improving the known n^(3/2) construction is a parallel goal.\n\n**Sources checked.**\n\n- Paul Erdős, András Hajnal, and Endre Szemerédi, On almost bipartite large chromatic graphs, Annals of Discrete Mathematics 12 / North-Holland Mathematics Studies 60 (1982), 117-123, DOI 10.1016/S0304-0208(08)73497-2. (primary): https://doi.org/10.1016/S0304-0208(08)73497-2\n Evidence used: This is the originating paper for the almost-bipartite large-chromatic problem and its baseline bounds.\n- Thomas F. Bloom, Erdős Problem #111, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/111\n Evidence used: The maintained record, crawled in May 2026, marks the exact question open and lists no solution or partial claim in its comments.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1936, "problem_number": "EP-112", "title": "Erdős Problem #112", "statement": "Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament of size $m$. Determine $k(n,m)$.", "background": "A problem of Erd\\H{o}s and Rado \\cite{ErRa67}, who showed $k(n,m) \\ll_m n^{m-1}$, or more precisely, $ k(n,m) \\leq \\frac{2^{m-1}(n-1)^m+n-2}{2n-3}. $ Larson and Mitchell \\cite{LaMi97} improved the dependence on $m$, establishing in particular that $k(n,3)\\leq n^{2}$. Zach Hunter has observed that $ R(n,m) \\leq k(n,m)\\leq R(n,m,m), $ which in particular proves the upper bound $k(n,m)\\leq 3^{n+2m}$.\nSee also the entry in the graphs problem collection - on this site the problem replaces transitive tournament with directed path, but Zach Hunter and Raphael Steiner have a simple argument that proves, for this alternative definition, that $k(n,m)=(n-1)(m-1)$.\nReferences\n\n\n[ErRa67] Erd\\H{o}s, P. and Rado, R., Partition relations and transitivity domains of binary\nrelations. J. London Math. Soc. (1967), 624-633.\n\n[LaMi97] Larson, Jean A. and Mitchell, William J., On a problem of Erd\\H{o}s and Rado. Ann. Comb. (1997), 245-252.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The extremal function for independent sets versus transitive tournaments in directed graphs has not been determined. Erdős-Rado and Larson-Mitchell provide upper bounds, and a Ramsey sandwich is known, but the exact directed-path formula in the background concerns a different variant.\n\n**Verified partial progress.**\n\n- Erdős and Rado proved k(n,m)<<_m n^(m-1), including the explicit bound reproduced in the input background.\n- Larson and Mitchell improved the dependence on m and established k(n,3)<=n^2.\n- The comparison R(n,m)<=k(n,m)<=R(n,m,m) gives general Ramsey bounds.\n- Replacing transitive tournament by directed path yields (n-1)(m-1), but that does not solve the imported statement.\n\n**Full solution or refutation.**\n\nNo exact or asymptotically sharp general determination was located. Confidence is medium because the intended convention for a directed graph should be checked against the original binary-relation paper before proof work.\n\n**What remains.**\n\nFix the digraph convention from the 1967 source, then determine exact values or sharp asymptotics for the transitive-tournament version, especially small m.\n\n**Sources checked.**\n\n- Paul Erdős and Richard Rado, Partition relations and transitivity domains of binary relations, Journal of the London Mathematical Society s1-42 (1967), 624-633, DOI 10.1112/jlms/s1-42.1.624. (primary): https://doi.org/10.1112/jlms/s1-42.1.624\n Evidence used: The originating paper proves the classical polynomial upper bound for fixed m.\n- Jean A. Larson and William J. Mitchell, On a problem of Erdős and Rado, Annals of Combinatorics 1 (1997), 245-252, DOI 10.1007/BF02558478. (primary): https://doi.org/10.1007/BF02558478\n Evidence used: The paper improves the m-dependence and is the source cited for k(n,3)<=n^2.\n- Thomas F. Bloom, Erdős Problem #112, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/112\n Evidence used: The maintained record marks the transitive-tournament problem open and explicitly separates the solved directed-path variant.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1937, "problem_number": "EP-114", "title": "Erdős Problem #114", "statement": "If $p(z)\\in\\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\\{ z\\in \\mathbb{C} : \\lvert p(z)\\rvert=1\\}$ maximised when $p(z)=z^n-1$?", "background": "A problem of Erd\\H{o}s, Herzog, and Piranian \\cite{EHP58}. It is also listed as Problem 4.10 in \\cite{Ha74}, where it is attributed to Erd\\H{o}s.\nLet the maximal length of such a curve be denoted by $f(n)$.\n{UL}\n{LI}The length of the curve when $p(z)=z^n-1$ is $2n+O(1)$, and hence the conjecture implies in particular that $f(n)=2n+O(1)$.{/LI}\n{LI}Dolzhenko \\cite{Do61} proved $f(n) \\leq 4\\pi n$, but few were aware of this work.{/LI}\n{LI}Pommerenke \\cite{Po61} proved $f(n)\\ll n^2$.{/LI}\n{LI}Borwein \\cite{Bo95} proved $f(n)\\ll n$ (Borwein was unaware of Dolzhenko's earlier work). The prize of \\$250 is reported by Borwein \\cite{Bo95}.{/LI}\n{LI}Eremenko and Hayman \\cite{ErHa99} proved the full conjecture when $n=2$, and $f(n)\\leq 9.173n$ for all $n$.{/LI}\n{LI}Danchenko \\cite{Da07} proved $f(n)\\leq 2\\pi n$.{/LI}\n{LI}Fryntov and Nazarov \\cite{FrNa09} proved that $z^n-1$ is a local maximiser, and solved this problem asymptotically, proving that $ f(n)\\leq 2n+O(n^{7/8}). $ {/LI}\n{LI} Tao \\cite{Ta25} has proved that $p(z)=z^n-1$ is the unique (up to rotation and translation) maximiser for all sufficiently large $n$.\n{/UL}\nErd\\H{o}s, Herzog, and Piranian \\cite{EHP58} also ask whether the length is at least $2\\pi$ if $\\{ z: \\lvert f(z)\\rvert<1\\}$ is connected (which $z^n$ shows is the best possible). This was proved by Pommerenke \\cite{Po59}.\nReferences\n\n\n[Bo95] Borwein, Peter, The arc length of the lemniscate {$\\{|p(z)|=1\\}$}. Proc. Amer. Math. Soc. (1995), 797--799.\n\n[Da07] Danchenko, V. I., The lengths of lemniscates. {V}ariations of rational\nfunctions. Mat. Sb. (2007), 51--58.\n\n[Do61] Dol\\v zenko, E. P., Some estimates concerning algebraic hypersurfaces and\nderivatives of rational functions. Dokl. Akad. Nauk SSSR (1961), 1287--1290.\n\n[EHP58] Erd\\H{o}s, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.\n\n[ErHa99] Eremenko, Alexandre and Hayman, Walter, On the length of lemniscates. Michigan Math. J. (1999), 409--415.\n\n[FrNa09] Fryntov, Alexander and Nazarov, Fedor, New estimates for the length of the {E}rd\\H\nos-{H}erzog-{P}iranian lemniscate. (2009), 49--60.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Po59] Pommerenke, Ch., On some problems by Erd\\H{o}s, Herzog and Piranian. Michigan Math. J. (1959), 221-225.\n\n[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.\n\n[Ta25] T. Tao, The maximal length of the Erd\\H{o}s-Herzog-Piranian leminscate length in high degree. arXiv:2512.12455 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Tao proved the Erdős--Herzog--Piranian extremal conjecture for every sufficiently large degree in 2025. Earlier work proves degree 2, local maximality of z^n-1, and an asymptotically sharp global bound. A conventionally verified argument covering every remaining finite degree was not located, so the full all-degree question remains only partially solved.\n\n**Verified partial progress.**\n\n- Eremenko and Hayman proved the extremal assertion in degree 2.\n- Fryntov and Nazarov proved local maximality of z^n-1 and the asymptotically sharp bound f(n) <= 2n+O(n^(7/8)).\n- Tao proved that z^n-1 is the unique extremizer up to rotation and translation for all sufficiently large n.\n- The maintained tracker discusses recent finite-degree certificates, including a claimed verification through degree 14, but these were not independently certified in this audit.\n\n**Full solution or refutation.**\n\nTao's high-degree argument establishes the exact conjectured maximizer, rather than only its leading asymptotic length, once n exceeds an effective threshold. Together with the degree-two theorem and recent tracker-linked low-degree work, this reduces the original universal conjecture to a finite range. The located primary literature does not close that entire range.\n\n**What remains.**\n\nSettle every degree outside the established low-degree cases and Tao's sufficiently-large-degree theorem. The recent computer-assisted interval certificates linked from the tracker need an accessible, publication-quality independent audit before they can be used to shrink or eliminate the finite gap.\n\n**Sources checked.**\n\n- Terence Tao, The maximal length of the Erdős--Herzog--Piranian lemniscate in high degree, arXiv:2512.12455 (2025). (primary): https://arxiv.org/abs/2512.12455\n Evidence used: The abstract states that the exact extremal conjecture holds for all sufficiently large degrees.\n- Alexander Fryntov and Fedor Nazarov, New estimates for the length of the Erdős--Herzog--Piranian lemniscate, arXiv:0808.0717; St. Petersburg Math. J. 20 (2009), 49--60. (primary): https://arxiv.org/abs/0808.0717\n Evidence used: Proves local maximality of the conjectured polynomial and the asymptotically sharp global upper estimate.\n- Alexandre Eremenko and Walter Hayman, On the length of lemniscates, Michigan Math. J. 46 (1999), 409--415; arXiv:0805.2295. (primary): https://arxiv.org/abs/0805.2295\n Evidence used: Includes the complete degree-two extremal result and a linear general upper bound.\n- Thomas F. Bloom, Erdős Problem #114 and discussion, Erdős Problems (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/114\n Evidence used: Records the current partially resolved/falsifiable status, Tao's theorem, and links to recent finite-degree claims.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1938, "problem_number": "EP-117", "title": "Erdős Problem #117", "statement": "Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.\nEstimate $h(n)$ as well as possible.", "background": "Pyber \\cite{Py87} has proved there exist constants $c_2>c_1>1$ such that $c_1^n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Pyber established that the universal abelian-cover function h(n) has exponential upper and lower bounds, but its sharper growth remains undetermined. The maintained January 2026 page reports no later solution or partial claim improving the status.\n\n**Verified partial progress.**\n\n- The hypothesis is equivalently a bound n on the size of a pairwise noncommuting subset.\n- Pyber proved an exponential upper bound through an exponential bound on the index of the centre.\n- An Isaacs example gives exponential lower behavior, so constants c2>c1>1 satisfy c1^n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1939, "problem_number": "EP-119", "title": "Erdős Problem #119", "statement": "Let $z_i$ be an infinite sequence of complex numbers such that $\\lvert z_i\\rvert=1$ for all $i\\geq 1$, and for $n\\geq 1$ let $ p_n(z)=\\prod_{i\\leq n} (z-z_i). $ Let $M_n=\\max_{\\lvert z\\rvert=1}\\lvert p_n(z)\\rvert$.\nIs it true that $\\limsup M_n=\\infty$?\nIs it true that there exists $c>0$ such that for infinitely many $n$ we have $M_n > n^c$?\nIs it true that there exists $c>0$ such that, for all large $n$, $ \\sum_{k\\leq n}M_k > n^{1+c}? $ ", "background": "This is Problem 4.1 in \\cite{Ha74} where it is attributed to Erd\\H{o}s.\nThe weaker conjecture that $\\limsup M_n=\\infty$ was proved by Wagner \\cite{Wa80}, who show that there is some $c>0$ with $M_n>(\\log n)^c$ infinitely often.\nThe second question was answered by Beck \\cite{Be91}, who proved that there exists some $c>0$ such that $ \\max_{n\\leq N} M_n > N^c. $ Erd\\H{o}s (e.g. see \\cite{Ha74}) gave a construction of a sequence with $M_n\\leq n+1$ for all $n$. Linden \\cite{Li77} improved this to give a sequence with $M_n\\ll n^{1-c}$ for some $c>0$.\nThe third question seems to remain open.\nReferences\n\n\n[Be91] Beck, J., The modulus of polynomials with zeros on the unit circle: A problem of Erd\\H{o}s. Annals of Math. (1991), 609-651.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Li77] Linden, C. N., The modulus of polynomials with zeros on the unit circle. Bull. London Math. Soc. (1977), 65--69.\n\n[Wa80] Wagner, Gerold, On a problem of {E}rd\\H{o}s in {D}iophantine approximation. Bull. London Math. Soc. (1980), 81--88.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The first two questions were settled classically by Wagner and Beck. A current Formal Conjectures record marks the third question solved as well and credits GPT-5.6 and Samuel Korsky with the stronger estimate sum_{k<=n} M_k >> n^(5/4)/sqrt(log n). No stable primary manuscript for this newest bound was located, so the current solved classification carries medium confidence and requires expert review.\n\n**Verified partial progress.**\n\n- Wagner proved M_n>(log n)^c infinitely often for some c>0, settling unbounded limsup.\n- Beck proved max_{n<=N} M_n>N^c for some c>0, settling the second question.\n- The current Formal Conjectures status record states the cumulative bound sum_{k<=n} M_k >> n^(5/4)/sqrt(log n), which implies the third question for every fixed c<1/4.\n\n**Full solution or refutation.**\n\nThe newest claimed theorem gives a cumulative lower bound with exponent 5/4, up to a square-root logarithmic loss. Since n^(1/4-c)/sqrt(log n) tends to infinity for every fixed c<1/4, this is stronger than the requested eventual n^(1+c) bound. The status record identifies a proof claim and checker, but its Lean file contains an admitted statement and is not itself the proof.\n\n**What remains.**\n\nUnder the current maintained classification, all three questions are closed. Bibliographically, the proof of the cumulative bound should be placed in a stable primary manuscript or accessible proof transcript and independently checked by a specialist; the linked forum proof-claim page was inaccessible during this audit.\n\n**Sources checked.**\n\n- Google DeepMind Formal Conjectures, FormalConjectures/ErdosProblems/119.lean (accessed 2026-08-17). (authoritative_secondary): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/119.lean\n Evidence used: Explicitly marks all three research assertions solved, credits the newest proof, and records the stronger cumulative bound; the file is a status/formalization record rather than a completed proof.\n- József Beck, The modulus of polynomials with zeros on the unit circle: A problem of Erdős, Annals of Mathematics 134 (1991), 609--651. (primary): https://annals.math.princeton.edu/1991/134-3/p03\n Evidence used: Classical primary source for the power lower bound that settles the second question.\n- Gerold Wagner, On a problem of Erdős in Diophantine approximation, Bulletin of the London Mathematical Society 12 (1980), 81--88. (primary): https://doi.org/10.1112/blms/12.2.81\n Evidence used: Classical primary source for an infinitely-often logarithmic-power lower bound, settling the first question.\n- Thomas F. Bloom, Erdős Problem #119, Erdős Problems (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/119\n Evidence used: Maintained public status trail for the three-part problem; some cached tracker text predates the newest third-part proof claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1940, "problem_number": "EP-120", "title": "Erdős Problem #120", "statement": "Let $A\\subseteq\\mathbb{R}$ be an infinite set. Must there be a set $E\\subset \\mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\\in\\mathbb{R}$ and $a\neq 0$?", "background": "The Erd\\H{o}s similarity problem.\nThis is true if $A$ is unbounded or dense in some interval. It therefore suffices to prove this when $A=\\{a_1>a_2>\\cdots\\}$ is a countable strictly monotone sequence which converges to $0$.\nSteinhaus \\cite{St20} has proved this is false whenever $A$ is a finite set.\nThis conjecture is known in many special cases (but, for example, it is open when $A=\\{1,1/2,1/4,\\ldots\\}$, which is Problem 94 on Green's open problems list). For an overview of progress we recommend a nice survey by Svetic \\cite{Sv00} on this problem. A survey of more recent progress was written by Jung, Lai, and Mooroogen \\cite{JLM24}.\nReferences\n\n\n[JLM24] Y. Jung and C.-K. Lai and Y. Mooroogen, Some recent progress on the Erd\\H{o}s similarity conjecture. arXiv:2412.11062 (2024).\n\n[St20] Steinhaus, Hugo, Sur les distances des points dans les ensembles de measure positive. Fund. Math. (1920), 93-104.\n\n[Sv00] Svetic, R. E., The Erd\\H{o}s similarity problem: a survey. Real Anal. Exchange (2000/01), 525-539.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The original Erdős similarity conjecture remains open. A 2024 survey explicitly says it is unresolved for exponentially decaying sequences, including the central geometric-sequence regime; solved topological or in-the-large variants do not imply the imported positive-measure statement.\n\n**Verified partial progress.**\n\n- The conjecture is known when A is unbounded or dense in an interval, reducing the difficult case to sequences decreasing to zero.\n- Many special sequence classes are known, but the geometric sequence {1,1/2,1/4,...} remains open in the original formulation.\n- Recent topological, bi-Lipschitz, and in-the-large variants provide new perspectives but are logically distinct.\n\n**Full solution or refutation.**\n\nNo accepted full solution is known. Although arXiv:2001.02395 is titled A proof of the Erdös similarity conjecture, the later 2024 survey, coauthored by Chun-Kit Lai from that preprint, explicitly treats the conjecture as open; the preprint is therefore not sufficient solution evidence.\n\n**What remains.**\n\nProve positive-measure affine-copy avoidance for every infinite sequence decreasing to zero, notably the geometric sequence, or produce an infinite universal set refuting the conjecture.\n\n**Sources checked.**\n\n- Yeonwook Jung, Chun-Kit Lai, and Yuveshen Mooroogen, Fifty years of the Erdős similarity conjecture, arXiv:2412.11062 (2024). (primary): https://arxiv.org/abs/2412.11062\n Evidence used: The abstract explicitly states that the conjecture remains open for exponentially decaying sequences and certain Cantor sets.\n- Angel Cruz, Chun-Kit Lai, and Malabika Pramanik, A proof of the Erdös similarity conjecture, arXiv:2001.02395 (2020). (primary): https://arxiv.org/abs/2001.02395\n Evidence used: This is a conflicting earlier solution claim; it is treated conservatively because a later survey by a coauthor still declares the conjecture open.\n- Thomas F. Bloom, Erdős Problem #120, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/120\n Evidence used: The maintained June 2026 record marks the original similarity problem open and lists the geometric-sequence case as unresolved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1941, "problem_number": "EP-122", "title": "Erdős Problem #122", "statement": "For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $f(n)/F(n)\\to 0$ for almost all $n$, there are infinitely many $x$ such that $ \\frac{\\#\\{ n\\in \\mathbb{N} : n+f(n)\\in (x,x+F(x))\\}}{F(x)}\\to \\infty? $ ", "background": "Asked by Erd\\H{o}s, Pomerance, and S\\'{a}rk\"{o}zy \\cite{EPS97} who prove that this is true when $f$ is the divisor function or the number of distinct prime divisors of $n$, but Erd\\H{o}s believed it is false when $f(n)=\\phi(n)$ or $\\sigma(n)$.\nReferences\n\n\n[EPS97] Erd\\H{o}s, Paul and Pomerance, Carl and S\\'{a}rk\"{o}zy, Andr\\'{a}s, On locally repeated values of certain arithmetic functions. IV. Ramanujan J. (1997), 227-241.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported statement cannot be safely classified because it uses f(n)/F(n)->0, while the maintained page was corrected in April 2026 to F(n)/f(n)->0, and its displayed conclusion does not specify the limiting variable for the arrow to infinity. The intended classification problem remains open after statement repair.\n\n**Verified partial progress.**\n\n- Erdős, Pomerance, and Sárközy proved a strong concentration theorem for n+omega(n), with omega the number of distinct prime divisors.\n- Erdős reported the intended general property for tau(n) and omega(n), and predicted failure for phi(n) and sigma(n).\n- The live tracker now restricts the historical setting to slowly growing arithmetic functions and reverses the imported ratio to F/f.\n\n**Full solution or refutation.**\n\nNo classification of arithmetic functions is known. More importantly, the exact imported wording is not a stable mathematical proposition: the scale condition is reversed relative to the current source, and 'infinitely many x such that [expression] tends to infinity' requires a limsup or an explicit sequence of x-values.\n\n**What remains.**\n\nRecover the intended condition and limiting quantifier from the 1997 sources, then decide the classification question and in particular the predicted negative cases phi and sigma.\n\n**Sources checked.**\n\n- Paul Erdős, Carl Pomerance, and András Sárközy, On locally repeated values of certain arithmetic functions. IV, Ramanujan Journal 1 (1997), 227-241, DOI 10.1023/A:1009723712317. (primary): https://doi.org/10.1023/A:1009723712317\n Evidence used: The paper proves the cited concentration result for omega(n) and supplies primary context for the intended local-repetition property.\n- Thomas F. Bloom, Erdős Problem #122 and revision history, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/122\n Evidence used: The April 2026 history documents the correction from f/F to F/f and the expanded statement context.\n- Carl Pomerance, publication list, entry 113. (bibliographic_index): https://math.dartmouth.edu/~carlp/\n Evidence used: The author-maintained bibliography confirms the paper title, authors, journal, year, and pages.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1942, "problem_number": "EP-123", "title": "Erdős Problem #123", "statement": "Let $a,b,c\\geq 1$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\\geq 0$), none of which divide any other?", "background": "A sequence is said to be $d$-complete if every large integer is the sum of distinct integers from the sequence, none of which divide any other. This particular case of $d$-completeness was conjectured by Erd\\H{o}s and Lewin \\cite{ErLe96}, who (among other related results) prove this when $a=3$, $b=5$, and $c=7$.\nAs a partial record of progress so far, the sequence $\\{a^kb^lc^m\\}$ is known to be $d$-complete when:\n{UL}\n{LI}$a=3$, $b=5$, $c=7$ (Erd\\H{o}s and Lewin \\cite{ErLe96}).{/LI}\n{LI}$a=2$, $b=5$, $c\\in \\{7,11,13,17,19\\}$ (Erd\\H{o}s and Lewin \\cite{ErLe96}).{/LI}\n{LI}$a=2$, $b=5$, $c\\in \\{9,21,23,27,29,31\\}$ - more generally, $a=2$, $b=5$, and any $c>6$ with $(c,10)=1$ such that there exists $N$ where every integer in $(N,25cN)$ is the sum of distinct elements of $\\{2^k3^lc^m\\}$, none of which divide any other (Ma and Chen \\cite{MaCh16}).{/LI}\n{LI} $a=2$, $b=5$, $3\\leq c\\leq 87$ with $(c,10)=1$, or $a=2$, $b=7$, $3\\leq c\\leq 33$ with $(c,14)=1$, or $a=3$, $b=5$, $2\\leq c\\leq 14$ with $(c,15)=1$ (Chen and Yu \\cite{ChYu23b}).{/LI}\n{/UL}\nIn \\cite{Er92b} Erd\\H{o}s makes the stronger conjecture (for $a=2$, $b=3$, and $c=5$) that, for any $\\epsilon>0$, all large integers $n$ can be written as the sum of distinct integers $b_1<\\cdots \n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A commit-pinned Lean development proves the intended nondegenerate statement: for every pairwise-coprime a,b,c>1, the set of a^k b^l c^m is d-complete. The supplied dataset says a,b,c>=1, so literal degenerate cases are not covered by that theorem and the corpus wording needs correction or separate treatment.\n\n**Verified partial progress.**\n\n- Erdős and Lewin proved several triples, including (3,5,7).\n- Ma and Chen supplied a conditional criterion and further explicit triples.\n- Chen and Yu verified additional finite ranges of triples.\n- The recent Lean theorem proves the general pairwise-coprime case for bases greater than one.\n\n**Full solution or refutation.**\n\nThe formal theorem says that every sufficiently large natural number is the sum of a finite set of distinct three-base smooth numbers no two of which are comparable under divisibility. The proof develops eventual semigroup density, primitive correction gadgets on exponent levels, residue representatives, scaling-and-correction lemmas preserving the antichain condition, and a final permutation argument. Inspection of the pinned file found no `sorry`; `#print axioms` lists only propext, Classical.choice, and Quot.sound.\n\n**What remains.**\n\nCorrect or clarify the dataset's a,b,c>=1 wording, independently audit the recent formal proof, and produce a conventional human-readable publication. Erdős's stronger (2,3,5) narrow-interval variant remains open and is not implied by the d-completeness theorem.\n\n**Sources checked.**\n\n- Star Fleet Math, Claude Fable 5, Colin Snyder, and Formal Conjectures authors, Erdős Problem 123 Lean proof, commit a28a04b6b8ce43d5260a7466677c1f23833bfc38 (2026). (primary): https://github.com/plby/lean-proofs/blob/a28a04b6b8ce43d5260a7466677c1f23833bfc38/src/latest/ErdosProblems/Erdos123.lean\n Evidence used: Commit-pinned formal proof of d-completeness for pairwise-coprime bases greater than one; the inspected file has no admitted goals and prints a standard axiom footprint.\n- Google DeepMind Formal Conjectures, FormalConjectures/ErdosProblems/123.lean (accessed 2026-08-17). (authoritative_secondary): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/123.lean\n Evidence used: Records the affirmative resolution, distinguishes the intended nondegenerate statement, links the proof artifact, and notes the stronger open variant.\n- P. Erdős and M. Lewin, d-complete sequences of integers, Mathematics of Computation 65 (1996), 837--840. (primary): https://www.brand.site.co.il/riddles/201507a_files/2153618.pdf\n Evidence used: Original-source context, definition of d-completeness, and early solved triples.\n- M.-M. Ma and Y.-G. Chen, On d-complete sequences of integers, Journal of Number Theory 164 (2016), 1--12. (primary): https://www.sciencedirect.com/science/article/pii/S0022314X16000342\n Evidence used: Provides a conditional criterion and additional examples before the general proof.\n- Y.-G. Chen and W.-X. Yu, On d-complete sequences of integers, II, Acta Arithmetica 208 (2023), 161--181. (primary): https://doi.org/10.4064/aa220818-20-1\n Evidence used: Further pre-resolution progress for explicit families of triples.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1943, "problem_number": "EP-124", "title": "Erdős Problem #124", "statement": "For any $d\\geq 1$ and $k\\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\\geq k$. Let $3\\leq d_10$, of an infinite set of $d_i$ for which every sufficiently large integer can be written as a finite sum of the shape $\\sum_i c_ia_i$ where $c_i\\in \\{0,1\\}$ and $a_i\\in P(d_i,0)$ and yet $\\sum_{i}\\frac{1}{d_i-1}<\\epsilon$.\nSee also [125].\nReferences\n\n\n[BEGL96] Burr, S. A. and Erd\\H{o}s, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.\n\n[Er97] Erd\\H{o}s, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160.\n\n[Er97e] Erd\\H{o}s, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[Me04] Melfi, Giuseppe, On certain positive integer sequences. Riv. Mat. Univ. Parma (7) (2004), 253--260.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The corrected k=0 version has a public Lean-checked proof, in fact in a stronger form, while the gcd-one k>=1 conjecture of Burr-Erdős-Graham-Li remains open beyond special cases. The imported first premise has d_r rather than d_i and is impossible under 3<=d_1<...=r+2, it can never be at least one.\n- After correcting the denominator to d_i-1 and allowing powers d^0, a public Lean artifact proves that every nonnegative integer is representable, under weaker base hypotheses than the input.\n- Burr, Erdős, Graham, and Li formulated the powers-of-exponent-at-least-k conjecture and proved the special base set {3,4,7}.\n- The public Lean file also corrects a separate equality-versus-inequality typo in an earlier formal statement.\n\n**Full solution or refutation.**\n\nThis is not a full solution to the multipart imported record. The first part is vacuous literally and solved only under the documented intended d_i correction; the second, genuinely different k>=1 problem remains open. Treating the formal proof as solving the second question would conflate the availability of the units digit with the truncated-power setting.\n\n**What remains.**\n\nResolve the gcd(d_1,...,d_r)=1 conjecture for arbitrary k>=1 and general finite base sets; separately correct d_r to d_i in the corpus with a provenance note.\n\n**Sources checked.**\n\n- S. A. Burr, Paul Erdős, Ronald L. Graham, and W. Wen-Ching Li, Complete sequences of sets of integer powers, Acta Arithmetica 77 (1996), 133-138, DOI 10.4064/aa-77-2-133-138. (primary): https://doi.org/10.4064/aa-77-2-133-138\n Evidence used: The paper states the k>=1 conjecture with conditions sum 1/(a-1)>=1 and gcd(A)=1, and gives the special-case progress.\n- plby/lean-proofs, Erdos124b.lean, Lean 4.24.0 / Mathlib commit f897ebc, accessed 2026-08-17. (formal_verification): https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos124b.lean\n Evidence used: The artifact states that the proof is verified by Lean and contains both a corrected >=1 theorem and a strengthened k=0 representation theorem.\n- Erdős Problem #124 discussion thread, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/124\n Evidence used: The discussion records the source-variant distinction, the impossible d_r denominator, and why the formal k=0 result does not settle the BEGL k>=1 conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1944, "problem_number": "EP-125", "title": "Erdős Problem #125", "statement": "Let $A = \\{ \\sum\\epsilon_k3^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\\{ \\sum\\epsilon_k4^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $4$.\nDoes $A+B$ have positive density?", "background": "A problem of Burr, Erd\\H{o}s, Graham, and Li \\cite{BEGL96}. More generally, if $n_1<\\cdots1 $ and $A_i$ is the set of integers with only the digits $0,1$ in base $n_i$ then does $A_1+\\cdots+A_k$ have positive density? Melfi \\cite{Me01} noted this is false as written, with a counterexample given by $\\{3,9,81\\}$, but suggests it is true if we further insist that the $n_k$ are pairwise coprime.\nIf $C=A+B$ then Melfi \\cite{Me01} showed $\\lvert C\\cap[1,x]\\rvert \\gg x^{0.965}$ and Hasler and Melfi \\cite{HaMe24} improved this to $\\lvert C\\cap [1,x]\\rvert \\gg x^{0.9777}$. Hasler and Melfi also show that the lower density of $C$ is at most $ \\frac{1015}{1458}\\approx 0.69616. $ See also [124].\nReferences\n\n\n[BEGL96] Burr, S. A. and Erd\\H{o}s, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.\n\n[HaMe24] M. Hasler and G. Melfi, On sums of distinct powers of $3$ and $4$. Combinatorics and Number Theory (2024).\n\n[Me01] Melfi, Giuseppe, An additive problem about powers of fixed integers. Rend. Circ. Mat. Palermo (2) (2001), 239--246.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A 2026 primary preprint and companion Lean artifact prove that the lower asymptotic density of A+B is zero, refuting the positive-lower-density conjecture. The original literature also asks about positive upper density; that distinct question is not refuted by the lower-density theorem and remains open in the sources checked.\n\n**Verified partial progress.**\n\n- Hasler and Melfi proved a counting lower bound of order x^0.97777 and a nontrivial upper bound on the liminf density.\n- The 2026 proof obtains a uniform multiplicative density loss on a sequence of suitably aligned ternary and quaternary scales.\n- Iteration of the scale-density loss proves that the lower asymptotic density is zero.\n\n**Full solution or refutation.**\n\nThe proof selects comparable blocks of powers of 3 and 4 using the irrationality of log 4/log 3, decomposes allowed digit choices into upper and lower blocks, and shows that at each selected scale the possible sums occupy a fixed proportion strictly below one after accounting for the previous scale. Iterating this contraction forces liminf |(A+B) intersect [0,N]|/N to equal zero. The companion repository supplies the formal proof artifact.\n\n**What remains.**\n\nDetermine the upper asymptotic density of A+B, in particular whether it is positive or zero, and ensure historical statements do not conflate density, lower density, and upper density. The lower-density conjecture itself is closed negatively.\n\n**Sources checked.**\n\n- George Tsoukalas et al., Advancing Mathematics Research with AI-Driven Formal Proof Search, arXiv:2605.22763v2 (2026). (primary): https://arxiv.org/abs/2605.22763\n Evidence used: Contains a human-readable theorem and proof that the lower density of the restricted ternary-plus-quaternary sumset is zero.\n- Google DeepMind, AlphaProof Nexus results, Erdős Problem 125 formal artifact (2026). (primary): https://github.com/google-deepmind/alphaproof-nexus-results\n Evidence used: Companion primary repository for the paper's Lean developments and natural-language proof material, including Problem 125.\n- J.-C. Hasler and G. Melfi, On sums of distinct powers of 3 and 4, Combinatorics and Number Theory 13 (2024), 141--156. (primary): https://doi.org/10.2140/cnt.2024.13.141\n Evidence used: Published pre-refutation quantitative bounds, including a nonzero counting exponent and a liminf-density upper bound.\n- Thomas F. Bloom, Erdős Problem #125 and discussion, Erdős Problems (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/125\n Evidence used: Classifies the lower-density assertion as disproved in Lean and records the distinction from the upper-density question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1945, "problem_number": "EP-126", "title": "Erdős Problem #126", "statement": "Let $f(n)$ be maximal such that if $A\\subseteq\\mathbb{N}$ has $\\lvert A\\rvert=n$ then $\\prod_{a\neq b\\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\\log n\\to\\infty$?", "background": "Investigated by Erd\\H{o}s and Tur\\'{a}n \\cite{ErTu34} (prompted by a question of L\\'{a}z\\'{a}r and Gr\"{u}nwald) in their first joint paper, where they proved that $ \\log n \\ll f(n) \\ll n/\\log n $ (the upper bound is trivial, taking $A=\\{1,\\ldots,n\\}$). Erd\\H{o}s says that $f(n)=o(n/\\log n)$ has never been proved, but perhaps never seriously attacked.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[ErTu34] Erd\\H{o}s, Paul and Turan, Paul, On a Problem in the Elementary Theory of Numbers. Amer. Math. Monthly (1934), 608-611.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The superlogarithmic lower-bound question remains open. Erdős and Turán proved log n<infinity was located. The ordered product duplicates every unordered sum, but this has no effect on the number of distinct prime factors. Formalized status only indicates that a Lean statement exists.\n\n**What remains.**\n\nProve f(n)=omega(log n) or construct arbitrarily large n-element sets with only O(log n) distinct primes dividing all pair-sums; improving the upper bound is a separate target.\n\n**Sources checked.**\n\n- Paul Erdős and Paul Turán, On a Problem in the Elementary Theory of Numbers, American Mathematical Monthly 41 (1934), 608-611, DOI 10.1080/00029890.1934.11987659. (primary): https://doi.org/10.1080/00029890.1934.11987659\n Evidence used: The paper proves the finite-set theorem underlying the logarithmic lower bound.\n- Paul Erdős author archive, 1934-03 PDF. (source_collection): https://combinatorica.hu/~p_erdos/1934-03.pdf\n Evidence used: The scanned primary text states that 3*2^(k-1) positive integers cannot have all two-term sums composed from only k fixed primes.\n- Thomas F. Bloom, Erdős Problem #126, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/126\n Evidence used: The maintained page marks the problem open, records the log n and n/log n bounds, and distinguishes formalization from solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1946, "problem_number": "EP-129", "title": "Erdős Problem #129", "statement": "Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that $ R(n;3,r) < C^{\\sqrt{n}}. $ ", "background": "Conjectured by Erd\\H{o}s and Gy\\'{a}rf\\'{a}s, who proved the existence of some $C>1$ such that $R(n;3,r)>C^{\\sqrt{n}}$. Note that when $r=k=2$ we recover the classic Ramsey numbers. Erd\\H{o}s thought it likely that for all $r,k\\geq 2$ there exists some $C_1,C_2>1$ (depending only on $r$) such that $ C_1^{n^{1/k-1}}< R(n;k,r) < C_2^{n^{1/k-1}}. $ Antonio Girao has pointed out that this problem as written is easily disproved, and indeed $R(n;3,2) \\geq C^{n}$:\nThe obvious probabilistic construction (randomly colour the edges red/blue independently uniformly at random) yields a 2-colouring of the edges of $K_N$ such every set on $n$ vertices contains a red triangle and a blue triangle (using that every set of $n$ vertices contains $\\gg n^2$ edge-disjoint triangles), provided $N \\leq C^n$ for some absolute constant $C>1$. This implies $R(n;3,2) \\geq C^{n}$, contradicting the conjecture.\nPerhaps Erd\\H{o}s had a different problem in mind, but it is not clear what that might be. It would presumably be one where the natural probabilistic argument would deliver a bound like $C^{\\sqrt{n}}$ as Erd\\H{o}s and Gy\\'{a}rf\\'{a}s claim to have achieved via the probabilistic method.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The exact displayed assertion is false already for r=2. A standard random red-blue colouring gives R(n;3,2)>=c^n for some c>1, contradicting any upper bound C^(sqrt(n)). The maintained page remains labelled open only because the intended historical Erdős-Gyárfás statement is ambiguous.\n\n**Verified partial progress.**\n\n- Antonio Girão observed the contradiction recorded by the maintained tracker.\n- Every n-vertex set contains Omega(n^2) edge-disjoint triangles; in a random two-colouring the probability it misses one colour's triangle is exponentially small in n^2.\n- A union bound over n-subsets works for N<=c^n, yielding a colouring in which every n-set contains both a red and a blue triangle.\n- The imported background's broader exponent n^(1/k-1) is negative as typeset and cannot reliably identify the intended replacement conjecture.\n\n**Full solution or refutation.**\n\nThe corpus statement is refuted in the literature record, not merely open. What remains open or uncertain is historical reconstruction: Erdős and Gyárfás apparently intended a different generalized Ramsey parameter or exponent, but the source has not been unambiguously recovered.\n\n**What remains.**\n\nRecover the intended 1997 statement and formulate a noncontradictory replacement. No proof obligation remains for the exact imported upper bound because it is false for r=2.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #129, with disproof attributed to Antonio Girão, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/129\n Evidence used: The maintained record gives the random-colouring lower bound R(n;3,2)>=C^n, explicitly says the problem as written is disproved, and flags the original source as ambiguous.\n- Maintained bibliography entry [Er97b] linked from Erdős Problem #129. (source_collection): https://www.erdosproblems.com/latex/129\n Evidence used: The source metadata preserves the historical attribution but does not disambiguate the intended statement; this supports separating refutation of the exact text from reconstruction of the original problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1947, "problem_number": "EP-130", "title": "Erdős Problem #130", "statement": "Let $A\\subset\\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in $A$, where two vertices are joined by an edge if and only if they are an integer distance apart.\nHow large can the chromatic number and clique number of this graph be? In particular, can the chromatic number be infinite?", "background": "Asked by Andr\\'{a}sfai and Erd\\H{o}s. Erd\\H{o}s \\cite{Er97b} also asked where such a graph could contain an infinite complete graph, but this is impossible by an earlier result of Anning and Erd\\H{o}s \\cite{AnEr45}.\nSee also [213].\nReferences\n\n\n[AnEr45] Anning, Norman H. and Erd\\H{o}s, Paul, Integral distances. Bull. Amer. Math. Soc. (1945), 598-600.\n\n[Er97b] Erd\\H{o}s, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The possible chromatic and clique numbers of the integer-distance graph remain open; Anning-Erdos only rules out an infinite clique.\n\n**Verified partial progress.**\n\n- Anning-Erdos excludes an infinite complete graph with all pairwise distances integral.\n- This leaves the infinite-chromatic-number question open.\n\n**Full solution or refutation.**\n\nNo resolution was located.\n\n**What remains.**\n\nBound or construct the graph's chromatic and clique numbers under the general-position hypotheses.\n\n**Sources checked.**\n\n- Norman Anning and Paul Erdos, Integral distances, Bulletin of the AMS 51 (1945), 598-600. (primary): https://www.erdosproblems.com/latex/130\n Evidence used: The source record cites its no-infinite-integral-distance-set consequence.\n- Thomas F. Bloom, Erdos Problem #130, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/130\n Evidence used: Lists the literal graph questions as open.\n\n**Review notes.** Infinite clique and infinite chromatic number are distinct questions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1948, "problem_number": "EP-131", "title": "Erdős Problem #131", "statement": "Let $F(N)$ be the maximal size of $A\\subseteq\\{1,\\ldots,N\\}$ such that no $a\\in A$ divides the sum of any distinct elements of $A\\backslash\\{a\\}$. Estimate $F(N)$. In particular, is it true that $ F(N) > N^{1/2-o(1)}? $ ", "background": "This was studied by Erd\\H{o}s, Lev, Rauzy, S\\'{a}ndor, and S\\'{a}rk\"{o}zy \\cite{ELRSS99}, where they call such a property 'non-dividing', and prove the explicit bound $ F(N)<3N^{1/2}+1. $ In \\cite{Er97b} Erd\\H{o}s credits Csaba with a construction that proves $F(N) \\gg N^{1/5}$. Such a construction was also given in \\cite{ELRSS99}, where it is linked to the problem of non-averaging sets (see [186]).\nIndeed, every such set is non-averaging, and hence the result of Pham and Zakharov \\cite{PhZa24} implies $ F(N) \\leq N^{1/4+o(1)}. $ This shows the answer to the original question is no, but the general question of the correct growth of $F(N)$ remains open.\nIn \\cite{Er75b} Erd\\H{o}s writes that he originally thought $F(N) <(\\log N)^{O(1)}$, but that Straus proved that $ F(N) > \\exp((\\sqrt{\\tfrac{2}{\\log 2}}+o(1))\\sqrt{\\log N}). $ See also [13].\nThis is discussed in problem C16 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[ELRSS99] Erd\\H{o}s, P. and Lev, V. and Rauzy, G. and S\\'andor, C. and\nS\\'ark\"ozy, A., Greedy algorithm, arithmetic progressions, subset sums and\ndivisibility. Discrete Math. (1999), 119--135.\n\n[Er75b] Erd\\H{o}s, Paul, Problems and results in combinatorial number theory. Journ\\'{e}es Arithm\\'{e}tiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310.\n\n[Er97b] Erd\\H{o}s, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[PhZa24] Pham, H. T. and Zakharov, D., Sharp bound for the Erd\\H{o}s-Straus non-averaging set problem. arXiv:2410.14624 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The proposed F(N)>N^(1/2-o(1)) lower bound is false because F(N)<=N^(1/4+o(1)); the correct order of F(N) remains open.\n\n**Verified partial progress.**\n\n- Erdos, Lev, Rauzy, Sandor, and Sarkozy proved F(N)<3 sqrt(N)+1 and constructed a power lower bound.\n- Every non-dividing set is non-averaging, so Pham-Zakharov gives F(N)<=N^(1/4+o(1)).\n- This refutes the explicit near-square-root subquestion.\n\n**Full solution or refutation.**\n\nA named subquestion is disproved, but the request to estimate F(N) is unresolved.\n\n**What remains.**\n\nDetermine the true power/order of F(N) between the known construction and N^(1/4+o(1)).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #131, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/131\n Evidence used: States the non-averaging reduction, the N^(1/4+o(1)) consequence, and that the original lower-bound question is no.\n\n**Review notes.** Mixed-status record: only its explicit lower-bound example is refuted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1949, "problem_number": "EP-132", "title": "Erdős Problem #132", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points? Must the number of such distances $\\to \\infty$ as $n\\to \\infty$?", "background": "Asked by Erd\\H{o}s and Pach. Hopf and Pannowitz \\cite{HoPa34} proved that the largest distance between points of $A$ can occur at most $n$ times, but it is unknown whether a second such distance must occur.\nIt may be true that there are at least $n^{1-o(1)}$ many such distances. In \\cite{Er97e} Erd\\H{o}s offers \\$100 for 'any nontrivial result'.\nErd\\H{o}s \\cite{Er84c} believed that for $n\\geq 5$ there must always exist at least two such distances. This is false for $n=4$, as witnessed by two equilateral triangles of the same side-length glued together. Erd\\H{o}s and Fishburn \\cite{ErFi95} proved this is true for $n=5$ and $n=6$.\nClemen, Dumitrescu, and Liu \\cite{CDL25} have proved that there always at least two such distances if $A$ is in convex position (that is, no point lies inside the convex hull of the others). They also prove it is true if the set $A$ is 'not too convex', in a specific technical sense.\nSee also [223], [756], and [957].\nReferences\n\n\n[CDL25] F. Clemen, A. Dumitrescu, and D. Liu, On multiplicities of interpoint distances. arXiv:2505.04283 (2025).\n\n[Er84c] Erd\\H{o}s, Paul, Some old and new problems in combinatorial geometry. Convexity and graph theory (Jerusalem, 1981) (1984), 129-136.\n\n[Er97e] Erd\\H{o}s, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[ErFi95] Erd\\H{o}s, Paul and Fishburn, Peter C., Multiplicities of interpoint distances in finite planar sets. Discrete Appl. Math. (1995), 141--147.\n\n[HoPa34] Hopf, H. and Pannwitz, E., Aufgabe 167. Jber. Deutsch. Math. Verein. (1934), 114.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Only the diameter is known always to occur at most n times; a second sparse multiplicity distance and a diverging count remain open.\n\n**Verified partial progress.**\n\n- Hopf-Pannowitz proves the largest distance has multiplicity at most n.\n- No general second distance theorem was located.\n\n**Full solution or refutation.**\n\nNo resolution was located.\n\n**What remains.**\n\nForce a second distance of multiplicity at most n and quantify their number.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #132, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/132\n Evidence used: Lists the problem as open and records the Hopf-Pannowitz first-distance theorem.\n\n**Review notes.** The known first distance does not supply the requested second one.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1950, "problem_number": "EP-137", "title": "Erdős Problem #137", "statement": "We say that $N$ is powerful if whenever $p\\mid N$ we also have $p^2\\mid N$. Let $k\\geq 3$. Can the product of any $k$ consecutive positive integers ever be powerful?", "background": "Conjectured by Erd\\H{o}s and Selfridge. There are infinitely many $n$ such that $n(n+1)$ is powerful (see [364]). Erd\\H{o}s and Selfridge \\cite{ErSe75} proved that the product of $k\\geq 3$ consecutive positive integers can never be a perfect power. Erd\\H{o}s remarked that this 'seems hopeless at present'.\nIn \\cite{Er82c} he further conjectures that, if $k$ is fixed and $n$ is sufficiently large, then, for all $m$, there must be at least $k$ distinct primes $p$ such that $ p\\mid m(m+1)\\cdots (m+n) $ and yet $p^2$ does not divide the right-hand side.\nSee also [364].\nReferences\n\n\n[Er82c] Erd\\H{o}s, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.\n\n[ErSe75] Erd\\H{o}s, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether a product of k>=3 consecutive integers can be powerful remains open; the never-a-perfect-power theorem is weaker.\n\n**Verified partial progress.**\n\n- Erdos-Selfridge proves the product of at least three consecutive positive integers is never a perfect power.\n- Powerful products need not be perfect powers, so this does not settle EP-137.\n\n**Full solution or refutation.**\n\nNo proof or example for the powerful-product question was located.\n\n**What remains.**\n\nProve that every such product has a prime divisor of exponent exactly one, or find a powerful example.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #137, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/137\n Evidence used: Lists the literal problem as open and distinguishes the perfect-power theorem.\n\n**Review notes.** Do not conflate powerful with perfect power.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1951, "problem_number": "EP-138", "title": "Erdős Problem #138", "statement": "Let the van der Waerden number $W(k)$ be such that whenever $N\\geq W(k)$ and $\\{1,\\ldots,N\\}$ is $2$-coloured there must exist a monochromatic $k$-term arithmetic progression. Improve the bounds for $W(k)$ - for example, prove that $W(k)^{1/k}\\to \\infty$.", "background": "When $p$ is prime Berlekamp \\cite{Be68} has proved $W(p+1)\\geq p2^p$. Gowers \\cite{Go01} has proved $ W(k) \\leq 2^{2^{2^{2^{2^{k+9}}}}}. $ The best general lower bound is $W(k)\\gg 2^k$, due to Kozik and Shabanov \\cite{KoSh16}.\nIn \\cite{Er81} Erd\\H{o}s further asks whether $W(k+1)/W(k)\\to \\infty$, or $W(k+1)-W(k)\\to \\infty$.\nIn \\cite{Er80} Erd\\H{o}s asks whether $W(k)/2^k\\to \\infty$, and offers \\$500 for a proof or disproof of $W(k)^{1/k}\\to \\infty$.\nReferences\n\n\n[Be68] Berlekamp, E. R., A construction for partitions which avoid long arithmetic progressions. Canad. Math. Bull. (1968), 409-414.\n\n[Er80] Erd\\H{o}s, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Go01] Gowers, W. T., A new proof of Szemer\\'{e}di's theorem. Geom. Funct. Anal. (2001), 465-588.\n\n[KoSh16] Kozik, Jakub and Shabanov, Dmitry, Improved algorithms for colorings of simple hypergraphs and\napplications. J. Combin. Theory Ser. B (2016), 312--332.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The proposed W(k)^(1/k)->infinity remains open; an ancillary difference question W(k+1)-W(k)->infinity is now resolved by W(k+1)>=W(k)+k.\n\n**Verified partial progress.**\n\n- The best general lower bound remains exponential in k.\n- A 2026 Lean proof establishes W(k+1)>=W(k)+k.\n- Fox-Hunter establish superexponential growth in the analogous three-colour problem, not the two-colour target.\n\n**Full solution or refutation.**\n\nThe main example remains open; only a related question in the background is solved.\n\n**What remains.**\n\nProve or disprove superexponential growth for two-colour W(k).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #138 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/138\n Evidence used: Lists the root-growth problem as open and records the difference-bound resolution.\n\n**Review notes.** Formal proof of the difference inequality is not evidence for the root-growth conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1952, "problem_number": "EP-141", "title": "Erdős Problem #141", "statement": "Let $k\\geq 3$. Are there $k$ consecutive primes in arithmetic progression?", "background": "Green and Tao \\cite{GrTa08} have proved that there must always exist some $k$ primes in arithmetic progression, but these need not be consecutive. Erd\\H{o}s called this conjecture 'completely hopeless at present'.\nThe existence of such progressions for small $k$ has been verified for $k\\leq 10$, see the Wikipedia page. It is open, even for $k=3$, whether there are infinitely many such progressions.\nSee also [219].\nThis is discussed in problem A6 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[GrTa08] Green, Ben and Tao, Terence, The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) (2008), 481-547.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L3\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Arbitrarily long consecutive-prime arithmetic progressions remain open, even the infinitude of consecutive prime triples in arithmetic progression.\n\n**Verified partial progress.**\n\n- Green-Tao proves arbitrarily long arithmetic progressions of primes without consecutiveness.\n- Small lengths up to ten have computational examples.\n- Standard prime-tuple hypotheses conditionally imply the assertion.\n\n**Full solution or refutation.**\n\nNo unconditional resolution was located.\n\n**What remains.**\n\nProve infinitely many consecutive prime APs for k=3 and then arbitrary fixed k.\n\n**Sources checked.**\n\n- Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics 167 (2008), 481-547. (primary): https://annals.math.princeton.edu/2008/167-2/p01\n Evidence used: Proves the nonconsecutive comparison theorem.\n- Thomas F. Bloom, Erdos Problem #141, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/141\n Evidence used: Lists the consecutive version as open.\n\n**Review notes.** A progression of primes is not necessarily a progression of consecutive primes.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1953, "problem_number": "EP-142", "title": "Erdős Problem #142", "statement": "Let $r_k(N)$ be the largest possible size of a subset of $\\{1,\\ldots,N\\}$ that does not contain any non-trivial $k$-term arithmetic progression. Prove an asymptotic formula for $r_k(N)$.", "background": "Erd\\H{o}s remarked this is 'probably unattackable at present'. In \\cite{Er97c} Erd\\H{o}s offered \\$1000, but given that he elsewhere offered \\$5000 just for (essentially) showing that $r_k(N)=o_k(N/\\log N)$, that value seems odd. In \\cite{Er81} he offers \\$10000, stating it is 'probably enormously difficult'.\nThe best known upper bounds for $r_k(N)$ are due to Kelley and Meka \\cite{KeMe23} for $k=3$, Green and Tao \\cite{GrTa17} for $k=4$, and Leng, Sah, and Sawhney \\cite{LSS24} for $k\\geq 5$. An asymptotic formula is still far out of reach, even for $k=3$.\nSee also [3] and [139].\nReferences\n\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er97c] Erd\\H{o}s, Paul, Some of my favorite problems and results. The mathematics of Paul Erd\\H{o}s, I (1997), 47-67.\n\n[GrTa17] Green, Ben and Tao, Terence, New bounds for Szemer\\'{e}di's theorem, III: a polylogarithmic bound for $r_4(N)$. Mathematika (2017), 944-1040.\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\n\n[LSS24] Leng, J., Sah, A. and Sawhney, M., Improved bounds for Szemer\\'{e}di's theorem. arXiv:2402.17995 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** An asymptotic formula for r_k(N) is open even at k=3; modern bounds establish quantitatively stronger density decay without identifying an asymptotic.\n\n**Verified partial progress.**\n\n- Kelley-Meka advanced the k=3 upper bound.\n- Green-Tao advanced k=4.\n- Leng-Sah-Sawhney prove r_k(N)<=5.\n\n**Full solution or refutation.**\n\nNo asymptotic formula was located.\n\n**What remains.**\n\nDetermine the order of r_k(N), much less an asymptotic formula, beginning with k=3.\n\n**Sources checked.**\n\n- James Leng, Ashwin Sah, and Mehtaab Sawhney, Improved Bounds for Szemeredi's Theorem, arXiv:2402.17995 (2024). (primary): https://arxiv.org/abs/2402.17995\n Evidence used: Abstract states the k>=5 exp(-(log log N)^c_k) upper bound.\n- Thomas F. Bloom, Erdos Problem #142 LaTeX source, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/142\n Evidence used: Lists the asymptotic question as open and maps the best-known regimes.\n\n**Review notes.** Szemeredi's qualitative theorem does not supply the requested asymptotic formula.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1954, "problem_number": "EP-143", "title": "Erdős Problem #143", "statement": "Let $A\\subset (1,\\infty)$ be a countably infinite set such that for all $x\neq y\\in A$ and integers $k\\geq 1$ we have $ \\lvert kx -y\\rvert \\geq 1. $ Does this imply that $A$ is sparse? In particular, does this imply that $ \\sum_{x\\in A}\\frac{1}{x\\log x}<\\infty $ or $ \\sum_{\\substack{x \n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Koukoulopoulos-Lamzouri-Lichtman prove the harmonic sparsity alternative sum_{x", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1955, "problem_number": "EP-145", "title": "Erdős Problem #145", "statement": "Let $s_1\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Existence of all squarefree-gap moments remains open unconditionally; Chan proved it for every 0<=alpha<3.75 and Granville proved it conditionally on abc.\n\n**Verified partial progress.**\n\n- Erdos and Hooley covered alpha up to 3.\n- Greaves-Harman-Huxley reached 11/3.\n- Chan reached every alpha below 3.75.\n\n**Full solution or refutation.**\n\nNo unconditional theorem for every nonnegative alpha was located.\n\n**What remains.**\n\nExtend the moment asymptotic beyond alpha<3.75, ultimately to all alpha>=0 without abc.\n\n**Sources checked.**\n\n- Tsz Ho Chan, On moments of gaps between consecutive square-free numbers, Moscow Journal of Combinatorics and Number Theory 12 (2023), 287-295. (primary): https://doi.org/10.2140/moscow.2023.12.287\n Evidence used: Establishes the asymptotic for 0<=alpha<3.75.\n- Thomas F. Bloom, Erdos Problem #145, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/145\n Evidence used: Lists the all-alpha assertion as open and the abc conditional theorem.\n\n**Review notes.** The source's alpha<=3.75 wording is normalized conservatively to the paper's alpha<3.75 assertion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1956, "problem_number": "EP-146", "title": "Erdős Problem #146", "statement": "If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\\leq r$, then $ \\mathrm{ex}(n;H) \\ll n^{2-1/r}. $ ", "background": "Conjectured by Erd\\H{o}s and Simonovits \\cite{ErSi84}. Open even for $r=2$. Alon, Krivelevich, and Sudakov \\cite{AKS03} have proved $ \\mathrm{ex}(n;H) \\ll n^{2-1/4r}. $ They also prove the full Erd\\H{o}s-Simonovits conjectured bound if $H$ is bipartite and the maximum degree in one side of the bipartition is $r$.\nSee also [113] and [147].\nThis problem is #43 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[AKS03] Alon, Noga and Krivelevich, Michael and Sudakov, Benny, Tur\\'{a}n numbers of bipartite graphs and related\nRamsey-type questions. Combin. Probab. Comput. (2003), 477-494.\n\n[ErSi84] Erd\\H{o}s, P. and Simonovits, M., Cube-supersaturated graphs and related problems. Progress in graph theory (Waterloo, Ont., 1982) (1984), 203-218.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Erdos-Simonovits n^(2-1/r) extremal bound for every r-degenerate bipartite H remains open, even r=2.\n\n**Verified partial progress.**\n\n- Alon-Krivelevich-Sudakov proved ex(n,H)<", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1957, "problem_number": "EP-148", "title": "Erdős Problem #148", "statement": "Let $F(k)$ be the number of solutions to $ 1= \\frac{1}{n_1}+\\cdots+\\frac{1}{n_k}, $ where $1\\leq n_1<\\cdots0$ is some absolute constant and $c_0=1.26408\\cdots$ is the 'Vardi constant'. The lower bound is due to Konyagin \\cite{Ko14} and the upper bound to Elsholtz and Planitzer \\cite{ElPl21}.\nReferences\n\n\n[ElPl21] Elsholtz, Christian and Planitzer, Stefan, Sums of four and more unit fractions and approximate parametrizations. Bull. Lond. Math. Soc. (2021), 695-709.\n\n[Ko14] Konyagin, S. V., Double exponential lower bound for the number of representations of unity by Egyptian fractions. Math. Notes (2014), 277-281.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Estimating the number F(k) of distinct-denominator Egyptian-fraction representations of 1 remains open between widely separated double-exponential bounds.\n\n**Verified partial progress.**\n\n- The tracker reports 2^{c^{k/log k}} <= F(k) <= c0^{(1/5+o(1))2^k}.\n- Konyagin acknowledged and corrected a false displayed identity in the printed 2014 proof while maintaining the theorem.\n- Elsholtz independently proves the same lower-bound scale even with all denominators odd, corroborating the unrestricted lower bound.\n\n**Full solution or refutation.**\n\nNo asymptotically matching estimate or resolution was located.\n\n**What remains.**\n\nClose the large gap between the lower and upper double-exponential estimates for F(k).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #148 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/148?embed=1\n Evidence used: Lists the open problem and bounds, identifies the printed error, and records Konyagin's correction.\n- S. V. Konyagin, Double Exponential Lower Bound for the Number of Representations of Unity by Egyptian Fractions, Mathematical Notes 95 (2014), 277-281, DOI 10.1134/S0001434614010295. (primary): https://doi.org/10.1134/S0001434614010295\n Evidence used: Original source of the claimed lower bound; its printed identity requires the correction recorded by the tracker.\n- Christian Elsholtz, Egyptian Fractions with odd denominators, arXiv:1606.02117 (2016). (primary): https://arxiv.org/abs/1606.02117\n Evidence used: The abstract gives exp(exp(c1 k/log k)) distinct odd-denominator representations for odd k.\n\n**Review notes.** The imported background ends in serialization noise; the exact statement was not altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1958, "problem_number": "EP-149", "title": "Erdős Problem #149", "statement": "Let $G$ be a graph with maximum degree $\\Delta$. Is $G$ the union of at most $\\tfrac{5}{4}\\Delta^2$ sets of strongly independent edges (sets such that the induced subgraph is the union of vertex-disjoint edges)?", "background": "Asked by Erd\\H{o}s and Ne\\v{s}et\\v{r}il in 1985 (see \\cite{FGST89}). This is equivalent to asking whether the chromatic number of the square of the line graph $L(G)^2$ is at most $\\frac{5}{4}\\Delta^2$.\nThis bound would be the best possible, as witnessed by a blowup of $C_5$. The minimum number of such sets required is sometimes called the strong chromatic index of $G$.\nThe weaker conjecture that there exists some $c>0$ such that $(2-c)\\Delta^2$ sets suffice was proved by Molloy and Reed \\cite{MoRe97}, who proved that $1.998\\Delta^2$ sets suffice (for $\\Delta$ sufficiently large). This was improved to $1.93\\Delta^2$ by Bruhn and Joos \\cite{BrJo18} and to $1.835\\Delta^2$ by Bonamy, Perrett, and Postle \\cite{BPP22}. The best bound currently available is $ 1.772\\Delta^2, $ proved by Hurley, de Joannis de Verclos, and Kang \\cite{HJK22}. Mahdian has, in their Masters' thesis, proved an upper bound of $(2+o(1))\\frac{\\Delta^2}{\\log \\Delta}$ under the additional assumption that $G$ is $C_4$-free.\nErd\\H{o}s and Ne\\v{s}et\\v{r}il also asked the easier problem of whether $G$ containing at least $\\tfrac{5}{4}\\Delta^2$ many edges implies $G$ containing two strongly independent edges. This was proved by Chung, Gy\\'{a}rf\\'{a}s, Tuza, and Trotter \\cite{CGTT90}.\nIt is still open even whether the clique number of $L(G)^2$ at most $\\frac{5}{4}\\Delta^2$. Let $\\omega=\\omega(L(G)^2)$ be this clique number. \\'{S}leszy\\'{n}ska-Nowak \\cite{Sl15} proved $\\omega \\leq \\frac{3}{2}\\Delta^2$. Faron and Postle \\cite{FaPo19} proved $\\omega\\leq \\frac{4}{3}\\Delta^2$. Cames van Batenburg, Kang, and Pirot \\cite{CKP20} have proved $\\omega\\leq \\frac{5}{4}\\Delta^2$ under the additional assumption that $G$ is triangle-free (and $\\omega\\leq \\Delta^2$ if $G$ is $C_5$-free).\nReferences\n\n\n[BPP22] Bonamy, Marthe and Perrett, Thomas and Postle, Luke, Colouring graphs with sparse neighbourhoods: bounds and\napplications. J. Combin. Theory Ser. B (2022), 278-317.\n\n[BrJo18] Bruhn, Henning and Joos, Felix, A stronger bound for the strong chromatic index. Combin. Probab. Comput. (2018), 21-43.\n\n[CGTT90] Chung, F. R. K. and Gy\\'arf\\'as, A. and Tuza, Z. and Trotter,\nW. T., The maximum number of edges in {$2K_2$}-free graphs of bounded\ndegree. Discrete Math. (1990), 129--135.\n\n[CKP20] Cames van Batenburg, Wouter and Kang, Ross J. and Pirot,\nFran\\c cois, Strong cliques and forbidden cycles. Indag. Math. (N.S.) (2020), 64--82.\n\n[FGST89] Faudree, R. J. and Gy\\'{a}rf\\'{a}s, A. and Schelp, R. H. and Tuza,\nZs., Induced matchings in bipartite graphs. Discrete Math. (1989), 83-87.\n\n[FaPo19] Faron, Maxime and Postle, Luke, On the clique number of the square of a line graph and its\nrelation to maximum degree of the line graph. J. Graph Theory (2019), 261--274.\n\n[HJK22] Hurley, Eoin and de Joannis de Verclos, R\\'{e}mi and Kang, Ross\nJ., An improved procedure for colouring graphs of bounded local\ndensity. Adv. Comb. (2022), Paper No. 7, 33.\n\n[MoRe97] Molloy, Michael and Reed, Bruce, A bound on the strong chromatic index of a graph. J. Combin. Theory Ser. B (1997), 103-109.\n\n[Sl15] No reference found.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Erdos-Nesetril 5Delta^2/4 strong chromatic index conjecture remains open; the best general asymptotic upper bound located is 1.772Delta^2.\n\n**Verified partial progress.**\n\n- Hurley, de Joannis de Verclos, and Kang prove a 1.772Delta^2 upper bound for sufficiently large maximum degree.\n- The general strong-clique bound is still only 4Delta^2/3.\n- The conjectured 5Delta^2/4 strong-clique bound holds for triangle-free graphs, with sharper cycle-free special cases.\n\n**Full solution or refutation.**\n\nNo proof of the unrestricted 5Delta^2/4 coloring bound was located.\n\n**What remains.**\n\nLower the unrestricted chromatic bound to 5Delta^2/4; even the matching unrestricted clique-number bound remains open.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #149, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/149\n Evidence used: Lists the conjecture as open and summarizes the general and special-case bounds.\n- Eoin Hurley, Remi de Joannis de Verclos, and Ross J. Kang, An improved procedure for colouring graphs of bounded local density, Advances in Combinatorics (2022), Paper 7, DOI 10.19086/aic.2022.7. (primary): https://arxiv.org/abs/2007.07874\n Evidence used: Proves that the strong chromatic index is at most 1.772Delta^2 for sufficiently large Delta.\n- Wouter Cames van Batenburg, Ross J. Kang, and Francois Pirot, Strong cliques and forbidden cycles, Indagationes Mathematicae 31 (2020), 64-82, DOI 10.1016/j.indag.2019.09.003. (primary): https://arxiv.org/abs/1903.06087\n Evidence used: Proves the 5Delta^2/4 strong-clique bound for triangle-free graphs and sharper forbidden-cycle cases.\n\n**Review notes.** Special-case clique bounds do not settle the unrestricted coloring conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1959, "problem_number": "EP-151", "title": "Erdős Problem #151", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ on at least two vertices (sometimes called the clique transversal number).\nLet $H(n)$ be maximal such that every triangle-free graph on $n$ vertices contains an independent set on $H(n)$ vertices.\nIf $G$ is a graph on $n$ vertices then is $ \\tau(G)\\leq n-H(n)? $ ", "background": "It is easy to see that $\\tau(G) \\leq n-\\sqrt{n}$. Note also that if $G$ is triangle-free then trivially $\\tau(G)\\leq n-H(n)$.\nThis is listed in \\cite{Er88} as a problem of Erd\\H{o}s and Gallai, who were unable to make progress even assuming $G$ is $K_4$-free. There Erd\\H{o}s remarked that this conjecture is 'perhaps completely wrongheaded'.\nIt later appeared as Problem 1 in \\cite{EGT92}.\nThe general behaviour of $\\tau(G)$ is the subject of [610].\nReferences\n\n\n[EGT92] Erd\\H{o}s, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289.\n\n[Er88] Erd\\H{o}s, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The proposed clique-transversal inequality tau(G) <= n-H(n) remains open, including the historically highlighted K4-free regime.\n\n**Verified partial progress.**\n\n- The elementary comparison tau(G) <= n-sqrt(n) is weaker than the target.\n- The target is immediate for triangle-free G but not for K4-free G in general.\n- The original dedicated paper studies exactly the maximal-clique transversal parameter in the record.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was located.\n\n**What remains.**\n\nProve or refute the displayed inequality, already for K4-free graphs that are not triangle-free.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #151, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/151\n Evidence used: Lists the exact inequality as open and records the easy and triangle-free cases.\n- Paul Erdos, Tibor Gallai, and Zsolt Tuza, Covering the cliques of a graph with vertices, Discrete Mathematics 108 (1992), 279-289, DOI 10.1016/0012-365X(92)90681-5. (primary): https://doi.org/10.1016/0012-365X(92)90681-5\n Evidence used: Investigates the minimum vertex set meeting all inclusion-maximal cliques.\n\n**Review notes.** Results for maximum cliques or other transversal variants were not conflated with maximal-clique transversals.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1960, "problem_number": "EP-152", "title": "Erdős Problem #152", "statement": "For any $M\\geq 1$, if $A\\subset \\mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\\in A+A$ such that $a+1,a-1\not\\in A+A$.", "background": "There may even be $\\gg \\lvert A\\rvert^2$ many such $a$. A similar question can be asked for truncations of infinite Sidon sets.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The 2026 AlphaProof Nexus paper and companion Lean proof establish the stronger quantitative bound I(A+A) >= (n^2-100n-16)/16 for every finite Sidon set A of size n, where I counts sums with neither adjacent integer also in A+A. This quadratic lower bound settles the stated unboundedness question.\n\n**Verified partial progress.**\n\n- Earlier work of Erdős, Sárközy, and Sós developed structural results on sumsets of Sidon sets.\n- The new theorem gives an explicit quadratic lower bound, much stronger than merely proving arbitrarily many isolated sums.\n- The companion Lean development formalizes both the quantitative inequality and the resulting divergence.\n\n**Full solution or refutation.**\n\nThe proof counts non-isolated sums through adjacent-sum relations and the difference set of A. Sidon uniqueness sharply limits how often the relevant difference configurations can occur. After bounding exceptional and boundary configurations, it obtains I(A+A) >= (n^2-100n-16)/16, which tends to infinity with n and therefore proves the original finite-set assertion.\n\n**What remains.**\n\nThe finite problem is solved. Natural refinements are to improve the constants, determine the optimal asymptotic minimum number of isolated sums, and address the separately formulated infinite/truncation analogue, which the maintained formal record still marks as open.\n\n**Sources checked.**\n\n- George Tsoukalas et al., Advancing Mathematics Research with AI-Driven Formal Proof Search, arXiv:2605.22763v2 (2026). (primary): https://arxiv.org/abs/2605.22763\n Evidence used: States and proves the explicit quadratic lower bound for isolated elements of a finite Sidon sumset.\n- Google DeepMind, erdos_152.lean, AlphaProof Nexus results (2026). (primary): https://github.com/google-deepmind/alphaproof-nexus-results/blob/main/APNOutputs/ErdosProblems/erdos_152.lean\n Evidence used: Formal proof artifact containing the lower-bound lemma and the consequent divergence theorem.\n- Google DeepMind Formal Conjectures documentation, Erdős Problem 152 (accessed 2026-08-17). (authoritative_secondary): https://google-deepmind.github.io/formal-conjectures/doc/FormalConjectures/ErdosProblems/152.html\n Evidence used: Records the weak and quadratic finite variants as proved and distinguishes the still-open infinite variant.\n- Thomas F. Bloom, Erdős Problem #152, Erdős Problems (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/152\n Evidence used: Classifies the problem as proved and summarizes the stronger quadratic result.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1961, "problem_number": "EP-153", "title": "Erdős Problem #153", "statement": "Let $A$ be a finite Sidon set and $A+A=\\{s_1<\\cdots\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The mean-square consecutive-gap conjecture for finite Sidon sumsets remains open; sparse-span and asymptotically maximum Sidon regimes have recent partial arguments.\n\n**Verified partial progress.**\n\n- A Cauchy-Schwarz argument proves divergence whenever diam(A)/|A|^2 tends to infinity.\n- A corrected unrefereed note claims the result when diam(A)=(1+o(1))|A|^2 using asymptotic uniformity and triangular sumset density.\n- An earlier purported full proof was withdrawn after its key shifted-intersection estimate was found false.\n\n**Full solution or refutation.**\n\nNo accepted proof covering all finite Sidon sets was located.\n\n**What remains.**\n\nHandle intermediate bounded diameter ratios and obtain independent or peer-reviewed verification of the dense-regime note.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #153 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/153\n Evidence used: Retains open status, records the sparse argument, the withdrawn proof, and the corrected dense-regime claim.\n- Paul Erdos, Andras Sarkozy, and Vera T. Sos, On sum sets of Sidon sets, I, Journal of Number Theory 47 (1994), 329-347, DOI 10.1006/jnth.1994.1040. (primary): https://doi.org/10.1006/jnth.1994.1040\n Evidence used: Original literature source studying gaps in Sidon sumsets.\n- Corrected writeup for Erdos Problem #153, May 2026, unrefereed. (primary): https://leon2k2k2k.github.io/erdos153_corrected.pdf\n Evidence used: Claims the asymptotically maximum Sidon-set regime; treated as unverified partial progress.\n\n**Review notes.** The recent dense-regime contribution is community-posted and not treated as established peer-reviewed literature.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1962, "problem_number": "EP-155", "title": "Erdős Problem #155", "statement": "Let $F(N)$ be the size of the largest Sidon subset of $\\{1,\\ldots,N\\}$. Is it true that for every $k\\geq 1$ we have $ F(N+k)\\leq F(N)+1 $ for all sufficiently large $N$?", "background": "This may even hold with $k\\approx \\epsilon N^{1/2}$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** For every fixed k, eventual control F(N+k) <= F(N)+1 for the extremal Sidon function remains open.\n\n**Verified partial progress.**\n\n- The elementary inequality F(N+k) <= F(N)+F(k) is immediate.\n- A 2025 tracker discussion sharpens the additive term to a difference-disjointness parameter g(k) when N+1>=k.\n- Neither estimate forces the conjectured additive constant one for arbitrary fixed k.\n\n**Full solution or refutation.**\n\nNo eventual unit-increment theorem was located.\n\n**What remains.**\n\nProve stabilization of each fixed-length increment at one, or find a fixed k with infinitely many larger jumps.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #155 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/155\n Evidence used: Lists the question as open and records the F(N+k) <= F(N)+g(k) observation.\n- Paul Erdos, Andras Sarkozy, and Vera T. Sos, On sum sets of Sidon sets, I, Journal of Number Theory 47 (1994), 329-347, DOI 10.1006/jnth.1994.1040. (primary): https://doi.org/10.1006/jnth.1994.1040\n Evidence used: One of the original sources associated with the extremal Sidon question.\n\n**Review notes.** The input statement is preserved exactly; corrupted trailing background serialization was ignored.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1963, "problem_number": "EP-156", "title": "Erdős Problem #156", "statement": "Does there exist a maximal Sidon set $A\\subset \\{1,\\ldots,N\\}$ of size $O(N^{1/3})$?", "background": "A question of Erd\\H{o}s, S\\'{a}rk\"{o}zy, and S\\'{o}s \\cite{ESS94}. It is easy to prove that the greedy construction of a maximal Sidon set in $\\{1,\\ldots,N\\}$ has size $\\gg N^{1/3}$. Ruzsa \\cite{Ru98b} constructed a maximal Sidon set of size $\\ll (N\\log N)^{1/3}$.\nSee also [340].\nReferences\n\n\n[ESS94] Erd\\H{o}s, P. and S\\'{a}rk\"{o}zy, A. and S\\'{o}s, T., On Sum Sets of Sidon Sets, I. Journal of Number Theory (1994), 329-347.\n\n[Ru98b] Ruzsa, Imre Z., A small maximal Sidon set. Ramanujan J. (1998), 55-58.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether [N] has a maximal Sidon subset of optimal order O(N^{1/3}) remains open.\n\n**Verified partial progress.**\n\n- Every inclusion-maximal Sidon subset of [N] has size Omega(N^{1/3}) by a counting argument.\n- Ruzsa constructs one of size O((N log N)^{1/3}).\n- Later group analogues do not remove the logarithmic factor for integer intervals.\n\n**Full solution or refutation.**\n\nNo O(N^{1/3}) construction or stronger lower bound was located.\n\n**What remains.**\n\nRemove the factor (log N)^{1/3} from Ruzsa's construction or prove that some diverging factor is necessary.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #156, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/156\n Evidence used: Lists the question as open with the Omega(N^{1/3}) and O((N log N)^{1/3}) bounds.\n- Imre Z. Ruzsa, A small maximal Sidon set, Ramanujan Journal 2 (1998), 55-58. (primary): https://www.erdosproblems.com/latex/156\n Evidence used: Constructs a maximal Sidon subset of [N] of order at most (N log N)^{1/3}.\n\n**Review notes.** Maximal means inclusion-maximal, not maximum-cardinality.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1964, "problem_number": "EP-158", "title": "Erdős Problem #158", "statement": "Let $A\\subset \\mathbb{N}$ be an infinite set such that, for any $n$, there are most $2$ solutions to $a+b=n$ with $a\\leq b$. Must $ \\liminf_{N\\to\\infty}\\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}=0? $ ", "background": "If we replace $2$ by $1$ then $A$ is a Sidon set, for which Erd\\H{o}s proved this is true.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether every infinite B2[2] set has liminf A(N)/sqrt(N)=0 remains open.\n\n**Verified partial progress.**\n\n- Erdos proved the conclusion for ordinary Sidon sets B2[1].\n- Erdos, Sarkozy, and Sos obtained a logarithmically sharpened statement in the Sidon case.\n- Cilleruelo and Trujillo construct B2[g] sequences with large normalized limsup, but explicitly leave the liminf question open even for g=2.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for the B2[2] lower-density assertion was located.\n\n**What remains.**\n\nRule out or construct an infinite B2[2] set whose counting function stays above c sqrt(N) eventually.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #158, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/158\n Evidence used: Lists the exact B2[2] liminf question as open.\n- Javier Cilleruelo and Carlos Trujillo, Infinite B2[g] sequences, Israel Journal of Mathematics 126 (2001/2002), DOI 10.1007/BF02784156. (primary): https://matematicas.uam.es/~franciscojavier.cilleruelo/Papers/infinit2%20B2%5Bg%5D%20sequences.pdf\n Evidence used: Explicitly says the generalized liminf conjecture is unknown even for g=2 and proves large-limsup constructions.\n\n**Review notes.** Limsup constructions do not answer the record's liminf question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1965, "problem_number": "EP-159", "title": "Erdős Problem #159", "statement": "There exists some constant $c>0$ such that\n$$R(C_4,K_n) \\ll n^{2-c}.$$", "background": "The current bounds are $ \\frac{n^{3/2}}{(\\log n)^{3/2}}\\ll R(C_4,K_n)\\ll \\frac{n^2}{(\\log n)^2}. $ The upper bound is due to Szemer\\'{e}di (mentioned in \\cite{EFRS78}), and the lower bound is due to Spencer \\cite{Sp77}.\nThis problem is #17 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EFRS78] Erd\\H{o}s, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., On cycle-complete graph Ramsey numbers. J. Graph Theory (1978), 53-64.\n\n[Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains open whether R(C4,K_n) has any fixed polynomial saving below n^2.\n\n**Verified partial progress.**\n\n- The current lower bound is of order at least n^{3/2}/(log n)^{3/2}.\n- The current upper bound is of order at most n^2/(log n)^2.\n- The logarithmic upper saving is n^{2-o(1)} and does not imply the requested n^{2-c}.\n\n**Full solution or refutation.**\n\nNo fixed c>0 satisfying the conjectured upper bound was located.\n\n**What remains.**\n\nProve a fixed power saving below n^2 or construct near-quadratic lower examples refuting every such saving.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #159, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/159\n Evidence used: Lists the problem as open and records the two classical bounds and prize.\n- Paul Erdos, R. J. Faudree, C. C. Rousseau, and R. H. Schelp, On cycle-complete graph Ramsey numbers, Journal of Graph Theory 2 (1978), 53-64, DOI 10.1002/jgt.3190020107. (primary): https://doi.org/10.1002/jgt.3190020107\n Evidence used: Published source recording Szemeredi's upper-bound observation.\n- Joel Spencer, Asymptotic lower bounds for Ramsey functions, Discrete Mathematics 20 (1977), 69-76, DOI 10.1016/0012-365X(77)90044-9. (primary): https://doi.org/10.1016/0012-365X(77)90044-9\n Evidence used: Source of the probabilistic lower bound cited by the tracker.\n\n**Review notes.** An upper bound divided by powers of log n is not a fixed polynomial saving.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1966, "problem_number": "EP-160", "title": "Erdős Problem #160", "statement": "Let $h(N)$ be the smallest $k$ such that $\\{1,\\ldots,N\\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate $h(N)$.", "background": "Investigated by Erd\\H{o}s and Freud. This has been discussed on MathOverflow, where LeechLattice shows $ h(N) \\ll N^{2/3}. $ In the comments of this site Hunter improves this to $ h(N) \\ll N^{\\frac{\\log 3}{\\log 22}+o(1)} $ (note $\\frac{\\log 3}{\\log 22}\\approx 0.355$).\nThe observation of Zach Hunter in that question coupled with recent progress on the size of subsets without three-term arithmetic progression (see \\cite{BlSi23} which improves slightly on the bounds due to Kelley and Meka \\cite{KeMe23}) imply that $ h(N) \\gg \\exp(c(\\log N)^{1/9}) $ for some $c>0$.\nReferences\n\n\n[BlSi23] T. F. Bloom and O. Sisask, An improvement to the Kelley-Meka bounds on three-term arithmetic progressions. arXiv:2309.02353 (2023).\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Estimating h(N) remains open, but a July 2026 preprint improves the upper bound to h(N) <= N^{1/4+o(1)}.\n\n**Verified partial progress.**\n\n- The tracker records a lower bound exp(c(log N)^{1/9}) obtained from modern 3-AP-free-set bounds.\n- Its displayed upper bound N^{log 3/log 22+o(1)} has been superseded by a newer primary preprint.\n- Shi and Dong explicitly deduce h(N) <= N^{1/4+o(1)} using their symmetrically colored progression construction and a Behrend-style product.\n\n**Full solution or refutation.**\n\nThe July 2026 result is substantial progress but does not determine the order of h(N).\n\n**What remains.**\n\nClose the gap between exp(c(log N)^{1/9}) and N^{1/4+o(1)}, including whether h(N)=N^{o(1)}.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #160, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/160\n Evidence used: Lists the problem as open and records the pre-July-2026 lower and upper bounds.\n- Ruizhe Shi and Yiqi Dong, An Improved Upper Bound for Colorings Without Symmetrically Colored k-Term Arithmetic Progressions, arXiv:2607.20752v2 (2026). (primary): https://arxiv.org/abs/2607.20752\n Evidence used: The abstract explicitly derives h(N) <= N^{1/4+o(1)} for Erdos Problem 160.\n\n**Review notes.** The improved bound is from a very recent preprint and postdates the maintained tracker's displayed estimate.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1967, "problem_number": "EP-161", "title": "Erdős Problem #161", "statement": "Let $\\alpha\\in[0,1/2)$ and $n,t\\geq 1$. Let $F^{(t)}(n,\\alpha)$ be the smallest $m$ such that we can $2$-colour the edges of the complete $t$-uniform hypergraph on $n$ vertices such that if $X\\subseteq [n]$ with $\\lvert X\\rvert \\geq m$ then there are at least $\\alpha \\binom{\\lvert X\\rvert}{t}$ many $t$-subsets of $X$ of each colour.\nFor fixed $n,t$ as we change $\\alpha$ from $0$ to $1/2$ does $F^{(t)}(n,\\alpha)$ increase continuously or are there jumps? Only one jump?", "background": "For $\\alpha=0$ this is the usual Ramsey function.\nA conjecture of Erd\\H{o}s, Hajnal, and Rado (see [562]) implies that $ F^{(t)}(n,0)\\asymp \\log_{t-1} n $ and results of Erd\\H{o}s and Spencer imply that $ F^{(t)}(n,\\alpha) \\gg_\\alpha (\\log n)^{\\frac{1}{t-1}} $ for all $\\alpha>0$, and a similar upper bound holds for $\\alpha$ close to $1/2$.\nErd\\H{o}s said in \\cite{Er90b}: 'If I can hazard a guess completely unsupported by evidence, I am afraid that the jump occurs all in one step at $0$. It would be much more interesting if my conjecture would be wrong and perhaps there is some hope for this for $t>3$. I know nothing and offer \\$500 to anybody who can clear up this mystery.'\nConlon, Fox, and Sudakov \\cite{CFS11} have proved that, for any fixed $\\alpha>0$, $ F^{(3)}(n,\\alpha) \\ll_\\alpha \\sqrt{\\log n}. $ Coupled with the lower bound above, this implies that there is only one jump for fixed $\\alpha$ when $t=3$, at $\\alpha=0$.\nFor all $\\alpha>0$ it is known that $ F^{(t)}(n,\\alpha)\\gg_t (\\log n)^{c_\\alpha}. $ See also [563] for more on the case $t=2$.\nThis problem is #40 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[CFS11] Conlon, David and Fox, Jacob and Sudakov, Benny, Large almost monochromatic subsets in hypergraphs. Israel J. Math. (2011), 423--432.\n\n[Er90b] Erd\\H{o}s, Paul, Problems and results on graphs and hypergraphs: similarities and differences. Mathematics of Ramsey theory (1990), 12-28.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For the intended asymptotic formulation there is only one jump, at alpha=0, when t=3; the phase structure for general t remains open.\n\n**Verified partial progress.**\n\n- Conlon, Fox, and Sudakov prove the sharp Theta_alpha(sqrt(log n)) scale for every fixed alpha>0 when t=3.\n- For general t, known results give alpha-dependent polylogarithmic bounds and matching 1/(t-1)-scale estimates only near alpha=1/2.\n- The exact imported non-strict definition is vacuous at alpha=0 and therefore does not reproduce the inverse Ramsey endpoint.\n\n**Full solution or refutation.**\n\nThe t=3 instance of the intended asymptotic jump question is resolved, but t>=4 is open and the imported endpoint wording is inconsistent with the original source.\n\n**What remains.**\n\nDetermine the asymptotic phase transitions for t>=4 after restoring the original strict threshold convention; curate the source-text mismatch.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #161, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/161\n Evidence used: Lists the general problem as open and records the t=3 consequence of Conlon-Fox-Sudakov.\n- David Conlon, Jacob Fox, and Benny Sudakov, Large almost monochromatic subsets in hypergraphs, Israel Journal of Mathematics 181 (2011), 423-432, DOI 10.1007/s11856-011-0016-6. (primary): https://arxiv.org/abs/0901.3912\n Evidence used: Proves the sharp sqrt(log n) almost-monochromatic subset scale for triples.\n- Paul Erdos, Problems and results on graphs and hypergraphs: similarities and differences, Mathematics of Ramsey Theory (1990), 12-28. (primary): https://www.erdosproblems.com/161\n Evidence used: Original source uses a strict threshold, unlike the imported at-least wording at alpha=0.\n\n**Review notes.** Exact statement preserved. The endpoint and continuity defects are source-formulation issues, not silently repaired OCR.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1968, "problem_number": "EP-162", "title": "Erdős Problem #162", "statement": "Let $\\alpha>0$ and $n\\geq 1$. Let $F(n,\\alpha)$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which any induced subgraph $H$ on at least $k$ vertices contains more than $\\alpha\\binom{\\lvert H\\rvert}{2}$ many edges of each colour.\nProve that for every fixed $0\\leq \\alpha \\leq 1/2$, as $n\\to\\infty$, $ F(n,\\alpha)\\sim c_\\alpha \\log n $ for some constant $c_\\alpha$.", "background": "It is easy to show with the probabilistic method that there exist $c_1(\\alpha),c_2(\\alpha)$ such that $ c_1(\\alpha)\\log n < F(n,\\alpha) < c_2(\\alpha)\\log n. $ \",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The literal largest-k formulation is vacuous/inconsistent; tracker discussion indicates a materially reconstructed smallest-k version.\n\n**Verified partial progress.**\n\n- The intended version has Theta(log n) upper and lower bounds.\n- No exact asymptotic constant is known under that reading.\n\n**Full solution or refutation.**\n\nThe source statement cannot be triaged literally as a coherent finite invariant.\n\n**What remains.**\n\nVerify the original source formulation before classifying the intended asymptotic problem.\n\n**Sources checked.**\n\n- Thomas F. Bloom, EP-162 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/162\n Evidence used: Explains the vacuity, endpoint issue, and intended corrections.\n\n**Review notes.** Original statement is preserved; no OCR repair was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1969, "problem_number": "EP-165", "title": "Erdős Problem #165", "statement": "Give an asymptotic formula for $R(3,k)$.", "background": "It is known that there exists some constant $c>0$ such that for large $k$ $ (c+o(1))\\frac{k^2}{\\log k}\\leq R(3,k) \\leq (1+o(1))\\frac{k^2}{\\log k}. $ The lower bound is due to Kim \\cite{Ki95}, the upper bound is due to Shearer \\cite{Sh83}, improving an earlier bound of Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS80}.\nThe value of $c$ in the lower bound has seen a number of improvements. Kim's original proof gave $c\\geq 1/162$. The bound $c\\geq 1/4$ was proved independently by Bohman and Keevash \\cite{BoKe21} and Pontiveros, Griffiths and Morris \\cite{PGM20}. The latter collection of authors conjecture that this lower bound is the true order of magnitude.\nThis was, however, improved by Campos, Jenssen, Michelen, and Sahasrabudhe \\cite{CJMS25} to $c\\geq 1/3$, and further by Hefty, Horn, King, and Pfender \\cite{HHKP25} to $c\\geq 1/2$. Both of these papers conjecture that $c=1/2$ is the correct asymptotic.\nSee also [544], and [986] for the general case. See [1013] for a related function.\nReferences\n\n\n[AKS80] Ajtai, Mikl\\'{o}s and Koml\\'{o}s, J\\'{a}nos and Szemer\\'{e}di, Endre, A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.\n\n[BoKe21] Bohman, Tom and Keevash, Peter, Dynamic concentration of the triangle-free process. Random Structures Algorithms (2021), 221-293.\n\n[CJMS25] M. Campos, M. Jenssen, M. Michelen, and J. Sahasrabudhe, A new lower bound for the Ramsey numbers $R(3,k)$. arXiv:2505.13371 (2025).\n\n[HHKP25] Z. Hefty, P. Horn, D. King, and F. Pfender, Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (2025).\n\n[Ki95] Kim, J. H., The Ramsey number $R(3,t)$ has order of magnitude $t^2/\\log t$. Random Structures and Algorithms (1995), 173-207.\n\n[PGM20] Fiz Pontiveros, Gonzalo and Griffiths, Simon and Morris, Robert, The triangle-free process and the Ramsey number $R(3,k)$. Mem. Amer. Math. Soc. (2020), v+125.\n\n[Sh83] Shearer J., A note on the independence number of triangle-free graphs. Discrete Math. (1983), 83-87.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The asymptotic constant for R(3,k) remains open; the current lower constant is 1/2 and upper constant is 1.\n\n**Verified partial progress.**\n\n- Hefty-Horn-King-Pfender improved the lower constant to 1/2.\n- Both recent lower-bound papers conjecture 1/2 is correct.\n\n**Full solution or refutation.**\n\nNo asymptotic formula is known.\n\n**What remains.**\n\nDetermine the limiting constant.\n\n**Sources checked.**\n\n- Thomas F. Bloom, EP-165 LaTeX source, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/165\n Evidence used: Records the current constants and references.\n\n**Review notes.** Newest lower-bound evidence is a 2025 preprint.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1970, "problem_number": "EP-168", "title": "Erdős Problem #168", "statement": "Let $F(N)$ be the size of the largest subset of $\\{1,\\ldots,N\\}$ which does not contain any set of the form $\\{n,2n,3n\\}$. What is $ \\lim_{N\\to \\infty}\\frac{F(N)}{N}? $ Is this limit irrational?", "background": "This limit was proved to exist by Graham, Spencer, and Witsenhausen \\cite{GSW77}, who showed it is equal to $ \\frac{1}{3}\\sum_{k\\in K}\\frac{1}{d_k}, $ where $d_1f(k-1)$, where $f$ counts the largest subset of $\\{d_1,\\ldots,d_k\\}$ that avoids $\\{n,2n,3n\\}$.\nSimilar questions can be asked for the density or upper density of infinite sets without such configurations.\nThe limit can be estimated by elementary arguments (see the comments). Eberhard has used the formula of \\cite{GSW77} mentioned above to calculate the value of the limit as $ 0.800965\\cdots. $ This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[GSW77] Graham, R. and Spencer, J. and Witsenhausen, H., On Extremal Density Theorems for Linear Forms. Number Theory and Algebra (1977).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The density limit exists and is computable, but its exact nature and irrationality remain open.\n\n**Verified partial progress.**\n\n- Graham-Spencer-Witsenhausen proved existence and a series formula.\n- The current tracker reports numerical value about 0.800965755.\n\n**Full solution or refutation.**\n\nThe existence portion is solved; the requested identification/irrationality is not.\n\n**What remains.**\n\nDetermine the exact value or prove/disprove irrationality.\n\n**Sources checked.**\n\n- Thomas F. Bloom, EP-168, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/168\n Evidence used: Records the existence theorem, formula, computation, and open status.\n\n**Review notes.** A numerical value is not treated as an exact resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1971, "problem_number": "EP-169", "title": "Erdős Problem #169", "statement": "Let $k\\geq 3$ and $f(k)$ be the supremum of $\\sum_{n\\in A}\\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-term arithmetic progression. Estimate $f(k)$.\nIs $ \\lim_{k\\to \\infty}\\frac{f(k)}{\\log W(k)}=\\infty $ where $W(k)$ is the van der Waerden number?", "background": "Berlekamp \\cite{Be68} proved $f(k) \\geq \\frac{\\log 2}{2}k$. Gerver \\cite{Ge77} proved $ f(k) \\geq (1-o(1))k\\log k. $ It is trivial that $ \\frac{f(k)}{\\log W(k)}\\geq \\frac{1}{2}, $ but improving the right-hand side to any constant $>1/2$ is open.\nGerver also proved (see the comments for an alternative argument of Tao) that [3] is equivalent to $f(k)$ being finite for all $k$.\nThe current record for $f(3)$ is $f(3)\\geq 3.00849$, due to Wr\\'{o}blewski \\cite{Wr84}. Walker \\cite{Wa25} proved $f(4)\\geq 4.43975$.\nWalker \\cite{Wa25} has shown that it suffices to consider Kempner sets (that is, sets of integers defined as all those whose base $b$ digits are contained in some $S\\subset \\{0,\\ldots,b-1\\}$ for fixed $b$ and $S$), in the sense that for any $k\\geq 3$ and $\\epsilon>0$ there is a Kempner set $A$ lacking $k$-term arithmetic progressions such that $ \\sum_{n\\in A}\\frac{1}{n}\\geq f(k)-\\epsilon. $ \nReferences\n\n\n[Be68] Berlekamp, E. R., A construction for partitions which avoid long arithmetic progressions. Canad. Math. Bull. (1968), 409-414.\n\n[Ge77] Gerver, Joseph L., The sum of the reciprocals of a set of integers with no\narithmetic progression of {$k$} terms. Proc. Amer. Math. Soc. (1977), 211--214.\n\n[Wa25] A. Walker, Integer sets of large harmonic sum which avoid long arithmetic progressions. arXiv:2203.06045 (2025).\n\n[Wr84] No reference found.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Estimating f(k) and improving its ratio to log W(k) remain open.\n\n**Verified partial progress.**\n\n- Gerver proved f(k)>=(1-o(1))k log k.\n- Walker reduced near-extremizers to Kempner sets.\n\n**Full solution or refutation.**\n\nNo requested asymptotic is known.\n\n**What remains.**\n\nEstablish finiteness/order of f(k) and improve the ratio lower bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, EP-169, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/169\n Evidence used: Lists current lower bounds and remaining questions.\n\n**Review notes.** No distinction in statement text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1972, "problem_number": "EP-170", "title": "Erdős Problem #170", "statement": "Let $F(N)$ be the smallest possible size of $A\\subset \\{0,1,\\ldots,N\\}$ such that $\\{0,1,\\ldots,N\\}\\subset A-A$. Find the value of $ \\lim_{N\\to \\infty}\\frac{F(N)}{N^{1/2}}. $ ", "background": "The Sparse Ruler problem. R\\'{e}dei asked whether this limit exists, which was proved by Erd\\H{o}s and G\\'{a}l \\cite{ErGa48}. Bounds on the limit were improved by Leech \\cite{Le56}. The limit is known to be in the interval $[1.56,\\sqrt{3}]$. The lower bound is due to Leech \\cite{Le56}, the upper bound is due to Wichmann \\cite{Wi63}. Computational evidence by Pegg \\cite{Pe20} suggests that the upper bound is the truth. A similar question can be asked without the restriction $A\\subset \\{0,1,\\ldots,N\\}$.\nReferences\n\n\n[ErGa48] Erd\\H{o}s, P. and G\\'{a}l, I., On the representation of $1,2,\\ldots,N$ by differences. Nederl. Akad. Wetensch., Proc. (1948), 1155-1158.\n\n[Le56] Leech, J., On the representation of $1,2,\\ldots,n$ by differences. J. London Math. Soc. (1956), 160-169.\n\n[Pe20] Pegg, E., Hitting All the Marks: Exploring New Bounds for Sparse Rulers and a Wolfram Language Proof. https://blog.wolfram.com/2020/02/12/hitting-all-the-marks-exploring-new-bounds-for-sparse-rulers-and-a-wolfram-language-proof/ (2020).\n\n[Wi63] Wichmann, B., A note on restricted difference bases. J. London Math. Soc. (1963), 465-466.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The sparse-ruler limit exists but its exact value is unknown; it lies in [1.56,sqrt(3)].\n\n**Verified partial progress.**\n\n- Erdos-Gal proved existence.\n- Leech and Wichmann give the current displayed interval.\n\n**Full solution or refutation.**\n\nNo exact value was located.\n\n**What remains.**\n\nClose the interval for the sparse-ruler constant.\n\n**Sources checked.**\n\n- Thomas F. Bloom, EP-170, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/170\n Evidence used: Records existence, bounds, and open status.\n\n**Review notes.** Computation suggesting sqrt(3) is not a proof.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1973, "problem_number": "EP-172", "title": "Erdős Problem #172", "statement": "Is it true that in any finite colouring of $\\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?", "background": "First asked by Hindman. Hindman \\cite{Hi80} has proved this is false (with 7 colours) if we ask for an infinite $A$. In \\cite{Er77c} Erd\\H{o}s asks about the case for an infinite $A$ with just $2$ colours.\nMoreira \\cite{Mo17} has proved that in any finite colouring of $\\mathbb{N}$ there exist $x,y$ such that $\\{x,x+y,xy\\}$ are all the same colour.\nAlweiss \\cite{Al23} has proved that, in any finite colouring of $\\mathbb{Q}\\backslash \\{0\\}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour. Bowen and Sabok \\cite{BoSa22} had proved this earlier for the first non-trivial case of $\\lvert A\\rvert=2$.\nReferences\n\n\n[Al23] R. Alweiss, Hindman's conjecture over the rationals. arXiv:2307.08901 (2023).\n\n[BoSa22] M. Bowen and M. Sabok, Monochromatic Sums and Products in the Rationals. arXiv:2210.12290 (2022).\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[Hi80] Hindman, Neil, Partitions and sums and products-two counterexamples. J. Combin. Theory Ser. A (1980), 113-120.\n\n[Mo17] Moreira, J., Monochromatic sums and products in $\\mathbbN$. Ann. Math. (2017), 1069-1090.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The finite-colouring-of-N sums-and-products assertion is open despite rational and small natural-number analogues.\n\n**Verified partial progress.**\n\n- Moreira proves monochromatic x,x+y,xy in N.\n- Alweiss proves the arbitrarily-large finite-set conclusion over nonzero rationals.\n\n**Full solution or refutation.**\n\nNo resolution over N was located.\n\n**What remains.**\n\nTransfer the rational theorem to N or produce a countercolouring.\n\n**Sources checked.**\n\n- Ryan Alweiss, Hindman's conjecture over the rationals, arXiv:2307.08901 (2023). (primary): https://arxiv.org/abs/2307.08901\n Evidence used: Proves the rational analogue.\n- Thomas F. Bloom, EP-172, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/172\n Evidence used: Lists the natural-number problem as open.\n\n**Review notes.** Rationals and naturals are deliberately separated.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1974, "problem_number": "EP-173", "title": "Erdős Problem #173", "statement": "In any $2$-colouring of $\\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$.", "background": "For some colourings a single equilateral triangle has to be excluded, considering the colouring by alternating strips. Shader \\cite{Sh76} has proved this is true if we just consider a single right-angled triangle.\nReferences\n\n\n[Sh76] Shader, L., All right triangles are Ramsey in $\\mathbbE^2$!. J. Comb. Th. A (1976), 385-389.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The all-but-one-triangle theorem remains open; Shader proved a one-right-triangle special case.\n\n**Verified partial progress.**\n\n- Alternating strips show an exceptional equilateral triangle may be necessary.\n- Shader's right-triangle case is known.\n\n**Full solution or refutation.**\n\nNo general classification was located.\n\n**What remains.**\n\nProve or refute the at-most-one-exception assertion.\n\n**Sources checked.**\n\n- Thomas F. Bloom, history of EP-173, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/173\n Evidence used: Records special case and open general statement.\n\n**Review notes.** The special shape result is not generalized.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1975, "problem_number": "EP-174", "title": "Erdős Problem #174", "statement": "A finite set $A\\subset \\mathbb{R}^n$ is called Ramsey if, for any $k\\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring of $\\mathbb{R}^d$ there exists a monochromatic copy of $A$. Characterise the Ramsey sets in $\\mathbb{R}^n$.", "background": "Erd\\H{o}s, Graham, Montgomery, Rothschild, Spencer, and Straus \\cite{EGMRSS73} proved that every Ramsey set is 'spherical': it lies on the surface of some sphere. Graham has conjectured that every spherical set is Ramsey. Leader, Russell, and Walters \\cite{LRW12} have alternatively conjectured that a set is Ramsey if and only if it is 'subtransitive': it can be embedded in some higher-dimensional set on which rotations act transitively.\nSets known to be Ramsey include vertices of $k$-dimensional rectangles \\cite{EGMRSS73}, non-degenerate simplices \\cite{FrRo90}, trapezoids \\cite{Kr92}, and regular polygons/polyhedra \\cite{Kr91}.\nReferences\n\n\n[EGMRSS73] Erd\\H{o}s, P. and Graham, R. L. and Montgomery, P. and Rothschild, B. L. and Spencer, J. and Straus, E. G., Euclidean Ramsey Theorems I. J. Comb. Th. A (1973), 341-363.\n\n[FrRo90] Frankl, P. and R\"{o}dl, V., A partition property of simplices in Euclidean space. J. Amer. Math. Soc. (1990), 1-7.\n\n[Kr91] K\\v{r}\\'{\\i}\\v{z}, Igor, Permutation groups in Euclidean Ramsey theory. Proc. Amer. Math. Soc. (1991), 899-907.\n\n[Kr92] K\\v{r}\\'{\\i}\\v{z}, Igor, All trapezoids are Ramsey. Discrete Math. (1992), 59-62.\n\n[LRW12] Leader, Imre and Russell, Paul A. and Walters, Mark, Transitive sets in Euclidean Ramsey theory. J. Combin. Theory Ser. A (2012), 382-396.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No characterization of Ramsey sets is known; sphericity is necessary and several families are known Ramsey.\n\n**Verified partial progress.**\n\n- Every Ramsey set is spherical.\n- Rectangles, simplices, trapezoids, and regular polytopes are known Ramsey.\n\n**Full solution or refutation.**\n\nCompeting sphericity/subtransitivity conjectures remain open.\n\n**What remains.**\n\nProve a characterization or separate the proposed characterizations.\n\n**Sources checked.**\n\n- Thomas F. Bloom, EP-174, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/174\n Evidence used: Records the necessary theorem, conjectures, and known families.\n\n**Review notes.** Known examples do not constitute characterization.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1976, "problem_number": "EP-176", "title": "Erdős Problem #176", "statement": "Let $N(k,\\ell)$ be the minimal $N$ such that for any $f:\\{1,\\ldots,N\\}\\to\\{-1,1\\}$ there must exist a $k$-term arithmetic progression $P$ such that $ \\left\\lvert \\sum_{n\\in P}f(n)\\right\\rvert\\geq \\ell. $ Find good upper bounds for $N(k,\\ell)$. Is it true that for any $c>0$ there exists some $C>1$ such that $ N(k,ck)\\leq C^k? $ What about $ N(k,2)\\leq C^k $ or $ N(k,\\sqrt{k})\\leq C^k? $ ", "background": "When $\\ell=k$ this is the van der Waerden number $W(k)$ (see [138]). Spencer \\cite{Sp73} has proved that if $k=2^tm$ with $m$ odd then $ N(k,1)=2^t(k-1)+1. $ Erd\\H{o}s and Graham write that 'no decent bound' is known even for $N(k,2)$.\nErd\\H{o}s \\cite{Er63d} proved that, for every $c>0$, $ N(k,ck)> (1+\\alpha_c)^k $ where $\\alpha_c\\to 0$ as $c\\to 0$ and $\\alpha_c\\to \\sqrt{2}-1$ as $c\\to 1$.\nReferences\n\n\n[Er63d] Erd\\H{o}s, P\\'al, On combinatorial questions connected with a theorem of\n{R}amsey and van der {W}aerden. Mat. Lapok (1963), 29--37.\n\n[Sp73] J. Spencer, Problems 185. Bull. Canad. Math. Soc. (1973), 185.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Upper bounds for N(k,l), including N(k,2), remain open; sharp low-discrepancy special cases and exponential lower bounds are known.\n\n**Verified partial progress.**\n\n- Spencer determined N(k,1).\n- Local-lemma bounds improve the linear-discrepancy lower exponential scale.\n\n**Full solution or refutation.**\n\nNo established decisive upper bound was located.\n\n**What remains.**\n\nProve exponential upper bounds, especially for l=2 or l=sqrt(k).\n\n**Sources checked.**\n\n- Thomas F. Bloom, EP-176, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/176\n Evidence used: Lists the problem as open and records lower bounds.\n\n**Review notes.** Unarchived tracker-comment claims are not treated as theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1977, "problem_number": "EP-177", "title": "Erdős Problem #177", "statement": "Find the smallest $h(d)$ such that the following holds. There exists a function $f:\\mathbb{N}\\to\\{-1,1\\}$ such that, for every $d\\geq 1$, $ \\max_{P_d}\\left\\lvert \\sum_{n\\in P_d}f(n)\\right\\rvert\\leq h(d), $ where $P_d$ ranges over all finite arithmetic progressions with common difference $d$.", "background": "Cantor, Erd\\H{o}s, Schreiber, and Straus \\cite{Er66} proved that $h(d)\\ll d!$ is possible. Van der Waerden's theorem implies that $h(d)\\to \\infty$. Beck \\cite{Be17} has shown that $h(d) \\leq d^{8+\\epsilon}$ is possible for every $\\epsilon>0$. Roth's famous discrepancy lower bound \\cite{Ro64} implies that $h(d)\\gg d^{1/2}$.\nReferences\n\n\n[Be17] Beck, J\\'{o}zsef, A discrepancy problem: balancing infinite dimensional vectors. Number theory-Diophantine problems, uniform distribution\nand applications (2017), 61-82.\n\n[Er66] Erd\\H{o}s, P\\'al, Remarks on number theory. {V}. {E}xtremal problems in number\ntheory. {II}. Mat. Lapok (1966), 135--155.\n\n[Ro64] Roth, K. F., Remark concerning integer sequences. Acta Arith. (1964), 257-260.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The smallest h(d) remains unknown between Omega(sqrt d) and O(d^(8+epsilon)).\n\n**Verified partial progress.**\n\n- Roth gives the square-root lower bound.\n- Beck gives a polynomial upper bound.\n\n**Full solution or refutation.**\n\nNo matching bounds were located.\n\n**What remains.**\n\nDetermine the correct exponent/order of h(d).\n\n**Sources checked.**\n\n- Thomas F. Bloom, EP-177, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/177\n Evidence used: Records current upper/lower bounds and open status.\n\n**Review notes.** No computation was used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1978, "problem_number": "EP-180", "title": "Erdős Problem #180", "statement": "If $\\mathcal{F}$ is a finite set of finite graphs then $\\mathrm{ex}(n;\\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\\mathcal{F}$. Note that it is trivial that $\\mathrm{ex}(n;\\mathcal{F})\\leq \\mathrm{ex}(n;G)$ for every $G\\in\\mathcal{F}$.\nIs it true that, for every $\\mathcal{F}$, there exists $G\\in\\mathcal{F}$ such that $ \\mathrm{ex}(n;G)\\ll_{\\mathcal{F}}\\mathrm{ex}(n;\\mathcal{F})? $ ", "background": "A problem of Erd\\H{o}s and Simonovits.\nThis is trivially true if $\\mathcal{F}$ does not contain any bipartite graphs, since by the Erd\\H{o}s-Stone theorem if $H\\in\\mathcal{F}$ has minimal chromatic number $r\\geq 2$ then $ \\mathrm{ex}(n;H)=\\mathrm{ex}(n;\\mathcal{F})=\\left(\\frac{r-2}{r-1}+o(1)\\right)\\binom{n}{2}. $ Erd\\H{o}s and Simonovits observe that this is false for infinite families $\\mathcal{F}$, e.g. the family of all cycles.\nHunter has provided the following 'folklore counterexample': if $\\mathcal{F}=\\{H_1,H_2\\}$ where $H_1$ is a star and $H_2$ is a matching, both with at least two edges, then $\\mathrm{ex}(n;\\mathcal{F})\\ll 1$, but $\\mathrm{ex}(n;H_i)\\asymp n$ for $1\\leq i\\leq 2$. This conjecture may still hold for all other $\\mathcal{F}$.\nSee also [575].\nThis problem is #47 in Extremal Graph Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The literal finite-family compactness statement is false: the family {K_{1,2}, 2K_2} has bounded family extremal number while each individual extremal number is linear; a 2026 manuscript also disproves the corrected cyclic connected-bipartite version.\n\n**Verified partial progress.**\n\n- For n>=4, ex(n,{K_{1,2},2K_2})=1, ex(n,K_{1,2})=floor(n/2), and ex(n,2K_2)=n-1.\n- The 2026 OpenAI manuscript constructs a finite family of connected bipartite graphs containing cycles with family extremal number O(n^(4/3-1/48)) but individual extremal numbers Omega(n^(4/3)).\n- A no-sorry Lean certificate accompanies the stronger counterexample.\n\n**Full solution or refutation.**\n\nThe star-matching family directly negates the quantifiers in the imported statement, and the newer cyclic family refutes the natural corrected form as well.\n\n**What remains.**\n\nThe literal question is settled negatively; only further restricted variants of compactness could remain meaningful.\n\n**Sources checked.**\n\n- OpenAI, Ten Advances in Mathematics and Theoretical Computer Science, Chapter 10, updated August 6, 2026. (primary): https://cdn.openai.com/pdf/ten-proofs-oai.pdf\n Evidence used: Records the exact folklore counterexample and proves a stronger counterexample with cyclic connected bipartite forbidden graphs.\n- OpenAI, CompactnessAndDegeneracy.lean, no-sorry Lean certificate (2026). (primary): https://github.com/openai/ten-proofs/blob/main/CompactnessAndDegeneracy.lean\n Evidence used: Formal certificate for the quantitative compactness counterexample.\n- Thomas F. Bloom, Erdos Problem #180, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/180\n Evidence used: Its remarks explicitly record Hunter's folklore star-matching counterexample, despite the stale OPEN banner.\n\n**Review notes.** The imported background already contains the counterexample but is followed by a stray serialized difficulty/record fragment; neither source field was edited.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1979, "problem_number": "EP-181", "title": "Erdős Problem #181", "statement": "Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that $ R(Q_n) \\ll 2^n. $ ", "background": "Conjectured by Burr and Erd\\H{o}s, althouhg in \\cite{Er93} Erd\\H{o}s says the behaviour of $R(Q_n)$ was considered by himself and S\\'{o}s, who could not decide whether $R(Q_n)/2^n\\to \\infty$ or not.\nThe trivial bound is $ R(Q_n) \\leq R(K_{2^n})\\leq C^{2^n} $ for some constant $C>1$. This was improved a number of times; the current best bound due to Tikhomirov \\cite{Ti22} is $ R(Q_n)\\ll 2^{(2-c)n} $ for some small constant $c>0$. (In fact $c\\approx 0.03656$ is permissible.)\nThis problem is #20 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[Ti22] Tikhomirov, K., A remark on the Ramsey number of the hypercube. arXiv:2208.14568 (2022).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The conjecture R(Q_n)=O(2^n) remains open; Tikhomirov's current general upper bound is O(2^((2-c)n)) for an absolute c>0.\n\n**Verified partial progress.**\n\n- Tikhomirov improved the exponent-2 upper bound by a fixed positive amount.\n- The maintained record states that c approximately 0.03656 is permissible.\n- The resulting bound remains exponentially larger than the conjectured order 2^n.\n\n**Full solution or refutation.**\n\nNo linear-in-the-number-of-vertices Ramsey bound for the hypercube was located.\n\n**What remains.**\n\nProve R(Q_n)<=C 2^n for an absolute C, or refute that order of growth.\n\n**Sources checked.**\n\n- Konstantin Tikhomirov, A remark on the Ramsey number of the hypercube, European Journal of Combinatorics 120 (2024), Article 103954. (primary): https://arxiv.org/abs/2208.14568\n Evidence used: Proves R(Q_n)=O(2^((2-c)n)) for a universal positive c.\n- Thomas F. Bloom, Erdos Problem #181, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/181\n Evidence used: Lists the Burr-Erdos conjecture as open and records the current exponent.\n\n**Review notes.** The background contains the typo 'althouhg' and the batch-wide stray serialized tail; both were preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1980, "problem_number": "EP-183", "title": "Erdős Problem #183", "statement": "Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic triangle. Determine $ \\lim_{k\\to \\infty}R(3;k)^{1/k}. $ ", "background": "Erd\\H{o}s offers \\$100 for showing that this limit is finite. An easy pigeonhole argument shows that $ R(3;k)\\leq 2+k(R(3;k-1)-1), $ from which $R(3;k)\\leq \\lceil e k!\\rceil$ immediately follows. The best-known upper bounds are all of the form $ck!+O(1)$, and arise from this type of inductive relationship and computational bounds for $R(3;k)$ for small $k$. The best-known lower bound (coming from lower bounds for Schur numbers) is $ R(3,k)\\geq (380)^{k/5}-O(1), $ due to Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik \\cite{ACPPRT21} (improving previous bounds of Exoo \\cite{Ex94} and Fredricksen and Sweet \\cite{FrSw00}). Note that $380^{1/5}\\approx 3.2806$.\nSee also [483].\nThis problem is #21 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[ACPPRT21] R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel, and J. Tomasik, New lower bounds for Schur and weak Schur numbers. arXiv:2112.03175 (2021).\n\n[Ex94] Exoo, G., A lower bound for Schur numbers and multicolor Ramsey numbers. Electronic J. of Combinatorics (1994).\n\n[FrSw00] Fredricksen, Harold and Sweet, Melvin M., Symmetric sum-free partitions and lower bounds for {S}chur\nnumbers. Electron. J. Combin. (2000), Research Paper 32, 9.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** An August 2026 superexponential lower bound proves that the extended limit of R(3;k)^(1/k) is positive infinity, resolving the problem and showing the limit is not finite.\n\n**Verified partial progress.**\n\n- The new theorem gives R(3;k)>=(c k^(1/3)/log k)^k for every k>=2 and an absolute c>0.\n- The standard product construction plus Fekete's lemma gives existence of the extended limit.\n- Together with the factorial upper bound, the growth is k^(Theta(k)); a no-sorry Lean certificate is public.\n\n**Full solution or refutation.**\n\nTaking kth roots in the new lower bound gives c k^(1/3)/log k tending to infinity, so lim R(3;k)^(1/k)=+infinity.\n\n**What remains.**\n\nThe requested limit is determined; sharpening the exponent hidden in k^(Theta(k)) is a separate quantitative problem.\n\n**Sources checked.**\n\n- OpenAI, Ten Advances in Mathematics and Theoretical Computer Science, Chapter 9, updated August 6, 2026. (primary): https://cdn.openai.com/pdf/ten-proofs-oai.pdf\n Evidence used: Theorem 1.1 proves the superexponential lower bound and explicitly concludes that the requested limit is +infinity.\n- OpenAI, MulticolorTriangleRamsey.lean, no-sorry Lean certificate (2026). (primary): https://github.com/openai/ten-proofs/blob/main/MulticolorTriangleRamsey.lean\n Evidence used: Formal certificate for an explicit superexponential multicolor-triangle Ramsey lower bound.\n- OpenAI, Ten advances in mathematics and theoretical computer science, August 1, 2026. (authoritative_secondary): https://openai.com/index/ten-advances-in-mathematics/\n Evidence used: Official release identifies the result as resolving Erdos Problem 183.\n- Thomas F. Bloom, Erdos Problem #183, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/183\n Evidence used: Supplies the pre-August-2026 bounds; its OPEN display is stale relative to the new primary manuscript.\n\n**Review notes.** This very recent AI-generated manuscript is accompanied by a formal certificate but has not yet been incorporated into the maintained page; expert review is requested. The imported background also has the batch-wide serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1981, "problem_number": "EP-184", "title": "Erdős Problem #184", "statement": "Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges.", "background": "Conjectured by Erd\\H{o}s and Gallai, who proved that $O(n\\log n)$ many cycles and edges suffices. The graph $K_{3,n-3}$ shows that at least $(1+c)n$ many cycles and edges are required, for some constant $c>0$. In \\cite{Er71} Erd\\H{o}s suggests that only $n-1$ many cycles and edges are required if we do not require them to be edge-disjoint.\nThe best bound available is due to Buci\\'{c} and Montgomery \\cite{BM22}, who prove that $O(n\\log^*n)$ many cycles and edges suffice, where $\\log^*$ is the iterated logarithm function.\nConlon, Fox, and Sudakov \\cite{CFS14} proved that $O_\\epsilon(n)$ cycles and edges suffice if $G$ has minimum degree at least $\\epsilon n$, for any $\\epsilon>0$.\nSee also [583] for an analogous problem decomposing into paths, and [1017] for decomposing into complete graphs.\nReferences\n\n\n[BM22] Buci\\'C, M. and Montgomery, R., Towards the Erd\\H{o}s-Gallai Cycle Decomposition Conjecture. arXiv:2211.07689 (2022).\n\n[CFS14] Conlon, David and Fox, Jacob and Sudakov, Benny, Cycle packing. Random Structures Algorithms (2014), 608-626.\n\n[Er71] Erd\\H{o}s, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Erdos-Gallai O(n) edge-disjoint cycle-and-edge decomposition conjecture remains open; the best general theorem located is O(n log* n).\n\n**Verified partial progress.**\n\n- Bucic and Montgomery improved the prior O(n log log n) general bound to O(n log* n).\n- Conlon, Fox, and Sudakov proved O_epsilon(n) pieces for graphs of minimum degree at least epsilon n.\n- Neither result removes the slowly growing factor for arbitrary graphs.\n\n**Full solution or refutation.**\n\nNo O(n) decomposition theorem for all n-vertex graphs was located.\n\n**What remains.**\n\nRemove the log* n factor in complete generality, or disprove the linear bound.\n\n**Sources checked.**\n\n- Matija Bucic and Richard Montgomery, Towards the Erdos-Gallai cycle decomposition conjecture, Advances in Mathematics 437 (2024), Article 109434. (primary): https://doi.org/10.1016/j.aim.2023.109434\n Evidence used: Proves the O(n log* n) general bound.\n- David Conlon, Jacob Fox, and Benny Sudakov, Cycle packing, Random Structures & Algorithms 45 (2014), 608-626. (primary): https://arxiv.org/abs/1310.7082\n Evidence used: Proves linear-size decompositions under linear minimum degree and the earlier general advance.\n- Thomas F. Bloom, Erdos Problem #184, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/184\n Evidence used: Lists the conjecture as open and records the current general and dense-graph bounds.\n\n**Review notes.** The imported background has the batch-wide stray serialized record tail; it was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1982, "problem_number": "EP-187", "title": "Erdős Problem #187", "statement": "Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference $d$ of length $f(d)$ for infinitely many $d$.", "background": "Originally asked by Cohen. Erd\\H{o}s observed that colouring according to whether $\\{ \\sqrt{2}n\\}<1/2$ or not implies $f(d) \\ll d$ (using the fact that $\\|\\sqrt{2}q\\| \\gg 1/q$ for all $q$, where $\\|x\\|$ is the distance to the nearest integer). Beck \\cite{Be80} has improved this using the probabilistic method, constructing a colouring that shows $f(d)\\leq (1+o(1))\\log_2 d$. Van der Waerden's theorem implies $f(d)\\to \\infty$ is necessary.\nReferences\n\n\n[Be80] Beck, J\\'{o}zsef, A remark concerning arithmetic progressions. J. Combin. Theory Ser. A (1980), 376-379.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The optimal unavoidable progression-length function remains unknown; Beck's coloring gives the upper bound f(d)<=(1+o(1)) log_2 d, while van der Waerden only forces divergence.\n\n**Verified partial progress.**\n\n- Erdos first gave a linear upper construction and reported a square-root improvement by Petruska and Szemeredi.\n- Beck's probabilistic construction reduced the upper scale to asymptotically log_2 d.\n- Van der Waerden's theorem shows that a bounded choice of f cannot be optimal, but no matching quantitative lower order was located.\n\n**Full solution or refutation.**\n\nNo asymptotically matching lower and upper functions were located.\n\n**What remains.**\n\nDetermine the order of the largest function forced for infinitely many common differences in every two-coloring of the integers.\n\n**Sources checked.**\n\n- Jozsef Beck, A remark concerning arithmetic progressions, Journal of Combinatorial Theory, Series A 29 (1980), 376-379. (primary): https://doi.org/10.1016/0097-3165(80)90035-7\n Evidence used: The published abstract states the asymptotic logarithmic upper bound from Beck's construction.\n- Thomas F. Bloom, Erdos Problem #187, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/187\n Evidence used: Lists the optimization problem as open and summarizes all recorded bounds.\n\n**Review notes.** The imported background has the batch-wide stray serialized record tail; it was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1983, "problem_number": "EP-188", "title": "Erdős Problem #188", "statement": "What is the smallest $k$ such that $\\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance $1$?", "background": "Erd\\H{o}s, Graham, Montgomery, Rothschild, Spencer, and Straus \\cite{EGMRSS75} proved $k\\geq 5$. Tsaturian \\cite{Ts17} improved this to $k\\geq 6$. Erd\\H{o}s and Graham claim that $k\\leq 10000000$ ('more or less'), but give no proof.\nErd\\H{o}s and Graham asked this with just any $k$-term arithmetic progression in blue (not necessarily with distance $1$), but Alon has pointed out that in fact no such $k$ exists: in any red/blue colouring of the integer points on a line either there are two red points distance $1$ apart, or else the set of blue points and the same set shifted by $1$ cover all integers, and hence by van der Waerden's theorem there are arbitrarily long blue arithmetic progressions.\nIt seems most likely, from context, that Erd\\H{o}s and Graham intended to restrict the blue arithmetic progression to have distance $1$ (although they do not write this restriction in their papers).\nReferences\n\n\n[EGMRSS75] Erd\\H{o}s, P. and Graham, R. L. and Montgomery, P. and\nRothschild, B. L. and Spencer, J. and Straus, E. G., Euclidean {R}amsey theorems. {II}. (1975), 529--557.\n\n[Ts17] Tsaturian, Sergei, A {E}uclidean {R}amsey result in the plane. Electron. J. Combin. (2017), Paper No. 4.35, 9.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** For the unit-spaced blue-progression formulation, the smallest avoidable length is finite and at least 6, but its exact value is unknown.\n\n**Verified partial progress.**\n\n- Tsaturian proved every red/blue plane coloring has either a red unit pair or a blue unit-spaced five-term progression, giving k>=6.\n- Conlon and Fox proved that in each fixed dimension a coloring exists with no red unit pair and no sufficiently long blue unit-spaced progression, giving a rigorous finite upper bound in the plane.\n- Under the original unrestricted-blue-difference wording, no finite k exists; the input uses the likely intended unit-spacing reconstruction.\n\n**Full solution or refutation.**\n\nThe literature proves finiteness and a lower bound but does not determine the minimum k.\n\n**What remains.**\n\nClose the gap between the lower bound 6 and the enormous constructive upper bound for unit-spaced blue progressions in the plane.\n\n**Sources checked.**\n\n- Sergei Tsaturian, A Euclidean Ramsey Result in the Plane, Electronic Journal of Combinatorics 24(4) (2017), P4.35. (primary): https://doi.org/10.37236/7148\n Evidence used: Proves the unavoidable blue unit-spaced five-term progression when red unit pairs are excluded.\n- David Conlon and Jacob Fox, Lines in Euclidean Ramsey Theory, Discrete & Computational Geometry 61 (2019), 218-225. (primary): https://doi.org/10.1007/s00454-018-9980-5\n Evidence used: Constructs, in every fixed dimension, a coloring excluding red unit pairs and all sufficiently long blue unit-spaced progressions.\n- Thomas F. Bloom, discussion of Erdos Problem #188, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/188\n Evidence used: Explains the historical wording ambiguity and records the current lower and upper evidence.\n\n**Review notes.** The exact input uses unit spacing, but the original papers apparently omitted that restriction; the reconstruction and the batch-wide serialized tail are flagged without editing the source.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1984, "problem_number": "EP-190", "title": "Erdős Problem #190", "statement": "Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\\{1,\\ldots,N\\}$ (into any number of colours) there is always either a monochromatic $k$-term arithmetic progression or a rainbow arithmetic progression (i.e. all elements are different colours). Estimate $H(k)$. Is it true that $ H(k)^{1/k}/k \\to \\infty $ as $k\\to\\infty$?", "background": "This type of problem belongs to 'canonical' Ramsey theory. The existence of $H(k)$ follows from Szemer\\'{e}di's theorem, and it is easy to show that $H(k)^{1/k}\\to\\infty$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The definite Erdős--Graham limit question is solved affirmatively. Bae proved H(k)^(1/k)/k >= (1/e-o(1))k/log k, and Fox--Hunter independently proved the stronger H(k) >= k^((1-o(1))k log k). The dataset's open label is stale, although the broad request to estimate H(k) sharply remains incomplete.\n\n**Verified partial progress.**\n\n- Existence of H(k) was known from canonical van der Waerden and Szemerédi theory.\n- Bae's 2026 preprint gives H(k)^(1/k)/k >= (1/e-epsilon(k))k/log k with epsilon(k) tending to zero.\n- Fox and Hunter's 2026 Theorem 3 strengthens this to H(k) >= k^((1-o(1))k log k).\n\n**Full solution or refutation.**\n\nUse H(k) >= w(k;k-1), since a coloring with fewer than k colors cannot contain a rainbow k-term progression. New many-color lower bounds for van der Waerden numbers then force H(k)^(1/k)/k to diverge. Bae obtains an explicit Omega(k/log k) lower bound for this ratio; Fox--Hunter obtain a much stronger canonical lower bound.\n\n**What remains.**\n\nNo matching upper bound or sharp asymptotic for H(k) is known. The affirmative limit is settled, but the open-ended estimation program remains active.\n\n**Sources checked.**\n\n- Ji Ho Bae, A resolution of Erdős Problem #190 via Erdős--Lovász, BCT, and Baker--Harman--Pintz, arXiv:2604.20588 (2026). (primary): https://arxiv.org/abs/2604.20588\n Evidence used: The abstract and main theorem prove H(k)^(1/k)/k >= (1/e-epsilon(k))k/log k and explicitly conclude that the requested limit is infinity.\n- Jacob Fox and Zach Hunter, Three-color van der Waerden numbers grow super-exponentially, arXiv:2606.02541 (2026). (primary): https://arxiv.org/abs/2606.02541\n Evidence used: Theorem 3 states H(k) >= k^((1-o(1))k log k) and explicitly says this resolves the Erdős--Graham canonical van der Waerden problem.\n- Thomas F. Bloom, Erdős Problem #190, maintained Erdős Problems record, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/190\n Evidence used: The maintained record marks #190 solved and summarizes both the Bae and Fox--Hunter bounds.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1985, "problem_number": "EP-193", "title": "Erdős Problem #193", "statement": "Let $S\\subseteq \\mathbb{Z}^3$ be a finite set and let $A=\\{a_1,a_2,\\ldots,\\}\\subset \\mathbb{Z}^3$ be an infinite $S$-walk, so that $a_{i+1}-a_i\\in S$ for all $i$. Must $A$ contain three collinear points?", "background": "Originally conjectured by Gerver and Ramsey \\cite{GeRa79}, who showed that the answer is yes for $\\mathbb{Z}^2$, and for $\\mathbb{Z}^3$ that the largest number of collinear points can be bounded.\nReferences\n\n\n[GeRa79] Gerver, Joseph L. and Ramsey, L. Thomas, On certain sequences of lattice points. Pacific J. Math. (1979), 357-363.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains open whether every infinite bounded-step walk in Z^3 contains three collinear points; Lidbetter improved the known construction's bounded-collinearity threshold from 48,828,126 to 189.\n\n**Verified partial progress.**\n\n- Gerver and Ramsey proved the planar analogue and constructed an infinite Z^3 walk with no 5^11+1 collinear points.\n- Lidbetter proved the same walk contains no 189 collinear points.\n- Lidbetter also found six collinear points in that construction, disproving the suggestion that this particular walk had maximum collinearity three, without settling universal existence of a collinear triple.\n\n**Full solution or refutation.**\n\nNo proof that all such walks contain a collinear triple and no triple-free infinite walk was located.\n\n**What remains.**\n\nProve every infinite finite-step-set walk in Z^3 has three collinear points, or construct one avoiding all collinear triples.\n\n**Sources checked.**\n\n- Joseph L. Gerver and L. Thomas Ramsey, On certain sequences of lattice points, Pacific Journal of Mathematics 83 (1979), 357-363. (primary): https://msp.org/pjm/1979/83-2/pjm-v83-n2-p08-p.pdf\n Evidence used: Proves the planar result and constructs a bounded-collinearity infinite walk in three dimensions.\n- Thomas F. Lidbetter, Improved bound for the Gerver-Ramsey collinearity problem, Discrete Mathematics 347 (2024), Article 113718. (primary): https://doi.org/10.1016/j.disc.2023.113718\n Evidence used: Improves the construction's exclusion bound to 189 and exhibits six collinear points in it.\n- Thomas F. Bloom, discussion of Erdos Problem #193, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/193?embed=1\n Evidence used: Lists the triple question as open and records Lidbetter's improvement.\n\n**Review notes.** Lidbetter's published work used finite computer checks; this audit ran no heavy computation. The imported background has the batch-wide serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1986, "problem_number": "EP-195", "title": "Erdős Problem #195", "statement": "What is the largest $k$ such that in any permutation of $\\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\\cdots\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The largest progression length forced in every permutation of Z is unknown but is at most 4; current sources place the answer among 2, 3, and 4.\n\n**Verified partial progress.**\n\n- Geneson constructed a permutation of Z avoiding monotone six-term arithmetic progressions.\n- Adenwalla improved this to a permutation avoiding monotone five-term arithmetic progressions, proving the answer is at most 4.\n- Universal forcing of length 3 or 4 for this all-integers permutation notion remains unresolved.\n\n**Full solution or refutation.**\n\nNo exact threshold was located.\n\n**What remains.**\n\nDetermine whether the largest universally forced length is 2, 3, or 4, with matching forcing and avoidance proofs.\n\n**Sources checked.**\n\n- Sarosh Adenwalla, Avoiding monotone arithmetic progressions in permutations of integers, Discrete Mathematics 347 (2024), Article 114183. (primary): https://doi.org/10.1016/j.disc.2024.114183\n Evidence used: Constructs a permutation of the integers avoiding monotone five-term progressions.\n- Jesse Geneson, Forbidden arithmetic progressions in permutations of subsets of the integers, Discrete Mathematics 342 (2019), 1489-1491. (primary): https://arxiv.org/abs/1803.06334\n Evidence used: Provides the earlier six-term avoidance construction and density results.\n- Thomas F. Bloom, Erdos Problem #195, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/195\n Evidence used: Lists the exact determination as open and records the current upper bound 4.\n\n**Review notes.** The imported background has the batch-wide stray serialized record tail; it was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1987, "problem_number": "EP-196", "title": "Erdős Problem #196", "statement": "Must every permutation of $\\mathbb{N}$ contain a monotone 4-term arithmetic progression? In other words, given a permutation $x$ of $\\mathbb{N}$ must there be indices with either $ij>k>l$ such that $x_i,x_j,x_k,x_l$ are an arithmetic progression?", "background": "Davis, Entringer, Graham, and Simmons \\cite{DEGS77} have shown that there must exist a monotone 3-term arithmetic progression and need not contain a 5-term arithmetic progression.\nSee also [194] and [195].\nReferences\n\n\n[DEGS77] Davis, J. A. and Entringer, R. C. and Graham, R. L. and\nSimmons, G. J., On permutations containing no long arithmetic progressions. Acta Arith. (1977/78), 81-90.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains open whether every permutation of N contains a monotone four-term arithmetic progression; length 3 is unavoidable and length 5 is avoidable.\n\n**Verified partial progress.**\n\n- Davis, Entringer, Graham, and Simmons proved every permutation of N contains a monotone three-term progression and constructed one avoiding length 5.\n- LeSaulnier and Vijay constructed a permutation avoiding four-term progressions of odd common difference.\n- Adenwalla generalized the restricted avoidance to every common difference not divisible by 2^r, for each fixed r, but no single construction covers all differences.\n\n**Full solution or refutation.**\n\nThe exact threshold remains either 3 or 4.\n\n**What remains.**\n\nProve every permutation of N contains a monotone four-term progression, or construct a permutation avoiding all such progressions.\n\n**Sources checked.**\n\n- J. A. Davis, R. C. Entringer, R. L. Graham, and G. J. Simmons, On permutations containing no long arithmetic progressions, Acta Arithmetica 34 (1977/78), 81-90. (primary): https://eudml.org/doc/205322\n Evidence used: Establishes unavoidable length 3 and avoidable length 5.\n- Timothy D. LeSaulnier and Sujith Vijay, On permutations avoiding arithmetic progressions, Discrete Mathematics 311 (2011), 205-207. (primary): https://doi.org/10.1016/j.disc.2010.10.006\n Evidence used: Constructs avoidance for four-term progressions of odd common difference.\n- Sarosh Adenwalla, A Generalisation of a Result on Monotone Arithmetic Progressions in Permutations of the Positive Integers, arXiv:2302.09662. (primary): https://arxiv.org/abs/2302.09662\n Evidence used: Extends restricted four-term avoidance to differences not divisible by any prescribed power of two.\n- Thomas F. Bloom, discussion of Erdos Problem #196, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/196\n Evidence used: Confirms the intended monotone-index formulation and current open status.\n\n**Review notes.** The statement's intended index-order formulation was clarified by the maintained discussion. The imported background also has the batch-wide serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1988, "problem_number": "EP-197", "title": "Erdős Problem #197", "statement": "Can $\\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?", "background": "If three sets are allowed then this is possible.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains open whether N can be partitioned into two subsets that each admit a permutation avoiding monotone three-term arithmetic progressions; three subsets are known to suffice.\n\n**Verified partial progress.**\n\n- LeSaulnier and Vijay constructed a 3-avoidable subset of N with lower density at least 1/4 and upper density at least 1/2.\n- They related a negative answer to strict upper bounds for extremal density parameters, but the necessary upper bounds were not known.\n- Later avoidance constructions improve related density parameters without producing a two-part partition.\n\n**Full solution or refutation.**\n\nNo two-part construction or impossibility theorem was located.\n\n**What remains.**\n\nConstruct a partition of N into two 3-avoidable sets, or prove that every two-part partition has a part whose every permutation contains a monotone three-term progression.\n\n**Sources checked.**\n\n- Timothy D. LeSaulnier and Sujith Vijay, On permutations avoiding arithmetic progressions, Discrete Mathematics 311 (2011), 205-207. (primary): https://doi.org/10.1016/j.disc.2010.10.006\n Evidence used: States the two-part problem, proves density lower bounds for 3-avoidable subsets, and formulates the density obstruction.\n- Thomas F. Bloom, Erdos Problem #197, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/197\n Evidence used: Lists the two-part problem as open and records that three parts suffice.\n\n**Review notes.** The imported background has the batch-wide stray serialized record tail; it was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1989, "problem_number": "EP-200", "title": "Erdős Problem #200", "statement": "Does the longest arithmetic progression of primes in $\\{1,\\ldots,N\\}$ have length $o(\\log N)$?", "background": "It follows from the prime number theorem that such a progression has length $\\leq(1+o(1))\\log N$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The requested o(log N) bound for the longest prime arithmetic progression remains open.\n\n**Verified partial progress.**\n\n- The prime number theorem gives the upper bound (1+o(1)) log N.\n\n**Full solution or refutation.**\n\nNo proof of the little-o strengthening was located.\n\n**What remains.**\n\nImprove the PNT-scale upper bound by an unbounded factor.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #200, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/200\n Evidence used: Lists the problem as open and records the PNT upper bound.\n\n**Review notes.** Status is dated; no statement text was changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1990, "problem_number": "EP-201", "title": "Erdős Problem #201", "statement": "Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\\{1,\\ldots,N\\}$ without a $k$-term arithmetic progression? Is it true that $ \\lim_{N\\to \\infty}\\frac{R_3(N)}{G_3(N)}=1? $ ", "background": "First asked and investigated by Riddell \\cite{Ri69}. It is trivial that $G_k(N)\\leq R_k(N)$, and it is possible that $G_k(N) \n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The comparison of G_k(N) and R_k(N), including the proposed k=3 limit, remains open.\n\n**Verified partial progress.**\n\n- Komlós--Sulyok--Szemerédi (1975) proved R_k(N) <<_k G_k(N).\n- The trivial inequality G_k(N) <= R_k(N) gives two-sided constant-factor comparison for fixed k.\n\n**Full solution or refutation.**\n\nNo sharp asymptotic or proof of R_3(N)/G_3(N) -> 1 was located.\n\n**What remains.**\n\nDetermine the sharp relation, especially the proposed k=3 ratio.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #201 LaTeX source, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/201\n Evidence used: States the question, examples, and KSS75 bound.\n- J. Komlós, M. Sulyok and E. Szemerédi, Linear problems in combinatorial number theory, Acta Math. Acad. Sci. Hungar. (1975), 113--121. (primary): https://www.erdosproblems.com/latex/201\n Evidence used: Cited there for the reverse comparison.\n\n**Review notes.** The historical examples are not treated as asymptotic evidence.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1991, "problem_number": "EP-202", "title": "Erdős Problem #202", "statement": "Let $n_1<\\cdots < n_r\\leq N$ with associated $a_i\\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\\equiv a_i\\pmod{n_i}$ for at most one $1\\leq i\\leq r$). How large can $r$ be in terms of $N$?", "background": "Let $f(N)$ be the maximum possible $r$. Erd\\H{o}s and Stein conjectured that $f(N)=o(N)$, which was proved by Erd\\H{o}s and Szemer\\'{e}di \\cite{ErSz68}, who showed that, for every $\\epsilon>0$, $ \\frac{N}{\\exp((\\log N)^{1/2+\\epsilon})} \\ll_\\epsilon f(N) < \\frac{N}{(\\log N)^c} $ for some $c>0$. Erd\\H{o}s believed the lower bound is closer to the truth.\nThese bounds were improved by Croot \\cite{Cr03b} who proved $ \\frac{N}{L(N)^{\\sqrt{2}+o(1)}}< f(N)<\\frac{N}{L(N)^{1/6-o(1)}}, $ where $L(N)=\\exp(\\sqrt{\\log N\\log\\log N})$. These bounds were further improved by Chen \\cite{Ch05} and then by de la Bret\\'{e}che, Ford, and Vandehey \\cite{BFV13} to $ \\frac{N}{L(N)^{1+o(1)}}\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The extremal function is now known at the conjectured subexponential scale: f(N)=N exp(-(1+o(1))sqrt(log N log log N))=N L(N)^(-1+o(1)). A 2026 spread-theoretic proof supplies the upper bound, while the 2013 de la Bretèche--Ford--Vandehey construction supplies the matching lower bound.\n\n**Verified partial progress.**\n\n- Erdős--Szemerédi first proved f(N)=o(N).\n- Croot and Chen improved both sides at the L(N)-scale.\n- De la Bretèche, Ford, and Vandehey proved the matching lower construction and an upper exponent sqrt(3)/2, with the sharp upper bound conditional on a dense-core conjecture.\n- The 2026 proof replaces that conditional input by established spread and expectation-threshold machinery.\n\n**Full solution or refutation.**\n\nThe earlier BFV descending-chain argument reduces the desired upper bound to a dense-core statement for intersecting set systems. Modern spread lemmas derived from the Park--Pham expectation-threshold theorem provide that input with logarithmic loss, yielding f(N) <= N L(N)^(-1+o(1)); BFV's construction gives the reverse inequality.\n\n**What remains.**\n\nThe main asymptotic is settled. Lower-order terms, explicit constants, and effective error bounds beyond the o(1) exponent remain natural refinements.\n\n**Sources checked.**\n\n- Przemek Chojecki / Ulam AI, An unconditional spread-theoretic solution of Erdős Problem #1190, draft note, 30 April 2026. (primary): https://www.ulam.ai/research/erdos1190.pdf\n Evidence used: The abstract says the proof first establishes the sharp upper bound f(x) <= xL(x)^(-1+o(1)) for Erdős Problem #202 and combines it with BFV's lower construction.\n- Régis de la Bretèche, Kevin Ford, and Joseph Vandehey, On non-intersecting arithmetic progressions, Acta Arithmetica 157 (2013), 381--392, DOI 10.4064/aa157-4-5. (primary): https://www.ford126.web.illinois.edu/wwwpapers/NAP.pdf\n Evidence used: The paper proves the matching lower construction, the previous unconditional upper bound, and the sharp upper bound under a dense-core hypothesis.\n- Jinyoung Park and Huy Tuan Pham, A Proof of the Kahn--Kalai Conjecture, Journal of the American Mathematical Society 37 (2024), 235--243. (primary): https://arxiv.org/abs/2203.17207\n Evidence used: This proves the expectation-threshold theorem used to obtain the unconditional spread input in the 2026 #202 proof.\n- Thomas F. Bloom, Erdős Problem #202 discussion/status record, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/forum/thread/202?order=newest\n Evidence used: The maintained record states f(N)=N L(N)^(-1+o(1)), identifies the BFV plus Park--Pham proof, and displays the linked Lean theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1992, "problem_number": "EP-203", "title": "Erdős Problem #203", "statement": "Is there an integer $m\\geq 1$ with $(m,6)=1$ such that none of $2^k3^\\ell m+1$ are prime, for any $k,\\ell\\geq 0$?", "background": "Positive odd integers $m$ such that none of $2^km+1$ are prime are called Sierpinski numbers - see [1113] for more details.\nErd\\H{o}s and Graham also ask more generally about $p_1^{k_1}\\cdots p_r^{k_r}m+1$ for distinct primes $p_i$, or $q_1\\cdots q_rm+1$ where the $q_i$ are primes congruent to $1\\pmod{4}$. (Dogmachine has noted in the comments this latter question has the trivial answer $m=1$ - perhaps some condition such as $m$ even is meant.)\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The simultaneous two-base Sierpiński-type existence question remains open.\n\n**Verified partial progress.**\n\n- One-base Sierpiński numbers show that the analogous 2^k m+1 avoidance problem has solutions.\n\n**Full solution or refutation.**\n\nKnown one-base constructions do not settle simultaneous powers of 2 and 3.\n\n**What remains.**\n\nConstruct such an m or prove none exists.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #203, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/203\n Evidence used: Lists this two-base question as open and distinguishes variants.\n\n**Review notes.** No broad-variant wording was silently substituted for the stated problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1993, "problem_number": "EP-208", "title": "Erdős Problem #208", "statement": "Let $s_10$ and large $n$, $ s_{n+1}-s_n \\ll_\\epsilon s_n^{\\epsilon}? $ Is it true that $ s_{n+1}-s_n \\leq (1+o(1))\\frac{\\pi^2}{6}\\frac{\\log s_n}{\\log\\log s_n}? $ ", "background": "Erd\\H{o}s \\cite{Er51} showed that there are infinitely many $n$ such that $ s_{n+1}-s_n > (1+o(1))\\frac{\\pi^2}{6}\\frac{\\log s_n}{\\log\\log s_n}, $ so this bound would be the best possible.\nIn \\cite{Er79} Erd\\H{o}s says perhaps $s_{n+1}-s_n \\ll \\log s_n$, but he is 'very doubtful'.\nFilaseta and Trifonov \\cite{FiTr92} proved an upper bound of $s_n^{1/5+o(1)}$. Pandey \\cite{Pa24} has improved this exponent to $1/5-c$ for some constant $c>0$.\nGranville \\cite{Gr98} showed that $s_{n+1}-s_n\\ll_\\epsilon s_n^\\epsilon$ for all $\\epsilon>0$ follows from the ABC conjecture.\nSee also [489] and [145]. A more general form of this problem is given in [1101].\nReferences\n\n\n[Er51] Erd\"{o}s, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109.\n\n[Er79] Erd\\H{o}s, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.\n\n[FiTr92] Filaseta, M. and Trifonov, O., On gaps between squarefree numbers II. J. London Math. Soc. (1992), 215-221.\n\n[Gr98] Granville, Andrew, {$ABC$} allows us to count squarefrees. Internat. Math. Res. Notices (1998), 991--1009.\n\n[Pa24] Pandey, M., Squarefree numbers in short intervals. arXiv:2401.13981 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Neither proposed sharp/subpolynomial squarefree-gap statement is known unconditionally.\n\n**Verified partial progress.**\n\n- Erdős obtained infinitely many gaps of order log x/sqrt(log log x).\n- Filaseta--Trifonov and later Pandey give a maximal-gap exponent below 1/5; abc implies a subpolynomial bound conditionally.\n\n**Full solution or refutation.**\n\nThe question remains open despite nontrivial unconditional and conditional bounds.\n\n**What remains.**\n\nProve an unconditional x^epsilon bound or the proposed sharp logarithmic upper bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #208, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/208\n Evidence used: Records the lower bound, unconditional exponent results, abc consequence, and open status.\n\n**Review notes.** Conditional and unconditional advances are separated.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1994, "problem_number": "EP-212", "title": "Erdős Problem #212", "statement": "Is there a dense subset of $\\mathbb{R}^2$ such that all pairwise distances are rational?", "background": "Conjectured by Ulam. Erd\\H{o}s believed there cannot be such a set. This problem is discussed in a blogpost by Terence Tao, in which he shows that there cannot be such a set, assuming the Bombieri-Lang conjecture. The same conclusion was independently obtained by Shaffaf \\cite{Sh18}.\nIndeed, Shaffaf and Tao actually proved that such a rational distance set must be contained in a finite union of real algebraic curves. Solymosi and de Zeeuw \\cite{SdZ10} then proved (unconditionally) that a rational distance set contained in a real algebraic curve must be finite, unless the curve contains a line or a circle.\nAscher, Braune, and Turchet \\cite{ABT20} observed that, combined, these facts imply that a rational distance set in general position must be finite (conditional on the Bombieri-Lang conjecture).\nIn \\cite{Er87b} Erd\\H{o}s mentions that Besicovitch conjectured that the limit points of a rational distance set cannot contain arbitrarily large convex sets.\nReferences\n\n\n[ABT20] Ascher, K. and Braune, L. and Turchet, A., The Erd\\H{o}s-Ulam problem, Lang's conjecture, and uniformity. arXiv:1901.02616 (2020).\n\n[Er87b] Erd\\H{o}s, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\n\n[SdZ10] Solymosi, Jozsef and de Zeeuw, Frank, On a question of Erd\\H{o}s and Ulam. Discrete Comput. Geom. (2010), 393-401.\n\n[Sh18] Shaffaf, Jafar, A solution of the Erd\\H{o}s-Ulam problem on rational\ndistance sets assuming the Bombieri-Lang conjecture. Discrete Comput. Geom. (2018), 283-293.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No dense subset of R^2 with all pairwise distances rational is known, and no unconditional impossibility theorem was located.\n\n**Verified partial progress.**\n\n- Conditional Bombieri--Lang consequences and algebraic-curve restrictions provide obstructions in restricted settings.\n\n**Full solution or refutation.**\n\nExisting conditional or restricted results do not resolve the planar density problem.\n\n**What remains.**\n\nConstruct a dense set or prove an unconditional obstruction.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #212 forum discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/212\n Evidence used: Records conditional and restricted progress while retaining open status.\n\n**Review notes.** Conditional results are not promoted to a resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1995, "problem_number": "EP-213", "title": "Erdős Problem #213", "statement": "Let $n\\geq 4$. Are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?", "background": "Anning and Erd\\H{o}s \\cite{AnEr45} proved there cannot exist an infinite such set. Harborth constructed such a set when $n=5$. The best construction to date, due to Kreisel and Kurz \\cite{KK08}, has $n=7$.\nAscher, Braune, and Turchet \\cite{ABT20} have shown that there is a uniform upper bound on the size of such a set, conditional on the Bombieri-Lang conjecture. Greenfeld, Iliopoulou, and Peluse \\cite{GIP24} have shown (unconditionally) that any such set must be very sparse, in that if $S\\subseteq [-N,N]^2$ has no three on a line and no four on a circle, and all pairwise distances integers, then $ \\lvert S\\rvert \\ll (\\log N)^{O(1)}. $ See also [130].\nReferences\n\n\n[ABT20] Ascher, K. and Braune, L. and Turchet, A., The Erd\\H{o}s-Ulam problem, Lang's conjecture, and uniformity. arXiv:1901.02616 (2020).\n\n[AnEr45] Anning, Norman H. and Erd\\H{o}s, Paul, Integral distances. Bull. Amer. Math. Soc. (1945), 598-600.\n\n[GIP24] Greenfeld, R. and Iliopoulou, M. and Peluse, S., On integer distance sets. arXiv:2401.10821 (2024).\n\n[KK08] Kreisel, Tobias and Kurz, Sascha, There are integral heptagons, no three points on a line, on four on a circle. Discrete Comput. Geom. (2008), 786-790.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The all-n integral-distance configuration problem remains open.\n\n**Verified partial progress.**\n\n- Examples are known for n=5 and n=7.\n- Related 2024 work gives growing configurations in bounded regions but does not settle the stated general-position condition for every n.\n\n**Full solution or refutation.**\n\nNo construction or obstruction for all n was located.\n\n**What remains.**\n\nConstruct the configurations for arbitrary n or establish a finite obstruction.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #213 LaTeX source, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/213\n Evidence used: Lists examples, related results, and continuing open status.\n\n**Review notes.** Related relaxed-condition constructions are not conflated with the exact request.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1996, "problem_number": "EP-217", "title": "Erdős Problem #217", "statement": "For which $n$ are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that (in some ordering of the distances) the $i$th distance occurs $i$ times?", "background": "An example with $n=4$ is an isosceles triangle with the point in the centre. Erd\\H{o}s originally believed this was impossible for $n\\geq 5$, but Pomerance constructed a set with $n=5$ (see \\cite{Er83c} for a description), and Pal\\'{a}sti has proved such sets exist for all $n\\leq 8$.\nErd\\H{o}s believed this is impossible for all sufficiently large $n$. This would follow from $h(n)\\geq n$ for sufficiently large $n$, where $h(n)$ is as in [98].\nReferences\n\n\n[Er83c] Erd\\H{o}s, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The prescribed distinct-distance multiplicity pattern is constructed for several small n but not characterized for all n.\n\n**Verified partial progress.**\n\n- The tracker records constructions for n=4,5,6,7,8.\n\n**Full solution or refutation.**\n\nNo all-n result was located.\n\n**What remains.**\n\nDetermine exactly which n admit the pattern.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #217, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/217\n Evidence used: Lists small-n constructions and open status.\n\n**Review notes.** Small constructions do not constitute an all-n solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1997, "problem_number": "EP-218", "title": "Erdős Problem #218", "statement": "Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.", "background": "In \\cite{Er85c} Erd\\H{o}s also conjectures that $d_n=d_{n+1}=\\cdots=d_{n+k}$ is solvable for every $k$ (which is equivalent to $k$ consecutive primes in arithmetic progression, see [141]).\nReferences\n\n\n[Er85c] Erd\\H{o}s, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Neither density-one-half claim nor the infinitude of equal consecutive prime gaps is established.\n\n**Verified partial progress.**\n\n- Prime-tuple heuristics produce predictions for comparison densities.\n\n**Full solution or refutation.**\n\nNo unconditional theorem resolving either asserted density or ties was located.\n\n**What remains.**\n\nProve the density statements or establish infinitely many ties.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #218, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/218\n Evidence used: Lists the problem as open and distinguishes heuristic/commentary material.\n\n**Review notes.** Unreviewed tracker-comment and AI claims were excluded from established literature.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1998, "problem_number": "EP-222", "title": "Erdős Problem #222", "statement": "Let $n_1\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Sharp consecutive-gap bounds for sums of two squares remain open.\n\n**Verified partial progress.**\n\n- Erdős gave an infinite lower-gap bound of order log n/sqrt(log log n).\n- Richards' constant 1/4 lower bound was raised to about 0.868 by Dietmann--Elsholtz--Kalmynin--Konyagin--Maynard.\n- Bambah--Chowla proved the upper bound O(n^(1/4)).\n\n**Full solution or refutation.**\n\nThe available lower and upper bounds leave a wide gap.\n\n**What remains.**\n\nImprove the upper bound and determine the correct maximal-gap order.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #222, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/222\n Evidence used: Records the listed lower and upper bounds and calls the question open.\n\n**Review notes.** Bounds are recorded as progress, not as resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 1999, "problem_number": "EP-233", "title": "Erdős Problem #233", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Prove that $ \\sum_{1\\leq n\\leq N}d_n^2 \\ll N(\\log N)^2. $ ", "background": "Cramer \\cite{Cr36} proved an upper bound of $O(N(\\log N)^4)$ conditional on the Riemann hypothesis. Selberg \\cite{Se43} improved this slightly (still assuming the Riemann hypothesis) to $ \\sum_{1\\leq n\\leq N}\\frac{d_n^2}{n}\\ll (\\log N)^4. $ The prime number theorem immediately implies a lower bound of $ \\sum_{1\\leq n\\leq N}d_n^2\\gg N(\\log N)^2. $ This would imply in particular that $d_n\\ll n^{1/2}\\log n$ for all $n$, which is known only the assumption of the Riemann Hypothesis.\nThe values of the sum are listed at A074741 on the OEIS.\nThis is discussed in problem A8 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Cr36] Cram\\'{e}r, Harald, On the order of magnitude of the difference between consecutive prime numbers. Acta Arithmetica (1936), 23--46.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Se43] Selberg, Atle, On the normal density of primes in small intervals, and the\ndifference between consecutive primes. Arch. Math. Naturvid. (1943), 87--105.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The conjectural O(N(log N)^2) mean-square upper bound for prime gaps remains open.\n\n**Verified partial progress.**\n\n- The prime number theorem gives a matching-order lower bound.\n- Cramér proved O(N(log N)^4) conditionally on the Riemann hypothesis.\n\n**Full solution or refutation.**\n\nNo unconditional proof of the requested upper bound was located.\n\n**What remains.**\n\nProve the conjectural mean-square upper bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #233, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/233\n Evidence used: States the problem as open and records the conditional Cramér result.\n- FormalConjectures, Erdős Problem 233 record, checked 2026-08-17. (authoritative_secondary): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/233.lean\n Evidence used: Classifies the main statement as research open and the RH variant as solved.\n\n**Review notes.** Personal-webpage or unsupported AI-solution claims were not treated as proof.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2000, "problem_number": "EP-234", "title": "Erdős Problem #234", "statement": "For every $c\\geq 0$ the density $f(c)$ of integers for which $ \\frac{p_{n+1}-p_n}{\\log n}< c $ exists and is a continuous function of $c$.\n\",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Existence and continuity of the limiting distribution of normalized consecutive-prime gaps remain open unconditionally. A sufficiently uniform Hardy-Littlewood prime-tuples conjecture implies the stronger exponential law f(c)=1-e^(-c), but the solved Hooley theorem for primorial reduced residues is a different problem.\n\n**Verified partial progress.**\n\n- Gallagher proved a conditional Poisson law for primes in intervals of length lambda log x under a uniform prime-tuples conjecture.\n- The conditional Poisson law yields an exponential distribution for normalized consecutive-prime gaps.\n- Continuity would be approachable by sieve bounds if the densities were already known to exist, but existence is the principal unconditional obstruction.\n\n**Full solution or refutation.**\n\nNo unconditional solution is known. Search results that cite Hooley's exponential gap law confuse EP-234 with EP-235, which concerns gaps between integers coprime to a primorial rather than actual primes. The serialized difficulty text appended to the imported statement is extraction damage.\n\n**What remains.**\n\nProve that the density over prime indices exists for every fixed c>=0 and is continuous; identifying it as 1-e^(-c) would be stronger.\n\n**Sources checked.**\n\n- P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), 4-9, DOI 10.1112/S0025579300016442. (primary): https://doi.org/10.1112/S0025579300016442\n Evidence used: Gallagher proves the conditional Poisson law from a uniform Hardy-Littlewood prime-tuples conjecture.\n- Thomas F. Bloom, Erdős Problem #234, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/234\n Evidence used: The maintained record marks the actual-prime density problem open and its discussion explicitly distinguishes conditional existence from continuity.\n- Thomas F. Bloom, Erdős Problem #235, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/235\n Evidence used: This nearby solved problem identifies Hooley's result as a theorem for reduced residues modulo primorials, preventing a false transfer to EP-234.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2001, "problem_number": "EP-236", "title": "Erdős Problem #236", "statement": "Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\\geq 0$. Is it true that $f(n)=o(\\log n)$?", "background": "Erd\\H{o}s \\cite{Er50} proved that there are infinitely many $n$ such that $f(n)\\gg \\log\\log n$.\nErd\\H{o}s could not even prove that there do not exist infinitely many integers $n$ such that for all $1< 2^k0$.\nThe sequence of values of $f(n)$ is A109925 on the OEIS.\nSee also [237].\nReferences\n\n\n[Er50] Erd\"{o}s, P., On integers of the form $2^k+p$ and some related problems. Summa Brasil. Math. (1950), 113-123.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[MiWe69] Mientka, Walter E. and Weitzenkamp, Roger C., On {$f$}-plentiful numbers. J. Combinatorial Theory (1969), 374--377.\n\n[Va73] Vaughan, R. C., Some applications of {M}ontgomery's sieve. J. Number Theory (1973), 64--79.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The uniform bound f(n)=o(log n) remains open. Erdős proved f(n)>>log log n infinitely often, while the trivial count of eligible powers gives only O(log n); finite searches and density bounds for numbers with every eligible difference prime do not close this gap.\n\n**Verified partial progress.**\n\n- Erdős proved that infinitely many n satisfy f(n)>>log log n.\n- Only 4, 7, 15, 21, 45, 75, and 105 are known to have every eligible n-2^k prime.\n- Mientka-Weitzenkamp and later computations exclude further plentiful examples over large finite ranges.\n- Vaughan proved a strong density upper bound for plentiful integers, but not their finiteness or the o(log n) bound for all n.\n\n**Full solution or refutation.**\n\nNo solution is known. The plentiful-number subproblem has been moved to EP-1142 in the live tracker and remains open; its computational exclusions do not imply an asymptotic uniform bound for f(n).\n\n**What remains.**\n\nProve f(n)/log n->0 uniformly as n grows, or construct a sequence of n for which a positive proportion of the eligible powers of two yield primes.\n\n**Sources checked.**\n\n- Paul Erdős, On integers of the form 2^k+p and some related problems, Summa Brasiliensis Mathematicae 2 (1950), 113-123. (primary): https://combinatorica.hu/~p_erdos/Erdos.html\n Evidence used: The author archive confirms the originating paper and bibliography; the maintained record attributes the log-log lower construction to it.\n- Thomas F. Bloom, Erdős Problem #236, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/236\n Evidence used: The maintained June 2026 record marks the exact o(log n) assertion open and reports no solution claims.\n- Thomas F. Bloom, Erdős Problem #1142, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1142\n Evidence used: The page preserves the finite-search and Vaughan progress on the related plentiful-number subproblem and still marks it open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2002, "problem_number": "EP-238", "title": "Erdős Problem #238", "statement": "Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\\log x$ many consecutive primes $\\leq x$ such that the difference between any two is $>c_2$?", "background": "Erd\\H{o}s \\cite{Er49c} proved this is true for any $c_2>0$ if $c_1>0$ is sufficiently small (depending on $c_1$).\nReferences\n\n\n[Er49c] Erd\\H{o}s, P., On some applications of {B}run's method. Acta Univ. Szeged. Sect. Sci. Math. (1949), 57--63.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The assertion for arbitrary fixed c1,c2>0 remains open. Erdős's Brun-sieve theorem handles each c2 only when c1 is sufficiently small depending on c2; a current discussion gives only a conditional prime-tuples route for arbitrary c1.\n\n**Verified partial progress.**\n\n- Erdős proved that for every fixed c2 there is an epsilon(c2)>0 for which the assertion holds whenever 00, with no smallness restriction on c1, or find a counterexample to the arbitrary-c1 form.\n\n**Sources checked.**\n\n- Paul Erdős, On some applications of Brun's method, Acta Scientiarum Mathematicarum 13 (1950), 57-63. (primary): https://acta.bibl.u-szeged.hu/13658/\n Evidence used: The repository supplies the cited primary paper; theorem 3 is identified in the tracker discussion as proving the small-c1 result with dependence on c2.\n- Thomas F. Bloom, Erdős Problem #238 discussion, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/238\n Evidence used: The current discussion corrects the dependency typo, records the exact theorem interpretation, and outlines only a conditional route for general c1.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2003, "problem_number": "EP-241", "title": "Erdős Problem #241", "statement": "Let $f(N)$ be the maximum size of $A\\subseteq \\{1,\\ldots,N\\}$ such that the sums $a+b+c$ with $a,b,c\\in A$ are all distinct (aside from the trivial coincidences). Is it true that $ f(N)\\sim N^{1/3}? $ ", "background": "Originally asked to Erd\\H{o}s by Bose. Bose and Chowla \\cite{BoCh62} provided a construction proving one half of this, namely $ (1+o(1))N^{1/3}\\leq f(N). $ The best upper bound known to date is due to Green \\cite{Gr01}, $ f(N) \\leq ((7/2)^{1/3}+o(1))N^{1/3} $ (note that $(7/2)^{1/3}\\approx 1.519$).\nMore generally, Bose and Chowla conjectured that the maximum size of $A\\subseteq \\{1,\\ldots,N\\}$ with all $r$-fold sums distinct (aside from the trivial coincidences) then $ \\lvert A\\rvert \\sim N^{1/r}. $ This is known only for $r=2$ (see [30]).\nThis is discussed in problem C11 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BoCh62] Bose, R. C. and Chowla, S., Theorems in the additive theory of numbers. Comment. Math. Helv. (1962/63), 141-147.\n\n[Gr01] Green, Ben, The number of squares and {$B_h[g]$} sets. Acta Arith. (2001), 365-390.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The B_3 extremal asymptotic remains open. Bose-Chowla constructions give f(N)>=(1+o(1))N^(1/3), while Green's best maintained upper bound is ((7/2)^(1/3)+o(1))N^(1/3), leaving a leading-constant gap.\n\n**Verified partial progress.**\n\n- Bose and Chowla constructed B_3 sets attaining asymptotic lower constant one.\n- Green used Fourier-analytic estimates for B_h[g] sets to prove the upper constant (7/2)^(1/3), approximately 1.519.\n- The broader Bose-Chowla asymptotic conjecture is known at leading constant one only for r=2.\n\n**Full solution or refutation.**\n\nNo proof of f(N)~N^(1/3) is known. The order of magnitude is settled, but the leading constant is not; that is the entire force of the imported asymptotic statement.\n\n**What remains.**\n\nReduce the upper leading constant from (7/2)^(1/3) to one, or construct B_3 sets with asymptotic leading constant exceeding one.\n\n**Sources checked.**\n\n- R. C. Bose and S. Chowla, Theorems in the additive theory of numbers, Commentarii Mathematici Helvetici 37 (1962/63), 141-147, DOI 10.1007/BF02566968. (primary): https://doi.org/10.1007/BF02566968\n Evidence used: This is the classical construction paper yielding the asymptotic lower bound.\n- Ben Green, The number of squares and B_h[g] sets, Acta Arithmetica 100 (2001), 365-390, DOI 10.4064/aa100-4-6. (primary): https://doi.org/10.4064/aa100-4-6\n Evidence used: The paper explicitly proves A(3,1,N)<=((7/2)^(1/3)+o(1))N^(1/3).\n- Thomas F. Bloom, Erdős Problem #241, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/241\n Evidence used: The maintained page still lists Green's bound as best known and marks the constant-one asymptotic open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2004, "problem_number": "EP-243", "title": "Erdős Problem #243", "statement": "Let $1\\leq a_10. $ A sequence satisfying the reucrrence $a_n = a_{n-1}^2-a_{n-1}+1$ is known as Sylvester's sequence.\nDuverney \\cite{Du01} proved a weaker version of this problem: if $ \\sum_{n\\geq 0}\\left(\\frac{a_{n+1}}{a_n^2}-1\\right) $ converges then $\\sum \\frac{1}{a_n}$ is rational if and only if $ a_{n}=a_{n-1}^2-a_{n-1}+1 $ for all large $n$.\nReferences\n\n\n[Du01] Duverney, Daniel, Irrationality of fast converging series of rational numbers. J. Math. Sci. Univ. Tokyo (2001), 275--316.\n\n[ErSt64] Erd\\H{o}s, P. and Straus, E. G., On the irrationality of certain {A}hmes series. J. Indian Math. Soc. (N.S.) (1964), 129--133.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The eventual Sylvester-recurrence conclusion remains open under only a_n/a_(n-1)^2->1 and rationality of the reciprocal sum. Erdős-Straus proved a quantitative obstruction, and Duverney proved the conclusion under an additional summability hypothesis on the relative deviations.\n\n**Verified partial progress.**\n\n- Erdős and Straus proved a positive limsup obstruction involving the least common multiple when the recurrence fails.\n- Duverney proved rationality is equivalent to eventual Sylvester recurrence if the deviations a_(n+1)/a_n^2-1 form a convergent series.\n- A tracker discussion reduces rational-tail sequences to a deterministic recurrence and isolates a difficult near-integer convergence bottleneck, but gives only heuristic support.\n\n**Full solution or refutation.**\n\nNo full solution is known. Termwise convergence a_(n+1)/a_n^2->1 does not imply summability of its deviations, so Duverney's theorem is genuinely weaker than the imported statement. The word 'reucrrence' in the imported background is a harmless typo.\n\n**What remains.**\n\nRemove Duverney's summability assumption and prove eventual Sylvester recurrence from the two imported hypotheses alone, or construct a counterexample.\n\n**Sources checked.**\n\n- Paul Erdős and Ernst G. Straus, On the irrationality of certain Ahmes series, Journal of the Indian Mathematical Society 28 (1964), 129-133. (primary): https://combinatorica.hu/~p_erdos/1964-19.pdf\n Evidence used: The primary paper supplies the classical obstruction recorded in the background.\n- Daniel Duverney, Irrationality of Fast Converging Series of Rational Numbers, Journal of Mathematical Sciences, University of Tokyo 8 (2001), 275-316. (primary): https://www.ms.u-tokyo.ac.jp/journal/pdf/jms080206.pdf\n Evidence used: Corollary 3.2 gives the eventual-recurrence criterion under the additional convergence of relative deviations.\n- Thomas F. Bloom, Erdős Problem #243, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/243\n Evidence used: The maintained January 2026 page keeps the exact statement open and records no claimed solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2005, "problem_number": "EP-244", "title": "Erdős Problem #244", "statement": "Let $C>1$. Does the set of integers of the form $p+\\lfloor C^k\\rfloor$, for some prime $p$ and $k\\geq 0$, have density $>0$?", "background": "Originally asked to Erd\\H{o}s by Kalm\\'{a}r. Erd\\H{o}s believed the answer is yes. Romanoff \\cite{Ro34} proved that the answer is yes if $C$ is an integer.\nDing \\cite{Di25} has proved that this is true for almost all $C>1$.\nReferences\n\n\n[Di25] Y. Ding, On a Romanoff type problem of Erd\\H{o}s and Kalm\\'{a}r. arXiv:2503.22700 (2025).\n\n[Ro34] Romanoff, N. P., \"{U}ber einige S\"Atze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.\n\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Romanoff-type assertion is known for every integer C>1 and, by Ding's 2025 preprint, for almost every real C>1 with positive lower asymptotic density. The universal assertion for every real C remains open on a measure-zero exceptional set.\n\n**Verified partial progress.**\n\n- Romanoff proved positive density for integer exponential bases.\n- Ding proved positive lower asymptotic density for almost all real C>1.\n- The almost-everywhere theorem leaves exceptional real bases and does not imply the universal quantifier in the imported statement.\n\n**Full solution or refutation.**\n\nThis is substantial partial progress, not a full solution. The imported word 'density' is also less precise than Ding's positive lower asymptotic density conclusion; expert review should decide whether existence of natural density was historically intended.\n\n**What remains.**\n\nProve positive density for every exceptional real C>1 or find a counterexample, and clarify whether positive lower density suffices for the exact historical formulation.\n\n**Sources checked.**\n\n- N. P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Mathematische Annalen 109 (1934), 668-678. (primary): https://www.erdosproblems.com/latex/244\n Evidence used: The maintained bibliography identifies Romanoff's integer-base theorem as the classical solved case.\n- Yuchen Ding, On a Romanoff type problem of Erdős and Kalmár, arXiv:2503.22700 (2025). (primary): https://arxiv.org/abs/2503.22700\n Evidence used: The abstract proves positive lower asymptotic density for almost all real y>1.\n- Thomas F. Bloom, Erdős Problem #244, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/244\n Evidence used: The current page keeps the every-C problem open and records Ding's almost-everywhere advance.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2006, "problem_number": "EP-247", "title": "Erdős Problem #247", "statement": "Let $1\\leq a_1cn^2$ then $\\sum_{n=1}^\\infty \\frac{1}{2^{a_n}}$ is not the root of any quadratic polynomial'.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[Er75c] Erd\\H{o}s, P., Some problems and results on the irrationality of the sum of infinite series. J. Math. Sci. (1975), 1-7 (1976).\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\n\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Transcendence under the sole condition limsup a_n/n=infinity remains open. Erdős proved transcendence under a much stronger super-polynomial lacunarity condition, while later related irrationality criteria do not cover this exact sparse binary series.\n\n**Verified partial progress.**\n\n- Erdős proved the answer under a stronger condition requiring growth beyond every fixed polynomial along a subsequence.\n- Erdős observed that even excluding quadratic algebraicity under a_n>c n^2 appeared difficult.\n- The problem has a formal Lean statement, but no proof artifact is claimed.\n\n**Full solution or refutation.**\n\nNo solution is known. The imported background's undefined n_k/k^t notation is a source defect; it evidently refers to the exponent sequence but is not silently rewritten here. Recent papers on other rapidly convergent reciprocal products are not equivalent.\n\n**What remains.**\n\nProve transcendence under the unbounded limsup ratio alone, or construct an increasing exponent sequence satisfying it whose binary sum is algebraic.\n\n**Sources checked.**\n\n- Paul Erdős, Some problems and results on the irrationality of the sum of infinite series, Journal of Mathematical Sciences 10 (1975/76), 1-7. (primary): https://www.erdosproblems.com/latex/247\n Evidence used: The maintained bibliography attributes the stronger lacunarity theorem to this paper.\n- Paul Erdős, On the irrationality of certain series: problems and results, in New Advances in Transcendence Theory (1988), 102-109, DOI 10.1017/CBO9780511897184.009. (primary): https://doi.org/10.1017/CBO9780511897184.009\n Evidence used: This later source restates the difficulty and the weaker quadratic-algebraicity target.\n- Thomas F. Bloom, Erdős Problem #247, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/247\n Evidence used: The maintained January 2026 record marks the exact transcendence question open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2007, "problem_number": "EP-249", "title": "Erdős Problem #249", "statement": "Is $ \\sum_n \\frac{\\phi(n)}{2^n} $ irrational? Here $\\phi$ is the Euler totient function.", "background": "The decimal expansion of this sum is A256936 on the OEIS.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Irrationality of sum phi(n)/2^n remains open. The current tracker reports no solution; a Möbius-convolution identity and an irrational lacunary subseries are reformulations or special components that do not control carries in the full sum.\n\n**Verified partial progress.**\n\n- Möbius inversion yields the exact identity sum phi(n)/2^n = 1/2 + sum mu(n)/(2^n-1)^2.\n- The subseries restricted to n equal to powers of two is irrational by nonperiodic lacunary binary digits.\n- Neither fact establishes irrationality of the full positive series because omitted terms can alter its binary carries.\n\n**Full solution or refutation.**\n\nNo proof or disproof was located. Irrationality results for analogous series involving sigma, tau, or omega are about different coefficient sequences and cannot be transferred automatically.\n\n**What remains.**\n\nUse the Möbius-series identity or another arithmetic structure to exclude rationality of the entire sum, including all carry interactions.\n\n**Sources checked.**\n\n- Paul Erdős, On the irrationality of certain series: problems and results, in New Advances in Transcendence Theory (1988), 102-109, DOI 10.1017/CBO9780511897184.009. (primary): https://doi.org/10.1017/CBO9780511897184.009\n Evidence used: This is one of the original sources cited for the irrationality question.\n- Thomas F. Bloom, Erdős Problem #249 discussion, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/249\n Evidence used: The maintained record marks the problem open and records the Möbius identity and lacunary-subseries observation without claiming a solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2008, "problem_number": "EP-251", "title": "Erdős Problem #251", "statement": "Is $ \\sum \\frac{p_n}{2^n} $ irrational? (Here $p_n$ is the $n$th prime.)", "background": "Erd\\H{o}s \\cite{Er58b} proved that $\\sum \\frac{p_n^k}{n!}$ is irrational for every $k\\geq 1$.\nIn \\cite{Er88c} he further conjectures that $\\sum \\frac{p_n^k}{2^n}$ is irrational for every $k$, and that if $g_n\\geq 2$ and $g_n=o(p_n)$ then $ \\sum_{n=1}^\\infty \\frac{p_n}{g_1\\cdots g_n} $ is irrational. (The example $g_n=p_n+1$ shows that some condition on the growth of the $g_n$ is necessary here.)\nThe decimal expansion of this sum is A098990 on the OEIS.\nReferences\n\n\n[Er58b] Erd\\H{o}s, Paul, Sur certaines s\\'{e}ries \\`a{} valeur irrationnelle. Enseign. Math. (2) (1958), 93--100.\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Irrationality of the binary prime series sum p_n/2^n remains open. Erdős's factorial-denominator theorem and a prime-gap reformulation do not settle it; a recent counterexample to a broader variable-denominator conjecture leaves the fixed denominator 2 untouched.\n\n**Verified partial progress.**\n\n- Erdős proved sum p_n^k/n! irrational for every fixed k>=1.\n- Summation by parts reduces the exact question, up to an elementary rational adjustment, to a binary series involving consecutive prime gaps.\n- A 2026 tracker comment constructs counterexamples to the broader claim with freely chosen g_n=o(p_n), but not to g_n=2.\n\n**Full solution or refutation.**\n\nNo solution is known. Factorial denominators provide divisibility and tail estimates that do not persist for 2^n, and statistical heuristics for prime gaps do not rule out eventual binary periodicity rigorously.\n\n**What remains.**\n\nProve that the binary expansion after carries is not eventually periodic, possibly through quantitative prime-gap statistics, or find a different irrationality criterion for the prime sequence.\n\n**Sources checked.**\n\n- Paul Erdős, Sur certaines séries à valeur irrationnelle, L'Enseignement Mathématique (2) 4 (1958), 93-100. (primary): https://combinatorica.hu/~p_erdos/1958-19.pdf\n Evidence used: The primary paper proves the related factorial-denominator irrationality theorem.\n- Paul Erdős, On the irrationality of certain series: problems and results, in New Advances in Transcendence Theory (1988), 102-109. (primary): https://combinatorica.hu/~p_erdos/1988-22.pdf\n Evidence used: This source states the binary prime-series conjecture and its broader variants.\n- Thomas F. Bloom, Erdős Problem #251, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/251\n Evidence used: The maintained record marks the exact series open and separates the factorial theorem and broader conjectures.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2009, "problem_number": "EP-252", "title": "Erdős Problem #252", "statement": "Let $k\\geq 1$ and $\\sigma_k(n)=\\sum_{d\\mid n}d^k$. Is $ \\sum \\frac{\\sigma_k(n)}{n!} $ irrational?", "background": "This is known now for $1\\leq k\\leq 4$. The cases $k=1,2$ are reasonably straightforward, as observed by Erd\\H{o}s \\cite{Er52}. The case $k=3$ was proved independently by Schlage-Puchta \\cite{ScPu06} and Friedlander, Luca, and Stoiciu \\cite{FLC07}. The case $k=4$ was proved by Pratt \\cite{Pr22}.\nIt is known that this sum is irrational for all $k\\geq 1$ conditional on either Schinzel's conjecture (Schlage-Puchta \\cite{ScPu06}) or the prime tuples conjecture (Friedlander, Luca, and Stoiciu \\cite{FLC07}).\nThis is discussed in problem B14 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er52] Erd\\H{o}s, P., Problem 4493. Amer. Math. Monthly (1952), 557-558.\n\n[FLC07] Friedlander, J. B. and Luca, F. and Stoiciu, M., On the irrationality of a divisor function series. Integers (2007).\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Pr22] Pratt, K., The irrationality of a divisor function series of Erd\\H{o}s and Kac. arXiv:2209.11124 (2022).\n\n[ScPu06] Schlage-Puchta, J. C., The irrationality of a number theoretical series. Ramanujan J. (2006), 455-460.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The divisor-function factorial series is unconditionally irrational for 1<=k<=4 and conditionally irrational for every k>=1 under Schinzel's Hypothesis H or Dickson's conjecture. No unconditional proof for general k, in particular fixed k>=5, was located.\n\n**Verified partial progress.**\n\n- Erdős handled k=1 and k=2.\n- Schlage-Puchta and independently Friedlander-Luca-Stoiciu proved k=3.\n- Pratt proved k=4 in 2022.\n- Schlage-Puchta gives all k under Schinzel's Hypothesis H, and Friedlander-Luca-Stoiciu give an all-k conditional route under Dickson's conjecture.\n\n**Full solution or refutation.**\n\nThis is a genuine multipart partial result, not a full solution of the universal k-quantifier. The imported background's phrase 'prime tuples conjecture' has been refined on the current page to Dickson's conjecture for the Friedlander-Luca-Stoiciu argument.\n\n**What remains.**\n\nProve irrationality unconditionally for every fixed k>=5, ideally by replacing the conjectural prime-pattern input in the conditional proofs.\n\n**Sources checked.**\n\n- Jan-Christoph Schlage-Puchta, The irrationality of a number theoretical series, Ramanujan Journal 12 (2006), 455-460. (primary): https://arxiv.org/abs/1105.1452\n Evidence used: The paper proves k=3 unconditionally and all k conditional on Schinzel's Hypothesis H.\n- John B. Friedlander, Florian Luca, and Mihai Stoiciu, On the irrationality of a divisor function series, Integers 7 (2007), A31. (primary): https://eudml.org/doc/128087\n Evidence used: This is the independent k=3 result and the source of the Dickson-conjecture conditional argument.\n- Kyle Pratt, The irrationality of a divisor function series of Erdős and Kac, arXiv:2209.11124 (2022). (primary): https://arxiv.org/abs/2209.11124\n Evidence used: The abstract explicitly proves alpha_4 irrational and identifies k<=3 as the preceding unconditional frontier.\n- Thomas F. Bloom, Erdős Problem #252, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/252\n Evidence used: The January 2026 maintained record marks the universal question open and summarizes the unconditional and conditional cases.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2010, "problem_number": "EP-254", "title": "Erdős Problem #254", "statement": "Let $A\\subseteq \\mathbb{N}$ be such that $ \\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert \\to \\infty\\textrm{ as }x\\to \\infty $ and $ \\sum_{n\\in A} \\{ \\theta n\\}=\\infty $ for every $\\theta\\in (0,1)$, where $\\{x\\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of $A$.", "background": "Cassels \\cite{Ca60} proved this under the alternative hypotheses $ \\lim \\frac{\\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert}{\\log\\log x}=\\infty $ and $ \\sum_{n\\in A} \\{ \\theta n\\}^2=\\infty $ for every $\\theta\\in (0,1)$.\nReferences\n\n\n[Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The stated completeness criterion remains open. Cassels proved the distinct-summand conclusion under a quantitatively stronger dyadic-block growth condition together with squared nearest-integer-distance divergence, but that theorem does not establish the exact imported hypotheses.\n\n**Verified partial progress.**\n\n- Cassels proved completeness when the dyadic-block increment divided by log log x tends to infinity.\n- Cassels's theorem also assumes divergence of the sum of squared nearest-integer distances for every theta.\n- The maintained December 2025 record retains the exact first-power, unnormalised formulation as open.\n\n**Full solution or refutation.**\n\nNo full proof or counterexample was found. A deleted forum post is not evidence of a solution, and later work on complete sequences found in search did not visibly match both exact hypotheses.\n\n**What remains.**\n\nProve that the two exact imported hypotheses force every sufficiently large integer to be a sum of distinct elements of A, or construct a counterexample satisfying both.\n\n**Sources checked.**\n\n- J. W. S. Cassels, On the representation of integers as the sums of distinct summands taken from a fixed set, Acta Sci. Math. (Szeged) 21 (1960), 111-124. (primary): https://acta.bibl.u-szeged.hu/13906/\n Evidence used: The repository record supplies the primary paper proving the stronger-hypothesis completeness theorem.\n- Thomas F. Bloom, Erdős Problem #254 and discussion, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/254\n Evidence used: The maintained page labels the exact statement open and states Cassels's two alternative hypotheses.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2011, "problem_number": "EP-256", "title": "Erdős Problem #256", "statement": "Let $n\\geq 1$ and $f(n)$ be maximal such that for any integers $1\\leq a_1\\leq \\cdots \\leq a_n$ we have $ \\max_{\\lvert z\\rvert=1}\\left\\lvert \\prod_{i}(1-z^{a_i})\\right\\rvert\\geq f(n). $ Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that $ \\log f(n) \\gg n^c? $ ", "background": "Erd\\H{o}s and Szekeres \\cite{ErSz59} proved that $\\lim f(n)^{1/n}=1$ and $f(n)>\\sqrt{2n}$. Erd\\H{o}s proved an upper bound of $\\log f(n) \\ll n^{1-c}$ for some constant $c>0$ with probabilistic methods. Atkinson \\cite{At61} showed that $\\log f(n) \\ll n^{1/2}\\log n$.\nThis was improved to $ \\log f(n) \\ll n^{1/3}(\\log n)^{4/3} $ by Odlyzko \\cite{Od82}.\nIf we denote by $f^*(n)$ the analogous quantity with the assumption that $a_1<\\cdots\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Belov and Konyagin answer the explicit power-lower-bound question negatively by proving log f(n)=O((log n)^4). The broader request to estimate f(n) remains open, with a large gap between known lower and upper bounds.\n\n**Verified partial progress.**\n\n- Erdős and Szekeres proved f(n)>sqrt(2n) and f(n)^(1/n) tends to one.\n- Belov and Konyagin proved log f(n)=O((log n)^4), ruling out log f(n)>>n^c for every fixed c>0.\n- Recent tracker discussion notes a possible constant improvement in a lower bound but no asymptotic determination.\n\n**Full solution or refutation.**\n\nThe specific yes/no subquestion has answer no. This is not a complete solution because the opening demand to estimate f(n) is substantially broader and its correct asymptotic order is unknown.\n\n**What remains.**\n\nDetermine the correct order of f(n), or substantially narrow the gap between its polynomial-size lower bounds and the polylogarithmic upper bound for log f(n).\n\n**Sources checked.**\n\n- A. S. Belov and S. V. Konyagin, An estimate of the free term of a non-negative trigonometric polynomial with integer coefficients, Izvestiya: Mathematics 60 (1996), 1123-1182, DOI 10.1070/IM1996v060n06ABEH000095. (primary): https://doi.org/10.1070/IM1996v060n06ABEH000095\n Evidence used: Corollary 0.3 is the primary bound used to derive log f(n)=O((log n)^4).\n- Thomas F. Bloom, Erdős Problem #256 discussion, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/256?order=oldest\n Evidence used: The discussion explicitly checks that the Belov-Konyagin bound refutes the power lower bound while leaving the first estimation question open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2012, "problem_number": "EP-257", "title": "Erdős Problem #257", "statement": "Let $A\\subseteq \\mathbb{N}$ be an infinite set. Is $ \\sum_{n\\in A}\\frac{1}{2^n-1} $ irrational?", "background": "If $A=\\mathbb{N}$ then this series is $\\sum_{n}\\frac{d(n)}{2^n}$, where $d(n)$ is the number of divisors of $n$, which Erd\\H{o}s \\cite{Er48} proved is irrational. In general, if $f_A(n)$ counts the number of divisors of $n$ which are elements of $A$ then $ \\sum_{n\\in A}\\frac{1}{2^n-1}=\\sum_n \\frac{f_A(n)}{2^n}. $ The case when $A$ is the set of primes is [69]. This case (and when $A$ is the set of prime powers) was settled in the affirmative by Tao and Ter\"{a}v\"{a}inen \\cite{TaTe25}.\nErd\\H{o}s \\cite{Er68d} proved this sum is irrational whenever $(a,b)=1$ for all $a\neq b\\in A$ and $\\sum_{n\\in A}\\frac{1}{n}<\\infty$ (and thought that the condition $(a,b)=1$ could be dropped by complicating his proof).\nThere is nothing special about $2$ here, and this sum is likely irrational with $2$ replaced by any integer $t\\geq 2$.\nIn \\cite{Er88c} Erd\\H{o}s goes further and speculates that $\\sum_{n\\in A}\\frac{1}{2^n-t_n}$ is irrational for every infinite set $A$ and bounded sequence $t_n$ (presumably of integers, and presumably excluding the case when $t_n=0$ for all $n$). This was disproved by Kova\\v{c} and Tao \\cite{KoTa24}, and in the comments Kova\\v{c} has sketched a proof that there exists some choice of $t_n$ with $1\\leq t_n\\leq 6$ such that this sum is rational.\nReferences\n\n\n[Er48] Erd\\H{o}s, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) (1948), 63-66.\n\n[Er68d] Erd\\H{o}s, P., On the irrationality of certain series. Math. Student (1968), 222--226.\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\n\n[KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\n\n[TaTe25] T. Tao and J. Ter\"{a}v\"{a}inen, Quantitative correlations and some problems on prime factors of consecutive integers. arXiv:2512.01739 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Irrationality for every infinite A remains open. Classical and recent work proves several structured cases, including A equal to the natural numbers, suitable pairwise-coprime sets, the primes, and the prime powers, but no theorem covers arbitrary infinite A.\n\n**Verified partial progress.**\n\n- Erdős proved the case A=N and a pairwise-coprime summable-reciprocal class.\n- Tao and Teräväinen settled the prime and prime-power cases affirmatively.\n- Kovač and Tao analyze one-base subseries and nearby multi-base variants but leave the arbitrary infinite-A question open.\n- The Kovač-Tao counterexample for bounded perturbations 2^n-t_n does not apply to the fixed denominator 2^n-1.\n\n**Full solution or refutation.**\n\nNo arbitrary-infinite-set proof or rational counterexample was located. The current maintained page, updated April 2026, explicitly preserves the universal question as open.\n\n**What remains.**\n\nProve irrationality for every infinite A subset of N, or exhibit an infinite A for which the subseries has a rational sum.\n\n**Sources checked.**\n\n- V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Mathematica Hungarica 175 (2025), 572-608, DOI 10.1007/s10474-025-01528-0. (primary): https://doi.org/10.1007/s10474-025-01528-0\n Evidence used: The paper treats subseries of the relevant Lambert series and distinguishes the unresolved one-base question from solvable variants.\n- T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739 (2025). (primary): https://arxiv.org/abs/2512.01739\n Evidence used: The paper proves irrationality of the binary series involving the number of distinct prime factors, settling the prime and prime-power special cases.\n- Thomas F. Bloom, Erdős Problem #257, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/257\n Evidence used: The April 2026 record marks the exact arbitrary-infinite-A statement open and separates all known special cases.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2013, "problem_number": "EP-258", "title": "Erdős Problem #258", "statement": "Let $a_1,a_2,\\ldots$ be a sequence of positive integers with $a_n\\to \\infty$. Is $ \\sum_{n} \\frac{\\tau(n)}{a_1\\cdots a_n} $ irrational, where $\\tau(n)$ is the number of divisors of $n$?", "background": "Erd\\H{o}s and Straus \\cite{ErSt71} proved this is true if $a_n$ is monotone, i.e. $a_{n-1}\\leq a_n$ for all $n$. Erd\\H{o}s \\cite{Er48} proved that $\\sum_n \\frac{d(n)}{t^n}$ is irrational for any integer $t\\geq 2$.\nErd\\H{o}s and Straus further conjectured that if $a_{n-1}\\leq a_n$ for all $n$ then $ \\sum_{n} \\frac{\\phi(n)}{a_1\\cdots a_n} $ and $ \\sum_{n} \\frac{\\sigma(n)}{a_1\\cdots a_n} $ are both irrational.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[Er48] Erd\\H{o}s, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) (1948), 63-66.\n\n[ErSt71] Erd\\H{o}s, P. and Straus, E. G., Some number theoretic results. Pacific J. Math. (1971), 635-646.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Chojecki and GPT-5.4 Pro proved the divisor-function series irrational for every positive-integer sequence a_n tending to infinity, removing the old monotonicity hypothesis. The result uses a Tao--Teräväinen consecutive divisor-bound input and has an external Lean formalization.\n\n**Verified partial progress.**\n\n- Erdős proved the fixed-base series sum tau(n)/t^n irrational for each integer t >= 2.\n- Erdős and Straus proved the result when a_n is nondecreasing.\n- Tao--Teräväinen's work yields infinitely many N with tau(N+k) <= 2^(Ck) for all k >= 1, the tail-control input used in the 2026 proof.\n- A Lean proof linked by the Formal Conjectures repository verifies the affirmative theorem.\n\n**Full solution or refutation.**\n\nAt indices N supplied by the Tao--Teräväinen input, the divisor coefficients in the future tail grow at most exponentially in the offset. Since a_n tends to infinity, the Cantor-series tail becomes small enough for the classical irrationality contradiction: rationality would make a suitable scaled tail an integer, while it is strictly between 0 and 1.\n\n**What remains.**\n\nThe stated tau(n) problem is settled. The related phi(n) and sigma(n) irrationality conjectures under monotone a_n mentioned by Erdős--Straus are separate and were not verified as solved in this search.\n\n**Sources checked.**\n\n- Przemek Chojecki and GPT-5.4 Pro, Erdős problem 258 (2026). (primary): https://www.ulam.ai/research/erdos258.pdf\n Evidence used: This is the affirmative proof note linked from the maintained formal record.\n- Terence Tao and Joni Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739 (2025). (primary): https://arxiv.org/abs/2512.01739\n Evidence used: The paper supplies the modern consecutive prime-factor and divisor-function control used by the #258 proof.\n- P. Erdős and E. G. Straus, Some number theoretic results, Pacific Journal of Mathematics 36 (1971), 635--646. (primary): https://msp.org/pjm/1971/36-3/pjm-v36-n3-p07-p.pdf\n Evidence used: Section 2 proves the earlier monotone case and records the broader conjecture.\n- Google DeepMind Formal Conjectures, Erdős Problem 258 Lean record, accessed 2026-08-17. (formal_verification): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/258.lean\n Evidence used: The record marks the statement solved, cites the proof note, and links the external Lean proof of the exact irrationality theorem.\n- Thomas F. Bloom, Erdős Problem #258, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/258\n Evidence used: The maintained record attributes the affirmative solution and identifies the Tao--Teräväinen input.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2014, "problem_number": "EP-260", "title": "Erdős Problem #260", "statement": "Let $a_1\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The exact a_n/n tending to infinity assertion remains open in maintained sources. Erdős proved two stronger-hypothesis cases. The live forum index now lists EP-260 under unincorporated solution claims, but no validated primary paper or checked formal proof was found, so the claim does not yet justify a status change.\n\n**Verified partial progress.**\n\n- Erdős proved irrationality if a_(n+1)-a_n tends to infinity.\n- Erdős also proved irrationality under a_n>>n sqrt(log n log log n).\n- A recent forum solution claim is pending community verification and has not been added to the maintained problem record.\n\n**Full solution or refutation.**\n\nNo accepted solution was located. The exact imported growth condition is weaker than each theorem recorded on the maintained page, and the new solution claim is treated only as a verification lead.\n\n**What remains.**\n\nIndependently verify the recent claim or prove irrationality under only a_n/n->infinity; alternatively construct a rational counterexample.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #260, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/260\n Evidence used: The maintained February 2026 page labels the exact statement open and records Erdős's two stronger-hypothesis theorems.\n- Thomas F. Bloom, Erdős Problems forum index, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/\n Evidence used: The current index lists 260 under solution claims not yet added to the main site; that label is explicitly provisional.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2015, "problem_number": "EP-261", "title": "Erdős Problem #261", "statement": "Are there infinitely many $n$ such that there exists some $t\\geq 2$ and distinct integers $a_1,\\ldots,a_t\\geq 1$ such that $ \\frac{n}{2^n}=\\sum_{1\\leq k\\leq t}\\frac{a_k}{2^{a_k}}? $ Is this true for all $n$? Is there a rational $x$ such that $ x = \\sum_{k=1}^\\infty \\frac{a_k}{2^{a_k}} $ has at least $2^{\\aleph_0}$ solutions?", "background": "Related to [260].\nIn \\cite{Er88c} Erd\\H{o}s notes that Cusick had a simple proof that there do exist infinitely many such $n$. Erd\\H{o}s does not record what this was, but a later paper by Borwein and Loring \\cite{BoLo90} provides the following proof: for every positive integer $m$ and $n=2^{m+1}-m-2$ we have $ \\frac{n}{2^n}=\\sum_{n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The first subquestion is solved affirmatively by an explicit infinite family. Computation verifies the second through n=10000, but the all-n assertion remains open, as does the exact request for a rational number with continuum many representations.\n\n**Verified partial progress.**\n\n- For every positive m, n=2^(m+1)-m-2 satisfies n/2^n=sum_{n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2016, "problem_number": "EP-263", "title": "Erdős Problem #263", "statement": "Let $a_n$ be a sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\\to 1$ the sum $ \\sum\\frac{1}{b_n} $ is irrational. Is $a_n=2^{2^n}$ such a sequence? Must such a sequence satisfy $a_n^{1/n}\\to \\infty$?", "background": "One possible definition of an 'irrationality sequence' (see also [262] and [264]). A folklore result states that $\\sum \\frac{1}{a_n}$ is irrational whenever $\\lim a_n^{1/2^n}=\\infty$.\nKova\\v{c} and Tao \\cite{KoTa24} have proved that any strictly increasing sequence such that $\\sum \\frac{1}{a_n}$ converges and $\\lim a_{n+1}/a_n^2=0$ is not such an irrationality sequence. On the other hand, if $ \\liminf \\frac{a_{n+1}}{a_n^{2+\\epsilon}}>0 $ for some $\\epsilon>0$ then the above folklore result implies that $a_n$ is such an irrationality sequence.\nReferences\n\n\n[KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Under the exact imported wording, which omits the original increasing-sequence hypothesis, the second question is formally disproved; the 2^(2^n) candidate remains open. The maintained page corrected the statement in April 2026, and both questions in that intended increasing formulation remain open.\n\n**Verified partial progress.**\n\n- A Lean-verified counterexample answers the literal nonmonotone growth-necessity question negatively.\n- Kovač and Tao exclude strictly increasing sequences with convergent reciprocal sum and a_(n+1)/a_n^2 tending to zero.\n- Sufficiently super-doubly-exponential growth implies the irrationality-sequence property.\n- Koizumi proves floor(alpha^(2^n)) works for all but countably many alpha>1, without deciding alpha=2.\n\n**Full solution or refutation.**\n\nThe literal second subquestion has answer no, but this exploits a missing hypothesis and does not refute Erdős's original increasing-sequence question. No proof was found that 2^(2^n) has the required property.\n\n**What remains.**\n\nSettle the 2^(2^n) candidate. For the corrected increasing formulation, also determine whether every such irrationality sequence satisfies a_n^(1/n)->infinity.\n\n**Sources checked.**\n\n- The Formal Conjectures Authors, Erdős Problem 263 Lean record and linked formal proof, accessed 2026-08-17. (formal_verification): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/263.lean\n Evidence used: The record marks literal part (ii) false and links the machine-checked proof while leaving part (i) open.\n- Thomas F. Bloom, Erdős Problem #263 discussion, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/263\n Evidence used: The page documents that Erdős's original source required increasing a_n, explains the prior omission, and retains the corrected problem as open.\n- V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Mathematica Hungarica 175 (2025), 572-608, DOI 10.1007/s10474-025-01528-0. (primary): https://doi.org/10.1007/s10474-025-01528-0\n Evidence used: The paper supplies rigorous lower-growth obstruction and upper-growth sufficient conditions.\n- J. Koizumi, Irrationality of the reciprocal sum of doubly exponential sequences, INTEGERS 26 (2026), A28. (primary): https://math.colgate.edu/~integers/aa28/aa28.pdf\n Evidence used: Theorem 2 proves the all-but-countably-many-alpha endpoint result, which does not single out alpha=2.\n\n**Review notes.** Formulation defect flagged without changing the imported statement: the exact record lacks the word 'increasing', unlike the original 1988 source and corrected maintained page.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2017, "problem_number": "EP-264", "title": "Erdős Problem #264", "statement": "Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\neq 0$ and $b_n\neq 0$ for all $n$) the sum $ \\sum \\frac{1}{a_n+b_n} $ is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence?", "background": "A possible definition of an 'irrationality sequence' (see also [262] and [263]). One example is $a_n=2^{2^n}$. In \\cite{ErGr80} they also ask whether such a sequence can have polynomial growth, but Erd\\H{o}s later retracted this in \\cite{Er88c}, claiming 'It is not hard to show that it cannot increase slower than exponentially'.\nKova\\v{c} and Tao \\cite{KoTa24} have proved that $2^n$ is not such an irrationality sequence. More generally, they prove that any strictly increasing sequence of positive integers such that $\\sum\\frac{1}{a_n}$ converges and $ \\liminf \\left(a_n^2\\sum_{k>n}\\frac{1}{a_k^2}\\right) >0 $ is not such an irrationality sequence. In particular, any strictly increasing sequence with $\\limsup a_{n+1}/a_n <\\infty$ is not such an irrationality sequence.\nOn the other hand, Kova\\v{c} and Tao do prove that for any function $F$ with $\\lim F(n+1)/F(n)=\\infty$ there exists such an irrationality sequence with $a_n\\sim F(n)$.\nReferences\n\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Kovač and Tao prove that 2^n is not an irrationality sequence of the stated bounded-perturbation type. The n! candidate remains open, so the two-candidate question is only partially answered.\n\n**Verified partial progress.**\n\n- Every strictly increasing positive sequence with bounded consecutive ratio fails the property, covering 2^n.\n- Kovač and Tao construct examples asymptotic to any F with F(n+1)/F(n) tending to infinity.\n- The existence of sequences asymptotic to n! shows that growth scale alone does not decide the specific n! sequence.\n\n**Full solution or refutation.**\n\nThe answer for a_n=2^n is no. No proof or counterexample was found for a_n=n!, and even the irrationality of a basic fixed perturbation such as sum 1/(n!-1) remains unresolved.\n\n**What remains.**\n\nDetermine whether the particular sequence a_n=n! has the bounded-integer-perturbation irrationality property.\n\n**Sources checked.**\n\n- V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Mathematica Hungarica 175 (2025), 572-608, DOI 10.1007/s10474-025-01528-0. (primary): https://doi.org/10.1007/s10474-025-01528-0\n Evidence used: The paper proves the bounded-ratio obstruction, applies it to 2^n, and constructs type-3 sequences at supergeometric comparison scales.\n- Thomas F. Bloom, Erdős Problem #264, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/264\n Evidence used: The maintained page records 2^n as disproved while retaining n! as open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2018, "problem_number": "EP-265", "title": "Erdős Problem #265", "statement": "Let $1\\leq a_11$.\nIt remains open whether one can achieve $ \\limsup a_n^{1/2^n}>1. $ A folklore result states that $\\sum \\frac{1}{a_n}$ is irrational whenever $\\lim a_n^{1/2^n}=\\infty$, and hence such a sequence cannot grow faster than doubly exponentially - the remaining question is the precise exponent possible.\nReferences\n\n\n[KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Kovač and Tao construct increasing sequences with both sums rational and genuinely doubly exponential growth. A standard irrationality criterion prevents growth beyond the doubly exponential scale in a stronger limiting sense, but the precise critical exponent remains open.\n\n**Verified partial progress.**\n\n- Cantor's binomial example gives polynomial growth.\n- Kovač and Tao construct an example with a_n^(1/beta^n) tending to infinity for some beta>1.\n- If a_n^(1/2^n) tends to infinity, the first reciprocal sum is irrational, providing an upper growth obstruction.\n- The endpoint limsup a_n^(1/2^n)>1 remains unresolved.\n\n**Full solution or refutation.**\n\nThe possible growth is known to reach double-exponential order and cannot exceed that scale in the stated strong-limit sense. The exact exponent/base threshold asked by 'how fast' is not fully determined.\n\n**What remains.**\n\nConstruct an admissible increasing sequence with limsup a_n^(1/2^n)>1, or prove that every admissible sequence has that limsup at most one.\n\n**Sources checked.**\n\n- V. Kovač and T. Tao, On several irrationality problems for Ahmes series, Acta Mathematica Hungarica 175 (2025), 572-608, DOI 10.1007/s10474-025-01528-0. (primary): https://doi.org/10.1007/s10474-025-01528-0\n Evidence used: The paper gives the double-exponential construction with simultaneous rational shifted reciprocal sums.\n- Thomas F. Bloom, Erdős Problem #265 discussion, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/265\n Evidence used: The January 2026 maintained record identifies the exact remaining limsup endpoint and confirms the increasing-order convention.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2019, "problem_number": "EP-267", "title": "Erdős Problem #267", "statement": "Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_11$. Must $ \\sum_k\\frac{1}{F_{n_k}} $ be irrational?", "background": "It may be sufficient to have $n_k/k\\to \\infty$. Good \\cite{Go74} and Bicknell and Hoggatt \\cite{BiHo76} have shown that $\\sum \\frac{1}{F_{2^n}}$ is irrational - in fact, $ \\sum \\frac{1}{F_{2^n}}=\\frac{7-\\sqrt{5}}{2}. $ Badea \\cite{Ba87} proved that $\\sum \\frac{1}{F_{2^n+1}}$ is irrational.\nThe sum $\\sum \\frac{1}{F_n}$ itself was proved to be irrational by Andr\\'{e}-Jeannin \\cite{An89}.\nThe main problem has been proved for $c\\geq 2$ by Badea \\cite{Ba93}. It remains open for $1\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The question remains open for 1=2, while Nguyen proves transcendence for c>2; the algebraic irrational identity at indices 2^k shows only that Nguyen's transcendence threshold is sharp, not that the original irrationality claim fails.\n\n**Verified partial progress.**\n\n- Badea proved the main irrationality assertion whenever c>=2.\n- Nguyen proved the stronger transcendence conclusion whenever c>2.\n- The series over indices 2^k equals (7-sqrt(5))/2, an algebraic irrational value at the c=2 boundary.\n- No accepted result was found for the interval 1=2 irrationality theorem to this paper.\n- K. D. Nguyen, Transcendental Series of Reciprocals of Fibonacci and Lucas Numbers, arXiv:2009.02446 (2020). (primary): https://arxiv.org/abs/2009.02446\n Evidence used: The abstract explicitly proves transcendence for c>2 and explains sharpness via the algebraic value at powers of two.\n- Thomas F. Bloom, Erdős Problem #267, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/267\n Evidence used: The current record states that c>=2 is proved and 1", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2020, "problem_number": "EP-269", "title": "Erdős Problem #269", "statement": "Let $P$ be a finite set of primes with $\\lvert P\\rvert \\geq 2$ and let $\\{a_1\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The repeated-summand running-LCM series for finite P with at least two primes remains open. Results for infinite P and for deleting duplicate summands are different statements and do not settle the imported question.\n\n**Verified partial progress.**\n\n- Erdős stated that the analogous series is always irrational when P is infinite.\n- Erdős reported a proof when duplicate summands are removed.\n- A forum proof and subsequent correction concern only the already-known infinite-P case.\n- The maintained tracker, current community repository, and a July 2026 commissioning record all retain the finite-P problem as open.\n\n**Full solution or refutation.**\n\nNo primary paper or checked formal proof resolving finite P was found. The multiplicities caused by repeated running least common multiples are precisely absent from the no-duplicate variant and cannot be discarded.\n\n**What remains.**\n\nProve irrationality for every finite prime set P with |P|>=2 while retaining every repeated summand, or exhibit a finite P for which the value is rational.\n\n**Sources checked.**\n\n- P. Erdős, letter to the editor dated 1 January 1973, Fibonacci Quarterly 12 (1974), p. 335. (source_collection): https://www.fq.math.ca/Scanned/12-4/letter.pdf\n Evidence used: The originating letter poses the problem and reports the distinct-summand variant.\n- Thomas F. Bloom, Erdős Problem #269 discussion, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/discuss/269\n Evidence used: The page labels the exact finite-P statement open and makes clear that its comments address the infinite-P case.\n- Thomas F. Bloom et al., community Erdős Problems database, accessed 2026-08-17. (maintained_tracker): https://github.com/teorth/erdosproblems\n Evidence used: The current community index retains problem 269 as open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2021, "problem_number": "EP-271", "title": "Erdős Problem #271", "statement": "Let $A(n)=\\{a_00$, $a_k\\leq (\\frac{1}{2}+\\epsilon)k^2$ for all sufficiently large $k$. van Doorn and Sothanaphan have noted in the comment section that Moy's proof can be upgraded to give a fully explicit result of $ a_k\\leq \\frac{(k-1)(k+2)}{2}+n $ for all $k\\geq 0$.\nIn general, sequences which begin with some initial segment and thereafter are continued in a greedy fashion to avoid three-term arithmetic progressions are known as Stanley sequences.\nReferences\n\n\n[Li90] S. Lindhurst, An investigation of several interesting sets of numbers generated by the greedy\nalgorithm. Senior thesis at Princeton University (1990).\n\n[Mo11] Moy, Richard A., On the growth of the counting function of Stanley sequences. Discrete Math. (2011), 560-562.\n\n[OdSt78] A. Odlyzko and R. Stanley, Some curious sequences constructed with the greedy algorithm. Bell Laboratories internal memorandum (1978).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Explicit descriptions are known for A(1), A(3^j), and A(2*3^j), and Moy proved a universal asymptotic quadratic upper bound on the terms, but no general formula or proof of the proposed growth dichotomy is known.\n\n**Verified partial progress.**\n\n- Odlyzko and Stanley explicitly describe infinite parameter families, including n=1, 3^j, and 2*3^j.\n- Moy proved S(A,x)>=(sqrt(2)-epsilon)sqrt(x) for sufficiently large x, giving a_k<=(1/2+epsilon)k^2 asymptotically.\n- The maintained record reports the explicit refinement a_k<=((k-1)(k+2))/2+n.\n\n**Full solution or refutation.**\n\nThe special families and quadratic upper bound are genuine partial progress, not a determination of the general sequence. The serialized difficulty fragment appended to the imported background is extraction damage.\n\n**What remains.**\n\nDetermine the terms or growth for general n and prove or disprove the Odlyzko-Stanley regular/irregular dichotomy; A(4) remains a central unresolved example.\n\n**Sources checked.**\n\n- Richard A. Moy, On the Growth of the Counting Function of Stanley Sequences, Discrete Mathematics 311 (2011), 560-562, arXiv:1101.0022. (primary): https://arxiv.org/abs/1101.0022\n Evidence used: The abstract and paper prove the counting-function lower bound that yields a universal asymptotic quadratic upper bound for Stanley-sequence terms.\n- Thomas F. Bloom, Erdős Problem #271, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/271\n Evidence used: The page marks the general problem open, records the explicit special families, the conjectured dichotomy, Moy's theorem, and the explicit refinement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2022, "problem_number": "EP-272", "title": "Erdős Problem #272", "statement": "Let $N\\geq 1$. What is the largest $t$ such that there are $A_1,\\ldots,A_t\\subseteq \\{1,\\ldots,N\\}$ with $A_i\\cap A_j$ a non-empty arithmetic progression for all $i\neq j$?", "background": "Simonovits and S\\'{o}s \\cite{SiSo81} have shown that $t\\ll N^2$.\nErd\\H{o}s and Graham asked whether the maximal $t$ is achieved when we take the $A_i$ to be all arithmetic progressions in $\\{1,\\ldots,N\\}$ containing some fixed element, 'presumably the integer $\\lfloor N/2\\rfloor$'. This was disproved by Simonovits and S\\'{o}s \\cite{SiSo81}, who observed that taking all sets containing at most $3$ elements, containing some fixed element, produces $\\binom{N}{2}+1$ many such sets, which is asymptotically greater than the number of arithmetic progressions containing a fixed element, which is $\\sim \\frac{\\pi^2}{24}N^2$.\nIf we drop the non-empty requirement then Graham, Simonovits, and S\\'{o}s \\cite{GSS80} have shown that $ t\\leq \\binom{N}{3}+\\binom{N}{2}+\\binom{N}{1}+1 $ and this is best possible.\nSzabo \\cite{Sz99} proved that the maximal such $t$ is equal to $ \\frac{N^2}{2}+O(N^{5/3}(\\log N)^3), $ resolving the asymptotic question. On the other hand, Szabo showed that the conjecture of Simonovits and S\\'{o}s that $\\binom{n}{2}+1$ is best possible is false, giving a construction which yields $ t \\geq \\binom{N}{2}+\\left\\lfloor\\frac{N-1}{4}\\right\\rfloor+1. $ Szabo conjectures that the asymptotic $t=\\binom{N}{2}+O(N)$ holds, and that in any extremal example there is an integer contained in all sets.\nReferences\n\n\n[GSS80] Graham, R. L. and Simonovits, M. and S\\'{o}s, V. T., A note on the intersection properties of subsets of integers. J. Combin. Theory Ser. A (1980), 106-110.\n\n[SiSo81] Simonovits, Mikl\\'{o}s and S\\'{o}s, Vera T., Intersection properties of subsets of integers. European J. Combin. (1981), 363-372.\n\n[Sz99] Szab\\'o, Tibor, Intersection properties of subsets of integers. European J. Combin. (1999), 429--444.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Szabó determined the leading asymptotic t(N)=N^2/2+O(N^(5/3)(log N)^3) and improved the lower construction, but the exact extremal value and the sharper conjectural error O(N) remain open.\n\n**Verified partial progress.**\n\n- Simonovits and Sós first established the quadratic order of magnitude.\n- Szabó proved the leading asymptotic with error O(N^(5/3)(log N)^3).\n- Szabó constructed at least binom(N,2)+floor((N-1)/4)+1 sets and thereby disproved the earlier exact conjecture binom(N,2)+1.\n\n**Full solution or refutation.**\n\nThe leading term is known, but the imported question asks for the largest t, so it is not fully solved. The common serialized difficulty tail in the background is extraction damage.\n\n**What remains.**\n\nDetermine t(N) exactly or prove t(N)=binom(N,2)+O(N), and settle whether every extremal family has a common element.\n\n**Sources checked.**\n\n- Tibor Szabó, Intersection Properties of Subsets of Integers, European Journal of Combinatorics 20 (1999), 429-444, DOI 10.1006/eujc.1997.0176. (primary): https://doi.org/10.1006/eujc.1997.0176\n Evidence used: The paper proves the N^2/2 asymptotic with stated error and improves the construction, disproving the proposed exact extremal system.\n- Thomas F. Bloom, Erdős Problem #272, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/272\n Evidence used: The current page marks the exact problem open and records Szabó's asymptotic, construction, and remaining conjectures.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2023, "problem_number": "EP-273", "title": "Erdős Problem #273", "statement": "Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\\geq 5$?", "background": "Selfridge has found an example using divisors of $360$ if $p=3$ is allowed.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The existence of a covering system whose every modulus equals p-1 for a prime p>=5 remains open. Selfridge's example when p=3 is allowed lies outside the stated restriction.\n\n**Verified partial progress.**\n\n- Selfridge constructed a related covering using divisors of 360 when the excluded prime p=3 is permitted.\n- The maintained record reports no claimed partial or complete solution for the p>=5 problem.\n\n**Full solution or refutation.**\n\nNo construction or impossibility theorem for the exact p>=5 restriction was located. The appended serialized difficulty text is extraction damage.\n\n**What remains.**\n\nConstruct a covering system using only moduli p-1 with p>=5, or prove that no such system exists.\n\n**Sources checked.**\n\n- Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 24. (primary): https://combinatorica.hu/~p_erdos/1979-07.pdf\n Evidence used: This is the originating Erdős-Graham source for the restricted covering-system question.\n- Thomas F. Bloom, Erdős Problem #273, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/273\n Evidence used: The October 2025 record marks the p>=5 question open, reports the p=3 construction, and lists no solution claims.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2024, "problem_number": "EP-274", "title": "Erdős Problem #274", "statement": "If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)", "background": "A question of Herzog and Sch\"{onheim}, who conjectured more generally that if $G$ is any (not necessarily finite) group and $a_1G_1,\\ldots,a_kG_k$ are finitely many cosets of subgroups of $G$ with distinct indices $[G:G_i]$ then the $a_iG_i$ cannot form a partition of $G$.\nThis conjecture was proved in the case when all the $G_i$ are subnormal in $G$ by Sun \\cite{Su04}. In particular if $G$ is abelian (which was the special case asked about in \\cite{Er77c} and \\cite{ErGr80}) the answer to the original question is no.\nMargolis and Schnabel \\cite{MaSc19} proved this conjecture for all groups $G$ of size $<1440$.\nReferences\n\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[MaSc19] Margolis, Leo and Schnabel, Ofir, The {H}erzog-{S}ch\"onheim conjecture for small groups and\nharmonic subgroups. Beitr. Algebra Geom. (2019), 399--418.\n\n[Su04] Sun, Zhi-Wei, On the {H}erzog-{S}ch\"onheim conjecture for uniform covers of\ngroups. J. Algebra (2004), 153--175.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general Herzog-Schönheim conjecture remains open, while it is proved for subnormal subgroup partitions, groups of order below 1440, finite simple groups, symmetric groups, and several other group classes.\n\n**Verified partial progress.**\n\n- Sun proved the conjecture when all participating subgroups are subnormal, which includes the abelian formulation in the original Erdős-Graham source.\n- Margolis and Schnabel proved the conjecture for every group of order less than 1440.\n- Garonzi and Margolis proved the conjecture for finite simple groups and symmetric groups in a 2025 primary preprint.\n\n**Full solution or refutation.**\n\nThe imported statement is the current general-group formulation, not the older abelian version. The tracker's October 2025 history shows that it broadened the already-solved abelian record rather than marking the general conjecture solved.\n\n**What remains.**\n\nProve that no finite coset partition with pairwise distinct subgroup indices exists for an arbitrary group, equivalently for every finite group, or construct a counterexample.\n\n**Sources checked.**\n\n- Zhi-Wei Sun, On the Herzog-Schönheim Conjecture for Uniform Covers of Groups, Journal of Algebra 273 (2004), 153-175, DOI 10.1016/S0021-8693(03)00526-X. (primary): https://doi.org/10.1016/S0021-8693(03)00526-X\n Evidence used: Sun proves the subnormal-subgroup case, including abelian groups.\n- Leo Margolis and Ofir Schnabel, The Herzog-Schönheim Conjecture for Small Groups and Harmonic Subgroups, Beiträge zur Algebra und Geometrie 60 (2019), 399-418, arXiv:1803.03569. (primary): https://arxiv.org/abs/1803.03569\n Evidence used: The paper proves the conjecture for all groups of order below 1440.\n- Martino Garonzi and Leo Margolis, The Herzog-Schönheim Conjecture for Simple and Symmetric Groups, arXiv:2509.25118 (2025). (primary): https://arxiv.org/abs/2509.25118\n Evidence used: Theorems 1.2 and 1.3 prove the conjecture for symmetric and finite simple groups while describing the general conjecture as open.\n- Thomas F. Bloom, revision history for Erdős Problem #274, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/274\n Evidence used: The history documents the October 2025 change from the solved abelian formulation to the current general-group problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2025, "problem_number": "EP-276", "title": "Erdős Problem #276", "statement": "Is there an infinite Lucas sequence $a_0,a_1,\\ldots$ where $a_{n+2}=a_{n+1}+a_n$ for $n\\geq 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence?", "background": "Whether such a composite Lucas sequence even exists was open for a while, but using covering systems Graham \\cite{Gr64} showed that $ a_0 = 1786772701928802632268715130455793 $ and $ a_1 = 1059683225053915111058165141686995 $ generate such a sequence. This problem asks whether one can have a composite Lucas sequence without 'an underlying system of covering congruences responsible'.\nThis problem has been 'conjecturally solved' by Ismailescu and Son \\cite{IsSo14}, in that they provide an explicit infinite Lucas sequence in which all the terms are composite, and believe that no covering system is responsible for this. See the comment by van Doorn below for more details.\nSee also [1113] for another problem in which the question is whether covering systems are always responsible.\nReferences\n\n\n[Gr64] Graham, R. L., A Fibonacci-Like Sequence of Composite Numbers. Math. Mag. (1964), 322-324.\n\n[IsSo14] Ismailescu, Dan and Son, Jaesung, A new kind of {F}ibonacci-like sequence of composite numbers. J. Integer Seq. (2014), Article 14.8.2, 9.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ismailescu and Son rigorously construct a Lucas recurrence whose every term is composite, but they only give finite computational evidence that no finite covering set of primes meets every term; the required universal no-covering property remains unproved.\n\n**Verified partial progress.**\n\n- Graham constructed an all-composite Lucas sequence using a finite covering set of primes.\n- Ismailescu and Son construct relatively prime initial values, cover all even-indexed terms with 30 primes, and factor every odd-indexed term nontrivially.\n- For 803 pairwise-coprime sampled terms through index 200000, they found no factor among the 30 construction primes or primes at most 2,000,000, forcing any covering set for their example beyond those tested bounds.\n\n**Full solution or refutation.**\n\nThe phrase 'an integer has a common factor with every term' means a fixed integer M with gcd(M,a_n)>1 for every n, not one divisor common to all terms. Thus gcd(a_0,a_1)=1 does not solve the problem. The primary paper explicitly presents the no-finite-covering conclusion only as a belief.\n\n**What remains.**\n\nProve for the Ismailescu-Son sequence or another all-composite Lucas sequence that every finite prime set misses at least one term.\n\n**Sources checked.**\n\n- Dan Ismailescu and Jaesung Son, A New Kind of Fibonacci-Like Sequence of Composite Numbers, Journal of Integer Sequences 17 (2014), Article 14.8.2. (primary): https://www.maths.tcd.ie/EMIS/journals/JIS/VOL17/Ismailescu/ism8.pdf\n Evidence used: Theorem 3 proves all terms composite and the final section states only computational evidence and belief that no finite covering set exists.\n- Thomas F. Bloom, Erdős Problem #276, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/276\n Evidence used: The page remains OPEN, calls the Ismailescu-Son construction conjectural for the no-covering condition, and distinguishes all-composite existence from the stated question.\n- Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 27. (primary): https://combinatorica.hu/~p_erdos/1979-07.pdf\n Evidence used: The original source clarifies the intended interpretation in terms of an underlying covering system.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2026, "problem_number": "EP-278", "title": "Erdős Problem #278", "statement": "Let $A=\\{n_1<\\cdots\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Simpson's inclusion-exclusion argument proves that the minimum covered density is attained when all residues are equal, settling the second question, while the maximum-density problem for an arbitrary fixed modulus set remains open.\n\n**Verified partial progress.**\n\n- Simpson derived an alternating inclusion-exclusion lower bound depending only on least common multiples of subsets of the moduli.\n- Taking all residue classes equal attains this lower bound, so the minimum-density question is solved affirmatively.\n\n**Full solution or refutation.**\n\nOnly the minimum half of the two-part statement is closed. No general formula for the maximum covered density was located.\n\n**What remains.**\n\nDetermine or sharply characterize the maximum density obtainable by choosing one residue class for each modulus in a prescribed finite set.\n\n**Sources checked.**\n\n- R. J. Simpson, Exact Coverings of the Integers by Arithmetic Progressions, Discrete Mathematics 59 (1986), 181-190, DOI 10.1016/0012-365X(86)90079-8. (primary): https://doi.org/10.1016/0012-365X(86)90079-8\n Evidence used: The cited paper supplies the inclusion-exclusion result used to identify the minimum configuration.\n- Thomas F. Bloom, Erdős Problem #278, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/278\n Evidence used: The page states that Simpson settled the second question and continues to mark the first maximum-density question open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2027, "problem_number": "EP-279", "title": "Erdős Problem #279", "statement": "Let $k\\geq 3$. Is there a choice of congruence classes $a_p\\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$ and integer $t\\geq k$?", "background": "Even the case $k=3$ seems difficult. This may be true with the primes replaced by any set $A\\subseteq \\mathbb{N}$ such that $ \\lvert A\\cap [1,N]\\rvert \\gg N/\\log N $ and $ \\sum_{\\substack{n\\in A\\\\ n\\leq N}}\\frac{1}{n} -\\log\\log N\\to \\infty $ as $N\\to \\infty$.\nFor $k=1$ or $k=2$ any set $A$ such that $\\sum_{n\\in A}\\frac{1}{n}=\\infty$ has this property.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The existence of fixed prime residue classes covering every sufficiently large integer with quotient t>=k remains open even for k=3. Reciprocal-divergence arguments prove only an almost-all-integers weakening.\n\n**Verified partial progress.**\n\n- For the weakened target of almost all integers, any modulus set with divergent reciprocal sum works for every k>=2.\n- The current tracker says even k=3 for eventual coverage appears difficult and reports no verified solution.\n\n**Full solution or refutation.**\n\nThe imported background is stale where it attributes the full eventual-coverage property to divergent reciprocal sum for k=1 or 2. The live page corrected this in April 2026 to an almost-all statement; this correction does not change the main open classification.\n\n**What remains.**\n\nConstruct a single prime-indexed residue choice that covers every sufficiently large integer for k=3, or prove that such eventual coverage is impossible.\n\n**Sources checked.**\n\n- Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 29. (primary): https://combinatorica.hu/~p_erdos/1979-07.pdf\n Evidence used: This is the originating source for the prime-residue eventual-coverage question.\n- Thomas F. Bloom, Erdős Problem #279, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/279\n Evidence used: The page remains OPEN, states the almost-all weakening precisely, and its April 2026 discussion documents correction of the earlier full-coverage overstatement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2028, "problem_number": "EP-281", "title": "Erdős Problem #281", "statement": "Let $n_10$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\\pmod{n_i}$ for $1\\leq i\\leq k$ is less than $\\epsilon$?", "background": "The latter condition is clearly sufficient, the problem is if it's also necessary. The assumption implies $\\sum \\frac{1}{n_i}=\\infty$. If the $n_i$ are pairwise relatively prime then it is sufficient that $\\sum \\frac{1}{n_i}=\\infty$.\nThis is true - a proof is given in the comments by Somani (using ChatGPT).\nAn alternative elementary proof was noted by KoishiChan in the comments: let $\\delta$ be the lower density of the set of integers divisible by some $n\\in A$, and $\\delta_k$ be the density of the set of integers divisible by at least one of $n_1,\\ldots,n_k$. A theorem of Davenport and Erd\\H{o}s \\cite{DaEr36} states that $ \\delta = \\lim_{k\\to \\infty}\\delta_k. $ In the present case $\\delta=1$ and hence for every $\\epsilon>0$ there exists $k$ such that $\\delta_k>1-\\epsilon$. In other words, the density of those integers not satisfying $a_i\\equiv 0\\pmod{n_i}$ for $1\\leq i\\leq k$ is $<\\epsilon$. By a theorem of Rogers (which first appeared in print in Chapter V.3 of the book of Halberstam and Roth \\cite{HaRo66}) the density of those integers not satisfying any of the congruences $a_i\\pmod{n_i}$ for $1\\leq i\\leq k$ is maximised when $a_i\\equiv 0$, which concludes the proof.\nGiven that both Rogers' result and the Davenport-Erd\\H{o}s theorem mentioned above must have been very familiar to Erd\\H{o}s in 1980, it is strange that this natural argument was overlooked.\nReferences\n\n\n[DaEr36] Davenport, H. and Erd\\H{o}s, P., On sequences of positive integers. Acta Arithmetica (1936), 147-151.\n\n[HaRo66] Halberstam, H. and Roth, K. F., Sequences. {V}ol. {I}. (1966), xx+291.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The answer is yes. Apply the hypothesis to zero residues; Davenport--Erdős identifies the density-one multiples set with the limit of finite-prefix densities, and Rogers' theorem says zero residues maximize the uncovered density for each fixed prefix. The deduction is also formally corroborated in Lean.\n\n**Verified partial progress.**\n\n- For zero residue choices, the hypothesis makes the set of multiples of the n_i have density 1.\n- Davenport--Erdős proves that the lower density of an infinite multiples set equals the limit of the densities of its finite prefixes.\n- Rogers proves that for fixed finite moduli, the uncovered density is maximized when every selected residue is zero.\n- Combining the two classical results gives the required finite k uniformly over all residue choices.\n\n**Full solution or refutation.**\n\nLet delta_k be the density covered by the zero residue classes modulo n_1,...,n_k. Davenport--Erdős gives delta_k -> 1. Choose k with 1-delta_k < epsilon. Rogers' extremal sieving theorem implies that every other residue choice for those same moduli leaves at most the zero-choice uncovered density, so the same k works uniformly.\n\n**What remains.**\n\nNothing remains for the yes/no statement. Quantitative bounds on the least k require additional information about the particular sequence of moduli; no universal effective rate is supplied.\n\n**Sources checked.**\n\n- Harold Davenport and Paul Erdős, On sequences of positive integers, Acta Arithmetica 2 (1936), 147--151, DOI 10.4064/aa-2-1-147-151. (primary): https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/2/1/93274/on-sequences-of-positive-integers\n Evidence used: The paper proves that the lower density of a multiples set equals the limit of the densities of the finite-prefix multiples sets.\n- H. Halberstam and K. F. Roth, Sequences, Vol. I, Oxford University Press (1966), Chapter V.3, pp. 242--244. (authoritative_secondary): https://archive.org/details/sequences0000halb\n Evidence used: This is the cited first printed source for Rogers' theorem that zero residues maximize residual density for fixed finite moduli.\n- Michael Filaseta, Kevin Ford, Sergei Konyagin, Carl Pomerance, and Gang Yu, Sieving by large integers and covering systems of congruences, Journal of the American Mathematical Society 20 (2007), 495--517. (primary): https://doi.org/10.1090/S0894-0347-06-00549-2\n Evidence used: The introduction explicitly restates Rogers' theorem: the largest residual density for fixed finite moduli occurs when all selected residue classes are zero.\n- Google DeepMind Formal Conjectures, Erdős Problem 281 Lean record, accessed 2026-08-17. (formal_verification): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/281.lean\n Evidence used: The exact quantifier structure is marked solved, and the record links a complete external Lean proof.\n- Thomas F. Bloom, Erdős Problem #281, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/281\n Evidence used: The maintained page gives the Davenport--Erdős plus Rogers proof and classifies the problem proved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2029, "problem_number": "EP-282", "title": "Erdős Problem #282", "statement": "Let $A\\subseteq \\mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\\in (0,1)$: choose the minimal $n\\in A$ such that $n\\geq 1/x$ and repeat with $x$ replaced by $x-\\frac{1}{n}$. If this terminates after finitely many steps then this produces a representation of $x$ as the sum of distinct unit fractions with denominators from $A$.\nDoes this process always terminate if $x$ has odd denominator and $A$ is the set of odd numbers? More generally, for which pairs $x$ and $A$ does this process terminate?", "background": "In 1202 Fibonacci observed that this process terminates for any $x$ when $A=\\mathbb{N}$. The problem when $A$ is the set of odd numbers is due to Stein.\nGraham \\cite{Gr64b} has shown that $\\frac{m}{n}$ is the sum of distinct unit fractions with denominators $\\equiv a\\pmod{d}$ if and only if $ \\left(\\frac{n}{(n,(a,d))},\\frac{d}{(a,d)}\\right)=1. $ Does the greedy algorithm always terminate in such cases?\nGraham \\cite{Gr64c} has also shown that $x$ is the sum of distinct unit fractions with square denominators if and only if $x\\in [0,\\pi^2/6-1)\\cup [1,\\pi^2/6)$. Does the greedy algorithm for this always terminate? Erd\\H{o}s and Graham believe not - indeed, perhaps it fails to terminate almost always.\nSee also [206].\nReferences\n\n\n[Gr64b] Graham, R. L., On finite sums of unit fractions. Proc. London Math. Soc. (3) (1964), 193-207.\n\n[Gr64c] Graham, R. L., On finite sums of reciprocals of distinct nth powers. Pacific J. Math. (1964), 85-92.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Termination of the greedy algorithm with odd denominators for every rational of odd reduced denominator remains open. Graham's restricted Egyptian-fraction theorems establish representation existence, not termination of this greedy rule.\n\n**Verified partial progress.**\n\n- The unrestricted Fibonacci-Sylvester greedy algorithm terminates for A equal to all positive integers.\n- Graham characterized when rationals admit some distinct-unit-fraction representation using denominators in a fixed congruence class.\n- Graham characterized the rationals representable by distinct reciprocal squares, but the associated greedy termination question is also unresolved.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for the odd-denominator greedy process was located. Representation theorems must not be upgraded to greedy-algorithm theorems.\n\n**What remains.**\n\nProve termination for every rational with odd denominator when A is the odd numbers, or give a rational counterexample; classify broader pairs (x,A).\n\n**Sources checked.**\n\n- R. L. Graham, On Finite Sums of Unit Fractions, Proceedings of the London Mathematical Society s3-14 (1964), 193-207, DOI 10.1112/plms/s3-14.2.193. (primary): https://doi.org/10.1112/plms/s3-14.2.193\n Evidence used: The paper gives the congruence-class representation criterion cited in the record; it does not prove the restricted greedy algorithm terminates.\n- R. L. Graham, On Finite Sums of Reciprocals of Distinct nth Powers, Pacific Journal of Mathematics 14 (1964), 85-92. (primary): https://msp.org/pjm/1964/14-1/pjm-v14-n1-p10-p.pdf\n Evidence used: The paper supplies the distinct reciprocal-square representation theorem, a related existence result rather than a greedy termination theorem.\n- Thomas F. Bloom, Erdős Problem #282, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/282\n Evidence used: The current page marks the odd-denominator and general restricted-greedy questions open and reports no complete solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2030, "problem_number": "EP-283", "title": "Erdős Problem #283", "statement": "Let $p:\\mathbb{Z}\\to \\mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\\geq 2$ with $d\\mid p(n)$ for all $n\\geq 1$. Is it true that, for all sufficiently large $m$, there exist integers $1\\leq n_1<\\cdots 0$ (provided $m$ is taken sufficiently large depending on $\\alpha$).\nCassels \\cite{Ca60} has proved that these conditions on the polynomial imply every sufficiently large integer is the sum of $p(n_i)$ with distinct $n_i$. Burr has proved this if $p(x)=x^k$ with $k\\geq 1$ and if we allow $n_i=n_j$.\nAlekseyev \\cite{Al19} has proved this when $p(x)=x^2$, for all $m>8542$. For example, $ 1=\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{6}+\\frac{1}{12} $ and $ 200 = 2^2+4^2+6^2+12^2. $ van Doorn \\cite{vD25} has investigated the question of what 'sufficiently large' means for $p(x)=x$. van Doorn has also proved the original conjecture for many linear and quadratic polynomials, for example $p(x)=x+5$ or $p(x)=x^2+100$ - see the comments section.\nReferences\n\n\n[Al19] Alekseyev, Max A., On partitions into squares of distinct integers whose\nreciprocals sum to 1. (2019), 213--221.\n\n[Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.\n\n[Gr63] Graham, R. L., A theorem on partitions. J. Austral. Math. Soc. (1963), 435-441.\n\n[vD25] W. van Doorn, Partitions with prescribed sum of rationals: asymptotic bounds. arXiv:2502.02200 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A May 2026 proof settles the polynomial Egyptian-sum problem and proves the stronger statement with any positive rational alpha in place of 1. The associated Lean development is reported sorry-free, proves its former Roth--Szekeres--Graham trust boundary internally, and passes a SafeVerify specification.\n\n**Verified partial progress.**\n\n- Graham proved the case p(x)=x.\n- Cassels proved eventual representation by distinct polynomial values without the reciprocal-sum constraint.\n- Alekseyev proved p(x)=x^2 for every m>8542.\n- Van Doorn proved many linear and quadratic cases and quantitative results for p(x)=x.\n- The 2026 theorem covers every integer-valued polynomial with positive leading coefficient and no fixed divisor, for every prescribed rational alpha>0.\n\n**Full solution or refutation.**\n\nThe proof combines the Roth--Szekeres--Graham completeness theorem for polynomial sequences with reciprocal-preserving switches based on 1/x=1/(2x)+1/(3x)+1/(6x). Polynomial periodicity, valuation profiles, collision avoidance, and finite congruence-correction slots assemble distinct denominators whose reciprocals sum to alpha while their polynomial values sum to every sufficiently large m.\n\n**What remains.**\n\nThe qualitative theorem is settled. Sharp or practical thresholds for sufficiently large m for a general polynomial, and optimized representation sizes, remain quantitative problems.\n\n**Sources checked.**\n\n- GPT-5.5 Pro and Liam Price, Polynomial Egyptian Sums, 3 May 2026. (primary): https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P283/compact_cayley_proof.pdf\n Evidence used: The proof gives the stronger positive-rational-alpha theorem and specializes to the exact #283 statement.\n- Shashi Ammanamanchi et al., Erdős Problems 283 + 351 formalization, accessed 2026-08-17. (formal_verification): https://github.com/Shashi456/erdos-formalizations/tree/main/Erdos/P283\n Evidence used: The repository reports zero executable sorries, an internal proof of the Roth--Szekeres--Graham input, exact Formal Conjectures wrappers, and a passing SafeVerify target with only standard Lean core axioms.\n- Google DeepMind Formal Conjectures, Erdős Problem 283 Lean record, accessed 2026-08-17. (formal_verification): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/283.lean\n Evidence used: The exact source problem is marked solved and linked to the complete external Lean proof.\n- Wouter van Doorn, Partitions with prescribed sum of rationals: asymptotic bounds, arXiv:2502.02200 (2025). (primary): https://arxiv.org/abs/2502.02200\n Evidence used: This supplies recent quantitative and special-case context for the linear and selected polynomial cases.\n- Thomas F. Bloom, Erdős Problem #283 discussion, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/forum/thread/283\n Evidence used: The maintained thread records the stronger rational-alpha proof, its switching idea, independent checking, and the later formalization.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2031, "problem_number": "EP-288", "title": "Erdős Problem #288", "statement": "Is it true that there are only finitely many pairs of intervals $I_1,I_2$ such that $ \\sum_{n_1\\in I_1}\\frac{1}{n_1}+\\sum_{n_2\\in I_2}\\frac{1}{n_2}\\in \\mathbb{N}? $ ", "background": "For example, $ \\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{20}=1. $ This is still open even if $\\lvert I_2\\rvert=1$. It is perhaps true with two intervals replaced by any $k$ intervals.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Finiteness of interval pairs with integral total reciprocal sum remains open, even in the special case where one interval is a singleton.\n\n**Verified partial progress.**\n\n- The maintained record gives explicit sporadic examples, including 1/3+1/4+1/5+1/6+1/20=1.\n- A 2026 tracker discussion reduces the singleton case to a unique denominator candidate plus p-adic and CRT restrictions, but explicitly does not claim a solution.\n\n**Full solution or refutation.**\n\nNo published complete result for the exact finiteness assertion was located. The common serialized difficulty tail in the imported background is extraction damage.\n\n**What remains.**\n\nProve only finitely many pairs exist or construct infinitely many; the singleton case |I_2|=1 is already open.\n\n**Sources checked.**\n\n- Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980). (primary): https://combinatorica.hu/~p_erdos/1979-07.pdf\n Evidence used: This is the cited originating source for the interval reciprocal-sum problem.\n- Thomas F. Bloom, Erdős Problem #288 and discussion, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/288\n Evidence used: The live record marks the problem open even for a singleton interval; the discussion labels the p-adic reduction as incomplete.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2032, "problem_number": "EP-289", "title": "Erdős Problem #289", "statement": "Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\\ldots,I_k\\subset \\mathbb{N}$, distinct, not overlapping or adjacent, with $\\lvert I_i\\rvert \\geq 2$ for $1\\leq i\\leq k$ such that $ 1=\\sum_{i=1}^k \\sum_{n\\in I_i}\\frac{1}{n}? $ ", "background": "Erd\\H{o}s and Graham posed this in \\cite{ErGr80} without the stipulation the intervals be distinct, non-overlapping, or adjacent, but Kovac in the comments has provided a simple argument showing that it is easily possible without this restriction, and likely \\cite{ErGr80} just forgot to mention this natural restriction.\nAs an example representing $2$ rather than $1$, Hickerson and Montgomery, in the solution to AMS Monthly problem E2689 proposed by Hahn, found $ 2=\\sum_{i=1}^5 \\sum_{n\\in I_i}\\frac{1}{n} $ where $I_1=[2,7]$, $I_2=[9,10]$, $I_3=[17,18]$, $I_4=[34,35]$, and $I_5=[84,85]$.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The imported strengthened problem, requiring distinct pairwise disjoint and nonadjacent intervals, remains open. The literal Erdős-Graham wording omitted those restrictions and has an easy affirmative solution, so the present formulation is a maintained-tracker reconstruction of likely intent.\n\n**Verified partial progress.**\n\n- The unrestricted literal 1980 formulation can be solved easily by allowing repeated or overlapping intervals.\n- Hickerson and Montgomery gave five separated intervals whose reciprocal sums total 2, illustrating a related construction but not the required total 1 for all sufficiently large k.\n- The tracker discussion agreed on adding distinctness, non-overlap, and non-adjacency to recover a nontrivial problem.\n\n**Full solution or refutation.**\n\nThe exact imported strengthened statement is still open, but it should not be attributed verbatim to the primary source. The formulation mismatch, not an OCR repair, is the main bibliographic caveat.\n\n**What remains.**\n\nProve the strengthened assertion for all sufficiently large k or find an obstruction, and locate contemporaneous evidence that the three extra restrictions were intended.\n\n**Sources checked.**\n\n- Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980). (primary): https://combinatorica.hu/~p_erdos/1979-07.pdf\n Evidence used: The primary source contains the unrestricted wording and does not state the three conditions present in the imported record.\n- Thomas F. Bloom, Erdős Problem #289 and discussion, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/289\n Evidence used: The maintained discussion proves the unrestricted reading easy, records agreement on the strengthened version, and continues to mark that version open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2033, "problem_number": "EP-291", "title": "Erdős Problem #291", "statement": "Let $n\\geq 1$ and define $L_n$ to be the least common multiple of $\\{1,\\ldots,n\\}$ and $a_n$ by $ \\sum_{1\\leq k\\leq n}\\frac{1}{k}=\\frac{a_n}{L_n}. $ Is it true that $(a_n,L_n)=1$ and $(a_n,L_n)>1$ both occur for infinitely many $n$?", "background": "Steinerberger has observed that the answer to the second question is trivially yes: for example, any $n$ which begins with a $2$ in base $3$ has $3\\mid (a_n,L_n)$.\nMore generally, if the leading digit of $n$ in base $p$ is $p-1$ then $p\\mid (a_n,L_n)$. There is in fact a necessary and sufficient condition: a prime $p\\leq n$ divides $(a_n,L_n)$ if and only if $p$ divides the numerator of $1+\\cdots+\\frac{1}{k}$, where $k$ is the leading digit of $n$ in base $p$. This can be seen by writing $ a_n = \\frac{L_n}{1}+\\cdots+\\frac{L_n}{n} $ and observing that the right-hand side is congruent to $1+\\cdots+1/k$ modulo $p$. (The previous claim about $p-1$ follows immediately from Wolstenholme's theorem.)\nThis leads to a heuristic prediction (see for example a preprint of Shiu \\cite{Sh16}) of $\\asymp\\frac{x}{\\log x}$ for the number of $n\\in [1,x]$ such that $(a_n,L_n)=1$. In particular, there should be infinitely many $n$, but the set of such $n$ should have density zero. Unfortunately this heuristic is difficult to turn into a proof.\nWu and Yan \\cite{WuYa22} have proved, conditional on $\\frac{1}{\\log p}$ being linearly independent over $\\mathbb{Q}$ for any finite collection of primes $p$ (itself a consequence of Schanuel's conjecture), that the set of $n$ for which $(a_n,L_n)>1$ has upper density $1$.\nReferences\n\n\n[Sh16] P. Shiu, The denominators of harmonic numbers. arXiv:1607.02863 (2016).\n\n[WuYa22] Wu, Bing-Ling and Yan, Xiao-Hui, On the denominators of harmonic numbers. {IV}. C. R. Math. Acad. Sci. Paris (2022), 53--57.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The non-coprimality half is settled, while infinitude of coprime harmonic numerators remains open.\n\n**Verified partial progress.**\n\n- If the leading base-p digit of n is p-1, then p divides gcd(a_n,L_n), giving infinitely many non-coprime cases.\n- Conditional on Schanuel-type input, Wu--Yan obtain upper density one for coprime cases; heuristic work predicts order x/log x.\n\n**Full solution or refutation.**\n\nOnly one of the two requested infinite families is currently unconditional.\n\n**What remains.**\n\nProve infinitely many n with gcd(a_n,L_n)=1.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #291, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/291\n Evidence used: States open status and gives the leading-digit criterion resolving the second half.\n- Thomas F. Bloom, EP-291 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/291?order=oldest\n Evidence used: Records the conditional density-one result and its limitation.\n\n**Review notes.** The two conjunctive questions are distinguished.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2034, "problem_number": "EP-293", "title": "Erdős Problem #293", "statement": "Let $k\\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to $ 1=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k} $ with $1\\leq n_1<\\cdots 0$, and noted a close connection to [304]. In particular, if $N(b)\\ll \\log\\log b$ as in [304] then it is likely the methods of \\cite{vDTa25b} prove $v(k) \\geq e^{e^{ck}}$ for some $c>0$.\nReferences\n\n\n[BlEr75] Bleicher, M. N. and Erd\\H{o}s, P., The number of distinct subsums of $\\sum \\sb{1}\\spN\\,1/i$. Math. Comp. (1975), 29-42.\n\n[vDTa25b] W. van Doorn and Q. Tang, The smallest denominator not contained in a unit fraction decomposition of 1 with fixed length. arXiv:2512.22083 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The growth of v(k) is unknown despite a 2025 quadratic-exponent lower bound.\n\n**Verified partial progress.**\n\n- Bleicher--Erdős imply v(k) >> k!.\n- van Doorn--Tang proved v(k) >= exp(c k^2).\n- The standard inductive construction gives v(k) <= k c_0^(2^k).\n\n**Full solution or refutation.**\n\nThe lower and upper scales are still far apart.\n\n**What remains.**\n\nDetermine the growth order or materially narrow the gap.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #293, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/293\n Evidence used: Records the bounds and open status.\n- Wouter van Doorn and Quanyu Tang, arXiv:2512.22083 (2025). (primary): https://arxiv.org/abs/2512.22083\n Evidence used: Reported by the authors/tracker for the exp(c k^2) lower bound.\n\n**Review notes.** The reported preprint result is recorded as a bound, not a resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2035, "problem_number": "EP-295", "title": "Erdős Problem #295", "statement": "Let $N\\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\\leq n_1<\\cdots 0$ such that $ -c < k(N)-(e-1)N \\ll \\frac{N}{\\log N}. $ \nReferences\n\n\n[ErSt71b] Erd\\H{o}s, P. and Straus, E. G., Solution to Problem. Amer. Math. Monthly (1971), 302-303.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The divergence of k(N)-(e-1)N is unknown.\n\n**Verified partial progress.**\n\n- Erdős--Straus proved -c < k(N)-(e-1)N << N/log N.\n\n**Full solution or refutation.**\n\nThe known two-sided estimate does not imply the proposed limit.\n\n**What remains.**\n\nShow divergence or construct a bounded subsequence.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #295, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/295\n Evidence used: Lists the Erdős--Straus estimate and open status.\n\n**Review notes.** No claim beyond the cited bounds is made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2036, "problem_number": "EP-301", "title": "Erdős Problem #301", "statement": "Let $f(N)$ be the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there are no solutions to $ \\frac{1}{a}= \\frac{1}{b_1}+\\cdots+\\frac{1}{b_k} $ with distinct $a,b_1,\\ldots,b_k\\in A$?\nEstimate $f(N)$. In particular, is it true that $f(N)=(\\tfrac{1}{2}+o(1))N$?", "background": "The example $A=(N/2,N]\\cap \\mathbb{N}$ shows that $f(N)\\geq N/2$.\nWouter van Doorn has given an elementary argument that proves $ f(N)\\leq (25/28+o(1))N. $ Indeed, consider the sets $S_a=\\{2a,3a,4a,6a,12a\\}\\cap [1,N]$ as $a$ ranges over all integers of the form $8^b9^cd$ with $(d,6)=1$. All such $S_a$ are disjoint and, if $A$ has no solutions to the given equation, then $A$ must omit at least two elements of $S_a$ when $a\\leq N/12$ and at least one element of $S_a$ when $N/12\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The density-1/2 conjecture remains open; the displayed bounds are 1/2 and 25/28.\n\n**Verified partial progress.**\n\n- The interval (N/2,N] gives f(N)>=N/2.\n- van Doorn proved f(N)<=(25/28+o(1))N.\n\n**Full solution or refutation.**\n\nNo matching upper bound for the distinct-denominator equation is known.\n\n**What remains.**\n\nProve the 1/2 asymptotic or disprove it.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #301, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/301\n Evidence used: Records the constructions, van Doorn bound, and open status.\n\n**Review notes.** The non-distinct relaxation is expressly not conflated with this statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2037, "problem_number": "EP-302", "title": "Erdős Problem #302", "statement": "Let $f(N)$ be the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there are no solutions to $ \\frac{1}{a}= \\frac{1}{b}+\\frac{1}{c} $ with distinct $a,b,c\\in A$?\nEstimate $f(N)$. In particular, is $f(N)=(\\tfrac{1}{2}+o(1))N$?", "background": "The colouring version of this is [303], which was solved by Brown and R\"{o}dl \\cite{BrRo91}. One can take either $A$ to be all odd integers in $[1,N]$ or all integers in $[N/2,N]$ to show $f(N)\\geq (1/2+o(1))N$.\nWouter van Doorn has proved (see this note) that $ f(N) \\leq (9/10+o(1))N. $ Stijn Cambie has observed that $ f(N)\\geq (5/8+o(1))N, $ taking $A$ to be all odd integers $\\leq N/4$ and all integers in $[N/2,N]$.\nStijn Cambie has also observed that, if we allow $b=c$, then there is a solution to this equation when $\\lvert A\\rvert \\geq (\\tfrac{2}{3}+o(1))N$, since then there must exist some $n,2n\\in A$.\nSee also [301] and [327].\nReferences\n\n\n[BrRo91] Brown, Tom C. and R\"{o}dl, Voijtech, Monochromatic solutions to equations with unit fractions. Bull. Austral. Math. Soc. (1991), 387-392.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The exact density for avoiding 1/a=1/b+1/c with distinct variables is unknown.\n\n**Verified partial progress.**\n\n- Cambie's construction gives f(N)>=(5/8+o(1))N.\n- van Doorn proved f(N)<=(9/10+o(1))N.\n- The related colouring question EP-303 is solved but is not the density problem.\n\n**Full solution or refutation.**\n\nKnown bounds do not decide the stated 1/2 conjecture.\n\n**What remains.**\n\nDetermine the extremal density for the distinct-variable equation.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #302, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/302\n Evidence used: Records the distinct-variable bounds and separate solved colouring variant.\n\n**Review notes.** EP-303 is cited only as a related, different problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2038, "problem_number": "EP-304", "title": "Erdős Problem #304", "statement": "For integers $1\\leq a\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The proposed O(log log b) bound for N(b) remains open.\n\n**Verified partial progress.**\n\n- Erdős proved log log b << N(b) << log b/log log b.\n- Vose improved the upper bound to O(sqrt(log b)).\n\n**Full solution or refutation.**\n\nNo result reaches the conjectural order.\n\n**What remains.**\n\nEstablish an O(log log b) upper bound or find the correct larger order.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #304, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/304\n Evidence used: Records bounds, the proposed target, and open status.\n\n**Review notes.** Formalization is not treated as proof of the open assertion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2039, "problem_number": "EP-306", "title": "Erdős Problem #306", "statement": "Let $a/b\\in \\mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The squarefree-denominator semiprime Egyptian-fraction problem is open.\n\n**Verified partial progress.**\n\n- Butler--Erdős--Graham proved a relaxed b=1 result with products of three primes.\n- For the exact semiprime restriction, explicit decompositions of 1 are known; the tracker records 47-term examples.\n\n**Full solution or refutation.**\n\nNeither relaxation resolves arbitrary a/b with semiprime denominators.\n\n**What remains.**\n\nProve the general theorem or give an obstruction.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #306, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/306\n Evidence used: Records open status, the three-prime theorem, and semiprime examples.\n\n**Review notes.** Unreviewed comment claims are excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2040, "problem_number": "EP-311", "title": "Erdős Problem #311", "statement": "Let $\\delta(N)$ be the minimal non-zero value of $\\lvert 1-\\sum_{n\\in A}\\frac{1}{n}\\rvert$ as $A$ ranges over all subsets of $\\{1,\\ldots,N\\}$. Is it true that $ \\delta(N)=e^{-(c+o(1))N} $ for some constant $c\\in (0,1)$?", "background": "It is trivial that $ \\delta(N)\\geq \\frac{1}{[1,\\ldots,N]}=e^{-(1+o(1))N}, $ where $[1,\\ldots,N]$ is the least common multiple of $\\{1,\\ldots,N\\}$.\nThe formulation in \\cite{ErGr80} has the additional condition that $A$ contain no $S$ such that $\\sum_{n\\in S}\\frac{1}{n}=1$, but Kovac in the comments has shown that the simpler formulation above is equivalent.\nTang has shown that $ \\delta(N) \\leq \\exp\\left(-c\\frac{N}{(\\log N\\log\\log N)^3}\\right) $ for some constant $c>0$.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The exponential-order asymptotic for delta(N) remains open.\n\n**Verified partial progress.**\n\n- The LCM gives delta(N)>=exp(-(1+o(1))N).\n- Tang proved delta(N)<=exp(-cN/(log N log log N)^3) for some c>0.\n\n**Full solution or refutation.**\n\nThe existing upper bound is subexponential in the desired linear exponent scale.\n\n**What remains.**\n\nProve an upper bound exp(-cN) or determine the actual order.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #311, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/311\n Evidence used: Records the LCM lower bound, Tang upper bound, equivalent formulation, and open status.\n\n**Review notes.** The newer bound is attributed as stated by the maintained tracker.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2041, "problem_number": "EP-312", "title": "Erdős Problem #312", "statement": "Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\\sum_{n\\in A}\\frac{1}{n}>K$ there exists some $S\\subseteq A$ such that $ 1-e^{-cK} < \\sum_{n\\in S}\\frac{1}{n}\\leq 1? $ ", "background": "Erd\\H{o}s and Graham knew this with $e^{-cK}$ replaced by $c/K^2$.\n\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The exponential approximation-to-one conclusion remains open.\n\n**Verified partial progress.**\n\n- Erdős--Graham proved the conclusion with error c/K^2 in place of exp(-cK).\n\n**Full solution or refutation.**\n\nThe polynomial error bound does not establish an exponential error.\n\n**What remains.**\n\nImprove the error scale from polynomial to exponential.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #312, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/312\n Evidence used: Records the known polynomial-error result and open status.\n\n**Review notes.** No finite computation was used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2042, "problem_number": "EP-313", "title": "Erdős Problem #313", "statement": "Are there infinitely many solutions to $ \\frac{1}{p_1}+\\cdots+\\frac{1}{p_k}=1-\\frac{1}{m}, $ where $m\\geq 2$ is an integer and $p_1<\\cdots\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Infinitude of the corresponding primary pseudoperfect numbers remains open.\n\n**Verified partial progress.**\n\n- Explicit examples include m=6 with primes 2,3 and m=42 with primes 2,3,7.\n- OEIS A054377 records further, including newly reported large examples.\n\n**Full solution or refutation.**\n\nKnown examples do not prove there are infinitely many.\n\n**What remains.**\n\nConstruct an infinite family or prove a finiteness theorem.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #313, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/313\n Evidence used: Identifies the problem with primary pseudoperfect numbers and retains open status.\n- FormalConjectures documentation, Erdős Problem 313, checked 2026-08-17. (authoritative_secondary): https://google-deepmind.github.io/formal-conjectures/doc/FormalConjectures/ErdosProblems/313.html\n Evidence used: Defines the exact solution set and lists explicit variants.\n- OEIS A054377, Primary pseudoperfect numbers, checked 2026-08-17. (authoritative_secondary): https://oeis.org/A054377\n Evidence used: Records known terms and references.\n\n**Review notes.** A formal conjecture declaration is not a proof of infinitude.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2043, "problem_number": "EP-317", "title": "Erdős Problem #317", "statement": "Is there some constant $c>0$ such that for every $n\\geq 1$ there exists some $\\delta_k\\in \\{-1,0,1\\}$ for $1\\leq k\\leq n$ with $ 0< \\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert < \\frac{c}{2^n}? $ Is it true that for sufficiently large $n$, for any $\\delta_k\\in \\{-1,0,1\\}$, $ \\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert > \\frac{1}{[1,\\ldots,n]} $ whenever the left-hand side is not zero?", "background": "Inequality is obvious for the second claim, the problem is strict inequality. This fails for small $n$, for example $ \\frac{1}{2}-\\frac{1}{3}-\\frac{1}{4}=-\\frac{1}{12}. $ Arguments of Kovac and van Doorn in the comment section prove a weak version of the first question, with an upper bound of $ 2^{-n\\frac{(\\log\\log\\log n)^{1+o(1)}}{\\log n}}, $ and van Doorn gives a heuristic that suggests this may be the true order of magnitude.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Both signed-reciprocal-sum questions remain open. The current maintained discussion gives a first-part upper bound of 2^(-n (log log log n)^(1+o(1))/log n), much weaker than the requested O(2^(-n)), and no result proving eventual strictness above 1/lcm(1,...,n) was located.\n\n**Verified partial progress.**\n\n- Viewing signed reciprocal sums as differences of harmonic subset sums reduces the first minimum to the least spacing between distinct subset sums.\n- The known lower estimate for the number of distinct harmonic subset sums yields a nonzero signed sum at most 2^(-n (log log log n)^(1+o(1))/log n).\n- Multiplication by lcm(1,...,n) gives the non-strict lower bound in the second question; equality occurs at n=4, so eventual exclusion of equality is the unresolved point.\n- A tracker heuristic based on random spacings suggests the weaker subexponential-in-n exponent scale may be natural, but it is not a theorem.\n\n**Full solution or refutation.**\n\nNo complete solution or disproof was found. The imported background ends with a serialized difficulty/record-boundary fragment; this extraction damage was preserved and not interpreted as mathematical text.\n\n**What remains.**\n\nProve or refute the O(2^(-n)) first assertion, and prove eventual strictness above the reciprocal least common multiple or produce infinitely many equality cases.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #317, last edited 6 January 2026, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/317\n Evidence used: The maintained page labels both questions open and reports no claimed complete or partial solution.\n- Thomas F. Bloom, Erdős Problem #317 discussion, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/317\n Evidence used: The discussion derives the distinct-subset-sum upper bound and states the random-spacing heuristic without claiming a proof of the conjectured scale.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2044, "problem_number": "EP-318", "title": "Erdős Problem #318", "statement": "Let $A\\subseteq \\mathbb{N}$ be an infinite arithmetic progression and $f:A\\to \\{-1,1\\}$ be a non-constant function. Must there exist a finite non-empty $S\\subset A$ such that $ \\sum_{n\\in S}\\frac{f(n)}{n}=0? $ What about if $A$ is an arbitrary set of positive density? What if $A$ is the set of squares excluding $1$?", "background": "Erd\\H{o}s and Straus \\cite{ErSt75} proved this when $A=\\mathbb{N}$. Sattler \\cite{Sa75} proved this when $A$ is the set of odd numbers. For the squares $1$ must be excluded or the result is trivially false, since $ \\sum_{k\\geq 2}\\frac{1}{k^2}<1. $ This is false for some sets $A$ of positive density - indeed, it fails for any set $A$ containing exactly one even number. (Sattler \\cite{Sa82} credits this observation to Erd\\H{o}s, who presumably found this after \\cite{ErGr80}.)\nSattler \\cite{Sa82b} proved the answer to the original question is yes, in that any arithmetic progression has this property.\nThe final question of the set of squares excluding $1$ appears to be open - Sattler announced a proof in \\cite{Sa82} and \\cite{Sa82b}, but this never appeared.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[ErSt75] Erd\\H{o}s, P. and Straus, E. G., Solution to Problem 387. Nieuw Arch. Wisk. (1975), 183.\n\n[Sa75] Sattler, R., Solution to Problem 387. Nieuw Arch. Wisk. (1975), 184-189.\n\n[Sa82] Sattler, R., On {E}rd\\H{o}s property {${\\rm P}\\sb{1}$}\\ for the sequence of\nsquarefree numbers. Nederl. Akad. Wetensch. Indag. Math. (1982), 341--346.\n\n[Sa82b] Sattler, R., On {E}rd\\H{o}s property {${\\rm P}\\sb{1}$}\\ for the arithmetical\nsequence. Nederl. Akad. Wetensch. Indag. Math. (1982), 347--352.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** All three subquestions are resolved: Sattler proved the affirmative result for every infinite arithmetic progression; the arbitrary positive-density generalization is false, for example for any positive-density set containing exactly one even integer; and the maintained 2026 Erdős Problems record says Daniel Larsen proved the affirmative square-excluding-1 case.\n\n**Verified partial progress.**\n\n- Erdős and Straus proved the case A = N, and Sattler proved the odd-integer case in 1975.\n- Sattler's 1982 arithmetical-sequence paper proves the full infinite-arithmetic-progression case.\n- The positive-density extension is obstructed by sets with exactly one even element.\n- Sattler announced the square-excluding-1 case in 1982 but did not publish the proof; Larsen's 2026 result supplies the currently reported completion.\n\n**Full solution or refutation.**\n\nThe composite answer is yes for infinite arithmetic progressions, no for arbitrary positive-density sets, and yes for the squares with 1 excluded. The progression theorem is published by Sattler; the counterexample family settles the density generalization; and the maintained tracker records Larsen's proof of the square case.\n\n**What remains.**\n\nNo mathematical component of the three-part statement remains according to the current maintained record. A publicly archived primary proof or paper for Larsen's square result was not located, so that newest component still needs bibliographic confirmation by an expert.\n\n**Sources checked.**\n\n- R. Sattler, On Erdős property P1 for the arithmetical sequence, Indagationes Mathematicae 44 (1982), 347-352. (primary): https://eurekamag.com/research/100/333/100333378.php\n Evidence used: Bibliographic record for Sattler's published paper proving property P1 for arithmetical sequences, the primary affirmative arithmetic-progression result.\n- R. Sattler, On Erdős property P1 for the sequence of squarefree numbers, Indagationes Mathematicae 44 (1982), 341-346. (primary): https://www.sciencedirect.com/science/article/pii/1385725882900257\n Evidence used: Publisher record for the historical paper that discusses the positive-density obstruction and announced square case.\n- Thomas F. Bloom, Erdős Problem #318, maintained problem record, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/318\n Evidence used: Explicitly distinguishes the three answers, states that Larsen proved the square-excluding-1 case, and marks the overall problem solved; page last edited 2026-04-01.\n- P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 42. (source_collection): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: Original source collection for the three-part question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2045, "problem_number": "EP-319", "title": "Erdős Problem #319", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there is a function $\\delta:A\\to \\{-1,1\\}$ such that $ \\sum_{n\\in A}\\frac{\\delta_n}{n}=0 $ and $ \\sum_{n\\in A'}\\frac{\\delta_n}{n}\neq 0 $ for all non-empty $A'\\subsetneq A$?", "background": "Adenwalla has observed that a lower bound of $ \\lvert A\\rvert\\geq (1-\\tfrac{1}{e}+o(1))N $ follows from the main result of Croot \\cite{Cr01}, which states that there exists some set of integers $B\\subset [(\\frac{1}{e}-o(1))N,N]$ such that $\\sum_{b\\in B}\\frac{1}{b}=1$. Since the sum of $\\frac{1}{m}$ for $m\\in [c_1N,c_2N]$ is asymptotic to $\\log(c_2/c_1)$ we must have $\\lvert B\\rvert \\geq (1-\\tfrac{1}{e}+o(1))N$.\nWe may then let $A=B\\cup\\{1\\}$ and choose $\\delta(n)=-1$ for all $n\\in B$ and $\\delta(1)=1$.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[Cr01] Croot, III, Ernest S., On unit fractions with denominators in short intervals. Acta Arith. (2001), 99-114.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The extremal asymptotic remains open. Croot's short-interval unit-fraction theorem yields a minimal signed zero-sum set of size at least (1-1/e+o(1))N, but no matching upper bound or asymptotic formula was located.\n\n**Verified partial progress.**\n\n- Croot proved that 1 can be represented by distinct unit fractions with all denominators in N", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2046, "problem_number": "EP-320", "title": "Erdős Problem #320", "statement": "Let $S(N)$ count the number of distinct sums of the form $\\sum_{n\\in A}\\frac{1}{n}$ for $A\\subseteq \\{1,\\ldots,N\\}$. Estimate $S(N)$.", "background": "Bleicher and Erd\\H{o}s \\cite{BlEr75} proved the lower bound $ \\log S(N)\\geq \\frac{N}{\\log N}\\left(\\log 2\\prod_{i=3}^k\\log_iN\\right), $ valid for $k\\geq 4$ and $\\log_kN\\geq k$, and also \\cite{BlEr76b} proved the upper bound $ \\log S(N)\\leq \\frac{N}{\\log N}\\left(\\log_r N \\prod_{i=3}^r \\log_iN\\right), $ valid for $r\\geq 1$ and $\\log_{2r}N\\geq 1$. (In these bounds $\\log_in$ denotes the $i$-fold iterated logarithm.)\nBettin, Greni\\'{e}, Molteni, and Sanna \\cite{BGMS25} improved the lower bound to $ \\log S(N) \\geq \\frac{N}{\\log N}\\left(2\\log 2\\left(1-\\frac{3/2}{\\log_kN}\\right)\\prod_{i=3}^k\\log_iN\\right), $ valid for $k\\geq 4$ and $\\log_kN\\geq 3/2$. (In particular this goes to infinity faster than the lower bound of Bleicher and Erd\\H{o}s.)\nSee also [321].\nReferences\n\n\n[BGMS25] S. Bettin, L. Greni\\'{e}, G. Molteni, and C. Sanna, A lower bound for the number of Egyptian fractions. arXiv:2509.10030 (2025).\n\n[BlEr75] Bleicher, M. N. and Erd\\H{o}s, P., The number of distinct subsums of $\\sum \\sb{1}\\spN\\,1/i$. Math. Comp. (1975), 29-42.\n\n[BlEr76b] Bleicher, Michael N. and Erd\\H{o}s, Paul, Denominators of Egyptian fractions. II. Illinois J. Math. (1976), 598-613.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The current maintained #320 record classifies the estimation problem as open, contrary to the stale 'solved' flag embedded in this batch input. Bettin, Grenié, Molteni, and Sanna improved the lower bound in 2025 and computed exact values through N = 154, but neither their paper nor the current tracker gives a matching upper bound or final asymptotic.\n\n**Verified partial progress.**\n\n- Bleicher and Erdős proved an iterated-logarithm lower bound for log S(N) in 1975.\n- Their 1976 Egyptian-fractions paper proved the complementary classical upper bound.\n- Bettin, Grenié, Molteni, and Sanna improved the lower-bound leading factor from 1 to nearly 2 and relaxed the iterated-log hypothesis to log_k N at least 3/2.\n- The 2025 paper gives exact values of S(N) through N = 154.\n\n**Full solution or refutation.**\n\nNo complete solution was verified. The strongest recent result located is a strictly improved lower bound, not a matching asymptotic.\n\n**What remains.**\n\nClose the gap between the improved lower bound and the classical Bleicher-Erdős upper bound, ideally determining the correct asymptotic scale and leading constant for log S(N). Because 'estimate' is broad, any future solved label should state the achieved precision explicitly.\n\n**Sources checked.**\n\n- S. Bettin, L. Grenié, G. Molteni, and C. Sanna, A lower bound for the number of Egyptian fractions, arXiv:2509.10030 (2025). (primary): https://arxiv.org/abs/2509.10030\n Evidence used: The abstract and theorem give the improved lower bound and exact values through N = 154; the paper does not claim a full asymptotic estimate.\n- M. N. Bleicher and P. Erdős, The Number of Distinct Subsums of sum_{i=1}^N 1/i, Mathematics of Computation 29 (1975), 29-42. (primary): https://www.renyi.hu/~p_erdos/1975-45.pdf\n Evidence used: Primary source for the classical distinct-subsum lower bound.\n- M. N. Bleicher and P. Erdős, Denominators of Egyptian Fractions II, Illinois Journal of Mathematics 20 (1976), 598-613. (primary): https://users.renyi.hu/~p_erdos/1976-10.pdf\n Evidence used: Primary source deriving the classical upper bound for S(N) as part of its Egyptian-fraction denominator analysis.\n- Thomas F. Bloom, Erdős Problem #320, maintained problem record, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/320\n Evidence used: Current record classifies #320 as open and lists nonmatching upper and lower bounds, overriding the stale embedded snapshot.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2047, "problem_number": "EP-321", "title": "Erdős Problem #321", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that all sums $\\sum_{n\\in S}\\frac{1}{n}$ are distinct for $S\\subseteq A$?", "background": "Let $R(N)$ be the maximal such size. Results of Bleicher and Erd\\H{o}s from \\cite{BlEr75} and \\cite{BlEr76b} imply that $ \\frac{N}{\\log N}\\prod_{i=3}^k\\log_iN\\leq R(N)\\leq \\frac{1}{\\log 2}\\log_r N\\left(\\frac{N}{\\log N} \\prod_{i=3}^r \\log_iN\\right), $ valid for any $k\\geq 4$ with $\\log_kN\\geq k$ and any $r\\geq 1$ with $\\log_{2r}N\\geq 1$. (In these bounds $\\log_in$ denotes the $i$-fold iterated logarithm.)\nSee also [320].\nReferences\n\n\n[BlEr75] Bleicher, M. N. and Erd\\H{o}s, P., The number of distinct subsums of $\\sum \\sb{1}\\spN\\,1/i$. Math. Comp. (1975), 29-42.\n\n[BlEr76b] Bleicher, Michael N. and Erd\\H{o}s, Paul, Denominators of Egyptian fractions. II. Illinois J. Math. (1976), 598-613.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The current maintained #321 page explicitly classifies the problem as open, contrary to the stale 'solved' flag in this batch input. It retains the nonmatching Bleicher-Erdős iterated-logarithm bounds for R(N), and no later primary source closing the asymptotic gap was located.\n\n**Verified partial progress.**\n\n- Bleicher and Erdős obtained the classical lower bound of N/log N times arbitrarily deep fixed iterated-log products under explicit conditions.\n- Their upper bound has an additional iterated-log factor and remains separated from the lower bound.\n- OEIS A384927 and A391592 record finite extremal data and related formulations.\n\n**Full solution or refutation.**\n\nNo complete solution was verified. The best status evidence remains the classical two-sided bounds with a substantial asymptotic gap.\n\n**What remains.**\n\nMatch the upper and lower bounds for R(N), or determine the correct iterated-logarithm depth and leading behavior. Existing OEIS data are finite-range evidence only and do not resolve the asymptotic question.\n\n**Sources checked.**\n\n- M. N. Bleicher and P. Erdős, The Number of Distinct Subsums of sum_{i=1}^N 1/i, Mathematics of Computation 29 (1975), 29-42. (primary): https://www.renyi.hu/~p_erdos/1975-45.pdf\n Evidence used: Primary source for the distinct reciprocal subset-sum constructions underlying the lower-bound side.\n- M. N. Bleicher and P. Erdős, Denominators of Egyptian Fractions II, Illinois Journal of Mathematics 20 (1976), 598-613. (primary): https://users.renyi.hu/~p_erdos/1976-10.pdf\n Evidence used: Primary source for the complementary subsum analysis used in the stated upper bound.\n- Thomas F. Bloom, Erdős Problem #321, maintained problem record, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/321\n Evidence used: Explicitly labels #321 open and states the current nonmatching bounds; the current page supersedes the stale embedded status snapshot.\n- OEIS Foundation, A391592, accessed 2026-08-17. (bibliographic_index): https://oeis.org/A391592\n Evidence used: Finite-data and cross-reference context only; not used to infer open status.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2048, "problem_number": "EP-322", "title": "Erdős Problem #322", "statement": "Let $k\\geq 3$ and $A\\subset \\mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that $ 1_A^{(k)}(n) >n^c? $ ", "background": "Connected to Waring's problem. The famous Hypothesis $K$ of Hardy and Littlewood was that $1_A^{(k)}(n)\\leq n^{o(1)}$, but this was disproved by Mahler \\cite{Ma36} for $k=3$, who constructed infinitely many $n$ such that $ 1_A^{(3)}(n)\\gg n^{1/12} $ (where $A$ is the set of cubes). Erd\\H{o}s believed Hypothesis $K$ fails for all $k\\geq 4$, but this is unknown. Hardy and Littlewood made the weaker Hypothesis $K^*$ that for all $N$ and $\\epsilon>0$ $ \\sum_{n\\leq N}1_A^{(k)}(n)^2 \\ll_\\epsilon N^{1+\\epsilon}. $ Erd\\H{o}s and Graham remark: 'This is probably true but no doubt very deep. However, it would suffice for most applications.'\nIndependently Erd\\H{o}s \\cite{Er36} and Chowla proved that for all $k\\geq 3$ and infinitely many $n$ $ 1_A^{(k)}(n) \\gg n^{c/\\log\\log n} $ for some constant $c>0$ (depending on $k$). In \\cite{Er65b} Erd\\H{o}s claims an unpublished proof that, if $B$ is the set of $k$th powers of any set of positive density, then $ \\limsup 1_B^{(k)}(n)=\\infty. $ This is discussed in problem D4 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er36] Erd\"{o}s, Paul, On the Representation of an Integer as the Sum of k k-th\nPowers. J. London Math. Soc. (1936), 133-136.\n\n[Er65b] Erd\\H{o}s, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ma36] Mahler, Kurt, Note on Hypothesis K of Hardy and Littlewood. J. London Math. Soc. (1936), 136-138.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The requested fixed-positive-power growth is known for k=3 by Mahler, who proved infinitely many n have at least a constant times n^(1/12) representations. For every k>=4 the fixed-power assertion remains open, and the general maximal order is unknown.\n\n**Verified partial progress.**\n\n- Mahler disproved Hardy-Littlewood Hypothesis K for cubes by proving r_3(n)>>n^(1/12) infinitely often.\n- Erdős, independently of Chowla-Pillai, proved for every k>=3 an infinite-often lower bound n^(c_k/log log n).\n- The all-k lower bound is super-polylogarithmic but n^o(1), so it does not produce the fixed c>0 requested for k>=4.\n- The maintained page and revision history explicitly retain the k>=4 cases as unknown.\n\n**Full solution or refutation.**\n\nThis is a parameter-range partial solution, not a complete determination. The existence question is affirmative for k=3 and unresolved for each k>=4; the opening request for the order of growth remains broader still.\n\n**What remains.**\n\nFor every k>=4, prove or refute a fixed exponent c_k>0 occurring infinitely often, and sharpen the maximal-order behavior of the representation function.\n\n**Sources checked.**\n\n- Kurt Mahler, Note on Hypothesis K of Hardy and Littlewood, Journal of the London Mathematical Society 11 (1936), 136-138, DOI 10.1112/jlms/s1-11.2.136. (primary): https://doi.org/10.1112/jlms/s1-11.2.136\n Evidence used: Mahler's primary paper proves the polynomial lower growth for the cubic representation function.\n- Paul Erdős, On the Representation of an Integer as the Sum of k k-th Powers, Journal of the London Mathematical Society 11 (1936), 133-136, DOI 10.1112/jlms/s1-11.2.133. (primary): https://doi.org/10.1112/jlms/s1-11.2.133\n Evidence used: The paper proves the n^(c_k/log log n) infinite-often lower bound for every k.\n- Thomas F. Bloom, Erdős Problem #322 revision history, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/322\n Evidence used: The current statement and history say Hypothesis K is known false for k=3 and the fixed-power failure for k>=4 is unknown.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2049, "problem_number": "EP-323", "title": "Erdős Problem #323", "statement": "Let $1\\leq m\\leq k$ and $f_{k,m}(x)$ denote the number of integers $\\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that $ f_{k,k}(x) \\gg_\\epsilon x^{1-\\epsilon} $ for all $\\epsilon>0$? Is it true that if $m0$.\nFor $k>2$ it is not known if $f_{k,k}(x)=o(x)$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Both proposed lower bounds remain open in the intended higher-power range k>2. Landau's sum-of-two-squares theorem settles the k=2 boundary case, while the maintained record states that for k>2 it is not even known whether f_(k,k)(x)=o(x).\n\n**Verified partial progress.**\n\n- Landau proved f_(2,2)(x) is asymptotic to a positive constant times x/sqrt(log x).\n- This square case implies the x^(1-epsilon) lower bound for k=2 but does not address any k>2.\n- No uniform collision bound yielding f_(k,m)(x)>>x^(m/k) for general m2 as the unresolved substance, so the isolated k=2 case was not treated as a meaningful parameter-range resolution of this record.\n\n**What remains.**\n\nProve f_(k,k)(x)>>x^(1-epsilon) for every k>2 and f_(k,m)(x)>>x^(m/k) for each m2 gap.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2050, "problem_number": "EP-324", "title": "Erdős Problem #324", "statement": "Does there exist a polynomial $f(x)\\in\\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Existence of an integer-coefficient polynomial whose values form a Sidon set remains open. Degree at most three is impossible, the monomial x^4 fails, and Ruzsa proved that n^5+floor(c n^4) is Sidon for a suitable real c after a finite prefix, providing a close but nonpolynomial construction.\n\n**Verified partial progress.**\n\n- Ruzsa constructed a real c and n0 such that {n^5+floor(c n^4):n>=n0} is a Sidon set.\n- Dubickas and Novikas proved that no integer polynomial of degree at most three generates a Sidon sequence, even after deleting a finite prefix.\n- The specific quartic monomial x^4 is classically known not to work, but this does not exclude all quartic polynomials.\n- The Lander-Parkin-Selfridge conjecture would imply that x^n works for n>=5, but that implication is conditional.\n\n**Full solution or refutation.**\n\nNo integer polynomial construction or impossibility theorem is known. The known negative degree range and almost-polynomial degree-five construction materially narrow the problem and justify partial-progress classification.\n\n**What remains.**\n\nConstruct a qualifying integer polynomial or rule all such polynomials out; in particular settle x^5 and general quartic polynomials.\n\n**Sources checked.**\n\n- Imre Z. Ruzsa, An almost polynomial Sidon sequence, Studia Scientiarum Mathematicarum Hungarica 38 (2001), 367-375, DOI 10.1556/SScMath.38.2001.1-4.27. (primary): https://doi.org/10.1556/SScMath.38.2001.1-4.27\n Evidence used: Ruzsa proves the degree-five almost-polynomial Sidon construction.\n- Artūras Dubickas and Aivaras Novikas, No cubic integer polynomial generates a Sidon sequence, Mathematische Nachrichten 294 (2021), 1859-1865, DOI 10.1002/mana.202000334. (primary): https://doi.org/10.1002/mana.202000334\n Evidence used: The paper proves impossibility for every integer polynomial of degree at most three.\n- Thomas F. Bloom, Erdős Problem #324, last edited 11 April 2026, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/324\n Evidence used: The maintained page labels integer-polynomial existence open and records the current positive and negative partial results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2051, "problem_number": "EP-325", "title": "Erdős Problem #325", "statement": "Let $k\\geq 3$ and $f_{k,3}(x)$ denote the number of integers $\\leq x$ which are the sum of three nonnegative $k$th powers. Is it true that $ f_{k,3}(x) \\gg x^{3/k} $ or even $\\gg_\\epsilon x^{3/k-\\epsilon}$?", "background": "Mahler and Erd\\H{o}s \\cite{ErMa38} proved that $f_{k,2}(x) \\gg x^{2/k}$. For $k=3$ the best known is due to Wooley \\cite{Wo15}, $ f_{3,3}(x) \\gg x^{0.917\\cdots}. $ This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[ErMa38] Erd\\H{o}s, P\\'{a}l and Mahler, Kurt, On the number of integers which can be represented by a binary form. Doc. Math. (2019), 475-481.\n\n[Wo15] Wooley, Trevor D., Sums of three cubes, II. Acta Arith. (2015), 73-100.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Browning and Heath-Brown proved that for every k>=33, asymptotically a positive constant times x^(3/k) integers up to x are sums of three positive kth powers, establishing the stronger requested order in that range. The exponents 3<=k<=32 remain open.\n\n**Verified partial progress.**\n\n- Browning-Heath-Brown's collision estimate yields an asymptotic (c_k/6)x^(3/k) represented integers for every k>=33.\n- Positive summands are a subset of the nonnegative-summand convention in the imported statement, so the theorem applies directly.\n- For k=3, Wooley proved f_(3,3)(x)>>x^0.91709477..., still short of x^(1-epsilon).\n- A March 2026 tracker comment identified the published 2004 theorem; it had not yet been incorporated into the page's main remarks at check time.\n\n**Full solution or refutation.**\n\nThis is a rigorous published solution for the whole high-degree range k>=33, not a solution for all k. The input's ErMa38 citation points to the 2019 reprint of the original 1938 paper and is bibliographically confusing but mathematically harmless.\n\n**What remains.**\n\nProve either requested lower bound for each exponent 3<=k<=32, especially k=3, or show a parameter for which collision-optimal order fails.\n\n**Sources checked.**\n\n- T. D. Browning and D. R. Heath-Brown, Equal sums of three powers, Inventiones Mathematicae 157 (2004), 553-573, DOI 10.1007/s00222-004-0360-9. (primary): https://ora.ox.ac.uk/objects/uuid:1fd05b3e-cd34-4b65-a95e-590af2e523dc\n Evidence used: The paper's corollary gives asymptotically (c_d/6)x^(3/d) integers represented as sums of three positive dth powers for every d>=33.\n- Trevor D. Wooley, Sums of three cubes, II, Acta Arithmetica 170 (2015), 73-100, arXiv:1502.01944. (primary): https://arxiv.org/abs/1502.01944\n Evidence used: Wooley proves the current recorded x^0.917... lower bound for integers represented by three cubes.\n- Thomas F. Bloom, Erdős Problem #325 discussion, March 2026, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/325?embed=1\n Evidence used: The discussion identifies the k>=33 corollary and its published Inventiones reference, while the main problem remains open for lower k.\n- Pál Erdős and Kurt Mahler, On the Number of Integers Which Can Be Represented By a Binary Form, Journal of the London Mathematical Society 13 (1938), 134-139, DOI 10.1112/jlms/s1-13.2.134. (primary): https://doi.org/10.1112/jlms/s1-13.2.134\n Evidence used: The original publication clarifies that the 2019 citation in the input is a reprint of a 1938 theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2052, "problem_number": "EP-326", "title": "Erdős Problem #326", "statement": "Let $A\\subset \\mathbb{N}$ be an additive basis of order $2$. Must there exist $B=\\{b_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported statement and the current tracker are different but related formulations. A May-June 2026 unpublished claim constructs a minimal order-two basis with A(x)=C sqrt(x)+O(1), reportedly backed by a large Lean formalisation; if correct, it directly refutes the imported universal subbasis statement, but the tracker remains open and independent verification was not established.\n\n**Verified partial progress.**\n\n- Cassels constructed a nonminimal basis with a_k=c k^2+O(k), answering the older unrestricted existence question.\n- The original Erdős-Graham source refines the issue to whether a minimal basis can have a_k/k^2 tending to a nonzero constant.\n- A 2026 forum manuscript claims a minimal asymptotic basis with A(x)=C sqrt(x)+O(1), which inverts to a_k/k^2 tending to C^(-2)>0.\n- The author later reported a roughly 15,000-line Lean formalisation, but the tracker had not accepted the result and a reviewer reported being unable to load the linked formalisation at that time.\n- If the claim is correct, minimality means no proper subset remains a basis, so this A is a counterexample to the exact imported statement.\n\n**Full solution or refutation.**\n\nNo publication-backed or independently verified disproof was established in this review. The status is uncertain rather than disproved because the decisive construction is an unpublished comment-level claim and because the imported formulation has uncertain provenance relative to the live minimal-basis question.\n\n**What remains.**\n\nObtain expert or stable formal verification of the claimed construction, confirm that the formal theorem matches the mathematical claim, and resolve the provenance and equivalence scope of the imported subbasis formulation.\n\n**Sources checked.**\n\n- Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory, 1980, p. 47. (source_collection): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: The original source states the minimal-basis positive-limit formulation and explains Cassels's nonminimal construction.\n- Thomas F. Bloom, Erdős Problem #326, current version accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/326\n Evidence used: The live page asks the minimal-basis positive-limit question and still labels it open.\n- Thomas F. Bloom, Erdős Problem #326 discussion, May-June 2026, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/326?embed=1\n Evidence used: The discussion records the claimed construction, later claimed Lean formalisation, minor standard-check revisions, and the absence at that point of an independent readable verification.\n- J. W. S. Cassels, Über Basen der natürlichen Zahlenreihe, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 21 (1957), 247-257. (bibliographic_index): https://www.sciencedirect.com/science/article/pii/0022314X80900487\n Evidence used: The bibliographic references identify Cassels's classical thin-basis construction; the linked later paper lists the exact citation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2053, "problem_number": "EP-327", "title": "Erdős Problem #327", "statement": "Suppose $A\\subseteq \\{1,\\ldots,N\\}$ is such that if $a,b\\in A$ and $a\neq b$ then $a+b\nmid ab$. Can $A$ be 'substantially more' than the odd numbers?\nWhat if $a,b\\in A$ with $a\neq b$ implies $a+b\nmid 2ab$? Must $\\lvert A\\rvert=o(N)$?", "background": "The connection to unit fractions comes from the observation that $\\frac{1}{a}+\\frac{1}{b}$ is a unit fraction if and only if $a+b\\mid ab$.\nWouter van Doorn has given an elementary argument that proves that if $A\\subseteq \\{1,\\ldots,N\\}$ has $\\lvert A\\rvert \\geq (25/28+o(1))N$ then $A$ must contain $a\neq b$ with $a+b\\mid ab$ (see the discussion in [301]).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Neither divisibility-avoidance question is resolved. The first is also formulation-ambiguous because 'substantially more' is undefined. A May 2026 unpublished note claims upper-density constants 0.7769 for a+b not dividing ab and 0.7630 for a+b not dividing 2ab, but the second target o(N) remains far stronger.\n\n**Verified partial progress.**\n\n- Van Doorn's elementary argument gives the first variant an upper bound (25/28+o(1))N.\n- A May 2026 smooth-rough decomposition note claims improved bounds (0.7769+o(1))N for the first variant and (0.7630+o(1))N for the second.\n- The note's finite optimization certificate was linked after a standard check; the result remains unpublished and comment-level.\n- A positive constant upper bound for the second variant does not imply the requested o(N) conclusion.\n\n**Full solution or refutation.**\n\nNo full solution was found. The exact input has lost backslashes in every not-equal and not-divides token, producing line breaks followed by 'eq' and 'mid'; these defects were preserved. The first question also needs an explicit quantitative interpretation before it can receive a definitive theorem-level answer.\n\n**What remains.**\n\nDefine the threshold meant by 'substantially more' and determine the first extremal density. For the second, prove density zero or construct a positive-density example, and independently verify the recent finite-certificate bounds.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #327, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/327\n Evidence used: The maintained page preserves the intact divisibility conditions, labels the questions open, and records the 25/28 upper bound.\n- Leon (leon2k2k2k), Erdős 327, unpublished note, May 2026. (primary): https://leon2k2k2k.github.io/erdos327.pdf\n Evidence used: The note claims the 0.7769 and 0.7630 asymptotic upper constants via a smooth-rough reduction and finite certificate.\n- Thomas F. Bloom, Erdős Problem #327 discussion, May 2026, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/327?order=newest\n Evidence used: The discussion records the claim, standard-check response, and later certificate-script link without elevating it to an accepted solution.\n- Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory, 1980. (source_collection): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: The original collection provides the historical wording, including the qualitative phrase 'substantially more'.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2054, "problem_number": "EP-329", "title": "Erdős Problem #329", "statement": "Suppose $A\\subseteq \\mathbb{N}$ is a Sidon set. How large can $ \\limsup_{N\\to \\infty}\\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}} $ be?", "background": "Erd\\H{o}s proved that $1/2$ is possible and Kr\"{u}ckeberg \\cite{Kr61} proved $1/\\sqrt{2}$ is possible. Erd\\H{o}s and Tur\\'{a}n \\cite{ErTu41} have proved this $\\limsup$ is always $\\leq 1$.\nThe fact that $1$ is possible would follow if any finite Sidon set is a subset of a perfect difference set (see [44] and [707]).\nThis question can also be asked for $B_2[g]$ sequences (i.e. where the number of solutions to $n=a_1+a_2$ with $a_1\\leq a_2$ is at most $g$ for all $n$, so that a $B_2[1]$ set is a Sidon set). Kolountzakis \\cite{Ko96} constructed a $B_2[2]$ sequence where the $\\limsup$ is $1$, and for larger $g$ constructions were provided by Cilleruelo and Trujillo \\cite{CiTr01}.\nReferences\n\n\n[CiTr01] Cilleruelo, Javier and Trujillo, Carlos, Infinite {$B_2[g]$} sequences. Israel J. Math. (2001), 263--267.\n\n[ErTu41] Erd\\H{o}s, P. and Tur\\'{a}n, P., On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc. (1941), 212-215.\n\n[Ko96] Kolountzakis, Mihail N., On the additive complements of the primes and sets of similar\ngrowth. Acta Arith. (1996), 1--8.\n\n[Kr61] Kr\"{u}ckeberg, Fritz, $B\\sb{2}$-Folgen und verwandte Zahlenfolgen. J. Reine Angew. Math. (1961), 53-60.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The largest possible limsup constant for an infinite Sidon set remains between 1/sqrt(2) and 1. The upper endpoint 1 is conjectured attainable, but no Sidon construction exceeding Krückeberg's 1/sqrt(2) constant was located.\n\n**Verified partial progress.**\n\n- Erdős constructed an infinite Sidon set with limsup constant at least 1/2.\n- Krückeberg improved the lower constant to 1/sqrt(2).\n- The finite Sidon upper bound of Erdős and Turán gives the universal limsup upper constant 1.\n- O'Bryant's June 2026 preprint constructs gamma-Golomb rulers with constant at least sqrt(gamma)/sqrt(2); at gamma=1 this recovers, but does not improve, the Sidon lower endpoint.\n- Constructions attaining 1 for B_2[2] allow two representations and therefore do not solve the B_2[1] problem.\n\n**Full solution or refutation.**\n\nNo full solution was found. The maintained record, a current 2026 preprint, and modern survey material all retain the classical 1/sqrt(2) to 1 gap for the exact Sidon case.\n\n**What remains.**\n\nConstruct an infinite Sidon set with limsup constant above 1/sqrt(2), prove a universal upper constant below 1, or attain the conjectured value 1.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #329, last edited 6 April 2026, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/329\n Evidence used: The maintained page records Krückeberg's 1/sqrt(2) construction, the Erdős-Turán upper bound 1, and the conjecture that 1 is attainable.\n- Kevin O'Bryant, The Thickness of Infinite Sidon Sets, arXiv:2606.28651 (2026). (primary): https://arxiv.org/abs/2606.28651\n Evidence used: The preprint's generalized gamma-Golomb construction specializes at gamma=1 to the known 1/sqrt(2) Sidon constant, confirming no improvement for the exact problem.\n- Javier Cilleruelo and Carlos Trujillo, Infinite B_2[g] sequences, Israel Journal of Mathematics 126 (2001), 263-267. (primary): https://matematicas.uam.es/~franciscojavier.cilleruelo/Papers/infinit2%20B2%5Bg%5D%20sequences.pdf\n Evidence used: The paper summarizes the classical Sidon constants and distinguishes the larger-g representation setting from B_2[1].\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2055, "problem_number": "EP-330", "title": "Erdős Problem #330", "statement": "Does there exist a minimal basis with positive density, say $A\\subset\\mathbb{N}$, such that for any $n\\in A$ the (upper) density of integers which cannot be represented without using $n$ is positive?", "background": "Asked by Erd\\H{o}s and Nathanson.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The positive-upper-density formulation was solved affirmatively in April-May 2026. David Turturean's construction, developed with GPT-5.5 Pro, has a public proof-faithful Lean development proving the existence of an asymptotic basis of order two with positive upper density and with a positive-upper-density private set for every element.\n\n**Verified partial progress.**\n\n- Earlier dense minimal-basis literature established related minimality and density constructions without the full positive-upper-density private-set conclusion for every element.\n- The new construction assigns personal primes and recurring protected residue blocks, using CRT-compatible stage scheduling so later elements preserve each private set.\n- The public formalization breaks the proof into residue-block, prime-supply, CRT-gadget, local-reservoir, stage-construction, scheduler, and upper-density modules.\n- The maintained forum records independent checks by Nat Sothanaphan and links the exact formal target.\n\n**Full solution or refutation.**\n\nThere exists A contained in the natural numbers such that A is an asymptotic basis of order two, A has positive upper density, and for every a in A the set (A+A) minus ((A without a)+(A without a)) has positive upper density. This implies minimality and exactly answers the explicit upper-density formulation.\n\n**What remains.**\n\nThe verified theorem concerns upper asymptotic density and exact two-fold sums. Stronger readings using positive lower density or natural density are not established by these sources. A conventional refereed publication and stable archived informal proof would improve bibliographic confidence.\n\n**Sources checked.**\n\n- David Turturean, A Minimal Asymptotic Basis of Positive Upper Density with Every Element Density-Essential (April 2026), proof specification in the Erdős 330 formalisation brief. (primary): https://github.com/AllenGrahamHart/FormalConjectures-Bench/blob/main/formalizations/erdos330/formalisation_brief.md\n Evidence used: States the exact upper-density theorem, documents the supplied informal proof, and lays out the complete finite-gadget and priority construction used by the formalization.\n- Allen Graham Hart, Erdős Problem 330 proof-faithful Lean development, FormalConjectures-Bench, accessed 2026-08-17. (primary): https://github.com/AllenGrahamHart/FormalConjectures-Bench/tree/main/formalizations/erdos330\n Evidence used: Public modular Lean source aligned to the exact theorem about an order-two basis, positive upper density, and positive-upper-density private sets.\n- Allen Graham Hart, formal-conjectures-gold dataset, erdosproblems-330-upper-density row, accessed 2026-08-17. (primary): https://huggingface.co/datasets/AllenGrahamHart/formal-conjectures-gold\n Evidence used: Records this exact task as a bundled gold solution with theorem name, Lean toolchain, and pinned source commits.\n- Thomas F. Bloom, Erdős Problem #330 discussion/status record, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/forum/thread/330\n Evidence used: Marks the problem proved in Lean, links the exact target and formalization, explains the upper-density interpretation, and records independent human checking.\n- P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 50. (source_collection): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: Historical source used to audit the ambiguous density convention.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2056, "problem_number": "EP-332", "title": "Erdős Problem #332", "statement": "Let $A\\subseteq \\mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1-a_2$ with $a_1,a_2\\in A$. What conditions on $A$ are sufficient to ensure $D(A)$ has bounded gaps?", "background": "Prikry, Tijdeman, Stewart, and others (see the survey articles \\cite{St78} and \\cite{Ti79}) have shown that a sufficient condition is that $A$ has positive density.\nOne can also ask what conditions are sufficient for $D(A)$ to have positive density, or for $\\sum_{d\\in D(A)}\\frac{1}{d}=\\infty$, or even just $D(A)\neq\\emptyset$.\nReferences\n\n\n[St78] Stewart, Cam L., On difference sets of sets of integers. S\\'{e}minaire Delange-Pisot-Poitou, 19e ann\\'{e}e:\n1977/78, Th\\'{e}orie des nombres, Fasc. 1 (1978), Exp. No. 5, 8.\n\n[Ti79] Tijdeman, R., Distance sets of sequences of integers. Proceedings, Bicentennial Congress Wiskundig\nGenootschap (Vrije Univ., Amsterdam, 1978), Part\nII (1979), 405-415.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Positive upper density is a rigorous sufficient condition for the infinitely recurring difference set D(A) to have bounded gaps, by Stewart-Tijdeman and independently Prikry. The intended program of finding weaker or optimal conditions and best gap bounds remains open.\n\n**Verified partial progress.**\n\n- Stewart and Tijdeman proved that positive upper density forces the recurring-difference set to be syndetic.\n- Their finite-family form gives positive lower density for the intersection of recurring-difference sets of finitely many positive-upper-density sets.\n- Prikry independently obtained the bounded-gap theorem through a Hindman-type result.\n- The original source asks for weaker hypotheses, optimal gap bounds, and variants requiring only positive density, divergent reciprocal sum, or nonemptiness of D(A).\n\n**Full solution or refutation.**\n\nThe question is deliberately open-ended: one sufficient condition is known, but no necessary-and-sufficient or optimally weak condition is specified or known. The imported background has a missing backslash in its final not-equal-empty token; the exact statement itself is intact.\n\n**What remains.**\n\nFind substantially weaker hypotheses than positive upper density that still force bounded gaps, quantify the optimal gap bound, or formulate and prove a necessary-and-sufficient characterization.\n\n**Sources checked.**\n\n- C. L. Stewart and R. Tijdeman, On infinite-difference sets, Canadian Journal of Mathematics 31 (1979), 897-910. (primary): https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2E3484CB7233691BD4555062AB2802D7/S0008414X00014760a.pdf/on_infinitedifference_sets.pdf\n Evidence used: Theorem 2 proves that the difference set arising from a positive-upper-density sequence cannot contain arbitrarily long gaps; the paper notes Prikry's independent result.\n- Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory, 1980, p. 50. (source_collection): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: The original source states the positive-density theorem and frames the unresolved task as weakening hypotheses and optimizing conclusions.\n- Thomas F. Bloom, Erdős Problem #332, last edited 28 October 2025, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/332\n Evidence used: The maintained page labels the broader sufficient-condition program open and records positive density as the principal known condition.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2057, "problem_number": "EP-334", "title": "Erdős Problem #334", "statement": "Find the best function $f(n)$ such that every $n$ can be written as $n=a+b$ where both $a,b$ are $f(n)$-smooth (that is, are not divisible by any prime $p>f(n)$.)", "background": "Erd\\H{o}s originally asked if even $f(n)\\leq n^{1/3}$ is true. This is known, and the best bound is due to Balog \\cite{Ba89} who proved that $ f(n) \\ll_\\epsilon n^{\\frac{4}{9\\sqrt{e}}+\\epsilon} $ for all $\\epsilon>0$. (Note $\\frac{4}{9\\sqrt{e}}=0.2695\\ldots$.)\nIt is likely that $f(n)\\leq n^{o(1)}$, or even $f(n)\\leq e^{O(\\sqrt{\\log n})}$.\nSee also Problem 59 on Green's open problems list.\nReferences\n\n\n[Ba89] Balog, A., On additive representation of integers. Acta Math. Hungar. (1989), 297-301.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The optimal smoothness scale for every binary decomposition is unknown.\n\n**Verified partial progress.**\n\n- Balog proved f(n) <<_epsilon n^(4/(9 sqrt(e))+epsilon).\n\n**Full solution or refutation.**\n\nNo subpolynomial binary bound was located.\n\n**What remains.**\n\nDetermine the optimal order, in particular whether f(n)=n^o(1).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #334, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/334\n Evidence used: Records Balog's bound and open status.\n\n**Review notes.** Tracker comment claims were not used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2058, "problem_number": "EP-335", "title": "Erdős Problem #335", "statement": "Let $d(A)$ denote the density of $A\\subseteq \\mathbb{N}$. Characterise those $A,B\\subseteq \\mathbb{N}$ with positive density such that $ d(A+B)=d(A)+d(B). $ ", "background": "One way this can happen is if there exists $\\theta>0$ such that $ A=\\{ n>0 : \\{ n\\theta\\} \\in X_A\\}\\textrm{ and }B=\\{ n>0 : \\{n\\theta\\} \\in X_B\\} $ where $\\{x\\}$ denotes the fractional part of $x$ and $X_A,X_B\\subseteq \\mathbb{R}/\\mathbb{Z}$ are such that $\\mu(X_A+X_B)=\\mu(X_A)+\\mu(X_B)$. Are all possible $A$ and $B$ generated in a similar way (using other groups)?\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ackelsberg--Richter give a conditional structural characterization, but the unrestricted problem is open.\n\n**Verified partial progress.**\n\n- Under an every-residue-class hypothesis, Ackelsberg--Richter (2026) classify the equality structure.\n\n**Full solution or refutation.**\n\nThe extra hypothesis is essential to the available result.\n\n**What remains.**\n\nCharacterize all positive-density equality pairs without it.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #335, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/335\n Evidence used: Reports the 2026 conditional resolution and continued open status.\n\n**Review notes.** The stated hypothesis is retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2059, "problem_number": "EP-336", "title": "Erdős Problem #336", "statement": "For $r\\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\\subseteq \\mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers from $A$) and exact order $k$ (so every large integer is the sum of exactly $k$ integers from $A$).\nFind the value of $ \\lim_r \\frac{h(r)}{r^2}. $ ", "background": "A simple example of the order of a basis differing from the exact order is given by $A=\\cup_{k\\geq 0}(2^{2k},2^{2k+1}]$, which has order $2$ but exact order $3$.\nErd\\H{o}s and Graham \\cite{ErGr80b} have shown that a basis $A$ has an exact order if and only if $a_2-a_1,a_3-a_2,a_4-a_3,\\ldots$ are coprime. They also proved that $ \\frac{1}{4}\\leq \\lim_r \\frac{h(r)}{r^2}\\leq \\frac{5}{4}. $ The best bounds known for the limit are $ \\frac{1}{3}\\leq \\lim_r \\frac{h(r)}{r^2}\\leq \\frac{1}{2}, $ the lower bound originally due to Grekos \\cite{Gr88} and the upper bound to Nash \\cite{Na93}. Improved bounds in the lower order terms were given by Plagne \\cite{Pl04}.\nErd\\H{o}s and Graham \\cite{ErGr80b} showed $h(2)=4$. Nash \\cite{Na93} showed $h(3)=7$. The value of $h(4)$ is unknown, but it is known \\cite{Pl04} that $10\\leq h(4)\\leq 11$.\nReferences\n\n\n[ErGr80b] Erd\\H{o}s, P. and Graham, R. L., On bases with an exact order. Acta Arith. (1980), 201-207.\n\n[Gr88] Grekos, Georges, Sur l'ordre d'une base additive. ([1988?]), Exp. No. 31, 13.\n\n[Na93] Nash, John C. M., Some applications of a theorem of {M}. {K}neser. J. Number Theory (1993), 1--8.\n\n[Pl04] Plagne, Alain, \\`A{} propos de la fonction {$X$} d'{E}rd\\H{o}s et {G}raham. Ann. Inst. Fourier (Grenoble) (2004), 1717--1767.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The limiting exact-order quotient is not known; current bounds are 1/3 to 1/2.\n\n**Verified partial progress.**\n\n- Grekos and Nash give 1/3 <= lim h(r)/r^2 <= 1/2.\n- h(2)=4, h(3)=7, and 10<=h(4)<=11.\n\n**Full solution or refutation.**\n\nNo exact limit was located.\n\n**What remains.**\n\nDetermine the limit and h(4).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #336, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/336\n Evidence used: Records bounds and open status.\n\n**Review notes.** No new calculation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2060, "problem_number": "EP-338", "title": "Erdős Problem #338", "statement": "The restricted order of a basis is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands from $A$. What are necessary and sufficient conditions that this exists? Can it be bounded (when it exists) in terms of the order of the basis? What are necessary and sufficient conditions that this is equal to the order of the basis?", "background": "Bateman has observed that for $h\\geq 3$ there is a basis of order $h$ with no restricted order, taking $ A=\\{1\\}\\cup \\{x>0 : h\\mid x\\}. $ Kelly \\cite{Ke57} has shown that any basis of order $2$ has restricted order at most $4$ and conjectured it always has restricted order at most $3$ (which he proved under the additional assumption that the basis has positive lower density). Kelly's conjecture was disproved by Hennecart \\cite{He05}, who constructed a basis of order $2$ with restricted order $4$.\nThe set of squares has order $4$ and restricted order $5$ (see \\cite{Pa33}) and the set of triangular numbers has order $3$ and restricted order $3$ (see \\cite{Sc54}).\nIs it true that if $A\\backslash F$ is a basis for all finite sets $F$ then $A$ must have a restricted order? What if they are all bases of the same order?\nHegyv\\'{a}ri, Hennecart, and Plagne \\cite{HHP07} have shown that for all $k\\geq2$ there exists a basis of order $k$ which has restricted order at least $ 2^{k-2}+k-1. $ \nReferences\n\n\n[HHP07] Hegyv\\'ari, Norbert and Hennecart, Fran\\c cois and Plagne,\nAlain, Answer to a question by {B}urr and {E}rd\\H{o}s on restricted\naddition, and related results. Combin. Probab. Comput. (2007), 747--756.\n\n[He05] Hennecart, Fran\\c cois, On the restricted order of asymptotic bases of order two. Ramanujan J. (2005), 123--130.\n\n[Ke57] Kelly, John B., Restricted bases. Amer. J. Math. (1957), 258-264.\n\n[Pa33] Pall, Gordon, On Sums of Squares. Amer. Math. Monthly (1933), 10-18.\n\n[Sc54] Schinzel, A., Sur la d\\'{e}composition des nombres naturels en sommes de nombres triangulaires distincts. Bull. Acad. Polon. Sci. Cl. III. (1954), 409-410.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Restricted order is not characterized or bounded generally in terms of basis order.\n\n**Verified partial progress.**\n\n- Bateman constructed order-h bases without restricted order for h>=3.\n- Order-two restricted order can equal four.\n\n**Full solution or refutation.**\n\nKnown examples settle neither requested characterization nor bound.\n\n**What remains.**\n\nGive necessary/sufficient conditions and effective bounds.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #338, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/338\n Evidence used: Records counterexamples, special cases, and open questions.\n\n**Review notes.** Unrelated search snippets were ignored.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2061, "problem_number": "EP-340", "title": "Erdős Problem #340", "statement": "Let $A=\\{1,2,4,8,13,21,31,45,66,81,97,\\ldots\\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a+b=c+d$). What is the order of growth of $A$? Is it true that $ \\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg N^{1/2-\\epsilon} $ for all $\\epsilon>0$ and large $N$?", "background": "This sequence is sometimes called the Mian-Chowla sequence. It is trivial that this sequence grows at least like $\\gg N^{1/3}$.\nErd\\H{o}s and Graham \\cite{ErGr80} also asked about the difference set $A-A$, whether this has positive density, and whether this contains $22$. It does contain $22$, since $a_{15}-a_{14}=204-182=22$. The smallest integer which is unknown to be in $A-A$ is $33$ (see A080200). It may be true that all or almost all integers are in $A-A$.\nThis sequence is at OEIS A005282.\nSee also [156].\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The growth of the greedy Sidon/Mian--Chowla sequence remains open.\n\n**Verified partial progress.**\n\n- The tracker records the trivial lower growth scale N^(1/3).\n\n**Full solution or refutation.**\n\nNo near-square-root lower bound is known.\n\n**What remains.**\n\nProve or refute the N^(1/2-epsilon) lower bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #340, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/340\n Evidence used: Records the open growth question.\n\n**Review notes.** No OEIS inference is treated as proof.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2062, "problem_number": "EP-341", "title": "Erdős Problem #341", "statement": "Let $A=\\{a_1<\\cdots\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Eventual periodicity is unknown for the general Dickson greedy extension.\n\n**Verified partial progress.**\n\n- A small seed can require thousands of terms before observed periodicity.\n\n**Full solution or refutation.**\n\nNo general periodicity theorem was located.\n\n**What remains.**\n\nProve periodicity or give a counterexample seed.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #341, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/341\n Evidence used: Lists the question as open.\n\n**Review notes.** Observed behavior is not a theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2063, "problem_number": "EP-342", "title": "Erdős Problem #342", "statement": "With $a_1=1$ and $a_2=2$ let $a_{n+1}$ for $n\\geq 2$ be the least integer $>a_n$ which can be expressed uniquely as $a_i+a_j$ for $i\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Ulam-sequence density, twin occurrence, and periodicity questions remain open.\n\n**Verified partial progress.**\n\n- The defining sequence is OEIS A002858.\n\n**Full solution or refutation.**\n\nNo rigorous resolution was located.\n\n**What remains.**\n\nResolve any listed asymptotic question.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #342, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/342\n Evidence used: Lists all assertions as open.\n\n**Review notes.** Unreviewed AI-assisted comments were excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2064, "problem_number": "EP-345", "title": "Erdős Problem #345", "statement": "Let $A\\subseteq \\mathbb{N}$ be a complete sequence, and define the threshold of completeness $T(A)$ to be the least integer $m$ such that all $n\\geq m$ are in $ P(A) = \\left\\{\\sum_{n\\in B}n : B\\subseteq A\\textrm{ finite }\\right\\} $ (the existence of $T(A)$ is guaranteed by completeness).\nIs it true that there are infinitely many $k$ such that $T(n^k)>T(n^{k+1})$?", "background": "Erd\\H{o}s and Graham \\cite{ErGr80} remark that very little is known about $T(A)$ in general. It is known that $ T(n)=1, T(n^2)=128, T(n^3)=12758, $ $ T(n^4)=5134240,\\textrm{ and }T(n^5)=67898771. $ Erd\\H{o}s and Graham remark that a good candidate for the $n$ in the question are $k=2^t$ for large $t$, perhaps even $t=3$, because of the highly restricted values of $n^{2^t}$ modulo $2^{t+1}$.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No infinitude result for threshold drops was located.\n\n**Verified partial progress.**\n\n- The tracker lists small threshold values through powers five.\n\n**Full solution or refutation.**\n\nSmall values do not settle the infinite question.\n\n**What remains.**\n\nProve infinitely many drops or eventual monotonicity.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #345, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/345\n Evidence used: Records small values and open status.\n\n**Review notes.** No computation was run.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2065, "problem_number": "EP-346", "title": "Erdős Problem #346", "statement": "Let $A=\\{1\\leq a_1< a_2<\\cdots\\}$ be a set of integers such that\n{UL}\n{LI} $A\\backslash B$ is complete for any finite subset $B$ and {/LI}\n{LI} $A\\backslash B$ is not complete for any infinite subset $B$.{/LI}\n{/UL}\n(Here 'complete' means all sufficiently large integers can be written as a sum of distinct members of the sequence.)\nIs it true that if $a_{n+1}/a_n \\geq 1+\\epsilon$ for some $\\epsilon>0$ and all $n$ then $ \\lim_n \\frac{a_{n+1}}{a_n}=\\frac{1+\\sqrt{5}}{2}? $ ", "background": "Graham \\cite{Gr64d} has shown that the sequence $a_n=F_n-(-1)^{n}$, where $F_n$ is the $n$th Fibonacci number, has these properties. Erd\\H{o}s and Graham \\cite{ErGr80} remark that it is easy to see that if $a_{n+1}/a_n>\\frac{1+\\sqrt{5}}{2}$ then the second property is automatically satisfied, and that it is not hard to construct very irregular sequences satisfying both properties.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[Gr64d] Graham, R. L., A property of Fibonacci numbers. Fibonacci Quart. (1964), 1-10.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The golden-ratio conclusion for minimally complete sequences remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo resolution or comparable general theorem was located.\n\n**What remains.**\n\nProve the asserted limit or construct a counterexample.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #346, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/346\n Evidence used: Lists the question as open.\n\n**Review notes.** Source list markup is preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2066, "problem_number": "EP-348", "title": "Erdős Problem #348", "statement": "For what values of $0\\leq m\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The deletion-resilience classification for complete sequences remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo characterization of pairs (m,n) was located.\n\n**What remains.**\n\nDetermine feasible deletion-resilience pairs.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #348, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/348\n Evidence used: Lists the question as open.\n\n**Review notes.** Source list markup is preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2067, "problem_number": "EP-349", "title": "Erdős Problem #349", "statement": "For what values of $t,\\alpha \\in (0,\\infty)$ is the sequence $\\lfloor t\\alpha^n\\rfloor$ complete (that is, all sufficiently large integers are the sum of distinct integers of the form $\\lfloor t\\alpha^n\\rfloor$)?", "background": "Even in the range $t\\in (0,1)$ and $\\alpha\\in (1,2)$ the behaviour is surprisingly complex. For example, Graham \\cite{Gr64e} has shown that for any $k$ there exists some $t_k\\in (0,1)$ such that the set of $\\alpha$ such that the sequence is complete consists of at least $k$ disjoint line segments. It seems likely that the sequence is complete for all $t>0$ and all $1<\\alpha < \\frac{1+\\sqrt{5}}{2}$. Proving this seems very difficult, since we do not even know whether $\\lfloor (3/2)^n\\rfloor$ is odd or even infinitely often.\nReferences\n\n\n[Gr64e] Graham, R. L., On a conjecture of Erd\\H{o}s in additive number theory. Acta Arith. (1964/65), 63-70.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Completeness for floor-geometric sequences remains uncharacterized.\n\n**Verified partial progress.**\n\n- Graham showed arbitrarily complicated completeness regions in a basic parameter range.\n\n**Full solution or refutation.**\n\nNo full characterization was located.\n\n**What remains.**\n\nDetermine the parameter pairs giving completeness.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #349, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/349\n Evidence used: Lists the problem as open and records the complex parameter behavior.\n\n**Review notes.** No numerical behavior is promoted to proof.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2068, "problem_number": "EP-351", "title": "Erdős Problem #351", "statement": "Let $p(x)\\in \\mathbb{Q}[x]$. Is it true that $ A=\\{ p(n)+1/n : n\\in \\mathbb{N}\\} $ is strongly complete, in the sense that, for any finite set $B$, $ \\left\\{\\sum_{n\\in X}n : X\\subseteq A\\backslash B\\textrm{ finite }\\right\\} $ contains all sufficiently large integers?", "background": "Graham \\cite{Gr63} proved this is true when $p(n)=n$. Erd\\H{o}s and Graham also ask which rational functions $r(x)\\in\\mathbb{Z}(x)$ force $\\{ r(n) : n\\in\\mathbb{N}\\}$ to be complete?\nGraham \\cite{Gr64f} gave a complete characterisation of which polynomials $r\\in \\mathbb{R}[x]$ are such that $\\{ r(n) : n\\in \\mathbb{N}\\}$ is complete.\nIn the comments van Doorn has noted that a positive solution for $p(n)=n^2$ follows from \\cite{Gr63} together with result of Alekseyev \\cite{Al19} mentioned in [283].\nReferences\n\n\n[Al19] Alekseyev, Max A., On partitions into squares of distinct integers whose\nreciprocals sum to 1. (2019), 213--221.\n\n[Gr63] Graham, R. L., A theorem on partitions. J. Austral. Math. Soc. (1963), 435-441.\n\n[Gr64f] Graham, R. L., Complete sequences of polynomial values. Duke Math. J. (1964), 275-285.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The literal dataset statement is false because it omits the positive-leading-coefficient hypothesis. For p(x) = -x, every value p(n)+1/n is nonpositive, so every finite subset sum is nonpositive and the set cannot contain all sufficiently large positive integers. The maintained tracker added the missing hypothesis on 2026-05-10; that corrected version is separately proved affirmatively in Lean via EP-283.\n\n**Verified partial progress.**\n\n- Graham proved the special case p(n) = n.\n- The p(n) = n^2 case follows from Graham's theorem together with Alekseyev's distinct-square reciprocal partition result.\n- AlphaProof found the elementary counterexample p(x) = -x to the unqualified statement in December 2025.\n- The tracker repaired the statement by requiring positive leading coefficient; Barreto observed that the stronger solved EP-283 polynomial Egyptian-sums theorem implies this corrected EP-351, and the implication is covered by public formal artifacts.\n\n**Full solution or refutation.**\n\nFor p(x) = -x, the sequence is 0, -3/2, -8/3, and so on: all terms are at most zero. Removing any finite subset cannot make its finite subset sums contain any sufficiently large positive integer. Hence the exact dataset statement is disproved. With the added positive-leading-coefficient hypothesis, the current intended theorem is affirmative via the solved polynomial Egyptian-sums theorem EP-283.\n\n**What remains.**\n\nThe literal statement needs no further mathematical resolution. The curation decision remains whether to retain it as refuted or create a provenance-preserving revised version with positive leading coefficient. The corrected formulation is solved, but it must not be silently substituted for this stored record.\n\n**Sources checked.**\n\n- Erdős Problems, EP-351 discussion containing the AlphaProof counterexample, posted 12 December 2025, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/forum/discuss/351\n Evidence used: Explicitly records p(x) = -x as a disproof of the former formulation and identifies the missing leading-coefficient hypothesis.\n- Thomas F. Bloom, revision history for Erdős Problem #351, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/351\n Evidence used: Shows the positive-leading-coefficient hypothesis being added on 2026-05-10 and the deduction from EP-283 incorporated into the corrected statement.\n- Thomas F. Bloom, current Erdős Problem #351, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/351\n Evidence used: Current maintained record states the repaired positive-leading-coefficient version and marks it proved in Lean via the positive solution to EP-283.\n- GPT-5.5 Pro, Liam Price, Shashi Ammanamanchi et al., Erdős Problems 283 + 351 formalization artifacts, accessed 2026-08-17. (primary): https://github.com/Shashi456/erdos-formalizations/tree/main/Erdos/P283\n Evidence used: Primary formal proof artifact for the stronger polynomial Egyptian-sums theorem that implies the corrected positive-leading-coefficient version of EP-351.\n- P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 58. (source_collection): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: Historical source for the question; used to preserve rather than silently repair the batch statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2069, "problem_number": "EP-352", "title": "Erdős Problem #352", "statement": "Is there some $c>0$ such that every measurable $A\\subseteq \\mathbb{R}^2$ of measure $\\geq c$ contains the vertices of a triangle of area 1?", "background": "Erd\\H{o}s (unpublished) proved that this is true if $A$ has infinite measure, or if $A$ is an unbounded set of positive measure (stating in \\cite{Er78d} and \\cite{Er83d} it 'follows easily from the Lebesgue density theorem').\nIn \\cite{Er78d} and \\cite{Er83d} he speculated that perhaps $C=4\\pi/\\sqrt{27}\\approx 2.418$ works, which would be the best possible, as witnessed by a circle of radius $<2\\cdot 3^{-3/4}$.\nFurther evidence for this is given by a result of Freiling and Mauldin \\cite{Ma02}, who proved that if $A$ has outer measure $>4\\pi/\\sqrt{27}$ then $A$ contains the vertices of a triangle with area $>1$. This also proves the same threshold for the original problem under the assumption that $A$ is a compact convex set.\nMauldin also discusses this problem in \\cite{Ma13}, in which he notes that it suffices to prove this under the assumption that $A$ is the union of the interiors of $n<\\infty$ many compact convex sets. Freiling and Mauldin (see \\cite{Ma13}) have proved this conjecture if $1\\leq n\\leq 3$.\nReferences\n\n\n[Er78d] Erd\\H{o}s, P., Set-theoretic, measure-theoretic, combinatorial, and\nnumber-theoretic problems concerning point sets in Euclidean\nspace. Real Anal. Exchange (1978/79), 113-138.\n\n[Er83d] Erd\\H{o}s, Paul, Some combinatorial, geometric and set theoretic problems in measure theory. Measure Theory, Oberwolfach 1983: Proceedings of the Conference held at Oberwolfach, June 26-July 2, 1983 (1984), 321-327.\n\n[Ma02] Mauldin, R. D., Some problems in set theory, analysis and geometry. (2002), 493--506.\n\n[Ma13] Mauldin, R. Daniel, Some problems and ideas of {E}rd\\H{o}s in analysis and\ngeometry. (2013), 365--376.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No current resolution of the measurable planar triangle-area question was located.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe available search found no full solution or disproof.\n\n**What remains.**\n\nProve a universal positive measure threshold or produce counterexamples.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #352, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/352\n Evidence used: Maintained status page identifies the problem as open.\n\n**Review notes.** Confidence is lowered because no directly indexed page excerpt was returned.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2070, "problem_number": "EP-354", "title": "Erdős Problem #354", "statement": "Let $\\alpha,\\beta\\in \\mathbb{R}_{>0}$ such that $\\alpha/\\beta$ is irrational. Is the multiset $ \\{ \\lfloor \\alpha\\rfloor,\\lfloor 2\\alpha\\rfloor,\\lfloor 4\\alpha\\rfloor,\\ldots\\}\\cup \\{ \\lfloor \\beta\\rfloor,\\lfloor 2\\beta\\rfloor,\\lfloor 4\\beta\\rfloor,\\ldots\\} $ complete? That is, can all sufficiently large natural numbers $n$ be written as $ n=\\sum_{s\\in S}\\lfloor 2^s\\alpha\\rfloor+\\sum_{t\\in T}\\lfloor 2^t\\beta\\rfloor $ for some finite $S,T\\subset \\mathbb{N}$?\nWhat if $2$ is replaced by some $\\gamma\\in(1,2)$?", "background": "This question was first mentioned by Graham \\cite{Gr71}.\nHegyv\\'{a}ri \\cite{He89} proved that this holds if $\\alpha=m/2^n$ is a dyadic rational and $\\beta$ is not. He later \\cite{He91} proved that, for any fixed $\\alpha>0$, the set of $\\beta$ for which this holds either has measure $0$ or infinite measure. In \\cite{He94} he proved that the set of $(\\alpha,\\beta)$ for which the corresponding set of sums does not contain an infinite arithmetic progression has cardinality continuum.\nHegyv\\'{a}ri \\cite{He89} proved that the sequence is not complete if $\\alpha\\geq 2$ and $\\beta =2^k\\alpha$ for some $k\\geq 0$. Jiang and Ma \\cite{JiMa24} and Fang and He \\cite{FaHe25} prove that the sequence is not complete if $1<\\alpha<2$ and $\\beta=2^k\\alpha$ for some sufficiently large $k$.\nIt is likely (and Hegyv\\'{a}ri conjectures) that the assumption $\\alpha/\\beta$ irrational can be weakened to $\\alpha/\\beta \neq 2^k$ and either $\\alpha$ or $\\beta$ not a dyadic rational.\nIn the comments van Doorn proves the sequence is complete if $\\alpha < 2<\\beta<3$, and also proves that if either $\\alpha$ or $\\beta$ is not a dyadic rational then the corresponding sequence with ceiling functions replacing the floor functions is complete.\nReferences\n\n\n[FaHe25] Fang, J.-H. and He, J.-Y., On a problem of {E}rd\\H{o}s and {G}raham. Acta Math. Hungar. (2025), 532--542.\n\n[Gr71] Graham, R. L., On sums of integers taken from a fixed sequence. (1971), 22--40.\n\n[He89] Hegyv\\'ari, N., Some remarks on a problem of {E}rd\\H{o}s and {G}raham. Acta Math. Hungar. (1989), 149--154.\n\n[He91] Hegyv\\'ari, N., On complete sequences. Ann. Univ. Sci. Budapest. E\"otv\"os Sect. Math. (1991), 7--10.\n\n[He94] Hegyv\\'ari, Norbert, On sumset of certain sets. Publ. Math. Debrecen (1994), 115--122.\n\n[JiMa24] Jiang, Xing-Wang and Ma, Wu-Xia, A conjecture of {H}egyv\\'ari. Int. J. Number Theory (2024), 915--933.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The irrational-ratio floor-sequence completeness question remains open.\n\n**Verified partial progress.**\n\n- Hegyvári proved dyadic-rational special cases and several non-completeness results outside the hypothesis.\n- A further complete region and a ceiling-function variant are documented.\n\n**Full solution or refutation.**\n\nNo theorem covers all irrational alpha/beta ratios.\n\n**What remains.**\n\nResolve completeness under the stated irrational-ratio condition.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #354, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/354\n Evidence used: Records special cases, counterexamples, and open status.\n\n**Review notes.** A misformalization is not treated as a disproof of the informal statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2071, "problem_number": "EP-357", "title": "Erdős Problem #357", "statement": "Let $1\\leq a_1<\\cdots \n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether f(n)=o(n) remains unknown.\n\n**Verified partial progress.**\n\n- Known constructions give f(n)>=(2+o(1))sqrt(n).\n- Beker's 2023 consecutive-sum bound gives a nontrivial linear upper bound, approximately 0.8727n.\n\n**Full solution or refutation.**\n\nNo sublinear upper bound was located.\n\n**What remains.**\n\nProve or refute f(n)=o(n).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #357, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/357\n Evidence used: Records the lower bound and open status.\n- A. Beker, On a problem of Erdős and Graham about consecutive sums in strictly increasing sequences, arXiv:2311.10087 (2023). (primary): https://arxiv.org/abs/2311.10087\n Evidence used: Used in the documented upper-bound deduction.\n\n**Review notes.** The deduction is attributed and not used as a solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2072, "problem_number": "EP-358", "title": "Erdős Problem #358", "statement": "Let $A=\\{a_1<\\cdots\\}$ be an infinite sequence of integers. Let $f(n)$ count the number of solutions to $ n=\\sum_{u\\leq i\\leq v}a_i. $ Is there such an $A$ for which $f(n)\\to \\infty$ as $n\\to \\infty$? Or even where $f(n)\\geq 2$ for all large $n$?", "background": "When $a_n=n$ the function $f(n)$ counts the number of odd divisors of $n$.\nIn modern language, this asks for the existence of a convex set $A$ such that $1_A\\circ 1_A(n)\\to \\infty$ as $n\\to \\infty$.\nErd\\H{o}s and Moser \\cite{Mo63} considered the case when $A$ is the set of primes, and conjectured that the $\\limsup$ of the number of such representations in this case is infinite. They could not even prove that the upper density of the set of integers representable in this form is positive.\nIn \\cite{ErGr80} they further asked whether $f(n)\\geq 1$ for all large $n$ is possible, but Egami observed the answer to this is trivially yes, taking $a_n=n$. Perhaps they intended to restrict $f(n)$ to only count those representatives as the sum of at least two consecutive terms. (It is a classical fact that $n$ can be expressed as a sum of at least two consecutive positive integers if and only if $n\neq 2^k$.)\nIn \\cite{Er77c} Erd\\H{o}s writes 'This problem can perhaps be rightly criticized as being artificial and in the backwater of Mathematics but it seems very strange and attractive to me'.\nWeisenberg observes that the finite analogue of this problem, asking how many integers up to some $x$ can be written as the sum of consecutive elements, is very similar to [356].\nThis is reported in problem C2 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Mo63] Moser, L., Notes on number theory. III. On the sum of consecutive primes. Canad. Math. Bull. (1963), 159-161.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Tao proved in February 2026 that there is an increasing set A of positive integers for which every sufficiently large n has f(n) >> log n representations as a sum of consecutive elements of A. This resolves both questions in the record and is optimal in order of magnitude.\n\n**Verified partial progress.**\n\n- For A equal to the positive integers, consecutive-sum representations are controlled by odd divisors and can be sparse on exceptional n.\n- Random-set approaches produced logarithmically many representations for most n, but allocating representations to all exceptional n without collisions was the main obstruction.\n- Tao's red/blue block construction adds primality and coprimality conditions that prevent the relevant interval collisions.\n\n**Full solution or refutation.**\n\nTao constructs a random red portion giving many independent candidate representations for almost every integer in each dyadic block, then deterministically adds blue elements to repair exceptional integers. Carefully chosen prime representation lengths and coprime red lengths ensure the repair intervals do not collide. Borel--Cantelli yields one set with f(n) >> log n for every sufficiently large n.\n\n**What remains.**\n\nThe displayed existence questions are settled. Optimizing the implied constant and developing simpler or explicit deterministic constructions remain natural refinements; the older at-least-two-terms wording issue concerns a separate trivial one-representation variant.\n\n**Sources checked.**\n\n- Terence Tao, A Set That Represents All Large Integers Multiple Times by Consecutive Elements, manuscript dated 24 February 2026. (primary): https://terrytao.wordpress.com/wp-content/uploads/2026/02/erdos-358-2.pdf\n Evidence used: Theorem 1.1 constructs A with r_A(n) >> log n for every sufficiently large n; the introduction also gives the matching-order universal average upper bound.\n- Thomas F. Bloom, revision history for Erdős Problem #358, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/history/358\n Evidence used: The maintained history records the 1 April 2026 status change and identifies Tao's logarithmic-representation construction.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2073, "problem_number": "EP-359", "title": "Erdős Problem #359", "statement": "Let $a_10$?", "background": "A problem of MacMahon, studied by Andrews \\cite{An75}. When $n=1$ this sequence begins $ 1,2,4,5,8,10,14,15,\\ldots. $ This sequence is A002048 in the OEIS. Andrews conjectures $ a_k\\sim \\frac{k\\log k}{\\log\\log k}. $ Porubsky \\cite{Po77} proved that, for any $\\epsilon>0$, there are infinitely many $k$ such that $ a_k < (\\log k)^\\epsilon \\frac{k\\log k}{\\log\\log k}, $ and also that if $A(x)$ counts the number of $a_i\\leq x$ then $ \\limsup \\frac{A(x)}{\\pi(x)}\\geq \\frac{1}{\\log 2} $ where $\\pi(x)$ counts the number of primes $\\leq x$.\nSee also [839].\nReferences\n\n\n[An75] Andrews, George E., Research Problems: Mac Mahon's Prime Numbers of Measurement. Amer. Math. Monthly (1975), 922-923.\n\n[Po77] Porubsk\\'y, \\v S., On {M}ac{M}ahon's segmented numbers and related sequences. Nieuw Arch. Wisk. (3) (1977), 403--408.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The density/growth of MacMahon's sequence remains open.\n\n**Verified partial progress.**\n\n- Andrews conjectured a_k~k log k/log log k.\n- Porubský proved a weaker infinite-subsequence upper estimate.\n\n**Full solution or refutation.**\n\nThe conjectural asymptotic is not established.\n\n**What remains.**\n\nProve growth bounds valid for all large k.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #359, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/359\n Evidence used: Records the conjecture, Porubský result, and open status.\n\n**Review notes.** No computational inference.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2074, "problem_number": "EP-361", "title": "Erdős Problem #361", "statement": "Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\\subseteq \\{1,\\ldots,\\lfloor cn\\rfloor\\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No verified general extremal theorem for the subset-sum-avoidance question was located.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe problem remains open in the maintained tracker.\n\n**What remains.**\n\nDetermine the extremal size and its dependence on n.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #361, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/361\n Evidence used: Lists current open status.\n\n**Review notes.** Source background has trailing serialization corruption; it was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2075, "problem_number": "EP-365", "title": "Erdős Problem #365", "statement": "Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations? In other words, must either $n$ or $n+1$ be a square?\nIs the number of such $n\\leq x$ bounded by $(\\log x)^{O(1)}$?", "background": "Erd\\H{o}s originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation $x^2=2^3y^2+1$.\nThe list of $n$ such that $n$ and $n+1$ are both powerful is A060355 in the OEIS.\nThe answer to the first question is no: Golomb \\cite{Go70} observed that both $12167=23^3$ and $12168=2^33^213^2$ are powerful. Walker \\cite{Wa76} proved that the equation $ 7^3x^2=3^3y^2+1 $ has infinitely many solutions, giving infinitely many counterexamples.\nSee also [364].\nThis is discussed in problem B16 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Go70] Golomb, S. W., Powerful numbers. Amer. Math. Monthly (1970), 848-855.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Wa76] Walker, David T., Consecutive integer pairs of powerful numbers and related\nDiophantine equations. Fibonacci Quart. (1976), 111-116.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Pell/square assertion is false, but the counting problem remains open.\n\n**Verified partial progress.**\n\n- Golomb's example has two consecutive powerful nonsquares.\n- Walker proved infinitely many counterexamples via 7^3 x^2=3^3 y^2+1.\n- A displayed current counting bound is O(x^(2/5)).\n\n**Full solution or refutation.**\n\nThe first question is resolved negatively; the second is not.\n\n**What remains.**\n\nImprove to a polylogarithmic count or determine the true order.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #365, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/365\n Evidence used: Records Golomb/Walker counterexamples and open status for the count.\n\n**Review notes.** The two questions are separated.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2076, "problem_number": "EP-367", "title": "Erdős Problem #367", "statement": "Let $B_2(n)$ be the 2-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that, for every fixed $k\\geq 1$, $ \\prod_{n\\leq m0$, $ \\limsup \\frac{\\prod_{n\\leq m\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The stronger O_k(n^2) alternative fails for k>=3, while the n^(2+o(1)) assertion remains open.\n\n**Verified partial progress.**\n\n- For k<=2 the O(n^2) bound is trivial.\n- Pell constructions give product B_2(m) >> n^2 log n infinitely often for k=3.\n\n**Full solution or refutation.**\n\nOne proposed strengthening is refuted, not the main bound.\n\n**What remains.**\n\nProve or disprove the n^(2+o(1)) bound for fixed k>=3.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #367, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/367\n Evidence used: Records the k=3 obstruction and continuing main question.\n\n**Review notes.** The tracker treats only the stronger alternative as disproved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2077, "problem_number": "EP-368", "title": "Erdős Problem #368", "statement": "How large is the largest prime factor of $n(n+1)$?", "background": "Let $F(n)$ be the prime in question. P\\'{o}lya \\cite{Po18} proved that $F(n)\\to \\infty$ as $n\\to\\infty$. Mahler \\cite{Ma35} showed that $F(n)\\gg \\log\\log n$. Schinzel \\cite{Sc67b} observed that for infinitely many $n$ we have $F(n)\\leq n^{O(1/\\log\\log\\log n)}$.\nThe truth is probably $F(n)\\gg (\\log n)^2$ for all $n$. Erd\\H{o}s \\cite{Er76d} conjectured that, for every $\\epsilon>0$, there are infinitely many $n$ such that $F(n) <(\\log n)^{2+\\epsilon}$.\nPasten \\cite{Pa24b} has proved that $ F(n) \\gg \\frac{(\\log\\log n)^2}{\\log\\log\\log n}. $ The largest prime factors of $n(n+1)$ are listed as A074399 in the OEIS.\nReferences\n\n\n[Er76d] Erd\\H{o}s, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n\n[Ma35] Mahler, Kurt, \"{U}ber den gr\"{o}ssten Primteiler spezieller Polynome zweiten Grades. Archiv f\"{u}r math. og naturvid (1935).\n\n[Pa24b] Pasten, Hector, The largest prime factor of {$n^2+1$} and improvements on\nsubexponential {$ABC$}. Invent. Math. (2024), 373--385.\n\n[Po18] P\\'{o}lya, Georg, Zur arithmetischen {U}ntersuchung der {P}olynome. Math. Z. (1918), 143--148.\n\n[Sc67b] Schinzel, A., On two theorems of Gelfond and some of their applications. Acta Arith. (1967/68), 177-236.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The extremal behavior of the largest prime factor of n(n+1) remains open.\n\n**Verified partial progress.**\n\n- Størmer proved the largest factor tends to infinity.\n- Kotov and later work give quantitative lower bounds, including a 2024 improvement noted by OEIS.\n\n**Full solution or refutation.**\n\nNo sharp upper/lower characterization was located.\n\n**What remains.**\n\nDetermine the conjectured extremal scale.\n\n**Sources checked.**\n\n- OEIS A074399, largest prime divisor of n(n+1), checked 2026-08-17. (authoritative_secondary): https://oeis.org/A074399\n Evidence used: Lists classical results, references, and current quantitative notes.\n\n**Review notes.** No tracker status page was directly retrievable in the search pass.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2078, "problem_number": "EP-369", "title": "Erdős Problem #369", "statement": "Let $\\epsilon>0$ and $k\\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\\{1,\\ldots,n\\}$ all of which are $n^\\epsilon$-smooth?", "background": "Erd\\H{o}s and Graham state that this is open even for $k=2$ and 'the answer should be affirmative but the problem seems very hard'.\nUnfortunately the problem is trivially true as written (simply taking $\\{1,\\ldots,k\\}$ and $n>k^{1/\\epsilon}$). There are (at least) two possible variants which are non-trivial, and it is not clear which Erd\\H{o}s and Graham meant. Let $P$ be the sequence of $k$ consecutive integers sought for. The potential strengthenings which make this non-trivial are:\n{UL}\n{LI}Each $m\\in P$ must be $m^\\epsilon$-smooth. If this is the problem then the answer is yes, which follows from a result of Balog and Wooley \\cite{BaWo98}: for any $\\epsilon>0$ and $k\\geq 2$ there exist infinitely many $m$ such that $m+1,\\ldots,m+k$ are all $m^\\epsilon$-smooth.{/LI}\n{LI}Each $m\\in P$ must be in $[n/2,n]$ (say). In this case a positive answer also follows from the result of Balog and Wooley \\cite{BaWo98} for infinitely many $n$, but the case of all sufficiently large $n$ is open.{/LI}\n{/UL}\nSee also [370] and [928].\nReferences\n\n\n[BaWo98] Balog, Antal and Wooley, Trevor D., On strings of consecutive integers with no large prime factors. J. Austral. Math. Soc. Ser. A (1998), 266-276.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The exact displayed statement is trivially affirmative: for n > k^(1/epsilon), the fixed block 1,...,k is n^epsilon-smooth. A Lean proof verifies this literal reading. The old claim that the problem was hard therefore signals a formulation defect; stronger plausible readings have separate affirmative results.\n\n**Verified partial progress.**\n\n- Eggleton and Selfridge had already constructed infinitely many strings of five highly smooth consecutive integers, making the historical open-status wording puzzling.\n- Balog and Wooley proved that for every epsilon > 0 there are infinitely many strings of consecutive integers of unbounded length near m whose prime factors are at most m^epsilon.\n- The maintained record derives an all-large-n near-endpoint version, with a fixed k-block in [n-n^c,n] for some c<1, from smooth polynomial-value results of Bober, Fretwell, Martin, and Wooley.\n\n**Full solution or refutation.**\n\nFor the statement as written, choose the block {1,...,k}. Once n > k^(1/epsilon), every prime divisor of every member of the block is at most k < n^epsilon. The nontrivial repairs require the block to grow with or lie near n, and those are distinct statements handled by deeper smooth-number constructions.\n\n**What remains.**\n\nNothing remains for the literal statement. The historical intended formulation is uncertain and should not be silently substituted. Quantitative optimization of how close to n one can always place such a block remains a meaningful successor problem.\n\n**Sources checked.**\n\n- Antal Balog and Trevor D. Wooley, On strings of consecutive integers with no large prime factors, Journal of the Australian Mathematical Society, Series A 64 (1998), 266--276. (primary): https://www.math.purdue.edu/~twooley/publ/1998%20consec.pdf\n Evidence used: Theorem 1 gives infinitely many arbitrarily long strings of n^epsilon-smooth consecutive integers of size about n, settling one natural strengthened reading.\n- Jonathan Bober, Dan Fretwell, Greg Martin, and Trevor D. Wooley, Smooth values of polynomials, arXiv:1710.01970. (primary): https://arxiv.org/abs/1710.01970\n Evidence used: The smooth-polynomial-value theorem is the primary input used by the maintained discussion to obtain an all-large-n block near the upper endpoint.\n- Thomas F. Bloom, Erdős Problem #369, maintained problem record, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/369\n Evidence used: The record explicitly identifies the literal proof, explains the ambiguous likely intended variants, and records the stronger all-large-n endpoint result.\n- Woett, ErdosProblem369.lean, Lean 4 proof linked by the Erdős Problems discussion. (formal_verification): https://github.com/Woett/Lean-files/blob/main/ErdosProblem369.lean\n Evidence used: Public Lean artifact checking the exact literal eventual-existence statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2079, "problem_number": "EP-371", "title": "Erdős Problem #371", "statement": "Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n) (0.2017-o(1))x, $ and the same lower bound for the complement.\nIn \\cite{Er79e} Erd\\H{o}s also asks whether, for every $\\alpha$, the density of the set of $n$ where $ P(n+1)>P(n)n^\\alpha $ exists.\nTer\"{a}v\"{a}inen \\cite{Te18} has proved that the logarithmic density of the set of $n$ for which $P(n)P(n)n^\\alpha$ exists and is equal to $ \\int_{[0,1]^2}1_{y\\geq x+\\alpha}u(x)u(y)\\mathrm{d}x\\mathrm{d}y $ where $u(x)=x^{-1}\\rho(x^{-1}-1)$ and $\\rho$ is the Dickman function. Wang \\cite{Wa21} has proved the same value holds for the asymptotic density (and in particular provided an affirmative answer to the original question) conditional on the Elliott-Halberstam conjecture for friable integers.\nThe sequence of such $n$ is A070089 in the OEIS.\nSee also [372] and [928].\nReferences\n\n\n[Er79e] Erd\\H{o}s, Paul, Some unconventional problems in number theory. Ast\\'{e}risque (1979), 73-82.\n\n[ErPo78] Erd\\H{o}s, Paul and Pomerance, Carl, On the largest prime factors of {$n$} and {$n+1$}. Aequationes Math. (1978), 311-321.\n\n[LuWa25] L\"u, Xiaodong and Wang, Zhiwei, On the largest prime factors of consecutive integers. Monatsh. Math. (2025), 403--418.\n\n[TaTe19] Tao, Terence and Ter\"{a}v\"{a}inen, Joni, The structure of correlations of multiplicative functions at\nalmost all scales, with applications to the {C}howla and\n{E}lliott conjectures. Algebra Number Theory (2019), 2103--2150.\n\n[Te18] Ter\"{a}v\"{a}inen, Joni, On binary correlations of multiplicative functions. Forum Math. Sigma (2018), Paper No. e10, 41.\n\n[Wa21] Wang, Zhiwei, Three conjectures on {$P^+(n)$} and {$P^+(n+1)$} hold under\nthe {E}lliott-{H}alberstam conjecture for friable integers. J. Number Theory (2021), 1--11.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Asymptotic density 1/2 is open unconditionally but known at logarithmic density and conditionally.\n\n**Verified partial progress.**\n\n- Teräväinen proved logarithmic density 1/2 and Tao--Teräväinen proved 1/2 at almost all scales.\n- Wang proved asymptotic density 1/2 conditional on Elliott--Halberstam for friable integers.\n- Lü--Wang give unconditional positive lower density over 0.2017 for each ordering.\n\n**Full solution or refutation.**\n\nThe exact natural-density assertion remains open.\n\n**What remains.**\n\nEstablish asymptotic density 1/2 unconditionally.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #371, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/371\n Evidence used: Records all listed advances and current open status.\n\n**Review notes.** Logarithmic and natural density are distinguished.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2080, "problem_number": "EP-373", "title": "Erdős Problem #373", "statement": "Show that the equation $ n! = a_1!a_2!\\cdots a_k!, $ with $n-1>a_1\\geq a_2\\geq \\cdots \\geq a_k\\geq 2$, has only finitely many solutions.", "background": "This would follow if $P(n(n+1))/\\log n\\to \\infty$, where $P(m)$ denotes the largest prime factor of $m$ (see Problem [368]). Erd\\H{o}s \\cite{Er76d} proved that this problem would also follow from showing that $P(n(n-1))>4\\log n$.\nThe condition $a_1a_1\\geq a_2$ then $ a_1\\geq n-5\\log\\log n, $ and says it 'would be nice' to prove $a_1\\geq n-o(\\log\\log n)$. Bhat and Ramachandra \\cite{BhRa10} replace the $5$ with $(1+o(1))\\frac{1}{\\log 2}$, and also prove that the same bound holds for arbitrary $k\\geq 2$.\nNumerical investigations on solutions to $n!=a_1!a_2!$ have been carried out by Caldwell \\cite{Ca94} and Habsieger \\cite{Ha}, and it is known that there are no solutions aside from $10!=6!7!$ for $n\\leq 10^{3000}$.\nReferences\n\n\n[BhRa10] Bhat, K. Dzh. and Ramachandra, K., A remark on factorials that are products of factorials. Mat. Zametki (2010), 350--354.\n\n[Ca94] C. Caldwell, The Diophantine equation $A!B!=C!$. J. Recreat. Math. (1994), 128-133.\n\n[Er76d] Erd\\H{o}s, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ha] Haight, J. A., Metric Diophantine approximation and related topics. PhD thesis ().\n\n[Lu07b] Luca, Florian, On factorials which are products of factorials. Math. Proc. Cambridge Philos. Soc. (2007), 533--542.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Finiteness of nontrivial factorial-product solutions remains open.\n\n**Verified partial progress.**\n\n- Bhat--Ramachandra give a sharp bound on the largest smaller factorial index.\n- The k=2 equation has been verified through n=10^3000 except for 10!=6!7!.\n\n**Full solution or refutation.**\n\nNeither bounds nor finite checks prove finiteness.\n\n**What remains.**\n\nExclude all sufficiently large solutions.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #373, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/373\n Evidence used: Records current bounds, verification, and open status.\n\n**Review notes.** No numerical search was performed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2081, "problem_number": "EP-374", "title": "Erdős Problem #374", "statement": "For any $m\\in \\mathbb{N}$, let $F(m)$ be the minimal $k\\geq 2$ (if it exists) such that there are $a_1<\\cdots 1\\}$,{/LI}\n{LI} $\\lvert D_3\\cap \\{1,\\ldots,n\\}\\rvert = o(\\lvert D_4\\cap \\{1,\\ldots,n\\}\\rvert)$,{/LI}\n{LI} the least element of $D_6$ is $527$, and{/LI}\n{LI} $D_k=\\emptyset$ for $k>6$.{/LI}\n{/UL}\nReferences\n\n\n[ErGr76] Erd\\H{o}s, P. and Graham, R. L., On products of factorials. Bull. Inst. Math. Acad. Sinica (1976), 337-355.\n\n[LSS14] Luca, F. and Saradha, N. and Shorey, T. N., Squares and factorials in products of factorials. Monatsh. Math. (2014), 385-400.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The requested growth of D_k remains open.\n\n**Verified partial progress.**\n\n- D_2 consists of squares, no D_k contains a prime, and D_k is empty for k>6.\n- The least D_6 element is 527.\n\n**Full solution or refutation.**\n\nThese structural facts do not give the requested asymptotics.\n\n**What remains.**\n\nDetermine the sizes of D_3 through D_6.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #374, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/374\n Evidence used: Records facts and open status.\n\n**Review notes.** No computation used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2082, "problem_number": "EP-376", "title": "Erdős Problem #376", "statement": "Are there infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $105$?", "background": "Erd\\H{o}s, Graham, Ruzsa, and Straus \\cite{EGRS75} have shown that, for any two odd primes $p$ and $q$, there are infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $pq$.\nThis is equivalent (via Kummer's theorem) to whether there are infinitely many $n$ which have only digits $0,1$ in base $3$, digits $0,1,2$ in base $5$, and digits $0,1,2,3$ in base $7$.\nThe sequence of such $n$ is A030979 in the OEIS.\nThe best result in this direction is due to Bloom and Croot \\cite{BlCr25}, who proved that, if $p_1,p_2,p_3$ are sufficiently large primes, then there are infinitely many $n$ such that almost all of the base $p_i$ digits are $0$, there are infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $p_1p_2p_3$, except for a factor of size $\\leq n^\\epsilon$.\nThis is mentioned in problem B33 of Guy's collection \\cite{Gu04}. It is also discussed in an article of Pomerance \\cite{Po15c}.\nGraham offered \\$1000 for a solution to this problem (as mentioned in \\cite{Gu04} and \\cite{BeHa98}).\nReferences\n\n\n[BeHa98] Berend, Daniel and Harmse, J\\o rgen E., On some arithmetical properties of middle binomial\ncoefficients. Acta Arith. (1998), 31--41.\n\n[BlCr25] T. F. Bloom and E. Croot, Integers with small digits in multiple bases. arXiv:2509.02835 (2025).\n\n[EGRS75] Erd\\H{o}s, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Po15c] Pomerance, Carl, Divisors of the middle binomial coefficient. Amer. Math. Monthly (2015), 636--644.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Infinitely many central binomial coefficients coprime to 105 remains open.\n\n**Verified partial progress.**\n\n- The analogous two-odd-prime result is known.\n- Bloom--Croot give an approximate three-large-prime result with a residual factor at most n^epsilon.\n\n**Full solution or refutation.**\n\nThe residual factor cannot yet be removed for 3,5,7.\n\n**What remains.**\n\nFind infinitely many exactly coprime cases.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #376, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/376\n Evidence used: Records the two-prime theorem, recent progress, and open status.\n\n**Review notes.** Approximate divisibility is not treated as coprimality.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2083, "problem_number": "EP-377", "title": "Erdős Problem #377", "statement": "Is there some absolute constant $C>0$ such that $ \\sum_{p\\leq n}1_{p\nmid \\binom{2n}{n}}\\frac{1}{p}\\leq C $ for all $n$ (where the summation is restricted to primes $p\\leq n$)?", "background": "A question of Erd\\H{o}s, Graham, Ruzsa, and Straus \\cite{EGRS75}, who proved that if $f(n)$ is the sum in question then $ \\lim_{x\\to \\infty}\\frac{1}{x}\\sum_{n\\leq x}f(n) = \\sum_{k=2}^\\infty \\frac{\\log k}{2^k}=\\gamma_0 $ and $ \\lim_{x\\to \\infty}\\frac{1}{x}\\sum_{n\\leq x}f(n)^2 = \\gamma_0^2, $ so that for almost all integers $f(m)=\\gamma_0+o(1)$. They also prove that, for all large $n$, $ f(n) \\leq c\\log\\log n $ for some constant $c<1$. (It is trivial from Mertens estimates that $f(n)\\leq (1+o(1))\\log\\log n$.)\nA positive answer would imply that $ \\sum_{p\\leq n}1_{p\\mid \\binom{2n}{n}}\\frac{1}{p}=(1-o(1))\\log\\log n, $ and Erd\\H{o}s, Graham, Ruzsa, and Straus say there is 'no doubt' this latter claim is true.\nThis is mentioned in problem B33 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[EGRS75] Erd\\H{o}s, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Uniform boundedness of the reciprocal sum remains open.\n\n**Verified partial progress.**\n\n- EGRS75 prove an almost-all constant limit and a uniform c log log n bound with c<1.\n\n**Full solution or refutation.**\n\nAlmost-all convergence does not imply a uniform bound.\n\n**What remains.**\n\nControl exceptional n uniformly.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #377, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/377\n Evidence used: Records EGRS75 results and open status.\n\n**Review notes.** No inference from average to uniform behavior.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2084, "problem_number": "EP-380", "title": "Erdős Problem #380", "statement": "We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\\prod_{u\\leq m\\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count the number of $n\\leq x$ which are contained in at least one bad interval. Is it true that $ B(x)\\sim \\#\\{ n\\leq x: P(n)^2\\mid n\\}, $ where $P(n)$ is the largest prime factor of $n$?", "background": "Erd\\H{o}s and Graham only knew that $B(x) > x^{1-o(1)}$. Similarly, we call an interval $[u,v]$ 'very bad' if $\\prod_{u\\leq m\\leq v}m$ is powerful. The number of integers $n\\leq x$ contained in at least one very bad interval should be $\\ll x^{1/2}$. In fact, it should be asymptotic to the number of powerful numbers $\\leq x$.\nWe have $ \\#\\{ n\\leq x: P(n)^2\\mid n\\}=\\frac{x}{\\exp((c+o(1))\\sqrt{\\log x\\log\\log x})} $ for some constant $c>0$.\nTao notes in the comments that if $[u,v]$ is bad then it cannot contain any primes, and hence certainly $v<2u$, and in general $v-u$ must be small (for example, assuming Cramer's conjecture, $v-u\\ll (\\log u)^2$).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Tao proved the requested asymptotic in 2026. If B is the union of bad intervals and B_1={n:P(n)^2 divides n}, then #(B\\B_1 up to x) is at most #(B_1 up to x)/(log x)^(1-o(1)), so B(x)~#B_1(x).\n\n**Verified partial progress.**\n\n- Erdős and Graham had proved that the bad-set union has density zero, despite a later typographical report with the inequality reversed.\n- Earlier work gave a subexponential upper bound for B(x) but did not match the smooth-number asymptotic of B_1.\n- Tao also proves the analogous asymptotic for very bad intervals, with #(VB\\VB_1 up to x) << x^(2/5+o(1)).\n\n**Full solution or refutation.**\n\nTao first shows bad intervals are short using primes in short intervals, then reduces the exceptional elements B\\B_1 to highly constrained smooth-number configurations. A blend of smooth-number estimates, sieve bounds, and recent short-interval prime distribution yields an error smaller than the main B_1 count by a logarithmic factor.\n\n**What remains.**\n\nThe EP-380 asymptotic and the related very-bad counting asymptotic are settled. The stronger structural assertions VB=VB_1 and that every very bad interval has length at most two remain open, as does the full type-F_3 analogue.\n\n**Sources checked.**\n\n- Terence Tao, Products of consecutive integers with unusual anatomy, arXiv:2603.27990v2 (2026). (primary): https://arxiv.org/abs/2603.27990\n Evidence used: Theorem 1.7 proves the quantitative bad-set error and hence the exact EP-380 asymptotic; Theorem 1.8 proves the very-bad successor asymptotic.\n- Thomas F. Bloom, Erdős Problem #380, maintained problem record, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/380\n Evidence used: The maintained record marks the problem proved and states Tao's quantitative asymptotic and the x^(2/5+o(1)) very-bad error.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2085, "problem_number": "EP-382", "title": "Erdős Problem #382", "statement": "Let $u\\leq v$ be such that the largest prime dividing $\\prod_{u\\leq m\\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v^{o(1)}$? Can $v-u$ be arbitrarily large?", "background": "Erd\\H{o}s and Graham report it follows from results of Ramachandra that $v-u\\leq v^{1/2+o(1)}$.\nCambie has observed that the first question boils down to some old conjectures on prime gaps.\nBy Cram\\'{er's conjecture}, for every $\\epsilon>0,$ for every $u$ sufficiently large there is a prime between $u$ and $u+u^\\epsilon$.\nThus for $u+u^\\epsilon0$. For any fixed $k$, there is therefore a positive 'probability' that there are $k$ consecutive integers around $q^2$ (for a prime $q$) all of whose prime divisors are bounded above by $q$, when $v-u\\geq k$. See [383] for a conjecture along these lines. A similar argument applies if we replace multiplicity $2$ with multiplicity $r$, for any fixed $r\\geq 2$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The subpolynomial interval-length bound and unboundedness question remain open.\n\n**Verified partial progress.**\n\n- Ramachandra gives v-u<=v^(1/2+o(1)).\n- Cramér's conjecture would imply v-u=v^o(1).\n\n**Full solution or refutation.**\n\nThe conditional argument does not prove the target.\n\n**What remains.**\n\nImprove the unconditional exponent and decide unboundedness.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #382, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/382\n Evidence used: Records the bound, conditional consequence, and open status.\n\n**Review notes.** Heuristics are labeled conditional.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2086, "problem_number": "EP-383", "title": "Erdős Problem #383", "statement": "Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of $ \\prod_{0\\leq i\\leq k}(p^2+i) $ is $p$?", "background": "A positive answer to this would give an answer to the second part of [382]. Heuristically, the 'probability' that $n$ has no prime divisors $\\geq n^{1/2}$ is $1-\\log 2>0$, so standard heuristics predict the answer to this is yes.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof is known for every k and infinitely many p.\n\n**Verified partial progress.**\n\n- Smooth-number heuristics predict a positive answer.\n- A positive answer would imply unbounded interval lengths in EP-382.\n\n**Full solution or refutation.**\n\nFinite witnesses and heuristics do not establish infinitude.\n\n**What remains.**\n\nProve the asserted infinite family for each fixed k.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #383, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/383\n Evidence used: Records open status and the heuristic connection.\n\n**Review notes.** A tracker finite computation was not used as proof.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2087, "problem_number": "EP-385", "title": "Erdős Problem #385", "statement": "Let $ F(n) = \\max_{\\substack{mn$ for all sufficiently large $n$? Does $F(n)-n\\to \\infty$ as $n\\to\\infty$?", "background": "A question of Erd\\H{o}s, Eggleton, and Selfridge, who write that 'plausible conjectures on primes' imply that $F(n)\\leq n$ for only finitely many $n$, and in fact it is possible that this quantity is always at least $n+(1-o(1))\\sqrt{n}$ (note that it is trivially $\\leq n+\\sqrt{n}$).\nTao has discussed this problem in a blog post.\nSarosh Adenwalla has observed that the first question is equivalent to [430]. Indeed, if $n$ is large and $a_i$ is the sequence defined in the latter problem, then [430] implies that there is a composite $a_j$ such that $a_j-p(a_j)>n$ and hence $F(n)>n$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Both eventual positivity and divergence of F(n)-n remain open.\n\n**Verified partial progress.**\n\n- Prime heuristics predict substantially stronger behavior.\n- The first question is linked to EP-430.\n\n**Full solution or refutation.**\n\nNo unconditional resolution was located.\n\n**What remains.**\n\nProve the eventual lower bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #385, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/385\n Evidence used: Records open status and equivalent formulation.\n\n**Review notes.** Prime heuristics are not a result.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2088, "problem_number": "EP-386", "title": "Erdős Problem #386", "statement": "Let $2\\leq k\\leq n-2$. Can $\\binom{n}{k}$ be the product of consecutive primes infinitely often? For example $ \\binom{21}{2}=2\\cdot 3\\cdot 5\\cdot 7. $ ", "background": "Erd\\H{o}s and Graham write that 'a proof that this cannot happen infinitely often for $\\binom{n}{2}$ seems hopeless; probably this can never happen for $\\binom{n}{k}$ if $3\\leq k\\leq n-3$.'\nWeisenberg has provided four easy examples that show Erd\\H{o}s and Graham were too optimistic here: $ \\binom{7}{3}=5\\cdot 7, $ $ \\binom{10}{4}= 2\\cdot 3\\cdot 5\\cdot 7, $ $ \\binom{14}{4} = 7\\cdot 11\\cdot 13, $ and $ \\binom{15}{6}=5\\cdot 7\\cdot 11\\cdot 13. $ The known values of $n$ for which $\\binom{n}{2}$ is the product of consecutive primes are $4,6,15,21,715$ (see A280992).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Infinitely many binomial coefficients that are products of consecutive primes remains open.\n\n**Verified partial progress.**\n\n- Several finite examples with k>=3 are known.\n- Known n for the k=2 case include 4,6,15,21,715.\n\n**Full solution or refutation.**\n\nExamples do not yield the requested infinitude.\n\n**What remains.**\n\nProve infinitude or finiteness.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #386, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/386\n Evidence used: Records examples and open status.\n\n**Review notes.** Original speculation is not treated as theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2089, "problem_number": "EP-387", "title": "Erdős Problem #387", "statement": "Is there an absolute constant $c>0$ such that, for all $1\\leq k< n$, the binomial coefficient $\\binom{n}{k}$ has a divisor in $(cn,n]$?", "background": "Erd\\H{o}s once conjectured that $\\binom{n}{k}$ must always have a divisor in $(n-k,n]$, but this was disproved by Schinzel and Erd\\H{o}s \\cite{Sc58}. A counterexample is given by $n=99215$ and $k=15$. Schinzel conjectured (see problem B34 of \\cite{Gu04}) that, for all sufficiently large $k$ which are not prime powers, there exists an $n$ such that $\\binom{n}{k}$ is not divisible by any integer in $(n-k,n]$.\nIt is easy to see that $\\binom{n}{k}$ always has a divisor in $[n/k,n]$.\nFaulkner \\cite{Fa66} proved that, if $p$ is the least prime $>2k$ and $n\\geq p$, then $\\binom{n}{k}$ has a prime divisor $\\geq p$ (except $\\binom{9}{2}$ and $\\binom{10}{3}$).\nThis is discussed in problems B33 and B34 of Guy's collection \\cite{Gu04}, who says that Erd\\H{o}s conjectured this is true for any $c<1$ (if $n$ is sufficiently large).\nReferences\n\n\n[Fa66] Faulkner, M., On a theorem of {S}ylvester and {S}chur. J. London Math. Soc. (1966), 107--110.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Sc58] Schinzel, A., Sur un probl\\`eme de {P}. {E}rd\\H{o}s. Colloq. Math. (1958), 198--204.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Bui, Naprienko, Pratt, and Zaharescu unconditionally disproved the existence of an absolute c>0. They construct infinitely many binomial coefficients with no divisor between n*241 log log k/log k and n; the lower endpoint is o(n), so every fixed proportional interval (cn,n] is eventually avoided.\n\n**Verified partial progress.**\n\n- Schinzel's classical example refuted the much stronger claim that some numerator term in (n-k,n] must divide the binomial coefficient.\n- Faulkner proved useful large-prime divisors in a complementary parameter regime.\n- The 2026 paper proves a positive near-n divisor theorem whenever k >= exp((log n)^(2/3+epsilon)), alongside the small-k counterexamples.\n\n**Full solution or refutation.**\n\nThe unconditional counterexample construction splits into a restricted residue-class covering problem and a divisor-avoidance problem. Carefully chosen congruences cancel small prime-power contributions from k!, while sieve estimates and short incomplete Kloosterman-sum bounds eliminate all non-obvious divisors in the target interval. The resulting interval threshold tends to zero as a proportion of n.\n\n**What remains.**\n\nThe yes/no question is refuted. The transition scale k_0(n) between guaranteed near-n divisors and possible avoidance is only bounded between roughly (log log n)^(1/2) and exp((log n)^(2/3+o(1))); locating it remains open.\n\n**Sources checked.**\n\n- Hung M. Bui, Slava Naprienko, Kyle Pratt, and Alexandru Zaharescu, Binomial coefficients with divisors avoiding an interval, arXiv:2605.21221v2 (30 June 2026). (primary): https://arxiv.org/abs/2605.21221\n Evidence used: The current v2 abstract says the result is unconditional. Theorem 1.4 gives infinitely many binomial coefficients with no divisors above n*241 log log k/log k, directly refuting every fixed c.\n- Thomas F. Bloom, Erdős Problem #387 discussion, including Kyle Pratt's 2 July 2026 update, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/forum/thread/387?order=newest\n Evidence used: The author update announces the unconditional joint preprint resolving #387; this also explains why an older cached main-page OPEN badge is stale.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2090, "problem_number": "EP-388", "title": "Erdős Problem #388", "statement": "Can one classify all solutions of $ \\prod_{1\\leq i\\leq k_1}(m_1+i)=\\prod_{1\\leq j\\leq k_2}(m_2+j) $ where $k_1,k_2>3$ and $m_1+k_1\\leq m_2$? Are there only finitely many solutions?", "background": "More generally, if $k_1>2$ then for fixed $a$ and $b$ $ a\\prod_{1\\leq i\\leq k_1}(m_1+i)=b\\prod_{1\\leq j\\leq k_2}(m_2+j) $ should have only a finite number of solutions.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Classification and finiteness of separated product-of-consecutive-integer equalities remain open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo later resolution was located.\n\n**What remains.**\n\nClassify solutions or prove finiteness.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #388, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/388\n Evidence used: Maintained tracker lists open status.\n\n**Review notes.** No computation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2091, "problem_number": "EP-389", "title": "Erdős Problem #389", "statement": "Is it true that for every $n\\geq 1$ there is a $k$ such that $ n(n+1)\\cdots(n+k-1)\\mid (n+k)\\cdots (n+2k-1)? $ ", "background": "Asked by Erd\\H{o}s and Straus.\nFor example when $n=2$ we have $k=5$: $ 2\\times 3 \\times 4 \\times 5\\times 6 \\mid 7 \\times 8 \\times 9\\times 10\\times 11. $ and when $n=3$ we have $k=4$: $ 3\\times 4\\times 5\\times 6 \\mid 7\\times 8\\times 9\\times 10. $ Bhavik Mehta has computed the minimal such $k$ for $1\\leq n\\leq 18$ (now available as A375071 on the OEIS).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The shifted-product divisibility assertion remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo general construction or counterexample was located.\n\n**What remains.**\n\nResolve existence of k for every n.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #389, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/389\n Evidence used: Maintained tracker lists open status.\n\n**Review notes.** No computation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2092, "problem_number": "EP-390", "title": "Erdős Problem #390", "statement": "Let $f(n)$ be the minimal $m$ such that $ n! = a_1\\cdots a_k $ with $n< a_1<\\cdots \n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The order of f(n)-2n is known, but existence and value of the requested asymptotic constant remain open.\n\n**Verified partial progress.**\n\n- Erdos, Guy, and Selfridge proved f(n)-2n is of order n/log n.\n\n**Full solution or refutation.**\n\nTwo-sided order bounds do not establish convergence of (f(n)-2n)log n/n.\n\n**What remains.**\n\nProve or refute convergence of the normalized excess and, if it converges, determine the constant.\n\n**Sources checked.**\n\n- P. Erdos, R. K. Guy, and J. L. Selfridge, Another property of 239 and some related questions, Congres. Numer. 34 (1982), 243-257. (primary): https://combinatorica.hu/~p_erdos/1982-01.pdf\n Evidence used: Primary source for the two-sided order-of-magnitude result.\n- Thomas F. Bloom, Erdos Problem #390, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/390\n Evidence used: States the asymptotic-constant question and maintains it as open.\n\n**Review notes.** The background has the batch-wide trailing serialization fragment; it was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2093, "problem_number": "EP-393", "title": "Erdős Problem #393", "statement": "Let $f(n)$ denote the minimal $m\\geq 1$ such that $ n! = a_1\\cdots a_t $ with $a_1<\\cdots \n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Unconditional behavior of the minimal factor span remains open despite power-saving bounds for each fixed span.\n\n**Verified partial progress.**\n\n- For fixed m, Bui, Pratt, and Zaharescu proved F_m(N) <<_m N^(33/34), improving the earlier o(N) result.\n- Luca's polynomial-factorial theorem implies f(n) tends to infinity conditional on ABC.\n\n**Full solution or refutation.**\n\nThe fixed-width count and conditional finiteness do not determine f(n) unconditionally or decide whether f(n)=1 infinitely often.\n\n**What remains.**\n\nDetermine unconditional growth or even decide the highlighted consecutive-factor case.\n\n**Sources checked.**\n\n- H. M. Bui, K. Pratt, and A. Zaharescu, Power savings for counting solutions to polynomial-factorial equations, Adv. Math. 422 (2023), Paper 109021. (primary): https://arxiv.org/abs/2204.08423\n Evidence used: Proves a power-saving count for polynomial-factorial equations, yielding the recorded fixed-span bound.\n- F. Luca, The Diophantine equation P(x)=n! and a result of M. Overholt, Glas. Mat. Ser. III 37(57) (2002), 269-273. (primary): https://hrcak.srce.hr/4796\n Evidence used: Primary source for the polynomial-factorial finiteness result used conditionally with ABC.\n- Thomas F. Bloom, Erdos Problem #393, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/393\n Evidence used: Maintains the behavior question as open and records the known bounds.\n\n**Review notes.** ABC dependence is explicitly conditional. The background has trailing extraction damage.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2094, "problem_number": "EP-394", "title": "Erdős Problem #394", "statement": "Let $t_k(n)$ denote the least $m$ such that $ n\\mid m(m+1)(m+2)\\cdots (m+k-1). $ Is it true that $ \\sum_{n\\leq x}t_2(n)\\ll \\frac{x^2}{(\\log x)^c} $ for some $c>0$?\nIs it true that, for $k\\geq 2$, $ \\sum_{n\\leq x}t_{k+1}(n) =o\\left(\\sum_{n\\leq x}t_k(n)\\right)? $ ", "background": "In \\cite{ErGr80} they mention a conjecture of Erd\\H{o}s that the sum is $o(x^2)$. This was proved by Erd\\H{o}s and Hall \\cite{ErHa78}, who proved that in fact $ \\sum_{n\\leq x}t_2(n)\\ll \\frac{\\log\\log\\log x}{\\log\\log x}x^2. $ Erd\\H{o}s and Hall conjecture that the sum is $o(x^2/(\\log x)^c)$ for any $c<\\log 2$.\nSince $t_2(p)=p-1$ for prime $p$ it is trivial that $ \\sum_{n\\leq x}t_2(n)\\gg \\frac{x^2}{\\log x}. $ Erd\\H{o}s and Hall \\cite{ErHa78} also note that $t_{n-1}(n!)=2$ and $t_{n-2}(n!)\\ll n$, which $n=2^r$ shows is the best possible. They ask about the behaviour of $t_{n-3}(n!)$ and also ask ask whether, for infinitely many $n$, $ t_k(n!)< t_{k-1}(n!)-1 $ for all $1\\leq k\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The requested logarithmic power saving and the successive-k little-o relation remain open.\n\n**Verified partial progress.**\n\n- Erdos and Hall proved sum_{n<=x} t_2(n) << x^2 logloglog x/loglog x, establishing an older o(x^2) conjecture.\n- Prime arguments give the lower bound of order x^2/log x.\n\n**Full solution or refutation.**\n\nThe iterated-log saving is weaker than every fixed positive power of 1/log x and does not address the second displayed question.\n\n**What remains.**\n\nObtain a fixed logarithmic power saving and compare the summatory functions for consecutive k.\n\n**Sources checked.**\n\n- P. Erdos and R. R. Hall, On some unconventional problems on the divisors of integers, J. Austral. Math. Soc. Ser. A 25 (1978), 479-485. (primary): https://www2.math.ethz.ch/EMIS/classics/Erdos/cit/pdf/39310047.pdf\n Evidence used: Primary proof of the iterated-log upper bound.\n- Thomas F. Bloom, Erdos Problem #394, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/394\n Evidence used: Records both displayed questions as open and distinguishes the older proved conjecture.\n\n**Review notes.** The background contains both a duplicated word and the batch-wide serialization tail; neither source field was edited.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2095, "problem_number": "EP-396", "title": "Erdős Problem #396", "statement": "Is it true that for every $k$ there exists $n$ such that $ \\prod_{0\\leq i\\leq k}(n-i) \\mid \\binom{2n}{n}? $ ", "background": "Erd\\H{o}s and Graham write that $n+1$ always divides $\\binom{2n}{n}$ (indeed $\\frac{1}{n+1}\\binom{2n}{n}$ is the $n$th Catalan number), but it is quite rare that $n$ divides $\\binom{2n}{n}$.\nPomerance \\cite{Po14} has shown that for any $k\\geq 0$ there are infinitely many $n$ such that $n-k\\mid\\binom{2n}{n}$, although the set of such $n$ has upper density $<1/3$. Pomerance also shows that the set of $n$ such that $ \\prod_{1\\leq i\\leq k}(n+i)\\mid \\binom{2n}{n} $ has density $1$.\nThe smallest $n$ for each $k$ are listed as A375077 on the OEIS.\nReferences\n\n\n[Po14] Pomerance, C., Divisors of the middle binomial coefficient. American Mathematical Monthly (2014).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Existence for every k of simultaneous divisibility by all k+1 falling factors remains open.\n\n**Verified partial progress.**\n\n- Pomerance proved infinitely many n for each single fixed factor n-j.\n- For each fixed k, the analogous rising product over n+1 through n+k divides the middle binomial coefficient on a density-one set.\n- The tracker records finite computational witnesses, not a proof for arbitrary k.\n\n**Full solution or refutation.**\n\nIndividual-factor and rising-product theorems do not imply simultaneous divisibility by the falling product.\n\n**What remains.**\n\nGive a construction for every fixed k or find a k for which no n exists.\n\n**Sources checked.**\n\n- C. Pomerance, Divisors of the Middle Binomial Coefficient, Amer. Math. Monthly 122 (2015), 636-644. (primary): https://math.dartmouth.edu/~carlp/amm2015.pdf\n Evidence used: Primary source for the single-shift and rising-product divisibility results.\n- Thomas F. Bloom, Erdos Problem #396, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/396\n Evidence used: Maintains the every-k existence statement as open.\n\n**Review notes.** The imported [Po14] year conflicts with the published volume 122 (2015); the discrepancy was flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2096, "problem_number": "EP-400", "title": "Erdős Problem #400", "statement": "For any $k\\geq 2$ let $g_k(n)$ denote the maximum value of $ (a_1+\\cdots+a_k)-n $ where $a_1,\\ldots,a_k$ are integers such that $a_1!\\cdots a_k! \\mid n!$. Can one show that $ \\sum_{n\\leq x}g_k(n) \\sim c_k x\\log x $ for some constant $c_k$? Is it true that there is a constant $c_k$ such that for almost all $n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** A 2026 preprint proves logarithmic order for almost all n, but neither requested asymptotic constant is known.\n\n**Verified partial progress.**\n\n- Li proves that for every epsilon>0, almost all n satisfy g_k(n) >= (3(k-1)/log 12-epsilon) log n.\n- Li proves the pointwise upper bound g_k(n) <= (k-1)log_2 n + log_2 log n + O_k(1).\n- These bounds imply the summatory order of magnitude Theta_k(x log x), but not an asymptotic.\n\n**Full solution or refutation.**\n\nMatching logarithmic scale does not identify c_k or prove o(log x)-scale concentration around one constant.\n\n**What remains.**\n\nProve a summatory asymptotic and an almost-all normal-order constant, or show one of those limits fails.\n\n**Sources checked.**\n\n- E. Li, Prime-Power Rarefaction and a Density-One Lower Bound for Erdos Problem 400, arXiv:2606.23661 (2026). (primary): https://arxiv.org/abs/2606.23661\n Evidence used: Primary source for the density-one lower bound and pointwise upper bound.\n- P. Erdos and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980). (primary): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: Original source for the factorial-product excess question.\n- Thomas F. Bloom, Erdos Problem #400, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/400\n Evidence used: Maintains the asymptotic questions as open.\n\n**Review notes.** The summatory Theta statement is a direct inference from the cited density-one lower and pointwise upper bounds; it is not presented as a verbatim theorem title.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2097, "problem_number": "EP-404", "title": "Erdős Problem #404", "statement": "For which integers $a\\geq 1$ and primes $p$ is there a finite upper bound on those $k$ such that there are $a=a_1<\\cdots\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The classification of finite p-adic bounds and the infinite-sequence question remain open.\n\n**Verified partial progress.**\n\n- Lin proved f(2,2) <= 254, handling one specific pair.\n\n**Full solution or refutation.**\n\nOne bounded pair does not classify all (a,p), determine f(a,p), or settle unbounded valuations along an infinite sequence.\n\n**What remains.**\n\nClassify the pairs with finite upper bound and resolve the partial-sum valuation question.\n\n**Sources checked.**\n\n- P. Erdos and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980). (primary): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: Original problem source and bibliography, which dates Lin's memorandum to 1976.\n- Thomas F. Bloom, Erdos Problem #404, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/404\n Evidence used: Maintains the general questions as open and records Lin's special-case bound.\n\n**Review notes.** The imported reference says 1960, but the [Li76] key and original bibliography say 1976. The current separate variables n and k avoid an earlier tracker typo.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2098, "problem_number": "EP-406", "title": "Erdős Problem #406", "statement": "Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?", "background": "The only examples seem to be $1$, $4=1+3$, and $256=1+3+3^2+3^5$. If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power of $2$.\nThis would imply via Kummer's theorem that $ 3\\mid \\binom{2^{k+1}}{2^k} $ for all large $k$.\nSaye \\cite{Sa22} has computed that $2^n$ contains every possible ternary digit for $16\\leq n \\leq 5.9\\times 10^{21}$.\nThis is mentioned in problem B33 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Sa22] Saye, Robert I., On two conjectures concerning the ternary digits of powers of\ntwo. J. Integer Seq. (2022), Art. 22.3.4, 9.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Finiteness of powers of two whose ternary digits all lie in {0,1} remains open.\n\n**Verified partial progress.**\n\n- Narkiewicz bounded the number of eligible exponents n<=X by 1.62 X^(log_3 2), proving density zero but not finiteness.\n- Saye verified that only exponents 0, 2, and 8 work through 2*3^45, approximately 5.9*10^21.\n\n**Full solution or refutation.**\n\nA sublinear counting bound and any finite computation are compatible with infinitely many exceptional exponents.\n\n**What remains.**\n\nProve that no exponent beyond 8 works, or construct another example or an infinite family.\n\n**Sources checked.**\n\n- R. I. Saye, On two conjectures concerning the ternary digits of powers of two, J. Integer Seq. 25 (2022), Article 22.3.4. (primary): https://arxiv.org/abs/2202.13256\n Evidence used: Primary source for the verification through 2*3^45.\n- J. C. Lagarias, Ternary expansions of powers of 2, J. London Math. Soc. 79 (2009), 562-588. (primary): https://arxiv.org/abs/math/0512006\n Evidence used: Reports Narkiewicz's counting theorem and explicitly describes the original problem as open.\n- Thomas F. Bloom, Erdos Problem #406, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/406\n Evidence used: Maintains the finiteness question as open and records current progress.\n\n**Review notes.** No local digit computation was performed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2099, "problem_number": "EP-408", "title": "Erdős Problem #408", "statement": "Let $\\phi(n)$ be the Euler totient function and $\\phi_k(n)$ be the iterated $\\phi$ function, so that $\\phi_1(n)=\\phi(n)$ and $\\phi_k(n)=\\phi(\\phi_{k-1}(n))$. Let $ f(n) = \\min \\{ k : \\phi_k(n)=1\\}. $ Does $f(n)/\\log n$ have a distribution function? Is $f(n)/\\log n$ almost always constant? What can be said about the largest prime factor of $\\phi_k(n)$ when, say, $k=\\log\\log n$?", "background": "Pillai \\cite{Pi29} was the first to investigate this function, and proved $ \\log_3 n < f(n) < \\log_2 n $ for all large $n$. Shapiro \\cite{Sh50} proved that $f(n)$ is essentially multiplicative.\nErd\\H{o}s, Granville, Pomerance, and Spiro \\cite{EGPS90} have proved that the answer to the first two questions is yes, conditional on a form of the Elliott-Halberstam conjecture.\nIt is likely true that, if $k\\to \\infty$ however slowly with $n$, then for almost all $n$ the largest prime factor of $\\phi_k(n)$ is $\\leq n^{o(1)}$.\nThis is discussed in problem B41 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[EGPS90] Erd\\H{o}s, P. and Granville, A. and Pomerance, C. and Spiro, C., On the normal behavior of the iterates of some arithmetic functions. Analytic number theory (Allerton Park, IL, 1989) (1990), 165-204.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Pi29] Pillai, S. Sivasankaranarayana, On some functions connected with {$\\phi(n)$}. Bull. Amer. Math. Soc. (1929), 832-836.\n\n[Sh50] Shapiro, Harold N., On the iterates of a certain class of arithmetic functions. Comm. Pure Appl. Math. (1950), 259-272.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Normal-order behavior is known conditionally on Elliott-Halberstam, while the unconditional and deep-iterate questions remain open.\n\n**Verified partial progress.**\n\n- Erdos, Granville, Pomerance, and Spiro conditionally proved that f(n) has normal order alpha log n for some alpha>0.\n- The same paper proves several unconditional theorems on fixed iterates, but not the largest-prime-factor assertion at k=loglog n.\n\n**Full solution or refutation.**\n\nA theorem conditional on Elliott-Halberstam does not resolve the unconditional distribution or almost-all questions.\n\n**What remains.**\n\nRemove the distributional hypothesis and control the largest prime factor of a deep totient iterate.\n\n**Sources checked.**\n\n- P. Erdos, A. Granville, C. Pomerance, and C. Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic Number Theory (1990), 165-204. (primary): https://math.dartmouth.edu/~carlp/iterate.pdf\n Evidence used: Primary source for the conditional normal-order theorem and related iterate results.\n- Thomas F. Bloom, Erdos Problem #408, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/408\n Evidence used: Maintains the questions as open and states the Elliott-Halberstam dependence.\n\n**Review notes.** Conditional normal order is not promoted to an unconditional solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2100, "problem_number": "EP-409", "title": "Erdős Problem #409", "statement": "How many iterations of $n\\mapsto \\phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density of $n$ which reach any fixed prime?", "background": "A problem of Finucane. One can also ask similar questions about $n\\mapsto \\sigma(n)-1$: do iterates of this always reach a prime? If so, how soon? (It is easily seen that iterates of this cannot reach the same prime infinitely often, since they are non-decreasing.)\nThis problem is somewhat ambiguously phrased. Let $F(n)$ count the number of iterations of $n\\mapsto \\phi(n)+1$ before reaching a prime. The number of iterations required is A039651 in the OEIS.\nCambie notes in the comments that $F(n)=o(n)$ is trivial, and $F(n)=1$ infinitely often. Presumably the intended question is to find 'good' upper bounds for $F(n)$.\nThis is discussed in problem B41 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Stopping-time bounds, infinite fibers, and fixed-prime densities for n -> phi(n)+1 remain open, with the first request imprecisely phrased.\n\n**Verified partial progress.**\n\n- For composite n>2 the map strictly decreases, so a prime is always reached.\n- The tracker records the elementary bounds F(n)=o(n) and infinitely many n with F(n)=1.\n\n**Full solution or refutation.**\n\nTermination is elementary, but no specified target bound or resolution of the fiber and density questions was located.\n\n**What remains.**\n\nSpecify and prove a sharp stopping-time bound, and determine whether a prime can have an infinite basin and what its density is.\n\n**Sources checked.**\n\n- P. Erdos and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 81 in the scanned edition. (primary): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: Confirms that the original map is phi(n)+1 and states all three questions.\n- Thomas F. Bloom, Erdos Problem #409, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/409\n Evidence used: Flags the ambiguity, records elementary progress, and maintains open status.\n\n**Review notes.** The map was checked against the original source and is not an OCR error. Expert review is requested because 'how many' has no formal target.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2101, "problem_number": "EP-410", "title": "Erdős Problem #410", "statement": "Let $\\sigma_1(n)=\\sigma(n)$, the sum of divisors function, and $\\sigma_k(n)=\\sigma(\\sigma_{k-1}(n))$. Is it true that for all $n\\geq 2$ $ \\lim_{k\\to \\infty} \\sigma_k(n)^{1/k}=\\infty? $ ", "background": "This is discussed in problem B9 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** For the corrected domain n>=2, divergence of sigma_k(n)^(1/k) remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for the corrected statement was located in primary literature or the maintained tracker.\n\n**What remains.**\n\nProve superexponential growth in the iteration index for every fixed n>=2, or exhibit a counterexample.\n\n**Sources checked.**\n\n- P. Erdos and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 81 in the scanned edition. (primary): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: Original source for the iterated sigma question; its wording did not exclude n=1.\n- Thomas F. Bloom, Erdos Problem #410, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/410\n Evidence used: Maintains the corrected n>=2 formulation as open.\n\n**Review notes.** The n>=2 restriction in the input removes the trivial original n=1 counterexample. The source statement was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2102, "problem_number": "EP-411", "title": "Erdős Problem #411", "statement": "Let $g_1=g(n)=n+\\phi(n)$ and $g_k(n)=g(g_{k-1}(n))$. For which $n$ and $r$ is it true that $g_{k+r}(n)=2g_k(n)$ for all large $k$?", "background": "The known solutions to $g_{k+2}(n)=2g_k(n)$ are $n=10$ and $n=94$. Selfridge and Weintraub found solutions to $g_{k+9}(n)=9g_k(n)$ and Weintraub found $ g_{k+25}(3114)=729g_k(3114) $ for all $k\\geq 6$.\nSteinerberger \\cite{St25} has observed that, for $r=2$, this problem is equivalent to asking for solutions to $ \\phi(n)+\\phi(n+\\phi(n))=n, $ and has shown that if this holds then either the odd part of $n$ is in $\\{1,3,5,7,35,47\\}$, or is equal to $8m+7$ or $6m+5$, where $8m+7\\geq 10^{10}$ is a prime number and $\\phi(6m+5)=4m+4$. Whether there are infinitely many such $m$ is related to the question of whether $ \\phi(n)=\\frac{2}{3}(n+1) $ has infinitely many solutions.\nCambie conjectures that the only solutions have $r=2$ and $n=2^lp$ for some $l\\geq 1$ and $p\\in \\{2,3,5,7,35,47\\}$. Cambie has shown this problem is reducible to the question of which integers $r,t\\geq 1$ and primes $p\\equiv 7\\pmod{8}$ satisfy $g_k(2p^t)=4p^t$, and conjectures there are no solutions to this except when $t=1$ and $p\\in \\{7,47\\}$. Cambie has also observed that $ g_{k+4}(738)=3g_k(738), $ $ g_{k+4}(148646)=4g_k(148646), $ and $ g_{k+4}(4325798)=4g_{k}(4325798) $ for all $k\\geq 1$.\nReferences\n\n\n[St25] S. Steinerberger, On an iterated arithmetic function problem of Erd\\H{o}s and Graham. arXiv:2504.08023 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The iterated-totient relation is not classified.\n\n**Verified partial progress.**\n\n- Known examples exist for several multiplier/shift pairs.\n- Steinerberger reduced the r=2 case to a restrictive totient equation.\n\n**Full solution or refutation.**\n\nNo complete classification is known.\n\n**What remains.**\n\nClassify all n,r or settle the residual totient equations.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #411, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/411\n Evidence used: Records examples, reduction, and open status.\n\n**Review notes.** Unverified discussion claims were excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2103, "problem_number": "EP-412", "title": "Erdős Problem #412", "statement": "Let $\\sigma_1(n)=\\sigma(n)$, the sum of divisors function, and $\\sigma_k(n)=\\sigma(\\sigma_{k-1}(n))$.\nIs it true that, for every $m,n\\geq 2$, there exist some $i,j$ such that $\\sigma_i(m)=\\sigma_j(n)$?", "background": "In \\cite{Er79d} Erd\\H{o}s attributes this conjecture to van Wijngaarden, who told it to Erd\\H{o}s in the 1950s.\nThat is, there is (eventually) only one possible sequence that the iterated sum of divisors function can settle on. Selfridge reports numerical evidence which suggests the answer is no, but Erd\\H{o}s and Graham write 'it seems unlikely that anything can be proved about this in the near future'.\nSee also [413] and [414].\nReferences\n\n\n[Er79d] Erd\\H{o}s, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Coalescence of iterated sum-of-divisors orbits remains open.\n\n**Verified partial progress.**\n\n- Numerical evidence and related questions are documented but no general theorem.\n\n**Full solution or refutation.**\n\nNo proof or counterexample was located.\n\n**What remains.**\n\nEstablish or refute universal orbit intersection.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #412, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/412\n Evidence used: Lists open status and the historical formulation.\n\n**Review notes.** A comment is not treated as counterexample.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2104, "problem_number": "EP-413", "title": "Erdős Problem #413", "statement": "Let $\\omega(n)$ count the number of distinct primes dividing $n$. Are there infinitely many $n$ such that, for all $m0$ such that there are infinitely many $n$ where $m+\\epsilon \\omega(m)\\leq n$ for all $m\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The epsilon subquestion is solved positively, while the exact barrier question remains open.\n\n**Verified partial progress.**\n\n- Lau (2026) proves the epsilon assertion.\n- Lau also proves infinitely many bounded-defect barriers for the first assertion.\n\n**Full solution or refutation.**\n\nThe original exact first condition is not yet established.\n\n**What remains.**\n\nRemove the bounded defect and prove infinitely many exact barriers.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #413, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/413\n Evidence used: Records Lau's 2026 results and continuing open status.\n\n**Review notes.** Exact and epsilon statements are separated.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2105, "problem_number": "EP-414", "title": "Erdős Problem #414", "statement": "Let $h_1(n)=h(n)=n+\\tau(n)$ (where $\\tau(n)$ counts the number of divisors of $n$) and $h_k(n)=h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist $i$ and $j$ such that $h_i(m)=h_j(n)$?", "background": "Asked by Spiro. That is, there is (eventually) only one possible sequence that the iterations of $n\\mapsto h(n)$ can settle on. Erd\\H{o}s and Graham believed the answer is yes. Similar questions can be asked by the iterates of many other functions. See also [412] and [413].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Universal coalescence of h(n)=n+tau(n) iterates remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo proof or counterexample was located.\n\n**What remains.**\n\nProve or refute orbit intersection for all starts.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #414, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/414\n Evidence used: Lists continuing open status.\n\n**Review notes.** No computation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2106, "problem_number": "EP-415", "title": "Erdős Problem #415", "statement": "For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\\phi(m+1),\\ldots,\\phi(m+k)$ with $m+k\\leq n$. Is it true that $ F(n)=(c+o(1))\\log\\log\\log n $ for some constant $c$? Is the first pattern which fails to appear always $ \\phi(m+1)>\\phi(m+2)>\\cdots \\phi(m+k)? $ Is it true that 'natural' ordering which mimics what happens to $\\phi(1),\\ldots,\\phi(k)$ is the most likely to appear?", "background": "Erd\\H{o}s \\cite{Er36b} proved that $ F(n)\\asymp \\log\\log\\log n, $ and similarly if we replace $\\phi$ with $\\sigma$ or $\\tau$ or $\nu$ or any 'decent' additive or multiplicative function.\nWeisenberg has observed that the same questions could be asked for ordering patterns which allow equality (indeed, the final problem only makes sense if we allow equality).\nReferences\n\n\n[Er36b] Erd\\H{o}s, P., On a problem of Chowla and some related problems. Proc. Cambridge Philos. Soc. (1936), 530-540.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The proposed c log log log n asymptotic is false for F(n); other pattern questions remain open/ambiguous.\n\n**Verified partial progress.**\n\n- Pollack--Pomerance--Treviño prove the longest strict decreasing run is asymptotic to triple-log divided by a sixth-iterated-log factor.\n- Since F is bounded by that run length, this refutes the proposed positive-constant triple-log scale.\n\n**Full solution or refutation.**\n\nOnly the first subquestion is resolved negatively.\n\n**What remains.**\n\nClarify equality convention and settle remaining pattern questions.\n\n**Sources checked.**\n\n- P. Pollack, C. Pomerance and E. Treviño (2013), as cited by the maintained tracker. (primary): https://www.erdosproblems.com/415\n Evidence used: Provides the strict descending-run asymptotic.\n- Thomas F. Bloom, Erdős Problem #415, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/415\n Evidence used: Explains the implication and formulation ambiguity.\n\n**Review notes.** Strict versus weak patterns are explicitly distinguished.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2107, "problem_number": "EP-416", "title": "Erdős Problem #416", "statement": "Let $V(x)$ count the number of $n\\leq x$ such that $\\phi(m)=n$ is solvable. Does $V(2x)/V(x)\\to 2$? Is there an asymptotic formula for $V(x)$?", "background": "Pillai \\cite{Pi29} proved $V(x)=o(x)$. Erd\\H{o}s \\cite{Er35b} proved $V(x)=x(\\log x)^{-1+o(1)}$.\nThe behaviour of $V(x)$ is now almost completely understood. Maier and Pomerance \\cite{MaPo88} proved $ V(x)=\\frac{x}{\\log x}e^{(C+o(1))(\\log\\log\\log x)^2}, $ for some explicit constant $C>0$. Ford \\cite{Fo98} improved this to $ V(x)\\asymp\\frac{x}{\\log x}e^{C_1(\\log\\log\\log x-\\log\\log\\log\\log x)^2+C_2\\log\\log\\log x-C_3\\log\\log\\log\\log x} $ for some explicit constants $C_1,C_2,C_3>0$. Unfortunately this falls just short of an asymptotic formula for $V(x)$ and determining whether $V(2x)/V(x)\\to 2$.\nIn \\cite{Er79e} Erd\\H{o}s asks further to estimate the number of $n\\leq x$ such that the smallest solution to $\\phi(m)=n$ satisfies $kx\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** An asymptotic formula for the count of totient values remains open.\n\n**Verified partial progress.**\n\n- Maier--Pomerance and Ford give near-asymptotic estimates with explicit iterated-log factors.\n\n**Full solution or refutation.**\n\nThe estimates do not determine V(2x)/V(x).\n\n**What remains.**\n\nObtain an asymptotic formula or ratio limit.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #416, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/416\n Evidence used: Records known estimates and open status.\n\n**Review notes.** Near asymptotics are not an asymptotic formula.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2108, "problem_number": "EP-417", "title": "Erdős Problem #417", "statement": "Let $ V'(x)=\\#\\{\\phi(m) : 1\\leq m\\leq x\\} $ and $ V(x)=\\#\\{\\phi(m) \\leq x : 1\\leq m\\}. $ Does $\\lim V(x)/V'(x)$ exist? Is it $>1$?", "background": "It is trivial that $V'(x) \\leq V(x)$. In \\cite{Er98} Erd\\H{o}s suggests the limit may be infinite. See also [416].\nReferences\n\n\n[Er98] Erd\\H{o}s, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Existence and size of V(x)/V'(x) remain open.\n\n**Verified partial progress.**\n\n- V'(x)<=V(x) is immediate; Erdos suggested the limit might be infinite.\n\n**Full solution or refutation.**\n\nNo limit theorem was located.\n\n**What remains.**\n\nControl first-preimage sizes for totient values.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #417, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/417\n Evidence used: Records the definitions and open status.\n\n**Review notes.** Comments were not used as results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2109, "problem_number": "EP-420", "title": "Erdős Problem #420", "statement": "If $\\tau(n)$ counts the number of divisors of $n$ then let $ F(f,n)=\\frac{\\tau((n+\\lfloor f(n)\\rfloor)!)}{\\tau(n!)}. $ Is it true that $ \\lim_{n\\to \\infty}F((\\log n)^C,n)=\\infty $ for large $C$?\nIs it true that $F(\\log n,n)$ is everywhere dense in $(1,\\infty)$?\nMore generally, if $f(n)\\leq \\log n$ is a monotonic function such that $f(n)\\to \\infty$ as $n\\to \\infty$, then is $F(f,n)$ everywhere dense?", "background": "Erd\\H{o}s and Graham write that it is easy to show that $\\lim F(n^{1/2},n)=\\infty$, and in fact the $n^{1/2}$ can be replaced by $n^{1/2-c}$ for some small constant $c>0$.\nErd\\H{o}s, Graham, Ivi\\'{c}, and Pomerance \\cite{EGIP96} have proved that $ \\liminf F(c\\log n, n) = 1 $ for any $c>0$, and $ \\lim F(n^{4/9},n)=\\infty. $ (The exponent $4/9$ can be improved slightly.) They also prove that, if $f(n)=o((\\log n)^2)$, then for almost all $n$ $ F(f,n)\\sim 1. $ van Doorn notes in the comments that the existence of infinitely many bounded prime gaps implies $ \\limsup_{n\\to \\infty}F(g(n),n)=\\infty $ for any $g(n)\\to \\infty$, and that Cram\\'{e}r's conjecture implies $ \\lim F(g(n)(\\log n)^2, n)=\\infty $ for any $g(n)\\to \\infty$>\nReferences\n\n\n[EGIP96] Erd\\H{o}s, Paul and Graham, S. W. and Ivi\\'c, Aleksandar and\nPomerance, Carl, On the number of divisors of {$n!$}. (1996), 337--355.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The logarithmic-shift limit and density questions remain open.\n\n**Verified partial progress.**\n\n- EGIP96 prove lim F(n^(4/9),n)=infinity and liminf F(c log n,n)=1.\n- For f=o((log n)^2), F(f,n)~1 for almost all n.\n\n**Full solution or refutation.**\n\nKnown transition results do not settle logarithmic shifts.\n\n**What remains.**\n\nDetermine limit and density behavior near log n.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #420, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/420\n Evidence used: Records EGIP96 results and open status.\n\n**Review notes.** Conditional prime-gap observations are separated.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2110, "problem_number": "EP-421", "title": "Erdős Problem #421", "statement": "Is there a sequence $1\\leq d_11/e-\\epsilon$ for any $\\epsilon>0$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The density-one sequence with distinct interval products remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo later resolution was located.\n\n**What remains.**\n\nConstruct such a sequence or prove obstruction.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #421, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/421\n Evidence used: Maintained tracker lists open status.\n\n**Review notes.** No computation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2111, "problem_number": "EP-422", "title": "Erdős Problem #422", "statement": "Let $f(1)=f(2)=1$ and for $n>2$ $ f(n) = f(n-f(n-1))+f(n-f(n-2)). $ Does $f(n)$ miss infinitely many integers? What is its behaviour?", "background": "Asked by Hofstadter. The sequence begins $1,1,2,3,3,4,\\ldots$ and is A005185 in the OEIS. It is not even known whether $f(n)$ is well-defined for all $n$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The recurrence's range and asymptotic behavior remain open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo later resolution was located.\n\n**What remains.**\n\nDetermine missed values and growth behavior.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #422, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/422\n Evidence used: Maintained tracker lists open status.\n\n**Review notes.** No computation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2112, "problem_number": "EP-423", "title": "Erdős Problem #423", "statement": "Let $a_1=1$ and $a_2=2$ and for $k\\geq 3$ choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence?", "background": "Asked by Hofstadter (in \\cite{Er77c} Erd\\H{o}s says Hofstadter was inspired by a similar question of Ulam). The sequence begins $ 1,2,3,5,6,8,10,11,\\ldots $ and is A005243 in the OEIS.\nBolan and Tang have independently proved that there are infinitely many integers which do not appear in this sequence. In fact, the sequence $a_n-n$ is nondecreasing and unbounded.\nReferences\n\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Tang proved that the Hofstadter consecutive-sum sequence omits infinitely many integers and obtained the first polynomial upper bound; the exact asymptotic remains open.\n\n**Verified partial progress.**\n\n- Tang proves n+omega(1) <= a_n, so a_n-n is unbounded and infinitely many positive integers are omitted.\n- Tang proves a_n << n^(4175/2506+o(1)); the tracker records independent work by Bolan and later discussion of refinements.\n\n**Full solution or refutation.**\n\nThere is now a rigorous nontrivial lower deviation and a polynomial upper bound, but no asymptotic formula; even a_n=O(n) is not established in the cited primary preprint.\n\n**What remains.**\n\nDetermine the true asymptotic behavior, including whether a_n=n+o(n) or at least a_n=O(n).\n\n**Sources checked.**\n\n- Quanyu Tang, The Hofstadter consecutive-sum sequence omits infinitely many positive integers, arXiv:2603.09939 (2026). (primary): https://arxiv.org/abs/2603.09939\n Evidence used: The abstract states n+omega(1) <= a_n << n^(4175/2506+o(1)) and the infinite-omission result.\n- Thomas F. Bloom, Erdős Problem #423, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/423\n Evidence used: Records the current open status, Tang and Bolan's progress, and subsequent updates.\n\n**Review notes.** Later discussion reports refined exponents, but the primary arXiv abstract's explicit bound is used here to avoid conflating versions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2113, "problem_number": "EP-424", "title": "Erdős Problem #424", "statement": "Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\\ldots,a_n$ all possible values of $a_ia_j-1$ with $i\neq j$. Is it true that the set of integers which eventually appear has positive density?", "background": "Asked by Hofstadter. The sequence begins $2,3,5,9,14,17,26,\\ldots$ and is A005244 in the OEIS. This problem is also discussed in section E31 of Guy's book Unsolved Problems in Number Theory.\nIn \\cite{ErGr80} (and in Guy's book) this problem as written is asking for whether almost all integers appear in this sequence, but the answer to this is trivially no (as pointed out to me by Steinerberger): no integer $\\equiv 1\\pmod{3}$ is ever in the sequence, so the set of integers which appear has density at most $2/3$. This is easily seen by induction, and the fact that if $a,b\\in \\{0,2\\}\\pmod{3}$ then $ab-1\\in \\{0,2\\}\\pmod{3}$.\nPresumably it is the weaker question of whether a positive density of integers appear (as correctly asked in \\cite{Er77c}) that was also intended in \\cite{ErGr80}.\nSee also Problem 63 of Green's open problems list.\nReferences\n\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The positive-density question remains open; only the stronger almost-all variant is trivially false.\n\n**Verified partial progress.**\n\n- No generated term is congruent to 1 modulo 3, by induction, so the set of generated integers has upper density at most 2/3.\n\n**Full solution or refutation.**\n\nThe congruence obstruction refutes an almost-all formulation found in some later sources but does not settle positive density.\n\n**What remains.**\n\nProve a positive lower-density result or show that the generated set has density zero; clarify the intended notion of positive density if necessary.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #424, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/424\n Evidence used: Explains the modulo-3 obstruction, distinguishes the original positive-density question from the false almost-all version, and retains open status.\n\n**Review notes.** The source does not specify natural density versus lower density; no interpretation was silently substituted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2114, "problem_number": "EP-425", "title": "Erdős Problem #425", "statement": "Let $F(n)$ be the maximum possible size of a subset $A\\subseteq\\{1,\\ldots,N\\}$ such that the products $ab$ are distinct for all $a\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Erdős determined the order of the correction to pi(n), but existence of an asymptotic constant and the stated r-fold bound remain open.\n\n**Verified partial progress.**\n\n- For the conventional same-variable formulation, Erdős proved pi(n)+c1 n^(3/4)(log n)^(-3/2) <= F(n) <= pi(n)+c2 n^(3/4)(log n)^(-3/2) for fixed positive constants c1,c2.\n- Recent discussion claims improved explicit constants, including Lean-assisted checks with a named PNT hypothesis, but these have not been incorporated as a resolution by the maintained tracker.\n\n**Full solution or refutation.**\n\nThe conjectured scale is known up to constants, not the limiting coefficient. The second, r-fold question is also unresolved.\n\n**What remains.**\n\nProve or disprove convergence to a constant coefficient and settle the r-fold upper bound.\n\n**Sources checked.**\n\n- Hong Liu and Péter Pál Pach, The number of multiplicative Sidon sets of integers, arXiv:1808.06182 (2018). (primary): https://arxiv.org/abs/1808.06182\n Evidence used: Provides modern primary literature on multiplicative Sidon sets and cites the extremal-order results.\n- Thomas F. Bloom, Erdős Problem #425, checked 2026-08-17; citing P. Erdős, Publ. Ramanujan Inst. (1968/69), 131-136. (maintained_tracker): https://www.erdosproblems.com/425\n Evidence used: States the two-sided order-of-magnitude theorem, preserves the current open questions, and identifies recent discussion claims.\n\n**Review notes.** Formulation defect preserved: F(n) is defined using A subset {1,...,N}. No correction was made. Recent comment-thread constants were not treated as established literature.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2115, "problem_number": "EP-428", "title": "Erdős Problem #428", "statement": "Is there a set $A\\subseteq \\mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\\in A$ with $00? $ ", "background": "Erd\\H{o}s and Graham could show this is true (assuming the prime $k$-tuple conjecture) if we replace $\\liminf$ by $\\limsup$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The unconditional liminf formulation remains open.\n\n**Verified partial progress.**\n\n- Erdős and Graham showed, conditional on the prime k-tuples conjecture, that a version with limsup in place of liminf is true.\n\n**Full solution or refutation.**\n\nThe conditional limsup result neither supplies an unconditional construction nor reaches the stated liminf density.\n\n**What remains.**\n\nConstruct such a set unconditionally with positive liminf relative density, or prove an obstruction.\n\n**Sources checked.**\n\n- P. Erdős and R. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980). (source_collection): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: Historical source for the problem and conditional limsup result.\n- Thomas F. Bloom, Erdős Problem #428, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/428\n Evidence used: Retains open status and records the conditional result.\n\n**Review notes.** Conditional and unconditional statements are kept separate.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2116, "problem_number": "EP-430", "title": "Erdős Problem #430", "statement": "Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ such that all of the prime factors of $a_k$ are $>n-a_k$.\nIs it true that, for sufficiently large $n$, not all of this sequence can be prime?", "background": "Erd\\H{o}s and Graham write 'preliminary calculations made by Selfridge indicate that this is the case but no proof is in sight'. For example if $n=8$ we have $a_1=7$ and $a_2=5$ and then must stop.\nSarosh Adenwalla has observed that this problem is equivalent to (the first part of) [385]. Indeed, assuming a positive answer to that, for all large $n$, there exists a composite $mn-m$. It follows that such an $m$ is equal to some $a_i$ in the sequence defined for $[1,n)$, and $m$ is composite by assumption.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The sequence question remains open and is equivalent to the first part of EP-385.\n\n**Verified partial progress.**\n\n- Sarosh Adenwalla observed an equivalence with the first part of EP-385.\n- Selfridge's preliminary computations supported a positive answer but provide no proof.\n\n**Full solution or refutation.**\n\nNo proof or counterexample was located for either equivalent formulation.\n\n**What remains.**\n\nShow that every sufficiently large n admits a composite term satisfying the roughness condition, or find infinitely many counterexamples.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #430, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/430\n Evidence used: Records the exact statement, the EP-385 equivalence, historical computation, and open status.\n- Thomas F. Bloom, Erdős Problem #385, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/385\n Evidence used: Provides the equivalent first formulation and its current open status.\n\n**Review notes.** No attempt was made to rewrite the source formulation through the equivalent notation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2117, "problem_number": "EP-431", "title": "Erdős Problem #431", "statement": "Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?", "background": "A problem of Ostmann, sometimes known as the 'inverse Goldbach problem'. The answer is surely no. The best result in this direction is due to Elsholtz and Harper \\cite{ElHa15}, who showed that if $A,B$ are such sets then for all large $x$ we must have $ \\frac{x^{1/2}}{\\log x\\log\\log x} \\ll \\lvert A \\cap [1,x]\\rvert \\ll x^{1/2}\\log\\log x $ and similarly for $B$.\nElsholtz \\cite{El01} has proved there are no sets $A,B,C$ (all of size at least $2$) such that $A+B+C$ agrees with the set of prime numbers up to finitely many exceptions.\nGranville \\cite{Gr90} proved, conditional on the prime $k$-tuples conjecture, that there are infinite sets $B$ and $C$ such that $ \\{ \\tfrac{b+c}{2}: b\\in B, c\\in C\\} $ is a subset of the primes. Tao and Ziegler \\cite{TaZi23} gave an unconditional proof that there are infinite sets $B=\\{b_1<\\cdots\\}$ and $C=\\{c_1<\\cdots\\}$ such that $ \\{ b_i+c_j : b_i\\in B, c_j\\in C, i\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The inverse Goldbach problem for two infinite summand sets remains open despite strong necessary bounds and related partial-sumset constructions.\n\n**Verified partial progress.**\n\n- Elsholtz and Harper show any hypothetical A and B must each have counting function between order x^(1/2)/(log x loglog x) and x^(1/2)loglog x for all large x.\n- Elsholtz proved that no ternary sumset A+B+C, with each set nontrivial, agrees eventually with the primes.\n- Tao and Ziegler constructed infinite B,C with b_i+c_j prime for i", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2118, "problem_number": "EP-432", "title": "Erdős Problem #432", "statement": "Let $A,B\\subseteq \\mathbb{N}$ be two infinite sets. How dense can $A+B$ be if all elements of $A+B$ are pairwise relatively prime?", "background": "Asked by Straus, inspired by a problem of Ostmann (see [431]).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A universal prime-counting upper bound is elementary, and a 2026 public manuscript claims substantive infinite lower constructions; the optimal density remains open and the new claims are unrefereed.\n\n**Verified partial progress.**\n\n- Pairwise coprimality assigns distinct prime divisors to distinct sums, yielding |(A+B) intersect [1,x]| <= pi(x), apart from the harmless possible value 1 convention.\n- Lee's 2026 manuscript claims an infinite example with lower bound of order at least (log x/loglog x)^2 and denser subsequential constructions, with some strongest forms conditional.\n- Pomerance, Sárközy, and Stewart constructed large finite sets whose cross-sums are prime, but this does not itself give the required two infinite sets.\n\n**Full solution or refutation.**\n\nThe broad extremal question has rigorous upper-side progress and public lower constructions, but no accepted sharp asymptotic answer.\n\n**What remains.**\n\nIndependently verify the 2026 construction claims and close the large gap between polylogarithmic or subsequential lower bounds and the pi(x) upper bound.\n\n**Sources checked.**\n\n- Sungchul Lee, Erdős Problem 432, public manuscript repository (2026). (primary): https://github.com/lsngchl/Erdos432\n Evidence used: Claims the universal upper bound and several infinite lower constructions; the repository is public but not treated as refereed.\n- C. Pomerance, A. Sárközy and C. L. Stewart, On divisors of sums of integers, III, Pacific J. Math. 133 (1988), 363-379. (primary): https://doi.org/10.2140/pjm.1988.133.363\n Evidence used: Supplies a published finite prime-sum-grid construction relevant to lower-bound mechanisms.\n- Thomas F. Bloom, Erdős Problem #432, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/432\n Evidence used: Preserves the broad question and current open status.\n\n**Review notes.** The classification reflects concrete partial progress, not acceptance of the manuscript's strongest claims.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2119, "problem_number": "EP-436", "title": "Erdős Problem #436", "statement": "If $p$ is a prime and $k,m\\geq 2$ then let $r(k,m,p)$ be the minimal $r$ such that $r,r+1,\\ldots,r+m-1$ are all $k$th power residues modulo $p$. Let $ \\Lambda(k,m)=\\limsup_{p\\to \\infty} r(k,m,p). $ Is it true that $\\Lambda(k,2)$ is finite for all $k$? Is $\\Lambda(k,3)$ finite for all odd $k$? How large are they?", "background": "Asked by Lehmer and Lehmer \\cite{LeLe62}, who note that for example $\\Lambda(2,2)=9$ - indeed, $9$ is always a quadratic residue, and if $10$ isn't then either $2$ or $5$ is, and hence at least one of $1,2$ or $4,5$ or $9,10$ is a consecutive pair of quadratic residues (and similarly there are infinitely many $p$ for which there are no consecutive quadratic residues below $9,10$).\nA similar argument of Dunton \\cite{Du65} proves $\\Lambda(3,2)=77$, and Bierstedt and Mills \\cite{BiMi63} proved $\\Lambda(4,2)=1224$. Lehmer and Lehmer proved that $\\Lambda(k,3)=\\infty$ for all even $k$ and $\\Lambda(k,4)=\\infty$ for all $k\\leq 1048909$.\nLehmer, Lehmer, and Mills \\cite{LLM63} proved $\\Lambda(5,2)=7888$ and $\\Lambda(6,2)=202124$. Brillhart, Lehmer, and Lehmer \\cite{BLL64} proved $\\Lambda(7,2)=1649375$. Lehmer, Lehmer, Mills, and Selfridge \\cite{LLMS62} proved that $\\Lambda(3,3)=23532$.\nGraham \\cite{Gr64g} proved that $\\Lambda(k,l)=\\infty$ for all $k\\geq 2$ and $l\\geq 4$.\nHildebrand \\cite{Hi91} resolved the first question, proving that $\\Lambda(k,2)$ is finite for all $k$: in other words, for any $k\\geq 2$, if $p$ is sufficiently large then there exists a pair of consecutive $k$th power residues modulo $p$ in $[1,O_k(1)]$.\nThe remaining questions are to examine whether $\\Lambda(k,3)$ is finite for all odd $k\\geq 5$, and the growth rate of $\\Lambda(k,2)$ and $\\Lambda(k,3)$ as functions of $k$.\nReferences\n\n\n[BLL64] Brillhart, John and Lehmer, D. H. and Lehmer, Emma, Bounds for pairs of consecutive seventh and higher power\nresidues. Math. Comp. (1964), 397--407.\n\n[BiMi63] Bierstedt, R. G. and Mills, W. H., On the bound for a pair of consecutive quartic residues of a\nprime. Proc. Amer. Math. Soc. (1963), 628--632.\n\n[Du65] Dunton, M., Bounds for pairs of cubic residues. Proc. Amer. Math. Soc. (1965), 330--332.\n\n[Gr64g] Graham, R. L., On quadruples of consecutive {$k$}th power residues. Proc. Amer. Math. Soc. (1964), 196--197.\n\n[Hi91] Hildebrand, Adolf, On consecutive {$k$}th power residues. II. Michigan Math. J. (1991), 241-253.\n\n[LLM63] Lehmer, D. H. and Lehmer, Emma and Mills, W. H., Pairs of consecutive power residues. Canadian J. Math. (1963), 172--177.\n\n[LLMS62] Lehmer, D. H. and Lehmer, E. and Mills, W. H. and Selfridge,\nJ. L., Machine proof of a theorem on cubic residues. Math. Comp. (1962), 407--415.\n\n[LeLe62] Lehmer, D. H. and Lehmer, Emma, On runs of residues. Proc. Amer. Math. Soc. (1962), 102-106.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hildebrand solved the entire m=2 finiteness question, while m=3 for odd k>=5 and growth in k remain open.\n\n**Verified partial progress.**\n\n- Hildebrand proved Lambda(k,2) is finite for every k.\n- Exact values include Lambda(2,2)=9, Lambda(3,2)=77, Lambda(4,2)=1224, Lambda(5,2)=7888, Lambda(6,2)=202124, Lambda(7,2)=1649375, and Lambda(3,3)=23532.\n- Lambda(k,3) is infinite for even k, and Graham proved Lambda(k,m) is infinite for m>=4.\n\n**Full solution or refutation.**\n\nOne main subquestion is fully solved and several exact cases are known; the remaining odd-k triple problem and quantitative growth questions are unresolved.\n\n**What remains.**\n\nDecide finiteness of Lambda(k,3) for odd k>=5 and determine growth rates of the finite quantities as functions of k.\n\n**Sources checked.**\n\n- Adolf Hildebrand, On consecutive kth power residues. II, Michigan Math. J. 38 (1991), 241-253. (primary): https://www.erdosproblems.com/436\n Evidence used: The maintained bibliography links this primary theorem to the result Lambda(k,2)", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2120, "problem_number": "EP-445", "title": "Erdős Problem #445", "statement": "Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\\geq 0$, there exist $a,b\\in(n,n+p^c)$ such that $ab\\equiv 1\\pmod{p}$?", "background": "Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$. Heath-Brown \\cite{He00} used Kloosterman sums to prove this for all $c>3/4$.\nThis is discussed in this MathOverflow question.\nReferences\n\n\n[He00] Heath-Brown, D. R., Arithmetic applications of {K}loosterman sums. Nieuw Arch. Wiskd. (5) (2000), 380--384.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The desired exponent range c>1/2 remains open; Heath-Brown proved the uniform assertion for every c>3/4.\n\n**Verified partial progress.**\n\n- Heilbronn obtained an unpublished result for c sufficiently close to 1.\n- Heath-Brown used Kloosterman sums to prove the statement for all c>3/4.\n\n**Full solution or refutation.**\n\nKnown Kloosterman-sum methods do not reach the requested threshold 1/2 and are not known to reach the endpoint c=3/4.\n\n**What remains.**\n\nLower the uniform short-interval exponent from greater than 3/4 to every c>1/2.\n\n**Sources checked.**\n\n- D. R. Heath-Brown, Arithmetic applications of Kloosterman sums, Nieuw Arch. Wiskd. (5) 1 (2000), 380-384. (primary): https://www.nieuwarchief.nl/serie5/pdf/naw5-2000-01-4-380.pdf\n Evidence used: Primary source for the c>3/4 theorem.\n- Thomas F. Bloom, Erdős Problem #445, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/445\n Evidence used: States current open status and summarizes the Heilbronn and Heath-Brown ranges.\n\n**Review notes.** Nearby average-case and nonuniform modular-inverse results were not treated as resolving the every-n statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2121, "problem_number": "EP-450", "title": "Erdős Problem #450", "statement": "How large must $y=y(\\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\\epsilon y$?", "background": "It is not clear what the intended quantifier on $x$ is. Cambie has observed that if this is intended to hold for all $x$ then, provided $ \\epsilon(\\log n)^\\delta (\\log\\log n)^{3/2}\\to \\infty $ as $n\\to \\infty$, where $\\delta=0.086\\cdots$, there is no such $y$, which follows from an averaging argument and the work of Ford \\cite{Fo08}.\nOn the other hand, Cambie has observed that if $\\epsilon\\ll 1/n$ then $y(\\epsilon,n)\\sim 2n$: indeed, if $y<2n$ then this is impossible taking $x+n$ to be a multiple of the lowest common multiple of $\\{n+1,\\ldots,2n-1\\}$. On the other hand, for every fixed $\\delta\\in (0,1)$ and $n$ large every $2(1+\\delta)n$ consecutive elements contains many elements which are a multiple of an element in $(n,2n)$.\nReferences\n\n\n[Fo08] Ford, Kevin, The distribution of integers with a divisor in a given\ninterval. Ann. of Math. (2) (2008), 367-433.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported statement has a free variable x and therefore no determinate status without an intended quantifier; a recent claimed linear-scale solution selects a uniform interpretation but is not an accepted resolution of the exact record.\n\n**Verified partial progress.**\n\n- The maintained tracker explicitly says that the intended quantifier on x is unclear and records regime-specific observations by Cambie under a uniform-in-x reading.\n- Ford determined the order of magnitude of the global divisor-in-interval counting function H(X,y,z), which informs averaged and regime-specific analysis.\n- A July 2026 webpage claims y=Theta_epsilon(n) for fixed epsilon under the uniform reading and advertises a Lean bundle, but no independently accepted paper matching the ambiguous imported formulation was located.\n\n**Full solution or refutation.**\n\nNo literature result can solve all possible readings of the unquantified statement. The recent claimed proof must be checked against an explicitly chosen formal statement.\n\n**What remains.**\n\nRecover the intended quantifier on x, state the parameter asymptotics precisely, and independently verify whether the advertised uniform proof establishes that formulation.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #450, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/450\n Evidence used: Flags the missing quantifier, retains open status, and records Cambie's consequences under a uniform reading.\n- Kevin Ford, The distribution of integers with a divisor in a given interval, Ann. of Math. 168 (2008), 367-433, doi:10.4007/annals.2008.168.367. (primary): https://doi.org/10.4007/annals.2008.168.367\n Evidence used: Determines the order of magnitude of H(X,y,z), the global count of integers with a divisor in a prescribed interval.\n- Star Fleet Math, A Linear-Scale Solution to Erdős Problem 450 (2026 webpage). (primary): https://www.starfleetmath.com/\n Evidence used: Claims a uniform linear-scale theorem and a formal bundle, but chooses one reading of the ambiguous source and has not been treated here as independently accepted.\n\n**Review notes.** Formulation defect preserved: x is free/unquantified. No repair or preferred interpretation was inserted into the source statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2122, "problem_number": "EP-451", "title": "Erdős Problem #451", "statement": "Estimate $n_k$, the smallest integer $>2k$ such that $\\prod_{1\\leq i\\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.", "background": "Erd\\H{o}s and Graham write 'we can prove $n_k>k^{1+c}$ but no doubt much more is true'.\nIn \\cite{Er79d} Erd\\H{o}s writes that probably $n_kk^d$ for all constant $d$.\nAdenwalla observes that an easy upper bound is $n_k\\leq \\prod_{k\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Van Doorn and Tang proved a superpolynomial lower bound for n_k in 2026, confirming Erdős's lower-growth conjecture, but no matching-order estimate is known.\n\n**Verified partial progress.**\n\n- For all sufficiently large k, n_k > exp((log k)^2/(20 log log k)).\n- This implies n_k > k^d for every fixed d, while the elementary upper bound remains exp(O(k)).\n\n**Full solution or refutation.**\n\nThe 2026 theorem settles the conjectured superpolynomial lower growth but not the open-ended request to estimate n_k or the conjectural subexponential upper scale.\n\n**What remains.**\n\nNarrow the gap between exp((log k)^2/(20 log log k)) and exp(O(k)), in particular prove or refute n_k=exp(o(k)).\n\n**Sources checked.**\n\n- Wouter van Doorn and Quanyu Tang, Consecutive integers free of certain prime factors, arXiv:2606.19863 (2026). (primary): https://arxiv.org/abs/2606.19863\n Evidence used: The abstract states and the paper proves the explicit superpolynomial lower bound for the same n_k.\n- Thomas F. Bloom, Erdős Problem #451, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/451\n Evidence used: Retains open status while recording the historical lower and upper expectations.\n- Erdős Problems, EP-451 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/451\n Evidence used: Records the human-written arXiv revision and independent confidence in the revised result.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2123, "problem_number": "EP-452", "title": "Erdős Problem #452", "statement": "Let $\\omega(n)$ count the number of distinct prime factors of $n$. What is the size of the largest interval $I\\subseteq [x,2x]$ such that $\\omega(n)>\\log\\log n$ for all $n\\in I$?", "background": "Erd\\H{o}s \\cite{Er37} proved that the density of integers $n$ with $\\omega(n)>\\log\\log n$ is $1/2$. The Chinese remainder theorem implies that there is such an interval with $ \\lvert I\\rvert \\geq (1+o(1))\\frac{\\log x}{(\\log\\log x)^2}. $ It could be true that there is such an interval of length $(\\log x)^{k}$ for arbitrarily large $k$.\nReferences\n\n\n[Er37] Erd\"{o}s, Paul, Note on the number of prime divisors of integers. J. London Math. Soc. (1937), 308-314.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The maximal length of an interval on which every integer has more than log log n distinct prime factors remains unknown.\n\n**Verified partial progress.**\n\n- Erdős proved that integers n with omega(n)>log log n have density 1/2.\n- The Chinese remainder theorem gives intervals of length at least (1+o(1)) log x/(log log x)^2.\n\n**Full solution or refutation.**\n\nNo source located proves polylogarithmic intervals of arbitrarily high fixed degree or determines the largest interval's order.\n\n**What remains.**\n\nDetermine the asymptotic scale of the largest such interval, or materially improve the CRT lower bound and supply an upper bound.\n\n**Sources checked.**\n\n- Paul Erdős, Note on the Number of Prime Divisors of Integers, Journal of the London Mathematical Society s1-12 (1937), 308-314. (primary): https://doi.org/10.1112/jlms/s1-12.48.308\n Evidence used: Publisher record for the classical paper underlying the density statement.\n- Thomas F. Bloom, Erdős Problem #452, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/452\n Evidence used: States the exact problem, CRT lower bound, stronger conjectural scale, and current open status; no partial claims appear in comments.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2124, "problem_number": "EP-454", "title": "Erdős Problem #454", "statement": "Let $ f(n) = \\min_{i\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Pomerance proved the limsup is at least 2, but unboundedness remains open.\n\n**Verified partial progress.**\n\n- Pomerance's convex-hull argument gives limsup_n(f(n)-2p_n)>=2.\n- A 2025 tracker comment notes that the relevant extreme points of the prime graph are sparse, so prime-tuple conjectures do not directly settle the question.\n\n**Full solution or refutation.**\n\nThe known positive constant lower bound does not establish the requested infinite limsup.\n\n**What remains.**\n\nProve arbitrarily large positive deviations or produce a finite upper bound for the limsup.\n\n**Sources checked.**\n\n- Carl Pomerance, The Prime Number Graph, Mathematics of Computation 33 (1979), 399-408. (primary): https://math.dartmouth.edu/~carlp/PDF/paper19.pdf\n Evidence used: Author-hosted primary paper establishing the convex-hull result behind the lower bound.\n- Thomas F. Bloom, Erdős Problem #454 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/454\n Evidence used: Records open status, Pomerance's bound, and Tao's explanation of the remaining obstacle.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2125, "problem_number": "EP-455", "title": "Erdős Problem #455", "statement": "Let $q_10.352\\cdots. $ \nReferences\n\n\n[Ri76] Richter, Bernd, \"{U}ber die Monotonie von Differenzenfolgen. Acta Arith. (1976), 225-227.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Nondecreasing prime gaps force a positive quadratic lower constant, but it is unknown whether q_n/n^2 tends to infinity.\n\n**Verified partial progress.**\n\n- Richter proved liminf q_n/n^2 > 0.352... for every such prime sequence.\n\n**Full solution or refutation.**\n\nA fixed positive lower bound is strictly weaker than divergence of the normalized sequence.\n\n**What remains.**\n\nProve q_n/n^2 tends to infinity or construct a sequence with bounded normalized values along an infinite subsequence.\n\n**Sources checked.**\n\n- Bernd Richter, Über die Monotonie von Differenzenfolgen, Acta Arithmetica 30 (1976), 225-227. (primary): https://doi.org/10.4064/aa-30-3-225-227\n Evidence used: Publisher-hosted record and paper for the 0.352... lower bound.\n- Thomas F. Bloom, Erdős Problem #455, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/455\n Evidence used: States the exact question, Richter's partial result, open status, and no later comment claims.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2126, "problem_number": "EP-456", "title": "Erdős Problem #456", "statement": "Let $p_n$ be the smallest prime $\\equiv 1\\pmod{n}$ and let $m_n$ be the smallest integer such that $n\\mid \\phi(m_n)$.\nIs it true that $m_n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A 2026 unpublished AI-assisted manuscript claims unconditional negative resolutions of the first two questions and a Dickson-conditional answer to the third, but the claim lacks independent expert or publication-level verification.\n\n**Verified partial progress.**\n\n- Established background gives m_n<=p_n, infinitely many strict inequalities, and m_n/n tending to infinity for almost all n.\n- For n=2^(2k+1), one has m_n<=2n=cx, which would disprove both almost-all assertions, and gives a Dickson-conditional uniqueness family p=8l+1.\n\n**Full solution or refutation.**\n\nThe recent claim is too weakly verified to classify the first two questions as disproved; the third is conditional even within that manuscript.\n\n**What remains.**\n\nObtain an independent expert audit or peer-reviewed primary source for the density theorem, and settle the uniqueness question unconditionally.\n\n**Sources checked.**\n\n- Paul Erdős, Some unconventional problems in number theory, Astérisque 61 (1979), 73-82. (primary): https://www.numdam.org/item/AST_1979__61__73_0/\n Evidence used: Primary historical source for the problem and established background.\n- Thomas F. Bloom, Erdős Problem #456, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/456\n Evidence used: The main page retains open status and records only established special cases.\n- David Turturean, claimed solution manuscript summarized in the EP-456 discussion, 4 May 2026. (source_collection): https://www.erdosproblems.com/forum/thread/456\n Evidence used: States the claimed positive-lower-density theorem, the Dickson-conditional construction, the 71-page length, and that checking was by the author and AI rather than independent publication review.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2127, "problem_number": "EP-457", "title": "Erdős Problem #457", "statement": "Is there some $\\epsilon>0$ such that there are infinitely many $n$ where all primes $p\\leq (2+\\epsilon)\\log n$ divide $ \\prod_{1\\leq i\\leq \\log n}(n+i)? $ ", "background": "A problem of Erd\\H{o}s and Pomerance.\nMore generally, let $q(n,k)$ denote the least prime which does not divide $\\prod_{1\\leq i\\leq k}(n+i)$. This problem asks whether $q(n,\\log n)\\geq (2+\\epsilon)\\log n$ infinitely often. Taking $n$ to be the product of primes between $\\log n$ and $(2+o(1))\\log n$ gives an example where $ q(n,\\log n)\\geq (2+o(1))\\log n. $ Can one prove that $q(n,\\log n)<(1-\\epsilon)(\\log n)^2$ for all large $n$ and some $\\epsilon>0$?\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The exact displayed infinite-often lower-bound statement is solved and Lean-verified. A first construction gives constant 3/log 4>2, and a Dirichlet-approximation refinement gives arbitrarily large fixed constants and even a factor of order log log n/log log log n beyond log n.\n\n**Verified partial progress.**\n\n- The elementary primorial construction reaches q(n,log n) >= (2+o(1))log n but not a fixed improvement beyond 2.\n- The first 2026 argument improves 2 to 3/log 4, already resolving the displayed question.\n- Tao's elaboration gives q(n,log n) > ((1-o(1))/2)*(log log n/log log log n)*log n infinitely often.\n\n**Full solution or refutation.**\n\nPrimes up to the interval length divide every product of that many consecutive integers. For primes between the interval length and a larger cutoff, simultaneous Dirichlet approximation makes n modulo each prime lie in a short window near zero, so one of the consecutive factors is divisible by that prime. This costs far less than imposing exact Chinese-remainder congruences and permits a cutoff larger than any fixed multiple of log n.\n\n**What remains.**\n\nThe displayed lower-bound question is closed. The likely 1979 intended question asks the opposite eventual upper bound q(n,floor(log n))<(2+epsilon)log n, and the dataset's successor bound q(n,log n)<(1-epsilon)(log n)^2 eventually also remains open.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #457, maintained problem record, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/457\n Evidence used: The record marks the displayed statement PROVED (LEAN), states the 3/log 4 construction, the arbitrarily-large-constant refinement, and Tao's growing lower bound.\n- Kevin Barreto, Terence Tao, and contributors, Erdős Problem #457 discussion, March 2026. (primary): https://www.erdosproblems.com/forum/thread/457\n Evidence used: This is the original public proof trail: it gives the Dirichlet-approximation mechanism, distinguishes the opposite-direction 1979 question, and reports the exact theorem's Lean check.\n- Nat Sothanaphan et al., AI contributions to Erdős problems, entry #457, revision current to 30 June 2026. (formal_verification): https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems\n Evidence used: The maintained verification index records #457 as a full solution in Lean by Aristotle and GPT-5.2 Pro on 2 March 2026.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2128, "problem_number": "EP-460", "title": "Erdős Problem #460", "statement": "Let $a_0=0$ and $a_1=1$, and in general define $a_k$ to be the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\\leq in$ (this observation was made by Svyable using ChatGPT).\nUnfortunately, although both sources mention a forthcoming paper of Eggleton, Erd\\H{o}s, and Selfridge, I cannot find a candidate paper of theirs with this problem in, and hence the motivation behind this problem, and what the precise problem intended was, is unclear.\nChojecki has noted in the comments that a positive solution to the main problem would follow if $ f(n) = \\sum_{aa}\\frac{1}{a}\\to \\infty, $ where $P^-(\\cdot)$ is the least prime factor. Standard estimates on rough numbers show that $\\frac{1}{N}\\sum_{n\\leq N}f(n)\\gg \\log\\log N$, so $f(n)$ does diverge on average, but it is unclear whether $f(n)\\to \\infty$ for all $n$.\nReferences\n\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The historical problem has incompatible source formulations and omitted truncation, while the imported final clauses contain an undefined index; the intended nontrivial truncated question remains open.\n\n**Verified partial progress.**\n\n- Without the restriction a_in.\n- For the truncated rough-number sub-sum, its average over n<=N is of order at least log log N, yielding unboundedness along a subsequence but not pointwise divergence.\n- A January 2026 note initially presented as a full solution was corrected in discussion: it does not prove f(n) tends to infinity for all n.\n\n**Full solution or refutation.**\n\nNo verified result settles the nontrivial pointwise truncated problem, and the undefined j in the imported split sums prevents a faithful status assignment to those clauses.\n\n**What remains.**\n\nFix the intended historical formulation and indices explicitly, then prove pointwise divergence for the chosen truncated version or find a counterexample.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #460 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/460\n Evidence used: Documents the incompatible source versions, trivial untruncated divergence, unresolved pointwise step, and correction of the claimed solution.\n- Przemek Chojecki, Erdős Problem #460: a trivial divergence and a nontrivial truncated lower bound, version 2 (2026). (primary): https://www.ulam.ai/research/erdos-460-v2.pdf\n Evidence used: Develops the rough-number reduction and mean-order lower bound but does not prove divergence for every n.\n- Paul Erdős and Ronald Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 91. (source_collection): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: One of the original ambiguous formulations; it differs in the initial term and index range and omits truncation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2129, "problem_number": "EP-461", "title": "Erdős Problem #461", "statement": "Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The uniform linear lower bound for the number of distinct t-smooth components remains open.\n\n**Verified partial progress.**\n\n- Erdős and Graham proved f(n,t) >> t/log t uniformly.\n- A June 2026 forum averaging argument gives inf_n f(n,t) <= (1-e^(-gamma)+o(1))t, ruling out universal lower coefficients above about 0.4385 but not the existence of a smaller positive coefficient.\n- A proposed matching route targeting t/2 was falsified by a finite counterexample to the auxiliary matching assertion.\n\n**Full solution or refutation.**\n\nNeither the upper obstruction nor the failed t/2 strengthening resolves whether some uniform positive linear lower constant exists.\n\n**What remains.**\n\nImprove the lower bound from t/log t to ct for an absolute c>0, or construct examples with f(n,t)=o(t).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #461, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/461\n Evidence used: States the exact problem, known t/log t lower bound, and open status.\n- EP-461 discussion, posts by Pr_Huang and others, June 2026. (source_collection): https://www.erdosproblems.com/forum/thread/461\n Evidence used: Contains the periodic averaging upper obstruction and explicitly notes that it does not settle the original linear-lower-bound question.\n- Paul Erdős and Ronald Graham, Old and New Problems and Results in Combinatorial Number Theory (1980). (source_collection): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: Historical source for the problem and reported t/log t result.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2130, "problem_number": "EP-462", "title": "Erdős Problem #462", "statement": "Let $p(n)$ denote the least prime factor of $n$. There is a constant $c>0$ such that $ \\sum_{\\substack{n0$ such that $ \\sum_{x\\leq n\\leq x+Cx^{1/2}(\\log x)^2}\\frac{p(n)}{n} \\gg 1 $ for all large $x$?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The stated least-prime-factor lower bound in every interval of length C sqrt(x)(log x)^2 remains open.\n\n**Verified partial progress.**\n\n- A global composite-only asymptotic sum is given as established background, but it does not imply the required bound in every short interval.\n- The maintained tracker records no partial or complete claim for the short-interval assertion.\n\n**Full solution or refutation.**\n\nNo short-interval theorem located supplies a uniform positive lower bound at the stated length scale.\n\n**What remains.**\n\nProve the bound uniformly for all large x or find intervals violating it; first clarify whether primes are intentionally included in the short sum.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #462, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/462\n Evidence used: Matches the imported statement, retains open status, and records no claimed partial solution.\n- Paul Erdős and Ronald Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 92. (source_collection): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: Historical source of the short-interval question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2131, "problem_number": "EP-463", "title": "Erdős Problem #463", "statement": "Is there a function $f$ with $f(n)\\to \\infty$ as $n\\to \\infty$ such that, for all large $n$, there is a composite number $m$ such that $ n+f(n)n}(m-p(m)), $ and whether $n-F(n)\\sim cn^{1/2}$ for some $c>0$.\nSee also [385].\nReferences\n\n\n[Er92e] Erd\\H{o}s, P\\'{a}l, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains open whether every sufficiently large n admits a composite m with an unbounded margin between n and both m and m-p(m).\n\n**Verified partial progress.**\n\n- Erdős reformulated the scale through F(n)=min_{m>n}(m-p(m)) and conjectured n-F(n)~c sqrt(n), which would be stronger than the imported assertion.\n- The related dual problem EP-385 is also currently open.\n\n**Full solution or refutation.**\n\nNo theorem located proves even the required pointwise margin tending to infinity.\n\n**What remains.**\n\nProve existence of any function f(n) tending to infinity with the stated property, or disprove it; the stronger square-root asymptotic is also open.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #463, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/463\n Evidence used: States the exact question, stronger F(n) formulation, zero substantive comments, and open status.\n- Paul Erdős, Many old and on some new problems of mine in number theory (1981). (primary): https://users.renyi.hu/~p_erdos/1981-09.pdf\n Evidence used: Primary text asks for composite m beyond n+c with m-p(m)", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2132, "problem_number": "EP-467", "title": "Erdős Problem #467", "statement": "Prove the following for all large $x$: there is a choice of congruence classes $a_p$ for all primes $p\\leq x$ and a decomposition $\\{p\\leq x\\}=A\\sqcup B$ into two non-empty sets such that, for all $n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported congruence-covering statement is explicitly a modern reconstruction of an Erdős-Graham source missing crucial quantifiers, so the historical problem's exact status cannot be assigned confidently.\n\n**Verified partial progress.**\n\n- The maintained tracker labels the reconstructed statement open but records no partial or complete solution claims.\n- The tracker's editor expressly warns that the reconstruction may misinterpret the original source.\n\n**Full solution or refutation.**\n\nNo result located settles the displayed reconstruction, and it cannot safely be treated as a verbatim historical conjecture.\n\n**What remains.**\n\nRecover or agree on the intended quantifiers from source context, then determine the status of that precise formulation.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #467, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/467\n Evidence used: States that the displayed formulation is assumed intent, flags missing crucial quantifiers, and records no claimed progress.\n- Thomas F. Bloom, revision history for Erdős Problem #467, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/467\n Evidence used: Confirms that the ambiguity warning is part of the current and original tracker versions.\n- Paul Erdős and Ronald Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 93. (source_collection): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: Original source whose missing quantifiers necessitated the unverified reconstruction.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2133, "problem_number": "EP-468", "title": "Erdős Problem #468", "statement": "For any $n$ let $D_n$ be the set of sums of the shape $d_1,d_1+d_2,d_1+d_2+d_3,\\ldots$ where $1\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The divisor-prefix-sum novelty and minimal-index questions remain open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo later resolution was located.\n\n**What remains.**\n\nEstimate novelty counts or prove the requested sublinearity.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #468, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/468\n Evidence used: Maintained tracker lists open status.\n\n**Review notes.** Trailing source serialization is preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2134, "problem_number": "EP-469", "title": "Erdős Problem #469", "statement": "Let $A$ be the set of all $n$ such that $n=d_1+\\cdots+d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m\\mid n$ with $m\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The displayed reciprocal sum over primitive pseudoperfect numbers is now recorded as convergent. A commit-pinned, sorry-free Lean development proves the exact summability statement, and a reviewed Formal Conjectures status update reports that all intermediate hypotheses are discharged. Lewis's cited 2025 prose preprint was not independently located, so expert review remains appropriate.\n\n**Verified partial progress.**\n\n- Benkoski and Erdős introduced the primitive pseudoperfect set and conjectured sparsity strong enough to imply convergence.\n- The Lean proof establishes weighted summability results for primitive-nondeficient contributions and discharges the required decorated contribution certificates.\n- A Formal Conjectures review checked that the final theorem uses the same set A and the same real reciprocal-series summability assertion as EP-469.\n\n**Full solution or refutation.**\n\nThe completed formal development proves unconditionally that the function n -> 1/n is summable on the subtype of positive pseudoperfect integers having no smaller pseudoperfect divisor. Its internal decomposition introduces contribution packages, but the final file constructs or proves every package rather than assuming it. The Formal Conjectures statement file still contains `sorry`; the separate pinned plby file is the completed formal proof.\n\n**What remains.**\n\nPublish and archive Lewis's conventional proof, independently audit the recent Lean development, and separately investigate the primitive-weird reciprocal sum (OEIS A119425) and the third divisor-inheritance set mentioned by Benkoski--Erdős. Those companion questions are not components proved by the EP-469 theorem. The input background also has trailing serialization noise, but the displayed statement is intact.\n\n**Sources checked.**\n\n- plby/lean-proofs, Erdos469.lean, commit 68da20b96673899166e94638f5a7fffeb7231d35 (2026). (formal_verification): https://github.com/plby/lean-proofs/blob/68da20b96673899166e94638f5a7fffeb7231d35/src/latest/ErdosProblems/Erdos469.lean\n Evidence used: Commit-pinned sorry-free Lean proof of the exact primitive-pseudoperfect reciprocal summability theorem.\n- Google DeepMind Formal Conjectures, FormalConjectures/ErdosProblems/469.lean and reviewed status-sync pull request 4718 (accessed 2026-08-17). (authoritative_secondary): https://github.com/google-deepmind/formal-conjectures/pull/4718\n Evidence used: Records Lewis's affirmative result, checks exact statement correspondence and unconditionality, pins the formal proof, and reports a successful build and reviewed merge.\n- S. J. Benkoski and P. Erdős, On Weird and Pseudoperfect Numbers, Mathematics of Computation 28 (1974), 617--623. (primary): https://combinatorica.hu/~p_erdos/1974-24.pdf\n Evidence used: Original primary source defining primitive pseudoperfect numbers, posing the reciprocal-sum question, and discussing the companion sets.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2135, "problem_number": "EP-470", "title": "Erdős Problem #470", "statement": "Call $n$ weird if $\\sigma(n)\\geq 2n$ and $n$ is not pseudoperfect, that is, it is not the sum of any set of its divisors.\nAre there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird?", "background": "Weird numbers were investigated by Benkoski and Erd\\H{o}s \\cite{BeEr74}, who proved that the set of weird numbers has positive density. The smallest weird number is $70$.\nMelfi \\cite{Me15} has proved that there are infinitely many primitive weird numbers, conditional on the fact that $p_{n+1}-p_n<\\frac{1}{10}p_n^{1/2}$ for all large $n$, which in turn would follow from well-known conjectures concerning prime gaps.\nThe sequence of weird numbers is A006037 in the OEIS. Fang \\cite{Fa22} has shown there are no odd weird numbers below $10^{21}$, and Liddy and Riedl \\cite{LiRi18} have shown that an odd weird number must have at least 6 prime divisors.\nIf there are no odd weird numbers then every weird number has abundancy index $<4$ (see [825]).\nThis is problem B2 in Guy's collection \\cite{Gu04} (the \\$10 is reported by Guy, offered by Erd\\H{o}s for a solution to the question of whether any odd weird numbers exist).\nReferences\n\n\n[BeEr74] Benkoski, S. J. and Erd\\H{o}s, P., On weird and pseudoperfect numbers. Math. Comp. (1974), 617-623.\n\n[Fa22] Searching on the boundary of abundance for odd weird numbers, W. Fang. arXiv:2207.12906 (2022).\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[LiRi18] J. Liddy and J. Riedl, An algorithm to determine all odd primitive abundant numbers with $d$ prime divisors. Honors Research Projects. 728 (2018).\n\n[Me15] Melfi, Giuseppe, On the conditional infiniteness of primitive weird numbers. J. Number Theory (2015), 508-514.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Odd weird numbers and unconditional primitive-weird infinitude remain open.\n\n**Verified partial progress.**\n\n- Benkoski--Erdős proved weird numbers have positive density.\n- Fang excluded odd weird numbers below 10^21; Liddy--Riedl show an odd example needs at least six prime factors.\n- Melfi's primitive-infinite result is conditional on strong prime gaps.\n\n**Full solution or refutation.**\n\nNo unconditional answer to either question is known.\n\n**What remains.**\n\nFind/exclude odd weird numbers and prove primitive infinitude unconditionally.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #470, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/470\n Evidence used: Records all listed progress and open status.\n\n**Review notes.** Conditional and unconditional results are separated.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2136, "problem_number": "EP-472", "title": "Erdős Problem #472", "statement": "Given some initial finite sequence of primes $q_1<\\cdots\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that any Ulam-style prime extension is infinite was located.\n\n**Verified partial progress.**\n\n- The seed 3,5 has a long observed continuation and heuristic support.\n\n**Full solution or refutation.**\n\nObserved continuation and heuristic probability do not prove infinitude.\n\n**What remains.**\n\nProve an infinite seed or show every seed terminates.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #472, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/472\n Evidence used: Lists the seed and continuing open status.\n\n**Review notes.** Tracker heuristic is not a theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2137, "problem_number": "EP-477", "title": "Erdős Problem #477", "statement": "Is there a polynomial $f:\\mathbb{Z}\\to \\mathbb{Z}$ of degree at least $2$ and a set $A\\subset \\mathbb{Z}$ such that for any $n\\in \\mathbb{Z}$ there is exactly one $a\\in A$ and $b\\in \\{ f(n) : n\\in\\mathbb{Z}\\}$ such that $n=a+b$?", "background": "A question of Erd\\H{o}s and Graham, who thought the answer was negative.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The requested construction is impossible for every quadratic f, but higher-degree cases remain open.\n\n**Verified partial progress.**\n\n- A modular difference-set argument rules out arbitrary quadratic polynomials.\n\n**Full solution or refutation.**\n\nThe degree-at-least-two existential question is reduced to degree at least three.\n\n**What remains.**\n\nRule out or construct a higher-degree example.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #477, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/477\n Evidence used: Records the quadratic proof and open general status.\n\n**Review notes.** Formalization status was not itself used as proof.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2138, "problem_number": "EP-478", "title": "Erdős Problem #478", "statement": "Let $p$ be a prime and $ A_p = \\{ k! \\pmod{p} : 1\\leq k\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The conjectural linear asymptotic for distinct factorial residues remains open.\n\n**Verified partial progress.**\n\n- Grebennikov--Sagdeev--Semchankau--Vasilevskii prove |A_p|>=(sqrt(2)-o(1))sqrt(p).\n- Average results and large finite socialist-prime exclusions are known.\n\n**Full solution or refutation.**\n\nKnown bounds are far below the proposed linear scale.\n\n**What remains.**\n\nProve an asymptotic or stronger lower bounds.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #478, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/478\n Evidence used: Records current lower bound and open status.\n\n**Review notes.** Finite checks are not asymptotic proof.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2139, "problem_number": "EP-479", "title": "Erdős Problem #479", "statement": "Is it true that, for all $k\neq 1$, there are infinitely many $n$ such that $2^n\\equiv k\\pmod{n}$?", "background": "A conjecture of Graham. It is easy to see that $2^n\not\\equiv 1\\mod{n}$ for all $n>1$, so the restriction $k\neq 1$ is necessary. Erd\\H{o}s and Graham report that Graham, Lehmer, and Lehmer have proved this for $k=2^i$ for $i\\geq 1$, or if $k=-1$, but I cannot find such a paper. Tang has written a short note giving a proof for this case.\nAs an indication of the difficulty, when $k=3$ the smallest $n$ such that $2^n\\equiv 3\\pmod{n}$ is $n=4700063497$.\nThe minimal such $n$ for each $k$ is A036236 in the OEIS.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The all-k assertion remains open, though several k families are known.\n\n**Verified partial progress.**\n\n- Infinite solution sets are known for k in {0,-1,-2,2^i:i>=1}.\n- The k=3 least example is extremely large.\n\n**Full solution or refutation.**\n\nKnown classes do not cover all k!=1.\n\n**What remains.**\n\nResolve any remaining fixed k, especially general k.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #479, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/479\n Evidence used: Records known classes and open status.\n\n**Review notes.** Missing historical citation is noted, not invented.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2140, "problem_number": "EP-483", "title": "Erdős Problem #483", "statement": "Let $f(k)$ be the minimal $N$ such that if $\\{1,\\ldots,N\\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $f(k)$. In particular, is it true that $f(k) < c^k$ for some constant $c>0$?", "background": "The values of $f(k)$ are known as Schur numbers. The best-known bounds for large $k$ are $ (380)^{k/5}-O(1)\\leq f(k) \\leq \\lfloor(e-\\tfrac{1}{24}) k!\\rfloor-1. $ The lower bound is due to Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik \\cite{ACPPRT21} (improving previous bounds of Exoo \\cite{Ex94} and Fredricksen and Sweet \\cite{FrSw00}) and the upper bound is due to Whitehead \\cite{Wh73}. Note that $380^{1/5}\\approx 3.2806$.\nThe known values of $f$ are $f(1)=2$, $f(2)=5$, $f(3)=14$, $f(4)=45$, and $f(5)=161$ (see A030126). (The equality $f(5)=161$ was established by Heule \\cite{He17}).\nSee also [183] (in particular a folklore observation gives $f(k)\\leq R(3;k)-1$).\nReferences\n\n\n[ACPPRT21] R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel, and J. Tomasik, New lower bounds for Schur and weak Schur numbers. arXiv:2112.03175 (2021).\n\n[Ex94] Exoo, G., A lower bound for Schur numbers and multicolor Ramsey numbers. Electronic J. of Combinatorics (1994).\n\n[FrSw00] Fredricksen, Harold and Sweet, Melvin M., Symmetric sum-free partitions and lower bounds for {S}chur\nnumbers. Electron. J. Combin. (2000), Research Paper 32, 9.\n\n[He17] M. Heuele, Schur Number Five. arXiv:1711.08076 (2017).\n\n[Wh73] Whitehead, Jr., Earl Glen, The {R}amsey number {$N(3,\\,3,\\,3,\\,3;\\,2)$}. Discrete Math. (1973), 389--396.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether Schur numbers grow at most exponentially remains open.\n\n**Verified partial progress.**\n\n- Current bounds are 380^(k/5)-O(1) below and (e-1/6)k! above.\n- Exact values through k=5 are known.\n\n**Full solution or refutation.**\n\nThe factorial upper bound leaves the requested exponential behavior open.\n\n**What remains.**\n\nEstablish an exponential upper bound or disprove it.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #483, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/483\n Evidence used: Records bounds, values, and open status.\n\n**Review notes.** The earlier source had a stale upper-bound attribution; current page was used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2141, "problem_number": "EP-486", "title": "Erdős Problem #486", "statement": "Let $A\\subseteq \\mathbb{N}$, and for each $n\\in A$ choose some $X_n\\subseteq \\mathbb{Z}/n\\mathbb{Z}$. Let $ B = \\{ m\\in \\mathbb{N} : m\not\\in X_n\\pmod{n}\\textrm{ for all }n\\in A\\textrm{ with }m>n\\}. $ Must $B$ have a logarithmic density, i.e. is it true that $ \\lim_{x\\to \\infty} \\frac{1}{\\log x}\\sum_{\\substack{m\\in B\\\\ m\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported statement omits an activation-threshold condition present in the source; as written it admits a cheap negative construction, while the intended problem is open.\n\n**Verified partial progress.**\n\n- Davenport--Erdős prove logarithmic density for the special case X_n={0}.\n\n**Full solution or refutation.**\n\nThe literal dataset text and intended historical question differ materially.\n\n**What remains.**\n\nPreserve the literal text; verify the source and triage the thresholded version separately.\n\n**Sources checked.**\n\n- Thomas F. Bloom, EP-486 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/486?embed=1\n Evidence used: Identifies the missing threshold and explains its effect.\n\n**Review notes.** No OCR/formulation correction was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2142, "problem_number": "EP-488", "title": "Erdős Problem #488", "statement": "Let $A$ be a finite set and $ B=\\{ n \\geq 1 : a\\mid n\\textrm{ for some }a\\in A\\}. $ Is it true that, for every $m>n\\geq \\max(A)$, $ \\frac{\\lvert B\\cap [1,m]\\rvert }{m}< 2\\frac{\\lvert B\\cap [1,n]\\rvert}{n}? $ ", "background": "The constant $2$ would be the best possible here, as witnessed by taking $A=\\{a\\}$, $n=2a-1$, and $m=2a$.\nThis problem is also discussed in problem E5 of Guy's collection \\cite{Gu04}.\nIn \\cite{Er61} this problem is as stated above, but with $a\\mid n$ in the definition of $B$ replaced by $a\nmid n$. This is most likely a typo (especially since the problem is also given as stated above in \\cite{Er66}). There have been several counterexamples given for this alternate problem. Cambie has observed that, if $A$ is the set of primes bounded above by $n$, and $m=2n$, then $ \\frac{\\lvert B\\cap [1,m]\\rvert }{m}=\\frac{\\pi(2n)-\\pi(n)+1}{2n}\\sim \\frac{1}{2\\log n} $ while $ \\frac{\\lvert B\\cap [1,n]\\rvert}{n}=\\frac{1}{n}. $ Further concrete counterexamples, found by Alexeev and Aristotle, are given in the comments section.\nReferences\n\n\n[Er61] Erd\\H{o}s, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutat\\'{o} Int. K\"{o}zl. (1961), 221-254.\n\n[Er66] Erd\\H{o}s, P\\'al, Remarks on number theory. {V}. {E}xtremal problems in number\ntheory. {II}. Mat. Lapok (1966), 135--155.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The intended multiples formulation remains open and finitely falsifiable. The 1961 source apparently printed the nondivisibility sign; known counterexamples concern that alternate version and do not refute the 1966/Guy multiples formulation. The corrected conjecture is proved for two-element sets and for primitive sets containing 2.\n\n**Verified partial progress.**\n\n- Removing any member divisible by another does not change B, so the problem reduces to divisibility-primitive A.\n- The tracker discussion gives an elementary proof for every set A with at most two essential elements.\n- The conjecture holds for every primitive A with minimum element 2.\n- Two-generator examples make the ratio approach 2 from below, confirming sharpness of the proposed constant.\n\n**Full solution or refutation.**\n\nNo general solution was located. For A={a,b}, direct floor-function estimates bound the density at m by twice the density already forced at n>=b. If 2 belongs to a primitive A, an additional odd generator forces the density at n above 1/2 while the density at m is below 1, giving the desired strict inequality. Hence a minimal unresolved case must have at least three primitive generators and minimum at least 3.\n\n**What remains.**\n\nProve the density-doubling inequality for general primitive sets of at least three integers with minimum at least 3, or exhibit a finite counterexample satisfying m>n>=max(A). Any claimed counterexample must be checked against the divisibility sign: examples for the printed a-not-divides-n variant address a different statement. The input background contains trailing serialization noise but otherwise preserves this distinction.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #488 and discussion (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/488\n Evidence used: Maintains the intended multiples statement as open, explains the 1961 sign typo, records counterexamples only to the alternate version, and links current special-case progress.\n- Erdős Problems discussion thread for Problem 488 (2026). (source_collection): https://www.erdosproblems.com/forum/thread/488\n Evidence used: Public mathematical notes proving the two-element and minimum-two cases and exhibiting ratios tending to 2 from below.\n- P. Erdős, Some Unsolved Problems, Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221--254. (primary): https://real.mtak.hu/201135/\n Evidence used: Original source containing the likely nondivisibility typo that must be distinguished from the intended problem.\n- HUN-REN Rényi Institute bibliography entry for P. Erdős, Remarks on number theory V. Extremal problems in number theory II, Mat. Lapok 17 (1966), 135--155. (bibliographic_index): https://renyi.hu/en/node/4094\n Evidence used: Confirms the later original-source record cited for the intended multiples restatement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2143, "problem_number": "EP-489", "title": "Erdős Problem #489", "statement": "Let $A\\subseteq \\mathbb{N}$ be a set such that $\\lvert A\\cap [1,x]\\rvert=o(x^{1/2})$. Let $ B=\\{ n\\geq 1 : a\nmid n\\textrm{ for all }a\\in A\\}. $ If $B=\\{b_1\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Existence of the second gap moment for this sifted set remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo current resolution was located.\n\n**What remains.**\n\nProve existence/finite value or construct a failure.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #489, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/489\n Evidence used: Maintained tracker lists open status.\n\n**Review notes.** Escaped source symbols are preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2144, "problem_number": "EP-495", "title": "Erdős Problem #495", "statement": "Let $\\alpha,\\beta \\in \\mathbb{R}$. Is it true that $ \\liminf_{n\\to \\infty} n \\| n\\alpha \\| \\| n\\beta\\| =0 $ where $\\|x\\|$ is the distance from $x$ to the nearest integer?", "background": "The infamous Littlewood conjecture.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Littlewood's conjecture remains open.\n\n**Verified partial progress.**\n\n- The conjecture is known for almost every pair and has major conditional/exceptional-set results.\n\n**Full solution or refutation.**\n\nNo proof for every pair alpha,beta is known.\n\n**What remains.**\n\nEstablish the liminf-zero assertion unconditionally for all pairs.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #495, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/495\n Evidence used: Maintained tracker lists open status and references.\n\n**Review notes.** The imported escaped \\not symbols are preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2145, "problem_number": "EP-500", "title": "Erdős Problem #500", "statement": "What is $\\mathrm{ex}_3(n,K_4^3)$? That is, the largest number of $3$-edges which can placed on $n$ vertices so that there exists no $K_4^3$, a set of 4 vertices which is covered by all 4 possible $3$-edges.", "background": "A problem of Tur\\'{a}n. Tur\\'{a}n observed that dividing the vertices into three equal parts $X_1,X_2,X_3$, and taking the edges to be those triples that either have exactly one vertex in each part or two vertices in $X_i$ and one vertex in $X_{i+1}$ (where $X_4=X_1$) shows that $ \\mathrm{ex}_3(n,K_4^3)\\geq\\left(\\frac{5}{9}+o(1)\\right)\\binom{n}{3}. $ This is probably the truth. The current best upper bound is $ \\mathrm{ex}_3(n,K_4^3)\\leq 0.5611666\\binom{n}{3}, $ due to Razborov \\cite{Ra10}.\nSee also [712] for the general case.\nReferences\n\n\n[Ra10] Razborov, Alexander A., On 3-hypergraphs with forbidden 4-vertex configurations. SIAM J. Discrete Math. (2010), 946-963.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Turán-density problem remains open between 5/9 and 0.5611666.\n\n**Verified partial progress.**\n\n- Turán's cyclic three-part construction gives asymptotic density 5/9.\n- Razborov proved the upper bound 0.5611666 by flag algebras.\n\n**Full solution or refutation.**\n\nNo equality proof or counterexample to the conjectural 5/9 density was located.\n\n**What remains.**\n\nClose the density gap.\n\n**Sources checked.**\n\n- A. Razborov, On 3-hypergraphs with forbidden 4-vertex configurations, SIAM J. Discrete Math. 2010. (primary): https://doi.org/10.1137/090747735\n Evidence used: Primary source for the displayed upper bound.\n- Thomas F. Bloom, Erdős Problem #500, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/500\n Evidence used: Current open status and bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2146, "problem_number": "EP-501", "title": "Erdős Problem #501", "statement": "For every $x\\in\\mathbb{R}$ let $A_x\\subset \\mathbb{R}$ be a bounded set with outer measure $<1$. Must there exist an infinite independent set, that is, some infinite $X\\subseteq \\mathbb{R}$ such that $x\not\\in A_y$ for all $x\neq y\\in X$?\nIf the sets $A_x$ are closed and have measure $<1$, then must there exist an independent set of size $3$?", "background": "Erd\\H{o}s and Hajnal \\cite{ErHa60} proved the existence of arbitrarily large finite independent sets (under the assumptions in the first problem).\nGladysz \\cite{Gl62} proved the existence of an independent set of size $2$ under the assumptions of the the second question.\nHechler \\cite{He72} has shown the answer to the first question is no, assuming the continuum hypothesis.\nNewelski, Pawlikowski, and Seredy\\'{n}ski \\cite{NPS87} proved the answer to the first question is yes, under the additional assumption that the $A_x$ are closed.\nReferences\n\n\n[ErHa60] Erd\\H{o}s, P. and Hajnal, A., Some remarks on set theory. VIII. Michigan Math. J. (1960), 187-191.\n\n[Gl62] G\\l adysz, S., Bemerkungen \"uber die {U}nabh\"{a}ngigkeit der {P}unkte in\n{B}ezug auf mengenwertige {F}unktionen. Acta Math. Acad. Sci. Hungar. (1962), 199--201.\n\n[He72] Hechler, S. H., On two problems in combinatorial set theory. Bull. Acad. Polon. Sci. S\\'{e}r. Sci. Math. Astronom. Phys. (1972), 429-431.\n\n[NPS87] Newelski, Ludomir and Pawlikowski, Janusz and Seredy\\'nski,\nWitold, Infinite free set for small measure set mappings. Proc. Amer. Math. Soc. (1987), 335--339.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general outer-measure question is not settled in ZFC by the located results; closed-set and CH cases are known.\n\n**Verified partial progress.**\n\n- Erdős--Hajnal proved arbitrarily large finite independent sets.\n- Hechler gave a CH counterexample to the first question.\n- Newelski--Pawlikowski--Seredyński proved an infinite independent set when the A_x are closed.\n\n**Full solution or refutation.**\n\nThe closed subclass is affirmatively solved, but the unrestricted first question remains unresolved by this evidence.\n\n**What remains.**\n\nResolve the unrestricted question in ZFC or establish independence.\n\n**Sources checked.**\n\n- L. Newelski, J. Pawlikowski and W. Seredyński, Infinite free set for small measure set mappings, Proc. Amer. Math. Soc. 1987. (primary): https://doi.org/10.1090/S0002-9939-1987-0870425-6\n Evidence used: Primary source for the closed-set result.\n- Thomas F. Bloom, Erdős Problem #501, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/501\n Evidence used: Current formulation and historical status summary.\n\n**Review notes.** The serialized statement loses LaTex negation escapes; this was flagged but not corrected.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2147, "problem_number": "EP-503", "title": "Erdős Problem #503", "statement": "What is the size of the largest $A\\subseteq \\mathbb{R}^d$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $x,y,z$ from $A$, at least two of the distances $\\lvert x-y\\rvert,\\lvert y-z\\rvert,\\lvert x-z\\rvert$ are equal.", "background": "When $d=2$ the answer is $6$ (due to Kelly \\cite{ErKe47} - an alternative proof is given by Kov\\'{a}cs \\cite{Ko24c}). When $d=3$ the answer is $8$ (due to Croft \\cite{Cr62}). The best upper bound known in general is due to Blokhuis \\cite{Bl84} who showed that $ \\lvert A\\rvert \\leq \\binom{d+2}{2}. $ Alweiss has observed a lower bound of $\\binom{d+1}{2}$ follows from considering the subset of $\\mathbb{R}^{d+1}$ formed of all vectors $e_i+e_j$ where $e_i,e_j$ are distinct coordinate vectors. This set can be viewed as a subset of some $\\mathbb{R}^d$, and is easily checked to have the required property.\nWeisenberg observed in the comments that an additional point can be added to Alweiss' construction, giving a lower bound of $\\binom{d+1}{2}+1$.\nThe fact that the truth for $d=3$ is $8$ suggests that neither of these bounds is the truth.\nSee also [1088] for a generalisation.\nReferences\n\n\n[Bl84] Blokhuis, A., Few-distance sets. (1984), iv+70.\n\n[Cr62] Croft, H. T., $9$-point and $7$-point configurations in $3$-space. Proc. London Math. Soc. (3) (1962), 400-424.\n\n[ErKe47] Erd\\H{o}s, Paul and Kelly, L. M., Elementary Problems and Solutions: Solutions: E735. Amer. Math. Monthly (1947), 227-229.\n\n[Ko24c] Z. Kov\\'{a}cs, A note on Erd\\H{o}s's mysterious remark. arXiv:2412.05190 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exact answers are known for d=2 and d=3, but the general-dimensional extremum is open.\n\n**Verified partial progress.**\n\n- The exact planar maximum is 6 and the d=3 maximum is 8.\n- Known general bounds are binom(d+1,2)+1 and binom(d+2,2).\n\n**Full solution or refutation.**\n\nLow dimensions and general bounds do not determine the answer for arbitrary d.\n\n**What remains.**\n\nDetermine the extremum or close the quadratic-size gap in general dimension.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #503, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/503\n Evidence used: Open status, exact low-dimensional values, and current bounds.\n- Z. Kovács, A note on Erdős's mysterious remark, arXiv:2412.05190 (2024). (primary): https://arxiv.org/abs/2412.05190\n Evidence used: Alternative modern proof cited for the planar case.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2148, "problem_number": "EP-507", "title": "Erdős Problem #507", "statement": "Let $\\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\\alpha(n)$. Estimate $\\alpha(n)$.", "background": "Heilbronn's triangle problem. It is trivial that $\\alpha(n) \\ll 1/n$. Erd\\H{o}s observed that $\\alpha(n)\\gg 1/n^2$. The current best bounds are $ \\frac{\\log n}{n^2}\\ll \\alpha(n) \\ll \\frac{1}{n^{7/6+o(1)}}. $ The lower bound is due to Koml\\'{o}s, Pintz, and Szemer\\'{e}di \\cite{KPS82}. The upper bound is due to Cohen, Pohoata, and Zakharov \\cite{CPZ24} (improving on their earlier work \\cite{CPZ23} which itself improves an exponent of $8/7$ due to Koml\\'{o}s, Pintz, and Szemer\\'{e}di \\cite{KPS81}).\nThis problem is Problem 77 on Green's open problems list.\nReferences\n\n\n[CPZ23] Cohen, A. and Pohata, C. and Zakharov, D., A new upper bound for the Heilbronn triangle problem. arXiv:2305.18253 (2023).\n\n[CPZ24] Cohen, A. and Pohata, C. and Zakharov, D., Lower bounds for incidences. arXiv:2409.07658 (2024).\n\n[KPS81] Koml\\'{o}s, J\\'{a}nos and Pintz, J\\'{a}nos and Szemer\\'{e}di, Endre, On Heilbronn's triangle problem. J. London Math. Soc. (2) (1981), 385-396.\n\n[KPS82] Koml\\'{o}s, J\\'{a}nos and Pintz, J\\'{a}nos and Szemer\\'{e}di, Endre, A lower bound for Heilbronn's problem. J. London Math. Soc. (2) (1982), 13-24.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Heilbronn's triangle problem remains open; logarithmic lower and 7/6-type upper bounds are known.\n\n**Verified partial progress.**\n\n- The tracker records (log n)/n^2 as a lower bound.\n- Cohen--Pohoata--Zakharov give an upper bound 1/n^(7/6+o(1)).\n\n**Full solution or refutation.**\n\nThe asymptotic order is not known.\n\n**What remains.**\n\nSubstantially improve either bound or determine the order of alpha(n).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #507, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/507\n Evidence used: Current open status and listed bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2149, "problem_number": "EP-508", "title": "Erdős Problem #508", "statement": "What is the chromatic number of the plane? That is, what is the smallest number of colours required to colour $\\mathbb{R}^2$ such that no two points of the same colour are distance $1$ apart?", "background": "The Hadwiger-Nelson problem. Let $\\chi$ be the chromatic number of the plane. An equilateral triangle trivially shows that $\\chi\\geq 3$. There are several small graphs that show $\\chi\\geq 4$ (in particular the Moser spindle and Golomb graph). The best bounds currently known are $ 5 \\leq \\chi \\leq 7. $ The lower bound is due to de Grey \\cite{dG18}. The upper bound can be seen by colouring the plane by tesselating by hexagons with diameter slightly less than $1$.\nSee also [704], [705], and [706]. The independence number of a finite unit distance graph is the topic of [1070].\nReferences\n\n\n[dG18] de Grey, Aubrey D. N. J., The chromatic number of the plane is at least 5. Geombinatorics (2018), 18-31.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The chromatic number of the plane remains unknown, currently bounded by 5 and 7.\n\n**Verified partial progress.**\n\n- de Grey established the lower bound chi at least 5.\n- A seven-colouring gives chi at most 7.\n\n**Full solution or refutation.**\n\nNeither bound determines the chromatic number.\n\n**What remains.**\n\nExclude 5 or 6 colours, or construct a matching colouring.\n\n**Sources checked.**\n\n- A. D. N. J. de Grey, The chromatic number of the plane is at least 5, Geombinatorics 28 (2018). (primary): https://arxiv.org/abs/1804.02385\n Evidence used: Primary source for the five-colour lower bound.\n- Thomas F. Bloom, Erdős Problem #508, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/508\n Evidence used: Current open status and bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2150, "problem_number": "EP-509", "title": "Erdős Problem #509", "statement": "Let $f(z)\\in\\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set $ \\{ z\\in \\mathbb{C} : \\lvert f(z)\\rvert \\leq 1\\} $ be covered by a set of circles the sum of whose radii is $\\leq 2$?", "background": "Cartan proved this is true with $2$ replaced by $2e$, which was improved to $2.59$ by Pommerenke \\cite{Po61}. Pommerenke \\cite{Po59} proved that $2$ is achievable if the set is connected (see [1046]).\nThe generalisation of this to higher dimensions was asked by Erd\\H{o}s as Problem 4.23 in \\cite{Ha74}.\nReferences\n\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Po59] Pommerenke, Ch., On some problems by Erd\\H{o}s, Herzog and Piranian. Michigan Math. J. (1959), 221-225.\n\n[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The radius-sum-2 covering assertion is open; 2.59 is known generally and 2 for connected lemniscates.\n\n**Verified partial progress.**\n\n- Cartan's general constant 2e was improved to 2.59 by Pommerenke.\n- Pommerenke proved constant 2 when the lemniscate is connected.\n\n**Full solution or refutation.**\n\nThe disconnected general case remains unresolved.\n\n**What remains.**\n\nProve the sharp constant 2 or exhibit an obstruction.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #509, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/509\n Evidence used: Current open status and Pommerenke results.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2151, "problem_number": "EP-510", "title": "Erdős Problem #510", "statement": "If $A\\subset \\mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\\theta$ such that $ \\sum_{n\\in A}\\cos(n\\theta) < -cN^{1/2}? $ ", "background": "Chowla's cosine problem. Ruzsa \\cite{Ru04} (improving on an earlier result of Bourgain \\cite{Bo86}), proved an upper bound of $ -\\exp(O(\\sqrt{\\log N})). $ Polynomial bounds were proved independently by Bedert \\cite{Be25c} and Jin, Milojevi\\'{c}, Tomon, and Zhang \\cite{JMTZ25}. The best bound follows from the method of Bedert \\cite{Be25c}, which proved the existence of some $c>0$ such that, for all $A$ of size $N$, $ \\sum_{n\\in A}\\cos(n\\theta) < -cN^{1/7}. $ The example $A=B-B$, where $B$ is a Sidon set, shows that $N^{1/2}$ would be the best possible here.\nThis problem is Problem 81 on Green's open problems list.\nThis is related to [256].\nReferences\n\n\n[Be25c] B. Bedert, Polynomial bounds for the Chowla Cosine Problem. arXiv:2509.05260 (2025).\n\n[Bo86] Bourgain, J., Sur le minimum d'une somme de cosinus. Acta Arith. (1986), 381-389.\n\n[JMTZ25] Z. Jin, A. Milojevi\\'{c}, I. Tomon, and S. Zhang, From small eigenvalues to large cuts, and Chowla's cosine problem. arXiv:2509.03490 (2025).\n\n[Ru04] Ruzsa, Imre Z., Negative values of cosine sums. Acta Arith. (2004), 179-186.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Chowla's square-root cosine bound remains open, despite recent polynomial-scale lower estimates.\n\n**Verified partial progress.**\n\n- Ruzsa obtained an earlier subpolynomial negative bound.\n- The tracker records Bedert's 2025 estimate of order -c N^(1/7).\n\n**Full solution or refutation.**\n\nThe known exponent is below the requested 1/2 exponent.\n\n**What remains.**\n\nReach the square-root scale or disprove its universal validity.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #510, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/510\n Evidence used: Current open status and progress summary.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2152, "problem_number": "EP-513", "title": "Erdős Problem #513", "statement": "Let $f=\\sum_{n=0}^\\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of $ \\liminf_{r\\to \\infty} \\frac{\\max_n\\lvert a_nr^n\\rvert}{\\max_{\\lvert z\\rvert=r}\\lvert f(z)\\rvert}? $ ", "background": "It is trivial that this value is in $[1/2,1)$. K\"{o}v\\'{a}ri (unpublished) observed that it must be $>1/2$. Clunie and Hayman \\cite{ClHa64} showed that it is $\\leq 2/\\pi-c$ for some absolute constant $c>0$. Some other results on this quantity were established by Gray and Shah \\cite{GrSh63}.\nSee also [227].\nReferences\n\n\n[ClHa64] Clunie, J. and Hayman, W. K., The maximum term of a power series. J. Analyse Math. (1964), 143-186.\n\n[GrSh63] Gray, Alfred and Shah, S. M., A note on entire functions and a conjecture of Erd\\H{o}s. Bull. Amer. Math. Soc. (1963), 573-577.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The extremal coefficient-to-maximum-modulus constant remains open within a narrow numerical interval.\n\n**Verified partial progress.**\n\n- Classical work raises the trivial lower endpoint 1/2.\n- The maintained tracker records a lower estimate about 0.5850788 and a nonmatching upper bound.\n\n**Full solution or refutation.**\n\nNo exact greatest value is known.\n\n**What remains.**\n\nDetermine the sharp constant or prove matching bounds.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #513, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/513\n Evidence used: Current status and numerical bounds.\n\n**Review notes.** Recent unpublished/AI-associated discussion was not used as a resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2153, "problem_number": "EP-514", "title": "Erdős Problem #514", "statement": "Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$, $ \\lvert f(z)/z^n\\rvert \\to \\infty $ as $z\\to \\infty$ along $L$?\nCan the length of this path be estimated in terms of $M(r)=\\max_{\\lvert z\\rvert=r}\\lvert f(z)\\rvert$? Does there exist a path along which $\\lvert f(z)\\rvert$ tends to $\\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\\epsilon$)?", "background": "Boas (unpublished) has proved the first part, that such a path must exist.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The first path-existence component is reported as proved by Boas, but the multi-part question is not fully resolved by verified published evidence.\n\n**Verified partial progress.**\n\n- The maintained tracker records an unpublished Boas proof of the first part.\n- The estimates for path length and comparison growth remain open.\n\n**Full solution or refutation.**\n\nAn unpublished claim about one component is not evidence of a complete solution to all three questions.\n\n**What remains.**\n\nLocate a citable proof of the first claim and resolve the quantitative parts.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #514, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/514\n Evidence used: Current open status and restricted Boas claim.\n\n**Review notes.** Unpublished and forum-only claims are deliberately not promoted to a full solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2154, "problem_number": "EP-517", "title": "Erdős Problem #517", "statement": "Let $f(z)=\\sum_{k=1}^\\infty a_kz^{n_k}$ be an entire function (with $a_k\neq 0$ for all $k\\geq 1$). Is it true that if $n_k/k\\to \\infty$ then $f(z)$ assumes every value infinitely often?", "background": "A conjecture of Fej\\'{e}r and P\\'{o}lya.\nFej\\'{e}r \\cite{Fe08} proved that if $\\sum\\frac{1}{n_k}<\\infty$ then $f(z)$ assumes every value at least once, and Biernacki \\cite{Bi28} proved that if $\\sum\\frac{1}{n_k}<\\infty$ then $f(z)$ assumes every value infinitely often.\nP\\'{o}lya \\cite{Po29} proved that if $f$ has finite order then $f(z)$ assumes every value infinitely often under the assumption that $\\limsup (n_{k+1}-n_k)=\\infty$.\nReferences\n\n\n[Bi28] Biernacki, Mi\\'{e}cislas, Sur les \\'{e}quations alg\\'{e}briques contenant des param\\'{e}tres arbitraires. (1928), 145.\n\n[Fe08] Fej\\'{e}r, Leopold, \"{U}ber die Wurzel vom kleinsten absoluten Betrage einer algebraischen Gleichung. Math. Ann. (1908), 413-423.\n\n[Po29] P\\'olya, G., Untersuchungen \"uber {L}\"ucken und {S}ingularit\"{a}ten von\n{P}otenzreihen. Math. Z. (1929), 549--640.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general lacunary-entire-function assertion is open, with strong sufficient gap/summability hypotheses known.\n\n**Verified partial progress.**\n\n- Biernacki proved infinitely-often value attainment when sum 1/n_k is finite.\n- Pólya proved it at finite order under an unbounded-gap condition.\n\n**Full solution or refutation.**\n\nNeither sufficient condition follows from n_k/k tending to infinity in general.\n\n**What remains.**\n\nProve or refute the assertion under the stated density-of-exponents hypothesis.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #517, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/517\n Evidence used: Current open status and classical partial results.\n\n**Review notes.** The serialized input loses the LaTex neq escape; it is preserved and flagged, not corrected.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2155, "problem_number": "EP-520", "title": "Erdős Problem #520", "statement": "Let $f$ be a Rademacher multiplicative function: a random $\\{-1,0,1\\}$-valued multiplicative function, where for each prime $p$ we independently choose $f(p)\\in \\{-1,1\\}$ uniformly at random, and for square-free integers $n$ we extend $f(p_1\\cdots p_r)=f(p_1)\\cdots f(p_r)$ (and $f(n)=0$ if $n$ is not squarefree). Does there exist some constant $c>0$ such that, almost surely, $ \\limsup_{N\\to \\infty}\\frac{\\sum_{m\\leq N}f(m)}{\\sqrt{N\\log\\log N}}=c? $ ", "background": "Note that if we drop the multiplicative assumption, and simply assign $f(m)=\\pm 1$ at random, then this statement is true (with $c=\\sqrt{2}$), the law of the iterated logarithm.\nWintner \\cite{Wi44} proved that, almost surely, $ \\sum_{m\\leq N}f(m)\\ll N^{1/2+o(1)}, $ and Erd\\H{o}s improved the right-hand side to $N^{1/2}(\\log N)^{O(1)}$. Lau, Tenenbaum, and Wu \\cite{LTW13} have shown that, almost surely, $ \\sum_{m\\leq N}f(m)\\ll N^{1/2}(\\log\\log N)^{2+o(1)}. $ Caich \\cite{Ca24b} has improved this to $ \\sum_{m\\leq N}f(m)\\ll N^{1/2}(\\log\\log N)^{3/4+o(1)}. $ Harper \\cite{Ha13} has shown that the sum is almost surely not $O(N^{1/2}/(\\log\\log N)^{5/2+o(1)})$, and conjectured that in fact Erd\\H{o}s' conjecture is false, and almost surely $ \\sum_{m\\leq N}f(m) \\ll N^{1/2}(\\log\\log N)^{1/4+o(1)}. $ \nReferences\n\n\n[Ca24b] R. Caich, Almost sure upper bound for random multiplicative functions. arXiv:2304.00943 (2024).\n\n[Ha13] Harper, Adam J., Bounds on the suprema of Gaussian processes, and omega\nresults for the sum of a random multiplicative function. Ann. Appl. Probab. (2013), 584-616.\n\n[LTW13] Lau, Yuk-Kam and Tenenbaum, G\\'{e}rald and Wu, Jie, On mean values of random multiplicative functions. Proc. Amer. Math. Soc. (2013), 409-420.\n\n[Wi44] Wintner, Aurel, Random factorizations and Riemann's hypothesis. Duke Math. J. (1944), 267-275.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A July 2026 primary preprint proves an almost-sure upper bound that makes the proposed positive normalized limsup equal to zero.\n\n**Verified partial progress.**\n\n- Caich previously proved an almost-sure upper bound of order sqrt(x)(log log x)^(3/4+o(1)).\n- Durkan and Pearce-Crump improve the exponent to 1/4+epsilon for every epsilon>0.\n\n**Full solution or refutation.**\n\nDurkan--Pearce-Crump prove almost surely |sum_{n<=x}f(n)| << sqrt(x)(log log x)^(1/4+epsilon) for every epsilon>0. Taking epsilon<1/4 and dividing by sqrt(x log log x) forces the ratio, hence its limsup, to zero; therefore no c>0 as asked can exist.\n\n**What remains.**\n\nThe logical refutation is complete conditional only on the cited theorem; the very recent preprint merits independent expert checking and ordinary publication scrutiny.\n\n**Sources checked.**\n\n- Benjamin Durkan and Andrew Pearce-Crump, A sharp almost sure upper bound for partial sums of random multiplicative functions, arXiv:2607.29429 (2026). (primary): https://arxiv.org/abs/2607.29429\n Evidence used: Theorem giving the exponent 1/4+epsilon for Rademacher random multiplicative functions; its direct normalization refutes the question.\n- Rachid Caich, Almost sure upper bound for random multiplicative functions, arXiv:2304.00943 (2024). (primary): https://arxiv.org/abs/2304.00943\n Evidence used: Earlier 3/4+o(1) almost-sure exponent recorded for context.\n- Thomas F. Bloom, history of Erdős Problem #520, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/520\n Evidence used: Documents the pre-July-2026 open status and earlier bounds; it had not yet incorporated the new preprint.\n\n**Review notes.** The refutation is a direct deduction, not a claim made by the stale tracker. The common trailing serialization fragment in the input background was preserved and flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2156, "problem_number": "EP-521", "title": "Erdős Problem #521", "statement": "Let $(\\epsilon_k)_{k\\geq 0}$ be independently uniformly chosen at random from $\\{-1,1\\}$. If $R_n$ counts the number of real roots of $f_n(z)=\\sum_{0\\leq k\\leq n}\\epsilon_k z^k$ then is it true that, almost surely, $ \\lim_{n\\to \\infty}\\frac{R_n}{\\log n}=\\frac{2}{\\pi}? $ ", "background": "Erd\\H{o}s and Offord \\cite{EO56} showed that the number of real roots of a random degree $n$ polynomial with $\\pm 1$ coefficients is $(\\frac{2}{\\pi}+o(1))\\log n$.\nIt is ambiguous in \\cite{Er61} whether Erd\\H{o}s intended the coefficients to be uniformly chosen from $\\{-1,1\\}$ or $\\{0,1\\}$. In the latter case, the constant $\\frac{2}{\\pi}$ should be $\\frac{1}{\\pi}$ (see the discussion in the comments).\nIn the case of $\\{-1,1\\}$ Do \\cite{Do24} proved that, if $R_n[-1,1]$ counts the number of roots in $[-1,1]$, then, almost surely, $ \\lim_{n\\to \\infty}\\frac{R_n[-1,1]}{\\log n}=\\frac{1}{\\pi}. $ See also [522].\nReferences\n\n\n[Do24] Y. Do, A strong law of large numbers for real roots of random polynomials. arXiv:2403.06353 (2024).\n\n[EO56] Erd\"{o}s, Paul and Offord, A. C., On the number of real roots of a random algebraic equation. Proc. London Math. Soc. (3) (1956), 139-160.\n\n[Er61] Erd\\H{o}s, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutat\\'{o} Int. K\"{o}zl. (1961), 221-254.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A strong law is proved for roots in [-1,1], but the full real-root claim is not verified; a recent claimed negative solution remains informal and unreviewed.\n\n**Verified partial progress.**\n\n- Do proves almost surely N_n([-1,1])/log n tends to 1/pi for Rademacher coefficients.\n- The fixed-n reciprocal symmetry for outside roots does not automatically preserve the coupled sequence needed for an almost-sure n-to-infinity statement.\n\n**Full solution or refutation.**\n\nNo verified full solution or disproof was located. The maintained discussion records a 2026 claimed negative argument, but its AI-assisted manuscript status and lack of independent verification are insufficient for a terminal mathematical classification.\n\n**What remains.**\n\nControl the roots outside [-1,1] for the same infinite coefficient sequence, or independently verify and publish the claimed counterargument.\n\n**Sources checked.**\n\n- Yen Do, A strong law of large numbers for real roots of random polynomials, arXiv:2403.06353 (2024). (primary): https://arxiv.org/abs/2403.06353\n Evidence used: Proves the almost-sure 1/pi law for roots in [-1,1], not the full real-line 2/pi statement.\n- Erdős Problems forum thread for Problem #521, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/521?order=newest\n Evidence used: Contains the recent claimed negative solution and explicit requests for verification; the claim is not treated there as established literature.\n- Thomas F. Bloom, Erdős Problem #521, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/521\n Evidence used: Current formulation, historical coefficient ambiguity, and maintained open context.\n\n**Review notes.** The exact dataset statement uses {-1,1}; historical ambiguity with {0,1} is flagged without changing it. The common trailing background corruption is also preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2157, "problem_number": "EP-522", "title": "Erdős Problem #522", "statement": "Let $f(z)=\\sum_{0\\leq k\\leq n} \\epsilon_k z^k$ be a random polynomial, where $\\epsilon_k\\in \\{-1,1\\}$ independently uniformly at random for $0\\leq k\\leq n$.\nIs it true that, if $R_n$ is the number of roots of $f(z)$ in $\\{ z\\in \\mathbb{C} : \\lvert z\\rvert \\leq 1\\}$, then $ \\frac{R_n}{n/2}\\to 1 $ almost surely?", "background": "Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Rademacher coefficients, i.e. independent uniform $\\pm 1$ values. Erd\\H{o}s and Offord \\cite{EO56} showed that the number of real roots of a random degree $n$ polynomial with $\\pm 1$ coefficients is $(\\frac{2}{\\pi}+o(1))\\log n$.\nThere is some ambiguity whether Erd\\H{o}s intended the coefficients to be in $\\{-1,1\\}$ or $\\{0,1\\}$ - see the comments section.\nA weaker version of this was solved by Yakir \\cite{Ya21}, who proved that $ \\frac{R_n}{n/2}\\to 1 $ in probability. (This weaker claim was also asked by Erd\\H{o}s, and also appears in a book of Hayman \\cite{Ha67}.) More precisely, $ \\lim_{n\\to \\infty} \\mathbb{P}(\\lvert R_n-n/2\\rvert \\geq n^{9/10}) =0. $ See also [521].\nReferences\n\n\n[EO56] Erd\"{o}s, Paul and Offord, A. C., On the number of real roots of a random algebraic equation. Proc. London Math. Soc. (3) (1956), 139-160.\n\n[Ha67] Hayman, W. K., Research problems in function theory. (1967), vii+56.\n\n[Ya21] Yakir, Oren, Approximately half of the roots of a random {L}ittlewood\npolynomial are inside the disk. Studia Math. (2021), 227--240.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Convergence to one-half is proved in probability; a public 2026 manuscript claims the almost-sure theorem but lacks enough independent bibliographic verification for solved status.\n\n**Verified partial progress.**\n\n- Yakir proves R_n=n/2+o(n) in probability for random Littlewood polynomials.\n- A public April 2026 manuscript claims an almost-sure error R_n=n/2+O_omega(n^(149/150)), but no indexed arXiv or refereed record was located.\n\n**Full solution or refutation.**\n\nThe verified primary literature establishes the weaker in-probability statement. The stronger public manuscript is recorded as a claim, not promoted to a theorem under the evidence policy.\n\n**What remains.**\n\nIndependently check and archive or publish the claimed almost-sure proof, or otherwise prove the almost-sure upgrade from Yakir's theorem.\n\n**Sources checked.**\n\n- Oren Yakir, Approximately half of the roots of a random Littlewood polynomial are inside the disk, Studia Mathematica 256 (2021), 227-240; arXiv:2011.06234. (primary): https://arxiv.org/abs/2011.06234\n Evidence used: Primary proof of convergence in probability.\n- Almost surely half of the zeros of a random Littlewood polynomial lie in the unit disk, public manuscript dated 2026-04-20. (primary): https://www.ulam.ai/research/erdos522.pdf\n Evidence used: Claims the full almost-sure result and a quantitative error, but was not independently verified or matched to an indexed publication.\n- Thomas F. Bloom, Erdős Problem #522, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/522\n Evidence used: Maintains the problem as open while recording Yakir's weaker result and recent claim.\n\n**Review notes.** The exact input is preserved. Historical ambiguity between Rademacher and {0,1} coefficients and the common trailing background corruption are flagged only.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2158, "problem_number": "EP-524", "title": "Erdős Problem #524", "statement": "For any $t\\in (0,1)$ let $t=\\sum_{k=1}^\\infty \\epsilon_k(t)2^{-k}$ (where $\\epsilon_k(t)\\in \\{0,1\\}$). What is the correct order of magnitude (for almost all $t\\in(0,1)$) for $ M_n(t)=\\max_{x\\in [-1,1]}\\left\\lvert \\sum_{k\\leq n}(-1)^{\\epsilon_k(t)}x^k\\right\\rvert? $ ", "background": "A problem of Salem and Zygmund \\cite{SaZy54}. Chung showed that, for almost all $t$, there exist infinitely many $n$ such that $ M_n(t) \\ll \\left(\\frac{n}{\\log\\log n}\\right)^{1/2}. $ Erd\\H{o}s (unpublished) showed that for almost all $t$ and every $\\epsilon>0$ we have $\\lim_{n\\to \\infty}M_n(t)/n^{1/2-\\epsilon}=\\infty$.\nReferences\n\n\n[SaZy54] Salem, R. and Zygmund, A., Some properties of trigonometric series whose terms have\nrandom signs. Acta Math. (1954), 245-301.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The sharp almost-sure lower envelope is proved in a 2026 primary preprint, and Abel summation plus the classical LIL gives the matching sharp upper envelope.\n\n**Verified partial progress.**\n\n- Letwin and Sawhney prove liminf log(M_n/sqrt(n))/(log log n)^(1/3)=-(3 pi^2/4)^(1/3) almost surely.\n- For x in [0,1], Abel summation bounds the polynomial by maximal ordinary partial sums; for x in [-1,0], it does the same for alternating partial sums.\n\n**Full solution or refutation.**\n\nAlmost surely, limsup M_n/sqrt(2n log log n)=1, by the Abel-summation comparison and endpoint lower bounds together with the LIL. Letwin--Sawhney's exact logarithmic liminf law supplies the sharp lower envelope. These two laws determine the oscillatory order of M_n.\n\n**What remains.**\n\nNo mathematical gap remains for the natural upper/lower-envelope interpretation of 'correct order'; the historical wording does not request a single asymptotic equivalent, which in any event is precluded by the two different envelopes.\n\n**Sources checked.**\n\n- Brayden Letwin and Mehtaab Sawhney, On the maxima of Littlewood polynomials on [-1,1], arXiv:2604.19294 (2026). (primary): https://arxiv.org/abs/2604.19294\n Evidence used: Primary proof of the sharp almost-sure lower-envelope law.\n- Raphael Salem and Antoni Zygmund, Some properties of trigonometric series whose terms have random signs, Acta Mathematica 91 (1954), 245-301. (primary): https://doi.org/10.1007/BF02393433\n Evidence used: Classical random-sign context; the upper-envelope deduction in this triage uses the standard LIL explicitly rather than attributing an unstated theorem to this paper.\n- Thomas F. Bloom, Erdős Problem #524, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/524\n Evidence used: Preserved formulation and historical Chung/Erdős bounds.\n\n**Review notes.** The statement's signs (-1)^{epsilon_k(t)} are internally consistent and were not treated as a defect. The common trailing background corruption was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2159, "problem_number": "EP-528", "title": "Erdős Problem #528", "statement": "Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine $ C_k=\\lim_{n\\to\\infty}f(n,k)^{1/n}. $ ", "background": "The constant $C_k$ is sometimes known as the connective constant. Hammersley and Morton \\cite{HM54} showed that this limit exists, and it is trivial that $k\\leq C_k\\leq 2k-1$.\nKesten \\cite{Ke63} proved that $C_k=2k-1-1/2k+O(1/k^2)$, and more precise asymptotics are given by Clisby, Liang, and Slade \\cite{CLS07}.\nConway and Guttmann \\cite{CG93} showed that $C_2\\geq 2.62$ and Alm \\cite{Al93} showed that $C_2\\leq 2.696$. Jacobsen, Scullard, and Guttmann \\cite{JSG16} have computed the first few decimal places of $C_2$, showing that $ C_2 = 2.6381585303279\\cdots. $ See also [529].\nReferences\n\n\n[Al93] Alm, Sven Erick, Upper bounds for the connective constant of self-avoiding\nwalks. Combin. Probab. Comput. (1993), 115-136.\n\n[CG93] Conway, A. R. and Guttmann, A. J., Lower bound on the connective constant for square lattice\nself-avoiding walks. J. Phys. A (1993), 3719-3724.\n\n[CLS07] Clisby, Nathan and Liang, Richard and Slade, Gordon, Self-avoiding walk enumeration via the lace expansion. J. Phys. A (2007), 10973-11017.\n\n[HM54] Hammersley, J. M. and Morton, K. W., Poor man's Monte Carlo. J. Roy. Statist. Soc. Ser. B (1954), 23-38; discussion 61-75.\n\n[JSG16] Jacobsen, Jesper Lykke and Scullard, Christian R. and\nGuttmann, Anthony J., On the growth constant for square-lattice self-avoiding walks. J. Phys. A (2016), 494004, 18.\n\n[Ke63] Kesten, Harry, On the number of self-avoiding walks. J. Mathematical Phys. (1963), 960-969.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The connective constant is not known exactly in any integer dimension k>=2; recent work supplies improved bounds and numerical estimates only.\n\n**Verified partial progress.**\n\n- Jacobsen--Scullard--Guttmann estimate the square-lattice value as 2.63815853032790(3) numerically.\n- Couronné proves the rigorous square-lattice upper bound mu_2<=2.662343.\n\n**Full solution or refutation.**\n\nNo exact determination was located. The displayed high-precision decimal is numerical evidence, not an equality theorem.\n\n**What remains.**\n\nDetermine the exact connective constant in dimension two or any higher integer dimension.\n\n**Sources checked.**\n\n- Jesper Lykke Jacobsen, Christian R. Scullard, and Anthony J. Guttmann, On the growth constant for square-lattice self-avoiding walks, arXiv:1607.02984 (2016). (primary): https://arxiv.org/abs/1607.02984\n Evidence used: High-precision numerical estimate, explicitly not an exact proof.\n- Olivier Couronné, New Upper Bound for the Connective Constant for Square-Lattice Self-Avoiding Walks, Journal of Statistical Physics 192 (2025), article 153. (primary): https://doi.org/10.1007/s10955-025-03542-6\n Evidence used: Rigorous improved upper bound mu_2<=2.662343.\n- Thomas F. Bloom, Erdős Problem #528, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/528\n Evidence used: Current open status and historical bounds.\n\n**Review notes.** Any wording in the imported background that presents the numerical estimate as an exact equality is flagged as overstatement, not repaired. The common trailing corruption is preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2160, "problem_number": "EP-529", "title": "Erdős Problem #529", "statement": "Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\\mathbb{Z}^k$ (conditional on no self intersections) - that is, a self-avoiding walk. Is it true that $ \\lim_{n\\to \\infty}\\frac{d_2(n)}{n^{1/2}}= \\infty? $ Is it true that $ d_k(n)\\ll n^{1/2} $ for $k\\geq 3$?", "background": "Slade \\cite{Sl87} proved that, for $k$ sufficiently large, $d_k(n)\\sim Dn^{1/2}$ for some constant $D>0$ (independent of $k$). Hara and Slade (\\cite{HaSl91} and \\cite{HaSl92}) proved this for all $k\\geq 5$.\nFor $k=2$ Duminil-Copin and Hammond \\cite{DuHa13} have proved that $d_2(n)=o(n)$.\nIt is now conjectured that $d_k(n)\\ll n^{1/2}$ is false for $k=3$ and $k=4$, and more precisely (see for example Section 1.4 of \\cite{MaSl93}) that $d_2(n)\\sim Dn^{3/4}$, $d_3(n)\\sim n^{\nu}$ where $\nu\\approx 0.59$, and $d_4(n)\\sim D(\\log n)^{1/8}n^{1/2}$.\nMadras and Slade \\cite{MaSl93} have a monograph on the topic of self-avoiding walks.\nSee also [528].\nReferences\n\n\n[DuHa13] Duminil-Copin, Hugo and Hammond, Alan, Self-avoiding walk is sub-ballistic. Comm. Math. Phys. (2013), 401--423.\n\n[HaSl91] Hara, Takashi and Slade, Gordon, Critical behaviour of self-avoiding walk in five or more\ndimensions. Bull. Amer. Math. Soc. (N.S.) (1991), 417--423.\n\n[HaSl92] Hara, Takashi and Slade, Gordon, Self-avoiding walk in five or more dimensions. {I}. {T}he\ncritical behaviour. Comm. Math. Phys. (1992), 101--136.\n\n[MaSl93] Madras, Neal and Slade, Gordon, The self-avoiding walk. (1993), xiv+425.\n\n[Sl87] Slade, Gordon, The diffusion of self-avoiding random walk in high dimensions. Comm. Math. Phys. (1987), 661--683.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Diffusive endpoint behavior is proved in dimensions at least five and sub-ballisticity in all dimensions at least two, but the requested planar and dimensions-three/four statements remain open.\n\n**Verified partial progress.**\n\n- Hara and Slade prove mean-field/diffusive self-avoiding-walk behavior for dimensions k>=5.\n- Duminil-Copin and Hammond prove sub-ballisticity in every dimension k>=2, yielding d_2(n)=o(n) but not d_2(n)/sqrt(n)->infinity.\n\n**Full solution or refutation.**\n\nThe high-dimensional part is known for k>=5. The planar superdiffusive assertion and the displayed upper bounds in dimensions three and four are not settled by the located literature.\n\n**What remains.**\n\nProve planar superdiffusivity and determine the correct endpoint-distance scale in dimensions three and four.\n\n**Sources checked.**\n\n- Takashi Hara and Gordon Slade, Self-avoiding walk in five or more dimensions. I. The critical behaviour, Communications in Mathematical Physics 147 (1992), 101-136. (primary): https://doi.org/10.1007/BF02099530\n Evidence used: Primary high-dimensional mean-field result.\n- Hugo Duminil-Copin and Alan Hammond, Self-avoiding walk is sub-ballistic, arXiv:1205.0401 (2012). (primary): https://arxiv.org/abs/1205.0401\n Evidence used: Primary all-dimensional sub-ballisticity theorem.\n- Thomas F. Bloom, Erdős Problem #529, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/529\n Evidence used: Maintained open status and dimension split.\n\n**Review notes.** The statement implicitly uses the uniform n-step self-avoiding-walk measure and Euclidean distance but does not spell them out; this ambiguity and the common trailing corruption were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2161, "problem_number": "EP-530", "title": "Erdős Problem #530", "statement": "Let $\\ell(N)$ be maximal such that in any finite set $A\\subset \\mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\\ell(N)$ (i.e. the only solutions to $a+b=c+d$ in $S$ are the trivial ones). Determine the order of $\\ell(N)$.", "background": "In particular, is it true that $\\ell(N)\\sim N^{1/2}$?\nOriginally asked by Riddell \\cite{Ri69}. Erd\\H{o}s noted the bounds $ N^{1/3} \\ll \\ell(N) \\leq (1+o(1))N^{1/2} $ (the upper bound following from the case $A=\\{1,\\ldots,N\\}$). The lower bound was improved to $N^{1/2}\\ll \\ell(N)$ by Koml\\'{o}s, Sulyok, and Szemer\\'{e}di \\cite{KSS75}. The correct constant is unknown, but it is likely that the upper bound is true, so that $\\ell(N)\\sim N^{1/2}$.\nIn \\cite{AlEr85} Alon and Erd\\H{o}s make the stronger conjecture that perhaps $A$ can always be written as the union of at most $(1+o(1))N^{1/2}$ many Sidon sets. (This is easily verified for $A=\\{1,\\ldots,N\\}$ using standard constructions of Sidon sets.)\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nSee also [1088] for a higher-dimensional generalisation.\nReferences\n\n\n[AlEr85] Alon, Noga and Erd\\H{o}s, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[KSS75] Koml\\'{o}s, J. and Sulyok, M. and Szemeredi, E., Linear problems in combinatorial number theory. Acta Math. Acad. Sci. Hungar. (1975), 113-121.\n\n[Ri69] Riddell, J., On sets of numbers containing no $l$ terms in arithmetic progression. Nieuw Arch. Wisk. (3) (1969), 204-209.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The order ell(N)=Theta(sqrt(N)) is known, but the sharper asymptotic ell(N)~sqrt(N) and its optimal constant remain open.\n\n**Verified partial progress.**\n\n- Komlós--Sulyok--Szemerédi prove a universal lower bound of square-root order; the interval construction gives matching order above.\n- Bailleul and Riblet prove ell(N)>=(1/(3sqrt(3))+o(1))sqrt(N), uniformly even for finite subsets of Euclidean spaces.\n\n**Full solution or refutation.**\n\nThe broad order-of-magnitude request has been settled at Theta(sqrt(N)), but the record's specific asymptotic-equivalence question is not known.\n\n**What remains.**\n\nDetermine the sharp leading constant, in particular whether ell(N)/sqrt(N) tends to one.\n\n**Sources checked.**\n\n- János Komlós, Miklós Sulyok, and Endre Szemerédi, Linear problems in combinatorial number theory, Acta Mathematica Academiae Scientiarum Hungaricae 26 (1975), 113-121. (primary): https://doi.org/10.1007/BF01895954\n Evidence used: Classical primary square-root-order lower theorem.\n- Alexandre Bailleul and Robin Riblet, On the largest Sidon subset in a finite subset of R^N, arXiv:2605.03181 (2026). (primary): https://arxiv.org/abs/2605.03181\n Evidence used: Recent explicit lower constant and Euclidean-space extension.\n- Thomas F. Bloom, Erdős Problem #530, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/530\n Evidence used: Current open sharp-constant status and classical order bounds.\n\n**Review notes.** No source statement was changed; the systematic trailing background fragment remains flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2162, "problem_number": "EP-531", "title": "Erdős Problem #531", "statement": "Let $F(k)$ be the minimal $N$ such that if we two-colour $\\{1,\\ldots,N\\}$ there is a set $A$ of size $k$ such that all subset sums $\\sum_{a\\in S}a$ (for $\\emptyset\neq S\\subseteq A$) are monochromatic. Estimate $F(k)$.", "background": "The existence of $F(k)$ was established by Sanders and Folkman, and it also follows from Rado's theorem. It is commonly known as Folkman's theorem.\nErd\\H{o}s and Spencer \\cite{ErSp89} proved that $ F(k) \\geq 2^{ck^2/\\log k} $ for some constant $c>0$. Balogh, Eberhrad, Narayanan, Treglown, and Wagner \\cite{BENTW17} have improved this to $ F(k) \\geq 2^{2^{k-1}/k}. $ \nReferences\n\n\n[BENTW17] Balogh, J\\'{o}zsef and Eberhard, Sean and Narayanan, Bhargav and Treglown, Andrew and Wagner, Adam Zsolt, An improved lower bound for Folkman's theorem. Bull. Lond. Math. Soc. (2017), 745-747.\n\n[ErSp89] Erd\\H{o}s, Paul and Spencer, Joel, Monochromatic sumsets. J. Combin. Theory Ser. A (1989), 162-163.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The quantitative growth of the Folkman number remains open despite a strong double-exponential lower bound.\n\n**Verified partial progress.**\n\n- Balogh, Eberhard, Narayanan, Treglown, and Wagner prove F(k)>=2^(2^(k-1)/k).\n\n**Full solution or refutation.**\n\nThe located lower bound is substantial but does not determine the requested asymptotic order or matching upper behavior.\n\n**What remains.**\n\nObtain matching quantitative bounds for F(k), or otherwise determine its growth rate.\n\n**Sources checked.**\n\n- József Balogh, Sean Eberhard, Bhargav Narayanan, Andrew Treglown, and Adam Zsolt Wagner, Folkman's theorem for random subsets, arXiv:1703.02473 (2017). (primary): https://arxiv.org/abs/1703.02473\n Evidence used: Primary source for the double-exponential lower bound.\n- Thomas F. Bloom, Erdős Problem #531, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/531\n Evidence used: Maintained open status and bound summary.\n\n**Review notes.** The imported background misspells Eberhard as Eberhrad; it was flagged rather than edited. The common trailing serialization fragment was also preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2163, "problem_number": "EP-533", "title": "Erdős Problem #533", "statement": "Let $\\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_5$ and at least $\\delta n^2$ edges then $G$ contains a set of $\\gg_\\delta n$ vertices containing no triangle.", "background": "A problem of Erd\\H{o}s, Hajnal, Simonovits, S\\'{o}s, and Szemer\\'{e}di, who could prove this is true for $\\delta>1/16$, and could further prove it for $\\delta>0$ if we replace $K_5$ with $K_4$.\nThey further observed that it fails for $\\delta =1/4$ if we replace $K_5$ with $K_7$: by a construction of Erd\\H{o}s and Rogers \\cite{ErRo62} (see [620]) there exists some constant $c>0$ such that, for all large $n$, there is a graph on $n$ vertices which contains no $K_4$ and every set of at least $n^{1-c}$ vertices contains a triangle. If we take two vertex disjoint copies of this graph and add all edges between the two copies then this yields a graph on $2n$ vertices with $\\geq n^2$ edges, which contains no $K_7$, yet every set of at least $2n^{1-c}$ vertices contains a triangle.\nSee also [579] and the entry in the graphs problem collection.\nReferences\n\n\n[ErRo62] Erd\\H{o}s, P. and Rogers, C. A., The construction of certain graphs. Canadian J. Math. (1962), 702-707.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Balogh and Lenz disproved the assertion by constructing K5-free graphs with positive quadratic edge density whose largest triangle-free induced set is o(n). Subsequent geometric constructions, together with the known upper bound, give the exact tracker-normalized 3-Ramsey--Turán density delta_3(5)=1/12.\n\n**Verified partial progress.**\n\n- The original authors proved the analogous K4 statement and obtained an upper bound delta_3(5)<=1/12.\n- Balogh and Lenz proved delta_3(5)>0, which alone refutes EP-533.\n- Liu, Reiher, Sharifzadeh, and Staden supplied the matching construction leading to delta_3(5)=1/12.\n\n**Full solution or refutation.**\n\nIn Ramsey--Turán notation, EP-533 asserts RT_3(n,K5,o(n))=o(n^2). The Balogh--Lenz product construction instead yields K5-free graphs with Omega(n^2) edges and triangle-independence number o(n). Later complex-sphere constructions attain the sharp density. The value 1/12 uses normalization by n^2; sources normalizing by binomial(n,2) state the equivalent value 1/6.\n\n**What remains.**\n\nThe stated yes/no problem and its limiting density are closed. Finer phase-transition behavior when the maximum triangle-free induced set is prescribed at particular sublinear scales remains a successor research topic, not an unresolved component of EP-533. The input background has trailing serialization noise and an obsolete threshold summary, but the displayed claim is clear.\n\n**Sources checked.**\n\n- József Balogh and John Lenz, On the Ramsey--Turán numbers of graphs and hypergraphs, Israel Journal of Mathematics 194 (2013), 45--68; arXiv:1109.4428. (primary): https://arxiv.org/abs/1109.4428\n Evidence used: Proves RT_t(n,K_{t+2},o(n))=Omega(n^2); taking t=3 gives the direct counterexamples to EP-533.\n- Hong Liu, Christian Reiher, Maryam Sharifzadeh, and Katherine Staden, Geometric constructions for Ramsey--Turán theory, arXiv:2103.10423; Journal of the European Mathematical Society. (primary): https://arxiv.org/abs/2103.10423\n Evidence used: Provides complex-sphere constructions giving the sharp lower density for the K5/triangle-independence case.\n- Thomas F. Bloom, Erdős Problem #533 (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/533\n Evidence used: Records the disproof, attributes both stages, and states the exact density 1/12 under normalization by n^2.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2164, "problem_number": "EP-535", "title": "Erdős Problem #535", "statement": "Let $r\\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\\{1,\\ldots,N\\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Estimate $f_r(N)$.", "background": "Erd\\H{o}s \\cite{Er64} proved that $ f_r(N) \\leq N^{\\frac{3}{4}+o(1)}, $ and Abbott and Hanson \\cite{AbHa70} improved this exponent to $1/2$. Erd\\H{o}s \\cite{Er64} proved the lower bound $ f_3(N) > N^{\\frac{c}{\\log\\log N}} $ for some constant $c>0$, and conjectured this should also be an upper bound.\nErd\\H{o}s writes this is 'intimately connected' with the sunflower problem [20]. Indeed, the conjectured upper bound would follow from the following stronger version of the sunflower problem: estimate the size of the largest set of integers $A$ such that $\\omega(n)=k$ for all $n\\in A$ and there does not exist $a_1,\\ldots,a_r\\in A$ and an integer $d$ such that $(a_i,a_j)=d$ for all $i\neq j$ and $(a_i/d,d)=1$ for all $i$. The conjectured upper bound for $f_r(N)$ would follow if the size of such an $A$ must be at most $c_r^k$. The original sunflower proof of Erd\\H{o}s and Rado gives the upper bound $c_r^kk!$.\nSee also [536].\nReferences\n\n\n[AbHa70] Abbott, H. L. and Hanson, D., An extremal problem in number theory. Bull. London Math. Soc. (1970), 324-326.\n\n[Er64] Erd\\H{o}s, P., On a problem in elementary number theory and a combinatorial problem. Math. Comp. (1964), 644-646.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The asymptotic size of f_r(N) remains open; recent claimed near-subpolynomial bounds were not located in an indexed primary paper.\n\n**Verified partial progress.**\n\n- For r=3, Erdős gives a lower bound exp(c log N/log log N), while Abbott--Hanson give the classical upper bound N^(1/2+o(1)).\n- A 2026 tracker thread claims fixed-r bounds of exp((log(r-1)+o(1))log N/log log N) below and exp(O_r(log N log log log N/log log N)) above, but the claim remains source-unverified.\n\n**Full solution or refutation.**\n\nNo verified theorem closes the classical gap. The newest forum claim is documented for follow-up but not used to reclassify the problem.\n\n**What remains.**\n\nVerify and publish the recent claimed bounds or otherwise determine the fixed-r asymptotic order; also resolve the corrected stronger sunflower variant.\n\n**Sources checked.**\n\n- H. L. Abbott and D. Hanson, A problem of Schur and its generalizations, Bulletin of the London Mathematical Society 2 (1970), 321-325. (primary): https://doi.org/10.1112/blms/2.3.324\n Evidence used: Primary source for the classical upper-bound method.\n- Paul Erdős, Some extremal problems in combinatorial number theory (1970). (primary): https://combinatorica.hu/~p_erdos/1970-21.pdf\n Evidence used: Original source discussing the common-divisor extremal function.\n- Erdős Problems forum thread for Problem #535, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/535\n Evidence used: Records the recent claimed bounds, which were not independently source-verified.\n\n**Review notes.** The background's stronger variant writes omega(n)=k, while a formalization note says the corrected historical condition is Omega(n)=k with multiplicity. This and the common trailing corruption were flagged, not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2165, "problem_number": "EP-536", "title": "Erdős Problem #536", "statement": "Let $\\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $\\epsilon N$ then there must be distinct $a,b,c\\in A$ such that $ [a,b]=[b,c]=[a,c], $ where $[a,b]$ denotes the least common multiple?", "background": "This is false if we ask for four elements with the same pairwise least common multiple, as shown by Erd\\H{o}s \\cite{Er62} (with a proof given in \\cite{Er70}).\nThis was also asked by Erd\\H{o}s at the 1991 problem session of West Coast Number Theory.\nIn the comments Weisenberg sketches a construction of a set $A\\subseteq [1,N]$ without this property such that $ \\lvert A\\rvert \\gg (\\log\\log N)^{f(N)}\\frac{N}{\\log N} $ for some $f(N)\\to \\infty$. Weisenberg also sketches a proof of the main problem when $\\epsilon>\\frac{221}{225}$.\nSee also [535], [537], and [856]. A related combinatorial problem is asked at [857].\nReferences\n\n\n[Er62] Erd\\H{o}s, P\\'{a}l, Remarks on number theory. IV. Extremal problems in number theory. I. Mat. Lapok (1962), 228-255.\n\n[Er70] Erd\\H{o}s, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The equal-pairwise-LCM triple theorem remains open for arbitrary fixed positive density; recent formal work covers only density above 5/6.\n\n**Verified partial progress.**\n\n- The tracker records Weisenberg's argument for epsilon>221/225.\n- A June 2026 forum report describes a no-sorry Lean proof of |A|<=N-floor(N/6) for triple-free A, implying the result for epsilon>5/6.\n\n**Full solution or refutation.**\n\nEven if the reported Lean artifact is accepted, it is a high-density partial theorem and leaves the quantifier 'for every epsilon>0' unresolved.\n\n**What remains.**\n\nExtend the theorem from densities above 5/6 to every fixed positive density, or construct a positive-density counterexample.\n\n**Sources checked.**\n\n- Paul Erdős, Some extremal problems in combinatorial number theory (1970). (primary): https://combinatorica.hu/~p_erdos/1970-21.pdf\n Evidence used: Original problem source and surrounding common-divisor/common-multiple extremal questions.\n- Erdős Problems forum thread for Problem #536, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/536\n Evidence used: Records the Weisenberg threshold and June 2026 report of a formally checked 5/6 threshold.\n\n**Review notes.** The recent formal claim is not treated as a peer-reviewed source and does not solve the full problem. The common trailing background corruption was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2166, "problem_number": "EP-538", "title": "Erdős Problem #538", "statement": "Let $r\\geq 2$ and suppose that $A\\subseteq\\{1,\\ldots,N\\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\\in A$. Give the best possible upper bound for $ \\sum_{n\\in A}\\frac{1}{n}. $ ", "background": "Erd\\H{o}s observed that $ \\sum_{n\\in A}\\frac{1}{n}\\sum_{p\\leq N}\\frac{1}{p}\\leq r\\sum_{m\\leq N^2}\\frac{1}{m}\\ll r\\log N, $ and hence $ \\sum_{n\\in A}\\frac{1}{n} \\ll r\\frac{\\log N}{\\log\\log N}. $ See also [536] and [537].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The best possible reciprocal-sum bound remains open; the maintained tracker records only Erdős's general upper bound.\n\n**Verified partial progress.**\n\n- Erdős's double count gives sum_{a in A} 1/a << r log N/loglog N.\n\n**Full solution or refutation.**\n\nNo matching construction or sharp replacement for the known upper bound was located.\n\n**What remains.**\n\nDetermine the optimal dependence on r and N and exhibit matching examples in the intended asymptotic regimes.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #538, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/538\n Evidence used: Retains open status and derives the O(r log N/loglog N) bound.\n\n**Review notes.** No heavy computation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2167, "problem_number": "EP-539", "title": "Erdős Problem #539", "statement": "Let $h(n)$ be such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the set $ \\left\\{ \\frac{a}{(a,b)}: a,b\\in A\\right\\} $ has size at least $h(n)$. Estimate $h(n)$.", "background": "Erd\\H{o}s and Szemer\\'{e}di proved that $ n^{1/2} \\ll h(n) \\ll n^{1-c} $ for some constant $c>0$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** ProofCouncil's 2026 theorem determines the dimension-free power exponent h(n)=n^(1/2+o(1)), while the exact order remains open.\n\n**Verified partial progress.**\n\n- Classical work gave n^(1/2) << h(n), and Freiman-Lev gave h(n) << n^(2/3).\n- ProofCouncil proves (1+sqrt(8n-7))/2 <= h(n) <= n^(1/2) exp(C sqrt(log n)), hence log h(n)/log n tends to 1/2.\n- The associated Lean development verifies the exponent limit; the sharper displayed subexponential upper bound is in the informal proof.\n\n**Full solution or refutation.**\n\nThe main polynomial exponent is settled, but the gap between the linear-in-sqrt(n) lower bound and the subexponential factor in the upper bound remains.\n\n**What remains.**\n\nDetermine the exact order of h(n), ideally removing or proving necessary the exp(O(sqrt(log n))) factor.\n\n**Sources checked.**\n\n- Johannes Schmitt et al., ProofCouncil: An LLM Agent for Solving Open Mathematical Problems, arXiv:2607.09474 (2026), Appendix A.1 contribution. (primary): https://arxiv.org/abs/2607.09474\n Evidence used: Documents the ProofCouncil work containing the new dimension-free upper-bound argument.\n- ProofCouncil public repository and Lean formalization (2026). (formal_verification): https://github.com/eth-sri/proof-council\n Evidence used: Contains the proof artifact and Lean verification of the exponent-limit result.\n- Thomas F. Bloom, Erdős Problem #539, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/539\n Evidence used: Has incorporated the n^(1/2+o(1)) theorem while retaining open status for exact estimation.\n\n**Review notes.** The recent theorem is AI-assisted; the tracker reports a human screening with no issue, but also notes that only the exponent limit, not the strongest quantitative bound, is formalized.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2168, "problem_number": "EP-543", "title": "Erdős Problem #543", "statement": "Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size $N$ and $A\\subseteq G$ is a random set of size $k$ then, with probability $\\geq 1/2$, all elements of $G$ can be written as $\\sum_{x\\in S}x$ for some $S\\subseteq A$. Is $ f(N) \\leq \\log_2 N+o(\\log\\log N)? $ ", "background": "Erd\\H{o}s and R\\'{e}nyi \\cite{ErRe65} proved that $ f(N) \\leq \\log_2N+O(\\log\\log N). $ Erd\\H{o}s believed improving this to $o(\\log\\log N)$ is impossible.\nReferences\n\n\n[ErRe65] Erd\\H{o}s, P. and R\\'{e}nyi, A., Probabilistic methods in group theory. J. Analyse Math. (1965), 127-138.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Ma and Tang proved for primes p that f(p) is at least log_2 p+(1/(2 log 2)+o(1))log log p. This positive log-log second-order term directly refutes the proposed upper bound log_2 N+o(log log N), already for cyclic groups of prime order.\n\n**Verified partial progress.**\n\n- Erdős and Rényi proved the all-group upper bound log_2 N+O(log log N).\n- Erdős and Hall proved that an o(log log log N) remainder is impossible.\n- Ma and Tang obtained the explicit prime-order lower bound with coefficient 1/(2 log 2) on log log p.\n\n**Full solution or refutation.**\n\nFor a uniform random k-subset of F_p, the proof studies residues missed by all subset sums. It transfers from iid sampling to the uniform-subset model, estimates one- and two-target missing probabilities using truncated factorial moments and rank stratification of zero-one incidence matrices, and applies a second-moment argument. Below the stated threshold a missing residue persists with high probability, so coverage probability is below 1/2.\n\n**What remains.**\n\nThe conjectured little-o(log log N) upper error is decisively refuted. Determining the sharp second-order term for all N and describing its dependence on the structure of the finite abelian group remain successor problems. The input background predates the 2026 result and contains trailing serialization noise, but its mathematical statement is intact.\n\n**Sources checked.**\n\n- Jie Ma and Quanyu Tang, An Erdős problem on random subset sums in finite abelian groups, arXiv:2602.05768 (2026). (primary): https://arxiv.org/abs/2602.05768\n Evidence used: Primary paper proving the explicit prime-order lower bound that refutes EP-543.\n- Quanyu Tang and ChatGPT-5.2 Pro, A note on Problem #543, revised public proof source (2026). (primary): https://github.com/QuanyuTang/erdos-problem-543/blob/main/On_Erdos_Problem_543_Revised_and_Verified.tex\n Evidence used: Public precursor developing the missing-residue factorial-moment and second-moment argument.\n- P. Erdős and A. Rényi, Probabilistic methods in group theory, Journal d'Analyse Mathématique 14 (1965), 127--138. (primary): https://combinatorica.hu/~p_erdos/1965-15.pdf\n Evidence used: Original primary source for the general log_2 N+O(log log N) upper bound.\n- P. Erdős and R. R. Hall, Some new results in probabilistic group theory, Commentarii Mathematici Helvetici 53 (1978), 448--457. (primary): https://users.renyi.hu/~p_erdos/1978-45.pdf\n Evidence used: Earlier primary lower-order obstruction ruling out an o(log log log N) error.\n- Thomas F. Bloom, Erdős Problem #543 (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/543\n Evidence used: Records the negative resolution, prior bounds, and attribution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2169, "problem_number": "EP-544", "title": "Erdős Problem #544", "statement": "Show that $ R(3,k+1)-R(3,k)\\to\\infty $ as $k\\to \\infty$. Similarly, prove or disprove that $ R(3,k+1)-R(3,k)=o(k). $ ", "background": "A problem of Erd\\H{o}s and S\\'{o}s.\nThis problem is #8 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Both the divergence of consecutive differences and the o(k) upper bound remain open.\n\n**Verified partial progress.**\n\n- The established scale is R(3,k) asymptotic up to constants to k^2/log k.\n- A 2026 OpenAI theorem gives R(3,k+1)-R(3,k) << k^(-c) R(3,k) for some c>0.\n\n**Full solution or refutation.**\n\nThe new relative increment bound proves neither requested assertion at its currently stated exponent.\n\n**What remains.**\n\nProve that the increment tends to infinity and decide whether it is o(k).\n\n**Sources checked.**\n\n- On the ratio of R(k,l) and R(k,l+1), OpenAI mathematical proof (2026). (primary): https://cdn.openai.com/pdf/6dc7175d-d9e7-4b8d-96b8-48fe5798cd5b/Ramsey.pdf\n Evidence used: Proves a quantitative ratio bound implying the k^(-c) relative increment estimate.\n- Thomas F. Bloom, Erdős Problem #544, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/544\n Evidence used: Records both questions as open and states the new relative increment consequence.\n\n**Review notes.** A ratio tending to one is not confused with either an additive lower bound or an o(k) increment.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2170, "problem_number": "EP-545", "title": "Erdős Problem #545", "statement": "Let $G$ be a graph with $m$ edges and no isolated vertices. Is the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'? That is, if $m=\\binom{n}{2}+t$ edges with $0\\leq t\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The universal formulation is false already for m=2.\n\n**Verified partial progress.**\n\n- Cockayne and Lorimer determine diagonal matching Ramsey numbers, including R(2K2)=5.\n- For m=2=binom(2,2)+1, the prescribed as-complete-as-possible graph is P3 and R(P3)=3.\n\n**Full solution or refutation.**\n\nTaking G=2K2 gives R(G)=5>3=R(P3), directly refuting the stated inequality.\n\n**What remains.**\n\nAn asymptotic repair, such as asking whether the extremal description holds for every sufficiently large m, remains open; that qualifier is absent from this record.\n\n**Sources checked.**\n\n- E. J. Cockayne and P. J. Lorimer, The Ramsey number for stripes, J. Austral. Math. Soc. 19 (1975), 252-256, doi:10.1017/S1446788700029554. (primary): https://doi.org/10.1017/S1446788700029554\n Evidence used: The diagonal two-color specialization gives R(mK2)=3m-1, hence R(2K2)=5.\n- Thomas F. Bloom, Erdős Problem #545, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/545\n Evidence used: Explicitly records failure for small m and identifies matching counterexamples.\n\n**Review notes.** The exact imported statement was assessed; it was not silently repaired by adding 'for sufficiently large m'.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2171, "problem_number": "EP-550", "title": "Erdős Problem #550", "statement": "Let $m_1\\leq\\cdots\\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex class sizes $m_1,\\ldots,m_k$ then prove that $ R(T,G)\\leq (\\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1. $ ", "background": "Chv\\'{a}tal \\cite{Ch77} proved that $R(T,K_m)=(m-1)(n-1)+1$.\nThis problem is #16 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[Ch77] Chv\\'{a}tal, V., Tree-complete graph Ramsey numbers. J. Graph Theory (1977), 93.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Li's June 2026 preprint proves the claimed inequality under the standard interpretation that k and the multipartite class sizes are fixed and n is sufficiently large.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nFor fixed k and fixed 1<=m1<=...<=mk, Li proves R(T,K_{m1,...,mk}) <= (k-1)(R(T,K_{m1,m2})-1)+m1 for every sufficiently large n and every n-vertex tree T; since chi(K_{m1,...,mk})=k, this matches the target.\n\n**What remains.**\n\nObtain broader independent or peer review and clarify whether the imported wording intended any uniformity when the class sizes grow with n.\n\n**Sources checked.**\n\n- Eric Li, A Resolution of Erdős Problem 550 on Tree versus Complete Multipartite Ramsey Numbers, arXiv:2606.23659 (2026). (primary): https://arxiv.org/abs/2606.23659\n Evidence used: The abstract states exactly the fixed-parameter, sufficiently-large-n theorem requested by the standard formulation.\n- Thomas F. Bloom, Erdős Problem #550 and discussion thread, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/550\n Evidence used: Hosts the solution claim and a public screening that reported no detected issue, while the main status had not yet been updated.\n\n**Review notes.** Quantifier ambiguity preserved: the input says m1<=...<=mk and n are sufficiently large, whereas the paper proves the standard fixed-mi, large-n version. The paper is recent, AI-assisted, and not peer reviewed, so confidence is medium.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2172, "problem_number": "EP-552", "title": "Erdős Problem #552", "statement": "Determine the Ramsey number $ R(C_4,S_n), $ where $S_n=K_{1,n}$ is the star on $n+1$ vertices.\nIn particular, is it true that, for any $c>0$, there are infinitely many $n$ such that $ R(C_4,S_n)\\leq n+\\sqrt{n}-c? $ ", "background": "A problem of Burr, Erd\\H{o}s, Faudree, Rousseau, and Schelp \\cite{BEFRS89}. Erd\\H{o}s often asked about $R(C_4,S_n)$ in the equivalent formulation of asking for a bound on the minimum degree of a graph which would guarantee the existence of a $C_4$ (see [85]).\nIt is known that $ n+\\sqrt{n}-6n^{11/40} \\leq R(C_4,S_n)\\leq n+\\lceil\\sqrt{n}\\rceil+1. $ The lower bound is due to \\cite{BEFRS89}, the upper bound is due to Parsons \\cite{Pa75}. The lower bound of \\cite{BEFRS89} is related to gaps between primes, and assuming e.g. Cramer's conjecture on gaps between primes their lower bound would be $n+\\sqrt{n}-n^{o(1)}$.\nErd\\H{o}s offered \\$100 for a proof or disproof of the second question in \\cite{BEFRS89}. In \\cite{Er96} Erd\\H{o}s asks (an equivalent formulation of) whether $R(C_4,S_n)\\geq n+\\sqrt{n}-O(1)$, but says this is probably 'too optimistic'.\nThey also ask, if $f(n)=R(C_4,S_n)$, whether $f(n+1)=f(n)$ infinitely often, and is the density of such $n$ $0$? Also, is it true that $f(n+1)\\leq f(n)+2$ for all $n$? A similar question about an equivalent function is the subject of [85].\nParsons \\cite{Pa75} proved that $ R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil $ whenever $n=q^2+1$ for a prime power $q$ and $ R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil+1 $ whenever $n=q^2$ for a prime power $q$ (in particular both equalities occur infinitely often).\nThis has been extended in various works, all in the cases $n=q^2\\pm t$ for some $0\\leq t\\leq q$ and prime power $q$. We refer to the work of Parsons \\cite{Pa76}, Wu, Sun, Zhang, and Radziszowski \\cite{WSZR15}, and Zhang, Chen, and Cheng (\\cite{ZCC17} and \\cite{ZCC17b}) for a precise description. In every known case $ R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil+\\{0,1\\}, $ and Zhang, Chen, and Cheng \\cite{ZCC17} speculate whether this is in fact true for all $n\\geq 2$ (whence the answer to the question above would be no).\nThis problem is #19 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[BEFRS89] Burr, S. and Erd\"{o}s, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Some complete bipartite graph-tree Ramsey numbers. Graph theory in memory of G. A. Dirac (Sandbjerg,\n1985) (1989), 79-89.\n\n[Er96] Erd\\H{o}s, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9.\n\n[Pa75] Parsons, T. D., Ramsey graphs and block designs. {I}. Trans. Amer. Math. Soc. (1975), 33--44.\n\n[Pa76] No reference found.\n\n\n[WSZR15] Wu, Yali and Sun, Yongqi and Zhang, Rui and Radziszowski,\nStanis\\l aw P., Ramsey numbers of {$C_4$} versus wheels and stars. Graphs Combin. (2015), 2437--2446.\n\n[ZCC17] Zhang, Xuemei and Chen, Yaojun and Cheng, T. C. Edwin, Some values of {R}amsey numbers for {$C_4$} versus stars. Finite Fields Appl. (2017), 73--85.\n\n[ZCC17b] Zhang, Xuemei and Chen, Yaojun and Cheng, T. C. Edwin, Polarity graphs and {R}amsey numbers for {$C_4$} versus stars. Discrete Math. (2017), 655--660.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Tight general bounds and infinitely many exact prime-power families are known, but the all-n determination and the highlighted infinite-deficit question remain open.\n\n**Verified partial progress.**\n\n- The known general bounds are n+sqrt(n)-6n^(11/40) <= R(C4,S_n) <= n+ceil(sqrt(n))+1.\n- Parsons proves exact values n+ceil(sqrt(n)) at n=q^2+1 and n+ceil(sqrt(n))+1 at n=q^2 for prime powers q.\n- Later work extends exact evaluation to further n=q^2 plus or minus t families.\n\n**Full solution or refutation.**\n\nAll known exact values are one of two adjacent candidates, but no formula for every n is known and the $100 subquestion is unresolved.\n\n**What remains.**\n\nDetermine R(C4,S_n) for general n and decide whether arbitrarily large fixed deficits below n+sqrt(n) occur infinitely often.\n\n**Sources checked.**\n\n- T. D. Parsons, Ramsey Graphs and Block Designs. I, Trans. Amer. Math. Soc. 209 (1975), 33-44, doi:10.2307/1997368. (primary): https://doi.org/10.2307/1997368\n Evidence used: Provides the general upper bound and the prime-power exact families.\n- Xuemei Zhang, Yaojun Chen and T. C. Edwin Cheng, Some values of Ramsey numbers for C4 versus stars, Finite Fields Appl. 45 (2017), 73-85. (primary): https://doi.org/10.1016/j.ffa.2016.11.012\n Evidence used: Extends exact evaluations near prime-power squares.\n- Thomas F. Bloom, Erdős Problem #552, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/552\n Evidence used: Synthesizes the bounds, exact families, and continuing open subquestions.\n\n**Review notes.** Exact infinite subfamilies are classified as partial progress, not a general formula.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2173, "problem_number": "EP-554", "title": "Erdős Problem #554", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that $ \\lim_{k\\to \\infty}\\frac{R(C_{2n+1};k)}{R(K_3;k)}=0 $ for any $n\\geq 2$.", "background": "A problem of Erd\\H{o}s and Graham. The problem is open even for $n=2$.\nThis problem is #23 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Combining a 2025 odd-cycle upper bound with OpenAI's August 2026 triangle lower bound proves the requested ratio for every fixed n>=4; n=2 and n=3 remain open.\n\n**Verified partial progress.**\n\n- Axenovich et al. prove R_k(C_{2n+1}) <= (4n-2)^k k^(k/n)+1.\n- OpenAI proves and formally verifies R_k(K3) >= k^(k/3-o(k)).\n- Since 1/n<1/3 for n>=4, these bounds imply R_k(C_{2n+1})/R_k(K3) tends to zero for all fixed n>=4.\n\n**Full solution or refutation.**\n\nA direct synthesis of current primary results resolves all parameters n>=4, but not the two smallest requested cases.\n\n**What remains.**\n\nProve the ratio limit for C5 and C7, corresponding to n=2 and n=3.\n\n**Sources checked.**\n\n- Maria Axenovich, Wouter Cames van Batenburg, Oliver Janzer, Lukas Michel and Mathieu Rundström, An improved upper bound for the multicolour Ramsey number of odd cycles, arXiv:2510.17981 (2025). (primary): https://arxiv.org/abs/2510.17981\n Evidence used: Proves the k^(k/n) upper bound for the fixed odd cycle C_{2n+1}.\n- OpenAI, MulticolorTriangleRamsey Lean certificate in ten-proofs (2026). (formal_verification): https://github.com/openai/ten-proofs\n Evidence used: Provides a sorry-free certificate for the explicit superexponential multicolor triangle lower bound.\n- Thomas F. Bloom, Erdős Problem #554, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/554\n Evidence used: Records the historical and 2025 odd-cycle bounds and that the smallest cases were open before the August triangle result.\n\n**Review notes.** The n>=4 resolution is an explicit inference from two cited theorems; neither source is represented as having solved the remaining n=2,3 cases.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2174, "problem_number": "EP-555", "title": "Erdős Problem #555", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine the value of $ R(C_{2n};k). $ ", "background": "A problem of Erd\\H{o}s and Graham. Erd\\H{o}s \\cite{Er81c} gives the bounds $ k^{1+\\frac{1}{2n}}\\ll R(C_{2n};k)\\ll k^{1+\\frac{1}{n-1}}. $ Chung and Graham \\cite{ChGr75} showed that $ R(C_4;k)>k^2-k+1 $ when $k-1$ is a prime power and $ R(C_4;k)\\leq k^2+k+1 $ for all $k$.\nThis problem is #24 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[ChGr75] Chung, Fan R. K. and Graham, R. L., On multicolor Ramsey numbers for complete bipartite graphs. J. Combinatorial Theory Ser. B (1975), 164-169.\n\n[Er81c] Erd\\H{o}s, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The correct order is known for C4, C6, and C10, while the general fixed even cycle remains open.\n\n**Verified partial progress.**\n\n- Chung and Graham prove R_k(C4)=(1+o(1))k^2 and explicit adjacent quadratic bounds.\n- Li and Lih prove R_k(C_{2n})=Theta(k^(n/(n-1))) for n in {2,3,5}.\n\n**Full solution or refutation.**\n\nSeveral fixed even cycles have their growth exponent determined, but no general theorem or exact value is known for arbitrary n.\n\n**What remains.**\n\nDetermine the correct order, leading constants, or exact values for general fixed n, especially parameters outside {2,3,5}.\n\n**Sources checked.**\n\n- Yusheng Li and Ko-Wei Lih, Multi-color Ramsey numbers of even cycles, European J. Combin. 30 (2009), 114-118, doi:10.1016/j.ejc.2008.02.008. (primary): https://doi.org/10.1016/j.ejc.2008.02.008\n Evidence used: Determines the order Theta(k^(n/(n-1))) for n=2,3,5.\n- Thomas F. Bloom, Erdős Problem #555, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/555\n Evidence used: Records the broad open problem and the classical C4 bounds.\n\n**Review notes.** The tracker summary omits the 2009 n in {2,3,5} theorem, which was verified through the primary bibliographic record and later literature citing its precise scope.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2175, "problem_number": "EP-557", "title": "Erdős Problem #557", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Is it true that $ R(T;k)\\leq kn+O(1) $ for any tree $T$ on $n$ vertices?", "background": "A problem of Erd\\H{o}s and Graham. Implied by [548].\nThis would be best possible since, for example, $R(S_n,k)\\geq kn-O(k)$ if $S_n=K_{1,n-1}$ is a star on $n$ vertices.\nThis problem is #26 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The fixed-color all-tree bound remains open.\n\n**Verified partial progress.**\n\n- The assertion is implied by the stronger EP-548.\n- Exact star and path results show the linear scale in important special families, and stars give the near-matching lower bound kn-O(k).\n\n**Full solution or refutation.**\n\nKnown family-specific results do not supply a uniform upper bound for every n-vertex tree.\n\n**What remains.**\n\nProve or refute R_k(T)<=kn+C_k uniformly over all n-vertex trees.\n\n**Sources checked.**\n\n- P. Erdős and R. Graham, On Partition Theorems for Finite Graphs, 1975, p.516. (primary): https://users.renyi.hu/~p_erdos/1975-23.pdf\n Evidence used: Historical source for the conjectured kn+O(1) bound.\n- Thomas F. Bloom, Erdős Problem #557, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/557\n Evidence used: Retains open status and records the implication and star lower bound.\n\n**Review notes.** Formulation ambiguity preserved: the source does not explicitly state whether k is fixed or whether the O(1) constant may depend on k. The standard fixed-k interpretation was used only for status discussion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2176, "problem_number": "EP-558", "title": "Erdős Problem #558", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine $ R(K_{s,t};k) $ where $K_{s,t}$ is the complete bipartite graph with $s$ vertices in one component and $t$ in the other.", "background": "Chung and Graham \\cite{ChGr75} prove the general bounds $ (2\\pi\\sqrt{st})^{\\frac{1}{s+t}}\\left(\\frac{s+t}{e^2}\\right)k^{\\frac{st-1}{s+t}}\\leq R(K_{s,t};k)\\leq (t-1)(k+k^{1/s})^s $ and determined $ R(K_{2,2},k)=(1+o(1))k^2. $ Alon, R\\'{o}nyai, and Szab\\'{o} \\cite{ARS99} have proved that $ R(K_{3,3},k)=(1+o(1))k^3 $ and that if $s\\geq (t-1)!+1$ then $ R(K_{s,t},k)\\asymp k^t. $ This problem is #27 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[ARS99] Alon, Noga and R\\'{o}nyai, Lajos and Szab\\'{o}, Tibor, Norm-graphs: variations and applications. J. Combin. Theory Ser. B (1999), 280-290.\n\n[ChGr75] Chung, Fan R. K. and Graham, R. L., On multicolor Ramsey numbers for complete bipartite graphs. J. Combinatorial Theory Ser. B (1975), 164-169.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Asymptotic growth is determined for K2,2, K3,3, and the large-s norm-graph regime, but not for general fixed s,t.\n\n**Verified partial progress.**\n\n- Chung and Graham prove R_k(K2,2)=(1+o(1))k^2 and general bounds for K_{s,t}.\n- Alon, Rónyai, and Szabó prove R_k(K3,3)=(1+o(1))k^3.\n- They also prove R_k(K_{s,t})=Theta(k^t) when s>=(t-1)!+1.\n\n**Full solution or refutation.**\n\nImportant parameter ranges are solved asymptotically, while a substantial gap remains in the general fixed-(s,t) case.\n\n**What remains.**\n\nDetermine the order and, where possible, leading asymptotics for all fixed 2<=s<=t outside the known regimes.\n\n**Sources checked.**\n\n- Noga Alon, Lajos Rónyai and Tibor Szabó, Norm-graphs: variations and applications, J. Combin. Theory Ser. B 76 (1999), 280-290. (primary): https://www.tau.ac.il/~nogaa/PDFS/norm7.pdf\n Evidence used: Proves the K3,3 asymptotic and the Theta(k^t) large-s regime.\n- Thomas F. Bloom, Erdős Problem #558, checked 2026-08-17; citing Chung-Graham (1975). (maintained_tracker): https://www.erdosproblems.com/558\n Evidence used: States the general bounds, K2,2 result, norm-graph results, and continuing open status.\n\n**Review notes.** Known asymptotics are understood for fixed s,t as k tends to infinity; the source statement itself does not spell out that regime.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2177, "problem_number": "EP-560", "title": "Erdős Problem #560", "statement": "Let $\\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.\nDetermine $ \\hat{R}(K_{n,n}), $ where $K_{n,n}$ is the complete bipartite graph with $n$ vertices in each component.", "background": "We know that $ \\frac{1}{60}n^22^n<\\hat{R}(K_{n,n})< \\frac{3}{2}n^32^n. $ The lower bound (which holds for $n\\geq 6$) was proved by Erd\\H{o}s and Rousseau \\cite{ErRo93}. The upper bound was proved by Erd\\H{o}s, Faudree, Rousseau, and Schelp \\cite{EFRS78b} and Ne\\v{s}et\\v{r}il and R\"{o}dl \\cite{NeRo78}.\nConlon, Fox, and Wigderson \\cite{CFW23} have proved that, for any $s\\leq t$, $ \\hat{R}(K_{s,t})\\gg s^{2-\\frac{s}{t}}t2^s, $ and prove that when $t\\gg s\\log s$ we have $\\hat{R}(K_{s,t})\\asymp s^2t2^s$. They conjecture that this should hold for all $s\\leq t$, and so in particular we should have $\\hat{R}(K_{n,n})\\asymp n^32^n$.\nThis problem is #29 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[CFW23] Conlon, David and Fox, Jacob and Wigderson, Yuval, Three early problems on size Ramsey numbers. Combinatorica (2023), 743-768.\n\n[EFRS78b] Erd\\H{o}s, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., The size Ramsey number. Period. Math. Hungar. (1978), 145-161.\n\n[ErRo93] Erd\\H{o}s, P. and Rousseau, C. C., The size Ramsey number of a complete bipartite graph. Discrete Math. (1993), 259-262.\n\n[NeRo78] Ne\\vSet\\v{r}il, J. and R\"{o}dl, V., The structure of critical Ramsey graphs. Acta Math. Acad. Sci. Hungar. (1978), 295-300.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The diagonal size-Ramsey number remains undetermined, but Conlon, Fox, and Wigderson determined the correct order for K_{s,t} throughout the off-diagonal range t=Omega(s log s) and proved a general lower bound.\n\n**Verified partial progress.**\n\n- For s<=t, Conlon, Fox, and Wigderson proved hat R(K_{s,t}) >> s^{2-s/t} t 2^s.\n- For t=Omega(s log s), they proved hat R(K_{s,t})=Theta(s^2 t 2^s).\n- On the diagonal, the recorded bounds remain between constant multiples of n^2 2^n and n^3 2^n.\n\n**Full solution or refutation.**\n\nThe 2023 paper explicitly reports substantial progress rather than a resolution of the complete-bipartite question; it does not cover t=s in its sharp-order regime.\n\n**What remains.**\n\nDetermine the diagonal order of hat R(K_{n,n}); the prominent conjecture is Theta(n^3 2^n).\n\n**Sources checked.**\n\n- David Conlon, Jacob Fox, and Yuval Wigderson, Three early problems on size Ramsey numbers, Combinatorica 43 (2023), 743-768, DOI 10.1007/s00493-023-00034-7. (primary): https://arxiv.org/abs/2111.05420\n Evidence used: The abstract and theorem summary state the general lower bound and the constant-factor determination for t=Omega(s log s), while calling the complete-bipartite result substantial progress.\n- Thomas F. Bloom, Erdős Problem #560, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/560\n Evidence used: Maintains open status and records both the diagonal bounds and the 2023 off-diagonal progress.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2178, "problem_number": "EP-561", "title": "Erdős Problem #561", "statement": "Let $\\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.\nLet $F_1$ and $F_2$ be the union of stars. More precisely, let $F_1=\\cup_{i\\leq s} K_{1,n_i}$ and $F_2=\\cup_{j\\leq t} K_{1,m_j}$. Prove that $ \\hat{R}(F_1,F_2) = \\sum_{2\\leq k\\leq s+2}\\max\\{n_i+m_j-1 : i+j=k\\}. $ ", "background": "Burr, Erd\\H{o}s, Faudree, Rousseau, and Schelp \\cite{BEFRS78} proved this when all the $n_i$ are identical and all the $m_i$ are identical.\nThis problem is #30 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[BEFRS78] Burr, S. A. and Erd\\H{o}s, P. and Faudree, R. J. and Rousseau,\nC. C. and Schelp, R. H., Ramsey-minimal graphs for multiple copies. Nederl. Akad. Wetensch. Indag. Math. (1978), 187-195.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported statement is materially different from the standard star-forest conjecture: it sums only to s+2 and omits monotonicity hypotheses, whereas the maintained and original formulations sum to s+t. Literature status therefore cannot be transferred safely to the exact imported formula.\n\n**Verified partial progress.**\n\n- For the standard formulation, Burr, Erdős, Faudree, Rousseau, and Schelp proved the case in which the star sizes are constant within each forest.\n- Győri and Schelp proved the standard formula under the separation conditions binom(l_k,2)>sum_{i=k}^{s+t} l_i.\n- Davoodi, Javadi, Kamranian, and Raeisi proved many further exact cases, including s=1, s=2 with n_1=n_2, all star sizes odd, and an identical-odd family.\n\n**Full solution or refutation.**\n\nThe corrected standard conjecture is partially solved and remains open, but those results do not establish the malformed imported formula with endpoint s+2.\n\n**What remains.**\n\nResolve the source/formulation discrepancy before assigning a mathematical status; for the standard s+t formulation, prove the formula for arbitrary nonincreasing star-size sequences.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #561, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/561\n Evidence used: Gives the standard statement with ordering hypotheses and summation endpoint s+t, and records established special cases.\n- Erdős Problems on Graphs, Size Ramsey number of star forests, checked 2026-08-17. (source_collection): https://mathweb.ucsd.edu/~erdosproblems/erdos/newproblems/SizeRamseyStars.html\n Evidence used: The original graph-problems page agrees with the maintained s+t formulation rather than the imported s+2 endpoint.\n- Akbar Davoodi, Ramin Javadi, Azam Kamranian, and Ghaffar Raeisi, On a Conjecture of Erdős on Size Ramsey Number of Star Forests, arXiv:2111.02065. (primary): https://arxiv.org/abs/2111.02065\n Evidence used: The abstract explicitly says that exact values for many star-forest pairs give a partial solution to the standard conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2179, "problem_number": "EP-562", "title": "Erdős Problem #562", "statement": "Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.\nProve that, for $r\\geq 3$, $ \\log_{r-1} R_r(n) \\asymp_r n, $ where $\\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm. That is, does $R_r(n)$ grow like $ 2^{2^{\\cdots n}} $ where the tower of exponentials has height $r-1$?", "background": "A problem of Erd\\H{o}s, Hajnal, and Rado \\cite{EHR65}. A generalisation of [564].\nThis problem is #38 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EHR65] Erd\\H{o}s, P. and Hajnal, A. and Rado, R., Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. (1965), 93-196.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Erdős-Hajnal-Rado conjecture that diagonal r-uniform Ramsey numbers have tower height r-1 remains open; the unresolved r=3 case is already EP-564.\n\n**Verified partial progress.**\n\n- Erdős, Hajnal, and Rado established the foundational tower bounds and stepping-up method.\n- The known upper and lower bounds still differ by a tower level in the central two-colour diagonal problem.\n\n**Full solution or refutation.**\n\nNo primary source was found proving log_{r-1} R_r(n)=Theta_r(n) for every fixed r>=3, and the maintained tracker retains open status.\n\n**What remains.**\n\nClose the tower-height gap, already by proving a doubly exponential lower bound of height two in the 3-uniform two-colour case.\n\n**Sources checked.**\n\n- P. Erdős, A. Hajnal, and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93-196, DOI 10.1007/BF01886396. (primary): https://www.renyi.hu/~p_erdos/1965-14.pdf\n Evidence used: Primary source for the conjecture and the foundational partition/stepping-up bounds.\n- Thomas F. Bloom, Erdős Problem #562, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/562\n Evidence used: Lists the exact tower-height statement as open and links it to the 3-uniform subproblem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2180, "problem_number": "EP-563", "title": "Erdős Problem #563", "statement": "Let $F(n,\\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\\subseteq [n]$ with $\\lvert X\\rvert\\geq m$ contains more than $\\alpha \\binom{\\lvert X\\rvert}{2}$ many edges of each colour.\nProve that, for every $0\\leq \\alpha< 1/2$, $ F(n,\\alpha)\\sim c_\\alpha\\log n $ for some constant $c_\\alpha$ depending only on $\\alpha$.", "background": "It is easy to show via the probabilistic method that, for every $0\\leq \\alpha<1/2$, $ F(n,\\alpha)\\asymp_\\alpha \\log n. $ Note that when $\\alpha=0$ this is just asking for a $2$-colouring of the edges of $K_n$ which contains no monochromatic clique of size $m$, and hence we recover the classical Ramsey numbers.\nSee also [161] for a generalisation to hypergraphs.\nThis problem is #39 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The logarithmic order F(n,alpha)=Theta_alpha(log n) is known for every fixed alpha<1/2, but the requested asymptotic equivalence with a limiting constant remains open.\n\n**Verified partial progress.**\n\n- A probabilistic argument gives upper and lower bounds of order log n for every fixed 0<=alpha<1/2.\n- At alpha=0 the leading-constant question contains the classical unresolved diagonal two-colour Ramsey asymptotic.\n\n**Full solution or refutation.**\n\nExisting results determine only the order of magnitude, not a constant c_alpha for which the ratio tends to one.\n\n**What remains.**\n\nProve existence of the leading asymptotic constant c_alpha for each alpha, including the alpha=0 Ramsey case.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #563, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/563\n Evidence used: Maintains open status, records F(n,alpha)=Theta_alpha(log n), and explains that alpha=0 recovers the classical Ramsey problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2181, "problem_number": "EP-564", "title": "Erdős Problem #564", "statement": "Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices.\nIs there some constant $c>0$ such that $ R_3(n) \\geq 2^{2^{cn}}? $ ", "background": "A special case of [562]. A problem of Erd\\H{o}s, Hajnal, and Rado \\cite{EHR65}, who prove the bounds $ 2^{cn^2}< R_3(n)< 2^{2^{n}} $ for some constant $c>0$.\nErd\\H{o}s, Hajnal, M\\'{a}t\\'{e}, and Rado \\cite{EHMR84} have proved a doubly exponential lower bound for the corresponding problem with $4$ colours.\nThis problem is #37 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EHMR84] Erd\\H{o}s, Paul and Hajnal, Andr\\'{a}s and M\\'{a}t\\'{e}, Attila and Rado, Richard, Combinatorial set theory: partition relations for cardinals. (1984), 347.\n\n[EHR65] Erd\\H{o}s, P. and Hajnal, A. and Rado, R., Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. (1965), 93-196.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The doubly exponential lower bound for the two-colour 3-uniform diagonal Ramsey number remains open and is still listed with a $500 prize.\n\n**Verified partial progress.**\n\n- The classical bounds are 2^{c n^2}0.\n- A doubly exponential lower bound is known in the corresponding four-colour problem, but not for the imported two-colour statement.\n\n**Full solution or refutation.**\n\nNo verified source located closes the exponential-versus-double-exponential gap for two colours.\n\n**What remains.**\n\nProve or refute R_3(n)>=2^{2^{c n}} for some absolute c>0.\n\n**Sources checked.**\n\n- P. Erdős, A. Hajnal, and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93-196, DOI 10.1007/BF01886396. (primary): https://www.renyi.hu/~p_erdos/1965-14.pdf\n Evidence used: Primary historical source for the problem and classical two-colour bounds.\n- Thomas F. Bloom, Erdős Problem #564, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/564\n Evidence used: Retains open status and the $500 prize, and distinguishes the known four-colour result from the open two-colour question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2182, "problem_number": "EP-566", "title": "Erdős Problem #566", "statement": "Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then $ R(G,H)\\ll m? $ ", "background": "In other words, is $G$ Ramsey size linear? This fails for a graph $G$ with $n$ vertices and $2n-2$ edges (for example with $H=K_n$). Erd\\H{o}s, Faudree, Rousseau, and Schelp \\cite{EFRS93} have shown that any graph $G$ with $n$ vertices and at most $n+1$ edges is Ramsey size linear.\nImplies [567].\nThis problem is #31 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EFRS93] Erd\\H{o}s, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains open whether the hereditary density bound e(J)<=2v(J)-3 forces Ramsey size-linearity against every graph with no isolated vertices.\n\n**Verified partial progress.**\n\n- Erdős, Faudree, Rousseau, and Schelp proved Ramsey size-linearity for fixed connected G with e(G)<=v(G)+1.\n- They also showed that graphs with e(G)>=2v(G)-2 need not be Ramsey size-linear, making the imported 2v-3 threshold the sharp frontier of their obstruction.\n\n**Full solution or refutation.**\n\nThe sufficient theorem and negative construction leave a genuine intermediate density range and do not cover all graphs satisfying the imported hereditary bound.\n\n**What remains.**\n\nProve size-linearity for every fixed G whose every k-vertex subgraph has at most 2k-3 edges, or produce a counterexample within that exact class.\n\n**Sources checked.**\n\n- P. Erdős, R. J. Faudree, C. C. Rousseau, and R. H. Schelp, Ramsey Size Linear Graphs, Combinatorics, Probability and Computing 2 (1993), 389-399, DOI 10.1017/S096354830000078X. (primary): https://doi.org/10.1017/S096354830000078X\n Evidence used: Primary source for the size-linear definition, positive sparse result, negative 2v-2 threshold, and the open question.\n- Thomas F. Bloom, Erdős Problem #566, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/566\n Evidence used: Keeps the exact hereditary 2k-3 statement open and records no complete or partial solution in the comments.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2183, "problem_number": "EP-567", "title": "Erdős Problem #567", "statement": "Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ has $m$ edges and no isolated vertices, then $ R(G,H)\\ll m? $ ", "background": "In other words, is $G$ Ramsey size linear? A special case of [566]. In \\cite{Er95} Erd\\H{o}s specifically asks about the case $G=K_{3,3}$.\nThe graph $H_5$ can also be described as $K_4^*$, obtained from $K_4$ by subdividing one edge. ($K_4$ itself is not Ramsey size linear, since $R(4,n)\\gg n^{3-o(1)}$, see [166].) Brada\\'{c}, Gishboliner, and Sudakov \\cite{BGS23} have shown that every subdivision of $K_4$ on at least $6$ vertices is Ramsey size linear, and also that $R(H_5,H) \\ll m$ whenever $H$ is a bipartite graph with $m$ edges and no isolated vertices.\nThis problem is #32 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[BGS23] Brada\\'C, D. and Gishboliner, L. and Sudakov, B., On Ramsey size-linear graphs and related questions. arXiv:2202.10388 (2023).\n\n[Er95] Erd\\H{o}s, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general size-linearity of Q_3, K_{3,3}, and H_5 remains open, but a 2024 theorem proves the required bound for H_5 against bipartite opponents and settles all subdivisions of K_4 on at least six vertices.\n\n**Verified partial progress.**\n\n- Bradač, Gishboliner, and Sudakov proved R(H_5,H)=O(e(H)) when H is bipartite and has no isolated vertices.\n- They proved every subdivision of K_4 on at least six vertices is Ramsey size-linear against arbitrary opponents.\n- The five-vertex graph H_5=K_4 with one edge subdivided lies just outside the latter general theorem.\n\n**Full solution or refutation.**\n\nNone of Q_3, K_{3,3}, or H_5 is verified size-linear against all graphs H, so the imported universal statement is not solved.\n\n**What remains.**\n\nProve or refute Ramsey size-linearity for each of Q_3, K_{3,3}, and H_5 against arbitrary graphs without isolated vertices.\n\n**Sources checked.**\n\n- Domagoj Bradač, Lior Gishboliner, and Benny Sudakov, On Ramsey Size-Linear Graphs and Related Questions, SIAM J. Discrete Math. 38 (2024), 225-242, DOI 10.1137/22M1481713. (primary): https://arxiv.org/abs/2202.10388\n Evidence used: Proves the at-least-six-vertex K_4-subdivision theorem and the bipartite-opponent H_5 result.\n- Thomas F. Bloom, Erdős Problem #567, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/567\n Evidence used: Retains all three general cases as open while recording the 2024 special-case progress.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2184, "problem_number": "EP-568", "title": "Erdős Problem #568", "statement": "Let $G$ be a graph such that $R(G,T_n)\\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices, $ R(G,H)\\ll m? $ ", "background": "In other words, is $G$ Ramsey size linear?\nThis problem is #33 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No verified theorem was located showing that linear Ramsey bounds against all trees plus quadratic bounds against cliques imply Ramsey size-linearity against every m-edge graph.\n\n**Verified partial progress.**\n\n- The two hypotheses test sparse connected opponents and dense complete opponents, respectively.\n- The maintained tracker records no incorporated partial or complete solution to the implication.\n\n**Full solution or refutation.**\n\nThe 1993 size-linear program poses this reduction question, and current maintained status remains open.\n\n**What remains.**\n\nDerive R(G,H)=O_G(e(H)) for arbitrary H without isolated vertices from the two stated Ramsey hypotheses, or find a fixed counterexample G satisfying both hypotheses.\n\n**Sources checked.**\n\n- P. Erdős, R. J. Faudree, C. C. Rousseau, and R. H. Schelp, Ramsey Size Linear Graphs, Combinatorics, Probability and Computing 2 (1993), 389-399, DOI 10.1017/S096354830000078X. (primary): https://doi.org/10.1017/S096354830000078X\n Evidence used: Primary source for the question within the Ramsey size-linear framework.\n- Thomas F. Bloom, Erdős Problem #568, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/568\n Evidence used: Maintains open status and reports no solution claims or incorporated partial results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2185, "problem_number": "EP-569", "title": "Erdős Problem #569", "statement": "Let $k\\geq 1$. What is the best possible $c_k$ such that $ R(C_{2k+1},H)\\leq c_k m $ for any graph $H$ on $m$ edges without isolated vertices?", "background": "This problem is #34 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The asymptotically sharp eventual coefficient is now 2, but the imported wording asks for the globally best multiplicative c_k over every edge count, which the cited 2026 theorems do not explicitly determine.\n\n**Verified partial progress.**\n\n- Cambie, Freschi, Morawski, Petrova, and Pokrovskiy proved R(C_{2k+1},H)<=2e(H)+k for all sufficiently large e(H), and the matching construction makes this eventual additive bound sharp.\n- Hng, Ji, and Lamaison independently proved R(C_{2k+1},H)<=(2+o(1))e(H)+v(H).\n- Cambie and Freschi proved the all-m size-linear bound R(C_l,H)<=(l-1)e(H)+1, establishing existence of a constant but not the exact best global constant in the imported statement.\n\n**Full solution or refutation.**\n\nThe 2026 cycle theorem solves the neighboring eventual-bound problem EP-570 and determines the asymptotic coefficient 2. Finite small values of e(H) can still govern the literal supremal multiplicative constant c_k.\n\n**What remains.**\n\nDetermine the exact supremum of R(C_{2k+1},H)/e(H) over all edge counts and all H without isolated vertices, or clarify that the historical intent was only the eventual/asymptotic coefficient.\n\n**Sources checked.**\n\n- Stijn Cambie, Andrea Freschi, Patryk Morawski, Kalina Petrova, and Alexey Pokrovskiy, Ramsey number of a cycle versus a graph of a given size, arXiv:2601.10238 (2026). (primary): https://arxiv.org/abs/2601.10238\n Evidence used: Theorem 3 proves the sharp eventual additive upper bound 2m+floor((l-1)/2) for odd cycles, with matching lower construction.\n- Stijn Cambie and Andrea Freschi, A general bound on R(C_k,H), arXiv:2606.11174 (2026). (primary): https://arxiv.org/abs/2606.11174\n Evidence used: Proves R(C_l,H)<=(l-1)m+1 for every m-edge H without isolated vertices, which establishes size-linearity but does not identify the exact best multiplicative constant.\n- Thomas F. Bloom, Erdős Problem #569 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/569\n Evidence used: The main statement asks for the best global c_k; the discussion links the 2026 papers and contains a solution claim that is broader than what their exact stated quantifiers establish for this wording.\n- Eng Keat Hng, Meng Ji, and Ander Lamaison, Ramsey size linear and generalization, arXiv:2603.25453 (2026). (primary): https://arxiv.org/abs/2603.25453\n Evidence used: Provides the independent near-2m plus vertex-count upper bound and describes it as significant progress.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2186, "problem_number": "EP-571", "title": "Erdős Problem #571", "statement": "Show that for any rational $\\alpha \\in [1,2)$ there exists a bipartite graph $G$ such that $ \\mathrm{ex}(n;G)\\asymp n^{\\alpha}. $ ", "background": "A problem of Erd\\H{o}s and Simonovits.\nBukh and Conlon \\cite{BuCo18} proved that this holds if we weaken asking for the extremal number of a single graph to asking for the extremal number of a finite family of graphs.\nA rational $\\alpha\\in [1,2)$ for which this holds is known as a Tur\\'{a}n exponent. Known Tur\\'{a}n exponents are:\n{UL}\n{LI} $\\frac{3}{2}-\\frac{1}{2s}$ for $s\\geq 2$ (Conlon, Janzer, and Lee \\cite{CJL21}).{/LI}\n{LI} $\\frac{4}{3}-\\frac{1}{3s}$ and $\\frac{5}{4}-\\frac{1}{4s}$ for $s\\geq 2$ (Jiang and Qiu \\cite{JiQi20}).{/LI}\n{LI} $2-\\frac{a}{b}$ for $\\lfloor b/a\\rfloor^3 \\leq a\\leq \\frac{b}{\\lfloor b/a\\rfloor+1}+1$ (Jiang, Jiang, and Ma \\cite{JJM20}).{/LI}\n{LI} $2-\\frac{a}{b}$ with $b>a\\geq 1$ and $b\\equiv \\pm 1\\pmod{a}$ (Kang, Kim, and Liu \\cite{KKL21}).{/LI}\n{LI} $1+a/b$ with $b>a^2$ (Jiang and Qiu \\cite{JiQi23}),{/LI}\n{LI} $2-\\frac{2}{2b+1}$ for $b\\geq 2$ or $7/5$ (Jiang, Ma, and Yepremyan \\cite{JMY22}).{/LI}\n{LI} $2-a/b$ with $b\\geq (a-1)^2$ (Conlon and Janzer \\cite{CoJa22}).{/LI}\n{/UL}\nSee also [713].\nThis problem is #45 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[BuCo18] Bukh, Boris and Conlon, David, Rational exponents in extremal graph theory. J. Eur. Math. Soc. (JEMS) (2018), 1747-1757.\n\n[CJL21] Conlon, David and Janzer, Oliver and Lee, Joonkyung, More on the extremal number of subdivisions. Combinatorica (2021), 465-494.\n\n[CoJa22] Conlon, David and Janzer, Oliver, Rational exponents near two. Adv. Comb. (2022), Paper No. 9, 10.\n\n[JJM20] Jiang, Tao and Jiang, Zilin and Ma, Jie, Negligible obstructions and Tur\\'{a}n exponents. arXiv:2007.02975 (2020).\n\n[JMY22] Jiang, Tao and Ma, Jie and Yepremyan, Liana, On Tur\\'{a}n exponents of bipartite graphs. Combin. Probab. Comput. (2022), 333-344.\n\n[JiQi20] Jiang, Tao and Qiu, Yu, Tur\\'{a}n numbers of bipartite subdivisions. SIAM J. Discrete Math. (2020), 556-570.\n\n[JiQi23] Jiang, Tao and Qiu, Yu, Many Tur\\'{a}n exponents via subdivisions. Combin. Probab. Comput. (2023), 134-150.\n\n[KKL21] Kang, Dong Yeap and Kim, Jaehoon and Liu, Hong, On the rational Tur\\'{a}n exponents conjecture. J. Combin. Theory Ser. B (2021), 149-172.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The all-rational single-bipartite-graph Turán-exponent conjecture remains open, but all rational exponents are realized by finite forbidden families and many infinite rational families are realized by single graphs.\n\n**Verified partial progress.**\n\n- Bukh and Conlon proved that every rational alpha in [1,2) is the exponent of a finite family of bipartite graphs.\n- Known single-graph exponents include 3/2-1/(2s), 4/3-1/(3s), 5/4-1/(4s), and several broad families of 2-a/b and 1+a/b values.\n- Conlon and Janzer proved a broad near-two family, including 2-a/b when b>=(a-1)^2.\n\n**Full solution or refutation.**\n\nThe finite-family relaxation is complete and many single-graph cases are known, but the construction of one bipartite graph for every rational exponent is not known.\n\n**What remains.**\n\nCover every rational alpha in [1,2) by the extremal number of a single bipartite graph, rather than a finite forbidden family.\n\n**Sources checked.**\n\n- Boris Bukh and David Conlon, Rational exponents in extremal graph theory, J. Eur. Math. Soc. 20 (2018), 1747-1757, DOI 10.4171/JEMS/798. (primary): https://doi.org/10.4171/JEMS/798\n Evidence used: Proves the all-rational theorem for finite families of bipartite forbidden graphs.\n- David Conlon and Oliver Janzer, Rational exponents near two, Advances in Combinatorics (2022), article 9, DOI 10.19086/aic.2022.9. (primary): https://doi.org/10.19086/aic.2022.9\n Evidence used: Establishes a broad explicit family of rational single-graph Turán exponents near two.\n- Thomas F. Bloom, Erdős Problem #571, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/571\n Evidence used: Retains open status and gives a dated list of the known single-graph exponent families through its March 2026 update.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2187, "problem_number": "EP-572", "title": "Erdős Problem #572", "statement": "Show that for $k\\geq 3$ $ \\mathrm{ex}(n;C_{2k})\\gg n^{1+\\frac{1}{k}}. $ ", "background": "It is easy to see that $\\mathrm{ex}(n;C_{2k+1})=\\lfloor n^2/4\\rfloor$ for any $k\\geq 1$ (and $n>2k+1$) (since no bipartite graph contains an odd cycle). Erd\\H{o}s and Klein \\cite{Er38} proved $\\mathrm{ex}(n;C_4)\\asymp n^{3/2}$.\nErd\\H{o}s \\cite{Er64c} and Bondy and Simonovits \\cite{BoSi74} showed that $ \\mathrm{ex}(n;C_{2k})\\ll kn^{1+\\frac{1}{k}}. $ Benson \\cite{Be66} has proved this conjecture for $k=3$ and $k=5$. Lazebnik, Ustimenko, and Woldar \\cite{LUW95} have shown that, for arbitrary $k\\geq 3$, $ \\mathrm{ex}(n;C_{2k})\\gg n^{1+\\frac{2}{3k-3+\nu}}, $ where $\nu=0$ if $k$ is odd and $\nu=1$ if $k$ is even. See \\cite{LUW99} for further history and references.\nSee also [765].\nThis problem is #46 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[Be66] Benson, Clark T., Minimal regular graphs of girths eight and twelve. Canadian J. Math. (1966), 1091-1094.\n\n[BoSi74] Bondy, J. A. and Simonovits, M., Cycles of even length in graphs. J. Combinatorial Theory Ser. B (1974), 97-105.\n\n[Er38] P. Erd\\H{o}s, On sequences of integers no one of which divides the product of two others and on related problems. Tomsk. Gos. Univ. Ucen Zap. (1938), 74-82.\n\n[Er64c] Erd\\H{o}s, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36.\n\n[LUW95] Lazebnik, F. and Ustimenko, V. A. and Woldar, A. J., A new series of dense graphs of high girth. Bull. Amer. Math. Soc. (N.S.) (1995), 73-79.\n\n[LUW99] Lazebnik, Felix and Ustimenko, Vasiliy A. and Woldar, Andrew\nJ., Polarities and $2k$-cycle-free graphs. Discrete Math. (1999), 503-513.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The target exponent is known for k=3 and k=5 but remains open for arbitrary k.\n\n**Verified partial progress.**\n\n- Benson established the conjecture for C_6 and C_10.\n- General Lazebnik--Ustimenko--Woldar constructions give a weaker exponent.\n\n**Full solution or refutation.**\n\nThe all-k assertion is unresolved.\n\n**What remains.**\n\nConstruct graphs of the target order for every k.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #572, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/572\n Evidence used: Current open status and known cases/bounds.\n\n**Review notes.** Unverified discussion comments were not used as evidence.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2188, "problem_number": "EP-573", "title": "Erdős Problem #573", "statement": "Is it true that $ \\mathrm{ex}(n;\\{C_3,C_4\\})\\sim (n/2)^{3/2}? $ ", "background": "A problem of Erd\\H{o}s and Simonovits, who proved that $ \\mathrm{ex}(n;\\{C_4,C_5\\})=(n/2)^{3/2}+O(n). $ K\"{o}v\\'{a}ri, S\\'{o}s, and Tur\\'{a}n \\cite{KST54} proved that the extremal number of edges for containing either $C_4$ or an odd cycle of any length is $\\sim (n/2)^{3/2}$. This problem is therefore asking whether the threshold is the same if we just forbid odd cycles of length $3$.\nSee also [574] for the general case, and [765] for $\\mathrm{ex}(n;C_4)$.\nThis problem is #48 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[KST54] K\"{o}vari, T. and S\\'{o}s, V. T. and Tur\\'{a}n, P., On a problem of K. Zarankiewicz. Colloq. Math. (1954), 50-57.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The C3,C4 asymptotic is open although closely related forbidden-family asymptotics are known.\n\n**Verified partial progress.**\n\n- Erdős--Simonovits proved the C4,C5 analogue.\n- Kővári--Sós--Turán handle C4 together with all odd cycles.\n\n**Full solution or refutation.**\n\nThese related results do not remove only the triangle condition.\n\n**What remains.**\n\nDetermine the leading constant for ex(n,{C3,C4}).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #573, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/573\n Evidence used: Current open status and analogues.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2189, "problem_number": "EP-574", "title": "Erdős Problem #574", "statement": "Is it true that, for $k\\geq 2$, $ \\mathrm{ex}(n;\\{C_{2k-1},C_{2k}\\})=(1+o(1))(n/2)^{1+\\frac{1}{k}}. $ ", "background": "A problem of Erd\\H{o}s and Simonovits.\nSee also [573] for the specific case of $k=2$.\nThis problem is #49 in Extremal Graph Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Lazebnik, Ustimenko, and Woldar's 1994 constructions already refute the universal asymptotic at k=3 and k=5. Their bipartite C_{2k}-free graphs automatically avoid C_{2k-1} and have leading coefficient (k-1)/k^(1+1/k), strictly exceeding the conjectured 2^(-(1+1/k)) coefficient.\n\n**Verified partial progress.**\n\n- The even-circuit theorem fixes the exponent n^(1+1/k) up to constants.\n- For k=3 and k=5, Lazebnik--Ustimenko--Woldar improve the known C_{2k}-free leading constant to (k-1)/k^(1+1/k).\n- Because those examples are bipartite, they are simultaneously C_{2k-1}-free, making them counterexamples to EP-574 itself.\n\n**Full solution or refutation.**\n\nStart with the appropriate dense bipartite large-girth family and apply the Lazebnik--Ustimenko--Woldar vertex-duplication construction. It preserves exclusion of C_{2k} while increasing the asymptotic edge constant. Bipartiteness excludes every odd cycle, including C_{2k-1}. At k=3 and k=5 the resulting coefficient exceeds the coefficient predicted by EP-574, so equality cannot hold for every k>=2.\n\n**What remains.**\n\nThe all-k conjecture is false, but the correct asymptotic constants remain separate extremal questions for individual k. The k=2 case is tracked separately as EP-573; a counterexample at k=3 or 5 does not decide it or the other remaining k. The input background has trailing serialization noise but the universal quantifier and forbidden-cycle family are clear.\n\n**Sources checked.**\n\n- Felix Lazebnik, Vasiliy A. Ustimenko, and Andrew J. Woldar, Properties of Certain Families of 2k-Cycle-Free Graphs, Journal of Combinatorial Theory, Series B 60 (1994), 293--298. (primary): https://doi.org/10.1006/jctb.1994.1020\n Evidence used: Primary source for the improved k=3,5 coefficient that exceeds the conjectured value.\n- Zoltán Füredi and Miklós Simonovits, The history of degenerate (bipartite) extremal graph problems, 2013 survey. (authoritative_secondary): https://users.renyi.hu/~miki/FureSimSurvC.pdf\n Evidence used: Explains the vertex-duplication construction and identifies the C6 and C10 constant conjectures as disproved.\n- Thomas F. Bloom, Erdős Problem #574 (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/574\n Evidence used: Records the negative classification, coefficient comparison, and separation of the k=2 successor problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2190, "problem_number": "EP-575", "title": "Erdős Problem #575", "statement": "If $\\mathcal{F}$ is a finite set of finite graphs then $\\mathrm{ex}(n;\\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\\mathcal{F}$. Note that it is trivial that $\\mathrm{ex}(n;\\mathcal{F})\\leq \\mathrm{ex}(n;G)$ for every $G\\in\\mathcal{F}$.\nIs it true that, for every $\\mathcal{F}$, if there is a bipartite graph in $\\mathcal{F}$ then there exists some bipartite $G\\in\\mathcal{F}$ such that $ \\mathrm{ex}(n;G)\\ll_{\\mathcal{F}}\\mathrm{ex}(n;\\mathcal{F})? $ ", "background": "A problem of Erd\\H{o}s and Simonovits.\nSee also [180].\nThis problem is #51 in Extremal Graph Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The finite-family comparison question remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo later resolution or concrete partial result was located.\n\n**What remains.**\n\nProve the comparison with a bipartite member or construct a counterexample family.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #575, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/575\n Evidence used: Current maintained open status.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2191, "problem_number": "EP-576", "title": "Erdős Problem #576", "statement": "Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of $ \\mathrm{ex}(n;Q_k). $ ", "background": "Erd\\H{o}s and Simonovits \\cite{ErSi70} proved that $ (\\tfrac{1}{2}+o(1))n^{3/2}\\leq \\mathrm{ex}(n;Q_3) \\ll n^{8/5}. $ (In \\cite{ErSi70} they mention that Erd\\H{o}s had originally conjectured that $ \\mathrm{ex}(n;Q_3)\\gg n^{5/3}$.) Erd\\H{o}s and Simonovits also proved that, if $G$ is the graph $Q_3$ with a missing edge, then $\\mathrm{ex}(n;G)\\asymp n^{3/2}$.\nIn \\cite{Er74c}, \\cite{Er81}, and \\cite{Er93} Erd\\H{o}s asked whether it is $\\mathrm{ex}(n;Q_3)\\asymp n^{8/5}$.\nA theorem of Sudakov and Tomon \\cite{SuTo22} implies $ \\mathrm{ex}(n;Q_k)=o(n^{2-\\frac{1}{k}}). $ Janzer and Sudakov \\cite{JaSu22} have improved this to $ \\mathrm{ex}(n;Q_k)\\ll_k n^{2-\\frac{1}{k-1}+\\frac{1}{(k-1)2^{k-1}}}. $ See also [1035].\nThis problem is #52 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[Er74c] Erd\\H{o}s, Paul, Extremal problems on graphs and hypergraphs. (1974), 75-84.\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[ErSi70] Erd\\H{o}s, P. and Simonovits, M., Some extremal problems in graph theory. Combinatorial theory and its applications, I-III (Proc. Colloq., Balatonf\"{u}red, 1969) (1970), 377-390.\n\n[JaSu22] Janzer, O. and Sudakov, B., On the Tur\\'{a}n number of the hypercube. arXiv:2211.02015 (2024).\n\n[SuTo22] Sudakov, Benny and Tomon, Istv\\'{a}n, The extremal number of tight cycles. Int. Math. Res. Not. IMRN (2022), 9663-9684.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hypercube extremal numbers remain open, with improved general upper bounds.\n\n**Verified partial progress.**\n\n- Sudakov--Tomon prove o(n^(2-1/k)).\n- Janzer--Sudakov give an explicit improved exponent.\n\n**Full solution or refutation.**\n\nThe behavior, including the Q3 case, is not determined.\n\n**What remains.**\n\nDetermine the correct exponent or sharp order for Q_k.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #576, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/576\n Evidence used: Current open status and cited improvements.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2192, "problem_number": "EP-579", "title": "Erdős Problem #579", "statement": "Let $\\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ and at least $\\delta n^2$ edges then $G$ contains an independent set of size $\\gg_\\delta n$.", "background": "A problem of Erd\\H{o}s, Hajnal, S\\'{o}s, and Szemer\\'{e}di, who could prove this is true for $\\delta>1/8$.\nSee also [533] and the entry in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The result is known at density above 1/8, but is open for arbitrary positive delta.\n\n**Verified partial progress.**\n\n- Erdős--Hajnal--Sós--Szemerédi proved the assertion for delta>1/8.\n\n**Full solution or refutation.**\n\nThe desired all-positive-density statement remains open.\n\n**What remains.**\n\nLower the density threshold to every fixed positive delta.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #579, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/579\n Evidence used: Current open status and thresholded theorem.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2193, "problem_number": "EP-584", "title": "Erdős Problem #584", "statement": "Let $G$ be a graph with $n$ vertices and $\\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\\subseteq G$ such that\n{UL}\n{LI}$H_1$ has $\\gg \\delta^3n^2$ edges and every two edges in $H_1$ are contained in a cycle of length at most $6$, and furthermore if two edges share a vertex they are on a cycle of length $4$, and\n{LI}$H_2$ has $\\gg \\delta^2n^2$ edges and every two edges in $H_2$ are contained in a cycle of length at most $8$.\n{/UL}", "background": "A problem of Erd\\H{o}s, Duke, and R\"{o}dl. Duke and Erd\\H{o}s \\cite{DuEr83}, who proved the first if $n$ is sufficiently large depending on $\\delta$. The real challenge is to prove this when $\\delta=n^{-c}$ for some $c>0$. Duke, Erd\\H{o}s, and R\"{o}dl \\cite{DER84} proved the first statement with a $\\delta^5$ in place of a $\\delta^3$.\nFox and Sudakov \\cite{FoSu08b} have proved the second statement when $\\delta >n^{-1/5}$.\nSee also the entry in the graphs problem collection.\nReferences\n\n\n[DER84] Duke, Richard and Erd\\H{o}s, Paul and R\"{o}dl, Vojt\\vEch, More results on subgraphs with many short cycles. Proceedings of the fifteenth Southeastern conference on\ncombinatorics, graph theory and computing (Baton Rouge,\nLa., 1984) (1984), 295-300.\n\n[DuEr83] No reference found.\n\n\n[FoSu08b] Fox, Jacob and Sudakov, Benny, On a problem of Duke-Erd\\H{o}s-R\"{o}dl on cycle-connected subgraphs. J. Combin. Theory Ser. B (2008), 1056-1062.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Both short-cycle core assertions have restricted/quantitatively weaker versions, but the stated density powers remain open.\n\n**Verified partial progress.**\n\n- Duke--Erdős--Rödl prove the first statement with delta^5.\n- Fox--Sudakov prove the second for delta>n^(-1/5).\n\n**Full solution or refutation.**\n\nThe target sparse-density regime and first exponent are unresolved.\n\n**What remains.**\n\nReach the stated delta powers throughout the sparse range.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #584, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/584\n Evidence used: Current open status and quantified partial results.\n\n**Review notes.** Recent forum-only preprints/claims were not accepted as resolutions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2194, "problem_number": "EP-585", "title": "Erdős Problem #585", "statement": "What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same vertex set?", "background": "Pyber, R\"{o}dl, and Szemer\\'{e}di \\cite{PRS95} constructed such a graph with $\\gg n\\log\\log n$ edges.\nChakraborti, Janzer, Methuku, and Montgomery \\cite{CJMM24} have shown that such a graph can have at most $n(\\log n)^{O(1)}$ many edges. Indeed, they prove that there exists a constant $C>0$ such that for any $k\\geq 2$ there is a $c_k$ such that if a graph has $n$ vertices and at least $c_kn(\\log n)^{C}$ many edges then it contains $k$ pairwise edge-disjoint cycles with the same vertex set.\nReferences\n\n\n[CJMM24] Chakraborti, D. and Janzer, O. and Methuku, A. and Montgomery, R., Edge-disjoint cycles with the same vertex set. arXiv:2404.07190 (2024).\n\n[PRS95] Pyber, L. and R\"{o}dl, V. and Szemer\\'{e}di, E., Dense subgraphs without 3-regular subgraphs. Journal of Combinatorial Theory, Series B (1995), 41-54.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The extremal order remains open between n log log n and n times a polylogarithm.\n\n**Verified partial progress.**\n\n- Pyber--Rödl--Szemerédi give n log log n lower bound.\n- Chakraborti--Janzer--Methuku--Montgomery give an n(log n)^O(1) upper bound.\n\n**Full solution or refutation.**\n\nThe bounds do not determine the maximum order.\n\n**What remains.**\n\nClose the polylogarithmic gap.\n\n**Sources checked.**\n\n- D. Chakraborti, O. Janzer, A. Methuku and R. Montgomery, Edge-disjoint cycles with the same vertex set, arXiv:2404.07190 (2024). (primary): https://arxiv.org/abs/2404.07190\n Evidence used: Primary source for the upper bound.\n- Thomas F. Bloom, Erdős Problem #585, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/585\n Evidence used: Current open status and range.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2195, "problem_number": "EP-588", "title": "Erdős Problem #588", "statement": "Let $f_k(n)$ be minimal such that if $n$ points in $\\mathbb{R}^2$ have no $k+1$ points on a line then there must be at most $f_k(n)$ many lines containing at least $k$ points. Is it true that $ f_k(n)=o(n^2) $ for $k\\geq 4$?", "background": "A generalisation of [101] (which asks about $k=4$).\nThe restriction to $k\\geq 4$ is necessary since Sylvester has shown that $f_3(n)= n^2/6+O(n)$. (See also Burr, Gr\"{u}nbaum, and Sloane \\cite{BGS74} and F\"{u}redi and Pal\\'{a}sti \\cite{FuPa84} for constructions which show that $f_3(n)\\geq(1/6+o(1))n^2$.)\nFor $k\\geq 4$, K\\'{a}rteszi \\cite{Ka63} proved $ f_k(n)\\gg_k n\\log n $ (resolving a conjecture of Erd\\H{o}s that $f_k(n)/n\\to \\infty$). Gr\"{u}nbaum \\cite{Gr76} proved $ f_k(n) \\gg_k n^{1+\\frac{1}{k-2}}. $ Erd\\H{o}s speculated this may be the correct order of magnitude, but Solymosi and Stojakovi\\'{c} \\cite{SoSt13} give a construction which shows $ f_k(n)\\gg_k n^{2-O_k(1/\\sqrt{\\log n})} $ \nReferences\n\n\n[BGS74] Burr, Stefan A. and Gr\"{u}nbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.\n\n[FuPa84] F\"{u}redi, Z. and Pal\\'{a}sti, I., Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. (1984), 561-566.\n\n[Gr76] Gr\"{u}nbaum, Branko, New views on some old questions of combinatorial geometry. Colloquio Internazionale sulle Teorie Combinatorie\n(Roma, 1973), Tomo I (1976), 451-468.\n\n[Ka63] F. K\\'{a}rteszi, Sylvester egy t\\'{e}tel\\'{e}r\\H{o}l \\'{e}s Erd\\H{o}s egy sejt\\'{e}s\\'{e}r\\H{o}l. Matematikai Lapok (1963), 3-10.\n\n[SoSt13] Solymosi, J\\'{o}zsef and Stojakovi\\'C, Milo\\vS, Many collinear {$k$}-tuples with no {$k+1$} collinear points. Discrete Comput. Geom. (2013), 811-820.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The o(n^2) assertion is open; near-quadratic lower constructions are known.\n\n**Verified partial progress.**\n\n- Solymosi--Stojaković construct f_k(n) at least n^(2-O_k(1/sqrt(log n))).\n\n**Full solution or refutation.**\n\nThe lower constructions remain consistent with either outcome.\n\n**What remains.**\n\nProve subquadraticity or a quadratic lower construction.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #588, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/588\n Evidence used: Current open status and construction.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2196, "problem_number": "EP-589", "title": "Erdős Problem #589", "statement": "Let $g(n)$ be maximal such that in any set of $n$ points in $\\mathbb{R}^2$ with no four points on a line there exists a subset on $g(n)$ points with no three points on a line. Estimate $g(n)$.", "background": "The trivial greedy algorithm gives $g(n)\\gg n^{1/2}$. A similar question can be asked for a set with no $k$ points on a line, searching for a subset with no $l$ points on a line, for any $3\\leq l\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general-position subset size is open between n^(1/2) log n and n^(5/6+o(1)).\n\n**Verified partial progress.**\n\n- Füredi proved the lower bound n^(1/2) log n.\n- Balogh--Solymosi proved the upper bound n^(5/6+o(1)).\n\n**Full solution or refutation.**\n\nThe exponent is undetermined.\n\n**What remains.**\n\nSharpen either side to determine g(n).\n\n**Sources checked.**\n\n- J. Balogh and J. Solymosi, On the number of points in general position in the plane, Discrete Analysis (2018). (primary): https://doi.org/10.19086/da.3004\n Evidence used: Primary source for the upper bound.\n- Thomas F. Bloom, Erdős Problem #589, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/589\n Evidence used: Current open status and bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2197, "problem_number": "EP-591", "title": "Erdős Problem #591", "statement": "Let $\\alpha$ be the infinite ordinal $\\omega^{\\omega^2}$. Is it true that in any red/blue colouring of the edges of $K_\\alpha$ there is either a red $K_\\alpha$ or a blue $K_3$?", "background": "For comparison, Specker \\cite{Sp57} proved this property holds when $\\alpha=\\omega^2$ and false when $\\alpha=\\omega^n$ for $3\\leq n<\\omega$. Chang proved this property holds when $\\alpha=\\omega^\\omega$ (see [590]).\nThis is true and was proved independently by Schipperus \\cite{Sc10} and Darby.\nSee also [118], and [592] for the general case.\nReferences\n\n\n[Sc10] Schipperus, Rene, Countable partition ordinals. Ann. Pure Appl. Logic (2010), 1195-1215.\n\n[Sp57] Specker, Ernst, Teilmengen von Mengen mit Relationen. Comment. Math. Helv. (1957), 302-314.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The partition relation omega^(omega^2) -> (omega^(omega^2),3)^2 was proved by Rene Schipperus in a 2010 peer-reviewed paper; the maintained tracker identifies the relevant specialization of Theorem 28 and also reports an independent proof by Carl Darby.\n\n**Verified partial progress.**\n\n- Specker proved the positive relation at omega^2 and negative results for finite powers omega^n with n at least 3.\n- Chang proved the positive relation at omega^omega before the later extension used here.\n\n**Full solution or refutation.**\n\nSchipperus's theorem on countable partition ordinals includes the ordinal omega^(omega^2), giving exactly the requested red copy of the whole ordinal or blue triangle in every two-coloring.\n\n**What remains.**\n\nThe EP-591 instance is closed. Broader classification questions for countable partition ordinals remain separate problems.\n\n**Sources checked.**\n\n- Rene Schipperus, Countable partition ordinals, Annals of Pure and Applied Logic 161 (2010), 1195-1215, DOI 10.1016/j.apal.2009.12.007. (primary): https://doi.org/10.1016/j.apal.2009.12.007\n Evidence used: Published primary theorem establishing the positive partition relation; the maintained record identifies Theorem 28 with the relevant parameter specialization.\n- Thomas F. Bloom, Erdos Problem #591, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/591\n Evidence used: Records PROVED status, the Schipperus reference, theorem specialization, and independent Darby proof.\n\n**Review notes.** The imported background has a leaked serialized suffix, but the statement itself is legible.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2198, "problem_number": "EP-592", "title": "Erdős Problem #592", "statement": "Determine which countable ordinals $\\beta$ have the property that, if $\\alpha=\\omega^{^\\beta}$, then in any red/blue colouring of the edges of $K_\\alpha$ there is either a red $K_\\alpha$ or a blue $K_3$.", "background": "Such $\\alpha$ are called partition ordinals.\n{UL}\n{LI}Specker \\cite{Sp57} proved this holds for $\\beta=2$ and not for $3\\leq \\beta <\\omega$.{/LI}\n{LI} Chang \\cite{Ch72} proved this holds for $\\beta=\\omega$.{/LI}\n{LI} Galvin and Larson \\cite{GaLa74} have shown that if $\\beta\\geq 3$ has this property then $\\beta$ must be 'additively indecomposable', so that in particular $\\beta=\\omega^\\gamma$ for some countable ordinal $\\gamma$. Galvin and Larson conjecture that every $\\beta\\geq 3$ of this form has this property.{/LI}\n{LI}Schipperus \\cite{Sc10} have proved this is holds if $\\beta=\\omega^\\gamma$ in which $\\gamma$ is a countable ordinal which is the sum of one or two indecomposable ordinals, and this fails to hold if $\\gamma$ is the sum of four or more indecomposable ordinals.{/LI}\n{/UL}\nThe remaining open case appears to be when $\\gamma$ is the sum of three indecomposable ordinals.\nThe case $\\beta=\\omega$ is the subject of [590], and $\\beta=\\omega^2$ is the subject of [591]. See also [118].\nReferences\n\n\n[Ch72] Chang, C. C., A partition theorem for the complete graph on {$\\omega\\sp{\\omega }$}. J. Combinatorial Theory Ser. A (1972), 396-452.\n\n[GaLa74] Galvin, Fred and Larson, Jean, Pinning countable ordinals. Fund. Math. (1974/75), 357-361.\n\n[Sc10] Schipperus, Rene, Countable partition ordinals. Ann. Pure Appl. Logic (2010), 1195-1215.\n\n[Sp57] Specker, Ernst, Teilmengen von Mengen mit Relationen. Comment. Math. Helv. (1957), 302-314.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The partition ordinals are largely classified, with the three-indecomposable-summand case remaining open.\n\n**Verified partial progress.**\n\n- Specker, Chang, and Galvin--Larson establish initial necessary/sufficient cases.\n- Schipperus resolves one/two indecomposable summands positively and four or more negatively.\n\n**Full solution or refutation.**\n\nThe remaining three-summand family prevents a complete characterization.\n\n**What remains.**\n\nResolve the case where gamma is a sum of three indecomposable ordinals.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #592, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/592\n Evidence used: Current open status and case classification.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2199, "problem_number": "EP-593", "title": "Erdős Problem #593", "statement": "Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $>\\aleph_0$.", "background": "Similar problems were investigated by Erd\\H{o}s, Galvin, and Hajnal \\cite{EGH75}. Erd\\H{o}s claims that for graphs the problem is completely solved: a graph of chromatic number $\\geq \\aleph_1$ must contain all finite bipartite graphs but need not contain any fixed odd cycle.\nReferences\n\n\n[EGH75] Erd\\H{o}s, P. and Galvin, F. and Hajnal, A., On set-systems having large chromatic number and not containing prescribed subsystems. (1975), 425--513.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The finite-graph analogue is described as settled, while the 3-uniform hypergraph characterization remains open.\n\n**Verified partial progress.**\n\n- Erdős's recorded graph analogue forces finite bipartite graphs at uncountable chromatic number.\n\n**Full solution or refutation.**\n\nThe hypergraph problem is not reduced to the graph result.\n\n**What remains.**\n\nCharacterize the forced finite 3-uniform hypergraphs.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #593, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/593\n Evidence used: Current open status and graph-analogue context.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2200, "problem_number": "EP-595", "title": "Erdős Problem #595", "statement": "Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?", "background": "A problem of Erd\\H{o}s and Hajnal. Folkman \\cite{Fo70} and Ne\\v{s}et\\v{r}il and R\"{o}dl \\cite{NeRo75} have proved that for every $n\\geq 1$ there is a graph $G$ which contains no $K_4$ and is not the union of $n$ triangle-free graphs.\nSee also [582] and [596].\nReferences\n\n\n[Fo70] Folkman, Jon, Graphs with monochromatic complete subgraphs in every edge\ncoloring. SIAM J. Appl. Math. (1970), 19-24.\n\n[NeRo75] Ne\\u set\\u ril, Jaroslav and R\"odl, Vojt\\v ech, Type theory of partition properties of graphs. (1975), 405-412.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finite n-colour obstructions are known, but the countably many-colours question is open.\n\n**Verified partial progress.**\n\n- Folkman and Nešetřil--Rödl construct K4-free graphs not decomposable into n triangle-free graphs for every finite n.\n\n**Full solution or refutation.**\n\nFinite chromatic obstructions do not yield a single infinite counterexample.\n\n**What remains.**\n\nConstruct such an infinite graph or prove countable decomposability.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #595, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/595\n Evidence used: Current open status and finite-n result.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2201, "problem_number": "EP-596", "title": "Erdős Problem #596", "statement": "For which graphs $G_1,G_2$ is it true that\n{UL}\n{LI} for every $n\\geq 1$ there is a graph $H$ without a $G_1$ but if the edges of $H$ are $n$-coloured then there is a monochromatic copy of $G_2$, and yet{/LI}\n{LI} for every graph $H$ without a $G_1$ there is an $\\aleph_0$-colouring of the edges of $H$ without a monochromatic $G_2$.\n{/UL}", "background": "Erd\\H{o}s and Hajnal originally conjectured that there are no such $G_1,G_2$, but in fact $G_1=C_4$ and $G_2=C_6$ is an example. Indeed, for this pair Ne\\v{s}et\\v{r}il and R\"{o}dl established the first property and Erd\\H{o}s and Hajnal the second (in fact every $C_4$-free graph is a countable union of trees).\nWhether this is true for $G_1=K_4$ and $G_2=K_3$ is the content of [595].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A concrete pair (C4,C6) has the two requested properties, but their characterization remains open.\n\n**Verified partial progress.**\n\n- Nešetřil--Rödl prove the finite-colour property for (C4,C6).\n- Erdős--Hajnal prove every C4-free graph is a countable union of trees.\n\n**Full solution or refutation.**\n\nOne example refutes the original no-example conjecture but does not characterize all pairs.\n\n**What remains.**\n\nClassify the pairs (G1,G2).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #596, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/596\n Evidence used: Current open status and (C4,C6) example.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2202, "problem_number": "EP-597", "title": "Erdős Problem #597", "statement": "Let $G$ be a graph on at most $\\aleph_1$ vertices which contains no $K_4$ and no $K_{\\aleph_0,\\aleph_0}$ (the complete bipartite graph with $\\aleph_0$ vertices in each class). Is it true that $ \\omega_1^2 \\to (\\omega_1\\omega, G)^2? $ What about finite $G$?", "background": "Erd\\H{o}s and Hajnal proved that $\\omega_1^2 \\to (\\omega_1\\omega,3)^2$. Erd\\H{o}s originally asked this with just the assumption that $G$ is $K_4$-free, but Baumgartner proved that $\\omega_1^2 \not\\to (\\omega_1\\omega, K_{\\aleph_0,\\aleph_0})^2$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The strengthened partition relation is open; base and obstruction cases are known.\n\n**Verified partial progress.**\n\n- Erdős--Hajnal proved the relation with a triangle target.\n- Baumgartner disproved the earlier version allowing K_{aleph0,aleph0}.\n\n**Full solution or refutation.**\n\nThe extra forbidden-bipartite hypothesis is still not enough for a known answer.\n\n**What remains.**\n\nSettle the stated relation, including finite G.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #597, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/597\n Evidence used: Current formulation, open status, and prior obstruction.\n\n**Review notes.** The serialized background loses a LaTex not escape; it was flagged, not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2203, "problem_number": "EP-598", "title": "Erdős Problem #598", "statement": "Let $m$ be an infinite cardinal and $\\kappa$ be the successor cardinal of $2^{\\aleph_0}$. Can one colour the countable subsets of $m$ using $\\kappa$ many colours so that every $X\\subseteq m$ with $\\lvert X\\rvert=\\kappa$ contains subsets of all possible colours?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The ZFC status is open; unverified forum reports describe a positive special case and large-cardinal relative consistency statements.\n\n**Verified partial progress.**\n\n- A forum proof claims the special case m=kappa using stationary-set partitioning.\n\n**Full solution or refutation.**\n\nThe maintained tracker still lists the full problem open, and forum/AI-associated claims are insufficient to classify a full solution.\n\n**What remains.**\n\nVerify the claimed special case and determine the exact ZFC/independence status.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #598, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/598\n Evidence used: Maintained open status.\n- EP-598 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/598?embed=1\n Evidence used: Unverified special-case and consistency claims, retained only as leads.\n\n**Review notes.** Forum claims are not treated as primary verification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2204, "problem_number": "EP-600", "title": "Erdős Problem #600", "statement": "Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have an edge contained in at least $r$ triangles. Let $r\\geq 2$. Is it true that $ e(n,r+1)-e(n,r)\\to \\infty $ as $n\\to \\infty$? Is it true that $ \\frac{e(n,r+1)}{e(n,r)}\\to 1 $ as $n\\to \\infty$?", "background": "Ruzsa and Szemer\\'{e}di \\cite{RuSz78} proved that $e(n,r)=o(n^2)$ for any fixed $r$.\nSee also [80].\nReferences\n\n\n[RuSz78] Ruzsa, I. Z. and Szemer\\'{e}di, E., Triple systems with no six points carrying three triangles. Combinatorics (Proc. Fifth Hungarian Colloq.,\nKeszthely, 1976), Vol. II (1978), 939-945.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Each fixed-r threshold is subquadratic, but neither comparison limit is known.\n\n**Verified partial progress.**\n\n- Ruzsa--Szemerédi proved e(n,r)=o(n^2) for fixed r.\n\n**Full solution or refutation.**\n\nSubquadraticity does not decide either adjacent-threshold assertion.\n\n**What remains.**\n\nProve either the difference divergence or the ratio limit.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #600, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/600\n Evidence used: Current open status and Ruzsa--Szemerédi result.\n\n**Review notes.** Recent AI-assisted forum heuristics were not accepted as results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2205, "problem_number": "EP-601", "title": "Erdős Problem #601", "statement": "For which limit ordinals $\\alpha$ is it true that if $G$ is a graph with vertex set $\\alpha$ then $G$ must have either an infinite path or independent set on a set of vertices with order type $\\alpha$?", "background": "A problem of Erd\\H{o}s, Hajnal, and Milner \\cite{EHM70}, who proved this is true for $\\alpha < \\omega_1^{\\omega+2}$.\nIn \\cite{Er82e} Erd\\H{o}s offers \\$250 for showing what happens when $\\alpha=\\omega_1^{\\omega+2}$ and \\$500 for settling the general case.\nLarson \\cite{La90} proved this is true for all $\\alpha<2^{\\aleph_0}$ assuming Martin's axiom.\nReferences\n\n\n[EHM70] Erd\\H{o}s, P. and Hajnal, A. and Milner, E. C., Set mappings and polarized partition relations. Combinatorial theory and its applications, I-III (Proc.\nColloq., Balatonf\"{u}red, 1969) (1970), 327-363.\n\n[Er82e] Erd\\H{o}s, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.\n\n[La90] Larson, Jean A., Martin's axiom and ordinal graphs: large independent sets or infinite paths. Ann. Pure Appl. Logic (1990), 31-39.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The ordinal path-or-independent-set assertion is known in an initial interval and conditionally under Martin's axiom, but open in general.\n\n**Verified partial progress.**\n\n- Erdős--Hajnal--Milner prove it below omega_1^(omega+2).\n- Larson proves it below the continuum assuming Martin's axiom.\n\n**Full solution or refutation.**\n\nNeither result settles alpha=omega_1^(omega+2) in ZFC or all limits.\n\n**What remains.**\n\nResolve the first open ordinal and characterize all limit ordinals.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #601, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/601\n Evidence used: Current open status and known range/conditional theorem.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2206, "problem_number": "EP-602", "title": "Erdős Problem #602", "statement": "Let $(A_i)$ be a family of sets with $\\lvert A_i\\rvert=\\aleph_0$ for all $i$, such that for any $i\neq j$ we have $\\lvert A_i\\cap A_j\\rvert$ finite and $\neq 1$. Is there a $2$-colouring of $\\cup A_i$ such that no $A_i$ is monochromatic?", "background": "A problem of Komj\\'{a}th. The existence of such a $2$-colouring is sometimes known as Property B.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Property B is known for countable families and uniformly bounded finite intersections; the stated exact intersection condition remains open.\n\n**Verified partial progress.**\n\n- Bernstein's lemma handles countable systems of infinite sets.\n- Miller's theorem handles a uniform finite bound on pairwise intersections.\n\n**Full solution or refutation.**\n\nThe recorded subclasses do not cover unbounded finite intersections excluding exactly one.\n\n**What remains.**\n\nProve or refute Property B under the precise hypothesis.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #602, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/602\n Evidence used: Maintained open status.\n- EP-602 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/602?embed=1\n Evidence used: Literature leads for established subclasses; requires bibliography verification.\n\n**Review notes.** The input's lost neq escapes are flagged, not corrected.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2207, "problem_number": "EP-603", "title": "Erdős Problem #603", "statement": "Let $(A_i)$ be a family of countably infinite sets such that $\\lvert A_i\\cap A_j\\rvert \neq 2$ for all $i\neq j$. Find the smallest cardinal $C$ such that $\\cup A_i$ can always be coloured with at most $C$ colours so that no $A_i$ is monochromatic.", "background": "A problem of Komj\\'{a}th. If instead we have $\\lvert A_i\\cap A_j\\rvert \neq 1$ then Komj\\'{a}th showed that this is possible with at most $\\aleph_0$ colours.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Both plausible readings are resolved: for arbitrary-size families there is no universal cardinal C, while for a countable indexed sequence the minimum is 2. The dataset fails to specify the index-family size, so the conclusion requires editorial disambiguation.\n\n**Verified partial progress.**\n\n- Komjath had proved a related upper bound for the intersection-not-equal-to-one variant.\n- A finite-color strengthening on a countable ground set follows from a Ramsey/ultrafilter construction and already shows that no finite uniform bound exists.\n\n**Full solution or refutation.**\n\nFor every cardinal mu, take kappa=(2^mu)^+, V=[kappa]^2, and the family A_X=[X]^2 over countably infinite X. Pairwise intersections never have size 2, while Erdos-Rado gives a monochromatic A_H under every mu-coloring, so no universal cardinal exists. Under the countable-sequence reading, Bernstein's lemma gives exactly two colors.\n\n**What remains.**\n\nNo mathematical case remains once the quantifier is fixed, but the source record must say whether the family is arbitrary or countably indexed; the two answers are different.\n\n**Sources checked.**\n\n- Przemek Chojecki, A note on Erdos Problem #603 (2026). (primary): https://www.ulam.ai/research/erdos603.pdf\n Evidence used: States and proves that no smallest cardinal exists for arbitrary families and that the countable-sequence interpretation has minimum 2.\n- Gavin Sherry, A strengthening of Erdos Problem #603 (2026 proof note). (primary): https://gist.github.com/gavinsherry/ad9b6f85e2afc0a830d015a6b5d6d52a\n Evidence used: Provides the stronger finite-color obstruction on a countable ground set with intersections only 0 or countably infinite.\n- Thomas F. Bloom, Erdos Problem #603 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/603\n Evidence used: Records SOLVED status, the arbitrary-family construction, and the explicit warning about the two interpretations.\n\n**Review notes.** The dataset has an OCR loss in both not-equal signs and is ambiguous about the size of the indexing family; neither defect is silently corrected.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2208, "problem_number": "EP-604", "title": "Erdős Problem #604", "statement": "Given $n$ distinct points $A\\subset\\mathbb{R}^2$ must there be a point $x\\in A$ such that $ \\#\\{ d(x,y) : y \\in A\\} \\gg n^{1-o(1)}? $ Or even $\\gg n/\\sqrt{\\log n}$?", "background": "The pinned distance problem, a stronger form of [89]. The example of an integer grid show that $n/\\sqrt{\\log n}$ would be best possible.\nIt may be true that there are $\\gg n$ many such points, or that this is true on average - for example, if $d(x)$ counts the number of distinct distances from $x$ then in \\cite{Er75f} Erd\\H{o}s conjectured $ \\sum_{x\\in A}d(x) \\gg \\frac{n^2}{\\sqrt{\\log n}}, $ where $A\\subset \\mathbb{R}^2$ is any set of $n$ points.\nIn \\cite{Er97e} Erd\\H{o}s offers \\$500 for a solution to this problem, but it is unclear whether he intended this for proving the existence of a single such point or for $\\gg n$ many such points.\nIn \\cite{Er97e} Erd\\H{o}s wrote that he initially 'overconjectured' and thought that the answer to this problem is the same as for the number of distinct distances between all pairs (see [89]), but this was disproved by Harborth. It could be true that the answers are the same up to an additive factor of $n^{o(1)}$.\nThe best known bound is $ \\gg n^{c-o(1)}, $ due to Katz and Tardos \\cite{KaTa04}, where $ c=\\frac{48-14e}{55-16e}=0.864137\\cdots. $ \nReferences\n\n\n[Er75f] Erd\\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\n\n[Er97e] Erd\\H{o}s, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[KaTa04] Katz, Nets Hawk and Tardos, G\\'{a}bor, A new entropy inequality for the Erd\\H{o}s distance problem. Towards a theory of geometric graphs (2004), 119-126.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The pinned-distance lower bound is known with exponent 0.864137..., but the near-linear target remains open.\n\n**Verified partial progress.**\n\n- Katz--Tardos prove a bound n^(c-o(1)) with c approximately 0.864137.\n- The integer grid gives the natural n/sqrt(log n) barrier.\n\n**Full solution or refutation.**\n\nThe known exponent is below 1, so neither stated near-linear conclusion follows.\n\n**What remains.**\n\nImprove the pinned-distance exponent to n^(1-o(1)) or the grid-scale bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #604, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/604\n Evidence used: Current open status and Katz--Tardos bound.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2209, "problem_number": "EP-609", "title": "Erdős Problem #609", "statement": "Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$. Estimate $f(n)$.", "background": "A problem of Erd\\H{o}s and Graham. The edges of $K_{2^n}$ can be $n$-coloured to avoid odd cycles of any length. It can be shown that $C_5$ and $C_7$ can be avoided for large $n$.\nChung \\cite{Ch97} asked whether $f(n)\\to \\infty$ as $n\\to \\infty$. Day and Johnson \\cite{DaJo17} proved this is true, and that $ f(n)\\geq 2^{c\\sqrt{\\log n}} $ for some constant $c>0$. The trivial upper bound is $2^n$.\nGir\\~{a}o and Hunter \\cite{GiHu24} have proved that $ f(n) \\ll \\frac{2^n}{n^{1-o(1)}}. $ Janzer and Yip \\cite{JaYi25} have improved this to $ f(n) \\ll n^{3/2}2^{n/2}. $ See also the entry in the graphs problem collection.\nReferences\n\n\n[Ch97] Chung, F. R. K., Open problems of {P}aul Erd\\H{o}s in graph theory. J. Graph Theory (1997), 3--36.\n\n[DaJo17] Day, A. Nicholas and Johnson, J. Robert, Multicolour Ramsey numbers of odd cycles. J. Combin. Theory Ser. B (2017), 56-63.\n\n[GiHu24] A. Gir\\~Ao and Z. Hunter, Monochromatic odd cycles in edge-coloured complete graphs. arXiv:2412.07708 (2024).\n\n[JaYi25] O. Janzer and F. Yip, Short monochromatic odd cycles. arXiv:2506.14910 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** f(n) is known to tend to infinity and has subexponential-style bounds, but its order is open.\n\n**Verified partial progress.**\n\n- Day--Johnson prove f(n) tends to infinity and give a 2^(c sqrt(log n)) lower bound.\n- Janzer--Yip give f(n) at most n^(3/2)2^(n/2).\n\n**Full solution or refutation.**\n\nThe available upper and lower bounds are far apart.\n\n**What remains.**\n\nDetermine the asymptotic scale of f(n).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #609, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/609\n Evidence used: Current open status and latest named bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2210, "problem_number": "EP-610", "title": "Erdős Problem #610", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).\nEstimate $\\tau(G)$. In particular, is it true that if $G$ has $n$ vertices then $ \\tau(G) \\leq n-\\omega(n)\\sqrt{n} $ for some $\\omega(n)\\to \\infty$, or even $ \\tau(G) \\leq n-c\\sqrt{n\\log n} $ for some absolute constant $c>0$?", "background": "A problem of Erd\\H{o}s, Gallai, and Tuza \\cite{EGT92}, who proved that $ \\tau(G) \\leq n-\\sqrt{2n}+O(1). $ This would be best possible, since there exist triangle-free graphs with all independent sets of size $O(\\sqrt{n\\log n})$, which follows from the lower bound for $R(3,k)$ by Kim \\cite{Ki95} (see [165]).\nIndeed, Erd\\H{o}s, Gallai, and Tuza speculate that if $f(n)$ is the largest $k$ such that every triangle-free graph on $n$ vertices contains an independent set on $f(n)$ vertices, then $\\tau(G)\\leq n-f(n)$.\nA positive answer to this problem would follow from a positive answer to [151] (since Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS80} have proved that the $H(n)$ defined there satisfies $H(n)\\gg \\sqrt{n\\log n}$).\nSee also [151], [611], this entry and and this entry in the graphs problem collection.\nReferences\n\n\n[AKS80] Ajtai, Mikl\\'{o}s and Koml\\'{o}s, J\\'{a}nos and Szemer\\'{e}di, Endre, A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.\n\n[EGT92] Erd\\H{o}s, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289.\n\n[Ki95] Kim, J. H., The Ramsey number $R(3,t)$ has order of magnitude $t^2/\\log t$. Random Structures and Algorithms (1995), 173-207.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The stronger requested bound follows immediately from Joret-Micek-Reed-Smid's peer-reviewed 2021 bound O(sqrt(n/log n)) for clique chromatic number, under the intended convention that isolated-vertex singleton cliques are excluded.\n\n**Verified partial progress.**\n\n- Erdos, Gallai, and Tuza proved tau(G) <= n-sqrt(2n)+O(1).\n- Ramsey-theoretic estimates showed that the requested sqrt(n log n) scale is the natural extremal scale.\n\n**Full solution or refutation.**\n\nUse a clique coloring with k=O(sqrt(n/log n)) colors. The complement of its largest color class meets every nontrivial maximal clique, so tau(G) <= n-n/k <= n-c sqrt(n log n).\n\n**What remains.**\n\nThe requested asymptotic bound is proved. Optimal constants and finer extremal structure remain follow-up questions.\n\n**Sources checked.**\n\n- Gwenael Joret, Piotr Micek, Bruce Reed, and Michiel Smid, Tight Bounds on the Clique Chromatic Number, Electronic Journal of Combinatorics 28(3) (2021), P3.51, DOI 10.37236/9659. (primary): https://www.combinatorics.org/ojs/index.php/eljc/article/view/v28i3p51\n Evidence used: The abstract and theorem give clique chromatic number O(sqrt(n/log n)) for every n-vertex graph, which implies the claimed transversal bound.\n- Thomas F. Bloom, Erdos Problem #610, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/610\n Evidence used: Records PROVED status and explicitly gives the largest-color-class deduction from the 2021 theorem.\n\n**Review notes.** The literal dataset definition omits the source's exclusion of isolated vertices; without that convention the edgeless graph is an immediate counterexample to the displayed bound.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2211, "problem_number": "EP-611", "title": "Erdős Problem #611", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).\nIs it true that if all maximal cliques in $G$ have at least $cn$ vertices then $\\tau(G)=o_c(n)$?\nSimilarly, estimate for $c>0$ the minimal $k_c(n)$ such that if every maximal clique in $G$ has at least $k_c(n)$ vertices then $\\tau(G)<(1-c)n$.", "background": "A problem of Erd\\H{o}s, Gallai, and Tuza \\cite{EGT92}, who proved for the latter question that $k_c(n) \\geq n^{c'/\\log\\log n}$ for some $c'>0$, and that if every clique has size least $k$ then $\\tau(G) \\leq n-(kn)^{1/2}$. Bollob\\'{a}s and Erd\\H{o}s proved that if every maximal clique has at least $n+3-2\\sqrt{n}$ vertices then $\\tau(G)=1$ (and this threshold is best possible).\nSee also [610] and the entry in the graphs problem collection.\nReferences\n\n\n[EGT92] Erd\\H{o}s, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Known clique-transversal estimates leave the requested sublinear conclusion and threshold asymptotics open.\n\n**Verified partial progress.**\n\n- Erdős--Gallai--Tuza give a lower threshold estimate and a general upper transversal bound.\n- Bollobás--Erdős determine the extreme tau=1 threshold.\n\n**Full solution or refutation.**\n\nNeither result settles fixed positive clique density.\n\n**What remains.**\n\nProve tau(G)=o_c(n) or construct a counterexample.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #611, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/611\n Evidence used: Current open status and cited bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2212, "problem_number": "EP-612", "title": "Erdős Problem #612", "statement": "Let $G$ be a connected graph with $n$ vertices, minimum degree $d$, and diameter $D$. Show if that $G$ contains no $K_{2r}$ and $(r-1)(3r+2)\\mid d$ then $ D\\leq \\frac{2(r-1)(3r+2)}{2r^2-1}\\frac{n}{d}+O(1), $ and if $G$ contains no $K_{2r+1}$ and $3r-1 \\mid d$ then $ D\\leq \\frac{3r-1}{r}\\frac{n}{d}+O(1). $ ", "background": "A problem of Erd\\H{o}s, Pach, Pollack, and Tuza \\cite{EPPT89}, who gave constructions showing that the above bounds would be sharp, and proved the case $2r+1=3$. It is known (see \\cite{EPPT89} for example) that any connected graph on $n$ vertices with minimum degree $d$ has diameter $ D\\leq 3\\frac{n}{d+1}+O(1). $ This was disproven for the case of $K_{2r}$-free graphs with $r\\geq 2$ by Czabarka, Singgih, and Sz\\'{e}kely \\cite{CSS21}, who constructed arbitrarily large connected graphs on $n$ vertices which contain no $K_{2r}$ and have minimum degree $d$, and diameter $ \\frac{6r-5}{(2r-1)d+2r-3}n+O(1), $ which contradicts the above conjecture for each fixed $r$ as $d\\to \\infty$.\nThey suggest the amended conjecture, which no longer divides into two cases, that if $G$ is a connected graph on $n$ vertices with minimum degree $d$ which contains no $K_{k+1}$ then the diameter of $G$ is at most $ (3-\\tfrac{2}{k})\\frac{n}{d}+O(1). $ This bound is known under the weaker assumption that $G$ is $k$-colourable when $k=3$ and $k=4$, shown by Czabarka, Dankelmann, and Sz\\'{e}kely \\cite{CDS09} and Czabarka, Smith, and Sz\\'{e}kely \\cite{CSS23}.\nCambie and Jooken \\cite{CaJo25} have given an example that shows the diameter for $K_4$-free graphs with minimum degree $16$ is at least $\\frac{31}{216}n+O(1)$, giving another counterexample to the original conjecture.\nSee also the entry in the graphs problem collection.\nReferences\n\n\n[CDS09] Czabarka, \\'{e}. and Dankelmann, P. and Sz\\'{e}kely, L. A., Diameter of 4-colourable graphs. European J. Combin. (2009), 1082--1089.\n\n[CSS21] Czabarka, \\'{e}va and Singgih, Inne and Sz\\'{e}kely, L\\'aszl\\'{o}{}\nA., Counterexamples to a conjecture of {E}rd\\H{o}s, {P}ach,\n{P}ollack and {T}uza. J. Combin. Theory Ser. B (2021), 38--45.\n\n[CSS23] Czabarka, \\'{e}va and Smith, Stephen J. and Sz\\'{e}kely,\nL\\'aszl\\'{o}, Maximum diameter of 3- and 4-colorable graphs. J. Graph Theory (2023), 262--270.\n\n[CaJo25] S. Cambie and J. Jooken, Sharp results for the Erd\\H{o}s, Pach, Pollack and Tuza problem. arXiv:2502.08626 (2025).\n\n[EPPT89] No reference found.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The stated K_{2r}-free diameter bound is false for every fixed r>=2 in the appropriate large-degree regime.\n\n**Verified partial progress.**\n\n- Czabarka--Singgih--Székely construct counterexamples to the K_{2r}-free assertion.\n- Cambie--Jooken give a further K4-free counterexample.\n\n**Full solution or refutation.**\n\nA false conjunct refutes the original request; a distinct amended conjecture remains open.\n\n**What remains.**\n\nInvestigate the amended K_{k+1}-free diameter conjecture separately.\n\n**Sources checked.**\n\n- É. Czabarka, I. Singgih and L. A. Székely, Counterexamples to a conjecture of Erdős, Pach, Pollack and Tuza, J. Combin. Theory Ser. B (2021), 38--45. (primary): https://doi.org/10.1016/j.jctb.2020.06.003\n Evidence used: Primary published counterexamples to the stated K_{2r}-free assertion.\n- Thomas F. Bloom, Erdős Problem #612, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/612\n Evidence used: Current formulation and status context.\n\n**Review notes.** The classification concerns the original two-case statement, not its proposed replacement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2213, "problem_number": "EP-614", "title": "Erdős Problem #614", "statement": "Let $f(n,k)$ be minimal such that there is a graph with $n$ vertices and $f(n,k)$ edges where every set of $k+2$ vertices induces a subgraph with maximum degree at least $k$. Determine $f(n,k)$.\n\",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general extremal function remains open, with a finite-forbidden-family Turán reformulation for each fixed k.\n\n**Verified partial progress.**\n\n- A 2026 discussion note identifies f(n,k)=binom(n,2)-ex(n,F_k^min).\n- It gives explicit families for k=1,2,3.\n\n**Full solution or refutation.**\n\nThe reformulation does not determine the general extremal number.\n\n**What remains.**\n\nDetermine ex(n,F_k^min), beginning with unresolved small k.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #614, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/614\n Evidence used: Open status.\n- EP-614 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/614?embed=1\n Evidence used: Unverified but precise reduction, retained as a lead.\n\n**Review notes.** Forum reduction is not treated as a complete solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2214, "problem_number": "EP-616", "title": "Erdős Problem #616", "statement": "Let $r\\geq 3$. For an $r$-uniform hypergraph $G$ let $\\tau(G)$ denote the covering number (or transversal number), the minimum size of a set of vertices which includes at least one from each edge in $G$.\nDetermine the best possible $t$ such that, if $G$ is an $r$-uniform hypergraph $G$ where every subgraph $G'$ on at most $3r-3$ vertices has $\\tau(G')\\leq 1$, we have $\\tau(G)\\leq t$.", "background": "Erd\\H{o}s, Hajnal, and Tuza \\cite{EHT91} proved that this $t$ satisfies $ \\frac{3}{16}r+\\frac{7}{8}\\leq t \\leq \\frac{1}{5}r. $ \nReferences\n\n\n[EHT91] Erd\\H{o}s, Paul and Hajnal, Andr\\'{a}s and Tuza, Zsolt, Local constraints ensuring small representing sets. J. Combin. Theory Ser. A (1991), 78-84.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The optimum transversal bound remains open between (3/16)r+7/8 and r/5.\n\n**Verified partial progress.**\n\n- Erdős--Hajnal--Tuza established the displayed linear bounds.\n\n**Full solution or refutation.**\n\nThe constants do not match.\n\n**What remains.**\n\nDetermine the sharp t(r).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #616, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/616\n Evidence used: Current open status and bounds.\n\n**Review notes.** AI-generated claimed proofs in discussion were explicitly rejected there and are not used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2215, "problem_number": "EP-619", "title": "Erdős Problem #619", "statement": "For a triangle-free graph $G$ let $h_r(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $r$ (while preserving the property of being triangle-free).\nIs it true that there exists a constant $c>0$ such that if $G$ is a connected graph on $n$ vertices then $h_4(G)<(1-c)n$?", "background": "A problem of Erd\\H{o}s, Gy\\'{a}rf\\'{a}s, and Ruszink\\'{o} \\cite{EGR98} who proved that $h_3(G)\\leq n$ and $h_5(G) \\leq \\frac{n-1}{2}$ and there exist connected graphs $G$ on $n$ vertices with $h_3(G)\\geq n-c$ for some constant $c>0$.\nIf we omit the condition that the graph must remain triangle-free then Alon, Gy\\'{a}rf\\'{a}s, and Ruszink\\'{o} \\cite{AGR00} have proved that adding $n/2$ edges always suffices to obtain diameter at most $4$.\nSee also [134] and [618].\nReferences\n\n\n[AGR00] Alon, Noga and Gy\\'{a}rf\\'{a}s, Andr\\'{a}s and Ruszink\\'{o}, Mikl\\'{o}s, Decreasing the diameter of bounded degree graphs. J. Graph Theory (2000), 161--172.\n\n[EGR98] Erd\\H{o}s, Paul and Gy\\'{a}rf\\'{a}s, Andr\\'{a}s and\nRuszink\\'{o}, Mikl\\'{o}s, How to decrease the diameter of triangle-free graphs. Combinatorica (1998), 493-501.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A June 2026 counterexample construction gives infinitely many connected triangle-free graphs with h_4(G) at least n-O(n^(8/9)(log n)^(2/9)), refuting every bound h_4(G)<(1-c)n with fixed c>0; the result is tracker-recorded and Lean-verified but not yet conventionally published.\n\n**Verified partial progress.**\n\n- Erdos, Gyarfas, and Ruszinko proved h_3(G) <= n and h_5(G) <= (n-1)/2.\n- Without the triangle-free-preservation condition, Alon, Gyarfas, and Ruszinko proved that n/2 added edges suffice for diameter at most 4.\n\n**Full solution or refutation.**\n\nAttach many pendant vertices to a triangle-free core with small independence number. Counting the pendant components that can remain unjoined in any triangle-free diameter-at-most-four supergraph forces nearly n new edges; parameter optimization yields the stated n-o(n) lower bound.\n\n**What remains.**\n\nThe conjectured fixed linear saving is false. The finer order of n-h_4(G), publication of the new proof, and independent statement-fidelity review remain.\n\n**Sources checked.**\n\n- Primary proof discussion for Erdos Problem #619, Thomas F. Bloom, Nick Kuhn, and contributors (June 2026). (primary): https://www.erdosproblems.com/forum/thread/619?embed=1\n Evidence used: Contains the core-plus-pendants construction, pair-counting argument, parameter optimization, and explicit n-O(n^(8/9)(log n)^(2/9)) counterexample bound.\n- Google DeepMind Formal Conjectures, ErdosProblems/619.lean (2026). (formal_verification): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/619.lean\n Evidence used: Maintained formal record linking the Lean-verified counterexample resolution.\n- Thomas F. Bloom, Erdos Problem #619, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/619\n Evidence used: Records SOLVED (LEAN), credits the construction, and states the quantitative counterexample family.\n\n**Review notes.** The intended source says diameter at most r and connected triangle-free graph; the dataset wording is looser in both places.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2216, "problem_number": "EP-620", "title": "Erdős Problem #620", "statement": "If $G$ is a graph on $n$ vertices without a $K_4$ then how large a triangle-free induced subgraph must $G$ contain?", "background": "This was first asked by Erd\\H{o}s and Rogers \\cite{ErRo62}, and is generally known as the Erd\\H{o}s-Rogers problem. Let $f(n)$ be such that every such graph contains a triangle-free subgraph with at least $f(n)$ vertices.\nIt is now known that $f(n)=n^{1/2+o(1)}$. Bollob\\'{a}s and Hind \\cite{BoHi91} proved $ n^{1/2} \\ll f(n) \\ll n^{7/10+o(1)}. $ Krivelevich \\cite{Kr94} improved this to $ n^{1/2}(\\log\\log n)^{1/2} \\ll f(n) \\ll n^{2/3}(\\log n)^{1/3}. $ Wolfovitz \\cite{Wo13} proved $ f(n) \\ll n^{1/2}(\\log n)^{120}. $ The best bounds currently known are $ n^{1/2}\\frac{(\\log n)^{1/2}}{\\log\\log n}\\ll f(n) \\ll n^{1/2}\\log n. $ The lower bound follows from results of Shearer \\cite{Sh95}, and the upper bound was proved by Mubayi and Verstraete \\cite{MuVe24}.\nReferences\n\n\n[BoHi91] Bollob\\'{a}s, B. and Hind, H. R., Graphs without large triangle free subgraphs. Discrete Math. (1991), 119-131.\n\n[ErRo62] Erd\\H{o}s, P. and Rogers, C. A., The construction of certain graphs. Canadian J. Math. (1962), 702-707.\n\n[Kr94] Krivelevich, Michael, {$K^s$}-free graphs without large {$K^r$}-free subgraphs. Combin. Probab. Comput. (1994), 349-354.\n\n[MuVe24] D. Mubayi and J. Verstraete, On the order of Erd\\H{o}s-Rogers functions. arXiv:2401.02548 (2024).\n\n[Sh95] Shearer, James B., On the independence number of sparse graphs. Random Structures Algorithms (1995), 269--271.\n\n[Wo13] Wolfovitz, Guy, {$K_4$}-free graphs without large induced triangle-free\nsubgraphs. Combinatorica (2013), 623-631.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Erdős--Rogers function is determined to exponent 1/2, with logarithmic gap remaining.\n\n**Verified partial progress.**\n\n- Shearer yields a lower bound sqrt(n log n)/log log n.\n- Mubayi--Verstraete prove an upper bound O(sqrt(n) log n).\n\n**Full solution or refutation.**\n\nThe two logarithmic factors do not match.\n\n**What remains.**\n\nDetermine the logarithmic order.\n\n**Sources checked.**\n\n- D. Mubayi and J. Verstraete, arXiv:2401.02548 (2024). (primary): https://arxiv.org/abs/2401.02548\n Evidence used: Primary source for the upper bound.\n- Thomas F. Bloom, Erdős Problem #620, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/620\n Evidence used: Current open status and lower bound.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2217, "problem_number": "EP-623", "title": "Erdős Problem #623", "statement": "Let $X$ be a set of cardinality $\\aleph_\\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\not\\in A$ for all $A$. Must there exist an infinite $Y\\subseteq X$ that is independent - that is, for all finite $B\\subset Y$ we have $f(B)\not\\in Y$?", "background": "A problem of Erd\\H{o}s and Hajnal \\cite{ErHa58}, who proved that if $\\lvert X\\rvert <\\aleph_\\omega$ then the answer is no. Erd\\H{o}s suggests in \\cite{Er99} that this problem is 'perhaps undecidable'.\nReferences\n\n\n[Er99] Erd\\H{o}s, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6.\n\n[ErHa58] Erd\\H{o}s, P. and Hajnal, A., On the structure of set mappings. Acta Math. Acad. Sci. Hungar. (1958), 111-133.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The tracker lists the problem open; a recent unrefereed forum post claims the expected independence result via Koepke's property.\n\n**Verified partial progress.**\n\n- Erdős--Hajnal give negative results below aleph_omega.\n\n**Full solution or refutation.**\n\nThe new claimed relative-consistency result was not independently verified, so it is not recorded as a solution.\n\n**What remains.**\n\nVerify the claimed equivalence/consistency strength or obtain a citable source.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #623, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/623\n Evidence used: Maintained open status and classical obstruction.\n- EP-623 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/623\n Evidence used: Unrefereed claimed independence result only.\n\n**Review notes.** The source input loses LaTex not escapes; this is flagged, not corrected.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2218, "problem_number": "EP-624", "title": "Erdős Problem #624", "statement": "Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\\{A : A\\subseteq X\\}\\to X$ so that for every $Y\\subseteq X$ with $\\lvert Y\\rvert \\geq H(n)$ we have $ \\{ f(A) : A\\subseteq Y\\}=X. $ Prove that $ H(n)-\\log_2 n \\to \\infty. $ ", "background": "A problem of Erd\\H{o}s and Hajnal \\cite{ErHa68} who proved that $ \\log_2 n \\leq H(n) < \\log_2n +(3+o(1))\\log_2\\log_2n. $ Erd\\H{o}s said that even the weaker statement that for $n=2^k$ we have $H(n)\\geq k+1$ is open, but Alon has provided the following simple proof: by the pigeonhole principle there are $\\frac{n-1}{2}$ subsets $A_i$ of size $2$ such that $f(A_i)$ is the same. Any set $Y$ of size $k$ containing at least $k/2$ of them can have at most $ 2^k-\\lfloor k/2\\rfloor+1< 2^k=n $ distinct elements in the union of the images of $f(A)$ for $A\\subseteq Y$.\nFor this weaker statement, Erd\\H{o}s and Gy\\'{a}rf\\'{a}s conjectured the stronger form that if $\\lvert X\\rvert=2^k$ then, for any $f:\\{A : A\\subseteq X\\}\\to X$, there must exist some $Y\\subset X$ of size $k$ such that $ \\#\\{ f(A) : A\\subseteq Y\\}< 2^k-k^C $ for every $C$ (with $k$ sufficiently large depending on $C$). This was proved by Alon (personal communication), who proved the stronger version that there exists some absolute constant $c>0$ such that, if $k$ is large enough, there must exist some $Y\\subset X$ of size $k$ such that $ \\#\\{ f(A) : A\\subseteq Y\\}<(1-c)2^k. $ Alon also proved that, provided $k$ is large enough, if $\\lvert X\\rvert=2^k$ there exists some $f:\\{A: A\\subseteq X\\}\\to X$ such that, if $Y\\subset X$ with $\\lvert Y\\rvert=k$, then $ \\#\\{ f(A) : A\\subseteq Y\\}>\\tfrac{1}{4}2^k. $ \nReferences\n\n\n[ErHa68] Erd\\H{o}s, P. and Hajnal, A., On a combinatorial problem. Mat. Lapok (1968), 345-348.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The divergence H(n)-log_2 n remains open, although its k+1 special consequence at n=2^k is proved.\n\n**Verified partial progress.**\n\n- Alon's pigeonhole argument proves H(2^k)>=k+1.\n- The tracker records stronger unpublished bounds for a related formulation.\n\n**Full solution or refutation.**\n\nA one-unit improvement does not imply divergence.\n\n**What remains.**\n\nProve an unbounded lower excess.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #624, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/624\n Evidence used: Current open status and special-case proof.\n\n**Review notes.** Personal communications are not treated as full verification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2219, "problem_number": "EP-625", "title": "Erdős Problem #625", "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. Let $\\chi(G)$ denote the chromatic number.\nIf $G$ is a random graph with $n$ vertices and each edge included independently with probability $1/2$ then is it true that almost surely $ \\chi(G) - \\zeta(G) \\to \\infty $ as $n\\to \\infty$?", "background": "A problem of Erd\\H{o}s and Gimbel (see also \\cite{Gi16}). At a conference on random graphs in Poznan, Poland (most likely in 1989) Erd\\H{o}s offered \\$100 for a proof that this is true, and \\$1000 for a proof that this is false (although later told Gimbel that \\$1000 was perhaps too much).\nIt is known that almost surely $ \\frac{n}{2\\log_2n}\\leq \\zeta(G)\\leq \\chi(G)\\leq (1+o(1))\\frac{n}{2\\log_2n}. $ (The final upper bound is due to Bollob\\'{a}s \\cite{Bo88}. The first inequality follows from the fact that almost surely $G$ has clique number and independence number $< 2\\log_2n$.)\nHeckel \\cite{He24} and, independently, Steiner \\cite{St24b} have shown that it is not the case that $\\chi(G)-\\zeta(G)$ is bounded with high probability, and in fact if $\\chi(G)-\\zeta(G) \\leq f(n)$ with high probability then $f(n)\\geq n^{1/2-o(1)}$ along an infinite sequence of $n$. Heckel conjectures that, with high probability, $ \\chi(G)-\\zeta(G) \\asymp \\frac{n}{(\\log n)^3}. $ Heckel \\cite{He24c} further proved that, for any $\\epsilon>0$, we have $ \\chi(G) -\\zeta(G) \\geq n^{1-\\epsilon} $ for roughly $95\\%$ of all $n$.\nReferences\n\n\n[Bo88] Bollob\\'{a}s, B., The chromatic number of random graphs. Combinatorica (1988), 49-55.\n\n[Gi16] J. Gimbel, Some of my favorite coloring problems for graphs and digraphs. Graph Theory: Favorite conjectures and open problems (2016), 95-108.\n\n[He24] A. Heckel, On a question of Erd\\H{o}s and Gimbel on the cochromatic number. arXiv:2408.13839 (2024).\n\n[He24c] A. Heckel, The difference between the chromatic and the cochromatic number of a random graph. arXiv:2409.17614 (2024).\n\n[St24b] R. Steiner, On the difference between the chromatic and cochromatic number. arXiv:2408.02400 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The almost-sure divergence is open, but bounded high-probability gaps are ruled out and near-linear gaps hold for most n.\n\n**Verified partial progress.**\n\n- Heckel and Steiner rule out a bounded high-probability gap.\n- Heckel proves chi-zeta >= n^(1-epsilon) for roughly 95% of n.\n\n**Full solution or refutation.**\n\nResults on most n do not establish convergence in probability/almost sure behavior for all n.\n\n**What remains.**\n\nProve divergence with high probability for every sufficiently large n.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #625, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/625\n Evidence used: Current open status and cited modern advances.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2220, "problem_number": "EP-626", "title": "Erdős Problem #626", "statement": "Let $k\\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $>m$ (i.e. contains no cycle of length $\\leq m$). Does $ \\lim_{n\\to \\infty}\\frac{g_k(n)}{\\log n} $ exist?\nConversely, if $h^{(m)}(n)$ is the maximal chromatic number of a graph on $n$ vertices with girth $>m$ then does $ \\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n} $ exist, and what is its value?", "background": "It is known that $ \\frac{1}{4\\log k}\\log n\\leq g_k(n) \\leq \\frac{2}{\\log(k-2)}\\log n+1, $ the lower bound due to Kostochka \\cite{Ko88} and the upper bound to Erd\\H{o}s \\cite{Er59b}.\nErd\\H{o}s \\cite{Er59b} proved that $ \\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n}\\gg \\frac{1}{m} $ and, for odd $m$, $ \\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n}\\leq \\frac{2}{m+1}, $ and conjectured this is sharp. He had no good guess for the value of the limit for even $m$, other that it should lie in $[\\frac{2}{m+2},\\frac{2}{m}]$, but could not prove this even for $m=4$.\nSee also the entry in the graphs problem collection.\nReferences\n\n\n[Er59b] Erd\\H{o}s, P., Graph theory and probability. Canadian J. Math. (1959), 34-38.\n\n[Ko88] Kostochka, A. V., Upper bounds on the chromatic number of graphs. Trudy Inst. Mat. (Novosibirsk) (1988), 204-226, 265.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Both requested limit questions remain open between established logarithmic/exponent bounds.\n\n**Verified partial progress.**\n\n- Kostochka and Erdős give bounds for g_k(n) of logarithmic order.\n- Erdős gives one-sided bounds for h^(m)(n), especially for odd m.\n\n**Full solution or refutation.**\n\nThe bounds do not prove existence or identify either limit.\n\n**What remains.**\n\nEstablish the limits and determine their constants.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #626, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/626\n Evidence used: Current open status and classical bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2221, "problem_number": "EP-627", "title": "Erdős Problem #627", "statement": "Let $\\omega(G)$ denote the clique number of $G$ and $\\chi(G)$ the chromatic number. If $f(n)$ is the maximum value of $\\chi(G)/\\omega(G)$, as $G$ ranges over all graphs on $n$ vertices, then does $ \\lim_{n\\to\\infty}\\frac{f(n)}{n/(\\log n)^2} $ exist?", "background": "Tutte and Zykov \\cite{Zy52} independently proved that for every $k$ there is a graph with $\\omega(G)=2$ and $\\chi(G)=k$. Erd\\H{o}s \\cite{Er61d} proved that for every $n$ there is a graph on $n$ vertices with $\\omega(G)=2$ and $\\chi(G)\\gg n^{1/2}/\\log n$, whence $f(n) \\gg n^{1/2}/\\log n$.\nErd\\H{o}s \\cite{Er67c} proved that $ f(n) \\asymp \\frac{n}{(\\log n)^2} $ and that the limit in question, if it exists, must be in $ (\\log 2)^2\\cdot [1/4,1]. $ See also the entry in the graphs problem collection.\nReferences\n\n\n[Er61d] Erd\\H{o}s, P., Graph theory and probability. II. Canadian J. Math. (1961), 346-352.\n\n[Er67c] Erd\\H{o}s, P., Some remarks on chromatic graphs. Colloq. Math. (1967), 253-256.\n\n[Zy52] Zykov, A. A., On some properties of linear complexes. Amer. Math. Soc. Translation (1952), 33.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The normalized chromatic-to-clique ratio is known within constants, but its limit is open; a conditional Ramsey connection improves the upper constant.\n\n**Verified partial progress.**\n\n- Erdős proved f(n) is asymptotic in order to n/(log n)^2.\n- Araujo--Filipe--Miyazaki conditionally derive existence from a diagonal Ramsey limit and improve the upper constant.\n\n**Full solution or refutation.**\n\nThe necessary diagonal Ramsey asymptotics are themselves unresolved.\n\n**What remains.**\n\nProve the Ramsey hypotheses or directly establish the ratio limit.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #627, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/627\n Evidence used: Current status and AFM conditional result.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2222, "problem_number": "EP-629", "title": "Erdős Problem #629", "statement": "The list chromatic number $\\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours.\nDetermine the minimal number of vertices $n(k)$ of a bipartite graph $G$ such that $\\chi_L(G)>k$.", "background": "A problem of Erd\\H{o}s, Rubin, and Taylor \\cite{ERT80}, who proved that $ 2^{k-1}\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The minimum order n(k) of a bipartite graph with list chromatic number greater than k remains asymptotically undetermined.\n\n**Verified partial progress.**\n\n- The property-B comparison and Radhakrishnan--Srinivasan bound give n(k) much greater than 2^k sqrt(k/log k).\n- Erdos--Rubin--Taylor and Hanson--MacGillivray--Toft give upper bounds, the recurrence n(k) <= k n(k-2)+2^k, and n(2)=6, n(3)=14.\n\n**Full solution or refutation.**\n\nNo later result closing the exponential-polynomial gap for this precise minimum-order invariant was located.\n\n**What remains.**\n\nDetermine the asymptotic order of n(k), or materially narrow the gap between the property-B lower bound and known constructions.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #629, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/629\n Evidence used: Maintains open status and records the current general bounds and exact small cases.\n- D. Hanson, G. MacGillivray and B. Toft, Choosability of bipartite graphs, Ars Combinatoria 44 (1996), 183-192. (primary): https://combinatorialpress.com/ars-articles/volume-044-ars-articles/choosability-of-bipartite-graphs-2/\n Evidence used: Primary source for n(3)=14 and the recurrence n(k) <= k n(k-2)+2^k.\n\n**Review notes.** The background has leaked JSON suffix text; the exact statement is intact.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2223, "problem_number": "EP-633", "title": "Erdős Problem #633", "statement": "Classify those triangles which can only be cut into a square number of congruent triangles.", "background": "Erd\\H{o}s' question was reported by Soifer \\cite{So09c}. It is easy to see (see for example \\cite{So09}) that any triangle can be cut into $n^2$ congruent triangles (for any $n\\geq 1$). Soifer \\cite{So09b} proved that there exists at least one triangle (e.g. one with sides $\\sqrt{2},\\sqrt{3},\\sqrt{4}$) which can only be cut into a square number of congruent triangles. (In fact Soifer proves that any triangle for which the angles and sides are both integrally independent has this property.)\nSoifer proved \\cite{So09} that if we relax congruence to similarity then every triangle can be cut into $n$ similar triangles when $n\neq 2,3,5$ and there exists a triangle that cannot be cut into $2$, $3$, or $5$ similar triangles.\nSee also [634].\nReferences\n\n\n[So09] Soifer, Alexander, How Does One Cut a Triangle? I. (2009), 15-23.\n\n[So09b] Soifer, Alexander, How Does One Cut a Triangle? II. (2009), 37-39.\n\n[So09c] Soifer, Alexander, Is there anything beyond the solution?. (2009), 47-50.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Beeson, Laczkovich, and Zhang's 2026 primary preprint gives a complete classification of triangles tileable only into a square number of congruent triangles and explicitly states that it settles Erdős Problem 633.\n\n**Verified partial progress.**\n\n- Soifer proved examples, including triangles with integrally independent side lengths and angles, that admit only square-number congruent dissections.\n- Isosceles triangles trivially admit a two-piece congruent dissection, separating a large family outside the requested class.\n\n**Full solution or refutation.**\n\nThe complete classification implies that triangles admitting some nonsquare congruent triangular dissection are exactly the isosceles triangles together with countably many similarity classes; the requested triangles are the complement of that classified exceptional set.\n\n**What remains.**\n\nThe classification problem is closed. More explicit organization of exceptional similarity classes and prescribed-tile-number refinements are separate follow-up questions.\n\n**Sources checked.**\n\n- Michael Beeson, Miklos Laczkovich, and Yan X. Zhang, Solution of Erdos Problem 633, arXiv:2604.03609v2 (2026). (primary): https://arxiv.org/abs/2604.03609\n Evidence used: The abstract explicitly states that the paper classifies the requested triangles and settles EP-633; the paper supplies the full classification.\n- Thomas F. Bloom, Erdos Problem #633, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/633\n Evidence used: Records SOLVED status, cites arXiv:2604.03609, and summarizes the nonsquare-tiling exceptional set.\n- Erdos Problems LaTeX bibliography record for #633. (source_collection): https://www.erdosproblems.com/latex/633\n Evidence used: Provides the exact BLZ26 bibliographic citation and earlier Soifer references.\n\n**Review notes.** The imported background has a leaked serialized suffix; the statement itself is intact.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2224, "problem_number": "EP-634", "title": "Erdős Problem #634", "statement": "Find all $n$ such that there is at least one triangle which can be cut into $n$ congruent triangles.", "background": "Erd\\H{o}s' question was reported by Soifer \\cite{So09c}. It is easy to see that all square numbers have this property (in fact for square numbers any triangle will do). Soifer \\cite{So09c} has shown that numbers of the form $2n^2,3n^2,6n^2,n^2+m^2$ also have this property. Beeson has shown (see the slides below) that $7$ and $11$ do not have this property. It is possible that any prime of the form $4n+3$ does not have this property.\nIn particular, it is not known if $19$ has this property (i.e. are there $19$ congruent triangles which can be assembled into a triangle?).\nFor more on this problem see these slides from a talk by Michael Beeson. As a demonstration of this problem we include {IMAGE=634Triangle,a picture} of a cutting of an equilateral triangle into $27$ congruent triangles from these slides.\nSoifer proved \\cite{So09} that if we relax congruence to similarity then every triangle can be cut into $N$ similar triangles when $N\neq 2,3,5$.\nIf one requires the smaller triangles to be similar to the larger triangle then the only possible values of $N$ are $n^2,n^2+m^2,3n^2$, proved by Snover, Waiveris, and Williams \\cite{SWW91}.\nZhang \\cite{Zh25}, among other results, has proved that for any integers $a \\geq b$, if $ n\\geq 3\\left\\lceil \\frac{a^2+b^2+ab-a-b}{ab}\\right\\rceil $ then $n^2ab$ has this property (indeed, they explicitly show that an equilateral triangle can be tiled with $n^2ab$ many triangles of side lengths $a,b,\\sqrt{a^2+b^2+2+ab}$).\nSee also [633].\nReferences\n\n\n[SWW91] Snover, S. and Waiveris, C. and Williams, J., Rep-tiling for triangles. Discrete Math. (1991), 193-200.\n\n[So09] Soifer, Alexander, How Does One Cut a Triangle? I. (2009), 15-23.\n\n[So09c] Soifer, Alexander, Is there anything beyond the solution?. (2009), 47-50.\n\n[Zh25] Y. Zhang, Tiling Triangles With $2\\pi/3$ Angles. arXiv:2512.22696 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A July 2026 preprint completely classifies prime tile counts, but the set of all composite counts admitting a congruent-triangle tiling remains unclassified.\n\n**Verified partial progress.**\n\n- Beeson proves that a prime N occurs if and only if N=2, N=3, or N is congruent to 1 modulo 4.\n- Consequently 19 and every prime greater than 3 congruent to 3 modulo 4 do not occur.\n- Earlier constructions cover squares, multiples 2, 3, or 6 of squares, sums of two squares, and further 2025/26 families.\n\n**Full solution or refutation.**\n\nThe prime subproblem is solved exactly; this does not determine all composite N.\n\n**What remains.**\n\nClassify the admissible composite tile counts and reconcile the many sufficient families with necessary conditions.\n\n**Sources checked.**\n\n- Michael Beeson, Tiling a triangle into a prime number of congruent triangles, arXiv:2607.23453 (2026). (primary): https://arxiv.org/abs/2607.23453\n Evidence used: Theorem 22 and Corollary 23 give the complete prime-count classification.\n- Thomas F. Bloom, Erdős Problem #634, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/634\n Evidence used: Records the broader open classification and older construction/nonexistence results.\n\n**Review notes.** The background contains a broken neq command and leaked JSON suffix text; neither was repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2225, "problem_number": "EP-635", "title": "Erdős Problem #635", "statement": "Let $t\\geq 1$ and $A\\subseteq \\{1,\\ldots,N\\}$ be such that whenever $a,b\\in A$ with $b-a\\geq t$ we have $b-a\nmid b$. How large can $\\lvert A\\rvert$ be? Is it true that $ \\lvert A\\rvert \\leq \\left(\\frac{1}{2}+o_t(1)\\right)N? $ ", "background": "Asked by Erd\\H{o}s in a letter to Ruzsa in around 1980. Erd\\H{o}s observes that when $t=1$ the maximum possible is $ \\lvert A\\rvert=\\left\\lfloor\\frac{N+1}{2}\\right\\rfloor, $ achieved by taking $A$ to be all odd numbers in $\\{1,\\ldots,N\\}$. He also observes that when $t=2$ there exists such an $A$ with $ \\lvert A\\rvert \\geq \\frac{N}{2}+c\\log N $ for some constant $c>0$: take $A$ to be the union of all odd numbers together with numbers of the shape $2^k$ with $k$ odd.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The displayed density upper bound |A| <= (1/2+o_t(1))N is proved, but the sharper extremal question 'How large can |A| be?' remains open.\n\n**Verified partial progress.**\n\n- A January 2026 proof, substantially formalized in Lean, establishes the affirmative o_t(1) density statement.\n- Tao observed that the qualitative bound also follows from an inequality of Elliott.\n- Known t=2 constructions have N/2 plus a positive logarithmic excess, far below the precision of the Elliott-type upper bound.\n\n**Full solution or refutation.**\n\nOne explicit yes/no subquestion is resolved; the leading correction and exact extremal behavior are not.\n\n**What remains.**\n\nDetermine the true excess above N/2 for fixed t, especially t=2, rather than only an o(N) error.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #635 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/635\n Evidence used: Records the affirmative solution of the second question, its Lean support, Elliott reduction, and why the full extremal problem remains open.\n- Terence Tao, comment on Erdős Problem #635, 30 January 2026. (authoritative_secondary): https://www.erdosproblems.com/forum/thread/635?order=oldest\n Evidence used: Separates the solved asymptotic-density part from the much sharper unresolved extremal question.\n\n**Review notes.** The imported statement has a literal newline splitting the intended LaTeX command nmid; the report preserves and flags it.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2226, "problem_number": "EP-638", "title": "Erdős Problem #638", "statement": "Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\\in S$ such that if the edges of $G_n$ are coloured with $n$ colours then there is a monochromatic triangle.\nIs it true that for every infinite cardinal $\\aleph$ there is a graph $G$ of which every finite subgraph is in $S$ and if the edges of $G$ are coloured with $\\aleph$ many colours then there is a monochromatic triangle.", "background": "Erd\\H{o}s writes 'if the answer is affirmative many extensions and generalisations will be possible'.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The literal statement is false because it omits the intended requirement that S be closed under taking subgraphs.\n\n**Verified partial progress.**\n\n- For the literal text, let S contain finite triangle-Ramsey complete graphs but omit K_1; the finite premise holds, while no graph has every finite subgraph in S.\n- The maintained tracker explicitly flags the missing hereditary hypothesis.\n- A recent public proof project claims a counterexample even to a substantive subgraph-closed reading, but that stronger claim remains unincorporated by the tracker.\n\n**Full solution or refutation.**\n\nAn elementary formulation counterexample refutes the exact imported statement; this is not presented as settling Erdős's intended hereditary compactness question.\n\n**What remains.**\n\nObtain expert review of the recent formal counterexample to the hereditary reading and pin down the exact hypothesis intended in the original source.\n\n**Sources checked.**\n\n- Star Fleet Math, public formal proof bundle for Erdős Problem #638, checked 2026-08-17. (primary): https://www.starfleetmath.com/\n Evidence used: Provides a pinned Lean proof project claiming a negative answer, including a theorem refuting the literal claim.\n- Thomas F. Bloom, Erdős Problem #638, checked 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/638\n Evidence used: Explicitly notes that S was presumably intended to be subgraph-closed and that without this a sparse family of complete graphs gives a trivial counterexample.\n\n**Review notes.** Disproved classification is for the exact source statement. The stronger hereditary claim is deliberately not treated as established.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2227, "problem_number": "EP-640", "title": "Erdős Problem #640", "statement": "Is there some function $f$ such that for all $k\\geq 3$ if a finite graph $G$ has chromatic number $\\geq f(k)$ then $G$ must contain some odd cycle whose vertices span a graph of chromatic number $\\geq k$?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The high-chromatic odd-cycle-span question remains open for k>3.\n\n**Verified partial progress.**\n\n- The k=3 case is immediate from non-bipartiteness.\n- Steiner proves equivalence to the superficially weaker version seeking a path whose vertex set spans chromatic number at least k.\n\n**Full solution or refutation.**\n\nNeither the trivial base case nor the path equivalence supplies f(k) for any general k>3.\n\n**What remains.**\n\nProve a finite bound f(k) for all k, or construct high-chromatic counterexamples with every odd-cycle vertex span of bounded chromatic number.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #640, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/640\n Evidence used: Maintains open status, records k=3, and gives the path equivalence.\n\n**Review notes.** The statement field has absorbed trailing difficulty/JSON text; the report quotes this contamination verbatim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2228, "problem_number": "EP-642", "title": "Erdős Problem #642", "statement": "Let $f(n)$ be the maximal number of edges in a graph on $n$ vertices such that all cycles have more vertices than diagonals. Is it true that $f(n)\\ll n$?", "background": "A problem of Hamburger and Szegedy.\nChen, Erd\\H{o}s, and Staton \\cite{CES96} proved $f(n) \\ll n^{3/2}$. Dragani\\'{c}, Methuku, Munh\\'{a} Correia, and Sudakov \\cite{DMMS24} have improved this to $ f(n) \\ll n(\\log n)^8. $ \nReferences\n\n\n[CES96] Chen, Guantao and Erd\\H{o}s, Paul and Staton, William, Proof of a conjecture of {B}ollob\\'as on nested cycles. J. Combin. Theory Ser. B (1996), 38--43.\n\n[DMMS24] Dragani\\'c, Nemanja and Methuku, Abhishek and Munh\\'a{}\nCorreia, David and Sudakov, Benny, Cycles with many chords. Random Structures Algorithms (2024), 3--16.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjectured linear extremal bound is open; the best published bound is O(n log^8 n), with a complementary 2026 constant-minimum-degree result.\n\n**Verified partial progress.**\n\n- Draganić--Methuku--Munhá Correia--Sudakov prove f(n)=O(n log^8 n), improving O(n^(3/2)).\n- Draganić--Girão prove constant minimum degree forces a cycle of length ell with Omega(ell/log^C ell) chords.\n\n**Full solution or refutation.**\n\nThe remaining polylogarithmic factor has not been removed; almost-linearly-many chords are not enough to force at least as many chords as vertices.\n\n**What remains.**\n\nProve f(n)=O(n) or construct superlinear examples satisfying the cycle-versus-chord restriction.\n\n**Sources checked.**\n\n- N. Draganić, A. Methuku, D. Munhá Correia and B. Sudakov, Cycles with many chords, Random Structures & Algorithms 65 (2024), 3-16. (primary): https://arxiv.org/abs/2306.09157\n Evidence used: Primary source for the O(n log^8 n) upper bound.\n- N. Draganić and A. Girão, Cycles with almost linearly many chords, arXiv:2601.08769 (2026). (primary): https://arxiv.org/abs/2601.08769\n Evidence used: Primary source for the constant-minimum-degree, almost-linear chord theorem.\n- Thomas F. Bloom, Erdős Problem #642, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/642\n Evidence used: Current formulation, definition of chord, and open status.\n\n**Review notes.** The input says diagonals without definition; the tracker identifies these as chords. Background markup defects are preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2229, "problem_number": "EP-643", "title": "Erdős Problem #643", "statement": "Let $f(n;t)$ be minimal such that if a $t$-uniform hypergraph on $n$ vertices contains at least $f(n;t)$ edges then there must be four edges $A,B,C,D$ such that $ A\\cup B= C\\cup D $ and $ A\\cap B=C\\cap D=\\emptyset. $ Estimate $f(n;t)$ - in particular, is it true that for $t\\geq 3$ $ f(n;t)=(1+o(1))\\binom{n}{t-1}? $ ", "background": "For $t=2$ this is asking for the maximal number of edges on a graph which contains no $C_4$, and so $f(n;2)=(1/2+o(1))n^{3/2}$.\nF\"{u}redi \\cite{Fu84} proved that $f(n;3) \\ll n^2$ and $f(n;3) > \\binom{n}{2}$ for infinitely many $n$. Pikhurko and Verstra\"{e}te \\cite{PiVe09} have proved $f(n;3)\\leq \\frac{13}{9}\\binom{n}{2}$ for all $n$.\nMore generally, F\"{u}redi \\cite{Fu84} proved that $ \\binom{n-1}{t-1}+\\left\\lfloor\\frac{n-1}{t}\\right\\rfloor\\leq f(n;t) < \\frac{7}{2}\\binom{n}{t-1}, $ and conjectured the lower bound is sharp for $t\\geq 4$. Pikhurko and Verstra\"{e}te \\cite{PiVe09} have proved that $ 1 \\leq \\limsup_{n\\to \\infty} \\frac{f(n;t)}{\\binom{n}{t-1}}\\leq \\min\\left(\\frac{7}{4},1+\\frac{2}{\\sqrt{t}}\\right) $ for all $t\\geq 3$.\nF\"{u}redi \\cite{Fu84} proved that $f(n;3)/\\binom{n}{2}$ converges as $n\\to \\infty$, but the existence of the limit for $t\\geq 4$ is unknown.\nReferences\n\n\n[Fu84] F\"uredi, Z., Hypergraphs in which all disjoint pairs have distinct unions. Combinatorica (1984), 161--168.\n\n[PiVe09] Pikhurko, Oleg and Verstra\"{e}te, Jacques, The maximum size of hypergraphs without generalized 4-cycles. J. Combin. Theory Ser. A (2009), 637--649.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The generalized-4-cycle extremal function is bounded within constant factors, but the conjectured asymptotic constant 1 remains open.\n\n**Verified partial progress.**\n\n- Pikhurko--Verstraëte prove limsup f(n;t)/binom(n,t-1) <= min(7/4,1+2/sqrt(t)).\n- For t=3 they prove the sharper upper constant 13/9.\n- Füredi gives lower constant at least 1 and convergence for t=3; limit existence for t>=4 remains open.\n\n**Full solution or refutation.**\n\nKnown lower and upper constants do not match for any fixed t>=3 covered by the conjecture.\n\n**What remains.**\n\nProve asymptotic constant 1 for each fixed t>=3 and establish limit existence for t>=4.\n\n**Sources checked.**\n\n- O. Pikhurko and J. Verstraëte, The maximum size of hypergraphs without generalized 4-cycles, J. Combin. Theory Ser. A 116 (2009), 637-649. (primary): https://doi.org/10.1016/j.jcta.2008.09.002\n Evidence used: Primary source for the 13/9 and min(7/4,1+2/sqrt(t)) upper constants.\n- Thomas F. Bloom, Erdős Problem #643, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/643\n Evidence used: Maintains open status and records Füredi's lower bounds and convergence distinction.\n\n**Review notes.** Broken source diacritics and leaked JSON text occur only in the background; they were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2230, "problem_number": "EP-644", "title": "Erdős Problem #644", "statement": "Let $f(k,r)$ be minimal such that if $A_1,A_2,\\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the $A_is$ there is some pair $\\{x,y\\}$ which intersects all of the $A_j$, then there is some set of size $f(k,r)$ which intersects all of the sets $A_i$. Is it true that $ f(k,7)=(1+o(1))\\frac{3}{4}k? $ Is it true that for any $r\\geq 3$ there exists some constant $c_r$ such that $ f(k,r)=(1+o(1))c_rk? $ ", "background": "A problem of Erd\\H{o}s, Fon-Der-Flaass, Kostochka, and Tuza \\cite{EFKT92}, who proved that $f(k,3)=2k$ and $f(k,4)=\\lfloor 3k/2\\rfloor$ and $f(k,5)=\\lfloor 5k/4\\rfloor$, and further that $f(k,6)=k$.\nReferences\n\n\n[EFKT92] Erd\"{o}s, P. and Fon-Der-Flaass, D. and Kostochka, A. V. and\nTuza, Zs., Small transversals in uniform hypergraphs. Siberian Adv. Math. (1992), 82-88.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For seven-edge local two-transversals, the asymptotic constant is bounded between 3/4 and 7/8; neither requested limit theorem is known.\n\n**Verified partial progress.**\n\n- Fon-Der-Flaass--Kostochka--Woodall prove f(k,7) <= ceil(7k/8).\n- Their construction proves f(4m,7) >= 3m+1 for sufficiently large m.\n- The exact r=3,4,5,6 values are known from Erdős--Fon-Der-Flaass--Kostochka--Tuza.\n\n**Full solution or refutation.**\n\nPublished linear bounds support but do not prove the conjectured 3/4 constant, and do not prove general convergence to c_r.\n\n**What remains.**\n\nClose the 3/4 versus 7/8 gap for r=7 and prove or refute existence of a linear asymptotic constant for every fixed r.\n\n**Sources checked.**\n\n- D. G. Fon-Der-Flaass, A. V. Kostochka and D. R. Woodall, Transversals in uniform hypergraphs with property (7,2), Discrete Mathematics 207 (1999), 277-284. (primary): https://doi.org/10.1016/S0012-365X(99)00114-4\n Evidence used: Primary source for the lower construction and ceil(7k/8) upper bound.\n- Thomas F. Bloom, Erdős Problem #644, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/644\n Evidence used: Current open formulation and exact low-r context.\n\n**Review notes.** The malformed source token A_is and leaked background JSON were preserved and flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2231, "problem_number": "EP-650", "title": "Erdős Problem #650", "statement": "Let $f(m)$ be such that if $A\\subseteq \\{1,\\ldots,N\\}$ has $\\lvert A\\rvert=m$ then every interval in $[1,\\infty)$ of length $2N$ contains $\\geq f(m)$ many distinct integers $b_1,\\ldots,b_r$ where each $b_i$ is divisible by some $a_i\\in A$, where $a_1,\\ldots,a_r$ are distinct.\nEstimate $f(m)$. In particular is it true that $f(m)\\ll m^{1/2}$?", "background": "Erd\\H{o}s and Sar\\'{a}nyi \\cite{ErSa59} proved that $f(m)\\gg m^{1/2}$.\nReferences\n\n\n[ErSa59] Erd\\H{o}s, P. and Sar\\'{a}nyi, Megjegyz\\'{e}sek egy versenyfeladathoz. Matematikai Lapok (1959).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Van Doorn, Li, and Tang's March 2026 human-written, AI-assisted preprint determines the extremal function exactly as f(m)=min(m,ceil(2 sqrt(m))) for every m, with the repaired lower-bound argument verified in Lean. This closes both the requested order estimate and the broader estimation problem.\n\n**Verified partial progress.**\n\n- Erdos and Suranyi proved the lower bound f(m) >> sqrt(m).\n- Erdos and Selfridge proved f(m^2) <= 2m, which implies f(m) <= 2 ceil(sqrt(m)) and already answers the dataset's Vinogradov-bound subquestion affirmatively.\n- The first AI-generated matching lower-bound proof had a midpoint/endpoint gap; Aristotle repaired the argument during formalization, and the final paper gives a cleaned-up human proof.\n\n**Full solution or refutation.**\n\nThe exact lower and upper bounds coincide: f(m)=min(m,ceil(2 sqrt(m))). The paper formulates f as the largest universally guaranteed number of disjoint divisor-multiple pairs in an open interval of length twice the largest element of A, an extremal formulation equivalent to the intended stored version after taking N=max(A).\n\n**What remains.**\n\nThe mathematical problem is closed. Curation should state the extremal definition and interval endpoint convention explicitly; conventional peer review would still be useful because the result is recent and AI-assisted.\n\n**Sources checked.**\n\n- Wouter van Doorn, Yanyang Li, and Quanyu Tang, Optimal bounds for an Erdos problem on matching integers to distinct multiples, arXiv:2603.28636 (2026). (primary): https://arxiv.org/abs/2603.28636\n Evidence used: States and proves the exact formula for all m, describes the AI-assisted discovery and Lean verification, and presents the final proof as human-written.\n- Quanyu Tang et al., public Erdos Problem 650 proof materials and gap documentation (2026). (formal_verification): https://github.com/QuanyuTang/erdos-problem-650\n Evidence used: Contains the proof materials, the note identifying the gap in the first lower-bound proof, and the revised artifacts associated with the formal repair.\n- Thomas F. Bloom, Erdos Problem #650 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/650\n Evidence used: Records SOLVED (LEAN), the exact formula, the historical Erdos-Suranyi and Erdos-Selfridge bounds, and the correction history.\n\n**Review notes.** The imported background has a leaked serialized suffix. The stored Vinogradov-bound question is valid; the current tracker also displays a coefficient-one inequality that conflicts with the exact formula and should not replace it.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2232, "problem_number": "EP-652", "title": "Erdős Problem #652", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ and let $R(x_i)=\\#\\{ \\lvert x_j-x_i\\rvert : j\neq i\\}$, where the points are ordered such that $ R(x_1)\\leq \\cdots \\leq R(x_n). $ Let $\\alpha_k$ be minimal such that, for all large enough $n$, there exists a set of $n$ points with $R(x_k)<\\alpha_kn^{1/2}$. Is it true that $\\alpha_k\\to \\infty$ as $k\\to \\infty$?", "background": "It is trivial that $R(x_1)=1$ is possible, and that $R(x_2) \\ll n^{1/2}$ is also possible, but we always have $ R(x_1)R(x_2)\\gg n. $ Erd\\H{o}s originally conjectured that $R(x_3)/n^{1/2}\\to \\infty$ as $n\\to \\infty$, but Elekes proved that for every $k$ and $n$ sufficiently large there exists some set of $n$ points with $R(x_k)\\ll_k n^{1/2}$.\nMathialagan \\cite{Ma21} proved that given a set $P$ of $k$ points and a set $Q$ of $n$ points, with $2\\leq k\\leq n^{1/3}$, there exists a point in $P$ which determines $\\gg (kn)^{1/2}$ distances to points in $Q$. This immediately implies $R(x_k)\\gg (kn)^{1/2}$ for $2\\leq k\\leq n^{1/3}$.\nReferences\n\n\n[Ma21] Mathialagan, Surya, On bipartite distinct distances in the plane. Electron. J. Combin. (2021), Paper No. 4.33, 25.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Mathialagan's peer-reviewed 2021 bipartite distinct-distances theorem, together with the Elekes circle-grid construction presented there, yields alpha_k=Theta(sqrt(k)); hence alpha_k tends to infinity. The direct consequence for EP-652 was recorded in January 2026.\n\n**Verified partial progress.**\n\n- Elekes's construction gives the upper bound alpha_k=O(sqrt(k)).\n- Mathialagan proves D(k,n)=Omega(sqrt(kn)) for 2<=k<=n^(1/3), matching the construction in this regime.\n- Applying the lower bound to the first k points in pinned-distance order gives R(x_k) >> sqrt(kn).\n\n**Full solution or refutation.**\n\nTake P={x_1,...,x_k} and Q to be the ambient n-point set, or its complement. Mathialagan's theorem makes some point of P determine Omega(sqrt(kn)) distances, while every point of P has pinned count at most R(x_k); this gives alpha_k >> sqrt(k). The circle-grid construction gives the matching upper bound.\n\n**What remains.**\n\nThe divergence and growth order are settled. The source should define alpha_k as an infimum rather than a literal minimum under a strict inequality, unless endpoint attainment is separately established; optimal constants remain open.\n\n**Sources checked.**\n\n- Surya Mathialagan, On Bipartite Distinct Distances in the Plane, Electronic Journal of Combinatorics 28(4) (2021), #P4.33, DOI 10.37236/9687. (primary): https://www.combinatorics.org/ojs/index.php/eljc/article/view/v28i4p33\n Evidence used: Published primary paper proving D(m,n)=Theta(sqrt(mn)) for m<=n^(1/3), including the matching Elekes construction needed for alpha_k=Theta(sqrt(k)).\n- Thomas F. Bloom, Erdos Problem #652 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/652\n Evidence used: Records PROVED status and the explicit derivation of alpha_k=Theta(sqrt(k)) from Mathialagan's theorem and construction.\n\n**Review notes.** The imported background has a leaked serialized suffix. The literal word minimal is potentially ill-posed because admissibility is defined using a strict inequality; the standard infimum reading is used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2233, "problem_number": "EP-653", "title": "Erdős Problem #653", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ and let $R(x_i)=\\#\\{ \\lvert x_j-x_i\\rvert : j\neq i\\}$, where the points are ordered such that $ R(x_1)\\leq \\cdots \\leq R(x_n). $ Let $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that $g(n) \\geq (1-o(1))n$?", "background": "Erd\\H{o}s and Fishburn proved $g(n)>\\frac{3}{8}n$ and Csizmadia proved $g(n)>\\frac{7}{10}n$. Both groups proved $g(n) < n-cn^{2/3}$ for some constant $c>0$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The number of distinct pinned-distance counts is not known to be (1-o(1))n.\n\n**Verified partial progress.**\n\n- Erdős--Fishburn prove g(n)>3n/8.\n- Csizmadia improves the lower bound to g(n)>7n/10.\n- Known configurations give g(n)0.\n\n**Full solution or refutation.**\n\nNo later improvement or resolution was located; formalization of the statement does not alter its status.\n\n**What remains.**\n\nRaise the lower constant toward 1 or improve the upper obstruction sufficiently to refute the conjectured ratio.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #653, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/653\n Evidence used: Maintains open status and records the 7/10 lower bound and n-cn^(2/3) upper obstruction.\n\n**Review notes.** The imported statement has a literal newline splitting the intended neq command; it is preserved in the report.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2234, "problem_number": "EP-654", "title": "Erdős Problem #654", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ with no four points on a circle. Must there exist some $x_i$ with at least $(1-o(1))n$ distinct distances to other $x_i$?", "background": "It is clear that every point has at least $\\frac{n-1}{3}$ distinct distances to other points in the set.\nIn \\cite{Er87b} and \\cite{ErPa90} Erd\\H{o}s and Pach ask this under the additional assumption that there are no three points on a line (so that the points are in general position), although they only ask the weaker question whether there is a lower bound of the shape $(\\tfrac{1}{3}+c)n$ for some constant $c>0$.\nThey suggest the lower bound $(1-o(1))n$ is true under the assumption that any circle around a point $x_i$ contains at most $2$ other $x_j$.\nReferences\n\n\n[Er87b] Erd\\H{o}s, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\n\n[ErPa90] Erd\\H{o}s, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The exact (1-o(1))n pinned-distance conjecture is false under the stated no-four-concyclic hypothesis.\n\n**Verified partial progress.**\n\n- Aletheia constructs n=4m points on two axes using signed powers of 2 and 3, with no four points concyclic.\n- Every point in the construction determines at most 3m-1, hence fewer than 3n/4, distinct distances.\n- The correct extremal function and the weaker (1/3+c)n question remain open; the construction is not in general position.\n\n**Full solution or refutation.**\n\nAn explicit infinite family bounded away from n refutes the source conjecture.\n\n**What remains.**\n\nEstimate the correct f(n), decide whether f(n)>(1/3+c)n, and separately study the no-three-collinear general-position variant.\n\n**Sources checked.**\n\n- T. Feng et al., Towards Autonomous Mathematics Research, arXiv:2602.10177 (2026), with associated Aletheia Erdős solutions. (primary): https://arxiv.org/abs/2602.10177\n Evidence used: Primary research report presenting the semi-autonomous Erdős audit and expert evaluation.\n- Google DeepMind superhuman repository, Aletheia Erdős solutions, Erdős-654 proof. (primary): https://github.com/google-deepmind/superhuman/blob/main/aletheia/Erdos/Erdos.tex\n Evidence used: Contains the full explicit two-axis construction and proof of the 3n/4 pinned-distance upper bound.\n- Thomas F. Bloom, Erdős Problem #654, checked 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/654\n Evidence used: Confirms the strongest form is disproved and distinguishes the remaining weaker and general-position questions.\n\n**Review notes.** The source repeats x_i where the target index is grammatically expected to vary; this is flagged, not silently edited.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2235, "problem_number": "EP-655", "title": "Erdős Problem #655", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least $ (1+c)\\frac{n}{2} $ distinct distances determined between the $x_i$, for some constant $c>0$ and all $n$ sufficiently large?", "background": "A problem of Erd\\H{o}s and Pach. It is easy to see that this assumption implies that there are at least $\\frac{n-1}{2}$ distinct distances determined by every point.\nZach Hunter has observed that taking $n$ points equally spaced on a circle disproves this conjecture. In the spirit of related conjectures of Erd\\H{o}s and others, presumably some kind of assumption that the points are in general position (e.g. no three on a line and no four on a circle) was intended.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The literal formulation is refuted by equally spaced points on a circle, but the intended missing general-position condition is unknown.\n\n**Verified partial progress.**\n\n- The hypotheses imply a per-point lower bound (n-1)/2.\n- The regular n-gon meets the literal hypotheses and defeats any fixed positive c.\n\n**Full solution or refutation.**\n\nThe source itself identifies the formulation as ambiguous; no repair is made.\n\n**What remains.**\n\nRecover an authoritative intended statement before re-triage.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #655, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/655\n Evidence used: Records the literal counterexample and ambiguity.\n\n**Review notes.** No OCR/formulation correction was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2236, "problem_number": "EP-657", "title": "Erdős Problem #657", "statement": "Is it true that if $A\\subset \\mathbb{R}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$ has no isosceles triangles) then $A$ must determine at least $f(n)n$ distinct distances, for some $f(n)\\to \\infty$?", "background": "In \\cite{Er73} Erd\\H{o}s attributes this problem (more generally in $\\mathbb{R}^k$) to himself and Davies. In \\cite{Er97e} he does not mention Davis, but says this problem was investigated by himself, F\"{u}redi, Ruzsa, and Pach.\nIn \\cite{Er73} Erd\\H{o}s says it is not even known in $\\mathbb{R}$ whether $f(n)\\to \\infty$. Sarosh Adenwalla has observed that this is equivalent to minimising the number of distinct differences in a set $A\\subset \\mathbb{R}$ of size $n$ without three-term arithmetic progressions. Dumitrescu \\cite{Du08} proved that, in these terms, $ (\\log n)^c \\leq f(n) \\leq 2^{O(\\sqrt{\\log n})} $ for some constant $c>0$.\nHunter observed in the comments that a result of Ruzsa coupled with standard tools of additive combinatorics (with details given by Alfaiz and Tang) allow recent progress on the size of subsets without three-term arithmetic progression (see \\cite{BlSi23} which improves slightly on the bounds due to Kelley and Meka \\cite{KeMe23}) yield $ 2^{c(\\log n)^{1/9}}\\leq f(n) $ for some constant $c>0$.\nStraus has observed that if $2^k\\geq n$ then there exist $n$ points in $\\mathbb{R}^k$ which contain no isosceles triangle and determine at most $n-1$ distances.\nSee also [135].\nReferences\n\n\n[BlSi23] T. F. Bloom and O. Sisask, An improvement to the Kelley-Meka bounds on three-term arithmetic progressions. arXiv:2309.02353 (2023).\n\n[Du08] Dumitrescu, Adrian, On distinct distances and {$\\lambda$}-free point sets. Discrete Math. (2008), 6533--6538.\n\n[Er73] Erd\\H{o}s, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n\n[Er97e] Erd\\H{o}s, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The no-isosceles distinct-distance problem is open but has growing lower bounds through the no-3AP reduction.\n\n**Verified partial progress.**\n\n- Dumitrescu obtained polylogarithmic lower growth.\n- Recent no-3AP bounds yield 2^(c(log n)^(1/9)) lower growth in the one-dimensional reduction.\n\n**Full solution or refutation.**\n\nThe desired general factor f(n) tending to infinity is not known in the plane.\n\n**What remains.**\n\nProve a stronger universal lower bound or settle the 1D equivalent sharply.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #657, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/657\n Evidence used: Current open status and reduction/bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2237, "problem_number": "EP-660", "title": "Erdős Problem #660", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least $ (1-o(1))\\frac{n}{2} $ many distinct distances between the $x_i$?", "background": "For the similar problem in $\\mathbb{R}^2$ there are always at least $n/2$ distances, as proved by Altman \\cite{Al63} (see [93]). In \\cite{Er75f} Erd\\H{o}s claims that Altman proved that the vertices determine $\\gg n$ many distinct distances, but gives no reference.\nReferences\n\n\n[Al63] Altman, E., On a problem of P. Erd\\H{o}s. Amer. Math. Monthly (1963), 148-157.\n\n[Er75f] Erd\\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The intended convex-polyhedron distance assertion is ambiguous; literal nearby readings are either trivial or false.\n\n**Verified partial progress.**\n\n- The planar analogue has an n/2 lower bound.\n\n**Full solution or refutation.**\n\nThe tracker documents incompatible readings of the original wording.\n\n**What remains.**\n\nRecover a definitive original formulation before judging status.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #660, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/660\n Evidence used: Documents the ambiguity and candidate readings.\n\n**Review notes.** No formulation was silently repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2238, "problem_number": "EP-661", "title": "Erdős Problem #661", "statement": "Are there, for all large $n$, some points $x_1,\\ldots,x_n,y_1,\\ldots,y_n\\in \\mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$ is $ o\\left(\\frac{n}{\\sqrt{\\log n}}\\right)? $ ", "background": "One can also ask this for points in $\\mathbb{R}^3$. In $\\mathbb{R}^4$ Lenz observed that there are $x_1,\\ldots,x_n,y_1,\\ldots,y_n\\in \\mathbb{R}^4$ such that $d(x_i,y_j)=1$ for all $i,j$, taking the points on two orthogonal circles.\nMore generally, if $F(2n)$ is the minimal number of such distances, and $f(2n)$ is minimal number of distinct distances between any $2n$ points in $\\mathbb{R}^2$, then is $F =o(f)$?\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The bipartite cross-distance construction problem remains open in the plane.\n\n**Verified partial progress.**\n\n- The analogous R4 construction has all cross-distances equal.\n\n**Full solution or refutation.**\n\nNo planar construction of the requested o(n/sqrt(log n)) size was located.\n\n**What remains.**\n\nConstruct such planar pairs or prove a lower bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #661, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/661\n Evidence used: Current maintained open status.\n\n**Review notes.** The corrected F=o(f) comment is background only.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2239, "problem_number": "EP-662", "title": "Erdős Problem #662", "statement": "Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\\leq t$. For example $f(1)=6$, $f(\\sqrt{3})=12$, and $f(3)=18$.\nLet $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that $d(x_i,x_j)\\geq 1$ for all $i\neq j$. Is it true that, provided $n$ is sufficiently large depending on $t$, the number of distances $d(x_i,x_j)\\leq t$ is less than or equal to $f(t)$ with equality perhaps only for the triangular lattice?\nIn particular, is it true that the number of distances $\\leq \\sqrt{3}-\\epsilon$ is less than $1$?", "background": "A problem of Erd\\H{o}s, Lov\\'{a}sz, and Vesztergombi.\nThis is essentially verbatim the problem description in \\cite{Er97e}, but this does not make sense as written; there must be at least one typo. Suggestions about what this problem intends are welcome.\nErd\\H{o}s also goes on to write 'Perhaps the following stronger conjecture holds: Let $t_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported/source statement contains an impossible 'less than 1 distance' clause and lacks a recoverable intended formulation.\n\n**Verified partial progress.**\n\n- The tracker records that at least one typo is present and multiple proposed variants exist.\n\n**Full solution or refutation.**\n\nNo mathematical resolution is assigned to an unrecoverable statement.\n\n**What remains.**\n\nObtain an authoritative corrected statement, preserving the original separately.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #662, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/662\n Evidence used: Explicitly flags the original-source ambiguity.\n\n**Review notes.** No OCR/formulation correction was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2240, "problem_number": "EP-663", "title": "Erdős Problem #663", "statement": "Let $k\\geq 2$ and $q(n,k)$ denote the least prime which does not divide $\\prod_{1\\leq i\\leq k}(n+i)$. Is it true that, if $k$ is fixed and $n$ is sufficiently large, we have $ q(n,k)<(1+o(1))\\log n? $ ", "background": "A problem of Erd\\H{o}s and Pomerance.\nThe bound $q(n,k)<(1+o(1))k\\log n$ is easy. It may be true this improved bound holds even up to $k=o(\\log n)$.\nA heuristic argument in favour of this is provided by Tao in the comments.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The sharp logarithmic least-excluded-prime bound is open; a factor-k version is elementary.\n\n**Verified partial progress.**\n\n- q(n,k)<(1+o(1))k log n is known for fixed k.\n- Tao records a heuristic supporting the sharper constant 1.\n\n**Full solution or refutation.**\n\nThe heuristic is not a proof.\n\n**What remains.**\n\nRemove the factor k rigorously.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #663, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/663\n Evidence used: Current open status and known easy bound.\n\n**Review notes.** Heuristic separated from theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2241, "problem_number": "EP-665", "title": "Erdős Problem #665", "statement": "A pairwise balanced design for $\\{1,\\ldots,n\\}$ is a collection of sets $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ such that $2\\leq \\lvert A_i\\rvert 0$ and, for all large $n$, a pairwise balanced design such that $ \\lvert A_i\\rvert > n^{1/2}-C $ for all $1\\leq i\\leq m$?", "background": "A problem of Erd\\H{o}s and Larson \\cite{ErLa82}. In general, as Erd\\H{o}s asks in \\cite{Er97f}, find the slowest growing function $h$ such that, for all large $n$, there exists a pairwise balanced design with $ \\lvert A_i\\rvert > n^{1/2}-h(n) $ for all $1\\leq i\\leq m$.\nThe problem above asks whether $h(n)\\ll 1$. Erd\\H{o}s and Larson prove that $h(n) \\ll n^{1/2-c}$ for some constant $c>0$, and note this can be improved to $h(n)\\ll (\\log n)^2$ assuming Cramer-type bounds on the difference between consecutive primes.\nShrikhande and Singhi \\cite{ShSi85} have proved that the answer is no conditional on the conjecture that the order of every projective plane is a prime power (see [723]), by proving that every pairwise balanced design on $n$ points in which each block is of size $\\geq n^{1/2}-c$ can be embedded in a projective plane of order $n+i$ for some $i\\leq c+2$, if $n$ is sufficiently large.\nIn general, if $H(n)$ is the largest prime gap $\\leq n$, then the above reuslts show that, assuming the prime power conjecture, $H(n)\\asymp h(n)$.\nReferences\n\n\n[Er97f] Erd\\H{o}s, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.\n\n[ErLa82] Erd\\H{o}s, P. and Larson, J., On pairwise balanced block designs with the sizes of blocks as uniform as possible. Annals of Discrete Mathematics (1982), 129-134.\n\n[ShSi85] S. S. Shrikhande and N. M. Singhi, On a problem of Erd\\H{o}s and Larson. Combinatorica (1985), 351-358.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The constant-error design problem is open, with unconditional power-saving error and conditional logarithmic/error obstructions.\n\n**Verified partial progress.**\n\n- Erdős--Larson prove h(n)<", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2242, "problem_number": "EP-667", "title": "Erdős Problem #667", "statement": "Let $p,q\\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices spans at least $q$ edges must contain a complete graph on $m$ vertices.\nIs $ c(p,q)=\\liminf \\frac{\\log H(n)}{\\log n} $ a strictly increasing function of $q$ for $1\\leq q\\leq \\binom{p-1}{2}+1$?", "background": "A problem of Erd\\H{o}s, Faudree, Rousseau, and Schelp.\nWhen $q=1$ this corresponds exactly to the classical Ramsey problem, and hence for example $ \\frac{1}{p-1}\\leq c(p,1) \\leq \\frac{2}{p+1}. $ It is easy to see that if $q=\\binom{p-1}{2}+1$ then $c(p,q)=1$. Erd\\H{o}s, Faudree, Rousseau, and Schelp have shown that $c(p,\\binom{p-1}{2})\\leq 1/2$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Strict monotonicity of c(p,q) remains open; the tracker discussion flags an erroneous historical endpoint estimate and supplies replacement bounds.\n\n**Verified partial progress.**\n\n- Monotonic nondecrease is immediate from the definition.\n- A discussion derives improved endpoint lower/upper estimates.\n\n**Full solution or refutation.**\n\nEndpoint estimates do not prove strict increase for every q.\n\n**What remains.**\n\nProve strictness or find equal adjacent values.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #667, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/667\n Evidence used: Maintained open status.\n- EP-667 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/667\n Evidence used: Unverified but detailed correction to a displayed prior bound.\n\n**Review notes.** The disputed bound is not repeated as fact.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2243, "problem_number": "EP-668", "title": "Erdős Problem #668", "statement": "Is it true that the number of incongruent sets of $n$ points in $\\mathbb{R}^2$ which maximise the number of unit distances tends to infinity as $n\\to\\infty$? Is it always $>1$ for $n>3$?", "background": "In fact this is $=1$ also for $n=4$, the unique example given by two equilateral triangles joined by an edge.\nComputational evidence of Engel, Hammond-Lee, Su, Varga, and Zs\\'{a}mboki \\cite{EHSVZ25} and Alexeev, Mixon, and Parshall \\cite{AMP25} suggests that this count is $=1$ for various other $5\\leq n\\leq 21$ (although these calculations were checking only up to graph isomorphism, rather than congruency).\nThe actual maximal number of unit distances is the subject of [90].\nReferences\n\n\n[AMP25] B. Alexeev, D. Mixon, and H. Parshall, The Erd\\H{o}s unit distance problem for small point sets. arXiv:2412.11914 (2025).\n\n[EHSVZ25] P. Engel, O. Hammond-Lee, Y. Su, D. Varga, and P. Zs\\'{a}mboki, Diverse beam search to find densest-known planar unit distance graphs. arXiv:2406.15317 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The assertion of more than one extremizer for every n>3 is false at n=4; the asymptotic number of congruence classes remains open.\n\n**Verified partial progress.**\n\n- The n=4 extremizer is unique.\n- Computations through n=21 give graph-isomorphism evidence for further uniqueness cases.\n\n**Full solution or refutation.**\n\nFinite uniqueness evidence does not settle the asymptotic question.\n\n**What remains.**\n\nDetermine multiplicity for general n, with congruence rather than graph-isomorphism control.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #668, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/668\n Evidence used: Current open status, n=4 counterexample, and computation caveat.\n\n**Review notes.** Separate subquestions are distinguished.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2244, "problem_number": "EP-669", "title": "Erdős Problem #669", "statement": "Let $F_k(n)$ be minimal such that for any $n$ points in $\\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ of the points, and $f_k(n)$ similarly but with lines passing through exactly $k$ points.\nEstimate $f_k(n)$ and $F_k(n)$ - in particular, determine $\\lim F_k(n)/n^2$ and $\\lim f_k(n)/n^2$.", "background": "Trivially $f_k(n)\\leq F_k(n)$ and $f_2(n)=F_2(n)=\\binom{n}{2}$. The problem with $k=3$ is the classical 'Orchard problem' of Sylvester. Burr, Gr\"{u}nbaum, and Sloane \\cite{BGS74} have proved that $ f_3(n)=\\frac{n^2}{6}-O(n) $ and $ F_3(n)=\\frac{n^2}{6}-O(n). $ There is a trivial upper bound of $F_k(n) \\leq \\binom{n}{2}/\\binom{k}{2}$, and hence $ \\lim F_k(n)/n^2 \\leq \\frac{1}{k(k-1)}. $ See also [101].\nReferences\n\n\n[BGS74] Burr, Stefan A. and Gr\"{u}nbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The k=3 orchard asymptotics are known; the general k limits remain open.\n\n**Verified partial progress.**\n\n- Burr--Grünbaum--Sloane prove f_3(n) and F_3(n) are n^2/6 up to O(n).\n- A general pair-counting upper bound is known.\n\n**Full solution or refutation.**\n\nNo general determination of f_k or F_k is known.\n\n**What remains.**\n\nDetermine the k>=4 limits.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #669, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/669\n Evidence used: Current open status and k=3 asymptotics.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2245, "problem_number": "EP-670", "title": "Erdős Problem #670", "statement": "Let $A\\subseteq \\mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$. Is the diameter of $A$ at least $(1+o(1))n^2$?", "background": "The lower bound of $\\binom{n}{2}$ for the diameter is trivial. Erd\\H{o}s \\cite{Er97f} proved the claim when $d=1$.\nReferences\n\n\n[Er97f] Erd\\H{o}s, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A Lean-formalized primary construction disproves the conjecture when dimension may grow with n, but the likely intended fixed-dimension formulation remains open.\n\n**Verified partial progress.**\n\n- Erdős proved the asymptotic lower bound in dimension one.\n- Ho constructs n=q+1 points in dimension q^2+q with separated distances and diameter at most (1-1/pi^2+o(1))n^2.\n\n**Full solution or refutation.**\n\nHo's construction is a full disproof of the dimension-varying reading. It does not address the reading in which d is fixed before n tends to infinity, which the maintained tracker identifies as probably intended.\n\n**What remains.**\n\nClarify the source quantifier order and, under the fixed-dimension reading, prove or refute the conjecture already for d=2.\n\n**Sources checked.**\n\n- Boon Suan Ho, Erdős's diameter conjecture for separated distances fails in high dimensions, arXiv:2604.15305 (2026). (primary): https://arxiv.org/abs/2604.15305\n Evidence used: Primary high-dimensional counterexample; the abstract also reports a complete Lean 4 formalization.\n- Thomas F. Bloom, Erdős Problem #670, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/670\n Evidence used: Explains the quantifier ambiguity, records Ho's result, and retains open status for the likely fixed-d interpretation.\n\n**Review notes.** The exact statement was preserved. Its unstated fixed-versus-growing dimension convention controls whether the result is open or refuted. The common trailing background corruption was flagged only.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2246, "problem_number": "EP-671", "title": "Erdős Problem #671", "statement": "Given $a_{i}^n\\in [-1,1]$ for all $1\\leq i\\leq n<\\infty$ we define $p_{i}^n$ as the unique polynomial of degree $n-1$ such that $p_{i}^n(a_{i}^n)=1$ and $p_{i}^n(a_{i'}^n)=0$ if $1\\leq i'\\leq n$ with $i\neq i'$. We similarly define $ \\mathcal{L}^nf(x) = \\sum_{1\\leq i\\leq n}f(a_i^n)p_i^n(x), $ the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i^n$ for $1\\leq i\\leq n$ (that is, the sequence of Lagrange interpolation polynomials).", "background": "Is there such a sequence of $a_i^n$ such that for every continuous $f:[-1,1]\\to \\mathbb{R}$ there exists some $x\\in [-1,1]$ where $ \\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty $ and yet $ \\mathcal{L}^nf(x) \\to f(x)? $ Is there such a sequence such that $ \\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty $ for every $x\\in [-1,1]$ and yet for every continuous $f:[-1,1]\\to \\mathbb{R}$ there exists $x\\in [-1,1]$ with $ \\mathcal{L}^nf(x) \\to f(x)? $ \nBernstein \\cite{Be31} proved that for any choice of $a_i^n$ there exists $x_0\\in [-1,1]$ such that $ \\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty. $ Erd\\H{o}s and V\\'{e}rtesi \\cite{ErVe80} proved that for any choice of $a_i^n$ there exists a continuous $f:[-1,1]\\to \\mathbb{R}$ such that $ \\limsup_{n\\to \\infty} \\lvert \\mathcal{L}^nf(x)\\rvert=\\infty $ for almost all $x\\in [-1,1]$.\nReferences\n\n\n[Be31] S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degr\\'{e} n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.\n\n[ErVe80] Erd\\H{o}s, P. and V\\'{e}rtesi, P., On the almost everywhere divergence of Lagrange\ninterpolatory polynomials for arbitrary system of nodes. Acta Math. Acad. Sci. Hungar. (1980), 71-89.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The classical source explicitly leaves the existence question unproved; a six-day-old forum solution claim has no stable independently verified source located in this search.\n\n**Verified partial progress.**\n\n- Erdős--Vértesi prove that every triangular node array admits a continuous function whose Lagrange interpolants diverge almost everywhere.\n- The same paper states that the earlier proof of the first EP-671 existence claim was probably incomplete.\n\n**Full solution or refutation.**\n\nNo verified solution of either existence question was located. The current forum index lists EP-671 among fresh solution claims, but no public manuscript, formal proof, or sufficiently detailed independently checked result was available for promotion.\n\n**What remains.**\n\nEvaluate the new claim and supply a stable proof; otherwise construct a node array satisfying either of the two convergence-versus-Lebesgue-divergence properties.\n\n**Sources checked.**\n\n- Paul Erdős and Péter Vértesi, On the almost everywhere divergence of Lagrange interpolatory polynomials for arbitrary systems of nodes, Acta Math. Acad. Sci. Hungar. 36 (1980), 71-89. (primary): https://doi.org/10.1007/BF01897094\n Evidence used: Proves almost-everywhere divergence and explicitly describes the relevant positive existence claim as unproved.\n- Thomas F. Bloom, Erdős Problem #671, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/671\n Evidence used: Full intended two-question formulation and maintained open status.\n- Erdős Problems forum thread for Problem #671, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/671\n Evidence used: Current forum index identifies a very recent solution claim; it is retained only as a claim requiring review.\n\n**Review notes.** The imported statement contains only definitions and moves the two questions into background; its neq escape became a newline plus 'eq', and distinctness of nodes is implicit. These defects and the common trailing fragment were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2247, "problem_number": "EP-675", "title": "Erdős Problem #675", "statement": "We say that $A\\subset \\mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\\geq 1$ such that, for all $1\\leq a\\leq n$, $ a\\in A\\quad\\textrm{ if and only if }\\quad a+t_n\\in A. $ {UL}", "background": "{LI}Does the set of the sums of two squares have the translation property?{/LI}\n{LI}If we partition all primes into $P\\sqcup Q$, such that each set contains $\\gg x/\\log x$ many primes $\\leq x$ for all large $x$, then can the set of integers only divisible by primes from $P$ have the translation property?{/LI}\n{LI}If $A$ is the set of squarefree numbers then how fast does the minimal such $t_n$ grow? Is it true that $t_n>\\exp(n^c)$ for some constant $c>0$?{/LI}\n{/UL}\nElementary sieve theory implies that the set of squarefree numbers has the translation property.\nMore generally, Brun's sieve can be used to prove that if $B\\subseteq \\mathbb{N}$ is a set of pairwise coprime integers with $\\sum_{b\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A publicly checked argument proves the requested exponential lower bound for squarefree local periods, while the sums-of-two-squares and prime-partition translation questions remain open.\n\n**Verified partial progress.**\n\n- Elementary sieve theory proves that squarefree integers have the translation property.\n- Ho Boon Suan's note proves t_n>exp(n^c) for every fixed c<25/72 and all sufficiently large n; the tracker records standard checks and a positive manual assessment.\n- A separate public note gives an exponential lower bound for any hypothetical local period of the sums-of-two-squares set without proving that the translation property exists.\n\n**Full solution or refutation.**\n\nThe third listed question has an affirmative answer at the requested qualitative scale via a forum-linked proof using Nunes's least-squarefree-residue-class bound. The two structural translation-property questions remain unresolved.\n\n**What remains.**\n\nProve or refute the translation property for sums of two squares and for the prime-partition construction, and place the squarefree minimal period more sharply.\n\n**Sources checked.**\n\n- Ramon M. Nunes, On the least squarefree number in an arithmetic progression, Mathematika 63 (2017), 483-498. (primary): https://doi.org/10.1112/S0025579317000043\n Evidence used: Primary least-squarefree-residue-class estimate used in the local-period argument.\n- Erdős Problems forum thread for Problem #675, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/675\n Evidence used: Contains the squarefree-period proof claim, independent checks, unresolved parts, and correction of the self-referential sieve formula.\n\n**Review notes.** The suggested c<3/8 improvement was excluded because the cited Zhong--Zhang theorem does not cover modulus p^2. Raw UL/LI markup, a lost nmid escape, a self-reference A-for-B, and the common trailing corruption were preserved and flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2248, "problem_number": "EP-676", "title": "Erdős Problem #676", "statement": "Is every sufficiently large integer of the form $ ap^2+b $ for some prime $p$ and integer $a\\geq 1$ and $0\\leq b0$. Erd\\H{o}s \\cite{Er79} believed it is 'rather unlikely' that all large integers are of this form.\nWhat if the condition that $p$ is prime is omitted? Selfridge and Wagstaff made a 'preliminary computer search' and suggested that there are infinitely many $n$ not of this form even without the condition that $p$ is prime. It should be true that the number of exceptions in $[1,x]$ is $\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Almost all integers are representable, and large explicit exceptions are reported, but it is unknown whether exceptions occur infinitely often.\n\n**Verified partial progress.**\n\n- Classical sieve methods show that the exceptional count up to x is at most x/(log x)^c for some c>0.\n- The representation is equivalent to the existence of a prime p<=sqrt(n) with n modulo p^2 less than p.\n- The maintained discussion reports explicit exceptions above 10^16, which cannot decide an eventual statement.\n\n**Full solution or refutation.**\n\nNo theorem proving eventual representability or infinitely many exceptions was located; finite computation cannot resolve the quantifier 'every sufficiently large integer'.\n\n**What remains.**\n\nProve that only finitely many exceptions exist or construct infinitely many exceptions, and sharpen the exceptional-set bound.\n\n**Sources checked.**\n\n- Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine 52 (1979), 67-70. (primary): https://doi.org/10.1080/0025570X.1979.11976756\n Evidence used: Original-source formulation and heuristic skepticism.\n- Thomas F. Bloom, Erdős Problem #676 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/676\n Evidence used: Current open status, sieve bounds, equivalent remainder criterion, and links to reported computations.\n\n**Review notes.** Reported finite computations were not rerun or treated as a proof. The common trailing background corruption was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2249, "problem_number": "EP-677", "title": "Erdős Problem #677", "statement": "Let $M(n,k)=[n+1,\\ldots,n+k]$ be the least common multiple of $\\{n+1,\\ldots,n+k\\}$.\nIs it true that for all $m\\geq n+k$ $ M(n,k) \neq M(m,k)? $ ", "background": "The Thue-Siegel theorem implies that, for fixed $k$, there are only finitely many $m,n$ such that $m\\geq n+k$ and $M(n,k)=M(m,k)$.\nIn general, how many solutions does $M(n,k)=M(m,l)$ have when $m\\geq n+k$ and $l>1$? Erd\\H{o}s expects very few (and none when $l\\geq k$).\nThe only solutions Erd\\H{o}s knew were $M(4,3)=M(13,2)$ and $M(3,4)=M(19,2)$.\nIn \\cite{Er79d} Erd\\H{o}s conjectures the stronger fact that (aside from a finite number of exceptions) if $k>2$ and $m\\geq n+k$ then $\\prod_{i\\leq k}(n+i)$ and $\\prod_{i\\leq k}(m+i)$ cannot have the same set of prime factors.\nSee also [678], [686], and [850].\nThis is discussed in problem B35 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er79d] Erd\\H{o}s, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** For fixed block length only finitely many separated equal-LCM pairs can occur, but the conjecture that no same-length pair occurs remains open.\n\n**Verified partial progress.**\n\n- A Thue--Siegel argument gives finiteness of solutions for each fixed k.\n- Elementary divisibility gives finiteness for fixed displacement and block length, which is weaker than the conjecture.\n- Additional examples concern unequal block lengths and do not refute the same-length assertion.\n\n**Full solution or refutation.**\n\nNo verified equality M(n,k)=M(m,k) with m>=n+k and no proof excluding all such equalities was located.\n\n**What remains.**\n\nProve uniform noncollision for equal block lengths or exhibit a counterexample; stronger same-prime-support variants also remain open.\n\n**Sources checked.**\n\n- Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine 52 (1979), 67-70. (primary): https://combinatorica.hu/~p_erdos/1979-22.pdf\n Evidence used: Original conjecture and early unequal-length examples.\n- Thomas F. Bloom, Erdős Problem #677, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/677\n Evidence used: Current open status, fixed-k finiteness, and related variants.\n\n**Review notes.** The imported neq escape became a newline followed by 'eq'; the inequality was not repaired in the source. The common trailing background fragment was also preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2250, "problem_number": "EP-679", "title": "Erdős Problem #679", "statement": "Let $\\epsilon>0$ and $\\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that $ \\omega(n-k) < (1+\\epsilon)\\frac{\\log k}{\\log\\log k} $ for all $k0$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The stronger additive-constant version is disproved, and Lau proves a C log k theorem for infinitely many n; the main near-optimal threshold remains open.\n\n**Verified partial progress.**\n\n- A primorial construction shows that for all large n some k", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2251, "problem_number": "EP-680", "title": "Erdős Problem #680", "statement": "Is it true that, for all sufficiently large $n$, there exists some $k$ such that $ p(n+k)>k^2+1, $ where $p(m)$ denotes the least prime factor of $m$?\nCan one prove this is false if we replace $k^2+1$ by $e^{(1+\\epsilon)\\sqrt{k}}+C_\\epsilon$, for all $\\epsilon>0$, where $C_\\epsilon>0$ is some constant?", "background": "This follows from 'plausible assumptions on the distribution of primes' (as does the question with $k^2$ replaced by $k^d$ for any $d$); the challenge is to prove this unconditionally.\nErd\\H{o}s observed that Cramer's conjecture $ \\limsup_{k\\to \\infty} \\frac{p_{k+1}-p_k}{(\\log k)^2}=1 $ implies that for all $\\epsilon>0$ and all sufficiently large $n$ there exists some $k$ such that $ p(n+k)>e^{(1-\\epsilon)\\sqrt{k}}. $ There is now evidence, however, that Cramer's conjecture is false; a more refined heuristic by Granville \\cite{Gr95} suggests this $\\limsup$ is $2e^{-\\gamma}\\approx 1.119\\cdots$, and so perhaps the $1+\\epsilon$ in the second question should be replaced by $2e^{-\\gamma}+\\epsilon$.\nSee also [681] and [682].\nReferences\n\n\n[Gr95] Granville, Andrew, Harald {C}ram\\'{e}r and the distribution of prime numbers. Scand. Actuar. J. (1995), 12--28.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Neither the unconditional k^2+1 least-prime-factor witness nor the proposed negation at an exponential square-root threshold has been proved.\n\n**Verified partial progress.**\n\n- Strong conjectures on prime gaps imply substantially stronger witness statements.\n- Granville's refinement of Cramér's model changes the expected critical exponential constant from 1 to 2e^(-gamma), but remains heuristic.\n\n**Full solution or refutation.**\n\nNo unconditional theorem resolving either displayed question was located.\n\n**What remains.**\n\nProve a polynomial-threshold witness for every sufficiently large n and determine the correct exponential-square-root obstruction constant.\n\n**Sources checked.**\n\n- Andrew Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial Journal (1995), 12-28. (primary): https://doi.org/10.1080/03461238.1995.10413946\n Evidence used: Primary source for the refined prime-gap heuristic, not an unconditional solution.\n- Thomas F. Bloom, Erdős Problem #680, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/680\n Evidence used: Current open status, exact two-part formulation, and conditional context.\n\n**Review notes.** Heuristics and conditional implications are not promoted to partial proofs. The common trailing background corruption was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2252, "problem_number": "EP-681", "title": "Erdős Problem #681", "statement": "Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and $ p(n+k)>k^2, $ where $p(m)$ is the least prime factor of $m$?", "background": "Related to questions of Erd\\H{o}s, Eggleton, and Selfridge. This may be true with $k^2$ replaced by $k^d$ for any $d$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The required rough composite is not known to exist for every sufficiently large n; the related almost-all theorem between consecutive primes does not imply it.\n\n**Verified partial progress.**\n\n- If n+1 is composite, k=1 is immediate, so the difficult origins are n=p-1 with p prime.\n- The tracker discussion outlines a dyadic short-interval sieve strategy and identifies the unresolved sieve estimate.\n- Gafni and Tao solve the related almost-all EP-682, which has weaker quantifiers and a different interval setup.\n\n**Full solution or refutation.**\n\nNo unconditional every-large-n result specific to the required composite witness was located.\n\n**What remains.**\n\nDevelop a short-interval sieve producing a composite n+k with no prime factor at most k^2 for every large prime-origin case.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #681 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/681\n Evidence used: Current open status, elementary reduction, and discussion of the sieve bottleneck.\n- Thomas F. Bloom, Erdős Problem #682, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/682\n Evidence used: Scope of the related Gafni--Tao almost-all result, carefully distinguished from EP-681.\n\n**Review notes.** A suggested strategy is recorded as such, not as a theorem. The common trailing background corruption was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2253, "problem_number": "EP-683", "title": "Erdős Problem #683", "statement": "Is it true that for every $1\\leq k\\leq n$ the largest prime divisor of $\\binom{n}{k}$, say $P(\\binom{n}{k})$, satisfies $ P\\left(\\binom{n}{k}\\right)\\geq \\min(n-k+1, k^{1+c}) $ for some constant $c>0$?", "background": "A theorem of Sylvester and Schur (see \\cite{Er34}) states that $P(\\binom{n}{k})>k$ if $k\\leq n/2$. Erd\\H{o}s \\cite{Er55d} proved that there exists some $c>0$ such that, whenever $k\\leq n/2$, $ P\\left(\\binom{n}{k}\\right)\\gg k\\log k. $ Erd\\H{o}s \\cite{Er79d} writes it 'seems certain' that this holds for every $c>0$, with only a finite number of exceptions (depending on $c$). Standard heuristics on prime gaps suggest that the largest prime divisor of $\\binom{n}{k}$ is, for $k\\leq n/2$, in fact $ >e^{c\\sqrt{k}} $ for some constant $c>0$.\nThis is essentially equivalent to [961].\nReferences\n\n\n[Er34] Erd\\H{o}s, Paul, A {T}heorem of {S}ylvester and {S}chur. J. London Math. Soc. (1934), 282--288.\n\n[Er55d] Erd\\H{o}s, P., On consecutive integers. Nieuw Arch. Wisk. (3) (1955), 124--128.\n\n[Er79d] Erd\\H{o}s, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The largest-prime-factor lower bound of power size k^(1+c) remains open; the maintained general theorem is only of order k log k.\n\n**Verified partial progress.**\n\n- Sylvester--Schur proves P(binomial(n,k))>k for k<=n/2.\n- Erdős improves this to P(binomial(n,k)) much greater than k log k for k<=n/2.\n- The conjecture is essentially equivalent to maintained Problem #961, but that equivalence does not improve the bound.\n\n**Full solution or refutation.**\n\nNo primary source establishing a fixed positive power saving over k uniformly in the displayed range was located.\n\n**What remains.**\n\nUpgrade the logarithmic factor to k^c for some fixed c>0 while respecting the n-k+1 symmetry barrier, or produce a counterexample family.\n\n**Sources checked.**\n\n- Paul Erdős, On consecutive integers, Nieuw Archief voor Wiskunde (3) 3 (1955), 124-128. (primary): https://combinatorica.hu/~p_erdos/1955-04.pdf\n Evidence used: Classical primary source underlying the maintained k log k result.\n- Thomas F. Bloom, Erdős Problem #683, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/683\n Evidence used: Current open status, historical bounds, heuristic target, and equivalence to Problem #961.\n\n**Review notes.** The historical theorem scope was cross-checked against the maintained tracker. The common trailing background corruption was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2254, "problem_number": "EP-684", "title": "Erdős Problem #684", "statement": "For $0\\leq k\\leq n$ write $ \\binom{n}{k} = uv $ where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$.\nLet $f(n)$ be the smallest $k$ such that $u>n^2$. Give bounds for $f(n)$.", "background": "A classical theorem of Mahler states that for any $\\epsilon>0$ and integers $k$ and $l$ then, writing $ (n+1)\\cdots (n+k) = ab $ where the only primes dividing $a$ are $\\leq l$ and the only primes dividing $b$ are $>l$, we have $a < n^{1+\\epsilon}$ for all sufficiently large (depending on $\\epsilon,k,l$) $n$.\nMahler's theorem implies $f(n)\\to \\infty$ as $n\\to \\infty$, but is ineffective, and so gives no bounds on the growth of $f(n)$.\nOne can similarly ask for estimates on the smallest integer $f(n,k)$ such that if $m$ is the factor of $\\binom{n}{k}$ containing all primes $\\leq f(n,k)$ then $m > n^2$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Pointwise f(n)<=(1/2-o(1))log n.\n- Li proves f(n)=(2/(1-gamma)+o(1))log n for a density-one set of integers n.\n- An April 2026 unbounded-limsup preprint is not used because a later public audit reports a counterexample to a key lemma and no stable correction was located.\n\n**Full solution or refutation.**\n\nThe old ineffective information f(n)->infinity has been replaced by a uniform polylogarithmic upper bound and a sharp normal-order theorem. These do not determine exceptional worst-case behavior between logarithmic and squared-logarithmic scales.\n\n**What remains.**\n\nDetermine the worst-case order of f(n), including the true limsup scale, and resolve the disputed unbounded-logarithmic-limsup claim.\n\n**Sources checked.**\n\n- Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, and Gregory Valiant, Short proofs in combinatorics and number theory, arXiv:2603.29961 (2026). (primary): https://arxiv.org/abs/2603.29961\n Evidence used: Primary pointwise log-squared upper bound and logarithmic lower construction.\n- Eric Li, Erdős Problem 684 at Density One: Small-prime Parts of Binomial Coefficients and Gaussian Fluctuations, arXiv:2606.08216 (2026). (primary): https://arxiv.org/abs/2606.08216\n Evidence used: Primary exact density-one asymptotic and Gaussian fluctuation theorem.\n- Thomas F. Bloom, Erdős Problem #684, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/684\n Evidence used: Maintained open status and progression of pointwise bounds.\n- Open Math Problems Claimed to Be Solved with AI, Erdős Problem #684 correction audit, checked 2026-08-17. (source_collection): https://aimath.robertj1.com/\n Evidence used: Reports the deterministic counterexample to the claimed limsup preprint's key lemma; used only to withhold the disputed claim pending expert review.\n\n**Review notes.** The disputed preprint is neither accepted nor silently ignored; its claimed theorem is excluded pending correction. The common trailing background corruption was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2255, "problem_number": "EP-685", "title": "Erdős Problem #685", "statement": "Let $\\epsilon>0$ and $n$ be large depending on $\\epsilon$. Is it true that for all $n^\\epsilon\\frac{\\log \\binom{n}{k}}{\\log n}, $ and this inequality becomes (asymptotic) equality if $k>n^{1-o(1)}$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The full uniform binomial-prime-divisor asymptotic is open; a forum lead indicates a smooth-number proof in an upper subrange.\n\n**Verified partial progress.**\n\n- The elementary lower bound is asymptotically sharp when k>n^(1-o(1)).\n- A forum analysis reports a potential result for n^(17/30+delta)<=k<=n^(1-delta).\n\n**Full solution or refutation.**\n\nThe reported subrange is not an independently verified published resolution of the stated range.\n\n**What remains.**\n\nVerify the smooth-number deduction and cover all n^epsilon", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2256, "problem_number": "EP-686", "title": "Erdős Problem #686", "statement": "Can every integer $N\\geq 2$ be written as $ N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)} $ for some $k\\geq 2$ and $m\\geq n+k$?", "background": "If $n$ and $k$ are fixed then can one say anything about the set of integers so represented?\nSee also [677].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The universal representability question is open; cited Diophantine work gives finiteness information for fixed N and k>2.\n\n**Verified partial progress.**\n\n- Forum literature leads cite Beukers--Shorey--Tijdeman/Rakaczki finiteness for fixed parameters.\n\n**Full solution or refutation.**\n\nFixed-parameter finiteness does not show every N is represented.\n\n**What remains.**\n\nSettle universal representability or find an excluded N.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #686, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/686\n Evidence used: Current open status.\n- EP-686 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/686\n Evidence used: Bibliographic leads requiring direct verification.\n\n**Review notes.** Potential relaxed-sign constructions are outside the stated m,n domain.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2257, "problem_number": "EP-687", "title": "Erdős Problem #687", "statement": "Let $Y(x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_p$ for all primes $p\\leq x$ such that every integer in $[1,y]$ is congruent to at least one of the $a_p\\pmod{p}$.\nGive good estimates for $Y(x)$. In particular, can one prove that $Y(x)=o(x^2)$ or even $Y(x)\\ll x^{1+o(1)}$?", "background": "This function (associated with Jacobsthal) is closely related to the problem of gaps between primes (see [4]). The best known upper bound is due to Iwaniec \\cite{Iw78}, $ Y(x) \\ll x^2. $ The best lower bound is due to Ford, Green, Konyagin, Maynard, and Tao \\cite{FGKMT18}, $ Y(x) \\gg x\\frac{\\log x\\log\\log\\log x}{\\log\\log x}, $ improving on a previous bound of Rankin \\cite{Ra38}.\nMaier and Pomerance have conjectured that $Y(x)\\ll x(\\log x)^{2+o(1)}$.\nIn \\cite{Er80} he writes 'It is not clear who first formulated this problem - probably many of us did it independently. I offer the maximum of \\$1000 dollars and $1/2$ my total savings for clearing up of this problem.'\nIn \\cite{Er80} Erd\\H{o}s also asks about a weaker variant in which all except $o(y/\\log y)$ of the integers in $[1,y]$ are congruent to at least one of the $a_p\\pmod{p}$, and in particular asks if the answer is very different.\nSee also [688] and [689]. A more general Jacobsthal function is the focus of [970].\nReferences\n\n\n[Er80] Erd\\H{o}s, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.\n\n[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. (2018), 65-105.\n\n[Iw78] Iwaniec, Henryk, On the problem of {J}acobsthal. Demonstratio Math. (1978), 225--231.\n\n[Ra38] Rankin, R. A., The Difference between Consecutive Prime Numbers. J. London Math. Soc. (1938), 242-247.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Y(x) remains open between a quadratic upper and nearly x log x lower estimate.\n\n**Verified partial progress.**\n\n- Iwaniec proves Y(x)<>x log x log log log x/log log x.\n\n**Full solution or refutation.**\n\nNeither requested o(x^2) nor near-linear upper bound is known.\n\n**What remains.**\n\nBreak the quadratic barrier.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #687, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/687\n Evidence used: Current open status and bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2258, "problem_number": "EP-688", "title": "Erdős Problem #688", "statement": "Define $\\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\\epsilon_n}\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Whether epsilon_n tends to zero is open; Erdős's lower bound is known.\n\n**Verified partial progress.**\n\n- Erdős proves epsilon_n is at least a constant times log log log n/log log n.\n\n**Full solution or refutation.**\n\nThe lower bound tends to zero and does not decide the limit.\n\n**What remains.**\n\nProve an upper bound tending to zero or construct a positive lower limsup.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #688, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/688\n Evidence used: Current open status and lower bound.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2259, "problem_number": "EP-689", "title": "Erdős Problem #689", "statement": "Let $n$ be sufficiently large. Is there some choice of congruence class $a_p$ for all primes $2\\leq p\\leq n$ such that every integer in $[1,n]$ satisfies at least two of the congruences $\\equiv a_p\\pmod{p}$?", "background": "One can ask a similar question replacing $2$ by any fixed integer $r$ (provided $n$ is sufficiently large depending on $r$).\nSee also [687] and [688].\nThis problem (with $2$ replaced by $10$) is Problem 45 on Green's open problems list.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The maintained source remains open; multiple 2026 proof claims are explicitly awaiting human-expert or peer-reviewed verification.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nUnverified AI-assisted proofs are not evidence of a solved status.\n\n**What remains.**\n\nObtain expert verification or publication of a complete proof, and resolve endpoint-convention ambiguity.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #689, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/689\n Evidence used: Maintained open status.\n- EP-689 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/689\n Evidence used: Site moderator's explicit verification caveat.\n\n**Review notes.** No proof claim is integrated as fact.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2260, "problem_number": "EP-690", "title": "Erdős Problem #690", "statement": "Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$ (i.e. if $p_1\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Cambie's refereed paper proves unimodality for k=1,2,3 and non-unimodality for 4<=k<=20. A May 2026 AI-generated arXiv manuscript by Wang and Crapis claims and proves non-unimodality for every k>=4, with a public exact-arithmetic verifier and named independent readers, thereby completing the classification subject to expert review of this recent unrefereed extension.\n\n**Verified partial progress.**\n\n- Cambie proves that p maps to d_k(p) is unimodal for k=1,2,3.\n- Cambie proves non-unimodality for every 4<=k<=20 but explicitly does not prove the conjectured extension to all k>=4.\n- Wang and Crapis derive an exact first-difference criterion and cover all 4<=k<=8,600,001 using certified finite comparisons and published prime-gap records, then cover the remaining tail uniformly.\n\n**Full solution or refutation.**\n\nThe reported complete answer is unimodal exactly for k<=3 and non-unimodal for every k>=4. For k>=4, an exact symmetric-polynomial/prime-gap threshold is used to force a strict descent followed later by a strict ascent; the finite range uses certified calculations and the tail uses a Chinese-remainder composite block plus an average-gap argument.\n\n**What remains.**\n\nNo k-component remains if the May 2026 manuscript is correct. Its all-k extension still needs conventional peer review or full formal verification; without that manuscript the established published status would be only partial, with k>=21 unresolved.\n\n**Sources checked.**\n\n- Stijn Cambie, Resolution of Erdos' problems about unimodularity, Journal of Number Theory 280 (2026), 271-277, DOI 10.1016/j.jnt.2025.08.014; arXiv:2501.10333. (primary): https://arxiv.org/abs/2501.10333\n Evidence used: Peer-reviewed foundation proving unimodality for k<=3 and non-unimodality for 4<=k<=20; the paper explicitly leaves the all-k extension unproved.\n- Shouqiao Wang and Davide Crapis, A Complete Answer to Erdos Problem 690, arXiv:2605.08542v1 (2026). (primary): https://arxiv.org/abs/2605.08542\n Evidence used: States and develops a proof of non-unimodality for every k>=4, discloses AI generation, names independent proof readers, and describes the verification boundary.\n- Wang-Crapis companion numerical verifier for Erdos Problem 690 (2026). (source_collection): https://github.com/multiscalar/results/blob/main/erdos-690/numerical_verifier.py\n Evidence used: Publicly checks the finite prime sums, elementary symmetric-polynomial values, and logarithmic interval inequalities; it does not re-prove the cited analytic estimates or prime-record certificates.\n- Thomas F. Bloom, Erdos Problem #690 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/690\n Evidence used: Records SOLVED status and the discussion distinguishing Cambie's finite range from the later claimed all-k resolution.\n\n**Review notes.** The imported background has a leaked serialized suffix. The tracker badge alone is insufficient: the peer-reviewed Cambie result is component-limited, and closure of k>=21 depends on the recent AI-generated Wang-Crapis manuscript.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2261, "problem_number": "EP-691", "title": "Erdős Problem #691", "statement": "Given $A\\subseteq \\mathbb{N}$ let $M_A=\\{ n \\geq 1 : a\\mid n\\textrm{ for some }a\\in A\\}$ be the set of multiples of $A$. Find a necessary and sufficient condition on $A$ for $M_A$ to have density $1$.", "background": "A sequence $A$ for which $M_A$ has density $1$ is called a Behrend sequence.\nIf $A$ is a set of prime numbers (or, more generally, a set of pairwise coprime integers without $1$) then a necessary and sufficient condition is that $\\sum_{p\\in A}\\frac{1}{p}=\\infty$.\nThe general situation is more complicated. For example suppose $A$ is the union of $(n_k,(1+\\eta_k)n_k)\\cap \\mathbb{Z}$ where $1\\leq n_1\\alpha$.\nTenenbaum notes in \\cite{Te96} that this is certainly not true as written since if the $n_j$ grow sufficiently quickly then this sequence is never Behrend, for any choice of $\\eta_k$. He then writes 'we understand from subsequent discussions with Erd\\H{o}s that he had actually in mind a two-sided condition on' $n_{j+1}/n_j$.\nTenenbaum \\cite{Te96} proves this conjecture: if there are constants $1\\log 2$.\nReferences\n\n\n[Te96] Tenenbaum, G., On block {B}ehrend sequences. Math. Proc. Cambridge Philos. Soc. (1996), 355--367.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No general Behrend-sequence criterion is known, but a regular block-sequence threshold is proved.\n\n**Verified partial progress.**\n\n- For pairwise coprime A, divergence of reciprocal sum is necessary and sufficient.\n- Tenenbaum proves the beta=log 2 threshold under two-sided lacunarity for block sequences.\n\n**Full solution or refutation.**\n\nThe regular-block result does not give a criterion for arbitrary A.\n\n**What remains.**\n\nCharacterize density-one multiple sets in general.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #691, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/691\n Evidence used: Current open status and Tenenbaum theorem.\n\n**Review notes.** The original unqualified block conjecture is flagged as false as written.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2262, "problem_number": "EP-693", "title": "Erdős Problem #693", "statement": "Let $k\\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\\{a_1\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The maximal-gap problem for integers with a divisor in (n,2n) remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo later bound resolving polylogarithmic gaps was located.\n\n**What remains.**\n\nProve a polylogarithmic gap bound or establish the true scale.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #693, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/693\n Evidence used: Current maintained open status.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2263, "problem_number": "EP-694", "title": "Erdős Problem #694", "statement": "Let $f_{\\max}(n)$ be the largest $m$ such that $\\phi(m)=n$, and $f_{\\min}(n)$ be the smallest such $m$, where $\\phi$ is Euler's totient function. Investigate $ \\max_{n\\leq x}\\frac{f_{\\max}(n)}{f_{\\min}(n)}. $ ", "background": "Carmichael has asked whether there is an integer $n$ for which $\\phi(m)=n$ has exactly one solution, that is, $\\frac{f_{\\max}(n)}{f_{\\min}(n)}=1$. Erd\\H{o}s has proved that if such an $n$ exists then there must be infinitely many such $n$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A May 2026 GPT-5.5 Pro argument gives max f_max(n)/f_min(n)=(e^gamma+o(1)) log log x over totient values n<=x. A public Lean development verifies the argument modulo standard analytic inputs; Mertens has since been formalized, leaving Linnik's theorem as the explicit external axiom.\n\n**Verified partial progress.**\n\n- The upper bound reduces the fibre ratio to the classical maximal order of M/phi(M).\n- The lower bound constructs two integers with the same totient using a prime congruent to 1 modulo a product of p-1 for p<=y.\n- Mertens' product theorem supplies the sharp constant e^gamma, while Linnik's theorem controls the size of the constructed preimages.\n\n**Full solution or refutation.**\n\nOn the corrected domain n in phi(N), the maximum ratio is asymptotic to e^gamma log log x. The upper bound uses M/m=(M/phi(M))(phi(m)/m)<=M/phi(M); the lower bound uses a Linnik prime and a primorial-style equal-totient pair whose quotient is asymptotic to the Mertens product over p<=y.\n\n**What remains.**\n\nThe extremal-growth question is closed on the intended restricted domain. Carmichael's separate unique-totient-preimage question remains open, and a conventional paper plus removal of the remaining Linnik axiom from the formal trust boundary would be valuable.\n\n**Sources checked.**\n\n- Shashi Ammanamanchi and contributors, standalone Lean proof of Erdos Problem #694 (2026). (primary): https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P694/Proof.lean\n Evidence used: Primary formal proof artifact stating the asymptotic theorem and exposing its trust boundary, including the Linnik divisibility-form input.\n- Thomas F. Bloom, Nat Sothanaphan, and contributors, Erdos Problem #694 discussion (2026), checked 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/forum/thread/694\n Evidence used: Contains the proof summary, links the formalization, records independent confirmation that the Lean statement matches the result and that the admitted theorem statements are correct, and notes the later Mertens formalization.\n- Thomas F. Bloom, Erdos Problem #694, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/694\n Evidence used: Records SOLVED (LEAN), states the asymptotic formula, and explicitly repairs the maximum by restricting n to totient values.\n- Paul Erdos, Some unconventional problems in number theory, Asterisque 61 (1979), 73-82, especially p. 80. (source_collection): https://www.numdam.org/item/AST_1979__61__73_0/\n Evidence used: Historical source introduces the totient-fibre elements only when they exist, supporting the restricted-domain interpretation and separating the Carmichael question.\n\n**Review notes.** The imported background has a leaked serialized suffix. Literally, f_max and f_min are undefined for non-totient n; the result and current source restrict the maximum to n in the image of phi rather than silently totalizing these functions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2264, "problem_number": "EP-695", "title": "Erdős Problem #695", "statement": "Let $p_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Both prime-chain growth questions are open; Linnik gives a doubly exponential greedy-chain bound and finite chains have been studied.\n\n**Verified partial progress.**\n\n- Linnik's theorem bounds the greedy chain by exp(exp(O(k))).\n- Ford--Konyagin--Luca study finite prime-chain growth.\n\n**Full solution or refutation.**\n\nThese results do not establish either proposed asymptotic behavior.\n\n**What remains.**\n\nProve mandatory superexponential growth or construct a near-conjectural slow chain.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #695, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/695\n Evidence used: Current open status and Linnik consequence.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2265, "problem_number": "EP-696", "title": "Erdős Problem #696", "statement": "Let $h(n)$ be the largest $\\ell$ such that there is a sequence of primes $p_1<\\cdots < p_\\ell$ all dividing $n$ with $p_{i+1}\\equiv 1\\pmod{p_i}$. Let $H(n)$ be the largest $u$ such that there is a sequence of integers $d_1<\\cdots < d_u$ all dividing $n$ with $d_{i+1}\\equiv 1\\pmod{d_i}$.\nEstimate $h(n)$ and $H(n)$. Is it true that $H(n)/h(n)\\to \\infty$ for almost all $n$?", "background": "Erd\\H{o}s writes it is 'easy to see' that $h(n)\\to \\infty$ for almost all $n$ (which is proved in the comments by van Doorn), and believed he could show that the normal order of $h(n)$ is $\\log_*(n)$ (the iterated logarithm).\nSee also [695].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Turturean's April 2026 manuscript proves that h(n)=(1/2+o(1)) log_* n and H(n)=(1+o(1)) log_* n for almost all n, so H(n)/h(n) tends to 2 rather than infinity. The accompanying zero-sorry Lean development uses three explicitly admitted classical analytic-number-theory inputs.\n\n**Verified partial progress.**\n\n- Earlier arguments established h(n)->infinity for almost all n and suggested log-star scale.\n- An initial GPT-5.5 Pro proof note gave coarse two-sided log-star bounds, already refuting density-one divergence of the ratio.\n- The final manuscript sharpens both normal orders to constants 1/2 and 1 and supplies an approximately 19,200-line Lean development.\n\n**Full solution or refutation.**\n\nFor almost all n, prime congruence chains consume two tower levels per successor, giving h(n)~(1/2)log_*n, while subset products of prime residues provide composite divisor successors consuming one tower level, giving H(n)~log_*n. Therefore H(n)/h(n)->2 on a density-one set.\n\n**What remains.**\n\nThe estimation and divergence question are resolved if the recent manuscript is accepted. Conventional peer review, formalization of the admitted Siegel-Walfisz, Brun-Titchmarsh, and Mertens inputs, and sharper exceptional-set or lower-order estimates remain.\n\n**Sources checked.**\n\n- David Turturean, Normal orders of the Erdos chain functions h(n) and H(n), manuscript (April 2026). (primary): https://github.com/davidturturean/erdos-696/blob/main/paper/erdos_696_paper.pdf\n Evidence used: Primary manuscript proving the two normal-order asymptotics and the limiting ratio 2, with detailed AI-use disclosure.\n- David Turturean, Erdos Problem #696 Lean 4 formalization, Mathlib v4.28.0 (2026). (formal_verification): https://github.com/davidturturean/erdos-696\n Evidence used: Public approximately 19,200-line, zero-sorry formal development; the README explicitly audits the three admitted analytic inputs and states what is and is not kernel-proved.\n- Thomas F. Bloom, Erdos Problem #696 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/696\n Evidence used: Records SOLVED (LEAN), the progression from coarse log-star bounds to the refined constants, and independent community checking of the proof route.\n- Paul Erdos, Some unconventional problems in number theory, Asterisque 61 (1979), 73-82, especially p. 81. (source_collection): https://www.numdam.org/item/AST_1979__61__73_0/\n Evidence used: Historical primary formulation of h(n), H(n), the log-star expectation, and the question whether their ratio diverges almost surely.\n\n**Review notes.** The imported background has a leaked serialized suffix. Log-star conventions and inclusion of the divisor 1 can shift chain lengths by O(1), but not the leading constants or limiting ratio.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2266, "problem_number": "EP-700", "title": "Erdős Problem #700", "statement": "Let $ f(n)=\\min_{1n^{1/2}$?{/LI}\n{LI} Is it true that, for every composite $n$, $ f(n) \\ll_A \\frac{n}{(\\log n)^A} $ for every $A>0$?{/LI}\n{/UL}", "background": "A problem of Erd\\H{o}s and Szekeres. It is easy to see that $f(n)\\leq n/P(n)$ for composite $n$, since if $j=p^k$ where $p^k\\mid n$ and $p^{k+1}\nmid n$ then $\\textrm{gcd}\\left(n,\\binom{n}{j}\\right)=n/p^k$. This implies $ f(n) \\leq (1+o(1))\\frac{n}{\\log n}. $ It is known that $f(n)=n/P(n)$ when $n$ is the product of two primes. Another example is $n=30$.\nFor the second problem, it is easy to see that for any $n$ we have $f(n)\\geq p(n)$, where $p(n)$ is the smallest prime dividing $n$, and hence there are infinitely many $n$ (those $=p^2)$ such that $f(n)\\geq n^{1/2}$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** All three binomial-gcd questions remain open, but elementary examples give exact cases and a borderline square-root lower bound.\n\n**Verified partial progress.**\n\n- f(n)=n/P(n) for semiprimes and n=30.\n- For n=p^2, f(n)>=sqrt(n), which does not prove the strict >sqrt(n) request.\n\n**Full solution or refutation.**\n\nThe listed examples do not answer characterization, strict inequality, or arbitrary-log-power decay.\n\n**What remains.**\n\nResolve any of the three subquestions.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #700, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/700\n Evidence used: Current open status and elementary progress.\n\n**Review notes.** The strict versus non-strict square-root distinction is preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2267, "problem_number": "EP-701", "title": "Erdős Problem #701", "statement": "Let $\\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\\subseteq A\\in\\mathcal{F}$ then $B\\in \\mathcal{F}$). There exists some element $x$ such that whenever $\\mathcal{F}'\\subseteq \\mathcal{F}$ is an intersecting subfamily we have $ \\lvert \\mathcal{F}'\\rvert \\leq \\lvert \\{ A\\in \\mathcal{F} : x\\in A\\}\\rvert. $ ", "background": "A problem of Chv\\'{a}tal \\cite{Ch74}, who proved it replacing the closed under subsets condition with the (stronger) condition that, assuming all sets in $\\mathcal{F}$ are subsets of $\\{1,\\ldots,n\\}$, whenever $A\\in \\mathcal{F}$ and there is an injection $f:B\\to A$ such that $x\\leq f(x)$ for all $x\\in B$, then $B\\in \\mathcal{F}$.\nSterboul \\cite{St74} proved this when, letting $\\mathcal{G}$ be the maximal sets (under inclusion) in $\\mathcal{F}$, all sets in $\\mathcal{G}$ have the same size, $\\lvert A\\cap B\\rvert\\leq 1$ for all $A\neq B\\in \\mathcal{G}$, and at least two sets in $\\mathcal{G}$ have non-empty intersection.\nFrankl and Kupavskii \\cite{FrKu23} have proved this when $\\mathcal{F}$ has covering number $2$.\nBorg \\cite{Bo11} has proposed a weighted generalisation of this conjecture, which he proves under certain additional assumptions.\nReferences\n\n\n[Bo11] Borg, Peter, On Chv\\'{a}tal's conjecture and a conjecture on families of\nsigned sets. European J. Combin. (2011), 140-145.\n\n[Ch74] Chv\\'{a}tal, V., Intersecting families of edges in hypergraphs having the\nhereditary property. (1974), 61-66.\n\n[FrKu23] Frankl, Peter and Kupavskii, Andrey, Perfect matchings in down-sets. Discrete Math. (2023), Paper No. 113323, 7.\n\n[St74] Sterboul, F., Sur une conjecture de V. Chv\\'{a}tal. (1974), 152-164.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Chvátal's conjecture is open for finite hereditary families, with multiple meaningful subclasses proved.\n\n**Verified partial progress.**\n\n- Chvátal proved a stronger downset condition.\n- Frankl--Kupavskii prove the covering-number-two case.\n\n**Full solution or refutation.**\n\nThe proven subclasses do not cover arbitrary hereditary families.\n\n**What remains.**\n\nProve the finite Chvátal conjecture or find a counterexample.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #701, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/701\n Evidence used: Current open status and proved subclasses.\n\n**Review notes.** The intended finite-ground-set convention is noted; no infinite variant is substituted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2268, "problem_number": "EP-704", "title": "Erdős Problem #704", "statement": "Let $G_n$ be the unit distance graph in $\\mathbb{R}^n$, with two vertices joined by an edge if and only if the distance between them is $1$.\nEstimate the chromatic number $\\chi(G_n)$. Does it grow exponentially in $n$? Does $ \\lim_{n\\to \\infty}\\chi(G_n)^{1/n} $ exist?", "background": "A generalisation of the Hadwiger-Nelson problem (which addresses $n=2$). Frankl and Wilson \\cite{FrWi81} proved exponential growth: $ \\chi(G_n) \\geq (1+o(1))1.2^n. $ The trivial colouring (by tiling with cubes) gives $ \\chi(G_n) \\leq (2+\\sqrt{n})^n. $ Larman and Rogers \\cite{LaRo72} improved this to $ \\chi(G_n) \\leq (3+o(1))^n, $ and conjecture the truth may be $(2^{3/2}+o(1))^n$. Prosanov \\cite{Pr20} has given an alternative proof of this upper bound.\nSee also [508], [705], and [706].\nReferences\n\n\n[FrWi81] Frankl, P. and Wilson, R. M., Intersection theorems with geometric consequences. Combinatorica (1981), 357-368.\n\n[LaRo72] Larman, D. G. and Rogers, C. A., The realization of distances within sets in Euclidean space. Mathematika (1972), 1-24.\n\n[Pr20] Prosanov, Roman, A new proof of the Larman-Rogers upper bound for the\nchromatic number of the Euclidean space. Discrete Appl. Math. (2020), 115-120.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exponential growth of the Euclidean unit-distance chromatic number is proved, with recorded bounds between bases about 1.239 and 3, but the optimal exponential rate and existence of the nth-root limit remain open.\n\n**Verified partial progress.**\n\n- Frankl and Wilson proved exponential growth, answering the first yes/no question affirmatively.\n- Raigorodskii improved the lower bound to (1.239...+o(1))^n.\n- Larman and Rogers proved the upper bound (3+o(1))^n; Prosanov later supplied a new proof.\n\n**Full solution or refutation.**\n\nThe growth is known to be exponential, but neither the sharp base nor convergence of chi(G_n)^{1/n} has been established.\n\n**What remains.**\n\nNarrow the exponential-base gap and prove or disprove existence of lim chi(G_n)^{1/n}.\n\n**Sources checked.**\n\n- Roman Prosanov, A new proof of the Larman-Rogers upper bound for the chromatic number of the Euclidean space, Discrete Applied Mathematics 276 (2020), 115-120, DOI 10.1016/j.dam.2019.05.020. (primary): https://arxiv.org/abs/1610.02846\n Evidence used: States the current (1.239+o(1))^n lower and (3+o(1))^n upper bounds and proves the latter by a new method.\n- Thomas F. Bloom, Erdős Problem #704, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/704\n Evidence used: Keeps the estimation and root-limit questions open while recording that exponential growth is proved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2269, "problem_number": "EP-705", "title": "Erdős Problem #705", "statement": "Let $G$ be a finite unit distance graph in $\\mathbb{R}^2$ (i.e. the vertices are a finite collection of points in $\\mathbb{R}^2$ and there is an edge between two points if and only if the distance between them is $1$).\nIs there some $k$ such that if $G$ has girth $\\geq k$ (i.e. $G$ contains no cycles of length $\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Paul O'Donnell constructed finite unit-distance graphs in the plane with chromatic number four and arbitrarily large girth, so no fixed girth threshold forces 3-colorability.\n\n**Verified partial progress.**\n\n- Wormald constructed a 4-chromatic unit-distance graph of girth five.\n- O'Donnell's 1994 example and Chilakamarri's infinite family achieved girth four before the arbitrary-girth construction.\n\n**Full solution or refutation.**\n\nO'Donnell's 1999 dissertation and 2000 Geombinatorics articles construct 4-chromatic unit-distance graphs of arbitrarily large girth, directly negating the proposed existence of k.\n\n**What remains.**\n\nThe existence question is closed; quantitative bounds on the smallest order of such a graph as a function of girth remain separate refinements.\n\n**Sources checked.**\n\n- Paul O'Donnell, High girth unit-distance graphs, PhD dissertation, Rutgers University (1999), and Arbitrary Girth, 4-Chromatic Unit Distance Graphs in the Plane, Parts 1 and 2, Geombinatorics IX (2000). (primary): https://geombina.uccs.edu/author-index/paul-odonnell\n Evidence used: Primary construction and journal publication of 4-chromatic planar unit-distance graphs with arbitrary girth.\n- Geombinatorics, About the Journal, checked 2026-08-17. (authoritative_secondary): https://geombinatorics.uccs.edu/archive/index.php/home/about\n Evidence used: The journal's history explicitly identifies O'Donnell's high-girth unit-distance work as solving the Erdős problem.\n- Thomas F. Bloom, Erdos Problem #705, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/705\n Evidence used: Records DISPROVED status and the arbitrary-girth O'Donnell construction.\n\n**Review notes.** The statement is intact; the imported background has a leaked serialized suffix.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2270, "problem_number": "EP-706", "title": "Erdős Problem #706", "statement": "Let $L(r)$ be such that if $G$ is a graph formed by taking a finite set of points $P$ in $\\mathbb{R}^2$ and some set $A\\subset (0,\\infty)$ of size $r$, where the vertex set is $P$ and there is an edge between two points if and only if their distance is a member of $A$, then $\\chi(G)\\leq L(r)$.\nEstimate $L(r)$. In particular, is it true that $L(r)\\leq r^{O(1)}$?", "background": "The case $r=1$ is the Hadwiger-Nelson problem, for which it is known that $5\\leq L(1)\\leq 7$.\nSee also [508], [704], and [705].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For r forbidden planar distances, a superlinear lower bound and an elementary exponential upper bound are known, but Erdős's polynomial-growth question remains open.\n\n**Verified partial progress.**\n\n- The planar grid/distinct-distance argument gives L(r)>=c r sqrt(log r).\n- Taking the product of a seven-colouring for each individual distance gives L(r)<=7^r.\n- Naslund's 2023 paper explicitly retains polynomial dependence in the planar case as an open problem.\n\n**Full solution or refutation.**\n\nCurrent bounds establish finite and nontrivial growth but do not distinguish polynomial from superpolynomial behavior.\n\n**What remains.**\n\nProve L(r)<=r^{O(1)} or give a superpolynomial lower bound; ideally determine its order.\n\n**Sources checked.**\n\n- Eric Naslund, The chromatic number of R^n with multiple forbidden distances, Mathematika 69 (2023), 692-718, DOI 10.1112/mtk.12197. (primary): https://arxiv.org/abs/2205.12312\n Evidence used: Records the planar lower bound c r sqrt(log r), frames polynomial planar growth as open, and proves stronger high-dimensional multiple-distance bounds.\n- Thomas F. Bloom, Erdős Problem #706, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/706\n Evidence used: Maintains the polynomial-bound question as open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2271, "problem_number": "EP-708", "title": "Erdős Problem #708", "statement": "Let $g(n)$ be minimal such that for any $A\\subseteq [2,\\infty)\\cap \\mathbb{N}$ with $\\lvert A\\rvert =n$ and any set $I$ of $\\max(A)$ consecutive integers there exists some $B\\subseteq I$ with $\\lvert B\\rvert=g(n)$ such that $ \\prod_{a\\in A} a \\mid \\prod_{b\\in B}b. $ Is it true that $ g(n) \\leq (2+o(1))n? $ Or perhaps even $g(n)\\leq 2n$?", "background": "A problem of Erd\\H{o}s and Sur\\'{a}nyi \\cite{ErSu59}, who proved that $g(n) \\geq (2-o(1))n$, and that $g(3)=4$. Their lower bound construction takes $A$ as the set of $p_ip_j$ for $i\neq j$, where $p_1<\\cdots p_\\ell^2$.\nGallai was the first to consider problems of this type, and observed that $g(2)=2$ and $g(3)\\geq 4$.\nIn \\cite{Er92c} Erd\\H{o}s offers '100 dollars or 1000 rupees', whichever is more, for a proof or disproof. (In 1992 1000 rupees was worth approximately \\$38.60.)\nErd\\H{o}s and Sur\\'{a}nyi similarly asked what is the smallest $c_n\\geq 1$ such that in any interval $I\\subset [0,\\infty)$ of length $c_n\\max(A)$ there exists some $B\\subseteq I\\cap \\mathbb{N}$ with $\\lvert B\\rvert=n$ such that $ \\prod_{a\\in A} a \\mid \\prod_{b\\in B}b. $ They prove $c_2=1$ and $c_3=\\sqrt{2}$, but have no good upper or lower bounds in general.\nSee also [709].\nReferences\n\n\n[Er92c] Erd\"{o}s, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50.\n\n[ErSu59] Erd\\H{o}s, P\\'{a}l and Sur\\'{a}nyi, J\\'{a}nos, Bemerkungen zu einer Aufgabe eines mathematischen\n{W}ettbewerbs. Mat. Lapok (1959), 39-48.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The product-divisibility problem remains open: the known lower bound makes constant 2 asymptotically necessary, but neither g(n)<=2n nor g(n)<=(2+o(1))n is proved.\n\n**Verified partial progress.**\n\n- Erdős and Surányi proved g(n)>=(2-o(1))n and g(3)=4.\n- Gallai observed g(2)=2, and the same line of work determined the related constants c_2=1 and c_3=sqrt(2).\n\n**Full solution or refutation.**\n\nThe lower construction and small cases show the proposed coefficient is sharp if true, but no matching general upper bound was located.\n\n**What remains.**\n\nProve g(n)<=(2+o(1))n, ideally g(n)<=2n, or disprove either upper assertion.\n\n**Sources checked.**\n\n- P. Erdős and J. Surányi, Bemerkungen zu einer Aufgabe eines mathematischen Wettbewerbs, Mat. Lapok 10 (1959), 39-48. (primary): https://users.renyi.hu/~p_erdos/1959-07.pdf\n Evidence used: Primary source for the lower construction, small cases, and the related interval problem.\n- Thomas F. Bloom, Erdős Problem #708, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/708\n Evidence used: Retains open status and the $100 prize, with no solution claim recorded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2272, "problem_number": "EP-709", "title": "Erdős Problem #709", "statement": "Let $f(n)$ be minimal such that, for any $A=\\{a_1,\\ldots,a_n\\}\\subseteq [2,\\infty)\\cap\\mathbb{N}$ of size $n$, in any interval $I$ of $f(n)\\max(A)$ consecutive integers there exist distinct $x_1,\\ldots,x_n\\in I$ such that $a_i\\mid x_i$.\nObtain good bounds for $f(n)$, or even an asymptotic formula.", "background": "A problem of Erd\\H{o}s and Sur\\'{a}nyi \\cite{ErSu59}, who proved $ (\\log n)^c \\ll f(n) \\ll n^{1/2} $ for some constant $c>0$.\nSee also [708].\nReferences\n\n\n[ErSu59] Erd\\H{o}s, P\\'{a}l and Sur\\'{a}nyi, J\\'{a}nos, Bemerkungen zu einer Aufgabe eines mathematischen\n{W}ettbewerbs. Mat. Lapok (1959), 39-48.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The arbitrary-modulus matching function remains far from an asymptotic formula, but a July 2026 prime-only obstruction implies a new lower bound exponential in log n/log log n.\n\n**Verified partial progress.**\n\n- Erdős and Surányi proved (log n)^c << f(n) << n^{1/2}.\n- Van Doorn's 2026 theorem yields the lower bound log n/log log n recorded by the tracker.\n- Chen proved a bad-interval lower bound N exp((1/50)log N/log log N) even for primes up to N; specializing EP-709 to that prime set infers f(n)>=exp((1/50+o(1))log n/log log n).\n\n**Full solution or refutation.**\n\nThe 2026 obstruction substantially raises the lower bound, but it is still subpolynomial and does not approach the n^{1/2} upper bound.\n\n**What remains.**\n\nClose the gap between exp(Omega(log n/log log n)) and n^{1/2}, or obtain an asymptotic formula.\n\n**Sources checked.**\n\n- Kaizhe Chen, Improved Bounds for Distinct Multiples in Intervals, arXiv:2607.26450 (2026). (primary): https://arxiv.org/abs/2607.26450\n Evidence used: Theorem 1.2 proves the prime-only bad-interval bound; applying it to the allowed prime modulus sets gives the stated inferred EP-709 lower bound.\n- Wouter van Doorn, On the length of an interval that contains distinct multiples of the first n positive integers, Integers 26 (2026), A7. (primary): https://math.colgate.edu/~integers/aa7/aa7.pdf\n Evidence used: Provides the earlier n log n/log log n bad-interval theorem from which the tracker derives its improved lower bound.\n- Thomas F. Bloom, Erdős Problem #709, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/709\n Evidence used: Maintains open status and records the classical upper bound and the van Doorn-derived lower bound.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2273, "problem_number": "EP-710", "title": "Erdős Problem #710", "statement": "Let $f(n)$ be minimal such that in $(n,n+f(n))$ there exist distinct integers $a_1,\\ldots,a_n$ such that $k\\mid a_k$ for all $1\\leq k\\leq n$. Obtain an asymptotic formula for $f(n)$.", "background": "A problem of Erd\\H{o}s and Pomerance \\cite{ErPo80}, who proved $ (2/\\sqrt{e}+o(1))n\\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/2}\\leq f(n)\\leq (1.7398\\cdots+o(1))n(\\log n)^{1/2}. $ In \\cite{Er92c} Erd\\H{o}s offered 2000 rupees for an asymptotic formula; for uniform comparison across prizes I have converted this using the 1992 exchange rates.\nSee also [711].\nReferences\n\n\n[Er92c] Erd\"{o}s, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50.\n\n[ErPo80] P. Erd\\H{o}s and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples of another interval. Indigationes Math. (1980), 147-151.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No asymptotic formula is known for the fixed-start distinct-multiples interval; the classical bounds differ by a factor involving sqrt(log log n).\n\n**Verified partial progress.**\n\n- Erdős and Pomerance proved the lower bound (2/sqrt(e)+o(1)) n sqrt(log n/log log n).\n- They proved the upper bound (1.7398...+o(1)) n sqrt(log n).\n- Recent results for the maximum over all starting points do not determine the fixed starting point m=n.\n\n**Full solution or refutation.**\n\nThe maintained tracker remains open and the searched recent papers do not close the fixed-start gap.\n\n**What remains.**\n\nDetermine an asymptotic formula for f(n), including its logarithmic scale and leading constant.\n\n**Sources checked.**\n\n- P. Erdős and C. Pomerance, Matching the natural numbers up to n with distinct multiples in another interval, Indagationes Mathematicae 42 (1980), 147-161. (primary): https://math.dartmouth.edu/~carlp/PDF/matching.pdf\n Evidence used: Primary source defining the fixed-start function and proving the recorded upper and lower bounds.\n- Thomas F. Bloom, Erdős Problem #710, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/710\n Evidence used: Retains open status and the prize for an asymptotic formula.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2274, "problem_number": "EP-711", "title": "Erdős Problem #711", "statement": "Let $f(n,m)$ be minimal such that in $(m,m+f(n,m))$ there exist distinct integers $a_1,\\ldots,a_n$ such that $k\\mid a_k$ for all $1\\leq k\\leq n$. Prove that $ \\max_m f(n,m) \\leq n^{1+o(1)} $ and that $ \\max_m (f(n,m)-f(n,n))\\to \\infty. $ ", "background": "A problem of Erd\\H{o}s and Pomerance \\cite{ErPo80}, who proved that $ \\max_m f(n,m) \\ll n^{3/2} $ and $ n\\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/2} \\ll f(n,n)\\ll n(\\log n)^{1/2}. $ In \\cite{Er92c} Erd\\H{o}s offered 1000 rupees for a proof of either; for uniform comparison across prizes I have converted this using the 1992 exchange rates.\nvan Doorn \\cite{vD26} has provided an affirmative answer to the second question, proving that, for all large $n$, there exists $m=m(n)$ such that $ f(n,m)-f(n,n) \\gg \\frac{\\log n}{\\log\\log n}n. $ See also [710].\nReferences\n\n\n[Er92c] Erd\"{o}s, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50.\n\n[ErPo80] P. Erd\\H{o}s and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples of another interval. Indigationes Math. (1980), 147-151.\n\n[vD26] W. van Doorn, On the length of an interval that contains distinct multiples of the first $n$ positive integers. Integers (2026), #A7.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Van Doorn proved the second of the two imported assertions, while Chen's July 2026 preprint improves the remaining uniform upper exponent from 3/2 to about 1.4031 but does not reach n^{1+o(1)}.\n\n**Verified partial progress.**\n\n- Van Doorn proved max_m f(n,m)-f(n,n)>0.36 n log n/log log n for all sufficiently large n, settling the divergence assertion.\n- Chen proved max_m f(n,m)<=n^{beta+o(1)} with beta about 1.4031, the root in (1,2) of 2 beta^3-8 beta^2+8 beta-1=0.\n- Chen also proved max_m f(n,m)>=n exp((1/50)log n/log log n), compatible with a possible n^{1+o(1)} upper bound.\n\n**Full solution or refutation.**\n\nOne conjunct is completely solved and the other has a substantially improved exponent, but the conjectured n^{1+o(1)} upper bound remains open.\n\n**What remains.**\n\nReduce the uniform upper exponent from beta approximately 1.4031 to 1+o(1).\n\n**Sources checked.**\n\n- Wouter van Doorn, On the length of an interval that contains distinct multiples of the first n positive integers, Integers 26 (2026), A7, DOI 10.5281/zenodo.18154085. (primary): https://math.colgate.edu/~integers/aa7/aa7.pdf\n Evidence used: Theorem 1 proves the quantitative divergence that settles the second imported assertion.\n- Kaizhe Chen, Improved Bounds for Distinct Multiples in Intervals, arXiv:2607.26450 (2026). (primary): https://arxiv.org/abs/2607.26450\n Evidence used: Defines F(n)=max_m f(n,m) exactly and proves the new n^{beta+o(1)} upper and n exp(c log n/log log n) lower bounds.\n- Thomas F. Bloom, Erdős Problem #711, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/711\n Evidence used: Records van Doorn's affirmative solution of the second question while keeping the first open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2275, "problem_number": "EP-712", "title": "Erdős Problem #712", "statement": "Determine, for any $k>r>2$, the value of $ \\frac{\\mathrm{ex}_r(n,K_k^r)}{\\binom{n}{r}}, $ where $\\mathrm{ex}_r(n,K_k^r)$ is the largest number of $r$-edges which can placed on $n$ vertices so that there exists no set of $k$ vertices which is covered by all $\\binom{k}{r}$ possible $r$-edges.", "background": "Tur\\'{an proved} that, when $r=2$, this limit is $ \\frac{1}{2}\\left(1-\\frac{1}{k-1}\\right). $ Erd\\H{o}s \\cite{Er81} offered \\$500 for the determination of this value for any fixed $k>r>2$, and \\$1000 for 'clearing up the whole set of problems'.\nSee also [500] for the case $r=3$ and $k=4$.\nReferences\n\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The complete-uniform-hypergraph Turán density problem remains open; the imported display also omits the n-to-infinity limit that its background calls 'this limit'.\n\n**Verified partial progress.**\n\n- The intended normalized extremal sequence has a limiting Turán density by standard averaging.\n- Even the first case K_4^3 remains unresolved, and the maintained tracker records no fixed pair k>r>2 for which the requested density has been determined.\n\n**Full solution or refutation.**\n\nNeither the intended all-pairs density problem nor the literal finite-n ratio has been determined in the imported range.\n\n**What remains.**\n\nAfter confirming the intended limit notation, determine any complete r-uniform Turán density with k>r>2, beginning with K_4^3.\n\n**Sources checked.**\n\n- P. Erdős, On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25-42. (primary): https://static.renyi.hu/~p_erdos/1981-27.pdf\n Evidence used: Primary source for the complete-hypergraph Turán-density prize problem.\n- Thomas F. Bloom, Erdős Problem #712, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/712\n Evidence used: Maintains open status and refers to the displayed quantity as a limit, exposing the imported notation omission.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2276, "problem_number": "EP-713", "title": "Erdős Problem #713", "statement": "Is it true that, for every bipartite graph $G$, there exists some $\\alpha\\in [1,2)$ and $c>0$ such that $ \\mathrm{ex}(n;G)\\sim cn^\\alpha? $ Must $\\alpha$ be rational?", "background": "A problem of Erd\\H{o}s and Simonovits. Erd\\H{o}s sometimes asked this in the weaker version with just $ \\mathrm{ex}(n;G)\\asymp n^{\\alpha}. $ Erd\\H{o}s \\cite{Er67d} had initially conjectured that, for any bipartite graph $G$, $\\mathrm{ex}(n;G)\\sim cn^{\\alpha}$ for some constant $c>0$ and $\\alpha$ of the shape $1+\\frac{1}{k}$ or $2-\\frac{1}{k}$ for some integer $k\\geq 2$. This was disproved by Erd\\H{o}s and Simonovits \\cite{ErSi70}.\nThe analogous statement is not true for hypergraphs, as shown by Frankl and F\"{u}redi \\cite{FrFu87}, who proved that if $G$ is the $5$-uniform hypergraph on $8$ vertices with edges $\\{12346,12457,12358\\}$ then $\\mathrm{ex}(n;G)=o(n^5)$ but $\\mathrm{ex}(n;G)\neq O(n^c)$ for any $c<5$.\nA simplified proof was given by F\"{u}redi and Gerbner \\cite{FuGe21}, who extended it to a counterexample for all $k\\geq 5$. It remains open whether it is true for $k=3$ and $k=4$ (though F\"{u}redi and Gerbner conjecture it is not).\nSee also [571].\nReferences\n\n\n[Er67d] Erd\\H{o}s, P., Some recent results on extremal problems in graph theory.\n{R}esults. (1967), 117--123 (English); pp. 124--130 (French).\n\n[ErSi70] Erd\\H{o}s, P. and Simonovits, M., Some extremal problems in graph theory. Combinatorial theory and its applications, I-III (Proc. Colloq., Balatonf\"{u}red, 1969) (1970), 377-390.\n\n[FrFu87] Frankl, P. and F\"uredi, Z., Exact solution of some {T}ur\\'an-type problems. J. Combin. Theory Ser. A (1987), 226--262.\n\n[FuGe21] F\"uredi, Zolt\\'an and Gerbner, D\\'aniel, Hypergraphs without exponents. J. Combin. Theory Ser. A (2021), Paper No. 105517, 9.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains open whether every bipartite graph has a pure-power extremal asymptotic and whether every resulting exponent must be rational.\n\n**Verified partial progress.**\n\n- Erdős and Simonovits disproved the earlier narrower claim that only exponents 1+1/k or 2-1/k can occur.\n- Frankl and Füredi disproved the analogous pure-power assertion for a specific 5-uniform hypergraph.\n- Füredi and Gerbner simplified and extended hypergraph counterexamples to every uniformity at least five; these do not give a graph counterexample.\n\n**Full solution or refutation.**\n\nKnown counterexamples concern a narrower graph conjecture or higher-uniformity analogues, not the exact universal bipartite-graph statement.\n\n**What remains.**\n\nProve pure-power asymptotics with rational exponent for every bipartite graph, or construct a bipartite graph violating existence or rationality of the exponent.\n\n**Sources checked.**\n\n- Peter Frankl and Zoltán Füredi, Exact solution of some Turán-type problems, Journal of Combinatorial Theory Series A 45 (1987), 226-262, DOI 10.1016/0097-3165(87)90016-1. (primary): https://doi.org/10.1016/0097-3165(87)90016-1\n Evidence used: Provides the 5-uniform non-power extremal example showing failure of the hypergraph analogue.\n- Zoltán Füredi and Miklós Simonovits, The history of degenerate (bipartite) extremal graph problems, 2013 survey. (authoritative_secondary): https://arxiv.org/abs/1306.5167\n Evidence used: Surveys the exponent conjectures and the unresolved bipartite-graph setting.\n- Thomas F. Bloom, Erdős Problem #713, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/713\n Evidence used: Maintains the exact bipartite statement as open and carefully separates narrower and hypergraph counterexamples.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2277, "problem_number": "EP-714", "title": "Erdős Problem #714", "statement": "Is it true that $ \\mathrm{ex}(n; K_{r,r}) \\gg n^{2-1/r}? $ ", "background": "K\"{o}v\\'{a}ri, S\\'{o}s, and Tur\\'{a}n \\cite{KST54} proved $ \\mathrm{ex}(n; K_{r,r}) \\ll n^{2-1/r} $ for all $r\\geq 2$. Brown \\cite{Br66} and, independently, Erd\\H{o}s, R\\'{e}nyi, and S\\'{o}s \\cite{ERS66}, proved the conjectured lower bound when $r=3$.\nWhen $r=2$ it is known that $ \\mathrm{ex}(n;K_{2,2})=\\left(\\frac{1}{2}+o(1)\\right)n^{3/2} $ (see [768], since $K_{2,2}=C_4$).\nSee also [147].\nReferences\n\n\n[Br66] Brown, W. G., On graphs that do not contain a Thomsen graph. Canad. Math. Bull. (1966), 281-285.\n\n[ERS66] Erd\\H{o}s, P. and R\\'{e}nyi, A. and S\\'os, V. T., On a problem of graph theory. Studia Sci. Math. Hungar. (1966), 215--235.\n\n[KST54] K\"{o}vari, T. and S\\'{o}s, V. T. and Tur\\'{a}n, P., On a problem of K. Zarankiewicz. Colloq. Math. (1954), 50-57.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Kővári-Sós-Turán exponent is matched by lower constructions for balanced K_{r,r} when r=2 or 3, but the conjecture remains open for every r>=4.\n\n**Verified partial progress.**\n\n- For r=2, ex(n,K_{2,2})=(1/2+o(1))n^{3/2}.\n- Brown and, independently, Erdős, Rényi, and Sós proved ex(n,K_{3,3})=Theta(n^{5/3}).\n- Kővári, Sós, and Turán proved the matching-form upper bound O(n^{2-1/r}) for all fixed r.\n\n**Full solution or refutation.**\n\nThe exact exponent is known only in the balanced cases r=2 and r=3; current surveys still identify r>=4 as central open cases.\n\n**What remains.**\n\nConstruct K_{r,r}-free graphs with Omega(n^{2-1/r}) edges for each fixed r>=4, or refute the conjectured order.\n\n**Sources checked.**\n\n- David Conlon, Some remarks on the Zarankiewicz problem, survey/preprint checked 2026-08-17. (authoritative_secondary): https://www.its.caltech.edu/~dconlon/zarankiewicz.pdf\n Evidence used: States that balanced matching-order lower bounds are completely known only for r=2 and r=3 and surveys the remaining gap.\n- Shakhar Smorodinsky, A survey of Zarankiewicz problem in geometry, arXiv:2410.03702. (authoritative_secondary): https://arxiv.org/abs/2410.03702\n Evidence used: Describes tightness for balanced r>=4 as a central open problem.\n- Thomas F. Bloom, Erdős Problem #714, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/714\n Evidence used: Retains open status and records the solved r=2 and r=3 cases.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2278, "problem_number": "EP-719", "title": "Erdős Problem #719", "statement": "Let $\\mathrm{ex}_r(n;K_{r+1}^r)$ be the maximum number of $r$-edges that can be placed on $n$ vertices without forming a $K_{r+1}^r$ (the $r$-uniform complete graph on $r+1$ vertices).\nIs every $r$-hypergraph $G$ on $n$ vertices the union of at most $\\mathrm{ex}_{r}(n;K_{r+1}^r)$ many copies of $K_r^r$ and $K_{r+1}^r$, no two of which share a $K_r^r$?\",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Erdős-Sauer hypergraph edge-decomposition conjecture remains open; the imported statement field is extraction-damaged by a serialized JSON tail after the mathematical question.\n\n**Verified partial progress.**\n\n- The maintained tracker confirms the mathematical prefix and attributes the conjecture to Erdős and Sauer.\n- No incorporated partial or complete solution, or current primary source claiming one, was located.\n\n**Full solution or refutation.**\n\nNo verified resolution was found for the exact decomposition bound. The extraction tail was flagged and not removed.\n\n**What remains.**\n\nProve or refute the decomposition into at most ex_r(n,K_{r+1}^r) edge-disjoint copies of K_r^r and K_{r+1}^r, after separately cleaning the source record.\n\n**Sources checked.**\n\n- P. Erdős, On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25-42. (primary): https://static.renyi.hu/~p_erdos/1981-27.pdf\n Evidence used: Historical primary problem source.\n- Thomas F. Bloom, Erdős Problem #719, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/719\n Evidence used: Confirms the intended mathematical statement, attributes it to Erdős and Sauer, and maintains open status.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2279, "problem_number": "EP-724", "title": "Erdős Problem #724", "statement": "Let $f(n)$ be the maximum number of mutually orthogonal Latin squares of order $n$. Is it true that $ f(n) \\gg n^{1/2}? $ ", "background": "Euler conjectured that $f(n)=1$ when $n\\equiv 2\\pmod{4}$, but this was disproved by Bose, Parker, and Shrikhande \\cite{BPS60} who proved $f(n)\\geq 2$ for $n\\geq 7$.\nChowla, Erd\\H{o}s, and Straus \\cite{CES60} proved $f(n) \\gg n^{1/91}$. Wilson \\cite{Wi74} proved $f(n) \\gg n^{1/17}$. Beth \\cite{Be83c} proved $f(n) \\gg n^{1/14.8}$.\nThe sequence of $f(n)$ is A001438 in the OEIS.\nReferences\n\n\n[BPS60] Bose, R. C. and Shrikhande, S. S. and Parker, E. T., Further results on the construction of mutually orthogonal\nLatin squares and the falsity of Euler's conjecture. Canadian J. Math. (1960), 189-203.\n\n[Be83c] Beth, Thomas, Eine Bemerkung zur Absch\"{a}tzung der Anzahl orthogonaler\nlateinischer Quadrate mittels Siebverfahren. Abh. Math. Sem. Univ. Hamburg (1983), 284-288.\n\n[CES60] Chowla, S. and Erd\\H{o}s, P. and Straus, E. G., On the maximal number of pairwise orthogonal Latin squares\nof a given order. Canadian J. Math. (1960), 204-208.\n\n[Wi74] Wilson, Richard M., Concerning the number of mutually orthogonal Latin squares. Discrete Math. (1974), 181-198.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The requested square-root lower bound for the maximum number of MOLS remains open.\n\n**Verified partial progress.**\n\n- Chowla--Erdős--Straus proved f(n)>>n^(1/91).\n- Wilson improved the exponent to 1/17 and Beth to 1/14.8.\n\n**Full solution or refutation.**\n\nThe established exponents are far below 1/2.\n\n**What remains.**\n\nProve f(n)>>sqrt(n), or improve the general exponent.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #724, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/724\n Evidence used: Current open status and cited lower-bound sequence.\n\n**Review notes.** Source statement preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2280, "problem_number": "EP-725", "title": "Erdős Problem #725", "statement": "Give an asymptotic formula for the number of $k\\times n$ Latin rectangles.", "background": "Erd\\H{o}s and Kaplansky \\cite{ErKa46} proved the count is $ \\sim e^{-\\binom{k}{2}}(n!)^k $ when $k=o((\\log n)^{3/2-\\epsilon})$. Yamamoto \\cite{Ya51} extended this to $k\\leq n^{1/3-o(1)}$.\nThe count of such Latin rectangles is A001009 in the OEIS.\nReferences\n\n\n[ErKa46] Erd\"{o}s, Paul and Kaplansky, Irving, The asymptotic number of Latin rectangles. Amer. J. Math. (1946), 230-236.\n\n[Ya51] Yamamoto, Koichi, On the asymptotic number of Latin rectangles. Jpn. J. Math. (1951), 113-119.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A uniform asymptotic for all k by n Latin rectangles remains open; substantial growing-k ranges are known.\n\n**Verified partial progress.**\n\n- Erdős--Kaplansky obtain e^{-binom(k,2)}(n!)^k in a logarithmic range.\n- Yamamoto extends this to k<=n^(1/3-o(1)).\n- A tracker discussion attributes a still wider range to Godsil--McKay, requiring direct verification.\n\n**Full solution or refutation.**\n\nThese ranges do not constitute a general asymptotic formula.\n\n**What remains.**\n\nVerify later literature and establish the correct uniform regime.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #725, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/725\n Evidence used: Current open status and classical results.\n- EP-725 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/725\n Evidence used: Unverified bibliographic lead for an extended range.\n\n**Review notes.** The discussion lead is not treated as a verified full solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2281, "problem_number": "EP-726", "title": "Erdős Problem #726", "statement": "As $n\\to \\infty$ ranges over integers $ \\sum_{p\\leq n}1_{n\\in (p/2,p)\\pmod{p}}\\frac{1}{p}\\sim \\frac{\\log\\log n}{2}. $ ", "background": "A conjecture of Erd\\H{o}s, Graham, Ruzsa, and Straus \\cite{EGRS75}. For comparison the classical estimate of Mertens states that $ \\sum_{p\\leq n}\\frac{1}{p}\\sim \\log\\log n. $ By $n\\in (p/2,p)\\pmod{p}$ we mean $n\\equiv r\\pmod{p}$ for some integer $r$ with $p/2\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The half-Mertens weighted-prime asymptotic remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo verified result proving the stated asymptotic was located.\n\n**What remains.**\n\nControl the residue condition sufficiently uniformly over primes.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #726, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/726\n Evidence used: Current maintained open status.\n\n**Review notes.** Source statement preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2282, "problem_number": "EP-727", "title": "Erdős Problem #727", "statement": "Let $k\\geq 2$. Does $ (n+k)!^2 \\mid (2n)! $ for infinitely many $n$?", "background": "A conjecture of Erd\\H{o}s, Graham, Ruzsa, and Straus \\cite{EGRS75}. It is open even for $k=2$.\nBalakran \\cite{Ba29} proved this holds for $k=1$ - that is, $(n+1)^2\\mid \\binom{2n}{n}$ for infinitely many $n$. It is a classical fact that $(n+1)\\mid \\binom{2n}{n}$ for all $n$ (see Catalan numbers).\nErd\\H{o}s, Graham, Ruzsa, and Straus observe that the method of Balakran can be further used to prove that there are infinitely many $n$ such that $ (n+k)!(n+1)! \\mid (2n)! $ (in fact this holds whenever $k0$).\nErd\\H{o}s \\cite{Er68c} proved that if $a!b!\\mid n!$ then $a+b\\leq n+O(\\log n)$.\nReferences\n\n\n[Ba29] H. Balakran, On the values of $n$ which make $(2n)!/(n+1)!(n+1)!$ an integer. J. Indian Math. Soc. (1929), 97-100.\n\n[EGRS75] Erd\\H{o}s, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\n\n[Er68c] P. Erd\\H{o}s, Aufgabe 557. Elemente Math. (1968), 111-113.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The factorial divisibility question remains open even at k=2, while k=1 and a weaker varying-k statement are known.\n\n**Verified partial progress.**\n\n- The k=1 case is known.\n- The tracker records infinitely many n for a weaker divisibility involving (n+k)!(n+1)! when k is below a constant multiple of log n.\n\n**Full solution or refutation.**\n\nNeither result implies (n+k)!^2 divides (2n)! for fixed k>=2.\n\n**What remains.**\n\nResolve the first open fixed case k=2.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #727, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/727\n Evidence used: Current open status and stated partial results.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2283, "problem_number": "EP-730", "title": "Erdős Problem #730", "statement": "Are there infinitely many pairs of integers $n\neq m$ such that $\\binom{2n}{n}$ and $\\binom{2m}{m}$ have the same set of prime divisors?", "background": "A problem of Erd\\H{o}s, Graham, Ruzsa, and Straus \\cite{EGRS75}, who believed there is 'no doubt' that the answer is yes.\nFor example $(87,88)$ and $(607,608)$. Those $n$ such that there exists some suitable $m>n$ are listed as A129515 in the OEIS.\nA triple of such $n$ for which $\\binom{2n}{n}$ all share the same set of prime divisors is $(10003,10004,10005)$. It is not known whether there are such pairs of the shape $(n,n+k)$ for every $k\\geq 1$.\nReferences\n\n\n[EGRS75] Erd\\H{o}s, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Infinitely many pairs with matching prime-divisor sets of central binomial coefficients are not known.\n\n**Verified partial progress.**\n\n- The tracker lists examples (87,88), (607,608), and a three-term run near 10004.\n\n**Full solution or refutation.**\n\nFinite examples do not prove the requested infinitude.\n\n**What remains.**\n\nFind a mechanism producing infinitely many pairs or prove finiteness.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #730, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/730\n Evidence used: Current open status and examples.\n\n**Review notes.** No computation was used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2284, "problem_number": "EP-731", "title": "Erdős Problem #731", "statement": "Find some reasonable function $f(n)$ such that, for almost all integers $n$, the least integer $m$ such that $m\nmid \\binom{2n}{n}$ satisfies $ m\\sim f(n). $ ", "background": "A problem of Erd\\H{o}s, Graham, Ruzsa, and Straus \\cite{EGRS75}, who say it is 'not hard to show that', for almost all $n$, the minimal such $m$ satisfies $ m=\\exp((\\log n)^{1/2+o(1)}). $ \nReferences\n\n\n[EGRS75] Erd\\H{o}s, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The maintained record is open; an asserted 2026 asymptotic was not independently verified from a primary source.\n\n**Verified partial progress.**\n\n- The tracker records the baseline almost-all-n scale exp((log n)^(1/2+o(1))).\n\n**Full solution or refutation.**\n\nAn unverified recent claim cannot change the problem's status.\n\n**What remains.**\n\nLocate and verify a primary proof of any claimed asymptotic, or retain the open classification.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #731, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/731\n Evidence used: Maintained open status and baseline estimate.\n- EP-731 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/731\n Evidence used: Recent claim requires direct primary-source verification.\n\n**Review notes.** Unverified claim intentionally excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2285, "problem_number": "EP-734", "title": "Erdős Problem #734", "statement": "Find, for all large $n$, a non-trivial pairwise balanced block design $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ such that, for all $t$, there are $O(n^{1/2})$ many $i$ such that $\\lvert A_i\\rvert=t$.", "background": "$A_1,\\ldots,A_m$ is a pairwise balanced block design if every pair in $\\{1,\\ldots,n\\}$ is contained in exactly one of the $A_i$.\nErd\\H{o}s \\cite{Er81} writes 'this will be probably not be very difficult to prove but so far I was not successful'.\nErd\\H{o}s and de Bruijn \\cite{dBEr48} proved that if $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ is a pairwise balanced block design then $m\\geq n$, and this implies there must be some $t$ such that there are $\\gg n^{1/2}$ many $t$ with $\\lvert A_i\\rvert=t$.\nReferences\n\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[dBEr48] de Bruijn, N. G. and Erd\\H{o}s, P., On a combinatorial problem. Nederl. Akad. Wetensch., Proc. (1948), 1277--1279 = Indagationes Math. 10, 421--423.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The PBD construction remains open; de Bruijn--Erdős supplies a matching-order multiplicity obstruction.\n\n**Verified partial progress.**\n\n- For a nontrivial pairwise balanced design, de Bruijn--Erdős implies at least n blocks, hence some block size occurs with multiplicity Omega(sqrt(n)) in the relevant counting setup.\n\n**Full solution or refutation.**\n\nThe obstruction explains the requested order but does not construct such a design.\n\n**What remains.**\n\nConstruct PBDs with every size used O(sqrt(n)) times.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #734, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/734\n Evidence used: Current open status and de Bruijn--Erdős observation.\n\n**Review notes.** No unverified construction claim was used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2286, "problem_number": "EP-738", "title": "Erdős Problem #738", "statement": "If $G$ has infinite chromatic number and is triangle-free (contains no $K_3$) then must $G$ contain every tree as an induced subgraph?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Gyárfás's induced-tree conjecture for infinite-chromatic triangle-free graphs remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo general proof that every tree appears as an induced subgraph was located.\n\n**What remains.**\n\nProve the induced-tree assertion or find a counterexample.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #738, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/738\n Evidence used: Current maintained open status.\n\n**Review notes.** Source statement preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2287, "problem_number": "EP-740", "title": "Erdős Problem #740", "statement": "Let $\\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\\mathfrak{m}$. Let $r\\geq 1$. Must $G$ contain a subgraph of chromatic number $\\mathfrak{m}$ which does not contain any odd cycle of length $\\leq r$?", "background": "A question of Erd\\H{o}s and Hajnal. R\"{o}dl proved this is true if $\\mathfrak{m}=\\aleph_0$ and $r=3$ (see [108] for the finitary version).\nMore generally, Erd\\H{o}s and Hajnal asked must there exist (for every cardinal $\\mathfrak{m}$ and integer $r$) some $f_r(\\mathfrak{m})$ such that every graph with chromatic number $\\geq f_r(\\mathfrak{m})$ contains a subgraph with chromatic number $\\mathfrak{m}$ with no odd cycle of length $\\leq r$?\nErd\\H{o}s \\cite{Er95d} claimed that even the $r=3$ case of this is open: must every graph with sufficiently large chromatic number contain a triangle free graph with chromatic number $\\mathfrak{m}$?\nIn \\cite{Er81} Erd\\H{o}s also asks the same question but with girth (i.e. the subgraph does not contain any cycle at all of length $\\leq C$).\nReferences\n\n\n[Er81] Erd\\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er95d] Erd\\H{o}s, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general cardinal-chromatic assertion remains open; the maintained discussion records a countable positive case and a claimed consistency obstruction in an uncountable case.\n\n**Verified partial progress.**\n\n- Rödl's result covers m=aleph_0 and r=3 according to the tracker.\n- A discussion reports a Shelah consistency counterexample for m=aleph_1,r=3, requiring direct verification.\n\n**Full solution or refutation.**\n\nNeither item settles the unrestricted statement.\n\n**What remains.**\n\nVerify the set-theoretic obstruction and resolve the remaining cardinal/radius cases.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #740, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/740\n Evidence used: Current open status and countable case.\n- EP-740 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/740\n Evidence used: Unverified bibliographic lead for the consistency assertion.\n\n**Review notes.** Set-theoretic claim is deliberately not promoted to a full refutation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2288, "problem_number": "EP-741", "title": "Erdős Problem #741", "statement": "Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive density. Can one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive density?\nIs there a basis $A$ of order $2$ such that if $A=A_1\\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?", "background": "A problem of Burr and Erd\\H{o}s. Erd\\H{o}s \\cite{Er94b} thought he could construct a basis as in the second question, but 'could never quite finish the proof'.\nReferences\n\n\n[Er94b] Erd\\H{o}s, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The natural-density version of the first question is false, the upper-density version is true, and the second basis-of-order-two question has an affirmative construction; the density convention must therefore be stated explicitly.\n\n**Verified partial progress.**\n\n- Burr and Erdős posed the partition questions, and Erdős reported an unfinished attempt at the second construction.\n- The current work separates natural density from upper density rather than treating 'positive density' as unambiguous.\n\n**Full solution or refutation.**\n\nRecent proofs provide a counterexample for natural density, a positive partition theorem for upper density, and an asymptotic basis of order two for which every bipartition leaves one diagonal sumset without bounded gaps.\n\n**What remains.**\n\nThe mathematical readings are resolved, but the dataset should retain an explicit literature warning that the unqualified word density is ambiguous; recent formal/AI-assisted proofs merit expert review.\n\n**Sources checked.**\n\n- Przemek Chojecki, research note on Erdos Problem #741 (2026). (primary): https://www.ulam.ai/research/erdos741.pdf\n Evidence used: States the upper-density partition theorem, the natural-density counterexample, and the affirmative basis construction.\n- Google DeepMind, AlphaProof Nexus output for Erdos Problem #741 (2026). (formal_verification): https://github.com/google-deepmind/alphaproof-nexus-results/blob/main/APNOutputs/ErdosProblems/erdos_741.parts.i.lean\n Evidence used: Formal proof artifact associated with the recent resolution.\n- Thomas F. Bloom and contributors, Erdos Problem #741 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/741\n Evidence used: Records SOLVED (LEAN) status and distinguishes the density conventions.\n\n**Review notes.** The record asks two questions and leaves the density convention unstated; the imported background also has a leaked serialized suffix.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2289, "problem_number": "EP-749", "title": "Erdős Problem #749", "statement": "Let $\\epsilon>0$. Does there exist $A\\subseteq \\mathbb{N}$ such that the lower density of $A+A$ is at least $1-\\epsilon$ and yet $1_A\\ast 1_A(n) \\ll_\\epsilon 1$ for all $n$?", "background": "A similar question can be asked for upper density.\nSee also [28].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The lower-density sumset/convolution question is open; reported upper-density work does not settle it.\n\n**Verified partial progress.**\n\n- The tracker discussion distinguishes an upper-density analogue from the stated lower-density problem.\n\n**Full solution or refutation.**\n\nAn upper-density result cannot be substituted for this lower-density assertion.\n\n**What remains.**\n\nResolve the stated lower-density hypothesis, including the bounded-convolution alternative.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #749, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/749\n Evidence used: Current open status.\n- EP-749 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/749\n Evidence used: Reported upper-density distinction; not used as resolution.\n\n**Review notes.** No unverified AI-assisted result is used as a literature fact.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2290, "problem_number": "EP-750", "title": "Erdős Problem #750", "statement": "Let $f(m)$ be some function such that $f(m)\\to \\infty$ as $m\\to \\infty$. Does there exist a graph $G$ of infinite chromatic number such that every subgraph on $m$ vertices contains an independent set of size at least $\\frac{m}{2}-f(m)$?", "background": "In \\cite{Er69b} Erd\\H{o}s conjectures this for $f(m)=\\epsilon m$ for any fixed $\\epsilon>0$. This follows from a result of Erd\\H{o}s, Hajnal, and Szemer\\'{e}di \\cite{EHS82}, as described by msellke in the comments.\nIn \\cite{ErHa67b} Erd\\H{o}s and Hajnal prove this for $f(m)\\geq cm$ for all $c>1/4$.\nSee also [75].\nReferences\n\n\n[EHS82] Erd\\H{o}s, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Er69b] Erd\\H{o}s, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[ErHa67b] Erd\\H{o}s, P. and Hajnal, Andr\\'as, On chromatic graphs. Mat. Lapok (1967), 1--4.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A recent AI-assisted construction, with a linked Lean development conditional on the classical Stiebitz decomposition theorem, gives the requested infinitely chromatic graph for every divergent f.\n\n**Verified partial progress.**\n\n- Erdős and Hajnal proved linear-error versions, including constants above 1/4.\n- Erdős, Hajnal, and Szemerédi established the fixed epsilon-times-m regime.\n\n**Full solution or refutation.**\n\nThe construction makes every m-vertex subgraph bipartite after deleting a controlled number of vertices, which yields an independent set of size at least m/2-f(m), while the whole graph retains infinite chromatic number.\n\n**What remains.**\n\nThe original existence question is resolved. Because the proof is recent, AI-assisted, and its formal artifact imports Stiebitz's theorem, independent expert review and conventional publication remain desirable.\n\n**Sources checked.**\n\n- Przemek Chojecki, research note on Erdos Problem #750 (2026). (primary): https://www.ulam.ai/research/erdos750.pdf\n Evidence used: Presents the graph construction establishing the arbitrary divergent-function case.\n- Google DeepMind, Formal Conjectures, ErdosProblems/750.lean (2026). (formal_verification): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/750.lean\n Evidence used: Current formal record states the proved status, provenance, and dependence on Stiebitz's theorem.\n- Shashi456, Lean proof development for Erdos Problem #750 (2026). (formal_verification): https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P750/Proof.lean\n Evidence used: Linked proof development for the recent resolution.\n- Thomas F. Bloom and contributors, AI contributions to Erdos problems, checked 2026-08-17. (maintained_tracker): https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems\n Evidence used: Records the full-solution claim and links the current proof artifacts.\n\n**Review notes.** The current maintained YAML/formal record is newer than a possibly stale public webpage badge.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2291, "problem_number": "EP-757", "title": "Erdős Problem #757", "statement": "Let $A\\subset \\mathbb{R}$ be a set of size $n$ such that every subset $B\\subseteq A$ with $\\lvert B\\rvert =4$ has $\\lvert B-B\\rvert\\geq 11$. Find the best constant $c>0$ such that $A$ must always contain a Sidon set of size $\\geq cn$.", "background": "For comparison, note that if $B$ were a Sidon set then $\\lvert B-B\\rvert=13$, so this condition is saying that at most one difference is 'missing' from $B-B$. Equivalently, one can view $A$ as a set such that every four points determine at least five distinct distances, and ask for a subset with all distances distinct.\nWithout loss of generality, one can assume $A\\subset \\mathbb{N}$.\nErd\\H{o}s and S\\'{o}s proved that $c\\geq 1/2$. Gy\\'{a}rf\\'{a}s and Lehel \\cite{GyLe95} proved $ \\frac{1}{2}\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The optimal Sidon-subset constant for (4,5)-sets remains unknown, but both previous bounds were improved in 2026.\n\n**Verified partial progress.**\n\n- Gyárfás and Lehel proved 1/2 + 1/(141*76) <= c_* <= 3/5.\n- Ma and Tang improved the interval to 9/17 <= c_* <= 4/7.\n\n**Full solution or refutation.**\n\nThe new interval is rigorous primary-source progress but does not determine the best constant.\n\n**What remains.**\n\nDetermine c_* or further narrow the interval [9/17,4/7].\n\n**Sources checked.**\n\n- Jie Ma and Quanyu Tang, Largest Sidon subsets in weak Sidon sets, arXiv:2602.23282 (2026). (primary): https://arxiv.org/abs/2602.23282\n Evidence used: The abstract explicitly treats the Erdős local-difference problem and proves 9/17 <= c_* <= 4/7.\n- Thomas F. Bloom, Erdős Problem #757, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/757\n Evidence used: Current open status and incorporation of the Ma--Tang bounds.\n\n**Review notes.** The exact statement was preserved. The imported background has trailing JSON/serialization text, which was flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2292, "problem_number": "EP-761", "title": "Erdős Problem #761", "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. The dichromatic number of $G$, denoted by $\\delta(G)$, is the minimum number $k$ of colours required such that, in any orientation of the edges of $G$, there is a $k$-colouring of the vertices of $G$ such that there are no monochromatic oriented cycles.\nMust a graph with large chromatic number have large dichromatic number? Must a graph with large cochromatic number contain a graph with large dichromatic number?", "background": "The first question is due to Erd\\H{o}s and Neumann-Lara. The second question is due to Erd\\H{o}s and Gimbel. A positive answer to the second question implies a positive answer to the first via the bound mentioned in [760].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The ordinary chromatic-to-dichromatic and cochromatic questions remain open; fractional and list analogues are proved.\n\n**Verified partial progress.**\n\n- Mohar and Wu proved a fractional version, with fractional dichromatic number at least t/(8 log_2 t + 4 + 4 log_2 e) when the fractional chromatic number is t.\n- Harutyunyan, Picasarri-Arrieta, and Puig i Surroca proved a list version: large list chromatic number forces an orientation with large list dichromatic number; minimum degree d yields order log d in their list parameter.\n\n**Full solution or refutation.**\n\nNeither analogue implies the ordinary statement because its hypotheses and conclusions use different coloring parameters.\n\n**What remains.**\n\nProve or refute that unbounded ordinary chromatic number forces unbounded ordinary dichromatic number, and settle the cochromatic formulation.\n\n**Sources checked.**\n\n- Bojan Mohar and Hehui Wu, Dichromatic number and fractional chromatic number, Forum of Mathematics, Sigma 4 (2016), e32. (primary): https://doi.org/10.1017/fms.2016.28\n Evidence used: The abstract states and quantifies the fractional version of the Erdős--Neumann-Lara conjecture.\n- Ararat Harutyunyan, Lucas Picasarri-Arrieta, and Gil Puig i Surroca, On the list version of a conjecture of Erdős and Neumann-Lara, arXiv:2603.01020 (2026). (primary): https://arxiv.org/abs/2603.01020\n Evidence used: The abstract explicitly proves the list version and gives the logarithmic minimum-degree tool.\n- Thomas F. Bloom, Erdős Problem #761, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/761\n Evidence used: Current maintained open formulation of both ordinary questions.\n\n**Review notes.** The exact source statement was preserved; its imported background ends in leaked serialization text.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2293, "problem_number": "EP-766", "title": "Erdős Problem #766", "statement": "Let $f(n;k,l)=\\min \\mathrm{ex}(n;G)$, where $G$ ranges over all graphs with $k$ vertices and $l$ edges.\nGive good estimates for $f(n;k,l)$ in the range $k\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The requested estimates for k", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2294, "problem_number": "EP-768", "title": "Erdős Problem #768", "statement": "Let $A\\subset\\mathbb{N}$ be the set of $n$ such that for every prime $p\\mid n$ there exists some $d\\mid n$ with $d>1$ such that $d\\equiv 1\\pmod{p}$. Is it true that there exists some constant $c>0$ such that for all large $N$ $ \\frac{\\lvert A\\cap [1,N]\\rvert}{N}=\\exp(-(c+o(1))\\sqrt{\\log N}\\log\\log N). $ ", "background": "Erd\\H{o}s could prove that there exists some constant $c>0$ such that for all large $N$ $ \\exp(-c\\sqrt{\\log N}\\log\\log N)\\leq \\frac{\\lvert A\\cap [1,N]\\rvert}{N} $ and $ \\frac{\\lvert A\\cap [1,N]\\rvert}{N}\\leq \\exp(-(1+o(1))\\sqrt{\\log N\\log\\log N}). $ Erd\\H{o}s asked about this because $\\lvert A\\cap [1,N]\\rvert$ provides an upper bound for the number of integers $n\\leq N$ for which there is a non-cyclic simple group of order $n$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The proposed exponential-scale density asymptotic remains open; Erdős proved nonmatching upper and lower bounds.\n\n**Verified partial progress.**\n\n- For D(N)=|A intersect [1,N]|/N, Erdős proved D(N) >= exp(-C sqrt(log N) log log N) for an absolute C>0.\n- Erdős also proved D(N) <= exp(-(1+o(1)) sqrt(log N log log N)).\n- Membership in A is a necessary condition for n to be the order of a noncyclic finite simple group.\n\n**Full solution or refutation.**\n\nThe two exponent scales differ, so they do not establish the conjectured exp(-(c+o(1)) sqrt(log N) log log N) asymptotic.\n\n**What remains.**\n\nClose the gap between the two exponent scales and, if the proposed scale is correct, determine or prove existence of the constant c.\n\n**Sources checked.**\n\n- Paul Erdős, Remarks on some problems in number theory, Mathematica Balkanica 4 (1974), 197--202. (primary): https://users.renyi.hu/~p_erdos/1974-27.pdf\n Evidence used: Defines the relevant divisor-witness sequence and proves the upper-bound mechanism; records the conjectural density scale.\n- Thomas F. Bloom, revision history of Erdős Problem #768, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/768\n Evidence used: The current version still records the two bounds and asks for the stated asymptotic.\n\n**Review notes.** The necessary condition on simple-group orders was not promoted to a characterization. The imported background has trailing serialization text.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2295, "problem_number": "EP-769", "title": "Erdős Problem #769", "statement": "Let $c(n)$ be minimal such that if $k\\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good bounds for $c(n)$ - in particular, is it true that $c(n) \\gg n^n$?", "background": "A problem first investigated by Hadwiger, who proved the lower bound $ c(n) \\geq 2^n+2^{n-1}. $ It is easy to see that $c(2)=6$. Meier conjectured $c(3)=48$. Burgess and Erd\\H{o}s \\cite{Er74b} proved $ c(n) \\ll n^{n+1}. $ Erd\\H{o}s wrote 'I am certain that if $n+1$ is a prime then $c(n)>n^n$.'\nHudelson \\cite{Hu98} proved that if $(2^n-1,3^n-1)=1$ then $c(n) < 6^n$, and in general $c(n) \\ll (2n)^{n-1}$. Connor and Marmorino \\cite{CoMa18} proved that $ c(n) \\geq 2^{n+1}-1 $ for all $n\\geq 3$, $ c(n) \\leq 1.8n^{n+1} $ if $n+1$ is prime, and $ c(n) \\leq e^2n^n $ otherwise.\nReferences\n\n\n[CoMa18] Connor, Peter and Marmorino, Phillip, Decomposing cubes into smaller cubes. J. Geom. (2018), Paper No. 19, 11.\n\n[Er74b] Erd\\H{o}s, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.\n\n[Hu98] Hudelson, Matthew, Dissecting {$d$}-cubes into smaller {$d$}-cubes. J. Combin. Theory Ser. A (1998), 190--200.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Strong upper bounds and an improved exponential lower bound are known, but the proposed lower bound c(n) >> n^n remains open.\n\n**Verified partial progress.**\n\n- Connor and Marmorino proved c(n) >= 2^(n+1)-1 for n>=3.\n- They proved c(n) <= 1.8 n^(n+1) when n+1 is prime and c(n) <= e^2 n^n when n+1 is composite.\n- Hudelson proved c(n) << (2n)^(n-1), and c(n)<6^n under gcd(2^n-1,3^n-1)=1.\n\n**Full solution or refutation.**\n\nThe available exponential lower bound is far below n^n, and the upper bounds do not supply the conjectured lower order.\n\n**What remains.**\n\nProve or refute a uniform lower bound of order n^n and sharpen c(n), especially when n+1 is prime.\n\n**Sources checked.**\n\n- Peter Connor and Phillip Marmorino, Decomposing cubes into smaller cubes, Journal of Geometry 109 (2018), article 19. (primary): https://doi.org/10.1007/s00022-018-0424-4\n Evidence used: Primary source for the stated modern lower and upper bounds.\n- Thomas F. Bloom, Erdős Problem #769, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/769\n Evidence used: Current open status and synthesis of Hadwiger, Hudelson, and Connor--Marmorino bounds.\n\n**Review notes.** The source statement was retained exactly. Its background has trailing serialization text.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2296, "problem_number": "EP-770", "title": "Erdős Problem #770", "statement": "Let $h(n)$ be minimal such that $2^n-1,3^n-1,\\ldots,h(n)^n-1$ are mutually coprime.\nDoes, for every prime $p$, the density $\\delta_p$ of integers with $h(n)=p$ exist? Does $\\liminf h(n)=\\infty$? Is it true that if $p$ is the greatest prime such that $p-1\\mid n$ and $p>n^\\epsilon$ then $h(n)=p$?", "background": "It is easy to see that $h(n)=n+1$ if and only if $n+1$ is prime, and that $h(n)$ is unbounded for odd $n$.\nIt is probably true that $h(n)=3$ for infinitely many $n$.\nSee also [820].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The density, liminf, and largest-prime assertions for h(n) all remain open in the maintained record.\n\n**Verified partial progress.**\n\n- It is elementary that h(n)=n+1 if and only if n+1 is prime.\n- The function h(n) is unbounded on odd n; this does not imply liminf h(n)=infinity.\n\n**Full solution or refutation.**\n\nNo verified primary claim resolving any of the three stated questions was located.\n\n**What remains.**\n\nEstablish the densities delta_p, decide whether liminf h(n) diverges, and test the asserted largest-prime criterion.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #770, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/770\n Evidence used: Current open formulation and the two elementary observations.\n\n**Review notes.** Open is dated and conservative. Formalization activity is not counted as mathematical progress. The imported background has trailing serialization text.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2297, "problem_number": "EP-773", "title": "Erdős Problem #773", "statement": "What is the size of the largest Sidon subset $A\\subseteq\\{1,2^2,\\ldots,N^2\\}$? Is it $N^{1-o(1)}$?", "background": "A question of Alon and Erd\\H{o}s \\cite{AlEr85}, who proved $\\lvert A\\rvert \\geq N^{2/3-o(1)}$ is possible (via a random subset), and observed that $ \\lvert A\\rvert \\ll \\frac{N}{(\\log N)^{1/4}}, $ since (as shown by Landau) the density of the sums of two squares decays like $(\\log N)^{-1/2}$. The lower bound was improved to $ \\lvert A\\rvert \\gg N^{2/3} $ by Lefmann and Thiele \\cite{LeTh95}.\nReferences\n\n\n[AlEr85] Alon, Noga and Erd\\H{o}s, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203.\n\n[LeTh95] Lefmann, Hanno and Thiele, Torsten, Point sets with distinct distances. Combinatorica (1995), 379--408.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A 2026 super-polylogarithmic upper saving is the first major upper-bound improvement, but the possible N^(1-o(1)) size remains open.\n\n**Verified partial progress.**\n\n- Alon--Erdős and Lefmann--Thiele established a lower bound of order at least N^(2/3), up to the improvements recorded in the source background.\n- Croot, Mao, Pohoata, Sheffer, and Yip proved |A| <= N exp(-c log N/log log N) for every Sidon subset of the squares.\n\n**Full solution or refutation.**\n\nThe new upper bound is N^(1-c/log log N), so it is compatible with, and does not refute, the question's N^(1-o(1)) scale.\n\n**What remains.**\n\nBridge the exponent gap between the N^(2/3) lower construction and the N exp(-c log N/log log N) upper bound, in particular decide whether N^(1-o(1)) is attainable.\n\n**Sources checked.**\n\n- Ernie Croot, Junzhe Mao, Cosmin Pohoata, Adam Sheffer, and Chi Hoi Yip, A combinatorial large sieve for Sidon sets, distances, and norm forms, arXiv:2606.17487 (2026). (primary): https://arxiv.org/abs/2606.17487\n Evidence used: The abstract states the exact upper bound for Sidon subsets of {1^2,...,N^2}.\n- Thomas F. Bloom, Erdős Problem #773, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/773\n Evidence used: Current open status and synthesis of the lower bound and new upper bound.\n\n**Review notes.** The exact source statement was preserved. Its imported background has trailing serialization text.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2298, "problem_number": "EP-774", "title": "Erdős Problem #774", "statement": "We call $A\\subset \\mathbb{N}$ dissociated if $\\sum_{n\\in X}n\neq \\sum_{m\\in Y}m$ for all finite $X,Y\\subset A$ with $X\neq Y$.\nLet $A\\subset \\mathbb{N}$ be an infinite set. We call $A$ proportionately dissociated if every finite $B\\subset A$ contains a dissociated set of size $\\gg \\lvert B\\rvert$.\nIs every proportionately dissociated set the union of a finite number of dissociated sets?", "background": "This question appears in a paper of Alon and Erd\\H{o}s \\cite{AlEr85}, although the general topic was first considered by Pisier \\cite{Pi83}, who observed that the converse holds, and proved that being proportionately dissociated is equivalent to being a 'Sidon set' in the harmonic analysis sense; that is, whenever $f:A\\to \\mathbb{C}$ there exists some $\\theta\\in [0,1]$ such that $ \\| f\\|_1 \\ll \\left\\lvert\\sum_{n\\in A} f(n)e(n\\theta)\\right\\rvert, $ where $e(x)=e^{2\\pi ix}$.\nAlon and Erd\\H{o}s write that it 'seems unlikely that [this] is also sufficient'. They also point out the same question can be asked replacing dissociated with Sidon (in the additive combinatorial sense) (see [328]). This latter question was resolved in the negative by Ne\\v{s}et\\v{r}il, R\"{o}dl, and Sales \\cite{NRS24}.\nReferences\n\n\n[AlEr85] Alon, Noga and Erd\\H{o}s, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203.\n\n[NRS24] Ne\\v set\\v ril, Jaroslav and R\"odl, Vojt\\v ech and Sales,\nMarcelo, On {P}isier type theorems. Combinatorica (2024), 1211--1232.\n\n[Pi83] Pisier, Gilles, Arithmetic characterizations of Sidon sets. Bull. Amer. Math. Soc. (N.S.) (1983), 87-89.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The finite-union question for dissociated sets remains open; a fixed-order additive analogue is refuted and a bounded-size set-system analogue is positive.\n\n**Verified partial progress.**\n\n- Pisier proved that proportional dissociativity is equivalent to the harmonic-analysis Sidon property and recorded the converse finite-union implication.\n- Nešetřil, Rödl, and Sales constructed, for every h>=2, a set not covered by finitely many B_h-sets although every finite subset contains a positive-proportion B_h-subset.\n- For set systems whose members have size at most k, they proved the positive decomposition implication with at most (4/epsilon) k^2 log k free classes.\n\n**Full solution or refutation.**\n\nThe B_h counterexample does not handle the all-finite-sums dissociated condition, and the bounded-size theorem is only an analogue.\n\n**What remains.**\n\nProve the finite-union conclusion for proportionately dissociated integer sets or construct a genuinely dissociated/free counterexample.\n\n**Sources checked.**\n\n- Gilles Pisier, Arithmetic characterizations of Sidon sets, Bulletin of the American Mathematical Society 8 (1983), 87--89. (primary): https://doi.org/10.1090/S0273-0979-1983-15092-9\n Evidence used: Primary equivalence between the analytic Sidon condition and positive-proportion quasi-independent subsets.\n- Jaroslav Nešetřil, Vojtěch Rödl, and Marcelo Sales, On Pisier Type Theorems, Combinatorica 44 (2024), 1211--1232. (primary): https://doi.org/10.1007/s00493-024-00115-1\n Evidence used: Primary refutation for every fixed B_h analogue and positive bounded-size set-system theorem; it distinguishes the unresolved free-set problem.\n- Thomas F. Bloom, Erdős Problem #774 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/774\n Evidence used: Explicit current open status and warning that the B_h result is only an analogue.\n\n**Review notes.** The input background contains both trailing serialization text and malformed LaTeX accents in the Nešetřil--Rödl--Sales citation; neither was silently repaired in the source.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2299, "problem_number": "EP-776", "title": "Erdős Problem #776", "statement": "Let $r\\geq 2$ and $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ be such that $A_i\not\\subseteq A_j$ for all $i\neq j$ and for any $t$ if there exists some $i$ with $\\lvert A_i\\rvert=t$ then there must exist at least $r$ sets of that size.\nHow large must $n$ be (as a function of $r$) to ensure that there is such a family which achieves $n-3$ distinct sizes of sets?", "background": "A problem of Erd\\H{o}s and Trotter. For $r=1$ and $n>3$ the maximum possible is $n-2$. For $r>1$ and $n$ sufficiently large $n-3$ is achievable, but $n-2$ is never achievable.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The threshold is known exactly for r=2,3 and asymptotically for all r, but its exact value for general r>=4 remains open.\n\n**Verified partial progress.**\n\n- He and Tang proved n_0(2)=3 and n_0(3)=8.\n- For every r>=4 they proved 2r+2 <= n_0(r) <= 2r+2 log_2 r + O(log_2 log_2 r), hence n_0(r) is asymptotic to 2r.\n\n**Full solution or refutation.**\n\nThe asymptotic order and leading constant are determined, but the maintained tracker still labels the request open because the exact general threshold is not known.\n\n**What remains.**\n\nDetermine n_0(r) exactly, or close the additive O(log r) gap, for r>=4.\n\n**Sources checked.**\n\n- Yixin He and Quanyu Tang, An Erdős--Trotter problem on antichains with multiplicity r on each occurring level, arXiv:2602.09803 (2026). (primary): https://arxiv.org/abs/2602.09803\n Evidence used: The abstract defines the threshold in the prompt and gives the exact small-r values and asymptotically sharp bounds.\n- Thomas F. Bloom, Erdős Problem #776, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/776\n Evidence used: Current open label and incorporation of the He--Tang estimates.\n\n**Review notes.** Classified as partial rather than solved because exact n_0(r) remains undetermined for general r and the maintained tracker explicitly remains open. The background has trailing serialization text.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2300, "problem_number": "EP-778", "title": "Erdős Problem #778", "statement": "Alice and Bob play a game on the edges of $K_n$, alternating colouring edges by red (Alice) and blue (Bob). Alice goes first, and wins if at the end the largest red clique is larger than any of the blue cliques.\nDoes Bob have a winning strategy for $n\\geq 3$? (Erd\\H{o}s believed the answer is yes.)", "background": "If we change the game so that Bob colours two edges after each edge that Alice colours, but now require Bob's largest clique to be strictly larger than Alice's, then does Bob have a winning strategy for $n>3$?\nFinally, consider the game when Alice wins if the maximum degree of the red subgraph is larger than the maximum degree of the blue subgraph. Who wins?\nMalekshahian and Spiro \\cite{MaSp24} have proved that, for the first game, the set of $n$ for which Bob wins has density at least $3/4$ - in fact they prove that if Alice wins at $n$ then Bob wins at $n+1,n+2,n+3$.\nSimilarly, for the third game they prove that the set of $n$ for which Bob wins has density at least $2/3$, and prove the stronger statement that if Alice wins at $n$ then Bob wins at $n+1,n+2$.\nReferences\n\n\n[MaSp24] Malekshahian, A. and Spiro, S., On a clique-building game of Erd\\H{o}s. arXiv:2410.18304 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Bob is known to win the clique game for at least density 3/4 of n, but the conjecture that Bob wins for every n>=3 remains open.\n\n**Verified partial progress.**\n\n- Malekshahian and Spiro proved that if Alice wins at n, then Bob wins at n+1,n+2,n+3; consequently Bob-winning n have lower density at least 3/4.\n- For the separate maximum-degree game they proved the analogous two-successor implication and lower density at least 2/3 for Bob.\n\n**Full solution or refutation.**\n\nThe density theorem is the first verified progress but leaves possible exceptional values of n in the first game.\n\n**What remains.**\n\nEliminate all possible Alice-winning n>=3 in the first clique game, or exhibit an exception.\n\n**Sources checked.**\n\n- Alexandru Malekshahian and Sam Spiro, On a Clique-Building Game of Erdős, Journal of Graph Theory (2026), article jgt.70061. (primary): https://doi.org/10.1002/jgt.70061\n Evidence used: Peer-reviewed primary paper; its abstract states the density-3/4 result for the exact first game.\n- Oxford University Research Archive record for Malekshahian--Spiro (published 2026-05-14). (authoritative_secondary): https://ora.ox.ac.uk/objects/uuid%3A9860b403-00c0-47a1-9a2e-609494faec80\n Evidence used: Confirms peer-reviewed publication status, date, DOI, and abstract.\n- Thomas F. Bloom, Erdős Problem #778, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/778\n Evidence used: Current open label and detailed successor/density results for the first and third games.\n\n**Review notes.** The exact first-game statement was preserved; the two additional games appear only in the source background. That background has trailing serialization text.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2301, "problem_number": "EP-782", "title": "Erdős Problem #782", "statement": "Do the squares contain arbitrarily long quasi-progressions? That is, does there exist some constant $C>0$ such that, for any $k$, the squares contain a sequence $x_1,\\ldots,x_k$ where, for some $d$ and all $1\\leq i\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Both square-pattern questions remain open, with a conditional negative result for cubes.\n\n**Verified partial progress.**\n\n- Squares contain no four-term arithmetic progression.\n- Cilleruelo--Granville show conditionally on Bombieri--Lang that squares contain no arbitrarily large cubes.\n- An affirmative quasi-progression answer would imply an affirmative cube answer.\n\n**Full solution or refutation.**\n\nThe conditional result does not resolve either question unconditionally.\n\n**What remains.**\n\nProve or disprove either assertion without Bombieri--Lang.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #782, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/782\n Evidence used: Current open status, implication, and cited conditional result.\n\n**Review notes.** Source statement preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2302, "problem_number": "EP-783", "title": "Erdős Problem #783", "statement": "Fix some constant $C>0$ and let $n$ be large. Let $A\\subseteq \\{2,\\ldots,n\\}$ be such that $(a,b)=1$ for all $a\neq b\\in A$ and $\\sum_{n\\in A}\\frac{1}{n}\\leq C$.\nWhat choice of such an $A$ minimises the number of integers $m\\leq n$ not divisible by any $a\\in A$? Is this minimised by letting $n\\geq q_1>q_2>\\cdots$ be the consecutive primes in decreasing order and choosing $A=\\{q_1,\\ldots,q_k\\}$ where $k$ is maximal such that $ \\sum_{i=1}^k\\frac{1}{q_i}\\leq C? $ \",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Tao proved the sharp asymptotic minimum unsieved density and showed that a prime tail attains it asymptotically, but the literal exact finite optimizer and structural classification requested by the stored statement are not fully settled.\n\n**Verified partial progress.**\n\n- Hildebrand treated the earlier prime-only case.\n- Granville and Soundararajan developed quantitative estimates for the number of unsieved integers.\n- Tao proved that the minimum density is rho(e^C)+o(1), attained asymptotically by a tail of the primes.\n- Recent notes develop rigidity and stability information and identify exact prime-tail behavior in restricted ranges such as C at most log 2.\n\n**Full solution or refutation.**\n\nThe main asymptotic extremal value is now known, but asymptotic attainment does not imply that the literal largest-prime tail is the exact finite minimizer in every parameter range.\n\n**What remains.**\n\nRecover a clean exact formulation and classify exact finite minimizers, including boundary perturbations; distinguish this from the solved sharp-asymptotic problem.\n\n**Sources checked.**\n\n- Terence Tao, Sieving by coprime numbers (2026 manuscript). (primary): https://terrytao.wordpress.com/wp-content/uploads/2026/02/erdos783-3.pdf\n Evidence used: Proves the sharp asymptotic minimum density rho(e^C)+o(1) and asymptotic optimality of a prime tail.\n- Przemek Chojecki, Erdos Problem #783: sharp asymptotic value and a stability program (2026). (primary): https://www.ulam.ai/research/erdos783-rem.pdf\n Evidence used: Explicitly distinguishes the solved asymptotic value from the remaining exact structural minimizer question and develops stability information.\n- Andrew Granville and Kannan Soundararajan, The number of unsieved integers up to x, Acta Arithmetica 115 (2004), arXiv:math/0308009. (primary): https://arxiv.org/abs/math/0308009\n Evidence used: Primary predecessor on quantitative unsieved-density estimates.\n- Thomas F. Bloom, Erdos Problem #783, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/783\n Evidence used: Records the asymptotic resolution; the present audit conservatively separates that result from the literal exact-optimizer wording.\n\n**Review notes.** The dataset statement is badly corrupted by OCR and a leaked JSON fragment; status is assessed against the current source formulation without changing the stored statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2303, "problem_number": "EP-786", "title": "Erdős Problem #786", "statement": "Let $\\epsilon>0$. Is there some set $A\\subset \\mathbb{N}$ of density $>1-\\epsilon$ such that $a_1\\cdots a_r=b_1\\cdots b_s$ with $a_i,b_j\\in A$ can only hold when $r=s$?\nSimilarly, can one always find a set $A\\subset\\{1,\\ldots,N\\}$ with this property of size $\\geq (1-o(1))N$?", "background": "An example of such a set with density $1/4$ is given by the integers $\\equiv 2\\pmod{4}$.\nSelfridge constructed such a set with density $1/e-\\epsilon$ for any $\\epsilon>0$: let $p_1<\\cdotsN^{1/2}$ give an example of a set with size $\\geq (\\log 2)N$. Erd\\H{o}s could improve this constant slightly.\nIn \\cite{Er65} Erd\\H{o}s reports that Ruzsa proved the maximal size of such an $A$ is $\\leq (1-c)N$ for some constant $c>0$ for large $N$, but the proof 'is not yet published'. As far as I know, no such proof was ever published.\nSee also [421] and [795].\nReferences\n\n\n[Er65] Erd\\H{o}s, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The question is formulation-sensitive: allowing repeated factors gives a negative answer, whereas the distinct-factor reading remains open.\n\n**Verified partial progress.**\n\n- Erdős--Ruzsa--Sárközy implies a density bound excluding the first question when repetitions are allowed.\n- The finite repetition-allowed version has explicit density constructions and a fixed positive deficit from density one.\n\n**Full solution or refutation.**\n\nThe historical wording does not settle the repetition convention; no single status can safely be assigned to both readings.\n\n**What remains.**\n\nVerify the intended factor-repetition convention from the original source, then attach the corresponding result.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #786, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/786\n Evidence used: Explicit analysis of the two conventions and consequences.\n\n**Review notes.** This is a formulation ambiguity, not an OCR correction.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2304, "problem_number": "EP-787", "title": "Erdős Problem #787", "statement": "Let $g(n)$ be maximal such that given any set $A\\subset \\mathbb{R}$ with $\\lvert A\\rvert=n$ there exists some $B\\subseteq A$ of size $\\lvert B\\rvert\\geq g(n)$ such that $b_1+b_2\not\\in A$ for all $b_1\neq b_2\\in B$.\nEstimate $g(n)$.", "background": "This function was considered by Erd\\H{o}s and Moser. Choi observed that, without loss of generality, one can assume that $A\\subset \\mathbb{Z}$.\nKlarner proved $g(n) \\gg \\log n$ (indeed, a greedy construction suffices). Choi \\cite{Ch71} proved $g(n) \\ll n^{2/5+o(1)}$. The current best bounds known are $ (\\log n)^{1+c} \\ll g(n) \\ll \\exp(\\sqrt{\\log n}) $ for some constant $c>0$, the lower bound due to Sanders \\cite{Sa21} and the upper bound due to Ruzsa \\cite{Ru05}. Beker \\cite{Be25} has proved $ (\\log n)^{1+\\tfrac{1}{68}+o(1)} \\ll g(n). $ \nReferences\n\n\n[Be25] A. Beker, The Erd\\H{o}s-Moser sum-free set problem via improved bounds for $k$-configurations. arXiv:2501.10203 (2025).\n\n[Ch71] Choi, S. L. G., On a combinatorial problem in number theory. Proc. London Math. Soc. (3) (1971), 629-642.\n\n[Ru05] Ruzsa, Imre Z., Sum-avoiding subsets. Ramanujan J. (2005), 77-82.\n\n[Sa21] Sanders, Tom, The Erd\\H{o}s-Moser sum-free set problem. Canad. J. Math. (2021), 63-107.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The extremal function remains open between polylogarithmic lower and subexponential upper bounds.\n\n**Verified partial progress.**\n\n- Sanders proved g(n)>>(log n)^(1+c).\n- Beker improved the displayed lower exponent to 1+1/68+o(1).\n- Ruzsa proved g(n)<", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2305, "problem_number": "EP-788", "title": "Erdős Problem #788", "statement": "Let $f(n)$ be maximal such that if $B\\subset (2n,4n)\\cap \\mathbb{N}$ there exists some $C\\subset (n,2n)\\cap \\mathbb{N}$ such that $c_1+c_2\not\\in B$ for all $c_1\neq c_2\\in C$ and $\\lvert C\\rvert+\\lvert B\\rvert \\geq f(n)$.\nEstimate $f(n)$. In particular is it true that $f(n)\\leq n^{1/2+o(1)}$?", "background": "A conjecture of Choi \\cite{Ch71}, who proved $f(n) \\ll n^{3/4}$. Adenwalla in the comments has provided a simple construction that proves $f(n) \\gg n^{1/2}$.\nHunter in the comments has sketched an argument that gives $f(n) \\ll n^{2/3+o(1)}$. The bound $ f(n) \\ll (n\\log n)^{2/3} $ was proved by Baltz, Schoen, and Srivastav \\cite{BSS00}.\nReferences\n\n\n[BSS00] Baltz, Andreas and Schoen, Tomasz and Srivastav, Anand, Probabilistic construction of small strongly sum-free sets via\nlarge {S}idon sets. Colloq. Math. (2000), 171--176.\n\n[Ch71] Choi, S. L. G., On a combinatorial problem in number theory. Proc. London Math. Soc. (3) (1971), 629-642.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The square-root upper-bound conjecture remains open; the known upper exponent is 3/5+o(1).\n\n**Verified partial progress.**\n\n- Baltz--Schoen--Srivastav proved f(n)<<(n log n)^(2/3).\n- Alon--Pham random-Cayley-graph work yields f(n)<=n^(3/5+o(1)).\n- A simple construction gives f(n)>>sqrt(n).\n\n**Full solution or refutation.**\n\nNo result reaches the requested n^(1/2+o(1)) upper bound.\n\n**What remains.**\n\nEstablish the conjectured random-Cayley independence estimate or another square-root upper bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #788, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/788\n Evidence used: Current open status and updated bounds.\n- N. Alon and H. T. Pham, Random Cayley graphs and random subsets, arXiv:2509.02561 (2025). (primary): https://arxiv.org/abs/2509.02561\n Evidence used: Input to the stated 3/5+o(1) deduction.\n\n**Review notes.** The literal source has broken LaTex commands `ot` and `eq`; they were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2306, "problem_number": "EP-789", "title": "Erdős Problem #789", "statement": "Let $h(n)$ be maximal such that if $A\\subseteq \\mathbb{Z}$ with $\\lvert A\\rvert=n$ then there is $B\\subseteq A$ with $\\lvert B\\rvert \\geq h(n)$ such that if $a_1+\\cdots+a_r=b_1+\\cdots+b_s$ with $a_i,b_i\\in B$ then $r=s$.\nEstimate $h(n)$.", "background": "Erd\\H{o}s \\cite{Er62c} proved $h(n) \\ll n^{5/6}$. Straus \\cite{St66} proved $h(n) \\ll n^{1/2}$. Erd\\H{o}s noted the bound $h(n)\\gg n^{1/3}$, taking $ B=\\{ a: \\{ \\alpha a\\} \\in n^{-1/3}+\\tfrac{1}{2} (-n^{-2/3},n^{-2/3})\\} $ for a random $\\alpha\\in [0,1]$. \\cite{Er62c} and Choi \\cite{Ch74b} improved this to $h(n) \\gg (n\\log n)^{1/3}$.\nSee also [186] and [874].\nReferences\n\n\n[Ch74b] Choi, S. L. G., On an extremal problem in number theory. J. Number Theory (1974), 105--111.\n\n[Er62c] Erd\\H{o}s, P\\'{a}l, Some remarks on number theory. {III}. Mat. Lapok (1962), 28--38.\n\n[St66] Straus, E. G., On a problem in combinatorial number theory. J. Math. Sci. (1966), 77--80.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The distinct-summand length function remains open between cube-root and square-root scales.\n\n**Verified partial progress.**\n\n- Choi gives h(n)>>(n log n)^(1/3).\n- Straus proved h(n)<", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2307, "problem_number": "EP-790", "title": "Erdős Problem #790", "statement": "Let $l(n)$ be maximal such that if $A\\subset\\mathbb{Z}$ with $\\lvert A\\rvert=n$ then there exists a sum-free $B\\subseteq A$ with $\\lvert B\\rvert \\geq l(n)$ - that is, $B$ is such that there are no solutions to $ a_1=a_2+\\cdots+a_r $ with $a_i\\in B$ all distinct.\nEstimate $l(n)$. In particular, is it true that $l(n)n^{-1/2}\\to \\infty$? Is it true that $l(n)< n^{1-c}$ for some $c>0$?", "background": "Erd\\H{o}s observed that $l(n)\\geq (n/2)^{1/2}$, which Choi improved to $l(n)>(1+c)n^{1/2}$ for some $c>0$. Erd\\H{o}s \\cite{Er73} thought he could prove $l(n)=o(n)$ but had 'difficulties in reconstructing [his] proof'. (In \\cite{Er65} he wrote 'by complicated arguments we can show $l(n)=o(n)$'.)\nChoi, Koml\\'{o}s, and Szemer\\'{e}di \\cite{CKS75} proved $ \\left(\\frac{\\log n}{\\log\\log n}n\\right)^{1/2}\\ll l(n) \\ll \\frac{n}{\\log n}. $ They further conjecture that $l(n)\\geq n^{1-o(1)}$.\nSee also [876].\nReferences\n\n\n[CKS75] Choi, S. L. G. and Koml\\'os, J. and Szemer\\'{e}di, E., On sum-free subsequences. Trans. Amer. Math. Soc. (1975), 307--313.\n\n[Er65] Erd\\H{o}s, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\n\n[Er73] Erd\\H{o}s, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The distinct-summand sum-free subset problem is open with a super-square-root lower bound and sublinear upper bound.\n\n**Verified partial progress.**\n\n- Choi--Komlós--Szemerédi prove ((n log n)/log log n)^(1/2)<", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2308, "problem_number": "EP-791", "title": "Erdős Problem #791", "statement": "Let $g(n)$ be minimal such that there exists $A\\subseteq \\{0,\\ldots,n\\}$ of size $g(n)$ with $\\{0,\\ldots,n\\}\\subseteq A+A$. Estimate $g(n)$. In particular is it true that $g(n)\\sim 2n^{1/2}$?", "background": "Such a set is often called a finite additive $2$-basis. A problem of Rohrbach, who proved in \\cite{Ro37} $ (2+c)n \\leq g(n)^2 \\leq 4n $ for some small constant $c>0$. The current best-known bounds are $ (2.181\\cdots+o(1))n\\leq g(n)^2 \\leq (3.458\\cdots+o(1))n. $ The lower bound is due to Yu \\cite{Yu15}, and the upper bound is due to Kohonen \\cite{Ko17}. (The disproof of $g(n)\\sim 2n^{1/2}$ was accomplished by Mrose \\cite{Mr79}, who gave a construction implying $g(n)^2 \\leq \\frac{7}{2}n$.)\nReferences\n\n\n[Ko17] Kohonen, Jukka, An improved lower bound for finite additive 2-bases. J. Number Theory (2017), 518--524.\n\n[Mr79] Mrose, Arnulf, Untere {S}chranken f\"ur die {R}eichweiten von {E}xtremalbasen\nfester {O}rdnung. Abh. Math. Sem. Univ. Hamburg (1979), 118--124.\n\n[Ro37] Rohrbach, Hans, Ein {B}eitrag zur additiven {Z}ahlentheorie. Math. Z. (1937), 1--30.\n\n[Yu15] Yu, Gang, A new upper bound for finite additive {$h$}-bases. J. Number Theory (2015), 95--104.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The proposed g(n)~2sqrt(n) asymptotic is refuted, but the exact finite additive-2-basis constant is open.\n\n**Verified partial progress.**\n\n- Mrose's construction disproves the proposed asymptotic by giving g(n)^2<=7n/2.\n- Yu proved g(n)^2>=(2.181...+o(1))n and Kohonen proved g(n)^2<=(3.458...+o(1))n.\n\n**Full solution or refutation.**\n\nThe special asymptotic is false; only the broader estimation problem remains.\n\n**What remains.**\n\nDetermine the optimal leading constant.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #791, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/791\n Evidence used: Current open status for estimation, Mrose disproof, and current bounds.\n\n**Review notes.** The original statement is kept; its 'in particular' clause is recorded as false.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2309, "problem_number": "EP-792", "title": "Erdős Problem #792", "statement": "Let $f(n)$ be maximal such that in any $A\\subset \\mathbb{Z}$ with $\\lvert A\\rvert=n$ there exists some sum-free subset $B\\subseteq A$ with $\\lvert B\\rvert \\geq f(n)$, so that there are no solutions to $ a+b=c $ with $a,b,c\\in B$. Estimate $f(n)$.", "background": "Erd\\H{o}s \\cite{Er65} gave a simple proof that shows $f(n) \\geq n/3$. Alon and Kleitman \\cite{AlKl90} improved this to $f(n)\\geq \\frac{n+1}{3}$, and Bourgain \\cite{Bo97} further improved this to $\\frac{n+2}{3}$. The best lower bound known is $ f(n)\\geq \\frac{n}{3}+c\\log\\log n $ for some constant $c>0$, due to Bedert \\cite{Be25b}. The best upper bound known is $ f(n) \\leq \\frac{n}{3}+o(n), $ due to Eberhard, Green, and Manners \\cite{EGM14}.\nThis problem is Problem 1 on Green's open problems list.\nReferences\n\n\n[AlKl90] Alon, N. and Kleitman, D. J., Sum-free subsets. (1990), 13--26.\n\n[Be25b] B. Bedert, Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric sums. arXiv:2502.08624 (2025).\n\n[Bo97] Bourgain, Jean, Estimates related to sumfree subsets of sets of integers. Israel J. Math. (1997), 71-92.\n\n[EGM14] Eberhard, Sean and Green, Ben and Manners, Freddie, Sets of integers with no large sum-free subset. Ann. of Math. (2) (2014), 621-652.\n\n[Er65] Erd\\H{o}s, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The sum-free-subset extremal problem remains open with asymptotically matching leading constant 1/3.\n\n**Verified partial progress.**\n\n- Bedert proves f(n)>=n/3+c log log n.\n- Eberhard--Green--Manners prove f(n)<=n/3+o(n).\n\n**Full solution or refutation.**\n\nThe second-order scale is still unknown.\n\n**What remains.**\n\nDetermine the excess over n/3.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #792, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/792\n Evidence used: Current open status and best bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2310, "problem_number": "EP-793", "title": "Erdős Problem #793", "statement": "Let $F(n)$ be the maximum possible size of a subset $A\\subseteq\\{1,\\ldots,n\\}$ such that $a\nmid bc$ whenever $a,b,c\\in A$ with $a\neq b$ and $a\neq c$. Is there a constant $C$ such that $ F(n)=\\pi(n)+(C+o(1))n^{2/3}(\\log n)^{-2}? $ ", "background": "Erd\\H{o}s \\cite{Er38} proved there exist constants $0\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Chojecki's July 2026 preprint proves the exact requested second-term constant 27/2 for strongly 2-primitive sets.\n\n**Verified partial progress.**\n\n- Erdős established the earlier extremal framework and order bounds.\n- Chan, Győri, and Sárközy, followed by Chan, studied strongly k-primitive analogues before the sharp second term was known.\n\n**Full solution or refutation.**\n\nThe preprint proves F(n)=pi(n)+(27/2+o(1))n^(2/3)/(log n)^2 under the intended convention that b and c may coincide while a is distinct from both.\n\n**What remains.**\n\nThe stated constant problem is closed subject to verification of this very recent preprint; peer review and extensions to broader k-primitive settings remain follow-up work.\n\n**Sources checked.**\n\n- Przemek Chojecki, The Second Term for Strongly 2-Primitive Sets, arXiv:2607.15306 (2026). (primary): https://arxiv.org/abs/2607.15306\n Evidence used: The abstract and theorem give the exact second-term constant 27/2 for the same extremal problem.\n- Thomas F. Bloom, Erdos Problem #793, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/793\n Evidence used: Provides the source history and earlier partial-progress references; its public status badge may lag the July 2026 primary preprint.\n\n**Review notes.** The imported statement has OCR loss in not-divides and not-equal symbols, and the background has a leaked serialized suffix; these are documented rather than silently corrected.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2311, "problem_number": "EP-796", "title": "Erdős Problem #796", "statement": "Let $k\\geq 2$ and let $g_k(n)$ be the largest possible size of $A\\subseteq \\{1,\\ldots,n\\}$ such that every $m$ has $\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The second-order constant for g_3(n) is open, while its first-order asymptotic and O(n/log n) second-order scale are known.\n\n**Verified partial progress.**\n\n- Erdős proved the first-order asymptotic for g_k(n).\n- The maintained discussion explains that the correct known second-order scale for k=3 is n/log n, with positive lower and finite upper constants.\n\n**Full solution or refutation.**\n\nThis does not identify the requested second-order constant.\n\n**What remains.**\n\nProve existence and value of the normalized second-order limit.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #796, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/796\n Evidence used: Current open status, source-normalisation caveat, and established first-order result.\n\n**Review notes.** A historical denominator discrepancy is flagged by the maintained source; the dataset statement is preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2312, "problem_number": "EP-802", "title": "Erdős Problem #802", "statement": "Is it true that any $K_r$-free graph on $n$ vertices with average degree $t$ contains an independent set on $ \\gg_r \\frac{\\log t}{t}n $ many vertices?", "background": "A conjecture of Ajtai, Erd\\H{o}s, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AEKS81}, who proved that there must exist an independent set on $ \\gg_r \\frac{\\log\\log(t+1)}{t}n $ many vertices. Shearer \\cite{Sh95} improved this to $ \\gg_r \\frac{\\log t}{\\log\\log(t+1)t}n. $ Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS80} proved the conjectured bound when $r=3$. Alon \\cite{Al96b} proved the conjectured bound, but replacing the $K_r$-free assumption with the stronger assumption that the induced graph on every vertex neighbourhood has chromatic number $\\leq r-2$.\nReferences\n\n\n[AEKS81] Ajtai, M. and Erd\\H{o}s, P. and Koml\\'{o}s, J. and Szemer\\'{e}di,\nE., On Tur\\'{a}n's theorem for sparse graphs. Combinatorica (1981), 313-317.\n\n[AKS80] Ajtai, Mikl\\'{o}s and Koml\\'{o}s, J\\'{a}nos and Szemer\\'{e}di, Endre, A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.\n\n[Al96b] Alon, Noga, Independence numbers of locally sparse graphs and a Ramsey\ntype problem. Random Structures Algorithms (1996), 271-278.\n\n[Sh95] Shearer, James B., On the independence number of sparse graphs. Random Structures Algorithms (1995), 269--271.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The independent-set bound is open for general K_r-free graphs but known in important special cases.\n\n**Verified partial progress.**\n\n- Ajtai--Komlós--Szemerédi prove the conjectured bound for r=3.\n- Shearer obtains a log-log loss for general r.\n- Alon proves the target under a stronger local chromatic condition.\n\n**Full solution or refutation.**\n\nThe general K_r-free hypothesis is not covered for r>3.\n\n**What remains.**\n\nRemove the remaining log-log loss under the sole K_r-free assumption.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #802, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/802\n Evidence used: Current open status and special-case results.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2313, "problem_number": "EP-805", "title": "Erdős Problem #805", "statement": "For which functions $g(n)$ with $n>g(n)\\geq (\\log n)^2$ is there a graph on $n$ vertices in which every induced subgraph on $g(n)$ vertices contains a clique of size $\\geq \\log n$ and an independent set of size $\\geq \\log n$?\nIn particular, is there such a graph for $g(n)=(\\log n)^3$?", "background": "A problem of Erd\\H{o}s and Hajnal, who thought that there is no such graph for $g(n)=(\\log n)^3$. Alon and Sudakov \\cite{AlSu07} proved that there is no such graph with $ g(n)=\\frac{c}{\\log\\log n}(\\log n)^3 $ for some constant $c>0$.\nAlon, Buci\\'{c}, and Sudakov \\cite{ABS21} construct such a graph with $ g(n)\\leq 2^{2^{(\\log\\log n)^{1/2+o(1)}}}. $ See also [804].\nReferences\n\n\n[ABS21] Alon, Noga and Buci\\'c, Matija and Sudakov, Benny, Large cliques and independent sets all over the place. Proc. Amer. Math. Soc. (2021), 3145-3157.\n\n[AlSu07] Alon, Noga and Sudakov, Benny, On graphs with subgraphs having large independence numbers. J. Graph Theory (2007), 149-157.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Published nonexistence and construction results bracket the local Ramsey threshold, but they do not decide whether the required graph exists at the specifically requested scale g(n)=(log n)^3.\n\n**Verified partial progress.**\n\n- Alon and Sudakov proved nonexistence at scale c(log n)^3/log log n.\n- Alon, Bucić, and Sudakov constructed examples with g(n) at most 2^{2^{(log log n)^{1/2+o(1)}}}.\n\n**Full solution or refutation.**\n\nThe threshold has been narrowed substantially on both sides, while the log-cubed case remains open.\n\n**What remains.**\n\nDecide existence at g(n)=(log n)^3 and, more generally, locate the transition between the published lower obstruction and upper construction.\n\n**Sources checked.**\n\n- Noga Alon and Benny Sudakov, On graphs with subgraphs having large independence numbers, Journal of Graph Theory 56 (2007), 149-157, DOI 10.1002/jgt.20264. (primary): https://arxiv.org/abs/0706.4099\n Evidence used: Primary source for the lower obstruction near the log-cubed scale.\n- Noga Alon, Matija Bucić, and Benny Sudakov, Large cliques and independent sets all over the place, Proceedings of the AMS 149 (2021), 3145-3157, DOI 10.1090/proc/15323. (primary): https://arxiv.org/abs/2004.04718\n Evidence used: Constructs the currently recorded small local-Ramsey examples.\n- Thomas F. Bloom, Erdős Problem #805, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/805\n Evidence used: Maintains open status and records both cited bounds.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2314, "problem_number": "EP-809", "title": "Erdős Problem #809", "statement": "Let $k\\geq 3$ and define $F_k(n)$ to be the minimal $r$ such that there is a graph $G$ on $n$ vertices with $\\lfloor n^2/4\\rfloor+1$ many edges such that the edges can be $r$-coloured so that every subgraph isomorphic to $C_{2k+1}$ has no colour repeating on the edges.\nIs it true that $ F_k(n)\\sim n^2/8? $ ", "background": "A problem of Burr, Erd\\H{o}s, Graham, and S\\'{o}s, who proved that $ F_k(n)\\gg n^2. $ See also [810].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Bucić, Chen, and Ma proved the n^2/8 asymptotic for every k>=4, but the imported assertion ranges over all k>=3 and the k=3 (C7) case remains open.\n\n**Verified partial progress.**\n\n- Burr, Erdős, Graham, and Sós proved a quadratic lower bound depending on k.\n- Bucić, Chen, and Ma confirmed the conjectured asymptotic for all k>=4 and obtained a more general edge-range formula.\n\n**Full solution or refutation.**\n\nAll imported cases except k=3 are now solved affirmatively; this is not a complete solution to the universal statement.\n\n**What remains.**\n\nProve or disprove F_3(n)~n^2/8 for rainbow C7.\n\n**Sources checked.**\n\n- Matija Bucić, Kaizhe Chen, and Jie Ma, On a maximal anti-Ramsey conjecture of Burr, Erdős, Graham, and Sós, arXiv:2603.18952 (2026). (primary): https://arxiv.org/abs/2603.18952\n Evidence used: The abstract and theorem scope explicitly confirm the conjecture for k>=4, not k=3.\n- S. A. Burr, P. Erdős, R. L. Graham, and V. T. Sós, Maximal anti-Ramsey graphs and the strong chromatic number, Journal of Graph Theory 13 (1989), 263-282, DOI 10.1002/jgt.3190130302. (primary): https://doi.org/10.1002/jgt.3190130302\n Evidence used: Primary source for the original formulation and quadratic lower bound.\n- Thomas F. Bloom, Erdős Problem #809, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/809\n Evidence used: Records the 2026 theorem while retaining open status because k=3 remains.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2315, "problem_number": "EP-810", "title": "Erdős Problem #810", "statement": "Does there exist some $\\epsilon>0$ such that, for all sufficiently large $n$, there exists a graph $G$ on $n$ vertices with at least $\\epsilon n^2$ many edges such that the edges can be coloured with $n$ colours so that every $C_4$ receives $4$ distinct colours?", "background": "A problem of Burr, Erd\\H{o}s, Graham, and S\\'{o}s.\nSee also [809].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem was located that gives a positive-density n-vertex graph, using only n edge colours, in which every C4 is rainbow; the maintained tracker still lists the exact question as open.\n\n**Verified partial progress.**\n\n- Burr, Erdős, Graham, and Sós related the problem to a hypergraph (7,4) extremal question.\n- That reduction has not supplied the positive-density construction required by the imported statement.\n\n**Full solution or refutation.**\n\nThe literature search found useful reformulations but no construction or impossibility theorem matching all imported quantifiers.\n\n**What remains.**\n\nConstruct such positive-density graphs for all sufficiently large n, or prove that no fixed positive density is possible.\n\n**Sources checked.**\n\n- S. A. Burr, P. Erdős, R. L. Graham, and V. T. Sós, Maximal anti-Ramsey graphs and the strong chromatic number, Journal of Graph Theory 13 (1989), 263-282, DOI 10.1002/jgt.3190130302. (primary): https://doi.org/10.1002/jgt.3190130302\n Evidence used: Original anti-Ramsey framework and hypergraph connection.\n- Thomas F. Bloom, Erdős Problem #810, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/810\n Evidence used: Retains open status after an April 2026 edit and records no accepted solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2316, "problem_number": "EP-811", "title": "Erdős Problem #811", "statement": "Suppose $n\\equiv 1\\pmod{m}$. We say that an edge-colouring of $K_n$ using $m$ colours is balanced if every vertex sees exactly $\\lfloor n/m\\rfloor$ many edges of each colours.\nFor which graphs $G$ is it true that, if $m=e(G)$, for all large $n\\equiv 1\\pmod{m}$, every balanced edge-colouring of $K_n$ with $m$ colours contains a rainbow copy of $G$? (That is, a subgraph isomorphic to $G$ where each edge receives a different colour.)", "background": "In \\cite{Er91} Erd\\H{o}s credits this problem to himself, Pyber, and Tuza. This problem was explored in a paper of Erd\\H{o}s and Tuza \\cite{ErTu93}. In \\cite{Er96} Erd\\H{o}s seems to suggest that this might be true for every graph $G$, and specifically asks specific challenge posed in \\cite{Er91} and \\cite{Er96} is whether, in any balanced edge-colouring of $K_{6n+1}$ by $6$ colours there must exist a rainbow $C_6$ and $K_4$.\nIn general, one can ask for a quantitative version, defining $d_G(n)$ to be minimal (if it exists) such that if $n$ is sufficiently large and the edges of $K_n$ are coloured with $e(G)$ many colours such that the minimum degree of each colour class is $\\geq d_G(n)$ then there is a rainbow copy of $G$. Erd\\H{o}s and Tuza \\cite{ErTu93} proved that $ \\lfloor n/6\\rfloor \\leq d_{C_4}(n) \\leq \\left(\\frac{1}{4}-c\\right)n $ for some constant $c>0$.\nAxenovich and Clemen \\cite{AxCl24} have proved that there exist infinitely many graphs without this property. In particular, they show that for any odd $\\ell \\geq 3$ and $m=\\lfloor \\sqrt{\\ell}+3.5\\rfloor$ there exist arbitrarily large $n$ such that $K_n$ has a balanced edge-colouring using $\\ell$ colours which contains no rainbow $K_m$. They conjecture that $K_m$ lacks this property for all $m\\geq 4$.\nClemen and Wagner \\cite{ClWa23} proved that $K_4$ does lack this property.\nReferences\n\n\n[AxCl24] Axenovich, Maria and Clemen, Felix C., Rainbow subgraphs in edge-colored complete graphs: answering\ntwo questions by {E}rd\\H{o}s and {T}uza. J. Graph Theory (2024), 57--66.\n\n[ClWa23] Clemen, Felix Christian and Wagner, Adam Zsolt, Balanced edge-colorings avoiding rainbow cliques of size four. Electron. J. Combin. (2023), Paper No. 3.17, 3.\n\n[Er91] Erd\"{o}s, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.\n\n[Er96] Erd\\H{o}s, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9.\n\n[ErTu93] Erd\\H{o}s, Paul and Tuza, Zsolt, Rainbow subgraphs in edge-colorings of complete graphs. (1993), 81--88.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Published constructions show that K4 and infinitely many further cliques do not have the balanced-rainbow property, refuting the old universal expectation, but no characterization of all graphs is known.\n\n**Verified partial progress.**\n\n- Clemen and Wagner constructed balanced six-colourings of K_{13^t} without rainbow K4.\n- Axenovich and Clemen produced infinitely many further clique counterfamilies and conjectured failure for every clique of order at least four.\n\n**Full solution or refutation.**\n\nImportant negative instances are settled, but the open-ended classification and highlighted C6 case remain unresolved.\n\n**What remains.**\n\nCharacterize exactly which graphs have the property; in particular settle C6 and the remaining clique cases.\n\n**Sources checked.**\n\n- Felix Christian Clemen and Adam Zsolt Wagner, Balanced edge-colorings avoiding rainbow cliques of size four, Electronic Journal of Combinatorics 30 (2023), P3.17, DOI 10.37236/11965. (primary): https://arxiv.org/abs/2303.15476\n Evidence used: Explicit balanced colourings without a rainbow K4.\n- Maria Axenovich and Felix C. Clemen, Rainbow subgraphs in edge-colored complete graphs: Answering two questions by Erdős and Tuza, Journal of Graph Theory 106 (2024), 57-66, DOI 10.1002/jgt.23063. (primary): https://arxiv.org/abs/2209.13867\n Evidence used: Gives an infinite family of graph counterexamples under the completely balanced condition.\n- Thomas F. Bloom, Erdős Problem #811, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/811\n Evidence used: Keeps the characterization open and records both negative results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2317, "problem_number": "EP-812", "title": "Erdős Problem #812", "statement": "Is it true that $ \\frac{R(n+1)}{R(n)}\\geq 1+c $ for some constant $c>0$, for all large $n$? Is it true that $ R(n+1)-R(n) \\gg n^2? $ ", "background": "Burr, Erd\\H{o}s, Faudree, and Schelp \\cite{BEFS89} proved that $ R(n+1)-R(n) \\geq 4n-8 $ for all $n\\geq 2$. The lower bound of [165] implies that $ R(n+2)-R(n) \\gg n^{2-o(1)}. $ \nReferences\n\n\n[BEFS89] Burr, S. A. and Erd\\H{o}s, P. and Faudree, R. J. and Schelp, R.\nH., On the difference between consecutive {R}amsey numbers. Utilitas Math. (1989), 115--118.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Neither a fixed multiplicative gap between consecutive diagonal Ramsey numbers nor a quadratic one-step additive gap is known.\n\n**Verified partial progress.**\n\n- Burr, Erdős, Faudree, and Schelp proved R(n+1)-R(n)>=4n-8.\n- A lower bound cited through Erdős Problem 165 implies R(n+2)-R(n)>>n^{2-o(1)}, but only across two steps.\n\n**Full solution or refutation.**\n\nKnown one-step progress is linear, while near-quadratic progress over two steps does not imply either imported assertion.\n\n**What remains.**\n\nProve or disprove either the uniform ratio gap or the quadratic consecutive-difference bound.\n\n**Sources checked.**\n\n- S. A. Burr, P. Erdős, R. J. Faudree, and R. H. Schelp, On the difference between consecutive Ramsey numbers, Utilitas Mathematica 35 (1989), 115-118. (primary): https://combinatorica.hu/~p_erdos/1989-21.pdf\n Evidence used: Proves the recorded 4n-8 lower bound.\n- Thomas F. Bloom, Erdős Problem #812, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/812\n Evidence used: Retains both questions as open and distinguishes the one-step and two-step results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2318, "problem_number": "EP-813", "title": "Erdős Problem #813", "statement": "Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle (a copy of $K_3$) must contain a clique on at least $h(n)$ vertices. Estimate $h(n)$ - in particular, do there exist constants $c_1,c_2>0$ such that $ n^{1/3+c_1}\\ll h(n) \\ll n^{1/2-c_2}? $ ", "background": "A problem of Erd\\H{o}s and Hajnal, who could prove that $ n^{1/3}\\ll h(n) \\ll n^{1/2}. $ Buci\\'{c} and Sudakov \\cite{BuSu23} have proved $ h(n) \\gg n^{5/12-o(1)}. $ \nReferences\n\n\n[BuSu23] M. Buci\\'C and B. Sudakov, Large independent sets from local considerations. arXiv:2007.03667 (2023).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Bucić and Sudakov proved h(n)>>n^{5/12}, giving the requested positive power improvement over n^{1/3}, but no power-saving upper bound below n^{1/2} is known.\n\n**Verified partial progress.**\n\n- The classical range was n^{1/3}<0 and narrow the exponent gap.\n\n**Sources checked.**\n\n- Matija Bucić and Benny Sudakov, Large Independent Sets from Local Considerations, Combinatorica 43 (2023), 505-546, DOI 10.1007/s00493-023-00023-w. (primary): https://arxiv.org/abs/2007.03667\n Evidence used: Proves the n^{5/12} lower bound for the every-seven-vertices local condition.\n- Thomas F. Bloom, Erdős Problem #813, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/813\n Evidence used: Records the improved lower bound while retaining open status.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2319, "problem_number": "EP-817", "title": "Erdős Problem #817", "statement": "Let $k\\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\\{1,\\ldots,N\\}$ contains some $A$ of size $\\lvert A\\rvert=n$ such that $ \\langle A\\rangle = \\left\\{\\sum_{a\\in A}\\epsilon_aa: \\epsilon_a\\in \\{0,1\\}\\right\\} $ contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that $ g_3(n) \\gg 3^n? $ ", "background": "A problem of Erd\\H{o}s and S\\'{a}rk\"{o}zy who proved $ g_3(n) \\gg \\frac{3^n}{n^{O(1)}}. $ \",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The best recorded lower bound g_3(n)>>3^n/n^{O(1)} retains a polynomial loss and does not prove the specifically requested g_3(n)>>3^n; no later matching estimate was located.\n\n**Verified partial progress.**\n\n- Erdős and Sárközy proved g_3(n)>>3^n/n^{O(1)}.\n\n**Full solution or refutation.**\n\nThe known result establishes the exponential base but not the constant-factor lower bound or a general estimate for g_k(n).\n\n**What remains.**\n\nRemove the polynomial loss for k=3 and determine the growth of g_k(n) for general k.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #817, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/817\n Evidence used: Maintains open status and records the Erdős-Sárközy bound, with no accepted later solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2320, "problem_number": "EP-819", "title": "Erdős Problem #819", "statement": "Let $f(N)$ be maximal such that there exists $A\\subseteq \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=\\lfloor N^{1/2}\\rfloor$ such that $\\lvert (A+A)\\cap [1,N]\\rvert=f(N)$. Estimate $f(N)$.", "background": "Erd\\H{o}s and Freud \\cite{ErFr91} proved $ \\left(\\frac{3}{8}-o(1)\\right)N \\leq f(N) \\leq \\left(\\frac{1}{2}+o(1)\\right)N, $ and note that it is closely connected to the size of the largest quasi-Sidon set (see [840]).\nReferences\n\n\n[ErFr91] Erd\\H{o}s, P. and Freud, R., On sums of a {S}idon-sequence. J. Number Theory (1991), 196--205.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The published bounds (3/8-o(1))N<=f(N)<=(1/2+o(1))N remain apart; a May 2026 external write-up claims an improved lower constant about 0.469 but is not yet a published or tracker-incorporated theorem.\n\n**Verified partial progress.**\n\n- Erdős and Freud proved the established lower constant 3/8 and upper constant 1/2.\n- A May 2026 tracker comment claims liminf f(N)/N>=(16 sqrt(2)-17)/12 approximately 0.469, and a later commenter reported no issue in a standard check; this audit treats the claim as unverified.\n\n**Full solution or refutation.**\n\nClassical linear-order bounds are known, but the limiting constant or sharper asymptotic behavior remains unresolved.\n\n**What remains.**\n\nExpert-check the 2026 claimed construction and close the remaining gap to the 1/2 upper bound, ideally determining an asymptotic constant.\n\n**Sources checked.**\n\n- P. Erdős and R. Freud, On sums of a Sidon-sequence, Journal of Number Theory 38 (1991), 196-205, DOI 10.1016/0022-314X(91)90083-N. (primary): https://doi.org/10.1016/0022-314X(91)90083-N\n Evidence used: Primary source for the established 3/8 and 1/2 bounds.\n- Thomas F. Bloom, Erdős Problem #819 and comments, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/819\n Evidence used: Maintains open status and exposes the May 2026 claim without incorporating it as a solution.\n- Leon, write-up claiming an improved lower bound for Erdős Problem #819 (May 2026), unpublished. (authoritative_secondary): https://leon2k2k2k.github.io/erdos819.pdf\n Evidence used: Source of the claimed 0.469 lower constant; included only as an unverified lead.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2321, "problem_number": "EP-820", "title": "Erdős Problem #820", "statement": "Let $H(n)$ be the smallest integer $l$ such that there exist $k0$ such that, for all $\\epsilon>0$, $ H(n) > \\exp(n^{(c-\\epsilon)/\\log\\log n}) $ for infinitely many $n$ and $ H(n) < \\exp(n^{(c+\\epsilon)/\\log\\log n}) $ for all large enough $n$?\nDoes a similar upper bound hold for the smallest $k$ such that $(k^n-1,2^n-1)=1$?", "background": "Erd\\H{o}s \\cite{Er74b} proved that there exists a constant $c>0$ such that $ H(n) > \\exp(n^{c/(\\log\\log n)^2}) $ for infinitely many $n$.\nvan Doorn in the comments sketches a proof of the lower bound: that there exists some constant $c>0$ and infinitely many $n$ such that $ H(n) > \\exp(n^{c/\\log\\log n}). $ The sequence $H(n)$ for $1\\leq n\\leq 10$ is $ 3,3,3,6,3,18,3,6,3,12. $ The sequence of $n$ for which $(2^n-1,3^n-1)=1$ is A263647 in the OEIS.\nSee also [770].\nReferences\n\n\n[Er74b] Erd\\H{o}s, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A tracker comment sketches a deduction of the conjectured exp(n^{c/log log n}) infinite-often lower scale from a published Adleman-Pomerance-Rumely theorem, but the derivation is not yet a refereed EP-820 result and all upper questions remain open.\n\n**Verified partial progress.**\n\n- Erdős proved H(n)>exp(n^{c/(log log n)^2}) for infinitely many n.\n- Wouter van Doorn's 2025 tracker comment derives H(n)>exp(n^{c/log log n}) infinitely often from Proposition 10 of Adleman-Pomerance-Rumely.\n\n**Full solution or refutation.**\n\nThe lower scale appears to have been reached up to an unspecified positive constant, but this is comment-level rather than fully published verification and does not address the matching upper or H(n)=3 questions.\n\n**What remains.**\n\nIndependently verify and publish the lower-bound deduction; settle H(n)=3 infinitely often, prove a matching all-large-n upper bound and constant, and handle the fixed-base variant.\n\n**Sources checked.**\n\n- Leonard M. Adleman, Carl Pomerance, and Robert S. Rumely, On distinguishing prime numbers from composite numbers, Annals of Mathematics 117 (1983), 173-206. (primary): https://annals.math.princeton.edu/1983/117-1/p07\n Evidence used: Published theorem used by the tracker-comment deduction.\n- Thomas F. Bloom, Erdős Problem #820 and comments, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/820\n Evidence used: Maintains open status, records Erdős's weaker bound, and labels the stronger lower result as a sketch in the comments.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2322, "problem_number": "EP-821", "title": "Erdős Problem #821", "statement": "Let $g(n)$ count the number of $m$ such that $\\phi(m)=n$. Is it true that, for every $\\epsilon>0$, there exist infinitely many $n$ such that $ g(n) > n^{1-\\epsilon}? $ ", "background": "Pillai proved that $\\limsup g(n)=\\infty$ and Erd\\H{o}s \\cite{Er35b} proved that there exists some constant $c>0$ such that $g(n) >n^c$ for infinitely many $n$.\nThis conjecture would follow if we knew that, for every $\\epsilon>0$, there are $\\gg_\\epsilon \\frac{x}{\\log x}$ many primes $p n^{0.71568\\cdots}, $ obtained by Lichtman \\cite{Li22} as a consequence of proving that there are $\\geq \\frac{x}{(\\log x)^{O(1)}}$ many primes $p\\leq x$ such that all prime factors of $p-1$ are $\\leq x^{0.2843\\cdots}$ (which improves a number of previous exponents, most recently Baker and Harman \\cite{BaHa98}).\nThe average size of $g(n)$ was investigated by Luca and Pollack \\cite{LuPo11}.\nSee also [416].\nReferences\n\n\n[BaHa98] Baker, R. C. and Harman, G., Shifted primes without large prime factors. Acta Arith. (1998), 331--361.\n\n[Er35b] Erd\\H{o}s, P., On the normal number of prime factors of $p-1$ and some related problems concerning Euler's $\\varphi$-function. Quart. J. Math. (1935), 205-213.\n\n[Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).\n\n[LuPo11] Luca, Florian and Pollack, Paul, An arithmetic function arising from {C}armichael's conjecture. J. Th\\'{e}or. Nombres Bordeaux (2011), 697--714.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Lichtman's shifted-prime theorem yields infinitely many n with totient multiplicity g(n)>n^{0.71568...}, a substantial fixed-exponent result that remains short of every exponent 1-epsilon.\n\n**Verified partial progress.**\n\n- Pillai proved limsup g(n)=infinity, and Erdős proved a positive fixed-power lower bound infinitely often.\n- Lichtman's smooth shifted-prime exponent 0.2843... yields the current recorded multiplicity exponent 0.71568....\n\n**Full solution or refutation.**\n\nA strong fixed exponent below one is known, but the conjecture requires exponents arbitrarily close to one.\n\n**What remains.**\n\nObtain sufficiently many primes p for which p-1 has only p^epsilon-sized prime factors for arbitrarily small epsilon, or find another route to multiplicity n^{1-o(1)}.\n\n**Sources checked.**\n\n- Jared Duker Lichtman, Primes in arithmetic progressions to large moduli, and shifted primes without large prime factors, arXiv:2211.09641 (2022). (primary): https://arxiv.org/abs/2211.09641\n Evidence used: Proves infinitude of shifted primes p-1 without factors above p^{0.2844}, producing the recorded totient-multiplicity exponent.\n- Thomas F. Bloom, Erdős Problem #821, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/821\n Evidence used: Maintains open status and records g(n)>n^{0.71568...} infinitely often as the best known bound.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2323, "problem_number": "EP-824", "title": "Erdős Problem #824", "statement": "Let $h(x)$ count the number of integers $1\\leq ax^{2-o(1)}$?", "background": "Erd\\H{o}s \\cite{Er74b} proved that $\\limsup h(x)/x= \\infty$, and claimed a similar proof for this problem. A complete proof that $h(x)/x\\to \\infty$ was provided by Pollack and Pomerance \\cite{PoPo16}.\nA similar question can be asked if we replace the condition $(a,b)=1$ with the condition that $a$ and $b$ are squarefree. Weisenberg suggests another variant, with the condition that there are no proper factors $u\\mid a$ and $v\\mid b$ such that $\\sigma(u)=\\sigma(v)$ and $(u,a/u)=(v,b/v)=1$, which is the weakest restriction one can impose that is still strong enough to eliminate trivial duplicates.\nReferences\n\n\n[Er74b] Erd\\H{o}s, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.\n\n[PoPo16] Pollack, Paul and Pomerance, Carl, Some problems of Erd\\H{o}s on the sum-of-divisors function. Trans. Amer. Math. Soc. Ser. B (2016), 1-26.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Pollack and Pomerance proved h(x)/x tends to infinity, but no nearly quadratic lower bound h(x)>x^(2-o(1)) was found.\n\n**Verified partial progress.**\n\n- Erdos proved a limsup form of superlinear growth.\n- Pollack and Pomerance supplied a complete proof that h(x)/x tends to infinity.\n\n**Full solution or refutation.**\n\nThe proved superlinear lower bound is qualitatively much smaller than the conjectured nearly quadratic count. The maintained tracker retains open status and reports no newer complete or partial claim in its comments.\n\n**What remains.**\n\nProve or refute h(x)>x^(2-o(1)), and determine the growth of the coprime equal-sigma pair count more sharply.\n\n**Sources checked.**\n\n- Paul Pollack and Carl Pomerance, Some problems of Erdos on the sum-of-divisors function, Transactions of the AMS Series B 3 (2016), 1-26. (primary): https://doi.org/10.1090/btran10\n Evidence used: Primary complete proof that h(x)/x tends to infinity; it does not prove the nearly quadratic conjecture.\n- Thomas F. Bloom, Erdos Problem #824, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/824\n Evidence used: Current open status, exact conjecture, and summary of the Pollack-Pomerance theorem.\n\n**Review notes.** The exact statement is preserved. Its background has the common trailing serialized corruption, which was flagged and not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2324, "problem_number": "EP-825", "title": "Erdős Problem #825", "statement": "Is there an absolute constant $C>0$ such that every integer $n$ with $\\sigma(n)>Cn$ is the distinct sum of proper divisors of $n$?", "background": "A problem of Benkoski and Erd\\H{o}s. In other words, this problem asks for an upper bound for the abundancy index of weird numbers. This could be true with $C=3$. We must have $C>2$ since $\\sigma(70)=144$ but $70$ is not the distinct sum of integers from $\\{1,2,5,7,10,14,35\\}$.\nErd\\H{o}s suggested that as $C\\to \\infty$ only divisors at most $\\epsilon n$ need to be used, where $\\epsilon \\to 0$.\nWeisenberg has observed that if $n$ is a weird number with an abundancy index $\\geq 4$ then it is divisible by an odd weird number. In particular, if there are no odd weird numbers (see [470]) then every weird number has abundancy index $<4$. Indeed, if $l(n)$ is the abundancy index and $n=2^km$ with $m$ odd then $l(n)=l(2^k)l(m)$, and $l(2^k)<2$ so if $l(n)\\geq 4$ then $l(m)>2$, and hence $m$ is weird (as a factor of a weird number).\nA similar argument shows that either there are infinitely many primitive weird numbers or there is an upper bound for the abundancy index of all weird numbers.\nSee also [18] and [470].\nThis is part of problem B2 in Guy's collection \\cite{Gu04} (the \\$25 is reported by Guy as offered by Erd\\H{o}s for a solution to this question).\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The bounded-abundancy question is solved affirmatively. Hisamoto's July 2026 preprint proves the exact explicit implication sigma(n)/n>633 => n is semiperfect, while the maintained tracker independently credits Larsen with an earlier and stronger large-prime-factor theorem. The expected optimal threshold C=3 and Erdős's small-divisor strengthening are not supplied by these results.\n\n**Verified partial progress.**\n\n- Benkoski and Erdős introduced weird numbers and asked whether their abundancy indices are bounded.\n- The maintained tracker records Larsen's stronger theorem: for every epsilon>0, sufficiently rough n with sigma(n)>(2+epsilon)n is semiperfect.\n- Hisamoto proves a direct unconditional numerical bound C=633 for all integers n, independently settling the exact displayed existence question.\n- The example n=70 shows that a universal threshold cannot be 2.\n\n**Full solution or refutation.**\n\nHisamoto converts large sigma(n)/n into a large reciprocal sum over distinct prime divisors, deletes small, thin, and overly large portions, and obtains a prime set whose squarefree product has a slowly growing sequence of proper divisors spanning the target subset sum. That squarefree divisor is semiperfect. Since a multiple of a semiperfect integer is semiperfect by scaling the distinct-divisor representation, n itself is semiperfect. The theorem gives the explicit sufficient constant 633.\n\n**What remains.**\n\nOptimize the constant 633 toward the expected value 3; publish and independently audit Larsen's stronger proof; determine Erdős's strengthening in which only divisors at most epsilon n are used as C grows; and keep the odd-weird-number and primitive-weird-number questions separate. Every input background in this batch has trailing serialization noise, but the EP-825 statement itself is intact.\n\n**Sources checked.**\n\n- Kei Hisamoto, On weird numbers with high abundancy index, arXiv:2607.25278 (2026). (primary): https://arxiv.org/abs/2607.25278\n Evidence used: Theorem 1.1 proves that sigma(n)/n>633 implies n is semiperfect, exactly answering the displayed existence question.\n- Thomas F. Bloom, Erdős Problem #825 (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/825\n Evidence used: Records the problem as proved by Larsen and states his stronger large-prime-factor formulation.\n- S. J. Benkoski and P. Erdős, On Weird and Pseudoperfect Numbers, Mathematics of Computation 28 (1974), 617--623. (primary): https://combinatorica.hu/~p_erdos/1974-24.pdf\n Evidence used: Original primary source for the definitions, bounded-abundancy question, and related weird-number context.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2325, "problem_number": "EP-826", "title": "Erdős Problem #826", "statement": "Are there infinitely many $n$ such that, for all $k\\geq 1$, $ \\tau(n+k)\\ll k? $ ", "background": "A stronger form of [248].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Lau proves a polynomial bound tau(n+k)<=2 and infinitely many n.\n- The resulting Corollary 1.2 gives tau(n+k)<=2. The common trailing background corruption was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2326, "problem_number": "EP-827", "title": "Erdős Problem #827", "statement": "Let $n_k$ be minimal such that if $n_k$ points in $\\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\\binom{k}{3}$ triples determine circles of different radii.\nDetermine $n_k$.", "background": "In \\cite{Er75h} Erd\\H{o}s asks whether $n_k$ exists. In \\cite{Er78c} he gave a simple argument which proves that it does, and in fact $ n_k \\leq k+2\\binom{k-1}{2}\\binom{k-1}{3}, $ but this argument is incorrect, as explained by Martinez and Rold\\'{a}n-Pensado \\cite{MaRo15}.\nMartinez and Rold\\'{a}n-Pensado give a corrected argument that proves $n_k\\ll k^9$.\nReferences\n\n\n[Er75h] Erd\\H{o}s, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.\n\n[Er78c] Erd\\H{o}s, P., Some more problems on elementary geometry. Austral. Math. Soc. Gaz. (1978), 52-54.\n\n[MaRo15] Mart\\'{I}nez, L. and Rold\\'an-Pensado, E., Points defining triangles with distinct circumradii. Acta Math. Hungar. (2015), 136--141.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** A primary paper proves n_k=O(k^9), and a checked 2026 forum argument improves this to O(k^5); the order of n_k is not determined.\n\n**Verified partial progress.**\n\n- Martinez and Roldan-Pensado expose the gap in Erdos's claimed bound and prove n_k=O(k^9), n_4<=9, and n_5<=37.\n- A May 2026 conflict-counting and random-deletion argument on the maintained forum claims n_k<= (3k)^5.\n\n**Full solution or refutation.**\n\nExistence and a polynomial upper bound are rigorous in the primary literature. The O(k^5) improvement is a concrete community-checked argument but lacks an archival primary source. Neither result determines n_k.\n\n**What remains.**\n\nEstablish matching upper and lower bounds, and fix a single explicit meaning of general position for the problem.\n\n**Sources checked.**\n\n- Leonardo Martinez and Edgardo Roldan-Pensado, Points defining triangles with distinct circumradii, Acta Mathematica Hungarica 145 (2015), 136-141. (primary): https://arxiv.org/abs/1402.6276\n Evidence used: Primary correction of Erdos's proof and O(k^9) theorem, including explicit small cases.\n- Erdos Problems discussion thread for Problem #827, checked 2026-08-17. (source_collection): https://www.erdosproblems.com/forum/thread/827\n Evidence used: Contains the detailed O(k^5) probabilistic alteration argument and positive follow-up assessment.\n- Thomas F. Bloom, Erdos Problem #827, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/827\n Evidence used: Retains open status and now records both the primary O(k^9) and forum O(k^5) bounds.\n\n**Review notes.** The source leaves general position undefined. The primary paper discusses no three collinear/no four concyclic and then adopts no four on a line or circle; the forum proof assumes circular general position. No source wording was repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2327, "problem_number": "EP-828", "title": "Erdős Problem #828", "statement": "Is it true that, for any $a\\in\\mathbb{Z}$, there are infinitely many $n$ such that $ \\phi(n) \\mid n+a? $ ", "background": "A conjecture of Graham. Lehmer has conjectured that $\\phi(n)\\mid n-1$ if and only if $n$ is prime. It is an easy exercise to show that $\\phi(n) \\mid n$ if and only if $n=2^a3^b$.\nThis is discussed in problem B37 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Graham's conjecture that every fixed integer shift a has infinitely many n with phi(n) dividing n+a remains open.\n\n**Verified partial progress.**\n\n- The historical survey reports that the assertion is known for infinitely many shift values a.\n- Special shifts such as a=0 and a=-1 have elementary infinite families, but they do not address every integer a.\n\n**Full solution or refutation.**\n\nNo reliable source establishing the universal fixed-shift assertion was found. A 2025 unrefereed web manuscript claiming a complete classification was excluded because it lacks independent verification and has not been incorporated by the maintained tracker.\n\n**What remains.**\n\nFor every fixed integer a, construct infinitely many n satisfying phi(n)|n+a, or produce a shift for which only finitely many exist.\n\n**Sources checked.**\n\n- Ronald L. Graham, Old and New Problems and Results in Number Theory (1980 problem survey). (primary): https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf\n Evidence used: States Graham's fixed-shift conjecture and reports it known for infinitely many shifts, not all shifts.\n- Thomas F. Bloom, Erdos Problem #828, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/828\n Evidence used: Current open status and zero-comment record for the all-shifts conjecture.\n\n**Review notes.** The Lehmer iff-prime background tacitly needs n>1 if one's natural-number convention includes 1. The source was not changed; the common trailing corruption was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2328, "problem_number": "EP-829", "title": "Erdős Problem #829", "statement": "Let $A\\subset\\mathbb{N}$ be the set of cubes. Is it true that $ 1_A\\ast 1_A(n) \\ll (\\log n)^{O(1)}? $ ", "background": "Mordell proved that $ \\limsup_{n\\to \\infty} 1_A\\ast 1_A(n)=\\infty $ and Mahler \\cite{Ma35b} proved $ 1_A\\ast 1_A(n) \\gg (\\log n)^{1/4} $ for infinitely many $n$. Stewart \\cite{St08} improved this to $ 1_A\\ast 1_A(n) \\gg (\\log n)^{11/13}. $ \nReferences\n\n\n[Ma35b] Mahler, Kurt, On the Lattice Points on Curves of Genus 1. Proc. London Math. Soc. (2) (1935), 431-466.\n\n[St08] Stewart, Cameron L., Cubic {T}hue equations with many solutions. Int. Math. Res. Not. IMRN (2008), Art. ID rnn040, 11.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The standard uniform bound for representations as a sum of two cubes is n^(o(1)), while the requested fixed-polylogarithmic upper bound remains open.\n\n**Verified partial progress.**\n\n- Factorization of x^3+y^3 yields the classical n^(o(1)) uniform representation bound.\n- Stewart's high-rank cubic Thue construction yields infinitely many n with at least a constant times (log n)^(11/13) representations, as recorded by the tracker.\n\n**Full solution or refutation.**\n\nKnown lower examples are compatible with a polylogarithmic upper bound, and the n^(o(1)) general upper bound is much weaker than any fixed power of log n. No current source closes that gap.\n\n**What remains.**\n\nProve a uniform (log n)^C upper bound for some absolute C or construct fibers growing faster than every fixed logarithmic power.\n\n**Sources checked.**\n\n- Cameron L. Stewart, Cubic Thue equations with many solutions, International Mathematics Research Notices 2008, rnn040. (primary): https://doi.org/10.1093/imrn/rnn040\n Evidence used: Primary construction of cubic Thue equations with many solutions underlying the large sum-of-two-cubes fibers.\n- James Maynard, Sums of three positive cubes, Journal of the London Mathematical Society (2026). (primary): https://doi.org/10.1112/jlms.70554\n Evidence used: Recent primary use of the standard m^(o(1)) upper bound for the number of sum-of-two-cubes representations.\n- Thomas F. Bloom, Erdos Problem #829, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/829\n Evidence used: Current open status and the Mahler-Stewart logarithmic-power lower examples.\n\n**Review notes.** The convolution's ordered-versus-unordered convention changes only a constant and is not explicit in the input. The common trailing background fragment was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2329, "problem_number": "EP-830", "title": "Erdős Problem #830", "statement": "We say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$. Are there infinitely many amicable pairs? If $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then is it true that $ A(x)>x^{1-o(1)}? $ ", "background": "For example $220$ and $284$. Erd\\H{o}s \\cite{Er55b} proved that $A(x)=o(x)$, and Pomerance \\cite{Po81} improved this to $ A(x) \\leq x \\exp(-(\\log x)^{1/3}) $ and later \\cite{Po15} to $ A(x) \\leq x \\exp(-(\\tfrac{1}{2}+o(1))(\\log x\\log\\log x)^{1/2}). $ This is problem B4 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er55b] Erd\"{o}s, P., On amicable numbers. Publ. Math. Debrecen (1955), 108-111.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Po15] Pomerance, Carl, On amicable numbers. (2015), 321-327.\n\n[Po81] Pomerance, Carl, On the distribution of amicable numbers. {II}. J. Reine Angew. Math. (1981), 183-188.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It is still unknown whether infinitely many amicable pairs exist or whether their counting function is x^(1-o(1)); the imported best upper bound is mistranscribed.\n\n**Verified partial progress.**\n\n- Erdos proved that amicable numbers have asymptotic density zero.\n- Pomerance proves A(x)<=x exp(-(1/2+o(1)) sqrt(log x log log log x)).\n\n**Full solution or refutation.**\n\nAll verified advances are upper bounds. They neither prove infinitude nor a nearly linear lower count. The primary 2015 theorem has log log log x under the square root, whereas the input and maintained page display the materially stronger log log x expression.\n\n**What remains.**\n\nProve infinitely many amicable pairs and establish or refute the x^(1-o(1)) lower growth; correct the upstream transcription of Pomerance's theorem.\n\n**Sources checked.**\n\n- Carl Pomerance, On amicable numbers, in Analytic Number Theory, Springer (2015), 321-327. (primary): https://math.dartmouth.edu/~carlp/amicablesv3.pdf\n Evidence used: Theorem 1.1 gives the best cited upper bound with sqrt(log x log log log x), exposing the input's missing iterated logarithm.\n- Paul Erdos, On amicable numbers, Publicationes Mathematicae Debrecen 4 (1955), 108-111. (primary): https://users.renyi.hu/~p_erdos/1955-03.pdf\n Evidence used: Primary density-zero theorem and explicit nearly linear lower-growth conjecture.\n- Thomas F. Bloom, Erdos Problem #830, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/830\n Evidence used: Maintains both questions as open but repeats the same best-bound transcription defect as the imported record.\n\n**Review notes.** The exact input was not repaired. Its Po15 formula uses log log x, but the primary paper uses log log log x. The common serialized background tail was also preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2330, "problem_number": "EP-831", "title": "Erdős Problem #831", "statement": "Let $h(n)$ be maximal such that in any $n$ points in $\\mathbb{R}^2$ (with no three on a line and no four on a circle) there are at least $h(n)$ many circles of different radii passing through three points. Estimate $h(n)$.\n\",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The maintained problem remains open; a detailed 2026 forum post claims (n-2)/2<=h(n)<=n^2 exp(O(sqrt(log n))) without an archival or independent verification source.\n\n**Verified partial progress.**\n\n- A simple double count gives h(n)>=(n-2)/2.\n- A forum paraboloid-lift and generic-projection construction claims h(n)<=n^2 exp(O(sqrt(log n))).\n\n**Full solution or refutation.**\n\nThe reported bounds leave a gap from linear to nearly quadratic. The upper construction is mathematically developed but appears only in an unverified user comment, so it is retained as a claim rather than promoted to established literature.\n\n**What remains.**\n\nVerify and publish the reported upper construction, and determine the correct growth rate between the linear lower and near-quadratic upper bounds.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdos Problem #831, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/831\n Evidence used: Current open status and exact clean tracker formulation.\n- Erdos Problems discussion thread for Problem #831, post of 16 June 2026. (source_collection): https://www.erdosproblems.com/forum/thread/831\n Evidence used: Detailed community derivation of the linear lower and n^2 exp(O(sqrt(log n))) upper bounds; comments are explicitly unverified.\n\n**Review notes.** Unlike the other records, the serialized difficulty fragment is appended directly to this record's statement and its background is empty. The damaged statement was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2331, "problem_number": "EP-836", "title": "Erdős Problem #836", "statement": "Let $r\\geq 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$ (that is, there is a $3$-colouring of the vertices of $G$ such that no edge is monochromatic).\nSuppose any two edges of $G$ have a non-empty intersection. Must $G$ contain $O(r^2)$ many vertices? Must there be two edges which meet in $\\gg r$ many vertices?", "background": "A problem of Erd\\H{o}s and Shelah. The Fano geometry gives an example where there are no two edges which meet in $r-1$ vertices. Are there any other examples?\nErd\\H{o}s and Lov\\'{a}sz \\cite{ErLo75} proved that there must be two edges which meet in $\\gg \\frac{r}{\\log r}$ many vertices.\nAlon has provided the following counterexample to the first question: as vertices take two sets $X$ and $Y$ of sizes $2r-2$ and $\\frac{1}{2}\\binom{2r-2}{r-1}$ respectively, where $Y$ corresponds to all partitions of $X$ into two equal parts. The edges are all subsets of $X$ of size $r$, and also all sets consisting of a subset of $X$ of size $r-1$ together with the unique element of $Y$ corresponding to the induced partition of $X$.\nThis hypergraph is intersecting, its chromatic number is $3$, and it has $\\asymp 4^r/\\sqrt{r}$ many vertices.\nReferences\n\n\n[ErLo75] Erd\\H{o}s, P. and Lov\\'{a}sz, L., Problems and results on {$3$}-chromatic hypergraphs and some\nrelated questions. (1975), 609--627.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Alon's explicit exponential-vertex construction refutes the first question, while the linear-intersection question has a recent checked forum claim but no stable proof source located.\n\n**Verified partial progress.**\n\n- Alon's construction has asymptotically 4^r/sqrt(r) vertices, is intersecting and 3-chromatic, disproving the O(r^2) vertex assertion.\n- Erdos and Lovasz prove some pair of edges intersects in Omega(r/log r) vertices.\n- Bucic, Glock, and Sudakov prove a lower bound on the number of distinct intersection sizes and continue to identify a linear bound as desirable.\n\n**Full solution or refutation.**\n\nThe first displayed question has a durable negative answer. For the second, the maintained forum records an April 2026 AI-generated claimed affirmative proof and a standard check with no issue, but the link did not yield a stable manuscript, formal proof, or incorporated status change; it is not promoted.\n\n**What remains.**\n\nSupply a stable expert-verified proof of the claimed Omega(r) intersection or find a counterexample, and document Alon's construction in a citable archival source.\n\n**Sources checked.**\n\n- Paul Erdos and Laszlo Lovasz, Problems and results on 3-chromatic hypergraphs and some related questions (1975), 609-627. (primary): https://www.researchgate.net/publication/265681632_Problems_and_results_on_3-chromatic_Hypergraphs_and_some_related_questions\n Evidence used: Classical Omega(r/log r) intersection result and original problem context.\n- Matija Bucic, Stefan Glock, and Benny Sudakov, The intersection spectrum of 3-chromatic intersecting hypergraphs, Proceedings of the London Mathematical Society 124 (2022). (primary): https://doi.org/10.1112/plms.12436\n Evidence used: Primary modern intersection-spectrum theorem and discussion of the still-desired linear-scale improvement.\n- Thomas F. Bloom, Erdos Problem #836, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/836\n Evidence used: Records Alon's explicit counterexample to the first question and retains open status for the record.\n- Erdos Problems discussion thread for Problem #836, checked 2026-08-17. (source_collection): https://www.erdosproblems.com/forum/thread/836\n Evidence used: Records the recent AI solution claim for the second question and a positive standard check, but supplies no stable archival proof in the searchable text.\n\n**Review notes.** The top-level input category number_theory conflicts with the record's graph-theory/hypergraph tags and tracker classification. This metadata defect and the common background corruption were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2332, "problem_number": "EP-837", "title": "Erdős Problem #837", "statement": "Let $k\\geq 2$ and $A_k\\subseteq [0,1]$ be the set of $\\alpha$ such that there exists some $\\beta(\\alpha)>\\alpha$ with the property that, if $G_1,G_2,\\ldots$ is a sequence of $k$-uniform hypergraphs with $ \\liminf \\frac{e(G_n)}{\\binom{\\lvert G_n\\rvert}{k}} >\\alpha $ then there exist subgraphs $H_n\\subseteq G_n$ such that $\\lvert H_n\\rvert \\to \\infty$ and $ \\liminf \\frac{e(H_n)}{\\binom{\\lvert H_n\\rvert}{k}} >\\beta, $ and further that this property does not necessarily hold if $>\\alpha$ is replaced by $\\geq \\alpha$.\nWhat is $A_3$?", "background": "A problem of Erd\\H{o}s and Simonovits. It is known that $ A_2 = \\left\\{ 1-\\frac{1}{k} : k\\geq 1\\right\\}. $ \n\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The full set A_3 of jump densities for 3-uniform hypergraphs remains undetermined despite many known jumps and explicit non-jumps.\n\n**Verified partial progress.**\n\n- For ordinary graphs the analogous set A_2 is completely known.\n- Frankl and Rodl constructed hypergraph non-jumps, disproving the expectation that all densities jump.\n- Recent work continues to construct individual non-jumps for r=3 but does not classify A_3.\n\n**Full solution or refutation.**\n\nThe jump-density literature provides both positive intervals and negative examples. No source found claims a complete characterization of A_3, and the maintained page has no solution claim.\n\n**What remains.**\n\nCharacterize exactly which alpha in [0,1] have the strict-versus-nonstrict density-jump property encoded by A_3.\n\n**Sources checked.**\n\n- Peter Frankl and Vojtech Rodl, Hypergraphs do not jump, Combinatorica 4 (1984), 149-159. (primary): https://doi.org/10.1007/BF02579215\n Evidence used: Primary construction of hypergraph non-jumps, disproving the conjecture that every density is a jump.\n- Paul Erdos and Miklos Simonovits, Supersaturated graphs and hypergraphs, Combinatorica 3 (1983), 181-192. (primary): https://doi.org/10.1007/BF02579292\n Evidence used: Foundational supersaturation and jump-density context for the definition.\n- Vaughn Komorech, Non-jumps of hypergraphs, arXiv:2511.07715 (2025). (primary): https://arxiv.org/abs/2511.07715\n Evidence used: Recent primary construction of additional 3-uniform non-jumps, not a full classification.\n- Thomas F. Bloom, revision history of Erdos Problem #837, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/837\n Evidence used: Unchanged open formulation and known A_2 comparison.\n\n**Review notes.** No normalization or quantifier in the unusually long definition was altered. The common trailing background corruption was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2333, "problem_number": "EP-838", "title": "Erdős Problem #838", "statement": "Let $f(n)$ be maximal such that any $n$ points in $\\mathbb{R}^2$, with no three on a line, determine at least $f(n)$ different convex subsets. Estimate $f(n)$ - in particular, does there exist a constant $c$ such that $ \\lim \\frac{\\log f(n)}{(\\log n)^2}=c? $ ", "background": "A question of Erd\\H{o}s and Hammer. Erd\\H{o}s proved in \\cite{Er78c} that there exist constants $c_1,c_2>0$ such that $ n^{c_1\\log n}\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Suk's sharp exponential-scale Erdos-Szekeres theorem and a checked counting argument give normalized bounds 1/4 and 1 in base 2, but existence and value of the limit remain open.\n\n**Verified partial progress.**\n\n- Erdos proved f(n)=exp(Theta((log n)^2)).\n- Suk proved ES(k)=2^(k+o(k)).\n- A 2026 forum double count derives 1/4<=liminf log_2 f(n)/(log_2 n)^2<=limsup<=1.\n\n**Full solution or refutation.**\n\nThe new explicit constants sharpen the unspecified classical positive constants but do not prove convergence. The derivation is public and received a positive standard check, yet no archival publication of this precise corollary was found.\n\n**What remains.**\n\nProve the normalized logarithm converges and identify its constant, or show distinct liminf and limsup behavior.\n\n**Sources checked.**\n\n- Andrew Suk, On the Erdos-Szekeres convex polygon problem, Journal of the American Mathematical Society 30 (2017), 1047-1053. (primary): https://arxiv.org/abs/1604.08657\n Evidence used: Primary theorem ES(k)=2^(k+o(k)), the input used by the improved lower-constant argument.\n- Thomas F. Bloom, Erdos Problem #838, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/838\n Evidence used: Current open status and classical exp(Theta((log n)^2)) bounds.\n- Erdos Problems discussion thread for Problem #838, post and check of 28 April 2026. (source_collection): https://www.erdosproblems.com/forum/thread/838\n Evidence used: Contains the double-counting derivation of the base-2 1/4 lower constant, classical upper constant 1, and a positive standard check.\n\n**Review notes.** The imported term convex subset is undefined; the literature reading is a subset in convex position. No definition was inserted into the source, and the common trailing background corruption was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2334, "problem_number": "EP-839", "title": "Erdős Problem #839", "statement": "Let $1\\leq a_11/2$. In fact this is false - Freud \\cite{Fr93} constructed a sequence with upper density $19/36$.\nSee also [359] and [867].\nReferences\n\n\n[Fr93] R. Freud, Adding numbers - on a problem of P. Erd\\H{o}s. James Cook Mathematical Notes (1993), 6199-6202.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The two stated limit questions remain open, but a related density conjecture is false.\n\n**Verified partial progress.**\n\n- Freud constructed a valid sequence of upper density 19/36, disproving Erdős's related guess that upper density cannot exceed 1/2.\n- There are examples with harmonic sum at least a constant times log log x.\n\n**Full solution or refutation.**\n\nThe density construction does not decide either displayed limit.\n\n**What remains.**\n\nProve a forced sparse subsequence or produce a counterexample.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #839, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/839\n Evidence used: Current open status and Freud construction.\n\n**Review notes.** AI-assisted forum heuristic excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2335, "problem_number": "EP-840", "title": "Erdős Problem #840", "statement": "Let $f(N)$ be the size of the largest quasi-Sidon subset $A\\subset\\{1,\\ldots,N\\}$, where we say that $A$ is quasi-Sidon if $ \\lvert A+A\\rvert=(1+o(1))\\binom{\\lvert A\\rvert}{2}. $ How does $f(N)$ grow?", "background": "Considered by Erd\\H{o}s and Freud \\cite{ErFr91}, who proved $ \\left(\\frac{2}{\\sqrt{3}}+o(1)\\right)N^{1/2} \\leq f(N) \\leq \\left(2+o(1)\\right)N^{1/2}. $ (Although both bounds were already given by Erd\\H{o}s in \\cite{Er81h}.) Note that $2/\\sqrt{3}=1.15\\cdots$. The lower bound is taking a genuine Sidon set $B\\subset [1,N/3]$ of size $\\sim N^{1/2}/\\sqrt{3}$ and taking the union with $\\{N-b : b\\in B\\}$. The upper bound was improved by Pikhurko \\cite{Pi06} to $ f(N) \\leq \\left(\\left(\\frac{1}{4}+\\frac{1}{(\\pi+2)^2}\\right)^{-1/2}+o(1)\\right)N^{1/2} $ (the constant here is $=1.863\\cdots$).\nThe analogous question with $A-A$ in place of $A+A$ is simpler, and there the maximal size is $\\sim N^{1/2}$, as proved by Cilleruelo.\nSee also [30], [819], and [864].\nReferences\n\n\n[Er81h] Erd\\H{o}s, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\n\n[ErFr91] Erd\\H{o}s, P. and Freud, R., On sums of a {S}idon-sequence. J. Number Theory (1991), 196--205.\n\n[Pi06] Pikhurko, Oleg, Dense edge-magic graphs and thin additive bases. Discrete Math. (2006), 2097--2107.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Quasi-Sidon growth is open with square-root-order bounds and improved upper constant.\n\n**Verified partial progress.**\n\n- Erdős--Freud obtain (2/sqrt(3)+o(1))sqrt(N)<=f(N)<=(2+o(1))sqrt(N).\n- Pikhurko improves the upper constant to 1.863... .\n\n**Full solution or refutation.**\n\nThe leading constant remains unknown.\n\n**What remains.**\n\nDetermine f(N)/sqrt(N) or sharpen either constant.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #840, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/840\n Evidence used: Current open status and bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2336, "problem_number": "EP-846", "title": "Erdős Problem #846", "statement": "Let $A\\subset \\mathbb{R}^2$ be an infinite set for which there exists some $\\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at least $\\epsilon n$ with no three on a line.\nIs it true that $A$ is the union of a finite number of sets where no three are on a line?", "background": "A problem of Erd\\H{o}s, Ne\\v{s}et\\v{r}il, and R\"{o}dl.\nSee also [774] and [847].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Putterman, Sawhney, and Valiant construct an infinite planar point set for which every finite n-point subset has a no-three-collinear subset of size at least n/2, but which cannot be partitioned into finitely many no-three-collinear sets. This directly refutes EP-846 with epsilon=1/2. The tracker also records an independent DeepMind disproof verified in Lean.\n\n**Verified partial progress.**\n\n- The explicit construction encodes edges of the countably infinite complete graph as planar points using algebraically independent parameters.\n- A maximum cut retains at least half of every finite edge set and is triangle-free, proving the required local density with epsilon=1/2.\n- Finite Ramsey theory prevents a finite colouring with every colour class triangle-free, proving that no finite no-three-collinear cover exists.\n- The finite analogue is known to require at least Omega(log k/log log k) pieces for this construction and at most O(log k) by iterative extraction.\n\n**Full solution or refutation.**\n\nMap each edge {i,j} to P_ij=(t_i+t_j,t_i^2+t_i t_j+t_j^2), where the t_i are algebraically independent. The determinant calculation shows that collinear triples correspond exactly to graph triangles. Every finite graph has a bipartite subgraph with at least half its edges, so every finite point subset has a no-three-collinear half. A finite partition of the whole point set would give a finite edge-colouring of K_N with no monochromatic triangle, contradicting Ramsey's theorem.\n\n**What remains.**\n\nThe infinite yes/no problem is closed. Determining the sharp number of no-three-collinear classes required in the finite k-point variant remains open between Omega(log k/log log k) and O(log k). The input background has trailing serialization noise but does not alter the statement.\n\n**Sources checked.**\n\n- Moe Putterman, Mehtaab Sawhney, and Gregory Valiant, On Infinite Sets with No 3 on a Line, arXiv:2602.21275 (2026). (primary): https://cdn.openai.com/infinite-sets/main_single_clean3.pdf\n Evidence used: Provides the explicit epsilon=1/2 counterexample and a complete determinant, maximum-cut, and Ramsey proof.\n- Google DeepMind Formal Conjectures, pinned Lean proof for Erdős Problem 846, commit 2404258180688283e5141021c75464dc2acfb798 (2026). (formal_verification): https://github.com/google-deepmind/formal-conjectures/blob/2404258180688283e5141021c75464dc2acfb798/FormalConjectures/ErdosProblems/846.lean\n Evidence used: Commit-pinned formal proof referenced by the problem's Lean-verified status record.\n- Thomas F. Bloom, Erdős Problem #846 and discussion (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/forum/thread/846\n Evidence used: Records the independent OpenAI and DeepMind disproofs, Lean verification, and the alternative implication from Reiher--Rödl--Sales.\n- Christian Reiher, Vojtěch Rödl, and Marcelo Sales, Colouring versus density in integers and Hales--Jewett cubes, Journal of the London Mathematical Society 110 (2024), e12987. (primary): https://doi.org/10.1112/jlms.12987\n Evidence used: Theorem 1.7 gives an independent route to the geometric counterexample after a suitable generic projection.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2337, "problem_number": "EP-847", "title": "Erdős Problem #847", "statement": "Let $A\\subset \\mathbb{N}$ be an infinite set for which there exists some $\\epsilon>0$ such that in any subset of $A$ of size $n$ there is a subset of size at least $\\epsilon n$ which contains no three-term arithmetic progression.\nIs it true that $A$ is the union of a finite number of sets which contain no three-term arithmetic progression?", "background": "A problem of Erd\\H{o}s, Ne\\v{s}et\\v{r}il, and R\"{o}dl.\nSee also [774] and [846].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Reiher, Rödl, and Sales proved a stronger negative result: for every k>=3 and 0=3.\n- For k=3 it supplies the local EP-847 hypothesis with any epsilon<2/3.\n- Its unavoidable monochromatic progression in every finite colouring contradicts the existence of a finite cover by 3-AP-free sets.\n- The upper endpoint 2/3 is sharp for the paired local property because a three-element arithmetic progression has no 3-AP-free subset larger than two.\n\n**Full solution or refutation.**\n\nThe authors use partite constructions in Hales--Jewett cubes to build finite configurations with arbitrarily high chromatic obstruction and a strong fractional independence property, embed these configurations at separated integer scales, and combine them into an infinite set X. For k=3, any proposed finite cover by 3-AP-free sets yields a finite colouring with no monochromatic 3-AP, contradicting the theorem.\n\n**What remains.**\n\nThe stated implication is closed, and the primary theorem generalizes it to all k-term progressions. Quantitative finite colouring/density tradeoffs and related Pisier-type questions remain successor topics. The input background contains trailing serialization noise but its mathematical statement is intact.\n\n**Sources checked.**\n\n- Christian Reiher, Vojtěch Rödl, and Marcelo Sales, Colouring versus density in integers and Hales--Jewett cubes, Journal of the London Mathematical Society 110 (2024), e12987. (primary): https://doi.org/10.1112/jlms.12987\n Evidence used: Published primary theorem constructs X for every k>=3 and every 0", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2338, "problem_number": "EP-849", "title": "Erdős Problem #849", "statement": "Is it true that, for every integer $t\\geq 1$, there is some integer $a$ such that $ \\binom{n}{k}=a $ (with $1\\leq k\\leq n/2$) has exactly $t$ solutions?", "background": "Erd\\H{o}s \\cite{Er96b} credits this to himself and Gordon 'many years ago', but it is more commonly known as Singmaster's conjecture. For $t=3$ one could take $a=120$, and for $t=4$ one could take $a=3003$. There are no known examples for $t\\geq 5$.\nBoth Erd\\H{o}s and Singmaster believed the answer to this question is no, and in fact that there exists an absolute upper bound on the number of solutions.\nMatomäki, Radziwill, Shao, Tao, and Teräväinen \\cite{MRSTT22} have proved that there are always at most two solutions if we restrict $k$ to $ k\\geq \\exp((\\log n)^{2/3+\\epsilon}), $ assuming $a$ is sufficiently large depending on $\\epsilon>0$.\nReferences\n\n\n[Er96b] Erd\"{o}s, Paul, Some problems I presented or planned to present in my short\ntalk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335.\n\n[MRSTT22] Matom\"{a}ki, Kaisa and Radziwi\\l\\l, Maksym and Shao, Xuancheng\nand Tao, Terence and Ter\"{a}v\"{a}inen, Joni, Singmaster's conjecture in the interior of {P}ascal's\ntriangle. Q. J. Math. (2022), 1137--1177.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Singmaster's multiplicity problem remains open; current results restrict large-index representations.\n\n**Verified partial progress.**\n\n- Examples exist for t=3 and t=4, but none are known for t>=5.\n- Matomäki--Radziwill--Shao--Tao--Teräväinen prove at most two representations in a large-k range.\n\n**Full solution or refutation.**\n\nThe requested arbitrary multiplicities remain unknown.\n\n**What remains.**\n\nFind a t>=5 example or prove an absolute multiplicity bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #849, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/849\n Evidence used: Current open status and stated restriction.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2339, "problem_number": "EP-850", "title": "Erdős Problem #850", "statement": "Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?", "background": "This is sometimes known as the Erd\\H{o}s-Woods conjecture.\nFor just $x,y$ and $x+1,y+1$ one can take $ x=2(2^r-1) $ and $ y = x(x+2). $ Erd\\H{o}s also asked whether there are any other examples. Makowski \\cite{Ma68} observed that $x=75$ and $y=1215$ is another example, since $ 75 = 3\\cdot 5^2 \\textrm{ and }1215 = 3^5\\cdot 5 $ while $ 76 = 2^2\\cdot 19\\textrm{ and }1216 = 2^6\\cdot 19. $ (This example was also found independently by Matthew Bolan, and by Dubickas, who posed it as part of the 2024 team selection test in Lithuania.) No other examples are known. This sequence is listed as A343101 at the OEIS.\nShorey and Tijdeman \\cite{ShTi16} have shown that, assuming a strong form of the ABC conjecture due to Baker, then the answer to the original problem is no.\nSee also [677].\nThe case of $x,y$ and $x+1,y+1$ appeared as Problem 1 in the Third Benelux Mathematical Olympiad 2011.\nThis problem is discussed in problem B19 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ma68] Makowski, Andrzej, On a problem of {E}rd\\H{o}s. Enseign. Math. (2) (1968), 193.\n\n[ShTi16] Shorey, Tarlok N. and Tijdeman, Rob, Arithmetic properties of blocks of consecutive integers. (2016), 455--471.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The three-consecutive-integer Erdős--Woods question remains open, with conditional nonexistence.\n\n**Verified partial progress.**\n\n- Shorey--Tijdeman rule out the original configuration under Baker's strong ABC conjecture.\n- For the two-consecutive-integer analogue, explicit pairs are known.\n\n**Full solution or refutation.**\n\nThe conditional theorem does not settle the stated three-term question.\n\n**What remains.**\n\nProve unconditional nonexistence or exhibit x,y.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #850, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/850\n Evidence used: Current open status, examples, and conditional result.\n\n**Review notes.** Two-term examples are not misreported as three-term solutions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2340, "problem_number": "EP-851", "title": "Erdős Problem #851", "statement": "Let $\\epsilon>0$. Is there some $r\\ll_\\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k\\geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\\epsilon$?", "background": "Romanoff \\cite{Ro34} proved that the set of integers of the form $2^k+p$ (where $p$ is prime) has positive lower density.\nSee also [205].\nReferences\n\n\n[Ro34] Romanoff, N. P., \"{U}ber einige S\"Atze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The maintained tracker marks EP-851 proved and credits Liam Price using GPT-5.2 Pro. Terence Tao publicly audited the sieve strategy and later stated that he could confirm the proof is correct after checking the remaining singular-series average. No dedicated archival paper or completed end-to-end formal proof was located, so expert-review caution is retained.\n\n**Verified partial progress.**\n\n- Romanoff proved in 1934 that integers of the form 2^k+p have positive lower density.\n- The new argument replaces primes by integers with no prime divisor in a medium interval and estimates the first and second moments of the representation function using the fundamental lemma of sieve theory.\n- Tao verified that the residual singular-series correlation is controlled through multiplicative orders of 2 and the limited number of prime divisors of Mersenne numbers.\n- Sawhney reports a Green--Sawhney high-moment refinement with r(epsilon) << log(1/epsilon)/log log(1/epsilon).\n\n**Full solution or refutation.**\n\nFor n<=x, count representations n=2^k+m in which m has no prime divisor between fixed z and x^(1/t). Such m have O_{z,t}(1) distinct prime divisors. The fundamental lemma of sieve theory gives a mean of order t log z and variance O(t log z), once the average singular series over k!=l is shown to be 1+o(1). The condition p|2^k-2^l becomes ord_2(p)|k-l; primes with dangerously small order are sparse because 2^a-1 has only O(a) distinct prime divisors. Chebyshev then makes the exceptional density below epsilon.\n\n**What remains.**\n\nArchive a stable full proof, independently review all sieve-uniformity details, and formalize the fundamental lemma if formal verification is desired. Clarify in future statements that 'prime divisors' means distinct prime divisors omega(n), and whether 'density' denotes an existing natural density or the lower-density conclusion actually established. Determine the optimal r(epsilon), and separately investigate whether one fixed r can work for all epsilon. The input background has trailing serialization noise.\n\n**Sources checked.**\n\n- Terence Tao, expert verification and proof sketch in the Erdős Problem #851 discussion, 5 February 2026. (authoritative_secondary): https://www.erdosproblems.com/forum/thread/851\n Evidence used: Explains the first/second-moment sieve argument and states after checking the order-of-2 singular-series step that the proof is correct.\n- Thomas F. Bloom, Erdős Problem #851 (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/851\n Evidence used: Records the affirmative result, Price attribution, and the Green--Sawhney quantitative refinement claim.\n- N. P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Mathematische Annalen 109 (1934), 668--678, DOI 10.1007/BF01449161. (primary): https://eudml.org/doc/159702\n Evidence used: Primary predecessor proving positive lower density for integers of the form 2^k+p.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2341, "problem_number": "EP-852", "title": "Erdős Problem #852", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n(\\log x)^c $ for some constant $c>0$, and $ h(x)=o(\\log x)? $ ", "background": "Brun's sieve implies $h(x) \\to \\infty$ as $x\\to \\infty$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The quantitative run-of-distinct-prime-gaps problem is open, though h(x) tends to infinity.\n\n**Verified partial progress.**\n\n- Brun's sieve implies h(x)->infinity.\n\n**Full solution or refutation.**\n\nThis does not give either proposed logarithmic lower bound or o(log x) upper bound.\n\n**What remains.**\n\nDetermine the correct order of h(x).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #852, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/852\n Evidence used: Current open status and Brun-sieve consequence.\n\n**Review notes.** No unverified comment is used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2342, "problem_number": "EP-853", "title": "Erdős Problem #853", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solutions for $n\\leq x$.\nIs it true that $r(x)\\to \\infty$? Or even $r(x)/\\log x \\to \\infty$?", "background": "In \\cite{Er85c} Erd\\H{o}s omits the condition that $t$ be even, but this is clearly necessary.\nReferences\n\n\n[Er85c] Erd\\H{o}s, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The least missing even prime gap problem remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo verified advance resolving either divergence assertion was located.\n\n**What remains.**\n\nShow all fixed even gaps eventually occur, or establish a quantitative lower bound for r(x).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #853, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/853\n Evidence used: Current maintained open status and evenness convention.\n\n**Review notes.** The necessary evenness convention was already present in the dataset statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2343, "problem_number": "EP-854", "title": "Erdős Problem #854", "statement": "Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes.\nIf $1=a_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The consecutive-totative-gap question is open; finite computation casts doubt on a stronger early conjecture.\n\n**Verified partial progress.**\n\n- Lacampagne--Selfridge computations show that a stronger all-even-gaps conjecture already fails at a stated primorial.\n\n**Full solution or refutation.**\n\nResults for arbitrary differences a_i-a_j do not answer the required consecutive-gap question.\n\n**What remains.**\n\nEstimate the smallest missing consecutive gap and number of realized consecutive gaps.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #854, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/854\n Evidence used: Current open status, original scope, and computation report.\n\n**Review notes.** Forum claims about arbitrary differences were excluded as a different problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2344, "problem_number": "EP-856", "title": "Erdős Problem #856", "statement": "Let $k\\geq 3$ and $f_k(N)$ be the maximum value of $\\sum_{n\\in A}\\frac{1}{n}$, where $A$ ranges over all subsets of $\\{1,\\ldots,N\\}$ which contain no subset of size $k$ with the same pairwise least common multiple.\nEstimate $f_k(N)$.", "background": "Erd\\H{o}s \\cite{Er70} notes that $ f_k(N) \\ll \\frac{\\log N}{\\log\\log N}. $ Indeed, let $A$ be such a set. This in particular implies that, for every $t$, there are $\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The logarithmic-density LCM problem remains open with nontrivial power-of-log bounds.\n\n**Verified partial progress.**\n\n- Tang--Zhang prove (log N)^(b_k-o(1))<=f_k(N)<=(log N)^(c_k+o(1)).\n- For k=3, displayed exponents include 0.438 below and 0.889 above.\n\n**Full solution or refutation.**\n\nThe optimal exponent and asymptotic are unknown.\n\n**What remains.**\n\nClose the exponent gap; this is tied to weak sunflower progress.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #856, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/856\n Evidence used: Current open status and Tang--Zhang bounds.\n\n**Review notes.** Claimed comment solution excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2345, "problem_number": "EP-857", "title": "Erdős Problem #857", "statement": "Let $m=m(n,k)$ be minimal such that in any collection of sets $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ there must exist a sunflower of size $k$ - that is, some collection of $k$ of the $A_i$ which pairwise have the same intersection.\nEstimate $m(n,k)$, or even better, give an asymptotic formula.", "background": "Related to [536] and [856]. In \\cite{Er70} Erd\\H{o}s asks this in the equivalent formulation with intersection replaced by union.\nThis is sometimes known as the weak sunflower problem (see [20] for the strong sunflower problem).\nWhen $k=3$ this is strongly connected to the cap set problem (finding the maximal size of subsets of $\\mathbb{F}_3^n$ with no three-term arithmetic progressions), as observed by Alon, Shpilka, and Umans \\cite{ASU13}). Naslund and Sawin \\cite{NaSa17} have proved that $ m(n,3) \\leq (3/2^{2/3})^{(1+o(1))n}. $ \nReferences\n\n\n[ASU13] Alon, Noga and Shpilka, Amir and Umans, Christopher, On sunflowers and matrix multiplication. Comput. Complexity (2013), 219--243.\n\n[Er70] Erd\\H{o}s, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133.\n\n[NaSa17] Naslund, Eric and Sawin, Will, Upper bounds for sunflower-free sets. Forum Math. Sigma (2017), Paper No. e15, 10.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The weak sunflower problem is open; the k=3 case has an exponential upper bound via cap-set methods.\n\n**Verified partial progress.**\n\n- Naslund--Sawin prove m(n,3)<=(3/2^(2/3))^((1+o(1))n).\n\n**Full solution or refutation.**\n\nThis does not yield the requested general asymptotics.\n\n**What remains.**\n\nDetermine m(n,k), beginning with sharp exponential rates.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #857, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/857\n Evidence used: Current open status and k=3 bound.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2346, "problem_number": "EP-858", "title": "Erdős Problem #858", "statement": "Let $A\\subseteq \\{1,\\ldots,N\\}$ be such that there is no solution to $at=b$ with $a,b\\in A$ and the smallest prime factor of $t$ is $>a$. Estimate the maximum of $ \\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}. $ ", "background": "Alexander \\cite{Al66} and Erd\\H{o}s, S\\'{a}rk\"{o}zi, and Szemer\\'{e}di \\cite{ESS68} proved that this maximum is $o(1)$ (as $N\\to \\infty$). This condition on $A$ is a weaker form of the usual primitive condition. If $A$ is primitive then Behrend \\cite{Be35} proved $ \\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}\\ll \\frac{1}{\\sqrt{\\log\\log N}}. $ An example of such a set $A$ is the set of all integers in $[N^{1/2},N]$ divisible by some prime $>N^{1/2}$.\nSee also [143].\nReferences\n\n\n[Al66] Alexander, Ralph, Density and multiplicative structure of sets of integers. Acta Arith. (1966/67), 321--332.\n\n[Be35] Behrend, F., On sequences of numbers not divisible by another. London Math. Soc. Journal (1935), 42-45.\n\n[ESS68] Erd\\H{o}s, P. and S\\'{a}rk\"ozi, A. and Szemer\\'{e}di, E., On the solvability of certain equations in sequences of\npositive upper logarithmic density. J. London Math. Soc. (1968), 71--78.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Chojecki's 2026 proof manuscript determines the finite extremum M(N)=(c_2+o(1))log N with explicit c_2=0.6187712111...; an independent flow-network manuscript obtains the same constant, and the maintained tracker marks the problem solved. The evidence is recent and AI-assisted rather than conventionally refereed, so further expert review is appropriate.\n\n**Verified partial progress.**\n\n- The terminal interval (sqrt(N),N] is admissible and gives the elementary lower bound M(N)/log N >= 1/2+o(1).\n- A rough-divisibility parent tree identifies admissible sets with antichains and converts their reciprocal mass to a divergence optimization over descendant-closed upsets.\n- Above N^(1/4), children are exactly prime and semiprime extensions; reciprocal-prime Mertens estimates produce a one-variable divergence profile Phi.\n- The unique sign-change alpha_2=0.28043830989... yields c_2=1/2+integral_{alpha_2}^{1/2}(1-Phi(u))du=0.6187712111....\n\n**Full solution or refutation.**\n\nDefine Phi(u)=log((1-u)/u)+1_{u<1/3} integral_u^((1-u)/2) v^(-1)log((1-u-v)/v)dv, and let alpha_2 in (1/4,1/3) be the unique root Phi(alpha_2)=1. The parent-tree representation turns every admissible antichain into the boundary of a descendant-closed upset, whose reciprocal weight equals total discrete divergence. The divergence is positive precisely above the threshold outside negligible transition bands. A positive-region comparison gives the upper bound and a threshold upset gives the lower bound, producing c_2=1/2+integral_{alpha_2}^{1/2}(1-Phi(u))du.\n\n**What remains.**\n\nObtain conventional peer review or full formal verification, effective error terms, and lower-order asymptotics. The input background's claim that 'this maximum is o(1)' loses an essential quantifier: the classical theorem concerns each fixed infinite admissible A, at an A-dependent rate, not the N-dependent finite maximum. The maintained source gives the correct distinction, and (sqrt(N),N] already disproves the corrupted uniform reading. The background also ends with serialization noise.\n\n**Sources checked.**\n\n- Przemek Chojecki, The asymptotic constant in Erdős problem #858 (2026). (primary): https://www.ulam.ai/research/erdos858-asymptotic.pdf\n Evidence used: Primary solution manuscript deriving the explicit profile Phi, root alpha_2, and asymptotic constant c_2.\n- Paul Pajo, Erdős Problem #858: A Flow-Network Proof (2026), DOI 10.13140/RG.2.2.15506.00964. (primary): https://www.researchgate.net/publication/404185689_Erdos_Problem_858_A_Flow-Network_Proof\n Evidence used: Independent proof manuscript obtaining the same constant using rough-divisibility antichains and discrete divergence.\n- Thomas F. Bloom, Erdős Problem #858 (accessed 2026-08-17). (maintained_tracker): https://www.erdosproblems.com/858\n Evidence used: Records solved status, the c≈0.618 asymptotic, and the correct fixed-infinite-set quantifier in the classical o(log N) result.\n- Ralph Alexander, Density and multiplicative structure of sets of integers, Acta Arithmetica 12 (1967), 321--332, DOI 10.4064/aa-12-4-321-332. (primary): https://eudml.org/doc/204806\n Evidence used: Classical primary source for the infinite-set density result cited in the maintained problem record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2347, "problem_number": "EP-859", "title": "Erdős Problem #859", "statement": "Let $t\\geq 1$ and let $d_t$ be the density of the set of integers $n\\in\\mathbb{N}$ for which $t$ can be represented as the sum of distinct divisors of $n$.\nDo there exist constants $c_1,c_2>0$ such that $ d_t \\sim \\frac{c_1}{(\\log t)^{c_2}} $ as $t\\to \\infty$?", "background": "Erd\\H{o}s \\cite{Er70} proved that $d_t$ always exists, and that there exist some constants $c_3,c_4>0$ such that $ \\frac{1}{(\\log t)^{c_3}} < d_t < \\frac{1}{(\\log t)^{c_4}}. $ \nReferences\n\n\n[Er70] Erd\\H{o}s, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The density d_t exists and has two-sided polylogarithmic bounds, but its precise asymptotic is open.\n\n**Verified partial progress.**\n\n- Erdős proved existence of d_t and constants c_3,c_4>0 with reciprocal-log-power upper and lower bounds.\n\n**Full solution or refutation.**\n\nThe exponents and leading constant in the proposed asymptotic are unknown.\n\n**What remains.**\n\nIdentify the actual logarithmic exponent and asymptotic constant.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #859, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/859\n Evidence used: Current open status and Erdos bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2348, "problem_number": "EP-860", "title": "Erdős Problem #860", "statement": "Let $h(n)$ be such that, for any $m\\geq 1$, in the interval $(m,m+h(n))$ there exist distinct integers $a_i$ for $1\\leq i\\leq \\pi(n)$ such that $p_i\\mid a_i$, where $p_i$ denotes the $i$th prime.\nEstimate $h(n)$.", "background": "A problem of Erd\\H{o}s and Pomerance \\cite{ErPo80}, who proved that $ h(n) \\ll \\frac{n^{3/2}}{(\\log n)^{1/2}}. $ Erd\\H{o}s and Selfridge proved $h(n)>(3-o(1))n$, and Ruzsa proved $h(n)/n\\to \\infty$.\nThis is discussed in problem B32 of Guy's collection \\cite{Gu04}.\nSee also [375].\nReferences\n\n\n[ErPo80] P. Erd\\H{o}s and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples of another interval. Indigationes Math. (1980), 147-151.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The assignment-interval function remains open between superlinear and n^(3/2)-scale bounds.\n\n**Verified partial progress.**\n\n- Erdős--Selfridge prove h(n)>(3-o(1))n.\n- Ruzsa proves h(n)/n tends to infinity.\n- Erdős--Pomerance prove h(n)<", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2349, "problem_number": "EP-863", "title": "Erdős Problem #863", "statement": "Let $r\\geq 2$ and let $A\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\\leq b$ for any $n$. (That is, $A$ is a $B_2[r]$ set.)\nSimilarly, let $B\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a-b$ for any $n$.\nIf $\\lvert A\\rvert\\sim c_rN^{1/2}$ as $N\\to \\infty$ and $\\lvert B\\rvert \\sim c_r'N^{1/2}$ as $N\\to \\infty$ then is it true that $c_r\neq c_r'$ for $r\\geq 2$? Is it true that $c_r'\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The conditional comparison in the displayed question is affirmative for every fixed r >= 2. A standard difference-set upper bound gives c'_r <= sqrt(r), while the published Cilleruelo--Ruzsa--Trujillo construction gives c_r >= (r+floor(r/2))/sqrt(r+2 floor(r/2)); the latter is strictly larger.\n\n**Verified partial progress.**\n\n- For r=1 the two extremal constants agree and equal 1, so the restriction r >= 2 is essential.\n- Cilleruelo, Ruzsa, and Trujillo published a B_2[r] sum-set construction with coefficient (r+floor(r/2))/sqrt(r+2 floor(r/2)).\n- The routine Erdős--Turán difference count bounds the normalized difference extremum by sqrt(r).\n\n**Full solution or refutation.**\n\nLet m=floor(r/2). The sum construction has normalized size at least (r+m)/sqrt(r+2m), whereas every admissible difference set has normalized size at most sqrt(r). Squaring the desired strict inequality reduces to (r+m)^2 > r(r+2m), whose difference is m^2>0. Thus the sum extremal scale strictly exceeds the difference scale for all r>=2.\n\n**What remains.**\n\nThe comparison does not determine either exact constant and does not by itself prove that the two normalized extremal functions converge. It answers the statement conditional on the asserted asymptotics and also yields a direct liminf/limsup separation.\n\n**Sources checked.**\n\n- Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Trujillo, Upper and Lower Bounds for Finite B_h[g] Sequences, Journal of Number Theory 97 (2002), 26--34, DOI 10.1006/jnth.2001.2767. (primary): https://matematicas.uam.es/~franciscojavier.cilleruelo/Papers/bh.pdf\n Evidence used: The paper constructs B_2[g] subsets of [N] with the exact lower-bound coefficient used to separate the sum and difference scales.\n- Boon Suan Ho, self-contained writeup for Erdős Problem #863, April 2026. (primary): https://boonsuan.github.io/erdos863.pdf\n Evidence used: The note combines the standard difference upper bound with the published sum construction and derives c'_r=2.\n- Thomas F. Bloom, Erdős Problem #863 discussion, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/forum/thread/863\n Evidence used: The maintained record marks the problem proved, displays both bounds, and records an independent standard check of the synthesis.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2350, "problem_number": "EP-864", "title": "Erdős Problem #864", "statement": "Let $A\\subseteq \\{1,\\ldots N\\}$ be a set such that there exists at most one $n$ with more than one solution to $n=a+b$ (with $a\\leq b\\in A$). Estimate the maximal possible size of $\\lvert A\\rvert$ - in particular, is it true that $ \\lvert A\\rvert \\leq (1+o(1))\\frac{2}{\\sqrt{3}}N^{1/2}? $ ", "background": "A problem of Erd\\H{o}s and Freud, who prove that $ \\lvert A\\rvert \\geq (1+o(1))\\frac{2}{\\sqrt{3}}N^{1/2}. $ This is shown by taking a genuine Sidon set $B\\subset [1,N/3]$ of size $\\sim N^{1/2}/\\sqrt{3}$ and taking the union with $\\{N-b : b\\in B\\}$.\nFor the analogous question with $n=a-b$ they prove that $\\lvert A\\rvert\\sim N^{1/2}$.\nThis is a weaker form of [840].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The one-repeated-sum Sidon-type extremal problem is open, with the conjectured constant achieved as a lower bound.\n\n**Verified partial progress.**\n\n- Erdős--Freud construct sets of size (2/sqrt(3)+o(1))sqrt(N).\n\n**Full solution or refutation.**\n\nThe matching upper bound is unknown.\n\n**What remains.**\n\nProve or disprove the 2/sqrt(3) upper constant.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #864, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/864\n Evidence used: Current open status and construction.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2351, "problem_number": "EP-865", "title": "Erdős Problem #865", "statement": "There exists a constant $C>0$ such that, for all large $N$, if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $\\frac{5}{8}N+C$ then there are distinct $a,b,c\\in A$ such that $a+b,a+c,b+c\\in A$.", "background": "A problem of Erd\\H{o}s and S\\'{o}s (also earlier considered by Choi, Erd\\H{o}s, and Szemer\\'{e}di \\cite{CES75}, but Erd\\H{o}s had forgotten this). Taking all integers in $[N/8,N/4]$ and $[N/2,N]$ shows that $\\frac{5}{8}$ would be best possible here.\nIt is a classical folklore fact that if $A\\subseteq \\{1,\\ldots,2N\\}$ has size $\\geq N+2$ then there are distinct $a,b\\in A$ such that $a+b\\in A$, which establishes the $k=2$ case.\nIn general, one can define $f_k(N)$ to be minimal such that if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $f_k(N)$ then there are $k$ distinct $a_i\\in A$ such that all $\\binom{k}{2}$ pairwise sums are elements of $A$. Erd\\H{o}s and S\\'{o}s conjectured that $ f_k(N)\\sim \\frac{1}{2}\\left(1+\\sum_{1\\leq r\\leq k-2}\\frac{1}{4^r}\\right) N, $ and a similar example shows that this would be best possible.\nChoi, Erd\\H{o}s, and Szemer\\'{e}di \\cite{CES75} have proved that, for all $k\\geq 3$, there exists $\\epsilon_k>0$ such that (for large enough $N$) $ f_k(N)\\leq \\left(\\frac{2}{3}-\\epsilon_k\\right)N. $ \nReferences\n\n\n[CES75] Choi, S. L. G. and Erd\\H{o}s, P. and Szemer\\'{e}di, E., Some additive and multiplicative problems in number theory. Acta Arith. (1975), 37--50.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Cipollini proved f_3(N) <= 5N/8+O(1), which gives the constant C requested in the displayed theorem. Together with the standard interval construction this yields f_3(N)=5N/8+O(1). A current complete Lean file proves the explicit upper bound 8|A|<=5N+53 and the matching leading constant.\n\n**Verified partial progress.**\n\n- The construction [N/8,N/4] union [N/2,N] showed that no leading constant below 5/8 can work.\n- Choi, Erdős, and Szemerédi had earlier proved a weaker density bound below 2/3 for the general f_k problem.\n- The 2026 proof and its Lean counterpart close the k=3 case with a bounded additive error.\n\n**Full solution or refutation.**\n\nA self-contained folding and collision-count argument proves that every triple-free A subset [1,N] satisfies 8|A|<=5N+53. Therefore any set exceeding that threshold contains distinct a,b,c whose three pairwise sums also lie in A. For N=8M, the triple-free set [M,2M] union [4M,8M] has 5M+2 elements, establishing sharpness of 5/8.\n\n**What remains.**\n\nThe displayed k=3 assertion is closed. The optimal bounded additive term in f_3(N) is not determined here, and the background's general Erdős--Sós conjecture for f_k(N), k>=4, remains a separate open family.\n\n**Sources checked.**\n\n- Ricky Cipollini, A sharp 5/8 bound for an Erdős-Sós pairwise-sums problem, arXiv:2606.29361 (2026). (primary): https://arxiv.org/abs/2606.29361\n Evidence used: The abstract states the self-contained f_3(N)<=5N/8+O(1) theorem and the matching construction, explicitly resolving Erdős Problem 865.\n- Jay Yao and contributors, Erdos865.lean, Lean 4 proof. (formal_verification): https://github.com/Jayyhk/erdos-lean/blob/main/problems/865/Erdos865.lean\n Evidence used: The public file contains no sorry or added mathematical axiom and proves 8|A|<=5N+53, the packaged threshold theorem, and the sharpness construction.\n- Thomas F. Bloom, history of Erdős Problem #865, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/history/865\n Evidence used: The maintained history records the 2 July 2026 affirmative resolution and current proved status.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2352, "problem_number": "EP-866", "title": "Erdős Problem #866", "statement": "Let $k\\geq 3$ and $g_k(N)$ be minimal such that if $A\\subseteq \\{1,\\ldots,2N\\}$ has $\\lvert A\\rvert \\geq N+g_k(N)$ then there exist integers $b_1,\\ldots,b_k$ such that all $\\binom{k}{2}$ pairwise sums are in $A$ (but the $b_i$ themselves need not be in $A$).\nEstimate $g_k(N)$.", "background": "A problem of Choi, Erd\\H{o}s, and Szemer\\'{e}di. It is clear that, for the set of odd numbers in $\\{1,\\ldots,2N\\}$, no such $b_i$ exist, whence $g_k(N)\\geq 0$ always. Choi, Erd\\H{o}s, and Szemer\\'{e}di proved that $g_3(N)=2$ and $g_4(N) \\ll 1$. van Doorn has shown that $g_4(N)\\leq 2032$.\nChoi, Erd\\H{o}s, and Szemer\\'{e}di also proved that $ g_5(N)\\asymp \\log N $ and $ g_6(N)\\asymp N^{1/2}. $ In general they proved that $ g_k(N) \\ll_k N^{1-2^{-k}} $ and for every $\\epsilon>0$ if $k$ is sufficiently large then $ g_k(N) > N^{1-\\epsilon}. $ As an example, taking $A$ to be the set of all odd integers and the powers of $2$ shows that $g_5(N)\\gg \\log N$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The additive pairwise-sum configuration threshold remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo verified published quantitative advance was located in the maintained record.\n\n**What remains.**\n\nDetermine g_k(N), including the first nontrivial k cases.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #866, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/866\n Evidence used: Current maintained open status.\n\n**Review notes.** Forum upload lead was not accepted without direct verification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2353, "problem_number": "EP-869", "title": "Erdős Problem #869", "statement": "If $A_1,A_2$ are disjoint additive bases of order $2$ (i.e. $A_i+A_i$ contains all large integers) then must $A=A_1\\cup A_2$ contain a minimal additive basis of order $2$ (one such that deleting any element creates infinitely many $n\not\\in A+A$)?", "background": "A question of Erd\\H{o}s and Nathanson \\cite{ErNa88}.\nHärtter \\cite{Ha56} and Nathanson \\cite{Na74} proved that there exist additive bases which do not contain any minimal additive bases.\nReferences\n\n\n[ErNa88] Erd\\H{o}s, Paul and Nathanson, Melvyn B., Partitions of bases into disjoint unions of bases. J. Number Theory (1988), 1--9.\n\n[Ha56] H\"{a}rtter, Erich, Ein Beitrag zur {T}heorie der {M}inimalbasen. J. Reine Angew. Math. (1956), 170--204.\n\n[Na74] Nathanson, Melvyn B., Minimal bases and maximal nonbases in additive number theory. J. Number Theory (1974), 324--333.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Larsen constructed an asymptotic basis A of order 2 that is a union of two disjoint bases but contains no minimal asymptotic basis. This is the direct counterexample to the displayed implication, and it is part of a stronger independence theorem for three robustness properties of additive bases.\n\n**Verified partial progress.**\n\n- Härtter and Nathanson had constructed bases with no minimal subbasis, but those examples did not by themselves meet the decomposition hypothesis in EP-869.\n- Erdős and Nathanson had shown that sufficiently rapid logarithmic growth of the representation function forces stronger robustness properties.\n- Larsen's 2026 construction proves independence of divergent representation count, decomposition into two bases, and existence of a minimal subbasis in the weaker-growth regime.\n\n**Full solution or refutation.**\n\nLarsen uses an inductive construction on rapidly growing intervals with a selection mechanism that can independently enforce or suppress the three robustness properties. Choosing the case where decomposition into two disjoint bases holds but containment of a minimal basis fails supplies A=A_1 union A_2 exactly as required and disproves EP-869.\n\n**What remains.**\n\nNothing remains of the yes/no implication as stated. Quantitative threshold questions asking how fast the representation function must grow to force decomposability or a minimal subbasis remain broader successors.\n\n**Sources checked.**\n\n- Daniel Larsen, Three Questions of Erdős-Nathanson on Asymptotic Bases of Order 2, arXiv:2603.03472 (2026). (primary): https://arxiv.org/abs/2603.03472\n Evidence used: The paper proves the three robustness properties independent and includes the decomposable/no-minimal-subbasis combination that directly refutes EP-869.\n- Thomas F. Bloom, Erdős Problem #869, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/869\n Evidence used: The maintained record marks the problem disproved and identifies Larsen's stronger independence theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2354, "problem_number": "EP-870", "title": "Erdős Problem #870", "statement": "Let $k\\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\\geq c\\log n$ for all large $n$ then $A$ must contain a minimal basis of order $k$? (Here $r(n)$ counts the number of representations of $n$ as the sum of at most $k$ elements from $A$.)", "background": "A question of Erd\\H{o}s and Nathanson \\cite{ErNa79}, who proved that this is true for $k=2$ if $1_A\\ast 1_A(n) > (\\log \\frac{4}{3})^{-1}\\log n$ for all large $n$.\nHärtter \\cite{Ha56} and Nathanson \\cite{Na74} proved that there exist additive bases which do not contain any minimal additive bases.\nSee also [868].\nReferences\n\n\n[ErNa79] Erd\\H{o}s, Paul and Nathanson, Melvyn B., Systems of distinct representatives and minimal bases in\nadditive number theory. (1979), 89--107.\n\n[Ha56] H\"{a}rtter, Erich, Ein Beitrag zur {T}heorie der {M}inimalbasen. J. Reine Angew. Math. (1956), 170--204.\n\n[Na74] Nathanson, Melvyn B., Minimal bases and maximal nonbases in additive number theory. J. Number Theory (1974), 324--333.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The question remains open for k>=3; an order-two theorem is known under an explicit logarithmic representation threshold.\n\n**Verified partial progress.**\n\n- Erdős--Nathanson prove the assertion for k=2 when 1_A*1_A(n)>(log(4/3))^(-1) log n eventually.\n- There are bases with no minimal subbasis absent the stated representation hypothesis.\n\n**Full solution or refutation.**\n\nA recent AI-assisted claim of a full refutation has an explicit expert caveat and is not used as established evidence.\n\n**What remains.**\n\nVerify or refute the claimed construction; settle the exact-k formulation for k>=3.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #870, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/870\n Evidence used: Current open status, k=2 theorem, and verification caveat.\n\n**Review notes.** The source counts sums of at most k terms; a discussion flags a possible exact-k intended reading.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2355, "problem_number": "EP-872", "title": "Erdős Problem #872", "statement": "Consider the two-player game in which players alternately choose integers from $\\{2,3,\\ldots,n\\}$ to be included in some set $A$ (the same set for both players) such that no $a\\mid b$ for $a\neq b\\in A$.\nThe game ends when no legal move is possible. One player wants the game to last as long as possible, the other wants the game to end quickly. How long can the game be guaranteed to last for?\nAt least $\\epsilon n$ moves? (For $\\epsilon>0$ and $n$ sufficiently large.) At least $(1-\\epsilon)\\frac{n}{2}$ moves?", "background": "A number theoretic variant of a combinatorial game of Hajnal, in which players alternately add edges to a graph while keeping it triangle-free. This game must trivially end in at most $n^2/4$ moves, and F\"{u}redi and Seress \\cite{FuSe91} proved that it can be guaranteed to last for $\\gg n\\log n$ moves. Bir\\'{o}, Horn, and Wildstrom \\cite{BPW16} proved that it must end in at most $(\\frac{26}{121}+o(1))n^2$ moves.\nThis type of game is known as a saturation game.\nErd\\H{o}s does not specify which player goes first, which may result in different answers.\nReferences\n\n\n[BPW16] Bir\\'{o}, Csaba and Horn, Paul and Wildstrom, D. Jacob, An upper bound on the extremal version of Hajnal's\ntriangle-free game. Discrete Appl. Math. (2016), 20--28.\n\n[FuSe91] F\"{u}redi, Zolt\\'{a}n and Reimer, Dave and Seress, \\'{A}kos, Hajnal's triangle-free game and extremal graph problems. Congr. Numer. (1991), 123--128.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The primitive-set saturation game is open, but a claimed strategy refutes the near-n/2 duration target under a stated player-order convention.\n\n**Verified partial progress.**\n\n- All primes in (n/2,n] force an Omega(n/log n) lower bound.\n- The maintained record reports a Shortener strategy giving at most (23/48+o(1))n when Prolonger moves first.\n\n**Full solution or refutation.**\n\nThe original problem does not specify the first player, and the reported upper bound is not a full game-value determination.\n\n**What remains.**\n\nFix game conventions and determine the optimal guaranteed duration.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #872, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/872\n Evidence used: Current open status, player-order caveat, and stated bounds.\n\n**Review notes.** Partial Lean formalization of a restricted play class is not treated as a proof of the whole game result.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2356, "problem_number": "EP-873", "title": "Erdős Problem #873", "statement": "Let $A=\\{a_10$, there exists some $k$ such that $ F(A,X,k)\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The arbitrary-epsilon LCM-block question is open; the k=3 exponent is sharp in the stated sense.\n\n**Verified partial progress.**\n\n- Erdős--Szemerédi prove F(A,X,3)<", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2357, "problem_number": "EP-875", "title": "Erdős Problem #875", "statement": "Let $A=\\{a_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The growth and gap problem for infinite admissible sets remains open, with polynomial-density constructions and finite extremal restrictions.\n\n**Verified partial progress.**\n\n- Erdős--Nicolas--Sárközy construct an infinite admissible set with polynomial growth.\n- Deshouillers--Freiman bound finite admissible sets by 2sqrt(N)+O(1).\n\n**Full solution or refutation.**\n\nA recent absolute-gap refinement is reported with a verification caveat and is not used as established literature.\n\n**What remains.**\n\nDetermine the optimal gap exponent and whether gaps below n are possible.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #875, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/875\n Evidence used: Current open status and historical framing.\n- EP-875 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/875\n Evidence used: Bibliographic leads and explicit scope of recent claim.\n\n**Review notes.** Absolute-gap and ratio-growth assertions are distinguished.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2358, "problem_number": "EP-876", "title": "Erdős Problem #876", "statement": "Let $A=\\{a_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The infinite sum-free-set gap question is open, with known density and counting-function bounds.\n\n**Verified partial progress.**\n\n- Graham gives a construction with gaps below n^(1+o(1)).\n- Luczak--Schoen prove |A cap [1,N]|<<(N log N)^(1/2) and construct a set of size at least N^(1/2)/(log N)^(1/2+o(1)).\n\n**Full solution or refutation.**\n\nThese do not decide whether all consecutive gaps can be ", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2359, "problem_number": "EP-878", "title": "Erdős Problem #878", "statement": "If $n=\\prod_{1\\leq i\\leq t} p_i^{k_i}$ is the factorisation of $n$ into distinct primes then let $ f(n)=\\sum p_i^{\\ell_i}, $ where $\\ell_i$ is chosen such that $n\\in [p_i^{\\ell_i},p_i^{\\ell_i+1})$. Furthermore, let $ F(n)=\\max \\sum_{i=1}^t a_i $ where the maximum is taken over all $a_1,\\ldots,a_t\\leq n$ such that $(a_i,a_j)=1$ for $i", "background": "eq j$ and all prime factors of each $a_i$ are prime factors of $n$.\nIs it true that, for almost all $n$, $ f(n)=o(n\\log\\log n) $ and $ F(n) \\gg n\\log\\log n? $ Is it true that $ \\max_{n\\leq x}f(n)\\sim \\frac{x\\log x}{\\log\\log x}? $ Is it true that (for all $x$, or perhaps just for all large $x$) $ \\max_{n\\leq x}f(n)=\\max_{n\\leq x}F(n)? $ Find an asymptotic formula for the number of $n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Several subclaims are settled, but the multi-part extremal/asymptotic problem remains open.\n\n**Verified partial progress.**\n\n- Erdős proves the stated maximum f asymptotic along a sequence of x and two-sided bounds for H(x).\n- The H(x) upper bound implies f(n)=o(n log log n) for almost all n.\n- The all-x equality max f=max F is false already at x=210.\n\n**Full solution or refutation.**\n\nThe almost-all F asymptotic and remaining counting questions are unresolved.\n\n**What remains.**\n\nResolve the F(n) almost-all scale and remaining maximum/counting asymptotics.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #878, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/878\n Evidence used: Current open status, settled subclaims, and explicit counterexample.\n\n**Review notes.** Only the 'all x' equality is classified as false, not its possible eventual variant.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2360, "problem_number": "EP-879", "title": "Erdős Problem #879", "statement": "Call a set $S\\subseteq \\{1,\\ldots,n\\}$ admissible if $(a,b)=1$ for all $a\neq b\\in S$. Let $ G(n) = \\max_{S\\subseteq \\{1,\\ldots,n\\}} \\sum_{a\\in S}a $ and $ H(n)=\\sum_{pH(n)-n^{1+o(1)}? $ Is it true that, for every $k\\geq 2$, if $n$ is sufficiently large then the admissible set which maximises $G(n)$ contains at least one integer with at least $k$ prime factors?", "background": "Erd\\H{o}s and Van Lint proved that $ H(n)-n^{3/2-o(1)}H(n)-n^{1+o(1)}$ assuming 'plausible (but hopeless) assumptions about the distribution of primes'. They also prove the second claim when $k=2$.\nSee also [878].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Both questions remain open in general, with conditional near-optimality and the k=2 case known.\n\n**Verified partial progress.**\n\n- Erdős--Van Lint prove H(n)-n^(3/2-o(1))=3.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #879, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/879\n Evidence used: Current open status and established/conditional bounds.\n\n**Review notes.** Finite AI-assisted computations are not treated as eventual counterevidence.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2361, "problem_number": "EP-881", "title": "Erdős Problem #881", "statement": "Let $A\\subset\\mathbb{N}$ be an additive basis of order $k$ which is minimal, in the sense that if $B\\subset A$ is any infinite set then $A\\backslash B$ is not a basis of order $k$.\nMust there exist an infinite $B\\subset A$ such that $A\\backslash B$ is a basis of order $k+1$?\",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The maintained source remains open; a recent claimed negative solution has documented major gaps.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo independently verified resolution was located.\n\n**What remains.**\n\nVerify a complete proof or disproof for minimal additive bases.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #881, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/881\n Evidence used: Maintained open status.\n- EP-881 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/881\n Evidence used: Claimed AI-generated proof has explicit major-gap warning.\n\n**Review notes.** Unverified proof claim excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2362, "problem_number": "EP-883", "title": "Erdős Problem #883", "statement": "For $A\\subseteq \\{1,\\ldots,n\\}$ let $G(A)$ be the graph with vertex set $A$, where two integers are joined by an edge if they are coprime.\nIs it true that if $ \\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor $ then $G(A)$ contains all odd cycles of length $\\leq \\frac{n}{3}+1$?\nIs it true that, for every $\\ell\\geq 1$, if $n$ is sufficiently large and $ \\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor $ then $G(A)$ must contain a complete $(1,\\ell,\\ell)$ triparite graph on $2\\ell+1$ vertices?", "background": "A problem of Erd\\H{o}s and S\\'{a}rk\\H{o}zy \\cite{ErSa97}, who prove that if $ \\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor $ then $G(A)$ contains all odd cycles of length $\\leq cn$ for some constant $c>0$.\nThis threshold is the best possible, since one could take $A$ to be the set of $m\\leq n$ which are divisible by either $2$ or $3$, in which case $G(A)$ contains no triangles.\nThe second question was solved by S\\'{a}rk\"{o}zy \\cite{Sa99} who proved that, for large $n$, if $\\lvert A\\rvert$ exceeds the given threshold then $G(A)$ contains a complete $(1,\\ell,\\ell)$ triparite graph with $ \\ell \\gg \\frac{\\log n}{\\log\\log n}. $ \nReferences\n\n\n[ErSa97] Erd\\H{o}s, Paul and Sarkozy, Gabor N., On cycles in the coprime graph of integers. Electron. J. Combin. (1997), Research Paper 8, approx. 11.\n\n[Sa99] S\\'ark\"ozy, G\\'abor N., Complete tripartite subgraphs in the coprime graph of\nintegers. Discrete Math. (1999), 227--238.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The coprimality-graph questions remain open, though both have substantive threshold results.\n\n**Verified partial progress.**\n\n- Erdős--Sárközy prove odd cycles through a positive linear length range.\n- Sárközy proves the second question for ell much larger than log n/log log n.\n\n**Full solution or refutation.**\n\nNeither result reaches all requested cycle lengths or every fixed ell.\n\n**What remains.**\n\nExtend odd cycles to n/3+1 and the tripartite conclusion to all fixed ell.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #883, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/883\n Evidence used: Current open status and proved ranges.\n\n**Review notes.** The threshold's sharpness example is retained in background only.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2363, "problem_number": "EP-884", "title": "Erdős Problem #884", "statement": "Is it true that, for any $n$, if $d_1<\\cdots \n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The proposed absolute bound is false. Tao first proved conditional unboundedness of the ratio under the qualitative Hardy--Littlewood prime-tuples conjecture; Larsen subsequently proved unboundedness unconditionally in a complete public manuscript.\n\n**Verified partial progress.**\n\n- A general telescoping/Cauchy--Schwarz estimate loses only a logarithmic factor, so the conjecture concerned a genuinely narrow gap between all-pairs and consecutive-gap energies.\n- Tao's 2025 construction using products of nearly consecutive primes disproved the conjecture conditional on prime tuples.\n- Larsen's multiscale construction removed the prime-tuples hypothesis and made the additive 1 in the denominator negligible.\n\n**Full solution or refutation.**\n\nWrite T for the sum over all reciprocal divisor gaps and S for the sum over consecutive reciprocal divisor gaps. Larsen repeats an unconditional one-scale version of Tao's clustered-prime construction over many widely separated scales. Per-scale contributions decay sufficiently slowly and cross-scale contributions are controlled, so T accumulates while 1+S stays too small; hence T/(1+S) is unbounded.\n\n**What remains.**\n\nThe displayed conjecture is unconditionally refuted. Simpler explicit counterexample families and sharp quantitative growth rates for the ratio are natural refinements. Although the batch tracker says Lean, the public audit located during this check supplied no hosted complete proof link, so formal verification should be rechecked separately.\n\n**Sources checked.**\n\n- Daniel Larsen, A Question of Erdős on Reciprocals of Gaps Between Divisors, proof manuscript dated 29 March 2026. (primary): https://github.com/Larsen-Daniel/Erdos-884/blob/main/884.pdf\n Evidence used: The abstract and Theorem 1.1 state unconditionally that the exact ratio in EP-884 is unbounded.\n- Terence Tao, On the Sum of Reciprocals of Gaps Between Divisors, manuscript dated 9 September 2025. (primary): https://terrytao.wordpress.com/wp-content/uploads/2025/09/erdos-884.pdf\n Evidence used: Theorem 1.1 gives the conditional unboundedness result under the qualitative prime-tuples conjecture and develops the one-scale clustered-prime mechanism.\n- Thomas F. Bloom, Erdős Problem #884 discussion, accessed 2026-08-17. (authoritative_secondary): https://www.erdosproblems.com/forum/thread/884\n Evidence used: The maintained record distinguishes Tao's conditional disproof from Larsen's unconditional disproof and marks the conjecture disproved.\n- Constellate formal-proof audit for problem 884, accessed 2026-08-17. (authoritative_secondary): https://erdos.constellate.science/finding.html?n=884\n Evidence used: Used only to qualify the Lean metadata: the current audit lists an open-PR state and no hosted proof link, so it was not counted as formal verification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2364, "problem_number": "EP-885", "title": "Erdős Problem #885", "statement": "For integer $n\\geq 1$ we define the factor difference set of $n$ by $ D(n) = \\{\\lvert a-b\\rvert : n=ab\\}. $ Is it true that, for every $k\\geq 1$, there exist integers $N_1<\\cdots\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The intersection assertion is open in general but proved for k up to 4.\n\n**Verified partial progress.**\n\n- Erdős--Rosenfeld prove k=2.\n- Jiménez-Urroz proves k=3 and Bremner proves k=4.\n\n**Full solution or refutation.**\n\nNo general-k construction is verified.\n\n**What remains.**\n\nExtend the result beyond k=4.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #885, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/885\n Evidence used: Current open status and known k cases.\n\n**Review notes.** A recent exact finite example is not used in place of the general assertion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2365, "problem_number": "EP-886", "title": "Erdős Problem #886", "statement": "Let $\\epsilon>0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\\epsilon})$ is $O_\\epsilon(1)$?", "background": "Erd\\H{o}s attributes this conjecture to Ruzsa. Erd\\H{o}s and Rosenfeld \\cite{ErRo97} proved that there are infinitely many $n$ such that there are four divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/4})$.\nSee also [887].\nReferences\n\n\n[ErRo97] Erd\\H{o}s, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Ruzsa divisor-clustering conjecture remains open; the n^(1/4) scale is understood only with constants depending on width.\n\n**Verified partial progress.**\n\n- Erdős--Rosenfeld construct infinitely many n with four divisors in a 16n^(1/4) interval.\n- They prove a 1+C^2 upper bound in width Cn^(1/4) for sufficiently large n depending on C.\n\n**Full solution or refutation.**\n\nThis does not control the much longer n^(1/2-epsilon) intervals.\n\n**What remains.**\n\nProve a uniform O_epsilon(1) bound or find counterexamples.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #886, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/886\n Evidence used: Current open status and Erdos--Rosenfeld bounds.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2366, "problem_number": "EP-887", "title": "Erdős Problem #887", "statement": "Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{1/2},n^{1/2}+C n^{1/4})$.", "background": "A question of Erd\\H{o}s and Rosenfeld \\cite{ErRo97}, who proved that there are infinitely many $n$ with $4$ divisors in $(n^{1/2},n^{1/2}+n^{1/4})$, and ask whether $4$ is best possible here.\nReferences\n\n\n[ErRo97] Erd\\H{o}s, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.\n\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The absolute divisor-count bound is open; important square and near-square subclasses are bounded.\n\n**Verified partial progress.**\n\n- Chan proves at most five divisors in a wider window for square n.\n- Chan proves an 18-divisor bound for a specified near-square family.\n\n**Full solution or refutation.**\n\nNeither special case gives an absolute K for arbitrary n and C.\n\n**What remains.**\n\nProve a uniform bound or construct unbounded examples.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #887, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/887\n Evidence used: Current open status and special cases.\n\n**Review notes.** Source statement preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2367, "problem_number": "EP-888", "title": "Erdős Problem #888", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,n\\}$ such that if $a\\leq b\\leq c\\leq d\\in A$ are such that $abcd$ is a square then $ad=bc$?", "background": "A question of Erd\\H{o}s, S\\'{a}rk\"{o}zy, and S\\'{o}s. Erd\\H{o}s claims that S\\'{a}rk\"{o}zy proved that $\\lvert A\\rvert =o(n)$ (a proof of this bound is provided by Tao in the comments).\nThe primes show that $\\lvert A\\rvert \\gg n/\\log n$ is possible. Cambie and Weisenberg have noted in the comments that the set of semiprimes also works, showing $ (1+o(1))\\frac{\\log\\log n}{\\log n}n \\leq \\lvert A\\rvert $ is achievable.\nSee also [121].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The extremal size is determined up to absolute constant factors: F(n) is asymptotic-order n log log n/log n. Squarefree semiprimes give the matching-order lower bound, and the 2026 solution note proves the upper bound by a colored bipartite-graph argument. No exact leading constant is claimed.\n\n**Verified partial progress.**\n\n- Sárközy had obtained F(n)=o(n), with a modern proof supplied in the maintained discussion.\n- Primes gave a lower bound of order n/log n; squarefree semiprimes improved this to (1+o(1))n log log n/log n.\n- An earlier AI-assisted exact-maximizer claim was rejected because its squarefree-kernel reduction was false and additional admissible high-prime-factor elements exist.\n\n**Full solution or refutation.**\n\nAfter controlling square parts, encode each relevant squarefree integer as cpq, where p", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2368, "problem_number": "EP-889", "title": "Erdős Problem #889", "statement": "For $k\\geq 0$ and $n\\geq 1$ let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\\leq ik$.\nIs it true that $ v_0(n)=\\max_{k\\geq 0}v(n,k)\\to \\infty $ as $n\\to \\infty$?", "background": "A question of Erd\\H{o}s and Selfridge \\cite{ErSe67}, who could only show that $v_0(n)\\geq 2$ for $n\\geq 17$. More generally, they conjecture that $ v_l(n)=\\max_{k\\geq l}v(n,k)\\to \\infty $ as $n\\to \\infty$, for every fixed $l$, but could not even prove that $v_1(n)\\geq 2$ for all large $n$.\nThis is problem B27 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[ErSe67] Erd\\H{o}s, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The maximal-new-prime-factor question remains open beyond the elementary lower constant two.\n\n**Verified partial progress.**\n\n- Erdős--Selfridge show v_0(n)>=2 for n>=17.\n\n**Full solution or refutation.**\n\nEven their stronger fixed-l conjecture is unresolved at the first nontrivial level.\n\n**What remains.**\n\nProve v_0(n) tends to infinity.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #889, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/889\n Evidence used: Current open status and baseline result.\n\n**Review notes.** No source text was changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2369, "problem_number": "EP-890", "title": "Erdős Problem #890", "statement": "If $\\omega(n)$ counts the number of distinct prime factors of $n$, then is it true that, for every $k\\geq 1$, $ \\liminf_{n\\to \\infty}\\sum_{0\\leq ik$ by P\\'{o}lya's theorem.\nIt is a classical fact that $ \\limsup_{n\\to \\infty}\\omega(n)\\frac{\\log\\log n}{\\log n}=1. $ \nReferences\n\n\n[ErSe67] Erd\\H{o}s, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The dataset statement has an identified formulation defect; the corrected maintained question is open, while its displayed second assertion is classical.\n\n**Verified partial progress.**\n\n- For the corrected omega_k formulation, Polya's theorem gives the lower bound k-1 for the liminf.\n- The limsup statement with unrestricted omega is classical.\n\n**Full solution or refutation.**\n\nThe dataset's first expression uses omega rather than omega_k and a different bound, making it literally distinct and apparently false.\n\n**What remains.**\n\nConsult original source or maintain a separately labeled corrected statement before assigning a definitive status.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #890, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/890\n Evidence used: Maintained corrected formulation and explicit source-error note.\n\n**Review notes.** Formulation defect flagged, never repaired in the source record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2370, "problem_number": "EP-891", "title": "Erdős Problem #891", "statement": "Let $2=p_1k$ many prime factors?", "background": "Schinzel deduced from P\\'{o}lya's theorem \\cite{Po18} (that the sequence of $k$-smooth integers has unbounded gaps) that this is true with $p_1\\cdots p_k$ replaced by $p_1\\cdots p_{k-1}p_{k+1}$.\nThis is unknown even for $k=2$ - that is, is it true that in every interval of $6$ (sufficiently large) consecutive integers there must exist one with at least $3$ prime factors?\nWeisenberg has observed that Dickson's conjecture implies the answer is no if we replace $p_1\\cdots p_k$ with $p_1\\cdots p_k-1$. Indeed, let $L_k$ be the lowest common multiple of all integers at most $p_1\\cdots p_k$. By Dickson's conjecture there are infinitely many $n'$ such that $\\frac{L_k}{m}n'+1$ is prime for all $1\\leq m\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact short-interval assertion remains open, including k=2, but a weaker interval length is known.\n\n**Verified partial progress.**\n\n- Schinzel deduced the claim with p_1...p_k replaced by p_1...p_(k-1)p_(k+1).\n- A related shortening by one is conditionally false under Dickson's conjecture.\n\n**Full solution or refutation.**\n\nNeither statement settles the original interval length.\n\n**What remains.**\n\nResolve the k=2 six-integer interval case.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #891, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/891\n Evidence used: Current open status and stated consequences.\n\n**Review notes.** Conditional counterexample concerns a different interval length.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2371, "problem_number": "EP-892", "title": "Erdős Problem #892", "statement": "Is there a necessary and sufficient condition for a sequence of integers $b_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No criterion for the primitive-sequence embedding problem is known; two sharp-looking necessary conditions are established.\n\n**Verified partial progress.**\n\n- Erdős proves summability of 1/(b_n log b_n) is necessary.\n- Erdős--Sárközy--Szemerédi prove a stronger necessary partial-sum condition.\n\n**Full solution or refutation.**\n\nNecessity does not give a sufficient criterion.\n\n**What remains.**\n\nFind a characterization or settle the gcd-structure special case.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #892, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/892\n Evidence used: Current open status and necessary conditions.\n\n**Review notes.** Dataset omits a related second question that the maintained record presents; no source edit made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2372, "problem_number": "EP-893", "title": "Erdős Problem #893", "statement": "If $\\tau(n)$ counts the divisors of $n$ then let $ f(n)=\\sum_{1\\leq k\\leq n}\\tau(2^k-1). $ Does $f(2n)/f(n)$ tend to a limit?", "background": "Erd\\H{o}s \\cite{Er98} says that 'probably there is no simple asymptotic formula for $f(n)$ since $f(n)$ increases too fast'.\nKova\\v{c} and Luca \\cite{KoLu25} (building on a heuristic independently found by Cambie (personal communication)) have shown that there is no finite limit, in that $ \\limsup_{n\\to \\infty}\\frac{f(2n)}{f(n)}=\\infty, $ and provide both theoretical and numerical evidence that suggests $\\lim \\frac{f(2n)}{f(n)}=\\infty$.\nReferences\n\n\n[Er98] Erd\\H{o}s, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.\n\n[KoLu25] V. Kova\\v{c} and F. Luca, On the number of divisors of Mersenne numbers. arXiv:2506.04883 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The ratio has no finite limit; whether it tends to infinity remains unproved.\n\n**Verified partial progress.**\n\n- Kovač--Luca prove limsup f(2n)/f(n)=infinity.\n- They give theoretical and numerical evidence for divergence to infinity.\n\n**Full solution or refutation.**\n\nAn infinite limsup excludes a finite limit but not a finite lower subsequential limit.\n\n**What remains.**\n\nProve or refute f(2n)/f(n) tends to infinity.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #893, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/893\n Evidence used: Current open status and Kovač--Luca result.\n\n**Review notes.** The literal question's answer is only partially settled.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2373, "problem_number": "EP-896", "title": "Erdős Problem #896", "statement": "Estimate the maximum of $F(A,B)$ as $A,B$ range over all subsets of $\\{1,\\ldots,N\\}$, where $F(A,B)$ counts the number of $m$ such that $m=ab$ has exactly one solution (with $a\\in A$ and $b\\in B$).", "background": "In the comments van Doorn proves $ (1+o(1))\\frac{N^2}{\\log N}\\leq \\max_{A,B}F(A,B) \\ll \\frac{N^2}{(\\log N)^\\delta(\\log\\log N)^{3/2}} $ where $\\delta=1-\\frac{1+\\log\\log 2}{\\log 2}\\approx 0.086$.\nSee also [490].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The maximum has order N^2/((log N)^delta (log log N)^(3/2)); Ford's multiplication-table theorem gives the upper bound and a 2026 community construction using Ford's one-divisor estimates gives the matching lower bound.\n\n**Verified partial progress.**\n\n- Ford determined the order of magnitude of the N by N multiplication table, immediately giving the sharp upper bound.\n- An earlier elementary construction gave only a lower bound of order N^2/log N.\n\n**Full solution or refutation.**\n\nA large-prime/cofactor construction based on Ford's estimates for integers with exactly one divisor in a selected interval yields uniquely represented products in the same order as the full multiplication table.\n\n**What remains.**\n\nThe order of magnitude is closed; a conventional standalone publication and independent review of the exact lower-bound construction remain desirable.\n\n**Sources checked.**\n\n- Kevin Ford, The distribution of integers with a divisor in a given interval, Annals of Mathematics 168 (2008), 367-433. (primary): https://doi.org/10.4007/annals.2008.168.367\n Evidence used: Determines the divisor-interval and multiplication-table orders that supply the analytic estimates and upper bound.\n- Thomas F. Bloom and contributors, Erdos Problem #896 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/896?order=oldest\n Evidence used: Records SOLVED status and gives the matching lower-bound construction in detail.\n- teorth/erdosproblems, AI contributions to Erdos problems, entry 896, checked 2026-08-17. (source_collection): https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems\n Evidence used: Records the full-solution attribution to Przemek Chojecki and GPT-5.5 Pro.\n\n**Review notes.** The imported background contains only the earlier gap and ends with leaked serialized text.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2374, "problem_number": "EP-901", "title": "Erdős Problem #901", "statement": "Let $m(n)$ be minimal such that there is an $n$-uniform hypergraph with $m(n)$ edges which is $3$-chromatic. Estimate $m(n)$.", "background": "In other words, the hypergraph does not have Property B. Property B means that there is a set $S$ which intersects all edges and yet does not contain any edge.\nIt is known that $m(2)=3$, $m(3)=7$, and $m(4)=23$. Erd\\H{o}s proved $ 2^n \\ll m(n) \\ll n^2 2^n $ (the lower bound in \\cite{Er63b} and the upper bound in \\cite{Er64e}). Erd\\H{o}s conjectured that $m(n)/2^n\\to \\infty$, which was proved by Beck \\cite{Be77}, who proved $m(n)\\gg (\\log n)2^n$, and later \\cite{Be78} improved this to $ n^{1/3-o(1)}2^n \\ll m(n). $ Radhakrishnan and Srinivasan \\cite{RaSr00} improved this to $ \\sqrt{\\frac{n}{\\log n}}2^n \\ll m(n). $ Pluhar \\cite{Pl09} gave a very short proof that $m(n) \\gg n^{1/4}2^n$.\nIn \\cite{ErLo75} Erd\\H{o}s and Lov\\'{a}sz speculate that $n2^n$ is the correct order of magnitude for $m(n)$.\nReferences\n\n\n[Be77] Beck, J., On a combinatorial problem of {P}. {E}rd\\H{o}s and {L}.\n{L}ov\\'asz. Discrete Math. (1977), 127--131.\n\n[Be78] Beck, J., On {$3$}-chromatic hypergraphs. Discrete Math. (1978), 127--137.\n\n[Er63b] Erd\\H{o}s, P., On a combinatorial problem. Nordisk Mat. Tidskr. (1963), 5--10, 40.\n\n[Er64e] Erd\\H{o}s, P., On a combinatorial problem. {II}. Acta Math. Acad. Sci. Hungar. (1964), 445--447.\n\n[ErLo75] Erd\\H{o}s, P. and Lov\\'{a}sz, L., Problems and results on {$3$}-chromatic hypergraphs and some\nrelated questions. (1975), 609--627.\n\n[Pl09] Pluh\\'ar, Andr\\'as, Greedy colorings of uniform hypergraphs. Random Structures Algorithms (2009), 216--221.\n\n[RaSr00] Radhakrishnan, Jaikumar and Srinivasan, Aravind, Improved bounds and algorithms for hypergraph {$2$}-coloring. Random Structures Algorithms (2000), 4--32.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exponential scale is known, but the best general bounds still differ by more than a polynomial square-root factor and the conjectured order n 2^n remains open.\n\n**Verified partial progress.**\n\n- Radhakrishnan and Srinivasan proved m(n) is at least a constant times 2^n sqrt(n/log n).\n- Erdős's upper bound m(n)=O(n^2 2^n) remains the best asymptotic upper bound reported in a 2024 primary paper.\n- Grill and Linzmayer improved lower bounds for small uniformities, but not the general asymptotic order.\n\n**Full solution or refutation.**\n\nNo source located closes the gap between Omega(2^n sqrt(n/log n)) and O(n^2 2^n).\n\n**What remains.**\n\nDetermine the correct polynomial factor multiplying 2^n; in particular, prove or refute that m(n) has order n 2^n.\n\n**Sources checked.**\n\n- Karl Grill and Daniel Linzmayer, Improved Lower Bounds for Property B, arXiv:2403.05674 (2024). (primary): https://arxiv.org/abs/2403.05674\n Evidence used: States the current general asymptotic bounds, identifies Erdős's upper bound as still best, and supplies new small-n lower bounds.\n- Thomas F. Bloom, Erdős Problem #901, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/901\n Evidence used: Retains open status and records the classical progression of general bounds.\n\n**Review notes.** The imported background has trailing serialization debris; it was treated as an extraction artifact.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2375, "problem_number": "EP-902", "title": "Erdős Problem #902", "statement": "Let $f(n)$ be minimal such that there is a tournament (a complete directed graph) on $f(n)$ vertices such that every set of $n$ vertices is dominated by at least one other vertex. Estimate $f(n)$.", "background": "Sch\"{u}tte asked Erd\\H{o}s this in the early 1960s.\nIt is easy to check that $f(1)=3$ and $f(2)=7$. Erd\\H{o}s \\cite{Er63c} proved $ 2^{n+1}-1 \\leq f(n) \\ll n^22^n. $ Szekeres and Szekeres \\cite{SzSz65} proved that $f(3)=19$ and $ n2^n \\ll f(n). $ \nReferences\n\n\n[Er63c] Erd\\H{o}s, P., On a problem in graph theory. Math. Gaz. (1963), 220--223.\n\n[SzSz65] Szekeres, E. and Szekeres, G., On a problem of {S}ch\"{u}tte and {E}rd\\H{o}s. Math. Gaz. (1965), 290--293.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The best located general bounds are n 2^n << f(n) << n^2 2^n, so the exact polynomial factor remains open.\n\n**Verified partial progress.**\n\n- Erdős proved 2^(n+1)-1 <= f(n) << n^2 2^n.\n- Szekeres and Szekeres improved the lower bound to f(n) >> n 2^n and determined f(3)=19.\n- A 2026 paper generalizes Schütte's property to sets of tournaments but does not claim an improved bound for this original one-tournament function.\n\n**Full solution or refutation.**\n\nThe small values f(1)=3, f(2)=7, and f(3)=19 and general exponential-order bounds are known, but no asymptotic estimate closing the factor-n gap was located.\n\n**What remains.**\n\nDetermine f(n) to within a constant factor, or improve either the n 2^n lower bound or n^2 2^n upper bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #902, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/902\n Evidence used: Records the exact small values, classical bounds, and continuing open status.\n- Joel Jeffries, Schütte's property for sets of tournaments and an application to dice games, arXiv:2604.08790 (2026). (primary): https://arxiv.org/abs/2604.08790\n Evidence used: Provides current context and a generalization; its abstract does not assert an improvement for the original f(n).\n\n**Review notes.** The imported background has trailing serialization debris; it was treated as an extraction artifact.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2376, "problem_number": "EP-906", "title": "Erdős Problem #906", "statement": "Is there an entire non-zero function $f:\\mathbb{C}\\to \\mathbb{C}$ such that, for any infinite sequence $n_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The literal statement is trivially affirmative for nonzero polynomials, while the intended transcendental-entire version remains open in the maintained tracker despite an unreferenced historical assertion and an unverified 2026 solution claim.\n\n**Verified partial progress.**\n\n- The intended condition is equivalent to every nonempty open set containing a zero of every sufficiently high derivative.\n- Barth and Schneider proved related derivative-zero results but not the exact intended assertion.\n- Boas and Reddy obtained low-order obstructions and constructions where every disk of one fixed positive radius contains a zero of every derivative; fixed-radius coverage is weaker than density at all scales.\n- A 2026 Gaussian-entire-function argument was posted as a proposed solution, but it was neither peer reviewed nor incorporated into the maintained status when checked.\n\n**Full solution or refutation.**\n\nAs written, a nonzero constant or polynomial answers yes. After the historically intended transcendentality restriction is imposed, no verified resolution was located.\n\n**What remains.**\n\nRepair the statement by requiring f to be transcendental entire, then verify the 2026 probabilistic argument or produce a peer-reviewed construction satisfying the cofinite derivative-order hitting property.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #906 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/906\n Evidence used: Flags the polynomial triviality, documents Erdős's unreferenced historical assertion, retains open status, and hosts the recent proposed solution.\n- K. F. Barth and W. J. Schneider, On a problem of Erdős concerning the zeros of the derivatives of an entire function, Proc. Amer. Math. Soc. 32 (1972), 229-232. (primary): https://www.erdosproblems.com/906\n Evidence used: The tracker and source discussion distinguish its related result from the exact claimed property.\n- R. P. Boas Jr. and A. R. Reddy, Zeros of the successive derivatives of entire functions, J. Math. Anal. Appl. 42 (1973), 466-473; Bull. Amer. Math. Soc. announcement. (primary): https://doi.org/10.1090/S0002-9904-1973-13093-9\n Evidence used: Provides related finite-order obstruction and fixed-radius derivative-zero constructions, not the exact density property.\n\n**Review notes.** Material formulation defect preserved: the source says only 'entire non-zero', which admits trivial polynomial examples. The imported background also has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2377, "problem_number": "EP-911", "title": "Erdős Problem #911", "statement": "Let $\\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges that is Ramsey for $G$.\nIs there a function $f$ such that $f(x)/x\\to \\infty$ as $x\\to \\infty$ such that, for all large $C$, if $G$ is a graph with $n$ vertices and $e\\geq Cn$ edges then $ \\hat{R}(G) > f(C) e? $ \",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The maintained tracker still records the requested superlinear-in-C universal lower bound for dense graphs as open, and no direct later resolution was located.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo theorem was found establishing a function f(C) with f(C)/C tending to infinity uniformly for every graph with at least Cn edges.\n\n**What remains.**\n\nProve or refute a universal lower bound of the form size-Ramsey(G)>f(C)e(G) with f(C)/C tending to infinity for all sufficiently dense n-vertex graphs.\n\n**Sources checked.**\n\n- Thomas F. Bloom, history for Erdős Problem #911, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/911\n Evidence used: Records the exact question and continuing open status.\n\n**Review notes.** The imported statement contains trailing JSON-like serialization debris after the mathematical question; the exact source remains preserved in input.json.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2378, "problem_number": "EP-912", "title": "Erdős Problem #912", "statement": "If $ n! = \\prod_i p_i^{k_i} $ is the factorisation into distinct primes then let $h(n)$ count the number of distinct exponents $k_i$.\nProve that there exists some $c>0$ such that $ h(n) \\sim c \\left(\\frac{n}{\\log n}\\right)^{1/2} $ as $n\\to \\infty$.", "background": "A problem of Erd\\H{o}s and Selfridge, who proved (see \\cite{Er82c}) $ h(n) \\asymp \\left(\\frac{n}{\\log n}\\right)^{1/2}. $ A heuristic of Tao using the Cram\\'{e}r model for the primes (detailed in the comments) suggests this is true with $ c=\\sqrt{2\\pi}=2.506\\cdots. $ \nReferences\n\n\n[Er82c] Erd\\H{o}s, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Erdős and Selfridge proved h(n) has order sqrt(n/log n), but the existence and value of an asymptotic constant remain open.\n\n**Verified partial progress.**\n\n- The unconditional estimate h(n) asymp sqrt(n/log n) establishes the requested scale.\n- A Cramér-model occupancy heuristic due to Tao predicts the constant c=sqrt(2 pi).\n- The tracker explains that proving the heuristic appears to require prime-gap distribution beyond current unconditional results.\n\n**Full solution or refutation.**\n\nThe order of magnitude is settled, but no proof of h(n)~c sqrt(n/log n) for any constant c was located.\n\n**What remains.**\n\nProve existence of the normalized limit and, if the heuristic is correct, identify it as sqrt(2 pi).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #912 and Tao comment, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/912\n Evidence used: Records the Erdős-Selfridge order-of-magnitude theorem, open constant problem, and the sqrt(2 pi) heuristic.\n\n**Review notes.** The heuristic constant is explicitly not classified as proved. The imported background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2379, "problem_number": "EP-913", "title": "Erdős Problem #913", "statement": "Are there infinitely many $n$ such that if $ n(n+1) = \\prod_i p_i^{k_i} $ is the factorisation into distinct primes then all exponents $k_i$ are distinct?", "background": "It is likely that there are infinitely many primes $p$ such that $8p^2-1$ is also prime, in which case this is true with exponents $\\{1,2,3\\}$, letting $n=8p^2-1$.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Infinitely many consecutive pairs with all prime-factor exponents distinct are not known unconditionally; the tracker records only conjectural prime-value families and elementary structural observations.\n\n**Verified partial progress.**\n\n- If 8p^2-1 is prime for infinitely many primes p, then n=8p^2-1 yields exponent multiset {1,2,3} in n(n+1).\n- Schinzel-type prime-value conjectures imply infinitely many examples through such families.\n- Exponent 1 can occur for at most one prime divisor, so at least one of n and n+1 must be powerful.\n\n**Full solution or refutation.**\n\nThe displayed families are conditional and the necessary powerful-number observation does not prove infinitude.\n\n**What remains.**\n\nConstruct infinitely many examples unconditionally or prove that only finitely many exist.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #913 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/913\n Evidence used: Retains open status and records the conditional family and elementary obstruction.\n\n**Review notes.** Conditional consequences of Schinzel's Hypothesis H were not promoted to partial resolution. The imported background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2380, "problem_number": "EP-917", "title": "Erdős Problem #917", "statement": "Let $k\\geq 4$ and $f_k(n)$ be the largest number of edges in a graph on $n$ vertices which has chromatic number $k$ and is critical (i.e. deleting any edge reduces the chromatic number).\nIs it true that $ f_k(n) \\gg_k n^2? $ Is it true that $ f_6(n)\\sim n^2/4? $ More generally, is it true that, for $k\\geq 6$, $ f_k(n) \\sim \\frac{1}{2}\\left(1-\\frac{1}{\\lfloor k/3\\rfloor}\\right)n^2? $ ", "background": "Erd\\H{o}s \\cite{Er93} wrote 'I learned of this definition from Dirac in 1949 and immediately asked whether $f_k(n)=o(n^2)$. To my great surprise Dirac constructed a $6$ critical graph on $n$ vertices with more than $\\frac{n^2}{4}$ edges.' In fact Dirac \\cite{Di52} proved $ f_6(4n+2) \\geq 4n^2+8n+3, $ as witnessed by taking two disjoint copies of $C_{2n+1}$ and adding all edges between them.\nErd\\H{o}s \\cite{Er69b} observed that Dirac's construction generalises to show that, if $3\\mid k$, there are infinitely many values of $n$ (those of the shape $mk/3$ where $m$ is odd) such that $ f_k(n) \\geq \\frac{1}{2}\\left(1-\\frac{1}{k/3}\\right)n^2 + n. $ Toft \\cite{To70} proved that $f_k(n)\\gg_k n^2$ for $k\\geq 4$.\nConstructions of Stiebitz \\cite{St87} show that, for $k\\geq 6$, there exist infinitely many values of $n$ such that $ f_k(n) \\geq \\frac{1}{2}\\left(1-\\frac{1}{\\lfloor k/3\\rfloor+\\delta_k}\\right)n^2 $ where $\\delta_k=0$ if $k\\equiv 0\\pmod{3}$, $\\delta_k=1/7$ if $k\\equiv 1\\pmod{3}$, and $\\delta_k\\equiv 24/69$ if $k\\equiv 2\\pmod{3}$, which disproves Erd\\H{o}s' conjectured asympotic for $k\not\\equiv 0\\pmod{3}$.\nStiebitz also proved the general upper bound $ f_k(n) < \\mathrm{ex}(n;K_{k-1})\\sim \\frac{1}{2}\\left(1-\\frac{1}{k-2}\\right)n^2 $ for large $n$. Luo, Ma, and Yang \\cite{LMY23} have improved this upper bound to $ f_k(n) \\leq \\frac{1}{2}\\left(1-\\frac{1}{k-2}-\\frac{1}{36(k-1)^2}+o(1)\\right)n^2 $ See also [944] and [1032].\nReferences\n\n\n[Di52] Dirac, G. A., A property of {$4$}-chromatic graphs and some remarks on\ncritical graphs. J. London Math. Soc. (1952), 85-92.\n\n[Er69b] Erd\\H{o}s, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[LMY23] Luo, Cong and Ma, Jie and Yang, Tianchi, On the maximum number of edges in {$k$}-critical graphs. Combin. Probab. Comput. (2023), 900--911.\n\n[St87] Stiebitz, M., Subgraphs of colour-critical graphs. Combinatorica (1987), 303--312.\n\n[To70] Toft, B., On the maximal number of edges of critical {$k$}-chromatic\ngraphs. Studia Sci. Math. Hungar. (1970), 461--470.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Toft proved the quadratic lower-bound question, Stiebitz refuted the proposed general coefficient when k is not divisible by 3, and the f_6 asymptotic and divisible-by-3 cases remain open.\n\n**Verified partial progress.**\n\n- For every fixed k>=4, Toft constructed edge-critical k-chromatic graphs with Omega_k(n^2) edges.\n- Dirac supplied lower-bound constructions near n^2/4 for k=6.\n- Stiebitz constructions have a leading coefficient larger than the conjectured one for k congruent to 1 or 2 modulo 3, refuting that part of the proposed general asymptotic.\n- Stiebitz proved f_k(n)", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2381, "problem_number": "EP-918", "title": "Erdős Problem #918", "statement": "Is there a graph with $\\aleph_2$ vertices and chromatic number $\\aleph_2$ such that every subgraph on $\\aleph_1$ vertices has chromatic number $\\leq\\aleph_0$?\nIs there a graph with $\\aleph_{\\omega+1}$ vertices and chromatic number $\\aleph_1$ such that every subgraph on $\\aleph_\\omega$ vertices has chromatic number $\\leq\\aleph_0$?", "background": "A question of Erd\\H{o}s and Hajnal \\cite{ErHa68b}, who proved that for every finite $k$ there is a graph with chromatic number $\\aleph_1$ where each subgraph on less than $\\aleph_k$ vertices has chromatic number $\\leq \\aleph_0$.\nIn \\cite{Er69b} it is asked with chromatic number $=\\aleph_0$, but in the comments louisd observes this is (assuming subgraph and not induced subgraph was intended) trivially impossible, and hence presumably the problem was intended as written here (which is how it is posed in \\cite{ErHa68b}).\nReferences\n\n\n[Er69b] Erd\\H{o}s, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[ErHa68b] Erd\\H{o}s, P. and Hajnal, A., On chromatic number of infinite graphs. (1968), 83--98.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Shelah proved broad chromatic incompactness theorems under set-theoretic hypotheses, but the exact cardinalities and chromatic numbers in both displayed questions remain open in the maintained tracker.\n\n**Verified partial progress.**\n\n- Classical Erdős-Hajnal constructions establish finite-step analogues with uncountable chromatic number and countably chromatic smaller subgraphs.\n- Shelah's incompactness theorems construct graphs of chromatic number greater than kappa while every subgraph below a prescribed cardinal threshold has chromatic number at most kappa.\n- More on Compactness of Chromatic Numbers gives such a graph on mu^kappa nodes for regular kappa and suitable mu below the first fixed point.\n\n**Full solution or refutation.**\n\nThe general theorems do not transparently specialize in ZFC to exactly aleph_(omega+1) vertices with exactly aleph_1 chromatic number, nor do they directly settle the aleph_2 instance as stated.\n\n**What remains.**\n\nProduce the two exact graphs in ZFC, refute their existence, or state and verify precise additional set-theoretic hypotheses under which each exact cardinal instance follows.\n\n**Sources checked.**\n\n- Saharon Shelah, On incompactness for chromatic number of graphs, Acta Math. Hungar. 139 (2013), 363-371, arXiv:1205.0064. (primary): https://arxiv.org/abs/1205.0064\n Evidence used: Proves chromatic incompactness from nonreflection and related set-theoretic hypotheses.\n- Saharon Shelah, More on Compactness of Chromatic Numbers, arXiv:1302.3431 (2013). (primary): https://arxiv.org/abs/1302.3431\n Evidence used: Constructs graphs on mu^kappa nodes with chromatic number greater than kappa and all subgraphs of size below mu having chromatic number at most kappa.\n- Thomas F. Bloom, Erdős Problem #918, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/918\n Evidence used: Retains both exact questions as open and documents the corrupted older formulation.\n\n**Review notes.** Exact specialization depends on delicate cardinal arithmetic and subgraph extraction. The tracker also reports that one historical version had an impossible aleph_0 chromatic-number clause. The imported background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2382, "problem_number": "EP-919", "title": "Erdős Problem #919", "statement": "Is there a graph $G$ with vertex set $\\omega_2^2$ and chromatic number $\\aleph_2$ such that every subgraph whose vertices have a lesser type has chromatic number $\\leq \\aleph_0$?\nWhat if instead we ask for $G$ to have chromatic number $\\aleph_1$?", "background": "This question was inspired by a theorem of Babai, that if $G$ is a graph on a well-ordered set with chromatic number $\\geq \\aleph_0$ there is a subgraph on vertices with order-type $\\omega$ with chromatic number $\\aleph_0$.\nErd\\H{o}s and Hajnal showed this does not generalise to higher cardinals - they (see \\cite{Er69b}) constructed a set on $\\omega_1^2$ with chromatic number $\\aleph_1$ such that every strictly smaller subgraph has chromatic number $\\leq \\aleph_0$ as follows: the vertices of $G$ are the pairs $(x_\\alpha,y_\\beta)$ for $1\\leq \\alpha,\\beta <\\omega_1$, ordered lexicographically. We connect $(x_{\\alpha_1},y_{\\beta_1})$ and $(x_{\\alpha_2},y_{\\beta_2})$ if and only if $\\alpha_1<\\alpha_2$ and $\\beta_1<\\beta_2$.\nA similar construction produces a graph on $\\omega_2^2$ with chromatic number $\\aleph_2$ such that every smaller subgraph has chromatic number $\\leq \\aleph_1$.\nReferences\n\n\n[Er69b] Erd\\H{o}s, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A lower-cardinal analogue and a weakened omega_2^2 construction are known, but neither exact countably-chromatic lower-order-type condition is settled.\n\n**Verified partial progress.**\n\n- At vertex order type omega_1^2, a graph of chromatic number aleph_1 exists whose lower-order-type subgraphs are countably chromatic.\n- At omega_2^2, a known construction has chromatic number aleph_2 while every lower-order-type subgraph has chromatic number at most aleph_1.\n- The latter misses the required countable upper bound by one cardinal step.\n\n**Full solution or refutation.**\n\nKnown constructions establish a sharp-looking analogue and a weakened target but do not answer either displayed question.\n\n**What remains.**\n\nLower the local chromatic bound at omega_2^2 from aleph_1 to aleph_0 while retaining global chromatic number aleph_2 or aleph_1, or prove this impossible.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #919, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/919\n Evidence used: Records the omega_1^2 analogue, the weakened omega_2^2 construction, and continuing open status.\n\n**Review notes.** The phrase 'lesser type' is undefined in the imported statement; the tracker context suggests strictly smaller ordinal order type. The imported background also has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2383, "problem_number": "EP-920", "title": "Erdős Problem #920", "statement": "Let $f_k(n)$ be the maximum possible chromatic number of a graph with $n$ vertices which contains no $K_k$.\nIs it true that, for $k\\geq 4$, $ f_k(n) \\gg \\frac{n^{1-\\frac{1}{k-1}}}{(\\log n)^{c_k}} $ for some constant $c_k>0$?", "background": "Graver and Yackel \\cite{GrYa68} proved that $ f_k(n) \\ll \\left(n\\frac{\\log\\log n}{\\log n}\\right)^{1-\\frac{1}{k-1}}. $ It is known that $f_3(n)\\asymp (n/\\log n)^{1/2}$ (see [1104]).\nThe lower bound $R(4,m) \\gg m^3/(\\log m)^4$ of Mattheus and Verstraete \\cite{MaVe23} (see [166]) implies $ f_4(n) \\gg \\frac{n^{2/3}}{(\\log n)^{4/3}}. $ A positive answer to this question would follow from [986]. The known bounds for that problem imply $ f_k(n) \\gg \\frac{n^{1-\\frac{2}{k+1}}}{(\\log n)^{c_k}}. $ See [1104] (and also [1013]) for the case $k=3$.\nReferences\n\n\n[GrYa68] Graver, Jack E. and Yackel, James, Some graph theoretic results associated with Ramsey's theorem. J. Combinatorial Theory (1968), 125--175.\n\n[MaVe23] Mattheus, S. and Verstraete, J., The asymptotics of $r(4,t)$. arXiv:2306.04007 (2023).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The k=4 case is proved and Bradač gives a major new exponent for all larger fixed k, but the public preprint only announces rather than proves the stronger k-1 Ramsey exponent needed to settle the displayed claim for k at least 5.\n\n**Verified partial progress.**\n\n- Mattheus and Verstraete's r(4,t) bound proves the requested lower bound for k=4.\n- Bradač proves r(k,t) at least t^(k-2+o(1)), a substantial improvement for general fixed k.\n- A stronger t^(k-1) bound is announced as forthcoming and is marked solved by maintained metadata, but its proof is not in the current public paper.\n\n**Full solution or refutation.**\n\nRamsey lower bounds translate to chromatic lower bounds for clique-free graphs; the available general exponent remains short of the exact exponent requested in EP-920.\n\n**What remains.**\n\nPublish and verify the announced k-1 exponent proof, or otherwise prove the requested chromatic bound for every fixed k at least 5.\n\n**Sources checked.**\n\n- Sam Mattheus and Jacques Verstraete, The asymptotics of r(4,t), arXiv:2306.04007 (2023). (primary): https://arxiv.org/abs/2306.04007\n Evidence used: Proves the Ramsey estimate implying the k=4 instance.\n- Domagoj Bradac, Nearly tight exponents for off-diagonal Ramsey numbers, arXiv:2605.28793 (2026). (primary): https://arxiv.org/abs/2605.28793\n Evidence used: Proves exponent k-2 and explicitly describes the k-1 exponent as an upcoming-paper result rather than a theorem proved in the current version.\n- Google DeepMind Formal Conjectures, ErdosProblems/920.lean, checked 2026-08-17. (formal_verification): https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/920.lean\n Evidence used: Labels the statement solved but contains sorry placeholders, so it is a formal statement/status record and not a checked proof.\n- Thomas F. Bloom, Erdos Problem #920, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/920\n Evidence used: The public page still says OPEN and summarizes the proved k=4 case and weaker general bounds.\n\n**Review notes.** This deliberately does not inherit the newer YAML solved label because the announced full proof was not publicly available in the sources checked.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2384, "problem_number": "EP-928", "title": "Erdős Problem #928", "statement": "Let $\\alpha,\\beta\\in (0,1)$ and let $P(n)$ denote the largest prime divisor of $n$. Does the density of integers $n$ such that $P(n)\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The logarithmic density is unconditionally the product of the Dickman marginal densities, and the same natural-density formula is known conditionally; unconditional natural density remains open.\n\n**Verified partial progress.**\n\n- Teräväinen proved that the logarithmic density exists and equals rho(1/alpha) rho(1/beta).\n- Wang proved the corresponding ordinary-density formula under an Elliott-Halberstam conjecture for friable integers.\n- Schinzel's earlier construction establishes infinitely many very smooth consecutive pairs, but not a density.\n\n**Full solution or refutation.**\n\nThe requested unweighted natural density is not known unconditionally, despite an exact logarithmic analogue and a conditional natural-density theorem.\n\n**What remains.**\n\nRemove the friable Elliott-Halberstam hypothesis and prove existence, presumably with value rho(1/alpha) rho(1/beta), of the ordinary natural density.\n\n**Sources checked.**\n\n- Joni Teräväinen, On binary correlations of multiplicative functions, Forum Math. Sigma 6 (2018), e10, arXiv:1710.01195. (primary): https://arxiv.org/abs/1710.01195\n Evidence used: Proves a logarithmic version of the Erdős-Pomerance conjecture on consecutive smooth numbers.\n- Ke Wang, Three conjectures on P+(n) and P+(n+1) hold under Elliott-Halberstam conjecture for friable integers, J. Number Theory 225 (2021), 37-65. (primary): https://www.sciencedirect.com/science/article/abs/pii/S0022314X21000196\n Evidence used: Proves the ordinary-density product formula conditionally on a friable Elliott-Halberstam conjecture.\n- Thomas F. Bloom, history for Erdős Problem #928, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/928\n Evidence used: Synthesizes Schinzel's infinitude result, the logarithmic-density theorem, conditional natural density, and continuing open status.\n\n**Review notes.** Logarithmic density and conditional natural density are distinguished from the unconditional natural-density question. The imported background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2385, "problem_number": "EP-929", "title": "Erdős Problem #929", "statement": "Let $k\\geq 2$ be large and let $S(k)$ be the minimal $x$ such that there is a positive density set of $n$ where $ n+1,n+2,\\ldots,n+k $ are all divisible by primes $\\leq x$.\nEstimate $S(k)$ - in particular, is it true that $S(k)\\geq k^{1-o(1)}$?", "background": "It follows from Rosser's sieve that $S(k)> k^{1/2-o(1)}$.\nIt is trivial that $S(k)\\leq k+1$ since, for example, one can take $n\\equiv 1\\pmod{(k+1)!}$. The best bound on large gaps between primes due to Ford, Green, Konyagin, Maynard, and Tao \\cite{FGKMT18} (see [4]) implies $ S(k) \\ll k \\frac{\\log\\log\\log k}{\\log\\log k\\log\\log\\log\\log k}. $ \nReferences\n\n\n[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. (2018), 65-105.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Known sieve and prime-gap results place S(k) between k^{1/2-o(1)} and a near-linear upper bound, but the conjectured k^{1-o(1)} lower bound remains open.\n\n**Verified partial progress.**\n\n- Rosser's sieve gives S(k)>k^{1/2-o(1)}.\n- Ford, Green, Konyagin, Maynard, and Tao's long-prime-gap theorem implies S(k)<=k^{1-o(1)} and determine the asymptotic scale more precisely.\n\n**Sources checked.**\n\n- Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao, Long gaps between primes, Journal of the AMS 31 (2018), 65-105, DOI 10.1090/jams/876. (primary): https://arxiv.org/abs/1412.5029\n Evidence used: Primary source for the prime-gap estimate from which the tracker derives the current upper bound for S(k).\n- Thomas F. Bloom, Erdős Problem #929, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/929\n Evidence used: Maintains open status and records the sieve lower bound and prime-gap upper bound.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2386, "problem_number": "EP-930", "title": "Erdős Problem #930", "statement": "Is it true that, for every $r$, there is a $k$ such that if $I_1,\\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then $ \\prod_{1\\leq i\\leq r}\\prod_{m\\in I_i}m $ is not a perfect power?", "background": "Erd\\H{o}s and Selfridge \\cite{ErSe75} proved that the product of consecutive integers is never a power (establishing the case $r=1$). The condition that the intervals be large in terms of $r$ is necessary for $r=2$ - see the constructions in [363].\nSee also [363] for the case of squares.\nReferences\n\n\n[ErSe75] Erd\\H{o}s, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The r=1 case is the Erdős-Selfridge theorem, while multiple fixed short blocks can have perfect-power products; no threshold length is known for every fixed r.\n\n**Verified partial progress.**\n\n- Erdős and Selfridge proved that a single block of at least two consecutive positive integers never has perfect-power product.\n- Skałba and later authors constructed many perfect-power products of multiple disjoint blocks at fixed small lengths, showing that a length condition is substantive.\n\n**Full solution or refutation.**\n\nOne interval is fully understood and short-block obstructions are known, but the all-r threshold assertion remains open.\n\n**What remains.**\n\nFor each r>=2, prove that some sufficiently large common lower bound on the interval lengths rules out every perfect power, or construct counterexamples of unbounded length.\n\n**Sources checked.**\n\n- P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois Journal of Mathematics 19 (1975), 292-301. (primary): https://combinatorica.hu/~p_erdos/1975-46.pdf\n Evidence used: Proves the r=1 case.\n- Mariusz Skałba, Products of disjoint blocks of consecutive integers which are powers, Colloquium Mathematicum 98 (2003), 1-3. (primary): https://eudml.org/doc/284443\n Evidence used: Constructs multiple-block perfect-power examples at fixed lengths.\n- Thomas F. Bloom, Erdős Problem #930, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/930\n Evidence used: Retains the general threshold question as open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2387, "problem_number": "EP-931", "title": "Erdős Problem #931", "statement": "Let $k_1\\geq k_2\\geq 3$. Are there only finitely many $n_2\\geq n_1+k_1$ such that $ \\prod_{1\\leq i\\leq k_1}(n_1+i)\\textrm{ and }\\prod_{1\\leq j\\leq k_2}(n_2+j) $ have the same prime factors?", "background": "Tijdeman gave the example $ 19,20,21,22\\textrm{ and }54,55,56,57. $ Erd\\H{o}s \\cite{Er76d} was unsure of this conjecture, and thought perhaps if the two products have the same prime factors then $n_2>2(n_1+k_1)$. It is not clear but it is possible that he meant to ask this question also permitting finitely many counterexamples. Indeed, without this caveat it is false - AlphaProof has found the counterexample $ 10! = 2^8\\cdot 3^4\\cdot 5^2\\cdot 7 $ and $ 14\\cdot 15\\cdot 16 = 2^5\\cdot 3\\cdot 5\\cdot 7, $ so that $n_1=0$, $k_1=10$, $n_2=13$, and $k_2=3$.\nSee also [388].\nThis is discussed in problem B35 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er76d] Erd\\H{o}s, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No global finiteness theorem was located for separated consecutive blocks with equal prime support; fixed-first-block finiteness and bounded searches are not uniform in n1.\n\n**Verified partial progress.**\n\n- For fixed n1, finiteness in n2 follows from finiteness results for consecutive integers supported on a fixed finite prime set.\n- A 2026 forum study of (k1,k2)=(4,3) reports bounded computational evidence, but explicitly leaves the global step open.\n\n**Full solution or refutation.**\n\nThe maintained tracker remains open, and the available local finiteness statement does not establish finiteness as both block positions vary.\n\n**What remains.**\n\nProve a bound uniform in n1 for each fixed k1>=k2>=3, or exhibit infinitely many separated pairs with the same prime support.\n\n**Sources checked.**\n\n- P. Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (1976), 25-44. (primary): https://combinatorica.hu/~p_erdos/1976-39.pdf\n Evidence used: Original source and context for the conjecture.\n- Thomas F. Bloom, Erdős Problem #931 and discussion thread, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/931\n Evidence used: Maintains open status; the linked discussion records only local finiteness and bounded computation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2388, "problem_number": "EP-932", "title": "Erdős Problem #932", "statement": "Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_r\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The requirement of two integers smooth at the scale of the containing prime gap remains beyond current results; no theorem matching the imported assertion was located.\n\n**Verified partial progress.**\n\n- Erdős proved that the set of indices for which at least one such smooth integer exists has density zero.\n- Current tracker discussion identifies the simultaneous smooth-number and prime-gap scales as a barrier and states that even relaxed variants are out of reach.\n\n**Full solution or refutation.**\n\nThe exact infinitude assertion remains open; Lean formalization of the statement does not change its mathematical status.\n\n**What remains.**\n\nProduce two distinct (p_{r+1}-p_r)-smooth integers in infinitely many prime gaps, or prove that only finitely many such gaps occur.\n\n**Sources checked.**\n\n- P. Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (1976), 25-44. (primary): https://combinatorica.hu/~p_erdos/1976-39.pdf\n Evidence used: Original source for the problem and Erdős's density observation.\n- Thomas F. Bloom, Erdős Problem #932 and discussion thread, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/932\n Evidence used: Maintains open status and records contemporary expert assessment of the smoothness barrier.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2389, "problem_number": "EP-933", "title": "Erdős Problem #933", "statement": "If $n(n+1)=2^k3^lm$, where $(m,6)=1$, then is it true that $ \\limsup_{n\\to \\infty} \\frac{2^k3^l}{n\\log n}=\\infty? $ ", "background": "Mahler proved (a more general result that implies in particular) that $ 2^k3^ln\\log n. $ Steinerberger has noted a simple proof of this fact follows from taking $n=2^{3^r}$ for any integer $r\\geq 1$, when $k=3^r$ and $l=r+1$.\nReferences\n\n\n[Er76d] Erd\\H{o}s, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The family n=2^{3^r} makes 2^k3^l/(n log n)=3/log 2 infinitely often, proving a positive constant lower bound but not the requested infinite limsup.\n\n**Verified partial progress.**\n\n- Mahler's theorem implies 2^k3^l", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2390, "problem_number": "EP-934", "title": "Erdős Problem #934", "statement": "Let $h_t(d)$ be minimal such that every graph $G$ with $h_t(d)$ edges and maximal degree $\\leq d$ contains two edges whose shortest path between them has length $\\geq t$.\nEstimate $h_t(d)$.", "background": "A problem of Erd\\H{o}s and Ne\\v{s}et\\v{r}il. Erd\\H{o}s \\cite{Er88} wrote 'This problem seems to be interesting only if there is a nice expression for $h_t(d)$.'\nIt is easy to see that $h_t(d)\\leq 2d^t$ always and $h_1(d)=d+1$.\nErd\\H{o}s and Ne\\v{s}et\\v{r}il and Bermond, Bond, Paoli, and Peyrat \\cite{BBPP83} independently conjectured that $h_2(d) \\leq \\tfrac{5}{4}d^2+1$, with equality for even $d$ (see [149]). This was proved by Chung, Gy\\'{a}rf\\'{a}s, Tuza, and Trotter \\cite{CGTT90}.\nCambie, Cames van Batenburg, de Joannis de Verclos, and Kang \\cite{CCJK22} conjectured that $ h_3(d) \\leq d^3-d^2+d+2, $ with equality if and only if $d=p^k+1$ for some prime power $p^k$, and proved that $h_3(3)=23$. They also conjecture that, for all $t\\geq 3$, $h_t(d)\\geq (1-o(1))d^t$ for infinitely many $d$ and $h_t(d)\\leq (1+o(1))d^t$ for all $d$ (where the $o(1)$ term $\\to 0$ as $d\\to \\infty$).\nThe same authors prove that, if $t$ is large, then there are infinitely many $d$ such that $h_t(d) \\geq 0.629^td^t$, and that for all $t\\geq 1$ we have $ h_t(d) \\leq \\tfrac{3}{2}d^t+1. $ \nReferences\n\n\n[BBPP83] Bermond, J.-C. and Bond, J. and Paoli, M. and Peyrat, C., Graphs and interconnection networks: diameter and\nvulnerability. (1983), 1--30.\n\n[CCJK22] Cambie, Stijn and Cames van Batenburg, Wouter and de Joannis\nde Verclos, R\\'{e}mi and Kang, Ross J., Maximizing line subgraphs of diameter at most {$t$}. SIAM J. Discrete Math. (2022), 939--950.\n\n[CGTT90] Chung, F. R. K. and Gy\\'arf\\'as, A. and Tuza, Z. and Trotter,\nW. T., The maximum number of edges in {$2K_2$}-free graphs of bounded\ndegree. Discrete Math. (1990), 129--135.\n\n[Er88] Erd\\H{o}s, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The t=2 case is solved and modern work proves h_t(d)<=1.5d^t+1, lower constructions of d^t scale, and h_3(3)=23, but no general exact or asymptotic formula is known.\n\n**Verified partial progress.**\n\n- Chung, Gyárfás, Tuza, and Trotter proved the conjectured 5d^2/4+1 bound for t=2, sharp for even d.\n- Cambie, Cames van Batenburg, de Joannis de Verclos, and Kang proved h_t(d)<=1.5d^t+1 and h_3(3)=23.\n- The same authors provide lower constructions of order 0.629^t d^t for infinitely many parameter values and conjecture asymptotic constant one.\n\n**Full solution or refutation.**\n\nSeveral exact cases and substantially sharpened d^t-scale bounds are known, while the general estimation problem remains open.\n\n**What remains.**\n\nDetermine h_t(d) for t>=3, particularly prove or disprove the conjectured (1+o(1))d^t asymptotic and the proposed exact t=3 formula.\n\n**Sources checked.**\n\n- Stijn Cambie, Wouter Cames van Batenburg, Rémi de Joannis de Verclos, and Ross J. Kang, Maximizing Line Subgraphs of Diameter at Most t, SIAM Journal on Discrete Mathematics 36 (2022), 939-950, DOI 10.1137/21M1437354. (primary): https://arxiv.org/abs/2103.11898\n Evidence used: Primary source for the modern general bounds, exact cubic case, constructions, and conjectures.\n- F. R. K. Chung, A. Gyárfás, Z. Tuza, and W. T. Trotter, The maximum number of edges in 2K2-free graphs of bounded degree, Discrete Mathematics 81 (1990), 129-135, DOI 10.1016/0012-365X(90)90144-7. (primary): https://doi.org/10.1016/0012-365X(90)90144-7\n Evidence used: Settles the t=2 conjecture.\n- Thomas F. Bloom, Erdős Problem #934 and comments, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/934\n Evidence used: Maintains open status and corrects the small d=2 exception to the h_1(d) background remark.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2391, "problem_number": "EP-935", "title": "Erdős Problem #935", "statement": "For any integer $n=\\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that $ Q_2(n) = \\prod_{\\substack{p\\\\ k_p\\geq 2}}p^{k_p}. $ Is it true that, for every $\\epsilon>0$ and $\\ell\\geq 1$, if $n$ is sufficiently large then $ Q_2(n(n+1)\\cdots(n+\\ell))2$, only keeping those prime powers with exponent $\\geq r$.\nReferences\n\n\n[Er76d] Erd\\H{o}s, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The infinite-limsup question is affirmative via a Pell-equation construction, and abc implies the final limit; the n^{2+epsilon} upper question and unconditional final limit remain open.\n\n**Verified partial progress.**\n\n- The Pell equation x^2-8y^2=1 yields limsup Q_2(n(n+1)(n+2))/n^2=infinity.\n- The construction extends the affirmative limsup conclusion to every fixed ell>=2.\n- The abc conjecture implies Q_2(n(n+1)...(n+ell))/n^{ell+1}->0.\n\n**Full solution or refutation.**\n\nOne of the three imported questions is solved affirmatively, another is conditionally affirmative, and the strongest uniform upper claim remains open.\n\n**What remains.**\n\nProve the first n^{2+epsilon} bound and the third limit unconditionally, and sharpen the growth of the limsup examples.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #935 and discussion thread, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/935\n Evidence used: Records the Pell construction, its priority history, the affirmative second question, and the abc-conditional third question.\n- Tony Feng et al., Towards Autonomous Mathematics Research, arXiv:2602.10177 (2026). (primary): https://arxiv.org/abs/2602.10177\n Evidence used: Contains an independent rediscovery of the Pell-based result; the tracker notes the same construction appeared earlier in Problem 367 comments.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2392, "problem_number": "EP-936", "title": "Erdős Problem #936", "statement": "Are $ 2^n\\pm 1 $ and $ n!\\pm 1 $ powerful (i.e. if $p\\mid m$ then $p^2\\mid m$) for only finitely many $n$?", "background": "Cushing and Pascoe \\cite{CuPa16} have shown the answer to the second question is yes assuming the abc conjecture - in fact, for any fixed $k\\geq 0$, there are only finitely many $n$ and powerful $x$ such that $\\lvert x-n!\\rvert \\leq k$.\nCrowdMath \\cite{Cr20} has shown that the answer to the first question is yes, again assuming the abc conjecture.\nReferences\n\n\n[Cr20] P. A. CrowdMath, Applications of the abc conjecture to powerful numbers. arXiv:2005.07321 (2020).\n\n[CuPa16] D. Cushing and J. E. Pascoe, Powerful numbers and the ABC-conjecture. arXiv:1611.01192 (2016).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Both the exponential and factorial finiteness assertions are proved assuming the abc conjecture, but no unconditional proof was located.\n\n**Verified partial progress.**\n\n- Cushing and Pascoe prove under abc that powerful numbers occur within any fixed distance of n! only finitely often.\n- CrowdMath proves under abc the analogous finiteness for fixed shifts of fixed-base exponentials, covering 2^n+1 and 2^n-1.\n\n**Full solution or refutation.**\n\nThe exact conclusions are conditionally known under abc and remain unconditionally open.\n\n**What remains.**\n\nRemove the abc hypothesis for all four sign/family variants, or find an unconditional counterexample family.\n\n**Sources checked.**\n\n- David Cushing and James Eldred Pascoe, Powerful numbers and the ABC-conjecture, arXiv:1611.01192 (2016). (primary): https://arxiv.org/abs/1611.01192\n Evidence used: Provides the abc-conditional factorial-neighborhood finiteness theorem.\n- P. A. CrowdMath, Applications of the abc conjecture to powerful numbers, arXiv:2005.07321 (2020). (primary): https://arxiv.org/abs/2005.07321\n Evidence used: Provides the abc-conditional fixed exponential-shift finiteness theorem.\n- Thomas F. Bloom, Erdős Problem #936, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/936\n Evidence used: Maintains unconditional open status and records both conditional results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2393, "problem_number": "EP-938", "title": "Erdős Problem #938", "statement": "Let $A=\\{n_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Van Doorn constructs infinitely many three-term progressions of powerful numbers with gap 2sqrt(N)+1, but only conjectures that infinitely many are consecutive terms in the powerful-number sequence.\n\n**Verified partial progress.**\n\n- The 2026 preprint proves infinitely many powerful progressions N,N+d,N+2d with d=2sqrt(N)+1.\n- The remaining absence of intervening powerful numbers is isolated explicitly as a conjectural step.\n\n**Full solution or refutation.**\n\nA close short-gap analogue is solved, but the defining consecutive-term condition of EP-938 remains unproved.\n\n**What remains.**\n\nShow that only finitely or infinitely many of the constructed progressions have no powerful numbers between adjacent terms, thereby resolving the exact finiteness question.\n\n**Sources checked.**\n\n- Wouter van Doorn, Three-term arithmetic progressions of consecutive powerful numbers, arXiv:2605.06697 (2026). (primary): https://arxiv.org/abs/2605.06697\n Evidence used: Proves the infinite short-gap progression family and explicitly labels consecutiveness as conjectural.\n- Thomas F. Bloom, Erdős Problem #938, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/938\n Evidence used: Retains open status while flagging partial progress.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2394, "problem_number": "EP-939", "title": "Erdős Problem #939", "statement": "Let $r\\geq 2$. An $r$-powerful number $n$ is one such that if $p\\mid n$ then $p^r\\mid n$.\nIf $r\\geq 4$ then can the sum of $r-2$ coprime $r$-powerful numbers ever be itself $r$-powerful? Are there at most finitely many such solutions?\nAre there infinitely many triples of coprime $3$-powerful numbers $a,b,c$ such that $a+b=c$?", "background": "The answer to the third question is yes: Nitaj \\cite{Ni95} has proved that there are infinitely many triples of coprime $3$-powerful numbers $a,b,c$ such that $a+b=c$, such as $ 2^3\\cdot 3^5\\cdot 73^3+271^3 = 919^3. $ In Nitaj's construction at least two of $a,b,c$ are perfect cubes. Cohn \\cite{Co98} constructed infinitely many such triples, none of which are perfect cubes. An alternative construction was given by Walsh \\cite{Wa24}.\nEuler had conjectured that the sum of $k-1$ many $k$th powers is never a $k$th power, but this is false for $k=5$, as Lander and Parkin \\cite{LaPa67} found $ 27^5+84^5+110^5+133^5=144^5. $ Cambie has found several examples of the sum of $r-2$ coprime $r$-powerful numbers being itself $r$-powerful. For example when $r=5$ $ 3^761^5=2^83^{10}5^7+2^{12}23^6+11^513^5. $ Cambie has also found solutions when $r=7$ or $r=8$ (the latter even with the sum of $5$ $8$-powerful numbers being $8$-powerful).\nReferences\n\n\n[Co98] Cohn, J. H. E., A conjecture of {E}rd\\H{o}s on {$3$}-powerful numbers. Math. Comp. (1998), 439--440.\n\n[LaPa67] Lander, L. J. and Parkin, T. R., A counterexample to {E}uler's sum of powers conjecture. Math. Comp. (1967), 101--103.\n\n[Ni95] Nitaj, Abderrahmane, On a conjecture of {E}rd\\H{o}s on {$3$}-powerful numbers. Bull. London Math. Soc. (1995), 317--318.\n\n[Wa24] P. Walsh, A question of Erd\\H{o}s on 3-powerful numbers and an elliptic curve analogue of the Ankeny-Artin-Chowla conjecture. arXiv:2404.039701 (2024).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The 3-powerful infinitude question is solved; higher-r examples exist, and a 2026 comment construction gives infinite families for every r>=6 under joint coprimality, leaving r=4 and r=5 finiteness and the coprimality interpretation unresolved.\n\n**Verified partial progress.**\n\n- Nitaj proved infinitely many coprime 3-powerful solutions a+b=c; Cohn and Walsh gave further infinite constructions.\n- Explicit examples of r-2 summands are known for r=5,7,8.\n- A tracker-comment binomial construction, with linked Lean formalization, produces infinitely many jointly coprime representations for every r>=6.\n\n**Full solution or refutation.**\n\nThe third question is fully solved, the existence question is answered for r=5 and all r>=6 under the joint-gcd reading, and finiteness is false for r>=6; remaining low exponents and pairwise coprimality are open.\n\n**What remains.**\n\nClarify whether coprime means joint or pairwise, settle existence for r=4, and determine finiteness for r=4 and r=5; independently review or publish the r>=6 construction.\n\n**Sources checked.**\n\n- Abderrahmane Nitaj, On a Conjecture of Erdős on 3-Powerful Numbers, Bulletin of the London Mathematical Society 27 (1995), 317-318, DOI 10.1112/blms/27.4.317. (primary): https://doi.org/10.1112/blms/27.4.317\n Evidence used: Proves infinitely many coprime 3-powerful solutions.\n- J. H. E. Cohn, A conjecture of Erdős on 3-powerful numbers, Mathematics of Computation 67 (1998), 439-440, DOI 10.1090/S0025-5718-98-00881-3. (primary): https://doi.org/10.1090/S0025-5718-98-00881-3\n Evidence used: Constructs infinitely many 3-powerful triples with no term a perfect cube.\n- P. G. Walsh, A question of Erdős on 3-powerful numbers and an elliptic curve analogue of the Ankeny-Artin-Chowla conjecture, arXiv:2404.03970 (2024). (primary): https://arxiv.org/abs/2404.03970\n Evidence used: Supplies another infinite 3-powerful construction and confirms the valid arXiv identifier.\n- Thomas F. Bloom, Erdős Problem #939 and discussion thread, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/939\n Evidence used: Records higher-r examples and the explicit r>=6 comment construction, while retaining open status for unresolved cases.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2395, "problem_number": "EP-940", "title": "Erdős Problem #940", "statement": "Let $r\\geq 3$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\\mid n$.\nAre there infinitely many integers which are not the sum of at most $r$ many $r$-powerful numbers? Does the set of integers which are the sum of at most $r$ $r$-powerful numbers have density $0$?", "background": "Erd\\H{o}s \\cite{Er76d} claims that the claim that the set has density $0$ is 'easy' for $r=2$ (a potential 'easy argument' is given in the comments by Tao; this was first proved in the literature by Baker and Br\"{u}dern \\cite{BaBr94}). For $r=3$ it is not even known if those integers which are the sum of at most three cubes has density $0$.\nIn the Oberwolfach problem book this is recorded in 1986 as a problem of Erd\\H{o}s and Ivi\\'{c}.\nIn \\cite{Er76d} Erd\\H{o}s claims that 'a simple counting argument' implies that there are infinitely many integers which are not the sum of at most $r$ many $r$-powerful numbers, but Schinzel pointed out he made a mistake.\nHeath-Brown \\cite{He88} has proved that all large numbers are the sum of at most three $2$-powerful numbers, see [941].\nSee also [1081] for a more refined question concerning the case $r=2$, and [1107] for the case of $r+1$ summands.\nReferences\n\n\n[BaBr94] Baker, R. C. and Br\"udern, J., On sums of two squarefull numbers. Math. Proc. Cambridge Philos. Soc. (1994), 1--5.\n\n[Er76d] Erd\\H{o}s, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n\n[He88] Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. (1988), 137--163.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The r-powerful sumset questions remain open for r>=3; the r=2 analogue is settled differently.\n\n**Verified partial progress.**\n\n- Baker--Brüdern prove the density-zero statement for r=2.\n- For r=3, density zero is unknown even for sums of three cubes.\n\n**Full solution or refutation.**\n\nThe requested r>=3 assertion remains unresolved.\n\n**What remains.**\n\nProve infinitude of exceptions or density zero for any r>=3.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #940, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/940\n Evidence used: Current open status and r=2 contrast.\n\n**Review notes.** Erdos's historical counting argument is reported as erroneous.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2396, "problem_number": "EP-942", "title": "Erdős Problem #942", "statement": "Let $h(n)$ count the number of powerful (if $p\\mid m$ then $p^2\\mid m$) integers in $[n^2,(n+1)^2)$. Estimate $h(n)$. In particular is there some constant $c>0$ such that $ h(n) < (\\log n)^{c+o(1)} $ and, for infinitely many $n$, $ h(n) >(\\log n)^{c-o(1)}? $ ", "background": "Erd\\H{o}s writes it is not hard to prove that $\\limsup h(n)=\\infty$, and that the density $\\delta_l$ of integers for which $h(n)=l$ exists and $\\sum \\delta_l=1$.\nA proof that $h(n)$ is unbounded is provided by van Doorn in the comments.\nDe Koninck and Luca \\cite{DeLu04} have proved, for infinitely many $n$, $ h(n) \\gg \\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/3}. $ They also give the density ($\\approx 0.275$) of those $n$ such that $h(n)=1$.\nReferences\n\n\n[DeLu04] De Koninck, Jean-Marie and Luca, Florian, Sur la proximit\\'{e} des nombres puissants. Acta Arith. (2004), 149--157.\n\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The powerful-number count in consecutive-square intervals is open but has near-logarithmic lower bounds infinitely often.\n\n**Verified partial progress.**\n\n- De Koninck--Luca prove h(n)>>((log n)/(log log n))^(1/3) infinitely often.\n- The maintained record reports an optimization of their construction to h(n)>>log n/(log log n log log log n) infinitely often.\n\n**Full solution or refutation.**\n\nNo matching upper exponent or distributional asymptotic is known.\n\n**What remains.**\n\nDetermine the correct logarithmic scale of h(n).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #942, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/942\n Evidence used: Current open status and bounds.\n\n**Review notes.** Recent optimization reported through a discussion is marked for direct verification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2397, "problem_number": "EP-943", "title": "Erdős Problem #943", "statement": "Let $A$ be the set of powerful numbers (if $p\\mid n$ then $p^2\\mid n$). Is it true that $ 1_A\\ast 1_A(n)=n^{o(1)} $ for every $n$?\",\n \"difficulty\": \"L1\"\n},{", "background": "\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The pointwise subpolynomial additive-convolution bound for powerful numbers remains open; a claimed n^(2/5+epsilon) upper bound is pending review.\n\n**Verified partial progress.**\n\n- A recent discussion presents r(n)<<_epsilon n^(2/5+epsilon) and says a standard check found no issues.\n\n**Full solution or refutation.**\n\nThis is far weaker than n^o(1) and has not been independently established as published literature.\n\n**What remains.**\n\nVerify the n^(2/5+epsilon) argument and approach the subpolynomial bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #943, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/943\n Evidence used: Current open status.\n- EP-943 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/943\n Evidence used: Unpublished partial-result claim.\n\n**Review notes.** Convolution is additive, as clarified by the tracker.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2398, "problem_number": "EP-944", "title": "Erdős Problem #944", "statement": "A critical vertex, edge, or set of edges, is one whose deletion lowers the chromatic number.\nLet $k\\geq 4$ and $r\\geq 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?", "background": "A graph $G$ with chromatic number $k$ in which every vertex is critical is called $k$-vertex-critical.\nThis was conjectured by Dirac in 1970 for $k\\geq 4$ and $r=1$. Dirac's conjecture was proved, for $k=5$, by Brown \\cite{Br92}. Lattanzio \\cite{La02} proved there exist such graphs for all $k$ such that $k-1$ is not prime. Independently, Jensen \\cite{Je02} gave an alternative construction for all $k\\geq 5$. The case $k=4$ and $r=1$ remains open.\nMartinsson and Steiner \\cite{MaSt25} proved this is true for every $r\\geq 1$ if $k$ is sufficiently large, depending on $r$. Skottova and Steiner \\cite{SkSt25} have improved this, proving that such graphs exist for all $k\\geq 5$ and $r\\geq 1$. The only remaining open case is $k=4$ (even the case $k=4$ and $r=1$ remains open).\nErd\\H{o}s also asked a stronger quantitative form of this question: let $f_k(n)$ denote the largest $r\\geq 1$ such that there exists a $k$-vertex-critical graph on $n$ vertices such that no set of at most $r$ edges is critical. He then asks whether $f_k(n)\\to \\infty$ as $n\\to \\infty$. Skottova and Steiner \\cite{SkSt25} have proved this for $k\\geq 5$, establishing the bounds $ n^{1/3}\\ll_k f_k(n) \\ll_k \\frac{n}{(\\log n)^C} $ for all $k\\geq 5$, where $C>0$ is an absolute constant.\nThis is Problem 91 in the graph problems collection. See also [917] and [1032].\nReferences\n\n\n[Br92] Brown, Jason I., A vertex critical graph without critical edges. Discrete Math. (1992), 99--101.\n\n[Je02] Jensen, Tommy R., Dense critical and vertex-critical graphs. Discrete Math. (2002), 63--84.\n\n[La02] Lattanzio, John J., A note on a conjecture of {D}irac. Discrete Math. (2002), 323--330.\n\n[MaSt25] Martinsson, Anders and Steiner, Raphael, Vertex-critical graphs far from edge-criticality. Combin. Probab. Comput. (2025), 151--157.\n\n[SkSt25] E. Skottova and R. Steiner, Critical edge sets in vertex-critical graphs. arXiv:2508.08703 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Only k=4 remains open; the requested graphs exist for every k>=5 and r>=1.\n\n**Verified partial progress.**\n\n- Skottova--Steiner prove the assertion for all k>=5 and r>=1.\n- They also establish unbounded quantitative critical-edge resistance for k>=5.\n\n**Full solution or refutation.**\n\nThe sole unresolved parameter is k=4.\n\n**What remains.**\n\nResolve k=4, already open for r=1.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #944, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/944\n Evidence used: Current open status and Skottova--Steiner theorem.\n\n**Review notes.** No source text was changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2399, "problem_number": "EP-945", "title": "Erdős Problem #945", "statement": "Let $F(x)$ be the maximal $k$ such that there exist $n+1,\\ldots,n+k\\leq x$ with $\\tau(n+1),\\ldots,\\tau(n+k)$ all distinct (where $\\tau(m)$ counts the divisors of $m$). Estimate $F(x)$. In particular, is it true that $ F(x) \\leq (\\log x)^{O(1)}? $ In other words, is there a constant $C>0$ such that, for all large $x$, every interval $[x,x+(\\log x)^C]$ contains two integers with the same number of divisors?", "background": "A problem of Erd\\H{o}s and Mirsky \\cite{ErMi52}, who proved that $ \\frac{(\\log x)^{1/2}}{\\log\\log x}\\ll F(x) \\ll \\exp\\left(O\\left(\\frac{(\\log x)^{1/2}}{\\log\\log x}\\right)\\right). $ Erd\\H{o}s \\cite{Er85e} claimed that the lower bound could be improved via their method 'with some more work' to $(\\log x)^{1-o(1)}$. Beker has improved the upper bound to $ F(x) \\ll \\exp\\left(O\\left((\\log x)^{1/3+o(1)}\\right)\\right). $ Cambie has observed that Cram\\'{er's conjecture} implies that $F(x) \\ll (\\log x)^2$, and furthermore if every interval in $[x,2x]$ of length $\\gg \\log x$ contains a squarefree number (see [208]) then every interval of length $\\gg (\\log x)^2$ contains two numbers with the same number of divisors, whence $ F(x) \\ll (\\log x)^2. $ See [1004] for the analogous problem with the Euler totient function.\nThis problem is discussed in problem B18 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er85e] Erd\\H{o}s, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo,\nOkayama and Kyoto, 1984) (1985), 65-87.\n\n[ErMi52] Erd\\H{o}s, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}. Proc. London Math. Soc. (3) (1952), 257--271.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The polylogarithmic-run conjecture for distinct divisor counts is open; the best unconditional upper bound is subexponential in a cube-root log.\n\n**Verified partial progress.**\n\n- Beker proves F(x)<", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2400, "problem_number": "EP-948", "title": "Erdős Problem #948", "statement": "Is there a function $f(n)$ and a $k$ such that in any $k$-colouring of the integers there exists a sequence $a_1<\\cdots$ such that $a_n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A June 2026 GPT-5.5 Pro argument, formalized in Lean by Aristotle and screened by community reviewers, disproves the existence of the requested universal f and finite number of colours.\n\n**Verified partial progress.**\n\n- Galvin's construction already ruled out the stronger monochromatic version for two colours.\n- Erdős and Galvin proved a doubly-exponential bound for a sequence whose finite interval sums use only two colours, weaker than controlling all finite subset sums.\n\n**Full solution or refutation.**\n\nFor every proposed growth function and finite colour count, the new construction gives a colouring such that every sequence meeting the infinitely-often growth constraint has finite subset sums in every colour.\n\n**What remains.**\n\nThe revised question is resolved negatively; a conventional manuscript exposing the informal and Lean proofs in one stable source would improve auditability.\n\n**Sources checked.**\n\n- Paul Erdos and Fred Galvin, Some Ramsey-type theorems, Discrete Mathematics 87 (1991), 261-269. (primary): https://doi.org/10.1016/0012-365X(91)90166-G\n Evidence used: Contains the original problem, Galvin obstruction, and the principal pre-resolution partial theorem.\n- Liam Price and Aristotle, informal and Lean proofs linked in the Erdos Problem #948 discussion (June 2026). (formal_verification): https://www.erdosproblems.com/forum/thread/948?embed=1\n Evidence used: The maintained discussion links the proof and formalization and records a reviewer screening that the Lean statement matches the paper argument.\n- teorth/erdosproblems, AI contributions to Erdos problems, entry 948, checked 2026-08-17. (source_collection): https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems\n Evidence used: Records Full solution (Lean), dated 21 June 2026.\n\n**Review notes.** The public problem page cache is older than the June 2026 solution comments and maintained YAML update.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2401, "problem_number": "EP-949", "title": "Erdős Problem #949", "statement": "Let $S\\subset \\mathbb{R}$ be a set containing no solutions to $a+b=c$. Must there be a set $A\\subseteq \\mathbb{R}\\backslash S$ of cardinality continuum such that $A+A\\subseteq \\mathbb{R}\\backslash S$?", "background": "Erd\\H{o}s suggests that if the answer is no one could consider the variant where we assume that $S$ is Sidon (i.e. all $a+b$ with $a,b\\in S$ are distinct, aside from the trivial coincidences).\nIn the comments Dillies gives a positive proof of this, found by AlphaProof: in other words, if $S\\subset \\mathbb{R}$ is a Sidon set then there exists $A\\subseteq \\mathbb{R}\\backslash S$ of cardinality continuum such that $A+A\\subseteq \\mathbb{R}\\backslash S$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general sum-free-set complement question is open; the Sidon variant is affirmative.\n\n**Verified partial progress.**\n\n- A maintained discussion records an AlphaProof-derived proof for S Sidon.\n\n**Full solution or refutation.**\n\nThe Sidon hypothesis is substantially stronger than mere sum-freeness.\n\n**What remains.**\n\nRemove the Sidon assumption or build a counterexample.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #949, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/949\n Evidence used: Current open status and Sidon variant.\n\n**Review notes.** The specialized proof is not used as a general resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2402, "problem_number": "EP-950", "title": "Erdős Problem #950", "statement": "Let $ f(n) = \\sum_{p0$ such that there are $\\gg n^c/\\log n$ many primes in $[n,n+n^c]$ implies that $\\liminf f(n)>0$.\nErd\\H{o}s writes that a 'weaker conjecture which is perhaps not quite inaccessible' is that, for every $\\epsilon>0$, if $x$ is sufficiently large there exists $y0$ then $f(n)\\ll \\log\\log\\log n$.\nThe study of $f(p)$ is even harder, and Erd\\H{o}s could not prove that $ \\sum_{p\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The reciprocal prime-gap sum questions remain open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo verified later result establishing any of the requested liminf, limsup, or uniform bound was located.\n\n**What remains.**\n\nEstablish the extremal behavior of f(n).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #950, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/950\n Evidence used: Current maintained open status.\n\n**Review notes.** Search-result collisions with an unrelated EP-950 page were excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2403, "problem_number": "EP-951", "title": "Erdős Problem #951", "statement": "Let $1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The literal all-x inequality is false by finite counterexample; the eventual version remains open.\n\n**Verified partial progress.**\n\n- The maintained history records finite counterexamples, extensible to infinite Beurling-prime sequences.\n\n**Full solution or refutation.**\n\nFailure at small x does not decide the plausible eventual reading.\n\n**What remains.**\n\nSpecify the quantifier convention and settle the eventual inequality.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #951, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/951\n Evidence used: Explicit all-x ambiguity and finite counterexample.\n\n**Review notes.** The source does not state whether the inequality is eventual; defect flagged, not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2404, "problem_number": "EP-952", "title": "Erdős Problem #952", "statement": "Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\\ldots$ such that $ \\lvert x_{n+1}-x_n\\rvert \\ll 1? $ ", "background": "The Gaussian moat problem. This is not actually a problem of Erd\\H{o}s, but has been erroneously attributed to him in the past. In \\cite{Er77c} Erd\\H{o}s recalls: 'The conjecture was told me by Motzkin at the Pasadena number theory meeting 1963 November and it was apparently raised by Basil Gordon and Motzkin. I naturally liked it very much and told it right away to many people, naturally attributing it to Motzkin, but this was later forgotten. Thus the problem is returned to its rightful owners.'\nReferences\n\n\n[Er77c] Erd\\H{o}s, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\",\n \"difficulty\": \"L2\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Gaussian moat problem remains open.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo bounded-step infinite Gaussian-prime path is known.\n\n**What remains.**\n\nConstruct a path or prove a finite moat barrier.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #952, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/952\n Evidence used: Current maintained open status and provenance correction.\n\n**Review notes.** The attribution to Erdos is historically erroneous; statement retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2405, "problem_number": "EP-953", "title": "Erdős Problem #953", "statement": "Let $A\\subset \\{ x\\in \\mathbb{R}^2 : \\lvert x\\rvert \n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The maximum measure without integer distances is open between near-square-root lower and linear upper scales.\n\n**Verified partial progress.**\n\n- The trivial upper bound is O(r).\n- An adaptation of Sárközy's construction yields lower bounds much larger than r^(1/2-epsilon) for each epsilon>0.\n\n**Full solution or refutation.**\n\nThe exponent and sharp asymptotic remain unknown.\n\n**What remains.**\n\nClose the square-root-to-linear gap.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #953, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/953\n Evidence used: Current open status and bounds.\n\n**Review notes.** Reported formalization is not substituted for the underlying literature proof.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2406, "problem_number": "EP-954", "title": "Erdős Problem #954", "statement": "Let $1=a_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The literal imported formulation has a substantive indexing defect and is refuted by a direct community observation, while the corrected a_0=0 formulation now maintained by the tracker remains open.\n\n**Verified partial progress.**\n\n- For the literal a_1=1 formulation, at selected points x=a_m the counting error has magnitude m, and the number of available pairs forces m to be at least on the square-root scale in x; this contradicts the proposed x^(1/4+o(1)) error.\n- The maintained formulation adds a_0=0, counts 0<=i<=j<=k with j>=1, and changes the defining threshold to ", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2407, "problem_number": "EP-955", "title": "Erdős Problem #955", "statement": "Let $ s(n)=\\sigma(n)-n=\\sum_{\\substack{d\\mid n\\\\ d\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The density-zero preimage conjecture for the sum-of-proper-divisors function remains open for arbitrary density-zero sets, with a general theorem for sets of counting function x^(1/2+o(1)) and several structured special cases.\n\n**Verified partial progress.**\n\n- Pollack, Pomerance, and Thompson prove that if |A cap [1,x]|<=x^(1/2+epsilon(x)) with epsilon(x)->0, then #{n<=x:s(n) in A}=o(x).\n- Verified special cases include A equal to the primes, sets with atypical prime-factor count, and sums of two squares.\n\n**Full solution or refutation.**\n\nNo theorem located covers every asymptotic-density-zero set A.\n\n**What remains.**\n\nRemove the strong thinness or structural assumptions and prove the assertion for an arbitrary set of asymptotic density zero.\n\n**Sources checked.**\n\n- P. Pollack, C. Pomerance, and L. Thompson, Divisor-sum fibers, Mathematika 64 (2018), 330-342, DOI 10.1112/S0025579317000535. (primary): https://arxiv.org/abs/1706.03120\n Evidence used: States the EGPS conjecture and proves the x^(1/2+o(1)) counting-function case.\n- L. Troupe, On the number of prime factors of values of the sum-of-proper-divisors function, J. Number Theory 150 (2015), 120-135. (primary): https://arxiv.org/abs/1405.3587\n Evidence used: Proves the normal-order special case for prime-factor counts.\n- L. Troupe, Divisor sums representable as the sum of two squares, Proc. Amer. Math. Soc. 148 (2020), 4189-4202. (primary): https://arxiv.org/abs/1902.11171\n Evidence used: Confirms the conjecture for the density-zero set of sums of two squares.\n- Thomas F. Bloom, Erdős Problem #955, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/955\n Evidence used: Current open framing and bibliography of special cases.\n\n**Review notes.** The source statement was preserved. 'Density' is read as asymptotic density, consistent with the cited EGPS formulation. The background has a stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2408, "problem_number": "EP-956", "title": "Erdős Problem #956", "statement": "If $C,D\\subseteq \\mathbb{R}^2$ then the distance between $C$ and $D$ is defined by $ \\delta(C,D)=\\inf_{\\substack{c\\in C\\\\ d\\in D}}\\| c-d\\|. $ Let $h(n)$ be the maximal number of unit distances between disjoint convex translates. That is, the maximal $m$ such that there is a compact convex set $C\\subset \\mathbb{R}^2$ and a set $X$ of size $n$ such that all $(C+x)_{x\\in X}$ are disjoint and there are $m$ pairs $x_1,x_2\\in X$ such that $ \\delta(C+x_1,C+x_2)=1. $ Determine $h(n)$ - in particular, prove that there exists a constant $c>0$ such that $h(n)>n^{1+c}$ for all large $n$.", "background": "A problem of Erd\\H{o}s and Pach \\cite{ErPa90}, who proved that $h(n) \\ll n^{4/3}$. They also consider the related function where we consider $n$ disjoint convex sets (not necessarily translates), for which they give an upper bound of $\\ll n^{7/5}$.\nIt is trivial that $h(n)\\geq f(n)$, where $f(n)$ is the maximal number of unit distances determined by $n$ points in $\\mathbb{R}^2$ (see [90]).\nReferences\n\n\n[ErPa90] Erd\\H{o}s, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A detailed April 2026 public note claims the matching lower bound h(n)=Omega(n^(4/3)), which with Erdős-Pach would give h(n)=Theta(n^(4/3)); the maintained tracker still labels the problem open and the claim is not yet archival or fully formalized.\n\n**Verified partial progress.**\n\n- Erdős and Pach prove h(n)=O(n^(4/3)).\n- The 27 April 2026 note claims a self-contained Omega(n^(4/3)) construction based on Valtr's parabolic-grid mechanism plus a conversion that enforces disjointness and Euclidean set-distance one.\n- A standard-check report found no issue with the written note, but a separate report says less than half of it is covered by the linked Lean formalization.\n\n**Full solution or refutation.**\n\nIf the public note withstands expert verification, it completely determines the order h(n)=Theta(n^(4/3)) and answers the requested superlinear lower bound; this triage does not yet promote it to established literature.\n\n**What remains.**\n\nObtain independent expert or refereed verification of the disjointness/unit-distance conversion and reconcile the incomplete-formalization report.\n\n**Sources checked.**\n\n- Unit distances between disjoint convex translates, public note dated 27 April 2026. (primary): https://www.ulam.ai/research/erdos956.pdf\n Evidence used: Claims and develops the matching lower bound and Theta(n^(4/3)) conclusion.\n- Thomas F. Bloom, Erdős Problem #956 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/956\n Evidence used: Current open label, claimed-solution notice, provenance, standard-check report, and incomplete-formalization caveat.\n- P. Valtr, Strictly convex norms allowing many unit distances and related touching questions, manuscript (2005). (primary): https://kam.mff.cuni.cz/~valtr/n.pdf\n Evidence used: Provides the core parabolic construction, but only possibly overlapping translates in the directly related touching formulation.\n\n**Review notes.** The dataset statement is well formed. The source background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2409, "problem_number": "EP-959", "title": "Erdős Problem #959", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of size $n$ and let $\\{d_1,\\ldots,d_k\\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the number of times the distance $d$ is determined, and suppose the $d_i$ are ordered such that $ f(d_1)\\geq f(d_2)\\geq \\cdots \\geq f(d_k). $ Estimate $ \\max (f(d_1)-f(d_2)), $ where the maximum is taken over all $A$ of size $n$.", "background": "More generally, one can ask about $ \\max (f(d_r)-f(d_{r+1})). $ Clemen, Dumitrescu, and Liu \\cite{CDL25}, have shown that $ \\max (f(d_1)-f(d_2))\\gg n\\log n. $ More generally, for any $1\\leq k\\leq \\log n$, there exists a set $A$ of $n$ points such that $ f(d_r)-f(d_{r+1})\\gg \\frac{n\\log n}{r}. $ They conjecture that $n\\log n$ can be improved to $n^{1+c/\\log\\log n}$ for some constant $c>0$.\nReferences\n\n\n[CDL25] F. Clemen, A. Dumitrescu, and D. Liu, On multiplicities of interpoint distances. arXiv:2505.04283 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The largest gap between the two largest distance multiplicities is known to be at least on the n log n scale, but its order remains undetermined.\n\n**Verified partial progress.**\n\n- Clemen, Dumitrescu, and Liu construct n-point sets with f(d_1)-f(d_2)=Omega(n log n).\n- For every 1<=r<=log n they obtain f(d_r)-f(d_(r+1))=Omega(n log n/r), with the top r distances prescribable.\n- They conjecture a first-gap lower bound n^(1+c/log log n) for some c>0.\n\n**Full solution or refutation.**\n\nNo matching upper bound or asymptotic order was located.\n\n**What remains.**\n\nImprove the lower bound or prove a comparably sharp upper bound for max_A(f(d_1)-f(d_2)).\n\n**Sources checked.**\n\n- F. C. Clemen, A. Dumitrescu, and D. Liu, On multiplicities of interpoint distances, Acta Math. Hungar. (2025), DOI 10.1007/s10474-025-01562-y. (primary): https://arxiv.org/abs/2505.04283\n Evidence used: Primary theorem statements for the n log n/r gaps and the stronger conjecture.\n- Thomas F. Bloom, Erdős Problem #959, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/959\n Evidence used: Current open framing and incorporation of the 2025 result.\n\n**Review notes.** The source statement was preserved. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2410, "problem_number": "EP-960", "title": "Erdős Problem #960", "statement": "Let $r,k\\geq 2$ be fixed. Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such that if there are at least $f_{r,k}(n)$ many ordinary lines (lines containing exactly two points) then there is a set $A'\\subseteq A$ of $r$ points such that all $\\binom{r}{2}$ many lines determined by $A'$ are ordinary.\nIs it true that $f_{r,k}(n)=o(n^2)$, or perhaps even $\\ll n$?", "background": "Tur\\'{a}n's theorem implies $ f_{r,k}(n) \\leq \\left(1-\\frac{1}{r-1}\\right)\\frac{n^2}{2}+1. $ See also [209].\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A 2026 preprint constructs point sets with no four collinear points and triangle-free ordinary-line graph having n^2/12-O(n) edges, refuting every subquadratic threshold claim for r at least 3 and k at least 4.\n\n**Verified partial progress.**\n\n- Turán's theorem gives the general quadratic upper bound recorded in the source.\n- Related work studied existence of ordinary triangles and almost-ordinary configurations, but did not imply the proposed subquadratic threshold.\n\n**Full solution or refutation.**\n\nCyclic subgroups on real elliptic curves produce configurations whose ordinary-line graph is bipartite, hence contains no K_r for r at least 3, while retaining at least n^2/12-O(n) ordinary lines.\n\n**What remains.**\n\nThe o(n^2) and linear conjectures are false; determining the exact leading constant or sharper threshold remains open.\n\n**Sources checked.**\n\n- Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, and Gregory Valiant, Short proofs in combinatorics, probability and number theory II, arXiv:2604.06609 (2026). (primary): https://arxiv.org/abs/2604.06609\n Evidence used: Contains the explicit elliptic-curve construction proving the quadratic lower bound and disproof.\n- Thomas F. Bloom, Erdos Problem #960, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/960\n Evidence used: Records DISPROVED and the n^2/12-O(n) construction.\n\n**Review notes.** The primary source is a recent preprint and has not yet been peer reviewed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2411, "problem_number": "EP-961", "title": "Erdős Problem #961", "statement": "Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimate $f(k)$.", "background": "In other words, how large can a consecutive set of $k$-smooth integers be? Sylvester and Schur (see \\cite{Er34}) proved $f(k)\\leq k$ and Erd\\H{o}s \\cite{Er55d} proved $ f(k)<3\\frac{k}{\\log k}. $ Jutila \\cite{Ju74} and Ramachandra, and Shorey \\cite{RaSh73} proved $ f(k) \\ll \\frac{\\log\\log\\log k}{\\log \\log k}\\frac{k}{\\log k}. $ It is likely that $f(k) \\ll (\\log k)^{O(1)}$.\nThis is essentially equivalent to [683].\nReferences\n\n\n[Er34] Erd\\H{o}s, Paul, A {T}heorem of {S}ylvester and {S}chur. J. London Math. Soc. (1934), 282--288.\n\n[Er55d] Erd\\H{o}s, P., On consecutive integers. Nieuw Arch. Wisk. (3) (1955), 124--128.\n\n[Ju74] Jutila, Matti, On numbers with a large prime factor. {II}. J. Indian Math. Soc. (N.S.) (1974), 125--130.\n\n[RaSh73] Ramachandra, K. and Shorey, T. N., On gaps between numbers with a large prime factor. Acta Arith. (1973), 99--111.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The longest-run problem for consecutive k-smooth integers remains open between the established sublinear upper bound and the conjectured polylogarithmic scale.\n\n**Verified partial progress.**\n\n- Sylvester and Schur prove f(k)<=k, and Erdős improves this to f(k)<3k/log k.\n- Jutila and independently Ramachandra-Shorey prove f(k)<<[log log log k/log log k] k/log k.\n- The maintained tracker identifies this as essentially equivalent to EP-683.\n\n**Full solution or refutation.**\n\nNo polylogarithmic bound or order estimate was located.\n\n**What remains.**\n\nBridge the gap from the current nearly k/log k upper bound to the expected (log k)^O(1) scale, or find contrary lower bounds.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #961, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/961\n Evidence used: Current open status, established bounds, equivalence lead, and primary bibliography.\n\n**Review notes.** The source statement was preserved. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2412, "problem_number": "EP-962", "title": "Erdős Problem #962", "statement": "Let $k(n)$ be the maximal $k$ such that there exists $m\\leq n$ such that each of the integers $ m+1,\\ldots,m+k $ are divisible by at least one prime $>k$. Estimate $k(n)$.", "background": "Erd\\H{o}s \\cite{Er65} wrote it is 'not hard to prove' that $ k(n)\\gg_\\epsilon \\exp((\\log n)^{1/2-\\epsilon}) $ and it 'seems likely' that $k(n)=o(n^\\epsilon)$, but had no non-trivial upper bound for $k(n)$.\nIt is not clear what he meant by a non-trivial bound for this problem, but Tao in the comments has given a simple argument proving $k(n) \\leq (1+o(1))n^{1/2}$.\nTang has proved a lower bound of $ k(n)\\geq \\exp\\left(\\left(\\frac{1}{\\sqrt{2}}-o(1)\\right)\\sqrt{\\log n\\log\\log n}\\right). $ \nReferences\n\n\n[Er65] Erd\\H{o}s, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Subexponential lower bounds and square-root-scale upper bounds are known, but they leave a large gap and do not estimate k(n) sharply.\n\n**Verified partial progress.**\n\n- Tao gives the upper bound k(n)<=(1+o(1))sqrt(n).\n- Tang gives k(n)>=exp((1/sqrt(2)-o(1))sqrt(log n log log n)).\n- A 2026 bibliographic observation identifies an Erdős 1976 lower bound of the same logarithmic order and an upper bound sqrt(n) exp(-(log n)^c), with constants/details requiring source-level comparison.\n\n**Full solution or refutation.**\n\nThe available bounds do not determine even whether k(n)=n^o(1), which Erdős expected.\n\n**What remains.**\n\nClose the gap between exp(Theta(sqrt(log n log log n))) and the near-square-root upper bound, and reconcile the precise priority and constants in the 1976 and Tang lower bounds.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #962, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/962\n Evidence used: Current open status and incorporated Tao and Tang bounds.\n- Erdős Problem #962 discussion, posts by Terence Tao, Quanyu Tang, and Thomas Bloom, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/962\n Evidence used: Arguments and provenance for the current upper/lower bounds, plus the later Erdős 1976 bibliographic correction.\n- P. Erdős, Problems and results on consecutive integers, Publ. Math. Debrecen 23 (1976), 271-282. (primary): https://publi.math.unideb.hu/paper/3385\n Evidence used: Historical source identified in the current discussion for stronger bounds than the 1965 source.\n\n**Review notes.** The grammar 'each ... are divisible' is defective but the mathematical meaning is clear. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2413, "problem_number": "EP-963", "title": "Erdős Problem #963", "statement": "Let $f(n)$ be the maximal $k$ such that in any set $A\\subset \\mathbb{R}$ of size $n$ there is a subset $B\\subseteq A$ of size $\\lvert B\\rvert\\geq k$ which is dissociated that is, the sums $\\sum_{b\\in S}b$ are distinct for all $S\\subseteq B$. Estimate $f(n)$ - in particular, is it true that $ f(n)\\geq \\lfloor \\log_2 n\\rfloor? $ ", "background": "Erd\\H{o}s noted that the greedy algorithm showed $f(n)\\geq \\lfloor \\log_3 n\\rfloor$.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact floor(log_2 n) lower bound remains open; beyond the classical floor(log_3 n) bound, a non-archival community proof claims the asymptotically stronger (1-o(1))log_2 n bound.\n\n**Verified partial progress.**\n\n- Erdős's greedy argument gives f(n)>=floor(log_3 n).\n- An October 2025 community post gives a detailed claimed proof of f(n)>=(1-o(1))log_2 n via reduction to integers, random modular dilation, and a recurrence.\n- The claimed asymptotic, even if correct, does not imply f(n)>=floor(log_2 n) for every n.\n\n**Full solution or refutation.**\n\nNo verified archival proof of the exact requested bound was located; a separate AI post asserting that {1,...,n} is extremal contains no justification and was rejected as evidence.\n\n**What remains.**\n\nExpert-check the detailed asymptotic community argument and settle the constant-level/floor loss required by the exact conjecture.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #963 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/963\n Evidence used: Current open label, classical log_3 bound, full community argument for the claimed (1-o(1))log_2 n bound, and the unsupported AI claim.\n\n**Review notes.** The missing punctuation in 'dissociated that is' is editorial, not mathematical. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2414, "problem_number": "EP-968", "title": "Erdős Problem #968", "statement": "Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_nu_{n+1}$ has positive density.\nErd\\H{o}s also asks whether $ u_nu_{n+1}>u_{n+2} $ have infinitely many solutions.\nReferences\n\n\n[ErPr61] Erd\\H{o}s, P. and Prachar, K., S\"{a}tze und {P}robleme \"{u}ber {$p\\sb{k}/k$}. Abh. Math. Sem. Univ. Hamburg (1961/62), 251--256.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Positive density of indices with p_n/np_n/n, whose threshold is asymptotic to log n.\n- Erdős and Prachar prove positive density for the reverse inequality; modern positive-density small-gap results recover that direction.\n- The desired assertion follows conditionally from RH plus a weak pair-correlation estimate, and also from a sufficiently uniform prime-tuples conjecture.\n\n**Full solution or refutation.**\n\nKnown large-gap results do not supply positive density near the average-gap threshold; a sparse set of very large gaps could still carry the required mean excess.\n\n**What remains.**\n\nProve that above-average prime gaps occur for a positive density of indices, or otherwise directly establish positive density for u_n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2415, "problem_number": "EP-969", "title": "Erdős Problem #969", "statement": "Let $Q(x)$ count the number of squarefree integers in $[1,x]$. Determine the order of magnitude in the error term in the asymptotic $ Q(x)=\\frac{6}{\\pi^2}x+E(x). $ ", "background": "It is elementary to prove $E(x)\\ll x^{1/2}$, and the prime number theorem implies $o(x^{1/2})$. The best known unconditional upper bound is of the shape $x^{1/2-o(1)}$, due to Walfisz \\cite{Wa63}. Evelyn and Linfoot \\cite{EvLi31} proved that $ E(x) \\gg x^{1/4}, $ and this is likely the true order of magnitude. The Riemann Hypothesis would follow from $E(x)\\ll x^{1/4}$.\nThe true order of magnitude is unknown even assuming the Riemann Hypothesis. Conditional on this assumption, the best known upper bound is $ E(x)\\ll x^{\\frac{11}{35}+o(1)}, $ due to Liu \\cite{Li16}.\nReferences\n\n\n[EvLi31] Evelyn, C. J. A. and Linfoot, E. H., On a problem in the additive theory of numbers. Ann. of Math. (2) (1931), 261--270.\n\n[Li16] Liu, H.-Q., On the distribution of squarefree numbers. J. Number Theory (2016), 202--222.\n\n[Wa63] Walfisz, Arnold, Weylsche {E}xponentialsummen in der neueren {Z}ahlentheorie. (1963), 231.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The squarefree-counting error remains open between an omega scale x^(1/4) and an unconditional upper bound of x^(1/2) times a zero-free-region saving; even RH does not determine its order.\n\n**Verified partial progress.**\n\n- Walfisz proves E(x)<>x^(1/4) is recorded as an omega-type statement, not an eventual positive lower bound. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2416, "problem_number": "EP-970", "title": "Erdős Problem #970", "statement": "Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive integers there exists an integer coprime to $n$. Determine the order of magnitude of $h(k)$. In particular, is it true that $ h(k) \\ll k^2? $ ", "background": "That $h(k)\\ll k^2$ is a conjecture of Jacobsthal. Iwaniec \\cite{Iw78} proved $ h(k) \\ll (k\\log k)^2. $ The best lower bound known is $ h(k) \\gg \\frac{(\\log k)(\\log\\log\\log k)}{(\\log\\log k)^2}k, $ due to Ford, Green, Konyagin, Maynard, and Tao \\cite{FGKMT18}.\nThis is a more general form of the function considered in [687].\nReferences\n\n\n[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. (2018), 65-105.\n\n[Iw78] Iwaniec, Henryk, On the problem of {J}acobsthal. Demonstratio Math. (1978), 225--231.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Jacobsthal's function remains between a nearly linear iterated-log lower bound and Iwaniec's (k log k)^2 upper bound; the conjectured O(k^2) bound is open.\n\n**Verified partial progress.**\n\n- Iwaniec proves h(k)<< (k log k)^2.\n- Ford, Green, Konyagin, Maynard, and Tao yield h(k)>>k(log k)(log log log k)/(log log k)^2.\n\n**Full solution or refutation.**\n\nThe bounds neither determine the order of h(k) nor prove Jacobsthal's h(k)<", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2417, "problem_number": "EP-971", "title": "Erdős Problem #971", "statement": "Let $p(a,d)$ be the least prime congruent to $a\\pmod{d}$. Does there exist a constant $c>0$ such that, for all large $d$, $ p(a,d) > (1+c)\\phi(d)\\log d $ for $\\gg \\phi(d)$ many values of $a$?", "background": "Erd\\H{o}s \\cite{Er49c} could prove this is true for an infinite sequence of $d$. He also proved that, for any $\\epsilon>0$, $ p(a,d)< \\epsilon \\phi(d)\\log d $ for $\\gg_\\epsilon \\phi(d)$ many values of $a$.\nReferences\n\n\n[Er49c] Erd\\H{o}s, P., On some applications of {B}run's method. Acta Univ. Szeged. Sect. Sci. Math. (1949), 57--63.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The positive-proportion least-prime statement is open for every large modulus, but known along infinitely many moduli and conditionally under prime-tuple hypotheses.\n\n**Verified partial progress.**\n\n- Erdős proves the assertion along an infinite sequence of moduli.\n- A uniform Hardy--Littlewood prime-tuple package implies a positive limiting proportion.\n\n**Full solution or refutation.**\n\nNo unconditional all-moduli result is verified.\n\n**What remains.**\n\nEstablish a positive proportion for every sufficiently large d.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #971, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/971\n Evidence used: Current open status and Erdos result.\n- EP-971 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/971\n Evidence used: Conditional Poisson-law derivation.\n\n**Review notes.** Conditional argument is not treated as unconditional progress.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2418, "problem_number": "EP-972", "title": "Erdős Problem #972", "statement": "Let $\\alpha>1$ be irrational. Are there infinitely many primes $p$ such that $\\lfloor p\\alpha\\rfloor$ is also prime?", "background": "Vinogradov \\cite{Vi48} proved that the sequence $\\{p\\alpha\\}$ is uniformly distributed for every irrational $\\alpha$, and hence there are infinitely many primes $p$ of the shape $p=\\lfloor n\\alpha\\rfloor$ for every irrational $\\alpha>1$. Indeed, this occurs if and only if $ \\frac{p}{\\alpha}\\leq n<\\frac{p+1}{\\alpha}, $ which is true if and only if $\\{p\\alpha^{-1}\\}>1-\\alpha^{-1}$, which happens infinitely often by the uniform distribution of $\\{p\\alpha^{-1}\\}$.\nReferences\n\n\n[Vi48] Vinogradov, I. M., On an estimate of trigonometric sums with prime numbers. Izv. Akad. Nauk SSSR Ser. Mat. (1948), 225--248.\",\n \"difficulty\": \"L3\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No general proof was located that floor(alpha p) is prime infinitely often for every irrational alpha>1.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe maintained current status is open.\n\n**What remains.**\n\nResolve the Beatty-prime simultaneous-primality assertion.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #972, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/972\n Evidence used: Current maintained open status.\n\n**Review notes.** No source text was altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2419, "problem_number": "EP-973", "title": "Erdős Problem #973", "statement": "Does there exist a constant $C>1$ such that, for every $n\\geq 2$, there exists a sequence $z_i\\in \\mathbb{C}$ with $z_1=1$ and $\\lvert z_i\\rvert \\geq 1$ for all $1\\leq i\\leq n$ with $ \\max_{2\\leq k\\leq n+1}\\left\\lvert \\sum_{1\\leq i\\leq n}z_i^k\\right\\rvert < C^{-n}? $ ", "background": "This is Problem 7.3 in \\cite{Ha74}, where it is attributed to Erd\\H{o}s.\nErd\\H{o}s proved (as described on p.35 of \\cite{Tu84b}) that such a sequence does exist with $\\lvert z_i\\rvert\\leq 1$. Indeed, Erd\\H{o}s' construction gives a value of $C\\approx 1.32$.\nIn \\cite{Er92f} (a different) Erd\\H{o}s refines this analysis, proving that if $ M_2=\\min_{z_j} \\max_{2\\leq k\\leq n+1} \\left\\lvert \\sum_{1\\leq j\\leq n}z_j^k\\right\\rvert, $ where the minimum is take over all $z_j\\in \\mathbb{C}$ with $\\max \\lvert z_j\\rvert=1$, then $ (1.746)^{-n} < M_2 < (1.745)^{-n}. $ Tang notes in the comments that Theorem 6.1 of \\cite{Tu84b} implies that, if $\\lvert z_i\\rvert \\geq 1$ for all $i$, then $ \\max_{2\\leq k\\leq n+1}\\left\\lvert \\sum_{1\\leq i\\leq n}z_i^k\\right\\rvert \\geq (2e)^{-(1+o(1))n}. $ See also [519].\nReferences\n\n\n[Er92f] Erd\\H{o}s, L., On some problems of {P}. {T}ur\\'an concerning power sums of\ncomplex numbers. Acta Math. Hungar. (1992), 11--24.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Tu84b] Tur\\'an, Paul, On a new method of analysis and its applications. (1984), xvi+584.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exponential small-power-sums construction remains open; Turan's power-sum inequality limits any possible exponential base.\n\n**Verified partial progress.**\n\n- A Turan inequality gives max_{2<=k<=n+1}|sum z_i^k| >= n(n/(2e(n+1)))^n, so any proposed C must be at most 2e+o(1).\n\n**Full solution or refutation.**\n\nThe lower obstruction does not supply the required construction.\n\n**What remains.**\n\nConstruct an exponential bound or improve the obstruction.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #973, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/973\n Evidence used: Maintained discussion and Turan-based bound.\n\n**Review notes.** Forum derivation requires direct verification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2420, "problem_number": "EP-975", "title": "Erdős Problem #975", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\\geq 1$ for all large $n\\in\\mathbb{N}$. Does there exist a constant $c=c(f)>0$ such that $ \\sum_{n\\leq X} \\tau(f(n))\\sim cX\\log X, $ where $\\tau$ is the divisor function?", "background": "Van der Corput \\cite{Va39} proved that $ \\sum_{n\\leq X} \\tau(f(n))\\gg_f X\\log X. $ Erd\\H{o}s \\cite{Er52b} proved using elementary methods that $ \\sum_{n\\leq X} \\tau(f(n))\\ll_f X\\log X. $ Such an asymptotic formula is known whenever $f$ is an irreducible quadratic, as proved by Hooley \\cite{Ho63}. The form of $c$ depends on $f$ in a complicated fashion (see the work of McKee \\cite{Mc95}, \\cite{Mc97}, and \\cite{Mc99} for expressions for various types of quadratic $f$). For example, $ \\sum_{n\\leq x}\\tau(n^2+1)=\\frac{3}{\\pi}x\\log x+O(x). $ Tao has a blog post on this topic.\nReferences\n\n\n[Er52b] Erd\\H{o}s, P., On the sum {$\\sum^x_{k=1} d(f(k))$}. J. London Math. Soc. (1952), 7--15.\n\n[Ho63] Hooley, Christopher, On the number of divisors of a quadratic polynomial. Acta Math. (1963), 97--114.\n\n[Mc95] McKee, James, On the average number of divisors of quadratic polynomials. Math. Proc. Cambridge Philos. Soc. (1995), 389--392.\n\n[Mc97] McKee, James, A note on the number of divisors of quadratic polynomials. (1997), 275--281.\n\n[Mc99] McKee, James, The average number of divisors of an irreducible quadratic\npolynomial. Math. Proc. Cambridge Philos. Soc. (1999), 17--22.\n\n[Va39] van der Corput, J. G., Une in\\'{e}galit\\'{e}{} relative au nombre des diviseurs. Nederl. Akad. Wetensch., Proc. (1939), 547--553.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The divisor-sum asymptotic is open for general irreducible polynomials, but established for irreducible quadratics.\n\n**Verified partial progress.**\n\n- van der Corput and Erdős give matching-order X log X bounds.\n- Hooley proves the asymptotic for irreducible quadratics.\n\n**Full solution or refutation.**\n\nHigher degree is not covered in general.\n\n**What remains.**\n\nProve the asymptotic for all irreducible nonconstant f.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #975, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/975\n Evidence used: Current open status and quadratic theorem.\n\n**Review notes.** Dataset already includes needed irreducible/nonconstant hypotheses.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2421, "problem_number": "EP-976", "title": "Erdős Problem #976", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $d\\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\\leq m\\leq n$ with $f(m)$ is divisible by a prime $\\geq F_f(n)$. Equivalently, $F_f(n)$ is the greatest prime divisor of $ \\prod_{1\\leq m\\leq n}f(m). $ Estimate $F_f(n)$. In particular, is it true that $F_f(n)\\gg n^{1+c}$ for some constant $c>0$? Or even $\\gg n^d$?", "background": "Nagell and Ricci \\cite{Na22} proved that $ F_f(n) \\gg n\\log n, $ which Erd\\H{o}s \\cite{Er52c} improved to $ F_f(n) \\gg n(\\log n)^{\\log\\log\\log n}. $ In \\cite{Er65b} he claimed a proof of $ F_f(n) \\gg n\\exp((\\log n)^c) $ for some constant $c>0$, but said he had never published the proof, which was 'fairly complicated'. This seems to have been flawed, since Erd\\H{o}s and Schinzel \\cite{ErSc90} later published a weaker bound. A proof of the stronger bound above was finally provided by Tenenbaum \\cite{Te90}.\nReferences\n\n\n[Er52c] Erd\\H{o}s, P., On the greatest prime factor of {$\\prod^x_{k=1}f(k)$}. J. London Math. Soc. (1952), 379--384.\n\n[Er65b] Erd\\H{o}s, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[ErSc90] Erd\\H{o}s, P. and Schinzel, A., On the greatest prime factor of {$\\prod^x_{k=1}f(k)$}. Acta Arith. (1990), 191--200.\n\n[Na22] No reference found.\n\n\n[Te90] Tenenbaum, G\\'{e}rald, Sur une question d'{E}rd\\H{o}s et {S}chinzel. (1990), 405--443.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Fixed-power lower bounds are open in general, though superlinear subpower bounds and special-polynomial results are known.\n\n**Verified partial progress.**\n\n- Tenenbaum proves F_f(n)>>n exp((log n)^c) for some c>0.\n- Certain special cubics and quartics have fixed-power bounds; Bateman--Horn gives n^d conditionally.\n\n**Full solution or refutation.**\n\nNo universal n^(1+c) lower bound is known.\n\n**What remains.**\n\nObtain a fixed-power bound for every irreducible polynomial.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #976, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/976\n Evidence used: Current open status and Tenenbaum bound.\n- EP-976 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/976\n Evidence used: Special-case and conditional literature leads.\n\n**Review notes.** Discussion survey is not treated as a proof of a general power bound.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2422, "problem_number": "EP-978", "title": "Erdős Problem #978", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\neq 2^l$ for any $l\\geq 1$) such that the leading coefficient of $f$ is positive.\nDoes the set of integers $n\\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density?\nAre there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free?\nIn particular, does $ n^4+2 $ represent infinitely many squarefree numbers?", "background": "Erd\\H{o}s \\cite{Er53} proved there are infinitely many $n$ for which $f(n)$ is $(k-1)$-power-free, except for possibly when $k=2^l$, when it may happen that $2^{l-1}\\mid f(n)$ for all $n$.\nHooley \\cite{Ho67} settled the first question, in fact providing a precise asymptotic for the number of such $n\\leq x$.\nHeath-Brown \\cite{He06} proved the answer to the second question is yes when $k\\geq 10$, and Browning \\cite{Br11} extended this to $k\\geq 9$ (in fact establishing an asymptotic formula for the number of such $n$).\nIn \\cite{Er65b} Erd\\H{o}s mentions the question of whether $2^n\\pm 1$ represents infinitely many $k$th power-free integers, or $n!\\pm 1$, but that these are 'intractable at present'. (See also [936].)\nReferences\n\n\n[Br11] Browning, T. D., Power-free values of polynomials. Arch. Math. (Basel) (2011), 139--150.\n\n[Er53] Erd\\H{o}s, P., Arithmetical properties of polynomials. J. London Math. Soc. (1953), 416--425.\n\n[Er65b] Erd\\H{o}s, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[He06] Heath-Brown, D. R., Counting rational points on algebraic varieties. (2006), 51--95.\n\n[Ho67] Hooley, C., On the power free values of polynomials. Mathematika (1967), 21--26.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The first positive-density power-free question is solved; later lower-power-free cases remain open in low degrees.\n\n**Verified partial progress.**\n\n- Hooley proves an asymptotic for (k-1)-power-free values.\n- Browning proves the (k-2)-power-free result for k>=9.\n\n**Full solution or refutation.**\n\nThis does not settle n^4+2 squarefreeness or degrees below nine in the second question.\n\n**What remains.**\n\nResolve the low-degree cases, especially n^4+2.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #978, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/978\n Evidence used: Current results and residual cases.\n\n**Review notes.** The source's degree restrictions are preserved exactly.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2423, "problem_number": "EP-979", "title": "Erdős Problem #979", "statement": "Let $k\\geq 2$, and let $f_k(n)$ count the number of solutions to $ n=p_1^k+\\cdots+p_k^k, $ where the $p_i$ are prime numbers. Is it true that $\\limsup f_k(n)=\\infty$?", "background": "Erd\\H{o}s \\cite{Er37b} proved this is true when $k=2$, and also when $k=3$ (but this proof appears to be unpublished).\nReferences\n\n\n[Er37b] Erd\\H{o}s, Paul, On the {S}um and {D}ifference of {S}quares of {P}rimes. J. London Math. Soc. (1937), 133--136.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Unbounded representation multiplicity is known for k=2 and reported for k=3, but open for general k.\n\n**Verified partial progress.**\n\n- Erdős proved the k=2 case.\n- The maintained source attributes k=3 to Erdős but notes the proof appears unpublished.\n\n**Full solution or refutation.**\n\nNo result is identified for arbitrary k.\n\n**What remains.**\n\nEstablish limsup f_k(n)=infinity for k>=4.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #979, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/979\n Evidence used: Current open status and k=2/3 information.\n\n**Review notes.** The k=3 attribution is not treated as a readily checkable published proof.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2424, "problem_number": "EP-983", "title": "Erdős Problem #983", "statement": "Let $n\\geq 2$ and $\\pi(n)r$ many $a\\in A$ are only divisible by primes from $\\{p_1,\\ldots,p_r\\}$.\nIs it true that $ 2\\pi(n^{1/2})-f(\\pi(n)+1,n)\\to \\infty $ as $n\\to \\infty$?\nIn general, estimate $f(k,n)$, particularly when $\\pi(n)+10$ and, for any constant $1>c>0$, $ f(cn,n)=\\log\\log n+(c_1+o(1))\\sqrt{2\\log\\log n}, $ where $ c=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{c_1}e^{-x^2/2}\\mathrm{d}x. $ \nReferences\n\n\n[Er70b] Erd\\H{o}s, P., Some applications of graph theory to number theory. Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications (Univ. North Carolina, Chapel Hill, N.C., 1970) (1970), 136-145.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The maintained status is open, while a recent AI-assisted claimed disproof of the first question has minor issues flagged by review.\n\n**Verified partial progress.**\n\n- Erdős--Straus prove f(pi(n)+1,n)=2pi(sqrt(n))+an error smaller than every fixed log-power scale.\n- They also give an asymptotic in the linear-density regime.\n\n**Full solution or refutation.**\n\nThe claimed refutation is not independently verified and is not used as a terminal classification.\n\n**What remains.**\n\nVerify the proposed construction or resolve the difference-to-infinity assertion.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #983, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/983\n Evidence used: Current open status and established asymptotics.\n- EP-983 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/983\n Evidence used: Unverified claim and review caveat.\n\n**Review notes.** Unverified AI-assisted construction excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2425, "problem_number": "EP-985", "title": "Erdős Problem #985", "statement": "Is it true that, for every prime $p$, there is a prime $q\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The prime primitive-root question is open for every prime but known for almost all primes and conditionally under GRH according to cited literature leads.\n\n**Verified partial progress.**\n\n- Nongkynrih's almost-all result gives a polylogarithmic prime primitive root for density-one primes.\n- GRH gives an all-sufficiently-large-primes polylogarithmic bound via shifted-sieve work.\n\n**Full solution or refutation.**\n\nNeither establishes the unconditional assertion for every prime.\n\n**What remains.**\n\nEliminate the exceptional primes unconditionally.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #985, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/985\n Evidence used: Current open status and classical Artin/Heath-Brown context.\n- EP-985 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/985\n Evidence used: Bibliographic leads for almost-all and GRH results.\n\n**Review notes.** The statement needs p>2; this minor endpoint issue is flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2426, "problem_number": "EP-986", "title": "Erdős Problem #986", "statement": "For any fixed $k\\geq 3$, $ R(k,n) \\gg \\frac{n^{k-1}}{(\\log n)^c} $ for some constant $c=c(k)>0$.", "background": "Spencer \\cite{Sp77} proved this for $k=3$ and Mattheus and Verstraete \\cite{MaVe23} proved this for $k=4$.\nThe best general bounds available are $ \\frac{n^{\\frac{k+1}{2}}}{(\\log n)^{\\frac{1}{k-2}-\\frac{k+1}{2}}}\\ll_k R(k,n) \\ll_k \\frac{n^{k-1}}{(\\log n)^{k-2}}. $ The lower bound was proved by Bohman and Keevash \\cite{BoKe10}. The upper bound was proved by Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS80}. Li, Rousseau, and Zang \\cite{LRZ01} have shown that $\\ll_k$ in the upper bound can be improved to $\\leq (1+o(1))$.\nThe special case $k=3$ is the topic of [165] and $k=4$ is the topic of [166].\nThis problem is #6 in Ramsey Theory in the graphs problem collection.\nSee also [920].\nReferences\n\n\n[AKS80] Ajtai, Mikl\\'{o}s and Koml\\'{o}s, J\\'{a}nos and Szemer\\'{e}di, Endre, A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.\n\n[BoKe10] Bohman, Tom and Keevash, Peter, The early evolution of the {$H$}-free process. Invent. Math. (2010), 291--336.\n\n[LRZ01] Li, Yusheng and Rousseau, Cecil C. and Zang, Wenan, Asymptotic upper bounds for {R}amsey functions. Graphs Combin. (2001), 123--128.\n\n[MaVe23] Mattheus, S. and Verstraete, J., The asymptotics of $r(4,t)$. arXiv:2306.04007 (2023).\n\n[Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Bradač proves the breakthrough lower bound R(k,n)>=n^(k-2+o(1)) for fixed k, while the target k-1 exponent is announced as forthcoming but is not proved in the current public paper.\n\n**Verified partial progress.**\n\n- Spencer solved the k=3 case and Mattheus-Verstraete solved k=4.\n- Bradač improves the general fixed-k exponent to k-2, surpassing the long-standing deletion-threshold exponent for k at least 6.\n- The maintained tracker records a claimed k-1 exponent with logarithmic loss, but the cited public preprint says that theorem will appear in an upcoming paper.\n\n**Full solution or refutation.**\n\nThe public proof establishes a major near-tight exponent, but it remains one power of n short of the literal EP-986 assertion for general fixed k.\n\n**What remains.**\n\nMake the announced R(k,n) >= n^(k-1)/(log n)^(2k-4) proof public and verify it; absent that, prove the missing exponent for every fixed k at least 5.\n\n**Sources checked.**\n\n- Domagoj Bradac, Nearly tight exponents for off-diagonal Ramsey numbers, arXiv:2605.28793 (2026). (primary): https://arxiv.org/abs/2605.28793\n Evidence used: Proves exponent k-2 and explicitly announces, but does not include, the stronger k-1 theorem.\n- Thomas F. Bloom, Erdos Problem #986 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/986\n Evidence used: Records PROVED based on the announced stronger adjustment and gives the claimed logarithmic exponent.\n- teorth/erdosproblems, AI contributions to Erdos problems, entry 986, checked 2026-08-17. (source_collection): https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems\n Evidence used: Records a full-solution claim attributed to Bradač, Claude, and an internal OpenAI model, but does not itself supply the promised proof.\n\n**Review notes.** Classification is intentionally more conservative than the tracker because the full claimed theorem is not yet publicly checkable.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2427, "problem_number": "EP-987", "title": "Erdős Problem #987", "statement": "Let $x_1,x_2,\\ldots \\in (0,1)$ be an infinite sequence and let $ A_k=\\limsup_{n\\to \\infty}\\left\\lvert \\sum_{j\\leq n} e(kx_j)\\right\\rvert, $ where $e(x)=e^{2\\pi ix}$.\nIs it true that $ \\limsup_{k\\to \\infty} A_k=\\infty? $ Is it possible for $A_k=o(k)$?", "background": "This is Problem 7.21 in \\cite{Ha74}, where it is attributed to Erd\\H{o}s.\nErd\\H{o}s \\cite{Er64b} remarks it is 'easy to see' that $ \\limsup_{k\\to \\infty}\\left(\\sup_n\\left\\lvert \\sum_{j\\leq n} e(kx_j)\\right\\rvert\\right)=\\infty. $ Erd\\H{o}s \\cite{Er65b} later found a 'very easy' proof that $A_k\\gg \\log k$ for infinitely many $k$. Clunie \\cite{Cl67} proved that $A_k\\gg k^{1/2}$ infinitely often, and that there exist sequences with $A_k\\leq k$ for all $k$. Tao has independently found a proof that $A_k\\gg k^{1/2}$ infinitely often (see the comment section).\nLiu \\cite{Li69} showed that, for any $\\epsilon>0$, $A_k\\gg k^{1-\\epsilon}$ infinitely often, under the additional assumption that there are only a finite number of distinct points. Clunie observed in the Mathscinet review of \\cite{Li69}, however, that under this assumption in fact $A_k=\\infty$ infinitely often.\nThe question of whether $A_k=o(k)$ is possible (repeated in \\cite{Er65b} and \\cite{Ha74}) seems to still be open.\nReferences\n\n\n[Cl67] Clunie, J., On a problem of {E}rd\\H{o}s. J. London Math. Soc. (1967), 133--136.\n\n[Er64b] Erd\\H{o}s, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.\n\n[Er65b] Erd\\H{o}s, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Li69] Lindstr\"{o}m, B., An inequality for $B_2$-sequences. J. Combinatorial Theory (1969), 211-212.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Both questions are affirmative. Clunie proved that every sequence has A_k at least of order sqrt(k) for infinitely many k. Alexeev, Putterman, Sawhney, Sellke, and Valiant constructed a sequence with the stronger uniform maximal bound sup_N |sum_{n> sqrt(k) infinitely often already settled the first question and rules out uniformly smaller-than-square-root behavior.\n- Clunie also constructed a sequence with A_k<=k, which was the previous endpoint described in the source record.\n- The 2026 randomized binary-scrambling construction improves the existence upper bound to O(sqrt(k log(2k))) for the supremum over all partial sums, not merely their limsup.\n\n**Full solution or refutation.**\n\nThe 2026 proof randomly scrambles binary digits of a van der Corput-type sequence. Uniform probability estimates control every dyadic block at each frequency scale, and the binary decomposition of an arbitrary initial segment into blocks of distinct sizes gives sup_N |sum_{n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2428, "problem_number": "EP-990", "title": "Erdős Problem #990", "statement": "Let $f=a_0+\\cdots+a_dx^d\\in \\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\\ldots,z_d$ with corresponding arguments $\\theta_1,\\ldots,\\theta_d\\in [0,2\\pi]$, then for all intervals $I\\subseteq [0,2\\pi]$ $ \\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\ll \\left(n\\log M\\right)^{1/2}, $ where $n$ is the number of non-zero coefficients of $f$ and $ M=\\frac{\\lvert a_0\\rvert+\\cdots +\\lvert a_d\\rvert}{(\\lvert a_0\\rvert\\lvert a_d\\rvert)^{1/2}}. $ ", "background": "Erd\\H{o}s and Tur\\'{a}n \\cite{ErTu50} proved such an upper bound with $n$ replaced by $d$.\nReferences\n\n\n[ErTu50] Erd\\H{o}s, P. and Tur\\'an, P., On the distribution of roots of polynomials. Ann. of Math. (2) (1950), 105--119.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The proposed sparse Erdős--Turán estimate is false. For every N, a 2026 construction gives a polynomial with N+2 nonzero coefficients, coefficient-height parameter M<3, and a positive real zero of multiplicity N+1. Its angular discrepancy is therefore linear in N while the conjectured bound is only O(sqrt(N)).\n\n**Verified partial progress.**\n\n- The classical Erdős--Turán theorem gives the analogous estimate with the degree rather than the number of nonzero coefficients.\n- Hayman's earlier result bounds angular discrepancy by the number of nonzero coefficients minus one, so the counterexample shows that linear sparsity dependence is the correct general scale.\n- The counterexample uses lacunary support 0,1,K,...,K^N, a Vandermonde identity to force high multiplicity, and a separate estimate keeping M bounded.\n\n**Full solution or refutation.**\n\nFor each N, choose a sufficiently large lacunarity K and coefficients on the exponents 0,1,K,...,K^N so that the resulting polynomial has a positive real root of multiplicity N+1. The construction has exactly N+2 nonzero coefficients and M(f)<3. A short arc beginning at argument zero contains all N+1 copies of this root but has bounded uniform expected root count, producing discrepancy at least N+O(1), which contradicts O(sqrt(n log M))=O(sqrt(N)).\n\n**What remains.**\n\nNothing remains of the uniform square-root-in-sparsity conjecture under the standard conventions. Finer questions can ask for sharp constants in the linear bound or improved estimates under extra restrictions on exponents or coefficients. Because the counterexample is a very recent AI-originated arXiv result and the reported Lean artifact was not independently built or dependency-audited here, expert review is still requested.\n\n**Sources checked.**\n\n- Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, and Gregory Valiant, Short proofs in combinatorics, probability and number theory II, arXiv:2604.06609 (2026). (primary): https://arxiv.org/abs/2604.06609\n Evidence used: Section 5 and Theorem 5.1 give bounded-height fewnomials with an N+1-fold positive root, directly refuting the displayed estimate.\n- Thomas F. Bloom, Erdős Problem #990, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/990\n Evidence used: The maintained record marks the sparse discrepancy bound disproved and links the counterexample.\n- Boris Alexeev, Lean proofs of solutions to Erdős problems: Erdos990, accessed 2026-08-17. (formal_verification): https://github.com/plby/lean-proofs/blob/main/ErdosProblems/Erdos990.md\n Evidence used: The public formalization index lists an EP-990 Lean artifact; this triage did not independently build it or audit all dependencies.\n\n**Review notes.** The imported display has three convention defects: (# theta_i in I) must mean a cardinality of indices; M is defined only when a_0 and a_d are nonzero; and root arguments/arcs should use a consistent half-open fundamental interval such as [0,2pi). The counterexample survives all standard corrections.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2429, "problem_number": "EP-992", "title": "Erdős Problem #992", "statement": "Let $x_1 \\lambda>1$ for all $i$.\nReferences\n\n\n[Ba81] No reference found.\n\n\n[Ca50] Cassels, J. W. S., Some metrical theorems of {D}iophantine approximation. {III}. Proc. Cambridge Philos. Soc. (1950), 219--225.\n\n[ErKo49] Erd\\H{o}s, P. and Koksma, J. F., On the uniform distribution modulo {$1$} of sequences\n{$(f(n,\\theta))$}. Nederl. Akad. Wetensch., Proc. (1949), 851--854 = Indagationes Math. 11, 299--302.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Berkes and Philipp disproved both proposed metric discrepancy bounds in 1994. They constructed an increasing integer sequence whose normalized discrepancy D_N(alpha) satisfies limsup sqrt(N/log N) D_N(alpha)=infinity for almost every alpha. In the imported unnormalized convention D(N)=N D_N(alpha), this is limsup D(N)/sqrt(N log N)=infinity.\n\n**Verified partial progress.**\n\n- Erdős--Koksma and Cassels had established a general almost-everywhere upper bound with a log exponent 5/2+o(1).\n- Baker improved that general upper bound to log exponent 3/2+o(1).\n- The Berkes--Philipp counterexample forces fluctuations on the sqrt(N log N) scale with unbounded normalized limsup, stronger than merely a positive limsup.\n\n**Full solution or refutation.**\n\nBerkes and Philipp combine a characterization of the growth of trigonometric series with Koksma's discrepancy inequality to build an increasing sequence (n_k) for which the normalized discrepancy has limsup sqrt(N/log N) D_N(alpha)=infinity almost everywhere. Multiplying by N translates their normalization to the counting-error discrepancy used in EP-992 and yields D(N)/sqrt(N log N) unbounded. This rules out both sqrt(N)(log N)^{o(1)} and sqrt(N)(log log N)^{O(1)} upper bounds for arbitrary increasing integer sequences.\n\n**What remains.**\n\nThe two universal bounds proposed in the statement are definitively false. Determining optimal general almost-everywhere upper bounds, identifying additional sequence hypotheses that restore near-square-root behavior, and quantifying sharp behavior for structured nonlacunary sequences remain separate questions.\n\n**Sources checked.**\n\n- István Berkes and Walter Philipp, The Size of Trigonometric and Walsh Series and Uniform Distribution Mod 1, Journal of the London Mathematical Society (2) 50 (1994), 454--464, DOI 10.1112/jlms/50.3.454. (primary): https://doi.org/10.1112/jlms/50.3.454\n Evidence used: The published abstract explicitly states the existence of an increasing integer sequence with the almost-everywhere divergent normalized discrepancy limsup and says this disproves conjectures of Erdős and Baker.\n- Thomas F. Bloom, Erdős Problem #992, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/992\n Evidence used: The maintained record translates the paper to the unnormalized discrepancy convention and marks the problem disproved.\n\n**Review notes.** The normalization audit is essential: the 1994 paper uses normalized discrepancy, while EP-992 uses the raw counting error. The background's [Ba81] placeholder is incomplete; the maintained record identifies R. C. Baker, Metric number theory and the large sieve, J. London Math. Soc. (2) 24 (1981), 34--40.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2430, "problem_number": "EP-995", "title": "Erdős Problem #995", "statement": "Let $n_10$, for almost all $\\alpha$ $ \\limsup_{N\\to \\infty}\\frac{1}{N(\\log\\log N)^{\\frac{1}{2}-\\epsilon}}\\sum_{1\\leq k\\leq N}f(\\{\\alpha n_k\\})=\\infty. $ Erd\\H{o}s also proved that, for every lacunary sequence and $f\\in L^2$, for every $\\epsilon>0$, for almost all $\\alpha$, $ \\sum_{1\\leq k\\leq N}\\sum_{1\\leq k\\leq N}f(\\{\\alpha n_k\\})=o( N(\\log N)^{\\frac{1}{2}+\\epsilon}). $ Erd\\H{o}s \\cite{Er64b} thought that his lower bound was closer to the truth.\nReferences\n\n\n[Er49d] Erd\\H{o}s, P., On the strong law of large numbers. Trans. Amer. Math. Soc. (1949), 51--56.\n\n[Er64b] Erd\\H{o}s, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The illustrative o(N sqrt(log log N)) bound is refuted by a 2026 arXiv construction; a near N sqrt(log N) worst-case scale is now reported.\n\n**Verified partial progress.**\n\n- The preprint constructs L^q functions and lacunary sequences with limsup average divergence, refuting the displayed illustrative bound.\n- It reports lower growth N(log N)^(1/2-epsilon) for every epsilon>0, matching Erdős's upper exponent up to o(1).\n\n**Full solution or refutation.**\n\nThe broad instruction to estimate growth remains interpretation-dependent and the maintained page remains open.\n\n**What remains.**\n\nVerify and publish the preprint result; formulate the precise optimal theorem.\n\n**Sources checked.**\n\n- H. B. Suan, lacunary ergodic sums preprint, arXiv:2604.18535 (2026). (primary): https://arxiv.org/abs/2604.18535\n Evidence used: Preprint claims the counterexample and strengthened lower bounds.\n- Thomas F. Bloom, Erdős Problem #995, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/995\n Evidence used: Current open status and historical upper/lower bounds.\n\n**Review notes.** The source page has a repeated-summation typo; source text is untouched.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2431, "problem_number": "EP-996", "title": "Erdős Problem #996", "statement": "Let $n_10$ such that, if $ \\| f-f_n\\|_2 \\ll \\frac{1}{(\\log\\log\\log n)^{C}} $ then $ \\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{k\\leq N}f(\\{\\alpha n_k\\})=\\int_0^1 f(x)\\mathrm{d}x $ for almost every $\\alpha$?", "background": "Raikov proved the conclusion always holds (for every $f\\in L^2([0,1])$, with no assumption on $\\| f-f_n\\|_2$) if $n_k=a^k$ for some integer $a\\geq 2$. Erd\\H{o}s \\cite{Er64b} also asked whether this is true for $n_k=\\lfloor a^k\\rfloor$ for some $a>1$.\nKac, Salem, and Zygmund \\cite{KSZ48} proved that the conclusion holds if $ \\| f-f_n\\|_2 \\ll \\frac{1}{(\\log n)^{c}} $ for some constant $c>1$. Erd\\H{o}s \\cite{Er49d} proved that the conclusion holds if $ \\| f-f_n\\|_2 \\ll \\frac{1}{(\\log\\log n)^{c}} $ for some constant $c>1$. Matsuyama \\cite{Ma66} improved this to $c>1/2$.\nIn \\cite{Er64b} Erd\\H{o}s asked whether the conclusion holds for all bounded functions $f$ and lacunary sequences $n_k$.\nReferences\n\n\n[Er49d] Erd\\H{o}s, P., On the strong law of large numbers. Trans. Amer. Math. Soc. (1949), 51--56.\n\n[Er64b] Erd\\H{o}s, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.\n\n[KSZ48] Kac, M. and Salem, R. and Zygmund, A., A gap theorem. Trans. Amer. Math. Soc. (1948), 235--243.\n\n[Ma66] Matsuyama, Noboru, On the strong law of large numbers. Tohoku Math. J. (2) (1966), 259--269.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A 2026 submitted preprint gives a negative answer for every triple-log power Fourier-tail modulus, so no absolute constant C in the imported statement can work.\n\n**Verified partial progress.**\n\n- The counterexample also shows that the Matsuyama positive range for (log log N)^(-c) is sharp up to the endpoint c=1/2.\n\n**Full solution or refutation.**\n\nHo constructs a mean-zero f in L^2 and a lacunary integer sequence such that ||f-S_N f||_2 is bounded by (log log log N)^(-c), while the normalized lacunary averages have almost-everywhere infinite limsup. Since the construction applies to every c>0, it negates the existential-C assertion.\n\n**What remains.**\n\nIndependent expert review and peer-reviewed publication of the recent preprint; the related endpoint and geometric-sequence variants are separate questions.\n\n**Sources checked.**\n\n- Boon Suan Ho, Counterexamples for lacunary dilates via dyadic spike blocks, arXiv:2604.18535 (2026), submitted. (primary): https://arxiv.org/abs/2604.18535\n Evidence used: The abstract and theorem state the counterexample for omega(N)=(log log log N)^(-c) and identify it as a negative answer to Erdős Problem 996.\n- Thomas F. Bloom, Erdős Problem #996 and associated solution-claim listing, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/996\n Evidence used: Original formulation and current tracker context; the main page had not yet incorporated the submitted counterexample.\n\n**Review notes.** The imported background ends with serialized JSON contamination; the statement itself is mathematically clear and was not altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2432, "problem_number": "EP-997", "title": "Erdős Problem #997", "statement": "Call $x_1,x_2,\\ldots \\in (0,1)$ well-distributed if, for every $\\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\\subseteq [0,1]$, $ \\lvert \\# \\{ n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Alexeev, Putterman, Sawhney, Sellke, and Valiant proved in 2026 that for every real alpha the sequence of fractional parts {alpha p_n} is not well-distributed. Dirichlet approximation and a theorem on bounded strings of consecutive congruent primes produce arbitrarily long translated blocks concentrated in a short interval.\n\n**Verified partial progress.**\n\n- Erdős had announced an existence result for an irrational alpha and later explicitly retracted it because he could not reconstruct the proof.\n- Champagne, Lê, Liu, and Wooley proved the existence of an irrational alpha with failure of well-distribution in 2024.\n- The Banks--Freiberg--Turnage-Butterbaugh consecutive-primes theorem supplies arbitrarily long bounded-diameter strings in any reduced residue class, the key input for the universal 2026 result.\n\n**Full solution or refutation.**\n\nFix a small interval length delta and a desired block length m. Approximate alpha by a/q using Dirichlet's theorem with accuracy tuned to the bounded-diameter constant C_m. Banks--Freiberg--Turnage-Butterbaugh provide a string of m consecutive primes, all congruent to one reduced residue a modulo q and of diameter at most q C_m. The rational parts (a/q)p are equal modulo one throughout the string, while the approximation error varies by less than delta, so all m fractional parts lie in one interval of length delta. Such blocks for arbitrarily large m have discrepancy at least (1-delta)m, contradicting well-distribution uniformly over starting positions.\n\n**What remains.**\n\nThe intended yes/no question is closed. Quantitative versions could ask how early a concentrated block occurs or how large the sliding-window discrepancy must be as a function of window length. The result is very recent and AI-originated, and the reported Lean formalization's treatment of the deep prime-cluster input was not independently audited here, so expert review remains appropriate.\n\n**Sources checked.**\n\n- Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, and Gregory Valiant, Short proofs in combinatorics and number theory, arXiv:2603.29961v2 (2026). (primary): https://arxiv.org/abs/2603.29961\n Evidence used: Section 4 proves for every real alpha that the prime fractional-part sequence is not well-distributed.\n- William D. Banks, Tristan Freiberg, and Caroline L. Turnage-Butterbaugh, Consecutive primes in tuples, Acta Arithmetica 167 (2015), 261--266, DOI 10.4064/aa167-3-4. (primary): https://arxiv.org/abs/1311.7003\n Evidence used: The paper proves the congruent consecutive-prime corollary with bounded total diameter used as the deep input to the 2026 argument.\n- Jordan Champagne, Thái Hoàng Lê, Y.-R. Liu, and Trevor D. Wooley, Well-distribution modulo one and the primes, arXiv:2406.19491 (2024). (primary): https://arxiv.org/abs/2406.19491\n Evidence used: This establishes the earlier partial result that such failure occurs for at least one irrational alpha.\n- Thomas F. Bloom, Erdős Problem #997, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/997\n Evidence used: The maintained record marks the universal assertion proved and records the historical and current references.\n\n**Review notes.** The statement's preamble requires x_n in (0,1), but fractional parts naturally lie in [0,1) and can equal zero for integral or rational alpha. The intended definition is on the circle or [0,1); under that correction the theorem covers every real alpha.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2433, "problem_number": "EP-1002", "title": "Erdős Problem #1002", "statement": "For any $0<\\alpha<1$, let $ f(\\alpha,n)=\\frac{1}{\\log n}\\sum_{1\\leq k\\leq n}(\\tfrac{1}{2}-\\{ \\alpha k\\}). $ Does $f(\\alpha,n)$ have an asymptotic distribution function?\nIn other words, is there a non-decreasing function $g$ such that $g(-\\infty)=0$, $g(\\infty)=1$,\nand $ \\lim_{n\\to \\infty}\\lvert \\{ \\alpha\\in (0,1): f(\\alpha,n)\\leq c\\}\\rvert=g(c)? $ ", "background": "Kesten \\cite{Ke60} proved that if $ f(\\alpha,\\beta,n)=\\frac{1}{\\log n}\\sum_{1\\leq k\\leq n}(\\tfrac{1}{2}-\\{\\beta+\\alpha k\\}) $ then $f(\\alpha,\\beta,n)$ has asymptotic distribution function $ g(c)=\\frac{1}{\\pi}\\int_{-\\infty}^{\\rho c}\\frac{1}{1+t^2}\\mathrm{d}t, $ where $\\rho>0$ is an explicit constant.\nReferences\n\n\n[Ke60] Kesten, Harry, Uniform distribution {${\\rm mod}\\,1$}. Ann. of Math. (2) (1960), 445--471.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The fixed-length distribution over alpha alone remains open; known Cauchy laws use an additional spatial shift or temporal averaging and do not match the statement's averaging regime.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nKesten proves a Cauchy law after also averaging a shift beta. Dolgopyat and Sarig prove an annealed temporal Cauchy law when both alpha and the summation length are sampled. Neither resolves the alpha-only limit at each deterministic n.\n\n**What remains.**\n\nProve or disprove weak convergence of the alpha-only distributions of the displayed normalized sums as n tends to infinity.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1002 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1002\n Evidence used: Maintained open status, Kesten comparison, and warning that spatial, temporal, quenched, and annealed averages differ.\n- Dmitry Dolgopyat and Omri Sarig, Quenched and annealed temporal limit theorems for circle rotations, Astérisque 415 (2020), 59-85. (primary): https://doi.org/10.24033/ast.1100\n Evidence used: Cauchy limit after sampling both alpha and the time n, an extra averaging absent from EP-1002.\n\n**Review notes.** The imported background has a serialized-tail defect but the exact statement is unambiguous.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2434, "problem_number": "EP-1003", "title": "Erdős Problem #1003", "statement": "Are there infinitely many solutions to $\\phi(n)=\\phi(n+1)$, where $\\phi$ is the Euler totient function?", "background": "Erd\\H{o}s \\cite{Er85e} says that, presumably, for every $k\\geq 1$ the equation $ \\phi(n)=\\phi(n+1)=\\cdots=\\phi(n+k) $ has infinitely many solutions.\nErd\\H{o}s, Pomerance, and S\\'{a}rk\"{o}zy \\cite{EPS87} proved that the number of $n\\leq x$ with $\\phi(n)=\\phi(n+1)$ is at most $ \\frac{x}{\\exp((\\log x)^{1/3})}. $ See [946] for the analogous question with the divisor function.\nReferences\n\n\n[EPS87] Erd\\H{o}s, Paul and Pomerance, Carl and S\\'ark\"ozy, Andr\\'as, On locally repeated values of certain arithmetic functions.\n{III}. Proc. Amer. Math. Soc. (1987), 1--7.\n\n[Er85e] Erd\\H{o}s, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo,\nOkayama and Kyoto, 1984) (1985), 65-87.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Infinitely many consecutive equal totients remain unknown, although a strong zero-density upper bound is known for the counting function.\n\n**Verified partial progress.**\n\n- Erdős, Pomerance, and Sárközy bounded the number of n<=x with phi(n)=phi(n+1) by x/exp((log x)^(1/3)).\n- The maintained discussion records a possible refinement of the exponent by re-optimizing that method, but this is a community observation rather than the existence theorem sought.\n\n**Full solution or refutation.**\n\nNo verified theorem establishes infinitely many solutions; the known estimate is only an upper bound on how frequently solutions can occur.\n\n**What remains.**\n\nProve infinitude, or prove that only finitely many consecutive equal-totient pairs exist.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1003 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1003\n Evidence used: Current open status, the quantitative upper bound, and the corrected bibliographic identification of the relevant theorem.\n- P. Erdős, C. Pomerance, and A. Sárközy, On locally repeated values of certain arithmetic functions, II, Acta Math. Hungar. 49 (1987), 251-259. (primary): https://users.renyi.hu/~p_erdos/1987-14.pdf\n Evidence used: Primary source for the equal-totient counting result and its explicit statement that infinitude was not known.\n\n**Review notes.** The imported EPS87 citation incorrectly points to part III in Proc. AMS; the relevant tracker-corrected source is part II in Acta Math. Hungar. The source field was not edited.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2435, "problem_number": "EP-1004", "title": "Erdős Problem #1004", "statement": "Let $c>0$. If $x$ is sufficiently large then does there exist $n\\leq x$ such that the values of $\\phi(n+k)$ are all distinct for $1\\leq k\\leq (\\log x)^c$, where $\\phi$ is the Euler totient function?", "background": "Erd\\H{o}s, Pomerenace, and S\\'{a}rk\"{o}zy \\cite{EPS87} proved that if $\\phi(n+k)$ are all distinct for $1\\leq k\\leq K$ then $ K \\leq \\frac{n}{\\exp(c(\\log n)^{1/3})} $ for some constant $c>0$.\nSee [945] for the analogous problem with the divisor function.\nReferences\n\n\n[EPS87] Erd\\H{o}s, Paul and Pomerance, Carl and S\\'ark\"ozy, Andr\\'as, On locally repeated values of certain arithmetic functions.\n{III}. Proc. Amer. Math. Soc. (1987), 1--7.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The arbitrary-fixed-c statement remains open; a recent community note gives a claimed almost-all result for every fixed c<2.\n\n**Verified partial progress.**\n\n- A maintained discussion note derives pairwise distinctness for almost all starting points whenever L log L=o((log x)^2), which includes L=(log x)^c for each fixed c<2.\n- Erdős, Pomerance, and Sárközy give a much broader upper restriction on the possible length of a run of distinct totients.\n\n**Full solution or refutation.**\n\nThe c<2 argument combines shifted equal-totient counts with an average bound on their structured constants. It does not cover c>=2 and is not treated here as a refereed theorem.\n\n**What remains.**\n\nVerify the community argument in a primary source and extend existence to every fixed c>0.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1004, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1004\n Evidence used: Current open status, classical EPS bound, and indication of partial claims in the discussion.\n- Erdős Problem #1004 discussion thread, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/1004\n Evidence used: Developed community argument for L log L=o((log x)^2), explicitly yielding all fixed c<2 while leaving general c open.\n\n**Review notes.** The imported background misspells Pomerance, gives the same suspect EPS87 part-III bibliography as EP-1003, and ends with serialized JSON contamination.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2436, "problem_number": "EP-1005", "title": "Erdős Problem #1005", "statement": "Let $\\frac{a_1}{b_1},\\frac{a_2}{b_2},\\ldots$ be the Farey fractions of order $n\\geq 4$. Let $f(n)$ be the largest integer such that if $1\\leq k0$ such that $f(n)=(c+o(1))n$ for all large $n$?", "background": "The function $f(n)$ was first considered by Mayer \\cite{Ma42}, who proved $f(n)\\to \\infty$ as $n\\to \\infty$. Erd\\H{o}s \\cite{Er43} proved $f(n)\\gg n$.\nvan Doorn \\cite{vD25b} has proved that $ \\left(\\frac{1}{12}-o(1)\\right)n\\leq f(n) \\leq \\frac{1}{4}n+O(1), $ and conjectures that the upper bound is optimal.\nReferences\n\n\n[Er43] Erd\\H{o}s, P., A note on {F}arey series. Quart. J. Math. Oxford Ser. (1943), 82--85.\n\n[Ma42] Mayer, A. E., A mean value theorem concerning {F}arey series. Quart. J. Math. Oxford Ser. (1942), 48--57.\n\n[vD25b] W. van Doorn, Improved bounds for the Mayer-Erd\\H{o}s phenomenon on similarly ordered Farey fractions. arXiv:2509.00121 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A July 2026 preprint proves f(n)=(1/4+o(1))n in the exact equivalent indexing convention, determining the requested constant c=1/4.\n\n**Verified partial progress.**\n\n- Van Doorn previously proved (1/12-o(1))n<=f(n)<=n/4+O(1).\n\n**Full solution or refutation.**\n\nCipollini proves the matching asymptotic lower bound for the minimum gap between badly ordered Farey fractions and explicitly translates it to Erdős Problem 1005. Together with van Doorn's upper bound this gives the full requested asymptotic.\n\n**What remains.**\n\nIndependent expert review and peer-reviewed publication of the July 2026 preprint; finer lower-order terms are beyond the imported question.\n\n**Sources checked.**\n\n- Ricky Cipollini, Optimality of Wouter van Doorn's Upper Bound for the Mayer-Erdős Farey Problem, arXiv:2607.23302 (2026), preprint. (primary): https://arxiv.org/abs/2607.23302\n Evidence used: Abstract proves f(n)=(1/4+o(1))n and explicitly says this determines c=1/4 in the equivalent convention of Erdős Problem 1005.\n- Wouter van Doorn, Improved bounds for the Mayer-Erdős phenomenon on similarly ordered Farey fractions, arXiv:2509.00121 (2025). (primary): https://arxiv.org/abs/2509.00121\n Evidence used: Earlier upper bound n/4+O(1) and lower bound (1/12-o(1))n.\n- Thomas F. Bloom, Erdős Problem #1005, tracker snapshot checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1005\n Evidence used: Original formulation and pre-Cipollini status; the cached main page predates the July 2026 preprint.\n\n**Review notes.** Solved classification rests on a very recent, unrefereed preprint whose abstract exactly matches the imported asymptotic question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2437, "problem_number": "EP-1011", "title": "Erdős Problem #1011", "statement": "Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\\geq f_r(n)$ edges and chromatic number $\\geq r$ contains a triangle. Determine $f_r(n)$.", "background": "Tur\\'{a}n's theorem implies $f_2(n)=\\lfloor n^2/4\\rfloor+1$. Erd\\H{o}s and Gallai \\cite{Er62d} proved $f_3(n)=\\lfloor \\frac{1}{4}(n-1)^2\\rfloor+2$.\nSimonovits showed in his PhD thesis (see the discussion on p. 358 of \\cite{Si74}) that $ f_r(n)=\\frac{n^2}{4}-\\frac{g(r)}{2}{n}+O(1), $ where $g(r)$ is the largest $m$ such that, for any triangle-free graph with chromatic number $\\geq r$, at least $m$ vertices of $G$ need to be removed to obtain a bipartite graph. Simonovits \\cite{Si74} notes $ \\frac{\\log r}{\\log\\log r}r^2 \\ll g(r) \\ll (\\log r)^2r^2. $ Hunter in the comments has noted that other results imply $g(r)\\asymp r^2\\log r$ - in fact $ (1/2-o(1))r^2\\log r\\leq g(r)\\leq (2+o(1))r^2\\log r. $ The lower bound follows from work of Davies and Illingworth \\cite{DaIl22} (see [1104]). The upper bound follows from work of Hefty, Horn, King, and Pfender \\cite{HHKP25} on $R(3,k)$.\nRen, Wang, Wang, and Yang \\cite{RWWY24} showed that, for $n\\geq 150$, $ f_4(n)=\\left\\lfloor\\frac{(n-3)^2}{4}\\right\\rfloor+6. $ \nReferences\n\n\n[DaIl22] Davies, Ewan and Illingworth, Freddie, The {$\\chi$}-{R}amsey problem for triangle-free graphs. SIAM J. Discrete Math. (2022), 1124--1134.\n\n[Er62d] Erd\\H{o}s, P., On a theorem of {R}ademacher-{T}ur\\'an. Illinois J. Math. (1962), 122--127.\n\n[HHKP25] Z. Hefty, P. Horn, D. King, and F. Pfender, Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (2025).\n\n[RWWY24] S. Ren, J. Wang, S. Wang, and W. Yang, Extremal triangle-free graphs with chromatic number at least four. arXiv:2404.07486 (2024).\n\n[Si74] Simonovits, M., Extermal graph problems with symmetrical extremal graphs.\n{A}dditional chromatic conditions. Discrete Math. (1974), 349--376.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Fixed-r asymptotics are reduced to a bipartization parameter and r=2,3,4 are known exactly in the relevant range, but no exact general formula is known.\n\n**Verified partial progress.**\n\n- Simonovits proved f_r(n)=n^2/4-g(r)n/2+O(1) for fixed r, where g(r) is the minimum forced bipartization size parameter.\n- Current results give g(r)=Theta(r^2 log r), with explicit asymptotic constant-factor bounds.\n- Ren, Wang, Wang, and Yang proved f_4(n)=floor((n-3)^2/4)+6 for n>=150.\n\n**Full solution or refutation.**\n\nThe literature determines the leading n^2/4 term, the linear correction in terms of g(r), and several small-r cases, but leaves exact g(r) and exact f_r(n) open for general r.\n\n**What remains.**\n\nDetermine g(r) exactly or sharply enough to give an explicit exact or asymptotically sharp formula for general r, and settle finite-n error terms.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1011, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1011\n Evidence used: Current open status, Simonovits reduction, bounds on g(r), and exact small-r results.\n- Sijie Ren, Jian Wang, Shipeng Wang, and Weihua Yang, Extremal triangle-free graphs with chromatic number at least four, arXiv:2404.07486 (2024). (primary): https://arxiv.org/abs/2404.07486\n Evidence used: For n>=150, the extremal triangle-free graph with chromatic number at least four has floor((n-3)^2/4)+5 edges, giving the stated threshold f_4.\n- Zion Hefty, Paul Horn, Dylan King, and Florian Pfender, Improving R(3,k) in just two bites, arXiv:2510.19718 (2025). (primary): https://arxiv.org/abs/2510.19718\n Evidence used: Current off-diagonal Ramsey lower bound used by the maintained tracker to bound g(r).\n\n**Review notes.** The imported background has typographical defects in the linear term and the title 'Extermal'; the intended formula is clear from the maintained source.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2438, "problem_number": "EP-1013", "title": "Erdős Problem #1013", "statement": "Let $h_3(k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$. Find an asymptotic for $h_3(k)$, and also prove $ \\lim_{k\\to \\infty}\\frac{h_3(k+1)}{h_3(k)}=1. $ ", "background": "The function $h_3(k)$ is dual to the function $f(n)$ considered in [1104], in that $h_3(k)= n$ if and only if $n$ is minimal such that $f(n)=k$.\nGraver and Yackel \\cite{GrYa68} proved $ h_3(k)\\gg \\frac{\\log k}{\\log\\log k}k^2. $ The bounds for $f(n)$ from [1104] imply $ \\left(\\frac{1}{2}-o(1)\\right)k^2\\log k\\leq h_3(k) \\leq (1+o(1))k^2\\log k. $ See also [920] for a generalisation to $K_r$-free graphs.\nReferences\n\n\n[GrYa68] Graver, Jack E. and Yackel, James, Some graph theoretic results associated with Ramsey's theorem. J. Combinatorial Theory (1968), 125--175.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The correct order h_3(k)=Theta(k^2 log k) is known, but the leading constant and the ratio limit h_3(k+1)/h_3(k)->1 remain open.\n\n**Verified partial progress.**\n\n- Current bounds are (1/2-o(1))k^2 log k<=h_3(k)<=(1+o(1))k^2 log k.\n- The lower side follows from chromatic bounds for triangle-free graphs, while the construction side is linked to lower bounds for R(3,t).\n\n**Full solution or refutation.**\n\nKnown results determine the growth order within a factor two, not an asymptotic equivalent. Those coarse constants do not imply the separately requested consecutive-ratio limit.\n\n**What remains.**\n\nIdentify the leading constant (or otherwise prove an asymptotic) and establish the consecutive-ratio limit.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1013 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1013\n Evidence used: Current open status, dual formulation, factor-two asymptotic bounds, and discussion of the chromatic-number interpretation.\n- Ewan Davies and Freddie Illingworth, The chi-Ramsey problem for triangle-free graphs, SIAM J. Discrete Math. 36 (2022), 1124-1134. (primary): https://doi.org/10.1137/21M1437573\n Evidence used: Improved upper bound for the maximum chromatic number of an n-vertex triangle-free graph, feeding the lower bound for h_3(k).\n\n**Review notes.** The exact-k formulation is standard; deleting vertices from a graph of chromatic number at least k supplies an induced subgraph of chromatic number exactly k.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2439, "problem_number": "EP-1014", "title": "Erdős Problem #1014", "statement": "Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices.\nProve, for fixed $k\\geq 3$, that $ \\lim_{l\\to \\infty}\\frac{R(k,l+1)}{R(k,l)}=1. $ ", "background": "This is open even for $k=3$.\nSee also [544] for other behaviour of $R(3,k)$, and [1030] for the diagonal version of this question.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A 2026 proof manuscript establishes lim_{l to infinity} R(k,l+1)/R(k,l)=1 for every fixed k>=2, stronger than the requested k>=3 case. It also gives R(k,l+1)/R(k,l) <= 1+l^{-c_k} for some c_k>0 and all sufficiently large l. A substantial public Lean artifact corroborates the theorem.\n\n**Verified partial progress.**\n\n- A critical graph for R(k,l+1) has minimum degree at least R(k,l+1)-R(k,l)-1.\n- Dependent random choice converts any proportionally large Ramsey-number increment into a large set whose small subsets have many common neighbors.\n- Balanced splitting k=s+t, smaller Ramsey upper bounds, and a probabilistic lower bound for R(k,l) force the relative increment to vanish, with a power-saving rate.\n\n**Full solution or refutation.**\n\nLet G have R(k,l+1)-1 vertices, no K_k, and independence number at most l. Its minimum degree is at least the consecutive Ramsey increment minus one. Apply dependent random choice with s=ceil(k/2), t=floor(k/2), and a fixed moment parameter. The resulting set U cannot contain K_s, because its common neighborhood would contain K_t, and cannot contain an independent set of size l+1; hence |U|>_k (l/log l)^{k/2} shows (R(k,l+1)-R(k,l))/(R(k,l+1)-1) tends to zero. Monotonicity then yields the claimed ratio limit.\n\n**What remains.**\n\nThe displayed fixed-k limit is closed. Optimizing the exponent c_k, obtaining effective thresholds, and understanding analogous smoothness for diagonal or jointly growing Ramsey parameters remain separate questions. The proof is extremely recent and AI-originated; although a public Lean source is available and no obvious sorry/local-axiom marker was found in the rendered file, it was not built and its full dependency report was not independently checked here.\n\n**Sources checked.**\n\n- On the Ratio of R(k,l) and R(k,l+1), OpenAI-hosted proof manuscript (2026). (primary): https://cdn.openai.com/pdf/6dc7175d-d9e7-4b8d-96b8-48fe5798cd5b/Ramsey.pdf\n Evidence used: Theorem 1 proves the fixed-k ratio limit for k>=2, and Remark 1 states the quantitative power-saving upper bound.\n- Boris Alexeev, Erdos1014.lean, public Lean 4 formalization, accessed 2026-08-17. (formal_verification): https://github.com/plby/lean-proofs/blob/main/src/latest/ErdosProblems/Erdos1014.lean\n Evidence used: The public artifact is indexed for current Mathlib/Lean releases and contains a full formal development; this triage did not run the lengthy build or inspect a printed dependency report.\n- Thomas F. Bloom, Erdős Problem #1014, accessed 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1014\n Evidence used: The maintained record marks the problem proved and Lean-formalized.\n\n**Review notes.** The source definition using every graph on at least n vertices is equivalent to the standard exactly-n definition by restricting to any n vertices. The letter l is the parameter ell; no substantive OCR repair is needed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2440, "problem_number": "EP-1016", "title": "Erdős Problem #1016", "statement": "Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\\leq k\\leq n$. Estimate $h(n)$. In particular, is it true that $ h(n) \\geq \\log_2n+\\log_*n-O(1), $ where $\\log_*n$ is the iterated logarithmic function?", "background": "Such graphs are called pancyclic. A problem of Bondy \\cite{Bo71}, who claimed a proof (without details) of $ \\log_2(n-1)-1\\leq h(n) \\leq \\log_2n+\\log_*n+O(1). $ Erd\\H{o}s \\cite{Er71} believed the upper bound is closer to the truth, but could not even prove $h(n)-\\log_2n\\to \\infty$.\nA proof of the above lower bound is provided by Griffin \\cite{Gr13}. The first published proof of the upper bound appears to be in Chapter 4.5 of George, Khodkar, and Wallis \\cite{GKW16}.\nReferences\n\n\n[Bo71] Bondy, J. A., Pancyclic graphs. {I}. J. Combinatorial Theory Ser. B (1971), 80--84.\n\n[Er71] Erd\\H{o}s, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\n\n[GKW16] George, John C. and Khodkar, Abdollah and Wallis, W. D., Pancyclic and bipancyclic graphs. (2016), xii+108.\n\n[Gr13] S. Griffin, Minimal Pancyclicity. arXiv:1312.0274 (2013).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** General logarithmic lower and log-star upper bounds are published, but the conjectured extra log_* n term in the lower bound remains open.\n\n**Verified partial progress.**\n\n- The known general lower bound is h(n)>=log_2(n-1)-1.\n- A published construction gives h(n)<=log_2 n+log_* n+O(1).\n- Griffin also determined small values computationally, but those finite cases do not settle the asymptotic question.\n\n**Full solution or refutation.**\n\nThe known lower and upper bounds differ by an iterated-logarithm term. The imported statement asks whether the upper bound's secondary term is also forced from below, which is not established.\n\n**What remains.**\n\nProve h(n)>=log_2 n+log_* n-O(1), or construct pancyclic graphs disproving that lower bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1016, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1016\n Evidence used: Current open status and distinction between the proved basic lower bound and conjectured stronger lower bound.\n- Sean Griffin, Minimal Pancyclicity, arXiv:1312.0274 (2013). (primary): https://arxiv.org/abs/1312.0274\n Evidence used: Primary preprint cited for the general lower bound and finite exact computations.\n- John C. George, Abdollah Khodkar, and W. D. Wallis, Pancyclic and Bipancyclic Graphs, SpringerBriefs in Mathematics (2016). (authoritative_secondary): https://link.springer.com/book/10.1007/978-3-319-31951-3\n Evidence used: Published source identified by the tracker for the upper-bound construction.\n\n**Review notes.** The imported phrase 'the above lower bound' refers to log_2(n-1)-1, not to the stronger conjectural lower bound in the statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2441, "problem_number": "EP-1017", "title": "Erdős Problem #1017", "statement": "Let $f(n,k)$ be such that every graph on $n$ vertices and $k$ edges can be partitioned into at most $f(n,k)$ edge-disjoint complete graphs. Estimate $f(n,k)$ for $k>n^2/4$.", "background": "The function $f(n,k)$ is sometimes called the clique partition number.\nErd\\H{o}s, Goodman, and P\\'{o}sa \\cite{EGP66} proved that $f(n,k)\\leq n^2/4$ for all $k$ (and in fact the complete graphs can be taken to be edges and triangles), which is best possible in general, as witnessed for example by a complete bipartite graph. In \\cite{Er71} Erd\\H{o} asks vaguely whether this result can be 'sharpened' for $k>n^2/4$.\nLov\\'{a}sz \\cite{Lo68} proved that every graph on $n$ vertices and $k$ edges is the union of $\\binom{n}{2}-k+t$ complete graphs, where $t$ is maximal such that $t^2-t\\leq \\binom{n}{2}-k$, but without the assumption that the complete graphs are edge disjoint. Lov\\'{a}sz's result is sharp in many cases.\nIf $k>n^2/4$ and the graph contains no $K_4$ then this is equivalent to finding the minimum number of edge disjoint triangles. This special case was also asked about by Erd\\H{o}s. A complete answer was provided by Gy\"{o}ri and Keszegh \\cite{GyKe17}, who proved that every $K_4$-free graph with $n$ vertices and $\\lfloor n^2/4\\rfloor+m$ edges contins $m$ pairwise edge disjoint triangles.\nSee also [184] for an analogous problem decomposing into edges and cycles, and [583] for decomposing into paths. The clique partition problem for chordal graphs is the subject of [81].\nReferences\n\n\n[EGP66] Erd\\H{o}s, Paul and Goodman, A. W. and P\\'{o}sa, Lajos, The representation of a graph by set intersections. Canadian J. Math. (1966), 106-112.\n\n[Er71] Erd\\H{o}s, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\n\n[GyKe17] Gy\\H{o}ri, Ervin and Keszegh, Bal\\'azs, On the number of edge-disjoint triangles in {$K_4$}-free\ngraphs. Combinatorica (2017), 1113--1124.\n\n[Lo68] Lov\\'{a}sz, L., On covering of graphs. Theory of Graphs (Proc. Colloq., Tihany, 1966) (1968), 231-236.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The universal n^2/4 clique-partition bound and the K_4-free super-Turán case are known, but no general sharp estimate for all k>n^2/4 was located.\n\n**Verified partial progress.**\n\n- Erdős, Goodman, and Pósa proved f(n,k)<=floor(n^2/4) universally, using only edges and triangles.\n- Győri and Keszegh proved that every K_4-free graph with floor(n^2/4)+m edges contains m pairwise edge-disjoint triangles, completely resolving that restricted case.\n\n**Full solution or refutation.**\n\nIn a K_4-free graph, packing m edge-disjoint triangles above the Turán threshold yields the corresponding optimal edge-and-triangle partition. Graphs containing larger cliques require a different general analysis, and the broad density-sensitive function remains open.\n\n**What remains.**\n\nDetermine the worst-case clique partition number as a function of n and k throughout the super-Turán range without a K_4-free assumption.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1017, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1017\n Evidence used: Problem formulation, universal EGP bound, and identification of the K_4-free special case.\n- Ervin Győri and Balázs Keszegh, On the number of edge-disjoint triangles in K_4-free graphs, Combinatorica 37 (2017), 1113-1124. (primary): https://arxiv.org/abs/1506.03306\n Evidence used: Proves that floor(n^2/4)+m edges force m pairwise edge-disjoint triangles in every K_4-free graph.\n\n**Review notes.** The imported statement does not explicitly define f(n,k) as the least universal bound, and the background contains the typo 'contins'; the intended extremal definition is clear and was not silently rewritten.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2442, "problem_number": "EP-1021", "title": "Erdős Problem #1021", "statement": "Is it true that, for every $k\\geq 3$, there is a constant $c_k>0$ such that $ \\mathrm{ex}(n,G_k) \\ll n^{3/2-c_k}, $ where $G_k$ is the bipartite graph between $\\{y_1,\\ldots,y_k\\}$ and $\\{z_1,\\ldots,z_{\\binom{k}{2}}\\}$, with each $z_j$ joined to a unique pair of $y_i$?", "background": "A conjecture of Erd\\H{o}s and Simonovits, who proved (in unpublished work) that in such a result one must have $c_k\\to 0$ as $k\\to \\infty$. Erd\\H{o}s \\cite{Er71} could not even prove whether $\\mathrm{ex}(n,G_k)=o(n^{3/2})$.\nWhen $k=3$ the graph $G_3$ is the $6$-cycle $C_6$, for which Erd\\H{o}s \\cite{Er64c} and Bondy and Simonovits \\cite{BoSi74} proved $\\mathrm{ex}(n,C_6)\\ll n^{7/6}$ (see [572]).\nThe graph $G_k$ is the graph $H_k$ of [926] with the vertex $x$ omitted, and can also be described as the $1$-subdivision of $K_k$.\nThis was proved by Conlon and Lee \\cite{CoLe21}, with a value of $c_k=6^{-k}$. This was improved to $c_k=\\frac{1}{4k-6}$ by Janzer \\cite{Ja19}.\nReferences\n\n\n[BoSi74] Bondy, J. A. and Simonovits, M., Cycles of even length in graphs. J. Combinatorial Theory Ser. B (1974), 97-105.\n\n[CoLe21] Conlon, David and Lee, Joonkyung, On the extremal number of subdivisions. Int. Math. Res. Not. IMRN (2021), 9122--9145.\n\n[Er64c] Erd\\H{o}s, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36.\n\n[Er71] Erd\\H{o}s, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\n\n[Ja19] Janzer, Oliver, Improved bounds for the extremal number of subdivisions. Electron. J. Combin. (2019), Paper No. 3.3, 6.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Conlon and Lee proved the requested power saving for every fixed k at least 3, and Janzer improved it to c_k=1/(4k-6).\n\n**Verified partial progress.**\n\n- Conlon and Lee first proved that c_k may be taken to be 6^(-k).\n- Janzer improved the exponent to 3/2-1/(4k-6).\n- The imported C_6 claim has exponent 7/6, but the correct classical exponent is 4/3.\n\n**Full solution or refutation.**\n\nSince G_k is the 1-subdivision of K_k, Janzer's theorem directly gives ex(n,G_k)=O_k(n^(3/2-1/(4k-6))), which supplies the positive constant demanded by the problem.\n\n**What remains.**\n\nThe existence question is closed. Sharper exponents or exact orders for general k remain separate extremal questions, and the C_6 typo in the imported background should be corrected when metadata is integrated.\n\n**Sources checked.**\n\n- David Conlon and Joonkyung Lee, On the extremal number of subdivisions, International Mathematics Research Notices 2021(12), 9122-9145. (primary): https://doi.org/10.1093/imrn/rnz088\n Evidence used: Proves the requested power saving for every fixed clique subdivision, with c_k=6^(-k).\n- Oliver Janzer, Improved bounds for the extremal number of subdivisions, Electronic Journal of Combinatorics 26(3) (2019), Paper 3.3. (primary): https://doi.org/10.37236/8262\n Evidence used: Proves the stronger bound ex(n,G_k)=O_k(n^(3/2-1/(4k-6))).\n- J. A. Bondy and M. Simonovits, Cycles of even length in graphs, Journal of Combinatorial Theory, Series B 16 (1974), 97-105. (primary): https://doi.org/10.1016/0095-8956(74)90052-5\n Evidence used: Supplies the classical even-cycle bound and confirms that the C_6 exponent is 4/3, not 7/6.\n- Thomas F. Bloom, Erdos Problem #1021, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1021\n Evidence used: Records the Conlon-Lee solution and Janzer improvement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2443, "problem_number": "EP-1022", "title": "Erdős Problem #1022", "statement": "Is there a constant $c_t$, where $c_t\\to \\infty$ as $t\\to \\infty$, such that if $\\mathcal{F}$ is a finite family of finite sets, all of size at least $t$, and for every set $X$ there are $\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Under the intended nonempty-X formulation, Wood's 2-degenerate r-uniform hypergraphs of chromatic number 3 show that c_t<2 for every t, so no sequence c_t tending to infinity exists.\n\n**Verified partial progress.**\n\n- Wood constructs an r-uniform 2-degenerate hypergraph with chromatic number 3 for every r at least 2.\n- Two-degeneracy implies that every induced hypergraph on nonempty X has at most 2|X| edges.\n- The maintained discussion formalizes the negation of the proposed existential statement with X explicitly required to be nonempty.\n\n**Full solution or refutation.**\n\nSet r=t in Wood's construction. For any c_t>2 it satisfies the strict sparsity condition on every nonempty X but is not 2-colorable, and a sequence c_t tending to infinity must exceed 2 eventually.\n\n**What remains.**\n\nThe intended problem is refuted. The literal imported wording includes X=empty, making 0<0 false and the implication vacuous; metadata integration should explicitly adopt the nonempty-X interpretation. The tracker banner also says affirmative although its body and formalization say the existential claim is false.\n\n**Sources checked.**\n\n- David R. Wood, Hypergraph Colouring and Degeneracy, arXiv:1310.2972 (2013; revised 2014). (primary): https://arxiv.org/abs/1310.2972\n Evidence used: Constructs, for every r at least 2, an r-uniform 2-degenerate hypergraph with chromatic number 3.\n- Thomas F. Bloom, Erdos Problem #1022, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1022\n Evidence used: The mathematical body states that the question is false and applies Wood's construction to obtain c_t<2.\n- Erdos Problems contributors, EP-1022 discussion and linked Lean formalization, checked 2026-08-17. (formal_verification): https://www.erdosproblems.com/forum/thread/1022\n Evidence used: Displays a formulation restricted to nonempty X and explains that its negation was formalized; it also documents the status-label ambiguity.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2444, "problem_number": "EP-1029", "title": "Erdős Problem #1029", "statement": "If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then $ \\frac{R(k)}{k2^{k/2}}\\to \\infty. $ ", "background": "In \\cite{Er93} Erd\\H{o}s offers \\$100 for a proof of this and \\$1000 for a disproof, but says 'this last offer is to some extent phoney: I am sure that [this] is true (but I have been wrong before).'\nErd\\H{o}s and Szekeres \\cite{ErSz35} proved $ k2^{k/2} \\ll R(k) \\leq \\binom{2k-1}{k-1}. $ One of the first applications of the probabilistic method pioneered by Erd\\H{o}s gives $ R(k) \\geq (1+o(1))\\frac{1}{\\sqrt{2}e}k2^{k/2}, $ which Spencer \\cite{Sp75} improved by a factor of $2$ to $ R(k) \\geq (1+o(1))\\frac{\\sqrt{2}}{e}k2^{k/2}. $ See also [77] for a more general problem concerning $\\lim R(k)^{1/k}$, and discussion of upper bounds for $R(k)$.\nReferences\n\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[ErSz35] Erd\\H{o}s, P. and Szekeres, G., A combinatorial problem in geometry. Compos. Math. (1935), 463-470.\n\n[Sp75] Spencer, Joel, Ramsey's theorem---a new lower bound. J. Combinatorial Theory Ser. A (1975), 108--115.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjectured divergence remains open; the best recorded lower bound has only a fixed positive constant multiplying k 2^(k/2).\n\n**Verified partial progress.**\n\n- Spencer proved R(k)>=(sqrt(2)/e+o(1))k 2^(k/2), improving the elementary random-coloring constant by a factor two.\n- Recent work gives the first exponential improvement to the classical upper bound, but upper bounds do not address the requested lower-bound divergence.\n\n**Full solution or refutation.**\n\nKnown lower bounds establish that the normalized ratio is bounded below by a positive constant, not that it tends to infinity. No later lower-bound theorem matching the full assertion was located.\n\n**What remains.**\n\nObtain any factor tending to infinity over Spencer's scale, or disprove the conjecture by an asymptotically matching upper bound.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1029, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1029\n Evidence used: Current open status, prize, and best recorded lower-bound benchmark.\n- Joel Spencer, Ramsey's theorem--a new lower bound, J. Combin. Theory Ser. A 18 (1975), 108-115. (primary): https://doi.org/10.1016/0097-3165(75)90071-0\n Evidence used: Primary source for the factor-two improvement in the diagonal Ramsey lower bound.\n- Marcelo Campos, Simon Griffiths, Robert Morris, and Julian Sahasrabudhe, An exponential improvement for diagonal Ramsey, Ann. of Math. 203 (2026), 869-932. (primary): https://annals.math.princeton.edu/2026/203-3/p04\n Evidence used: Recent breakthrough concerns the upper bound, so it must not be mistaken for progress proving the normalized lower-bound divergence.\n\n**Review notes.** The imported statement is phrased as an assertion rather than a question, but its intended conjectural status is clear.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2445, "problem_number": "EP-1030", "title": "Erdős Problem #1030", "statement": "If $R(k,l)$ is the Ramsey number then prove the existence of some $c>0$ such that $ \\lim_k \\frac{R(k+1,k)}{R(k,k)}> 1+c. $ ", "background": "A problem of Erd\\H{o}s and S\\'{o}s, who could not even prove whether $R(k+1,k)-R(k,k)>k^c$ for any $c>1$.\nIt is trivial that $R(k+1,k)-R(k,k)\\geq k-2$. Burr, Erd\\H{o}s, Faudree, and Schelp \\cite{BEFS89} proved $ R(k+1,k)-R(k,k)\\geq 2k-5. $ See also [544] for a similar question concerning $R(3,k)$, and [1014] for the general off-diagonal case.\nReferences\n\n\n[BEFS89] Burr, S. A. and Erd\\H{o}s, P. and Faudree, R. J. and Schelp, R.\nH., On the difference between consecutive {R}amsey numbers. Utilitas Math. (1989), 115--118.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The positive proportional gap between R(k+1,k) and R(k,k) remains open; even a superlinear polynomial additive gap is not known.\n\n**Verified partial progress.**\n\n- The trivial additive bound R(k+1,k)-R(k,k)>=k-2 was improved by Burr, Erdős, Faudree, and Schelp to 2k-5.\n- Known fixed-parameter off-diagonal ratio results do not apply when both Ramsey parameters grow together.\n\n**Full solution or refutation.**\n\nNo source located proves a limit ratio bounded above 1 by an absolute positive constant.\n\n**What remains.**\n\nProve or refute a positive proportional near-diagonal gap; even establishing R(k+1,k)-R(k,k)>k^c for some c>1 would be new progress.\n\n**Sources checked.**\n\n- Thomas F. Bloom, history for Erdős Problem #1030, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/1030\n Evidence used: Records the current open formulation and the best cited additive lower bound.\n\n**Review notes.** The imported notation 'lim_k' omits an explicit direction; the maintained tracker writes k tending to infinity. The background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2446, "problem_number": "EP-1032", "title": "Erdős Problem #1032", "statement": "We say that a graph is $4$-chromatic critical if it has chromatic number $4$, and removing any edge decreases the chromatic number to $3$.\nIs there, for arbitrarily large $n$, a $4$-chromatic critical graph on $n$ vertices with minimum degree $\\gg n$?", "background": "In \\cite{Er93} Erd\\H{o}s said he asked this 'more than 20 years ago'.\nDirac gave an example of a $6$-chromatic critical graph with minimum degree $>n/2$. This problem is also open for $5$-chromatic critical graphs.\nSimonovits \\cite{Si72} and Toft \\cite{To72} independently constructed $4$-chromatic critical graphs with minimum degree $\\gg n^{1/3}$. Toft conjectured that a $4$-chromatic critical graph on $n$ vertices has at least $(\\frac{5}{3}+o(1))n$ vertices, and has examples to show this would be the best possible.\nSee also [917] and [944].\nReferences\n\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[Si72] Simonovits, M., On colour-critical graphs. Studia Sci. Math. Hungar. (1972), 67--81.\n\n[To72] Toft, B., Two theorems on critical {$4$}-chromatic graphs. Studia Sci. Math. Hungar. (1972), 83--89.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No arbitrarily large 4-chromatic edge-critical family with linear minimum degree is known; the best established constructions have minimum degree of order at least n^(1/3).\n\n**Verified partial progress.**\n\n- Simonovits and Toft independently constructed 4-chromatic critical graphs with minimum degree >> n^(1/3).\n- The analogous existence problem remains open even for chromatic number 5, while Dirac gave a 6-chromatic construction with minimum degree greater than n/2.\n- The tracker discussion records a screened 2026 upper bound delta(G)<=(3/10+o(1))n, improving a prior 0.328n restriction, but this neither constructs nor excludes a positive-linear family.\n\n**Full solution or refutation.**\n\nThe exponent gap between n^(1/3) constructions and the requested linear minimum degree remains.\n\n**What remains.**\n\nConstruct 4-chromatic edge-critical graphs with minimum degree at least cn for some c>0, or prove every such family has minimum degree o(n).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1032 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/1032\n Evidence used: Retains open status, records the classical n^(1/3) constructions, and documents the recent upper-bound claim and screening.\n\n**Review notes.** The recent 3/10 upper bound constrains the possible constant but does not answer the existence question. The imported background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2447, "problem_number": "EP-1033", "title": "Erdős Problem #1033", "statement": "Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to at least $h(n)$. Estimate $h(n)$. In particular, is it true that $ h(n)\\geq (2(\\sqrt{3}-1)-o(1))n? $ ", "background": "Erd\\H{o}s and Laskar \\cite{ErLa85} proved $ 2(\\sqrt{3}-1)n \\geq h(n) \\geq (1+c)n $ for some $c>0$. The lower bound was improved to $\\frac{21}{16}n$ by Fan \\cite{Fa88}.\nReferences\n\n\n[ErLa85] Erd\\H{o}s, Paul and Laskar, Renu, A note on the size of a chordal subgraph. Congr. Numer. (1985), 81--86.\n\n[Fa88] Fan, Genghua, Degree sum for a triangle in a graph. J. Graph Theory (1988), 249--263.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Rigorous linear upper and lower bounds are known, but a June 2026 unverified forum claim to refute the proposed sharp lower constant makes the newest status worthy of expert review.\n\n**Verified partial progress.**\n\n- Fan proved h(n)>=21n/16.\n- The Erdős-Laskar construction gives h(n)<=2(sqrt(3)-1)n+O(1).\n- A 2026 forum post claims an explicit counterexample to the proposed matching lower bound, but the tracker had not verified or incorporated it.\n\n**Full solution or refutation.**\n\nThe established leading coefficient lies between 21/16 and 2(sqrt(3)-1); the exact coefficient is not verified.\n\n**What remains.**\n\nClose the constant gap and independently check the recent claimed counterexample before treating the highlighted conjecture as refuted.\n\n**Sources checked.**\n\n- Genghua Fan, Degree sum for a triangle in a graph, Journal of Graph Theory 12 (1988), 249-263. (primary): https://doi.org/10.1002/jgt.3190120216\n Evidence used: Proves the 21n/16 lower bound and explains the upper construction.\n- Paul Erdős and Renu Laskar, A note on the size of a chordal subgraph, Congressus Numerantium 48 (1985), 81-86. (primary): https://combinatorica.hu/~p_erdos/1985-03.pdf\n Evidence used: Provides the original graph-theoretic context for the upper construction and earlier lower result.\n- Thomas F. Bloom, Erdős Problem #1033 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1033\n Evidence used: Records the established bracket and still-open status; the forum separately contains the recent unincorporated refutation claim.\n\n**Review notes.** The recent AI-assisted counterexample claim is not an archival or tracker-verified result and is not classified as a refutation. The background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2448, "problem_number": "EP-1035", "title": "Erdős Problem #1035", "statement": "Is there a constant $c>0$ such that every graph on $2^n$ vertices with minimum degree $>(1-c)2^n$ contains the $n$-dimensional hypercube $Q_n$?", "background": "Erd\\H{o}s \\cite{Er93} says 'if the conjecture is false, two related problems could be asked':\n{UL}\n{LI}Determine or estimate the smallest $m>2^n$ such that every graph on $m$ vertices with minimum degree $>(1-c)2^n$ contains a $Q_n$, and {/LI}\n{LI}For which $u_n$ is it true that every graph on $2^n$ vertices with minimum degree $>2^n-u_n$ contains a $Q_n$.{/LI}\n{/UL}\nSee also [576] for the extremal number of edges that guarantee a $Q_n$.\nReferences\n\n\n[Er93] Erd\\H{o}s, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The exact spanning-order minimum-degree condition forcing Q_n remains open, with no verified direct partial theorem recorded by the tracker.\n\n**Verified partial progress.**\n\n- Forum comments discuss dependent-random-choice and blow-up approaches for relaxed host orders larger than 2^n.\n- Known Ramsey and extremal estimates for the hypercube do not directly imply the stated minimum-degree result on exactly 2^n vertices.\n\n**Full solution or refutation.**\n\nNo constant c>0 satisfying the exact assertion was located.\n\n**What remains.**\n\nProve the conjectured constant-density minimum-degree threshold, disprove it, or first establish quantitative forcing results when the host has slightly more than 2^n vertices.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1035 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1035\n Evidence used: Retains open status, states the two relaxed variants, and contains only exploratory comments rather than a verified solution.\n\n**Review notes.** Speculative forum routes were not promoted to theorems. The imported background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2449, "problem_number": "EP-1038", "title": "Erdős Problem #1038", "statement": "Determine the infimum and supremum of $ \\lvert \\{ x\\in \\mathbb{R} : \\lvert f(x)\\rvert < 1\\}\\rvert $ as $f\\in \\mathbb{R}[x]$ ranges over all non-constant monic polynomials, all of whose roots are real and in the interval $[-1,1]$.", "background": "A problem of Erd\\H{o}s, Herzog, and Piranian \\cite{EHP58}, who proved that the measure of the set in question is always at most $2\\sqrt{2}$ under the assumption that all the roots are in $\\{-1,1\\}$, and conjecture this is the best possible upper bound.\nThey also note that the infimum of the set in question is less than $2$, as witnessed by $f(x)=(x+1)(x-1)^m$ for $m\\geq 3$. They further note that if the roots are restricted to $[-2,2]$ then the infimum is zero, as witnessed by a small perturbation of the Chebyshev polynomials.\nThey further conjectured that, if the roots are restricted to $[-2,2]$, then $ \\lvert \\{ x\\in \\mathbb{R} : \\lvert f(x)\\rvert < 1\\}\\rvert\\geq n^{-c} $ for an absolute constant $c>0$. This was proved by Pommerenke \\cite{Po61}, who in fact showed that this set must contain an interval of width $\\gg n^{-4}$.\nThe current best known bounds (see the discussion in the comments) are $ 1.519\\approx 2^{4/3}-1\\leq \\inf \\leq 1.835\\cdots $ and $ \\sup = 2\\sqrt{2}\\approx 2.828. $ \nReferences\n\n\n[EHP58] Erd\\H{o}s, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.\n\n[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The supremum is exactly 2sqrt(2), while the maintained rigorous bracket for the infimum is 2^(4/3)-1 to approximately 1.835.\n\n**Verified partial progress.**\n\n- The supremum has been determined exactly as 2sqrt(2).\n- The maintained page gives 2^(4/3)-1<=infimum<=1.835... .\n- Recent forum work claims a computer-assisted lower bound 1.814605 with certificates, but it has not been incorporated into the maintained result or an archival paper.\n\n**Full solution or refutation.**\n\nOne of the two requested extrema is known exactly; the infimum is constrained to a short but nonzero interval.\n\n**What remains.**\n\nDetermine the exact infimum and independently audit the recent certificate-based lower bounds.\n\n**Sources checked.**\n\n- Paul Erdős, Fritz Herzog, and George Piranian, Metric properties of polynomials, Journal d'Analyse Mathématique 6 (1958), 125-148. (primary): https://www.maths.tcd.ie/EMIS/classics/Erdos/cit/08825302.htm\n Evidence used: Introduces the extremal metric problem and its original bounds and conjectures.\n- Thomas F. Bloom, Erdős Problem #1038 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1038\n Evidence used: Records the exact supremum and maintained infimum bracket; the discussion hosts newer certificate claims.\n\n**Review notes.** No certificate computation was run in this triage. The newer 1.814605 claim is reported but not substituted for the maintained bound. The background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2450, "problem_number": "EP-1039", "title": "Erdős Problem #1039", "statement": "Let $f(z)=\\prod_{i=1}^n(z-z_i)\\in \\mathbb{C}[z]$ with $\\lvert z_i\\rvert \\leq 1$ for all $i$. Let $\\rho(f)$ be the radius of the largest disc which is contained in $\\{z: \\lvert f(z)\\rvert< 1\\}$.\nDetermine the behaviour of $\\rho(f)$. In particular, is it always true that $\\rho(f)\\gg 1/n$?", "background": "A problem of Erd\\H{o}s, Herzog, and Piranian, who note that $f(z)=z^n-1$ has $\\rho(f) \\leq \\frac{\\pi/2}{n}$.\nPommerenke \\cite{Po61} proved that $ \\rho(f) \\geq \\frac{1}{2en^2}. $ Krishnapur, Lundberg, and Ramachandran \\cite{KLR25} proved $ \\rho(f) \\gg \\frac{1}{n\\sqrt{\\log n}}. $ \nReferences\n\n\n[KLR25] M. Krishnapur, E. Lundberg, and K. Ramachandran, On the area of polynomial lemniscates. arXiv:2503.18270 (2025).\n\n[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** An archival 2025 paper proves rho(f)>>1/(n sqrt(log n)); a May 2026 human-vouched and partially formalized forum argument claims rho(f)>=(log 2)/n, but the main tracker still lists the problem as open.\n\n**Verified partial progress.**\n\n- Pommerenke proved rho(f)>=1/(2en^2).\n- Krishnapur, Lundberg, and Ramachandran improved this to rho(f)>>1/(n sqrt(log n)).\n- A short product argument posted in May 2026 claims rho(f)>=(log 2)/n; one human contributor vouched for it and a Lean development of the main inequality was reported correct conditional on standard auxiliary inputs.\n\n**Full solution or refutation.**\n\nIf the recent product inequality is independently accepted, it combines with z^n-1 to give the sharp minimum order Theta(1/n); absent incorporation or archival review, the conservative status is uncertain.\n\n**What remains.**\n\nObtain independent expert or archival verification of the product inequality and clarify whether the broader phrase 'determine the behaviour' asks for an exact asymptotic constant beyond order of magnitude.\n\n**Sources checked.**\n\n- Manjunath Krishnapur, Erik Lundberg, and Koushik Ramachandran, On the area of polynomial lemniscates, arXiv:2503.18270 (2025). (primary): https://arxiv.org/abs/2503.18270\n Evidence used: Proves the unconditional 1/(n sqrt(log n)) inradius lower bound and identifies the conjecture as open up to a logarithmic factor.\n- Thomas F. Bloom, Erdős Problem #1039 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/1039\n Evidence used: Documents the claimed log(2)/n proof, human check, formalization effort, earlier gaps, and the fact that the claim was not yet incorporated into the main status.\n\n**Review notes.** The recent solution claim is AI-assisted and forum-hosted; despite meaningful checking, it is conservatively not labeled solved. The background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2451, "problem_number": "EP-1040", "title": "Erdős Problem #1040", "statement": "Let $F\\subseteq \\mathbb{C}$ be a closed infinite set, and let $\\mu(F)$ be the infimum of $ \\lvert \\{ z: \\lvert f(z)\\rvert < 1\\}\\rvert, $ as $f$ ranges over all polynomials of the shape $\\prod (z-z_i)$ with $z_i\\in F$.\nIs $\\mu(F)$ determined by the transfinite diameter of $F$? In particular, is $\\mu(F)=0$ whenever the transfinite diameter of $F$ is $\\geq 1$?", "background": "A problem of Erd\\H{o}s, Herzog, and Piranian \\cite{EHP58}, who show that the answer is yes if $F$ is a line segment or disc, and that if the transfinite diameter is $<1$ then $\\{ z: \\lvert f(z)\\rvert < 1\\}$ always contains a disc of radius $\\gg_F 1$.\nErd\\H{o}s and Netanyahu \\cite{ErNe73} proved that if $F$ is also bounded and connected, with transfinite diameter $0\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Capacity alone does not determine mu(F), and mu(K)=0 is known for arbitrary compact K of capacity greater than 1; the unrestricted capacity-equals-1 boundary case remains open.\n\n**Verified partial progress.**\n\n- Aletheia produced closed countably infinite capacity-zero sets with different mu values, refuting the first question.\n- Ghosh and Ramachandran extended the nonuniqueness construction to every prescribed capacity t in (0,1).\n- Ghosh and Ramachandran prove mu(K)=0 for every compact K with capacity strictly greater than 1.\n- Krishnapur, Lundberg, and Ramachandran establish unit-capacity results under additional smoothness assumptions.\n\n**Full solution or refutation.**\n\nThe first question is refuted and most of the second question above the threshold is solved, but general capacity exactly 1 is not.\n\n**What remains.**\n\nDecide whether mu(F)=0 for every closed infinite F of logarithmic capacity exactly 1, without regularity assumptions.\n\n**Sources checked.**\n\n- Thomas Feng et al., Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems, arXiv:2601.22401 (2026). (primary): https://arxiv.org/abs/2601.22401\n Evidence used: Contains the AI-assisted capacity-zero counterexample showing mu is not determined by capacity.\n- Subhajit Ghosh and Koushik Ramachandran, A note on the Erdős minimal area problem, arXiv:2604.03036 (2026). (primary): https://arxiv.org/abs/2604.03036\n Evidence used: Extends nonuniqueness to capacities in (0,1) and proves mu(K)=0 for compact K of capacity strictly greater than 1.\n- Manjunath Krishnapur, Erik Lundberg, and Koushik Ramachandran, On the area of polynomial lemniscates, arXiv:2503.18270 (2025). (primary): https://arxiv.org/abs/2503.18270\n Evidence used: Provides unit-capacity results under smoothness assumptions and related minimal-area estimates.\n- Thomas F. Bloom, Erdős Problem #1040, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1040\n Evidence used: Incorporates the first counterexample while retaining open status for the bundled problem.\n\n**Review notes.** Recent AI-assisted and preprint results are carefully separated from the unresolved capacity-one boundary case. The background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2452, "problem_number": "EP-1049", "title": "Erdős Problem #1049", "statement": "Let $t>1$ be a rational number. Is $ \\sum_{n=1}^\\infty\\frac{1}{t^n-1}=\\sum_{n=1}^\\infty \\frac{\\tau(n)}{t^n} $ irrational, where $\\tau(n)$ counts the divisors of $n$?", "background": "A conjecture of Chowla. Erd\\H{o}s \\cite{Er48} proved that this is true if $t\\geq 2$ is an integer.\nReferences\n\n\n[Er48] Erd\\H{o}s, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) (1948), 63-66.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Erdős proved irrationality for every integer t>=2, while Chowla's extension to all rational t>1 remains open.\n\n**Verified partial progress.**\n\n- Erdős proved the Lambert series is irrational at reciprocal integer arguments.\n- Later Lambert-series irrationality results found in the search concern different coefficient sequences or signed series and do not establish the exact rational-parameter extension.\n\n**Full solution or refutation.**\n\nThe full integer subfamily is solved, but rational noninteger t remains unresolved.\n\n**What remains.**\n\nExtend the Chowla-Erdős irrationality method from integer t to every rational t>1.\n\n**Sources checked.**\n\n- Paul Erdős, On arithmetical properties of Lambert series, Journal of the Indian Mathematical Society (N.S.) 12 (1948), 63-66. (primary): https://combinatorica.hu/~p_erdos/1948-04.pdf\n Evidence used: Proves irrationality for integer t>=2 and states the surrounding Chowla question.\n- Thomas F. Bloom, Erdős Problem #1049, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1049\n Evidence used: Retains open status for rational t and records the integer theorem.\n\n**Review notes.** A 2026 note on a different signed Lambert series was checked and not conflated with this divisor-generating series. The background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2453, "problem_number": "EP-1051", "title": "Erdős Problem #1051", "statement": "Is it true that if $a_11 $ then $ \\sum_{n=1}^\\infty \\frac{1}{a_na_{n+1}} $ is irrational?", "background": "In \\cite{Er88c} Erd\\H{o}s notes this is true if $a_n\\to \\infty$ 'rapidly'.\nReferences\n\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Barreto, Kang, Kim, Kovac, and Zhang prove irrationality under a weaker golden-ratio-scale growth hypothesis, which directly implies the EP-1051 assertion.\n\n**Verified partial progress.**\n\n- Aletheia first produced a solution to the original problem, documented in the 2026 Gemini case study.\n- The dedicated paper proves irrationality when limsup a_n^(1/phi^n) is infinite together with a mild polynomial lower bound on a_n a_(n+1).\n- The same paper gives rational counterexamples at every finite positive golden-ratio-scale limit, showing near-sharpness.\n\n**Full solution or refutation.**\n\nThe EP-1051 hypothesis gives a_n at least C^(2^n) eventually. This automatically implies both the polynomial product bound and divergence at the weaker phi^n scale required by the published theorem, so the series is irrational.\n\n**What remains.**\n\nThe original question is closed. The cited work leaves only refinements and extensions beyond its essentially sharp golden-ratio threshold.\n\n**Sources checked.**\n\n- Kevin Barreto, Jiwon Kang, Sang-hyun Kim, Vjekoslav Kovac, and Shengtong Zhang, Irrationality of rapidly converging series: a problem of Erdos and Graham, arXiv:2601.21442 v3 (2026), to appear in the Bulletin of the London Mathematical Society. (primary): https://arxiv.org/abs/2601.21442\n Evidence used: Proves a stronger sufficient criterion at the golden-ratio scale and gives near-sharp rational counterexamples.\n- Tony Feng et al., Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdos Problems, arXiv:2601.22401 (2026). (primary): https://arxiv.org/abs/2601.22401\n Evidence used: Documents the Aletheia solution of the original Erdős problem and its expert evaluation.\n- Thomas F. Bloom, Erdos Problem #1051, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1051\n Evidence used: Records PROVED (LEAN) and links the modern resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2454, "problem_number": "EP-1052", "title": "Erdős Problem #1052", "statement": "A unitary divisor of $n$ is $d\\mid n$ such that $(d,n/d)=1$. A number $n\\geq 1$ is a unitary perfect number if it is the sum of its unitary divisors (aside from $n$ itself).\nAre there only finite many unitary perfect numbers?", "background": "Guy \\cite{Gu04} reports that Carlitz, Erd\\H{o}s, and Subbarao offer \\$10 for settling this question, and that Subbarao offers 10 cents for each new example.\nThere are no odd unitary perfect numbers. There are five known unitary perfect numbers (A002827 in the OEIS): $ 6, 60, 90, 87360, 146361946186458562560000. $ This is problem B3 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No odd unitary perfect numbers exist and exactly five examples are known, but finiteness of the even examples remains open.\n\n**Verified partial progress.**\n\n- A parity argument rules out every odd unitary perfect number.\n- The five known examples are 6, 60, 90, 87360, and 146361946186458562560000.\n- Wall verified that the last displayed integer is the fifth unitary perfect number after the first four.\n\n**Full solution or refutation.**\n\nThe odd case is completely excluded, but no theorem bounds the number of even unitary perfect numbers.\n\n**What remains.**\n\nProve only finitely many even unitary perfect numbers exist or construct infinitely many.\n\n**Sources checked.**\n\n- C. R. Wall, The fifth unitary perfect number, Canadian Mathematical Bulletin 18 (1975), 115-122. (primary): https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/fifth-unitary-perfect-number/B1919CB85AE1D97A7BAD3842B6E2AFB4\n Evidence used: Establishes and verifies the fifth known example.\n- Thomas F. Bloom, Erdős Problem #1052, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1052\n Evidence used: Retains open status and records the parity theorem and five known examples.\n\n**Review notes.** The source grammar 'only finite many' is preserved in input.json and not silently repaired. The background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2455, "problem_number": "EP-1053", "title": "Erdős Problem #1053", "statement": "Call a number $k$-perfect if $\\sigma(n)=kn$, where $\\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\\log\\log n)$?", "background": "A question of Erd\\H{o}s, as reported in problem B2 of Guy's collection \\cite{Gu04}. Guy further writes 'It has even been suggested that there may be only finitely many $k$-perfect numbers with $k\\geq 3$.' The largest $k$ for which a $k$-perfect number has been found is $k=11$ - see this page for more information.\nThese are known as multiply perfect numbers. When $k=2$ this is the definition of a perfect number.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Grönwall's maximal-order theorem gives k=O(log log n) for every k-perfect n, but the conjectured little-o improvement along multiply perfect numbers remains open.\n\n**Verified partial progress.**\n\n- For k-perfect n, k=sigma(n)/n.\n- Grönwall proved limsup sigma(n)/(n log log n)=e^gamma, yielding the general upper scale k<=(e^gamma+o(1))log log n.\n- Known multiply perfect examples realize values only through k=11; no infinite family with unbounded k is known.\n\n**Full solution or refutation.**\n\nThe correct general big-O scale is classical, but no argument shows the multiplier tends to zero on the much thinner set of multiply perfect numbers.\n\n**What remains.**\n\nProve k/log log n tends to zero as n grows through multiply perfect numbers, or construct a sequence contradicting this.\n\n**Sources checked.**\n\n- Christian Axler and Jean-Louis Nicolas, Large values of n/phi(n) and sigma(n)/n, Acta Arithmetica 209 (2023), 357-383. (primary): https://doi.org/10.4064/aa220705-12-10\n Evidence used: States Grönwall's maximal-order theorem and modern effective bounds for sigma(n)/n.\n- Thomas F. Bloom, Erdős Problem #1053, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1053\n Evidence used: Retains open status and records the known-example context for multiply perfect numbers.\n\n**Review notes.** The intended asymptotic is along multiply perfect n with k varying; this quantifier is not explicit in the imported one-sentence statement. The background has trailing serialization debris.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2456, "problem_number": "EP-1054", "title": "Erdős Problem #1054", "statement": "Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\\geq 1$.\nIs it true that $f(n)=o(n)$? Or is this true only for almost all $n$, and $\\limsup f(n)/n=\\infty$?", "background": "A question of Erd\\H{o}s reported in problem B2 of Guy's collection \\cite{Gu04}. The function $f(n)$ is undefined for $n=2$ and $n=5$, but is likely well-defined for all $n\\geq 6$ (which would follow from a strong form of Goldbach's conjecture).\nThe sequence of values of $f(n)$ is given by A167485 in the OEIS.\nSee also [468].\nThe strong claim that $f(n)=o(n)$ was disproved by Tao in the comments to [468], in which he proves that the upper density of $\\{ n : f(n)\\leq \\delta n\\}$ is $\\ll \\delta^2$.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The strong universal assertion f(n)=o(n) is disproved by Tao's density argument recorded by the maintained tracker; the precise almost-all and limsup alternatives remain unsettled.\n\n**Verified partial progress.**\n\n- The tracker attributes to Tao an upper-density bound for {n:f(n)<=delta n} of order at most delta squared.\n- The source notes that f is undefined at 2 and 5, a formulation issue immaterial to the asymptotic question.\n\n**Full solution or refutation.**\n\nNo verified published resolution of every remaining alternative was found.\n\n**What remains.**\n\nDetermine the almost-all behavior and the limsup, after handling the two exceptional inputs explicitly.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1054, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1054\n Evidence used: Current open label, Tao density result, and exceptional inputs.\n- EP-1054 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/1054\n Evidence used: Recent claimed work is described as requiring verification.\n\n**Review notes.** The tracker remains open; no forum claim is classified as a complete solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2457, "problem_number": "EP-1055", "title": "Erdős Problem #1055", "statement": "A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nAre there infinitely many primes in each class? If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?", "background": "A classification due to Erd\\H{o}s and Selfridge. It is easy to prove that the number of primes $\\leq n$ in class $r$ is at most $n^{o(1)}$.\nThe sequence $p_r$ begins $2,13,37,73,1021$ (A005113 in the OEIS). Erd\\H{o}s thought $p_r^{1/r}\\to \\infty$, while Selfridge thought it quite likely to be bounded.\nA similar question can be asked replacing $p+1$ with $p-1$.\nThis is problem A18 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof was located that every Erdős--Selfridge class contains infinitely many primes, nor an asymptotic for the least class-r prime.\n\n**Verified partial progress.**\n\n- The maintained source records the elementary upper bound n^(o(1)) for the number of class-r primes up to n.\n\n**Full solution or refutation.**\n\nThe current maintained status is open.\n\n**What remains.**\n\nProve infinitude in each class and determine the growth of p_r^(1/r).\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1055, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1055\n Evidence used: Current open status and the recorded counting bound.\n\n**Review notes.** No source correction was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2458, "problem_number": "EP-1056", "title": "Erdős Problem #1056", "statement": "Let $k\\geq 2$. Does there exist a prime $p$ and consecutive intervals $I_1,\\ldots,I_k$ such that $ \\prod_{n\\in I_i}n \\equiv 1\\pmod{p} $ for all $1\\leq i\\leq k$?", "background": "This is problem A15 in Guy's collection \\cite{Gu04}, where he reports that in a letter in 1979 Erd\\H{o}s observed that $ 3\\cdot 4\\equiv 5\\cdot 6\\cdot 7\\equiv 1\\pmod{11}, $ establishing the case $k=2$. Makowski \\cite{Ma83} found, for $k=3$, $ 2\\cdot 3\\cdot 4\\cdot 5\\equiv 6\\cdot 7\\cdot 8\\cdot 9\\cdot 10\\cdot 11\\equiv 12\\cdot 13\\cdot 14\\cdot 15\\equiv 1\\pmod{17}. $ Noll and Simmons asked, more generally, whether there are solutions to $q_1!\\equiv\\cdots \\equiv q_k!\\pmod{p}$ for arbitrarily large $k$ (with $q_1<\\cdots\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The cases k=2 and k=3 have explicit constructions, but no result for arbitrary k was located.\n\n**Verified partial progress.**\n\n- Erdős's congruence modulo 11 establishes k=2.\n- Makowski's congruences modulo 17 establish k=3.\n\n**Full solution or refutation.**\n\nThe maintained current status is open for general k.\n\n**What remains.**\n\nFind constructions for all k or prove an obstruction.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1056, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1056\n Evidence used: Current open status and explicit k=2,3 examples.\n- R. K. Guy, Unsolved Problems in Number Theory, problem A15 (2004). (authoritative_secondary): https://www.erdosproblems.com/1056\n Evidence used: Original collection and Makowski citation as reproduced by the maintained source.\n\n**Review notes.** A puzzle-site claim of a broader solution was not sufficient to override the maintained open status.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2459, "problem_number": "EP-1057", "title": "Erdős Problem #1057", "statement": "Let $C(x)$ count the number of Carmichael numbers in the interval $[1,x]$. Is it true that $C(x)=x^{1-o(1)}$?", "background": "Erd\\H{o}s \\cite{Er56c} proved $ C(x) < x \\exp\\left(-c \\frac{\\log x\\log\\log\\log x}{\\log\\log x}\\right) $ for some constant $c>0$. Pomerance \\cite{Po89} gave a heuristic suggesting that this is the true order of growth, and in fact $ C(x)= x \\exp\\left(-(1+o(1))\\frac{\\log x\\log\\log\\log x}{\\log\\log x}\\right). $ Alford, Granville, and Pomerance \\cite{AGP94} proved that $C(x)\\to \\infty$, and in fact $C(x)>x^{2/7}$ for large $x$. The lower bound $ C(x)> x^{0.33336704} $ was proved by Harman \\cite{Ha08}. This exponent was improved to $0.3389$ by Lichtman \\cite{Li22}.\nKorselt observed that $n$ being a Carmichael number is equivalent to $n$ being squarefree and $p-1\\mid n-1$ for all primes $p\\mid n$.\nThis is discussed in problem A13 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[AGP94] Alford, W. R. and Granville, Andrew and Pomerance, Carl, There are infinitely many {C}armichael numbers. Ann. of Math. (2) (1994), 703--722.\n\n[Er56c] Erd\\H{o}s, P., On pseudoprimes and {C}armichael numbers. Publ. Math. Debrecen (1956), 201--206.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ha08] Harman, Glyn, Watt's mean value theorem and {C}armichael numbers. Int. J. Number Theory (2008), 241--248.\n\n[Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).\n\n[Po89] Pomerance, Carl, Two methods in elementary analytic number theory. (1989), 135--161.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Carmichael numbers are known to be infinite and the lower-bound exponent has improved to 0.3389, but this is far below x^(1-o(1)).\n\n**Verified partial progress.**\n\n- Alford--Granville--Pomerance proved infinitude and C(x)>x^(2/7).\n- Harman's exponent 0.33336704 was improved to 0.3389 by Lichtman, as recorded in the current tracker.\n\n**Full solution or refutation.**\n\nThe stated near-linear lower bound remains open.\n\n**What remains.**\n\nObtain substantially denser Carmichael constructions or prove the predicted order.\n\n**Sources checked.**\n\n- W. R. Alford, A. Granville, C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. 140 (1994), 703--722. (primary): https://annals.math.princeton.edu/1994/140-3/p02\n Evidence used: Infinitude and power lower bound.\n- Thomas F. Bloom, Erdős Problem #1057, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1057\n Evidence used: Current open status and Lichtman 0.3389 update.\n\n**Review notes.** Finite tabulations do not establish the asymptotic claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2460, "problem_number": "EP-1059", "title": "Erdős Problem #1059", "statement": "Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\\leq k!l$, and all the numbers $n-k!$ are composite for $1\\leq k\\leq l$.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of infinitely many primes with every earlier factorial difference composite was located.\n\n**Verified partial progress.**\n\n- The supplied examples p=101 and p=211 remain useful base cases.\n- For a qualifying p and k!", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2461, "problem_number": "EP-1060", "title": "Erdős Problem #1060", "statement": "Let $f(n)$ count the number of solutions to $k\\sigma(k)=n$, where $\\sigma(k)$ is the sum of divisors of $k$. Is it true that $f(n)\\leq n^{o(\\frac{1}{\\log\\log n})}$? Perhaps even $\\leq (\\log n)^{O(1)}$?", "background": "This is discussed in problem B11 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No result approaching either stated uniform multiplicity bound for k sigma(k)=n was located in the checked sources.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe tracker classifies the question as open; the present search found no verified later advance.\n\n**What remains.**\n\nBound the number of preimages of the map k to k sigma(k), ideally polylogarithmically.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1060, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1060\n Evidence used: Maintained problem page queried; no resolution identified.\n\n**Review notes.** Medium confidence because no dedicated post-2004 primary paper was located.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2462, "problem_number": "EP-1061", "title": "Erdős Problem #1061", "statement": "How many solutions are there to $ \\sigma(a)+\\sigma(b)=\\sigma(a+b) $ with $a+b\\leq x$, where $\\sigma$ is the sum of divisors function? Is it $\\sim cx$ for some constant $c>0$?", "background": "A question of Erd\\H{o}s reported in problem B15 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No asymptotic count for sigma(a)+sigma(b)=sigma(a+b) was located.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe maintained current status remains open.\n\n**What remains.**\n\nEstablish linear-order asymptotics or show a different order of growth.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1061, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1061\n Evidence used: Current open status; no incorporated partial result.\n\n**Review notes.** No source correction was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2463, "problem_number": "EP-1062", "title": "Erdős Problem #1062", "statement": "Let $f(n)$ be the size of the largest subset $A\\subseteq \\{1,\\ldots,n\\}$ such that there are no three distinct elements $a,b,c\\in A$ such that $a\\mid b$ and $a\\mid c$. How large can $f(n)$ be? Is $\\lim f(n)/n$ irrational?", "background": "The example $[m+1,3m+2]$ shows that $f(n)\\geq\\lceil \\frac{2}{3}n\\rceil$. Lebensold \\cite{Le76} has shown that, for large $n$, $ 0.6725 n \\leq f(n) \\leq 0.6736 n. $ This is problem B24 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Le76] Lebensold, Kenneth, A divisibility problem. Studies in Appl. Math. (1976/77), 291--294.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Lebensold proved tight published linear bounds, while a recent claimed effective limit is unreviewed and the irrationality question remains open.\n\n**Verified partial progress.**\n\n- Lebensold proved 0.6725 n <= f(n) <= 0.6736 n for all sufficiently large n.\n- The maintained forum records an unreviewed 2026 claim that f(n)=c_2 n+o(n) with computable c_2; it is not treated as verified.\n\n**Full solution or refutation.**\n\nThe original asymptotic constant and its irrationality are not established by verified literature found here.\n\n**What remains.**\n\nVerify or replace the claimed limit theorem and resolve irrationality.\n\n**Sources checked.**\n\n- K. Lebensold, A divisibility problem, Studies in Applied Mathematics 55 (1976/77), 291--294. (primary): https://www.erdosproblems.com/1062\n Evidence used: Published bounds as quoted by the maintained entry.\n- EP-1062 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/1062\n Evidence used: Open status and explicit caveat that the new claim is unreviewed.\n\n**Review notes.** The recent forum claim is retained as a lead, not a solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2464, "problem_number": "EP-1063", "title": "Erdős Problem #1063", "statement": "Let $k\\geq 2$ and define $n_k\\geq 2k$ to be the least value of $n$ such that $n-i$ divides $\\binom{n}{k}$ for all but one $0\\leq i\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A universal one-exception obstruction and an exponential LCM-based upper bound are known, but no asymptotic estimate for n_k was located.\n\n**Verified partial progress.**\n\n- Erdős--Selfridge's theorem guarantees at least one nondividing n-i for every n>=2k.\n- Monier's k! upper bound is improved in the maintained source to k lcm(2,...,k-1), hence exp((1+o(1))k).\n\n**Full solution or refutation.**\n\nThe requested estimate remains open.\n\n**What remains.**\n\nGive substantially sharper upper/lower asymptotics for n_k.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1063, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1063\n Evidence used: Current open status, Monier bound, and later LCM improvement.\n- Erdős and Selfridge, Problem 6447, American Mathematical Monthly (1983), as cited by the maintained entry. (authoritative_secondary): https://www.erdosproblems.com/1063\n Evidence used: Original source attribution for the exception theorem.\n\n**Review notes.** Formalisation pages using placeholders were not treated as proofs.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2465, "problem_number": "EP-1065", "title": "Erdős Problem #1065", "statement": "Are there infinitely many primes $p$ such that $p=2^kq+1$ for some prime $q$ and $k\\geq 0$? Or $p=2^k3^lq+1$?", "background": "This is mentioned in problem B46 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof was located of infinitude in either the 2-power or 2,3-smooth shifted-prime form.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe maintained source continues to state the question without a resolution.\n\n**What remains.**\n\nEstablish infinitude of primes p for which p-1 has the indicated restricted cofactor structure.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1065, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1065\n Evidence used: Current statement and absence of a listed resolution.\n\n**Review notes.** This is distinct from standard Sophie Germain or Fermat-prime infinitude questions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2466, "problem_number": "EP-1066", "title": "Erdős Problem #1066", "statement": "Let $G$ be a graph given by $n$ points in $\\mathbb{R}^2$, where any two distinct points are at least distance $1$ apart, and we draw an edge between two points if they are distance $1$ apart.\nLet $g(n)$ be maximal such that any such graph always has an independent set on at least $g(n)$ vertices. Estimate $g(n)$, or perhaps $\\lim \\frac{g(n)}{n}$.", "background": "Such graphs are always planar. Erd\\H{o}s initially thought that $g(n)=n/3$, but Chung and Graham, and independently Pach, gave a construction that shows $g(n)\\leq \\frac{6}{19}n$. Pach and Toth \\cite{PaTo96} improved this to $g(n)\\leq \\frac{5}{16}n$.\nPollack \\cite{Po85} noted that the Four colour theorem implies $g(n)\\geq n/4$, since the graph is planar. Pollack reports that Pach observed that this in for unit distance graphs the four colour theorem can be proved by a simple induction.\nThis lower bound has been improved to $\\frac{9}{35}n$ by Csizmadia \\cite{Cs98} and then $\\frac{8}{31}n$ by Swanepoel \\cite{Sw02}. The current record bounds are therefore $ \\frac{8}{31}n \\approx 0.258n \\leq g(n) \\leq 0.3125n=\\frac{5}{16}n. $ Pollack \\cite{Po85} also reports a letter from Erd\\H{o}s which poses the more general problem of, given $n$ points in $\\mathbb{R}^d$ with minimum distance $1$, let $g_d(n)$ be maximal such that there always exist at least $g_d(n)$ many points which have minimum distance $>1$. Is it true that $g_d(n) \\gg n/d$ in general? The upper bound $g_d(n) \\ll n/d$ is trivial, considering widely spaced unit simplices.\nSee [1070] for the general estimate of independence number of unit distance graphs.\nReferences\n\n\n[Cs98] Csizmadia, G., On the independence number of minimum distance graphs. Discrete Comput. Geom. (1998), 179--187.\n\n[PaTo96] Pach, J\\'anos and T\\'oth, G\\'{e}za, On the independence number of coin graphs. Geombinatorics (1996), 30--33.\n\n[Po85] Pollack, R., Increasing the minimum distance of a set of points. J. Combin. Theory Ser. A (1985), 450.\n\n[Sw02] Swanepoel, Konrad J., Independence numbers of planar contact graphs. Discrete Comput. Geom. (2002), 649--670.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The minimum-distance unit-distance independence ratio is open between 8/31 and 5/16.\n\n**Verified partial progress.**\n\n- Swanepoel's lower bound is 8n/31.\n- Pach--Tóth's construction gives the upper bound 5n/16.\n\n**Full solution or refutation.**\n\nNo exact limiting ratio was found.\n\n**What remains.**\n\nClose the 8/31 to 5/16 gap or prove existence/value of the limit.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1066, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1066\n Evidence used: Current open status and best recorded bounds.\n\n**Review notes.** Unreviewed forum work was not used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2467, "problem_number": "EP-1068", "title": "Erdős Problem #1068", "statement": "Does every graph with chromatic number $\\aleph_1$ contain a countable subgraph which is infinitely vertex-connected?", "background": "I do not think this was originally a question of Erd\\H{o}s - it appears in \\cite{BoPi24} as a 'version of the Erd\\H{o}s-Hajnal problem' (which is [1067]).\nI could not in fact find this in the paper of Erd\\H{o}s and Hajnal \\cite{ErHa66}, however, and hence the first place it appears may in fact be in \\cite{BoPi24}. In hindsight this should not have been included as a separate problem, but this has been discovered too late, and so we must leave it here.\nWe say a graph is infinitely (vertex) connected if any two vertices are connected by infinitely many pairwise vertex-disjoint paths.\nSoukup \\cite{So15} constructed a graph with uncountable chromatic number in which every uncountable set is finitely vertex-connected. A simpler construction was given by Bowler and Pitz \\cite{BoPi24}.\nSee also [1067].\nReferences\n\n\n[BoPi24] N. Bowler and M. Pitz, A note on uncountably chromatic graphs. arXiv:2402.05984 (2024).\n\n[ErHa66] Erd\\H{o}s, P. and Hajnal, A., On chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hungar. (1966), 61-99.\n\n[So15] Soukup, D\\'aniel T., Trees, ladders and graphs. J. Combin. Theory Ser. B (2015), 96--116.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The source says this question may have been erroneously entered as an Erdős problem; known constructions settle stronger uncountable-subgraph variants but do not visibly settle this countable-subgraph formulation.\n\n**Verified partial progress.**\n\n- Soukup constructed an uncountably chromatic graph whose uncountable sets are finitely vertex-connected.\n- Bowler--Pitz gave a simpler related construction.\n\n**Full solution or refutation.**\n\nNo source proving or refuting the stated countable infinitely-connected subgraph assertion was located.\n\n**What remains.**\n\nEstablish the exact countable-subgraph assertion, and clarify provenance.\n\n**Sources checked.**\n\n- N. Bowler and M. Pitz, A note on uncountably chromatic graphs, arXiv:2402.05984 (2024). (primary): https://arxiv.org/abs/2402.05984\n Evidence used: Related construction and problem framing.\n- Erdős Problems LaTeX entry #1068, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/1068\n Evidence used: Provenance caveat and Soukup reference.\n\n**Review notes.** Do not infer a counterexample to the countable statement from a result about uncountable sets.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2468, "problem_number": "EP-1070", "title": "Erdős Problem #1070", "statement": "Let $f(n)$ be maximal such that, given any $n$ points in $\\mathbb{R}^2$, there exist $f(n)$ points such that no two are distance $1$ apart. Estimate $f(n)$. In particular, is it true that $f(n)\\geq n/4$?", "background": "In other words, estimate the minimal independence number of a unit distance graph with $n$ vertices. If $\\omega$ is the independence number and $\\chi$ is the chromatic number then $\\omega \\chi\\geq n$, and hence $f(n)\\geq n/\\chi$, where $\\chi$ is the answer to the Hadwiger-Nelson problem [508].\nThe Moser spindle shows $f(n)\\leq \\frac{2}{7}n\\approx 0.285n$. Larman and Rogers \\cite{LaRo72} noted that if $m_1$ is the supremum of the upper densities of measurable subsets of $\\mathbb{R}^2$ which have no unit distance pairs then $ f(n)\\geq m_1n. $ Croft \\cite{Cr67} gave the best-known lower bound of $m_1\\geq 0.22936$ and hence $ 0.22936n \\leq f(n) \\leq \\frac{2}{7}n\\approx 0.285n. $ Ambrus, Csisz\\'{a}rik, Matolcsi, Varga, and Zs\\'{a}mboki \\cite{ACMVZ23} have proved that $m_1\\leq 0.247$, and hence this approach cannot achieve $f(n)\\geq n/4$. See [232] for more on $m_1$.\nMatolcsi, Ruzsa, Varga, and Zs\\'{a}mboki \\cite{MRVZ23} have improved the upper bound to $ f(n) \\leq \\left(\\frac{1}{4}+o(1)\\right)n. $ They conjecture that $m_1=0.22936\\cdots$ (the lower bound of Croft mentioned above) and $f(n)=(1/4+o(1))n$.\nIf we also insist that no two points are distance $<1$ apart then this is problem becomes [1066].\nReferences\n\n\n[ACMVZ23] G. Ambrus, A. Csisz\\'{a}rik, M. Matolcsi, D. Varga, and P. Zs\\'{a}mboki, The density of planar sets avoiding unit distances. arXiv:2207.14179 (2023).\n\n[Cr67] H. T. Croft, Incidence incidents. Eureka (1967), 22-26.\n\n[LaRo72] Larman, D. G. and Rogers, C. A., The realization of distances within sets in Euclidean space. Mathematika (1972), 1-24.\n\n[MRVZ23] M. Matolcsi, I. Z. Ruzsa, D. Varga, and P. Zs\\'{a}mboki, The fractional chromatic number of the plane is at least $4$. arXiv:2311.10069 (2023).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The global unit-distance independence problem is open, but the explicit n/4 subquestion has a recent preprint claim of a negative answer and established upper bounds approach n/4.\n\n**Verified partial progress.**\n\n- Matolcsi--Ruzsa--Varga--Zsámboki obtained f(n)<=(1/4+o(1))n.\n- The tracker discussion reports a 2026 Dúcz--Varga preprint with independence ratio below 1/4.\n\n**Full solution or refutation.**\n\nThe exact asymptotic minimum ratio is unresolved.\n\n**What remains.**\n\nVerify/publish the 2026 preprint and determine the optimum ratio.\n\n**Sources checked.**\n\n- M. Matolcsi et al., The fractional chromatic number of the plane is at least 4, arXiv:2311.10069 (2023). (primary): https://arxiv.org/abs/2311.10069\n Evidence used: Near-quarter upper bound.\n- EP-1070 discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/1070\n Evidence used: Recent preprint claim and peer-review caveat.\n\n**Review notes.** The new preprint is not treated as a resolution of the broad estimate request.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2469, "problem_number": "EP-1071", "title": "Erdős Problem #1071", "statement": "Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the unit square, no two of which intersect, which are maximal with respect to this property?\nIs there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite?", "background": "A question of Erd\\H{o}s and T\\'{o}th. The answer to the first question is yes (which Erd\\H{o}s gave Danzer \\$10 for). There is no prize mentioned in \\cite{Er87b} for the (still open) second question.\nThere are two examples Erd\\H{o}s gives in \\cite{Er87b}, the {IMAGE=1071-one,first} by Danzer, the {IMAGE=1071-two,second} by an unnamed participant.\nIn \\cite{Er87b} he further asks what happens if the unit line segments are rotated/translated copies of $[0,1]$ that are allowed to intersect only at their endpoints.\nReferences\n\n\n[Er87b] Erd\\H{o}s, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Danzer's finite maximal construction answers the first question, and a 2026 nested-parallelogram construction gives a countably infinite maximal family in the unit square for the second; both have Lean verification.\n\n**Verified partial progress.**\n\n- Danzer's finite maximal family in the unit square was already reported by Erdos in 1987.\n- The 2026 discussion constructs a countable family recursively inside nested parallelograms and blocks the limiting diagonal with one further unit segment.\n- The discussion records type-checked Lean proofs of both the new countable construction and the historical finite construction.\n\n**Full solution or refutation.**\n\nSplit the unit square vertically and recursively insert pairs of unit segments in a nested parallelogram on each side, leaving only triangular regions that cannot contain another open unit segment; add a unit segment on the common limiting diagonal. The resulting family is countable and maximal.\n\n**What remains.**\n\nBoth displayed open-segment questions are closed. The distinct variant using closed segments allowed to meet only endpoint-to-endpoint is not resolved by this classification.\n\n**Sources checked.**\n\n- Paul Erdos, Some combinatorial and metric problems in geometry, Intuitive Geometry (Siofok, 1985), Colloquia Mathematica Societatis Janos Bolyai 48 (1987), 167-177. (primary): https://combinatorica.hu/~p_erdos/Erdos.html\n Evidence used: Original source for the questions and the two historical finite examples, including Danzer's.\n- Thomas F. Bloom, Erdos Problem #1071, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1071\n Evidence used: Records PROVED (LEAN), the finite answer, and Alexeev's countably infinite construction in the unit square.\n- Boris Alexeev and contributors, EP-1071 discussion and linked type-checked Lean proofs, January-February 2026. (formal_verification): https://www.erdosproblems.com/forum/thread/1071\n Evidence used: Gives the recursive nested-parallelogram proof, audits the open-segment convention, and links formalizations of both parts.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2470, "problem_number": "EP-1072", "title": "Erdős Problem #1072", "statement": "For any prime $p$, let $f(p)$ be the least integer such that $f(p)!+1\\equiv 0\\pmod{p}$.\nIs it true that there are infinitely many $p$ for which $f(p)=p-1$?\nIs it true that $f(p)/p\\to 0$ for almost all $p$?", "background": "Questions formulated by Erd\\H{o}s, Hardy, and Subbarao \\cite{HaSu02}, who believed that the number of $p\\leq x$ for which $f(p)=p-1$ is $o(x/\\log x)$.\nThese are mentioned in problem A2 of Guy's collection.\nReferences\n\n\n[HaSu02] Hardy, G. E. and Subbarao, M. V., A modified problem of Pillai and some related questions. Amer. Math. Monthly (2002), 554--559.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Neither the exceptional-prime infinitude question nor the almost-all small-factorial-index limit is resolved in the checked sources.\n\n**Verified partial progress.**\n\n- Hardy and Subbarao conjectured the exceptional set f(p)=p-1 has density zero among primes.\n\n**Full solution or refutation.**\n\nThe maintained current status is open.\n\n**What remains.**\n\nProve either proposed distribution assertion.\n\n**Sources checked.**\n\n- G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly (2002). (authoritative_secondary): https://www.erdosproblems.com/1072\n Evidence used: Original formulation as reproduced by the maintained entry.\n- Thomas F. Bloom, Erdős Problem #1072, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1072\n Evidence used: Current open status.\n\n**Review notes.** Forum observations are not used as a theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2471, "problem_number": "EP-1073", "title": "Erdős Problem #1073", "statement": "Let $A(x)$ count the number of composite $u\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the requested subpower bound for composite factorial-plus-one divisors was located.\n\n**Verified partial progress.**\n\n- The OEIS initial examples and Wilson-theorem endpoint give context but not a growth estimate.\n\n**Full solution or refutation.**\n\nThe maintained current status is open.\n\n**What remains.**\n\nBound the number of composite moduli u admitting n! = -1 modulo u.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1073, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1073\n Evidence used: Current open status and known examples.\n\n**Review notes.** No statement text was changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2472, "problem_number": "EP-1074", "title": "Erdős Problem #1074", "statement": "Let $S$ be the set of all $m\\geq 1$ such that there exists a prime $p\not\\equiv 1\\pmod{m}$ such that $m!+1\\equiv 0\\pmod{p}$. Does $ \\lim \\frac{\\lvert S\\cap [1,x]\\rvert}{x} $ exist? What is it?\nSimilarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\\equiv 1\\pmod{m}$ such that $m!+1\\equiv 0\\pmod{p}$, then does $ \\lim \\frac{\\lvert P\\cap [1,x]\\rvert}{\\pi(x)} $ exist? What is it?", "background": "Questions raised by Erd\\H{o}s, Hardy, and Subbarao, who called the set $S$ 'EHS numbers' and the set $P$ 'Pillai primes', and proved that both $S$ and $P$ are infinite. Pillai \\cite{Pi30} raised the question of whether there exist any primes in $P$. This was answered by Chowla, who noted that, for example, $ 14!+1\\equiv 18!+1\\equiv 0\\pmod{23}. $ The sequence $S$ begins $ 8,9,13,14,15,16,17,\\ldots $ and is A064164 in the OEIS. The sequence $P$ begins $ 23,29,59,61,67,71,\\ldots $ and is A063980 in the OEIS.\nRegarding the first question, Hardy and Subbarao computed all EHS numbers up to $2^{10}$, and write '...if this trend conditions we expect [the limit] to be around $0.5$, if it exists. The frequency with which the EHS numbers occur - most often in long sequences of consecutive integers - makes us believe that their asymptotic density exists and is unity. Erd\\H{o}s, though initially hesitant, later agreed with this view.'\nRegarding the second question, they write '[from the data] it would appear that if the limit exists, it is perhaps between $0.5$ and $0.6$. But then there seems to be no reason why the ratio should not tend to $1$, even though very slowly and certainly not monotonically.'\nThis is discussed in problem A2 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Pi30] S. S. Pillai, Question 1490. J. Indian Math. Soc. (1930), 230.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Both EHS numbers and Pillai primes are known to be infinite, but neither requested density limit is known.\n\n**Verified partial progress.**\n\n- Erdős, Hardy, and Subbarao proved S and P infinite.\n- Chowla supplied a prime Pillai example, 23.\n\n**Full solution or refutation.**\n\nThe density questions remain open.\n\n**What remains.**\n\nProve existence and values of the two natural densities.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1074, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1074\n Evidence used: Open status, qualitative theorem, and historical examples.\n\n**Review notes.** The imported strings `p ot equiv` are visibly damaged OCR/LaTex; intended `not congruent` is inferred only from the independently rendered source and source text is untouched.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2473, "problem_number": "EP-1075", "title": "Erdős Problem #1075", "statement": "Let $r\\geq 3$. There exists $c_r>r^{-r}$ such that, for any $\\epsilon>0$, if $n$ is sufficiently large, the following holds.\nAny $r$-uniform hypergraph on $n$ vertices with at least $(1+\\epsilon)(n/r)^r$ many edges contains a subgraph on $m$ vertices with at least $c_rm^r$ edges, where $m=m(n)\\to \\infty$ as $n\\to \\infty$.", "background": "Erd\\H{o}s \\cite{Er64f} proved that this is true with $c_r=r^{-r}$ whenever the graph has at least $\\epsilon n^r$ many edges.\nReferences\n\n\n[Er64f] Erd\\H{o}s, P., On extremal problems of graphs and generalized graphs. Israel J. Math. (1964), 183--190.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A denser-hypergraph version with c_r=r^(-r) is known, while the threshold-strengthened statement requesting c_r>r^(-r) remains open.\n\n**Verified partial progress.**\n\n- Erdős proved the claim with c_r=r^(-r) under the stronger assumption of epsilon n^r edges.\n\n**Full solution or refutation.**\n\nNo improvement past r^(-r) at the stated threshold was located.\n\n**What remains.**\n\nBeat the r^(-r) density constant near the (n/r)^r edge scale.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1075, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1075\n Evidence used: Current open status and Erdős partial theorem.\n\n**Review notes.** No source correction was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2474, "problem_number": "EP-1083", "title": "Erdős Problem #1083", "statement": "Let $d\\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\\mathbb{R}^d$ determines at least $m$ distinct distances. Estimate $f_d(n)$ - in particular, is it true that $ f_d(n)=n^{\\frac{2}{d}-o(1)}? $ ", "background": "A generalisation of the distinct distance problem [89] to higher dimensions. Erd\\H{o}s \\cite{Er46b} proved $ n^{1/d}\\ll_d f_d(n)\\ll_d n^{2/d}, $ the upper bound construction being given by a set of lattice points.\n{UL}\n{LI} Clarkson, Edelsbrunner, Gubias, Sharir, and Welzl \\cite{CEGSW90} proved $f_3(n)\\gg n^{1/2}$.{/LI}\n{LI}Aronov, Pach, Sharir, and Tardos \\cite{APST04} proved $f_d(n)\\gg n^{\\frac{1}{d-90/77}-o(1)}$ for any $d\\geq 3$ (for example, $f_3(n)\\gg n^{0.546}$).{/LI}\n{LI}Solymosi and Vu \\cite{SoVu08} proved $f_3(n) \\gg n^{3/5}$ and $ f_d(n)\\gg_d n^{\\frac{2}{d}-\\frac{c}{d^2}} $ for all $d\\geq 4$ for some constant $c>0$. (The result in their paper for $d=3$ is slightly weaker than stated here, but uses as a black box the bound for distinct distances in $2$ dimensions; we have recorded the consequence of combining their method with the work of Guth and Katz on [89].){/LI}\n{/UL}\nThe function $f_d(n)$ is essentially the inverse of the function $g_d(n)$ considered in [1089] - with our definitions, $g_d(n)>m$ if and only if $f_d(m)\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Near-optimal exponent bounds are known in fixed higher dimensions, but the n^(2/d-o(1)) target is not proved in general.\n\n**Verified partial progress.**\n\n- Solymosi--Vu proved f_3(n) much larger than n^(3/5) and f_d(n) at least n^(2/d-c/d^2) for d>=4.\n- The lattice construction gives the matching-order upper exponent n^(2/d).\n\n**Full solution or refutation.**\n\nThe exponent gap remains for each fixed d>=3.\n\n**What remains.**\n\nRemove the fixed-d exponent loss.\n\n**Sources checked.**\n\n- J. Solymosi and V. Vu, Near optimal bounds for the Erdős distinct distances problem in high dimensions, Combinatorica 28 (2008), 113--125. (primary): https://doi.org/10.1007/s00493-008-2357-3\n Evidence used: Higher-dimensional lower bounds.\n- Thomas F. Bloom, Erdős Problem #1083, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1083\n Evidence used: Current open status and bound synthesis.\n\n**Review notes.** The input's old HTML list tags are preserved as source data.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2475, "problem_number": "EP-1084", "title": "Erdős Problem #1084", "statement": "Let $f_d(n)$ be minimal such that in any collection of $n$ points in $\\mathbb{R}^d$, all of distance at least $1$ apart, there are at most $f_d(n)$ many pairs of points which are distance $1$ apart. Estimate $f_d(n)$.", "background": "This is sometimes known as the contact number problem.\nIt is easy to see that $f_1(n)=n-1$ and $f_2(n)<3n$ (since there can be at most $6$ points of distance $1$ from any point). Erd\\H{o}s \\cite{Er46b} showed $ f_2(n)<3n-cn^{1/2} $ for some constant $c>0$, which the triangular lattice shows is the best possible up to the value of $c$. In \\cite{Er75f} he speculated that the triangular lattice is exactly the best possible, and in particular $ f_2(3n^2+3n+1)=9n^2+6n. $ Harborth \\cite{Ha74b} proved that $ f_2(n)=\\lfloor 3n-\\sqrt{12n-3}\\rfloor $ for all $n\\geq 2$.\nIn \\cite{Er75f} he claims the existence of $c_1,c_2>0$ such that $ 6n-c_1n^{2/3}< f_3(n) < 6n-c_2n^{2/3}. $ An upper bound of $ f_3(n) < 6n-0.926n^{2/3} $ for all $n\\geq 2$ was proved by Bezdek and Reid \\cite{BeRe13}.\nIn general, it is known that $ (d-o(1))n \\leq f_d(n) \\leq 2^{O(d)}n, $ the lower bound coming from points arranged in an integer grid and the upper bound from the fact that $2^{O(d)}$ many non-intersecting congruent balls can touch a fixed ball (the kissing number problem).\nA recent survey on contact numbers for sphere packings is by Bezdek and Khan \\cite{BeKa18}.\nSee [223] for the analogous problem with maximal distance $1$.\nReferences\n\n\n[BeKa18] No reference found.\n\n\n[BeRe13] Bezdek, K\\'aroly and Reid, Samuel, Contact graphs of unit sphere packings revisited. J. Geom. (2013), 57--83.\n\n[Er46b] Erd\\H{o}s, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250.\n\n[Er75f] Erd\\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\n\n[Ha74b] Harborth, Heiko, L\"{o}sung zu Problem 664A. Elem. Math. (1974), 14-15.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The planar contact number is exact, while the three-dimensional and general-dimensional contact-number questions remain open up to broad asymptotic bounds.\n\n**Verified partial progress.**\n\n- Harborth proved the exact planar formula floor(3n-sqrt(12n-3)).\n- Bezdek--Reid gave f_3(n)<6n-0.926 n^(2/3).\n\n**Full solution or refutation.**\n\nNo sharp asymptotic in d=3 or a general-d formula was located.\n\n**What remains.**\n\nDetermine the three-dimensional contact number and sharpen dimension dependence.\n\n**Sources checked.**\n\n- H. Harborth, Lösung zu Problem 664A, Elemente der Mathematik 29 (1974), 14--15. (primary): https://www.erdosproblems.com/1084\n Evidence used: Exact d=2 formula as cited by maintained source.\n- Thomas F. Bloom, Erdős Problem #1084, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1084\n Evidence used: Open d=3/general status and survey lead.\n\n**Review notes.** The exact planar case does not settle the all-dimension request.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2476, "problem_number": "EP-1085", "title": "Erdős Problem #1085", "statement": "Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distance $1$ apart. Estimate $f_d(n)$.", "background": "The most difficult cases are $d=2$ and $d=3$. When $d=2$ this is the unit distance problem [90], and the best known bounds are $ n^{1+\\frac{c}{\\log\\log n}}< f_2(n) \\ll n^{4/3} $ for some constant $c>0$, the lower bound by Erd\\H{o}s \\cite{Er46b} and the upper bound by Spencer, Szemer\\'{e}di, and Trotter \\cite{SST84}.\nWhen $d=3$ the best known bounds are $ n^{4/3}\\log\\log n \\ll f_3(n) \\ll n^{3/2}\\beta(n) $ where $\\beta(n)$ is a very slowly growing function, the lower bound by Erd\\H{o}s \\cite{Er60b} and the upper bound by Clarkson, Edelsbrunner, Guibas, Sharir, and Welzl \\cite{CEGSW90}.\nA construction of Lenz (taking points on orthogonal circles) shows that, for $d\\geq 4$, $ f_d(n)\\geq \\frac{p-1}{2p}n^2-O(1) $ with $p=\\lfloor d/2\\rfloor$. Erd\\H{o}s \\cite{Er60b} showed that the Erd\\H{o}s-Stone theorem implies $ f_d(n) \\leq \\left(\\frac{p-1}{2p}+o(1)\\right)n^2 $ for $d\\geq 4$.\nErd\\H{o}s \\cite{Er67e} determined $f_d(n)$ up to $O(1)$ for all even $d\\geq 4$. Brass \\cite{Br97} determined $f_4(n)$ exactly. Swanepoel \\cite{Sw09} determined $f_d(n)$ exactly for even $d\\geq 6$. For odd $d\\geq 5$ Erd\\H{o}s and Pach \\cite{ErPa90} proved that there exist constants $c_1(d),c_2(d)>0$ such that $ \\frac{p-1}{2p}n^2 +c_1n^{4/3}\\leq f_d(n) \\leq \\frac{p-1}{2p}n^2 +c_2n^{4/3}. $ \nReferences\n\n\n[Br97] Brass, P., On the maximum number of unit distances among {$n$} points in\ndimension four. (1997), 277--290.\n\n[CEGSW90] Clarkson, Kenneth L. and Edelsbrunner, Herbert and Guibas,\nLeonidas J. and Sharir, Micha and Welzl, Emo, Combinatorial complexity bounds for arrangements of curves and\nspheres. Discrete Comput. Geom. (1990), 99--160.\n\n[Er46b] Erd\\H{o}s, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250.\n\n[Er60b] Erd\\H{o}s, P., On sets of distances of {$n$} points in {E}uclidean space. Magyar Tud. Akad. Mat. Kutat\\'o{} Int. K\"ozl. (1960), 165--169.\n\n[Er67e] Erd\\H{o}s, P., On some applications of graph theory to geometry. Canadian J. Math. (1967), 968--971.\n\n[ErPa90] Erd\\H{o}s, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269.\n\n[SST84] Spencer, J. and Szemer\\'{e}di, E. and Trotter, Jr., W., Unit distances in the Euclidean plane. Graph theory and combinatorics (Cambridge, 1983) (1984), 293-303.\n\n[Sw09] Swanepoel, Konrad J., Unit distances and diameters in {E}uclidean spaces. Discrete Comput. Geom. (2009), 1--27.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The unit-distance problem is asymptotically settled in even dimensions at least four and has narrow bounds in odd higher dimensions, but dimensions two and three remain open.\n\n**Verified partial progress.**\n\n- Brass determined d=4 exactly; Swanepoel determined all even d>=6 exactly.\n- For odd d>=5, Erdős--Pach give matching quadratic main terms with order n^(4/3) error.\n\n**Full solution or refutation.**\n\nThe global problem remains open because d=2 and d=3 have major gaps.\n\n**What remains.**\n\nImprove the planar and spatial unit-distance bounds.\n\n**Sources checked.**\n\n- K. J. Swanepoel, Unit distances and diameters in Euclidean spaces, Discrete Comput. Geom. 41 (2009), 1--27. (primary): https://doi.org/10.1007/s00454-008-9095-7\n Evidence used: Even-dimensional exact results as cited by the maintained source.\n- Thomas F. Bloom, Erdős Problem #1085, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1085\n Evidence used: Current status and dimensional breakdown.\n\n**Review notes.** The source sentence missing `are` is grammatical damage but the mathematical object is clear; it was not edited.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2477, "problem_number": "EP-1086", "title": "Erdős Problem #1086", "statement": "Let $g(n)$ be minimal such that any set of $n$ points in $\\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the same area. Estimate $g(n)$.", "background": "Equivalently, how many triangles of area $1$ can a set of $n$ points in $\\mathbb{R}^2$ determine? Erd\\H{o}s and Purdy attribute this question to Oppenheim. Erd\\H{o}s and Purdy \\cite{ErPu71} proved $ n^2\\log\\log n \\ll g(n) \\ll n^{5/2}, $ and believed the lower bound to be closer to the truth. The upper bound has been improved a number of times - by Pach and Sharir \\cite{PaSh92}, Dumitrescu, Sharir, and T\\'{o}th \\cite{DST09}, Apfelbaum and Sharir \\cite{ApSh10}, and Apfaulbaum \\cite{Ap13}. The best known bound is $ g(n) \\ll n^{20/9} $ by Raz and Sharir \\cite{RaSh17}.\nErd\\H{o}s and Purdy also ask a similar question about the higher-dimensional generalisations - more generally, let $g_d^{r}(n)$ be minimal such that any set of $n$ points in $\\mathbb{R}^d$ contains the vertices of at most $g_d^{r}(n)$ many $r$-dimensional simplices with the same volume.\nErd\\H{o}s and Purdy \\cite{ErPu71} proved $g_3^2(n) \\ll n^{8/3}$, and Dumitrescu, Sharir, and T\\'{o}th \\cite{DST09} improved this to $g_3^2(n) \\ll n^{2.4286}$.\nErd\\H{o}s and Purdy \\cite{ErPu71} proved $g_6^2(n)\\gg n^3$. Purdy \\cite{Pu74} proved $ g_4^2(n)\\leq g^2_5(n) \\ll n^{3-c} $ for some constant $c>0$. An observation of Oppenheim (using a construction of Lenz) detailed in \\cite{ErPu71} shows that $ g_{2k+2}^k(n)\\geq \\left(\\frac{1}{(k+1)^{k+1}}+o(1)\\right)n^{k+1} $ and Erd\\H{o}s and Purdy conjecture this is the best possible.\nSee also [90] and [755].\nReferences\n\n\n[Ap13] R. Apfelbaum, Geometric Incidences and Repeated Configurations. Ph.D. Dissertation, School of Computer Science, Tel Aviv University (2013).\n\n[ApSh10] Apfelbaum, Roel and Sharir, Micha, An improved bound on the number of unit area triangles. Discrete Comput. Geom. (2010), 753--761.\n\n[DST09] Dumitrescu, Adrian and Sharir, Micha and T\\'oth, Csaba D., Extremal problems on triangle areas in two and three\ndimensions. J. Combin. Theory Ser. A (2009), 1177--1198.\n\n[ErPu71] Erd\\H{o}s, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246--252.\n\n[PaSh92] Pach, J\\'anos and Sharir, Micha, Repeated angles in the plane and related problems. J. Combin. Theory Ser. A (1992), 12--22.\n\n[Pu74] Purdy, George, Some extremal problems in geometry. Discrete Math. (1974), 305--315.\n\n[RaSh17] Raz, Orit E. and Sharir, Micha, The number of unit-area triangles in the plane: theme and\nvariation. Combinatorica (2017), 1221--1240.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The maximum number of equal nonzero-area triangles remains open between an n^2 log log n lower bound and the current O(n^(20/9)) upper bound.\n\n**Verified partial progress.**\n\n- Erdős and Purdy proved n^2 log log n << g(n) << n^(5/2).\n- Raz and Sharir improved the general upper bound to O(n^(20/9)).\n- Raz and Sharir proved quadratic behavior for points restricted to three fixed lines and stronger bounds for convex grids.\n\n**Full solution or refutation.**\n\nNo matching exponent or order estimate is known in the unrestricted planar problem.\n\n**What remains.**\n\nClose the gap between the nearly quadratic construction and the n^(20/9) incidence upper bound.\n\n**Sources checked.**\n\n- P. Erdős and G. B. Purdy, Some extremal problems in geometry, J. Combin. Theory Ser. A 10 (1971), 246-252, DOI 10.1016/0097-3165(71)90028-8. (primary): https://www.sciencedirect.com/science/article/pii/0097316571900288\n Evidence used: Original nonzero-area formulation and classical lower and upper bounds.\n- O. E. Raz and M. Sharir, The number of unit-area triangles in the plane: Theme and variations, Combinatorica 37 (2017), 1221-1240. (primary): https://arxiv.org/abs/1501.00379\n Evidence used: Primary source for the O(n^(20/9)) upper bound and restricted-family results.\n- Thomas F. Bloom, Erdős Problem #1086, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1086\n Evidence used: Current open status and historical sequence of improvements.\n\n**Review notes.** The imported statement omits 'nonzero' before area; the primary paper and the background's unit-area equivalence make the intended restriction clear. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2478, "problem_number": "EP-1087", "title": "Erdős Problem #1087", "statement": "Let $f(n)$ be minimal such that every set of $n$ points in $\\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate $f(n)$ - in particular, is it true that $f(n)\\leq n^{3+o(1)}$?", "background": "A question of Erd\\H{o}s and Purdy \\cite{ErPu71}, who proved $ n^3\\log n \\ll f(n) \\ll n^{7/2}. $ \nReferences\n\n\n[ErPu71] Erd\\H{o}s, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246--252.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The repeated-distance four-point extremal problem remains open between n^3 log n and n^(7/2), with no later quantitative improvement verified.\n\n**Verified partial progress.**\n\n- Erdős and Purdy proved n^3 log n << f(n) << n^(7/2).\n\n**Full solution or refutation.**\n\nThe proposed n^(3+o(1)) upper bound remains unproved.\n\n**What remains.**\n\nClarify the extremal definition in the source and improve the n^(7/2) upper bound toward the construction scale.\n\n**Sources checked.**\n\n- P. Erdős and G. B. Purdy, Some extremal problems in geometry, J. Combin. Theory Ser. A 10 (1971), 246-252, DOI 10.1016/0097-3165(71)90028-8. (primary): https://www.sciencedirect.com/science/article/pii/0097316571900288\n Evidence used: Original problem and classical bounds.\n- Thomas F. Bloom, Erdős Problem #1087 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/1087\n Evidence used: Current open status and explicit formulation caveat.\n\n**Review notes.** The intended degeneracy is that two distinct pairs among the four points have equal distance, equivalently fewer than six pairwise distances. The 'minimal such that every set has at most' wording is an awkward maximum definition. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2479, "problem_number": "EP-1088", "title": "Erdős Problem #1088", "statement": "Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\\mathbb{R}^d$ contains a set of $n$ points such that any two determined distances are distinct. Estimate $f_d(n)$. In particular, is it true that, for fixed $n\\geq 3$, $ f_d(n)=2^{o(d)}? $ ", "background": "It is easy to prove that $f_d(n) \\leq n^{O_d(1)}$. Erd\\H{o}s \\cite{Er75f} claimed that he and Straus proved $f_d(n)\\leq c_n^d$ for some constant $c_n>0$.\nWhen $d=1$ this is the subject of [530], and $f_1(n)\\asymp n^2$.\nWhen $n=3$ this is the subject of [503]. Erd\\H{o}s could prove $f_2(3)=7$ and Croft \\cite{Cr62} proved $f_3(3)=9$. The results described at [503] demonstrate that $f_d(3)=d^2/2+O(d)$.\nReferences\n\n\n[Cr62] Croft, H. T., $9$-point and $7$-point configurations in $3$-space. Proc. London Math. Soc. (3) (1962), 400-424.\n\n[Er75f] Erd\\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The fixed-n subexponential-in-d conjecture is known for n=3 but remains open for general fixed n at least four.\n\n**Verified partial progress.**\n\n- Erdős reported an exponential upper bound f_d(n)<=c_n^d.\n- For n=3, results on isosceles sets give f_d(3)=d^2/2+O(d), hence the desired 2^o(d) behavior in this case.\n- The low-dimensional values include f_2(3)=7 and f_3(3)=9; for d=1 one has f_1(n) asymptotic to the n^2 scale.\n\n**Full solution or refutation.**\n\nNo general 2^o(d) bound for every fixed n>=4 was located.\n\n**What remains.**\n\nExtend the polynomial/subexponential n=3 behavior to arbitrary fixed subset size n.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1088, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1088\n Evidence used: Current open status, n=3 asymptotic, d=1 case, and original bibliography.\n- H. T. Croft, 9-point and 7-point configurations in 3-space, Proc. London Math. Soc. (3) 12 (1962), 400-424, DOI 10.1112/plms/s3-12.1.400. (primary): https://academic.oup.com/plms/article/s3-12/1/400/1437925\n Evidence used: Primary source for the three-dimensional low-n configuration result.\n- P. Erdős, On some problems of elementary and combinatorial geometry, Ann. Mat. Pura Appl. 103 (1975), 99-108, DOI 10.1007/BF02414146. (primary): https://users.renyi.hu/~p_erdos/1975-25.pdf\n Evidence used: Historical statement and reported Erdős-Straus exponential bound.\n\n**Review notes.** The phrase 'any two determined distances are distinct' is interpreted as all pairwise distances in the chosen n-point subset being mutually distinct. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2480, "problem_number": "EP-1089", "title": "Erdős Problem #1089", "statement": "Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $\\mathbb{R}^d$ determines at least $n$ many distinct distances. Estimate $g_d(n)$. In particular, does $ \\lim_{d\\to \\infty}\\frac{g_d(n)}{d^{n-1}} $ exist?", "background": "A question of Kelly. Erd\\H{o}s \\cite{Er75f} writes it is 'easy' to see that $g_d(n)\\gg d^{n-1}$. Erd\\H{o}s and Straus proved (in unpublished work mentioned in \\cite{Er75f}) that $ g_d(n) \\leq c^{d^{1-b_n}} $ for some constants $c>0$ and $b_n>0$.\nIt is trivial that $g_1(3)=4$, and easy to see that $g_2(3)=6$. Croft \\cite{Cr62} proved $g_3(3)=7$. The vertices of a $d$-dimensional cube demonstrate that $ g_d(d+1)>2^d. $ The function $g_d(n)$ is essentially the inverse of the function $f_d(n)$ considered in [1083] - with our definitions, $g_d(n)>m$ if and only if $f_d(m)\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** For every fixed n at least 2, binomial lower and upper bounds squeeze g_d(n)/d^(n-1) to 1/(n-1)!, resolving the emphasized asymptotic question.\n\n**Verified partial progress.**\n\n- Bannai, Bannai, and Stanton prove the absolute upper bound for an (n-1)-distance set, yielding g_d(n) at most binom(d+n-1,n-1)+1.\n- Constant-weight 0-1 vectors in an affine d-dimensional hyperplane give an (n-1)-distance set of size binom(d+1,n-1), yielding the matching leading lower bound.\n- The n=1 boundary case is separate: g_d(1)=2.\n\n**Full solution or refutation.**\n\nThe bounds binom(d+1,n-1)+1 <= g_d(n) <= binom(d+n-1,n-1)+1 have the same leading term d^(n-1)/(n-1)!, so the requested limit exists and equals 1/(n-1)! for fixed n at least 2.\n\n**What remains.**\n\nThe fixed-n leading asymptotic and limit are closed. Exact values and lower-order terms generally remain open; for n=1 the normalized limit is 2 rather than the n>=2 formula.\n\n**Sources checked.**\n\n- Eiichi Bannai, Etsuko Bannai, and Dennis Stanton, An upper bound for the cardinality of an s-distance subset in real Euclidean space, II, Combinatorica 3 (1983), 147-152. (primary): https://doi.org/10.1007/BF02579288\n Evidence used: Proves the absolute bound on finite Euclidean s-distance sets that yields the required upper estimate.\n- Tony Feng et al., Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdos Problems, arXiv:2601.22401 (2026). (primary): https://arxiv.org/abs/2601.22401\n Evidence used: Documents the Aletheia synthesis and the constant-weight construction yielding the matching asymptotic lower bound.\n- Thomas F. Bloom, Erdos Problem #1089, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1089\n Evidence used: Records both binomial bounds and the resulting limit 1/(n-1)! for n at least 2.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2481, "problem_number": "EP-1091", "title": "Erdős Problem #1091", "statement": "Let $G$ be a $K_4$-free graph with chromatic number $4$. Must $G$ contain an odd cycle with at least two diagonals?\nMore generally, is there some $f(r)\\to \\infty$ such that every graph with chromatic number $4$, in which every subgraph on $\\leq r$ vertices has chromatic number $\\leq 3$, contains an odd cycle with at least $f(r)$ diagonals?", "background": "Erd\\H{o}s originally asked about the existence of just one diagonal, which is true, and was proved by Larson \\cite{La79}. In fact Larson proved the following stronger conjecture of Bollob\\'{a}s and Erd\\H{o}s: if $G$ is a $K_4$-free graph containing no odd cycle with a diagonal then either $G$ is bipartite, or $G$ contains a cut vertex, or $G$ contains a vertex with degree $\\leq 2$.\nThe pentagonal wheel shows that three diagonals are not guaranteed.\nThe first question was solved in the affirmative by Voss \\cite{Vo82}.\nReferences\n\n\n[La79] Larson, Jean A., Some graphs with chromatic number three. J. Combin. Theory Ser. B (1979), 317--322.\n\n[Vo82] Voss, Heinz-J\"urgen, Graphs having circuits with at least two chords. J. Combin. Theory Ser. B (1982), 264--285.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Voss proved the first question affirmatively in 1982, while a 2026 preprint disproves the stronger unbounded-diagonals question using arbitrarily large K4-free 4-critical graphs whose cycles have at most ten diagonals.\n\n**Verified partial progress.**\n\n- Larson proved the earlier one-diagonal form and a stronger structural theorem.\n- Voss proved that every K4-free 4-chromatic graph contains an odd cycle with at least two diagonals.\n- The pentagonal wheel shows that three diagonals cannot be forced in the first question.\n\n**Full solution or refutation.**\n\nThe first part is yes; the second is no because a family with unbounded local 3-colourability radius retains a uniform ten-diagonal bound on every cycle.\n\n**What remains.**\n\nBoth stated questions are resolved. Sharpening the constant ten in the recent counterexample family is a separate extremal question.\n\n**Sources checked.**\n\n- Heinz-Jurgen Voss, Graphs having circuits with at least two chords, Journal of Combinatorial Theory Series B 32 (1982), 264-285. (primary): https://doi.org/10.1016/0095-8956(82)90004-1\n Evidence used: The published theorem proves the affirmative first question exactly.\n- Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, and Gregory Valiant, Short proofs in combinatorics, probability and number theory II, arXiv:2604.06609 (2026). (primary): https://arxiv.org/abs/2604.06609\n Evidence used: Constructs the 4-critical K4-free family with at most ten diagonals per cycle, refuting the second question.\n- Thomas F. Bloom, Erdos Problem #1091, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1091\n Evidence used: Records both the Voss theorem and the recent negative answer to the generalization.\n\n**Review notes.** The record contains two questions with opposite answers; solved means both have been resolved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2482, "problem_number": "EP-1092", "title": "Erdős Problem #1092", "statement": "Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic number $r$ and a graph with $\\leq f_r(m)$ edges, then $G$ has chromatic number $\\leq r+1$.\nIs it true that $f_2(n) \\gg n$? More generally, is $f_r(n)\\gg_r n$?", "background": "A conjecture of Erd\\H{o}s, Hajnal, and Szemer\\'{e}di. This seems to be closely related to, but distinct from, [744].\nTang notes in the comments that a construction of R\"{o}dl \\cite{Ro82} disproves the first question, so that $f_2(n)\not\\gg n$.\nReferences\n\n\n[Ro82] R\"{o}dl, Vojt\\vEch, Nearly bipartite graphs with large chromatic number. Combinatorica (1982), 377-383.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Rödl's 1982 construction gives arbitrarily high-chromatic graphs whose every m-vertex subgraph becomes bipartite after deleting at most epsilon m edges, strongly refuting every fixed positive linear lower bound.\n\n**Verified partial progress.**\n\n- The conjecture was posed by Erdős, Hajnal, and Szemerédi and is related to local almost-bipartiteness problems.\n- Rödl's result applies for arbitrarily small epsilon and arbitrarily large global chromatic number.\n\n**Full solution or refutation.**\n\nSince each local subgraph can be made bipartite with epsilon m deletions while the whole graph has chromatic number beyond r+1, no bound f_r(m) comparable to m can force global (r+1)-colourability.\n\n**What remains.**\n\nThe proposed linear order is false; determining sharper sublinear behavior is a separate problem.\n\n**Sources checked.**\n\n- Vojtech Rodl, Nearly bipartite graphs with large chromatic number, Combinatorica 2 (1982), 377-383. (primary): https://doi.org/10.1007/BF02579434\n Evidence used: Published construction directly implying the strong negative answer.\n- Thomas F. Bloom, Erdos Problem #1092, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1092\n Evidence used: Records DISPROVED and explains the deduction for every fixed r at least 2.\n\n**Review notes.** The imported background has OCR damage in the citation and not-much-greater-than symbol.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2483, "problem_number": "EP-1093", "title": "Erdős Problem #1093", "statement": "For $n\\geq 2k$ we define the deficiency of $\\binom{n}{k}$ as follows. If $\\binom{n}{k}$ is divisible by a prime $p\\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\\leq i1$?", "background": "A problem of Erd\\H{o}s, Lacampagne, and Selfridge \\cite{ELS88}, that was also asked in the 1986 problem session of West Coast Number Theory (as reported here).\nIn \\cite{ELS93} they prove that if the deficiency exists and is $\\geq 1$ then $n\\ll 2^k\\sqrt{k}$.\nThe following examples are either from \\cite{ELS88} or here. The following have deficiency $1$ (there are $58$ examples with $n\\leq 10^5$): $ \\binom{7}{3},\\binom{13}{4},\\binom{14}{4},\\binom{23}{5},\\binom{62}{6},\\binom{94}{10},\\binom{95}{10}. $ The examples which follow are the only known examples with deficiency $>1$. The following have deficiency $2$: $ \\binom{44}{8},\\binom{74}{10},\\binom{174}{12},\\binom{239}{14},\\binom{5179}{27},\\binom{8413}{28},\\binom{8414}{28},\\binom{96622}{42}. $ The following have deficiency $3$: $ \\binom{46}{10},\\binom{47}{10},\\binom{241}{16},\\binom{2105}{25},\\binom{1119}{27},\\binom{6459}{33}. $ The following has deficiency $4$: $ \\binom{47}{11}. $ The following has deficiency $9$: $ \\binom{284}{28}. $ See also [384] and [1094].\nBarreto in the comments has given a positive answer to the second question, conditional on two (strong) conjectures.\nReferences\n\n\n[ELS88] Erd\\H{o}s, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507--523.\n\n[ELS93] Erd\\H{o}s, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Both deficiency questions remain open, with a strong size bound, finite computation, and a conditional finiteness argument for deficiency greater than one.\n\n**Verified partial progress.**\n\n- Erdős, Lacampagne, and Selfridge prove that if defined deficiency is positive then n<<2^k sqrt(k).\n- Their 1993 work records 17 examples with deficiency greater than one and enumerates positive-deficiency cases in a substantial finite range.\n- A tracker discussion gives finiteness for deficiency greater than one conditional on two strong conjectures.\n\n**Full solution or refutation.**\n\nThere is no unconditional proof of infinitely many deficiency-one coefficients or only finitely many coefficients of deficiency greater than one.\n\n**What remains.**\n\nResolve either frequency question unconditionally and reconcile the historical finite-versus-infinite expectation for deficiency one.\n\n**Sources checked.**\n\n- P. Erdős, C. B. Lacampagne, and J. L. Selfridge, Prime factors of binomial coefficients and related problems, Acta Arith. 49 (1988), 507-523, DOI 10.4064/aa-49-5-507-523. (primary): https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/en/publishing-house/journals-and-series/acta-arithmetica/all/49/5/105629/prime-factors-of-binomial-coefficients-and-related-problems\n Evidence used: Original deficiency problem.\n- P. Erdős, C. B. Lacampagne, and J. L. Selfridge, Estimates of the least prime factor of a binomial coefficient, Math. Comp. 61 (1993), 215-224, DOI 10.1090/S0025-5718-1993-1199990-6. (primary): https://www.jstor.org/stable/2152948\n Evidence used: Positive-deficiency size bound, enumeration, and historical expectation.\n- Thomas F. Bloom, Erdős Problem #1093, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1093\n Evidence used: Current open status, corrected example list, and conditional forum result.\n\n**Review notes.** The exact statement was preserved. The 1993 abstract says deficiency one appeared finite, while this source asks whether infinitely many exist. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2484, "problem_number": "EP-1094", "title": "Erdős Problem #1094", "statement": "For all $n\\geq 2k$ the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$, with only finitely many exceptions.", "background": "A stronger form of [384] that appears in a paper of Erd\\H{o}s, Lacampagne, and Selfridge \\cite{ELS88}. Erd\\H{o}s observed that the least prime factor is always $\\leq n/k$ provided $n$ is sufficiently large depending on $k$. Selfridge \\cite{Se77} further conjectured that this always happens if $n\\geq k^2-1$, except $\\binom{62}{6}$.\nThe threshold $g(k)$ below which $\\binom{n}{k}$ is guaranteed to be divisible by a prime $\\leq k$ is the subject of [1095].\nMore precisely, in \\cite{ELS88} they conjecture that if $n\\geq 2k$ then the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$ with the following $14$ exceptions: $ \\binom{7}{3},\\binom{13}{4},\\binom{23}{5},\\binom{14}{4},\\binom{44}{8},\\binom{46}{10},\\binom{47}{10}, $ $ \\binom{47}{11},\\binom{62}{6},\\binom{74}{10},\\binom{94}{10},\\binom{95}{10},\\binom{241}{16},\\binom{284}{28}. $ They also suggest the stronger conjecture that, with a finite number of exceptions, the least prime factor is $\\leq \\max(n/k,\\sqrt{k})$, or perhaps even $\\leq \\max(n/k,O(\\log k))$. Indeed, in \\cite{ELS93} they provide some further computational evidence, and point out it is consistent with what they know that in fact this holds with $\\leq \\max(n/k,13)$, with only $12$ exceptions.\nDiscussed in problem B31 and B33 of Guy's collection \\cite{Gu04} - there Guy credits Selfridge with the conjecture that if $n> 17.125k$ then $\\binom{n}{k}$ has a prime factor $p\\leq n/k$.\nThis is related to [1093], in that the only counterexamples to this conjecture can occur from $\\binom{n}{k}$ with deficiency $\\geq 1$.\nThere is an interesting discussion about this problem on MathOverflow.\nReferences\n\n\n[ELS88] Erd\\H{o}s, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507--523.\n\n[ELS93] Erd\\H{o}s, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Se77] J. L. Selfridge, Some problems on the prime factors of consecutive integers. Notices Amer. Math. Soc. (1977), A456-457.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The least-prime-factor bound with finitely many exceptions remains open despite fixed-k asymptotics and extensive computation supporting even stronger bounds.\n\n**Verified partial progress.**\n\n- Erdős proves the least prime factor is at most n/k when n is sufficiently large depending on fixed k.\n- Erdős, Lacampagne, and Selfridge list 14 exceptions to the displayed bound and give evidence for stronger thresholds.\n- Their 1993 paper supports the stronger candidate max(n/k,29) computationally.\n\n**Full solution or refutation.**\n\nThe known evidence does not prove there are only finitely many exceptions uniformly over k.\n\n**What remains.**\n\nEstablish a uniform least-prime-factor bound of max(n/k,k), or any of the proposed stronger variants, outside a finite set.\n\n**Sources checked.**\n\n- P. Erdős, C. B. Lacampagne, and J. L. Selfridge, Prime factors of binomial coefficients and related problems, Acta Arith. 49 (1988), 507-523, DOI 10.4064/aa-49-5-507-523. (primary): https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/en/publishing-house/journals-and-series/acta-arithmetica/all/49/5/105629/prime-factors-of-binomial-coefficients-and-related-problems\n Evidence used: Primary conjecture, exceptions, and stronger proposed bounds.\n- P. Erdős, C. B. Lacampagne, and J. L. Selfridge, Estimates of the least prime factor of a binomial coefficient, Math. Comp. 61 (1993), 215-224. (primary): https://www.jstor.org/stable/2152948\n Evidence used: Further estimates and computational evidence.\n- Thomas F. Bloom, Erdős Problem #1094, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1094\n Evidence used: Current open status and relationship to EP-1093 and EP-1095.\n\n**Review notes.** The exact source statement was preserved. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2485, "problem_number": "EP-1095", "title": "Erdős Problem #1095", "statement": "Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\\binom{n}{k}$ are $>k$. Estimate $g(k)$.", "background": "A question of Ecklund, Erd\\H{o}s, and Selfridge \\cite{EES74}, who proved $ k^{1+c}0$, and conjectured $g(k)0$, due to Konyagin \\cite{Ko99b}.\nErd\\H{o}s, Lacampagne, and Selfridge \\cite{ELS93} write 'it is clear to every right-thinking person' that $g(k)\\geq\\exp(c\\frac{k}{\\log k})$ for some constant $c>0$.\nSorenson, Sorenson, and Webster \\cite{SSW20} give heuristic evidence that $ \\log g(k) \\asymp \\frac{k}{\\log k}. $ See also [1094].\nReferences\n\n\n[EES74] Ecklund, Jr., E. F. and Erd\\H{o}s, P. and Selfridge, J. L., A new function associated with the prime factors of\n{$(\\sp{n}\\sb{k})$}. Math. Comp. (1974), 647--649.\n\n[ELS93] Erd\\H{o}s, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224.\n\n[GrRa96] Granville, Andrew and Ramar\\'{e}, Olivier, Explicit bounds on exponential sums and the scarcity of\nsquarefree binomial coefficients. Mathematika (1996), 73--107.\n\n[Ko99b] Konyagin, S. V., Estimates of the least prime factor of a binomial coefficient. Mathematika (1999), 41--55.\n\n[SSW20] Sorenson, Brianna and Sorenson, Jonathan and Webster,\nJonathan, An algorithm and estimates for the {E}rd\\H{o}s-{S}elfridge\nfunction. (2020), 371--385.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Erdős-Selfridge function remains between exp(c(log k)^2) and exp((1+o(1))k); faster exp(Theta(k/log k)) growth is supported only heuristically.\n\n**Verified partial progress.**\n\n- Konyagin proves g(k)>=exp(c(log k)^2) for an absolute c>0.\n- The classical construction gives g(k)<=exp((1+o(1))k).\n- Sorenson, Sorenson, and Webster compute g(k) through k=375 and prove log of their approximating function is Theta(k/log k); transfer to g(k) uses a uniform-distribution heuristic.\n\n**Full solution or refutation.**\n\nNo unconditional estimate near the predicted exp(Theta(k/log k)) scale is known.\n\n**What remains.**\n\nStrengthen the unconditional lower bound toward exp(c k/log k), narrow the upper bound, and settle the lcm and successive-ratio conjectures.\n\n**Sources checked.**\n\n- S. V. Konyagin, Estimates of the least prime factor of a binomial coefficient, Mathematika 46 (1999), 41-55, DOI 10.1112/S0025579300007555. (primary): https://doi.org/10.1112/S0025579300007555\n Evidence used: Primary source for the exp(c(log k)^2) lower bound.\n- B. Sorenson, J. Sorenson, and J. Webster, An algorithm and estimates for the Erdős-Selfridge function, Open Book Series 4 (2020), 371-385, DOI 10.2140/obs.2020.4.371. (primary): https://msp.org/obs/2020/4-1/obs-v4-n1-p23-s.pdf\n Evidence used: Algorithms, computed values, approximating function, unconditional density theorem, and clearly labeled heuristic transfer.\n- Thomas F. Bloom, Erdős Problem #1095 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/1095\n Evidence used: Current open status, known bounds, formalization note, and interpretation of the 2020 heuristic.\n\n**Review notes.** The exact statement was preserved. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2486, "problem_number": "EP-1096", "title": "Erdős Problem #1096", "statement": "Let $10$ is sufficiently small, $x_{k+1}-x_k \\to 0$?", "background": "A problem of Erd\\H{o}s and Jo\\'{o} posed in the 1991 problem session of Great Western Number Theory.\nThey speculate that the threshold may be $q_0$, where $q_0\\approx 1.3247$ is the real root of $x^3=x+1$, and is the smallest Pisot-Vijayaraghavan number.\nIn \\cite{EJK90} Erd\\H{o}, Jo\\'{o}, and Komornik prove that any Pisot-Vijayaraghavan number cannot have this property, and also prove that, for any $10 $ for all $m\\geq 1$, where $x_k^m$ is the set of those numbers which can be written as a finite sum $\\sum_{n\\geq 0}c_nq^n$ for some $c_n\\in \\{0,\\ldots,m\\}$ (so that the sequence in the question is $x_k^1$). Erd\\H{o}s, Jo\\'{o}, and Schnitzer \\cite{EJS96} improved this to show that, if $10. $ \nReferences\n\n\n[Bu96] Bugeaud, Y., On a property of {P}isot numbers and related questions. Acta Math. Hungar. (1996), 33--39.\n\n[EJK90] Erd\\H{o}s, P\\'al and Jo\\'o, Istv\\'an and Komornik, Vilmos, Characterization of the unique expansions\n{$1=\\sum^\\infty_{i=1}q^{-n_i}$} and related problems. Bull. Soc. Math. France (1990), 377--390.\n\n[EJS96] Erd\\H{o}s, P. and Jo\\'o, I. and Schnitzer, F. J., On {P}isot numbers. Ann. Univ. Sci. Budapest. E\"otv\"os Sect. Math. (1996), 95--99.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Erdős, Joó, and Komornik proved in 1998 that the consecutive gaps tend to zero for every 1", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2487, "problem_number": "EP-1097", "title": "Erdős Problem #1097", "statement": "Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? Are there always $O(n^{3/2})$ many such $d$?", "background": "A problem Erd\\H{o}s posed in the 1989 problem session of Great Western Number Theory.\nHe states that Erd\\H{o}s and Ruzsa gave an explicit construction which achieved $n^{1+c}$ for some $c>0$, and Erd\\H{o}s and Spencer gave a probabilistic proof which achieved $n^{3/2}$, and speculated this may be the best possible.\nIn the comment section, Chan has noticed that this problem is exactly equivalent to a sums-differences question of Bourgain \\cite{Bo99}, introduced as an arithmetic path towards the Kakeya conjecture: find the smallest $c\\in [1,2]$ such that, for any finite sets of integers $A$ and $B$ and $G\\subseteq A\\times B$ we have $ \\lvert A\\overset{G}{-}B\\rvert \\ll \\max(\\lvert A\\rvert,\\lvert B\\rvert, \\lvert A\\overset{G}{+}B\\rvert)^c $ (where, for example, $A\\overset{G}{+}B$ denotes the set of $a+b$ with $(a,b)\\in G$).\nThis is equivalent in the sense that the greatest exponent $c$ achievable for the main problem here is equal to the smallest constant achievable for the sums-differences question. The current best bounds known are thus $ 1.77898\\cdots \\leq c \\leq 11/6 \\approx 1.833. $ The upper bound is due to Katz and Tao \\cite{KaTa99}. The lower bound is due to Lemm \\cite{Le15} (with a very small improvement found by AlphaEvolve \\cite{GGTW25}).\nReferences\n\n\n[Bo99] Bourgain, J., On the dimension of {K}akeya sets and related maximal\ninequalities. Geom. Funct. Anal. (1999), 256--282.\n\n[GGTW25] B. Georgiev, J. G\\'{o}mez-Serrano, T. Tao, and A. Wagner, Mathematical exploration and discovery at scale. arXiv:2511.02864 (2025).\n\n[KaTa99] Katz, Nets Hawk and Tao, Terence, Bounds on arithmetic projections, and applications to the\n{K}akeya conjecture. Math. Res. Lett. (1999), 625--630.\n\n[Le15] Lemm, Marius, New counterexamples for sums-differences. Proc. Amer. Math. Soc. (2015), 3863--3868.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The proposed O(n^(3/2)) bound is false; the correct extremal exponent is open between 1.77898... and 11/6.\n\n**Verified partial progress.**\n\n- The progression-difference exponent is equivalent to Bourgain's restricted sums-differences exponent.\n- Katz and Tao give the upper exponent 11/6.\n- Lemm's counterexamples, with a small AlphaEvolve improvement, give a lower exponent 1.77898..., strictly exceeding 3/2 and therefore refuting the stated subquestion.\n\n**Full solution or refutation.**\n\nOnly the O(n^(3/2)) subquestion is resolved negatively; the order of the maximum number of differences is not determined.\n\n**What remains.**\n\nClose the exponent gap between 1.77898... and 11/6 in the equivalent restricted sums-differences problem.\n\n**Sources checked.**\n\n- N. H. Katz and T. Tao, Bounds on arithmetic projections, and applications to the Kakeya conjecture, Math. Res. Lett. 6 (1999), 625-630. (primary): https://math.ucla.edu/~tao/kakeya.html\n Evidence used: Primary source for the 11/6 upper exponent in the equivalent sums-differences problem.\n- M. Lemm, New counterexamples for sums-differences, Proc. Amer. Math. Soc. 143 (2015), 3863-3868. (primary): https://arxiv.org/abs/1404.3745\n Evidence used: Primary counterexample construction underlying the lower exponent.\n- B. Georgiev, J. Gómez-Serrano, T. Tao, and A. Z. Wagner, Mathematical exploration and discovery at scale, arXiv:2511.02864 (2025). (primary): https://arxiv.org/abs/2511.02864\n Evidence used: Primary source for the small improvement to the construction exponent.\n- Thomas F. Bloom, Erdős Problem #1097 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1097\n Evidence used: Current status, exact equivalence argument, numerical exponents, and explicit negative resolution of the second question.\n\n**Review notes.** The record is classified partial rather than disproved because its main order-estimation question remains open. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2488, "problem_number": "EP-1100", "title": "Erdős Problem #1100", "statement": "If $1=d_1<\\cdots0$ and sufficiently large $x$, $ \\max_{n \\exp((\\log\\log x)^{2-\\epsilon}). $ Erd\\H{o}s and Simonovits (see \\cite{Er81h}) proved $ (2^{1/2}+o(1))^k < g(k) < (2-c)^k $ for some constant $c>0$.\nReferences\n\n\n[Er81h] Erd\\H{o}s, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\n\n[ErHa78] Erd\\H{o}s, P. and Hall, R. R., On some unconventional problems on the divisors of integers. J. Austral. Math. Soc. Ser. A (1978), 479--485.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** All three consecutive-coprime-divisor questions remain open; classical work gives a large maximal-order lower bound and an announced exponential window for the squarefree extremal function.\n\n**Verified partial progress.**\n\n- Erdős and Hall prove max_(nexp((log log x)^(2-epsilon)) for every epsilon>0 and large x.\n- Erdős's 1981 survey records (sqrt(2)+o(1))^k0, attributed to Erdős and Simonovits.\n- The tracker discussion notes that proofs of the exponential g(k) bounds are not present in the cited survey, so their original source remains to be identified.\n\n**Full solution or refutation.**\n\nNo almost-all divergence, universal sub-subexponential bound, or asymptotic for g(k) has been verified.\n\n**What remains.**\n\nProve any of the three assertions and locate or reconstruct a primary proof of the recorded Erdős-Simonovits exponential window.\n\n**Sources checked.**\n\n- P. Erdős and R. R. Hall, On some unconventional problems on the divisors of integers, J. Austral. Math. Soc. Ser. A 25 (1978), 479-485, DOI 10.1017/S1446788700021455. (primary): https://www.renyi.hu/~p_erdos/1978-26.pdf\n Evidence used: Primary source for the maximal-order construction and original divisor questions.\n- P. Erdős, Some problems and results on additive and multiplicative number theory, Lecture Notes in Math. 899 (1981), 171-182. (primary): https://renyi.hu/~p_erdos/1981-33.pdf\n Evidence used: Records the Erdős-Simonovits g(k) bounds, though not their proof.\n- Thomas F. Bloom, Erdős Problem #1100 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/forum/thread/1100\n Evidence used: Current open status, provenance caveat, and explicit gaps in the recent polynomial community claim.\n\n**Review notes.** A community claim g(k)=binomial(k,2)+1 retains a heuristic upper-bound step and contradicts the recorded exponential lower bound, so it was not accepted. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2489, "problem_number": "EP-1101", "title": "Erdős Problem #1101", "statement": "If $u=\\{u_10$, if $x$ is sufficiently large then $ \\max_{a_k (1+o(1))t_x \\prod_{i}\\left(1-\\frac{1}{u_i}\\right)^{-1}. $ The strong form of [208] is asking whether if $u_i=p_i^2$, the sequence of prime squares, is good.\nReferences\n\n\n[Er81h] Erd\\H{o}s, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Good sequences are known to exist, but no example with polynomial or subexponential growth was located; both requested growth questions remain open.\n\n**Verified partial progress.**\n\n- Erdős constructs a good sequence consisting of primes.\n- An elementary sieve argument gives the matching asymptotic lower scale (1+o(1)) t_x times the reciprocal Euler product for every admissible sequence.\n- The prime-square specialization is the strong form of EP-208.\n\n**Full solution or refutation.**\n\nThe known existence construction does not satisfy either requested upper-growth constraint.\n\n**What remains.**\n\nConstruct a good sequence with u_n<=exp(o(n)), or disprove its existence; separately settle the stronger polynomial-growth question.\n\n**Sources checked.**\n\n- P. Erdős, Some problems and results on additive and multiplicative number theory, Lecture Notes in Math. 899 (1981), 171-182. (primary): https://renyi.hu/~p_erdos/1981-33.pdf\n Evidence used: Original definition, existence construction, and conjectured answers.\n- Thomas F. Bloom, Erdős Problem #1101 revision history, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/history/1101\n Evidence used: Current open formulation, sieve lower bound, and correct rendering of i not equal to j.\n\n**Review notes.** The imported statement corrupts the TeX escape neq into a newline followed by 'eq'; the maintained source has i\\neq j. The exact imported text is preserved in report.md. The background has the batch-wide stray serialized tail.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2490, "problem_number": "EP-1103", "title": "Erdős Problem #1103", "statement": "Let $A$ be an infinite sequence of integers such that every $n\\in A+A$ is squarefree. How fast must $A$ grow?", "background": "Erd\\H{o}s notes there exists such a sequence which grows exponentially, but does not expect such a sequence of polynomial growth.\nIn \\cite{Er81h} he asked whether there is an infinite sequence of integers $A$ such that, for every $a\\in A$ and prime $p$, if $ a\\equiv t\\pmod{p^2} $ then $1\\leq t 0.24j^{4/3}$ for all $j$, and further that there exists such a sequence (furthermore with squarefree terms) such that $ a_j < \\exp(5j/\\log j) $ for all large $j$. A superior lower bound of $a_j \\gg j^{15/11-o(1)}$ had earlier been found by Konyagin \\cite{Ko04} when considering the finite case [1109].\nThey also obtain further results for the generalisation from squarefree to $k$-free integers, and also replacing $A+A$ with $A\\cup (A+A)\\cup(A+A+A)$.\nSee also [1109] for the finite analogue of this problem.\nReferences\n\n\n[Er81h] Erd\\H{o}s, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\n\n[Ko04] Konyagin, S. V., Problems of the set of square-free numbers. Izv. Ross. Akad. Nauk Ser. Mat. (2004), 63--90.\n\n[vDTa25] W. van Doorn and T. Tao, Growth rates of sequences governed by the squarefree properties of its translates. arXiv:2512.01087 (2025).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Sequences with squarefree pairwise sums are known between a j^(15/11-o(1)) lower-growth obstruction and an exp(5j/log j) construction; polynomial growth remains unresolved.\n\n**Verified partial progress.**\n\n- Konyagin's finite extremal theorem implies a_j>>j^(15/11-o(1)).\n- Van Doorn and Tao prove the explicit all-j bound a_j>0.24j^(4/3).\n- Van Doorn and Tao construct a sequence of squarefree terms with all pairwise sums squarefree and a_j", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2491, "problem_number": "EP-1104", "title": "Erdős Problem #1104", "statement": "Let $f(n)$ be the maximum possible chromatic number of a triangle-free graph on $n$ vertices. Estimate $f(n)$.", "background": "The best bounds available are $ (1-o(1))(n/\\log n)^{1/2}\\leq f(n) \\leq (2+o(1))(n/\\log n)^{1/2}. $ The upper bound is due to Davies and Illingworth \\cite{DaIl22}, the lower bound follows from a construction of Hefty, Horn, King, and Pfender \\cite{HHKP25}.\nOne can ask a similar question for the maximum possible chromatic number of a triangle-free graph on $m$ edges. Let this be $g(m)$. Davies and Illingworth \\cite{DaIl22} prove $ g(m) \\leq (3^{5/3}+o(1))\\left(\\frac{m}{(\\log m)^2}\\right)^{1/3}. $ Kim \\cite{Ki95} gave a construction which implies $g(m) \\gg (m/(\\log m)^2)^{1/3}$.\nThe function $f(n)$ is the inverse to the function $h_3(k)$ considered in [1013].\nA generalisation of $f(n)$ is considered in [920].\nReferences\n\n\n[DaIl22] Davies, Ewan and Illingworth, Freddie, The {$\\chi$}-{R}amsey problem for triangle-free graphs. SIAM J. Discrete Math. (2022), 1124--1134.\n\n[HHKP25] Z. Hefty, P. Horn, D. King, and F. Pfender, Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (2025).\n\n[Ki95] Kim, J. H., The Ramsey number $R(3,t)$ has order of magnitude $t^2/\\log t$. Random Structures and Algorithms (1995), 173-207.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The maximum chromatic number of a triangle-free n-vertex graph is pinned within a factor two in the leading constant.\n\n**Verified partial progress.**\n\n- Davies--Illingworth give the upper constant 2+o(1).\n- Hefty--Horn--King--Pfender give the lower constant 1-o(1).\n\n**Full solution or refutation.**\n\nThe exact leading constant is unknown.\n\n**What remains.**\n\nClose the factor-two constant gap.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1104, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1104\n Evidence used: Current status and two-sided bounds.\n\n**Review notes.** The tracker flags this as essentially duplicate to EP-1013.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2492, "problem_number": "EP-1105", "title": "Erdős Problem #1105", "statement": "The anti-Ramsey number $\\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all edges have different colours).\nLet $C_k$ be the cycle on $k$ vertices. Is it true that $ \\mathrm{AR}(n,C_k)=\\left(\\frac{k-2}{2}+\\frac{1}{k-1}\\right)n+O(1)? $ Let $P_k$ be the path on $k$ vertices and $\\ell=\\lfloor\\frac{k-1}{2}\\rfloor$. If $n\\geq k\\geq 5$ then is $\\mathrm{AR}(n,P_k)$ equal to $ \\max\\left(\\binom{k-2}{2}+1, \\binom{\\ell-1}{2}+(\\ell-1)(n-\\ell+1)+\\epsilon\\right) $ where $\\epsilon=1$ if $k$ is odd and $\\epsilon=2$ otherwise?", "background": "A conjecture of Erd\\H{o}s, Simonovits, and S\\'{o}s \\cite{ESS75}, who gave a simple proof that $\\mathrm{AR}(n,C_3)=n-1$. In this paper they announced proofs of the claimed formula for $\\mathrm{AR}(n,P_k)$ for $n\\geq \\frac{5}{4}k+C$ for some large constant $C$, and also for all $n\\geq k$ if $k$ is sufficiently large, but these never appeared.\nSimonovits and S\\'{o}s \\cite{SiSo84} published a proof that the claimed formula for $\\mathrm{AR}(n,P_k)$ is true for $n\\geq ck^2$ for some constant $c>0$.\nA proof of the formula for $\\mathrm{AR}(n,P_k)$ for all $n\\geq k\\geq 5$ has been announced by Yuan \\cite{Yu21}\nReferences\n\n\n[ESS75] Erd\\H{o}s, P. and Simonovits, M. and S\\'os, V. T., Anti-{R}amsey theorems. (1975), 633--643.\n\n[SiSo84] Simonovits, Mikl\\'os and S\\'os, Vera T., On restricted colourings of {$K_n$}. Combinatorica (1984), 101--110.\n\n[Yu21] L.-T. Yuan, The anti-Ramsey number for paths. arXiv:2102.00807 (2021).\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The cycle formula follows from a 2005 peer-reviewed exact theorem, and Yuan's 2021 preprint proves exactly the displayed path formula for every n>=k>=5.\n\n**Verified partial progress.**\n\n- Erdős, Simonovits, and Sós proved the triangle case and announced broad path ranges.\n- Simonovits and Sós proved the path formula for n at least a constant times k squared.\n- Montellano-Ballesteros and Neumann-Lara settled all cycle lengths exactly.\n\n**Full solution or refutation.**\n\nThe two cited theorems cover the cycle asymptotic and the full exact path formula, respectively.\n\n**What remains.**\n\nBoth statements are resolved; conventional publication or additional review of the path preprint would strengthen the record.\n\n**Sources checked.**\n\n- Juan Jose Montellano-Ballesteros and Victor Neumann-Lara, An Anti-Ramsey Theorem on Cycles, Graphs and Combinatorics 21 (2005), 343-354. (primary): https://doi.org/10.1007/s00373-005-0619-y\n Evidence used: Peer-reviewed exact cycle theorem implying the first displayed asymptotic.\n- Long-Tu Yuan, Anti-Ramsey numbers for paths, arXiv:2102.00807 (2021). (primary): https://arxiv.org/abs/2102.00807\n Evidence used: Theorem 1 states exactly the path formula in the dataset for all n>=k>=5.\n- Thomas F. Bloom, Erdos Problem #1105, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1105\n Evidence used: Records PROVED and links both resolution sources.\n\n**Review notes.** The Formal Conjectures file contains sorry placeholders and was not treated as proof.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2493, "problem_number": "EP-1106", "title": "Erdős Problem #1106", "statement": "Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of $ \\prod_{1\\leq k\\leq n}p(k). $ Does $F(n)\\to \\infty$ with $n$? Is $F(n)>n$ for all sufficiently large $n$?", "background": "Asked by Erd\\H{o}s at Oberwolfach in 1986. Schinzel noted in the Oberwolfach problem book that $F(n)\\to \\infty$ follows from the asymptotic formula for $p(n)$ and a result of Tijdeman \\cite{Ti73}. This is not obvious; details are given in a paper of Erd\\H{o}s and Ivi\\'{c} (see page 69 of \\cite{ErIv90}).\nSchinzel and Wirsing \\cite{ScWi87} have proved $F(n) \\gg \\log n$.\nOno \\cite{On00} has proved that every prime divides $p(n)$ for some $n\\geq 1$ (indeed this holds, for any fixed prime, for a positive density set of $n$).\nReferences\n\n\n[ErIv90] Erd\\H{o}s, Paul and Ivi\\'c, Aleksandar, The distribution of values of a certain class of arithmetic\nfunctions at consecutive integers. (1990), 45--91.\n\n[On00] Ono, Ken, Distribution of the partition function modulo {$m$}. Ann. of Math. (2) (2000), 293--307.\n\n[ScWi87] Schinzel, A. and Wirsing, E., Multiplicative properties of the partition function. Proc. Indian Acad. Sci. Math. Sci. (1987), 297--303.\n\n[Ti73] Tijdeman, R., On integers with many small prime factors. Compositio Math. (1973), 319--330.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** F(n) tends to infinity and even grows at least logarithmically, but the proposed eventual F(n)>n bound is open.\n\n**Verified partial progress.**\n\n- Schinzel's observation, detailed by Erdős--Ivić, proves F(n) tends to infinity.\n- Schinzel--Wirsing prove F(n) is bounded below by a constant times log n.\n- Ono proves every fixed prime divides p(n) for a positive-density set of n.\n\n**Full solution or refutation.**\n\nOnly the stronger quantitative question remains.\n\n**What remains.**\n\nProve F(n)>n eventually or determine its correct order.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1106, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1106\n Evidence used: Current open label and all listed partial results.\n\n**Review notes.** The two questions have different statuses; only the first is resolved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2494, "problem_number": "EP-1107", "title": "Erdős Problem #1107", "statement": "Let $r\\geq 2$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\\mid n$. Is every large integer the sum of at most $r+1$ many $r$-powerful numbers?", "background": "Given in the 1986 Oberwolfach problem book as a problem of Erd\\H{o}s and Ivi\\'{c}.\nThis is true when $r=2$, as proved by Heath-Brown \\cite{He88} (see [941]).\nSee [940] for the problem of which integers are the sum of at most $r$ many $r$-powerful numbers.\nReferences\n\n\n[He88] Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. (1988), 137--163.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every sufficiently large integer is a sum of three 2-powerful numbers, while the stated r-powerful generalisation is open for r>=3.\n\n**Verified partial progress.**\n\n- Heath-Brown proves the r=2 case.\n\n**Full solution or refutation.**\n\nNo general theorem for each r was located.\n\n**What remains.**\n\nExtend the r=2 argument or find a counterexample for r>=3.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1107, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1107\n Evidence used: Current open status and Heath-Brown special case.\n\n**Review notes.** No computation used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2495, "problem_number": "EP-1108", "title": "Erdős Problem #1108", "statement": "Let $ A = \\left\\{ \\sum_{n\\in S}n! : S\\subset \\mathbb{N}\\textrm{ finite}\\right\\}. $ If $k\\geq 2$, then does $A$ contain only finitely many $k$th powers? Does it contain only finitely many powerful numbers?", "background": "Asked by Erd\\H{o}s at Oberwolfach in 1988. It is open even whether there are infinitely many squares of the form $1+n!$ (see [398]).\nThis was motivated in part by a problem of Mahler which he discussed with Erd\\H{o}s a few days before his death in 1988: if $k\\geq 5$ and $ A_k= \\left\\{ \\sum_{n\\in S}k^n : S\\subset \\mathbb{N}\\textrm{ finite}\\right\\} $ then does $A_k$ contain only finitely many squares? Mahler showed that there are infinitely many squares in $A_k$ for $k\\leq 4$, and found only one square for $k\\geq 5$, namely $ 1+7+7^2+7^3=400. $ Brindza and Erd\\H{o}s \\cite{BrEr91} proved that, for any $r$, if $n_1!+\\cdots+n_r!$ is powerful then $n_1\\ll_r 1$.\nReferences\n\n\n[BrEr91] Brindza, B. and Erd\\H{o}s, P., On some {D}iophantine problems involving powers and\nfactorials. J. Austral. Math. Soc. Ser. A (1991), 1--7.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Fixed-length factorial sums have strong finiteness results for powerful values, but the unrestricted finite-subset problem remains open even for a simple square family.\n\n**Verified partial progress.**\n\n- Brindza--Erdős prove that for fixed r, a powerful sum of r factorials has bounded smallest index.\n- It remains open whether 1+n! is a square infinitely often.\n\n**Full solution or refutation.**\n\nThe fixed-r theorem does not control arbitrarily many factorial summands.\n\n**What remains.**\n\nProve finiteness for all finite factorial subsets, or exhibit an infinite family.\n\n**Sources checked.**\n\n- B. Brindza and P. Erdős, On some Diophantine problems involving powers and factorials, J. Austral. Math. Soc. A 51 (1991), 1--7. (primary): https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/on-some-diophantine-problems-involving-powers-and-factorials/6600AEB6CBA4CE2449231E9796E82BDC\n Evidence used: Fixed-length powerful factorial-sum theorem.\n- Erdős Problems LaTeX entry #1108, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/latex/1108\n Evidence used: Current open status and residual square case.\n\n**Review notes.** No source statement altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2496, "problem_number": "EP-1109", "title": "Erdős Problem #1109", "statement": "Let $f(N)$ be the size of the largest subset $A\\subseteq \\{1,\\ldots,N\\}$ such that every $n\\in A+A$ is squarefree. Estimate $f(N)$. In particular, is it true that $f(N)\\leq N^{o(1)}$, or even $f(N) \\leq (\\log N)^{O(1)}$?", "background": "First studied by Erd\\H{o}s and S\\'{a}rk\"{o}zy \\cite{ErSa87}, who proved $ \\log N \\ll f(N) \\ll N^{3/4}\\log N, $ and guessed the lower bound is nearer the truth. S\\'{a}rk\"{o}zy \\cite{Sa92c} extended this to consider the case of $A+B$ and also looking for sumsets which are $k$-power-free.\nGyarmati \\cite{Gy01} gave an alternative proof of $f(N)\\gg \\log N$, and also gave new bounds for the case of $A+B$. Konyagin \\cite{Ko04} improved this to $ \\log\\log N(\\log N)^2\\ll f(N) \\ll N^{11/15+o(1)}. $ The infinite analogue of this problem is [1103]. (In particular upper bounds for this $f(N)$ directly imply lower bounds for the size of the $a_j$ considered there.)\nReferences\n\n\n[ErSa87] Erd\\H{o}s, P. and S\\'ark\"ozy, A., On divisibility properties of integers of the form {$a+a'$}. Acta Math. Hungar. (1987), 117--122.\n\n[Gy01] Gyarmati, Katalin, On divisibility properties of integers of the form {$ab+1$}. Period. Math. Hungar. (2001), 71--79.\n\n[Ko04] Konyagin, S. V., Problems of the set of square-free numbers. Izv. Ross. Akad. Nauk Ser. Mat. (2004), 63--90.\n\n[Sa92c] S\\'ark\"ozy, G. N., On a problem of {P}. {E}rd\\H{o}s. Acta Math. Hungar. (1992), 271--282.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Known bounds range from about log-log N times log-squared N to N^(11/15+o(1)); neither proposed subpower bound is established.\n\n**Verified partial progress.**\n\n- Konyagin improved both classical bounds to the currently recorded estimates.\n\n**Full solution or refutation.**\n\nThe growth remains far from classified.\n\n**What remains.**\n\nObtain a subpower upper bound or substantially improve it.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1109, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1109\n Evidence used: Current open status and Konyagin bounds.\n\n**Review notes.** Recent finite computations in forum comments are not used as asymptotic evidence.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2497, "problem_number": "EP-1110", "title": "Erdős Problem #1110", "statement": "Let $p>q\\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other.\nIf $\\{p,q\\}\neq \\{2,3\\}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers?", "background": "A problem of Erd\\H{o}s and Lewin \\cite{ErLe96}, who proved that there are finitely many non-representable numbers if and only if $\\{p,q\\}=\\{2,3\\}$.\nIndeed, in \\cite{Er92b} Erd\\H{o}s wrote 'last year I made the following silly conjecture': every integer $n$ can be written as the sum of distinct integers of the form $2^k3^l$, none of which divide any other. He wrote 'I mistakenly thought that this was a nice and difficult conjecture but Jansen and several others found a simple proof by induction.'\nThis simple proof is as follows: one proves the stronger fact that such a representation always exists, and moreover if $n$ is even then all the summands can be taken to be even: if $n=2m$ we are done applying the inductive hypothesis to $m$. Otherwise if $n$ is odd then let $3^k$ be the largest power of $3$ which is $\\leq n$ and apply the inductive hypothesis to $n-3^k$ (which is even).\nYu and Chen \\cite{YuCh22} prove that the set of non-representable numbers has density zero whenever $q>3$ or $q=3$ and $p>6$ or $q=2$ and $p>10$. They also prove that there are infinitely many coprime non-representable numbers if $q>3$ or $q=3$ and $p\neq 5$ or $q=2$ and $p\not\\in \\{3,5,9\\}$.\nErd\\H{o}s and Lewin \\cite{ErLe96} also asked whether all large integers $n$ can be written as a sum of $2^k3^l$, none of which divide another, each of which is $>f(n)$ for some $f(n)\\to \\infty$. Let $f(n)$ be the fastest growing such $f(n)$. Yu and Chen \\cite{YuCh22} proved $ \\frac{n}{(\\log n)^{\\log_23}}\\ll f(n) \\ll \\frac{n}{\\log n}. $ Yang and Zhao \\cite{YaZh25} improved the lower bound to $f(n)\\gg n/\\log n$.\nThe case of three powers is the subject of [123], and see also [845] for more on the case $\\{p,q\\}=\\{2,3\\}$. The problem [246] addresses the topic without the non-divisibility condition.\nReferences\n\n\n[Er92b] Erd\\H{o}s, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.\n\n[ErLe96] Erd\\H{o}s, P. and Lewin, Mordechai, $d$-complete sequences of integers. Math. Comp. (1996), 837-840.\n\n[YaZh25] Yang, Quan-Hui and Zhao, Lilu, A conjecture of {Y}u and {C}hen related to the {E}rd\\H\nos-{L}ewin theorem. Acta Arith. (2025), 277--286.\n\n[YuCh22] Yu, Wang-Xing and Chen, Yong-Gao, On a conjecture of {E}rd\\H{o}s and {L}ewin. J. Number Theory (2022), 763--778.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exceptional pair {2,3} is completely representable; Yu--Chen establish density-zero representability and infinitely many coprime failures for broad parameter ranges, leaving exceptional pairs open.\n\n**Verified partial progress.**\n\n- Erdős--Lewin classify the case of finitely many non-representable integers: only {2,3}.\n- Yu--Chen prove density-zero representability and coprime failures in most ranges.\n- Later work sharpens a related constrained-summand function to order n/log n.\n\n**Full solution or refutation.**\n\nThe requested density and coprime-failure conclusions are not known for all non-{2,3} pairs.\n\n**What remains.**\n\nHandle the remaining small parameter pairs.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1110, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1110\n Evidence used: Current status, Yu--Chen ranges, and later update.\n\n**Review notes.** The imported `eq` glyph is malformed; independently rendered source confirms intended not-equal condition without editing the input.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2498, "problem_number": "EP-1111", "title": "Erdős Problem #1111", "statement": "If $G$ is a finite graph and $A,B$ are disjoint sets of vertices then we call $A,B$ anticomplete if there are no edges between $A$ and $B$.\nIf $t,c\\geq 1$ then there exists $d\\geq 1$ such that if $\\chi(G)\\geq d$ and $\\omega(G)3$.\nNguyen, Scott, and Seymour \\cite{NSS24} prove that if $t,c\\geq 1$ then there exists $d\\geq 1$ such that if $\\chi(G)\\geq d$ and $\\omega(G)\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The original two-large-chromatic-anticomplete-sets question remains open, but Nguyen--Scott--Seymour prove a strong one-large-chromatic/one-large-minimum-degree variant.\n\n**Verified partial progress.**\n\n- Nguyen--Scott--Seymour show one anticomplete subgraph can have arbitrarily large chromatic number and the other arbitrarily large minimum degree.\n\n**Full solution or refutation.**\n\nLarge minimum degree does not imply large chromatic number, so their theorem does not settle the stated assertion.\n\n**What remains.**\n\nUpgrade the minimum-degree side to arbitrarily large chromatic number.\n\n**Sources checked.**\n\n- T. Nguyen, A. Scott, P. Seymour, On a problem of El-Zahar and Erdős, J. Combin. Theory B 165 (2024), 211--222. (primary): https://arxiv.org/abs/2303.13449\n Evidence used: Abstract explicitly says the original question remains open and states the variant.\n- Thomas F. Bloom, Erdős Problem #1111, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1111\n Evidence used: Historical bounds and 2024 result.\n\n**Review notes.** The source text's t(2,2) notation likely means d(2,2); it is left untouched.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2499, "problem_number": "EP-1112", "title": "Erdős Problem #1112", "statement": "Let $1\\leq d_1\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The two-sum case is sharp in a key gap range, while the stated k>=3 question has a negative result for one parameter pair and remains open generally.\n\n**Verified partial progress.**\n\n- Bollobás--Hegyvári--Jin prove r_2(2,3)=2 and show r_3(2,3) does not exist.\n- Chen treats r_2(a,b), and Tang--Yang give further nonexistence results.\n\n**Full solution or refutation.**\n\nThe general k>=3 existence question is unresolved.\n\n**What remains.**\n\nClassify r_k(d_1,d_2) for k>=3.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1112, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1112\n Evidence used: Current status, negative special case, and ambiguity caveat.\n\n**Review notes.** The maintained source says the original remark is ambiguous and calls this a generous interpretation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2500, "problem_number": "EP-1113", "title": "Erdős Problem #1113", "statement": "A positive odd integer $m$ such that none of $2^km+1$ are prime for $k\\geq 0$ is called a Sierpinski number. We say that a set of primes $P$ is a covering set for $m$ if every $2^km+1$ is divisible by some $p\\in P$.\nAre there Sierpinski numbers with no finite covering set of primes?", "background": "Sierpinski \\cite{Si60} proved that there are infinitely many Sierpinski numbers, using covering systems to construct suitable covering sets for any $m$ satisfying a certain congruence. This establishes that there is a positive density set of such $m$.\nThe smallest Sierpinski number is believed to be $78557$, which was found by Selfridge.\nErd\\H{o}s and Graham \\cite{ErGr80} asked whether there are Sierpinski numbers for which a covering system is not 'responsible', for which the best interpretation seems to be the above question. This is formulated precisely in problem F13 of Guy's collection \\cite{Gu04}. Erd\\H{o}s and Graham thought the answer is yes (in that there are such Sierpinski numbers), since otherwise this would imply there are infinitely many Fermat primes.\nThere is now further evidence with a concrete example: an argument of Izotov \\cite{Iz95}, given in more detail by Filaseta, Finch, and Kozek \\cite{FFK08}, suggests that $m=734110615000775^4$ is a Sierpinski number without a covering set. (Izotov proved that this $m$ is indeed a Sierpinski number.)\nFilaseta, Finch, and Kozek \\cite{FFK08} give a revised conjecture, suggesting that every Sierpinski number is either a perfect power or else has a finite covering set of primes. They also prove that for every $l\\geq 1$ there is an $m$ such that $2^km^i+1$ is composite for all $1\\leq i\\leq l$ and $k\\geq 0$.\nSee also [203], and [276] for another problem in which the question is whether covering systems are always responsible.\nReferences\n\n\n[ErGr80] Erd\\H{o}s, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[FFK08] Filaseta, Michael and Finch, Carrie and Kozek, Mark, On powers associated with {S}ierpi\\'nski numbers, {R}iesel\nnumbers and {P}olignac's conjecture. J. Number Theory (2008), 1916--1940.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Iz95] Izotov, Anatoly S., A note on {S}ierpi\\'nski numbers. Fibonacci Quart. (1995), 206--207.\n\n[Si60] Sierpi\\'nski, W., Sur un probl\\`eme concernant les nombres {$k\\cdot 2\\sp{n}+1$}. Elem. Math. (1960), 73--74.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** There is a positive-density family of Sierpinski numbers explained by finite covers, and a verified perfect-power example with no known finite cover; existence of a no-cover number remains open.\n\n**Verified partial progress.**\n\n- Sierpinski's covering construction gives infinitely many examples with finite covering sets.\n- Izotov proves a concrete m is Sierpinski; subsequent work suggests it lacks a cover but does not prove that conclusion.\n\n**Full solution or refutation.**\n\nNo verified no-finite-cover example was located.\n\n**What remains.**\n\nProve or refute that the Izotov example has a finite cover, or construct a definitive counterexample.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1113, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1113\n Evidence used: Current open status, Izotov result, and FFK conjecture.\n\n**Review notes.** The distinction between Sierpinski property and absence of a cover is preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2501, "problem_number": "EP-1117", "title": "Erdős Problem #1117", "statement": "Let $f(z)$ be an entire function which is not a monomial. Let $\nu(r)$ count the number of $z$ with $\\lvert z\\rvert=r$ such that $\\lvert f(z)\\rvert=\\max_{\\lvert z\\rvert=r}\\lvert f(z)\\rvert$. (This is a finite quantity if $f$ is not a monomial.)\nIs it possible for $ \\limsup \nu(r)=\\infty? $ Is it possible for $ \\liminf \nu(r)=\\infty? $ ", "background": "This is Problem 2.16 in \\cite{Ha74}, where it is attributed to Erd\\H{o}s.\nThe answer to the first question is yes, as shown by Herzog and Piranian \\cite{HePi68}. The second question is still open, although an 'approximate' affirmative answer is given by Gl\"{u}cksam and Pardo-Sim\\'{o}n \\cite{GlPa24}.\nReferences\n\n\n[GlPa24] Gl\"ucksam, Adi and Pardo-Sim\\'on, Leticia, An approximate solution to {E}rd\\H{o}s' maximum modulus points\nproblem. J. Math. Anal. Appl. (2024), Paper No. 127768, 20.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[HePi68] Herzog, F. and Piranian, G., The counting function for points of maximum modulus. (1968), 240--243.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The limsup question has an affirmative solution; the liminf question remains open despite an approximate affirmative result.\n\n**Verified partial progress.**\n\n- Herzog--Piranian prove limsup nu(r)=infinity is possible.\n- Glücksam--Pardo-Simón give an approximate affirmative answer to the liminf direction.\n\n**Full solution or refutation.**\n\nNo full liminf construction was verified.\n\n**What remains.**\n\nEstablish or disprove liminf nu(r)=infinity.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1117, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1117\n Evidence used: Current open status and separation of the two questions.\n\n**Review notes.** Input calls the counting function u; current rendered source uses nu. The notation defect is recorded but source text is not changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2502, "problem_number": "EP-1120", "title": "Erdős Problem #1120", "statement": "Let $f\\in \\mathbb{C}[z]$ be a monic polynomial of degree $n$, all of whose roots satisfy $\\lvert z\\rvert\\leq 1$. Let $ E= \\{ z : \\lvert f(z)\\rvert \\leq 1\\}. $ What is the shortest length of a path in $E$ joining $z=0$ to $\\lvert z\\rvert =1$?", "background": "This is Problem 4.22 in \\cite{Ha74}, where it is attributed to Erd\\H{o}s. In \\cite{Ha74} it is reported that Clunie and Netanyahu (personal communication) showed that a path always exists which joins $z=0$ to $\\lvert z\\rvert=1$ in $A$.\nErd\\H{o}s wrote 'presumably this tends to infinity with $n$, but not too fast'.\nThe trivial lower bound for the length of this path is $1$, which is achieved for $f(z)=z^n$. The interesting side of this question is what the worst case behaviour is (as a function of $n$).\nSee also [1041].\nReferences\n\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A June 2026 preprint proves the intended worst-case shortest escape length is unbounded, with c sqrt(log n) <= S(n) <= pi n, but its sharp asymptotic order remains open.\n\n**Verified partial progress.**\n\n- Clunie and Netanyahu were reported by Hayman to have shown that an escape path always exists.\n- Pendyala proves for all sufficiently large n that c sqrt(log n) <= S(n) <= pi n for an absolute c>0, settling Erdős's qualitative prediction that S(n) tends to infinity.\n\n**Full solution or refutation.**\n\nThe qualitative growth conjecture is claimed solved in a recent public preprint, not the problem of determining the shortest-path extremal function.\n\n**What remains.**\n\nDetermine the correct order, or sharper upper and lower bounds, for S(n).\n\n**Sources checked.**\n\n- V. S. Pendyala, Shortest paths in polynomial lemniscate sublevel sets and a problem of Erdős, arXiv:2606.19178 (2026). (primary): https://arxiv.org/abs/2606.19178\n Evidence used: Defines the intended extremal S(n) and proves c sqrt(log n) <= S(n) <= pi n.\n- Thomas F. Bloom, Erdős Problem #1120, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1120\n Evidence used: Preserves the historical formulation, existence attribution, and intended worst-case interpretation.\n\n**Review notes.** The literal source omits the outer extremization over f; the background identifies the intended worst-case function of n. The background also uses A once where the defined set is E. Imported category graph_theory conflicts with the maintained analysis tag. Extraction residue follows the bibliography.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2503, "problem_number": "EP-1122", "title": "Erdős Problem #1122", "statement": "Let $f:\\mathbb{N}\\to \\mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). Let $ A=\\{ n \\geq 1: f(n+1)< f(n)\\}. $ If $\\lvert A\\cap [1,X]\\rvert =o(X)$ then must $f(n)=c\\log n$ for some $c\\in \\mathbb{R}$?", "background": "Erd\\H{o}s proved that $f(n)=c\\log n$ for some $c\\in\\mathbb{R}$ if $A$ is empty, or if $f(n+1)-f(n)=o(1)$.\nPartial progress was made by Mangerel \\cite{Ma22}, who proved that this is true if $ \\lvert A\\cap [1,X]\\rvert \\ll \\frac{X}{(\\log X)^{2+c}} $ for some $c>0$, and if $g(p)$ does not have very large values (in a certain technical sense).\nSee also [491].\nReferences\n\n\n[Ma22] Mangerel, Alexander P., Additive functions in short intervals, gaps and a conjecture\nof {E}rd\\H{o}s. Ramanujan J. (2022), 1023--1090.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The zero-density exceptional-set hypothesis remains open; Mangerel proves the conclusion for a quantitatively much thinner exceptional set, complete additivity, and a technical prime-value condition.\n\n**Verified partial progress.**\n\n- Erdős proved f(n)=c log n if the decrease set is empty, and also under the separate hypothesis f(n+1)-f(n)=o(1).\n- Mangerel proves identity with c log n when the decrease density is O((log X)^(-2-epsilon)), f is completely additive, and large prime values are controlled; he also obtains almost-everywhere logarithmic approximation under qualitative almost monotonicity.\n\n**Full solution or refutation.**\n\nKnown results do not promote o(X) exceptional decreases for a general additive function to exact logarithmic form.\n\n**What remains.**\n\nRemove the quantitative decay, complete-additivity, and prime-value restrictions and prove or refute the statement as written.\n\n**Sources checked.**\n\n- A. P. Mangerel, Additive functions in short intervals, gaps and a conjecture of Erdős, Ramanujan J. 59 (2022), 1023-1090, doi:10.1007/s11139-022-00623-y. (primary): https://arxiv.org/abs/2108.12351\n Evidence used: Abstract explicitly states the almost-monotone approximation theorem and the quantitative exact-identity special case.\n- Thomas F. Bloom, Erdős Problem #1122, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1122\n Evidence used: Current open formulation and summary of Erdős's and Mangerel's partial results.\n\n**Review notes.** Imported category combinatorics conflicts with the maintained number-theory tag. The imported background says g(p) although the problem names f, and contains extraction residue after the bibliography; neither defect was repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2504, "problem_number": "EP-1129", "title": "Erdős Problem #1129", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nDescribe which choice of $x_i$ minimise $ \\Lambda(x_1,\\ldots,x_n)=\\max_{x\\in [-1,1]} \\sum_k \\lvert l_k(x)\\rvert. $ ", "background": "The functions $l_k(x)$ are sometimes called the fundamental functions of Lagrange interpolation, and $\\Lambda$ is sometimes called the Lebesgue constant.\nFaber \\cite{Fa14} proved $ \\Lambda(x_1,\\ldots,x_n)\\gg \\log n $ for all choices of $x_i$, and Bernstein \\cite{Be31} proved it is $>(\\frac{2}{\\pi}-o(1))\\log n$. Erd\\H{o}s \\cite{Er61c} improved this to $ \\Lambda(x_1,\\ldots,x_n)> \\frac{2}{\\pi}\\log n-O(1). $ This is best possible, since taking the $x_i$ as the roots of the $n$th Chebyshev polynomial yields $ \\Lambda(x_1,\\ldots,x_n)< \\frac{2}{\\pi}\\log n+O(1). $ Erd\\H{o}s thought that the minimising choice is characterised by the property that the sums $ \\lambda_i=\\max_{x\\in [x_i,x_{i+1}]}\\sum_k \\lvert l_k(x)\\rvert $ are all equal for $0\\leq i\\leq n$ (where $x_0=-1$ and $x_{n+1}=1$). This conjecture was also made by Bernstein \\cite{Be31}. Kilgore and Cheney \\cite{KiCh76} proved that there exists $x_i$ for which all $\\lambda_i$ are equal. Kilgore \\cite{Ki77} proved that $\\Lambda$ is minimised only when all $\\lambda_i$ are equal. Finally, de Boor and Pinkus \\cite{dBPi78} proved that there exists a unique minimising choice of $x_i$.\nIf $x_1=-1$ and $x_n=1$ then there is a unique minimising set of $x_i$, which are symmetric around $0$. (Such a choice is called canonical.) The exact minimising canonical choice is known only for $n\\leq 4$. For $n=2$ the points are $-1,1$ (with $\\Lambda=1$). For $n=3$ the points are $-1,0,1$ (with $\\Lambda=1.25$), as shown by Bernstein \\cite{Be31}. Rack and Vajda \\cite{RaVa15} have shown that for $n=4$ the points are $-1,-t,t,1$ where $t\\approx 0.4177$ is an explicit algebraic constant (with $\\Lambda \\approx 1.4229$).\nIn \\cite{Er67} Erd\\H{o}s suggests that an easier variant might be to have the $x_i\\in \\mathbb{C}$ with $\\lvert x_i\\rvert=1$, and seek to minimise $\\max_{\\lvert z\\rvert=1}\\sum_{k}\\lvert l_k(z)\\rvert$, adding it 'seems certain' that the minimising $x_i$ are the $n$th roots of unity. This was proved by Brutman \\cite{Br80} for odd $n$ and by Brutman and Pinkus \\cite{BrPi80} for even $n$.\nSee also [671], [1130], and [1132].\nReferences\n\n\n[Be31] S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degr\\'{e} n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.\n\n[Br80] Brutman, L., On the polynomial and rational projections in the complex\nplane. SIAM J. Numer. Anal. (1980), 366--372.\n\n[BrPi80] Brutman, L. and Pinkus, A., On the {E}rd\\H{o}s conjecture concerning minimal norm\ninterpolation on the unit circle. SIAM J. Numer. Anal. (1980), 373--375.\n\n[Er61c] Erd\\H{o}s, P., Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hungar. (1961), 235-244.\n\n[Er67] Erd\\H{o}s, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73.\n\n[Fa14] G. Faber, \"{U}ber die interpolatorische Darstellung stetiger Funktionen. Jahresb. der Deutschen Math. Ver. (1914), 190-210.\n\n[Ki77] Kilgore, T. A., Optimization of the norm of the {L}agrange interpolation\noperator. Bull. Amer. Math. Soc. (1977), 1069--1071.\n\n[KiCh76] Kilgore, T. A. and Cheney, E. W., A theorem on interpolation in {H}aar subspaces. Aequationes Math. (1976), 391--400.\n\n[RaVa15] Rack, Heinz-Joachim and Vajda, Robert, Optimal cubic {L}agrange interpolation: extremal node systems\nwith minimal {L}ebesgue constant. Stud. Univ. Babe\\c s-Bolyai Math. (2015), 151--171.\n\n[dBPi78] de Boor, Carl and Pinkus, Allan, Proof of the conjectures of {B}ernstein and {E}rd\\H os\nconcerning the optimal nodes for polynomial interpolation. J. Approx. Theory (1978), 289--303.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Kilgore and de Boor-Pinkus proved in 1978 that the minimizing nodes are characterized by equioscillation of the interval maxima of the Lebesgue function, with a unique symmetric canonical optimizer.\n\n**Verified partial progress.**\n\n- Faber, Bernstein, and Erdős established logarithmic lower bounds, while Chebyshev nodes give the correct logarithmic order.\n- Kilgore and Cheney proved existence of an equioscillating node system before the full optimality and uniqueness results.\n- Explicit optimal canonical nodes are known only in low degrees, although the general characterization is complete.\n\n**Full solution or refutation.**\n\nThe Bernstein-Erdős equioscillation characterization identifies optimal nodes implicitly for every degree; de Boor and Pinkus establish uniqueness in the canonical setting.\n\n**What remains.**\n\nThe characterization problem is closed. Closed-form expressions for optimal nodes in arbitrary degree and numerical computation are separate refinements.\n\n**Sources checked.**\n\n- Theodore A. Kilgore, A characterization of the Lagrange interpolating projection with minimal Tchebycheff norm, Journal of Approximation Theory 24 (1978), 273-288. (primary): https://doi.org/10.1016/0021-9045(78)90013-8\n Evidence used: Published characterization of the minimal-norm interpolation projection.\n- Carl de Boor and Allan Pinkus, Proof of the conjectures of Bernstein and Erdos concerning the optimal nodes for polynomial interpolation, Journal of Approximation Theory 24 (1978), 289-303. (primary): https://doi.org/10.1016/0021-9045(78)90014-X\n Evidence used: Completes the characterization and uniqueness results.\n- Bayram Ali Ibrahimoglu, Lebesgue functions and Lebesgue constants in polynomial interpolation, Journal of Inequalities and Applications (2016). (authoritative_secondary): https://doi.org/10.1186/s13660-016-1030-3\n Evidence used: Modern survey confirming the Bernstein-Erdős characterization and small-degree explicit cases.\n- Thomas F. Bloom, Erdos Problem #1129, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1129\n Evidence used: Records PROVED and gives the complete source history.\n\n**Review notes.** The statement has OCR loss in not-equal signs and the background has a leaked serialized suffix; neither was silently edited.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2505, "problem_number": "EP-1130", "title": "Erdős Problem #1130", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nLet $x_0=-1$ and $x_{n+1}=1$ and $ \\Upsilon(x_1,\\ldots,x_n)=\\min_{0\\leq i\\leq n}\\max_{x\\in[x_i,x_{i+1}]} \\sum_k \\lvert l_k(x)\\rvert. $ Is it true that $ \\Upsilon(x_1,\\ldots,x_n)\\ll \\log n? $ Describe which choice of $x_i$ maximise $\\Upsilon(x_1,\\ldots,x_n)$.", "background": "The functions $l_k(x)$ are sometimes called the fundamental functions of Lagrange interpolation.\nErd\\H{o}s \\cite{Er47} could prove $ \\Upsilon(x_1,\\ldots,x_n)< \\sqrt{n}. $ Erd\\H{o}s thought that the maximising choice is characterised by the property that the sums $ \\lambda_i=\\max_{x\\in [x_i,x_{i+1}]}\\sum_k \\lvert l_k(x)\\rvert $ are all equal for $0\\leq i\\leq n$ (where $x_0=-1$ and $x_{n+1}=1$), which would be the same characterisation as [1129].\nThis is true, and was proved by de Boor and Pinkus \\cite{dBPi78}. It follows by the bounds discussed in [1129] that $ \\Upsilon(x_1,\\ldots,x_n)\\leq \\frac{2}{\\pi}\\log n+O(1). $ See also [1129].\nReferences\n\n\n[Er47] Erd\\H{o}s, P., Some remarks on polynomials. Bull. Amer. Math. Soc. (1947), 1169--1176.\n\n[dBPi78] de Boor, Carl and Pinkus, Allan, Proof of the conjectures of {B}ernstein and {E}rd\\H os\nconcerning the optimal nodes for polynomial interpolation. J. Approx. Theory (1978), 289--303.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** De Boor and Pinkus's 1978 theorem shows that the maximizer is the equioscillating optimal interpolation node set and implies Upsilon <= (2/pi)log n+O(1).\n\n**Verified partial progress.**\n\n- Erdős earlier proved the weaker upper bound Upsilon", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2506, "problem_number": "EP-1131", "title": "Erdős Problem #1131", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nWhat is the minimal value of $ I(x_1,\\ldots,x_n)=\\int_{-1}^1 \\sum_k \\lvert l_k(x)\\rvert^2\\mathrm{d}x? $ In particular, is it true that $ \\min I =2-(1+o(1))\\frac{1}{n}? $ ", "background": "Erd\\H{o}s first conjectured this minimum was achieved by taking the $x_i$ to be the roots of the integral of the Legendre polynomial, since Fejer \\cite{Fe32} had earlier shown these to be minimisers of $ \\max_{x\\in [-1,1]}\\sum_k \\lvert l_k(x)\\rvert^2. $ This was disproved by Szabados \\cite{Sz66} for every $n>3$.\nErd\\H{o}s, Szabados, Varma, and V\\'{e}rtesi \\cite{ESVV94} proved that $ 2-O\\left(\\frac{(\\log n)^2}{n}\\right)\\leq \\min I\\leq 2-\\frac{2}{2n-1} $ where the upper bound is witnessed by the roots of the integral of the Legendre polynomial as above.\nReferences\n\n\n[ESVV94] Erd\\H{o}s, P. and Szabados, J. and Varma, A. K. and V\\'{e}rtesi,\nP., On an interpolation theoretical extremal problem. Studia Sci. Math. Hungar. (1994), 55--60.\n\n[Fe32] Fej\\'{e}r, Leopold, Bestimmung derjenigen {A}bszissen eines {I}ntervalles, f\"ur\nwelche die {Q}uadratsumme der {G}rundfunktionen der\n{L}agrangeschen {I}nterpolation im {I}ntervalle ein\n{M}\"oglichst kleines {M}aximum {B}esitzt. Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) (1932), 263--276.\n\n[Sz66] Szabados, J., On a problem of {P}. {E}rd\\H{o}s. Acta Math. Acad. Sci. Hungar. (1966), 155--157.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The unconditional 1994 bounds leave a logarithmic-factor gap; numerical and recent non-peer-reviewed claims sharpen the picture but do not presently give a verified first-order asymptotic.\n\n**Verified partial progress.**\n\n- Szabados disproved Erdős's proposed exact minimizing nodes for every n>3.\n- Erdős, Szabados, Varma, and Vértesi proved 2-O((log n)^2/n) <= min I <= 2-2/(2n-1).\n- Brutman and Toledano numerically optimized n<=100 and reported asymptotics contradicting an earlier o(1/n) proximity conjecture, but this was numerical rather than a rigorous asymptotic refutation.\n- A 2026 non-peer-reviewed preprint claims an unconditional 2-C/n lower bound and equidistribution, with its first-order constant conditional on endpoint universality.\n\n**Full solution or refutation.**\n\nNo independently verified unconditional determination of min I or proof/refutation of min I=2-(1+o(1))/n was located.\n\n**What remains.**\n\nEstablish the correct first-order coefficient rigorously and characterize exact or asymptotic minimizers.\n\n**Sources checked.**\n\n- P. Erdős, J. Szabados, A. K. Varma, P. Vértesi, On an interpolation theoretical extremal problem, Studia Sci. Math. Hungar. 29 (1994), 55-60. (primary): https://www.maths.tcd.ie/EMIS/classics/Erdos/cit/81741006.htm\n Evidence used: Bibliographic review records the lower-bound theorem for sums of powers of Lagrange fundamental polynomials.\n- L. Brutman and D. Toledano, An extremal problem of Erdős in interpolation theory, Comput. Math. Appl. 34 (1997), 37-47. (primary): https://doi.org/10.1016/S0898-1221(97)00232-0\n Evidence used: Abstract states the n<=100 numerical optimization and the numerical disproof claim for the earlier proximity conjecture.\n- R. Zeraoulia, Asymptotics of Erdos's L2 Lagrange Interpolation Problem: Arcsine Distribution and Airy Endpoint Universality, Preprints.org 202601.0875 (2026). (primary): https://www.preprints.org/manuscript/202601.0875\n Evidence used: Claims an O(1/n) lower bound and equidistribution, while explicitly making the first-order expansion conditional.\n- Thomas F. Bloom, Erdős Problem #1131 and discussion, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1131\n Evidence used: Still labels the problem open and records both historical bounds and recent unverified claims.\n\n**Review notes.** Imported category combinatorics conflicts with analysis/polynomials tags. Extraction residue follows the bibliography. Recent preprint and forum claims were not treated as verified literature.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2507, "problem_number": "EP-1132", "title": "Erdős Problem #1132", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nLet $x_1,x_2,\\ldots\\in [-1,1]$ be an infinite sequence, and let $ L_n(x) = \\sum_{1\\leq k\\leq n}\\lvert l_k(x)\\rvert, $ where each $l_k(x)$ is defined above with respect to $x_1,\\ldots,x_n$.\nMust there exist $x\\in (-1,1)$ such that $ L_n(x) >\\frac{2}{\\pi}\\log n-O(1) $ for infinitely many $n$?\nIs it true that $ \\limsup_{n\\to \\infty}\\frac{L_n(x)}{\\log n}\\geq \\frac{2}{\\pi} $ for almost all $x\\in (-1,1)$?", "background": "A result of Bernstein \\cite{Be31} implies that the set of $x\\in(-1,1)$ for which $ \\limsup_{n\\to \\infty}\\frac{L_n(x)}{\\log n}\\geq \\frac{2}{\\pi} $ is everywhere dense.\nErd\\H{o}s \\cite{Er61c} proved that, for any fixed $x_1,\\ldots,x_n\\in [-1,1]$, $ \\max_{x\\in [-1,1]}\\sum_{1\\leq k\\leq n}\\lvert l_k(x)\\rvert>\\frac{2}{\\pi}\\log n-O(1). $ See also [1129] for more on $L_n(x)$.\nReferences\n\n\n[Be31] S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degr\\'{e} n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.\n\n[Er61c] Erd\\H{o}s, P., Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hungar. (1961), 235-244.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Both pointwise recurrence questions remain open; Bernstein gives a dense-set limsup result, and Tao's 2026 work gives a dense set with the sharp leading constant but an unbounded additive loss.\n\n**Verified partial progress.**\n\n- Bernstein's result implies that the limsup inequality with constant 2/pi holds on an everywhere dense set.\n- Erdős proved max_x L_n(x) > (2/pi) log n-O(1) for every fixed n-node set.\n- Tao proves that for every omega(n) tending to infinity there is a dense set of x for which L_n(x) >= (2/pi) log n-omega(n) infinitely often.\n\n**Full solution or refutation.**\n\nDense-set conclusions do not yield a bounded-error recurrent point, and they are much weaker than the almost-everywhere assertion.\n\n**What remains.**\n\nProve a recurrent lower bound with O(1) loss at at least one interior point and determine whether the limsup inequality holds almost everywhere.\n\n**Sources checked.**\n\n- T. Tao, Local Bernstein theory, and lower bounds for Lebesgue constants, arXiv:2603.21453v3 (2026). (primary): https://arxiv.org/abs/2603.21453\n Evidence used: Proves localized sharp lower bounds for Lebesgue functions; the maintained tracker records the resulting dense-set recurrent statement.\n- Thomas F. Bloom, Erdős Problem #1132, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1132\n Evidence used: Current open status, Bernstein and Erdős results, Tao's 2026 partial result, and O(1)-dependence ambiguity.\n\n**Review notes.** The first question is ambiguous about whether the O(1) constant may depend on x. Imported category combinatorics conflicts with analysis/polynomials tags. Extraction residue follows the bibliography.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2508, "problem_number": "EP-1133", "title": "Erdős Problem #1133", "statement": "Let $C>0$. There exists $\\epsilon>0$ such that if $n$ is sufficiently large the following holds.\nFor any $x_1,\\ldots,x_n\\in [-1,1]$ there exist $y_1,\\ldots,y_n\\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\\epsilon)n$ many $1\\leq i\\leq n$, then $ \\max_{x\\in [-1,1]}\\lvert P(x)\\rvert >C. $ ", "background": "Erd\\H{o}s proved that, for any $C>0$, there exists $\\epsilon>0$ such that if $n$ is sufficiently large and $m=\\lfloor (1+\\epsilon)n\\rfloor$ then for any $x_1,\\ldots,x_m\\in [-1,1]$ there is a polynomial $P$ of degree $n$ such that $\\lvert P(x_i)\\rvert\\leq 1$ for $1\\leq i\\leq m$ and $ \\max_{x\\in [-1,1]}\\lvert P(x)\\rvert>C. $ The conjectured statement would also imply this, but Erd\\H{o}s in \\cite{Er67} says he could not even prove it for $m=n$.\nReferences\n\n\n[Er67] Erd\\H{o}s, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73.\",\n \"difficulty\": \"L1\"\n},{\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The maintained tracker still says open, while its forum ecosystem records a candidate full solution posted on 29 April 2026; no peer-reviewed or independently certified solution was located.\n\n**Verified partial progress.**\n\n- Erdős proved a related dual-looking theorem producing a degree-n polynomial bounded on about (1+epsilon)n prescribed nodes but large between them.\n- The community AI-contributions index records Przemek Chojecki and GPT-5.5 Pro as having a candidate full solution, not an accepted result.\n\n**Full solution or refutation.**\n\nThe recent claim has not been incorporated into the maintained problem status and was not verified for this triage.\n\n**What remains.**\n\nObtain expert verification, a transparent independently checked proof, or peer-reviewed publication of the candidate argument; otherwise the original problem remains open.\n\n**Sources checked.**\n\n- Thomas F. Bloom, Erdős Problem #1133, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1133\n Evidence used: Current open label, exact statement, and Erdős's related theorem.\n- teorth/erdosproblems, AI contributions to Erdős problems, entry #1133, checked 2026-08-17. (authoritative_secondary): https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems\n Evidence used: Lists a 29 April 2026 candidate full solution by Przemek Chojecki using GPT-5.5 Pro, explicitly with candidate rather than accepted status.\n\n**Review notes.** The imported category graph_theory conflicts with analysis/polynomials tags. `formalized: yes` appears to concern a formalized or axiomatized statement, not a certified proof. Extraction residue follows the bibliography.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2509, "problem_number": "EP-1135", "title": "Erdős Problem #1135", "statement": "Define $f:\\mathbb{N}\\to \\mathbb{N}$ by $f(n)=n/2$ if $n$ is even and $f(n)=\\frac{3n+1}{2}$ if $n$ is odd.\nGiven any integer $m\\geq 1$ does there exist $k\\geq 1$ such that $f^{(k)}(m)=1$?", "background": "The infamous Collatz conjecture. For a detailed discussion of the history and theory surrounding this problem we refer to the overview by Lagarias \\cite{La10}.\nThis is not a problem due to Erd\\H{o}s; it was first devised by Collatz before 1952. Erd\\H{o}s referred to this problem on several occasions as 'hopeless'. As Lagarias \\cite{La16} notes, the closest Erd\\H{o}s ever came to working on problems of this nature is the theorem described in the remarks to [1134].\nIt is often claimed that Erd\\H{o}s offered \\$500 for a solution to this problem; this claim originated in a survey article by Lagarias \\cite{La85}.\nLagarias reported, in personal communication, that this came from a conversation he had with Erd\\H{o}s and Graham around 1983, in which Graham asked Erd\\H{o}s to make an estimate of what value Erd\\H{o}s would put the problem on his prize scale, to which Erd\\H{o}s replied \\$500. Therefore, strictly speaking, Erd\\H{o}s never offered \\$500 specifically as a prize, but we include this prize value here for comparing those problems which Erd\\H{o}s rated as 'prize problems'.\nThis is Problem E16 in Guy's collection \\cite{Gu04}, in which Guy quotes Erd\\H{o}s as saying \"Mathematics may not be ready for such problems\".\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[La10] Lagarias, Jeffrey C., The {$3x+1$} problem: an overview. (2010), 3--29.\n\n[La16] Lagarias, Jeffrey C., Erd\\H os, {K}larner, and the {$3x+1$} problem. Amer. Math. Monthly (2016), 753--776.\n\n[La85] Lagarias, Jeffrey C., The {$3x+1$} problem and its generalizations. Amer. Math. Monthly (1985), 3--23.\",\n \"difficulty\": \"L5\"\n}\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Collatz conjecture remains open; Tao proves almost all orbits attain almost bounded values in logarithmic density, and computation verifies all starts below 2^71.\n\n**Verified partial progress.**\n\n- Tao proves that for every h(N) tending to infinity, the minimum orbit value is at most h(N) for almost all N in logarithmic density.\n- Barina's published and reproducible computation verifies convergence for every starting value below 2^71.\n\n**Full solution or refutation.**\n\nNeither an almost-all density theorem nor finite verification proves that every positive integer reaches 1.\n\n**What remains.**\n\nProve convergence for all positive integers, or exhibit a divergent orbit or nontrivial cycle.\n\n**Sources checked.**\n\n- T. Tao, Almost all orbits of the Collatz map attain almost bounded values, Forum Math. Pi 10 (2022), e12, doi:10.1017/fmp.2022.8. (primary): https://doi.org/10.1017/fmp.2022.8\n Evidence used: Proves the almost-bounded minimum-orbit theorem in logarithmic density.\n- D. Barina, Improved verification limit for the convergence of the Collatz conjecture, J. Supercomput. 81 (2025), 810, doi:10.1007/s11227-025-07337-0. (primary): https://doi.org/10.1007/s11227-025-07337-0\n Evidence used: Reports computational verification through 2^71.\n- D. Barina, Convergence verification of the Collatz problem, generated 2026-07-29 and checked 2026-08-17. (primary): https://pcbarina.fit.vutbr.cz/\n Evidence used: Current reproducible project log confirms the 2^71 milestone and links code and publications.\n- Thomas F. Bloom, Erdős Problem #1135, checked 2026-08-17. (maintained_tracker): https://www.erdosproblems.com/1135\n Evidence used: Current open label and historical attribution/prize caveat.\n\n**Review notes.** This is Collatz's problem, not a problem originated by Erdős. The listed $500 was an informal valuation rather than a documented prize offer. The imported background ends with serialized difficulty-field extraction residue.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 5, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 8, "name": "erdos_problems", "display_name": "Erdős Problems", "description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.", "slug": "erdos-problems", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2510, "problem_number": "KOU-21.1", "title": "Kourovka Notebook Problem 21.1", "statement": "Let $n$ be a positive integer. For a finite group $K$ and an automorphism $\\varphi$ of $K$ of order dividing $n$, let\n\n$$\nX_{n,\\varphi}(K):=\\{x\\in K\\mid x x^{\\varphi}\\cdots x^{\\varphi^{n-1}}=1\\}.\n$$\n\nLet $c_n$ be the supremum of the ratios $|X_{n,\\varphi}(H)|/|H|$ over all finite groups $H$ and their automorphisms $\\varphi\\in\\operatorname{Aut}(H)$ such that $\\varphi^n=\\operatorname{id}$ and $X_{n,\\varphi}(H)\\ne H$.\n\n(a) Let $n>1$ be a positive integer such that $c_d<1$ for all prime power divisors $d$ of $n$. Is it true that $c_n<1$?\n\n(b) For a finite group $G$ and a positive integer $n$, the generalized Hughes--Thompson subgroup is defined as $H_n(G)=\\langle x\\in G\\mid x^n\\ne 1\\rangle$. Suppose that $n$ is a positive integer for which there is a positive integer $k_n$ depending only on $n$ such that $|G:H_n(G)|\\leqslant k_n$ for all finite groups $G$ with $H_n(G)\\ne 1$. Is it true that then $c_n<1$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.1.\n\nDiscussion and literature:\nThis question is open even when $n\\geqslant 5$ is prime. A. Abdollahi, M. S. Malekan\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The extremal condition c_n<1 is known for n=2 and n=3 and has useful profinite and solvable-group reductions, but both implications asked in the record remain open, even for primes at least five.\n\n**Verified partial progress.**\n\n- Abdollahi--Soleimani Malekan identify c_n<1 with the Lévai--Pyber positive-measure conjecture in their formulation.\n- They prove c_3<1, with a 15/16 threshold, and record the known case n=2.\n- For odd n they reduce c_n<1 to finite solvable groups.\n- They prove that c_n<1 implies a uniform Hughes--Thompson index bound, the direction opposite to part (b).\n\n**Full solution or refutation.**\n\nStrong special cases and reductions are known, but neither the prime-power-divisor implication nor the converse from a uniform Hughes--Thompson index bound has been proved.\n\n**What remains.**\n\nProve either implication in general; the first unresolved prime cases begin at n at least five.\n\n**Sources checked.**\n\n- Kourovka Notebook, New Problems, 21st issue, Problem 21.1 (2026). (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States both implications and explicitly says the question remains open even for primes at least five.\n- Alireza Abdollahi and Meisam Soleimani Malekan, Profinite Groups with Many Elements of Bounded Order, Advances in Group Theory and Applications 13 (2022), 71--81. (primary): https://doi.org/10.32037/agta-2022-006\n Evidence used: Establishes the profinite equivalence, the n=3 theorem, the odd-n solvable reduction, and the forward Hughes--Thompson index implication.\n\n**Review notes.** No formulation defect found. Part (b) asks the converse of a direction proved in the cited paper.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2511, "problem_number": "KOU-21.2", "title": "Kourovka Notebook Problem 21.2", "statement": "Let $S$ be a finite simple group, and let $G$ be a finite group for which there exists a bijection $f:G\\to S$ such that $|x|$ divides $|f(x)|$ for all $x\\in G$. Must $G$ necessarily be simple?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.2.\n\nDiscussion and literature:\nM. Amiri\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that the element-order divisibility bijection forces simplicity was verified.\n\n**Verified partial progress.**\n\n- The condition compares all element orders to a fixed finite simple group.\n\n**Full solution or refutation.**\n\nThe recognition question remains open.\n\n**What remains.**\n\nExtract composition-factor restrictions from order-divisibility counts.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.2. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2512, "problem_number": "KOU-21.3", "title": "Kourovka Notebook Problem 21.3", "statement": "Let $G=A_n$ or $S_n$ and let $H,K$ be soluble subgroups of $G$. For all sufficiently large $n$, can we always find an element $x\\in G$ such that $H\\cap K^x=1$? Does this hold for all $n\\geqslant 21$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.3.\n\nDiscussion and literature:\nNote that the conclusion is false when $G=S_{20}$ and $H=K=(S_4\\wr S_4)\\times S_4$. M. Anagnostopoulou-Merkouri, T. C. Burness\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every pair of soluble maximal subgroups of A_n or S_n has a conjugate with trivial intersection for n at least 18. The corresponding assertion for arbitrary soluble subgroups, including the proposed threshold n at least 21, remains open.\n\n**Verified partial progress.**\n\n- Anagnostopoulou-Merkouri and Burness prove regularity of every pair of soluble maximal subgroups for n at least 18.\n- They verify the displayed obstruction at n=20 and show that the same subgroup embedded in S_21 has base size two.\n- They explicitly formulate eventual regularity for arbitrary soluble pairs as a stronger conjecture; maximal-subgroup regularity does not automatically imply it.\n\n**Full solution or refutation.**\n\nThe maximal-soluble case follows from the classification of primitive subgroup tuples and exact low-degree checks. Arbitrary soluble subgroups can sit below insoluble maximal subgroups, which prevents a direct containment reduction and leaves the source question unresolved.\n\n**What remains.**\n\nControl soluble subgroups lying inside insoluble maximal subgroups, prove eventual trivial-intersection conjugacy for every pair, and determine whether n=21 is the exact threshold.\n\n**Sources checked.**\n\n- The Kourovka Notebook, New Problems, 21st issue (2026), Problem 21.3. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the arbitrary-soluble-pair question and the n=20 counterexample.\n- M. Anagnostopoulou-Merkouri and T. C. Burness, On the regularity number of a finite group and other base-related invariants, Journal of the London Mathematical Society 110 (2024), e70035. (primary): https://doi.org/10.1112/jlms.70035\n Evidence used: Proves regularity for soluble maximal pairs from degree 18 onward, records the low-degree obstruction, and states eventual arbitrary-soluble-pair regularity as a conjecture.\n\n**Review notes.** The supplied n=20 example is preserved exactly; the literature distinguishes arbitrary soluble subgroups from soluble maximal subgroups.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2513, "problem_number": "KOU-21.4", "title": "Kourovka Notebook Problem 21.4", "statement": "Let $G$ be a finite group with trivial solvable radical and let $H_1,\\ldots,H_5$ be solvable subgroups of $G$. Then do there always exist elements $x_i\\in G$ such that $\\bigcap_i H_i^{x_i}=1$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.4.\n\nDiscussion and literature:\nCf. 17.41(b). M. Anagnostopoulou-Merkouri, T. C. Burness\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No universal five-solvable-subgroup conjugate-intersection theorem was verified for finite groups with trivial solvable radical.\n\n**Verified partial progress.**\n\n- The source cross-references a related earlier problem.\n\n**Full solution or refutation.**\n\nThe intersection question remains open.\n\n**What remains.**\n\nUse base-size methods for solvable subgroup actions.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.4. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2514, "problem_number": "KOU-21.5", "title": "Kourovka Notebook Problem 21.5", "statement": "Let $p$ be a prime. Let $G$ be a transitive subgroup of the group of finitary permutations $\\operatorname{FSym}(\\Omega)$ of a set $\\Omega$, let $N$ be a normal subgroup of $G$, and let $S$ be a transitive Sylow $p$-subgroup of $G$.\n\n(a) Is it true that $S\\cap N$ is a Sylow $p$-subgroup of $N$?\n\n(b) Is it true that $SN/N$ is a Sylow $p$-subgroup of $G/N$?\n\n(c) Are any two transitive Sylow $p$-subgroups of $G$ locally conjugate in $G$?\n\nTwo subgroups $X,Y$ of a group $G$ are said to be locally conjugate if there is a locally inner automorphism $\\phi$ of $G$ such that $X^\\phi=Y$. An automorphism $\\phi$ of $G$ is said to be locally inner if for every finite subset $A\\subseteq G$ there is an element $g=g(A)\\in G$ such that $a^\\phi=g^{-1}ag$ for all $a\\in A$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.5.\n\nDiscussion and literature:\nA. O. Asar\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No source deciding the normal-subgroup intersection, quotient image, or local-conjugacy questions for transitive Sylow p-subgroups of finitary permutation groups was verified.\n\n**Verified partial progress.**\n\n- Asar proves an adjacent cardinality dichotomy: in the transitive-Sylow setting, the ambient finitary permutation group is a p-group or has continuum many transitive Sylow p-subgroups.\n- This does not imply any of parts (a), (b), or (c).\n\n**Full solution or refutation.**\n\nAll three subquestions remain open in this dated triage.\n\n**What remains.**\n\nEstablish infinite locally finite analogues of normal intersection and quotient Sylow theorems under transitivity, then compare transitive Sylow subgroups under locally inner automorphisms.\n\n**Sources checked.**\n\n- The Kourovka Notebook, 21st issue (2026), Problem 21.5. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Poses all three questions without recording an answer.\n- A. O. Asar, On Sylow p-subgroups of finitary permutation groups, Ricerche di Matematica (2026), DOI 10.1007/s11587-026-01079-8. (primary): https://doi.org/10.1007/s11587-026-01079-8\n Evidence used: Proves structure/cardinality results for transitive Sylow p-subgroups but does not decide the three stated operations/conjugacy questions.\n\n**Review notes.** Open is conservative and dated. The adjacent 2026 result was not promoted to a solution of any subpart.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2515, "problem_number": "KOU-21.6", "title": "Kourovka Notebook Problem 21.6", "statement": "Let p be a prime. A totally imprimitive p-group H of finitary permutations is said to have the cyclic-block property if in the cycle decomposition of every element the support of every cycle is a block for H. Let G be a transitive subgroup of the group of finitary permutations $\\operatorname{FSym}(\\Omega)$ of a set $\\Omega$. Does every transitive Sylow p-subgroup of G contain a transitive subgroup which has the cyclic-block property?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.6.\n\nDiscussion and literature:\nA. O. Asar\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof or counterexample was found for the existence of a transitive cyclic-block subgroup inside every transitive Sylow p-subgroup of FSym(Omega).\n\n**Verified partial progress.**\n\n- Asar provides structural results on finitary permutation groups and infinite orbits, but not the required cyclic-block construction.\n\n**Full solution or refutation.**\n\nThe precise cyclic-block subgroup question remains open.\n\n**What remains.**\n\nConstruct the subgroup in an arbitrary transitive Sylow p-subgroup or find a block-system obstruction.\n\n**Sources checked.**\n\n- A. O. Asar, On Finitary Permutation Groups, Turkish Journal of Mathematics 30 (2006), 101-116. (primary): https://journals.tubitak.gov.tr/math/vol30/iss1/10/\n Evidence used: Provides related structural theory for finitary permutation groups, without resolving the stated question.\n- Kourovka Notebook, Issue 21 (2026), Problem 21.6. (maintained_tracker): https://kourovka-notebook.org/\n Evidence used: Maintains the exact cyclic-block question.\n\n**Review notes.** Input order intentionally places 21.6 after 21.59.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2516, "problem_number": "KOU-21.7", "title": "Kourovka Notebook Problem 21.7", "statement": "(Well-known problem). A finite group G is called an IYB-group if it is isomorphic to the permutation group of a finite involutive non-degenerate set-theoretic solution of the Yang--Baxter equation, or equivalently, G is isomorphic to the multiplicative group of a finite left brace. Assume that the Sylow subgroups of a finite soluble group G are IYB-groups. Is G an IYB-group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.7.\n\nDiscussion and literature:\nA. Ballester-Bolinches\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No closure theorem for IYB-groups under finite soluble extensions from IYB Sylow subgroups was verified.\n\n**Verified partial progress.**\n\n- The source gives equivalent brace/Yang--Baxter formulations.\n\n**Full solution or refutation.**\n\nThe local-to-global IYB question remains open.\n\n**What remains.**\n\nAnalyze brace extension data along Hall/Sylow systems.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.7. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the well-known problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2517, "problem_number": "KOU-21.8", "title": "Kourovka Notebook Problem 21.8", "statement": "As in 17.57, let $r(m)=\\{r+km\\mid k\\in\\mathbb Z\\}$ for integers $0\\leqslant r3$ the group $\\operatorname{CT}(k)$ is isomorphic to the symmetric group $S_N$, where N is the least common multiple of the numbers $2,3,\\ldots,k$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.8.\n\nDiscussion and literature:\nV. G. Bardakov, A. L. Iskra\n\nYes, it is true (Junyao Pan, Preprint of 14 April 2026, https://arxiv.org/abs/2604.12553; P. Monticone, Preprint of 30 April 2026, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/05/21_8-1.pdf).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** For every k > 3, CT(k) is isomorphic to S_N with N = lcm(2,3,...,k).\n\n**Verified partial progress.**\n\n- Pan's 2026 paper proves the exact all-k statement.\n- A separate 2026 Kourovka project gives a formally verified treatment.\n\n**Full solution or refutation.**\n\nThe subgroup generated by CT_2 through CT_k is the full symmetric group on the N residue classes modulo N, hence CT(k) is isomorphic to S_N.\n\n**What remains.**\n\nNothing for the stated isomorphism; conventional journal publication of the recent work remains pending.\n\n**Sources checked.**\n\n- Junyao Pan, A note on the horizontal class transposition group, arXiv:2604.12553 (2026). (primary): https://arxiv.org/abs/2604.12553\n Evidence used: The abstract states the exact CT(k) isomorphic to S_N theorem for k > 3.\n- Wouter van Doorn, Elias Judin, Pietro Monticone, and Daniel Morrison, On Some Problems from the Kourovka Notebook, arXiv:2607.17477 (2026). (primary): https://arxiv.org/abs/2607.17477\n Evidence used: Includes the class-transposition solution among the formally verified Kourovka results.\n- Pietro Monticone et al., Kourovka formalization repository, Problem 21.8 (2026). (primary): https://github.com/pitmonticone/Kourovka\n Evidence used: Contains the accompanying formal proof indexed to Problem 21.8.\n\n**Review notes.** No mathematical formulation defect found. The input's CT_k and CT(k) notations are distinct and were preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2518, "problem_number": "KOU-21.9", "title": "Kourovka Notebook Problem 21.9", "statement": "Let $F$ be a non-abelian free pro-$p$ group of finite rank. Can one find a finite collection $U_1,\\ldots,U_n$ of open subgroups of $F$, including $F$ itself, such that the only subgroup of $F$ which is contained in $U_i$ and is characteristic in $U_i$ for every $i$ is the trivial subgroup?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.9.\n\nDiscussion and literature:\nY. Barnea\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No finite characteristic-open-subgroup family with the stated trivial common characteristic-subgroup property was verified.\n\n**Verified partial progress.**\n\n- The source specifies a nonabelian finite-rank free pro-p group.\n\n**Full solution or refutation.**\n\nThe construction question remains open.\n\n**What remains.**\n\nUse characteristic subgroup growth and open-core intersections.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.9. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2519, "problem_number": "KOU-21.10", "title": "Kourovka Notebook Problem 21.10", "statement": "We call a group presentation finite if it represents a finite group. We say that a presentation is just finite if it is finite and is no longer finite on removal of any relation from it. Is it true that every finite group has a just finite presentation?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.10.\n\nDiscussion and literature:\nNote that if a group has a balanced presentation, then it is just finite. A similar argument can be applied for some p-groups using the Golod--Shafarevich inequality. Y. Barnea\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Marc Lackenby proved in May 2026 that every finite group admits a just finite presentation, giving a direct affirmative answer.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nFor every finite group G, Lackenby constructs a finite presentation of G such that deleting any relator makes the presented group infinite.\n\n**What remains.**\n\nThe stated existence problem is closed; optimization of presentation size is a separate question.\n\n**Sources checked.**\n\n- Marc Lackenby, Every finite group admits a just finite presentation, arXiv:2605.10402 (2026). (primary): https://arxiv.org/abs/2605.10402\n Evidence used: The abstract explicitly identifies Kourovka Problem 21.10 and states that it is resolved affirmatively for every finite group.\n- Kourovka Notebook, New Problems, 21st issue, Problem 21.10 (2026). (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Provides the exact original definition and question used by Lackenby.\n\n**Review notes.** Direct post-notebook solution; exact quantifiers match the input.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2520, "problem_number": "KOU-21.11", "title": "Kourovka Notebook Problem 21.11", "statement": "Can some or all groups of the following sorts be written as homomorphic images of nonprincipal ultraproducts of countable families of groups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.11.\n\nDiscussion and literature:\nThis is Question 18 in (G. M. Bergman, Pacific J. Math., 274 (2015) 451--495).\n\n(a) Infinite finitely generated groups of finite exponent.\n\n(b) For an infinite set X, the group of those permutations of X that move only finitely many elements.\n\nIt is known that no group of permutations containing an element with exactly one infinite orbit can be written as an image of such an ultraproduct (ibid.). G. M. Bergman\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Several image obstructions for nonprincipal ultraproducts are known, but the listed finite-exponent and finitary-permutation cases remain open.\n\n**Verified partial progress.**\n\n- Bergman records that a permutation group containing a one-infinite-orbit element cannot occur.\n\n**Full solution or refutation.**\n\nNo determination for all listed groups was verified.\n\n**What remains.**\n\nApply ultraproduct factorization obstructions to the two target classes.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.11. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Quotes the known obstruction and retains the questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2521, "problem_number": "KOU-21.12", "title": "Kourovka Notebook Problem 21.12", "statement": "Suppose that $U$ is a nonprincipal ultrafilter on $\\omega$, and $B$ is a group such that every element $b\\in B$ belongs to a subgroup of $B$ that is a homomorphic image of $\\mathbb Z^\\omega/U$. Must $B$ then be a homomorphic image of an ultraproduct group $(\\prod_{i\\in\\omega}G_i)/U$ for some groups $G_i$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.12.\n\nDiscussion and literature:\nThis is Question 19 in (G. M. Bergman, Pacific J. Math., 274 (2015) 451--495).\n\nAn affirmative answer would imply that every torsion group was such a homomorphic image for every $U$, and so would give positive answers to both parts of 21.11. G. M. Bergman\n\nNo, it need not (S. M. Corson, J. Algebra, 681 (2025), 306--317).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Corson's published counterexample gives a negative answer: the local homomorphic-image condition does not imply that the whole group is an ultraproduct image.\n\n**Verified partial progress.**\n\n- Corson studies a variant allowing the target ultraproduct to use a possibly different nonprincipal ultrafilter.\n- His counterexample is excluded from being a homomorphic image of any nonprincipal ultraproduct, which is stronger than needed for the input's same-ultrafilter formulation.\n\n**Full solution or refutation.**\n\nThe torsion group direct sum over n >= 1 of Z/nZ satisfies the local condition but is not a homomorphic image of any nonprincipal ultraproduct of groups.\n\n**What remains.**\n\nNothing for the stated yes/no question.\n\n**Sources checked.**\n\n- Samuel M. Corson, On homomorphic images of ultraproducts, Journal of Algebra 681 (2025), 306-317. (primary): https://doi.org/10.1016/j.jalgebra.2025.06.001\n Evidence used: The published paper states the ultraproduct-image question and constructs a torsion counterexample that is not an image of any nonprincipal ultraproduct.\n- Samuel M. Corson, On homomorphic images of ultraproducts, arXiv:2503.09228 (2025). (primary): https://arxiv.org/abs/2503.09228\n Evidence used: Author manuscript of the published counterexample theorem.\n\n**Review notes.** No statement defect found. The slight ultrafilter-quantifier difference in Corson's restatement is harmless because his non-representability conclusion is uniform over nonprincipal ultrafilters.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2522, "problem_number": "KOU-21.13", "title": "Kourovka Notebook Problem 21.13", "statement": "Does $\\mathbb Z^\\omega$ have a subgroup whose dual is free abelian of still larger rank (the largest possible being $2^{2^{\\aleph_0}}$)?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.13.\n\nDiscussion and literature:\nIt is known that the group $\\mathbb Z^\\omega$ has a subgroup whose dual is free abelian of rank $2^{\\aleph_0}$ (see 17.24 in Archive).\n\nThis is Question 11 in (G. M. Bergman, Portugaliae Math., 69 (2012) 69--84). G. M. Bergman\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A subgroup of Z^omega with free abelian dual of large rank is known, but the requested larger/maximal dual rank remains unresolved.\n\n**Verified partial progress.**\n\n- The source records an existing free-dual subgroup construction.\n\n**Full solution or refutation.**\n\nThe sharp rank question remains open.\n\n**What remains.**\n\nIncrease the dual rank construction toward 2^(2^aleph_0).\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.13. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States known nonmaximal construction and question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2523, "problem_number": "KOU-21.14", "title": "Kourovka Notebook Problem 21.14", "statement": "Suppose $\\alpha$ is an endomorphism of a group G such that for every group H and every homomorphism $f:G\\to H$, there exists an endomorphism $\\beta_f$ of $H$ such that $\\beta_f f=f\\alpha$. Must $\\alpha$ then be either an inner automorphism of G or the trivial endomorphism?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.14.\n\nDiscussion and literature:\nThis is Question 5 in (G. M. Bergman, Publ. Matem., 56 (2012), 91--126). G. M. Bergman\n\nYes, it must (F. Fournier-Facio, Preprint, 2026, https://arxiv.org/abs/2604.05728)\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Fournier-Facio's 2026 theorem implies that every endomorphism with the stated universal extension property is either trivial or an inner automorphism.\n\n**Verified partial progress.**\n\n- For each group G, the paper constructs an embedding G -> H that detects whether a nontrivial endomorphism of G is inner by whether it extends to H.\n\n**Full solution or refutation.**\n\nApply the assumed extension property to Fournier-Facio's detecting embedding. A nontrivial alpha must then be inner, while the trivial endomorphism is the stated alternative.\n\n**What remains.**\n\nIndependent peer review or publication of the recent preprint; no mathematical case remains in the stated dichotomy if the theorem is correct.\n\n**Sources checked.**\n\n- Francesco Fournier-Facio, An improved characterisation of inner automorphisms of groups, arXiv:2604.05728 (2026). (primary): https://arxiv.org/abs/2604.05728\n Evidence used: The abstract states an embedding theorem characterizing precisely the nontrivial endomorphisms that extend, and this directly gives the requested dichotomy.\n\n**Review notes.** The conclusion matches the universal quantifiers in the source statement. Confidence is moderated only because the exact-match source is a recent preprint.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2524, "problem_number": "KOU-21.15", "title": "Kourovka Notebook Problem 21.15", "statement": "Suppose B is a subgroup of the symmetric group $S_\\Omega$ on an infinite set $\\Omega$. Will the amalgamated free product $S_\\Omega *_B S_\\Omega$ of two copies of $S_\\Omega$ with amalgamation of B be embeddable in $S_\\Omega$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.15.\n\nDiscussion and literature:\nThis is a weakened form of the group case of Question 4.4 in (G. M. Bergman, Indag. Math., 18 (2007), 349--403).\n\nIt is known that $S_\\Omega *_B S_\\Omega$ need not be so embeddable by a map respecting B (Algebra Number Theory, 3 (2009), 847--879, \\S{} 10). G. M. Bergman\n\nNo, not necessarily: take $\\Omega$ countably infinite, $B$ a subgroup of $S_\\Omega$ which is not Borel, and $\\varphi:S_\\Omega *_B S_\\Omega\\to S_\\Omega$ an injective group homomorphism. Let $\\varphi_1$ be the restriction of $\\varphi$ to the first copy of $S_\\Omega$ and $\\varphi_2$ that to the second. Each $\\varphi_i$ is continuous (A. S. Kechris, C. Rosendal, Proc. Lond. Math. Soc., 94, no. 2 (2007), 302--350), so $\\operatorname{im}(\\varphi_i)$ is Borel in $S_\\Omega$ (as a continuous injective image of a Polish space). Now $B=\\varphi_1^{-1}(\\varphi(B))=\\varphi_1^{-1}(\\operatorname{im}(\\varphi_1)\\cap\\operatorname{im}(\\varphi_2))$ is Borel, a contradiction. (S. M. Corson, Letter of 26 January 2026; see also https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/03/solution-of-21.15.pdf.)\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** For countably infinite Omega, a non-Borel subgroup B of S_Omega gives a counterexample via automatic continuity.\n\n**Verified partial progress.**\n\n- Automatic continuity makes the restrictions of any hypothetical embedding to both copies of S_infinity continuous.\n- Lusin-Souslin makes both factor images Borel, while the normal-form intersection property then forces B itself to be Borel.\n\n**Full solution or refutation.**\n\nChoose a non-Borel B <= S_infinity. If S_infinity *_B S_infinity embedded in S_infinity, the two continuous injective factor maps would have Borel images and their intersection would be the image of B, contradicting that B is non-Borel.\n\n**What remains.**\n\nNothing for the existence question as stated; the counterexample already occurs for a countably infinite underlying set.\n\n**Sources checked.**\n\n- Samuel M. Corson, Solution of Kourovka Notebook Problem 21.15 (2026). (primary): https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/03/solution-of-21.15.pdf\n Evidence used: Gives the non-Borel-subgroup counterexample and the automatic-continuity proof.\n- Alexander S. Kechris and Christian Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proceedings of the London Mathematical Society 94 (2007), 302-350. (primary): https://arxiv.org/abs/math/0409567\n Evidence used: Provides the automatic-continuity input for homomorphisms from S_infinity used by the counterexample proof.\n\n**Review notes.** No formulation defect found. The proof uses the countable case, which suffices to refute the universal question over infinite Omega.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2525, "problem_number": "KOU-21.16", "title": "Kourovka Notebook Problem 21.16", "statement": "Let the width of a group (respectively, a monoid) H with respect to a generating set X mean the supremum over h $\\in$ H of the least length of a group word (respectively, a monoid word) in elements of X expressing h. A group (or monoid) is said to have finite width if its width with respect to every generating set is finite. (A common finite bound for these widths is not required.) Do there exist groups G having finite width as groups, but not as monoids?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.16.\n\nDiscussion and literature:\nThis is Question 9 in (G. M. Bergman, Bull. London Math. Soc., 38 (2006), 429--440). G. M. Bergman\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No group of finite group-width but infinite monoid-width was verified.\n\n**Verified partial progress.**\n\n- Bergman's formulation distinguishes group from monoid words.\n\n**Full solution or refutation.**\n\nThe separation question remains open.\n\n**What remains.**\n\nSeek generators whose inverse use is essential to bounded expression.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.16. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2526, "problem_number": "KOU-21.17", "title": "Kourovka Notebook Problem 21.17", "statement": "If $X$ is a class of groups, let $H(X)$ denote the class of homomorphic images of groups in $X$, let $S(X)$ denote the class of groups isomorphic to subgroups of groups in $X$, let $P(X)$ denote the class of groups isomorphic to (unrestricted) direct products of families of groups in $X$, and let $P_f(X)$ denote the class of groups isomorphic to direct products of finite families of groups in $X$. By Birkhoff's theorem, $H(S(P(X)))$ is the variety of groups generated by $X$. If $M$ is a class of metabelian groups, must $H(S(P_f(M)))\\subseteq S(H(P(S(M))))$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.17.\n\nDiscussion and literature:\nThis is Question 27 in (G. M. Bergman, Algebra Universalis, 26 (1989), 267--283). G. M. Bergman\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof or counterexample to the stated operator-class inclusion for metabelian groups was verified.\n\n**Verified partial progress.**\n\n- Birkhoff's theorem supplies the surrounding variety identity.\n\n**Full solution or refutation.**\n\nThe metabelian inclusion remains open.\n\n**What remains.**\n\nTest finitely generated metabelian varieties and product closures.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.17. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the precise inclusion problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2527, "problem_number": "KOU-21.18", "title": "Kourovka Notebook Problem 21.18", "statement": "Suppose that G is a finite group, and $A_1,A_2,A_3$ are subsets of G such that the multiplication map $A_1\\times A_2\\times A_3\\to G$ is bijective. Must the subgroup $\\langle A_2\\rangle$ generated by $A_2$ have order divisible by the cardinality $|A_2|$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.18.\n\nDiscussion and literature:\nThis is Question 8 in (G. M. Bergman, J. Iranian Math. Soc., 1 (2020), 157--161).\n\nIt is known (ibid.) that the corresponding statement is true for the subgroups $\\langle A_1\\rangle$ and $\\langle A_3\\rangle$. Moreover, $|A_2|$ will at least divide the order of the least subgroup containing $A_2$ and closed under conjugation by members of $A_1$, and similarly of the least subgroup containing $A_2$ and closed under conjugation by members of $A_3$. G. M. Bergman\n\nNo, it need not (M. I. Kabenyuk, Preprint, 2021, https://arxiv.org/abs/2102.08605).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The maintained Kourovka Notebook records a negative answer based on Kabenyuk's work on exact factorizations of finite groups.\n\n**Verified partial progress.**\n\n- Bergman's original note proves the corresponding divisibility statements for the first and third factors.\n- Those endpoint statements and the normal-closure divisibility observations do not imply the middle-factor assertion.\n\n**Full solution or refutation.**\n\nKabenyuk constructs a finite exact three-factor factorization for which |A_2| does not divide the order of ; the official issue-21 update accepts this as a negative solution.\n\n**What remains.**\n\nIndependently extract and check the exact finite counterexample from the latest version of Kabenyuk's evolving preprint, since the arXiv abstract does not expose the Problem 21.18 specialization.\n\n**Sources checked.**\n\n- Mikhail Kabenyuk, Factorizations of finite groups, arXiv:2102.08605 (2021; later revisions). (primary): https://arxiv.org/abs/2102.08605\n Evidence used: Primary preprint credited by the Notebook for the negative answer.\n- George M. Bergman, A note on factorizations of finite groups, Journal of the Iranian Mathematical Society 1 (2020), 157-161. (primary): https://math.berkeley.edu/~gbergman/papers/gp_factzn.pdf\n Evidence used: Contains the original Question 8 and proves the endpoint-factor divisibility results.\n- Kourovka Notebook, issue 21, Problem 21.18, checked 2026-08-17. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Records the answer as negative and attributes it to Kabenyuk's preprint.\n\n**Review notes.** No OCR defect found. Classification follows an authoritative maintained update plus the attributed primary paper, but the concrete witness should be checked before downstream theorem-level use.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2528, "problem_number": "KOU-21.19", "title": "Kourovka Notebook Problem 21.19", "statement": "Suppose that S and M are groups of finite Morley rank, S is an infinite group, and M is a non-trivial connected group definably and faithfully acting on S. This action is said to be irreducible if M does not leave invariant any definable non-trivial proper subgroup of S. Prove that if S is a simple group such that every proper definable subgroup of S is nilpotent, and the action of M on S is irreducible, then this action is equivalent to the action of S on itself by conjugation.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.19.\n\nDiscussion and literature:\nA. V. Borovik\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The finite-Morley-rank irreducible-action rigidity statement was not verified as proved.\n\n**Verified partial progress.**\n\n- The source spells out the minimal-simple/nilpotent-proper-subgroup hypotheses.\n\n**Full solution or refutation.**\n\nThe requested conjugation-action conclusion remains open.\n\n**What remains.**\n\nExploit definable representation and minimal-simple group structure.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.19. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the proof problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2529, "problem_number": "KOU-21.20", "title": "Kourovka Notebook Problem 21.20", "statement": "Prove that a simple group of finite Morley rank without involutions cannot act definably, faithfully, and irreducibly on a connected group other than on itself acting by conjugation.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.20.\n\nDiscussion and literature:\nA. V. Borovik\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The involution-free finite-Morley-rank action rigidity statement was not verified as proved.\n\n**Verified partial progress.**\n\n- The source specifies simple acting group and connected target.\n\n**Full solution or refutation.**\n\nThe requested conjugation-only conclusion remains open.\n\n**What remains.**\n\nDevelop signalizer/representation methods without involutions.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.20. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the proof problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2530, "problem_number": "KOU-21.21", "title": "Kourovka Notebook Problem 21.21", "statement": "Prove that a simple (that is, without proper non-trivial connected normal subgroups) algebraic group M over an algebraically closed field cannot act definably, faithfully, and irreducibly on a simple group of finite Morley rank other than M/Z(M).", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.21.\n\nDiscussion and literature:\nA. V. Borovik\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The algebraic-group-on-finite-Morley-rank rigidity statement was not verified as proved.\n\n**Verified partial progress.**\n\n- The source identifies M/Z(M) as the expected exception.\n\n**Full solution or refutation.**\n\nThe requested nonexistence statement remains open.\n\n**What remains.**\n\nCompare definable algebraic representations with simple target structure.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.21. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the proof problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2531, "problem_number": "KOU-21.22", "title": "Kourovka Notebook Problem 21.22", "statement": "Is the (standard, restricted) wreath product $G\\wr H$ of two finitely generated Hopfian groups Hopfian?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.22.\n\nDiscussion and literature:\nThe same question where G is assumed to be abelian or nilpotent is equivalent to Kaplansky's direct finiteness conjecture; see (H. Bradford, F. Fournier-Facio, Math. Z., 308, no. 4 (2024), Paper no. 58). H. Bradford, F. Fournier-Facio\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general Hopfian-wreath-product question remains open. For finitely generated abelian or nilpotent base groups, the universal assertion is equivalent to Kaplansky's stable-finiteness conjecture, yielding affirmative cases whenever stable finiteness is known for the acting group.\n\n**Verified partial progress.**\n\n- Bradford and Fournier-Facio characterize Hopficity for finitely generated abelian base groups through one-sided units in matrix algebras over group rings.\n- They prove that universal Hopficity for abelian bases, and equivalently for nilpotent bases, is equivalent to Kaplansky's stable-finiteness conjecture.\n- The conjecture is known for sofic acting groups, giving a large affirmative family; further positive structural results cover some centreless bases.\n\n**Full solution or refutation.**\n\nThe strongest reduction identifies a major special case with stable finiteness rather than resolving it: if the base is finitely generated nilpotent, its upper central series reduces Hopficity to the abelian layers, and those are governed by stable finiteness of the acting group's group rings.\n\n**What remains.**\n\nProve the unrestricted two-Hopfian-factor assertion or construct finitely generated Hopfian groups whose restricted wreath product is non-Hopfian. The universal abelian-base case already requires resolving stable finiteness beyond known classes.\n\n**Sources checked.**\n\n- The Kourovka Notebook, New Problems, 21st issue (2026), Problem 21.22. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the unrestricted question and records its connection with the Bradford--Fournier-Facio theorem.\n- H. Bradford and F. Fournier-Facio, Hopfian wreath products and the stable finiteness conjecture, Mathematische Zeitschrift 308 (2024), Paper 58. (primary): https://doi.org/10.1007/s00209-024-03589-3\n Evidence used: Formulates the general question, proves the abelian and nilpotent reductions to stable finiteness, and supplies additional positive cases.\n\n**Review notes.** The qualifier 'standard, restricted' fixes the wreath-product convention sufficiently; no statement repair was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2532, "problem_number": "KOU-21.23", "title": "Kourovka Notebook Problem 21.23", "statement": "A graph is called a cograph if it has no induced subgraph isomorphic to a path with 4 vertices. A graph is said to be chordal if it has no induced cycles with n vertices for every $n\\geqslant 4$. For a finite group G, the enhanced power graph E(G) is the graph with vertex set G and edges \\{x, y\\} for all $x\\ne y\\in G$ such that $\\langle x,y\\rangle$ is cyclic.\n\n(a) For a given integer $n\\geqslant 4$, determine the set of all finite nonabelian simple groups G such that E(G) has no induced cycles with n vertices.\n\n(b) Determine the set of all finite nonabelian simple groups G such that E(G) is chordal.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.23.\n\nDiscussion and literature:\nIn (Preprint, 2025, https://arxiv.org/abs/2510.18073) we proved that if the enhanced power graph of a given finite group is a cograph, then it is also chordal. Also the finite nonabelian simple groups whose enhanced power graph is a cograph are described, and additional information is obtained on finite nonabelian simple groups whose enhanced power graph has no induced cycles with 4 vertices. D. Bubboloni, F. Fumagalli, C. E. Praeger\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The finite nonabelian simple groups with cograph enhanced power graph are classified, and every such graph is chordal. Substantial bounds are known for the C4-free list, but neither the exact Cn-free lists nor the full chordal classification is complete.\n\n**Verified partial progress.**\n\n- The cograph cases are precisely PSL_2(q) for q at least 4, PSL_3(4), and Sz(q) for q=2^(2m+1) at least 8.\n- For finite groups, cograph enhanced power graph implies chordal enhanced power graph.\n- The C4-free simple groups are confined to an explicit short set of families and examples, with a smaller explicit list proved to occur; A_7 is chordal but not a cograph.\n\n**Full solution or refutation.**\n\nBubboloni, Fumagalli and Praeger convert cograph and C4-free conditions into restrictions on intersections of maximal cyclic subgroups and apply the classification of finite simple groups. This completely handles the cograph subclass and sharply narrows, but does not close, the requested classifications.\n\n**What remains.**\n\nDetermine the exact C4-free list, classify Cn-free enhanced power graphs for each n greater than four, and identify exactly which finite nonabelian simple groups have chordal enhanced power graph.\n\n**Sources checked.**\n\n- The Kourovka Notebook, New Problems, 21st issue (2026), Problem 21.23. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States both classification questions and records the 2025 cograph and C4-free progress.\n- D. Bubboloni, F. Fumagalli and C. E. Praeger, Enhanced power graphs of finite groups with cograph structure, arXiv:2510.18073 (2025). (primary): https://arxiv.org/abs/2510.18073\n Evidence used: Proves cograph implies chordal, classifies the simple cograph cases, bounds the C4-free cases, and explicitly poses the remaining questions.\n\n**Review notes.** The exact source typography leaves G and E(G) outside math delimiters in prose; it is preserved and merely flagged in report.md.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2533, "problem_number": "KOU-21.24", "title": "Kourovka Notebook Problem 21.24", "statement": "For a finite group G, the power graph P(G) is the graph with vertex set G and edges \\{x, y\\} for all $x\\ne y\\in G$ such that either $x\\in\\langle y\\rangle$ or $y\\in\\langle x\\rangle$. Is it true that, for every finite group G, if P(G) is a cograph, then P(G) is chordal?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.24.\n\nDiscussion and literature:\nCf. 21.23. This holds if every element of G has prime power order (D. Bubboloni, F. Fumagalli, C. E. Praeger, Preprint, 2025, https://arxiv.org/abs/2510.18073) and if G is a nonabelian simple group (J. Cameron, P. Manna, R. Mehatari, J. Algebra, 591 (2022), 59--74; J. Brachter, E. Kaja, J. Algebr. Comb., 58 (2023), 1095--1124). D. Bubboloni, F. Fumagalli, C. E. Praeger\n\nYes, it is true (M. Rundstr\\\"om, Preprint of 30 January 2026, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/04/21.24-runds.pdf; P. Monticone, Preprint of 29 March 2026, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/04/21_24-1.pdf)\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Every finite-group power graph that is a cograph is indeed chordal.\n\n**Verified partial progress.**\n\n- Earlier work handled groups in which every element has prime-power order and nonabelian finite simple groups.\n- The 2026 result covers all finite groups and is accompanied by a Lean formalization.\n- Independent manuscripts by Rundstrom and Monticone are linked from the Notebook.\n\n**Full solution or refutation.**\n\nThe full cograph-implies-chordal assertion is proved for the power graph of every finite group.\n\n**What remains.**\n\nNothing for the stated implication; journal publication of the recent manuscripts remains pending.\n\n**Sources checked.**\n\n- Wouter van Doorn, Elias Judin, Pietro Monticone, and Daniel Morrison, On Some Problems from the Kourovka Notebook, arXiv:2607.17477 (2026). (primary): https://arxiv.org/abs/2607.17477\n Evidence used: The abstract explicitly states the complete cograph-to-chordal theorem and formal verification.\n- Pietro Monticone et al., Kourovka formalization repository, Problem 21.24 (2026). (primary): https://github.com/pitmonticone/Kourovka\n Evidence used: Indexes the Lean proof to Problem 21.24.\n- M. Rundstrom, solution preprint for Kourovka Notebook Problem 21.24 (2026). (primary): https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/04/21.24-runds.pdf\n Evidence used: Independent manuscript proving the affirmative answer.\n\n**Review notes.** The background's text M. Rundstr\\\"om is a malformed LaTeX rendering of M. Rundstrom/Rundström; it does not affect the mathematical statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2534, "problem_number": "KOU-21.25", "title": "Kourovka Notebook Problem 21.25", "statement": "Let $G$ be a finite simple group and let $p_1,p_2$ be any (not necessarily distinct) prime divisors of $|G|$. Then can we always find Sylow $p_i$-subgroups $H_i$ such that $G=\\langle H_1,H_2\\rangle$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.25.\n\nDiscussion and literature:\n(T. Breuer, R. M. Guralnick).\n\nT. C. Burness\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The assertion is proved when one of the primes is 2, for all alternating and sporadic simple groups, and asymptotically for each fixed pair of primes. The remaining universal odd-prime cases in groups of Lie type are open.\n\n**Verified partial progress.**\n\n- Every finite nonabelian simple group is generated by a Sylow 2-subgroup together with a Sylow r-subgroup for any prime divisor r of its order.\n- Breuer and Guralnick verified the assertion for alternating and sporadic groups.\n- For fixed primes r and s, all sufficiently large finite simple groups divisible by both admit generating Sylow r- and s-subgroups.\n\n**Full solution or refutation.**\n\nGeneration by a Sylow 2-subgroup and any other requested Sylow subgroup follows from Burness--Guralnick's element-generation theorem. The later asymptotic theorem reduces the unresolved fixed odd-prime pairs to finitely many simple groups, but no uniform completion is yet known.\n\n**What remains.**\n\nResolve pairs of odd primes in the finite simple groups of Lie type left outside the existing family and asymptotic arguments.\n\n**Sources checked.**\n\n- The Kourovka Notebook, New Problems, 21st issue (2026), Problem 21.25. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Retains the universal Sylow-generation assertion as an open problem.\n- T. C. Burness and R. M. Guralnick, On the generation of simple groups by Sylow subgroups, Contemporary Mathematics 800 (2024), 81--100. (primary): https://arxiv.org/abs/2204.04311\n Evidence used: Proves the assertion whenever one selected prime is 2.\n- T. C. Burness, S. Gerhardt and R. M. Guralnick, Topological generation of simple algebraic groups, Journal of the European Mathematical Society 27 (2025), 2865--2951. (primary): https://doi.org/10.4171/JEMS/1425\n Evidence used: States the exact universal assertion as Conjecture 18 and proves its fixed-prime asymptotic form in Corollary 19.\n\n**Review notes.** If finite simple groups are taken to include cyclic prime-order groups, those cases are immediate; the substantive literature concerns nonabelian simple groups.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2535, "problem_number": "KOU-21.26", "title": "Kourovka Notebook Problem 21.26", "statement": "Let $G$ be a non-trivial finite group and let $p_1,\\ldots,p_k$ be the distinct prime divisors of $|G|$. For each $i$, let $H_i$ be a Sylow $p_i$-subgroup of $G$. Is it true that there exists an element $x\\in G$ such that for all $i$ the subgroup $H_i\\cap H_i^x$ is inclusion-minimal in $\\{H_i\\cap H_i^g:g\\in G\\}$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.26.\n\nDiscussion and literature:\n(F. Lisi, L. Sabatini).\n\nT. C. Burness\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The synchronized minimal-intersection conjecture is proved for all finite simple groups, all sufficiently large symmetric and alternating groups, and metanilpotent groups of odd order. It remains open for arbitrary finite groups, including general odd-order groups.\n\n**Verified partial progress.**\n\n- Lisi and Sabatini prove the conjecture for sufficiently large symmetric and alternating groups and for metanilpotent odd-order groups.\n- They prove the weaker general theorem that a finite group is not the union of proper Sylow normalizers corresponding to distinct primes.\n- Burness and Huang prove the conjecture for every non-alternating finite simple group, completing the simple-group case together with the alternating results.\n\n**Full solution or refutation.**\n\nFor simple groups the minimum possible Sylow intersection is trivial, and Burness--Huang show probabilistically that one element x can make all these intersections trivial simultaneously. Extension arguments needed for general finite groups remain unavailable.\n\n**What remains.**\n\nDevelop a normal-subgroup or extension mechanism that preserves the same conjugating element across all primes, thereby treating arbitrary soluble and nonsoluble finite groups.\n\n**Sources checked.**\n\n- The Kourovka Notebook, New Problems, 21st issue (2026), Problem 21.26. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the Lisi--Sabatini conjecture in the exact synchronized form.\n- F. Lisi and L. Sabatini, Sylow subgroups for distinct primes and intersection of nilpotent subgroups, Journal of Algebra 693 (2026), 824--837. (primary): https://doi.org/10.1016/j.jalgebra.2025.12.028\n Evidence used: Introduces the conjecture and proves it for sufficiently large symmetric and alternating groups and metanilpotent groups of odd order.\n- T. C. Burness and H. Y. Huang, On the intersections of nilpotent subgroups in simple groups, Proceedings of the London Mathematical Society 132 (2026), e70134. (primary): https://doi.org/10.1112/plms.70134\n Evidence used: Proves the Lisi--Sabatini conjecture for every non-alternating finite simple group and describes the general open range.\n\n**Review notes.** No OCR or mathematical formulation defect was found in the supplied statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2536, "problem_number": "KOU-21.27", "title": "Kourovka Notebook Problem 21.27", "statement": "A permutation on a set $\\Omega$ is called a derangement if it has no fixed points in $\\Omega$. Let G be a finite simple transitive permutation group. Is it true that every element in G is the product of two derangements?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.27.\n\nDiscussion and literature:\n(M. Larsen, A. Shalev, P. H. Tiep).\n\nT. C. Burness\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every element is a product of two derangements for sufficiently large finite simple transitive groups, for all alternating groups, and for all finite simple primitive groups with soluble point stabilizer. The unrestricted finite simple assertion remains open.\n\n**Verified partial progress.**\n\n- Larsen, Shalev and Tiep prove the assertion for all sufficiently large finite simple transitive permutation groups.\n- They prove it for every alternating group in every transitive permutation action, without a size restriction.\n- Burness and Fusari remove the size restriction for finite simple primitive actions with soluble point stabilizer and establish further explicit families.\n\n**Full solution or refutation.**\n\nThe known proofs use character estimates and products of conjugacy classes of derangements. They cover the asymptotic regime and several complete structural families, but do not uniformly eliminate the remaining small Lie-type and nonsoluble-stabilizer cases.\n\n**What remains.**\n\nProve derangement width two for the remaining finite simple groups and actions, particularly primitive actions with nonsoluble point stabilizers outside the existing explicit families.\n\n**Sources checked.**\n\n- The Kourovka Notebook, New Problems, 21st issue (2026), Problem 21.27. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Retains the unrestricted finite simple transitive assertion as open.\n- M. Larsen, A. Shalev and P. H. Tiep, Products of derangements in simple permutation groups, Forum of Mathematics, Sigma 10 (2022), e83. (primary): https://doi.org/10.1017/fms.2022.69\n Evidence used: Proves the sufficiently-large theorem and the all-alternating-groups theorem, and states the unrestricted conjecture.\n- T. C. Burness and M. Fusari, On derangements in simple permutation groups, Forum of Mathematics, Sigma 13 (2025), e98. (primary): https://doi.org/10.1017/fms.2025.10064\n Evidence used: Proves derangement width two for finite simple primitive groups with soluble point stabilizer and surveys the remaining conjecture.\n\n**Review notes.** The source leaves G outside math delimiters in the final two sentences; report.md preserves and flags this typography.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2537, "problem_number": "KOU-21.28", "title": "Kourovka Notebook Problem 21.28", "statement": "Let $G$ be a finite simple transitive permutation group, and let $\\delta(G)$ be the proportion of derangements in $G$. Is it true that $\\delta(G)\\geqslant 89/325$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.28.\n\nDiscussion and literature:\nNote that $\\delta(G)=89/325$ for the action of the Tits group $G={}^2F_4(2)'$ on the cosets of a maximal parabolic subgroup of the form $2^2.[2^8].S_3$ (Forum Math. Sigma, 13 (2025), paper no. e98, 62 pp.). T. C. Burness, M. Fusari\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The sharp bound 89/325 is proved for finite simple primitive groups with soluble point stabilizer, with equality in the stated Tits-group action. The universal bound for arbitrary simple transitive groups remains conjectural.\n\n**Verified partial progress.**\n\n- Burness and Fusari prove delta(G) at least 89/325 for every finite simple primitive group with soluble point stabilizer.\n- Equality occurs exactly for the Tits group acting on the cosets of the parabolic subgroup 2^2.[2^8].S_3, so the proposed constant cannot be improved.\n- A positive absolute lower bound is known for all sufficiently large finite simple transitive groups, but it is much smaller than 89/325.\n\n**Full solution or refutation.**\n\nClassification of simple groups with soluble maximal subgroups, together with character and fixed-point calculations, proves the optimal constant throughout that family. The remaining obstacle is not sharpness but coverage of nonsoluble point stabilizers.\n\n**What remains.**\n\nProve the 89/325 lower bound for primitive actions with nonsoluble point stabilizer; imprimitive transitive actions can be compared with their induced primitive block actions.\n\n**Sources checked.**\n\n- The Kourovka Notebook, New Problems, 21st issue (2026), Problem 21.28. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the universal bound and records the equality example.\n- T. C. Burness and M. Fusari, On derangements in simple permutation groups, Forum of Mathematics, Sigma 13 (2025), e98. (primary): https://doi.org/10.1017/fms.2025.10064\n Evidence used: Proves the sharp bound and equality characterization for primitive groups with soluble point stabilizer, while presenting the broader bound as conjectural.\n\n**Review notes.** No statement repair was made; the displayed equality example in the source background agrees with the sharp theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2538, "problem_number": "KOU-21.29", "title": "Kourovka Notebook Problem 21.29", "statement": "Let $G\\leqslant\\operatorname{Sym}(\\Omega)$ be a finite primitive permutation group with a regular suborbit (that is, $G$ has a trivial 2-point stabiliser). Then is it true that for all $\\alpha,\\beta\\in\\Omega$, there exists $\\gamma\\in\\Omega$ such that the 2-point stabilisers $G_{\\alpha,\\gamma}$ and $G_{\\beta,\\gamma}$ are both trivial?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.29.\n\nDiscussion and literature:\nT. C. Burness, M. Giudici\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The question is the Burness--Giudici common-neighbor conjecture for Saxl graphs. It is proved for several broad O'Nan--Scott families, including all almost simple primitive base-two groups with soluble point stabilizer, but remains open in general.\n\n**Verified partial progress.**\n\n- Burness and Giudici reformulate the condition as: any two vertices of the Saxl graph have a common neighbor.\n- Burness and Huang prove the conjecture for all almost simple primitive base-two groups with soluble point stabilizer and for the relevant L_2(q) families.\n- Additional papers establish product-type, diagonal-type, twisted-wreath, affine, and rank-one subfamilies, without exhausting all primitive groups.\n\n**Full solution or refutation.**\n\nIn the Saxl graph, an edge is exactly a pair with trivial two-point stabilizer, so the requested gamma is a common neighbor of alpha and beta. Existing probabilistic and family-specific analyses prove this graph property in major cases but not for every primitive base-two group.\n\n**What remains.**\n\nComplete the common-neighbor conjecture for the unresolved almost simple nonsoluble-stabilizer actions and remaining affine and other O'Nan--Scott families.\n\n**Sources checked.**\n\n- The Kourovka Notebook, New Problems, 21st issue (2026), Problem 21.29. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the common-neighbor property directly in two-point-stabilizer language.\n- T. C. Burness and M. Giudici, On the Saxl graph of a permutation group, Mathematical Proceedings of the Cambridge Philosophical Society 168 (2020), 219--248. (primary): https://doi.org/10.1017/S0305004118000610\n Evidence used: Introduces Saxl graphs and states the common-neighbor conjecture equivalent to this record.\n- T. C. Burness and H. Y. Huang, On the Saxl graphs of primitive groups with soluble stabilisers, Algebraic Combinatorics 5 (2022), 1053--1087. (primary): https://doi.org/10.5802/alco.238\n Evidence used: Proves the common-neighbor conjecture for almost simple primitive base-two groups with soluble point stabilizer and details the remaining general conjecture.\n\n**Review notes.** The parenthetical definition correctly encodes existence of a base of size two in the transitive primitive setting; it was preserved verbatim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2539, "problem_number": "KOU-21.30", "title": "Kourovka Notebook Problem 21.30", "statement": "(Well-known question). A discrete group G is said to have the Haagerup property (also known as Gromov's a-T-menability property) if there exists a metrically proper isometric action of G on a (possibly infinite-dimensional) Hilbert space. Are all 1-relator groups Haagerup groups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.30.\n\nDiscussion and literature:\nJ. O. Button\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Haagerup property is known for one-relator groups with torsion and for several geometric subclasses, but no proof or counterexample is known for all torsion-free one-relator groups. The 2026 Kourovka issue retains the general question.\n\n**Verified partial progress.**\n\n- Wise proves that every one-relator group with torsion has a quasiconvex Magnus--Moldavanskii hierarchy and is virtually compact special.\n- Virtual specialness supplies a proper action on a CAT(0) cube complex and hence the Haagerup property; finite-index passage preserves the property.\n- Free groups, surface groups, amenable one-relator groups, and suitable small-cancellation or cubulated one-relator classes are affirmative.\n\n**Full solution or refutation.**\n\nThe torsion case is handled geometrically by Wise's hierarchy and virtual-specialness theorem. This leaves the general torsion-free case, where the same hierarchy need not be quasiconvex and current cubulation criteria do not apply uniformly.\n\n**What remains.**\n\nConstruct proper affine Hilbert-space actions for all torsion-free one-relator groups or find a torsion-free one-relator counterexample to the Haagerup property.\n\n**Sources checked.**\n\n- The Kourovka Notebook, New Problems, 21st issue (2026), Problem 21.30. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Lists the all-one-relator-groups question as open in 2026.\n- D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy, Annals of Mathematics Studies 209, Princeton University Press, 2021. (primary): https://doi.org/10.1515/9780691213507\n Evidence used: Proves that the Magnus--Moldavanskii hierarchy is quasiconvex for one-relator groups with torsion and derives their virtual specialness.\n- B. Stucky, Cubulating one-relator products with torsion, Groups, Geometry, and Dynamics 15 (2021), 691--754. (primary): https://doi.org/10.4171/GGD/619\n Evidence used: Proves proper cocompact cubulations for a broad torsion one-relator-product class and records the full one-relator torsion virtual-specialness theorem.\n\n**Review notes.** The source definition says 'isometric' rather than the customary 'affine isometric' action and leaves G outside math delimiters. Report.md preserves and flags both points.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2540, "problem_number": "KOU-21.31", "title": "Kourovka Notebook Problem 21.31", "statement": "Conjecture: If N is a finite soluble group, then any regular subgroup in the holomorph Hol(N) of N is also soluble.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.31.\n\nDiscussion and literature:\nN. Byott\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No insoluble regular subgroup of the holomorph of a finite soluble group is known. Computations and order-specific theorems exclude broad ranges, and any irreducible counterexample has tightly constrained PSL_3(2)-type nonabelian composition factors.\n\n**Verified partial progress.**\n\n- Tsang and Qin exclude counterexamples of order at most 2000 and prove the conjecture for infinitely many orders, including several order-theoretic families.\n- Byott classifies irreducible insoluble transitive subgroups with soluble point stabilizers in holomorphs of finite soluble groups.\n- In any irreducible configuration, every nonabelian composition factor is PSL_3(2), and every maximal normal subgroup of the soluble group N has index 2.\n\n**Full solution or refutation.**\n\nThe minimal-counterexample strategy reduces the conjecture to an exceptionally narrow configuration controlled by the simple group of order 168. This is strong structural evidence but neither excludes nor constructs a regular insoluble subgroup.\n\n**What remains.**\n\nEliminate Byott's restricted minimal-counterexample configurations or realize one as a regular subgroup. Equivalently, decide whether a finite skew brace can have soluble additive group and insoluble multiplicative group.\n\n**Sources checked.**\n\n- The Kourovka Notebook, New Problems, 21st issue (2026), Problem 21.31. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Retains the soluble-holomorph conjecture as open.\n- N. P. Byott, On insoluble transitive subgroups in the holomorph of a finite soluble group, Journal of Algebra 638 (2024), 1--31. (primary): https://doi.org/10.1016/j.jalgebra.2023.10.001\n Evidence used: Classifies irreducible transitive configurations and proves the PSL_3(2) composition-factor and index-two restrictions.\n- C. Tsang and C. Qin, On the solvability of regular subgroups in the holomorph of a finite solvable group, International Journal of Algebra and Computation 30 (2020), 253--265. (primary): https://doi.org/10.1142/S0218196719500735\n Evidence used: Excludes counterexamples for broad order families, reports the finite computations, and formulates the direction still lacking examples.\n\n**Review notes.** The source leaves N and Hol(N) outside math delimiters; report.md preserves the statement and flags only the typography.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2541, "problem_number": "KOU-21.32", "title": "Kourovka Notebook Problem 21.32", "statement": "Is the following problem decidable, and if so, what is its complexity? Given a finite group G, is there a finite group H such that the derived subgroup of H is isomorphic to G?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.32.\n\nDiscussion and literature:\nP. J. Cameron\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No decision procedure or complexity classification for recognizing finite derived subgroups was verified.\n\n**Verified partial progress.**\n\n- The input/output formulation is explicit over finite groups.\n\n**Full solution or refutation.**\n\nThe decidability question remains open.\n\n**What remains.**\n\nTranslate the condition into finite presentation/cohomological extension data.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.32. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2542, "problem_number": "KOU-21.33", "title": "Kourovka Notebook Problem 21.33", "statement": "Does an analogue of Dunwoody's theorem hold for totally disconnected locally compact groups, that is, must a tdlc group of rational discrete cohomological dimension at most 1 be topologically isomorphic to the fundamental group of a graph of profinite groups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.33.\n\nDiscussion and literature:\nI. Castellano\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No tdlc analogue of Dunwoody's graph-of-profinite-groups theorem was verified.\n\n**Verified partial progress.**\n\n- The source identifies rational discrete cohomological dimension at most one as the hypothesis.\n\n**Full solution or refutation.**\n\nThe structural theorem remains open.\n\n**What remains.**\n\nDevelop tdlc accessibility and profinite vertex-group decompositions.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.33. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the analogue question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2543, "problem_number": "KOU-21.34", "title": "Kourovka Notebook Problem 21.34", "statement": "(Well-known problem). A group $G$ is a unique product group if, for any nonempty finite subsets $A,B$ of $G$, there exists an element of $G$ which can be written uniquely as $ab$ with $a\\in A$ and $b\\in B$. A group $G$ is locally invariant orderable if $G$ admits a partial order $<$ such that for all $g,h\\in G$ with $h\\ne 1$, we have either $gh>g$ or $gh^{-1}>g$. Does there exist a unique product group which is not locally invariant orderable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.34.\n\nDiscussion and literature:\nA. Clay\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No unique-product group failing local invariant orderability was verified.\n\n**Verified partial progress.**\n\n- The source supplies both order/combinatorial definitions.\n\n**Full solution or refutation.**\n\nThe separation question remains open.\n\n**What remains.**\n\nSearch among non-orderable unique-product constructions.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.34. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the well-known problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2544, "problem_number": "KOU-21.35", "title": "Kourovka Notebook Problem 21.35", "statement": "Let $G$ be a finite group, $w$ a multilinear commutator group-word, and $p$ a prime. Suppose that $p$ divides the order $|xy|$ whenever $x$ is a $w$-value of $p'$-order in $G$ and $y$ is a $w$-value in $G$ of order divisible by $p$. Is it true that then the verbal subgroup $w(G)$ must be $p$-nilpotent?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.35.\n\nDiscussion and literature:\nWithout the assumption that $w$ be multilinear, the answer is negative. An affirmative answer has been obtained in several special cases (J. Algebra, 609 (2022), 926--936). Y. Contreras Rojas, V. Grazian, C. Monetta\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Dropping multilinearity yields a negative answer, but no result for multilinear commutator words under the stated order condition was verified.\n\n**Verified partial progress.**\n\n- The source explicitly distinguishes the negative non-multilinear case.\n\n**Full solution or refutation.**\n\nThe multilinear p-nilpotence implication remains open.\n\n**What remains.**\n\nExploit multilinear commutator closure to force p-nilpotence.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.35. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records the counterexample outside the hypothesis.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2545, "problem_number": "KOU-21.36", "title": "Kourovka Notebook Problem 21.36", "statement": "Let a hierarchy of tdlc groups $\\mathbf H\\mathcal K$ be defined analogously to Kropholler's hierarchy in 15.45, with $\\mathcal K$ being the class of profinite groups and with the cell stabilisers of the admissible action required to be open. Is it true that $\\mathbf H\\mathcal K$ is closed under profinite extensions, that is, $(\\mathbf H_\\alpha\\mathcal K)\\mathcal K\\subseteq \\mathbf H_\\alpha\\mathcal K$ for every $\\alpha$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.36.\n\nDiscussion and literature:\nKropholler's hierarchy (see 15.45) is closed under finite extensions, that is, $(\\mathbf H_\\alpha\\mathcal F)\\mathcal F\\subseteq \\mathbf H_\\alpha\\mathcal F$ for every $\\alpha$ (P. Kropholler, J. Pure Appl. Algebra, 90 (1993), 55--67).\n\nG. C. Cook\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Kropholler's discrete hierarchy is closed under finite extensions, but closure of the tdlc/profinite hierarchy was not verified.\n\n**Verified partial progress.**\n\n- The source records the finite-extension analogue.\n\n**Full solution or refutation.**\n\nThe profinite-extension closure question remains open.\n\n**What remains.**\n\nInduct on hierarchy level while controlling open stabilizers.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.36. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Contrasts known discrete closure with tdlc problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2546, "problem_number": "KOU-21.37", "title": "Kourovka Notebook Problem 21.37", "statement": "By definition, a constructible totally disconnected, locally compact (tdlc) group is the result of a sequence of profinite extensions and ascending HNN-extensions starting from the trivial group. Are soluble tdlc groups of type $FP_\\infty$ constructible?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.37.\n\nDiscussion and literature:\nAs in the discrete case, soluble constructible tdlc groups have type $FP_\\infty$ (G. C. Cook, I. Castellano, J. Algebra, 543 (2020), 54--97).\n\nG. C. Cook\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Soluble constructible tdlc groups have the expected finiteness properties, but the converse from FP_infinity was not verified.\n\n**Verified partial progress.**\n\n- The source records the forward constructible implication.\n\n**Full solution or refutation.**\n\nThe characterization remains open.\n\n**What remains.**\n\nAdapt discrete soluble FP-infinity classification to tdlc groups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.37. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Contrasts the known direction with the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2547, "problem_number": "KOU-21.38", "title": "Kourovka Notebook Problem 21.38", "statement": "The spread of a group $G$ is the greatest nonnegative integer $k$ such that for all nontrivial elements $x_1,\\ldots,x_k\\in G$ there exists $y\\in G$ such that $\\langle x_1,y\\rangle=\\cdots=\\langle x_k,y\\rangle=G$, or is $\\infty$ in case there is no such maximum. Does there exist a group with spread equal to 1?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.38.\n\nDiscussion and literature:\n(S. Harper, C. Donoven).\n\nSuch a group must be infinite if it exists (T. C. Burness, R. M. Guralnick, S. Harper, Ann. Math., 193 (2021), 619--687). S. Corson\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A group of spread one must be infinite; no example or nonexistence theorem was verified.\n\n**Verified partial progress.**\n\n- The source records the necessary infinitude.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nConstruct an infinite 2-generated group with exact spread one.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.38. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the necessary condition and question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2548, "problem_number": "KOU-21.39", "title": "Kourovka Notebook Problem 21.39", "statement": "Are there any locally finite, characteristically simple groups with finitely many orbits under automorphisms that are not residually finite?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.39.\n\nDiscussion and literature:\nIt is known that there exist residually finite, locally finite, characteristically simple groups with finitely many orbits under automorphisms (A. B. Apps, J. Algebra, 81 (1983), 320--339).\n\nA. Dantas, E. de Melo\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Residually finite locally finite characteristically simple examples with finitely many automorphism orbits are known, but a non-residually-finite one was not verified.\n\n**Verified partial progress.**\n\n- The source records the positive residually finite examples.\n\n**Full solution or refutation.**\n\nThe requested non-residually-finite example remains open.\n\n**What remains.**\n\nStudy automorphism-orbit rigidity in non-residually-finite locally finite simple groups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.39. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Distinguishes known examples from target.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2549, "problem_number": "KOU-21.40", "title": "Kourovka Notebook Problem 21.40", "statement": "Let G be a subgroup of GL(n, Q) with finitely many orbits under automorphisms. Is G a virtually soluble group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.40.\n\nDiscussion and literature:\nA. Dantas, E. de Melo\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem that finite automorphism-orbit linear groups over Q are virtually soluble was verified.\n\n**Verified partial progress.**\n\n- The source gives the GL(n,Q) setting.\n\n**Full solution or refutation.**\n\nThe virtual-solubility question remains open.\n\n**What remains.**\n\nUse arithmetic linear-group structure and orbit-growth arguments.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.40. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2550, "problem_number": "KOU-21.41", "title": "Kourovka Notebook Problem 21.41", "statement": "A group is said to be self-similar if it admits a faithful state-closed representation by automorphisms of a regular one-rooted m-tree for some m. Can a torsion-free finitely presented metabelian group which is self-similar contain a subgroup isomorphic to the restricted wreath product H = $\\mathbb Z\\wr\\mathbb Z$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.41.\n\nDiscussion and literature:\nIt is known that $\\mathbb Z\\wr\\mathbb Z$ itself is self-similar (A. C. Dantas, T. M. G. Santos, S. N. Sidki, J. Algebra, 567 (2021), 564--581). A. Dantas, S. Sidki\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The restricted wreath product Z wr Z is self-similar in the intransitive sense used by the question, but no torsion-free finitely presented metabelian self-similar group containing it was verified.\n\n**Verified partial progress.**\n\n- Dantas, Santos, and Sidki construct a faithful, generally intransitive self-similar representation of Z wr Z.\n- Dantas and Sidki's earlier negative theorem concerns transitive self-similarity only, so the two results are compatible.\n\n**Full solution or refutation.**\n\nNo source deciding the existence of the ambient group with all four required properties was found.\n\n**What remains.**\n\nConstruct such an ambient torsion-free finitely presented metabelian self-similar group, or prove that self-similarity and finite presentability obstruct the embedding.\n\n**Sources checked.**\n\n- The Kourovka Notebook, 21st issue (2026), Problem 21.41. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Poses the embedding question and records self-similarity of Z wr Z.\n- Alex C. Dantas, Tulio M. G. Santos, Said N. Sidki, Intransitive Self-similar Groups, Journal of Algebra 567 (2021), 564-581, arXiv:2004.08941. (primary): https://arxiv.org/abs/2004.08941\n Evidence used: Constructs faithful self-similar representations for Z wr Z without imposing first-level transitivity.\n- Alex Dantas, Said Sidki, On self-similarity of wreath products of abelian groups, Groups, Geometry, and Dynamics 12 (2018), DOI 10.4171/GGD/462. (primary): https://doi.org/10.4171/GGD/462\n Evidence used: Rules out transitive self-similarity for the relevant wreath product, clarifying the scope of the later result.\n\n**Review notes.** The phrase 'regular one-rooted m-tree' is unusual but was preserved. Transitive and intransitive self-similarity were kept distinct.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2551, "problem_number": "KOU-21.42", "title": "Kourovka Notebook Problem 21.42", "statement": "Let $\\mathcal T_{d,c}$ denote the class of $d$-generated, torsion-free nilpotent groups having class $c$. Are there $\\mathcal T_{3,3}$-groups that are not self-similar?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.42.\n\nDiscussion and literature:\nIt is known that $\\mathcal T_{d,2}$-groups are self-similar for all $d$, that $\\mathcal T_{2,3}$-groups are self-similar, and that there are $\\mathcal T_{4,3}$-groups that are not self-similar (A. Berlatto, T. Santos, Preprint, 2025, https://arxiv.org/abs/2509.16947).\n\nA. Dantas, S. Sidki\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** All two-generated torsion-free nilpotent groups of class three are self-similar, while a four-generated class-three counterexample exists; the requested three-generator case remains open.\n\n**Verified partial progress.**\n\n- Every group in T_{2,3} has a faithful transitive self-similar representation.\n- There is a group in T_{4,3} which is not self-similar.\n\n**Full solution or refutation.**\n\nNo T_{3,3} counterexample or theorem ruling one out was verified.\n\n**What remains.**\n\nDetermine whether the rank-four obstruction can occur at generator rank three, or prove all T_{3,3} groups self-similar.\n\n**Sources checked.**\n\n- The Kourovka Notebook, 21st issue (2026), Problem 21.42. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the rank-three question and the known rank-two/rank-four bounds.\n- Adilson Berlatto, Tulio Santos, On self-similarity of finitely generated torsion-free nilpotent groups, arXiv:2509.16947 (2025). (primary): https://arxiv.org/abs/2509.16947\n Evidence used: Proves the rank-two class-three theorem and gives a rank-four class-three non-self-similar example.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2552, "problem_number": "KOU-21.43", "title": "Kourovka Notebook Problem 21.43", "statement": "Conjecture: Suppose that for a fixed positive integer $k$ at least half of the elements of a finite group $G$ have order $k$. Then $G$ is solvable.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.43.\n\nDiscussion and literature:\nM. Deaconescu\n\nThis is not always true. In a direct product of $N$ copies of $A_5$ there are $60^N$ elements, while the number of elements of order exactly 30 is $60^N-45^N-40^N-36^N+25^N+21^N+16^N-1$. As $N\\to\\infty$, the ratio of elements of order 30 converges to 1. (L. Tae Young, Letter of 9 January 2026).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Direct powers of the nonsolvable group A_5 have a proportion tending to one of elements of the single fixed order 30.\n\n**Verified partial progress.**\n\n- A_5 has 15, 20, and 24 elements of orders 2, 3, and 5 respectively, besides the identity.\n- In A_5^N, order 30 means that coordinate orders collectively include 2, 3, and 5; inclusion-exclusion gives the exact count displayed in the Notebook.\n\n**Full solution or refutation.**\n\nThe number of order-30 elements in A_5^N is 60^N - 45^N - 40^N - 36^N + 25^N + 21^N + 16^N - 1. Its ratio to 60^N tends to 1, so sufficiently large nonsolvable A_5^N violate the conjecture with fixed k=30.\n\n**What remains.**\n\nNothing for the stated conjecture; determining thresholds or classifying exceptional groups would be a different question.\n\n**Sources checked.**\n\n- Kourovka Notebook, issue 21, Problem 21.43, recording L. Tae Young's communication of 9 January 2026, checked 2026-08-17. (authoritative_secondary): https://alglog.org/21tkt.pdf\n Evidence used: Records the A_5^N counterexample and exact inclusion-exclusion formula.\n\n**Review notes.** The displayed formula was checked symbolically by inclusion-exclusion from the standard element-order counts in A_5; no computation was run.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2553, "problem_number": "KOU-21.44", "title": "Kourovka Notebook Problem 21.44", "statement": "Let $W_n=A_5\\wr\\cdots\\wr A_5$ be the $n$-times iterated permutational wreath product of $A_5$ in its natural action (so $W_n$ acts on $5^n$ points), and let $W=\\varprojlim W_n$ be the inverse limit (infinite iterated wreath product of $A_5$). Does $W$ contain a finitely generated dense subgroup of subexponential growth?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.44.\n\nDiscussion and literature:\nS. Eberhard\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The infinite iterated A5 wreath product is topologically finitely generated and therefore has finitely generated dense subgroups, but no such subgroup of subexponential growth was verified.\n\n**Verified partial progress.**\n\n- Bondarenko characterizes topological finite generation of infinite iterated wreath products through the product of factor abelianizations.\n- Because A5 is perfect, the criterion gives topological finite generation of W; abstract generators of a dense topologically generated subgroup supply finite generation and density.\n\n**Full solution or refutation.**\n\nThe remaining growth constraint was not decided by the sources found.\n\n**What remains.**\n\nConstruct topological generators whose abstract subgroup has subexponential growth, or prove every finitely generated dense subgroup has exponential growth.\n\n**Sources checked.**\n\n- The Kourovka Notebook, 21st issue (2026), Problem 21.44. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Poses the dense subgroup question with the subexponential-growth requirement.\n- Ievgen Bondarenko, Finite generation of iterated wreath products, arXiv:1002.0320 (2010). (primary): https://arxiv.org/abs/1002.0320\n Evidence used: Proves the topological finite-generation criterion; it applies because A5 has trivial abelianization.\n\n**Review notes.** The inference from topological finite generation to existence of a finitely generated dense abstract subgroup is immediate from the definition.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2554, "problem_number": "KOU-21.45", "title": "Kourovka Notebook Problem 21.45", "statement": "(Well-known problem). Does there exist a finitely presented (infinite) simple group requiring more than two generators?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.45.\n\nDiscussion and literature:\nCf. 6.44 in Archive. F. Fournier-Facio\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Large families of finitely generated infinite simple groups are proved two-generated, but no finitely presented infinite simple group with minimal generator number greater than two was verified.\n\n**Verified partial progress.**\n\n- Every finitely generated vigorous simple Cantor-homeomorphism group in the class studied by Bleak, Elliott, and Hyde is two-generated by torsion elements.\n- The theorem covers many standard simple families, including groups of Higman-Thompson, Brin-Thompson, Rover, and Nekrashevych type under the paper's hypotheses.\n\n**Full solution or refutation.**\n\nThe current Kourovka tracker continues to list the general existence problem, and the search found no qualifying example.\n\n**What remains.**\n\nProduce a finitely presented infinite simple group whose abelian generating rank is at least three, or prove a universal two-generation theorem.\n\n**Sources checked.**\n\n- The Kourovka Notebook, 21st issue (2026), Problem 21.45; cross-reference to archived Problem 6.44. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Lists the well-known problem as open in the current issue.\n- Collin Bleak, Luna Elliott, James Hyde, Sufficient conditions for a group of homeomorphisms of the Cantor set to be two-generated, Journal of the Institute of Mathematics of Jussieu 23 (2024), 2825-2858, DOI 10.1017/S1474748024000045. (primary): https://doi.org/10.1017/S1474748024000045\n Evidence used: Proves two-generation for a broad family of finitely generated simple groups.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2555, "problem_number": "KOU-21.46", "title": "Kourovka Notebook Problem 21.46", "statement": "(Well-known problem). Does there exist a finitely presented (infinite) simple group of finite cohomological dimension greater than 2?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.46.\n\nDiscussion and literature:\nF. Fournier-Facio\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finitely presented torsion-free infinite simple groups of cohomological dimension two are known, but no example of finite cohomological dimension greater than two was verified.\n\n**Verified partial progress.**\n\n- Rattaggi constructs a finitely presented torsion-free simple group acting cocompactly on a product of two trees.\n- The resulting finite aspherical square-complex model supplies the boundary case cohomological dimension two.\n\n**Full solution or refutation.**\n\nThe current tracker continues to pose the higher-dimensional existence question; no qualifying primary construction was found.\n\n**What remains.**\n\nConstruct a torsion-free finitely presented simple group with a finite-dimensional classifying space of dimension at least three, or prove a dimension-two obstruction.\n\n**Sources checked.**\n\n- The Kourovka Notebook, 21st issue (2026), Problem 21.46. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Lists the finite-cohomological-dimension-greater-than-two problem as open.\n- Diego Rattaggi, A finitely presented torsion-free simple group, Journal of Group Theory 10 (2007), 363-371, DOI 10.1515/JGT.2007.028. (primary): https://doi.org/10.1515/JGT.2007.028\n Evidence used: Constructs a torsion-free finitely presented simple cocompact lattice in a product of two regular trees.\n\n**Review notes.** Finite ordinary cohomological dimension forces torsion-freeness; torsion examples such as Thompson's T were not treated as qualifying.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2556, "problem_number": "KOU-21.47", "title": "Kourovka Notebook Problem 21.47", "statement": "(Well-known problem). Does there exist a finitely presented group $G$ such that $G\\cong G\\times H$ for some non-trivial group $H$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.47.\n\nDiscussion and literature:\nThe first finitely generated example was constructed in (J. M. Tyrer Jones, J. Austral. Math. Soc., 17 (1974), 174--196).\n\nA finitely presented group that surjects onto its own direct square was constructed in (G. Baumslag, C. F. Miller, III, Bull. London Math. Soc., 20, no. 3 (1988), 239--244). F. Fournier-Facio\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The requested absorption isomorphism exists for a finitely generated group, and finite presentation is known together with a surjection onto the direct square, but no finitely presented absorption example was verified.\n\n**Verified partial progress.**\n\n- Tyrer Jones constructs a finitely generated group isomorphic to a proper direct factor of itself.\n- Baumslag and Miller construct a finitely presented group which surjects onto its own direct square.\n\n**Full solution or refutation.**\n\nNo source combining finite presentability with an isomorphism G congruent to G x H for nontrivial H was found.\n\n**What remains.**\n\nUpgrade an absorption construction to finite presentation or prove finite presentability obstructs absorption of every nontrivial direct factor.\n\n**Sources checked.**\n\n- The Kourovka Notebook, 21st issue (2026), Problem 21.47. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Poses the finite-presentation problem and records both partial constructions.\n- J. M. Tyrer Jones, Direct products and the Hopf property, Journal of the Australian Mathematical Society 17 (1974), 174-196, DOI 10.1017/S144678870001675X. (primary): https://doi.org/10.1017/S144678870001675X\n Evidence used: Theorem A gives a finitely generated group isomorphic to a proper direct factor of itself.\n- Gilbert Baumslag, Charles F. Miller III, Some Odd Finitely Presented Groups, Bulletin of the London Mathematical Society 20 (1988), 239-244, DOI 10.1112/blms/20.3.239. (primary): https://doi.org/10.1112/blms/20.3.239\n Evidence used: Gives the finitely presented surjection-to-own-direct-square result cited by the tracker.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2557, "problem_number": "KOU-21.48", "title": "Kourovka Notebook Problem 21.48", "statement": "A quasimorphism on a group $G$ is a function $f:G\\to\\mathbb R$ such that the quantity $\\sup_{g,h}|f(g)+f(h)-f(gh)|$ is finite. A quasimorphism is homogeneous if it restricts to a homomorphism on every cyclic subgroup of $G$. Let $G$ be a group admitting an unbounded homogeneous quasimorphism $G\\to\\mathbb R$ that is not a homomorphism. Must $G$ contain a non-abelian free subgroup?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.48.\n\nDiscussion and literature:\nF. Fournier-Facio\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No general proof or counterexample was verified, but the implication cannot fail inside the broad class of subgroups of PL+(I): those groups have vanishing stable commutator length and hence no non-homomorphic homogeneous quasimorphisms.\n\n**Verified partial progress.**\n\n- Brin and Squier show the ambient finitely-broken piecewise-linear homeomorphism group contains no nonabelian free subgroup.\n- Calegari proves stable commutator length vanishes on [G,G] for every subgroup G of PL+(I); homogeneous-quasimorphism duality then forces every homogeneous quasimorphism on such G to be a homomorphism.\n\n**Full solution or refutation.**\n\nThis excludes a major free-subgroup-free class from furnishing a counterexample but does not settle arbitrary groups.\n\n**What remains.**\n\nEither construct a group without F2 whose kernel of the bounded-to-ordinary degree-two comparison map is nonzero, or prove such a kernel forces F2 in complete generality.\n\n**Sources checked.**\n\n- The Kourovka Notebook, 21st issue (2026), Problem 21.48. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the general implication question.\n- Matthew G. Brin, Craig C. Squier, Groups of piecewise linear homeomorphisms of the real line, Inventiones Mathematicae 79 (1985), 485-498, DOI 10.1007/BF01388519. (primary): https://doi.org/10.1007/BF01388519\n Evidence used: Proves the relevant piecewise-linear homeomorphism group contains no free subgroup of rank greater than one.\n- Danny Calegari, Stable commutator length in subgroups of PL+(I), Pacific Journal of Mathematics 232 (2007), 257-262, arXiv:math/0607482. (primary): https://arxiv.org/abs/math/0607482\n Evidence used: Proves stable commutator length vanishes on every element of the commutator subgroup of every subgroup of PL+(I).\n\n**Review notes.** The partial conclusion uses the standard Bavard duality between stable commutator length and homogeneous quasimorphisms modulo homomorphisms. Expert review is requested because no source explicitly frames these two PL results as a partial answer to KOU-21.48.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2558, "problem_number": "KOU-21.49", "title": "Kourovka Notebook Problem 21.49", "statement": "An isometric action of a group G on a metric space S is called acylindrical if for every $\\varepsilon>0$ there exist R, N > 0 such that for every two points x, y with $d(x,y)\\geqslant R$, there are at most N elements $g\\in G$ satisfying $d(x,gx)\\leqslant\\varepsilon$ and $d(y,gy)\\leqslant\\varepsilon$. A group is said to be acylindrically hyperbolic if it is not virtually cyclic and admits an acylindrical action on a hyperbolic space with unbounded orbits. Is the automorphism group of a finitely generated acylindrically hyperbolic group also acylindrically hyperbolic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.49.\n\nDiscussion and literature:\nA. Genevois\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The automorphism-group conjecture is proved for major families, including hyperbolic groups and broad graph-product cases, but the general finitely generated acylindrically hyperbolic case remains open on the proposer's maintained page.\n\n**Verified partial progress.**\n\n- Genevois proves Aut(G) acylindrically hyperbolic for one-ended hyperbolic G.\n- Genevois and Horbez prove Aut(G) acylindrically hyperbolic for every infinitely-ended finitely generated G, and prove a relative-hyperbolic extension.\n- Genevois proves the conclusion for broad non-join graph products of finitely generated irreducible groups, including corresponding right-angled Artin and Coxeter cases.\n\n**Full solution or refutation.**\n\nNo theorem covering every finitely generated acylindrically hyperbolic group was verified.\n\n**What remains.**\n\nExtend the known constructions of loxodromic WPD automorphisms beyond the hyperbolic, relatively hyperbolic, infinitely-ended, and graph-product regimes.\n\n**Sources checked.**\n\n- Anthony Genevois, Questions, maintained author page, checked 2026-08-17. (maintained_tracker): https://sites.google.com/view/agenevois/questions\n Evidence used: Still states as a conjecture that the automorphism group of a finitely generated acylindrically hyperbolic group is acylindrically hyperbolic.\n- The Kourovka Notebook, 21st issue (2026), Problem 21.49. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the general question and attributes it to Genevois.\n- Anthony Genevois, Negative curvature in automorphism groups of one-ended hyperbolic groups, Journal of Combinatorial Algebra 3 (2019), 305-329, DOI 10.4171/JCA/33. (primary): https://doi.org/10.4171/JCA/33\n Evidence used: Proves the one-ended hyperbolic case.\n- Anthony Genevois, Camille Horbez, Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups, arXiv:2002.01388; Journal of Topology 14 (2021). (primary): https://arxiv.org/abs/2002.01388\n Evidence used: Proves the infinitely-ended finitely generated case and a broad relatively hyperbolic case.\n- Anthony Genevois, Automorphisms of graph products of groups and acylindrical hyperbolicity, Memoirs of the AMS 301 (2024), no. 1509, arXiv:1807.00622. (primary): https://arxiv.org/abs/1807.00622\n Evidence used: Proves acylindrical hyperbolicity of automorphism groups for the stated broad non-join graph-product class.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2559, "problem_number": "KOU-21.50", "title": "Kourovka Notebook Problem 21.50", "statement": "Does every finite 3-group $T$ have a nontrivial characteristic subgroup $C$ such that if $T$ is a Sylow 3-subgroup of a finite group $G$, then $T\\cap G'=T\\cap H'$, where $H=N_G(C)$?\n\nThe group $S_4$ shows that no such characteristic subgroups can be found in some Sylow 2-subgroups.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.50.\n\nDiscussion and literature:\nSuch a characteristic subgroup is known to exist in $p$-groups for $p\\geqslant 5$ (G. Glauberman, Math. Z., 117 (1970), 46--56), and for $p=3$ there are two characteristic subgroups $K_1,K_2$ such that $T\\cap G'=(T\\cap H_1')(S\\cap H_2')$, where $H_i=N_G(K_i)$ (G. Glauberman, J. Algebra, 648 (2024), 62--86).\n\nG. Glauberman\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A single nontrivial characteristic subgroup controlling transfer is known for p at least 5, and two characteristic subgroups jointly control transfer for every odd prime; whether one always suffices for p=3 remains open.\n\n**Verified partial progress.**\n\n- Glauberman's 1970 result supplies a single characteristic subgroup for primes at least five.\n- Glauberman's 2024 theorem supplies two nonidentity characteristic subgroups whose normalizers jointly determine the largest abelian p-factor for every odd prime, including p=3.\n\n**Full solution or refutation.**\n\nThe two-subgroup theorem does not answer the requested one-subgroup statement at p=3, and no later reduction to one was found.\n\n**What remains.**\n\nShow that one of the two characteristic subgroups, or a canonical combination of them, alone enforces T intersect G' = T intersect N_G(C)' for every finite overgroup, or construct a 3-group obstruction.\n\n**Sources checked.**\n\n- The Kourovka Notebook, 21st issue (2026), Problem 21.50. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Poses the one-subgroup p=3 question and records the p at least 5 and two-subgroup odd-prime results.\n- George Glauberman, Prime-power factor groups of finite groups. II, Mathematische Zeitschrift 117 (1970), 46-56. (primary): https://www.math.uchicago.edu/~gg/ggbib.html\n Evidence used: Author-maintained bibliography identifies the 1970 primary paper cited for the p at least 5 theorem.\n- George Glauberman, Control of transfer for odd primes, Journal of Algebra 648 (2024), 62-86, DOI 10.1016/j.jalgebra.2024.01.003. (primary): https://doi.org/10.1016/j.jalgebra.2024.01.003\n Evidence used: Proves the joint two-characteristic-subgroup theorem for odd primes and gives a new single-subgroup construction for p at least 5.\n\n**Review notes.** The input background has an undefined S in '(T intersect H1')(S intersect H2')'. It was flagged rather than silently changed; the problem statement itself is intact.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2560, "problem_number": "KOU-21.51", "title": "Kourovka Notebook Problem 21.51", "statement": "Let $p$ be a prime, and $P$ a finite $p$-group.\n\n(a) Suppose that $P$ has an abelian subgroup of order $p^n$. For which $n$ does $P$ necessarily have a normal abelian subgroup of order $p^n$?\n\n(b) Suppose that $P$ has an elementary abelian subgroup of order $p^n$. For which $n$ does $P$ necessarily have a normal elementary abelian subgroup of order $p^n$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.51.\n\nDiscussion and literature:\nIt is easy to see that for $p=2$, the answer to (b) is \"yes\" only for $n=1$. The answer to both questions is \"yes\" for $n<(p+2)/2$ (G. G. Glauberman, J. Algebra, 319, no. 2 (2008), 800--805), as well as for $n\\leqslant 5$ when $p\\ne 2$ (M. Konvisser, D. Jonah, J. Algebra, 34 (1975), 309--330). The answer to both questions is \"no\" for $n\\geqslant (p+9)/2$ when $p\\geqslant 5$ (for $p\\geqslant 7$ due to G. Glauberman, Contemp. Math., 524 (2010), 61--65; for $p=3,5$ due to Ya. G. Berkovich, J. Algebra, 248, no. 2 (2002), 472--553). Thus, the only open cases for $p\\geqslant 5$ are $n=6$ for $p=5$, and $n=(p+3)/2,(p+5)/2,(p+7)/2$ for $p>5$. G. Glauberman\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Sharp positive and negative ranges are known, leaving only p=5,n=6 and n=(p+3)/2,(p+5)/2,(p+7)/2 for p>5; no closure of these boundary cases was verified.\n\n**Verified partial progress.**\n\n- Both abelian and elementary-abelian assertions hold for n<(p+2)/2, and for n<=5 when p is odd.\n- Both fail for n>=(p+9)/2 when p>=5; for p=2 the elementary-abelian assertion holds only for n=1.\n\n**Full solution or refutation.**\n\nThe notebook's explicit boundary cases remain open.\n\n**What remains.**\n\nDecide p=5,n=6 and the three parameterized boundary values for p>5, separately for parts (a) and (b).\n\n**Sources checked.**\n\n- Kourovka Notebook, Issue 21 (2026), Problem 21.51. (maintained_tracker): https://kourovka-notebook.org/\n Evidence used: Records the known positive/negative ranges and enumerates the remaining cases.\n- G. Glauberman, Contemporary Mathematics 524 (2010), 61-65. (primary): https://doi.org/10.1090/conm/524/10345\n Evidence used: Supplies the negative construction for p>=7 cited by the notebook.\n\n**Review notes.** Multi-part statement and exact residual cases retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2561, "problem_number": "KOU-21.52", "title": "Kourovka Notebook Problem 21.52", "statement": "Let $L$ be a finite non-abelian simple group, and let $D$ be a conjugacy class of involutions in $L$. Consider the complete graph $\\Gamma$ with vertex set $D$. Define an equivalence relation $\\sim$ (graph coloring) on the set of edges as follows: $(a,b)\\sim(c,d)$ if and only if $|ab|=|cd|$. An automorphism of the coloured graph $\\Gamma$ is a permutation $\\tau\\in S_D$ such that $(a,b)\\sim(a^\\tau,b^\\tau)$ for every edge $(a,b)$. Is it true that the automorphism group of $\\Gamma$ is a subgroup of $\\operatorname{Aut}(L)$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.52.\n\nDiscussion and literature:\nI. B. Gorshkov\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof or counterexample was found for reconstruction of the simple-group action from the involution product-order coloring.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe 2026 notebook remains the only located source directly addressing the exact question.\n\n**What remains.**\n\nProve all color-preserving permutations arise from Aut(L), or find an exceptional color symmetry.\n\n**Sources checked.**\n\n- Kourovka Notebook, Issue 21 (2026), Problem 21.52, I. B. Gorshkov. (maintained_tracker): https://kourovka-notebook.org/\n Evidence used: Primary maintained statement of the open problem.\n\n**Review notes.** No statement defect detected.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2562, "problem_number": "KOU-21.53", "title": "Kourovka Notebook Problem 21.53", "statement": "In the notation of 21.52, let $\\operatorname{Aut}_t(\\Gamma)$ be the set of permutations $\\tau\\in S_D$ such that $(a,b)\\sim(a^\\tau,b^\\tau)$ whenever $|ab|=t$ for $a,b\\in D$. Clearly, $\\operatorname{Aut}(\\Gamma)=\\bigcap_t\\operatorname{Aut}_t(\\Gamma)$. Is it true that for every finite simple group $G$ we have $\\operatorname{Aut}(\\Gamma)=\\operatorname{Aut}_2(\\Gamma)\\cap\\operatorname{Aut}_p(\\Gamma)$, where $\\{2,p\\}$ are the two minimal prime divisors of $|G|$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.53.\n\nDiscussion and literature:\nI. B. Gorshkov\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No source was found proving that the product-order coloring is recovered from the relations for the two smallest prime divisors.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe exact intersection identity remains unresolved in the literature checked.\n\n**What remains.**\n\nProve or refute Aut(Gamma)=Aut_2(Gamma) intersect Aut_p(Gamma), and clarify whether every involution class D is quantified.\n\n**Sources checked.**\n\n- Kourovka Notebook, Issue 21 (2026), Problem 21.53, I. B. Gorshkov. (maintained_tracker): https://kourovka-notebook.org/\n Evidence used: Primary maintained statement of the open problem.\n\n**Review notes.** Minor notation defect: the problem inherits L and D from 21.52 but switches to G and does not restate the quantifier over D.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2563, "problem_number": "KOU-21.54", "title": "Kourovka Notebook Problem 21.54", "statement": "Let $G$ be a finite soluble group with triality, which means that $G$ admits a group of automorphisms $S$ isomorphic to the symmetric group of degree 3 given by the presentation $S=\\langle\\sigma,\\rho\\mid \\sigma^2=\\rho^3=1;\\ \\sigma\\rho\\sigma=\\rho^2\\rangle$ such that $m\\cdot m^\\rho\\cdot m^{\\rho^2}=1$ for all $m$ in the set of commutators $M(G):=\\{[g,\\sigma]\\mid g\\in G\\}$. Suppose in addition that $G=[G,S]$, the group $G$ is generated by $d$ elements of $M(G)$ and their images under $S$, and $x^n=1$ for all $x\\in M(G)$. Is it true that the Fitting height of $G$ is bounded in terms of $d$ and $n$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.54.\n\nDiscussion and literature:\nAn affirmative answer would provide a reduction of the analogue of the Restricted Burnside Problem for Moufang loops to the nilpotent case. A. N. Grishkov, A. V. Zavarnitsine\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No Fitting-height bound in terms of the stated generator and exponent parameters was verified for finite soluble groups with triality.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe proposed reduction of the restricted Burnside problem for Moufang loops remains conditional on this bound.\n\n**What remains.**\n\nBound h(G) by d,n, or construct fixed-parameter examples of unbounded Fitting height.\n\n**Sources checked.**\n\n- Kourovka Notebook, Issue 21 (2026), Problem 21.54. (maintained_tracker): https://kourovka-notebook.org/\n Evidence used: States the Fitting-height problem and its Burnside-problem consequence.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2564, "problem_number": "KOU-21.55", "title": "Kourovka Notebook Problem 21.55", "statement": "Let $q$ be a power of a prime $p$, and let $m_n(q)$ be the maximum $p$-length of $p$-solvable subgroups of $\\operatorname{GL}(n,q)$. Is it true that $\\lim_{n\\to\\infty}m_n(q)/\\log_2 n=1$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.55.\n\nDiscussion and literature:\nI. G\\\"ulo\\u{g}lu\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof was found that the maximum p-length of p-soluble subgroups of GL(n,q) is asymptotic to log_2 n with leading constant one.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nClassical p-length theory gives context but does not settle the sharp limit in the record.\n\n**What remains.**\n\nProduce matching lower and upper asymptotics with leading constant one, or refute the limit.\n\n**Sources checked.**\n\n- Kourovka Notebook, Issue 21 (2026), Problem 21.55. (maintained_tracker): https://kourovka-notebook.org/\n Evidence used: Primary maintained statement of the asymptotic problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2565, "problem_number": "KOU-21.56", "title": "Kourovka Notebook Problem 21.56", "statement": "Let $\\ell(X)$ denote the composition length of a finite group $X$. Let $A$ be a finite nilpotent group acting by automorphisms on a finite soluble group $G$. Let $c(G,A)$ be the number of trivial $A$-modules in a given $A$-composition series of $G$. (Note that $c(G,A)=\\ell(C_G(A))$ if $(|A|,|G|)=1$.)\n\nConjecture: there are absolute constants $C_1$ and $C_2$ such that the Fitting height of $G$ is at most $C_1\\ell(A)+C_2c(G,A)$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.56.\n\nDiscussion and literature:\nI. G\\\"ulo\\u{g}lu\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The desired inequality holds with constants 2 and 1 when the acting group is cyclic, the soluble group has odd order, and the action normalizes a Sylow system; the general nilpotent-action case remains open.\n\n**Verified partial progress.**\n\n- Ercan-Guloglu prove h(G)<=2l(A)+c(G;A) under the cyclic/odd-order/Sylow-system hypotheses.\n- Their fixed-point-free corollary gives h(G)<=2l(A) in the same setting.\n\n**Full solution or refutation.**\n\nA strong special case verifies the conjectured linear form, but not the full hypotheses.\n\n**What remains.**\n\nRemove cyclicity, odd-order, and Sylow-system-normalization assumptions.\n\n**Sources checked.**\n\n- G. Ercan and I. S. Guloglu, Noncoprime action of a cyclic group, Journal of Algebra 643 (2024), 1-10. (primary): https://doi.org/10.1016/j.jalgebra.2023.12.020\n Evidence used: The main theorem gives h(G)<=2l(A)+c(G;A) in the stated cyclic special case.\n- Kourovka Notebook, Issue 21 (2026), Problem 21.56. (maintained_tracker): https://kourovka-notebook.org/\n Evidence used: States the general nilpotent-action conjecture.\n\n**Review notes.** The theorem is a special case, not a full solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2566, "problem_number": "KOU-21.57", "title": "Kourovka Notebook Problem 21.57", "statement": "Let X be a non-empty class of finite groups of odd order closed under taking subgroups, homomorphic images, and extensions. Let H be an X-maximal subgroup of a finite group G, and N a normal subgroup of G. Must H $\\cap$ N be an X-maximal subgroup of N?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.57.\n\nDiscussion and literature:\nW. Guo, D. O. Revin\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** General theory yields the weaker X-submaximal framework for normal intersections, while a recent odd-order nonpronormal example blocks a broad pronormality shortcut; the exact maximality question remains open.\n\n**Verified partial progress.**\n\n- X-submaximal subgroup theory provides a stable weaker replacement for X-maximality under normal intersections.\n- A relatively maximal nonpronormal odd-order subgroup is known in a finite simple group, though it does not itself answer the normal-intersection question.\n\n**Full solution or refutation.**\n\nNo proof or counterexample to the exact complete odd-order class assertion was verified.\n\n**What remains.**\n\nUpgrade submaximality to maximality using odd order, or build a nonsimple counterexample with a proper normal subgroup.\n\n**Sources checked.**\n\n- W. Guo, D. O. Revin and E. P. Vdovin, Finite groups in which X-maximal subgroups are conjugate, arXiv:1808.10107. (primary): https://arxiv.org/abs/1808.10107\n Evidence used: Develops the relevant complete-class and X-maximal/submaximal framework.\n- W. Zhang, N. Su and D. Revin, An Example of a Relatively Maximal Nonpronormal Subgroup of Odd Order in a Finite Simple Group, Siberian Mathematical Journal 65 (2024), 644-647. (primary): https://doi.org/10.1134/S0037446624030133\n Evidence used: Gives the adjacent nonpronormal odd-order example and delineates limits of pronormality methods.\n- Kourovka Notebook, Issue 21 (2026), Problem 21.57. (maintained_tracker): https://kourovka-notebook.org/\n Evidence used: Maintains the exact normal-intersection question.\n\n**Review notes.** Classified partial because the submaximality framework is a precise weakening; it is not claimed to settle maximality.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2567, "problem_number": "KOU-21.58", "title": "Kourovka Notebook Problem 21.58", "statement": "We say that a product $XY=\\{xy\\mid x\\in X,\\ y\\in Y\\}$ of two subsets $X,Y$ of a group $G$ is direct if for every $z\\in XY$ there are unique $x\\in X$, $y\\in Y$ such that $z=xy$. Is there an infinite group $G$ such that every subset $A\\subseteq G$ satisfies the following property: all the maximal subsets $B$ for which the product $AB$ is direct have the same cardinality?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.58.\n\nDiscussion and literature:\nNote that for checking the property for a given infinite group $G$, it suffices to consider only those subsets $A\\subseteq G$ for which $|A|=|G\\setminus A|$. Indeed, the property is equivalent to $A^{-1}A\\cap BB^{-1}=\\{1\\}$ and $A^{-1}AB=G$, and these imply $|G|=|A||B|$, since $G$ is infinite. Now, if $|A|<|G\\setminus A|$, then $|A|<|G|$, and so $|B|=|G|$; and if $|A|>|G\\setminus A|$, then $A^{-1}A=G$, and so $|B|=1$, for all $B$ satisfying the property. M. H. Hooshmand\n\nNo, there are no such groups (M. I. Kabenyuk, Preprint, 2026, https://arxiv.org/abs/2602.22876)\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Kabenyuk proves that no infinite group has the proposed equal-cardinality property for all maximal direct complements.\n\n**Verified partial progress.**\n\n- The preprint proves every infinite group is unstable: some subset has maximal direct complements of different cardinalities.\n- This statement is exactly the negation needed for the Notebook's existence question.\n\n**Full solution or refutation.**\n\nFor every infinite group G there exists a subset A and maximal subsets B making AB direct whose cardinalities differ; hence no requested group exists.\n\n**What remains.**\n\nPeer review or publication of the recent exact-match preprint; no case remains if its theorem is correct.\n\n**Sources checked.**\n\n- Mikhail Kabenyuk, Factors in infinite groups, arXiv:2602.22876 (2026). (primary): https://arxiv.org/abs/2602.22876\n Evidence used: The abstract states that every infinite group has a subset with maximal direct complements of different cardinalities and identifies this as the negative solution of Problem 21.58.\n\n**Review notes.** The input permits A to be empty, but the universal property is refuted by the nonempty witness supplied by the theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2568, "problem_number": "KOU-21.59", "title": "Kourovka Notebook Problem 21.59", "statement": "For a finite group $G$, let $\\chi_1(G)$ denote the totality of the degrees of all irreducible complex characters of $G$ with allowance for their multiplicities. Suppose that $H$ is a finite group with $\\chi_1(H)=\\chi_1(G)$.\n\na) If $G$ is an almost simple group, must $H$ be isomorphic to $G$?\n\nb) If $G$ is a quasisimple group, must $H$ be isomorphic to $G$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.59.\n\nDiscussion and literature:\nThis is true if $G$ is a simple group (see 11.8(a) in Archive). A. Iranmanesh, F. Shirjian\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Part (b) is already affirmative: every finite quasisimple group is determined by its complex group algebra, equivalently by the multiset of irreducible character degrees. Part (a) remains open in general.\n\n**Verified partial progress.**\n\n- Bessenrodt-Nguyen-Olsson-Tong-Viet complete complex-group-algebra recognition for all finite quasisimple groups.\n- The simple case and several almost-simple families are known, but no uniform theorem for all almost simple groups was verified.\n\n**Full solution or refutation.**\n\nThe source record combines a solved quasisimple subproblem with an unresolved almost-simple one.\n\n**What remains.**\n\nSettle part (a), especially almost simple groups with nontrivial outer automorphism structure not covered by family-specific results.\n\n**Sources checked.**\n\n- C. Bessenrodt, H. N. Nguyen, J. B. Olsson and H. P. Tong-Viet, Complex group algebras of the double covers of the symmetric and alternating groups, Algebra & Number Theory 9 (2015), arXiv:1308.4388. (primary): https://arxiv.org/abs/1308.4388\n Evidence used: The abstract states that this completes the proof that every finite quasisimple group is uniquely determined by its complex group algebra.\n- Kourovka Notebook, Issue 21 (2026), Problem 21.59. (maintained_tracker): https://kourovka-notebook.org/\n Evidence used: States both almost-simple and quasisimple parts, while mentioning only the simple case.\n\n**Review notes.** Dataset status defect: part (b) predates the 2026 problem statement as a completed theorem. Equality of degree multisets determines the complex semisimple group algebra by Wedderburn decomposition.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2569, "problem_number": "KOU-21.60", "title": "Kourovka Notebook Problem 21.60", "statement": "Let $G$ be a finite group, $\\mathbb Z_{(p)}$ the localization at $p$, and $\\mathbb F_p$ the field of $p$ elements. Let $\\mathcal X$ be the class of $\\mathbb F_pG$-modules obtained by reduction of simple $\\mathbb QG$-modules. Is it true that $\\mathbb Z_{(p)}G$ is semiperfect if and only if each projective indecomposable $\\mathbb F_pG$-module can be written as an $\\mathbb N$-linear combination of modules in $\\mathcal X$ inside the Grothendieck group of $\\mathbb F_pG$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.60.\n\nDiscussion and literature:\nThe \"only if\" direction is proved, and the affirmative answer is obtained if $p$ does not divide $|G|$ (D. Johnston, D. Rumynin, J. Algebra, 687, no. 1 (2026), 776--791). D. Johnston, D. Rumynin\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Grothendieck-group condition is necessary, and the equivalence holds when p does not divide |G|; sufficiency in the modular case remains a conjecture.\n\n**Verified partial progress.**\n\n- Johnston-Rumynin prove the only-if direction and formulate the converse in the modular setting.\n- The equivalence is affirmative in the nonmodular case p not dividing |G|.\n\n**Full solution or refutation.**\n\nThe 2026 Journal of Algebra paper establishes necessity but leaves the modular converse unresolved.\n\n**What remains.**\n\nProve sufficiency when p divides |G| or construct a counterexample satisfying the Grothendieck condition.\n\n**Sources checked.**\n\n- D. Johnston and D. Rumynin, On a question by Roggenkamp about group algebras, Journal of Algebra 687 (2026), 776-791; arXiv:2507.21316. (primary): https://doi.org/10.1016/j.jalgebra.2025.09.014\n Evidence used: Proves the necessary direction, treats the ordinary case, and states the modular converse as a conjecture.\n- Kourovka Notebook, Issue 21 (2026), Problem 21.60. (maintained_tracker): https://kourovka-notebook.org/\n Evidence used: Records the equivalence question and the known nonmodular case.\n\n**Review notes.** No post-publication resolution of the modular converse was located.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2570, "problem_number": "KOU-21.61", "title": "Kourovka Notebook Problem 21.61", "statement": "For a fixed (finitely generated free)-by-cyclic group $G=F_n\\rtimes\\mathbb Z$, is there an algorithm that, given a finite subset $S$ of $G$, finds a finite presentation for the subgroup $H=\\langle S\\rangle$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.61.\n\nDiscussion and literature:\nCf. 4.8 in Archive. I. Kapovich\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No uniform algorithm producing presentations for finitely generated subgroups of an arbitrary fixed free-by-cyclic group was verified.\n\n**Verified partial progress.**\n\n- The source cross-references the archive subgroup-presentation problem.\n\n**Full solution or refutation.**\n\nThe algorithmic question remains open.\n\n**What remains.**\n\nUse mapping-torus structure to compute subgroup cores.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.61. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2571, "problem_number": "KOU-21.62", "title": "Kourovka Notebook Problem 21.62", "statement": "Is the uniform subgroup membership problem decidable for (finitely generated free)-by-cyclic groups? That is, for a fixed group $G=F_n\\rtimes\\mathbb Z$, is there an algorithm that, given elements $w,h_1,\\ldots,h_k\\in G$, decides whether or not $w$ belongs to the subgroup $H=\\langle h_1,\\ldots,h_k\\rangle$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.62.\n\nDiscussion and literature:\nI. Kapovich\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No uniform subgroup-membership algorithm was verified for all fixed free-by-cyclic groups.\n\n**Verified partial progress.**\n\n- The source supplies the exact uniform membership formulation.\n\n**Full solution or refutation.**\n\nThe decidability question remains open.\n\n**What remains.**\n\nDevelop relative train-track/subgroup automata methods.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.62. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2572, "problem_number": "KOU-21.63", "title": "Kourovka Notebook Problem 21.63", "statement": "Let $F$ be a field of characteristic $p>0$, and let $\\Gamma$ be the principal congruence subgroup of $\\operatorname{Aut}(F[x_1,\\ldots,x_n])$ consisting of all automorphisms that send each variable $x_i$ to $x_i$ modulo terms of higher degree. Then $\\Gamma$ is a residually-$p$ group. Does $\\Gamma$ satisfy a pro-$p$ identity?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.63.\n\nDiscussion and literature:\n(E. Zelmanov).\n\nE. I. Khukhro\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The principal congruence automorphism group is residually p, but no pro-p identity was verified.\n\n**Verified partial progress.**\n\n- Residual p is stated as known.\n\n**Full solution or refutation.**\n\nThe pro-p identity question remains open.\n\n**What remains.**\n\nAnalyze graded automorphism Lie algebra for a potential identity.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.63. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records the known residual-p fact.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2573, "problem_number": "KOU-21.64", "title": "Kourovka Notebook Problem 21.64", "statement": "Is it true that if a normal subgroup $A$ of a Sylow $p$-subgroup of a $p$-soluble finite group $G$ has exponent $p^e$, then the normal closure of $A$ in $G$ has $(p,e)$-bounded (or even $e$-bounded) $p$-length?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.64.\n\nDiscussion and literature:\nE. I. Khukhro\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No p-length bound from the exponent of the normal Sylow-subgroup part was verified in the stated generality.\n\n**Verified partial progress.**\n\n- The question asks for p,e-bounded or e-bounded control.\n\n**Full solution or refutation.**\n\nThe boundedness assertion remains open.\n\n**What remains.**\n\nRelate normal closure layers to coprime action and Hall-Higman bounds.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.64. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2574, "problem_number": "KOU-21.65", "title": "Kourovka Notebook Problem 21.65", "statement": "Suppose that $\\phi$ is an automorphism of a finite soluble group $G$. Must $G$ contain a subgroup of index bounded in terms of $|\\phi|$ and $|C_G(\\phi)|$ whose Fitting height is bounded\n\n(a) in terms of $|\\phi|$?\n\n(b) or even in terms of the composition length of $\\langle\\phi\\rangle$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.65.\n\nDiscussion and literature:\nAn affirmative answer to part (b) is known when $|\\phi|$ is a prime power (B. Hartley--V. Turau), or when $(|G|,|\\phi|)=1$ (A. Turull, B. Hartley--I. M. Isaacs). Also cf. 19.43. E. I. Khukhro\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The stronger Fitting-height bound is known in a special automorphism-order regime, but not for arbitrary phi.\n\n**Verified partial progress.**\n\n- The source records an affirmative case for part (b) when |phi| is a specified special type.\n\n**Full solution or refutation.**\n\nThe general bounds remain open.\n\n**What remains.**\n\nExtend fixed-point automorphism methods beyond the known order case.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.65. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Distinguishes known special case and general problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2575, "problem_number": "KOU-21.66", "title": "Kourovka Notebook Problem 21.66", "statement": "Suppose that A is a nilpotent group of automorphisms of a finite soluble group G. Is the Fitting height of G bounded in terms of |A| and |CG(A)|?\n\nIsaacs), when A is cyclic (see 19.43), or when CG(A) = 1 (E. C. Dade).", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.66.\n\nDiscussion and literature:\nAn affirmative answer is known when (|G|, |A|) = 1 (J. G. Thompson, even for soluble A, with improved bounds in subsequent papers of H. Kurzweil, A. Turull, B. Hartley--I. M.\n\nNote that for any non-nilpotent finite group A there are finite soluble groups G of unbounded Fitting height with CG(A) = 1 (S. D. Bell--B. Hartley). E. I. Khukhro\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Fitting-height bound holds for coprime action, cyclic A, and fixed-point-free action, but not in the general nilpotent-automorphism setting.\n\n**Verified partial progress.**\n\n- Thompson/Dade/Isaacs-type special cases are recorded.\n\n**Full solution or refutation.**\n\nThe general bound remains open.\n\n**What remains.**\n\nReduce nilpotent A by central series while controlling noncoprime action.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.66. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Lists the positive special cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2576, "problem_number": "KOU-21.67", "title": "Kourovka Notebook Problem 21.67", "statement": "Suppose that $\\phi$ is an automorphism of a finite soluble group $G$, and let $r$ be the (Pr\\\"ufer) rank of the fixed-point subgroup $C_G(\\phi)$. Is the Fitting height of $G$ bounded in terms of $|\\phi|$ and $r$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.67.\n\nDiscussion and literature:\nThis is known to be true in the case where $|\\phi|$ is a product of at most two prime powers (B. Hartley, Preprint, 1994, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2025/08/hartley94-prepr-mims.pdf). An affirmative answer to this question would imply an affirmative answer to 13.8(b). E. I. Khukhro\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The fixed-point-rank Fitting-height bound is known when automorphism order has a stated restricted form, but no general theorem was verified.\n\n**Verified partial progress.**\n\n- The source records a positive restricted-order case.\n\n**Full solution or refutation.**\n\nThe general bound remains open.\n\n**What remains.**\n\nGeneralize automorphism-order reduction arguments.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.67. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Distinguishes known case from target.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2577, "problem_number": "KOU-21.68", "title": "Kourovka Notebook Problem 21.68", "statement": "A finite group $G$ is said to be semi-abelian if it has a sequence of subgroups $1=G_0\\leqslant G_1\\leqslant\\cdots\\leqslant G_n=G$ such that for every $i$ the subgroup $G_{i+1}$ is isomorphic to a quotient of a semidirect product $A_i\\rtimes G_i$ for some abelian group $A_i$.\n\nConjecture: Semi-abelian finite groups are monomial.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.68.\n\nDiscussion and literature:\nM. Kida\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that all finite semi-abelian groups are monomial was verified.\n\n**Verified partial progress.**\n\n- The source defines semi-abelian groups through quotient semidirect extensions.\n\n**Full solution or refutation.**\n\nThe monomiality conjecture remains open.\n\n**What remains.**\n\nInduct on the defining semidirect construction and character induction.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.68. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2578, "problem_number": "KOU-21.69", "title": "Kourovka Notebook Problem 21.69", "statement": "Is there an algorithm deciding if a given one-relator group is hyperbolic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.69.\n\nDiscussion and literature:\nD. Kielak\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No algorithm deciding hyperbolicity of arbitrary one-relator groups was verified.\n\n**Verified partial progress.**\n\n- The source records the direct decision problem.\n\n**Full solution or refutation.**\n\nThe algorithmic question remains open.\n\n**What remains.**\n\nCombine one-relator structure with effective small-cancellation/JSJ detection.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.69. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2579, "problem_number": "KOU-21.70", "title": "Kourovka Notebook Problem 21.70", "statement": "A group $G$ is called an orientable Poincar\\'e duality group of dimension $n$ over a ring $R$ if it is of type $FP$ over $R$ and $H^i(G;RG)=0$ for $i\\ne n$, while $H^n(G;RG)=R$ as an $RG$-module, where the action on $R$ is trivial. (Note that $G$ is not required to be finitely presented.) If $G$ is an orientable Poincar\\'e duality group of dimension $n$ over all fields, is it an orientable Poincar\\'e duality group over the integers?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.70.\n\nDiscussion and literature:\nD. Kielak\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No integral Poincare-duality conclusion from all-field Poincare duality was verified.\n\n**Verified partial progress.**\n\n- The source highlights the change of coefficients from all fields to Z.\n\n**Full solution or refutation.**\n\nThe coefficient-lifting question remains open.\n\n**What remains.**\n\nAnalyze torsion in a finite projective resolution.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.70. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2580, "problem_number": "KOU-21.71", "title": "Kourovka Notebook Problem 21.71", "statement": "For a ring $R$, we say that a group $G$ is of type $FL(R)$ if the trivial $RG$-module $R$ admits a finite resolution by finitely generated free modules. If $R$ is a field, we define the Euler characteristic of $G$ over $R$ to be the alternating sum of $R$-ranks of homology groups $H_i(G;R)$. Does there exist a group of type $FL(F_i)$ for two fields $F_1$ and $F_2$ such that the Euler characteristics of the group over the fields $F_1$ and $F_2$ differ?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.71.\n\nDiscussion and literature:\nD. Kielak\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No group of the stated two-field FL type with distinct Euler characteristics was verified.\n\n**Verified partial progress.**\n\n- The source gives the field-dependent finiteness hypothesis.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nSeek torsion-sensitive resolutions realizing field dependence.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.71. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2581, "problem_number": "KOU-21.72", "title": "Kourovka Notebook Problem 21.72", "statement": "We say that a group is a Tarski monster if it is finitely generated, not cyclic, and all of its proper non-trivial subgroups are isomorphic to each other.\n\na) Do there exist amenable torsion-free Tarski monsters?\n\nb) Do there exist amenable Tarski monsters of prime exponent?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.72.\n\nDiscussion and literature:\nD. Kielak\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No amenable torsion-free or prime-exponent Tarski monster was verified.\n\n**Verified partial progress.**\n\n- The source separates the two amenability questions.\n\n**Full solution or refutation.**\n\nBoth existence questions remain open.\n\n**What remains.**\n\nReconcile monster subgroup structure with amenable constructions.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.72. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2582, "problem_number": "KOU-21.73", "title": "Kourovka Notebook Problem 21.73", "statement": "Is the conjugacy problem in $\\operatorname{CT}(\\mathbb Z)$ algorithmically decidable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.73.\n\nDiscussion and literature:\nSee the definition of $\\operatorname{CT}(\\mathbb Z)$ in 17.57. S. Kohl\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No decision algorithm for conjugacy in CT(Z) was verified.\n\n**Verified partial progress.**\n\n- The source refers to the established CT(Z) definition.\n\n**Full solution or refutation.**\n\nThe conjugacy problem remains open.\n\n**What remains.**\n\nDevelop effective finite-state invariants for the action group.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.73. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2583, "problem_number": "KOU-21.74", "title": "Kourovka Notebook Problem 21.74", "statement": "Is it algorithmically decidable whether a given element $g\\in\\operatorname{CT}(\\mathbb Z)$\n\n(a) permutes a nontrivial partition of $\\mathbb Z$ into residue classes?\n\n(b) has only finite cycles?\n\n(c) has no finite cycles?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.74.\n\nDiscussion and literature:\nS. Kohl\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No algorithms deciding the three residue-class/cycle properties in CT(Z) were verified.\n\n**Verified partial progress.**\n\n- The source lists the three distinct decision tasks.\n\n**Full solution or refutation.**\n\nAll three remain open.\n\n**What remains.**\n\nUse orbit-structure algorithms for compatible transformations.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.74. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the subquestions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2584, "problem_number": "KOU-21.75", "title": "Kourovka Notebook Problem 21.75", "statement": "Given two distinct sets $P_1$ and $P_2$ of odd primes none of which is a subset of the other, is it true that $\\langle \\operatorname{CT}_{P_1}(\\mathbb Z),\\operatorname{CT}_{P_2}(\\mathbb Z)\\rangle \\lneq \\operatorname{CT}_{P_1\\cup P_2}(\\mathbb Z)$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.75.\n\nDiscussion and literature:\nSee the definition of $\\operatorname{CT}_P(\\mathbb Z)$ in 17.60. S. Kohl\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the strict subgroup-generation inequality for incomparable odd-prime sets was verified.\n\n**Verified partial progress.**\n\n- The source specifies the incomparability restriction.\n\n**Full solution or refutation.**\n\nThe strictness question remains open.\n\n**What remains.**\n\nFind a CT invariant absent from generators from the two smaller prime sets.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.75. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2585, "problem_number": "KOU-21.76", "title": "Kourovka Notebook Problem 21.76", "statement": "Let $\\sigma=(\\sigma_{ij})$, $1\\leqslant i\\ne j\\leqslant n$, be an irreducible elementary net (carpet) of order $n\\geqslant 3$ over a field $K$ (see 19.48). The net $\\sigma$ is said to be closed if the elementary net subgroup $E(\\sigma)$ does not contain new elementary transvections. The net $\\sigma$ is said to be completable if its diagonal can be supplemented with subgroups to a complete net. Do there exist irreducible closed elementary nets of order $n\\geqslant 3$ over a field of odd characteristic that are not completable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.76.\n\nDiscussion and literature:\nCompletable elementary nets are closed. It is known that over fields of characteristic 0 and 2 there exist irreducible closed elementary nets that are not completable (V. A. Koibaev, Trudy Inst. Mat. Mekh. Ural Div. Ross. Akad. Nauk, 17, no. 4 (2011), 134--141 (Russian); V. A. Koibaev, Siberian Math. J., 62, no. 2 (2021), 262--266).\n\nV. A. Koibaev\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Completable elementary nets have established structure theory, but existence of the requested irreducible closed noncompletable nets in odd characteristic was not verified.\n\n**Verified partial progress.**\n\n- The source contrasts closed and completable nets.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nConstruct a net with no new transvections yet a diagonal-completion obstruction.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.76. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Defines the target distinction.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2586, "problem_number": "KOU-21.77", "title": "Kourovka Notebook Problem 21.77", "statement": "Let $d$ be an integer that is not divisible by $n$-th powers of primes, let $x^n-d$ be an irreducible polynomial over $\\mathbb Q$, let $\\theta=\\sqrt[n]{d}$, and let $K=\\mathbb Q(\\theta)$ be the radical extension of degree $n$ of the field $\\mathbb Q$. The multiplicative group $K^*$ of the field $K$ is canonically embedded into the group $\\operatorname{Aut}_{\\mathbb Q}(K)$ of all invertible $\\mathbb Q$-linear mappings of the $\\mathbb Q$-space $K$; let $T$ be the image of $K^*$ under this embedding. In the natural basis $1,\\theta,\\theta^2,\\ldots,\\theta^{n-1}$ of the $\\mathbb Q$-space $K$ the group $\\operatorname{Aut}_{\\mathbb Q}(K)$ corresponds to $G=\\operatorname{GL}(n,\\mathbb Q)$, and the subgroup $T$ to a subgroup $T(d)$ (unsplit maximal torus). Every subgroup $H$ of $G$ containing $T(d)$ and some one-dimensional transformation is rich in elementary transvections and thus defines a net $\\sigma=\\sigma(H)$. Let $E(\\sigma)$ denote the subgroup generated by all transvections in the net group $G(\\sigma)$. Is it true that $H\\leqslant N_G(E(\\sigma))$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.77.\n\nDiscussion and literature:\nBackground on rich elementary transvections and the associated net: (V. A. Koibaev, St. Petersbg. Math. J., 21, no. 5 (2010), 731--742); (V. A. Koibaev, A. V. Shilov, J. Math. Sci. New York 171, no. 3 (2010), 380--385).\n\nThis inclusion was proved in the case $n=2$ (V. A. Koibaev, Dokl. Math., 41, no. 3 (1990), 414--416). V. A. Koibaev\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No resolution of the specified radical-extension linear-group question was verified.\n\n**Verified partial progress.**\n\n- The source gives the degree-n radical extension and canonical multiplication embedding.\n\n**Full solution or refutation.**\n\nThe stated problem remains open.\n\n**What remains.**\n\nAnalyze Zariski/arithmetic closure of the embedded multiplicative torus.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.77. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: The exact source formulation was retained; full display is long.\n\n**Review notes.** Long source statement retained verbatim in dataset.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2587, "problem_number": "KOU-21.78", "title": "Kourovka Notebook Problem 21.78", "statement": "Let $p$ be a prime and let $G$ be a pro-$p$ group. Suppose that all of the (continuous Galois) cohomology groups $H^n(G,\\mathbb F_p)$ of $G$ with coefficients in the field of $p$ elements are finite. Does it necessarily follow that the cohomology ring $H^*(G,\\mathbb F_p)$ is finitely generated?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.78.\n\nDiscussion and literature:\nThe answer is known to be `yes' if $G$ is abelian-by-($p$-adic analytic), as follows from (J. King, Commun. Algebra, 27, no. 10 (1999), 4969--4991). P. Kropholler\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The cohomology-ring finite-generation implication is known in important restricted pro-p cases, but not under only degreewise finite cohomology.\n\n**Verified partial progress.**\n\n- The source records known affirmative cases.\n\n**Full solution or refutation.**\n\nThe general implication remains open.\n\n**What remains.**\n\nRelate degreewise finiteness to finite generation via pro-p presentations.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.78. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Distinguishes known cases from the stated hypothesis.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2588, "problem_number": "KOU-21.79", "title": "Kourovka Notebook Problem 21.79", "statement": "Let $G$ be a finitely generated group with a fixed finite generating set $S$ and the corresponding word metric $L_S(*)$. An element $g$ is said to be distorted in $G$ if $L_S(g^n)/n\\to 0$ as $n\\to\\infty$; this notion is independent of the choice of the generating set $S$. For any, not necessarily finitely generated, group $H$, an element $g\\in H$ is said to be distorted if there is a finitely generated subgroup $G$ of $H$ containing $g$ in which $g$ is distorted. Do there exist finitely generated left-orderable groups in which every nontrivial element is distorted?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.79.\n\nDiscussion and literature:\nNote that it is straightforward to construct countable (not finitely generated) left-orderable groups with this property using HNN-extensions and applying results of V. V. Bludov and A. M. W. Glass. Y. Lodha, A. Navas\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No answer was verified to the source's distorted-element existence question for the stated finitely generated setting.\n\n**Verified partial progress.**\n\n- The source carefully fixes the intrinsic definition of distorted element.\n\n**Full solution or refutation.**\n\nThe question remains open.\n\n**What remains.**\n\nUse embedding/distortion constructions to test the precise target class.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.79. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: The full statement is retained in the source record.\n\n**Review notes.** Long source statement not rewritten.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2589, "problem_number": "KOU-21.80", "title": "Kourovka Notebook Problem 21.80", "statement": "Do there exist finitely generated left-orderable groups with only one nontrivial conjugacy class?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.80.\n\nDiscussion and literature:\nA positive answer to this question implies a positive answer to 21.78. Note that D. Osin constructed torsion-free finitely generated groups with only one nontrivial conjugacy class; see 9.10 in Archive. Y. Lodha, A. Navas\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No finitely generated left-orderable group with exactly one nontrivial conjugacy class was verified.\n\n**Verified partial progress.**\n\n- The source records implications of a positive answer.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nReconcile left orders with extreme conjugacy collapse.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.80. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2590, "problem_number": "KOU-21.81", "title": "Kourovka Notebook Problem 21.81", "statement": "Let $\\Gamma$ be a finite simple group and let $N_n(\\Gamma)$ denote the set of normal subgroups of the free group $F_n$ of rank $n$ whose quotient is isomorphic to $\\Gamma$.\n\nConjecture: $\\operatorname{Aut}(F_n)$ acts transitively on $N_n(\\Gamma)$ for $n\\geqslant 3$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.81.\n\nDiscussion and literature:\n(J. Wiegold).\n\nThis is not true for $n=2$ (B. H. Neumann, H. Neumann, Math. Nachr., 4 (1951), 106--125). A. Lubotzky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Aut(F_n) transitivity conjecture fails at n=2, but no resolution for n>=3 was verified.\n\n**Verified partial progress.**\n\n- The source records the n=2 failure.\n\n**Full solution or refutation.**\n\nThe n>=3 conjecture remains open.\n\n**What remains.**\n\nStudy Nielsen equivalence of epimorphisms to each finite simple quotient.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.81. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Distinguishes n=2 from the target range.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2591, "problem_number": "KOU-21.82", "title": "Kourovka Notebook Problem 21.82", "statement": "Conjecture: For $n\\geqslant 3$, there are no finite simple characteristic quotients of the free group $F_n$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.82.\n\nDiscussion and literature:\nThis is not true for $n=2$ (W. Y. Chen, A. Lubotzky, P. H. Tiep, to appear in Comment. Math. Helvetici, 2025). A. Lubotzky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finite simple characteristic quotients exist for F_2, but none or a classification for n>=3 was verified.\n\n**Verified partial progress.**\n\n- The source records the n=2 exception.\n\n**Full solution or refutation.**\n\nThe n>=3 nonexistence conjecture remains open.\n\n**What remains.**\n\nConnect characteristic quotients with Aut(F_n) orbit structure.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.82. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Distinguishes the exception from target.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2592, "problem_number": "KOU-21.83", "title": "Kourovka Notebook Problem 21.83", "statement": "Conjecture: Metabelian groups are permutation-stable.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.83.\n\nDiscussion and literature:\nThe function $d_n(\\sigma,\\tau)=(1/n)\\cdot|\\{x\\in\\{1,\\ldots,n\\}\\mid \\sigma(x)\\ne\\tau(x)\\}|$ is a distance on the symmetric group $S_n$. For a finitely generated group $G$, an almost-homomorphism is a sequence of set-theoretic maps $f_n:G\\to S_n$ satisfying $d_n(f_n(g)f_n(h),f_n(gh))\\to 0$ as $n\\to\\infty$ for all $g,h\\in G$. An almost-homomorphism $\\{f_n\\}$ is said to be close to a homomorphism if there is a sequence of group homomorphisms $\\rho_n:G\\to S_n$ such that $d_n(\\rho_n(g),f_n(g))\\to 0$ as $n\\to\\infty$ for all $g\\in G$. The group $G$ is said to be permutation stable if every almost-homomorphism of $G$ is close to a homomorphism.\n\nA. Lubotzky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that all metabelian groups are permutation-stable was verified.\n\n**Verified partial progress.**\n\n- The source records the conjecture in the normalized Hamming metric setting.\n\n**Full solution or refutation.**\n\nThe metabelian stability conjecture remains open.\n\n**What remains.**\n\nExtend known abelian/nilpotent stability methods through metabelian extensions.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.83. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2593, "problem_number": "KOU-21.84", "title": "Kourovka Notebook Problem 21.84", "statement": "For $\\sigma\\in S_n$ and $\\tau\\in S_m$, where $n\\leqslant m$, let $d_n^{\\mathrm{flex}}(\\sigma,\\tau)=(1/n)\\cdot(|\\{x\\in\\{1,\\ldots,n\\}\\mid \\sigma(x)\\ne\\tau(x)\\}|+(m-n))$. An almost-homomorphism $\\{f_n\\}$ is said to be flexibly close to a homomorphism if there is a sequence of group homomorphisms $\\rho_n:G\\to S_{m_n}$ with $n\\leqslant m_n$ such that $d_n^{\\mathrm{flex}}(\\rho_n(g),f_n(g))\\to 0$ as $n\\to\\infty$ for all $g\\in G$. The group $G$ is said to be flexibly permutation stable if every almost-homomorphism of $G$ is flexibly close to a homomorphism. Is $\\operatorname{SL}_n(\\mathbb Z)$ flexibly permutation-stable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.84.\n\nDiscussion and literature:\nA. Lubotzky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No resolution of the source's flexible permutation-stability problem was verified.\n\n**Verified partial progress.**\n\n- The source defines flexible closeness by allowing enlarged symmetric groups.\n\n**Full solution or refutation.**\n\nThe stated flexible-stability question remains open.\n\n**What remains.**\n\nCompare flexible approximations with standard permutation stability in key families.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.84. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Long source statement retained without reconstruction.\n\n**Review notes.** Long statement not rewritten.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2594, "problem_number": "KOU-21.85", "title": "Kourovka Notebook Problem 21.85", "statement": "Is a flexibly permutation-stable group always permutation-stable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.85.\n\nDiscussion and literature:\nA. Lubotzky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that flexible permutation stability implies ordinary permutation stability was verified.\n\n**Verified partial progress.**\n\n- The source gives the implication as a separate question.\n\n**Full solution or refutation.**\n\nThe implication remains open.\n\n**What remains.**\n\nControl the extra points in flexible approximations.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.85. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2595, "problem_number": "KOU-21.86", "title": "Kourovka Notebook Problem 21.86", "statement": "A group $G$ is said to be sofic if for every finite set $F\\subseteq G$ containing $1$ and every $\\varepsilon>0$ there exist $n\\in\\mathbb N$ and a map $\\phi:F\\to S_n$ such that $\\phi(1)=1$, $d(\\phi(gh),\\phi(g)\\phi(h))<\\varepsilon$ for all $g,h$ such that $gh\\in F$, and $\\phi(g)$ does not have fixed points for every $g\\in F\\setminus\\{1\\}$. Is every group sofic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.86.\n\nDiscussion and literature:\n(M. Gromov, B. Weiss).\n\nA. Lubotzky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether every group is sofic remains open.\n\n**Verified partial progress.**\n\n- The source gives the standard finite-approximation definition and attributes the problem to Gromov--Weiss.\n\n**Full solution or refutation.**\n\nNo proof or counterexample was verified.\n\n**What remains.**\n\nConstruct a nonsofic obstruction or prove universal approximability.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.86. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the foundational question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2596, "problem_number": "KOU-21.87", "title": "Kourovka Notebook Problem 21.87", "statement": "Assume that a finite group G has a family of d-generator subgroups whose indices have no common divisor. Is it true that G can be generated by d+1 elements?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.87.\n\nDiscussion and literature:\nThe answer is positive if G is solvable (L. G. Kov\\'acs, H.-S. Sim, Indag. Math., 2 (1991), 229--232). For an arbitrary finite group G it is proved that G can be generated by d + 2 elements (A. Lucchini, Commun. Algebra, 28, no. 4 (2000), 1875--1880). A. Lucchini\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The generator bound is positive for solvable finite groups, but no general finite-group theorem was verified.\n\n**Verified partial progress.**\n\n- The source records the solvable case.\n\n**Full solution or refutation.**\n\nThe nonsolvable case remains open.\n\n**What remains.**\n\nReduce using chief factors and generation of finite simple groups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.87. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Distinguishes known solvable case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2597, "problem_number": "KOU-21.88", "title": "Kourovka Notebook Problem 21.88", "statement": "Is there a finite non-abelian group $G$ of odd order, with $k(G)$ conjugacy classes, such that $k(G)/|G|=1/17$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.88.\n\nDiscussion and literature:\nD. MacHale\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No finite nonabelian odd-order group attaining conjugacy-class density 1/17 was verified.\n\n**Verified partial progress.**\n\n- The question is an exact finite search/existence problem.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nUse class-number constraints for odd-order groups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.88. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2598, "problem_number": "KOU-21.89", "title": "Kourovka Notebook Problem 21.89", "statement": "For $n>39$, is it true that the number of conjugacy classes in the symmetric group $S_n$ of degree $n$ is never a divisor of the order of $S_n$? In other words, is it true that, for $n>39$, the number $p(n)$ of integer partitions of $n$ is never a divisor of $n!$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.89.\n\nDiscussion and literature:\nD. MacHale\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that p(n) never divides n! for all n>39 was verified.\n\n**Verified partial progress.**\n\n- The source restates it as a partition-number divisibility problem.\n\n**Full solution or refutation.**\n\nThe universal divisibility claim remains open.\n\n**What remains.**\n\nCombine p-adic estimates for p(n) with n! valuations.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.89. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the equivalent question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2599, "problem_number": "KOU-21.90", "title": "Kourovka Notebook Problem 21.90", "statement": "Let $\\Gamma$ be a graph of diameter $d$. For $i\\in\\{1,2,\\ldots,d\\}$, let $\\Gamma_i$ be the graph on the same vertex set as $\\Gamma$ with vertices $u,w$ adjacent in $\\Gamma_i$ if and only if $d_\\Gamma(u,w)=i$. Does there exist a $Q$-polynomial distance-regular graph $\\Gamma$ of diameter 3 such that $\\Gamma_2$ and $\\Gamma_3$ are strongly regular?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.90.\n\nDiscussion and literature:\nA. A. Makhn\\\"ev\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No Q-polynomial distance-regular diameter-three graph satisfying both strong-regularity conditions was verified.\n\n**Verified partial progress.**\n\n- The source gives the exact distance-graph constraints.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nUse intersection-array feasibility and eigenvalue constraints.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.90. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2600, "problem_number": "KOU-21.91", "title": "Kourovka Notebook Problem 21.91", "statement": "Conjecture: The sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$ is at least the $p'$-part of $|G|$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.91.\n\nDiscussion and literature:\n(W. Willems).\n\nThis is known to be true for $p=2$ (G. Malle, Adv. Math., 380 (2021), Paper no. 107609, 15 pp.). G. Malle\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Brauer-character degree-square inequality is known for p=2, but no proof for arbitrary p was verified.\n\n**Verified partial progress.**\n\n- The source records the p=2 case.\n\n**Full solution or refutation.**\n\nThe general Willems conjecture remains open.\n\n**What remains.**\n\nExtend block-theoretic methods beyond p=2.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.91. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Distinguishes the known case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2601, "problem_number": "KOU-21.92", "title": "Kourovka Notebook Problem 21.92", "statement": "Conjecture: The number of irreducible $p$-Brauer characters of a finite group $G$ is bounded above by the maximum of the number of conjugacy classes $k(H)$ in $p'$-subgroups $H$ of $G$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.92.\n\nDiscussion and literature:\nG. Malle, G. Navarro, G. Robinson\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No general bound of the number of irreducible p-Brauer characters by the stated p-prime subgroup class maximum was verified.\n\n**Verified partial progress.**\n\n- The source records the Malle--Navarro--Robinson conjecture.\n\n**Full solution or refutation.**\n\nThe conjecture remains open.\n\n**What remains.**\n\nCompare blocks with p-prime local subgroups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.92. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2602, "problem_number": "KOU-21.93", "title": "Kourovka Notebook Problem 21.93", "statement": "Let $G$ be a group and let $k\\geqslant 2$. Let $H_1,\\ldots,H_k$ be subgroups of $G$, and $g_1,\\ldots,g_k$ elements of $G$ such that the cosets $g_1H_1,\\ldots,g_kH_k$ form a partition of $G$. Is it true that $|G:H_i|=|G:H_j|$ for some $i\\ne j$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.93.\n\nDiscussion and literature:\n(M. Herzog, J. Sch\\\"onheim).\n\nThis is known to be true for groups with a Sylow tower (M. A. Berger, A. Felzenbaum, A. Fraenkel, Fund. Math., 128, no. 3 (1987), 139--144). Also cf. 20.99. L. Margolis\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Herzog--Schonheim equal-index conclusion is known in important cases, but no general theorem for all coset partitions was verified.\n\n**Verified partial progress.**\n\n- The source records known affirmative cases.\n\n**Full solution or refutation.**\n\nThe general coset-partition question remains open.\n\n**What remains.**\n\nUse intersection of cores and finite quotient reductions.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.93. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Distinguishes known cases from full problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2603, "problem_number": "KOU-21.94", "title": "Kourovka Notebook Problem 21.94", "statement": "The Gruenberg--Kegel graph (or the prime graph) GK(G) of a finite group G is a labelled graph with vertex set consisting of all prime divisors of the order of G in which different vertices p and q are adjacent if and only if G contains an element of order pq. Let GK(G) denote the abstract graph obtained from GK(G) by removing all labels. A finite group G is said to be recognizable by the isomorphism type of its Gruenberg--Kegel graph if there are no finite groups H $\\ncong$ G with GK(H) isomorphic to GK(G). Are there infinitely many (pairwise non-isomorphic) finite groups which are recognizable by the isomorphism type of the Gruenberg--Kegel graph?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.94.\n\nDiscussion and literature:\nN. V. Maslova\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No infinite family recognizable by the abstract isomorphism type of its prime graph was verified.\n\n**Verified partial progress.**\n\n- The source distinguishes labelled and unlabelled prime graphs.\n\n**Full solution or refutation.**\n\nThe infinitude question remains open.\n\n**What remains.**\n\nConstruct groups with graph rigidity under all finite competitors.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.94. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the recognition question.\n\n**Review notes.** Long source statement retained without rewrite.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2604, "problem_number": "KOU-21.95", "title": "Kourovka Notebook Problem 21.95", "statement": "Is there an almost simple but not simple group which is recognizable by the isomorphism type of its Gruenberg--Kegel graph?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.95.\n\nDiscussion and literature:\nNote that if a group G is recognizable by the isomorphism type of its Gruenberg-- Kegel graph, then G is recognizable by its Gruenberg--Kegel graph, and therefore G is almost simple (P. J. Cameron, N. V. Maslova, J. Algebra, 607 (2022), 186--213). N. V. Maslova\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No almost simple nonsimple group recognizable by its abstract prime graph was verified.\n\n**Verified partial progress.**\n\n- The source notes the relationship to the preceding recognition question.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nAnalyze outer automorphism extensions of graph-rigid simple groups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.95. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2605, "problem_number": "KOU-21.96", "title": "Kourovka Notebook Problem 21.96", "statement": "Is it true that a periodic group containing an involution is locally finite if the centralizer of every element of even order is locally finite?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.96.\n\nDiscussion and literature:\nV. D. Mazurov\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that the specified periodic group is locally finite was verified.\n\n**Verified partial progress.**\n\n- The source assumes an involution and locally finite centralizers of every even-order element.\n\n**Full solution or refutation.**\n\nThe local-finiteness implication remains open.\n\n**What remains.**\n\nApply periodic-group centralizer techniques to finite even subgroups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.96. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2606, "problem_number": "KOU-21.97", "title": "Kourovka Notebook Problem 21.97", "statement": "Is it true that for every positive rational number $r$ there exists a finite group $G$ such that $|\\operatorname{Aut}(G)|/|G|=r$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.97.\n\nDiscussion and literature:\n(M. T\\u{a}rn\\u{a}uceanu).\n\nA similar question is answered in the positive for graphs, monoids, partial groups, and posets (R. Molinier, Preprint, 2025, https://arxiv.org/abs/2504.21059). It is also known that the set $\\{|\\operatorname{Aut}(G)|/|G|\\mid G\\text{ is a finite abelian group}\\}$ is dense in $[0,+\\infty)$ (M. T\\u{a}rn\\u{a}uceanu, Elemente Math. (2025), https://ems.press/journals/em/articles/14298544). R. Molinier\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No realization theorem for every positive rational automorphism-to-group order ratio was verified.\n\n**Verified partial progress.**\n\n- The source identifies a similar prior question.\n\n**Full solution or refutation.**\n\nThe rational-realization problem remains open.\n\n**What remains.**\n\nBuild finite groups with controlled automorphism groups and direct-product ratios.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.97. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2607, "problem_number": "KOU-21.98", "title": "Kourovka Notebook Problem 21.98", "statement": "Let w be a multilinear commutator word, and assume that G is a group where the set of w-values is covered by finitely many cyclic subgroups. Is it true that the verbal subgroup w(G) is finite-by-cyclic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.98.\n\nDiscussion and literature:\nThis is true for lower central words (G. Cutolo, C. Nicotera, J. Algebra, 324, no. 7 (2010), 1616--1624). M. Morigi\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The finite-by-cyclic conclusion is known for lower-central words, but no proof for every multilinear commutator word was verified.\n\n**Verified partial progress.**\n\n- Cutolo and collaborators prove the lower-central-word case.\n\n**Full solution or refutation.**\n\nThe general verbal-subgroup theorem remains open.\n\n**What remains.**\n\nReduce arbitrary multilinear words to lower-central behavior via outer commutator structure.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.98. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records the known special case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2608, "problem_number": "KOU-21.99", "title": "Kourovka Notebook Problem 21.99", "statement": "Conjecture: If $G$ is a transitive permutation group on a finite set $\\Omega$, then for any distinct $\\alpha,\\beta\\in\\Omega$ there is an element $g\\in G$ with $\\alpha^g=\\beta$ whose number of fixed points is different from 1.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.99.\n\nDiscussion and literature:\nP. M\\\"uller\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjecture remains open for arbitrary finite transitive groups, but the stronger derangement-transporter assertion is proved for every primitive almost-simple action with socle PSL_2(q), and further broad primitive families have regular subgroups that make the assertion immediate.\n\n**Verified partial progress.**\n\n- Muller proves that if q>=4 and G is almost simple with socle PSL_2(q) acting primitively, then every ordered pair of distinct points is joined by a derangement from the socle.\n- For most Aschbacher-O'Nan-Scott types, the stronger derangement-transporter property follows from the presence of regular subgroups.\n- The subgroup generated by all derangements is always transitive and contains every element whose number of fixed points differs from one, although this does not alone prove the required elementwise transporter claim.\n\n**Full solution or refutation.**\n\nMuller's April 2026 paper explicitly presents the exact statement as KOU-21.99 and as conjectural. Its counterexample to the older stronger derangement question does not refute KOU-21.99 because the exceptional transporter may contain elements with at least two fixed points.\n\n**What remains.**\n\nHandle arbitrary imprimitive actions and the remaining primitive families, or exhibit distinct alpha,beta for which every element in the transporter coset has exactly one fixed point. The exceptional ^3D_4(2) transporter from the stronger problem is a natural stress test.\n\n**Sources checked.**\n\n- Peter Muller, On transitive sets of derangements in primitive groups, Archiv der Mathematik 126 (2026), 551-556. (primary): https://doi.org/10.1007/s00013-026-02240-3\n Evidence used: Remark (f) states KOU-21.99 verbatim in substance and describes it as supported by evidence and partial results; the paper distinguishes it from the disproved stronger derangement conjecture.\n- Peter Muller, Transitive sets of derangements in primitive actions of PSL_2(q), arXiv:2512.19500 (2025). (primary): https://arxiv.org/abs/2512.19500\n Evidence used: Theorem 1.2 proves the stronger zero-fixed-point conclusion for primitive almost-simple groups with socle PSL_2(q).\n- R. A. Bailey, P. J. Cameron, M. Giudici and G. F. Royle, Groups generated by derangements, Journal of Algebra 572 (2021), 245-262. (primary): https://doi.org/10.1016/j.jalgebra.2020.12.020\n Evidence used: Proves that the derangement-generated subgroup is transitive and contains every element whose fixed-point number differs from one, supplying the closest general structural result.\n- The Kourovka Notebook: Unsolved Problems in Group Theory, Issue 21 (2026), Problem 21.99. (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Authoritative problem statement attributed to P. Muller.\n\n**Review notes.** No OCR or formulation defect detected. The classification is partial rather than open-only because a substantial infinite primitive family satisfies a strictly stronger theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2609, "problem_number": "KOU-21.100", "title": "Kourovka Notebook Problem 21.100", "statement": "Suppose that $A$ and $G$ are finite groups such that $A$ acts coprimely on $G$ by automorphisms. Let $C=C_G(A)$ be the fixed-point subgroup, and let $C'$ denote its derived subgroup. Is it true that the number of $A$-invariant irreducible characters $\\chi$ of $G$ whose restriction $\\chi_C$ is never zero is exactly $|C/C'|$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.100.\n\nDiscussion and literature:\nThis would follow if one could show that $\\chi_C$ is never zero if and only if the Glauberman--Isaacs correspondent $\\chi^*$ of $\\chi$ is linear. G. Navarro\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof or counterexample was found for the proposed count of nowhere-zero restrictions of A-invariant irreducible characters under coprime action.\n\n**Verified partial progress.**\n\n- The maintained problem entry reduces the desired equality to proving that chi restricted to C is nowhere zero exactly when its Glauberman--Isaacs correspondent is linear.\n- The equality is compatible with elementary trivial-action and abelian boundary cases.\n\n**Full solution or refutation.**\n\nThe Glauberman--Isaacs correspondence gives a precise proposed route, but the needed zero-versus-linearity equivalence remains unproved.\n\n**What remains.**\n\nRelate zeros of an A-invariant character on C_G(A) to the degree of its Glauberman--Isaacs correspondent, or find a counterexample.\n\n**Sources checked.**\n\n- Kourovka Notebook, New Problems, 21st issue, Problem 21.100 (2026). (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the exact character-counting problem and Navarro's proposed correspondence criterion.\n\n**Review notes.** No post-2026 resolution was located.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2610, "problem_number": "KOU-21.101", "title": "Kourovka Notebook Problem 21.101", "statement": "Which finite almost simple groups are the automorphism groups of regular polytopes of rank 3? In other words, which finite almost simple groups are generated by three involutions two of which commute?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.101.\n\nDiscussion and literature:\nThis question has been answered for finite simple groups; see 7.30 in Archive. Ya. N. Nuzhin\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The finite simple case is classified and important projective general linear families are settled, but no complete classification of all finite almost simple groups was verified.\n\n**Verified partial progress.**\n\n- Modulo CFSG, the finite simple classification is known; the maintained archive records four sporadic exceptions M11, M22, M23, and McL along with Lie-type and alternating results.\n- Markovskaya--Nuzhin give exact (2x2,2)-generation criteria for GL_n(q) and PGL_n(q).\n- Nuzhin's 2026 article gives conditions connecting involutory generating triples with regular rank-three polytope automorphism groups and surveys further open problems.\n\n**Full solution or refutation.**\n\nKnown work classifies the simple socles and some almost simple extensions, but does not settle every intermediate group between a simple socle and its automorphism group.\n\n**What remains.**\n\nClassify all extensions by outer automorphisms and clarify whether the target property is mere (2x2,2)-generation or a rank-three string C-group representation.\n\n**Sources checked.**\n\n- Kourovka Notebook, New Problems, 21st issue, Problem 21.101, and Archive Problem 7.30 (2026). (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the almost simple question and records the finite simple case as known.\n- I. A. Markovskaya and Ya. N. Nuzhin, On generation of the groups GL_n(q) and PGL_n(q) by three involutions, two of which commute, Trudy Instituta Matematiki i Mekhaniki 31 (2025). (primary): https://doi.org/10.21538/0134-4889-2025-31-4-fon-03\n Evidence used: Gives necessary and sufficient generation criteria for two major matrix-group families.\n- Ya. N. Nuzhin, Generating Sets of Involutions of Almost Simple Groups and Their Applications, Algebra and Logic (2026). (primary): https://doi.org/10.1007/s10469-026-09809-5\n Evidence used: Surveys progress and states extra conditions linking the generating-triple property to regular 3-polytopes.\n- Dimitri Leemans, String C-group representations of almost simple groups: a survey, Contemporary Mathematics 764 (2021), 157--178. (authoritative_secondary): https://doi.org/10.1090/CONM/764/15335\n Evidence used: Surveys the state of rank-three and higher string C-group representations for almost simple groups.\n\n**Review notes.** The phrase 'In other words' may omit the string C-group intersection condition required for regular-polytope automorphism groups.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2611, "problem_number": "KOU-21.102", "title": "Kourovka Notebook Problem 21.102", "statement": "Let $V$ be a variety generated by a finite group, and let $f(n)$ be the order of the free group in $V$ on $n$ generators. Is it true that the sequence $\\sqrt[n]{\\log f(n)}$ has a limit as $n\\to\\infty$, and this limit is an integer?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.102.\n\nDiscussion and literature:\nA. Yu. Olshanskii\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A June 2026 preprint gives the universal Birkhoff upper bound and a criterion for maximal growth, but does not settle convergence and integrality for every finite-group-generated variety.\n\n**Verified partial progress.**\n\n- Olshanskii proves |F_n| at most |A| raised to |A|^n for a finite generating algebra A.\n- Consequently the limsup of the nth root of log |F_n| is at most |A|.\n- For finite groups and finite nonassociative algebras, he gives a criterion for equality, equivalently maximal growth.\n\n**Full solution or refutation.**\n\nThe maximal-growth regime is characterized, but possible submaximal growth bases and equality of limsup and liminf remain uncontrolled in general.\n\n**What remains.**\n\nProve convergence and integrality in all cases or produce a finite group whose generated variety has a nonintegral or nonconvergent growth base.\n\n**Sources checked.**\n\n- Kourovka Notebook, New Problems, 21st issue, Problem 21.102 (2026). (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the convergence and integrality question.\n- Alexander Olshanskii, Finite groups and rings generating varieties with rapid growth, arXiv:2606.04577 (2026). (primary): https://arxiv.org/abs/2606.04577\n Evidence used: Proves the upper bound and characterizes equality in the maximal-growth case.\n\n**Review notes.** The base of the logarithm is immaterial after taking nth roots.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2612, "problem_number": "KOU-21.103", "title": "Kourovka Notebook Problem 21.103", "statement": "A Hausdorff topological group G is called minimal if it does not admit a strictly coarser Hausdorff group topology. A topological group is called Raikov complete if its two-sided uniform structure is complete. Is it true that an arbitrary Cartesian product of Raikov complete minimal topological groups remains minimal?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.103.\n\nDiscussion and literature:\n(V. V. Uspenskii).\n\nIt is known that a finite direct product of Raikov complete minimal topological groups is again minimal.\n\nIt is known that an arbitrary Cartesian product of centre-free minimal topological groups is minimal (M. Megrelishvili, Topology Appl., 62, no. 1 (1995), 1--19). D. Peng\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finite products and arbitrary products of centre-free complete minimal groups are known to be minimal, but arbitrary products with nontrivial centers remain open.\n\n**Verified partial progress.**\n\n- A finite direct product of Raikov-complete minimal groups is minimal.\n- Megrelishvili proves the arbitrary-product result for centre-free minimal groups.\n- Further product theorems hold under stronger total-minimality and h-completeness hypotheses.\n\n**Full solution or refutation.**\n\nThe conjecture holds in major subclasses, but completeness and minimality alone have not been shown sufficient for an arbitrary product.\n\n**What remains.**\n\nHandle factors with nontrivial centers or construct a Raikov-complete minimal counterexample family.\n\n**Sources checked.**\n\n- Kourovka Notebook, New Problems, 21st issue, Problem 21.103 (2026). (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the general question and the finite-product and centre-free cases.\n- Michael Megrelishvili, Group representations and construction of minimal topological groups, Topology and its Applications 62 (1995), 1--19. (primary): https://doi.org/10.1016/0166-8641(94)00043-3\n Evidence used: Supplies the centre-free product theorem cited by the maintained problem entry.\n\n**Review notes.** Raikov and Rajkov are alternate transliterations in the literature.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2613, "problem_number": "KOU-21.104", "title": "Kourovka Notebook Problem 21.104", "statement": "For a group word $w(x_1,\\ldots,x_n)$ on $n$ letters, define $e_0(x_1,\\ldots,x_n)=x_1$ and $e_{k+1}(x_1,\\ldots,x_n)=w(e_k(x_1,\\ldots,x_n),\\ldots,x_n)$ for all $k\\in\\mathbb N$. A group $G$ is said to satisfy the Engel type iterated identity $w$ if for all $x_1,\\ldots,x_n\\in G$ there exists $m\\in\\mathbb N$ such that $e_m(x_1,\\ldots,x_n)=1$.\n\nConjecture: For every non-trivial word $w$, if a finitely generated branch group $G$ (see 15.12) satisfies the iterated identity $w$, then $G$ is a torsion group.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.104.\n\nDiscussion and literature:\nThis is true in the case of the commutator word $w=[x_1,x_2]$ (G. Fern\\'andez-Alcober, M. Noce, G. Tracey, J. Algebra, 554 (2020), 54--77). M. Petschick\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The torsion conclusion is known for the commutator/Engel word, and finitely generated nonnilpotent Engel branch groups are known, but the conjecture for every nontrivial iterated word remains open.\n\n**Verified partial progress.**\n\n- Fernández-Alcober--Noce--Tracey prove the relevant periodicity restrictions for Engel elements in branch groups, yielding the commutator-word case.\n- Petschick constructs finitely generated nonnilpotent Engel branch groups, showing that torsion rather than nilpotence is the sharp known conclusion.\n- Petschick's contraction methods are designed to extend to some broader iterated identities.\n\n**Full solution or refutation.**\n\nThe classical Engel instance is settled, but no word-independent proof or noncommutator counterexample was found.\n\n**What remains.**\n\nProve torsion for every nontrivial word or construct a non-torsion finitely generated branch group satisfying one such iterated identity.\n\n**Sources checked.**\n\n- Kourovka Notebook, New Problems, 21st issue, Problem 21.104 (2026). (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the general conjecture and records the commutator-word case.\n- Gustavo A. Fernández-Alcober, Marialaura Noce, and Gareth M. Tracey, Engel elements in weakly branch groups, Journal of Algebra 554 (2020), 54--77. (primary): https://doi.org/10.1016/j.jalgebra.2020.03.003\n Evidence used: Proves structural and periodicity results for Engel elements in branch groups.\n- J. Moritz Petschick, On finitely generated Engel branch groups, Journal of the London Mathematical Society 110 (2024), e12980. (primary): https://doi.org/10.1112/jlms.12980\n Evidence used: Constructs finitely generated nonnilpotent Engel branch groups and discusses broader iterated identities.\n\n**Review notes.** The known Engel branch groups do not refute the conjecture because they are torsion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2614, "problem_number": "KOU-21.105", "title": "Kourovka Notebook Problem 21.105", "statement": "A group word $w$ is said to be concise in a class $\\mathcal C$ of groups if for every group $G$ in $\\mathcal C$ such that the set $G_w$ of word values of $w$ in $G$ is finite, the verbal subgroup $w(G)=\\langle G_w\\rangle$ is also finite. Is every word concise in the class of residually finite groups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.105.\n\nDiscussion and literature:\n(D. Segal).\n\nSee also Archive 2.45.\n\nSee (D. Segal, Words. Notes on verbal width in groups, Cambridge Univ. Press, 2009) for a partial solution. M. Petschick\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many broad word families are concise in residually finite groups, and the problem reduces to virtually pro-p groups, but the universal all-words question remains open.\n\n**Verified partial progress.**\n\n- Acciarri--Shumyatsky prove conciseness of w^q for a multilinear commutator w and prime-power q.\n- Their later work handles every positive q and also words [w^q,_n y].\n- The general residually finite problem is equivalent to the corresponding problem for virtually pro-p groups.\n- Further Engel-type words are known to be concise.\n\n**Full solution or refutation.**\n\nSubstantial classes are settled, but no proof for arbitrary words and no residually finite counterexample are known.\n\n**What remains.**\n\nExtend the methods to every word or build a residually finite counterexample to conciseness.\n\n**Sources checked.**\n\n- Kourovka Notebook, New Problems, 21st issue, Problem 21.105 (2026). (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: Retains Segal's universal question and points to the earlier partial theory.\n- Cristina Acciarri and Pavel Shumyatsky, On words that are concise in residually finite groups, Journal of Pure and Applied Algebra 218 (2014), 130--134. (primary): https://arxiv.org/abs/1212.0581\n Evidence used: Proves conciseness for prime-power powers of multilinear commutator words.\n- Cristina Acciarri and Pavel Shumyatsky, Varieties of groups and the problem on conciseness of words (2024). (primary): https://arxiv.org/abs/2308.02209\n Evidence used: Removes the prime-power restriction for key word families and gives the virtually pro-p reduction.\n- Eloisa Detomi, Marta Morigi, and Pavel Shumyatsky, Words of Engel type are concise in residually finite groups, Bulletin of Mathematical Sciences 9 (2019), 1950012. (primary): https://arxiv.org/abs/1711.04866\n Evidence used: Establishes additional Engel-type concise word families.\n\n**Review notes.** Counterexamples to conciseness in unrestricted groups are not known to be residually finite.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2615, "problem_number": "KOU-21.106", "title": "Kourovka Notebook Problem 21.106", "statement": "A first order formula $\\phi(x)$ in the group language with one free variable is said to be concise in a class $\\mathcal C$ of groups if for every group $G$ in $\\mathcal C$ such that the set $G_\\phi$ of elements in $G$ satisfying $\\phi$ is finite, the subgroup $\\phi(G)$ generated by $G_\\phi$ is also finite. Is every formula with one free variable concise in the class of residually finite groups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.106.\n\nDiscussion and literature:\nCf. 21.105 and Archive 2.45. M. Petschick\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** All formulas are concise in abelian groups and several broad formula classes are concise in residually finite groups, but the universal residually finite assertion remains open.\n\n**Verified partial progress.**\n\n- Conte--Petschick prove that every first-order formula is concise in abelian groups.\n- Every existential formula is concise in torsion-free locally class-two nilpotent groups.\n- Every residual weakly rational formula is concise in residually finite groups, yielding many concrete families.\n\n**Full solution or refutation.**\n\nThe new model-theoretic framework proves the claim under structural or syntactic hypotheses but explicitly leaves arbitrary formulas over arbitrary residually finite groups unresolved.\n\n**What remains.**\n\nRemove residuality and weak rationality or construct a parameter-free formula with finitely many values generating an infinite subgroup of a residually finite group.\n\n**Sources checked.**\n\n- Kourovka Notebook, New Problems, 21st issue, Problem 21.106 (2026). (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the universal first-order formula question.\n- Martina Conte and J. Moritz Petschick, Conciseness of first-order formulae, Monatshefte für Mathematik 209 (2026), 215--240. (primary): https://doi.org/10.1007/s00605-025-02127-5\n Evidence used: Introduces the exact framework, proves the abelian, nilpotent, and residual weakly rational cases, and retains the general question.\n\n**Review notes.** The paper and record use parameter-free first-order formulas.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2616, "problem_number": "KOU-21.107", "title": "Kourovka Notebook Problem 21.107", "statement": "A sequence $\\{F_n\\}$ of pairwise disjoint finite subsets of a topological group is called expansive if for every open subset $U$ there is a number $m$ such that $F_n\\cap U\\ne\\emptyset$ for all $n>m$. Suppose that a countable group $G$ can be partitioned into countably many dense subsets. Is it true that in $G$ there exists an expansive sequence?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.107.\n\nDiscussion and literature:\nCf. 15.80. I. V. Protasov\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The earlier unrestricted version is false by an uncountable box-product example, but the strengthened countable-group version in KOU-21.107 remains open.\n\n**Verified partial progress.**\n\n- Archive Problem 15.80 is negatively solved by the box-topology group product over the natural numbers of R, which is resolvable but has no countable dense subset and hence no expansive sequence.\n- That example is uncountable and therefore does not satisfy the new record's countability hypothesis.\n\n**Full solution or refutation.**\n\nThe added countability assumption evades the known counterexample; no countable counterexample or general construction was verified.\n\n**What remains.**\n\nConstruct expansive finite blocks in every countable resolvable topological group, or find a countable nonmetrizable counterexample.\n\n**Sources checked.**\n\n- Kourovka Notebook, New Problems, 21st issue, Problem 21.107 (2026). (maintained_tracker): https://alglog.org/21tkt.pdf\n Evidence used: States the strengthened problem with an explicit countability hypothesis and cites Archive 15.80.\n- Kourovka Notebook, updated archive, Problem 15.80 and the 19 May 2025 Corson/Protasov update. (maintained_tracker): https://alglog.org/20tkt.pdf\n Evidence used: Gives a negative answer to the unrestricted predecessor using an uncountable box-product group.\n\n**Review notes.** The countability qualifier must not be dropped: doing so would incorrectly classify the record as disproved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2617, "problem_number": "KOU-21.108", "title": "Kourovka Notebook Problem 21.108", "statement": "For a finite group $G$ let $\\operatorname{Cod}(G)$ denote the set of irreducible character codegrees of $G$ (see 20.78). Define $\\sigma(G)=\\max\\{|\\pi(m)|:m\\in\\operatorname{Cod}(G)\\}$, where $\\pi(m)$ denotes the set of prime divisors of an integer $m$. Can the constant $k$ be taken as 4?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.108.\n\nDiscussion and literature:\nIt is proved that there exists a constant $k$ such that $|\\pi(G)|\\leqslant k\\cdot\\sigma(G)$ for every finite group $G$ (Y. Yang, G. Qian, J. Algebra, 478 (2017), 215--219), but the estimate provided for $k$ is very crude.\n\nG. Qian\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A universal linear bound |pi(G)| <= k sigma(G) is proved, but whether k=4 suffices remains open.\n\n**Verified partial progress.**\n\n- Yang--Qian prove existence of a constant, with a nonsharp estimate.\n\n**Full solution or refutation.**\n\nThe sharp-constant question remains open.\n\n**What remains.**\n\nImprove the extremal analysis to k=4 or produce a counterexample.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.108. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records the proved existence theorem and asks k=4.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2618, "problem_number": "KOU-21.109", "title": "Kourovka Notebook Problem 21.109", "statement": "Conjecture: The derived length of a finite solvable group $G$ does not exceed $|\\operatorname{Cod}(G)|-1$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.109.\n\nDiscussion and literature:\nThe Fitting height of $G$ is known to be at most $\\min\\{|\\operatorname{Cod}(G)|-1,\\ |\\operatorname{Cod}(G)|/2+1\\}$ (G. Qian, Y. Zeng, J. Group Theory, to appear). G. Qian\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Fitting height has bounds in terms of the number of codegrees, but the proposed derived-length bound remains open.\n\n**Verified partial progress.**\n\n- Qian--Zeng bound the Fitting height by min(|Cod|-1, |Cod|/2+1).\n\n**Full solution or refutation.**\n\nNo proof of the derived-length conjecture was verified.\n\n**What remains.**\n\nRelate derived length directly to codegree strata.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.109. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records the Fitting-height progress and conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2619, "problem_number": "KOU-21.110", "title": "Kourovka Notebook Problem 21.110", "statement": "Let $S$ be a nonabelian finite simple group, and $x$ a nonidentity automorphism of $S$. Let $\\alpha(x)$ be the smallest number of conjugates of $x$ in $G=\\langle x,\\operatorname{Inn}S\\rangle$ that generate $G$.\n\n(a) Conjecture: If $S$ is an exceptional group of Lie type, then $\\alpha(x)\\leqslant 5$ for every nonidentity automorphism $x$ of $S$.\n\n(b) For each exceptional group $S$ of Lie type, find the largest value of $\\alpha(x)$.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.110.\n\nDiscussion and literature:\nThe values of $\\alpha(x)$ had been studied in (R. Guralnick, J. Saxl, J. Algebra, 268, no. 2 (2003), 519--571).\n\nPart (a) is attributed to R. Guralnick and J. Saxl. D. O. Revin\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of alpha(x)<=5 for every exceptional Lie-type simple group automorphism or complete maximum table was verified.\n\n**Verified partial progress.**\n\n- Guralnick--Saxl studied alpha values and formulated the conjecture.\n\n**Full solution or refutation.**\n\nBoth exceptional-type tasks remain open.\n\n**What remains.**\n\nAnalyze outer automorphism classes case by case using generation results.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.110. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records the prior study and conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2620, "problem_number": "KOU-21.111", "title": "Kourovka Notebook Problem 21.111", "statement": "Let $S$ be a finite simple nonabelian group that is not isomorphic to any group ${}^2B_2(q)$. A nonidentity automorphism $x$ of $S$ is called a $\\tau$-automorphism if every two conjugates of $x$ in $\\langle x,\\operatorname{Inn}(S)\\rangle$ generate a subgroup of order not divisible by 3. If $S$ admits a $\\tau$-automorphism, we call $S$ a $\\tau$-group.\n\n(a) List all $\\tau$-groups up to isomorphism.\n\n(b) Do $\\tau$-automorphisms of odd order exist?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.111.\n\nDiscussion and literature:\nD. O. Revin, N. Y. Yang\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No complete classification of tau-groups or determination of odd-order tau-automorphisms was verified.\n\n**Verified partial progress.**\n\n- The source supplies the exceptional Suzuki exclusion and definition.\n\n**Full solution or refutation.**\n\nBoth questions remain open.\n\n**What remains.**\n\nUse CFSG automorphism types to test the pairwise 3-divisibility condition.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.111. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Defines the class and retains both questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2621, "problem_number": "KOU-21.112", "title": "Kourovka Notebook Problem 21.112", "statement": "A nonempty class $\\mathcal X$ of finite groups is said to be complete if $\\mathcal X$ is closed under taking subgroups, homomorphic images, and extensions. The symmetric boundary of a complete class $\\mathcal X$ other than the class of all finite groups is defined as the largest integer $n$ such that $S_n\\in\\mathcal X$. For every positive integer $n\\ne 3$, let $f_+(n)$ and $f_-(n)$ be respectively the maximum and the minimum of $\\operatorname{BS}(\\mathcal X)$, where $\\mathcal X$ runs over all complete classes of symmetric boundary $n$.\n\n(a) Find $f_+(n)$ for $n=4,5,6$.\n\n(b) Is it true that $f_-(n)=n$ for all $n\\ne 3$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.112.\n\nDiscussion and literature:\nEvery positive integer $n\\ne 3$ coincides with the symmetric boundary of some complete class. It is proved (mod CFSG, D. O. Revin, Algebra i Analiz, 37, no. 1 (2025), 141--176 (Russian)) that, for every complete class $\\mathcal X$, there exists a nonnegative integer $m$ with the following property: for every finite group $G$ and each conjugacy class $D$ of $G$, if every $m$ elements of $D$ generate a subgroup belonging to $\\mathcal X$, then $\\langle D\\rangle\\in\\mathcal X$. The smallest such $m$ is called the Baer--Suzuki width of $\\mathcal X$, denoted by $\\operatorname{BS}(\\mathcal X)$. It is also proved (mod CFSG, ibid.) that, for a complete class $\\mathcal X$ of symmetric boundary $n$, the value of $\\operatorname{BS}(\\mathcal X)$ is at least $n$ and is bounded above in terms of $n$.\n\nIt is known that $f_+(1)=2$, $f_+(2)=3$, and $f_+(n)=2(n-1)$ for $n\\geqslant 7$.\n\nThis is known to be true for $n=1,2,4$. D. O. Revin, N. Y. Yang\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Existence of complete classes with each allowed symmetric boundary and a CFSG-based bounded-width theorem are known, but f-plus for 4,5,6 and the f-minus conjecture remain open.\n\n**Verified partial progress.**\n\n- Every positive n not 3 occurs as a symmetric boundary.\n- Revin proves a uniform bounded product property modulo CFSG.\n\n**Full solution or refutation.**\n\nThe requested extremal boundary values remain open.\n\n**What remains.**\n\nClassify complete classes at small boundaries and compare their BS values.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.112. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records the existence and bounded-width advances.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2622, "problem_number": "KOU-21.113", "title": "Kourovka Notebook Problem 21.113", "statement": "Let $G$ be a finite group and $p$ be a prime. Let $\\Psi_{p,G}$ be the class function of $G$ which vanishes on all $p$-singular elements of $G$ and whose value at each $p$-regular element $x$ of $G$ is the number of $p$-elements of $C_G(x)$.\n\n(a) Is it true that $\\Psi_{p,G}$ is a character of $G$?\n\n(b) If yes, can $\\Psi_{p,G}$ be afforded by a projective $RG$-module, where $R$ is a complete discrete valuation ring of characteristic zero such that the field of fractions of $R$ is a splitting field for $G$ and its subgroups, and the residue field $R/J(R)$ is a splitting field of characteristic $p$ for $G$ and its subgroups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.113.\n\nDiscussion and literature:\nIt is known that $\\Psi_{p,G}$ is a character when $G\\cong S_n$ for any positive integer $n$ and any prime $p$ (T. Scharf, J. Algebra, 139, no. 2 (1991), 446--457). G. Robinson\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The class function Psi_p,G is a character for every symmetric group, but no general character/projective-module theorem was verified.\n\n**Verified partial progress.**\n\n- Scharf proves the symmetric-group character case.\n\n**Full solution or refutation.**\n\nBoth general assertions remain open.\n\n**What remains.**\n\nTest character integrality and projective realization for finite simple/group extensions.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.113. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records Scharf's special case and asks the general questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2623, "problem_number": "KOU-21.114", "title": "Kourovka Notebook Problem 21.114", "statement": "A finite group G is called weakly ab-maximal if |H : [H, H]| $\\leqslant$ |G : [G, G]| for all H $\\leqslant$ G. Do weakly ab-maximal groups have bounded derived length?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.114.\n\nDiscussion and literature:\nIt is known that weakly ab-maximal groups are direct products of weakly ab-maximal p-groups (F. Lisi, L. Sabatini, J. Group Theory, 27 (2024), 1203--1217). L. Sabatini\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Weakly ab-maximal groups decompose as direct products of weakly ab-maximal p-groups, but a uniform derived-length bound remains unverified.\n\n**Verified partial progress.**\n\n- Lisi--Sabatini prove the direct-product reduction.\n\n**Full solution or refutation.**\n\nThe bounded-derived-length question remains open.\n\n**What remains.**\n\nBound derived length for weakly ab-maximal p-groups uniformly.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.114. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records the p-group reduction and retains the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2624, "problem_number": "KOU-21.115", "title": "Kourovka Notebook Problem 21.115", "statement": "Let $C_1,\\ldots,C_n$ be (left or right) cosets of a finite group $G$ such that $U:=C_1\\cup\\cdots\\cup C_n$ is not $G$. Is it always true that $|G\\setminus U|\\geqslant |G|/2^n$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.115.\n\nDiscussion and literature:\nAffirmative answers are known in some special cases (B. Sambale, M. T\\u{a}rn\\u{a}uceanu, J. Algebraic Combin., 55 (2022), 979--987). B. Sambale\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The half-to-the-n complement bound is known in special cases, but no general proof or counterexample was verified.\n\n**Verified partial progress.**\n\n- Sambale--Tarnauceanu establish affirmative special cases.\n\n**Full solution or refutation.**\n\nThe universal coset-union inequality remains open.\n\n**What remains.**\n\nReduce arbitrary coset covers to extremal subgroup configurations.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.115. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records special positive cases and asks the general bound.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2625, "problem_number": "KOU-21.116", "title": "Kourovka Notebook Problem 21.116", "statement": "A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Is every branch group boundedly acyclic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.116.\n\nDiscussion and literature:\nE. Schesler\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof or counterexample was verified for bounded acyclicity of every branch group.\n\n**Verified partial progress.**\n\n- The source records the bounded-cohomology formulation.\n\n**Full solution or refutation.**\n\nThe universal branch-group question remains open.\n\n**What remains.**\n\nCompute bounded cohomology for major branch-group families or find a nonvanishing class.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.116. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the new question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2626, "problem_number": "KOU-21.117", "title": "Kourovka Notebook Problem 21.117", "statement": "(a) Does there exist a finitely generated simple group that is of exponential growth but not of uniformly exponential growth?\n\n(b) Does there exist a finitely generated hereditarily just-infinite group that is of exponential growth but not of uniformly exponential growth?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.117.\n\nDiscussion and literature:\nE. Schesler\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No simple or hereditarily just-infinite example of exponential but non-uniform exponential growth was verified.\n\n**Verified partial progress.**\n\n- The source records both sharpened existence forms.\n\n**Full solution or refutation.**\n\nBoth questions remain open.\n\n**What remains.**\n\nConstruct a growth-critical simple/just-infinite quotient.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.117. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the two existence questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2627, "problem_number": "KOU-21.118", "title": "Kourovka Notebook Problem 21.118", "statement": "Is there any group which is not isomorphic to the quotient of a residually finite group by an amenable normal subgroup?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.118.\n\nDiscussion and literature:\n(A. Thom).\n\nE. Schesler\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No group outside all residually-finite-by-amenable quotients was verified.\n\n**Verified partial progress.**\n\n- The problem is attributed to Thom.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nFind an obstruction inherited by such quotients.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.118. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question without a claimed result.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2628, "problem_number": "KOU-21.119", "title": "Kourovka Notebook Problem 21.119", "statement": "Does there exist a group $G$ that contains a family $(G_n)_{n\\in\\mathbb N}$ of finite-index subgroups such that for every $n$ there is a homomorphism $f_n:G_n\\to\\mathbb Z$ whose kernel is of type $F_n$, but not of type $F_{n+1}$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.119.\n\nDiscussion and literature:\nE. Schesler\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No single group with the full family of finite-index kernels of successive finiteness types was verified.\n\n**Verified partial progress.**\n\n- The formulation targets a simultaneous hierarchy of F_n phenomena.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nBuild a group combining BNSR-type finiteness strata across finite-index subgroups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.119. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2629, "problem_number": "KOU-21.120", "title": "Kourovka Notebook Problem 21.120", "statement": "A pro-p group is (relatively) strictly finitely presented if it is the pro-p completion of a group that is finitely presented (respectively, finitely presented in some finitely-based variety of groups). A pro-p group is finitely axiomatizable if it is determined up to isomorphism by a single sentence in the first-order language of group theory.\n\n(a) Does there exist a (relatively) strictly finitely presented pro-p group that is not finitely axiomatizable in the class of all pro-p groups?\n\n(b) In particular, is every finitely generated free pro-p group finitely axiomatizable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.120.\n\nDiscussion and literature:\nSee (A. Nies, K. Tent, D. Segal, Proc. London Math. Soc. (3), 123 (2021), 597-- 635; D. Segal, Preprint, 2025, https://arxiv.org/abs/2505.04816). D. Segal\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Recent work studies finite axiomatizability of pro-p groups, but no resolution of strictly finitely presented examples or free pro-p groups was verified.\n\n**Verified partial progress.**\n\n- Nies--Tent--Segal and a 2025 Segal preprint provide the cited framework.\n\n**Full solution or refutation.**\n\nBoth finite-axiomatizability questions remain open.\n\n**What remains.**\n\nDetermine first-order rigidity of free pro-p groups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.120. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Cites the current relevant literature and retains both questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2630, "problem_number": "KOU-21.121", "title": "Kourovka Notebook Problem 21.121", "statement": "Let $p$ be a prime number. A group $\\Gamma$ is called $p$-Jordan if there exist constants $J$ and $e$ such that any finite subgroup $G\\subset\\Gamma$ contains a normal abelian subgroup of order coprime to $p$ and of index at most $J\\cdot |G(p)|^e$. (For example by the results of Brauer--Feit and Larsen--Pink, for any field $K$ of characteristic $p$ the group $\\operatorname{GL}_n(K)$ is $p$-Jordan with $e=3$.) Let the $p$-Jordan exponent $e(\\Gamma)$ of the group $\\Gamma$ be the infimum of all constants $e$ for which the above bound holds for some constant $J=J(e)$.\n\na) Is it true that this infimum is always attained?\n\nb) Is it true that $e(\\Gamma)\\leqslant 3$ for any $p$-Jordan group $\\Gamma$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.121.\n\nDiscussion and literature:\nC. Shramov\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Important linear groups are p-Jordan with exponent 3, but attainment of the infimum and the universal exponent-3 bound remain open.\n\n**Verified partial progress.**\n\n- Brauer--Feit and Larsen--Pink give GL_n examples with e=3.\n\n**Full solution or refutation.**\n\nBoth general questions remain open.\n\n**What remains.**\n\nFind an extremal p-Jordan group or prove a dimension-free reduction.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.121. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records the GL_n benchmark and asks both sharpness questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2631, "problem_number": "KOU-21.122", "title": "Kourovka Notebook Problem 21.122", "statement": "Let w be a group word, and G a profinite group. Is it true that the cardinality of the set of w-values in G is either finite or at least continuum?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.122.\n\nDiscussion and literature:\nAn affirmative answer is known for several important words; see (E. Detomi, B. Klopsch, P. Shumyatsky, J. London Math. Soc. (2), 102 (2020), 977--993). P. Shumyatsky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The finite-or-continuum alternative is known for several important words, but not for an arbitrary group word.\n\n**Verified partial progress.**\n\n- Detomi--Klopsch--Shumyatsky prove affirmative cases.\n\n**Full solution or refutation.**\n\nThe universal profinite-word question remains open.\n\n**What remains.**\n\nExtend word-map structure arguments to arbitrary w.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.122. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records known important-word cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2632, "problem_number": "KOU-21.123", "title": "Kourovka Notebook Problem 21.123", "statement": "Is it true that the extension of the A. Agrachev--R. Gamkrelidze construction of groups from pre-Lie rings suggested in Definition 66 produces groups from pre-Lie rings? If this extended construction does give a group, then it also gives a brace, and so an affirmative answer to this question would have consequences for the theory of set-theoretic solutions of the Yang--Baxter equation and for the theory of braces.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.123.\n\nDiscussion and literature:\nA. Agrachev--R. Gamkrelidze construction: (J. Soviet Math., 17 (1981), 1650--1675). Definition 66: (A. Smoktunowicz, J. Pure Appl. Algebra, 229, no. 12 (2025), 108128). A. Smoktunowicz\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that the proposed extension of the Agrachev--Gamkrelidze construction always yields a group was verified.\n\n**Verified partial progress.**\n\n- If valid, the construction would also yield braces and Yang--Baxter applications.\n\n**Full solution or refutation.**\n\nThe construction question remains open.\n\n**What remains.**\n\nCheck associativity/inverses under the 2025 pre-Lie extension.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.123. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Cites the original construction and proposed extension.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2633, "problem_number": "KOU-21.124", "title": "Kourovka Notebook Problem 21.124", "statement": "A group G is said to be virtually special if G has a finite-index subgroup isomorphic to the fundamental group of a special complex. A group G is called a CAT(0) group if it acts properly discontinuously and cocompactly by isometries on a CAT(0) metric space.\n\na) Is every CAT(0) free-by-cyclic group virtually special?\n\nb) A weaker question: does every CAT(0) free-by-cyclic group virtually embed into a right-angled Artin group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.124.\n\nDiscussion and literature:\nSpecial complex is meant in the sense of F. Haglund, D. T. Wise, Geom. Funct. Anal., 17, no. 5 (2008), 1551--1620; cf 20.60. I. Soroko\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem that every CAT(0) free-by-cyclic group is virtually special or even virtually RAAG-embeddable was verified.\n\n**Verified partial progress.**\n\n- Haglund--Wise special complexes supply the target notion.\n\n**Full solution or refutation.**\n\nBoth questions remain open.\n\n**What remains.**\n\nUse CAT(0) dynamics of free-by-cyclic groups to construct cubulations.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.124. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Defines the stronger and weaker questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2634, "problem_number": "KOU-21.125", "title": "Kourovka Notebook Problem 21.125", "statement": "Let $F_m$ be a free group of rank $m$ and let $\\varphi\\in\\operatorname{Aut}(F_m)$ be a polynomially growing automorphism of maximal degree $m-1$, which means that for some (equivalently, any) free basis $\\{x_1,\\ldots,x_m\\}$ of $F_m$, the sequence $\\max_i |\\varphi^n(x_i)|$ grows at the rate of $n^{m-1}$, where $|g|$ denotes the minimal length of $g$ in the $x_i$ and their inverses.\n\na) Is the free-by-cyclic group $F_m\\rtimes_\\varphi\\mathbb Z$ virtually special?\n\nb) In particular, are the Hydra groups $G_m=F_m\\rtimes\\mathbb Z=\\langle a_1,\\ldots,a_m,t\\mid t^{-1}a_1t=a_1,\\ t^{-1}a_it=a_ia_{i-1}\\text{ for all }i>1\\rangle$ virtually special?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.125.\n\nDiscussion and literature:\n(M. Bridson).\n\nI. Soroko\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No virtual-specialness theorem was verified for maximal-degree polynomial free-by-cyclic automorphisms or Hydra groups.\n\n**Verified partial progress.**\n\n- Hydra groups are explicitly identified as a key test family.\n\n**Full solution or refutation.**\n\nBoth questions remain open.\n\n**What remains.**\n\nCubulate the polynomial mapping tori or exhibit an obstruction.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.125. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the specialized questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2635, "problem_number": "KOU-21.126", "title": "Kourovka Notebook Problem 21.126", "statement": "Do there exist finitely presented subgroups of right-angled Artin groups whose Dehn functions are super-exponential, or sub-exponential but not polynomial?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.126.\n\nDiscussion and literature:\n(N. Brady).\n\nSuch subgroups are known to exist in general CAT(0) groups, whereas the only Dehn functions currently realized for subgroups of right-angled Artin groups are exponential and polynomial of arbitrary degree. I. Soroko\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** General CAT(0) groups have such subgroup Dehn functions, whereas known RAAG-subgroup examples realize only polynomial degrees and exponential functions.\n\n**Verified partial progress.**\n\n- The source records the current realized RAAG-subgroup spectrum.\n\n**Full solution or refutation.**\n\nBoth proposed gaps in the RAAG spectrum remain open.\n\n**What remains.**\n\nConstruct a RAAG subgroup with an intermediate/superexponential filling function.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.126. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Contrasts CAT(0) examples with the RAAG state of knowledge.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2636, "problem_number": "KOU-21.127", "title": "Kourovka Notebook Problem 21.127", "statement": "Let G be a right-angled Artin group. Is the stable commutator length scl(g) a rational number for every g $\\in$ [G, G]?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.127.\n\nDiscussion and literature:\nFor free groups this is true by Calegari's Rationality Theorem. (See 18.40 for the definition of scl(g).) I. Soroko\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Stable commutator length is rational in free groups, but rationality for every commutator in an arbitrary RAAG remains open.\n\n**Verified partial progress.**\n\n- Calegari's Rationality Theorem handles free groups.\n\n**Full solution or refutation.**\n\nThe RAAG rationality question remains open.\n\n**What remains.**\n\nExtend turn-graph/LP methods from free groups to RAAG normal forms.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.127. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Records the free-group theorem and asks the RAAG case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2637, "problem_number": "KOU-21.128", "title": "Kourovka Notebook Problem 21.128", "statement": "Two groups $G_1$ and $G_2$ are said to be commensurable if there exist finite-index subgroups $H_1\\leqslant G_1$ and $H_2\\leqslant G_2$ (not necessarily of the same index) such that $H_1\\cong H_2$. Let $A[F_4]$ and $A[H_4]$ denote the Artin groups of spherical types $F_4$ and $H_4$, respectively. Are these two groups commensurable?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.128.\n\nDiscussion and literature:\nThis is the most difficult case in the classification of Artin groups of spherical type up to commensurability. I. Soroko\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Commensurability of the spherical Artin groups A[F4] and A[H4] was not verified.\n\n**Verified partial progress.**\n\n- The source identifies it as the remaining difficult spherical-type case.\n\n**Full solution or refutation.**\n\nThe pairwise commensurability question remains open.\n\n**What remains.**\n\nCompare finite-index invariants of the two Artin groups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.128. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the specific question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2638, "problem_number": "KOU-21.129", "title": "Kourovka Notebook Problem 21.129", "statement": "If two Artin groups of spherical type are quasi-isometric, must they be commensurable? (This is not true for right-angled Artin groups.)", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.129.\n\nDiscussion and literature:\nI. Soroko\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that quasi-isometric spherical-type Artin groups must be commensurable was verified.\n\n**Verified partial progress.**\n\n- The statement explicitly separates the false RAAG analogue.\n\n**Full solution or refutation.**\n\nThe spherical Artin-group implication remains open.\n\n**What remains.**\n\nClassify quasi-isometry invariants and compare commensurability classes.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.129. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the open implication.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2639, "problem_number": "KOU-21.130", "title": "Kourovka Notebook Problem 21.130", "statement": "Conjecture: Let $G$ be a finite additive abelian group with $|G|$ odd. Then any subset $A$ of $G$ with $|A|=n>2$ can be written as $\\{a_1,\\ldots,a_n\\}$ in such a way that all the sums $a_1+a_2,a_2+a_3,\\ldots,a_{n-1}+a_n,a_n+a_1$ are distinct.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.130.\n\nDiscussion and literature:\nZ. W. Sun\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof or counterexample to the odd finite-abelian-group cyclic distinct-sums conjecture was verified.\n\n**Verified partial progress.**\n\n- The source records it as Sun's conjecture.\n\n**Full solution or refutation.**\n\nThe conjecture remains open.\n\n**What remains.**\n\nDevelop a Hamilton-cycle labeling avoiding repeated adjacent sums.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.130. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2640, "problem_number": "KOU-21.131", "title": "Kourovka Notebook Problem 21.131", "statement": "Construct a homomorphism of a subgroup of a Golod group onto an infinite AT-group.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.131.\n\nDiscussion and literature:\nGolod group: see 9.76. AT-group: (A. V. Rozhkov, Math. Notes, 40, no. 5 (1986), 827--836). Cf. 13.55 A. V. Timofeenko\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No homomorphism from a subgroup of a Golod group onto an infinite AT-group was verified.\n\n**Verified partial progress.**\n\n- The target uses the cited Rozhkov definition of AT-group.\n\n**Full solution or refutation.**\n\nThe construction problem remains open.\n\n**What remains.**\n\nAdapt Golod subgroup quotients to preserve the AT property.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.131. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the construction challenge.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2641, "problem_number": "KOU-21.132", "title": "Kourovka Notebook Problem 21.132", "statement": "Based on the development of E. S. Golod's construction, for each prime number p, construct a finitely generated residually finite p-group with a non-trivial finite centre.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.132.\n\nDiscussion and literature:\nDevelopment of E. S. Golod's construction: see, for example, Discrete Math. Appl., 23, no. 5--6 (2013), 491--501.\n\nSuch groups with infinite and trivial centres are known (see 9.76 and Archive 11.101). A. V. Timofeenko\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No Golod-based finitely generated residually finite p-group with nontrivial finite center was verified.\n\n**Verified partial progress.**\n\n- The source identifies Golod construction as the intended route.\n\n**Full solution or refutation.**\n\nThe construction question remains open.\n\n**What remains.**\n\nModify Golod relations while retaining a central finite p-subgroup.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.132. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the requested construction.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2642, "problem_number": "KOU-21.133", "title": "Kourovka Notebook Problem 21.133", "statement": "Does a group need to have a subnormal abelian series if every countable subgroup of it has such a series?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.133.\n\nDiscussion and literature:\nM. Trombetti\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No local-to-global theorem or counterexample for subnormal abelian series was verified.\n\n**Verified partial progress.**\n\n- The hypothesis is explicitly countable-subgroup local.\n\n**Full solution or refutation.**\n\nThe implication remains open.\n\n**What remains.**\n\nSeek a directed-union counterexample or a transfinite extension theorem.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.133. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the local-global question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2643, "problem_number": "KOU-21.134", "title": "Kourovka Notebook Problem 21.134", "statement": "For a finite group $G$, let the type of $G$ be the function on positive integers whose value at $n$ is the number of solutions of the equation $x^n=1$ in $G$.\n\na) Is it true that a group having the same type as a group with trivial solvable radical must also have trivial solvable radical?\n\nb) Is it true that a group having the same type as an almost simple group must be isomorphic to it?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.134.\n\nDiscussion and literature:\nNote that there are solvable and nonsolvable groups with the same type (see 12.37).\n\nThis is true for a group having the same type as a simple group, as follows from the affirmative answer to 12.39 (in Archive). A. V. Vasil'ev\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Known equal-type examples show type data is subtle, but neither radical-detection nor almost-simple recognition assertion was verified.\n\n**Verified partial progress.**\n\n- The source records solvable/nonsolvable groups with identical type.\n\n**Full solution or refutation.**\n\nBoth recognition questions remain open.\n\n**What remains.**\n\nExtract radical-sensitive statistics from solution-count sequences.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.134. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Notes existing ambiguity and asks stronger cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2644, "problem_number": "KOU-21.135", "title": "Kourovka Notebook Problem 21.135", "statement": "For a finite group $G$, let $\\chi_1(G)$ denote the totality of the degrees of all irreducible complex characters of $G$ with allowance for their multiplicities. Suppose that $H$ is a finite group with $\\chi_1(H)=\\chi_1(G)$. If $G$ has trivial solvable radical, must $H$ also have trivial solvable radical?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.135.\n\nDiscussion and literature:\nA. V. Vasil'ev\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem that full irreducible character-degree multiset detects trivial solvable radical was verified.\n\n**Verified partial progress.**\n\n- The problem refines character-degree recognition using multiplicities.\n\n**Full solution or refutation.**\n\nThe implication remains open.\n\n**What remains.**\n\nCombine degree multiplicities with radical constraints or find an extension counterexample.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.135. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the recognition question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2645, "problem_number": "KOU-21.136", "title": "Kourovka Notebook Problem 21.136", "statement": "Let G be a profinite group with fewer than $2^{\\aleph_0}$ conjugacy classes of elements of infinite order. Must G be a torsion group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.136.\n\nDiscussion and literature:\nThis holds in the case when G is finitely generated (J. S. Wilson, Arch. Math., 120 (2023), 557--563). John S. Wilson\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjugacy-class criterion forces torsion for finitely generated profinite groups, but no general profinite theorem was verified.\n\n**Verified partial progress.**\n\n- The source records the finitely generated case.\n\n**Full solution or refutation.**\n\nThe unrestricted question remains open.\n\n**What remains.**\n\nExtend finite generation arguments via open-subgroup structure.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.136. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the known case and general question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2646, "problem_number": "KOU-21.137", "title": "Kourovka Notebook Problem 21.137", "statement": "If the $p$-th powers in a finite $p$-group form a subgroup, must that subgroup be powerful? That is, for $p\\ne 2$, if the $p$-th powers in a $p$-group of exponent $p^2$ form a subgroup, must that subgroup be abelian? For a 2-group of exponent 8, if the squares form a subgroup, must that subgroup be abelian?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.137.\n\nDiscussion and literature:\nL. Wilson\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem that a subgroup of p-th powers/squares under the stated exponent hypotheses is powerful or abelian was verified.\n\n**Verified partial progress.**\n\n- The source provides the equivalent exponent-p-squared and exponent-eight forms.\n\n**Full solution or refutation.**\n\nThe powerful-subgroup question remains open.\n\n**What remains.**\n\nAnalyze power maps in small-class p-groups or construct a counterexample.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.137. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the equivalent forms.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2647, "problem_number": "KOU-21.138", "title": "Kourovka Notebook Problem 21.138", "statement": "Let G be an infinite finitely presented group such that every subgroup of infinite index is free. Must G be isomorphic to either a free group or a surface group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.138.\n\nDiscussion and literature:\nH. Wilton\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No classification of finitely presented groups whose infinite-index subgroups are free as only free or surface groups was verified.\n\n**Verified partial progress.**\n\n- The source frames this as a Wilton problem.\n\n**Full solution or refutation.**\n\nThe rigidity assertion remains open.\n\n**What remains.**\n\nUse ends/boundary/coherence to restrict possible groups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.138. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the classification question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2648, "problem_number": "KOU-21.139", "title": "Kourovka Notebook Problem 21.139", "statement": "Let G be a hyperbolic group which is virtually compact special in the sense of Haglund--Wise. Suppose that the set of second Betti numbers of the finite-index subgroups of G is bounded. Must G be virtually either a free group or a surface group?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.139.\n\nDiscussion and literature:\nH. Wilton\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No virtual free-or-surface classification was verified under bounded second Betti numbers for virtually compact special hyperbolic groups.\n\n**Verified partial progress.**\n\n- The source supplies the bounded-Betti hypothesis and specialness setting.\n\n**Full solution or refutation.**\n\nThe implication remains open.\n\n**What remains.**\n\nRelate virtual homology growth to cubical hierarchy structure.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.139. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2649, "problem_number": "KOU-21.140", "title": "Kourovka Notebook Problem 21.140", "statement": "Let G be a torsion-free group of type $F_\\infty$ of infinite cohomological dimension. Must G contain a copy of Thompson's group F?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.140.\n\nDiscussion and literature:\nS. Witzel, M. C. B. Zaremsky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem that every torsion-free F-infinity group of infinite cohomological dimension contains Thompson F was verified.\n\n**Verified partial progress.**\n\n- The question links finiteness properties and a copy of F.\n\n**Full solution or refutation.**\n\nThe containment assertion remains open.\n\n**What remains.**\n\nSeek type-F-infinity constructions avoiding F or prove a structural embedding theorem.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.140. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2650, "problem_number": "KOU-21.141", "title": "Kourovka Notebook Problem 21.141", "statement": "Let $G=G_1\\amalg_H G_2$ be a free pro-$p$ product of coherent pro-$p$ groups with polycyclic amalgamation. Is $G$ coherent?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.141.\n\nDiscussion and literature:\nFor abstract groups this is known to be true. A group is said to be coherent if each of its finitely generated subgroups is finitely presented, and in the question the coherency is used in the pro-$p$ sense. P. A. Zalesskii\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Coherence is known for the analogous abstract amalgamated products, but not for free pro-p products with polycyclic amalgamation.\n\n**Verified partial progress.**\n\n- The source records the abstract-group positive analogue.\n\n**Full solution or refutation.**\n\nThe pro-p coherence question remains open.\n\n**What remains.**\n\nAdapt subgroup decomposition methods to pro-p topology.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.141. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: Contrasts the abstract theorem with the pro-p question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2651, "problem_number": "KOU-21.142", "title": "Kourovka Notebook Problem 21.142", "statement": "A group $G$ is said to be invariably generated by $a$ and $b$ if $G$ is generated by the conjugates $a^g,b^h$ for every $g,h$. Let $p\\ne q$ be fixed primes. Does every finite group embed into a finite group invariably generated by an element of order $p$ and an element of order $q$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.142.\n\nDiscussion and literature:\nP. A. Zalesskii\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No universal embedding of finite groups into two-prime-order invariably generated finite groups was verified.\n\n**Verified partial progress.**\n\n- The source fixes arbitrary distinct primes p and q.\n\n**Full solution or refutation.**\n\nThe embedding question remains open.\n\n**What remains.**\n\nConstruct extensions with robust invariable generation while retaining a given subgroup.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.142. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2652, "problem_number": "KOU-21.143", "title": "Kourovka Notebook Problem 21.143", "statement": "(Well-known problem). Is Thompson's group F automatic?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.143.\n\nDiscussion and literature:\nM. C. B. Zaremsky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Automaticity of Thompson's group F remains open.\n\n**Verified partial progress.**\n\n- The source labels it well-known.\n\n**Full solution or refutation.**\n\nNo automatic structure or impossibility theorem was verified.\n\n**What remains.**\n\nFind a regular normal-form language with fellow-traveller property or obstruction.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.143. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the classic problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2653, "problem_number": "KOU-21.144", "title": "Kourovka Notebook Problem 21.144", "statement": "Conjecture: Every subgroup of Thompson's group F is either elementary amenable or else contains a subgroup isomorphic to F.", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.144.\n\nDiscussion and literature:\n(M. Brin, M. Sapir).\n\nM. C. B. Zaremsky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Brin--Sapir subgroup dichotomy for Thompson F remains open.\n\n**Verified partial progress.**\n\n- The source records the elementary-amenable versus contains-F conjecture.\n\n**Full solution or refutation.**\n\nNo general dichotomy proof or counterexample was verified.\n\n**What remains.**\n\nClassify non-elementary-amenable subgroups via diagram or action structure.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.144. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2654, "problem_number": "KOU-21.145", "title": "Kourovka Notebook Problem 21.145", "statement": "Is Thompson's group F quasi-isometric\n\n(a) to F $\\times$ Z?\n\n(b) to F $\\times$ F?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.145.\n\nDiscussion and literature:\n(M. Bridson).\n\nM. C. B. Zaremsky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No quasi-isometry classification distinguishing F from F times Z or F times F was verified.\n\n**Verified partial progress.**\n\n- The two comparison spaces are explicit.\n\n**Full solution or refutation.**\n\nBoth quasi-isometry questions remain open.\n\n**What remains.**\n\nDevelop quasi-isometry invariants sensitive to product directions.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.145. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States both questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2655, "problem_number": "KOU-21.146", "title": "Kourovka Notebook Problem 21.146", "statement": "(Well-known problem). A classifying space for a group $G$ is a connected CW-complex with fundamental group $G$ and all higher homotopy groups trivial. A group is of type $F_n$ if it has a classifying space with finite $n$-skeleton. For example, type $F_1$ is equivalent to finite generation, and type $F_2$ is equivalent to finite presentability. Type $F_\\infty$ means type $F_n$ for all $n$. For $n\\geqslant 3$, does every group of type $F_{n-1}$ embed as a subgroup of a group of type $F_n$? Or even in a group of type $F_\\infty$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.146.\n\nDiscussion and literature:\nM. C. B. Zaremsky\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No embedding theorem upgrading every F_(n-1) group to an F_n or F_infinity overgroup was verified.\n\n**Verified partial progress.**\n\n- The source records the standard finiteness-property formulation.\n\n**Full solution or refutation.**\n\nBoth embedding questions remain open.\n\n**What remains.**\n\nConstruct controlled overgroups preserving higher finiteness properties.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.146. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the well-known problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2656, "problem_number": "KOU-21.147", "title": "Kourovka Notebook Problem 21.147", "statement": "A subgroup H of a right-orderable group G is said to be right-relatively convex if it is convex under some right ordering on G. Is the lattice of right-relatively convex subgroups of a right-orderable group always a sublattice of the lattice of its subgroups?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.147.\n\nDiscussion and literature:\n(V. M. Kopytov, N. Ya. Medvedev).\n\nA. V. Zenkov\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that right-relatively convex subgroups are closed under meet and join as a sublattice was verified.\n\n**Verified partial progress.**\n\n- The source specifies the right-orderable setting.\n\n**Full solution or refutation.**\n\nThe lattice-closure question remains open.\n\n**What remains.**\n\nAnalyze compatible right orderings under subgroup operations.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.147. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2657, "problem_number": "KOU-21.148", "title": "Kourovka Notebook Problem 21.148", "statement": "Is it true that the lattice of right-relatively convex subgroups of a right-orderable group is distributive if and only if it is a chain?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.148.\n\nDiscussion and literature:\n(V. M. Kopytov, N. Ya. Medvedev).\n\nA. V. Zenkov\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No distributivity-if-and-only-if-chain characterization was verified for the lattice of right-relatively convex subgroups.\n\n**Verified partial progress.**\n\n- The source gives the exact lattice equivalence question.\n\n**Full solution or refutation.**\n\nThe characterization remains open.\n\n**What remains.**\n\nBuild nonchain distributive examples or prove a lattice obstruction.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.148. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2658, "problem_number": "KOU-21.149", "title": "Kourovka Notebook Problem 21.149", "statement": "Are there order automorphisms of Dlab groups that are not inner automorphisms?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.149.\n\nDiscussion and literature:\n(V. M. Kopytov, N. Ya. Medvedev).\n\nA. V. Zenkov\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No outer order automorphism of a Dlab group, nor a theorem excluding all such automorphisms, was verified.\n\n**Verified partial progress.**\n\n- The problem is recorded in the order-group setting.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nCompute automorphism groups of concrete Dlab groups.\n\n**Sources checked.**\n\n- Kourovka Notebook, 21st issue (2026), Problem KOU-21.149. (maintained_tracker): https://www.kourovka-notebook.org/\n Evidence used: States the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2659, "problem_number": "KOU-21.150", "title": "Kourovka Notebook Problem 21.150", "statement": "Let $G$ be an extension of a normal elementary abelian subgroup $A$ by an elementary abelian group $B\\cong G/A$ such that $A$ contains an element $a$ with $C_B(a)=1$. Is it true that the rank of the subgroup $Z(\\langle a,B\\rangle)\\cap\\langle a,B\\rangle'$ is at most the rank of $B$?", "background": "Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.150.\n\nDiscussion and literature:\nV. I. Zenkov\n\nNo, not always. Let $A=\\mathbb F_3[x,y]/I^3$ be the additive group of the quotient of the polynomial algebra $\\mathbb F_3[x,y]$ by the ideal $I^3$, where $I$ is the ideal generated by $x,y$. Let $B$ be the group of automorphisms of $A$ generated by multiplication by $1+x$ and $1+y$. Let $G=A\\rtimes B$. For $a=1\\in A$ we have $C_B(a)=1_B$. For $H=\\langle a,B\\rangle$ we have $Z(H)\\cap H'=\\langle x^2,xy,y^2\\rangle$, which has rank 3, while the rank of $B$ is 2. (P. Monticone, Letter of 21 March 2026; see also https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/03/solution_21_150-3.pdf).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Monticone's explicit characteristic-3 semidirect product has rank 3 in the central derived intersection but rank 2 in B.\n\n**Verified partial progress.**\n\n- With A = F_3[x,y]/(x,y)^3 additively, multiplication by 1+x and 1+y generates B isomorphic to C_3^2.\n- For a = 1, the stabilizer C_B(a) is trivial, while Z(H) intersect H' is generated by x^2, xy, and y^2.\n- The counterexample has an accompanying Lean formalization.\n\n**Full solution or refutation.**\n\nIn G = A semidirect B with H = <1,B>, one has rank(B)=2 but rank(Z(H) intersect H')=3, directly violating the proposed inequality.\n\n**What remains.**\n\nNothing for the stated inequality; possible corrected bounds would be a separate problem.\n\n**Sources checked.**\n\n- Wouter van Doorn, Elias Judin, Pietro Monticone, and Daniel Morrison, On Some Problems from the Kourovka Notebook, arXiv:2607.17477 (2026). (primary): https://arxiv.org/abs/2607.17477\n Evidence used: Explicitly reports the counterexample to the proposed rank inequality and the formal verification.\n- Pietro Monticone et al., Kourovka formalization repository, Problem 21.150 (2026). (primary): https://github.com/pitmonticone/Kourovka\n Evidence used: Contains the accompanying Lean formalization indexed to Problem 21.150.\n- Pietro Monticone, Solution of Kourovka Notebook Problem 21.150 (2026). (primary): https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/03/solution_21_150-3.pdf\n Evidence used: Presents the explicit F_3[x,y]/I^3 construction and rank computation.\n\n**Review notes.** No statement defect found; the explicit example matches all hypotheses.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "solved", "proposed_year": 2026, "category_id": 17, "set_id": 10, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 10, "name": "kourovka_new_problems_21", "display_name": "Kourovka Notebook - New Problems, Issue 21", "description": "New group theory problems from the 21st issue of the Kourovka Notebook, published in 2026.", "slug": "kourovka-new-problems-21", "order_index": 10, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2660, "problem_number": "KP-1.1", "title": "Kirby Problem 1.1", "statement": "Is the crossing number additive under connected sum:\n$c(K_{1}\\#K_{2}) = c(K_{1}) + c(K_{2})$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.1.\n\nLiterature notes:\nThis is Problem 1.65 in [Kir97]. Many special cases are known.\nThe most famous case is due to Kauffman [Kau87], Murasugi [Mur87], and\nThistlewaite [Thi87], who showed the conjecture for alternating knots (and more\ngenerally adequate knots, defined in [LT88]). Lackenby [Lac09] has shown in the\ngeneral case that if $K_{1},..., K_{n}$ are knots, then\n\n$$\n\\frac{c(K_{1}) + \\cdots + c(K_{n})}{152} \\leq c(K_{1}\\# \\cdots \\#K_{n}).\n$$\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Kau87] Louis H. Kauffman. State models and the Jones polynomial. Topology, 26(3):395– 407, 1987. doi:10.1016/0040-9383(87)90009-7.\n- [Mur87] Kunio Murasugi. Jones polynomials and classical conjectures in knot theory. Topology, 26(2):187–194, 1987. doi:10.1016/0040-9383(87)90058-9.\n- [Thi87] Morwen B. Thistlethwaite. A spanning tree expansion of the Jones polynomial. Topology, 26(3):297–309, 1987. doi:10.1016/0040-9383(87)90003-6.\n- [LT88] W. B. R. Lickorish and M. B. Thistlethwaite. Some links with nontrivial polynomials and their crossing-numbers. Comment. Math. Helv., 63(4):527–539, 1988. doi:10.1007/BF02566777.\n- [Lac09] Marc Lackenby. The crossing number of composite knots. J. Topol., 2(4):747–768, 2009. doi:10.1112/jtopol/jtp028.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Crossing-number additivity is known for alternating and adequate knots; in general Lackenby proved c(K1#...#Kn)>=(sum c(Ki))/152.\n\n**Verified partial progress.**\n\n- Alternating/adequate case proved by Kauffman, Murasugi and Thistlethwaite.\n- Lackenby's general quantitative lower bound has factor 152.\n\n**Full solution or refutation.**\n\nNo general equality theorem was verified.\n\n**What remains.**\n\nProve additivity or improve the universal factor to 1.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.1 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintained expert list states the special cases and Lackenby bound.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2661, "problem_number": "KP-1.2", "title": "Kirby Problem 1.2", "statement": "(a) Show that if $P$ is a nontrivial satellite operator and $K_{P}$ is a nontrivial\nsatellite of a knot $K$, then\n\n$$\nc(K_{P}) \\geq c(K),\n$$\n\nwhere $c$ denotes the crossing number.\n(b) More generally, show that if $P$ is a satellite operator then\n\n$$\nc(K_{P}) \\geq c(K) \\cdot w(P)^{2},\n$$\n\nwhere $w(P)$ denotes the geometric winding number.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.2.\n\nLiterature notes:\n(1) Part (a) is Problem 1.67 in [Kir97]. Part (b) appears as the Satellite\nCrossing Number Conjecture in [BM24a].\n(2) Note that a special case of this problem is whether\n\n$$\nc(K_{1}\\#K_{2}) \\geq c(K_{1}),\n$$\n\nwhich is a weaker form of Problem 1.1.\nIt is reasonable to expect equality in part (a) only when $P$ is the core\nof the solid torus.\nLackenby [Lac14] has shown the inequality\n\n$$\nc(K_{P}) \\geq c(K)/10^{13}.\n$$\n\nKalfagianni-Lee [KL23a] have shown that if $K$ is a nontrivial adequate\nknot that has an adequate diagram with writhe 0, then\n\n$$\nc(\\operatorname{Wh}_{\\pm}(K)) = 4c(K) + 2,\n$$\n\nwhere $\\operatorname{Wh}_{\\pm}(K)$ denote the Whitehead doubles of $K$. Kalfagianni-McConkey\n[KM24] have shown that if $K$ is adequate, then the cable knot $K_{p,q}$ sat-\nisfies\n\n$$\nc(K_{p,q}) \\geq q^{2} \\cdot c(K) + 1\n$$\n\nfor any coprime $p, q$.\nHere $q$, denotes the longitudinal winding of the\ncabling pattern. A recent preprint of Baker–Motegi claims that the more\ngeneral conjecture is true if one puts sufficiently many full twists in a\npattern with geometric winding number at least 2 [BM24a].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [BM24a] Kenneth L. Baker and Kimihiko Motegi. The stable crossing number of a twist family of knots and the satellite crossing number conjecture, 2024. arXiv:2404.05308.\n- [Lac14] Marc Lackenby. The crossing number of satellite knots. Algebr. Geom. Topol., 14(4):2379–2409, 2014. doi:10.2140/agt.2014.14.2379.\n- [KL23a] Efstratia Kalfagianni and Christine Ruey Shan Lee. Jones diameter and crossing number of knots. Adv. Math., 417:Paper No. 108937, 35, 2023. doi:10.1016/j.aim.2023.108937.\n- [KM24] Efstratia Kalfagianni and Rob Mcconkey. Crossing numbers of cable knots. Bulletin of the London Mathematical Society, 56(11):3400–3411, 2024. doi:10.1112/blms.13140.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Satellite Crossing Number Conjecture remains open in general. A universal lower bound with a large constant and sharp or stronger inequalities for several satellite families are known.\n\n**Verified partial progress.**\n\n- Lackenby proved c(K_P) >= c(K)/10^13 for satellite knots.\n- For adequate companions, Kalfagianni--Lee computed crossing numbers of writhe-zero Whitehead doubles and Kalfagianni--McConkey proved c(K_{p,q}) >= q^2 c(K)+1 for cables.\n- Baker--Motegi proved the conjectured behavior for specified sufficiently highly twisted satellite families.\n\n**Full solution or refutation.**\n\nThese theorems cover important patterns and give a general coarse bound, but do not prove either displayed inequality for every allowed satellite operator.\n\n**What remains.**\n\nProve c(K_P) >= c(K) in part (a) and the geometric-winding bound c(K_P) >= c(K)w(P)^2 in full generality.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.2 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Summarizes the general bound and the adequate, cable, Whitehead-double and twisted-family cases.\n- Marc Lackenby, The crossing number of satellite knots, Algebraic & Geometric Topology 14 (2014), 2379--2409. (primary): https://doi.org/10.2140/agt.2014.14.2379\n Evidence used: Establishes the general constant-factor lower bound.\n- Kenneth L. Baker and Kimihiko Motegi, The stable crossing number of a twist family of knots and the satellite crossing number conjecture, arXiv:2404.05308. (primary): https://arxiv.org/abs/2404.05308\n Evidence used: Proves satellite crossing results for sufficiently highly twisted families under its hypotheses.\n- Efstratia Kalfagianni and Rob McConkey, Crossing numbers of cable knots, Bulletin of the London Mathematical Society 56 (2024), 3400--3411. (primary): https://doi.org/10.1112/blms.13140\n Evidence used: Gives the q-squared lower bound for cables of adequate knots.\n\n**Review notes.** The geometric winding number in the source statement was not replaced by algebraic winding number.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2662, "problem_number": "KP-1.3", "title": "Kirby Problem 1.3", "statement": "How does unknotting number behave under connected sum and\nmutation?\n(a) Does the connected sum of $n$ nontrivial knots have unknotting number at\nleast $n$?\n(b) Is unknotting number invariant under mutation?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.3.\n\nLiterature notes:\n(1) Problem 1.69 (B) from [Kir97] asks if unknotting number is additive; a\nrecent preprint [BH25] claims an example where it fails to be additive,\nwhere the knots in question are the $(2, 7)$ torus knot and its mirror.\n(2) Question (a) is [Kir97, Problem 1.69 (A)], and is still open. It has a\npositive answer for $n = 2$ by Scharlemann [Sch85b].\n(3) With regard to Question (b), Gordon–Luecke [GL06] show that having\nunknotting number one is invariant under mutation.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [BH25] Mark Brittenham and Susan Hermiller. Unknotting number is not additive under connected sum, 2025. arXiv:2506.24088.\n- [Sch85b] Martin G. Scharlemann. Unknotting number one knots are prime. Invent. Math., 82(1):37–55, 1985. doi:10.1007/BF01394778.\n- [GL06] C. McA. Gordon and John Luecke. Knots with unknotting number 1 and essential Conway spheres. Algebr. Geom. Topol., 6:2051–2116, 2006. doi:10.2140/agt.2006.6.2051.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A 2025 preprint claims nonadditivity of unknotting number; the n-summand lower bound remains open, while unknotting number one is mutation invariant.\n\n**Verified partial progress.**\n\n- Scharlemann proves the n=2 lower-bound case.\n- Gordon--Luecke prove mutation invariance for unknotting number one.\n- Brittenham--Hermiller's preprint claims a nonadditivity example.\n\n**Full solution or refutation.**\n\nThe principal claims have differing status and the preprint needs expert review.\n\n**What remains.**\n\nVerify the preprint and settle the n-summand and general mutation questions.\n\n**Sources checked.**\n\n- M. Brittenham and S. Hermiller, Unknotting number is not additive under connected sum, arXiv:2506.24088 (2025). (primary): https://arxiv.org/abs/2506.24088\n Evidence used: Preprint claim, not treated as consensus resolution.\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.3 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the claim and known n=2/mutation results.\n\n**Review notes.** Preprint explicitly flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2663, "problem_number": "KP-1.4", "title": "Kirby Problem 1.4", "statement": "Let $P$ be a nontrivial satellite pattern with winding number\n$w(P) \\neq 0$. Then for any nontrivial knot $K$ and its satellite $K_{P}$ , one has\n\n$$\nu(K_{P}) \\geq w(P) + 1.\n$$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.4.\n\nLiterature notes:\n(1) This problem appears as [HLP22, Conjecture 1.10].\n(2) One might guess that $w(P)^{2}$ is a lower bound. However, it can be shown\nthat\n\n$$\nu((T_{2,3})_{2,1}) \\leq 3,\n$$\n\nwhere $K_{p,q}$ denotes the $(p, q)$ cable of the knot $K$.\n(3) It follows from [ST88] that if $w(P) \\neq 0$, then\n\n$$\nu(K_{P}) \\geq 2.\n$$\n\nProblem 1.4 would be a generalization of this result. More recently, Hom,\nLidman and Park [HLP22] have claimed that if $K$ is a nontrivial knot,\nthen\n\n$$\nu(K_{p,q}) \\geq p.\n$$\n\nIn the above, $p$ indicates the winding number of the satellite operation.\n\nReferences cited:\n- [HLP22] Jennifer Hom, Tye Lidman, and JungHwan Park. Unknotting number and cabling, 2022. arXiv:2206.04196.\n- [ST88] Martin Scharlemann and Abigail Thompson. Unknotting number, genus, and companion tori. Math. Ann., 280(2):191–205, 1988. doi:10.1007/BF01456051.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Nonzero-winding satellites have unknotting number at least two, and a cited preprint claims u(K_pq)>=p for cables; the stated w(P)+1 bound is open.\n\n**Verified partial progress.**\n\n- Scharlemann--Thompson prove u(KP)>=2 for nonzero winding.\n- Hom--Lidman--Park claim a cable lower bound u(K_pq)>=p.\n\n**Full solution or refutation.**\n\nNo proof of the stated universal satellite-pattern bound was verified.\n\n**What remains.**\n\nProve the w(P)+1 bound or give counterexamples.\n\n**Sources checked.**\n\n- J. Hom, T. Lidman, J. Park, Unknotting number and cabling, arXiv:2206.04196 (2022). (primary): https://arxiv.org/abs/2206.04196\n Evidence used: Cited preprint claim for cable knots.\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.4 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the proven lower bound and target conjecture.\n\n**Review notes.** Preprint status retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2664, "problem_number": "KP-1.5", "title": "Kirby Problem 1.5", "statement": "Is there a relationship between genus and unknotting number\nfor specific classes of knots? Here are two instances of classes of knots for which\nthere might be an interesting answer.\n(a) Conjecture (Murasugi-Przytycki). For every positive fibered knot $K$, the\ngenus of $K$ equals its unknotting number.\n(b) Question (Stoimenow). For every alternating fibered knot, is the genus\ngreater than or equal to the unknotting number?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.5.\n\nLiterature notes:\n(1) There are examples of knots where the genus of a knot is greater than the\nunknotting number, and vice versa. However, there could be an inequality\nin some particular classes of knots. In part (a) there is evidence for an\ninequality one direction; in a different setting, part (b), the evidence points\nto an inequality in the other direction.\n(2) The conjecture in (a) appears at the end of [MP89]. By work of Rudolph\n[Rud93, Rud99] it is now known that genus $\\leq$ unknotting number for\na positive knot. For a positive braid closure, manipulations with braid\ngroup relations [BW84] allow one to construct an unknotting sequence of\nlength = genus, so that genus = unknotting number for a positive braid\nknot. However, such arguments are not available for the slightly more\ngeneral case of a positive fibered knot.\n(3) A Knotinfo [LM23b] search shows that this holds for alternating fibered\nknots of at most 13 crossings, with strict inequality holding in all but 13\nof the 2106 such knots.\n\nReferences cited:\n- [MP89] Kunio Murasugi and Józef H. Przytycki. The skein polynomial of a planar star product of two links. Math. Proc. Cambridge Philos. Soc., 106(2):273–276, 1989. doi:10.1017/S0305004100078099.\n- [Rud93] Lee Rudolph. Quasipositivity as an obstruction to sliceness. Bull. Amer. Math. Soc. (N.S.), 29(1):51–59, 1993. doi:10.1090/S0273-0979-1993-00397-5.\n- [Rud99] Lee Rudolph. Positive links are strongly quasipositive. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 555–562. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.555.\n- [BW84] Michel Boileau and Claude Weber. Le problème de J. Milnor sur le nombre gordien des nœuds algébriques. Enseign. Math. (2), 30(3-4):173–222, 1984.\n- [LM23b] Charles Livingston and Allison H. Moore. Knotinfo: Table of knot invariants, November 2023. https://knotinfo.org.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjectured equality is proved for positive braid knots and one inequality holds for all positive knots, while the general positive-fibered and alternating-fibered questions remain open.\n\n**Verified partial progress.**\n\n- Rudolph's quasipositivity results imply that Seifert genus is at most unknotting number for every positive knot.\n- Positive braid knots satisfy equality of Seifert genus and unknotting number.\n- Kegel, Lewark, and Manikandan show that several standard optimal-unknotting mechanisms fail for positive fibered knots and propose a concrete genus-seven candidate with conjectural unknotting number nine.\n- K3 reports that the alternating-fibered inequality holds for every such knot through 13 crossings in KnotInfo.\n\n**Full solution or refutation.**\n\nNeither universal statement is settled. The proposed positive-fibered counterexample is conditional on an unproved unknotting-number computation and therefore is not a refutation.\n\n**What remains.**\n\nProve equality for all positive fibered knots or determine the unknotting number of the proposed candidate; and prove or disprove g(K) at least u(K) for every alternating fibered knot.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 1.5. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States both conjectural questions, records the positive and positive-braid results, and reports finite KnotInfo verification for alternating fibered knots.\n- Lee Rudolph, Positive links are strongly quasipositive, Geometry & Topology Monographs 2 (1999), 555-562. (primary): https://doi.org/10.2140/gtm.1999.2.555\n Evidence used: Provides the positivity/quasipositivity input used for the genus lower bound on unknotting number.\n- Marc Kegel, Lukas Lewark, and Naageswaran Manikandan, On unknotting fibered positive knots and braids, arXiv:2312.07339 (2023). (primary): https://arxiv.org/abs/2312.07339\n Evidence used: Proves the stronger positive-braid diagram result, identifies failures of proposed extensions, and gives a conjectural—not proved—counterexample in the positive-fibered class.\n\n**Review notes.** The unqualified word genus is interpreted as Seifert 3-genus from the source context. The exact stored wording is reproduced in report.md.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2665, "problem_number": "KP-1.6", "title": "Kirby Problem 1.6", "statement": "Suppose that $V_{1}$ and $V_{2}$ are $S$–equivalent Seifert forms. Does\nthere exist a fixed knot $K$ bounding Seifert surfaces $F_{1}$ and $F_{2}$ for which the as-\nsociated Seifert forms are $V_{1}$ and $V_{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.6.\n\nLiterature notes:\n(1) A Seifert form of a knot is a bilinear integral form $V$ for which $V - V^{*}$ is\nunimodular (with $V^{*}(\\alpha, \\beta) = V (\\beta, \\alpha)$). Every Seifert form $V$ arises as the\nSeifert form of a Seifert surface $F$ for some knot $K$. If $K$ bounds Seifert\nsurfaces $F_{1}$ and $F_{2}$, then the associated Seifert forms are $S$–equivalent.\nSee [Tro73] or [Kaw96, Chapter 5] for more details.\n(2) One way to think about this problem is as a comparison of two flavors\nof $S$–equivalence: that of $S$–equivalence of Seifert forms, versus that of\n$S$–equivalence of knots, with two knots being $S$–equivalent if they possess\nSeifert forms that are $S$–equivalent [Kea04, NS03]. It is shown in [NS03]\nthat two knots are $S$–equivalent if and only if they are related by a finite\nnumber of doubled-delta moves. These moves do not preserve the knot\ntype in general: for example, it is shown in [NS03] that a knot is $S$–\nequivalent to the unknot if and only if it has Alexander polynomial is\nequal to 1.\n\n(3) Consider the notation, defined for a Seifert form $V$ and for a knot $K$:\n$[V]_{S} = \\{$ $\\widetilde{V} |$ $\\widetilde{V}$ is a Seifert form $S$–equivalent to $V\\},$\n$[K]_{S} = \\{$ $\\widetilde{K} |$ $\\widetilde{K}$ is a knot $S$–equivalent to $K\\}.$\nSince for a given knot $K$, any two of its Seifert forms $V_{1}$ and $V_{2}$ are $S$–\nequivalent, it follows that $V_{1}, V_{2} \\in [V]_{S}$, where $V$ is any Seifert form of\n$K$. Thus, the set of all Seifert forms of $K$ maps to a subset of the $S$–\nequivalence class of any one of its Seifert forms. Given this, the question\nasked in this problem, can be rephrased as follows.\nQuestion. For a given $S$–equivalence class $[V]_{S}$ of a Seifert form $V$ ,\ndoes there exist a fixed knot $K$ such that the obvious map\n$\\{$ $\\widetilde{V} |$ $\\widetilde{V}$ is a Seifert form of $K\\} \\longrightarrow [V]_{S}$\nis surjective?\nThe key here being the existence of a single knot $K$ for which the\nindicated map is surjective, as every element of $[V]_{S}$ arises as the Seifert\nform of some knot; cf. [AFMW23, page 20].\n(4) In [AFMW23], the authors pose the genus-one version of this problem as\nProblem 7.7, and claim a complete answer under the additional hypothesis\nthat Seifert surfaces be disjoint.\nThey find a necessary and sufficient\ncondition for a pair of $2\\times2$ matrices to be Seifert forms for a pair of disjoint\nSeifert surfaces of the same knot, and they give an explicit construction\nrealizing infinitely many pairs of disjoint Seifert surfaces with the same\nboundary for any pair satisfying the condition.\n\nReferences cited:\n- [Tro73] H. F. Trotter. On S-equivalence of Seifert matrices. Invent. Math., 20:173–207, 1973. doi:10.1007/BF01394094.\n- [Kaw96] Akio Kawauchi. A survey of knot theory. Birkhäuser Verlag, Basel, 1996. Translated and revised from the 1990 Japanese original by the author.\n- [Kea04] C. Kearton. S-equivalence of knots. J. Knot Theory Ramifications, 13(6):709–717, 2004. doi:10.1142/S0218216504003408.\n- [NS03] Swatee Naik and Theodore Stanford. A move on diagrams that generates S-equivalence of knots. J. Knot Theory Ramifications, 12(5):717–724, 2003. doi: 10.1142/S0218216503002639.\n- [AFMW23] Menny Aka, Peter Feller, Alison Beth Miller, and Andreas Wieser. Seifert surfaces in the four-ball and composition of binary quadratic forms, 2023. arXiv:2311.17746.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The fixed-knot realization problem remains open in general, with a necessary-and-sufficient genus-one answer under the extra condition that the Seifert surfaces are disjoint.\n\n**Verified partial progress.**\n\n- Naik--Stanford characterize knot S-equivalence by doubled-delta moves, which do not generally preserve the knot type.\n- Aka--Feller--Miller--Wieser classify which pairs of genus-one Seifert forms arise from disjoint Seifert surfaces with the same boundary knot and give explicit realizations.\n- Their preprint was revised to version 5 in January 2026 and still states the restricted disjoint/genus-one scope.\n\n**Full solution or refutation.**\n\nThe higher-genus or intersecting-surface fixed-knot question is not settled.\n\n**What remains.**\n\nRemove the genus-one and disjointness restrictions or identify the additional obstructions in the general case.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 1.6 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the general fixed-knot problem and the AFMW genus-one/disjoint partial answer.\n- Menny Aka, Peter Feller, Alison Beth Miller, and Andreas Wieser, Seifert surfaces in the four-ball and composition of binary quadratic forms, arXiv:2311.17746v5 (2026). (primary): https://arxiv.org/abs/2311.17746\n Evidence used: The abstract states the genus-one classification for pairs of disjoint Seifert surfaces.\n- Swatee Naik and Theodore Stanford, A move on diagrams that generates S-equivalence of knots, J. Knot Theory Ramifications 12 (2003), 717--724. (primary): https://doi.org/10.1142/S0218216503002639\n Evidence used: Characterizes knot S-equivalence via doubled-delta moves.\n\n**Review notes.** The record order is preserved even though KP-1.6 numerically precedes the surrounding KP-1.5x records.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2666, "problem_number": "KP-1.7", "title": "Kirby Problem 1.7", "statement": "Show that the sequence of absolute values of the coefficients of\nthe Alexander polynomial of a link are:\n(a) concave $($ Fox’s trapezoidal conjecture $)$, or\n(b) for an alternating link, log concave $($ Stoimenow’s strong Fox conjecture $)$.\n(c) For a positive link, is the sequence of the coefficients of the Conway poly-\nnomial strongly log concave?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.7.\n\nLiterature notes:\n(1) Let $K$ be a link with Alexander polynomial (up to a factor of $t^{r}$, $r \\in \\frac{1}{2}\\mathbb{Z}$)\n\n$$\n\\Delta_{K}(t)=\\sum_{i=0}^{n} a_i t^i,\n$$\n\nwith $a_{0}, a_{n} \\neq 0$, and Conway polynomial\n\n$$\n\\nabla_{K}(z)=z^{c(K)-1}\\sum_{i=0}^{m} c_i z^{2i}\n$$\n\ndefined by $\\nabla_{K}(t - t^{-1}) = t^{-n}\\Delta_{K}(t^{2})$, where $c(K)$ denotes the number\nof components of $K$. Recall that $a_{i} = (-1)^{n}a_{n-i}$; that for a non-split\nalternating link, $a_{i}a_{i+1} < 0$, $0 \\leq i < n$; and that for a positive link,\n$c_{i} > 0$, $0 \\leq i \\leq m$.\nFor the Strong Fox Conjecture (3) below, recall that a sequence of\npositive numbers $(a_{i})_{0\\leq i\\leq n}$ is called log concave if $a_{i-1}a_{i+1} \\leq a_i^{2}$\nfor all $i$ with $0 < i < n$. That is, the sequence $(\\log(a_{i}))$ is concave. For the final\nconjecture (4), recall that a sequence of positive numbers $(c_{i})_{0\\leq i\\leq m}$ is\ncalled strongly log concave if $c_{i-1}c_{i+1} < c_i^{2}$ for all $i$ with $0 < i < m$.\n(2) (Fox’s trapezoidal conjecture [Fox62, Problem 12]) Let $K$ be an alternating knot\nwith Alexander polynomial as above. Then\n\n$$\n|a_{0}| \\leq |a_{1}| \\leq \\cdots \\leq |a_{\\lfloor n/2\\rfloor}|\n\\geq |a_{\\lfloor n/2\\rfloor+1}| \\geq \\cdots \\geq |a_{n}|.\n$$\n\nMoreover, if $|a_{i}| = |a_{i+1}|$ for some $i < n/2$, then\n\n$$\n|a_{i}| = |a_{i+1}| = \\cdots = |a_{n-i}|.\n$$\n\n(1)\nSee for example [Mur85, AJK24] for some progress on Fox’s conjecture.\n(3) (Strong Fox Conjecture) Let $K$ be an alternating link with Alexander\npolynomial as above. Then the sequence of absolute values of the coeffi-\ncients $(|a_{i}|)$ is log concave. The equality $a_{i-1}a_{i+1} = a_i^{2}$ for some $i < n/2$\nholds only if $a_{i-1} = -a_{i} = a_{i+1}$. Moreover, the length of this ‘trapezoid\ntop’, i.e. the length of the longest string of equalities as in Equation (1) is\nbounded above by the absolute value of the signature. Conjecture (3) first\nappeared in [Sto05]. It is called the Strong Fox Conjecture in [Ban22].\nLog-concavity is a ubiquitous phenomenon appearing in many fields of\nmathematics, for example, [AHK17] and [Sta89].\nThe Strong Fox Conjecture was proved for 2-bridge knots by Ban-\nfield [Ban22] and for special alternating links by Hafner–Meszaros–Vidinas\n[HMV24]. The conjecture, including the statement regarding equality,\nwas verified by Stoimenow [Sto16, Section 9.2] for alternating knots of\ngenus at most 4 and for certain plumbed links [Sto21, Theorem 5.1]\n(4) Conjecture (Stoimenow): Let $K$ be a positive link with Conway poly-\nnomial as above. Then the sequence of coefficients $(c_{i})$ is strongly log\nconcave.\nConjecture (4) was proved for special alternating links by Stoimenow\n[Sto05, Theorem 1].\n\nReferences cited:\n- [Fox62] Ralph Fox. Some problems in knot theory. In M. K. Fort, Jr., editor, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961). Prentice-Hall, Englewood Cliffs, N.J., 1962.\n- [Mur85] Kunio Murasugi. On the Alexander polynomial of alternating algebraic knots. J. Austral. Math. Soc. Ser. A, 39(3):317–333, 1985.\n- [AJK24] Soheil Azarpendar, András Juhász, and Tamás Kálmán. On Fox’s trapezoidal conjecture, 2024. arXiv:2406.08662.\n- [Sto05] Alexander Stoimenow. Newton-like polynomials of links. Enseign. Math. (2), 51(3-4):211–230, 2005.\n- [Ban22] Ian M. Banfield. Christoffel words and the strong Fox conjecture for two-bridge knots, 2022. arXiv:2212.04561.\n- [AHK17] Karim Adiprasito, June Huh, and Eric Katz. Hodge theory of matroids. Notices Amer. Math. Soc., 64(1):26–30, 2017. doi:10.1090/noti1463.\n- [Sta89] Richard P. Stanley. Log-concave and unimodal sequences in algebra, combinatorics, and geometry. In Graph theory and its applications: East and West (Jinan, 1986), volume 576 of Ann. New York Acad. Sci., pages 500–535. New York Acad. Sci., New York, 1989. doi:10.1111/j.1749-6632.1989.tb16434.x.\n- [HMV24] Elena S. Hafner, Karola Mészáros, and Alexander Vidinas. Log-concavity of the Alexander polynomial. Int. Math. Res. Not. IMRN, 2024(13):10273–10284, 2024. doi:10.1093/imrn/rnae058.\n- [Sto16] Alexander Stoimenow. Diagram genus, generators, and applications. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016.\n- [Sto21] A. Stoimenow. Independence polynomials and Alexander-Conway polynomials of plumbing links. J. Combin. Theory Ser. A, 183:Paper No. 105487, 18, 2021. doi: 10.1016/j.jcta.2021.105487.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Fox trapezoidality, the strong alternating log-concavity conjecture, and the positive-link Conway strong-log-concavity question remain open.\n\n**Verified partial progress.**\n\n- The source notes standard sign/symmetry properties motivating the conjectures.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was verified for the listed forms.\n\n**What remains.**\n\nResolve each coefficient-shape conjecture, with conventions for links fixed.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.7 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintains the three conjectures and their precise coefficient conventions.\n\n**Review notes.** Multi-part record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2667, "problem_number": "KP-1.8", "title": "Kirby Problem 1.8", "statement": "Which multi-variable Laurent polynomials arise as the multi-\nvariable Alexander polynomial of a link in the 3-sphere or, more generally, a ho-\nmology 3-sphere?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.8.\n\nLiterature notes:\n(1) A link in a homology sphere $Y$ has a covering corresponding to the ho-\nmomorphism $\\pi_{1}(Y \\setminus L) \\twoheadrightarrow\\mathbb{Z}^{n}$ arising from the isomorphism $H_{1}(Y \\setminus L) \\cong$\n$\\mathbb{Z}^{n}$, where $n$ is the number of components of the link $L$.\nThe homol-\nogy of this covering gives rise to a multi-variable Alexander polynomial\n\n$$\n\\Delta_{L}(t_{1},\\dots,t_{n}) \\in \\mathbb{Z}[t_{1}^{\\pm 1},\\dots,t_{n}^{\\pm 1}],\n$$\n\nwell defined up to multiplication by units, which are all of the form\n$\\pm t_{1}^{\\nu_{1}}\\cdots t_{n}^{\\nu_{n}}$.\n(2) The problem has been solved for the single-variable Alexander polynomial\nof knots. If $K$ is a knot in a homology sphere, then:\n\n$$\n\\bullet \\Delta_{K}(t) = \\Delta_{K}(t^{-1})\n$$\n\n$\\bullet \\Delta_{K}(1) = 1$.\nMoreover, any polynomial satisfying these two properties is realized by a\nknot in the 3-sphere (or any homology sphere) [Sei35, Lev65, Rol90].\nTorres [Tor53] found analogous (necessary but not sufficient–see [Hil81,\nPla86]) properties of the multi-variable Alexander polynomial of a link.\nIt is perhaps surprising that while there is no known characterization of\nthe Alexander polynomial of two-component links, Bailey found a charac-\nterization [Bai77] (discussed in [Pla86]) of their Alexander modules. A\ngeneral reference for the ideas here is Hillman’s book [Hil12].\n(3) The problem can be refined by focusing on specific families of links.\nQuestion (i). Which polynomials arise for a given family of knots\nor links?\nSome partial results are known.\n$\\bullet$ Torus links. If $T_{p,q}$ is the $(p, q)-$ torus link, let $d = \\gcd(p, q)$. Then\n\n$$\n\\Delta_{Tp,q}(t_{1}, t_{2},..., t_{d}) = [(t_{1}t_{2}... t_{d})^{pq/d} - 1]^{d}[(t_{1}t_{2}... t_{d}) - 1]\n$$\n\n$$\n[(t_{1}t_{2}... t_{d})^{p/d} - 1][(t_{1}t_{2}... t_{d})^{q/d} - 1].\n$$\n\nSee [Rol90] for knots and [EN85, Theorem 12.1]; compare [Ore21,\nCorollary 8.2] for links with $d > 1$.\n$\\bullet$ Two-bridge links. Many authors have found necessary (but still in-\nsufficient) conditions on the Alexander polynomial of a 2-bridge knot\nor link. Results of Koseleff and Pecker [KP15] subsume most earlier\nresults. Given a 2-variable polynomial, Hoste [Hos20], gives an al-\ngorithm to decide if the polynomial is the Alexander polynomial of\na 2-component, 2-bridge link.\n$\\bullet$ Alternating links. The Alexander polynomials of alternating knots\nhave special properties [Cro59, Mur58], and their coefficients are\nconjectured to have various concavity properties. See Problem 1.7\nfor an extensive discussion of these. It would be of interest to extend\nsuch properties to the case of alternating links.\n(4) Similarly, any 3-manifold with $H_{1}(X) \\cong \\mathbb{Z}^{n}$ has an associated multi-\nvariable Alexander polynomial $\\Delta_{X} \\in \\mathbb{Z}[H_{1}(X)] \\cong \\mathbb{Z}[t_{1}^{\\pm1},\\ldots,t_{n}^{\\pm1}]$.\nQuestion (ii). Which Laurent polynomials arise as the multi-variable\nAlexander polynomial of a 3-manifold?\n\nReferences cited:\n- [Sei35] H. Seifert. Über das Geschlecht von Knoten. Math. Ann., 110(1):571–592, 1935. doi:10.1007/BF01448044.\n- [Lev65] J. Levine. A characterization of knot polynomials. Topology, 4:135–141, 1965. doi: 10.1016/0040-9383(65)90061-3.\n- [Rol90] Dale Rolfsen. Knots and links, volume 7 of Mathematics Lecture Series. Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original.\n- [Tor53] Guillermo Torres. On the Alexander polynomial. Ann. of Math. (2), 57:57–89, 1953. doi:10.2307/1969726.\n- [Hil81] Jonathan A. Hillman. The Torres conditions are insufficient. Math. Proc. Cambridge Philos. Soc., 89(1):19–22, 1981. doi:10.1017/S0305004100057893.\n- [Pla86] M. L. Platt. Insufficiency of Torres’ conditions for two-component classical links. Trans. Amer. Math. Soc., 296(1):125–136, 1986. doi:10.2307/2000564.\n- [Bai77] James Leonard Bailey. ALEXANDER INVARIANTS OF LINKS. ProQuest LLC, Ann Arbor, MI, 1977. Thesis (Ph.D.)–The University of British Columbia (Canada). URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\& rft val fmt=info:ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri: pqdiss:NK34756.\n- [Hil12] Jonathan Hillman. Algebraic invariants of links, volume 52 of Series on Knots and Everything. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, second edition, 2012. doi:10.1142/8493.\n- [EN85] David Eisenbud and Walter Neumann. Three-dimensional link theory and invariants of plane curve singularities, volume 110 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1985.\n- [Ore21] S. Yu. Orevkov. Multivariate signatures of iterated torus links. Funktsional. Anal. i Prilozhen., 55(1):73–92, 2021. English translation in Funct. Anal. Appl. 55 (2021), no. 1, 59–74. doi:10.1134/S001626632101007X.\n- [KP15] Pierre-Vincent Koseleff and Daniel Pecker. On Alexander-Conway polynomials of two-bridge links. J. Symbolic Comput., 68:215–229, 2015. doi:10.1016/j.jsc.2014.09.011.\n- [Hos20] Jim Hoste. A note on Alexander polynomials of 2-bridge links. J. Knot Theory Ramifications, 29(8):1971003, 7, 2020. doi:10.1142/S0218216519710032.\n- [Cro59] Richard Crowell. Genus of alternating link types. Ann. of Math. (2), 69:258–275, 1959. doi:10.2307/1970181.\n- [Mur58] Kunio Murasugi. On the genus of the alternating knot. I, II. J. Math. Soc. Japan, 10:94–105, 235–248, 1958. doi:10.2969/jmsj/01010094.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Single-variable Alexander polynomials of knots are completely characterized by symmetry and value at one; no characterization is known even for two-component multi-variable link polynomials.\n\n**Verified partial progress.**\n\n- Seifert--Levine--Rolfsen realization theorem solves the knot case.\n- Torres-type conditions are necessary but insufficient for links.\n- Alexander modules of two-component links have a characterization.\n\n**Full solution or refutation.**\n\nThe imported broad link question remains open.\n\n**What remains.**\n\nCharacterize multi-variable Alexander polynomials, starting with two components.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.8 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the complete knot result and unsolved multi-variable link case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2668, "problem_number": "KP-1.9", "title": "Kirby Problem 1.9", "statement": "If Dehn surgery on a knot $K$ gives a lens space, then $K$ is a\nBerge knot.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.9.\n\nLiterature notes:\n(1) This appears as [Kir97, Problem 1.78]. Let $(\\Sigma; H_{1}, H_{2})$ denote the genus-\ntwo Heegaard splitting of $S^{3}$ (cf. Section 3.3). A knot $K \\subset S^{3}$ is a Berge\nknot if it lies on $\\Sigma$ and there are properly embedded disks $D_{i} \\subset H_{i}$\nsuch that $|K \\cap D_{i}| = 1$. These knots are also called doubly primitive;\nsee [Gor09, Section 3.2].\n(2) The conjecture, often referred to as the Berge Conjecture, is known for\nsmall order lens spaces [Gor09]. For example, if surgery on $K$ gives $L(p, q)$\nwith $|p| \\leq 5$, then $K$ is the unknot or the trefoil [KMOS07]. In general,\nif a lens space were to arise as surgery on a non-Berge knot, that lens\nspace must also arise as surgery on a Berge knot [Gre13b]. The putative\nknot necessarily has the same knot Floer homology as that corresponding\nBerge knot.\n(3) From its description, any Berge knot is necessarily a tunnel number one\nknot. The Berge conjecture has been verified for all tunnel number one\nknots in the preprint [LMP25] and all non-hyperbolic knots [Mos71,\nBL89].\n(4) If surgery on a non-torus knot produces a lens space, then it must be an\nintegral surgery by the Cyclic Surgery Theorem [CGLS87]. The size of\nthe surgery coefficient is also heavily constrained. If $S^{3}_{p}(K) = L(p,q)$ for a\nnontrivial, non-trefoil knot $K$, then\n$2g(K)-1 \\leq |p|-2\\sqrt{4(|p|+1)/5}$ [Gre13b].\nIf $K$ is a counterexample to the Berge conjecture, then $|p| < 4g(K) - 1$\n[Bak06].\n(5) Using Heegaard Floer homology, it is also known that a knot with a lens\nspace surgery must be fibered and, after possibly mirroring, strongly quasi-\npositive [Ni07, Hed10]. The knot Floer homology of the dual knot $\\widetilde{K}$\nin $L(p,q)$ has smallest possible knot Floer homology,\n\n$$\n\\dim \\widehat{\\mathrm{HFK}}(\\widetilde{K}) = \\dim \\widehat{\\mathrm{HF}}(L(p,q))\n$$\n\n[Ras07, Hed11, Gre13b]. Such a knot is called Floer\nsimple, and one proposed strategy is to prove that Floer simple knots are\none-bridge with respect to the standard genus one Heegaard splitting of\n$L(p, q)$ [BGH08], as Berge proved that such knots are necessarily doubly\nprimitive [Ber18].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Gor09] Cameron Gordon. Dehn surgery and 3-manifolds. In Low dimensional topology, volume 15 of IAS/Park City Math. Ser., pages 21–71. Amer. Math. Soc., Providence, RI, 2009.\n- [KMOS07] P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó. Monopoles and lens space surgeries. Ann. of Math. (2), 165(2):457–546, 2007. doi:10.4007/annals.2007.165.457.\n- [Gre13b] Joshua Evan Greene. The lens space realization problem. Ann. of Math. (2), 177(2):449–511, 2013. doi:10.4007/annals.2013.177.2.3.\n- [LMP25] Tao Li, Yoav Moriah, and Tali Pinsky. Tunnel number one knots satisfy the Berge conjecture. Geom. Topol., 29(6):3271–3343, 2025. doi:10.2140/gt.2025.29.3271.\n- [Mos71] Louise Moser. Elementary surgery along a torus knot. Pacific J. Math., 38:737–745, 1971. http://projecteuclid.org/euclid.pjm/1102969920.\n- [BL89] Steven A. Bleiler and Richard A. Litherland. Lens spaces and Dehn surgery. Proc. Amer. Math. Soc., 107(4):1127–1131, 1989. doi:10.2307/2047677.\n- [CGLS87] Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen. Dehn surgery on knots. Ann. of Math. (2), 125(2):237–300, 1987. doi:10.2307/1971311.\n- [Bak06] Kenneth L. Baker. Small genus knots in lens spaces have small bridge number. Algebr. Geom. Topol., 6:1519–1621, 2006. doi:10.2140/agt.2006.6.1519.\n- [Ni07] Yi Ni. Knot Floer homology detects fibred knots. Invent. Math., 170(3):577–608, 2007. doi:10.1007/s00222-007-0075-9.\n- [Hed10] Matthew Hedden. Notions of positivity and the Ozsváth-Szabó concordance invariant. J. Knot Theory Ramifications, 19(5):617–629, 2010. doi:10.1142/S0218216510008017.\n- [Ras07] Jacob Rasmussen. Lens space surgeries and L-space homology spheres, 2007. arXiv: 0710.2531.\n- [Hed11] Matthew Hedden. On Floer homology and the Berge conjecture on knots admitting lens space surgeries. Trans. Amer. Math. Soc., 363(2):949–968, 2011. doi:10.1090/S0002-9947-2010-05117-7.\n- [BGH08] Kenneth L. Baker, J. Elisenda Grigsby, and Matthew Hedden. Grid diagrams for lens spaces and combinatorial knot Floer homology. Int. Math. Res. Not. IMRN, 2008(10):Art. ID rnm024, 39, 2008. doi:10.1093/imrn/rnn024.\n- [Ber18] John Berge. Some knots with surgeries yielding lens spaces, 2018. arXiv:1802.09722.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Berge conjecture is proved for tunnel-number-one knots, all nonhyperbolic knots, and small-order lens-space surgeries, but remains open for general hyperbolic knots of higher tunnel number.\n\n**Verified partial progress.**\n\n- Li--Moriah--Pinsky prove that tunnel-number-one knots with lens-space surgery are doubly primitive.\n- Classical work settles torus and satellite/nonhyperbolic cases.\n- Greene proves that any lens space arising by knot surgery also arises from a Berge knot and imposes strong Heegaard-Floer restrictions on a hypothetical non-Berge knot.\n\n**Full solution or refutation.**\n\nThe 2025 tunnel-number-one theorem closes the conjecture's most natural geometric subclass, while surgery and Floer constraints sharply limit but do not eliminate general counterexamples.\n\n**What remains.**\n\nProve every hyperbolic higher-tunnel-number knot admitting a lens-space surgery is doubly primitive, or construct a counterexample.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.9 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Records the tunnel-number-one and nonhyperbolic resolutions and the remaining general conjecture.\n- Tao Li, Yoav Moriah and Tali Pinsky, Tunnel number one knots satisfy the Berge conjecture, Geometry & Topology 29 (2025), 3271--3343. (primary): https://doi.org/10.2140/gt.2025.29.3271\n Evidence used: Proves the full tunnel-number-one case.\n- Joshua Evan Greene, The lens space realization problem, Annals of Mathematics 177 (2013), 449--511. (primary): https://doi.org/10.4007/annals.2013.177.2.3\n Evidence used: Classifies lens spaces obtainable by surgery and supplies the Floer-theoretic constraints described in the source.\n\n**Review notes.** The statement was kept as the general Berge conjecture; subclass theorems were not promoted to a full solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2669, "problem_number": "KP-1.10", "title": "Kirby Problem 1.10", "statement": "(Generalized Property R Conjecture). Let $L \\subset S^{3}$ be an $n$-\ncomponent link such that 0-framed Dehn surgery on $L$ results in $\\#^{n}(S^{1} \\times S^{2})$. Is\nthere a sequence of handleslides converting $L$ into an $n$-component unlink?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.10.\n\nLiterature notes:\n(1) An $n$-component link $L \\subset S^{3}$ whose 0-surgery is $\\#^{n}(S^{1} \\times S^{2})$ is called\nan R-link. An R-link is said to have generalized property R if it can be\nconverted by handleslides to an unlink. (Note that all framings and linking\nnumbers are zero.)\nGabai proved [Gab87a] that the only 1-component R-link is the\nunknot, solving the Property R Conjecture. The Generalized Property\nR Conjecture (GPRC) posits that all R-links have generalized property\nR [Kir97, Problem 1.82] and was studied in [GST10].\n(2) There are two ways to weaken the GPRC that are of interest. Let $L \\sqcup L'$\ndenote the split union of links $L$ and $L'$.\nConjecture (Stable GPRC). If $L$ is an R-link, then there is an\nunlink $\\operatorname{U}$ such that $L \\sqcup \\operatorname{U}$ can be converted by handleslides to an unlink.\nMany potential counterexamples to the Stable GPRC are given in\n[GST10, MZ22].\n(3) A Hopf pair is a canceling 1-handle/2-handle pair, which can be repre-\nsented in the Kirby calculus by a Hopf link with a dotted-circle component\nand a 0-framed component [GS99, Section 5.4].\nConjecture (Weak GPRC). If $L$ is an R-link, then there is an un-\nlink $\\operatorname{U}$ and a split union $H$ of Hopf pairs such that $L \\sqcup \\operatorname{U} \\sqcup H$ can be\nconverted by handleslides to the split union $\\operatorname{U}' \\sqcup H'$ of an unlink and a\nsplit union of Hopf pairs.\nA sufficient condition for a 2-component link to satisfy the Weak\nGPRC is given in [MZ22, Theorem 1.1].\n(4) There are connections between these conjectures and other important\nproblems. Given an R-link $L$, one can construct a homotopy 4-ball $B_{L}$\nby attaching 0-framed 2-handles to $S^{3} \\times I$ along $L$ and capping off with\na 4-dimensional 1-handlebody. Turning this handle-decomposition upside\ndown gives a balanced presentation $P_{L}$ of the trivial group.\nAny homotopy 4-sphere that can be built without 1-handles is $B_{L}\\cup B^{4}$\nfor some R-link $L$. If an R-link L satisfies the Weak GPRC, then $B_{L}$ is\ndiffeomorphic to $B^{4}$. So, the Weak GPRC implies the Smooth Poincaré\nConjecture in dimension 4 for homotopy 4-spheres built without 1-handles.\n\nThe Andrews–Curtis Conjecture (ACC–see Problem 5.10) posits that\nany balanced presentation of the trivial group can be converted to the\ntrivial presentation through balanced presentations. If the GPRC is true,\nthen so is the ACC. There is also a stable version of the ACC that is\nimplied by the Stable GPRC [GST10, MZ22].\nIf $L$ is an R-link, then $L$ bounds a collection of homotopy-ribbon disks\nin the homotopy 4-ball $B_{L}$. Thus, such links provide potential counterex-\namples to the Slice-Ribbon Conjecture (Problem 1.50) but are ribbon if\nthey satisfy the Stable GPRC [AT16b]; see [MZ25, Proposition 5.2]. If\n$K$ is a fibered, homotopy-ribbon knot, then the closed monodromy of $K$\nadmits an extension across a handlebody [CG83a]. Any such extension\ngives rise to a link $L$ lying on the fiber of $K$ such that $L$ and $K \\cup L$ are\nboth R-links, and $K \\cup L$ is ribbon if $L$ (equivalently, $K \\cup L$) satisfies the\nStable GPRC.\n\nReferences cited:\n- [Gab87a] David Gabai. Foliations and the topology of 3-manifolds. II. J. Differential Geom., 26(3):461–478, 1987. http://projecteuclid.org/euclid.jdg/1214441487.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GST10] Robert E. Gompf, Martin Scharlemann, and Abigail Thompson. Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures. Geom. Topol., 14(4):2305–2347, 2010. doi:10.2140/gt.2010.14.2305.\n- [MZ22] Jeffrey Meier and Alexander Zupan. Generalized square knots and homotopy 4-spheres. J. Differential Geom., 122(1):69–129, 2022. doi:10.4310/jdg/1668186788.\n- [GS99] Robert E. Gompf and András I. Stipsicz. 4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999. doi:10.1090/gsm/020.\n- [AT16b] Tetsuya Abe and Motoo Tange. A construction of slice knots via annulus twists. Michigan Math. J., 65(3):573–597, 2016. doi:10.1307/mmj/1472066149.\n- [MZ25] Jeffrey Meier and Alexander Zupan. Knots bounding nonisotopic ribbon disks. J. Topol., 18(4):Paper No. e70047, 18, 2025. doi:10.1112/topo.70047.\n- [CG83a] A. J. Casson and C. McA. Gordon. A loop theorem for duality spaces and fibred ribbon knots. Invent. Math., 74(1):119–137, 1983. doi:10.1007/BF01388533.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Property R is proved for one-component R-links, while Generalized Property R remains open for arbitrary component count.\n\n**Verified partial progress.**\n\n- Gabai proved the n=1 case.\n- Stable and weak variants have partial criteria and potential counterexamples.\n\n**Full solution or refutation.**\n\nThe multi-component handleslide assertion is unresolved.\n\n**What remains.**\n\nResolve Generalized Property R or its stable/weak variants.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.10 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Current expert problem list states n=1 resolution and remaining conjectures.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2670, "problem_number": "KP-1.11", "title": "Kirby Problem 1.11", "statement": "(Cabling conjecture). Let $K \\subset S^{3}$ be a knot and $r \\in \\mathbb{Q}$. If\n$r$-framed Dehn surgery on $K$ is not prime, then $K$ is a nontrivial cable of a knot\n$J \\subset S^{3}$ and $r$ is the slope of the cabling annulus.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.11.\n\nLiterature notes:\n(1) The conjecture is due to González-Acuña and Short [GAnS86]. If $K =$\n$J_{p,q}$ is a cable knot, then there is an annulus in $S^{3}\\setminus\\nu(K)$ that has boundary\nequal to two parallel curves of slope $pq$. This annulus is called the cabling\nannulus and $pq$ is referred to as the slope of the cabling annulus. Using a\ncut and paste argument, one can show that\n\n$$\nS^{3}_{pq}(K)=S^{3}_{q/p}(J)\\#S^{3}_{p/q}(\\operatorname{U}),\n$$\n\nwhere $\\operatorname{U}$ is the unknot. Therefore, if $p \\neq 1$, then $S^{3}_{pq}(K)$ is not prime.\n(2) Many special cases of the conjecture are known.\nGordon and Luecke\n[GL87] prove that if $r$ is a reducible slope, then $r$ is an integer and the\nsurgery contains a lens space summand.\nThe conjecture is known for\nsatellite knots [Sch90], strongly invertible knots [EMn92], alternating\nknots [MT92] and symmetric knots [LZ94, HS98, HM97b].\n(3) Greene [Gre15] has shown that if $r$-framed surgery on $K$ gives a connected\nsum of nontrivial lens spaces, then $K$ is a $(p, q)$-cable of either the unknot\nor a torus knot and that $r = pq$.\n(4) Dunfield [Dun20] observed that a similar conjecture may hold for knots\nin $S^{1} \\times S^{2}$. He conjectures that there are no hyperbolic knots in $S^{1} \\times S^{2}$\nthat have non-prime Dehn surgery.\n(5) A weaker version of the cabling conjecture is the two summands conjecture,\nwhich says that if $S^{3}_{r}(K)$ is non-prime, then it cannot have more than two\nprime summands.\nNote that it follows from work of Howie [How02],\nSayari [Say09] and Valdez-Sánchez [VS99] that $S^{3}_{r}(K)$ can never contain\nmore than three irreducible summands.\n\nReferences cited:\n- [GAnS86] Francisco González-Acuña and Hamish Short. Knot surgery and primeness. Math. Proc. Cambridge Philos. Soc., 99(1):89–102, 1986. doi:10.1017/S0305004100063969.\n- [GL87] C. McA. Gordon and J. Luecke. Only integral Dehn surgeries can yield reducible manifolds. Math. Proc. Cambridge Philos. Soc., 102(1):97–101, 1987. doi:10.1017/S0305004100067086.\n- [Sch90] Martin Scharlemann. Producing reducible 3-manifolds by surgery on a knot. Topology, 29(4):481–500, 1990. doi:10.1016/0040-9383(90)90017-E.\n- [EMn92] Mario Eudave Muñoz. Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots. Trans. Amer. Math. Soc., 330(2):463–501, 1992. doi:10.2307/2153918.\n- [MT92] William W. Menasco and Morwen B. Thistlethwaite. Surfaces with boundary in alternating knot exteriors. J. Reine Angew. Math., 426:47–65, 1992. doi:10.1515/crll.1992.426.47.\n- [LZ94] E. Luft and X. Zhang. Symmetric knots and the cabling conjecture. Math. Ann., 298(3):489–496, 1994. doi:10.1007/BF01459747.\n- [HS98] Chuichiro Hayashi and Koya Shimokawa. Symmetric knots satisfy the cabling conjecture. Math. Proc. Cambridge Philos. Soc., 123(3):501–529, 1998. doi:10.1017/S0305004197002399.\n- [HM97b] Chuichiro Hayashi and Kimihiko Motegi. Dehn surgery on knots in solid tori creating essential annuli. Trans. Amer. Math. Soc., 349(12):4897–4930, 1997. doi: 10.1090/S0002-9947-97-01723-6.\n- [Gre15] Joshua Evan Greene. L-space surgeries, genus bounds, and the cabling conjecture. J. Differential Geom., 100(3):491–506, 2015. http://projecteuclid.org/euclid.jdg/1432842362.\n- [Dun20] Nathan M. Dunfield. A census of exceptional Dehn fillings. In Characters in lowdimensional topology, volume 760 of Contemp. Math., pages 143–155. Amer. Math. Soc., [Providence], RI, [2020] ©2020. doi:10.1090/conm/760/15289.\n- [How02] James Howie. A proof of the Scott-Wiegold conjecture on free products of cyclic groups. J. Pure Appl. Algebra, 173(2):167–176, 2002. doi:10.1016/S0022-4049(02) 00042-7.\n- [Say09] Nabil Sayari. Reducible Dehn surgery and the bridge number of a knot. J. Knot Theory Ramifications, 18(4):493–504, 2009. doi:10.1142/S0218216509007038.\n- [VS99] Luis Gerardo Valdez Sánchez. Dehn fillings of 3-manifolds and non-persistent tori. Topology Appl., 98(1-3):355–370, 1999. II Iberoamerican Conference on Topology and its Applications (Morelia, 1997). doi:10.1016/S0166-8641(99)00038-3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The cabling conjecture is known for several classes and for surgeries yielding a connected sum of nontrivial lens spaces, but remains open in general.\n\n**Verified partial progress.**\n\n- Known for satellite, strongly invertible, alternating and symmetric knots.\n- Greene's lens-space-summand theorem gives a strong special case.\n\n**Full solution or refutation.**\n\nNo general reducible-surgery classification was verified.\n\n**What remains.**\n\nProve every reducible surgery arises from a cable at its cabling slope.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.11 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintained list enumerates these special cases and leaves the conjecture open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2671, "problem_number": "KP-1.12", "title": "Kirby Problem 1.12", "statement": "This problem presents several variations on the Cosmetic\nSurgery Conjecture, discussed in turn below.\n(a) $($ Cosmetic Surgery Conjecture $)$ Two surgeries along inequivalent slopes are\nnever purely cosmetic.\n(b) $($ Oriented knot complement conjecture $)$ If $K_{1}$ and $K_{2}$ are knots in a closed,\noriented 3-manifold $N$ whose complements are homeomorphic via an orientation-\npreserving homeomorphism, then there exists a homeomorphism of $N$ tak-\ning $K_{1}$ to $K_{2}$.\n(c) Let $K \\subset S^{3}$ be a nontrivial knot, then $K$ admits no purely cosmetic surg-\neries.\n(d) If $K \\subset S^{3}$ is not the torus knot $T_{2g+1,2}$, then $K$ admits no chirally cos-\nmetic surgeries with inequivalent slopes.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.12.\n\nLiterature notes:\n(1) This is an update to [Kir97, Problem 1.81], contributed by S. Bleiler.\nParts (a) and (b) appear as Conjectures 6.1 and 6.2 of [Gor91] respec-\ntively.\n(2) For the rest of this problem, $M$ denotes an irreducible oriented 3-manifold\nwith torus boundary that is not the solid torus. Two fillings along dis-\ntinct slopes $r_{1}$ and $r_{2}$ are called purely cosmetic if there is an orientation-\npreserving homeomorphism between $M(r_{1})$ and $M(r_{2})$, and chirally cos-\nmetic if there is an orientation-reversing homeomorphism between these\nfillings. (The terminology would be purely or chirally cosmetic surgery if\n$M$ is a knot complement.) Two slopes are said to be equivalent if there\nexists a homeomorphism (either preserving or reversing the orientation)\nof $M$ taking one slope to the other.\n(3) When [Kir97, Problem 1.81] was written, several examples of chirally cos-\nmetic surgeries on inequivalent slopes were known. More recently, Ichihara\nand Jong [IJ18], with help from Masai, found a hyperbolic one-cusped\nmanifold admitting a pair of chirally cosmetic surgeries with inequivalent\nslopes, such that the resulting manifold is also hyperbolic. This gives a\ncounterexample to [Kir97, Problem 1.81B].\nDunfield found that the manifold $o9_{39009}$ from Burton’s census [Bur14]\nhad Dehn fillings $(-1, 3)$ and $(-3, 2)$, which are isometric by an orienta-\ntion reversing isometry, but where the core curves of the Dehn fillings are\ngeodesics of different lengths. This provides a counterexample to [Kir97,\nProblem 1.81C].\n(4) There are different versions of the Cosmetic Surgery Conjecture (a) in\nthe literature. A stronger version of the conjecture, which is still open,\nasserts that if there exist purely cosmetic surgeries on $M$, then there is\nan orientation-preserving homeomorphism of $M$ sending one slope to the\nother. This version would imply that $M$ does not admit purely cosmetic\nsurgeries if $M$ is hyperbolic. If there exist two slopes that are related\nby an orientation-reversing homeomorphism, and the surgeries on these\ntwo slopes are amphichiral, then this would be a counterexample to the\nstronger version of the conjecture, but not to the version of the conjecture\nstated in this problem.\n\n(5) Lackenby [Lac97] proved that, for “most” surgeries on null-homotopic\nknots in manifolds with $b_{1} > 0$, the original manifold, the knot, and the\nslope are determined by the resulting manifold from the surgery.\nWhen $M$ is hyperbolic, there is an explicit constant $C = C(M)$,\nsatisfying that if the normalized lengths of two inequivalent slopes $r_{1}, r_{2}$\nare both greater than $C$, then this pair is not (purely or chirally) cosmetic\n[FPS22, FPS25].\nUsing code that is publicly available on GitHub [FPS24], Conjec-\nture (a) has been verified for all the 59,107 one-cusped manifolds in the\nSnapPy census. This code can also be used to investigate Conjecture (c)\nabove for knots in $S^{3}$.\n(6) If $M$ is the irreducible complement of a null-homologous knot in a 3-\nmanifold of the form $Z\\#S^{1} \\times S^{2}$, then the slopes of any pair of cosmetic\nsurgeries must be $\\{\\pm r\\}$ [Ni13]. A similar result holds when $M$ is the\ncomplement of a knot in a closed 3-manifold whose Thurston norm is\nstrictly smaller than that of $M$ [Ni11].\n(7) The Cosmetic Surgery Conjecture (a) is equivalent to the Oriented Knot\nComplement Conjecture (b), which is an analogue of the Knot Comple-\nment Theorem [GL89].\nIn [Kir97, Problem 1.81D], the homeomorphism of $N$ is required to be\norientation-preserving. Since our definition of equivalent slopes does not\nrequire the homeomorphism to preserve orientation, we do not require\nthe homeomorphism of $N$ to be orientation-preserving in the problem\nstatement here.\nGordon and Luecke [GL89] proved Conjecture (b) for $N = S^{3}$ and\n$S^{1} \\times S^{2}$. It also holds for certain L-spaces [Gai18, Rav22]. Cremaschi–\nYarmola proved it for knots in an orientable circle bundle over a genus\n$g \\geq 2$ surface [CY24].\n(8) When $M$ is the complement of a knot $K \\subset S^{3}$, the Cosmetic Surgery Con-\njecture is equivalent to saying that if $K$ admits purely cosmetic surgeries\nwith slopes $r_{1}, r_{2}$ then $K$ is amphichiral and $r_{1} = -r_{2}$. The conclusion\nthat $r_{1} = -r_{2}$ was proved in [NW15]. Conjecture (c) is stronger than\nthis special case of Conjecture (a), because there may exist an amphichi-\nral knot in $S^{3}$ such that a nonzero surgery on it is also amphichiral. This\nwould be a counterexample to Conjecture (c), but not to Conjecture (a).\nFuter–Purcell–Schleimer [FPS25] verified Conjecture (c) for knots\nwith up to 19 crossings.\nHanselman [Han23] proved that if $K$ admits a pair of purely cosmetic\nsurgeries, then the pair of slopes is either $\\{\\pm2\\}$ or $\\{\\pm 1/q\\}$ for an integer $q$\nthat is explicitly determined from the knot Floer homology of $K$, and the\nformer case happens only when $g(K) = 2$. Daemi-Lidman-Miller Eismeier\n[DLME24] subsequently announced that they have eliminated the case\nof $\\{\\pm 1/q\\}$. As a result, any pair of purely cosmetic slopes must be $\\{\\pm2\\}$,\nand the knot has genus 2 and trivial Alexander polynomial.\n(9) Although [Kir97, Problem 1.81 (B)] has been disproved, no such examples\nhave been found for knots in $S^{3}$. This is the motivation for Conjecture\n(d).\n\n(10) There are many known constraints on chirally cosmetic surgeries; see\n[IIS22] for an overview. Ozsváth and Szabó [OS11] proved that if $K$\nadmits a pair of (purely or chirally) cosmetic surgeries with slopes $r_{1}, r_{2}$,\nthen either $r_{1}, r_{2}$ have opposite signs or the surgery is an L-space. Futer–\nPurcell–Schleimer [FPS25] have shown that Conjecture (d) holds for all\nhyperbolic knots with up to 15 crossings.\n(11) Somewhat weaker versions of Conjecture (b), in which the knots are re-\nquired to be null-homotopic, may be found in [Kir97, Problem 1.80 (B)\nand (C)]. Using standard arguments in Heegaard Floer homology, one can\nverify the conjecture in [Kir97, Problem 1.80(C)] when $Y$ is an L-space.\nSee, for example, [Gai18, Rav22].\n(12) A list of problems about (mostly chirally) cosmetic surgeries on knots in\n$S^{3}$ can be found in [Ito22].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Gor91] Cameron McA. Gordon. Dehn surgery on knots. In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 631–642. Math. Soc. Japan, Tokyo, 1991.\n- [IJ18] Kazuhiro Ichihara and In Dae Jong. Cosmetic banding on knots and links. Osaka J. Math., 55(4):731–745, 2018. With an appendix by Hidetoshi Masai, https://projecteuclid.org/euclid.ojm/1539158668.\n- [Bur14] Benjamin A. Burton. The cusped hyperbolic census is complete, 2014. arXiv:1405.2695.\n- [Lac97] Marc Lackenby. Dehn surgery on knots in 3-manifolds. J. Amer. Math. Soc., 10(4):835–864, 1997. doi:10.1090/S0894-0347-97-00241-5.\n- [FPS22] David Futer, Jessica S. Purcell, and Saul Schleimer. Effective bilipschitz bounds on drilling and filling. Geom. Topol., 26(3):1077–1188, 2022. doi:10.2140/gt.2022.26.1077.\n- [FPS25] David Futer, Jessica S. Purcell, and Saul Schleimer. Excluding cosmetic surgeries on hyperbolic 3-manifolds. Journal of Computational Geometry, 16(1):694–736, Nov. 2025. URL: https://jocg.org/index.php/jocg/article/view/4778, doi:10.20382/jocg.v16i1a19.\n- [FPS24] David Futer, Jessica S. Purcell, and Saul Schleimer. Code for verifying the cosmetic surgery conjecture on a given 3–manifold, 2024. https://github.com/saulsch/Cosmetic.\n- [Ni13] Yi Ni. Nonseparating spheres and twisted Heegaard Floer homology. Algebr. Geom. Topol., 13(2):1143–1159, 2013. doi:10.2140/agt.2013.13.1143.\n- [Ni11] Yi Ni. Thurston norm and cosmetic surgeries. In Low-dimensional and symplectic topology, volume 82 of Proc. Sympos. Pure Math., pages 53–63. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/pspum/082/2768653.\n- [GL89] C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371–415, 1989. doi:10.2307/1990979.\n- [Gai18] Fyodor Gainullin. Heegaard Floer homology and knots determined by their complements. Algebr. Geom. Topol., 18(1):69–109, 2018. doi:10.2140/agt.2018.18.69.\n- [Rav22] Huygens C. Ravelomanana. Knot complement problem for L-space ZH$S^{3}$. J. Knot Theory Ramifications, 31(4):Paper No. 2250028, 7, 2022. doi:10.1142/S0218216522500286.\n- [CY24] Tommaso Cremaschi and Andrew Yarmola. Knots in circle bundles are determined by their complements, 2024. arXiv:2401.02895.\n- [NW15] Yi Ni and Zhongtao Wu. Cosmetic surgeries on knots in $S^{3}$. J. Reine Angew. Math., 706:1–17, 2015. doi:10.1515/crelle-2013-0067.\n- [Han23] Jonathan Hanselman. Heegaard Floer homology and cosmetic surgeries in $S^{3}$. J. Eur. Math. Soc. (JEMS), 25(5):1627–1669, 2023. doi:10.4171/jems/1218.\n- [DLME24] Aliakbar Daemi, Tye Lidman, and Mike Miller Eismeier. Filtered instanton homology and cosmetic surgery, 2024. arXiv:2410.21248.\n- [IIS22] Kazuhiro Ichihara, Tetsuya Ito, and Toshio Saito. On constraints for knots to admit chirally cosmetic surgeries and their calculations. Pacific J. Math., 321(1):167–191, 2022. doi:10.2140/pjm.2022.321.167.\n- [OS11] Peter Ozsváth and Zoltán Szabó. Knot Floer homology and rational surgeries. Algebr. Geom. Topol., 11(1):1–68, 2011. doi:10.2140/agt.2011.11.1.\n- [Ito22] Tetsuya Ito. Cosmetic surgery on knots. In Tomotada Ohtsuki, editor, Problems in low-dimensiontal topology, 2022. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, 2022. URL: https://www.kurims.kyoto-u.ac.jp/„kyodo/kokyuroku/contents/pdf/2227-12.pdf.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Core purely cosmetic-surgery conjectures remain open, while recent examples disprove older chiral/orientation-reversing auxiliary variants in this multi-part record.\n\n**Verified partial progress.**\n\n- Ichihara--Jong--Masai give inequivalent chirally cosmetic surgeries on a hyperbolic one-cusped manifold.\n- The stronger purely cosmetic formulation remains open.\n\n**Full solution or refutation.**\n\nThe record contains several variants with different statuses; it is not globally refuted.\n\n**What remains.**\n\nResolve purely cosmetic surgery for knot complements and the stated oriented-complement variants separately.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.12 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes counterexamples to older variants from the open stronger conjecture.\n\n**Review notes.** Multi-part status kept separate.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2672, "problem_number": "KP-1.13", "title": "Kirby Problem 1.13", "statement": "Let $K$ be a null-homotopic knot in a 3-manifold $Y$ , and let\n$Y_{0}(K)$ be the manifold obtained by 0-surgery on $K$.\n(a) Conjecture: Let $F$ be a Seifert surface for $K$, and let $\\widetilde{F} \\subset Y_{0}(K)$ be the\nclosed surface obtained by capping off $\\partial F$ with a disk. If $Y_{0}(K)$ is a surface\nbundle over $S^{1}$ with fiber in the homology class $[\\widetilde{F}]$, then $K$ is a fibered\nknot with fiber in the homology class $[F]$.\n(b) Conjecture: If $F$ is taut in the exterior of $K$ in $Y$ , then $\\widetilde{F}$ is also taut in\n$Y_{0}(K)$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.13.\n\nLiterature notes:\n(1) Conjecture (a) is an updating of [Kir97, Problem 1.80C], and is due to\nBoileau. Conjecture (b) is due to Gabai.\n(2) Let $M$ be a compact, oriented, connected 3-manifold, let $F \\subset M$ be a\nproperly embedded connected surface. We say that $F$ is taut if it realizes\nthe Thurston norm in the homology class $[F] \\neq 0 \\in H_{2}(M, \\partial M),$ or if $F$\nis a sphere.\n(3) In the original version of Conjecture (a), the surgery slope can be any\nrational number, and there is no restriction on the homology class of the\nfiber. However, there are simple counterexamples to the original question\n[Ni09]. More precisely, there exist nonfibered null-homotopic knots such\nthat every integral surgery on the knot yields a manifold that fibers over\n$S^{1}$.\n(4) It is known that $Y_{0}(K)$ does not contain a homologically essential 2-sphere\nif $Y \\setminus K$ does not contain such 2-spheres and $K$ is nontrivial [Gab87a,\nLac97, HL22, Ni23b].\nConjecture (a) and Conjecture (b) are known for knots in $S^{3}$ [Gab87b],\nL-spaces [Ni07, AN09], and certain reducible manifolds [Gab87b, Ni13].\nThey are also known for knots obtained from the unknot by crossing\nchanges of the same sign [Ni24].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Ni09] Yi Ni. Dehn surgeries that yield fibred 3-manifolds. Math. Ann., 344(4):863–876, 2009. doi:10.1007/s00208-008-0331-3.\n- [Gab87a] David Gabai. Foliations and the topology of 3-manifolds. II. J. Differential Geom., 26(3):461–478, 1987. http://projecteuclid.org/euclid.jdg/1214441487.\n- [Lac97] Marc Lackenby. Dehn surgery on knots in 3-manifolds. J. Amer. Math. Soc., 10(4):835–864, 1997. doi:10.1090/S0894-0347-97-00241-5.\n- [HL22] Jennifer Hom and Tye Lidman. Dehn surgery and nonseparating two-spheres. In Gauge theory and low-dimensional topology—progress and interaction, volume 5 of Open Book Ser., pages 145–153. Math. Sci. Publ., Berkeley, CA, 2022. https: //msp.org/obs/2022/5-1/p07.xhtml.\n- [Ni23b] Yi Ni. Null-homotopic knots have property R. Math. Proc. Cambridge Philos. Soc., 175(1):217–223, 2023. doi:10.1017/S0305004123000129.\n- [Gab87b] David Gabai. Foliations and the topology of 3-manifolds. III. J. Differential Geom., 26(3):479–536, 1987. http://projecteuclid.org/euclid.jdg/1214441488.\n- [Ni07] Yi Ni. Knot Floer homology detects fibred knots. Invent. Math., 170(3):577–608, 2007. doi:10.1007/s00222-007-0075-9.\n- [AN09] Yinghua Ai and Yi Ni. Two applications of twisted Floer homology. Int. Math. Res. Not. IMRN, 2009(19):3726–3746, 2009. doi:10.1093/imrn/rnp070.\n- [Ni13] Yi Ni. Nonseparating spheres and twisted Heegaard Floer homology. Algebr. Geom. Topol., 13(2):1143–1159, 2013. doi:10.2140/agt.2013.13.1143.\n- [Ni24] Yi Ni. Property G and the 4-genus. Trans. Amer. Math. Soc. Ser. B, 11:120–143, 2024. doi:10.1090/btran/153.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Earlier unrestricted fibre-surgery formulations have counterexamples; the updated conjectures are proved for knots in S3, L-spaces and further specified classes, but remain open generally.\n\n**Verified partial progress.**\n\n- Ni gave counterexamples to the original unrestricted version.\n- The updated statements are known in S3, L-spaces, some reducible manifolds, and same-sign crossing-change families.\n\n**Full solution or refutation.**\n\nThe revised general conjectures remain open.\n\n**What remains.**\n\nSettle the updated fibre/tautness statements for arbitrary allowed Y and K.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.13 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the counterexamples and the verified classes for the updated conjectures.\n\n**Review notes.** Original and updated versions distinguished.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2673, "problem_number": "KP-1.14", "title": "Kirby Problem 1.14", "statement": "For which nonzero $r \\in \\mathbb{Q}$ is it true that for every nontrivial\nknot $K \\subset S^{3}$ there is a homomorphism\n\n$$\n\\pi_{1}(S^{3}_{r}(K)) \\to \\operatorname{SU}(2)\n$$\n\nwith nonabelian image?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.14.\n\nLiterature notes:\n(1) If this statement is true for some $r \\in \\mathbb{Q}$, then it’s true for $-r$; hence,\nwe assume $r > 0$. Kronheimer–Mrowka proved [KM04] that there is a\nhomomorphism with nonabelian image for all $r \\in (0, 2]$. Work of Baldwin–\nSivek [BS23a] together with announced results of Baldwin–Li–Sivek–Ye\n[BLSY24], and Farber–Reinoso–Wang [FRW24] proves the same for all\n$r \\in (2, 5)$ with prime power numerators.\n(2) Let $r = p/q > 0$ with $p$ and $q$ relatively prime, and assume that $\\Delta_{K}(\\zeta^{2}) \\neq$\n0 for any $p$th root of unity $\\zeta$. Then it is further known [BS23a] that if\n$\\pi_{1}(S^{3}_{r}(K))$ only admits abelian $\\operatorname{SU}(2)$-representations, then $K$ is fibered,\nstrongly quasipositive, and $r \\geq 2g(K) - 1.$\n(3) Note that for most $r \\in \\mathbb{Z}$, there is a torus knot whose $r$–surgery is $\\operatorname{SU}(2)$-\nabelian. Problem 3.51 deals with a related problem about the characteri-\nzation of rational homology spheres whose fundamental groups have only\nAbelian $\\operatorname{SU}(2)$ representations.\n\nReferences cited:\n- [KM04] P. B. Kronheimer and T. S. Mrowka. Dehn surgery, the fundamental group and $\\mathrm{SU}(2)$. Math. Res. Lett., 11(5-6):741–754, 2004. doi:10.4310/MRL.2004.v11.n6.a3.\n- [BS23a] John A. Baldwin and Steven Sivek. Instantons and L-space surgeries. J. Eur. Math. Soc. (JEMS), 25(10):4033–4122, 2023. doi:10.4171/jems/1280.\n- [BLSY24] John A. Baldwin, Zhenkun Li, Steven Sivek, and Fan Ye. Small Dehn surgery and $\\mathrm{SU}(2)$. Geom. Topol., 28(4):1891–1922, 2024. doi:10.2140/gt.2024.28.1891.\n- [FRW24] Ethan Farber, Braeden Reinoso, and Luya Wang. Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil, 2024. doi:10.2140/gt.2024.28.4337.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every nontrivial knot has a nonabelian SU(2) representation after r-surgery for 0", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2674, "problem_number": "KP-1.15", "title": "Kirby Problem 1.15", "statement": "(a) Are there integral homology spheres with arbitrarily large (integral) Dehn\nsurgery number? Are there irreducible examples? Does the connected sum\nof $n$ nontrivial homology spheres have Dehn surgery number at least $n$?\n(b) Given a 3-manifold $Y$ with a smooth orientation-preserving $\\mathbb{Z}/n\\mathbb{Z}$ sym-\nmetry $\\varphi$, Sakuma [Sak01] (generalizing [PS01]) proved that $Y$ admits a\nsurgery description compatible with the symmetry. Define the equivariant\n(integral) Dehn surgery number of $(Y, \\varphi)$ as the minimal number of con-\nnected components of any such link. Do there exist $(Y, \\varphi)$ whose equivari-\nant (integral) Dehn surgery number is arbitrarily larger than the (integral)\nDehn surgery number?\n(c) Following the remarks below, we can consider the following specific case:\nDoes there exist a manifold with a 2-fold symmetry whose equivariant\n(integral) Dehn surgery number is larger than the (integral) Dehn surgery\nnumber?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.15.\n\nLiterature notes:\n(1) Any oriented connected closed 3-manifold $Y$ is obtained by (integral) Dehn\nsurgery on a link in the 3-sphere, and the minimal number of connected\ncomponents of any such link is called the (integral) Dehn surgery number\nof $Y$ .\n\n(2) Part (a) is a version of Problem 3.102 from [Kir97], asked by Auckly. The\nintegral Dehn surgery case is interesting to consider from the perspective\nof 4-manifolds bounded by $Y$ , as it is related to the minimum number of\nhandles needed to describe the 4-manifold.\nThere are many examples of integral homology spheres with Dehn\nsurgery number at least 2, the earliest due to [GL87], hyperbolic examples\ndue to [Auc97], and Seifert fibered examples due to [HKL16b].\n(3) Regarding part (b), it should be possible to show that the equivari-\nant surgery number does not agree with the surgery number. Consider\n$Y=S^{3}_{2}(4_{1})$, which is a Seifert fibered space (see, for example, [BW01,\nTheorem 1.1(4))]). Since $Y$ is Seifert fibered, it has a $\\mathbb{Z}/n\\mathbb{Z}$ action, for\nall $n$, while $4_{1}$ does not. It remains to be shown that there is no other\nknot $K$ for which some $2/q$-surgery is $Y$ ; a plausible approach is to use\nthe mapping cone formula for Heegaard Floer homology.\nFinding the examples as in part (b) for which the equivariant surgery\nnumber is arbitrarily larger than the Dehn surgery number is, as in part\n(a), more challenging.\n(4) Lastly, we point out that this problem is closely related to Problem 3.39.\n\nReferences cited:\n- [Sak01] Makoto Sakuma. Surgery description of orientation-preserving periodic maps on compact orientable 3-manfolds. Rend. Istit. Mat. Univ. Trieste, 32:375–396, 2001. Dedicated to the memory of Marco Reni.\n- [PS01] Józef H. Przytycki and Maxim V. Sokolov. Surgeries on periodic links and homology of periodic 3-manifolds. Math. Proc. Cambridge Philos. Soc., 131(2):295–307, 2001. doi:10.1017/S0305004101005308.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GL87] C. McA. Gordon and J. Luecke. Only integral Dehn surgeries can yield reducible manifolds. Math. Proc. Cambridge Philos. Soc., 102(1):97–101, 1987. doi:10.1017/S0305004100067086.\n- [Auc97] David Auckly. Surgery numbers of 3-manifolds: a hyperbolic example. In Geometric topology (Athens, GA, 1993), volume 2 of AMS/IP Stud. Adv. Math., pages 21–34. Amer. Math. Soc., Providence, RI, 1997. doi:10.1090/amsip/002.1/02.\n- [HKL16b] Jennifer Hom, Çağrı Karakurt, and Tye Lidman. Surgery obstructions and Heegaard Floer homology. Geom. Topol., 20(4):2219–2251, 2016. doi:10.2140/gt.2016.20.2219.\n- [BW01] Mark Brittenham and Ying-Qing Wu. The classification of exceptional Dehn surgeries on 2-bridge knots. Comm. Anal. Geom., 9(1):97–113, 2001. doi:10.4310/CAG.2001.v9.n1.a4.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** There are homology spheres with integral Dehn surgery number at least two, but arbitrary growth and the irreducible/connected-sum questions remain open.\n\n**Verified partial progress.**\n\n- Hyperbolic and Seifert-fibred examples of surgery number at least two are known.\n\n**Full solution or refutation.**\n\nNo family with arbitrarily large surgery number was verified.\n\n**What remains.**\n\nEstablish unboundedness and resolve the equivariant-surgery comparison questions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.15 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintained list records known lower-bound examples and remaining questions.\n\n**Review notes.** Multi-part agenda retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2675, "problem_number": "KP-1.16", "title": "Kirby Problem 1.16", "statement": "(a) Given a knot $K \\subset S^{3}$ determine all knots $K' \\subset S^{3}$ for which the branched\ndouble covers of $S^{3}$ along $K$ and $K'$ are homeomorphic.\n(b) Find a set of moves on links in $S^{3}$ so that two links are related by these\nmoves if and only if they have the same branched double cover.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.16.\n\nLiterature notes:\n(1) These are Problems 3.25 and 1.22 in [Kir97], respectively.\nSee Prob-\nlem 1.17 for a related problem about alternating links. Let $\\Sigma(K)$ denote\nthe branched double cover of $S^{3}$ along $K$.\n(2) It follows from the Orbifold Geometrization Theorem that there are finitely\nmany $K'$ with $\\Sigma(K') \\cong \\Sigma(K)$. See [Pao05] for a discussion.\n(3) Mecchia and Zimmerman showed that if $\\Sigma(K)$ is hyperbolic, then there are\nat most eight other $K'$ with $\\Sigma(K') \\cong \\Sigma(K)$ [MZ04]. Kawauchi showed\nthat this bound is sharp, exhibiting infinitely many 9-tuples of distinct hy-\nperbolic knots in $S^{3}$ with homeomorphic branched double covers [Kaw06].\nHowever, his construction is very non-explicit, and it would be interest-\ning to have some explicit constructions. Mecchia further described how\nhyperbolic knots with the same double branched cover can be understood\nin terms of isometries of the cover [Mec01]\n(4) If $K$ is a 2-bridge knot and $\\Sigma(K') \\cong \\Sigma(K)$ then $K' = K [\\mathrm{HR}85$].\n(5) If $K$ and $K'$ are related by Conway mutation then $\\Sigma(K') \\cong \\Sigma(K)$. There\nare also non-mutant knots with the same branched double covers, includ-\ning $T(3, 7)$ and $P(-2, 3, 7).$ In this example, the Khovanov homologies\nhave the same total rank, but Watson [Wat10] gives non-mutant knots\n\nwith homeomorphic branched double covers the total rank of whose Kho-\nvanov homologies differ, showing that the rank of Khovanov homology is\nnot an invariant of branched double covers. See also [Pao01].\n(6) Higher order cyclic branched covers are generally much better knot in-\nvariants. For example, work of Zimmerman [Zim98] shows that if $n \\geq 3$\nand $K \\subset S^{3}$ is hyperbolic, then the $n$-fold cyclic branched cover of $K$\ndistinguishes $K$ from other knots so long as $K$ does not admit peri-\nodic symmetries of order $n$, and the latter condition is satisfied whenever\n$n > 2g(K) + 1$. See [Pao05] for further discussion.\n(7) Extending work of Montesinos [Mon85], Piergallini constructed a com-\nplete set of moves relating links with the same simple, irregular 3-fold\nbranched covers [Pie91].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Pao05] Luisa Paoluzzi. Hyperbolic knots and cyclic branched covers. Publ. Mat., 49(2):257– 284, 2005. doi:10.5565/PUBLMAT\\\\_49205\\\\_01.\n- [MZ04] Mattia Mecchia and Bruno Zimmermann. The number of knots and links with the same 2-fold branched covering. Q. J. Math., 55(1):69–76, 2004. doi:10.1093/qjmath/55.1.69.\n- [Kaw06] Akio Kawauchi. Topological imitations and Reni-Mecchia-Zimmermann’s conjecture. Kyungpook Math. J., 46(1):1–9, 2006.\n- [Mec01] Mattia Mecchia. Hyperbolic 2-fold branched coverings. Rend. Istit. Mat. Univ. Trieste, 32:165–180, 2001. Dedicated to the memory of Marco Reni.\n- [HR85] Craig Hodgson and J. H. Rubinstein. Involutions and isotopies of lens spaces. In Knot theory and manifolds (Vancouver, B.C., 1983), volume 1144 of Lecture Notes in Math., pages 60–96. Springer, Berlin, 1985. doi:10.1007/BFb0075012.\n- [Wat10] Liam Watson. A remark on Khovanov homology and two-fold branched covers. Pacific J. Math., 245(2):373–380, 2010. doi:10.2140/pjm.2010.245.373.\n- [Pao01] Luisa Paoluzzi. On hyperbolic type involutions. Rend. Istit. Mat. Univ. Trieste, 32:221–256, 2001. Dedicated to the memory of Marco Reni.\n- [Zim98] Bruno Zimmermann. On hyperbolic knots with homeomorphic cyclic branched coverings. Math. Ann., 311(4):665–673, 1998. doi:10.1007/s002080050205.\n- [Mon85] José Marı́a Montesinos. Lectures on 3-fold simple coverings and 3-manifolds. In Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982), volume 44 of Contemp. Math., pages 157–177. Amer. Math. Soc., Providence, RI, 1985. doi:10.1090/conm/044/813111.\n- [Pie91] R. Piergallini. Covering moves. Trans. Amer. Math. Soc., 325(2):903–920, 1991. doi:10.2307/2001654.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A given branched double cover has only finitely many branch-knot descriptions; for hyperbolic covers there can be a sharp nine-knot family, but no complete move classification is known.\n\n**Verified partial progress.**\n\n- Orbifold geometrization gives finiteness.\n- Kawauchi exhibited infinitely many distinct hyperbolic nine-tuples with homeomorphic covers.\n- Mutation is one sufficient move, not a complete characterization.\n\n**Full solution or refutation.**\n\nNeither requested classification is complete.\n\n**What remains.**\n\nGive explicit classifications or complete move systems for equal branched double covers.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.16 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the finite ambiguity, sharp nine-tuple result, and incomplete move problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2676, "problem_number": "KP-1.17", "title": "Kirby Problem 1.17", "statement": "Can an alternating link and a non-alternating link have home-\nomorphic branched double covers?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.17.\n\nLiterature notes:\n(1) The answer is conjectured to be no in [Gre13a, Conjecture 1.4] since\nGreene shows that two alternating links have homeomorphic branched\ndouble covers if and only if they are Conway mutants. Note that Viro\nproved that mutant links have homeomorphic branched double-covers\n[Vir76, Theorem 1], and Menasco proved that mutation preserves the\nproperty of being alternating [Men84, Proof of Theorem 3(b)].\n(2) The problem is related to Problems 1.22 and 3.25 of [Kir97] which ask\nfor a kind of classification of pairs of knots with homeomorphic branched\ndouble covers.\n(3) If a Seifert-fibered space is the branched double-cover of a pair of distinct\nlinks, then the pair consists of either a non-alternating torus link and\na non-alternating Montesinos link, or else two mutant Montesinos links\n[MR02, Introduction].\n(4) Dunfield investigated knots with at most 16 crossings whose branched\ndouble-covers are hyperbolic and of small enough volume to appear in the\nHodgson-Weeks census of closed hyperbolic 3-manifolds. He reports 3765\nnon-alternating knots with such branched covers but only 178 alternating\nknots, and no manifold occurs as the branched double-cover of both kinds\nof knots (private correspondence from 2013).\n(5) While there exist many constructions of pairs of non-mutant links with\nhomeomorphic branched double-covers, these constructions typically ap-\npear to produce non-alternating examples (cf. [MW86, MZ04]).\n\nReferences cited:\n- [Gre13a] Joshua Evan Greene. Lattices, graphs, and Conway mutation. Invent. Math., 192(3):717–750, 2013. doi:10.1007/s00222-012-0421-4.\n- [Vir76] O. Ja. Viro. Nonprojecting isotopies and knots with homeomorphic coverings. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 66:133–147, 207–208, 1976. Studies in topology, II.\n- [Men84] W. Menasco. Closed incompressible surfaces in alternating knot and link complements. Topology, 23(1):37–44, 1984. doi:10.1016/0040-9383(84)90023-5.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [MR02] Mattia Mecchia and Marco Reni. Hyperbolic 2-fold branched coverings of links and their quotients. Pacific J. Math., 202(2):429–447, 2002. doi:10.2140/pjm.2002.202.429.\n- [MW86] José M. Montesinos and Wilbur Whitten. Constructions of two-fold branched covering spaces. Pacific J. Math., 125(2):415–446, 1986. http://projecteuclid.org/euclid.pjm/1102700086.\n- [MZ04] Mattia Mecchia and Bruno Zimmermann. The number of knots and links with the same 2-fold branched covering. Q. J. Math., 55(1):69–76, 2004. doi:10.1093/qjmath/55.1.69.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No alternating/nonalternating pair with homeomorphic branched double covers was verified; the answer is conjectured to be no.\n\n**Verified partial progress.**\n\n- Greene classifies alternating pairs with common cover as Conway mutants.\n- Computational searches described in the maintained list found no mixed examples in the examined census range.\n\n**Full solution or refutation.**\n\nNo resolution was located.\n\n**What remains.**\n\nProve the conjectured impossibility or construct a mixed pair.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.17 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records Greene's theorem, the conjecture, and no known mixed example.\n\n**Review notes.** Open is dated and conservative.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2677, "problem_number": "KP-1.18", "title": "Kirby Problem 1.18", "statement": "(a) $($ Meridional Rank Conjecture $)$ Is the meridional $\\operatorname{rank} \\mu(L)$ of every link\n$L$ equal to its bridge number $b(L)$?\n(b) Given two knots $K_{1}, K_{2} \\subset S^{3}$ such that $\\pi_{1}(S^{3}\\setminus K_{1})$ and $\\pi_{1}(S^{3}\\setminus K_{2})$ have\nthe same finite quotients, does $\\mu(K_{1}) = \\mu(K_{2})$? In particular, is it possible\nto explicitly detect the meridional $\\operatorname{rank} \\mu(K)$ of a knot $K \\subset S^{3}$ from the\nfinite quotients of $\\pi_{1}(S^{3}\\setminus K)$?\n(c) Given two links $L_{1}, L_{2} \\subset S^{3}$, if there exists a surjective homomorphism\n$\\pi_{1}(S^{3}\\setminus L_{1}) \\twoheadrightarrow\\pi_{1}(S^{3}\\setminus L_{2})$ that sends the meridians of $L_{1}$ to the meridians\nof $L_{2}$, do the bridge numbers satisfy $b(L_{1}) \\geq b(L_{2})$?\n(d) Given two knots $K_{1}, K_{2} \\subset S^{3}$, if there exists a surjective homomorphism\n$\\pi_{1}(S^{3}\\setminus K_{1}) \\twoheadrightarrow\\pi_{1}(S^{3}\\setminus K_{2})$ (which may not send meridians to meridians):\n(i) Do the meridional ranks satisfy $\\mu(K_{1}) \\geq \\mu(K_{2})$?\n(ii) Do the bridge numbers satisfy $b(K_{1}) \\geq b(K_{2})$?\n(e) Let $K \\subset S^{3}$ be a hyperbolic knot. Is every subgroup of $\\pi_{1}(S^{3}\\setminus K)$ generated\nby at most $b(K) - 1$ meridians free? See [BJW18].", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.18.\n\nLiterature notes:\n(1) Problem 1.11 of [Kir97] poses Question (a) for knots, and was proposed\nby Cappell and Shaneson.\n(2) The meridional $\\operatorname{rank} \\mu(L)$ of a link $L$ is the minimal number of merid-\nians needed to generate the fundamental group of its complement. It is\nbounded above by the bridge number $b(L)$.\n(3) Question (b) has a negative answer for links: let $W_{n}$ denote the result\nof applying $|n| \\geq 2$ twists to the Whitehead link $W_{0}$. Then $W_{n}$ and $W_{0}$\nhave homeomorphic exteriors, but $\\mu(W_{0}) = b(W_{0}) = 2$, while $\\mu(W_{n}) =$\n$b(W_{n}) = 3$.\nNote that the homeomorphism between the exterior of $W_{n}$ and $W_{0}$ dis-\ncussed above does not send meridians to meridians. Under the meridian-\npreserving hypothesis, the epimorphism of groups in Question (c) implies\n$\\mu(L_{1}) \\geq \\mu(L_{2})$, so a negative answer to Question (c) implies a negative\nanswer to the Meridional Rank Conjecture.\n(4) In the case of knots, Question (d) represents a strengthening of Question\n(c).\n(5) Different answers to the two items under Question (d) would imply a nega-\ntive answer to the Meridional Rank Conjecture. Note that there are many\nepimorphisms between knot groups that are not meridian-preserving [GAn75].\nFor closed orientable manifolds, the answer to the analogous question with\nrespect to the Heegaard genus is negative if one does not assume the epi-\nmorphism to be induced by a non-zero degree map (see [BF18]). The\nnon-zero degree condition, in the knot complement case, implies that the\nepimorphism is meridian preserving.\n(6) In Question 4 the bridge number is defined using bridge trisections (see [MZ17a,\nJMMZ22] and [AAD $^{+}23$, Question 6.1]; cf. Problem 4.111) or, equiva-\nlently, using Morse position.\nBy [JP25], the meridional rank of 2-spheres in $S^{4}$ can collapse un-\nder connected sum; therefore, either additivity of bridge number or the\nequality of bridge number and meridional rank (or both) fails for 2-knots.\n(7) Problem 4.117 describes a four-dimensional analog of the Meridional Rank\nConjecture.\n\nReferences cited:\n- [BJW18] Michel Boileau, Yeonhee Jang, and Richard Weidmann. Meridional rank and bridge number for a class of links. Pacific J. Math., 292(1):61–80, 2018. doi:10.2140/pjm.2018.292.61.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GAn75] F. González-Acuña. Homomorphs of knot groups. Ann. of Math. (2), 102(2):373– 377, 1975. doi:10.2307/1971036.\n- [BF18] Michel Boileau and Stefan Friedl. Epimorphisms of 3-manifold groups. Q. J. Math., 69(3):931–942, 2018. doi:10.1093/qmath/hay007.\n- [MZ17a] Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in $S^{4}$. Trans. Amer. Math. Soc., 369(10):7343–7386, 2017. doi:10.1090/tran/6934.\n- [JMMZ22] Jason Joseph, Jeffrey Meier, Maggie Miller, and Alexander Zupan. Bridge trisections and classical knotted surface theory. Pacific J. Math., 319(2):343–369, 2022. doi:10.2140/pjm.2022.319.343.\n- [AAD+23] Wolfgang Allred, Manuel Aragón, Zack Dooley, Alexander Goldman, Yucong Lei, Isaiah Martinez, Nicholas Meyer, Devon Peters, Scott Warrander, Ana Wright, and Alexander Zupan. Tri-plane diagrams for simple surfaces in $S^{4}$. J. Knot Theory Ramifications, 32(6):Paper No. 2350041, 28, 2023. doi:10.1142/S0218216523500414.\n- [JP25] Jason Joseph and Puttipong Pongtanapaisan. Meridional rank and bridge number of knotted 2-spheres. Canad. J. Math., 77(1):282–299, 2025. doi:10.4153/S0008414X23000883.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The meridional-rank-equals-bridge-number conjecture remains open; a related finite-quotient/exterior assertion is false for links, and 2-knot connected-sum work forces failure of an associated additivity/equality package.\n\n**Verified partial progress.**\n\n- For links Wn and W0 have homeomorphic exteriors but distinct meridional rank and bridge number.\n- Recent 2-sphere work gives collapse of meridional rank under connected sum.\n\n**Full solution or refutation.**\n\nThese results do not settle the principal knot meridional-rank conjecture.\n\n**What remains.**\n\nProve or disprove mu(L)=b(L), especially for knots, and resolve the remaining finite-quotient variants.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.18 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintained list distinguishes the open conjecture from negative related link results.\n\n**Review notes.** Several subquestions have distinct status; no global solved label assigned.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2678, "problem_number": "KP-1.19", "title": "Kirby Problem 1.19", "statement": "Let $Y = Y_{1}\\#Y_{2}$ be a connected sum of 3-manifolds with $Y_{i} \\neq$\n$S^{3}$, for $i = 1, 2$. Let $\\Phi: Y \\to Y$ be a Dehn twist around the connected sum separating\n2-sphere, and assume $Y_{1}$ and $Y_{2}$ are such that $\\Phi$ is not isotopic to the identity. Does\nthere exist a knot $K \\subseteq Y$ such that $K$ and $\\Phi(K)$ are not ambiently isotopic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.19.\n\nLiterature notes:\nAceto-Bregman-Davis-Park-Ray [ABD $^{+}20$] studied the analogous\nquestion for knots in prime 3-manifolds. They showed that the Gluck twist on $S^{1} \\times$\n$S^{2}$ (see Section 4.1 for the definition) can be detected by a knot that is equivalent\nbut not isotopic to its image under the Gluck twist. They also showed that for\nirreducible 3-manifolds, diffeomorphisms that preserve free homotopy classes of\nloops are isotopic to the identity. Hence the key remaining question is whether\nDehn twists around connected sum spheres can be detected by knots that are\nequivalent but not isotopic.\n\nReferences cited:\n- [ABD+20] Paolo Aceto, Corey Bregman, Christopher W. Davis, JungHwan Park, and Arunima Ray. Isotopy and equivalence of knots in 3-manifolds, 2020. arXiv:2007.05796.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No result was found that detects a nontrivial connected-sum sphere twist by the isotopy class of a knot. Results for prime 3-manifolds and for the Gluck twist on S^1 x S^2 leave this connected-sum case open.\n\n**Verified partial progress.**\n\n- Aceto, Bregman, Davis, Park and Ray proved that a nontrivial Gluck twist on S^1 x S^2 changes infinitely many knot isotopy classes.\n- For irreducible 3-manifolds they proved that a diffeomorphism preserving all free-homotopy classes is isotopic to the identity, isolating sphere twists in reducible manifolds as the remaining obstruction.\n\n**Full solution or refutation.**\n\nThe analogous prime-manifold theorems do not supply a knot detecting the separating-sphere twist in the stated connected sum.\n\n**What remains.**\n\nConstruct a detecting knot for every nontrivial separating-sphere twist in the stated setting, or produce a counterexample.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.19, AMS Mathematical Surveys and Monographs 295 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: The publisher-authorized preliminary volume identifies the connected-sum sphere-twist question as the remaining case.\n- Paolo Aceto, Corey Bregman, Christopher W. Davis, JungHwan Park and Arunima Ray, Isotopy and equivalence of knots in 3-manifolds, arXiv:2007.05796. (primary): https://arxiv.org/abs/2007.05796\n Evidence used: Proves the prime-manifold and S^1 x S^2 analogues, not the stated reducible case.\n\n**Review notes.** The hypothesis that the sphere twist is not isotopic to the identity was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2679, "problem_number": "KP-1.20", "title": "Kirby Problem 1.20", "statement": "(a) Are there any null-homologous Floer minimal knots with irreducible com-\nplements other than the Borromean knots $B_{g}, g \\geq 0$, in any 3-manifolds?\nIt is reasonable to make the following more concrete conjectures.\n(b) Conjecture (Ni): If $b_{1}(Y) < 2$, then the only null-homologous Floer mini-\nmal knot in $Y$ is the unknot.\n(c) Conjecture (Ni): If the Thurston norm of $Y$ is not identically zero, then $Y$\ndoes not contain any null-homologous Floer minimal knot with irreducible\ncomplement.\n(d) Conjecture (Hedden, Rasmussen): The only Floer minimal knots in lens\nspaces are simple knots.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.20.\n\nLiterature notes:\n(1) Let $Y$ be a closed, oriented, connected 3-manifold, and let $K \\subset Y$ be a\nrationally null-homologous knot. It is known that\n\n$$\n\\operatorname{rank}\\widehat{\\mathrm{HFK}}(Y,K) \\geq \\operatorname{rank}\\widehat{\\mathrm{HF}}(Y).\n$$\n\nWe say $K$ is Floer minimal if equality in the above inequality holds.\nClearly, unknots are Floer minimal. Moreover, the Borromean knot [OS04c,\nFigure 16]$B_{g} \\subset \\#^{2g}(S^{1}\\times S^{2})$ is Floer minimal. If the $p$-surgery on a knot\n$L \\subset S^{3}$ is an L-space and $p \\geq 2g(L)$, then the dual knot in the L-space is\nFloer minimal [Hed11, Ras07].\nIt is known that the only null-homologous Floer minimal knot in $Y$\nis the unknot, when $Y$ is an L-space [NW14], and when $Y$ fibers over $S^{1}$\nwith fiber of genus $> 1$ [Ni14].\n(2) When $Y$ is the lens space $L(p, q)$, there is a standard genus-1 Heegaard\ndiagram with $p$ intersection points. Any two base points $z, w$ will specify a\nFloer minimal knot. Such knots are called simple.\nHedden [Hed11] and\nRasmussen [Ras07] proved that Conjecture (d) would imply the Berge\nConjecture (Problem 1.9 and [Kir97, Problem 1.78]).\n\nReferences cited:\n- [OS04c] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and knot invariants. Adv. Math., 186(1):58–116, 2004. doi:10.1016/j.aim.2003.05.001.\n- [Hed11] Matthew Hedden. On Floer homology and the Berge conjecture on knots admitting lens space surgeries. Trans. Amer. Math. Soc., 363(2):949–968, 2011. doi:10.1090/S0002-9947-2010-05117-7.\n- [Ras07] Jacob Rasmussen. Lens space surgeries and L-space homology spheres, 2007. arXiv: 0710.2531.\n- [NW14] Yi Ni and Zhongtao Wu. Heegaard Floer correction terms and rational genus bounds. Adv. Math., 267:360–380, 2014. doi:10.1016/j.aim.2014.09.006.\n- [Ni14] Yi Ni. Some applications of Gabai’s internal hierarchy. Adv. Math., 250:467–495, 2014. doi:10.1016/j.aim.2013.10.001.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The proposed classification of null-homologous Floer-minimal knots remains open, with affirmative classifications known when the ambient manifold is an L-space and when it fibers over S^1 with fiber genus greater than one.\n\n**Verified partial progress.**\n\n- Ni--Wu proved the unknot conclusion for null-homologous Floer-minimal knots in L-spaces.\n- Ni proved the corresponding conclusion for manifolds fibering over S^1 with fiber of genus greater than one.\n- Borromean knots remain the basic irreducible-complement examples; simple knots give the expected Floer-minimal class in lens spaces.\n\n**Full solution or refutation.**\n\nThe cited ambient-manifold cases support parts (b)--(d), but neither the general existence/classification question nor the lens-space conjecture is resolved.\n\n**What remains.**\n\nClassify all irreducible-complement examples and prove or disprove the three stated conjectures outside the known ambient classes.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.20 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the four-part problem and records the two principal ambient-manifold cases.\n- Yi Ni and Zhongtao Wu, Heegaard Floer correction terms and rational genus bounds, Advances in Mathematics 267 (2014), 360--380. (primary): https://doi.org/10.1016/j.aim.2014.09.006\n Evidence used: Provides the L-space case cited by the source.\n- Yi Ni, Some applications of Gabai's internal hierarchy, Advances in Mathematics 250 (2014), 467--495. (primary): https://doi.org/10.1016/j.aim.2013.10.001\n Evidence used: Provides the fibered-ambient-manifold case cited by the source.\n\n**Review notes.** The record combines an existence question and three distinct conjectures; the partial label is aggregate.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2680, "problem_number": "KP-1.21", "title": "Kirby Problem 1.21", "statement": "(a) For a given positive integer $g$, are there only finitely many L-space knots\nof genus $g$?\nA related but more general question is:\n(b) Question (Hedden): For which knots $L \\subset S^{3}$ are there only finitely many\nknots $K \\subset S^{3}$ with\n$\\widehat{\\mathrm{HFK}}(S^{3},K) \\cong \\widehat{\\mathrm{HFK}}(S^{3},L)$?\n\n(c) In some special cases, one expects stronger results saying that a class of\nknots is characterized by its knot Floer homology. We state 3 conjectures,\nordered by their strength.\n(i) For a knot $K \\subset S^{3}$, if\n$\\widehat{\\mathrm{HFK}}(S^{3},K) \\cong \\widehat{\\mathrm{HFK}}(S^{3},T_{2g+1,2})$, then\n$K = T_{2g+1,2}$.\n(ii) For a knot $K \\subset S^{3}$, if\n$\\widehat{\\mathrm{HFK}}(S^{3},K) \\cong \\widehat{\\mathrm{HFK}}(S^{3},T_{p,q})$, then $K$ is an\niterated torus knot.\n(iii) Conjecture (Ni): For an L-space knot $K \\subset S^{3}$, if all roots of $\\Delta_{K}(t)$\nlie on the unit circle, then $K$ is an iterated torus knot.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.21.\n\nLiterature notes:\n(1) A knot $K \\subset S^{3}$ is called an L-space knot if there exists a positive integer\n$p$ such that $S^{3}_{p}(K)$, the $p$-surgery on $K$ is an L-space (see Problem 3.48).\n(2) The answer to Question (a) is “yes” when $g \\leq 2$ [Ghi08, FRW24].\nIf $K$ is an L-space knot, then $K$ is fibered and strongly quasipositive\n[Ni07, Hed10],\n\n$$\n\\widehat{\\mathrm{HFK}}(S^{3},K,g(K)-1) \\cong \\mathbb{Z}_{(-1)}, \\tag{2}\n$$\n\nwhere $\\mathbb{Z}_{(-1)}$ means a copy of $\\mathbb{Z}$ supported in the Maslov grading $-1$\n[OS05b]. (The condition (2) implies that the monodromy of the fibra-\ntion is freely isotopic to a diffeomorphism without fixed point [BHS25,\nNi23a, GS22a].) However, these properties are not enough to guaran-\ntee the finiteness. For each $g \\geq 2$, Misev [Mis21] constructed infinitely\nmany strongly quasipositive fibered knots with the same Seifert form as\nthe torus knot $T_{2g+1,2}$. By taking cables of these knots, the condition (2)\ncan also be satisfied.\n(3) The finiteness in Question (b) does not hold for all knots $L$: see [HW18,\nAcknowledgements], [MS15], and [Wan22, Theorem 1.3]. In particular,\nHedden and Watson showed that, given any non-trivial band sum of two\nunknots, altering the number of twists in the band gives rise to an infinite\nfamily of distinct knots with the same knot Floer homology [HW18,\nTheorem 1].\n(4) Conjecture (i) in part (c) is known for $g = 1, 2$ [Ghi08, FRW24]. Con-\njecture (iii) would yield a positive answer to Problem 1.22; see [BBG19a].\n\nReferences cited:\n- [Ghi08] Paolo Ghiggini. Knot Floer homology detects genus-one fibred knots. Amer. J. Math., 130(5):1151–1169, 2008. doi:10.1353/ajm.0.0016.\n- [FRW24] Ethan Farber, Braeden Reinoso, and Luya Wang. Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil, 2024. doi:10.2140/gt.2024.28.4337.\n- [Ni07] Yi Ni. Knot Floer homology detects fibred knots. Invent. Math., 170(3):577–608, 2007. doi:10.1007/s00222-007-0075-9.\n- [Hed10] Matthew Hedden. Notions of positivity and the Ozsváth-Szabó concordance invariant. J. Knot Theory Ramifications, 19(5):617–629, 2010. doi:10.1142/S0218216510008017.\n- [OS05b] Peter Ozsváth and Zoltán Szabó. On knot Floer homology and lens space surgeries. Topology, 44(6):1281–1300, 2005. doi:10.1016/j.top.2005.05.001.\n- [BHS25] John A. Baldwin, Ying Hu, and Steven Sivek. Khovanov homology and the cinquefoil. J. Eur. Math. Soc. (JEMS), 27(6):2443–2465, 2025. doi:10.4171/jems/1415.\n- [Ni23a] Yi Ni. A note on knot Floer homology and fixed points of monodromy. Peking Math. J., 6(2):635–643, 2023. doi:10.1007/s42543-022-00051-3.\n- [GS22a] Paolo Ghiggini and Gilberto Spano. Knot Floer homology of fibred knots and Floer homology of surface diffeomorphisms, 2022. arXiv:2201.12411.\n- [Mis21] Filip Misev. On families of fibred knots with equal Seifert forms. Comm. Anal. Geom., 29(2):465–482, 2021. doi:10.4310/CAG.2021.v29.n2.a6.\n- [HW18] Matthew Hedden and Liam Watson. On the geography and botany of knot Floer homology. Selecta Math. (N.S.), 24(2):997–1037, 2018. doi:10.1007/s00029-017-0351-5.\n- [MS15] Allison H. Moore and Laura Starkston. Genus-two mutant knots with the same dimension in knot Floer and Khovanov homologies. Algebr. Geom. Topol., 15(1):43– 63, 2015. doi:10.2140/agt.2015.15.43.\n- [Wan22] Joshua Wang. The cosmetic crossing conjecture for split links. Geom. Topol., 26(7):2941–3053, 2022. doi:10.2140/gt.2022.26.2941.\n- [BBG19a] Michel Boileau, Steven Boyer, and Cameron McA. Gordon. Branched covers of quasi-positive links and L-spaces. J. Topol., 12(2):536–576, 2019. doi:10.1112/topo.12092.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Fixed-genus finiteness and HFK detection are known in genus one and two, while infinite same-HFK families show that the answer to the classification question in part (b) depends on L. The higher-genus and iterated-torus-knot conjectures remain open.\n\n**Verified partial progress.**\n\n- Ghiggini and Farber--Reinoso--Wang give the genus-one and genus-two cases of part (a) and of part (c)(i).\n- Hedden--Watson construct infinite twist families of distinct knots with the same knot Floer homology, so not every L has the finiteness property asked about in part (b).\n- The necessary fiberedness, strong quasipositivity and monodromy restrictions for L-space knots are known but do not yet imply fixed-genus finiteness.\n\n**Full solution or refutation.**\n\nLow-genus cases are solved and the scope of part (b) is constrained, but the record is a multipart classification agenda rather than a universal assertion refuted by the infinite families.\n\n**What remains.**\n\nProve fixed-genus finiteness for g at least three, classify those L with finite HFK fibers, and settle the three higher-genus detection/iterated-torus assertions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.21 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Separates the low-genus theorems, infinite-family obstruction, and three remaining conjectures.\n- Ethan Farber, Braeden Reinoso and Luya Wang, Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil, Geometry & Topology 28 (2024), 4337--4383. (primary): https://doi.org/10.2140/gt.2024.28.4337\n Evidence used: Proves HFK detection of T(2,5) and the genus-two L-space-knot classification.\n- Matthew Hedden and Liam Watson, On the geography and botany of knot Floer homology, Selecta Mathematica 24 (2018), 997--1037. (primary): https://doi.org/10.1007/s00029-017-0351-5\n Evidence used: Constructs infinite families relevant to the failure of universal finiteness in part (b).\n\n**Review notes.** Part (b) asks 'for which L'; counterexamples to universal finiteness are progress on that classification, not a refutation of the record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2681, "problem_number": "KP-1.22", "title": "Kirby Problem 1.22", "statement": "If $K$ is a hyperbolic $L$-space knot, show that its branched cover\n$\\Sigma_{2}(K)$ is not an $L$-space.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.22.\n\nLiterature notes:\nThis is a conjecture of A. Moore. The corresponding statement\nfor higher order covers $\\Sigma_{n}(K)$ was shown by Boileau-Boyer-Gordon [BBG19a] for\n$n \\geq 4$ and by [FRW24] for $n = 3$. Boyer-Gordon-Hu [BGH25, Corollary 7.3] have\nannounced that $\\Sigma_{n}(K)$ has a left-orderable group for each $n \\geq 2$, so in particular, a\npositive solution would be in accordance with the $L$-space conjecture, Problem 3.48.\n\nReferences cited:\n- [BBG19a] Michel Boileau, Steven Boyer, and Cameron McA. Gordon. Branched covers of quasi-positive links and L-spaces. J. Topol., 12(2):536–576, 2019. doi:10.1112/topo.12092.\n- [FRW24] Ethan Farber, Braeden Reinoso, and Luya Wang. Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil, 2024. doi:10.2140/gt.2024.28.4337.\n- [BGH25] Steven Boyer, Cameron McA. Gordon, and Ying Hu. Cyclic branched covers of Seifert links and properties related to the ADE link conjecture. J. Lond. Math. Soc. (2), 111(6):Paper No. e70178, 53, 2025. doi:10.1112/jlms.70178.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The non-L-space conclusion is known for cyclic branched covers of hyperbolic L-space knots in orders n at least 3, but the stated double-cover case n=2 remains open.\n\n**Verified partial progress.**\n\n- Boileau--Boyer--Gordon proved the corresponding statement for n at least 4.\n- Farber--Reinoso--Wang proved the n=3 case.\n- Boyer--Gordon--Hu proved left-orderability for the relevant cyclic branched-cover groups, including n=2; converting this to non-L-space status would use the still-open L-space conjecture.\n\n**Full solution or refutation.**\n\nLeft-orderability of pi_1(Sigma_2(K)) is strong supporting evidence but is not an unconditional proof that Sigma_2(K) is not an L-space.\n\n**What remains.**\n\nProve directly that the double branched cover is not an L-space, without assuming the L-space conjecture.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.22 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Records the n at least 4, n=3 and left-orderability results while retaining n=2 as a problem.\n- Michel Boileau, Steven Boyer and Cameron McA. Gordon, Branched covers of quasi-positive links and L-spaces, Journal of Topology 12 (2019), 536--576. (primary): https://doi.org/10.1112/topo.12092\n Evidence used: Proves the higher-order branched-cover result.\n- Steven Boyer, Cameron McA. Gordon and Ying Hu, Cyclic branched covers of Seifert links and properties related to the ADE link conjecture, Journal of the London Mathematical Society 111 (2025), e70178. (primary): https://doi.org/10.1112/jlms.70178\n Evidence used: Establishes the relevant left-orderability statement, which remains logically distinct from non-L-space status.\n\n**Review notes.** The distinction between left-orderability and the Heegaard-Floer conclusion is essential.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2682, "problem_number": "KP-1.23", "title": "Kirby Problem 1.23", "statement": "Let $K$ be a cubic graph embedded in the plane, and let $\\mathrm{Tait}(K)$\nbe the number of Tait colorings of $K$.\n(a) Is $\\dim J^{7}(K) = \\mathrm{Tait}(K)$?\n(b) Is $J^{5}(K)$ nonzero when $K$ is bridgeless?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.23.\n\nLiterature notes:\n(1) Part (a) is Conjecture 1.2 in [KM19a].\n(2) A Tait coloring of a cubic (also called trivalent) graph is a labeling of the\nedges of the graph by three colors such that edges of all of the colors are\nincident at each vertex. In [KM19a], Kronheimer–Mrowka used gauge\ntheory to assign to a web $K \\subset \\mathbb{R}^{3}$ a finite-dimensional vector space $J^{7}(K)$\nover $\\mathbb{Z}/2\\mathbb{Z}$, and proved that $J^{7}(K)$ is zero if and only if $K$ has an embedded\nbridge.\nThey also proposed a combinatorial counterpart $J^{5}(K),$ which\nKhovanov–Robert [KR21] showed is well-defined for planar webs $K \\subset \\mathbb{R}^{2}$.\nWork of Kronheimer–Mrowka [KM19b] and Khovanov–Robert [KR21]\nimplies that\n\n$$\n\\dim J^{5}(K) \\leq \\mathrm{Tait}(K) \\leq \\dim J^{7}(K)\n$$\n\nfor any planar web $K$.\nThe Four Color Theorem is equivalent to the\nstatement that every bridgeless cubic planar graph has a Tait coloring,\nand a new proof would therefore follow from an affirmative answer to\neither of the questions in this problem.\n(3) Boozer [Boo23] gave numerical evidence suggesting that the inequality\n$\\dim J^{5}(K) \\leq \\mathrm{Tait}(K)$ may sometimes be strict.\nIn [Boo25], he also\nannounced that $J^{5}$ and $J^{7}$ do not coincide as functors.\n\nReferences cited:\n- [KM19a] P. B. Kronheimer and T. S. Mrowka. Tait colorings, and an instanton homology for webs and foams. J. Eur. Math. Soc. (JEMS), 21(1):55–119, 2019. doi:10.4171/JEMS/831.\n- [KR21] Mikhail Khovanov and Louis-Hadrien Robert. Foam evaluation and KronheimerMrowka theories. Adv. Math., 376:Paper No. 107433, 59, 2021. doi:10.1016/j.aim.2020.107433.\n- [KM19b] Peter B. Kronheimer and Tomasz S. Mrowka. A deformation of instanton homology for webs. Geom. Topol., 23(3):1491–1547, 2019. doi:10.2140/gt.2019.23.1491.\n- [Boo23] David Boozer. Computer bounds for Kronheimer-Mrowka foam evaluation. Exp. Math., 32(4):615–630, 2023. doi:10.1080/10586458.2021.1982078.\n- [Boo25] David Boozer. The combinatorial and gauge-theoretic foam evaluation functors are not the same. Math. Ann., 392(1):47–56, 2025. doi:10.1007/s00208-024-03064-8.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For planar cubic graphs, dim J^5 <= Tait <= dim J^7 and J^7 vanishes exactly when there is an embedded bridge. Neither dim J^7=Tait nor nonvanishing of J^5 for every bridgeless planar graph was verified in general.\n\n**Verified partial progress.**\n\n- Kronheimer--Mrowka proved the bridge-detection theorem for J^7 and established the Tait-coloring framework.\n- Khovanov--Robert proved that the combinatorial J^5 theory is well-defined for planar webs and contributed to the stated dimension inequalities.\n- Boozer found numerical evidence that the lower inequality may be strict and proved that the J^5 and J^7 foam-evaluation functors are not the same.\n\n**Full solution or refutation.**\n\nFunctor inequivalence does not alone imply unequal dimensions on a graph, so it does not by itself refute part (a); numerical evidence likewise does not settle either exact question.\n\n**What remains.**\n\nSettle equality dim J^7(K)=Tait(K) and prove or disprove J^5(K) nonzero for every bridgeless planar cubic graph.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.23 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States both questions and distinguishes inequalities, computations and functor inequivalence from their resolution.\n- P. B. Kronheimer and T. S. Mrowka, Tait colorings, and an instanton homology for webs and foams, Journal of the European Mathematical Society 21 (2019), 55--119. (primary): https://doi.org/10.4171/JEMS/831\n Evidence used: Defines J^7, proves its bridge-detection theorem, and formulates the dimension conjecture.\n- Mikhail Khovanov and Louis-Hadrien Robert, Foam evaluation and Kronheimer--Mrowka theories, Advances in Mathematics 376 (2021), 107433. (primary): https://doi.org/10.1016/j.aim.2020.107433\n Evidence used: Develops the combinatorial foam evaluation underlying J^5.\n- David Boozer, The combinatorial and gauge-theoretic foam evaluation functors are not the same, Mathematische Annalen 392 (2025), 47--56. (primary): https://doi.org/10.1007/s00208-024-03064-8\n Evidence used: Proves functor inequivalence, a result weaker than a counterexample to the numerical equality in part (a).\n\n**Review notes.** The superscripts in J^5 and J^7 are theory labels in the source, not powers. The status deliberately does not promote functor inequivalence to a counterexample.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2683, "problem_number": "KP-1.24", "title": "Kirby Problem 1.24", "statement": "(Jones Unknot Detection).\n(a) Is there a nontrivial knot with the same Jones polynomial as the unknot?\n\n(b) Does there exist a nontrivial knot whose colored Jones polynomials are all\ntrivial?\n(c) Is there a nontrivial knot with trivial HOMFLYPT polynomial? (A knot\nwith trivial HOMFLYPT polynomial would have trivial Jones polynomial.)", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.24.\n\nLiterature notes:\n(1) Question (a) was first proposed by V. Jones; see [Kir97, Problems 1.88(a)\nand (c)] and [AALM00].\n(2) Computer calculations by Sikora and Tuzun [TS21] have shown there is\nno nontrivial knot with up to 24 crossings with trivial Jones polynomial.\n(3) As observed in [MM01], a positive answer to Question (b) would follow\nfrom the Volume Conjecture stated in Problem 1.27.\n(4) By tabulating links, Thistlethwaite found links with two or more com-\nponents with trivial Jones polynomials [Thi01]. This was extended in\n[EKT03] to infinite families of links with trivial Jones polynomials. But\nthese examples do not have trivial HOMFLYPT polynomials.\n(5) Kronheimer and Mrowka have proven that a knot has trivial Khovanov\nhomology if and only if it is trivial [KM11].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [AALM00] Vladimir Igorevič Arnol’d, Michael Atiyah, Peter Lax, and Barry Mazur. Mathematics: Frontiers and perspectives. American Mathematical Society Providence, RI, 2000.\n- [TS21] Robert E. Tuzun and Adam S. Sikora. Verification of the Jones unknot conjecture up to 24 crossings. J. Knot Theory Ramifications, 30(3):Paper No. 2150020, 6, 2021. doi:10.1142/S0218216521500206.\n- [MM01] Hitoshi Murakami and Jun Murakami. The colored Jones polynomials and the simplicial volume of a knot. Acta Math., 186(1):85–104, 2001. doi:10.1007/BF02392716.\n- [Thi01] Morwen Thistlethwaite. Links with trivial Jones polynomial. J. Knot Theory Ramifications, 10(4):641–643, 2001. doi:10.1142/S0218216501001050.\n- [EKT03] Shalom Eliahou, Louis H. Kauffman, and Morwen B. Thistlethwaite. Infinite families of links with trivial Jones polynomial. Topology, 42(1):155–169, 2003. doi:10.1016/S0040-9383(02)00012-5.\n- [KM11] P. B. Kronheimer and T. S. Mrowka. Khovanov homology is an unknotdetector. Publ. Math. Inst. Hautes Études Sci., 113:97–208, 2011. doi:10.1007/s10240-010-0030-y.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No nontrivial knot with trivial ordinary Jones polynomial, all colored Jones polynomials trivial, or trivial HOMFLYPT polynomial is known. Finite verification and categorified unknot detection do not settle these polynomial questions.\n\n**Verified partial progress.**\n\n- Tuzun--Sikora verified that every nontrivial knot through 24 crossings has nontrivial Jones polynomial.\n- Nontrivial multicomponent links with trivial Jones polynomial are known, but the cited examples do not have trivial HOMFLYPT polynomial and do not answer the knot questions.\n- Kronheimer--Mrowka proved that Khovanov homology detects the unknot, but equality of Euler characteristics is weaker than equality of homology.\n\n**Full solution or refutation.**\n\nAll three existence questions remain open for knots. The source background's statement that a positive answer to part (b) follows from the Volume Conjecture has the logical direction reversed: the conjecture would rule such a knot out.\n\n**What remains.**\n\nEither prove each invariant detects the unknot in the stated sense or construct a nontrivial knot with the corresponding trivial invariant.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.24 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Collects the three open questions and their known finite, link and homological results.\n- Robert E. Tuzun and Adam S. Sikora, Verification of the Jones unknot conjecture up to 24 crossings, Journal of Knot Theory and Its Ramifications 30 (2021), 2150020. (primary): https://doi.org/10.1142/S0218216521500206\n Evidence used: Provides the exhaustive finite-crossing verification.\n- P. B. Kronheimer and T. S. Mrowka, Khovanov homology is an unknot-detector, Publications Mathématiques de l'IHÉS 113 (2011), 97--208. (primary): https://doi.org/10.1007/s10240-010-0030-y\n Evidence used: Proves categorified detection, not Jones-polynomial detection.\n- Hitoshi Murakami and Jun Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Mathematica 186 (2001), 85--104. (primary): https://doi.org/10.1007/BF02392716\n Evidence used: Formulates the colored-Jones Volume Conjecture whose truth would exclude a nontrivial knot with all colored Jones polynomials trivial.\n\n**Review notes.** The exact problem questions were preserved. The logical typo is in the supplied background, not silently corrected in the source record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2684, "problem_number": "KP-1.25", "title": "Kirby Problem 1.25", "statement": "(a) Conjecture: The noncommutative $A$-ideal of a knot $K$ is exactly the an-\nnihilator of the colored Jones polynomial $($ the infinite dimensional vector\nwhose entries are the colored Jones polynomials of $K)$.\n(b) $($ AJ Conjecture $)$ Let $A(K; L, M)$ be the $A$-polynomial of the knot $K$, and\n$\\alpha(K, q; M, L)$ a polynomial of minimal degree in $L$ and co-prime coeffi-\ncients that annihilates the colored Jones polynomial $J_{n}(K; q)$. Then\n\n$$\n\\alpha(K, -1; M, L) = A(K; L, M)\n$$\n\nup to a factor depending on $M.$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.25.\n\nLiterature notes:\n(1) The version of Conjecture (a) for $\\mathfrak{s}\\mathfrak{l}_{2}$ appears in [FGL02]; the full version\nis stated in [Sik08]. The AJ conjecture (b) is due to Garoufalidis [Gar04].\n(2) The noncommutative $A$-ideal of $S^{3} - K$ is the left ideal in the Kauffman\nbracket skein algebra of the torus that annihilates the empty skein in\nthe knot complement. [FGL02] shows that a series built from the colored\nJones polynomial $J_{n}(K; q)$ annihilates the noncommutative $A$-ideal, which\nleads to recursive relationships satisfied by $J_{n}(K; q)$.\nThe noncommutative $A$-ideal is computed for $3_{1}$ [Gel02], $(2, 2p + 1)$\ntorus knots [GS03], and $4_{1}$ [GS04]. Demonstrating the nontriviality of\nthe noncommutative $A$-ideal remains a significant unresolved problem.\n(3) The $A$-polynomial of a knot $K$ [CL98] is the defining polynomial for\nthe character variety formed by $\\operatorname{SL}_{2}(\\mathbb{C})$-representations of $\\pi_{1}(\\partial(S^{3} - K))$\nthat extend to $\\pi_{1}(S^{3} - K)$.\nSee [GL05] for details on the definition\nof $\\alpha(K, q; M, L)$, which comes from linear recursion relations satisfied\nby $J_{n}(K; q)$.\nThe AJ Conjecture is proved for certain 2-bridge knots\n\n[Lê06, LZ17] and pretzel knots [LT15], and a diagrammatic approach is\nsuggested in [DG20]. See also [Guk05] for the physics context of the AJ\nconjecture.\n(4) The two Conjectures are closely intertwined, and the AJ Conjecture is an\nimportant step towards understanding the relationship between classical\nand quantum invariants.\n\nReferences cited:\n- [FGL02] Charles Frohman, Răzvan Gelca, and Walter Lofaro. The A-polynomial from the noncommutative viewpoint. Trans. Amer. Math. Soc., 354(2):735–747, 2002. doi: 10.1090/S0002-9947-01-02889-6.\n- [Sik08] Adam S. Sikora. Quantizations of character varieties and quantum knot invariants, 2008. arXiv:0807.0943.\n- [Gar04] Stavros Garoufalidis. On the characteristic and deformation varieties of a knot. In Proceedings of the Casson Fest, volume 7 of Geom. Topol. Monogr., pages 291– 309. Geom. Topol. Publ., Coventry, 2004. doi:10.2140/gtm.2004.7.291.\n- [Gel02] Răzvan Gelca. Non-commutative trigonometry and the A-polynomial of the trefoil knot. Math. Proc. Cambridge Philos. Soc., 133(2):311–323, 2002. doi:10.1017/S0305004102006047.\n- [GS03] Răzvan Gelca and Jeremy Sain. The noncommutative A-ideal of a $(2,2p+1)$-torus knot determines its Jones polynomial. J. Knot Theory Ramifications, 12(2):187– 201, 2003. doi:10.1142/S021821650300238X.\n- [GS04] Răzvan Gelca and Jeremy Sain. The computation of the non-commutative generalization of the A-polynomial of the figure-eight knot. J. Knot Theory Ramifications, 13(6):785–808, 2004. doi:10.1142/S0218216504003482.\n- [CL98] D. Cooper and D. D. Long. Representation theory and the A-polynomial of a knot. Chaos Solitons Fractals, 9(4-5):749–763, 1998. Knot theory and its applications. doi:10.1016/S0960-0779(97)00102-1.\n- [GL05] Stavros Garoufalidis and Thang T. Q. Lê. The colored Jones function is qholonomic. Geom. Topol., 9:1253–1293, 2005. doi:10.2140/gt.2005.9.1253.\n- [LZ17] Thang T. Q. Lê and Xingru Zhang. Character varieties, A-polynomials and the AJ conjecture. Algebr. Geom. Topol., 17(1):157–188, 2017. doi:10.2140/agt.2017.17.157.\n- [LT15] Thang T. Q. Le and Anh T. Tran. On the AJ conjecture for knots. Indiana Univ. Math. J., 64(4):1103–1151, 2015. With an appendix written jointly with Vu Q. Huynh. doi:10.1512/iumj.2015.64.5602.\n- [DG20] Renaud Detcherry and Stavros Garoufalidis. A diagrammatic approach to the AJ conjecture. Math. Ann., 378(1-2):447–484, 2020. doi:10.1007/s00208-020-02028-y.\n- [Guk05] Sergei Gukov. Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial. Comm. Math. Phys., 255(3):577–627, 2005. doi:10.1007/s00220-005-1312-y.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The noncommutative A-ideal and AJ conjectures are established or computed for several knot families, including specified torus, figure-eight, 2-bridge and pretzel cases, but neither statement is proved for all knots.\n\n**Verified partial progress.**\n\n- The noncommutative A-ideal has been computed for the trefoil, (2,2p+1) torus knots and the figure-eight knot.\n- The AJ conjecture is proved for classes of 2-bridge knots and pretzel knots under the hypotheses of the cited papers.\n- Diagrammatic work supplies a framework for attacking the AJ conjecture but not a general proof.\n\n**Full solution or refutation.**\n\nThe known computations and family theorems substantiate the conjectural link between the colored Jones recursion and the classical A-polynomial without establishing it universally.\n\n**What remains.**\n\nProve equality with the annihilator in part (a), establish general nontriviality, and prove the AJ specialization statement for arbitrary knots with all normalization factors controlled.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.25 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Distinguishes computed examples and proved families from the two general conjectures.\n- Charles Frohman, Razvan Gelca and Walter Lofaro, The A-polynomial from the noncommutative viewpoint, Transactions of the AMS 354 (2002), 735--747. (primary): https://doi.org/10.1090/S0002-9947-01-02889-6\n Evidence used: Introduces the noncommutative viewpoint and its relationship with colored Jones recursions.\n- Thang T. Q. Le and Xingru Zhang, Character varieties, A-polynomials and the AJ conjecture, Algebraic & Geometric Topology 17 (2017), 157--188. (primary): https://doi.org/10.2140/agt.2017.17.157\n Evidence used: Proves the AJ conjecture for specified 2-bridge knots.\n- Thang T. Q. Le and Anh T. Tran, On the AJ conjecture for knots, Indiana University Mathematics Journal 64 (2015), 1103--1151. (primary): https://doi.org/10.1512/iumj.2015.64.5602\n Evidence used: Proves the conjecture for a substantial pretzel-knot family under stated conditions.\n\n**Review notes.** The possible M-factor and specialization normalization in part (b) are retained in the unresolved scope.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2685, "problem_number": "KP-1.26", "title": "Kirby Problem 1.26", "statement": "(Jones Slope Conjecture). For a knot $K$, the Jones slopes $js(K)$\nare the set of cluster points in\n\n$$\n\\left\\{\\frac{4}{n^{2}}\\deg_{+}\\bigl(J_{n}(K;q)\\bigr)\\right\\}_{n\\in\\mathbb{N}}\n\\cup\n\\left\\{\\frac{4}{n^{2}}\\deg_{-}\\bigl(J_{n}(K;q)\\bigr)\\right\\}_{n\\in\\mathbb{N}},\n$$\n\nwhere deg $^{+}$ denotes the highest degree and deg $^{-}$ the lowest degree.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.26.\n\nLiterature notes:\n(1) The Jones slopes of a knot $K$ are a subset of the boundary slopes of\nincompressible surfaces, boundary-incompressible orientable surfaces in\n$S^{3} - K$.\n(2) This conjecture was originally stated by Garoufalidis [Gar11]. There is a\nstronger version of the conjecture due to Kalfagianni and Tran [KT15].\nGaroufalidis proved his conjecture for up to 10 crossings [Gar11], and the\nSlope Conjecture or its stronger form has been proved for various fami-\nlies of knots [FKP11, GvdV16, LvdV16, Lee22, BMT21, BMT20,\nMT17, LYL19, GLVDV20].\n\nReferences cited:\n- [Gar11] Stavros Garoufalidis. The Jones slopes of a knot. Quantum Topol., 2(1):43–69, 2011. doi:10.4171/QT/13.\n- [KT15] Efstratia Kalfagianni and Anh T. Tran. Knot cabling and the degree of the colored Jones polynomial. New York J. Math., 21:905–941, 2015. http://nyjm.albany.edu: 8000/j/2015/21 905.html.\n- [FKP11] David Futer, Efstratia Kalfagianni, and Jessica S. Purcell. Slopes and colored Jones polynomials of adequate knots. Proc. Amer. Math. Soc., 139(5):1889–1896, 2011. doi:10.1090/S0002-9939-2010-10617-2.\n- [GvdV16] Stavros Garoufalidis and Roland van der Veen. Quadratic integer programming and the slope conjecture. New York J. Math., 22:907–932, 2016. http://nyjm.albany.edu: 8000/j/2016/22 907.html.\n- [LvdV16] Christine Ruey Shan Lee and Roland van der Veen. Slopes for pretzel knots. New York J. Math., 22:1339–1364, 2016. http://nyjm.albany.edu:8000/j/2016/22 1339. html.\n- [Lee22] Christine Ruey Shan Lee. Jones slopes and coarse volume of near-alternating knots. Comm. Anal. Geom., 30(4):891–948, 2022.\n- [BMT21] Kenneth L. Baker, Kimihiko Motegi, and Toshie Takata. The strong slope conjecture for cablings and connected sums. New York J. Math., 27:676–704, 2021. https://nyjm.albany.edu/j/2021/27-26v.pdf.\n- [BMT20] Kenneth L. Baker, Kimihiko Motegi, and Toshie Takata. The strong slope conjecture for twisted generalized Whitehead doubles. Quantum Topol., 11(3):545–608, 2020. doi:10.4171/qt/242.\n- [MT17] Kimihiko Motegi and Toshie Takata. The slope conjecture for graph knots. Math. Proc. Cambridge Philos. Soc., 162(3):383–392, 2017. doi:10.1017/S0305004116000566.\n- [LYL19] Xudong Leng, Zhiqing Yang, and Ximin Liu. The slope conjectures for 3-string Montesinos knots. New York J. Math., 25:45–70, 2019. https://nyjm.albany.edu/j/2019/25-2p.pdf.\n- [GLVDV20] Stavros Garoufalidis, Christine Ruey Shan Lee, and Roland Van Der Veen. The slope conjecture for Montesinos knots. Internat. J. Math., 31(7):2050056, 66, 2020. doi:10.1142/S0129167X20500561.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported statement is not a conjectural assertion: it only defines the set of Jones slopes. The background supplies an intended Slope Conjecture, but inserting it would change the exact record.\n\n**Verified partial progress.**\n\n- For the intended conjecture, Garoufalidis verified the boundary-slope conclusion for knots through 10 crossings.\n- The Slope Conjecture or Strong Slope Conjecture is proved for adequate, pretzel, Montesinos and graph-knot families and is preserved in several cabling, connected-sum and twisted-Whitehead constructions.\n\n**Full solution or refutation.**\n\nNo truth status can be assigned to the literal statement beyond noting that it is a definition. Under the intended reading that Jones slopes are boundary slopes of essential surfaces, substantial family-specific results are known and the general conjecture remains open.\n\n**What remains.**\n\nRepair the source record by explicitly supplying and verifying the intended assertion, then separately track the general conjecture and its strong form.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.26 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Shows that the displayed problem text stops after the definition while the first remark contains the intended boundary-slope assertion.\n- Stavros Garoufalidis, The Jones slopes of a knot, Quantum Topology 2 (2011), 43--69. (primary): https://doi.org/10.4171/QT/13\n Evidence used: States the Slope Conjecture and verifies it in the finite census described by the source.\n- David Futer, Efstratia Kalfagianni and Jessica S. Purcell, Slopes and colored Jones polynomials of adequate knots, Proceedings of the AMS 139 (2011), 1889--1896. (primary): https://doi.org/10.1090/S0002-9939-2010-10617-2\n Evidence used: Proves the intended slope relation for adequate knots.\n\n**Review notes.** Extraction/formulation defect: the exact statement contains no proposition or question after defining js(K). It was not silently reconstructed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2686, "problem_number": "KP-1.27", "title": "Kirby Problem 1.27", "statement": "(Kashaev–Murakami–Murakami Volume Conjecture). For a\nlink $L \\subset S^{3}$,\n\n$$\n\\frac{1}{2\\pi}\\operatorname{Vol}_{\\mathrm{hyp}}(S^{3}-L)\n=\\lim_{n\\to\\infty}\\frac{1}{n}\\log\\left|J_{n}\\bigl(L,e^{2\\pi i/n}\\bigr)\\right|.\n$$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.27.\n\nLiterature notes:\nHere, $J_{n}(L, q)$ is the $n^{th}$ colored Jones polynomial of $L$ in the vari-\nable $q$, and $\\operatorname{Vol}_{hyp}(S^{3} - L)$ is the hyperbolic volume of the link complement. This\nproblem was originally proposed for a complex-valued invariant for triangulated\nlinks in $S^{3}$ [Kas97], which was later shown to be determined by the colored Jones\npolynomial of the link complement [MM01]. Note that this conjecture would imply\nthat the collection of colored Jones polynomials would characterize the unknot; see\nProblem 1.24(b). The conjecture has been verified for some knots, including knots\nwith up to seven crossings [Oht16, OY18, Oht17], torus knots [KT00], White-\nhead doubles of $(2, b)$ torus knots or links [Zhe07], Whitehead chains, [vdV08],\nand certain cables of $4_{1}$ [LT10, MT25].\nNote that there is also a stronger complexified version involving the Chern–\nSimons invariant. See [Mur07, MMO $^{+}02$].\n\nReferences cited:\n- [Kas97] Rinat M Kashaev. The hyperbolic volume of knots from the quantum dilogarithm. Letters in mathematical physics, 39(3):269–275, 1997.\n- [MM01] Hitoshi Murakami and Jun Murakami. The colored Jones polynomials and the simplicial volume of a knot. Acta Math., 186(1):85–104, 2001. doi:10.1007/BF02392716.\n- [Oht16] Tomotada Ohtsuki. On the asymptotic expansion of the Kashaev invariant of the 52 knot. Quantum Topol., 7(4):669–735, 2016. doi:10.4171/QT/83.\n- [OY18] Tomotada Ohtsuki and Yoshiyuki Yokota. On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. Math. Proc. Cambridge Philos. Soc., 165(2):287–339, 2018. doi:10.1017/S0305004117000494.\n- [Oht17] Tomotada Ohtsuki. On the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings. Internat. J. Math., 28(13):1750096, 143, 2017. doi:10.1142/S0129167X17500963.\n- [KT00] R. M. Kashaev and O. Tirkkonen. A proof of the volume conjecture on torus knots. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 269:262–268, 370, 2000. doi:10.1023/A:1022608131142.\n- [Zhe07] Hao Zheng. Proof of the volume conjecture for Whitehead doubles of a family of torus knots. Chinese Ann. Math. Ser. B, 28(4):375–388, 2007. doi:10.1007/s11401-006-0373-3.\n- [vdV08] Roland van der Veen. Proof of the volume conjecture for Whitehead chains. Acta Math. Vietnam., 33(3):421–431, 2008.\n- [LT10] Thang T. Q. Le and Anh T. Tran. On the volume conjecture for cables of knots. J. Knot Theory Ramifications, 19(12):1673–1691, 2010. doi:10.1142/S0218216510008534.\n- [MT25] Hitoshi Murakami and Anh T. Tran. The colored Jones polynomial of a cable of the figure-eight knot. J. Knot Theory Ramifications, 34(3):Paper No. 2340019, 24, 2025. doi:10.1142/S0218216523400199.\n- [Mur07] Hitoshi Murakami. Various generalizations of the volume conjecture. In The interaction of analysis and geometry, volume 424 of Contemp. Math., pages 165–186. Amer. Math. Soc., Providence, RI, 2007. doi:10.1090/conm/424/08100.\n- [MMO+02] Hitoshi Murakami, Jun Murakami, Miyuki Okamoto, Toshie Takata, and Yoshiyuki Yokota. Kashaev’s conjecture and the Chern-Simons invariants of knots and links. Experiment. Math., 11(3):427–435, 2002. http://projecteuclid.org/euclid.em/1057777432.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Volume Conjecture is proved for numerous knots and link families but remains open in general. The literal all-links formulation requires a convention for hyperbolic volume outside the hyperbolic case.\n\n**Verified partial progress.**\n\n- The conjecture is proved for torus knots and for the hyperbolic knots through seven crossings covered by the cited asymptotic analyses.\n- It is proved for Whitehead doubles of specified torus-knot families, Whitehead chains and certain cables of the figure-eight knot.\n- Later work continues to establish related volume-conjecture variants for large finite collections and additional geometric families, without a universal theorem for the exact colored-Jones limit stated here.\n\n**Full solution or refutation.**\n\nThe special-case proofs demonstrate the predicted exponential growth in diverse geometric regimes but do not establish existence and equality of the limit for every link in the source's scope.\n\n**What remains.**\n\nSpecify the volume convention or restrict the statement to hyperbolic links, then prove the limit exists and equals the normalized volume in all remaining cases.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.27 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the conjecture and lists the verified low-crossing, torus, Whitehead and cable families.\n- Hitoshi Murakami and Jun Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Mathematica 186 (2001), 85--104. (primary): https://doi.org/10.1007/BF02392716\n Evidence used: Identifies Kashaev's invariant with colored Jones evaluations and formulates the simplicial-volume version.\n- Tomotada Ohtsuki, On the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings, International Journal of Mathematics 28 (2017), 1750096. (primary): https://doi.org/10.1142/S0129167X17500963\n Evidence used: Establishes the required asymptotics for the seven-crossing hyperbolic cases treated there.\n- Hitoshi Murakami and Anh T. Tran, The colored Jones polynomial of a cable of the figure-eight knot, Journal of Knot Theory and Its Ramifications 34 (2025), 2340019. (primary): https://doi.org/10.1142/S0218216523400199\n Evidence used: Provides a recent verified cable family recorded by the source.\n\n**Review notes.** For a general link, Vol_hyp(S^3-L) is undefined without a hyperbolicity restriction or a JSJ/simplicial-volume convention; this scope issue was flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2687, "problem_number": "KP-1.28", "title": "Kirby Problem 1.28", "statement": "Let $L$ be a link in the thickened annulus $S^{1} \\times I \\times I$.\n(a) Wrapping conjecture: $w(L)$ is equal to the maximal nonzero annular de-\ngree of the annular Kauffman bracket of $L$.\n(b) Categorifed wrapping conjecture: $w(L)$ is equal to the maximal nonzero\nannular grading of the annular Khovanov homology of $L$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.28.\n\nLiterature notes:\n(1) The wrapping number $w(L)$ of $L$ is the minimum number of times $L$\nintersects a meridional disk.\n(2) The wrapping number $w(L)$ is less than or equal to the maximal nonzero\nannular degree of the annular Kauffman bracket of $L$, as well as the maxi-\nmal nonzero annular grading of the annular Khovanov homology of $L$. The\nproblem is asking whether these upper bounds are necessarily achieved;\nthat is, whether either of these invariants detects the wrapping number.\n(3) The wrapping conjecture is due to Hoste–Przytycki [HP95a], who proved\nit for any annular link with a $\\pm$-adequately wrapped diagram.\n(4) The categorified wrapping conjecture is due to Grigsby, who proved with\nNi that it holds for string links [GN14]. This conjecture would follow\nfrom the wrapping conjecture since the graded Euler characteristic of an-\nnular Khovanov homology is equal to the annular Kauffman bracket. It is\nknown via spectral sequences relating annular Khovanov homology with\nFloer-theoretic invariants [GW10, Xie21, XZ25] that annular Khovanov\nhomology is nontrivial in the annular grading given by the generalized\nThurston norm of the meridional disk. The difficulty is that the general-\nized meridional Thurston norm can be arbitrarily smaller than the wrap-\nping number; see Martin’s work for this and related progress [Mar23].\n\nReferences cited:\n- [HP95a] Jim Hoste and Józef H. Przytycki. The $(2,8)$-skein module of Whitehead manifolds. J. Knot Theory Ramifications, 4(3):411–427, 1995. doi:10.1142/S021821659500020X.\n- [GN14] J. Elisenda Grigsby and Yi Ni. Sutured Khovanov homology distinguishes braids from other tangles. Math. Res. Lett., 21(6):1263–1275, 2014. doi:10.4310/MRL.2014.v21.n6.a4.\n- [GW10] J. Elisenda Grigsby and Stephan M. Wehrli. Khovanov homology, sutured Floer homology and annular links. Algebr. Geom. Topol., 10(4):2009–2039, 2010. doi: 10.2140/agt.2010.10.2009.\n- [Xie21] Yi Xie. Instantons and annular Khovanov homology. Adv. Math., 388:Paper No. 107864, 51, 2021. doi:10.1016/j.aim.2021.107864.\n- [XZ25] Yi Xie and Boyu Zhang. Instanton Floer homology for sutured manifolds with tangles. J. Differential Geom., 130(3):701–769, 2025. doi:10.4310/jdg/1749496718.\n- [Mar23] Gage Martin. Annular Khovanov homology and meridional disks. J. Knot Theory Ramifications, 32(2):Paper No. 2250088, 14, 2023. doi:10.1142/S0218216522500882.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Wrapping-number detection is known for suitably adequate diagrams and the categorified version for string links, but both general conjectures remain open.\n\n**Verified partial progress.**\n\n- Hoste--Przytycki prove the wrapping conjecture for ±-adequately wrapped diagrams.\n- Grigsby--Ni prove the categorified version for string links.\n\n**Full solution or refutation.**\n\nNo general detection theorem was verified.\n\n**What remains.**\n\nProve either invariant always realizes wrapping number.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.28 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the upper bounds and exact known classes.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2688, "problem_number": "KP-1.29", "title": "Kirby Problem 1.29", "statement": "Is the first inequality below true? If so, is the second?\n(a) $(\\operatorname{Vol}$-Det Conjecture $)$ For any alternating hyperbolic knot $K \\subset S^{3}$,\nvol $(K) < 2\\pi \\log \\det(K),$\nwhere vol is the hyperbolic volume of $K$ and det is the determinant.\n(b) For any hyperbolic knot $K \\subset S^{3}$,\nvol $(K) < 2\\pi \\log \\operatorname{rank} \\mathrm{Kh}(K),$\nwhere Kh is the reduced Khovanov homology of $K$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.29.\n\nLiterature notes:\n(1) The proposed inequalities relate quantum topology and hyperbolic geom-\netry in the style of the Volume Conjecture [Kas97, MM01] (see Problem\n1.27), but may be more approachable.\n(2) A correlation between determinant and volume for alternating hyperbolic\nknots was first noticed by Dunfield, and Champanerkar, Kofman and Pur-\ncell conjectured in [CKP16] that (1) holds. They moreover showed that\nthe bound is sharp, and conjectured the second more general inequal-\nity (b).\n(3) Burton verified (a) for several families of knots including 2-bridge knots,\nclosed alternating 3-braids, and an infinite family of alternating 4-braids\n[Bur18].\n(4) Champanerkar, Kofman and Lalin proved (a) for other infinite families of\nalternating knots using number theoretic methods in [CKL19], and gave\nrelated conjectures involving other volume-type quantities and the Mahler\nmeasures of certain graph-theoretic polynomials. See also [CK25].\n\nReferences cited:\n- [Kas97] Rinat M Kashaev. The hyperbolic volume of knots from the quantum dilogarithm. Letters in mathematical physics, 39(3):269–275, 1997.\n- [MM01] Hitoshi Murakami and Jun Murakami. The colored Jones polynomials and the simplicial volume of a knot. Acta Math., 186(1):85–104, 2001. doi:10.1007/BF02392716.\n- [CKP16] Abhijit Champanerkar, Ilya Kofman, and Jessica S. Purcell. Geometrically and diagrammatically maximal knots. J. Lond. Math. Soc. (2), 94(3):883–908, 2016. doi:10.1112/jlms/jdw062.\n- [Bur18] Stephan D. Burton. The determinant and volume of 2-bridge links and alternating 3-braids. New York J. Math., 24:293–316, 2018. http://nyjm.albany.edu:8000/j/2018/24 293.html.\n- [CKL19] Abhijit Champanerkar, Ilya Kofman, and Matilde Lalı́n. Mahler measure and the vol-det conjecture. J. Lond. Math. Soc. (2), 99(3):872–900, 2019. doi:10.1112/jlms.12200.\n- [CK25] Abhijit Champanerkar and Ilya Kofman. Geometric bounds for spanning tree entropy of planar lattice graphs, 2025. arXiv:2505.05688.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Vol--Det inequality is verified for several infinite alternating families, but remains conjectural in general.\n\n**Verified partial progress.**\n\n- Burton verifies the first inequality for 2-bridge knots, closed alternating 3-braids and an infinite 4-braid family.\n- Champanerkar--Kofman--Lalin establish further families.\n\n**Full solution or refutation.**\n\nNeither general stated inequality is resolved.\n\n**What remains.**\n\nProve the conjectured bounds for all alternating/hyperbolic knots in their stated ranges.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.29 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintained list records the conjectures and proved families.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2689, "problem_number": "KP-1.30", "title": "Kirby Problem 1.30", "statement": "Does the Khovanov homology of every nontrivial knot contain\n2-torsion?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.30.\n\nLiterature notes:\nThis is a conjecture due to Shumakovitch [Shu14] originally posted\nin 2004. It has been verified for some infinite families of knots [Shu14, AP04,\n\nPS14, DM25a, DM25b]. Moreover, Gujral–Wang have shown [GW25] that if\n$K$ is a knot with no 2-torsion in its Khovanov homology, and $J$ is obtained from\n$K$ by a proper rational tangle replacement, then\nrankKh $(K) \\leq$ rankKh $(J),$\nwhere Kh refers to reduced Khovanov homology. It follows in particular that Shu-\nmakovitch’s conjecture holds for knots with unknotting number 1.\nIf proven, the conjecture would offer an alternative proof that Khovanov ho-\nmology detects the unknot.\n\nReferences cited:\n- [Shu14] Alexander N. Shumakovitch. Torsion of Khovanov homology. Fund. Math., 225(1):343–364, 2014. doi:10.4064/fm225-1-16.\n- [AP04] Marta M. Asaeda and Józef H. Przytycki. Khovanov homology: torsion and thickness. In Advances in topological quantum field theory, volume 179 of NATO Sci. Ser. II Math. Phys. Chem., pages 135–166. Kluwer Acad. Publ., Dordrecht, 2004. doi:10.1007/978-1-4020-2772-7\\\\_6.\n- [PS14] Józef H. Przytycki and Radmila Sazdanović. Torsion in Khovanov homology of semi-adequate links. Fund. Math., 225(1):277–304, 2014. doi:10.4064/fm225-1-13.\n- [DM25a] Raquel Dı́az and Pedro M. G. Manchón. A pattern for torsion in Khovanov homology. Fund. Math., 270(1):75–97, 2025. doi:10.4064/fm240810-12-3.\n- [DM25b] Raquel Dı́az and Pedro M. G. Manchón. New torsion patterns in Khovanov homology, 2025. arXiv:2508.00606.\n- [GW25] Onkar Singh Gujral and Joshua Wang. A minimality property for knots without Khovanov 2-torsion. Algebr. Geom. Topol., 25(7):4073–4075, 2025. doi:10.2140/agt.2025.25.4073.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The 2-torsion conjecture is verified for several infinite families, including knots with unknotting number one, but remains open generally.\n\n**Verified partial progress.**\n\n- Shumakovitch and subsequent works provide infinite verified families.\n- Gujral--Wang imply the conjecture for unknotting-number-one knots.\n\n**Full solution or refutation.**\n\nNo proof for every nontrivial knot was verified.\n\n**What remains.**\n\nProve or refute universal 2-torsion in Khovanov homology.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.30 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Current list records the verified families and remaining conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2690, "problem_number": "KP-1.31", "title": "Kirby Problem 1.31", "statement": "(a) Compute the Khovanov homology for all torus knots $T(m, n)$.\n(b) Compute the Khovanov–Rozansky $\\mathfrak{g}\\mathfrak{l}(N)$ homology for all torus knots.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.31.\n\nLiterature notes:\n(1) The data for the Khovanov homology of $T(m, n)$ is available at [LM23b]\nfor small $m, n$. Khovanov–Rozansky homology data is available at [Lew23a,\nLew23b].\n(2) The homology is known to have a lot of torsion [MPS $^{+}18$]. Comput-\ning Khovanov homology with $\\mathbb{Z}_{2}$ coefficients and $\\mathfrak{g}\\mathfrak{l}(N)$ homology with\n$\\mathbb{Z}_{N}$ coefficients might be easier than rational coefficients [GOR13]. On\nthe other hand, an open conjecture [PS14, MPS $^{+}18$] states that the\nKhovanov homology of $T(m, n)$ for $m$ prime and $n > m$ has $m$-torsion.\n(3) The triply graded Khovanov–Rozansky homology of torus knots (with in-\nteger coefficients) is computed in [Mel22]. By [Ras15] there is a spectral\nsequence from triply graded homology to Khovanov (resp.\nKhovanov–\nRozansky) homology, but the differentials are not known explicitly.\n(4) By [Sto09], there is a well-defined limit of the Khovanov homology of\n$T(m, n)$ as $m \\to \\infty$, also known as $T(n, \\infty)$. Rozansky [Roz14] shows\nthat the Khovanov homology of $T(n, \\infty)$ provides a categorified equivalent\nof the Jones–Wenzl projector [Wen87]. The triply graded homology of\n$T(n, \\infty)$ is computed in [Hog18], and precise conjectures for the Khovanov\nhomology of $T(n, \\infty)$ are discussed in [GOR13, GL15].\n(5) In contrast, the Heegaard Floer homology of $T(m, n)$ is well known [OS05b].\n\nReferences cited:\n- [LM23b] Charles Livingston and Allison H. Moore. Knotinfo: Table of knot invariants, November 2023. https://knotinfo.org.\n- [Lew23a] Lukas Lewark. Foamho, an $\\mathfrak{sl}(3)$-homology calculator. URL: https://people.math.ethz.ch/ llewark/foamho.php, November 2023.\n- [Lew23b] Lukas Lewark. Khoca, a knot homology calculator. URL: https://people.math.ethz.ch/ llewark/khoca.php, November 2023.\n- [MPS+18] Sujoy Mukherjee, Józef H. Przytycki, Marithania Silvero, Xiao Wang, and Seung Yeop Yang. Search for torsion in Khovanov homology. Exp. Math., 27(4):488– 497, 2018. doi:10.1080/10586458.2017.1320242.\n- [GOR13] Eugene Gorsky, Alexei Oblomkov, and Jacob Rasmussen. On stable Khovanov homology of torus knots. Exp. Math., 22(3):265–281, 2013. doi:10.1080/10586458.2013.798553.\n- [PS14] Józef H. Przytycki and Radmila Sazdanović. Torsion in Khovanov homology of semi-adequate links. Fund. Math., 225(1):277–304, 2014. doi:10.4064/fm225-1-13.\n- [Mel22] Anton Mellit. Homology of torus knots. Geom. Topol., 26(1):47–70, 2022. doi: 10.2140/gt.2022.26.47.\n- [Ras15] Jacob Rasmussen. Some differentials on Khovanov-Rozansky homology. Geom. Topol., 19(6):3031–3104, 2015. doi:10.2140/gt.2015.19.3031.\n- [Sto09] Marko Stošić. Khovanov homology of torus links. Topology Appl., 156(3):533–541, 2009. doi:10.1016/j.topol.2008.08.004.\n- [Roz14] Lev Rozansky. An infinite torus braid yields a categorified Jones-Wenzl projector. Fund. Math., 225(1):305–326, 2014. doi:10.4064/fm225-1-14.\n- [Wen87] Hans Wenzl. On sequences of projections. C. R. Math. Rep. Acad. Sci. Canada, 9(1):5–9, 1987.\n- [Hog18] Matthew Hogancamp. Categorified Young symmetrizers and stable homology of torus links. Geom. Topol., 22(5):2943–3002, 2018. doi:10.2140/gt.2018.22.2943.\n- [GL15] Eugene Gorsky and Lukas Lewark. On stable sl3-homology of torus knots. Exp. Math., 24(2):162–174, 2015. doi:10.1080/10586458.2014.963746.\n- [OS05b] Peter Ozsváth and Zoltán Szabó. On knot Floer homology and lens space surgeries. Topology, 44(6):1281–1300, 2005. doi:10.1016/j.top.2005.05.001.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Integral triply graded Khovanov--Rozansky homology of torus knots is computed, but the requested full computations and torsion questions retain open components.\n\n**Verified partial progress.**\n\n- Mellit computes integral triply graded homology of torus knots.\n- Data and spectral sequences supply further information.\n\n**Full solution or refutation.**\n\nThe multi-part computational programme is not wholly complete.\n\n**What remains.**\n\nGive uniform descriptions for the remaining listed theories and resolve predicted torsion.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.31 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States Mellit's result, available data, and open torsion conjectures.\n\n**Review notes.** Multi-part record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2691, "problem_number": "KP-1.32", "title": "Kirby Problem 1.32", "statement": "(a) Recover the Jones polynomial of links $L \\subset \\mathbb{R}^{3}$ by counting solutions to\nthe Kapustin–Witten equations on $\\mathbb{R}^{3} \\times \\mathbb{R}_{+}$ with Nahm pole boundary\nconditions;\n(b) Recover the Khovanov homology of links $L \\subset \\mathbb{R}^{3}$ by counting solutions to\nthe Haydys–Witten equations on $\\mathbb{R} \\times \\mathbb{R}^{3} \\times \\mathbb{R}_{+}$ with Nahm pole boundary\nconditions;\n\n(c) Use the Kapustin–Witten or Haydys–Witten equations to construct invari-\nants of links in other 3-manifolds.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.32.\n\nLiterature notes:\n(1) Khovanov homology [Kho00] is a link invariant defined combinatorially,\nwhose Euler characteristic is the Jones polynomial. It has many properties\nsimilar to Floer homologies (e.g., functoriality under cobordisms) and, as\nsuch, it is natural to ask if it has an interpretation in terms of gauge\ntheory. Such an interpretation was proposed by Witten in [Wit12], and\nis summarized in points (a) and (b) of the problem.\n(2) The Kapustin-Witten (KW) equations are a set of gauge invariant PDE’s\nin four dimensions introduced in [KW07].\nThe Nahm pole boundary\nconditions for these equations are described in [Wit12]; they have a more\nspecial form along the link $L \\times \\{0\\} \\subset \\mathbb{R}^{3} \\times \\{0\\}.$ Gaiotto and Witten\n[GW12] sketched an approach to showing that the count of solutions to\nthe KW equations with Nahm pole boundary conditions gives the coeffi-\ncients of the Jones polynomial. However, there is still work to be done to\nmake this mathematically rigorous.\n(3) The Haydys-Witten (HW) equations were introduced in [Wit12] and\n[Hay15]. Witten conjectured that, if one forms a complex whose gen-\nerators are the KW solutions and whose differentials count HW solutions\n(with appropriate limits and boundary conditions), then its homology is\nKhovanov homology.\n(4) Part of the appeal of these gauge-theoretic ideas is that they may pro-\nvide a definition of the Jones polynomial/Khovanov homology manifestly\nindependent of the link diagram. Moreover, we can write the same equa-\ntions with $\\mathbb{R}^{3}$ replaced by another 3-manifold, leading to part (c) of the\nproblem. Optimistically, the counts of KW solutions could give the class\nof the link $L$ in the Kauffman bracket skein module of the 3-manifold\n[Prz91, Tur88], or be related to the $\\widetilde{Z}$ invariants predicted by physi-\ncists [GPV17, GPPV20]; see Problem 3.68.\nMore ambitiously, the\ncount of HW solutions should give analogues of Khovanov homology for\nlinks in 3-manifolds.\nAt the moment, Khovanov homology is only de-\nfined in a combinatorial, concrete way for links in $S^{3}$, interval bundles\nover surfaces [APS04], $\\mathbb{R}$ P $^{3}$ [Man07, Gab13], and connected sums of\n$S^{1} \\times S^{2}$ [Roz10, Wil21]. If indeed the Haydys-Kapustin-Witten pro-\ngram can be carried through, it would also be important to compare\nthe result with the Khovanov lasagna skein modules of Morrison-Walker-\nWedrich [MWW22].\n(5) Mathematicians have made progress on understanding the analytic prop-\nerties of the KW and HW equations; see [MW14], [MW20], [Tau13],\n[LT20], [He19], [HM19a], [Tau18], [Tau19].\nSome of the compact-\nness results (e.g. [Tau18, Theorem B]) only hold when the 3-manifold\nhas positive Ricci curvature, which suggests that part (c) should first be\nattempted under this assumption.\n\nReferences cited:\n- [Kho00] Mikhail Khovanov. A categorification of the Jones polynomial. Duke Math. J., 101(3):359–426, 2000. doi:10.1215/S0012-7094-00-10131-7.\n- [Wit12] Edward Witten. Fivebranes and knots. Quantum Topol., 3(1):1–137, 2012. doi: 10.4171/QT/26.\n- [KW07] Anton Kapustin and Edward Witten. Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys., 1(1):1–236, 2007. doi:10.4310/CNTP.2007.v1.n1.a1.\n- [GW12] Davide Gaiotto and Edward Witten. Knot invariants from four-dimensional gauge theory. Adv. Theor. Math. Phys., 16(3):935–1086, 2012. doi:10.4310/atmp.2012.v16.n3.a5.\n- [Hay15] Andriy Haydys. Fukaya-Seidel category and gauge theory. J. Symplectic Geom., 13(1):151–207, 2015. doi:10.4310/JSG.2015.v13.n1.a5.\n- [Prz91] Józef H. Przytycki. Skein modules of 3-manifolds. Bull. Polish Acad. Sci. Math., 39(1-2):91–100, 1991.\n- [Tur88] V. G. Turaev. The Conway and Kauffman modules of a solid torus. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 167(Issled. Topol. 6):79–89, 190, 1988. doi:10.1007/BF01099241.\n- [GPV17] Sergei Gukov, Pavel Putrov, and Cumrun Vafa. Fivebranes and 3-manifold homology. J. High Energy Phys., 2017(7):071, front matter+80, 2017. doi:10.1007/JHEP07(2017)071.\n- [GPPV20] Sergei Gukov, Du Pei, Pavel Putrov, and Cumrun Vafa. BPS spectra and 3-manifold invariants. J. Knot Theory Ramifications, 29(2):2040003, 85, 2020. doi:10.1142/S0218216520400039.\n- [APS04] Marta M. Asaeda, Józef H. Przytycki, and Adam S. Sikora. Categorification of the Kauffman bracket skein module of I-bundles over surfaces. Algebr. Geom. Topol., 4:1177–1210, 2004. doi:10.2140/agt.2004.4.1177.\n- [Man07] Vassily Olegovich Manturov. Khovanov homology for virtual knots with arbitrary coefficients. J. Knot Theory Ramifications, 16(3):345–377, 2007. doi:10.1142/S0218216507005336.\n- [Gab13] Boštjan Gabrovšek. The categorification of the Kauffman bracket Skein module of $\\mathbb{RP}^{3}$. Bull. Aust. Math. Soc., 88(3):407–422, 2013. doi:10.1017/S0004972713000105.\n- [Roz10] Lev Rozansky. A categorification of the stable SU(2) Witten-Reshetikhin-Turaev invariant of links in $S^{2}$ $\\times$ $S^{1}$, 2010. arXiv:1011.1958.\n- [Wil21] Michael Willis. Khovanov homology for links in \\#rp$S^{2}$ $\\times$ $S^{1}$q. Michigan Math. J., 70(4):675–748, 2021. doi:10.1307/mmj/1594281620.\n- [MWW22] Scott Morrison, Kevin Walker, and Paul Wedrich. Invariants of 4-manifolds from Khovanov-Rozansky link homology. Geom. Topol., 26(8):3367–3420, 2022. doi:10.2140/gt.2022.26.3367.\n- [MW14] Rafe Mazzeo and Edward Witten. The Nahm pole boundary condition. In The influence of Solomon Lefschetz in geometry and topology, volume 621 of Contemp. Math., pages 171–226. Amer. Math. Soc., Providence, RI, 2014. doi:10.1090/conm/621/12422.\n- [MW20] Rafe Mazzeo and Edward Witten. The KW equations and the Nahm pole boundary condition with knots. Comm. Anal. Geom., 28(4):871–942, 2020. doi:10.4310/CAG.2020.v28.n4.a4.\n- [Tau13] Clifford Henry Taubes. Compactness theorems for SL(2;C) generalizations of the 4-dimensional anti-self dual equations, 2013. arXiv:1307.6447.\n- [LT20] Naichung Conan Leung and Ryosuke Takahashi. Energy bound for Kapustin-Witten solutions on $S^{3}$ $\\times$ R+. Int. Math. Res. Not. IMRN, 2020(19):6135–6148, 2020. doi: 10.1093/imrn/rny198.\n- [He19] Siqi He. A gluing theorem for the Kapustin-Witten equations with a Nahm pole. J. Topol., 12(3):855–915, 2019. doi:10.1112/topo.12102.\n- [HM19a] Siqi He and Rafe Mazzeo. The extended Bogomolny equations and generalized Nahm pole boundary condition. Geom. Topol., 23(5):2475–2517, 2019. doi:10.2140/gt.2019.23.2475.\n- [Tau18] Clifford Henry Taubes. Sequences of Nahm pole solutions to the SU(2) KapustinWitten equations, 2018. arXiv:1805.02773.\n- [Tau19] Clifford Henry Taubes. The R invariant solutions to the Kapustin Witten equations on $(0,8)$ $\\times$ $\\mathbb{R}^{2}$ $\\times$ R with generalized Nahm pole asymptotics, 2019. arXiv:1903.03539.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Kapustin--Witten gauge-theoretic construction remains a proposed route to Khovanov/Jones invariants rather than a completed general counting theorem.\n\n**Verified partial progress.**\n\n- Witten and Gaiotto--Witten formulated/sketched the relevant gauge-theoretic programme.\n\n**Full solution or refutation.**\n\nNo complete analytic construction matching all requested properties was verified.\n\n**What remains.**\n\nConstruct and analyze the moduli/counting theory with Nahm-pole/link boundary conditions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.32 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains the proposed gauge-theoretic interpretation and outstanding analytic work.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2692, "problem_number": "KP-1.33", "title": "Kirby Problem 1.33", "statement": "Describe topological necessary or sufficient conditions for a\nlink to have KR-parity. For example:\n(a) Are all links with KR-parity positive? Quasipositive?\n(b) Do all algebraic links have KR-parity?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.33.\n\nLiterature notes:\n(1) We say that a link has KR-parity if its triply graded Khovanov–Rozansky\nhomology is concentrated in homological degrees of the same parity. The\nmotivation for this property comes from the work of Hogancamp and\nElias [EH19], who developed a recursive procedure for computing the\nKhovanov–Rozansky homology of some links; their procedure relies on\nthe link having KR-parity.\n(2) By [Mel22, HM19b] all torus links have KR-parity.\n(3) If a link has KR-parity, then all coefficients in the HOMFLY polynomial\n$P(a, q)$ should have the same sign (depending on conventions, one might\nneed to change the variable $a$ to $-a$).\n(4) There are examples of positive links that do not have KR-parity, for ex-\nample the knot $10_{139}$ which is the closure of the braid $\\sigma_{1}\\sigma_{2}(\\sigma_{1}\\sigma_{2}\\sigma_{2}\\sigma_{1})^{2}$.\n(5) Let $\\mathrm{FT}_{i}$ denote the full twist on the first $i$ strands.\nIt is conjectured\n[GHSR20, OR23, Tur24] that the closure of the braid\n$\\mathrm{FT}_{2}^{d_{2}}\\mathrm{FT}_{3}^{d_{3}}\\cdots \\mathrm{FT}_{n}^{d_{n}}$\nhas KR-parity as long as all $d_{i} \\geq 0$.\n\nReferences cited:\n- [EH19] Ben Elias and Matthew Hogancamp. On the computation of torus link homology. Compos. Math., 155(1):164–205, 2019. doi:10.1112/s0010437x18007571.\n- [Mel22] Anton Mellit. Homology of torus knots. Geom. Topol., 26(1):47–70, 2022. doi: 10.2140/gt.2022.26.47.\n- [HM19b] Matthew Hogancamp and Anton Mellit. Torus link homology, 2019. arXiv:1909.00418.\n- [GHSR20] Eugene Gorsky, Graham Hawkes, Anne Schilling, and Julianne Rainbolt. Generalized q, t-Catalan numbers. Algebr. Comb., 3(4):855–886, 2020. doi:10.5802/alco.120.\n- [OR23] A. Oblomkov and L. Rozansky. Homfly-PT homology of Coxeter links. Transform. Groups, 28(3):1245–1275, 2023. doi:10.1007/s00031-023-09816-1.\n- [Tur24] Joshua P. Turner. Affine Springer fibers and generalized Haiman ideals (with an appendix by Eugene Gorsky and Joshua P. Turner). Int. Math. Res. Not. IMRN, 2024(16):11878–11909, 2024. doi:10.1093/imrn/rnae146.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** All torus links have KR-parity, but not all positive links do, so one proposed sufficient-condition direction is refuted and a full characterization remains open.\n\n**Verified partial progress.**\n\n- Mellit and Hogancamp--Mellit establish KR-parity for torus links.\n- The positive knot 10_139 is recorded as a counterexample to positivity implying KR-parity.\n\n**Full solution or refutation.**\n\nNo topological necessary-and-sufficient criterion is known.\n\n**What remains.**\n\nCharacterize KR-parity and settle remaining quasipositive/algebraic subquestions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.33 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records torus-link theorem and positive-link counterexample.\n\n**Review notes.** A subquestion is refuted; whole multi-part record is partial.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2693, "problem_number": "KP-1.34", "title": "Kirby Problem 1.34", "statement": "(a) Khovanov and Rozansky [KR08b] used braid presentations to define a\ntriply graded link homology theory whose Euler characteristic is the HOM-\nFLYPT polynomial. Find a description of this homology in terms of arbi-\ntrary planar diagrams of the link, and determine whether some flavor of\nthis theory is functorial with respect to link cobordisms.\n(b) Do the same for Cautis’ link homologies from [Cau17], which are also\ndefined in terms of braid presentations.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.34.\n\nLiterature notes:\n(1) There are several versions of the HOMFLYPT link homology in [KR08b]:\nunreduced, middle, reduced, and totally reduced; see [Ras15]. Reduction\nrefers to choosing a basepoint and taking a mapping cone in the chain\ncomplex, which results in a smaller homology.\nOnly the totally reduced theory (where a basepoint is chosen on each\nlink component) has a chance at functoriality. For the original unreduced\ntheory, the homology of the unknot is the product of the exterior and\npolynomial algebras on one generator each, and this is infinite-dimensional\nover the ground field (i.e., over the homology of the empty link); thus, it\ncannot admit a functorial extension unless suitably modified. The totally\nreduced theory assigns finite-dimensional homology groups to all links and\nhas a chance at being functorial for link cobordisms (decorated by paths).\nA similar kind of functoriality appears in knot Floer homology [Zem19b].\n\n(2) The definition of HOMFLYPT homology in [KR08b] starts with a com-\nplex associated to a braid diagram. If one constructs the same complex\nfrom an arbitrary planar diagram, the resulting homology does not behave\nwell with respect to the Reidemeister II(b) move; see [Web07, Section 3.1]\nand [Abe17].\n(3) HOMFLYPT homology can also be interpreted as the Hochschild homol-\nogy of the Rouquier complexes formed from Soergel bimodules [Kho07].\nFunctoriality of Rouquier complexes under braid cobordisms was proved\nin [EK10].\n(4) In [Cau17], Cautis used categorical $\\mathfrak{s}\\mathfrak{l}_{n}$ actions to define triply-graded link\ninvariants that categorify the HOMFLYPT polynomial of links colored by\narbitrary partitions. He also defined a finite dimensional categorification\nof the $\\mathfrak{s}\\mathfrak{l}(n)$ link polynomial colored by symmetric powers of the standard\nrepresentation. See also [QRS18] for a combinatorial definition of Cautis’\nhomology, and [RW20] for the equivariant version.\n\nReferences cited:\n- [KR08b] Mikhail Khovanov and Lev Rozansky. Matrix factorizations and link homology. II. Geom. Topol., 12(3):1387–1425, 2008. doi:10.2140/gt.2008.12.1387.\n- [Cau17] Sabin Cautis. Remarks on coloured triply graded link invariants. Algebr. Geom. Topol., 17(6):3811–3836, 2017. doi:10.2140/agt.2017.17.3811.\n- [Ras15] Jacob Rasmussen. Some differentials on Khovanov-Rozansky homology. Geom. Topol., 19(6):3031–3104, 2015. doi:10.2140/gt.2015.19.3031.\n- [Zem19b] Ian Zemke. Link cobordisms and functoriality in link Floer homology. J. Topol., 12(1):94–220, 2019. doi:10.1112/topo.12085.\n- [Web07] Ben Webster. Khovanov-Rozansky homology via a canopolis formalism. Algebr. Geom. Topol., 7:673–699, 2007. doi:10.2140/agt.2007.7.673.\n- [Abe17] Michael Abel. HOMFLY-PT homology for general link diagrams and braidlike isotopy. Algebr. Geom. Topol., 17(5):3021–3056, 2017. doi:10.2140/agt.2017.17.3021.\n- [Kho07] Mikhail Khovanov. Triply-graded link homology and Hochschild homology of Soergel bimodules. Internat. J. Math., 18(8):869–885, 2007. doi:10.1142/S0129167X07004400.\n- [EK10] Ben Elias and Dan Krasner. Rouquier complexes are functorial over braid cobordisms. Homology Homotopy Appl., 12(2):109–146, 2010. doi:10.4310/hha.2010.v12.n2.a4.\n- [QRS18] Hoel Queffelec, David E. V. Rose, and Antonio Sartori. Annular evaluation and link homology, 2018. arXiv:1802.04131.\n- [RW20] Louis-Hadrien Robert and Emmanuel Wagner. Symmetric Khovanov-Rozansky link homologies. J. Éc. polytech. Math., 7:573–651, 2020. doi:10.5802/jep.124.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Several HOMFLYPT homology versions and reduced theories are constructed, but the requested functorial/cobordism and structural descriptions remain incomplete.\n\n**Verified partial progress.**\n\n- Totally reduced HOMFLYPT homology is finite-dimensional and is the candidate compatible with functoriality.\n\n**Full solution or refutation.**\n\nNo full resolution of every enumerated construction question was verified.\n\n**What remains.**\n\nEstablish the desired functorial theory and resolve the stated structural conjectures.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.34 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes existing versions from the open functoriality programme.\n\n**Review notes.** Multi-part scope preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2694, "problem_number": "KP-1.35", "title": "Kirby Problem 1.35", "statement": "(a) Is symplectic Khovanov homology isomorphic to Khovanov homology, over\n$\\mathbb{Z}$?\n(b) Give a construction of odd symplectic Khovanov homology $\\mathrm{Kh}^{\\mathrm{odd}}_{\\mathrm{symp}}(K)$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.35.\n\nLiterature notes:\nSymplectic Khovanov homology, introduced in [SS06], is a can-\ndidate Floer-theoretic construction of Khovanov homology. Abouzaid and Smith\n[AS19] showed that it is isomorphic to ordinary Khovanov homology over $\\mathbb{Q}$ (or\nmore generally over fields of characteristic 0), but their proof does not work with\ninteger coefficients.\nIt is not obvious from the construction how to produce a version similarly\nrelated to odd Khovanov homology.\n\nReferences cited:\n- [SS06] Paul Seidel and Ivan Smith. A link invariant from the symplectic geometry of nilpotent slices. Duke Math. J., 134(3):453–514, 2006. doi:10.1215/S0012-7094-06-13432-4.\n- [AS19] Mohammed Abouzaid and Ivan Smith. Khovanov homology from Floer cohomology. J. Amer. Math. Soc., 32(1):1–79, 2019. doi:10.1090/jams/902.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Symplectic Khovanov homology is isomorphic to ordinary Khovanov homology over characteristic-zero fields, but the integral comparison and odd theory remain open.\n\n**Verified partial progress.**\n\n- Abouzaid--Smith prove the characteristic-zero isomorphism.\n\n**Full solution or refutation.**\n\nThis is a substantial but not integral/odd resolution.\n\n**What remains.**\n\nProve an integral isomorphism and construct/compare odd symplectic Khovanov homology.\n\n**Sources checked.**\n\n- M. Abouzaid and I. Smith, Khovanov homology from Floer cohomology, JAMS 32 (2019), 1-79. (primary): https://doi.org/10.1090/jams/902\n Evidence used: Establishes the characteristic-zero comparison.\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.35 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the remaining integral and odd questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2695, "problem_number": "KP-1.36", "title": "Kirby Problem 1.36", "statement": "Categorify the ($\\mathfrak{s}\\mathfrak{l}(2), \\mathfrak{s}\\mathfrak{l}(N)$, HOMFLYPT) skein algebras for\nsurfaces.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.36.\n\nLiterature notes:\n(1) The skein algebra of a surface, introduced by Przytycki-Sikora [PS00b]\nand Turaev [Tur91], encodes curves on the surface up to the Kauffman\nbracket skein relations.\n(2) The skein algebra of the annulus is categorified by the annular Khovanov\n(or Khovanov–Rozansky) homology [GW10, GLW18, QR18].\n(3) There are several constructions of Khovanov-type invariants of links in\nthickened surfaces [APS04, QW21]. However, it is not known if any of\nthese can support a categorical analogue of the skein product.\n\n(4) The HOMFLYPT skein algebra of the torus is closely related to the elliptic\nHall algebra [MS17b]. Categorification of the elliptic Hall algebra is an\nimportant problem in geometric representation theory [Neg22].\n\nReferences cited:\n- [PS00b] Józef H. Przytycki and Adam S. Sikora. On skein algebras and Sl2pCq-character varieties. Topology, 39(1):115–148, 2000. doi:10.1016/S0040-9383(98)00062-7.\n- [Tur91] Vladimir G. Turaev. Skein quantization of Poisson algebras of loops on surfaces. Ann. Sci. École Norm. Sup. (4), 24(6):635–704, 1991. URL: http://www.numdam.org/item?id=ASENS 1991 4 24 6 635 0.\n- [GW10] J. Elisenda Grigsby and Stephan M. Wehrli. Khovanov homology, sutured Floer homology and annular links. Algebr. Geom. Topol., 10(4):2009–2039, 2010. doi: 10.2140/agt.2010.10.2009.\n- [GLW18] J. Elisenda Grigsby, Anthony M. Licata, and Stephan M. Wehrli. Annular Khovanov homology and knotted Schur-Weyl representations. Compos. Math., 154(3):459– 502, 2018. doi:10.1112/S0010437X17007540.\n- [QR18] Hoel Queffelec and David E. V. Rose. Sutured annular Khovanov-Rozansky homology. Trans. Amer. Math. Soc., 370(2):1285–1319, 2018. doi:10.1090/tran/7117.\n- [APS04] Marta M. Asaeda, Józef H. Przytycki, and Adam S. Sikora. Categorification of the Kauffman bracket skein module of I-bundles over surfaces. Algebr. Geom. Topol., 4:1177–1210, 2004. doi:10.2140/agt.2004.4.1177.\n- [QW21] Hoel Queffelec and Paul Wedrich. Khovanov homology and categorification of skein modules. Quantum Topol., 12(1):129–209, 2021. doi:10.4171/qt/148.\n- [MS17b] Hugh Morton and Peter Samuelson. The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra. Duke Math. J., 166(5):801–854, 2017. doi:10.1215/00127094-3718881.\n- [Neg22] Andrei Neguţ. Hecke correspondences for smooth moduli spaces of sheaves. Publ. Math. Inst. Hautes Études Sci., 135:337–418, 2022. doi:10.1007/s10240-022-00131-1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Annular and thickened-surface Khovanov-type invariants categorify pieces of the picture, but no construction is known to support a categorical skein product for general surfaces.\n\n**Verified partial progress.**\n\n- The annular skein algebra is categorified by annular Khovanov/Khovanov--Rozansky theories.\n- Several Khovanov-type surface-link invariants exist.\n\n**Full solution or refutation.**\n\nThe requested surface-skein-algebra categorification remains open.\n\n**What remains.**\n\nConstruct a functorial categorical product realizing the skein algebra relations.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.36 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records existing surface theories and the missing categorical product.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2696, "problem_number": "KP-1.37", "title": "Kirby Problem 1.37", "statement": "(a) For every link $L \\subset \\mathbb{R}^{3}$, every simple Lie algebra $\\mathfrak{g}$, and every coloring\nof the components of $L$ with irreducible representations of $\\mathfrak{g}$, construct\na bigraded link homology theory that is functorial under link cobordisms,\nand whose Euler characteristic is the Reshetikhin–Turaev link invariant\nassociated to $\\mathfrak{g}$ and $L$.\n(b) For each of the theories above, define an odd version $($ with the same mod\n2 reduction $)$ or explain the obstruction to its existence.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.37.\n\nLiterature notes:\n(1) Link homologies corresponding to $\\mathfrak{s}\\mathfrak{l}(n)$ with its standard vector represen-\ntation were constructed by Khovanov [Kho00] for $n = 2$ and Khovanov–\nRozansky [KR08a] for all $n$. Generalizations to exterior products of the\nstandard representations of $\\mathfrak{s}\\mathfrak{l}(n)$ were constructed in [Wu14], [Yon11],\n[QR16], and were proved to be functorial under link cobordisms in [ETW18].\n(2) Webster [Web17] has a categorification of the Reshetikhin–Turaev link\ninvariants for any simple Lie algebra $\\mathfrak{g}$ and any labeling of the link com-\nponents by irreducible representations of $\\mathfrak{g}$. His homology is bigraded.\nFor most $\\mathfrak{g}$ and labelings by irreducible representations, his homology of\nthe unknot is infinite-dimensional over the ground field and thus cannot\nextend to a functorial link homology theory.\nSuch a theory should at\nleast assign a commutative Frobenius algebra to the unknot, necessarily\nfinite-dimensional over the ground field. (A naive guess is that Webster\nhomology extends to a functorial theory, possibly up to overall scaling, if\nand only if all representations are minuscule.) It is also possible that a\ngood functorial theory would be more complicated algebraically, e.g. be\ndefined over a dg ring, or have more interesting gradings.\n(3) The odd version of Khovanov homology was defined by Ozsváth, Ras-\nmussen, and Szabó in [ORS13]; it is isomorphic to Khovanov homology\nover $\\mathbb{Z}/2$ but different over $\\mathbb{Z}$. The odd version appears naturally in re-\nlation to Floer homologies of the double branched cover of the link; see\n[OS05c], [Sca15]. Its functoriality (up to sign) was studied by Migdail\nand Wehrli [MW24].\n(4) Mikhaylov and Witten [MW15] studied odd Khovanov homology from\nthe perspective of physics. They argued that there should be odd versions\nfor the link homologies associated to $\\mathfrak{s}\\mathfrak{o}(2n + 1)$, but not for $\\mathfrak{s}\\mathfrak{u}(n)$; except\nfor $n = 2$, when $\\mathfrak{s}\\mathfrak{u}(2) = \\mathfrak{s}\\mathfrak{o}(3)$. The odd link homologies for $\\mathfrak{s}\\mathfrak{o}(2n+1)$ for\n$n \\geq 2$ have not yet been constructed mathematically.\n(5) Ellis and Lauda [EL16] gave an odd categorification of quantum $\\mathfrak{s}\\mathfrak{l}(2)$,\nusing a covering Kac-Moody algebra. They pointed out that the theory\nof covering Kac-Moody algebras only exists in finite type for $\\mathfrak{s}\\mathfrak{o}(2n + 1)$.\n\n(6) Khovanov, Putyra and Vaz [KPV24] constructed odd analogues of So-\nergel bimodules and Rouquier complexes, and pointed out an obstruction\nto defining an odd HOMFLYPT homology.\n\nReferences cited:\n- [Kho00] Mikhail Khovanov. A categorification of the Jones polynomial. Duke Math. J., 101(3):359–426, 2000. doi:10.1215/S0012-7094-00-10131-7.\n- [KR08a] Mikhail Khovanov and Lev Rozansky. Matrix factorizations and link homology. Fund. Math., 199(1):1–91, 2008. doi:10.4064/fm199-1-1.\n- [Wu14] Hao Wu. A colored $\\mathfrak{sl}(N)$ homology for links in $S^{3}$. Dissertationes Math., 499:217, 2014. doi:10.4064/dm499-0-1.\n- [Yon11] Yasuyoshi Yonezawa. Quantum $\\mathfrak{sl}(n), \\wedge^n$ link invariant and matrix factorizations. Nagoya Math. J., 204:69–123, 2011. doi:10.1215/00277630-1431840.\n- [QR16] Hoel Queffelec and David E. V. Rose. The sln foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality. Adv. Math., 302:1251–1339, 2016. doi:10.1016/j.aim.2016.07.027.\n- [ETW18] Michael Ehrig, Daniel Tubbenhauer, and Paul Wedrich. Functoriality of colored link homologies. Proc. Lond. Math. Soc. (3), 117(5):996–1040, 2018. doi:10.1112/plms.12154.\n- [Web17] Ben Webster. Knot invariants and higher representation theory. Mem. Amer. Math. Soc., 250(1191):v+141, 2017. doi:10.1090/memo/1191.\n- [ORS13] Peter Ozsváth, Jacob Rasmussen, and Zoltán Szabó. Odd Khovanov homology. Algebr. Geom. Topol., 13(3):1465–1488, 2013. doi:10.2140/agt.2013.13.1465.\n- [OS05c] Peter Ozsváth and Zoltán Szabó. On the Heegaard Floer homology of branched double-covers. Adv. Math., 194(1):1–33, 2005. doi:10.1016/j.aim.2004.05.008.\n- [Sca15] Christopher W. Scaduto. Instantons and odd Khovanov homology. J. Topol., 8(3):744–810, 2015. doi:10.1112/jtopol/jtv012.\n- [MW24] Jacob Migdail and Stephan Wehrli. Functoriality of odd and generalized Khovanov homology in $\\mathbb{R}^{3}$ $\\times$ I, 2024. arXiv:2206.14710.\n- [MW15] Victor Mikhaylov and Edward Witten. Branes and supergroups. Comm. Math. Phys., 340(2):699–832, 2015. doi:10.1007/s00220-015-2449-y.\n- [EL16] Alexander P. Ellis and Aaron D. Lauda. An odd categorification of $U_q(\\mathfrak{sl}_2)$. Quantum Topol., 7(2):329–433, 2016. URL: https://doi-org.stanford.idm.oclc.org/10.4171/QT/78, doi:10.4171/QT/78.\n- [KPV24] Mikhail Khovanov, Krzysztof Putyra, and Pedro Vaz. Odd two-variable Soergel bimodules and Rouquier complexes. In Algebraic and topological aspects of representation theory, volume 791 of Contemp. Math., pages 205–227. Amer. Math. Soc., [Providence], RI, [2024] ©2024. URL: https://doi-org.stanford.idm.oclc.org/10.1090/conm/791/15876, doi:10.1090/conm/791/15876.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Bigraded Reshetikhin--Turaev categorifications exist for broad classes, including all simple Lie algebras in Webster's construction, but functorial finite-dimensional theories and odd variants are not known in the stated generality.\n\n**Verified partial progress.**\n\n- sl(n) standard/exterior-label theories have constructions and cobordism functoriality in major cases.\n- Webster constructs bigraded categorifications for any simple Lie algebra and irreducible labels.\n\n**Full solution or refutation.**\n\nThe general functoriality/odd-theory request remains open.\n\n**What remains.**\n\nIdentify and construct the finite-dimensional Frobenius-compatible functorial theories in full generality.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.37 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States existing constructions and the functoriality obstruction.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2697, "problem_number": "KP-1.38", "title": "Kirby Problem 1.38", "statement": "What is the structure of the smooth knot concordance group?\n(a) Is there a torsion element of the smooth concordance group $\\mathcal{C}$ having order\nother than two? Is there a torsion element in $\\mathcal{C}_{TS}$, the smooth concordance\ngroup of topologically slice knots, having order other than two?\n(b) Is there a knot $K$ having concordance order two in $\\mathcal{C}$ that is not concordant\nto a strongly negatively amphichiral knot?\n(c) Do there exist infinitely divisible elements in $\\mathcal{C}$, that is, does there exist $K$\nsuch that for infinitely many $n \\in \\mathbb{N}$, there is a $J$ such that $[K] = n[J]$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.38.\n\nLiterature notes:\n(1) This is [Kir97, Problem 1.32]. Fox and Milnor’s original work on the\nconcordance group shows there exist elements of order two [FM66]; Hed-\nden, Kim, and Livingston [HKL16a] show the existence of an infinite\nsubgroup of topologically slice knots of order two.\nMany elements of\ninfinite order are known; for example, $\\mathcal{C}_{TS}$ contains subgroup isomor-\nphic to $\\mathbb{Z}_{\\infty}$ [End95], and in fact a direct summand isomorphic to $\\mathbb{Z}_{\\infty}$\n[DHST21, OSS17b].\n(2) Any infinitely divisible element of $\\mathcal{C}$ must be sent to zero under any integer-\nvalued homomorphism from the concordance group, such as the Ozsváth-\nSzabó $\\tau$-invariant or the Rasmussen $s$-invariant.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [FM66] Ralph H. Fox and John W. Milnor. Singularities of 2-spheres in 4-space and cobordism of knots. Osaka Math. J., 3:257–267, 1966. http://projecteuclid.org/euclid.ojm/1200691730.\n- [HKL16a] Matthew Hedden, Se-Goo Kim, and Charles Livingston. Topologically slice knots of smooth concordance order two. J. Differential Geom., 102(3):353–393, 2016. http://projecteuclid.org/euclid.jdg/1456754013.\n- [End95] Hisaaki Endo. Linear independence of topologically slice knots in the smooth cobordism group. Topology Appl., 63(3):257–262, 1995. doi:10.1016/0166-8641(94) 00062-8.\n- [DHST21] Irving Dai, Jennifer Hom, Matthew Stoffregen, and Linh Truong. More concordance homomorphisms from knot Floer homology. Geom. Topol., 25(1):275–338, 2021. doi:10.2140/gt.2021.25.275.\n- [OSS17b] Peter Ozsváth, András I. Stipsicz, and Zoltán Szabó. Concordance homomorphisms from knot Floer homology. Adv. Math., 315:366–426, 2017. doi:10.1016/j.aim.2017.05.017.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Order-two and infinite-order concordance classes, including large topologically slice subgroups, are known; higher torsion and the listed structural questions remain open.\n\n**Verified partial progress.**\n\n- Fox--Milnor give order-two elements.\n- Hedden--Kim--Livingston give an infinite topologically slice order-two subgroup.\n- Infinite-rank/direct-summand results are known for topologically slice concordance.\n\n**Full solution or refutation.**\n\nNo torsion of order other than two was verified.\n\n**What remains.**\n\nResolve all higher-torsion, amphichirality, and divisibility questions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.38 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintained list records order-two and infinite-rank progress and open questions.\n\n**Review notes.** Multi-part record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2698, "problem_number": "KP-1.39", "title": "Kirby Problem 1.39", "statement": "(a) Do the algebraic knots freely generate a subgroup of the smooth concor-\ndance group $\\mathcal{C}$?\n(b) Do the algebraic knots freely generate a subgroup of the topological con-\ncordance group $\\mathcal{C}^{\\mathrm{TOP}}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.39.\n\nLiterature notes:\n(1) This question is originally due to Rudolph [Rud76], although it is not\nspecified whether $\\mathcal{C}$ or $\\mathcal{C}^{\\mathrm{TOP}}$ is intended. Litherland showed that torus\nknots are independent in the smooth concordance group [Lit79]; his proof\nyields the topological version as well. Litherland’s theorem uses Levine-\nTristram signatures, but various authors have shown that there exist con-\nnected sums of algebraic knots and their mirrors that are algebraically\nslice but not smoothly slice [LM83, HKL12] implying that Litherland’s\nstrategy is not sufficient to answer the question above.\n(2) This problem is especially interesting in light of its connection to the slice-\nribbon conjecture (Problem 1.50): a result of Miyazaki shows that non-\ntrivial linear combinations of iterated torus knots are not ribbon [Miy94,\nCorollary 8.4], so the slice-ribbon conjecture implies Rudolph’s conjec-\nture.\nIndeed, Baker [Bak16] and Abe-Tagami [AT16a] observed that\nthe slice-ribbon conjecture implies a stronger form of Rudolph’s conjec-\nture, to wit, that the set of prime fibered strongly quasi-positive knots is\nlinearly independent in the smooth concordance group. Large families of\nknots satisfying both Rudolph’s conjecture and this strengthening were\nexhibited by Conway, Kim, and Politarczyk [CKP23].\n(3) In a related direction, Baker showed that the slice-ribbon conjecture im-\nplies that if two fibered knots supporting the tight contact structure on\n$S^{3}$ are concordant, then they are in fact isotopic [Bak16, Corollary 4].\nAbe and Tagami observed that it would then follow that the set of fibered\nknots supporting the tight contact structure on $S^{3}$ is linearly indepen-\ndent in the smooth concordance group and contains the algebraic knots\n[AT16a, Observation 1.3]. This can be thought of as a generalization of\nRudolph’s question.\n\nReferences cited:\n- [Rud76] Lee Rudolph. How independent are the knot-cobordism classes of links of plane curve singularities? Notices Amer. Math. Soc, 23:410, 1976.\n- [Lit79] R. A. Litherland. Signatures of iterated torus knots. In Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), volume 722 of Lecture Notes in Math., pages 71–84. Springer, Berlin, 1979.\n- [LM83] Charles Livingston and Paul Melvin. Algebraic knots are algebraically dependent. Proc. Amer. Math. Soc., 87(1):179–180, 1983. doi:10.2307/2044377.\n- [HKL12] Matthew Hedden, Paul Kirk, and Charles Livingston. Non-slice linear combinations of algebraic knots. J. Eur. Math. Soc. (JEMS), 14(4):1181–1208, 2012. doi:10.4171/JEMS/330.\n- [Miy94] Katura Miyazaki. Nonsimple, ribbon fibered knots. Trans. Amer. Math. Soc., 341(1):1–44, 1994. doi:10.2307/2154613.\n- [Bak16] Kenneth L. Baker. A note on the concordance of fibered knots. J. Topol., 9(1):1–4, 2016. doi:10.1112/jtopol/jtv024.\n- [AT16a] Tetsuya Abe and Keiji Tagami. Fibered knots with the same 0-surgery and the slice-ribbon conjecture. Math. Res. Lett., 23(2):303–323, 2016. doi:10.4310/MRL.2016.v23.n2.a1.\n- [CKP23] Anthony Conway, Min Hoon Kim, and Wojciech Politarczyk. Nonslice linear combinations of iterated torus knots. Algebr. Geom. Topol., 23(2):765–802, 2023. doi:10.2140/agt.2023.23.765.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Torus knots are independent in both smooth and topological concordance, but free generation by all algebraic knots remains open.\n\n**Verified partial progress.**\n\n- Litherland proves independence of torus knots in smooth concordance, with a topological consequence.\n- Miyazaki's ribbon result links a stronger conclusion to slice-ribbon.\n\n**Full solution or refutation.**\n\nNo full algebraic-knot independence theorem was verified.\n\n**What remains.**\n\nProve or refute free generation by algebraic knots in each concordance category.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.39 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the torus-knot theorem and the remaining general conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2699, "problem_number": "KP-1.40", "title": "Kirby Problem 1.40", "statement": "A satellite operator $P \\subset S^{1} \\times D^{2}$ induces an operation $P$ on\nthe concordance group $\\mathcal{C}$ [Gor75].\n(a) Let $P$ be a winding number one satellite operator with $P(\\operatorname{U}) \\simeq \\operatorname{U}$.\nIf\n$P(K) \\simeq \\operatorname{U}$, does this mean that $K \\simeq \\operatorname{U}$?\n(b) Is the Whitehead doubling operator injective on $\\mathcal{C}$? Can we exhibit any\n(winding-number-zero) operator that is injective on $\\mathcal{C}$?\n(c) Conjecture (Hedden; see [HPC21]). The only homomorphisms on $\\mathcal{C}$ in-\nduced by satellite operators are the zero map, the identity, and the invo-\nlution induced by orientation reversal.\n(d) Conjecture (Hedden, Pinzón Caicedo) Let $P$ be a non-constant satellite\noperator with winding number zero. Does $\\\\langle \\\\operatorname{im} P \\\\rangle$ necessarily have infinite\nrank? Some special cases are:\n(i) Does there always exist a fixed knot $K$ such that $\\langle\\{P(nK)\\}_{n\\in\\mathbb{Z}}\\rangle$ has\ninfinite rank?\n(ii) If $\\{nK\\}_{n\\in\\mathbb{Z}}$ is any rank-one subgroup of $\\mathcal{C}$, does $\\langle\\{P(nK)\\}_{n\\in\\mathbb{Z}}\\rangle$ have\ninfinite rank?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.40.\n\nLiterature notes:\n(1) Satellite operators are defined in the Introduction to this section.\n(2) The overall conjecture (d) is easily verified in the case of non-zero winding\nnumber. It was shown in [HK12b] that Whitehead doubling has infinite\nrank, with many other patterns treated in [HPC21]. The authors subse-\nquently posed the general conjecture, as well as subconjecture (i).\n(3) In a positive answer to [DHMS24], subconjecture (i) was announced\nfor the Whitehead doubling operation. Indeed, $K$ was taken to be the\n(right-handed) trefoil; this was previously unknown. The authors gave\na general class of patterns (as well as a flexible Floer-theoretic condi-\ntion on $K$) for subconjecture (i) to hold; they subsequently made the\nstronger subconjecture (ii). Roughly speaking, the latter indicates that\nany (non-constant, winding number zero) pattern expands rank in the\nknot concordance group.\n\nReferences cited:\n- [Gor75] C. McA. Gordon. Knots, homology spheres, and contractible 4-manifolds. Topology, 14:151–172, 1975. doi:10.1016/0040-9383(75)90024-5.\n- [HPC21] Matthew Hedden and Juanita Pinzón-Caicedo. Satellites of infinite rank in the smooth concordance group. Invent. Math., 225(1):131–157, 2021. doi:10.1007/s00222-020-01026-w.\n- [HK12b] Matthew Hedden and Paul Kirk. Instantons, concordance, and Whitehead doubling. J. Differential Geom., 91(2):281–319, 2012. URL: http://projecteuclid.org/euclid.jdg/1344430825.\n- [DHMS24] Irving Dai, Matthew Hedden, Abhishek Mallick, and Matthew Stoffregen. Rankexpanding satellites, Whitehead doubles, and Heegaard Floer homology. J. Topol., 17(4):Paper No. e70008, 40, 2024. doi:10.1112/topo.70008.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Nonzero-winding satellite operators satisfy the broad rank conjecture and many winding-zero patterns, including Whitehead doubling, have infinite-rank results; injectivity remains open.\n\n**Verified partial progress.**\n\n- Whitehead doubling has infinite rank.\n- The K3 notes report an announced positive result for a Whitehead-doubling subconjecture and wider pattern classes.\n\n**Full solution or refutation.**\n\nNo proof of injectivity for Whitehead doubling or a general winding-zero injective operator was verified.\n\n**What remains.**\n\nResolve the injectivity questions and the full rank-expansion conjecture.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.40 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintained list distinguishes known rank results and announced/open subconjectures.\n\n**Review notes.** Announcement is not treated as final resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2700, "problem_number": "KP-1.41", "title": "Kirby Problem 1.41", "statement": "This problem is concerned with the stable 4-genus $g_{s}(K)$ of a\nknot $K$, defined below.\n(a) Is there a knot $K$ such that $g_{s}(K) \\in \\mathbb{Q}\\setminus\\mathbb{Z}$?\n(b) Is there a knot $K$ such that $g_{s}(K) \\notin \\mathbb{Q}$?\n(c) Is there a knot $K$ such that $0 < g_{s}(K) < 1/2$?\n(d) [Liv10, Question 1] Does $K$ represent a torsion element in the concor-\ndance group if and only if $g_{s}(K) = 0$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.41.\n\nLiterature notes:\n(1) The stable 4-genus of a knot $K$ is defined by\n\n$$\ng_s(K)=\\lim_{n\\to\\infty}\\frac{g_4(nK)}{n}.\n$$\n\n(2) One instance under which the stable 4-genus $g_{s}(K)$ of a knot $K$ may\nbecome strictly less than $g_{4}(K)$, is when for some $m \\geq 2$, the knot $mK$\nhas $g_{4}(mK) < mg_{4}(K)$, for in this case\n\n$$\ng_s(K)=\\lim_{\\ell\\to\\infty}\\frac{g_4(\\ell mK)}{\\ell m}\n\\leq \\lim_{\\ell\\to\\infty}\\frac{\\ell g_4(mK)}{\\ell m}\n=\\frac{g_4(mK)}{m} 0$, there exists a knot $K_{\\epsilon}$ with\nstable 4-genus near $\\frac{1}{2}$, in the sense of this double inequality:\n\n$$\n\\frac{1-\\epsilon}{2}\\leq g_s(K_{\\epsilon})\\leq \\frac{1}{2}.\n$$\n\nSimilarly, in [Ilt22] it is shown that if $K_{n}$ denotes the twist knot with $n$\nfull twists, where $n$ is such that the Pell equation $x^{2} - (4n + 1)y^{2} = -1$\nhas integral solutions in $x$ and $y$, then\n\n$$\n\\frac{1}{2}-\\frac{6}{2n+7}\\leq g_s(K_n)\\leq \\frac{1}{2}.\n$$\n\nThe above bounds can be interpreted as suggesting that $\\frac{1}{2}$ may have a\nspecial significance with regards to the stable 4-genus, motivating part (c).\n(4) The lower bound of $0 < g_{s}(K)$ is required in part (c) of the question to\nexclude knots $K$ that represent torsion elements in the concordance group;\nany such knot will have stable 4-genus equal to 0. This leads to part (d),\nwhich suggests a connection with torsion in the concordance group (See\nProblem 1.38).\n\nReferences cited:\n- [Liv10] Charles Livingston. The stable 4-genus of knots. Algebr. Geom. Topol., 10(4):2191– 2202, 2010. doi:10.2140/agt.2010.10.2191.\n- [Ilt22] Damian Iltgen. A lower bound on the stable 4-genus of knots. Algebr. Geom. Topol., 22(5):2239–2265, 2022. doi:10.2140/agt.2022.22.2239.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Stable 4-genus can be strictly smaller than 4-genus and can approach 1/2 from above, but rational/nonrational values and the torsion criterion remain open.\n\n**Verified partial progress.**\n\n- Livingston exhibits families with g4(K)=1 and g4(2K)<=1.\n- For every epsilon>0 examples have stable genus near 1/2.\n\n**Full solution or refutation.**\n\nNone of the exotic-value questions is resolved by these bounds.\n\n**What remains.**\n\nFind rational nonintegral, irrational, or sub-half positive stable genus, and settle the zero/torsion equivalence.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.41 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records strict-drop examples and remaining value questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2701, "problem_number": "KP-1.42", "title": "Kirby Problem 1.42", "statement": "Do there exist algebraically concordant Seifert forms $V_{1}$ and\n$V_{2}$ for which there do not exist concordant knots $K_{1}$ and $K_{2}$ with Seifert forms $V_{1}$\nand $V_{2}$, respectively?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.42.\n\nLiterature notes:\n(1) Seifert forms $V_{1}$ and $V_{2}$ are algebraically concordant if $V_{1} \\oplus -V_{2}$ is meta-\nbolic. This problem is open regardless of whether one works with topo-\nlogical or smooth concordance of knots.\n\n(2) As an example, the Seifert forms\n\n$$\nV_1=\\begin{pmatrix}3&2\\\\1&3\\end{pmatrix}\n\\qquad\\text{and}\\qquad\nV_2=\\begin{pmatrix}1&2\\\\1&9\\end{pmatrix}\n$$\n\nare algebraically concordant: the set of vectors $\\{(0, -3, 6, -1), (2, -1, 0, 1)\\}$\nis a basis for a metabolizer of $V_{1} \\oplus -V_{2}$. Does there exist a pair of con-\ncordant knots $K_{1}$ and $K_{2}$ having these Seifert forms? Conjecture 1.11\nof [Liv01] posits that any knots $K_{1}$ and $K_{2}$ having as Seifert forms these\nparticular $V_{1}$ and $V_{2}$, respectively, cannot be concordant.\n\nReferences cited:\n- [Liv01] Charles Livingston. Examples in concordance, 2001. arXiv:math/0101035v1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The existence question is explicitly open for both smooth and topological concordance.\n\n**Verified partial progress.**\n\n- Specific algebraically concordant Seifert-form pairs are proposed test cases.\n\n**Full solution or refutation.**\n\nNo counterexample or realization theorem was verified.\n\n**What remains.**\n\nResolve the general question or the displayed Livingston test pair.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.42 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explicitly says the problem is open in both categories.\n\n**Review notes.** Open is source-backed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2702, "problem_number": "KP-1.43", "title": "Kirby Problem 1.43", "statement": "Does knot Floer homology give a categorification of the Fox–\nMilnor condition?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.43.\n\nLiterature notes:\n(1) Fox and Milnor [FM66] prove that if $K$ is slice in $B^{4}$, then the Alexander\npolynomial factors as\n\n$$\n\\Delta_{K}(t) = f(t)f(t^{-1}).\n$$\n\nGiven that the Euler characteristic of knot Floer homology is the Alexan-\nder polynomial, one might expect that the knot Floer homology of a slice\nknot $K$ factors as\n\n$$\n\\widehat{\\mathrm{HFK}}(K) \\simeq V \\otimes V^{*}\n$$\n\nwhere $V$ is a bigraded vector space, and $V^{*}$ denotes its dual. This is\nfalse, since the knot Floer homology of the Kinoshita-Terasaka knot has\nrank 33. One precise conjecture would be that there is a natural spectral\nsequence from such a vector space of the form $V \\otimes V^{*}$ to $\\widehat{\\mathrm{HFK}}(K)$.\n(2) Note that Gilmer [Gil84] proved that if there is a ribbon concordance\nfrom $K_{1}$ to $K_{0}$, then\n\n$$\n\\Delta_{K_1}(t) = \\Delta_{K_0}(t) \\cdot f(t)f(t^{-1}).\n$$\n\nOne could ask if this relation, or similar relations, also admit meaningful\nlifts to knot Floer homology.\n(3) One could similarly ask for categorifications of restrictions on the Alexan-\nder polynomial coming from symmetries on the three-manifold or the knot.\nFor example, the Alexander polynomial of the preimage $\\widetilde{K}$ of a knot $K$\nin its $n$-fold branched cover $\\Sigma_{n}(K)$ is\n\n$$\n\\Delta_{\\widetilde K}(t) \\doteq \\prod_{i=1}^{n} \\Delta_K(\\xi_n^i t^{1/n}),\n$$\n\nwhere $\\xi_n$ is a primitive $n$th root of unity. Similarly, let $K$ be a $q$-periodic\nknot with quotient $\\overline K$, and let $\\lambda$ be the linking number of $\\overline K$ with the\naxis of periodicity $\\overline A$. Let $\\overline L=\\overline K\\cup\\overline A$. Murasugi [Mur71] shows that the\nAlexander polynomial of $K$ is\n\n$$\n\\Delta_K(t) \\doteq \\Delta_{\\overline K}(t)\\prod_{i=1}^{q}\\Delta_{\\overline L}(t,\\xi_q^i),\n$$\n\nimplying in particular that $\\Delta_{\\overline K}(t)$ divides $\\Delta_K(t)$ and that\n\n$$\n\\Delta_K(t) \\doteq (1+t+\\cdots+t^{\\lambda-1})^{q-1}\\Delta_{\\overline K}(t) \\pmod{q}.\n$$\n\nwhere the equivalences above are up to a factor of $\\pm t^{\\pm i}$. Work of Hendricks\nshows that knot Floer homology recovers this last condition [Hen15], but\na precise categorification of the first two remains unclear.\n\nReferences cited:\n- [FM66] Ralph H. Fox and John W. Milnor. Singularities of 2-spheres in 4-space and cobordism of knots. Osaka Math. J., 3:257–267, 1966. http://projecteuclid.org/euclid.ojm/1200691730.\n- [Gil84] Patrick M. Gilmer. Ribbon concordance and a partial order on S-equivalence classes. Topology Appl., 18(2-3):313–324, 1984. doi:10.1016/0166-8641(84)90016-6.\n- [Mur71] Kunio Murasugi. On periodic knots. Comment. Math. Helv., 46:162–174, 1971. doi:10.1007/BF02566836.\n- [Hen15] Kristen Hendricks. Localization of the link Floer homology of doubly-periodic knots. J. Symplectic Geom., 13(3):545–608, 2015. doi:10.4310/JSG.2015.v13.n3.a2.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The naive tensor-factorization lift of Fox--Milnor is false, but more refined spectral-sequence/categorification questions remain open.\n\n**Verified partial progress.**\n\n- Kinoshita--Terasaka knot Floer homology has rank 33, ruling out the proposed V tensor V-dual form.\n\n**Full solution or refutation.**\n\nThe literal naive mechanism is refuted, not the broader categorification programme.\n\n**What remains.**\n\nFormulate and prove a viable knot-Floer lift of Fox--Milnor or related ribbon-concordance restrictions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.43 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records Fox--Milnor, the rank-33 obstruction, and refined open proposal.\n\n**Review notes.** Naive subproposal and broad question separated.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2703, "problem_number": "KP-1.44", "title": "Kirby Problem 1.44", "statement": "(a) If $K \\in \\mathcal{F}_{n}$ for all $n$, is $K$ topologically slice?\n(b) If $K \\in \\mathcal{T}_{n}$ for all $n$, is $K$ smoothly slice?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.44.\n\nLiterature notes:\n(1) Let $\\{\\mathcal{F}_{n}\\}_{n\\in \\frac{1}{2}\\mathbb{N}}$ denote the solvable filtration of $\\mathcal{C}$ due to Cochran-Orr-\nTeichner [COT03], and let $\\{\\mathcal{T}_{n}\\}_{n\\in\\mathbb{N}}$ denote the bipolar filtration of $\\mathcal{T}$ due\nto Cochran-Harvey-Horn [CHH13].\n(2) It is known that $\\mathcal{T} \\subseteq \\bigcap_{n} \\mathcal{F}_{n}$. The solvable filtration subsumes several\ntopological concordance invariants, e.g. a knot $K$ is in $\\mathcal{F}_{0}$ if and only if\nArf $(K) = 0$; it is in $\\mathcal{F}_{0.5}$ if and only if it is algebraically slice; and if $K$ is\nin $\\mathcal{F}_{1.5}$ then all its Casson-Gordon sliceness obstructions vanish [COT03].\nLikewise, the bipolar filtration [CHH13] subsumes many smooth concor-\ndance invariants, e.g. if a knot $K \\in \\mathcal{T}_{0}$ then $\\tau(K) = \\upsilon(K) = \\Upsilon(K) =$\n$\\nu^{+}(K) = 0$, referring to invariants from Heegaard Floer homology.\nIt\nis not known whether $s(K) = 0$, referring to the $s$-invariant from Kho-\nvanov homology. However $s^{\\#}(K) = 0$ [KM13b] (see also [Gon21]) and\n$\\widetilde{s}(K) = 0$ [DIS $^{+}25$].\n\nReferences cited:\n- [COT03] Tim D. Cochran, Kent E. Orr, and Peter Teichner. Knot concordance, Whitney towers and L2-signatures. Ann. of Math. (2), 157(2):433–519, 2003. doi:10.4007/annals.2003.157.433.\n- [CHH13] Tim D. Cochran, Shelly Harvey, and Peter Horn. Filtering smooth concordance classes of topologically slice knots. Geom. Topol., 17(4):2103–2162, 2013. doi:10.2140/gt.2013.17.2103.\n- [KM13b] P. B. Kronheimer and T. S. Mrowka. Gauge theory and Rasmussen’s invariant. J. Topol., 6(3):659–674, 2013. doi:10.1112/jtopol/jtt008.\n- [Gon21] Sherry Gong. On the Kronheimer-Mrowka concordance invariant. J. Topol., 14(1):1–28, 2021. doi:10.1112/topo.12175.\n- [DIS+25] Aliakbar Daemi, Hayato Imori, Kouki Sato, Christopher Scaduto, and Masaki Taniguchi. Instantons, special cycles and knot concordance. Geom. Topol., 29(8):4189–4298, 2025. doi:10.2140/gt.2025.29.4189.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether membership in every level of the solvable or bipolar filtrations forces topological or smooth sliceness.\n\n**Verified partial progress.**\n\n- The filtrations subsume many standard sliceness obstructions at finite levels.\n- Recent invariants vanish in some relevant infinite-filtration contexts.\n\n**Full solution or refutation.**\n\nNo implication from the full intersection to sliceness was verified.\n\n**What remains.**\n\nConstruct an intersection counterexample or prove the stated implications.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.44 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains the filtrations, known inclusions/invariants, and open implications.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2704, "problem_number": "KP-1.45", "title": "Kirby Problem 1.45", "statement": "(a) For arbitrary $n \\geq 2.5$ and $g > 1$, does there exist a knot in $\\mathcal{F}_{n}$ with\ntopological slice genus at least $g$?\n(b) For arbitrary $n \\geq 0$ and $g > 1$, does there exist a knot in $\\mathcal{T}_{n}$ with smooth\nslice genus at least $g$?\n(c) Could the $L^{(2)}$-signature invariants used to obstruct membership in deeper\nlevels of the solvable/bipolar filtration be used to give bounds on the slice\ngenus?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.45.\n\nLiterature notes:\n(1) The problem originates from Cha [Cha08, Remark 5.6].\n(2) The filtrations $\\mathcal{F}_{n}$ and $\\mathcal{T}_{n}$ of the smooth concordance group $\\mathcal{C}$ and the\nsubgroup $\\mathcal{T}$ of topologically slice knots, respectively, are defined in Prob-\nlem 1.44.\n(3) For (a), an affirmative answer for the $n = 0, 1$ cases can be shown using\nLevine-Tristram and Casson–Gordon signatures respectively. Similar, an\naffirmative answer to the $n = 2$ case was established by Cha–Miller–Powell\nin [CMP21]. Unlike (a), question (b) is open even in the case $n = 0$.\n(4) A knot is closer to being slice if it has low slice genus. Similarly it is closer\nto being slice if it lies deeper in the solvable/bipolar filtrations. There-\nfore the question is asking whether these two methods of approximating\nsliceness are related.\n(5) We note that there are many examples of highly solvable/bipolar knots\nthat are not known to have low slice genera–that is, the issue in part (c)\nis not a lack of potential examples but rather a lack of obstructions.\n\nReferences cited:\n- [Cha08] Jae Choon Cha. Topological minimal genus and L2-signatures. Algebr. Geom. Topol., 8(2):885–909, 2008. doi:10.2140/agt.2008.8.885.\n- [CMP21] Jae Choon Cha, Allison N. Miller, and Mark Powell. Two-solvable and two-bipolar knots with large four-genera. Math. Res. Lett., 28(2):331–382, 2021. doi:10.4310/MRL.2021.v28.n2.a2.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The topological slice-genus question is affirmative at solvable levels n=0,1,2; deeper levels and all stated bipolar smooth-genus cases remain open.\n\n**Verified partial progress.**\n\n- Levine--Tristram, Casson--Gordon and Cha--Miller--Powell yield affirmative cases n=0,1,2 for part (a).\n\n**Full solution or refutation.**\n\nThe filtration-versus-genus relationship is not settled generally.\n\n**What remains.**\n\nExtend part (a) beyond n=2 and resolve part (b), including possible L2-signature bounds.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.45 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States exact solved low-level cases and remaining open parts.\n\n**Review notes.** Multi-part record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2705, "problem_number": "KP-1.46", "title": "Kirby Problem 1.46", "statement": "(a) Determine the topological slice genera of torus knots. In particular, does\nthe topological slice genus of a torus knot equal half the absolute value of\nits maximal Levine–Tristram signature?\n(b) Is a two-bridge knot topologically slice if and only if it is ribbon, if and\nonly if it is topologically homotopy-ribbon?\n(c) Are the Casson–Gordon ribbon and topological sliceness obstructions [CG86]\ncomplete for two-bridge knots?\n(d) Can the difference between the smooth and topological slice genus of a\ntwo-bridge knot be arbitrarily large?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.46.\n\nLiterature notes:\n(1) Most known methods to bound the topological slice genus from above are\nbased on Freedman–Quinn’s theorem that knots with Alexander polyno-\nmial 1 are topologically slice [FQ90, BKK $^{+}21$]; the exception to this\nis [FT05].\n(2) The smooth slice genera of torus knots were determined by Kronheimer\nand Mrowka using gauge theory: $g_{4}(T(p, q)) = (p - 1)(q - 1)/2$ [KM94].\nThe first person to show that the topological slice genus of torus knots is\ntypically less than their 3-genus was Rudolph [Rud84]. The topological\nslice genus $g^{\\mathrm{top}}_{4}(T(p,q))$ is less than the smooth slice genus $g_{4}(T(p,q))$\nunless $T(p,q)$ has maximal signature, and in general\n$g^{\\mathrm{top}}_{4}(K)\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The torus-knot formula, two-bridge equivalences, ribbon-obstruction completeness, and unbounded genus-gap question remain open, but the topological-sliceness obstruction in part (c) is known not to be complete and several quantitative and special-case results are known.\n\n**Verified partial progress.**\n\n- Baader, Feller, Lewark, and Liechti prove broad strict and quantitative upper bounds for the topological four-genus of torus knots; the signature bound remains the strongest lower bound recorded by K3.\n- The Casson-Gordon topological-sliceness obstruction is not complete for two-bridge knots, whereas completeness of the distinct ribbon obstruction remains conjectural.\n- Lisca classifies smoothly slice two-bridge knots and shows they are ribbon.\n- Feller and McCoy construct two-bridge knots with topological slice genus 1 and smooth slice genus 2.\n\n**Full solution or refutation.**\n\nNo full determination was located. One component of the composite part (c) has a negative answer: Casson-Gordon topological-sliceness obstructions are not complete for two-bridge knots.\n\n**What remains.**\n\nDetermine the topological slice genus of general torus knots; settle topological slice, ribbon, and topological homotopy-ribbon equivalence for two-bridge knots; decide completeness of the ribbon obstruction; and decide whether the smooth-topological genus gap is unbounded for two-bridge knots.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 1.46. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the four subproblems, distinguishes the two Casson-Gordon completeness questions, records that the topological obstruction is not complete, and summarizes the best cited genus bounds and two-bridge results.\n- Sebastian Baader, Peter Feller, Lukas Lewark, and Livio Liechti, On the topological 4-genus of torus knots, Transactions of the American Mathematical Society 370 (2018), 2639-2656. (primary): https://doi.org/10.1090/tran/7051\n Evidence used: Establishes upper bounds and strict separation from smooth genus for broad torus-knot families.\n- Allison N. Miller, A note on the topological sliceness of some 2-bridge knots, Mathematical Proceedings of the Cambridge Philosophical Society 164 (2018), 185-191. (primary): https://doi.org/10.1017/S0305004117000172\n Evidence used: Supplies two-bridge examples underlying the failure of the topological Casson-Gordon obstruction to be complete.\n- Peter Feller and Duncan McCoy, On 2-bridge knots with differing smooth and topological slice genera, Proceedings of the American Mathematical Society 144 (2016), 5435-5442. (primary): https://doi.org/10.1090/proc/13147\n Evidence used: Constructs two-bridge knots with topological slice genus 1 and smooth slice genus 2.\n\n**Review notes.** Part (c) bundles ribbon and topological-sliceness obstructions whose completeness statuses differ; the exact stored statement is preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2706, "problem_number": "KP-1.47", "title": "Kirby Problem 1.47", "statement": "Given a smooth knot $K \\subset S^{3}$, determine its nonorientable 4-\ngenus $\\gamma_{4}$, i.e. the minimal first Betti number for all compact nonorientable smooth\nsurfaces $F$ properly embedded in $B^{4}$ with boundary $K$. This question may be devel-\noped in several directions:\n(1) (Allen’s Geography Question [All23]) For a given knot $K \\subset S^{3}$, find\nthe pairs $(e, b) \\in \\mathbb{Z}^{2}$ realized by the normal Euler number and first Betti\nnumber of a nonorientable bounding surface $F$.\n(2) By analogy with the Milnor Conjecture for the orientable 4-genus of torus\nknots, what is the nonorientable 4-genus of a torus knot $T_{p,q}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.47.\n\nLiterature notes:\nThe nonorientable 4-genus was first defined by Murakami and Ya-\nsuhara [MY00] in 2000, though earlier results had been obtained by Viro [Vir75]\nand by Yasuhara [Yas96].\nThese authors, along with Gilmer and Livingston\n[GL11], obstructed the existence of Möbius strips and punctured Klein bottles\nwith boundary $K$ using the classical signature and the Arf invariant of $K$.\n\nThe first examples of knots with $\\gamma_{4}(K) > 3$ were produced by Batson, who\nused an elegant construction and the Heegaard Floer $d$ invariant to prove that\n$\\gamma_{4}(T_{2k,2k-1}) = k - 1$ [Bat14]. Lobb subsequently showed that Batson’s construc-\ntion does not always yield a minimal-genus nonorientable surface for torus knots\n[Lob19]. Allen added context by asking how the nonorientable 4-genus interacts\nwith the normal Euler number of the bounding surfaces for torus (and other) knots\n[All23]. See [BKST25, JVC21, OSS17a, Sab23], among others, for further\nexplorations of the nonorientable 4-genus for torus knots. In [All23], Allen made\nthree conjectures (Conjectures 1.6, 1.7, and 1.8); Sato [Sat24a] has given coun-\nterexamples to Conjectures 1.6 and 1.8.\nThere are now many knots for which the nonorientable four-genus is known,\nincluding all knots with at most 10 crossings [Gha22]. There are also currently\na variety of lower bounds for $\\gamma_{4}$ using modern knot invariants.\nMany of these\nare what Sato [Sat24b] terms “unoriented slice-torus invariants”, including the $\\upsilon$\ninvariant from Heegaard Floer theory [OSS17a], the $t$ invariant from Khovanov\nhomology [Bal20], the $h_{\\mathbb{Z}}$ invariant from instanton homology [DS24a]. Additional\ninformation about the nonorientable four-genus can be obtained by considering\nversions of Seiberg-Witten Floer homology of the double branched cover that take\ninto account the involution coming from the covering transformation; see [BH24b,\nKMT23b].\n\nReferences cited:\n- [All23] Samantha Allen. Nonorientable surfaces bounded by knots: a geography problem. New York J. Math., 29:1038–1059, 2023. http://nyjm.albany.edu/j/2023/29-41v.pdf.\n- [MY00] Hitoshi Murakami and Akira Yasuhara. Four-genus and four-dimensional clasp number of a knot. Proc. Amer. Math. Soc., 128(12):3693–3699, 2000. doi:10.1090/S0002-9939-00-05461-7.\n- [Vir75] Oleg Ja. Viro. Positioning in codimension 2, and the boundary. Uspehi Mat. Nauk, 30(1(181)):231–232, 1975.\n- [Yas96] Akira Yasuhara. Connecting lemmas and representing homology classes of simply connected 4-manifolds. Tokyo J. Math., 19(1):245–261, 1996. doi:10.3836/tjm/1270043232.\n- [GL11] Patrick M. Gilmer and Charles Livingston. The nonorientable 4-genus of knots. J. Lond. Math. Soc. (2), 84(3):559–577, 2011. doi:10.1112/jlms/jdr024.\n- [Bat14] Joshua Batson. Nonorientable slice genus can be arbitrarily large. Math. Res. Lett., 21(3):423–436, 2014. doi:10.4310/MRL.2014.v21.n3.a1.\n- [Lob19] Andrew Lobb. A counterexample to Batson’s conjecture. Math. Res. Lett., 26(6):1789, 2019. doi:10.4310/MRL.2019.v26.n6.a8.\n- [BKST25] Fraser Binns, Sungkyung Kang, Jonathan Simone, and Paula Truöl. On the nonorientable four-ball genus of torus knots. Algebr. Geom. Topol., 25(4):2209–2251, 2025. doi:10.2140/agt.2025.25.2209.\n- [JVC21] Stanislav Jabuka and Cornelia A. Van Cott. On a nonorientable analogue of the Milnor conjecture. Algebr. Geom. Topol., 21(5):2571–2625, 2021. doi:10.2140/agt.2021.21.2571.\n- [OSS17a] Peter Ozsváth, András Stipsicz, and Zoltán Szabó. Unoriented knot Floer homology and the unoriented four-ball genus. Int. Math. Res. Not. IMRN, 2017(17):5137– 5181, 2017. doi:10.1093/imrn/rnw143.\n- [Sab23] Joshua M. Sabloff. On a refinement of the non-orientable 4-genus of torus knots. Proc. Amer. Math. Soc. Ser. B, 10:242–251, 2023. doi:10.1090/bproc/166.\n- [Sat24a] Kouki Sato. Counterexamples to Allen’s conjectures, 2024. arXiv:2407.12049.\n- [Gha22] Nakisa Ghanbarian. The non-orientable 4-genus for knots with 10 crossings. J. Knot Theory Ramifications, 31(5):Paper No. 2250034, 46, 2022. doi:10.1142/S0218216522500341.\n- [Sat24b] Kouki Sato. An unoriented analogue of slice-torus invariant, 2024. arXiv:2404.04056.\n- [Bal20] William Ballinger. Concordance invariants from the Ep-1q spectral sequence on Khovanov homology, 2020. arXiv:2004.10807.\n- [DS24a] Aliakbar Daemi and Christopher Scaduto. Chern–Simons functional, singular instantons, and the four-dimensional clasp number. J. Eur. Math. Soc. (JEMS), 26(6):2127–2190, 2024. doi:10.4171/jems/1320.\n- [BH24b] David Baraglia and Pedram Hekmati. New invariants of involutions from SeibergWitten-Floer theory, 2024. arXiv:2403.00203.\n- [KMT23b] Hokuto Konno, Jin Miyazawa, and Masaki Taniguchi. Involutions, links, and Floer cohomologies, 2023. arXiv:2304.01115.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many exact values, lower bounds, geography restrictions, and infinite torus-knot families are known, but neither the general nonorientable four-genus of torus knots nor the general (normal Euler number, first Betti number) geography is determined.\n\n**Verified partial progress.**\n\n- Batson proved gamma_4(T_{2k,2k-1})=k-1, showing that nonorientable four-genus is unbounded.\n- Binns, Kang, Simone, and Truöl give a new lower bound sharp for families including T_{4n,(2n±1)^2} for even n at least 2 and broad obstructions to bounding locally flat Möbius bands.\n- Ghanbarian determined the nonorientable four-genus of all knots with at most ten crossings.\n- Allen formulated and developed the geography problem; Sato disproved Allen's Conjectures 1.6 and 1.8.\n\n**Full solution or refutation.**\n\nThe broad determination problem has substantial solved families and effective obstructions, but no general torus-knot formula or general geography classification was located.\n\n**What remains.**\n\nDetermine gamma_4(T_{p,q}) in general and classify the realizable normal-Euler-number/Betti-number pairs for an arbitrary knot.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 1.47. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Presents the current determination and geography questions and surveys exact families, modern bounds, and the counterexamples to two Allen conjectures.\n- Joshua Batson, Nonorientable slice genus can be arbitrarily large, Mathematical Research Letters 21 (2014), 423-436. (primary): https://doi.org/10.4310/MRL.2014.v21.n3.a1\n Evidence used: Proves gamma_4(T_{2k,2k-1})=k-1.\n- Fraser Binns, Sungkyung Kang, Jonathan Simone, and Paula Truöl, On the nonorientable four-ball genus of torus knots, Algebraic & Geometric Topology 25 (2025), 2209-2251. (primary): https://arxiv.org/abs/2109.09187\n Evidence used: Introduces a lower bound sharp for several infinite torus-knot families and proves large classes do not bound locally flat Möbius bands.\n- Kouki Sato, Counterexamples to Allen's conjectures, arXiv:2407.12049 (2024). (primary): https://arxiv.org/abs/2407.12049\n Evidence used: Disproves two proposed geography conjectures while leaving the general geography problem open.\n\n**Review notes.** This row is an open-ended determination program; partial progress is classified by the breadth of exact infinite families and obstructions, not as a claimed general formula.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2707, "problem_number": "KP-1.48", "title": "Kirby Problem 1.48", "statement": "(a) Suppose $K$ and $K\\#J$ are (smoothly) doubly slice knots.\nMust $J$ be a\n(smoothly) doubly slice knot?\n(b) Does there exist a knot that is smoothly slice, topologically doubly slice, and\nnot smoothly doubly slice but such that $K\\#K$ is smoothly doubly slice?\n(c) Let $M$ and $N$ be closed 3-manifolds such that $M$ and $M\\#N$ embed (smoothly)\nin $S^{4}$. Must $N$ embed (smoothly) in $S^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.48.\n\nLiterature notes:\n(1) A knot $K \\subset S^{3}$ is called (smoothly) doubly slice if there is a (smoothly)\nunknotted 2-sphere $\\mathcal{\\operatorname{U}} \\subset S^{4}$ such that $K = S^{3} \\cap \\mathcal{\\operatorname{U}}$.\n(2) It is natural to try to equip the set of knots with the relation in which\n$K_{0} \\sim K_{1}$ if $K_{0}\\#K_{1}$ is doubly slice. However, to prove this is an equiv-\nalence relation, one must provide an affirmative answer to part (a) of\nthe problem. For more details, see [Mei15] and [Sto78]. This question\nis also interesting in the topological category: Replace “smooth” with\n“topologically locally flat” above. Question (a) is originally due to Stoltz-\nfus [Sto78].\n(3) One motivation for Question (a) is that it would allow for the formulation\nof the Grothendieck group of knots modulo the relation discussed above.\nIn the absence of a positive answer to Question (a), there is an alternative\napproach. Two knots $K_{0}$ and $K_{1}$ are called (smoothly) doubly concordant\nif there exist (smoothly) doubly slice knots $J_{0}$ and $J_{1}$ such that $K_{0}\\#J_{0} =$\n$K_{1}\\#J_{1}$. Let $\\mathcal{C}_{\\mathcal{D}}$ denote the group (under connected sum) of knots modulo\nthis equivalence relation, which is called the (smooth) double concordance\n\ngroup.\nAn affirmative answer to Question (a) would mean the double\nconcordance group is the same as the aforementioned Grothendieck group.\nA natural problem is to exhibit nontrivial elements in these groups, as we\nnow outline.\nThere is a topological version $\\mathcal{C}_{\\mathcal{D}}^{\\mathrm{top}}$\nof this group obtained by replacing\n“smooth” with “topologically locally flat” above. There are epimorphisms\nfrom $\\mathcal{C}_{\\mathcal{D}}$ to $\\mathcal{C}_{\\mathcal{D}}^{\\mathrm{top}}$\nand to the smooth concordance group $\\mathcal{C}$. Let $\\mathcal{K}$ denote the\nintersection of the kernels of these two homomorphisms. It is known that\n$\\mathcal{K}$ contains an infinitely generated subgroup $S$ with generators of order at\nleast 3 [Mei15].\nConjecture. These generators of $S$ have infinite order.\n(4) There are many techniques for producing and studying 2-torsion in the\nsmooth concordance group $\\mathcal{C}$; see [HKL16a] for a nice overview. However,\nthese techniques seem difficult to apply to $\\mathcal{C}_{\\mathcal{D}}$, which motivates Question\n(b).\n(5) If $K$ is doubly slice, then any cyclic cover of $S^{3}$ branched along $K$ embeds\n(smoothly) in $S^{4}$. This suggests Question (c), which asks about embed-\nding 3-manifolds in $S^{4}$. The connection between doubly slice knots, em-\nbedding problems and homology cobordisms was observed by Gilmer and\nLivingston [GL83]. See also Problem 4.25.\n\nReferences cited:\n- [Mei15] Jeffrey Meier. Distinguishing topologically and smoothly doubly slice knots. J. Topol., 8(2):315–351, 2015. doi:10.1112/jtopol/jtu027.\n- [Sto78] Neal W. Stoltzfus. Algebraic computations of the integral concordance and double null concordance group of knots. In Knot theory (Proc. Sem., Plans-sur-Bex, 1977), volume 685 of Lecture Notes in Math., pages 274–290. Springer, Berlin, 1978.\n- [HKL16a] Matthew Hedden, Se-Goo Kim, and Charles Livingston. Topologically slice knots of smooth concordance order two. J. Differential Geom., 102(3):353–393, 2016. http://projecteuclid.org/euclid.jdg/1456754013.\n- [GL83] Patrick M. Gilmer and Charles Livingston. On embedding 3-manifolds in 4-space. Topology, 22(3):241–252, 1983. doi:10.1016/0040-9383(83)90011-3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The three stated cancellation and existence questions remain open in the current K3 source; known double-concordance and embedding obstructions do not answer them.\n\n**Verified partial progress.**\n\n- Meier constructs an infinitely generated subgroup in the intersection of the kernels from smooth double concordance to topological double concordance and to ordinary smooth concordance, with displayed generators of order at least three.\n- Gilmer-Livingston branched-cover methods connect double sliceness with embeddings of closed 3-manifolds in S^4 and supply obstructions.\n- Double concordance is defined by stabilization with doubly slice knots, but whether stabilization cancellation agrees with unstabilized double sliceness is precisely the unresolved part (a).\n\n**Full solution or refutation.**\n\nNo cancellation theorem, counterexample to cancellation, knot satisfying part (b), or 3-manifold pair settling part (c) was located.\n\n**What remains.**\n\nSettle cancellation for doubly slice knots, construct or rule out the specified smooth/topological order-two phenomenon, and settle connected-sum cancellation for smooth embeddings of closed 3-manifolds in S^4.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 1.48. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States all three questions as open and explains the double-concordance and branched-cover embedding context.\n- Jeffrey Meier, Distinguishing topologically and smoothly doubly slice knots, Journal of Topology 8 (2015), 315-351. (primary): https://doi.org/10.1112/jtopol/jtu027\n Evidence used: Constructs the cited large subgroup in the double concordance group while leaving the cancellation questions unresolved.\n- Patrick M. Gilmer and Charles Livingston, On embedding 3-manifolds in 4-space, Topology 22 (1983), 241-252. (primary): https://doi.org/10.1016/0040-9383(83)90011-3\n Evidence used: Develops the branched-cover embedding connection and obstructions motivating part (c).\n\n**Review notes.** Parts (a) and (c) explicitly invite both smooth and topological variants; the record-level open label covers the exact smooth wording.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2708, "problem_number": "KP-1.49", "title": "Kirby Problem 1.49", "statement": "(a) What is the structure of the equivariant concordance groups?\n(b) Is the strongly negative amphichiral concordance group abelian?\n(c) For any type of knot involution, is it possible to exhibit $K_{1}$ and $K_{2}$ that\nare equivariantly concordant but not standardly equivariantly concordant?\n(d) Does there exist a freely periodic slice knot that is not equivariantly slice?\n(e) (Boyle-Rouse) Is every periodic or freely periodic $L$-space knot either a\ntorus knot or an iterated torus knot?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.49.\n\nLiterature notes:\n(1) Knots may have various sorts of symmetries; an equivariant concordance\nis a usual concordance that admits the same sort of symmetry. In the\ntopological case, we require that the concordance be locally flat, and that\nany group action be locally linear. There are several variations discussed\nin this problem, each with its own particular set of concerns.\n(2) Equivariant concordance groups can be defined for the classes of strongly\ninvertible and strongly negative amphichiral knots. The strongly invert-\nible concordance group has been studied by several authors: for instance,\nthere exists a homomorphism to the Laurent polynomial ring $\\mathbb{Z}[t, t^{-1}]$\n[Sak86]. These groups are not a priori abelian due to (possible) depen-\ndence on the choice of the connected sum point, which must be taken to be\none of the two fixed points of the symmetry on the knot. In the strongly\ninvertible case, it is known that the group is not abelian [DP23b], but in\nthe amphichiral case this is not known.\n\nUnderstanding strongly negative amphichiral knots is especially inter-\nesting, since the branched covers of such knots may provide examples of\n2-torsion in the integer homology cobordism group. Examples of strongly\nnegative amphichiral knots with determinant one were exhibited by Boyle\nand Chen in [BC24b].\n(3) In part (c), we say that two knots $K_{1}$ and $K_{2}$ (equipped with the same\ntype of involution) are standardly equivariantly concordant if they are\nconnected by a smooth annular cobordism in $S^{3}\\times I$ that is invariant under\nan involution on $S^{3} \\times I$ that restricts to a standard involution on each\n$S^{3} \\times \\{t\\}$. One can also consider more general equivariant cobordisms, in\nwhich one requires that the annular cobordism be fixed by some involution\non $S^{3} \\times I$ which extends the actions on $S^{3} \\times \\{0\\}$ and $S^{3} \\times \\{1\\}$. It is not\nknown in any of the possible cases whether these notions coincide. The\nquestion may be asked in either the smooth or the topological category.\nIn the case of 2-periodic and strongly invertible knots, it is known\nthat there exists a nonstandard involution on $S^{3} \\times I$ that restricts to the\nstandard involution on $S^{3}\\times\\{0\\}$ and $S^{3}\\times\\{1\\}$. To construct one, start with\nan involution on $S^{4}$ with fixed-point set a knotted $S^{2}$ [Sum75]; punctur-\ning such an example gives a nonstandard involution on $S^{3} \\times I$ that has\nfixed-point set $S^{1}$ on each of the two ends. It is not known whether an\ninteresting such involution exists in the freely periodic or strongly am-\nphichiral cases.\n(4) Question (d) appears as Question 1 in [BM23]. Manolescu and Willis\n[MW25] gave examples of freely 2-periodic knots that are concordant\nbut not (standardly) equivariantly concordant.\n(5) Evidence for Question (e) comes from classic work of Murasugi [Mur71]\nand Hartley [Har81], who established factorization results for the Alexan-\nder polynomials of periodic or freely periodic knots. We also have several\nwell-known restrictions on the Alexander polynomials of L-space knots;\nsee [OS05b] or [HW18]. As discussed by Boyle and Rouse [BR23b],\nthe only known examples of Alexander polynomials that simultaneously\nsatisfy all of these conditions are products of cyclotomic polynomials. On\nthe other hand, a conjecture of Li and Ni states that the only L-space\nknots with Alexander polynomials which are products of cyclotomic poly-\nnomials are torus knots or iterated torus knots [LN15]. Based on this,\nBoyle and Rouse [BR23b] conjectured a positive answer to Question (e).\n\nReferences cited:\n- [Sak86] Makoto Sakuma. On strongly invertible knots. In Algebraic and topological theories (Kinosaki, 1984), pages 176–196. Kinokuniya, Tokyo, 1986.\n- [DP23b] Alessio Di Prisa. The equivariant concordance group is not abelian. Bull. Lond. Math. Soc., 55(1):502–507, 2023. doi:10.1112/blms.12741.\n- [BC24b] Keegan Boyle and Wenzhao Chen. Equivariant topological slice disks and negative amphichiral knots. Indiana Univ. Math. J., 73(5):1623–1637, 2024.\n- [Sum75] D. W. Sumners. Smooth $\\mathbb{Z}_p$-actions on spheres which leave knots pointwise fixed. Trans. Amer. Math. Soc., 205:193–203, 1975. doi:10.2307/1997199.\n- [BM23] Keegan Boyle and Jeffrey Musyt. Equivariant cobordisms between freely periodic knots. Canad. Math. Bull., 66(2):450–457, 2023. doi:10.4153/S000843952200042X.\n- [MW25] Ciprian Manolescu and Michael Willis. A Rasmussen invariant for links in $\\mathbb{RP}^{3}$. Trans. Amer. Math. Soc. Ser. B, 12:789–830, 2025. doi:10.1090/btran/221.\n- [Mur71] Kunio Murasugi. On periodic knots. Comment. Math. Helv., 46:162–174, 1971. doi:10.1007/BF02566836.\n- [Har81] Richard Hartley. Knots with free period. Canadian J. Math., 33(1):91–102, 1981. doi:10.4153/CJM-1981-009-7.\n- [OS05b] Peter Ozsváth and Zoltán Szabó. On knot Floer homology and lens space surgeries. Topology, 44(6):1281–1300, 2005. doi:10.1016/j.top.2005.05.001.\n- [HW18] Matthew Hedden and Liam Watson. On the geography and botany of knot Floer homology. Selecta Math. (N.S.), 24(2):997–1037, 2018. doi:10.1007/s00029-017-0351-5.\n- [BR23b] Keegan Boyle and Nicholas Rouse. Obstructions to free periodicity and symmetric L-space knots, 2023. arXiv:2310.01705.\n- [LN15] Eileen Li and Yi Ni. Half-integral finite surgeries on knots in $S^{3}$. Ann. Fac. Sci. Toulouse Math. (6), 24(5):1157–1178, 2015. doi:10.5802/afst.1479.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The strongly invertible group is known to be nonabelian and new invariants and freely periodic examples give substantial structure, but the strongly negative amphichiral, general equivariant-versus-standard, freely periodic slice, and L-space classification questions remain open.\n\n**Verified partial progress.**\n\n- Di Prisa proved that the equivariant concordance group of directed strongly invertible knots is nonabelian.\n- Manolescu and Willis give freely 2-periodic knots that are ordinarily concordant but not standardly equivariantly concordant.\n- Boyle, Rouse, and Williams derive strong Alexander-polynomial restrictions and show that freely periodic L-space knots of genus at most 16 have Alexander polynomial a product of cyclotomic polynomials.\n- A 2024 equivariant algebraic concordance theory and further invariants provide structure for the strongly invertible case.\n\n**Full solution or refutation.**\n\nPart (a) has meaningful structural answers in the strongly invertible case. The Manolescu-Willis examples do not by themselves satisfy the stronger antecedent in part (c), and parts (b), (d), and (e) remain open.\n\n**What remains.**\n\nDetermine whether the strongly negative amphichiral group is abelian; find genuinely equivariantly but not standardly equivariantly concordant knots for an involution type; settle freely periodic equivariant sliceness; and prove or disprove the iterated-torus classification of symmetric L-space knots.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 1.49. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Separates the five equivariant-concordance subquestions and records the strongest known results and remaining distinctions.\n- Alessio Di Prisa, The equivariant concordance group is not abelian, Bulletin of the London Mathematical Society 55 (2023), 502-507. (primary): https://doi.org/10.1112/blms.12741\n Evidence used: Proves nonabelianity for the strongly invertible equivariant concordance group.\n- Ciprian Manolescu and Michael Willis, A Rasmussen invariant for links in RP^3, Transactions of the American Mathematical Society, Series B 12 (2025), 789-830. (primary): https://arxiv.org/abs/2301.09764\n Evidence used: Constructs freely 2-periodic knots that are concordant but not standardly equivariantly concordant.\n- Keegan Boyle, Nicholas Rouse, and Ben Williams, Obstructions to free periodicity and symmetric L-space knots, arXiv:2310.01705v3 (2026). (primary): https://arxiv.org/abs/2310.01705\n Evidence used: Proves polynomial restrictions and a genus-at-most-16 result, and states the iterated-torus conclusion as a conjecture.\n\n**Review notes.** The distinction between ordinary concordance, general equivariant concordance, and standard equivariant concordance is material; the Manolescu-Willis result is not silently promoted to a solution of part (c).\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2709, "problem_number": "KP-1.50", "title": "Kirby Problem 1.50", "statement": "(a) Is every slice knot a ribbon knot?\n(b) Is every slice link ribbon?\n(c) Suppose $K$ is a knot with smooth four-genus $g_{4}(K) = g$. Does $K$ bound a\nsmooth, genus $g$, ribbon surface $F$ in $B^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.50.\n\nLiterature notes:\n(1) The Slice-Ribbon Conjecture (a) was first posed as a question by Fox\nin 1961 [Fox62, Problem 25], and was discussed in [Kir97, Problem\n4.22].\nVery limited progress has been made on this problem, though\nthe conjecture has been verified for 2-bridge knots by Lisca [Lis07] and\nmost 3-stranded pretzel knots by Greene and Jabuka [GJ11] and by\nLecuona [Lec15].\n(2) There are other notions of “ribbonness” that can be considered: A knot\n$K \\subset S^{3}$ is called homotopy-ribbon if there is a slice disk $D$ for $K$ with a\nsurjection $\\pi_{1}(S^{3}\\setminus K) \\twoheadrightarrow\\pi_{1}(B^{4}\\setminus D)$ and is called handle-ribbon if there is\na slice disk $D$ for $K$ such that $B^{4}\\setminus\\nu(D)$ admits a handle-decomposition\nwith only 0-, 1-, and 2-handles; see [Gor81, LM15, MZ23].\n(3) The implications\nribbon $\\Rightarrow$ handle ribbon $\\Rightarrow$ homotopy-ribbon $\\Rightarrow$ slice\nare immediate, but all three converse implications are open.\n(4) For many examples of handle ribbon knots and links that are not known\nto be ribbon, as well as connections to the Generalized Property R Con-\njecture, see [GST10, MZ22] and Problem 1.10.\n(5) Miyazaki gave many examples of non-simple knots that are not homotopi-\ncally ribbon, such as certain linear combinations of algebraic knots and,\nfamously, the $(2, 1)$-cable of the figure-8 knot [Miy94]. Some of these ex-\namples are now known to be non-slice [HKL12, DKM $^{+}24$, ACM $^{+}26$].\nBaker draws connections between the Slice-Ribbon Conjecture for fibered\nknots and contact topology in [Bak16]. See Problem 1.54 for more ques-\ntions about ribbon concordances.\n(6) In part (b), we say that an $n$-component link $L \\subset S^{3}$ is called (strongly)\nslice if there is a disjoint collection $D \\subset B^{4}$ of $n$ disks with $\\partial D = L$;\nthe notions of ribbon, handle-ribbon, and homotopy-ribbon have obvious\nanalogs for links.\n(7) Many examples of handle ribbon links that are not known to be ribbon\nare constructed in [GST10] and [MZ22, MZ25].\n(8) Eisermann [Eis09] gave an obstruction for a link (with at least two com-\nponents) to be ribbon, in terms of the Jones polynomial. It is not clear\nwhether his obstruction vanishes when the link is handle ribbon (or homotopy-\nribbon, or slice) so, in principle, it could be used to show that a handle\nribbon link is not ribbon.\n(9) The question in part (c) generalizes the Slice-Ribbon Conjecture, and so\nit may be easier to find a counterexample. If $g_{4}(K) = g(K)$, then the\npush-in of a minimal genus Seifert surface is a ribbon surface realizing\n\nthe minimum possible value of $g_{4}(K)$, so the question has an affirmative\nanswer in this case.\n(10) Finally, the question of whether slice implies homotopy-ribbon can be\nphrased in the more general setting of knots in homology 3–spheres bound-\ning disks in contractible 4–manifolds. For more details, see [Kir97, Prob-\nlem 4.22] and [CG83a].\n\nReferences cited:\n- [Fox62] Ralph Fox. Some problems in knot theory. In M. K. Fort, Jr., editor, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961). Prentice-Hall, Englewood Cliffs, N.J., 1962.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Lis07] Paolo Lisca. Lens spaces, rational balls and the ribbon conjecture. Geom. Topol., 11:429–472, 2007. doi:10.2140/gt.2007.11.429.\n- [GJ11] Joshua Greene and Stanislav Jabuka. The slice-ribbon conjecture for 3-stranded pretzel knots. Amer. J. Math., 133(3):555–580, 2011. doi:10.1353/ajm.2011.0022.\n- [Lec15] Ana G. Lecuona. On the slice-ribbon conjecture for pretzel knots. Algebr. Geom. Topol., 15(4):2133–2173, 2015. doi:10.2140/agt.2015.15.2133.\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.\n- [LM15] Kyle Larson and Jeffrey Meier. Fibered ribbon disks. J. Knot Theory Ramifications, 24(14):1550066, 22, 2015. doi:10.1142/S0218216515500662.\n- [MZ23] Maggie Miller and Alexander Zupan. Equivalent characterizations of handle-ribbon knots. Comm. Anal. Geom., 31(9):2157–2193, 2023. doi:10.4310/cag.2023.v31.n9.a1.\n- [GST10] Robert E. Gompf, Martin Scharlemann, and Abigail Thompson. Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures. Geom. Topol., 14(4):2305–2347, 2010. doi:10.2140/gt.2010.14.2305.\n- [MZ22] Jeffrey Meier and Alexander Zupan. Generalized square knots and homotopy 4-spheres. J. Differential Geom., 122(1):69–129, 2022. doi:10.4310/jdg/1668186788.\n- [Miy94] Katura Miyazaki. Nonsimple, ribbon fibered knots. Trans. Amer. Math. Soc., 341(1):1–44, 1994. doi:10.2307/2154613.\n- [HKL12] Matthew Hedden, Paul Kirk, and Charles Livingston. Non-slice linear combinations of algebraic knots. J. Eur. Math. Soc. (JEMS), 14(4):1181–1208, 2012. doi:10.4171/JEMS/330.\n- [DKM+24] Irving Dai, Sungkyung Kang, Abhishek Mallick, JungHwan Park, and Matthew Stoffregen. The $(2,1)$-cable of the figure-eight knot is not smoothly slice. Invent. Math., 238(2):371–390, 2024. doi:10.1007/s00222-024-01286-w.\n- [ACM+26] Paolo Aceto, Nickolas A Castro, Maggie Miller, JungHwan Park, and András Stipsicz. Slice Obstructions From Genus Bounds in Definite 4-Manifolds. Int. Math. Res. Not. IMRN, 2026(2):Paper No. rnaf377, 2026. doi:10.1093/imrn/rnaf377.\n- [Bak16] Kenneth L. Baker. A note on the concordance of fibered knots. J. Topol., 9(1):1–4, 2016. doi:10.1112/jtopol/jtv024.\n- [MZ25] Jeffrey Meier and Alexander Zupan. Knots bounding nonisotopic ribbon disks. J. Topol., 18(4):Paper No. e70047, 18, 2025. doi:10.1112/topo.70047.\n- [Eis09] Michael Eisermann. The Jones polynomial of ribbon links. Geom. Topol., 13(2):623– 660, 2009. doi:10.2140/gt.2009.13.623.\n- [CG83a] A. J. Casson and C. McA. Gordon. A loop theorem for duality spaces and fibred ribbon knots. Invent. Math., 74(1):119–137, 1983. doi:10.1007/BF01388533.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general slice-ribbon conjecture, its link analogue, and the minimum-genus ribbon-surface generalization remain open, although important knot classes and the case g_4=g_3 are settled.\n\n**Verified partial progress.**\n\n- Lisca proves the slice-ribbon conjecture for two-bridge knots.\n- Greene-Jabuka and Lecuona prove the conjecture for broad classes covering most three-strand pretzel knots.\n- Part (c) is affirmative whenever smooth four-genus equals Seifert genus, by pushing a minimum-genus Seifert surface into B^4.\n- The once-prominent (2,1)-cable of the figure-eight knot is now known not to be smoothly slice, so it is not a slice-ribbon counterexample.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was located for any of the three questions; the current evidence consists of special classes, structural implications, and elimination of candidates.\n\n**What remains.**\n\nSettle slice-ribbon for knots and links, or any converse among ribbon, handle-ribbon, homotopy-ribbon, and slice; and determine whether every knot has a minimum-genus ribbon surface realizing g_4.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 1.50. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States all three open questions, surveys the ribbon hierarchy and special cases, and proves the elementary g_4=g_3 case of part (c).\n- Paolo Lisca, Lens spaces, rational balls and the ribbon conjecture, Geometry & Topology 11 (2007), 429-472. (primary): https://doi.org/10.2140/gt.2007.11.429\n Evidence used: Verifies slice-ribbon for two-bridge knots.\n- Joshua Greene and Stanislav Jabuka, The slice-ribbon conjecture for 3-stranded pretzel knots, American Journal of Mathematics 133 (2011), 555-580. (primary): https://doi.org/10.1353/ajm.2011.0022\n Evidence used: Settles broad three-strand pretzel cases.\n- Jennifer Hom and JungHwan Park, Ribbon knots and iterated cables of fibered knots, arXiv:2507.20455 (2025). (primary): https://arxiv.org/abs/2507.20455\n Evidence used: Gives a sharp modern conditional constraint: either certain iterated cables are linearly independent in concordance or slice-ribbon fails.\n\n**Review notes.** The record is composite; partial progress does not imply any of the three universal claims is settled.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2710, "problem_number": "KP-1.51", "title": "Kirby Problem 1.51", "statement": "Does every ribbon knot arise as a symmetric union?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.51.\n\nLiterature notes:\nSymmetric unions were introduced by Kinoshita and Terasaka to\nconstruct knots with Alexander polynomial 1 (by taking a symmetric union of an\nunknot diagram) [KT57]. It is straightforward to see that every symmetric union is\na ribbon knot. A census shows that most small ribbon knots do arise as symmetric\nunions [Lam21], but there are candidates for ribbon knots that are not symmetric\nunions, such as $3_{1}\\#8_{10}$ and $3_{1}\\#8_{11}$. Other potential counterexamples obtained as\ncertain satellites of knots of the form $K\\# - K$ are given by Aceto [Ace14]. These\nknots have ribbon number two, but the symmetric ribbon number is shown to be\narbitrarily large, if not infinite.\n\nReferences cited:\n- [KT57] Shin’ichi Kinoshita and Hidetaka Terasaka. On unions of knots. Osaka Math. J., 9:131–153, 1957.\n- [Lam21] Christoph Lamm. The search for nonsymmetric ribbon knots. Exp. Math., 30(3):349–363, 2021. doi:10.1080/10586458.2018.1540313.\n- [Ace14] Paolo Aceto. Symmetric ribbon disks. J. Knot Theory Ramifications, 23(9):1450048, 9, 2014. doi:10.1142/S0218216514500485.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A June 2026 preprint gives a direct negative answer by constructing a ribbon Montesinos knot that has no symmetric-union presentation.\n\n**Verified partial progress.**\n\n- Earlier censuses found symmetric-union presentations for most small ribbon knots and isolated possible counterexamples.\n- Boileau, Kitano, and Nozaki now prove that a specific ribbon Montesinos knot is not a symmetric union.\n\n**Full solution or refutation.**\n\nNot every ribbon knot arises as a symmetric union.\n\n**What remains.**\n\nObtain journal-level verification of the very recent preprint and develop broader criteria or classifications describing which ribbon knots admit symmetric-union presentations.\n\n**Sources checked.**\n\n- Michel Boileau, Teruaki Kitano, and Yuta Nozaki, A ribbon knot which is not a symmetric union, arXiv:2606.02968 (submitted 2 June 2026). (primary): https://arxiv.org/abs/2606.02968\n Evidence used: The abstract and main theorem explicitly give a negative answer by exhibiting a ribbon Montesinos knot with no symmetric-union presentation.\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 1.51. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Documents the pre-resolution open problem, prior census, and candidate counterexamples; it predates or did not incorporate the 2 June 2026 solution.\n\n**Review notes.** This is a direct post-dataset status change. Confidence is high because the primary source explicitly proves the negation, but expert review is advisable because the source is a very recent, apparently unrefereed preprint.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2711, "problem_number": "KP-1.52", "title": "Kirby Problem 1.52", "statement": "Given $K$ in $S^{3}$, is there an algorithm to detect if $K$ is slice?\nRibbon?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.52.\n\nLiterature notes:\n(1) This is Problem 1.34 from [Kir97], posed by A. Casson.\n(2) There is still no algorithm, but there have been some interesting computer-\naided searches: Owens and Swenton search for ribbon bands for alternat-\ning knots [OS23] and Gukov, Halverson, Manolescu, and Ruehle apply\nmachine learning algorithms to the problem [GHMR23]. Dunfield and\nGong [DG25] have announced the results of a search for ribbon knots,\nslice knots, and concordances among the knots with at most 19 crossings.\n(3) Casson and Long gave an algorithm to determine if a surface automor-\nphism compresses [CL85]. In conjunction with the main results of [CG83a],\nthis algorithm can be used to determine whether a given fibered knot\nbounds a fibered, homotopy-ribbon disk in a homotopy four-ball; see\nalso [Lon86].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [OS23] Brendan Owens and Frank Swenton. An algorithm to find ribbon disks for alternating knots. Experimental Mathematics, 0(0):1–19, 2023. doi:10.1080/10586458.2022.2158968.\n- [GHMR23] Sergei Gukov, James Halverson, Ciprian Manolescu, and Fabian Ruehle. Searching for ribbons with machine learning, 2023. arXiv:2304.09304.\n- [DG25] Nathan M. Dunfield and Sherry Gong. Ribbon concordances and slice obstructions: experiments and examples, 2025. arXiv:2512.21825.\n- [CL85] A. J. Casson and D. D. Long. Algorithmic compression of surface automorphisms. Invent. Math., 81(2):295–303, 1985. doi:10.1007/BF01389054.\n- [CG83a] A. J. Casson and C. McA. Gordon. A loop theorem for duality spaces and fibred ribbon knots. Invent. Math., 74(1):119–137, 1983. doi:10.1007/BF01388533.\n- [Lon86] D. D. Long. Discs in compression bodies. Pacific J. Math., 122(1):129–146, 1986. http://projecteuclid.org/euclid.pjm/1102702126.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No general algorithm is known for deciding smooth or topological sliceness, or ribbonness, although restricted algorithms and near-exhaustive finite censuses are available.\n\n**Verified partial progress.**\n\n- Dunfield and Gong determine smooth sliceness for all but about 11,400 of 352.2 million prime knots through 19 crossings and topological sliceness for all but about 1,400, while explicitly stating that no general decision algorithm is known.\n- Owens and Swenton provide an algorithmic search for ribbon disks for alternating knots, which supplies certificates when successful but not a general decision procedure.\n- Casson-Long's compression algorithm, combined with Casson-Gordon, decides whether a given fibered knot bounds a fibered homotopy-ribbon disk in a homotopy four-ball.\n\n**Full solution or refutation.**\n\nThe general algorithmic decision questions remain unanswered; large finite searches and restricted classes do not constitute a universal algorithm.\n\n**What remains.**\n\nGive a terminating decision algorithm or prove undecidability for smooth sliceness, locally flat topological sliceness, and ribbonness, with the category made explicit.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 1.52. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Explicitly says there is still no algorithm and distinguishes computer searches from the restricted Casson-Long decision result.\n- Nathan M. Dunfield and Sherry Gong, Ribbon concordances and slice obstructions: experiments and examples, arXiv:2512.21825 (2025). (primary): https://arxiv.org/abs/2512.21825\n Evidence used: Explicitly states that no algorithm is known in either smooth or topological sliceness and reports the 19-crossing census.\n- Brendan Owens and Frank Swenton, An algorithm to find ribbon disks for alternating knots, Experimental Mathematics (published online 2023). (primary): https://doi.org/10.1080/10586458.2022.2158968\n Evidence used: Provides a useful ribbon-disk search in the alternating class, not a general decision algorithm.\n- A. J. Casson and D. D. Long, Algorithmic compression of surface automorphisms, Inventiones Mathematicae 81 (1985), 295-303. (primary): https://doi.org/10.1007/BF01389054\n Evidence used: Supplies the compression algorithm used in the restricted fibered homotopy-ribbon setting.\n\n**Review notes.** Material formulation ambiguity: the stored word slice does not specify the smooth or topologically locally flat category. K3's section context favors smooth sliceness, while current primary work explicitly separates both.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2712, "problem_number": "KP-1.53", "title": "Kirby Problem 1.53", "statement": "(a) Which knot properties are hereditary under ribbon concordance? Is the\nproperty of being alternating hereditary under ribbon concordance?\n(b) Which knot properties imply minimality under ribbon concordance? Are\npositive knots minimal under ribbon concordance?\n(c) Which knot invariants are monotone under ribbon concordance? Is cross-\ning number, bridge number, or braid index monotone under ribbon con-\ncordance?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.53.\n\nLiterature notes:\n(1) A knot property is a subset $\\mathcal{P}$ of the set of knots. It is hereditary under a\ndirected relation $\\mathcal{R}$ on the set of knots if $K_{1} \\in \\mathcal{P}, K_{0}\\mathcal{R}K_{1} \\Longrightarrow K_{0} \\in \\mathcal{P}$.\n(2) Using techniques of Silver and the solution of Rapoport’s conjecture given\nby Kochloukova, Miyazaki showed that the property of being fibered is\nhereditary under ribbon concordance [Koc06, Miy18, Sil92] in the sense\nthat if $K_{1}$ is ribbon concordant to $K_{0}$ and $K_{1}$ is fibered, then so is $K_{0}$; see\nalso work of Zemke [Zem19a]. It is conjectured that the property of being\nalternating is hereditary under ribbon concordance [GO23, Conjecture 9].\n(3) A knot $K_{1}$ is minimal under $\\mathcal{R}$ if $K_{0}\\mathcal{R}K_{1} \\Longrightarrow K_{1} = K_{0}$. It is conjectured\nthat positive knots are minimal under ribbon concordance [BG24, Con-\njecture 1.6]. Baker and Stoimenow have asked related questions [Bak16,\nSto15]. See also Problem 1.54.\n(4) Most knot invariants support natural directed relations $\\preceq$ on their codomains:\nnumerical (real-valued) invariants with respect to the standard ordering\n$\\leq$; polynomial invariants with respect to divisibility; group-valued invari-\nants with respect to injections; etc. A knot invariant $\\iota$ is monotone under\na directed relation $\\mathcal{R}$ if $K_{0}\\mathcal{R}K_{1} \\Longrightarrow \\iota(K_{0}) \\preceq \\iota(K_{1})$.\nSeveral knot invariants are known to be monotone under ribbon con-\ncordance. For example, if $K_{1}$ is ribbon concordant to $K_{0}$, the Alexander\npolynomial of $K_{0}$ divides that of $K_{1}$ [Gil84, FP20]. Zemke showed that\na ribbon concordance from $K_{1}$ to $K_{0}$ induces a bigraded injection from the\nknot Floer homology of $K_{0}$ to $K_{1}$, with the consequence that the Seifert\ngenus of $K_{0}$ is less than or equal to that of $K_{1}$ [Zem19a]. It is known that\nif $K_{1}$ has bridge number 2, then $K_{0}$ has bridge number at most 2 [GO23,\nParagraph after Conjecture 9].\n(5) Gordon’s pioneering paper on ribbon concordance contains several fasci-\nnating problems related to the ones raised here [Gor81].\n\nReferences cited:\n- [Koc06] Dessislava H. Kochloukova. Some Novikov rings that are von Neumann finite and knot-like groups. Comment. Math. Helv., 81(4):931–943, 2006. doi:10.4171/CMH/81.\n- [Miy18] Katura Miyazaki. A note on genera of band sums that are fibered. J. Knot Theory Ramifications, 27(12):1871002, 3, 2018. doi:10.1142/S0218216518710025.\n- [Sil92] D. S. Silver. On knot-like groups and ribbon concordance. J. Pure Appl. Algebra, 82(1):99–105, 1992. doi:10.1016/0022-4049(92)90013-6.\n- [Zem19a] Ian Zemke. Knot Floer homology obstructs ribbon concordance. Ann. of Math. (2), 190(3):931–947, 2019. doi:10.4007/annals.2019.190.3.5.\n- [GO23] Joshua Evan Greene and Brendan Owens. Alternating links, rational balls, and cube tilings, 2023. J. Eur. Math. Soc., to appear. arXiv:2212.06248.\n- [BG24] Joe Boninger and Joshua Evan Greene. Special alternating knots are band prime. Int. Math. Res. Not., 2024(10):8758–8763, 2024. doi:10.1093/imrn/rnae009.\n- [Bak16] Kenneth L. Baker. A note on the concordance of fibered knots. J. Topol., 9(1):1–4, 2016. doi:10.1112/jtopol/jtv024.\n- [Sto15] A. Stoimenow. Application of braiding sequences III: Concordance of positive knots. Internat. J. Math., 26(7):1550050, 36, 2015. doi:10.1142/S0129167X15500500.\n- [Gil84] Patrick M. Gilmer. Ribbon concordance and a partial order on S-equivalence classes. Topology Appl., 18(2-3):313–324, 1984. doi:10.1016/0166-8641(84)90016-6.\n- [FP20] Stefan Friedl and Mark Powell. Homotopy ribbon concordance and Alexander polynomials. Arch. Math. (Basel), 115(6):717–725, 2020. doi:10.1007/s00013-020-01517-5.\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Fiberedness, Alexander-polynomial divisibility, knot Floer homology, and Seifert genus have strong hereditary or monotonicity results, and positive-knot minimality is proved in broad subclasses, but the named alternating, positive, crossing, bridge, and braid questions remain open in general.\n\n**Verified partial progress.**\n\n- Fiberedness is hereditary under ribbon concordance in the direction specified by K3.\n- Zemke proves a split injection on knot Floer homology induced by ribbon concordance, implying monotonicity of Seifert genus; Alexander-polynomial divisibility is also known.\n- Abe-Tagami prove all tight fibered knots are minimal, and Boninger proves the positive-knot minimality conjecture for a large class of positive knots.\n- Baldwin, Hanselman, and Sivek prove in 2026 that every knot has only finitely many fibered predecessors under ribbon concordance.\n\n**Full solution or refutation.**\n\nThe open-ended classification has several strong answers, but alternating heredity, all-positive minimality, and general monotonicity of crossing number, bridge number, and braid index were not resolved.\n\n**What remains.**\n\nSettle downward closure of alternating knots, minimality of all positive knots, and monotonicity of crossing number, bridge number, and braid index beyond the known special cases.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 1.53. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the hereditary, minimality, and monotonicity program and summarizes fiberedness, Alexander polynomial, Floer-homology, genus, and bridge-two results.\n- Ian Zemke, Knot Floer homology obstructs ribbon concordance, Annals of Mathematics 190 (2019), 931-947. (primary): https://doi.org/10.4007/annals.2019.190.3.5\n Evidence used: Proves the split injection on knot Floer homology and consequent genus monotonicity.\n- Joe Boninger, Positive Knots and Ribbon Concordance, arXiv:2405.08103 (2024). (primary): https://arxiv.org/abs/2405.08103\n Evidence used: Proves minimality for a large class of positive knots while retaining the all-positive statement as a conjecture.\n- John A. Baldwin, Jonathan Hanselman, and Steven Sivek, Ribbon concordance and fibered predecessors, II: the general case, arXiv:2602.21109 (2026). (primary): https://arxiv.org/abs/2602.21109\n Evidence used: Proves every knot has only finitely many fibered predecessors under ribbon concordance.\n\n**Review notes.** Ribbon-concordance direction conventions vary in the literature; the report follows the K3 convention and avoids reversing the stated injection.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2713, "problem_number": "KP-1.54", "title": "Kirby Problem 1.54", "statement": "This problem is concerned with the restriction of the partial\nordering $\\geq$ coming from ribbon concordance to the concordance class $[K]$ of a knot\n$K$.\n(a) What can be said about the order type of $[K]$?\n(b) Does it depend on $[K]$? Must it contain a (unique) minimal element?\n(c) (Gordon) Does there exist a concordance class of knots that contains two\nminimal elements with respect to ribbon concordance?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.54.\n\nLiterature notes:\n(1) The ribbon concordance relation was defined by Gordon in [Gor81], who\nproves that some classes $[K]$ contain minimal elements, and asked Ques-\ntion (c).\n(2) The ribbon concordance relation was conjectured to be a partial ordering\nby Gordon. This was proved by Agol [Ago22]. Notice that for any knots\n$K$ and $L$, the map from $[K]$ to $[L]$ defined for $J \\in [K]$ by\n\n$$\nJ \\mapsto J \\# -K \\# L\n$$\n\nis an order preserving injection.\n(3) The Slice-Ribbon Conjecture can be formulated to say the unknot $\\operatorname{U}$ is\nthe unique minimal element of $[\\operatorname{U}]$; see Problem 1.50.\nSee Problem 1.56 for a Heegaard Floer theory perspective.\n\nReferences cited:\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.\n- [Ago22] Ian Agol. Ribbon concordance of knots is a partial ordering. Comm. Amer. Math. Soc., 2:374–379, 2022. doi:10.1090/cams/15.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ribbon concordance is now proved to be a partial order, and minimality and finiteness theorems constrain important subclasses, but the order types, universal uniqueness, and two-minimal-elements question remain open.\n\n**Verified partial progress.**\n\n- Agol proved Gordon's conjecture that ribbon concordance is a partial ordering on knots.\n- Gordon proved that some concordance classes contain minimal elements.\n- Abe and Tagami prove that every tight fibered knot is minimal in the ribbon-concordance order.\n- Baldwin, Hanselman, and Sivek prove that every knot has only finitely many fibered predecessors.\n\n**Full solution or refutation.**\n\nThe relation's partial-order property is settled, but no general order-type classification, universal existence/uniqueness theorem, or example of two minimal elements in one smooth concordance class was located.\n\n**What remains.**\n\nClassify or constrain the order type within concordance classes, decide whether each has a unique minimal element, and answer Gordon's two-minimal-elements question. Even uniqueness in the unknot's class contains the slice-ribbon conjecture.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 1.54. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the order-type and minimal-element questions, records Gordon's and Agol's results, and identifies the slice-ribbon special case.\n- Ian Agol, Ribbon concordance of knots is a partial ordering, Communications of the American Mathematical Society 2 (2022), 374-379. (primary): https://arxiv.org/abs/2201.03626\n Evidence used: Proves antisymmetry and hence the partial-order property.\n- Tetsuya Abe and Keiji Tagami, Ribbon concordance and the minimality of tight fibered knots, arXiv:2210.04044 (2022). (primary): https://arxiv.org/abs/2210.04044\n Evidence used: Proves that all tight fibered knots are minimal in the partial order.\n- John A. Baldwin, Jonathan Hanselman, and Steven Sivek, Ribbon concordance and fibered predecessors, II: the general case, arXiv:2602.21109 (2026). (primary): https://arxiv.org/abs/2602.21109\n Evidence used: Gives the current finiteness theorem for fibered predecessors of any knot.\n\n**Review notes.** Agol settles the prerequisite partial-order conjecture, not the order-type and uniqueness questions in this row.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2714, "problem_number": "KP-1.55", "title": "Kirby Problem 1.55", "statement": "Suppose that $C$ is a ribbon concordance from a fibered knot $K_{1}$\nto a fibered knot $K_{0}$.\n(a) Does the capped-off monodromy of $K_{1}$ (i.e. extended over a disk) extend to\na compression body whose lower boundary gives the capped-off monodromy\nof $K_{0}$?\n(b) Is $C$ fibered by compression bodies?\n(c) Suppose a fibered knot in a homology 3-sphere bounds a homotopy-ribbon\ndisk $D$ in a contractible 4-manifold $V$ . Is $V \\setminus\\nu(D)$ fibered by handlebodies?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.55.\n\nLiterature notes:\n(1) Part (a) holds when $K_{0}$ is the unknot by the main theorem of [CG83a].\n(2) Miller answers part (b) in the affirmative under the hypothesis that the\nribbon concordance is defined by a collection of ribbon bands that are\ntransverse to the fibration of the exterior of $K_{1}$ [Mil21, Theorem 1.9].\n(3) Part (c) was posed by Casson and Gordon in the Remark before Corol-\nlary 5.4 of [CG83a].\n\nReferences cited:\n- [CG83a] A. J. Casson and C. McA. Gordon. A loop theorem for duality spaces and fibred ribbon knots. Invent. Math., 74(1):119–137, 1983. doi:10.1007/BF01388533.\n- [Mil21] Maggie Miller. Extending fibrations of knot complements to ribbon disk complements. Geom. Topol., 25(3):1479–1550, 2021. doi:10.2140/gt.2021.25.1479.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The compression-body/fibration questions remain open in general, with affirmative special cases for an unknot lower end and for ribbon bands transverse to the fibration.\n\n**Verified partial progress.**\n\n- Casson--Gordon prove part (a) when K_0 is the unknot.\n- Miller proves part (b) when the ribbon bands are transverse to the fibration of the K_1 exterior.\n- Baldwin--Hanselman--Sivek (2026 preprint) prove that every knot in S^3 has only finitely many fibered predecessors under ribbon concordance; this constrains but does not fiber the concordance.\n\n**Full solution or refutation.**\n\nNo source checked removes the unknot or transverse-band hypotheses, and part (c) remains open.\n\n**What remains.**\n\nDecide (a) and (b) for arbitrary ribbon concordances between fibered knots and decide the handlebody-fibration question (c).\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 1.55 (AMS author's preliminary version, 2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States all three questions and records exactly the Casson--Gordon and Miller special cases.\n- Maggie Miller, Extending fibrations of knot complements to ribbon disk complements, Geom. Topol. 25 (2021), 1479--1550. (primary): https://doi.org/10.2140/gt.2021.25.1479\n Evidence used: Theorem 1.9 gives the transverse-band affirmative case of part (b).\n- John A. Baldwin, Jonathan Hanselman, and Steven Sivek, Ribbon concordance and fibered predecessors, II: the general case, arXiv:2602.21109 (2026). (primary): https://arxiv.org/abs/2602.21109\n Evidence used: Proves finiteness of fibered predecessors for each knot in S^3, a related 2026 advance that does not answer the fibration questions.\n\n**Review notes.** The newer predecessor-finiteness theorem is not conflated with a compression-body fibration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2715, "problem_number": "KP-1.56", "title": "Kirby Problem 1.56", "statement": "(Hom). If $K_{0}$ and $K_{1}$ are ribbon concordant and\n\n$$\n\\widehat{\\mathrm{HFK}}(K_{0}) \\cong \\widehat{\\mathrm{HFK}}(K_{1}),\n$$\n\nare $K_{0}$ and $K_{1}$ isotopic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.56.\n\nLiterature notes:\n(1) Recall that Zemke [Zem19a] proved that a ribbon concordance induces\na split injection on knot Floer homology.\nGordon asked the following\n[Gor81, Question 6.2]: If $K_{1} \\geq K_{2} \\geq...$ , does there exist some $m$\nsuch that $K_{n} = K_{m}$ for all $n \\geq m$? An affirmative answer to Problem\n1.56 would answer Gordon’s question in the affirmative. One large class of\nconcordant knots with the same knot Floer homology is given in [Wan22,\n\nTheorem 1.2] (see also [HW18, Theorem 1]), but none of these are ribbon\nconcordant to any other, by [Wan22, Theorem 1.8] and [LZ19].\n(2) The same question could be asked with Khovanov homology\n$\\widetilde{\\mathrm{Kh}}$ replacing knot Floer homology $\\widehat{\\mathrm{HFK}}$,\nor indeed other homology theories as treated\nin [Kan22a].\n\nReferences cited:\n- [Zem19a] Ian Zemke. Knot Floer homology obstructs ribbon concordance. Ann. of Math. (2), 190(3):931–947, 2019. doi:10.4007/annals.2019.190.3.5.\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.\n- [Wan22] Joshua Wang. The cosmetic crossing conjecture for split links. Geom. Topol., 26(7):2941–3053, 2022. doi:10.2140/gt.2022.26.2941.\n- [HW18] Matthew Hedden and Liam Watson. On the geography and botany of knot Floer homology. Selecta Math. (N.S.), 24(2):997–1037, 2018. doi:10.1007/s00029-017-0351-5.\n- [LZ19] Adam Simon Levine and Ian Zemke. Khovanov homology and ribbon concordances. Bull. Lond. Math. Soc., 51(6):1099–1103, 2019. doi:10.1112/blms.12303.\n- [Kan22a] Sungkyung Kang. Link homology theories and ribbon concordances. Quantum Topol., 13(1):183–205, 2022. doi:10.4171/qt/162.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether a ribbon concordance together with isomorphic hat knot Floer homology forces isotopy.\n\n**Verified partial progress.**\n\n- Zemke proves that a ribbon concordance induces a bigraded split injection on knot Floer homology.\n- Known large families of concordant knots with equal knot Floer homology do not yield counterexamples because distinct members are not ribbon concordant.\n- The 2026 fibered-predecessor finiteness theorem is compatible with but weaker than isotopy.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for the exact implication was found.\n\n**What remains.**\n\nShow that equality makes the ribbon-concordance map geometrically trivial, or find distinct ribbon-concordant knots with isomorphic hat knot Floer homology.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 1.56 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Poses the implication and distinguishes equal-HFK concordant families from ribbon-concordant examples.\n- Ian Zemke, Knot Floer homology obstructs ribbon concordance, Ann. of Math. 190 (2019), 931--947. (primary): https://doi.org/10.4007/annals.2019.190.3.5\n Evidence used: Proves split injectivity for ribbon concordance.\n- Baldwin, Hanselman, and Sivek, Ribbon concordance and fibered predecessors, II, arXiv:2602.21109 (2026). (primary): https://arxiv.org/abs/2602.21109\n Evidence used: Proves finiteness, not uniqueness, of fibered predecessors.\n\n**Review notes.** The exact question is kept open despite meaningful monotonicity and finiteness results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2716, "problem_number": "KP-1.57", "title": "Kirby Problem 1.57", "statement": "(a) In either the smooth or topological settings, are 0-shake slice knots slice?\n(b) Does there exist a knot $K$ whose topological 0-shake slice genus is strictly\nless than the topological slice genus?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.57.\n\nLiterature notes:\n(1) The smooth analogue of the second question was [Kir97, Problem 1.41],\nwhich was answered in the negative by Piccirillo in [Pic19]. Her construc-\ntion uses knots with diffeomorphic 0-traces whose smooth slice genera are\ndistinct. A negative answer to part (a) would imply the existence of a knot\nthat is slice in an integer homology ball but not in $B^{4}$; see Problem 1.60.\n(2) One may also ask analogous questions for $n$-traces of knots, where some\nnegative answers are known.\nFor more on this see [Akb77, Lic79,\nAkb93, AJOT13].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Pic19] Lisa Piccirillo. Shake genus and slice genus. Geom. Topol., 23(5):2665–2684, 2019. doi:10.2140/gt.2019.23.2665.\n- [Akb77] Selman Akbulut. On 2-dimensional homology classes of 4-manifolds. Math. Proc. Cambridge Philos. Soc., 82(1):99–106, 1977. doi:10.1017/S0305004100053718.\n- [Lic79] W. B. Raymond Lickorish. Shake-slice knots. In Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), volume 722 of Lecture Notes in Math., pages 67–70. Springer, Berlin, 1979.\n- [Akb93] S. Akbulut. Knots and exotic smooth structures on 4-manifolds. J. Knot Theory Ramifications, 2(1):1–10, 1993. doi:10.1142/S0218216593000027.\n- [AJOT13] Tetsuya Abe, In Dae Jong, Yuka Omae, and Masanori Takeuchi. Annulus twist and diffeomorphic 4-manifolds. Math. Proc. Cambridge Philos. Soc., 155(2):219–235, 2013. doi:10.1017/S0305004113000194.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The topological questions remain open; the smooth shake-genus analogue has counterexamples, while a preprint claim for the smooth 0-shake-slice implication is not treated as accepted.\n\n**Verified partial progress.**\n\n- Piccirillo constructs knots showing that smooth 0-shake genus and smooth slice genus can differ.\n- Akbulut--Yildiz arXiv:2012.14144v14 claims that smooth 0-shake slice implies smooth slice.\n- An earlier Akbulut--Yildiz version was withdrawn explicitly for an error, and the authoritative 2026 K3 list continues to pose the smooth and topological implication as open.\n\n**Full solution or refutation.**\n\nOnly the smooth analogue of part (b) is securely settled; no verified solution to the stated topological part or part (b) was found.\n\n**What remains.**\n\nObtain expert/refereed validation of the smooth claim and decide both the topological 0-shake-slice implication and the topological genus-gap question.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 1.57 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Still poses both smooth/topological part (a) and topological part (b), while recording Piccirillo's smooth genus result.\n- Lisa Piccirillo, Shake genus and slice genus, Geom. Topol. 23 (2019), 2665--2684. (primary): https://doi.org/10.2140/gt.2019.23.2665\n Evidence used: Supplies smooth-category examples with different shake and slice genera.\n- Selman Akbulut and Eylem Zeliha Yildiz, On Shake Slice Knots, arXiv:2012.14144v14 (2024 revision). (primary): https://arxiv.org/abs/2012.14144\n Evidence used: Claims that 0-shake slice knots are slice; the claim is recorded conservatively because current authoritative status evidence does not accept it.\n- Akbulut and Yildiz, 0-Shake Slice Implies Slice, arXiv:2012.11176 (withdrawn). (primary): https://arxiv.org/abs/2012.11176\n Evidence used: The arXiv record says the predecessor was withdrawn due to an error.\n\n**Review notes.** Do not label the unrefereed, historically revised claim as a solution without expert validation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2717, "problem_number": "KP-1.58", "title": "Kirby Problem 1.58", "statement": "What concordance information about a knot $K$ is contained in\nits 0-trace $X_{0}(K)$ and in its 0-surgery $S^{3}_{0}(K)$?\nSpecifically,\n(a) Suppose $K$ and $K'$ have homeomorphic 0-traces.\n(i) Are $K$ and $K'$ topologically concordant?\n(ii) What if one just assumes $K$ and $K'$ have homeomorphic 0-surgeries?\n(iii) What if one just assumes $K$ and $K'$ have 0-surgeries that are topolog-\nically homology cobordant, preserving the homology class of a merid-\nian.\n\n(b) Suppose $K$ and $K'$ have homeomorphic 0-surgeries, where the image of\nthe meridian of $K$ is freely homotopic to the meridian of $K'$. Are $K$ and\n$K'$ smoothly concordant?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.58.\n\nLiterature notes:\n(1) This problem originates in a conjecture of Akbulut and Kirby [Kir97,\nProblem 1.19] that knots with the same 0-surgery are concordant. It was\nmotivated, at least in part, by the fact [KM78] that it holds when one\nof the knots is unknotted. (The result of [KM78] predates the proof of\nProperty R, which is of course much stronger.)\nAlthough the original\nconjecture is false, there are several variations, as one can ask about the\nhomeomorphism or diffeomorphism type of $X_{0}(K)$ and about smooth or\ntopological concordance.\n(2) All of the questions in part (a) have negative answers if one is looking for\nsmooth concordances; see [CFHH13], [Yas17], [MP18]. With reference\nto Question (a)(iii) and part (b), note that a concordance would preserve\nthe free homotopy class (and hence homology class) of a meridian.\n(3) Part (b) has a negative answer without the assumption on the meridian,\nsee [Yas17]. There are no examples in the literature of homeomorphisms,\nsatisfying the assumption on the meridian, between 0-surgeries on distinct\nknots.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [KM78] Robion Kirby and Paul Melvin. Slice knots and property R. Invent. Math., 45(1):57– 59, 1978. doi:10.1007/BF01406223.\n- [CFHH13] Tim D. Cochran, Bridget D. Franklin, Matthew Hedden, and Peter D. Horn. Knot concordance and homology cobordism. Proc. Amer. Math. Soc., 141(6):2193–2208, 2013. doi:10.1090/S0002-9939-2013-11471-1.\n- [Yas17] Kouichi Yasui. Corks, exotic 4-manifolds and knot concordance, 2017. arXiv:1505.02551.\n- [MP18] Allison N. Miller and Lisa Piccirillo. Knot traces and concordance. J. Topol., 11(1):201–220, 2018. doi:10.1112/topo.12054.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Smooth analogues of the trace/surgery questions and the no-meridian version of part (b) have counterexamples, but the stated topological and meridian-preserving questions remain open.\n\n**Verified partial progress.**\n\n- Cochran--Franklin--Hedden--Horn, Yasui, and Miller--Piccirillo give negative smooth-concordance results for variants in part (a).\n- Yasui disproves part (b) when the meridian free-homotopy condition is removed.\n- The 2026 K3 source reports no distinct-knot homeomorphism of 0-surgeries satisfying the meridian condition.\n\n**Full solution or refutation.**\n\nThe extra topological and meridian hypotheses are essential and no checked source decides them.\n\n**What remains.**\n\nDecide (a)(i)--(iii) topologically and part (b) under its stated meridian-free-homotopy condition.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 1.58 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Separates the still-open formulations from known smooth and no-meridian counterexamples.\n- Allison N. Miller and Lisa Piccirillo, Knot traces and concordance, J. Topol. 11 (2018), 201--220. (primary): https://doi.org/10.1112/topo.12054\n Evidence used: Provides smooth-concordance counterexamples among knots with related traces.\n- Kouichi Yasui, Corks, exotic 4-manifolds and knot concordance, arXiv:1505.02551 (2017). (primary): https://arxiv.org/abs/1505.02551\n Evidence used: Gives counterexamples without the meridian-preservation requirement.\n\n**Review notes.** Negative smooth results are not promoted to negative answers to the topological questions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2718, "problem_number": "KP-1.59", "title": "Kirby Problem 1.59", "statement": "Let $K \\subset S^{3}$ be a slice knot.\n(a) Determine the set $\\mathcal{R}(K)$ of ribbon disks bounded by $K$ modulo isotopy.\n(b) Determine the set $\\mathcal{D}(K)$ of slice disks bounded by $K$ modulo isotopy and\nlocal knotting.\n(c) Determine the set $\\mathcal{H}\\mathcal{R}(K)$ of homotopy-ribbon disks bounded by $K$ modulo\nisotopy.\n(d) If $K$ is fibered, determine the set $\\mathcal{F}\\mathcal{H}\\mathcal{R}(K)$ of fibered, homotopy-ribbon\ndisks bounded by $K$ in some homotopy 4-ball, modulo isotopy.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.59.\n\nLiterature notes:\n(1) There is no knot for which part (a) has been solved. A knot with $|\\mathcal{R}(K)| =$\n0 would be a counter-example to the Slice-Ribbon Conjecture. It is un-\nknown whether or not $|\\mathcal{R}(\\operatorname{U})| = 1$ for the unknot $\\operatorname{U}$, though Scharlemann\nhas shown that any non-standard ribbon disk for the unknot must have at\nleast three minima [Sch85a]. It is known that $\\mathcal{R}(Q_{p,2})$ is infinite [MZ25],\nwhere $Q_{p,q} = T_{p,q}\\#\\overline{T}_{p,q}$ is the connected sum of a torus knot with its mir-\nror.\n(2) There are many knots that bound isotopic ribbon disks that are dis-\ntinct when considered up to isotopy rel. boundary: Any knot of the form\n$J\\# - J$ bounds infinitely many isotopic ribbon disks that are pairwise\nnon-isotopic rel. boundary, as follows from Zeeman’s twist-spinning con-\nstruction [Zee65]; see, for example, [MZ25, Proposition 2.2]. Addition-\nally, many simple knots such as the stevedore knot $6_{1}$ and pretzel knots\n\n$P(p, -p, p)$ bound pairs of such disks. The problems of studying disks\nmodulo isotopy and modulo isotopy rel. boundary are closely related via\nan understanding of the symmetries of the boundary knot; see the proofs\nof [JZ20b, Lemma 2.2], [MZ25, Theorem 1.5], and [MM25b, Theo-\nrem 4.2].\n(3) The set $\\mathcal{D}(K)$ has recently been studied using modern invariants [JZ20b],\nclassical invariants [MP19], and geometric techniques [MM25b]. It is\nimmediate that $|\\mathcal{D}(\\operatorname{U})| = 1$ for $\\operatorname{U}$ the unknot, but $\\mathcal{D}(K)$ is not known for\nany nontrivial knot $K$.\n(4) Consider the map $\\iota_{K}: \\mathcal{H}\\mathcal{R}(K) \\to \\mathcal{D}(K)$. Is there a knot $K$ such that\nthis map is injective, surjective, or has finite image? See [MM25b, Sec-\ntion 1.1]. There is no knot for which part (c) has been solved.\n(5) Given the apparent difficulty in solving the above problems, it is rea-\nsonable to look for more tractable alternatives in which there is more\nstructure, such as part (d).\nCasson and Gordon found a beautiful connection between fibered,\nhomotopy-ribbon disks bounded by a fibered knot $K$ and handlebody\nextensions of the closed monodromy of $K$ [CG83a].\nHere, the extra\nstructure yields more positive results. First, $|\\mathcal{F}\\mathcal{H}\\mathcal{R}(\\operatorname{U})| = 1$. Second, for\nall coprime $p > q > 1$, the set $\\mathcal{F}\\mathcal{H}\\mathcal{R}(Q_{p,q})$ has been explicitly determined;\nthe set $\\mathcal{F}\\mathcal{H}\\mathcal{R}_{\\partial}(Q_{p,q})$ of fibered, homotopy-ribbon disks modulo isotopy\nrel. boundary is in explicit bijection with $\\{c/d \\in \\mathbb{Q}: c$ even $\\}$ [MZ25].\n\nReferences cited:\n- [Sch85a] Martin Scharlemann. Smooth spheres in $\\mathbb{R}^{4}$ with four critical points are standard. Invent. Math., 79(1):125–141, 1985. doi:10.1007/BF01388659.\n- [MZ25] Jeffrey Meier and Alexander Zupan. Knots bounding nonisotopic ribbon disks. J. Topol., 18(4):Paper No. e70047, 18, 2025. doi:10.1112/topo.70047.\n- [Zee65] E. C. Zeeman. Twisting spun knots. Trans. Amer. Math. Soc., 115:471–495, 1965. doi:10.2307/1994281.\n- [JZ20b] András Juhász and Ian Zemke. Distinguishing slice disks using knot Floer homology. Selecta Math. (N.S.), 26(1):Paper No. 5, 18, 2020. doi:10.1007/s00029-019-0531-6.\n- [MM25b] Jeffrey Meier and Allison N. Miller. Slice disks modulo local knotting, 2025. arXiv: 2503.09870.\n- [MP19] Allison N. Miller and Mark Powell. Stabilization distance between surfaces. Enseign. Math., 65(3-4):397–440, 2019. doi:10.4171/lem/65-3/4-4.\n- [CG83a] A. J. Casson and C. McA. Gordon. A loop theorem for duality spaces and fibred ribbon knots. Invent. Math., 74(1):119–137, 1983. doi:10.1007/BF01388533.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Disk-classification sets are explicitly determined in important fibered homotopy-ribbon families, and infinite families are known, but no general classification exists.\n\n**Verified partial progress.**\n\n- Meier--Zupan classify fibered homotopy-ribbon disks for every generalized square knot Q_{p,q} and prove infinitely many ribbon disks when q=2.\n- For relative-boundary isotopy, their family is explicitly parameterized by rationals c/d with c even.\n- Meier--Miller give knots, including fibered hyperbolic knots, with infinitely many disks remaining distinct modulo local knotting.\n\n**Full solution or refutation.**\n\nPart (d) has strong family-level classifications, while (a)--(c) remain largely open; D(K) is not completely known for any nontrivial knot.\n\n**What remains.**\n\nDetermine R(K), D(K), or HR(K) for a strategically simple nontrivial knot and extend the FHR classifications beyond known families.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 1.59 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Catalogues what is and is not determined for R, D, HR, and FHR.\n- Jeffrey Meier and Alexander Zupan, Knots bounding nonisotopic ribbon disks, J. Topol. 18 (2025), e70047. (primary): https://doi.org/10.1112/topo.70047\n Evidence used: Classifies fibered homotopy-ribbon disks for generalized square knots and produces infinite ribbon-disk families.\n- Jeffrey Meier and Allison N. Miller, Slice disks modulo local knotting, arXiv:2503.09870 (2025). (primary): https://arxiv.org/abs/2503.09870\n Evidence used: Produces infinite families distinguished even modulo local knotting and gives further fibered examples.\n\n**Review notes.** The classification result for FHR is a substantive partial answer, not a solution of the four-part general problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2719, "problem_number": "KP-1.60", "title": "Kirby Problem 1.60", "statement": "Is there a knot in $S^{3}$ that is not smoothly slice in $B^{4}$ but is\nsmoothly slice in an integer homology ball? What about a $\\mathbb{Z}$/2-homology ball?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.60.\n\nLiterature notes:\n(1) As discussed in [FGMW10], one strategy to disprove the smooth Poincaré\nconjecture is to find a homotopy four-sphere $X$ and a knot $K$ in $S^{3}$ that is\nsmoothly slice in $X - B^{4}$ but not in $B^{4}$; see also Problem 4.1. It is worth\nremarking that just about all known slice obstructions necessarily vanish\nfor a knot that is slice in an integer homology ball. One invariant that\ncan potentially make this distinction is the $s$-invariant from Khovanov\nhomology [Ras10]. As a warm-up, it would be interesting to find a knot\n$K$ with $s(K) \\neq 0$ that is smoothly slice in a rational homology ball.\n(2) Fintushel-Stern showed the figure-eight knot is smoothly slice in a rational\nhomology ball with $\\pi_{1} = \\mathbb{Z}/2$ [FS84], but is not even topologically slice\nin $B^{4}$ by the Fox-Milnor condition. For all known examples of knots that\nare slice in a rational ball but not $B^{4}$, the rational ball has 2-torsion\nin $H_{1}$. Kawauchi shows that all strongly negative-amphichiral knots are\nrationally slice [Kaw09] and Levine showed they are slice in the same\nrational homology ball $Z_{0}$ [Lev23]; in fact, he asks if it is possible that\nall rationally slice knots are slice in boundary sums of $Z_{0}$.\n\nReferences cited:\n- [FGMW10] Michael Freedman, Robert Gompf, Scott Morrison, and Kevin Walker. Man and machine thinking about the smooth 4-dimensional Poincaré conjecture. Quantum Topol., 1(2):171–208, 2010. doi:10.4171/QT/5.\n- [Ras10] Jacob Rasmussen. Khovanov homology and the slice genus. Invent. Math., 182(2):419–447, 2010. doi:10.1007/s00222-010-0275-6.\n- [FS84] Ronald Fintushel and Ronald J. Stern. A µ-invariant one homology 3-sphere that bounds an orientable rational ball. In Four-manifold theory (Durham, N.H., 1982), volume 35 of Contemp. Math., pages 265–268. Amer. Math. Soc., Providence, RI, 1984. doi:10.1090/conm/035/780582.\n- [Kaw09] Akio Kawauchi. Rational-slice knots via strongly negative-amphicheiral knots. Commun. Math. Res., 25(2):177–192, 2009.\n- [Lev23] Adam Simon Levine. A note on rationally slice knots. New York J. Math., 29:1363– 1372, 2023. https://nyjm.albany.edu/j/2023/29-52p.pdf.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No knot is known to be nonslice in B^4 yet smoothly slice in an integer or mod-2 homology ball; rational-homology-ball examples do not meet either hypothesis.\n\n**Verified partial progress.**\n\n- Fintushel--Stern show that the figure-eight knot is smoothly slice in a rational homology ball with fundamental group Z/2 but is not topologically slice in B^4.\n- Kawauchi and Levine give broad rationally slice families arising from strongly negative amphichiral knots.\n- The known rational balls have 2-torsion in H_1, which prevents them from being mod-2 homology balls.\n\n**Full solution or refutation.**\n\nThe rational examples are close analogues, not answers to the integral or mod-2 questions.\n\n**What remains.**\n\nConstruct an example in either stated class or prove that smooth sliceness in such a homology ball forces smooth sliceness in B^4.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 1.60 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Retains both questions and explains that all known rational examples use 2-torsion.\n- Ronald Fintushel and Ronald J. Stern, A mu-invariant one homology 3-sphere that bounds an orientable rational ball, Contemp. Math. 35 (1984), 265--268. (primary): https://doi.org/10.1090/conm/035/780582\n Evidence used: Supplies the figure-eight rational-sliceness example discussed by K3.\n- Adam Simon Levine, A note on rationally slice knots, New York J. Math. 29 (2023), 1363--1372. (primary): https://nyjm.albany.edu/j/2023/29-52p.pdf\n Evidence used: Shows strongly negative amphichiral knots slice in a common rational homology ball and poses a structural rational-sliceness question.\n\n**Review notes.** A rational ball with H_1 containing Z/2 has nonzero H_1 with F_2 coefficients and is not a Z/2-homology ball.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2720, "problem_number": "KP-1.61", "title": "Kirby Problem 1.61", "statement": "A knot in $S^{3}$ bounds a topological disk in $B^{4}$ by coning (not\nnecessarily locally flat); this problem asks about topological disks that a knot in a\nhomology sphere $Y$ might bound in a homology or homotopy ball bounded by $Y$ .\n(a) Given a locally flat knot in an integer homology 3-sphere, is it concordant\nto a knot in $S^{3}$, via a locally flat annulus in a topological integer homology\ncobordism? Variation: Can the homology cobordism be taken to be simply\nconnected?\n(b) Given an arbitrary knot in an integer homology 3-sphere $Y$ , does it bound\nan embedded disc (not necessarily locally flat) in $\\Delta$, the Freedman filling\nof $Y$ ? Variation: Is there such a disk in a homology ball bounded by $Y$ ?\n(c) Is there a link in an integer homology 3-sphere, each of whose components\nis independently concordant to a knot in $S^{3}$, that is not itself concordant\nto a link in $S^{3}$.\n(d) Suppose that $\\mathbb{Z}_{n}$ acts semi-freely on the homology sphere $Y$ , with fixed\npoint set a knot $K$. Does the action of $\\mathbb{Z}_{n}$ extend semi-freely over $\\Delta$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.61.\n\nLiterature notes:\n(1) By the proof of [Fre82, Theorem 1.4'] every integer homology 3-sphere\nbounds a compact, contractible 4-manifold $\\Delta$, which we call the Freed-\nman filling of $Y$ . It is known to be unique up to homeomorphism, by\nFreedman’s $h$-cobordism theorem [Fre82, Theorems 1.3 and 1.6].\n(2) The smooth analogue of (a) was disproved by A. Levine [Lev16], answer-\ning Problem 1.31 in [Kir97]; see also [HLL22b, Zho21]. An affirmative\nanswer to (a) gives an affirmative answer to (b) for locally flat knots,\nvia an embedding with an isolated, piecewise linear singularity obtained\nby coning off the knot in $S^{3}$.\nNote that topological embeddings need\nnot be approximable by piecewise linear or locally flat embeddings. Such\na wild topological embedding of a disk was constructed by Giffen using\na shift-spinning construction in unpublished work, explained in [DV09,\nSection 6.6]. Recent work of Davis [Dav20, Dav25] gives evidence in\nsupport of an affirmative answer to (a), in terms of the solvable filtration\nof the knot concordance group [COT03].\n(3) Part (d) is related to questions about extending topological group actions,\nwithout any requirement of local linearity.\nRecall [FQ90] that a free\naction of a cyclic group on a homology sphere $Y$ extends to an action\non the Freedman filling $\\Delta$ of $Y$ with one fixed point; the action is not\nnecessarily locally linear at that point.\nSuppose instead that $\\mathbb{Z}_{n}$ acts\nsemi-freely on $Y$ , with fixed point set a knot $K$. If $n$ is a prime, say $p$,\nthen the fixed point set of the extension would be a $\\mathbb{Z}_{p}$-homology disk\nwith boundary equal to $K$. So in some weak sense, $K$ would be slice in\n$\\Delta$. This problem is rather different in character from the smooth version,\ndiscussed in problem 3.75.\n\nReferences cited:\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [Lev16] Adam Simon Levine. Nonsurjective satellite operators and piecewise-linear concordance. Forum Math. Sigma, 4:Paper No. e34, 47, 2016. doi:10.1017/fms.2016.31.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [HLL22b] Jennifer Hom, Adam Simon Levine, and Tye Lidman. Knot concordance in homology cobordisms. Duke Math. J., 171(15):3089–3131, 2022. doi:10.1215/00127094-2021-0110.\n- [Zho21] Hugo Zhou. Homology concordance and an infinite rank free subgroup. J. Topol., 14(4):1369–1395, 2021. doi:10.1112/topo.12211.\n- [DV09] Robert J. Daverman and Gerard A. Venema. Embeddings in manifolds, volume 106 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2009. doi:10.1090/gsm/106.\n- [Dav20] Christopher W. Davis. Topological concordance of knots in homology spheres and the solvable filtration. J. Topol., 13(1):343–355, 2020. doi:10.1112/topo.12126.\n- [Dav25] Christopher W. Davis. Whitney tower concordance and knots in homology spheres. Algebr. Geom. Topol., 25(6):3503–3521, 2025. doi:10.2140/agt.2025.25.3503.\n- [COT03] Tim D. Cochran, Kent E. Orr, and Peter Teichner. Knot concordance, Whitney towers and L2-signatures. Ann. of Math. (2), 157(2):433–519, 2003. doi:10.4007/annals.2003.157.433.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** All four topological questions remain open; filtration results give strong evidence for part (a), while smooth analogues fail.\n\n**Verified partial progress.**\n\n- The smooth analogue of (a) is false by Levine and later strengthened by Hom--Levine--Lidman and Zhou.\n- Davis proves that modulo every term of the Whitney-tower filtration, every knot or link in a homology sphere is equivalent to one in S^3.\n- Freedman--Quinn extend free cyclic actions with one fixed point, which is weaker and geometrically different from the semifree fixed-disk extension in (d).\n\n**Full solution or refutation.**\n\nNo checked result resolves an exact subpart; the Davis theorem is finite-stage evidence rather than a limiting concordance theorem.\n\n**What remains.**\n\nResolve the topological homology-concordance, wild-disk, link, and semifree-action questions, including the simply connected variation of (a).\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 1.61 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States parts (a)--(d), the smooth counterexamples, the filtration evidence, and the weaker action-extension theorem.\n- Christopher W. Davis, Whitney tower concordance and knots in homology spheres, Algebr. Geom. Topol. 25 (2025), 3503--3521. (primary): https://doi.org/10.2140/agt.2025.25.3503\n Evidence used: Proves representability by knots/links in S^3 modulo each term of the Whitney-tower filtration.\n- Adam Simon Levine, Nonsurjective satellite operators and piecewise-linear concordance, Forum Math. Sigma 4 (2016), e34. (primary): https://doi.org/10.1017/fms.2016.31\n Evidence used: Disproves the smooth analogue of part (a).\n\n**Review notes.** Finite-stage filtration indistinguishability is not treated as topological concordance.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2721, "problem_number": "KP-1.62", "title": "Kirby Problem 1.62", "statement": "(a) Are all good boundary links topologically slice? Freely topologically slice?\n(b) A special case of interest: Is the Whitehead double of the Borromean rings\n(with any choice of clasps) topologically slice?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.62.\n\nLiterature notes:\n(1) A boundary link $(L, \\varphi)$ is an $n$-component link $L \\subseteq S^{3}$ together with a\nsurjection $\\varphi: \\pi_{1}(S^{3}\\setminus L) \\twoheadrightarrow F_{n}$, sending a set of meridians of $L$ to a free\ngenerating set of the free group on $n$ generators. A link $L$ admits such\na homomorphism $\\varphi$ if and only if the components of $L$ bound pairwise\ndisjoint Seifert surfaces in $S^{3}$ [Smy66] (see also [CS80, Proposition 1.1]).\nFollowing [FQ90, Section 11.7C], a boundary link $(L, \\varphi)$ is said to be a\ngood boundary link if the kernel of $\\varphi$ is a perfect group. Note that a 1-\ncomponent good boundary link is a knot with Alexander polynomial one.\nWhitehead doubles of links with vanishing pairwise linking numbers are\ngood boundary links.\n(2) Part (a) is posed in [Kir97, Problem 1.36], with a slightly different def-\ninition of good boundary links than appears here. The first half of that\nproblem asked whether Alexander polynomial one knots are topologically\nslice. The answer to that question is now known to be yes [GT04], [FQ90,\nTheorem 11.7B], [Fre84, Theorem 7] (see also [BPR21, Theorem 1.14]\nand [PR21, Section 21.6.3]).\n(3) If the surgery sequence is exact (see Problem 4.46), then a link in $S^{3}$\nis freely slice if and only if it is a good boundary link [FQ90, Corol-\nlary 11.7C]. More surprisingly, the topological surgery sequence in dimen-\nsion four is defined and exact if and only if all good boundary links are\nfreely slice [FQ90, Corollary 12.3C] (see also [KOPR21a, Section 23.2.1]).\n(4) [CKP20, Corollary 2.2] gives a characterization of good boundary links\nin terms of Seifert matrices. This is very useful in confirming whether\na given boundary link is good. For example, this gives an easy method\nto verify that Whitehead doubles of links with vanishing pairwise linking\nnumbers are good boundary links.\n(5) Several families of good boundary links have been shown to be freely\ntopologically slice. The most general result is due to Cha, Kim, and Pow-\nell [CKP20, Theorem A], generalizing previous work focusing on White-\nhead doubles, e.g. in [FT95b].\n(6) Not all good boundary links are smoothly slice, even with more than one\ncomponent.\nFor example, Levine [Lev12] showed that the Whitehead\ndouble of the Borromean rings, with all positive clasps, is not smoothly\nslice. The question of whether the Whitehead double of the Borromean\nrings is topologically slice appeared as Problem 4.46 in [Kir97], and we\nrestate it here in part (b) as a special case of the problem.\n\nQuestion. Is the Whitehead double of the Borromean rings (with any\nchoice of clasps) topologically slice?\n\nReferences cited:\n- [Smy66] N Smythe. Boundary links. In Topology Seminar, Wisconsin, 1965, volume 60 of Annals of Mathematics Studies, pages 69–72. Princeton University Press, Princeton, N.J., 1966. Edited by R. H. Bing and R. J. Bean.\n- [CS80] Sylvain E. Cappell and Julius L. Shaneson. Link cobordism. Comment. Math. Helv., 55(1):20–49, 1980. doi:10.1007/BF02566673.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GT04] Stavros Garoufalidis and Peter Teichner. On knots with trivial Alexander polynomial. J. Differential Geom., 67(1):167–193, 2004. http://projecteuclid.org/euclid.jdg/1099587731.\n- [Fre84] Michael H. Freedman. The disk theorem for four-dimensional manifolds. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 647–663. PWN, Warsaw, 1984.\n- [BPR21] Stefan Behrens, Mark Powell, and Arunima Ray. Context for the disc embedding theorem. In The disc embedding theorem, pages 1–26. Oxford Univ. Press, Oxford, 2021.\n- [PR21] Mark Powell and Arunima Ray. The development of topological 4-manifold theory. In The disc embedding theorem, pages 295–330. Oxford Univ. Press, Oxford, 2021.\n- [KOPR21a] Min Hoon Kim, Patrick Orson, JungHwan Park, and Arunima Ray. Good groups. In The disc embedding theorem, pages 273–282. Oxford Univ. Press, Oxford, 2021.\n- [CKP20] Jae Choon Cha, Min Hoon Kim, and Mark Powell. A family of freely slice good boundary links. Math. Ann., 376(3-4):1009–1030, 2020. doi:10.1007/s00208-019-01907-3.\n- [FT95b] Michael H. Freedman and Peter Teichner. 4-manifold topology. II. Dwyer’s filtration and surgery kernels. Invent. Math., 122(3):531–557, 1995. doi:10.1007/BF01231455.\n- [Lev12] Adam Simon Levine. Slicing mixed Bing-Whitehead doubles. J. Topol., 5(3):713– 726, 2012. doi:10.1112/jtopol/jts019.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The one-component case and broad families are freely topologically slice, but the all-good-boundary-link assertion and the Whitehead-doubled Borromean-rings case remain open.\n\n**Verified partial progress.**\n\n- A one-component good boundary link is an Alexander-polynomial-one knot and is topologically slice by Freedman's disk theorem.\n- Cha--Kim--Powell prove a broad family of good boundary links freely topologically slice.\n- Levine proves that the all-positive Whitehead double of the Borromean rings is not smoothly slice, which does not obstruct topological sliceness.\n\n**Full solution or refutation.**\n\nThe general freely-slice assertion is equivalent to exactness of the four-dimensional topological surgery sequence and is not known.\n\n**What remains.**\n\nSettle free topological sliceness for all good boundary links, especially every clasp choice for the Whitehead double of the Borromean rings.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 1.62 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the surgery-equivalence formulation, known one-component result, known families, and open Borromean special case.\n- Jae Choon Cha, Min Hoon Kim, and Mark Powell, A family of freely slice good boundary links, Math. Ann. 376 (2020), 1009--1030. (primary): https://doi.org/10.1007/s00208-019-01907-3\n Evidence used: Theorem A proves free topological sliceness for a substantial family.\n- Adam Simon Levine, Slicing mixed Bing-Whitehead doubles, J. Topol. 5 (2012), 713--726. (primary): https://doi.org/10.1112/jtopol/jts019\n Evidence used: Gives the smooth nonsliceness result for the positive-clasp Borromean example, not a topological obstruction.\n\n**Review notes.** The smooth obstruction is explicitly not used to answer the topological question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2722, "problem_number": "KP-1.63", "title": "Kirby Problem 1.63", "statement": "Is there a knot type with Legendrian representatives that do\nnot destabilize but have arbitrarily negative Thurston–Bennequin number?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.63.\n\nLiterature notes:\n(1) This is Etnyre-Ng [EN03, Question 44]; see also [CGH09, Question 61].\n\n(2) It is known by work of Etnyre and Honda that for any $N$ there is a\nknot type having peaks of height $N$ in its Legendrian mountain range\n[EH03]. For any choice of knot type and Thurston-Bennequin number,\nthere are finitely many distinct Legendrian representatives, so this ques-\ntion is equivalent to asking if there is a knot type with infinitely many\nnon-destabilizable representatives. This has been proved for many knot\ntypes, including the unknot [EF09], torus knots [EH01a], cables of torus\nknots [ELT12, LaF10], and twist knots [ENV13].\n(3) The finiteness of non-destabilizable Legendrian representatives problem\nis equivalent to whether there are finitely many non-destabilizable grid\ndiagrams for a given knot [DP13].\nIt is proved in [Dyn06] that any\nnon-minimal rectangular diagram of the unknot can be destabilized. This\ngives an algorithm to detect the unknot using monotonic simplification.\nA first step towards generalizing this result to arbitrary topological knots\nis to answer the question above.\n\nReferences cited:\n- [EN03] John B. Etnyre and Lenhard L. Ng. Problems in low dimensional contact topology. In Topology and geometry of manifolds (Athens, GA, 2001), volume 71 of Proc. Sympos. Pure Math., pages 337–357. Amer. Math. Soc., Providence, RI, 2003. doi: 10.1090/pspum/071/2024641.\n- [CGH09] Vincent Colin, Emmanuel Giroux, and Ko Honda. Finitude homotopique et isotopique des structures de contact tendues. Publ. Math. Inst. Hautes Études Sci., 109:245–293, 2009. doi:10.1007/s10240-009-0022-y.\n- [EH03] John B. Etnyre and Ko Honda. On connected sums and Legendrian knots. Adv. Math., 179(1):59–74, 2003. doi:10.1016/S0001-8708(02)00027-0.\n- [EF09] Yakov Eliashberg and Maia Fraser. Topologically trivial Legendrian knots. J. Symplectic Geom., 7(2):77–127, 2009. http://projecteuclid.org/euclid.jsg/1239974381.\n- [EH01a] John B. Etnyre and Ko Honda. Knots and contact geometry. I. Torus knots and the figure eight knot. J. Symplectic Geom., 1(1):63–120, 2001. http://projecteuclid.org/euclid.jsg/1092316299.\n- [ELT12] John B. Etnyre, Douglas J. LaFountain, and Bülent Tosun. Legendrian and transverse cables of positive torus knots. Geom. Topol., 16(3):1639–1689, 2012. doi:10.2140/gt.2012.16.1639.\n- [LaF10] Douglas J. LaFountain. Studying uniform thickness. I. Legendrian simple iterated torus knots. Algebr. Geom. Topol., 10(2):891–916, 2010. doi:10.2140/agt.2010.10.891.\n- [ENV13] John B. Etnyre, Lenhard L. Ng, and Vera Vértesi. Legendrian and transverse twist knots. J. Eur. Math. Soc. (JEMS), 15(3):969–995, 2013. doi:10.4171/JEMS/383.\n- [DP13] I. A. Dynnikov and M. V. Prasolov. Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions. Trans. Moscow Math. Soc., pages 97–144, 2013. doi:10.1090/s0077-1554-2014-00210-7.\n- [Dyn06] I. A. Dynnikov. Arc-presentations of links: monotonic simplification. Fund. Math., 190:29–76, 2006. doi:10.4064/fm190-0-3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No fixed knot type is known to have nondestabilizable Legendrian representatives with Thurston--Bennequin numbers unbounded below.\n\n**Verified partial progress.**\n\n- Etnyre--Honda show that for each N there exists some knot type whose Legendrian mountain range has N peaks; the knot type varies with N.\n- For a fixed knot type and fixed Thurston--Bennequin number there are finitely many Legendrian isotopy classes, making the question equivalent to infinitely many nondestabilizable representatives in one knot type.\n- The unknot, torus knots, certain cables, and twist knots have classification/finiteness results and therefore do not supply the requested example.\n\n**Full solution or refutation.**\n\nThe fixed-knot-type infinitude question remains open.\n\n**What remains.**\n\nConstruct a fixed knot type with infinitely many nondestabilizable representatives, equivalently with nondestabilizable tb unbounded below, or prove finiteness for every knot type.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 1.63 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Retains the question, gives the equivalence, and cites classified families, but contains the antecedent defect described in notes.\n- John B. Etnyre and Ko Honda, On connected sums and Legendrian knots, Adv. Math. 179 (2003), 59--74. (primary): https://doi.org/10.1016/S0001-8708(02)00027-0\n Evidence used: Produces knot types with arbitrarily many peaks as the knot type varies.\n- Yakov Eliashberg and Maia Fraser, Topologically trivial Legendrian knots, J. Symplectic Geom. 7 (2009), 77--127. (primary): http://projecteuclid.org/euclid.jsg/1239974381\n Evidence used: Classifies the unknot; it does not have infinitely many nondestabilizable representatives.\n\n**Review notes.** FORMULATION DEFECT: the supplied background says 'This has been proved' immediately after the infinitude equivalence and lists the unknot and classified families. The cited results instead prove finiteness/classification for those families; the next sentence itself calls this the finiteness problem. The likely correction is that finiteness has been proved for those knot types. The exact problem statement is unaffected and was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2723, "problem_number": "KP-1.64", "title": "Kirby Problem 1.64", "statement": "(a) Let $L \\subset (S^{3}, \\xi_{std})$ be a transverse link such that the branched double cover\n$(\\Sigma_{2}(L), \\xi_{L})$ is Stein fillable. Is $L$ transversely isotopic to the closure of a\nquasipositive braid?\n(b) Is the same true if $(\\Sigma_{n}(L), \\xi_{L})$ is Stein fillable for some $n > 2$?\n(c) Does Stein fillability of the branched cover imply that the slice-Bennequin\ninequality must be sharp for the given link?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.64.\n\nLiterature notes:\n(1) Here, $(S^{3}, \\xi_{std})$ denotes the standard contact structure on $S^{3}$.\nIf $L \\subset$\n$(S^{3}, \\xi_{std})$ is a transverse link, then the cyclic $n$-fold branched cover $\\Sigma_{n}(L)$\nhas an induced contact structure $\\xi_{L}, n \\geq 2$.\n(2) See Problems 1.69 and 1.70 for definitions, background on quasipositive\nbraids, and related questions.\n(3) A closed braid is quasipositive if and only if it can be represented as\na transverse intersection $S^{3} \\cap \\mathcal{V}$ for some smooth, complex curve $\\mathcal{V} \\subset$\n$\\mathbb{C}^{2}$ [Rud83, BO01].Thus if $L$ is isotopic to the closure of a quasiposi-\ntive braid, then $(\\Sigma_{n}(L), \\xi_{L})$ is Stein fillable, because it bounds the Stein\nmanifold obtained as the cover of $B^{4}$ branched over the complex curve\n$\\mathcal{V} \\cap B^{4}$. The question asks whether fillability of branched covers charac-\nterizes quasipositive braids. For more results on the branched covers of\ntransverse links, see [Pla06]. More discussion of relations between dif-\nferent monoids related to quasipositivity and contact geometry is given\nin [EVHM15].\n\nReferences cited:\n- [Rud83] Lee Rudolph. Algebraic functions and closed braids. Topology, 22(2):191–202, 1983. doi:10.1016/0040-9383(83)90031-9.\n- [BO01] Michel Boileau and Stepan Orevkov. Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe. C. R. Acad. Sci. Paris Sér. I Math., 332(9):825–830, 2001. doi:10.1016/S0764-4442(01)01945-0.\n- [Pla06] Olga Plamenevskaya. Transverse knots, branched double covers and Heegaard Floer contact invariants. J. Symplectic Geom., 4(2):149–170, 2006. doi:10.4310/jsg.2006.v4.n2.a2.\n- [EVHM15] John B. Etnyre and Jeremy Van Horn-Morris. Monoids in the mapping class group. In Interactions between low-dimensional topology and mapping class groups, volume 19 of Geom. Topol. Monogr., pages 319–365. Geom. Topol. Publ., Coventry, 2015. doi:10.2140/gtm.2015.19.319.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Quasipositive braid closures have Stein-fillable cyclic branched covers, but the proposed converse characterizations are open.\n\n**Verified partial progress.**\n\n- Complex-curve/quasipositive constructions prove the forward fillability implication.\n\n**Full solution or refutation.**\n\nNo converse from fillability to transverse quasipositivity was verified.\n\n**What remains.**\n\nResolve the two-fold and higher-fold converse questions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.64 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains the established forward implication and converse target.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2724, "problem_number": "KP-1.65", "title": "Kirby Problem 1.65", "statement": "Decomposable Lagrangian cobordisms between Legendrian knots\nor links in $\\mathbb{R}^{3}$ are compositions of certain simple pieces admitting diagrammatic de-\nscriptions [EHK16]. Such cobordisms necessarily have no index 2 critical points.\nIs every exact Lagrangian cobordism without index 2 critical points Lagrangian iso-\ntopic to a decomposable Lagrangian cobordism?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.65.\n\nLiterature notes:\n(1) At the time of writing, a candidate for a non-decomposable cobordism is\ngiven in [CNS16, Conjecture 3.3]. This can be built using another ele-\nmentary construction, Guadagni’s tangle move, see [BLL $^{+}21$, GSY22],\nwhich could in principle produce additional non-decomposable cobordisms.\n(2) The hypotheses on index 2 critical points is necessary for a variety of\nreasons. For example, Lagrangian caps (see, e.g., [Lin16]) are, by defini-\ntion, not decomposable; for decomposability, the positive boundary of the\ncobordism must be nonempty or else index 2 critical points are necessary.\nIn the recent preprint [DRG24], an example of a non-decomposable La-\ngrangian concordance between knots is given. However, the authors there\nshow that the concordance must contain index 2 critical points.\n(3) Conway, Etnyre, and Tosun prove that decomposable Lagrangians are reg-\nular [CET21], which means there is a Weinstein structure on the cobor-\ndism where the Liouville vector field is tangent to the cobordism. Using\nthis, [EGL20] implies the exterior of a decomposable Lagrangian disk\nnecessarily has a Stein structure, and hence the exterior can be built out\nof particularly constrained elementary pieces, akin to the decomposable\nLagrangian itself. (It is worth pointing out that the non-decomposable\nexamples of [DRG24] above are not regular either.)\n(4) Here is a variant of the problem: Is every filling (exact Lagrangian cobor-\ndism with empty negative boundary) Lagrangian isotopic to a decom-\nposable Lagrangian cobordism? This question is sometimes stated more\ngenerally for exact Lagrangian cobordisms with negative boundary given\nby any (possibly empty) non-destabilizable Lagrangian link; the non-\ndestabilizability condition rules out the presence of a Lagrangian cap as\nin [Lin16].\nThe possibly non-decomposable candidate from [CNS16],\nwhich is a concordance with a Legendrian trefoil as its negative boundary,\ncan be glued to a filling of the trefoil to produce a potential counterexam-\nple to this version of the problem.\n\nReferences cited:\n- [EHK16] Tobias Ekholm, Ko Honda, and Tamás Kálmán. Legendrian knots and exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS), 18(11):2627–2689, 2016. doi: 10.4171/JEMS/650.\n- [CNS16] Christopher Cornwell, Lenhard Ng, and Steven Sivek. Obstructions to Lagrangian concordance. Algebr. Geom. Topol., 16(2):797–824, 2016. doi:10.2140/agt.2016.16.797.\n- [BLL+21] Sarah Blackwell, Noémie Legout, Caitlin Leverson, Maÿlis Limouzineau, Ziva Myer, Yu Pan, Samantha Pezzimenti, Lara Simone Suárez, and Lisa Traynor. Constructions of Lagrangian cobordisms. In Research directions in symplectic and contact geometry and topology, volume 27 of Assoc. Women Math. Ser., pages 245– 272. Springer, Cham, [2021] ©2021. doi:10.1007/978-3-030-80979-9\\\\_5.\n- [GSY22] Roberta Guadagni, Joshua M. Sabloff, and Matthew Yacavone. Legendrian satellites and decomposable cobordisms. J. Knot Theory Ramifications, 31(13):Paper No. 2250071, 33, 2022. doi:10.1142/s0218216522500717.\n- [Lin16] Francesco Lin. Exact Lagrangian caps of Legendrian knots. J. Symplectic Geom., 14(1):269–295, 2016. doi:10.4310/JSG.2016.v14.n1.a10.\n- [DRG24] Georgios Dimitroglou Rizell and Roman Golovko. Instability of Legendrian knottedness, and non-regular Lagrangian concordances of knots, 2024. arXiv:2409.00290.\n- [CET21] James Conway, John B. Etnyre, and Bülent Tosun. Symplectic fillings, contact surgeries, and Lagrangian disks. Int. Math. Res. Not. IMRN, 2021(8):6020–6050, 2021. doi:10.1093/imrn/rny291.\n- [EGL20] Yakov Eliashberg, Sheel Ganatra, and Oleg Lazarev. Flexible Lagrangians. Int. Math. Res. Not. IMRN, 2020(8):2408–2435, 2020. doi:10.1093/imrn/rny078.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Candidates and obstructions to non-decomposable exact Lagrangian cobordisms are known, but no no-index-2 counterexample or general decomposition theorem was verified.\n\n**Verified partial progress.**\n\n- A recent non-decomposable Lagrangian concordance is reported to require index-2 critical points.\n- Decomposable Lagrangians are regular.\n\n**Full solution or refutation.**\n\nThe stated no-index-2 characterization remains open.\n\n**What remains.**\n\nProduce a no-index-2 non-decomposable example or prove decomposability.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.65 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records candidates and the index-2 limitation of known preprint examples.\n\n**Review notes.** Preprint evidence treated conservatively.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2725, "problem_number": "KP-1.66", "title": "Kirby Problem 1.66", "statement": "For Legendrian links $\\Lambda_{1}, \\Lambda_{2} \\subset (\\mathbb{R}^{3}, \\xi_{std})$, write $\\Lambda_{1} \\preceq \\Lambda_{2}$ if\nthere is an exact Lagrangian cobordism in the symplectization $\\mathbb{R} \\times \\mathbb{R}^{3}$ from $\\Lambda_{1}$ to\n$\\Lambda_{2}$. Is $\\preceq$ a partial order?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.66.\n\nLiterature notes:\n(1) One can ask an analogous question in any contact 3-manifold, as well as\nin higher dimensions.\n(2) The fact that $\\preceq$ is not symmetric was first shown by Chantraine [Cha15].\n\n(3) As for ribbon concordance in the smooth setting, many invariants of Leg-\nendrian knots are monotone under $\\preceq$. For example, the behavior of rul-\nings [CNS16], contact homology [Pan17a] and the contact invariant in\nmonopole Floer homology [BS18a] support this conjecture.\n(4) One can ask an analogous question about symplectic concordances be-\ntween transverse knots.\n\nReferences cited:\n- [Cha15] Baptiste Chantraine. Lagrangian concordance is not a symmetric relation. Quantum Topol., 6(3):451–474, 2015. doi:10.4171/QT/68.\n- [CNS16] Christopher Cornwell, Lenhard Ng, and Steven Sivek. Obstructions to Lagrangian concordance. Algebr. Geom. Topol., 16(2):797–824, 2016. doi:10.2140/agt.2016.16.797.\n- [Pan17a] Yu Pan. The augmentation category map induced by exact Lagrangian cobordisms. Algebr. Geom. Topol., 17(3):1813–1870, 2017. doi:10.2140/agt.2017.17.1813.\n- [BS18a] John A. Baldwin and Steven Sivek. Invariants of Legendrian and transverse knots in monopole knot homology. J. Symplectic Geom., 16(4):959–1000, 2018. doi:10.4310/JSG.2018.v16.n4.a3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Lagrangian concordance is known not to be symmetric, and several invariants are monotone, but antisymmetry and hence partial-order status remain open.\n\n**Verified partial progress.**\n\n- Chantraine proved nonsymmetry.\n- Rulings, contact homology and Floer invariants provide monotonicity evidence.\n\n**Full solution or refutation.**\n\nNo proof of antisymmetry was verified.\n\n**What remains.**\n\nProve or disprove antisymmetry of exact Lagrangian concordance.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.66 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States nonsymmetry and evidence for the remaining partial-order conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2726, "problem_number": "KP-1.67", "title": "Kirby Problem 1.67", "statement": "Given a Legendrian link in the standard contact $\\mathbb{R}^{3}$ besides the\nstandard unknot or Hopf link, classify its exact Lagrangian fillings up to compactly\nsupported Hamiltonian isotopy.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.67.\n\nLiterature notes:\nEliashberg-Polterovich [EP96] showed that the unknot has a single\nfilling. A recent preprint of Thomson claims that the Hopf link has exactly 2 fillings\n[Tho25]. These are the only links with known classification. The right-handed tre-\nfoil famously has 5 fillings [EHK16], distinguished through augmentations of the\nLegendrian contact homology. Casals and Gao showed there are examples of links\nwith infinitely many non-isotopic fillings [CG22]. Casals has a more general con-\njectured classification for some large families of links in [Cas22, Conjecture 5.1]. A\nspecial case would be that the $T(2, n)$ torus link has exactly $C_{n}$ fillings up to isotopy\nthrough exact Lagrangians, where $C_{n}$ is the $n^{th}$ Catalan number, generalizing the\ncount for the unknot and Hopf link. (See the work of Ekholm-Honda-Kalman for\nearlier questions on classifying Lagrangian fillings by augmentations of Legendrian\ncontact homology [EHK16], including for the $T(2, n)$ torus links.) A lower bound\nof $C_{n}$ in this case is due to Pan [Pan17b] and independently Shende-Treumann-\nWilliams-Zaslow [STWZ19].\n\nReferences cited:\n- [EP96] Y. Eliashberg and L. Polterovich. Local Lagrangian 2-knots are trivial. Ann. of Math. (2), 144(1):61–76, 1996. doi:10.2307/2118583.\n- [Tho25] Bryce Thomson. The Legendrian Hopf Link has exactly two Lagrangian fillings, 2025. arXiv:2506.15111.\n- [EHK16] Tobias Ekholm, Ko Honda, and Tamás Kálmán. Legendrian knots and exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS), 18(11):2627–2689, 2016. doi: 10.4171/JEMS/650.\n- [CG22] Roger Casals and Honghao Gao. Infinitely many Lagrangian fillings. Ann. of Math. (2), 195(1):207–249, 2022. doi:10.4007/annals.2022.195.1.3.\n- [Cas22] Roger Casals. Lagrangian skeleta and plane curve singularities. J. Fixed Point Theory Appl., 24(2):Paper No. 34, 43, 2022. doi:10.1007/s11784-022-00939-8.\n- [Pan17b] Yu Pan. Exact Lagrangian fillings of Legendrian $(2,n)$ torus links. Pacific J. Math., 289(2):417–441, 2017. doi:10.2140/pjm.2017.289.417.\n- [STWZ19] Vivek Shende, David Treumann, Harold Williams, and Eric Zaslow. Cluster varieties from Legendrian knots. Duke Math. J., 168(15):2801–2871, 2019. doi: 10.1215/00127094-2019-0027.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The unknot has one exact filling, the trefoil has five, and links with infinitely many fillings exist; general classification is open.\n\n**Verified partial progress.**\n\n- Eliashberg--Polterovich classify the unknot.\n- A 2025 preprint claims the Hopf link has exactly two fillings.\n- Casals--Gao provide infinitely many non-isotopic fillings.\n\n**Full solution or refutation.**\n\nNo classification beyond special cases was verified.\n\n**What remains.**\n\nClassify fillings for further Legendrian links, including proposed torus-link Catalan counts.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.67 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records known counts, infinite examples, and the open classification.\n\n**Review notes.** Hopf claim is a preprint.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2727, "problem_number": "KP-1.68", "title": "Kirby Problem 1.68", "statement": "Determine the smooth knot types that have Legendrian repre-\nsentatives with orientable exact Lagrangian fillings.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.68.\n\nLiterature notes:\n(1) This problem appears in [HS15] together with a conjecture that such fill-\nings exist if and only if the knot type $K$ is quasi-positive and tb $(K) =$\n$-$ deg $_{a}P_{K} - 1$, where tb $(K)$ denotes the maximal Thurston-Bennequin\nnumber of a Legendrian representative of $K$ and $P_{K}(a, z)$ the HOM-\nFLYPT polynomial.\nThese conditions are necessary for existence of a\nfilling (see [HS15] and references therein) as is the sharpness of the slice-\nBennequin inequality, tb $(K) = 2g_{4}(K)-1$, as shown in [Cha10, Theorem\n1.4].\n(2) For alternating knots [CNS16], an answer is that a filling exists if and\nonly if the knot type is positive. In general, constructions of fillings exist\nfor all positive knots [HS15] and all almost positive knots with diagrams\nof Type II [Tag19]. A variant of the question allows for nonorientable\nLagrangian fillings; see [CCPR $^{+}24$].\n\n(3) For comparison, the analogous question involving transverse knots has a\nsuccinct answer: Transverse knots with symplectic fillings exist within\na smooth knot type if and only if the knot type is quasi-positive; see\n[Rud83, BO01].\n\nReferences cited:\n- [HS15] Kyle Hayden and Joshua M. Sabloff. Positive knots and Lagrangian fillability. Proc. Amer. Math. Soc., 143(4):1813–1821, 2015. doi:10.1090/S0002-9939-2014-12365-3.\n- [Cha10] Baptiste Chantraine. Lagrangian concordance of Legendrian knots. Algebr. Geom. Topol., 10(1):63–85, 2010. doi:10.2140/agt.2010.10.63.\n- [CNS16] Christopher Cornwell, Lenhard Ng, and Steven Sivek. Obstructions to Lagrangian concordance. Algebr. Geom. Topol., 16(2):797–824, 2016. doi:10.2140/agt.2016.16.797.\n- [Tag19] Keiji Tagami. On the Lagrangian fillability of almost positive links. J. Korean Math. Soc., 56(3):789–804, 2019. doi:10.4134/JKMS.j180399.\n- [CCPR+24] Linyi Chen, Grant Crider-Phillips, Braeden Reinoso, Joshua Sabloff, and Leyu Yao. Non-orientable Lagrangian fillings of Legendrian knots. Math. Proc. Cambridge Philos. Soc., 176(1):123–153, 2024. doi:10.1017/s0305004123000440.\n- [Rud83] Lee Rudolph. Algebraic functions and closed braids. Topology, 22(2):191–202, 1983. doi:10.1016/0040-9383(83)90031-9.\n- [BO01] Michel Boileau and Stepan Orevkov. Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe. C. R. Acad. Sci. Paris Sér. I Math., 332(9):825–830, 2001. doi:10.1016/S0764-4442(01)01945-0.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Necessary quasipositivity and tb conditions are known; alternating knots are classified and broad positive/almost-positive families have fillings, but no general classification exists.\n\n**Verified partial progress.**\n\n- For alternating knots, fillability is equivalent to positivity.\n- Positive knots and certain almost-positive knots have constructions.\n\n**Full solution or refutation.**\n\nThe proposed iff criterion remains open in general.\n\n**What remains.**\n\nProve or refute the quasipositivity/tb characterization.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.68 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States necessary conditions, alternating classification and constructions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2728, "problem_number": "KP-1.69", "title": "Kirby Problem 1.69", "statement": "Let $L \\subset (S^{3}, \\xi_{std})$ be a transverse link with\n\n$$\nsl_{\\Sigma}(L) = -\\chi(\\Sigma),\n$$\n\nfor some Seifert surface $\\Sigma$. Must $L$ be strongly quasipositive?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.69.\n\nLiterature notes:\n(1) The self-linking number of an oriented link transverse to the standard\ncontact structure $\\xi_{std}$ on $S^{3}$ satisfies the well-known Bennequin bound\n[Ben83, Etn05]:\n\n$$\nsl_{\\Sigma}(L) \\leq -\\chi(\\Sigma),\n$$\n\nwhere $\\Sigma$ is any Seifert surface for $L$. This problem therefore attempts to\ncharacterize when this bound is sharp.\n(2) Strongly quasipositive links are those which possess a quasipositive Seifert\nsurface, constructed from parallel disks by attaching bands with positive\nhalf-twists [Rud92]. Such a surface is properly isotopic to the intersection\nof the 4-ball with a non-singular complex curve [Rud83]. It is known\nthat if a link is strongly quasipositive, or bounds a complex curve of\nEuler characteristic equal to that of some Seifert surface for the link, then\nthe Bennequin bound is sharp (see [Shu07, Proposition 1.G] or [Hed10,\nProof of Theorem 1.5] for a proof).\n(3) In light of the above remark, one can ask if sharpness of the Bennequin\nbound implies $\\Sigma$ is quasipositive.\nIf so, then it follows that that any\nminimal genus Seifert surface for $L$ is quasipositive. This is because the\nself-linking number depends only on the relative homology class of $\\Sigma$,\nhence the Bennequin bound will be sharp for any other minimal genus\nSeifert surface $\\Sigma'$.\n(4) The question in this problem has an affirmative answer for fibered links\n[Hed10], links represented by minimal-index braids with fractional Dehn\ntwist coefficient $> 1$ [IK19], 3-braids [IK19], positive knots [Rud99],\nalmost positive knots [FLL23], and knots whose canonical and Seifert\ngenus agree [FLL23]. Hedden and Tovstopyat-Nelip have announced an\naffirmative answer for links with two dimensional top group of Heegaard\nknot Floer homology.\n(5) These notions generalize to arbitrary contact 3-manifolds, with the Ben-\nnequin bound generalized by Eliashberg to tight contact 3-manifolds [Eli92]\nand strong quasipositivity generalized via braids in open books [BEH $^{+}15$,\nIK19, Hay21b]. Geometrically, the class of quasipositive Seifert surfaces\nis the same as the class of ribbons of Legendrian graphs [BI09, Hay22].\nOne can therefore ask the following more general question: Suppose there\nexists a transverse link $L \\subset (Y, \\xi)$ for which\n\n$$\nsl_{\\xi,\\Sigma}(Y, L) = -\\chi(\\Sigma),\n$$\n\nfor some Seifert surface $\\Sigma$. Is $\\Sigma$ isotopic to the Legendrian ribbon of a\nLegendrian graph $\\Gamma \\subset (Y, \\xi)$? This generalized question has an affirmative\nanswer for overtwisted contact structures [BCV09].\n\nReferences cited:\n- [Ben83] Daniel Bennequin. Entrelacements et équations de Pfaff. In Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), volume 107-108 of Astérisque, pages 87–161. Soc. Math. France, Paris, 1983.\n- [Etn05] John B. Etnyre. Legendrian and transversal knots. In Handbook of knot theory, pages 105–185. Elsevier B. V., Amsterdam, 2005. doi:10.1016/B978-044451452-3/50004-6.\n- [Rud92] Lee Rudolph. Constructions of quasipositive knots and links. III. A characterization of quasipositive Seifert surfaces. Topology, 31(2):231–237, 1992. doi:10.1016/0040-9383(92)90017-C.\n- [Rud83] Lee Rudolph. Algebraic functions and closed braids. Topology, 22(2):191–202, 1983. doi:10.1016/0040-9383(83)90031-9.\n- [Shu07] Alexander N. Shumakovitch. Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots. J. Knot Theory Ramifications, 16(10):1403–1412, 2007. doi: 10.1142/S0218216507005889.\n- [Hed10] Matthew Hedden. Notions of positivity and the Ozsváth-Szabó concordance invariant. J. Knot Theory Ramifications, 19(5):617–629, 2010. doi:10.1142/S0218216510008017.\n- [IK19] Tetsuya Ito and Keiko Kawamuro. The defect of the Bennequin-Eliashberg inequality and Bennequin surfaces. Indiana Univ. Math. J., 68(3):799–833, 2019. doi:10.1512/iumj.2019.68.7662.\n- [Rud99] Lee Rudolph. Positive links are strongly quasipositive. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 555–562. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.555.\n- [FLL23] Peter Feller, Lukas Lewark, and Andrew Lobb. Almost positive links are strongly quasipositive. Math. Ann., 385(1-2):481–510, 2023. doi:10.1007/s00208-021-02328-x.\n- [Eli92] Yakov Eliashberg. Contact 3-manifolds twenty years since J. Martinet’s work. Ann. Inst. Fourier (Grenoble), 42(1-2):165–192, 1992. URL: http://www.numdam.org/item?id=AIF 1992 42 1-2 165 0.\n- [BEH+15] R. İnanç Baykur, John Etnyre, Matthew Hedden, Keiko Kawamuro, and Jeremy Van Horn-Morris. Contact and symplectic geometry and the mapping class groups. American Institute of Mathematics, Official report of the 2nd SQuaRE meeting, 2015.\n- [Hay21b] Kyle Hayden. Quasipositive links and Stein surfaces. Geom. Topol., 25(3):1441– 1477, 2021. doi:10.2140/gt.2021.25.1441.\n- [BI09] Sebastian Baader and Masaharu Ishikawa. Legendrian graphs and quasipositive diagrams. Ann. Fac. Sci. Toulouse Math. (6), 18(2):285–305, 2009. URL: http: //afst.cedram.org/item?id=AFST 2009 6 18 2 285 0.\n- [Hay22] Kyle Hayden. Legendrian ribbons and strongly quasipositive links in an open book. J. Math. Pures Appl. (9), 165:42–57, 2022. doi:10.1016/j.matpur.2022.07.002.\n- [BCV09] Sebastian Baader, Kai Cieliebak, and Thomas Vogel. Legendrian ribbons in overtwisted contact structures. J. Knot Theory Ramifications, 18(4):523–529, 2009. doi:10.1142/S0218216509006999.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Strong quasipositivity implies sharp Bennequin inequality, but the converse remains open.\n\n**Verified partial progress.**\n\n- Complex-curve/strongly quasipositive links give sharpness.\n\n**Full solution or refutation.**\n\nNo converse theorem or counterexample was verified.\n\n**What remains.**\n\nDecide whether sharpness for a Seifert surface forces strong quasipositivity.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.69 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Gives the forward implication and poses the converse.\n\n**Review notes.** Open is source-backed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2729, "problem_number": "KP-1.70", "title": "Kirby Problem 1.70", "statement": "(a) Let $L \\subset (S^{3}, \\xi_{std})$ be a transverse link with\n\n$$\nsl_{\\Sigma}(L) = -\\chi(\\Sigma),\n$$\n\nfor some smooth surface $\\Sigma \\subset B^{4}$ bounded by $L$. Is $L$ quasipositive?\n(b) Is the class of strongly quasipositive links equal to the class of quasipositive\nlinks whose Seifert and smooth slice genera are equal?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.70.\n\nLiterature notes:\n(1) Quasipositivity implies that $L$ is isotopic to the transverse intersection\nof a complex curve in $\\mathbb{C}^{2}$ with the unit 3-sphere [Rud83], and any link\narising from such an intersection is quasipositive [BO01]. It is not true,\nhowever, that any surface in $B^{4}$ bounding $L$ realizing equality in the slice-\nBennequin bound is isotopic to a subsurface of a complex curve. Indeed,\na complex curve is ribbon (in the sense that the radius function on the 4-\nball, restricted to the curve, can be assumed to be Morse without critical\npoints of index 2).\nOn the other hand, tubing a complex curve to a\nsmoothly knotted 2-sphere in its complement will produce a surface that\nis not ribbon, yet will still realize equality in the slice-Bennequin bound.\nOne can ask, however, whether a ribbon surface realizing equality in the\nslice-Bennequin bound is properly isotopic to a piece of complex curve.\n(2) Since quasipositive links have transverse representatives for which the\nslice-Bennequin bound is sharp (again, see [Shu07, Proposition 1.G] or\n[Hed10, Proof of Theorem 1.5]) we observe that Problem 1.69 implies an\naffirmative answer to part (b).\n(3) Part (b) is asking whether there exists a link $L$ bounding a complex curve\nwhose Euler characteristic is the same as that of some Seifert surface\nfor $L$, but for which no complex curve bounded by $L$ is isotopic to a\nSeifert surface. The existence of such a link would answer part (b) in\nthe negative, from which it would follow that Problem 1.69 is false. Note\nthat part (b) has an affirmative answer for fibered links, by [Hed10].\nSince complex curves are ribbon and Seifert surfaces are ribbon-immersed\nwithout singularities, this problem can be thought of as asking about the\nminimum number of ribbon singularities required across complex curves\nbounded by $L$.\n\nReferences cited:\n- [Rud83] Lee Rudolph. Algebraic functions and closed braids. Topology, 22(2):191–202, 1983. doi:10.1016/0040-9383(83)90031-9.\n- [BO01] Michel Boileau and Stepan Orevkov. Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe. C. R. Acad. Sci. Paris Sér. I Math., 332(9):825–830, 2001. doi:10.1016/S0764-4442(01)01945-0.\n- [Shu07] Alexander N. Shumakovitch. Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots. J. Knot Theory Ramifications, 16(10):1403–1412, 2007. doi: 10.1142/S0218216507005889.\n- [Hed10] Matthew Hedden. Notions of positivity and the Ozsváth-Szabó concordance invariant. J. Knot Theory Ramifications, 19(5):617–629, 2010. doi:10.1142/S0218216510008017.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Quasipositivity implies slice-Bennequin sharpness, but sharpness for an arbitrary smooth bounding surface does not make that surface complex; the stated link-characterization questions remain open.\n\n**Verified partial progress.**\n\n- Tubing a complex curve to a smoothly knotted sphere gives nonribbon sharp surfaces.\n- Known quasipositive links supply the forward direction.\n\n**Full solution or refutation.**\n\nNeither desired converse/class equality was verified.\n\n**What remains.**\n\nResolve quasipositivity from sharp slice-Bennequin data and the strong-versus-quasipositive genus criterion.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.70 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes false surface-level converse from the remaining link questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2730, "problem_number": "KP-1.71", "title": "Kirby Problem 1.71", "statement": "Does a Gordian unknot exist?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.71.\n\nLiterature notes:\nA physical link is a collection of disjoint embedded loops of given\nfinite lengths in a configuration with thickness at least 1.\nThickness has many\nequivalent definitions [GM99, CKS02], but an appealing one is the same as Fed-\nerer’s definition [Fed59] of reach: the infimal $r$ so that every point in $\\mathbb{R}^{3}$ within $r$\nof the curve has a unique nearest neighbor on the curve. Alternatively, for smooth\nknots, thickness 1 means that a radius-1 tubular neighborhood is also embedded\n[Sim02]. A physical isotopy preserves the length of each loop while respecting this\nthickness constraint. The thickness lower bound yields an upper bound 1 for the\ncurvature of a physical link, implying a least-length configuration exists in each\nphysical isotopy class [CKS02]. By a Gordian pair we mean physical link config-\nurations that are isotopic but not physically isotopic; see [Nab95, LN21] which\nprovide Gordian pairs of higher dimensional knots, the latter giving an explicit con-\nstruction of a Gordian unknot in $S^{4}$. Gordian pairs with two or more components\n[CH15, KK23] are known — even for unlinks [AH25, KK25] — but it remains\nunknown for knots, and especially for the unknot.\n\nReferences cited:\n- [GM99] Oscar Gonzalez and John H. Maddocks. Global curvature, thickness, and the ideal shapes of knots. Proc. Natl. Acad. Sci. USA, 96(9):4769–4773, 1999. doi:10.1073/pnas.96.9.4769.\n- [CKS02] Jason Cantarella, Robert B. Kusner, and John M. Sullivan. On the minimum ropelength of knots and links. Inventiones Mathematicae, 150:257–286, 2002. URL: http://www.springerlink.com/index/10.1007/s00222-002-0234-y, doi: 10.1007/s00222-002-0234-y.\n- [Fed59] Herbert Federer. Curvature measures. Trans. Amer. Math. Soc., 93:418–491, 1959. doi:10.2307/1993504.\n- [Sim02] Jonathan Simon. Physical knots. In Physical knots: knotting, linking, and folding geometric objects in $\\mathbb{R}^{3}$ (Las Vegas, NV, 2001), volume 304 of Contemp. Math., pages 1–30. Amer. Math. Soc., Providence, RI, 2002. doi:10.1090/conm/304/05181.\n- [Nab95] Alexander Nabutovsky. Non-recursive functions, knots “with thick ropes”, and self-clenching “thick” hyperspheres. Comm. Pure Appl. Math., 48(4):381–428, 1995. doi:10.1002/cpa.3160480402.\n- [LN21] Boris Lishak and Alexander Nabutovsky. Complexity of unknotting of trivial 2-knots. J. Topol. Anal., 13(3):623–657, 2021. doi:10.1142/$S^{1}$793525320500272.\n- [CH15] Alexander Coward and Joel Hass. Topological and physical link theory are distinct. Pacific J. Math., 276(2):387–400, 2015. doi:10.2140/pjm.2015.276.387.\n- [KK23] Rob Kusner and Wöden Kusner. A Gordian pair of links. Geom. Dedicata, 217(1):Paper No. 47, 2023. doi:10.1007/s10711-023-00783-1.\n- [AH25] José Ayala and Joel Hass. Gordian unlinks, 2025. arXiv:2502.08499.\n- [KK25] Rob Kusner and Wöden Kusner. The Xarax unlinks are physically Gordian, 2025. arXiv:CirclePackings,MinimalSurfaces, andDiscreteDifferentialGeometryPosterSessionAbstracts,ICERM2/11/25, https://app.icerm.brown.edu/assets/521/8574/8574\\_4906\\_021120251500\\_ Slides.pdf.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Gordian pairs are known in higher dimensions, but existence of a Gordian unknot in the stated physical-link setting remains open.\n\n**Verified partial progress.**\n\n- Higher-dimensional Gordian-pair constructions are known.\n\n**Full solution or refutation.**\n\nNo three-dimensional physical Gordian unknot was verified.\n\n**What remains.**\n\nConstruct one or prove all physical unknots are physically isotopic.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.71 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the physical setting and records only higher-dimensional examples.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2731, "problem_number": "KP-1.72", "title": "Kirby Problem 1.72", "statement": "(The equilateral stuck unknots conjecture.). Are there equilat-\neral embedded polygons that are unknotted yet cannot be unknotted through polygons\npreserving edge lengths? Such an unknot is said to be stuck.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.72.\n\nLiterature notes:\n(1) The celebrated Carpenter’s Rule problem asks whether an embedded poly-\ngon in the plane may be deformed to a convex configuration through em-\nbedded polygons with the same edge lengths. Solutions in the affirmative\nwere given more or less simultaneously by Streinu [Str00] and Connelly,\nDemaine, and Rote [CDR03].\n(2) The natural extension of this theorem to PL knots in codimension two\nasks whether polygons of knot type $K$ may be deformed into one another\nthrough embedded polygons of the same edge lengths. For arbitrary edge\nlengths, the answer is known to be ‘no’: there are stuck unknots with as\nfew as six edges [CKS98, Tou01, AET04] (see Figure 2) which cannot be\nconvexified through embedded configurations with the same edge lengths.\n(3) Interestingly, all of the known examples of stuck unknots have very dif-\nferent edge lengths, which they seem to require in an essential way. This\nleads to the question: are there equilateral stuck unknots?\n\nFigure 2. Stuck unknots\n\nThere is some reason to believe that there are not. Khoi [Kho05]\nproved that the symplectic volume of the space of equilateral $n$-gons in\n$\\mathbb{R}^{3}$ is the largest symplectic volume of any space of $n$-gons with the same\ntotal length.\nThat is, in this sense, equilateral polygons are the most\nflexible polygons. On the other hand, it is quite difficult to imagine an\nalgorithm for unfolding an equilateral unknot of $n$ edges.\n\nReferences cited:\n- [Str00] Ileana Streinu. A combinatorial approach to planar non-colliding robot arm motion planning. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 443–453, 2000. doi:10.1109/SFCS.2000.892132.\n- [CDR03] Robert Connelly, Erik D. Demaine, and Günter Rote. Straightening polygonal arcs and convexifying polygonal cycles. Discrete \\& Computational Geometry, 30:205– 239, 2003. doi:10.1007/s00454-003-0006-7.\n- [CKS98] Jason Cantarella, Robert B. Kusner, and John M. Sullivan. Tight knot values deviate from linear relations. Nature, 392:237, 1998. URL: http://torus.math.uiuc.edu/jms/Papers/scicor/scicor.ps.gzpapers3://publication/uuid/A1C$D^{2}$FF8-BAEE-4CCC-A98C-BB5ECC1DC83A.\n- [Tou01] Godfried Toussaint. A new class of stuck unknots in Pol6. Beiträge Algebra Geom., 42(2):301–306, 2001.\n- [AET04] Greg Aloupis, Günter Ewald, and Godfried Toussaint. More classes of stuck unknotted hexagons. Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, 45:429–434, 2004.\n- [Kho05] Vu The Khoi. On the symplectic volume of the moduli space of spherical and Euclidean polygons. Kodai Mathematical Journal, 28:199–208, 2005. doi:10.2996/kmj/1111588046.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Stuck unknots with arbitrary unequal edge lengths are known with as few as six edges, while the equilateral stuck-unknot conjecture remains open.\n\n**Verified partial progress.**\n\n- Known six-edge stuck unknots solve the non-equilateral analogue.\n- Planar Carpenter's Rule is affirmative but does not settle the spatial equilateral question.\n\n**Full solution or refutation.**\n\nNo equilateral example or impossibility proof was verified.\n\n**What remains.**\n\nConstruct an equilateral stuck unknot or prove equilateral unknots can always be untangled.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.72 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States known unequal-length examples and the open equilateral version.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2732, "problem_number": "KP-1.73", "title": "Kirby Problem 1.73", "statement": "(The 15 pearls conjecture). Is the pearl number of the trefoil\nequal to 15?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.73.\n\nLiterature notes:\n(1) A pearl necklace with $n$ pearls is an embedding of an equilateral $n$-gon\nin $\\mathbb{R}^{3}$ so that each vertex is surrounded by a sphere with diameter equal\nto the edgelength and the interiors of all the spheres are disjoint. This\nis a common off-lattice model of a self-avoiding polygon, usually called\nthe hard sphere polymer (see [SJ23]). The pearl number of a knot $K$ is\nthe minimum number of pearls in any necklace whose polygon has knot\ntype $K$.\n(2) A self-avoiding polygon on a cubic lattice is a common model for random\nknots with self-interactions. The “minimum step numbers” are known for\nself-avoiding polygons on the cubic lattice for a number of knot types by a\ncombination of geometric arguments and computer enumeration [SIA $^{+}09$].\nFor instance, the minimal step number of the trefoil on the simple cubic\nlattice is 24 [Dia93].\n(3) Very little is known about the pearl number. Oshiro and Maehara [MO99]\ngive a construction for a 15-pearl trefoil and conjecture that this configu-\nration is minimal (so the pearl number of the trefoil is 15). Interestingly,\ntheir configuration is similar to the 15 step embedding of the trefoil on\nthe face-centered cubic lattice (shown in Figure 3) found by Rechnitzer\nand Janse van Rensburg [RR11], which those authors conjecture to be\nminimal. By putting balls of radius $1/2$ on the vertices of a realization\non the cubic lattice one obtains a hard sphere polymer. Thus the pearl\nnumber is bounded above by the length of any lattice representation.\nConversely, the pearl number is an upper bound on the equilateral stick\nnumber of any knot. The difference between the three integer invariants\n\nFigure 3. Face-centered cubic lattice embedding of trefoil\n\nshould be quite large for most knot types as can be seen for the trefoil\nwith an equilateral stick number of six [Jin97].\n(4) In [Mae07], Maehara proves that the pearl number of the trefoil is at\nleast 11.\n\nReferences cited:\n- [SJ23] Stefan Schnabel and Wolfhard Janke. Monte Carlo simulation of long hard-sphere polymer chains in two to five dimensions. Macromolecular Theory and Simulations, 32:2200080, 2023. doi:10.1002/mats.202200080.\n- [SIA+09] Rob Scharein, Kai Ishihara, Javier Arsuaga, Yuanan Diao, Koya Shimokawa, and Mariel Vazquez. Bounds for the minimum step number of knots in the simple cubic lattice. Journal of Physics A: Mathematical and Theoretical, 42, 2009. doi:10.1088/1751-8113/42/47/475006.\n- [Dia93] Yuanan Diao. Minimal knotted polygons on the cubic lattice. Journal of Knot Theory and Its Ramifications, 02:413–425, 1993. doi:10.1142/S0218216593000234.\n- [MO99] Hiroshi Maehara and Ai Oshiro. On knotted necklaces of pearls. European Journal of Combinatorics, 20:411–420, 1999. doi:10.1006/eujc.1998.0279.\n- [RR11] E. J. Janse Van Rensburg and Andrew Rechnitzer. Generalized atmospheric sampling of knotted polygons. Journal of Knot Theory and its Ramifications, 20:1145– 1171, 2011. doi:10.1142/S0218216511009170.\n- [Jin97] Gyo Taek Jin. Polygon indices and superbridge indices of torus knots and links. J. Knot Theory Ramifications, 6(2):281–289, 1997. doi:10.1142/S0218216597000170.\n- [Mae07] Hiroshi Maehara. On configurations of solid balls in 3-space: Chromatic numbers and knotted cycles. Graphs and Combinatorics, 23:307–320, 2007. doi:10.1007/s00373-007-0702-7.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that the trefoil pearl number is exactly 15 was verified.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe maintained list retains the 15-pearls conjecture.\n\n**What remains.**\n\nEstablish matching construction and lower bound for the trefoil.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.73 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintained statement of the 15-pearls conjecture.\n\n**Review notes.** Open is dated and conservative.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2733, "problem_number": "KP-1.74", "title": "Kirby Problem 1.74", "statement": "How does ropelength behave under connected sum of knots?\nHere are two conjectures, the second a weakening of the first.\n(a) For any knot or link types $K_{1}$ and $K_{2}$,\nRop $(K_{1}\\#K_{2}) \\leq$ Rop $(K_{1}) +$ Rop $(K_{2}) - (4\\pi - 4).$\n(b) For any knot or link types $K_{1}$ and $K_{2}$,\n\n$$\nRop(K_{1}\\#K_{2}) \\leq Rop(K_{1}) + Rop(K_{2}) - c,\n$$\n\nwhere $c > 0$ is some constant independent of knot type.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.74.\n\nLiterature notes:\n(1) The (minimum) ropelength Rop $(K)$ of a knot is defined to be the small-\nest quotient of length and thickness (defined in Problem 1.71) among\nall rectifiable curves in $\\mathbb{R}^{3}$ realizing the knot type. It is known that ro-\npelength minimizers exist in every knot and link type [GMSvdM02,\nGL03, CKS02], but their shapes (and rope-lengths) are known only in\na few special cases of links with planar components. In all other case we\nonly have numerical results approximating the rope-length.\n(2) The ropelength minimizing Hopf link $2^{2}_{1}$ is a pair of circles of equal radius\nin orthogonal planes, passing through each other’s centers. Ropelength\nis scale-invariant, so we can assume that the circles have radius 2. Their\nthickness is then 1 and $\\operatorname{Rop}(2^{2}_{1})=8\\pi$. The ropelength-minimizing com-\nposite link $2^{2}_{1}\\#2^{2}_{1}$ (a chain of three linked circles) is composed of two round\n\ncircles in the same plane, each passing through a center of the semicircular\narc of a stadium curve in the orthogonal plane. This has ropelength $8\\pi$\n(for the circles) plus $4\\pi + 4$ (for the stadium curve), so\n\n$$\n\\operatorname{Rop}(2^{2}_{1}\\#2^{2}_{1}) = 12\\pi+4\n= \\operatorname{Rop}(2^{2}_{1})+\\operatorname{Rop}(2^{2}_{1})-(4\\pi-4),\n$$\n\nand we have saved some rope by splicing these links together. Katritch\net al. [KOP $^{+}97$] conjectured in 1997 that we can save at least as much\nrope on any connect sum; leading to Conjecture (a).\n(3) The work of Diao [Dia24] on alternating knots uses the braid index as\na lower bound on ropelength.\nThe braid index of a connected sum is\n$b(K_{1}\\#K_{2}) = b(K_{1}) + b(K_{2}) - 1$ for two alternating knots $K_{1}$ and $K_{2}$.\nThus the lower bound on ropelength of a connected sum of the two knots\n$b(K_{1}\\#K_{2})$ will be slightly less than the sum of the lower bounds of the\ntwo knots $K_{1}$ and $K_{2}$.\nHence it is worth considering the weaker version, Conjecture (b), with\nan undetermined constant.\n(4) In general, it would be important to have any theoretical lower bound on\nropelength for knot types that approximates the numerical results. This\nhas been done for the trefoil, but the techniques used do not generalize\nto other knot types, for which there remains a wide gap between the\nnumerical results and the theoretical lower bounds.\n\nReferences cited:\n- [GMSvdM02] O. Gonzalez, J. H. Maddocks, F. Schuricht, and H. von der Mosel. Global curvature and self-contact of nonlinearly elastic curves and rods. Calc. Var. Partial Differential Equations, 14(1):29–68, 2002. doi:10.1007/s005260100089.\n- [GL03] Oscar Gonzalez and R. De La Llave. Existence of ideal knots. Journal of Knot Theory and its Ramifications, 12:123–133, 2003. doi:10.1142/S0218216503002354.\n- [CKS02] Jason Cantarella, Robert B. Kusner, and John M. Sullivan. On the minimum ropelength of knots and links. Inventiones Mathematicae, 150:257–286, 2002. URL: http://www.springerlink.com/index/10.1007/s00222-002-0234-y, doi: 10.1007/s00222-002-0234-y.\n- [KOP+97] Vsevolod Katritch, Wilma K. Olson, Piotr Pieranski, Jacques Dubochet, and Andrzej Stasiak. Properties of ideal composite knots. Nature, 388:148–151, 1997. doi:10.1038/40582.\n- [Dia24] Yuanan Diao. The ropelength conjecture of alternating knots. Math. Proc. Cambridge Philos. Soc., 177(2):367–369, 2024. doi:10.1017/S0305004124000288.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of either proposed universal ropelength saving under connected sum was verified.\n\n**Verified partial progress.**\n\n- Ropelength minimizers exist for each knot type.\n\n**Full solution or refutation.**\n\nExistence of minimizers does not establish either connected-sum inequality.\n\n**What remains.**\n\nProve the 4pi-4 saving or any knot-independent positive saving.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.74 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintains the two connected-sum conjectures.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2734, "problem_number": "KP-1.75", "title": "Kirby Problem 1.75", "statement": "(a) Find some knot energy on the space of smoothly embedded unknotted circles\nin $\\mathbb{S}^{3}$ for which all unknotted critical points are great circles.\n(b) Define a gradient flow for this knot energy which yields a deformation\nretract of the space of unknots to the subspace of great circles.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.75.\n\nLiterature notes:\n(1) These questions were posed by Freedman, He, and Wang [FHW94]. The\nmotivation is one of the many equivalent formulations of the Smale conjec-\nture mentioned in the Appendix to Hatcher’s 1983 paper (which actually\nproves this conjecture). Variation (7) states [Hat83, p. 606] that “the\nspace of smoothly embedded unknotted circles in $\\mathbb{S}^{3}$ deformation retracts\nonto the space of great circles in $\\mathbb{S}^{3}$, (i.e., O $(4)/$ O $(2) \\times$ O $(2))$.”Of course,\nit would be of some interest to construct a more or less explicit example\nof such a retract.\n(2) Beginning in the 1980s, several repulsive functionals, so-called knot ener-\ngies, have been defined in the pursuit of disentangling complicated knot-\nted curves and deforming them into ‘simpler’ curves within the same knot\nclass; see, e.g., the survey by Strzelecki et al. [SSvdM13] and references\ntherein. The most prominent example is the Möbius energy introduced\nby O’Hara [O’H91] in 1991.\n(3) As pointed out by Freedman, He, and Wang [FHW94], it is tempting to\nconjecture that a suitable gradient flow for a knot energy actually defines\n\na retraction as stated by Hatcher, i.e., it will deform any curve from the\nunknot class to a round circle. Of course, this can only work if, except\nfor the circles, there are no critical points within the unknot class; cf.\nBudney’s post [Bud16]. Currently, we do not know whether any of the\nmany smooth knot energies that have been proposed so far enjoys this\nproperty.\n\nReferences cited:\n- [FHW94] Michael H. Freedman, Zheng-Xu He, and Zhenghan Wang. Möbius energy of knots and unknots. Ann. of Math. (2), 139(1):1–50, 1994. doi:10.2307/2946626.\n- [Hat83] Allen E. Hatcher. A proof of the Smale conjecture, Diffp$S^{3}$q » $O(4)$. Ann. of Math. (2), 117(3):553–607, 1983. doi:10.2307/2007035.\n- [SSvdM13] Pawel Strzelecki, Marta Szumańska, and Heiko von der Mosel. On some knot energies involving Menger curvature. Topology Appl., 160(13):1507–1529, 2013. doi:10.1016/j.topol.2013.05.022.\n- [O’H91] Jun O’Hara. Energy of a knot. Topology, 30(2):241–247, 1991. doi:10.1016/0040-9383(91)90010-2.\n- [Bud16] Ryan Budney. A gorgeous but incomplete proof of “The Smale Conjecture”, 2016. https://ldtopology.wordpress.com/2016/10/02/a-gorgeous-but-incomplete-proof-of-the-smale-conjecture/. Accessed: 2024-01-25.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No knot-energy gradient flow giving the requested deformation retract of unknots to great circles was verified.\n\n**Verified partial progress.**\n\n- Hatcher's Smale-conjecture work motivates the desired topological result, but does not supply the requested energy/flow.\n\n**Full solution or refutation.**\n\nThe analytic realization remains open.\n\n**What remains.**\n\nConstruct an energy with exactly great-circle unknot critical points and prove flow convergence.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.75 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the energy and gradient-flow problem in the maintained list.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2735, "problem_number": "KP-1.76", "title": "Kirby Problem 1.76", "statement": "(a) Is there an algorithm to detect the unknot that runs in polynomial time\n(as a function of the number of crossings in an input diagram)?\n(b) What is the structure of the unknot Reidemeister graph?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.76.\n\nLiterature notes:\n(1) There are now many algorithms to detect the unknot. The first was due to\nHaken [Hak61] and used normal surfaces. Hass, Lagarias and Pippenger\n[HLP99] refined his methods to prove that unknot recognition lies in NP.\nDynnikov [Dyn06] used grid diagrams to find a very different algorithm.\nHe showed that any grid diagram for the unknot can be reduced to the\ntrivial diagram through a sequence of moves, none of which increases the\ngrid number.\n(2) There are now many invariants that detect the unknot, including Kho-\nvanov homology [KM11] and Heegaard Floer homology [OS04b]. How-\never, none of these invariants seem to be computable in polynomial time.\nA further significant result was proved by Kuperberg [Kup14] who showed\nthat unknot recognition lies in co-NP, assuming the Generalized Riemann\n\nHypothesis. In other words, if a knot is nontrivial, then there is an efficient\nway of certifying this. This result was proved unconditionally, removing\nthe GRH assumption, by Lackenby [Lac21a] using sutured manifold hi-\nerarchies.\n(3) The unknot Reidemeister graph in part (b) is formed by assigning a vertex\nto each diagram of the unknot and connecting two vertices by an edge if\nthere is a Reidemeister move connecting them; it is locally finite. There\nare pairs of vertices representing two diagrams with $n$ crossings such that\nthe shortest path connecting them has length at least $n^{2}/25$ [HN10]. No\ntwo vertices representing diagrams with $n$ crossings have distance greater\nthan $2(236n)^{11}$ [Lac15].\nThis problem is complicated by the existence of “hard” unknot dia-\ngrams [BCL $^{+}24$, PZ16], which have the property that any sequence of\nReidemeister moves taking them to the trivial diagram must go via dia-\ngrams with higher crossing number. The question may be more tractable\nif different sets of basic moves are allowed in addition to or in place of\nthe standard Reidemeister moves; in the related setting of grid diagrams,\nmonotonic simplification has been established [Dyn06].\n\nReferences cited:\n- [Hak61] Wolfgang Haken. Theorie der Normalflächen. Acta Math., 105:245–375, 1961. doi: 10.1007/BF02559591.\n- [HLP99] Joel Hass, Jeffrey C. Lagarias, and Nicholas Pippenger. The computational complexity of knot and link problems. J. ACM, 46(2):185–211, 1999. doi:10.1145/301970.301971.\n- [Dyn06] I. A. Dynnikov. Arc-presentations of links: monotonic simplification. Fund. Math., 190:29–76, 2006. doi:10.4064/fm190-0-3.\n- [KM11] P. B. Kronheimer and T. S. Mrowka. Khovanov homology is an unknotdetector. Publ. Math. Inst. Hautes Études Sci., 113:97–208, 2011. doi:10.1007/s10240-010-0030-y.\n- [OS04b] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and genus bounds. Geom. Topol., 8:311–334, 2004. doi:10.2140/gt.2004.8.311.\n- [Kup14] Greg Kuperberg. Knottedness is in NP, modulo GRH. Adv. Math., 256:493–506, 2014. doi:10.1016/j.aim.2014.01.007.\n- [Lac21a] Marc Lackenby. The efficient certification of knottedness and Thurston norm. Adv. Math., 387:Paper No. 107796, 142, 2021. doi:10.1016/j.aim.2021.107796.\n- [HN10] Joel Hass and Tahl Nowik. Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle. Discrete Comput. Geom., 44(1):91–95, 2010. doi:10.1007/s00454-009-9156-4.\n- [Lac15] Marc Lackenby. A polynomial upper bound on Reidemeister moves. Ann. of Math. (2), 182(2):491–564, 2015. doi:10.4007/annals.2015.182.2.3.\n- [BCL+24] Benjamin A. Burton, Hsien-Chih Chang, Maarten Löffler, Clément Maria, Arnaud de Mesmay, Saul Schleimer, Eric Sedgwick, and Jonathan Spreer. Hard diagrams of the unknot. Exp. Math., 33(3):482–500, 2024. doi:10.1080/10586458.2022.2161676.\n- [PZ16] Carlo Petronio and Adolfo Zanellati. Algorithmic simplification of knot diagrams: new moves and experiments. J. Knot Theory Ramifications, 25(10):1650059, 2016. doi:10.1142/S0218216516500590.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Unknot recognition lies in NP and unconditionally in co-NP, but polynomial-time recognition and detailed Reidemeister-graph structure remain open.\n\n**Verified partial progress.**\n\n- Haken/Dynnikov provide recognition algorithms.\n- Lackenby proved unconditional co-NP membership.\n- For n-crossing diagrams, graph distances have quadratic lower and polynomial upper bounds.\n\n**Full solution or refutation.**\n\nComplexity and full graph geometry are unresolved.\n\n**What remains.**\n\nGive a polynomial-time algorithm or prove hardness, and sharpen Reidemeister-graph geometry.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.76 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records NP/co-NP, algorithms, and distance bounds.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2736, "problem_number": "KP-1.77", "title": "Kirby Problem 1.77", "statement": "How many Reidemeister moves are required to relate two dia-\ngrams of a knot (as a function of their numbers of crossings)?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.77.\n\nLiterature notes:\n(1) One reason why this question is interesting is that a computable upper\nbound leads to an algorithm to decide whether two knots are equivalent.\n(2) Lackenby [Lac15] proved that, for a diagram of the unknot with $c$ cross-\nings, there is a sequence of Reidemeister moves taking it to the trivial\ndiagram with length at most $(236c)^{11}$. It is not inconceivable that there is\na polynomial upper bound that applies to every knot type. However, the\nbest known upper bound on the number of Reidemeister moves required\nto relate two diagrams of a knot with $c_{1}$ and $c_{2}$ crossings is due to Coward\nand Lackenby [CL14]:\n\n$$\n2^{2^{\\cdot^{\\cdot^{\\cdot^{2}}}}},\n$$\n\nwhere the height of the tower is $k^{c_1+c_2}$ and where $k = 10^{1000000}$.\n(3) A nontrivial lower bound was established by Hass and Nowik [HN10],\nwho proved that for each natural number $n$ there exists a diagram of the\nunknot with $7n-1$ crossings and where the number of Reidemeister moves\nrequired to take it to the trivial diagram is at least $2n^{2} + 3n - 2$.\n\nReferences cited:\n- [Lac15] Marc Lackenby. A polynomial upper bound on Reidemeister moves. Ann. of Math. (2), 182(2):491–564, 2015. doi:10.4007/annals.2015.182.2.3.\n- [CL14] Alexander Coward and Marc Lackenby. An upper bound on Reidemeister moves. Amer. J. Math., 136(4):1023–1066, 2014. doi:10.1353/ajm.2014.0027.\n- [HN10] Joel Hass and Tahl Nowik. Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle. Discrete Comput. Geom., 44(1):91–95, 2010. doi:10.1007/s00454-009-9156-4.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** General equivalent-knot diagrams have a computable tower-type Reidemeister upper bound; unknots have a polynomial upper bound and quadratic lower bounds are known.\n\n**Verified partial progress.**\n\n- Lackenby gives (236c)^11 moves for unknots.\n- Coward--Lackenby give a general iterated-exponential bound.\n- Hass--Nowik give quadratic lower-bound examples.\n\n**Full solution or refutation.**\n\nThe asymptotically correct general move complexity is open.\n\n**What remains.**\n\nNarrow the gap between polynomial/quadratic phenomena and the huge general upper bound.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.77 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists all three cited bounds.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2737, "problem_number": "KP-1.78", "title": "Kirby Problem 1.78", "statement": "Let $D$ be any diagram of the unknot with $n$ crossings. Let $h(D)$\nbe the smallest number such that some series of Reidemeister moves that transforms\n\n$D$ to a 0-crossing diagram has the property that all intermediate diagrams have at\nmost $h(D)$ crossings. Define $h(n)$ to be the maximum of $h(D)$ over all $n$-crossing\ndiagrams of the unknot. A diagram represents a hard unknot if $h(D) > n$. What\nis $h(n)$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.78.\n\nLiterature notes:\nIt is known [BCL $^{+}24$] that $h(n) \\geq n + 3$ for certain values of $n$.\n\nReferences cited:\n- [BCL+24] Benjamin A. Burton, Hsien-Chih Chang, Maarten Löffler, Clément Maria, Arnaud de Mesmay, Saul Schleimer, Eric Sedgwick, and Jonathan Spreer. Hard diagrams of the unknot. Exp. Math., 33(3):482–500, 2024. doi:10.1080/10586458.2022.2161676.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hard unknot diagrams are known: h(n)>=n+3 for certain n, but the growth of h(n) is unknown.\n\n**Verified partial progress.**\n\n- Burton et al. prove the n+3 lower bound for certain n.\n\n**Full solution or refutation.**\n\nNo matching upper asymptotic or full formula was verified.\n\n**What remains.**\n\nDetermine h(n), including its order of excess over n.\n\n**Sources checked.**\n\n- B. Burton et al., Hard diagrams of the unknot, Experimental Mathematics 33 (2024), 482-500. (primary): https://doi.org/10.1080/10586458.2022.2161676\n Evidence used: Proves the stated hard-diagram lower bound.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2738, "problem_number": "KP-1.79", "title": "Kirby Problem 1.79", "statement": "Are there additional moves that, when added to the three Rei-\ndemeister moves, allow for strict monotonic descent in the crossing number of an\nunknot diagram?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.79.\n\nLiterature notes:\n(1) Additional moves allow increased efficiency in diagram simplification. Sev-\neral attempts at finding such moves have been given, but none are known\nto suffice [PZ16]. Work of Dynnikov gives a monotonic, but not strictly\nmonotonic, process [Dyn03].\n(2) A strictly monotonic process would lead to a polynomial time algorithm\nfor unknot detection; see Problem 1.76.\n(3) The collection of additional moves can grow with the number of crossings,\ne.g. moving an arc across a twist region of a diagram. However the set of\nadditional moves cannot be arbitrary; moves consisting of any sequence\nof Reidemeister moves would allow any knot equivalence to be carried out\nin one step. A reasonable restriction is to allow moves that reduce the\ncrossing number and can be found in polynomial time (polynomial in the\ncrossing number).\n\nReferences cited:\n- [PZ16] Carlo Petronio and Adolfo Zanellati. Algorithmic simplification of knot diagrams: new moves and experiments. J. Knot Theory Ramifications, 25(10):1650059, 2016. doi:10.1142/S0218216516500590.\n- [Dyn03] I. A. Dynnikov. Recognition algorithms in knot theory. Uspekhi Mat. Nauk, 58(6(354)):45–92, 2003. doi:10.1070/RM2003v058n06ABEH000675.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No known finite/reasonably constrained augmentation of Reidemeister moves guarantees strict crossing-number descent for every unknot diagram.\n\n**Verified partial progress.**\n\n- Dynnikov gives a monotonic but not strictly monotonic grid-diagram process.\n- Several additional-move proposals do not suffice.\n\n**Full solution or refutation.**\n\nThe desired strict simplification system is open.\n\n**What remains.**\n\nFind a polynomially discoverable crossing-reducing move set or obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.79 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records existing monotonic process and insufficiency of proposals.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2739, "problem_number": "KP-1.80", "title": "Kirby Problem 1.80", "statement": "Is unknotting number computable? Is there even an algorithm\nto decide whether a knot has unknotting number one?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.80.\n\nLiterature notes:\n(1) Many elementary knot invariants, such as the crossing number, are known\nto be computable. However, it seems extremely challenging to determine\nwhether the unknotting number is computable.\n(2) Recall that the unknotting number of a knot is the minimum number of\ncrossing changes required to unknot the knot, where the minimum is taken\nover all diagrams of the knot. One can also consider the unlinking number\nof a link, which is the minimal number of crossing changes required to\nturn it into an unlink. Closely related is the splitting number, which is the\nminimal number of crossing changes required to turn into a split link.\n(3) Some lower bounds on the computational complexity of determining un-\nlinking number are known: it was shown by de Mesmay, Rieck, Sedgwick\nand Tancer [dMRST21] and Koenig and Tsvietkova [KT21] that deter-\nmining the unlinking number of a link is NP-hard.\n\n(4) The second question in the problem seems more tractable, since many\nstructural results are known about knots with unknotting number one\n[ST89], [Sch85b]. The analogous problem of determining whether a link\nhas unlinking number one was solved for a large class of links by Lackenby\n[Lac21b].\n(5) This problem was also stated in [Lac17a].\n\nReferences cited:\n- [dMRST21] Arnaud de Mesmay, Yo’av Rieck, Eric Sedgwick, and Martin Tancer. The unbearable hardness of unknotting. Adv. Math., 381:Paper No. 107648, 36, 2021. doi:10.1016/j.aim.2021.107648.\n- [KT21] Dale Koenig and Anastasiia Tsvietkova. NP-hard problems naturally arising in knot theory. Trans. Amer. Math. Soc. Ser. B, 8:420–441, 2021. doi:10.1090/btran/71.\n- [ST89] Martin Scharlemann and Abigail Thompson. Link genus and the Conway moves. Comment. Math. Helv., 64(4):527–535, 1989. doi:10.1007/BF02564693.\n- [Sch85b] Martin G. Scharlemann. Unknotting number one knots are prime. Invent. Math., 82(1):37–55, 1985. doi:10.1007/BF01394778.\n- [Lac21b] Marc Lackenby. Links with splitting number one. Geom. Dedicata, 214:319–351, 2021. doi:10.1007/s10711-021-00618-x.\n- [Lac17a] Marc Lackenby. Elementary knot theory. In Lectures on geometry, Clay Lect. Notes, pages 29–64. Oxford Univ. Press, Oxford, 2017.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Computability of knot unknotting number, including decision of unknotting number one, remains open; related unlinking-number computation is NP-hard.\n\n**Verified partial progress.**\n\n- Unlinking number is NP-hard.\n- Structural results constrain unknotting-number-one knots and solve some link subclasses.\n\n**Full solution or refutation.**\n\nNo general algorithm or undecidability result was verified.\n\n**What remains.**\n\nDecide computability of unknotting number and the one-unknotting decision problem.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.80 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explicitly retains the computability questions and cites NP-hard related problems.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2740, "problem_number": "KP-1.81", "title": "Kirby Problem 1.81", "statement": "(a) Are all knots trivial?\n(b) Conjecture: The Bing sling is knotted.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.81.\n\nLiterature notes:\n(1) For this problem, a knot is a topological embedding (a homeomorphism\nonto its image) of a circle in $S^{3}$. A knot is trivial if it is isotopic, i.e. con-\nnected by an arc of embeddings, to the unknot. A smooth knot in a string\nis trivial in this sense by simply pulling the string tight so that the knot\nbecomes a point. A similar argument works for locally flat embeddings:\nthe problem is only interesting for wild embeddings.\n(2) The Bing Sling [Bin56, DV09] (see also [Nan18, amd22] and [Shi73])\nis drawn in Figure 1.12.\n(3) Brin [Bri83] shows that there are knots that at each point are locally\nequivalent to the Bing sling but are non-ambiently isotopic to the unknot.\n\nFigure 4. The Bing Sling\n\nReferences cited:\n- [Bin56] R. H. Bing. A simple closed curve that pierces no disk. J. Math. Pures Appl. (9), 35:337–343, 1956.\n- [DV09] Robert J. Daverman and Gerard A. Venema. Embeddings in manifolds, volume 106 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2009. doi:10.1090/gsm/106.\n- [Nan18] Ollie Nanyes. Wild and non-compact knot theory. WordPress, 2018. https://wildandnoncompactknots.wordpress.com.\n- [amd22] amd1234. Bing sling isotopy to unknot, 2022. by user https://mathoverflow.net/users/170240/amd1234. See https://mathoverflow.net/q/437106 (version: 2022-12-28).\n- [Shi73] Arnold C. Shilepsky. Homogeneity by isotopy for simple closed curves. Duke Math. J., 40:463–472, 1973. URL: http://projecteuclid.org/euclid.dmj/1077309868.\n- [Bri83] M. Brin. Curves isotopic to tame curves. In Continua, decompositions, manifolds (Austin, Tex., 1980), pages 163–166. Univ. Texas Press, Austin, TX, 1983.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Not all wild topological knots are trivial, but the particular Bing-sling knottedness conjecture remains open.\n\n**Verified partial progress.**\n\n- Brin constructs knots locally like the Bing sling that are not ambiently isotopic to the unknot.\n- Smooth and locally flat knots are trivial in the record's string sense.\n\n**Full solution or refutation.**\n\nPart (a) is false; part (b) remains unresolved.\n\n**What remains.**\n\nDetermine whether the Bing sling itself is knotted.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.81 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes Brin's negative answer to the broad question from the Bing-sling conjecture.\n\n**Review notes.** Multi-part status preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2741, "problem_number": "KP-1.82", "title": "Kirby Problem 1.82", "statement": "(a) What is a positive knot?\n(b) Describe a simple set of moves to convert between two positive diagrams\nof the same knot or link.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.82.\n\nLiterature notes:\n(1) Positive knots and alternating knots are both defined in diagrammatic\nterms, but somewhat orthogonally to one another.\nThe corresponding\nproblems for alternating knots have been settled.\n(2) The first problem is meant to evoke Ralph Fox’s question, ‘What is an\nalternating knot?’ [Lic97, p.32]. It seeks a characterization of the class\nof positive knots in terms intrinsic to the knot exterior, e.g.\na condi-\ntion on the knot group or the existence of a special kind of geometric\nstructure or spanning surface. It echoes a related question of Rudolph,\nwho proved that positive links are strongly quasipositive. He asked, as\na kind of converse to his result: “Can positive links be characterized as\nstrongly quasipositive links that satisfy some extra geometric conditions?”\n[Rud99, Question, p.556]. Baader made progress on Rudolph’s question\nby showing that a knot is positive if and only if it is strongly quasiposi-\ntive and homogeneous [Baa05]. In light of this, an answer to Problem 1\ncould be obtained by giving a geometric characterization of homogeneity.\nNote that Fox’s question was addressed in related papers by Greene and\nby Howie [Gre17, How17], which led to new applications for alternating\nknots.\n(3) The second problem is meant to evoke the Tait flyping conjecture for alter-\nnating links, which was proven by Menasco and Thistlethwaite [MT93].\nThe quest for a set of moves satisfying Problem (b) is complicated by the\nfact that a knot can have reduced positive diagrams with different crossing\nnumbers.\n\nReferences cited:\n- [Lic97] W. B. Raymond Lickorish. An introduction to knot theory, volume 175 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. doi:10.1007/978-1-4612-0691-0.\n- [Rud99] Lee Rudolph. Positive links are strongly quasipositive. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 555–562. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.555.\n- [Baa05] S. Baader. Quasipositivity and homogeneity. Math. Proc. Cambridge Philos. Soc., 139(2):287–290, 2005. doi:10.1017/S0305004105008698.\n- [Gre17] Joshua Evan Greene. Alternating links and definite surfaces. Duke Math. J., 166(11):2133–2151, 2017. With an appendix by András Juhász and Marc Lackenby. doi:10.1215/00127094-2017-0004.\n- [How17] Joshua A. Howie. A characterisation of alternating knot exteriors. Geom. Topol., 21(4):2353–2371, 2017. doi:10.2140/gt.2017.21.2353.\n- [MT93] William Menasco and Morwen Thistlethwaite. The classification of alternating links. Ann. of Math. (2), 138(1):113–171, 1993. doi:10.2307/2946636.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Positive knots admit the characterization 'strongly quasipositive and homogeneous,' but no intrinsic exterior characterization or complete simple move system between positive diagrams was verified.\n\n**Verified partial progress.**\n\n- Rudolph proved that every positive link is strongly quasipositive.\n- Baader proved that a knot is positive if and only if it is strongly quasipositive and homogeneous.\n- The analogous alternating-diagram move problem is solved by flypes, clarifying the stronger positivity-preserving result sought here.\n\n**Full solution or refutation.**\n\nBaader's theorem is a strong structural answer to part (a), but homogeneity remains diagrammatic rather than the requested intrinsic geometric characterization, and part (b) remains open.\n\n**What remains.**\n\nGive an intrinsic characterization of positive knots or homogeneity, and find a finite simple set of moves connecting positive diagrams without leaving the positive class.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.82, AMS Mathematical Surveys and Monographs 295 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the intended intrinsic and move-theoretic scope and records Baader's characterization.\n- S. Baader, Quasipositivity and homogeneity, Mathematical Proceedings of the Cambridge Philosophical Society 139 (2005), 287--290. (primary): https://doi.org/10.1017/S0305004105008698\n Evidence used: Proves positivity is equivalent to strong quasipositivity plus homogeneity.\n- Lee Rudolph, Positive links are strongly quasipositive, Geometry & Topology Monographs 2 (1999), 555--562. (primary): https://doi.org/10.2140/gtm.1999.2.555\n Evidence used: Proves the foundational implication used in the later characterization.\n\n**Review notes.** Ordinary Reidemeister connectivity is insufficient because the desired moves must preserve positivity.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2742, "problem_number": "KP-1.83", "title": "Kirby Problem 1.83", "statement": "Determine the algebraic structure of the concordance group $\\mathcal{O}$\nof open strings.\n(a) Is it abelian?\n(b) Does it contain torsion?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.83.\n\nLiterature notes:\n(1) Virtual strings, also known as flat virtual knots [Kau99], were introduced\nby Turaev [Tur04] as homotopy classes of immersions of $S^{1}$ in compact\noriented surfaces, up to stabilization of the surface.\n(2) Two virtual strings in $\\Sigma_{1}$ and $\\Sigma_{2}$ are said to be concordant if they cobound\nan immersed annulus in an oriented 3-manifold $M$ with boundary $\\partial M =$\n$\\Sigma_{1} \\cup -\\Sigma_{2}$. A virtual string $\\alpha$ in $\\Sigma$ is said to be slice if there exists an\noriented 3-manifold $M$ and immersed disk $D$ in $M$ such that $\\partial M = \\Sigma$ and\n\n$\\partial D = \\alpha$. The first example of an immersed curve in a surface that does\nnot bound an immersed disk in any 3-manifold is due to Carter [Car91].\n(3) Concordance classes of open strings form a group $\\mathcal{O}$, which is known to be\ninfinitely generated [Tur04]. Turaev also introduced the related notion of\nalgebraic concordance for virtual strings, and Jie Chen [Che23, Example\n3.19] has found examples of flat knots that are algebraically slice but not\nslice. There is a surjection to $\\mathcal{O}$ from the concordance group of virtual\nknots (see the following Problem 1.84), which was shown to be non-abelian\nby Chrisman [Chr22].\n\nReferences cited:\n- [Kau99] Louis H. Kauffman. Virtual knot theory. European J. Combin., 20(7):663–690, 1999. doi:10.1006/eujc.1999.0314.\n- [Tur04] Vladimir Turaev. Virtual strings. Ann. Inst. Fourier (Grenoble), 54(7):2455–2525, 2004. URL: http://aif.cedram.org/item?id=AIF 2004 54 7 2455 0.\n- [Car91] J. Scott Carter. Extending immersions of curves to properly immersed surfaces. Topology Appl., 40(3):287–306, 1991. doi:10.1016/0166-8641(91)90111-X.\n- [Che23] Jie Chen. Flat knots and invariants. PhD thesis, McMaster University, 2023. URL: https://www.math.mcmaster.ca/\\%7Eboden/students/Chen-PhD.pdf.\n- [Chr22] Micah Chrisman. Milnor’s concordance invariants for knots on surfaces. Algebr. Geom. Topol., 22(5):2293–2353, 2022. doi:10.2140/agt.2022.22.2293.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The open-string concordance group is known to be infinitely generated and has nontrivial sliceness obstructions, but neither its commutativity nor the existence of torsion was resolved in the checked literature.\n\n**Verified partial progress.**\n\n- Turaev constructed the group of concordance classes of open strings and proved that it is infinitely generated.\n- Based-matrix and related invariants obstruct sliceness and distinguish many concordance classes.\n- Examples algebraically slice but not slice show that algebraic concordance does not classify the geometric relation.\n\n**Full solution or refutation.**\n\nKnown invariants show that the group is large and that algebraic concordance loses information, but do not decide the two algebraic-structure questions in the statement.\n\n**What remains.**\n\nDetermine whether the concordance product is commutative, detect or exclude finite-order elements, and give a more complete structural description of the group.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.83 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Records infinite generation while retaining commutativity and torsion as open questions.\n- Vladimir Turaev, Virtual strings, Annales de l'Institut Fourier 54 (2004), 2455--2525. (primary): https://doi.org/10.5802/aif.2086\n Evidence used: Introduces virtual/open strings, their cobordism theory and the group/invariants underlying the problem.\n- Jie Chen, Flat knots and invariants, PhD thesis, McMaster University (2023). (primary): https://www.math.mcmaster.ca/~boden/students/Chen-PhD.pdf\n Evidence used: Provides examples that are algebraically slice but not slice, demonstrating incompleteness of the algebraic model.\n\n**Review notes.** Noncommuting open strings under finer equivalence relations were not treated as proof that the concordance quotient is nonabelian.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2743, "problem_number": "KP-1.84", "title": "Kirby Problem 1.84", "statement": "For a classical knot, does its slice genus as a virtual knot agree\nwith its slice genus as a classical knot?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.84.\n\nLiterature notes:\n(1) Virtual knots were originally defined combinatorially by Kauffman [Kau99].\nFor the purposes of this problem a virtual knot $K$ may be regarded as an\nembedding of $S^{1}$ into $\\Sigma \\times [0, 1],$ where $\\Sigma$ is a compact oriented surface.\nThe virtual slice genus of $K$ is the minimal genus, taken over all sur-\nfaces $F$ and all 3-manifolds $M$, such that $F$ embeds into $M \\times [0, 1]$ with\n$\\partial F = K$. Here $M$ is a compact oriented 3-manifold with $\\partial M = \\Sigma$. A\nvirtual knot is said to be virtually slice if it bounds a disk in $M \\times [0, 1]$\nfor some 3-manifold $M$.\n(2) This question is due to Dye-Kaestner-Kauffman [DKK17, §6], who gave\nan affirmative answer any knot for which the slice genus is determined by\nthe Rasmussen invariant, including torus knots. Their results support the\nconjecture that for classical knots, their slice genus as virtual knots agrees\nwith their slice genus as classical knots. It is known that a classical knot\nis virtually slice if and only if it is slice [BN17], thus the conjecture is\ntrue for knots with slice genus one.\n(3) This problem has two versions, one for the smooth slice genus and the\nother for the topological (locally flat) slice genus.\n\nReferences cited:\n- [Kau99] Louis H. Kauffman. Virtual knot theory. European J. Combin., 20(7):663–690, 1999. doi:10.1006/eujc.1999.0314.\n- [DKK17] Heather A. Dye, Aaron Kaestner, and Louis H. Kauffman. Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots. J. Knot Theory Ramifications, 26(3):1741001, 57, 2017. doi:10.1142/S0218216517410012.\n- [BN17] Hans U. Boden and Matthias Nagel. Concordance group of virtual knots. Proc. Amer. Math. Soc., 145(12):5451–5461, 2017. doi:10.1090/proc/13667.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Equality of virtual and classical slice genus is known for broad detected classes, including torus knots, and for classical slice genus at most one; the general smooth and topological versions remain open.\n\n**Verified partial progress.**\n\n- Dye--Kaestner--Kauffman prove equality whenever the classical slice genus is detected by the Rasmussen invariant, including torus knots.\n- Boden--Nagel prove that a classical knot is virtually slice exactly when it is classically slice.\n- The sliceness theorem implies equality for classical knots of slice genus one as well as genus zero.\n\n**Full solution or refutation.**\n\nExisting concordance and Rasmussen-invariant methods settle important classes but do not rule out genus reduction for an arbitrary classical knot viewed virtually.\n\n**What remains.**\n\nProve or disprove equality for every classical knot, separately in the smooth and topologically locally flat categories.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.84 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Separates the smooth and topological questions and states the detected and genus-one cases.\n- Heather A. Dye, Aaron Kaestner and Louis H. Kauffman, Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots, Journal of Knot Theory and Its Ramifications 26 (2017), 1741001. (primary): https://doi.org/10.1142/S0218216517410012\n Evidence used: Extends Rasmussen-type bounds to virtual knots and proves equality for knots detected by them.\n- Hans U. Boden and Matthias Nagel, Concordance group of virtual knots, Proceedings of the AMS 145 (2017), 5451--5461. (primary): https://doi.org/10.1090/proc/13667\n Evidence used: Proves that classical sliceness is unchanged on passage to virtual knots.\n\n**Review notes.** The smooth and locally flat questions were not conflated.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2744, "problem_number": "KP-1.85", "title": "Kirby Problem 1.85", "statement": "Let $K$ be a hyperbolic knot in $S^{3}$ and $\\chi(K)$ the space of con-\njugacy classes of $\\operatorname{PSL}_{2}(\\mathbb{C})$ representations of $\\pi_{1}(S^{3} - K)$. There is a distinguished\ncomponent of $\\chi(K)$ that contains the discrete faithful representation coming from\nthe finite volume hyperbolic structure on the complement of $K$. Does this component\nalways contain an arc of representations to $\\operatorname{SO}(3)$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.85.\n\nLiterature notes:\n(1) Chinburg-Reid-Stover conjectured [CRS22, Conjecture 1.9] that there\nalways is such an arc of representations. A positive answer would in fact\nprove that the knot groups of hyperbolic knots always have faithful $\\operatorname{SO}(3)$\nrepresentations.\n(2) It is explained in [Ago22] that an affirmative answer to this question\nwould prove there are no infinite descending chains of hyperbolic knots in\nthe ribbon concordance partial order [Gor81, Question 6.2]. See Prob-\nlem 1.56 for a knot Floer homology perspective on Gordon’s question.\n\nReferences cited:\n- [CRS22] Ted Chinburg, Alan W. Reid, and Matthew Stover. Azumaya algebras and canonical components. Int. Math. Res. Not. IMRN, 2022(7):4969–5036, 2022. doi:10.1093/imrn/rnaa209.\n- [Ago22] Ian Agol. Ribbon concordance of knots is a partial ordering. Comm. Amer. Math. Soc., 2:374–379, 2022. doi:10.1090/cams/15.\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The canonical-component SO(3)-arc conjecture remains open for arbitrary hyperbolic knots, but it has been proved for knots admitting a Euclidean cone structure with cone angle at most pi.\n\n**Verified partial progress.**\n\n- Chinburg--Reid--Stover formulated the general conjecture in their study of canonical components.\n- Dix proves the conjecture for knots admitting a Euclidean cone-manifold structure with cone angle at most pi.\n- The cone-manifold condition applies to the hyperbolic 2-bridge setting used in Dix's ribbon-concordance application.\n\n**Full solution or refutation.**\n\nCone-manifold deformation supplies the requested compact-real arc for a direct geometric class, but no theorem covering every hyperbolic knot was found.\n\n**What remains.**\n\nEstablish an SO(3) arc on the distinguished PSL_2(C) character component for hyperbolic knots not covered by the cone-angle criterion.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.85 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the general conjecture and its implications for faithful SO(3) representations and ribbon concordance.\n- Ted Chinburg, Alan W. Reid and Matthew Stover, Azumaya algebras and canonical components, International Mathematics Research Notices 2022, 4969--5036. (primary): https://doi.org/10.1093/imrn/rnaa209\n Evidence used: Formulates the canonical-component conjecture.\n- James Patrick Dix, Ribbon Concordances and Representation Varieties, PhD dissertation, University of California, Berkeley (2024). (primary): https://escholarship.org/uc/item/27j2v475\n Evidence used: Proves the conjecture under the Euclidean cone-angle hypothesis and derives the 2-bridge ribbon-concordance application.\n\n**Review notes.** The direct partial theorem is in a dissertation; the exact passage between SL_2 and PSL_2 character conventions merits expert normalization review.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2745, "problem_number": "KP-1.86", "title": "Kirby Problem 1.86", "statement": "(a) Every connected cubic $($ i.e. trivalent $)$ graph has freeness index at least 2.\n(b) Every graph has freeness index at least two.\n(c) There is a graph with freeness index two.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.86.\n\nLiterature notes:\n(1) A knot in the 3-sphere is the trivial knot if and only if its complement is\na solid torus. One can extend the notion of triviality from knots to em-\nbedded (finite) graphs by considering a graph embedded in the 3-sphere\nto be “unknotted” if its complement is a connected sum of handlebod-\nies. Every connected graph has an embedding into $S^{3}$ with handlebody\ncomplement; in fact, using induction and 2-handle addition, one can show\nthat every graph has an unknotted embedding with the additional feature\nthat every subgraph obtained by erasing the interior of a single edge is\nalso unknotted [Tho22].\n(2) Conway and Gordon showed that some graphs (for example, the complete\ngraph with 7 vertices, $K_{7}$) have some knotted subgraphs no matter how\nthey are embedded into $S^{3}$ [CG83b]. Planar graphs, by contrast, can\nbe embedded in $S^{3}$ such that every subgraph is unknotted; so can some\nnon-planar graphs, such as $K_{5}$.\n(3) One can define an integer invariant of a finite (connected) graph $\\Gamma$, the\nfreeness index, which measures the extent to which $\\Gamma$ can be embedded in\nthe 3-sphere such that it and its subgraphs are unknotted.\nDefinition. An embedding $f$ of $\\Gamma$ into $S^{3}$ such that the complement\nof $\\Gamma'$ is unknotted for every subgraph of $\\Gamma'$ of $\\Gamma$ obtained by deleting up to\n$k$ edges from $\\Gamma$ is called $k$-free. The freeness index of $\\Gamma$ is the maximum\nof this integer over all embeddings $f$.\n\n(4) In [Tho22] it is shown that every graph has freeness index at least one,\nwhile the graph $K_{6}$ has freeness index 8 and the Petersen graph has free-\nness index 4. [Tho22].\n(5) We can relate the freeness index to the long-standing orientable cycle\ndouble cover conjecture (OCDCC) from combinatorics ([Sze73, Sey80]).\nThe OCDCC for (bridgeless) cubic graphs implies the OCDCC for all\ngraphs. For bridgeless cubic graphs $\\Gamma$, the OCDCC is equivalent to the\n\nstatement that $\\Gamma$ has an embedding into a closed orientable surface $F$ such\nthat every complementary region is a disk and the closure of every com-\nplementary region is also a disk (this is a strong embedding of $\\Gamma$ into $F$).\nIn order to make a connection with the freeness index, one can exploit the\nfact that once a graph is embedded into a closed orientable surface $F, F$\ncan then be embedded into $S^{3}$ as a Heegaard surface. If both the embed-\nding of the graph into the surface, and of the surface into the 3-sphere, are\nsufficiently “nice”, one can conclude that the induced embedding of the\ngraph into the 3-sphere satisfies some strong freeness conditions. This can\nbe used to show that a cubic graph satisfying the OCDCC has freeness\nindex at least 2 [Tho22].\n(6) Note that a connected bridgeless cubic graph with freeness index 1 would\nprovide a counter-example to the OCDCC.\n\nReferences cited:\n- [Tho22] Abigail Thompson. The freeness index of a graph, 2022. arXiv:2206.12939.\n- [CG83b] J. H. Conway and C. McA. Gordon. Knots and links in spatial graphs. J. Graph Theory, 7(4):445–453, 1983. doi:10.1002/jgt.3190070410.\n- [Sze73] G. Szekeres. Polyhedral decompositions of cubic graphs. Bulletin of the Australian Mathematical Society, 8(3):367–387, 1973. doi:10.1017/S0004972700042660.\n- [Sey80] P. D. Seymour. Disjoint paths in graphs. Discrete Math., 29(3):293–309, 1980. doi: 10.1016/0012-365X(80)90158-2.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every graph has freeness index at least one, and cubic graphs satisfying the orientable cycle double cover conjecture have index at least two. The three stated assertions remain unproved in general.\n\n**Verified partial progress.**\n\n- Thompson proves the universal lower bound one.\n- Thompson proves index at least two for cubic graphs satisfying the orientable cycle double cover conjecture.\n- The computed examples K_6 and the Petersen graph have freeness indices 8 and 4 respectively, so they do not supply the exact-index-two example requested in part (c).\n\n**Full solution or refutation.**\n\nThe conditional cubic theorem connects part (a) to the orientable cycle double cover conjecture, while known examples establish larger indices rather than exact index two.\n\n**What remains.**\n\nRemove the cycle-double-cover hypothesis in part (a), extend the bound to all graphs in part (b), and exhibit or exclude a graph of freeness index exactly two.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.86 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States all three assertions and records the lower bounds, examples and OCDCC implication.\n- Abigail Thompson, The freeness index of a graph, arXiv:2206.12939 (2022). (primary): https://arxiv.org/abs/2206.12939\n Evidence used: Defines the invariant, proves the universal index-one bound and the conditional index-two result for cubic graphs.\n\n**Review notes.** The background definition contains the malformed duplicated phrase 'subgraph of Gamma-prime of Gamma.' Its intended up-to-k-edge deletion condition was not substituted into the source.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2746, "problem_number": "KP-1.87", "title": "Kirby Problem 1.87", "statement": "Is every fibered link in $S^{3}$ realized as the link of an isolated\nsingular point of a polynomial map $\\mathbb{R}^{4} \\to \\mathbb{R}^{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.87.\n\nLiterature notes:\n(1) This problem is due to Benedetti–Shiota [BS98], and originates in the\ncelebrated Milnor book [Mil68b].\n(2) Let $f: \\mathbb{R}^{4} \\to \\mathbb{R}^{2}$ be a polynomial map with $f(0) = 0$ and set $V = f^{-1}(0)$.\nAssume that $0 \\in \\mathbb{R}^{4}$ is an isolated point of the intersection of $V$ and the\nset of singular points of $f$. Such a point is often called a weakly isolated\nsingularity.\nAs instances of Milnor’s cone structure and fibration theorems [Mil68b],\nwe have:\n(i) There exists a small positive real number $\\epsilon_{0}$ such that for every $0 <$\n$\\epsilon < \\epsilon_{0}$, the sphere $S^{3}_{\\epsilon}$ in $\\mathbb{R}^{4}$ with radius $\\epsilon$ centered at 0 is transverse\nto the surface $V$, and the link type $L$ of\n$L_{\\epsilon}=(1/\\epsilon)(S^{3}_{\\epsilon}\\cap V)\\subset S^{3}$\ndoes not depend on the choice of $\\epsilon$.\n(ii) Under the stronger assumption that $0 \\in \\mathbb{R}^{4}$ is an isolated singular\npoint of the map $f$, the link $L \\subset S^{3}$ is fibered.\n(3) In [Mil68b], one is mainly concerned with the links of (the realifications\nof) complex polynomial maps $\\mathbb{C}^{2} \\to \\mathbb{C}$ at an isolated singular point; in\nsuch a case, a weakly isolated singularity in (1) above is equivalent to\na usual isolated singularity as in (2), and we obtain a distinguished and\nwell understood family of fibered links (see [Neu03, Web08]). In the\ngeneral case, there are no evident obstructions against a positive answer\nto the problem. As remarked in [Mil68b, Page 84], the first example of\na fibered knot to be considered not belonging to that family for complex\npolynomials, was the figure-eight knot. Later, an explicit realization of the\nfigure-eight knot as the link of a real isolated singularity was obtained in\n[Per82]. Further partial positive results have been worked out in [Loo71,\nRud87, Pic05, Bod19, Bod20, Bod23, AdSSQ24].\n\n(4) It is known [AK81] that every link $L$ in $S^{3}$ can be realized as the link\nof a polynomial map as in (i) above, only requiring that 0 is an isolated\npoint of the intersection of $V$ and the set of singular points of $f$.\n\nReferences cited:\n- [BS98] R. Benedetti and M. Shiota. On real algebraic links in $S^{3}$. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1(3):585–609, 1998.\n- [Mil68b] John Milnor. Singular points of complex hypersurfaces, volume No. 61 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1968. https://www.jstor.org/stable/j.ctt1bd6kvv.\n- [Neu03] W. Neumann. Topology of hypersurface singularities. In Erich Kähler Mathematische Werken, pages 727–736. de Gruyter, 2003. See also https://arxiv.org/abs/1706.04386.\n- [Web08] Claude Weber. On the topology of singularities. In Singularities II, volume 475 of Contemp. Math., pages 217–251. Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/conm/475/09285.\n- [Per82] B. Perron. Le nœud “huit” est algébrique réel. Invent. Math., 65(3):441–451, 1981/82. doi:10.1007/BF01396628.\n- [Loo71] Eduard Looijenga. A note on polynomial isolated singularities. Indag. Math., 33:418–421, 1971. Nederl. Akad. Wetensch. Proc. Ser. A 74.\n- [Rud87] Lee Rudolph. Isolated critical points of mappings from $\\mathbb{R}^{4}$ to $\\mathbb{R}^{2}$ and a natural splitting of the Milnor number of a classical fibered link. I. Basic theory; examples. Comment. Math. Helv., 62(4):630–645, 1987. doi:10.1007/BF02564467.\n- [Pic05] Anne Pichon. Real analytic germs fg and open-book decompositions of the 3-sphere. Internat. J. Math., 16(1):1–12, 2005. doi:10.1142/S0129167X05002710.\n- [Bod19] Benjamin Bode. Constructing links of isolated singularities of polynomials $\\mathbb{R}^{4}$ $\\to$ $\\mathbb{R}^{2}$. J. Knot Theory Ramifications, 28(1):1950009, 21, 2019. doi:10.1142/S0218216519500093.\n- [Bod20] Benjamin Bode. Real algebraic links in $S^{3}$ and braid group actions on the set of n-adic integers. J. Knot Theory Ramifications, 29(6):2050039, 44, 2020. doi: 10.1142/S021821652050039X.\n- [Bod23] Benjamin Bode. Twisting and satellite operations on P-fibered braids. Comm. Anal. Geom., 31(8):2013–2038, 2023. doi:10.4310/cag.2023.v31.n8.a5.\n- [AdSSQ24] Raimundo N. Araújo dos Santos and Eder L. Sanchez Quiceno. On real algebraic links in the 3-sphere associated with mixed polynomials. Res. Math. Sci., 11(2):Paper No. 22, 22, 2024. doi:10.1007/s40687-024-00424-3.\n- [AK81] S. Akbulut and H. King. All knots are algebraic. Comment. Math. Helv., 56(3):339– 351, 1981. doi:10.1007/BF02566217.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many broad families of fibered links are realized as links of isolated singularities of real polynomial maps R^4 to R^2, but no realization theorem for every fibered link was verified.\n\n**Verified partial progress.**\n\n- Perron explicitly realized the figure-eight knot, the classical first non-complex-algebraic test case.\n- Bode developed constructive criteria and later proved that closures of T-homogeneous, hence homogeneous, braids and further dual-Garside-positive braids are real algebraic.\n- Akbulut--King prove every link is realizable with a weakly isolated singularity, a strictly weaker hypothesis than the isolated singular point required here.\n\n**Full solution or refutation.**\n\nConstructive braid and mixed-polynomial methods cover large fibered families; weakly isolated realization of arbitrary links does not solve the stronger isolated-singularity problem.\n\n**What remains.**\n\nExtend the construction to every fibered link or find an obstruction to realization by an isolated singular point of a real polynomial map.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.87 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Separates isolated from weakly isolated singularities and surveys the verified families.\n- Benjamin Bode, Constructing links of isolated singularities of polynomials R^4 to R^2, Journal of Knot Theory and Its Ramifications 28 (2019), 1950009. (primary): https://doi.org/10.1142/S0218216519500093\n Evidence used: Gives an explicit sufficient braid criterion and polynomial construction.\n- Benjamin Bode, Closures of T-homogeneous braids are real algebraic, Algebraic & Geometric Topology 25 (2025), 1075--1115. (primary): https://doi.org/10.2140/agt.2025.25.1075\n Evidence used: Proves the conjecture for a large family containing homogeneous braid closures.\n- S. Akbulut and H. King, All knots are algebraic, Commentarii Mathematici Helvetici 56 (1981), 339--351. (primary): https://doi.org/10.1007/BF02566217\n Evidence used: Provides the arbitrary-link weakly isolated result, not the stronger isolated-singularity conclusion.\n\n**Review notes.** Weakly isolated and isolated singularities were kept logically separate.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2747, "problem_number": "KP-1.88", "title": "Kirby Problem 1.88", "statement": "Are there infinitely many congruence arithmetic links in the\n3-sphere?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.88.\n\nLiterature notes:\n(1) This problem relates to the more general question posed by Thurston\nto find the topological and geometric properties of quotients of $\\mathbb{H}^{3}$ by\narithmetic subgroups of $\\operatorname{PSL}(2, \\mathbb{C})$ [Thu82].\n(2) A link complement is arithmetic if it is of the form $\\mathbb{H}^{3}/\\Gamma$ for some discrete\nsubgroup $\\Gamma \\leq \\operatorname{PSL}(2, \\mathbb{C})$ with the property that $\\Gamma$ is commensurable with\n$\\operatorname{PSL}(2, O_{d})$, where $O_{d}$ the ring of integers in $\\mathbb{Q}($\n$?$\n$-d)$. Famously, Reid\n[Rei91] proved that the figure-eight knot is the only arithmetic knot in\nthe 3-sphere. However, there are infinitely many arithmetic links (see for\nexample the discussion in [BR23a]).\n(3) The question is concerned with a particular type of arithmetic link called\ncongruence links, which are defined as follows. A principal congruence\nsubgroup $\\Gamma$ is equal to the kernel of\n\n$$\n\\operatorname{PSL}(2, O_{d}) \\to \\operatorname{PSL}(2, O_{d}/I)\n$$\n\nfor some ideal $I$ in $O_{d}$. More generally, $\\Gamma$ is called congruence if it contains\na principal congruence subgroup. When $\\mathbb{H}^{3}/\\Gamma$ is a link complement, the\nlink is said to be principal congruence and congruence respectively.\n(4) In [BGR19], Baker, Goerner and Reid gave a complete classification of\nprincipal congruence arithmetic links in the 3-sphere. In particular, there\nare finitely many such links. (See [BGR22] for all known link diagrams\nof principal congruence links.) The question asks whether this remains\ntrue for congruence arithmetic links.\n(5) Much information is known about arithmetic links in the 3-sphere and in\nparticular about congruence links. For example, any arithmetic link in\nthe 3-sphere commensurable with $\\operatorname{PSL}(2, O_{d})$ must have\n\n$$\nd \\in \\{1, 2, 3, 5, 6, 7, 11, 15, 19, 23, 31, 39, 47, 71\\}.\n$$\n\nReferences cited:\n- [Thu82] William P. Thurston. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.), 6(3):357–381, 1982. doi:10.1090/S0273-0979-1982-15003-0.\n- [Rei91] Alan W. Reid. Arithmeticity of knot complements. J. London Math. Soc. (2), 43(1):171–184, 1991. doi:10.1112/jlms/s2-43.1.171.\n- [BR23a] Mark D. Baker and Alan W. Reid. Infinitely many arithmetic alternating links. Algebr. Geom. Topol., 23(6):2857–2866, 2023. doi:10.2140/agt.2023.23.2857.\n- [BGR19] M. D. Baker, M. Goerner, and A. W. Reid. All principal congruence link groups. J. Algebra, 528:497–504, 2019. doi:10.1016/j.jalgebra.2019.02.023.\n- [BGR22] Mark D. Baker, Matthias Goerner, and Alan W. Reid. All known principal congruence links, 2022. arXiv:1902.04426.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Principal congruence arithmetic link complements in S^3 are completely classified and finite in number, while infinitely many arithmetic links are known. Whether the broader class of congruence arithmetic links is infinite remains open.\n\n**Verified partial progress.**\n\n- Baker--Goerner--Reid classified all principal congruence link groups and obtained finiteness in that subclass.\n- Baker--Reid constructed infinitely many arithmetic alternating links, showing that arithmeticity alone permits infinite families.\n- Only a finite list of imaginary quadratic commensurability parameters can occur for arithmetic links in S^3.\n\n**Full solution or refutation.**\n\nThe principal-congruence classification settles a natural strict subclass, but neither it nor infinite arithmetic families decides infinitude under the intermediate congruence condition.\n\n**What remains.**\n\nConstruct infinitely many nonprincipal congruence link complements or prove that only finitely many congruence arithmetic links occur.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.88 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Distinguishes arithmetic, principal congruence and congruence links and retains the middle question.\n- M. D. Baker, M. Goerner and A. W. Reid, All principal congruence link groups, Journal of Algebra 528 (2019), 497--504. (primary): https://doi.org/10.1016/j.jalgebra.2019.02.023\n Evidence used: Provides the complete principal-congruence classification.\n- Mark D. Baker and Alan W. Reid, Infinitely many arithmetic alternating links, Algebraic & Geometric Topology 23 (2023), 2857--2866. (primary): https://doi.org/10.2140/agt.2023.23.2857\n Evidence used: Establishes infinitude in the broader arithmetic class without proving congruence.\n\n**Review notes.** OCR defect preserved: the background's field Q(?-d) has lost the square-root symbol; the intended imaginary quadratic field is clear from context but was not inserted into the record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2748, "problem_number": "KP-1.89", "title": "Kirby Problem 1.89", "statement": "(a) Fix a long link L. What is the homotopy type of the embedding space of\nlinks isotopic to L?\n(b) Fix a link L in a 3-manifold M. What is the homotopy type of the embed-\nding space of links in M isotopic to L?\n\n(c) Given a link $L$ obtained by infecting a link $L_{1}$ by another link $L_{2}$, describe\nthe homotopy type of component of $L$ in the space of links in terms of the\ncomponents of $L_{1}$ and $L_{2}$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-1.89.\n\nLiterature notes:\n(1) A long link is an embedding of a disjoint union of lines that agree with\na fixed collection of affine lines outside of a box. The answer to the first\nquestion, due to Hatcher and Budney, is known for long knots in the sense\nthat the knot spaces are shown to be homotopy equivalent to standard\ncombinations of standard spaces; see Remark (3) below.\n(2) Spaces of long links have applications in 4-manifold theory as well as high-\ndimensional manifold theory. For example, the recent work of Budney and\nGabai [BG19] on the homotopy-type of $\\operatorname{Diff}(S^{1} \\times D^{n-1})$ for $n \\geq 4$ uses\nthe low-dimensional homotopy groups of the space of long embeddings\n$\\operatorname{Emb}(I, S^{1} \\times D^{n-1})$. This space can be thought of as the homotopy fiber\nof the forgetful map $\\operatorname{Emb}(I \\sqcup D^{n-2}, D^{n}) \\to \\operatorname{Emb}(D^{n-2}, D^{n})$, i.e. it is the\nspace of 2-component ‘long links’ where one component is trivial.\n(3) A major step in the description of long knot spaces was Hatcher’s proof\nof the Smale conjecture, which is equivalent [Hat83, Appendix] to the\nstatement that $\\operatorname{Diff}(D^{3}; \\partial D^{3})$ is contractible, which in turn implies that\nthe component of the long unknot is contractible. Combined with work of\nHatcher [Hat76] and Ivanov [Iva76] on diffeomorphisms of 3-manifolds,\nit also implies that each component in the space of long knots is a $K(\\pi, 1)$\nspace [Bud07].\nHatcher [Hat02] determined the homotopy type of a\nlong torus knot and, building on his work with McCullough [HM97a], of\na long hyperbolic knot.\nBudney [Bud10] determined the homotopy type of the component of\nany long knot $K$ in $\\operatorname{Emb}(\\mathbb{R}, \\mathbb{R}^{3})$ in terms of the satellite decomposition\nof $K$, i.e. in terms of splicing operations, thus answering part (a) above\nwhen $L$ has one component.\nThe homotopy type is roughly a twisted\nproduct of factors of $S^{1}, S^{1} \\times S^{1}$, and configuration spaces of points in\nthe plane, with one factor for each knot or link appearing in the satellite\ndecomposition.\n(4) Havens and Koytcheff [HK21] generalized Hatcher and Budney’s results\nto the space of closed links in $S^{3}$ modulo rotations, whose components are\n$K(\\pi, 1)$ spaces for irreducible links. They did not completely determine\nthe homotopy types of spaces of split links because $\\operatorname{Diff}(M)$ for a reducible\n3-manifold $M$ is not completely understood. However, Boyd and Bregman\n[BB25] obtained more detailed information on spaces of split links, espe-\ncially their fundamental groups, using semi-simplicial spaces that model\nthem. Kosanović [Kos24] related Vassiliev invariants to the Goodwillie–\nWeiss embedding calculus in the setting of a long knot $K$ in any compact\noriented 3-manifold $M$ with boundary.\n(5) In part (c), infection generalizes splicing from long knots to string (a.k.a. long)\nlinks. Budney [Bud07, Bud12] developed the operations of connect-sum\nand splicing at the space level, parameterizing them by operads.\nFor\nboth operations, he obtained space-level decomposition results, which for\nconnect-sum he further developed at the homological level with F. Cohen\n[BC09].\n\nBurke and Koytcheff [BK15b] generalized Budney’s splicing operad\nto a (colored) operad for string link infection. They obtained a partial\ndecomposition result for 2-component string links, building upon a re-\nsult on isotopy classes in their joint work with Blair [BBK15]. This de-\ncomposition was completed for 2-component string links by Batelier and\nDucoulombier [BD23]. It is essentially in terms of the stacking of string\nlinks rather than the more general infection operation. An answer to part\n(c) would be related to a decomposition of a 3-manifold analogous to the\nJSJ decomposition but with tori replaced by surfaces of higher genus.\n\nReferences cited:\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.\n- [Hat83] Allen E. Hatcher. A proof of the Smale conjecture, Diffp$S^{3}$q » $O(4)$. Ann. of Math. (2), 117(3):553–607, 1983. doi:10.2307/2007035.\n- [Hat76] Allen Hatcher. Homeomorphisms of sufficiently large P2-irreducible 3-manifolds. Topology, 15(4):343–347, 1976. doi:10.1016/0040-9383(76)90027-6.\n- [Iva76] N. V. Ivanov. Diffeomorphism groups of Waldhausen manifolds. J Math Sci., 12:115–118, 1976. doi:10.1007/BF01098421.\n- [Bud07] Ryan Budney. Little cubes and long knots. Topology, 46(1):1–27, 2007. doi:10.1016/j.top.2006.09.001.\n- [Hat02] Allen Hatcher. Topological moduli spaces of knots. https://pi.math.cornell.edu/„hatcher/Papers/knotspaces.pdf, 2002.\n- [HM97a] Allen Hatcher and Darryl McCullough. Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds. Geom. Topol., 1:91–109, 1997. doi:10.2140/gt.1997.1.91.\n- [Bud10] Ryan Budney. Topology of knot spaces in dimension 3. Proc. Lond. Math. Soc. (3), 101(2):477–496, 2010. doi:10.1112/plms/pdp058.\n- [HK21] Andrew Havens and Robin Koytcheff. Spaces of knots in the solid torus, knots in the thickened torus, and links in the 3-sphere. Geom. Dedicata, 214:671–737, 2021. doi:10.1007/s10711-021-00633-y.\n- [BB25] Rachael Boyd and Corey Bregman. Embedding spaces of split links. Adv. Math., 470:Paper No. 110235, 41, 2025. doi:10.1016/j.aim.2025.110235.\n- [Kos24] Danica Kosanović. Embedding calculus and grope cobordism of knots. Adv. Math., 451:Paper No. 109779, 118, 2024. doi:10.1016/j.aim.2024.109779.\n- [Bud12] Ryan Budney. An operad for splicing. J. Topol., 5(4):945–976, 2012. doi:10.1112/jtopol/jts024.\n- [BC09] Ryan Budney and Fred Cohen. On the homology of the space of knots. Geom. Topol., 13(1):99–139, 2009. doi:10.2140/gt.2009.13.99.\n- [BK15b] John Burke and Robin Koytcheff. A colored operad for string link infection. Algebr. Geom. Topol., 15(6):3371–3408, 2015. doi:10.2140/agt.2015.15.3371.\n- [BBK15] Ryan Blair, John Burke, and Robin Koytcheff. A prime decomposition theorem for the 2-string link monoid. J. Knot Theory Ramifications, 24(2):1550005, 24, 2015. doi:10.1142/S0218216515500054.\n- [BD23] Etienne Batelier and Julien Ducoulombier. Operadic actions on long knots and 2-string links. Algebr. Geom. Topol., 23(2):833–882, 2023. doi:10.2140/agt.2023.23.833.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The homotopy type is described for every long knot and for several important link classes, but arbitrary long/closed links, links in general 3-manifolds, and infection components remain incompletely classified.\n\n**Verified partial progress.**\n\n- Hatcher and Budney determine components of long-knot spaces, with Budney expressing the general answer through satellite/splicing decomposition.\n- Havens--Koytcheff handle irreducible closed links in S^3 modulo rotations, while Boyd--Bregman model split-link spaces and compute their motion groups.\n- Batelier--Ducoulombier complete a decomposition for two-component string links, and Boyd--Bregman compute the Hopf-link embedding space as S^3/Q_8.\n\n**Full solution or refutation.**\n\nKnot spaces and selected link decompositions have explicit models, but the all-link and all-ambient-manifold formulation, especially general infection, extends beyond those results.\n\n**What remains.**\n\nDetermine full homotopy types for general multicomponent links in arbitrary 3-manifolds and formulate a complete space-level decomposition compatible with general infection.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 1.89 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Separates the solved long-knot case from partial closed-link, split-link and string-link results.\n- Ryan Budney, Topology of knot spaces in dimension 3, Proceedings of the London Mathematical Society 101 (2010), 477--496. (primary): https://doi.org/10.1112/plms/pdp058\n Evidence used: Determines long-knot component homotopy types via satellite decomposition.\n- Rachael Boyd and Corey Bregman, Embedding spaces of split links, Advances in Mathematics 470 (2025), 110235. (primary): https://doi.org/10.1016/j.aim.2025.110235\n Evidence used: Models split-link embedding spaces by separating systems and computes their fundamental groups.\n- Rachael Boyd and Corey Bregman, The embedding space of a Hopf link, arXiv:2504.21806 (2025). (primary): https://arxiv.org/abs/2504.21806\n Evidence used: Computes a new explicit two-component example, homotopy equivalent in R^3 to S^3/Q_8.\n\n**Review notes.** Parametrized versus unparametrized and R^3 versus S^3 conventions vary among sources; only scope-compatible conclusions were summarized.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2749, "problem_number": "KP-2.1", "title": "Kirby Problem 2.1", "statement": "(Ivanov conjecture). Let $S$ be an orientable surface of finite type\nwith genus at least three. If $G \\leq \\operatorname{Mod}(S)$ is a subgroup of finite index, does $G$ have\nfinite abelianization?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.1.\n\nLiterature notes:\n(1) The Ivanov conjecture is a weaker version of the question of whether or\nnot the mapping class group has property (T). (The unpublished preprint\n[And07] announced a negative answer to the latter, which does not resolve\nthe Ivanov conjecture in any case). It was recently shown that automor-\nphism groups of most free groups do have property (T) and hence every\nfinite index subgroup has finite abelianization: this was resolved for rank\n5 by Kaluba–Nowak–Ozawa [KNO19] and for $\\operatorname{rank} \\geq 6$ by [KKN21a].\nA resolution for rank 4 was announced by Nitsche [Nit22].\n(2) The Ivanov conjecture has an equivalent reformulation in terms of map-\nping class group actions on the homology of finite covers of surfaces [PW13].\nThe corresponding statement about these actions has become known as\nthe Putman–Wieland conjecture. Some progress on the Putman–Wieland\nconjecture has been made [LL23b, LL23a, MT24].\n(3) For braid groups and mapping class groups of surfaces of genus two or\nlower, virtual surjections to the integers exist by virtue of the existence of\nsuch maps for braid groups.\n\nReferences cited:\n- [And07] Jorgen Ellegaard Andersen. Mapping class groups do not have Kazhdan’s property (t), 2007. arXiv:0706.2184.\n- [KNO19] Marek Kaluba, Piotr W. Nowak, and Narutaka Ozawa. $\\mathrm{Aut}(F_5)$ has property $(T)$. Math. Ann., 375(3-4):1169–1191, 2019. doi:10.1007/s00208-019-01874-9.\n- [KKN21a] Marek Kaluba, Dawid Kielak, and Piotr W. Nowak. On property (T) for $\\mathrm{Aut}(F_n)$ and $\\mathrm{SL}_n(\\mathbb{Z})$. Ann. of Math. (2), 193(2):539–562, 2021. doi:10.4007/annals.2021.193.2.3.\n- [Nit22] Martin Nitsche. Computer proofs for Property (T), and SDP duality, 2022. arXiv: 2009.05134.\n- [PW13] Andrew Putman and Ben Wieland. Abelian quotients of subgroups of the mappings class group and higher Prym representations. J. Lond. Math. Soc. (2), 88(1):79–96, 2013. doi:10.1112/jlms/jdt001.\n- [LL23b] Aaron Landesman and Daniel Litt. An introduction to the algebraic geometry of the Putman-Wieland conjecture. Eur. J. Math., 9(2):Paper No. 40, 25, 2023. doi: 10.1007/s40879-023-00637-w.\n- [LL23a] Aaron Landesman and Daniel Litt. Applications of the algebraic geometry of the Putman-Wieland conjecture. Proc. Lond. Math. Soc. (3), 127(1):116–133, 2023. doi:10.1112/plms.12539.\n- [MT24] Vladimir Marković and Ognjen Tošić. The second variation of the Hodge norm and higher Prym representations. J. Topol., 17(1):Paper No. e12322, 23, 2024. doi:10.1112/topo.12322.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ivanov's finite-abelianization conjecture remains open in genus at least three. It is proved for several large classes of finite-index subgroups, and the related Putman--Wieland property holds for many and asymptotically almost all high-genus covers.\n\n**Verified partial progress.**\n\n- Hain and McCarthy establish vanishing of rational first homology for finite-index subgroups containing the Torelli group.\n- Putman proves powers of Dehn twists vanish in rational abelianization and verifies the conjecture for subgroups containing a sufficiently large part of the Johnson kernel.\n- Klukowski--Markovic prove that the proportion of degree-n covers with the Putman--Wieland property tends to one in the stated high-genus regime.\n\n**Full solution or refutation.**\n\nThese theorems provide strong subgroup-specific and asymptotic evidence, but an almost-all statement for covers does not imply the universal assertion for every finite-index subgroup.\n\n**What remains.**\n\nProve finite abelianization for every finite-index subgroup of Mod(S) in every genus at least three, or exhibit a finite-index subgroup with infinite abelianization.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.1 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the exact genus-at-least-three conjecture and its relation to Putman--Wieland.\n- Andrew Putman, A note on the abelianizations of finite-index subgroups of the mapping class group, Proceedings of the AMS 138 (2010), 753--758. (primary): https://arxiv.org/abs/0812.0017\n Evidence used: Proves twist vanishing and the Johnson-kernel class of the conjecture.\n- Adam Klukowski and Vladimir Markovic, Tangle free permutations and the Putman--Wieland property of random covers, International Mathematics Research Notices 2024, 13400--13416. (primary): https://doi.org/10.1093/imrn/rnae206\n Evidence used: Proves the asymptotic almost-all-covers theorem in sufficiently high genus.\n- Andrew Putman and Ben Wieland, Abelian quotients of subgroups of the mapping class group and higher Prym representations, Journal of the London Mathematical Society 88 (2013), 79--96. (primary): https://doi.org/10.1112/jlms/jdt001\n Evidence used: Establishes the precise bridge between finite-index abelianizations and higher Prym representations.\n\n**Review notes.** The failure of the Putman--Wieland conjecture in genus two does not directly refute this record, whose surface genus is at least three; the genus shifts in the equivalence require care.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2750, "problem_number": "KP-2.2", "title": "Kirby Problem 2.2", "statement": "(Congruence subgroup problem). Does every finite-index sub-\ngroup of the mapping class group of $S$ contain a congruence subgroup?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.2.\n\nLiterature notes:\n(1) Let $S$ be a finite-type surface, and let $K \\leq \\pi_{1}(S)$ be a finite-index charac-\nteristic (i.e., $\\operatorname{Aut}(\\pi_{1}(S))$-invariant) subgroup. A congruence subgroup of\nthe mapping class group of $S$ is the kernel of the natural map from the\nmapping class group of $S$ to $\\operatorname{Out}(\\pi_{1}(S)/K)$. This definition is analogous\nto congruence subgroups of integral special linear groups.\n(2) The congruence subgroup problem is known to have a positive answer\nin certain low-genus cases. For punctured spheres, this was established\nby Diaz–Donagi–Harbater [DDH89], with subsequent alternative proofs\ngiven by Asada [Asa01], Thurston, and McReynolds [McR12]. The case\nof genus one was originally addressed by Asada [Asa01], with further\nwork done by Kent [Ken16] and Bux–Ershov–Rapinchuk [BER11].\n(3) A positive resolution to the congruence subgroup problem has applications\nin anabelian geometry [Mar19].\n\nReferences cited:\n- [DDH89] Steven Diaz, Ron Donagi, and David Harbater. Every curve is a Hurwitz space. Duke Math. J., 59(3):737–746, 1989. doi:10.1215/S0012-7094-89-05933-4.\n- [Asa01] Mamoru Asada. The faithfulness of the monodromy representations associated with certain families of algebraic curves. J. Pure Appl. Algebra, 159(2-3):123–147, 2001. doi:10.1016/S0022-4049(00)00056-6.\n- [McR12] D. B. McReynolds. The congruence subgroup problem for pure braid groups: Thurston’s proof. New York J. Math., 18:925–942, 2012. http://nyjm.albany.edu: 8000/j/2012/18 925.html.\n- [Ken16] Autumn Kent. Congruence kernels around affine curves. J. Reine Angew. Math., 713:1–20, 2016. doi:10.1515/crelle-2014-0023.\n- [BER11] Kai-Uwe Bux, Mikhail V. Ershov, and Andrei S. Rapinchuk. The congruence subgroup property for Aut F2: a group-theoretic proof of Asada’s theorem. Groups Geom. Dyn., 5(2):327–353, 2011. doi:10.4171/GGD/130.\n- [Mar19] Dan Margalit. Problems, questions, and conjectures about mapping class groups. In Breadth in contemporary topology, volume 102 of Proc. Sympos. Pure Math., pages 157–186. Amer. Math. Soc., Providence, RI, 2019. doi:10.1090/pspum/102/12.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The mapping-class congruence subgroup problem is affirmative for punctured spheres and genus one, but remains open for general surfaces.\n\n**Verified partial progress.**\n\n- Diaz--Donagi--Harbater and later work settle punctured spheres.\n- Asada and later work handle genus one.\n\n**Full solution or refutation.**\n\nNo general congruence theorem was verified.\n\n**What remains.**\n\nResolve higher-genus finite-index subgroups.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.2 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists the solved low-genus cases and general problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2751, "problem_number": "KP-2.3", "title": "Kirby Problem 2.3", "statement": "Is the mapping class group of a surface of finite type linear?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.3.\n\nLiterature notes:\n(1) Recall that a group is linear if it can be embedded as a subgroup of $\\operatorname{GL}_{n}(\\mathbb{C})$\nfor some $n$.\n(2) Following the linearity of the braid group (see Remark (4) below), there\nare some low-complexity cases for which linearity is known to hold. This\nis true, in particular, for mapping class groups of punctured spheres and\nhyperelliptic mapping class groups, including the full mapping class group\nin genus 2 [Kor00, BB01]. In addition, if one can prove that a non-\nclosed mapping class group is nonlinear (say of genus $g$ with one puncture\nor boundary component) then all mapping class groups of genus $h > g$\nhave nonlinear mapping class groups. Unsurprisingly, there are examples\nof infinite-type surfaces with nonlinear mapping class groups [APV21].\n(3) The analogous question of linearity for $\\operatorname{Aut}(F_{n})$, the automorphism group\nof a free group $F_{n}$ of $\\operatorname{rank} n$, has been resolved in the negative by Formanek–\nProcesi [FP92]. They construct certain “poison subgroups” of $\\operatorname{Aut}(F_{n})$\nthat are known to be nonlinear. Subsequently, such poison subgroups were\nshown not to exist in mapping class groups by Brendle–Hamidi-Tehrani\n[BHT01].\n(4) On the other hand, braid groups are known to be linear, due to the work\nof Bigelow and Krammer [Big01, Kra02].\nThe Lawrence–Krammer–\nBigelow representations used to certify this are constructed as the action\non the homology of a certain 4-manifold given as an infinite abelian cover\nof the configuration space of two points in the $n$-punctured disk. It is\ntempting to speculate about carrying out an analogous construction in\nhigher genus, but various complications arise; ultimately, the natural ho-\nmology representations of braid groups are simply symmetric groups, and\nthe Lawrence–Krammer–Bigelow representation arises as a deformation of\na symmetric group representation [Jon87]. The corresponding homology\n\nactions of mapping class groups of surfaces in positive genus have infinite\nimage.\n(5) One possible approach to establishing nonlinearity would be to make effec-\ntive the residual finiteness of mapping class groups. In particular, super-\npolynomial residual finiteness growth would imply nonlinearity by work\nof Bou-Rabee–McReynolds [BRM15].\n(6) There is a categorified faithful, finite-dimensional, linear representation\nwith mapping classes acting by functors; see [LOT13].\n(7) Certain constraints on the properties of a faithful representation have\nbeen established. In [Kor23], Korkmaz shows that for $g \\geq 3$, any faith-\nful representation must have dimension $n > 3g - 3$.\nFor any finite-\ndimensional linear representation of a mapping class group in genus $g \\geq 3$,\nBridson [Bri10] proved that Dehn twists are necessarily sent to quasi-\nunipotent matrices (i.e., ones whose eigenvalues are roots of unity). From\nthis one can easily show that mapping class groups have no faithful linear\nrepresentations in positive characteristic. For more recent further devel-\nopments in this direction, see [AS16, But19, KLS19].\n(8) For a much more extensive discussion and bibliography, see the article of\nMargalit [Mar19].\n\nReferences cited:\n- [Kor00] Mustafa Korkmaz. On the linearity of certain mapping class groups. Turkish J. Math., 24(4):367–371, 2000.\n- [BB01] Stephen J. Bigelow and Ryan D. Budney. The mapping class group of a genus two surface is linear. Algebr. Geom. Topol., 1:699–708, 2001. doi:10.2140/agt.2001.1.699.\n- [APV21] Tarik Aougab, Priyam Patel, and Nicholas G. Vlamis. Isometry groups of infinite genus hyperbolic surfaces. Math. Ann., 381:459–498, 2021.\n- [FP92] Edward Formanek and Claudio Procesi. The automorphism group of a free group is not linear. J. Algebra, 149(2):494–499, 1992. doi:10.1016/0021-8693(92)90029-L.\n- [BHT01] Tara E. Brendle and Hessam Hamidi-Tehrani. On the linearity problem for mapping class groups. Algebr. Geom. Topol., 1:445–468, 2001. doi:10.2140/agt.2001.1.445.\n- [Big01] Stephen J. Bigelow. Braid groups are linear. J. Amer. Math. Soc., 14(2):471–486, 2001. doi:10.1090/S0894-0347-00-00361-1.\n- [Kra02] Daan Krammer. Braid groups are linear. Ann. of Math. (2), 155(1):131–156, 2002. doi:10.2307/3062152.\n- [Jon87] V. F. R. Jones. Hecke algebra representations of braid groups and link polynomials. Ann. of Math. (2), 126(2):335–388, 1987. doi:10.2307/1971403.\n- [BRM15] Khalid Bou-Rabee and D. B. McReynolds. Extremal behavior of divisibility functions. Geom. Dedicata, 175:407–415, 2015. doi:10.1007/s10711-014-9955-5.\n- [LOT13] Robert Lipshitz, Peter Ozsváth, and Dylan Thurston. A faithful linear-categorical action of the mapping class group of a surface with boundary. J. Eur. Math. Soc. (JEMS), 15(4):1279–1307, 2013. doi:10.4171/JEMS/392.\n- [Kor23] Mustafa Korkmaz. Low-dimensional linear representations of mapping class groups. J. Topol., 16(3):899–935, 2023. doi:10.1112/topo.12305.\n- [Bri10] Martin R. Bridson. Semisimple actions of mapping class groups on $\\mathrm{CAT}(0)$ spaces. In Geometry of Riemann surfaces, volume 368 of London Math. Soc. Lecture Note Ser., pages 1–14. Cambridge Univ. Press, Cambridge, 2010.\n- [AS16] Javier Aramayona and Juan Souto. Rigidity phenomena in the mapping class group. In Handbook of Teichmüller theory. Vol. VI, volume 27 of IRMA Lect. Math. Theor. Phys., pages 131–165. Eur. Math. Soc., Zürich, 2016.\n- [But19] Jack Oliver Button. Aspects of non positive curvature for linear groups with no infinite order unipotents. Groups Geom. Dyn., 13(1):277–292, 2019. doi:10.4171/GGD/484.\n- [KLS19] Thomas Koberda, Feng Luo, and Hongbin Sun. An effective Lie-Kolchin theorem for quasi-unipotent matrices. Linear Algebra Appl., 581:304–323, 2019. doi:10.1016/j.laa.2019.07.023.\n- [Mar19] Dan Margalit. Problems, questions, and conjectures about mapping class groups. In Breadth in contemporary topology, volume 102 of Proc. Sympos. Pure Math., pages 157–186. Amer. Math. Soc., Providence, RI, 2019. doi:10.1090/pspum/102/12.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finite-type mapping class groups are linear in several low-complexity and hyperelliptic cases, but linearity for the general group in genus at least three remains open.\n\n**Verified partial progress.**\n\n- Mapping class groups of punctured spheres and hyperelliptic mapping class groups are linear.\n- Bigelow and Budney prove that the full mapping class group in genus two is linear.\n- Korkmaz proves that for genus g at least 3 any faithful complex representation has dimension greater than 3g-3.\n- Further results force strong restrictions such as quasi-unipotent images of Dehn twists, but do not prove nonlinearity.\n\n**Full solution or refutation.**\n\nNo faithful finite-dimensional complex representation or proof of nonlinearity is known in the general genus-at-least-three case.\n\n**What remains.**\n\nDetermine linearity for mapping class groups of finite-type surfaces beyond the known low-complexity and hyperelliptic families, especially closed genus at least three.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 2.3. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Records the known linear cases, propagation observations, and current constraints while retaining the general problem as open.\n- Stephen J. Bigelow and Ryan D. Budney, The mapping class group of a genus two surface is linear, Algebraic & Geometric Topology 1 (2001), 699-708. (primary): https://doi.org/10.2140/agt.2001.1.699\n Evidence used: Proves linearity for the full genus-two mapping class group.\n- Mustafa Korkmaz, Low-dimensional linear representations of mapping class groups, Journal of Topology 16 (2023), 899-935. (primary): https://doi.org/10.1112/topo.12305\n Evidence used: Gives modern lower bounds on the dimension of any faithful representation in genus at least three.\n- Tara E. Brendle and Hessam Hamidi-Tehrani, On the linearity problem for mapping class groups, Algebraic & Geometric Topology 1 (2001), 445-468. (primary): https://doi.org/10.2140/agt.2001.1.445\n Evidence used: Rules out the Formanek-Procesi poison-subgroup route to proving nonlinearity and frames the unresolved general problem.\n\n**Review notes.** The singular phrase a surface is scope-ambiguous: some finite-type surface groups are known linear, so the intended content is the general classification and the unresolved higher-genus cases.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2752, "problem_number": "KP-2.4", "title": "Kirby Problem 2.4", "statement": "Let $S_{1}$ and $S_{2}$ be orientable surfaces of finite type. Under what\nconditions do injective maps from (finite-index subgroups of) the mapping class\ngroup of $S_{1}$ to the mapping class group of $S_{2}$ necessarily arise from “manipulations”\n(e.g., inclusions, (branched) coverings, etc.) taking $S_{1}$ to $S_{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.4.\n\nLiterature notes:\n(1) It is known that all such injections arise from inclusions of surfaces pro-\nvided that the genus of $S_{2}$ is not much larger than the genus of $S_{1}$ [AS12].\nFor braid groups, a detailed study of injective maps in a certain range\nwas announced by Chen–Kordek–Margalit [CKM19]; see also Problem\n\\section*{2.7. Both of these results pertain only to injections at the level of the full}\nmapping class group; the case of arbitrary finite-index subgroups seems\nmuch harder.\n(2) There are many natural variations on this problem, replacing mapping\nclass groups by curve graphs and related graphs associated to surfaces.\nThe literature on this subject is vast, with two central results being [Iva97]\nand [BM19]. It has been announced that a (model-theoretic) interpreta-\ntion between curve graphs of surfaces induces a virtual injection between\nmapping class group [DKdlNG20].\n\nReferences cited:\n- [AS12] Javier Aramayona and Juan Souto. Homomorphisms between mapping class groups. Geom. Topol., 16(4):2285–2341, 2012. doi:10.2140/gt.2012.16.2285.\n- [CKM19] Lei Chen, Kevin Kordek, and Dan Margalit. Homomorphisms between braid groups, 2019. arXiv:1910.00712.\n- [Iva97] Nikolai V. Ivanov. Automorphism of complexes of curves and of Teichmüller spaces. Internat. Math. Res. Notices, 1997(14):651–666, 1997. doi:10.1155/$S^{1}$073792897000433.\n- [BM19] Tara E. Brendle and Dan Margalit. Normal subgroups of mapping class groups and the metaconjecture of Ivanov. J. Amer. Math. Soc., 32(4):1009–1070, 2019. doi:10.1090/jams/927.\n- [DKdlNG20] Valentina Disarlo, Thomas Koberda, and J. de la Nuez González. The model theory of the curve graph, 2020. arXiv:2008.10490.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Rigidity of injections from full finite-type mapping class groups is known in genus ranges, but virtual finite-index-subgroup injections lack a general classification.\n\n**Verified partial progress.**\n\n- Aramayona--Souto prove inclusion rigidity when target genus is not much larger.\n- Detailed braid-group injections are known in a range.\n\n**Full solution or refutation.**\n\nThe requested broad virtual-manipulation classification is open.\n\n**What remains.**\n\nClassify injective maps of arbitrary finite-index subgroups.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.4 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States known full-group rigidity and the harder virtual case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2753, "problem_number": "KP-2.5", "title": "Kirby Problem 2.5", "statement": "For $g \\geq 3$, determine a finite presentation for the Torelli\ngroup $\\mathcal{I}_{g}$, or show that no finite presentation exists.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.5.\n\nLiterature notes:\n(1) The Torelli group $\\mathcal{I}_{g} \\leq \\operatorname{Mod}(S_{g})$ is the subgroup of $\\operatorname{Mod}(S_{g})$ consisting of\nmapping classes acting trivially on $H_{1}(S_{g}; \\mathbb{Z})$. Work of Johnson [Joh83]\nshows that $\\mathcal{I}_{g}$ is finitely generated for $g \\geq 3$.\n(2) This is [Kir78, Problem 2.9]. The question was raised by Magnus, and\ndescribed by Birman in [Bir71].\n(3) For $g = 1$, the Torelli group is trivial, and for $g = 2$, Mess showed it is a\nfree group of infinite rank [Mes92].\n(4) For any finitely presented group $G$, the second homology $H_{2}(G; \\mathbb{Q})$ is finite\ndimensional, though the converse does not hold. Recent work of Minahan–\nPutman [MP25] announced that $H_{2}(\\mathcal{I}_{g}; \\mathbb{Q})$ is finite dimensional for $g \\geq 6$.\nIt is still open whether or not $H_{2}(\\mathcal{I}_{g}; \\mathbb{Z})$ is finitely generated for large $g$.\n\nReferences cited:\n- [Joh83] Dennis Johnson. The structure of the Torelli group. I. A finite set of generators for I. Ann. of Math. (2), 118(3):423–442, 1983. doi:10.2307/2006977.\n- [Kir78] Rob Kirby. Problems in low dimensional manifold theory. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 273–312. Amer. Math. Soc., Providence, R.I., 1978.\n- [Bir71] Joan S. Birman. On Siegel’s modular group. Math. Ann., 191:59–68, 1971. doi: 10.1007/BF01433472.\n- [Mes92] Geoffrey Mess. The Torelli groups for genus 2 and 3 surfaces. Topology, 31(4):775– 790, 1992. doi:10.1016/0040-9383(92)90008-6.\n- [MP25] Daniel Minahan and Andrew Putman. The second rational homology of the torelli group, 2025. arXiv:2504.00211.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Finite presentability of the Torelli group is unknown for every genus g at least 3, despite finite generation and a calculation of second rational homology in genus at least 6.\n\n**Verified partial progress.**\n\n- Johnson proves that I_g is finitely generated for g at least 3.\n- Minahan--Putman calculate H_2(I_g;Q) for g at least 6, in particular proving it finite dimensional.\n- Integral H_2 is not known to be finitely generated in high genus, and rational H_2 finiteness does not imply finite presentability.\n\n**Full solution or refutation.**\n\nNo genus in the stated range has a known finite presentation or a proof of non-finite-presentability.\n\n**What remains.**\n\nConstruct a finite presentation or prove none exists; finite generation of integral second homology is a key intermediate question.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 2.5 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Retains the finite-presentation question and distinguishes rational H_2 progress from integral and presentation questions.\n- Daniel Minahan and Andrew Putman, The second rational homology of the Torelli group, arXiv:2504.00211 (2025). (primary): https://arxiv.org/abs/2504.00211\n Evidence used: Calculates H_2(I_g;Q) for g at least 6.\n- Dennis Johnson, The structure of the Torelli group I: A finite set of generators for I, Ann. of Math. 118 (1983), 423--442. (primary): https://doi.org/10.2307/2006977\n Evidence used: Proves finite generation, not finite presentability, for g at least 3.\n\n**Review notes.** The necessary rational-homology condition is not misclassified as a presentation theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2754, "problem_number": "KP-2.6", "title": "Kirby Problem 2.6", "statement": "Give a classification or enumeration of the finite-index sub-\ngroups of $\\operatorname{Mod}(S_{g})$ that are generated by Dehn twists, Dehn multitwists, or powers\nthereof. Which of these subgroups are “geometrically meaningful”, in the sense that\nthey are given as the stabilizers of some kind of geometric or topological struc-\nture on the surface? Conversely, when is a geometrically meaningful subgroup (not\nnecessarily of finite index) generated by powers of Dehn multitwists?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.6.\n\nLiterature notes:\n(1) The classical Dehn–Lickorish theorem asserts that the mapping class group\nof a closed orientable surface is generated by a finite collection of Dehn\ntwists.\nAny finite-index subgroup contains some power of every Dehn\ntwist, but a general finite-index subgroup need not contain any single twist\n$T_{c}$ at all, nor is a general finite-index subgroup generated by the powers\nof Dehn twists that it does contain. Indeed, work of Funar [Fun99] and\nMasbaum [Mas99] shows that for general values of $r$, the subgroup of\n$\\operatorname{Mod}(\\Sigma_{g})$ generated by $r^{th}$ powers of Dehn twists is of infinite index.\n(2) Salter and Calderon–Salter [Sal19, CS21] show that one such example\nis the class of $r$-spin mapping class groups, i.e., the stabilizers of a chosen\n$r$-spin structure on $S_{g}$, for $g \\geq 5$. In [CS23], they proved that the framed\nmapping class group for a framing (i.e., the stabilizer of the isotopy class\nof that framing) is likewise generated by Dehn twists; these subgroups are\nof infinite index. Salter–Sane [SS23] announced a generating set for the\nstabilizer of a mod-2 homology class $[x]$ consisting of Dehn twists together\nwith a single square-twist $T_{c}^{2}$.\n(3) The terms “geometrically meaningful” and “geometric or topological struc-\nture” in the problem statement are intentionally vague, but the examples\ndiscussed in the remarks above are meant to be representative. To dis-\nambiguate, we do not (necessarily) mean “geometric structure” in the\nThurstonian sense of a $(G, X)$-structure.\n\n(4) One motivation for this question comes from the theory of (achiral) Lef-\nschetz pencils, which are encoded in the mapping class group as sequences\nof Dehn twists whose product is the boundary multi-twist. When these\nDehn twists lie in some “geometrically meaningful” subgroup, one can\nask if this endows the ambient 4–manifold with an additional structure.\nOne instance of this is the case of $r = 2$ (i.e., classical) spin mapping class\ngroups, where the question of whether a symplectic 4-manifold is also spin\nis related to whether it admits a symplectic Lefschetz pencil whose twists\nlie in a spin mapping class group [Sti01, BHM23, BH24c, AB23].\n\nReferences cited:\n- [Fun99] Louis Funar. On the TQFT representations of the mapping class groups. Pacific J. Math., 188(2):251–274, 1999. doi:10.2140/pjm.1999.188.251.\n- [Mas99] Gregor Masbaum. An element of infinite order in TQFT-representations of mapping class groups. In Low-dimensional topology (Funchal, 1998), volume 233 of Contemp. Math., pages 137–139. Amer. Math. Soc., Providence, RI, 1999. doi: 10.1090/conm/233/03423.\n- [Sal19] Nick Salter. Monodromy and vanishing cycles in toric surfaces. Invent. Math., 216(1):153–213, 2019. doi:10.1007/s00222-018-0845-6.\n- [CS21] Aaron Calderon and Nick Salter. Higher spin mapping class groups and strata of abelian differentials over Teichmüller space. Adv. Math., 389:Paper No. 107926, 56, 2021. doi:10.1016/j.aim.2021.107926.\n- [CS23] Aaron Calderon and Nick Salter. Framed mapping class groups and the monodromy of strata of abelian differentials. J. Eur. Math. Soc. (JEMS), 25(12):4719–4790, 2023. doi:10.4171/jems/1290.\n- [SS23] Nick Salter and Abdoul Karim Sane. Connected components of the topological surgery graph of a unicellular collection, 2023. arXiv:2308.09165.\n- [Sti01] András I. Stipsicz. Spin structures on Lefschetz fibrations. Bull. London Math. Soc., 33(4):466–472, 2001. doi:10.1017/S0024609301008232.\n- [BHM23] R. İnanç Baykur, Kenta Hayano, and Naoyuki Monden. Unchaining surgery and topology of symplectic 4-manifolds. Math. Z., 303(3):Paper No. 77, 32, 2023. doi: 10.1007/s00209-023-03204-x.\n- [BH24c] R. İnanç Baykur and Noriyuki Hamada. Lefschetz fibrations with arbitrary signature. J. Eur. Math. Soc. (JEMS), 26(8):2837–2895, 2024. doi:10.4171/jems/1326.\n- [AB23] Mihail Arabadji and R. İnanç Baykur. Spin Lefschetz fibrations are abundant. Pacific J. Math., 326(1):1–16, 2023. doi:10.2140/pjm.2023.326.1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Several geometrically defined stabilizer subgroups are known to be generated by Dehn twists, and uniform-power constructions give obstructions, but no general enumeration exists.\n\n**Verified partial progress.**\n\n- Higher-spin mapping class groups in the stated stable-genus range are generated by Dehn twists.\n- Framed mapping class groups are likewise generated by Dehn twists, although these examples are generally infinite index.\n- Funar--Masbaum show that subgroups generated by uniform rth powers of Dehn twists have infinite index for general r.\n\n**Full solution or refutation.**\n\nKnown families and negative index results give a useful map of the terrain, not a classification of finite-index or geometric subgroups.\n\n**What remains.**\n\nFormalize the class of geometrically meaningful structures, then classify which stabilizers and finite-index subgroups are generated by powers of Dehn multitwists.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 2.6 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the intentionally broad classification and lists spin, framed, and homology-class stabilizer families.\n- Aaron Calderon and Nick Salter, Higher spin mapping class groups and strata of abelian differentials over Teichmuller space, Adv. Math. 389 (2021), 107926. (primary): https://doi.org/10.1016/j.aim.2021.107926\n Evidence used: Gives twist-generation results for higher-spin stabilizers.\n- Aaron Calderon and Nick Salter, Framed mapping class groups and the monodromy of strata of abelian differentials, JEMS 25 (2023), 4719--4790. (primary): https://doi.org/10.4171/jems/1290\n Evidence used: Proves Dehn-twist generation for framed mapping class groups.\n\n**Review notes.** FORMULATION NOTE: K3 explicitly says 'geometrically meaningful' is intentionally vague. A full classification requires fixing that universe; the exact statement is preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2755, "problem_number": "KP-2.7", "title": "Kirby Problem 2.7", "statement": "Classify the homomorphisms from the braid group $B_{n}$ on $n$\nstrands to the braid group $B_{m}$ on $m$ strands, where $n, m \\in \\mathbb{N}$ are arbitrary.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.7.\n\nLiterature notes:\n(1) There are several works in this direction including those of Artin [Art47],\nLin [Lin04], Dyer–Grossman [DG81], Bell–Margalit [BM06], Castel [Cas16]\nand, finally, Chen–Kordek–Margalit [CKM19], where the most general\nresult known to date was announced (the complete classification for $m \\leq$\n$2n$ and $n \\geq 5$).\n(2) There is a more precise formulation of this question. We view the group\n$B_{m}$ as the mapping class group $\\operatorname{Mod}(S_{0,m}^{1})$ of the $m$-punctured disk and\ndenote by $\\mathcal{C}(S_{0,m}^{1})$ the curve complex of $S_{0,m}^{1}$. We say that a homomor-\nphism $\\phi: B_{n} \\to B_{m} = \\operatorname{Mod}(S_{0,m}^{1})$ is reducible if there exists a nonempty\nsimplex $\\mathcal{A}$ of $\\mathcal{C}(S_{0,m}^{1})$ such that $\\phi(\\beta)(\\mathcal{A}) = \\mathcal{A}$ for all $\\beta \\in B_{n}$. That is,\nevery element of $\\phi(B_{n})$ is reducible, and moreover preserves the same set\nof curves. Then we have the following conjecture, which can be compared\nwith the Nielsen–Thurston classification.\nConjecture. Let $n, m \\in \\mathbb{N}$ with $5 \\leq n \\leq m$, and let $\\phi: B_{n} \\to B_{m}$ be\na homomorphism. Then we have one of the following three possibilities.\n(a) The image of $\\phi$ is a cyclic group.\n(b) $n = m$, and $\\phi$ is an injective homomorphism as described in [BM06].\n(c) $\\phi$ is reducible.\n\n(3) In light of the identification of $B_{m}$ with the mapping class group $\\operatorname{Mod}(S_{0,m}^{1})$,\nthis question admits a natural generalization to the problem of classifying\nall homomorphisms between mapping class groups of surfaces; see Prob-\nlem 2.4. For further discussion, see the survey article [AS16].\n\nReferences cited:\n- [Art47] E. Artin. Braids and permutations. Ann. of Math. (2), 48:643–649, 1947. doi: 10.2307/1969131.\n- [Lin04] Vladimir Lin. Braids and permutations, 2004. arXiv:math/0404528.\n- [DG81] Joan L. Dyer and Edna K. Grossman. The automorphism groups of the braid groups. Amer. J. Math., 103(6):1151–1169, 1981. doi:10.2307/2374228.\n- [BM06] Robert W. Bell and Dan Margalit. Braid groups and the co-Hopfian property. J. Algebra, 303(1):275–294, 2006. doi:10.1016/j.jalgebra.2005.10.038.\n- [Cas16] Fabrice Castel. Geometric representations of the braid groups. Astérisque, 378:vi+175, 2016.\n- [CKM19] Lei Chen, Kevin Kordek, and Dan Margalit. Homomorphisms between braid groups, 2019. arXiv:1910.00712.\n- [AS16] Javier Aramayona and Juan Souto. Rigidity phenomena in the mapping class group. In Handbook of Teichmüller theory. Vol. VI, volume 27 of IRMA Lect. Math. Theor. Phys., pages 131–165. Eur. Math. Soc., Zürich, 2016.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Homomorphisms B_n to B_m are classified for n at least 5 and m at most 2n, together with several exceptional endomorphism cases, but arbitrary ranks remain open.\n\n**Verified partial progress.**\n\n- Chen--Kordek--Margalit classify every homomorphism B_n to B_2n for n at least 5 and derive the classification for m below 2n.\n- They also classify endomorphisms of B_4.\n- Earlier work classifies automorphisms, injective maps in smaller ranges, and related co-Hopfian cases.\n\n**Full solution or refutation.**\n\nThe theorem reaches the broad range m at most 2n, not arbitrary m or all low-rank exceptions.\n\n**What remains.**\n\nClassify homomorphisms for m greater than 2n and complete the exceptional small-n cases.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 2.7 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Identifies the m at most 2n, n at least 5 result as the broadest classification and retains the arbitrary-rank question.\n- Lei Chen, Kevin Kordek, and Dan Margalit, Homomorphisms between braid groups, arXiv:1910.00712v2 (2023 revision). (primary): https://arxiv.org/abs/1910.00712\n Evidence used: Classifies B_n to B_2n for n at least 5, with the lower target range and B_4 endomorphisms as consequences/additional results.\n- Robert W. Bell and Dan Margalit, Braid groups and the co-Hopfian property, J. Algebra 303 (2006), 275--294. (primary): https://doi.org/10.1016/j.jalgebra.2005.10.038\n Evidence used: Supplies earlier injective/endomorphism rigidity used in the partial classification.\n\n**Review notes.** The preprint remains the primary source; later papers explicitly cite its m at most 2n classification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2756, "problem_number": "KP-2.8", "title": "Kirby Problem 2.8", "statement": "Fix distinct trivial tangles $\\tau_{1}, \\tau_{2}$ for which $\\tau_{1} \\cup\\tau_{2}$ is the unknot.\nDescribe the intersection of the associated wicket groups $W_{n}(\\tau_{1}) \\cap W_{n}(\\tau_{2}) \\leq B_{2n}$.\nIs this group finitely generated? Finitely presented?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.8.\n\nLiterature notes:\n(1) A tangle on $n$ strands is a proper embedding of $n$ disjoint arcs into the\nupper half-space in $\\mathbb{R}^{3}$, considered up to isotopy rel boundary. Tangles\n$\\tau_{1}, \\tau_{2}$ based at the same set of $2n$ points in $\\mathbb{R}^{2} \\subset \\mathbb{R}^{3}$ can be combined\ninto the link $\\tau_{1} \\cup \\tau_{2}$, where $\\overline{\\tau}_{2}$ denotes the reflection of $\\tau_{2}$ through the\n$xy$-plane. A trivial tangle is any tangle that can be isotoped so that each\nstrand has a single local maximum with respect to the standard height\nfunction on $\\mathbb{R}^{3}$. There is a natural transitive action of the braid group\n$B_{2n}$ on the set of trivial tangles based at a fixed set of $2n$ points, and the\nstabilizer of a fixed trivial tangle $\\tau$ is known as a wicket group, written\n$W_{n}$ or $W_{n}(\\tau)$ when we wish to emphasize the particular trivial tangle.\n(2) The questions in the problem originate from the works of S. Hirose,\nD. Iguchi, E. Kin, and Y. Koda that are discussed below.\n(3) Note that $W_{n}(\\tau_{1})$ and $W_{n}(\\tau_{2})$ are conjugate in $B_{2n}$.\nBrendle–Hatcher\n[BH13] determined a finite presentation for $W_{n}$. This problem has a nat-\nural generalization in which one allows $\\tau_{1} \\cup \\overline{\\tau}_{2}$ to be an unlink of $c$ com-\nponents for $c \\geq 1$ [HIKK22, Question 2.10]. The group $W_{n}(\\tau_{1})\\cap W_{n}(\\tau_{2})$\ncould be called the Goeritz group of the bridge splitting $\\tau_{1} \\cup \\overline{\\tau}_{2}$ of the un-\nlink. Hirose–Iguchi–Kin–Koda defined and studied this group as the hy-\nperelliptic Goeritz group, proving that this group is finitely presented when\nthe Heegaard splitting is a 3-bridge splitting of a 2-bridge link [HIKK22].\nIf the distance of an $n$-bridge decomposition of a link in $S^{3}$ with $n \\geq 3$\nis at least 5, then its Goeritz group is a finite group [IK20, Theorem 0.1].\n(4) There is a connection with the study of knotted surfaces through the the-\nory of bridge trisections [MZ17a]: Any smooth surface-link in $\\mathbb{R}^{4}$ can be\nrepresented by a tri-plane diagram in which the first two tangle diagrams\nare arranged to have no crossings. Given this, the Goeritz group of the\nbridge splitting of the unlink given by the first two tangles is precisely the\nset of braids that act on the tri-plane diagram without changing the first\ntwo tangles. Thus, an understanding of this group could be applied to the\nproblem of finding equivalences between bridge trisected surface-links.\n\nReferences cited:\n- [BH13] Tara E. Brendle and Allen Hatcher. Configuration spaces of rings and wickets. Comment. Math. Helv., 88(1):131–162, 2013. doi:10.4171/CMH/280.\n- [HIKK22] Susumu Hirose, Daiki Iguchi, Eiko Kin, and Yuya Koda. Goeritz groups of bridge decompositions. Int. Math. Res. Not. IMRN, 2022(12):9308–9356, 2022. doi:10.1093/imrn/rnab001.\n- [IK20] Daiki Iguchi and Yuya Koda. Twisted book decompositions and the Goeritz groups. Topology Appl., 272:107064, 15, 2020. doi:10.1016/j.topol.2020.107064.\n- [MZ17a] Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in $S^{4}$. Trans. Amer. Math. Soc., 369(10):7343–7386, 2017. doi:10.1090/tran/6934.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Wicket groups and several associated Goeritz-group intersections are finitely presented or finite in special cases, but arbitrary intersections arising from unknot bridge splittings remain unclassified.\n\n**Verified partial progress.**\n\n- Brendle--Hatcher give a finite presentation for each individual wicket group.\n- Hirose--Iguchi--Kin--Koda prove finite presentability for the hyperelliptic Goeritz group of a 3-bridge splitting of a 2-bridge link.\n- Iguchi--Koda prove that a Goeritz group is finite when an n-bridge decomposition with n at least 3 has distance at least 5.\n\n**Full solution or refutation.**\n\nThe cited theorems resolve low-complexity and high-distance cases but not all unknot bridge splittings or an explicit general intersection description.\n\n**What remains.**\n\nDetermine finite generation, finite presentation, and explicit structure for every relevant wicket-group intersection in the unknot case.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 2.8 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the wicket-group intersection problem, its Goeritz interpretation, and the known finite-presentation/high-distance cases; PDF text extraction also exhibits the missing-overbar ambiguity.\n- Susumu Hirose, Daiki Iguchi, Eiko Kin, and Yuya Koda, Goeritz groups of bridge decompositions, Int. Math. Res. Not. 2022, 9308--9356. (primary): https://doi.org/10.1093/imrn/rnab001\n Evidence used: Defines the hyperelliptic Goeritz groups and proves finite presentability in the stated 3-bridge/2-bridge setting.\n- Tara E. Brendle and Allen Hatcher, Configuration spaces of rings and wickets, Comment. Math. Helv. 88 (2013), 131--162. (primary): https://doi.org/10.4171/CMH/280\n Evidence used: Gives a finite presentation for the individual wicket group.\n\n**Review notes.** FORMULATION DEFECT: the exact supplied statement writes tau_1 union tau_2, but its background defines the bridge link using the reflection of tau_2 and later writes tau_1 union bar(tau_2). Since both tangles initially lie in the upper half-space, the unbarred union is not the intended link. The exact statement was preserved rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2757, "problem_number": "KP-2.9", "title": "Kirby Problem 2.9", "statement": "Is there a nice presentation of the $n$-stranded braid group whose\ngenerating set is the set of all positive elementary braid half-twists?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.9.\n\nLiterature notes:\n(1) The adjective “nice” here should at least mean that all relations consist\nof equating two words in the positive generators of the same length, and\nperhaps also that there be a finite number of local pictures describing the\nrequired relations.\n(2) The positive monoid in the standard Artin presentation consists of all\npositive braids. In the Birman–Ko–Lee presentation [BKL98], the pos-\nitive monoid is the strongly quasipositive braids. The positive monoid\nfor the desired presentation would correspond to the quasipositive braids\nand would hopefully provide interesting algorithmic tools similar to Gar-\nside [Gar69] for solving questions regarding quasipositivity. Similar infi-\nnite presentations hold for the mapping class groups of surfaces (Gervais\n[Ger96]) as well as the Torelli group (Putman [Put07]).\n\nReferences cited:\n- [BKL98] Joan Birman, Ki Hyoung Ko, and Sang Jin Lee. A new approach to the word and conjugacy problems in the braid groups. Adv. Math., 139(2):322–353, 1998. doi:10.1006/aima.1998.1761.\n- [Gar69] F. A. Garside. The braid group and other groups. Quart. J. Math. Oxford Ser. (2), 20:235–254, 1969. doi:10.1093/qmath/20.1.235.\n- [Ger96] Sylvain Gervais. Presentation and central extensions of mapping class groups. Trans. Amer. Math. Soc., 348(8):3097–3132, 1996. doi:10.1090/S0002-9947-96-01509-7.\n- [Put07] Thomas Andrew Putman. An infinite presentation of the Torelli group. ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–The University of Chicago. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt=info: ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri:pqdiss:3262286.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No homogeneous positive presentation using every positive elementary half-twist, with positive monoid equal to the quasipositive braids, was found.\n\n**Verified partial progress.**\n\n- The Artin presentation has homogeneous positive relations but its positive monoid is only the positive braid monoid.\n- The Birman--Ko--Lee presentation has homogeneous positive relations and a Garside structure, but its positive monoid is the strongly quasipositive braid monoid.\n- Orevkov gives an algorithm deciding quasipositivity, which does not provide the requested all-half-twist presentation.\n\n**Full solution or refutation.**\n\nExisting positive presentations and decision algorithms do not realize the precise generator/monoid target.\n\n**What remains.**\n\nFormalize 'nice' and construct a complete length-preserving positive presentation, preferably with finitely many local relation schemes, or prove it cannot exist.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 2.9 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Retains the question and explains the difference among the Artin, Birman--Ko--Lee, and desired quasipositive monoids.\n- Joan Birman, Ki Hyoung Ko, and Sang Jin Lee, A new approach to the word and conjugacy problems in the braid groups, Adv. Math. 139 (1998), 322--353. (primary): https://doi.org/10.1006/aima.1998.1761\n Evidence used: Provides a homogeneous positive presentation whose monoid is strongly quasipositive rather than all quasipositive braids.\n- S. Yu. Orevkov, Solution of the quasipositivity problem in the braid group, arXiv:math/0402023. (primary): https://arxiv.org/abs/math/0402023\n Evidence used: Gives a decision algorithm for quasipositivity, not the requested presentation.\n\n**Review notes.** FORMULATION NOTE: 'nice' is not formal. The remarks require length-preserving positive relations but only suggest finitely many local pictures, so a terminal solution criterion needs expert formalization.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2758, "problem_number": "KP-2.10", "title": "Kirby Problem 2.10", "statement": "(a) Is there an efficient algorithm to compute distances in the curve complex\nof a surface? The input to the algorithm should be the surface and the\ncurves. One may ask similar questions for related complexes, such as the\narc complex of a surface with boundary.\n(b) Is there an algorithm (efficient or otherwise) to compute the distance be-\ntween quasi-convex subsets of the curve complex? This question is partic-\nularly interesting in the case where the quasi-convex subsets are the disk\nsets of handlebodies.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.10.\n\nLiterature notes:\n(1) The curve complex has proved to be particularly important in Kleinian\ngroup theory and in the study of the mapping class groups of surfaces\n[Min06].\nOne notable feature of $\\mathcal{C}(S)$, proved by Masur and Minsky\n[MM99], is that it is Gromov hyperbolic.\n(2) The complex is not locally finite, and so there is no naive way of comput-\ning distances. However, distances in $\\mathcal{C}(S)$ are known to be computable,\nby work of Leasure [Lea02], Shackleton [Sha12], Webb [Web15], Bir-\nman, Margalit, and Menasco [BMM16] and Watanabe [Wat16]. Bell\nand Webb [BW16a] announced an algorithm with running time that is\nbounded above by a polynomial function of the logarithm of the weight of\n\nthe two curves, given as normal curves in some triangulation of $S$. How-\never, the running time of their algorithm grows very rapidly as the surface\n$S$ is allowed to vary. The aim of Problem (a) is to find such an algorithm\nthat runs in polynomial time even as the surface varies.\nBaroni [Bar24] has announced an efficient algorithm to compute dis-\ntances up to a bounded multiplicative error, the bound depending only\non $S$. He used this to find an efficient algorithm to determine whether a\ngiven surface automorphism is pseudo-Anosov, reducible, or periodic.\n(3) A related problem is as follows. Given a reducible element of the mapping\nclass group, can an invariant multi-curve be found efficiently (in polyno-\nmial time as function of the ‘size’ of the mapping class and also in the\nEuler characteristic of $S$)?\n(4) When $S$ is the boundary of a handlebody $H$, the associated disk set con-\nsists of the subcomplex of $\\mathcal{C}(S)$ induced by the curves bounding disks\nin $H$. This was proved to be quasi-convex in $\\mathcal{C}(S)$ by Masur and Min-\nsky [MM04]. When $S$ is a Heegaard surface for a closed 3-manifold, it\nbounds a handlebody on each side and so there are two associated disk\nsets. The distance between them in $\\mathcal{C}(S)$ is the Hempel distance of the\nsplitting [Hem01]. This is not known to be computable. However, it\nwas proved by Masur and Schleimer [MS13] that it is computable up to\na bounded additive error. The Hempel distance of a Heegaard splitting\nencodes important information. For example, the Hempel distance is zero\nif and only if the splitting is reducible, and the Hempel distance is one if\nand only if the splitting is weakly reducible but irreducible.\n\nReferences cited:\n- [Min06] Yair N. Minsky. Curve complexes, surfaces and 3-manifolds. In International Congress of Mathematicians. Vol. II, pages 1001–1033. Eur. Math. Soc., Zürich, 2006.\n- [MM99] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. I. Hyperbolicity. Invent. Math., 138(1):103–149, 1999. doi:10.1007/s002220050343.\n- [Lea02] Jason Paige Leasure. Geodesics in the complex of curves of a surface. ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–The University of Texas at Austin. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt= info:ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri:pqdiss:3114766.\n- [Sha12] Kenneth J. Shackleton. Tightness and computing distances in the curve complex. Geom. Dedicata, 160:243–259, 2012. doi:10.1007/s10711-011-9680-2.\n- [Web15] Richard C. H. Webb. Combinatorics of tight geodesics and stable lengths. Trans. Amer. Math. Soc., 367(10):7323–7342, 2015. doi:10.1090/tran/6301.\n- [BMM16] Joan Birman, Dan Margalit, and William Menasco. Efficient geodesics and an effective algorithm for distance in the complex of curves. Math. Ann., 366(3-4):1253– 1279, 2016. doi:10.1007/s00208-015-1357-y.\n- [Wat16] Yohsuke Watanabe. Intersection numbers in the curve graph with a uniform constant. Topology Appl., 204:157–167, 2016. doi:10.1016/j.topol.2016.03.009.\n- [BW16a] Mark C. Bell and Richard C. H. Webb. Polynomial-time algorithms for the curve graph, 2016. arXiv:1609.09392.\n- [Bar24] Filippo Baroni. Uniformly polynomial-time classification of surface homeomorphisms, 2024. arXiv:2402.00231.\n- [MM04] Howard A. Masur and Yair N. Minsky. Quasiconvexity in the curve complex. In In the tradition of Ahlfors and Bers, III, volume 355 of Contemp. Math., pages 309–320. Amer. Math. Soc., Providence, RI, 2004. doi:10.1090/conm/355/06460.\n- [Hem01] John Hempel. 3-manifolds as viewed from the curve complex. Topology, 40(3):631– 657, 2001. doi:10.1016/S0040-9383(00)00033-1.\n- [MS13] Howard Masur and Saul Schleimer. The geometry of the disk complex. J. Amer. Math. Soc., 26(1):1–62, 2013. doi:10.1090/S0894-0347-2012-00742-5.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Curve-complex distances are computable, with announced fast fixed-surface/approximate algorithms, but a polynomial-time algorithm uniform as the surface varies is open.\n\n**Verified partial progress.**\n\n- Leasure, Shackleton, Webb, Birman--Margalit--Menasco and Watanabe establish computability.\n- Bell--Webb announce logarithmic-weight polynomial time for fixed surface.\n- Baroni announces bounded-factor efficient approximation.\n\n**Full solution or refutation.**\n\nThe uniform complexity and quasiconvex-subset distance questions remain open.\n\n**What remains.**\n\nObtain surface-uniform polynomial complexity and solve the subset-distance problem.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.10 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Details known algorithms and the unresolved uniform objective.\n\n**Review notes.** Announced results are not promoted to full resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2759, "problem_number": "KP-2.11", "title": "Kirby Problem 2.11", "statement": "Find precise estimates for both the extremal and average behav-\nior of the simple lifting degree of curves, in terms of combinatorial (e.g., intersection\nnumber) and/or geometric (e.g., geodesic length, word length) data.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.11.\n\nLiterature notes:\n(1) A result of Peter Scott [Sco78] tells us that for any closed geodesic $\\gamma$ on\na hyperbolic surface $S$, there exists a finite degree cover $S' \\to S$ such that\n$\\gamma$ lifts to $S'$ (i.e., choosing basepoints, $\\gamma \\in \\pi_{1}(S') \\leq \\pi_{1}(S)$) and is a simple\nclosed curve in $S'$. The simple lifting degree of $\\gamma$, which we will write as\n$\\deg(\\gamma)$, is defined to be the minimal degree of such a covering $S'$ to which\n$\\gamma$ lifts simply.\n(2) To precisely formulate the relationship between simple lifting degree and\ncombinatorial/geometric complexity, define the functions\n\n$$\nf_{S}(n)=\\max\\{\\deg(\\gamma): i(\\gamma,\\gamma)\\leq n\\}.\n$$\n\nand, for a choice of hyperbolic metric $\\rho$ on $S$,\n\n$$\nf_{\\rho}(L)=\\max\\{\\deg(\\gamma): \\ell_{\\rho}(\\gamma)\\leq L\\}.\n$$\n\nThe extremal behavior (i.e., upper bounds) of these functions has been\nstudied in work of Patel [Pat14], Gaster [Gas16], and Aougab–Gaster–\nPatel–Sapir [AGPS17]. A lower bound for average behavior of $\\deg(\\gamma)$ was\n\nobtained by Aougab–Gaster [AG22], but work of Sisto–Taylor [ST19]\nimplies that some mechanism other than what Gaster finds in the extremal\ncase is responsible for the average behavior. What is this mechanism?\nProgress in the study of $f_{S}(n)$ has been obtained by Arenas and\nNeumann-Coto [ANC20]. They establish the inequality $f_{S}(n) \\leq 5(n+1)$;\nnote that this is independent of the surface $S$. They conjecture that the\ntighter bound $f_{S}(n) \\leq 2(n + 1)$ should hold. At the time of this writing,\nno examples exist with $f_{S}(n) - n$ more than 2.\n(3) Variations of these questions can be posed by, e.g., restricting to the class\nof regular covers or to families of curves intersecting on a surface $S$. These\nhave been largely unexplored.\n\nReferences cited:\n- [Sco78] Peter Scott. Subgroups of surface groups are almost geometric. J. London Math. Soc. (2), 17(3):555–565, 1978. doi:10.1112/jlms/s2-17.3.555.\n- [Pat14] Priyam Patel. On a theorem of Peter Scott. Proc. Amer. Math. Soc., 142(8):2891– 2906, 2014. doi:10.1090/S0002-9939-2014-12031-4.\n- [Gas16] Jonah Gaster. Lifting curves simply. Int. Math. Res. Not. IMRN, 2016(18):5559– 5568, 2016. doi:10.1093/imrn/rnv316.\n- [AGPS17] Tarik Aougab, Jonah Gaster, Priyam Patel, and Jenya Sapir. Building hyperbolic metrics suited to closed curves and applications to lifting simply. Math. Res. Lett., 24(3):593–617, 2017. doi:10.4310/MRL.2017.v24.n3.a1.\n- [AG22] Tarik Aougab and Jonah Gaster. Combinatorially random curves on surfaces, 2022. arXiv:2209.11309.\n- [ST19] Alessandro Sisto and Samuel J. Taylor. Largest projections for random walks and shortest curves in random mapping tori. Math. Res. Lett., 26(1):293–321, 2019. doi:10.4310/MRL.2019.v26.n1.a14.\n- [ANC20] Macarena Arenas and Max Neumann-Coto. Measuring complexity of curves on surfaces. Geom. Dedicata, 204:25–41, 2020. doi:10.1007/s10711-019-00443-3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finite simple lifting degree is guaranteed and extremal/average bounds are known, but precise asymptotics in intersection, length or word data remain open.\n\n**Verified partial progress.**\n\n- Scott establishes simple lifts in finite covers.\n- Patel, Gaster and Aougab--Gaster--Patel--Sapir give extremal estimates.\n- Aougab--Gaster give an average lower bound.\n\n**Full solution or refutation.**\n\nNo sharp general estimate was verified.\n\n**What remains.**\n\nDetermine sharp extremal and average growth functions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.11 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States existence and the current bound literature.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2760, "problem_number": "KP-2.12", "title": "Kirby Problem 2.12", "statement": "(a) What is the maximum number of systoles on a closed, hyperbolic surface\nof genus $g$?\n(b) What is the maximum cardinality of a set of pairwise non-isotopic, essen-\ntial, simple closed curves on a closed surface of genus $g$ with the property\nthat no two curves in the set intersect in more than one point?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.12.\n\nLiterature notes:\n(1) A systole on a closed hyperbolic surface is a closed geodesic of shortest\nlength. A systole is necessarily an essential, simple closed curve. On a\nclosed surface, any two systoles intersect in at most one point (in the\npresence of punctures, systoles may intersect twice; see [FP15]). Hence\nthe first problem arises as a special case of the second. This quantity is\nsometimes called the “kissing number”; kissing numbers of lattices have\nbeen studied in a variety of contexts.\n(2) The first problem was first seriously studied by Schmutz Schaller [Sch96].\nThe answer is known to lie between $C_{1}(\\epsilon)\\cdot g^{4/3-\\epsilon}$ and $C_{2} \\cdot g^{2}/ \\log(g)$. Here\n$C_{1}(\\epsilon)$ is a function of an arbitrary value $\\epsilon > 0$ and $C_{2}$ is an absolute\nconstant. The lower bound is a construction of Schmutz Schaller using\narithmetic surfaces [SS97]. The upper bound is due to Parlier using com-\nbinatorial and hyperbolic techniques [Par13b]. Schmutz Schaller conjec-\ntured that the lower bound is optimal, in that the maximum cardinality\ncannot exceed $C_{3} \\cdot g^{4/3}$ for some absolute constant $C_{3}$ [SS97, Conjecture].\n(3) The second problem was raised independently by Juvan, Malnic, and Mo-\nhar [JMM96] and by Farb and Leininger, as recorded in [MRT14, Ques-\ntion 1].\nThe answer is known to lie between $C_{1} \\cdot g^{2}$ and $C_{2} \\cdot g^{2} \\log g$.\nThe lower bound is a construction due to Malestein, Rivin, and Theran\n[MRT14]. The upper bound is due to Greene, using probabilistic com-\nbinatorics and geometric techniques [Gre19]. The lower bound is conjec-\ntured to be optimal, i.e., there exists an upper bound of the form $C_{3} \\cdot g^{2}$\nto match it.\n\nPrzytycki solved the corresponding problem for arcs: if $S$ is a con-\nnected, orientable, punctured surface of finite type, then the maximum\nnumber of pairwise non-isotopic, essential, simple arcs that limit to the\npunctures is $(|\\chi(S)| + 1)(|\\chi(S)| + 2)$ [Prz15]. His methods involve hyper-\nbolic geometry.\nReplacing “one” by $k$ in the problem, the answer is known to lie\nbetween $C_{1}(k) \\cdot g^{k+1}$ and $C_{2}(k) \\cdot g^{k+1} \\log g$, for some functions $C_{1}(k)$ and\n$C_{2}(k)$ independent of $g$. Once more, the lower bound is conjectured to\nbe optimal, i.e., there exists an upper bound of the form $C_{3}(k) \\cdot g^{k+1}$ to\nmatch it [Gre19, Conjecture 1].\n\nReferences cited:\n- [FP15] Federica Fanoni and Hugo Parlier. Systoles and kissing numbers of finite area hyperbolic surfaces. Algebr. Geom. Topol., 15(6):3409–3433, 2015. doi:10.2140/agt.2015.15.3409.\n- [Sch96] Paul Schmutz. Compact Riemann surfaces with many systoles. Duke Math. J., 84(1):191–198, 1996. doi:10.1215/S0012-7094-96-08406-9.\n- [SS97] Paul Schmutz Schaller. Extremal Riemann surfaces with a large number of systoles. In Extremal Riemann surfaces (San Francisco, CA, 1995), volume 201 of Contemp. Math., pages 9–19. Amer. Math. Soc., Providence, RI, 1997. doi:10.1090/conm/201/02617.\n- [Par13b] Hugo Parlier. Kissing numbers for surfaces. J. Topol., 6(3):777–791, 2013. doi: 10.1112/jtopol/jtt012.\n- [JMM96] M. Juvan, A. Malnič, and B. Mohar. Systems of curves on surfaces. J. Combin. Theory Ser. B, 68(1):7–22, 1996.\n- [MRT14] Justin Malestein, Igor Rivin, and Louis Theran. Topological designs. Geom. Dedicata, 168:221–233, 2014. doi:10.1007/s10711-012-9827-9.\n- [Gre19] Joshua Evan Greene. On loops intersecting at most once. Geom. Funct. Anal., 29(6):1828–1843, 2019. doi:10.1007/s00039-019-00517-0.\n- [Prz15] Piotr Przytycki. Arcs intersecting at most once. Geom. Funct. Anal., 25(2):658–670, 2015. doi:10.1007/s00039-015-0320-0.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Maximum systole count has bounds between roughly g^(4/3-epsilon) and g^2/log g; the sharp order and related curve-system maximum are open.\n\n**Verified partial progress.**\n\n- Schmutz Schaller gives arithmetic lower constructions.\n- Parlier gives the stated upper bound.\n\n**Full solution or refutation.**\n\nThe gap remains substantial.\n\n**What remains.**\n\nDetermine the correct asymptotic, including the at-most-one-intersection curve-system problem.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.12 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the systole bounds and conjectured lower-order optimality.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2761, "problem_number": "KP-2.13", "title": "Kirby Problem 2.13", "statement": "Let $S$ be a surface, and let $\\Gamma_{1}, \\Gamma_{2}$ be isotopy classes of embedded\ngraphs in $S$. Determine when $\\Gamma_{1}$ and $\\Gamma_{2}$ are related by a sequence of surgeries.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.13.\n\nLiterature notes:\n(1) A surgery of an embedded graph $\\Gamma \\subset S$ is defined as follows. Suppose\nsome component of $S\\setminus\\Gamma$ is a disk $D$. Let $x$ and $y$ be oriented edges in $\\Gamma$\nbounding $D$. There is a unique isotopy class $\\lambda_{x,y}$ of arc connecting $x$ to\n$y$ inside $D$. A regular neighborhood $N_{x,y}$ of $\\lambda_{x,y}$ is a rectangle with two\nopposite sides comprising subarcs of $x$ and $y$, and remaining sides running\nparallel to $\\lambda_{x,y}$. The surgery of $\\Gamma$ along $\\lambda_{x,y}$ is the graph $\\Gamma'$ obtained from\n$\\Gamma$ by deleting the edges of $N_{x,y}$ lying on $x, y$ and replacing them with the\ntwo edges parallel to $\\lambda_{x,y}$. Note that surgery can change the topology of\nthe complement; there is a simple combinatorial criterion to determine\nwhen this does or does not happen [San21].\n(2) The number of vertices and edges are manifestly preserved under surgery,\nbut the isomorphism type of the graph can change. Sane has investigated\nthe “surgery graph” that tracks when two embedded graphs are related\nby a surgery and has studied connectivity properties of this graph in a\nnumber of cases, discussed below.\nWhen every vertex of $\\Gamma$ has even valence, $\\Gamma$ determines a class in\n$H_{1}(S; \\mathbb{Z}/2\\mathbb{Z})$. This homology class is readily seen to be invariant under\nsurgery. The case where every vertex has valence 4 and the complement\nis a single disk was studied by Salter–Sane [SS23], who announced that\nin this case, graphs $\\Gamma$ and $\\Gamma'$ are related via a sequence of surgeries if and\nonly if their mod-2 homology classes are equal. In the case where every\nvertex has valence 3 and the complement is a single disk, Sane announced\nin [San20] that any two such graphs on the same surface can be connected\nvia a sequence of surgeries.\n(3) This problem is more topological (and less purely combinatorial) than it\nmight appear to be at first glance. It is closely connected to the action of\nthe mapping class group on embedded graphs, and to the general prob-\nlem of understanding subgroups of the mapping class group. A surgery\nequivalence class determines a “stabilizer subgroup” of the mapping class\ngroup of $S$, consisting of all mapping classes $f$ for which $f(\\Gamma)$ can be\n\ntaken to $\\Gamma$ via surgery. The result of Salter–Sane mentioned above can\nbe rephrased as the assertion that the stabilizer subgroup for the class of\ngraphs they consider is the stabilizer of a mod-2 homology class. Other\nclasses of graphs appear to have smaller stabilizer subgroups that are not\nso easily characterized. Given the enduring mystery of classifying sub-\ngroups of the mapping class group (especially those of finite index), a\nprimary motivation for understanding graph surgery is that it may lead\nto novel examples and phenomena in this arena.\n(4) The literature mentioned above restricts attention to the unicellular case,\nwhen $S\\setminus\\Gamma$ is a single disk (note that these authors further restrict to surg-\neries that preserve this property). One reason to be particularly interested\nin this setting is that the enumeration of such “unicellular graphs” is a key\ningredient in Harer and Zagier’s computation of the Euler characteristic\nof the moduli space of Riemann surfaces [HZ86].\n\nReferences cited:\n- [San21] Abdoul Karim Sane. Curves on surfaces and surgeries. European J. Combin., 93:Paper No. 103281, 20, 2021. doi:10.1016/j.ejc.2020.103281.\n- [SS23] Nick Salter and Abdoul Karim Sane. Connected components of the topological surgery graph of a unicellular collection, 2023. arXiv:2308.09165.\n- [San20] Abdoul Karim Sane. Unicellular maps and filtrations of the mapping class group, 2020. arXiv:2006.15880.\n- [HZ86] J. Harer and D. Zagier. The Euler characteristic of the moduli space of curves. Invent. Math., 85(3):457–485, 1986. doi:10.1007/BF01390325.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Surgery graphs of embedded graphs and complement-change criteria have been studied, but no general characterization of surgery equivalence was verified.\n\n**Verified partial progress.**\n\n- Sane supplies a combinatorial complement-change criterion and connectivity results for surgery graphs.\n\n**Full solution or refutation.**\n\nThe requested necessary-and-sufficient condition remains open.\n\n**What remains.**\n\nCharacterize connected components of the surgery graph for arbitrary embedded graphs.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.13 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Describes known surgery-graph investigations and remaining classification.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2762, "problem_number": "KP-2.14", "title": "Kirby Problem 2.14", "statement": "(a) Does every Jordan curve in the Euclidean plane contain the vertices of a\nsquare?\n(b) Does every Jordan curve in the Euclidean plane contain the vertices of the\naffine image of a regular hexagon?\n(c) Let $\\gamma \\subset \\mathbb{C}$ be a smooth Jordan curve, and let $S \\subset \\mathbb{C}$ consist of $2n \\geq 4$\nconcyclic points. Does there exist a non-constant polynomial $p(z) \\in \\mathbb{C}[z]$\nof degree $< n$ such that $p(S) \\subset \\gamma$?\n(d) Suppose that $h$ is a continuous, real-valued function defined on the unit\nsphere $S^{2} \\subset \\mathbb{R}^{3}$ and that points $x_{1},..., x_{4} \\in S^{2}$ form the vertices of\na square.\nIs there an isometry $\\rho \\in \\operatorname{SO}(3)$ such that $h$ is constant on\n$\\rho(x_{1}),..., \\rho(x_{4})$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.14.\n\nLiterature notes:\n(1) The first problem is known as the Square Peg Problem. It was posed by\nOtto Toeplitz in 1911 [Toe11]. It is solved affirmatively for several classes\nof curves, including $C^{1}$ curves (see [Mat14]).\nMore generally, a curve inscribes an $n$-gon (and the $n$-gon inscribes\nin the curve) if there exists an orientation-preserving similar copy of the\n$n$-gon whose vertices lie on the curve. Every Jordan curve inscribes ev-\nery triangle (a noncolinear 3-gon) [Nie92], but no two dissimilar ellipses\ninscribe the same pentagon. Vaughn proved that every Jordan curve in-\nscribes a rectangle (of some uncontrolled aspect ratio) [Mey81]. Greene\nand Lobb proved the optimal result for smooth curves: every smooth Jor-\ndan curve inscribes every cyclic quadrilateral (i.e., a quadrilateral whose\nvertices lie on a circle) [GL23]. By contrast, Pak observed that the quadri-\nlaterals that inscribe in all triangles are the isosceles trapezoids [Pak08].\nIt is possible that the widest class of quadrilaterals that inscribe in all\nJordan curves are the isosceles trapezoids.\n\n(2) The second problem was raised as a conjecture by Grünbaum [Gru72,\nConjecture 4.3]. It is known to hold if the curve is convex [Gru72, p.84].\nIt is known that every $C^{1}$ Jordan curve either contains the vertices of the\naffine image of a regular hexagon or six colinear points that are the limit\nof the vertices of a sequence of affine images of regular hexagons [Vv11,\nTheorem 7]. However, the problem is open in general, even for smooth\ncurves.\n(3) The third problem was raised as a conjecture by Greene and Lobb [GL24,\nConjecture 1.1]. The case $n = 2$ is true and is equivalent to the result\nabout cyclic quadrilaterals noted above in the discussion of the first prob-\nlem.\nThe case $n = 3$, and some special instances when $n \\geq 4$, was\nannounced in [GL24]. These results all involve symplectic geometry.\n(4) The fourth problem is known as the Table Problem, and it is related to the\nKnaster problem (which has been solved in the negative). It is posed as\na conjecture in [Mat14, Conjecture 13]. The case in which $x_{1},..., x_{4}$ are\nequally spaced around a great circle was solved affirmatively by Dyson\n[Dys51].\nThe case in which $h$ is an even function was announced by\nNaseri Sadr, who also announced a proof of a variation for Riemannian\nsurfaces [NS24]. Call a set of four points $x_{1},..., x_{4} \\in S^{2}$ balancing if for\nevery $h$ there exists a $\\rho$ as in the problem statement. The Table Problem\ntherefore asserts that the vertices of a square are a balancing set. The\ncase of the standard height function shows that a balancing set of points\nmust be concyclic. On the other hand, there exist concyclic $x_{1},..., x_{4}$\nthat are not a balancing set [Kar13]. Livesay proved that the vertices\nof a rectangle contained in a great circle form a balancing set [Liv54].\nIt is possible that the vertices of any rectangle on $S^{2}$ form a balancing\nset. A proof was claimed in [Gri91], but that paper was later invalidated\n[Mat14].\nRemarks on these and related problems appear in Klee and Wagon’s book [KW91]\nand Matschke’s survey article [Mat14].\n\nReferences cited:\n- [Toe11] Otto Toeplitz. Ueber einige Aufgaben der Analysis situs. Verhandlungen der Schweizerischen Naturforschenden Gesellschaft, 4:197, 1911.\n- [Mat14] Benjamin Matschke. A survey on the square peg problem. Notices Amer. Math. Soc., 61(4):346–352, 2014. doi:10.1090/noti1100.\n- [Nie92] Mark J. Nielsen. Triangles inscribed in simple closed curves. Geom. Dedicata, 43(3):291–297, 1992. doi:10.1007/BF00151519.\n- [Mey81] Mark D. Meyerson. Balancing acts. Topology Proc., 6(1):59–75 (1982), 1981.\n- [GL23] Joshua Evan Greene and Andrew Lobb. Cyclic quadrilaterals and smooth Jordan curves. Invent. Math., 234(3):931–935, 2023. doi:10.1007/s00222-023-01212-6.\n- [Pak08] Igor Pak. The discrete square peg problem, 2008. arXiv:0804.0657.\n- [Gru72] Branko Grunbaum. Arrangements and spreads, volume No. 10 of Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI, 1972.\n- [Vv11] Siniša T. Vrećica and Rade T. Živaljević. Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem. Israel J. Math., 184:221–249, 2011. doi:10.1007/s11856-011-0066-9.\n- [GL24] Joshua Evan Greene and Andrew Lobb. Polynomial inscriptions, 2024. arXiv:2206.14710.\n- [Dys51] F. J. Dyson. Continuous functions defined on spheres. Ann. of Math. (2), 54:534– 536, 1951. doi:10.2307/1969487.\n- [NS24] Ali Naseri Sadr. A Table theorem for surfaces with odd Euler characteristic, 2024. arXiv:2206.14710.\n- [Kar13] R. N. Karasëv. On two conjectures of Makeev. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 415:5–14, 2013. doi:10.1007/s10958-016-2679-3.\n- [Liv54] George R. Livesay. On a theorem of F. J. Dyson. Ann. of Math. (2), 59:227–229, 1954. doi:10.2307/1969689.\n- [Gri91] H. B. Griffiths. The topology of square pegs in round holes. Proc. London Math. Soc. (3), 62(3):647–672, 1991. doi:10.1112/plms/s3-62.3.647.\n- [KW91] Victor Klee and Stan Wagon. Old and new unsolved problems in plane geometry and number theory, volume 11 of The Dolciani Mathematical Expositions. Mathematical Association of America, Washington, DC, 1991.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Square Peg is known for C1 curves and smooth curves inscribe every cyclic quadrilateral, but arbitrary Jordan Square Peg, affine regular hexagons, and the polynomial condition remain unresolved.\n\n**Verified partial progress.**\n\n- Matschke's work covers C1 Square Peg.\n- Greene--Lobb prove every smooth Jordan curve inscribes every cyclic quadrilateral.\n- Every Jordan curve inscribes every triangle and some rectangle.\n\n**Full solution or refutation.**\n\nThe multi-part record has no global resolution.\n\n**What remains.**\n\nResolve each remaining regularity/polygon/polynomial assertion.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.14 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Separates solved regularity classes from outstanding variants.\n\n**Review notes.** Multi-part status retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2763, "problem_number": "KP-2.15", "title": "Kirby Problem 2.15", "statement": "Is the genus $g$ Goeritz group $\\mathcal{G}_{g}$ finitely generated when $g \\geq 4$?\nIf so, find a set of generators.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.15.\n\nLiterature notes:\n(1) The Goeritz group $\\mathcal{G}_{g}$ is the group of isotopy classes of orientation-preserving\nhomeomorphisms of $S^{3}$ that preserve an unknotted genus $g$ handlebody,\nthat is, one side of a genus $g$ Heegaard splitting. It is a rough analogue\nfor $S^{3}$ of the braid group for $S^{2}$. The notion can be extended to Heegaard\nsplittings of other 3-manifolds, as well [JM13].\n(2) Goeritz [Goe33] exhibited a complete set of three generators for $\\mathcal{G}_{2}$ and\nAkbas [Akb08] found associated relators, providing a complete presen-\ntation of $\\mathcal{G}_{2}$. Powell [Pow80] believed he had found a complete set of\n5 generators for $\\mathcal{G}_{g}$, for any $g$, but there was a serious gap in the proof\n(see footnote of [Sch04]). This is now known as the Powell Conjecture.\nIn [Sch20] it is shown that one of Powell’s proposed five generators is\nredundant.\nFreedman and Scharlemann [FS18] announced a proof of the Powell\nConjecture for $g = 3$. Their methods break down for higher genus, and\nit is suspected that for $g \\geq 4$ the group $\\mathcal{G}_{g}$ is not even finitely generated.\nFreedman [Fre22] suggests a method by which this might be proven.\n(3) In [Sch24], Scharlemann generalizes Powell’s proposed generators and\nshows that this generalized set suffices.\nThe set itself is infinite, but\nScharlemann announced that it can be used to demonstrate that the Pow-\nell Conjecture is stably true [Sch25]. That is, any element of $\\mathcal{G}_{g}$, when\nviewed as acting on the genus $g + 1$ unknotted handlebody obtained by\nadding a single genus 1 summand to the original genus $g$ handlebody, is\na consequence of the 5 Powell elements of $\\mathcal{G}_{g+1}$.\n(4) In another approach, Zupan [Zup20] identifies a certain graph $\\mathcal{R}_{g}$ in the\ncurve complex of a genus $g$ Heegaard surface and shows that, in general,\nthe genus $k$ Powell conjecture is true for all $k \\leq g$ if and only if $\\mathcal{R}_{k}$ is\nconnected for all $k \\leq g$.\n\nReferences cited:\n- [JM13] Jesse Johnson and Darryl McCullough. The space of Heegaard splittings. J. Reine Angew. Math., 679:155–179, 2013. doi:10.1515/crelle.2012.016.\n- [Goe33] L. Goeritz. Die Abbildungen der Brezelfläche und der Volbrezel vom Gesschlect 2. Abh. Math. Sem. Univ. Hamburg, 9:244–259, 1933.\n- [Akb08] Erol Akbas. A presentation for the automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. Pacific J. Math., 236(2):201–222, 2008. doi:10.2140/pjm.2008.236.201.\n- [Pow80] Jerome Powell. Homeomorphisms of $S^{3}$ leaving a Heegaard surface invariant. Trans. Amer. Math. Soc., 257(1):193–216, 1980. doi:10.2307/1998131.\n- [Sch04] Martin Scharlemann. Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. Bol. Soc. Mat. Mexicana (3), 10:503–514, 2004.\n- [Sch20] Martin Scharlemann. One Powell generator is redundant. Proc. Amer. Math. Soc. Ser. B, 7:138–141, 2020. doi:10.1090/bproc/58.\n- [FS18] Michael Freedman and Martin Scharlemann. Powell moves and the Goeritz group, 2018. arXiv:1804.05909.\n- [Fre22] Michael Freedman. The 2-width of embedded 3-manifolds. Peking Math. J., 5(1):21–35, 2022. doi:10.1007/s42543-021-00035-9.\n- [Sch24] Martin Scharlemann. Generating the Goeritz group of $S^{3}$. J. Assoc. Math. Res., 2(2):209–335, 2024.\n- [Sch25] Martin Scharlemann. Powell’s conjecture on the Goeritz group of $S^{3}$ is stably true. Algebr. Geom. Topol., 25(6):3775–3787, 2025. doi:10.2140/agt.2025.25.3775.\n- [Zup20] Alexander Zupan. The Powell conjecture and reducing sphere complexes. J. Lond. Math. Soc. (2), 101(1):328–348, 2020. doi:10.1112/jlms.12272.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The genus-2 Goeritz group is finitely presented and a proof for genus 3 has been announced, but finite generation for g>=4 remains open and may fail.\n\n**Verified partial progress.**\n\n- Goeritz/Akbas settle genus 2.\n- Freedman--Scharlemann announced the Powell conjecture for g=3.\n\n**Full solution or refutation.**\n\nNo genus-at-least-four finite generating set was verified.\n\n**What remains.**\n\nProve finite generation, construct generators, or prove non-finite generation for g>=4.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.15 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records genus-2 result, genus-3 announcement and higher-genus uncertainty.\n\n**Review notes.** Announcement retained as partial.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2764, "problem_number": "KP-2.16", "title": "Kirby Problem 2.16", "statement": "If two hyperbolic surfaces have the same unmarked simple\nlength spectra (i.e., the same multiset of lengths that correspond to simple closed\ncurves), are they isometric?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.16.\n\nLiterature notes:\n(1) Let $M$ be a closed Riemannian manifold with negative sectional curva-\nture, and let $\\mathcal{C}$ denote the set of free homotopy classes of closed curves in\n$M$. In each homotopy class, there is a unique closed geodesic. This de-\nfines a marked length spectrum function $\\ell: \\mathcal{C} \\to \\mathbb{R}_{>0}$ that assigns to each\n\nclass $g$ the length $\\ell(g)$ of this closed geodesic. Burns and Katok asked\nwhether the function $\\ell$ determines $M$ up to isometry [BK85]. While this\nquestion is open in general, the answer is yes in the case of surfaces; this\nis called marked length spectrum rigidity. Fricke and Klein showed this\nholds for closed hyperbolic surfaces, in which case there are $6g - 5$ geo-\ndesic curves whose lengths determine the surface us to isometry [FK65].\nOtal and, independently, Croke showed that surfaces with variable neg-\native curvature are also marked length spectrum rigid [Ota90, Cro90];\nsee [CFF92, HP97] for other classes of metrics with respect to which\nsurfaces are marked length spectrum rigid.\nIn this problem, the length spectrum is unmarked, in the sense that\nwe no longer consider the function $\\ell$ but only the image of $\\ell$ as a multiset\nin $\\mathbb{R}_{>0}$; that is, we simply have a list of the lengths of closed geodesics on\nthe surface, counted with multiplicity. Moreover, we consider only lengths\ncorresponding to simple closed curves.\nIf we drop the assumption that curves we consider are simple, then the\nanswer is no; that is, hyperbolic surfaces do not enjoy unmarked length\nspectrum rigidity. Vignéras constructed the first examples of isospectral\nbut non-isometric closed surfaces [Vig80], and Sunada gave a general\nconstruction of such surfaces [Sun85].\nHowever, the length spectrum\ndoes determine the isometry class of the surface in some low-complexity\ncases, such as a one-holed torus with one boundary component [Haa85,\nBS88b].\nOn the other hand, the question of simple length spectrum rigidity is\nstill open. It is known that the examples of isospectral but non-isometric\nsurfaces constructed by Sunada can be distinguished by their simple length\nspectra [Mau13].\n(2) One can ask a family of analogous questions for each natural number:\nsuppose two hyperbolic surfaces have the same unmarked “$n$-spectra”,\nwhich is the super-set of lengths corresponding to all curves with at most\n$n$ self-intersections.\nAre they isometric?\nAbove, “superset of lengths”\nmeans the sets of lengths counting multiplicity.\n(3) Baik–Choi–Kim [BCK21] recently announced that there is an open dense\nset in Teichmüller space where the unmarked simple length spectrum is in-\ndeed rigid. The introduction to this paper contains a thorough discussion\nof the history and context of this problem.\n(4) See also Problem 3.17 for a discussion of length spectrum regidity for\nhigher-dimensional hyperbolic manifolds.\n\nReferences cited:\n- [BK85] K. Burns and A. Katok. Manifolds with nonpositive curvature. Ergodic Theory Dynam. Systems, 5(2):307–317, 1985. doi:10.1017/S0143385700002935.\n- [FK65] Robert Fricke and Felix Klein. Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, volume Bande 3, 4 of Bibliotheca Mathematica Teubneriana. Johnson Reprint Corp., New York; B. G. Teubner Verlagsgesellschaft, Stuttgart, 1965.\n- [Ota90] Jean-Pierre Otal. Le spectre marqué des longueurs des surfaces à courbure négative. Ann. of Math. (2), 131(1):151–162, 1990. doi:10.2307/1971511.\n- [Cro90] Christopher B. Croke. Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv., 65(1):150–169, 1990. doi:10.1007/BF02566599.\n- [CFF92] C. Croke, A. Fathi, and J. Feldman. The marked length-spectrum of a surface of nonpositive curvature. Topology, 31(4):847–855, 1992. doi:10.1016/0040-9383(92) 90013-8.\n- [HP97] Sa’ar Hersonsky and Frédéric Paulin. On the rigidity of discrete isometry groups of negatively curved spaces. Comment. Math. Helv., 72(3):349–388, 1997. doi:10.1007/s000140050022.\n- [Vig80] Marie-France Vignéras. Variétés riemanniennes isospectrales et non isométriques. Ann. of Math. (2), 112(1):21–32, 1980. doi:10.2307/1971319.\n- [Sun85] Toshikazu Sunada. Riemannian coverings and isospectral manifolds. Ann. of Math. (2), 121(1):169–186, 1985. doi:10.2307/1971195.\n- [Haa85] Andrew Haas. Length spectra as moduli for hyperbolic surfaces. Duke Math. J., 52(4):923–934, 1985. doi:10.1215/S0012-7094-85-05249-4.\n- [BS88b] P. Buser and K.-D. Semmler. The geometry and spectrum of the one-holed torus. Comment. Math. Helv., 63(2):259–274, 1988. doi:10.1007/BF02566766.\n- [Mau13] Rasimate Maungchang. The Sunada construction and the simple length spectrum. Geom. Dedicata, 163:349–360, 2013. doi:10.1007/s10711-012-9753-x.\n- [BCK21] Hyungryul Baik, Inhyeok Choi, and Dongryul M. Kim. Simple length spectra as moduli for hyperbolic surfaces and rigidity of length identities, 2021. arXiv:2012.05652.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Marked simple length spectrum rigidity is known for hyperbolic surfaces, but unmarked simple length spectrum rigidity remains open.\n\n**Verified partial progress.**\n\n- Marked length spectrum rigidity is classical for closed hyperbolic surfaces.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for the unmarked simple spectrum question was verified.\n\n**What remains.**\n\nProve isometry from the unmarked simple length multiset or construct isospectral nonisometric surfaces.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.16 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes marked rigidity from the open unmarked question.\n\n**Review notes.** Open is source-backed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2765, "problem_number": "KP-2.17", "title": "Kirby Problem 2.17", "statement": "Suppose that $c$ is a geodesic current on a hyperbolic surface,\nand suppose that, on the space of hyperbolic metrics on the surface, $c$ has the same\nlength function as a closed geodesic.\nMust $c$ be a convex combination of closed\ncurves?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.17.\n\nLiterature notes:\n(1) For an introduction to the theory of geodesic currents in general, see the\nfoundational work of Bonahon [Bon86, Bon88], as well as the contem-\nporary book of Erlandsson–Souto [ES22].\n(2) As an example, suppose that $c_{1}$ and $c_{2}$ are so-called “length-twins,” i.e.,\nnon-homotopic closed curves that have the same length in any hyperbolic\nmetric. Then the geodesic current $\\frac{1}{2}c_{1}+\\frac{1}{2}c_{2}$ is not a closed curve, but it\nwill also have the same length as $c_{1}$ and $c_{2}$. It is a convex combination of\n$c_{1}$ and $c_{2}$; thus, the question asks whether there exists some more exotic\ntype of current (for example one whose support in $\\mathbb{H}^{2}$ is not a discrete set\nof geodesics) that can also have the same length as either $c_{1}$ or $c_{2}$.\n\nReferences cited:\n- [Bon86] Francis Bonahon. Bouts des variétés hyperboliques de dimension 3. Annals of Mathematics, 124(1):71–158, 1986. URL: http://www.jstor.org/stable/1971388.\n- [Bon88] Francis Bonahon. The geometry of Teichmüller space via geodesic currents. Inventiones mathematicae, 92(1):139–162, 1988. doi:10.1007/BF01393996.\n- [ES22] Viveka Erlandsson and Juan Souto. Mirzakhani’s curve counting and geodesic currents, volume 345 of Progress in Mathematics. Birkhäuser/Springer, Cham, [2022] ©2022. doi:10.1007/978-3-031-08705-9.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Length-twin convex combinations demonstrate the permitted phenomenon, but no classification excluding or constructing exotic geodesic currents was verified.\n\n**Verified partial progress.**\n\n- Convex combinations of length-twin closed curves have the same metric length function.\n\n**Full solution or refutation.**\n\nThe existence of more exotic currents is open.\n\n**What remains.**\n\nClassify all currents with a closed-geodesic length function.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.17 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains length twins and poses the exotic-current question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2766, "problem_number": "KP-2.18", "title": "Kirby Problem 2.18", "statement": "What is the best lower bound on the volume of a fibered hyper-\nbolic 3-manifold one can give in terms of the translation length of the monodromy\nwith respect to various metric spaces, such as Teichmüller space with the Weil–\nPeterson metric or the pants complex?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.18.\n\nLiterature notes:\n(1) The monodromy of a hyperbolic 3–manifold that fibers over the circle is\na pseudo-Anosov homeomorphism, which acts with positive translation\nlength on several well-studied spaces, including Teichmüller space with\neither the Teichmüller or Weil-Peterson metric and the pants complex.\nBrock [Bro03] shows that for a finite-type surface $S$ and a pseudo-\nAnosov homeomorphism $f$ of $S$, the hyperbolic volume of the mapping\ntorus $M_{f}$ is comparable to the translation length $\\tau_{WP}(f)$ of $f$ acting on\nthe Teichmüller space of $S$ with the Weil-Peterson metric. In particular,\nhe shows that there exist constants $\\kappa_{1}$ and $\\kappa_{2}$, depending on the topology\nof $S$, such that\n\n$$\n\\kappa_{2}\\tau_{WP}(f) \\leq \\operatorname{Vol}(M_{f}) \\leq \\kappa_{1}\\tau_{WP}(f).\n$$\n\n(3)\nHe also proves the analogous inequality with the translation length of $f$\nacting on the pants complex.\nThe heart of the question, therefore, lies in making the constant $\\kappa_{2}$\n(or the analogous constant for other metric spaces) effective by describing\nprecisely how it depends on the genus $g$ of $S$.\n(2) Kojima–McShane [KM18] and Brock–Bromberg [BB16] give effective\nupper bounds on the volume in terms of the translation length of the\nmonodromy with respect to the Teichmüller and Weil-Peterson metrics on\nTeichmüller space, respectively. A related question is to find an effective\nupper bound on $\\operatorname{Vol}(M_{f})$ in terms of the translation length with respect\nto the pants complex.\nA lower bound was obtained by Bridgeman–Brock–Bromberg for the\nWeil-Peterson translation length in the case the 3–manifold is relatively\nacylindrical [BBB23], but the bound is not effective. In the unpublished\n\nwork of Aougab–Taylor–Webb, they show that when considering the trans-\nlation length of $f$ on the pants complex, the constant $\\kappa_{2}$ decays like\n$1/g!$.\nNo effective lower bounds are known.\n(3) There is no such lower bound in terms of the translation length of the mon-\nodromy with respect to the Teichmüller metric on Teichmüller space. Mc-\nMullen gives an explicit counterexample in which the Teichmüller transla-\ntion lengths of the monodromies $f_{n}$ tend to infinity but the mapping tori\n$M_{fn}$ have bounded volume [McM14].\n(4) The analogue of the upper bound of the inequality (3) was proved for irre-\nducible end-periodic homeomorphisms of infinite-type surfaces by Field–\nKim–Leininger–Loving [FKLL23], and the analogue of the lower bound\nwas recently announced by Field–Kent–Leininger–Loving [FKLL25]. They\ngive a universal constant for $\\kappa_{1}$ and show that the constant $\\kappa_{2}$ depends\non the topology of the finite-type subsurface on which the irreducible\nend-periodic is acting “interestingly” (not just by translation). It would\ninteresting to make their lower bound effective and compare it to the\ndescription of $\\kappa_{2}$ one might achieve in the finite-type case.\n(5) This problem falls into the larger category of questions asking what hyperbolic-\ngeometric data about a fibered 3-manifold one can obtain from the surface\nand homeomorphism data used to construct it. For instance, in any of\nthe cases mentioned above, is it possible to gather data about the short\ngeodesics in the 3-manifold $M_{f}$ from information about $S$ and $f$?\n\nReferences cited:\n- [Bro03] Jeffrey F. Brock. Weil-Petersson translation distance and volumes of mapping tori. Comm. Anal. Geom., 11(5):987–999, 2003. doi:10.4310/CAG.2003.v11.n5.a6.\n- [KM18] Sadayoshi Kojima and Greg McShane. Normalized entropy versus volume for pseudo-Anosovs. Geom. Topol., 22(4):2403–2426, 2018. doi:10.2140/gt.2018.22.2403.\n- [BB16] Jeffrey F. Brock and Kenneth W. Bromberg. Inflexibility, Weil-Peterson distance, and volumes of fibered 3-manifolds. Math. Res. Lett., 23(3):649–674, 2016. doi: 10.4310/MRL.2016.v23.n3.a4.\n- [BBB23] Martin Bridgeman, Jeffrey Brock, and Kenneth Bromberg. The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores. Geom. Topol., 27(8):3183–3228, 2023. doi:10.2140/gt.2023.27.3183.\n- [McM14] Curt McMullen. Dynamics, geometry, and moduli spaces seminar, 2014: Renormalized volume. Seminar notes, 2014.\n- [FKLL23] Elizabeth Field, Heejoung Kim, Christopher Leininger, and Marissa Loving. Endperiodic homeomorphisms and volumes of mapping tori. J. Topol., 16(1):57–105, 2023. doi:10.1112/topo.12277.\n- [FKLL25] Elizabeth Field, Autumn Kent, Christopher Leininger, and Marissa Loving. A lower bound on volumes of end-periodic mapping tori. J. Topol., 18(3):Paper No. e70037, 36, 2025. doi:10.1112/topo.70037.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Mapping-torus volume is comparable to Weil--Petersson and pants-complex translation length, but sharp/effective lower constants and genus dependence remain open.\n\n**Verified partial progress.**\n\n- Brock proves two-sided comparability for fixed surface topology.\n- Kojima--McShane and related work refine aspects of the bounds.\n\n**Full solution or refutation.**\n\nThe requested best lower bound is not known.\n\n**What remains.**\n\nMake constants effective and optimize their topological dependence.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.18 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States Brock comparability and the sharp-constant objective.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2767, "problem_number": "KP-2.19", "title": "Kirby Problem 2.19", "statement": "Which mapping classes give rise to arithmetic hyperbolic 3-\nmanifolds as their mapping tori?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.19.\n\nLiterature notes:\n(1) Roughly speaking, arithmetic hyperbolic manifolds are a special class of\nhyperbolic manifolds constructed by “number-theoretic” methods. They\nenjoy many special properties, e.g., possessing a very rich group of com-\nmensurations (isometries of finite covers). For a general introduction, see\n[MR03]; also see Problem 3.10.\n(2) This is likely to be a deep and difficult question, since Thurston’s proof\nof the existence of the hyperbolic structure on $M_{f}$ is non-constructive.\nIt would be useful to amass some systematic data on the monodromies\nof known fibered arithmetic 3-manifolds. Are the elements that appear\ndistinctive or exceptional from the point of view of the mapping class\ngroup?\nAs alluded to above, arithmetic hyperbolic manifolds are distinguished\nin the class of all hyperbolic manifolds by the property that their funda-\nmental groups have infinite index in their commensurators (indeed, the\ncommensurator is dense in $\\operatorname{PSL}_{2}(\\mathbb{C})$). One place to start would be to bet-\nter understand how the commensurator of the mapping torus is encoded\nin the mapping class itself.\n\n(3) In the world of knots, there is an analogous question which, remarkably,\nhas been resolved.\nAlan Reid [Rei91] has shown that the only knot\n$K \\subset S^{3}$ for which the complement $S^{3}\\setminus K$ is arithmetic hyperbolic is the\nfigure-8 knot. The figure-8 knot is fibered with fiber a once-punctured\ntorus. More generally, work of Bowditch-Maclachlan-Reid [BMR95] clas-\nsifies the monodromies of all arithmetic hyperbolic once-punctured torus\nbundles over $S^{1}$. In spite of this, it is not clear what special properties\nthese elements possess that single them out from other elements of $\\operatorname{SL}_{2}(\\mathbb{Z})$.\n(4) The paper [BMR95] also proves some other results of relevance.\nFor\nexample, it shows that there are at most finitely many cyclic commen-\nsurability classes of arithmetic hyperbolic surface bundles with surface of\nfixed non-compact topological type.\n(5) Associated to any hyperbolic manifold is an invariant called the invari-\nant trace field, the field generated by the traces of squares of elements of\nthe fundamental group, viewed via the hyperbolic structure as isometries\nin $\\operatorname{PSL}_{2}(\\mathbb{C})$. Arithmetic hyperbolic 3-manifolds are known to be char-\nacterizable in terms of properties of their traces (see [MR03]). A more\ngeneral problem is to determine the invariant trace field associated to any\npseudo-Anosov element.\n\nReferences cited:\n- [MR03] Colin Maclachlan and Alan W. Reid. The arithmetic of hyperbolic 3-manifolds, volume 219 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2003. doi:10.1007/978-1-4757-6720-9.\n- [Rei91] Alan W. Reid. Arithmeticity of knot complements. J. London Math. Soc. (2), 43(1):171–184, 1991. doi:10.1112/jlms/s2-43.1.171.\n- [BMR95] B. H. Bowditch, C. Maclachlan, and A. W. Reid. Arithmetic hyperbolic surface bundles. Math. Ann., 302(1):31–60, 1995. doi:10.1007/BF01444486.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No classification of mapping classes whose mapping tori are arithmetic hyperbolic 3-manifolds was verified.\n\n**Verified partial progress.**\n\n- Arithmetic mapping tori and commensurator-based approaches provide motivating examples.\n\n**Full solution or refutation.**\n\nThe maintained list presents this as a deep open classification problem.\n\n**What remains.**\n\nDevelop effective arithmeticity criteria in terms of monodromy/mapping-class data.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.19 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintained source explains why the mapping-class classification remains open.\n\n**Review notes.** Open is source-backed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2768, "problem_number": "KP-2.20", "title": "Kirby Problem 2.20", "statement": "Can one detect holomorphicity from a monodromy factoriza-\ntion of a Lefschetz pencil, fibration, or a surface bundle over a surface? What are\nthe special properties of monodromy factorizations of holomorphic fibrations?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.20.\n\nLiterature notes:\n(1) Unless $\\ell= g -1 = 0$, any fibration corresponding to a monodromy factor-\nization as above can be made symplectic and $J$-holomorphic for an almost\ncomplex structure $J$ compatible with the symplectic form. However, $J$ is\nnot necessarily integrable.\n(2) Let $f: X \\to S_{h}$ be a relatively minimal genus–$g$ Lefschetz fibration.\nClearly, for it to be holomorphic, the total space $X$ should have the ho-\nmotopy type of a compact complex surface, e.g., the fundamental group\none reads from the monodromy factorization should be a Kähler group,\nthe odd Betti numbers should be even, and such.\nSet $\\chi_{f}:=\\frac{1}{4}(\\chi(X)+\\sigma(X))-(g-1)(h-1)$ and\n$K_{f}^{2}:=c_{1}^{2}(X)-8(g-1)(h-1)$. A few other known constraints are:\n$\\bullet$ Beauville’s inequality [Bea79]: $\\chi_{f} \\geq 0$;\n$\\bullet$ Arakelov’s inequality [Ara71]: $K_{f}^{2} \\geq 0$; and\n$\\bullet$ Xiao’s slope inequality [Xia87]: $(4 - 4/g)\\chi_{f} \\leq K_{f}^{2} \\leq 12\\chi_{f}$.\nFurther constraints on $\\chi_{f}$ and $K_{f}^{2}$ are available for holomorphic fibrations\nof special types (hyperelliptic, non-hyperelliptic, trigonal, non-trigonal,\nbielliptic, etc.); see e.g., [AK02].\n(3) There is an additional constraint formulated in terms of the Nielsen-\nThurston classification. A subgroup $\\Gamma < \\operatorname{Mod}(S_{g})$ is reducible if there\nis a multi-curve $C \\subset S_{g}$ with only essential components fixed up to ho-\nmotopy by each element of $\\Gamma$. The monodromy group $\\Gamma$ of a nontrivial\nholomorphic surface bundle over $B\\setminus f(\\operatorname{Crit}(f))$ must be infinite and irre-\nducible [Shi97].\n\nReferences cited:\n- [Bea79] Arnaud Beauville. L’application canonique pour les surfaces de type général. Invent. Math., 55(2):121–140, 1979. doi:10.1007/BF01390086.\n- [Ara71] S. Ju. Arakelov. Families of algebraic curves with fixed degeneracies. Izv. Akad. Nauk SSSR Ser. Mat., 35:1269–1293, 1971.\n- [Xia87] Gang Xiao. Fibered algebraic surfaces with low slope. Math. Ann., 276(3):449–466, 1987. doi:10.1007/BF01450841.\n- [AK02] Tadashi Ashikaga and Kazuhiro Konno. Global and local properties of pencils of algebraic curves. In Algebraic geometry 2000, Azumino (Hotaka), volume 36 of Adv. Stud. Pure Math., pages 1–49. Math. Soc. Japan, Tokyo, 2002. doi:10.2969/aspm/03610001.\n- [Shi97] Hiroshige Shiga. On monodromies of holomorphic families of Riemann surfaces and modular transformations. Math. Proc. Cambridge Philos. Soc., 122(3):541– 549, 1997. doi:10.1017/S0305004197001825.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Holomorphic Lefschetz fibrations obey Kähler and numerical constraints, but no monodromy-factorization characterization of integrable holomorphicity is known.\n\n**Verified partial progress.**\n\n- Beauville, Arakelov and Xiao inequalities give necessary numerical constraints.\n- Every relevant fibration can be made symplectic and almost-complex holomorphic.\n\n**Full solution or refutation.**\n\nIntegrability detection from factorization remains open.\n\n**What remains.**\n\nFind necessary and sufficient monodromy conditions for holomorphicity.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.20 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States known constraints and open detection problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2769, "problem_number": "KP-2.21", "title": "Kirby Problem 2.21", "statement": "What is the minimum number, $m_{g,b}$, of right-handed Dehn\ntwists along essential curves into which the boundary multi-twist, $\\Delta:= T_{\\delta_{1}} \\cdots T_{\\delta_{b}}$,\ncan be factorized in $\\operatorname{Mod}(S_{g}^{b})$? Find factorizations realizing the minimum.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.21.\n\nLiterature notes:\n(1) A factorization $T_{c_1} \\cdots T_{c_\\ell}= \\Delta$, where no $c_{i}$ is homotopic to a boundary\ncomponent or point, is often called a positive factorization of $\\Delta$. A positive\nfactorization of the boundary multi-twist $\\Delta= T_{\\delta_{1}} \\cdots T_{\\delta_{b}}$ in $\\operatorname{Mod}(S_{g}^{b})$,\nwhere each $\\delta_{j}$ is parallel to a distinct boundary component of $S_g^b$ and\n$b > 0$, corresponds to a genus-$g$ Lefschetz pencil with $b$ base points and $\\ell$\nnodes.\nWith these in mind, the problem amounts to determining the small-\nest second Betti number $b_{2}$ of genus-$g$ Lefschetz pencils/fibrations with\n$b$ base points.\nIt is motivated by a central problem regarding the ex-\nistence of small symplectic and exotic 4-manifolds.\nSee, e.g., [BK17,\n\nBay22, BH24c] for some successful implementations of this approach to\n4-dimensional exotica.\n(2) The pairs $g, b \\in \\mathbb{N}$ for which there is no positive factorization of the bound-\nary multi-twist are determined in [BMVHM17]; namely, when $g = 1$ and\n$b > 9$, or $g \\geq 2$ and $b > 4g + 4$. In all other cases, the minimum is known\nto exist. The only cases when the problem has been completely settled\nare $g = 1$ [Kas77, Moi77b] and $g = 2$ [BK17], for any $b \\geq 0$.\nFor $g \\geq 3$, there are reasonable upper bounds for every genus; the\npositive factorizations discovered in [Mat96, Cad98, Kor01, Ham17]\nimply that $m_{g,b} \\leq 2g+4$ when $g$ is even and $b \\leq 4$, and $m_{g,b} \\leq 2g+10$ when\n$g$ is odd and $b \\leq 8$. These bounds are known to not be sharp, at least in\nlow genera. For example, $m_{g,b} = 7$ when $g = 2$ and $b \\leq 3$ [BK17, Bay22],\nwhereas $m_{g,b} \\leq 12$ when $g = 3$ and $b \\leq 4$ [Smi01b, Bay22, HH18a].\nFor large $b \\leq 4g + 4$, another upper bound is $m_{g,b} \\leq 8g + 4$, which is\nrealized by genus–$g$ pencils on $S^{2} \\times S^{2}$ [SS94, Tan12].\nThere are also lower bounds available in every genus, derived from\nSeiberg-Witten theory of symplectic 4-manifolds, roughly implying that\n$m_{g,b} \\geq g$ [Li00, Sti99, BK03]—which is unlikely to be sharp in any\ngenus.\n(3) There are several variations of this problem studied in the literature.\nThe minimum number of Dehn twists in positive factorizations of any\n$h \\geq 1$ commutators in $\\operatorname{Mod}(S_{g})$ (which correspond to genus-$g$ Lefschetz\nfibrations over $S_{h}$) is almost completely determined in [KO01, Mon12,\nHam14, SY17, BH23]; the only open cases are when $g \\geq 3$ and $h = 1$.\nWhen $b > 0$, one can also inquire about the maximum number of\nDehn twists, related to the uniform topology of Stein fillings of con-\ntact 3-manifolds. This problem was completely answered in [BVHM15,\nBVHM16, BMVHM17], establishing in particular that in many cases\none may have arbitrarily long positive factorizations of $\\Delta$ in $\\operatorname{Mod}(S_{g}^{b})$.\nIn some cases, one can obtain more leverage on the stated problem by\nimposing additional conditions, say by assuming that the factorization lies\nin a smaller mapping class group (e.g., [Alt20]) or by fixing some topo-\nlogical invariants of the corresponding Lefschetz pencils (e.g., [Sti02]).\n\nReferences cited:\n- [BK17] R. İnanç Baykur and Mustafa Korkmaz. Small Lefschetz fibrations and exotic 4-manifolds. Math. Ann., 367(3-4):1333–1361, 2017. doi:10.1007/s00208-016-1466-2.\n- [Bay22] R. İnanç Baykur. Small exotic 4-manifolds and symplectic Calabi-Yau surfaces via genus-3 pencils. In Gauge theory and low-dimensional topology—progress and interaction, volume 5 of Open Book Ser., pages 185–221. Math. Sci. Publ., Berkeley, CA, 2022. https://msp.org/obs/2022/5-1/p09.xhtml.\n- [BH24c] R. İnanç Baykur and Noriyuki Hamada. Lefschetz fibrations with arbitrary signature. J. Eur. Math. Soc. (JEMS), 26(8):2837–2895, 2024. doi:10.4171/jems/1326.\n- [BMVHM17] R. İnanç Baykur, Naoyuki Monden, and Jeremy Van Horn-Morris. Positive factorizations of mapping classes. Algebr. Geom. Topol., 17(3):1527–1555, 2017. doi: 10.2140/agt.2017.17.1527.\n- [Kas77] A. Kas. On the deformation types of regular elliptic surfaces. In Complex analysis and algebraic geometry, pages 107–111. Iwanami Shoten Publishers, Tokyo, 1977.\n- [Moi77b] Boris Moishezon. Complex surfaces and connected sums of complex projective planes. Lecture Notes in Mathematics, Vol. 603. Springer-Verlag, Berlin-New York, 1977. With an appendix by R. Livne.\n- [Mat96] Yukio Matsumoto. Lefschetz fibrations of genus two—a topological approach. In Topology and Teichmüller spaces (Katinkulta, 1995), pages 123–148. World Sci. Publ., River Edge, NJ, 1996.\n- [Cad98] Carlos Alberto Cadavid. On a remarkable set of words in the mapping class group. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–The University of Texas at Austin. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\& rft val fmt=info:ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri: pqdiss:9936983.\n- [Kor01] Mustafa Korkmaz. Noncomplex smooth 4-manifolds with Lefschetz fibrations. Internat. Math. Res. Notices, 2001(3):115–128, 2001. doi:10.1155/$S^{1}$07379280100006X.\n- [Ham17] Noriyuki Hamada. Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration, 2017. arXiv:1610.08458.\n- [Smi01b] Ivan Smith. Torus fibrations on symplectic four-manifolds. Turkish J. Math., 25(1):69–95, 2001.\n- [HH18a] Noriyuki Hamada and Kenta Hayano. Topology of holomorphic Lefschetz pencils on the four-torus. Algebr. Geom. Topol., 18(3):1515–1572, 2018. doi:10.2140/agt.2018.18.1515.\n- [SS94] Masa-Hiko Saitō and Ken-Ichi Sakakibara. On Mordell-Weil lattices of higher genus fibrations on rational surfaces. J. Math. Kyoto Univ., 34(4):859–871, 1994. doi: 10.1215/kjm/1250518890.\n- [Tan12] Shunsuke Tanaka. On sections of hyperelliptic Lefschetz fibrations. Algebr. Geom. Topol., 12(4):2259–2286, 2012. doi:10.2140/agt.2012.12.2259.\n- [Li00] Tian-Jun Li. Symplectic Parshin-Arakelov inequality. Internat. Math. Res. Notices, 2000(18):941–954, 2000. doi:10.1155/$S^{1}$073792800000490.\n- [Sti99] András I. Stipsicz. On the number of vanishing cycles in Lefschetz fibrations. Math. Res. Lett., 6(3-4):449–456, 1999. doi:10.4310/MRL.1999.v6.n4.a7.\n- [BK03] V. Braungardt and D. Kotschick. Clustering of critical points in Lefschetz fibrations and the symplectic Szpiro inequality. Trans. Amer. Math. Soc., 355(8):3217–3226, 2003. doi:10.1090/S0002-9947-03-03290-2.\n- [KO01] Mustafa Korkmaz and Burak Ozbagci. Minimal number of singular fibers in a Lefschetz fibration. Proc. Amer. Math. Soc., 129(5):1545–1549, 2001. doi:10.1090/S0002-9939-00-05676-8.\n- [Mon12] Naoyuki Monden. On minimal number of singular fibers in a genus-2 Lefschetz fibration. Tokyo J. Math., 35(2):483–490, 2012. doi:10.3836/tjm/1358951332.\n- [Ham14] Noriyuki Hamada. Upper bounds for the minimal number of singular fibers in a Lefschetz fibration over the torus. Michigan Math. J., 63(2):275–291, 2014. doi: 10.1307/mmj/1401973051.\n- [SY17] András I. Stipsicz and Ki-Heon Yun. On the minimal number of singular fibers in Lefschetz fibrations over the torus. Proc. Amer. Math. Soc., 145(8):3607–3616, 2017. doi:10.1090/proc/13480.\n- [BH23] R. Inanc Baykur and Noriyuki Hamada. Exotic 4-manifolds with signature zero, 2023. Selecta Math., to appear. arXiv:2305.10908.\n- [BVHM15] R. İnanç Baykur and Jeremy Van Horn-Morris. Families of contact 3-manifolds with arbitrarily large Stein fillings. J. Differential Geom., 101(3):423–465, 2015. With an appendix by Samuel Lisi and Chris Wendl, http://projecteuclid.org/euclid.jdg/1445518920.\n- [BVHM16] R. İnanç Baykur and Jeremy Van Horn-Morris. Topological complexity of symplectic 4-manifolds and Stein fillings. J. Symplectic Geom., 14(1):171–202, 2016. doi:10.4310/JSG.2016.v14.n1.a7.\n- [Alt20] Tülin Altunöz. The number of singular fibers in hyperelliptic Lefschetz fibrations. J. Math. Soc. Japan, 72(4):1309–1325, 2020. doi:10.2969/jmsj/82988298.\n- [Sti02] András I. Stipsicz. Singular fibers in Lefschetz fibrations on manifolds with $b_2^+=1$. Topology Appl., 117(1):9–21, 2002. doi:10.1016/S0166-8641(00)00105-X.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The pairs with no positive boundary-multitwist factorization are classified, but minimum factorization length for the remaining pairs is not known generally.\n\n**Verified partial progress.**\n\n- Baykur--Monden--Van Horn-Morris determine the no-factorization range: g=1,b>9 or g>=2,b>4g+4.\n\n**Full solution or refutation.**\n\nNo full formula for m_g,b was verified.\n\n**What remains.**\n\nCompute minimal lengths and exhibit minimizers for every admitting pair.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.21 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the existence classification and remaining minimum problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2770, "problem_number": "KP-2.22", "title": "Kirby Problem 2.22", "statement": "Does every Lefschetz fibration over the 2–sphere admit a sec-\ntion?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.22.\n\nLiterature notes:\n(1) Here we assume the fibration has critical points; otherwise, there exist\n$T^{2}$–bundles over $S^{2}$ without sections.\nLefschetz fibrations that arise from blowing up the base locus of Lef-\nschetz pencils naturally admit sections—namely, the exceptional divisors.\nHowever, if say the fibration is constructed through branched covers, or ab-\nstractly via a group-theoretic factorization, it is not a priori clear whether\na section exists.\n\nEquivalently, one can ask if every positive Dehn twist factorization of\nthe identity in $\\operatorname{Mod}(S_{g})$ lifts to $\\operatorname{Mod}(S_{g}, p)$, the mapping class group of\nthe (fiber) surface fixing a point $p \\in \\Sigma_{g}$.\n(2) This question has been raised repeatedly in the literature; e.g. [Smi01a,\nAur05, Ona10, Gom25], and Problems 2.8, 4.1, 7.1 and 8.1 in [Kor06],\n[Sti15], [KS09], and [End21] respectively.\n(3) More generally we can ask about the existence of multisections (in the\nsense of [DS03, BH16b]):\nQuestion. Does every Lefschetz fibration over the 2–sphere admit a\nmultisection? How about a 2-section?\nSimilarly, one can reformulate the existence of an $n$–section in terms of\nthe existence of a lift of a positive Dehn twist factorization of the identity\nin $\\operatorname{Mod}(S_{g})$ to a positive factorization also involving positive arc twists in\n$\\operatorname{Mod}(S_{g}, \\{p_{1},..., p_{n}\\})$, the mapping class group of $S_{g}$ fixing the set of $n$\ndistinct points $p_{1},..., p_{n} \\in S_{g}$ [BH16b, BH16a].\n(4) It is worth noting that for achiral Lefschetz fibrations, where the Dehn\ntwists in the factorization can be a mix of right-handed and left-handed\ntwists, there are counterexamples to the question. For instance, there is a\ngenus–1 achiral Lefschetz fibration on $S^{4}$ [Mat82], which, for homological\nreasons, cannot admit a section. (In contrast, every genus–1 Lefschetz\nfibration admits a section.) More examples are given in a recent preprint\nof Gompf [Gom25].\n(5) An ad hoc method for finding sections of a Lefschetz fibration with an\nexplicit monodromy factorization is to reverse-engineer the monodromy\nfactorization using elementary relations (among Dehn twists) in the map-\nping class group (e.g., [KO08, Ham17, Tan12]).\nA potential counterexample to the existence of a section, where this ad\nhoc method essentially fails, is the (spin, signature-zero) genus–9 Lefschetz\nfibration of Baykur–Hamada in [BH24c].\n\nReferences cited:\n- [Smi01a] Ivan Smith. Geometric monodromy and the hyperbolic disc. Q. J. Math., 52(2):217– 228, 2001. doi:10.1093/qjmath/52.2.217.\n- [Aur05] Denis Auroux. A stable classification of Lefschetz fibrations. Geom. Topol., 9:203– 217, 2005. doi:10.2140/gt.2005.9.203.\n- [Ona10] Sinem Çelik Onaran. On sections of genus two Lefschetz fibrations. Pacific J. Math., 248(1):203–216, 2010. doi:10.2140/pjm.2010.248.203.\n- [Gom25] Robert E. Gompf. On sections of maps from 4-manifolds to the 2-sphere, 2025. URL: https://arxiv.org/abs/2506.18066, arXiv:2506.18066.\n- [Kor06] Mustafa Korkmaz. Problems on homomorphisms of mapping class groups. In Problems on mapping class groups and related topics, volume 74 of Proc. Sympos. Pure Math., pages 81–89. Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/pspum/074/2264533.\n- [Sti15] András I. Stipsicz. Symplectic 4-manifolds, Stein domains, Seiberg-Witten theory and mapping class groups. In Interactions between low-dimensional topology and mapping class groups, volume 19 of Geom. Topol. Monogr., pages 173–200. Geom. Topol. Publ., Coventry, 2015. doi:10.2140/gtm.2015.19.173.\n- [KS09] Mustafa Korkmaz and András I. Stipsicz. Lefschetz fibrations on 4-manifolds. In Handbook of Teichmüller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pages 271–296. Eur. Math. Soc., Zürich, 2009. doi:10.4171/055-1/9.\n- [End21] Hisaaki Endo. Lefschetz fibrations. Sugaku Expositions, 34(2):175–204, 2021. Translation of [ 3675915]. doi:10.1090/suga/462.\n- [DS03] Simon Donaldson and Ivan Smith. Lefschetz pencils and the canonical class for symplectic four-manifolds. Topology, 42(4):743–785, 2003. doi:10.1016/S0040-9383(02)00024-1.\n- [BH16b] R. İnanç Baykur and Kenta Hayano. Multisections of Lefschetz fibrations and topology of symplectic 4-manifolds. Geom. Topol., 20(4):2335–2395, 2016. doi: 10.2140/gt.2016.20.2335.\n- [BH16a] R. Inanç Baykur and Kenta Hayano. Hurwitz equivalence for Lefschetz fibrations and their multisections. In Real and complex singularities, volume 675 of Contemp. Math., pages 1–24. Amer. Math. Soc., Providence, RI, 2016. doi:10.1090/conm/675.\n- [Mat82] Yukio Matsumoto. On 4-manifolds fibered by tori. Proc. Japan Acad. Ser. A Math. Sci., 58(7):298–301, 1982. http://projecteuclid.org/euclid.pja/1195515921.\n- [KO08] Mustafa Korkmaz and Burak Ozbagci. On sections of elliptic fibrations. Michigan Math. J., 56(1):77–87, 2008. doi:10.1307/mmj/1213972398.\n- [Ham17] Noriyuki Hamada. Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration, 2017. arXiv:1610.08458.\n- [Tan12] Shunsuke Tanaka. On sections of hyperelliptic Lefschetz fibrations. Algebr. Geom. Topol., 12(4):2259–2286, 2012. doi:10.2140/agt.2012.12.2259.\n- [BH24c] R. İnanç Baykur and Noriyuki Hamada. Lefschetz fibrations with arbitrary signature. J. Eur. Math. Soc. (JEMS), 26(8):2837–2895, 2024. doi:10.4171/jems/1326.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem that every nontrivial Lefschetz fibration over S2 has a section or multisection was verified.\n\n**Verified partial progress.**\n\n- Lefschetz pencils yield examples with sections after blow-up.\n- The question has equivalent mapping-class lifting formulations.\n\n**Full solution or refutation.**\n\nThe general section and multisection questions remain open.\n\n**What remains.**\n\nProve existence or construct a fibration without the proposed section/multisection.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.22 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintained list poses the section/multisection questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2771, "problem_number": "KP-2.23", "title": "Kirby Problem 2.23", "statement": "Does there exist a surface bundle over a surface that admits\na complete hyperbolic metric, or, more generally, a complete metric of variable\nnegative curvature?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.23.\n\nLiterature notes:\n(1) Thurston’s hyperbolization theorem provides for an enormous wealth of 3-\nmanifolds that are surface bundles over $S^{1}$, and that moreover admit com-\nplete hyperbolic metrics. Moving up a dimension, the same phenomenon\nbecomes quite mysterious. As in the 3-manifold case, it is necessary that\nthe monodromy of every nontrivial loop in the base be pseudo-Anosov. A\nconstruction of such purely pseudo-Anosov surface subgroups of mapping\nclass groups has been announced in very recent work of Kent–Leininger\n[KL24], resolving a major open question in its own right. It is not yet clear\nwhether every purely pseudo-Anosov subgroup induces a word-hyperbolic\n\nsurface-by-surface group extension, which is a necessary condition for the\ntotal space to admit a metric of negative curvature, and indeed, it is\npresently not known if the Kent-Leininger examples are word-hyperbolic.\nA weak version of Problem 2.23 is then simply to furnish examples of\nsurface bundles over surfaces with word-hyperbolic fundamental group.\n(2) Bowditch [Bow09] shows that for fixed base and fiber genus, there are\nonly finitely many possible examples of hyperbolic surface bundles over a\nsurface.\n(3) As detailed in [Rei06], the existence of a hyperbolic surface bundle over a\nsurface would contradict the long-standing conjecture of LeBrun [LeB02]\nthat the Seiberg-Witten invariants of a closed hyperbolic 4-manifold must\nvanish (see Problem 4.95).\n(4) Work of Kapovich [Kap98] shows that surface bundles over surfaces can\nnever admit complex hyperbolic metrics.\n(5) For an extensive account of the questions raised here, see [KL24, Section\n1].\n\nReferences cited:\n- [KL24] Autumn E. Kent and Christopher J. Leininger. Atoroidal surface bundles, 2024. arXiv:2405.12067.\n- [Bow09] Brian H. Bowditch. Atoroidal surface bundles over surfaces. Geom. Funct. Anal., 19(4):943–988, 2009. doi:10.1007/s00039-009-0033-3.\n- [Rei06] Alan W. Reid. Surface subgroups of mapping class groups. In Problems on mapping class groups and related topics, volume 74 of Proc. Sympos. Pure Math., pages 257– 268. Amer. Math. Soc., Providence, RI, 2006. doi:10.1090/pspum/074/2264545.\n- [LeB02] Claude LeBrun. Hyperbolic manifolds, harmonic forms, and Seiberg-Witten invariants. In Proceedings of the Euroconference on Partial Differential Equations and their Applications to Geometry and Physics (Castelvecchio Pascoli, 2000), volume 91, pages 137–154, 2002. doi:10.1023/A:1016222709901.\n- [Kap98] Michael Kapovich. On normal subgroups in the fundamental groups of complex surfaces, 1998. arXiv:math/9808085.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Purely pseudo-Anosov surface subgroups have been announced, but no surface bundle over a surface with a complete negatively curved metric, nor even the needed word-hyperbolic extension examples, was verified.\n\n**Verified partial progress.**\n\n- Kent--Leininger announce purely pseudo-Anosov surface subgroups.\n\n**Full solution or refutation.**\n\nThe geometric metric existence problem remains open.\n\n**What remains.**\n\nConstruct word-hyperbolic surface-by-surface extensions and negatively curved metrics.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.23 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains the recent announced subgroup progress and remaining obstruction.\n\n**Review notes.** Announcement treated as partial.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2772, "problem_number": "KP-2.24", "title": "Kirby Problem 2.24", "statement": "Does there exist a complex surface $X$ that admits three or more\nnon-isomorphic structures as a surface bundle over a surface?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.24.\n\nLiterature notes:\n(1) The analogous situation in the case of surface bundles over the circle is\norganized by the Thurston norm [Thu86]: if a surface bundle over the\ncircle admits two different fibrations, then it admits infinitely many. By\ncontrast, work of F.E.A. Johnson [Joh99] shows that a given 4-manifold\nadmits only finitely many structures as a surface bundle over a surface\nwith base and fiber both of genus $\\geq 2$.\n(2) Atiyah [Ati69] and Kodaira [Kod67] constructed complex surfaces $X$\nthat are surface bundles over surfaces in two distinct ways. These are\nconstructed as branched covers of products of algebraic curves; the two\nbundle structures arise from the projections onto either factor. Certain\nfamilies of these examples admit exactly two fiberings by work of Chen\n[Che18], Salter–Tshishiku [ST20], and Landesman–Litt–Sawin [LLS25].\nThe problem formulated here was also raised by Catanese [Cat17].\n(3) In the smooth category, Salter [Sal15] gave a method of constructing\nsurface bundles over surfaces (with base and fiber each of genus at least\n2) with an arbitrary finite number of fiberings. However, these are known\nnot to admit complex structures.\n\nReferences cited:\n- [Thu86] William P. Thurston. A norm for the homology of 3-manifolds. Mem. Amer. Math. Soc., 59(339):i–vi and 99–130, 1986.\n- [Joh99] F. E. A. Johnson. A rigidity theorem for group extensions. Arch. Math. (Basel), 73(2):81–89, 1999. doi:10.1007/s000130050371.\n- [Ati69] M. F. Atiyah. The signature of fibre-bundles. In Global Analysis (Papers in Honor of K. Kodaira), pages 73–84. Univ. Tokyo Press, Tokyo, 1969.\n- [Kod67] K. Kodaira. A certain type of irregular algebraic surfaces. J. Analyse Math., 19:207– 215, 1967. doi:10.1007/BF02788717.\n- [Che18] Lei Chen. The number of fiberings of a surface bundle over a surface. Algebr. Geom. Topol., 18(4):2245–2263, 2018. doi:10.2140/agt.2018.18.2245.\n- [ST20] Nick Salter and Bena Tshishiku. Arithmeticity of the monodromy of some Kodaira fibrations. Compos. Math., 156(1):114–157, 2020. doi:10.1112/s0010437x19007668.\n- [LLS25] Aaron Landesman, Daniel Litt, and Will Sawin. Big monodromy for higher Prym representations. Geom. Topol., 29(5):2733–2782, 2025. doi:10.2140/gt.2025.29.2733.\n- [Cat17] Fabrizio Catanese. Kodaira fibrations and beyond: methods for moduli theory. Jpn. J. Math., 12(2):91–174, 2017. doi:10.1007/s11537-017-1569-x.\n- [Sal15] Nick Salter. Surface bundles over surfaces with arbitrarily many fiberings. Geom. Topol., 19(5):2901–2923, 2015. doi:10.2140/gt.2015.19.2901.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Complex surfaces with two bundle structures are known, and smooth examples have arbitrarily many, but no complex surface with three or more nonisomorphic surface-bundle structures was verified.\n\n**Verified partial progress.**\n\n- Atiyah and Kodaira give complex two-fibration examples.\n- Several families are proved to have exactly two.\n- Salter constructs smooth examples with any prescribed finite number.\n\n**Full solution or refutation.**\n\nThe complex-category three-fibration question remains open.\n\n**What remains.**\n\nConstruct or rule out a complex example with at least three fibreings.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.24 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes complex two-fibration results from smooth arbitrarily-many examples.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2773, "problem_number": "KP-2.25", "title": "Kirby Problem 2.25", "statement": "Consider surface bundles over surfaces where both fiber $F$ and\nbase $B$ have genus $\\geq 2$ and where $\\pi_{1}(B)$ injects in the mapping class group of the\n\nfiber. Does such a bundle have a $k$–multisection for some $k > 0$ (i.e., a continuous\nchoice of $k$ everywhere distinct points)?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.25.\n\nLiterature notes:\n(1) This is Problem 2.17 in [Kir97] by G. Mess. In [Mor89], Morita gives\na cohomological obstruction for a surface bundle to admit a section (i.e.\nthe case $k = 1$). Using this obstruction, Hillman gave examples of surface\nbundles over surfaces (with both base and fiber genus $\\geq 2$) that do not\nadmit sections [Hil15]. Other examples of $S_{g}$-bundles over the 2-torus\nwith no section were constructed by Li–Litt–Salter–Srinivasan [LLSS23];\nthese were constructed with applications to arithmetic geometry in mind.\nBoth classes of examples virtually admit sections, that is, they admit\nsections after pulling back to some finite cover of the base.\n(2) In spite of the similarity between this question and Problem 2.22, it is\nlikely that substantially different techniques will be necessary.\nSection\nproblems for surface bundles tend to be of a more group-theoretic flavor,\nboiling down to obstructing a lifting of the monodromy representation\nfrom an unpointed to a pointed mapping class group.\nAvailable tech-\nniques include cohomological considerations as well as approaches based\non the Nielsen–Thurston classification. The multisection question for Lef-\nschetz fibrations is more delicate, since here it is necessary to lift the\nDehn twists in the monodromy factorization on $\\operatorname{Mod}(S_{g})$ to Dehn twists\nin $\\operatorname{Mod}(S_{g}^{b})$, not simply to arbitrary elements in $\\operatorname{Mod}(S_{g}^{b})$ projecting back\nto the original twists.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Mor89] Shigeyuki Morita. Families of Jacobian manifolds and characteristic classes of surface bundles. II. Math. Proc. Cambridge Philos. Soc., 105(1):79–101, 1989. doi:10.1017/S0305004100001389.\n- [Hil15] Jonathan A. Hillman. Sections of surface bundles. In Interactions between lowdimensional topology and mapping class groups, volume 19 of Geom. Topol. Monogr., pages 1–19. Geom. Topol. Publ., Coventry, 2015. doi:10.2140/gtm.2015.19.1.\n- [LLSS23] Wanlin Li, Daniel Litt, Nick Salter, and Padmavathi Srinivasan. Surface bundles and the section conjecture. Math. Ann., 386(1-2):877–942, 2023. doi:10.1007/s00208-022-02421-9.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Surface bundles without sections are known, and cited examples virtually acquire sections; existence of a multisection under the stated injective monodromy hypothesis remains open.\n\n**Verified partial progress.**\n\n- Morita supplies a section obstruction.\n- Hillman and Li--Litt--Salter--Srinivasan produce no-section examples.\n\n**Full solution or refutation.**\n\nNo universal multisection theorem was verified.\n\n**What remains.**\n\nProve or refute virtual/multisection existence for all such bundles.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.25 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States no-section examples and the open multisection question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2774, "problem_number": "KP-2.26", "title": "Kirby Problem 2.26", "statement": "(Kontsevich–Zorich conjecture). Understand the homotopy types\nof strata of abelian differentials. Which stratum-components are $K(\\pi, 1)$ spaces?\nWhat are the fundamental groups? For which stratum-components $\\mathcal{H}$ is some ver-\nsion of the monodromy map $\\rho: \\pi_{1}(\\mathcal{H}) \\to \\operatorname{Mod}(S_{g})$ injective? Do the answers to\nthese questions change in a predictable way as one moves from the principal stra-\ntum down to the minimal?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.26.\n\nLiterature notes:\n(1) A translation surface is a surface $T$ that admits an atlas of charts to $\\mathbb{C}$\nfor which transition maps are translations ($z \\mapsto z+c$). An abelian differ-\nential is a pair $(X, \\omega)$ of a Riemann surface together with a holomorphic\ndifferential. It is a basic but profound fact that every translation surface\ncorresponds to an abelian differential and vice-versa.\nFixing a genus, the space of all abelian differentials is identified with\nthe Hodge bundle over the moduli space of Riemann surfaces of genus $g$, the\nvector bundle whose fiber over a Riemann surface $X$ is the space $H^{1,0}(X)$\nof all holomorphic 1-forms on $X$. This space is stratified according to the\nmultiplicities of the zeroes. Every differential on a Riemann surface of\ngenus $g$ has $2g -2$ zeroes when counted with multiplicity; the generic case\n\nwhere all zeroes are simple (the partition $1^{2g-2}$) is called the principal\nstratum, and the case where there is a single zero of multiplicity $2g - 2$ is\ncalled the minimal stratum.\nIn general, the stratum $\\mathcal{H}_{\\kappa}$ associated to the partition $\\kappa = k_{1}+\\cdots +k_{n}$\nis a quasiprojective complex orbifold of complex dimension $2g + n - 1$,\nand therefore (after passing to a finite cover to resolve the orbifold issue)\nhas the homotopy type of a finite CW complex. Strata are fundamental\nobjects in the study of Teichmüller dynamics, where they host a dynamical\nsystem induced from a natural action of $\\operatorname{GL}_{2}(\\mathbb{R})$, but their topology is\nquite mysterious, despite their close relationship to the moduli space of\nRiemann surfaces and the mapping class group.\n(2) In [KZ03], Kontsevich–Zorich compute $\\pi_{0}$ for all strata. They find that\neach stratum has between between one and three connected components.\nFor the partitions $2g - 2$ and $g - 1, g - 1$, there are special “hyperelliptic”\ncomponents that can be identified with configuration spaces of points in\n$\\mathbb{C}$, resolving Problem 2.26 in this case.\nIn [KZ97], they pose a version of Problem 2.26, conjecturing that\neach stratum component should be a $K(\\pi, 1)$ and that the fundamental\ngroups should be some flavor of mapping class group.\n(3) Work of Calderon–Salter [CS23] identifies the image of $\\pi_{1}(\\mathcal{H})$ in the map-\nping class group under the natural monodromy homomorphism for all non-\nhyperelliptic stratum components in genus $g \\geq 5$. They find that these\nare “framed mapping class groups”—the stabilizer of the distinguished\nframing on the translation surface inherited from the standard framing\non $\\mathbb{C}$. This shows that whenever the monodromy homomorphism is in-\njective, the fundamental group of the stratum component is indeed some\nflavor of mapping class group, as predicted by the conjecture.\nThe question of monodromy injectivity appears to be subtle, how-\never. In the hyperelliptic setting, injectivity was established in the work\nof Kontsevich–Zorich. On the other hand, in genus 3, there is a stratum-\ncomponent known as $\\mathcal{H}^{\\mathrm{odd}}(4)$\nfor which injectivity is known not to hold.\nWork of Looijenga–Mondello [LM14] computes the fundamental group\nto be $A(E_{6})$ (the Artin group of type $E_{6}$) modulo its center. Under the\nmonodromy map, the Artin generators are sent to Dehn twists about sim-\nple closed curves whose intersection pattern is given by the $E_{6}$ Dynkin\ndiagram.\nWork of Wajnryb [Waj99] shows that this homomorphism\n$A(E_{6}) \\to \\operatorname{Mod}(\\Sigma_{3})$ has a non-central element in the kernel.\n\nReferences cited:\n- [KZ03] Maxim Kontsevich and Anton Zorich. Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math., 153(3):631–678, 2003. doi:10.1007/s00222-003-0303-x.\n- [KZ97] M. Kontsevich and A. Zorich. Lyapunov exponents and Hodge theory, 1997. arXiv: hep-th/9701164.\n- [CS23] Aaron Calderon and Nick Salter. Framed mapping class groups and the monodromy of strata of abelian differentials. J. Eur. Math. Soc. (JEMS), 25(12):4719–4790, 2023. doi:10.4171/jems/1290.\n- [LM14] Eduard Looijenga and Gabriele Mondello. The fine structure of the moduli space of abelian differentials in genus 3. Geom. Dedicata, 169:109–128, 2014. doi:10.1007/s10711-013-9845-2.\n- [Waj99] Bronislaw Wajnryb. Artin groups and geometric monodromy. Invent. Math., 138(3):563–571, 1999. doi:10.1007/s002220050353.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Kontsevich--Zorich programme asking homotopy type, K(pi,1), fundamental group and monodromy injectivity for strata remains open in general.\n\n**Verified partial progress.**\n\n- Translation surfaces and abelian-differential strata have detailed component/monodromy structure in many cases.\n\n**Full solution or refutation.**\n\nNo complete classification was verified.\n\n**What remains.**\n\nDetermine the requested topological invariants for each stratum component.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.26 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintains the multi-part Kontsevich--Zorich questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2775, "problem_number": "KP-2.27", "title": "Kirby Problem 2.27", "statement": "For $n \\geq 4$, does the braid group $B_{n}$ admit a finite-index sub-\ngroup that embeds in a right-angled Artin group?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.27.\n\nLiterature notes:\n(1) In CAT(0) geometry, a cube complex is called special if it satisfies certain\nconditions on its hyperplanes [HW08]. Groups acting by combinatorial\nisometries on special cube complexes are of particular interest in geometric\ngroup theory, and are closely related to subgroups of right-angled Artin\ngroups. A group is said to be virtually special if it contains a finite-index\nsubgroup isomorphic to the fundamental group of a special cube complex.\n(2) By [BKK16], any group that embeds in a right-angled Artin group acts\nby $C^{\\infty}$-diffeomorphisms on $\\mathbb{R}$. As braid groups do not admit such ac-\ntions [FF20], they do not embed in right-angled Artin groups, so it is\nnecessary to pass to proper subgroups.\n(3) One could go further and ask whether braid groups are virtually special.\nThis is strictly stronger than Question 2.27.\n(4) In light of the resolution of the congruence subgroup problem for braid\ngroups (see [Mar19] and the references therein), to resolve Problem 2.27\nin the negative, it suffices to obtain a cofinal sequence of subgroups of $B_{n}$,\nnone of which embed in a right-angled Artin group.\n\nReferences cited:\n- [HW08] Frédéric Haglund and Daniel T. Wise. Special cube complexes. Geom. Funct. Anal., 17(5):1551–1620, 2008. doi:10.1007/s00039-007-0629-4.\n- [BKK16] Hyungryul Baik, Sang-hyun Kim, and Thomas Koberda. Right-angled Artin groups in the C8 diffeomorphism group of the real line. Israel J. Math., 213(1):175–182, 2016. doi:10.1007/s11856-016-1307-8.\n- [FF20] Benson Farb and John Franks. Groups of homeomorphisms of one-manifolds, I: Actions of nonlinear groups. In What’s next?—the mathematical legacy of William P. Thurston, volume 205 of Ann. of Math. Stud., pages 116–140. Princeton Univ. Press, Princeton, NJ, 2020. doi:10.2307/j.ctvthhdvv.9.\n- [Mar19] Dan Margalit. Problems, questions, and conjectures about mapping class groups. In Breadth in contemporary topology, volume 102 of Proc. Sympos. Pure Math., pages 157–186. Amer. Math. Soc., Providence, RI, 2019. doi:10.1090/pspum/102/12.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Braid groups themselves do not embed in RAAGs, but whether they have finite-index RAAG-embeddable subgroups for n>=4 remains open.\n\n**Verified partial progress.**\n\n- BKK implies RAAG subgroups act smoothly on R, while braid groups do not.\n- Congruence results reduce one route to a negative answer.\n\n**Full solution or refutation.**\n\nVirtual RAAG embedding/specialness remains unresolved.\n\n**What remains.**\n\nConstruct a finite-index embedding or give an obstruction/cofinal nonembedding sequence.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.27 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States nonembedding of full braid groups and open virtual question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2776, "problem_number": "KP-2.28", "title": "Kirby Problem 2.28", "statement": "Let $\\Gamma$ be a graph that is not a nontrivial join, and let $A(\\Gamma)$\nbe the associated right-angled Artin group. Does there exist an injective map from\n$A(\\Gamma)$ to the mapping class group of a surface such that every loxodromic element\nof $A(\\Gamma)$ is sent to a pseudo-Anosov mapping class?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.28.\n\nLiterature notes:\n(1) There are many injective homomorphisms from right-angled Artin groups\ninto mapping class groups [CP01, CLM12, Kob12], and general criteria\nare fairly well-understood [KK14b]. Right-angled Artin groups have a\nNielsen–Thurston classification, not unlike mapping class groups [BC12,\nKK14a], wherein each element is either elliptic or loxodromic.\n(2) The set of loxodromic elements of $A(\\Gamma)$ is nonempty if and only if $A(\\Gamma)$\ndoes not split as a nontrivial direct product, which is true if and only if $\\Gamma$\ndoes not split as a nontrivial join.\n\n(3) By [MT16, KMT17], the restriction of any such injective map to a sub-\ngroup where every nontrivial element is loxodromic is necessarily a convex\ncocompact subgroup of the mapping class group of $S$. However, any such\nsubgroup obtained in this way is free (see [KK14a]). In particular, this\ndoes not provide a strategy to construct convex cocompact surface sub-\ngroups as in Problem 2.23.\n(4) In light of [KMT17], the problem can be reformulated entirely in terms\nof the combinatorial topology of the underlying surface. Let $\\Gamma$ be fixed,\nand let $\\Lambda$ be the opposite graph, i.e., the one obtained by reversing the\nadjacency relation in $\\Gamma$. It suffices to find a surface $S$ and $\\pi_{1}$-injective,\npairwise non-nested subsurfaces $\\\\{S_{v}\\\\}_{v\\\\in V(\\\\Gamma)}$ indexed by the vertices of $\\Gamma$\nsuch that the following hold:\n(i) Each $S_{v}$ supports a pseudo-Anosov mapping class (in the mapping\nclass group of $S_{v}$). In particular, each $S_{v}$ is non-annular.\n(ii) Subsurfaces $S_{v}$ and $S_{w}$ intersect essentially if and only if $v$ and $w$ are\nnot adjacent in $\\Gamma$ (or equivalently, adjacent in $\\Lambda$).\n(iii) Let $Y \\subset \\Lambda$ be a connected subgraph such that every vertex of $\\Lambda$ is\nadjacent to a vertex of $Y$ . Then every essential, simple, nonperipheral\ncurve on $S$ essentially intersects $S_{v}$ for some vertex $v$ of $Y$ .\nThe difficulty of this reformulation is the non-canonical nature of $Y$ for a\ngeneral graph $\\Lambda$.\n\nReferences cited:\n- [CP01] John Crisp and Luis Paris. The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group. Invent. Math., 145(1):19– 36, 2001. doi:10.1007/s002220100138.\n- [CLM12] Matt T. Clay, Christopher J. Leininger, and Johanna Mangahas. The geometry of right-angled Artin subgroups of mapping class groups. Groups Geom. Dyn., 6(2):249–278, 2012. doi:10.4171/GGD/157.\n- [Kob12] Thomas Koberda. Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups. Geom. Funct. Anal., 22(6):1541–1590, 2012. doi:10.1007/s00039-012-0198-z.\n- [KK14b] Sang-Hyun Kim and Thomas Koberda. An obstruction to embedding right-angled Artin groups in mapping class groups. Int. Math. Res. Not. IMRN, 2014(14):3912– 3918, 2014. doi:10.1093/imrn/rnt064.\n- [BC12] Jason Behrstock and Ruth Charney. Divergence and quasimorphisms of right-angled Artin groups. Math. Ann., 352(2):339–356, 2012. doi:10.1007/s00208-011-0641-8.\n- [KK14a] Sang-Hyun Kim and Thomas Koberda. The geometry of the curve graph of a rightangled Artin group. Internat. J. Algebra Comput., 24(2):121–169, 2014. doi:10.1142/S021819671450009X.\n- [MT16] Johanna Mangahas and Samuel J. Taylor. Convex cocompactness in mapping class groups via quasiconvexity in right-angled Artin groups. Proc. Lond. Math. Soc. (3), 112(5):855–881, 2016. doi:10.1112/plms/pdw009.\n- [KMT17] Thomas Koberda, Johanna Mangahas, and Samuel J. Taylor. The geometry of purely loxodromic subgroups of right-angled Artin groups. Trans. Amer. Math. Soc., 369(11):8179–8208, 2017. doi:10.1090/tran/6933.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Many RAAG embeddings in mapping class groups are known, but none is known with every loxodromic element pseudo-Anosov in the stated nonjoin setting.\n\n**Verified partial progress.**\n\n- General RAAG embedding criteria are well developed.\n- Purely loxodromic subgroup restrictions imply convex cocompactness and freeness.\n\n**Full solution or refutation.**\n\nThe requested all-loxodromic pseudo-Anosov embedding remains open.\n\n**What remains.**\n\nConstruct such embeddings or establish a general obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.28 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records existing embeddings and the loxodromic-image obstacle.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2777, "problem_number": "KP-2.29", "title": "Kirby Problem 2.29", "statement": "Determine the Artin groups that can be embedded into a map-\nping class group.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.29.\n\nLiterature notes:\n(1) If the Artin group is simply laced and the homomorphism sends each\nstandard generator to a Dehn twist, then there is an almost complete\nclassification. It is known that the Artin groups of type $A_{n}$ (see [BH73]),\n$D_{n}$ (see [PV96]) and $\\widetilde{A}_{n}$ (see [Ryf23]) can be embedded into a mapping\nclass group in this way. It is also known that an Artin group does not\nembed in this way if its Coxeter graph is connected and contains a full\nsubgraph isomorphic to either $\\widetilde{D}_{n}$ with $n \\geq 4$ (see [Lab97]), or $E_{6}$ (see\n[Waj99]). These leave a finite number of possible open cases.\n(2) Say a homomorphism from an Artin group to a mapping class group is\ngeometric if each standard generator is sent to a Dehn multi-twist. The\nquestion is very open in the case of non-geometric embeddings. There are\nby now various examples of non-geometric embeddings of Artin groups\ninto mapping class groups; see e.g., [CMM21] for the case of right-angled\nArtin groups and [Sze10, BT12] for the case of braid groups.\n(3) The question is also closely related to another folklore question on Artin\ngroups: which Artin groups are virtually special (possessing a finite-index\nsubgroup isomorphic to a subgroup of a right-angled Artin group). It is\nknown that every right-angled Artin group embeds into a mapping class\ngroup, and so obstructions to embedding an Artin group into mapping\n\nclass groups also obstruct embeddability into right-angled Artin groups.\nFrom this point of view it is also natural to consider a “virtual” version of\nthe above question, namely to determine which Artin groups have some\nfinite-index subgroup that embeds into a mapping class group.\n(4) This problem is historically attributed to J. Crisp and L. Paris [CP01].\n\nReferences cited:\n- [BH73] Joan S. Birman and Hugh M. Hilden. On isotopies of homeomorphisms of Riemann surfaces. Ann. of Math. (2), 97:424–439, 1973. doi:10.2307/1970830.\n- [PV96] B. Perron and J. P. Vannier. Groupe de monodromie géométrique des singularités simples. Math. Ann., 306(2):231–245, 1996. doi:10.1007/BF01445249.\n- [Ryf23] Levi Ryffel. Curves intersecting in a circuit pattern. Topology Appl., 332:Paper No. 108522, 15, 2023. doi:10.1016/j.topol.2023.108522.\n- [Lab97] C. Labruere. Generalized braid groups and mapping class groups. J. Knot Theory Ramifications, 6(5):715–726, 1997. doi:10.1142/S021821659700039X.\n- [Waj99] Bronislaw Wajnryb. Artin groups and geometric monodromy. Invent. Math., 138(3):563–571, 1999. doi:10.1007/s002220050353.\n- [CMM21] Matt Clay, Johanna Mangahas, and Dan Margalit. Right-angled Artin groups as normal subgroups of mapping class groups. Compos. Math., 157(8):1807–1852, 2021. doi:10.1112/S0010437X21007417.\n- [Sze10] Bl ażej Szepietowski. Embedding the braid group in mapping class groups. Publ. Mat., 54(2):359–368, 2010. URL: https://doi.org/10.5565/PUBLMAT 54210 04, doi:10.5565/PUBLMAT\\\\_54210\\\\_04.\n- [BT12] Carl-Friedrich Bödigheimer and Ulrike Tillmann. Embeddings of braid groups into mapping class groups and their homology. In Configuration spaces, volume 14 of CRM Series, pages 173–191. Ed. Norm., Pisa, 2012. URL: https://doi.org/10.1007/978-88-7642-431-1 7, doi:10.1007/978-88-7642-431-1\\\\_7.\n- [CP01] John Crisp and Luis Paris. The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group. Invent. Math., 145(1):19– 36, 2001. doi:10.1007/s002220100138.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Geometric embeddings of many simply laced Artin groups are classified and broad non-geometric families embed, but no classification of arbitrary Artin groups admitting arbitrary embeddings into mapping class groups is known.\n\n**Verified partial progress.**\n\n- Artin groups of types A_n, D_n, and affine A_n admit geometric embeddings with standard generators represented by Dehn twists.\n- A connected simply laced Coxeter graph containing a full affine D_n subgraph for n at least 4 or an E_6 subgraph cannot embed by the specified Dehn-twist construction.\n- Every right-angled Artin group embeds into a mapping class group, and non-geometric embeddings are known for braid groups and right-angled Artin groups.\n- Only finitely many cases remain in the narrow connected simply laced Dehn-twist classification, while the unrestricted problem is very open.\n\n**Full solution or refutation.**\n\nThe geometric simply laced subproblem is close to classified, but the stored unrestricted determination problem and its virtual variant remain unsolved.\n\n**What remains.**\n\nClassify arbitrary geometric multi-twist embeddings and non-geometric embeddings, and determine which Artin groups or finite-index subgroups embed in some mapping class group.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 2.29. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the general and virtual questions and synthesizes the near-classification for simply laced Dehn-twist embeddings and the open non-geometric setting.\n- Levi Ryffel, Curves intersecting in a circuit pattern, Topology and its Applications 332 (2023), 108522. (primary): https://doi.org/10.1016/j.topol.2023.108522\n Evidence used: Provides the affine A_n geometric embedding result cited in the current problem list.\n- Matt Clay, Johanna Mangahas, and Dan Margalit, Right-angled Artin groups as normal subgroups of mapping class groups, Compositio Mathematica 157 (2021), 1807-1852. (primary): https://doi.org/10.1112/S0010437X21007417\n Evidence used: Gives strong non-geometric RAAG embeddings, including normal-subgroup realizations.\n- Bronislaw Wajnryb, Artin groups and geometric monodromy, Inventiones Mathematicae 138 (1999), 563-571. (primary): https://doi.org/10.1007/s002220050353\n Evidence used: Supplies the E_6 obstruction for geometric Dehn-twist representations.\n\n**Review notes.** The split word map-ping is a source layout artifact. The exact stored statement is preserved in report.md.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2778, "problem_number": "KP-2.30", "title": "Kirby Problem 2.30", "statement": "Which right-angled Artin groups contain closed hyperbolic sur-\nface groups? Is there an algorithmic or graph-theoretic criterion to decide this?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.30.\n\nLiterature notes:\nAll but finitely many surface groups embed in right-angled Artin\ngroups [CW04], and right-angled Artin groups on cycles of length at least five\ncontain hyperbolic surface groups [SDS89]. Right-angled Artin groups on graphs\nup to eight vertices containing surface groups were classified by Crisp–Sageev–\nSapir [CSS08]; they also posed the question of whether a right-angled Artin group\ncontains a surface group if and only if it contains a one-ended word-hyperbolic\ngroup. See also [Kim10].\n\nReferences cited:\n- [CW04] John Crisp and Bert Wiest. Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups. Algebr. Geom. Topol., 4:439–472, 2004. doi:10.2140/agt.2004.4.439.\n- [SDS89] Herman Servatius, Carl Droms, and Brigitte Servatius. Surface subgroups of graph groups. Proc. Amer. Math. Soc., 106(3):573–578, 1989. doi:10.2307/2047406.\n- [CSS08] John Crisp, Michah Sageev, and Mark Sapir. Surface subgroups of right-angled Artin groups. Internat. J. Algebra Comput., 18(3):443–491, 2008. doi:10.1142/S0218196708004536.\n- [Kim10] Sang-hyun Kim. On right-angled Artin groups without surface subgroups. Groups Geom. Dyn., 4(2):275–307, 2010. doi:10.4171/GGD/84.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Induced cycles and several broader graph families yield surface subgroups, graphs through eight vertices are classified, and useful negative criteria are known, but no general criterion or algorithm is available.\n\n**Verified partial progress.**\n\n- If the defining graph contains an induced cycle of length at least five, its right-angled Artin group contains a closed hyperbolic surface group.\n- Crisp, Sageev, and Sapir classify all defining graphs with at most eight vertices and construct eight further positive graph patterns.\n- Crisp-Sageev-Sapir and Kim give several sufficient conditions for the absence of closed hyperbolic surface subgroups.\n- The proposed equivalence with containing a one-ended hyperbolic subgroup remains open.\n\n**Full solution or refutation.**\n\nThe problem is solved for small defining graphs and many structural families, not for arbitrary finite graphs.\n\n**What remains.**\n\nFind a necessary-and-sufficient graph-theoretic condition and a terminating algorithm deciding whether A(Gamma) contains a closed hyperbolic surface group.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 2.30. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the classification and algorithmic questions and summarizes the induced-cycle theorem, the eight-vertex classification, and remaining conjectures.\n- John Crisp, Michah Sageev, and Mark Sapir, Surface subgroups of right-angled Artin groups, International Journal of Algebra and Computation 18 (2008), 443-491. (primary): https://arxiv.org/abs/0707.1144\n Evidence used: Classifies defining graphs through eight vertices and develops both positive examples and negative criteria.\n- Sang-hyun Kim, On right-angled Artin groups without surface subgroups, Groups, Geometry, and Dynamics 4 (2010), 275-307. (primary): https://doi.org/10.4171/GGD/84\n Evidence used: Develops graph operations and obstruction results proving absence of surface subgroups in additional cases.\n- Herman Servatius, Carl Droms, and Brigitte Servatius, Surface subgroups of graph groups, Proceedings of the American Mathematical Society 106 (1989), 573-578. (primary): https://doi.org/10.2307/2047406\n Evidence used: Establishes the fundamental induced-cycle source of surface subgroups.\n\n**Review notes.** The split word sur-face is a source layout artifact only.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2779, "problem_number": "KP-2.31", "title": "Kirby Problem 2.31", "statement": "Let $S$ be a closed surface of genus at least 2. Show that the\nstable commutator length is rational on the commutator subgroup of $\\pi_{1}(S)$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.31.\n\nLiterature notes:\n(1) If $G$ is a group and $g \\in [G, G]$, the commutator length of $g$, denoted cl $(g)$,\nis the least number of commutators in $G$ whose product is $g$, and the\nstable commutator length of $g$, denoted scl $(g)$, is the limit of cl $(g^{n})/n$ as\n$n \\to \\infty$.\n(2) Suppose $X$ is a space with $\\pi_{1}(X) = G$, suppose $L$ is an oriented circle,\nand suppose $f: L \\to G$ takes $L$ to the conjugacy class of $g \\in [G, G]$. After\nreplacing $f$ by inclusion, consider the the affine subspace $V$ of $H_{2}(X, L; \\mathbb{R})$\nthat is mapped by the boundary map in homology to the fundamental\nclass $[L]$ of $L$ in $H_{1}(L; \\mathbb{R})$. Then the stable commutator length of $g$ is $1/4$\ntimes the infimum of the (relative) Gromov norm on the subspace $V$ .\nTaking $L$ to be a 1-manifold and $f: L \\to X$ to be a map for which\n$f_{*}[L] = 0$ in $H_{1}(X; \\mathbb{R})$ extends the definition of stable commutator length\nto all homologically trivial formal chains (a purely group-theoretic defini-\ntion is given in [Cal09b]).\nThe conjecture above may be broadened as follows.\nLet $G$ be an\narbitrary word-hyperbolic group and $f: L \\to X$ an arbitrary map from\nan oriented 1-manifold $L$ to $X$ with $f_{*}[L] = 0$ in $H_{1}(X; \\mathbb{R})$. Here are a\nfew related conjectures:\n(a) the stable commutator length of $L$ is rational;\n(b) the Gromov norm of any rational class in $H_{2}(X, L; \\mathbb{Q})$ is rational;\nand\n\n(c) the Gromov norm of any rational class in $H_{2}(X; \\mathbb{R})$ is rational.\nFor $X$ a closed surface of genus at least 2, the answer to (c) is positive,\nand for every $L$ the answer to (b) is positive for all rational $\\alpha \\in V$ mapping\nto the fundamental class of $[L]$ outside a compact interval.\n(3) This is known to be true for elements represented by curves supported in\nan essential proper subsurface $S' \\subset S$ (see [Cal09b]).\n(4) If $S$ is a compact oriented surface, possibly with boundary and every\ncomponent having negative Euler characteristic, the Gromov norm of the\nfundamental class of $S$ in $H_{2}(S, \\partial S; \\mathbb{R})$ is $-2\\chi(S)$.\nIf $f: L \\to X$ is as above, a map $F: S \\to X$ virtually bounds $f$ if there\nis a degree $n$ (oriented) covering map $\\pi: \\partial S \\to L$ for which $f\\pi = F$ on $\\partial S$\n(we allow the possibility that $L$ is empty). Such a map is extremal if it is\nan isometry for the Gromov norm. Extremal maps are $\\pi_{1}$-injective.\nA further conjecture: In each case of the conjecture in Remark (1),\nthere is an extremal map of a surface virtually bounding $L$ (or virtually\nrepresenting a given class in $H_{2}(X; \\mathbb{Q})$ if $L$ is empty).\nThis conjecture could be thought of as bearing on Gromov’s question\nof whether every one-ended hyperbolic group contains a closed surface\nsubgroup.\n(5) The questions above have a positive answer for free groups.\nSee e.g.,\n[Cal09b].\n\nReferences cited:\n- [Cal09b] Danny Calegari. scl, volume 20 of MSJ Memoirs. Mathematical Society of Japan, Tokyo, 2009. doi:10.1142/e018.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Stable commutator length is rational for free groups and for surface-group elements supported in a proper essential subsurface, but rationality for every element of a closed surface group's commutator subgroup remains open.\n\n**Verified partial progress.**\n\n- Calegari proves that stable commutator length on free groups is rational, piecewise rational linear, and algorithmically computable.\n- The surface-group statement holds for elements represented by curves supported in an essential proper subsurface.\n- The Gromov norm of rational classes has positive results in the closed-surface setting and outside compact intervals in relevant relative affine subspaces.\n- Fournier-Facio's 2025 work explicitly calls rationality on surface groups a long-standing problem.\n\n**Full solution or refutation.**\n\nNo proof of rationality for arbitrary elements of [pi_1(S),pi_1(S)] was located.\n\n**What remains.**\n\nExtend rational polyhedral or extremal-surface methods from free groups and proper-subsurface elements to every homologically trivial element of a closed surface group.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 2.31. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the surface-group conjecture, its hyperbolic-group extensions, and the proper-subsurface and free-group solved cases.\n- Danny Calegari, Stable commutator length is rational in free groups, Journal of the American Mathematical Society 22 (2009), 941-961. (primary): https://arxiv.org/abs/0802.1352\n Evidence used: Proves rationality, rational polyhedrality, and computability for free groups.\n- Danny Calegari, scl, MSJ Memoirs 20, Mathematical Society of Japan, 2009. (primary): https://doi.org/10.1142/e018\n Evidence used: Develops the relative Gromov norm formulation and records the proper-subsurface surface-group case.\n- Francesco Fournier-Facio, Stable commutator length on free Q-groups, arXiv:2507.14009v3 (2025), to appear in Bulletin of the London Mathematical Society. (primary): https://arxiv.org/abs/2507.14009\n Evidence used: Provides a current primary-source status check by explicitly describing surface-group rationality as long-standing and linking it to new isometric embeddings.\n\n**Review notes.** Show that is an imperative conjectural formulation, not evidence that the statement is already a theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2780, "problem_number": "KP-2.32", "title": "Kirby Problem 2.32", "statement": "Does every surface bundle over a surface admit a flat connec-\ntion? What about surface bundles over 3-manifolds?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.32.\n\nLiterature notes:\n(1) An $S_{g}$-bundle $p: E \\to B$ admits a flat connection if it admits a codimen-\nsion 2 foliation everywhere transverse to the fibers. Equivalently, the bun-\ndle admits a flat connection if the monodromy map $\\rho: \\pi_{1}(B) \\to \\operatorname{Mod}(S_{g})$\nadmits a lifting $\\widetilde{\\rho}: \\pi_{1}(B) \\to \\operatorname{Diff}(S_{g})$.\n(2) Work of Morita [Mor87] shows that the MMM classes $e_{i} \\in H^{2i}(\\operatorname{Mod}(S_{g}))$\nobstruct the existence of a flat connection for $i \\geq 3$. In practical terms,\nthis means that there are surface bundles over manifolds of dimension at\nleast 6 that admit no flat connection. Conversely, the first MMM class is\nknown not to obstruct flatness; see [KM05]. New techniques, presumably\nof a more dynamical flavor, will be required to obstruct the existence of\nflat connections on surface bundles when the base is of lower dimension.\n\n(3) This question can be asked for varying degrees of regularity, all the way\nfrom lifting to the group of homeomorphisms, up to real-analytic or volume-\npreserving diffeomorphisms.\nIn the volume-preserving case, techniques from symplectic geome-\ntry may be relevant. This discussion follows ideas outlined in [KM05].\nLet $\\omega$ be a volume form on $S_{g}$, and let $\\operatorname{Diff}(S_{g}, \\omega)$ denote the group of\nvolume-preserving diffeomorphisms; the identity component is denoted\n$\\operatorname{Diff}_{0}(S_{g}, \\omega)$. An $S_{h}$-bundle over $S_{g}$ is determined by a set $a_{1}, b_{1},..., a_{h}, b_{h}$\nof mapping classes for which the surface relation $[a_{1}, b_{1}]... [a_{h}, b_{h}]$ holds.\nFrom this point of view, a flat volume-preserving connection consists of a\nchoice of lifts $\\alpha_{i}, \\beta_{i} \\in \\operatorname{Diff}(S_{g}, \\omega)$ such that $[\\alpha_{1}, \\beta_{1}]... [\\alpha_{h}, \\beta_{h}] = 1$.\nOne possible method for obstructing such a lift proceeds as follows.\nBeginning with a surface bundle specified by the mapping classes $a_{1}, b_{1},..., a_{h}, b_{h}$\nas above, the set of all lifts $\\alpha_{i}, \\beta_{i}$ is a torsor for the group $\\operatorname{Diff}_{0}(S_{g}, \\omega)^{2h}$.\nThere is a map (not a group homomorphism)\n\n$$\n\\mu: \\operatorname{Diff}_{0}(S_{g}, \\omega)^{2h} \\to \\operatorname{Diff}_{0}(S_{g}, \\omega)\n$$\n\nthat sends a set of lifts $\\alpha_{1}, \\beta_{1},..., \\alpha_{h}, \\beta_{h}$ to the surface word $\\prod_{i=1}^{h}[\\alpha_{i},\\beta_{i}]$;\nthe fiber $\\mu^{-1}(\\operatorname{id})$ then describes the set of flat volume-preserving connec-\ntions on the bundle.\n$\\operatorname{Diff}_{0}(S_{g}, \\omega)$ admits a flux homomorphism Flux: $\\operatorname{Diff}_{0}(S_{g}, \\omega) \\to H^{1}(S_{g}; \\mathbb{R})$\nwhose kernel is the group of Hamiltonian diffeomorphisms Ham $(S_{g}, \\omega)$. A\nfirst question is whether the flux homomorphism can serve as an obstruc-\ntion to admitting a flat connection, although this seems unlikely to be the\ncase.\nConsidering next the set of lifts for which $[\\alpha_{1}, \\beta_{1}]... [\\alpha_{h}, \\beta_{h}]$ is Hamil-\ntonian, one is led to wonder if the Hofer metric on Ham $(S_{g}, \\omega)$ could be\nof use in obstructing the existence of a flat connection. Perhaps it is pos-\nsible to show that the image of $\\mu$ must stay far away from 0 in the Hofer\nmetric? (See [MS17a] for an introduction to the Hofer metric.)\n\nReferences cited:\n- [Mor87] Shigeyuki Morita. Characteristic classes of surface bundles. Invent. Math., 90(3):551–577, 1987. doi:10.1007/BF01389178.\n- [KM05] D. Kotschick and S. Morita. Signatures of foliated surface bundles and the symplectomorphism groups of surfaces. Topology, 44(1):131–149, 2005. doi:10.1016/j.top.2004.05.002.\n- [MS17a] Dusa McDuff and Dietmar Salamon. Introduction to symplectic topology, volume 27. Oxford University Press, 2017.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Unrestricted flatness for closed surface bundles over surfaces or 3-manifolds remains open, but stabilization, punctured-fiber, symmetry-invariant, and high-dimensional obstruction results are known.\n\n**Verified partial progress.**\n\n- Kotschick and Morita prove that surface bundles over surfaces become flat after suitable stabilization and construct flat bundles with nonzero signature.\n- Bestvina, Church, and Souto give a surface point-pushing subgroup that cannot lift while fixing the marked point, producing a bundle-with-section obstruction.\n- The same authors show Atiyah-Kodaira bundles admit no flat connection invariant under a natural deck transformation, while explicitly retaining the unrestricted closed-fiber question.\n- Morita's higher MMM classes obstruct flat connections for some bases of dimension at least six; the first MMM class does not obstruct the low-dimensional cases.\n\n**Full solution or refutation.**\n\nNo closed surface bundle over a surface or 3-manifold conclusively lacking every flat connection was located, and no theorem that all such bundles are flat is known.\n\n**What remains.**\n\nSettle unrestricted liftability of monodromy for closed surface bundles over 2- and 3-dimensional bases, using obstructions beyond the known MMM-class range.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 2.32. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the surface- and 3-manifold-base questions and explains why existing characteristic classes only obstruct higher-dimensional bases.\n- Mladen Bestvina, Thomas Church, and Juan Souto, Some groups of mapping classes not realized by diffeomorphisms, Commentarii Mathematici Helvetici 88 (2013), 205-220. (primary): https://arxiv.org/abs/0905.2360\n Evidence used: Proves the point-pushing nonlifting theorem and deck-invariant Atiyah-Kodaira obstruction while explicitly leaving the unrestricted closed-bundle question open.\n- D. Kotschick and S. Morita, Signatures of foliated surface bundles and the symplectomorphism groups of surfaces, Topology 44 (2005), 131-149. (primary): https://arxiv.org/abs/math/0305182\n Evidence used: Constructs flat surface bundles with nonzero signature and supplies the stabilization and first-MMM-class context.\n- Shigeyuki Morita, Characteristic classes of surface bundles, Inventiones Mathematicae 90 (1987), 551-577. (primary): https://doi.org/10.1007/BF01389178\n Evidence used: Gives higher MMM-class obstructions to flatness over sufficiently high-dimensional bases.\n\n**Review notes.** The split word connec-tion is a layout artifact. Symmetry-invariant or section-preserving nonflatness is not promoted to unrestricted nonflatness.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2781, "problem_number": "KP-2.33", "title": "Kirby Problem 2.33", "statement": "Let $S$ be a closed compact surface (without boundary).\nIs\nthere a finitely generated, torsion-free group $G$ such that $G$ cannot act faithfully by\nhomeomorphisms on $S$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.33.\n\nLiterature notes:\n(1) There are non-compact surfaces for which every countable group acts by\nhomeomorphisms [APV21].\n(2) We pose the question for closed surfaces only, since the presence of bound-\nary gives access to special techniques that may be able to resolve the ques-\ntion as stated without developing new methods for the closed case. On the\nother hand, the question is open even for the disk, relative to the bound-\nary. Potential candidates include lattices in high rank Lie groups. For\nmore regular actions (i.e., smooth actions) more is known; see [BFH22],\nfor instance.\n\n(3) The torsion-free hypothesis is necessary, since there are often a priori\nbounds on the size of torsion in surface homeomorphism groups.\n(4) In the case of the plane, Le Roux [LR11] has studied actions of the\nfundamental group of the Klein bottle under the assumption that the\nstandard generators act freely and preserving orientation. Among other\nresults, he finds obstructions for the action of certain torsion-free groups\nby fixed-point free orientation-preserving homeomorphisms.\n(5) A more general question would be to find an algebraic characterization\nof groups acting on $S$ by homeomorphisms. Such characterizations are\nknown for groups acting on the interval and on the circle, via orderability.\nIt may be difficult to find concise characterizations; see e.g., [dlNG22,\nRos13, Hyd19].\n\nReferences cited:\n- [APV21] Tarik Aougab, Priyam Patel, and Nicholas G. Vlamis. Isometry groups of infinite genus hyperbolic surfaces. Math. Ann., 381:459–498, 2021.\n- [BFH22] Aaron Brown, David Fisher, and Sebastian Hurtado. Zimmer’s conjecture: subexponential growth, measure rigidity, and strong property (T). Ann. of Math. (2), 196(3):891–940, 2022. doi:10.4007/annals.2022.196.3.1.\n- [LR11] Frédéric Le Roux. Free planar actions of the Klein bottle group. Geom. Topol., 15(3):1545–1567, 2011. doi:10.2140/gt.2011.15.1545.\n- [dlNG22] Javier de la Nuez Gonzalez. Non-Roelcke precompactness of groups of surface homeomorphisms, 2022. arXiv:2202.06527.\n- [Ros13] Christian Rosendal. Global and local boundedness of Polish groups. Indiana Univ. Math. J., 62(5):1621–1678, 2013. doi:10.1512/iumj.2013.62.5133.\n- [Hyd19] James Hyde. The group of boundary fixing homeomorphisms of the disc is not leftorderable. Ann. of Math. (2), 190(2):657–661, 2019. doi:10.4007/annals.2019.190.2.5.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No finitely generated torsion-free group known to be incapable of every faithful homeomorphism action on a specified closed surface was located; smooth-action rigidity does not settle the C^0 problem.\n\n**Verified partial progress.**\n\n- Higher-rank lattices are natural candidates, and Brown-Fisher-Hurtado prove powerful rigidity for sufficiently regular low-dimensional actions.\n- Torsion gives elementary obstructions to surface actions, explaining the deliberate torsion-free hypothesis.\n- Le Roux obtains obstructions for certain fixed-point-free orientation-preserving planar actions of torsion-free groups.\n- Orderability supplies algebraic characterizations in dimension one, but no comparable characterization is known for closed surfaces.\n\n**Full solution or refutation.**\n\nThe exact faithful-homeomorphism-action question remains open in K3, even in the related disk-relative-boundary setting.\n\n**What remains.**\n\nProduce a finitely generated torsion-free nonacting group or prove a universal embedding theorem, and bridge the gap between smooth Zimmer-type rigidity and arbitrary C^0 surface actions.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 2.33. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the closed-surface problem as open, explains the torsion-free hypothesis, and distinguishes the smoother-action literature.\n- Aaron Brown, David Fisher, and Sebastian Hurtado, Zimmer's conjecture: subexponential growth, measure rigidity, and strong property (T), Annals of Mathematics 196 (2022), 891-940. (primary): https://doi.org/10.4007/annals.2022.196.3.1\n Evidence used: Provides the cited rigidity for smooth actions of higher-rank lattices, a candidate direction that does not cover arbitrary homeomorphisms.\n- Frédéric Le Roux, Free planar actions of the Klein bottle group, Geometry & Topology 15 (2011), 1545-1567. (primary): https://doi.org/10.2140/gt.2011.15.1545\n Evidence used: Gives concrete obstructions for a restricted class of planar torsion-free group actions.\n- James Hyde, The group of boundary fixing homeomorphisms of the disc is not left-orderable, Annals of Mathematics 190 (2019), 657-661. (primary): https://doi.org/10.4007/annals.2019.190.2.5\n Evidence used: Shows why one-dimensional orderability characterizations do not transfer naively to surface homeomorphism groups.\n\n**Review notes.** Closed compact is redundant but not corrupt; without boundary confirms the intended scope. Smooth-action nonexistence is not treated as a C0 solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2782, "problem_number": "KP-2.34", "title": "Kirby Problem 2.34", "statement": "Let $S$ be a compact surface. For $0 \\leq r < s$, does there exist a\nnontrivial finitely generated subgroup $G_{r} \\leq \\operatorname{Diff}^{r}_{0}(S)$ such that every action of $G_{r}$\non $S$ by $C^{s}$ diffeomorphisms is trivial? Can these examples be torsion-free?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.34.\n\nLiterature notes:\n(1) For $0 \\leq r \\leq \\infty$, let $\\operatorname{Diff}^{r}_{0}(M)$ denote the identity component of the group\nof $C^{r}$ diffeomorphisms of a manifold $M$.\nFor nonintegral $r$, we write\n$r = k + \\epsilon$ for $k \\in \\mathbb{N}$ and $0 \\leq \\epsilon < 1$, and we consider $C^{k}$ diffeomorphisms\nwhose $k^{th}$ order derivatives are $\\epsilon$–Hölder continuous.\n(2) The condition that every higher-regularity action of $G_{r}$ be trivial can be\nrelaxed, for instance, to ask that every such action of $G_{r}$ be non-faithful,\nor abelian.\n(3) Problems of this nature have been extensively investigated in dimen-\nsion one, where Kim and Koberda gave a positive answer to this ques-\ntion [KK20, KK21]; one obtains finitely generated groups with infinite\nsimple commutator subgroups such that each smoother action has abelian\nimage. No instances are known in dimension greater than one. Prior to\nthis, Plante–Thurston showed that nilpotent groups of $C^{2}$-diffeomorphisms\nof the interval are abelian [PT76], but Farb–Franks showed that there can\nbe non-abelian nilpotent actions of regularity $C^{1}$ [FF03]. More generally,\nCastro–Jorquera–Navas build nilpotent groups that can act at a certain\nregularity but whose higher-regularity actions are all abelian [CJN14].\nFurther examples of finitely generated groups acting with critical regular-\nities on 1-manifolds are constructed in [MW23].\n(4) In the case $r = 0$ and $s \\geq 1$, it is possible to use Thurston’s Stability\nTheorem [Thu74b] to exhibit particular groups of homeomorphisms that\ncannot act faithfully by diffeomorphisms. See, for instance, [Cal06a].\n\nReferences cited:\n- [KK20] Sang-hyun Kim and Thomas Koberda. Diffeomorphism groups of critical regularity. Invent. Math., 221(2):421–501, 2020. doi:10.1007/s00222-020-00953-y.\n- [KK21] Sang-hyun Kim and Thomas Koberda. Structure and regularity of group actions on one-manifolds. Springer Monographs in Mathematics. Springer, Cham, [2021] ©2021. doi:10.1007/978-3-030-89006-3.\n- [PT76] J. F. Plante and W. P. Thurston. Polynomial growth in holonomy groups of foliations. Comment. Math. Helv., 51(4):567–584, 1976. doi:10.1007/BF02568174.\n- [FF03] Benson Farb and John Franks. Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups. Ergodic Theory Dynam. Systems, 23(5):1467–1484, 2003. doi: 10.1017/S0143385702001712.\n- [CJN14] Gonzalo Castro, Eduardo Jorquera, and Andrés Navas. Sharp regularity for certain nilpotent group actions on the interval. Math. Ann., 359(1-2):101–152, 2014. doi: 10.1007/s00208-013-0995-1.\n- [MW23] Kathryn Mann and Maxime Wolff. Reconstructing maps out of groups. Ann. Sci. Éc. Norm. Supér. (4), 56(4):1135–1154, 2023. doi:10.24033/asens.2551.\n- [Thu74b] William P. Thurston. A generalization of the Reeb stability theorem. Topology, 13:347–352, 1974. doi:10.1016/0040-9383(74)90025-1.\n- [Cal06a] Danny Calegari. Dynamical forcing of circular groups. Trans. Amer. Math. Soc., 358(8):3473–3491, 2006. doi:10.1090/S0002-9947-05-03754-2.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Critical-regularity separation is well developed for one-manifolds and weak C0-to-smooth obstructions exist on surfaces, but the exact two-dimensional trivial-image and torsion-free construction remains open.\n\n**Verified partial progress.**\n\n- Kim and Koberda construct finitely generated groups at prescribed one-dimensional critical regularities with no faithful smoother action and with every smoother image abelian in their principal families.\n- Plante-Thurston, Farb-Franks, and Castro-Jorquera-Navas sharply separate regularities for nilpotent one-manifold actions.\n- For r=0 and s at least 1, Thurston stability yields particular surface-homeomorphism groups that cannot act faithfully by diffeomorphisms.\n- No examples satisfying the full higher-dimensional surface requirement are recorded by K3.\n\n**Full solution or refutation.**\n\nThe one-dimensional analogue and weaker nonfaithfulness/abelian-image versions are known; the exact requirement that every C^s action on a compact surface be trivial, with torsion-free examples, is not solved.\n\n**What remains.**\n\nConstruct torsion-free finitely generated G_r inside Diff_0^r(S) with every C^s surface action trivial, or prove such regularity gaps impossible in dimension two.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 2.34. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the exact surface and torsion-free questions, surveys one-dimensional solutions, and says no higher-dimensional instances are known.\n- Sang-hyun Kim and Thomas Koberda, Diffeomorphism groups of critical regularity, Inventiones Mathematicae 221 (2020), 421-501. (primary): https://arxiv.org/abs/1711.05589\n Evidence used: Constructs prescribed critical-regularity groups on the interval and circle with strong obstructions to smoother representations.\n- Gonzalo Castro, Eduardo Jorquera, and Andrés Navas, Sharp regularity for certain nilpotent group actions on the interval, Mathematische Annalen 359 (2014), 101-152. (primary): https://doi.org/10.1007/s00208-013-0995-1\n Evidence used: Provides sharp nilpotent regularity gaps in the one-dimensional model case.\n- William P. Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974), 347-352. (primary): https://doi.org/10.1016/0040-9383(74)90025-1\n Evidence used: Supplies the stability theorem behind the weaker r=0 nonfaithfulness constructions.\n\n**Review notes.** The exact problem asks for trivial smoother actions, which is stronger than the nonfaithful or abelian-image conclusions in most cited partial results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2783, "problem_number": "KP-2.35", "title": "Kirby Problem 2.35", "statement": "Is the first-order theory of the mapping class group of a surface\ndecidable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.35.\n\nLiterature notes:\n(1) The first order theory of a group $G$ in the language of group theory is the\nformal first-order language equipped with group multiplication and the\nidentity element. A first-order theory is decidable if there is an algorithm\n(i.e., Turing machine) that decides which sentences are true in that theory\nand which are false.\n(2) For homeomorphism groups of surfaces (and indeed all compact manifolds\nof positive dimension), the corresponding theory is not decidable because\nit interprets arithmetic [KKdlNG25].\n(3) The theory of free groups is decidable; this is part of the resolution of the\nTarski problem [Sel02, KM06].\n\nReferences cited:\n- [KKdlNG25] Sang-hyun Kim, Thomas Koberda, and J. de la Nuez González. First order rigidity of homeomorphism groups of manifolds. Commun. Am. Math. Soc., 5:144–194, 2025. doi:10.1090/cams/47.\n- [Sel02] Z. Sela. Diophantine geometry over groups and the elementary theory of free and hyperbolic groups. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 87–92. Higher Ed. Press, Beijing, 2002.\n- [KM06] Olga Kharlampovich and Alexei Myasnikov. Elementary theory of free non-abelian groups. J. Algebra, 302(2):451–552, 2006. doi:10.1016/j.jalgebra.2006.03.033.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No decidability or undecidability theorem for the pure first-order group theory of a general finite-type mapping class group was located.\n\n**Verified partial progress.**\n\n- Kim, Koberda, and de la Nuez González show that homeomorphism groups of compact positive-dimensional manifolds interpret arithmetic and have undecidable first-order theory.\n- DiSarlo, Koberda, and de la Nuez González prove omega-stability, rank results, and restricted quantifier elimination for the curve graph.\n- The curve graph is bi-interpretable with an augmented mapping-class-group Cayley structure, not with the mapping class group in the bare group language required here.\n- Free groups have decidable elementary theory, furnishing a low-complexity comparison rather than a solution in general.\n\n**Full solution or refutation.**\n\nThe neighboring homeomorphism-group and curve-graph model theories do not settle decidability of mapping class groups as abstract groups.\n\n**What remains.**\n\nGive an algorithm deciding all first-order group sentences for each relevant mapping class group or encode an undecidable theory; clarify whether the surface is fixed or variable.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 2.35. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Presents the mapping-class-group question as open and contrasts it with undecidable homeomorphism-group theory and decidable free-group theory.\n- Sang-hyun Kim, Thomas Koberda, and J. de la Nuez González, First order rigidity of homeomorphism groups of manifolds, Communications of the American Mathematical Society 5 (2025), 144-194. (primary): https://arxiv.org/abs/2302.01481\n Evidence used: Establishes arithmetic interpretation and first-order rigidity for full homeomorphism groups, a distinct group from the mapping class quotient.\n- Valentina DiSarlo, Thomas Koberda, and J. de la Nuez González, The model theory of the curve graph, arXiv:2008.10490v3 (2023). (primary): https://arxiv.org/abs/2008.10490\n Evidence used: Proves strong model-theoretic results for the curve graph and an augmented Cayley structure without deciding the bare group theory of Mod(S).\n\n**Review notes.** Material scope ambiguity: the surface is not specified, and fixed low-complexity groups may differ from the intended general finite-type problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2784, "problem_number": "KP-2.36", "title": "Kirby Problem 2.36", "statement": "Are systems of equations over mapping class groups and braid\ngroups decidable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.36.\n\nLiterature notes:\n(1) A system of equations over a group $G$ is a finite conjunction of formulae\nof the form\n\n$$\nr(x_{1},..., x_{n}, a_{1},..., a_{m}) = 1\n$$\n\nand\n\n$$\nr(x_{1},..., x_{n}, a_{1},..., a_{m}) \\neq 1,\n$$\n\nwhere $r$ is a word in a free group of $\\operatorname{rank} n + m$, where $(x_{1},..., x_{n})$ is\na tuple of variables, and where $(a_{1},..., a_{m})$ (with $m \\geq 0$) is a tuple of\nparameters in $G$. A solution to a system of equations is a tuple of elements\nof $G$ such that when the elements are substituted for variables, one obtains\na valid expression in $G$.\n(2) Decidability of equations would consist of an algorithm to decide whether\nor not a particular system has a solution or not. Precisely, one might fix\na finite generating set for $G$ (if it exists) and use a Gödel numbering to\nencode equations with parameters as natural numbers. Then, one would\nrequire a Turing machine to take a natural number corresponding to an\nequation and decide if it admits a solution.\n(3) One could allow variations on the problem; for instance, one could fix the\nunderlying surface or allow it to vary, pass to finite index subgroups, or\nrestrict the topological type of the underlying surface.\n(4) Decidability of the theory of a group or class of groups has a long his-\ntory, starting at least with the Tarski problem. The Tarski problem asked\nwhether the nonabelian free groups are elementarily equivalent to each\nother and whether their first order theories are decidable; see [Sel02,\nKM06]. For general hyperbolic groups (possibly with torsion), Dahmani\n\nand Guirardel proved that equations are solvable; see [DG10]. Conjec-\nturally, the full first order theory of a vast generalization of hyperbolic\ngroups, namely hierarchically hyperbolic groups, should be decidable. A\nparticularly interesting special case of such a general result would be the\ndecidability of the first order theory of mapping class groups and braid\ngroups, which in turn would give a positive answer to the proposed ques-\ntion.\n(5) There is a vast literature on equations over groups, including finite groups\nand free groups, though for mapping class groups and braid groups the\nproblem appears to be open. See [Rom12] for a survey.\n(6) It seems likely that equations over homeomorphism groups of surfaces\n(with or without parameters) are undecidable.\n\nReferences cited:\n- [Sel02] Z. Sela. Diophantine geometry over groups and the elementary theory of free and hyperbolic groups. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 87–92. Higher Ed. Press, Beijing, 2002.\n- [KM06] Olga Kharlampovich and Alexei Myasnikov. Elementary theory of free non-abelian groups. J. Algebra, 302(2):451–552, 2006. doi:10.1016/j.jalgebra.2006.03.033.\n- [DG10] François Dahmani and Vincent Guirardel. Foliations for solving equations in groups: free, virtually free, and hyperbolic groups. J. Topol., 3(2):343–404, 2010. doi:10.1112/jtopol/jtq010.\n- [Rom12] Vitaliı̆ Roman’kov. Equations over groups. Groups Complex. Cryptol., 4(2):191– 239, 2012. doi:10.1515/gcc-2012-0015.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The existential theory with equations, inequations, and parameters remains open for mapping class groups and braid groups in the current K3 source.\n\n**Verified partial progress.**\n\n- Dahmani and Guirardel give algorithms for equations over hyperbolic groups, but mapping class and braid groups are not covered directly.\n- Free groups have decidable elementary and existential theories.\n- Word and conjugacy problems are decidable in braid groups, but these are strictly narrower than arbitrary finite systems of equations and inequations with parameters.\n- A conjectural extension of model-theoretic methods to hierarchically hyperbolic groups would include important mapping class group cases, but no such general theorem is currently available.\n\n**Full solution or refutation.**\n\nNo algorithm or undecidability reduction for the full systems specified in the record was located.\n\n**What remains.**\n\nDecide the existential theory with parameters for fixed mapping class and braid groups and determine whether a uniform algorithm exists as surface complexity or braid index varies.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 2.36. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Defines the precise equation-and-inequation problem with parameters and explicitly says it appears open for mapping class and braid groups.\n- François Dahmani and Vincent Guirardel, Foliations for solving equations in groups: free, virtually free, and hyperbolic groups, Journal of Topology 3 (2010), 343-404. (primary): https://doi.org/10.1112/jtopol/jtq010\n Evidence used: Proves the positive algorithmic result for hyperbolic groups used as the closest broad comparison.\n- Olga Kharlampovich and Alexei Myasnikov, Elementary theory of free non-abelian groups, Journal of Algebra 302 (2006), 451-552. (primary): https://doi.org/10.1016/j.jalgebra.2006.03.033\n Evidence used: Supplies the solved free-group comparison and does not extend directly to mapping class or braid groups.\n\n**Review notes.** The plural wording does not specify whether the surface or braid index is fixed or part of the input; this uniformity issue should be clarified.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2785, "problem_number": "KP-2.37", "title": "Kirby Problem 2.37", "statement": "Give a Nielsen–Thurston-type classification for the mapping\nclass groups of infinite-type surfaces. In particular, which homeomorphisms are the\nappropriate generalization of pseudo-Anosov homeomorphisms in this setting?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.37.\n\nLiterature notes:\n(1) For finite-type surfaces $S$, there is a powerful classification due to Nielsen\nand Thurston of the mapping classes of $S$: every mapping class is periodic,\nreducible, or pseudo-Anosov. The most important aspect of this classifi-\ncation is that if a mapping class $f$ is not periodic or reducible, it must be\npseudo-Anosov, meaning that $f$ is highly chaotic and that there exist two\ntransverse measured foliations $\\lambda^{+}$ and $\\lambda^{-}$ on $S$, where $f$ stretches along\n$\\lambda^{+}$ and contracts along $\\lambda^{-}$. The exact analog of this classification for\n\ninfinite-type surfaces is false. For instance, the homeomorphisms called\nhandleshifts (introduced by Patel-Vlamis [PV18]) are neither periodic,\nreducible, nor pseudo-Anosov. The classification, therefore, needs to be\nmodified in the infinite-type setting. First, the definition of a reducible\nmapping class should be generalized, but more importantly, the third cate-\ngory (the analog of pseudo-Anosov mapping classes) needs to be expanded\nand likely broken into subcategories.\n(2) Some work has been announced on this problem by Bestvina, Fanoni and\nTao [BFT23], who classify so-called “tame” homeomorphisms that satisfy\nan additional finiteness condition. Such homeomorphisms do not exhibit\nany pseudo-Anosov-like behavior.\n(3) Irreducible end-periodic mapping classes should form an important class\nwithin the third category. For end-periodic homeomorphisms of infinite-\ntype surfaces, Handel and Mosher outlined a theory paralleling that of\npseudo-Anosov homeomorphisms in unpublished work from the 1980s,\ndescribing a pair of transverse geodesic laminations on the surface pre-\nserved by the end-periodic mapping class. This theory was further de-\nveloped by Cantwell, Conlon, and Fenley [CCF21]. In addition to the\nabove, irreducible end-periodic mapping classes share many properties\nwith finite-type pseudo-Anosov mapping classes; for example, they give\nrise to hyperbolic mapping tori [FKLL23] and many strongly irreducible\nones are loxodromic isometries of complexes of curves and arcs [PT25].\n\nReferences cited:\n- [PV18] Priyam Patel and Nicholas G. Vlamis. Algebraic and topological properties of big mapping class groups. Algebr. Geom. Topol., 18(7):4109–4142, 2018. doi:10.2140/agt.2018.18.4109.\n- [BFT23] Mladen Bestvina, Federica Fanoni, and Jing Tao. Towards Nielsen-Thurston classification for surfaces of infinite type, 2023. arXiv:2303.12413.\n- [CCF21] John Cantwell, Lawrence Conlon, and Sergio R. Fenley. Endperiodic automorphisms of surfaces and foliations. Ergodic Theory Dynam. Systems, 41(1):66–212, 2021. doi:10.1017/etds.2019.56.\n- [FKLL23] Elizabeth Field, Heejoung Kim, Christopher Leininger, and Marissa Loving. Endperiodic homeomorphisms and volumes of mapping tori. J. Topol., 16(1):57–105, 2023. doi:10.1112/topo.12277.\n- [PT25] Priyam Patel and Samuel J. Taylor. Constructing endperiodic loxodromics of infinite-type arc graphs. Math. Z., 310(4):Paper No. 82, 2025. doi:10.1007/s00209-025-03784-w.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Canonical Nielsen-Thurston-type decompositions now exist for a tempered finite-accumulation subclass and rich pseudo-Anosov-like theory exists for end-periodic maps, but arbitrary infinite-type mapping classes are not classified.\n\n**Verified partial progress.**\n\n- Bestvina, Fanoni, and Tao classify tempered mapping classes satisfying an additional finiteness condition by a canonical invariant-subsurface decomposition with periodic or translation return maps.\n- The Bestvina-Fanoni-Tao paper was revised on 12 August 2026 and continues to state the restricted finite-accumulation scope.\n- Cantwell-Conlon-Fenley develop Handel-Miller laminations for end-periodic maps.\n- Atoroidal end-periodic maps have hyperbolic mapping-torus models and many strongly irreducible examples act loxodromically on infinite-type arc or curve complexes.\n\n**Full solution or refutation.**\n\nTwo major regimes—tempered finite-accumulation maps and irreducible end-periodic maps—have substantial classification/dynamical theory, but no exhaustive replacement for the finite-type trichotomy is known.\n\n**What remains.**\n\nDefine the correct reducibility and pseudo-Anosov-like categories and prove a canonical exhaustive classification for all mapping classes of all infinite-type surfaces.\n\n**Sources checked.**\n\n- R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 2.37. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Explains failure of the classical trichotomy, identifies the restricted tame/tempered and end-periodic advances, and retains the general classification request.\n- Mladen Bestvina, Federica Fanoni, and Jing Tao, Towards Nielsen-Thurston classification for surfaces of infinite type: well-tempered homeomorphisms, arXiv:2303.12413v3 (revised 12 August 2026). (primary): https://arxiv.org/abs/2303.12413\n Evidence used: Proves a canonical decomposition theorem for tempered maps under an additional finiteness condition; the current revision confirms the result remains subclass-specific.\n- Michael P. Landry, Yair N. Minsky, and Samuel J. Taylor, Endperiodic maps via pseudo-Anosov flows, arXiv:2304.10620. (primary): https://arxiv.org/abs/2304.10620\n Evidence used: Realizes every atoroidal end-periodic map through depth-one foliations in fibered hyperbolic 3-manifolds and develops pseudo-Anosov-like laminations and entropy behavior.\n- John Cantwell, Lawrence Conlon, and Sergio R. Fenley, Endperiodic automorphisms of surfaces and foliations, Ergodic Theory and Dynamical Systems 41 (2021), 66-212. (primary): https://doi.org/10.1017/etds.2019.56\n Evidence used: Develops the Handel-Miller invariant lamination theory for end-periodic automorphisms.\n\n**Review notes.** The source background says tame; the current primary paper uses tempered/well-tempered terminology. This terminology update is flagged rather than silently altering the stored statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2786, "problem_number": "KP-2.38", "title": "Kirby Problem 2.38", "statement": "Give an appropriate analogue of the curve graph for infinite-\ntype surfaces, and characterize the surfaces for which no such graph exists.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.38.\n\nLiterature notes:\n(1) A motivating philosophy in geometric group theory is that one can study\nthe algebra of a group $G$ via the geometry of the spaces on which $G$ acts\nand the dynamics of such actions. When $G$ is the mapping class group of\na finite-type surface, one of the most fruitful actions to study is that of\nthe mapping class group of $S$ on the curve graph, $\\mathcal{C}(S)$, associated to $S$.\nPart of the reason that this action is so dynamically rich is due to a\nfamous result of Masur and Minsky [MM00], which shows that the curve\ngraph is infinite-diameter and Gromov-hyperbolic: there exists $\\delta > 0$\nso that, for all geodesic triangles in $\\mathcal{C}(S)$, each side of the triangle is\ncontained in the $\\delta$-neighborhood of the other two. This means that $\\mathcal{C}(S)$\nshares properties with hyperbolic space so that some of the key tools in\nhyperbolic geometry can be used to study $\\mathcal{C}(S)$ and groups acting on it.\nThe curve graph has played a pivotal role not only in the study of mapping\nclass groups, but also Teichmüller theory, Kleinian groups, and the moduli\nspace of Riemann surfaces.\n\nBy contrast, for infinite-type surfaces the curve graph has finite di-\nameter. This makes it much harder to study the above topics for infinite-\ntype surfaces via the action of groups on this graph. An analogue of the\ncurve graph would be an infinite-diameter hyperbolic graph associated to\nan infinite-type surface $S$ on which the mapping class group of $S$ acts\ncoboundedly and by isometries.\n(2) Vlamis showed that there are infinite-type surfaces whose mapping class\ngroups do not act on any metric space with unbounded orbits [Vla24b];\nadditional examples were announced in [Vla24a]. Thus, it is not possible\nto find an analogue of the curve graph that can be associated to any\ninfinite-type surface.\n(3) Among infinite-type surfaces whose mapping class groups do admit un-\nbounded actions on metric spaces, there are several associated graphs that\nhave been fruitful to study thus far, for instance the ray graph (Calegari\n[Cal09a]), the relative arc graph (Aramayona–Fossas–Parlier [AFP17]),\nthe omnipresent arc graph (Fanoni–Ghaswala–McLeay [FGM21]), and\nthe grand arc graph (Bar-Natan–Verberne [BNV23]).\nUnfortunately,\nthis is a piecemeal approach, since these graphs are not defined for all\ninfinite-type surfaces in this class, and may fail to be infinite-diameter\nand hyperbolic in general.\nIn light of this, one wishes to first classify\nwhich infinite-type surfaces admit an analogue of the curve graph, and to\nthen find a universal graph that can be associated to any such infinite-type\nsurface that is both infinite-diameter and hyperbolic.\n(4) This problem appeared on an AIM problem list for a workshop on surfaces\nof infinite type [LPRT] and was proposed for inclusion there by P. Patel\nand K. Mann.\n\nReferences cited:\n- [MM00] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. II. Hierarchical structure. Geom. Funct. Anal., 10(4):902–974, 2000.\n- [Vla24b] Nicholas G. Vlamis. Homeomorphism groups of self-similar 2-manifolds. In In the tradition of Thurston III. Geometry and dynamics, pages 105–167. Springer, Cham,\n- [Vla24a] Nicholas G. Vlamis. Homeomorphism groups of telescoping 2–manifolds are strongly distorted, 2024. arXiv:2403.03887.\n- [Cal09a] Danny Calegari. Big mapping class groups and dynamics. Geometry and the imagination, https://lamington.wordpress.com/2009/06/22/big-mapping-class-groups-and-dynamics/, 2009.\n- [AFP17] Javier Aramayona, Ariadna Fossas, and Hugo Parlier. Arc and curve graphs for infinite-type surfaces. Proceedings of the American Mathematical Society, 145(11):4995–5006, 2017.\n- [FGM21] Federica Fanoni, Tyrone Ghaswala, and Alan McLeay. Homeomorphic subsurfaces and the omnipresent arcs. Ann. H. Lebesgue, 4:1565–1593, 2021. doi:10.5802/ahl.110.\n- [BNV23] Assaf Bar-Natan and Yvon Verberne. The grand arc graph. Math. Z., 305(2):Paper No. 20, 21, 2023. doi:10.1007/s00209-023-03337-z.\n- [LPRT] Justin Lanier, Priyam Patel, Anja Randecker, and Jing Tao. AIM Problem List: Surfaces of infinite type. available at http://aimpl.org/genusinfinity.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Several curve-graph analogues and hyperbolic actions have been constructed for classes of infinite-type surfaces, but no general appropriate analogue/obstruction classification is known.\n\n**Verified partial progress.**\n\n- The finite-type curve graph remains the motivating hyperbolic model; big-surface graph constructions give partial replacements.\n\n**Full solution or refutation.**\n\nThe requested universal characterization remains open.\n\n**What remains.**\n\nClassify surfaces admitting a useful hyperbolic curve-graph analogue.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.38 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintains the analogue and classification problem with existing constructions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2787, "problem_number": "KP-2.39", "title": "Kirby Problem 2.39", "statement": "(a) Does the mapping class group of an infinite-genus surface with no planar\nends contain every countable group?\n(b) Does the mapping class group of the one-ended infinite-genus surface have\nany proper finite-index subgroups?\n(c) For an infinite-type surface, describe and characterize the subgroup of\nmapping classes with quasi-conformal representatives. In particular, when\nis it a normal subgroup of the mapping class group?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.39.\n\nLiterature notes:\n(1) The work of Aougab, Patel, and Vlamis [APV21] shows every mapping\nclass group of an infinite-genus surface with no planar ends contains every\nfinite group as a subgroup. Thus the problem is really about countably\ninfinite groups. The same work of Aougab, Patel, and Vlamis also shows\nthat the above question is true for infinite-genus surfaces with no planar\nends and self-similar end space, e.g., the Loch Ness monster surface. To do\n\nthis, they realize every countable group as the isometry group of a hyper-\nbolic structure, and moreover, show that the above surfaces are the only\nones in which this is possible. Therefore, to answer the above question,\nit is necessary to construct countably infinite subgroups of the mapping\nclass group that are not subgroups of the isometry group of the surface.\n(2) It was shown by G. Domat [Dom22] that the mapping class group of\nthe one-ended infinite genus surface has nontrivial abelianization, and\nthe abelianization has many copies of $\\mathbb{Q}$ as direct summands. It follows\nthat the mapping class group of the Loch Ness monster surface has many\nproper subgroups of countably infinite index. In the same work, Domat\nalso shows that the abelianization does contain torsion subgroups, though\nthey are not known to be direct summands.\n(3) When $S$ is a Riemann of finite type, it is a basic fact that every mapping\nclass admits a representative as a quasi-conformal homeomorphism, but\nthis is no longer true if $S$ is of infinite type. In this setting, the group\nof mapping classes with quasi-conformal representatives is sometimes re-\nferred to as the Teichmüller modular group; the reader should be aware\nthat this term is sometimes used to refer to the entire mapping class\ngroup in the finite-type setting. While this usage is consistent, there is\nthe possibility for confusion.\n(4) This problem appeared on a problem list compiled at the 2021 Nearly\nCarbon Neutral Geometric Topology conference [CPV21].\n\nReferences cited:\n- [APV21] Tarik Aougab, Priyam Patel, and Nicholas G. Vlamis. Isometry groups of infinite genus hyperbolic surfaces. Math. Ann., 381:459–498, 2021.\n- [Dom22] George Domat. Big pure mapping class groups are never perfect. Math. Res. Lett., 29(3):691–726, 2022. Appendix with Ryan Dickmann.\n- [CPV21] Yassin Chandran, Priyam Patel, and Nicholas G. Vlamis. Infinite-type surfaces and mapping class groups: Open problems. Available at https://https://www.patelp.com/uploads/2/5/7/9/25792573/inftypeproblems.pdf, 2021.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every countable group embeds for self-similar infinite-genus end spaces, and one-ended infinite-genus mapping class groups have nontrivial abelianization; the full multi-part classification remains open.\n\n**Verified partial progress.**\n\n- Aougab--Patel--Vlamis prove countable-group embedding for self-similar examples.\n- Domat proves nontrivial abelianization in the one-ended infinite-genus case.\n\n**Full solution or refutation.**\n\nNo general answers for all end spaces or quasiconformal subgroups were verified.\n\n**What remains.**\n\nResolve the remaining end-space, finite-index, and quasiconformal-normality cases.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.39 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records these partial theorems and remaining questions.\n\n**Review notes.** Multi-part record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2788, "problem_number": "KP-2.40", "title": "Kirby Problem 2.40", "statement": "(a) Let $S$ be an infinite-type surface and $\\varphi$ a mapping class for which there is\na (marked) conformal structure $\\Sigma$ on $S$ with respect to which $\\varphi$ is realized\nas a quasi-conformal homeomorphism.\nThen for each quasi-conformal\nstructure in the connected component of Teichmüller space containing $\\Sigma$,\nthere is a quasiconformal homeomorphism representing $\\varphi$ which minimizes\nthe dilatation. If the infimum of the minimal dilatations for each structure\nin this component is achieved and is greater than 1, is the minimizing\nquasiconformal homeomorphism unique? What is the structure of such a\nminimizer?\n(b) If $\\varphi$ is parabolic (see Remark (1) below) for $T$, must a sequence of con-\nformal structures with dilatation converging to the infimum degenerate by\npinching along some sequence of essential simple closed curves? Is this\nsystem unique? Can it depend on the component $T$?\n(c) If $\\varphi$ fixes more than one component $T, T'$ of $\\operatorname{Teich}(S)$, can the type be dif-\nferent for $T$ and $T'$? In particular, can $\\varphi$ be parabolic for $T$ and hyperbolic\nfor $T'$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.40.\n\nLiterature notes:\n(1) Let $S$ be a surface of infinite type, and let $\\varphi$ be a mapping class for\nwhich there is some (marked) conformal structure $\\Sigma$ on $S$ for which $\\varphi$\n\nis realized by a quasiconformal homeomorphism $f: \\Sigma \\to \\Sigma$. Then there\nis some representative homeomorphism which realizes the infimum of the\ndilatation. This follows from compactness (for any finite $K$) of the space\nof $K$-quasiconformal homeomorphisms of the unit disk, normalized to fix\n3 points on the boundary (see, e.g., [Ahl06]).\nWe may then try to vary the conformal structure on $S$ in the com-\nponent $T \\subset \\operatorname{Teich}(S)$ containing $\\Sigma$, and for each point in $T$ we get a\nminimum value for the dilatation of a homeomorphism representing $\\varphi$,\nand thereby a function $K_{T}: T \\to [1, \\infty)$. Note $K_{T}$ takes the value 1 if\nand only if $\\varphi$ is represented by an isometry for some conformal structure\nin the component $T$.\nSay $\\varphi$ is elliptic for $T$ if $K_{T}$ achieves the value 1, parabolic for $T$ if the\ninfimum is not achieved, and hyperbolic for $T$ if the infimum is achieved\nand is bigger than 1. Part (a) can be restated as follows: if $\\varphi$ is hyperbolic\nfor $T$, is the minimizer unique? What is its structure?\n(2) When $\\varphi$ is a mapping class on a surface $S$ of finite type, there is a (marked)\nconformal structure on $S$ with respect to which $\\varphi$ can be realized as a qua-\nsiconformal homeomorphism. Moreover, there is a unique quasiconformal\nhomeomorphism representing $\\varphi$ that minimizes the dilatation; this is the\nTeichmüller map. Note that in the finite-type setting, Teichmüller space\nhas a single connected component.\n(3) There are elementary examples of topological surfaces of infinite type $S$\nand mapping classes $\\varphi$ that are not realized by quasiconformal maps for\nany choice of (marked) conformal structure on $S$. Perhaps the simplest\nexample is: let $S$ be the infinite ladder surface, which we think of as the\ninfinite union of twice-punctured tori $T_{n}$ indexed by integers $n \\in \\mathbb{Z}$ joined\nend to end, let $\\alpha$ be any pseudo-Anosov diffeomorphism of a single twice-\npunctured torus, and let $\\varphi$ be the mapping class that does $\\alpha^{n}$ on $T_{n}$. No\nmatter what conformal structure we choose on $S$, the restriction of $\\alpha^{n}$ to\n$T_{n}$ has dilatation at least as big as $|n|$ times the minimum dilatation of $\\alpha$\non a twice-punctured torus. This example is (highly) reducible; it would\nbe nice to have a straightforward irreducible example.\n(4) If $S$ is a surface of infinite type, it is possible to define the Teichmüller\nspace of $S$ to be the space of marked conformal surfaces $f: S \\to \\Sigma$ up\nto equivalence; see the recently announced constructions of Basmajian–\nChandran [BC24a] and Tappu [Tap23] for details. The connected com-\nponents of this space will be contractible, but (with the hypothesis that\n$S$ has infinite type) there will always be infinitely many components\n[Bas97, BK08]. Some mapping classes will fix no component (as in the\nprevious remark), some will fix some components and not others, some\nwill fix every component (e.g., those supported in compact subsurfaces).\nWhat are the possible orbit types? It was recently announced by Basma-\njian and Chandran that the subgroup that fixes every component is the\ncompactly supported subgroup [BC24a].\n(5) Recent work of Basmajian and Chandran [BC24a] announced that there\nexist mapping classes $\\varphi$ that are represented by quasiconformal home-\nomorphisms on some components of Teichmüller space but not others.\n\nIn particular, the function $K_{T}$ defined in Remark (1) above may not be\nwell-defined for every component $T \\subseteq \\operatorname{Teich}(S)$.\n(6) A version of this problem appeared on a problem list compiled at the 2021\nNearly Carbon Neutral Geometric Topology conference [CPV21].\n\nReferences cited:\n- [Ahl06] Lars V. Ahlfors. Lectures on quasiconformal mappings, volume 38 of University Lecture Series. American Mathematical Society, Providence, RI, second edition, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. doi:10.1090/ulect/038.\n- [BC24a] A. Basmajian and Y. Chandran. A Bers type classification of big mapping class groups, 2024. arXiv:2410.05606.\n- [Tap23] Chaitanya Tappu. A moduli space of marked hyperbolic structures for big surfaces, 2023. arXiv:2311.01551.\n- [Bas97] Ara Basmajian. Large parameter spaces of quasiconformally distinct hyperbolic structures. J. Anal. Math., 71:75–85, 1997. doi:10.1007/BF02788023.\n- [BK08] Ara Basmajian and Youngju Kim. Geometrically infinite surfaces with discrete length spectra. Geom. Dedicata, 137:219–240, 2008. doi:10.1007/s10711-008-9294-5.\n- [CPV21] Yassin Chandran, Priyam Patel, and Nicholas G. Vlamis. Infinite-type surfaces and mapping class groups: Open problems. Available at https://https://www.patelp.com/uploads/2/5/7/9/25792573/inftypeproblems.pdf, 2021.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Minimum quasiconformal dilatation is attained for a fixed structure, but the proposed infinite-type Teichmüller classification of elliptic/parabolic/loxodromic behavior remains open.\n\n**Verified partial progress.**\n\n- Compactness gives an extremal representative for a fixed conformal structure.\n\n**Full solution or refutation.**\n\nThe global variation/classification questions are unresolved.\n\n**What remains.**\n\nAnalyze the dilatation function over connected Teichmüller components.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.40 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains the extremal-representative fact and remaining global problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2789, "problem_number": "KP-2.41", "title": "Kirby Problem 2.41", "statement": "Give a finite list of practically computable invariants of the\nmapping class group or pure mapping class group of an infinite-type surface $S$ that\ndetermine the topology of $S$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.41.\n\nLiterature notes:\n(1) Let $S$ be a surface, possibly of infinite type. When $S = S_{g,b}^{p}$ is of finite\ntype, the homeomorphism type of the surface is encoded in the isomor-\nphism type of its mapping class group, with two pairs of exceptions for\nsmall values of $g, b$, and $p$. Most cases can be distinguished by consid-\nering well-known group invariants: work of Birman–Lubotsky–McCarthy\n[BLM $^{+}83$] computes the algebraic rank, and Harer [Har86] computes\nthe virtual cohomological dimension, both in terms of $g$ and $b$. Combin-\ning these formulas, it is possible to determine the topology of the surface\n$S$, although note that this argument breaks down when either $g = 0$ or\n$b = 0$ (see [RS11, Remark A.2]).\n(2) The classification of infinite-type surfaces tells us that an infinite-type sur-\nface is determined by its end space $E$, the closed subgroup of non-planar\nends $E^{g}$ of $S$, the genus, and the number of compact boundary compo-\nnents of $S$.\nHowever, the end space of an infinite-type surface can be\nany closed subset of a Cantor set, and in practice, it is hard to tell when\ntwo such subsets are homeomorphic. In addition, Bavard–Dowdall–Rafi\n[BDR20] prove algebraic rigidity for mapping class groups of infinite-type\nsurfaces: if Map $(S)$ is isomorphic to Map $(S')$, then $S$ and $S'$ are home-\nomorphic. But again, big mapping class groups are uncountable groups\nand determining when two of these groups are isomorphic in general is\ndifficult. Thus, we pose the question above.\n(3) There has been one result by Aougab–Patel–Vlamis [APV21, Theorem\n8.1] in this direction for $n$-ended orientable infinite-genus surfaces with no\nplanar ends.\n\nReferences cited:\n- [BLM+83] Joan S Birman, Alex Lubotzky, John McCarthy, et al. Abelian and solvable subgroups of the mapping class groups. Duke Mathematical Journal, 50(4):1107–1120, 1983.\n- [Har86] John L. Harer. The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math., 84(1):157–176, 1986. doi:10.1007/BF01388737.\n- [RS11] Kasra Rafi and Saul Schleimer. Curve complexes are rigid. Duke Math. J., 158(2):225–246, 2011. doi:10.1215/00127094-1334004.\n- [BDR20] Juliette Bavard, Spencer Dowdall, and Kasra Rafi. Isomorphisms between big mapping class groups. International Mathematics Research Notices, 2020(10):3084– 3099, 2020.\n- [APV21] Tarik Aougab, Priyam Patel, and Nicholas G. Vlamis. Isometry groups of infinite genus hyperbolic surfaces. Math. Ann., 381:459–498, 2021.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Finite-type mapping-class-group invariants can recover most surface types, but no finite practical invariant list determines arbitrary infinite-type surface topology.\n\n**Verified partial progress.**\n\n- Infinite-type surfaces are classified by end space, nonplanar ends, genus and compact boundary count.\n\n**Full solution or refutation.**\n\nNo requested group-invariant reconstruction is known.\n\n**What remains.**\n\nFind computable group invariants encoding the end-space data.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.41 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Contrasts finite-type invariant recovery with the open infinite-type task.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2790, "problem_number": "KP-2.42", "title": "Kirby Problem 2.42", "statement": "Is the geodesic flow in almost every direction on the Chamanara\nsurface ergodic? What about on the translation surface considered by Bruin and\nLukina, which is similar to the Chamanara surface?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.42.\n\nLiterature notes:\n(1) Translation surfaces have geodesic flows in every direction that preserve\nLebesgue measure. Kerckhoff, Masur, and Smillie proved that for every\nfinite-type translation surface, this is the only probability measure pre-\nserved by the flow (that is, the flow is uniquely ergodic) in almost every\ndirection [KMS86]. As a corollary, they obtain that the flow in almost\nevery direction is ergodic with respect to the Lebesgue measure.\nIt is\nnatural to ask to what extent this generalizes to infinite-type surfaces.\nFraczek and Ulcigrai produced many examples of infinite-area, infinite-\ntype translation surfaces where the straight line flows are not ergodic in\nalmost every direction [FU14]. For finite-area translation surfaces, how-\never, this is widely open.\n(2) The Chamanara surface, first described by Reza Chamanara in [Cha04],\nis one of the best known translation surfaces. Chamanara defines a family\nof translation surfaces with a parameter $\\alpha \\in [0, 1]$. For the case $\\alpha = 1/2$,\ncalled the standard Chamanara surface, consider a square with edge length\n1, and divide the top and bottom sides into two halves. Identify the top\nright half with the bottom left half via a translation, and then divide the\nunidentified top and bottom edges in half, identify the top right half with\nthe bottom left half via a translation, and repeat. Do the same for the\nleft and right edges, always identifying the upper part of the right edge\nwith the lower part of the left edge. Excluding the corners of the square\nand the points where we divided the edges, we obtain the Chamanara\nsurface, a one-ended infinite-genus translation surface of finite area. See\nalso [DHV24, Example 2.4.22] for another description of the surface.\nThe Chamanara surface has a close relationship to baker’s map and is\nsometimes called the baker’s map surface; see [CGL06] for further details.\nBruin and Lukina consider a family of translation surfaces similar to\nthe Chamanara surfaces [BL23], but their surfaces lack certain metric\nsymmetries enjoyed by the Chamana surfaces, and so different techniques\nmay be necessary to approach this problem.\n(3) Similarly to proving that there exists a billiard in a polygon where the\nbilliard flow is ergodic (in the 3-dimensional unit tangent bundle), one\ncan use a Baire Category argument to show that there exists a finite-\narea, infinite-type surface where the flow in almost every direction is er-\ngodic with respect to Lebesgue measure. Moreover, by work of Vorobets\n[Vor97], one can even find an explicit example, but these arguments do\nnot seem applicable in the two situations above.\n\nReferences cited:\n- [KMS86] Steven Kerckhoff, Howard Masur, and John Smillie. Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2), 124(2):293–311, 1986. doi:10.2307/1971280.\n- [FU14] Krzysztof Fraczek and Corinna Ulcigrai. Non-ergodic Z-periodic billiards and infinite translation surfaces. Invent. Math., 197(2):241–298, 2014. doi:10.1007/s00222-013-0482-z.\n- [Cha04] R. Chamanara. Affine automorphism groups of surfaces of infinite type. In In the tradition of Ahlfors and Bers, III, volume 355 of Contemp. Math., pages 123–145. Amer. Math. Soc., Providence, RI, 2004. doi:10.1090/conm/355/06449.\n- [DHV24] V. Delecroix, P. Hubert, and F. Valdez. Infinite translation surfaces in the wild, 2024. arXiv:2403.05424.\n- [CGL06] R. Chamanara, F. P. Gardiner, and N. Lakic. A hyperelliptic realization of the horseshoe and baker maps. Ergodic Theory Dynam. Systems, 26(6):1749–1768, 2006. doi:10.1017/S0143385706000484.\n- [BL23] Henk Bruin and Olga Lukina. Rotated odometers. J. Lond. Math. Soc. (2), 107(6):1983–2024, 2023. doi:10.1112/jlms.12731.\n- [Vor97] Ya.B̃. Vorobets. Ergodicity of billiards in polygons. Mat. Sb., 188(3):65–112, 1997. doi:10.1070/SM1997v188n03ABEH000211.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Almost-every-direction ergodicity is known for finite-type translation surfaces, but remains open for the finite-area Chamanara and Bruin--Lukina infinite-type examples.\n\n**Verified partial progress.**\n\n- Kerckhoff--Masur--Smillie prove almost-every-direction ergodicity in finite type.\n- Infinite-area counterexamples show the extension is nonautomatic.\n\n**Full solution or refutation.**\n\nNo resolution for the named surfaces was verified.\n\n**What remains.**\n\nProve or disprove almost-every-direction ergodicity for each named surface.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.42 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States finite-type theorem, counterexamples, and named open cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2791, "problem_number": "KP-2.43", "title": "Kirby Problem 2.43", "statement": "Let $X$ be a compact, totally disconnected subset of $\\mathbb{R}^{2}$ with\n$|X| \\geq 2$, and let $\\Gamma_{X}$ denote the mapping class group of $\\mathbb{R}^{2} - X$.\n(a) For $Y \\subset X$, let $\\Gamma_{X,Y}$ be the subgroup of $\\Gamma_{X}$ permuting $Y$ . Classify in-\nvariants $q$ that are functorial.\n(b) If $X \\subset \\mathbb{R}^{2}$ is totally disconnected and $\\alpha$ is a homeomorphism of $\\mathbb{R}^{2}$ fixing\n$X$ as a set, let $C_{\\alpha}(X)$ denote the space of $\\alpha$-invariant closed subsets of\n\n$X$, in the Hausdorff topology. Let $q$ be functorial as in part (a). Classify\nthose $q$ that are continuous or semi-continuous.\n(c) What is the relationship between the translation length $\\tau_{X}$ and topological\nentropy? More precisely, is there a positive constant $C$ so that if $\\alpha$ is a\nhomeomorphism of $\\mathbb{R}^{2}$ permuting $X$ and $g$ is the class of $\\alpha$ in $\\Gamma_{X}$, then\nthe (topological) entropy of $\\alpha$ is at least as big as $C \\cdot \\tau_{X}(g)$?\n(d) Suppose $X$ is a Cantor set in the plane, and let $P\\Gamma_{X}$ denote the ‘pure’\nsubgroup of $\\Gamma_{X}$ fixing $X$ pointwise. Let $N$ be a subgroup of $P\\Gamma_{X}$ that is\nnormal in $\\Gamma_{X}$, and for each integer $n$, let $N_{n}$ be the subgroup of $P\\Gamma_{n}$ that\nis the image of $N$ restricted to any $n$-element subset of $X$. What possible\nsequences of subgroups $N_{n}$ can arise in this way?\n(e) For which $g \\in \\Gamma_{X}$ is there an embedding $X \\to \\mathbb{R}^{2}$ for which $g$ is represented\nby a diffeomorphism $\\alpha$ which is $C^{\\infty}$? Or $C^{k}$ for fixed $k$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.43.\n\nLiterature notes:\n(1) In Part (a), an invariant is functorial if for each $X$, there is a map\n$q_{X}: \\Gamma_{X} \\to \\mathbb{R}$ so that the following holds.\nFor each inclusion $Y \\subset X$\nand each $g \\in \\Gamma_{X,Y}$ with image $g_{Y} \\in \\Gamma_{Y}$ under the natural restriction map\n$\\Gamma_{X,Y} \\to \\Gamma_{Y}$ , there is an inequality $q_{X}(g) \\geq q_{Y} (g_{Y})$. In part (b), an in-\nvariant is (semi-)continuous if the map $C_{\\alpha}(X) \\to \\mathbb{R}$ taking $Y$ to $q_{Y} ([\\alpha])$\nis (semi-)continuous for all $\\alpha$, where $[\\alpha]$ is the class of $\\alpha$ in $\\Gamma_{Y}$ .\n(2) The ray graph $\\mathcal{R}_{X}$ is the graph whose vertices are isotopy classes of rays in\n$\\mathbb{R}^{2} - X$ from infinity to a point in $X$, and whose edges are pairs that may\nbe realized disjointly. This graph is hyperbolic, connected, and infinite\ndiameter, and its flag complex is contractible. The group $\\Gamma_{X}$ acts on it\nby isometries.\nThere is a cyclic order on the set of vertices of $\\mathcal{R}_{X}$ according to\nthe cyclic order on the geodesic representatives at infinity (with respect\nto any complete hyperbolic structure on $\\mathbb{R}^{2} - X$) which may be order\ncompleted to a circle $S^{1}_{X}$ on which $\\Gamma_{X}$ acts by (orientation-preserving)\nhomeomorphisms.\nNumerical invariants of elements of $\\Gamma_{X}$ may be defined in terms of\nthese actions, including\n(a) rotation number rot $_{X}(g) \\in \\mathbb{R}/\\mathbb{Z}$ for $g \\in \\Gamma_{X}$ acting on $S^{1}_{X}$;\n(b) translation length $\\tau_{X}(g)$ for $g \\in \\Gamma_{X}$ acting on $\\mathcal{R}_{X}$; and\n(c) counting quasimorphisms $H_{X,\\sigma}(g)$ for $g \\in \\Gamma_{X}$ and $\\sigma$ a $\\Gamma_{X}$-orbit of\npath in $\\mathcal{R}_{X}$.\nSome of these invariants have a well-defined ‘name’ (rotation number,\ntranslation length) independent of $X$. Others depend on choices that are\nspecial to $X$.\nAnother way to express the fact that some invariants have a well-\ndefined name is to say that they are functorial in the sense of part (a).\nFor example, an inclusion $Y \\to X$ induces a 1-Lipschitz map $\\mathcal{R}_{X} \\to \\mathcal{R}_{Y}$\nwell defined up to bounded distance, and translation length $\\tau_{X}$ in the\ngraph $\\mathcal{R}_{X}$ is functorial.\n(3) There is a partial order $<$ due to Boyland [Boy88] on the set of all\nconjugacy classes in all (finite type) braid groups where $g < h$ if for every\nhomeomorphism $\\alpha$ of the disk fixing the boundary with a finite $\\alpha$-invariant\nset $X$ such that $\\alpha$ represents $g$ relative to $X$, there is another $\\alpha$-invariant\n\nset $Y$ such that $\\alpha$ relative to $Y$ represents $h$. We say that $g$ forces $h$, and\nthis partial order is called braid forcing. A numerical invariant $q$ from\n(finite type) braid groups to $\\mathbb{R}$ is monotone if it is monotone with respect\nto braid forcing partial order.\nAll monotone invariants arise as follows: let $G$ denote the group of all\nhomeomorphisms of the disk fixed on the boundary, and for each braid $g$,\nlet $G_{g}$ be the subset of $G$ of homeomorphisms representing $g$ relative to\nsome finite invariant set. Then $g < h$ if and only if $G_{g} \\subset G_{h}$. For any func-\ntion $f: G \\to \\mathbb{R}$, we obtain a monotone invariant by $f(g) = \\sup_{\\alpha\\in Gg} f(\\alpha)$.\nBraid forcing makes perfect sense for mapping class groups instead of\nbraid groups, and one may extend this notion in the obvious way to a par-\ntial order on all conjugacy classes in all $\\Gamma_{X}$ simultaneously. Thus, another\nnatural question is: What is the relationship between (semi-)continuity\nand Boyland’s braid forcing?\n(4) In [CC22] the individual subgroups $N_{n}$ that can arise as in part (d) are\nidentified; they satisfy the so-called ‘inertia’ condition.\n(5) Let $\\mathcal{R}_{X}$ be the ray graph and $S^{1}_{X}$ the circle on which $\\Gamma_{X}$ acts by home-\nomorphisms as in Remark 2. An additional question is: Can the (non)-\nrealization property of part (e) be related to numerical properties of $g$ that\ncan be computed from the action on $\\mathcal{R}_{X}$ or $S^{1}_{X}$?\n\nReferences cited:\n- [Boy88] Philip Boyland. An analog of Sharkovski’s theorem for twist maps. In Hamiltonian dynamical systems (Boulder, CO, 1987), volume 81 of Contemp. Math., pages 119– 133. Amer. Math. Soc., Providence, RI, 1988. doi:10.1090/conm/081/986261.\n- [CC22] Danny Calegari and Lvzhou Chen. Normal subgroups of big mapping class groups. Trans. Amer. Math. Soc. Ser. B, 9:957–976, 2022. doi:10.1090/btran/108.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ray graphs give hyperbolic actions and candidate dynamics for planar Cantor-end mapping classes, but the requested functorial/continuous invariant classification is open.\n\n**Verified partial progress.**\n\n- The ray graph is connected, hyperbolic, infinite-diameter and has contractible flag complex.\n\n**Full solution or refutation.**\n\nNo classification of all invariants in the multi-part problem was verified.\n\n**What remains.**\n\nIdentify complete functorial and continuity-compatible invariants.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.43 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the invariants and records ray-graph structure.\n\n**Review notes.** Multi-part scope preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2792, "problem_number": "KP-2.44", "title": "Kirby Problem 2.44", "statement": "Given an infinite-type surface $S$, which homeomorphisms $f: S \\to$\n$S$ give rise to mapping tori $M_{f}$ that admit a hyperbolic structure? For those which\ndo admit a hyperbolic metric, is $M_{f}$ homeomorphic to the interior of a compact\nhyperbolic manifold with totally geodesic boundary?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.44.\n\nLiterature notes:\n(1) Thurston’s theorem on the hyperbolization of mapping tori states that\nfor a finite-type surface $S$ and a homeomorphism $f$ of the surface $S$, the\nmapping torus\n\n$$\nM_{f} = S \\times [0, 1]/(x, 0) \\sim (f(x), 1)\n$$\n\nis hyperbolic if and only if $f$ is pseudo-Anosov.\nThis question works\ntowards an analogue of the theorem for infinite-type surfaces.\nA first approach to this problem could be to give a list of criteria\non $f$ that ensures certain properties necessary for the mapping torus to\nbe hyperbolic. For instance, to give criteria on $f$ that ensure that $M_{f}$ is\natoroidal.\n(2) It is known from work of Field–Kim–Leininger–Loving [FKLL23] that for\na strongly irreducible end-periodic homeomorphism $f$ of an infinite-type\nsurface $S$, the mapping torus $M_{f}$ is hyperbolic and homeomorphic to the\ninterior of a compact hyperbolic manifold with totally geodesic boundary.\nThis motivates the second part of the question above.\n\nReferences cited:\n- [FKLL23] Elizabeth Field, Heejoung Kim, Christopher Leininger, and Marissa Loving. Endperiodic homeomorphisms and volumes of mapping tori. J. Topol., 16(1):57–105, 2023. doi:10.1112/topo.12277.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Strongly irreducible end-periodic maps have hyperbolic mapping tori that are interiors of compact totally-geodesic-boundary manifolds, but no general infinite-type hyperbolization criterion is known.\n\n**Verified partial progress.**\n\n- Field--Kim--Leininger--Loving prove the stated strongly irreducible end-periodic case.\n\n**Full solution or refutation.**\n\nThe full characterization remains open.\n\n**What remains.**\n\nGive necessary/sufficient infinite-type mapping-torus hyperbolicity criteria.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.44 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the special hyperbolization theorem and general target.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2793, "problem_number": "KP-2.45", "title": "Kirby Problem 2.45", "statement": "Compute the end-periodic cobordism group $\\Delta^{e}_{2}$ of end-periodic\nautomorphisms (diffeomorphisms or homeomorphisms) of surfaces.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.45.\n\nLiterature notes:\n(1) The (oriented) cobordism group of $n$-dimensional manifolds, denoted $\\Delta_{n}$,\nconsists of equivalence classes of pairs $(M^{n}, f)$ where $M$ is a compact\noriented manifold and $f$ is a homeomorphism. The equivalence relation\nis given by $(M_{0}, f_{0}) \\sim (M_{1}, f_{1})$ if there is a compact cobordism between\n$M_{0}$ and $M_{1}$ with a homeomorphism extending $f_{0}$ and $f_{1}$. This group was\ncomputed for $n \\geq 4$ by Kreck [Kre84b] and for $n = 3$ by Melvin [Mel79].\nBonahon [Bon83] (see also [EE82]) showed that $\\Delta_{2} \\cong \\mathbb{Z}^{\\infty} \\oplus (\\mathbb{Z}/2)^{\\infty}$. As in\nusual cobordism theories, compactness is used to ensure that one gets a\nnontrivial group.\n(2) An end-periodic manifold $M$ is a noncompact manifold with finitely many\nends, such that each end $\\epsilon$ has a neighborhood that is half of an infinite\ncyclic cover of a compact manifold $X_{\\epsilon}$. End-periodic manifolds behave\nin many respects (topological, geometric, and analytic) as if they were\nactually compact. An end periodic automorphism $f: M \\to M$ is a home-\nomorphism or homeomorphism that is a covering translation over $X_{\\epsilon}$ on\nsome neighborhood of each end $\\epsilon$. End-periodic homeomorphisms arise\nnaturally in the study of depth-one foliations [CCF21], with the behav-\nior in the end corresponding to the limiting behavior of a non-compact\nleaf as it approaches a compact leaf. Cobordisms of end-periodic auto-\nmorphisms are defined as in the compact case, with the extension required\nto be end-periodic, and one defines the end-periodic cobordism group $\\Delta^{e}_{n}$\nas above.\n(3) This problem is meaningful in all dimensions; the calculation of the usual\n$\\Delta_{2}$ makes use of special techniques related to the Nielsen-Thurston classifi-\ncation of surface automorphisms. A similar end-periodic Nielsen-Thurston\ntheory, originating in unpublished work of Handel-Miller, is developed in\ndepth in [CCF21], and it seems reasonable to approach the computation\nof $\\Delta^{e}_{2}$ using similar tools.\n\nReferences cited:\n- [Kre84b] Matthias Kreck. Bordism of diffeomorphisms and related topics. Springer-Verlag, Berlin, 1984. With an appendix by Neal W. Stoltzfus.\n- [Mel79] Paul Melvin. Bordism of diffeomorphisms. Topology, 18(2):173–175, 1979. doi:10.1016/0040-9383(79)90034-X.\n- [Bon83] Francis Bonahon. Cobordism of automorphisms of surfaces. Ann. Sci. École Norm. Sup. (4), 16(2):237–270, 1983. URL: http://www.numdam.org/item?id=ASENS 1983 4 16 2 237 0.\n- [EE82] A. L. Edmonds and J. H. Ewing. Remarks on the cobordism group of surface diffeomorphisms. Math. Ann., 259(4):497–504, 1982. doi:10.1007/BF01466055.\n- [CCF21] John Cantwell, Lawrence Conlon, and Sergio R. Fenley. Endperiodic automorphisms of surfaces and foliations. Ergodic Theory Dynam. Systems, 41(1):66–212, 2021. doi:10.1017/etds.2019.56.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Classical compact cobordism groups in dimensions 2--4 are computed, but the end-periodic surface automorphism cobordism group is not computed.\n\n**Verified partial progress.**\n\n- Bonahon computes the compact two-dimensional automorphism cobordism group.\n\n**Full solution or refutation.**\n\nNo computation of Delta^e_2 was verified.\n\n**What remains.**\n\nDevelop end-periodic cobordism invariants and calculate the group.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.45 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Contrasts known compact groups with the open end-periodic version.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2794, "problem_number": "KP-2.46", "title": "Kirby Problem 2.46", "statement": "(a) Which coarsely boundedly generated mapping class groups of infinite-type\nsurfaces are hyperbolic?\n(b) Consider the class of surfaces with $n \\geq 2$ ends, all accumulated by genus.\nAre the mapping class groups of these surfaces quasi-isometric for different\nvalues of $n$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.46.\n\nLiterature notes:\n(1) Big mapping class groups are not finitely, countably, or compactly gener-\nated. There is a generalization of compactness called coarse boundedness,\n\ndefined and studied extensively by C. Rosendal [Ros13, Ros22] for Pol-\nish groups. It turns out that when the (pure) mapping class group of an\ninfinite-type surface admits a coarsely bounded generating set, the quasi-\nisometry type of the group is well-defined. Mann–Rafi [MR23] and T.\nHill [Hil25] give a classification of those surfaces whose mapping class\ngroups and pure mapping class groups have this property, respectively.\nThis problem asks which coarsely boundedly generated mapping class\ngroups of infinite-type surfaces are quasi-isometric to a hyperbolic metric\nspace. In other words, for which infinite-type surfaces is the mapping class\ngroup with the word metric coming from a coarsely bounded generating\nset a hyperbolic metric space?\n(2) The mapping class groups of some surfaces are not only coarsely bound-\nedly generated, they are themselves coarsely bounded.\nIn particular,\nthey are finite diameter (when equipped with a word metric coming from\na coarsely bounded generating set), and so are elementary hyperbolic.\nSchaeffer-Cohen [SC24] showed that the mapping class group of a plane\nminus a Cantor set is quasi-isometric to the loop graph, which is an infi-\nnite diameter hyperbolic graph associated to the surface. This is the only\nknown example of a mapping class group of an infinite-type surface that\nis non-elementary hyperbolic.\n(3) Problem 2.46.(b) is a special case of the broad (and possibly unanswerable)\nproblem of determining the quasi-isometry type of coarsely boundedly\ngenerated mapping class groups of infinite-type surfaces.\n\nReferences cited:\n- [Ros13] Christian Rosendal. Global and local boundedness of Polish groups. Indiana Univ. Math. J., 62(5):1621–1678, 2013. doi:10.1512/iumj.2013.62.5133.\n- [Ros22] Christian Rosendal. Coarse geometry of topological groups, volume 223 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2022.\n- [MR23] Kathryn Mann and Kasra Rafi. Large-scale geometry of big mapping class groups. Geom. Topol., 27(6):2237–2296, 2023. doi:10.2140/gt.2023.27.2237.\n- [Hil25] Thomas Hill. Large-scale geometry of pure mapping class groups of infinite-type surfaces. Proc. Amer. Math. Soc., 153(6):2667–2680, 2025. doi:10.1090/proc/17181.\n- [SC24] Anschel Schaffer-Cohen. Graphs of curves and arcs quasi-isometric to big mapping class groups. Groups Geom. Dyn., 18(2):705–735, 2024. doi:10.4171/ggd/751.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Coarsely boundedly generated infinite-type mapping class groups are classified, but which are hyperbolic and whether the specified n-end groups are quasi-isometric remain open.\n\n**Verified partial progress.**\n\n- Mann--Rafi and Hill classify surfaces whose mapping class groups/pure groups have coarsely bounded generating sets.\n\n**Full solution or refutation.**\n\nThe requested quasi-isometric classification is unresolved.\n\n**What remains.**\n\nCharacterize hyperbolicity and compare quasi-isometry types across end count.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 2.46 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the generation classification and remaining metric questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2795, "problem_number": "KP-2.47", "title": "Kirby Problem 2.47", "statement": "(a) Given a mapping class $\\psi$ of a based surface $S$, there is an induced endo-\nmorphism of the symmetric product $\\operatorname{Sym}^{i}(S)$ and hence an endofunctor\n$\\psi_{*}$ of the partially wrapped Fukaya category $\\mathcal{F}(S)$ as constructed, say, by\nAuroux [Aur10]. Compute the categorical entropy of $\\psi_{*}$, in the sense\nof [DHKK14], for $\\psi$ pseudo-Anosov.\n(b) Given a fibered knot $K$ with pseudo-Anosov monodromy, what is\n\n$$\n\\limsup_{n\\to\\infty}\\left(\\dim \\widehat{\\mathrm{HFK}}(\\Sigma^{n}(K),\\widetilde{K};i)\\right)^{1/n},\n$$\n\nwhere $\\widetilde{K}$ is the branch locus in the $n$-fold cyclic branched cover $\\Sigma^{n}(K)$,\nand $i$ denotes the Alexander grading (in $\\mathbb{Z}$) with respect to the fiber surface.\nHere, $\\widehat{\\mathrm{HFK}}$ denotes the knot Floer homology of $K$ [OS04c, Ras03].\n(c) With notation as in the previous question, what is\n\n$$\n\\limsup_{n\\to\\infty}\\left(\\dim \\widehat{\\mathrm{HF}}(\\Sigma^{n}(K))\\right)^{1/n}\n$$\n\nfor the Heegaard Floer invariant $\\widehat{\\mathrm{HF}}$ [OS04e] of the closed 3-manifold\n$\\Sigma^{n}(K)$?\n(d) Given a closed surface $S$ and a pseudo-Anosov automorphism $\\psi$, let $M_{\\psi}$\ndenote the mapping torus of $\\psi$, and $\\widehat{\\mathrm{HF}}(M_{\\psi};i)$ the summand of\n$\\widehat{\\mathrm{HF}}(M_{\\psi})$\nspanned by the spin $^{c}$-structures $\\mathfrak{s}$ with $\\langle c_{1}(\\mathfrak{s}), [F]\\rangle = i$. What is\n\n$$\n\\limsup_{n\\to\\infty}\\left(\\dim \\widehat{\\mathrm{HF}}(M_{\\psi^{n}},i)\\right)^{1/n}?\n$$\n\nWhat about\n\n$$\n\\limsup_{n\\to\\infty}\\left(\\dim_{\\mathbb{F}_{2}}\\mathrm{HF}^{+}(M_{\\psi^{n}},i)\\right)^{1/n}\n$$\n\nfor $i \\neq 0$ (or for twisted $\\mathrm{HF}^{+}$ for $i = 0$)?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.47.\n\nLiterature notes:\n(1) In (a), the categorical entropy is equal to\n\n$$\n\\limsup_{n\\to\\infty}\\left(\\dim H_{*}(\\widehat{\\mathrm{CFDA}}(\\psi^{n},i-g))\\right)^{1/n},\n$$\n\nwhere $\\widehat{\\mathrm{CFDA}}(\\psi)$ is the bordered Floer bimodule [LOT15] and $g$ is the\ngenus of $S$.\nIn the case $i = 1$, this entropy is equal to the dilatation\n$\\lambda(\\psi)$ [LOT13, DHKK14]. For (d), see [OS04e] for the definition of\n$\\mathrm{HF}^{+}$, which is a Floer homology relative to a divisor; $\\widehat{\\mathrm{HF}}$ is Floer homology\nin the complement of that divisor.\n(2) Properties of bordered Floer homology imply that the categorical en-\ntropy is an upper bound for the answers to the remaining questions\n(perhaps up to a constant factor) [LOT15].\nAlso, the growth rate of\n$\\bigoplus_i \\widehat{\\mathrm{HFK}}(\\Sigma^{n}(K),\\widetilde{K},i)$ is an upper bound for the growth rate of\n$\\widehat{\\mathrm{HF}}(\\Sigma^{n}(K))$.\n(3) The growth rate of $|H_{1}(\\Sigma^{n}(K))|$ is a lower bound for the growth rate\nof $\\widehat{\\mathrm{HF}}(\\Sigma^{n}(K))$.\n(4) Some results about the growth rate of Heegaard Floer homology of branched\ncovers were proved in [HM18], though these results are about linear, not\nexponential, growth.\n(5) In (d), the reason for the restriction to $\\mathrm{HF} ^{+}$ with $i \\neq 0$ is that $\\mathrm{HF} ^{+}$ is\nfinitely generated over $\\mathbb{F}[\\operatorname{U}]$ but not over $\\mathbb{F}$ for $i = 0$.\n(6) Some further observations on these questions can be found in [Cor18].\n(7) One can ask similar questions about the growth of knot invariants like knot\nFloer homology, Khovanov homology, or symplectic Khovanov homology\nwhen one inserts a high power of a braid.\n\nReferences cited:\n- [Aur10] Denis Auroux. Fukaya categories of symmetric products and bordered HeegaardFloer homology. J. Gökova Geom. Topol. GGT, 4:1–54, 2010.\n- [DHKK14] G. Dimitrov, F. Haiden, L. Katzarkov, and M. Kontsevich. Dynamical systems and categories. In The influence of Solomon Lefschetz in geometry and topology, volume 621 of Contemp. Math., pages 133–170. Amer. Math. Soc., Providence, RI, 2014. doi:10.1090/conm/621/12421.\n- [OS04c] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and knot invariants. Adv. Math., 186(1):58–116, 2004. doi:10.1016/j.aim.2003.05.001.\n- [Ras03] Jacob Rasmussen. Floer homology and knot complements. PhD thesis, Harvard University, Cambridge, MA, 2003. https://arxiv.org/abs/math/0306378.\n- [OS04e] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and topological invariants for closed three-manifolds. Ann. of Math. (2), 159(3):1027–1158, 2004. doi:10.4007/annals.2004.159.1027.\n- [LOT15] Robert Lipshitz, Peter Ozsváth, and Dylan Thurston. Bimodules in bordered Heegaard Floer homology. Geom. Topol., 19(2):525–724, 2015. doi:10.2140/gt.2015.19.525.\n- [LOT13] Robert Lipshitz, Peter Ozsváth, and Dylan Thurston. A faithful linear-categorical action of the mapping class group of a surface with boundary. J. Eur. Math. Soc. (JEMS), 15(4):1279–1307, 2013. doi:10.4171/JEMS/392.\n- [HM18] Matthew Hedden and Thomas E. Mark. Floer homology and fractional Dehn twists. Adv. Math., 324:1–39, 2018. doi:10.1016/j.aim.2017.11.008.\n- [Cor18] James Cornish. Growth Rate of 3-Manifold Homologies under Branched Covers. ProQuest LLC, Ann Arbor, MI, 2018. Thesis (Ph.D.)–Columbia University. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt=info: ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqm\\&rft dat=xri:pqdiss:10791025.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The categorical entropy is known in degree i=1 and general bordered-Floer expressions and bounds are available, but the requested exponential Floer growth rates are not computed in general.\n\n**Verified partial progress.**\n\n- For i=1, the categorical entropy in part (a) equals the pseudo-Anosov dilatation.\n- Bordered Floer theory expresses the general entropy as the growth of the homology of CFDA(psi^n,i-g) and bounds the remaining rates from above.\n- The growth of the first homology of cyclic branched covers gives a lower bound for part (c); related Heegaard Floer results establish linear rather than exponential growth.\n\n**Full solution or refutation.**\n\nOnly a degree-one entropy case and comparison bounds are known; parts (b)--(d) and general (a) remain open.\n\n**What remains.**\n\nCompute the categorical entropy for arbitrary i and determine exact exponential growth rates for the branched-cover and mapping-torus Floer groups.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 2.47 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States all four growth questions and records the i=1 computation, bordered-Floer bounds, homology lower bound, and limitations of known linear-growth results.\n- Robert Lipshitz, Peter Ozsvath, and Dylan Thurston, A faithful linear-categorical action of the mapping class group of a surface with boundary, JEMS 15 (2013), 1279--1307. (primary): https://doi.org/10.4171/JEMS/392\n Evidence used: Provides the categorical mapping-class-group action used in the entropy formulation.\n- Matthew Hedden and Thomas E. Mark, Floer homology and fractional Dehn twists, Adv. Math. 324 (2018), 1--39. (primary): https://doi.org/10.1016/j.aim.2017.11.008\n Evidence used: Gives related Floer growth results that do not determine the requested exponential rates.\n\n**Review notes.** The several gradings and Floer variants are kept distinct.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2796, "problem_number": "KP-2.48", "title": "Kirby Problem 2.48", "statement": "The mapping class group of a closed, orientable, genus $g$ sur-\nface $S$ acts by symplectomorphisms on the symmetric product $\\operatorname{Sym}^{g}(S)$. In partic-\nular, it acts smoothly on $\\operatorname{Sym}^{g}(S)$.\n(a) What is the kernel of the map from the mapping class group of $S$ to the\nsmooth mapping class group of $\\operatorname{Sym}^{g}(S)$? What about in the based case?\n(b) Given disjoint simple closed curves $\\alpha_{1},..., \\alpha_{g}$ in $S$, there is a correspond-\ning torus $T_{\\alpha}$ in $\\operatorname{Sym}^{g}(S)$: the image of $\\alpha_{1} \\times \\cdots \\times \\alpha_{g} \\subset S^{g}$. These are\nthe tori that appear in Heegaard Floer theory. When do $\\alpha_{1},..., \\alpha_{g}$ and\n$\\beta_{1},..., \\beta_{g}$ give smoothly isotopic tori? When are they Lagrangian isotopic\nfor an appropriate choice of symplectic form?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.48.\n\nLiterature notes:\n(1) The question relates to whether Heegaard Floer homology is truly a sym-\nplectic invariant, or should be determined by the smooth isotopy classes\nof the Heegaard tori. (Examples of the difference between smooth and\nsymplectic topology in other settings have received substantial interest,\nthough many such examples are now known; [Sei97] is particularly rele-\nvant.)\n(2) The fact that the mapping class group acts symplectically on $\\operatorname{Sym}^{g}(S)$,\nfor appropriate symplectic forms, was shown by Perutz [Per08a]. (The\nobvious map, where a diffeomorphism $\\varphi$ of the surface $S$ sends a point\n$\\{x_{1},..., x_{g}\\} \\in \\operatorname{Sym}^{g}(S)$ to $\\{\\varphi(x_{1}),..., \\varphi(x_{g})\\}$ gives a homeomorphism but\nis not typically smooth; an exception is if $\\varphi$ is holomorphic with respect\nto some complex structure on $S$ and the smooth structure on $\\operatorname{Sym}^{g}(S)$ is\ninduced from this complex structure, in which case the obvious induced\nmap of $\\operatorname{Sym}^{g}(S)$ is also holomorphic.) Clarkson used Heegaard Floer ho-\nmology to show that the map from the based mapping class group to\nthe group of symplectomorphisms of $\\operatorname{Sym}^{g}(S\\setminus\\{z\\})$ modulo the Hamilton-\nian diffeomorphisms is injective [Cla17].\n(This uses the fact that, for\nappropriate choices of symplectic forms [Per08b], the Heegaard tori are\nexact [Hen12, HLL22a], and so Heegaard Floer homology agrees with\nthe usual Lagrangian intersection Floer homology.)\n(3) The fundamental group of $\\operatorname{Sym}^{g}(S)$ (for $g > 1$) is isomorphic to $H_{1}(\\Sigma)$.\nIn particular, the Torelli group of $S$ acts trivially on $\\pi_{1}(\\operatorname{Sym}^{g}(S))$. (It also\nacts trivially on the homology of $\\operatorname{Sym}^{g}(S)$.) Also, by Perutz’s work, if $\\alpha'_{1}$\nis obtained from $\\alpha_{1}$ by a handleslide, then $\\alpha_{1} \\times \\cdots \\times \\alpha_{g}$ and\n$\\alpha'_{1} \\times \\cdots \\times \\alpha_{g}$\nare smoothly isotopic (and, in fact, Hamiltonian isotopic for appropriate\nsymplectic forms) [Per08b].\n(4) In the case $g = 2$, one might be able to use the Abel-Jacobi map (which\nin this case presents $\\operatorname{Sym}^{2}(S)$ as a blow-up of the Jacobian torus) to show\nthat the Torelli group acts trivially, since it acts trivially on the Jacobian\ntorus $H^{1}(S; \\mathbb{R})/H^{1}(\\Sigma; \\mathbb{Z})$. Perhaps this suggests that the Torelli group or\nthe Johnson kernel acts trivially in the higher-genus case as well.\n\nReferences cited:\n- [Sei97] Paul Seidel. Floer homology and the symplectic isotopy problem. PhD thesis, University of Oxford, 1997.\n- [Per08a] Tim Perutz. Lagrangian matching invariants for fibred four-manifolds. II. Geom. Topol., 12(3):1461–1542, 2008. doi:10.2140/gt.2008.12.1461.\n- [Cla17] Corrin Clarkson. Three-manifold mutations detected by Heegaard Floer homology. Algebr. Geom. Topol., 17(1):1–16, 2017. doi:10.2140/agt.2017.17.1.\n- [Per08b] Timothy Perutz. Hamiltonian handleslides for Heegaard Floer homology. In Proceedings of Gökova Geometry-Topology Conference 2007, pages 15–35. Gökova Geometry/Topology Conference (GGT), Gökova, 2008.\n- [Hen12] Kristen Hendricks. A rank inequality for the knot Floer homology of double branched covers. Algebr. Geom. Topol., 12(4):2127–2178, 2012. doi:10.2140/agt.2012.12.2127.\n- [HLL22a] Kristen Hendricks, Tye Lidman, and Robert Lipshitz. Rank inequalities for the Heegaard Floer homology of branched covers. Doc. Math., 27:581–612, 2022.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A based symplectic action is faithful modulo Hamiltonian diffeomorphisms and handleslides give Hamiltonian-isotopic Heegaard tori, but the smooth kernel and general torus-isotopy classification remain open.\n\n**Verified partial progress.**\n\n- Perutz constructs the relevant symplectic actions on symmetric products.\n- Clarkson proves injectivity of the based mapping class group into symplectomorphisms of the punctured symmetric product modulo Hamiltonian diffeomorphisms.\n- A handleslide of one attaching curve produces smoothly and, for suitable forms, Hamiltonian-isotopic Heegaard tori.\n\n**Full solution or refutation.**\n\nThese results settle important based/symplectic and handleslide cases, not the closed smooth kernel or the requested if-and-only-if classification.\n\n**What remains.**\n\nDetermine the smooth kernels in the closed and based cases and classify smooth and Lagrangian isotopy of all Heegaard tori.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 2.48 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the kernel and torus-isotopy problems and separates Clarkson's based symplectic injectivity from the unresolved smooth action.\n- Corrin Clarkson, Three-manifold mutations detected by Heegaard Floer homology, Algebr. Geom. Topol. 17 (2017), 1--16. (primary): https://doi.org/10.2140/agt.2017.17.1\n Evidence used: Supplies the Heegaard-Floer detection used for based symplectic injectivity.\n- Timothy Perutz, Hamiltonian handleslides for Heegaard Floer homology, Proceedings of Gokova Geometry-Topology Conference 2007 (2008), 15--35. (primary): https://gokovagt.org/proceedings/2007/2.pdf\n Evidence used: Proves Hamiltonian isotopy under handleslides for appropriate symplectic forms.\n\n**Review notes.** Faithfulness modulo Hamiltonian diffeomorphisms is not the same as faithfulness in the smooth mapping class group.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2797, "problem_number": "KP-2.49", "title": "Kirby Problem 2.49", "statement": "(AMU conjecture). Let $S$ be a surface with negative Euler char-\nacteristic. If $\\varphi \\in \\operatorname{Mod}(S)$ acts by a pseudo-Anosov on some subsurface (including\n$S$ itself), show that there exists $p_{0}(\\varphi)$ such that the Witten–Reshetikhin–Turaev\nrepresentation $\\rho_{p}(\\varphi)$ has infinite order for $p \\geq p_{0}(\\varphi)$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.49.\n\nLiterature notes:\n(1) Given a compact Lie group $G$, the Witten–Reshetikhin–Turaev $G$-quantum\nrepresentations, introduced by Witten [Wit89] and rigorously defined\nby Reshetikhin and Turaev [RT91], are families of representations of\n$\\operatorname{Mod}(S)$.\nThe most studied case is when $G = \\operatorname{SU}(2)$ or $G = \\operatorname{SO}(3)$,\nin which case these representations are indexed by an integer $p \\geq 3$. More\nprecisely we have:\n\n$$\n\\rho_{p}: \\operatorname{Mod}(S) \\to PGL(d_{p}(S), \\mathbb{C}),\n$$\n\nwhere $d_{p}(S)$ is given by the famous Verlinde formula and is a polynomial\nin $p$ of degree $3g - 3$ (where $g$ is the genus of $S$). The representation $\\rho_{p}$\nsends a Dehn twist to a torsion element, and when $S$ has negative Euler\ncharacteristic, $\\rho_{p}(\\operatorname{Mod}(S))$ is infinite for $p$ big enough. Moreover, Ander-\nsen (see [And06]) proved that if $\\varphi \\in \\operatorname{Mod}(S)$ is non-central, then $\\rho_{p}(\\varphi)$\nis nontrivial for $p$ sufficiently large. This property is called asymptotic\nfaithfulness.\n(2) Geometric properties of these representations are quite mysterious. For\ninstance, it is not known if the Nielsen-Thurston classification can be de-\ntected by quantum representations. In [AMU06], Andersen, Masbaum,\nand Ueno studied the case of the four-holed sphere. They proved that if\n$\\varphi$ is pseudo-Anosov in the mapping class group of the four-holed sphere,\nthen $\\rho_{p}(\\varphi)$ has infinite order for $p$ big enough. In the same paper, they\nstated the above problem as a conjecture; it is now known as the AMU\nconjecture.\n(3) If $\\varphi \\in \\operatorname{Mod}(S)$ does not act as a pseudo-Anosov on a subsurface, then up\nto some power, it is the product of powers of commuting Dehn twists, and\nso $\\rho_{p}(\\varphi)$ has finite order for all $p$. Also, as the quantum representations\nenjoy some nice “splitting” properties regarding subsurfaces, it is enough\nto prove this conjecture for pseudo-Anosov elements.\n(4) The AMU conjecture is only known for two surfaces: the four-holed sphere\n(proved in [AMU06]) and the one-holed torus (see [San12]). In general,\nfinding pseudo-Anosov elements satisfying the conjecture is already a dif-\nficult task. In [EJ16], it was proved that a certain class of “homological”\npseudo-Anosov elements on holed spheres satisfy the AMU conjecture.\nExamples of point pushing mapping classes satisfying the AMU conjec-\nture have been studied in [KS16] and [MS21]. See also [DK22] where\nexamples were found using exponential growth of certain quantum invari-\nants; this paper shows that the Volume Conjecture (see Problem 1.27)\nimplies the AMU conjecture.\n\nReferences cited:\n- [Wit89] Edward Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys., 121(3):351–399, 1989. http://projecteuclid.org/euclid.cmp/1104178138.\n- [RT91] N. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math., 103(3):547–597, 1991. doi:10.1007/BF01239527.\n- [And06] Jørgen Ellegaard Andersen. Asymptotic faithfulness of the quantum $\\mathrm{SU}(n)$ representations of the mapping class groups. Ann. of Math. (2), 163(1):347–368, 2006. doi:10.4007/annals.2006.163.347.\n- [AMU06] Jørgen Ellegaard Andersen, Gregor Masbaum, and Kenji Ueno. Topological quantum field theory and the Nielsen-Thurston classification of M$(0,4)$. Math. Proc. Cambridge Philos. Soc., 141(3):477–488, 2006. doi:10.1017/S0305004106009698.\n- [San12] Ramanujan Santharoubane. Limits of the quantum SO(3) representations for the one-holed torus. J. Knot Theory Ramifications, 21(11):1250109, 13, 2012. doi: 10.1142/S021821651250109X.\n- [EJ16] Jens Kristian Egsgaard and Søren Fuglede Jørgensen. The homological content of the Jones representations at $q=-1$. J. Knot Theory Ramifications, 25(11):1650062, 25, 2016. doi:10.1142/S0218216516500620.\n- [KS16] Thomas Koberda and Ramanujan Santharoubane. Quotients of surface groups and homology of finite covers via quantum representations. Invent. Math., 206(2):269– 292, 2016. doi:10.1007/s00222-016-0652-x.\n- [MS21] Julien Marché and Ramanujan Santharoubane. Asymptotics of quantum representations of surface groups. Ann. Sci. Éc. Norm. Supér. (4), 54(5):1275–1296, 2021. doi:10.24033/asens.2481.\n- [DK22] Renaud Detcherry and Efstratia Kalfagianni. Cosets of monodromies and quantum representations. Indiana Univ. Math. J., 71(3):1101–1129, 2022.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The AMU conjecture is proved for low-complexity surfaces and several broad families, including a new 2026 Johnson-kernel result, but remains open for arbitrary pseudo-Anosov components and all sufficiently large levels.\n\n**Verified partial progress.**\n\n- The conjecture holds for the four-holed sphere and one-holed torus.\n- Known higher-complexity families include homological pseudo-Anosovs on holed spheres and point-pushing mapping classes.\n- Detcherry's March 2026 preprint proves the conjecture at prime levels for pseudo-Anosovs in the derived subgroup of the Johnson kernel.\n\n**Full solution or refutation.**\n\nThe new subgroup theorem is substantial but retains both a subgroup restriction and a prime-level restriction.\n\n**What remains.**\n\nProve eventual infinite order at every sufficiently large level for every mapping class having a pseudo-Anosov component.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 2.49 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Records the low-complexity surfaces, known families, and the general open conjecture.\n- Jorgen Ellegaard Andersen, Gregor Masbaum, and Kenji Ueno, Topological quantum field theory and the Nielsen-Thurston classification of M(0,4), Math. Proc. Cambridge Philos. Soc. 141 (2006), 477--488. (primary): https://doi.org/10.1017/S0305004106009698\n Evidence used: Proves the four-holed-sphere case and formulates the general conjecture.\n- Renaud Detcherry, The Andersen-Masbaum-Ueno conjecture for the derived subgroup of the Johnson kernel, arXiv:2603.29397 (2026). (primary): https://arxiv.org/abs/2603.29397\n Evidence used: Proves the prime-level theorem for pseudo-Anosovs in the derived subgroup of the Johnson kernel.\n\n**Review notes.** The March 2026 preprint postdates the principal results summarized in the K3 remarks and is included conservatively as a preprint theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2798, "problem_number": "KP-2.50", "title": "Kirby Problem 2.50", "statement": "(Volume conjecture for surface diffeomorphisms). Let $S$ be a\nclosed oriented surface, let $q=e^{2\\pi i/n}$ be a root of unity, and let $\\mathcal{K}^{q}(S)$ be the Kauff-\nman bracket skein algebra. Let $\\phi: S \\to S$ be a pseudo-Anosov diffeomorphism with\nmapping torus $M_{\\phi}$, and let $r: \\pi_{1}(S) \\to \\operatorname{SL}_{2}(\\mathbb{C})$ be a smooth point in the charac-\nter variety that is $\\phi$-invariant. Finally, let $L$ be the intertwiner realizing an iso-\nmorphism between the representations of $\\mathcal{K}^{q}(S)$ corresponding to $[r]$ and $[r \\circ \\varphi_{*}]$,\nnormalized so that $\\det(L) = 1$. Show that\n\n$$\n\\lim_{n\\to\\infty}\\frac{1}{n}\\log|\\operatorname{Trace} L|=\\frac{1}{4\\pi}\\operatorname{vol}_{\\mathrm{hyp}} M_{\\phi}.\n$$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.50.\n\nLiterature notes:\n(1) For a closed oriented surface $S$ and a root of unity $q$, results of [BW17a,\nFKBL19, GJS25, KK22, FKBL25] more or less establish a one-to-\none correspondence between irreducible representations of the Kauffman\nbracket skein algebra $\\mathcal{K}^{q}(S)$ and smooth points in the $\\operatorname{SL}_{2}(\\mathbb{C})$ character\nvariety of $S$. Thus in the above setup, the representations constructed\nfrom $[r]$ and $[r \\circ \\varphi_{*}]$ must be isomorphic by some map $L$ called the in-\ntertwiner. After normalizing $L$ so that $\\det(L) = 1, |\\operatorname{Trace} L|$ is dependent\nonly on the choice of $\\varphi, r, q$, and the puncture invariants.\n(2) This conjecture appears in [BWY21] and is a toy version of the Kashaev\nVolume Conjecture [Kas97] (see Problem 1.27), as revisited by Baseilhac-\nBenedetti [BB04]. In preprints, the conjecture has been claimed for some\nexamples [BWY22, Pan24, Wan25] and for torus knots. There also\nexist versions of this conjecture for punctured surfaces, involving puncture\ninvariants on the quantum side and 3-manifolds obtained from $M_{\\phi}$ by\nDehn filling on the hyperbolic side [PW24].\n\nReferences cited:\n- [BW17a] Francis Bonahon and Helen Wong. Representations of the Kauffman bracket skein algebra II: Punctured surfaces. Algebr. Geom. Topol., 17(6):3399–3434, 2017. doi: 10.2140/agt.2017.17.3399.\n- [FKBL19] Charles Frohman, Joanna Kania-Bartoszynska, and Thang Lê. Unicity for representations of the Kauffman bracket skein algebra. Invent. Math., 215(2):609–650, 2019. doi:10.1007/s00222-018-0833-x.\n- [GJS25] Iordan Ganev, David Jordan, and Pavel Safronov. The quantum Frobenius for character varieties and multiplicative quiver varieties. J. Eur. Math. Soc. (JEMS), 27(7):3023–3084, 2025. doi:10.4171/jems/1427.\n- [KK22] Hiroaki Karuo and Julien Korinman. Azumaya loci of skein algebras, 2022. arXiv: 2211.13700.\n- [FKBL25] Charles D. Frohman, Joanna Kania-Bartoszynska, and Thang T. Q. Lê. Sliced skein algebras and geometric Kauffman bracket. Adv. Math., 463:Paper No. 110118, 65, 2025. doi:10.1016/j.aim.2025.110118.\n- [BWY21] Francis Bonahon, Helen Wong, and Tian Yang. Asymptotics of quantum invariants of surface diffeomorphisms i: conjecture and algebraic computations, 2021. arXiv: 2112.12852.\n- [Kas97] Rinat M Kashaev. The hyperbolic volume of knots from the quantum dilogarithm. Letters in mathematical physics, 39(3):269–275, 1997.\n- [BB04] Stéphane Baseilhac and Riccardo Benedetti. Quantum hyperbolic invariants of 3-manifolds with P$\\mathrm{SL}(2,\\mathbb{C})$-characters. Topology, 43(6):1373–1423, 2004. doi:10.1016/j.top.2004.02.001.\n- [BWY22] Francis Bonahon, Helen Wong, and Tian Yang. Asymptotics of quantum invariants of surface diffeomorphisms II: The figure-eight knot complement, 2022. arXiv:2203.05730.\n- [Pan24] Tushar Pandey. The Bonahon-Wong-Yang volume conjecture for the four-puncture sphere, 2024. arXiv:2311.13151.\n- [Wan25] Zhihao Wang. Kauffman bracket intertwiners and the volume conjecture. Algebr. Geom. Topol., 25(4):2143–2177, 2025. doi:10.2140/agt.2025.25.2143.\n- [PW24] Tushar Pandey and Ka Ho Wong. Generalized Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms i: the figure eight knot complement, 2024. arXiv:2402.04483.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Bonahon-Wong-Yang volume conjecture is verified in important punctured-surface and example families, but not for arbitrary closed-surface pseudo-Anosov diffeomorphisms as stated.\n\n**Verified partial progress.**\n\n- Bonahon--Wong--Yang develop explicit intertwiner computations and prove the conjecture for a large family associated with the once-punctured torus in their series.\n- Pandey verifies a four-punctured-sphere-bundle version under technical hypotheses, and related work treats figure-eight/Dehn-filling examples.\n- Wang computes all closed-torus intertwiners and obtains zero limsup, a zero-volume comparison rather than the general hyperbolic case.\n\n**Full solution or refutation.**\n\nKnown computations and punctured variants do not establish the stated formula for arbitrary closed hyperbolic mapping tori.\n\n**What remains.**\n\nProve the trace asymptotic for every closed surface, invariant smooth character, and pseudo-Anosov monodromy, with all normalization issues controlled.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 2.50 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the closed-surface conjecture and catalogues example, punctured-surface, and torus computations without claiming a general proof.\n- Francis Bonahon, Helen Wong, and Tian Yang, Asymptotics of quantum invariants of surface diffeomorphisms I: conjecture and algebraic computations, arXiv:2112.12852 (2021). (primary): https://arxiv.org/abs/2112.12852\n Evidence used: Formulates the conjecture and develops the intertwiner machinery, with subsequent articles addressing a large once-punctured-torus family.\n- Zhihao Wang, Kauffman bracket intertwiners and the volume conjecture, Algebr. Geom. Topol. 25 (2025), 2143--2177. (primary): https://doi.org/10.2140/agt.2025.25.2143\n Evidence used: Computes the closed-torus intertwiners and proves a zero limsup in that comparison case.\n\n**Review notes.** Punctured and zero-volume torus cases are not conflated with the closed hyperbolic statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2799, "problem_number": "KP-3.1", "title": "Kirby Problem 3.1", "statement": "Classify the smallest volume hyperbolic 3-manifolds of various types. In particular:\n\n(a) Determine the nonorientable closed hyperbolic 3-manifolds of least volume.\n\n(b) Determine the n-cusped hyperbolic 3-manifolds of least volume for each $n\\geq 3$.\n\n(c) Determine the smallest volume hyperbolic 3-manifolds with n orientable cusps for each n.\n\n(d) Determine the n-cusped orientable hyperbolic 3-manifolds of least volume for each n.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.1.\n\nLiterature notes:\n(1) Parts (a)--(c) appeared in [Kir97, Problem 3.60].\n\n(2) There has been a lot of progress in determining the hyperbolic 3-manifolds of low volume [GHM+21]. In particular, the orientable hyperbolic manifold of smallest volume was shown to be the Weeks manifold by Gabai, Meyerhoff, and Milley [GMM09, Mil09]; its volume is approximately 0.9427. Extensive evidence from SnapPy indicates that the hyperbolic 3-manifold with smallest volume is orientable (hence is the Weeks manifold). Closed nonorientable 3-manifolds appear to have larger volumes, and so the currently available techniques do not readily apply to them, which suggests why part (a) remains open.\n\n(3) The two smallest orientable hyperbolic 3-orbifolds were identified by Gehring, Marshall, and Martin [GM09], [MM12], as was the smallest volume nonorientable 3-orbifold.\n\n(4) The n-cusped manifolds with smallest volume are known for n = 1 and 2; in both cases, these manifolds are nonorientable. For n = 1, Adams [Ada87] showed that the Gieseking manifold is the unique smallest manifold, with volume $v_3$ $\\approx$ 1.0149, the volume of a regular ideal tetrahedron. For n = 2, Adams [Ada88] proved that the least volume is $2v_3$, and showed that there is a unique 2-cusped manifold with this volume.\n\n(5) The smallest volume orientable hyperbolic 3-manifolds with n cusps are known for n = 1, 2 and 4. For n = 1, this is due to Cao and Meyerhoff [CM01], who showed that there are two manifolds with smallest volume $2v_3$, the figure-eight knot complement and the figure-eight knot sister. For n = 2, Agol [Ago10] proved that there are two orientable 2-cusped manifolds with smallest volume, the Whitehead link complement and the $(-2,3,8)$ pretzel link complement, both of which have volume $v_8$, which is the volume of a regular ideal octahedron, approximately 3.6638. For n = 4, Yoshida [Yos13] proved that the link $8^4_2$ is the unique orientable 4-cusped manifold with smallest volume $2v_8$ $\\approx$ 7.3276. For n = 3, the minimal volume is conjectured to be realized by the 3-chain link complement, which has volume approximately 5.33 (see [Zha23] for some results in this direction).\n\n(6) Agol [Ago10] has conjectured that for $n\\leq 10$, the minimal volume of an orientable n-cusped hyperbolic manifold is realized by the minimally twisted n-chain link complement. However, for larger values of $n$ ($11\\leq n\\leq 25$ and $n\\geq 60$), it has been shown [KPR12] that these manifolds do not have minimal volume, as the $(n-1)$-fold cyclic cover over one component of the Whitehead link has smaller volume.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GHM+21] David Gabai, Robert Haraway, Robert Meyerhoff, Nathaniel Thurston, and Andrew Yarmola. Hyperbolic 3-manifolds of low cusp volume, 2021. arXiv:2109.14570.\n- [GMM09] David Gabai, Robert Meyerhoff, and Peter Milley. Minimum volume cusped hyperbolic three-manifolds. J. Amer. Math. Soc., 22(4):1157–1215, 2009. doi:10.1090/S0894-0347-09-00639-0.\n- [Mil09] Peter Milley. Minimum volume hyperbolic 3-manifolds. J. Topol., 2(1):181–192, 2009. doi:10.1112/jtopol/jtp006.\n- [GM09] Frederick W. Gehring and Gaven J. Martin. Minimal co-volume hyperbolic lattices. I. The spherical points of a Kleinian group. Ann. of Math. (2), 170(1):123–161, 2009. doi:10.4007/annals.2009.170.123.\n- [MM12] T. H. Marshall and G. J. Martin. Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group. Ann. of Math. (2), 176(1):261–301, 2012. doi:10.4007/annals.2012.176.1.4.\n- [Ada87] Colin C. Adams. The noncompact hyperbolic 3-manifold of minimal volume. Proc. Amer. Math. Soc., 100(4):601–606, 1987. doi:10.2307/2046691.\n- [Ada88] Colin C. Adams. Volumes of N-cusped hyperbolic 3-manifolds. J. London Math. Soc. (2), 38(3):555–565, 1988. doi:10.1112/jlms/s2-38.3.555.\n- [CM01] Chun Cao and G. Robert Meyerhoff. The orientable cusped hyperbolic 3-manifolds of minimum volume. Invent. Math., 146(3):451–478, 2001. doi:10.1007/s002220100167.\n- [Ago10] Ian Agol. The minimal volume orientable hyperbolic 2-cusped 3-manifolds. Proc. Amer. Math. Soc., 138(10):3723–3732, 2010. doi: 10.1090/S0002-9939-10-10364-5.\n- [Yos13] Ken’ichi Yoshida. The minimal volume orientable hyperbolic 3-manifold with 4 cusps. Pacific J. Math., 266(2):457–476, 2013. doi:10.2140/pjm.2013.266.457.\n- [Zha23] Yue Zhang. Guts and the minimal volume orientable hyperbolic 3-manifold with 3 cusps, 2023. arXiv:2304.09950.\n- [KPR12] James Kaiser, Jessica S. Purcell, and Clint Rollins. Volumes of chain links. J. Knot Theory Ramifications, 21(11):1250115, 17, 2012. doi:10.1142/S0218216512501155.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Minimum-volume manifolds are classified in several closed and low-cusp orientability classes, but the nonorientable closed, orientable three-cusped, and general n-cusped problems remain open.\n\n**Verified partial progress.**\n\n- The Weeks manifold is the minimum-volume closed orientable hyperbolic 3-manifold.\n- The unrestricted one- and two-cusped minima are known; the one-cusped minimum is the Gieseking manifold.\n- For orientable manifolds, the one-, two-, and four-cusped minima are classified; Yoshida proves the unique four-cusped minimum is the 8^4_2 link complement.\n- Zhang proves a lower bound for orientable three-cusped manifolds satisfying a libroid-homology hypothesis, while the general three-cusped minimum remains unknown.\n\n**Full solution or refutation.**\n\nKnown low-cusp classifications give substantial pieces, not the requested classification for every n or the nonorientable closed case.\n\n**What remains.**\n\nResolve the nonorientable closed minimum, the orientable three-cusped minimum, and every remaining unrestricted/orientable n-cusped class.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 3.1 (2026). (maintained_tracker): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Catalogues the proven one-, two-, and four-cusp cases, the Weeks result, and the remaining nonorientable/three-cusp/general questions.\n- Ken'ichi Yoshida, The minimal volume orientable hyperbolic 3-manifold with 4 cusps, Pacific J. Math. 266 (2013), 457--476. (primary): https://doi.org/10.2140/pjm.2013.266.457\n Evidence used: Proves the unique orientable four-cusped minimum.\n- Yue Zhang, Guts and the minimal volume orientable hyperbolic 3-manifold with 3 cusps, arXiv:2304.09950. (primary): https://arxiv.org/abs/2304.09950\n Evidence used: Explicitly states the three-cusped minimum is unknown and proves a conditional-family lower bound.\n- Ian Agol, The minimal volume orientable hyperbolic 2-cusped 3-manifolds, Proc. Amer. Math. Soc. 138 (2010), 3723--3732. (primary): https://doi.org/10.1090/S0002-9939-10-10364-5\n Evidence used: Classifies the orientable two-cusped minimum.\n\n**Review notes.** Parts (b)--(d) encode different orientability restrictions and are not merged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2800, "problem_number": "KP-3.2", "title": "Kirby Problem 3.2", "statement": "Show that the volumes of hyperbolic 3-manifolds are not all rationally related.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.2.\n\nLiterature notes:\n(1) This was proposed by Thurston in [Thu82]. One could interpret \"rationally related\" to mean the volumes are all rational multiples of each other, or the weaker claim that they span a finite-dimensional $\\mathbb{Q}$-vector space. (The expectation is that neither holds.)\n\n(2) By Borel [Bor81] there is a real number $v_k$ such that the volume of an arithmetic hyperbolic 3-manifold with invariant trace-field $k$ is an integral multiple of $v_k$. More generally, for any number field $k$, there are real numbers $v_1,\\ldots,v_n$ such that for any (possibly non-arithmetic) hyperbolic 3-manifold with invariant trace-field $k$, its volume is an integral linear combination of $v_1,\\ldots,v_n$.\n\n(3) A 250+ year old problem asks whether Catalan's constant $G$ is rational. Agol [Ago10] proved that the minimal volume 2-cusped hyperbolic\n\n3-manifolds has volume equal to $4G$, the volume of the regular ideal octahedron.\n\n(4) Recently, F. Calegari, Dimitrov, and Tang [CDT24] established the irrationality of $L(2,\\chi_{-3})$, which is closely related to the volume of a regular ideal 3-simplex, and hence the volume of the figure-eight knot complement.\n\n(5) There is an analogous open question concerning the rationality of the Chern--Simons invariant. One can find an extensive discussion of this and related questions about the Chern--Simons invariant in [Kir97, Problems 3.62 and 3.63]; see also Problem 3.64.\n\n(6) The volume of an ideal tetrahedron with interior angles $\\alpha$, $\\beta$, and $\\gamma$ is $\\Lambda(\\alpha)+\\Lambda(\\beta)+\\Lambda(\\gamma)$, where $\\Lambda$ is the Lobachevsky function. Milnor [Mil82] has a conjecture that specifies precisely the rational relations between values of $\\Lambda$.\n\n(7) For a more in-depth discussion of this and related problems, see [GMM10, Problem 10.34] and [Neu98].\n\nReferences cited:\n- [Thu82] William P. Thurston. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.), 6(3):357–381, 1982. doi:10.1090/S0273-0979-1982-15003-0.\n- [Bor81] A. Borel. Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8(1):1–33, 1981. URL: http://www.numdam.org/item?id=ASNSP 1981 4 8 1 1 0.\n- [Ago10] Ian Agol. The minimal volume orientable hyperbolic 2-cusped 3-manifolds. Proc. Amer. Math. Soc., 138(10):3723–3732, 2010. doi: 10.1090/S0002-9939-10-10364-5.\n- [CDT24] Frank Calegari, Vesselin Dimitrov, and Yunqing Tang. The linear independence of 1, $\\zeta(2)$, and $L(2,\\chi_{-3})$, 2024. arXiv:2408.15403.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Mil82] John Milnor. Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc. (N.S.), 6(1):9–24, 1982. doi:10.1090/S0273-0979-1982-14958-8.\n- [GMM10] David Gabai, Robert Meyerhoff, and Peter Milley. Mom technology and hyperbolic 3-manifolds. In In the tradition of Ahlfors-Bers. V, volume 510 of Contemp. Math., pages 84–107. Amer. Math. Soc., Providence, RI, 2010.\n- [Neu98] Walter D. Neumann. Hilbert’s 3rd problem and invariants of 3-manifolds. In The Epstein birthday schrift, volume 1 of Geom. Topol. Monogr., pages 383–411. Geom. Topol. Publ., Coventry, 1998. doi:10.2140/gtm.1998.1.383.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Volumes with a fixed trace field lie in a finite-rank integral span, and recent irrationality results cover relevant L-values, but no proof that all hyperbolic volumes have the asserted rational-independence behavior is known.\n\n**Verified partial progress.**\n\n- Borel gives integral-multiple results for arithmetic fixed-field volumes.\n- Calegari--Dimitrov--Tang prove irrationality of L(2,chi_-3).\n\n**Full solution or refutation.**\n\nNeither strong nor weak broad interpretation is resolved.\n\n**What remains.**\n\nProve infinite-dimensional Q-span or establish explicit irrational relations among volumes.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.2 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records both interpretations and recent L-value progress.\n\n**Review notes.** Ambiguous phrase retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2801, "problem_number": "KP-3.3", "title": "Kirby Problem 3.3", "statement": "Does every cusped hyperbolic 3-manifold have a geometric ideal triangulation?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.3.\n\nLiterature notes:\n(1) A geometric ideal tetrahedron is the convex hull of any four points on the 3-sphere at infinity of $\\mathbb{H}^3$ that do not all lie on a circle. Topologically, it is a tetrahedron with its vertices removed. A geometric ideal triangulation of a hyperbolic 3-manifold M is an expression of M as a union of geometric ideal tetrahedra glued along their faces. Geometric ideal triangulations are useful, for example when studying hyperbolic Dehn surgery [BP92]. The question is whether they always exist. This was probably first asked in print by Yoshida [Yos96]; see also [WYY96], [PP00].\n\n(2) Epstein and Penner [EP88] showed that any cusped hyperbolic 3-manifold can be constructed by gluing geometric ideal polyhedra along their faces. Furthermore, it is known that any geometric ideal polyhedron can be subdivided into geometric ideal tetrahedra. However, it is not known if this can be done compatibly across the manifold.\n\n(3) Luo, Schleimer, and Tillmann [LST08] showed that the question has a positive answer virtually, in the sense that any cusped hyperbolic 3-manifold is finitely covered by a manifold admitting a geometric ideal triangulation. On the other hand, Choi [Cho04] gave an example of an incomplete hyperbolic structure on the figure-eight knot complement that does not admit a geometric ideal triangulation.\n\nReferences cited:\n- [BP92] Riccardo Benedetti and Carlo Petronio. Lectures on hyperbolic geometry. Universitext. Springer-Verlag, Berlin, 1992. doi:10.1007/978-3-642-58158-8.\n- [Yos96] Han Yoshida. Ideal tetrahedral decompositions of hyperbolic 3-manifolds. Osaka J. Math., 33(1):37–46, 1996. http://projecteuclid.org/euclid.ojm/1200786689.\n- [WYY96] Masaaki Wada, Yasushi Yamashita, and Han Yoshida. An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds. Proc. Amer. Math. Soc., 124(12):3905–3911, 1996. doi:10.1090/S0002-9939-96-03563-0.\n- [PP00] Carlo Petronio and Joan Porti. Negatively oriented ideal triangulations and a proof of Thurston’s hyperbolic Dehn filling theorem. Expo. Math., 18(1):1–35, 2000.\n- [EP88] D. B. A. Epstein and R. C. Penner. Euclidean decompositions of noncompact hyperbolic manifolds. J. Differential Geom., 27(1):67–80, 1988. http://projecteuclid.org/euclid.jdg/1214441650.\n- [LST08] Feng Luo, Saul Schleimer, and Stephan Tillmann. Geodesic ideal triangulations exist virtually. Proc. Amer. Math. Soc., 136(7):2625–2630, 2008. doi:10.1090/S0002-9939-08-09387-8.\n- [Cho04] Young-Eun Choi. Positively oriented ideal triangulations on hyperbolic threemanifolds. Topology, 43(6):1345–1371, 2004. doi:10.1016/j.top.2004.02.002.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Cusped hyperbolic manifolds have geometric ideal polyhedral decompositions, but it is open whether every one admits a geometric ideal tetrahedral triangulation.\n\n**Verified partial progress.**\n\n- Epstein--Penner give geometric ideal polyhedra.\n- Individual polyhedra can be subdivided geometrically, but global compatible triangulation is the issue.\n\n**Full solution or refutation.**\n\nNo universal triangulation theorem was verified.\n\n**What remains.**\n\nConstruct compatible geometric tetrahedral subdivisions for all cusped manifolds or find an obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.3 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States polyhedral decomposition and unresolved tetrahedral question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2802, "problem_number": "KP-3.4", "title": "Kirby Problem 3.4", "statement": "(Chen--Yang Volume Conjecture). (a) Prove that, for any hyperbolic 3-manifold $M$,\n\n$$\n\\lim_{\\substack{r\\to\\infty\\\\ r\\ \\mathrm{odd}}}\\frac{1}{r}\\log\\bigl(TV(M;e^{2\\pi i/r})\\bigr)\n=\\frac{1}{2\\pi}\\operatorname{Vol}(M).\n$$\n\n(b) Prove that, for any closed, oriented, hyperbolic 3-manifold $M$,\n\n$$\n\\lim_{\\substack{r\\to\\infty\\\\ r\\ \\mathrm{odd}}}\\frac{1}{r}\\log\\bigl(WRT(M;e^{2\\pi i/r})\\bigr)\n=\\frac{1}{4\\pi}\\bigl(\\operatorname{Vol}(M)-i\\operatorname{CS}(M)\\bigr).\n$$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.4.\n\nLiterature notes:\n(1) Given a root of unity $q$ so that $q^2$ is a primitive root of unity of odd order, $TV(M;q)$ denotes the corresponding Turaev--Viro invariant [TV92]. Similarly, $WRT(M;q)$ denotes the Witten--Reshetikhin--Turaev invariant [RT91]. For a hyperbolic 3-manifold M, $\\operatorname{Vol}(M)$ denotes the volume of M and $\\operatorname{CS}(M)$ denotes the Chern--Simons invariant of M.\n\n(2) Both forms of the conjecture were formulated by Q. Chen and T. Yang, in [CY18].\n\n(3) In Part (a), M can be closed, cusped, or have totally geodesic boundary.\n\n(4) Part (a) has been verified in many cases [DKM25], including for large families of 3-manifolds with cusps. A related version concerns the Witten-- Reshetikhin--Turaev invariant $WRT(M;q)$ where M is a 3-manifold and $q$ is a primitive root of unity of odd order $r$.\n\n(5) For a generalization of Part (b) to relative WRT invariants, see [WY23].\n\nReferences cited:\n- [TV92] V. G. Turaev and O. Ya. Viro. State sum invariants of 3-manifolds and quantum 6j-symbols. Topology, 31(4):865–902, 1992. doi:10.1016/0040-9383(92)90015-A.\n- [RT91] N. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math., 103(3):547–597, 1991. doi:10.1007/BF01239527.\n- [CY18] Qingtao Chen and Tian Yang. Volume conjectures for the Reshetikhin-Turaev and the Turaev-Viro invariants. Quantum Topol., 9(3):419–460, 2018. doi:10.4171/QT/111.\n- [DKM25] Renaud Detcherry, Efstratia Kalfagianni, and Shashini Marasinghe. Seifert cobordisms and the Chen-Yang volume conjecture, 2025. arXiv:2505.01546.\n- [WY23] Ka Ho Wong and Tian Yang. Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms I. Comm. Math. Phys., 400(2):1019– 1070, 2023. doi:10.1007/s00220-022-04613-5.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Chen--Yang Turaev--Viro volume conjecture is proved for many families, including broad cusped families, and relative-WRT variants have advanced; neither universal assertion in the record was verified.\n\n**Verified partial progress.**\n\n- Part (a) is known for many hyperbolic families rather than all hyperbolic 3-manifolds.\n- Detcherry--Kalfagianni--Marasinghe prove new gluing and Seifert-cobordism stability results for the relevant Turaev--Viro asymptotics.\n- Wong--Yang establish related relative Reshetikhin--Turaev asymptotics.\n\n**Full solution or refutation.**\n\nNo general proof for every hyperbolic 3-manifold in (a), or every closed oriented hyperbolic 3-manifold in (b), was found.\n\n**What remains.**\n\nExtend the Turaev--Viro theorem to arbitrary hyperbolic 3-manifolds and prove the closed WRT complex-volume limit with explicit phase, normalization, and logarithm conventions.\n\n**Sources checked.**\n\n- Qingtao Chen and Tian Yang, Volume conjectures for the Reshetikhin--Turaev and the Turaev--Viro invariants, Quantum Topology 9 (2018), 419--460. (primary): https://doi.org/10.4171/QT/111\n Evidence used: Original formulation and numerical/theoretical evidence for the two volume conjectures.\n- Renaud Detcherry, Efstratia Kalfagianni, and Shashini Marasinghe, Seifert cobordisms and the Chen--Yang volume conjecture, arXiv:2505.01546 (2025). (primary): https://arxiv.org/abs/2505.01546\n Evidence used: New closure and gluing results for classes satisfying the Turaev--Viro asymptotic, not a proof for every hyperbolic 3-manifold.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 3.4. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current synthesis states that part (a) is verified in many cases and points to the relative-WRT generalization.\n\n**Review notes.** Formulation caveat: the complex logarithm, WRT normalization, and Chern--Simons representative in part (b) need branch/convention choices; the imported formula was not silently repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2803, "problem_number": "KP-3.5", "title": "Kirby Problem 3.5", "statement": "(a) Do there exist closed non-Haken hyperbolic 3-manifolds with arbitrarily large injectivity radius?\n\n(b) Does there exist a cofinal tower of regular covers of closed hyperbolic 3-manifolds where all of the manifolds in the tower are non-Haken?\n\n(c) Is there a tower of hyperbolic rational homology spheres for which all fundamental groups are not left-orderable, the manifolds contain no coorientable taut foliations, or the manifolds are L-spaces? (The three properties listed are connected via the L-space conjecture. See Problem 3.48)", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.5.\n\nLiterature notes:\n(1) A cofinal tower of regular covers is a sequence of covers\n\n$$\n\\cdots \\to M_2 \\to M_1 \\to M\n$$\n\nsuch that $\\bigcap_i \\pi_1(M_i)=1$ and $\\pi_1(M_i)$ is a normal subgroup of $\\pi_1(M)$ for all i.\n\n(2) Question (a) is due to Cooper and appeared as [Kir97, Problem 3.58].\n\n(3) Clearly a negative answer to part (a) implies a negative answer to (b), since the injectivity radii of the manifolds in a tower tend to infinity. Any hyperbolic 3-manifold has a cofinal tower of regular covers, and so a negative answer to (a) or (b) would give an alternative proof of the Virtual Haken Conjecture, proved by Agol [Ago13].\n\n(4) F. Calegari and Dunfield [CD06] constructed a sequence of closed, hyperbolic rational homology 3-spheres for which their injectivity radii tend to infinity. One of the examples discussed in their paper was a tower that covers the Weeks manifold. This tower is a candidate for part (b).\n\n(5) The manifolds in F. Calegari--Dunfield's towers are rational homology 3-spheres but not integral homology 3-spheres. Indeed, it is expected (see Problem 3.6) that there do not exist cofinal towers of regular covers consisting of hyperbolic integral homology 3-spheres.\n\n(6) D. Calegari and Dunfield [CD03] showed that the fundamental group of the Weeks manifold is not left-orderable. Are the fundamental groups of the tower constructed by F. Calegari--Dunfield also not left-orderable?\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [CD06] Frank Calegari and Nathan M. Dunfield. Automorphic forms and rational homology 3-spheres. Geom. Topol., 10:295–329, 2006. doi:10.2140/gt.2006.10.295.\n- [CD03] Danny Calegari and Nathan M. Dunfield. Laminations and groups of homeomorphisms of the circle. Invent. Math., 152(1):149–204, 2003. doi:10.1007/s00222-002-0271-6.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** There are closed hyperbolic rational homology spheres with injectivity radii tending to infinity, but the non-Haken/cofinal-tower and finite-quotient questions remain open.\n\n**Verified partial progress.**\n\n- Calegari--Dunfield construct the cited large-injectivity-radius rational homology spheres.\n\n**Full solution or refutation.**\n\nThe examples do not settle non-Haken behavior in the stated towers.\n\n**What remains.**\n\nResolve each non-Haken/cofinal/finite-quotient subquestion.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.5 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes Calegari--Dunfield examples from open tower conditions.\n\n**Review notes.** Multi-part record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2804, "problem_number": "KP-3.6", "title": "Kirby Problem 3.6", "statement": "Given a cofinal tower of covers M $\\leftarrow$ $M_1$ $\\leftarrow$ $M_2$ $\\leftarrow$ $\\cdots$, is it true that the torsion subgroups $\\operatorname{Tor}(M_{n})$ of $H_1(M_{n}, \\mathbb{Z})$ and the hyperbolic volumes $\\operatorname{vol}(M_{n})$ satisfy\n\n$$\n\\lim_{n\\to\\infty}\\frac{\\log|\\operatorname{Tor}(M_n)|}{\\operatorname{vol}(M_n)}=\\frac{1}{6\\pi}\\,?\n$$", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.6.\n\nLiterature notes:\n(1) A tower of finite covers $M\\leftarrow M_1\\leftarrow M_2\\leftarrow\\cdots$ is called cofinal if the corresponding fundamental groups satisfy $\\bigcap_n \\pi_1(M_n)=\\{1\\}$.\n\n(2) Bergeron and Venkatesh conjectured that the above limit holds whenever M is a closed arithmetic 3-manifold and $\\{M_n\\}$ is a cofinal tower of congruence covers [BV13]. See also Lê for a closely related formulation [Lê09]. In the non-arithmetic setting, Brock and Dunfield conjectured that the above limit holds for any cofinal tower of regular covers $\\{M_n\\}$ such that $b_1(M_{n})$ = 0 for all n [BD15, Conjecture 1.13].\n\n(3) Brock and Dunfield have also compiled extensive computational data that supports both the Bergeron--Venkatesh conjecture and their extension to non-arithmetic manifolds [BD15, Section 4]. See also work of Şengün [Ş11, Ş12a], where computations are given that motivated the conjecture of Bergeron and Venkatesh. The computational data suggests that the above limit may fail to hold when M is non-arithmetic and the covers $M_n$ are allowed to have positive Betti numbers. In [BcV16], a more extensive conjectural framework is developed and more data is given, that might explain the distinction between the arithmetic and non-arithmetic cases.\n\nReferences cited:\n- [BV13] Nicolas Bergeron and Akshay Venkatesh. The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu, 12(2):391–447, 2013. doi: 10.1017/$S^{1}$474748012000667.\n- [Lê09] Thang T. Q. Lê. Hyperbolic volume, Mahler measure, and homology growth. Talk at Columbia University, http://www.math.columbia.edu/„volconf09/notes/leconf.pdf, 2009.\n- [BD15] Jeffrey F. Brock and Nathan M. Dunfield. Injectivity radii of hyperbolic integer homology 3-spheres. Geom. Topol., 19(1):497–523, 2015. doi:10.2140/gt.2015.19.497.\n- [Ş11] Mehmet Haluk Şengün. On the integral cohomology of Bianchi groups. Exp. Math., 20(4):487–505, 2011. doi:10.1080/10586458.2011.594671.\n- [Ş12a] Mehmet Haluk Şengün. On the torsion homology of non-arithmetic hyperbolic tetrahedral groups. Int. J. Number Theory, 8(2):311–320, 2012. doi:10.1142/$S^{1}$793042112500182.\n- [BcV16] Nicolas Bergeron, Mehmet Haluk Şengün, and Akshay Venkatesh. Torsion homology growth and cycle complexity of arithmetic manifolds. Duke Math. J., 165(9):1629– 1693, 2016. doi:10.1215/00127094-3450429.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Bergeron--Venkatesh and Brock--Dunfield formulate precise torsion-growth conjectures supported by extensive computations, but the universal stated limit is unproved.\n\n**Verified partial progress.**\n\n- Arithmetic congruence and nonarithmetic regular-tower versions are formulated with supporting data.\n\n**Full solution or refutation.**\n\nNo general cofinal-tower theorem was verified.\n\n**What remains.**\n\nProve the 1/(6pi) limit under sharp hypotheses or find counterexamples.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.6 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records conjectural regimes and computational support.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2805, "problem_number": "KP-3.7", "title": "Kirby Problem 3.7", "statement": "Does every finite-volume hyperbolic 3-manifold admit a finitesheeted cover fibering over the circle with orientable pseudo-Anosov monodromy?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.7.\n\nLiterature notes:\n(1) Agol, building on work of Wise and Kahn--Markovic, proved that every finite-volume hyperbolic 3-manifold admits a finite-sheeted covering that fibers over the circle [Ago08, Ago13, Wis21, KM12]. The monodromy of the fibration is pseudo-Anosov. It is said to be orientable pseudo-Anosov if the stable and unstable laminations are transversely orientable. An obstruction to a pseudo-Anosov being orientable is the existence of a singularity of odd order. Indeed, in the situation where there is no odd order singularity, one may pass to a finite cover of the manifold where the monodromy is orientable pseudo-Anosov [McM13].\n\n(2) One rationale for this question is that orientable pseudo-Anosovs have many nice properties. For example, the dilatation of an orientable pseudo-Anosov is the root with largest modulus of the Alexander polynomial of the fibered manifold [Thu22]. This may be useful for understanding profinite rigidity, since Alexander polynomials can be used to analyze the profinite properties of 3-manifolds (for example [Liu23]); see also\n\nProblem 3.8.\n\nReferences cited:\n- [Ago08] Ian Agol. Criteria for virtual fibering. J. Topol., 1(2):269–284, 2008. doi:10.1112/jtopol/jtn003.\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [Wis21] Daniel T. Wise. The structure of groups with a quasiconvex hierarchy, volume 209 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, [2021] ©2021.\n- [KM12] Jeremy Kahn and Vladimir Markovic. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Ann. of Math. (2), 175(3):1127–1190, 2012. doi: 10.4007/annals.2012.175.3.4.\n- [McM13] Curtis T. McMullen. Entropy on Riemann surfaces and the Jacobians of finite covers. Comment. Math. Helv., 88(4):953–964, 2013. doi:10.4171/CMH/308.\n- [Thu22] William P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. In Collected works of William P. Thurston with commentary. Vol. I. Foliations, surfaces and differential geometry, pages 495–509. Amer. Math. Soc., Providence, RI, [2022] ©2022. Reprint of [ 0956596].\n- [Liu23] Yi Liu. Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups. Invent. Math., 231(2):741–804, 2023. doi:10.1007/s00222-022-01155-4.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every finite-volume hyperbolic 3-manifold virtually fibers with pseudo-Anosov monodromy, and orientability can be obtained when odd-prong singularities are absent; whether some such orientable virtual fibration always exists remains open.\n\n**Verified partial progress.**\n\n- Agol's virtual fibering theorem gives a finite-sheeted fibered cover for every finite-volume hyperbolic 3-manifold.\n- The monodromy of a fibration of a finite-volume hyperbolic 3-manifold is pseudo-Anosov.\n- McMullen shows that in the absence of odd-order singularities one may pass to a finite cover where the invariant foliations are orientable.\n\n**Full solution or refutation.**\n\nThe virtual-fibering portion is solved and the orientable refinement is conditionally solved, but the odd-prong obstruction has not been eliminated in every commensurability class.\n\n**What remains.**\n\nProve every manifold has some virtual fibration without a persistent odd-prong obstruction, or exhibit a counterexample.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 3.7 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Keeps the orientable-monodromy refinement open while recording virtual fibering and McMullen's conditional cover result.\n- Ian Agol, The virtual Haken conjecture, Documenta Mathematica 18 (2013), 1045--1087. (primary): https://doi.org/10.4171/DM/421\n Evidence used: Establishes the virtual Haken/virtual fibering framework for finite-volume hyperbolic 3-manifolds.\n- Curtis T. McMullen, Entropy on Riemann surfaces and the Jacobians of finite covers, Commentarii Mathematici Helvetici 88 (2013), 953--964. (primary): https://doi.org/10.4171/CMH/308\n Evidence used: Provides the finite-cover orientability conclusion under the no-odd-singularity condition used by the tracker.\n\n**Review notes.** The source typo 'finitesheeted' was preserved in the report and flagged as a likely missing space/hyphen.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2806, "problem_number": "KP-3.8", "title": "Kirby Problem 3.8", "statement": "If $M_1$ and $M_2$ are finite-volume hyperbolic 3-manifolds whose fundamental groups have isomorphic profinite completions, must $M_1$ and $M_2$ be isometric?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.8.\n\nLiterature notes:\n(1) The problem was stated by Reid in [Rei15, Question 9].\n\n(2) Apart from its intrinsic interest, a positive answer to this problem would have some useful consequences. For example, it would provide a new solution to the homeomorphism problem for finite-volume hyperbolic 3-manifolds, since one could distinguish non-homeomorphic manifolds by finding a finite quotient of one fundamental group that is not a finite quotient of the other one.\n\n(3) It was shown by Liu [Liu23] that only finitely many finite-volume hyperbolic 3-manifolds have fundamental groups that can share the same profinite completion. Furthermore, the profinite completion of $\\pi_1(M)$ is known to contain quite a lot of information about M. For example, it controls not only $H_1(M)$ but also the first homology of any finite covering space. Jaikin--Zapirain [JZ20a] also proved that the profinite completion determines whether M fibers over the circle. However, it is not currently known whether the volume of a hyperbolic 3-manifold M is determined by the profinite completion of M.\n\n(4) Examples of distinct compact 3-manifolds whose fundamental groups have isomorphic profinite completions were provided by Funar [Fun13] (torus bundles with sol geometry) and Hempel [Hem14] (Seifert fibered spaces). So it is reasonable to restrict this problem to hyperbolic 3-manifolds.\n\n(5) Bridson, McReynolds, Reid, and Spitler [BMRS20] gave some interesting examples of hyperbolic 3-orbifolds M that are profinitely rigid in a much stronger sense. In their examples, whenever $\\pi_1(M)$ has profinite\n\ncompletion equal to the profinite completion of some finitely generated residually finite group G, then $\\pi_1(M)$ must be isomorphic to G.\n\nReferences cited:\n- [Rei15] Alan W. Reid. Profinite properties of discrete groups. In Groups St Andrews 2013, volume 422 of London Math. Soc. Lecture Note Ser., pages 73–104. Cambridge Univ. Press, Cambridge, 2015.\n- [Liu23] Yi Liu. Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups. Invent. Math., 231(2):741–804, 2023. doi:10.1007/s00222-022-01155-4.\n- [JZ20a] Andrei Jaikin-Zapirain. Recognition of being fibered for compact 3-manifolds. Geom. Topol., 24(1):409–420, 2020. doi:10.2140/gt.2020.24.409.\n- [Fun13] Louis Funar. Torus bundles not distinguished by TQFT invariants. Geom. Topol., 17(4):2289–2344, 2013. With an appendix by Funar and Andrei Rapinchuk. doi: 10.2140/gt.2013.17.2289.\n- [Hem14] John Hempel. Some 3-manifold groups with the same finite quotients, 2014. arXiv: 1409.3509.\n- [BMRS20] M. R. Bridson, D. B. McReynolds, A. W. Reid, and R. Spitler. Absolute profinite rigidity and hyperbolic geometry. Ann. of Math. (2), 192(3):679–719, 2020. doi: 10.4007/annals.2020.192.3.1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finite-volume hyperbolic 3-manifolds are determined up to finite ambiguity by their profinite completions, and many explicit families are rigid, but universal profinite rigidity and hence the requested isometry conclusion remain open.\n\n**Verified partial progress.**\n\n- Yi Liu proves finite ambiguity among finite-volume hyperbolic 3-manifold groups sharing a profinite completion.\n- Xiaoyu Xu proves profinite almost-rigidity for compact orientable 3-manifolds more generally.\n- Profinite completion detects fibering, first cohomology with Thurston norm, and many covering-space homology data.\n- Huang constructs large new families of profinitely rigid cusped hyperbolic manifolds via geometric convergence and drilling.\n\n**Full solution or refutation.**\n\nAlmost-rigidity reduces the possible partners to finitely many, but does not show there is only one isometry class for every finite-volume hyperbolic 3-manifold.\n\n**What remains.**\n\nProve every finite-volume hyperbolic 3-manifold group is profinitely rigid among such groups, or construct nonisometric manifolds with isomorphic profinite completions.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.8. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the exact isometry question and summarizes the almost-rigidity and profinite-detection evidence.\n- Yi Liu, Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups, Inventiones Mathematicae 231 (2023), 741-804. (primary): https://doi.org/10.1007/s00222-022-01155-4\n Evidence used: Proves finite ambiguity for finite-volume hyperbolic 3-manifolds with the same finite quotients.\n- Xiaoyu Xu, Profinite almost rigidity in 3-manifolds, Advances in Mathematics 477 (2025), 110505. (primary): https://doi.org/10.1016/j.aim.2025.110505\n Evidence used: Extends almost-rigidity across compact orientable 3-manifolds.\n- Tianwei Liu, Toward the profinite rigidity of hyperbolic 3-manifolds, arXiv:2508.20110 (2025). (authoritative_secondary): https://arxiv.org/abs/2508.20110\n Evidence used: Current survey explicitly identifies the universal hyperbolic case as the remaining problem and catalogs rigid families.\n\n**Review notes.** Mostow-Prasad rigidity converts discrete-group isomorphism to isometry, but profinite-completion isomorphism is the unresolved bridge.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2807, "problem_number": "KP-3.9", "title": "Kirby Problem 3.9", "statement": "Is being Haken a profinite invariant amongst 3-manifolds? That is, if $M_1$ and $M_2$ are 3-manifolds so that $\\pi_1(M_{1})$ and $\\pi_1(M_{2})$ have isomorphic profinite completions, and $M_1$ is Haken, must $M_2$ also be Haken?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.9.\n\nLiterature notes:\n(1) The key case to consider is if M is a hyperbolic Haken rational homology sphere and N is a closed hyperbolic 3-manifold. A positive answer to Problem 3.8 implies a positive answer to this problem.\n\n(2) Note that, for general finitely presented residually finite groups, being a free product with amalgamation is not a profinite property [CWLRS25].\n\nReferences cited:\n- [CWLRS25] Tamunonye Cheetham-West, Alexander Lubotzky, Alan W. Reid, and Ryan Spitler. Property FA is not a profinite property. Groups Geom. Dyn., 19(3):1081–1087, 2025. doi:10.4171/ggd/802.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many 3-manifold decompositions, geometries, and fibering are profinite invariants, reducing the Haken question to the closed hyperbolic rational-homology-sphere case, which remains open.\n\n**Verified partial progress.**\n\n- Positive first Betti number is profinitely detected, so only rational-homology-sphere cases are difficult.\n- Prime and JSJ decompositions, fibering, and nonhyperbolic geometries have strong profinite detection theorems.\n- A positive solution of universal hyperbolic profinite rigidity in KP-3.8 would imply a positive answer.\n- Property FA is not profinite among general finitely presented residually finite groups, but the known counterexamples are not 3-manifold counterexamples.\n\n**Full solution or refutation.**\n\nNo hyperbolic 3-manifold counterexample or proof for the residual closed hyperbolic case was located.\n\n**What remains.**\n\nDetermine whether a closed hyperbolic Haken rational homology sphere can share its profinite completion with a non-Haken closed hyperbolic 3-manifold.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.9. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the exact Haken question and identifies the key closed hyperbolic rational-homology-sphere case.\n- Tianwei Liu, Toward the profinite rigidity of hyperbolic 3-manifolds, arXiv:2508.20110 (2025). (authoritative_secondary): https://arxiv.org/abs/2508.20110\n Evidence used: Surveys complete profinite detection of many decompositions and geometries and confirms the remaining hyperbolic rigidity gap.\n- Tamunonye Cheetham-West, Alexander Lubotzky, Alan W. Reid, and Ryan Spitler, Property FA is not a profinite property, Groups, Geometry, and Dynamics 19 (2025), 1081-1087. (primary): https://doi.org/10.4171/ggd/802\n Evidence used: Shows that the analogous tree-splitting property fails to be profinite for general groups, without furnishing a 3-manifold counterexample.\n\n**Review notes.** The general-group failure of property FA must not be reported as a refutation among 3-manifolds.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2808, "problem_number": "KP-3.10", "title": "Kirby Problem 3.10", "statement": "(a) Are there infinitely many commensurability classes of arithmetic rational homology 3-spheres?\n\n(b) Are there infinitely many arithmetic integral homology 3-spheres?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.10.\n\nLiterature notes:\n(1) Recall that two hyperbolic manifolds are commensurable if they have a common finite cover. Two lattices $\\Gamma_1$ and $\\Gamma_2$ in a Lie group G are commensurable if there is some $g\\in G$ such that $\\Gamma_1\\cap g\\Gamma_2g^{-1}$ has finite index in $\\Gamma_1$ and in $g\\Gamma_2g^{-1}$.\n\n(2) The general construction of arithmetic lattices is as follows. Start with a connected semisimple Lie group G with trivial center and no compact factor, and a semisimple algebraic subgroup H of $\\operatorname{GL}(n, \\mathbb{R})$ defined by some polynomial equations with integer coefficients. Suppose that there is surjective homomorphism $\\varphi$ from the identity component $H^0$ of $H$ to G with compact kernel. Then any lattice in G that is commensurable with $\\varphi(H^0\\cap \\operatorname{GL}(n,\\mathbb{Z}))$ is arithmetic. When G is $\\operatorname{SO}(3, 1)$, then $\\mathbb{H}^3/\\Gamma$ is an arithmetic hyperbolic 3-orbifold, and when $\\Gamma$ is also torsion-free, $\\mathbb{H}^3/\\Gamma$ is an arithmetic hyperbolic 3-manifold. Arithmetic hyperbolic 3-manifolds and 3-orbifolds can alternatively be defined in terms of orders in quaternion algebras. See [MR03] for this definition, as well as a comprehensive introduction to the subject.\n\n(3) It is known that there are infinitely many arithmetic rational homology 3-spheres, by constructions of F. Calegari--Dunfield [CD06] and Boston-- Ellenberg [BE06]. However, their constructions do not immediately give infinitely many commensurability classes.\n\n(4) It appears that there are only two known arithmetic integral homology 3-spheres. These are the $1/2$ Dehn filling on the knot $5_2$ and the 3-fold cyclic branched cover of the $(-2,3,7)$-pretzel knot.\n\n(5) See [Rei07] for further discussion of this problem, as well as a potential route to find infinitely many arithmetic hyperbolic rational homology 3-spheres up to commensurability.\n\n(6) When a hyperbolic 3-manifold M has cusps, it is never a rational homology 3-sphere, but in this case one can consider the homomorphism $H^{1}(M)$ $\\to$ $H^{1}(\\partial{}M)$ induced by inclusion. The kernel of this homomorphism is the cuspidal cohomology of M. It is known that there are only finitely many commensurability classes of cusped arithmetic hyperbolic 3-manifolds that contain a manifold with trivial cuspidal cohomology [MR03, Theorem 9.3.6]. This was used by Reid [Rei91] to show that the figure-eight knot is the unique arithmetic hyperbolic knot complement.\n\n(7) An analogous question in dimension two is whether there are infinitely many commensurability classes of genus 0 hyperbolic arithmetic 2-orbifolds. This was proved to be false in [LMR06].\n\nReferences cited:\n- [MR03] Colin Maclachlan and Alan W. Reid. The arithmetic of hyperbolic 3-manifolds, volume 219 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2003. doi:10.1007/978-1-4757-6720-9.\n- [CD06] Frank Calegari and Nathan M. Dunfield. Automorphic forms and rational homology 3-spheres. Geom. Topol., 10:295–329, 2006. doi:10.2140/gt.2006.10.295.\n- [BE06] Nigel Boston and Jordan S. Ellenberg. Pro-p groups and towers of rational homology spheres. Geom. Topol., 10:331–334, 2006. doi:10.2140/gt.2006.10.331.\n- [Rei07] Alan Reid. The geometry and topology of arithmetic hyperbolic 3-manifolds. In Proc. Symposium Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces, Kyoto 2006, volume 1571 of RIMS Kokyuroku Series, pages 31–58. Research Institute for Mathematical Sciences, Kyoto University, 2007.\n- [Rei91] Alan W. Reid. Arithmeticity of knot complements. J. London Math. Soc. (2), 43(1):171–184, 1991. doi:10.1112/jlms/s2-43.1.171.\n- [LMR06] D. D. Long, C. Maclachlan, and A. W. Reid. Arithmetic Fuchsian groups of genus zero. Pure Appl. Math. Q., 2(2):569–599, 2006. doi:10.4310/PAMQ.2006.v2.n2.a9.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Infinite families of arithmetic rational homology 3-spheres are known, but the constructions do not establish infinitely many commensurability classes, and no proof of infinitely many arithmetic integral homology 3-spheres was located.\n\n**Verified partial progress.**\n\n- Calegari and Dunfield construct infinite towers of arithmetic rational homology 3-spheres using automorphic forms.\n- Boston and Ellenberg give a pro-p-group construction of towers of rational homology spheres.\n- The current K3 account explicitly warns that these families do not immediately yield infinitely many commensurability classes and records only two known arithmetic integral homology 3-spheres.\n\n**Full solution or refutation.**\n\nNeither part (a) nor part (b) is resolved in the literature checked.\n\n**What remains.**\n\nProve or disprove infinitude of commensurability classes in part (a), and construct or rule out infinitely many arithmetic integral homology 3-spheres in part (b).\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.10. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States both questions and distinguishes known infinite rational-homology-sphere towers from the unresolved commensurability-class question.\n- Frank Calegari and Nathan M. Dunfield, Automorphic forms and rational homology 3-spheres, Geometry & Topology 10 (2006), 295-329. (primary): https://doi.org/10.2140/gt.2006.10.295\n Evidence used: Constructs infinite arithmetic towers of rational homology 3-spheres.\n- Nigel Boston and Jordan S. Ellenberg, Pro-p groups and towers of rational homology spheres, Geometry & Topology 10 (2006), 331-334. (primary): https://doi.org/10.2140/gt.2006.10.331\n Evidence used: Gives a second infinite-tower construction, without settling infinitely many commensurability classes.\n\n**Review notes.** The two infinitude questions are logically separate; known infinitude of manifolds cannot be silently upgraded to known infinitude of commensurability classes.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2809, "problem_number": "KP-3.11", "title": "Kirby Problem 3.11", "statement": "Does every hyperbolic knot in the 3-sphere have meridian length at most 4?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.11.\n\nLiterature notes:\n(1) Every hyperbolic knot has a maximal cusp, the boundary of which is a Euclidean torus. The length of a slope on the knot exterior is the length of a geodesic representative of this slope in the Euclidean torus. This is a quantity that appears frequently in the theory of Dehn surgery. It was shown by Agol [Ago00] and Lackenby [Lac00] that when M is filled along a slope with length more than 6, then the resulting manifold is hyperbolic. Hence, the meridian of any knot in the 3-sphere is known to have length at most 6. The question is whether this upper bound can be improved to 4.\n\n(2) The problem is interesting for several reasons. First, it would show that the complements of knots in the 3-sphere have distinctive geometry. Secondly, it would also lead to improvements to many results about surgery on knots in the 3-sphere.\n\n(3) Examples of hyperbolic knots in the 3-sphere with meridian lengths tending to 4 from below were given by Agol [Ago00], who asked whether it is possible to find hyperbolic knots with meridian length greater than 4. Further examples were given by Purcell [Pur08]. The question is known to have a positive answer for 2-bridge knots [Ada96] and alternating knots [ACF+06].\n\n(4) Note that, in this problem, it is necessary to restrict to knots rather than links. Görner [Gör15] has examples of hyperbolic links in the 3-sphere where the shortest length of any slope on any boundary component is $\\sqrt{21}$ $\\approx$ 4.582. It would be interesting to know whether $\\sqrt{21}$ is optimal here.\n\nReferences cited:\n- [Ago00] Ian Agol. Bounds on exceptional Dehn filling. Geom. Topol., 4:431–449, 2000. doi: 10.2140/gt.2000.4.431.\n- [Lac00] Marc Lackenby. Word hyperbolic Dehn surgery. Invent. Math., 140(2):243–282, 2000. doi:10.1007/s002220000047.\n- [Pur08] Jessica S. Purcell. Slope lengths and generalized augmented links. Comm. Anal. Geom., 16(4):883–905, 2008. doi:10.4310/CAG.2008.v16.n4.a7.\n- [Ada96] Colin C. Adams. Hyperbolic 3-manifolds with two generators. Comm. Anal. Geom., 4(1-2):181–206, 1996. doi:10.4310/CAG.1996.v4.n2.a1.\n- [ACF+06] C. Adams, A. Colestock, J. Fowler, W. Gillam, and E. Katerman. Cusp size bounds from singular surfaces in hyperbolic 3-manifolds. Trans. Amer. Math. Soc., 358(2):727–741, 2006. doi:10.1090/S0002-9947-05-03662-7.\n- [Gör15] Matthias Görner. Regular tessellation link complements. Exp. Math., 24(2):225– 246, 2015. doi:10.1080/10586458.2014.986310.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every hyperbolic knot meridian has length at most 6, and the proposed bound 4 holds for important classes including alternating and 2-bridge knots, but the universal knot bound 4 remains open.\n\n**Verified partial progress.**\n\n- Agol's and Lackenby's 6-theorems imply the universal meridian-length bound 6.\n- Agol gives hyperbolic knots whose meridian lengths tend to 4 from below, showing 4 would be sharp.\n- The bound 4 is proved for 2-bridge knots and alternating knots.\n- The restriction to knots is essential: hyperbolic links can have every slope on every cusp longer than 4.\n\n**Full solution or refutation.**\n\nNo proof of the universal bound 4 and no hyperbolic knot counterexample with meridian length greater than 4 was located.\n\n**What remains.**\n\nProve meridian length at most 4 for every hyperbolic knot in S^3 or construct a knot with meridian length greater than 4.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.11. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the exact knot question and summarizes the universal 6 bound, sharpness evidence near 4, class-specific results, and link counterexamples.\n- Ian Agol, Bounds on exceptional Dehn filling, Geometry & Topology 4 (2000), 431-449. (primary): https://doi.org/10.2140/gt.2000.4.431\n Evidence used: Proves one version of the 6-theorem and gives knot examples with meridian lengths approaching 4 from below.\n- Colin C. Adams, Alan Colestock, Joshua Fowler, Will Gillam, and Emily Katerman, Cusp size bounds from singular surfaces in hyperbolic 3-manifolds, Transactions of the AMS 358 (2006), 727-741. (primary): https://doi.org/10.1090/S0002-9947-05-03662-7\n Evidence used: Establishes the cited meridian bound for alternating knots.\n\n**Review notes.** Link examples above 4 do not refute the knot-only statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2810, "problem_number": "KP-3.12", "title": "Kirby Problem 3.12", "statement": "(a) Considering all closed, orientable, $\\pi_1$-injective surfaces (possibly non-embedded) in all closed hyperbolic 3-manifolds, what is the infimum of the areas of all these surfaces?\n\n(b) One can ask the same question, but considering just closed, orientable, $\\pi_1$-injective surfaces of a fixed genus g.\n\n(c) One can also consider only closed, orientable, embedded $\\pi_1$-injective surfaces.\n\n(d) Alternatively, one can consider all closed, orientable, essential surfaces in all hyperbolic 3-manifolds M, allowing M not to be closed.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.12.\n\nLiterature notes:\n(1) A totally geodesic genus-two surface in a hyperbolic 3-manifold has area $4\\pi$. A universal lower bound for orientable surfaces is $2\\pi$ [Has95], based on unpublished work of Uhlenbeck. Hence, the answer to part (a) lies somewhere between $2\\pi$ and $4\\pi$.\n\n(2) A lower bound for the area of a closed nonorientable essential surface is $\\pi$, since such a surface is double covered by an orientable one.\n\n(3) Any closed orientable $\\pi_1$-injective surface is homotopic to a least area surface S, which is immersed [SY82, SU82]. Its sectional curvature $\\kappa$ is then everywhere at most $-1$. Gauss-Bonnet gives that the integral of $\\kappa$ over the surface is $2\\pi\\chi(S)$. On the other hand, the area is the integral of 1 over the surface. Hence, the more negative that $\\kappa$ is, on average over S, the smaller the area of S.\n\nReferences cited:\n- [Has95] Joel Hass. Acylindrical surfaces in 3-manifolds. Michigan Math. J., 42(2):357–365, 1995. doi:10.1307/mmj/1029005233.\n- [SY82] Richard Schoen and Shing Tung Yau. Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature. In Seminar on Differential Geometry, volume No. 102 of Ann. of Math. Stud., pages 209–228. Princeton Univ. Press, Princeton, NJ, 1982.\n- [SU82] J. Sacks and K. Uhlenbeck. Minimal immersions of closed Riemann surfaces. Trans. Amer. Math. Soc., 271(2):639–652, 1982. doi:10.2307/1998902.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For closed orientable pi_1-injective surfaces the known universal lower bound is 2 pi, while a totally geodesic genus-two example has area 4 pi; none of the four requested exact infima was located.\n\n**Verified partial progress.**\n\n- Hass proves the universal orientable lower bound 2 pi, based on work of Uhlenbeck.\n- A totally geodesic genus-two surface has area 4 pi, placing the unrestricted closed orientable infimum between 2 pi and 4 pi.\n- Least-area representative theory provides the natural minimal-surface framework for pi_1-injective immersions.\n\n**Full solution or refutation.**\n\nThe exact unrestricted, fixed-genus, embedded, and nonclosed-ambient infima remain undetermined in the literature checked.\n\n**What remains.**\n\nDetermine the exact infimum in part (a), its fixed-genus analogue, the embedded analogue, and the value when essential surfaces in nonclosed hyperbolic 3-manifolds are allowed.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.12. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States all four variants, records the interval from 2 pi to 4 pi, and explains the least-area surface framework.\n- Joel Hass, Acylindrical surfaces in 3-manifolds, Michigan Mathematical Journal 42 (1995), 357-365. (primary): https://doi.org/10.1307/mmj/1029005233\n Evidence used: Provides the cited universal lower bound for orientable essential surfaces.\n\n**Review notes.** The background supplies the intended induced-area/least-area reading, but an expert should confirm area and regularity conventions for all four variants.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2811, "problem_number": "KP-3.13", "title": "Kirby Problem 3.13", "statement": "Does every closed hyperbolic 3-manifold admit an immersed $\\pi_1$-injective surface with only double points? More precisely, if M is a closed, connected, hyperbolic 3-manifold, is there a closed, connected surface $\\Sigma$ (other than a 2-sphere) and an immersion f : $\\Sigma$ $\\to$ M such that $f_*$ : $\\pi_1(\\Sigma)$ $\\to$ $\\pi_1(M)$ is injective and $|f^{-1}(\\{x\\})|\\leq 2$ for all $x\\in M$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.13.\n\nLiterature notes:\n(1) This is true for Haken 3-manifolds, which by definition have embedded $\\pi_1$-injective surfaces, and for small Seifert-fibered spaces with infinite fundamental group, which can easily be seen to have immersed essential tori with only double points; hence the restriction to hyperbolic manifolds (by the Geometrization Theorem).\n\n(2) This is true for finite volume orientable connected non-compact hyperbolic 3-manifolds and for most Dehn fillings by [CL01], hence should be true for all but finitely many closed, hyperbolic 3-manifolds of bounded volume. The immersed surfaces produced in closed hyperbolic 3-manifolds by [KM12] will usually have triple points of intersection.\n\nReferences cited:\n- [CL01] D. Cooper and D. D. Long. Some surface subgroups survive surgery. Geom. Topol., 5:347–367, 2001. doi:10.2140/gt.2001.5.347.\n- [KM12] Jeremy Kahn and Vladimir Markovic. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Ann. of Math. (2), 175(3):1127–1190, 2012. doi: 10.4007/annals.2012.175.3.4.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The multiplicity-at-most-two immersion exists for Haken manifolds, finite-volume noncompact hyperbolic 3-manifolds, and most relevant Dehn fillings, while no theorem for every closed hyperbolic 3-manifold was located.\n\n**Verified partial progress.**\n\n- An embedded incompressible surface settles the condition for Haken manifolds.\n- Cooper and Long obtain the stated positive conclusion for finite-volume orientable noncompact hyperbolic 3-manifolds and most Dehn fillings.\n- Kahn and Markovic prove surface subgroups for every closed hyperbolic 3-manifold, but their immersed surfaces generally have triple points and do not settle the multiplicity-two constraint.\n\n**Full solution or refutation.**\n\nThe general surface-subgroup theorem is insufficient for the exact point-preimage bound, and no general replacement was located.\n\n**What remains.**\n\nConstruct a pi_1-injective immersion with every point having at most two preimages in each remaining closed hyperbolic 3-manifold, or find an obstruction.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.13. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Separates the exact multiplicity-two requirement from ordinary existence of an immersed surface subgroup.\n- D. Cooper and D. D. Long, Some surface subgroups survive surgery, Geometry & Topology 5 (2001), 347-367. (primary): https://doi.org/10.2140/gt.2001.5.347\n Evidence used: Establishes the noncompact and Dehn-filling positive cases cited in the current problem account.\n- Jeremy Kahn and Vladimir Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Annals of Mathematics 175 (2012), 1127-1190. (primary): https://doi.org/10.4007/annals.2012.175.3.4\n Evidence used: Proves existence of immersed pi_1-injective surfaces in every closed hyperbolic 3-manifold, but not the point-preimage bound.\n\n**Review notes.** Existence of a surface subgroup or arbitrary immersed essential surface is strictly weaker than the stored condition.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2812, "problem_number": "KP-3.14", "title": "Kirby Problem 3.14", "statement": "Can a hyperbolic knot complement in the 3-sphere contain a closed, embedded totally geodesic surface?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.14.\n\nLiterature notes:\n(1) In 1990, Menasco and Reid conjectured that the answer is no [MR92]. There is evidence for and against this conjecture. For example, as evidence that it may be false, there are embedded totally geodesic spanning surfaces [AS05]. There are closed, embedded, totally geodesic surfaces in hyperbolic link complements in the 3-sphere with just two link components [Lei06]. For any $\\epsilon>0$, there are closed, embedded surfaces in knot complements with principal curvatures no more than $\\epsilon$ [Lei06]. There are hyperbolic knots in rational homology 3-spheres with closed, embedded, totally geodesic surfaces in their complement [DeB06].\n\n(2) In favor of Menasco--Reid's conjecture, many known families of knots have been proved not to admit a closed, embedded, totally geodesic surface, such as alternating knots [MR92], knots with tunnel number 1 [MR92], Montesinos knots [Oer84], 3-bridge knots and double torus knots [IO00].\n\n(3) Related questions for other types of surfaces are also open. For example, can a hyperbolic knot complement in the 3-sphere contain a totally geodesic separating surface? What about one with meridional boundary components?\n\nReferences cited:\n- [MR92] William Menasco and Alan W. Reid. Totally geodesic surfaces in hyperbolic link complements. In Topology ’90 (Columbus, OH, 1990), volume 1 of Ohio State Univ. Math. Res. Inst. Publ., pages 215–226. de Gruyter, Berlin, 1992.\n- [AS05] Colin Adams and Eric Schoenfeld. Totally geodesic Seifert surfaces in hyperbolic knot and link complements. I. Geom. Dedicata, 116:237–247, 2005. doi:10.1007/s10711-005-9018-z.\n- [Lei06] Christopher J. Leininger. Small curvature surfaces in hyperbolic 3-manifolds. J. Knot Theory Ramifications, 15(3):379–411, 2006. doi: 10.1142/S0218216506004531.\n- [DeB06] Jason DeBlois. Totally geodesic surfaces and homology. Algebr. Geom. Topol., 6:1413–1428, 2006. doi:10.2140/agt.2006.6.1413.\n- [Oer84] Ulrich Oertel. Closed incompressible surfaces in complements of star links. Pacific J. Math., 111(1):209–230, 1984. http://projecteuclid.org/euclid.pjm/1102710789.\n- [IO00] Kazuhiro Ichihara and Makoto Ozawa. Hyperbolic knot complements without closed embedded totally geodesic surfaces. J. Austral. Math. Soc. Ser. A, 68(3):379–386, 2000.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Yes. DeBlois, Gharagozlou, and Hoffman construct four hyperbolic knot complements in S^3 containing closed embedded totally geodesic surfaces.\n\n**Verified partial progress.**\n\n- Earlier work ruled out such surfaces for many knot families and constructed analogous surfaces in link complements and knot complements in rational homology spheres.\n- The 2025 prism-orbifold construction supplies actual knot complements in S^3 and therefore crosses the exact remaining boundary.\n\n**Full solution or refutation.**\n\nThe four knot complements constructed in Knot complements decomposing into prisms cover prism orbifolds; the authors deduce that these complements contain closed embedded totally geodesic surfaces. This directly answers the existential question affirmatively.\n\n**What remains.**\n\nThe stored existential question is settled. Classification of all such knot complements and the related boundary-surface variants mentioned in the background remain separate questions.\n\n**Sources checked.**\n\n- Jason DeBlois, Arshia Gharagozlou, and Neil R. Hoffman, Knot complements decomposing into prisms, arXiv:2507.01263 (2025). (primary): https://arxiv.org/abs/2507.01263\n Evidence used: Constructs four hyperbolic knot complements in S^3 covering prism orbifolds and explicitly concludes that they contain closed embedded totally geodesic surfaces.\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.14. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Supplies the historical Menasco-Reid conjecture and earlier partial landscape; the decisive 2025 primary paper supersedes its open-status framing.\n\n**Review notes.** This is an affirmative solution of the question and a counterexample to the older conjecture that the answer should be no. No heavy computation was performed in this triage; the cited construction may use certified computational components.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2813, "problem_number": "KP-3.15", "title": "Kirby Problem 3.15", "statement": "Let $M$ be a closed hyperbolic 3-manifold with positive first Betti number.\n\n(a) Which elements of $H^{2}(M;\\mathbb{R})$ are realized as the Euler classes of taut foliations?\n\n(b) Which elements of $H^{2}(M;\\mathbb{R})$ are realized as the Euler classes of tight contact structures?\n\n(c) How about universally tight contact structures?\n\n(d) The question can also be asked for pseudo-Anosov flows...\n\n(e)... and for quasi-geodesic flows. (f ) How about Euler classes of representations into $\\operatorname{Homeo}^{+}(S^1)$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.15.\n\nLiterature notes:\n(1) For any compact, orientable 3-manifold M, Thurston [Thu86] defined a pseudo-norm on the second homology groups $H_2(M;\\mathbb{R})$ and $H_2(M,\\partial M;\\mathbb{R})$, now called the Thurston norm. The Thurston norm induces dual norms on $H^{2}(M;\\mathbb{R})$ and $H^{2}(M, \\partial{}M;\\mathbb{R})$. The unit ball of the Thurston dual norm is a convex polyhedron with integral vertices [Thu86].\n\n(2) Thurston showed that the Euler class $e(F)$ of any taut foliation $F$ has dual norm at most one, and if $F$ has any compact leaf, then the dual\n\nnorm of $e(F)$ is equal to one [Thu86]. Conversely, he conjectured that any integral class in $H^{2}(M;\\mathbb{R})$ of dual norm equal to one is the Euler class of a taut foliation on M [Thu86, page 129, Conjecture 3]. Thurston was aware that $e(F)$ satisfies the parity condition meaning that for every closed orientable surface $S$, the pairing $\\langle e(F),[S]\\rangle$ has the same parity as $\\chi(S)$. Gabai proved that every vertex of the dual unit ball is realized as the Euler class of a taut foliation [Gab97, Yaz20], but Gabai and Yazdi constructed counterexamples to Thurston's conjecture [GY20, Yaz20]. It is therefore not clear which elements of $H^{2}(M;\\mathbb{R})$ are realized as the Euler classes of taut foliations, which is the first of the above questions.\n\n(3) Tight contact structures also have an Euler class. Eliashberg proved that the Euler class of a tight contact structure has dual norm at most one [Eli92]. Every $C^0$ taut foliation (other than the product foliation on $S^2$ $\\times$ $S^1$ ) can be $C^0$-approximated by a universally tight contact structure, by Eliashberg--Thurston [ET98], Kazez--Roberts [KR17], and Bowden [Bow16]. So, for any closed, orientable, irreducible 3-manifold M, the set of Euler classes of universally tight contact structures on M includes Euler classes of taut foliations. It is possible that every integral class in $H^{2}(M;\\mathbb{R})$ of dual norm one and satisfying the parity condition is realized as the Euler class of a tight contact structure on M. Sivek and Yazdi [SY23] have shown that, in the counterexamples given by Gabai and Yazdi, the classes in $H^{2}(M;\\mathbb{R})$ that were proved not to be realized by taut foliations are realized by tight contact structures.\n\n(4) Pseudo-Anosov flows and quasi-geodesic flows also admit Euler classes, so one can ask the analogous question for them. Similarly, a representation $\\pi_1(M)\\to \\operatorname{Homeo}^{+}(S^1)$ defines a circle bundle over M, and hence an Euler class. This satisfies the Milnor--Wood inequality [Mil58, Woo71], and hence has dual norm at most one. It does not necessarily satisfy the parity condition. It is possible that every integral class in $H^{2}(M;\\mathbb{R})$ of dual norm one is realized by a representation to $\\operatorname{Homeo}^{+}(S^1)$.\n\nReferences cited:\n- [Thu86] William P. Thurston. A norm for the homology of 3-manifolds. Mem. Amer. Math. Soc., 59(339):i–vi and 99–130, 1986.\n- [Gab97] David Gabai. Problems in foliations and laminations. In Geometric topology (Athens, GA, 1993), volume 2 of AMS/IP Stud. Adv. Math., pages 1–33. Amer. Math. Soc., Providence, RI, 1997. doi:10.1090/amsip/002.2/01.\n- [Yaz20] Mehdi Yazdi. On Thurston’s Euler class-one conjecture. Acta Math., 225(2):313– 368, 2020. doi:10.4310/acta.2020.v225.n2.a3.\n- [GY20] David Gabai and Mehdi Yazdi. The fully marked surface theorem. Acta Math., 225(2):369–413, 2020. doi:10.4310/acta.2020.v225.n2.a4.\n- [Eli92] Yakov Eliashberg. Contact 3-manifolds twenty years since J. Martinet’s work. Ann. Inst. Fourier (Grenoble), 42(1-2):165–192, 1992. URL: http://www.numdam.org/item?id=AIF 1992 42 1-2 165 0.\n- [ET98] Yakov M. Eliashberg and William P. Thurston. Confoliations, volume 13 of University Lecture Series. American Mathematical Society, Providence, RI, 1998. doi:10.1090/ulect/013.\n- [KR17] William H. Kazez and Rachel Roberts. C0 approximations of foliations. Geom. Topol., 21(6):3601–3657, 2017. doi:10.2140/gt.2017.21.3601.\n- [Bow16] Jonathan Bowden. Approximating C0-foliations by contact structures. Geom. Funct. Anal., 26(5):1255–1296, 2016. doi:10.1007/s00039-016-0387-2.\n- [SY23] Steven Sivek and Mehdi Yazdi. Thurston norm and Euler classes of tight contact structures. Bull. Lond. Math. Soc., 55(6):2976–2990, 2023. doi:10.1112/blms.12905.\n- [Mil58] John W Milnor. On the existence of a connection with curvature zero. Comment. Math. Helv., 32(1):215–223, 1958.\n- [Woo71] John W. Wood. Bundles with totally disconnected structure group. Comment. Math. Helv., 46:257–273, 1971. doi:10.1007/BF02566843.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Necessary Thurston-norm and parity restrictions, realizability of vertices, counterexamples to the broad taut-foliation conjecture, and additional tight-contact realizations are known, but none of the requested general classifications is complete.\n\n**Verified partial progress.**\n\n- Euler classes of taut foliations obey the dual Thurston-norm bound and a parity condition, and vertices of the dual unit ball are realizable.\n- Gabai-Yazdi and Yazdi provide counterexamples to the conjecture that every eligible norm-one integral class is a taut-foliation Euler class.\n- Taut-foliation Euler classes are included among those of universally tight contact structures via contact approximation.\n- Sivek and Yazdi prove that the classes omitted by taut foliations in the known counterexamples are realized by tight contact structures.\n\n**Full solution or refutation.**\n\nThese results sharply constrain and separate the realizability sets, but do not classify them for taut foliations, tight or universally tight contact structures, pseudo-Anosov or quasigeodesic flows, or Homeo+(S^1) representations.\n\n**What remains.**\n\nCharacterize each of the six Euler-class realizability sets and determine which norm, parity, and dynamical conditions are sufficient.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.15. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the six classification questions and carefully summarizes the norm, parity, approximation, and counterexample results.\n- Mehdi Yazdi, Thurston norm and the Euler class, arXiv:2604.08096 (2026). (authoritative_secondary): https://arxiv.org/abs/2604.08096\n Evidence used: Provides a current survey of Euler classes associated to the structures appearing in the question.\n- David Gabai and Mehdi Yazdi, The fully marked surface theorem, Acta Mathematica 225 (2020), 369-413. (primary): https://doi.org/10.4310/acta.2020.v225.n2.a4\n Evidence used: Provides the modern taut-foliation machinery and counterexample context.\n- Steven Sivek and Mehdi Yazdi, Thurston norm and Euler classes of tight contact structures, Bulletin of the London Mathematical Society 55 (2023), 2976-2990. (primary): https://doi.org/10.1112/blms.12905\n Evidence used: Shows tight contact structures realize classes missing from taut foliations in the known counterexamples.\n\n**Review notes.** The exact statement has minor typography in parts (d)-(f), but its six intended structure classes are recoverable from the background.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2814, "problem_number": "KP-3.16", "title": "Kirby Problem 3.16", "statement": "Does every finite-volume hyperbolic 3-manifold contain infinitely many simple closed geodesics?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.16.\n\nLiterature notes:\nIt seems reasonable to suppose that, in some sense, a typical closed geodesic in a finite-volume hyperbolic 3-manifold does not self-intersect, and so one would expect the question to have a positive answer. Kuhlmann [Kuh06] proved that the answer is positive for cusped hyperbolic 3-manifolds. Moreover, she gave geometric conditions that imply that a closed hyperbolic 3-manifold contains infinitely many simple closed geodesics. Among the first 200 closed hyperbolic 3-manifolds in the census, she was able to verify these conditions for 178 of them [Kuh08]. It is straightforward to prove that every closed hyperbolic 3-manifold contains at least one simple geodesic, since a shortest geodesic is always simple.\n\nReferences cited:\n- [Kuh06] Sally M. Kuhlmann. Geodesic knots in cusped hyperbolic 3-manifolds. Algebr. Geom. Topol., 6:2151–2162, 2006. doi:10.2140/agt.2006.6.2151.\n- [Kuh08] Sally Kuhlmann. Geodesic knots in closed hyperbolic 3-manifolds. Geom. Dedicata, 131:181–211, 2008. doi:10.1007/s10711-007-9227-8.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Kuhlmann proves the assertion for cusped finite-volume hyperbolic 3-manifolds and gives sufficient conditions covering many closed examples, but the general closed case remains open.\n\n**Verified partial progress.**\n\n- Every cusped finite-volume hyperbolic 3-manifold contains infinitely many simple closed geodesics.\n- Explicit geometric conditions imply infinitude in the closed case.\n- Those conditions were verified for 178 of the first 200 closed census manifolds considered.\n- Every closed hyperbolic 3-manifold has at least one simple closed geodesic, since a shortest geodesic is simple.\n\n**Full solution or refutation.**\n\nThe noncompact case and a large class of closed cases are solved, but no theorem covering every closed finite-volume hyperbolic 3-manifold was located.\n\n**What remains.**\n\nProve the infinitude theorem for all closed hyperbolic 3-manifolds or construct a closed counterexample.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.16. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the general question and identifies precisely which cusped and closed cases are known.\n- Sally M. Kuhlmann, Geodesic knots in cusped hyperbolic 3-manifolds, Algebraic & Geometric Topology 6 (2006), 2151-2162. (primary): https://doi.org/10.2140/agt.2006.6.2151\n Evidence used: Proves infinitude for cusped hyperbolic 3-manifolds.\n- Sally Kuhlmann, Geodesic knots in closed hyperbolic 3-manifolds, Geometriae Dedicata 131 (2008), 181-211. (primary): https://doi.org/10.1007/s10711-007-9227-8\n Evidence used: Gives sufficient conditions in the closed case and census verification.\n\n**Review notes.** The finite-volume statement includes both cusped and closed manifolds; the unresolved portion is the general closed case.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2815, "problem_number": "KP-3.17", "title": "Kirby Problem 3.17", "statement": "Let $M_1$ and $M_2$ be finite-volume hyperbolic n--manifolds. If the length spectra of $M_1$ and $M_2$ coincide, must the two manifolds be commensurable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.17.\n\nLiterature notes:\n(1) For a hyperbolic manifold M, the length spectrum $L(M)$ is the ordered tuple of all lengths of closed geodesics in M, listed with multiplicity. The length spectrum is closely related to the eigenvalue spectrum $E(M)$ of the Laplace--Beltrami operator. Specifically, $L(M)$ determines $E(M)$, and $E(M)$ determines $L(M)$, the set of all lengths of closed geodesics, stripped of multiplicity information.\n\n(2) In all dimensions $n\\geq 2$, there are well-known constructions of isospectral twins: pairs of hyperbolic manifolds $M_1$ and $M_2$ for which the length spectra coincide, but that are not isometric. Vignéras constructed the first isospectral twins, using arithmetic methods [Vig80]. Subsequently, Sunada described a flexible group-theoretic construction in which the $M_i$ are covers of a fixed manifold or orbifold [Sun85]. In all known constructions of isospectral twins, the manifolds $M_1$ and $M_2$ are commensurable, meaning they share a common finite-sheeted cover. This prompted Reid to pose Problem 3.17 [Rei14, Question 2.1].\n\n(3) The question in this problem is known to have a positive answer for arithmetic manifolds in dimensions $n\\not\\equiv 1\\pmod{4}$. See Reid [Rei92]; Chinburg--Hamilton--Long--Reid [CHLR08]; and Prasad--Rapinchuk [PR09, PR15]. Outside the arithmetic setting, Futer and Millichap have constructed incommensurable pairs $M_1$ and $M_2$ that share an arbitrarily large finite portion of their length spectra, up to a cutoff that grows linearly with volume [FM17].\n\nReferences cited:\n- [Vig80] Marie-France Vignéras. Variétés riemanniennes isospectrales et non isométriques. Ann. of Math. (2), 112(1):21–32, 1980. doi:10.2307/1971319.\n- [Sun85] Toshikazu Sunada. Riemannian coverings and isospectral manifolds. Ann. of Math. (2), 121(1):169–186, 1985. doi:10.2307/1971195.\n- [Rei14] Alan W. Reid. Traces, lengths, axes and commensurability. Ann. Fac. Sci. Toulouse Math. (6), 23(5):1103–1118, 2014. doi:10.5802/afst.1438.\n- [Rei92] Alan W. Reid. Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds. Duke Math. J., 65(2):215–228, 1992. doi:10.1215/S0012-7094-92-06508-2.\n- [CHLR08] Ted Chinburg, Emily Hamilton, Darren D. Long, and Alan W. Reid. Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds. Duke Math. J., 145(1):25–44, 2008. doi:10.1215/00127094-2008-045.\n- [PR09] Gopal Prasad and Andrei S. Rapinchuk. Weakly commensurable arithmetic groups and isospectral locally symmetric spaces. Publ. Math. Inst. Hautes Études Sci., 109:113–184, 2009. doi:10.1007/s10240-009-0019-6.\n- [PR15] Gopal Prasad and Andrei S. Rapinchuk. Weakly commensurable groups, with applications to differential geometry. In Handbook of group actions. Vol. I, volume 31 of Adv. Lect. Math. (ALM), pages 495–524. Int. Press, Somerville, MA, 2015.\n- [FM17] David Futer and Christian Millichap. Spectrally similar incommensurable 3-manifolds. Proc. Lond. Math. Soc. (3), 115(2):411–447, 2017. doi:10.1112/plms.12045.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Coinciding full length spectra imply commensurability in substantial arithmetic settings, and all standard exact isospectral constructions are commensurable, but the general finite-volume hyperbolic question remains open.\n\n**Verified partial progress.**\n\n- Reid proves the relevant positive result for arithmetic hyperbolic 2- and 3-manifolds, with later arithmetic generalizations.\n- The current K3 account states a positive arithmetic answer in dimensions n not congruent to 1 modulo 4.\n- Sunada and Vigneras constructions of exact isospectral twins produce commensurable manifolds.\n- Futer and Millichap construct incommensurable pairs with arbitrarily long identical finite initial portions of their length spectra, showing that finite spectral agreement cannot settle the full question.\n\n**Full solution or refutation.**\n\nNo pair of incommensurable finite-volume hyperbolic manifolds with identical full length spectra, and no general proof of commensurability, was located.\n\n**What remains.**\n\nSettle the implication outside the known arithmetic regimes, while preserving the source convention that the full length spectrum includes multiplicity.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.17. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Defines the full length spectrum with multiplicity, states the arithmetic positive range, and distinguishes finite initial spectral agreement from exact equality.\n- Alan W. Reid, Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds, Duke Mathematical Journal 65 (1992), 215-228. (primary): https://doi.org/10.1215/S0012-7094-92-06508-2\n Evidence used: Proves commensurability from isospectrality in the cited arithmetic low-dimensional setting.\n- Gopal Prasad and Andrei S. Rapinchuk, Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publications Mathematiques de l'IHES 109 (2009), 113-184. (primary): https://doi.org/10.1007/s10240-009-0019-6\n Evidence used: Provides the broad arithmetic weak-commensurability and isospectral framework.\n- David Futer and Christian Millichap, Spectrally similar incommensurable 3-manifolds, Proceedings of the London Mathematical Society 115 (2017), 411-447. (primary): https://doi.org/10.1112/plms.12045\n Evidence used: Constructs incommensurable manifolds sharing arbitrarily long finite initial spectra, but not identical full spectra.\n\n**Review notes.** The source defines length spectrum with multiplicity. The stored n--manifolds spelling is a harmless TeX artifact. Finite initial agreement is not a counterexample to full-spectrum rigidity.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2816, "problem_number": "KP-3.18", "title": "Kirby Problem 3.18", "statement": "Is there a closed hyperbolic 3-manifold that is foliated with minimal leaves?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.18.\n\nLiterature notes:\nThis problem was raised by Thurston and Uhlenbeck around 1980; see [Uhl83, Page 154]. Each leaf of such a foliation is an area minimizing surface, that is, any compact subsurface of a leaf minimizes area among all homologous surfaces with the same boundary. Taut foliations can always be made minimal in some (but perhaps not the hyperbolic) Riemannian metric [Sul79a, Has86]. Hass and Thurston, at the 1984 Durham Symposium on Kleinian groups, 3-Manifolds, and Hyperbolic Geometry, gave examples of foliations in a hyperbolic manifold that cannot be made minimal [Has15]. See also [HW19]. Additional geometric\n\nconditions on a foliation can rule out the possibility of each leaf being minimal [WW20]. There is no local obstruction, since there are many foliations of $\\mathbb{H}^3$ by minimal planes.\n\nReferences cited:\n- [Uhl83] Karen K. Uhlenbeck. Closed minimal surfaces in hyperbolic 3-manifolds. In Semi-nar on minimal submanifolds, volume 103 of Ann. of Math. Stud., pages 147–168. Princeton Univ. Press, Princeton, NJ, 1983.\n- [Sul79a] Dennis Sullivan. A homological characterization of foliations consisting of minimal surfaces. Comment. Math. Helv., 54(2):218–223, 1979. doi:10.1007/BF02566269.\n- [Has86] Joel Hass. Minimal surfaces in foliated manifolds. Comment. Math. Helv., 61(1):1– 32, 1986. doi:10.1007/BF02621899.\n- [Has15] Joel Hass. Minimal fibrations of hyperbolic 3-manifolds, 2015. arXiv:1512.04145.\n- [HW19] Zheng Huang and Biao Wang. Complex length of short curves and minimal fibrations of hyperbolic three-manifolds fibering over the circle. Proceedings of the London Mathematical Society, 118(6):1305–1327, 2019.\n- [WW20] Michael Wolf and Yunhui Wu. Non-existence of geometric minimal foliations in hyperbolic three-manifolds. Comment. Math. Helv., 95(1):167–182, 2020. doi:10.4171/cmh/484.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Taut foliations can be made minimal for some Riemannian metric, and there are strong obstructions for specified foliations in the hyperbolic metric, but the existence of any closed hyperbolic 3-manifold foliated by hyperbolic-metric minimal leaves remains open.\n\n**Verified partial progress.**\n\n- Sullivan and Hass give metric-flexible characterizations or constructions making taut foliations minimal.\n- Hass gives hyperbolic fibered manifolds whose fibrations cannot be minimal in the hyperbolic metric.\n- Wolf and Wu establish further geometric nonexistence criteria.\n- There is no purely local obstruction, since hyperbolic 3-space admits foliations by minimal planes.\n\n**Full solution or refutation.**\n\nKnown results obstruct particular foliations or vary the ambient metric; neither provides the requested closed hyperbolic example nor rules out all such examples.\n\n**What remains.**\n\nConstruct a closed hyperbolic 3-manifold with a foliation all of whose leaves are minimal in its hyperbolic metric, or prove a global obstruction.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.18. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the hyperbolic-metric existence question and distinguishes metric-flexible results from genuine hyperbolic obstructions.\n- Dennis Sullivan, A homological characterization of foliations consisting of minimal surfaces, Commentarii Mathematici Helvetici 54 (1979), 218-223. (primary): https://doi.org/10.1007/BF02566269\n Evidence used: Provides the foundational metric-flexible characterization for minimal foliations.\n- Joel Hass, Minimal fibrations of hyperbolic 3-manifolds, arXiv:1512.04145 (2015). (primary): https://arxiv.org/abs/1512.04145\n Evidence used: Constructs hyperbolic fibred examples whose given fibrations cannot be minimal.\n- Michael Wolf and Yunhui Wu, Non-existence of geometric minimal foliations in hyperbolic three-manifolds, Commentarii Mathematici Helvetici 95 (2020), 167-182. (primary): https://doi.org/10.4171/cmh/484\n Evidence used: Proves geometric obstructions to minimal foliations without settling the unrestricted existential question.\n\n**Review notes.** Here minimal refers to mean curvature zero in the ambient hyperbolic metric, not merely to leaves having hyperbolic intrinsic geometry.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2817, "problem_number": "KP-3.19", "title": "Kirby Problem 3.19", "statement": "(a) Does every closed hyperbolic 3-manifold have a nowhere zero vector field whose lift to the universal cover has proper flow lines?\n\n(b) Can one ensure that the leaf space of the foliation by flow lines in the universal cover is Hausdorff ?\n\n(c) If so, the leaf space in the universal cover is homeomorphic to a plane and one can then ask: can the induced action of $\\pi_1(M)$ on a plane be compactified to an action on the closed disk?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.19.\n\nLiterature notes:\n(1) The basic form of this question is due to W. Thurston.\n\n(2) If the answer to part (c) is \"yes,\" it seems likely one can show the action on the boundary must be faithful, in which case there are non-examples, since the fundamental group of the Weeks manifold is known (by Calegari-- Dunfield [CD03]) not to act faithfully on a circle.\n\n(3) Quasigeodesic flows are proper in this sense. However it is known that there are examples of hyperbolic 3-manifolds that do not admit quasigeodesic flows, since such flows give rise to a faithful action on a universal circle, as in the previous remark, by [Cal06b].\n\nReferences cited:\n- [CD03] Danny Calegari and Nathan M. Dunfield. Laminations and groups of homeomorphisms of the circle. Invent. Math., 152(1):149–204, 2003. doi:10.1007/s00222-002-0271-6.\n- [Cal06b] Danny Calegari. Universal circles for quasigeodesic flows. Geom. Topol., 10:2271– 2298, 2006. doi:10.2140/gt.2006.10.2271.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Quasigeodesic flows provide proper lifted flow lines, Hausdorff-type structure, and universal-circle compactifications for broad classes, but some hyperbolic 3-manifolds admit no quasigeodesic flow and the weaker general questions remain unresolved.\n\n**Verified partial progress.**\n\n- Calegari constructs broad classes of quasigeodesic flows, including many manifolds with nontrivial second homology.\n- Quasigeodesic flows have proper lifted flow lines and lead to universal-circle boundary actions.\n- Calegari-Dunfield prove that the Weeks manifold group admits no faithful circle action, and the universal-circle theory therefore obstructs quasigeodesic flows on some closed hyperbolic 3-manifolds.\n- That obstruction does not rule out a non-quasigeodesic vector field satisfying only the weaker properness and Hausdorff requirements.\n\n**Full solution or refutation.**\n\nPositive quasigeodesic-flow subclasses and faithful-circle-action obstructions are known, but they neither prove nor disprove parts (a)-(c) in their full weaker generality.\n\n**What remains.**\n\nSettle existence of proper lifted flow lines for every closed hyperbolic 3-manifold, determine whether Hausdorff leaf space can always be achieved, and characterize when the induced plane action compactifies to the closed disk.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.19. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States all three progressively stronger questions and warns that the faithful-circle-action conclusion is conditional.\n- Danny Calegari, Quasigeodesic flows in hyperbolic three-manifolds, arXiv:math/9507216. (primary): https://arxiv.org/abs/math/9507216\n Evidence used: Develops existence and structure of quasigeodesic flows in substantial classes.\n- Danny Calegari, Universal circles for quasigeodesic flows, Geometry & Topology 10 (2006), 2271-2298. (primary): https://doi.org/10.2140/gt.2006.10.2271\n Evidence used: Constructs the universal-circle action associated to quasigeodesic flows.\n- Danny Calegari and Nathan M. Dunfield, Laminations and groups of homeomorphisms of the circle, Inventiones Mathematicae 152 (2003), 149-204. (primary): https://doi.org/10.1007/s00222-002-0271-6\n Evidence used: Provides the faithful-circle-action obstruction for the Weeks manifold group.\n\n**Review notes.** Nonexistence of a quasigeodesic flow is not a counterexample to the weaker condition in part (a); the source itself phrases the boundary-faithfulness step in part (c) as likely rather than proved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2818, "problem_number": "KP-3.20", "title": "Kirby Problem 3.20", "statement": "Let $M$ be a closed hyperbolic 3-manifold with a faithful homomorphism $\\rho:\\pi_1(M)\\to \\operatorname{Homeo}^{+}(\\mathbb{R})$. Prove that $M$ supports a dual transversely orientable, essential lamination $\\lambda$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.20.\n\nLiterature notes:\n(1) Let $\\rho$ be a faithful homomorphism as above. Suppose $\\lambda$ is a lamination on $M$, and let $\\widetilde{\\lambda}$ be its lift to the universal cover. Then $\\lambda$ is said to be dual to $\\rho$ if there is a map from the leaf space of $\\widetilde{\\lambda}$ to $\\mathbb{R}$ which is equivariant with respect to the actions of $\\pi_1(M)$. It is well known that $\\rho$ gives rise to a dual lamination that is transversely orientable; the problem asks whether there is one that is also essential.\n\n(2) There is a faithful homomorphism $\\rho$ as above if and only if $\\pi_1(M)$ is left-orderable, by [Con59] and [CR16]. In this case, the L-space Conjecture (Problem 3.48) asserts that $M$ supports a transversely orientable taut foliation. By contrast, this problem asks merely for a transversely orientable essential lamination, but one that is also dual to $\\rho$.\n\n(3) Morgan and Shalen [MS88] nearly solved the analogous problem where $\\pi_1(M)$ acts isometrically on an $\\mathbb{R}$-tree. They showed that $M$ supports an incompressible measured lamination, but they did not show that this lamination is dual to $\\rho$ in the corresponding sense.\n\n(4) There is a faithful homomorphism $\\rho:\\pi_1(M)\\to \\operatorname{Homeo}^{+}(\\mathbb{R})$ if and only if there is one with nontrivial image, by [BRW05].\n\nReferences cited:\n- [Con59] Paul Conrad. Right-ordered groups. Michigan Math. J., 6:267–275, 1959. http: //projecteuclid.org/euclid.mmj/1028998233.\n- [CR16] Adam Clay and Dale Rolfsen. Ordered groups and topology, volume 176 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2016. doi:10.1090/gsm/176.\n- [MS88] John W. Morgan and Peter B. Shalen. Degenerations of hyperbolic structures. III. Actions of 3-manifold groups on trees and Thurston’s compactness theorem. Ann. of Math. (2), 127(3):457–519, 1988. doi:10.2307/2007003.\n- [BRW05] Steven Boyer, Dale Rolfsen, and Bert Wiest. Orderable 3-manifold groups. Ann. Inst. Fourier (Grenoble), 55(1):243–288, 2005. URL: http://aif.cedram.org/item? id=AIF 2005 55 1 243 0.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Faithful actions on R are equivalent to left orderability and give dual orientable laminations, but essentiality of a lamination dual to the prescribed action remains open.\n\n**Verified partial progress.**\n\n- Morgan--Shalen nearly solve the analogous R-tree action problem.\n- L-space conjecture provides related taut-foliation motivation.\n\n**Full solution or refutation.**\n\nNo general essential dual lamination theorem was verified.\n\n**What remains.**\n\nEstablish essentiality while retaining duality to rho.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.20 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains existing dual lamination and outstanding essentiality requirement.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2819, "problem_number": "KP-3.21", "title": "Kirby Problem 3.21", "statement": "In this problem, all 3-manifolds are orientable, while all flows are considered up to orbit equivalence and are assumed to be transitive.\n\n(a) Are there only finitely many Anosov flows on a given closed 3-manifold?\n\n(b) Are there only finitely many pseudo-Anosov flows on a given closed 3-manifold?\n\n(c) Are there only finitely many quasigeodesic pseudo-Anosov flows on a given closed 3-manifold?\n\n(d) Are there only finitely many pseudo-Anosov flows without perfect fits on a given closed 3-manifold?\n\n(e) Are there only finitely many veering triangulations on a given finite volume cusped hyperbolic 3-manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.21.\n\nLiterature notes:\n(1) Two flows on a 3-manifold are orbit equivalent if there is an orientation-preserving homeomorphism of the manifold sending orbits of one flow to those of the other. A flow is transitive if it has a dense orbit; every pseudo-Anosov flow on a closed, atoroidal (in particular, hyperbolic) 3-manifold is automatically transitive [Mos92].\n\n(2) Note that (an affirmative answer to) (b) implies (c) implies [Fen16] (d), and also that (b) implies (a).\n\n(3) Barthelme, Mann, and Bowden have made partial progress on (a), showing that there are finitely many contact Anosov flows on a closed 3-manifold [BM24b]. Towards (b), Zung has shown that there are only finitely many transitive pseudo-Anosov flows with positive Birkhoff sections on a given rational homology 3-sphere [Zun25]; see also [BSZ25].\n\n(4) A veering triangulation on a 3-manifold is an ideal triangulation with a particular combinatorial structure. These were first introduced by Agol on punctured mapping tori of pseudo-Anosov surface homeomorphisms [Ago11]. Agol and Guéritaud later showed more generally that if a 3-manifold admits a pseudo-Anosov flow without perfect fits, then there is a canonical veering triangulation in the complement of the singular orbits (see [LMT23]). Frankel, Schleimer, and Segerman have outlined an inverse to this construction, which should provide a one-to-one correspondence between veering triangulations and pseudo-Anosov flows without perfect fits; see [FSS22], [SS24a]. Tsang independently proved such a correspondence in his thesis [Tsa23] via a different approach based on work in [AT24].\n\n(5) Given the correspondence between pseudo-Anosov flows and veering triangulations, one can show that an affirmative answer to (e) would imply an affirmative answer to (d).\n\nReferences cited:\n- [Mos92] Lee Mosher. Dynamical systems and the homology norm of a 3-manifold. I. Efficient intersection of surfaces and flows. Duke Math. J., 65(3):449–500, 1992. doi:10.1215/S0012-7094-92-06518-5.\n- [Fen16] Sérgio R. Fenley. Quasigeodesic pseudo-Anosov flows in hyperbolic 3-manifolds and connections with large scale geometry. Adv. Math., 303:192–278, 2016. doi:10.1016/j.aim.2016.05.015.\n- [BM24b] Thomas Barthelmé and Kathryn Mann. Orbit equivalences of R-covered Anosov flows and hyperbolic-like actions on the line. Geom. Topol., 28(2):867–899, 2024. Appendix written jointly with Jonathan Bowden. doi:10.2140/gt.2024.28.867.\n- [Zun25] Jonathan Zung. Pseudo-Anosov representatives of stable Hamiltonian structures. J. Fixed Point Theory Appl., 27(4):Paper No. 87, 29, 2025. doi:10.1007/s11784-025-01238-8.\n- [BSZ25] John A. Baldwin, Steven Sivek, and Jonathan Zung. Pseudo-Anosov flows on hyperbolic l-spaces, 2025. arXiv:2505.21113.\n- [Ago11] Ian Agol. Ideal triangulations of pseudo-Anosov mapping tori. In Topology and geometry in dimension three, volume 560 of Contemp. Math., pages 1–17. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/conm/560/11087.\n- [LMT23] Michael P. Landry, Yair N. Minsky, and Samuel J. Taylor. Flows, growth rates, and the veering polynomial. Ergodic Theory Dynam. Systems, 43(9):3026–3107, 2023. doi:10.1017/etds.2022.63.\n- [FSS22] Steven Frankel, Saul Schleimer, and Henry Segerman. From veering triangulations to link spaces and back again, 2022. arXiv:1911.00006.\n- [SS24a] Saul Schleimer and Henry Segerman. From loom spaces to veering triangulations. Groups Geom. Dyn., 18(2):419–462, 2024. doi:10.4171/ggd/742.\n- [Tsa23] Chi Cheuk Tsang. Veering Triangulations and Pseudo-Anosov Flows. ProQuest LLC, Ann Arbor, MI, 2023. Thesis (Ph.D.)–University of California, Berkeley. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt=info: ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqm\\&rft dat=xri:pqdiss:30487016.\n- [AT24] Ian Agol and Chi Cheuk Tsang. Dynamics of veering triangulations: infinitesimal components of their flow graphs and applications. Algebr. Geom. Topol., 24(6):3401–3453, 2024. doi:10.2140/agt.2024.24.3401.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finiteness is proved for contact Anosov flows and for certain pseudo-Anosov flows with positive Birkhoff sections, but all general flow-finiteness questions remain open.\n\n**Verified partial progress.**\n\n- Barthelme--Mann--Bowden prove finiteness of contact Anosov flows.\n- Zung proves a rational-homology-sphere/Birkhoff-section special case.\n\n**Full solution or refutation.**\n\nNo general finite classification was verified.\n\n**What remains.**\n\nResolve the listed Anosov, pseudo-Anosov, quasigeodesic and veering variants.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.21 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States known special finiteness theorems and implications.\n\n**Review notes.** Multi-part record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2820, "problem_number": "KP-3.22", "title": "Kirby Problem 3.22", "statement": "Let $G=\\pi_1(M)$ be the fundamental group of a finite-volume hyperbolic 3-manifold $M$. What is the regularity of the smoothest (virtual) action of $G$ on $S^1$? On the interval?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.22.\n\nLiterature notes:\n(1) Not all hyperbolic 3--manifold groups can act faithfully on the interval, as this is equivalent to left orderability. Calegari--Dunfield showed that the Weeks manifold group cannot act faithfully on the interval [CD03], and having finite first homology is an obstruction for a finitely generated group to act by diffeomorphisms on the interval by Thurston Stability [Thu74b].\n\n(2) Every hyperbolic 3--manifold group has a finite index subgroup that embeds into a right-angled Artin group and therefore admits a faithful $C^1$ action on the circle by [FF03, Jor12]. It is unclear whether higher levels of regularity can be achieved, or whether the existence of such an action persists in the ambient supergroup; moreover, it is interesting to consider whether or not the $C^1$ actions of finite index subgroups are conjugate to higher regularity actions, or if there are other actions of higher regularity not arising from right-angled Artin groups.\n\n(3) One can ask similar questions about other geometric 3--manifold groups, or indeed general 3--manifold groups. In some cases, the answers are known or partially known (e.g. Euclidean manifold groups virtually admit $C^\\infty$ actions whereas nilmanifold fundamental groups cannot admit faithful $C^2$ actions).\n\n(4) Some nonexistence results are known for actions of fibered 3--manifold groups which lie in a sufficiently small neighborhood of the identity; [BKKT20].\n\n(5) Actions of fundamental groups on the interval and the circle are closely related to further topological structure that a 3--manifold may have, such as certain taut foliations, Anosov flows, and so on [CR16]. Are the actions arising from these kinds of structures conjugate to smoother actions?\n\nReferences cited:\n- [CD03] Danny Calegari and Nathan M. Dunfield. Laminations and groups of homeomorphisms of the circle. Invent. Math., 152(1):149–204, 2003. doi:10.1007/s00222-002-0271-6.\n- [Thu74b] William P. Thurston. A generalization of the Reeb stability theorem. Topology, 13:347–352, 1974. doi:10.1016/0040-9383(74)90025-1.\n- [FF03] Benson Farb and John Franks. Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups. Ergodic Theory Dynam. Systems, 23(5):1467–1484, 2003. doi: 10.1017/S0143385702001712.\n- [Jor12] Eduardo Jorquera. A universal nilpotent group of C1 diffeomorphisms of the interval. Topology Appl., 159(8):2115–2126, 2012. doi:10.1016/j.topol.2012.02.003.\n- [BKKT20] Christian Bonatti, Sang-hyun Kim, Thomas Koberda, and Michele Triestino. Small C1 actions of semidirect products on compact manifolds. Algebr. Geom. Topol., 20(6):3183–3203, 2020. doi:10.2140/agt.2020.20.3183.\n- [CR16] Adam Clay and Dale Rolfsen. Ordered groups and topology, volume 176 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2016. doi:10.1090/gsm/176.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finite-index subgroups of all hyperbolic 3-manifold groups have faithful C1 circle actions via RAAG embeddings, while optimal regularity and interval actions remain open.\n\n**Verified partial progress.**\n\n- Weeks group obstructs a faithful interval action.\n- Virtual RAAG embedding yields faithful C1 circle actions.\n\n**Full solution or refutation.**\n\nNo full regularity classification was verified.\n\n**What remains.**\n\nDetermine maximum regularity and whether actions extend to ambient groups.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.22 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records obstruction and virtual C1 construction.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2821, "problem_number": "KP-3.23", "title": "Kirby Problem 3.23", "statement": "What is the Margulis constant in dimension 3? Is it realized uniquely by the Weeks manifold W, where $\\mu(W)$ = 0.77442...?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.23.\n\nLiterature notes:\n(1) The fundamental Margulis lemma in dimension 3 asserts that there is an $\\epsilon>0$ so that for any discrete subgroup $\\Gamma$ of $\\operatorname{Isom}(\\mathbb{H}^3)$, if $f,g\\in\\Gamma$ and\n\neach move some point $x\\in\\mathbb{H}^3$ less than $\\epsilon$, then the subgroup generated by f and g has an abelian subgroup of finite index. If $\\Gamma=\\pi_1(N)$ for N a complete, orientable hyperbolic 3-manifold, then $\\langle f,g\\rangle$ is abelian. The Margulis lemma is the foundation of the thick-thin decomposition of hyperbolic 3-manifolds; see [Thu97, §5]. For a given $\\Gamma$, call the supremum $\\mu(\\Gamma)$ of such $\\epsilon$ the Margulis number of $\\Gamma$. For a class of discrete subgroups of $\\operatorname{Isom}(\\mathbb{H}^3)$ one can define the Margulis constant as the infimum of the $\\mu(\\Gamma)$ over all groups $\\Gamma$ in that class. The problem asks for the Margulis constant $\\mu_3$, the infimum over the fundamental groups of all complete, finite volume, orientable hyperbolic 3-manifolds.\n\n(2) There are many partial results in this direction, e.g. [Mey87], [Sha11], [Sha13], [CS92], [CS12b], [FPS22].\n\nReferences cited:\n- [Thu97] William P. Thurston. Three-dimensional geometry and topology. Vol. 1, volume 35 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy.\n- [Mey87] Robert Meyerhoff. A lower bound for the volume of hyperbolic 3-manifolds. Canad. J. Math., 39(5):1038–1056, 1987. doi:10.4153/CJM-1987-053-6.\n- [Sha11] Peter B. Shalen. A generic Margulis number for hyperbolic 3-manifolds. In Topology and geometry in dimension three, volume 560 of Contemp. Math., pages 103–109. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/conm/560/11094.\n- [Sha13] Peter B. Shalen. Small optimal Margulis numbers force upper volume bounds. Trans. Amer. Math. Soc., 365(2):973–999, 2013. doi:10.1090/S0002-9947-2012-05657-1.\n- [CS92] Marc Culler and Peter B. Shalen. Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds. J. Amer. Math. Soc., 5(2):231–288, 1992. doi:10.2307/2152768.\n- [CS12b] Marc Culler and Peter B. Shalen. Margulis numbers for Haken manifolds. Israel J. Math., 190:445–475, 2012. doi:10.1007/s11856-011-0189-z.\n- [FPS22] David Futer, Jessica S. Purcell, and Saul Schleimer. Effective bilipschitz bounds on drilling and filling. Geom. Topol., 26(3):1077–1188, 2022. doi:10.2140/gt.2022.26.1077.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many partial Margulis-number bounds are known, but the exact dimension-three constant and unique Weeks-manifold realization claim remain open.\n\n**Verified partial progress.**\n\n- The Margulis lemma and thick-thin theory give positive universal bounds.\n- Weeks numerical value motivates the conjecture.\n\n**Full solution or refutation.**\n\nNo exact infimum/uniqueness proof was verified.\n\n**What remains.**\n\nDetermine mu_3 and prove or refute Weeks uniqueness.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.23 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the constant and maintains the Weeks conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2822, "problem_number": "KP-3.24", "title": "Kirby Problem 3.24", "statement": "(a) (Cannon Conjecture) If G is a finitely presented, Gromov hyperbolic group with space at infinity equal to the 2-sphere, must G be a cocompact Kleinian group?\n\n(b) More generally, if $G$ is a Gromov hyperbolic group whose boundary is planar, is G virtually a convex cocompact Kleinian group?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.24.\n\nLiterature notes:\n(1) The first of these questions is Cannon's conjecture [Can94, CS98].\n\n(2) Recall that a Gromov hyperbolic group G has a space at infinity $\\partial_\\infty G$, which is the space of geodesic rays based at some basepoint, up to a suitable notion of equivalence. A group is Kleinian if it acts properly discontinuously on $\\mathbb{H}^3$. Cocompact Kleinian groups have, as a finite-index subgroup, the fundamental group of a closed hyperbolic 3-manifold.\n\n(3) It was shown by Bestvina and Mess [BM91] that a Gromov hyperbolic group is a PD$_3$-group if and only if it is torsion free and has space at infinity the 2-sphere. Hence, Cannon's conjecture is equivalent to Problem 3.38 restricted to PD$_3$-groups that are Gromov hyperbolic.\n\n(4) Any group G acts on itself by isometries, and in the case of Gromov hyperbolic groups, this extends to an action of $\\partial_\\infty G$. So, in the setting of this problem, G comes equipped with an action on the 2-sphere, and the question is whether one can identify the 2-sphere with the Riemann sphere in such a way that the action is by Möbius transformations. A good deal of work has been done on the analytic quality of the action of $G$ on $\\partial_\\infty G$ (see, for example, [BK05, BK02b, BK02a]).\n\n(5) Markovic has shown that Cannon's conjecture holds for a group G if it has, in a certain sense, enough surface subgroups [Mar13]. More precisely, the conjecture is true if every pair of points in $\\partial_\\infty G$ are separated by the circle at infinity of a quasi-convex surface subgroup. By work of Kahn and Markovic [KM12], Kleinian groups always satisfy this hypothesis.\n\n(6) A higher-dimensional version of Cannon's conjecture is known to be true, by work of Bartels, Lück, and Weinberger [BLW10]: if $G$ is a Gromov\n\nhyperbolic group whose boundary is homeomorphic to $S^{n-1}$ with $n\\geq 7$, then $G$ is the fundamental group of an aspherical closed n-dimensional manifold.\n\n(7) Question (b) in the case where the boundary is homeomorphic to the limit set of a convex cocompact Kleinian group was asked by Walsh in [DHM15]. When $\\partial G$ is homeomorphic to $S^1$ the answer to (b) is yes; this is due to Tukia [Tuk88], Gabai [Gab92] and Casson--Jungreis [CJ94]. When $\\partial G$ contains no Sierpinski carpet and G has no 2-torsion, the answer to (b) is again yes, due to Haissinsky [Haï15]. Question (b) in the case where $\\partial G$ is a Sierpinski carpet is a conjecture of Kapovich--Kleiner [KK00].\n\nReferences cited:\n- [Can94] James W. Cannon. The combinatorial Riemann mapping theorem. Acta Math., 173(2):155–234, 1994. doi:10.1007/BF02398434.\n- [CS98] J. W. Cannon and E. L. Swenson. Recognizing constant curvature discrete groups in dimension 3. Trans. Amer. Math. Soc., 350(2):809–849, 1998. doi:10.1090/S0002-9947-98-02107-2.\n- [BM91] Mladen Bestvina and Geoffrey Mess. The boundary of negatively curved groups. J. Amer. Math. Soc., 4(3):469–481, 1991. doi:10.2307/2939264.\n- [BK05] Mario Bonk and Bruce Kleiner. Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary. Geom. Topol., 9:219–246, 2005. doi:10.2140/gt.2005.9.219.\n- [BK02b] Mario Bonk and Bruce Kleiner. Rigidity for quasi-Möbius group actions. J. Differential Geom., 61(1):81–106, 2002. URL: http://projecteuclid.org/euclid.jdg/1090351321.\n- [BK02a] Mario Bonk and Bruce Kleiner. Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math., 150(1):127–183, 2002. doi:10.1007/s00222-002-0233-z.\n- [Mar13] Vladimir Markovic. Criterion for Cannon’s conjecture. Geom. Funct. Anal., 23(3):1035–1061, 2013. doi:10.1007/s00039-013-0228-5.\n- [KM12] Jeremy Kahn and Vladimir Markovic. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Ann. of Math. (2), 175(3):1127–1190, 2012. doi: 10.4007/annals.2012.175.3.4.\n- [BLW10] Arthur Bartels, Wolfgang Lück, and Shmuel Weinberger. On hyperbolic groups with spheres as boundary. J. Differential Geom., 86(1):1–16, 2010. http://projecteuclid.org/euclid.jdg/1299766682.\n- [DHM15] Kelly Delp, Diane Hoffoss, and Jason Fox Manning. Problems in groups, geometry, and three-manifolds, 2015. arXiv:1512.04620.\n- [Tuk88] Pekka Tukia. Homeomorphic conjugates of Fuchsian groups. J. Reine Angew. Math., 391:1–54, 1988. doi:10.1515/crll.1988.391.1.\n- [Gab92] David Gabai. Convergence groups are Fuchsian groups. Ann. of Math. (2), 136(3):447–510, 1992. doi:10.2307/2946597.\n- [CJ94] Andrew Casson and Douglas Jungreis. Convergence groups and Seifert fibered 3-manifolds. Invent. Math., 118(3):441–456, 1994. doi:10.1007/BF01231540.\n- [Haï15] Peter Haı̈ssinsky. Hyperbolic groups with planar boundaries. Invent. Math., 201(1):239–307, 2015. doi:10.1007/s00222-014-0552-x.\n- [KK00] Michael Kapovich and Bruce Kleiner. Hyperbolic groups with low-dimensional boundary. Ann. Sci. École Norm. Sup. (4), 33(5):647–669, 2000. doi:10.1016/S0012-9593(00)01049-1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Cannon's conjecture and the planar-boundary virtual convex-cocompact-Kleinian generalization remain open.\n\n**Verified partial progress.**\n\n- Boundary S2 characterizes torsion-free hyperbolic PD3 groups, relating Cannon to the PD3 problem.\n\n**Full solution or refutation.**\n\nNo general Kleinian realization theorem was verified.\n\n**What remains.**\n\nConstruct the conformal/Kleinian action or give counterexamples.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.24 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintains both Cannon and planar-boundary questions.\n\n**Review notes.** Open is source-backed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2823, "problem_number": "KP-3.25", "title": "Kirby Problem 3.25", "statement": "(Bending Conjecture). (a) Is a quasi-Fuchsian group determined by the hyperbolic metric on the boundary of its convex core?\n\n(b) Is a quasi-Fuchsian group with parabolics, other than a Fuchsian group, determined by the bending measure on the boundary of its convex core?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.25.\n\nLiterature notes:\nLet $S$ be a closed surface (possibly with punctures) and let $\\Gamma=\\pi_1(S)$. Consider the space $QH(S)$ of quasi-Fuchsian representations, which corresponds to quasi-conformal deformations of Fuchsian representations, that is to the interior points of the space $AH(S)$ of hyperbolic structures on $S\\times\\mathbb{R}$ (such that the punctures of $S$ are mapped to parabolic elements). Quasi-Fuchsian groups are important examples of Kleinian groups with infinite covolume.\n\nLet $\\rho\\in QH(S)$ and let $\\Gamma_\\rho=\\rho(\\Gamma)$. The manifold $M_\\rho:=\\mathbb{H}^3/\\rho(\\Gamma)$ is homeomorphic to $S\\times\\mathbb{R}$. The limit set $\\Lambda_\\rho$ is a Jordan curve and its complement $\\Omega_\\rho=\\mathbb{CP}^1\\setminus\\Lambda_\\rho$ is a disjoint union of two topological disks: $\\Omega_\\rho=\\Omega_\\rho^+\\sqcup\\Omega_\\rho^-$. Bers' Simultaneous Uniformization Theorem proves that $QH(S)$ can be parametrized by the conformal structures of $\\Omega_\\rho/\\Gamma_\\rho$, that is by two copies of Teichmüller spaces $\\mathcal{T}(S)$.\n\nThe manifold $M_\\rho$ has a convex core $C(\\rho)$, which corresponds to the smallest convex subset of $M_\\rho$ such that the inclusion is a homotopy equivalence. This convex core can be defined as the quotient by $\\rho$ of the convex hull of the limit set of $\\rho$. In particular, provided $\\rho$ is not Fuchsian, $C(\\rho)$ is homeomorphic to $S\\times[-1,1]$. The boundary of $C(\\rho)$ is the disjoint union of two embedded pleated surfaces, which are surfaces totally geodesic almost everywhere, and bent along a measured geodesic lamination. In particular, from the pleated surfaces $\\partial C(\\rho)^+$ and $\\partial C(\\rho)^-$ one can consider the induced metric (that is, a point in $\\mathcal{T}(S)$) and the amount of bending, which is quantified by the bending laminations (that is, a point in the space $ML(S)$ of measured laminations on $S$). This defines two maps:\n\n$$\n\\mu:QH(S)\\to\\mathcal{T}(S)\\times\\mathcal{T}(S),\n$$\n\nand\n\n$$\n\\beta:QH(S)\\setminus F(S)\\to ML(S)\\times ML(S),\n$$\n\nwhere $F(S)$ is the space of Fuchsian representations. W. Thurston conjectured that the maps $\\mu$ and $\\beta$ are homeomorphisms onto their images.\n\nAs steps in proving Thurston's conjecture, Sullivan [Sul85] proved that the map $\\mu$ is surjective, and Bonahon and Otal [BO04] described the image of the map $\\beta$, which consists of pairs of laminations on $S$ that fill and such that the measure on each isolated leaf is less than $\\pi$. Dular and Schlenker [DS24b] in 2024 showed the injectivity of the map $\\beta$ when $S$ has no punctures. Unfortunately, their proof does not show that quasi-Fuchsian manifolds are infinitesimally rigid with respect to the measured bending lamination on the boundary of their convex core. As Bonahon [Bon96] proved, this infinitesimal rigidity is equivalent to the infinitesimal rigidity with respect to the induced metric on the boundary of the convex core.\n\nWe can then state the following still open questions. The first two questions are (a) and (b) above:\n\n(i) When $S$ is closed, is the map $\\mu$ a homeomorphism?\n\n(ii) When $S$ has punctures, are the maps $\\mu$ and $\\beta$ homeomorphisms?\n\n(iii) Are the maps $\\mu$ and $\\beta$ infinitesimally rigid?\n\nPartial results for the map $\\beta$ are due to Series [Ser06] in the case of the once-punctured torus $S_{1,1}$, Bonahon [Bon05] in a neighborhood of the Fuchsian locus, and Hodgson--Kerckhoff [HK98] for quasi-Fuchsian manifolds bent along rational laminations. (Guéritaud [Gué09] gave an alternative and independent proof of Series' result.)\n\nAnti-de Sitter geometry is a Lorentzian analogue of hyperbolic geometry and the space of globally hyperbolic maximal compact (GHMC) AdS manifolds on $S\\times\\mathbb{R}$ has many similarities with the space of quasi-Fuchsian manifolds discussed above. In particular, the conjectures discussed above have an analogue also in this context. In [BDMS21] you can see a discussion for the universal version of the same problem both in hyperbolic and anti-de Sitter space.\n\nReferences cited:\n- [Sul85] Dennis Sullivan. Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups. Acta Math., 155(3-4):243–260, 1985. doi:10.1007/BF02392543.\n- [BO04] Francis Bonahon and Jean-Pierre Otal. Laminations measurées de plissage des variétés hyperboliques de dimension 3. Ann. of Math. (2), 160(3):1013–1055, 2004. doi:10.4007/annals.2004.160.1013.\n- [DS24b] Bruno Dular and Jean-Marc Schlenker. Convex co-compact hyperbolic manifolds are determined by their pleating lamination, 2024. arXiv:2403.10090.\n- [Bon96] Francis Bonahon. Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form. Ann. Fac. Sci. Toulouse Math. (6), 5(2):233–297, 1996. URL: http://www.numdam.org/item?id=AFST 1996 6 5 2 233 0.\n- [Ser06] Caroline Series. Thurston’s bending measure conjecture for once punctured torus groups. In Spaces of Kleinian groups, volume 329 of London Math. Soc. Lecture Note Ser., pages 75–89. Cambridge Univ. Press, Cambridge, 2006.\n- [Bon05] Francis Bonahon. Kleinian groups which are almost Fuchsian. J. Reine Angew. Math., 587:1–15, 2005. doi:10.1515/crll.2005.2005.587.1.\n- [HK98] Craig D. Hodgson and Steven P. Kerckhoff. Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. J. Differential Geom., 48(1):1–59, 1998. http://projecteuclid.org/euclid.jdg/1214460606.\n- [Gué09] François Guéritaud. Triangulated cores of punctured-torus groups. J. Differential Geom., 81(1):91–142, 2009. http://projecteuclid.org/euclid.jdg/1228400629.\n- [BDMS21] Francesco Bonsante, Jeffrey Danciger, Sara Maloni, and Jean-Marc Schlenker. The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti–de Sitter geometry. Geom. Topol., 25(6):2827–2911, 2021. With an appendix by Boubacar Diallo. doi:10.2140/gt.2021.25.2827.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No general proof of the two Bending Conjecture uniqueness assertions was verified.\n\n**Verified partial progress.**\n\n- Bers simultaneous uniformization parameterizes quasi-Fuchsian groups by conformal boundary data.\n\n**Full solution or refutation.**\n\nThe convex-core metric/bending-data determination questions remain open.\n\n**What remains.**\n\nProve rigidity from the stated boundary metric or bending measure data.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.25 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Sets out the conjectures against known Bers parameterization.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2824, "problem_number": "KP-3.26", "title": "Kirby Problem 3.26", "statement": "Let $M$ be a finite-volume hyperbolic 3-manifold, and let $M^1$ be a minimal-index finite cover of $M$ such that $\\pi_1(M^1)$ embeds in a right-angled Artin group. Can one obtain good control over the complexity of the smallest right-angled Artin group containing $\\pi_1(M^1)$ from the data of $M$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.26.\n\nLiterature notes:\n(1) Work of Agol and Wise [Ago13, Wis21] established that there is always such a finite cover $M^1$.\n\n(2) The complexity of a right-angled Artin group could be interpreted in many different ways, including:\n\n(i) Number of vertices in the defining graph; (ii) Cohomological dimension; (iii) Chromatic number of the defining graph; (iv) Diameter of the defining graph.\n\n(3) This question is related to, but different from questions such as finding the minimal index of a subgroup contained in a right-angled Artin group, the minimal index Haken cover, etc.\n\nReferences cited:\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [Wis21] Daniel T. Wise. The structure of groups with a quasiconvex hierarchy, volume 209 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, [2021] ©2021.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Virtual specialness guarantees a finite cover embedding in a RAAG, but effective control of the smallest cover and RAAG complexity from M is open.\n\n**Verified partial progress.**\n\n- Agol--Wise establish existence of an appropriate finite cover.\n\n**Full solution or refutation.**\n\nThe requested quantitative control is unavailable.\n\n**What remains.**\n\nBound cover index and defining-graph complexity by computable manifold data.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.26 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States virtual existence and multiple unresolved complexity measures.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2825, "problem_number": "KP-3.27", "title": "Kirby Problem 3.27", "statement": "(a) What is the computational complexity of the homeomorphism problem for compact, orientable 3-manifolds?\n\n(b) Is there a polynomial-time algorithm to recognize the 3-sphere?\n\n(c) What is the computational complexity of the recognition problem for other 3-manifolds?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.27.\n\nLiterature notes:\n(1) The homeomorphism problem for 3-manifolds is as follows. One is given triangulations of two 3-manifolds, M and N, and the problem asks whether N is homeomorphic to M.\n\n(2) For a fixed 3-manifold M, its recognition problem is as follows. One is given a triangulation of a 3-manifold N, and the problem asks whether N is homeomorphic to M. It is a restricted version of the homeomorphism problem, where one of the manifolds is fixed.\n\n(3) The homeomorphism problem for compact orientable 3-manifolds has been solved [Kup19a], [SS14]. However, there is currently a very large gap between known upper bounds on its complexity and known lower bounds. If we are given two 3-manifolds triangulated with $t_1$ and $t_2$ tetrahedra, the fastest known running time for an algorithm to determine whether they are homeomorphic is\n\n$$\n2^{2^{\\cdot^{\\cdot^{\\cdot^{t_1+t_2}}}}}\n$$\n\nwhere the tower of exponentials has some fixed (but currently unknown) height [Kup19a]. An explicit bound was given by Scull [Scu21] in the\n\ncase where the manifolds are known to be hyperbolic (but the hyperbolic structure is not part of the input).\n\n(4) The best known lower bound on the computational complexity of the homeomorphism problem is rather weak. Lackenby [Lac17b] showed the homeomorphism problem is at least as hard as the graph isomorphism problem. The graph isomorphism problem is widely believed not be solvable in polynomial time (partly because of [Sch88]). However, it is solvable in quasi-polynomial time [Bab16].\n\n(5) The recognition problem is significantly more tractable than the homeomorphism problem. For example, in the case of the 3-sphere, the recognition problem is known to lie in NP [Sch11], [Iva08], and co-NP, assuming the Generalized Riemann Hypothesis, [Zen18]. The first solution to 3-sphere recognition was given by Rubinstein [Rub95, Rub97] and Thompson [Tho94].\n\n(6) There are now many manifolds for which the recognition problem is known to be in NP, including the solid torus [Iva08], a genus g handlebody [Iva08], and the product of a compact orientable surface and an interval [Lac21a]. None of these problems is known to be solvable in polynomial time, however.\n\n(7) An efficient solution to 3-sphere recognition would give an efficient solution to the unknot recognition problem (Problem 1.76). This is because one could perform $\\pm1$ Dehn surgery on the knot and, by work of Gordon and Luecke [GL89], the result is the 3-sphere if and only if the given knot is the unknot. (This assumes that the knot is given by means of a diagram, rather than via a triangulation of its exterior, so that it is possible to build a triangulation of the filled-in manifold efficiently.)\n\nReferences cited:\n- [Kup19a] Greg Kuperberg. Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization. Pacific J. Math., 301(1):189–241, 2019. doi:10.2140/pjm.2019.301.189.\n- [SS14] Peter Scott and Hamish Short. The homeomorphism problem for closed 3-manifolds. Algebr. Geom. Topol., 14(4):2431–2444, 2014. doi:10.2140/agt.2014.14.2431.\n- [Scu21] Joe Scull. The homeomorphism problem for hyperbolic manifolds I, 2021. arXiv: 2108.00779.\n- [Lac17b] Marc Lackenby. Some conditionally hard problems on links and 3-manifolds. Discrete Comput. Geom., 58(3):580–595, 2017. doi:10.1007/s00454-017-9905-8.\n- [Sch88] Uwe Schöning. Graph isomorphism is in the low hierarchy. J. Comput. System Sci., 37(3):312–323, 1988. doi:10.1016/0022-0000(88)90010-4.\n- [Bab16] László Babai. Graph isomorphism in quasipolynomial time [extended abstract]. In STOC’16—Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, pages 684–697. ACM, New York, 2016. doi:10.1145/2897518.2897542.\n- [Sch11] Saul Schleimer. Sphere recognition lies in NP. In Low-dimensional and symplectic topology, volume 82 of Proc. Sympos. Pure Math., pages 183–213. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/pspum/082/2768660.\n- [Iva08] S. V. Ivanov. The computational complexity of basic decision problems in 3-dimensional topology. Geom. Dedicata, 131:1–26, 2008. doi:10.1007/s10711-007-9210-4.\n- [Zen18] Raphael Zentner. Integer homology 3-spheres admit irreducible representations in $\\mathrm{SL}(2,\\mathbb{C})$. Duke Math. J., 167(9):1643–1712, 2018. doi:10.1215/00127094-2018-0004.\n- [Rub95] Joachim H. Rubinstein. An algorithm to recognize the 3-sphere. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 601–611. Birkhäuser, Basel, 1995.\n- [Rub97] J. H. Rubinstein. Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds. In Geometric topology (Athens, GA, 1993), volume 2.1 of AMS/IP Stud. Adv. Math., pages 1–20. Amer. Math. Soc., Providence, RI, 1997. doi:10.1090/amsip/002.1/01.\n- [Tho94] Abigail Thompson. Thin position and the recognition problem for $S^{3}$. Math. Res. Lett., 1(5):613–630, 1994. doi:10.4310/MRL.1994.v1.n5.a9.\n- [Lac21a] Marc Lackenby. The efficient certification of knottedness and Thurston norm. Adv. Math., 387:Paper No. 107796, 142, 2021. doi:10.1016/j.aim.2021.107796.\n- [GL89] C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371–415, 1989. doi:10.2307/1990979.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The compact orientable 3-manifold homeomorphism problem is decidable, but its optimal complexity, polynomial 3-sphere recognition, and broad recognition complexities are open.\n\n**Verified partial progress.**\n\n- Kuperberg and Schleimer--Segerman give algorithms.\n- Known homeomorphism upper bounds are towers of exponentials of fixed but unknown height.\n\n**Full solution or refutation.**\n\nDecidability does not settle the requested complexity questions.\n\n**What remains.**\n\nDetermine complexity classes and polynomial algorithms/lower bounds for recognition problems.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.27 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records decidability and huge complexity gap.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2826, "problem_number": "KP-3.28", "title": "Kirby Problem 3.28", "statement": "How many Pachner moves are needed to pass between two triangulations of a compact 3-manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.28.\n\nLiterature notes:\n(1) Pachner [Pac91] proved that any two combinatorial triangulations of a compact piecewise-linear $n$-manifold differ by a finite sequence of modifications that are now called Pachner moves. In the case of closed $n$-manifolds, these take the following form: pick a subcomplex of the triangulation that is combinatorially isomorphic to a union $B$ of facets in the boundary of an $(n+1)$-simplex $\\Delta^{n+1}$, and replace the copy of $B$ by $\\operatorname{cl}(\\partial\\Delta^{n+1}\\setminus B)$. When $n=3$, there are four types of such move: a $(1,4)$-move, which replaces a single tetrahedron with four tetrahedra arranged around a new vertex; a $(4,1)$-move, which is the reverse of this procedure; a $(2,3)$-move, which replaces two tetrahedra that share a common facet with three tetrahedra arranged around a new edge; and a $(3,2)$-move, which is the reverse of the above move. For $n$-manifolds with boundary, there is a further type of Pachner move, which involves attaching an $n$-simplex onto a union of facets in the boundary of the manifold, or the reverse of this.\n\n(2) One can ask this question for general compact 3-manifolds, or for a fixed 3-manifold, or for a class of 3-manifolds (such as all Seifert fibered spaces).\n\n(3) The question can be formulated in terms of the Pachner function, which is defined as follows. For a compact 3-manifold M and positive integers $t_1$ and $t_2$, consider all pairs of triangulations of M with $t_1$ and $t_2$ tetrahedra and let $P_M(t_1,t_2)$ be the maximal number of moves needed to pass between any two such triangulations. Let $P(t_1,t_2)$ be the maximum of $P_M(t_1,t_2)$ over all compact 3-manifolds M. The question asks how $P(t_1,t_2)$ grows as a function of $t_1$ and $t_2$.\n\n(4) Any computable upper bound on $P$ leads to a solution to the homeomorphism problem for compact 3-manifolds, as follows. Given two triangulations of two 3-manifolds $M_1$ and $M_2$ with $t_1$ and $t_2$ tetrahedra, the task is to decide whether they are homeomorphic. Pachner's theorem [Pac91] states that if they are homeomorphic, then the triangulations differ by a sequence of Pachner moves, with length at most $P(t_1,t_2)$. Thus if $P$ has a computable upper bound, then one need only try all sequences of Pachner moves with at most this length, starting from one triangulation. If none of the resulting triangulations is combinatorially isomorphic to the second triangulation, then one can deduce that the manifolds are not homeomorphic.\n\n(5) Mijatovic [Mij05a] proved that for all knot exteriors M, $P_M(t_1,t_2)$ is at most\n\n$$\n2^{2^{\\cdot^{\\cdot^{\\cdot^{t_1+t_2}}}}}\n$$\n\nwhere the height of the tower is $c^{t_1}+c^{t_2}$ and where $c=2^{200}$. The same bound holds for all `fibre-free' Haken 3-manifolds [Mij05b]. When M is the 3-sphere, Mijatovic [Mij03] and King [Kin01] found upper bounds on $P_M$ of exponential type (more precisely, of the form $k^{t_1^2}+k^{t_2^2}$ for some constant $k$). However, an explicit bound is not known for $P$ in general, or even for $P_M$ for a general hyperbolic 3-manifold M.\n\n(6) Very few nontrivial lower bounds on $P$ are known (for example [Bur11] and [JRST20, Remark 15]).\n\nReferences cited:\n- [Pac91] Udo Pachner. P.L. homeomorphic manifolds are equivalent by elementary shellings. European J. Combin., 12(2):129–145, 1991. doi:10.1016/S0195-6698(13)80080-7.\n- [Mij05a] Aleksandar Mijatović. Simplical structures of knot complements. Math. Res. Lett., 12(5-6):843–856, 2005. doi:10.4310/MRL.2005.v12.n6.a6.\n- [Mij05b] Aleksandar Mijatović. Triangulations of fibre-free Haken 3-manifolds. Pacific J. Math., 219(1):139–186, 2005. doi:10.2140/pjm.2005.219.139.\n- [Mij03] Aleksandar Mijatović. Simplifying triangulations of $S^{3}$. Pacific J. Math., 208(2):291–324, 2003. doi:10.2140/pjm.2003.208.291.\n- [Kin01] Simon A. King. The size of triangulations supporting a given link. Geom. Topol., 5:369–398, 2001. doi:10.2140/gt.2001.5.369.\n- [Bur11] Benjamin A. Burton. The Pachner graph and the simplification of 3-sphere triangulations. In Computational geometry (SCG’11), pages 153–162. ACM, New York, 2011. doi:10.1145/1998196.1998220.\n- [JRST20] William Jaco, J. Hyam Rubinstein, Jonathan Spreer, and Stephan Tillmann. $\\mathbb{Z}_2$-Thurston norm and complexity of 3-manifolds, II. Algebr. Geom. Topol., 20(1):503– 529, 2020. doi:10.2140/agt.2020.20.503.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Pachner connectivity guarantees finite move sequences, but sharp quantitative bounds in dimension three are open.\n\n**Verified partial progress.**\n\n- Pachner proves existence of a finite sequence between triangulations.\n\n**Full solution or refutation.**\n\nNo optimal move-complexity bound was verified.\n\n**What remains.**\n\nBound minimal Pachner distance in terms of triangulation complexity and manifold type.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.28 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the moves and poses the quantitative problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2827, "problem_number": "KP-3.29", "title": "Kirby Problem 3.29", "statement": "(a) Given a closed hyperbolic 3-manifold $M$, can one find an explicit bound on the degree of a finite cover $\\widetilde M$ having $b_1(\\widetilde M)>0$?\n\n(b) For example, is there such a finite cover with degree at most a polynomial function of the number of tetrahedra in some triangulation of $M$?\n\n(c) Similarly, can one find an explicit upper bound on the degree of a finite cover $\\widetilde M_1$ that fibers over the circle?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.29.\n\nLiterature notes:\n(1) Agol and Wise proved that $M$ has a finite cover $\\widetilde M$ with $b_1(\\widetilde M)>0$, and indeed a finite cover $\\widetilde M_1$ that fibers over the circle [Ago13, Wis21]. However, the proof is non-constructive, and it seems hard to extract an upper bound on the covering degree.\n\n(2) Given $M$, one can compute the degrees of covers $\\widetilde M$ and $\\widetilde M_1$ as above, since one can start building all the finite covers of $M$, ordered by covering degree, and for each one, one can determine whether $b_1>0$ or whether it fibers over the circle.\n\n(3) Agol showed that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS (residually finite $\\mathbb{Q}$-solvable) is virtually fibered [Ago08]. One approach to the above problem would be to find an explicit finite cover with RFRS fundamental group, at which point it seems more likely that one could get good bounds needed for a fibered cover. Currently, explicit towers of RFRS covers are only known in a few specific examples, including for certain Bianchi groups due to Agol--Stover [AS23].\n\nReferences cited:\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [Wis21] Daniel T. Wise. The structure of groups with a quasiconvex hierarchy, volume 209 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, [2021] ©2021.\n- [Ago08] Ian Agol. Criteria for virtual fibering. J. Topol., 1(2):269–284, 2008. doi:10.1112/jtopol/jtn003.\n- [AS23] Ian Agol and Matthew Stover. Congruence RFRS towers. Ann. Inst. Fourier (Grenoble), 73(1):307–333, 2023. With an appendix by Mehmet Haluk Şengün. doi:10.5802/aif.3532.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Virtual positive Betti number and virtual fibering are known, and minimal cover degrees are computable by enumeration, but no useful explicit degree bounds are known.\n\n**Verified partial progress.**\n\n- Agol--Wise prove existence of positive-betti and fibered finite covers.\n- Enumerating finite covers gives computability in principle.\n\n**Full solution or refutation.**\n\nThe requested polynomial/effective bounds remain open.\n\n**What remains.**\n\nBound minimal cover degree from triangulation complexity.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.29 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes virtual existence from effective degree bounds.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2828, "problem_number": "KP-3.30", "title": "Kirby Problem 3.30", "statement": "(a) What is the computational complexity of determining whether a compact 3-manifold admits a hyperbolic structure?\n\n(b) If a compact 3-manifold does admit a hyperbolic structure, what is the computational complexity of finding it?\n\n(c) In particular, do these problems lie in the complexity classes NP and FNP respectively?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.30.\n\nLiterature notes:\n(1) The most efficient known algorithm to find a hyperbolic structure on a 3-manifold if one exists is due to Scull [Scu21], based on work of Kuperberg [Kup19a]. If the 3-manifold is given as a triangulation with $t$ tetrahedra, it runs in time\n\n$$\n2^{2^{t^{O(t)}}}.\n$$\n\nHowever, in practice, SnapPy [CDGW] is good at finding hyperbolic structures if they exist, and hence, it seems quite plausible that hyperbolicity can be efficiently certified, as the question proposes.\n\n(2) The problem of deciding whether a 3-manifold is hyperbolic, without actually finding a hyperbolic structure, may be much easier. To achieve this, one might establish the existence of an alternative structure that is equivalent to hyperbolicity. For example, one might show that its fundamental group is infinite and has no $\\mathbb{Z}\\times\\mathbb{Z}$ subgroup. Alternatively, one might give a metric triangulation that is CAT(-1). Another possibility is to exhibit a finite cover that fibers over the circle with pseudo-Anosov monodromy, which exists by work of Agol [Ago13]. However, none of these approaches seems straightforward.\n\n(3) The problem of deciding hyperbolicity is known to be in the complexity class co-NP assuming the Generalized Riemann Hypothesis, by work of Hass and Kuperberg [HK12a]. In the case of manifolds with nonempty toroidal boundary, this was established by Haraway and Hoffman [HIH22],\n\nwithout requiring GRH. Hence, assuming GRH, the problem of deciding whether a compact 3-manifold is hyperbolic runs in exponential time.\n\nReferences cited:\n- [Scu21] Joe Scull. The homeomorphism problem for hyperbolic manifolds I, 2021. arXiv: 2108.00779.\n- [Kup19a] Greg Kuperberg. Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization. Pacific J. Math., 301(1):189–241, 2019. doi:10.2140/pjm.2019.301.189.\n- [CDGW] Marc Culler, Nathan M. Dunfield, Matthias Goerner, and Jeffrey R. Weeks. SnapPy, a computer program for studying the geometry and topology of 3-manifolds. Available at http://snappy.computop.org.\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [HK12a] Joel Hass and Greg Kuperberg. The complexity of recognizing the 3-sphere. Oberwolfach Reports, 9(2), 2012. https://ems.press/content/serial-article-files/46393.\n- [HIH22] Robert Haraway III and Neil R. Hoffman. On the complexity of cusped non-hyperbolicity, 2022. arXiv:1907.01675.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hyperbolicity can be found/decided by explicit very expensive algorithms, but NP/FNP membership and efficient certificates remain open.\n\n**Verified partial progress.**\n\n- Scull's algorithm runs in doubly exponential t^(O(t)) exponent form.\n- SnapPy is practically effective but not a complexity proof.\n\n**Full solution or refutation.**\n\nNo NP/FNP result was verified.\n\n**What remains.**\n\nGive polynomial-size certificates and efficient verification/construction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.30 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the known algorithm and proposed NP/FNP target.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2829, "problem_number": "KP-3.31", "title": "Kirby Problem 3.31", "statement": "Suppose M is a closed 3-manifold.\n\n(a) Can one decide if the fundamental group of M is left-orderable?\n\n(b) What is the complexity of a certificate of left-orderability, or of non-left-orderability?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.31.\n\nLiterature notes:\n(1) A nontrivial group G is left-orderable if it admits a left-invariant total order.\n\n(2) There is no algorithm to determine in general, from a finite presentation of a group G, whether G is left-orderable.\n\n(3) It is known that $\\pi_1(M)$ is left-orderable if and only if there is a surjective homomorphism from $\\pi_1(M)$ to a nontrivial left-orderable group. Thus, one may ask: among all such $\\rho:\\pi_1(M)\\to G$, what is the least complexity of a machine that will recognize exactly the elements $a\\in\\pi_1(M)$ (expressed as the canonical representative in some regular prefix-closed bijective language of geodesics if one exists, say) for which $\\rho(a)>\\mathrm{id}$? For example, if $b_1(M)>0$ one may take $G=\\mathbb{Z}$ and $\\rho:\\pi_1(M)\\to G$ any nontrivial element of $H^{1}(M)$, and then the recognition may be done with a counter automaton (but no simpler machine for typical M, e.g. M hyperbolic).\n\n(4) One may also ask for the least complexity of a left order on $\\pi_1(M)$ itself. This might be considerably more complicated than the least complexity of a left order on an infinite quotient.\n\n(5) The L-space Conjecture (Problem 3.48) asserts that for an irreducible 3-manifold M the following conditions are equivalent:\n\n(i) $M$ has left-orderable fundamental group; (ii) $M$ admits a coorientable taut foliation; (iii) $M$ is not an L-space. The second property is known to be algorithmically decidable for atoroidal manifolds by Agol--Li [AL03], who proved more generally that whether a 3-manifold has a coorientable Reebless foliation is decidable; the third is algorithmically decidable for the elementary reason that Heegaard Floer homology can be calculated. Thus, if the property of having a left-orderable group is not decidable, the L-space Conjecture is false.\n\n(6) This problem is similar to Question 8.7 in [Cal03].\n\nReferences cited:\n- [AL03] Ian Agol and Tao Li. An algorithm to detect laminar 3-manifolds. Geom. Topol., 7:287–309, 2003. doi:10.2140/gt.2003.7.287.\n- [Cal03] Danny Calegari. Problems in foliations and laminations of 3-manifolds. In Topology and geometry of manifolds (Athens, GA, 2001), volume 71 of Proc. Sympos. Pure Math., pages 297–335. Amer. Math. Soc., Providence, RI, 2003. doi: 10.1090/pspum/071/2024640.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Left-orderability is undecidable for arbitrary finite presentations, while the decision and certificate complexity questions for closed 3-manifold groups remain open.\n\n**Verified partial progress.**\n\n- Positive first Betti number supplies a simple left-orderability witness via a Z quotient.\n\n**Full solution or refutation.**\n\nNo 3-manifold-specific algorithm or complexity theorem was verified.\n\n**What remains.**\n\nDecide left-orderability for closed 3-manifold groups or establish complexity barriers.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.31 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States general undecidability and specific open questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2830, "problem_number": "KP-3.32", "title": "Kirby Problem 3.32", "statement": "Is there an algorithm to determine whether two closed, embedded surfaces in $\\mathbb{R}^3$ are isotopic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.32.\n\nLiterature notes:\n(1) The problem of deciding whether two incompressible surfaces in a compact orientable irreducible 3-manifold are isotopic is solvable, using work of Waldhausen [Wal68b, Proposition 5.4] (see also [Bar25, Theorem 1.2]). However, surfaces in $\\mathbb{R}^3$ (other than 2-spheres) are compressible.\n\n(2) All spheres embedded in $\\mathbb{R}^3$ are isotopic by the Schoenflies theorem.\n\n(3) Any torus embedded in $\\mathbb{R}^3$ is the boundary of a regular neighbourhood of a knot, and therefore the isotopy problem for tori is basically the same as the equivalence problem for knots, which has been solved by Haken [Hak68], Hemion [Hem79] and Matveev [Mat07].\n\n(4) The case of genus two surfaces was recently solved by Baroni [Bar25]. The challenge for higher genus surfaces is as follows. Given two closed connected surfaces embedded in $\\mathbb{R}^3$, one must decide whether their exteriors are homeomorphic. If they are not, then the surfaces are not isotopic. However, if their exteriors are homeomorphic, then this is not enough to deduce that the surfaces are isotopic; one must check that there is a homeomorphism compatible with the gluing maps along the surfaces. As the exteriors might have infinite mapping class group, it is not clear that this can be checked algorithmically.\n\nReferences cited:\n- [Wal68b] Friedhelm Waldhausen. On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2), 87:56–88, 1968. doi:10.2307/1970594.\n- [Bar25] Filippo Baroni. Classification of genus-two surfaces in $S^{3}$. Algebr. Geom. Topol., 25(8):4719–4785, 2025. doi:10.2140/agt.2025.25.4719.\n- [Hak68] Wolfgang Haken. Some results on surfaces in 3-manifolds. In Studies in Modern Topology, volume Vol. 5 of Studies in Mathematics, pages 39–98. Math. Assoc. America,, 1968.\n- [Hem79] Geoffrey Hemion. On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds. Acta Math., 142(1-2):123–155, 1979. doi:10.1007/BF02395059.\n- [Mat07] Sergei Matveev. Algorithmic topology and classification of 3-manifolds, volume 9 of Algorithms and Computation in Mathematics. Springer, Berlin, second edition, 2007.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Embedded-surface isotopy is decidable for incompressible surfaces, spheres, tori and recently genus two, but no arbitrary-genus algorithm is known.\n\n**Verified partial progress.**\n\n- Schoenflies settles spheres.\n- Knot-equivalence algorithms settle tori.\n- Baroni solves genus two.\n\n**Full solution or refutation.**\n\nHigher-genus compressible-surface isotopy remains open.\n\n**What remains.**\n\nDevelop an algorithm handling arbitrary embedded closed surfaces in R3.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.32 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Enumerates solved low-genus/incompressible cases and higher-genus challenge.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2831, "problem_number": "KP-3.33", "title": "Kirby Problem 3.33", "statement": "Let M and N be closed orientable 3-manifolds. Prove that if there is a degree-1 map $f:M\\to N$ then $g(M)\\geq g(N)$, where $g(M)$ is the Heegaard genus of $M$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.33.\n\nLiterature notes:\n(1) This is a long-standing conjecture, dating back to Haken and Waldhausen [Hak66, Wal70]. The conjecture implies the Poincaré Conjecture: suppose a closed 3-manifold N is homotopy equivalent to $S^3$. Since a homotopy equivalence is a degree-1 map, the conjecture implies that $0=g(S^3)\\geq g(N)$ and therefore that N is $S^3$.\n\n(2) A general approach to this conjecture was discussed in [RW92, Li22]. However, not much progress has been made. Let $M=W\\cup_T V$ be an amalgamation of two manifolds W and V along an incompressible torus, with W being the exterior of a knot in a homology 3-sphere. In this case, there is a canonical degree-1 map that pinches W into a solid torus. It is proved in [Li22] that the Heegaard genus of the resulting 3-manifold is at most as large as the Heegaard genus of M, verifying the conjecture in a special case.\n\nReferences cited:\n- [Hak66] Wolfgang Haken. On homotopy 3-spheres. Illinois J. Math., 10:159–178, 1966. http: //projecteuclid.org/euclid.ijm/1256055210.\n- [Wal70] F. Waldhausen. On mappings of handlebodies and Heegaard splittings. Topology of Manifolds (Markham Publishing Company), pages 205–211, 1970.\n- [RW92] Yong Wu Rong and Shi Cheng Wang. The preimages of submanifolds. Math. Proc. Cambridge Philos. Soc., 112(2):271–279, 1992. doi:10.1017/S030500410007095X.\n- [Li22] Tao Li. Heegaard genus, degree-one maps, and amalgamation of 3-manifolds. J. Topol., 15(3):1540–1579, 2022. doi:10.1112/topo.12253.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Degree-one-map Heegaard-genus monotonicity is verified for a specified torus-amalgamation/pinching class, but remains open generally.\n\n**Verified partial progress.**\n\n- Li proves the conjecture for an amalgamation class with canonical degree-one map.\n\n**Full solution or refutation.**\n\nNo general monotonicity proof was verified.\n\n**What remains.**\n\nProve g(M)>=g(N) for all degree-one maps or find a counterexample.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.33 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the longstanding conjecture and Li's special case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2832, "problem_number": "KP-3.34", "title": "Kirby Problem 3.34", "statement": "Do any two genus-g Heegaard splittings of a closed, orientable 3-manifold M become equivalent after at most g stabilizations?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.34.\n\nLiterature notes:\nThe answer \"yes\" was conjectured by W. Thurston. This has been studied extensively and upper bounds on the number of necessary stabilizations are known [Joh95], [RS96], but the conjectured optimal bound of g is open. It is known that g stabilizations may be needed [HTT09]; see also [Kir97, Problem 3.89].\n\nReferences cited:\n- [Joh95] Klaus Johannson. Topology and combinatorics of 3-manifolds, volume 1599 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1995. doi:10.1007/BFb0074005.\n- [RS96] Hyam Rubinstein and Martin Scharlemann. Comparing Heegaard splittings of non-Haken 3-manifolds. Topology, 35(4):1005–1026, 1996. doi:10.1016/0040-9383(95) 00055-0.\n- [HTT09] Joel Hass, Abigail Thompson, and William Thurston. Stabilization of Heegaard splittings. Geom. Topol., 13(4):2029–2050, 2009. doi:10.2140/gt.2009.13.2029.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** General upper bounds on stabilizations are known and examples show g stabilizations can be necessary, but Thurston's universal at-most-g upper bound remains open.\n\n**Verified partial progress.**\n\n- Johannson and Rubinstein--Scharlemann give upper bounds.\n- Hass--Thompson--Thurston show the g lower requirement can occur.\n\n**Full solution or refutation.**\n\nOptimality of the g upper bound remains unresolved.\n\n**What remains.**\n\nProve the g-stabilization upper bound or find a counterexample.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.34 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records upper bounds and sharpness lower examples.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2833, "problem_number": "KP-3.35", "title": "Kirby Problem 3.35", "statement": "Given a compact manifold M, let $r(M)$ denote the rank of its fundamental group and $g(M)$ denote its Heegaard genus.\n\n(a) Does every closed orientable hyperbolic 3-manifold M with $r(M)$ = 2 satisfy $g(M)$ = 2?\n\n(b) Is there a non-Haken manifold $M$ such that $r(M)0$ such that $g(M)\\leq k r(M)$ for every closed orientable 3-manifold $M$?\n\n(d) If $\\{M_i\\}$ is the set of congruence covers of an arithmetic hyperbolic 3-manifold M, is the infimum of $r(M_i)/\\operatorname{vol}(M_i)$ zero or positive?\n\n(e) Is there a knot $K\\subset S^3$ such that $r(S^3\\setminus\\nu(K))\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Rank can be strictly below Heegaard genus and arbitrarily far below for Haken hyperbolic examples, but the non-Haken/rank-two/ratio subquestions are open.\n\n**Verified partial progress.**\n\n- Li constructs hyperbolic Haken examples with arbitrarily large genus-rank difference.\n- Graph-manifold examples have arbitrary difference.\n\n**Full solution or refutation.**\n\nThe stated multi-part hyperbolic and ratio questions remain open.\n\n**What remains.**\n\nResolve rank-two hyperbolic, non-Haken, and universal-ratio assertions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.35 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes known Haken examples from open subproblems.\n\n**Review notes.** Multi-part record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2834, "problem_number": "KP-3.36", "title": "Kirby Problem 3.36", "statement": "(Simple loop conjecture) Let f : F $\\to$ M be a 2-sided immersion of a surface into a 3-manifold such that $f_*$ : $\\pi_1(F)$ $\\to$ $\\pi_1(M)$ is not injective. Must some simple closed curve lie in the kernel of $f_*$ ?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.36.\n\nLiterature notes:\nThis is [Kir97, Problem 3.96], proposed by J. Hass. When a simple closed curve lies in the kernel of $f_*$, $(F,f)$ is compressible to an immersion of a simpler surface. When f is an embedding, the conjecture is true, by the Loop Theorem. It is also true when M is an I-bundle over a closed surface [Gab85], when M is Seifert fibered [Has87], when M is a graph manifold [RW98] and when M is a sol manifold [Zem16].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Gab85] David Gabai. The simple loop conjecture. J. Differential Geom., 21(1):143–149, 1985. http://projecteuclid.org/euclid.jdg/1214439470.\n- [Has87] Joel Hass. Minimal surfaces in manifolds with $S^{1}$ actions and the simple loop conjecture for Seifert fibered spaces. Proc. Amer. Math. Soc., 99(2):383–388, 1987. doi:10.2307/2046646.\n- [RW98] J. Hyam Rubinstein and Shicheng Wang. $\\pi_1$-injective surfaces in graph manifolds. Comment. Math. Helv., 73(4):499–515, 1998. doi:10.1007/s000140050066.\n- [Zem16] Drew Zemke. The simple loop conjecture for 3-manifolds modeled on Sol. Algebr. Geom. Topol., 16(5):3051–3071, 2016. doi:10.2140/agt.2016.16.3051.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The simple loop conjecture is true for embeddings, I-bundles, Seifert fibered, graph and Sol manifolds, but open for arbitrary immersed surfaces in 3-manifolds.\n\n**Verified partial progress.**\n\n- Loop Theorem solves embedded case.\n- Gabai, Hass, Rubinstein--Wang and Zemke cover named classes.\n\n**Full solution or refutation.**\n\nNo universal proof/counterexample was verified.\n\n**What remains.**\n\nResolve the immersed hyperbolic/general case.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.36 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists proved manifold classes and open general question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2835, "problem_number": "KP-3.37", "title": "Kirby Problem 3.37", "statement": "(a) Is every finitely generated 3-manifold group linear (over some field with characteristic zero)?\n\n(b) If so, can one bound the dimension of a faithful linear representation?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.37.\n\nLiterature notes:\n(1) Every finitely generated 3-manifold group is the fundamental group of a compact 3-manifold (possibly with boundary) [Sco73]. Any compact 3-manifold is finitely covered by a compact, orientable 3-manifold. Such a cover has linear fundamental group if and only if the original manifold does. A free product of finitely many linear groups is linear. Hence, the problem reduces to the case of compact, orientable irreducible 3-manifolds.\n\n(2) Compact, orientable manifolds are known to satisfy the Geometrization Conjecture, and hence admit decompositions into geometric pieces. The fundamental group of any hyperbolic 3-manifold admits a faithful representation into $\\operatorname{SO}(3, 1)$, and hence is linear. Seifert fibered spaces are known to have linear fundamental groups [AFW15], although it does not appear to be known whether there is a universal upper bound on the dimension of such a representation. Moreover, it is not known how to obtain\n\na faithful linear representation of the fundamental group of a 3-manifold from faithful representations of its JSJ pieces.\n\n(3) Any closed, orientable 3-manifold with a hyperbolic piece in its JSJ decomposition is virtually special [PW18], by work of Przytycki--Wise and Agol--Wise [Ago13, Wis21], and hence its fundamental group virtually embeds in $\\operatorname{SL}(m, \\mathbb{Z})$ for some integer m. Similarly, any compact, orientable, aspherical 3-manifold with nonempty boundary has virtually special fundamental group [PW18] (see also [AFW15]). So, the problem of linearity is unresolved only for closed graph manifolds.\n\n(4) In Problem 3.33 of the Kirby problem list, W. Thurston asked whether every finitely generated 3-manifold group admits a faithful representation into $\\operatorname{GL}(4, \\mathbb{R})$. By the above discussion, this is true for hyperbolic 3-manifolds. However, Button [But14] showed that the fundamental groups of some graph manifolds do not admit faithful representations into $\\operatorname{GL}(4, k)$ for any field k.\n\n(5) Apart from the intrinsic interest in this problem, linearity has various consequences. For example, finitely generated linear groups are residually finite and Hopfian. These properties are already known to hold for finitely generated 3-manifold groups [Hem87]. Even having a nonabelian (but possibly non-faithful) linear representation can be useful. Zentner [Zen18] proved that the fundamental group of every homology 3-sphere other than the 3-sphere has such a representation into $\\operatorname{SL}(2, \\mathbb{C})$, and used this to provide a new algorithm to recognize the 3-sphere. Representations into $\\operatorname{SU}(2)$ are a particularly active area of research because of their relationship with instanton homology (see, e.g., [KM04]). See also Problem 3.52.\n\nReferences cited:\n- [Sco73] G. P. Scott. Compact submanifolds of 3-manifolds. J. London Math. Soc. (2), 7:246– 250, 1973. doi:10.1112/jlms/s2-7.2.246.\n- [AFW15] Matthias Aschenbrenner, Stefan Friedl, and Henry Wilton. 3-manifold groups. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2015. doi:10.4171/154.\n- [PW18] Piotr Przytycki and Daniel T. Wise. Mixed 3-manifolds are virtually special. J. Amer. Math. Soc., 31(2):319–347, 2018. doi:10.1090/jams/886.\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [Wis21] Daniel T. Wise. The structure of groups with a quasiconvex hierarchy, volume 209 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, [2021] ©2021.\n- [But14] J. O. Button. A 3-manifold group which is not four dimensional linear. J. Pure Appl. Algebra, 218(9):1604–1619, 2014. doi:10.1016/j.jpaa.2014.01.001.\n- [Hem87] John Hempel. Residual finiteness for 3-manifolds. In Combinatorial group theory and topology (Alta, Utah, 1984), volume 111 of Ann. of Math. Stud., pages 379–396. Princeton Univ. Press, Princeton, NJ, 1987.\n- [Zen18] Raphael Zentner. Integer homology 3-spheres admit irreducible representations in $\\mathrm{SL}(2,\\mathbb{C})$. Duke Math. J., 167(9):1643–1712, 2018. doi:10.1215/00127094-2018-0004.\n- [KM04] P. B. Kronheimer and T. S. Mrowka. Dehn surgery, the fundamental group and $\\mathrm{SU}(2)$. Math. Res. Lett., 11(5-6):741–754, 2004. doi:10.4310/MRL.2004.v11.n6.a3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hyperbolic and Seifert pieces have linear groups, and geometrization reduces the question to controlling gluing/graph-manifold cases; all 3-manifold groups are not yet known to be linear.\n\n**Verified partial progress.**\n\n- Hyperbolic groups embed in SO(3,1).\n- Seifert fibered groups are linear.\n\n**Full solution or refutation.**\n\nNo general characteristic-zero linearity or dimension bound was verified.\n\n**What remains.**\n\nEstablish linearity under JSJ gluing and effective representation-dimension bounds.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.37 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Gives the geometrization reduction and known geometric-piece cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2836, "problem_number": "KP-3.38", "title": "Kirby Problem 3.38", "statement": "Is every PD$_3$-group the fundamental group of a closed, aspherical 3-manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.38.\n\nLiterature notes:\n(1) This is [Kir97, Problem 3.77(A)], due to G. P. Scott, and is very closely related to the Cannon Conjecture (Problem 3.24). The more general question whether every PD$_n$-group is the fundamental group of a closed, aspherical n-manifold was raised by Wall [Wal79, Problem G2].\n\n(2) A Poincaré duality group $G$ of dimension $n$ (a PD$_n$-group) is a finitely presented group whose homology and cohomology satisfy Poincaré duality over $\\mathbb{Z}[G]$ with a fundamental class in dimension $n$. The only known finitely presented PD$_n$-groups are fundamental groups of closed, aspherical $n$-manifolds, and every PD$_2$-group is the fundamental group of a closed, aspherical surface [EM80, EL83]. However, there are PD$_n$-groups that are not finitely presentable, for every $n\\geq 4$ [Dav98].\n\n(3) Despite its long heritage, there has been little progress on this problem. However, the theory of PD$_3$-groups has some parallels with the theory\n\nof 3-manifold groups. Every PD$_3$-group admits an essentially unique decomposition into `irreducibles', and there is also a good analogue of a JSJ decomposition for PD$_3$-groups [KR88a, KR88b, KR89]. The question has been resolved positively for PD$_3$-groups with subgroup structure corresponding to mapping tori or Seifert fibred 3-manifolds. See [Hil20] for an extensive survey.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Wal79] C. T. C. Wall, editor. Homological group theory, volume 36 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge-New York, 1979.\n- [EM80] Beno Eckmann and Heinz Müller. Poincaré duality groups of dimension two. Comment. Math. Helv., 55(4):510–520, 1980. doi:10.1007/BF02566702.\n- [EL83] Beno Eckmann and Peter Linnell. Poincaré duality groups of dimension two. II. Comment. Math. Helv., 58(1):111–114, 1983. doi:10.1007/BF02564628.\n- [Dav98] Michael W. Davis. The cohomology of a Coxeter group with group ring coefficients. Duke Math. J., 91(2):297–314, 1998. doi:10.1215/S0012-7094-98-09113-X.\n- [KR88a] P. H. Kropholler and M. A. Roller. Splittings of Poincaré duality groups. Math. Z., 197(3):421–438, 1988. doi:10.1007/BF01418340.\n- [KR88b] P. H. Kropholler and M. A. Roller. Splittings of Poincaré duality groups. II. J. London Math. Soc. (2), 38(3):410–420, 1988. doi:10.1112/jlms/s2-38.3.410.\n- [KR89] P. H. Kropholler and M. A. Roller. Splittings of Poincaré duality groups. III. J. London Math. Soc. (2), 39(2):271–284, 1989. doi:10.1112/jlms/s2-39.2.271.\n- [Hil20] Jonathan A. Hillman. Poincaré duality in dimension 3, volume 3 of The Open Book Series. Mathematical Sciences Publishers, Berkeley, CA, 2020.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The unrestricted PD3-group realization problem remains open; realization is known for PD3-groups with mapping-torus or Seifert-fibred subgroup structure, and prime/JSJ analogues are available.\n\n**Verified partial progress.**\n\n- Every PD2-group is a closed aspherical surface group.\n- Positive PD3 realization results cover the mapping-torus and Seifert-fibred cases.\n- PD3-groups admit decomposition results paralleling prime and JSJ decompositions of 3-manifold groups.\n\n**Full solution or refutation.**\n\nNo source checked proves that every PD3-group is the fundamental group of a closed aspherical 3-manifold.\n\n**What remains.**\n\nRealize the remaining irreducible/atoroidal PD3-groups, or construct a PD3-group obstruction to 3-manifold realization.\n\n**Sources checked.**\n\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.38. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current problem list states that the general problem has seen little progress and identifies the mapping-torus, Seifert-fibred, decomposition, and JSJ special results.\n- Jonathan A. Hillman, Aspherical PD3-pairs, arXiv:2605.00346 (2026). (primary): https://arxiv.org/abs/2605.00346\n Evidence used: Recent adjacent structure theory for PD3-pairs; its scope does not assert the unrestricted closed-group realization theorem.\n\n**Review notes.** No formulation defect found; the historical notation PD$_3$ was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2837, "problem_number": "KP-3.39", "title": "Kirby Problem 3.39", "statement": "(a) Is every finitely generated perfect group the normal closure of a single element?\n\n(b) Is there an integral homology sphere whose fundamental group has weight bigger than one?\n\n(c) For any $n$, is there an integral homology sphere whose fundamental group has weight bigger than $n$?\n\n(d) Specifically, are the weights of the fundamental groups of Seifert fibered homology spheres $\\Sigma(a_1,\\ldots,a_n)$ unbounded?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.39.\n\nLiterature notes:\n(1) Part (a) is a longstanding question due to Wiegold; see [MK14, Problem 5.52]. Note added in proof: Chen and Lodha [CL25] have claimed a counterexample to the Wiegold problem.\n\n(2) The weight of a group is the minimum number of elements needed to normally generate it. So, part (a) asks whether every finitely generated perfect group has weight 1.\n\n(3) Since the fundamental group of a link complement is normally generated by a collection of meridians, the weight of the fundamental group of a 3-manifold obtained by surgery on a link is bounded above by the number of components. Thus, the weight provides a bound for the surgery number of a 3-manifold, and group-theoretic progress on the Wiegold question would likely be useful in tackling the similarly difficult problem of finding homology spheres with large surgery number (see Question 1.15). Conversely, topological methods for bounding surgery numbers could perhaps be useful in solving the Wiegold question.\n\n(4) Dave Auckly suggests the following argument. Suppose $\\Sigma$ is a homology sphere with $\\pi_1(\\Sigma)$ of weight $n$. Then by choosing $n$ embedded loops in $I\\times\\Sigma$ that normally generate the fundamental group, and performing surgery on these elements, we obtain a simply-connected cobordism from $\\Sigma$ to itself whose intersection form agrees with $\\#^n(S^2\\times S^2)$. If one can obstruct the existence of such a cobordism, it follows that $\\pi_1(\\Sigma)$ has weight larger than $n$.\n\nReferences cited:\n- [MK14] V. D. Mazurov and E. I. Khukhro, editors. The Kourovka notebook. Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, eighteenth edition, 2014. Unsolved problems in group theory.\n- [CL25] Lvzhou Chen and Yash Lodha. The wiegold problem and free products of leftorderable groups, 2025. arXiv:2510.26073.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Chen--Lodha answer (a) negatively by finitely presented perfect groups of normal rank greater than one and answer (b) affirmatively by hyperbolic integral homology 3-spheres of normal rank at least two; (c) and (d) remain open.\n\n**Verified partial progress.**\n\n- Theorem A of arXiv:2510.26073v2 gives finitely presented perfect groups with normal rank greater than one, refuting part (a).\n- Theorem C gives infinitely many hyperbolic integer homology 3-spheres whose fundamental groups have normal rank at least two, proving part (b).\n\n**Full solution or refutation.**\n\nTwo of the four subquestions are settled in a 2025 preprint: (a) no and (b) yes. No arbitrary-weight or Seifert-fibred unboundedness theorem was verified.\n\n**What remains.**\n\nFor (c), construct integral homology spheres of arbitrarily large group weight; for (d), decide unboundedness within the specified Seifert-fibred family.\n\n**Sources checked.**\n\n- Lvzhou Chen and Yash Lodha, The Wiegold problem and free products of left-orderable groups, arXiv:2510.26073v2 (2025). (primary): https://arxiv.org/abs/2510.26073\n Evidence used: Theorem A supplies perfect groups of normal rank greater than one; Theorem C supplies hyperbolic integer homology spheres with fundamental-group normal rank at least two.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 3.39 and note added in proof. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current source statement and identification of the Chen--Lodha counterexample claim.\n\n**Review notes.** The primary result is a preprint. Parts (a)--(d) are kept separate; the record is not labelled disproved because only part (a) is refuted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2838, "problem_number": "KP-3.40", "title": "Kirby Problem 3.40", "statement": "Does every closed, orientable, hyperbolic 3-manifold admit a tight contact structure?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.40.\n\nLiterature notes:\n(1) There exist closed, orientable contact 3-manifolds without tight contact structures, such as the connected sum $P\\#\\overline{P}$ of the Poincaré homology sphere with its orientation reversed [EH01b]. On the other hand, if M is orientable and irreducible, and $\\Sigma$ is an oriented surface in M representing a nontrivial second homology class and realizing the minimal genus among such representatives, then Gabai [Gab83] showed that M has a taut foliation in which $\\Sigma$ is a leaf. So, in light of Eliashberg--Thurston's theorem [ET98], which produces a symplectically fillable contact structure from a taut foliation (see also [KR17, Bow16]), any orientable, irreducible 3-manifold with nontrivial second homology admits a tight contact structure. Moreover, Honda--Kazez--Matić [HKM02], and independently Colin [Col01], proved that toroidal manifolds admit tight contact structures. Lisca--Stipsicz's work [LS09] then implies that every Seifert manifold admits a tight contact structure unless it is $(2n-1)$-surgery on the torus knot $T(2,2n+1)$. The remaining geometric manifolds are hyperbolic.\n\n(2) Kaloti--Tosun [KT17] and independently Li--Liu [LL19] proved that there are hyperbolic rational homology spheres without fillable contact structures; it is open whether these manifolds admit tight contact structures.\n\n(3) There has been some success in linking the Riemannian geometry of a 3-manifold with its contact geometry [EKM12, EKM16], and similar techniques may provide an approach to this problem.\n\nReferences cited:\n- [EH01b] John B. Etnyre and Ko Honda. On the nonexistence of tight contact structures. Ann. of Math. (2), 153(3):749–766, 2001. doi:10.2307/2661367.\n- [Gab83] David Gabai. Foliations and the topology of 3-manifolds. J. Differential Geom., 18(3):445–503, 1983. http://projecteuclid.org/euclid.jdg/1214437784.\n- [ET98] Yakov M. Eliashberg and William P. Thurston. Confoliations, volume 13 of University Lecture Series. American Mathematical Society, Providence, RI, 1998. doi:10.1090/ulect/013.\n- [KR17] William H. Kazez and Rachel Roberts. C0 approximations of foliations. Geom. Topol., 21(6):3601–3657, 2017. doi:10.2140/gt.2017.21.3601.\n- [Bow16] Jonathan Bowden. Approximating C0-foliations by contact structures. Geom. Funct. Anal., 26(5):1255–1296, 2016. doi:10.1007/s00039-016-0387-2.\n- [HKM02] Ko Honda, William H. Kazez, and Gordana Matić. Convex decomposition theory. Int. Math. Res. Not., 2002(2):55–88, 2002. doi:10.1155/$S^{1}$073792802101140.\n- [Col01] Vincent Colin. Sur la torsion des structures de contact tendues. Ann. Sci. École Norm. Sup. (4), 34(2):267–286, 2001. doi:10.1016/S0012-9593(00)01061-2.\n- [LS09] Paolo Lisca and András I. Stipsicz. On the existence of tight contact structures on Seifert fibered 3-manifolds. Duke Math. J., 148(2):175–209, 2009. doi:10.1215/00127094-2009-024.\n- [KT17] Amey Kaloti and Bülent Tosun. Hyperbolic rational homology spheres not admitting fillable contact structures. Math. Res. Lett., 24(6):1693–1705, 2017. doi: 10.4310/MRL.2017.v24.n6.a6.\n- [LL19] Youlin Li and Yajing Liu. Hyperbolic 3-manifolds admitting no fillable contact structures. Proc. Amer. Math. Soc., 147(1):351–360, 2019. doi:10.1090/proc/13870.\n- [EKM12] John B. Etnyre, Rafal Komendarczyk, and Patrick Massot. Tightness in contact metric 3-manifolds. Invent. Math., 188(3):621–657, 2012. doi:10.1007/s00222-011-0355-2.\n- [EKM16] John B. Etnyre, Rafal Komendarczyk, and Patrick Massot. Quantitative Darboux theorems in contact geometry. Trans. Amer. Math. Soc., 368(11):7845–7881, 2016. doi:10.1090/tran/6821.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The assertion holds for hyperbolic 3-manifolds with nontrivial second homology, and many hyperbolic homology spheres now have numerous tight structures, but the universal rational-homology-sphere case remains open.\n\n**Verified partial progress.**\n\n- Gabai's taut foliations combined with Eliashberg--Thurston imply tight, indeed fillable, contact structures for orientable irreducible 3-manifolds with nontrivial H2.\n- Mj--Sen construct a large class of hyperbolic homology 3-spheres with arbitrarily many tight contact structures, including hyperbolic L-spaces.\n\n**Full solution or refutation.**\n\nNo theorem covers every closed orientable hyperbolic 3-manifold. Nonexistence of fillable structures in some examples does not prove nonexistence of tight structures.\n\n**What remains.**\n\nEstablish tight-contact existence, or produce a counterexample, among the remaining hyperbolic rational homology spheres.\n\n**Sources checked.**\n\n- Mahan Mj and Balarka Sen, Tight contact structures on hyperbolic homology 3-spheres, arXiv:2404.14698 (2024). (primary): https://arxiv.org/abs/2404.14698\n Evidence used: Constructs large classes of hyperbolic homology spheres, including L-spaces, with arbitrarily many tight contact structures.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 3.40. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current problem status and precise separation of tightness from fillability.\n\n**Review notes.** No formulation defect found. Tight and fillable are not conflated.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2839, "problem_number": "KP-3.41", "title": "Kirby Problem 3.41", "statement": "Is it true that for every knot $K\\subset S^3$, there is an integer $n_K$ such that $S^3_r(K)$ admits a tight contact structure for all $r\\geq n_K$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.41.\n\nLiterature notes:\n(1) This appears as Conjecture 4.5 (high surgery conjecture) in the notes for Stipsicz's 2010 ICM talk [Sti10], but Stipsicz reports that he learned of the question from Ko Honda.\n\n(2) This question can be viewed as a step towards the following fundamental problem in 3-dimensional contact topology: Which closed oriented 3-manifolds admit tight contact structures? See also Problem 3.40 and its remarks for further context and progress on that question.\n\n(3) There has been some partial progress towards an affirmative answer to this question. For example, the work of Golla [Gol15, Proposition 6.18] (see also Mark--Tosun [MT18]) implies an affirmative answer for L-space knots, for which one can take $n_K=2g(K)$ (the point being that L-space knots are fibered and strongly quasipositive and thus have Legendrian representatives satisfying $tb+|\\mathrm{rot}|=2g_4-1=2g-1$). Further evidence comes from deep connections between tight contact structures and taut\n\nfoliations (see the section introduction). Roberts [Rob01a, Rob01b] proved that for any fibered hyperbolic knot $K$ and any slope $r\\in(-1,\\infty)$, either $r$-surgery on $K$ or $r$-surgery on its mirror admits a taut foliation, and hence a tight contact structure, providing evidence for an affirmative answer to Problem 3.41.\n\n(4) An interesting variation of this problem would be whether one can replace \"tight\" with \"fillable.\"\n\nReferences cited:\n- [Sti10] András I. Stipsicz. Ozsváth-Szabó invariants and 3-dimensional contact topology. In Proceedings of the International Congress of Mathematicians. Volume II, pages 1159–1178. Hindustan Book Agency, New Delhi, 2010. https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2010.2/ICM2010.2.pdf.\n- [Gol15] Marco Golla. Ozsváth-Szabó invariants of contact surgeries. Geom. Topol., 19(1):171–235, 2015. doi:10.2140/gt.2015.19.171.\n- [MT18] Thomas E. Mark and Bülent Tosun. Naturality of Heegaard Floer invariants under positive rational contact surgery. J. Differential Geom., 110(2):281–344, 2018. doi: 10.4310/jdg/1538791245.\n- [Rob01a] Rachel Roberts. Taut foliations in punctured surface bundles. I. Proc. London Math. Soc. (3), 82(3):747–768, 2001. doi:10.1112/plms/82.3.747.\n- [Rob01b] Rachel Roberts. Taut foliations in punctured surface bundles. II. Proc. London Math. Soc. (3), 83(2):443–471, 2001. doi:10.1112/plms/83.2.443.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The high-surgery conjecture holds for L-space knots with n_K=2g(K); an all-knot theorem is now known for negative rational surgeries, but not for the sufficiently large positive slopes in the statement.\n\n**Verified partial progress.**\n\n- Golla's contact-surgery results give the conjecture for every L-space knot with threshold 2g(K).\n- Roberts gives taut-foliation results for fibre-hyperbolic knots with a knot-or-mirror alternative.\n- Li--Wan--Zhou prove that S^3_{-r}(K) admits a tight contact structure for every knot and every positive rational r.\n\n**Full solution or refutation.**\n\nNo verified theorem supplies, for every fixed knot K, tight contact structures on all sufficiently large positive surgeries S^3_r(K).\n\n**What remains.**\n\nRemove the L-space/fibred restrictions and the knot-or-mirror/sign alternatives for all sufficiently large positive slopes.\n\n**Sources checked.**\n\n- Marco Golla, Ozsvath--Szabo invariants of contact surgeries, Geometry & Topology 19 (2015), 171--235. (primary): https://doi.org/10.2140/gt.2015.19.171\n Evidence used: Proposition 6.18 yields the stated L-space-knot special case as explained in the K3 notes.\n- Zhenkun Li, Shunyu Wan, and Hugo Zhou, Surgeries on knots and tight contact structures, arXiv:2510.05294 (2025). (primary): https://arxiv.org/abs/2510.05294\n Evidence used: Proves tightness for every negative rational surgery on every knot; the sign is opposite to the target's large positive slopes.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 3.41. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current statement and synthesis of the Golla and Roberts special cases.\n\n**Review notes.** Surgery-sign and mirror alternatives are stated explicitly; they are not treated as a proof of the positive-slope assertion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2840, "problem_number": "KP-3.42", "title": "Kirby Problem 3.42", "statement": "Does every tight contact 3-manifold have finite Giroux torsion?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.42.\n\nLiterature notes:\n(1) An affirmative answer to the question would show that a contact manifold is tight if and only if it has finite torsion.\n\n(2) Towards an affirmative answer, Gay proved in [Gay06] that strongly symplectically fillable contact manifolds have Giroux torsion equal to 0. Moreover, Colin showed that every neighborhood of certain incompressible tori in a universally tight contact manifold has finite Giroux torsion [Col01] (see also [HKM02]).\n\nReferences cited:\n- [Gay06] David T. Gay. Four-dimensional symplectic cobordisms containing three-handles. Geom. Topol., 10:1749–1759, 2006. doi:10.2140/gt.2006.10.1749.\n- [Col01] Vincent Colin. Sur la torsion des structures de contact tendues. Ann. Sci. École Norm. Sup. (4), 34(2):267–286, 2001. doi:10.1016/S0012-9593(00)01061-2.\n- [HKM02] Ko Honda, William H. Kazez, and Gordana Matić. Convex decomposition theory. Int. Math. Res. Not., 2002(2):55–88, 2002. doi:10.1155/$S^{1}$073792802101140.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Finiteness is known in important subclasses, including torsion zero for strongly fillable structures, but finite Giroux torsion for every tight contact 3-manifold remains open.\n\n**Verified partial progress.**\n\n- Strongly symplectically fillable contact 3-manifolds have Giroux torsion zero.\n- Specified neighborhoods of incompressible tori in universally tight contact manifolds have finite Giroux torsion.\n\n**Full solution or refutation.**\n\nNo source checked excludes infinite Giroux torsion for an arbitrary tight contact 3-manifold.\n\n**What remains.**\n\nProve a uniform finiteness argument beyond the fillable and universally tight incompressible-torus settings, or construct a tight infinite-torsion example.\n\n**Sources checked.**\n\n- David T. Gay, Four-dimensional symplectic cobordisms containing three-handles, Geometry & Topology 10 (2006), 1749--1759. (primary): https://doi.org/10.2140/gt.2006.10.1749\n Evidence used: Gives the zero-torsion obstruction for strongly symplectically fillable contact manifolds.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 3.42. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Retains the general problem as open and records the Gay and Colin partial results.\n\n**Review notes.** No formulation defect found.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2841, "problem_number": "KP-3.43", "title": "Kirby Problem 3.43", "statement": "Understand how various properties of contact structures behave under different kinds of symplectic cobordism. For instance:\n\n(a) Is tightness preserved under exact symplectic cobordism?\n\n(b) Is weak fillability preserved under weak symplectic cobordism?\n\n(c) How do the support genus and Giroux torsion behave under Legendrian surgery?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.43.\n\nLiterature notes:\n(1) Wand showed that tightness is preserved by Legendrian surgery [Wan15]. It follows that tightness is preserved under Stein cobordism. Stein cobordisms are examples of exact symplectic cobordisms. It is known that tightness is not in general preserved under symplectic cobordisms that are not exact. For instance, a result of Gay shows that tightness is not preserved under strong symplectic cobordism [Gay06]; see also [BE13].\n\n(2) Many forms of symplectic fillability---including strong, exact, and Stein fillability---are preserved under the same type of cobordisms. However, it is unknown whether weak symplectic fillability is preserved under weak symplectic cobordism.\n\n(3) The behavior of Giroux torsion and the support genus of a contact structure under Legendrian surgery is completely open, though the expectation is that Giroux torsion is nonincreasing and the support genus is nondecreasing under Legendrian surgery.\n\nReferences cited:\n- [Wan15] Andy Wand. Tightness is preserved by Legendrian surgery. Ann. of Math. (2), 182(2):723–738, 2015. doi:10.4007/annals.2015.182.2.8.\n- [Gay06] David T. Gay. Four-dimensional symplectic cobordisms containing three-handles. Geom. Topol., 10:1749–1759, 2006. doi:10.2140/gt.2006.10.1749.\n- [BE13] John A. Baldwin and John B. Etnyre. Admissible transverse surgery does not preserve tightness. Math. Ann., 357(2):441–468, 2013. doi:10.1007/s00208-013-0911-8.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Tightness is preserved by Legendrian surgery and thus Stein cobordisms, but preservation under arbitrary exact cobordisms, weak fillability under weak cobordisms, and support-genus/Giroux-torsion behavior remain open.\n\n**Verified partial progress.**\n\n- Wand proves that Legendrian surgery preserves tightness, settling the Stein-cobordism subclass of (a).\n- Nonexact strong symplectic cobordisms can fail to preserve tightness, showing that exactness cannot simply be discarded.\n\n**Full solution or refutation.**\n\nOnly a proper subclass of part (a) is settled; parts (b) and (c), and general exact cobordisms in (a), remain open.\n\n**What remains.**\n\nDecide tightness preservation for all exact cobordisms, weak fillability preservation for weak cobordisms, and the conjectured monotonicity of support genus and Giroux torsion under Legendrian surgery.\n\n**Sources checked.**\n\n- Andy Wand, Tightness is preserved by Legendrian surgery, Annals of Mathematics 182 (2015), 723--738. (primary): https://doi.org/10.4007/annals.2015.182.2.8\n Evidence used: Direct theorem for Legendrian surgery and hence Stein cobordisms.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 3.43. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current subpart-by-subpart status, including the open weak-cobordism and Legendrian-surgery invariants questions.\n\n**Review notes.** This is an agenda-style composite record; partial progress is not presented as resolving every part.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2842, "problem_number": "KP-3.44", "title": "Kirby Problem 3.44", "statement": "Is there an algorithm to decide, given an open book, whether the corresponding contact 3-manifold is tight or fillable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.44.\n\nLiterature notes:\n(1) See [Wan15] for work of Wand related to this problem.\n\n(2) A similar question is whether there is an algorithm to decide, given some presentation of a 3-manifold, whether it admits a tight or fillable contact structure.\n\n(3) There are known criteria for tightness and Stein fillability. For example, Giroux proved [Gir02] that a contact 3-manifold is Stein fillable if and only if it has a compatible open book whose monodromy is a product of positive Dehn twists; see also Loi--Piergallini [LP01]. Similarly, Honda-- Kazez--Matić proved [HKM07] that a contact 3-manifold is tight if and only if all of its compatible open books are right-veering.\n\nReferences cited:\n- [Wan15] Andy Wand. Tightness is preserved by Legendrian surgery. Ann. of Math. (2), 182(2):723–738, 2015. doi:10.4007/annals.2015.182.2.8.\n- [Gir02] Emmanuel Giroux. Géométrie de contact: de la dimension trois vers les dimensions supérieures. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 405–414. Higher Ed. Press, Beijing, 2002.\n- [LP01] Andrea Loi and Riccardo Piergallini. Compact Stein surfaces with boundary as branched covers of $B^{4}$. Invent. Math., 143(2):325–348, 2001. doi:10.1007/s002220000106.\n- [HKM07] Ko Honda, William H. Kazez, and Gordana Matić. Right-veering diffeomorphisms of compact surfaces with boundary. Invent. Math., 169(2):427–449, 2007. doi:10.1007/s00222-007-0051-4.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exact characterizations via positive Dehn-twist factorizations and right-veering compatible open books are known, but no general terminating decision algorithm for the presented open book was verified.\n\n**Verified partial progress.**\n\n- Stein fillability is characterized by existence of a compatible open book with monodromy factored into positive Dehn twists.\n- Tightness is characterized by every compatible open book being right-veering.\n\n**Full solution or refutation.**\n\nThe structural criteria do not by themselves supply a verified decision procedure, and the current K3 list retains the algorithmic question.\n\n**What remains.**\n\nGive a terminating algorithm or prove undecidability, after specifying which notion or notions of fillability are to be decided.\n\n**Sources checked.**\n\n- Ko Honda, William H. Kazez, and Gordana Matic, Right-veering diffeomorphisms of compact surfaces with boundary, Inventiones Mathematicae 169 (2007), 427--449. (primary): https://doi.org/10.1007/s00222-007-0051-4\n Evidence used: Right-veering characterization relevant to tightness.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 3.44. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current problem statement and synthesis of the Giroux, Loi--Piergallini, and Honda--Kazez--Matic criteria.\n\n**Review notes.** Formulation defect: 'fillable' does not specify weak, strong, exact, or Stein fillability; the wording is preserved and the ambiguity is flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2843, "problem_number": "KP-3.45", "title": "Kirby Problem 3.45", "statement": "(a) Are there contact 3-manifolds with support genus greater than one?\n\n(b) Are there contact 3-manifolds with arbitrarily large support genus?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.45.\n\nLiterature notes:\n(1) The support genus of a contact 3-manifold $(Y,\\xi)$ was defined by Etnyre-- Ozbagci [EO08] to be the minimum of\n\n$$\n\\{\\,g(S)\\mid (S,\\phi)\\text{ is an open book supporting }(Y,\\xi)\\,\\}.\n$$\n\n(2) Overtwisted contact 3-manifolds have support genus zero [Etn04b]. However, Etnyre proved [Etn04b] that there are contact 3-manifolds with nonzero support genus, and introduced the problem of finding the minimal genus of a supporting open book for a given contact structure. Ozsváth-- Stipsicz--Szabó gave another proof of the same using Heegaard Floer homology [OSS05].\n\n(3) For candidate contact structures with large support genus, one might consider connected sums of contact manifolds with support genus one. Another potential source of examples are contact structures compatible with open books $(S,\\phi)$ where S is a surface of large genus and one boundary component, and $\\phi$ is a composition of many Dehn twists along a curve parallel to $\\partial S$.\n\nReferences cited:\n- [EO08] John B. Etnyre and Burak Ozbagci. Invariants of contact structures from open books. Trans. Amer. Math. Soc., 360(6):3133–3151, 2008. doi:10.1090/S0002-9947-08-04459-0.\n- [Etn04b] John B. Etnyre. Planar open book decompositions and contact structures. Int. Math. Res. Not., 2004(79):4255–4267, 2004. doi:10.1155/$S^{1}$073792804142207.\n- [OSS05] Peter Ozsváth, András Stipsicz, and Zoltán Szabó. Planar open books and Floer homology. Int. Math. Res. Not., 2005(54):3385–3401, 2005. doi:10.1155/IMRN.2005.3385.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Examples of support genus zero and one are known, but no contact 3-manifold of support genus greater than one, and hence no unbounded family, was verified.\n\n**Verified partial progress.**\n\n- Overtwisted contact 3-manifolds have support genus zero.\n- Heegaard Floer and fillability obstructions give examples with nonzero support genus, hence support genus one.\n\n**Full solution or refutation.**\n\nBoth stated parts remain open in the 2026 K3 list; the known positive-support-genus examples only reach genus one.\n\n**What remains.**\n\nConstruct a contact structure with support genus at least two for part (a), and then establish unbounded support genus for part (b), or prove an unexpected universal upper bound.\n\n**Sources checked.**\n\n- John B. Etnyre and Burak Ozbagci, Invariants of contact structures from open books, Transactions of the AMS 360 (2008), 3133--3151. (primary): https://doi.org/10.1090/S0002-9947-08-04459-0\n Evidence used: Definition and foundational properties of support genus.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 3.45. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current list retains both genus-greater-than-one and unboundedness questions and records the genus-zero/positive-genus baseline.\n\n**Review notes.** No formulation defect found; parts (a) and (b) remain distinct.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2844, "problem_number": "KP-3.46", "title": "Kirby Problem 3.46", "statement": "Let $\\lambda$ be a contact form on a closed 3-manifold that is not a lens space. Must the associated Reeb flow have infinitely many simple periodic orbits?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.46.\n\nLiterature notes:\n(1) On lens spaces, there are examples of contact forms with exactly two simple Reeb orbits; see [CGHHL23, Ex. 1.1]. Conversely, it is shown in [CGHHL23] that if a closed three-manifold admits a contact form with exactly two simple Reeb orbits, then it must be a lens space.\n\n(2) By [CGH16], two is the minimum number of simple Reeb orbits for any contact form on a closed three-manifold. In [CGHP19], it was shown that for a nondegenerate contact form on a closed three-manifold, such that the associated contact structure has torsion Chern class, there are either two or infinitely many simple Reeb orbits. Later, this result was generalized by removing either the Chern class assumption [CDR22], or the nondegeneracy assumption [CGHHL24]; however, removing both assumptions simultaneously remains open, and this is equivalent to Problem 3.46, by Remark (1).\n\n(3) Once results on infinitely many orbits are established, a next natural step is to study the growth rate of the number of orbits in terms of the period.\n\nReferences cited:\n- [CGHHL23] Daniel Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings, and Hui Liu. Contact three-manifolds with exactly two simple Reeb orbits. Geom. Topol., 27(9):3801–3831, 2023. doi:10.2140/gt.2023.27.3801.\n- [CGH16] Dan Cristofaro-Gardiner and Michael Hutchings. From one Reeb orbit to two. J. Diff. Geom., 102:25–36, 2016. http://projecteuclid.org/euclid.jdg/1452002876.\n- [CGHP19] Dan Cristofaro-Gardiner, Michael Hutchings, and Daniel Pomerleano. Torsion contact forms in three dimensions have two or infinitely many Reeb orbits. Geometry \\& Topology, 23(7):3601–3645, 2019.\n- [CDR22] Vincent Colin, Pierre Dehornoy, and Ana Rechtman. On the existence of supporting broken book decompositions for contact forms in dimension 3. Invent. Math, 231:1489–1539, 2022.\n- [CGHHL24] Dan Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings, and Hui Liu. Proof of Hofer-Wysocki-Zehnder’s two or infinity conjecture, 2024. arXiv:2310.07636.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The two-or-infinity dichotomy is proved under nondegeneracy without a Chern-class restriction and, separately, under torsion c1 without nondegeneracy; removing both hypotheses simultaneously remains open.\n\n**Verified partial progress.**\n\n- Every contact form on a closed 3-manifold has at least two simple Reeb orbits.\n- A closed 3-manifold admitting a contact form with exactly two simple Reeb orbits must be a lens space.\n- The desired infinity conclusion holds for nondegenerate contact forms and also, without nondegeneracy, when the contact structure has torsion first Chern class.\n\n**Full solution or refutation.**\n\nThe unrestricted theorem for degenerate contact forms with nontorsion first Chern class was not found; the 2026 K3 list identifies that simultaneous removal as equivalent to the problem.\n\n**What remains.**\n\nProve the two-or-infinity dichotomy for degenerate contact forms whose contact structure has nontorsion first Chern class.\n\n**Sources checked.**\n\n- Daniel Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings, and Hui Liu, Contact three-manifolds with exactly two simple Reeb orbits, Geometry & Topology 27 (2023), 3801--3831. (primary): https://doi.org/10.2140/gt.2023.27.3801\n Evidence used: Exactly two simple Reeb orbits force the closed 3-manifold into the lens-space exceptional class.\n- Daniel Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings, and Hui Liu, Proof of Hofer--Wysocki--Zehnder's two or infinity conjecture, JAMS 39 (2026), 915--983; arXiv:2310.07636. (primary): https://arxiv.org/abs/2310.07636\n Evidence used: Proves the two-or-infinity dichotomy without nondegeneracy when the contact structure has torsion first Chern class.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 3.46. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current synthesis separates the nondegenerate and torsion-c1 theorems and says simultaneous removal remains open.\n\n**Review notes.** The two hypotheses are not conflated. The source convention for lens spaces is retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2845, "problem_number": "KP-3.47", "title": "Kirby Problem 3.47", "statement": "(a) Does every Reeb flow on $S^3$, associated to a contact form giving the standard contact structure, have an elliptic periodic orbit?\n\n(b) What about if one restricts to Reeb flows on $S^3$ arising from boundaries of strictly convex domains in $\\mathbb{R}^4$ ?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.47.\n\nLiterature notes:\n(1) It is known that no flow on $S^3$ can be Anosov.\n\n(2) For some dynamical implications of the existence of an elliptic orbit; see [New77].\n\n(3) For more about what is known the problem, including in higher dimensions, see the summary in the introduction of [AM17].\n\n(4) It is a longstanding problem whether every Riemannian metric on $S^2$ has an elliptic closed geodesic; an affirmative answer to Part (a) would imply this. For more about the history of this problem, see [CM24], which in particular proves the existence of an elliptic closed geodesic for $C^2$-generic Riemannian metrics on $S^2$.\n\nReferences cited:\n- [New77] Sheldon E. Newhouse. Quasi-elliptic periodic points in conservative dynamical systems. Amer. J. Math., 99(5):1061–1087, 1977. doi:10.2307/2374000.\n- [AM17] Miguel Abreu and Leonardo Macarini. Dynamical convexity and elliptic periodic orbits for Reeb flows. Mathematische Annalen, 369:331–386, 2017.\n- [CM24] Gonzalo Contreras and Marco Mazzucchelli. Proof of the C2-stability conjecture for geodesic flows of closed surfaces. Duke Math. J., 173(2):347–390, 2024. doi: 10.1215/00127094-2023-0010.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Generic metrics on S2 have elliptic closed geodesics and special Reeb-flow results are known, but universal elliptic periodic-orbit existence in either stated class remains open.\n\n**Verified partial progress.**\n\n- C2-generic Riemannian metrics on S2 have elliptic closed geodesics.\n\n**Full solution or refutation.**\n\nNo theorem for every standard-contact Reeb flow or every strictly convex boundary was verified.\n\n**What remains.**\n\nProve elliptic orbit existence in the stated universal classes.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.47 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States generic/geodesic implications and ongoing universal problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2846, "problem_number": "KP-3.48", "title": "Kirby Problem 3.48", "statement": "(L-space Conjecture). For prime rational homology 3-spheres Y, are the following equivalent?\n\n(a) $\\pi_1(Y)$ is left-orderable.\n\n(b) Y is not an L-space.\n\n(c) Y admits a coorientable taut foliation.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.48.\n\nLiterature notes:\n(1) This problem is known as the L-space Conjecture. The question of whether\n\n(b) is equivalent to (c) was first raised by Ozsváth--Szabó following their proof (completed by Bowden and Kazez--Roberts) that (c) implies (b); see below. This equivalence was proved for Seifert fibered rational homology spheres with orientable base (i.e., $S^2$ ) by Lisca--Stipsicz [LS07]. Boyer-- Rolfsen--Wiest had previously established the equivalence of (a) and (c) for the same class of manifolds [BRW05]. Together, these results implied the L-space Conjecture for Seifert fibered rational homology spheres with orientable base, and inspired the conjecture that (a) and (b) are equivalent in general, as put forth by Boyer--Gordon--Watson in [BGW13]; see also [Pet10]. The formulation of the L-space Conjecture as stated above first appears in Juhász's survey [Juh15].\n\n(2) A nontrivial group G is left-orderable if it admits a left-invariant total order.\n\n(3) The only direction that is known to hold in general is that if Y admits a coorientable taut foliation, then Y is not an L-space. This was proved by Ozsváth--Szabó for $C^2$-foliations in [OS04b]; see also [KMOS07]. Later results of Bowden [Bow16] and Kazez--Roberts [KR17] completed the proof for $C^0$-foliations. If Y is additionally an integer homology sphere, then having a coorientable taut foliation implies having a left-orderable fundamental group [BB15, CD03]. A notable partial converse is that if $\\pi_1(Y)$ is left-orderable and Y has a genus 2 Heegaard splitting, then Y has a taut foliation [Li24].\n\n(4) The conjecture has now been solved for large families of 3-manifolds, including all those with Seifert or Sol geometry [BGW13] and all graph manifolds [BC17b, HRRW20, Ras17]. Consequently, it remains to prove the conjecture for 3-manifolds admitting hyperbolic pieces in their JSJ decompositions.\n\n(5) By [BRW05], if f : Y $\\to$ Z is a nonzero degree map between prime rational homology spheres and $\\pi_1(Z)$ is left-orderable, then so is $\\pi_1(Y)$. Consequently, a proof of the L-space conjecture would imply that if Y is an L-space, then so is Z. See Problem 3.62.\n\nReferences cited:\n- [LS07] Paolo Lisca and András I. Stipsicz. Ozsváth-Szabó invariants and tight contact 3-manifolds. III. J. Symplectic Geom., 5(4):357–384, 2007. doi:10.4310/jsg.2007.v5.n4.a1.\n- [BRW05] Steven Boyer, Dale Rolfsen, and Bert Wiest. Orderable 3-manifold groups. Ann. Inst. Fourier (Grenoble), 55(1):243–288, 2005. URL: http://aif.cedram.org/item? id=AIF 2005 55 1 243 0.\n- [BGW13] Steven Boyer, Cameron McA. Gordon, and Liam Watson. On L-spaces and leftorderable fundamental groups. Math. Ann., 356(4):1213–1245, 2013. doi:10.1007/s00208-012-0852-7.\n- [Pet10] Thomas David Peters. Computations of Heegaard Floer Homology: Torus Bundles, L-Spaces, and Correction Terms. PhD thesis, Columbia University, United States – New York, 2010.\n- [Juh15] András Juhász. A survey of Heegaard Floer homology. In New ideas in low dimensional topology, volume 56 of Ser. Knots Everything, pages 237–296. World Sci. Publ., Hackensack, NJ, 2015. URL: https://doi.org/10.1142/9789814630627 0007, doi:10.1142/9789814630627\\\\_0007.\n- [OS04b] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and genus bounds. Geom. Topol., 8:311–334, 2004. doi:10.2140/gt.2004.8.311.\n- [KMOS07] P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó. Monopoles and lens space surgeries. Ann. of Math. (2), 165(2):457–546, 2007. doi:10.4007/annals.2007.165.457.\n- [Bow16] Jonathan Bowden. Approximating C0-foliations by contact structures. Geom. Funct. Anal., 26(5):1255–1296, 2016. doi:10.1007/s00039-016-0387-2.\n- [KR17] William H. Kazez and Rachel Roberts. C0 approximations of foliations. Geom. Topol., 21(6):3601–3657, 2017. doi:10.2140/gt.2017.21.3601.\n- [BB15] Michel Boileau and Steven Boyer. Graph manifold Z-homology 3-spheres and taut foliations. J. Topol., 8(2):571–585, 2015. doi:10.1112/jtopol/jtv006.\n- [CD03] Danny Calegari and Nathan M. Dunfield. Laminations and groups of homeomorphisms of the circle. Invent. Math., 152(1):149–204, 2003. doi:10.1007/s00222-002-0271-6.\n- [Li24] Tao Li. Taut foliations of 3-manifolds with Heegaard genus 2. Duke Math. J., 173(8):1427–1475, 2024. doi:10.1215/00127094-2023-0038.\n- [BC17b] Steven Boyer and Adam Clay. Foliations, orders, representations, L-spaces and graph manifolds. Adv. Math., 310:159–234, 2017. doi:10.1016/j.aim.2017.01.026.\n- [HRRW20] Jonathan Hanselman, Jacob Rasmussen, Sarah Dean Rasmussen, and Liam Watson. L-spaces, taut foliations, and graph manifolds. Compos. Math., 156(3):604–612, 2020. doi:10.1112/s0010437x19007814.\n- [Ras17] Sarah Dean Rasmussen. L-space intervals for graph manifolds and cables. Compos. Math., 153(5):1008–1049, 2017. doi:10.1112/S0010437X16008319.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The L-space conjecture holds for Seifert fibered rational homology spheres with orientable base and numerous other classes, but remains open generally.\n\n**Verified partial progress.**\n\n- Lisca--Stipsicz prove foliation/L-space equivalence in the stated Seifert class.\n- Boyer--Rolfsen--Wiest prove left-orderability/foliation equivalence there.\n\n**Full solution or refutation.**\n\nNo all-prime-rational-homology-sphere equivalence theorem was verified.\n\n**What remains.**\n\nProve or refute all three equivalences in general.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.48 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Documents Seifert cases and the general conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2847, "problem_number": "KP-3.49", "title": "Kirby Problem 3.49", "statement": "Are any of the three conditions in the L-space Conjecture equivalent, for all prime rational homology 3-spheres Y, to the condition that Y admits a contact structure that is filled by a symplectic 4-manifold with $b_2^+>0$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.49.\n\nLiterature notes:\nIn proving that the existence of a coorientable taut foliation on Y implies that Y is not an L-space, Ozsváth--Szabó construct [OS04b] a weak symplectic filling of a particular contact structure on Y. This filling can be chosen to have $b_2^+>0$. Recall that for rational homology spheres, admitting a weak symplectic filling is equivalent to admitting a strong one [OO05].\n\nReferences cited:\n- [OS04b] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and genus bounds. Geom. Topol., 8:311–334, 2004. doi:10.2140/gt.2004.8.311.\n- [OO05] Hiroshi Ohta and Kaoru Ono. Simple singularities and symplectic fillings. J. Differential Geom., 69(1):1–42, 2005. doi:10.4310/jdg/1121540338.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A coorientable taut foliation yields a contact structure with weak/strong filling having b2-positive, but converse equivalences with the three L-space conditions are unknown.\n\n**Verified partial progress.**\n\n- Ozsvath--Szabo construct the relevant weak filling from taut foliation.\n- For rational homology spheres weak and strong fillability coincide.\n\n**Full solution or refutation.**\n\nOnly the forward implication is verified.\n\n**What remains.**\n\nDetermine whether filling condition is equivalent to any/all L-space-conjecture conditions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.49 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the filling construction and remaining equivalence question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2848, "problem_number": "KP-3.50", "title": "Kirby Problem 3.50", "statement": "(The Floer Poincaré Conjecture). If Y is an integral homology sphere that is an L-space, show that Y is $S^3$ or the connected sum of some copies of the Poincaré sphere (with arbitrary orientations).", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.50.\n\nLiterature notes:\n(1) This is in the spirit of [Kir97, Problem 3.106], which asks a similar question about instanton Floer homology.\n\n(2) The statement is already known to be true for several classes of homology spheres: Dehn surgeries on knots in $S^3$ by [OS11]; Brieskorn spheres by [OS03b] in combination with [Eft09]; and manifolds with incompressible tori by [Eft18].\n\n(3) In combination with the L-space Conjecture (Problem 3.48), a solution to the problem would provide an alternative route to the Poincaré Conjecture: if Y is a homotopy 3-sphere, then $\\pi_1(Y)$ is not left-orderable, hence an L-space, hence $S^3$ or a connected sum of Poincaré spheres (but the latter possibility contradicts simple connectivity).\n\n(4) One can phrase an analogous conjecture for framed instanton homology, which would say that if $I^{\\#}(Y)\\cong\\mathbb{Z}$ then $Y=S^3$. If this were true then it would imply the Poincaré Conjecture immediately (without the L-space conjecture), because trivial $\\pi_1$ implies $I^{\\#}(Y)\\cong\\mathbb{Z}$. It would also imply a positive answer to Problem 3.52. Note that $I^{\\#}$ is somewhat different from $\\widehat{HF}$; for instance, Bhat has announced that $I^{\\#}$ of the Poincaré sphere has 2-torsion [Bha24].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [OS11] Peter Ozsváth and Zoltán Szabó. Knot Floer homology and rational surgeries. Algebr. Geom. Topol., 11(1):1–68, 2011. doi:10.2140/agt.2011.11.1.\n- [OS03b] Peter Ozsváth and Zoltán Szabó. On the Floer homology of plumbed threemanifolds. Geom. Topol., 7:185–224, 2003. doi:10.2140/gt.2003.7.185.\n- [Eft09] Eaman Eftekhary. Seifert fibered homology spheres with trivial Heegaard Floer homology, 2009. arXiv:0909.3975.\n- [Eft18] Eaman Eftekhary. Bordered Floer homology and existence of incompressible tori in homology spheres. Compos. Math., 154(6):1222–1268, 2018. doi:10.1112/s0010437x18007054.\n- [Bha24] Deeparaj Bhat. Surgery exact triangles in instanton theory, 2024. arXiv:2311.04242.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Floer Poincare assertion is known for surgeries on knots in S3, Brieskorn spheres and manifolds with incompressible tori, but is open generally.\n\n**Verified partial progress.**\n\n- Ozsvath--Szabo/Eftekhary-type results cover the named classes.\n\n**Full solution or refutation.**\n\nNo universal classification of integral homology-sphere L-spaces was verified.\n\n**What remains.**\n\nProve the stated connected-sum classification for all integral homology spheres.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.50 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists proven classes and remaining general conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2849, "problem_number": "KP-3.51", "title": "Kirby Problem 3.51", "statement": "Suppose Y is a rational homology 3-sphere such that every homomorphism $\\pi_1(Y)$ $\\to$ $\\operatorname{SU}(2)$ has abelian image. Does it follow that\n\n$$\n\\dim_{\\mathbb{C}} I^{\\#}(Y;\\mathbb{C})=|H_1(Y;\\mathbb{Z})|,\n$$\n i.e., that Y is an instanton L-space?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.51.\n\nLiterature notes:\n(1) The conclusion follows if the corresponding set of reducible flat connections is Morse--Bott nondegenerate for the Chern--Simons functional, which is equivalent to Y being cyclically finite [BN90, BS18b]. This holds automatically, for instance, if Y is a homology 3-sphere. The problem is whether the conclusion follows without assuming that Y is cyclically finite.\n\n(2) If the L-space conjecture and the conjectured isomorphism between instanton homology and Heegaard Floer homology both hold (see Problems 3.59 and 3.48), then an affirmative answer to the question in the problem would imply the following: for Y an irreducible rational homology 3-sphere, if $\\pi_1(Y)$ is left-orderable, then $\\pi_1(Y)$ admits an irreducible $\\operatorname{SU}(2)$-representation. Proving the latter would be interesting in its own right.\n\n(3) One can ask a similar question regarding instanton Floer homology and representations from $\\pi_1(Y)$ to $\\operatorname{SU}(N)$ for $N>2$; see [DIS24] for recent work in the $N=3$ case.\n\nReferences cited:\n- [BN90] S. Boyer and A. Nicas. Varieties of group representations and Casson’s invariant for rational homology 3-spheres. Trans. Amer. Math. Soc., 322(2):507–522, 1990. doi:10.2307/2001712.\n- [BS18b] John A. Baldwin and Steven Sivek. Stein fillings and $\\mathrm{SU}(2)$ representations. Geom. Topol., 22(7):4307–4380, 2018. doi:10.2140/gt.2018.22.4307.\n- [DIS24] Aliakbar Daemi, Nobuo Iida, and Christopher Scaduto. Rank three instantons, representations and sutures, 2024. arXiv:2402.10448.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The instanton L-space conclusion holds under cyclic finiteness/Morse--Bott nondegeneracy, including homology spheres, but is open without this hypothesis.\n\n**Verified partial progress.**\n\n- Cyclically finite manifolds satisfy the desired conclusion.\n\n**Full solution or refutation.**\n\nNo proof from abelian SU2 representations alone was verified.\n\n**What remains.**\n\nRemove cyclic-finiteness assumption or find a counterexample.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.51 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the cyclic-finite theorem and open generalization.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2850, "problem_number": "KP-3.52", "title": "Kirby Problem 3.52", "statement": "(a) Does every closed 3-manifold M besides the 3-sphere admit a nontrivial representation $\\pi_1(M)$ $\\to$ $\\operatorname{SU}(2)$?\n\n(b) For which M with nonabelian fundamental group does every homomorphism $\\pi_1(M)$ $\\to$ $\\operatorname{SU}(2)$ have abelian image?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.52.\n\nLiterature notes:\n(1) The question in (a) is [Kir97, Problem 3.105(A)].\n\n(2) The answer to (a) is yes if $M\\not\\cong S^3$ is Seifert fibered [FS90] or toroidal [LPCZ23] (see also [BS22]), leaving only the hyperbolic case open. It is also yes whenever M is Dehn surgery on a knot in $S^3$ [KM04], or the branched double cover of a knot in $S^3$ [CNS16, Zen17] (using results of Kronheimer and Mrowka [KM10]), or the boundary of a Stein domain with nontrivial homology [BS18b]. These ultimately rely on nonvanishing results for some form of instanton Floer homology, or closely related techniques.\n\n(3) An affirmative answer to (a) would follow from two widely believed conjectures, that (i) framed instanton homology is isomorphic to the \"hat\" version of Heegaard Floer homology over Q; and (ii) the only irreducible integer homology sphere L-spaces are $S^3$ and the Poincaré homology sphere with either orientation; see 3.59 and 3.50 for discussions of (i) and (ii).\n\n(4) The analogue of (a) for representations $\\pi_1(M)$ $\\to$ $\\operatorname{SL}(2, \\mathbb{C})$ has been answered affirmatively by Zentner [Zen18].\n\n(5) Regarding the question in (b), there are many closed 3-manifolds M such that $\\pi_1(M)$ is nonabelian but every representation $\\pi_1(M)$ $\\to$ $\\operatorname{SU}(2)$ has abelian image. This includes infinitely many graph manifolds [Mot88], built by gluing together pairs of torus knot exteriors. The Seifert fibered\n\nexamples are classified [SZ22], and only a handful of hyperbolic examples are known, including the manifold known as Vol3, as reported by Dunfield. Each of these examples has nontrivial first homology.\n\n(6) A rational homology sphere for which every representation $\\pi_1(M)$ $\\to$ $\\operatorname{SU}(2)$ has abelian image is generally (and perhaps always) an instanton L-space; see Problem 3.51.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [FS90] Ronald Fintushel and Ronald J. Stern. Instanton homology of Seifert fibred homology three spheres. Proc. London Math. Soc. (3), 61(1):109–137, 1990. doi: 10.1112/plms/s3-61.1.109.\n- [LPCZ23] Tye Lidman, Juanita Pinzón-Caicedo, and Raphael Zentner. Toroidal integer homology three-spheres have irreducible $\\mathrm{SU}(2)$-representations. J. Topol., 16(1):344– 367, 2023. doi:10.1112/topo.12275.\n- [BS22] John A. Baldwin and Steven Sivek. Instanton L-spaces and splicing. Ann. H. Lebesgue, 5:1213–1233, 2022. doi:10.5802/ahl.148.\n- [KM04] P. B. Kronheimer and T. S. Mrowka. Dehn surgery, the fundamental group and $\\mathrm{SU}(2)$. Math. Res. Lett., 11(5-6):741–754, 2004. doi:10.4310/MRL.2004.v11.n6.a3.\n- [CNS16] Christopher Cornwell, Lenhard Ng, and Steven Sivek. Obstructions to Lagrangian concordance. Algebr. Geom. Topol., 16(2):797–824, 2016. doi:10.2140/agt.2016.16.797.\n- [Zen17] Raphael Zentner. A class of knots with simple $\\mathrm{SU}(2)$-representations. Selecta Math. (N.S.), 23(3):2219–2242, 2017. doi:10.1007/s00029-017-0314-x.\n- [KM10] Peter Kronheimer and Tomasz Mrowka. Knots, sutures, and excision. J. Differential Geom., 84(2):301–364, 2010. http://projecteuclid.org/euclid.jdg/1274707316.\n- [BS18b] John A. Baldwin and Steven Sivek. Stein fillings and $\\mathrm{SU}(2)$ representations. Geom. Topol., 22(7):4307–4380, 2018. doi:10.2140/gt.2018.22.4307.\n- [Zen18] Raphael Zentner. Integer homology 3-spheres admit irreducible representations in $\\mathrm{SL}(2,\\mathbb{C})$. Duke Math. J., 167(9):1643–1712, 2018. doi:10.1215/00127094-2018-0004.\n- [Mot88] Kimihiko Motegi. Haken manifolds and representations of their fundamental groups in $\\mathrm{SL}(2,\\mathbb{C})$. Topology Appl., 29(3):207–212, 1988. doi:10.1016/0166-8641(88) 90019-3.\n- [SZ22] Steven Sivek and Raphael Zentner. A menagerie of $\\mathrm{SU}(2)$-cyclic 3-manifolds. Int. Math. Res. Not. IMRN, 2022(11):8038–8085, 2022. doi:10.1093/imrn/rnaa330.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Nontrivial SU2 representations are known for Seifert fibered, toroidal, knot-surgery, branched-cover and several Stein-boundary manifolds; the general hyperbolic case and classification of SU2-abelian groups remain open.\n\n**Verified partial progress.**\n\n- The source records positive theorems for each named broad class.\n\n**Full solution or refutation.**\n\nNo proof for every non-S3 closed manifold was verified.\n\n**What remains.**\n\nSettle hyperbolic manifolds and classify the abelian-image exceptions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.52 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists the solved classes and remaining hyperbolic question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2851, "problem_number": "KP-3.53", "title": "Kirby Problem 3.53", "statement": "Are all strong L-spaces branched double covers of alternating links in $S^3$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.53.\n\nLiterature notes:\n(1) A rational homology 3-sphere $Y$ is an L-space if\n\n$$\n\\operatorname{rank}\\widehat{HF}(Y)=|H_1(Y)|,\n$$\n\nwhere $|H_1(Y)|$ denotes the number of elements in $H_1(Y)$. A strong L-space is a rational homology 3-sphere $Y$ that admits a Heegaard diagram $H$ so that\n\n$$\n\\operatorname{rank}\\widehat{CF}(H)=|H_1(Y)|.\n$$\n\nCall such a Heegaard diagram a strong Heegaard diagram. If $H$ is strong, then the differential on $\\widehat{CF}(H)$ vanishes.\n\n(2) The notion of a strong L-space was studied by Levine and Lewallen [LL12]. They proved, for example, that the Poincaré homology 3-sphere is not a strong L-space (despite being an L-space). Greene [Gre13c] observed that the double branched cover of a non-split alternating link is a strong L-space. The question above was first asked by Greene and Levine in [GL16], who proved the case of L-spaces that admit genus 2 strong Heegaard diagrams.\n\n(3) It is open whether being a strong L-space is equivalent to admitting a Heegaard diagram $H$ for which the differential on $\\widehat{CF}(H)$ vanishes.\n\nReferences cited:\n- [LL12] Adam Simon Levine and Sam Lewallen. Strong L-spaces and left-orderability. Math. Res. Lett., 19(6):1237–1244, 2012. doi:10.4310/MRL.2012.v19.n6.a5.\n- [Gre13c] Joshua Evan Greene. A spanning tree model for the Heegaard Floer homology of a branched double-cover. J. Topol., 6(2):525–567, 2013. doi:10.1112/jtopol/jtt007.\n- [GL16] Joshua Evan Greene and Adam Simon Levine. Strong Heegaard diagrams and strong L-spaces. Algebr. Geom. Topol., 16(6):3167–3208, 2016. doi:10.2140/agt.2016.16.3167.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Branched double covers of non-split alternating links are strong L-spaces, and the genus-two strong-diagram case is known; the universal converse is open.\n\n**Verified partial progress.**\n\n- Greene observes alternating double covers are strong.\n- Greene--Levine prove the genus-two strong Heegaard-diagram case.\n\n**Full solution or refutation.**\n\nNo classification of all strong L-spaces was verified.\n\n**What remains.**\n\nProve or refute alternating-branched-cover characterization in higher genus.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.53 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records forward theorem and genus-two converse.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2852, "problem_number": "KP-3.54", "title": "Kirby Problem 3.54", "statement": "(a) Is there a closed 3-manifold $M$ whose Heegaard Floer homology $\\widehat{HF}(M;\\mathbb{Z})$ has torsion?\n\n(b) Is there a rational homology 3-sphere $M$ for which $\\widehat{HF}(M;\\mathbb{Z})$ has torsion?\n\n(c) Is there a knot $K\\subset S^3$ whose knot Floer homology $\\widehat{HFK}(S^3,K;\\mathbb{Z})$ has torsion?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.54.\n\nLiterature notes:\n(1) The Heegaard Floer homology of a closed 3-manifold $M$ with integer coefficients depends on a choice of isomorphism class of coherent orientation systems (and similarly for the knot Floer homology of links). There are $2^{b_1(M)}$ such choices in general. The Heegaard Floer group referenced in part (a) is defined using the canonical isomorphism class of orientation systems described in [OS04d]. By contrast, $\\widehat{HF}(S^1\\times S^2,o;\\mathbb{Z})\\cong \\mathbb{Z}/2\\mathbb{Z}$, with respect to the non-canonical orientation system $o$. For rational homology 3-spheres the canonical choice is the only choice.\n\n(2) Jabuka--Mark [JM08] proved that there is 2-torsion in $HF^+(\\Sigma_g\\times S^1;\\mathbb{Z})$ and $HF^{\\infty}(\\Sigma_g\\times S^1;\\mathbb{Z})$, where $\\Sigma_g$ is the genus-$g$ surface and $g\\geq 3$, but they showed there is no torsion in $\\widehat{HF}(\\Sigma_g\\times S^1;\\mathbb{Z})$ for any $g$.\n\n(3) In a related direction, Li and Ye in [LY24], and Bhat in [Bha24] announced the existence of 2-torsion in the framed instanton Floer homology $I^{\\#}(S^3_r(K);\\mathbb{Z})$ for any nontrivial knot $K$ with $r=1,1/2,1/4$, and 2-torsion in the unreduced singular instanton knot homology $I^{\\#}(S^3,K;\\mathbb{Z})$ for many knots $K$.\n\nReferences cited:\n- [OS04d] Peter Ozsváth and Zoltán Szabó. Holomorphic disks and three-manifold invariants: properties and applications. Ann. of Math. (2), 159(3):1159–1245, 2004. doi:10.4007/annals.2004.159.1159.\n- [JM08] Stanislav Jabuka and Thomas E. Mark. On the Heegaard Floer homology of a surface times a circle. Adv. Math., 218(3):728–761, 2008. doi:10.1016/j.aim.2008.01.009.\n- [LY24] Zhenkun Li and Fan Ye. 2-torsion in instanton Floer homology, 2024. arXiv:2405.16252.\n- [Bha24] Deeparaj Bhat. Surgery exact triangles in instanton theory, 2024. arXiv:2311.04242.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Torsion occurs in other Heegaard Floer flavors/orientation choices, but existence of torsion in the canonical hat groups asked here remains open.\n\n**Verified partial progress.**\n\n- Jabuka--Mark find 2-torsion in HF-plus and HF-infinity for Sigma_g times S1.\n- Noncanonical orientation systems can produce hat torsion in S1 times S2.\n\n**Full solution or refutation.**\n\nThese do not answer the canonical hat/rational-homology-sphere/knot-Floer questions.\n\n**What remains.**\n\nConstruct canonical torsion or prove torsion-freeness for each requested theory.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.54 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Carefully distinguishes orientation choices and open canonical groups.\n\n**Review notes.** Orientation-system distinctions preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2853, "problem_number": "KP-3.55", "title": "Kirby Problem 3.55", "statement": "(a) For $K$ a nontrivial knot in $S^3$, does $HFK^-(K)$ always admit an $\\mathbb{F}_2$-summand, as an $\\mathbb{F}_2[U]$-module?\n\n(b) For $Y$ a rational homology sphere, if $HF_{\\mathrm{red}}(Y)$ is nontrivial, does $HF_{\\mathrm{red}}(Y)$ always admit an $\\mathbb{F}_2$-summand?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.55.\n\nLiterature notes:\n(1) The answer to Part (a) is \"yes\" for fibered knots [BVV18] and more generally for knots with $\\widehat{HFK}(K,g)$ supported in a single $\\mathbb{Z}/2\\mathbb{Z}$-grading [Ni22]. See also [HW18].\n\n(2) Regarding part (b), Lin [Lin24] proved that if a rational homology sphere $Y$ admits a taut foliation, then $HF_{\\mathrm{red}}(Y)$ admits an $\\mathbb{F}_2$-summand. In particular, if the taut foliation part of the L-space conjecture holds, then the answer to the second question is \"yes\"; and this question is implicitly raised in his paper.\n\n(3) The proofs of [BVV18] and [Ni22] use geometric perspectives (i.e., essential surfaces in knot exteriors) to deduce the existence of an $\\mathbb{F}_2$-summand, and [Lin24] uses taut foliations. Thus, it is natural to ask: does the generator of this summand hold special geometric meaning?\n\n(4) If $HF_{\\mathrm{red}}(Y)=\\mathbb{F}_2$, then [HKL19] implies the $\\mathbb{F}_2$-summand is either in grading $d$ or $d-1$, where $d$ denotes the correction term of $Y$.\n\n(5) Lastly, [AB24a, Question 1.10] asks the following refined version of the second question: If $HF_{\\mathrm{red}}(Y)$ contains an $\\mathbb{F}_2[U]/U^k$-summand, does $HF_{\\mathrm{red}}(Y)$ also contain an $\\mathbb{F}_2[U]/U^{\\ell}$-summand for all $1\\leq \\ell\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** F2 summands are known for fibered knots, a broader single-parity top-HFK class, and taut-foliated rational homology spheres; both universal questions remain open.\n\n**Verified partial progress.**\n\n- BVV and Ni cover the stated knot classes.\n- Lin proves the taut-foliation case for HF-red.\n\n**Full solution or refutation.**\n\nNo universal F2-summand theorem was verified.\n\n**What remains.**\n\nProve or refute each universal assertion and identify geometric meaning of the summand.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.55 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists proved classes and general questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2854, "problem_number": "KP-3.56", "title": "Kirby Problem 3.56", "statement": "(a) For $Y^3$ a rational homology sphere, is the Seiberg--Witten Floer spectrum $SWF(Y)$ always a wedge of spheres?\n\n(b) Is every monopole Floer homology L-space a Seiberg--Witten Floer homotopy L-space?\n\n(c) Find an example of two rational homology spheres with the same monopole Floer homology but with different Seiberg--Witten Floer spectra.\n\n(d) Develop an algorithm for computing Seiberg--Witten Floer spectra.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.56.\n\nLiterature notes:\n(1) There are some explicit computations of $SWF(Y)$. In certain cases, the critical points and gradient trajectories of the Chern--Simons--Dirac functional can be identified explicitly by work of Mrowka, Ozsváth, and Yu [MOY97]; in simple situations these suffice to determine the Floer homotopy type. (This holds for elliptic manifolds and a few families of Brieskorn spheres [Man14].) In [DSS23], $SWF(Y)$ is calculated for a general class of plumbed manifolds, including (for example) all Seifert fibered rational homology spheres with base orbifold $S^2$.\n\n(2) Problem 3.56 should be interpreted differently depending on whether $SWF(Y)$ is considered as a nonequivariant spectrum, an $S^1$-equivariant spectrum, or a $\\operatorname{Pin}(2)$-equivariant spectrum. Part (a) may be asked in the nonequivariant setting or the $S^1$-equivariant setting. In the nonequivariant setting, the meaning of Problem 3.56 is clear. In the $S^1$-equivariant setting, the simplest possible form of $SWF(Y)$ is given by the wedge of a single $S^1$-representation sphere $S^{n\\mathbb{C}}$ with a collection of (suspensions of) free $S^1$-spheres $S^{2m+1}_+$; the problem asks if every $SWF(Y)$ is of this form.\n\nCurrently, there are no known examples of rational homology spheres whose Seiberg--Witten Floer spectrum is not a wedge of spheres in the sense of (a). All examples in [Man03, Man14, Man16b] and [DSS23] are (up to suspension) homotopy equivalent to wedges of spheres, although in some cases this is not immediately obvious. For example, as discussed in [Man14, Section 5.2], the Seiberg--Witten Floer spectrum of $-\\Sigma(2,3,11)$ is (up to suspension) a torus. Nonequivariantly, the suspension of a torus is homotopy equivalent to $S^2\\vee S^2\\vee S^3$: suspending the torus gives a CW complex with a single 0-cell, two 2-cells, and one 3-cell; the attaching map for the 3-cell is nullhomotopic. As an $S^1$-space, the suspension of the torus is homotopy equivalent to $S^{\\mathbb{C}}\\vee \\Sigma\\mathbb{R}^2(S^1_+)$. One can check that (a) is closed under orientation reversal and connected sums.\n\nPart (b) may be asked in either the $S^1$- or $\\operatorname{Pin}(2)$-equivariant settings; the nonequivariant case is trivial. We say $Y$ is an $S^1$-homotopy L-space if the $S^1$-Seiberg--Witten--Floer spectrum is a suspension of $S^0$ in each $\\operatorname{spin}^c$-structure. If (in addition) the $\\operatorname{Pin}(2)$-homotopy type of $SWF(Y)$ is a suspension of $S^0$ in each self-conjugate $\\operatorname{spin}^c$-structure, we say that $Y$ is a $\\operatorname{Pin}(2)$-homotopy L-space.\n\nParts (c) and (d) may be interpreted in the nonequivariant, $S^1$-equivariant, or $\\operatorname{Pin}(2)$-equivariant settings.\n\nReferences cited:\n- [MOY97] Tomasz Mrowka, Peter Ozsváth, and Baozhen Yu. Seiberg-Witten monopoles on Seifert fibered spaces. Comm. Anal. Geom., 5(4):685–791, 1997. doi:10.4310/CAG.1997.v5.n4.a3.\n- [Man14] Ciprian Manolescu. On the intersection forms of spin four-manifolds with boundary. Math. Ann., 359(3-4):695–728, 2014. doi:10.1007/s00208-014-1010-1.\n- [DSS23] Irving Dai, Hirofumi Sasahira, and Matthew Stoffregen. Lattice homology and Seiberg-Witten-Floer spectra, 2023. arXiv:2309.01253.\n- [Man03] Ciprian Manolescu. Seiberg-Witten-Floer stable homotopy type of three-manifolds with $b_1=0$. Geom. Topol., 7:889–932, 2003. doi:10.2140/gt.2003.7.889.\n- [Man16b] Ciprian Manolescu. Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture. J. Amer. Math. Soc., 29(1):147–176, 2016. doi:10.1090/jams829.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** SWF is computed for elliptic manifolds, families of Brieskorn spheres and broad plumbed classes, but the proposed general homotopy-type questions remain open and depend on equivariant setting.\n\n**Verified partial progress.**\n\n- Dai--Stoffregen--type results compute general plumbed cases including Seifert rational homology spheres.\n\n**Full solution or refutation.**\n\nNo universal wedge-of-spheres/L-space-spectrum theorem was verified.\n\n**What remains.**\n\nResolve all four questions with S1/Pin2 equivariance specified.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.56 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists computations and distinguishes equivariant interpretations.\n\n**Review notes.** Equivariant ambiguity retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2855, "problem_number": "KP-3.57", "title": "Kirby Problem 3.57", "statement": "Construct an $S^1$ - or $\\operatorname{Pin}(2)$-equivariant lattice homotopy type that computes the Seiberg--Witten Floer homotopy type.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.57.\n\nLiterature notes:\nLattice homology is an invariant for plumbed 3-manifolds introduced by Némethi [Ném05, Ném08], based on earlier work by Ozsváth and Szabó [OS03b]. Zemke has announced that lattice homology is isomorphic to Heegaard Floer homology whenever Y is the boundary of a plumbing tree [Zem25]. Since lattice homology is completely combinatorial, this provides an effective calculation of the Heegaard Floer homology of these manifolds. In addition to computational applications for 3-manifolds (see for example [CK14, DM19, AKS20]), further uses of this relation include a determination of the link Floer complex of an algebraic link from its multivariable Alexander polynomial [BLZ24]. Many constructions in Floer theory have analogously been developed for lattice homology. For instance, one can define knot lattice homology [OSS14a] and construct a surgery exact sequence [Ném11, Gre13d]. Nevertheless, the parallel between lattice homology and Floer theory is incomplete. For instance, cobordism maps for lattice homology have not yet been defined in the literature. In [DSS23], it is indeed shown that in a restricted range of cases, Némethi's construction may be modified to define a lattice spectrum that is provably equal to the Seiberg--Witten Floer spectrum of Manolescu. However, it is unclear whether this is the correct construction to coincide with the Seiberg--Witten Floer spectrum\n\nin all cases. Such a result would allow for the possible computation of interesting Seiberg--Witten Floer spectra (compare Problem 3.56). Additional avenues of investigation include the calculation of Steenrod squares, or even equivariant cohomological operations, for such manifolds. These could lead to more refined numerical invariants (see for example [LS14] in the case of Khovanov homology) that capture more homology cobordism information for graph manifolds than current methods involving Floer homology.\n\nReferences cited:\n- [Ném05] András Némethi. On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds. Geom. Topol., 9:991–1042, 2005. doi:10.2140/gt.2005.9.991.\n- [Ném08] András Némethi. Lattice cohomology of normal surface singularities. Publ. Res. Inst. Math. Sci., 44(2):507–543, 2008. doi:10.2977/prims/1210167336.\n- [OS03b] Peter Ozsváth and Zoltán Szabó. On the Floer homology of plumbed threemanifolds. Geom. Topol., 7:185–224, 2003. doi:10.2140/gt.2003.7.185.\n- [Zem25] Ian Zemke. The equivalence of lattice and Heegaard Floer homology. Duke Math. J., 174(5):857–910, 2025. doi:10.1215/00127094-2024-0044.\n- [CK14] M. B. Can and Ç. Karakurt. Calculating Heegaard-Floer homology by counting lattice points in tetrahedra. Acta Math. Hungar., 144(1):43–75, 2014. doi:10.1007/s10474-014-0432-2.\n- [DM19] Irving Dai and Ciprian Manolescu. Involutive Heegaard Floer homology and plumbed three-manifolds. J. Inst. Math. Jussieu, 18(6):1115–1155, 2019. doi: 10.1017/s1474748017000329.\n- [AKS20] Antonio Alfieri, Sungkyung Kang, and András I. Stipsicz. Connected Floer homology of covering involutions. Math. Ann., 377(3-4):1427–1452, 2020. doi:10.1007/s00208-020-01992-9.\n- [BLZ24] Maciej Borodzik, Beibei Liu, and Ian Zemke. Lattice homology, formality, and plumbed l-space links, 2024. arXiv:2210.15792.\n- [OSS14a] Peter Ozsváth, András I. Stipsicz, and Zoltán Szabó. Knots in lattice homology. Comment. Math. Helv., 89(4):783–818, 2014. doi:10.4171/CMH/334.\n- [Ném11] András Némethi. Two exact sequences for lattice cohomology. In Noncommutative geometry and global analysis, volume 546 of Contemp. Math., pages 249–269. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/conm/546/10793.\n- [Gre13d] Joshua Evan Greene. A surgery triangle for lattice cohomology. Algebr. Geom. Topol., 13(1):441–451, 2013. doi:10.2140/agt.2013.13.441.\n- [DSS23] Irving Dai, Hirofumi Sasahira, and Matthew Stoffregen. Lattice homology and Seiberg-Witten-Floer spectra, 2023. arXiv:2309.01253.\n- [LS14] Robert Lipshitz and Sucharit Sarkar. A refinement of Rasmussen’s S-invariant. Duke Math. J., 163(5):923–952, 2014. doi:10.1215/00127094-2644466.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Lattice homology is a combinatorial model for plumbed manifolds and an announced theorem identifies it with Heegaard Floer homology, but no S1/Pin2-equivariant lattice homotopy type computing SWF was verified.\n\n**Verified partial progress.**\n\n- Zemke announces lattice/Heegaard Floer agreement for plumbing trees.\n\n**Full solution or refutation.**\n\nThe requested Floer homotopy refinement remains open.\n\n**What remains.**\n\nConstruct equivariant lattice spectrum and prove comparison to SWF.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.57 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records lattice homology and the outstanding equivariant refinement.\n\n**Review notes.** Announcement not treated as full solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2856, "problem_number": "KP-3.58", "title": "Kirby Problem 3.58", "statement": "Prove that there is an isomorphism relating Heegaard Floer homology and monopole Floer homology that commutes with the cobordism maps in the two settings. In particular, prove that the mixed invariant in Heegaard Floer homology is equal to the Seiberg--Witten invariant for closed 4-manifolds.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.58.\n\nLiterature notes:\nAs discussed in the section introduction, the Heegaard Floer homology of a closed 3-manifold is isomorphic to its monopole Floer homology, but it is not known whether these isomorphisms respect the TQFT-like structures of the theories.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Closed 3-manifold Heegaard and monopole Floer homologies are isomorphic, but naturality with respect to cobordism maps and equality of mixed/SW invariants remain open.\n\n**Verified partial progress.**\n\n- The underlying closed 3-manifold homology groups are known to be isomorphic.\n\n**Full solution or refutation.**\n\nThe requested TQFT-compatible comparison is unproved.\n\n**What remains.**\n\nConstruct an isomorphism commuting with all cobordism maps.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.58 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explicitly distinguishes group isomorphism from naturality.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2857, "problem_number": "KP-3.59", "title": "Kirby Problem 3.59", "statement": "Prove an isomorphism relating instanton Floer homology and Heegaard Floer homology.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.59.\n\nLiterature notes:\n(1) There is some context for this question in the section introduction.\n\n(2) Kronheimer--Mrowka conjectured that sutured instanton homology is isomorphic to sutured Heegaard Floer homology with coefficients in C,\n\n\\begin{equation}\\label{eq:shi-sfh}\\operatorname{SHI}(M,\\gamma;\\mathbb{C})\\cong \\operatorname{SFH}(M,\\gamma;\\mathbb{C}).\\tag{4}\\end{equation}\n\nfor every balanced sutured manifold $(M,\\gamma)$ [KM10]. If true, this would imply that the non-equivariant versions of the closed 3-manifold invariants and knot invariants in the two theories are also isomorphic over C, as these are special cases of the sutured invariants. In particular, it would show that the framed instanton homology of a closed 3-manifold Y is isomorphic to the hat version of its Heegaard Floer homology with coefficients in C,\n\n\\begin{equation}\\label{eq:instanton-hf}\\ I^{\\#}(Y;\\mathbb{C})\\cong \\widehat{HF}(Y;\\mathbb{C}).\\tag{5}\\end{equation}\n\nOne can interpret the problem above as asking for a proof of (4).\n\n(3) As evidence for the above conjecture, Baldwin--Li--Ye proved that the dimension of sutured instanton homology is bounded by the number of generators in the sutured Heegaard Floer chain complex; that is,\n\n$$\n\\dim_{\\mathbb{C}}\\operatorname{SHI}(M,\\gamma)\\leq \\dim_{\\mathbb{C}}\\operatorname{SFC}(H),\n$$\n\nwhere H is any admissible Heegaard diagram for $(M,\\gamma)$ [BLY23]. Moreover, the conjectured isomorphism (5) is known to hold for several classes of manifolds, including all Seifert fibered rational homology spheres [ABDS22].\n\n(4) More directly related to Witten's conjecture, one could also ask for a relationship between the equivariant instanton theory introduced by Austin-- Braam [AB95] and studied extensively by Miller [ME23] and Daemi-- Miller Eismeier [DME22], and the equivariant (plus or minus) versions of Heegaard Floer homology.\n\nReferences cited:\n- [KM10] Peter Kronheimer and Tomasz Mrowka. Knots, sutures, and excision. J. Differential Geom., 84(2):301–364, 2010. http://projecteuclid.org/euclid.jdg/1274707316.\n- [BLY23] John A. Baldwin, Zhenkun Li, and Fan Ye. Sutured instanton homology and Heegaard diagrams. Compos. Math., 159(9):1898–1915, 2023. doi:10.1112/s0010437x23007303.\n- [ABDS22] Antonio Alfieri, John A. Baldwin, Irving Dai, and Steven Sivek. Instanton Floer homology of almost-rational plumbings. Geom. Topol., 26(5):2237–2294, 2022. doi: 10.2140/gt.2022.26.2237.\n- [AB95] D. M. Austin and P. J. Braam. Morse-Bott theory and equivariant cohomology. In The Floer memorial volume, volume 133 of Progr. Math., pages 123–183. Birkhäuser, Basel, 1995.\n- [ME23] Mike Miller Eismeier. Equivariant instanton homology, 2023. To appear in Memoirs of the AMS. arXiv:1907.01091.\n- [DME22] Aliakbar Daemi and Mike Miller Eismeier. Instantons and rational homology spheres, 2022. arXiv:2210.14071.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The conjectural sutured instanton--sutured Heegaard Floer isomorphism, which would imply the stated closed/knot comparisons over C, remains open.\n\n**Verified partial progress.**\n\n- Kronheimer--Mrowka formulate the sutured comparison conjecture.\n\n**Full solution or refutation.**\n\nNo general instanton/Heegaard Floer isomorphism was verified.\n\n**What remains.**\n\nProve the sutured isomorphism and compatibility structures.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.59 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the conjectural sutured comparison and consequences.\n\n**Review notes.** Open is source-backed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2858, "problem_number": "KP-3.60", "title": "Kirby Problem 3.60", "statement": "Find an algorithm to compute instanton Floer homology of closed 3-manifolds and the Donaldson invariants of closed 4-manifolds.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.60.\n\nLiterature notes:\nThere are algorithms to compute the Heegaard Floer invariants of knots [MOS09], 3-manifolds [SW10, MOT25], cobordisms [LMW08, MOT25], and the Heegaard Floer mixed invariant of closed 4-manifolds [MOT25]. Per Problem 3.58 and Witten's Conjecture (see the section introduction), this should be closely related to the Donaldson invariant for manifolds of simple type.\n\nReferences cited:\n- [MOS09] Ciprian Manolescu, Peter Ozsváth, and Sucharit Sarkar. A combinatorial description of knot Floer homology. Ann. of Math. (2), 169(2):633–660, 2009. doi: 10.4007/annals.2009.169.633.\n- [SW10] Sucharit Sarkar and Jiajun Wang. An algorithm for computing some Heegaard Floer homologies. Ann. of Math. (2), 171(2):1213–1236, 2010. doi:10.4007/annals.2010.171.12s13.\n- [MOT25] Ciprian Manolescu, Peter Ozsváth, and Dylan Thurston. Grid diagrams and Heegaard Floer invariants. Ann. of Math. (2), 201(1):1–78, 2025. doi:10.4007/annals.2025.201.1.1.\n- [LMW08] Robert Lipshitz, Ciprian Manolescu, and Jiajun Wang. Combinatorial cobordism maps in hat Heegaard Floer theory. Duke Math. J., 145(2):207–247, 2008. doi: 10.1215/00127094-2008-050.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Algorithms exist for multiple Heegaard Floer invariants and mixed 4-manifold invariants, but no algorithm for the requested instanton Floer/Donaldson invariants was verified.\n\n**Verified partial progress.**\n\n- MOS, Sarkar--Wang and later work give computable Heegaard Floer invariants.\n\n**Full solution or refutation.**\n\nComparison conjectures motivate but do not produce the desired instanton/Donaldson algorithms.\n\n**What remains.**\n\nDevelop direct or comparison-based algorithms for instanton Floer and Donaldson theory.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.60 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists the Heegaard Floer algorithms and outstanding target.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2859, "problem_number": "KP-3.61", "title": "Kirby Problem 3.61", "statement": "Is the dimension of Heegaard Floer homology invariant under genus 2 mutation?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.61.\n\nLiterature notes:\n(1) Let $\\Sigma$ be a closed, genus 2 surface embedded in a 3-manifold $M$. The genus 2 mutant $M_\\Sigma$ is the 3-manifold obtained by cutting $M$ along $\\Sigma$ and regluing by the hyperelliptic involution of $\\Sigma$. This operation was introduced in [Rub87] as an analogue for 3-manifolds of Conway mutation for knots. If $K\\subset M$ is a knot and $\\Sigma$ is disjoint from $K$, then its image $K_\\Sigma\\subset M_\\Sigma$ is called the genus 2 mutant of $K$, and any Conway mutation can be achieved by a sequence of at most two genus 2 mutations.\n\n(2) Several invariants, like hyperbolic volume [Rub87] and the Alexander polynomial [CL99], are preserved by genus 2 mutation; see [DGST10] for a survey.\n\n(3) A broad interpretation of the problem is whether\n\n$$\n\\dim\\widehat{SFH}(M,\\gamma)=\\dim\\widehat{SFH}(M_{\\Sigma},\\gamma)\n$$\n\nover $\\mathbb{Z}/2\\mathbb{Z}$, for any balanced sutured manifold $(M,\\gamma)$ and genus 2 surface $\\Sigma\\subset M$. If true, this would imply that the dimensions of the Heegaard Floer invariants $\\widehat{HF}$ and $\\widehat{HFK}$ for closed 3-manifolds and knots are preserved by genus 2 mutation, as these are special cases of sutured Floer homology. It would also imply that the concordance invariant $\\tau$ is preserved by genus 2 mutation, and hence Conway mutation, as in [BS21, Proposition 1.30]. The latter is related to Problem 1.53 of [Kir97], which asked whether Conway mutation preserves the concordance class of a knot, and to which the answer is \"no\" [Kea89, KL99].\n\n(4) The $\\operatorname{spin}^c$-graded $\\widehat{HF}$ is not preserved by genus 2 mutation [Cla17]. Neither is the bigraded, or even the $\\delta$-graded, $\\widehat{HFK}$ [MS15], though $\\delta$-graded knot Floer homology is preserved by Conway mutation [Zib23].\n\n(5) The dimension of $\\widehat{HF}$ was shown to be invariant under genus-1 mutation in [HRW22].\n\n(6) Floer's original instanton homology with $\\mathbb{Z}/2\\mathbb{Z}$ coefficients is invariant under genus 2 mutation [Rub99a] (the paper claims the result over $\\mathbb{Z}$ but there is a gap in the proof).\n\nReferences cited:\n- [Rub87] Daniel Ruberman. Mutation and volumes of knots in $S^{3}$. Invent. Math., 90(1):189– 215, 1987. doi:10.1007/BF01389038.\n- [CL99] D. Cooper and W. B. R. Lickorish. Mutations of links in genus 2 handlebodies. Proc. Amer. Math. Soc., 127(1):309–314, 1999. doi:10.1090/S0002-9939-99-04871-6.\n- [DGST10] Nathan M. Dunfield, Stavros Garoufalidis, Alexander Shumakovitch, and Morwen Thistlethwaite. Behavior of knot invariants under genus 2 mutation. New York J. Math., 16:99–123, 2010. http://nyjm.albany.edu:8000/j/2010/16 99.html.\n- [BS21] John A. Baldwin and Steven Sivek. Framed instanton homology and concordance. J. Topol., 14(4):1113–1175, 2021. doi:10.1112/topo.12207.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Kea89] C. Kearton. Mutation of knots. Proc. Amer. Math. Soc., 105:206–208, 1989.\n- [KL99] Paul Kirk and Charles Livingston. Twisted knot polynomials: inversion, mutation and concordance. Topology, 38(3):663–671, 1999. doi:10.1016/S0040-9383(98) 00040-8.\n- [Cla17] Corrin Clarkson. Three-manifold mutations detected by Heegaard Floer homology. Algebr. Geom. Topol., 17(1):1–16, 2017. doi:10.2140/agt.2017.17.1.\n- [MS15] Allison H. Moore and Laura Starkston. Genus-two mutant knots with the same dimension in knot Floer and Khovanov homologies. Algebr. Geom. Topol., 15(1):43– 63, 2015. doi:10.2140/agt.2015.15.43.\n- [Zib23] Claudius Zibrowius. On symmetries of peculiar modules, or δ-graded link Floer homology is mutation invariant. J. Eur. Math. Soc. (JEMS), 25(8):2949–3006, 2023. doi:10.4171/jems/1201.\n- [HRW22] Jonathan Hanselman, Jacob Rasmussen, and Liam Watson. Heegaard Floer homology for manifolds with torus boundary: properties and examples. Proc. Lond. Math. Soc. (3), 125(4):879–967, 2022. doi:10.1112/plms.12473.\n- [Rub99a] Daniel Ruberman. Mutation and gauge theory. I. Yang-Mills invariants. Comment. Math. Helv., 74(4):615–641, 1999. doi:10.1007/s000140050108.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Several classical invariants are mutation invariant, but no general genus-two mutation invariance theorem for Heegaard Floer rank was verified.\n\n**Verified partial progress.**\n\n- Hyperbolic volume and Alexander polynomial are preserved by genus-two mutation.\n\n**Full solution or refutation.**\n\nThe Floer-rank question remains open in the generality stated.\n\n**What remains.**\n\nProve rank invariance or find a genus-two mutant counterexample.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.61 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Contrasts preserved invariants with open Floer question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2860, "problem_number": "KP-3.62", "title": "Kirby Problem 3.62", "statement": "How do Floer homological invariants behave under maps of nonzero degree? For instance, let $Y$ and $Z$ be closed, oriented 3-manifolds, and suppose that there is a degree-1 map $f:Y\\to Z$. Is it true that\n\\begin{equation}\\label{eq:hf-rank}\n\\operatorname{rk} HF(Y)\\geq \\operatorname{rk} HF(Z),\n\\tag{6}\n\\end{equation}\nwhere $HF$ is the $\\widehat{HF}$ or $HF_{\\mathrm{red}}$ version of Heegaard Floer homology?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.62.\n\nLiterature notes:\n(1) This question is inspired by the fact that many 3-manifold invariants, from the first Betti number to simplicial volume, exhibit monotonic behaviors under maps of nonzero degree; see [Wan02]. Perhaps more to the point, the order of first homology is monotonic under degree-1 maps between rational homology 3-spheres, and $\\widehat{HF}$ categorifies the order of $H_1$ for such manifolds.\n\n(2) If $M$ and $N$ are 3-manifolds and $d$ is a nonzero integer for which there exists a degree-$d$ map from $M$ to $N$, then $M$ is said to $d$-dominate $N$.\n\n(3) The inequality (6) is known to hold in the setting of $\\widehat{HF}$ for Seifert fibered homology spheres [KL15], and when $f$ is pinching a homology solid torus in $Y$ to a solid torus [HRW24].\n\n(4) If (6) holds for $\\widehat{HF}$ or $HF_{\\mathrm{red}}$, then one could prove that homotopy 3-spheres are L-spaces without using the Poincaré Conjecture, as every homotopy 3-sphere is 1-dominated by $S^3$.\n\n(5) If (6) holds for $HF_{\\mathrm{red}}$, then it would follow that if $Y$ is an L-space that 1-dominates $Z$, then $Z$ is also an L-space. This would also follow directly from the L-space Conjecture under the added assumption that $Y$ and $Z$ are prime, as discussed in Problem 3.48.\n\n(6) Gadgil [Gad07] proved that $Y$ 1-dominates $Z$ if and only if $Y$ embeds into\n\n$$\n(Z\\times[0,1])\\#^n(\\mathbb{CP}^2\\#\\overline{\\mathbb{CP}}^{2})\n$$\n\nas a hypersurface homologous to $Z\\times\\{0\\}$ for some $n$. In the case that $Y$ embeds in the same manner into $(Z\\times[0,1])\\#^n\\mathbb{CP}^{2}$, the inequality (6) follows from standard properties of Floer homology.\n\n(7) Gadgil [Gad07] also proved that $Y$ 1-dominates $Z$ if and only if $Y$ can be obtained by surgery on a link $L\\subset Z$ such that every component of $L$ is nullhomotopic. A special case is when $Y$ is obtained via $p/q$-surgery on a nullhomotopic knot $K\\subset Z$. In this case, one expects the stronger inequality\n\\begin{equation}\n\\operatorname{rk}\\widehat{HF}(Y)\\geq p\\cdot \\operatorname{rk}\\widehat{HF}(Z),\n\\tag{7}\n\\end{equation}\nas conjectured by Ni.\n\n(8) There are some known results for maps of degree greater than one. For example, if $f:Y\\to Z$ is a $p^n$-sheeted regular covering map between rational homology 3-spheres, for $p$ a prime, there is a Smith-type inequality relating the Heegaard and/or monopole Floer homologies of $Y$ and $Z$ with $\\mathbb{Z}/p\\mathbb{Z}$ coefficients [LT16, LM18b, Lar19]; see [HLL22a] for similar results on branched coverings.\n\n(9) It might also be interesting to study analogous questions for knot Floer homology, sutured Floer homology, or Khovanov homology. Suppose, for example, that $K$ and $L$ are knots in $S^3$ such that there is a proper degree-1 map from the exterior of $K$ to that of $L$. It is known in this case that the Alexander polynomial of $K$ divides that of $L$, and the genus behaves monotonically, $g(K)\\geq g(L)$. (One can find references for these and related facts in [BBRW16].) Ni has therefore asked whether knot Floer homology also behaves monotonically; i.e., does the rank inequality\n\n$$\n\\operatorname{rk}\\widehat{HFK}(S^3,K)\\geq \\operatorname{rk}\\widehat{HFK}(S^3,L)\n$$\n\nhold? Is there a rank inequality in each Alexander grading?\n\nReferences cited:\n- [Wan02] Shicheng Wang. Non-zero degree maps between 3-manifolds. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 457–468. Higher Ed. Press, Beijing, 2002.\n- [KL15] Çağrı Karakurt and Tye Lidman. Rank inequalities for the Heegaard Floer homology of Seifert homology spheres. Trans. Amer. Math. Soc., 367(10):7291–7322, 2015. doi:10.1090/S0002-9947-2014-06451-9.\n- [HRW24] Jonathan Hanselman, Jacob Rasmussen, and Liam Watson. Bordered Floer homology for manifolds with torus boundary via immersed curves. J. Amer. Math. Soc., 37(2):391–498, 2024. doi:10.1090/jams/1029.\n- [Gad07] Siddhartha Gadgil. Degree-one maps, surgery and four-manifolds. Bull. Lond. Math. Soc., 39(3):419–424, 2007. doi:10.1112/blms/bdm019.\n- [LT16] Robert Lipshitz and David Treumann. Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers. J. Eur. Math. Soc. (JEMS), 18(2):281–325, 2016. doi:10.4171/JEMS/590.\n- [LM18b] Tye Lidman and Ciprian Manolescu. Floer homology and covering spaces. Geom. Topol., 22(5):2817–2838, 2018. doi:10.2140/gt.2018.22.2817.\n- [Lar19] Tim Large. Equivariant Floer theory and double covers of three-manifolds, 2019. arXiv:1910.12119.\n- [HLL22a] Kristen Hendricks, Tye Lidman, and Robert Lipshitz. Rank inequalities for the Heegaard Floer homology of branched covers. Doc. Math., 27:581–612, 2022.\n- [BBRW16] M. Boileau, S. Boyer, D. Rolfsen, and S. C. Wang. One-domination of knots. Illinois J. Math., 60(1):117–139, 2016. URL: http://projecteuclid.org/euclid.ijm/1498032026.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Heegaard Floer rank monotonicity under degree-one maps is known for Seifert fibered homology spheres and pinching a homology solid torus, but open generally.\n\n**Verified partial progress.**\n\n- Karakhanyan--Lidman prove Seifert homology-sphere case.\n- Hanselman--Rasmussen--Watson prove the pinching case.\n\n**Full solution or refutation.**\n\nNo arbitrary degree-map Floer monotonicity theorem was verified.\n\n**What remains.**\n\nProve or disprove rank inequalities for all nonzero-degree maps and Floer variants.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.62 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records exact known cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2861, "problem_number": "KP-3.63", "title": "Kirby Problem 3.63", "statement": "Give a method for computing the $\\eta$ invariant for the Dirac operator, $\\eta_{\\mathrm{Dirac}}(Y, s)$, associated to a spin structure s on a hyperbolic 3-manifold Y.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.63.\n\nLiterature notes:\n(1) The Atiyah--Patodi--Singer $\\eta$-invariant [APS75a, Goe12] is a real number defined in terms of the spectrum of a first-order self-adjoint elliptic differential operator A on a closed odd-dimensional oriented Riemannian\n\nmanifold Y ; denote it by $\\eta_A(Y)$. When the operator $D=\\partial/\\partial t + A$ on $[0,1)\\times Y^3$ extends to an elliptic operator on a manifold X with boundary Y, then $\\eta_A(Y)$ serves as a boundary correction term for the Atiyah-Singer formula determining the index of D (with appropriate boundary conditions.) The main geometric examples are of `Dirac type', where D is the signature operator, Dolbeault operator, or the Dirac operator associated to a spin or spinc structure on X. In the last case the operator A is essentially a Dirac operator on Y. The $\\eta$-invariant is not a topological invariant: for the signature and Dirac operators, it varies with the metric. For hyperbolic manifolds, Mostow rigidity gives a canonical metric and so the $\\eta$-invariant is a topological invariant. The special case when Y is hyperbolic and D is the signature operator has been extensively studied, and in fact can be computed using formulas of [MN92, Ouy97, CGHN00]. This computation been implemented numerically in the program Snap [GN]. However, it seems that there has been little research into the case of the Dirac operator $\\eta_{\\mathrm{Dirac}}(Y, s)$ on a spin hyperbolic 3-manifold $(Y,s)$.\n\n(2) The only known examples of spin hyperbolic manifolds for which $\\eta_{\\mathrm{Dirac}}(Y, s)$ is known are those that admit an orientation-reversing (and spin-structure preserving, in an appropriate sense) isometry. For these, it follows that $\\eta_{\\mathrm{Dirac}}(Y, s)$ = 0. One might start by trying to prove cut/paste formulas analogous to [MR90a, MR90b] or to understand the reduction of $\\eta_{\\mathrm{Dirac}}(Y, s)$ modulo integers (in a suitable normalization). This is related to a version of the Chern--Simons invariant; see [APS76].\n\n(3) One can pose the same problem for manifolds with other geometries. Nicolaescu used adiabatic limit techniques to compute $\\eta_{\\mathrm{Dirac}}(Y, s)$ when Y is a circle bundle [Nic99] and more generally when Y is a Seifert-fibered 3-manifold with Seifert geometry [Nic00].\n\n(4) A related, and perhaps easier, problem is to calculate the reduced $\\eta$-invariant $\\xi_\\alpha(Y)\\in\\mathbb{R}$, defined in [APS75b], for a spin 3-manifold together with a unitary representation $\\alpha$. The metric dependence here is less impactful, as there can be at most integer jumps along a path of metrics.\n\nReferences cited:\n- [APS75a] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambridge Philos. Soc., 77:43–69, 1975. doi:10.1017/S0305004100049410.\n- [Goe12] Sebastian Goette. Computations and applications of η invariants. In Global differential geometry, volume 17 of Springer Proc. Math., pages 401–433. Springer, Heidelberg, 2012. URL: https://doi.org/10.1007/978-3-642-22842-1 13, doi:10.1007/978-3-642-22842-1\\\\_13.\n- [MN92] Robert Meyerhoff and Walter D. Neumann. An asymptotic formula for the eta invariants of hyperbolic 3-manifolds. Comment. Math. Helv., 67(1):28–46, 1992. doi:10.1007/BF02566487.\n- [Ouy97] Mingqing Ouyang. A simplicial formula for the η-invariant of hyperbolic 3-manifolds. Topology, 36(2):411–421, 1997. doi:10.1016/0040-9383(96)00013-4.\n- [CGHN00] David Coulson, Oliver A. Goodman, Craig D. Hodgson, and Walter D. Neumann. Computing arithmetic invariants of 3-manifolds. Experiment. Math., 9(1):127–152, 2000. http://projecteuclid.org/euclid.em/1046889596.\n- [GN] Oliver Goodman and Walter Neumann. Snap for hyperbolic 3-manifolds. Available at https://sourceforge.net/projects/snap-pari/.\n- [MR90a] R. Meyerhoff and D. Ruberman. Mutation and the η-invariant. J. Diff. Geo., 31:101–130, 1990. http://projecteuclid.org/euclid.jdg/1214444091.\n- [MR90b] Robert Meyerhoff and Daniel Ruberman. Cutting and pasting and the η-invariant. Duke Math. J., 61(3):747–761, 1990. doi:10.1215/S0012-7094-90-06127-7.\n- [APS76] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geometry. III. Math. Proc. Cambridge Philos. Soc., 79(1):71–99, 1976. doi:10.1017/S0305004100052105.\n- [Nic99] Liviu I. Nicolaescu. Eta invariants of Dirac operators on circle bundles over Riemann surfaces and virtual dimensions of finite energy Seiberg-Witten moduli spaces. Israel J. Math., 114:61–123, 1999. doi:10.1007/BF02785572.\n- [Nic00] Liviu I. Nicolaescu. Finite energy Seiberg-Witten moduli spaces on 4-manifolds bounding Seifert fibrations. Comm. Anal. Geom., 8(5):1027–1096, 2000. doi:10.4310/CAG.2000.v8.n5.a3.\n- [APS75b] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geometry. II. Math. Proc. Cambridge Philos. Soc., 78(3):405–432, 1975. doi:10.1017/S0305004100051872.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No general effective method for the Dirac eta invariant of spin hyperbolic 3-manifolds was verified.\n\n**Verified partial progress.**\n\n- APS theory supplies index-theoretic definitions and boundary correction formulas.\n\n**Full solution or refutation.**\n\nThe requested hyperbolic computation method remains open.\n\n**What remains.**\n\nRelate spectral eta values to effective hyperbolic/spin data.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.63 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the eta-invariant setup and unresolved computation problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2862, "problem_number": "KP-3.64", "title": "Kirby Problem 3.64", "statement": "Let A be a flat connection on the trivial $\\operatorname{SU}(2)$ bundle on a closed three-manifold M. The Chern--Simons invariant $\\operatorname{CS}(M, A)$ is defined to be\n\n$$\n\\operatorname{CS}(M,A)=\\frac{1}{4\\pi^2}\\int_M \\operatorname{Tr}\\!\\left(A\\wedge dA+\\frac{2}{3}A\\wedge A\\wedge A\\right).\n$$\n Is $\\operatorname{CS}(M, A)$ always rational?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.64.\n\nLiterature notes:\n(1) This was asked in [Auc94], and implicitly in [KK90]. It is related to Problems 3.62, 3.63, and 3.104 in [Kir97].\n\n(2) Rationality holds for all Seifert fibered spaces and graph manifolds [FS90, Fur90, Auc94].\n\n(3) One potential counterexample comes from +1\\{2-surgery on the mirror of 52. As described in [NST24], numerical calculations based on [KK90] suggest that a certain flat connection on this manifold has irrational Chern--Simons invariant. In addition to answering this problem, a proof of irrationality for this flat connection would imply that the integer homology cobordism group is not spanned by all graph manifold homology spheres. In general, a homology sphere with an irrational Chern--Simons invariant cannot admit a homology cobordism without 3-handles to a graph manifold.\n\n(4) On the other hand, Auckly has proposed some strategies to answer this question in the affirmative (see [Auc94] and [Kir97, Problem 3.104]). For example, he notes that if two flat connections are bordant, then the Chern--Simons invariants agree, and asks if one can show that any flat connection is bordant to one on a connected sum of lens spaces. This would then show all Chern--Simons invariants are rational.\n\n(5) It is natural to ask about rationality of Chern--Simons invariants for flat $G$-connections on all 3-manifolds, where G = $\\operatorname{SU}(n)$ or indeed any compact Lie group. A starting point might be to show rationality for all G when M is a graph manifold. See [Nis98] for some calculations of $\\operatorname{SU}(n)$ Chern-- Simons invariants for Seifert fibered 3-manifolds.\n\n(6) A hyperbolic 3-manifold M has two a priori different Chern--Simons invariants: $\\operatorname{CS}(M)$ associated to its Levi-Civita connection, and the complex Chern--Simons invariant $\\operatorname{CS}(M, A\\rho)$ associated to the flat $\\operatorname{PSL}(2, \\mathbb{C})$ connection coming from the discrete faithful representation $\\rho$ : $\\pi_1(M)$ $\\to$ $\\operatorname{PSL}(2, \\mathbb{C})$. These are related to the hyperbolic volume by Yoshida's formula [Yos85, KK93]\n\n$$\n\\operatorname{CS}(\\rho)=\\frac{1}{2}\\operatorname{CS}(M)-\\frac{i}{2\\pi}\\operatorname{vol}(M).\n$$\n Hence, for hyperbolic manifolds the rationality question here is closely tied to the rationality question for volumes discussed in Problem 3.2.\n\nReferences cited:\n- [Auc94] David R. Auckly. Topological methods to compute Chern-Simons invariants. Math. Proc. Cambridge Philos. Soc., 115(2):229–251, 1994. doi:10.1017/S0305004100072066.\n- [KK90] P. Kirk and E. Klassen. Chern-Simons invariants of 3-manifolds and representation spaces of knot groups. Math. Ann., 287:343–367, 1990.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [FS90] Ronald Fintushel and Ronald J. Stern. Instanton homology of Seifert fibred homology three spheres. Proc. London Math. Soc. (3), 61(1):109–137, 1990. doi: 10.1112/plms/s3-61.1.109.\n- [Fur90] Mikio Furuta. Homology cobordism group of homology 3-spheres. Invent. Math., 100(2):339–355, 1990. doi:10.1007/BF01231190.\n- [NST24] Yuta Nozaki, Kouki Sato, and Masaki Taniguchi. Filtered instanton Floer homology and the homology cobordism group. J. Eur. Math. Soc. (JEMS), 26(12):4699–4761, 2024. doi:10.4171/jems/1371.\n- [Nis98] Haruko Nishi. $\\mathrm{SU}(n)$-Chern-Simons invariants of Seifert fibered 3-manifolds. Internat. J. Math., 9(3):295–330, 1998. doi:10.1142/S0129167X98000130.\n- [Yos85] Tomoyoshi Yoshida. The η-invariant of hyperbolic 3-manifolds. Invent. Math., 81(3):473–514, 1985. doi:10.1007/BF01388583.\n- [KK93] Paul Kirk and Eric Klassen. Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space of T2. Comm. Math. Phys., 153(3):521–557, 1993. http://projecteuclid.org/euclid.cmp/1104252787.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Chern--Simons rationality holds for Seifert fibered spaces and graph manifolds, while a numerical candidate suggests irrationality for a hyperbolic surgery but is not a proof.\n\n**Verified partial progress.**\n\n- Fintushel--Stern, Furuta and Auckly establish rationality in the stated classes.\n- NST numerical calculations suggest a possible irrational counterexample.\n\n**Full solution or refutation.**\n\nNo all-manifold rationality theorem or rigorous irrational example was verified.\n\n**What remains.**\n\nProve rationality generally or rigorously establish an irrational flat connection.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 3.64 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes known classes from numerical counterexample evidence.\n\n**Review notes.** Numerical evidence not classified as a counterexample.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2863, "problem_number": "KP-3.65", "title": "Kirby Problem 3.65", "statement": "Let $S_{2,\\infty}(Y)$ denote the Kauffman bracket skein module of a closed, oriented 3-manifold $Y$; this is a module over $R=\\mathbb{Z}[A,A^{-1}]$. For $Y$ prime and connected, does $S_{2,\\infty}(Y)$ having rank one, in the sense that\n\n$$\nS_{2,\\infty}(Y)\\otimes_R \\mathbb{Q}(A)\\cong \\mathbb{Q}(A),\n$$\n\nimply that $Y$ is $S^3$ or $S^1\\times S^2$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.65.\n\nLiterature notes:\n(a) This problem appears as [DKS25, Question 10.3].\n\n(b) Przytycki [Prz00] showed that $S_{2,\\infty}(Y)\\otimes_R \\mathbb{Q}(A)$ is multiplicative under connected sum. Hence, the set of (closed, connected) 3-manifolds that satisfy $S_{2,\\infty}(Y)\\otimes_R \\mathbb{Q}(A)\\cong \\mathbb{Q}(A)$ is closed under connected sum. Thus, the problem can be alternatively formulated as asking whether this set consists only of $S^3$ and $\\#^k(S^1\\times S^2)$ for $k\\geq 1$.\n\n(c) The skein module is expected to be isomorphic to the degree 0 part of the framed $SL(2,\\mathbb{C})$ Floer homology $HP^{\\bullet}_{\\#}$ from [AM20]; see Problem 3.66. Zentner [Zen18] proved that every nontrivial integral homology sphere has an irreducible representation to $SL(2,\\mathbb{C})$; however, this does not immediately imply that $HP^{\\bullet}_{\\#}$ has rank greater than one.\n\n(d) The answer to the problem is affirmative if we add the condition that $S_{2,\\infty}(Y)\\otimes_R\\mathbb{Q}[A,A^{-1}]$ is finitely generated over $\\mathbb{Q}[A,A^{-1}]$. Indeed, in that case, the character variety $\\mathcal{X}(Y)$ is finite, because the skein module at $A=-1$ gives the ring of functions on $\\mathcal{X}(Y)$; see [Bul97], [PS00b]. A result of Detcherry, Kalfagianni, and Sikora [DKS25] implies that the rank of the skein module is greater or equal than the number of $SL(2,\\mathbb{C})$ representations. Thus, there must be a unique representation to $SL(2,\\mathbb{C})$, so $Y$ is an integral homology sphere, and Zentner's theorem mentioned above implies that $Y=S^3$.\n\n(e) The answer is also affirmative under the weaker condition that $S_{2,\\infty}(Y)\\otimes_R\\mathbb{Q}[A,A^{-1}]$ is tame, i.e., it is a direct sum of cyclic $\\mathbb{Q}[A^{\\pm1}]$-modules and does not contain any $\\mathbb{Q}[A,A^{-1}]/(\\phi_{2N})$ as a submodule, for some $N$ odd, where $\\phi_{2N}$ is the $(2N)$-th cyclotomic polynomial. For then, the results of [DKS25] and [GJS23] imply that $\\mathcal{X}(Y)$ is finite, and we conclude that $Y=S^3$, just as in (d).\n\n(f) The manifold $Y=S^1\\times S^2$ has $b_1(Y)>0$ and $S_{2,\\infty}(Y)$ of rank one; see [HP95b]. The skein module $S_{2,\\infty}(Y)\\otimes_R\\mathbb{Q}[A,A^{-1}]$ is not tame: by [HP95b], it contains $\\mathbb{Q}[A^{\\pm1}]/(\\phi_{2N})$ as a submodule, for all odd $N$.\n\n(g) The skein module of a rational homology sphere with coefficients in $\\mathbb{Q}(A)$ is at least 1-dimensional, by [DKS25].\n\nReferences cited:\n- [DKS25] Renaud Detcherry, Efstratia Kalfagianni, and Adam S. Sikora. Kauffman bracket skein modules of small 3-manifolds. Adv. Math., 467:Paper No. 110169, 45, 2025. doi:10.1016/j.aim.2025.110169.\n- [Prz00] Józef H. Przytycki. Kauffman bracket skein module of a connected sum of 3-manifolds. Manuscripta Math., 101(2):199–207, 2000. doi:10.1007/s002290050014.\n- [AM20] Mohammed Abouzaid and Ciprian Manolescu. A sheaf-theoretic model for $\\mathrm{SL}(2,\\mathbb{C})$ Floer homology. J. Eur. Math. Soc. (JEMS), 22(11):3641–3695, 2020. doi:10.4171/jems/994.\n- [Zen18] Raphael Zentner. Integer homology 3-spheres admit irreducible representations in $\\mathrm{SL}(2,\\mathbb{C})$. Duke Math. J., 167(9):1643–1712, 2018. doi:10.1215/00127094-2018-0004.\n- [Bul97] Doug Bullock. Rings of SL2pCq-characters and the Kauffman bracket skein module. Comment. Math. Helv., 72(4):521–542, 1997. doi:10.1007/s000140050032.\n- [PS00b] Józef H. Przytycki and Adam S. Sikora. On skein algebras and Sl2pCq-character varieties. Topology, 39(1):115–148, 2000. doi:10.1016/S0040-9383(98)00062-7.\n- [GJS23] Sam Gunningham, David Jordan, and Pavel Safronov. The finiteness conjecture for skein modules. Invent. Math., 232(1):301–363, 2023. doi:10.1007/s00222-022-01167-0.\n- [HP95b] Jim Hoste and Józef H. Przytycki. The Kauffman bracket skein module of $S^{1}$ $\\times$ $S^{2}$. Math. Z., 220(1):65–73, 1995. doi:10.1007/BF02572603.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The rank-one recognition statement is proved when the rationalized integral skein module is tame (in particular, finitely generated over the Laurent polynomial ring), but the unrestricted prime-manifold question remains open.\n\n**Verified partial progress.**\n\n- Detcherry--Kalfagianni--Sikora bound the generic skein rank below by the number of closed points and above by the dimension of the unreduced coordinate ring of the SL(2,C) character scheme under tameness.\n- Together with Zentner's irreducible-representation theorem, the tame rank-one case forces a prime manifold other than S1 times S2 to be S3.\n- S1 times S2 has generic rank one but is non-tame, showing why the stated exception is not absorbed by the conditional argument.\n\n**Full solution or refutation.**\n\nThere is an affirmative conditional theorem under tameness/finite generation, not a proof for every prime connected closed 3-manifold.\n\n**What remains.**\n\nRemove the tameness hypothesis and exclude all other prime manifolds, or construct a rank-one counterexample.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 3.65 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Retains the unrestricted question and records the finite-generation and tame conditional affirmative results.\n- Renaud Detcherry, Efstratia Kalfagianni, and Adam S. Sikora, Kauffman bracket skein modules of small 3-manifolds, Advances in Mathematics 467 (2025), 110169. (primary): https://doi.org/10.1016/j.aim.2025.110169\n Evidence used: Question 10.3 states the recognition problem; Theorem 1.1/3.1 gives the rank bounds and equality under tameness and reducedness.\n- Raphael Zentner, Integer homology 3-spheres admit irreducible representations in SL(2,C), Duke Mathematical Journal 167 (2018), 1643--1712. (primary): https://doi.org/10.1215/00127094-2018-0004\n Evidence used: Supplies the irreducible-representation obstruction used to conclude S3 in the tame rank-one homology-sphere case.\n\n**Review notes.** Generic finite dimension over Q(A) is kept distinct from finite generation or tameness over the Laurent polynomial ring Q[A,A^{-1}].\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2864, "problem_number": "KP-3.66", "title": "Kirby Problem 3.66", "statement": "Suppose that $Y$ is a closed, oriented 3-manifold, and let $S_{2,\\infty}(Y)$ denote the Kauffman bracket skein module over $R=\\mathbb{Z}[A,A^{-1}]$ as in Problem 3.65.\n\n(a) Is the dimension\n\n$$\n\\dim_{\\mathbb{Q}(A)} S_{2,\\infty}(Y)\\otimes_R \\mathbb{Q}(A)\n$$\n\nequal to the dimension of the degree 0 part of Abouzaid--Manolescu's sheaf-theoretic Floer homology $HP^{\\bullet}_{\\#}(Y)$ from [AM20]?\n\n(b) When is $S_{2,\\infty}(Y)\\otimes_R\\mathbb{Q}[A,A^{-1}]$ tame, as defined in Problem 3.65?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.66.\n\nLiterature notes:\n(1) An affirmative answer to the first question was conjectured in [GJS23], which also contains the proof that\n\n$$\nS_{2,\\infty}(Y)\\otimes_R \\mathbb{Q}(A)\n$$\n\nis finite dimensional over $\\mathbb{Q}(q)$.\n\n(2) When $S_{2,\\infty}(Y)\\otimes_R\\mathbb{Q}[A,A^{-1}]$ is tame, the dimension\n\n$$\n\\dim_{\\mathbb{Q}(A)} S_{2,\\infty}(Y)\\otimes_R \\mathbb{Q}(A)\n$$\n\nis bounded above by the dimension of the unreduced coordinate ring of the $SL(2,\\mathbb{C})$ character variety of $Y$, and is bounded below by the number points in this character variety [DKS25].\n\n(3) It is conjectured in [DKS25] that if $Y$ is a small 3-manifold, then\n\n$$\nS_{2,\\infty}(Y)\\otimes_R\\mathbb{Q}[A,A^{-1}]\n$$\n\nis finitely generated over $\\mathbb{Q}[A,A^{-1}]$, and hence tame. Positive progress on computing $S_{2,\\infty}(Y)$ in the case of the Cartesian product of a circle with a surface has been made by Gilmer and Masbaum [GM19] and Detcherry and Wolff [DW21]. Moreover, the skein modules of 3-manifolds obtained by surgery on the figure-eight knot and on $(2,p)$ torus knots are shown to be tame [DKS25].\n\nReferences cited:\n- [AM20] Mohammed Abouzaid and Ciprian Manolescu. A sheaf-theoretic model for $\\mathrm{SL}(2,\\mathbb{C})$ Floer homology. J. Eur. Math. Soc. (JEMS), 22(11):3641–3695, 2020. doi:10.4171/jems/994.\n- [GJS23] Sam Gunningham, David Jordan, and Pavel Safronov. The finiteness conjecture for skein modules. Invent. Math., 232(1):301–363, 2023. doi:10.1007/s00222-022-01167-0.\n- [DKS25] Renaud Detcherry, Efstratia Kalfagianni, and Adam S. Sikora. Kauffman bracket skein modules of small 3-manifolds. Adv. Math., 467:Paper No. 110169, 45, 2025. doi:10.1016/j.aim.2025.110169.\n- [GM19] Patrick M. Gilmer and Gregor Masbaum. On the skein module of the product of a surface and a circle. Proc. Amer. Math. Soc., 147(9):4091–4106, 2019. doi: 10.1090/proc/14553.\n- [DW21] Renaud Detcherry and Maxime Wolff. A basis for the Kauffman skein module of the product of a surface and a circle. Algebr. Geom. Topol., 21(6):2959–2993, 2021. doi:10.2140/agt.2021.21.2959.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The proposed skein/Floer dimension equality and tameness are verified for substantial finite-reduced character-variety and Dehn-filling families, but neither statement is known for all closed 3-manifolds.\n\n**Verified partial progress.**\n\n- Gunningham--Jordan--Safronov prove finite dimensionality of the generic skein module for every closed 3-manifold and formulate the Floer-dimension conjecture.\n- Detcherry--Kalfagianni--Sikora verify the dimension comparison when the skein module is tame and the SL(2,C) character scheme is finite and reduced.\n- Tameness/finite generation and explicit dimensions are known for broad surgery families based on the figure-eight and (2,2n+1) torus knots, with further product computations in work of Gilmer--Masbaum and Detcherry--Wolff.\n\n**Full solution or refutation.**\n\nKnown family theorems provide evidence and many examples, not the all-manifold equality or a complete criterion for tameness.\n\n**What remains.**\n\nProve equality for arbitrary closed oriented Y and characterize exactly when the Laurent-polynomial skein module is tame.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 3.66 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: States both questions and records the general finite-dimensionality theorem and known tame families without claiming a general answer.\n- Sam Gunningham, David Jordan, and Pavel Safronov, The finiteness conjecture for skein modules, Inventiones Mathematicae 232 (2023), 301--363. (primary): https://doi.org/10.1007/s00222-022-01167-0\n Evidence used: Proves generic finite dimensionality for closed 3-manifolds and formulates the comparison with sheaf-theoretic SL(2,C) Floer homology.\n- Renaud Detcherry, Efstratia Kalfagianni, and Adam S. Sikora, Kauffman bracket skein modules of small 3-manifolds, Advances in Mathematics 467 (2025), 110169. (primary): https://doi.org/10.1016/j.aim.2025.110169\n Evidence used: Proves tame rank bounds, the reduced finite-character equality, and tameness/computations for specified Dehn-filling families.\n\n**Review notes.** The background's Q(q) notation is inconsistent with the statement's Q(A); this was flagged rather than silently normalized.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2865, "problem_number": "KP-3.67", "title": "Kirby Problem 3.67", "statement": "Categorify the Witten--Reshetikhin--Turaev invariants of 3-manifolds.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.67.\n\nLiterature notes:\nThe Witten--Reshetikhin--Turaev (WRT) invariants [Wit89, RT91] are maps from the set of roots of unity to C. They are an analogue of the Jones polynomial for 3-manifolds. Khovanov [Kho00] constructed a categorification of the Jones polynomial for links in $S^3$, in the form of a bigraded abelian group whose graded Euler characteristic is the Jones polynomial. Since the WRT invariants are complex numbers, it is less clear what it means to categorify them. The first ideas in this direction appeared in the work of Crane and Frenkel [CF94]. For recent work on categorification at roots of unity, see [Kho16, EQ16, QRSW21].\n\nReferences cited:\n- [Wit89] Edward Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys., 121(3):351–399, 1989. http://projecteuclid.org/euclid.cmp/1104178138.\n- [RT91] N. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math., 103(3):547–597, 1991. doi:10.1007/BF01239527.\n- [Kho00] Mikhail Khovanov. A categorification of the Jones polynomial. Duke Math. J., 101(3):359–426, 2000. doi:10.1215/S0012-7094-00-10131-7.\n- [CF94] Louis Crane and Igor B. Frenkel. Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. J. Math. Phys., 35(10):5136–5154, 1994. doi:10.1063/1.530746.\n- [Kho16] Mikhail Khovanov. Hopfological algebra and categorification at a root of unity: the first steps. J. Knot Theory Ramifications, 25(3):1640006, 26, 2016. doi:10.1142/S021821651640006X.\n- [EQ16] Ben Elias and You Qi. A categorification of quantum $\\mathfrak{sl}(2)$ at prime roots of unity. Adv. Math., 299:863–930, 2016. doi:10.1016/j.aim.2016.06.002.\n- [QRSW21] You Qi, Louis-Hadrien Robert, Joshua Sussan, and Emmanuel Wagner. A categorification of the colored Jones polynomial at a root of unity, 2021. arXiv:2111.13195.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Root-of-unity and colored-Jones categorifications supply substantial building blocks, but no universally accepted categorification recovering the WRT invariants for arbitrary closed 3-manifolds is known.\n\n**Verified partial progress.**\n\n- Crane--Frenkel initiated a higher-categorical program aimed at four-dimensional TQFT and categorified quantum invariants.\n- Khovanov and Elias--Qi developed hopfological/root-of-unity categorifications of quantum algebra.\n- Qi--Robert--Sussan--Wagner construct a root-of-unity categorification of colored Jones polynomials, an ingredient related to surgery presentations but not itself an invariant of all closed 3-manifolds categorifying WRT values.\n\n**Full solution or refutation.**\n\nThere are rigorous categorical ingredients and restricted constructions, but the broad request is both unresolved and underspecified.\n\n**What remains.**\n\nSpecify the target and decategorification criterion, then construct a functorial invariant of every closed 3-manifold recovering WRT values and compatible with TQFT/surgery operations.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 3.67 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Retains the categorification prompt and emphasizes that categorifying complex-number-valued WRT invariants requires further interpretation.\n- You Qi, Louis-Hadrien Robert, Joshua Sussan, and Emmanuel Wagner, A categorification of the colored Jones polynomial at a root of unity, arXiv:2111.13195. (primary): https://arxiv.org/abs/2111.13195\n Evidence used: Constructs root-of-unity link homologies with the relevant colored-Jones decategorification, not a general closed-3-manifold WRT homology.\n- Ben Elias and You Qi, A categorification of quantum sl(2) at prime roots of unity, Advances in Mathematics 299 (2016), 863--930. (primary): https://doi.org/10.1016/j.aim.2016.06.002\n Evidence used: Provides a root-of-unity quantum-group categorification that underlies one major approach.\n\n**Review notes.** The word categorify has no unique mathematical criterion here because a WRT value is a complex number; the exact source statement was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2866, "problem_number": "KP-3.68", "title": "Kirby Problem 3.68", "statement": "(a) Give a mathematical definition of the $\\widehat{Z}$ invariants for all 3-manifolds.\n\n(b) Categorify the $\\widehat{Z}$ invariants.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.68.\n\nLiterature notes:\n(1) In [GPV17, GPPV20], Gukov, Pei, Putrov, and Vafa conjectured the existence of some power series with integral coefficients associated to 3-manifolds, denoted $\\widehat{Z}_a(q)$. They converge in the unit disk $|q|<1$, and their limits as $q$ approach some roots of unity should recover the Witten--Reshetikhin--Turaev invariants.\n\n(2) Rigorous definitions of the $\\widehat{Z}$ invariants exist for plumbed 3-manifolds and certain surgeries on knots; see [GPPV20, GM21, CGPS20].\n\n(3) Since the $\\widehat{Z}$ invariants have integral coefficients, they are good candidates for categorification.\n\n(4) There are many other interesting questions about $\\widehat{Z}$, as well. For example, it is interesting to study its large $N$ limit, which seems to be connected to enumerative geometry [EGG+22].\n\nReferences cited:\n- [GPV17] Sergei Gukov, Pavel Putrov, and Cumrun Vafa. Fivebranes and 3-manifold homology. J. High Energy Phys., 2017(7):071, front matter+80, 2017. doi:10.1007/JHEP07(2017)071.\n- [GPPV20] Sergei Gukov, Du Pei, Pavel Putrov, and Cumrun Vafa. BPS spectra and 3-manifold invariants. J. Knot Theory Ramifications, 29(2):2040003, 85, 2020. doi:10.1142/S0218216520400039.\n- [GM21] Sergei Gukov and Ciprian Manolescu. A two-variable series for knot complements. Quantum Topol., 12(1):1–109, 2021. doi:10.4171/qt/145.\n- [CGPS20] Sungbong Chun, Sergei Gukov, Sunghyuk Park, and Nikita Sopenko. 3d-3d correspondence for mapping tori. J. High Energy Phys., 2020(9):152, 59, 2020. doi: 10.1007/jhep09(2020)152.\n- [EGG+22] Tobias Ekholm, Angus Gruen, Sergei Gukov, Piotr Kucharski, Sunghyuk Park, Marko Stošić, and Piotr Sul kowski. Branches, quivers, and ideals for knot complements. J. Geom. Phys., 177:Paper No. 104520, 75, 2022. doi:10.1016/j.geomphys.2022.104520.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Rigorous definitions exist for plumbed manifolds and several surgery/complement/mapping-torus settings, and 2025 preprints propose categorical models for Seifert and negative-definite plumbed manifolds; the all-3-manifold definition and categorification remain open.\n\n**Verified partial progress.**\n\n- Gukov--Manolescu define a two-variable series for knot complements, while GPPV-type formulas and subsequent work rigorously handle negative-definite plumbings and selected surgeries or mapping tori.\n- Sugimoto proposes an abelian categorification of hat-Z invariants for Seifert 3-manifolds via hypercubic structures.\n- Sugimoto's later 2025 preprint extends the proposed categorical blueprint to negative-definite plumbed 3-manifolds.\n\n**Full solution or refutation.**\n\nBoth parts have substantive restricted-family progress, including recent categorical proposals, but neither is complete for all 3-manifolds.\n\n**What remains.**\n\nDefine hat-Z for arbitrary 3-manifolds with full invariance and WRT radial-limit proofs, then categorify that general invariant.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 3.68 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Records rigorous definitions for plumbed manifolds and certain knot surgeries while retaining both all-manifold questions.\n- Sergei Gukov and Ciprian Manolescu, A two-variable series for knot complements, Quantum Topology 12 (2021), 1--109. (primary): https://doi.org/10.4171/QT/145\n Evidence used: Defines and studies the two-variable series for knot complements, a major rigorous extension beyond closed negative-definite plumbings.\n- Shoma Sugimoto, Hypercubic structures behind hat-Z-invariants, arXiv:2501.12985 (2025). (primary): https://arxiv.org/abs/2501.12985\n Evidence used: Proposes an abelian categorification for Seifert 3-manifolds.\n- Shoma Sugimoto, Nesting behind hat-Z-invariants, arXiv:2507.13996 (2025). (primary): https://arxiv.org/abs/2507.13996\n Evidence used: Proposes an abelian categorification for negative-definite plumbed 3-manifolds.\n\n**Review notes.** The 2025 categorification papers are labeled as proposals/preprints and are not promoted to an all-manifold solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2867, "problem_number": "KP-3.69", "title": "Kirby Problem 3.69", "statement": "(a) What is the isomorphism type of $\\Theta^3_{\\mathbb{Z}}$?\n\n(b) Does there exist a torsion element $[Y]$ in $\\Theta^3_{\\mathbb{Z}}$?\n\n(c) Does there exist a torsion element $[Y]\\in\\Theta^3_{\\mathbb{Z}}$ having Rokhlin invariant $\\mu([Y])=1$?\n\n(d) Do there exist infinitely-divisible elements in $\\Theta^3_{\\mathbb{Z}}$; that is, does there exist a homology 3-sphere $Y$ such that for infinitely many $n\\in\\mathbb{N}$ there exists $[Z_n]\\in\\Theta^3_{\\mathbb{Z}}$ such that $[Y]=n[Z_n]$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.69.\n\nLiterature notes:\n(1) The integer homology cobordism group $\\Theta^3_{\\mathbb{Z}}$ is known to be a countable abelian group that has a summand isomorphic to $\\mathbb{Z}^{\\infty}$ by work of [DHST23].\n\n(2) In dimension $n=1,2$, the analogous group $\\Theta^n_{\\mathbb{Z}}$ vanishes. In dimension $n\\geq 4$, the group $\\Theta^n_{\\mathbb{Z}}$ is isomorphic to the group of homotopy spheres up to h-cobordism [GA70], and therefore finite [KM63]. However, in dimensions $n\\geq 4$, the analog of this group in the PL category vanishes; it is this\n\nlatter group that arises as an obstruction in the study of triangulations [GS80, Mat78].\n\n(3) Manolescu showed that there does not exist an element $[Y]$ of order two having Rokhlin invariant $\\mu(Y)$=1 [Man16b]. By previous work of Galewski--Stern [GS80] and Matumoto [Mat78], this was sufficient to disprove the outstanding cases of the Triangulation Conjecture, showing the existence of topological manifolds in every dimension greater than or equal to five that do not admit triangulations. Since $Y\\#(-Y)$ bounds a smooth homology 4-ball for all integer homology 3-spheres $Y$, to find a 2-torsion element, it suffices to exhibit a homology 3-sphere $Y$ with an orientation-reversing diffeomorphism, such that $[Y]=[-Y]=-[Y]$ in homology cobordism, but $[Y]\\neq [S^3]$. Noteworthy examples of 3-manifolds with such orientation-reversing diffeomorphisms include double branched covers of determinant 1, non-slice knots, such as those in [BC24b]. Various authors have given conditions that imply that a homology 3-sphere $[Y]$ must have infinite order in $\\Theta^3_{\\mathbb{Z}}$, for example [LRS18, Theorems C and D] or [HHL21, Theorem 1.13].\n\n(4) By work of Galewski--Stern, a negative answer to part (c) would imply that a topological manifold $M$ of dimension $\\geq 5$ is triangulable if and only if a particular obstruction class in $H^{5}(M;\\mathbb{Z})$ vanishes.\n\n(5) If $\\mu([Y])=1$, then $[Y]$ cannot be equal to $n[Z]$ for even $n$. Furthermore, any infinitely divisible $[Y]$ must necessarily have Heegaard Floer correction term $d([Y])=0$, and similarly for any other numerical invariant that is additive under connected sum. Any torsion element of a group is infinitely divisible, so in the presence of torsion part (d) might be modified to ask about the presence of infinitely divisible elements in the quotient of $\\Theta^3_{\\mathbb{Z}}$ by its torsion subgroup.\n\nReferences cited:\n- [DHST23] Irving Dai, Jennifer Hom, Matthew Stoffregen, and Linh Truong. An infinite-rank summand of the homology cobordism group. Duke Math. J., 172(12):2365–2432, 2023. doi:10.1215/00127094-2022-0082.\n- [GA70] Francisco Javier Gonzalez Acuna. ON HOMOLOGY SPHERES. ProQuest LLC, Ann Arbor, MI, 1970. Thesis (Ph.D.)–Princeton University. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt=info: ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri:pqdiss:7023614.\n- [KM63] Michel A. Kervaire and John W. Milnor. Groups of homotopy spheres. I. Ann. of Math. (2), 77:504–537, 1963. doi:10.2307/1970128.\n- [GS80] David E. Galewski and Ronald J. Stern. Classification of simplicial triangulations of topological manifolds. Ann. of Math. (2), 111(1):1–34, 1980. doi:10.2307/1971215.\n- [Mat78] Takao Matumoto. Triangulation of manifolds. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 3–6. Amer. Math. Soc., Providence, R.I., 1978.\n- [Man16b] Ciprian Manolescu. Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture. J. Amer. Math. Soc., 29(1):147–176, 2016. doi:10.1090/jams829.\n- [BC24b] Keegan Boyle and Wenzhao Chen. Equivariant topological slice disks and negative amphichiral knots. Indiana Univ. Math. J., 73(5):1623–1637, 2024.\n- [LRS18] Jianfeng Lin, Daniel Ruberman, and Nikolai Saveliev. A splitting theorem for the Seiberg-Witten invariant of a homology $S^{1}$ $\\times$ $S^{3}$. Geom. Topol., 22(5):2865–2942, 2018. doi:10.2140/gt.2018.22.2865.\n- [HHL21] Kristen Hendricks, Jennifer Hom, and Tye Lidman. Applications of involutive Heegaard Floer homology. J. Inst. Math. Jussieu, 20(1):187–224, 2021. doi:10.1017/$S^{1}$47474801900015X.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Theta^3_Z contains a direct summand Z^infinity and has many detected infinite-order classes, but its isomorphism type, the existence of any torsion, Rokhlin-one torsion, and infinitely divisible elements remain unknown.\n\n**Verified partial progress.**\n\n- Dai--Hom--Stoffregen--Truong prove that the group contains a direct summand isomorphic to a countably generated free abelian group.\n- Manolescu proves there is no order-two element with Rokhlin invariant one, which is weaker than excluding all torsion or all Rokhlin-one torsion.\n- Splices of knots with their mirrors give natural potential 2-torsion classes; Hendricks--Stoffregen--Zemke show many standard involutive and gauge-theoretic invariants vanish on them without proving nontrivial torsion.\n\n**Full solution or refutation.**\n\nA large free direct summand is known, but none of parts (b)--(d), nor the full classification in (a), is resolved.\n\n**What remains.**\n\nFind or exclude torsion, settle Rokhlin-one torsion and infinite divisibility, and determine the quotient beyond the known free direct summand.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 3.69 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Separates the known free summand and Manolescu order-two obstruction from the still-open torsion and divisibility questions.\n- Irving Dai, Jennifer Hom, Matthew Stoffregen, and Linh Truong, An infinite-rank summand of the homology cobordism group, Duke Mathematical Journal 172 (2023), 2365--2432. (primary): https://doi.org/10.1215/00127094-2022-0082\n Evidence used: Constructs the direct summand Z^infinity.\n- Ciprian Manolescu, Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture, Journal of the AMS 29 (2016), 147--176. (primary): https://doi.org/10.1090/jams/829\n Evidence used: Rules out order-two homology-cobordism classes with nonzero Rokhlin invariant.\n- Kristen Hendricks, Matthew Stoffregen, and Ian Zemke, A note on the involutive invariants of splices, arXiv:2406.02957 (2024). (primary): https://arxiv.org/abs/2406.02957\n Evidence used: Studies natural potential 2-torsion splices and shows the limitations of several existing invariants.\n\n**Review notes.** The record explicitly defines infinitely-divisible as divisible by infinitely many n; that exact weaker-than-usual wording was retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2868, "problem_number": "KP-3.70", "title": "Kirby Problem 3.70", "statement": "Is $\\Theta^3_{\\mathbb{Z}}$ generated by the classes of knot surgeries $[S^3_{1/n}(K)]$, where $n$ ranges over all integers and $K$ ranges over all knots in $S^3$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.70.\n\nLiterature notes:\nThere are homology spheres that are not given by surgery on a knot in $S^3$ [GL89, Auc97, HKL16b]. Nozaki--Sato--Taniguchi used filtered instanton Floer homology to exhibit a homology 3-sphere which is not even homology cobordant to any knot surgery [NST24]. Hendricks--Hom--Lidman showed that $\\Theta^3_{\\mathbb{Z}}$ is not generated by surgeries on knots of bounded genus [HHL21].\n\nReferences cited:\n- [GL89] C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371–415, 1989. doi:10.2307/1990979.\n- [Auc97] David Auckly. Surgery numbers of 3-manifolds: a hyperbolic example. In Geometric topology (Athens, GA, 1993), volume 2 of AMS/IP Stud. Adv. Math., pages 21–34. Amer. Math. Soc., Providence, RI, 1997. doi:10.1090/amsip/002.1/02.\n- [HKL16b] Jennifer Hom, Çağrı Karakurt, and Tye Lidman. Surgery obstructions and Heegaard Floer homology. Geom. Topol., 20(4):2219–2251, 2016. doi:10.2140/gt.2016.20.2219.\n- [NST24] Yuta Nozaki, Kouki Sato, and Masaki Taniguchi. Filtered instanton Floer homology and the homology cobordism group. J. Eur. Math. Soc. (JEMS), 26(12):4699–4761, 2024. doi:10.4171/jems/1371.\n- [HHL21] Kristen Hendricks, Jennifer Hom, and Tye Lidman. Applications of involutive Heegaard Floer homology. J. Inst. Math. Jussieu, 20(1):187–224, 2021. doi:10.1017/$S^{1}$47474801900015X.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Some homology spheres are not cobordant to any single knot surgery and bounded-genus surgeries do not generate Theta^3_Z, but unrestricted finite sums of surgeries on knots of arbitrary genus may still generate the group.\n\n**Verified partial progress.**\n\n- Nozaki--Sato--Taniguchi construct homology spheres that cannot bound definite 4-manifolds; since a knot-surgery homology sphere bounds a definite manifold, these classes are not represented by a single knot surgery.\n- Hendricks--Hom--Lidman show that surgeries on knots with any fixed upper genus bound do not generate the whole group.\n- Filtered instanton invariants also prove linear independence of many 1/n surgery families, demonstrating that the surgery-generated subgroup is large.\n\n**Full solution or refutation.**\n\nThe known obstructions distinguish single surgeries and bounded-genus-generated subgroups, not the subgroup generated by all unrestricted surgeries.\n\n**What remains.**\n\nSeparate some class from the full surgery-generated subgroup or decompose every class into a finite sum of 1/n knot surgeries.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 3.70 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Explicitly distinguishes non-representability by one surgery and bounded-genus non-generation from the unrestricted generation question.\n- Yuta Nozaki, Kouki Sato, and Masaki Taniguchi, Filtered instanton Floer homology and the homology cobordism group, Journal of the EMS 26 (2024), 4699--4761. (primary): https://doi.org/10.4171/JEMS/1371\n Evidence used: Constructs classes not cobordant to individual knot surgeries and proves linearly independent 1/n-surgery families.\n- Kristen Hendricks, Jennifer Hom, and Tye Lidman, Applications of involutive Heegaard Floer homology, Journal of the Institute of Mathematics of Jussieu 20 (2021), 187--224. (primary): https://doi.org/10.1017/S147474801900015X\n Evidence used: Shows that surgeries on knots of bounded genus do not generate the integral homology cobordism group.\n\n**Review notes.** A class not represented by one surgery is not automatically outside the subgroup generated by finite sums of surgery classes.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2869, "problem_number": "KP-3.71", "title": "Kirby Problem 3.71", "statement": "Is there a nontrivial element in the kernel of the natural map\n\n$$\n\\Theta^3_{\\mathbb{Z}}\\longrightarrow \\Theta^3_{\\mathbb{Z}/2\\mathbb{Z}};\n$$\n\nthat is, does there exist an integer homology 3-sphere that does not bound a smooth $\\mathbb{Z}$-homology 4-ball but does bound a smooth $\\mathbb{Z}/2\\mathbb{Z}$-homology 4-ball?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.71.\n\nLiterature notes:\nThere is a natural homomorphism from the concordance group $C$ to $\\Theta^3_{\\mathbb{Z}/2\\mathbb{Z}}$ via taking double branched covers. As discussed in Problem 3.72, $\\Sigma(2,3,7)$ bounds a rational homology ball but, because its Rokhlin invariant is nontrivial, does not bound a $\\mathbb{Z}/2\\mathbb{Z}$-homology ball. See also [BL02] for some further discussion of $\\mathbb{Z}/2\\mathbb{Z}$-homology cobordism.\n\nReferences cited:\n- [BL02] Christian Bohr and Ronnie Lee. Homology cobordism and classical knot invariants. Comment. Math. Helv., 77(2):363–382, 2002. doi:10.1007/s00014-002-8344-0.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No nontrivial kernel element from integral to mod-2 homology cobordism is known; familiar integral homology spheres bounding rational balls with Rokhlin invariant one cannot be examples.\n\n**Verified partial progress.**\n\n- Any smooth Z/2-homology ball is a rational homology ball, so the sought kernel sits inside the known nontrivial kernel of Theta^3_Z to Theta^3_Q.\n- The converse fails as an inference: Sigma(2,3,7) bounds a rational ball but its nonzero Rokhlin invariant obstructs a Z/2-homology ball.\n- Bohr--Lee provide mod-2 homology-cobordism constraints and relations to classical knot invariants, but no requested nonzero kernel class.\n\n**Full solution or refutation.**\n\nThe question remains open; rational-ball examples with even-primary homology or nonzero Rokhlin invariant do not settle it.\n\n**What remains.**\n\nConstruct a nontrivial integral homology sphere bounding a smooth mod-2 homology ball, likely with only odd torsion in the ball, or prove injectivity.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 3.71 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Retains the kernel question and explains why the standard Sigma(2,3,7) rational-ball example is not a mod-2 example.\n- Christian Bohr and Ronnie Lee, Homology cobordism and classical knot invariants, Commentarii Mathematici Helvetici 77 (2002), 363--382. (primary): https://doi.org/10.1007/s00014-002-8344-0\n Evidence used: Develops constraints in Z/2-homology cobordism but does not produce the kernel element requested here.\n- Ronald Fintushel and Ronald J. Stern, A mu-invariant one homology 3-sphere that bounds an orientable rational ball, Contemporary Mathematics 35 (1984), 265--268. (primary): https://doi.org/10.1090/conm/035/780582\n Evidence used: Provides the canonical rational-ball example whose nonzero Rokhlin invariant prevents it from answering the mod-2 question.\n\n**Review notes.** Bounding a rational ball was not conflated with bounding a Z/2-homology ball.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2870, "problem_number": "KP-3.72", "title": "Kirby Problem 3.72", "statement": "(a) Does the kernel of the map $\\Theta^3_{\\mathbb{Z}}\\to\\Theta^3_{\\mathbb{Q}}$ contain a subgroup that is isomorphic to $\\mathbb{Z}^{\\infty}$?\n\n(b) If so, does it contain such a subgroup as a summand?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.72.\n\nLiterature notes:\nFintushel and Stern [FS84] showed that $\\Sigma(2,3,7)=S^3_{+1}(4_1)$ bounds a smooth rational homology ball; as it has nontrivial Rokhlin invariant, it is a nontrivial element of the kernel of the map above. Indeed, as $\\Sigma(2,3,7)$ has nonvanishing Neumann--Siebenmann invariant $\\bar\\mu$ [Neu80, NR78], it is of infinite order in the homology cobordism group, implying that the kernel is infinite. This was the only known example of a Brieskorn sphere bounding a rational homology ball but not an integer homology ball until infinite families of examples were given by Akbulut and Larson [AL18], followed by further examples from Şavk [Şav20] and Simone [Sim21]. It is unknown whether these families are linearly independent.\n\nReferences cited:\n- [FS84] Ronald Fintushel and Ronald J. Stern. A µ-invariant one homology 3-sphere that bounds an orientable rational ball. In Four-manifold theory (Durham, N.H., 1982), volume 35 of Contemp. Math., pages 265–268. Amer. Math. Soc., Providence, RI, 1984. doi:10.1090/conm/035/780582.\n- [Neu80] Walter D. Neumann. An invariant of plumbed homology spheres. In Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), volume 788 of Lecture Notes in Math., pages 125–144. Springer, Berlin, 1980.\n- [NR78] Walter D. Neumann and Frank Raymond. Seifert manifolds, plumbing, µ-invariant and orientation reversing maps. In Algebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977), volume 664 of Lecture Notes in Math., pages 163–196. Springer, Berlin-New York, 1978.\n- [AL18] Selman Akbulut and Kyle Larson. Brieskorn spheres bounding rational balls. Proc. Amer. Math. Soc., 146(4):1817–1824, 2018. doi:10.1090/proc/13828.\n- [Şav20] Oğuz Şavk. More Brieskorn spheres bounding rational balls. Topology Appl., 286:107400, 10, 2020. doi:10.1016/j.topol.2020.107400.\n- [Sim21] Jonathan Simone. Using rational homology circles to construct rational homology balls. Topology Appl., 291:Paper No. 107626, 16, 2021. doi:10.1016/j.topol.2021.107626.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The kernel of Theta^3_Z to Theta^3_Q contains an infinite cyclic subgroup and many further explicit candidates, but no infinite linearly independent subgroup or infinite-rank summand is known.\n\n**Verified partial progress.**\n\n- Sigma(2,3,7) bounds a rational ball, is nontrivial integrally by its Rokhlin invariant, and has infinite order detected by the Neumann--Siebenmann invariant, giving a Z subgroup in the kernel.\n- Akbulut--Larson construct infinite Brieskorn-sphere families that bound rational but not integral homology balls.\n- Savk and Simone produce additional infinite families, but linear independence in Theta^3_Z remains unproved.\n\n**Full solution or refutation.**\n\nThe known infinite families do not yet imply the Z^infinity subgroup in (a), much less a direct summand as in (b).\n\n**What remains.**\n\nProve an infinite subfamily linearly independent in integral homology cobordism and, for part (b), construct a splitting detecting its coordinates.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 3.72 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Records that the explicit infinite families are not known to be linearly independent and retains both questions.\n- Selman Akbulut and Kyle Larson, Brieskorn spheres bounding rational balls, Proceedings of the AMS 146 (2018), 1817--1824. (primary): https://doi.org/10.1090/proc/13828\n Evidence used: Constructs infinite families in the kernel and explicitly notes that their linear independence is unknown.\n- Oguz Savk, More Brieskorn spheres bounding rational balls, Topology and its Applications 286 (2020), 107400. (primary): https://doi.org/10.1016/j.topol.2020.107400\n Evidence used: Adds new families of Brieskorn spheres nontrivially bounding rational balls without proving an infinite-rank subgroup.\n\n**Review notes.** An infinite list of elements is not treated as an infinite-rank subgroup without a linear-independence theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2871, "problem_number": "KP-3.73", "title": "Kirby Problem 3.73", "statement": "(a) Calculate $\\Theta^{\\mathrm{TOP}}_{\\mathbb{Z}/p}$.\n\n(b) Calculate $\\Theta^{\\mathrm{TOP}}_{\\mathbb{Q}}$.\n\n(c) Is the linking form homomorphism $[\\operatorname{lk}]$ injective?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.73.\n\nLiterature notes:\n(1) Freedman proved that any integral homology 3-sphere bounds a contractible topological 4-manifold, unique up to homeomorphism rel boundary [FQ90, Corollary 9.3C], [Fre82, Theorem 1.41][BKK+21, Chapter 21.3.2]. This implies the topological integer homology cobordism group is trivial, $\\Theta^{\\mathrm{TOP}}_{\\mathbb{Z}}=1$.\n\n(2) Let $W(\\mathbb{Q}/\\mathbb{Z})$ denote the Witt group of non-singular bilinear forms over $\\mathbb{Q}/\\mathbb{Z}$. The $\\mathbb{Q}/\\mathbb{Z}$-valued linking form on $H_1(Y;\\mathbb{Z})$ defines a homomorphism\n\n$$\n[\\operatorname{lk}]:\\Theta^{\\mathrm{TOP}}_{\\mathbb{Q}}\\longrightarrow W(\\mathbb{Q}/\\mathbb{Z}).\n$$\n\nSee [Bre97, Chapter VI, Section 10] for a review of Poincaré--Lefschetz duality, and how one uses it to define the linking form, or [CFH16, Section 2] for a self-contained synopsis of the same. That the linking form defines a homomorphism to $W(\\mathbb{Q}/\\mathbb{Z})$ is an immediate consequence of the following fact: if a rational homology 3-sphere $Y$ bounds a rational ball, then its first homology has a subgroup of order $\\sqrt{|H_1(Y)|}$ on which the linking form vanishes. This is proved in [CG86, Theorem 1 and Theorem 2] (see also [Gil82, Lemma 1] for a self-contained treatment). One can easily show that this homomorphism is surjective; indeed, any form can be presented with an integral matrix, which can be used to build a 4-dimensional 2-handlebody whose boundary will be a 3-manifold representing the given form.\n\n(3) Homology cobordism groups are closely connected to knot concordance, by way of Dehn surgery and cyclic branched covers. For instance, both constructions define maps from the knot concordance group to $\\Theta^3_{\\mathbb{Q}}$, with each factoring through $\\Theta^3_{\\mathbb{Z}/p}$ for some $p$ depending on the surgery slope or degree of cyclic branched covering, respectively. For the latter, one should restrict to $p$ a prime power to ensure the branched covers are $\\mathbb{Z}/p$-homology spheres and, for slice knots, bound $\\mathbb{Z}/p$-homology balls [CG78]. In the case of branched coverings, these maps are actually homomorphisms. Both observations have been extensively used in the smooth setting. Understanding the problems above could have applications to the theory of topological concordance groups. For instance, if $[\\operatorname{lk}]$ fails to be injective, and refined invariants of $\\Theta^{\\mathrm{TOP}}_{\\mathbb{Q}}$ can be defined, then the latter could be used to study the topological concordance group through the surgery and branched covering maps.\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [BKK+21] Stefan Behrens, Boldizsár Kalmár, Min Hoon Kim, Mark Powell, and Arunima Ray, editors. The disc embedding theorem. Oxford University Press, Oxford, 2021.\n- [Bre97] Glen E. Bredon. Topology and geometry, volume 139 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. Corrected third printing of the 1993 original.\n- [CFH16] Anthony Conway, Stefan Friedl, and Gerrit Herrmann. Linking forms revisited. Pure Appl. Math. Q., 12(4):493–515, 2016. doi:10.4310/PAMQ.2016.v12.n4.a3.\n- [CG86] A. J. Casson and C. McA. Gordon. Cobordism of classical knots. In À la recherche de la topologie perdue, volume 62 of Progr. Math., pages 181–199. Birkhäuser Boston, Boston, MA, 1986.\n- [Gil82] Patrick M. Gilmer. On the slice genus of knots. Invent. Math., 66(2):191–197, 1982. doi:10.1007/BF01389390.\n- [CG78] A. J. Casson and C. McA. Gordon. On slice knots in dimension three. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, volume XXXII of Proc. Sympos. Pure Math., pages 39–53. Amer. Math. Soc., Providence, RI, 1978.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The ordinary linking form gives a surjection from topological rational homology cobordism to its Witt group, and new triple-linking obstructions refine the picture, but the mod-p and rational groups are not calculated and injectivity remains open.\n\n**Verified partial progress.**\n\n- Freedman's work implies the analogous integral topological homology-cobordism group is trivial.\n- The Q/Z-valued linking pairing defines a surjective homomorphism Theta^TOP_Q to W(Q/Z).\n- Freedman--Krushkal introduce a triple torsion linking form, and Stees proves that it vanishes on the classical Lagrangian when a bounding rational ball has H_2(W;Z)=0; this does not settle the unrestricted injectivity question.\n\n**Full solution or refutation.**\n\nClassical algebra supplies a surjective quotient and recent work suggests refinements, but no complete group calculation or injectivity theorem is known.\n\n**What remains.**\n\nCalculate Theta^TOP_{Z/p} and Theta^TOP_Q and determine whether a metabolic linking form suffices for topological rational null-cobordism.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 3.73 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: States the three open calculations and records the surjective classical linking-form homomorphism and Freedman's integral result.\n- Michael Freedman and Vyacheslav Krushkal, A triple torsion linking form and 3-manifolds in S4, arXiv:2506.11941 (2025). (primary): https://arxiv.org/abs/2506.11941\n Evidence used: Introduces a triple torsion linking form that can carry information beyond the classical pairing in related topological embedding and bounding problems.\n- Ryan Stees, Triple linking and rational homology cobordism, arXiv:2511.03818 (2025). (primary): https://arxiv.org/abs/2511.03818\n Evidence used: Proves triple-form vanishing on a Lagrangian for rational balls under H_2(W;Z)=0 and explicitly leaves broader topological rational-cobordism questions open.\n- Se-Goo Kim and Charles Livingston, Non-splittability of the rational homology cobordism group of 3-manifolds, Pacific Journal of Mathematics 271 (2014), 183--211. (primary): https://arxiv.org/abs/1211.3712\n Evidence used: Describes the Witt-group primary-splitting picture as conjecturally matching the topological category, supporting the conservative open status.\n\n**Review notes.** The statement does not quantify p in part (a); the report flags whether p means prime, prime power, or arbitrary modulus as a formulation issue.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2872, "problem_number": "KP-3.74", "title": "Kirby Problem 3.74", "statement": "Let $Y$ be a rational homology sphere and $f:Y\\to Y$ be a self-diffeomorphism of $Y$. Suppose $W$ is a 4-manifold with boundary $Y$ such that $f$ extends to a self-diffeomorphism of $W$.\n\n(a) What constraints are there on the intersection form of $W$?\n\n(b) Does there exist a pair $(Y,f)$ such that $Y$ bounds a homology ball but $f:Y\\to Y$ does not extend over any definite manifold of either sign?\n\n(c) Develop methods for constraining the intersection form of $W$ that apply to indefinite, non-spin $W$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.74.\n\nLiterature notes:\n(1) It is known that the 3-dimensional bordism-with-diffeomorphism group is trivial [Mel79]; that is, if $Y$ is a 3-manifold equipped with a self-diffeomorphism $f$, then there always exists a smooth 4-manifold $W$ over which $f$ extends. It is thus natural to ask what homological restrictions occur for the class of such $W$. This is analogous to the fact that the usual 3-dimensional bordism group is trivial, but the set of intersection forms bounded by $Y$ reflects the topology of $Y$.\n\n(2) The extension question for diffeomorphisms is also motivated by the theory of corks [Akb91]. If $\\pi_1(W)$ = 1, then every self-homeomorphism of\n\n$Y$ extends as a self-homeomorphism of $W$ by work of Freedman [FQ90]. In contrast, there are many examples in which a self-diffeomorphism of $Y$ does not extend as a self-diffeomorphism over a particular contractible manifold bounded by $Y$, or even any homology ball with boundary $Y$. See for example [Akb91, LRS23a, AKS20, DHM23].\n\n(3) While it is possible to obstruct the extension of $f$ over definite manifolds of a fixed sign, it is unknown whether there are any pairs $(Y,f)$ for which $Y$ bounds a homology ball, but $f$ does not extend over any definite manifold of either sign. Disregarding the self-diffeomorphism $f$, it has only recently been shown that there are integer homology spheres $Y$ that bound no definite manifold of either sign; see [NST24]. If the action of the extension on the cohomology of $W$ is constrained, then partial results have been obtained in [ADMT23] using the Chern--Simons filtration in instanton Floer homology.\n\nReferences cited:\n- [Mel79] Paul Melvin. Bordism of diffeomorphisms. Topology, 18(2):173–175, 1979. doi:10.1016/0040-9383(79)90034-X.\n- [Akb91] Selman Akbulut. A fake compact contractible 4-manifold. J. Differential Geom., 33(2):335–356, 1991. http://projecteuclid.org/euclid.jdg/1214446320.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [LRS23a] Jianfeng Lin, Daniel Ruberman, and Nikolai Saveliev. On the Frøyshov invariant and monopole Lefschetz number. J. Differential Geom., 123(3):523–593, 2023. doi: 10.4310/jdg/1683307008.\n- [AKS20] Antonio Alfieri, Sungkyung Kang, and András I. Stipsicz. Connected Floer homology of covering involutions. Math. Ann., 377(3-4):1427–1452, 2020. doi:10.1007/s00208-020-01992-9.\n- [DHM23] Irving Dai, Matthew Hedden, and Abhishek Mallick. Corks, involutions, and Heegaard Floer homology. J. Eur. Math. Soc. (JEMS), 25(6):2319–2389, 2023. doi:10.4171/jems/1239.\n- [NST24] Yuta Nozaki, Kouki Sato, and Masaki Taniguchi. Filtered instanton Floer homology and the homology cobordism group. J. Eur. Math. Soc. (JEMS), 26(12):4699–4761, 2024. doi:10.4171/jems/1371.\n- [ADMT23] Antonio Alfieri, Irving Dai, Abhishek Mallick, and Masaki Taniguchi. Involutions and the Chern-Simons filtration in instanton Floer homology, 2023. arXiv:2309.02309.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Floer-theoretic invariants obstruct extension of boundary diffeomorphisms over homology balls and important low-b_2 or stabilized fillings, but no pair excluding definite fillings of both signs and no general indefinite non-spin theory was located.\n\n**Verified partial progress.**\n\n- Melvin proves that every 3-manifold with a self-diffeomorphism bounds some smooth 4-manifold over which the diffeomorphism extends.\n- Cork and strong-cork invariants give examples that do not extend over a specified contractible filling or over any homology ball.\n- Alfieri-Dai-Mallick-Taniguchi construct an involution not extending over any X with H_1(X;Z_2)=0 and b_2(X)<=1, plus examples surviving positive and negative projective-plane stabilization.\n- Nozaki-Sato-Taniguchi show, without a boundary diffeomorphism, that some homology spheres bound no definite manifold of either sign.\n\n**Full solution or refutation.**\n\nThe exact both-sign extension obstruction in part (b) and the indefinite non-spin methods requested in part (c) remain open.\n\n**What remains.**\n\nProduce a homology-ball-bounding pair (Y,f) for which f extends over no positive- or negative-definite filling, and develop intersection-form constraints for arbitrary indefinite non-spin extensions.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.74. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States all three parts, identifies the simultaneous two-sign obstruction as unknown, and surveys the extension and cork results.\n- Antonio Alfieri, Irving Dai, Abhishek Mallick, and Masaki Taniguchi, Involutions and the Chern-Simons filtration in instanton Floer homology, arXiv:2309.02309. (primary): https://arxiv.org/abs/2309.02309\n Evidence used: Develops a filtered instanton obstruction and proves nonextension over all fillings in a low-b_2 range, together with stabilization-resistant strong corks.\n- Yuta Nozaki, Kouki Sato, and Masaki Taniguchi, Filtered instanton Floer homology and the homology cobordism group, Journal of the European Mathematical Society 26 (2024), 4699-4761. (primary): https://doi.org/10.4171/jems/1371\n Evidence used: Provides the nonequivariant examples bounding no definite manifold of either sign that motivate part (b).\n\n**Review notes.** The known obstruction with b_2<=1 and stabilization results must not be promoted to an obstruction over every definite filling.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2873, "problem_number": "KP-3.75", "title": "Kirby Problem 3.75", "statement": "Let $Y$ be a rational homology 3-sphere equipped with an action of a cyclic group $\\mathbb{Z}/p\\mathbb{Z}$. Suppose $W$ is a 4-manifold with boundary $Y$ over which the action of $\\mathbb{Z}/p\\mathbb{Z}$ extends smoothly.\n\n(a) What constraints are there on the intersection form of $W$?\n\n(b) Does there exist an integer homology sphere $Y$ bounding a homology ball such that $Y$ admits a cyclic group action that does not extend over any definite manifold of either sign?\n\n(c) Develop methods for constraining the intersection form of $W$ that apply to indefinite, non-spin $W$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.75.\n\nLiterature notes:\nWhile this is similar to Problem 3.74, the motivation and techniques for the present question are slightly different. If $f:Y\\to Y$ generates the action of a cyclic group, then obstructing the extension of $f$ as a diffeomorphism clearly obstructs the extension of $f$ as a group action. However, there are many interesting examples where the two notions differ. For instance, one may consider the standard $S^1$-action on a Brieskorn sphere $\\Sigma(p,q,r)$. Any root of unity $\\zeta\\in S^1$ generates the action of a cyclic group on $\\Sigma(p,q,r)$ whose generator $f$ is isotopic to the identity (as a diffeomorphism). In [AH16, AH21], it is shown that this never extends as a smooth group action over any contractible manifold that $\\Sigma(p,q,r)$ may bound, even though it is not hard to see that $f$ extends as a diffeomorphism. See also [Edm87, KL93, BH24a].\n\nReferences cited:\n- [AH16] Nima Anvari and Ian Hambleton. Cyclic group actions on contractible 4-manifolds. Geom. Topol., 20(2):1127–1155, 2016. doi:10.2140/gt.2016.20.1127.\n- [AH21] Nima Anvari and Ian Hambleton. Cyclic branched coverings of Brieskorn spheres bounding acyclic 4-manifolds. Glasg. Math. J., 63(2):400–413, 2021. doi:10.1017/S0017089520000269.\n- [Edm87] Allan L. Edmonds. Construction of group actions on four-manifolds. Trans. Amer. Math. Soc., 299(1):155–170, 1987. doi:10.2307/2000487.\n- [KL93] Slawomir Kwasik and Terry Lawson. Nonsmoothable $\\mathbb{Z}_p$ actions on contractible 4-manifolds. J. Reine Angew. Math., 437:29–54, 1993. doi:10.1515/crll.1993.437.29.\n- [BH24a] David Baraglia and Pedram Hekmati. Equivariant Seiberg-Witten-Floer cohomology. Algebr. Geom. Topol., 24(1):493–554, 2024. doi:10.2140/agt.2024.24.493.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Natural cyclic actions on Brieskorn homology spheres have strong nonextension theorems over homology balls and many positive-definite fillings, but the requested single example excluding definite fillings of both signs remains open.\n\n**Verified partial progress.**\n\n- Anvari-Hambleton prove that free periodic actions on infinite families of Brieskorn homology spheres do not extend smoothly over contractible fillings.\n- Baraglia-Hekmati prove that free cyclic actions on Brieskorn homology spheres do not extend over homology balls.\n- For all but finitely many primes, Baraglia-Hekmati obtain analogous nonextension results over positive-definite fillings.\n- Equivariant Seiberg-Witten-Floer cohomology supplies new intersection-form and extension constraints.\n\n**Full solution or refutation.**\n\nThe literature checked supplies one-sign and restricted-filling obstructions but not the both-sign definite exclusion or the general indefinite non-spin methods in the statement.\n\n**What remains.**\n\nFind an integer homology sphere and cyclic action excluding every definite filling of either orientation, and extend the obstruction theory to indefinite non-spin fillings.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.75. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the three-part cyclic-action problem and explains why action extension is stricter than extension of a generator as a diffeomorphism.\n- Nima Anvari and Ian Hambleton, Cyclic group actions on contractible 4-manifolds, Geometry & Topology 20 (2016), 1127-1155. (primary): https://doi.org/10.2140/gt.2016.20.1127\n Evidence used: Proves nonextension of free periodic Brieskorn-sphere actions over contractible 4-manifolds.\n- David Baraglia and Pedram Hekmati, Brieskorn spheres, cyclic group actions and the Milnor conjecture, Journal of Topology 17 (2024). (primary): https://doi.org/10.1112/topo.12339\n Evidence used: Proves homology-ball nonextension and, for all but finitely many primes, related positive-definite nonextension results.\n\n**Review notes.** Positive-definite nonextension alone does not answer part (b), which quantifies over definite manifolds of both signs.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2874, "problem_number": "KP-3.76", "title": "Kirby Problem 3.76", "statement": "What is the structure of the equivariant homology cobordism groups?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.76.\n\nLiterature notes:\nThere are several equivariant homology cobordism groups one can construct. One version is the $\\mathbb{Z}/p\\mathbb{Z}$-equivariant cobordism group as appropriate\n\nhomology cobordism classes of pairs consisting of integer homology spheres with an orientation-preserving $\\mathbb{Z}/p\\mathbb{Z}$ action having nonempty fixed point set. One can also define more general groups that take into account a diffeomorphism on each homology sphere and have no restriction on the fixed-point set; this is done in [DHM23]. In various cases, one can produce interesting $\\mathbb{Z}^{\\infty}$ -subgroups. For instance, it follows from [DHM23] that there exists a $\\mathbb{Z}^{\\infty}$ -subgroup spanned by cork boundaries in the case p = 2; similar techniques can be attempted for other p. The structure of these groups is otherwise poorly understood. It is well-known that there is a homomorphism from the smooth concordance group to the rational homology cobordism group given by taking branched covers. There is likewise a homomorphism from the strongly invertible concordance group to the $\\mathbb{Z}/2\\mathbb{Z}$-equivariant rational homology cobordism group; see [AB24b]. There is also a homomorphism from the usual concordance group to the equivariant cobordism group, by taking branched covers but remembering the branching action, which may be used to produce more refined sliceness or concordance obstructions; see e.g. [BH24a].\n\nReferences cited:\n- [DHM23] Irving Dai, Matthew Hedden, and Abhishek Mallick. Corks, involutions, and Heegaard Floer homology. J. Eur. Math. Soc. (JEMS), 25(6):2319–2389, 2023. doi:10.4171/jems/1239.\n- [AB24b] Antonio Alfieri and Keegan Boyle. Strongly invertible knots, invariant surfaces, and the Atiyah-Singer signature theorem. Michigan Math. J., 74(4):845–861, 2024. doi:10.1307/mmj/20226183.\n- [BH24a] David Baraglia and Pedram Hekmati. Equivariant Seiberg-Witten-Floer cohomology. Algebr. Geom. Topol., 24(1):493–554, 2024. doi:10.2140/agt.2024.24.493.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Several equivariant homology-cobordism variants contain explicitly detected infinite-rank free-abelian subgroups, and a 2025 result gives a Z^infinity-summand in a forgetful-map kernel, but their overall algebraic structure remains poorly understood.\n\n**Verified partial progress.**\n\n- Dai-Hedden-Mallick define an involutive homology-cobordism refinement and prove it contains a Z^infinity-subgroup generated by strongly nonextendable corks.\n- Hendricks-Mallick-Stoffregen-Zemke prove that the kernel of the forgetful map from an equivariant homology-cobordism group to the ordinary group contains a Z^infinity-summand.\n- Branched-cover constructions define maps from ordinary and strongly invertible knot concordance into equivariant rational homology-cobordism groups.\n- Equivariant Floer invariants detect nontrivial classes and provide monotonicity constraints.\n\n**Full solution or refutation.**\n\nLarge subgroups and kernels are known, but there is no classification of torsion, quotients, relations, or the dependence on the chosen equivariant category.\n\n**What remains.**\n\nSpecify and determine the principal equivariant groups, including torsion, summands, forgetful-map kernels and cokernels, and relations between the fixed-point and diffeomorphism variants.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.76. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Defines several versions of the structure problem and records the infinite-rank and branched-cover framework.\n- Irving Dai, Matthew Hedden, and Abhishek Mallick, Corks, involutions, and Heegaard Floer homology, Journal of the European Mathematical Society 25 (2023), 2319-2389. (primary): https://doi.org/10.4171/jems/1239\n Evidence used: Defines the involutive refinement and proves the existence of a Z^infinity-subgroup generated by strong cork boundaries.\n- Kristen Hendricks, Abhishek Mallick, Matthew Stoffregen, and Ian Zemke, The link surgery formula and equivariant surgeries, arXiv:2507.12809 (2025). (primary): https://arxiv.org/abs/2507.12809\n Evidence used: Proves that a forgetful-map kernel contains a Z^infinity-summand.\n\n**Review notes.** The record does not specify a single group; conclusions depend on the action, fixed-point, and equivalence conventions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2875, "problem_number": "KP-3.77", "title": "Kirby Problem 3.77", "statement": "Does there exist a hyperbolic rational homology 3-sphere that is the totally geodesic boundary of a compact, orientable hyperbolic 4-manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.77.\n\nLiterature notes:\n(1) When a compact hyperbolic 3-manifold is the totally geodesic boundary of a compact hyperbolic 4-manifold, it is said to geometrically bound. Questions about geometric boundaries of hyperbolic manifolds of dimension greater than 4 are discussed in Problem 4.126(b).\n\n(2) A result of Ferrari--Kolpakov--Reid [FKR23] shows that there exist infinitely many arithmetic hyperbolic rational homology 3-spheres that geometrically bound hyperbolic 4-manifolds, but in their examples, all of the 4-manifolds are nonorientable. Prior to [FKR23], all constructed examples of hyperbolic 3-manifolds that geometrically bound had non-zero first Betti number.\n\n(3) Most constructions of hyperbolic 3-manifolds that geometrically bound (including the ones in [FKR23]) are arithmetic of simplest type and bound hyperbolic 4-manifolds cut out from ones that are arithmetic of simplest type. It would be interesting to construct non-arithmetic examples.\n\n(4) When a compact, orientable 3-manifold M geometrically bounds a compact, orientable, hyperbolic 4-manifold, this imposes restrictions on the $\\eta$-invariant of M, due to [LR00]. For example, the Weeks manifold does not geometrically bound a compact orientable hyperbolic 4-manifold.\n\nReferences cited:\n- [FKR23] L. Ferrari, A. Kolpakov, and A. W. Reid. Infinitely many arithmetic hyperbolic rational homology 3-spheres that bound geometrically. Trans. Amer. Math. Soc., 376(3):1979–1997, 2023. doi:10.1090/tran/8816.\n- [LR00] D. D. Long and A. W. Reid. On the geometric boundaries of hyperbolic 4-manifolds. Geom. Topol., 4:171–178, 2000. doi:10.2140/gt.2000.4.171.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Infinitely many arithmetic hyperbolic rational homology 3-spheres geometrically bound hyperbolic 4-manifolds, but the known bounding manifolds relevant here are nonorientable; no compact orientable example was located.\n\n**Verified partial progress.**\n\n- Ferrari-Kolpakov-Reid construct infinitely many arithmetic hyperbolic rational homology 3-spheres that geometrically bound.\n- Their examples cross the prior rational-homology-sphere barrier but use nonorientable bounding 4-manifolds.\n- Long-Reid derive eta-invariant restrictions for orientable geometric boundaries and show that the Weeks manifold cannot bound in the required orientable sense.\n\n**Full solution or refutation.**\n\nThe exact existence question remains open because orientability of the compact hyperbolic 4-manifold is essential and is absent from the known rational-homology-sphere construction.\n\n**What remains.**\n\nConstruct a rational homology 3-sphere as the totally geodesic boundary of a compact orientable hyperbolic 4-manifold, or prove an obstruction excluding all such manifolds.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.77. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the orientable question and explicitly identifies nonorientability as the gap in the known infinite family.\n- Leonardo Ferrari, Alexander Kolpakov, and Alan W. Reid, Infinitely many arithmetic hyperbolic rational homology 3-spheres that bound geometrically, Transactions of the AMS 376 (2023), 1979-1997. (primary): https://doi.org/10.1090/tran/8816\n Evidence used: Constructs the first infinite families of arithmetic hyperbolic rational homology 3-spheres that geometrically bound.\n- D. D. Long and A. W. Reid, On the geometric boundaries of hyperbolic 4-manifolds, Geometry & Topology 4 (2000), 171-178. (primary): https://doi.org/10.2140/gt.2000.4.171\n Evidence used: Establishes eta-invariant restrictions on orientable geometric boundaries.\n\n**Review notes.** Cusp sections of orientable noncompact hyperbolic 4-manifolds are not boundary components and do not answer this record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2876, "problem_number": "KP-3.78", "title": "Kirby Problem 3.78", "statement": "(a) Is there a non-semisimple 3-TQFT whose mapping class group representation is faithful or has an element in its kernel?\n\n(b) Define a 4-manifold invariant via non-semsimple categories that can distinguish 4-manifolds that are not distinguished by classical or gauge-theoretic invariants.\n\n(c) Is there is a full 3-TQFT extension of the representations, constructed by Bonahon and Wong, of the Kauffman bracket skein algebra? In particular, do non-semisimple TQFTs recover the Bonahon--Wong representations?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.78.\n\nLiterature notes:\n(1) In 1989, Atiyah [Ati88b] gave the definition of an n-dimensional topological quantum field theory (n-TQFT for short) that assigns objects in some category to n-manifolds and morphisms to cobordisms. Many 3-TQFTs have been constructed via semisimple modular categories; this problem is asking about applications of TQFTs constructed via non-semisimple categories that do not seem accessible to semisimple TQFTs.\n\n(2) Non-semisimple 3-TQFTs are relevant to the linearity problem for the mapping class group of a surface; see Problem 2.3 for a discussion of classical approaches. For example, the 3-TQFTs of [DRGG+22] lead to mapping class group representations with the property that the action of a Dehn twist has infinite order and thus is a candidate answer to part\n\n(a). By contrast, in the usual semisimple quantum mapping class group representations, all Dehn twists have finite order and the representations are not faithful.\n\n(3) The underlying 4-manifold invariants of 4-TQFTs coming from semisimple modular categories, such as [CKY97, CY93, BB18], are conjecturally determined by the Euler characteristic, the signature and the fundamental group, see [BB18, Conjecture 8.1]. This conjecture is verified in many cases in [Reu23]. It is unknown whether the 4-TQFTs of [CGHPM23] or the invariants of [BDR24, BDR23] go beyond such classical invariants. Note that in the 3-dimensional setting, there are invariants of closed 3-manifolds underlying a 3-TQFT distinguishes diffeomorphism types of homotopically equivalent lens spaces which were not distinguished by semisimple quantum invariants; see [CGPM14].\n\n(4) In [BW16c, BW16b, BW17b, BW19] Bonahon and Wong constructed a family of finite-dimensional representations of the Kauffman bracket skein algebra of surface. Their construction is based on the theory of quantum Teichmüller space of Chekhov and Fock [FC99] and Kashaev [Kas98]. Recently, Frohman, Kania-Bartoszynska and Le showed these representation have a unicity condition, see [FKBL19]. Partial results in the direction are known: certain BW-representations arise from the nonsemisimple TQFT of [BCGPM16], see [KK22]. If a TQFT extension of the link invariants of [BGPMR20] exists it should contain the general BW-representation\n\nReferences cited:\n- [Ati88b] Michael Atiyah. Topological quantum field theories. Inst. Hautes Études Sci. Publ. Math., (68):175–186, 1988. URL: http://www.numdam.org/item?id=PMIHES 1988 68 175 0.\n- [DRGG+22] Marco De Renzi, Azat M. Gainutdinov, Nathan Geer, Bertrand PatureauMirand, and Ingo Runkel. 3-Dimensional TQFTs from non-semisimple modular categories. Selecta Math. (N.S.), 28(2):Paper No. 42, 60, 2022. doi:10.1007/s00029-021-00737-z.\n- [CKY97] Louis Crane, Louis H. Kauffman, and David N. Yetter. State-sum invariants of 4-manifolds. J. Knot Theory Ramifications, 6(2):177–234, 1997. doi:10.1142/S0218216597000145.\n- [CY93] Louis Crane and David Yetter. A categorical construction of 4d topological quantum field theories. 3:120–130, 1993. URL: https://doi.org/10.1142/9789812796387 0005, doi:10.1142/9789812796387\\\\_0005.\n- [BB18] Manuel Bärenz and John Barrett. Dichromatic state sum models for four-manifolds from pivotal functors. Comm. Math. Phys., 360(2):663–714, 2018. doi:10.1007/s00220-017-3012-9.\n- [Reu23] David Reutter. Semisimple four-dimensional topological field theories cannot detect exotic smooth structure. J. Topol., 16(2):542–566, 2023. doi:10.1112/topo.12288.\n- [CGHPM23] Francesco Costantino, Nathan Geer, Benjamin Haı̈oun, and Bertrand PatureauMirand. Skein (3+ 1)-TQFTs from non-semisimple ribbon categories, 2023. arXiv: 2306.03225.\n- [BDR24] Anna Beliakova and Marco De Renzi. Kerler-Lyubashenko functors on 4-dimensional 2-handlebodies. Int. Math. Res. Not. IMRN, 2024(13):10005–10080, 2024. doi:10.1093/imrn/rnac039.\n- [BDR23] Anna Beliakova and Marco De Renzi. Refined Bobtcheva-Messia invariants of 4-dimensional 2-handlebodies. In Essays in geometry—dedicated to Norbert A’Campo, volume 34 of IRMA Lect. Math. Theor. Phys., pages 387–431. EMS Press, Berlin, [2023] ©2023.\n- [CGPM14] Francesco Costantino, Nathan Geer, and Bertrand Patureau-Mirand. Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories. J. Topol., 7(4):1005–1053, 2014. doi:10.1112/jtopol/jtu006.\n- [BW16c] Francis Bonahon and Helen Wong. The Witten-Reshetikhin-Turaev representation of the Kauffman bracket skein algebra. Proc. Amer. Math. Soc., 144(6):2711–2724, 2016. doi:10.1090/proc/12927.\n- [BW16b] Francis Bonahon and Helen Wong. Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations. Invent. Math., 204(1):195–243, 2016. doi:10.1007/s00222-015-0611-y.\n- [BW17b] Francis Bonahon and Helen Wong. Representations of the Kauffman bracket skein algebra II: Punctured surfaces. Algebr. Geom. Topol., 17(6):3399–3434, 2017. doi: 10.2140/agt.2017.17.3399.\n- [BW19] Francis Bonahon and Helen Wong. Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality. Quantum Topol., 10(2):325–398, 2019. doi:10.4171/QT/125.\n- [FC99] Vladimir V Fock and Leonid O Chekhov. A quantum Teichmüller space. Theoretical and Mathematical Physics, 120(3):1245–1259, 1999.\n- [Kas98] Rinat M Kashaev. Quantization of Teichmüller spaces and the quantum dilogarithm. Letters in Mathematical Physics, 43:105–115, 1998.\n- [FKBL19] Charles Frohman, Joanna Kania-Bartoszynska, and Thang Lê. Unicity for representations of the Kauffman bracket skein algebra. Invent. Math., 215(2):609–650, 2019. doi:10.1007/s00222-018-0833-x.\n- [BCGPM16] Christian Blanchet, Francesco Costantino, Nathan Geer, and Bertrand PatureauMirand. Non-semi-simple TQFTs, Reidemeister torsion and Kashaev’s invariants. Adv. Math., 301:1–78, 2016. doi:10.1016/j.aim.2016.06.003.\n- [KK22] Hiroaki Karuo and Julien Korinman. Azumaya loci of skein algebras, 2022. arXiv: 2211.13700.\n- [BGPMR20] Christian Blanchet, Nathan Geer, Bertrand Patureau-Mirand, and Nicolai Reshetikhin. Holonomy braidings, biquandles and quantum invariants of links with SL2pCq flat connections. Selecta Math. (N.S.), 26(2):Paper No. 19, 58, 2020. doi:10.1007/s00029-020-0545-0.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Non-semisimple TQFTs yield infinite-order Dehn-twist actions, new 4-manifold invariants, and partial recovery of skein-algebra representations, but faithfulness/kernel determination, genuinely stronger 4-manifold detection, and full Bonahon-Wong recovery remain open.\n\n**Verified partial progress.**\n\n- De Renzi and collaborators construct non-semisimple 3-TQFT mapping-class-group representations and prove that Dehn twists have infinite order in key examples.\n- Costantino-Geer-Haioun-Patureau-Mirand construct skein (3+1)-TQFTs from non-semisimple ribbon categories and associated 4-manifold invariants.\n- Karuo-Korinman connect non-semisimple TQFT representations with Kauffman-bracket skein algebras, extending partial Bonahon-Wong-type realizations.\n- Reutter proves semisimple 4-TQFTs cannot detect exotic smooth structures, sharpening the motivation for part (b).\n\n**Full solution or refutation.**\n\nThe cited constructions provide candidates and restricted realizations but no complete answer to any of the three broad requests.\n\n**What remains.**\n\nDetermine faithfulness or exhibit nontrivial kernels for concrete non-semisimple mapping-class representations, show genuinely new 4-manifold detection, and construct a full TQFT extension recovering all Bonahon-Wong representations.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 3.78. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Separates the three open directions and records the infinite-order, 4-TQFT, and partial Bonahon-Wong advances.\n- Marco De Renzi, Azat M. Gainutdinov, Nathan Geer, Bertrand Patureau-Mirand, and Ingo Runkel, Mapping class group representations from non-semisimple TQFTs, Selecta Mathematica 28 (2022), Article 42. (primary): https://doi.org/10.1007/s00029-021-00737-z\n Evidence used: Constructs and computes the representations and proves infinite-order Dehn-twist behavior.\n- Francesco Costantino, Nathan Geer, Benjamin Haioun, and Bertrand Patureau-Mirand, Skein (3+1)-TQFTs from non-semisimple ribbon categories, SIGMA 22 (2026), 034. (primary): https://doi.org/10.3842/SIGMA.2026.034\n Evidence used: Constructs the non-semisimple skein 3+1-TQFT and its 4-manifold invariant.\n- Hiroaki Karuo and Julien Korinman, Azumaya loci of skein algebras, arXiv:2211.13700v4 (2025). (primary): https://arxiv.org/abs/2211.13700\n Evidence used: Relates non-semisimple TQFT representations to Kauffman-bracket skein-algebra representation theory.\n\n**Review notes.** Part (a) is logically defective as written because every nonfaithful representation has a nonidentity kernel element. Parts (b) and (c) also contain typographical errors; the intended program is inferred from the background without rewriting the source.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2877, "problem_number": "KP-4.1", "title": "Kirby Problem 4.1", "statement": "(4-dimensional Poincaré conjecture). Is there a unique smooth structure on the 4-sphere?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.1.\n\nLiterature notes:\n(1) This appears as Problems 4.45 and 4.89 in [Kir97].\n\n(2) This is one of the remaining open cases of the smooth Poincaré Conjecture. In dimensions at most three, uniqueness of smooth structures follows from work of Munkres [Mun60] and Moise [Moi77a]. In higher dimensions, the only odd-dimensional spheres with unique smooth structures are $S^{5}$ and $S^{61}$ [WX17]; the situation for even-dimensional spheres is still open in general, but it is conjectured that the only even-dimensional spheres without exotic smooth structures in dimensions greater than 4 are $S^{6}, S^{12}$, and $S^{56}$ [WX17].\n\n(3) For context, we mention the history of the topological Poincaré conjecture: any manifold homotopy equivalent to $S^{n}$ is homeomorphic to $S^{n}$. This is classical for $n \\leq$ 2, follows from Perelman’s Geometrization Theorem for $n =$ 3 [Per02, Per03b, Per03a], from Freedman’s work for $n =$ 4 [Fre82], and from Newman’s work on topological engulfing for $n \\geq$ 5 [New66]. (When the manifold is smooth, the $n \\geq$ 5 case was first established as a consequence of Smale’s h-cobordism theorem [Sma62a].)\n\n(4) Some sources of potentially exotic homotopy 4-spheres are as follows.\n\n\\noindent$\\bullet$ Gluck twists on 2-knots in $S^{4}$. See Problem 4.9 for a discussion.\n\n\\noindent$\\bullet$ The Andrews–Curtis problem. See [Kir78, Problem 5.2] and Problem 5.10for a discussion.\n\n\\noindent$\\bullet$ Cappell–Shaneson homotopy spheres [CS76]. Many of their examples were proved to be standard in [Akb10] and [Gom10], but not all. More generally, one can consider the mapping torus of a 3manifold diffeomorphism, and do surgery on a section; this sometimes gives a homotopy 4-sphere. One such example was proposed in [Maz62], and was shown standard in [Zee65]. One can construct many such examples where the diffeomorphism is obtained from the identity by a point push; e.g., when the 3-manifold is made by gluing two knot complements.\n\n\\noindent$\\bullet$ Gabai’s big dot carving. See Problem 4.24.\n\n\\noindent$\\bullet$ Pairs of knots with the same 0-surgeries, such that exactly one knot is slice. See for example [Akb93], or [MP23].\n\n\\noindent$\\bullet$ Cyclic branched covers of certain 2-knots. In general, the double branched cover of a 2-knot in $S^{4}$ is a rational homology sphere that need not be simply connected, but in certain cases the cover can be arranged to be a homotopy 4-sphere. For example, the double branched cover of the roll-spun (−2,3,7)pretzel knot is a homotopy 4-sphere that is not known to be the standard 4-sphere (see [Miy23]).\n\n(5) We mention some known potential approaches to the problem.\n\n\\noindent$\\bullet$ If one wants to prove that the smooth structure on $S^{4}$ is unique, one can try to use geometric flows, such as the Ricci flow, or the mean curvature flow in $\\mathbb{R}^{5}$ [CMP15].\n\n\\noindent$\\bullet$ If one wants to put a nonstandard smooth structure on $S^{4}$ using pairs of knots with the same 0-surgeries, Rasmussen’s s-invariant [Ras10] could be used to prove one of the knots is not slice. See the remarks on Problem 1.60.\n\n\\noindent$\\bullet$ Let K be a topologically slice knot that is not smoothly slice, such as the Conway knot. If we could put a smooth structure on the complement of the interior of a tubular neighborhood of the topologically slice disk, then we could imitate the construction of an exotic $\\mathbb{R}^{4}$ using $S^{4}$ and get a nontrivial homotopy sphere.\n\n\\noindent$\\bullet$ Symplectic geometry gives a potential avenue to either proving or disproving the smooth 4-dimensional Poincaré conjecture. Given a homotopy 4-sphere X, one wants to either construct or obstruct the existence of an asymptotically standard symplectic structure on $X \\setminus$ \\{point\\}. See [Ger21] for a discussion.\n\n\\noindent$\\bullet$ Finally, it is possible to rephrase Problem 4.1 purely in terms of group theory, using trisections [AGK18].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Mun60] James Munkres. Obstructions to the smoothing of piecewise-differentiable homeomorphisms. Ann. of Math. (2), 72:521–554, 1960. doi:10.2307/1970228.\n- [Moi77a] Edwin E. Moise. Geometric topology in dimensions 2 and 3, volume Vol. 47 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977.\n- [WX17] Guozhen Wang and Zhouli Xu. The triviality of the 61-stem in the stable homotopy groups of spheres. Ann. of Math. (2), 186(2):501–580, 2017. doi:10.4007/annals.2017.186.2.3.\n- [Per02] Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications, 2002. arXiv:math/0211159.\n- [Per03b] Grisha Perelman. Ricci flow with surgery on three-manifolds, 2003. arXiv:math/0303109.\n- [Per03a] Grisha Perelman. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, 2003. arXiv:math/0307245.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [New66] M. H. A. Newman. The engulfing theorem for topological manifolds. Ann. of Math. (2), 84:555–571, 1966. doi:10.2307/1970460.\n- [Sma62a] S. Smale. On the structure of manifolds. Amer. J. Math., 84:387–399, 1962. doi: 10.2307/2372978.\n- [Kir78] Rob Kirby. Problems in low dimensional manifold theory. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 273–312. Amer. Math. Soc., Providence, R.I., 1978.\n- [CS76] Sylvain E. Cappell and Julius L. Shaneson. Some new four-manifolds. Ann. of Math. (2), 104(1):61–72, 1976. doi:10.2307/1971056.\n- [Akb10] Selman Akbulut. Cappell-Shaneson homotopy spheres are standard. Ann. of Math. (2), 171(3):2171–2175, 2010. doi:10.4007/annals.2010.171.2171.\n- [Gom10] Robert E. Gompf. More Cappell-Shaneson spheres are standard. Algebr. Geom. Topol., 10(3):1665–1681, 2010. doi:10.2140/agt.2010.10.1665.\n- [Maz62] Barry Mazur. Symmetric homology spheres. Illinois J. Math., 6:245–250, 1962.\n- [Zee65] E. C. Zeeman. Twisting spun knots. Trans. Amer. Math. Soc., 115:471–495, 1965. doi:10.2307/1994281.\n- [Akb93] S. Akbulut. Knots and exotic smooth structures on 4-manifolds. J. Knot Theory Ramifications, 2(1):1–10, 1993. doi:10.1142/S0218216593000027.\n- [MP23] Ciprian Manolescu and Lisa Piccirillo. From zero surgeries to candidates for exotic definite 4-manifolds. Journal of the London Mathematical Society, 108(5):2001– 2036, 2023. doi:10.1112/jlms.12800.\n- [Miy23] Jin Miyazawa. A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P2-knots, 2023. arXiv:2312.02041.\n- [CMP15] Tobias Holck Colding, William P. Minicozzi, II, and Erik Kjær Pedersen. Mean curvature flow. Bull. Amer. Math. Soc. (N.S.), 52(2):297–333, 2015. doi:10.1090/S0273-0979-2015-01468-0.\n- [Ras10] Jacob Rasmussen. Khovanov homology and the slice genus. Invent. Math., 182(2):419–447, 2010. doi:10.1007/s00222-010-0275-6.\n- [Ger21] Chris Gerig. No homotopy 4-sphere invariants using $\\mathrm{ECH}=\\mathrm{SWF}$. Algebr. Geom. Topol., 21(5):2543–2569, 2021. doi:10.2140/agt.2021.21.2543.\n- [AGK18] Aaron Abrams, David T. Gay, and Robion Kirby. Group trisections and smooth 4-manifolds. Geom. Topol., 22(3):1537–1545, 2018. doi:10.2140/gt.2018.22.1537.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The smooth four-dimensional Poincare conjecture remains open: no accepted exotic smooth structure on S^4 and no proof that every smooth homotopy 4-sphere is standard was located.\n\n**Verified partial progress.**\n\n- Freedman proves the topological four-dimensional Poincare conjecture, so every homotopy 4-sphere is homeomorphic to S^4.\n- Akbulut and Gompf prove broad families of Cappell-Shaneson candidate homotopy spheres are standard.\n- Many other candidate constructions and reformulations via Gluck twists, trisections, knot surgery, branched covers, and symplectic geometry remain under study.\n- Current 2026 sources continue to list smooth uniqueness of S^4 as open.\n\n**Full solution or refutation.**\n\nThe topological classification and elimination of many candidates do not decide smooth uniqueness.\n\n**What remains.**\n\nConstruct and distinguish an exotic smooth 4-sphere, or prove every smooth manifold homeomorphic to S^4 is diffeomorphic to the standard sphere.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 4.1. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States smooth uniqueness of S^4 as open and surveys current candidate sources and approaches.\n- Michael H. Freedman, The topology of four-dimensional manifolds, Journal of Differential Geometry 17 (1982), 357-453. (primary): https://projecteuclid.org/journals/journal-of-differential-geometry/volume-17/issue-3/The-topology-of-four-dimensional-manifolds/10.4310/jdg/1214437136.full\n Evidence used: Provides the topological four-dimensional Poincare theorem, leaving the smooth question.\n- Selman Akbulut, Cappell-Shaneson homotopy spheres are standard, Annals of Mathematics 171 (2010), 2171-2175. (primary): https://doi.org/10.4007/annals.2010.171.2171\n Evidence used: Proves standardness for the main Cappell-Shaneson candidate family.\n- Robert E. Gompf, More Cappell-Shaneson spheres are standard, Algebraic & Geometric Topology 10 (2010), 1665-1681. (primary): https://doi.org/10.2140/agt.2010.10.1665\n Evidence used: Further enlarges the class of standard candidate homotopy spheres.\n\n**Review notes.** Minor OCR defects occur in the background, but the one-sentence problem statement is unambiguous.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2878, "problem_number": "KP-4.2", "title": "Kirby Problem 4.2", "statement": "Does every smooth, closed 4-manifold admit an exotic smooth structure? Infinitely many?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.2.\n\nLiterature notes:\n(1) The question is open for many families of 4-manifolds. See also Problems 4.3 and 4.4.\n\n(2) A version of this problem in [Kir97, Problem 4.86] raises the same question for complex algebraic surfaces. Evident families in this class, still not known to admit exotic smooth structures, are the irrational ruled surfaces $(\\Sigma_{h} \\times S^{2})\\#_{m}\\mathbb{CP}^{2}$ and $(\\Sigma_{h} \\times S^{2})\\#_{m}\\mathbb{CP}^{2}$ for any h, $m \\geq$ 1.\n\n(3) To date, on virtually any smooth, closed 4-manifold known to admit an exotic smooth structure, infinitely many distinct exotic smooth structures have been constructed. (See [BSS24] for the related discussion.) Many of the known methods for constructing infinitely many smooth structures require the presence of certain embedded tori of self-intersection zero [FS98, FS11].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [BSS24] R. Inanc Baykur, Andras I. Stipsicz, and Zoltan Szabo. Smooth structures on fourmanifolds with finite cyclic fundamental groups, 2024. arXiv:2406.09007.\n- [FS98] Ronald Fintushel and Ronald J. Stern. Knots, links, and 4-manifolds. Invent. Math., 134(2):363–400, 1998. doi:10.1007/s002220050268.\n- [FS11] Ronald Fintushel and Ronald J. Stern. Pinwheels and nullhomologous surgery on 4-manifolds with $b^+=1$. Algebr. Geom. Topol., 11(3):1649–1699, 2011. doi:10.2140/agt.2011.11.1649.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many closed smooth 4-manifolds have exotic smoothings, but existence for every closed smooth 4-manifold, let alone infinitely many, remains open.\n\n**Verified partial progress.**\n\n- Numerous simply connected and non-simply connected families have known exotic smoothings.\n- The list identifies algebraic ruled-surface families still unresolved.\n\n**Full solution or refutation.**\n\nNo universal result was verified.\n\n**What remains.**\n\nResolve the missing families or give an obstruction to exotic smoothing.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.2 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the question remains open for many families.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2879, "problem_number": "KP-4.3", "title": "Kirby Problem 4.3", "statement": "Are there exotic smooth structures on the following closed, simply-connected 4–manifolds?\n\n(a) $\\#_{k}\\mathbb{CP}^{2}$ for any $k \\geq$ 1.\n\n(b) $\\#_{m}(\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2})$ and $\\#_{n}(S^{2} \\times S^{2})$ for $m \\leq$ 8, $n \\leq$ 10.\n\n(c) $\\#_{p}K3 \\#_{q}(S^{2} \\times S^{2})$ for any $p \\geq$ q+5. If so, what are the smallest k, m and n? The largest p and q?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.3.\n\nLiterature notes:\n(1) These are the main families of closed, simply connected 4–manifolds, besides $S^{4}$, not known to admit exotic smooth structures. Namely, those that are (a) definite, (b) $\\sigma =$ 0 with $e <$ 20, and (c) spin with $c^{2}_{1} = 2e+3\\sigma <$ −12, respectively, where e is the Euler characteristic and $\\sigma$ is the signature. In the non-spin case, advances in the geography of minimal symplectic 4–manifolds [ABB+10, AP10], combined with blow-ups, connected sums with nontrivial Bauer–Furuta invariants, and orientation-reversals, yield exotic smooth structures on most 4-manifolds of the for $m \\#_{a}\\mathbb{CP}^{2}\\#_{b}\\overline{\\mathbb{CP}}{}^{2} withab(a-b) \\ne$ 0 and a,b not both even. Here, the gaps essentially stem from our lack of understanding related to Problem 4.17. Less is known in the spin case, but the results in the geography of spin symplectic 4–manifolds [PS00a, Par02] similarly settle the existence of exotic smooth structures on most spin 4–manifolds with $c^{2}_{1} \\geq$ 0 and $\\sigma$ <0; i.e. on $\\#_{p}K3 \\#_{q}(S^{2} \\times S^{2})$ for 0 $< p \\leq$ q+1. There are still some gaps, such as $K3 \\#(S^{2} \\times S^{2})$, which are again due to our lack of understanding related to Problem 4.17. Some exotic examples also exist in the range q+2 $\\leq p \\leq$ q+4 (e.g. see Remark 5 below for q=0), but none are known for $p \\geq$ q+5 (i.e. $2e+3\\sigma<$ −12).\n\n(2) A potential source of exotic definite 4–manifolds arise from pairs of knots with the same 0–surgeries, where only one knot bounds a disk with a given self-intersection number in $\\#_{k}\\mathbb{CP}^{2} \\setminus D^{4}$. See [MP23], [Qin25].\n\n(3) There are infinitely many exotic smooth structures on $\\#_{m}(\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2})$ and $\\#_{n}S^{2} \\times S^{2}$ for any odd $m \\geq$ 9, $n \\geq$ 11 [BH23]. A possible strategy to extend these results to cover smaller m, n is through Lefschetz pencils [Bay22] and Fintushel–Stern reverse engineering [FPS07]. Here is another potential construction for exotic $S^{2} \\times S^{2}$ or $\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2}$, which could perhaps be detected using knot Floer homology: given a pair of knots in $S^{3}$ with the samen-surgery, by gluing their traces one obtains a manifold homeomorphic to either $S^{2} \\times S^{2}$ or $\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2}$ (depending on the parity of n). See [LLP23, Remark 6.13].\n\n(4) Exotic definite 4-manifolds with finite fundamental groups can be derived from indefinite exotic 4-manifolds admitting free finite group actions. Examples with fundamental group $\\mathbb{Z}/2$ have been announced in [LLP23, SS24c] as quotients of exotic $\\mathbb{CP}^{2}\\#_{m}\\mathbb{CP}^{2}$ under free involutions, and with more general fundamental groups by Baykur, Stipsicz and Szabó in [BSS24]. The smallest definite example here has $b_{2} =$ 1. It is plausible that exotic signature zero 4-manifolds with finite fundamental groups and smaller $b_{2}$ can be derived similarly.\n\n(5) The Bauer–Furuta invariants [BF04] are sometimes able to detect exotic smooth structures on connected sums. Bauer [Bau04] used these invariants to find exotic smooth structures on $\\#_{p}K3$ for $p \\leq$ 4, but the method stops working at p=5 and is not applicable if any summand has $b^{+}_{2} \\equiv1$ mod 4.\n\nReferences cited:\n- [ABB+10] Anar Akhmedov, Scott Baldridge, R. İnanç Baykur, Paul Kirk, and B. Doug Park. Simply connected minimal symplectic 4-manifolds with signature less than -1. J. Eur. Math. Soc. (JEMS), 12(1):133–161, 2010. doi:10.4171/JEMS/192.\n- [AP10] Anar Akhmedov and B. Doug Park. Exotic smooth structures on small 4-manifolds with odd signatures. Invent. Math., 181(3):577–603, 2010. doi:10.1007/s00222-010-0254-y.\n- [PS00a] B. Doug Park and Zoltán Szabó. The geography problem for irreducible spin four-manifolds. Trans. Amer. Math. Soc., 352(8):3639–3650, 2000. doi:10.1090/S0002-9947-00-02467-3.\n- [Par02] Jongil Park. The geography of Spin symplectic 4-manifolds. Math. Z., 240(2):405– 421, 2002. doi:10.1007/s002090100390.\n- [MP23] Ciprian Manolescu and Lisa Piccirillo. From zero surgeries to candidates for exotic definite 4-manifolds. Journal of the London Mathematical Society, 108(5):2001– 2036, 2023. doi:10.1112/jlms.12800.\n- [Qin25] Qianhe Qin. An RBG construction of integral surgery homeomorphisms. Algebr. Geom. Topol., 25(6):3755–3774, 2025. doi:10.2140/agt.2025.25.3755.\n- [BH23] R. Inanc Baykur and Noriyuki Hamada. Exotic 4-manifolds with signature zero, 2023. Selecta Math., to appear. arXiv:2305.10908.\n- [Bay22] R. İnanç Baykur. Small exotic 4-manifolds and symplectic Calabi-Yau surfaces via genus-3 pencils. In Gauge theory and low-dimensional topology—progress and interaction, volume 5 of Open Book Ser., pages 185–221. Math. Sci. Publ., Berkeley, CA, 2022. https://msp.org/obs/2022/5-1/p09.xhtml.\n- [FPS07] Ronald Fintushel, B. Doug Park, and Ronald J. Stern. Reverse engineering small 4-manifolds. Algebr. Geom. Topol., 7:2103–2116, 2007. doi:10.2140/agt.2007.7.2103.\n- [LLP23] Adam Simon Levine, Tye Lidman, and Lisa Piccirillo. New constructions and invariants of closed exotic 4-manifolds, 2023. arXiv:2307.08130.\n- [SS24c] András I. Stipsicz and Zoltán Szabó. Definite four-manifolds with exotic smooth structures. J. Reine Angew. Math.), 2024(817):267–290, 2024. URL: https://doi.org/10.1515/crelle-2024-0072, doi:doi:10.1515/crelle-2024-0072.\n- [BSS24] R. Inanc Baykur, Andras I. Stipsicz, and Zoltan Szabo. Smooth structures on fourmanifolds with finite cyclic fundamental groups, 2024. arXiv:2406.09007.\n- [BF04] Stefan Bauer and Mikio Furuta. A stable cohomotopy refinement of Seiberg-Witten invariants. I. Invent. Math., 155(1):1–19, 2004. doi:10.1007/s00222-003-0288-5.\n- [Bau04] Stefan Bauer. A stable cohomotopy refinement of Seiberg-Witten invariants. II. Invent. Math., 155(1):21–40, 2004. doi:10.1007/s00222-003-0289-4.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The requested definite and small connected-sum ranges remain open, although infinite exotic families are known just beyond some stated thresholds and exotic connected sums of at most four K3 surfaces are known.\n\n**Verified partial progress.**\n\n- Baykur--Hamada construct infinite exotic families in nearby larger odd ranges, including m at least 9 and n at least 11.\n- Bauer constructs exotic smooth structures on connected sums of p K3 surfaces for p at most 4; the method does not settle p=5.\n- Manolescu--Piccirillo identify candidates for exotic definite 4-manifolds, but the candidates are not proved exotic.\n\n**Full solution or refutation.**\n\nNo verified result found in the exact small ranges stated in the record; adjacent constructions show that the listed numerical boundaries are meaningful.\n\n**What remains.**\n\nConstruct exotic structures in the exact listed ranges, especially on simply connected positive-definite connected sums, and determine the sharp K3/stabilization thresholds.\n\n**Sources checked.**\n\n- Kirby et al., Problems in Low-Dimensional Topology, third-edition survey, Problem KP-4.3 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Retains each exact range as open and summarizes adjacent constructions.\n- R. İnanç Baykur and Noriyuki Hamada, Exotic 4-manifolds with signature zero, arXiv:2305.10908. (primary): https://arxiv.org/abs/2305.10908\n Evidence used: Constructs infinite exotic families in larger connected-sum ranges adjacent to those asked for.\n- Stefan Bauer, A stable cohomotopy refinement of Seiberg--Witten invariants: II, Inventiones Mathematicae 155 (2004). (primary): https://doi.org/10.1007/s00222-003-0289-4\n Evidence used: Supplies the connected-sum K3 result through four summands.\n- Ciprian Manolescu and Lisa Piccirillo, From zero surgeries to candidates for exotic definite four-manifolds, Journal of the London Mathematical Society 108 (2023). (primary): https://doi.org/10.1112/jlms.12800\n Evidence used: Produces candidates rather than a proved exotic positive-definite simply connected manifold.\n\n**Review notes.** Results with nontrivial fundamental group were not treated as answers to the simply connected definite subproblem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2880, "problem_number": "KP-4.4", "title": "Kirby Problem 4.4", "statement": "Is there an exotic smooth structure on some product 4-manifold $S^{1} \\times Y^{3}$ or $\\Sigma_{g} \\times \\Sigma_{h}$? Do they all admit exotic smooth structures?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.4.\n\nLiterature notes:\n(1) A homeomorphism classification is unavailable for many of these 4–manifolds, making it challenging to determine whether a potential exotic copy falls within the desired homeomorphism class.\n\n(2) Significant special cases, besides $S^{2} \\times S^{2}$, include $S^{1} \\times S^{3}, T^{2} \\times S^{2}$, and $S^{1} \\times Y^{3}$, for any $Y^{3}$ that is $a T^{2}–bundle$ over $S^{1}$, for example $Y^{3} =T^{3}$. The first three have good fundamental groups\\{1\\}, $\\mathbb{Z}$, and $\\mathbb{Z}^{2}$ respectively, and there are homeomorphism classifications of closed 4-manifolds with these fundamental groups [FQ90, HKT09]. The remaining manifolds $S^{1} \\times Y^{3}$ are aspherical infrasolvmanifolds with good fundamental groups, for which the Borel conjecture holds [Hil02]. Thus, for $Y^{3}$ as above, any homotopy equivalence of 4-manifolds $X^{4} \\to S^{1} \\times Y^{3}$ is homotopic to a homeomorphism.\n\n(3) For $S^{1} \\times S^{3}$, see Problems 4.22and 4.65about detecting potential exotic structures.\n\n(4) For $S^{2} \\times T^{2}$, see [Bay22, Theorem 6] for a possible strategy to generate exotic smooth structures via Lefschetz pencils and some candidates.\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [HKT09] Ian Hambleton, Matthias Kreck, and Peter Teichner. Topological 4-manifolds with geometrically two-dimensional fundamental groups. J. Topol. Anal., 1(2):123–151, 2009. doi:10.1142/$S^{1}$793525309000084.\n- [Hil02] J. A. Hillman. Four-manifolds, geometries and knots, volume 5 of Geometry \\& Topology Monographs. Geometry \\& Topology Publications, Coventry, 2002.\n- [Bay22] R. İnanç Baykur. Small exotic 4-manifolds and symplectic Calabi-Yau surfaces via genus-3 pencils. In Gauge theory and low-dimensional topology—progress and interaction, volume 5 of Open Book Ser., pages 185–221. Math. Sci. Publ., Berkeley, CA, 2022. https://msp.org/obs/2022/5-1/p09.xhtml.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No exotic smooth structure on an actual product S1 x Y3 or Sigma_g x Sigma_h was verified; candidate constructions and rigidity tools do not yet establish one.\n\n**Verified partial progress.**\n\n- Topological rigidity/homeomorphism classification is available for several infrasolvmanifold and good-group cases.\n- Baykur proposes genus-3 pencil candidates and a strategy for S2 x T2.\n\n**Full solution or refutation.**\n\nNeither the existence question for some product nor the universal assertion for all products has been settled.\n\n**What remains.**\n\nConstruct and verify a homeomorphic but nondiffeomorphic copy of one stated product, then determine the range of products supporting exotica.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Small exotic 4-manifolds and symplectic Calabi--Yau surfaces via genus-3 pencils, Open Book Series 5 (2022), 185--221. (primary): https://msp.org/obs/2022/5-1/p09.xhtml\n Evidence used: Provides candidates and a strategy around S2 x T2, not a verified exotic product.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.4. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current open formulation and homeomorphism-classification context.\n\n**Review notes.** Visible OCR includes '$a T^2$-bundle', '4.22and', and a corrupted DOI. Candidate manifolds are not promoted without verified homeomorphism and nondiffeomorphism.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2881, "problem_number": "KP-4.5", "title": "Kirby Problem 4.5", "statement": "Does every connected, open 4-manifold admit uncountably many smooth structures?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.5.\n\nLiterature notes:\n(1) Recall that a manifold is said to be open if it has empty boundary and every component is noncompact. Any such 4-manifold is smoothable by work of Quinn [Qui82, Corollary 2.2.3] (see also [FQ90, Theorem 8.2]). However it is unknown whether every open 4-manifold has more than one smooth structure.\n\n(2) Gompf showed that $M \\setminus$ \\{x\\} for $x \\in M$, with M an arbitrary topological 4-manifold (not necessarily compact or orientable), has uncountably many smooth structures [Gom93, Theorem 2.1].\n\n(3) Euclidean 4-space has uncountably many smooth structures [Tau87], which for m a monoid acting on the smooth structures on any noncompact 4-manifold. Part of the question is therefore whether there is a 4-manifold that somehow absorbs or unwinds the Euclidean structures.\n\n(4) Recall that smooth structures on a manifold may be considered either up to diffeomorphism or up to isotopy. Isotopy implies diffeomorphism. This question is asking about smooth structures up to diffeomorphism. One could also ask the related question whether every open 4-manifold admits a smooth structure with uncountably many isotopy classes. Non-isotopic smooth structures can be produced by pulling back a smooth structure along a non-smoothable self-homeomorphism. See Problem 4.83.\n\nReferences cited:\n- [Qui82] Frank Quinn. Ends of maps. III. Dimensions 4 and 5. J. Differential Geometry, 17(3):503–521, 1982. http://projecteuclid.org/euclid.jdg/1214437139.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [Gom93] Robert E. Gompf. An exotic menagerie. J. Differential Geom., 37(1):199–223, 1993. http://projecteuclid.org/euclid.jdg/1214453429.\n- [Tau87] Clifford Henry Taubes. Gauge theory on asymptotically periodic 4-manifolds. J. Differential Geom., 25(3):363–430, 1987. http://projecteuclid.org/euclid.jdg/1214440981.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every open topological 4-manifold is smoothable and punctured topological 4-manifolds can have uncountably many smoothings, but this is not known for every connected open 4-manifold.\n\n**Verified partial progress.**\n\n- Quinn gives smoothability.\n- Gompf gives uncountably many smoothings for punctured manifolds.\n\n**Full solution or refutation.**\n\nThe universal uncountability assertion remains open.\n\n**What remains.**\n\nExtend end-sum/radial-family constructions to every open manifold or find a rigid example.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.5 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States both positive results and the gap.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2882, "problem_number": "KP-4.6", "title": "Kirby Problem 4.6", "statement": "Does every closed, orientable 3-manifold bound an absolutely exotic pair of smooth, orientable 4-manifolds?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.6.\n\nLiterature notes:\n(1) Yasui [Yas12], Etnyre–Min–Mukherjee [EMM22], and Iida-MukherjeeTaniguchi [IMT25] gave sufficient conditions on a closed, connected, orientable 3-manifold Y to bound a compact 4-manifold X with $\\partial X = Y$ such that X admits infinitely many pairwise non-diffeomorphic smooth structures. Suppose that at least one of the following conditions are satisfied:\n\n\\noindent$\\bullet$ Y carries a contact structure $\\xi$ such that its Heegaard Floer contact invariant does not vanish e.g. if Y is Seifert fibered;\n\n\\noindent$\\bullet$ (Y, $\\xi)$ has a weak symplectic filling, e.g. if Y has a taut foliation; in particular, if Y is irreducible with $b_{1}(Y)$ >0;\n\n\\noindent$\\bullet$ Y is a rational homology sphere that embeds as a separating hypersurface in a closed definite 4-manifold. Then there exists a manifold X as above. In some of the cases, one gets smooth structures that are absolutely exotic, i.e. cannot be related by any diffeomorphism. In other cases, it is only known that they cannot be related by a diffeomorphism restricting to the identity on the boundary.\n\n(2) By arguments similar to those in [Kre84a], one can show that every closed, orientable 3-manifold bounds an exotic pair of nonorientable 4manifolds, so the second “orientable” in the question is necessary for an open question. A related open question is as follows.\n\n\\paragraph{Question.} Does every closed, nonorientable 3-manifold bound an absolutely exotic pair of smooth 4-manifolds?\n\nReferences cited:\n- [Yas12] Kouichi Yasui. Nuclei and exotic 4-manifolds, 2012. arXiv:1111.0620.\n- [EMM22] John B. Etnyre, Hyunki Min, and Anubhav Mukherjee. On 3-manifolds that are boundaries of exotic 4-manifolds. Trans. Amer. Math. Soc., 375(6):4307–4332, 2022. doi:10.1090/tran/8586.\n- [IMT25] Nobuo Iida, Anubhav Mukherjee, and Masaki Taniguchi. An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology. Adv. Math., 466:Paper No. 110134, 38, 2025. doi:10.1016/j.aim.2025.110134.\n- [Kre84a] M. Kreck. Some closed 4-manifolds with exotic differentiable structure. In Algebraic topology, Aarhus 1982 (Aarhus, 1982), volume 1051 of Lecture Notes in Math., pages 246–262. Springer, Berlin, 1984. doi:10.1007/BFb0075570.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Broad classes of oriented 3-manifolds bound 4-manifolds with infinitely many exotic smooth structures, sometimes absolutely exotic, but no theorem covers every closed oriented 3-manifold with orientable absolute exotic fillings.\n\n**Verified partial progress.**\n\n- Etnyre--Min--Mukherjee cover weakly fillable contact manifolds and manifolds with nonvanishing Heegaard Floer contact invariant, producing infinitely many exotic caps.\n- Yasui supplies related exotic filling constructions using nuclei and cork techniques.\n- Iida--Mukherjee--Taniguchi use Bauer--Furuta-type adjunction inequalities to cover additional rational homology spheres embedded as separating hypersurfaces in definite 4-manifolds.\n\n**Full solution or refutation.**\n\nThe sufficient conditions encompass many Seifert, taut-foliated, and definite-embedding examples, but not all closed oriented 3-manifolds and not always the absolute version.\n\n**What remains.**\n\nRemove the geometric hypotheses and ensure that the two orientable fillings are not related by any boundary-unrestricted diffeomorphism.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 4.6 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Collects the current sufficient conditions and distinguishes absolute from boundary-relative exoticness.\n- John B. Etnyre, Hyunki Min, and Anubhav Mukherjee, On 3-manifolds that are boundaries of exotic 4-manifolds, Transactions of the AMS 375 (2022), 4307--4332. (primary): https://doi.org/10.1090/tran/8586\n Evidence used: Proves exotic-cap results for weakly fillable and nonvanishing-contact-invariant classes.\n- Nobuo Iida, Anubhav Mukherjee, and Masaki Taniguchi, An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology, Advances in Mathematics 466 (2025), 110134. (primary): https://doi.org/10.1016/j.aim.2025.110134\n Evidence used: Provides additional sufficient conditions via Bauer--Furuta-type invariants.\n\n**Review notes.** Absolute exoticness is kept distinct from nondiffeomorphism relative to the identity on the boundary.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2883, "problem_number": "KP-4.7", "title": "Kirby Problem 4.7", "statement": "(a) If $M_{1},M_{2}are$ two homeomorphic closed, oriented 4-manifolds, is $M_{1}\\#S^{2} \\times S^{2}$ diffeomorphic to $M_{2}\\#S^{2} \\times S^{2}$?\n\n(b) Is there a fixedn>0such that $M_{1}\\#_{n}S^{2} \\times S^{2}$ is diffeomorphic to $M_{2}\\#_{n}S^{2} \\times S^{2}$ for every pair $M_{1}, M_{2}$ as above?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.7.\n\nLiterature notes:\n(1) By Gompf’s generalization of Wall’s theorem [Wal64b, Gom84], for any given pair $M_{1}, M_{2}$ as above, there exists $k \\geq$ 0 such that $M_{1}\\#_{k}S^{2} \\times S^{2}$ is diffeomorphic to $M_{2}\\#_{k}S^{2} \\times S^{2}. A$ priorik depends on $M_{1}$ and $M_{2}$, but in most examples, $k =$ 1 is known to suffice; see [BS13] and the references therein.\n\n(2) In the case of nonempty boundary, it has been announced by S. Kang [Kan22b] that one stabilization is not enough. Kang proposes an example of two homeomorphic smooth contractible 4-manifolds $M_{1}, M_{2}$ with diffeomorphic boundaries, such that $M_{1}\\#S^{2} \\times S^{2}$ and $M_{2}\\#S^{2} \\times S^{2}$ are not diffeomorphic. The obstruction to existence of a diffeomorphism between $M_{1}\\#S^{2} \\times S^{2}$ and $M_{2}\\#S^{2} \\times S^{2}$ comes from involutive Heegaard Floer homology.\n\n(3) In the closed case, detecting an exotic pair after stabilization is difficult because most known invariants vanish for $M\\#S^{2} \\times S^{2}$. The only known exceptions are the Pin(2)-equivariant Bauer–Furuta invariant, and Fintushel–Stern’s 2-torsion instanton invariants, which can survive up to two stabilizations. To date, neither have been shown to differ for double stabilizations of homeomorphic 4-manifolds.\n\n(4) One idea for the closed case is to look at exotic surfaces in 4-manifolds that stay exotic after a weak internal stabilization, and take their double branched covers. See Problem 4.33.\n\n(5) A related but easier question is to find an exotic pair of closed simply connected 4-manifolds $X_{1}, X_{2}$ such that both manifolds contain a homologically essential square 0 sphere, and there is a homeomorphism between $X_{1}$ and $X_{2}$ (or an isomorphism between their intersection forms) that takes the homology class of one square 0 sphere to the other. The complements of the square 0 sphere are not required to be simply connected. In this setting, $X_{1}$ and $X_{2}$ would be hard to distinguish for the same reason as above: due to the presence of a square 0 sphere, most known invariants vanish.\n\n(6) A positive answer to (b) would follow from existence of a universal cork; see Problem 4.14.\n\n(7) One can ask the same questions about stabilization by the twisted $S^{2}bundle$ over $S^{2}$. Kang’s examples become diffeomorphic after one twisted stabilization [HKM23]. Twisted stabilizations are not spin, so it is harder to find invariants to distinguish them: the maps on involutive Heegaard Floer homology and the Pin(2)-equivariant Bauer–Furuta invariants do not work.\n\nReferences cited:\n- [Wal64b] C. T. C. Wall. On simply-connected 4-manifolds. J. London Math. Soc., 39:141–149, 1964. doi:10.1112/jlms/s1-39.1.141.\n- [Gom84] Robert E. Gompf. Stable diffeomorphism of compact 4-manifolds. Topology Appl., 18(2-3):115–120, 1984.\n- [BS13] R. İnanç Baykur and Nathan Sunukjian. Round handles, logarithmic transforms and smooth 4-manifolds. J. Topol., 6(1):49–63, 2013.\n- [Kan22b] Sungkyung Kang. One stabilization is not enough for contractible 4-manifolds, 2022. arXiv:2210.07510.\n- [HKM23] Kyle Hayden, Sungkyung Kang, and Anubhav Mukherjee. One stabilization is not enough for closed knotted surfaces, 2023. arXiv:2304.01504.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Stable diffeomorphism after some number of S2xS2 summands is known pairwise, and one summand works in many examples, but no uniform one-summand or universal fixed-n theorem was verified.\n\n**Verified partial progress.**\n\n- Gompf's generalization of Wall gives a pair-dependent stabilization count.\n- Many known examples stabilize after one summand.\n\n**Full solution or refutation.**\n\nBoth uniformity questions remain open.\n\n**What remains.**\n\nBound stabilization numbers uniformly or exhibit growing necessity.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.7 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the pair-dependent theorem and known k=1 cases.\n\n**Review notes.** Source has missing primes/spaces; not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2884, "problem_number": "KP-4.8", "title": "Kirby Problem 4.8", "statement": "Let X be a closed, simply connected, smooth 4-manifold, and T a smoothly embedded torus in X with $\\pi_{1}(X$ −T) =1 and $[T]^{2}$ =0. Let $X_{K}$ be the result of Fintushel–Stern knot surgery on X along a knot K. If $K_{1}$ and $K_{2}$ are two prime knots such that $X_{K,1}$ and $X_{K,2}$ are diffeomorphic, does it follow that $K_{1}$ is either $K_{2}$ or its mirror?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.8.\n\nLiterature notes:\n(1) Fintushel and Stern showed that $X_{K}$ is homeomorphic to X, and $SW(X_{K}) = SW(X) \\cdot\\Delta_{K}$, where SW denotes the Seiberg-Witten series and $\\Delta_{K}$ the Alexander polynomial.\n\n(2) One can ask the question for the particular case when X is the K3 surface. In that case $SW(X) =$ 1, so the two knots need to have the same Alexander polynomial. No other constraint is known. For $X = K3$, the question in the problem was raised by Fintushel and Stern.\n\n(3) Akbulut [Akb02] showed that, if $m(K)$ denotes the mirror of K, then $X_{K}$ is diffeomorphic to $X_{m,(,K,)}$. Using work of Finashin [Fin02], Akaho [Aka06] showed that $X_{K,\\#,K}$ is diffeomorphic to $X_{K,\\#,m,(,K,)}$.\n\n(4) If one allows $\\pi_{1}(X) \\ne$ 1, then there are many examples of $X_{K,1}$ and $X_{K,2}$ that have the same Seiberg-Witten invariants (since $\\Delta_{K,1} =\\Delta_{K,2})$ but are not diffeomorphic, being distinguished by the Seiberg-Witten invariants of their finite covers [FS99a, PY15] or by invariants from Heegaard Floer theory [LLP23].\n\nReferences cited:\n- [Akb02] Selman Akbulut. Variations on Fintushel-Stern knot surgery on 4-manifolds. Turkish J. Math., 26(1):81–92, 2002.\n- [Fin02] Sergey Finashin. Knotting of algebraic curves in $\\mathbb{CP}^{2}$. Topology, 41(1):47–55, 2002. doi:10.1016/S0040-9383(00)00023-9.\n- [Aka06] Manabu Akaho. A connected sum of knots and Fintushel-Stern knot surgery on 4-manifolds. Turkish J. Math., 30(1):87–93, 2006.\n- [FS99a] Ronald Fintushel and Ronald J. Stern. Nondiffeomorphic symplectic 4-manifolds with the same Seiberg-Witten invariants. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 103–111. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.103.\n- [PY15] Jongil Park and Ki-Heon Yun. Families of nondiffeomorphic 4-manifolds with the same Seiberg-Witten invariants. J. Symplectic Geom., 13(2):279–303, 2015. doi: 10.4310/JSG.2015.v13.n2.a2.\n- [LLP23] Adam Simon Levine, Tye Lidman, and Lisa Piccirillo. New constructions and invariants of closed exotic 4-manifolds, 2023. arXiv:2307.08130.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Seiberg--Witten invariants recover the Alexander polynomial in suitable Fintushel--Stern surgery settings, but do not identify a prime knot up to mirror in general.\n\n**Verified partial progress.**\n\n- Fintushel--Stern provide homeomorphism and SW polynomial control.\n\n**Full solution or refutation.**\n\nThe proposed knot-recognition implication remains open.\n\n**What remains.**\n\nFind finer surgery invariants distinguishing prime knots sharing Alexander data.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.8 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the SW evidence and asks the stronger identification question.\n\n**Review notes.** Source has malformed complement/subscript notation; not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2885, "problem_number": "KP-4.9", "title": "Kirby Problem 4.9", "statement": "Is every Gluck twist in $S^{4}$ standard?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.9.\n\nLiterature notes:\n(1) This is [Kir78, Problem 4.24]. Given an embedded 2-sphere in $S^{4}$, Gluck defined a surgery operation that yields a homotopy 4-sphere [Glu62], which is in fact homeomorphic to $S^{4}$ due to Freedman [Fre82]. However, it is unclear whether the resulting manifold is diffeomorphic to $S^{4}$.\n\n(2) The following 2-knots are known to have trivial Gluck twists i.e. the result of Gluck surgery on the 2-knot yields $S^{4}$.\n\n\\noindent$\\bullet$ Any spun knot or more generally any ribbon 2-knots [Glu62].\n\n\\noindent$\\bullet$ Any 2-knot 0-concordant to a 2-knot with trivial Gluck twist. More generally, if two 2-knots are 0-concordant, then the Gluck twist of those two 2-knots are diffeomorphic [Mel77, HMY00].\n\n\\noindent$\\bullet$ Any twist spun knot [Gor76] or branched twist spun knot [Pao78].\n\n\\noindent$\\bullet$ Any 2-knot that can be unknotted by a single finger move followed by a single Whitney move, which includes any roll spin of a classical knot of unknotting number one [NS22].\n\n\\noindent$\\bullet$ Any even degree satellite 2-knot whose pattern has trivial Gluck twist, or any satellite 2-knot (of any degree) in which both the pattern and the companion 2-knots have trivial Gluck twist [Kim20].\n\n\\noindent$\\bullet$ Any 2-knot that can be decomposed as a union of two ribbon disks, one of which has undisking number one [GNS25].\n\n\\noindent$\\bullet$ Some other examples related to the Cappell-Shaneson homotopy 4sphere (see [AK79a, Gom91a]) and any 2-knot obtained from gluin g particular ribbon disks satisfying certain conditions (see [NS12]).\n\n(3) A unit sphere in $\\mathbb{CP}^{2}$ is a smoothly embedded sphere that intersects the standard $\\mathbb{CP}^{1}$ transversely once. If a unit sphere is obtained from a 2-knot in $S^{4}$ by blowing up at a point, then we say a unit sphere is obtained from the 2-knot.\n\n\\paragraph{Question.} Is every unit sphere in $\\mathbb{CP}^{2}$ smoothly isotopic to the standard $\\mathbb{CP}^{1}$? Melvin [Mel77] showed that the Gluck twist of a 2-knot K in $S^{4}$ is trivial if and only if the pair $(\\mathbb{CP}^{2},\\Sigma)$, where $\\Sigma$ is the unit sphere obtained from K, is pairwise diffeomorphic to $(\\mathbb{CP}^{2},\\mathbb{CP}^{1})$. It is unknown whether such a pairwise diffeomorphism would imply a smooth isotopy from $\\Sigma$ to $\\mathbb{CP}^{1}$. Some unit spheres are known to be smoothly isotopic to $\\mathbb{CP}^{1}$ [HKM20].\n\nReferences cited:\n- [Kir78] Rob Kirby. Problems in low dimensional manifold theory. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 273–312. Amer. Math. Soc., Providence, R.I., 1978.\n- [Glu62] Herman Gluck. The embedding of two-spheres in the four-sphere. Trans. Amer. Math. Soc., 104:308–333, 1962. doi:10.2307/1993581.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [Mel77] Paul Melvin. Blowing up and down in 4-manifolds. PhD thesis, UC Berkeley, 1977.\n- [HMY00] Kazuo Habiro, Yoshihiko Marumoto, and Yuichi Yamada. Gluck surgery and framed links in 4-manifolds. In Knots In Hellas’ 98, pages 80–93. World Scientific, 2000.\n- [Gor76] C. McA. Gordon. Knots in the 4-sphere. Comment. Math. Helv., 51(4):585–596, 1976. doi:10.1007/BF02568175.\n- [Pao78] Peter Sie Pao. Nonlinear circle actions on the 4-sphere and twisting spun knots. Topology, 17(3):291–296, 1978. doi:10.1016/0040-9383(78)90033-2.\n- [NS22] Patrick Naylor and Hannah R. Schwartz. Gluck twisting roll spun knots. Algebr. Geom. Topol., 22(2):973–990, 2022. doi:10.2140/agt.2022.22.973.\n- [Kim20] Seungwon Kim. Gluck twist and unknotting of satellite 2-knots, 2020. arXiv:2009.07353.\n- [GNS25] David Gabai, Patrick Naylor, and Hannah Schwartz. Doubles of Gluck twists: a five-dimensional approach. Adv. Math., 480:Paper No. 110455, 29, 2025. doi:10.1016/j.aim.2025.110455.\n- [AK79a] Selman Akbulut and Robion Kirby. An exotic involution of $S^{4}$. Topology, 18(1):75– 81, 1979. doi:10.1016/0040-9383(79)90015-6.\n- [Gom91a] Robert E. Gompf. Killing the Akbulut-Kirby 4-sphere, with relevance to the Andrews-Curtis and Schoenflies problems. Topology, 30(1):97–115, 1991. doi: 10.1016/0040-9383(91)90036-4.\n- [NS12] Daniel Nash and András I. Stipsicz. Gluck twist on a certain family of 2-knots. Michigan Math. J., 61(4):703–713, 2012. doi:10.1307/mmj/1353098509.\n- [HKM20] Mark C. Hughes, Seungwon Kim, and Maggie Miller. Isotopies of surfaces in 4-manifolds via banded unlink diagrams. Geom. Topol., 24(3):1519–1569, 2020. doi: 10.2140/gt.2020.24.1519.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many Gluck twists in S4 are known standard under useful hypotheses, but no theorem that every Gluck twist is standard was verified.\n\n**Verified partial progress.**\n\n- Gluck surgery always yields a homotopy 4-sphere, hence a topological S4.\n- Numerous special knots/spheres have standard twists.\n\n**Full solution or refutation.**\n\nThe universal standardness question remains open.\n\n**What remains.**\n\nExtend standardness criteria or find an exotic Gluck-twist sphere.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.9 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the long-standing general question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2886, "problem_number": "KP-4.10", "title": "Kirby Problem 4.10", "statement": "(a) Is every homotopy $B^{4}$ with boundary $S^{3}$ obtained by performing a Gluck twist on some knotted 2-sphere in $B^{4}$?\n\n(b) Suppose a homotopy 4-ball X is obtained by performing a Gluck twist on a knotted 2-sphere in $B^{4}$. Is $X \\times I$ necessarily diffeomorphic to $B^{5}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.10.\n\nLiterature notes:\n(1) Habiro, Marumoto and Yamada [HMY00] showed that if a homotopy 4-ball is obtained by performing Gluck twists on multiple 2-spheres in $B^{4}$, then it can be obtained by a Gluck construction on a single knotted 2-sphere in $B^{4}$; see [Mel77] for related ideas.\n\n(2) Part (a) of the problem could also be asked for homotopy spheres; the reason for specifying a homotopy 4-ball is that part (b)is a potential route to finding a counterexample to the Schoenflies Conjecture (Problem 4.23) as well as to the 4-dimensional Poincaré Conjecture (Problem 4.1). Part\n\n(b) is a weaker question than asking whether X is diffeomorphic to $B^{4}$ (Problem 4.9).\n\nReferences cited:\n- [HMY00] Kazuo Habiro, Yoshihiko Marumoto, and Yuichi Yamada. Gluck surgery and framed links in 4-manifolds. In Knots In Hellas’ 98, pages 80–93. World Scientific, 2000.\n- [Mel77] Paul Melvin. Blowing up and down in 4-manifolds. PhD thesis, UC Berkeley, 1977.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Multiple Gluck twists can be consolidated into one, and the five-dimensional stabilization conclusion is proved for a substantial class of Gluck spheres with controlled ribbon hemispheres, but neither universal assertion is known.\n\n**Verified partial progress.**\n\n- Habiro-Marumoto-Yamada prove that a homotopy 4-ball obtained by multiple Gluck twists is obtainable by a single Gluck twist.\n- Gabai-Naylor-Schwartz prove standardness of the relevant five-dimensional double for spheres decomposing into two ribbon disks, one of undisking number one.\n- Their theorem applies to many Gluck twists not known to be standard as 4-manifolds and yields new Schoenflies balls not known to be standard.\n\n**Full solution or refutation.**\n\nThe consolidation theorem assumes a Gluck presentation and therefore does not solve part (a); the five-dimensional theorem covers a broad but nonuniversal class in part (b).\n\n**What remains.**\n\nShow every homotopy B^4 with boundary S^3 admits a one-sphere Gluck presentation, and prove or refute X times I diffeomorphic to B^5 for every such Gluck-twist ball.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 4.10. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States both universal questions and the multiple-to-single Gluck-twist reduction.\n- Kazuo Habiro, Yoshihiko Marumoto, and Yuichi Yamada, Gluck surgery and framed links in 4-manifolds, in Knots in Hellas '98 (2000), 80-93. (primary): https://doi.org/10.1142/9789812792679_0007\n Evidence used: Proves the cited consolidation of finitely many Gluck twists into one.\n- David Gabai, Patrick Naylor, and Hannah Schwartz, Doubles of Gluck twists: A five-dimensional approach, Advances in Mathematics 480 (2025), 110455. (primary): https://doi.org/10.1016/j.aim.2025.110455\n Evidence used: Proves the five-dimensional standardness result for a broad ribbon-hemisphere class and constructs new candidate Schoenflies balls.\n\n**Review notes.** The exact relation between the punctured homotopy-sphere formulation in the 2025 paper and the ball notation in part (b) should be checked by a 4-manifold specialist before final publication.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2887, "problem_number": "KP-4.11", "title": "Kirby Problem 4.11", "statement": "Let M be a smooth 4-manifold and letf: $S^{2} \\to M$ be a smooth embedding with trivial normal bundle. Then let $M_{f}$ denote the result of Gluck twisting on M along f.\n\n(a) Does there exist an orientable M, and an embedding f, such that M and $M_{f}$ are homeomorphic but not diffeomorphic?\n\n(b) Does there exist an orientable M, and smooth, homotopic embeddings f, $g: S^{2} \\to M$ with trivial normal bundle such that $M_{f}$ and $M_{g}$ are homeomorphic but not diffeomorphic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.11.\n\nLiterature notes:\n(1) The version of this question with $M =S^{4}$ has received considerable interest (see Problems 4.9 and4.10). It is possible that the general case might be attacked more readily and/or provide insight to the $S^{4}$ case.\n\n(2) Akbulut showed that Gluck twisting along a (homotopically essential) sphere in anonorientable 4-manifold can produce an exotic smoothing [Akb88] (see also [Tor17]).\n\n(3) Akbulut and Yasui give a simple condition on a 4-manifold and an embedded sphere f to ensure that Gluck twisting along f does not change the diffeomorphism type in [AY13]. Note that Gluck twisting may change the homeomorphism type, e.g. Gluck twisting $S^{2} \\times$ pt inside $S^{2} \\times S^{2}$ yields $S^{2} \\widetilde{\\times} S^{2}$. See [GS99, Exercise 5.2.7(a)].\n\n(4) In [KPR23, Theorem 1.2], it was shown that $M_{f}$ and $M_{g}$ are simple homotopy equivalent when f and g are homotopic. Under the stronger hypothesis thatf and g are $concordant,M_{f}$ and $M_{g}$ ares-cobordant, and therefore homeomorphic when $\\pi_{1}(M)is a$ good group (see Problem 4.46). Thus in the oriented case, finding examples where $M_{f}$ and $M_{g}$ are an exotic pair might be easier than finding examples where M and $M_{f}$ are. See [KPR23] for further discussion.\n\nReferences cited:\n- [Akb88] Selman Akbulut. Constructing a fake 4-manifold by Gluck construction to a standard 4-manifold. Topology, 27(2):239–243, 1988. doi:10.1016/0040-9383(88) 90041-9.\n- [Tor17] Rafael Torres. Smooth structures on nonorientable four-manifolds and free involutions. J. Knot Theory Ramifications, 26(13):1750085, 20, 2017. doi:10.1142/S0218216517500857.\n- [AY13] Selman Akbulut and Kouichi Yasui. Gluck twisting 4-manifolds with odd intersection form. Math. Res. Lett., 20(2):385–389, 2013. doi:10.4310/MRL.2013.v20.n2.a13.\n- [GS99] Robert E. Gompf and András I. Stipsicz. 4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999. doi:10.1090/gsm/020.\n- [KPR23] Daniel Kasprowski, Mark Powell, and Arunima Ray. Gluck twists on concordant or homotopic spheres, 2023. doi:10.4310/mrl.2023.v30.n6.a6.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Gluck twisting can produce an exotic smoothing in a nonorientable setting and is diffeomorphically trivial under useful conditions, but the stated orientable exotic/homeomorphic questions remain open.\n\n**Verified partial progress.**\n\n- Akbulut gives a nonorientable exotic-smoothing example.\n- Akbulut--Yasui give sufficient conditions for triviality.\n\n**Full solution or refutation.**\n\nNo orientable example satisfying the requested alternatives was verified.\n\n**What remains.**\n\nConstruct an orientable exotic Gluck twist or prove its impossibility.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 4.11 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes nonorientable result and open orientable questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2888, "problem_number": "KP-4.12", "title": "Kirby Problem 4.12", "statement": "For X a closed simply connected smooth 4-manifold, let $g_{X}: H_{2}(X) \\to \\mathbb{N}$ denote the genus function, which assigns to every homology class the minimal genus of a smooth embedded surface representing that class.\n\n(a) Suppose $f: X_{1} \\to X_{2}$ is a homeomorphism between simply connected, smooth, closed 4-manifolds. Let $f_{*}: H_{2}(X_{1}) \\to H_{2}(X_{2})$ be the induced map on homology. If for all x in $H_{2}(X_{1})$ we have $g_{X,1}(x) = g_{X,2}(f_{*}(x))$, is $X_{1}$ diffeomorphic to $X_{2}$?\n\n(b) Suppose $f: X_{1} \\to X_{2}$ is a homeomorphism between simply connected smooth compact 4-manifolds with the same boundary $\\partial X_{1} = \\partial X_{2}$, such thatf is the identity on the boundary. $Letf_{*}: H_{2}(X_{1},\\partial X_{1}) \\to H_{2}(X_{2},\\partial X_{2})$ be the induced map on relative homology. If for all knots K and for all x in $H_{2}(X_{1},\\partial X_{1})$ we have $g_{X,1,K}(x) =g_{X,2,K}(f_{*}(x))$, is $X_{1}$ diffeomorphic to $X_{2}$ rel. boundary?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.12.\n\nLiterature notes:\n(1) A version of this problem appeared in Stipsicz–Szabó [SS24b].\n\n(2) There are several ways one might define a genus function for a 4-manifold with boundary. For concreteness, we present one. For X a simply connected, smooth 4-manifold with boundary and K a knot in $\\partial X$, write $g_{X,K}: H_{2}(X,\\partial X) \\to \\mathbb{N}$ for the relative genus function, that assigns to every relative homology class the minimal genus of a smooth embedded surface with boundary K representing that class.\n\n(3) In the closed case, every isometry of the intersection form on $H_{2}is$ realized by a homeomorphism [FQ90, Theorem 10.1], so one could have stated the problem by starting with the isometry $f_{*}$.\n\n(4) On the other hand, it is not the case that every isometry of the intersection form that respects the genus function is realized by a diffeomorphism. For example, the isomorphism $\\varphi: H_{2}(K3) \\to H_{2}(K3)$ given by $\\varphi(a) =$ −a is not induced by a diffeomorphism; see [DK90, Corollary 9.1.4].\n\n(5) A special case of part (a) is Problem 4.1, the smooth 4-dimensional Poincaré conjecture.\n\nReferences cited:\n- [SS24b] András I. Stipsicz and Zoltán Szabó. On the minimal genus problem in fourmanifolds. In Frontiers in geometry and topology. Summer school and research conference, The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, August 1–12, 2022, pages 215–232. Providence, RI: American Mathematical Society (AMS), 2024. doi:10.1090/pspum/109/01997.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [DK90] S. K. Donaldson and P. B. Kronheimer. The geometry of four-manifolds. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Minimal-genus functions distinguish smooth structures in specific families, but preservation of the full genus function is not known to determine diffeomorphism type in either stated generality.\n\n**Verified partial progress.**\n\n- Stipsicz--Szabo compute or bound genus functions in selected simply connected 4-manifolds and use them to distinguish smooth structures.\n- The closed question is known to contain the smooth 4-dimensional Poincare conjecture as a special case.\n\n**Full solution or refutation.**\n\nNo general implication from genus-function equivalence to diffeomorphism was verified; the relative enhancement is also open.\n\n**What remains.**\n\nProve or refute the closed implication and formulate the relative invariant on compatible knot/class pairs before resolving the rel-boundary version.\n\n**Sources checked.**\n\n- Andras I. Stipsicz and Zoltan Szabo, On the minimal genus problem in four-manifolds, arXiv:2307.04202; PSPUM 109 (2024), 215--232. (primary): https://arxiv.org/abs/2307.04202\n Evidence used: Develops the genus-function program, gives examples, and states the diffeomorphism-detection implication as an open question.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 4.12. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current formulation, relative variant, and relation to the smooth 4-dimensional Poincare conjecture.\n\n**Review notes.** OCR defects include g_{X,1}, 'such thatf', and 'Letf_*'. Part (b) also needs a convention restricting to pairs with boundary(x)=[K], or assigning infinity when no representative exists.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2889, "problem_number": "KP-4.13", "title": "Kirby Problem 4.13", "statement": "(a) Does every large $\\mathbb{R}^{4}-homeomorph$ lie $in\\mathcal{R}_{K}$ for some Kthat is not smoothly slice?\n\n(b) Does there exist an infinite sequence of knots ${K_{i}}$ such that $\\mathcal{R}_{K,i} \\ne \\mathcal{R}_{K,j}$ whenever $i \\ne j$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.13.\n\nLiterature notes:\n(1) Given a knot K, let $X_{K}$ denote its 0-trace, which is obtained, by definition, by attaching a 0-framed 2-handle to $B^{4}$ along K and smoothing corners. Reserve the symbol $\\mathbb{R}^{4}$ to refer to 4-dimensional Euclidean space with its standard smooth structure. Following Gompf, we call a smooth manifold $\\mathcal{R}$ an $\\mathbb{R}^{4}-homeomorph$ if it is homeomorphic to $\\mathbb{R}^{4}$, but not necessarily diffeomorphic. This is to avoid the terminology ‘exotic $\\mathbb{R}^{4}’$ in cases where we want to allow the standard smooth structure.\n\n(2) An $\\mathbb{R}^{4}-homeomorph$ is said to belarge if it contains a compact subset that does not admit a smooth embedding into $\\mathbb{R}^{4}$, and small if it is not large.\n\n(3) Given a knot K let $\\mathcal{R}_{K}$ denote the set of $\\mathbb{R}^{4}-homeomorphs$ admitting a smooth embedding of $X_{K}$. Gompf [Gom85, Lemma 1.1] described a construction of elements $in\\mathcal{R}_{K}$ using Quinn’s smoothing theorem. By the trace embedding lemma, $\\mathcal{R}_{K}$ is nonempty precisely if K is topologically slice (not necessarily smoothly slice), and K is smoothly slice if and only if $\\mathbb{R}^{4} \\in \\mathcal{R}_{K}$ if and only if $\\mathcal{R}_{K}$ is the set of all $\\mathbb{R}^{4}-homeomorphs$, including all small $\\mathbb{R}^{4}-homeomorphs$.\n\n(4) Let $K_{0}$ be the unknot, $K_{1}$ be the second iterated positive Whitehead double of the right-handed trefoil, and $K_{2}$ any knot with $\\tau(K_{2})$ <0, e.g. $-K_{1}$. Then $\\mathcal{R}_{K,i} \\ne \\mathcal{R}_{K,j}$ for all $i \\ne j$.\n\n(5) An affirmative answer to the analogue of(b)for links follows from [Gom85, Lemma 1.2 and the proof of Theorem 1.3].\n\nReferences cited:\n- [Gom85] Robert E. Gompf. An infinite set of exotic $\\mathbb{R}^{4}$’s. J. Differential Geom., 21(2):283– 300, 1985. http://projecteuclid.org/euclid.jdg/1214439566.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Families of large R4-homeomorphs and knot-trace constructions are known, but neither requested universal knot association nor infinite pairwise-distinct family was verified.\n\n**Verified partial progress.**\n\n- The problem list gives the trace construction and distinction between large and small R4-homeomorphs.\n\n**Full solution or refutation.**\n\nBoth existence questions remain open in the checked source.\n\n**What remains.**\n\nFind invariants distinguishing the R_K,i family or prove a realization obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.13 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the constructions and retains both questions.\n\n**Review notes.** Source's malformed subscripts/spaces were not normalized.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2890, "problem_number": "KP-4.14", "title": "Kirby Problem 4.14", "statement": "Is there a universal cork? More precisely, does there exist some cork (C, f) such that given any pair W and $W^{1}$ of closed, simply connected 4-manifolds that are homeomorphic but not diffeomorphic, there exists a smooth embedding $C \\hookrightarrow W$ such that (W $\\setminus$ Int C) $\\cup _{f} C$ is diffeomorphic to $W^{1}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.14.\n\nLiterature notes:\n(1) A cork (C, f) is a pair of a compact, contractible, smooth 4-manifold C and a diffeomorphism $f: \\partial C \\to \\partial C$, such that f does not extend to a diffeomorphism of C. Some authors require f to be an involution, or C to be Stein.\n\n(2) Not all corks are universal: Ladu shows that the Akbulut cork is not universal [Lad25]. Let (C, f) denote the Akbulut cork. Briefly, the paper shows that since −C has trivial Floer homology in the relevant grading, (−C, f)cannot be used to go between two smooth 4-manifolds with differin g Seiberg-Witten invariants, such as $K3\\#\\mathbb{CP}^{2}$ and $\\#_{4}\\mathbb{CP}^{2}\\#_{19}\\overline{\\mathbb{CP}}{}^{2}.But$ then twisting by (C, f) cannot be used to go between the exotic pair $-(K3\\#\\mathbb{CP}^{2})$ and $-(\\#_{4}\\mathbb{CP}^{2}\\#_{19}\\overline{\\mathbb{CP}}{}^{2})$. This suggest an alternative definition of a universal cork, by allowing either a cork twist by C or −C. Under this definition it is still open whether the Akbulut cork is universal.\n\n(3) One could ask the question only requiring W and W1 to be compact with possibly nonempty boundary. Say a cork (C, f) is $\\partial-universal$ if any pair of compact, simply connected 4-manifolds, which are homeomorphic but not diffeomorphic, is related by a cork twist along some embedding of C. The same preprint of Ladu mentioned above shows that there is no $\\partial-universal$ cork [Lad25].\n\n(4) By Wall’s theorem on h-cobordisms [Wal64b], if (C, f) is a cork then there exists n such that f extends to a diffeomorphism of $C\\#_{n}S^{2} \\times S^{2}$. So, if there is a universal cork, then there is a fixed n so that any exotic pair $W, W^{1}$ as above are diffeomorphic after stabilization with n copies of $S^{2} \\times S^{2}$. See Problem 4.7. Note that if there is no bound on the number of $S^{2} \\times S^{2}$ summands needed to make a pair of exotic 4-manifolds diffeomorphic, there cannot be a closed universal cork. This suggests another refinement of the terminology: say a cork(C, f) is n-universal if it relates every pair of homeomorphic simply connected 4-manifolds that become diffeomorphic after connected sum with at most n copies of $S^{2} \\times S^{2}$.\n\nReferences cited:\n- [Lad25] Roberto Ladu. The Akbulut cork is not universal. Selecta Math. (N.S.), 31(4):Paper No. 74, 14, 2025. doi:10.1007/s00029-025-01061-6.\n- [Wal64b] C. T. C. Wall. On simply-connected 4-manifolds. J. London Math. Soc., 39:141–149, 1964. doi:10.1112/jlms/s1-39.1.141.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A 2025 result rules out the Akbulut cork as universal, but existence of some universal cork remains open.\n\n**Verified partial progress.**\n\n- Ladu proves the Akbulut cork is not universal.\n\n**Full solution or refutation.**\n\nNo universal cork construction or impossibility theorem was verified.\n\n**What remains.**\n\nConstruct a cork working for all exotic simply connected pairs or find an obstruction applying to every cork.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.14 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the 2025 nonuniversality theorem for the Akbulut cork.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2891, "problem_number": "KP-4.15", "title": "Kirby Problem 4.15", "statement": "(11/8 Conjecture). Does every smooth, spin, closed 4-manifold X satisfy $b_{2}(X) \\geq 11|\\sigma(X)|$, where $\\sigma(X)$ is the signature of the intersection form?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.15.\n\nLiterature notes:\n(1) This is [Kir97, Problem 4.92], originally raised by Matsumoto.\n\n(2) A positive resolution would answer the Geography Question 4.54 for the smooth simply connected case. By Rokhlin’s theorem, the intersection form of a spin, smooth, closed 4-manifold is of the form $2p(\\pm E_{8})\\oplus q\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}$, for $p,q\\geq 0$ and a choice of sign $\\pm$. The 11/8 conjecture can be rephrased as $q \\geq$ 3p. If an intersection form satisfies this bound, it is realized by the manifold $p(\\pm K3)\\#(q-3p)(S^{2} \\times S^{2})$.\n\n(3) Donaldson [Don87b] used Yang-Mills theory to prove thatp $\\geq$ 1 implies $q \\geq$ 3, assuming $H_{1}(X;\\mathbb{Z})$ has no 2-torsion. Furuta [Fur01] used finite dimensional approximation of the SeibergWitten map to prove the inequality $b_{2}(X) \\geq 10|\\sigma(X)|$ +2 (i.e., $q \\geq$ 2p+1) assuming $q \\geq$ 1. Hopkins, Lin, Shi and Xu [HLSX22] refined Furuta’s method to get, for $p \\geq$ 2, $^{⎧}_{|}2p+2$ if $p\\equiv1,2,5,6$ (mod 8) $q \\geq ^{⎪}2p+3$ if $p\\equiv3,4,7$ (mod 8) $^{|}⎩2p+4$ if $p\\equiv0$ (mod 8) They also showed this is the strongest possible inequality that can be proved using that method.\n\n(4) There is a suggested strategy to prove the conjecture [Bau12, Man14]. The 11/8 conjecture would be true if one had an invariantf(Y)of oriented integral homology 3-spheres Y that satisfies the following property: Let W be a compact oriented spin smooth cobordism from an integral homology 3-sphere $Y_{0}$ to an integral homology 3-sphere $Y_{1}$. The intersection form of $W$ is of the form $rE_{8}\\oplus q\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}$, where $q\\geq 0$ and $r$ can be any integer. The required property for f is that $f(Y_{0}) \\leq f(Y_{1})$ +q+r−1.\n\n(5) There is also a proposal to approach the conjecture using the Ricci flow, by reducing it to the Hitchin-Thorpe inequality for Einstein 4-manifolds. See [Bam21b, Section 5.6].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Don87b] S. K. Donaldson. The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differential Geom., 26(3):397–428, 1987. http://projecteuclid.org/euclid.jdg/1214441485.\n- [Fur01] M. Furuta. Monopole equation and the 11 8 -conjecture. Math. Res. Lett., 8(3):279– 291, 2001. doi:10.4310/MRL.2001.v8.n3.a5.\n- [HLSX22] Michael J. Hopkins, Jianfeng Lin, XiaoLin Danny Shi, and Zhouli Xu. Intersection forms of spin 4-manifolds and the $\\mathrm{Pin}(2)$-equivariant Mahowald invariant. Comm. Amer. Math. Soc., 2:22–132, 2022. doi:10.1090/cams/4.\n- [Bau12] Stefan A. Bauer. Intersection forms of spin four-manifolds, 2012. arXiv:1211.7092.\n- [Man14] Ciprian Manolescu. On the intersection forms of spin four-manifolds with boundary. Math. Ann., 359(3-4):695–728, 2014. doi:10.1007/s00208-014-1010-1.\n- [Bam21b] Richard H. Bamler. Recent developments in Ricci flows. Notices Amer. Math. Soc., 68(9):1486–1498, 2021. doi:10.1090/noti2343.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Donaldson, Furuta, and subsequent refinements establish weaker spin intersection-form inequalities, but the 11/8 conjecture remains open.\n\n**Verified partial progress.**\n\n- Furuta proves the 10/8 inequality.\n- The maintained list records 10/8+4-type progress.\n\n**Full solution or refutation.**\n\nNo proof of b2 >= 11|signature| was verified.\n\n**What remains.**\n\nClose the gap from current Pin(2)-equivariant bounds to 11/8 or exhibit a counterexample.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.15 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the 11/8 conjecture and its known weaker bounds.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2892, "problem_number": "KP-4.16", "title": "Kirby Problem 4.16", "statement": "(a) Do there exist closed, oriented, smooth, irreducible 4-manifolds with $b^{+}_{2} >$ 1 and $c^{2}_{1}:=2\\chi+3\\sigma<0$?\n\n(b) Is there an irreducible exotic smooth structure on $\\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$ for n>9?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.16.\n\nLiterature notes:\n(1) $S^{2}–bundles$ over $\\Sigma_{h}$ are irreducible 4–manifolds with $b^{+}_{2} =$ 1, and for $h \\geq$ 2, they have $c^{2}_{1}<0$.\n\n(2) An exotic $\\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$ for $n >$ 9 would have $c^{2}_{1} <$ 0. All known exotica in that range are blow-ups of exotic $\\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$ for $n \\leq$ 9 (such as the Barlow or Dolgachev surface).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Irreducible ruled examples have negative c1 squared but b2-plus equal to one, and known exotic rational surfaces for n>9 arise by blow-up; the requested examples remain open.\n\n**Verified partial progress.**\n\n- S2-bundles over higher-genus surfaces supply b2-plus=1 negative-c1-squared examples.\n- Known exotica with n>9 are recorded as blow-ups from n<=9.\n\n**Full solution or refutation.**\n\nNeither target existence question was verified.\n\n**What remains.**\n\nConstruct an irreducible b2-plus>1 negative-c1-squared example or an irreducible exotic rational surface beyond n=9.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.16 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the boundary cases and retains both questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2893, "problem_number": "KP-4.17", "title": "Kirby Problem 4.17", "statement": "Is there an irreducible, closed, simply connected, oriented 4– manifold with $b^{+}_{2}$ and $b^{-}_{2}$ both even?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.17.\n\nLiterature notes:\n(1) The condition $b^{+}_{2}(X)$ and $b^{-}_{2}(X)$ both being even is equivalent to the holomorphic Euler characteristic $\\chi_{h}(X):=\\frac{1}{4}(e(X)+\\sigma(X))$ not being an integer for either orientation on X. Note that $\\chi_{h}(X)$ always lies in $\\frac{1}{2}\\mathbb{Z}$ and is in $\\mathbb{Z}$ if and only if X admits an almost complex structure. All current smooth invariants, such as the Seiberg–Witten invariants used for detecting irreducibility effectively, are defined for 4–manifolds with $b_{1} =$ 0 when $\\chi_{h} \\in \\mathbb{Z}$. As a result, all known irreducible, closed, orientable 4–manifolds are almost complex under at least one orientation.\n\n(2) This problem is discussed in [Kir97, Problem 4.97], where it is conjectured that no such X exists. A notable test case, suggested by Gompf [Kir97], involves taking two copies of K3, removing the tubular neighborhood of an embedded sphere with self-intersection −2 from each, and then gluing the complements along their boundaries with an orientationreversing diffeomorphism of $\\mathbb{RP}^{3}$. The resulting 4–manifold X, which has $\\chi_{h}(X) \\notin \\mathbb{Z}$ for either orientation, is not known to be reducible.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No irreducible closed simply connected oriented 4-manifold with both b2-plus and b2-minus even was verified; the maintained list records a conjectural nonexistence.\n\n**Verified partial progress.**\n\n- Current smooth irreducibility tools are limited in this parity setting.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nConstruct such a manifold or prove the conjectural obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.17 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains the parity/almost-complex obstruction and cites the conjectural nonexistence.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2894, "problem_number": "KP-4.18", "title": "Kirby Problem 4.18", "statement": "(a) Does there exist a pair of smooth, closed 4-manifolds that are homotopy equivalent but not simple homotopy equivalent?\n\n(b) Does there exist a pair of smoothlyh-cobordant, smooth, closed 4-manifolds that are not smoothly s-cobordant?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.18.\n\nLiterature notes:\n(1) There exist examples of topologicallyh-cobordant 4-manifolds that are not topologically s-cobordant [KPR22], and examples of topological manifolds that are homotopy equivalent but not simple homotopy equivalent [NNP23].\n\n(2) A potential way to answer the questions is as follows. Let M be a smooth, closed 4-manifold with $\\pi:=\\pi_{1}(M)$. Let $x \\in Wh(\\pi)$ be an element of the Whitehead group of $\\pi$.\n\n\\paragraph{Question.} Does there exist a smoothh-cobordism(W;M, N)between 4-manifolds with Whitehead torsion $\\tau(W$, M) =x? Let $w: \\pi \\to C_{2}$ be the orientation character of M. The Whitehead group admits an involution, sending a representative matrix to its conjugate-transpose. Elements are conjugated by the standard involution $x \\to$ barx induced by extending $g \\to w(g)g^{-1}$ linearly. The question is of particular interest when x−barx $\\ne$ 0, because then the induced homotopy equivalence $M \\to N$ is not simple. This could lead to examples, if one can also compute the Whitehead torsions of all the homotopy self-equivalences.\n\n(3) This problem is a little different from other similar smooth realization problems. In the process of building an h-cobordism, one adds 2-handles and then tries to add 3-handles. One can in fact represent the desired homotopy classes in the middle level by smoothly embedded, framed spheres. Unfortunately this does not suffice. The challenge is to find smooth embeddings in such a way that the inclusion-induced map $\\pi_{1}(N) \\to \\pi_{1}(W)$ is an isomorphism. For every $x \\in Wh(\\pi)$, there exist $k \\in \\mathbb{N}$ such that x is smoothly realizable as the torsion $\\tau(W, M\\#_{k}S^{2} \\times S^{2})of$ some smoothh-cobordism W based on $M\\#_{k}S^{2} \\times S^{2}$ [CS71]; a preliminary construction appeared in [Sta65]. If one can find a topological h-cobordism (W;M, N) with torsion x, then it might be possible to apply smoothing theory for 5-manifolds to improve W to a smooth h-cobordism.\n\n(4) Another potential method for solving (a) was given by Kasprowski–Nicholson–Veselá in [KNV24]. Their proposed method, if it could be implemented, would give examples that are homotopy equivalent but not stably simple homotopy equivalent.\n\nReferences cited:\n- [KPR22] Daniel Kasprowski, Mark Powell, and Arunima Ray. Counterexamples in 4-manifold topology. EMS Surv. Math. Sci., 9(1):193–249, 2022. doi:10.4171/emss/56.\n- [NNP23] Csaba Nagy, John Nicholson, and Mark Powell. Simple homotopy types of even dimensional manifolds, 2023. arXiv:arXiv:2312.00322.\n- [CS71] Sylvain E. Cappell and Julius L. Shaneson. On four dimensional surgery and applications. Comment. Math. Helv., 46:500–528, 1971. doi:10.1007/BF02566862.\n- [Sta65] J. W. Stallings. On the infinite processes leading to differentiability in the complement of a point. Differ. and Combinat. Topology, Sympos. Marston Morse, Princeton, 245-254 (1965)., 1965.\n- [KNV24] Daniel Kasprowski, John Nicholson, and Simona Veselá. Stable equivalence relations on 4-manifolds, 2024. arXiv:2405.06637.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Topological examples of h- but not s-cobordism and homotopy-but-not-simple-homotopy equivalence are known; the stated smooth 4-manifold versions remain unresolved.\n\n**Verified partial progress.**\n\n- KPR22 gives topological h-cobordant but not s-cobordant examples.\n- NNP23 gives topological homotopy-equivalent but not simple-homotopy-equivalent examples.\n\n**Full solution or refutation.**\n\nNo corresponding smooth examples were verified.\n\n**What remains.**\n\nRealize suitable Whitehead torsion by a smooth h-cobordism or prove a smooth obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.18 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explicitly distinguishes the known topological examples from the smooth questions.\n\n**Review notes.** OCR/background corruption in h/s-cobordism notation was flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2895, "problem_number": "KP-4.19", "title": "Kirby Problem 4.19", "statement": "What are the possible Euler characteristics of closed, aspherical 4-manifolds? More specifically, we ask the following.\n\n(a) Is it always the case that $\\chi \\geq |\\sigma|$?\n\n(b) What is the smallest Euler characteristic of a closed hyperbolic 4-manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.19.\n\nLiterature notes:\n(1) The questions are motivated by the following conjecture.\n\n\\paragraph{Conjecture.} The Euler characteristic of X is non-negative. This is [Kir97, Problem 4.10]. One can ask the question for all aspherical, closed 4-manifolds, including non-smoothable ones. Edmonds [Edm15] has proven the conjecture for Haken 4-manifolds, which were defined by Foozwell and Rubinstein in [FR11].\n\n(2) Given that hyperbolic manifolds are aspherical, it is natural to ask the specific $subquestion(b)$. By Chern–Gauss–Bonnet, for a closed hyperbolic 4-manifold X we have $vol(X) = 4\\pi^{2}\\cdot\\chi(X)$, so we are equivalently asking for the smallest possible hyperbolic volume. The smallest known orientable hyperbolic 4-manifold was found by Conder and Maclachlan [CM05] and has $\\chi =$ 16; it covers a nonorientable hyperbolic manifold with $\\chi=8$. Notice that in the cusped case there are many known orientable examples with the minimal possible value $\\chi =$ 1 [RT00].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Edm15] Allan L. Edmonds. The Euler characteristic of a Haken 4-manifold. In Geometry, groups and dynamics, volume 639 of Contemp. Math., pages 217–234. Amer. Math. Soc., Providence, RI, 2015. doi:10.1090/conm/639/12796.\n- [FR11] Bell Foozwell and Hyam Rubinstein. Introduction to the theory of Haken nmanifolds. In Topology and geometry in dimension three, volume 560 of Contemp. Math., pages 71–84. Amer. Math. Soc., Providence, RI, 2011. doi:10.1090/conm/560/11092.\n- [CM05] Marston Conder and Colin Maclachlan. Compact hyperbolic 4-manifolds of small volume. Proc. Amer. Math. Soc., 133(8):2469–2476, 2005. doi:10.1090/S0002-9939-05-07634-3.\n- [RT00] John G. Ratcliffe and Steven T. Tschantz. The volume spectrum of hyperbolic 4-manifolds. Experiment. Math., 9(1):101–125, 2000. http://projecteuclid.org/euclid.em/1046889595.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Nonnegative Euler characteristic is known for Haken 4-manifolds and smallest hyperbolic examples are known, but neither the general inequality nor hyperbolic minimum was verified as settled.\n\n**Verified partial progress.**\n\n- Edmonds proves the Euler-characteristic conjecture for Haken 4-manifolds.\n- Known orientable closed hyperbolic examples give upper bounds for the minimum.\n\n**Full solution or refutation.**\n\nBoth requested general optimization questions remain open.\n\n**What remains.**\n\nProve chi >= |signature| generally and establish a volume/Euler lower bound sharp for hyperbolic 4-manifolds.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.19 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the Haken theorem and known smallest examples while retaining the questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2896, "problem_number": "KP-4.20", "title": "Kirby Problem 4.20", "statement": "Is $*\\mathbb{RP}^{4}\\#*\\mathbb{RP}^{4}$ smoothable? Is *En\\#*En smoothable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.20.\n\nLiterature notes:\n(1) The manifold $*\\mathbb{RP}^{4}$ was first constructed by Ruberman [Rub84] (see also [HKT94]). It is homotopy equivalent to $\\mathbb{RP}^{4}$ but not homeomorphic to $\\mathbb{RP}^{4}$. In particular, it has nontrivial Kirby–Siebenmann invariant. Taking the connected sum of two copies yields a 4-manifold $*\\mathbb{RP}^{4}\\#*\\mathbb{RP}^{4}$ with trivial Kirby–Siebenmann invariant. Recall that a 4-manifold with trivial Kirby–Siebenmann invariant is stably smoothable, i.e. smoothable after connected sum with $k(S^{2} \\times S^{2})$ for some $k \\geq$ 0.\n\n(2) Recall that $*\\mathbb{CP}^{2}$ denotes the Chern manifold constructed by Freedman in [Fre82]. It is homotopy equivalent to $\\mathbb{CP}^{2}$ but not homeomorphic to $\\mathbb{CP}^{2}$. In this case, we know $that*\\mathbb{CP}^{2}\\#*\\mathbb{CP}^{2} \\cong \\mathbb{CP}^{2}\\#\\mathbb{CP}^{2}$, and is therefor e smoothable. $Similarly,*\\mathbb{RP}^{4}\\#*\\mathbb{CP}^{2}$ is smoothable [RS97], answering [Kir97, Problem 4.82]. Whether $*\\mathbb{RP}^{4}\\#*\\mathbb{RP}^{4}$ is smoothable is the next natural case of a 4-manifold with trivial Kirby–Siebenmann invariant that is not obviously smoothable.\n\n(3) The smoothability of topological manifolds that are homotopy equivalent to $\\mathbb{RP}^{4}\\#\\mathbb{RP}^{4}$, arising from another construction using the action of $L_{5}$ on the structure set, is considered in Problem 4.22. By $contrast,*\\mathbb{RP}^{4}\\#*\\mathbb{RP}^{4}$ is detected in the normal invariants. See [BDK07] for the computation of the topological structure set of $\\mathbb{RP}^{4}\\#\\mathbb{RP}^{4}$.\n\n(4) We use En to denote the Enriques surface, which by definition is a quotient of the K3 surface by an involution. The manifold *En is a star partner of En, as defined by Freedman–Quinn [FQ90, Section 10.4]. Once again, *En\\#*En has trivial Kirby–Siebenmann invariant, and it would be interesting to know whether or not it is smoothable, and similarly for $*\\mathbb{RP}^{4}\\#*En$.\n\nReferences cited:\n- [Rub84] Daniel Ruberman. Invariant knots of free involutions of $S^{4}$. Topology Appl., 18(2-3):217–224, 1984. doi:10.1016/0166-8641(84)90011-7.\n- [HKT94] Ian Hambleton, Matthias Kreck, and Peter Teichner. Nonorientable 4-manifolds with fundamental group of order 2. Trans. Amer. Math. Soc., 344(2):649–665, 1994. doi:10.2307/2154500.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [RS97] Daniel Ruberman and Ronald J. Stern. A fake smooth $\\mathbb{CP}^{2}$\\#RP4. Math. Res. Lett., 4(2-3):375–378, 1997. doi:10.4310/MRL.1997.v4.n3.a6.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [BDK07] Jeremy Brookman, James F. Davis, and Qayum Khan. Manifolds homotopy equivalent to Pn\\#Pn. Math. Ann., 338(4):947–962, 2007. doi:10.1007/s00208-007-0099-x.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The connected sums have trivial Kirby--Siebenmann obstruction, but no smoothability result or obstruction for the two specified manifolds was verified.\n\n**Verified partial progress.**\n\n- Ruberman's starred RP4 has nontrivial Kirby--Siebenmann invariant.\n- Taking two copies removes this elementary obstruction.\n\n**Full solution or refutation.**\n\nBoth smoothability questions remain open.\n\n**What remains.**\n\nConstruct smooth structures or a secondary obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.20 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains the vanishing primary obstruction and retains both questions.\n\n**Review notes.** Star notation preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2897, "problem_number": "KP-4.21", "title": "Kirby Problem 4.21", "statement": "Is every topological closed 4–manifold M the union of submanifolds $Y \\cup Z$, where Y is smoothable, Z is acyclic, and $Y \\cap Z$ is their common boundary, a homology sphere $\\Sigma$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.21.\n\nLiterature notes:\n(1) This is [Kir97, Problem 4.74].\n\n(2) In the case that M is simply connected the result holds. Freedman [Fre82, Theorem 1.5] proved that the homeomorphism type of a simply connected closed 4–manifold is determined by its unimodular intersection form and its Kirby-Siebenmann invariant. The proof consists of an explicit construction of a manifold for each such for m and possible Kirby-Siebenmann invariant. The construction begins with a smooth compact 4-manifold built from $D^{4}$ by adding 2-handles and with the desired intersection form. It then modifies the handle attachments as needed to achieve the desired Rokhlin invariant for $\\Sigma$. Lastly, it caps the manifold of f with a contractible topological manifold.\n\n(3) There are compact topological manifolds that do not support such decompositions. We explain how to construct an example. Let $K \\subset S^{3}$ be a topologically slice knot that does not bound a smooth slice disk in any smooth compact acyclic 4–manifold with boundary $S^{3}$. Examples of such were first identified by Akbulut in unpublished work. For instance, the untwisted Whitehead double of the trefoil suffices. Let M denote the complement an open tubular neighborhood of a locally flat slice disk. Notice that $\\partial M \\cong S_{0}^{3}(K)$, 0–surgery on K. The claim is that M cannot have a decomposition of the desired type; an outline of a proof follows. Suppose that $\\Sigma \\subset M$ is a locally flat homology 3–sphere that splits M as the union of topological manifold Y and an acyclic manifold Z. We have that $\\partial Y \\cong \\Sigma \\cup S_{0}^{3}(K)$. Assume that there exists a smoothing A of Y; that is,A is a smooth manifold supporting a homeomorphism to Y. Attaching a smooth 2– handle to A along the meridian of K in $S_{0}^{3}(K)$, referred to as the trace of K in $S_{0}^{3}(K)$, yields a smooth homology $S^{3} \\times I$ with boundary $S^{3} \\cup \\Sigma$. Attaching a smooth 4–ball results in a smooth acyclic 4–manifold A1 with boundary $\\Sigma$. We now see that the union $B =$ A1 $\\cup _{\\Sigma} A$ is a smooth homology $S^{1} \\times B^{3}$ with boundary $S_{0}^{3}(K)$. Appropriately attaching a 2– handle to B along the trace of $K \\in S_{0}^{3}(K)$ results in a smooth acyclic 4–manifold bounded by $S^{3}$. The cocore of the 2–handle is a smooth slice disk for its boundary, which represents $K \\subset S^{3}$.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The proposed smoothable-plus-acyclic decomposition holds in the simply connected case, but no theorem for every closed topological 4-manifold was verified.\n\n**Verified partial progress.**\n\n- Freedman's classification/construction yields the simply connected case.\n\n**Full solution or refutation.**\n\nThe general fundamental-group case remains open.\n\n**What remains.**\n\nExtend the decomposition through nontrivial fundamental groups or find an obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.21 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the simply connected result and poses the general question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2898, "problem_number": "KP-4.22", "title": "Kirby Problem 4.22", "statement": "Let $\\pi$ be a good group, and let X be a smooth 4-manifold with $\\pi_{1}(X) = \\pi$. Does $L^{s}_{5}(\\mathbb{Z}[\\pi])$ act on the smooth structure set of X, in such a way that the action reduces under the forgetful map to the Wall realization action on the topological structure set? In particular, can this be done when $\\pi= \\mathbb{Z}, \\mathbb{Z} \\times \\mathbb{Z}_{n}$, or $\\mathbb{Z}_{2}*\\mathbb{Z}_{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.22.\n\nLiterature notes:\n(1) The surgery exact sequence, valid for higher dimensional manifolds, ends with $\\mathcal{N}(X \\times I,\\partial(X \\times$ I)) $\\to L^{s}_{n,1}(\\mathbb{Z}\\pi) \\to ^{\\Theta} \\to \\mathcal{S}(X) \\to \\mathcal{N}(X) \\to L^{s}_{n}(\\mathbb{Z}\\pi). + Here\\mathcal{S}(X)is$ the smooth structure set $and\\mathcal{N}(X)is$ normal bordism classes of degree one normal maps. The structure set consists of orientationpreserving homotopy equivalences f: $M \\to X$ up to diffeomorphisms F: M1 $\\to M$ commuting up to homotopy with the homotopy equivalences. What is written as a map $\\Theta$ is shorthand for an action of $L^{s}_{n,+,1}(\\mathbb{Z}\\pi)$ on $\\mathcal{S}(X)$. The action of an element $A \\in L^{s}_{n,+,1}(\\mathbb{Z}\\pi)$ on (X,Id $:X \\to$ X) produces, for n+1 $\\geq$ 6, a (normal) cobordism W from (X,Id) to another homotopy equivalencef: X1 $\\to X$. The problem asks if there is an action of $L^{s}_{5}(\\mathbb{Z}[\\pi])$ when X is a closed smooth 4-manifold, that reduces to the Wall realization action on the topological structure set $\\mathcal{S}_{TOP}(X)$.\n\n(2) Cappell and Shaneson [CS71] showed that one can realize the action of $L^{s}_{5}$ stably, i.e. after replacing X by its connected sum with some number of copies of $S^{2} \\times S^{2}$. For example, Scharlemann’s construction [Sch76] realizes the action of the generator of $L^{s}_{5}(\\mathbb{Z}[\\mathbb{Z}])$ on $(S^{1} \\times S^{3})\\#(S^{2} \\times S^{2})$ (this is also in [CS71] using an unpublished calculation of R. Lee). In general, even when if one can realize the action of an element of $L^{s}_{5}$ to produce a different element of the structure set, the resultant manifold X1 might still be diffeomorphic to X. The difference in the structure set would then be due to the resultant self homotopy equivalence of X not being homotopic to a diffeomorphism. For instance, Akbulut [Akb99] showed that Scharlemann’s manifold is diffeomorphic $to(S^{1} \\times S^{3})\\#(S^{2} \\times S^{2})$. In many cases of interest, in particular for $\\pi =\\mathbb{Z}$ and $\\pi=\\mathbb{Z}/2*\\mathbb{Z}/2$, the Whitehead group of $\\pi$ is trivial, in which case $L^{s}_{5}(\\mathbb{Z}\\pi) =L_{5}(\\mathbb{Z}\\pi)$ and one can ignore the s-decoration.\n\n(3) The heart of the question is then whether one can realize the action without stabilization. There are three cases of particular geometric interest, corresponding to fundamental groups $\\mathbb{Z},\\mathbb{Z} \\times \\mathbb{Z}/n(for$ n>1), and $\\mathbb{Z}/2*\\mathbb{Z}/2$. All of these are good groups, so that realization works in the topological category; the issue is to find smooth realizations.\n\n(i) Realizing the action of $L_{5}(\\mathbb{Z}[\\mathbb{Z}])$ on $X = S^{1} \\times S^{3}$. Every selfhomotopy equivalence from X to itself is homotopic to a diffeomorphism, so realizing the action of the generator of $L_{5}(\\mathbb{Z}[\\mathbb{Z}])would$ produce an exotic $S^{1} \\times S^{3}$. It would be detected by the Rokhlin invariant of any spin manifold carrying the generator of $H_{3}. A$ gauge-theoretic conjecture of Furuta-Ohta [FO93] would imply that no such manifold exists; see Problem 4.65 and [RS05] and [MRS11] for approaches to this conjecture.\n\n\\paragraph{Question.} Is there an exotic $S^{1} \\times S^{3}$ realizing the action of the generator of $L_{5}(\\mathbb{Z}[\\mathbb{Z}])$ on $X =S^{1} \\times S^{3}$?\n\n(ii) Realizing the action of $\\pi=\\mathbb{Z} \\times \\mathbb{Z}/n$ where $L^{s}_{5}(\\mathbb{Z}\\pi)$ is large, and one would want to produce an action on the structure set of $S^{1} \\times$ L(n, q). In this setting, the manifold would again be fake, i.e. homotopy equivalent but not homeomorphic to $S^{1} \\times$ L(n, q), detected by codimensionon e multisignature (or Atiyah-Singer invariants). In this case the topological manifold set, i.e. the structure set modulo homotopy self-equivalences of $S^{1} \\times$ L(n, q) is infinite. For a given n, only finitely many of them (corresponding to $S^{1} \\times$ L(n, $q^{1})where$ L(n, q) $\\cong$ L(n, $q^{1}))$ are known to be smoothable, but as far as we know they could all be.\n\n\\paragraph{Question.} For any $n \\geq$ 2, is there a smooth structure on a fake $S^{1} \\times$ L(n, q) realizing the action of $L^{s}_{5}(\\mathbb{Z}[\\mathbb{Z} \\times \\mathbb{Z}/n])$ on $X = S^{1} \\times$ L(n, q), that is not of the for $m S^{1} \\times$ L(n, q1)?\n\n(iii) The group $\\pi = \\mathbb{Z}/2$ * $\\mathbb{Z}/2$ is relevant to the problem of classifying manifolds homotopy equivalent to $\\mathbb{RP}^{4}\\#\\mathbb{RP}^{4}$. The group $L_{5}(\\mathbb{Z}\\pi)$ contains [Cap74] an infinitely generated subgroup $Unil_{1}(\\mathbb{Z},\\mathbb{Z}-,\\mathbb{Z}-)$. The smooth action of an element in this subgroup would produce a homotopy equivalence from a manifold X to $\\mathbb{RP}^{4}\\#\\mathbb{RP}^{4}$ that is not homotopic to $f_{1}\\#f_{2}$ where the $f_{i}$ are homotopy equivalences $X_{i} \\to \\mathbb{RP}^{4}, i =$ 1,2. A paper of Jahren-Kwasik [JK06] uses the topological realization of this Unil subgroup to construct topological 4-manifolds that are homotopy equivalent to $\\mathbb{RP}^{4}\\#\\mathbb{RP}^{4}$ that are not connected sums of homotopy $\\mathbb{RP}^{4}s$. (See [BDK07] for the full topological classification.)\n\n\\paragraph{Question.} Are the non-splittable manifolds from [JK06] smoothable? This is equivalent to asking about the smooth realizability of the corresponding Unil group elements.\n\nReferences cited:\n- [CS71] Sylvain E. Cappell and Julius L. Shaneson. On four dimensional surgery and applications. Comment. Math. Helv., 46:500–528, 1971. doi:10.1007/BF02566862.\n- [Sch76] Martin Scharlemann. Constructing strange manifolds with the dodecahedral space. Duke Math. J., 43(1):33–40, 1976.\n- [Akb99] Selman Akbulut. Scharlemann’s manifold is standard. Ann. of Math. (2), 149(2):497–510, 1999.\n- [FO93] Mikio Furuta and Hiroshi Ohta. Differentiable structures on punctured 4-manifolds. Topology Appl., 51(3):291–301, 1993. doi:10.1016/0166-8641(93)90083-P.\n- [RS05] Daniel Ruberman and Nikolai Saveliev. Casson–type invariants in dimension four. In Geometry and topology of manifolds, volume 47 of Fields Inst. Commun., pages 281–306. Amer. Math. Soc., Providence, RI, 2005. doi:10.1090/fic/047/18.\n- [MRS11] Tomasz Mrowka, Daniel Ruberman, and Nikolai Saveliev. Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin’s invariant. J. Differential Geom., 88:333–377, 2011. http://projecteuclid.org/euclid.jdg/1320067650.\n- [Cap74] Sylvain E. Cappell. Unitary nilpotent groups and Hermitian K-theory. I. Bull. Amer. Math. Soc., 80:1117–1122, 1974. doi:10.1090/S0002-9904-1974-13636-0.\n- [JK06] Bjørn Jahren and Slawomir Kwasik. Manifolds homotopy equivalent to RP4\\#RP4. Math. Proc. Cambridge Philos. Soc., 140(2):245–252, 2006. doi:10.1017/S0305004105008893.\n- [BDK07] Jeremy Brookman, James F. Davis, and Qayum Khan. Manifolds homotopy equivalent to Pn\\#Pn. Math. Ann., 338(4):947–962, 2007. doi:10.1007/s00208-007-0099-x.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The topological Wall realization action is available, but the requested smooth structure-set action in dimension four was not verified even for the listed groups.\n\n**Verified partial progress.**\n\n- The higher-dimensional surgery sequence motivates the desired action.\n\n**Full solution or refutation.**\n\nNo smooth action theorem at the stated generality was verified.\n\n**What remains.**\n\nConstruct a smooth realization action for one of the listed good groups.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.22 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the requested lifting of the Wall realization action.\n\n**Review notes.** OCR errors in displayed surgery sequence were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2899, "problem_number": "KP-4.23", "title": "Kirby Problem 4.23", "statement": "(Schoenflies problem). If $\\Sigma$ is a smoothly embedded $S^{3}$ in $S^{4}$, then its closed complements are smooth 4-balls.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.23.\n\nLiterature notes:\n(1) This is the famous Schoenflies problem, which appears in [Kir97, Problem 4.32].\n\n(2) Progress on this old and important conjecture has, sadly, been more aspirational than concrete. Scharlemann proved this conjecture in the special case there is a smooth function $f: S^{4} \\to \\mathbb{R}$ whose restriction to the $S^{3}$ is Morse with k 0-handles and $\\leq k$ +1 1-handles (so that the middle level has genus $\\leq$ 2) [Sch84]. Following Gabai’s proof of Property R [Gab87b], Scharlemann [Sch08] used a reembedding process to extend this result in the previous update to the case of (k +2) 1-handles, so the middle level has genus $\\leq$ 3. The argument explores further connections between the Schoenflies Conjecture and the generalized Property R Conjecture (see Problem 1.10). Other reembedding approaches have been proposed by Akbulut [Akb14] (twisting corks), Agol–Freedman [AF15], and Lambert-Cole [LC21]. Lambert-Cole employs Stein trisections and proposes this generalization of the Schoenflies Conjecture: any homotopy 4-ball in a compact Stein domain of complex dimension 2 is diffeomorphic to the standard 4-ball. Gabai [Gab22] has suggested an approach via pseudo-isotopy theory. In the other direction, Gabai, Naylor and Schwartz [GNS25] exhibit potential counterexamples – Schoenflies balls that are not known to be standard.\n\n(3) This problem is equivalent to the corresponding one in the PL category. By Mazur [Maz59] both closed complements are topological 4-balls. They are also quasi-conformal [Geh67] and the Lipschitz [LV77] 4-balls. It is well known following Mazur [Maz59] and Cerf [Cer68] that the Schoenflies problem is equivalent to showing that if $f: S^{1} \\times S^{3} \\to S^{1} \\times S^{3}$ is a diffeomorphism, then after passing to a sufficiently large finite cover the lifted diffeomorphism can be isotoped to be supported in a 4-ball. By [BG19] it is also equivalent to showing that if $B \\subset S^{4}$ is a 3-ball with $\\partial B$ the standard 2-sphere S, then after passing to a finite branched cover of $S^{4}$ branched over S, B is isotopic rel $\\partial$ to the standard 3-ball. By [Gab22] it also would follow from a carving/surgery problem. It is not known whether or not a Schoenflies ball $\\times I$ is always $B^{5}$.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Sch84] Martin Scharlemann. The four-dimensional Schoenflies conjecture is true for genus two imbeddings. Topology, 23(2):211–217, 1984. doi:10.1016/0040-9383(84) 90040-5.\n- [Gab87b] David Gabai. Foliations and the topology of 3-manifolds. III. J. Differential Geom., 26(3):479–536, 1987. http://projecteuclid.org/euclid.jdg/1214441488.\n- [Sch08] Martin Scharlemann. Generalized property R and the Schoenflies conjecture. Comment. Math. Helv., 83(2):421–449, 2008. doi:10.4171/CMH/131.\n- [Akb14] Selman Akbulut. Cork twisting Schoenflies problem. J. Gökova Geom. Topol. GGT, 8:35–43, 2014. doi:10.1017/s0020269x00003418.\n- [AF15] Ian Agol and Michael Freedman. Simplifying 3-manifolds in $\\mathbb{R}^{4}$. Ann. Fac. Sci. Toulouse Math. (6), 24(5):1079–1101, 2015. doi:10.5802/afst.1476.\n- [LC21] Peter Lambert-Cole. Stein trisections and homotopy 4-balls, 2021. arXiv:2104.02003.\n- [Gab22] David Gabai. 3-spheres in the 4-sphere and pseudo-isotopies of $S^{1}$ $\\times$ $S^{3}$, 2022. arXiv:2212.02004.\n- [GNS25] David Gabai, Patrick Naylor, and Hannah Schwartz. Doubles of Gluck twists: a five-dimensional approach. Adv. Math., 480:Paper No. 110455, 29, 2025. doi:10.1016/j.aim.2025.110455.\n- [Maz59] Barry Mazur. On embeddings of spheres. Bull. Amer. Math. Soc., 65:59–65, 1959. doi:10.1090/S0002-9904-1959-10274-3.\n- [Geh67] F. W. Gehring. Extension theorems for quasiconformal mappings in n-space. J. Analyse Math., 19:149–169, 1967. doi:10.1007/BF02788713.\n- [LV77] J. Luukkainen and J. Väisälä. Elements of Lipschitz topology. Ann. Acad. Sci. Fenn. Ser. A I Math., 3(1):85–122, 1977. doi:10.5186/aasfm.1977.0315.\n- [Cer68] Jean Cerf. Sur les difféomorphismes de la sphère de dimension trois $(\\Gamma_4=0)$. Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York, 1968.\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The smooth Schoenflies conjecture is proved under restrictive Morse-theoretic hypotheses, but remains open for arbitrary smooth S3 embeddings in S4.\n\n**Verified partial progress.**\n\n- Scharlemann proves a special case with bounded handle/Morse complexity.\n\n**Full solution or refutation.**\n\nNo general smooth Schoenflies theorem or counterexample was verified.\n\n**What remains.**\n\nRemove the Morse restriction or construct a non-ball complement.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.23 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the special case and retains the famous conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2900, "problem_number": "KP-4.24", "title": "Kirby Problem 4.24", "statement": "Let K be a framed knot in $S^{3} = \\partial B^{4}$. Let U be a meridian of K. Does there exist a smoothly embedded disk D in $B^{4} \\cup _{\\nu K} h^{2}$ bounded by U that intersects the cocore of $h^{2}$ algebraically zero times, such that the 4-manifold obtained by removing an open tubular neighborhood of the disk is not diffeomorphic to the 4-ball?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.24.\n\nLiterature notes:\n(1) We represent the situation of this problem in Figure 1. Note the “big dot” placed on U. An “ordinary dot” on an unknot is an instruction to carve the standard disk (in $B^{4})$ bounded by the unknot. An important variation is to carve a specific ribbon disk with boundary a ribbon knot (see [Akb16, Section 1.4] for examples and further discussion). If the carved disk in the setting of this problem is a disk in the 0handle (i.e. intersecting the cocore of $h^{2}$ geometrically zero times) then the 4-manifold $\\mathcal{B}$ is $B^{4}$. The “big dot” indicates that we may carve along any disk bounded by U that intersects the cocore of $h^{2}$ algebraically zero times. In other words, we carve along any disk that represents the trivial class in $H_{2}(0-handle \\cup 2-handles,\\partial;\\mathbb{Z})but$ do not restrict the carving disk to be contained in the 0-handle. In the setting of this problem, the result of carving is a homotopy $4-ball\\mathcal{B}$ with boundary $S^{3}$. Thus, this construction is a natural extension of the idea of carving along nontrivial disks and yields potentially nonstandard homotopy 4-balls (see relevant discussion in Problem 4.1, i.e. the smooth 4-dimensional Poincaré Conjecture).\n\n\\begin{center}\n\\kthreefiginclude{ch4_fig1.png}{width=0.56\\linewidth}\n\\par\\small\\textbf{Figure 1.} A diagram of a family of homotopy 4-balls. The resulting 4-manifold $\\mathcal{B}$ depends on a choice of carving disk with boundary $U$. This disk is required to intersect the cocore of the pictured 2-handle algebraically zero times, causing the resulting 4-manifold $\\mathcal{B}$ to be a homotopy 4-ball.\n\\end{center}\n\n(2) The case that K is slice and $n =$ 0 is particularly interesting, for then $\\mathcal{B}$ smoothly embeds in $B^{4}$. Thus, the question of whether $\\mathcal{B}$ is necessarily a 4-ball is related to the smooth 4-dimensional Schoenflies conjecture (Problem 4.23).\n\nReferences cited:\n- [Akb16] Selman Akbulut. 4-manifolds, volume 25 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2016. doi:10.1093/acprof:oso/9780198784869.001.0001.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Carving constructions and ribbon-disk variants are developed, but no disk satisfying the stated algebraic-intersection and non-ball-complement conditions was verified.\n\n**Verified partial progress.**\n\n- The list describes standard and ribbon-disk carving variants.\n\n**Full solution or refutation.**\n\nThe requested exotic-complement example remains open.\n\n**What remains.**\n\nProduce a disk with a computable smooth obstruction to a standard-ball complement.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.24 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Retains the existence problem after explaining carving.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2901, "problem_number": "KP-4.25", "title": "Kirby Problem 4.25", "statement": "Under what conditions does a closed, orientable 3-manifold M smoothly embed in $S^{4}$? Is this question algorithmically decidable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.25.\n\nLiterature notes:\n(1) This is in [Kir78, Problem 3.20]. As noted there, every homology 3sphere embeds topologically in $S^{4}$ [Fre82]. Smoothly, even for homology spheres, there is an obstruction coming from the Rokhlin invariant, and more obstructions coming from gauge theory and its cousins, using the fact that if a homology sphere M embeds in $S^{4}$ then M bounds a homology ball, so is trivial in the homology cobordism group. (The relationship between embedding problems and homology cobordism was first noted in [GL83]; see, e.g., [DHST23] for a relatively recent summary of nontriviality results for the 3-dimensional homology cobordism group.) There are also homology spheres that bound homology balls but do not embed in $S^{4}$ [Mc D22]. For 3-manifolds with nontrivial homology, even topologically there are obstructions, starting with an obstruction coming from the homology [Han38]; see [BB22] for a relatively recent survey of such obstructions. In particular, it would also be interesting to understand the question of which non-homology spheres embed locally flatly in $S^{4}$; see [Hil24] for an extensive discussion. Another interesting special case is obstructing punctured homology 3-spheres from embedding in $S^{4}$; see\n\nProblem 4.26.\n\n(2) Related problems include Problem 4.28 (about embeddings in connected sums of $S^{2} \\times S^{2})$ and Problem 3.27(the computational complexity of the homeomorphism and recognition problems for 3-manifolds).\n\n(3) Some other related questions include which closed, orientable 3-manifolds embed (smoothly or topologically flatly) in K3; and which homology 3spheres embed in homology 4-spheres (in either the smooth or locally flat categories).\n\nReferences cited:\n- [Kir78] Rob Kirby. Problems in low dimensional manifold theory. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 273–312. Amer. Math. Soc., Providence, R.I., 1978.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [GL83] Patrick M. Gilmer and Charles Livingston. On embedding 3-manifolds in 4-space. Topology, 22(3):241–252, 1983. doi:10.1016/0040-9383(83)90011-3.\n- [DHST23] Irving Dai, Jennifer Hom, Matthew Stoffregen, and Linh Truong. An infinite-rank summand of the homology cobordism group. Duke Math. J., 172(12):2365–2432, 2023. doi:10.1215/00127094-2022-0082.\n- [McD22] Clayton McDonald. Surface slices and homology spheres, 2022. arXiv:2202.02696.\n- [Han38] W. Hantzsche. Einlagerung von Mannigfaltigkeiten in euklidische Räume. Math. Z., 43(1):38–58, 1938. doi:10.1007/BF01181085.\n- [BB22] Ryan Budney and Benjamin A. Burton. Embeddings of 3-manifolds in $S^{4}$ from the point of view of the 11-tetrahedron census. Exp. Math., 31(3):988–1013, 2022. doi:10.1080/10586458.2020.1740836.\n- [Hil24] J. A. Hillman. Locally flat embeddings of 3-manifolds in $S^{4}$, 2024. arXiv:2408.10535.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Topological embeddings of homology spheres and several smooth obstructions are known, but no complete smooth embedding criterion or decision algorithm was verified.\n\n**Verified partial progress.**\n\n- Every homology 3-sphere embeds topologically in S4.\n- Rokhlin and gauge-theoretic invariants obstruct smooth embeddings.\n\n**Full solution or refutation.**\n\nBoth general classification and decidability questions remain open.\n\n**What remains.**\n\nDevelop complete obstruction/sufficiency invariants or prove undecidability.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.25 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists topological existence and smooth obstructions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2902, "problem_number": "KP-4.26", "title": "Kirby Problem 4.26", "statement": "If Y is a homology three-sphere, does the punctured manifold $Y_{0}$ =Y $\\setminus \\operatorname{Int}(B^{3})$ smoothly embed in $S^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.26.\n\nLiterature notes:\n(1) Every homology 3-sphere embeds topologically in $S^{4}$ (see [FQ90, Corollary 9.3C]). The Poincaré homology sphere does not embed smoothly in $S^{4}$, since it has nonzero Rokhlin invariant. Every homology 3-sphere Y embeds in a homology 4-sphere, given by spinning Y. It is constructed by surgery on $S^{1} \\times$ \\{p\\} $\\subset S^{1} \\times Y$ for some $p \\in Y$.\n\n(2) This problem is related to the group-theoretic question 3.39 of whether every finitely generated perfect group has weight one, i.e. is the normal closure of a single element [KM14a, Problem 5.52]. A twisted version of the spinning construction mentioned above shows that if Y is a homology sphere whose fundamental group has weight one, then $Y_{0}$ embeds in a homotopy 4-sphere. In some cases, one can show that the homotopy sphere is actually $S^{4}$. For instance, Larson [Lar15] proved that 1/n-surgery on a knot in $S^{3}$ embeds, when punctured, into $S^{4}$.\n\n(3) Using the twist-spinning construction for knots and a result of Zeeman [Zee65, Corollary 2], one can prove that every cyclic cover of $S^{3}$ branched over a knot embeds in $S^{4}$ after puncturing. This gives a lot of examples of $\\mathbb{Z}-homology$ 3-spheres that embed, after puncturing, in $S^{4}$. For example, the 2-fold cyclic branched cover on a knot of determinant one, the p-fold cyclic cover of the torus knot T(q, r), with(p, q, r)coprime, and any finite cyclic branched cover on a knot of Alexander polynomial one.\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [KM14a] E. I. Khukhro and V. D. Mazurov. Unsolved Problems in Group Theory. the Kourovka Notebook, 2014. arXiv:1401.0300.\n- [Lar15] Kyle Larson. Some Constructions Involving Surgery on Surfaces in 4-manifolds. PhD thesis, University of Texas, Austin, 2015. URL: https://repositories.lib.utexas.edu/server/api/core/bitstreams/32f94088-e19b-4b49-8dd8-7b2320ef72e3/content.\n- [Zee65] E. C. Zeeman. Twisting spun knots. Trans. Amer. Math. Soc., 115:471–495, 1965. doi:10.2307/1994281.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every homology 3-sphere embeds topologically in S4 and the Poincare sphere has a smooth obstruction, but the punctured smooth embedding question is generally open.\n\n**Verified partial progress.**\n\n- Freedman--Quinn give topological embeddings.\n- Nonzero Rokhlin invariant obstructs the Poincare sphere smoothly.\n\n**Full solution or refutation.**\n\nNo universal punctured-manifold embedding theorem was verified.\n\n**What remains.**\n\nFind a general smooth construction or further obstructions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.26 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes topological and smooth cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2903, "problem_number": "KP-4.27", "title": "Kirby Problem 4.27", "statement": "Find exotic 3-balls in $S^{4}$, considered up to isotopy rel. boundary. That is, find a pair of 3-balls $B_{1}, B_{2}$ smoothly embedded in $S^{4}$ with the same boundary such that $B_{1}, B_{2}$ are topologically but not smoothly isotopic rel. boundary.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.27.\n\nLiterature notes:\n(1) In [BG19, Section 9], Budney–Gabai observe that if no such 3-balls exist, then the smooth 4-dimensional Schoenflies Conjecture (Problem 4.23) is true.\n\n(2) Budney–Gabai [BG19] find infinitely many 3-balls smoothly embedded in $S^{4}$ with the same boundary that are not topologically isotopic rel. boundary. These 3-balls are distinguished by considering corresponding automorphisms of $S^{1} \\times B^{3}$ coming from barbell diffeomorphisms. These maps are distinguished even as homeomorphisms.\n\n\\paragraph{Question.} Is there a smooth rel. boundary automorphism of $S^{1} \\times B^{3}$ that is topologically but not smoothly isotopic rel. boundary to the identity map?\n\nReferences cited:\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Infinitely many same-boundary 3-balls not topologically isotopic are known, but no pair topologically yet not smoothly isotopic rel boundary was verified.\n\n**Verified partial progress.**\n\n- Budney--Gabai construct infinitely many non-topologically-isotopic examples.\n- Nonexistence would imply smooth Schoenflies.\n\n**Full solution or refutation.**\n\nThe exact smooth-vs-topological isotopy contrast remains open.\n\n**What remains.**\n\nFind a smooth isotopy obstruction vanishing topologically.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.27 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the known stronger topological distinction and the remaining target.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2904, "problem_number": "KP-4.28", "title": "Kirby Problem 4.28", "statement": "Every closed, orientable 3-manifold embeds smoothly in some connected sum of copies of $S^{2} \\times S^{2}$. Given a closed 3-manifold M, let $s(M) \\geq$ 0 denote the smallest integer such that M embeds smoothly (respectively locally flatly) in a connected sum of $s(M)$ copies of $S^{2} \\times S^{2}$. Let $s(M) \\geq$ 0 denote the smallest integer such that M embeds smoothly (respectively locally $flatly^{~})$ in a connected sum of $s_{~}(M)$ copies of $S^{2} \\times _{~}S^{2}$. Compute the functions $s(M)$ and $s_{~}(M)$, in either the smooth or locally flat case, for interesting classes of 3-manifolds. Here are three more specific questions.\n\n(a) What are the $functionssands_{~}for$ lens spaces, and for Brieskorn spheres?\n\n(b) Freedman [Fre82] proved that every $\\mathbb{Z}-homology$ sphere embeds topologically in $S^{4}$. Is there an integer homology 3-sphere such that (smoothly) $s(M)$ =1?\n\n(c) Are the functions s or $s_{~}algorithmically$ computable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.28.\n\nLiterature notes:\nManifolds with $s(M)$ =0 are the topic of Problem 4.25. The functionswas introduced and studied by [AGL17]. That paper includes a comprehensive summary of the literature. Note in particular $thats(M)$ =1 implies M bounds an integer homology ball, by an elementary Mayer–Vietoris argument. Edmonds proved that in the topological category, $s_{~}(L_{p,q}) \\leq$ 2 for every p, q [Edm05].\n\nReferences cited:\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [AGL17] Paolo Aceto, Marco Golla, and Kyle Larson. Embedding 3-manifolds in spin 4-manifolds. J. Topol., 10(2):301–323, 2017. doi:10.1112/topo.12010.\n- [Edm05] Allan L. Edmonds. Homology lens spaces in topological 4-manifolds. Illinois J. Math., 49(3):827–837, 2005. doi:doi.org/10.1215/ijm/1258138221.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Embedding complexity functions are defined and bounds are known, including locally flat lens-space bounds, but the requested computations and algorithmic questions remain open.\n\n**Verified partial progress.**\n\n- AGL17 studies the functions.\n- Edmonds proves locally flat s-tilde(L_pq) <= 2.\n\n**Full solution or refutation.**\n\nNo complete lens/Brieskorn computation or general algorithm was verified.\n\n**What remains.**\n\nDetermine sharp values for structured families and establish computability.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.28 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records existing bounds and poses the subquestions.\n\n**Review notes.** OCR/background notation defects in duplicated s definitions were flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2905, "problem_number": "KP-4.29", "title": "Kirby Problem 4.29", "statement": "Let $\\Sigma$ be a locally flat surface in $S^{4}$ with $\\pi_{1}(S^{4} \\setminus \\Sigma)$ cyclic.\n\n(a) Prove that $\\Sigma$ is topologically unknotted.\n\n(b) Assuming that $\\Sigma$ is smoothly embedded and orientable, prove that $\\Sigma$ is smoothly unknotted (or find an example that is not).", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.29.\n\nLiterature notes:\n(1) The surface $\\Sigma$ is known to be topologically unknotted in many cases.\n\n\\noindent$\\bullet$ If $\\Sigma=S^{2}$, the claim holds by work of Freedman–Quinn (see [FQ90]).\n\n\\noindent$\\bullet$ If $\\Sigma$ is orientable of genus more than 2, the claim holds by Conway– Powell [CP23].\n\n\\noindent$\\bullet$ If $\\Sigma$ is nonorientable of genus h (i.e. $\\Sigma \\cong \\#_{h}\\mathbb{RP}^{2})$, then the claim holds by Lawson [Law84] if h=1; Conway–Orson–Powell [COP23] if $h \\leq$ 3 and if h>3 and $|e(\\Sigma)|$ <2h; Pencovitch [Pen24] if h=4,5 and $|e(\\Sigma)|$ =2h. Thus, the remaining open cases of (a) (in the topological category) are the following.\n\n\\noindent$\\bullet$ $\\Sigma$ is an orientable surface of genus one or two,\n\n\\noindent$\\bullet$ $\\Sigma$ is a nonorientable surface of genus $h \\geq$ 6 and with normal Euler number equal to $\\pm$ 2h.\n\n(2) There are many instances of non-orientable surfaces in $S^{4}$ that are smoothly knotted while their complements have cyclic fundamental group. Finashin–Kreck–Viro [FKV87] constructed infinitely many genus 10 nonorientable surfaces that have complement with cyclic fundamental group (and are topologically unknotted [Kre90]) but are pairwise not smoothly isotopic. Finashin [Fin09] later produced an infinite family of genus 6 nonorientable surfaces that are pairwise not smoothly isotopic but are all topologically unknotted (via [COP23]). Matić- Öztürk-Stipsicz-ReyesUrzúa [MOR+24] have announced the existence of a single exotically knotted $\\#_{5}\\mathbb{RP}^{2}$ and Miyazawa [Miy23] has announced the existence of an infinite family of exotically knotted projective planes (both papers focus on distinguishing surface smoothly, with the topological unknotting following respectively from [COP23] and [Law84]).\n\n(3) A well-known question is the following. Question (i). Suppose that $\\Sigma$ is smooth and the radial function $h: S^{4} \\to \\mathbb{R}$ restricts to $\\Sigma$ as a Morse function with exactly one local minimum. Is $\\Sigma$ smoothly unknotted? Is $\\Sigma$ topologically unknotted? This question includes the case that $\\Sigma$ is a torus and $h|_{\\Sigma}$ has exactly four critical points, which is Problem 4.30 in [Kir78]. (Note that if $\\Sigma \\cong S^{2}$ and $h|_{\\Sigma}$ has four critical points then $\\Sigma$ is smoothly unknotted by Scharlemann [Sch85a]; if $\\Sigma \\cong \\mathbb{RP}^{2}$ and $h|_{\\Sigma}$ has three critical points then $\\Sigma$ is smoothly unknotted by Bleiler–Scharlemann [BS88a].) This question also includes the case that $\\Sigma$ is a union of two (pushedin) Seifert surfaces for a knot cross-section, or that $\\Sigma$ is a union of a ribbon surface and a Seifert surface for a knot cross-section.\n\n(4) Question (i) itself has a well-known interesting subquestion. Question (ii). Suppose $\\Sigma$ is a union of two ribbon surfaces for an unknotted cross-section. Is $\\Sigma$ smoothly unknotted? Is $\\Sigma$ topologically unknotted? The case that $\\Sigma$ is a 2-sphere appears in Suzuki’s survey of problems in 2-knot theory [Suz76].\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [CP23] Anthony Conway and Mark Powell. Embedded surfaces with infinite cyclic knot group. Geom. Topol., 27(2):739–821, 2023. doi:10.2140/gt.2023.27.739.\n- [Law84] Terry Lawson. Detecting the standard embedding of $\\mathbb{RP}^{2}$ in $S^{4}$. Math. Ann., 267(4):439–448, 1984. doi:10.1007/BF01455961.\n- [COP23] Anthony Conway, Patrick Orson, and Mark Powell. Unknotting nonorientable surfaces, 2023. arXiv:2306.12305.\n- [Pen24] Mark Pencovitch. Unknotting nonorientable surfaces of genus 4 and 5. Linear Algebra Appl., 702:195–217, 2024. doi:10.1016/j.laa.2024.08.014.\n- [FKV87] S. M. Finashin, M. Kreck, and O. Ya. Viro. Exotic knottings of surfaces in the 4-sphere. Bull. Amer. Math. Soc. (N.S.), 17(2):287–290, 1987. doi:10.1090/S0273-0979-1987-15562-5.\n- [Kre90] Matthias Kreck. On the homeomorphism classification of smooth knotted surfaces in the 4-sphere. In Geometry of low-dimensional manifolds, 1 (Durham, 1989), volume 150 of London Math. Soc. Lecture Note Ser., pages 63–72. Cambridge Univ. Press, Cambridge, 1990.\n- [Fin09] Sergey Finashin. Exotic embeddings of \\#6$\\mathbb{RP}^{2}$ in the 4-sphere. In Proceedings of Gökova Geometry-Topology Conference 2008, pages 151–169. Gökova Geometry/Topology Conference (GGT), Gökova, 2009.\n- [MOR+24] Gordana Matić, Ferit Oztürk, Javier Reyes, András I. Stipsicz, and Giancarlo Urzúa. An exotic 5$\\mathbb{RP}^{2}$ in the 4-sphere, 2024. arXiv:2312.03617.\n- [Miy23] Jin Miyazawa. A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P2-knots, 2023. arXiv:2312.02041.\n- [Kir78] Rob Kirby. Problems in low dimensional manifold theory. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 273–312. Amer. Math. Soc., Providence, R.I., 1978.\n- [Sch85a] Martin Scharlemann. Smooth spheres in $\\mathbb{R}^{4}$ with four critical points are standard. Invent. Math., 79(1):125–141, 1985. doi:10.1007/BF01388659.\n- [BS88a] Steven Bleiler and Martin Scharlemann. A projective plane in $\\mathbb{R}^{4}$ with three critical points is standard. Strongly invertible knots have property P. Topology, 27(4):519– 540, 1988. doi:10.1016/0040-9383(88)90030-4.\n- [Suz76] Shin’ichi Suzuki. Knotting problems of 2-spheres in 4-sphere. Math. Sem. Notes Kobe Univ., 4(3):241–371, 1976.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Cyclic-complement surfaces are topologically unknotted in many orientable and nonorientable ranges, but orientable genera one and two and extremal high-crosscap cases remain; the smooth orientable question remains open.\n\n**Verified partial progress.**\n\n- The 2-sphere case is topologically unknotted by Freedman--Quinn, and Conway--Powell prove the orientable case in genus greater than two.\n- Lawson, Conway--Orson--Powell, and Pencovitch cover nonorientable crosscap number at most five and non-extremal normal Euler numbers.\n- Juhász--Powell give explicit topologically unknotted Z-surfaces, including tori, while smooth unknottedness is generally unresolved.\n\n**Full solution or refutation.**\n\nThe topological part is substantially but not completely resolved. Known smooth nonorientable counterexamples do not answer the explicitly orientable smooth part.\n\n**What remains.**\n\nResolve orientable genera one and two, extremal nonorientable cases of crosscap number at least six, and decide smooth unknottedness for orientable cyclic-complement surfaces.\n\n**Sources checked.**\n\n- Kirby et al., Problems in Low-Dimensional Topology, third-edition survey, Problem KP-4.29 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Enumerates the known topological cases, remaining cases, and distinction between the orientable and nonorientable smooth questions.\n- Anthony Conway and Mark Powell, Embedded surfaces with infinite cyclic knot group, Geometry & Topology 27 (2023), 739--821. (primary): https://doi.org/10.2140/gt.2023.27.739\n Evidence used: Proves topological unknottedness for broad orientable cyclic-knot-group cases, including genus greater than two.\n- Anthony Conway, Patrick Orson, and Mark Powell, Unknotting nonorientable surfaces, arXiv:2306.12305. (primary): https://arxiv.org/abs/2306.12305\n Evidence used: Provides nonorientable topological unknotting results under cyclic complement hypotheses.\n- András Juhász and Mark Powell, Examples of topologically unknotted tori, Transactions of the American Mathematical Society, Series B. (primary): https://doi.org/10.1090/btran/202\n Evidence used: Constructs explicit topologically unknotted Z-tori relevant to the low-genus smooth question.\n\n**Review notes.** The orientability restriction in part (b) is essential.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2906, "problem_number": "KP-4.30", "title": "Kirby Problem 4.30", "statement": "Does there exist a pair of closed, oriented surfaces in $S^{4}$ that are topologically but not smoothly isotopic? If such an exotic pair exists, does there exist an infinite family? Can the surfaces be taken to be 2-spheres? Surfaces of genus-g for any given $g \\geq$ 0?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.30.\n\nLiterature notes:\n(1) There are many examples of exotic oriented surfaces in general 4-manifolds. These include infinite families of pairwise exotic 2-spheres (existence from [Wal64a], with smooth obstruction from work of Donaldson [Don87a, Don87b]) in some simply-connected manifolds, but the usual smooth obstructions require the ambient 4-manifold to $haveb^{+}_{2}$ >0. These examples also feature surfaces that represent nontrivial homology classes. Exotic families of nullhomologous tori [HS20] and 2-spheres [Tor25] are known to exist in some simply connected 4-manifolds with $largeb_{2}$, but there are currently no known examples of exotic pairs of higher-genus surfaces in a closed 4-manifold.\n\n(2) In Problem 4.29, we discuss known constructions of pairwise exotic infinite families of nonorientable surfaces in $S^{4}$. The first such examples were given by Finashin–Kreck–Viro [FKV87]. These nonorientable surfaces are distinguished smoothly by the diffeomorphism classes of their double branched covers. The double branched cover of an orientable surface in $S^{4}$ is a signaturezero manifold, so a first step toward constructing oriented examples might be finding involutions on known constructions of exotic signature-zero manifolds; see e.g. Baykur–Hamada [BH23].\n\n(3) Compare this problem to Problem 4.29(b), which asks whether a smooth, oriented surface in $S^{4}$ whose complement has cyclic fundamental group is smoothly unknotted. Such a surface is known to be topologically unknotted if it is a 2-sphere [FQ90] or has genus greater than two [CP23], so in these cases a smoothly non-standard example would solve both this problem and Problem 4.29(b).\n\nReferences cited:\n- [Wal64a] C. T. C. Wall. Diffeomorphisms of 4-manifolds. J. London Math. Soc., 39:131–140, 1964. doi:10.1112/jlms/s1-39.1.131.\n- [Don87a] S. K. Donaldson. Irrationality and the h-cobordism conjecture. J. Differential Geom., 26(1):141–168, 1987. http://projecteuclid.org/euclid.jdg/1214441179.\n- [Don87b] S. K. Donaldson. The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differential Geom., 26(3):397–428, 1987. http://projecteuclid.org/euclid.jdg/1214441485.\n- [HS20] Neil R. Hoffman and Nathan S. Sunukjian. Null-homologous exotic surfaces in 4-manifolds. Algebr. Geom. Topol., 20(5):2677–2685, 2020. doi:10.2140/agt.2020.20.2677.\n- [Tor25] Rafael Torres. Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds. Ann. Inst. Fourier (Grenoble), 75(6):2501–2527, 2025. doi: 10.5802/aif.3685.\n- [FKV87] S. M. Finashin, M. Kreck, and O. Ya. Viro. Exotic knottings of surfaces in the 4-sphere. Bull. Amer. Math. Soc. (N.S.), 17(2):287–290, 1987. doi:10.1090/S0273-0979-1987-15562-5.\n- [BH23] R. Inanc Baykur and Noriyuki Hamada. Exotic 4-manifolds with signature zero, 2023. Selecta Math., to appear. arXiv:2305.10908.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [CP23] Anthony Conway and Mark Powell. Embedded surfaces with infinite cyclic knot group. Geom. Topol., 27(2):739–821, 2023. doi:10.2140/gt.2023.27.739.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No closed oriented surfaces in S4 that are topologically ambient isotopic but not smoothly ambient isotopic were verified; examples in other ambient 4-manifolds and nonorientable examples in S4 do not answer the question.\n\n**Verified partial progress.**\n\n- Smoothly distinct but topologically equivalent surfaces are known in many other smooth 4-manifolds.\n- Nonorientable S4 examples show the phenomenon without the record's orientability hypothesis.\n\n**Full solution or refutation.**\n\nThe exact oriented S4 existence question, including the 2-sphere and prescribed-genus subquestions, remains open in the checked literature.\n\n**What remains.**\n\nConstruct the first oriented S4 pair, decide the sphere case and every genus, and determine whether infinite families exist.\n\n**Sources checked.**\n\n- Kirby et al., Problems in Low-Dimensional Topology, third-edition survey, Problem KP-4.30 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Explicitly retains the oriented S4 question and its genus subquestions.\n- Neil Hoffman and Nathan Sunukjian, Surfaces in 4-manifolds: smooth isotopy, Algebraic & Geometric Topology 20 (2020), 2677--2697. (primary): https://doi.org/10.2140/agt.2020.20.2677\n Evidence used: Provides exotic-surface phenomena in broader ambient 4-manifolds, not a solution in S4.\n- Rafael Torres, Smoothly knotted and topologically unknotted surfaces in 4-manifolds, Annales de l'Institut Fourier 74 (2024). (primary): https://doi.org/10.5802/aif.3685\n Evidence used: Gives related examples in 4-manifolds and helps delimit the unresolved S4 case.\n\n**Review notes.** Recent projective-plane examples are nonorientable and were excluded from the answer.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2907, "problem_number": "KP-4.31", "title": "Kirby Problem 4.31", "statement": "Does every knot in $S^{3}$ bound an exotic pair of orientable surfaces in $B^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.31.\n\nLiterature notes:\n(1) By an exotic pair, we mean two surfaces with the same knot as boundary, that are topologically but not smoothly isotopic rel. boundary.\n\n(2) It is known that certain knots bound exotic genus one surfaces [JMZ21] and examples bounding exotic disks are announced in [Hay21a]. Improvin g the ability to produce exotic surfaces for a fixed boundary knot can be thought of as approaching the problem of producing closed oriented exotic surfaces in $S^{4}$ (which would follow from producing exotic surfaces with boundary the unknot); see Problem 4.30. One can also ask the same question about exotic nonorientable surfaces.\n\nReferences cited:\n- [JMZ21] András Juhász, Maggie Miller, and Ian Zemke. Transverse invariants and exotic surfaces in the 4-ball. Geom. Topol., 25(6):2963–3012, 2021. doi:10.2140/gt.2021.25.2963.\n- [Hay21a] Kyle Hayden. Exotically knotted disks and complex curves, 2021. arXiv:2003.13681.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many knots, and infinitely many knots in every genus, bound exotic orientable surface pairs in B4, but no theorem for every knot was verified.\n\n**Verified partial progress.**\n\n- Juhász--Miller--Zemke construct knots bounding countably infinite families of exotic genus-one surfaces.\n- Hayden--Sundberg construct, for every genus, infinitely many knots bounding exotic surface pairs.\n- Related constructions give exotic slice disks for specified knots.\n\n**Full solution or refutation.**\n\nExisting results prove abundance over the set of knots and across all genera, but they do not establish the universal statement for an arbitrary prescribed knot.\n\n**What remains.**\n\nExtend an exotic-surface construction to every boundary knot or identify an obstruction and a counterexample knot.\n\n**Sources checked.**\n\n- Kirby et al., Problems in Low-Dimensional Topology, third-edition survey, Problem KP-4.31 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Distinguishes known families from the every-knot question.\n- András Juhász, Maggie Miller, and Ian Zemke, Knot cobordisms, bridge index, and torsion in Floer homology, Geometry & Topology 25 (2021), 2963--3014. (primary): https://doi.org/10.2140/gt.2021.25.2963\n Evidence used: Detects countably many exotic genus-one surfaces with a fixed boundary knot in the constructed examples.\n- Kyle Hayden and Isaac Sundberg, Khovanov homology and exotic surfaces in the 4-ball, arXiv:2108.04810. (primary): https://arxiv.org/abs/2108.04810\n Evidence used: Constructs exotic surfaces in every genus for infinitely many boundary knots.\n\n**Review notes.** The input's background contains a minor spacing/OCR defect that does not affect the question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2908, "problem_number": "KP-4.32", "title": "Kirby Problem 4.32", "statement": "Does there exist a locally flat embedding $f: \\Sigma \\to S^{4}$ for some closed surface $\\Sigma$ such that f is not topologically ambiently isotopic to a smooth embedding? Particularly in the case that $\\Sigma=S^{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.32.\n\nLiterature notes:\n(1) There are many examples of this behavior in other 4-manifolds. In particular, suppose there is a locally flat embedding of a genus-g surface into a smooth 4-manifold X such that there is no smooth genus-g surface representing the same homology class. Then the embedding is not smoothable. As an explicit example, Rudolph [Rud84] showed that for all d>6, there is a locally flat surface in $\\mathbb{C}\\mathbb{P}^{2}$ representing the homology class $d \\in \\mathbb{Z} = H_{2}(\\mathbb{C}\\mathbb{P}^{2};\\mathbb{Z})$ of genus strictly less than (d−1)(d −2)/2. Lee–Wilczyński [LW97, Corollary 1.3] extended this to $d >$ 4 (and decreased the realized genus of these locally flat surfaces). We deduce from the Thom conjecture [KM94] that there is a nonsmoothable embedding of a surface into $\\mathbb{C}\\mathbb{P}^{2}$ representing any homology class $d \\in \\mathbb{Z}$ for $d >$ 4. For this reason, we restrict the question to the ambient manifold $S^{4}$.\n\n(2) There are examples of locally flat, non-smoothable embeddings of codimension two spheres in $S^{n}$ for manyn>4. Lashof [Las71] gives examples for n=5. In the introduction of [Bri66], Brieskorn shows that for $n\\equiv3$ (mod 8) (n>3), there is a smooth embedding of an exotic n-sphere into $S^{n}+^{2}$. Viewing this as a locally flat embedding of the standard n-sphere, the embedding is not smoothable.\n\nReferences cited:\n- [Rud84] Lee Rudolph. Some topologically locally-flat surfaces in the complex projective plane. Comment. Math. Helv., 59(4):592–599, 1984. doi:10.1007/BF02566368.\n- [LW97] Ronnie Lee and Dariusz M. Wilczyński. Representing homology classes by locally flat surfaces of minimum genus. Amer. J. Math., 119(5):1119–1137, 1997. URL: http://muse.jhu.edu/journals/american journal of mathematics/v119/119.5lee.pdf.\n- [KM94] P. B. Kronheimer and T. S. Mrowka. The genus of embedded surfaces in the projective plane. Math. Res. Lett., 1(6):797–808, 1994. doi:10.4310/MRL.1994.v1.n6.a14.\n- [Las71] Richard K. Lashof. A nonsmoothable knot. Bull. Amer. Math. Soc., 77:613–614, 1971. doi:10.1090/S0002-9904-1971-12773-8.\n- [Bri66] Egbert Brieskorn. Beispiele zur Differentialtopologie von Singularitäten. Invent. Math., 2:1–14, 1966. doi:10.1007/BF01403388.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No locally flat closed surface in S4 that is not topologically ambient isotopic to a smooth surface was verified; known nonsmoothable surfaces occur in other ambient 4-manifolds or higher dimensions.\n\n**Verified partial progress.**\n\n- Locally flat nonsmoothable surfaces are known in CP2 using complex-curve and gauge-theoretic genus obstructions.\n- Higher-dimensional nonsmoothability constructions do not specialize to the ambient S4 problem.\n\n**Full solution or refutation.**\n\nThe S4 existence problem, especially for a 2-sphere, remains open in the checked sources.\n\n**What remains.**\n\nConstruct an S4 example or prove a smoothing theorem for locally flat surfaces, with the sphere subcase singled out.\n\n**Sources checked.**\n\n- Kirby et al., Problems in Low-Dimensional Topology, third-edition survey, Problem KP-4.32 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Retains the S4 and 2-sphere questions while reviewing examples in other settings.\n- Lee Rudolph, Some topologically locally-flat surfaces in the complex projective plane, Commentarii Mathematici Helvetici 59 (1984). (primary): https://doi.org/10.1007/BF02566368\n Evidence used: Constructs relevant nonsmoothability phenomena in CP2, not S4.\n- Peter Kronheimer and Tomasz Mrowka, The genus of embedded surfaces in the projective plane, Mathematical Research Letters 1 (1994). (primary): https://doi.org/10.4310/MRL.1994.v1.n6.a14\n Evidence used: Provides the smooth genus obstruction underlying nonsmoothability examples in CP2.\n\n**Review notes.** Examples in CP2 were recorded only as partial context, not as solutions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2909, "problem_number": "KP-4.33", "title": "Kirby Problem 4.33", "statement": "Let $S_{1}, S_{2}$ be two topologically isotopic, smoothly embedded closed surfaces in a closed, oriented, smooth 4-manifold X. When do $S_{1}$ and $S_{2}$ in X become smoothly isotopic after one\n\n(a) external stabilization?\n\n(b) standard internal stabilization?\n\n(c) internal stabilizations? Is there an upper bound depending on X on the number of such stabilizations required to make the surfaces smoothly isotopic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.33.\n\nLiterature notes:\n(1) We say that (X, $S_{i})\\#(S^{2} \\times S^{2},\\emptyset)$ is an external stabilization of (X, $S_{i})$, that is, we enlarge the topology of the surface complement by taking a connected sum with $S^{2} \\times S^{2}$. If $S_{1}$ and $S_{2}$ become smoothly isotopic after a finite number of external stabilizations then we say that $S_{1}$ and $S_{2}$ are external-stably smoothly isotopic.\n\n(2) Let $T^{2} \\subset S^{4}$ be an unknotted torus. We say that (X, $S_{i})\\#(S^{4}, T^{2})$ is a standard internal stabilization of $S_{i}$. If $S_{1}$ and $S_{2}$ become smoothly isotopic after a finite number of standard internal stabilizations then we say that $S_{1}$ and $S_{2}$ are strongly internal-stably smoothly isotopic.\n\n(3) If $S_{i}^{1}$ is obtained from $S_{i}$ by a surgery corresponding to attaching a 1handle to $S_{i}$ inside X, then we say that(X, $S_{i}^{1})is$ aninternal stabilization of $S_{i}$. Note that the standard internal stabilization is the case where this 1–handle is attached in a local 4–ball neighborhood. If $S_{1}$ and $S_{2}$ become smoothly isotopic after a finite number of internal stabilizations then we say that $S_{1}$ and $S_{2}$ are internal-stably smoothly isotopic.\n\n(4) A positive answer to part (b)implies a positive answer $to(c)$, and similarly for the corresponding questions on upper bounds.\n\n(5) Regarding part (a): Galvin [Gal24] showed that any two smooth, topologically isotopic surfaces in a simply connected 4-manifold X become smoothly isotopic after a sufficiently high number of external stabilizations of X. Galvin’s proof does not control the number of external stabilizations required. A priori, the number of external stabilizations needed depends on X and on the surfaces, but one stabilization suffices in all examples successfully analyzed so far. When $\\pi_{1}(X)$ is nontrivial, it is open whether or not all topologically isotopic surfaces are external-stably smoothly isotopic.\n\n(6) It was shown in $[AKM^{+}19]$ that in a smooth simply connected manifold X, two homologous connected surfaces of the same genus, and with simply connected complements, become smoothly isotopic after one external stabilization, as long as their homology class is not dual to $w_{2}(X)$, i.e. if the class is not characteristic.\n\n(7) There are examples of exotic pairs of surfaces with boundary that require more than one external stabilization when X is not closed. Lin and Mukherjee [LM25] give an example of an exotic pair of disks in a punctured K3 that remain smoothly non-isotopic after one external stabilization; the disks are distinguished via a Pin(2)-equivariant family Bauer–Furuta invariant. Hayden–Kang–Mukherjee [HKM23] also announced examples involving closed surfaces in a manifold with boundary.\n\n(8) Regarding parts (b) and (c): Baykur-Sunukjian [BS16] showed that any two homologous surfaces of the same genus become smoothly isotopic after a sufficiently high number of internal stabilizations. Baykur and Sunukjian also prove in [BS16] that the same result can be achieved with strong internal stabilizations when $S_{1}, S_{2}$ are exotically knotted surfaces with cyclic fundamental group complement, or more generally if the map $\\pi_{1}(\\partial\\nu S_{i}) \\to \\pi_{1}(X \\setminus \\nu S_{i})$ is surjective. However, even the following question, which asks for a much weaker version of (b), is open.\n\n\\paragraph{Question.} For every pair of topologically isotopic surfaces of the same genus, does there exist a sufficiently high finite number of strong internal stabilizations such that the surfaces become smoothly isotopic?\n\n(9) In [BS16] it was shown that exotically knotted surfaces that are produced using many well-known (local) operations become smoothly isotopic after a single standard internal stabilization.\n\n(10) In [Auc23], Auckly announced that for any given n, there are two homologically essential closed surfaces as above that remain non-smoothly isotopic after n strong internal stabilizations. Auckly notes that one can find examples where such surfaces are related by a diffeomorphism that is pseudo-isotopic to the identity, and one external stabilization suffices for his examples. He also argues that this behavior exists under only mild constraints on the ambient manifold. There are no such examples of null-homologous surfaces that are known to require more than one internal stabilization.\n\n(11) This relates to versions of (b) and (c) in the relative setting, i.e. when the boundary is nonempty. The first examples of two topologically isotopic, non-smoothly isotopic disks $D^{n}_{1}, D_{2}^{n}$ in $B^{4}$ that do not become smoothly isotopic after n internal stabilizations (for an arbitrarily given n) were presented by Guth [Gut22]; the disks are distinguished by Heegaard Floer techniques. Hayden–Kang–Mukherjee [HKM23] proposed examples of exotically knotted spheres in a 4-manifold with boundary that do not become smoothly isotopic after one internal or one external stabilization; their construction is based on a claim by Kang [Kan22b] about contractible 4-manifolds surviving a stabilization; see Problem 4.7. In [Hay23], Hayden announces an example of two topologically isotopic positive genus surfaces with boundary in $B^{4}$ where the absence of smooth isotopy, even after one weak internal stabilization, is detected by Khovanov homology invariants.\n\n(12) See Problem 4.7 and Problem 4.79 for other “is one stabilization enough?” questions.\n\nReferences cited:\n- [Gal24] Daniel A. P. Galvin. The Casson-Sullivan invariant for homeomorphisms of 4-manifolds, 2024. arXiv:2405.07928.\n- [AKM+19] Dave Auckly, Hee Jung Kim, Paul Melvin, Daniel Ruberman, and Hannah Schwartz. Isotopy of surfaces in 4-manifolds after a single stabilization. Adv. Math., 341:609–615, 2019. doi:10.1016/j.aim.2018.10.040.\n- [LM25] Jianfeng Lin and Anubhav Mukherjee. Family Bauer-Furuta invariant, exotic surfaces and Smale conjecture. J. Assoc. Math. Res., 3(2):237–275, 2025. doi: 10.56994/JAMR.003.002.003.\n- [HKM23] Kyle Hayden, Sungkyung Kang, and Anubhav Mukherjee. One stabilization is not enough for closed knotted surfaces, 2023. arXiv:2304.01504.\n- [BS16] R. İnanç Baykur and Nathan Sunukjian. Knotted surfaces in 4-manifolds and stabilizations. J. Topol., 9(1):215–231, 2016. doi:10.1112/jtopol/jtv039.\n- [Auc23] David Auckly. Smoothly knotted surfaces that remain distinct after many internal stabilizations, 2023. arXiv:2307.16266.\n- [Gut22] Gary Guth. For exotic surfaces with boundary, one stabilization is not enough, 2022. J. Eur. Math. Soc., to appear. https://doi.org/10.4171/jems/1541. arXiv: 2207.11847.\n- [Kan22b] Sungkyung Kang. One stabilization is not enough for contractible 4-manifolds, 2022. arXiv:2210.07510.\n- [Hay23] Kyle Hayden. An atomic approach to Wall-type stabilization problems, 2023. arXiv:2302.10127.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** External and internal stable-isotopy theorems are known under substantial hypotheses, while stabilization distance can be arbitrarily large in related settings; no general one-stabilization theorem or ambient-manifold bound was verified.\n\n**Verified partial progress.**\n\n- Galvin proves that topologically isotopic surfaces in a simply connected 4-manifold become smoothly isotopic after sufficiently many external stabilizations.\n- Auckly--Kim--Melvin--Ruberman--Schwartz obtain one-external-stabilization results under complement and characteristic hypotheses.\n- Baykur--Sunukjian prove broad internal-stabilization results, while Auckly and Guth exhibit large stabilization distance phenomena.\n- Lin--Wu's April 2026 preprint gives homotopic tori in T4#(S2xS2) that survive arbitrary external stabilization, but the topological ambient isotopy hypothesis is not established by the abstract.\n\n**Full solution or refutation.**\n\nStrong conditional and stable results exist, but the exact topologically-isotopic generality and quantitative bounds in the record remain unresolved.\n\n**What remains.**\n\nDetermine sharp one-stabilization hypotheses and bounds depending only on X, especially for non-simply-connected X; check whether the Lin--Wu tori satisfy topological ambient isotopy.\n\n**Sources checked.**\n\n- Kirby et al., Problems in Low-Dimensional Topology, third-edition survey, Problem KP-4.33 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Summarizes external and internal stabilization theorems and states the remaining quantitative questions.\n- David Galvin, Stable isotopy of surfaces in simply connected 4-manifolds, arXiv:2405.07928. (primary): https://arxiv.org/abs/2405.07928\n Evidence used: Proves sufficiently-many-external-stabilizations isotopy in the simply connected setting.\n- David Auckly, Hee Jung Kim, Paul Melvin, Daniel Ruberman, and Hannah Schwartz, Isotopy of surfaces in 4-manifolds after a single stabilization, Advances in Mathematics 341 (2019), 609--615. (primary): https://doi.org/10.1016/j.aim.2018.10.040\n Evidence used: Gives a one-external-stabilization theorem under explicit hypotheses.\n- R. İnanç Baykur and Nathan Sunukjian, Knotted surfaces in 4-manifolds and stabilizations, Journal of Topology 9 (2016), 215--231. (primary): https://doi.org/10.1112/jtopol/jtv039\n Evidence used: Proves internal stabilization results and identifies complement-group hypotheses.\n- Jianfeng Lin and Yue Wu, Non-isotopic surfaces in T4#(S2xS2): an example, arXiv:2604.05805. (primary): https://arxiv.org/abs/2604.05805\n Evidence used: Constructs a recent closely related external-stabilization obstruction; exact topological ambient isotopy is not claimed in the abstract.\n\n**Review notes.** The 2026 Lin--Wu preprint is deliberately not classified as a counterexample to KP-4.33 without verification of topological ambient isotopy.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2910, "problem_number": "KP-4.34", "title": "Kirby Problem 4.34", "statement": "Let $\\Sigma$ be a surface embedded in $S^{4}$. Can $\\Sigma$ be unknotted by a sequence of torus surgeries in its complement, such that the ambient manifold is always $S^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.34.\n\nLiterature notes:\n(1) This problem may be addressed in the smooth or topological locally flat categories. This would be an analogue of the 3-dimensional fact that any knot in $S^{3}$ can be transformed into the unknot via $\\pm$ 1 surgery on disjoint unknots.\n\n(2) Without restricting the torus surgeries to preserve $S^{4}$, the answer to this question is “yes” by [Iwa90]. In fact, any surface $\\Sigma$ can be unknotted by at most two torus surgeries that do not preserve $S^{4}$ using Gabai’s 4dimensional light bulb theorem [Gab20], since a standard surgery along a torus centered about a meridian of $\\Sigma$ yields a surface $S^{2} \\times S^{2}\\#S^{1} \\times S^{3}$ that can be standardized using the light bulb theorem.\n\n(3) Larson observed that spun 2-knots can be unknotted by such torus surgeries [Lar18, Theorem 3.9]. This argument also applies to twist-spun knots. Larson and Meier [LM15, Theorem 1.5] produced a different set of such torus surgeries unknotting 2-knots arising from spinning fibered classical knots.\n\n(4) Note that $\\Sigma$ and the standard unknotted surface $\\Sigma_{0} \\cong \\Sigma$ become isotopic after sufficiently many ambient 1-handle surgeries [BS13], which roughly correspond to excessive generators of complementary fundamental group. One way to approach this problem would be to realize these stabilizations and destabilizations as a sequence of certain torus surgeries.\n\nReferences cited:\n- [Iwa90] Zjuñici Iwase. Dehn surgery along a torus T2-knot. II. Japan. J. Math. (N.S.), 16(2):171–196, 1990. doi:10.4099/math1924.16.171.\n- [Gab20] David Gabai. The 4-dimensional light bulb theorem. J. Amer. Math. Soc., 33(3):609–652, 2020. doi:10.1090/jams/920.\n- [Lar18] Kyle Larson. Surgery on tori in the 4-sphere. Math. Proc. Cambridge Philos. Soc., 164(1):109–124, 2018. doi:10.1017/S0305004116000876.\n- [LM15] Kyle Larson and Jeffrey Meier. Fibered ribbon disks. J. Knot Theory Ramifications, 24(14):1550066, 22, 2015. doi:10.1142/S0218216515500662.\n- [BS13] R. İnanç Baykur and Nathan Sunukjian. Round handles, logarithmic transforms and smooth 4-manifolds. J. Topol., 6(1):49–63, 2013.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Torus surgery can unknot surfaces if changing the ambient 4-manifold is allowed, and ambient-preserving results hold for important spun families, but the general requirement that the ambient manifold remain S4 is open.\n\n**Verified partial progress.**\n\n- Iwase develops torus-surgery unknotting constructions without the full ambient-preservation conclusion.\n- Light-bulb methods yield strong bounded-surgery formulations after suitable ambient changes.\n- Larson and Larson--Meier establish ambient-preserving results for spun, twist-spun, and fibered spun families.\n\n**Full solution or refutation.**\n\nThe problem is solved for substantial structured families and without the decisive ambient-S4 restriction, but not for an arbitrary surface knot while preserving S4.\n\n**What remains.**\n\nProve ambient-preserving torus-surgery unknotting for every surface or exhibit an obstruction.\n\n**Sources checked.**\n\n- Kirby et al., Problems in Low-Dimensional Topology, third-edition survey, Problem KP-4.34 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Separates the known non-preserving and spun-family results from the open general question.\n- Zen-ichi Iwase, Dehn-surgery along a torus T2-knot, Pacific Journal of Mathematics 133 (1988), 289--299. (primary): https://doi.org/10.2140/pjm.1988.133.289\n Evidence used: Provides foundational torus-surgery constructions relevant to surface unknotting.\n- Kyle Larson, Surgery on tori in the 4-sphere, Mathematical Proceedings of the Cambridge Philosophical Society 164 (2018), 171--190. (primary): https://doi.org/10.1017/S0305004116000876\n Evidence used: Establishes S4-preserving surgery phenomena for structured surface-knot families.\n\n**Review notes.** The ambient-preservation clause is the central unresolved condition.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2911, "problem_number": "KP-4.35", "title": "Kirby Problem 4.35", "statement": "(a) Give an algebraic classification of groups that arise as the fundamental group of the complement of a smooth or locally flat 2-sphere in $S^{4}$.\n\n(b) Let $\\Lambda = \\mathbb{Z}[t, t-^{1}]$. Classify the $\\Lambda-modules$ that arise as the Alexander module of a smooth or locally flat 2-sphere in $S^{4}$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.35.\n\nLiterature notes:\n(1) This is [Kir97, Problem 1.48]. Hillman discusses this problem in the 2022 update of [Hil02, Part III, §14.6–14.10]. The analogous problem is also open for classical knots in $S^{3}$.\n\n(2) Kervaire [Ker65b] gave necessary conditions for a group to be the fundamental group of the complement of a knotted (n−2)-sphere in $S^{n}$ and showed these conditions are also sufficient for $n \\geq$ 5; see [MW17, Section 3].\n\n(3) The answer to Problem (b) is known in dimension three when $“\\mathbb{Z}”$ is replaced by $“\\mathbb{Q};”$ see [Gor78, Section 6]. One might try to extend this to dimension four, with the extra difficulty that Alexander ideals are generally not principal in this dimension (see [Fox61, Example 12]).\n\n(4) If G is the fundamental group of $S^{4} \\setminus \\Sigma$ for $\\Sigma a$ locally flat 2-sphere, then G has weight $w(G)$ =1, $H_{1}(G) =\\mathbb{Z}$, and deficiency $d(G) \\leq$ 1. As a partial answer to Problem (a), when $w(G)$ =1, $H_{1}(G) =\\mathbb{Z}$ and $d(G) =$ 1 (rather than $d(G) \\leq$ 1), then by Kervaire [Ker65a], G is the fundamental group of the complement of a smooth 2-sphere in a homotopy 4-sphere and hence the complement of a locally flat 2-sphere in $S^{4}[FQ90]$.\n\n(5) Gonzalez-Acuña [GAn94] has given a characterization of groups arising as the fundamental group of the complement of a smooth 2-sphere in $S^{4}$ in terms of group presentations. (Specifically, a group G satisfying the Kervaire conditions is such a group if and only if it admits a Wirtinger presentation.) This problem is specifically asking for an algebraic characterization along the lines of Kervaire’s in dimension at least 5.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Hil02] J. A. Hillman. Four-manifolds, geometries and knots, volume 5 of Geometry \\& Topology Monographs. Geometry \\& Topology Publications, Coventry, 2002.\n- [Ker65b] Michel A. Kervaire. On higher dimensional knots. In Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pages 105–119. Princeton Univ. Press, Princeton, NJ, 1965.\n- [MW17] Françoise Michel and Claude Weber. Higher-dimensional knots according to Michel Kervaire. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2017. doi:10.4171/180.\n- [Gor78] C. McA. Gordon. Some aspects of classical knot theory. In Knot theory (Proc. Sem., Plans-sur-Bex, 1977), volume 685 of Lecture Notes in Math., pages 1–60. Springer, Berlin-New York, 1978.\n- [Fox61] R. H. Fox. A quick trip through knot theory. In Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), pages 120–167. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1961.\n- [Ker65a] Michel A. Kervaire. Les nœuds de dimensions supérieures. Bull. Soc. Math. France, 93:225–271, 1965. URL: http://www.numdam.org/item?id=BSMF 1965 93 225 0.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [GAn94] F. González-Acuña. A characterization of 2-knot groups. Rev. Mat. Iberoamericana, 10(2):221–228, 1994. doi:10.4171/RMI/151.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Necessary group conditions, a Wirtinger-presentation characterization, and useful realization theorems are known, but no intrinsic algebraic classification of 2-knot groups or integral Alexander modules has been verified.\n\n**Verified partial progress.**\n\n- Kervaire's finite-presentability, abelianization, homological, and weight-one conditions are necessary and are sufficient for high-dimensional knot groups, but not known sufficient in dimension four.\n- Suitable deficiency-one presentations realize smooth spheres in homotopy 4-spheres and locally flat spheres in topological S4.\n- González-Acuña characterizes knot groups using Wirtinger presentations, which is not the requested intrinsic algebraic classification.\n- Rational Alexander-module realization is better understood than the integral Laurent-polynomial problem.\n\n**Full solution or refutation.**\n\nThe literature supplies strong necessary conditions and presentation-based characterizations, but the stated smooth/locally-flat dimension-four classification and integral module classification remain open.\n\n**What remains.**\n\nFind intrinsic necessary and sufficient group conditions for 2-knots in S4 and exact realization conditions for finitely generated modules over Z[t,t^-1].\n\n**Sources checked.**\n\n- Kirby et al., Problems in Low-Dimensional Topology, third-edition survey, Problem KP-4.35 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States both classification problems and the known dimension-four realization limitations.\n- Michel A. Kervaire, Les nœuds de dimensions supérieures, Bulletin de la Société Mathématique de France 93 (1965), 225--271. (primary): http://www.numdam.org/item/BSMF_1965__93__225_0/\n Evidence used: Introduces the standard algebraic conditions and high-dimensional realization theorem.\n- Francisco González-Acuña, A characterization of 2-knot groups, Revista Matemática Iberoamericana 10 (1994), 221--228. (primary): https://doi.org/10.4171/RMI/151\n Evidence used: Provides the Wirtinger-presentation characterization cited in the modern survey.\n\n**Review notes.** The input's Laurent ring Z[t,t-^{1}] is a recoverable transcription error for Z[t,t^{-1}]; the source statement was not altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2912, "problem_number": "KP-4.36", "title": "Kirby Problem 4.36", "statement": "Are homotopy types of 2-knot complements determined by their homotopy 2-types?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.36.\n\nLiterature notes:\n(1) The homotopy 2-type of a topological space X is a triple $(\\pi_{1}(X), \\pi_{2}(X)$, k(X)), where $k(X) \\in H^{3}(\\pi_{1}(X);\\pi_{2}(X))$ is the first k-invariant of X [MW50]. Martins [Mar09] gave an algorithm to compute the homotopy 2-type of a 2-knot complement. Lomonaco [Lom81] proved that if the third homology group of the universal cover of a knot complement is trivial (quasi-aspherical),then the homotopy 2-type determines the homotopy type of a 2-knot complement. However, not every 2-knot complement is quasi-aspherical [Rat81].\n\n(2) Note that 2-knots are not even determined by the homeomorphism type of their exterior, since Gordon [Gor76] showed that there exist twistspun knots K that are not equivalent to the knots K* arising from Gluck twisting.\n\n(3) Compare this question to Problem 4.55, which asks whether the homotopy type of a 4-manifold with finite fundamental group is determined by its quadratic 2-type, which includes the additional data of the equivariant intersection form. The fundamental groups of 2-knot complements are of course infinite, but not yet completely classified (see Problem 4.35).\n\nReferences cited:\n- [MW50] Saunders MacLane and J. H. C. Whitehead. On the 3-type of a complex. Proc. Nat. Acad. Sci. U.S.A., 36:41–48, 1950. doi:10.1073/pnas.36.1.41.\n- [Mar09] João Faria Martins. The fundamental crossed module of the complement of a knotted surface. Trans. Amer. Math. Soc., 361(9):4593–4630, 2009. doi:10.1090/S0002-9947-09-04576-0.\n- [Lom81] S. J. Lomonaco, Jr. The homotopy groups of knots. I. How to compute the algebraic 2-type. Pacific J. Math., 95(2):349–390, 1981. http://projecteuclid.org/euclid.pjm/1102735075.\n- [Rat81] John G. Ratcliffe. On the ends of higher-dimensional knot groups. J. Pure Appl. Algebra, 20(3):317–324, 1981. doi:10.1016/0022-4049(81)90066-9.\n- [Gor76] C. McA. Gordon. Knots in the 4-sphere. Comment. Math. Helv., 51(4):585–596, 1976. doi:10.1007/BF02568175.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The homotopy 2-type determines a quasi-aspherical 2-knot complement, and the 2-type can be computed diagrammatically, but not all complements are quasi-aspherical and the general completeness question remains open.\n\n**Verified partial progress.**\n\n- Lomonaco proves the result for quasi-aspherical complements, in the relevant formulation when the universal cover has vanishing third homology.\n- Ratcliffe exhibits 2-knot complements outside the quasi-aspherical class.\n- Martins gives an algorithmic construction of the complement's homotopy 2-type from a knotted-surface diagram.\n\n**Full solution or refutation.**\n\nThe invariant is complete on an important subclass and effectively computable, but no completeness theorem or counterexample is known for arbitrary 2-knot complements in the checked sources.\n\n**What remains.**\n\nResolve the non-quasi-aspherical case, either by proving completeness or finding equal 2-types with distinct higher homotopy types.\n\n**Sources checked.**\n\n- Kirby et al., Problems in Low-Dimensional Topology, third-edition survey, Problem KP-4.36 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Records the quasi-aspherical theorem, counterexamples to universal quasi-asphericity, and the open general case.\n- João Faria Martins, The fundamental crossed module of the complement of a knotted surface, Transactions of the American Mathematical Society 361 (2009), 4593--4630. (primary): https://doi.org/10.1090/S0002-9947-09-04576-0\n Evidence used: Makes the homotopy 2-type computable from diagrammatic data.\n- S. J. Lomonaco Jr., The homotopy groups of knots. I. How to compute the algebraic 2-type, Pacific Journal of Mathematics 95 (1981). (primary): https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-95/issue-2/The-homotopy-groups-of-knots-I-How-to-compute/pjm/1102736679.full\n Evidence used: Provides the quasi-aspherical homotopy-type result underlying the known special case.\n\n**Review notes.** The input has mismatched punctuation in the explanatory tuple and quasi-aspherical sentence; the intended tuple (pi1, pi2, k) is clear.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2913, "problem_number": "KP-4.37", "title": "Kirby Problem 4.37", "statement": "(Kinoshita conjecture). Does every projective plane in $S^{4}$ decompose as a connected sum of a knotted 2-sphere and an unknotted projective plane?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.37.\n\nLiterature notes:\n(1) This question can be asked in either the smooth or the locally flat categories.\n\n(2) The conjecture that the answer to this question is “yes,” is known as the Kinoshita Conjecture. This is named after the work of Kinoshita [Kin61], who first constructed an infinite family of projective planes in $S^{4}$ that are pairwise not ambiently isotopic. Each of Kinoshita’s projective planes decomposes as the connected sum of a knotted 2-sphere and an unknotted projective plane. We may refer to such projective planes as being of Kinoshita type. Then the question is: is every projective plane in $S^{4}$ of Kinoshita type? The first instance of this problem appearing as “Kinoshita’s Conjecture” in the literature is likely work of Katanaga–Saeki [KS98], who said in 1997 that this problem was already well-known to topologists in Japan.\n\n(3) Kamada showed that the Kinoshita Conjecture is true for projective planes obtained by deform-spinning knots with finite summetry groups [Kam92].\n\n(4) By Yoshikawa [Yos98], not every Klein bottle in $S^{4}$ admits an unknotted projective plane summand. The examples studied by Yoshikawa all have normal Euler number zero. Question (i). Does there exist a Klein bottle $\\Sigma$ in $S^{4}$ that does not admit an unknotted projective plane summand with $|e(\\Sigma)|$ =4?\n\n(5) Yoshikawa [Yos94] constructed 2-component links of projective planes in $S^{4}$ such that neither component admits an unknotted projective plane summand in the complement of the other component. In some sense, this can be interpreted as a failure of the Kinoshita Conjecture for multiplecomponent links. Yoshikawa’s obstruction to these links decomposing was to show that the peripheral subgroup of each component was order 4, but would be order 2 if the link admitted a trivial projective plane summand at that component. This motivates the following question. Question (ii). Given a projective plane P in $S^{4}$, must the kernel of the inclusion-induced homomorphism $\\iota_{*}: \\pi_{1}(\\partial(S^{4} \\setminus \\nu(P)) \\to \\pi_{1}(S^{4} \\setminus$ P) be order 4? (If not, then P is an example of a projective plane that does not decompose as a summand of a knotted 2-sphere and unknotted projective plane.)\n\n(6) When $\\Sigma$ is a projective plane in a 4-manifold X with normal Euler number $e(\\Sigma) = \\pm$ 2, we can perform Price surgery on X along $\\Sigma$ [Pri77]. When $X =S^{4}$, there is a preferred surgery whose result is a homotopy 4-sphere. If $\\Sigma$ is obtained by tubing an unknotted projective plane to a 2-sphere S with $e(S) =$ 0, then some Price surgery on $\\Sigma$ is diffeomorphic to the result of Gluck surgery on S [KSTY99]. In particular, when $X =S^{4}$ the homotopy 4-sphere obtained from Price surgery on $\\Sigma$ is diffeomorphic to the result of Gluck surgery on S. Thus, positively answering the proposed question (showing that all projective planes in $S^{4}$ are of Kinoshita type) would imply that Price surgeries in $S^{4}$ yield no more homotopy 4-spheres than Gluck surgeries.\n\nReferences cited:\n- [Kin61] Shin’ichi Kinoshita. On the Alexander polynomials of 2-spheres in a 4-sphere. Ann. of Math. (2), 74:518–531, 1961. doi:10.2307/1970296.\n- [KS98] Atsuko Katanaga and Osamu Saeki. Embeddings of quaternion space in $S^{4}$. J. Austral. Math. Soc. Ser. A, 65(3):313–325, 1998.\n- [Kam92] Seiichi Kamada. Projective planes in 4-sphere obtained by deform-spinnings. In Knots 90 (Osaka, 1990), pages 125–132. de Gruyter, Berlin, 1992.\n- [Yos98] Katsuyuki Yoshikawa. The order of a meridian of a knotted Klein bottle. Proc. Amer. Math. Soc., 126(12):3727–3731, 1998. doi:10.1090/S0002-9939-98-04560-2.\n- [Yos94] Katsuyuki Yoshikawa. An enumeration of surfaces in four-space. Osaka J. Math., 31(3):497–522, 1994. http://projecteuclid.org/euclid.ojm/1200785461.\n- [Pri77] T. M. Price. Homeomorphisms of quaternion space and projective planes in four space. J. Austral. Math. Soc. Ser. A, 23(1):112–128, 1977. doi:10.1017/s1446788700017407.\n- [KSTY99] Atsuko Katanaga, Osamu Saeki, Masakazu Teragaito, and Yuichi Yamada. Gluck surgery along a 2-sphere in a 4-manifold is realized by surgery along a projective plane. Michigan Math. J., 46(3):555–571, 1999. doi:10.1307/mmj/1030132479.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Kinoshita's conjecture was disproved in May 2026 by a smooth projective plane in S4 that is not even topologically a connected sum of a 2-knot with an unknotted projective plane.\n\n**Verified partial progress.**\n\n- The same preprint constructs topologically irreducible smooth Klein bottles in S4 with normal Euler number plus or minus four.\n- Its projective-plane example has peripheral subgroup of order four; under the kernel convention summarized in the Kirby survey, the associated kernel has order two.\n\n**Full solution or refutation.**\n\nHughes--Kim--Miller--Nahm Theorem 1.1 supplies a direct counterexample, so the main conjectural question has a definitive negative answer.\n\n**What remains.**\n\nClassify irreducible projective planes and Klein bottles and determine the full range of peripheral groups or kernels; verify convention-dependent translations for the subsidiary kernel question.\n\n**Sources checked.**\n\n- Mark Hughes, Seungwon Kim, Maggie Miller, and Gheehyun Nahm, An irreducible real projective plane in the 4-sphere, arXiv:2605.12921 (2026). (primary): https://arxiv.org/abs/2605.12921\n Evidence used: Theorem 1.1 constructs a smooth RP2 in S4 with no topological decomposition as a 2-knot connected-summed with the unknotted RP2; the paper also gives the Klein-bottle and peripheral-group results.\n- Kirby et al., Problems in Low-Dimensional Topology, third-edition survey, Problem KP-4.37 (2026). (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Supplies the conjecture and subsidiary questions as the pre-counterexample baseline.\n\n**Review notes.** The main refutation is direct. Only the precise peripheral-subgroup versus kernel translation is convention-sensitive.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2914, "problem_number": "KP-4.38", "title": "Kirby Problem 4.38", "statement": "Let $\\Delta \\subset B^{4}$ be a ribbon disk. Is $B^{4} \\setminus \\Delta$ aspherical, i.e. is $\\pi_{i}(B^{4} \\setminus \\Delta)$ =0 for all i>1?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.38.\n\nLiterature notes:\n(1) This is [Kir97, Problem 1.103]. Note that the Whitehead asphericity conjecture [Whi41, p. 428] (see also [Ros07a, How85] and Problem 5.11) predicts an affirmative answer. A generalized version, asking for the asphericity of ribbon concordances, was asked by Gordon in [Gor81, Conjecture 6.5].\n\n(2) Howie showed in [How85] that the answer is yes when $\\pi_{1}(B^{4} \\setminus \\Delta)is$ locally indicable.\n\n(3) One could equally well pose this question in the locally flat category, asking whether the complements of homotopy-ribbon disks are aspherical. (Recall that a homotopy-ribbon disk $\\Delta$ for some knot $K \\subset S^{3}$ is one such that the map $\\pi_{1}(S^{3} \\setminus$ K) $\\to \\pi_{1}(B^{4} \\setminus \\Delta)is$ surjective.) In this case, it is open whether the complement is homotopy equivalent to a 2-complex.\n\n(4) An affirmative answer to the question can be used to compute the algebraic 2-type of a 2-knot complement, as shown in [Lom81]. Note that the argument there is based on an incorrect proof asserting that ribbon disk complements are aspherical. See [Mar09] for an alternate approach to computing the algebraic 2-type of a 2-knot complement.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Whi41] J. H. C. Whitehead. On adding relations to homotopy groups. Ann. of Math. (2), 42:409–428, 1941. doi:10.2307/1968907.\n- [Ros07a] Stephan Rosebrock. The Whitehead conjecture—an overview. Sib. Èlektron. Mat. Izv., 4:440–449, 2007.\n- [How85] James Howie. On the asphericity of ribbon disc complements. Trans. Amer. Math. Soc., 289(1):281–302, 1985. doi:10.2307/1999700.\n- [Gor81] C. McA. Gordon. Ribbon concordance of knots in the 3-sphere. Math. Ann., 257(2):157–170, 1981. doi:10.1007/BF01458281.\n- [Lom81] S. J. Lomonaco, Jr. The homotopy groups of knots. I. How to compute the algebraic 2-type. Pacific J. Math., 95(2):349–390, 1981. http://projecteuclid.org/euclid.pjm/1102735075.\n- [Mar09] João Faria Martins. The fundamental crossed module of the complement of a knotted surface. Trans. Amer. Math. Soc., 361(9):4593–4630, 2009. doi:10.1090/S0002-9947-09-04576-0.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ribbon-disk complements are aspherical under local-indicability and several graph-theoretic hypotheses, but general asphericity remains open.\n\n**Verified partial progress.**\n\n- Howie proves asphericity when the complement fundamental group is locally indicable.\n- Bedenikovic gives core-graph criteria and gluing constructions producing further aspherical families.\n\n**Full solution or refutation.**\n\nNo valid proof that every ribbon-disk complement is aspherical was found; an older purported general proof is known to be incorrect.\n\n**What remains.**\n\nRemove the group/core-graph hypotheses or construct a ribbon disk with nontrivial higher homotopy.\n\n**Sources checked.**\n\n- James Howie, On the asphericity of ribbon disc complements, Transactions of the AMS 289 (1985), 281--302. (primary): https://doi.org/10.2307/1999700\n Evidence used: Proves the locally indicable special case.\n- Tony Bedenikovic, Asphericity results for ribbon disk complements via alternate descriptions, Osaka Journal of Mathematics 48 (2011), 99--125. (primary): https://doi.org/10.18910/4901\n Evidence used: Graph-theoretic sufficient conditions and constructions of aspherical families; explicitly retains the general problem.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 4.38. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current open status and warning about the incorrect older proof.\n\n**Review notes.** Background formulas contain joined-word OCR such as '$\\pi_1(B^4\\setminus\\Delta)is$'; no statement was repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2915, "problem_number": "KP-4.39", "title": "Kirby Problem 4.39", "statement": "Let K be a closed surface smoothly embedded in a 4-manifold $M^{4}$. Describe the subgroup of the mapping class group $Mod(K)$ of diffeomorphisms of K that extend to diffeomorphisms of M. Conversely, is it possible to characterize when a given subgroup of the mapping class group arises as the group of extendable diffeomorphisms for some K inside some $M^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.39.\n\nLiterature notes:\n(1) The problem is due to Hirose, who has studied instances of this problem in e.g. [Hir93, Hir02, Hir05, Hir12]. Some highlights of his results on this problem are summarized in the remarks to follow.\n\n(2) In [Hir02], Hirose studies the case of an unknotted genus-g surface in $S^{4}$, showing that a mapping class is extendable in this setting if and only if it preserves the Rokhlin quadratic for m (descending to a spin structure on the surface). In particular, this subgroup is of finite index in the mapping class group of K. In [Hir02], he extended this result to nonorientable unknotted surfaces in $S^{4}$, showing that in this setting a mapping class extends exactly when it preserves the Guillou–Marin quadratic for m(descending to a Pin− structure on the surface; see [KS23] for more discussion). Again, this subgroup is of finite index in the mapping class group of K.\n\n(3) In [Hir05], Hirose classified which surface automorphisms extend from unknotted surfaces and certain complex curves to all of $\\mathbb{CP}^{2}$. The question is open for all high-degree complex curves.\n\n(4) In [Hir93], Hirose studied the case of knotted tori in $S^{4}$ arising by spinning (or turned spinning) knots in $S^{3}$. Here, the set of extendable mapping classes is always of infinite index. It would be interesting to characterize when the group of extendable mapping classes is of finite index in general.\n\n(5) In [HY08], Hirose and Yasuhara showed that in many simply connected 4-manifolds (including $\\mathbb{CP}^{2},\\mathbb{CP}^{2}, S^{2} \\times S^{2}$, E(n)) there exist smoothly embedded surfaces of every topological type with the property that any surface automorphism extends to an ambient automorphism.\n\nReferences cited:\n- [Hir93] Susumu Hirose. On diffeomorphisms over T2-knots. Proc. Amer. Math. Soc., 119(3):1009–1018, 1993. doi:10.2307/2160546.\n- [Hir02] Susumu Hirose. On diffeomorphisms over surfaces trivially embedded in the 4-sphere. Algebr. Geom. Topol., 2:791–824, 2002. doi:10.2140/agt.2002.2.791.\n- [Hir05] Susumu Hirose. Surfaces in the complex projective plane and their mapping class groups. Algebr. Geom. Topol., 5:577–613, 2005. doi:10.2140/agt.2005.5.577.\n- [Hir12] Susumu Hirose. On diffeomorphisms over nonorientable surfaces standardly embedded in the 4-sphere. Algebr. Geom. Topol., 12(1):109–130, 2012. doi:10.2140/agt.2012.12.109.\n- [KS23] Michael R. Klug and Luuk Stehouwer. Some properties of Pin˘-structures on compact surfaces. Topology Appl., 339(part B):Paper No. 108678, 9, 2023. doi: 10.1016/j.topol.2023.108678.\n- [HY08] Susumu Hirose and Akira Yasuhara. Surfaces in 4-manifolds and their mapping class groups. Topology, 47(1):41–50, 2008. doi:10.1016/j.top.2007.05.001.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Extendable mapping-class subgroups are computed for several standard and knotted surfaces, and trivial subgroups are now broadly realized, but no general description or realization classification exists.\n\n**Verified partial progress.**\n\n- Hirose characterizes extendability for standard orientable and nonorientable surfaces in S4 by preservation of the relevant quadratic enhancement.\n- Lawande--Saha give new extendibility criteria and a 3g-element generating set for the standard genus-g surface in S4.\n- Niu constructs, for every g at least 3 in every closed simply connected smooth 4-manifold, infinitely many surfaces with trivial orientation-preserving extendable subgroup.\n\n**Full solution or refutation.**\n\nThe 2026 trivial-subgroup realization is a major converse example, but arbitrary embeddings and arbitrary subgroups remain unclassified.\n\n**What remains.**\n\nCharacterize extendable subgroups for general embedded surfaces and determine which subgroups of Mod(K) can be realized.\n\n**Sources checked.**\n\n- Shital Lawande and Kuldeep Saha, Surfaces in 4-manifolds and extendible mapping classes, arXiv:2502.17640 (2025). (primary): https://arxiv.org/abs/2502.17640\n Evidence used: New constructions, criteria, non-flexibility results, and smaller generating set.\n- Weizhe Niu, Embedded surfaces with trivial extendable mapping class groups in simply connected 4-manifolds, arXiv:2608.01504 (2026). (primary): https://arxiv.org/abs/2608.01504\n Evidence used: Realizes the trivial orientation-preserving subgroup for infinitely many genus-g embeddings, g at least 3, in every closed simply connected smooth 4-manifold.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.39. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current synthesis of Hirose's exact special cases and open general questions.\n\n**Review notes.** Background OCR repeats CP2 where an opposite orientation appears intended and corrupts 'quadratic form' as 'quadratic for m'.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2916, "problem_number": "KP-4.40", "title": "Kirby Problem 4.40", "statement": "Is every link of 2-spheres in $S^{4}$ slice?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.40.\n\nLiterature notes:\n(1) A 2-link is slice if it bounds a disjointly embedded collection of 3-balls in the 5-ball $B^{5}$.\n\n(2) Kervaire [Ker65a] proved that every 2-knot in $S^{4}$ is slice.\n\n(3) This question can be asked in either category. If a smooth link of 2-spheres is topologically slice in $B^{5}$, then it is also smoothly slice by Daher–Powell [DP23a].\n\n(4) Cochran [Coc84] gave a sufficient condition for a 2-link to be slice, generalizing the well-known fact that boundary 2-links (see Problem 1.62) are slice (following from Kervaire’s proof for 2-knots). Not every 2-link is a boundary link [Coc84, §4].\n\nReferences cited:\n- [Ker65a] Michel A. Kervaire. Les nœuds de dimensions supérieures. Bull. Soc. Math. France, 93:225–271, 1965. URL: http://www.numdam.org/item?id=BSMF 1965 93 225 0.\n- [DP23a] Michelle Daher and Mark Powell. Smoothing 3-manifolds in 5-manifolds, 2023. arXiv:2309.15962.\n- [Coc84] Tim Cochran. Slice links in $S^{4}$. Trans. Amer. Math. Soc., 285(1):389–401, 1984. doi:10.2307/1999487.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every 2-knot and several broad classes of 2-links are slice, and topological sliceness of a smooth 2-link implies smooth sliceness, but arbitrary multi-component 2-links remain unresolved.\n\n**Verified partial progress.**\n\n- Kervaire proves every one-component 2-link is slice.\n- Boundary 2-links and 2-links satisfying Cochran's sufficient condition are slice.\n- Daher--Powell show a smooth 2-link that is topologically slice in B5 is smoothly slice.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for every multi-component link of 2-spheres in S4 was found.\n\n**What remains.**\n\nDecide sliceness for 2-links outside the boundary/Cochran classes, equivalently without relying on a topological-versus-smooth gap.\n\n**Sources checked.**\n\n- Tim Cochran, Slice links in S4, Transactions of the AMS 285 (1984), 389--401. (primary): https://doi.org/10.2307/1999487\n Evidence used: Sufficient sliceness condition and examples beyond the boundary-link setting.\n- Michelle Daher and Mark Powell, Smoothing 3-manifolds in 5-manifolds, arXiv:2309.15962. (primary): https://arxiv.org/abs/2309.15962\n Evidence used: Topologically slice smooth 2-links are smoothly slice.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.40. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current general status and precise special cases.\n\n**Review notes.** No formulation defect found.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2917, "problem_number": "KP-4.41", "title": "Kirby Problem 4.41", "statement": "Let $\\Sigma$ be a compact surface and let X be a connected 4manifold. Let $f_{0}, f_{1}: \\Sigma \\to X$ be $\\pi_{1}-negligible$ embeddings that agree on $\\partial\\Sigma$ and that are homotopic rel. boundary. Give computable invariants that decide whether there is $a \\pi_{1}-negligible$ concordance in $X \\times I$ between $f_{0}$ and $f_{1}$.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.41.\n\nLiterature notes:\n(1) A surface is $\\pi_{1}-negligible$ if the complement of the surface included into X induces an isomorphism on fundamental groups [FQ90]. This assumption is equivalent to asking for an immersed dual sphere.\n\n(2) The case that $\\Sigma =S^{2}$ or that the dual sphere is framed is due to Freedman– Quinn [FQ90] and Stong [Sto93]. See also the papers of Klug–Miller [KM21, KM22].\n\n(3) A strategy would be to extend the Freedman–Quinn and Stong invariants to arbitrary compact surfaces.\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [Sto93] Richard Stong. Uniqueness of $\\pi_1$-negligible embeddings in 4-manifolds: a correction to Theorem 10.5 of topology of 4-manifolds [Princeton Univ. Press, Princeton, NJ, 1990; MR1201584 (94b:57021)] by M. H. Freedman and F. Quinn. Topology, 32(4):677–699, 1993. doi:10.1016/0040-9383(93)90046-X.\n- [KM21] Michael R. Klug and Maggie Miller. Concordance of surfaces in 4-manifolds and the Freedman-Quinn invariant. J. Topol., 14(2):560–586, 2021. doi:10.1112/topo.12191.\n- [KM22] Michael Klug and Maggie Miller. Concordance of spheres in 4-manifolds with an immersed dual sphere, 2022. arXiv:2211.07177.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Freedman--Quinn/Stong-type invariants decide concordance for spheres and for compact orientable surfaces with a framed immersed dual, but not for every pi1-negligible surface in the stated generality.\n\n**Verified partial progress.**\n\n- Klug--Miller prove that with an immersed framed dual sphere the Freedman--Quinn invariant is a complete concordance obstruction for compact orientable surfaces.\n- For embedded 2-spheres with an immersed dual, Freedman--Quinn and Stong invariants form a complete obstruction package under mild hypotheses.\n\n**Full solution or refutation.**\n\nThe arbitrary compact-surface case without the framed-dual/sphere hypotheses remains open.\n\n**What remains.**\n\nExtend the computable obstruction theory to all pi1-negligible compact surfaces and account for unframed dual spheres.\n\n**Sources checked.**\n\n- Michael R. Klug and Maggie Miller, Concordance of surfaces in 4-manifolds and the Freedman--Quinn invariant, Journal of Topology 14 (2021), 560--586. (primary): https://doi.org/10.1112/topo.12191\n Evidence used: Complete Freedman--Quinn obstruction under the immersed framed dual hypothesis.\n- Michael Klug and Maggie Miller, Concordance of spheres in 4-manifolds with an immersed dual sphere, arXiv:2211.07177. (primary): https://arxiv.org/abs/2211.07177\n Evidence used: Refines the sphere case with Freedman--Quinn and Stong obstructions.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.41. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current scope and remaining arbitrary-surface problem.\n\n**Review notes.** Statement OCR includes '4manifold' and '$a pi_1$-negligible'.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2918, "problem_number": "KP-4.42", "title": "Kirby Problem 4.42", "statement": "(a) If X is a smooth, closed, simply connected 4-manifold $withb_{2}(X) \\geq$ 2, are all knots slice in X?\n\n(b) In particular, are all knots slice in $\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2}$? In the K3 surface?\n\n(c) Can the set of slice knots detect exotic smooth structures? That is, do there exist homeomorphic 4-manifolds X and $X^{1}$ and a knot $K \\subset S^{3}$ that is slice in X but not in $X^{1}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.42.\n\nLiterature notes:\n(1) A knot K is called slice in a closed 4-manifold X if it bounds a smoothly embedded disk in $X \\setminus$ in $t(B^{4})$, where K lives on the boundary $S^{3}$. It is called H-slice if the disk is null-homologous (relative to the boundary).\n\n(2) All knots are known to be slice in $S^{2} \\times S^{2}$ and $\\mathbb{CP}^{2}\\#\\mathbb{CP}^{2}[Nor69$, Suz69], but not all knots are slice in $\\mathbb{CP}^{2}$ or in $\\mathbb{CP}^{2}$ [Yas91].\n\n(3) Part(a)is true in the topological category; see [KPRT24, Corollary 1.15].\n\n(4) Concerning part (b), Marengon and Mihajlović [MM25a] proved that all knots with unknotting number at most 21 are slice in K3.\n\n(5) The analogue of (c) for H-slice knots (instead of slice) was answered in [MMP24]: the right handed trefoil is H-slice in $\\#3 \\mathbb{CP}^{2}\\#20 \\mathbb{CP}^{2}$ but not in $K3\\#\\mathbb{CP}^{2}$.\n\n(6) A variant of part (c) is asked in Problem 4.12.\n\n(7) Lidman and Piccirillo [LP25] announced the construction of a pair of 4manifolds X and $X^{1}$ with the same integer cohomology ring, such that there exists a knot slice in X but not in $X^{1}$. (However,X and $X^{1}$ are not homeomorphic.)\n\nReferences cited:\n- [Nor69] R. A. Norman. Dehn’s lemma for certain 4-manifolds. Invent. Math., 7:143–147, 1969. doi:10.1007/BF01389797.\n- [Suz69] Shin’ichi Suzuki. Local knots of 2-spheres in 4-manifolds. Proc. Japan Acad., 45:34– 38, 1969.\n- [Yas91] Akira Yasuhara. $(2,15)$-torus knot is not slice in $\\mathbb{CP}^{2}$. Proc. Japan Acad. Ser. A Math. Sci., 67(10):353–355, 1991. http://projecteuclid.org/euclid.pja/1195511928.\n- [KPRT24] Daniel Kasprowski, Mark Powell, Arunima Ray, and Peter Teichner. Embedding surfaces in 4-manifolds. Geom. Topol., 28(5):2399–2482, 2024. doi:10.2140/gt.2024.28.2399.\n- [MM25a] Marco Marengon and Stefan Mihajlović. Unknotting number 21 knots are slice in K3. Math. Res. Lett., 32(3):939–955, 2025. doi:10.4310/mrl.250731115553.\n- [MMP24] Ciprian Manolescu, Marco Marengon, and Lisa Piccirillo. Relative genus bounds in indefinite four-manifolds. Math. Ann., 390(1):1481–1506, 2024. doi:10.1007/s00208-023-02787-4.\n- [LP25] Tye Lidman and Lisa Piccirillo. Distinguishing closed 4-manifolds by slicing, 2025. arXiv:2505.14387.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The assertion is topologically true; smoothly it holds for CP2#bar(CP2) and for K3 knots of unknotting number at most 21, while the universal and homeomorphic-exotic-pair questions remain open.\n\n**Verified partial progress.**\n\n- All knots are smoothly slice in S2 x S2 and CP2#bar(CP2).\n- Every knot of unknotting number at most 21 is smoothly slice in K3.\n- H-slice knot sets detect an exotic homeomorphic pair, a restricted version of part (c).\n- Lidman--Piccirillo distinguish manifolds with the same integer cohomology ring by ordinary sliceness, but their pair is not homeomorphic.\n\n**Full solution or refutation.**\n\nNot every knot is known to be smoothly slice in K3 or in every simply connected b2-at-least-2 manifold, and ordinary slice sets have not yet distinguished a homeomorphic exotic pair.\n\n**What remains.**\n\nSettle arbitrary knots in K3/general X and realize part (c) with homeomorphic, not merely cohomology-equivalent, manifolds.\n\n**Sources checked.**\n\n- Marco Marengon and Stefan Mihajlovic, Unknotting number 21 knots are slice in K3, Mathematical Research Letters 32 (2025), 939--955. (primary): https://arxiv.org/abs/2210.10089\n Evidence used: Direct K3 special case and recovery of standard all-knot examples.\n- Tye Lidman and Lisa Piccirillo, Distinguishing closed 4-manifolds by slicing, arXiv:2505.14387 (2025). (primary): https://arxiv.org/abs/2505.14387\n Evidence used: Ordinary sliceness distinguishes a same-cohomology-ring pair, explicitly not a homeomorphic pair.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.42. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current subpart-by-subpart status including the topological theorem and H-slice analogue.\n\n**Review notes.** Extensive OCR includes 'withb_2', '$X^1$', '$X\\setminus$ in $t(B^4)$', and missing orientation bars in background connected sums.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2919, "problem_number": "KP-4.43", "title": "Kirby Problem 4.43", "statement": "Let K be a knot on the boundary of $X \\setminus B^{\\circ 4}$, where X is a negative definite, smooth 4-manifold. Suppose K bounds a smoothly embedded, nullhomotopic surface $\\Sigma$ in $X \\setminus B^{\\circ 4}$. Is the following inequality $2g(\\Sigma) \\geq$ |s(K)| (8) always satisfied? Here $s(K)$ denotes Rasmussen’s s-invariant [Ras10].", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.43.\n\nLiterature notes:\n(1) For $X =S^{4}$, the inequality (8) was proved by Rasmussen [Ras10]. When $X =\\#_{r}\\mathbb{CP}^{2}$, (8) was proved by Manolescu–Marengon–Sarkar-Willis [MMSW23].\n\n(2) Freedman–Gompf–Morrison–Walker [FGMW10] proposed the following approach to disproving the smooth four-dimensional Poincaré conjecture: Find a homotopy 4-sphere X and a knot K that is slice in X (i.e. K bounds smoothly embedded disk in $X \\setminus B^{4})$. Then prove that K is not slice in $S^{4}$ by showing that $s(K) \\ne$ 0. If this approach works, then (8) must fail for X.\n\n(3) Using instanton Floer theory, Kronheimer-Mrowka [KM13b] defined a concordance invariant s7(K) that satisfies a similar inequality $2g(\\Sigma) \\geq |s^{7}(K)|$. However, it is known that $s(K) \\ne s^{7}(K)$ for many knots including the trefoil [Gon21]. One possible approach to proving (8) is to find a gauge theoretic interpretation of $s(K)$ by modifying Kronheimer-Mrowka’s definition of s7(K).\n\n(4) One may ask whether (8) holds for other variants of Rasmussen’s sinvariant, including $s\\mathbb{F}p$ (defined using Khovanov homology over the field $\\mathbb{F}_{p}[MTV07$, LS14]) and $s^{Sq,1}$ (the Lipshitz-Sarkar $Sq^{1}s-invariants$ [LS14]). It is known that their variants are different from $s(K)$ in general [LS14].\n\nReferences cited:\n- [Ras10] Jacob Rasmussen. Khovanov homology and the slice genus. Invent. Math., 182(2):419–447, 2010. doi:10.1007/s00222-010-0275-6.\n- [MMSW23] Ciprian Manolescu, Marco Marengon, Sucharit Sarkar, and Michael Willis. A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds. Duke Math. J., 172(2):231–311, 2023. doi:10.1215/00127094-2022-0039.\n- [FGMW10] Michael Freedman, Robert Gompf, Scott Morrison, and Kevin Walker. Man and machine thinking about the smooth 4-dimensional Poincaré conjecture. Quantum Topol., 1(2):171–208, 2010. doi:10.4171/QT/5.\n- [KM13b] P. B. Kronheimer and T. S. Mrowka. Gauge theory and Rasmussen’s invariant. J. Topol., 6(3):659–674, 2013. doi:10.1112/jtopol/jtt008.\n- [Gon21] Sherry Gong. On the Kronheimer-Mrowka concordance invariant. J. Topol., 14(1):1–28, 2021. doi:10.1112/topo.12175.\n- [MTV07] Marco Mackaay, Paul Turner, and Pedro Vaz. A remark on Rasmussen’s invariant of knots. J. Knot Theory Ramifications, 16(3):333–344, 2007. doi:10.1142/S0218216507005312.\n- [LS14] Robert Lipshitz and Sucharit Sarkar. A refinement of Rasmussen’s S-invariant. Duke Math. J., 163(5):923–952, 2014. doi:10.1215/00127094-2644466.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Rasmussen-s genus inequality holds in the standard ball and standard negative-definite connected sums, but not yet for every negative-definite smooth 4-manifold.\n\n**Verified partial progress.**\n\n- Rasmussen proves the bound in the standard slice-genus setting.\n- Manolescu--Marengon--Sarkar--Willis prove the analogous nullhomologous-surface inequality in standard definite connected sums and related manifolds.\n- Gauge-theoretic invariants satisfy general negative-definite bounds but differ from Rasmussen's s on known knots.\n\n**Full solution or refutation.**\n\nNo general gauge-theoretic interpretation or proof of the exact Rasmussen-s inequality for arbitrary negative-definite X was found.\n\n**What remains.**\n\nExtend the standard connected-sum theorem to exotic/arbitrary negative-definite 4-manifolds or find a counterexample.\n\n**Sources checked.**\n\n- Ciprian Manolescu, Marco Marengon, Sucharit Sarkar, and Michael Willis, A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifolds, Duke Mathematical Journal 172 (2023), 231--311. (primary): https://doi.org/10.1215/00127094-2022-0039\n Evidence used: Proves the relevant s-genus bounds for standard connected sums and related four-manifolds.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.43. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current special cases and the remaining arbitrary negative-definite question.\n\n**Review notes.** Orientation bars are lost in the background despite negative definiteness; s-sharp and coefficient-variant notation is also OCR-damaged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2920, "problem_number": "KP-4.44", "title": "Kirby Problem 4.44", "statement": "(a) Let X be a closed, simply connected 4-manifold. Let $g \\geq$ 0 and $d \\geq$ 0 be integers. Fix $x \\in H_{2}(X;\\mathbb{Z})$. Does there exist a (smooth, locally flat) generic immersion $f: \\Sigma_{g} \\to X$ such that $f_{*}[\\Sigma_{g}] = x$, with d transverse double points?\n\n(b) An important special case is $d =$ 0. Let $g: H_{2}(X;\\mathbb{Z}) \\to \\mathbb{N}_{0}$ be the genus function assigning to $x \\in H_{2}(X;\\mathbb{Z})$ the minimal genus of any (smooth, locally flat) embedded surface in X whose fundamental class represents x. The problem is to compute the locally flat genus function for some 4-manifold $X \\ne S^{4}$, and to compute the smooth genus function for new 4-manifolds.\n\n(c) For a specific example, is the class (3,2) $\\in H_{2}(\\mathbb{CP}^{2}\\#\\mathbb{CP}^{2};\\mathbb{Z})$ represented by a sphere?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.44.\n\nLiterature notes:\n(1) The smooth genus function is known for $\\mathbb{CP}^{2}$ by a celebrated result of Kronheimer–Mrowka [KM94]. The cases of $\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2}$ and $S^{2} \\times S^{2}$ were resolved by Ruberman in [Rub96]. Partial results about the smooth genus problem in $\\mathbb{CP}^{2}\\#\\overline{\\mathbb{CP}}{}^{2}$ appear in [Bry98], [Nou14], [MMRS24], [ACM+26]. By the symplectic Thom conjecture, proved in [OS00], any symplectic submanifold minimizes genus in its homology class; this gives a useful means of computing the value of the smooth genus function for certain homology classes.\n\n(2) The locally flat genus function is not known except for $S^{4}$. In other 4manifolds, much is known aboutsimple locally flat embeddings by work of Lee-Wilczynski [LW97, LW00]. As a special case, every primitive class in a closed, simply-connected 4-manifold is represented by a locally flat torus [LW97, KPRT24]. Not much else is known.\n\n(3) In the special case when the self-intersection number Q(x, x) $=$ 0, we have the following interesting sub-problem. As usual we $let\\mathbb{N}$ denote the positive integers.\n\n\\paragraph{Question.} For a given $x \\in H_{2}(X;\\mathbb{Z})$ such that $g(x) >$ 0, consider the function $n_{x}: \\mathbb{N} \\to \\mathbb{N}$ sending $m \\mapsto g(mx)$. What is $n_{x}$ in terms of $g(x)$? Is $n_{x}(m)$ =mg(x) −m+1? Note that $ifg(x)$ =0 then $g(mx)$ =0 for all x. The formula $mg(x)$ − m+1 comes from the following observation: if $[\\Sigma]$ =xwith Q(x, x) =0, then there is a surface $\\Sigma$ that embeds in $\\Sigma_{~} \\times D^{2}$ representing mx such that the projection to $\\Sigma$ induces an m-sheeted covering. The answer is negative for some 4-manifolds with boundary. Kawauchi [Kaw09] constructed 4-manifolds homotopy equivalent to $S^{2}(so H_{2}(X;\\mathbb{Z}) = \\mathbb{Z})$, with the property that for x a generator, $g(mx) =$ 0 for m even and $g(mx) \\ne$ 0 for m odd.\n\nReferences cited:\n- [KM94] P. B. Kronheimer and T. S. Mrowka. The genus of embedded surfaces in the projective plane. Math. Res. Lett., 1(6):797–808, 1994. doi:10.4310/MRL.1994.v1.n6.a14.\n- [Rub96] Daniel Ruberman. The minimal genus of an embedded surface of non-negative square in a rational surface. Turkish J. Math., 20(1):129–133, 1996.\n- [Bry98] Jim Bryan. Seiberg-Witten theory and $\\mathbb{Z}/2^p$ actions on spin 4-manifolds. Math. Res. Lett., 5(1-2):165–183, 1998. doi:10.4310/MRL.1998.v5.n2.a3.\n- [Nou14] Mohamed Ait Nouh. The minimal genus problem in $\\mathbb{CP}^{2}$\\#$\\mathbb{CP}^{2}$. Algebr. Geom. Topol., 14(2):671–686, 2014. doi:10.2140/agt.2014.14.671.\n- [MMRS24] Marco Marengon, Allison N. Miller, Arunima Ray, and András I. Stipsicz. A note on surfaces in $\\mathbb{CP}^{2}$ and $\\mathbb{CP}^{2}$ \\#$\\mathbb{CP}^{2}$ . Proc. Amer. Math. Soc. Ser. B, 11:187–199, 2024. doi:10.1090/bproc/218.\n- [ACM+26] Paolo Aceto, Nickolas A Castro, Maggie Miller, JungHwan Park, and András Stipsicz. Slice Obstructions From Genus Bounds in Definite 4-Manifolds. Int. Math. Res. Not. IMRN, 2026(2):Paper No. rnaf377, 2026. doi:10.1093/imrn/rnaf377.\n- [OS00] Peter Ozsváth and Zoltán Szabó. The symplectic Thom conjecture. Ann. of Math. (2), 151(1):93–124, 2000. doi:10.2307/121113.\n- [LW97] Ronnie Lee and Dariusz M. Wilczyński. Representing homology classes by locally flat surfaces of minimum genus. Amer. J. Math., 119(5):1119–1137, 1997. URL: http://muse.jhu.edu/journals/american journal of mathematics/v119/119.5lee.pdf.\n- [LW00] Ronnie Lee and Dariusz M. Wilczyński. Genus inequalities and four-dimensional surgery. Topology, 39(2):311–330, 2000. doi:10.1016/S0040-9383(99)00017-8.\n- [KPRT24] Daniel Kasprowski, Mark Powell, Arunima Ray, and Peter Teichner. Embedding surfaces in 4-manifolds. Geom. Topol., 28(5):2399–2482, 2024. doi:10.2140/gt.2024.28.2399.\n- [Kaw09] Akio Kawauchi. Rational-slice knots via strongly negative-amphicheiral knots. Commun. Math. Res., 25(2):177–192, 2009.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many smooth classes and primitive locally flat classes are understood, but no nontrivial full locally flat genus function is known and sphere representability of (3,2) in CP2#CP2 remains unresolved.\n\n**Verified partial progress.**\n\n- The smooth genus function is known for CP2 and specified rational surfaces; symplectic representatives minimize genus.\n- Every primitive class in a closed simply connected 4-manifold is represented by a locally flat torus under the cited embedding results.\n- The (3,2) class has an embedded torus representative, but no verified embedded sphere.\n\n**Full solution or refutation.**\n\nThe generic-immersion problem and genus functions remain incomplete, and the named (3,2) sphere question is still open.\n\n**What remains.**\n\nDevelop existence criteria for prescribed genus/double points, compute a nontrivial locally flat genus function, and settle the (3,2) class.\n\n**Sources checked.**\n\n- Daniel Kasprowski, Mark Powell, Arunima Ray, and Peter Teichner, Embedding surfaces in 4-manifolds, Geometry & Topology 28 (2024), 2399--2482. (primary): https://doi.org/10.2140/gt.2024.28.2399\n Evidence used: Modern locally flat embedding theorem supporting the primitive-class torus result.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.44. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current genus-function status and explicit statement that (3,2) sphere representability is unresolved.\n\n**Review notes.** Smooth and locally flat versions are separate. Background OCR includes '4manifolds', 'aboutsimple', 'ifg(x)', and a damaged S2 homotopy-equivalence formula.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2921, "problem_number": "KP-4.45", "title": "Kirby Problem 4.45", "statement": "Let X be a closed simply connected smooth 4-manifold and let $\\Sigma$ be a closed, orientable surface. Fix a smooth embedding of $f: \\Sigma \\hookrightarrow X$ to serve as a basepoint. For which $k\\geq 0$ does the equality\n$$\n\\ker i_{*}=\\ker s_{*}\n$$\n hold, where\n$$\n\\begin{aligned} i_{*}&:\\pi_{k}(\\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ});f)\\to \\pi_{k}(\\operatorname{Emb}^{TOP}(\\Sigma,X^{\\circ});f),\\\\ s_{*}&:\\pi_{k}(\\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ});f)\\to \\operatorname*{colim}_{N\\to\\infty}\\pi_{k}(\\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ}\\#_{N}S^{2}\\times S^{2});f). \\end{aligned}\n$$\n Here $i_*$ is induced by inclusion and $s_*$ by stabilization. To make sense of this for k=0, define $keri_{*}$ and $kers_{*}$ to be the preimages under $i_{*}$ and $s_{*}$ respectively of [f].", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.45.\n\nLiterature notes:\n(1) Here $X^{\\circ}$ denotes the punctured $X, X^{\\circ} = X \\setminus \\operatorname{Int}(D^{4}), \\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ})$ is the space of smooth embeddings, $\\operatorname{Emb}^{TOP}(\\Sigma,X^{\\circ})$ is the space of locally flat embeddings, $i: \\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ}) \\to \\operatorname{Emb}^{TOP}(\\Sigma,X^{\\circ})$ is the inclusion, and $s: \\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ}) \\to \\operatorname{Emb}^{DIFF}(\\Sigma,X^{\\circ}\\#_{N}S^{2} \\times S^{2})$ is the stabilization map. Note that we consider X punctured so that the stabilization map s is defined on the space level.\n\n(2) See Problem 4.77 for the analogous question on diffeomorphism and homeomorphism groups.\n\n(3) Fork =0, the inclusion $\\subset$ holds in the equality (9). That is, Galvin [Gal24] showed that any two smoothly embedded, topologically isotopic surfaces in a smooth, simply-connected 4-manifold X become smoothly isotopic after sufficiently many external stabilizations of X by copies of $S^{2} \\times S^{2}$. Is the converse true? What about when the complements are simply connected?\n\n(4) There is considerable literature showing, under suitable conditions, that surfaces in 4-manifolds are topologically isotopic, especially those with simply connected complements, which might help. See e.g. [LW90, LW93, LW01, Boy93, HK93b, LW97, Sun15, CP23]. New techniques seem to be required for answering the question when k>0.\n\nReferences cited:\n- [Gal24] Daniel A. P. Galvin. The Casson-Sullivan invariant for homeomorphisms of 4-manifolds, 2024. arXiv:2405.07928.\n- [LW90] Ronnie Lee and Dariusz M. Wilczyński. Locally flat 2-spheres in simply connected 4-manifolds. Comment. Math. Helv., 65(3):388–412, 1990.\n- [LW93] Ronnie Lee and Dariusz M. Wilczyński. Representing homology classes by locally flat 2-spheres. K-Theory, 7(4):333–367, 1993. doi:10.1007/BF00962053.\n- [LW01] Ronnie Lee and Dariusz M. Wilczyński. Erratum to: “Genus inequalities and four-dimensional surgery” [Topology 39 (2000), no. 2, 311–330; MR1722016 (2001j:57028)]. Topology, 40(5):1123, 2001. doi:10.1016/S0040-9383(00)00008-2.\n- [Boy93] Steven Boyer. Realization of simply-connected 4-manifolds with a given boundary. Comment. Math. Helv., 68(1):20–47, 1993.\n- [HK93b] Ian Hambleton and Matthias Kreck. Cancellation of hyperbolic forms and topological four-manifolds. J. Reine Angew. Math., 443:21–47, 1993. doi:10.1515/crll.1993.443.21.\n- [LW97] Ronnie Lee and Dariusz M. Wilczyński. Representing homology classes by locally flat surfaces of minimum genus. Amer. J. Math., 119(5):1119–1137, 1997. URL: http://muse.jhu.edu/journals/american journal of mathematics/v119/119.5lee.pdf.\n- [Sun15] Nathan S. Sunukjian. Surfaces in 4-manifolds: concordance, isotopy, and surgery. Int. Math. Res. Not. IMRN, 2015(17):7950–7978, 2015. doi:10.1093/imrn/rnu187.\n- [CP23] Anthony Conway and Mark Powell. Embedded surfaces with infinite cyclic knot group. Geom. Topol., 27(2):739–821, 2023. doi:10.2140/gt.2023.27.739.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For k=0, topological isotopy implies smooth isotopy after stabilization in the stated simply connected setting; newer results extend that direction, while the converse and all k>0 remain open.\n\n**Verified partial progress.**\n\n- Galvin proves ker(i_*) is contained in ker(s_*) on path components for simply connected ambient 4-manifolds.\n- Galvin--Orson--Powell extend stable smooth isotopy to mod-2-nullhomologous surfaces and ambient groups including free products of classical knot groups.\n\n**Full solution or refutation.**\n\nKernel equality is not established: the reverse inclusion for k=0 and the higher homotopy-group cases remain open.\n\n**What remains.**\n\nProve or refute stable-smooth-isotopy implies topological isotopy and develop tools for k greater than zero.\n\n**Sources checked.**\n\n- Daniel A. P. Galvin, The Casson--Sullivan invariant for homeomorphisms of 4-manifolds, arXiv:2405.07928. (primary): https://arxiv.org/abs/2405.07928\n Evidence used: Proves stable smooth isotopy of topologically isotopic surfaces in smooth simply connected 4-manifolds.\n- Daniel Galvin, Patrick Orson, and Mark Powell, Smooth stable isotopy of topologically isotopic surfaces, arXiv:2606.06299 (2026). (primary): https://arxiv.org/abs/2606.06299\n Evidence used: Extends the same kernel-inclusion direction under mod-2 and group hypotheses; does not prove equality or k>0.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.45. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Exact embedding-space formulation and remaining directions.\n\n**Review notes.** The basepoint is stated in X while displayed spaces use punctured X, implicitly requiring a disjoint puncture. 'Fork =0', 'keri_*', and 'kers_*' are OCR spacing defects.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2922, "problem_number": "KP-4.46", "title": "Kirby Problem 4.46", "statement": "Are all groups good?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.46.\n\nLiterature notes:\n(1) The problem is due to Freedman and Quinn. The term ‘good’ group was first introduced in [Fre84] and [FQ90, p. 99] to refer to any group for which Freedman’s disk embedding theorem holds. In those two sources it was shown that the latter theorem holds for any group belonging to the smallest class of groups that contains $\\mathbb{Z}$ and finite groups and is closed under colimits, subgroups, extensions, and quotients. In [Kir97, Problem 4.6] a group is defined to be good if it belongs to this class of groups. The definition in terms of a capped grope was first formulated in [FT95a]. This has now become the accepted definition.\n\n(2) A group $\\Gamma$ is said to be good if for every height 1.5 disc-like capped grope G, with some choice of basepoint, and for every group homomorphism $\\varphi: \\pi_{1}(G) \\to \\Gamma$, there exists an immersed disk D in Gwhose framed boundary coincides with the attaching region of G, such that the double point loops of D, considered as fundamental group elements by making some choice of basing path, are mapped to the identity element of $\\Gamma$ by $\\varphi$. For more details on the definition, including the definition of a capped grope, see [FT95a, KOPR21a], cf. [Kir97, Problem 5.9] (these definitions differ in the height of a capped grope that one starts with, but the two definitions are equivalent by a method called grope height raising). We limit ourselves to saying that a capped grope is a smooth 4-manifold, and its attaching region is a specified solid torus in its boundary. The core of the solid torus is null-homotopic in the capped grope (regardless of height). Therefore some immersed disk D always exists – the additional fact about the double point loops is the key feature of the definition. It is also worth observing that good groups are defined by a property that can be entirely stated in the smooth category.\n\n(3) Almost every known result about topological 4-manifolds relies on the disk embedding theorem, and therefore the question of which groups are good is fundamental. Recall that any group is the fundamental group of some 4manifold, but fundamental groups of compact 4-manifolds are necessarily finitely presented. Therefore, one could choose to ask the question only for finitely presented groups. However, as we will see in item 7 below, if all finitely presented groups were good, then all groups would be good, so there is no benefit to this restriction.\n\n(4) By work of Freedman and Quinn [FQ90, Chapter 11] (see also [OPR21b]), the action of $L_{5}(\\mathbb{Z}[\\pi_{1}(X)])$ on the structure set $\\mathcal{S}(X)is$ defined, and the surgery sequence is known to be exact at $\\mathcal{S}(X)$ and $\\mathcal{N}(X)$, for every 4manifold X with $\\pi_{1}(X)$ good. Therefore an affirmative answer would give the exactness of the topological surgery sequence for 4-manifolds in full generality (cf. [Kir97, Problem 4.6]). This is known in dimensions five and higher in the smooth, piecewise-linear, and topological categories [Bro72, Nov64, Sul96, Wal99, KS77]. See Problem 4.22 for more background on the surgery sequence.\n\n(5) An affirmative answer would also imply the 5-dimensional s-cobordism theorem in full generality. The s-cobordism theorem for dimensions 6 and higher is true in both the smooth and topological settings [Sma62a, Bar63, Maz63,Sta67,KS77] (see also [Mil65, RS72]). The smooth 5dimensionals-cobordism theorem is false by work of Donaldson [Don87a]. See also Problem 4.56.\n\n(6) The exactness of the surgery sequence and the h-cobordism theorem is a key step in the classification of closed, simply connected 4-manifolds up to homeomorphism. Therefore the extension to general fundamental groups is likely to have several spectacular applications. We observe that even if all groups were good, classification results would not immediately follow. Nevertheless this would be a result of immense importance.\n\n(7) As mentioned above, it was shown in [FQ90] that finite groups and the infinite cyclic group are good. Groups of subexponential growth were shown to be good in [KQ00, FT95a]. The class of good groups is also known to be closed under subgroups, quotients, extensions, and direct limits [FT95a, KOPR21a]. It is an open question whether any nonabelian free group is good. Recall that every finitely generated group arises as a quotient of a subgroup of $\\mathbb{Z}*\\mathbb{Z}and$ every group is a colimit of its finitely generated subgroups. Therefore, all groups are good if and only if the free group on two generators is good. We state this as a question.\n\n\\paragraph{Question.} Is the group $\\mathbb{Z}*\\mathbb{Z}$ good?\n\n(8) At present it is also open whether amenable groups are good. Addressing this would be a substantial step towards the general question.\n\n\\paragraph{Question.} Are amenable groups good?\n\n(9) For further details on the relationships between the above problems and Problems 1.62and 4.47, see [KOPR21b].\n\nReferences cited:\n- [Fre84] Michael H. Freedman. The disk theorem for four-dimensional manifolds. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 647–663. PWN, Warsaw, 1984.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [FT95a] Michael H. Freedman and Peter Teichner. 4-manifold topology. I. Subexponential groups. Invent. Math., 122(3):509–529, 1995. doi:10.1007/BF01231454.\n- [KOPR21a] Min Hoon Kim, Patrick Orson, JungHwan Park, and Arunima Ray. Good groups. In The disc embedding theorem, pages 273–282. Oxford Univ. Press, Oxford, 2021.\n- [OPR21b] Patrick Orson, Mark Powell, and Arunima Ray. Surgery theory and the classification of closed, simply connected 4-manifolds. In The disc embedding theorem, pages 331–351. Oxford Univ. Press, Oxford, 2021.\n- [Bro72] William Browder. Surgery on simply-connected manifolds, volume 65 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York-Heidelberg, 1972.\n- [Nov64] S. P. Novikov. Homotopically equivalent smooth manifolds. I. Izv. Akad. Nauk SSSR Ser. Mat., 28:365–474, 1964.\n- [Sul96] D. P. Sullivan. Triangulating and smoothing homotopy equivalences and homeomorphisms. Geometric Topology Seminar Notes. In The Hauptvermutung book, volume 1 of K-Monogr. Math., pages 69–103. Kluwer Acad. Publ., Dordrecht, 1996. doi:10.1007/978-94-017-3343-4\\_3.\n- [Wal99] C. T. C. Wall. Surgery on compact manifolds, volume 69 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, 1999. Edited and with a foreword by A. A. Ranicki. doi:10.1090/surv/069.\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [Sma62a] S. Smale. On the structure of manifolds. Amer. J. Math., 84:387–399, 1962. doi: 10.2307/2372978.\n- [Bar63] D. Barden. The structure of manifolds. PhD thesis, Cambridge University, 1963.\n- [Maz63] Barry Mazur. Relative neighborhoods and the theorems of Smale. Ann. of Math. (2), 77:232–249, 1963. doi:10.2307/1970215.\n- [Sta67] John R. Stallings. Lectures on polyhedral topology, volume 43 of Tata Institute of Fundamental Research Lectures on Mathematics. Tata Institute of Fundamental Research, Bombay, 1967. Notes by G. Ananda Swarup.\n- [Mil65] J. W. Milnor. Lectures on the h-cobordism theorem. Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow.\n- [RS72] C. P. Rourke and B. J. Sanderson. Introduction to piecewise-linear topology. Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69.\n- [Don87a] S. K. Donaldson. Irrationality and the h-cobordism conjecture. J. Differential Geom., 26(1):141–168, 1987. http://projecteuclid.org/euclid.jdg/1214441179.\n- [KQ00] Vyacheslav S. Krushkal and Frank Quinn. Subexponential groups in 4-manifold topology. Geom. Topol., 4:407–430, 2000. doi:10.2140/gt.2000.4.407.\n- [KOPR21b] Min Hoon Kim, Patrick Orson, JungHwan Park, and Arunima Ray. Open problems. In The disc embedding theorem, pages 353–382. Oxford Univ. Press, Oxford, 2021.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Subexponential-growth groups and classes generated from them by standard closure operations are good, but goodness of F2, equivalently of all groups, remains open.\n\n**Verified partial progress.**\n\n- Finite, abelian, solvable, and more generally subexponential-growth groups are good.\n- Goodness is closed under subgroups, quotients, extensions, and direct limits.\n- The universal question reduces to the rank-two nonabelian free group.\n\n**Full solution or refutation.**\n\nNo proof that F2 or all amenable groups are good, and no bad group counterexample, was verified.\n\n**What remains.**\n\nDecide goodness of F2; a positive answer settles all groups, while a negative answer supplies the first bad group.\n\n**Sources checked.**\n\n- Michael H. Freedman and Peter Teichner, 4-manifold topology I: Subexponential groups, Inventiones Mathematicae 122 (1995), 509--529. (primary): https://doi.org/10.1007/BF01231454\n Evidence used: Foundational theorem proving goodness for subexponential groups and closure framework.\n- Vyacheslav S. Krushkal and Frank Quinn, Subexponential groups in 4-manifold topology, Geometry & Topology 4 (2000), 407--430. (primary): https://doi.org/10.2140/gt.2000.4.407\n Evidence used: Further proof and development of the subexponential-growth case.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.46. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current equivalence with F2 and open amenable-group subproblem.\n\n**Review notes.** Background contains joined-word OCR including 'Gin', '4manifold', '5dimensionals-cobordism', and '$Z*Zand$'.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2923, "problem_number": "KP-4.47", "title": "Kirby Problem 4.47", "statement": "(Round handle problem). Is there a link $L \\subset S^{3}$ with vanishing pairwise linking numbers that is not round handle slice?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.47.\n\nLiterature notes:\n(1) We define the terminology in the problem. A round handle is a copy of $D^{1} \\times D^{2} \\times S^{1}$, which is attached to the boundary of a 4-manifold along $S^{0} \\times D^{2} \\times S^{1}$. (So it is a 4–dimensional round 1–handle.) Given an m-component link $L \\subset S^{3}$ we construct a 4-manifold $R(L)$ by attaching m round handles to $D^{4}$ as follows. For the i-th component $L_{i}$ of L, let $\\lambda_{i}$ denote a 0-framed longitude and let $\\mu_{i}$ denote a meridian, chosen so that the linking number $lk(\\lambda_{i}, \\mu_{i}) =$ 0. To attach the ith round handle, identify \\{−1\\} $\\times S^{1} \\times D^{2}$ to $\\lambda_{i}$ and \\{+1\\} $\\times S^{1} \\times D^{2}$ to $\\mu_{i}$, using the trivial framing in both cases. The resulting 4-manifold contains the link L in the boundary $\\partial R(L). A$ link L is said to be round handle slice if $L \\subset \\partial R(L)$ is slice in $R(L)$, that is, if L is the boundary of a collection of locally flat pairwise disjoint embedded discs in $R(L)$.\n\n(2) The round handle problem was proposed by Freedman and Krushkal in [FK16, Section 5.1], and is presented in detail in [KPT21].\n\n(3) If the 4-dimensional topological surgery and s-cobordism conjectures hold for free groups, then every link with vanishing pairwise linking numbers is round handle slice. So this problem gives a way to potentially disprove the union of these conjectures, which are central open problems in 4-manifold topology. For more on those conjectures see Problems 4.46and 1.62.\n\nReferences cited:\n- [FK16] Michael Freedman and Vyacheslav Krushkal. Engel relations in 4-manifold topology. Forum Math. Sigma, 4:Paper No. e22, 57, 2016. doi:10.1017/fms.2016.20.\n- [KPT21] Min Hoon Kim, Mark Powell, and Peter Teichner. Round handle problem. Pure Appl. Math. Q., 17(1):237–247, 2021. doi:10.4310/PAMQ.2021.v17.n1.a6.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No vanishing-pairwise-linking link known not to be round-handle slice was verified.\n\n**Verified partial progress.**\n\n- The list gives a geometric reformulation using round 1-handles.\n\n**Full solution or refutation.**\n\nThe counterexample question remains open.\n\n**What remains.**\n\nFind an obstruction to round-handle sliceness beyond pairwise linking.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.47 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the construction and retains the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2924, "problem_number": "KP-4.48", "title": "Kirby Problem 4.48", "statement": "Let $\\Delta$ be a contractible, compact 4-manifold. Is the space of homeomorphisms of $\\Delta$ that fix the boundary pointwise, with the compact-open topology, a contractible space?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.48.\n\nLiterature notes:\nThis is true for $\\Delta = D^{4}$, by the Alexander trick. By PerronQuinn [Per86, Qui86], the space is connected. In dimensions at least 6, the space of homeomorphisms is contractible, as proven by Galatius–Randal-Williams [GRW24].\n\nReferences cited:\n- [Per86] B. Perron. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. Topology, 25(4):381–397, 1986. doi:10.1016/0040-9383(86)90018-2.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [GRW24] Sø ren Galatius and Oscar Randal-Williams. The Alexander trick for homology spheres. Int. Math. Res. Not. IMRN, 2024(24):14689–14703, 2024. doi:10.1093/imrn/rnae255.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The boundary-fixing homeomorphism space is contractible for D4 and connected in general, but general contractibility of a contractible topological 4-manifold was not verified.\n\n**Verified partial progress.**\n\n- Alexander trick settles D4.\n- Perron--Quinn prove connectedness.\n\n**Full solution or refutation.**\n\nNo general contractibility theorem was verified.\n\n**What remains.**\n\nControl higher homotopy groups of the relative homeomorphism space.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.48 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the D4 and connectedness results while posing the general question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2925, "problem_number": "KP-4.49", "title": "Kirby Problem 4.49", "statement": "Give an effective necessary and sufficient condition for an open 4-manifold to be homeomorphic to the interior of a compact 4-manifold.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.49.\n\nLiterature notes:\n(1) In his thesis [Sie65], Siebenmann gave necessary and sufficient conditions for an n-dimensional open PL manifold M, for $n \\geq$ 6, to be compactifiable by adding a manifold boundary (see also [BLL65]). Recall that a manifold is said to be open if it has empty boundary and only noncompact components. The proof extends to the case where M is noncompact but has compact boundary. The case of possibly noncompact boundary is addressed in [O’B83, GG20]. The foundational results of Kirby-Siebenmann [KS77] on high-dimensional topological manifolds imply that Siebenmann’s proof can also be applied in the purely topological setting.\n\n(2) The cases $n =$ 2,3 are also well understood [HP70], [Tuc74], [GG20, Section 2].\n\n(3) Note that if an n-manifold M is the interior of an n-manifold M, then there is a neighborhood of the end of M of the for m $Y \\times [0,\\infty)$ for some (n−1)-manifold Y. In this case we say that M has a collared end. Having a collared end imposes several homotopy-theoretical conditions on M. The prior results mentioned above roughly consist of showing that some collection of such homotopy-theoretic conditions is sufficient to conclude the existence of a collared end.\n\n(4) An end of a manifold M is said to betame if there is a closed neighborhood U of the end, and a proper map $U \\times [0,\\infty) \\to M$ which is the inclusion on $U \\times$ \\{0\\}. Suppose M is a manifold with a tame connected end with finitely presented fundamental group $\\pi$. Siebenmann defined an invariant $\\sigma(end(M)) \\in K_{~,0}(\\mathbb{Z}\\pi)$, related to Wall’s finiteness obstruction [Wal65]. Siebenmann’s result from [Sie65] mentioned above states that if an nmanifold M, for $n \\geq$ 6, has a tame connected end with finitely presented fundamental group $\\pi$, then $\\sigma(end(M)) \\in K_{~,0}(\\mathbb{Z}\\pi)$ vanishes if and only if M has a collared end. This result was extended by Quinn to the case that n=5 and $\\pi$ is good (see Problem 4.46). We emphasize that in the results above $\\pi$ is the fundamental group of the end of M, and not of M itself. See [FQ90, Section 11.9] for further details.\n\n(5) The direct topological analogue of Siebenmann’s theorem in dimension four fails by a result of Kwasik-Schultz [KS88, Theorem 2.1] and unpublished work of Weinberger. These latter authors constructed a class of 4-manifolds M such that the infinite cyclic cover $M_{\\infty}$ has an end with two components, each of which is tame, has good fundamental group, and for which Siebenmann’s obstruction vanishes, but such that $M_{\\infty}$is not homeomorphic to $Y \\times \\mathbb{R}$, for any compact 3-manifold Y.\n\n(6) By the result mentioned above, a characterization of when an open 4manifold has collared end must be necessarily more complicated than it is in higher dimensions. An ideal characterization would be purely homotopy-theoretic. Partial results in dimension four in the topological setting are given in [FQ90, Section 11.9B], but there is not yet a candidate for a necessary and sufficient condition.\n\nReferences cited:\n- [Sie65] Laurence Carl Siebenmann. THE OBSTRUCTION TO FINDING A BOUNDARY FOR AN OPEN MANIFOLD OF DIMENSION GREATER THAN FIVE. ProQuest LLC, Ann Arbor, MI, 1965. Thesis (Ph.D.)–Princeton University. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt=info: ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri:pqdiss:6605012.\n- [BLL65] W. Browder, J. Levine, and G. R. Livesay. Finding a boundary for an open manifold. Amer. J. Math., 87:1017–1028, 1965. doi:10.2307/2373259.\n- [O’B83] Gary O’Brien. The missing boundary problem for smooth manifolds of dimension greater than or equal to six. Topology Appl., 16(3):303–324, 1983. doi:10.1016/0166-8641(83)90027-5.\n- [GG20] Shijie Gu and Craig R. Guilbault. Compactifications of manifolds with boundary. J. Topol. Anal., 12(4):1073–1101, 2020. doi:10.1142/$S^{1}$793525319500754.\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [HP70] L. S. Husch and T. M. Price. Finding a boundary for a 3-manifold. Ann. of Math. (2), 91:223–235, 1970. doi:10.2307/1970605.\n- [Tuc74] Thomas W. Tucker. Non-compact 3-manifolds and the missing-boundary problem. Topology, 13:267–273, 1974. doi:10.1016/0040-9383(74)90019-6.\n- [Wal65] C. T. C. Wall. Finiteness conditions for CW-complexes. Ann. of Math. (2), 81:56– 69, 1965. doi:10.2307/1970382.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [KS88] Slawomir Kwasik and Reinhard Schultz. Desuspension of group actions and the ribbon theorem. Topology, 27(4):443–457, 1988. doi:10.1016/0040-9383(88)90023-7.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Siebenmann-type end criteria solve higher-dimensional PL compactification, but an effective necessary-and-sufficient topological 4-manifold criterion remains open.\n\n**Verified partial progress.**\n\n- Higher-dimensional end theorems establish the model result.\n- The list identifies dimension four as the unresolved case.\n\n**Full solution or refutation.**\n\nNo effective 4-dimensional criterion was verified.\n\n**What remains.**\n\nDevelop 4-dimensional end invariants sufficient for manifold-boundary compactification.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.49 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Contrasts higher-dimensional results with the requested four-dimensional condition.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2926, "problem_number": "KP-4.50", "title": "Kirby Problem 4.50", "statement": "Classify closed topological 4-manifolds (orientable and not) with finite fundamental group, up to homeomorphism. The following infinite families of fundamental groups are of particular interest.\n\n(a) Dihedral groups.\n\n(b) Quaternionic groups.\n\n(c) Abelian groups with two or three generators. A significant starting point would be to complete the classification for 4-manifolds with fundamental group of order at most eight.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.50.\n\nLiterature notes:\n(1) The problem is deemed solved if the classification is reduced to algebraic invariants. An interesting subproblem is to complete the classification for smooth 4-manifolds only.\n\n(2) The classification exists for trivial fundamental group by Freedman [Fre82], for cyclic fundamental groups in the orientable case by Hambleton-Kreck [HK88, HK93a, HK93c, HK93b], and for fundamental group $\\mathbb{Z}/2\\mathbb{Z}$ in the nonorientable case by Hambleton-Kreck-Teichner [HKT94]. See also [HH23].\n\n(3) The homotopy classification is known in some cases: [HK88, KPR24, KNR22, Hil23]. In these cases one strategy is therefore to upgrade this to a homeomorphism classification using surgery theory.\n\nReferences cited:\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [HK88] Ian Hambleton and Matthias Kreck. On the classification of topological 4-manifolds with finite fundamental group. Math. Ann., 280(1):85–104, 1988. doi:10.1007/BF01474183.\n- [HK93a] Ian Hambleton and Matthias Kreck. Cancellation, elliptic surfaces and the topology of certain four-manifolds. J. Reine Angew. Math., 444:79–100, 1993. doi:10.1515/crll.1993.444.79.\n- [HK93c] Ian Hambleton and Matthias Kreck. Cancellation of lattices and finite two-complexes. J. Reine Angew. Math., 442:91–109, 1993. doi:10.1515/crll.1993.442.91.\n- [HK93b] Ian Hambleton and Matthias Kreck. Cancellation of hyperbolic forms and topological four-manifolds. J. Reine Angew. Math., 443:21–47, 1993. doi:10.1515/crll.1993.443.21.\n- [HKT94] Ian Hambleton, Matthias Kreck, and Peter Teichner. Nonorientable 4-manifolds with fundamental group of order 2. Trans. Amer. Math. Soc., 344(2):649–665, 1994. doi:10.2307/2154500.\n- [HH23] Ian Hambleton and Jonathan A. Hillman. Quotients of $S^{2}$ $\\times$ $S^{2}$. J. Lond. Math. Soc. (2), 108(4):1393–1416, 2023. doi:10.1112/jlms.12783.\n- [KPR24] Daniel Kasprowski, Mark Powell, and Benjamin Ruppik. Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups. Math. Proc. Cambridge Philos. Soc., 177(2):263–283, 2024.\n- [KNR22] Daniel Kasprowski, John Nicholson, and Benjamin Ruppik. Homotopy classification of 4-manifolds whose fundamental group is dihedral. Algebr. Geom. Topol., 22(6):2915–2949, 2022. doi:10.2140/agt.2022.22.2915.\n- [Hil23] Jonathan A. Hillman. Homotopy types of 4-manifolds with 3-manifold fundamental groups, 2023. arXiv:2307.15292.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Topological classification is algebraically reduced for trivial, orientable cyclic, and nonorientable Z/2 fundamental groups, but the requested finite-group families remain open.\n\n**Verified partial progress.**\n\n- Freedman classifies the trivial group case.\n- Hambleton--Kreck(-Teichner) handle cyclic and nonorientable Z/2 cases.\n\n**Full solution or refutation.**\n\nNo classification for the listed general finite groups was verified.\n\n**What remains.**\n\nComplete algebraic classification for groups of order at most eight and specified families.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.50 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists the classified base cases and the remaining groups.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2927, "problem_number": "KP-4.51", "title": "Kirby Problem 4.51", "statement": "Let X be a closed, smooth 4-manifold with fundamental group isomorphic to $\\mathbb{Z}$. Is the $\\mathbb{Z}[\\mathbb{Z}]-valued$ intersection form on $\\pi_{2}(X)$ extended from $\\mathbb{Z}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.51.\n\nLiterature notes:\n(1) We say that a Hermitian for m on a free $\\mathbb{Z}[\\mathbb{Z}]-module$ isextended from $\\mathbb{Z}if$ it is represented by a matrix, with respect to some basis over $\\mathbb{Z}[\\mathbb{Z}]$, with integer entries.\n\n(2) Hambleton–Teichner [HT97] exhibited a unimodular Hermitian for m over $\\mathbb{Z}[\\mathbb{Z}]$ that is not extended over $\\mathbb{Z}$.\n\n(3) It was known from Freedman–Quinn [FQ90] that this for m can be realized as the intersection form of closed topological manifold with fundamental group $\\mathbb{Z}. A$ manifold with such a for m is not even homotopy equivalent to the connected sum of a homotopy $S^{1} \\times S^{3}$ with a simply connected 4-manifold.\n\n(4) Conversely, if the intersection form of a 4-manifold with fundamental group $\\mathbb{Z}$ is extended from $\\mathbb{Z}$, then that 4-manifold is homeomorphic to a connected sum $M\\#(S^{1} \\times S^{3})$, where $\\pi_{1}(M) =$ \\{1\\}. Thus an affirmative answer to the question would imply that every closed, smooth, oriented 4-manifold with fundamental group $\\mathbb{Z}$ splits topologically as such a connected sum.\n\n(5) Friedl–Hambleton–Melvin–Teichner [FHMT07] showed, by an application of Donaldson’s theorem [Don83] to the finite cyclic covers (of degree at least 3) of the manifold in [HT97] that no 4-manifold with this for m can be smooth. Conjecture 1.3 in [FHMT07] states that the intersection form of any closed, smooth manifold with fundamental group $\\mathbb{Z}$ is extended from the integers.\n\n(6) Fintushel–Stern [FS94] showed that there are smooth 4-manifolds with fundamental group $\\mathbb{Z}$ that do not have an $S^{1} \\times S^{3}$ connect summand smoothly. In fact, any symplectic 4-manifold will not smoothly decompose as a connected sum in this fashion.\n\nReferences cited:\n- [HT97] Ian Hambleton and Peter Teichner. A non-extended Hermitian form over ZrZs. Manuscripta Math., 93(4):435–442, 1997. doi:10.1007/BF02677483.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [FHMT07] Stefan Friedl, Ian Hambleton, Paul Melvin, and Peter Teichner. Non-smoothable four-manifolds with infinite cyclic fundamental group. Int. Math. Res. Not. IMRN, 2007(11):Art. ID rnm031, 20, 2007. doi:10.1093/imrn/rnm031.\n- [Don83] S. K. Donaldson. An application of gauge theory to four-dimensional topology. J. Differential Geom., 18(2):279–315, 1983.\n- [FS94] Ronald Fintushel and Ronald J. Stern. A fake 4-manifold with $\\pi_1=\\mathbb{Z}$ and $b^+=4$. Turkish J. Math., 18(1):1–6, 1994.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Non-extended unimodular hermitian forms over Z[Z] are known and realized topologically, while smooth realization as an intersection form remains unsettled.\n\n**Verified partial progress.**\n\n- Hambleton--Teichner exhibit a non-extended form.\n- Freedman--Quinn supply topological realization.\n\n**Full solution or refutation.**\n\nNo universal smooth extension theorem or smooth counterexample was verified.\n\n**What remains.**\n\nDetermine whether the non-extended form has a smooth realization.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.51 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes algebraic/topological examples from the smooth question.\n\n**Review notes.** OCR defects in the background's form notation were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2928, "problem_number": "KP-4.52", "title": "Kirby Problem 4.52", "statement": "Let M and N be closed, orientable topological 4-manifolds with $\\pi_{1}(M) \\cong \\pi_{1}(N)a$ good group. Suppose that M and N are simple homotopy equivalent and stably homeomorphic. Is there a simple homotopy equivalence $N \\to M$ that lies in the orbit of Id $:M \\to M$ under the Wall realization action of $L^{s}_{5}(\\mathbb{Z}[\\pi_{1}(M)])$ on the simple structure set of M?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.52.\n\nLiterature notes:\n(1) We can hope to classify 4-manifolds up to simple homotopy equivalence, and up to stable homeomorphism, for many groups. We can also hope to compute $L^{s}_{5}(\\mathbb{Z}[\\pi_{1}(M)])$. If the answer were yes, this would give a somewhat satisfying picture for the classification of 4-manifolds, for good groups. If the answer were no, this would uncover interesting new phenomena. The answer is yes for simply connected 4-manifolds, and for those with fundamental group $\\mathbb{Z}$ [FQ90] and $\\mathbb{Z}/n \\mathbb{Z}[HK88$, HK93a, HK93c, HK93b].\n\n(2) In the case of spin 4-manifolds, the program suggested by this question could potentially be made easier, only requiring the simple homotopy classification, if the answer to the following question is no.\n\n\\paragraph{Question.} Do there exist closed, spin 4-manifolds that are simple homotopy equivalent but not stably homeomorphic? The latter question was part of [Kir97, Problem 4.84], due to Teichner. Note that closed, spin 4-manifolds that are homotopy equivalent have the same Kirby–Siebenmann invariant, so that invariant cannot be used here.\n\n(3) Jim Davis [Dav05] proved that if the fundamental group is $\\pi$ and the map $\\kappa_{2}: H_{2}(\\pi;\\mathbb{Z}/2\\mathbb{Z}) \\to L_{4}(\\mathbb{Z}\\pi)is$ injective, then homotopy equivalent 4manifolds with that fundamental group and the same Kirby-Siebenmann invariant are stably homeomorphic.\n\n(4) There are counterexamples known for non-spin 4-manifolds, even assumin g they have the same Kirby-Siebenmann invariant [Tei97].\n\nReferences cited:\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [HK88] Ian Hambleton and Matthias Kreck. On the classification of topological 4-manifolds with finite fundamental group. Math. Ann., 280(1):85–104, 1988. doi:10.1007/BF01474183.\n- [HK93a] Ian Hambleton and Matthias Kreck. Cancellation, elliptic surfaces and the topology of certain four-manifolds. J. Reine Angew. Math., 444:79–100, 1993. doi:10.1515/crll.1993.444.79.\n- [HK93c] Ian Hambleton and Matthias Kreck. Cancellation of lattices and finite two-complexes. J. Reine Angew. Math., 442:91–109, 1993. doi:10.1515/crll.1993.442.91.\n- [HK93b] Ian Hambleton and Matthias Kreck. Cancellation of hyperbolic forms and topological four-manifolds. J. Reine Angew. Math., 443:21–47, 1993. doi:10.1515/crll.1993.443.21.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Dav05] James F. Davis. The Borel/Novikov conjectures and stable diffeomorphisms of 4-manifolds. In Geometry and topology of manifolds. Papers from the conference held at McMaster University, Hamilton, ON, USA, May 14–18, 2004, pages 63– 76. Providence, RI: American Mathematical Society (AMS), 2005.\n- [Tei97] Peter Teichner. On the star-construction for topological 4-manifolds. In Geometric topology. 1993 Georgia international topology conference, August 2–13, 1993, Athens, GA, USA, pages 300–312. Providence, RI: American Mathematical Society; Cambridge, MA: International Press, 1997.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Wall-orbit statement holds for simply connected manifolds and fundamental group Z, but no theorem for every good group was verified.\n\n**Verified partial progress.**\n\n- Known positive cases include trivial group and Z.\n\n**Full solution or refutation.**\n\nThe general good-group question remains open.\n\n**What remains.**\n\nExtend realization-action control to additional good groups or find a failure.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.52 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the positive cases and conditional classification programme.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2929, "problem_number": "KP-4.53", "title": "Kirby Problem 4.53", "statement": "Does there exist an algorithm that takes as input a closed, triangulated 4-manifold, and outputs in finite time whether or not that 4-manifold is homeomorphic to $S^{4}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.53.\n\nLiterature notes:\n(1) This is called the recognition problem. We say that we can recognize an n-manifold X if there exists an algorithm that takes as input a closed, triangulated n-manifold, and outputs in finite time whether or not that 4-manifold is homeomorphic to X.\n\n(2) We assume that the input 4-manifolds are represented by the finite data of a triangulation, and hence they are smooth, since triangulated 4-manifolds are smooth. We are not assured that the input 4-manifold is simplyconnected.\n\n(3) The corresponding question has a positive answer in dimensions $\\leq$ 3 [Tho94], [Rub95], and a negative answer in dimensions $\\geq$ 5 [VKF74].\n\n(4) Markov [Mar58] showed that, for some integerk, the corresponding question for the connected sum of k copies of $S^{2} \\times S^{2}$ has a negative answer. This raises the question of the minimal k for which this holds. It was shown that k could be taken to be 14 in [Sht05], 12 in [Gor22], and 9 in [Tan23]. See also the exposition in [Kir20].\n\n(5) By Markov [Mar58], there exist infinite lists of group presentations ${P_{i}}$ such that there is no algorithm taking as input one of the $P_{i}$, and outputting in finite time whether or not the group $G(P_{i})$ presented by $P_{i}$ is trivial. One possible strategy to solve the problem is to construct a corresponding list ${X_{i}}$ of closed, triangulated 4-manifolds, whose 2-skeleta give rise to the $P_{i}$, so in particular $\\pi_{1}(X_{i}) \\cong G(P_{i})$, with the property that $X_{i}$ is homeomorphic to $S^{4}$ if and only if $G(P_{i}) =$ \\{1\\}. The forward direction clearly holds, but it is not clear how to find ${P_{i}}$ and ${X_{i}}such$ that the backwards direction holds. This strategy does work for $\\#_{k}(S^{2} \\times S^{2})$ in place of $S^{4}$, and was the basis for the proofs of [Sht05, Gor22, Tan23] for $k =$ 14,12,9 respectively.\n\n(6) One can also ask the analogous smooth recognition problem with ‘diffeomorphic’ in place of ‘homeomorphic’. This is also open for $S^{4}$. As described in [Kir20], there exists aksuch that $\\#_{k}(S^{2} \\times S^{2})is$ not smoothly recognizable, but there are no known upper bounds on the minimal k for which this holds.\n\nReferences cited:\n- [Tho94] Abigail Thompson. Thin position and the recognition problem for $S^{3}$. Math. Res. Lett., 1(5):613–630, 1994. doi:10.4310/MRL.1994.v1.n5.a9.\n- [Rub95] Joachim H. Rubinstein. An algorithm to recognize the 3-sphere. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 601–611. Birkhäuser, Basel, 1995.\n- [VKF74] I. A. Volodin, V. E. Kuznecov, and A. T. Fomenko. The problem of the algorithmic discrimination of the standard three-dimensional sphere. Uspehi Mat. Nauk, 29:71– 168, 1974. Appendix by S. P. Novikov.\n- [Mar58] A. Markov. The insolubility of the problem of homeomorphy. Dokl. Akad. Nauk SSSR, 121:218–220, 1958.\n- [Sht05] M. A. Shtan’ko. On Markov’s theorem on the algorithmic nonrecognizability of manifolds. Fundam. Prikl. Mat., 11:257–259, 2005. doi:10.1007/s10958-007-0375-z.\n- [Gor22] Cameron McA. Gordon. On the homeomorphism problem for 4-manifolds. New Zealand J. Math., 52:821–826, 2021 [2021–2022]. doi:10.53733/205.\n- [Tan23] Martin Tancer. Simpler algorithmically unrecognizable 4-manifolds, 2023. arXiv: 2310.07421.\n- [Kir20] R. C. Kirby. Markov’s theorem on the nonrecognizablility of 4-manifolds: an exposition, 2020. Celebratio Mathematica: Martin Scharlemann, https://celebratio.org/Scharlemann M/article/785/.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No finite-time algorithm deciding whether an arbitrary closed triangulated 4-manifold is homeomorphic to S4 was verified.\n\n**Verified partial progress.**\n\n- The input convention restricts to finite triangulations and hence smooth manifolds.\n\n**Full solution or refutation.**\n\nThe 4-sphere recognition problem remains open in the checked source.\n\n**What remains.**\n\nFind a decidability method or reduce a known undecidable problem to recognition.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.53 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the recognition problem and retains the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2930, "problem_number": "KP-4.54", "title": "Kirby Problem 4.54", "statement": "The quadratic 2 type of a 4-manifold M is the data $(\\pi_{1}(M), \\pi_{2}(M), \\lambda_{M}, k_{M})$ of the fundamental group $\\pi_{1}(M)$, the second homotopy group $\\pi_{2}(M)$ considered as $a \\mathbb{Z}[\\pi_{1}(M)]-module$, the equivariant intersection for $m \\lambda_{M}$, and the k-invariant in $k_{M} \\in H^{3}(B\\pi_{1}(M);\\pi_{2}(M))$. Which quadratic 2-types are realized by closed, oriented topological 4-manifolds? Which are realized by closed, oriented, smooth 4-manifolds? Can this problem be solved for specific fundamental groups, for example for certain families of non-cyclic finite groups?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.54.\n\nLiterature notes:\n(1) This is in [Kir97, Problem 4.1].\n\n(2) We can also ask the question with the data of the Stiefel-Whitney classes $w_{1}$ and $w_{2}$. Which Stiefel-Whitney classes are realized within a given quadratic 2-type?\n\n(3) The problem builds on the geography problem for simply connected topological 4-manifolds, where it restricts to the question of which intersection forms occur. In the smooth category, this is answered by Donaldson’s diagonalizability theorem along with a positive resolution of the 11/8conjecture.\n\n(4) For the topological case, Freedman [Fre82] showed that every nonsingular symmetric bilinear for m is realized by a closed simply connected 4-manifold, so the problem is solved when $\\pi_{1} =$ 1. A similar result is known for $\\pi_{1} =\\mathbb{Z}$ [FQ90]. For other good fundamental groups, a possible strategy in the topological category was introduced by Hambleton-Kreck in [HK88, Lemma 4.1], the paper where the quadratic 2-type initially arose. First, classify the quadratic 2-types that are stably realizable, meaning that they are realizable by topological 4-manifolds after taking the orthogonal sum with the quadratic 2-type of $S^{2} \\times S^{2}$. If a quadratic 2-type is stably realizable then it is realizable unstably by a topological 4-manifold, which can be shown using the sphere embedding theorem [FQ90, $BKK^{+}21]$. Hambleton-Kreck [HK88, HK93a] used this strategy to solve the realization problem for finite cyclic groups, in the topological category.\n\n(5) For non-simply-connected, smooth, oriented 4-manifolds, Donaldson’s diagonalization theorem holds without any assumption on the fundamental group, so definite integral intersection forms must be diagonalizable. The sphere embedding theorem cannot be applied in the smooth case, so the strategy described above of stably realizing and then destabilizing is not currently viable.\n\n(6) The existence problem has a closely related uniqueness analogue. For finite cyclic groups [HK88], abelian groups with at most two generators [KPR24], dihedral groups [KNR22], and aspherical 3-manifold groups [Hil23], we know that the quadratic 2-type determines the 4manifold up to homotopy equivalence. Hence if we could also understand the image of the invariants in one of these cases, i.e. the realization problem, we would have a fairly complete homotopy classification, at least modulo the algebraic problem of being able to reliably distinguish or identify given quadratic 2-types.\n\n(7) Kirk and Livingston’s paper [KL09] contains many related references and its own list of interesting related problems.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [HK88] Ian Hambleton and Matthias Kreck. On the classification of topological 4-manifolds with finite fundamental group. Math. Ann., 280(1):85–104, 1988. doi:10.1007/BF01474183.\n- [BKK+21] Stefan Behrens, Boldizsár Kalmár, Min Hoon Kim, Mark Powell, and Arunima Ray, editors. The disc embedding theorem. Oxford University Press, Oxford, 2021.\n- [HK93a] Ian Hambleton and Matthias Kreck. Cancellation, elliptic surfaces and the topology of certain four-manifolds. J. Reine Angew. Math., 444:79–100, 1993. doi:10.1515/crll.1993.444.79.\n- [KPR24] Daniel Kasprowski, Mark Powell, and Benjamin Ruppik. Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups. Math. Proc. Cambridge Philos. Soc., 177(2):263–283, 2024.\n- [KNR22] Daniel Kasprowski, John Nicholson, and Benjamin Ruppik. Homotopy classification of 4-manifolds whose fundamental group is dihedral. Algebr. Geom. Topol., 22(6):2915–2949, 2022. doi:10.2140/agt.2022.22.2915.\n- [Hil23] Jonathan A. Hillman. Homotopy types of 4-manifolds with 3-manifold fundamental groups, 2023. arXiv:2307.15292.\n- [KL09] Paul Kirk and Charles Livingston. The geography problem for 4-manifolds with specified fundamental group. Trans. Amer. Math. Soc., 361(8):4091–4124, 2009. doi:10.1090/S0002-9947-09-04649-2.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Quadratic 2-type realization/classification is understood in important simply connected and selected group settings, but no general realization criterion for finite noncyclic groups was verified.\n\n**Verified partial progress.**\n\n- The simply connected restriction becomes intersection-form geography.\n- The list identifies added Stiefel--Whitney data and selected group families.\n\n**Full solution or refutation.**\n\nBoth topological and smooth general realization problems remain open.\n\n**What remains.**\n\nClassify realizable quadratic 2-types for a concrete noncyclic finite group.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.54 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Sets out the data and retains the realization programme.\n\n**Review notes.** OCR defects in module/form notation were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2931, "problem_number": "KP-4.55", "title": "Kirby Problem 4.55", "statement": "Let M and N be closed, orientable, connected 4-manifolds with isomorphic quadratic 2-types. If $\\pi_{1}(M) \\cong \\pi_{1}(N)are$ finite, are M and N homotopy equivalent?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.55.\n\nLiterature notes:\n(1) This is known for $\\pi_{1}$ trivial, finite cyclic groups [HK88], dihedral groups [KPR24], and abelian groups with at most two generators [KNR22]. It is known to be false in the nonorientable case, due to Kim-KojimaRaymond [KKR92].\n\n(2) A potentially interesting case is $\\pi_{1}(M) = \\mathbb{Z}/2 \\times \\mathbb{Z}/2 \\times \\mathbb{Z}/2$. As noted in [KPR24], in this case we can have torsion in $\\mathbb{Z}\\otimes_{\\mathbb{Z}[\\pi_{1}(M)]}\\Gamma(\\pi_{2}(M))$, which by [HK88] leads to polarized homotopically inequivalent Poincaré 4-complexes with the same quadratic 2-type. Are they homotopy equivalent? Do the homotopy types contain topological 4-manifolds?\n\n(3) Let $L_{p,q}$ and $L_{p,q,1}$ be lens spaces that are not homotopy equivalent. Then $S^{1} \\times L_{p,q}$ and $S^{1} \\times L_{p,q,1}$ are also not homotopy equivalent but they do have isomorphic quadratic 2-type. So the problem is not true in general for infinite fundamental groups.\n\nReferences cited:\n- [HK88] Ian Hambleton and Matthias Kreck. On the classification of topological 4-manifolds with finite fundamental group. Math. Ann., 280(1):85–104, 1988. doi:10.1007/BF01474183.\n- [KPR24] Daniel Kasprowski, Mark Powell, and Benjamin Ruppik. Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups. Math. Proc. Cambridge Philos. Soc., 177(2):263–283, 2024.\n- [KNR22] Daniel Kasprowski, John Nicholson, and Benjamin Ruppik. Homotopy classification of 4-manifolds whose fundamental group is dihedral. Algebr. Geom. Topol., 22(6):2915–2949, 2022. doi:10.2140/agt.2022.22.2915.\n- [KKR92] Myung Ho Kim, Sadayoshi Kojima, and Frank Raymond. Homotopy invariants of nonorientable 4-manifolds. Trans. Amer. Math. Soc., 333(1):71–81, 1992. doi: 10.2307/2154099.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The implication from quadratic 2-type to homotopy equivalence holds for trivial, finite cyclic, dihedral, and small-rank abelian groups, but is false nonorientably and unverified for all finite orientable groups.\n\n**Verified partial progress.**\n\n- Positive cases include finite cyclic, dihedral, and abelian groups with at most two generators.\n- The nonorientable analogue has counterexamples.\n\n**Full solution or refutation.**\n\nThe specified orientable finite-group question remains open.\n\n**What remains.**\n\nResolve the Z/2 cubed test case or construct an orientable counterexample.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.55 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists the positive cases, nonorientable failure, and open test groups.\n\n**Review notes.** Source display has missing spacing in group condition; not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2932, "problem_number": "KP-4.56", "title": "Kirby Problem 4.56", "statement": "(4D s-cobordism conjecture). Let $(W^{4};M_{0}^{3}, M_{1}^{3})$ be a smooth 4-dimensionals-cobordism between closed 3-manifolds. Is W diffeomorphic to $M_{0} \\times$ [0,1]?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.56.\n\nLiterature notes:\n(1) Matumoto and Siebenmann found counterexamples to the topological analogue in [MS78], where both $M_{0}$ and $M_{1}$ are $\\mathbb{RP}^{2} \\times S^{1}$ and W is not known to be smoothable. Specifically that paper showed that the scobordism theorem fails either in dimension four or five, providing specific s-cobordisms that would fail to be products. Later work of Freedman and Quinn [FQ90, Theorem 7.1A] showed that the potential 5-dimensional candidate is indeed a product. So the 4-dimensional candidate of Matumoto and Siebenmann must fail to be a product. Cappell and Shaneson later provided further topological counterexamples where $M_{0}$ and $M_{1}$ are orientable. In a subsequent paper [CS87b] they asserted these examples were smoothable, but this claim was later retracted [CS87a].\n\n(2) A weaker version of the question asks whether s-cobordant, or possibly even simple homotopy equivalent, 3-manifolds are necessarily homeomorphic. Kwasik and Schultz showed, assuming geometrization, that every topological h-cobordism between closed, orientable 3-manifolds is an s-cobordism, and that simple homotopy equivalence implies homeomorphism for closed, orientable 3-manifolds [KS92, Theorem and Theorem 1.1]. The questions appear to be open in the nonorientable setting; in particular geometrization is not yet known for nonorientable 3-manifolds. Whether there exists anh-cobordism between 3-manifolds with nontrivial Whitehead torsion appeared as Problem 4.9 on [Kir97].\n\n(3) The s-cobordism theorem for dimensions 6 and higher is true in the smooth, piecewise linear, and topological settings [Sma62a, Bar63,Maz63, Sta67, KS77] (see also [Mil65, RS72]). The s-cobordism theorem in dimension five is false in the smooth (and equivalently piecewise linear) settings, by work of Donaldson [Don87a], and known to be true in the topological setting for good fundamental groups [FQ90, Theorem 7.1A] (see also [OPR21a]). See Problem 4.46.\n\nReferences cited:\n- [MS78] T. Matumoto and L. Siebenmann. The topological s-cobordism theorem fails in dimension 4 or 5. Math. Proc. Cambridge Philos. Soc., 84(1):85–87, 1978. doi: 10.1017/S0305004100054918.\n- [FQ90] Michael H. Freedman and Frank Quinn. Topology of 4-manifolds, volume 39 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1990.\n- [CS87b] Sylvain E. Cappell and Julius L. Shaneson. Smooth nontrivial 4-dimensional scobordisms. Bull. Amer. Math. Soc. (N.S.), 17(1):141–143, 1987. doi:10.1090/S0273-0979-1987-15542-X.\n- [CS87a] Sylvain E. Cappell and Julius L. Shaneson. Corrigendum to: “Smooth nontrivial 4-dimensional s-cobordisms”. Bull. Amer. Math. Soc. (N.S.), 17(2):401, 1987. doi: 10.1090/S0273-0979-1987-15616-3.\n- [KS92] Slawomir Kwasik and Reinhard Schultz. Vanishing of Whitehead torsion in dimension four. Topology, 31(4):735–756, 1992. doi:10.1016/0040-9383(92)90005-3.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Sma62a] S. Smale. On the structure of manifolds. Amer. J. Math., 84:387–399, 1962. doi: 10.2307/2372978.\n- [Bar63] D. Barden. The structure of manifolds. PhD thesis, Cambridge University, 1963.\n- [Maz63] Barry Mazur. Relative neighborhoods and the theorems of Smale. Ann. of Math. (2), 77:232–249, 1963. doi:10.2307/1970215.\n- [Sta67] John R. Stallings. Lectures on polyhedral topology, volume 43 of Tata Institute of Fundamental Research Lectures on Mathematics. Tata Institute of Fundamental Research, Bombay, 1967. Notes by G. Ananda Swarup.\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [Mil65] J. W. Milnor. Lectures on the h-cobordism theorem. Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow.\n- [RS72] C. P. Rourke and B. J. Sanderson. Introduction to piecewise-linear topology. Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69.\n- [Don87a] S. K. Donaldson. Irrationality and the h-cobordism conjecture. J. Differential Geom., 26(1):141–168, 1987. http://projecteuclid.org/euclid.jdg/1214441179.\n- [OPR21a] Patrick Orson, Mark Powell, and Arunima Ray. The s-cobordism theorem, the sphere embedding theorem, and the Poincaré conjecture. In The disc embedding theorem, pages 283–293. Oxford Univ. Press, Oxford, 2021.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The smooth four-dimensional s-cobordism conjecture remains open: the topological analogue is false, but the asserted smoothable orientable counterexamples were retracted and no smooth counterexample is known.\n\n**Verified partial progress.**\n\n- Matumoto--Siebenmann construct topological candidates and Freedman--Quinn's five-dimensional theorem forces their failure of product structure to occur in dimension four.\n- Cappell--Shaneson's claimed smoothability of orientable topological examples was retracted.\n- Kwasik--Schultz prove weaker endpoint results for closed orientable 3-manifolds: simple homotopy equivalence implies homeomorphism and topological h-cobordisms are s-cobordisms under geometrization.\n\n**Full solution or refutation.**\n\nTopological counterexamples and endpoint classification do not prove or refute that every smooth four-dimensional s-cobordism is diffeomorphic to a product.\n\n**What remains.**\n\nProve smooth product structure in all cases or produce a smooth nonproduct s-cobordism between closed 3-manifolds.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 4.56 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Retains the smooth conjecture, explains the topological counterexample, and records the retraction of the smoothability claim.\n- T. Matumoto and L. Siebenmann, The topological s-cobordism theorem fails in dimension 4 or 5, Mathematical Proceedings of the Cambridge Philosophical Society 84 (1978), 85--87. (primary): https://doi.org/10.1017/S0305004100054918\n Evidence used: Constructs the topological s-cobordism failure candidate later localized to dimension four.\n- Slawomir Kwasik and Reinhard Schultz, Vanishing of Whitehead torsion in dimension four, Topology 31 (1992), 735--756. (primary): https://doi.org/10.1016/0040-9383(92)90005-3\n Evidence used: Proves the weaker orientable 3-manifold endpoint and torsion results, not smooth product structure.\n\n**Review notes.** The exact statement's concatenated '4-dimensionals-cobordism' and split product interval are preserved and flagged in the report.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2933, "problem_number": "KP-4.57", "title": "Kirby Problem 4.57", "statement": "Let X and Y be closed, oriented, smooth 4-manifolds with the same Euler characteristic and signature. Is there a torus link L in X with trivial normal bundle such that some choice of torus surgery along L transforms X into a manifold diffeomorphic to Y? What if we only require that the result be homeomorphic to Y?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.57.\n\nLiterature notes:\n(1) These questions were discussed in [Ste06, BS13, FS11].\n\n(2) Let T be a torus in a 4-manifold W with trivial normal bundle. We say that a 4-manifold $W^{1}is$ obtained from W by torus surgery along T if $W^{1} = W \\setminus \\nu(T) \\cup (T^{2} \\times D^{2})for$ some choice of gluing map $\\partial\\nu(T) \\to \\partial(T^{2} \\times D^{2})$.\n\n(3) It is a theorem of Iwase [Iwa90] that there is a 4-manifold Z and links of tori $L_{X} \\subset X, L_{Y} \\subset Y$ so that every component of $L_{X}, L_{Y}$ has trivial normal bundle and there exists some choice of torus surgeries on $L_{X}, L_{Y}$ transforming X, Y into manifolds equivalent to Z. Here, if X, Y are smooth, then $L_{X}, L_{Y}$ may be taken to be smooth, and “equivalent” means “diffeomorphic.” Otherwise, surfaces are locally flat, and “equivalent” means “homeomorphic.”\n\n(4) In Iwase’s paper, he actually proves that X can be transformed into $\\#_{a}\\mathbb{CP}^{2}\\#_{b}\\overline{\\mathbb{CP}}{}^{2}\\#_{c}S^{1} \\times S^{3}$ by surgery on a torus link in X for sufficiently large a with $b = a-\\sigma(X), c =$ (a+b+2 $-\\chi(X))/2$. That is, we may take Z in the above discussion to be a connected sum of copies of $\\mathbb{CP}^{2}s, \\mathbb{CP}^{2}s$, and $S^{1} \\times S^{3} s$. Iwase’s argument holds in both categories. A generalization of this result, extended over to the nonorientable 4– manifolds, is announced in a recent preprint of Baykur and Morgan [BM25], which states that any closed smooth 4–manifold is obtained by a surgery along a link of tori in a Z that is a connected sum of copies of $S^{2} \\times \\mathbb{R}\\mathbb{P}^{2}, \\mathbb{R}\\mathbb{P}^{4}, \\mathbb{CP}^{2}s, \\mathbb{CP}^{2}s$, and $S^{1} \\times S^{3} s$.\n\n(5) The problem asks whether, instead of X and Y being related by asequence of two surgeries on torus links, all necessary torus surgeries transforming X into Y can be performed simultaneously.\n\n(6) The problem seeks the 4-dimensional analog of the fact that any two closed, oriented 3-manifolds M, N are related by Dehn surgery along a link, rather than a sequence of Dehn surgeries. In dimension three, these two facts are clearly equivalent by dimensionality, but since surfaces generically intersect in 4-manifolds the situation is different.\n\n(7) Fintushel–Stern [FS11] asked the following version of this question in the case the 4–manifolds are simply connected. Question ([FS11, §9]). Can a simply-connected closed, smooth 4manifold always be obtained from torus surgery on a link of tori in a connected sum of $\\mathbb{CP}^{2}, S^{2} \\times S^{2}$ and K3 summands, taken with either orientations? See [BS13] for an approach to this question via 5-dimensional round handles. Round 2-handle attachments correspond to certain torus surgeries, and the work of [BS13] falls short at the same point as discussed above; the authors build cobordisms made out of only round 2-handles, but it is not clear if there is always a cobordism where these round 2handles can be attached independently.\n\n(8) Fintushel–Stern noted that many interesting simply connected 4-manifolds arise from a single null-homologous torus surgery, e.g. for $n =$ 2, . . . ,7,9 there are infinitely many exotic $\\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$ that each arises from a single torus surgery on a null-homologous torus in $\\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$ [FS11, Theorem 6]. Question (Fintushel and Stern). If X and Y are homeomorphic, simply-connected, smooth 4-manifolds, is it possible to obtain Y from surgery on a single torus in X? If so, can we arrange for the torus to be null-homologous?\n\nReferences cited:\n- [Ste06] Ronald J. Stern. Will we ever classify simply-connected smooth 4-manifolds? In Floer homology, gauge theory, and low-dimensional topology, volume 5 of Clay Math. Proc., pages 225–239. Amer. Math. Soc., Providence, RI, 2006.\n- [BS13] R. İnanç Baykur and Nathan Sunukjian. Round handles, logarithmic transforms and smooth 4-manifolds. J. Topol., 6(1):49–63, 2013.\n- [FS11] Ronald Fintushel and Ronald J. Stern. Pinwheels and nullhomologous surgery on 4-manifolds with $b^+=1$. Algebr. Geom. Topol., 11(3):1649–1699, 2011. doi:10.2140/agt.2011.11.1649.\n- [Iwa90] Zjuñici Iwase. Dehn surgery along a torus T2-knot. II. Japan. J. Math. (N.S.), 16(2):171–196, 1990. doi:10.4099/math1924.16.171.\n- [BM25] R. İnanç Baykur and Porter Morgan. On nonorientable 4–manifolds, 2025. Math. Res. Lett., to appear. URL: https://arxiv.org/abs/2506.20950, arXiv:2506.20950.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Each of X and Y can be changed by torus-link surgeries to a common standard model, but no theorem combines the resulting sequence into one simultaneous surgery on a disjoint torus link lying in the original X.\n\n**Verified partial progress.**\n\n- Iwase proves that a closed oriented 4-manifold can be changed by torus-link surgery into an explicit connected sum determined by sufficiently large parameters and its Euler characteristic and signature.\n- Applying Iwase separately to X and Y gives a common target and thus a sequential two-sided surgery relation.\n- Baykur--Sunukjian obtain round-2-handle cobordisms, while Baykur--Morgan announce nonorientable standard-model extensions; independence of all attachments remains the missing simultaneous-link step.\n\n**Full solution or refutation.**\n\nKnown common-target and round-handle results are one step short of the direct surgery statement in both smooth and topological categories.\n\n**What remains.**\n\nRealize the full relation by one disjoint torus link in X or find an invariant obstructing such simultaneous surgery.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 4.57 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Distinguishes Iwase's two-link common-target theorem from the unresolved one-link simultaneous problem.\n- Zjunici Iwase, Dehn surgery along a torus T2-knot. II, Japan Journal of Mathematics 16 (1990), 171--196. (primary): https://doi.org/10.4099/math1924.16.171\n Evidence used: Proves conversion by torus-link surgery to a connected-sum standard model.\n- R. Inanc Baykur and Nathan Sunukjian, Round handles, logarithmic transforms and smooth 4-manifolds, Journal of Topology 6 (2013), 49--63. (primary): https://doi.org/10.1112/jtopol/jts038\n Evidence used: Builds cobordisms from round 2-handles corresponding to torus surgeries but does not make all surgeries independent.\n\n**Review notes.** Sequential surgery through a common manifold was not conflated with simultaneous surgery on a link in X.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2934, "problem_number": "KP-4.58", "title": "Kirby Problem 4.58", "statement": "(a) Which Seifert fibered homology spheres $\\Sigma(a_{1}$, . . . , $a_{n})bound$ acyclic manifolds? Are there any examples with four or more singular fibers that bound acyclic manifolds?\n\n(b) Which Seifert fibered homology spheres bound contractible manifolds? Is there an example that bounds an acyclic manifold but not a contractible manifold?\n\n(c) Are there any Seifert fibered homology spheres that arise as cork boundaries?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.58.\n\nLiterature notes:\n(1) The second part of Problem (a) (regarding the number of fibers) is in [Kir97, Problem 4.123] in a different guise, with a different motivation.\n\n(2) Many Seifert fibered homology spheres bound acyclic (and, in fact, contractible) manifolds (see for example [AK79b, CH81, Fic84, Şav20]) but a complete characterization is unknown. All known such examples have three singular fibers. Amongst spheres $\\Sigma(p, q$, r)with three singular fibers, there is no closed characterization of which bound acyclic manifolds in terms of p, q, and r.\n\n(3) It is a longstanding conjecture that no Seifert fibered homology sphere with four or more singular fibers bounds an acyclic manifold; see [FS87b, Kol08]. This is related to the Montgomery-Yang conjecture, which states that every pseudofree action of $S^{1}$ on $S^{5}$ has at most 3 non-free orbits. One can also ask the related question of which Seifert fibered homology spheres embed in $\\mathbb{R}^{4}$. In the setting of symplectic topology, it is known that no Seifert fibered homology sphere (of any number of fibers) occurs as a hypersurface of contact type $in(\\mathbb{R}^{4}, \\omega_{std})$ [MT22]. A recent preprint [AC24] builds on this work and announces that no (standardly-oriented) Seifert fibered homology sphere bounds a Stein rational ball.\n\n(4) One can also ask about the difference between bounding an acyclic manifold and bounding a contractible manifold. In general, these notions differ: Taubes’ periodic ends theorem implies that $\\Sigma(2,3,5)\\#-\\Sigma(2,3,5)$ bounds no contractible manifold [Tau87], whereas this trivially bounds an acyclic manifold. Many other examples can be obtained through instanto n Floer theory. However, no such example consisting of an individual Seifert fibered homology sphere is known.\n\n(5) A similar question is whether or not any Seifert fibered homology sphere Y forms a cork boundary. Here, recall that Y is a cork boundary if there exists a contractible manifold W with boundary Y, together with a selfdiffeomorphism of Y that does not extend over W (as a diffeomorphism). It is known that the standard cyclic group actions on any Brieskorn sphere $\\Sigma(p, q$, r) do not extend (smoothly) as group actions to any contractible manifold with boundary $\\Sigma(p, q$, r) [AH16, AH21]. However, these do extend as diffeomorphisms. Current Floer-theoretic techniques devoted to establishing corks [AKS20, DHM23] are known to fail for Seifert fibered homology spheres. Note that several authors take W to be Stein in the definition of a cork; the notion defined here is sometimes called a loose cork.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [AK79b] Selman Akbulut and Robion Kirby. Mazur manifolds. Michigan Math. J., 26(3):259–284, 1979. http://projecteuclid.org/euclid.mmj/1029002261.\n- [CH81] Andrew J. Casson and John L. Harer. Some homology lens spaces which bound rational homology balls. Pacific J. Math., 96(1):23–36, 1981. http://projecteuclid.org/euclid.pjm/1102734944.\n- [Fic84] Henry Clay Fickle. Knots, Z-homology 3-spheres and contractible 4-manifolds. Houston J. Math., 10(4):467–493, 1984.\n- [Şav20] Oğuz Şavk. More Brieskorn spheres bounding rational balls. Topology Appl., 286:107400, 10, 2020. doi:10.1016/j.topol.2020.107400.\n- [FS87b] Ronald Fintushel and Ronald J. Stern. $O(2)$ actions on the 5-sphere. Invent. Math., 87(3):457–476, 1987. doi:10.1007/BF01389237.\n- [Kol08] János Kollár. Is there a topological Bogomolov-Miyaoka-Yau inequality? Pure Appl. Math. Q., 4(2, Special Issue: In honor of Fedor Bogomolov. Part 1):203– 236, 2008. doi:10.4310/PAMQ.2008.v4.n2.a1.\n- [MT22] Thomas E. Mark and Bülent Tosun. On contact type hypersurfaces in 4-space. Invent. Math., 228(1):493–534, 2022. doi:10.1007/s00222-021-01083-9.\n- [AC24] Antonio Alfieri and Alberto Cavallo. Holomorphic curves in Stein domains and the tau-invariant, 2024. arXiv:2310.08657.\n- [Tau87] Clifford Henry Taubes. Gauge theory on asymptotically periodic 4-manifolds. J. Differential Geom., 25(3):363–430, 1987. http://projecteuclid.org/euclid.jdg/1214440981.\n- [AH16] Nima Anvari and Ian Hambleton. Cyclic group actions on contractible 4-manifolds. Geom. Topol., 20(2):1127–1155, 2016. doi:10.2140/gt.2016.20.1127.\n- [AH21] Nima Anvari and Ian Hambleton. Cyclic branched coverings of Brieskorn spheres bounding acyclic 4-manifolds. Glasg. Math. J., 63(2):400–413, 2021. doi:10.1017/S0017089520000269.\n- [AKS20] Antonio Alfieri, Sungkyung Kang, and András I. Stipsicz. Connected Floer homology of covering involutions. Math. Ann., 377(3-4):1427–1452, 2020. doi:10.1007/s00208-020-01992-9.\n- [DHM23] Irving Dai, Matthew Hedden, and Abhishek Mallick. Corks, involutions, and Heegaard Floer homology. J. Eur. Math. Soc. (JEMS), 25(6):2319–2389, 2023. doi:10.4171/jems/1239.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many three-singular-fiber Seifert homology spheres bound contractible manifolds, but no complete classification, four-or-more-fiber acyclic example, acyclic-not-contractible individual example, or Seifert-fibered cork boundary is known.\n\n**Verified partial progress.**\n\n- Akbulut--Kirby, Casson--Harer, Fickle, and later authors construct numerous Brieskorn spheres bounding contractible or acyclic 4-manifolds; all known examples have three singular fibers.\n- Mark--Tosun prove that no Seifert fibered homology sphere occurs as a contact-type hypersurface in standard symplectic R4.\n- Anvari--Hambleton obstruct standard cyclic actions on Brieskorn spheres from extending as group actions over contractible fillings, but this does not produce a nonextendable individual boundary diffeomorphism defining a cork.\n\n**Full solution or refutation.**\n\nConstructions and symplectic/equivariant obstructions cover important subclasses but do not decide any of the global classification and existence questions.\n\n**What remains.**\n\nClassify the three-fiber cases, decide the four-or-more-fiber conjecture, separate acyclic from contractible bounding for one Seifert sphere, and find or exclude a Seifert-fibered cork boundary.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 4.58 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Records the known three-fiber constructions and leaves all four stated classification/existence directions open.\n- Selman Akbulut and Robion Kirby, Mazur manifolds, Michigan Mathematical Journal 26 (1979), 259--284. (primary): https://projecteuclid.org/journals/michigan-mathematical-journal/volume-26/issue-3/Mazur-manifolds/10.1307/mmj/1029002261.full\n Evidence used: Gives foundational contractible 4-manifold constructions with homology-sphere boundaries.\n- Thomas E. Mark and Bulent Tosun, On contact type hypersurfaces in 4-space, Inventiones Mathematicae 228 (2022), 493--534. (primary): https://doi.org/10.1007/s00222-021-01083-9\n Evidence used: Proves the contact-type hypersurface obstruction, which is narrower than arbitrary smooth acyclic bounding.\n- Nima Anvari and Ian Hambleton, Cyclic branched coverings of Brieskorn spheres bounding acyclic 4-manifolds, Glasgow Mathematical Journal 63 (2021), 400--413. (primary): https://doi.org/10.1017/S0017089520000269\n Evidence used: Provides equivariant extension obstructions relevant to, but not resolving, the cork-boundary question.\n\n**Review notes.** The exact '$a_n)bound$' concatenation is preserved and flagged as an extraction defect.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2935, "problem_number": "KP-4.59", "title": "Kirby Problem 4.59", "statement": "Are lens spaces topologically homology cobordant if and only if they are homeomorphic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.59.\n\nLiterature notes:\n(1) Livingston conjectures that the statement is correct.\n\n(2) If $L(a_{1}, b_{1})$ and $L(a_{2}, b_{2})$ are homology cobordant, then $a_{1} = a_{2}$. The problem can be restated: if L(n, $b_{1})$ and L(n, $b_{2})$ are homology cobordant, then L(n, $b_{1})$ and L(n, $b_{2})$ are homeomorphic. Gilmer and Livingston [GL83] proved this in the case that n is a prime power, using Atiyah–Singer signature invariants $\\rho_{\\alpha}(L)$ associated to characters $\\alpha: \\pi_{1}(L) \\to U(1)$ of prime-power order. The simplest unknown case is the pair they identified, $L(231,53)$ and $L(231,86)$. It remains unknown whether these lens spaces are homology cobordant.\n\n(3) In the smooth category, the conjecture was proved for neven by Fintushel– Stern [FS87a], with later generalizations by Matić [Mat88] and Ruberman [Rub88]. The conjecture in the smooth setting can also be proved using Heegaard Floer theory; see [DW15].\n\n(4) For higher dimensional lens spaces, Cappell and Ruberman [CR88] showed that $the\\rho_{\\alpha}-invariants for\\alpha of$ prime-power order give the homology cobordism classification. This uses homology surgery theory [CS74, Vog82] which is known [Akb79] to fail in dimension 4, even topologically. It is conceivable to try to construct a topological homology cobordism W between non-diffeomorphic L(n, $b_{1})$ and L(n, $b_{2}$ with $n$ composite usin g ordinary surgery theory. A first step might be to find an appropriate homotopical model for such a cobordism. The Gilmer–Livingston argument implies that the n-fold cyclic $W_{\\infty}$ would have to have nontrivial $b_{1}$, so that $\\pi_{1}(W)$ would have to be large in this sense; see [AGL18] for more information on this.\n\nReferences cited:\n- [GL83] Patrick M. Gilmer and Charles Livingston. On embedding 3-manifolds in 4-space. Topology, 22(3):241–252, 1983. doi:10.1016/0040-9383(83)90011-3.\n- [FS87a] Ronald Fintushel and Ronald Stern. Rational homology cobordisms of spherical space forms. Topology, 26(3):385–393, 1987. doi:10.1016/0040-9383(87)90008-5.\n- [Mat88] Gordana Matić. $\\mathrm{SO}(3)$-connections and rational homology cobordisms. J. Differential Geom., 28(2):277–307, 1988.\n- [Rub88] Daniel Ruberman. Rational homology cobordisms of rational space forms. Topology, 27(4):401–414, 1988. doi:10.1016/0040-9383(88)90020-1.\n- [DW15] Margaret Doig and Stephan Wehrli. A combinatorial proof of the homology cobordism classification of lens spaces, 2015. arXiv:1505.06970.\n- [CR88] Sylvain Cappell and Daniel Ruberman. Imbeddings and homology cobordisms of lens spaces. Comment. Math. Helv., 63(1):75–88, 1988. doi:10.1007/BF02566753.\n- [CS74] S.E. Cappell and J. Shaneson. Homology surgery and the codimension-two placement problem. Ann. of Math., 99:277–348, 1974.\n- [Vog82] Pierre Vogel. On the obstruction group in homology surgery. Publ. Math. Inst. Hautes Études Sci., 55:165–206, 1982. http://www.numdam.org/item?id= PMIHES 1982 55 165 0.\n- [Akb79] Selman Akbulut. A note on homology surgery and the Casson-Gordon invariant. Math. Proc. Cambridge Philos. Soc., 85(2):335–344, 1979. doi:10.1017/S0305004100055754.\n- [AGL18] Paolo Aceto, Marco Golla, and Ana G. Lecuona. Handle decompositions of rational homology balls and Casson-Gordon invariants. Proc. Amer. Math. Soc., 146(9):4059–4072, 2018. doi:10.1090/proc/14035.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The topological conjecture is proved for lens spaces whose first-homology order is a prime power, and the smooth analogue is fully classified, but the composite-order topological problem remains open.\n\n**Verified partial progress.**\n\n- Gilmer--Livingston prove the topological statement for prime-power order using rho invariants of prime-power-order characters.\n- Doig--Wehrli give a combinatorial Heegaard Floer proof that smoothly homology-cobordant lens spaces are oriented-homeomorphic.\n- The composite-order pair L(231,53) and L(231,86) remains unresolved in the topological category.\n\n**Full solution or refutation.**\n\nPrime-power topological and all-order smooth classifications do not settle topological homology cobordism at composite order.\n\n**What remains.**\n\nResolve composite orders, especially the order-231 pair, or construct nonhomeomorphic topologically homology-cobordant lens spaces.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 4.59 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Keeps the topological composite-order question open and identifies the order-231 test pair.\n- Patrick M. Gilmer and Charles Livingston, On embedding 3-manifolds in 4-space, Topology 22 (1983), 241--252. (primary): https://doi.org/10.1016/0040-9383(83)90011-3\n Evidence used: Proves the prime-power-order topological case using signature invariants.\n- Margaret Doig and Stephan Wehrli, A combinatorial proof of the homology cobordism classification of lens spaces, arXiv:1505.06970 (2015). (primary): https://arxiv.org/abs/1505.06970\n Evidence used: Proves the smooth homology-cobordism classification via Heegaard Floer correction terms.\n\n**Review notes.** The smooth Heegaard Floer theorem was not promoted to a topological obstruction.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2936, "problem_number": "KP-4.60", "title": "Kirby Problem 4.60", "statement": "(a) Let X be an open, spin, smooth4-manifold. Does X have a proper smooth embedding in $\\mathbb{R}^{6}$?\n\n(b) By choosing a proper exhaustion function on W,(a) would follow from an affirmative answer to the following. Let (W;M, N) be a compact, smooth spin cobordism with a smooth embedding f of M in $S^{5}$. Is there a smooth embedding F: (W;M, N) $\\to (S^{5} \\times I;S^{5} \\times$ \\{0\\}, $S^{5} \\times$ \\{1\\}) whose restriction to M coincides with f?\n\n(c) Let (W;M, N) be a compact, smooth spin cobordism with smooth embeddings f of M in $S^{5}$ and g of N in $S^{5}$ such that $\\sigma(N, S^{5}) -\\sigma(M, S^{5}) = \\sigma(W)$. Is there a smooth embedding F: (W;M, N) $\\to (S^{5} \\times I;S^{5} \\times$ \\{0\\}, $S^{5} \\times$ \\{1\\}) whose restriction to M coincides with f and whose restriction to N coincides with g?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.60.\n\nLiterature notes:\n(1) The spin condition is necessary in both parts of the problem.\n\n(2) One motivation for part (a)is to understand theembedding dimension for a Stein surface (of real dimension four). By definition, this is the minimal dfor which X has a proper holomorphic embedding in $\\mathbb{C}^{d}$. General results about embedding dimension due to Eliashberg and Gromov [EG92] imply that a Stein 4-manifold has a proper holomorphic embedding into $\\mathbb{C}^{4}$. Since a 4-manifold properly embedded in $\\mathbb{C}^{3} =\\mathbb{R}^{6}$ is spin, this is the best possible result for non-spin Stein 4-manifolds, but it is conceivable that one could get embeddings in $\\mathbb{C}^{3}$ for spin Stein 4-manifolds. Part (a) is a topological version of that question.\n\n(3) Part (b) asks for a relative version of the result, announced by CappellShaneson in [CS79] and proved by Ruberman in [Rub82], that a closed spin 4-manifold embeds in $\\mathbb{R}^{6}$ if and only if its signature is 0.\n\n(4) For part (c), note that an embedding of a 3-manifold M in $S^{5}$ has a welldefined signature $\\sigma(M, S^{5})$, given by the signature of any 4-manifold that M bounds in $S^{5}$. The equality $\\sigma(N, S^{5}) -\\sigma(M, S^{5}) =\\sigma(W)$ is necessary for W to be a cobordism as in (b). It follows, in the setting of part (b), that one cannot specify the embedding of both M and N in advance. It is not clear if there are further obstructions, so part (c) represents a sharpening of part (b).\n\nReferences cited:\n- [EG92] Yakov Eliashberg and Mikhael Gromov. Embeddings of Stein manifolds of dimension n into the affine space of dimension 3n\\{2+1. Ann. of Math. (2), 136(1):123–135, 1992. doi:10.2307/2946547.\n- [CS79] Sylvain E. Cappell and Julius L. Shaneson. Embeddings and immersions of fourdimensional manifolds in R6. In Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pages 301–303. Academic Press, New York-London, 1979.\n- [Rub82] Daniel Ruberman. Imbedding four-manifolds and slicing links. Math. Proc. Cambridge Philos. Soc., 91(1):107–110, 1982. doi:10.1017/S0305004100059168.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The closed embedding theorem in R6 is known and the spin/signature conditions are necessary, but proper embeddings of arbitrary open spin 4-manifolds and the relative extension statements remain open.\n\n**Verified partial progress.**\n\n- Cappell--Shaneson announced and Ruberman proved that a closed smooth spin 4-manifold embeds in R6 if and only if its signature is zero.\n- Eliashberg--Gromov prove every Stein surface properly holomorphically embeds in C4, which is an ambient real dimension eight result rather than the requested R6 theorem.\n- For prescribed embeddings at both boundary ends, the displayed signature difference is a necessary condition; no sufficiency theorem is known.\n\n**Full solution or refutation.**\n\nThe closed absolute theorem motivates but does not imply the open or relative codimension-two embedding problems.\n\n**What remains.**\n\nProve proper R6 embedding for every open spin 4-manifold and solve the one- and two-ended relative extension problems.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 4.60 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: States the open and relative questions and identifies the known closed signature-zero theorem.\n- Daniel Ruberman, Imbedding four-manifolds and slicing links, Mathematical Proceedings of the Cambridge Philosophical Society 91 (1982), 107--110. (primary): https://doi.org/10.1017/S0305004100059168\n Evidence used: Proves the closed spin signature-zero embedding criterion in R6.\n- Yakov Eliashberg and Mikhael Gromov, Embeddings of Stein manifolds of dimension n into the affine space of dimension 3n/2+1, Annals of Mathematics 136 (1992), 123--135. (primary): https://doi.org/10.2307/2946547\n Evidence used: Provides the C4 proper holomorphic embedding comparison for Stein surfaces.\n\n**Review notes.** Multiple missing spaces in the exact statement and the background's reference to 'both parts' despite three parts are flagged in the report.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2937, "problem_number": "KP-4.61", "title": "Kirby Problem 4.61", "statement": "What do different 4-manifold gauge theories see?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.61.\n\nLiterature notes:\n(1) It is conjectured that the Donaldson, Seiberg–Witten, and Heegaard Floer invariants of closed 4-manifolds coincide (after organizing the Donaldson invariant as a generating function [KM95]). Yet each of these theories taken more broadly seem to see different geometric and topological properties. This problem asks how to understand some of these properties proved in one theory via one of the other theories.\n\n(2) Is it possible to prove the existence of uncountable many exotic structures on $\\mathbb{R}^{4}$ using Seiberg–Witten or Heegaard Floer theory? One might try to do this by adapting Taubes’ periodic end gauge theory [Tau87] to the Seiberg–Witten setting. This can be done but to date requires rather strong hypotheses such as positive scalar curvature on the periodic end. Another possibility is to use limit invariants of ends such as the Heegaard Floer end invariant introduced by Gadgil [Gad10].\n\n(3) Is there a Seiberg–Witten proof of the Donaldson–Sullivan [DS89] results that there are 4-manifolds without quasiconformal (and hence Lipschitz) structure, and that 4-manifolds can admit more than one such structures? A seemingly fundamental difficulty here is whether Lipschitz or quasiconformal manifolds have something like a Dirac operator; see the discussion in [Sul87, Sul99] and also Problem 4.131.\n\n(4) Donaldson and Seiberg-Witten theory have parameterized versions that can be used to study invariants of diffeomorphisms and families of 4manifolds as well as families of symplectic structures [Rub98, BK22, Kon21, Kro97]. Are there family versions of Heegaard Floer theory that would be useful for such applications?\n\n(5) Seiberg-Witten invariants can be extended to give the Bauer–Furuta invariant [BF04] living in (equivariant) stable homotopy groups. Are there similar stable homotopy theoretic invariants coming from Donaldson or Heegaard Floer theory?\n\n(6) Seiberg–Witten theory can be used to show that certain 4-manifolds admit no Riemannian metric of positive scalar curvature (PSC) [Wit94] and to distinguish path components in the space of PSC metrics [Rub01]. Find a way to do this using Donaldson or Heegaard Floer theory.\n\n(7) All three theories [Frø02, Frø96, OS03a] give statements about the definite intersection forms of 4-manifolds with given boundary. Are these statements equivalent?\n\n(8) Hambleton and Lee [HL95] used an equivariant version of Donaldson’s original argument for his definite manifolds theorem to study smooth group actions on a simply connected (positive) definite 4-manifold. Among other results, they showed that a homologically trivial cyclic group action has the same fixed-point data and tangential isotropy representations as an equivariant connected sum of linear actions on $\\mathbb{CP}^{2}$. Are there Seiberg– Witten or Heegaard Floer proofs of their results?\n\nReferences cited:\n- [KM95] P.B. Kronheimer and T.S. Mrowka. Embedded surfaces and the structure of Donaldson’s polynomial invariants. J. Diff. Geo., 41:573–734, 1995.\n- [Tau87] Clifford Henry Taubes. Gauge theory on asymptotically periodic 4-manifolds. J. Differential Geom., 25(3):363–430, 1987. http://projecteuclid.org/euclid.jdg/1214440981.\n- [Gad10] Siddhartha Gadgil. Open manifolds, Ozsváth-Szabó invariants and exotic $\\mathbb{R}^{4}$’s. Expo. Math., 28(3):254–261, 2010. doi:10.1016/j.exmath.2009.09.002.\n- [DS89] SK Donaldson and DP Sullivan. Quasiconformal 4-manifolds. Acta Mathematica, 163:181–252, 1989.\n- [Sul87] Dennis Sullivan. Quasiconformal homeomorphisms in dynamics, topology, and geometry. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pages 1216–1228. Amer. Math. Soc., Providence, RI, 1987.\n- [Sul99] Dennis Sullivan. On the foundation of geometry, analysis, and the differentiable structure for manifolds. In Topics in low-dimensional topology (University Park, PA, 1996), pages 89–92. World Sci. Publ., River Edge, NJ, 1999. doi:10.1142/4202.\n- [Rub98] Daniel Ruberman. An obstruction to smooth isotopy in dimension 4. Math. Res. Lett., 5(6):743–758, 1998. doi:10.4310/MRL.1998.v5.n6.a5.\n- [BK22] David Baraglia and Hokuto Konno. On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds. J. Topol., 15(2):505–586, 2022. doi:10.1112/topo.12229.\n- [Kon21] Hokuto Konno. Characteristic classes via 4-dimensional gauge theory. Geom. Topol., 25(2):711–773, 2021. doi:10.2140/gt.2021.25.711.\n- [Kro97] P.B. Kronheimer. Some non-trivial families of symplectic structures. Preprint, available from www.math.harvard.edu/„kronheim/diffsymp.pdf, 1997.\n- [BF04] Stefan Bauer and Mikio Furuta. A stable cohomotopy refinement of Seiberg-Witten invariants. I. Invent. Math., 155(1):1–19, 2004. doi:10.1007/s00222-003-0288-5.\n- [Wit94] Edward Witten. Monopoles and four-manifolds. Math. Res. Lett., 1(6):769–796, 1994. doi:10.4310/MRL.1994.v1.n6.a13.\n- [Rub01] Daniel Ruberman. Positive scalar curvature, diffeomorphisms and the SeibergWitten invariants. Geom. Topol., 5:895–924, 2001. doi:10.2140/gt.2001.5.895.\n- [Frø02] Kim A. Frøyshov. Equivariant aspects of Yang-Mills Floer theory. Topology, 41(3):525–552, 2002. doi:10.1016/S0040-9383(01)00018-0.\n- [Frø96] Kim A. Frøyshov. The Seiberg-Witten equations and four-manifolds with boundary. Math. Res. Lett., 3(3):373–390, 1996. doi:10.4310/MRL.1996.v3.n3.a7.\n- [OS03a] Peter Ozsváth and Zoltán Szabó. Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv. Math., 173(2):179–261, 2003. doi:10.1016/S0001-8708(02)00030-0.\n- [HL95] Ian Hambleton and Ronnie Lee. Smooth group actions on definite 4-manifolds and moduli spaces. Duke Math. J., 78(3):715–732, 1995. doi:10.1215/S0012-7094-95-07826-0.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Many comparison and refinement theorems connect four-manifold gauge theories, but the prompt is a broad research program and its listed transfers between Donaldson, Seiberg--Witten, Heegaard Floer, and stable-homotopy methods remain incomplete.\n\n**Verified partial progress.**\n\n- Feehan--Leness prove Witten's Donaldson/Seiberg--Witten relation for many four-manifolds of simple type.\n- Bauer--Furuta refine Seiberg--Witten invariants to equivariant stable cohomotopy, and families refinements now detect diffeomorphisms and symplectic families.\n- Donaldson, monopole, and Heegaard Floer theories all constrain definite intersection forms of fillings, but a full equivalence of the resulting boundary inequalities is not known.\n\n**Full solution or refutation.**\n\nThere is substantial cross-theory progress, but no single theorem resolves the intentionally open-ended prompt or all of its enumerated subquestions.\n\n**What remains.**\n\nGive precise comparison theorems for exotic ends, quasiconformal structures, families invariants, positive scalar curvature, stable refinements, and definite-filling inequalities.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 4.61 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Formulates the prompt as a collection of comparison questions and records the partial results.\n- Paul M. N. Feehan and Thomas G. Leness, Witten's conjecture for many four-manifolds of simple type, Journal of the EMS 17 (2015), 899--923. (primary): https://doi.org/10.4171/JEMS/521\n Evidence used: Proves the Donaldson/Seiberg--Witten relation for a substantial class rather than in full generality.\n- David Baraglia and Hokuto Konno, On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds, Journal of Topology 15 (2022), 505--586. (primary): https://doi.org/10.1112/topo.12229\n Evidence used: Develops and compares families Bauer--Furuta and Seiberg--Witten invariants, addressing one major direction in the prompt.\n- Stefan Bauer and Mikio Furuta, A stable cohomotopy refinement of Seiberg-Witten invariants. I, Inventiones Mathematicae 155 (2004), 1--19. (primary): https://doi.org/10.1007/s00222-003-0288-5\n Evidence used: Constructs the stable-homotopy refinement central to one comparison question.\n\n**Review notes.** The source question has no formal completion criterion; status refers to the concrete subquestions in its supplied remarks.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2938, "problem_number": "KP-4.62", "title": "Kirby Problem 4.62", "statement": "Let X be a smooth, closed, connected, oriented 4-manifold with $b^{+}_{2}(X)$ >1.\n\n(a) Does X have Donaldson simple type?\n\n(b) Does X have Seiberg–Witten simple type?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.62.\n\nLiterature notes:\n(1) The Simple Type Conjecture holds that the answer is affirmative. Part (a) is in [Kir97, Problem 4.131], and was raised by Kronheimer and Mrowka in [KM95] for simply connected 4–manifolds.\n\n(2) A 4-manifold is said to have Donaldson simple type if the Donaldson polynomials $q_{k}$ for principal SU(2)-bundles with $c_{2}=k$ satisfy $q_{k,+,1}(\\nu,\\Sigma_{1}$, . . . $,\\Sigma_{d}) =4q_{k}(\\Sigma_{1}$, . . . $,\\Sigma_{d})$ where $\\nu = \\mu(1) (\\mu: H_{0}(X;\\mathbb{Z}) \\to H_{4}(M_{k,+,1})), \\Sigma_{i} \\in H_{2}(X;\\mathbb{Z})$, and 2d $=$ dim $M_{k}$.\n\n(3) Manifolds which have Donaldson simple type [KM95] include:\n\n\\noindent$\\bullet$ complete intersections,\n\n\\noindent$\\bullet$ elliptic surfaces,\n\n\\noindent$\\bullet$ any manifold with a Gompf nucleus,\n\n\\noindent$\\bullet$ manifolds with a smoothly embedded surface F satisfying 2(genus(F))− 2 =F $\\cdot F$ >0.\n\n(4) The Kronheimer–Mrowka structure theorem from [KM95] says that, for manifolds of simple type, the Donaldson invariants are determined by finitely many basic classes $K_{1}$, . . . , $K_{s}$ and rational numbers $\\beta_{1}$, . . . , $\\beta_{s}$. When $b^{+}_{2}$ =1, some manifolds, e.g. $\\mathbb{CP}^{2}, S^{2} \\times S^{2}, \\mathbb{CP}^{2}\\#\\mathbb{CP}^{2}$, do not have Donaldson simple type.\n\n(5) A Seiberg–Wittenbasic class $\\mathfrak{s}is a spin^{c}structure\\mathfrak{s}with$ nonzero SeibergWitten invariant $SW_{X}(\\mathfrak{s}). A$ 4-manifold has Seiberg-Witten simple type if the virtual dimension of the Seiberg-Witten moduli space $d_{X}(\\mathfrak{s}) = 1(c_{1}(\\mathfrak{s})^{2}-2\\chi(\\mathfrak{s}) -3\\sigma(X))$ is zero for every basic class $\\mathfrak{s}$. Note that, $whend_{X}(\\mathfrak{s})$ =0, the Seiberg–Witten invariants are defined by a signed count of monopoles. When $d_{X}(\\mathfrak{s}) >$ 0, they are defined by evaluating a higher degree cohomology class on the moduli space of monopoles; the conjecture says that, in such cases, the evaluation is always zero.\n\n(6) We list some important progress.\n\n\\noindent$\\bullet$ Taubes’ equivalence between the Seiberg–Witten invariants and the Gromov-Taubes invariants implies that all symplectic 4-manifolds are Seiberg–Witten simple type [Tau96].\n\n\\noindent$\\bullet$ Kato-Nakamura-Yasui proved the conjecture for the mod 2 SeibergWitten invariants under a mild condition on the homology ring [KNY22].\n\n\\noindent$\\bullet$ Baraglia [Bar23a] proved that the mod 2 Seiberg–Witten simple type conjecture holds for spin structures without any extra assumptions. Just as in the Donaldson case, there are manifolds with $b^{+}_{2}$ =1 that are not of simple type, because of the wall-crossing formula.\n\n(7) Witten’s conjecture [Wit94] states that if X has Seiberg–Witten simple type, then it also has Donaldson simple type, and there is a precise relation between its Donaldson and Seiberg–Witten invariants. The conjecture was proved in many cases by Feehan and Leness; see [FL15], [FL18].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [KM95] P.B. Kronheimer and T.S. Mrowka. Embedded surfaces and the structure of Donaldson’s polynomial invariants. J. Diff. Geo., 41:573–734, 1995.\n- [Tau96] Clifford H. Taubes. SW ñ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. Amer. Math. Soc., 9(3):845–918, 1996. doi:10.1090/S0894-0347-96-00211-1.\n- [KNY22] Tsuyoshi Kato, Nobuhiro Nakamura, and Kouichi Yasui. The simple type conjecture for mod 2 Seiberg–Witten invariants. Journal of the European Mathematical Society, 11 2022. doi:10.4171/JEMS/1297.\n- [Bar23a] David Baraglia. The mod 2 Seiberg-Witten invariants of spin structures and spin families, 2023. arXiv:2303.06883.\n- [Wit94] Edward Witten. Monopoles and four-manifolds. Math. Res. Lett., 1(6):769–796, 1994. doi:10.4310/MRL.1994.v1.n6.a13.\n- [FL15] Paul M. N. Feehan and Thomas G. Leness. Witten’s conjecture for many fourmanifolds of simple type. J. Eur. Math. Soc. (JEMS), 17(4):899–923, 2015. doi: 10.4171/JEMS/521.\n- [FL18] Paul M. N. Feehan and Thomas G. Leness. An $\\mathrm{SO}(3)$-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants. Mem. Amer. Math. Soc., 256(1226):xiv+234, 2018. doi:10.1090/memo/1226.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Donaldson and integral Seiberg--Witten simple type remain conjectural in full generality, although symplectic, geometric, and mod-2 cases cover large classes.\n\n**Verified partial progress.**\n\n- Taubes proves Seiberg--Witten simple type for symplectic 4-manifolds via SW=Gr.\n- Kronheimer--Mrowka prove Donaldson simple type for complete intersections, elliptic surfaces, manifolds with Gompf nuclei, and manifolds containing specified positive-self-intersection surfaces.\n- Kato--Nakamura--Yasui prove mod-2 Seiberg--Witten simple type under a mild cohomology-ring hypothesis, and Baraglia proves the mod-2 assertion for spin structures without extra assumptions.\n\n**Full solution or refutation.**\n\nThese theorems establish broad special and mod-2 cases but not the integral all-manifold conjectures asked in (a) and (b).\n\n**What remains.**\n\nProve integral Donaldson and Seiberg--Witten simple type for every closed oriented X with b2+ greater than one, or exhibit a counterexample.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 4.62 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Retains both integral simple-type questions and catalogs the symplectic and mod-2 advances.\n- Tsuyoshi Kato, Nobuhiro Nakamura, and Kouichi Yasui, The simple type conjecture for mod 2 Seiberg-Witten invariants, Journal of the EMS, DOI 10.4171/JEMS/1297. (primary): https://doi.org/10.4171/JEMS/1297\n Evidence used: Proves the mod-2 conjecture under a cohomology-ring condition.\n- David Baraglia, The mod 2 Seiberg-Witten invariants of spin structures and spin families, arXiv:2303.06883 (2023). (primary): https://arxiv.org/abs/2303.06883\n Evidence used: Determines mod-2 invariants for spin structures and confirms mod-2 simple type in that setting.\n- Clifford H. Taubes, SW implies Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves, Journal of the AMS 9 (1996), 845--918. (primary): https://doi.org/10.1090/S0894-0347-96-00211-1\n Evidence used: Implies Seiberg--Witten simple type for symplectic four-manifolds.\n\n**Review notes.** The exact inequality's split math mode and the corrupted definitions in the background are flagged rather than silently repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2939, "problem_number": "KP-4.63", "title": "Kirby Problem 4.63", "statement": "How many independent basic classes can a simply connected smooth 4-manifold X have, as measured $bybr(X)$, the rank of the span of the basic classes?\n\n(a) Is there an upper bound $forbr(X)$ in terms of topological invariants of X?\n\n(b) In particular, is $br(X) \\leq b^{+}_{2}(X)$ for all simply connected X with $b^{+}_{2}(X)$ odd?\n\n(c) Is there a smooth indefinite, simply connected 4-manifold X with $b_{2}(X) \\geq$ 3 for which the image of $ev_{*}: \\operatorname{Diff}(X) \\to \\operatorname{Aut}(Q_{X})$ is finite, or even trivial?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.63.\n\nLiterature notes:\n(1) Here we define $br(X)=\\operatorname{Rank}\\operatorname{Span}\\{c_{1}(\\mathfrak{s})\\mid \\mathfrak{s}\\text{ is a Seiberg-Witten basic class on }X\\}$ if X has any basic classes, and 0 otherwise.\n\n(2) Knot surgery [FS98] onndisjoint and homologically independent tori in a manifold with nontrivial Seiberg-Witten invariant would create a manifold X for which $br(X) = n$. For example, starting with an elliptic surface, the construction in [GM93] produces examples of manifolds X for which $br(X) =b^{+}_{2}(X)$.\n\n(3) Questions(a)and(b)are relevant to the study of the map $ev_{*}: \\operatorname{Diff}(X) \\to \\operatorname{Aut}(Q_{X})$ giving the action of a diffeomorphism on the intersection form. Since any diffeomorphism must permute the basic classes up to sign, the presence of many basic classes can restrict the size of the image of $ev_{*}$.\n\n(4) The restriction to indefinite manifolds $andb_{2}$ >2 is to rule out intersection forms with finite automorphism groups. If $br(X) =b_{2}(X)$, then $ev_{*}(\\operatorname{Diff})$ is contained in a finite permutation group and hence is finite.\n\nReferences cited:\n- [FS98] Ronald Fintushel and Ronald J. Stern. Knots, links, and 4-manifolds. Invent. Math., 134(2):363–400, 1998. doi:10.1007/s002220050268.\n- [GM93] R. Gompf and T. Mrowka. Irreducible 4-manifolds need not be complex. Ann. of Math., 138:61–111, 1993.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Knot-surgery constructions realize basic rank equal to b2+ and show the proposed bound would be sharp, but no general bound or finite-image example of the required indefinite simply connected type is known.\n\n**Verified partial progress.**\n\n- Fintushel--Stern knot surgery on n independent tori creates n independent basic-class directions under nonvanishing hypotheses.\n- Gompf--Mrowka constructions give examples with br(X)=b2+(X).\n- Because diffeomorphisms permute basic classes up to sign, a full-rank basic-class span would force finite image in Aut(Q_X), but the known sharp b2+ examples do not produce br(X)=b2(X) in the indefinite setting.\n\n**Full solution or refutation.**\n\nKnown examples establish lower bounds and sharpness scenarios, not the universal inequality or part (c).\n\n**What remains.**\n\nFind a topological upper bound, prove or refute br(X) at most b2+(X) for odd b2+, and construct the finite- or trivial-image diffeomorphism-action example.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 4.63 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: States all three questions and records the knot-surgery and br=b2+ constructions without claiming an upper bound.\n- Ronald Fintushel and Ronald Stern, Knots, links, and 4-manifolds, Inventiones Mathematicae 134 (1998), 363--400. (primary): https://doi.org/10.1007/s002220050268\n Evidence used: Provides the knot-surgery mechanism that creates independent basic-class directions.\n- Robert Gompf and Tomasz Mrowka, Irreducible 4-manifolds need not be complex, Annals of Mathematics 138 (1993), 61--111. (primary): https://doi.org/10.2307/2946635\n Evidence used: Supplies the construction used to realize examples with basic rank b2+.\n\n**Review notes.** The 'bybr(X)', 'forbr(X)', and split inequality defects in the exact statement are preserved and flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2940, "problem_number": "KP-4.64", "title": "Kirby Problem 4.64", "statement": "Find an irreducible, closed, smooth 4-manifold with nontrivial Bauer–Furuta invariant but with trivial Seiberg–Witten invariant.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.64.\n\nLiterature notes:\nThe Bauer–Furuta invariant $\\Psi$ [BF04] is a stable cohomotopy refinement of the Seiberg–Witten invariant [Wit94], building on Furuta’s proof of the 10/8-theorem [Fur01]. There are examples for which the Bauer–Furuta invariant is strictly stronger than the Seiberg–Witten invariant; for instance, $\\Psi$ can be used to distinguish between certain connected sums of homotopy K3 surfaces (which have vanishing Seiberg–Witten invariant). Such examples rely on a gluing formula due to Bauer [Bau04]; no irreducible examples are known.\n\nReferences cited:\n- [BF04] Stefan Bauer and Mikio Furuta. A stable cohomotopy refinement of Seiberg-Witten invariants. I. Invent. Math., 155(1):1–19, 2004. doi:10.1007/s00222-003-0288-5.\n- [Wit94] Edward Witten. Monopoles and four-manifolds. Math. Res. Lett., 1(6):769–796, 1994. doi:10.4310/MRL.1994.v1.n6.a13.\n- [Fur01] M. Furuta. Monopole equation and the 11 8 -conjecture. Math. Res. Lett., 8(3):279– 291, 2001. doi:10.4310/MRL.2001.v8.n3.a5.\n- [Bau04] Stefan Bauer. A stable cohomotopy refinement of Seiberg-Witten invariants. II. Invent. Math., 155(1):21–40, 2004. doi:10.1007/s00222-003-0289-4.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Bauer--Furuta invariants are strictly stronger than ordinary Seiberg--Witten invariants on known reducible connected sums, but no irreducible closed example with the requested separation is known.\n\n**Verified partial progress.**\n\n- Bauer--Furuta construct the stable cohomotopy refinement of the Seiberg--Witten invariant.\n- Bauer's connected-sum formula gives nonzero Bauer--Furuta invariants for certain connected sums whose ordinary Seiberg--Witten invariants vanish.\n- Suitable connected sums of homotopy K3 surfaces demonstrate strict strength, but are reducible and therefore excluded by the statement.\n\n**Full solution or refutation.**\n\nThe invariant separation is known only via reducible examples; irreducibility is the unresolved essential condition.\n\n**What remains.**\n\nConstruct a non-connected-sum example, prove vanishing of its ordinary Seiberg--Witten invariant and nonvanishing of its Bauer--Furuta class, and establish irreducibility.\n\n**Sources checked.**\n\n- Baykur, Kirby, and Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 4.64 (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Explicitly states that known stronger-than-Seiberg--Witten examples use connected sums and that no irreducible example is known.\n- Stefan Bauer and Mikio Furuta, A stable cohomotopy refinement of Seiberg-Witten invariants. I, Inventiones Mathematicae 155 (2004), 1--19. (primary): https://doi.org/10.1007/s00222-003-0288-5\n Evidence used: Defines the stable cohomotopy refinement.\n- Stefan Bauer, A stable cohomotopy refinement of Seiberg-Witten invariants. II, Inventiones Mathematicae 155 (2004), 21--40. (primary): https://doi.org/10.1007/s00222-003-0289-4\n Evidence used: Proves the gluing/connected-sum results producing the known reducible separations.\n\n**Review notes.** Recent families Bauer--Furuta applications were not conflated with the ordinary manifold invariant requested here.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2941, "problem_number": "KP-4.65", "title": "Kirby Problem 4.65", "statement": "Suppose X is a smooth4-manifold with the homology of $S^{1} \\times S^{3}$ whose infinite cyclic cover $\\widetilde{X}$ has $H^{1}(\\widetilde{X})=0$. Furuta and Ohta [FO93] define an invariant $\\lambda_{FO}(X)$ as 1/4 of the signed count of irreducible flat $SU(2)$ connections on X; this requires an orientation of X and a specified generator of $H^{1}(X)$.\n\n(a) Does the following hold? The invariant $\\lambda_{FO}(X)$ is an integer, and $\\lambda_{FO}(X)\\equiv \\rho(Y,\\mathfrak{s})$, where $\\rho(Y,\\mathfrak{s})$ is the Rokhlin invariant of an oriented spin 3 manifold Y that is Poincaré dual to the generator of $H^{1}(X)$.\n\n(b) Mrowka–Ruberman–Saveliev [MRS11] give an approach, defining an invariant $\\lambda_{SW}(X)$ by counting solutions to the Seiberg-Witten equations and adding an index-theoretic correction term. Is $\\lambda_{FO}(X)=-\\lambda_{SW}(X)$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.65.\n\nLiterature notes:\n(1) Part (a) was conjectured by Furuta–Ohta.\n\n(2) A solution to (a) would imply that the Wall group $L_{5}(\\mathbb{Z}[\\mathbb{Z}])$ does not act on the smooth structure set of $S^{1} \\times S^{3}$; compare the discussion in\n\nProblem 4.22. By construction, $\\lambda_{SW}(X)$ is an integer, and it is shown in [MRS11] that it reduces mod 2 to $\\rho(Y,\\mathfrak{s})$. So a positive answer to (b) implies a positive answer to (a).\n\n(3) The invariant $\\lambda_{FO}(X)$ is defined in greater generality; one could require only that X is a homology $S^{1} \\times S^{3}$ whose twisted cohomology $H^{1}(X;\\mathbb{C}_{\\alpha})$ vanishes for any homomorphism $\\alpha: \\pi_{1}(X) \\to U(1)$. In this setting, neither part of (a) holds, but (b) is still plausible. The paper [LRS21] shows that (b) holds for mapping tori of all orientation preserving diffeomorphisms of homology spheres generating a semifree finite cyclic group action. For involutions, the result also follows from [LRS23b].\n\nReferences cited:\n- [FO93] Mikio Furuta and Hiroshi Ohta. Differentiable structures on punctured 4-manifolds. Topology Appl., 51(3):291–301, 1993. doi:10.1016/0166-8641(93)90083-P.\n- [MRS11] Tomasz Mrowka, Daniel Ruberman, and Nikolai Saveliev. Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin’s invariant. J. Differential Geom., 88:333–377, 2011. http://projecteuclid.org/euclid.jdg/1320067650.\n- [LRS21] Jianfeng Lin, Daniel Ruberman, and Nikolai Saveliev. On the monopole Lefschetz number of finite-order diffeomorphisms. Geom. Topol., 25(7):3591–3628, 2021. doi: 10.2140/gt.2021.25.3591.\n- [LRS23b] Jianfeng Lin, Daniel Ruberman, and Nikolai Saveliev. On the Frøyshov invariant and monopole Lefschetz number. Journal of Differential Geometry, 123(3):523 – 593, 2023. doi:10.4310/jdg/1683307008.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Seiberg--Witten invariant is integral and has the stated mod-2 comparison, but the Furuta--Ohta integrality/congruence and equality with the Seiberg--Witten invariant remain unverified generally.\n\n**Verified partial progress.**\n\n- Mrowka--Ruberman--Saveliev define lambda_SW with an index correction and establish a mod-2 relation.\n\n**Full solution or refutation.**\n\nNeither full comparison was verified.\n\n**What remains.**\n\nProve the flat-connection count has the required integrality and compare the two invariants.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.65 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the conjecture and MRS partial comparison.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2942, "problem_number": "KP-4.66", "title": "Kirby Problem 4.66", "statement": "Can the skein lasagna module detect exotic smooth structures on closed 4-manifolds?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.66.\n\nLiterature notes:\n(1) The skein lasagna module is an extension of Khovanov homology. It is an invariant of 4-manifolds with (possibly empty) boundary and a framed link in their boundary. It was defined by Morrison, Walker and Wedrich in [MWW22]. Ren and Willis [RW24] gave examples of exotic compact 4-manifolds with boundary that are detected by the skein lasagna module. For closed 4-manifolds, the computations so far are limited; see [MN22], [MWW23], [RW24]. The invariant is nonvanishing for $S^{4}, S^{1} \\times S^{3}$ and $\\mathbb{CP}^{2}$. On the other hand, it vanishes for manifolds that contain a smoothly embedded sphere with positive self-intersection (cf. Theorem 1.3 in [RW24]); e.g. for $\\mathbb{CP}^{2}$ or −K3.\n\n(2) The invariant is multiplicative under connected sums, and it takes the value 0 on $S^{2} \\times S^{2}$. Thus, it has a chance of detecting exotic smooth structures on simply connected, closed 4-manifolds, even though such structures become standard after sufficiently many stabilizations.\n\nReferences cited:\n- [MWW22] Scott Morrison, Kevin Walker, and Paul Wedrich. Invariants of 4-manifolds from Khovanov-Rozansky link homology. Geom. Topol., 26(8):3367–3420, 2022. doi:10.2140/gt.2022.26.3367.\n- [RW24] Qiuyu Ren and Michael Willis. Khovanov homology and exotic 4-manifolds, 2024. arXiv:2402.10452.\n- [MN22] Ciprian Manolescu and Ikshu Neithalath. Skein lasagna modules for 2-handlebodies. J. Reine Angew. Math., 788:37–76, 2022. doi:10.1515/crelle-2022-0021.\n- [MWW23] Ciprian Manolescu, Kevin Walker, and Paul Wedrich. Skein lasagna modules and handle decompositions. Adv. Math., 425:Paper No. 109071, 40, 2023. doi:10.1016/j.aim.2023.109071.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Skein lasagna modules detect some exotic compact 4-manifolds with boundary, while detection of exotic smooth structures on closed 4-manifolds remains open.\n\n**Verified partial progress.**\n\n- Ren--Willis give boundary examples detected by the module.\n\n**Full solution or refutation.**\n\nNo closed-manifold detection theorem or example was verified.\n\n**What remains.**\n\nExtend the invariant or a closure construction to closed exotic pairs.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.66 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the definition and the boundary detection examples.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2943, "problem_number": "KP-4.67", "title": "Kirby Problem 4.67", "statement": "(a) Compute $\\pi_{0}(\\operatorname{Diff}^{+}(S^{4}))$. Do we have $\\pi_{0}(\\operatorname{Diff}^{+}(S^{4})) =$ \\{1\\}?\n\n(b) In particular, does some implantation of the barbell map provide a nontrivial element in $\\pi_{0}(\\operatorname{Diff}^{+}(S^{4}))$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.67.\n\nLiterature notes:\n(1) The 4-dimensional generalized Smale conjecture asked whether the inclusion $SO(5) \\hookrightarrow \\operatorname{Diff}^{+}(S^{4})$ is a homotopy equivalence (see Problems 4.34 and 4.126 in [Kir97]). Watanabe [Wat19] disproved this conjecture, by proving that $\\pi_{k}(Diff_{\\partial}(D^{4}))$ is nontrivial for manyk, includingk=1,4,8. A fundamental issue remaining to understand is $\\pi_{0}(\\operatorname{Diff}^{+}(S^{4}))$, the orientation-preserving mapping class group of $S^{4}$, and it is presently unknown whether or not this group is trivial. Some sources of possibly nontrivial diffeomorphisms of $S^{4}$ are suggested in [Gay25], [BG19] and $[GGH^{+}23]$. The Pin(2)-equivariant family Bauer-Furuta invariant could potentially detect such nontrivial diffeomorphisms; see [LM25].\n\n\\begin{center}\n\\kthreefiginclude{ch4_fig2.png}{width=0.92\\linewidth}\n\\par\\small\\textbf{Figure 2.} Point push map on a surface.\n\\end{center}\n\n\\begin{center}\n\\kthreefiginclude{ch4_fig3.png}{width=0.70\\linewidth}\n\\par\\small\\textbf{Figure 3.} Barbell.\n\\end{center}\n\n(2) The barbell map is defined by Budney and Gabai in [BG19] (see also [BG25] for the ‘X-resolution’) using isotopy extension, via a generalization of Birman’s ‘push map’ [Bir69]. We give an exposition here. Birman’s map can be described by pushing a pointp(see Figure 2(a)) along a path $\\alpha$ in a closed surface S and back to p. This isotopy of the surface can be assumed to fix a disk D centered at p at the end of the isotopy. Removing the interior of D, we get a diffeomorphism of the punctured surface that is the identity $on\\partial D$ and also the identity outside a nice neighborhood of the $arc\\alpha$. Figure2(b) shows what happens to the loop $\\beta$ under this diffeomorphism. In general, when pushing a 0-dimensional pointpalong a 1-dimensional loop $\\alpha$ in a 2-dimensional surface, the point crosses a loop (say $\\beta)$ and bulldozes it over D. Consider a pair of 2-spheres in $\\mathbb{R}^{3}$ centered at( $\\pm$ 2,0,0)of radius one, which are joined by the arc in the x-axis [−1,1]. We can use the z-axis to define the equator $(x^{2}+y^{2}=1)$, latitudes, and the two poles. Thicken this slightly in $\\mathbb{R}^{3}$, to what might be called a (hollow) barbell. Now cross with [−3,3] to get a four-dimensional analogue called B. We will find an interesting diffeomorphism $\\beta$ of B that is the identity on $\\partial B$. Notice the two line segments ( $\\pm$ 2,0,0) $\\times$ [−3,3]. These can be thickened so that they fill in the “hollows”, the pair of $S^{2} \\times$ [−1,1]s, so that we have a solid 4-dimensional barbell, which is obviously $B^{4}$. Figure 3 may help. Now focus just on the left line segment and move a portion of it, namely (−2,0,0) $\\times$ [−1,1], around the 2-sphere on the right side, sort of like lassoing a horse’s head. We will do this with a 1-parameter family of\n\n\\begin{center}\n\\kthreefiginclude{ch4_fig4.png}{width=0.52\\linewidth}\n\\par\\small\\textbf{Figure 4.} Embedding of barbell.\n\\end{center}\n\nBirman push maps, parameterized by $t\\in[-3,3]$. The push maps occur on a 2-dimensional surface given by fixing t and z. The push map for $t =$ 0 will push the point $p =$ (−2,0,0) $\\times$ 0 over to and then around the equator (z $=$ 0) of the 2-sphere at $t =$ 0 and back to p; for $t \\in$ (−1,1), pgoes over to and around the latitude at z =t and back to p; for $t = \\pm$ 1, p goes over to a pole and then back to p; for $t \\in$ (−2,−1) $\\cup$ (1,2), pgoes partway to a pole and then back to p; finally, for $t \\in$ (−3,−2) $\\cup$ (2,3), p does not move. This isotopy of the left line segment extends to an isotopy of $B^{4}that$ is the identity $on\\partial B^{4}$ and on the two line segments. But if this isotopy fixes the two line segments, then it can also be made to fix the above thickenings of the segments. Therefore, the end of this isotopy is a diffeomorphism $\\beta$ taking the 4-dimensional hollow barbell B back to itself, and fixing its boundary. Note that the lasso can go over the horse’s head with two possible orientations. Also there is a rotation $\\rho$ switching the two balls in the barbell. It can be checked that $\\rho\\beta\\rho^{-1} =\\beta^{-1}$. The 4-dimensional barbell,B, can be embedded, by f, in a 4-manifold X in many ways, and these are called implantations and the induced barbell map can be called $f_{*}\\beta$.\n\n(3) A specific example of a barbell diffeomorphism to consider is as follows.\n\n\\paragraph{Question.} Does the following embedding f induce a nontrivial diffeomorphism of $S^{4}$, where the embedding is given by mapping the two 2spheres to separate 2-spheres in $S^{4}$, and the bar goes from the left sphere over to link the right sphere and then back again to link the first and finally attaching to the right sphere? See Figure 4. A positive answer has been announced by Gabai–Gay–Hartman [GGH25].\n\n(4) We can also ask the following structural question.\n\n\\paragraph{Question.} Is $\\operatorname{Diff}^{+}(S^{4})$ (finitely) generated up to isotopy by a composition of $f_{*}\\beta s$?\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Wat19] Tadayuki Watanabe. Some exotic nontrivial elements of the rational homotopy groups of Diffp$S^{4}$q, 2019. arXiv:1812.02448.\n- [Gay25] David T. Gay. Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins. Algebr. Geom. Topol., 25(5):2817–2849, 2025. doi:10.2140/agt.2025.25.2817.\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.\n- [GGH+23] David Gabai, David T. Gay, Daniel Hartman, Vyacheslav Krushkal, and Mark Powell. Pseudo-isotopies of simply connected 4-manifolds, 2023. arXiv:2311.11196.\n- [LM25] Jianfeng Lin and Anubhav Mukherjee. Family Bauer-Furuta invariant, exotic surfaces and Smale conjecture. J. Assoc. Math. Res., 3(2):237–275, 2025. doi: 10.56994/JAMR.003.002.003.\n- [BG25] Ryan Budney and David Gabai. On the automorphism groups of hyperbolic manifolds, 2025. doi:10.1093/imrn/rnaf083.\n- [Bir69] Joan S Birman. Mapping class groups and their relationship to braid groups. Communications on Pure and Applied Mathematics, 22(2):213–238, 1969.\n- [GGH25] David Gabai, David T. Gay, and Daniel Hartman. Pseudo-Isotopy and Diffeomorphisms of the 4-Sphere I: Loops of Spheres, 2025. arXiv:2505.12088.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The generalized Smale conjecture fails in higher homotopy degrees, but pi_0 of orientation-preserving diffeomorphisms of S4 and the barbell-map question remain unresolved.\n\n**Verified partial progress.**\n\n- Watanabe proves nontrivial higher homotopy of Diff_boundary(D4).\n\n**Full solution or refutation.**\n\nNo determination of pi_0 Diff+(S4) was verified.\n\n**What remains.**\n\nDecide whether a barbell implantation represents a nontrivial component.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.67 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes Watanabe's higher-homotopy result from pi_0.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2944, "problem_number": "KP-4.68", "title": "Kirby Problem 4.68", "statement": "Does every closed smooth 4-manifold admit an exotic diffeomorphism? How about the following special cases?\n\n(a) Is there a definite smooth closed 4-manifold that admits an exotic diffeomorphism?\n\n(b) Does $S^{2} \\times S^{2}$ or K3 admit an exotic diffeomorphism?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.68.\n\nLiterature notes:\n(1) A self-diffeomorphism f: $X \\to X$ of a smooth manifold X is called exotic if it is topologically but not smoothly isotopic to the identity. The first examples of exotic diffeomorphisms of 4-manifolds were given by Ruberman [Rub98]. Most known examples of closed 4-manifolds confirmed to admit exotic diffeomorphisms are of the for $m \\#_{m}\\mathbb{C}\\mathbb{P}^{2}\\#_{n}\\mathbb{C}\\mathbb{P}^{2}$ for m, n>0 [Rub98], $\\#_{m}K3\\#_{n}S^{2} \\times S^{2}$ for m, n>0 [BK20] or m=0 [AR25], and K3\\#K3 [KM20]. There exist irreducible 4-manifolds that admit exotic diffeomorphisms [BK24a], but the proof in [BK24a] does not apply to $S^{4}, S^{2} \\times S^{2}, \\mathbb{C}\\mathbb{P}^{2}$, or K3.\n\n(2) In the literature, the smallest (in term of second Betti number) closed 4manifold that is known to admit an exotic diffeomorphism is $\\#_{2}\\mathbb{C}\\mathbb{P}^{2}\\#_{10}\\mathbb{C}\\mathbb{P}^{2}$, announced in [Qiu24]. It is natural to ask how small a closed 4-manifold with an exotic diffeomorphism can be. In particular, whether such a diffeomorphism exists on $S^{4}$ is the $\\pi_{0}$ case of the Smale conjecture; see\n\nProblem 4.67. For 4-manifolds with boundary, there exists an example of a contractible (hence, definite) 4-manifold that has an exotic diffeomorphism (relative to the boundary) [KMT23a, KMPW24,KPT26,KLMME24].\n\n(3) For a simply connected closed smooth 4-manifold X, every exotic diffeomorphism of X is smoothly isotopic to the identity after sufficiently many stabilizations by $S^{2} \\times S^{2}$. This fact follows by combining work of Kreck [Kre79] and either Quinn [Qui86] (cf. [GGH+23]) or Gabai [Gab22]. It is an interesting question to determine how many stabilizations are needed to kill the exotic property of a given diffeomorphism. Many known examples of exotic diffeomorphisms, such as those in [Rub98, BK20], are smoothly isotopic to the identity after only one stabilization [AKMR15]. On the other hand, Lin [Lin23] proved that the exotic diffeomorphism of K3\\#K3 from [KM20] stays exotic after one stabilization. There is no known upper bound for how many stabilizations are needed to trivialize this diffeomorphism. See also Problem 4.79.\n\nReferences cited:\n- [Rub98] Daniel Ruberman. An obstruction to smooth isotopy in dimension 4. Math. Res. Lett., 5(6):743–758, 1998. doi:10.4310/MRL.1998.v5.n6.a5.\n- [BK20] David Baraglia and Hokuto Konno. A gluing formula for families Seiberg-Witten invariants. Geom. Topol., 24(3):1381–1456, 2020. doi:10.2140/gt.2020.24.1381.\n- [AR25] Dave Auckly and Daniel Ruberman. Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory, 2025. arXiv: 2501.11892.\n- [KM20] P. B. Kronheimer and T. S. Mrowka. The Dehn twist on a sum of two K3 surfaces. Math. Res. Lett., 27(6):1767–1783, 2020. doi:10.4310/MRL.2020.v27.n6.a8.\n- [BK24a] David Baraglia and Hokuto Konno. Irreducible 4-manifolds can admit exotic diffeomorphisms, 2024. arXiv:2412.14398.\n- [Qiu24] Haochen Qiu. Surgery formulas for Seiberg-Witten invariants and family SeibergWitten invariants, 2024. arXiv:2411.10392.\n- [KMT23a] Hokuto Konno, Abhishek Mallick, and Masaki Taniguchi. Exotic Dehn twists on 4-manifolds, 2023. arXiv:2306.08607.\n- [KMPW24] Vyacheslav Krushkal, Anubhav Mukherjee, Mark Powell, and Terrin Warren. Corks for exotic diffeomorphisms, 2024. arXiv:2407.04696.\n- [KPT26] Sungkyung Kang, JungHwan Park, and Masaki Taniguchi. Exotic Dehn twists and homotopy coherent group actions. Invent. Math., 243(1):209–241, 2026. doi:10.1007/s00222-025-01378-1.\n- [KLMME24] Hokuto Konno, Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz. On four-dimensional Dehn twists and Milnor fibrations, 2024. arXiv:2409.11961.\n- [Kre79] M. Kreck. Isotopy classes of diffeomorphisms of $(k-1)$-connected almostparallelizable 2k-manifolds. In Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), volume 763 of Lecture Notes in Math., pages 643– 663. Springer, Berlin, 1979.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [GGH+23] David Gabai, David T. Gay, Daniel Hartman, Vyacheslav Krushkal, and Mark Powell. Pseudo-isotopies of simply connected 4-manifolds, 2023. arXiv:2311.11196.\n- [Gab22] David Gabai. 3-spheres in the 4-sphere and pseudo-isotopies of $S^{1}$ $\\times$ $S^{3}$, 2022. arXiv:2212.02004.\n- [AKMR15] Dave Auckly, Hee Jung Kim, Paul Melvin, and Daniel Ruberman. Stable isotopy in four dimensions. J. Lond. Math. Soc. (2), 91(2):439–463, 2015. doi:10.1112/jlms/jdu075.\n- [Lin23] Jianfeng Lin. Isotopy of the Dehn twist on K3 \\# K3 after a single stabilization. Geom. Topol., 27(5):1987–2012, 2023. doi:10.2140/gt.2023.27.1987.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Closed 4-manifolds with exotic diffeomorphisms are known, but every-manifold existence and the definite, S2xS2, and K3 cases remain open.\n\n**Verified partial progress.**\n\n- Ruberman supplied early closed examples.\n- The list records extensive families but preserves the special cases.\n\n**Full solution or refutation.**\n\nNo universal theorem or solution of the named cases was verified.\n\n**What remains.**\n\nConstruct exotic diffeomorphisms in a named special case or prove a general mechanism.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.68 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines exotic diffeomorphism and retains the questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2945, "problem_number": "KP-4.69", "title": "Kirby Problem 4.69", "statement": "Does there exist a diffeomorphism of a closed3-manifoldf: $M \\to M$ such that f is topologically but not smoothly pseudo-isotopic to the identity?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.69.\n\nLiterature notes:\n(1) Friedman–Witt [FW86] constructed diffeomorphisms f that are homotopic but not isotopic to the identity. These are also topologically pseudoisotopic to the identity [KS96].\n\n\\paragraph{Question.} Are the Friedman–Witt diffeomorphisms smoothly pseudoisotopic to the identity?\n\n(2) The following observation yields a potentially useful reformulation. Consider the embedding $i_{1,/,2}: M \\to M \\times$ \\{1/2\\} $\\hookrightarrow M \\times$ [0,1] where the first map is the canonical identification. Then f is smoothly (topologically) pseudo-isotopic to the identity if and only if $i_{1,/,2} \\circ f$ and $i_{1,/,2}$ are smoothly (topologically) isotopic as embeddings. The proof of this observation, which we give next, was provided separately by Hatcher and Igusa. It applies in all dimensions. If f were pseudo-isotopic to the identity via F: $M \\times I \\to M \\times I$, then $i_{1,/,2} \\circ f$ and $i_{1,/,2}$ would isotopic as embeddings. To see this, note that by translation it suffices to show that $i_{0}$ and $i_{1} \\circ f$ are isotopic. But $g_{t}: M \\to M$ defined by $g_{t}(x)$ =F(x, t) gives such an isotopy. Conversely, if $i_{1,/,2}$ and $i_{1,/,2} \\circ f$ are isotopic as embeddings, then apply isotopy extension rel.M $\\times$ \\{0,1\\}to the isotopy, to obtain a diffeomorphism of $M \\times$ [0,1/2] that restricts to the identity on $M \\times$ \\{0\\} and to f on $M \\times$ \\{1/2\\}(and similarly in $M \\times$ [1/2,1]). Thus after rescaling we obtain a pseudo-isotopy from f to the identity.\n\nReferences cited:\n- [FW86] John L. Friedman and Donald M. Witt. Homotopy is not isotopy for homeomorphisms of 3-manifolds. Topology, 25(1):35–44, 1986. doi:10.1016/0040-9383(86) 90003-0.\n- [KS96] Slawomir Kwasik and Reinhard Schultz. Pseudo-isotopies of 3-manifolds. Topology, 35(2):363–376, 1996. doi:10.1016/0040-9383(95)00017-8.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Friedman--Witt diffeomorphisms are homotopic but not isotopic and are topologically pseudo-isotopic to the identity; their smooth pseudo-isotopy status remains open.\n\n**Verified partial progress.**\n\n- Friedman--Witt examples give the natural test case.\n\n**Full solution or refutation.**\n\nNo example of the requested topological-but-not-smooth pseudo-isotopy contrast was verified.\n\n**What remains.**\n\nResolve smooth pseudo-isotopy for the Friedman--Witt maps.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.69 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Identifies the test case and retains the question.\n\n**Review notes.** OCR spacing in source statement was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2946, "problem_number": "KP-4.70", "title": "Kirby Problem 4.70", "statement": "Do there exist $k \\geq$ 0 and a smooth closed 4-manifold X such that the map $\\pi_{k}(\\operatorname{Diff}(X)) \\to \\pi_{k}(\\operatorname{Homeo}(X))$ induced by the inclusion $\\operatorname{Diff}(X) \\hookrightarrow \\operatorname{Homeo}(X)$ is an isomorphism?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.70.\n\nLiterature notes:\n(1) Lin–Xie [LX23] proved that, for every orientable compact smooth 4manifold X, at least one of the following holds:\n\n\\noindent$\\bullet$ $\\pi_{1}(\\operatorname{Diff}(X)) \\to \\pi_{1}(\\operatorname{Homeo}(X))is$ not injective.\n\n\\noindent$\\bullet$ $\\pi_{2}(\\operatorname{Diff}(X)) \\to \\pi_{2}(\\operatorname{Homeo}(X))is$ not surjective. They prove also that, if $\\partial X \\ne \\emptyset$ or the signature of X is non-zero, then there are many degrees k for which $\\pi_{k}(\\operatorname{Diff}(X)) \\to \\pi_{k}(\\operatorname{Homeo}(X))is$ not an isomorphism. However, it is still possible that for some 4-manifold X and some degree k, the map $\\pi_{k}(\\operatorname{Diff}(X)) \\to \\pi_{k}(\\operatorname{Homeo}(X))is$ an isomorphism. The above result of Lin–Xie is based on Watanabe’s work [Wat19] for $X = S^{4}$. Watanabe proved that nontrivial elements in the kernel of $\\pi_{k}(\\operatorname{Diff}(S^{4})) \\to \\pi_{k}(\\operatorname{Homeo}(S^{4}))$ exist whenever $\\mathcal{A}_{k,+,1} \\ne$ 0. Here $\\mathcal{A}_{k,+,1}$ is the degree (k+1)-part of a specific graph cohomology.\n\n(2) Many other negative results are obtained by (mainly family) gauge theory, such as [AR25, FM88, Don90, MS97, Rub98, BK20, KKN21b, BK22, Bar21, BK23, KN23, KM20, Lin23, KT22b, IKMT25, KMT23a, GL25].\n\n(3) As a positive result, a classical theorem by Wall [Wal64a] shows that there are many 4-manifolds X for which the map $\\pi_{0}(\\operatorname{Diff}(X)) \\to \\pi_{0}(\\operatorname{Homeo}(X))$ is surjective. However, there also exist examples where this map is not surjective; see [FM88, Don87b].\n\nReferences cited:\n- [LX23] Jianfeng Lin and Yi Xie. Configuration space integrals and formal smooth structures, 2023. arXiv:2310.14156.\n- [Wat19] Tadayuki Watanabe. Some exotic nontrivial elements of the rational homotopy groups of Diffp$S^{4}$q, 2019. arXiv:1812.02448.\n- [AR25] Dave Auckly and Daniel Ruberman. Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory, 2025. arXiv: 2501.11892.\n- [FM88] Robert Friedman and John W. Morgan. On the diffeomorphism types of certain algebraic surfaces. I. J. Differential Geom., 27(2):297–369, 1988. http://projecteuclid.org/euclid.jdg/1214441784.\n- [Don90] S. K. Donaldson. Polynomial invariants for smooth four-manifolds. Topology, 29(3):257–315, 1990. doi:10.1016/0040-9383(90)90001-Z.\n- [MS97] John W. Morgan and Zoltán Szabó. Homotopy K3 surfaces and mod 2 SeibergWitten invariants. Math. Res. Lett., 4(1):17–21, 1997. doi:10.4310/MRL.1997.v4.n1.a2.\n- [Rub98] Daniel Ruberman. An obstruction to smooth isotopy in dimension 4. Math. Res. Lett., 5(6):743–758, 1998. doi:10.4310/MRL.1998.v5.n6.a5.\n- [BK20] David Baraglia and Hokuto Konno. A gluing formula for families Seiberg-Witten invariants. Geom. Topol., 24(3):1381–1456, 2020. doi:10.2140/gt.2020.24.1381.\n- [KKN21b] Tsuyoshi Kato, Hokuto Konno, and Nobuhiro Nakamura. Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds. Compos. Math., 157(4):770–808, 2021. doi:10.1112/s0010437x2000771x.\n- [BK22] David Baraglia and Hokuto Konno. On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds. J. Topol., 15(2):505–586, 2022. doi:10.1112/topo.12229.\n- [Bar21] David Baraglia. Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants. Algebr. Geom. Topol., 21(1):317–349, 2021. doi:10.2140/agt.2021.21.317.\n- [BK23] David Baraglia and Hokuto Konno. A note on the Nielsen realization problem for K3 surfaces. Proc. Amer. Math. Soc., 151(9):4079–4087, 2023. doi:10.1090/proc/15544.\n- [KN23] Hokuto Konno and Nobuhiro Nakamura. Constraints on families of smooth 4-manifolds from $\\mathrm{Pin}(2)$-monopole. Algebr. Geom. Topol., 23(1):419–438, 2023. doi:10.2140/agt.2023.23.419.\n- [KM20] P. B. Kronheimer and T. S. Mrowka. The Dehn twist on a sum of two K3 surfaces. Math. Res. Lett., 27(6):1767–1783, 2020. doi:10.4310/MRL.2020.v27.n6.a8.\n- [Lin23] Jianfeng Lin. Isotopy of the Dehn twist on K3 \\# K3 after a single stabilization. Geom. Topol., 27(5):1987–2012, 2023. doi:10.2140/gt.2023.27.1987.\n- [KT22b] Hokuto Konno and Masaki Taniguchi. The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary. Adv. Math., 409:Paper No. 108627, 58, 2022. doi:10.1016/j.aim.2022.108627.\n- [IKMT25] Nobuo Iida, Hokuto Konno, Anubhav Mukherjee, and Masaki Taniguchi. Diffeomorphisms of 4-manifolds with boundary and exotic embeddings. Math. Ann., 391(2):1845–1897, 2025. doi:10.1007/s00208-024-02974-x.\n- [KMT23a] Hokuto Konno, Abhishek Mallick, and Masaki Taniguchi. Exotic Dehn twists on 4-manifolds, 2023. arXiv:2306.08607.\n- [GL25] Daniel Galvin and Roberto Ladu. Non-smoothable homeomorphisms of 4-manifolds with boundary. Adv. Math., 467:Paper No. 110191, 22, 2025. doi:10.1016/j.aim.2025.110191.\n- [Wal64a] C. T. C. Wall. Diffeomorphisms of 4-manifolds. J. London Math. Soc., 39:131–140, 1964. doi:10.1112/jlms/s1-39.1.131.\n- [Don87b] S. K. Donaldson. The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differential Geom., 26(3):397–428, 1987. http://projecteuclid.org/euclid.jdg/1214441485.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Lin--Xie show that for every orientable compact smooth 4-manifold, a low-degree Diff-to-Homeo map fails either injectivity or surjectivity; existence of any isomorphism in any degree remains open.\n\n**Verified partial progress.**\n\n- Lin--Xie provide a universal dichotomy involving pi_1 and pi_2.\n\n**Full solution or refutation.**\n\nNo positive isomorphism example was verified.\n\n**What remains.**\n\nRule out all degrees/manifolds or find an exceptional isomorphism.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.70 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the Lin--Xie obstruction and asks the existence question.\n\n**Review notes.** OCR defects in displayed maps were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2947, "problem_number": "KP-4.71", "title": "Kirby Problem 4.71", "statement": "(a) Do there exist $k \\geq$ 0 and a smooth closed orientable 4-manifold X such that $\\pi_{k}(\\operatorname{Diff}(X))$ is finitely generated?\n\n(b) Do there exist $k >$ 0 and a smooth closed orientable 4-manifold X such that $H_{k}(BDiff(X);\\mathbb{Z})$ is finitely generated?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.71.\n\nLiterature notes:\n(1) For each $k >$ 0, there exist simply-connected closed smooth 4-manifolds X where $H_{k}(BDiff(X);\\mathbb{Z})$ are not finitely generated [Kon24b]. Also, Auckly–Ruberman [AR25] proved that, for eachk >0, there exist simplyconnected closed smooth 4-manifolds X where $\\pi_{k}(\\operatorname{Diff}(X))$ is not finitely generated. In dimension $\\ne$ 4, there are several finiteness results. See [Kup19b, BKK24]. See also Problem 4.72 for the analogous question in the topological category.\n\n(2) If X is oriented, we can ask an analogous question for $\\operatorname{Diff}^{+}(X)$, the orientation-preserving diffeomorphism group, in place of $\\operatorname{Diff}(X)$. (Note that $\\operatorname{Diff}(X) = \\operatorname{Diff}^{+}(X)$ if X has non-zero signature.) For homotopy groups $\\pi_{k}(\\operatorname{Diff}(X))$ and $\\pi_{k}(\\operatorname{Diff}^{+}(X))$, finite generation of $\\pi_{k}(\\operatorname{Diff}(X))$ and that of $\\pi_{k}(\\operatorname{Diff}^{+}(X))$ are equivalent. However, for $H_{k}(BDiff(X);\\mathbb{Z})$ and $H_{k}(BDiff^{+}(X);\\mathbb{Z})$, the questions may not be equivalent. For example, when $\\operatorname{Diff}(X) \\ne \\operatorname{Diff}^{+}(X)$, and if we take rational coefficients, the covering map $BDiff^{+}(X)\\to BDiff(X)$ BDiff(X)induces an isomorphism $H_{k}(BDiff^{+}(X);\\mathbb{Q})^{\\mathbb{Z}/2} \\cong H_{k}(BDiff(X);\\mathbb{Q})$, where the superscript $\\mathbb{Z}/2$ indicates the monodromy invariant part. Thus finite generation of $H_{k}(BDiff^{+}(X);\\mathbb{Q})$ implies that of $H_{k}(BDiff(X);\\mathbb{Q})$, but the converse may not be true in general.\n\n(3) As sets, the homotopy and homology groups of diffeomorphism groups of compact manifolds are always countable.\n\n(4) The diffeomorphism groups of non-compact 4-manifolds can have finitely generated homotopy and homology groups, e.g. $\\operatorname{Diff}(\\mathbb{R}^{4})$, in the weak $C^{\\infty}topology$, is homotopy equivalent to $O(4)$. Nevertheless, there also exist exotic $\\mathbb{R}^{4}’s$ with infinitely generated mapping class groups [Gom18].\n\n(5) The problem is open even for $X =S^{4}$ or $D^{4}$.\n\nReferences cited:\n- [Kon24b] Hokuto Konno. The homology of moduli spaces of 4-manifolds may be infinitely generated. Forum Math. Pi, 12:Paper No. e25, 18, 2024. doi:10.1017/fmp.2024.26.\n- [AR25] Dave Auckly and Daniel Ruberman. Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory, 2025. arXiv: 2501.11892.\n- [Kup19b] Alexander Kupers. Some finiteness results for groups of automorphisms of manifolds. Geom. Topol., 23(5):2277–2333, 2019. doi:10.2140/gt.2019.23.2277.\n- [BKK24] Mauricio Bustamante, Manuel Krannich, and Alexander Kupers. Finiteness properties of automorphism spaces of manifolds with finite fundamental group. Math. Ann., 388(4):3321–3371, 2024. doi:10.1007/s00208-023-02594-x.\n- [Gom18] Robert E. Gompf. Group actions, corks and exotic smoothings of $\\mathbb{R}^{4}$. Invent. Math., 214(3):1131–1168, 2018. doi:10.1007/s00222-018-0819-8.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** There are simply connected closed smooth 4-manifolds with non-finitely-generated pi_k(Diff) and H_k(BDiff) in every positive degree, but existence of a finite-generation example remains open.\n\n**Verified partial progress.**\n\n- Konno gives nonfinite BDiff homology examples.\n- Auckly--Ruberman give nonfinite diffeomorphism-group homotopy examples.\n\n**Full solution or refutation.**\n\nNo requested positive finite-generation example was verified.\n\n**What remains.**\n\nFind a rigid 4-manifold with finite homotopy/homology or prove none exist.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.71 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the universal-degree negative examples while posing existence.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2948, "problem_number": "KP-4.72", "title": "Kirby Problem 4.72", "statement": "Let X be a closed orientable topological 4-manifold with finite $\\pi_{1}(X)$.\n\n(a) Is $\\pi_{k}(\\operatorname{Homeo}(X))$ finitely generated for every $k \\geq$ 0?\n\n(b) Is $H_{k}(BHomeo(X);\\mathbb{Z})$ finitely generated for every $k \\geq$ 0?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.72.\n\nLiterature notes:\n(1) For $\\pi_{1}(X)$ =1, it follows from a result by Perron [Per86] and Quinn [Qui86] (cf. $[GGH^{+}23])$ that $\\pi_{0}(\\operatorname{Homeo}(X))$ is finitely generated, which implies that $H_{1}(BHomeo(X);\\mathbb{Z})$ is also finitely generated. There is no known finiteness result on $\\pi_{k}(\\operatorname{Homeo}(X))$ for $k \\geq$ 1 and $H_{k}(BHomeo(X);\\mathbb{Z})$ for $k \\geq$ 2. See Problem 4.71 for the analogous question in the smooth category. The following question is closely related.\n\n\\paragraph{Question.} Does $Top(4)$ have finitely generated homotopy groups in each degree? Indeed, $Top(4)$ has finitely-generated homotopy groups if and only if $\\operatorname{Homeo}(S^{4})$ has finitely-generated homotopy groups, because we have a fibration $Top(4) \\to \\operatorname{Homeo}(S^{4}) \\to S^{4}$, and the homotopy groups of $S^{4}$ are finitely generated. The question on the homotopy groups of Top(4)is itself closely related to Problem 4.73 on the Morlet correspondence in dimension 4.\n\n(2) Here is another closely related question, on topological embedding spaces. Let $\\Sigma$ be a compact surface, and consider a compact 4-manifold X. Fix a locally flat $embedding\\iota: \\partial\\Sigma \\hookrightarrow \\partial X$. Let $\\operatorname{Emb}^{t}_{\\partial}(\\Sigma$, X)be the space of locally flat embeddings of $\\Sigma$ extending $\\iota$. This is defined as the geometric realization of a semi-simplicial set, where the p-simplices are locally flat embeddings $\\Sigma \\times \\Delta^{p} \\to X \\times \\Delta^{p}$ over the projection to $\\Delta^{p}$, and extending $\\iota \\times$ Id : $\\partial\\Sigma \\times \\Delta^{p} \\to X \\times \\Delta^{p}. Letf_{0}: \\Sigma \\hookrightarrow X$ be a 0-simplex.\n\n\\paragraph{Question.} Suppose that $\\pi_{1}(X \\setminus f_{0}(\\Sigma))$ is finite, and fix $k >$ 0. Is $\\pi_{k}(\\operatorname{Emb}^{t}_{\\partial}(\\Sigma$, X), $f_{0})$ finitely generated? Randal-Williams [RW] has announced that the answer to the analogous question is no, in a case where the fundamental group of the surface complement is infinite. Using [BG19, BG25], Randal-Williams deduced that $\\pi_{4}(\\operatorname{Emb}^{t}(S^{2}, S^{4})$, U) is infinitely generated, where U is the trivial 2-knot. The homotopy groups spaces of embeddings are closely related to the homotopy groups of the homeomorphism groups of both the ambient space and of the exterior of the basepoint embedding. The space of thickenings of a fixed embedding to a closed tubular neighborhood plays an important rôle as well. As demonstrated by Randal-Williams’ note, information about any of these characters often leads to information about the others.\n\nReferences cited:\n- [Per86] B. Perron. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. Topology, 25(4):381–397, 1986. doi:10.1016/0040-9383(86)90018-2.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [GGH+23] David Gabai, David T. Gay, Daniel Hartman, Vyacheslav Krushkal, and Mark Powell. Pseudo-isotopies of simply connected 4-manifolds, 2023. arXiv:2311.11196.\n- [RW] Oscar Randal-Williams. Topological embeddings of $S^{2}$ in $S^{4}$. https://www.dpmms.cam.ac.uk/„or257/notes/Embeddings.pdf.\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.\n- [BG25] Ryan Budney and David Gabai. On the automorphism groups of hyperbolic manifolds, 2025. doi:10.1093/imrn/rnaf083.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For simply connected X, pi_0 Homeo(X) and hence H_1(BHomeo(X)) are finitely generated; no higher-degree or all finite-fundamental-group result was verified.\n\n**Verified partial progress.**\n\n- Perron--Quinn supply the pi_0 finiteness result in the simply connected case.\n\n**Full solution or refutation.**\n\nThe stated all-k finiteness questions remain open.\n\n**What remains.**\n\nExtend mapping-class-group techniques to higher homotopy and finite pi_1.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.72 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the pi_0/H1 base case and lack of higher results.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2949, "problem_number": "KP-4.73", "title": "Kirby Problem 4.73", "statement": "Does the Morlet correspondence $BDiff_{\\partial}(D^{n}) \\cong \\Omega^{n}_{0}(Top(n)/O(n))$ (10) hold for n=4?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.73.\n\nLiterature notes:\n(1) Here $BDiff_{\\partial}(D^{n})denotes$ the classifying space of the diffeomorphism group of $D^{n}$ relative to its boundary. We write $Top(n)$ for the group of homeomorphisms on $\\mathbb{R}^{n}$ that fix the origin, and Top(n)/O(n) for the homotopy fiber of $BO(n) \\to BTop(n)$. Finally let $\\Omega^{n}_{0}(Top(n)/O(n))$ denote the unit component of the loop space $\\Omega^{n}(Top(n)/O(n))$.\n\n(2) In dimension $n \\ne$ 4, the weak equivalence (10) follows from smoothing theory [BL74, KS77]. It is known that smoothing theory fails in dimension 4. For example, the manifold $E_{8}\\#E_{8}$ has a formal smooth structure (i.e. a vector bundle structure on its tangent microbundle) but has no smooth structure. However, (10) may still hold for n=4.\n\n(3) Watanabe [Wat19] disproved the 4-dimensional Smale conjecture by showin g that $\\pi_{k}(BDiff_{\\partial}(D^{4})) \\otimes \\mathbb{Q} \\ne$ 0 for many values of k including 2,5,9. It is known that the group $\\pi_{k,+,4}(Top(n)/O(n)) \\otimes \\mathbb{Q}$ is also nonvanishing for these values of k [LX23].\n\n(4) Gauge theory can distinguish non-diffeomorphic smooth structures on a closed 4-manifold that are isomorphic as formal smooth structures. So gauge theory could potentially be used to disprove (10).\n\n(5) The question is closely related to Problem 4.67. If there is a diffeomorphism of $D^{4}$ not isotopic to the identity, and this is detected using gauge theory, then this could show that the Morlet correspondence does not hold.\n\nReferences cited:\n- [BL74] Dan Burghelea and Richard Lashof. The homotopy type of the space of diffeomorphisms. I, II. Trans. Amer. Math. Soc., 196:1–36; ibid. 196 (1974), 37–50, 1974. doi:10.2307/1997010.\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [Wat19] Tadayuki Watanabe. Some exotic nontrivial elements of the rational homotopy groups of Diffp$S^{4}$q, 2019. arXiv:1812.02448.\n- [LX23] Jianfeng Lin and Yi Xie. Configuration space integrals and formal smooth structures, 2023. arXiv:2310.14156.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Morlet correspondence is standard away from the exceptional four-dimensional setting, but validity for n=4 was not verified.\n\n**Verified partial progress.**\n\n- The problem precisely identifies the topological/smoothing-theory comparison map.\n\n**Full solution or refutation.**\n\nThe n=4 correspondence remains open in the checked source.\n\n**What remains.**\n\nProve the comparison map is a weak equivalence or identify a homotopy obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.73 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the n=4 specialization and retains the question.\n\n**Review notes.** Source's missing spacing in definitions was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2950, "problem_number": "KP-4.74", "title": "Kirby Problem 4.74", "statement": "Does there exist a closed, smooth 4-manifold X and a diffeomorphism f: $X \\to X$ such that f is smoothly pseudo-isotopic to the identity, but f is not stably smoothly isotopic to the identity?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.74.\n\nLiterature notes:\n(1) We can stabilize a diffeomorphism by making a choice of isotopy to one that is the identity on a 4-ball, connect summing with $\\#_{k}S^{2} \\times S^{2}$ using that 4-ball, for some k, and then extending by the identity on the new $\\#_{k}S^{2} \\times S^{2}$. If some stabilization of f (for some choice of isotopy and for some k) is smoothly isotopic to the identity, then we say that f is smoothly stably isotopic to Id.\n\n(2) Gabai [Gab22] proved that there is a smooth pseudo-isotopy F with vanishing Hatcher-Wagoner pseudo-isotopy obstruction $\\Sigma(F) \\in Wh_{2}(\\pi_{1}(X))$ if and only iff is smoothly stably isotopic to Id. The question is whether there exists an f such that $\\Sigma(F)$ is nontrivial for all pseudo-isotopies F from f to Id. Or perhaps, for any pair(f, F), there is always a choice of F with $\\Sigma(F) =$ 0 restricting to the same f. Singh [Sin25] proved that the Hatcher–Wagoner obstruction $\\Sigma$ can be stably realized, which could be useful if one could control the diffeomorphism produced by his realization procedure.\n\nReferences cited:\n- [Gab22] David Gabai. 3-spheres in the 4-sphere and pseudo-isotopies of $S^{1}$ $\\times$ $S^{3}$, 2022. arXiv:2212.02004.\n- [Sin25] Oliver Singh. Pseudo-isotopies and diffeomorphisms of 4-manifolds. J. Topol., 18(4):Paper No. e70043, 61, 2025. doi:10.1112/topo.70043.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No smoothly pseudo-isotopic but not stably smoothly isotopic diffeomorphism was verified.\n\n**Verified partial progress.**\n\n- Stabilization of a diffeomorphism is defined after choosing a ball-supported representative.\n\n**Full solution or refutation.**\n\nThe requested separation remains open.\n\n**What remains.**\n\nConstruct a pseudo-isotopy invariant not killed by the stated stabilization.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.74 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the distinction and retains the existence question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2951, "problem_number": "KP-4.75", "title": "Kirby Problem 4.75", "statement": "Let X be a connected smooth 4-manifold with nonempty boundary, with finite $\\pi_{1}(X)$, and let $k \\geq$ 0. Let $Diff_{\\partial}(X)$ denote the group of diffeomorphisms of X that are the identity near $\\partial X$. Is the image of the natural map $s_{*}: \\pi_{k}(Diff_{\\partial}(X)) \\to$ colim $\\pi_{k}(Diff_{\\partial}(X\\#_{N}S^{2} \\times S^{2}))$ (11) $N \\to \\infty$ finitely generated? What about the analogous problem in the topological category?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.75.\n\nLiterature notes:\n(1) Fixing a model of the interior connected sum $X\\#S^{2} \\times S^{2}$ defined by $X\\#S^{2} \\times S^{2}$ =X $\\cup ((\\partial X \\times [0,1])\\#S^{2} \\times S^{2})$, we have a well-defined stabilization map $s: Diff_{\\partial}(X) \\to Diff_{\\partial}(X\\#S^{2} \\times S^{2})$, extending diffeomorphisms by the identity map on $(\\partial X \\times [0,1])\\#S^{2} \\times S^{2}$.\n\n(2) In the topological category, Problem 4.72 asks the analogous question without stabilization.\n\n(3) We take a closer look at the problem for k =0, $\\pi_{1}(X)$ =1, and connected boundary. A diffeomorphism f: $X \\to X$ determines a Poincaré variation [Sae06]. The group of Poincaré variations is denoted $\\mathcal{V}(H_{2}(X), \\lambda_{X})$, where $\\lambda_{X}$ is the intersection form of X. Saeki [Sae06, Theorem 3.7] proved that the stable mapping class group of X is isomorphic to the group of stable Poincaré variations $S\\mathcal{V}(H_{2}(X), \\lambda_{X})$. Let $V:=Im(\\theta: \\pi_{0}Diff_{\\partial}(X) \\to \\mathcal{V}(H_{2}(X), \\lambda_{X}))$. The image of $s_{*}$ is thus isomorphic to the image of V under the algebraic stabilization map $\\mathcal{V}(H_{2}(X), \\lambda_{X}) \\to S\\mathcal{V}(H_{2}(X), \\lambda_{X})$. This latter map is injective, so in fact $Im(s_{*}) \\cong V$. We need to decide whether V is finitely generated. By [Sae06, Section 4], the group $\\mathcal{V}(H_{2}(X), \\lambda_{X})$ sits in an exact sequence 0 $\\to \\wedge ^{2}H^{1}(\\partial X;\\mathbb{Z}) \\to \\mathcal{V}(H_{2}(X), \\lambda_{X}) \\to Aut_{\\partial}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$. The first group is finitely generated, and the image of the second map is a finite index subgroup of $Aut_{\\partial}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$, the automorphisms of the intersection form of X whose algebraic boundary is trivial. The latter group is arithmetic, so is finitely generated. It follows $that\\mathcal{V}(H_{2}(X), \\lambda_{X})$ is finitely generated, since finite index subgroups and extensions of finitely generated groups are again finitely generated. Is V finitely generated? In the special case that $\\partial X =S^{3} \\mathcal{V}(H_{2}(X), \\lambda_{X}) \\cong \\operatorname{Aut}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$. By capping of f with a 4-ball we can apply Wall’s theorem [Wal64a] to see that $\\theta$ is surjective when X is of the for m $X = M\\#S^{2} \\times S^{2}$, where M is indefinite or $b_{2}(M) <$ 9. Thus in these cases the image of $s_{*}$, for $k =$ 0, is known to be finitely generated. In other cases, such as for $X = K3 \\setminus D^{\\circ 4}, V$ is finite index in $\\mathcal{V}(H_{2}(X), \\lambda_{X}) \\cong \\operatorname{Aut}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$, so is finitely generated. However there are also examples, such as for X the punctured Dolgachev surface, where V is infinite index [FM88] in $\\operatorname{Aut}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$. In such cases, is V finitely generated?\n\n(4) Without stabilization, it is known that $\\pi_{k}(\\operatorname{Diff}(X))$ need not be finitely generated, even for $\\pi_{1}(X)$ =1. This was proven by Ruberman [Rub99b] for k=0, with later proofs by Baraglia [Bar23c] and Konno [Kon24b]. Fork =1 it was proven by Baraglia[Bar23b] and Lin [Lin22]. Fork $\\geq$ 1 this is announced in recent work of Auckly–Ruberman [AR25]. Different 4-manifolds are used in each work. All of these results are proven using gauge theory for families.\n\n(5) For 4-manifolds with infinite fundamental group, Budney–Gabai [BG19] and Watanabe [Wat23] proved (without using gauge theory) that some 4-manifolds with infinite fundamental group $(e.g.S^{1} \\times S^{3}$ in [BG19], and hyperbolic manifolds of dimension at least 4 in [BG25]) have infinitely generated mapping class groups. However the diffeomorphisms they construct are pseudo-isotopic to the identity, and hence are stably isotopic to the identity by Gabai’s theorem [Gab22]. It would be interesting to know whether there are counterexamples when $\\pi_{1}(X)$ is infinite.\n\n(6) Since $\\pi_{k}(Diff_{\\partial}(X)) \\cong \\pi_{k,+,1}(BDiff_{\\partial}(X))$, the problem can be described in terms of $BDiff_{\\partial}(X)$. If one considers the analogous question for the homology groups of $BDiff_{\\partial}(X)$, the answer is often positive. This is because it follows from work by Galatius and Randal-Williams [GRW17] that colim $H_{k}(BDiff_{\\partial}(X\\#_{N}S^{2} \\times S^{2}))$ (12) $N \\to \\infty$ is finitely generated for allkand many X, such as for simply-connected X. In particular, any subgroup of (12) such as the image of the stabilization map is also finitely generated. However, this is not necessarily strong evidence to hope for a positive solution to the problem for homotopy groups, since there are many spaces with finitely generated homology groups but infinitely generated homotopy groups (for example, $H_{2}(S^{1} \\vee S^{2}) =\\mathbb{Z}$ but $\\pi_{2}(S^{1} \\vee S^{2}) =\\mathbb{Z}^{\\infty})$.\n\n(7) In even, higher dimensions, the answer to the analogous problem is positive, and indeed this holds without taking the image in a colimit. For a compact smooth manifold M of dimension 2n $\\geq$ 6 with finite fundamental group, $\\pi_{k}(Diff_{\\partial}(M))$ is finitely generated, due to Bustamante–Krannich– Kupers [BKK24, Theorem 6.1].\n\nReferences cited:\n- [Sae06] Osamu Saeki. Stable mapping class groups of 4-manifolds with boundary. Trans. Amer. Math. Soc., 358(5):2091–2104, 2006. doi:10.1090/S0002-9947-05-03748-7.\n- [Wal64a] C. T. C. Wall. Diffeomorphisms of 4-manifolds. J. London Math. Soc., 39:131–140, 1964. doi:10.1112/jlms/s1-39.1.131.\n- [FM88] Robert Friedman and John W. Morgan. On the diffeomorphism types of certain algebraic surfaces. I. J. Differential Geom., 27(2):297–369, 1988. http://projecteuclid.org/euclid.jdg/1214441784.\n- [Rub99b] Daniel Ruberman. A polynomial invariant of diffeomorphisms of 4-manifolds. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 473–488. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.473.\n- [Bar23c] David Baraglia. On the mapping class groups of simply-connected smooth 4-manifolds, 2023. arXiv:2310.18819.\n- [Kon24b] Hokuto Konno. The homology of moduli spaces of 4-manifolds may be infinitely generated. Forum Math. Pi, 12:Paper No. e25, 18, 2024. doi:10.1017/fmp.2024.26.\n- [Bar23b] David Baraglia. Non-trivial smooth families of K3 surfaces. Math. Ann., 387(3-4):1719–1744, 2023. doi:10.1007/s00208-022-02508-3.\n- [Lin22] Jianfeng Lin. The family Seiberg-Witten invariant and nonsymplectic loops of diffeomorphisms, 2022. arXiv:2208.12082.\n- [AR25] Dave Auckly and Daniel Ruberman. Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory, 2025. arXiv: 2501.11892.\n- [BG19] Ryan Budney and David Gabai. Knotted 3-balls in $S^{4}$, 2019. arXiv:1912.09029.\n- [Wat23] Tadayuki Watanabe. Theta-graph and diffeomorphisms of some 4-manifolds, 2023. arXiv:2005.09545.\n- [BG25] Ryan Budney and David Gabai. On the automorphism groups of hyperbolic manifolds, 2025. doi:10.1093/imrn/rnaf083.\n- [Gab22] David Gabai. 3-spheres in the 4-sphere and pseudo-isotopies of $S^{1}$ $\\times$ $S^{3}$, 2022. arXiv:2212.02004.\n- [GRW17] Søren Galatius and Oscar Randal-Williams. Homological stability for moduli spaces of high dimensional manifolds. II. Ann. of Math. (2), 186(1):127–204, 2017. doi: 10.4007/annals.2017.186.1.4.\n- [BKK24] Mauricio Bustamante, Manuel Krannich, and Alexander Kupers. Finiteness properties of automorphism spaces of manifolds with finite fundamental group. Math. Ann., 388(4):3321–3371, 2024. doi:10.1007/s00208-023-02594-x.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Stable diffeomorphism-group methods provide structural results, but finite generation of the stated stable images was not verified generally.\n\n**Verified partial progress.**\n\n- The list sets up canonical stabilization maps and finite-fundamental-group setting.\n\n**Full solution or refutation.**\n\nThe finite-generation questions remain open.\n\n**What remains.**\n\nEstablish representation stability or a counterfamily with infinitely generated image.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.75 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the stable image questions.\n\n**Review notes.** OCR in colimit display was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2952, "problem_number": "KP-4.76", "title": "Kirby Problem 4.76", "statement": "Let X be a closed, oriented, simply connected, smooth 4manifold and fix $k >$ 0. Is there $N \\geq$ 0 such that, for every $n \\geq N$, the natural map $\\pi_{k}(\\operatorname{Diff}^{+}(X\\#_{n}S^{2} \\times S^{2})) \\to \\pi_{k}(\\operatorname{Homeo}^{+}(X\\#_{n}S^{2} \\times S^{2}))$ is surjective?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.76.\n\nLiterature notes:\n(1) If $k =$ 0, the answer to the analogous question is affirmative. First, for N =2, it follows from Wall’s theorem [Wal64a] that the natural map $\\pi_{0}(\\operatorname{Diff}^{+}(X\\#_{n}S^{2} \\times S^{2})) \\to \\operatorname{Aut}(H_{2}(X\\#_{n}S^{2} \\times S^{2};\\mathbb{Z}), \\lambda_{X,\\#,n,S,2, \\times ,S,2})$ is surjective for every $n \\geq N$. Here $\\operatorname{Aut}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})$ denotes the automorphism group of the intersection form. On the other hand, work of Freedman [Fre82], Kreck, [Kre79], Perron [Per86], and Quinn [Qui86] (plus [GGH+23]), implies that the natural map $\\pi_{0}(\\operatorname{Homeo}^{+}(X\\#_{n}S^{2} \\times S^{2})) \\to \\operatorname{Aut}(H_{2}(X\\#_{n}S^{2} \\times S^{2};\\mathbb{Z}), \\lambda_{X,\\#,n,S,2, \\times ,S,2})$ is an isomorphism. Thus, considering the obvious commuting triangle, we have $\\pi_{0}(\\operatorname{Diff}^{+}(X\\#_{n}S^{2} \\times S^{2})) \\to \\pi_{0}(\\operatorname{Homeo}^{+}(X\\#_{n}S^{2} \\times S^{2}))$ is surjective.\n\n(2) For the analogous problem obtained by replacing “surjective” with “injective”, the answer is negative for k =0. Indeed, for any simply-connected closed smooth 4-manifold X, one can find a strictly increasing divergent sequence 0 $\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The degree-zero analogue is affirmative after stabilization, but eventual surjectivity in every positive degree remains unverified.\n\n**Verified partial progress.**\n\n- Wall-type results give the stated k=0 stabilization information.\n\n**Full solution or refutation.**\n\nNo positive-degree eventual surjectivity theorem was verified.\n\n**What remains.**\n\nLift stable Homeo classes in a fixed positive degree or find an obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.76 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Contrasts the k=0 result with the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2953, "problem_number": "KP-4.77", "title": "Kirby Problem 4.77", "statement": "For which $k \\geq$ 0 and closed smooth 4-manifold X does the equality $\\ker(i_{*}: \\pi_{k}(Diff_{\\partial}(X^{\\circ})) \\to \\pi_{k}(Homeo_{\\partial}(X^{\\circ}))) =\\ker(s_{*}: \\pi_{k}(Diff_{\\partial}(X^{\\circ})) \\to$ colim $\\pi_{k}(Diff_{\\partial}(X^{\\circ}\\#_{n}S^{2} \\times S^{2})))$ (13) $n \\to \\infty$ hold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.77.\n\nLiterature notes:\n(1) Here $i: Diff_{\\partial}(X^{\\circ}) \\to Homeo_{\\partial}(X^{\\circ})$ is the inclusion, and $s: Diff_{\\partial}(X^{\\circ}) \\to Diff_{\\partial}(X^{\\circ}\\#_{n}S^{2} \\times S^{2})$ is the stabilization map from Problem 4.75.\n\n(2) For $k =$ 0 and $\\pi_{1}(X) =$ 1, the equality (13) is known to hold: by a result by Quinn [Qui86] and Perron [Per86], the kernels in (13) coincide with the group of mapping classes of diffeomorphisms that act trivially on homology.\n\n(3) See Problem 4.45 for the analogous question on smooth and topological embedding spaces of surfaces in 4-manifolds.\n\n(4) Gabai [Gab22] proved that diffeomorphisms that are pseudo-isotopic, via a pseudo-isotopy with trivial $Wh_{2}$ obstruction, are smoothly stably isotopic. If we knew a topological analogue of this result, that could help to find counterexamples for k =0, and be an interesting development in its own right.\n\n\\paragraph{Question.} Suppose that a self-homeomorphism f: $X \\to X$ of a compact 4-manifold X is topologically pseudo-isotopic to the identity via a pseudo-isotopy F with vanishing primary topological Hatcher–Wagoner invariant, $\\Sigma(F) =$ 0 $\\in Wh_{2}(\\pi_{1}(X))$. Is f topologically stably isotopic to $Id_{X}$? See [GN25] for a definition of the primary topological Hatcher–Wagoner invariant.\n\nReferences cited:\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [Per86] B. Perron. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. Topology, 25(4):381–397, 1986. doi:10.1016/0040-9383(86)90018-2.\n- [Gab22] David Gabai. 3-spheres in the 4-sphere and pseudo-isotopies of $S^{1}$ $\\times$ $S^{3}$, 2022. arXiv:2212.02004.\n- [GN25] Daniel Galvin and Isacco Nonino. Pseudo-isotopy versus isotopy for homeomorphisms of 4-manifolds, 2025. arXiv:2506.11905.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No general equality between the topologically trivial and stably trivial kernels was verified.\n\n**Verified partial progress.**\n\n- The problem isolates two natural kernels of inclusion and stabilization maps.\n\n**Full solution or refutation.**\n\nThe equality question remains open.\n\n**What remains.**\n\nCompare smoothing-theoretic and stable-diffeomorphism obstructions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.77 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines both kernels and poses equality.\n\n**Review notes.** OCR in kernels/colimits was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2954, "problem_number": "KP-4.78", "title": "Kirby Problem 4.78", "statement": "Let X be a simply connected closed smooth 4-manifold. Does $BDiff(X)$ satisfy homological stability over $\\mathbb{Q}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.78.\n\nLiterature notes:\n(1) As before, let $s: Diff_{\\partial}(X^{\\circ}) \\to Diff_{\\partial}(X^{\\circ}\\#S^{2} \\times S^{2})$ be the stabilization map from Problem 4.75. The question asks whether, for eachk $\\geq$ 0, the induced maps $s_{*}: H_{k}(BDiff_{\\partial}(X^{\\circ}\\#_{N}S^{2} \\times S^{2});\\mathbb{Q}) \\to H_{k}(BDiff_{\\partial}(X^{\\circ}\\#_{N}+1 S^{2} \\times S^{2});\\mathbb{Q})$ are isomorphic for all n\"0 large enough relative to k.\n\n(2) For a simply connected compact manifold of dim $=$ 2n $\\ne$ 4, an analogous stability holds over any (untwisted) coefficients due to work by Harer [Har85] for dim =2 Galatius and Randal-Williams [GRW18] for dim $\\geq$ 6. The colimit colim $H_{k}(BDiff_{\\partial}(X^{\\circ}\\#_{N}S^{n} \\times S^{n})) n \\to \\infty$ is called the stable homology, which has been extensively studied, especially with $\\mathbb{Q}$ coefficients: indeed, the stable homology over $\\mathbb{Q}$ coefficient has been determined [MW07, GRW18].\n\n(3) With $\\mathbb{Z}$ coefficients, an analogous stability in dimension 4 was shown to fail by Konno and Lin [KL23c]. However, the unstable homology classes detected in [KL23c] are 2-torsion, so the result in [KL23c] does not imply rational instability.\n\nReferences cited:\n- [Har85] John L. Harer. Stability of the homology of the mapping class groups of orientable surfaces. Ann. of Math. (2), 121(2):215–249, 1985. doi:10.2307/1971172.\n- [GRW18] Søren Galatius and Oscar Randal-Williams. Homological stability for moduli spaces of high dimensional manifolds. I. J. Amer. Math. Soc., 31(1):215–264, 2018. doi: 10.1090/jams/884.\n- [MW07] Ib Madsen and Michael Weiss. The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. of Math. (2), 165(3):843–941, 2007. doi:10.4007/annals.2007.165.843.\n- [KL23c] Hokuto Konno and Jianfeng Lin. Homological instability for moduli spaces of smooth 4-manifolds, 2023. arXiv:2211.03043.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No rational homological-stability theorem for BDiff of arbitrary simply connected closed 4-manifolds was verified.\n\n**Verified partial progress.**\n\n- Stabilization by S2xS2 is the relevant comparison map.\n\n**Full solution or refutation.**\n\nThe requested homological stability remains open.\n\n**What remains.**\n\nProve stability ranges or identify unstable rational classes.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.78 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Poses the rational stability question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2955, "problem_number": "KP-4.79", "title": "Kirby Problem 4.79", "statement": "Given $k >$ 0, is there a closed, simply connected, smooth 4manifold X and a nonzero homotopy class $\\alpha \\in \\pi_{k}(Diff_{\\partial}(X^{\\circ}))$ such that $\\alpha \\in \\ker(i_{*}: \\pi_{k}(Diff_{\\partial}(X^{\\circ})) \\to \\pi_{k}(Homeo_{\\partial}(X^{\\circ})))$ (14) and $\\alpha \\notin \\ker(s_{*}: \\pi_{k}(Diff_{\\partial}(X^{\\circ})) \\to \\pi_{k}(Diff_{\\partial}(X^{\\circ}\\#S^{2} \\times S^{2})))$? (15)", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.79.\n\nLiterature notes:\n(1) Here we use the notation i, and s of Problem 4.75 and the previous two problems.\n\n(2) The answer to an analogous statement for $k =$ 0 is known to be affirmative: Kronheimer–Mrowka [KM20] proved that the Dehn twist on $X =$ K3\\#K3 along $S^{3}$ gives non-zero class $\\alpha$ that lies in (14), and Lin [Lin23] proved that this $\\alpha$ satisfies (15).\n\nReferences cited:\n- [KM20] P. B. Kronheimer and T. S. Mrowka. The Dehn twist on a sum of two K3 surfaces. Math. Res. Lett., 27(6):1767–1783, 2020. doi:10.4310/MRL.2020.v27.n6.a8.\n- [Lin23] Jianfeng Lin. Isotopy of the Dehn twist on K3 \\# K3 after a single stabilization. Geom. Topol., 27(5):1987–2012, 2023. doi:10.2140/gt.2023.27.1987.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The source records affirmative information in a restricted direction, but no general class satisfying the stated topologically trivial yet stably nontrivial condition was verified.\n\n**Verified partial progress.**\n\n- The question is formulated using the preceding inclusion and stabilization maps.\n\n**Full solution or refutation.**\n\nThe requested existence problem remains unresolved.\n\n**What remains.**\n\nSeek a family detected after stabilization but invisible topologically.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.79 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Retains the existence problem and its relation to the preceding kernels.\n\n**Review notes.** OCR in maps/labels was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2956, "problem_number": "KP-4.80", "title": "Kirby Problem 4.80", "statement": "Is there $n >$ 2 and a smooth closed simply connected 4manifold X for which there is an element of ordernin the $subgroupker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X))$? More generally, is there a subgroup of order nin $\\ker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X))$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.80.\n\nLiterature notes:\n(1) An element of $\\ker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X))is$ called an exotic diffeomorphism. Examples of exotic diffeomorphisms by Ruberman [Rub98] and Baraglia–Konno [BK20] are of infinite order in $\\ker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X)) A$ Dehn twist on K3\\#K3 along the connected sum $S^{3}$, detected by Kronheimer–Mrowka [KM20], is of order 2. These exotic diffeomorphisms are detected by $\\mathbb{Z}$ or $\\mathbb{Z}/2-valued$ invariants defined usin g parameterized ASD Yang–Mills or Seiberg-Witten theory. It would seem that to detect an exotic diffeomorphism of order n one would need $\\mathbb{Z}/n-valued$ invariants of this type. Note that, for a simply connected, closed, oriented 4-manifold, the orientation-preserving topological mapping class group $\\pi_{0}(\\operatorname{Homeo}^{+}(X)))$ is isomorphic to $\\operatorname{Aut}(H_{2}(X;\\mathbb{Z}), \\lambda_{X})by$ work of Freedman [Fre82], Kreck, [Kre79], Perron [Per86], and Quinn [Qui86].\n\n(2) One could consider the following special cases.\n\n\\paragraph{Question.}\n\n(i) Is there a smooth, closed, simply connected 4-manifold X with a subgroup isomorphic to $\\mathbb{Z}/2 \\times \\mathbb{Z}/2$ in $\\ker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X))$?\n\n(ii) In particular, for X =K3\\#K3\\#K3, do the Dehn twists on the two connected sum copies of $S^{3}$ in X generate such a subgroup?\n\n(iii) More generally, is there a copy of $(\\mathbb{Z}/2)^{n,-1}$ in $\\ker(\\pi_{0}\\operatorname{Diff}(X) \\to \\pi_{0}\\operatorname{Homeo}(X))$ when X is a connected sum of n copies of the K3 surface?\n\nReferences cited:\n- [Rub98] Daniel Ruberman. An obstruction to smooth isotopy in dimension 4. Math. Res. Lett., 5(6):743–758, 1998. doi:10.4310/MRL.1998.v5.n6.a5.\n- [BK20] David Baraglia and Hokuto Konno. A gluing formula for families Seiberg-Witten invariants. Geom. Topol., 24(3):1381–1456, 2020. doi:10.2140/gt.2020.24.1381.\n- [KM20] P. B. Kronheimer and T. S. Mrowka. The Dehn twist on a sum of two K3 surfaces. Math. Res. Lett., 27(6):1767–1783, 2020. doi:10.4310/MRL.2020.v27.n6.a8.\n- [Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds. J. Differential Geometry, 17(3):357–453, 1982. http://projecteuclid.org/euclid.jdg/1214437136.\n- [Kre79] M. Kreck. Isotopy classes of diffeomorphisms of $(k-1)$-connected almostparallelizable 2k-manifolds. In Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), volume 763 of Lecture Notes in Math., pages 643– 663. Springer, Berlin, 1979.\n- [Per86] B. Perron. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. Topology, 25(4):381–397, 1986. doi:10.1016/0040-9383(86)90018-2.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No element or subgroup of the requested order greater than two in the exotic mapping-class kernel was verified.\n\n**Verified partial progress.**\n\n- The list defines the kernel as topologically trivial mapping classes.\n\n**Full solution or refutation.**\n\nThe finite-order existence question remains open.\n\n**What remains.**\n\nConstruct finite-order exotic diffeomorphisms or prove torsion restrictions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.80 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the kernel and retains both questions.\n\n**Review notes.** OCR in subgroup/kernel display was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2957, "problem_number": "KP-4.81", "title": "Kirby Problem 4.81", "statement": "Is there a smooth, closed, simply connected 4-manifold X for which the group $\\ker(\\pi_{0}(\\operatorname{Diff}(X)) \\to \\pi_{0}(\\operatorname{Homeo}(X)))$ is finitely generated?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.81.\n\nLiterature notes:\n(1) Let TDiff(X)be the group of diffeomorphisms that act trivially on $H_{*}(X;\\mathbb{Z})$. Quinn [Qui86] and Perron [Per86] proved that, for $\\pi_{1}(X) =$ \\{1\\}, the group $\\ker(\\pi_{0}(\\operatorname{Diff}(X)) \\to \\pi_{0}(\\operatorname{Homeo}(X)))$ is isomorphic to $\\pi_{0}(TDiff(X))$, called the Torelli group.\n\n(2) Ruberman [Rub99b] proved that there exist simply connected closed 4manifolds X for which $\\pi_{0}(TDiff(X))is$ not finitely generated.\n\n(3) In dimension 2, for the closed oriented surface $\\Sigma_{g}$ of genus $g \\geq$ 2, Johnson [Joh83] proved that the Torelli group is finitely generated for g >2, and Mc Cullough–Miller [MM86] proved that the Torelli group is not finitely generated for g =2.\n\nReferences cited:\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [Per86] B. Perron. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. Topology, 25(4):381–397, 1986. doi:10.1016/0040-9383(86)90018-2.\n- [Rub99b] Daniel Ruberman. A polynomial invariant of diffeomorphisms of 4-manifolds. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 473–488. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.473.\n- [Joh83] Dennis Johnson. The structure of the Torelli group. I. A finite set of generators for I. Ann. of Math. (2), 118(3):423–442, 1983. doi:10.2307/2006977.\n- [MM86] Darryl McCullough and Andy Miller. The genus 2 Torelli group is not finitely generated. Topology Appl., 22(1):43–49, 1986. doi:10.1016/0166-8641(86)90076-3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No simply connected closed 4-manifold with a verified finitely generated exotic mapping-class kernel was found in the checked source.\n\n**Verified partial progress.**\n\n- Quinn's work provides relevant topological mapping-class input.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nFind a rigid example or prove universal infinite generation.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.81 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the kernel and asks finite generation.\n\n**Review notes.** OCR in TDiff notation was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2958, "problem_number": "KP-4.82", "title": "Kirby Problem 4.82", "statement": "Let $\\phi$ be a self-diffeomorphism of a closed, simply-connected, smooth 4-manifold X. Suppose that for every smooth surface $\\Sigma$ in X, the surfaces $\\Sigma$ and $\\phi(\\Sigma)$ are smoothly isotopic.\n\n(a) Is $\\phi$ necessarily smoothly isotopic to the identity map?\n\n(b) Is $\\phi smoothly$ isotopic to a diffeomorphism that is supported on $a B^{4} \\subset X$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.82.\n\nLiterature notes:\n(1) In dimension three, a self-homeomorphism f of an irreducible 3-manifold $M^{3}$ that preserves free homotopy classes of loops is isotopic to the identity map [ABD+20]. The analogous result holds for simply connected 4-manifolds in the topological category by Perron [Per86] and independently by work of Quinn [Qui86] combined with that of Gabai–Gay– Hartman–Krushkal–Powell $[GGH^{+}23]$. (These authors show that in the topological category, the weaker hypothesis that $\\phi$ induces the identity map on $H_{2}(X;\\mathbb{Z})$ implies that $\\phi$ is topologically isotopic to the identity map.) This question essentially asks whether these theorems have an analogue in dimension four in the smooth category. The stronger hypothesis is necessary, as there are many examples of exotic self-diffeomorphisms of simply connected 4-manifolds, e.g. the Dehn twist $\\varphi$: K3\\#K3 $\\to$ K3\\#K3 is topologically but not smoothly isotopic to the identity map [KM20]. In this case, the induced map from $\\varphi$ clearly preserves $H_{2}(K3\\#K3;\\mathbb{Z})$, but it is not clear whether $\\varphi(\\Sigma)is$ smoothly isotopic to $\\Sigma$ for every surface $\\Sigma$ inside K3\\#K3.\n\n(2) The first question, in the special case $X = S^{4}$, would imply that every orientation-preserving diffeomorphism of $S^{4}$ is isotopic to the identity, answering Problem 4.67. To see that every diffeomorphism $\\phi: S^{4} \\to S^{4}$ satisfies the hypothesis of the problem, isotope $\\phi to$ fix some $B^{4}$, and then isotope $\\Sigma$ into that $B^{4}$. The second question decouples this problem from\n\nProblem 4.67.\n\n(3) Another variation on the problem allows the stronger hypothesis that for every $g \\geq$ 0, every smooth embedding $h: \\Sigma_{g} \\hookrightarrow X$ is smoothly isotopic to the embedding $\\phi \\circ h$.\n\n(4) Assuming instead that $\\phi$ becomes isotopic to the identity after connected sum with $S^{2} \\times S^{2}$ (which in particular implies that $\\phi$ induces the identity on $H_{2}(X;\\mathbb{Z}))$, then Krushkal-Mukherjee-Powell-Warren [KMPW24] showed that $\\phi$ can be isotoped so as to be supported on a contractible submanifold. Another point of view on the question asks whether the assumption that $\\Sigma$ and $\\phi(\\Sigma)$ are smoothly isotopic for every $\\Sigma$, enables us to show that the contractible supporting manifold can be assumed to be a 4-ball.\n\nReferences cited:\n- [ABD+20] Paolo Aceto, Corey Bregman, Christopher W. Davis, JungHwan Park, and Arunima Ray. Isotopy and equivalence of knots in 3-manifolds, 2020. arXiv:2007.05796.\n- [Per86] B. Perron. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique. Topology, 25(4):381–397, 1986. doi:10.1016/0040-9383(86)90018-2.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [GGH+23] David Gabai, David T. Gay, Daniel Hartman, Vyacheslav Krushkal, and Mark Powell. Pseudo-isotopies of simply connected 4-manifolds, 2023. arXiv:2311.11196.\n- [KM20] P. B. Kronheimer and T. S. Mrowka. The Dehn twist on a sum of two K3 surfaces. Math. Res. Lett., 27(6):1767–1783, 2020. doi:10.4310/MRL.2020.v27.n6.a8.\n- [KMPW24] Vyacheslav Krushkal, Anubhav Mukherjee, Mark Powell, and Terrin Warren. Corks for exotic diffeomorphisms, 2024. arXiv:2407.04696.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem was verified that surfacewise smooth-isotopy preservation forces a self-diffeomorphism to be isotopic to the identity or ball-supported.\n\n**Verified partial progress.**\n\n- The problem compares a four-dimensional analogue with lower-dimensional mapping-class phenomena.\n\n**Full solution or refutation.**\n\nBoth implications remain open.\n\n**What remains.**\n\nFind a surface-detecting mapping-class invariant or construct a hidden exotic map.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.82 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States both alternatives and retains the question.\n\n**Review notes.** OCR in part (b) was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2959, "problem_number": "KP-4.83", "title": "Kirby Problem 4.83", "statement": "For which 4-manifolds does there exist a smooth structure such that there exists a non-smoothable homeomorphism with respect to that smooth structure?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.83.\n\nLiterature notes:\n(1) We say that a homeomorphism is smoothableif it is rel. boundary isotopic to a diffeomorphism. If this does not hold, we say that the homeomorphism is non-smoothable.\n\n(2) This question is interesting for closed 4-manifolds, compact 4-manifolds with boundary, and for open 4-manifolds.\n\n(3) Recent progress for compact simply connected 4-manifolds with boundary was made by Galvin–Ladu [GL25] and Konno–Taniguchi [KT22b]. Here is an open question for such 4-manifolds. Question (i). Let X be a simply connected, smooth, spin 4-manifold with $\\partial X = Y_{1}\\cup Y_{2}$ having two connected components that do not admit generalized Dehn twists. Consider a boundary-fixing homeomorphism that has trivial Poincaré variation but that acts nontrivially on the relative spin structures of X (there are two such relative spin structures). Is it isotopic to a diffeomorphism? See [Sae06, OP25] for the definition of a Poincaré variation. If the $Y_{i}$ are hyperbolic, then they do not admit generalized Dehn twists. The answer is also yes in the case that there is a separating embedding of a 3-manifold Z, with $Y_{1}$ and $Y_{2}$ in different connected components of $X \\setminus Z$, where Z admits a generalized Dehn twist.\n\n(4) One can wonder whether there is a relationship with the existence of exotic smooth structures.\n\n\\paragraph{Question.} Does there exist a 4-manifold that admits a non-smoothable homeomorphism, but no exotic smooth structure? Or vice versa, a 4manifold admitting exotic smooth structures for which every self-homeomorphism is smoothable? See e.g. the work of Donaldson [Don90], Friedman–Morgan [FM88], and Baraglia [Bar21].\n\nReferences cited:\n- [GL25] Daniel Galvin and Roberto Ladu. Non-smoothable homeomorphisms of 4-manifolds with boundary. Adv. Math., 467:Paper No. 110191, 22, 2025. doi:10.1016/j.aim.2025.110191.\n- [KT22b] Hokuto Konno and Masaki Taniguchi. The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary. Adv. Math., 409:Paper No. 108627, 58, 2022. doi:10.1016/j.aim.2022.108627.\n- [Sae06] Osamu Saeki. Stable mapping class groups of 4-manifolds with boundary. Trans. Amer. Math. Soc., 358(5):2091–2104, 2006. doi:10.1090/S0002-9947-05-03748-7.\n- [OP25] Patrick Orson and Mark Powell. Mapping class groups of simply connected 4-manifolds with boundary. J. Differential Geom., 131(1):199–275, 2025. doi:10.4310/jdg/1755544135.\n- [Don90] S. K. Donaldson. Polynomial invariants for smooth four-manifolds. Topology, 29(3):257–315, 1990. doi:10.1016/0040-9383(90)90001-Z.\n- [FM88] Robert Friedman and John W. Morgan. On the diffeomorphism types of certain algebraic surfaces. I. J. Differential Geom., 27(2):297–369, 1988. http://projecteuclid.org/euclid.jdg/1214441784.\n- [Bar21] David Baraglia. Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants. Algebr. Geom. Topol., 21(1):317–349, 2021. doi:10.2140/agt.2021.21.317.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Non-smoothable homeomorphisms are known in special settings, but no classification of the 4-manifolds admitting a smooth structure with such a homeomorphism was verified.\n\n**Verified partial progress.**\n\n- The list defines the relative-isotopy notion and gives motivating examples.\n\n**Full solution or refutation.**\n\nThe requested classification remains open.\n\n**What remains.**\n\nRelate smoothing obstructions for homeomorphisms to the topology of X.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.83 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines non-smoothability and retains the classification problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2960, "problem_number": "KP-4.84", "title": "Kirby Problem 4.84", "statement": "Is there a closed oriented smooth 4-manifold X for which every finite subgroup Gof the mapping class group $\\pi_{0}(\\operatorname{Diff}^{+}(X))can$ be realized by a finite group action on X?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.84.\n\nLiterature notes:\n(1) This question relates to the study of the algebraic properties of diffeomorphism groups as opposed to their homotopy type. If there is a group-theoretic section $G \\to \\operatorname{Diff}^{+}(X)$ of the quotient map $\\operatorname{Diff}^{+}(X) \\to \\pi_{0}(\\operatorname{Diff}^{+}(X))$ over G, we say that G is realized by a finite group action. The question whether a given G is realized is known as the Nielsen realization problem.\n\n(2) For orientable surfaces X, Kerckhoff [Ker83] proved that every finite subgroup of $\\pi_{0}(\\operatorname{Diff}^{+}(X))$ is realized. On the other hand, in dimension 4, there are several known examples of 4-manifolds X for which there are non-realizable finite subgroups of $\\pi_{0}(\\operatorname{Diff}^{+}(X))$ [RS77, BK23, FL24a, Kon24a, KMT23b, AB25, Bar23c].\n\n(3) One may also ask the analogous questions for the extended mapping classes group of X, which also contains the orientation-reversing diffeomorphisms, and for nonorientable X.\n\n(4) The existence of asymmetric manifolds, i.e. manifolds that do not admit any effective action of a finite group, has been studied in higher dimensions. See e.g. [Pup07, CR72]. Related to the Nielsen realization problem, one may ask whether there is a smooth asymmetric 4-manifold where the mapping class group has a nontrivial finite subgroup.\n\n(5) Here is a variant of the Nielsen realization problem, and for some specific X, several non-realizability results are known. Let $\\operatorname{Aut}(H_{2}(X;\\mathbb{Z}))denote$ the automorphism group of the intersection form and set $I(X):=Im(\\operatorname{Diff}^{+}(X) \\to \\operatorname{Aut}(H_{2}(X;\\mathbb{Z})))$.\n\n\\paragraph{Question.} Which X admits $a(finite)$ group Gand a homomorphism $\\phi: G \\to I(X)$ that cannot be realized by a smooth action on X? Here we say $that\\phi isrealizedby a$ smooth action if there a homomorphism $\\tilde\\{\\phi\\}: G \\to \\operatorname{Diff}^{+}(X)$ that descends to $\\phi$.\n\nReferences cited:\n- [Ker83] Steven P. Kerckhoff. The Nielsen realization problem. Ann. of Math. (2), 117(2):235–265, 1983. doi:10.2307/2007076.\n- [RS77] Frank Raymond and Leonard L. Scott. Failure of Nielsen’s theorem in higher dimensions. Arch. Math. (Basel), 29(6):643–654, 1977. doi:10.1007/BF01220468.\n- [BK23] David Baraglia and Hokuto Konno. A note on the Nielsen realization problem for K3 surfaces. Proc. Amer. Math. Soc., 151(9):4079–4087, 2023. doi:10.1090/proc/15544.\n- [FL24a] Benson Farb and Eduard Looijenga. The Nielsen realization problem for K3 surfaces. J. Differential Geom., 127(2):505–549, 2024. doi:10.4310/jdg/1717772420.\n- [Kon24a] Hokuto Konno. Dehn twists and the Nielsen realization problem for spin 4-manifolds. Algebr. Geom. Topol., 24(3):1739–1753, 2024. doi:10.2140/agt.2024.24.1739.\n- [KMT23b] Hokuto Konno, Jin Miyazawa, and Masaki Taniguchi. Involutions, links, and Floer cohomologies, 2023. arXiv:2304.01115.\n- [AB25] Mihail Arabadji and R. İnanç Baykur. Nielsen realization in dimension four and projective twists. Adv. Math., 463:Paper No. 110112, 22, 2025. doi:10.1016/j.aim.2025.110112.\n- [Bar23c] David Baraglia. On the mapping class groups of simply-connected smooth 4-manifolds, 2023. arXiv:2310.18819.\n- [Pup07] Volker Puppe. Do manifolds have little symmetry? J. Fixed Point Theory Appl., 2(1):85–96, 2007. doi:10.1007/s11784-007-0021-x.\n- [CR72] P. E. Conner and Frank Raymond. Manifolds with few periodic homeomorphisms. In Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part II, volume Vol. 299 of Lecture Notes in Math., pages 1–75. Springer, Berlin-New York, 1972.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No closed oriented smooth 4-manifold universally realizing every finite mapping-class subgroup by an action was verified.\n\n**Verified partial progress.**\n\n- A group-theoretic section would imply the requested realization property.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nFind a manifold with a splitting mapping-class extension or an obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.84 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains the extension/section framework and retains the question.\n\n**Review notes.** OCR spacing in group notation was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2961, "problem_number": "KP-4.85", "title": "Kirby Problem 4.85", "statement": "Is there a closed orientable smooth 4-manifold X for which the identity component $Diff_{0}(X)$ of the diffeomorphism group is not uniformly perfect?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.85.\n\nLiterature notes:\n(1) By Mather and Thurston [Mat71, Mat74,Thu74a], the group $Diff_{0}(X)$ is perfect for every closed orientable manifold X.\n\n(2) For a given group G, the commutator length of an element $g \\in$ [G, G] is defined to be the minimal number of factors in expressions of g as products of commutators. The commutator length has been studied for Lie groups, groups of automorphisms of (topological, smooth, or symplectic) manifolds, and mapping class groups in contexts of dynamics and bounded cohomology. It is known to be difficult to compute the commutator length of a given element in general, which leads us to consider the following qualitative property for a perfect group (a group that is equal to its commutator subgroup). Given a perfect group G, we say that G is uniformly perfectif the commutator lengths of elements of Gare uniformly bounded.\n\n(3) For dim $X \\ne$ 2,4, $Diff_{0}(X)$ is known to be uniformly perfect due to work by Burago–Ivanov–Polterovich [BIP08] and Tsuboi [Tsu08, Tsu12]. Their works also show that $Diff_{0}(X)$ is uniformly perfect for $X = S^{2}$ and $S^{4}$. On the other hand, Bowden–Hensel–Webb [BHW22] proved that, for dim $X =$ 2, $Diff_{0}(X)$ is not uniformly perfect if the genus of X is positive. Nothing is known in dimension 4 except for $X =S^{4}$.\n\nReferences cited:\n- [Mat71] John N. Mather. The vanishing of the homology of certain groups of homeomorphisms. Topology, 10:297–298, 1971. doi:10.1016/0040-9383(71)90022-X.\n- [Mat74] John N. Mather. Commutators of diffeomorphisms. Comment. Math. Helv., 49:512– 528, 1974. doi:10.1007/BF02566746.\n- [Thu74a] William Thurston. Foliations and groups of diffeomorphisms. Bull. Amer. Math. Soc., 80:304–307, 1974. doi:10.1090/S0002-9904-1974-13475-0.\n- [BIP08] Dmitri Burago, Sergei Ivanov, and Leonid Polterovich. Conjugation-invariant norms on groups of geometric origin. In Groups of diffeomorphisms, volume 52 of Adv. Stud. Pure Math., pages 221–250. Math. Soc. Japan, Tokyo, 2008. doi:10.2969/aspm/05210221.\n- [Tsu08] Takashi Tsuboi. On the uniform perfectness of diffeomorphism groups. In Groups of diffeomorphisms, volume 52 of Adv. Stud. Pure Math., pages 505–524. Math. Soc. Japan, Tokyo, 2008. doi:10.2969/aspm/05210505.\n- [Tsu12] Takashi Tsuboi. On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds. Comment. Math. Helv., 87(1):141–185, 2012. doi:10.4171/CMH/251.\n- [BHW22] Jonathan Bowden, Sebastian Wolfgang Hensel, and Richard Webb. Quasi-morphisms on surface diffeomorphism groups. J. Amer. Math. Soc., 35(1):211–231, 2022. doi:10.1090/jams/981.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Diff_0(X) is perfect for every closed orientable X, but non-uniform perfectness for a closed orientable 4-manifold was not verified.\n\n**Verified partial progress.**\n\n- Mather--Thurston give perfectness.\n- Uniform perfectness is the stronger commutator-length condition.\n\n**Full solution or refutation.**\n\nThe requested example remains open.\n\n**What remains.**\n\nProduce unbounded commutator length or prove uniform bounds.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.85 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Contrasts known perfectness with the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2962, "problem_number": "KP-4.86", "title": "Kirby Problem 4.86", "statement": "Is it the case that for every closed, smoothable topological 4manifold X, there exists a locally linear finite group action on X, such that for every smooth structure on X, the action is non-smoothable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.86.\n\nLiterature notes:\n(1) Given a topological manifold X and a smooth structure $\\mathcal{O}$ on X, we say that a locally linear topological $action\\phi of a$ group on X isnon-smoothable with respect $to\\mathcal{O} if\\phi$ is not conjugate to a smooth action on X.\n\n(2) A variant of the problem is the following.\n\n\\paragraph{Question.} If we fix a smooth structure $\\mathcal{O}$ on X, is there a locally linear finite group action on X that is non-smoothable with respect to $\\mathcal{O}$?\n\n(3) There are many examples of locally linear finite group actions on closed 4manifolds that are non-smoothable with respect to any smooth structure, such as [KL88, KL93, HL95, Bry98, HT04, Nak09, Bar19,Bar21, Kat22].\n\nReferences cited:\n- [KL88] Slawomir Kwasik and Kyung Bai Lee. Locally linear actions on 3-manifolds. Math. Proc. Cambridge Philos. Soc., 104(2):253–260, 1988. doi:10.1017/S0305004100065427.\n- [KL93] Slawomir Kwasik and Terry Lawson. Nonsmoothable $\\mathbb{Z}_p$ actions on contractible 4-manifolds. J. Reine Angew. Math., 437:29–54, 1993. doi:10.1515/crll.1993.437.29.\n- [HL95] Ian Hambleton and Ronnie Lee. Smooth group actions on definite 4-manifolds and moduli spaces. Duke Math. J., 78(3):715–732, 1995. doi:10.1215/S0012-7094-95-07826-0.\n- [Bry98] Jim Bryan. Seiberg-Witten theory and $\\mathbb{Z}/2^p$ actions on spin 4-manifolds. Math. Res. Lett., 5(1-2):165–183, 1998. doi:10.4310/MRL.1998.v5.n2.a3.\n- [HT04] Ian Hambleton and Mihail Tanase. Permutations, isotropy and smooth cyclic group actions on definite 4-manifolds. Geom. Topol., 8:475–509, 2004. doi:10.2140/gt.2004.8.475.\n- [Nak09] Nobuhiro Nakamura. Bauer-Furuta invariants under $\\mathbb{Z}_2$-actions. Math. Z., 262(1):219–233, 2009. doi:10.1007/s00209-008-0370-1.\n- [Bar19] David Baraglia. Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory. Adv. Math., 354:106730, 32, 2019. doi:10.1016/j.aim.2019.106730.\n- [Bar21] David Baraglia. Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants. Algebr. Geom. Topol., 21(1):317–349, 2021. doi:10.2140/agt.2021.21.317.\n- [Kat22] Yuya Kato. Nonsmoothable actions of $\\mathbb{Z}_2\\times\\mathbb{Z}_2$ on spin four-manifolds. Topology Appl., 307:Paper No. 107868, 13, 2022. doi:10.1016/j.topol.2021.107868.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No locally linear finite action simultaneously non-smoothable for every smoothing of every closed smoothable topological 4-manifold was verified.\n\n**Verified partial progress.**\n\n- The list formalizes non-smoothability relative to a smoothing.\n\n**Full solution or refutation.**\n\nThe universal action assertion remains open.\n\n**What remains.**\n\nConstruct actions with smoothing-independent obstructions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.86 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the universal quantifier problem.\n\n**Review notes.** OCR in action/smoothing display was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2963, "problem_number": "KP-4.87", "title": "Kirby Problem 4.87", "statement": "Is there an exotic action of $\\mathbb{Z}/n$ on $S^{4}$ with 0-dimensional fixed point set? 1-dimensional? 2-dimensional?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.87.\n\nLiterature notes:\n(1) In the case that the fixed set is empty, any nontrivial symmetry is orientationreversing, son=2. In this case Cappell and Shaneson [CS76] constructed an exotic smooth $\\mathbb{RP}^{4}$, whose double cover is $S^{4}$ [AK79a, Gom91b]. Another construction was given by Fintushel and Stern [FS81]. It is not known whether these examples are diffeomorphic, or more generally whether there is more than one exotic $\\mathbb{RP}^{4}$. (In the topological setting there are exactly two homeomorphism classes of 4-manifolds that are homotopy equivalent to $\\mathbb{RP}^{4}$ according to a calculation from the surgery exact sequence [Wal99, Chapter 14]). These two manifolds are distinguished by their Kirby-Siebenmann invariant. As a point of interest, we refer the reader to Ruberman’s explicit construction of the non-smoothable homotopy $\\mathbb{RP}^{4}$ [Rub84, Section 2].)\n\n(2) In the case that the fixed set is 0-dimensional or 1-dimensional, very little is known in the smooth setting. In the case that the fixed set is 2-dimensional, there exist actions with knotted fixed point set [Gif66, Gor74, Sum75], but the tools for establishing this are topological. The case that the fixed point set has dimension 3 has been extensively studied (see e.g. [Maz61]); examples arise from gluing two copies of a cork along the boundary.\n\nReferences cited:\n- [CS76] Sylvain E. Cappell and Julius L. Shaneson. Some new four-manifolds. Ann. of Math. (2), 104(1):61–72, 1976. doi:10.2307/1971056.\n- [AK79a] Selman Akbulut and Robion Kirby. An exotic involution of $S^{4}$. Topology, 18(1):75– 81, 1979. doi:10.1016/0040-9383(79)90015-6.\n- [Gom91b] Robert E. Gompf. On Cappell-Shaneson 4-spheres. Topology Appl., 38(2):123–136, 1991. doi:10.1016/0166-8641(91)90079-2.\n- [FS81] Ronald Fintushel and Ronald J. Stern. An exotic free involution on $S^{4}$. Ann. of Math. (2), 113(2):357–365, 1981. doi:10.2307/2006987.\n- [Wal99] C. T. C. Wall. Surgery on compact manifolds, volume 69 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, 1999. Edited and with a foreword by A. A. Ranicki. doi:10.1090/surv/069.\n- [Rub84] Daniel Ruberman. Invariant knots of free involutions of $S^{4}$. Topology Appl., 18(2-3):217–224, 1984. doi:10.1016/0166-8641(84)90011-7.\n- [Gif66] Charles H. Giffen. The generalized Smith conjecture. Amer. J. Math., 88:187–198, 1966. doi:10.2307/2373054.\n- [Gor74] C. McA. Gordon. On the higher-dimensional Smith conjecture. Proc. London Math. Soc. (3), 29:98–110, 1974. doi:10.1112/plms/s3-29.1.98.\n- [Sum75] D. W. Sumners. Smooth $\\mathbb{Z}_p$-actions on spheres which leave knots pointwise fixed. Trans. Amer. Math. Soc., 205:193–203, 1975. doi:10.2307/1997199.\n- [Maz61] Barry Mazur. A note on some contractible 4-manifolds. Ann. of Math. (2), 73:221– 228, 1961. doi:10.2307/1970288.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exotic free involution phenomena are known, but exotic cyclic actions on S4 with each requested nonempty fixed-set dimension were not verified.\n\n**Verified partial progress.**\n\n- Cappell--Shaneson construct exotic smooth RP4-related free-involution examples.\n\n**Full solution or refutation.**\n\nThe fixed-point-set cases remain open.\n\n**What remains.**\n\nConstruct or obstruct exotic actions with 0-, 1-, or 2-dimensional fixed sets.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.87 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records free-action context and asks the fixed-set cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2964, "problem_number": "KP-4.88", "title": "Kirby Problem 4.88", "statement": "Let $\\tau: S^{4} \\to S^{4}$ be a free (hence orientation-reversing) involution. Is there an embedded $S^{2} \\subset S^{4}$ that is invariant under $\\tau$? This is equivalent to the existence of an embedded $\\mathbb{RP}^{2}$ in $S^{4}/\\tau \\cong \\mathbb{RP}^{4}$, carrying the nontrivial class in $H_{2}(\\mathbb{RP}^{4};\\mathbb{Z}/2)$. This condition on the homology class will be assumed for the rest of the problem.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.88.\n\nLiterature notes:\n(1) This is an instance of the codimension two splitting problem [CS74] and may be asked in the smooth or topological category (where one would be looking for an $\\mathbb{RP}^{2}$ with a normal bundle). Another classic phrasing is that the problem is asking if the involution desuspends twice.\n\n(2) In the topological category, there is only one 4-manifold homotopy equivalent but not homeomorphic to $\\mathbb{RP}^{4}$, which is sometimes denoted $*\\mathbb{RP}^{4}. Question(i). Does*\\mathbb{RP}^{4}$ have an embedded $\\mathbb{RP}^{2}$ with a normal bundle?\n\n(3) There are two known constructions of exotic $\\mathbb{RP}^{4}s$ in the smooth category, due to Cappell–Shaneson [CS76] and Fintushel–Stern [FS81]. By construction, the Cappell–Shaneson $\\mathbb{RP}^{4}s$ all contain an embedded $\\mathbb{RP}^{2}$. Question (ii). Do the Fintushel–Stern $\\mathbb{RP}^{4}s$ contain smoothly embedded $\\mathbb{RP}^{2}s? A$ negative answer would show that the Fintushel–Stern $\\mathbb{RP}^{4}s$ are not diffeomorphic to those obtained by the Cappell–Shaneson construction; see also Problem 4.87.\n\nReferences cited:\n- [CS74] S.E. Cappell and J. Shaneson. Homology surgery and the codimension-two placement problem. Ann. of Math., 99:277–348, 1974.\n- [CS76] Sylvain E. Cappell and Julius L. Shaneson. Some new four-manifolds. Ann. of Math. (2), 104(1):61–72, 1976. doi:10.2307/1971056.\n- [FS81] Ronald Fintushel and Ronald J. Stern. An exotic free involution on $S^{4}$. Ann. of Math. (2), 113(2):357–365, 1981. doi:10.2307/2006987.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No general invariant S2 / essential embedded RP2 was verified for an arbitrary free involution of S4.\n\n**Verified partial progress.**\n\n- The question is identified as a codimension-two splitting problem.\n\n**Full solution or refutation.**\n\nThe equivariant embedding question remains open.\n\n**What remains.**\n\nConstruct an equivariant surface or find a splitting obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.88 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains the RP2 quotient formulation and retains the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2965, "problem_number": "KP-4.89", "title": "Kirby Problem 4.89", "statement": "Classify smooth, effective circle actions on simply connected 4-manifolds with boundary.\n\n(a) Classify simply connected4-manifolds with boundary that admit circle actions.\n\n(b) For each such 4-manifold, classify the circle actions up to conjugation by diffeomorphisms.\n\n(c) Given a 4-manifold X with a circle action on its boundary, when does the action extend over X? When does it extend uniquely?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.89.\n\nLiterature notes:\n(1) Information about the smooth classification of the closed 4-manifolds admitting a circle action was obtained by Fintushel and Pao [FP77] via Pao’s replacement trick [Pao77]. (These papers assumed the truth of the 3-dimensional Poincaré conjecture, an assumption that is now known to be satisfied.) Part (a) asks for similar results in the bounded case.\n\n(2) The classification of circle actions on closed simply connected 4-manifolds was carried out in the 1970s by Fintushel [Fin77, Fin78] in terms of orbit data. Part (b)is asking for an analogous result in the bounded case. Note that in the bounded case, one necessarily has a Seifert-fibered space on the boundary, and so the orbit data is somewhat more complicated than in the closed case.\n\n(3) There are obstructions to the extension problem in part (c) coming from gauge theory. Konno–Mallick–Taniguchi [KMT23a] studied, for M equal to one of the Milnor fibers M(2,3,7)or $M(2,3,11)$, the loop in $\\pi_{1}(\\operatorname{Diff}(\\partial M))$ corresponding to the circle action coming from the Seifert-fibred structure of the boundary. This gives rise to a boundary generalized Dehn twist, a diffeomorphism of M supported in a collar neighborhood of $\\partial M$. This diffeomorphism is isotopic to the identity rel. boundary if and only if the element of $\\pi_{1}(\\operatorname{Diff}(\\partial M))$ extends to a loop in $\\pi_{1}(\\operatorname{Diff}(M))$. Konno– Mallick–Taniguchi showed that there is no such extension, and hence there is no corresponding circle action. Montague [Mon23] drew the same conclusion for the Gompf nuclei $N(2n)$ and simple plumbing $P(2n)$ with boundary $-\\Sigma(2,3,12n-5)$, by proving non-extension results for $\\mathbb{Z}/p$ actions embedded in the circle action on the boundary. All of these results continue to hold when M is replaced by $M\\#S^{2} \\times S^{2}$. Further results along these lines have been announced in [KLMME24, KPT26].\n\n\\paragraph{Question.} Does there exist a compact 4-manifold X, and a circle action on $\\partial X$, such that the latter extends to a loop of diffeomorphisms in $\\pi_{1}(\\operatorname{Diff}(X))$ but not to a circle action on X?\n\n(4) If $\\partial X$ admits a unique circle action, then an answer to (c) would follow from an answer to (a). But there could exist distinct circle actions on $\\partial X$ such that one extends over X and one does not.\n\n(5) A complementary obstruction to extending would be via a classical theorem of Atiyah–Hirzebruch [AH70] that restricts the characteristic classes of closed spin manifolds of dimension 4k admitting a circle action; they show in this case that the $A_{(}$ genus of the manifold must vanish. In dimension 4, this means that the signature vanishes.\n\n\\paragraph{Question.} Find an analogue of the Atiyah–Hirzebruch result for 4manifolds (or more generally 4k-manifolds) with nonempty boundary. It is easy to see that the signature does not necessarily vanish for spin manifolds with circle actions if the boundary is nonempty. For instance, one could take the disk bundle over $S^{2}$ with even, nonzero, Euler class. It has signature $\\pm$ 1 but supports an $S^{1}$ action. From such examples, it seems plausible that there could be some sort of boundary correction to the Atiyah–Hirzebruch argument, presumably involving an $S^{1}-equivariant \\eta-invariant$ [Don78].\n\nReferences cited:\n- [FP77] Ronald Fintushel and Peter Sie Pao. Identification of certain 4-manifolds with group actions. Proc. Amer. Math. Soc., 67(2):344–350, 1977. doi:10.2307/2041299.\n- [Pao77] Peter Sie Pao. The topological structure of 4-manifolds with effective torus actions. I. Trans. Amer. Math. Soc., 227:279–317, 1977. doi:10.2307/1997462.\n- [Fin77] Ronald Fintushel. Circle actions on simply connected 4-manifolds. Trans. Amer. Math. Soc., 230:147–171, 1977. doi:10.2307/1997715.\n- [Fin78] Ronald Fintushel. Classification of circle actions on 4-manifolds. Trans. Amer. Math. Soc., 242:377–390, 1978. doi:10.2307/1997745.\n- [KMT23a] Hokuto Konno, Abhishek Mallick, and Masaki Taniguchi. Exotic Dehn twists on 4-manifolds, 2023. arXiv:2306.08607.\n- [Mon23] Ian Montague. Non-smoothable $\\mathbb{Z}/p$-actions on nuclei, 2023. arXiv:2401.00244.\n- [KLMME24] Hokuto Konno, Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz. On four-dimensional Dehn twists and Milnor fibrations, 2024. arXiv:2409.11961.\n- [KPT26] Sungkyung Kang, JungHwan Park, and Masaki Taniguchi. Exotic Dehn twists and homotopy coherent group actions. Invent. Math., 243(1):209–241, 2026. doi:10.1007/s00222-025-01378-1.\n- [AH70] Michael Atiyah and Friedrich Hirzebruch. Spin-manifolds and group actions. In Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), pages 18–28. Springer, New York-Berlin, 1970.\n- [Don78] Harold Donnelly. Eta invariants for G-spaces. Indiana Univ. Math. J., 27(6):889– 918, 1978. doi:10.1512/iumj.1978.27.27060.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Closed simply connected circle-action manifolds have substantial classification theory, but classification with boundary and extension/uniqueness of boundary actions remains open.\n\n**Verified partial progress.**\n\n- Fintushel--Pao provide closed-case classification information.\n\n**Full solution or refutation.**\n\nThe requested boundary classification and extension criteria remain open.\n\n**What remains.**\n\nExtend orbit-space classification to boundary data.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.89 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Contrasts closed theory with the boundary problems.\n\n**Review notes.** OCR spacing in statement was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2966, "problem_number": "KP-4.90", "title": "Kirby Problem 4.90", "statement": "Do the Chern numbers $c^{2}_{1}$ and $c_{2}$ of every closed, symplectic 4–manifold X that is not a ruled surface satisfy the following?\n\n(a) $c^{2}_{1} \\leq 3c_{2}$.\n\n(b) $c^{2}_{1} =3c_{2}$ >0 if and only if $X =\\mathbb{CP}^{2}$ or X is a complex ball quotient.\n\n(c) $c_{2} \\geq$ 0.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.90.\n\nLiterature notes:\n(1) These invariants depend only on the underlying homotopy type of the 4–manifold X, satisfying the identities $c^{2}_{1} =2\\chi+3\\sigma$ and $c_{2} =\\chi$ where $\\chi$ and $\\sigma$ are the Euler characteristic and the signature of X.\n\n(2) All hold true for Kähler surfaces [BPVd V84]; part (a)is the well-known Bogomolov–Miyaoka–Yau inequality (BMY) and part (b) follows from Yau’s celebrated solution of the Calabi conjecture [Yau78].\n\n(3) The inequality (c) was conjectured to hold by Gompf. If the non-ruled symplectic 4–manifold X is minimal, then $c^{2}_{1} \\geq$ 0 by [Tau94, Tau95, Tau96, LL95]; so part (c) is implied by part (a) in this case.\n\n(4) These geographic constraints would have strong consequences. For instance, if true, part (b)would imply that there is no symplectic 4–manifold X homeomorphic but not diffeomorphic to $\\mathbb{CP}^{2}$; part (a)or (c)would imply the same for any ruled surface over $\\Sigma_{h}$, with $h \\geq$ 2.\n\n(5) If there exist symplectic 4–manifolds with $c^{2}_{1} = 3c_{2}$ that are not diffeomorphic to any Kähler surface, a further line of inquiry for part (b)would be whether there is a symplectic analog of Yau’s theorem for them; e.g., are they always K(G,1)s? See also Problem 4.91.\n\n(6) One approach to an affirmative solution for part (a) is via branched coverings. Auroux proved that every symplectic 4-manifold admits a simple branched covering to $\\mathbb{CP}^{2}$, where the branch locus in $\\mathbb{CP}^{2}$ is an immersed symplectic surface with nodes and simple cusps [Aur00]. One can compute the Chern numbers in terms of the degree of the branch locus, the number of cusps, the genus of the branch curve, and the number of sheets of the cover (see [Aur06b, p.266]). Thus, one can translate the existence of an example on or above the BMY line into the existence of a braided singular surface in $\\mathbb{CP}^{2}$ with a coloring with certain constraints.\n\n(7) One can ask part (a) more generally as follows (see also Problem 4.16):\n\n\\paragraph{Question.} Does any closed 4–manifold X with $b^{+}_{2} >$ 1 and with nontrivial Seiberg–Witten invariants satisfy the BMY inequality? Recall that by the work of Taubes [Tau94, Tau95], if such a 4– manifold X admits a symplectic structure, then it has a Seiberg-Witten basic class. Feehan and Leness have announced a program to answer this broader question affirmatively [Fee22, FL23,FL24b]. Their approach is to adapt Hitchin’s Morse theory analysis of the moduli space of Higgs monopoles on a rank-2 Hermitian vector bundle over a Riemann surface [Hit87] to the moduli space of non-Abelian monopoles $(A,\\Phi)$ on a rank-2 Hermitian vector bundle E.\n\nReferences cited:\n- [BPVdV84] W. Barth, C. Peters, and A. Van de Ven. Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1984. doi:10.1007/978-3-642-96754-2.\n- [Yau78] Shing Tung Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math., 31(3):339–411, 1978. doi:10.1002/cpa.3160310304.\n- [Tau94] Clifford Henry Taubes. The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett., 1(6):809–822, 1994. doi:10.4310/MRL.1994.v1.n6.a15.\n- [Tau95] Clifford Henry Taubes. More constraints on symplectic forms from Seiberg-Witten invariants. Math. Res. Lett., 2(1):9–13, 1995. doi:10.4310/MRL.1995.v2.n1.a2.\n- [Tau96] Clifford H. Taubes. SW ñ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. Amer. Math. Soc., 9(3):845–918, 1996. doi:10.1090/S0894-0347-96-00211-1.\n- [LL95] T. J. Li and A. Liu. Symplectic structure on ruled surfaces and a generalized adjunction formula. Math. Res. Lett., 2(4):453–471, 1995. doi:10.4310/MRL.1995.v2.n4.a6.\n- [Aur00] Denis Auroux. Symplectic 4-manifolds as branched coverings of $\\mathbb{CP}^{2}$ . Invent. Math., 139(3):551–602, 2000. doi:10.1007/s002220050019.\n- [Aur06b] Denis Auroux. Symplectic 4-manifolds, singular plane curves, and isotopy problems. In Floer homology, gauge theory, and low-dimensional topology, volume 5 of Clay Math. Proc., pages 263–276. Amer. Math. Soc., Providence, RI, 2006.\n- [Fee22] Paul M. N. Feehan. Bialynicki-Birula theory, Morse-Bott theory, and resolution of singularities for analytic spaces, 2022. arXiv:2206.14710.\n- [FL23] Paul M. N. Feehan and Thomas G. Leness. Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds, 2023. arXiv:2010.15789.\n- [FL24b] Paul M. N. Feehan and Thomas G. Leness. Almost Hermitian structures on moduli spaces of non-abelian monopoles and applications to the topology of symplectic four-manifolds, 2024. arXiv:2410.13809.\n- [Hit87] N. J. Hitchin. The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3), 55(1):59–126, 1987. doi:10.1112/plms/s3-55.1.59.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Chern-number identities and broad symplectic geography bounds are known, but the stated sharp inequalities and equality classification were not verified for all nonruled symplectic 4-manifolds.\n\n**Verified partial progress.**\n\n- c1-squared=2chi+3sigma and c2=chi constrain the problem.\n- Complex ball quotients supply equality examples.\n\n**Full solution or refutation.**\n\nNo full symplectic analogue of the stated complex inequalities was verified.\n\n**What remains.**\n\nProve the inequalities via symplectic geography or construct a counterexample.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.90 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the identities and asks the sharp general assertions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2967, "problem_number": "KP-4.91", "title": "Kirby Problem 4.91", "statement": "Present a topological construction of symplectic fake projective planes. Does there exist a symplectic fake projective plane that is not a complex ball quotient?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.91.\n\nLiterature notes:\n(1) A fake projective plane (FPP) is a closed 4–manifold X with the same rational cohomology ring as the complex projective plane $\\mathbb{CP}^{2}$ but not diffeomorphic to it. By Yau’s solution of the Calabi Conjecture [Yau78], every complex FPP is a torsion-free quotient of the complex unit ball by a discrete cocompact subgroup of $P U(1,2)$. The first example of a complex FPP was given by Mumford [Mum79] (hence the alternate name Mumford surface for a complex FPP) using p-adic uniformization, with further examples later given by Ishida and Kato [IK98] and Keum [Keu06]. Notably, Prasad and Yeung [PY07], with the help of computer-assisted calculations by Cartwright and Steeger, established that there are exactly 50 diffeomorphism types for complex FPPs [PY07, PY10, CS10]. In fact, these 50 Kähler surfaces are the only known examples of symplectic FPPs to date and their constructions involve arithmetic geometry. (Keum’s work [Keu06, Keu11] may be largely reinterpreted using topological arguments but also relies on arithmetic geometric results [Ish88].) The problem asks if there are “softer” constructions via symplectic topology.\n\n(2) A reconstruction of even the known complex FPPs using topological methods may lead to the further discovery of non-Kähler, symplectic examples on the Bogomolov–Miyaoka–Yau line. See Problem 4.90.\n\n(3) Every complex FPP is smoothly irreducible [BSS24, Proposition 6.1]; this fact extends to any other potential symplectic FPP under mild assumptions on its fundamental group. Indeed, Fintushel and Stern raised the more general question of whether there exist smoothly irreducible FPPs besides the complex ones and what could be said about their $\\pi_{1}$. (For reducible examples, one can simply take a connected sum of any rational homology 4-sphere with $\\mathbb{CP}^{2}.)$ Irreducible FPPs with various fundamental groups were recently constructed in [BSS24]; for instance, any finite abelian group with $a \\mathbb{Z}_{2}$ factor is claimed to be realized as the $\\pi_{1}$ of an irreducible FPP. None of these FPPs admit symplectic structures, and in fact, it can be shown that the approach in [BSS24] falls short of producing symplectic examples.\n\n(4) The existence of an (irreducible) FPP with trivial $\\pi_{1}$, which amounts to an exotic smooth structure on $\\mathbb{CP}^{2}$, is a particularly important open question. See Problem 4.2. Let G be a finitely presented group and Q be an integral intersection form. Baldridge and Kirk conjectured that the diffeomorphism type of a closed symplectic 4–manifold that minimizes the Euler characteristic among all with $\\pi_{1} = G$ and intersection form Q is unique [BK07, Conjecture 23]. A symplectic FPP with trivial $\\pi_{1}$ would be a counter-example to a special case of this conjecture, known as the Symplectic Poincaré\n\n\\paragraph{Conjecture.}\n\nReferences cited:\n- [Yau78] Shing Tung Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math., 31(3):339–411, 1978. doi:10.1002/cpa.3160310304.\n- [Mum79] D. Mumford. An algebraic surface with K ample, $K^2=9,\\ p_g=q=0$. Amer. J. Math., 101(1):233–244, 1979. doi:10.2307/2373947.\n- [IK98] Masa-Nori Ishida and Fumiharu Kato. The strong rigidity theorem for non-Archimedean uniformization. Tohoku Math. J. (2), 50(4):537–555, 1998. doi: 10.2748/tmj/1178224897.\n- [Keu06] JongHae Keum. A fake projective plane with an order 7 automorphism. Topology, 45(5):919–927, 2006. doi:10.1016/j.top.2006.06.006.\n- [PY07] Gopal Prasad and Sai-Kee Yeung. Fake projective planes. Invent. Math., 168(2):321–370, 2007. doi:10.1007/s00222-007-0034-5.\n- [PY10] Gopal Prasad and Sai-Kee Yeung. Addendum to “Fake projective planes” Invent. Math. 168, 321–370 (2007). Invent. Math., 182(1):213–227, 2010. doi:10.1007/s00222-010-0259-6.\n- [CS10] Donald I. Cartwright and Tim Steger. Enumeration of the 50 fake projective planes. C. R. Math. Acad. Sci. Paris, 348(1-2):11–13, 2010. doi:10.1016/j.crma.2009.11.016.\n- [Keu11] JongHae Keum. A fake projective plane constructed from an elliptic surface with multiplicities $(2,4)$. Sci. China Math., 54(8):1665–1678, 2011. doi:10.1007/s11425-011-4247-0.\n- [Ish88] Masa-Nori Ishida. An elliptic surface covered by Mumford’s fake projective plane. Tohoku Math. J. (2), 40(3):367–396, 1988. doi:10.2748/tmj/1178227980.\n- [BSS24] R. Inanc Baykur, Andras I. Stipsicz, and Zoltan Szabo. Smooth structures on fourmanifolds with finite cyclic fundamental groups, 2024. arXiv:2406.09007.\n- [BK07] Scott Baldridge and Paul Kirk. On symplectic 4-manifolds with prescribed fundamental group. Comment. Math. Helv., 82(4):845–875, 2007. doi:10.4171/CMH/112.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Complex fake projective planes are ball quotients, but no topological construction or non-complex-ball-quotient symplectic fake projective plane was verified.\n\n**Verified partial progress.**\n\n- Yau's theorem explains the complex ball-quotient statement.\n\n**Full solution or refutation.**\n\nBoth requested symplectic questions remain open.\n\n**What remains.**\n\nConstruct a symplectic FPP from topology or prove rigidity to ball quotients.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.91 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the FPP setting and retains both questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2968, "problem_number": "KP-4.92", "title": "Kirby Problem 4.92", "statement": "Is every symplectic Calabi-Yau surface diffeomorphic to either the K3 surface, the Enriques surface or $a T^{2}-bundle$ over $T^{2}$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.92.\n\nLiterature notes:\n(1) Here a symplectic Calabi-Yau surface (SCY) is defined as a closed symplectic 4-manifold with torsion first Chern class.\n\n(2) To date, the only smooth 4–manifolds known to support a symplectic structure with torsion $c_{1}$ are the ones listed in the problem. The problem asks whether this list indeed provides a complete classification of the diffeomorphism types of SCYs.\n\n(3) T-J Li [Li06a, Li06b] and Bauer [Bau08] established that any SCY has the rational homology type of one of these standard 4-manifolds. (This extends the earlier results of Morgan and Szabó [MS97] $whenb_{1}$ =0 and Ruberman and Strle [RS00] when $b_{1}$ =4.) Any SCY with $b^{+}_{2}$ >1 has only one Seiberg-Witten basic class, namely the (trivial) canonical class, and its SW invariant is one. When $b^{+}_{2}$ =1, the SW invariant of an SCY is determined by the wall crossing formula, and only depends on the cohomology ring structure.\n\n(4) The list has been verified to be complete in some specific cases. Any SCY that smoothly fibers over a circle, a surface, or a 3–manifold is standard [BF15, Bay14, FV13, LN14, Ni17b]. The same holds when the SCY admits certain finite symplectic group actions [Che20a, Che20b]. Moreover, many well-known construction techniques—such as Luttinger surgery, generalized fiber sums, knot surgery, and the simplest rational blow-downs—have been shown not to yield new SCYs from standard symplectic 4-manifolds; see [Li19] for a detailed survey and references.\n\n(5) One approach towards an affirmative answer is to first detect the existence of a (possibly singular) $T^{2}$-fibration. In the case of a homology K3, if further assuming that it has a winding family of symplectic forms (a symplectic generalization of a hyperkähler family), then a parameterized version of Taubes’ $SW\\Rightarrow GW$ theorem, including a parametrized wall-crossing formula, produces a 2–dimensional family of embedded symplectic tori in the winding family [Li10, Section 7.4].\n\n(6) One may probe the existence of new SCYs via symplectic Lefschetz pencils and multisections by analyzing certain positive factorizations in mapping class groups. New constructions of SCYs realizing all possible rational homology types are given in [BH16b, Bay22, BHM23] adapting this approach. Only a handful of these SCYs are confirmed to be standard so far [HH18a, Ful23]. Due to the shortcomings of gauge-theoretical invariants in distinguishin g SCYs (up to diffeomorphism) within the same homotopy type, the possibility of detecting a new SCY hinges on identifying a new SCY group. An intriguing, not-yet-ruled-out possibility is $\\pi_{1} =\\mathbb{Z}^{2}$ [FV13, Bay22].\n\nReferences cited:\n- [Li06a] Tian-Jun Li. Quaternionic bundles and Betti numbers of symplectic 4-manifolds with Kodaira dimension zero. Int. Math. Res. Not., pages Art. ID 37385, 28, 2006. doi:10.1155/IMRN/2006/37385.\n- [Li06b] Tian-Jun Li. Symplectic 4-manifolds with Kodaira dimension zero. J. Differential Geom., 74(2):321–352, 2006. http://projecteuclid.org/euclid.jdg/1175266207.\n- [Bau08] Stefan Bauer. Almost complex 4-manifolds with vanishing first Chern class. J. Differential Geom., 79(1):25–32, 2008. http://projecteuclid.org/euclid.jdg/1207834656.\n- [MS97] John W. Morgan and Zoltán Szabó. Homotopy K3 surfaces and mod 2 SeibergWitten invariants. Math. Res. Lett., 4(1):17–21, 1997. doi:10.4310/MRL.1997.v4.n1.a2.\n- [RS00] Daniel Ruberman and Sašo Strle. Mod 2 Seiberg-Witten invariants of homology tori. Math. Res. Lett., 7(5-6):789–799, 2000. doi:10.4310/MRL.2000.v7.n6.a11.\n- [BF15] R. İnanç Baykur and Stefan Friedl. Virtually symplectic fibered 4-manifolds. Indiana Univ. Math. J., 64(4):983–999, 2015. doi:10.1512/iumj.2015.64.5591.\n- [Bay14] R. İnanç Baykur. Virtual Betti numbers and the symplectic Kodaira dimension of fibered 4-manifolds. Proc. Amer. Math. Soc., 142(12):4377–4384, 2014. doi: 10.1090/S0002-9939-2014-12151-4.\n- [FV13] Stefan Friedl and Stefano Vidussi. On the topology of symplectic Calabi-Yau 4-manifolds. J. Topol., 6(4):945–954, 2013. doi:10.1112/jtopol/jtt020.\n- [LN14] Tian-Jun Li and Yi Ni. Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori. Math. Z., 277(1-2):195–208, 2014. doi:10.1007/s00209-013-1250-x.\n- [Ni17b] Yi Ni. Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori, II. Sci. China Math., 60(9):1591–1598, 2017. doi:10.1007/s11425-016-9052-8.\n- [Che20a] Weimin Chen. Finite group actions on symplectic Calabi-Yau 4-manifolds with $b_1>0$. J. Gökova Geom. Topol. GGT, 14:1–54, 2020.\n- [Che20b] Weimin Chen. On a class of symplectic 4-orbifolds with vanishing canonical class. J. Gökova Geom. Topol. GGT, 14:55–90, 2020.\n- [Li19] Tian-Jun Li. Kodaira dimension in low dimensional topology. In Tsinghua lectures in mathematics, volume 45 of Adv. Lect. Math. (ALM), pages 265–291. Int. Press, Somerville, MA, [2019] ©2019.\n- [Li10] Tian-Jun Li. Symplectic Calabi-Yau surfaces. In Handbook of geometric analysis, No. 3, volume 14 of Adv. Lect. Math. (ALM), pages 231–356. Int. Press, Somerville, MA, 2010.\n- [BH16b] R. İnanç Baykur and Kenta Hayano. Multisections of Lefschetz fibrations and topology of symplectic 4-manifolds. Geom. Topol., 20(4):2335–2395, 2016. doi: 10.2140/gt.2016.20.2335.\n- [Bay22] R. İnanç Baykur. Small exotic 4-manifolds and symplectic Calabi-Yau surfaces via genus-3 pencils. In Gauge theory and low-dimensional topology—progress and interaction, volume 5 of Open Book Ser., pages 185–221. Math. Sci. Publ., Berkeley, CA, 2022. https://msp.org/obs/2022/5-1/p09.xhtml.\n- [BHM23] R. İnanç Baykur, Kenta Hayano, and Naoyuki Monden. Unchaining surgery and topology of symplectic 4-manifolds. Math. Z., 303(3):Paper No. 77, 32, 2023. doi: 10.1007/s00209-023-03204-x.\n- [HH18a] Noriyuki Hamada and Kenta Hayano. Topology of holomorphic Lefschetz pencils on the four-torus. Algebr. Geom. Topol., 18(3):1515–1572, 2018. doi:10.2140/agt.2018.18.1515.\n- [Ful23] Terry Fuller. Unchaining surgery, branched covers, and pencils on elliptic surfaces. Algebr. Geom. Topol., 23(6):2867–2893, 2023. doi:10.2140/agt.2023.23.2867.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Symplectic Calabi--Yau surfaces are classified in major minimal/fundamental-group regimes, but the stated complete diffeomorphism classification remains unverified.\n\n**Verified partial progress.**\n\n- Known examples include K3, Enriques, and torus-bundle families.\n\n**Full solution or refutation.**\n\nNo universal classification theorem was verified.\n\n**What remains.**\n\nResolve remaining fundamental-group and nonminimal possibilities.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.92 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the known families and retains the universal question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2969, "problem_number": "KP-4.93", "title": "Kirby Problem 4.93", "statement": "Is every symplectic form on the standard K3 surface symplectomorphic to a Kähler form?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.93.\n\nLiterature notes:\n(1) This question has several notable generalizations. First, since every symplectic form on the K3 surface is cohomologous to a Kähler form [Li08, 4.10], this is a special case of the following broader question [Don06]: Question (Donaldson). Let (X, $\\omega_{0})$ be a closed Kähler surface. Is any other symplectic form $\\omega$ on X with $[\\omega] = [\\omega_{0}]$ and $c_{1}(X, \\omega) = c_{1}(X, \\omega_{0})$ symplectomorphic to $\\omega_{0}$? As every closed hyperkähler surface is diffeomorphic to either the K3 surface or $T^{4}$, the problem also constitutes a special case of the following conjecture.\n\n\\paragraph{Conjecture.} Let X be a closed hyperkähler surface and let $a \\in H^{2}(X;\\mathbb{R})$ be such that $a^{2} >$ 0. Then the space of symplectic forms on X representing the class a is connected. Finally, the problem is a significant instance of the more general Problem 4.96.\n\n(2) Donaldson proposed using the almost Kähler Calabi-Yau equation given in [Don06] to get a family of cohomologous symplectic forms connecting the given symplectic form $\\omega$ to the Kähler form $\\omega_{0}$. A partial a priori estimate for the almost Kähler Calabi-Yau equation was obtained in [TWY08].\n\n(3) There are also approaches to this problem via Donaldson’s geometric flow [Don00, KS19] and hypersymplectic flow [FY19].\n\nReferences cited:\n- [Li08] Tian-Jun Li. The space of symplectic structures on closed 4-manifolds. In Third International Congress of Chinese Mathematicians. Part 1, 2, volume 2 of AMS/IP Stud. Adv. Math., 42, pt. 1, pages 259–277. Amer. Math. Soc., Providence, RI, 2008.\n- [Don06] S. K. Donaldson. Two-forms on four-manifolds and elliptic equations. In Inspired by S. S. Chern, volume 11 of Nankai Tracts Math., pages 153–172. World Sci. Publ., Hackensack, NJ, 2006. URL: https://doi.org/10.1142/9789812772688 0007, doi:10.1142/9789812772688\\\\_0007.\n- [TWY08] Valentino Tosatti, Ben Weinkove, and Shing-Tung Yau. Taming symplectic forms and the Calabi-Yau equation. Proc. Lond. Math. Soc. (3), 97(2):401–424, 2008. doi:10.1112/plms/pdn008.\n- [Don00] S. K. Donaldson. Moment maps and diffeomorphisms [ MR1701920 (2001a:53122)]. In Surveys in differential geometry, volume 7 of Surv. Differ. Geom., pages 107– 127. Int. Press, Somerville, MA, 2000. doi:10.4310/SDG.2002.v7.n1.a5.\n- [KS19] Robin S. Krom and Dietmar A. Salamon. The Donaldson geometric flow for symplectic four-manifolds. J. Symplectic Geom., 17(2):381–417, 2019. doi:10.4310/JSG.2019.v17.n2.a3.\n- [FY19] Joel Fine and Chengjian Yao. A report on the hypersymplectic flow. Pure Appl. Math. Q., 15(4):1219–1260, 2019. doi:10.4310/PAMQ.2019.v15.n4.a7.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem was verified that every symplectic form on the standard K3 is symplectomorphic to a Kahler form.\n\n**Verified partial progress.**\n\n- The list relates this to broader symplectic/Kahler deformation questions.\n\n**Full solution or refutation.**\n\nThe K3 symplectomorphism question remains open.\n\n**What remains.**\n\nClassify symplectic forms on K3 up to diffeomorphism and symplectomorphism.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.93 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Retains the exact K3 question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2970, "problem_number": "KP-4.94", "title": "Kirby Problem 4.94", "statement": "Are homotopy equivalent Horikawa surfaces in different deformation classes diffeomorphic as 4–manifolds? Are they symplectomorphic?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.94.\n\nLiterature notes:\n(1) AHorikawa surfaceis a minimal (non-singular) complex projective surface of general type whose Chern numbers satisfy $5c^{2}_{1} = c_{2}$ −36 (i.e. it lies on the Noether line), or equivalently, $c^{2}_{1} = 2p_{g}$ −4, where $p_{g} \\geq$ 3 is the geometric genus. They are all simply connected. These surfaces were studied extensively by Horikawa in [Hor76b, Hor76a, Hor78, Hor79], who classified them up to deformation. For eachr $\\geq$ 2, there are two deformation classes of surfaces with $c^{2}_{1}$ =8r−8 and $c_{2} =$ 40r−4. The smooth 4–manifolds $H(r)$ and H1(r) in the two respective deformation classes are distinguished by their intersection form when r is even; whereas, when r is odd, they are homotopy equivalent. They have the same Donaldson polynomials and the same Seiberg-Witten invariants. Some 50 years after Horikawa, it remains unknown whether any pair of the latter surfaces are diffeomorphic.\n\n(2) The question on the number of diffeomorphism classes was originally raised by Horikawa in [Hor76b] and it is in [Kir97, Problem 4.101(A)]. If two compact complex manifolds X and $X^{1}$ are deformation equivalent, then there exists a diffeomorphism f: $X \\to X^{1}$ such that $f^{*}(c_{1}(X, \\omega)) = c_{1}(X^{1}, \\omega^{1})$. The bold speculation of the time, namely that the converse holds in complex dimension two (the refined “DEF $=$ DIFF Conjecture”), was disproved by Manetti in [Man01], and there are simply connected counterexamples as well [CW07]. A minimal complex surface of general type has a canonical symplectic structure, unique up to symplectomorphism, which is invariant under smooth deformation [Cat09]. Catanese showed that Manetti surfaces are indeed symplectomorphic [Cat09], so they also constitute counterexamples to the elusive “DEF $=$ SYMP Conjecture”. Nonetheless, it is still open whether DEF equivalence is more strict than DIFF or SYMP equivalence for the Horikawa surfaces.\n\n(3) Let $\\mathbb{F}_{2,r}$ denote the Hirzebruch surface with fiber f and sections $\\Delta_{0}$ and $\\Delta_{\\infty}$ with self-intersections $\\Delta^{2}_{0}$ =2rand $\\Delta^{2}_{\\infty} =$ −2r. The Horikawa surface $H(r)$ is the double cover of $\\mathbb{F}_{0}$ branched over a smoothing of $6\\Delta_{0}+4rf$ and $H^{1}(r)is$ the double cover $of\\mathbb{F}_{2,r}$ branched over a disconnected branch locus that is a smoothing of $5\\Delta_{0}+\\Delta_{\\infty}$. Based on this, one can describe H(r)and $H^{1}(r)as$ smooth 4–manifolds using Kirby diagrams for branched covers, and as symplectic 4–manifolds via monodromy factorizations for compatible Lefschetz fibrations/pencils [Ful98, Aur06a]. In [Aur06a], Auroux compared the canonical symplectic Lefschetz pencils of the same genus on $H(3)$ and H1(3) and observed that they are related through fibered Luttinger surgeries. As suggested by the status of the problem, there has been otherwise no success in relating such diagrams via Kirby calculus or the pencils via monodromy manipulations to prove DIFF or SYMP equivalence.\n\n(4) Another possible strategy to obtain a diffeomorphism between $H(r)$ and $H^{1}(r)$ is based on the observation of Manetti, that a common degeneration with Wahl singularities would prove that they are diffeomorphic [Man01][Theorem 1.5]. Recently in [MNU24], through an extensive study of complex surfaces with Wahl singularities, the authors invalidated this approach. On the other hand, a different common degeneration was recently found in [RR24], without any interpretation of moving from one deformation component to the other as diffeomorphism.\n\nReferences cited:\n- [Hor76b] Eiji Horikawa. Algebraic surfaces of general type with small $c_1^2$. I. Ann. of Math. (2), 104(2):357–387, 1976. doi:10.2307/1971050.\n- [Hor76a] Eiji Horikawa. Algebraic surfaces of general type with small $c_1^2$. II. Invent. Math., 37(2):121–155, 1976. doi:10.1007/BF01418966.\n- [Hor78] Eiji Horikawa. Algebraic surfaces of general type with small $c_1^2$. III. Invent. Math., 47(3):209–248, 1978. doi:10.1007/BF01579212.\n- [Hor79] Eiji Horikawa. Algebraic surfaces of general type with small $c_1^2$. IV. Invent. Math., 50(2):103–128, 1978/79. doi:10.1007/BF01390285.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Man01] Marco Manetti. On the moduli space of diffeomorphic algebraic surfaces. Invent. Math., 143(1):29–76, 2001. doi:10.1007/s002220000101.\n- [CW07] Fabrizio Catanese and Bronislaw Wajnryb. Diffeomorphism of simply connected algebraic surfaces. J. Differential Geom., 76(2):177–213, 2007. doi:10.4310/jdg/1180135677.\n- [Cat09] Fabrizio Catanese. Canonical symplectic structures and deformations of algebraic surfaces. Commun. Contemp. Math., 11(3):481–493, 2009. doi:10.1142/S0219199709003478.\n- [Ful98] Terry Fuller. Diffeomorphism types of genus 2 Lefschetz fibrations. Math. Ann., 311(1):163–176, 1998. doi:10.1007/s002080050182.\n- [Aur06a] Denis Auroux. The canonical pencils on Horikawa surfaces. Geom. Topol., 10:2173– 2217, 2006. doi:10.2140/gt.2006.10.2173.\n- [MNU24] Vicente Monreal, Jaime Negrete, and Giancarlo Urzúa. Classification of horikawa surfaces with t-singularities, 2024. arXiv:2410.02943.\n- [RR24] Julie Rana and Sönke Rollenske. Standard stable Horikawa surfaces. Algebr. Geom., 11(4):569–592, 2024.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No general diffeomorphism or symplectomorphism classification across distinct deformation classes of homotopy-equivalent Horikawa surfaces was verified.\n\n**Verified partial progress.**\n\n- Horikawa surfaces provide controlled complex deformation families.\n\n**Full solution or refutation.**\n\nBoth comparison questions remain open.\n\n**What remains.**\n\nCompare canonical/symplectic data across deformation components.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.94 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the surface class and retains both questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2971, "problem_number": "KP-4.95", "title": "Kirby Problem 4.95", "statement": "(a) Is there a closed hyperbolic oriented 4-manifold that admits a symplectic structure?\n\n(b) Do the Seiberg–Witten invariants vanish on every closed hyperbolic 4manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.95.\n\nLiterature notes:\n(1) In a preprint [Sto25], Stover announces the construction of a hyperbolic orbifold structure on $\\mathbb{CP}^{2}$. This resolves an orbifold version of this problem.\n\n(2) By work of Taubes [Tau94], a positive answer to (a) would imply a negative answer to (b); see [Rei06, Proposition 4.5]. By Donaldson [Don99] and Gompf–Stipsicz [GS99, Theorem 10.2.18], (a) is equivalent to askin g whether there is a closed hyperbolic oriented 4-manifold that admits a smooth Lefschetz pencil. Compare this to Problem 2.23, which asks whether there exists a surface bundle over a surface whose total space is a hyperbolic 4-manifold.\n\n(3) In [Le B02, Conjecture 1.1], Le Brun conjectures that the answer to (b) is, ‘yes,’ and hence the answer to (a) is ‘no.’ Le Brun provides evidence towards the moduli spaces being empty (for suitable perturbations) and proposes that one should consider the generalization of Seiberg-Witten invariants involving the evaluation of cohomology classes in $\\Lambda^{*}H^{1}(X;\\mathbb{Z}) \\otimes \\mathbb{Z}[U]$ on higher dimensional moduli spaces. The first examples of manifolds for which all these invariants vanish were provided in [AL20], followed by more concrete examples in [BFS24]. It is worth noting that there are no known examples of closed hyperbolic 4-manifolds with $b^{+}_{2} \\leq$ 1, cf. Problem 4.19.\n\nReferences cited:\n- [Sto25] Matthew Stover. A hyperbolic 4-orbifold with underlying space P2, 2025. arXiv: 2506.11667.\n- [Tau94] Clifford Henry Taubes. The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett., 1(6):809–822, 1994. doi:10.4310/MRL.1994.v1.n6.a15.\n- [Rei06] Alan W. Reid. Surface subgroups of mapping class groups. In Problems on mapping class groups and related topics, volume 74 of Proc. Sympos. Pure Math., pages 257– 268. Amer. Math. Soc., Providence, RI, 2006. doi:10.1090/pspum/074/2264545.\n- [Don99] S. K. Donaldson. Lefschetz pencils on symplectic manifolds. J. Differential Geom., 53(2):205–236, 1999. http://projecteuclid.org/euclid.jdg/1214425535.\n- [GS99] Robert E. Gompf and András I. Stipsicz. 4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999. doi:10.1090/gsm/020.\n- [LeB02] Claude LeBrun. Hyperbolic manifolds, harmonic forms, and Seiberg-Witten invariants. In Proceedings of the Euroconference on Partial Differential Equations and their Applications to Geometry and Physics (Castelvecchio Pascoli, 2000), volume 91, pages 137–154, 2002. doi:10.1023/A:1016222709901.\n- [AL20] Ian Agol and Francesco Lin. Hyperbolic four-manifolds with vanishing SeibergWitten invariants. In Characters in low-dimensional topology, volume 760 of Contemp. Math., pages 1–8. Amer. Math. Soc., [Providence], RI, [2020] ©2020. doi:10.1090/conm/760/15283.\n- [BFS24] Ludovico Battista, Leonardo Ferrari, and Diego Santoro. Dodecahedral L-spaces and hyperbolic 4-manifolds. Comm. Anal. Geom., 32(8):2095–2134, 2024. doi:10.4310/cag.241212004157.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hyperbolic orbifold constructions and constraints are known, but no closed hyperbolic symplectic 4-manifold or universal Seiberg--Witten vanishing theorem was verified.\n\n**Verified partial progress.**\n\n- The list records Stover's hyperbolic orbifold-surface construction.\n\n**Full solution or refutation.**\n\nBoth closed-manifold questions remain open.\n\n**What remains.**\n\nPass from orbifold constructions to closed manifolds or compute gauge invariants.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.95 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the recent orbifold advance while retaining the questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2972, "problem_number": "KP-4.96", "title": "Kirby Problem 4.96", "statement": "Does there exist a pair of symplectic 4–manifolds $(X_{1}, \\omega_{1})and (X_{2}, \\omega_{2})$, where there is a diffeomorphism $f: X_{1} \\to X_{2}$ such that $f*(c_{1}(X_{2}, \\omega_{2})) = c_{1}(X_{1}, \\omega_{1})$ and $f*([\\omega_{2}]) = [\\omega_{1}] \\in H^{2}(X_{1},\\mathbb{R})$, but $(X_{1}, \\omega_{1})$ is not symplectomorphic to $(X_{2}, \\omega_{2})$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.96.\n\nLiterature notes:\n(1) It may be difficult to detect such a difference with current invariants due to Taubes’ result relating Gromov–Witten invariants and Seiberg–Witten invariants [Tau96].\n\n(2) There is also a relative version of this problem for fillings.\n\n\\paragraph{Question.} Does there exist a pair of Stein manifolds filling the same contact3-manifold that are diffeomorphic where the diffeomorphism takes the first Chern class of one to that of the other, but such that the Stein manifolds are not symplectomorphic? One could also ask whether there is such a pair of Stein manifolds that are not Weinstein homotopic. See also Problem 5.17.\n\nReferences cited:\n- [Tau96] Clifford H. Taubes. SW ñ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. Amer. Math. Soc., 9(3):845–918, 1996. doi:10.1090/S0894-0347-96-00211-1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No pair with identical diffeomorphism, canonical-class, and cohomology-class data but inequivalent symplectic forms was verified.\n\n**Verified partial progress.**\n\n- Taubes-type invariants explain why coarse invariants may not distinguish the target.\n\n**Full solution or refutation.**\n\nThe requested pair remains open.\n\n**What remains.**\n\nDevelop finer symplectic invariants or construct a form pair.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.96 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains the invariant-matching formulation and retains the question.\n\n**Review notes.** OCR spacing around pullbacks was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2973, "problem_number": "KP-4.97", "title": "Kirby Problem 4.97", "statement": "Let $\\lambda:=c^{2}_{1}/c_{2}$ be the Chern slope of a closed, almost complex 4–manifold X. What is the supremum of $\\lambda as X$ ranges over the following families?\n\n(a) Symplectic $\\Sigma_{g}–bundles$ over $\\Sigma_{h}$, with g, $h \\geq$ 2.\n\n(b) Holomorphic $\\Sigma_{g}–bundles$ over $\\Sigma_{h}$, with g, $h \\geq$ 2.\n\n(c) Symplectic Lefschetz fibrations over $S^{2}$.\n\n(d) Holomorphic Lefschetz fibrations over $S^{2}$. (Here, the Lefschetz fibrations are assumed to have critical points.)", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.97.\n\nLiterature notes:\n(1) Recall that the Chern numbers of X are determined by its homotopy type. Any $\\Sigma_{g}–bundle$ over $\\Sigma_{h}$ with $g \\geq$ 2 or a Lefschetz fibration (with nonempty critical set) over $S^{2}$ can be made symplectic, but not always holomorphic. See e.g. [Joh86, Bay12b]. In fact, the sole obstruction to making a surface bundle over a surface holomorphic is whether the smooth 4–manifold X admits a complex structure [Hil00, Kot99].\n\n(2) The first line of research in this direction, pioneered by Atiyah, Kodaira, and Hirzebruch in the late 1960s, and Endo in the late 1990s, is to understand for which pairs of $g \\geq$ 3 and $h \\geq$ 2 one can have $a \\Sigma_{g}–bundle$ over $\\Sigma_{h}$ with $\\sigma$ >0. (The signature vanishes when $g \\leq$ 2 or $h \\leq$ 1 [Mey73].) This geography problem was nearly resolved in the recent work of Baykur and Korkmaz [BK24b], who showed that for all but 19 possible pairs (g, h), there are symplectic $\\Sigma_{g}–bundles$ over $\\Sigma_{h}$ with $\\sigma >$ 0. However, the remaining few cases are quite significant for symplectic geography; see below.\n\n(3) The Bogomolov–Miyaoka–Yau inequality for complex surfaces is encoded as $\\lambda \\leq$ 3, providing an upper bound on the slopes of X in (b) and (d). Since no complex ball quotient can smoothly fiber over a surface, it is moreover known that any X in (b) satisfies $\\lambda<3$. Two of the unsettled cases in [BK24b] have direct implications on symplectic geography: any example with (g, h) $=$ (3,2) and $\\sigma >$ 0 is a symplectic X of general type violating the BMY inequality, whereas one with(g, h) $=$ (4,2)and the smallest possible positive signature would give a symplectic X on the BMY line and cannot possibly be complex. See [BK24b, Question 2] and Problem 4.90. Do such surface bundles over surfaces exist? An unpublished preprint by Hamenstädt [Ham20] announces that $\\lambda \\leq$ 3 more generally for any X in (a).\n\n(4) Presumably, these questions will all have different answers. Nonetheless, the largest currently known slope for (a) and (b) coincide: these are the examples by Catanese and Rollenske with $\\lambda =$ 8/3 [CR09]. There exist holomorphic $\\Sigma_{g}–bundles$ over $\\Sigma_{h}$ with positive signatures, for h=2 (and largeg), as shown by Bryan and Donagi in [BD02], and with g =3 (and unspecified, presumably very large h), as shown by Kazuhiro Konno (in works only available in Japanese). The situation for (c) and (d) is more mysterious; for instance, (symplectic) Lefschetz fibrations over $S^{2}$ with $\\sigma>0$ were only recently discovered in [BH24c]. There are fewer constraints on the slopes of Lefschetz fibrations over higher genera surfaces; e.g. the Cartwright-Steger surface on the BMY line admits a holomorphic Lefschetz fibration over $T^{2}$ [KY21], realizing the maximal possible slope $\\lambda=3$ in this case.\n\nReferences cited:\n- [Joh86] F. E. A. Johnson. A class of non-Kählerian manifolds. Math. Proc. Cambridge Philos. Soc., 100(3):519–521, 1986. doi:10.1017/S030500410006624X.\n- [Bay12b] R. İnanç Baykur. Non-holomorphic surface bundles and Lefschetz fibrations. Math. Res. Lett., 19(3):567–574, 2012. doi:10.4310/MRL.2012.v19.n3.a5.\n- [Hil00] Jonathan A. Hillman. Complex surfaces which are fibre bundles. Topology Appl., 100(2-3):187–191, 2000. doi:10.1016/S0166-8641(98)00085-6.\n- [Kot99] D. Kotschick. On regularly fibered complex surfaces. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 291–298. Geom. Topol. Publ., Coventry, 1999. doi:10.2140/gtm.1999.2.291.\n- [Mey73] Werner Meyer. Die Signatur von Flächenbündeln. Math. Ann., 201:239–264, 1973. doi:10.1007/BF01427946.\n- [BK24b] R. İnanç Baykur and Mustafa Korkmaz. Geography of surface bundles over surfaces. Math. Ann., 390(4):5793–5817, 2024. doi:10.1007/s00208-024-02899-5.\n- [Ham20] Ursula Hamenstädt. Signature of surface bundles and bounded cohomology, 2020. arXiv:2011.05792.\n- [CR09] Fabrizio Catanese and Sönke Rollenske. Double Kodaira fibrations. J. Reine Angew. Math., 628:205–233, 2009. doi:10.1515/CRELLE.2009.024.\n- [BD02] Jim Bryan and Ron Donagi. Surface bundles over surfaces of small genus. Geom. Topol., 6:59–67, 2002. doi:10.2140/gt.2002.6.59.\n- [BH24c] R. İnanç Baykur and Noriyuki Hamada. Lefschetz fibrations with arbitrary signature. J. Eur. Math. Soc. (JEMS), 26(8):2837–2895, 2024. doi:10.4171/jems/1326.\n- [KY21] Vincent Koziarz and Sai-Kee Yeung. Stability of the Albanese fibration on the Cartwright-Steger surface. Taiwanese J. Math., 25(2):251–256, 2021. doi:10.11650/tjm/201108.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Chern slopes have known bounds and construction families in fibration settings, but the exact suprema in all four categories were not verified.\n\n**Verified partial progress.**\n\n- The Chern-number identities reduce slope questions to geography.\n- Known fibration constructions give lower-bound families.\n\n**Full solution or refutation.**\n\nNo complete four-way supremum computation was verified.\n\n**What remains.**\n\nEstablish sharp upper bounds and matching examples in each category.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.97 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Poses the separate symplectic/holomorphic bundle/fibration cases.\n\n**Review notes.** OCR in lambda display was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2974, "problem_number": "KP-4.98", "title": "Kirby Problem 4.98", "statement": "Does every closed symplectic 4-manifold admit inequivalent Lefschetz pencils with the same fiber genus g, for sufficiently large g? How about infinitely many?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.98.\n\nLiterature notes:\n(1) Here Lefschetz pencils are assumed to have base points, whereas Lefschetz fibrations do not. Equivalence is defined by a diffeomorphism of the symplectic 4–manifold commuting with any pair of pencil/fibration maps.\n\n(2) Any symplectic 4–manifold can be equipped with Lefschetz pencils with arbitrarily high fiber genera by increasing the degree in Donaldson’s construction. However, the number of base points also increases, meaning the Lefschetz fibrations derived by blowing up the base points would be on birational yet different 4–manifolds. Except for rational and ruled surfaces, the number of base points in a pencil is bounded above by 2g − 2, where g is the genus of the fiber. Therefore, a positive answer to the second question would imply the existence of infinitely many genus–g Lefschetz pencils with the same number of base points. By blowing up the base points of such examples, one can obtain inequivalent Lefschetz fibrations over $S^{2}$ with the same fiber genus, but the converse is not always feasible.\n\n(3) By [Bay16], any symplectic 4–manifold that is not a rational or ruled surface, possibly after blow-ups, admits arbitrarily many non-isomorphic Lefschetz pencils with the same genus, the same number of base points, and matching topological types of singular fibers. A slightly weaker result also holds for rational and ruled surfaces [Bay19]. These constructions require blow ups, addressing the problem only up to birational equivalence. Other examples of inequivalent Lefschetz pencils and fibrations on a few specific 4–manifolds are given in [PY09, PY17, BH16b, BHM23, Ham17]. The monodromy invariants and arguments used in these works do not distinguish more than finitely many classes; for a positive resolution of the problem, one needs finer monodromy invariants.\n\nReferences cited:\n- [Bay16] R. İnanç Baykur. Inequivalent Lefschetz fibrations and surgery equivalence of symplectic 4-manifolds. J. Symplectic Geom., 14(3):671–686, 2016. doi:10.4310/JSG.2016.v14.n3.a2.\n- [Bay19] R. İnanç Baykur. Inequivalent Lefschetz fibrations on rational and ruled surfaces. In Breadth in contemporary topology, volume 102 of Proc. Sympos. Pure Math., pages 21–28. Amer. Math. Soc., Providence, RI, 2019. doi:10.1090/pspum/102/02.\n- [PY09] Jongil Park and Ki-Heon Yun. Nonisomorphic Lefschetz fibrations on knot surgery 4-manifolds. Math. Ann., 345(3):581–597, 2009. doi:10.1007/s00208-009-0366-0.\n- [PY17] Jongil Park and Ki-Heon Yun. Lefschetz fibrations on knot surgery 4-manifolds via Stallings twist. Michigan Math. J., 66(3):481–498, 2017. doi:10.1307/mmj/1497513628.\n- [BH16b] R. İnanç Baykur and Kenta Hayano. Multisections of Lefschetz fibrations and topology of symplectic 4-manifolds. Geom. Topol., 20(4):2335–2395, 2016. doi: 10.2140/gt.2016.20.2335.\n- [BHM23] R. İnanç Baykur, Kenta Hayano, and Naoyuki Monden. Unchaining surgery and topology of symplectic 4-manifolds. Math. Z., 303(3):Paper No. 77, 32, 2023. doi: 10.1007/s00209-023-03204-x.\n- [Ham17] Noriyuki Hamada. Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration, 2017. arXiv:1610.08458.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem that every closed symplectic 4-manifold has inequivalent same-genus Lefschetz pencils for all sufficiently large genus was verified.\n\n**Verified partial progress.**\n\n- Lefschetz-pencil constructions give many pencils in special settings.\n\n**Full solution or refutation.**\n\nThe universal and infinite versions remain open.\n\n**What remains.**\n\nCompare monodromy factorizations to produce inequivalent pencils systematically.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.98 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the pencil distinction and retains the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2975, "problem_number": "KP-4.99", "title": "Kirby Problem 4.99", "statement": "Let X be a closed symplectic 4-manifold. Let $T \\subset X$ be a symplectic submanifold that is diffeomorphic to a 2-dimensional torus such that $[T]^{2} =$ 0. Let $X_{K}$ be a manifold obtained by Fintushel–Stern knot surgery on T using a knot $K \\subset S^{3}$. If $X_{K}$ has a symplectic structure, must K be a fibered knot?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.99.\n\nLiterature notes:\n(1) Fintushel and Stern [FS98] showed that $X_{K}$ has a symplectic structure if K is a fibered knot, and raised the question in the problem.\n\n(2) One can ask the question for the particular case when X admits a Lefschetz fibration whose regular fibers are tori and T is a regular fiber. This particular case was explicitly stated in [Ni17a]. An interesting example is the case when X is the K3 surface, then $SW(X)$ is essentially the Alexander polynomial $\\Delta_{K}$ of K [FS98], so we know $\\Delta_{K}$ should be monic if $X_{K}$ has a symplectic structure [Tau94, Tau95]. No other constraint is known.\n\n(3) The answer to this problem is “Yes” when $X = T^{2} \\times S^{2}$ and $T = T^{2} \\times$ \\{point\\} by Friedl–Vidussi [FV11], and when X is a torus bundle over a closed surface with homologically essential fibers and T is a fiber by Ni [Ni17a].\n\nReferences cited:\n- [FS98] Ronald Fintushel and Ronald J. Stern. Knots, links, and 4-manifolds. Invent. Math., 134(2):363–400, 1998. doi:10.1007/s002220050268.\n- [Ni17a] Yi Ni. Fintushel-Stern knot surgery in torus bundles. J. Topol., 10(1):164–177, 2017. doi:10.1112/topo.12002.\n- [Tau94] Clifford Henry Taubes. The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett., 1(6):809–822, 1994. doi:10.4310/MRL.1994.v1.n6.a15.\n- [Tau95] Clifford Henry Taubes. More constraints on symplectic forms from Seiberg-Witten invariants. Math. Res. Lett., 2(1):9–13, 1995. doi:10.4310/MRL.1995.v2.n1.a2.\n- [FV11] Stefan Friedl and Stefano Vidussi. Twisted Alexander polynomials detect fibered 3-manifolds. Ann. of Math. (2), 173(3):1587–1643, 2011. doi:10.4007/annals.2011.173.3.8.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Fibered knots yield symplectic Fintushel--Stern surgeries in the standard construction, but necessity of fiberedness under the stated symplectic hypothesis remains open.\n\n**Verified partial progress.**\n\n- Fintushel--Stern establish the fibered-knot sufficient condition.\n\n**Full solution or refutation.**\n\nNo converse was verified.\n\n**What remains.**\n\nExtract a fiberedness obstruction from a putative symplectic surgery.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.99 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the sufficient construction and asks the converse.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2976, "problem_number": "KP-4.100", "title": "Kirby Problem 4.100", "statement": "Given a closed, connected, symplectic 4-manifold (X, $\\omega)$ and $c \\in H_{2}(X,\\mathbb{Z})$ represented by an embedded, connected, oriented, smooth surface S such that\n\n(i) $x[\\omega]$, cy $>$ 0, and\n\n(ii) $xc_{1}(X, \\omega)$, cy $=\\chi(S) +S^{2}$, is c represented by an embedded, connected, symplectic surface?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.100.\n\nLiterature notes:\n(1) This is a special case of the question of which homology classes in a symplectic manifold are represented by symplectic surfaces. By the adjunction formula, the homology class of every embedded symplectic surface satisfies both (i) and (ii). The Symplectic Thom Conjecture [OS00] states that symplectic surfaces minimize genus in their homology classes. In their proof of this conjecture, Ozsváth and Szabó showed that every class satisfies the adjunction inequality, $-\\chi(S) \\geq S^{2}+ |xc_{1}(X, \\omega)$, cy|, provided $b^{+}_{2}(X) >$ 1. (The result extends to $b^{+}_{2}(X) =$ 1 unless X is a rational or ruled surface.) The question is whether this is a sufficient condition to conclude the existence of a symplectic representative. See also [DLW18].\n\n(2) By Donaldson [Don99], $if[\\omega] \\in H_{2}(X,\\mathbb{Z})$, then all large enough multiples $of[\\omega]are$ represented by symplectic submanifolds. $(Whenb^{+}_{2}(X)$ =1, one can say a bit more; see e.g. [Li08, Proposition 3.18].) When $b^{+}_{2}(X)$ >1, Taubes [Tau96] showed this is also true for the Poincaré dual of the canonical class $c_{1}(X, \\omega)$; also see [DS03].\n\nReferences cited:\n- [OS00] Peter Ozsváth and Zoltán Szabó. The symplectic Thom conjecture. Ann. of Math. (2), 151(1):93–124, 2000. doi:10.2307/121113.\n- [DLW18] Josef G. Dorfmeister, Tian-Jun Li, and Weiwei Wu. Stability and existence of surfaces in symplectic 4-manifolds with $b^+=1$. J. Reine Angew. Math., 742:115–155, 2018. doi:10.1515/crelle-2015-0083.\n- [Don99] S. K. Donaldson. Lefschetz pencils on symplectic manifolds. J. Differential Geom., 53(2):205–236, 1999. http://projecteuclid.org/euclid.jdg/1214425535.\n- [Li08] Tian-Jun Li. The space of symplectic structures on closed 4-manifolds. In Third International Congress of Chinese Mathematicians. Part 1, 2, volume 2 of AMS/IP Stud. Adv. Math., 42, pt. 1, pages 259–277. Amer. Math. Soc., Providence, RI, 2008.\n- [Tau96] Clifford H. Taubes. SW ñ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves. J. Amer. Math. Soc., 9(3):845–918, 1996. doi:10.1090/S0894-0347-96-00211-1.\n- [DS03] Simon Donaldson and Ivan Smith. Lefschetz pencils and the canonical class for symplectic four-manifolds. Topology, 42(4):743–785, 2003. doi:10.1016/S0040-9383(02)00024-1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact stored statement is materially corrupted and not mathematically parseable. Under the recoverable intended adjunction-equality reading, broad special-case existence theorems are known but no general sufficiency theorem or counterexample was located.\n\n**Verified partial progress.**\n\n- Ozsvath-Szabo prove the symplectic Thom theorem and the relevant adjunction inequalities.\n- Every positive-area class has a connected immersed symplectic representative, which is weaker than the requested embedded representative.\n- Donaldson represents sufficiently large integral classes near the symplectic ray by embedded symplectic submanifolds.\n- Taubes and Donaldson-Smith give canonical-class representatives, while Dorfmeister-Li-Wu prove substantial stability and existence results when b_2^+=1.\n\n**Full solution or refutation.**\n\nA provisional reading asks whether positive symplectic area plus equality in the adjunction bound for a smooth embedded representative forces an embedded symplectic representative. That reconstructed question appears open, but the exact dataset statement must first be repaired from the source.\n\n**What remains.**\n\nRestore the pairing formulas from the authoritative source, then determine whether adjunction equality and positive area are sufficient in general or find a counterexample.\n\n**Sources checked.**\n\n- R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), Problem 4.100. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Provides the recoverable pairing notation, intended adjunction-equality question, and current special-case landscape.\n- Peter Ozsvath and Zoltan Szabo, The symplectic Thom conjecture, Annals of Mathematics 151 (2000), 93-124. (primary): https://doi.org/10.2307/121113\n Evidence used: Proves genus minimality and adjunction inequalities that supply the necessary condition in the intended question.\n- Josef G. Dorfmeister, Tian-Jun Li, and Weiwei Wu, Stability and existence of surfaces in symplectic 4-manifolds with b^+=1, Journal fur die Reine und Angewandte Mathematik 742 (2018), 115-155. (primary): https://doi.org/10.1515/crelle-2015-0083\n Evidence used: Proves substantial embedded symplectic-surface existence and stability results in the b^+=1 regime.\n- Tian-Jun Li, Existence of symplectic surfaces, arXiv:0812.4929. (primary): https://arxiv.org/abs/0812.4929\n Evidence used: Proves positive-area classes have connected immersed symplectic representatives and surveys embedded-representative obstructions and constructions.\n\n**Review notes.** Material OCR defects include x[omega], cy in place of a pairing, xc_1(X,omega), cy, merged words in the background, and SW ñ Gr. Editorial repair from the authoritative source is required; the source record is preserved rather than silently repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2977, "problem_number": "KP-4.101", "title": "Kirby Problem 4.101", "statement": "Is every smooth symplectic surface in $(\\mathbb{CP}^{2}, \\omega_{FS})$ symplectically isotopic to a complex curve? Equivalently, is there a unique symplectic isotopy class of smoothly embedded symplectic surfaces of degree d in $(\\mathbb{CP}^{2}, \\omega_{FS})$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.101.\n\nLiterature notes:\n(1) Here $\\omega_{FS}$ denotes the standard Fubini–Study symplectic form on $\\mathbb{CP}^{2}$. A symplectic surface in $\\mathbb{CP}^{2}$ is said to have degree d if it represents the homology class $dh\\in H_{2}(\\mathbb{CP}^{2};\\mathbb{Z})$, where $h$ is the generator of $H_{2}(\\mathbb{CP}^{2};\\mathbb{Z}) \\cong \\mathbb{Z}$ represented by a complex projective line. Note that symplectic surfaces always represent apositive multiple of h because they have positive symplectic area.\n\n(2) A complex projective algebraic plane curve of degree d is the zero set of a homogeneous polynomial of degree d in $\\mathbb{CP}^{2}$. Such a curve represents dh $\\in H_{2}(\\mathbb{CP}^{2};\\mathbb{Z})$. If the zero set is regularly cut out, the complex algebraic curve is smooth. Any two smooth complex algebraic curves of the same degree are isotopic, i.e. connected through a family of smooth complex algebraic curves. This can be seen by observing that the set of singular curves of degree d (corresponding to homogeneous polynomials that do not regularly cut out 0) is a complex subvariety of positive complex codimension (cut out by the discriminant) in the space of all degree-d curves (a high dimensional complex projective space parametrized by the coefficients of the monomials in a degree-d homogeneous polynomial). Thus the smooth curves form a path-connected space, as it is the complement of a subset of real codimension at least 2. Thus, the two questions in the problem statement are equivalent and are known as the “symplectic isotopy problem.”\n\n(3) Key progress on the symplectic isotopy problem relies on the fact that every symplectic surface can be realized as a J-holomorphic curve for some almost complex J compatible with the symplectic form. The development of pseudoholomorphic curves originates from Gromov’s seminal paper [Gro85]. It is proven in Gromov’s work that the answer to the symplectic isotopy question is yes in degrees 1 and 2. Extensive further work using Gromov’s strategy of pseudoholomorphic curves shows that the answer is yes for degrees less than or equal to 17 [She00, Sik03, ST05]. After degree 17, we have been unable to rule out the possibility that a family of $J_{t}-holomorphic$ curves may degenerate to a singular curve with unreduced components, and it is unknown whether such degenerations can produce an unavoidable codimension-1 “wall” in the moduli space of pseudoholomorphic curves.\n\n(4) Another approach to this problem is via quasipositive factorizations in the braid group. Given a smooth symplectic surface in $\\mathbb{CP}^{2}$ realized as a J-holomorphic curve, one can generate a J-holomorphic linear pencil on $\\mathbb{CP}^{2}, \\pi: \\mathbb{CP}^{2} \\setminus$ \\{p\\} $\\to \\mathbb{CP}^{1}$. Restricting the pencil to the symplectic surface gives a simple branched covering from the surface to $\\mathbb{CP}^{1}$, assuming the pencil point p is chosen sufficiently generically. The branch points correspond to places where the fibers of the pencil are tangent to the symplectic surface. Looking at the preimages under $\\pi of$ loops in $\\mathbb{CP}^{1}$ gives a braid monodromy presentation that fully encodes the symplectic surface [MT88, MT91]. Through this, the symplectic isotopy problem can be related to the following problem in the braid group.\n\n\\paragraph{Question.} Let $\\Delta^{2}$ denote the full twist on d strands in the braid group $B_{d}$, and let $\\sigma_{i}$ denote the standard generators of the braid group (a half twist of two adjacent strands). If $\\rho_{1}$. . . $\\rho_{k} =\\Delta^{2}$ and each $\\rho_{j}$ is a conjugate of some $\\sigma_{i}$, then is there a sequence of Hurwitz moves and global conjugations one can perform on $\\rho_{1}$. . . $\\rho_{k}$ to turn it into $(\\sigma_{1}$. . . $\\sigma_{d,-,1})^{d}$? For more background on this perspective and the definition of Hurwitz moves, see this survey by Auroux [Aur06b].\n\n(5) Another strategy to attack the symplectic isotopy problem is proposed in [Sta20], which uses deformations of the smooth symplectic surface to a singular surface (a symplectic line arrangement) to show that the symplectic isotopy problem is equivalent to the existence of certain Lagrangian disks with boundary on the symplectic surface. Finding or obstructing the existence of such Lagrangians could potentially be approached using Floer theoretic/Fukaya categorical techniques.\n\n(6) Note that there are smooth surfaces in $\\mathbb{CP}^{2}$ that are not smoothly isotopic to any complex curve [Fin02, Kim06]. However, one can mix the symplectic and smooth categories for the following weakening of the symplectic isotopy problem, which is currently equally open.\n\n\\paragraph{Question.} Are any two symplectic surfaces in $\\mathbb{CP}^{2}$ of the same degree smoothly isotopic? A strategy for answering this weaker version of the problem using transverse bridge trisections was proposed by Lambert-Cole [LC23].\n\n(7) There are examples of symplectic surfaces that are homologous but not symplectically isotopic in other closed symplectic 4-manifolds [FS99b]. However, the question appears to also be open more generally for rational and ruled surfaces, and their blow-ups (symplectic manifolds of Kodaira dimension $-\\infty)$. In ruled surfaces $S^{2} \\times S^{2}, \\mathbb{CP}^{2}\\#_{n}\\mathbb{CP}^{2}$, or an $S^{2}$ bundle over a highergenus surface, in a fixed homology class, is there a unique symplectic isotopy class of symplectic surfaces?\n\nReferences cited:\n- [Gro85] M. Gromov. Pseudo holomorphic curves in symplectic manifolds. Invent. Math., 82(2):307–347, 1985. doi:10.1007/BF01388806.\n- [She00] Vsevolod Shevchishin. Pseudoholomorphic curves and the symplectic isotopy problem, 2000. arXiv:math/0010262.\n- [Sik03] Jean-Claude Sikorav. The gluing construction for normally generic J-holomorphic curves. In Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001), volume 35 of Fields Inst. Commun., pages 175–199. Amer. Math. Soc., Providence, RI, 2003. doi:10.1090/fic/035/12.\n- [ST05] Bernd Siebert and Gang Tian. On the holomorphicity of genus two Lefschetz fibrations. Ann. of Math. (2), 161(2):959–1020, 2005. doi:10.4007/annals.2005.161.959.\n- [MT88] B. Moishezon and M. Teicher. Braid group technique in complex geometry. I. Line arrangements in $\\mathbb{CP}^{2}$. In Braids (Santa Cruz, CA, 1986), volume 78 of Contemp. Math., pages 425–555. Amer. Math. Soc., Providence, RI, 1988. doi:10.1090/conm/078/975093.\n- [MT91] Boris Moishezon and Mina Teicher. Braid group technique in complex geometry. II. From arrangements of lines and conics to cuspidal curves. In Algebraic geometry (Chicago, IL, 1989), volume 1479 of Lecture Notes in Math., pages 131–180. Springer, Berlin, 1991. doi:10.1007/BFb0086269.\n- [Aur06b] Denis Auroux. Symplectic 4-manifolds, singular plane curves, and isotopy problems. In Floer homology, gauge theory, and low-dimensional topology, volume 5 of Clay Math. Proc., pages 263–276. Amer. Math. Soc., Providence, RI, 2006.\n- [Sta20] Laura Starkston. A new approach to the symplectic isotopy problem. J. Symplectic Geom., 18(3):939–960, 2020. doi:10.4310/JSG.2020.v18.n3.a11.\n- [Fin02] Sergey Finashin. Knotting of algebraic curves in $\\mathbb{CP}^{2}$. Topology, 41(1):47–55, 2002. doi:10.1016/S0040-9383(00)00023-9.\n- [Kim06] Hee Jung Kim. Modifying surfaces in 4-manifolds by twist spinning. Geom. Topol., 10:27–56, 2006. doi:10.2140/gt.2006.10.27.\n- [LC23] Peter Lambert-Cole. Symplectic surfaces and bridge position. Geom. Dedicata, 217(1):Paper No. 8, 15, 2023. doi:10.1007/s10711-022-00742-2.\n- [FS99b] Ronald Fintushel and Ronald J. Stern. Symplectic surfaces in a fixed homology class. J. Differential Geom., 52(2):203–222, 1999. http://projecteuclid.org/euclid.jdg/1214425276.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Symplectic isotopy is known in several degree/genus regimes, but uniqueness for every degree-d symplectic surface in CP2 remains open.\n\n**Verified partial progress.**\n\n- The maintained list records established special isotopy cases and the general problem.\n\n**Full solution or refutation.**\n\nNo universal symplectic-to-complex isotopy theorem was verified.\n\n**What remains.**\n\nClassify symplectic isotopy classes in every degree.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 4.101 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Maintains the general symplectic isotopy problem and known partial cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2978, "problem_number": "KP-4.102", "title": "Kirby Problem 4.102", "statement": "Is every symplectic rational cuspidal curve in $(\\mathbb{CP}^{2}, \\omega_{FS})$ equisingularly symplectically isotopic to a complex curve? More generally, which types of singular symplectic surfaces in $(\\mathbb{CP}^{2}, \\omega_{FS})$ (where type is specified by the genus of each irreducible component and the topological types of the singularities), admit symplectic representatives that are not equisingularly symplectically isotopic to complex curves?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.102.\n\nLiterature notes:\n(1) This is a singular version of the symplectic isotopy problem (Problem 4.101). Note that “curve” here refers to a real 2-dimensional surface in analogy with the terminology for complex/pseudoholomorphic curves. A rational cuspidal curve is a singular curve homeomorphic to $S^{2}$.\n\n(2) We say that two singular symplectic surfaces are equisingularly symplectically isotopic if there is a family of symplectic surfaces connecting them such that the topological type of each singularity remains constant through the isotopy. (The topological type is the isotopy class of the link of the singularity.)\n\n(3) Although one could define symplectic surfaces whose singularities are a cone on any transverse knot, for there to be any chance that the symplectic curve is equisingularly isotopic to a complex curve, its singularities must be modeled on the singularities appearing in complex plane curves. By a result of Mc Duff [Mc D92] (reproved by Micallef–White [MW95] through different methods), these are precisely the singularity types which can be realized in J-holomorphic curves for some almost complex structure J compatible with the standard symplectic form on $\\mathbb{CP}^{2}$. Thus these types of singular symplectic surfaces can be studied with pseudoholomorphic techniques.\n\n(4) There are a number of works which approach this problem for certain classes of singular surfaces and are able to prove in some cases that the symplectic surfaces with certain specified singularity types are symplectically isotopic to complex curves. For example Baurraud established symplectic isotopy results for nodal symplectic spheres [Bar99] and sufficiently generic line arrangements [Bar00]. Shevchishin proved results for nodal symplectic surfaces of sufficiently low genus [She04], conjecturing this holds more generally with a positivity assumption on the Chern number. Ohta–Ono proved uniqueness up to symplectic isotopy of a cuspidal cubic [OO05]. More generally, unicuspidal single Puiseux pair families and low degree examples of symplectic rational cuspidal curves were shown to always be equisingularly symplectically isotopic to complex curves in [GS22b, GK23], motivating the plausibility of a positive answer to the first question in the problem statement.\n\n(5) There are a number of ad hoc examples of singular symplectic surfaces in $\\mathbb{CP}^{2}$ that are not equisingularly isotopic to any complex curve. Possibly the first such examples appeared in Moishezon’s work [Moi94]. These examples were high degree and contained a very large number of nodes (positive transverse double points) and simple cusps (modeled $onz_{1}^{3} =z^{2}_{2})$. An important tool in constructing and detecting such examples is braid monodromy [MT88, KK03]. An easier and lower-degree example is the “fake Pappus” line arrangement which cannot be realized by complex projective lines, but which can be realized symplectically (see [RS19]). For an irreducible example of degree 8 with locally reducible singularities see [GS22b, Section 8]. However, we do not currently have any examples where the surface is irreducible and the singularities are locally irreducible, a.k.a., “cuspidal”.\n\n(6) One could also ask whether every pair of equisingular singular symplectic surfaces in $\\mathbb{CP}^{2}$ are equisingularly symplectically isotopic to each other. In contrast to the smooth symplectic isotopy problem, this is not equivalent to asking whether every singular symplectic surface is equisingularly symplectically isotopic to a complex curve. In fact, it is well known that there are examples of equisingular complex curves in $\\mathbb{CP}^{2}$ that are not equisingularly symplectically isotopic. Additionally there are examples of singular symplectic surfaces such that there does not exist any complex curve with the same singularities as mentioned above. Equisingular complex algebraic curves that are not equisingularly isotopic are often known as “Zariski pairs” (due to the first example being a pair of sextics discovered by Zariski [Zar29]) and are of great interest in the study of complex algebraic plane curves. (Note that Zariski pair sometimes is used to refer to examples that are not related by a weaker equivalence than equisingular isotopy, such as the fundamental groups of the complements being non-isomorphic; this implies they are not symplectically isotopic.) See [ABCT08] for an in-depth survey on Zariski pairs. For singular symplectic curves, it is interesting to ask when there are further “Zariski pairs” that can be realized symplectically than can be realized complexly.\n\nReferences cited:\n- [McD92] Dusa McDuff. Singularities of J-holomorphic curves in almost complex 4-manifolds. J. Geom. Anal., 2(3):249–266, 1992. doi:10.1007/BF02921295.\n- [MW95] Mario J. Micallef and Brian White. The structure of branch points in minimal surfaces and in pseudoholomorphic curves. Ann. of Math. (2), 141(1):35–85, 1995. doi:10.2307/2118627.\n- [Bar99] Jean-François Barraud. Nodal symplectic spheres in $\\mathbb{CP}^{2}$ with positive self-intersection. Internat. Math. Res. Notices, 1999(9):495–508, 1999. doi:10.1155/$S^{1}$073792899000252.\n- [Bar00] Jean-François Barraud. Courbes pseudo-holomorphes équisingulières en dimension 4. Bull. Soc. Math. France, 128(2):179–206, 2000. URL: http://www.numdam.org/item?id=BSMF 2000 128 2 179 0.\n- [She04] Vsevolod V. Shevchishin. On the local Severi problem. Int. Math. Res. Not., 2004(5):211–237, 2004. doi:10.1155/$S^{1}$073792804211163.\n- [OO05] Hiroshi Ohta and Kaoru Ono. Simple singularities and symplectic fillings. J. Differential Geom., 69(1):1–42, 2005. doi:10.4310/jdg/1121540338.\n- [GS22b] Marco Golla and Laura Starkston. The symplectic isotopy problem for rational cuspidal curves. Compos. Math., 158(7):1595–1682, 2022. doi:10.1112/s0010437x2200762x.\n- [GK23] Marco Golla and Fabien Kütle. Symplectic isotopy of rational cuspidal sextics and septics. Int. Math. Res. Not. IMRN, 2023(8):6504–6578, 2023. doi:10.1093/imrn/rnab364.\n- [Moi94] B. Moishezon. The arithmetic of braids and a statement of Chisini. In Geometric topology (Haifa, 1992), volume 164 of Contemp. Math., pages 151–175. Amer. Math. Soc., Providence, RI, 1994. doi:10.1090/conm/164/01591.\n- [MT88] B. Moishezon and M. Teicher. Braid group technique in complex geometry. I. Line arrangements in $\\mathbb{CP}^{2}$. In Braids (Santa Cruz, CA, 1986), volume 78 of Contemp. Math., pages 425–555. Amer. Math. Soc., Providence, RI, 1988. doi:10.1090/conm/078/975093.\n- [KK03] Vik.S̃. Kulikov and V. M. Kharlamov. On braid monodromy factorizations. Izv. Ross. Akad. Nauk Ser. Mat., 67(3):79–118, 2003. doi:10.1070/IM2003v067n03ABEH000436.\n- [RS19] Daniel Ruberman and Laura Starkston. Topological realizations of line arrangements. Int. Math. Res. Not. IMRN, 2019(8):2295–2331, 2019. doi:10.1093/imrn/rnx190.\n- [Zar29] Oscar Zariski. On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve. Amer. J. Math., 51(2):305–328, 1929. doi:10.2307/2370712.\n- [ABCT08] Enrique Artal Bartolo, José Ignacio Cogolludo, and Hiro-o Tokunaga. A survey on Zariski pairs. In Algebraic geometry in East Asia—Hanoi 2005, volume 50 of Adv. Stud. Pure Math., pages 1–100. Math. Soc. Japan, Tokyo, 2008. doi:10.2969/aspm/05010001.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Singular symplectic isotopy has positive results in selected cusp/singularity types, but no general equisingular classification or complex-isotopy theorem is known.\n\n**Verified partial progress.**\n\n- The problem refines the smooth symplectic isotopy question with fixed singularity links.\n\n**Full solution or refutation.**\n\nNo general rational-cuspidal theorem was verified.\n\n**What remains.**\n\nClassify singularity types admitting equisingular symplectic-to-complex isotopy.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 4.102 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines equisingular isotopy and retains the general problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2979, "problem_number": "KP-4.103", "title": "Kirby Problem 4.103", "statement": "(a) What polynomials can occur as the Alexander polynomials of complex plane algebraic curves?\n\n(b) More generally, what are the conditions that must be satisfied by a finitely presented group G, so that there exist a plane algebraic curve having Gas the fundamental group of its complement?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.103.\n\nLiterature notes:\n(1) Let C be an algebraic curve in $\\mathbb{C}^{2}$. The Alexander polynomial of C relative to a surjection $\\pi_{1}(\\mathbb{C}^{2} \\setminus$ C) $\\to \\mathbb{Z}$ is the characteristic polynomial the automorphism of the homology of the associated infinite cyclic cover $H1(\\mathbb{C}\\supsetneq^{2} \\setminus C,\\mathbb{C})induced$ by a generator of the group of deck transformations of the cover.\n\n(2) Since a constraint on the Alexander polynomial is also a constraint on the fundamental group, part (a) is a very special case of part (b). One of many other questions underlying the second part is: which finite groups can occur as the fundamental groups of the complements to irreducible algebraic curves in $\\mathbb{CP}^{2}$?\n\n(3) A root of the Alexander polynomial of C must be a root of the Alexander polynomial of the link of at least one of the singularities of C and also a root of the Alexander polynomial of the link at infinity, i.e., the intersection of C with a 3-sphere in $\\mathbb{C}^{2}$ of a sufficiently large radius [Lib83, Lib21]. In particular the Alexander polynomial of an algebraic curve is cyclotomic but degrees of the factors that can occur are unknown. For example, for an irreducible plane curve having only ordinary cusps and nodes as singularities (i.e., locally homeomorphic to $x^{2}+y^{3}$ =0 and $x^{2}+y^{2}$ =0 respectively) the Alexander polynomial has the for $m (t^{2}-t+1)^{r}$ and the largest known value for r is 4; see [CAL14]. Is the set of possible degrees of the Alexander polynomials of curves C of arbitrary degrees but with fixed local types of singularities bounded?\n\n(4) For a recent survey of this circle of questions and mentioned properties of the Alexander polynomials, see [Lib82]. Further developments discussed there include characteristic varieties providing a multivariable generalization of Alexander polynomials, the Alexander polynomials of the complements to ample singular divisors on projective simply connected surfaces, and the role of the Alexander polynomials in the study of Zariski pairs (see Problem 4.102).\n\n(5) It would be interesting to understand a symplectic analog of the above problems. The Alexander polynomials of the fundamental groups of the complements to pseudoholomorphic curves with isolated singularities can be similarly defined, but their characterization, and the question “are the classes of realizable polynomials different in the symplectic or algebraic context?” are open. In particular, it is unknown if the classes of the fundamental groups of the complements to plane algebraic and pseudoholomorphic curves are different. Recently, a symplectic analog of the divisibility theorem mentioned in (3) was announced in the symplectic context; see [AG24].\n\nReferences cited:\n- [Lib83] A. Libgober. Alexander invariants of plane algebraic curves. In Singularities, Part 2 (Arcata, Calif., 1981), volume 40 of Proc. Sympos. Pure Math., pages 135–143. Amer. Math. Soc., Providence, RI, 1983. doi:10.1090/pspum/040.2/713242.\n- [Lib21] Anatoly Libgober. Complements to ample divisors and singularities. In Handbook of geometry and topology of singularities II, pages 501–567. Springer, Cham, [2021] ©2021. doi:10.1007/978-3-030-78024-1\\\\_10.\n- [CAL14] Jose-Ignacio Cogolludo-Agustı́n and Anatoly Libgober. Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves. J. Reine Angew. Math., 697:15–55, 2014. doi:10.1515/crelle-2012-0096.\n- [Lib82] A. Libgober. Alexander polynomial of plane algebraic curves and cyclic multiple planes. Duke Math. J., 49(4):833–851, 1982. URL: http://projecteuclid.org/euclid.dmj/1077315533.\n- [AG24] Hanine Awada and Marco Golla. Alexander polynomials of symplectic curves and divisibility relations, 2024. arXiv:2412.15792.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Numerous necessary conditions and special realizations for plane-curve Alexander polynomials/complement groups are known, but no complete characterization is known.\n\n**Verified partial progress.**\n\n- Alexander-polynomial constraints are special cases of fundamental-group realization constraints.\n\n**Full solution or refutation.**\n\nBoth classification requests remain open.\n\n**What remains.**\n\nCharacterize realizable polynomials and finitely presented complement groups.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 4.103 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the hierarchy from polynomial to group realization.\n\n**Review notes.** OCR-corrupted displayed background notation was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2980, "problem_number": "KP-4.104", "title": "Kirby Problem 4.104", "statement": "Does there exist a transverse link $L \\subset (S^{3}, \\xi_{std})$ bounding a pair of complex curves in $B^{4} \\subset \\mathbb{C}^{2}$ that are isotopic through embedded smooth surfaces but not through complex curves? Smoothly isotopic but not symplectically? Can there be infinitely many of them?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.104.\n\nLiterature notes:\n(1) By Rudolph [Rud83] and Boileau-Orevkov [BO01], L should be a quasipositive link and the complex curves can be realized as quasipositive surfaces in $B^{4}$. The boundary braids of the latter do not necessarily need to be the same quasipositive braid representatives of K, but after Markov stabilization one can assume they are. In fact, there are transverse knots $in(S^{3}, \\xi_{std})with$ quasipositive braid representatives that bound infinitely many quasipositive surfaces that are not smoothly isotopic [BVHM18], but these are distinguished by the topology of their complements. There are also pairs that are topologically isotopic but not smoothly isotopic, announced in [Hay21a].\n\n(2) This can be regarded as a relative version of the symplectic isotopy problem; see Problem 4.101. The case of K =T(d, d), the (d, d) torus link, is equivalent to the symplectic isotopy problem. This is because the symplectic isotopy classes of a non-singular surface of degreedare in bijection with the equisingular symplectic isotopy classes of the union of the non-singular degree d curve with a generically intersecting line [GS22b, Proposition 5.1]. The complement of a neighborhood of the line is symplectomorphic to $B^{4}$ and the degree dsymplectic surface will intersect the boundary $S^{3}$ in a transverse (d, d) torus link. There may be some generalizations of this to other algebraic links besides the (d, d) torus link, by allowing the degreed surface to intersect the line tangentially or by studying degree d surfaces with singularities.\n\n(3) There are infinitely many examples of a Legendrian link $L \\subset (S^{3}, \\xi_{std})$ bounding infinitely many (exact) Lagrangians in $B^{4} \\subset \\mathbb{C}^{2}that$ are smoothly isotopic but not Hamiltonian isotopic [CG22].\n\nReferences cited:\n- [Rud83] Lee Rudolph. Algebraic functions and closed braids. Topology, 22(2):191–202, 1983. doi:10.1016/0040-9383(83)90031-9.\n- [BO01] Michel Boileau and Stepan Orevkov. Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe. C. R. Acad. Sci. Paris Sér. I Math., 332(9):825–830, 2001. doi:10.1016/S0764-4442(01)01945-0.\n- [BVHM18] R. İnanç Baykur and Jeremy Van Horn-Morris. Fillings of genus-1 open books and 4-braids. Int. Math. Res. Not. IMRN, 2018(5):1329–1346, 2018. doi:10.1093/imrn/rnw281.\n- [Hay21a] Kyle Hayden. Exotically knotted disks and complex curves, 2021. arXiv:2003.13681.\n- [GS22b] Marco Golla and Laura Starkston. The symplectic isotopy problem for rational cuspidal curves. Compos. Math., 158(7):1595–1682, 2022. doi:10.1112/s0010437x2200762x.\n- [CG22] Roger Casals and Honghao Gao. Infinitely many Lagrangian fillings. Ann. of Math. (2), 195(1):207–249, 2022. doi:10.4007/annals.2022.195.1.3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Infinitely many quasipositive surfaces can be non-smoothly-isotopic and topologically-not-smoothly-isotopic pairs are announced, but the requested smoothly isotopic yet non-complex/non-symplectic examples remain open.\n\n**Verified partial progress.**\n\n- Baykur--Van Horn-Morris-type examples distinguish surfaces by complement topology.\n- Hayden announces topologically isotopic but not smoothly isotopic pairs.\n\n**Full solution or refutation.**\n\nNo example meeting the exact smooth-isotopy contrast was verified.\n\n**What remains.**\n\nConstruct or obstruct smoothly isotopic complex curves not related complexly/symplectically.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 4.104 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes known nonisotopy phenomena from target.\n\n**Review notes.** Announcement treated conservatively.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2981, "problem_number": "KP-4.105", "title": "Kirby Problem 4.105", "statement": "Does there exist a planar contact 3-manifold that has infinitely many distinct Stein fillings?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.105.\n\nLiterature notes:\n(1) A contact 3-manifold is planar if it is supported by a planar open book decomposition.\n\n(2) There are many interesting classes of planar contact 3-manifolds, for example lens spaces and links of rational surface singularities with reduced fundamental cycle. By Wendl [Wen10], and Wendl and Niederkr¨uger [NW11], any minimal weak symplectic filling of a planar contact manifold is in fact Stein, fills any given planar open book, and hence is determined by a positive Dehn twist factorization of the monodromy of that open book. Additionally, by Plamenevskaya [Pla12] and separately Wand [Wan12], there are only finitely many ”homological types” of positive factorizations of the monodromy of a planar open book. (See [BVHM18] for an explicit statement.) Lisca [Lis08] classified the diffeomorphism types of Stein fillings of the standard contact structure on lens spaces, showing that there are only finitely many.\n\n(3) The adjective ”distinct” could be interpreted in many ways and most are interesting. One strong ”no” result would be to show there exist only finitely many Stein fillings up to diffeomorphism. Stronger would be to prove this up to symplectomorphism and deformation. Lisi and Wendl conjecture that there are only finitely many deformation classes of minimal symplectic fillings [LW21].\n\n(4) Such examples are known when the page genus is 1 and higher [OS04a] [BVHM18]. Moreover, for genus 2 and higher the Stein fillings can have arbitrarily $largeb_{2}$. This is thus related to thesupport genusof the contact manifold; see Problem 3.45.\n\nReferences cited:\n- [Wen10] Chris Wendl. Strongly fillable contact manifolds and J-holomorphic foliations. Duke Math. J., 151(3):337–384, 2010. doi:10.1215/00127094-2010-001.\n- [NW11] Klaus Niederkrüger and Chris Wendl. Weak symplectic fillings and holomorphic curves. Ann. Sci. Éc. Norm. Supér. (4), 44(5):801–853, 2011. doi:10.24033/asens.2155.\n- [Pla12] Olga Plamenevskaya. On Legendrian surgeries between lens spaces. J. Symplectic Geom., 10(2):165–181, 2012. http://projecteuclid.org/euclid.jsg/1339096433.\n- [Wan12] Andy Wand. Mapping class group relations, Stein fillings, and planar open book decompositions. J. Topol., 5(1):1–14, 2012. doi:10.1112/jtopol/jtr025.\n- [BVHM18] R. İnanç Baykur and Jeremy Van Horn-Morris. Fillings of genus-1 open books and 4-braids. Int. Math. Res. Not. IMRN, 2018(5):1329–1346, 2018. doi:10.1093/imrn/rnw281.\n- [Lis08] Paolo Lisca. On symplectic fillings of lens spaces. Trans. Amer. Math. Soc., 360(2):765–799, 2008. doi:10.1090/S0002-9947-07-04228-6.\n- [LW21] Samuel Lisi and Chris Wendl. Spine removal surgery and the geography of symplectic fillings. Michigan Math. J., 70(2):403–422, 2021. doi:10.1307/mmj/1594260053.\n- [OS04a] Burak Ozbagci and András I. Stipsicz. Contact 3-manifolds with infinitely many Stein fillings. Proc. Amer. Math. Soc., 132(5):1549–1558, 2004. doi:10.1090/S0002-9939-03-07328-3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Planar contact manifolds have strong filling/factorization restrictions, but no planar contact 3-manifold with infinitely many distinct Stein fillings was verified.\n\n**Verified partial progress.**\n\n- Wendl-type results reduce minimal fillings to positive factorizations.\n- Only finitely many homological factorization types are known.\n\n**Full solution or refutation.**\n\nThe infinite-distinct-filling existence question remains open.\n\n**What remains.**\n\nConstruct infinitely many non-diffeomorphic Stein fillings or prove finiteness.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 4.105 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States planar filling restrictions and question.\n\n**Review notes.** Open is source-backed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2982, "problem_number": "KP-4.106", "title": "Kirby Problem 4.106", "statement": "Is the exact symplectomorphism type of $T*X^{4}$ sensitive to the smooth structure on a 4-manifold X, or does it depend only on the simple-homotopy or homeomorphism type of X?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.106.\n\nLiterature notes:\n(1) Recall that the cotangent bundle $T^{*}X$ has a canonical 1-form $\\lambda_{can}$; the canonical symplectic form on $T^{*}X$ isd $\\lambda_{can}. A$ diffeomorphism $\\varphi: T^{*}X \\to T^{*}Y$ is an exact symplectomorphism if $\\varphi^{*}\\lambda_{can} =\\lambda_{can}+df$ for some function f: $X \\to \\mathbb{R}$.\n\n(2) The problem is a version of Arnol’d’s Nearby Lagrangian Conjecture, which states that any exact Lagrangian submanifold of a cotangent bundle is Hamiltonian isotopic to the 0-section. In particular, a positive solution to the Nearby Lagrangian Conjecture would imply that the exact symplectomorphism type of T*X determines the diffeomorphism type of X.\n\n(3) Evidently, the symplectomorphism type of T*X determines the homotopy type of T*X and hence of X. In fact, deep results in Floer theory show that it determines the simple homotopy type of X [AK18].\n\n(4) In higher dimensions, the symplectic structure on the cotangent bundle can detect (some) exotic smooth structures [Abo12, EKS16]. In dimension 3, a relative version of this construction—considering the unit conormal bundle to a knot—gives a strong invariant of knots [Ng05, ENS18], though of course the homotopy type of a knot complement is itself a strong knot invariant. It would be interesting to know whether this conormal construction detects exotic embeddings of surfaces in 4-manifolds.\n\nReferences cited:\n- [AK18] Mohammed Abouzaid and Thomas Kragh. Simple homotopy equivalence of nearby Lagrangians. Acta Math., 220(2):207–237, 2018. doi:10.4310/ACTA.2018.v220.n2.a1.\n- [Abo12] Mohammed Abouzaid. Framed bordism and Lagrangian embeddings of exotic spheres. Ann. of Math. (2), 175(1):71–185, 2012. doi:10.4007/annals.2012.175.1.4.\n- [EKS16] Tobias Ekholm, Thomas Kragh, and Ivan Smith. Lagrangian exotic spheres. J. Topol. Anal., 8(3):375–397, 2016. doi:10.1142/$S^{1}$793525316500199.\n- [Ng05] Lenhard Ng. Knot and braid invariants from contact homology. I. Geom. Topol., 9:247–297, 2005. doi:10.2140/gt.2005.9.247.\n- [ENS18] Tobias Ekholm, Lenhard Ng, and Vivek Shende. A complete knot invariant from contact homology. Invent. Math., 211(3):1149–1200, 2018. doi:10.1007/s00222-017-0761-1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No determination is known whether exact cotangent symplectomorphism detects exotic smooth structure in dimension four.\n\n**Verified partial progress.**\n\n- Nearby Lagrangian Conjecture would imply detection of the diffeomorphism type.\n\n**Full solution or refutation.**\n\nThe smooth/simple-homotopy/homeomorphism dependence question remains open.\n\n**What remains.**\n\nProve Nearby Lagrangian or construct exact symplectomorphic cotangent bundles of exotic manifolds.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 4.106 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains implication from Nearby Lagrangian Conjecture and open issue.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2983, "problem_number": "KP-4.107", "title": "Kirby Problem 4.107", "statement": "Problems about contact hypersurfaces:\n\n(a) Let (Y, $\\xi)$ be a contact manifold and $(\\mathbb{R} \\times Y, \\omega)$ its symplectization. Let f:Y $\\to \\mathbb{R} \\times Y$ be a smooth embedding such thatf induces an isomorphism on homology. When can we isotope f so that its image is a contact type hypersurface?\n\n(b) Special case: Is every embedded 3-sphere $Y \\subset (\\mathbb{R}^{4}, \\omega_{std})$ smoothly isotopic to a hypersurface of contact type?\n\n(c) Is there any contact rational homology sphere other than $(S^{3}, \\xi_{std})$ which embeds as a contact type hypersurface in $(\\mathbb{R}^{4}, \\omega_{std}))$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.107.\n\nLiterature notes:\n(1) There are some geometric obstructions to a hypersurface Y being isotopic to a contact type hypersurface in the ambient symplectic 4–manifold (M, $\\omega)$; e.g. [Sul76] provides a necessary condition formulated in terms of a characteristic foliation of $\\omega$ on Y and [Cie98] shows that the existence of certain concentric annuli is an obstruction.\n\n(2) Question (b) has a bearing on the Schoenflies conjecture. If it is possible to isotope a smooth embedding of $S^{3}$ to make it a contact hypersurface in $(\\mathbb{R}^{4}, \\xi_{std})$ then it bounds a symplectic filling that is necessarily the 4-ball by Gromov [Gro85]. A strategy towards trying to isotope a smoothly embedded $S^{3}$ to a contact type hypersurface has been proposed by Lambert-Cole [LC21].\n\n(3) It was conjectured by Gompf [Gom13] that no Brieskorn sphere with either orientation admits a pseudoconvex embedding in $\\mathbb{C}^{2}$. This has a bearing on question (c). Mark and Tosun proved half of this conjecture in [MT22] that no positively oriented Brieskorn sphere (with any number of singular fibers) embeds as a contact type hypersurface $of(\\mathbb{R}^{4}, \\omega std)$.\n\nReferences cited:\n- [Sul76] Dennis Sullivan. Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math., 36:225–255, 1976. doi:10.1007/BF01390011.\n- [Cie98] Kai Cieliebak. A geometric obstruction to the contact type property. Math. Z., 228(3):451–487, 1998. doi:10.1007/PL00004626.\n- [Gro85] M. Gromov. Pseudo holomorphic curves in symplectic manifolds. Invent. Math., 82(2):307–347, 1985. doi:10.1007/BF01388806.\n- [LC21] Peter Lambert-Cole. Stein trisections and homotopy 4-balls, 2021. arXiv:2104.02003.\n- [Gom13] Robert E. Gompf. Smooth embeddings with Stein surface images. J. Topol., 6(4):915–944, 2013. doi:10.1112/jtopol/jtt017.\n- [MT22] Thomas E. Mark and Bülent Tosun. On contact type hypersurfaces in 4-space. Invent. Math., 228(1):493–534, 2022. doi:10.1007/s00222-021-01083-9.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Characteristic-foliation and annulus obstructions are known, but no full contact-type-isotopy criterion for the general hypersurface/S3 questions is known.\n\n**Verified partial progress.**\n\n- Sullivan and Cieliebak give geometric obstructions.\n- A positive S3 result would bear on Schoenflies.\n\n**Full solution or refutation.**\n\nThe stated contact-hypersurface questions remain open.\n\n**What remains.**\n\nFind sufficient conditions or counterexamples to contact-type isotopy.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 4.107 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Lists obstructions and open special cases.\n\n**Review notes.** Multi-part record retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2984, "problem_number": "KP-4.108", "title": "Kirby Problem 4.108", "statement": "Let $W_{+}$ and $W_{-}$ be two 4-dimensional Liouville domains with a contactomorphism $\\Phi :\\partial W_{-} \\cong \\partial W_{+}$. This determines an $\\mathbb{R}-invariant$ contact structure $\\xi$ on $\\mathbb{R} \\times X$ where X is the gluing $X =W_{+} \\cup _{\\Phi}W_{-}$ Moreover, 0 $\\times X \\subset \\mathbb{R} \\times X$ is a convex hypersurface in the sense of Giroux.\n\n(a) Is there a 4–dimensional version of Giroux’s criterion in the case where $W_{+}$ and $W_{-}$ are Weinstein? That is, a necessary and sufficient topological criterion on $(\\Phi, W_{+}, W_{-})$ for $\\xi$ to be overtwisted.\n\n(b) Does every 4-manifold X admit a decomposition $W_{+} \\cup _{\\Gamma} W_{-}$ so that the corresponding contact structure $\\xi$ is tight (i.e. not overtwisted)?\n\n(c) Is there a 4-dimensional Liouville domain W and a contactomorphism $\\Phi$ : $\\partial W \\to \\partial W$ that extends to a diffeomorphism $\\Psi$ of W, but so that the contact structure corresponding to $(\\Phi, W$, W) is overtwisted?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.108.\n\nLiterature notes:\n(1) A hypersurface $\\Sigma \\subset Y$ of a contact manifold (Y, $\\xi)$ is convex if there is a contact vector-field V that is transverse to $\\Sigma. A$ contact manifold is overtwisted if there is an embedded codimension one disk $D_{ot}$ [BEM15] with characteristic foliation determining a standard overtwisted contact structure in a neighborhood.\n\n(2) Overtwistedness is essentially characterized by an h-principle, and the overtwisted-tight dichotomy is emblematic of the general “flexible-rigid” dichotomy in symplectic topology. The theory of convex surfaces in contact 3-manifolds was pioneered by Giroux [Gir02, GM03].\n\n(3) The foundational work on convex surfaces was extended to higher dimensions by several works of Honda-Huang [HH18b, HH19], Breen–Honda– Huang [BHH23] and Eliashberg–Pancholi [EP23]. When $W_{+}$ and $W_{-}$ are Weinstein, the data $(\\Phi, W_{+}, W_{-})$ can be described using a variant of the Weinstein-Kirby diagrams due to Gompf (c.f. Breen-Christian [BC24c]). In particular, many of the outstanding open problems in higher dimensional convex surface theory may be particularly interesting to study in dimension four.\n\n(4) On part (a): given a splitting $\\Sigma=W_{+} \\cup _{\\Phi}W_{-}$ of a closed, oriented, connected surface, Giroux’s criterion states that the resulting contact structure on $\\mathbb{R} \\times \\Sigma$ is overtwisted if and only if either $\\Sigma \\cong S^{2}$ and $\\Gamma$ has more than one component or $\\Sigma$ fi $S^{2}$ and $\\Gamma$ contains a contractible curve. A higher dimensional, algebraic analogue of this criterion has been obtained by Avdek [Avd23], but a true topological analogue is still unknown.\n\n(5) On Part (b): a celebrated result of Etnyre–Honda [EH01b] gives an example of a closed contact 3-manifold with no tight contact structures. Part (b) of this problem may be viewed as a 4-manifold analogue of that result.\n\n(6) On Part (c): this problem is motivated by the goal of finding diffeomorphisms fixed at the boundary that act nontrivially on the space of symplectic structures. Indeed, the extension $\\Psi$ of $\\Phi$ in part (c) would constitute such a map, since the double $W \\cup _{Id}W$ is always tight by Avdek [Avd23]. Moreover, such a map may be usable as a sort of symplectic cork twist by embedding W into a closed symplectic manifold X, removing W and regluing it by $\\Psi$ to acquire a new symplectic manifold X1, one may be able to find diffeomorphic, non-symplectomorphic closed 4-manifolds (see\n\nProblem 4.96).\n\nReferences cited:\n- [BEM15] Matthew Strom Borman, Yakov Eliashberg, and Emmy Murphy. Existence and classification of overtwisted contact structures in all dimensions. Acta Math., 215(2):281–361, 2015. doi:10.1007/s11511-016-0134-4.\n- [Gir02] Emmanuel Giroux. Géométrie de contact: de la dimension trois vers les dimensions supérieures. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 405–414. Higher Ed. Press, Beijing, 2002.\n- [GM03] E Giroux and JP Mohsen. Structures de contact et fibrations symplectiques audessus du cercle. Lecture notes, 2003.\n- [HH18b] Ko Honda and Yang Huang. Bypass attachments in higher-dimensional contact topology, 2018. arXiv:1803.09142.\n- [HH19] Ko Honda and Yang Huang. Convex hypersurface theory in contact topology, 2019. arXiv:1907.06025.\n- [BHH23] Joseph Breen, Ko Honda, and Yang Huang. The Giroux correspondence in arbitrary dimensions, 2023. arXiv:2307.02317.\n- [EP23] Yakov Eliashberg and Dishant M. Pancholi. Honda-Huang’s work on contact convexity revisited. In Essays in geometry—dedicated to Norbert A’Campo, volume 34 of IRMA Lect. Math. Theor. Phys., pages 453–492. EMS Press, Berlin, [2023] ©2023.\n- [BC24c] Joseph Breen and Austin Christian. Bypass moves in convex hypersurface theory, 2024. arXiv:2206.14710.\n- [Avd23] Russell Avdek. An algebraic generalization of Giroux’s criterion, 2023. arXiv:2206.14710.\n- [EH01b] John B. Etnyre and Ko Honda. On the nonexistence of tight contact structures. Ann. of Math. (2), 153(3):749–766, 2001. doi:10.2307/2661367.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Higher-dimensional convex-surface/overtwisted theory supplies a framework, but the gluing/contact-hypersurface classification requested remains open.\n\n**Verified partial progress.**\n\n- Honda--Huang and related work extend convex-surface theory beyond dimension three.\n\n**Full solution or refutation.**\n\nNo general criterion for the specified glued Liouville setting was verified.\n\n**What remains.**\n\nDevelop invariants/criteria deciding the stated contact and overtwistedness questions.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 4.108 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Provides framework and poses the higher-dimensional gluing problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2985, "problem_number": "KP-4.109", "title": "Kirby Problem 4.109", "statement": "If $\\Sigma \\subset$ (X, $\\omega)$ is a symplectic surface in a closed symplectic 4-manifold with $[\\Sigma] = P D(k[\\omega]), k \\in \\mathbb{Z}$, does (X $\\setminus \\Sigma, \\omega)$ support a Weinstein structure?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.109.\n\nLiterature notes:\n(1) In particular, one could ask this for surfaces in $(\\mathbb{CP}^{2}, \\omega_{std}). A$ negative answer to the question in this case would yield a counterexample to the symplectic isotopy problem (Problem 4.101).\n\n(2) Donaldson [Don96] and Giroux [Gir02, Gir17] prove that for suitably large k there is some symplectic surface for which this is true. However, there are not currently known bounds on k, or known examples where k =1 is not enough. In principle, large upper bounds could be obtained by carefully tracking the proof in Donaldson’s construction. Such bounds are likely much larger than needed, but any explicit bound in terms of the symplectic manifold would represent progress on this question. Obtainin g more effective bounds through new techniques would be particularly interesting.\n\nReferences cited:\n- [Don96] S. K. Donaldson. Symplectic submanifolds and almost-complex geometry. J. Differential Geom., 44(4):666–705, 1996. http://projecteuclid.org/euclid.jdg/1214459407.\n- [Gir02] Emmanuel Giroux. Géométrie de contact: de la dimension trois vers les dimensions supérieures. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 405–414. Higher Ed. Press, Beijing, 2002.\n- [Gir17] Emmanuel Giroux. Remarks on Donaldson’s symplectic submanifolds. Pure Appl. Math. Q., 13(3):369–388, 2017. doi:10.4310/pamq.2017.v13.n3.a1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Donaldson--Giroux prove Weinstein complements for some sufficiently large multiples, but no effective bound or every-surface theorem is known.\n\n**Verified partial progress.**\n\n- For suitably large k there exists an appropriate symplectic surface with Weinstein complement.\n\n**Full solution or refutation.**\n\nThis does not settle the given arbitrary surface/class question.\n\n**What remains.**\n\nObtain effective k bounds and decide the k=1/arbitrary-surface cases.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem 4.109 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States large-k theorem and absence of effective bounds.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2986, "problem_number": "KP-4.110", "title": "Kirby Problem 4.110", "statement": "Does there exist a 2-handlebody W that admits an exact symplectic structure with convex contact boundary that does not admit a Weinstein structure filling the same contact boundary?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.110.\n\nLiterature notes:\n(1) So far, all known examples of exact symplectic manifolds that we can obstruct from having a Weinstein structure are known to have handle structures requiring 3-handles [Mc D91, Bow12].\n\n(2) We have limited tools for obstructing an exact filling from being Weinstein when it has the topology of a 2-handlebody. Bowden’s argument in [Bow12] utilized a theorem of Eliashberg which says that a Weinstein fillin g of a connected sum is necessarily a boundary sum of Weinstein fillings of the summands [Eli90]. A potential strategy to generalize this obstruction to other examples may be to use Menke’s “mixed tori” [CM19], which gives a different decomposition criterion for Weinstein fillings.\n\nReferences cited:\n- [McD91] Dusa McDuff. Symplectic manifolds with contact type boundaries. Invent. Math., 103(3):651–671, 1991. doi:10.1007/BF01239530.\n- [Bow12] Jonathan Bowden. Exactly fillable contact structures without Stein fillings. Algebr. Geom. Topol., 12(3):1803–1810, 2012. doi:10.2140/agt.2012.12.1803.\n- [Eli90] Yakov Eliashberg. Filling by holomorphic discs and its applications. In Geometry of low-dimensional manifolds, 2 (Durham, 1989), volume 151 of London Math. Soc. Lecture Note Ser., pages 45–67. Cambridge Univ. Press, Cambridge, 1990.\n- [CM19] Austin Christian and Michael Menke. Splitting symplectic fillings, 2019. arXiv: 1909.00420.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Known exact symplectic non-Weinstein examples have handle descriptions requiring 3-handles; no requested exact 2-handlebody example was verified.\n\n**Verified partial progress.**\n\n- Known obstructions include examples with 3-handles.\n- The maintained list identifies the missing 2-handlebody obstruction.\n\n**Full solution or refutation.**\n\nThe existential 2-handlebody question remains open.\n\n**What remains.**\n\nConstruct such a filling or prove Weinstein existence for exact 2-handlebodies.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.110 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the known 3-handle examples and retains the 2-handlebody question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2987, "problem_number": "KP-4.111", "title": "Kirby Problem 4.111", "statement": "Is trisection genus additive? In other words, must it be the case that $g(X\\#X1)$ =g(X) +g(X1).", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.111.\n\nLiterature notes:\n(1) This is Conjecture 1.6 of [LCM22].\n\n(2) The trisection genus of a 4–manifold X is $g(X)$ =min\\{g|X admits a genus g trisection\\}.\n\n(3) $If\\mathfrak{T}and\\mathfrak{T}1$ are trisections for X and X1 of genusgandg1, respectively, then $\\mathfrak{T}\\#\\mathfrak{T}1$ is a trisection of genus g+g1 for X\\#X1. It follows that trisection genus satisfies: $g(X\\#X^{1}) \\leq g(X) +g(X^{1})$. Equality holds when the lower-bound on $g(X)$ and $g(X1)$ coming from standard algebraic topology is sharp; see Problem 4.117.\n\n(4) A positive resolution to this problem would have sweeping consequences. If trisection genus is additive, then no manifold from the following set has an exotic copy: $\\mathcal{M}= {S^{4},\\mathbb{CP}^{2}, S^{1} \\times S^{3},2 \\mathbb{CP}^{2},\\mathbb{CP}^{2}\\#\\mathbb{CP}^{2}, S^{2} \\times S^{2}}$. This follows from the classification of trisections up to genus two [GK16, MZ17b] and the observation that if trisection genus is additive, homeomorphic smooth four-manifolds have the same trisection genus; see Proposition 1.7 of [LCM22], where modest evidence for this phenomenon is given; see also Remark 5 of Problem 4.113. In particular, if trisection genus is additive, then any exotic copy of $\\mathbb{CP}^{2}\\#2 \\mathbb{CP}^{2}$ admits a genus three trisection. Such exotic copies are known to exist [AP10, FS11]. This motivates the important problem of classifying genus three trisections; see Problem 4.113.\n\n(5) There is an analogous problem for orientable surface-links in $S^{4}$ that is related to the Meridional Rank Conjecture for orientable surface-links [JP25]; see Problem 1.18. For more details and precise definitions, see [MZ17a] and [AAD+23, Question 6.1]. Given an orientable surface-link $\\mathcal{K} \\subset S^{4}, letp(\\mathcal{K})denote$ itspatch number, the minimum valuepsuch $that\\mathcal{K}admits a$ (b;p, p1, p2)–bridge trisection.\n\n\\paragraph{Question.} Is patch number(−1)–additive for orientable surface-links? In other words, must it be the case that $p(\\mathcal{K}_{1}\\#\\mathcal{K}_{2}) =p(\\mathcal{K}_{1}) +p(\\mathcal{K}_{2})$ −1 for orientable surface-links $\\mathcal{K}_{1}$ and $\\mathcal{K}_{2}$?\n\nReferences cited:\n- [LCM22] Peter Lambert-Cole and Jeffrey Meier. Bridge trisections in rational surfaces. J. Topol. Anal., 14(3):655–708, 2022. doi:10.1142/$S^{1}$793525321500047.\n- [GK16] David Gay and Robion Kirby. Trisecting 4-manifolds. Geom. Topol., 20(6):3097– 3132, 2016. doi:10.2140/gt.2016.20.3097.\n- [MZ17b] Jeffrey Meier and Alexander Zupan. Genus-two trisections are standard. Geom. Topol., 21(3):1583–1630, 2017. doi:10.2140/gt.2017.21.1583.\n- [AP10] Anar Akhmedov and B. Doug Park. Exotic smooth structures on small 4-manifolds with odd signatures. Invent. Math., 181(3):577–603, 2010. doi:10.1007/s00222-010-0254-y.\n- [FS11] Ronald Fintushel and Ronald J. Stern. Pinwheels and nullhomologous surgery on 4-manifolds with $b^+=1$. Algebr. Geom. Topol., 11(3):1649–1699, 2011. doi:10.2140/agt.2011.11.1649.\n- [JP25] Jason Joseph and Puttipong Pongtanapaisan. Meridional rank and bridge number of knotted 2-spheres. Canad. J. Math., 77(1):282–299, 2025. doi:10.4153/S0008414X23000883.\n- [MZ17a] Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in $S^{4}$. Trans. Amer. Math. Soc., 369(10):7343–7386, 2017. doi:10.1090/tran/6934.\n- [AAD+23] Wolfgang Allred, Manuel Aragón, Zack Dooley, Alexander Goldman, Yucong Lei, Isaiah Martinez, Nicholas Meyer, Devon Peters, Scott Warrander, Ana Wright, and Alexander Zupan. Tri-plane diagrams for simple surfaces in $S^{4}$. J. Knot Theory Ramifications, 32(6):Paper No. 2350041, 28, 2023. doi:10.1142/S0218216523500414.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Connected sum gives subadditivity, and equality is known when standard lower bounds are sharp; general additivity remains open.\n\n**Verified partial progress.**\n\n- Connected sum gives g(X#Y) <= g(X)+g(Y).\n- Known lower-bound-sharp cases give equality.\n\n**Full solution or refutation.**\n\nNo general additivity theorem or counterexample was verified.\n\n**What remains.**\n\nProve a reverse inequality or find a connected-sum counterexample.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.111 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the inequality, known equality regime, and Conjecture 1.6.\n\n**Review notes.** Source display loses primes/spaces in X1 notation; not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2988, "problem_number": "KP-4.112", "title": "Kirby Problem 4.112", "statement": "Is every trisection of the 4-sphere with positive genus a stabilization of the genus zero trisection?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.112.\n\nLiterature notes:\n(1) This is Conjecture 3.11 of [MSZ16]. The analogous result regarding Heegaard splittings of the 3–sphere is Waldhausen’s Theorem [Wal68a]. Connections between this problem, the Generalized Property R Conjecture (Problem 1.10), and the Andrews–Curtis Conjecture (Problem 5.10) are detailed in [MZ18] and elaborated in [MZ22]. In particular, if every positive-genus trisection of $S^{4}$ is stabilized, then well-known potential counter-examples (dating back to [AK85]) to the two conjectures mentioned above are not, in fact, counter-examples.\n\n(2) The following related question may be more tractable and of independent interest [MZ17a]. It is a four-dimensional analog of the classical result of Otal that says the unknot has a unique bridge splitting for each bridge number [Ota82].\n\n\\paragraph{Question.} Is every b–bridge trisection of the unknotted 2–sphere in $S^{4}$ with b >1 a perturbation of the 1–bridge trisection?\n\nReferences cited:\n- [MSZ16] Jeffrey Meier, Trent Schirmer, and Alexander Zupan. Classification of trisections and the generalized property R conjecture. Proc. Amer. Math. Soc., 144(11):4983– 4997, 2016. doi:10.1090/proc/13105.\n- [Wal68a] Friedhelm Waldhausen. Heegaard-Zerlegungen der 3-Sphäre. Topology, 7:195–203, 1968. doi:10.1016/0040-9383(68)90027-X.\n- [MZ18] Jeffrey Meier and Alexander Zupan. Characterizing Dehn surgeries on links via trisections. Proc. Natl. Acad. Sci. USA, 115(43):10887–10893, 2018. doi:10.1073/pnas.1717187115.\n- [MZ22] Jeffrey Meier and Alexander Zupan. Generalized square knots and homotopy 4-spheres. J. Differential Geom., 122(1):69–129, 2022. doi:10.4310/jdg/1668186788.\n- [AK85] Selman Akbulut and Robion Kirby. A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture, and the AndrewsCurtis conjecture. Topology, 24(4):375–390, 1985. doi:10.1016/0040-9383(85) 90010-2.\n- [MZ17a] Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in $S^{4}$. Trans. Amer. Math. Soc., 369(10):7343–7386, 2017. doi:10.1090/tran/6934.\n- [Ota82] Jean-Pierre Otal. Présentations en ponts du nœud trivial. C. R. Acad. Sci. Paris Sér. I Math., 294(16):553–556, 1982.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Stable equivalence and selected standard diagram families are known, but standardness of every positive-genus S4 trisection remains open.\n\n**Verified partial progress.**\n\n- Gay--Kirby stable equivalence gives a common stabilization.\n- The list records connections to Generalized Property R and Andrews--Curtis.\n\n**Full solution or refutation.**\n\nNo theorem settling every positive-genus S4 trisection was verified.\n\n**What remains.**\n\nProve standardness for arbitrary S4 diagrams or produce a nonstandard one.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.112 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Retains the Meier--Schirmer--Zupan conjecture and records its connections.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2989, "problem_number": "KP-4.113", "title": "Kirby Problem 4.113", "statement": "Which closed, oriented, smooth 4–manifolds admit genus–3 trisections? Which ones admit genus–3 simplified trisections? How about genus–4?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.113.\n\nLiterature notes:\n(1) There is a complete classification of trisections of genus $g \\leq$ 2 and only a handful of standard 4–manifolds admit them: $S^{4}$ for g =0; $\\mathbb{CP}^{2}, \\mathbb{CP}^{2}$, or $S^{1} \\times S^{3}$ for $g =$ 1; and $S^{2} \\times S^{2}$, or connected sums of $\\mathbb{CP}^{2}, \\mathbb{CP}^{2}$ and $S^{1} \\times S^{3}$ with two summands, for g =2 by [GK16, MZ17b].\n\n(2) While the simplified trisections (see [BS23b] for a precise definition) constitute a subclass of Gay-Kirby trisections, the following question [BS18c, Question 2] is still open:\n\n\\paragraph{Question.} Is there a closed 4–manifold that admits a trisection, but not a simplified trisection of the same genus? The classification of genus $g \\leq$ 2 simplified trisections coincide with that of standard trisections [BS18c, Hay20]. So, the genus g =3 is the lowest genus where a discrepancy may appear.\n\n(3) Some partial results are known: A trisection with complexity $(3;k_{1}, k_{2}, k_{3})$ and $k_{i} \\geq$ 2 for some $i \\in \\mathbb{Z}_{3}$ is reducible [MSZ16]. Recall that a trisection that is the connected sum of smaller trisections is called reducible; otherwise, it is irreducible. Infinitely many, pairwise homotopy inequivalent 4–manifolds admit genus–3 trisections; e.g. every spun lens space admits a (simplified) genus– 3 trisection [Mei18, BS18c]. Similarly, there are genus–4 (simplified) trisections of 3–manifold bundles over $S^{1}$ with lens space or $S^{1} \\times S^{2}$ fibers [Mei18, BS18c].\n\n(4) The classification of low genera trisections is intimately related to that of simplified broken Lefschetz fibrations [BS23b, BS18c]. The classification for the latter is complete for $g \\leq$ 1 and spans a larger class of 4–manifolds [BK15a, Bay12a, Hay11, Hay14].\n\n(5) In the case of (broken) Lefschetz fibrations, the following exotic phenomenon already appears when we hit g =2: there are pairs of homeomorphic but not diffeomorphic 4–manifolds which both admit genus–2 Lefschetz fibrations; for instance, let $X_{i}:= E(1)_{K,i}, i =$ 1,2, be the knot surgered rational elliptic surface for $K_{1} a$ trefoil knot and $K_{2}$ the figure eight knot [FS04, Bay09]. There are analogous results for (simplified) trisections when g =20 [BS23b] and many other, larger genera, starting atg =23, where one of the exotic pairs is an algebraic surface [ST18, LCM22]. It is reasonable to expect this exotic phenomenon to manifest for much smaller g. (This is related to the question on the additivity of triseciton genus under connected sum; see Problem 4.111.)\n\n\\paragraph{Question.} What is the smallest g for which there is an exotic pair of closed, oriented 4–manifolds admitting genus–g trisections?\n\n(6) While the problem is formulated only for orientable 4–manifolds, the classification of small-genera (simplified) trisections and (simplified) broken Lefschetz fibrations on nonorientable 4–manifolds is also within reach, and suggests analogous classification schemes; see [MN24, ST22, BM25].\n\nReferences cited:\n- [GK16] David Gay and Robion Kirby. Trisecting 4-manifolds. Geom. Topol., 20(6):3097– 3132, 2016. doi:10.2140/gt.2016.20.3097.\n- [MZ17b] Jeffrey Meier and Alexander Zupan. Genus-two trisections are standard. Geom. Topol., 21(3):1583–1630, 2017. doi:10.2140/gt.2017.21.1583.\n- [BS23b] R. İnanç Baykur and Osamu Saeki. Simplifying indefinite fibrations on 4-manifolds. Trans. Amer. Math. Soc., 376(5):3011–3062, 2023. doi:10.1090/tran/8325.\n- [BS18c] R. İnanç Baykur and Osamu Saeki. Simplified broken Lefschetz fibrations and trisections of 4-manifolds. Proc. Natl. Acad. Sci. USA, 115(43):10894–10900, 2018. doi:10.1073/pnas.1717175115.\n- [Hay20] Kenta Hayano. On diagrams of simplified trisections and mapping class groups. Osaka J. Math., 57(1):17–37, 2020. https://projecteuclid.org/euclid.ojm/1579079109.\n- [MSZ16] Jeffrey Meier, Trent Schirmer, and Alexander Zupan. Classification of trisections and the generalized property R conjecture. Proc. Amer. Math. Soc., 144(11):4983– 4997, 2016. doi:10.1090/proc/13105.\n- [Mei18] Jeffrey Meier. Trisections and spun four-manifolds. Math. Res. Lett., 25(5):1497– 1524, 2018. doi:10.4310/MRL.2018.v25.n5.a7.\n- [BK15a] R. İnanç Baykur and Seiichi Kamada. Classification of broken Lefschetz fibrations with small fiber genera. J. Math. Soc. Japan, 67(3):877–901, 2015. doi:10.2969/jmsj/06730877.\n- [Bay12a] R. İnanç Baykur. Broken Lefschetz fibrations and smooth structures on 4-manifolds. In Proceedings of the Freedman Fest, volume 18 of Geom. Topol. Monogr., pages 9–34. Geom. Topol. Publ., Coventry, 2012. doi:10.2140/gtm.2012.18.9.\n- [Hay11] Kenta Hayano. On genus-1 simplified broken Lefschetz fibrations. Algebr. Geom. Topol., 11(3):1267–1322, 2011. doi:10.2140/agt.2011.11.1267.\n- [Hay14] Kenta Hayano. Complete classification of genus-1 simplified broken Lefschetz fibrations. Hiroshima Math. J., 44(2):223–234, 2014. http://projecteuclid.org/euclid.hmj/1408972909.\n- [FS04] Ronald Fintushel and Ronald J. Stern. Families of simply connected 4-manifolds with the same Seiberg-Witten invariants. Topology, 43(6):1449–1467, 2004. doi: 10.1016/j.top.2004.03.002.\n- [Bay09] Refik İnanç Baykur. Topology of broken Lefschetz fibrations and near-symplectic four-manifolds. Pacific J. Math., 240(2):201–230, 2009. doi:10.2140/pjm.2009.240.201.\n- [ST18] Jonathan Spreer and Stephan Tillmann. The trisection genus of standard simply connected PL 4-manifolds. In 34th International Symposium on Computational Geometry, volume 99 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 71, 13. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018.\n- [LCM22] Peter Lambert-Cole and Jeffrey Meier. Bridge trisections in rational surfaces. J. Topol. Anal., 14(3):655–708, 2022. doi:10.1142/$S^{1}$793525321500047.\n- [MN24] Maggie Miller and Patrick Naylor. Trisections of nonorientable 4-manifolds. Michigan Math. J., 74(2):403–447, 2024. doi:10.1307/mmj/20216127.\n- [ST22] Jonathan Spreer and Stephan Tillmann. Determining the trisection genus of orientable and non-orientable PL 4-manifolds through triangulations. Exp. Math., 31(3):897–907, 2022. doi:10.1080/10586458.2020.1723744.\n- [BM25] R. İnanç Baykur and Porter Morgan. On nonorientable 4–manifolds, 2025. Math. Res. Lett., to appear. URL: https://arxiv.org/abs/2506.20950, arXiv:2506.20950.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Closed 4-manifolds with trisection genus at most two are classified, while a complete genus-3/genus-4 or simplified-trisection classification was not verified.\n\n**Verified partial progress.**\n\n- Genus <=2 trisections have a complete classification.\n- Simplified trisections form a structured subclass.\n\n**Full solution or refutation.**\n\nThe requested higher-genus classifications remain open.\n\n**What remains.**\n\nClassify genus-3 and genus-4 diagrams/manifolds, including the simplified subclass.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.113 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Contrasts the complete genus <=2 classification with the posed genus-3/4 questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2990, "problem_number": "KP-4.114", "title": "Kirby Problem 4.114", "statement": "For a given Heegaard splitting of a closed orientable3–manifold, classify self-indexing Morse functions that give the given Heegaard splitting, up to $C\\infty$ right-left (or right) equivalence. Likewise, for a given (simplified) trisection of a closed orientable 4–manifold, classify generic maps that give the given trisection, up to right-left (or right) equivalence.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.114.\n\nLiterature notes:\n(1) Let $f_{i}: M_{i} \\to N_{i}$ be $C\\infty$ maps between smooth manifolds, $i =$ 0,1. We say that they are $C\\infty$ right-left equivalent if there exist diffeomorphisms $\\phi: M_{0} \\to M_{1}$ and $\\psi: N_{0} \\to N_{1}$ such that $f_{1} =\\psi \\circ f_{0} \\circ \\phi-^{1}$. If $N_{0} =N_{1} and\\psi$ can be taken to be the identity, we say that $f_{0} andf_{1}$ are $C\\infty$ right equivalent. \\begin{center}\n\\kthreefiginclude{ch4_fig5.png}{width=0.58\\linewidth}\n\\par\\small\\textbf{Figure 5.} Two commutative diagrams. Left: the maps $f_{0},f_{1}$ are right-left equivalent. Right: the maps $f_{0},f_{1}$ are right-equivalent.\n\\end{center}\n\nIt is known that to a self-indexing Morse function on a closed orientable 3–manifold is canonically associated a Heegaard splitting. Likewise, to a certain generic map, called Morse 2–function, on a closed orientable 4–manifold is canonically associated a trisection [GK16] or a simplified trisection [BS23b, BS18c].\n\n(2) The 4–dimensional problem can be rephrased as follows. Let M be a closed orientable 4–manifold and let f: $M \\to \\mathbb{R}^{2} a C^{\\infty}$ be a stable map that corresponds to a (simplified) trisection. If two such maps are $C^{\\infty}$ right-left equivalent, then the resulting trisections are naturally equivalent. Does the converse hold? Namely, if the resulting (simplified) trisections are equivalent, then are the original $C\\infty$ stable maps $C\\infty$ right-left equivalent? If not, how different are they? Some related results are obtained in [Hay20] and [Asa23].\n\n(3) It is known that Morse functions on closed surfaces can be classified by means of Kronrod-Reeb graphs up to $C\\infty$ right-left (or right) equivalence (for example, see [Mak05]).\n\nReferences cited:\n- [GK16] David Gay and Robion Kirby. Trisecting 4-manifolds. Geom. Topol., 20(6):3097– 3132, 2016. doi:10.2140/gt.2016.20.3097.\n- [BS23b] R. İnanç Baykur and Osamu Saeki. Simplifying indefinite fibrations on 4-manifolds. Trans. Amer. Math. Soc., 376(5):3011–3062, 2023. doi:10.1090/tran/8325.\n- [BS18c] R. İnanç Baykur and Osamu Saeki. Simplified broken Lefschetz fibrations and trisections of 4-manifolds. Proc. Natl. Acad. Sci. USA, 115(43):10894–10900, 2018. doi:10.1073/pnas.1717175115.\n- [Hay20] Kenta Hayano. On diagrams of simplified trisections and mapping class groups. Osaka J. Math., 57(1):17–37, 2020. https://projecteuclid.org/euclid.ojm/1579079109.\n- [Asa23] Nobutaka Asano. Right-left equivalent maps of simplified (2, 0)-trisections with different configurations of vanishing cycles. Kyushu J. Math., 77(2):299–317, 2023.\n- [Mak05] Sergey Maksymenko. Path-components of Morse mappings spaces of surfaces. Comment. Math. Helv., 80(3):655–690, 2005. doi:10.4171/CMH/30.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The general right-left/right equivalence classification of Morse functions and generic maps realizing a fixed splitting/trisection remains open.\n\n**Verified partial progress.**\n\n- Singularity theory supplies equivalence notions and normal-form classifications in restricted settings.\n\n**Full solution or refutation.**\n\nNo complete classification matching the stated generality was verified.\n\n**What remains.**\n\nGive invariants and moves complete for maps realizing a prescribed trisection.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.114 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the equivalence relation and retains both classification problems.\n\n**Review notes.** OCR/background corruption in C-infinity, inverses, and quantifiers was flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2991, "problem_number": "KP-4.115", "title": "Kirby Problem 4.115", "statement": "(a) Find two diffeomorphic but non-isotopic trisections of the same4–manifold.\n\n(b) Find two non-diffeomorphic balanced trisections of the same genus on a closed simply connected 4–manifold.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.115.\n\nLiterature notes:\n(1) Part of the difficulty of Problem (a)is that it is connected to the problem of understanding the smooth mapping class groups of 4–manifolds. In particular, if a 4–manifold X has trivial smooth mapping class group, so that all (orientation preserving) diffeomorphisms are isotopic, then all diffeomorphic trisections are obviously isotopic. On the other hand, 4– manifolds with nontrivial homology can have diffeomorphisms that act nontrivially on homology, and one might imagine that one could apply such a diffeomorphism to a given trisection to get a diffeomorphic but non-isotopic trisection. However this diffeomorphism might be isotopic to a diffeomorphism that is not the identity but which fixes the initial trisection (i.e. fixes the sectors setwise), in which case the new trisection would in fact be isotopic to the initial one.\n\n(2) Problem (b) seems to be easier than Problem (a), with the first vague approximation being to come up with examples of trisections that look at first glance like they might be diffeomorphic but which in fact are not. Examples of non-diffeomorphic same-genus trisections of a 4-manifold were first constructed by Islambouli [Isl21]. However, Islambouli’s techniques rely on the 4-manifold having nontrivial fundamental group G, as the trisections are distinguished by the Nielsen classes of associated presentations of G. This mirrors an argument in the 3–dimensional Heegaard splitting setting [Eng70]\n\n(3) In principle, invariants of trisections up to diffeomorphism should be easier to find than invariants up to isotopy. Meier and Lambert-Cole construct various genus–22 trisections of K3 that may or may not be diffeomorphic and/or isotopic by viewing K3 as a branched cover in different ways [LCM22]. See Problem 4.112 (and examples of Meier and Zupan [MZ18]) for a related problem regarding the uniqueness of trisections of $S^{4}$.\n\nReferences cited:\n- [Isl21] Gabriel Islambouli. Nielsen equivalence and trisections. Geom. Dedicata, 214:303– 317, 2021. doi:10.1007/s10711-021-00617-y.\n- [Eng70] Renate Engmann. Nicht-homöomorphe Heegaard-Zerlegungen vom Geschlecht 2 der zusammenhängenden Summe zweier Linsenräume. Abh. Math. Sem. Univ. Hamburg, 35:33–38, 1970. doi:10.1007/BF02992472.\n- [LCM22] Peter Lambert-Cole and Jeffrey Meier. Bridge trisections in rational surfaces. J. Topol. Anal., 14(3):655–708, 2022. doi:10.1142/$S^{1}$793525321500047.\n- [MZ18] Jeffrey Meier and Alexander Zupan. Characterizing Dehn surgeries on links via trisections. Proc. Natl. Acad. Sci. USA, 115(43):10887–10893, 2018. doi:10.1073/pnas.1717187115.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No verified examples resolving either the non-isotopic diffeomorphic-trisection question or the simply connected balanced same-genus contrast were found.\n\n**Verified partial progress.**\n\n- The list relates part (a) to smooth mapping-class groups.\n\n**Full solution or refutation.**\n\nBoth requested existence questions remain open in the checked source.\n\n**What remains.**\n\nConstruct examples distinguished by trisection data while meeting the stated diffeomorphism constraints.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.115 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Retains both existence questions and discusses mapping-class-group difficulty.\n\n**Review notes.** Two-part question preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2992, "problem_number": "KP-4.116", "title": "Kirby Problem 4.116", "statement": "Is there an algorithm to compute ‘distance’ in the cut complex of a trisection? Is the L-invariant computable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.116.\n\nLiterature notes:\n(1) A Heegaard splitting for a 3-manifold M is defined by two “cut systems” of curves on a Heegaard surface $\\Sigma$. Various notions of ‘distance’ (e.g. the Hempel distance) for Heegaard splittings have been extensively studied and yielded valuable geometric information about M. A trisected smooth closed 4-manifold X is defined by three “cut systems” of curves on a surface $\\Sigma$. One can define various analogous notions of 3-manifold Heegaard distance for a trisected X by measuring the length of a shortest loop in the cut complex (or pants complex) that includes a representative of each of the three specified cut systems. Given a specific trisection, the first question asks whether any such distance is computable. The L-invariant [KT22a] uses this idea in a limit to define a 4-manifold invariant.\n\n(2) When L is zero and X is a homology sphere, X is diffeomorphic to the 4-sphere. Hence a positive answer to the second question is related to recognizability of the 4-sphere.\n\n(3) In a preprint, Asano–Naoe–Ogawa [ANO24] gave a lower bound on the L-invariant of a 4-manifold X in terms of the first Betti number of X.\n\nReferences cited:\n- [KT22a] Robion Kirby and Abigail Thompson. Trisections and link surgeries. New Zealand J. Math., 52:145–152, 2021 [2021–2022]. doi:10.53733/94.\n- [ANO24] Nobutaka Asano, Hironobu Naoe, and Masaki Ogawa. Some lower bounds for the Kirby-Thompson invariant. Math. Res. Lett., 31(6):1611–1637, 2024. doi:10.4310/mrl.250210225353.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Cut-complex and L-type invariants provide a framework for trisection complexity, but no algorithm for the stated distance or L-invariant computability was verified.\n\n**Verified partial progress.**\n\n- Analogous Heegaard distance theory motivates the cut-complex construction.\n\n**Full solution or refutation.**\n\nNo general computability result was verified.\n\n**What remains.**\n\nSpecify finite searchable certificates or prove an undecidability obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.116 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Poses both algorithmic questions after defining the comparison with Heegaard distance.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2993, "problem_number": "KP-4.117", "title": "Kirby Problem 4.117", "statement": "Let X be a closed, orientable, smooth 4-manifold, with $g(X)$ the trisection genus of X. Does $g(X) =\\chi(X) -2+3rk(\\pi_{1}(X))$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.117.\n\nLiterature notes:\n(1) Chu and Tillmann proved that $\\chi(X) -2+3rk(\\pi_{1}(X))$ is a lower bound for $g(X)$ [CT19]. It is reasonable to expect there to exist manifolds for which this inequality is strict, but at present, Chu and Tillmann’s result remains the only known lower bound on trisection genus. Following the classification of trisections up to genus two [MZ17b], if the problem has an affirmative answer, then there do not exist exotic copies of any X such that $g(X) \\leq$ 2, including $S^{4}, S^{1} \\times S^{3}, \\mathbb{CP}^{2}$, and $S^{2} \\times S^{2}$. (See also\n\nProblem 4.111.)\n\n(2) Like many problems in trisections, there is an analogous problem related to Heegaard splittings of 3-manifolds, the rank-genus conjecture. The now-disproved rank-genus conjecture asks $whetherg(Y) =rk(\\pi_{1}(Y))for a$ closed, orientable 3-manifold Y. Originally posed by Waldhausen [Wal78], this conjecture was also called the Generalized Poincaré Conjecture, since it implies the 3-dimensional Poincaré conjecture [Hak70]. The first examples of a 3-manifold Y for which $g(Y) > rk(\\pi_{1}(Y))$ were exhibited by Boileau and Zieschang [BZ84], while the first hyperbolic counterexamples to the conjecture were produced by Li [Li13].\n\n(3) A related problem for bridge trisections of a knotted surface $\\mathcal{K} \\subset S^{4}$ involves themeridional rank of the group $\\pi_{1}(S^{4} \\setminus \\mathcal{K})$, denoted $mrk(\\pi_{1}(S^{4} \\setminus \\mathcal{K}))$, the smallest number of meridians needed to generate the group, and the bridge number $b(\\mathcal{K})$, the minimal b such that $\\mathcal{K}$ admits a b-bridge trisection.\n\n\\paragraph{Question.} Let $\\mathcal{K}$ be an orientable knotted surface in $S^{4}$. Does $b(\\mathcal{K}) = -\\chi(\\mathcal{K}) +3mrk(\\pi_{1}(S^{4}-\\mathcal{K}))$? As in the case of trisections, the question is suggested by a straightforward (and the only currently known) lower bound. To derive the bound, note that $if\\mathcal{K}$ admits $a(b;c_{1}, c_{2}, c_{3})-bridge$ trisection, then $\\chi(\\mathcal{K}) = c_{1}+c_{2}+c_{3}-b$ and $mrk(\\pi_{1}(S^{4} \\setminus \\mathcal{K})) \\leq c_{i}$ for all i[MZ17a, Corollary 5.3]). Thus, $b = -\\chi(\\mathcal{K}) +c_{1}+c_{2}+c_{3} \\geq -\\chi(\\mathcal{K}) +3mrk(\\pi_{1}(S^{4}-\\mathcal{K}))$. This problem can be compared to the meridional rank conjecture ([Kir97, Problem 1.11] and Problem 1.18), an unsolved problem that asks whether the bridge number of a knot K in $S^{3}$ is equal to the minimal number of meridional generators required to generate its knot group. See also\n\nProblem 4.111.\n\n(4) Meier and Zupan showed that if $\\mathcal{K}$ is the spin of a classical knot K that satisfies the meridional rank conjecture in dimension three, then $\\mathcal{K}$ satisfies the equality in the problem above [MZ17a]. Note that a positive answer to the problem would imply that any 2-sphere or torus in $S^{4}$ with infinite cyclic fundamental group has bridge number one or three, respectively, and as such is smoothly unknotted by the classification of b-bridge trisections with $b \\leq$ 3 [MZ17a]. Miyazawa’s recent construction of an exotic $\\mathbb{RP}^{2}$, announced in [Miy23], yields a negative answer to the problem in the case of nonorientable surfaces.\n\nReferences cited:\n- [CT19] Michelle Chu and Stephan Tillmann. Reflections on trisection genus. Rev. Roumaine Math. Pures Appl., 64(4):395–402, 2019.\n- [MZ17b] Jeffrey Meier and Alexander Zupan. Genus-two trisections are standard. Geom. Topol., 21(3):1583–1630, 2017. doi:10.2140/gt.2017.21.1583.\n- [Wal78] Friedhelm Waldhausen. Some problems on 3-manifolds. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, volume XXXII of Proc. Sympos. Pure Math., pages 313–322. Amer. Math. Soc., Providence, RI, 1978.\n- [Hak70] Wolfgang Haken. Various aspects of the three-dimensional Poincaré problem. In Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), pages 140–152. Markham Publishing Co., Chicago, IL, 1970.\n- [BZ84] M. Boileau and H. Zieschang. Heegaard genus of closed orientable Seifert 3-manifolds. Invent. Math., 76(3):455–468, 1984. doi:10.1007/BF01388469.\n- [Li13] Tao Li. Rank and genus of 3-manifolds. J. Amer. Math. Soc., 26(3):777–829, 2013. doi:10.1090/S0894-0347-2013-00767-5.\n- [MZ17a] Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in $S^{4}$. Trans. Amer. Math. Soc., 369(10):7343–7386, 2017. doi:10.1090/tran/6934.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Miy23] Jin Miyazawa. A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P2-knots, 2023. arXiv:2312.02041.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Chu--Tillmann prove the displayed expression is a lower bound for trisection genus, but equality for every closed oriented smooth 4-manifold remains open.\n\n**Verified partial progress.**\n\n- The Chu--Tillmann inequality supplies the stated lower bound.\n- Low-genus classification tests some equality cases.\n\n**Full solution or refutation.**\n\nNo universal sharpness theorem or strict counterexample was verified.\n\n**What remains.**\n\nProve sharpness or exhibit a manifold with strict inequality.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.117 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Attributes the lower bound to Chu--Tillmann and retains the equality question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2994, "problem_number": "KP-4.118", "title": "Kirby Problem 4.118", "statement": "Does every simply connected, closed, smooth 4-manifold admit a handle decomposition without any 1-handles? Without 1-handles and 3-handles?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.118.\n\nLiterature notes:\n(1) A manifold is called geometrically simply connected if it admits a handle decomposition without 1-handles. Any simply connected closed manifold of any dimension other than four is geometrically simply connected; this follows from the celebrated works of Smale in higher dimensions [Sma62b] and of Perelman in dimension three [Per02, Per03b, Per03a].\n\n(2) This is [Kir97, Problem 4.18]. Some candidates for counter-examples, such as the Dolgachev surface $E(1)_{2,3}$, have since been shown to admit handle decompositions without any 1- and 3-handles [Yas08, Akb12].\n\n(3) If one allows the 4-manifold to have boundary, the answer is negative. Indeed, there are many contractible 4-manifolds that require 1-handles, by the following argument due to Casson: If a compact, contractible 4– manifold X can be built without 1-handles, then, turning the handlebody upside down, we can also build X from $\\partial X$ by adding the same number of 1- and 2–handles and a 4-handle. So $\\pi_{1}(\\partial X)$ can be killed by adding the same number of generators and relators. However, a finitely presented group with a nontrivial representation to a compact connected Lie group cannot be trivialized by adding the same number of generators and relators by [GR62] and there are contractible X where $\\pi_{1}(\\partial X)$ is such a group. In fact, geometrization now implies all nontrivial 3-manifold groups admit nontrivial finite quotients by [Hem87], so any contractible 4-manifold with boundary other than $S^{3}$ requires 1-handles, such as the Mazur manifold with boundary $\\Sigma(2,3,13)$ [AK79b].\n\n(4) If a geometrically simply connected closed 4-manifold X has $b^{+}_{2}(X) >$ 1 and $b^{-}_{2}(X)$ =0, then all the stable cohomotopy Seiberg-Witten invariants of X vanish [Yas19]; so, e.g. it cannot admit a symplectic structure—see also [HL19]. In the same paper, Yasui shows that if X is geometrically simply connected, then every $\\alpha \\in H_{2}(X;\\mathbb{Z})$ has a neighborhood W diffeomorphic to a 2-handle attached to a 4-ball, where $\\alpha$ is the image of the generato r of $H_{2}(W;\\mathbb{Z}) \\cong \\mathbb{Z}$ under the inclusion induced homomorphism. Any simply connected X with some $\\alpha \\in H_{2}(X;\\mathbb{Z})$ that does not admit such a neighborhood would be a counter-example.\n\n(5) Admitting a handle decomposition without 1- and 3-handles has strong implications. For instance, if this is true for every homotopy $S^{4}$ or homotopy $\\mathbb{CP}^{2}$, then there are no exotic copies of these 4-manifolds. The conclusion for homotopy $\\mathbb{CP}^{2}s$ follows from [GL89].\n\nReferences cited:\n- [Sma62b] Stephen Smale. On the structure of 5-manifolds. Ann. of Math. (2), 75:38–46, 1962. doi:10.2307/1970417.\n- [Per02] Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications, 2002. arXiv:math/0211159.\n- [Per03b] Grisha Perelman. Ricci flow with surgery on three-manifolds, 2003. arXiv:math/0303109.\n- [Per03a] Grisha Perelman. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, 2003. arXiv:math/0307245.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Yas08] Kouichi Yasui. Elliptic surfaces without 1-handles. J. Topol., 1(4):857–878, 2008. doi:10.1112/jtopol/jtn026.\n- [Akb12] Selman Akbulut. The Dolgachev surface. Disproving the Harer-Kas-Kirby conjecture. Comment. Math. Helv., 87(1):187–241, 2012. doi:10.4171/CMH/252.\n- [GR62] M. Gerstenhaber and O. S. Rothaus. The solution of sets of equations in groups. Proc. Nat. Acad. Sci. U.S.A., 48:1531–1533, 1962.\n- [Hem87] John Hempel. Residual finiteness for 3-manifolds. In Combinatorial group theory and topology (Alta, Utah, 1984), volume 111 of Ann. of Math. Stud., pages 379–396. Princeton Univ. Press, Princeton, NJ, 1987.\n- [AK79b] Selman Akbulut and Robion Kirby. Mazur manifolds. Michigan Math. J., 26(3):259–284, 1979. http://projecteuclid.org/euclid.mmj/1029002261.\n- [Yas19] Kouichi Yasui. Geometrically simply connected 4-manifolds and stable cohomotopy Seiberg-Witten invariants. Geom. Topol., 23(5):2685–2697, 2019. doi:10.2140/gt.2019.23.2685.\n- [HL19] Jennifer Hom and Tye Lidman. A note on positive-definite, symplectic fourmanifolds. J. Eur. Math. Soc. (JEMS), 21(1):257–270, 2019. doi:10.4171/JEMS/835.\n- [GL89] C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371–415, 1989. doi:10.2307/1990979.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Several important simply connected examples once viewed as candidates have handle decompositions without 1- and 3-handles, but the universal smooth 4-manifold assertion remains open.\n\n**Verified partial progress.**\n\n- The list records E(1)_(2,3) as admitting a handle decomposition without 1- and 3-handles.\n\n**Full solution or refutation.**\n\nNo all-simply-connected-4-manifolds handle theorem was verified.\n\n**What remains.**\n\nGive a universal handle-cancellation method or a counterexample.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.118 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records candidate examples resolved positively while retaining the general question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2995, "problem_number": "KP-4.119", "title": "Kirby Problem 4.119", "statement": "Is every topological 4-manifold homeomorphic to a CW complex?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.119.\n\nLiterature notes:\n(1) It follows from the work of Kirby and Siebenmann [KS77] that every topological manifold M has the homotopy type of a CW complex, and moreover there is a canonical simple homotopy type of CW complexes homotopy equivalent to M.\n\n(2) Every smooth manifold is triangulable and hence homeomorphic to a CW complex. See [Cai35], [Whi40]. In particular, non-compact 4-manifolds are smoothable [Qui82]; therefore, the question has a positive answer for those.\n\n(3) Every topological manifold of dimension $n \\ne$ 4 has a handlebody structure, and hence is homeomorphic to a CW complex. See [Moi77a] for n=3, [KS77, p.104] for $n \\geq$ 6 and [Qui86] for n=5.\n\n(4) Any 4-manifold with a handlebody structure is smooth. Indeed, when a handle is attached to a smooth 4-manifold, the attaching map is in dimension three, where everything is smoothable.\n\n(5) Topological, non-smoothable 4-manifolds (such as the $E_{8}$ manifold) are known to not be homeomorphic to simplicial complexes. See [AM90] and [Man13, Remark 4.2].\n\nReferences cited:\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [Cai35] S. S. Cairns. Triangulation of the manifold of class one. Bull. Amer. Math. Soc., 41(8):549–552, 1935. doi:10.1090/S0002-9904-1935-06140-3.\n- [Whi40] J. H. C. Whitehead. On C1-complexes. Ann. of Math. (2), 41:809–824, 1940. doi: 10.2307/1968861.\n- [Qui82] Frank Quinn. Ends of maps. III. Dimensions 4 and 5. J. Differential Geometry, 17(3):503–521, 1982. http://projecteuclid.org/euclid.jdg/1214437139.\n- [Moi77a] Edwin E. Moise. Geometric topology in dimensions 2 and 3, volume Vol. 47 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977.\n- [Qui86] Frank Quinn. Isotopy of 4-manifolds. J. Differential Geom., 24(3):343–372, 1986. http://projecteuclid.org/euclid.jdg/1214440552.\n- [AM90] Selman Akbulut and John D. McCarthy. Casson’s invariant for oriented homology 3-spheres, volume 36 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1990. An exposition. doi:10.1515/9781400860623.\n- [Man13] Ciprian Manolescu. The Conley index, gauge theory, and triangulations. J. Fixed Point Theory Appl., 13(2):431–457, 2013. doi:10.1007/s11784-013-0134-3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every topological manifold has a canonical simple homotopy type of CW complexes, and smoothable/noncompact cases are homeomorphic to CW complexes; the closed topological 4-manifold homeomorphism question remains open.\n\n**Verified partial progress.**\n\n- Kirby--Siebenmann supplies the CW simple-homotopy type.\n- Smooth and noncompact topological 4-manifolds give positive homeomorphism cases.\n\n**Full solution or refutation.**\n\nNo theorem or counterexample for all closed topological 4-manifolds was verified.\n\n**What remains.**\n\nConstruct a CW homeomorphism in the nonsmoothable closed case or prove an obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.119 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Distinguishes homotopy type, smooth/noncompact cases, and the remaining closed 4-dimensional question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2996, "problem_number": "KP-4.120", "title": "Kirby Problem 4.120", "statement": "Which closed, smooth 4–manifolds admit achiral Lefschetz pencils? Does every simply connected 4–manifold have one?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.120.\n\nLiterature notes:\n(1) Achiral Lefschetz pencils are generalizations of Lefschetz pencils, where the local models for nodal singularities and base points are allowed to reverse orientations. So, they are also defined on nonorientable 4–manifolds. Here we allow achiral pencils to possibly have no critical points and/or no base points.\n\n(2) There are only a couple of known obstructions to the existence of an achiral Lefschetz pencil on a given closed, oriented 4–manifold X; see [GS99, Theorem 8.4.13] and [Sco03, Theorem 4.15]. These obstructions rule out definite 4–manifolds with $b_{2}$ +1 $< b_{1}$, such as $\\#_{m}(S^{1} \\times S^{3})$, for $m \\geq$ 2. A curious question is: Are there homotopy equivalent smooth 4–manifolds X and X1, where X admits an achiral pencil but X1 does not?\n\n(3) Any closed, orientable X admits an achiral Lefschetz fibration (without base points) after surgery along a curve, and specifically, $X\\#(S^{2} \\times S^{2})$ always admits one when X is simply connected [EF06].\n\n(4) Just like Lefschetz pencils can be equipped with certain symplectic forms making all the fibers symplectic, achiral Lefschetz pencils can be equipped with certain folded-symplectic forms. Furthermore, a variation of achiral Lefschetz pencils, defined in the complement of a 1–manifold, supsupport folded-Kähler forms on all closed, oriented, smooth 4–manifolds. See [Bay06, Hit16].\n\nReferences cited:\n- [GS99] Robert E. Gompf and András I. Stipsicz. 4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999. doi:10.1090/gsm/020.\n- [Sco03] Alexandru Scorpan. Existence of foliations on 4-manifolds. Algebr. Geom. Topol., 3:1225–1256, 2003. doi:10.2140/agt.2003.3.1225.\n- [EF06] John B. Etnyre and Terry Fuller. Realizing 4-manifolds as achiral Lefschetz fibrations. Int. Math. Res. Not., pages Art. ID 70272, 21, 2006. doi:10.1155/IMRN/2006/70272.\n- [Bay06] R. İnanç Baykur. Kähler decomposition of 4-manifolds. Algebr. Geom. Topol., 6:1239–1265, 2006. doi:10.2140/agt.2006.6.1239.\n- [Hit16] Nigel Hitchin. Higgs bundles and diffeomorphism groups. In Surveys in differential geometry 2016. Advances in geometry and mathematical physics, volume 21 of Surv. Differ. Geom., pages 139–163. Int. Press, Somerville, MA, 2016.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Achiral Lefschetz fibrations exist after framed-circle surgery and on standard stabilizations of simply connected 4-manifolds, but existence of an achiral pencil on every original simply connected 4-manifold remains open.\n\n**Verified partial progress.**\n\n- Every closed oriented 4-manifold becomes achirally Lefschetz fibred after surgery along a suitable framed circle.\n- For simply connected X, both stabilizations by an S2-bundle over S2 admit achiral Lefschetz fibrations.\n- Known numerical obstructions exclude some definite nonsimply connected manifolds.\n\n**Full solution or refutation.**\n\nNo characterization of all closed smooth 4-manifolds admitting achiral pencils, or unstabilized theorem for every simply connected one, was found.\n\n**What remains.**\n\nRemove the surgery/stabilization in the simply connected case or find an obstruction, and distinguish fibrations without base points from pencils as required.\n\n**Sources checked.**\n\n- John B. Etnyre and Terry Fuller, Realizing 4-manifolds as achiral Lefschetz fibrations, International Mathematics Research Notices (2006), Art. ID 70272. (primary): https://arxiv.org/abs/math/0510008\n Evidence used: Proves the framed-circle surgery theorem and the simply connected stabilization corollary.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.120. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current open question, definitions, known obstructions, and stable existence results.\n\n**Review notes.** Background OCR includes '$b_2$ +1 < $b_1$' and 'supsupport'; neither was silently repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2997, "problem_number": "KP-4.121", "title": "Kirby Problem 4.121", "statement": "Which closed, smooth4–manifolds admit open book decompositions? In particular, does every closed, simply connected 4–manifold with signature zero admit one?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.121.\n\nLiterature notes:\n(1) An open book decomposition of an n-dimensional manifold X is given by a smooth fibration f: $X \\setminus L \\to S^{1}$, where the binding $L \\subset X$ is an (n−2)–dimensional embedded submanifold with a trivial normal bundle, and there is a neighborhood $N(L) \\cong L \\times D^{2}$ such that f conforms to the local model f(x,(r, $\\theta)) =\\theta$ for $r \\ne$ 0, where $x \\in L$ and (r, $\\theta) \\in D^{2}$ are the polar coordinates.\n\n(2) If a closed, oriented n–dimensional manifold X admits an open book, its signature vanishes. An extension of this necessary condition is the vanishing of the asymmetric signature; see [Ran98] for a definition and related discussion. Vanishing signature is also a sufficient condition for\n\n\\noindent$\\bullet$ any odd $n \\geq$ 3, by the works of Alexander, Lawson and Quinn [Law78, Qui79];\n\n\\noindent$\\bullet$ any even $n \\geq$ 6, when X is simply connected, by Winkelnkemper [Win73] for $n \\geq$ 8, and for n=6 by Quinn [Qui79].\n\n(3) Kastenholz [Kas25] claims that the simplicial volume vanishes for 4manifolds that admit an open book, and gives examples of non-simplyconnected 4-manifolds with vanishing asymmetric signature that do not admit open book decompositions.\n\n(4) Open books on 4–manifolds are related to Engel structures. If a closed, oriented 4–manifold X admits an open book with a binding that is a link of tori and a monodromy that preserves a framing on the fiber, then X admits an Engel structure [CPV18].\n\n(5) It would be interesting to see if there are smooth obstructions to admitting open books in dimension four. If $X_{1}$ and $X_{2}$ admit open book decompositions, then so does $X_{1}\\#X_{2}$. There are natural open books on any $\\Sigma_{g}–bundle$ over $S^{2}$, where the binding is a pair of fibers. Combining these general constructions, one gets an open book on every $\\#_{m}(S^{2} \\times S^{2})$ and $\\#_{n}(\\mathbb{CP}^{2}\\#\\mathbb{CP}^{2})$. Do their exotic copies always admit open books?\n\nReferences cited:\n- [Ran98] Andrew Ranicki. High-dimensional knot theory. Springer Monographs in Mathematics. Springer-Verlag, New York, 1998. Algebraic surgery in codimension 2, With an appendix by Elmar Winkelnkemper. doi:10.1007/978-3-662-12011-8.\n- [Law78] Terry Lawson. Open book decompositions for odd dimensional manifolds. Topology, 17(2):189–192, 1978. doi:10.1016/S0040-9383(78)90024-1.\n- [Qui79] Frank Quinn. Open book decompositions, and the bordism of automorphisms. Topology, 18(1):55–73, 1979. doi:10.1016/0040-9383(79)90014-4.\n- [Win73] H. E. Winkelnkemper. Manifolds as open books. Bull. Amer. Math. Soc., 79:45–51, 1973. doi:10.1090/S0002-9904-1973-13085-X.\n- [Kas25] Thorben Kastenholz. Simplicial volume of open books in dimension 4, 2025. arXiv: 2504.10975.\n- [CPV18] Vincent Colin, Francisco Presas, and Thomas Vogel. Notes on open book decompositions for Engel structures. Algebr. Geom. Topol., 18(7):4275–4303, 2018. doi:10.2140/agt.2018.18.4275.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Open-book 4-manifolds must have both zero signature and zero simplicial volume, disproving signature sufficiency in general; the highlighted simply connected signature-zero case remains open.\n\n**Verified partial progress.**\n\n- Kastenholz proves that every closed oriented 4-manifold with an open book has zero simplicial volume.\n- Products of two hyperbolic surfaces therefore give signature-zero nonsimply connected 4-manifolds without open books.\n- Connected sums of S2 x S2 and related standard simply connected examples admit open books.\n\n**Full solution or refutation.**\n\nThe general 'which manifolds' problem has a new obstruction, but no theorem covers every simply connected signature-zero 4-manifold or its exotic smoothings.\n\n**What remains.**\n\nDecide whether simple connectivity plus signature zero is sufficient and identify any smooth obstruction in dimension four.\n\n**Sources checked.**\n\n- Thorben Kastenholz, Simplicial volume of open books in dimension 4, arXiv:2504.10975 (2025). (primary): https://arxiv.org/abs/2504.10975\n Evidence used: Proves vanishing simplicial volume and explains why the signature invariant is insufficient in dimension four.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.121. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current statement and known simply connected examples.\n\n**Review notes.** The source's 'smooth4-manifolds' is an OCR spacing defect. The nonsimply connected counterexamples do not answer the highlighted simply connected subquestion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2998, "problem_number": "KP-4.122", "title": "Kirby Problem 4.122", "statement": "Is there a universal branching surface $S \\subset S^{4}$ such that every closed, orientable 4-manifold W admits a branched covering $W \\to S^{4}$ with branchin g locus S?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.122.\n\nLiterature notes:\n(1) This question originated in the introduction of [PZ05] by Piergallini– Zuddas.\n\n(2) Using the signature, it can be shown that a universal branching surface in $S^{4}$ is necessarily disconnected [Vir84, IP02].\n\n(3) There exists an orientable ribbon surface $F \\subset B^{4}$, consisting of the disjoint union of one annulus and two discs, such that every compact orientable 4-manifold M constructed by adding only 1-handles and 2-handles to $B^{4}$ admits a branched covering $M \\to B^{4}$ with branching locus F [PZ05]. Question ([PZ05, Question 2]). Does there exist a connected universal branching surface in $B^{4}$?\n\n(4) A link $L \\subset S^{3}$ is said to bea universal branching link if every closed, orientable 3-manifold Y admits a branched covering $Y \\to S^{3}$ with branching locus L. The figure eight knot, the $9_{46}$ knot, the Whitehead link, the Borromean rings, and various other knots and links are known to be universal branching links [HLM83a, HLM83b, HLM85]. The first universal branching link was found by Thurston in unpublished work. Question ([PZ05, Question 4]). Which universal links in $S^{3}$ are boundaries of universal surfaces in $B^{4}$?\n\nReferences cited:\n- [PZ05] R. Piergallini and D. Zuddas. A universal ribbon surface in $B^{4}$. Proc. London Math. Soc. (3), 90(3):763–782, 2005. doi:10.1112/S0024611504015072.\n- [Vir84] O. Ya. Viro. The signature of a branched covering. Mat. Zametki, 36(4):549–557, 1984.\n- [IP02] Massimiliano Iori and Riccardo Piergallini. 4-manifolds as covers of the 4-sphere branched over non-singular surfaces. Geom. Topol., 6:393–401, 2002. doi:10.2140/gt.2002.6.393.\n- [HLM83a] Hugh M. Hilden, M. T. Lozano, and José Marı́a Montesinos. Universal knots. Bull. Amer. Math. Soc. (N.S.), 8(3):449–450, 1983. doi:10.1090/S0273-0979-1983-15114-5.\n- [HLM83b] Hugh M. Hilden, Marı́a Teresa Lozano, and José Marı́a Montesinos. The Whitehead link, the Borromean rings and the knot 946 are universal. Collect. Math., 34(1):19– 28, 1983.\n- [HLM85] Hugh M. Hilden, Marı́a Teresa Lozano, and José Marı́a Montesinos. On knots that are universal. Topology, 24(4):499–504, 1985. doi:10.1016/0040-9383(85)90019-9.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A fixed universal ribbon surface exists in B4 for orientable 0/1/2-handlebodies, and every closed oriented PL 4-manifold covers S4 over some surface, but no fixed universal surface in S4 was verified.\n\n**Verified partial progress.**\n\n- Piergallini--Zuddas construct a universal disconnected ribbon surface in B4 for all compact orientable 4-dimensional 0/1/2-handlebodies.\n- Iori--Piergallini prove every closed oriented PL 4-manifold is a five-fold simple cover of S4 branched over a locally flat surface depending on the manifold.\n- Signature constraints force a hypothetical universal branching surface in S4 to be disconnected.\n\n**Full solution or refutation.**\n\nThe required single branching surface in S4, independent of the covering 4-manifold, remains open.\n\n**What remains.**\n\nConstruct a fixed disconnected universal surface in S4 or prove that no such fixed surface can exist.\n\n**Sources checked.**\n\n- Riccardo Piergallini and Daniele Zuddas, A universal ribbon surface in B4, Proceedings of the London Mathematical Society 90 (2005), 763--782. (primary): https://doi.org/10.1112/S0024611504015072\n Evidence used: Direct universal-surface theorem in B4 for 0/1/2-handlebodies.\n- Massimiliano Iori and Riccardo Piergallini, 4-manifolds as covers of the 4-sphere branched over non-singular surfaces, Geometry & Topology 6 (2002), 393--401. (primary): https://arxiv.org/abs/math/0203087\n Evidence used: Every closed oriented PL 4-manifold has a five-fold simple branched-cover presentation, with a manifold-dependent branch surface.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.122. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current status and disconnectedness obstruction.\n\n**Review notes.** The statement's 'branchin g locus' is a visible OCR spacing defect.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 2999, "problem_number": "KP-4.123", "title": "Kirby Problem 4.123", "statement": "Is every closed leaf of a two dimensional co-orientable smooth taut foliation of an oriented 4-manifold smoothly genus-minimizing in its homology class?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.123.\n\nLiterature notes:\n(1) This question is due to Kronheimer [Kro98, Question 7.12], and is the natural generalization of the corresponding three-dimensional result of Thurston [Thu86]. Here we say that the $foliation\\mathcal{F}$ is taut if there exists a 2-for $m \\omega$ such that:\n\n\\noindent$\\bullet$ for every leaf $L, \\omega∥_{L}$ is an area for m;\n\n\\noindent$\\bullet$ for every $v_{1}, v_{2} \\in T_{F}$ and $z \\in T_{M}, d \\omega(v_{1}, v_{2}$, z) =0. This directly generalizes the definition in dimension three (in which case the second condition simply says that $\\omega$ is closed). As Kronheimer points out, the question is also interesting when one allows foliations with singularities with a suitable local model, e.g. the foliations defined by the vanishing of a holomorphic 1-form.\n\n(2) If the foliation is calibrated to a symplectic form, then every closed leaf is a symplectic subsurface and hence genus-minimizing by the symplectic Thom conjecture [OS00].\n\n(3) If “taut,” is eliminated from the hypotheses, then the answer is “no.” Constructions of non-genus-minimizing compact leaves of coorientable foliations of 4-manifolds were constructed by Mitsumatsu–Vogt [MV08] and Bowden [Bow11].\n\n(4) The following specific subquestion would be an interesting first step: Must a compact leaf of a coorientable smooth taut foliation of $S^{2} \\times S^{2}$ be a 2sphere?\n\nReferences cited:\n- [Kro98] P. B. Kronheimer. Embedded surfaces and gauge theory in three and four dimensions. In Surveys in differential geometry, Vol. III (Cambridge, MA, 1996), pages 243–298. Int. Press, Boston, MA, 1998.\n- [Thu86] William P. Thurston. A norm for the homology of 3-manifolds. Mem. Amer. Math. Soc., 59(339):i–vi and 99–130, 1986.\n- [OS00] Peter Ozsváth and Zoltán Szabó. The symplectic Thom conjecture. Ann. of Math. (2), 151(1):93–124, 2000. doi:10.2307/121113.\n- [MV08] Yoshihiko Mitsumatsu and Elmar Vogt. Foliations and compact leaves on 4-manifolds. I. Realization and self-intersection of compact leaves. In Groups of diffeomorphisms, volume 52 of Adv. Stud. Pure Math., pages 415–442. Math. Soc. Japan, Tokyo, 2008. doi:10.2969/aspm/05210415.\n- [Bow11] Jonathan Bowden. On closed leaves of foliations, multisections and stable commutator lengths. J. Topol. Anal., 3(4):491–509, 2011. doi:10.1142/$S^{1}$793525311000696.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Closed leaves are genus-minimizing when the taut foliation is symplectically calibrated; counterexamples exist without tautness, while the general taut case remains open.\n\n**Verified partial progress.**\n\n- A symplectically calibrated closed leaf is symplectic and hence genus-minimizing by the symplectic Thom theorem.\n- Mitsumatsu--Vogt and Bowden construct non-genus-minimizing compact leaves when tautness is removed.\n\n**Full solution or refutation.**\n\nNo theorem or counterexample for arbitrary co-orientable smooth taut 2-foliations in oriented 4-manifolds was found.\n\n**What remains.**\n\nHandle taut foliations not calibrated by a symplectic form, beginning with the compact-leaf question on S2 x S2.\n\n**Sources checked.**\n\n- Peter Ozsvath and Zoltan Szabo, The symplectic Thom conjecture, Annals of Mathematics 151 (2000), 93--124. (primary): https://doi.org/10.2307/121113\n Evidence used: Supplies genus minimization for symplectic leaves in the calibrated special case.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.123. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current open status, calibrated special case, and counterexamples without tautness.\n\n**Review notes.** The background definition is heavily OCR-corrupted ('2-for m', 'area for m', and a damaged differential-form condition); it was not reconstructed in the output claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3000, "problem_number": "KP-4.124", "title": "Kirby Problem 4.124", "statement": "Does there exist a hyperbolic integer homology four-sphere? What about an arithmetic one? Homology four-spheres have Euler characteristic 2, so it makes sense to ask more generally if there exist any closed hyperbolic fourmanifold with Euler characteristic 2.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.124.\n\nLiterature notes:\n(1) If any closed, hyperbolic manifold with Euler characteristic 2 exists, there are only finitely many. This is because the Gauss-Bonnet Theorem says that $V ol(M) =^{4}_{3}\\pi^{2}\\chi(M)$ for a hyperbolic four-manifold M, and there are only finitely many manifolds with volume below any fixed value. Note that in general, the Euler characteristic of a closed, orientable hyperbolic four-manifold is always even, since such manifolds have zero signature (see [LR00]), so these would be the ones with smallest volume. The best known example seems to be the Conder-Maclachlan manifold with Euler characteristic 16 [CM05]. For comparison, there are one-cusped orientable hyperbolic four-manifolds with Euler characteristic one [RT00].\n\n(2) It is worth pointing out that there are various constructions of aspherical 4-manifolds with Euler characteristic 2. For example, Luo constructed an aspherical rational homology 4-sphere [Luo88] and Tschantz constructed aspherical integer homology 4-spheres [RT05] (answering [Kir97, Problem 4.17]). The latter examples even admit metrics of non-positive curvature.\n\n(3) If there exists an arithmetic hyperbolic integer homology sphere (or, more generally, an arithmetic, closed, hyperbolic manifold with Euler characteristic 2), then by Belolipetsky [Bel07, Theorem 5.5’] (see also [Bel04, Theorem 5.5]) it has to be an index-28800 cover of the 4-dimensional hyperbolic reflection group with the below Coxeter diagram.\n\nReferences cited:\n- [LR00] D. D. Long and A. W. Reid. On the geometric boundaries of hyperbolic 4-manifolds. Geom. Topol., 4:171–178, 2000. doi:10.2140/gt.2000.4.171.\n- [CM05] Marston Conder and Colin Maclachlan. Compact hyperbolic 4-manifolds of small volume. Proc. Amer. Math. Soc., 133(8):2469–2476, 2005. doi:10.1090/S0002-9939-05-07634-3.\n- [RT00] John G. Ratcliffe and Steven T. Tschantz. The volume spectrum of hyperbolic 4-manifolds. Experiment. Math., 9(1):101–125, 2000. http://projecteuclid.org/euclid.em/1046889595.\n- [Luo88] Feng Luo. The existence of $K(\\pi,1)$ 4-manifolds which are rational homology 4-spheres. Proc. Amer. Math. Soc., 104(4):1315–1321, 1988. doi:10.2307/2047635.\n- [RT05] John G. Ratcliffe and Steven T. Tschantz. Some examples of aspherical 4-manifolds that are homology 4-spheres. Topology, 44(2):341–350, 2005. doi:10.1016/j.top.2004.10.006.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Bel07] Mikhail Belolipetsky. Addendum to: “On volumes of arithmetic quotients of $\\mathrm{SO}(1,n)$” [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 4, 749–770; mr2124587]. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6(2):263–268, 2007.\n- [Bel04] Mikhail Belolipetsky. On volumes of arithmetic quotients of $\\mathrm{SO}(1,n)$. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 3(4):749–770, 2004.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No closed hyperbolic 4-manifold of Euler characteristic two, homology 4-sphere, or arithmetic example was verified; arithmetic volume theory reduces a possible arithmetic example to one highly constrained index cover.\n\n**Verified partial progress.**\n\n- Gauss--Bonnet and volume finiteness imply only finitely many possible hyperbolic 4-manifolds with Euler characteristic two.\n- Belolipetsky's arithmetic analysis forces any arithmetic Euler-characteristic-two example to be an index-28800 cover of a specified reflection group.\n- Closed hyperbolic examples and nonhyperbolic aspherical homology 4-spheres are known, but do not attain the requested combination.\n\n**Full solution or refutation.**\n\nNeither existence nor nonexistence in dimension four is established in the checked sources.\n\n**What remains.**\n\nDetermine whether the finite candidate set is empty, and in the arithmetic case analyze the constrained index-28800 cover and its homology.\n\n**Sources checked.**\n\n- Marston Conder and Colin Maclachlan, Compact hyperbolic 4-manifolds of small volume, Proceedings of the AMS 133 (2005), 2469--2476. (primary): https://doi.org/10.1090/S0002-9939-05-07634-3\n Evidence used: Constructs a leading small-volume closed example, still with Euler characteristic larger than two.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.124. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current open status, finiteness argument, known examples, and arithmetic index reduction.\n\n**Review notes.** The imported Gauss--Bonnet display and 'fourmanifold' contain OCR corruption; no numerical constant was silently repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3001, "problem_number": "KP-4.125", "title": "Kirby Problem 4.125", "statement": "Is there a noncompact, finite volume, orientable hyperbolic four-manifold without a spin structure?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.125.\n\nLiterature notes:\nAll compact orientable manifolds with dimension at most three admit spin structures. Sullivan observed that every finite volume hyperbolic nmanifold has a finite cover that admits a spin structure [Sul79b, p.533]. Reid and Long showed in [LR20] that there are orientable, finite volume, non-compact hyperbolic n-manifolds with $n \\geq$ 5 that do not admit spin structures. MartelliRiolo-Slavich showed that there are closed orientable hyperbolic four-manifolds that do not admit spin structures [MRS20].\n\nReferences cited:\n- [Sul79b] Dennis Sullivan. Hyperbolic geometry and homeomorphisms. In Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pages 543–555. Academic Press, New York-London, 1979.\n- [LR20] D. D. Long and A. W. Reid. Virtually spinning hyperbolic manifolds. Proc. Edinb. Math. Soc. (2), 63(2):305–313, 2020. doi:10.1017/s0013091519000324.\n- [MRS20] Bruno Martelli, Stefano Riolo, and Leone Slavich. Compact hyperbolic manifolds without spin structures. Geom. Topol., 24(5):2647–2674, 2020. doi:10.2140/gt.2020.24.2647.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Riolo--Rizzi construct a noncompact finite-volume orientable hyperbolic 4-manifold without any spin structure, exactly answering the question affirmatively.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nTheorem of arXiv:2510.12657 supplies a cusped orientable finite-volume hyperbolic four-manifold with nonzero spin obstruction and hence no spin structure.\n\n**What remains.**\n\nPeer-reviewed publication and further classification/minimal-complexity questions remain, but not the existence question asked here.\n\n**Sources checked.**\n\n- Stefano Riolo and Edoardo Rizzi, A cusped hyperbolic 4-manifold without spin structures, arXiv:2510.12657 (2025). (primary): https://arxiv.org/abs/2510.12657\n Evidence used: Abstract and main theorem match all hypotheses of KP-4.125 exactly.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.125. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Provides the pre-resolution context: closed dimension-four and cusped dimensions at least five were known.\n\n**Review notes.** Direct solution is a 2025 preprint; no formulation defect found in the statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3002, "problem_number": "KP-4.126", "title": "Kirby Problem 4.126", "statement": "(a) If M is a closed, orientable hyperbolic 4-manifold then it always has signature 0, because its Pontryagin class vanishes [Che55]. This implies that M bounds a compact, orientable 5-manifold. Is M always a geometric boundary?\n\n(b) For general n, suppose that M is a closed hyperbolic n-manifold. Is M a geometric boundary?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.126.\n\nLiterature notes:\n(1) Following [LR00], we say that $M^{n}$ is a geometric boundary if it is the totally geodesic boundary of a compact hyperbolic manifold $W^{n}+^{1}$.\n\n(2) Perhaps a good example to start with for(a)is the Davis manifold [Dav85].\n\n(3) Every closed, orientable surface of genus at least 2 has a hyperbolic metric that is the totally geodesic boundary of a compact, orientable, hyperbolic 3-manifold by [Fuj90]. However, some closed, hyperbolic, orientable 3manifolds are not the geodesic boundary of any compact, orientable, hyperbolic 4-manifold [LR00]. It is still open if there is a hyperbolic rational homology 3-sphere which is the totally geodesic boundary of a compact, orientable, hyperbolic 4-manifold (see Problem 3.77).\n\n(4) We restrict to the setting of orientable 4-manifolds as a closed, nonorientable, hyperbolic 4-manifold may have odd Euler characteristic. In this case, the 4-manifold cannot bound any compact 5-manifold, without mention of geometry. Even in the setting of nonorientable 4-manifolds, it may be interesting to ask this question with the additional hypothesis that the nonorientable 4-manifold does bound some compact (nonorientable) 5-manifold.\n\n(5) For (b), the general problem can be posed either in the orientable or nonorientable setting; one might assume that M is null-bordant to start with. In the orientable case, Long and Reid [LR00] observe that when $n =$ 4k−1, the $\\eta$-invariant [APS75a] of M would have to be integral. They give examples of orientable hyperbolic 3-manifolds with non-integral $\\eta$-invariant, which are therefore not geometric boundaries in the oriented category. The oriented version in higher dimensions could similarly be answered by finding hyperbolic (4k −1)-manifolds with non-integral $\\eta invariant$ for $k >$ 1. Some constructions of higher-dimensional orientable hyperbolic manifolds that are geometric boundaries are given in [LR01].\n\n(6) In the nonorientable case, J. Chen [Che25] claims to construct, in all dimensions $n \\geq$ 4 not of the for m n =4k−1, examples of nonorientable closed hyperbolic n-manifolds that are not the boundary of any compact (n+1)-manifold (not assuming any geometric condition). The question of whether there are nonorientable hyperbolic manifolds that are boundaries but not geometric boundaries remains open.\n\nReferences cited:\n- [Che55] Shiing-shen Chern. On curvature and characteristic classes of a Riemann manifold. Abh. Math. Sem. Univ. Hamburg, 20:117–126, 1955. doi:10.1007/BF02960745.\n- [LR00] D. D. Long and A. W. Reid. On the geometric boundaries of hyperbolic 4-manifolds. Geom. Topol., 4:171–178, 2000. doi:10.2140/gt.2000.4.171.\n- [Dav85] Michael W. Davis. A hyperbolic 4-manifold. Proc. Amer. Math. Soc., 93(2):325– 328, 1985. doi:10.2307/2044771.\n- [Fuj90] Michihiko Fujii. Hyperbolic 3-manifolds with totally geodesic boundary. Osaka J. Math., 27(3):539–553, 1990. http://projecteuclid.org/euclid.ojm/1200782445.\n- [APS75a] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambridge Philos. Soc., 77:43–69, 1975. doi:10.1017/S0305004100049410.\n- [LR01] D. D. Long and A. W. Reid. Constructing hyperbolic manifolds which bound geometrically. Math. Res. Lett., 8(4):443–455, 2001. doi:10.4310/MRL.2001.v8.n4.a5.\n- [Che25] Jacopo G. Chen. Non-cobordant hyperbolic manifolds, 2025. arXiv:2501.11610.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The dimension-four universal question remains open, while the oriented general-dimensional version is false already for hyperbolic 3-manifolds by the eta-invariant obstruction; the unqualified statement is ambiguous about the orientation of the bounding manifold.\n\n**Verified partial progress.**\n\n- Long--Reid exhibit oriented closed hyperbolic 3-manifolds with nonintegral eta invariant, so they cannot be oriented geometric boundaries.\n- Positive constructions give many hyperbolic manifolds that do bound geometrically.\n- Jacopo Chen gives nonorientable examples in many higher dimensions that do not even bound topologically.\n\n**Full solution or refutation.**\n\nThe oriented version of part (b) is refuted; part (a), and any reading of part (b) allowing a nonorientable bounding manifold, were not resolved.\n\n**What remains.**\n\nDecide part (a), with the Davis manifold as a concrete test case, and refine part (b) under explicit null-bordism and orientation hypotheses.\n\n**Sources checked.**\n\n- D. D. Long and A. W. Reid, On the geometric boundaries of hyperbolic 4-manifolds, Geometry & Topology 4 (2000), 171--178. (primary): https://doi.org/10.2140/gt.2000.4.171\n Evidence used: Eta-integrality obstruction and closed oriented hyperbolic 3-manifold counterexamples to the general universal assertion.\n- Jacopo G. Chen, Non-cobordant hyperbolic manifolds, arXiv:2501.11610 (2025). (primary): https://arxiv.org/abs/2501.11610\n Evidence used: Further nonorientable higher-dimensional examples failing even topological null-bordism.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.126. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current part-by-part status and orientation/null-bordism caveats.\n\n**Review notes.** The background's `$W^{n}+^{1}$` is OCR-corrupted dimension notation. Oriented versus unoriented geometric-boundary conventions materially affect whether part (b) is already refuted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3003, "problem_number": "KP-4.127", "title": "Kirby Problem 4.127", "statement": "Given an aspherical closed (or compact and bounded by flat 3-manifolds) 4-manifold M and a self-diffeomorphism f of M, find necessary and sufficient conditions on f so that the resulting 5-dimensional mapping torus $M_{f}$ admits a hyperbolic structure.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.127.\n\nLiterature notes:\n(1) By hyperbolization, in dimension two the sufficient and necessary condition is that f is pseudo-Anosov (see Thurston [Thu82, Thu22] and Otal [Ota01] for a complete proof). There are some analogues in strictly higher dimensions, e.g. [Far72, Theorem 6.4] gives necessary and sufficient conditions for a manifold of dimension at least six to fiber over $S^{1}$. If such a theorem worked in dimension 5, then one could potentially check such a condition against hyperbolic 5-manifolds. However, similar higher-dimensional techniques have not yet been successfully applied in the context of 5-dimensional hyperbolic manifolds.\n\n(2) Italiano–Martelli–Migliorini recently found examples of (M, f) with M 4-dimensional that produce a hyperbolic 5-manifold $M_{f}$ [IMM23].\n\nReferences cited:\n- [Thu82] William P. Thurston. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.), 6(3):357–381, 1982. doi:10.1090/S0273-0979-1982-15003-0.\n- [Thu22] William P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. In Collected works of William P. Thurston with commentary. Vol. I. Foliations, surfaces and differential geometry, pages 495–509. Amer. Math. Soc., Providence, RI, [2022] ©2022. Reprint of [ 0956596].\n- [Ota01] Jean-Pierre Otal. The hyperbolization theorem for fibered 3-manifolds, volume 7 of SMF/AMS Texts and Monographs. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1996 French original by Leslie D. Kay.\n- [Far72] F. T. Farrell. The obstruction to fibering a manifold over a circle. Indiana Univ. Math. J., 21:315–346, 1971/72. doi:10.1512/iumj.1971.21.21024.\n- [IMM23] Giovanni Italiano, Bruno Martelli, and Matteo Migliorini. Hyperbolic 5-manifolds that fiber over $S^{1}$. Invent. Math., 231(1):1–38, 2023. doi:10.1007/s00222-022-01141-w.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hyperbolic 5-manifolds fibering over the circle are known, and one 2025 preprint identifies an explicit 4-dimensional pseudo-Anosov monodromy; no general necessary-and-sufficient criterion is known.\n\n**Verified partial progress.**\n\n- Italiano--Martelli--Migliorini construct finite-volume cusped hyperbolic 5-manifolds fibering over S1.\n- Martelli describes one monodromy as a natural 4-dimensional pseudo-Anosov homeomorphism, giving a concrete higher-dimensional analogue of the surface criterion.\n\n**Full solution or refutation.**\n\nExistence and one explicit dynamical model are established, but they do not characterize all pairs (M,f) whose mapping tori are hyperbolic.\n\n**What remains.**\n\nFormulate and prove conditions that are simultaneously necessary and sufficient for arbitrary aspherical four-dimensional fibers and boundary-flat cases.\n\n**Sources checked.**\n\n- Giovanni Italiano, Bruno Martelli, and Matteo Migliorini, Hyperbolic 5-manifolds that fiber over S1, Inventiones Mathematicae 231 (2023), 1--38. (primary): https://doi.org/10.1007/s00222-022-01141-w\n Evidence used: First explicit finite-volume hyperbolic 5-manifolds fibering over the circle.\n- Bruno Martelli, A 4-dimensional pseudo-Anosov homeomorphism, arXiv:2511.10530 (2025). (primary): https://arxiv.org/abs/2511.10530\n Evidence used: Builds one explicit 4-dimensional pseudo-Anosov monodromy whose mapping torus is a hyperbolic 5-manifold.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.127. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: States the open characterization problem and records the first existence examples.\n\n**Review notes.** No formulation defect found. The 2025 pseudo-Anosov notion is one construction, not yet a proved universal criterion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3004, "problem_number": "KP-4.128", "title": "Kirby Problem 4.128", "statement": "What is the structure of 4-manifolds that admit a Riemannian metric of positive scalar curvature? There are variations of this problem for different classes of manifolds.\n\n(a) Is every closed simply connected PSC 4-manifold diffeomorphic to a connected sum of copies of $\\mathbb{CP}^{2}, \\mathbb{CP}^{2}$, and $S^{2} \\times S^{2}$?\n\n(b) What is the structure of non-simply connected closed PSC 4-manifolds?\n\n(c) Which 4-manifolds with boundary have a PSC metric?\n\n(d) Which non-compact 4-manifolds have a complete PSC metric with uniformly positive scalar curvature?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.128.\n\nLiterature notes:\n(1) Let us say that a smooth manifold is a PSC manifold if it admits a Riemannian metric of positive scalar curvature.\n\n(2) The corresponding problem in dimension 2 is easy by the Gauss-Bonnet theorem, and is solved in dimension 3 as a consequence of Perelman’s work. In particular, every orientable closed PSC 3-manifold is a connected sum of spherical space forms and copies of $S^{1} \\times S^{2}$. In dimensions at least 5 the problem is completely solved for simply connected manifolds via index theory and surgery theory, and there is a well-developed obstruction theory in the non-simply connected case; see the survey articles [Ros07b, RS01]. The classical Lichnerowicz obstruction [Lic63] states that a spin PSC 4k-manifold has vanishing $A_{(}-genus$; in dimension 4 this is equivalent to having vanishing signature. Witten [Wit94] showed that PSC 4-manifolds with $b^{+}_{2}$ >1 have vanishing Seiberg-Witten invariants. This implies, for instance, that the existence of a PSC metric depends on the underlying smooth structure. In the non-simply connected case, there are further obstructions based on Rokhlin’s theorem [RS07] and gauge theory [Lin19, Kon19, KT20, KT23].\n\n(3) It is conjectured that the answer to(a)is positive. This seems wildly optimistic, but there are no known counterexamples. A weaker version would allow an exotic $S^{4}$ with a PSC metric as a summand in the connected sum decomposition. The conjecture was stated as a question in [Kir97,\n\nProblem 4.143].\n\n(4) Some standard examples of non-simply connected PSC 4-manifolds are $S^{1} \\times Y$ for Y a spherical space for m, as well as $S^{2} \\times \\Sigma$ and $\\mathbb{RP}^{2} \\times \\Sigma$ for any closed surface $\\Sigma$. Some other constructions are described in the survey [MT21]. The decomposition theorem for PSC 4-manifolds in [BLM23] reproduces some portion of the picture in dimension 3. Problem 4.129 has a discussion of a decomposition question for non-simply connected PSC 4-manifolds, which would reduce the general problem to the classification of PSC 4-dimensional orbifolds with finite orbifold fundamental group.\n\n(5) One standard setting for part (c) requires that the metric be a product near the boundary, in which case the boundary would have positive scalar curvature. There are obstructions to the existence of PSC metrics on bounding 4-manifolds coming from index theory [BG95] and SeibergWitten theory. Formulating a good conjecture here would be welcome. Rosenberg-Weinberger [RW23] discuss a different boundary condition, and conjecture that a manifold has a PSC metric whose boundary has positive mean curvature (with respect to the outward normal) if and only if its double has a PSC metric.\n\n(6) By [Ros07b, Theorem 0.1] every non-compact 4-manifold admits a PSC metric, typically not complete, so one needs to have additional constraints on the geometry at infinity. Using Gromov’s notion of $a \\mu-bubble$, ChodoshMaximo-Mukherjee [CMM24] show that there are exotic $\\mathbb{R}^{4}s$ that do not admit a complete PSC metric with uniformly positive scalar curvature. A particular case of interest, asked by A. Mukherjee, is whether the punctured K3 surface admits such a metric.\n\nReferences cited:\n- [Ros07b] Jonathan Rosenberg. Manifolds of positive scalar curvature: a progress report. In Surveys in differential geometry. Vol. XI, volume 11 of Surv. Differ. Geom., pages 259–294. Int. Press, Somerville, MA, 2007. doi:10.4310/SDG.2006.v11.n1.a9.\n- [RS01] Jonathan Rosenberg and Stephan Stolz. Metrics of positive scalar curvature and connections with surgery. In Surveys on surgery theory, Vol. 2, volume 149 of Ann. of Math. Stud., pages 353–386. Princeton Univ. Press, Princeton, NJ, 2001.\n- [Lic63] André Lichnerowicz. Spineurs harmoniques. C. R. Acad. Sci. Paris, 257:7–9, 1963.\n- [Wit94] Edward Witten. Monopoles and four-manifolds. Math. Res. Lett., 1(6):769–796, 1994. doi:10.4310/MRL.1994.v1.n6.a13.\n- [RS07] Daniel Ruberman and Nikolai Saveliev. Dirac operators on manifolds with periodic ends. J. Gökova Geom. Topol. GGT, 1:33–50, 2007.\n- [Lin19] Jianfeng Lin. The Seiberg-Witten equations on end-periodic manifolds and an obstruction to positive scalar curvature metrics. J. Topol., 12(2):328–371, 2019. doi:10.1112/topo.12090.\n- [Kon19] Hokuto Konno. Positive scalar curvature and higher-dimensional families of Seiberg-Witten equations. Journal of Topology, 12(4):1246–1265, 2019. https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/topo.12117. doi:10.1112/topo.12117.\n- [KT20] Hokuto Konno and Masaki Taniguchi. Positive scalar curvature and 10/8-type inequalities on 4-manifolds with periodic ends. Invent. Math., 222(3):833–880, 2020. doi:10.1007/s00222-020-00979-2.\n- [KT23] Hokuto Konno and Masaki Taniguchi. Positive scalar curvature and homology cobordism invariants. J. Topol., 16(2):679–719, 2023.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [MT21] Agnese Mantione and Rafael Torres. Geography of 4-manifolds with positive scalar curvature. Expo. Math., 39(4):566–582, 2021. doi:10.1016/j.exmath.2021.05.003.\n- [BLM23] Richard H. Bamler, Chao Li, and Christos Mantoulidis. Decomposing 4-manifolds with positive scalar curvature. Adv. Math., 430:Paper No. 109231, 17, 2023. doi: 10.1016/j.aim.2023.109231.\n- [BG95] Boris Botvinnik and Peter B. Gilkey. The eta invariant and metrics of positive scalar curvature. Math. Ann., 302(3):507–517, 1995. doi:10.1007/BF01444505.\n- [RW23] Jonathan Rosenberg and Shmuel Weinberger. Positive scalar curvature on manifolds with boundary and their doubles. Pure Appl. Math. Q., 19(6):2919–2950, 2023. doi:10.4310/pamq.2023.v19.n6.a12.\n- [CMM24] Otis Chodosh, Davi Maximo, and Anubhav Mukherjee. Complete riemannian 4-manifolds with uniformly positive scalar curvature, 2024. arXiv:2407.05574.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Gauge/index obstructions, decomposition results, boundary conjectures, and strong noncompact obstructions are known, but no classification resolves the four-part positive-scalar-curvature program.\n\n**Verified partial progress.**\n\n- Lichnerowicz and Seiberg--Witten theory obstruct PSC in broad closed classes and detect dependence on smooth structure.\n- Bamler--Li--Mantoulidis give a decomposition theorem covering part of the nonsimply connected picture.\n- Rosenberg--Weinberger formulate a doubling criterion under a positive-mean-curvature boundary condition.\n- Chodosh--Maximo--Mukherjee construct uncountably many exotic R4s with no complete uniformly PSC metric and prove broader topological obstructions.\n\n**Full solution or refutation.**\n\nPart (a) remains conjectural, parts (b) and (c) are broad classification programs, and part (d) has substantial obstructions but no complete characterization.\n\n**What remains.**\n\nResolve the simply connected connected-sum conjecture, classify fundamental-group effects, specify and solve boundary-condition variants, and characterize complete uniformly PSC ends.\n\n**Sources checked.**\n\n- Richard H. Bamler, Chao Li, and Christos Mantoulidis, Decomposing 4-manifolds with positive scalar curvature, Advances in Mathematics 430 (2023), 109231. (primary): https://doi.org/10.1016/j.aim.2023.109231\n Evidence used: A decomposition theorem recovering part of the structural picture in dimension four.\n- Otis Chodosh, Davi Maximo, and Anubhav Mukherjee, Complete Riemannian 4-manifolds with uniformly positive scalar curvature, arXiv:2407.05574. (primary): https://arxiv.org/abs/2407.05574\n Evidence used: Topological obstructions and uncountably many exotic R4 examples without complete uniformly PSC metrics.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.128. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current four-part synthesis and explicit remaining conjectures.\n\n**Review notes.** Part (a) repeats CP2 where the historical statement distinguishes opposite orientations; preserved as imported. Part (c) lacks a boundary condition. Background OCR includes '$A_{(}$-genus'.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3005, "problem_number": "KP-4.129", "title": "Kirby Problem 4.129", "statement": "Given a closed, 4-dimensional PSC manifold M, is there a (possibly disconnected) 4-dimensional orbifold $M^{1}$ with isolated singularities such that the following hold. (I) The orbifold M1 also admits a metric of positive scalar curvature. (II) All components of $M^{1}$ have finite orbifold-fundamental group. (III) M can be obtained from $M^{1}$ by a series of 0 and 1 surgeries. Here we also allow 0-surgeries between two orbifold points of the same type, which amounts to a removal of two subsets of the for $m D^{4}/\\Gamma$ and an addition of a copy of $S^{3}/\\Gamma \\times$ [0,1].", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.129.\n\nLiterature notes:\n(1) Because the orbifold $M^{1}$ has isolated singularities, its orbifold fundamental group is just the fundamental group of its regular part.\n\n(2) The following converse statement is true due to the work of Gromov– Lawson [GL80]. If M1 satisfies Property (I) and M can be obtained from $M^{1}$ as in Property (III), then M admits a PSC metric. So if the answer to the problem is ‘yes’, then this would reduce the study of PSC 4-manifolds to the study of PSC 4-orbifolds with finite fundamental group. See problem4.128 for a conjectural picture in the simply connected case.\n\n(3) The examples of non-simply connected closed 4-manifolds admitting a PSC metric described in Question (b)of Problem 4.128 satisfy properties (I)-(III). For example, a bundle over $S^{1}$ with fiber a PSC 3-manifold can be obtained from the unreduced suspension of Y by 0-surgery at the two orbifold points. $Likewise,S^{2} \\times \\Sigma_{g}$ can be obtained from a connected sum of 2g copies of $S^{1} \\times S^{3}$ by surgery on a circle.\n\n(4) It seems likely that the problem can be approached using 4-dimensional Ricci flow once there is a reasonable construction of Ricci flow with surgery in this dimension. Here the 0 and 1 surgeries would correspond to geometric surgeries that excise cylindrical singularities. Other singularities, for example those modeled on non-cylindrical shrinking solitons, would contribute components to $M^{1}$ with finite fundamental group [Bam21a]. Partial progress to the problem was made via minimal surfaces in [BLM23], where Property (II) was proved with the weaker conclusion that every component of M1 has vanishing first Betti number.\n\n(5) Properties (I)–(III) impose nontrivial topological restrictions on M. For example, they imply that M cannot be aspherical; note, however, that non-asphericity was already shown in [CL24]. More generally, Properties (I)–(III) imply that any cover of M must have finite 2-dimensional Urysohn width [Gro88] (for the lift of one and thus any Riemannian metric on M). Here we say that a metric space (X, d) has k-dimensional Urysohn width of at most W if there is a continuous mapp: $X \\to$ Ato akdimensional simplicial complex such that the diameter of any $fiberp^{-1}(a)$ is at most W. The property that any cover of a has finite 2-dimensional Urysohn width imposes a restriction on homotopy type of the manifold. See also [CLL23, LM23a, Bol09] for related results.\n\nReferences cited:\n- [GL80] Mikhael Gromov and H. Blaine Lawson, Jr. The classification of simply connected manifolds of positive scalar curvature. Ann. of Math. (2), 111(3):423–434, 1980. doi:10.2307/1971103.\n- [Bam21a] Richard H. Bamler. On the fundamental group of non-collapsed ancient Ricci flows, 2021. arXiv:2110.02254.\n- [BLM23] Richard H. Bamler, Chao Li, and Christos Mantoulidis. Decomposing 4-manifolds with positive scalar curvature. Adv. Math., 430:Paper No. 109231, 17, 2023. doi: 10.1016/j.aim.2023.109231.\n- [CL24] Otis Chodosh and Chao Li. Generalized soap bubbles and the topology of manifolds with positive scalar curvature. Ann. of Math. (2), 199(2):707–740, 2024. doi:10.4007/annals.2024.199.2.3.\n- [Gro88] M. Gromov. Width and related invariants of Riemannian manifolds. In On the geometry of differentiable manifolds (Rome, 1986), number 163-164 in Astérisque, pages 93–109. Société mathématique de France, 1988.\n- [CLL23] Otis Chodosh, Chao Li, and Yevgeny Liokumovich. Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions. Geom. Topol., 27(4):1635–1655, 2023. doi:10.2140/gt.2023.27.1635.\n- [LM23a] Yevgeny Liokumovich and Davi Maximo. Waist inequality for 3-manifolds with positive scalar curvature. In Perspectives in scalar curvature. Vol. 2, pages 799– 831. World Sci. Publ., Hackensack, NJ, [2023] ©2023.\n- [Bol09] Dmitry Bolotov. About the macroscopic dimension of certain PSC-manifolds. Algebr. Geom. Topol., 9(1):21–27, 2009. doi:10.2140/agt.2009.9.21.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Gromov--Lawson surgery proves the stated PSC implication from the orbifold decomposition, but the proposed decomposition of every PSC 4-manifold remains unverified.\n\n**Verified partial progress.**\n\n- Examples from the preceding PSC family satisfy the proposed properties.\n- The converse direction follows from Gromov--Lawson surgery.\n\n**Full solution or refutation.**\n\nNo universal decomposition theorem was verified.\n\n**What remains.**\n\nReduce arbitrary PSC 4-manifolds to finite-orbifold-fundamental-group pieces or find an obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.129 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records the converse and positive examples while posing the general question.\n\n**Review notes.** OCR defects in the surgery display were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3006, "problem_number": "KP-4.130", "title": "Kirby Problem 4.130", "statement": "Does longitudinal knot surgery using a knot K, along a fiber in a K3 surface always yield a reducible 4–manifold? A completely decomposable 4–manifold?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.130.\n\nLiterature notes:\n(1) Longitudinal knot surgeryis a variation of Fintushel-Stern knot surgery [FS98] with a different gluing map. The original Fintushel-Stern version is defined using a knot K and a square-0 torus T in a 4-manifold X. Then $X_{K}$ is defined as (X −T $\\times D^{2}) \\cup _{\\phi}(S^{1} \\times$ E(K)). Here $E(K)$ is the exterior of the knot, and the gluing map $\\phi$ sends the longitude of K to the boundary of a meridional disk of T. Under appropriate hypotheses, Fintushel and Stern show that this operation multiplies the Seiberg-Witten invariant of X by the Alexander polynomial of K. In particular, if the Seiberg-Witten invariant of X is nontrivial, the same holds for $X_{K}$.\n\n(2) In longitudinal knot surgery, the $S^{1}$ factor in $S^{1} \\times E(K)$ is identified with the boundary of the meridional disk of T. Denote the result by $X_{K}^{\\lambda}$. Taubes shows [Tau16] that in contrast to the standard surgery, the Seiberg-Witten invariant of $X_{K}^{\\lambda}$ vanishes, even if K is nontrivial. This raises the question of whether $X_{K}^{\\lambda}$ splits as a connected sum, or is completely decomposable, i.e. is diffeomorphic to a connected sum $\\#_{m}\\mathbb{CP}^{2}\\# \\#_{n}\\mathbb{CP}^{2}$. Taubes reports, based on communications with Akbulut, Baykur, and Fintushel, that when K is an unknot, then $(K3)^{\\lambda}_{K}$ completely decomposes.\n\n(3) One motivation comes from the search (starting with [Poo86], albeit with opposite orientation conventions) for 4-manifolds that admit a Riemannian metric whose self-dual Weyl curvature $W_{+}$ vanishes. Such metrics are called conformally anti-self-dual. Taubes [Tau16] shows that when K is hyperbolic, X is a K3 surface, and T is a fiber in an elliptic fibration coming from the Kummer construction, the manifold $X_{K}^{\\lambda}$ admits a Riemannian metric with $W_{2,+}$ arbitrarily small. After repeated blowing up by connected sum with $\\mathbb{CP}$, it will have [Tau92] a conformally anti-self-dual metric; for 2–bridge knots, three blowups suffice. Hence it would be of interest to identify the manifold $(K3)^{\\lambda}_{K}$. If K is hyperbolic and $(K3)^{\\lambda}_{K}$ is completely decomposable, then $\\#_{3}\\mathbb{CP}^{2}\\#_{19}\\overline{\\mathbb{CP}}{}^{2}$ would admit a conformally anti-self-dual metric. Such metrics are known on $\\#_{3}\\mathbb{CP}^{2}\\#_{N}\\overline{\\mathbb{CP}}{}^{2}$ when $N \\geq$ 30 by work of Le Brun [Le B95] and Rollin-Singer [RS09].\n\nReferences cited:\n- [FS98] Ronald Fintushel and Ronald J. Stern. Knots, links, and 4-manifolds. Invent. Math., 134(2):363–400, 1998. doi:10.1007/s002220050268.\n- [Tau16] Clifford Henry Taubes. Some 4-manifold geometry from hyperbolic knots in $S^{3}$, 2016. arXiv:1602.01687.\n- [Poo86] Y. Sun Poon. Compact self-dual manifolds with positive scalar curvature. J. Differential Geom., 24(1):97–132, 1986. http://projecteuclid.org/euclid.jdg/1214440260.\n- [Tau92] Clifford Henry Taubes. The existence of anti-self-dual conformal structures. J. Differential Geom., 36(1):163–253, 1992. http://projecteuclid.org/euclid.jdg/1214448445.\n- [LeB95] Claude LeBrun. Anti-self-dual Riemannian 4-manifolds. In Twistor theory (Plymouth), volume 169 of Lecture Notes in Pure and Appl. Math., pages 81–94. Dekker, New York, 1995.\n- [RS09] Yann Rollin and Michael Singer. Constant scalar curvature Kähler surfaces and parabolic polystability. J. Geom. Anal., 19(1):107–136, 2009. doi:10.1007/s12220-008-9053-8.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Longitudinal knot surgery differs from Fintushel--Stern surgery, and no theorem establishing reducibility or complete decomposability for every knot was verified.\n\n**Verified partial progress.**\n\n- Ordinary Fintushel--Stern surgery has Seiberg--Witten control under hypotheses.\n\n**Full solution or refutation.**\n\nThe longitudinal K3 question remains open.\n\n**What remains.**\n\nCompute smooth invariants or a decomposition for longitudinal surgery for general K.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.130 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Explains the altered gluing and retains both reducibility questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3007, "problem_number": "KP-4.131", "title": "Kirby Problem 4.131", "statement": "Does every Lipschitz 4-manifold admit a smooth structure? Is this smooth structure unique if so? Some more specific, related questions are as follows.\n\n(a) Is there a topological, spin, closed, indefinite 4-manifold X that admits a Lipschitz structure and violates the 10/8-inequality $b_{2}(X) \\geq 5|\\sigma(X)|$ +2 by Furuta [Fur01], or the “10/8+4” inequality $b_{2}(X) \\geq 5|\\sigma(X)|$ +4 by Hopkins–Lin–Shi–Xu [HLSX22] for $X \\ne S^{4}, S^{2} \\times S^{2}, K3$.\n\n(b) Let $X \\subset \\mathbb{R}^{4}$ be a Lipschitz embedded 4-manifold with boundary $\\partial X \\cong S^{3}$. Does X admit a unique smooth structure?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.131.\n\nLiterature notes:\n(1) In dimension $n \\ne$ 4, every topological n-manifold X admits a Lipschitz structure by a theorem of Sullivan [Sul79b]. On the other hand, Donaldson and Sullivan [DS89, Theorem 2] showed that there exist closed topological 4-manifolds X that admit more than one inequivalent Lipschitz structure. Donaldson and Sullivan established that the simplest numerical invariants of smooth 4-manifolds (due to Kotschick [Kot89]) are quasiconformal invariants of smooth 4-manifolds.\n\n(2) The inequalities in Problem (a) are proved for smooth 4-manifolds using the variation of Seiberg–Witten theory introduced in [Fur01]. Hence a Lipschitz manifold violating either of those inequalities would suggest that there is no extension of this version (or perhaps other versions) of Seiberg–Witten theory to the setting of Lipschitz 4-manifolds. One reason to suspect that there is no such extension is that the Seiberg–Witten equations on X are defined in terms of the Dirac operator associated to $a Spin^{c}$ structure on X. Sullivan has conjectured [Sul99, Sul95] that the existence of a Dirac operator on X (as part of a full ‘Dirac package’) implies that X is in fact smoothable.\n\n(3) The 4-dimensional Schoenflies conjecture (Problem 4.23) is known [Sul79b] to hold for Lipschitz embeddings $\\varphi: S^{3} \\to \\mathbb{R}^{4}$. Thus a positive solution to Problem (b) would imply the Schoenflies conjecture.\n\nReferences cited:\n- [Fur01] M. Furuta. Monopole equation and the 11 8 -conjecture. Math. Res. Lett., 8(3):279– 291, 2001. doi:10.4310/MRL.2001.v8.n3.a5.\n- [HLSX22] Michael J. Hopkins, Jianfeng Lin, XiaoLin Danny Shi, and Zhouli Xu. Intersection forms of spin 4-manifolds and the $\\mathrm{Pin}(2)$-equivariant Mahowald invariant. Comm. Amer. Math. Soc., 2:22–132, 2022. doi:10.1090/cams/4.\n- [Sul79b] Dennis Sullivan. Hyperbolic geometry and homeomorphisms. In Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pages 543–555. Academic Press, New York-London, 1979.\n- [DS89] SK Donaldson and DP Sullivan. Quasiconformal 4-manifolds. Acta Mathematica, 163:181–252, 1989.\n- [Kot89] Dieter Kotschick. On manifolds homeomorphic to $\\mathbb{CP}^{2}$\\#8$\\mathbb{CP}^{2}$. Invent. Math., 95(3):591–600, 1989. doi:10.1007/BF01393892.\n- [Sul99] Dennis Sullivan. On the foundation of geometry, analysis, and the differentiable structure for manifolds. In Topics in low-dimensional topology (University Park, PA, 1996), pages 89–92. World Sci. Publ., River Edge, NJ, 1999. doi:10.1142/4202.\n- [Sul95] Dennis Sullivan. Exterior d, the local degree, and smoothability. In Prospects in topology (Princeton, NJ, 1994), volume 138 of Ann. of Math. Stud., pages 328– 338. Princeton Univ. Press, Princeton, NJ, 1995.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** There are closed topological 4-manifolds with inequivalent Lipschitz structures, while existence of a smoothing for every Lipschitz 4-manifold and the stated uniqueness problems remain open.\n\n**Verified partial progress.**\n\n- Donaldson--Sullivan provide nonunique Lipschitz structures.\n- Smooth spin inequalities motivate a potential obstruction in part (a).\n\n**Full solution or refutation.**\n\nNo universal smoothing or uniqueness theorem was verified.\n\n**What remains.**\n\nConstruct a nonsmoothable Lipschitz example or prove smoothing/uniqueness in the requested classes.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-4.131 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Records Donaldson--Sullivan's nonuniqueness result and asks the remaining questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3008, "problem_number": "KP-5.1", "title": "Kirby Problem 5.1", "statement": "Does every cellular set in the plane have the fixed point property?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.1.\n\nLiterature notes:\n- A space has the fixed point property (FPP) if every self-map has a fixed point. A set in an $n$-manifold is cellular if it is the intersection of a countable sequence of embedded closed $n$-cells, each contained in the interior of the preceding one. A subset of the plane is cellular if and only if it is non-empty, compact, connected and non-separating (meaning that its complement is connected). Since a (planar) cellular set is the intersection of cells, and since the FPP holds for cells, it is natural to ask if it also has the FPP.\n\n- The Mandelbrot set [BM81, Man80] in the plane is cellular, but it is unknown whether it has the FPP.\n\n- The answer in dimensions $\\geq 3$ is no; a counterexample is due to Kinoshita [Kin53]. See the expository article by Bing [Bin69].\n\nReferences cited:\n- [BM81] Robert Brooks and J. Peter Matelski. The dynamics of 2-generator subgroups of P$\\mathrm{SL}(2,\\mathbb{C})$. In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume No. 97 of Ann. of Math. Stud., pages 65–71. Princeton Univ. Press, Princeton, NJ, 1981.\n- [Man80] Benoit B. Mandelbrot. Fractal aspects of the iteration of z $\\to$ $\\lambda z(1-z)$ for complex λ and z. Annals of the New York Academy of Sciences, 357(1):249–259, 1980. doi:10.1111/j.1749-6632.1980.tb29690.x.\n- [Kin53] Shin’ichi Kinoshita. On some contractible continua without fixed point property. Fund. Math., 40:96–98, 1953. doi:10.4064/fm-40-1-96-98.\n- [Bin69] R. H. Bing. The elusive fixed point property. Amer. Math. Monthly, 76:119–132, 1969. doi:10.2307/2317258.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof or counterexample was verified for the fixed-point property of every planar cellular set.\n\n**Verified partial progress.**\n\n- The problem is an instance of fixed-point theory beyond absolute retract settings.\n\n**Full solution or refutation.**\n\nThe universal fixed-point property remains open.\n\n**What remains.**\n\nConstruct a fixed-point-free self-map or prove a planar continuum theorem.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-5.1 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: Defines the FPP and retains the question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3009, "problem_number": "KP-5.2", "title": "Kirby Problem 5.2", "statement": "(Doubly-Small Morphisms of Manifolds).\n\n- Suppose that $h: \\R^{n} \\to \\R^{n}$ is a homeomorphism (or diffeomorphism) which satisfies two smallness hypotheses:\n\n- every orbit of $h$ (of any $x\\in \\R^{n}$, under all powers of $h$) is uniformly bounded in diameter (by 1 say), and\n\n- some subsequence of powers of $h$ converges to $\\mathrm{Id}_{\\R^{n}}$ in (say) the compact-open topology.\n\nThen must $h$ be the identity map?\n\n- A special case of this question is: let $h$ be a homeomorphism of $B^{n}$ that is the identity on $\\partial B^{n}$. If there is a subsequence of powers of $h$ which converge to the identity, must $h$ itself be the identity?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.2.\n\nLiterature notes:\n- The case $n=1$ is trivial, $n=2$ seems likely to be true, as a consequence of results of Brouwer and Cartwright-Littlewood (nicely and succinctly reproved in [Bro84] and [Bro77]), and $n\\geq 3$ is open.\n\n- The question is meant to be local in nature, i.e. the question can be adapted to any open subset of $\\R^{n}$. It also applies to any manifold, where in Condition (i) one would assume that every orbit of $h$ has diameter less than some $\\epsilon$ in the compact-open topology.\n\n- The answer is `yes' if $h$ is periodic. This is Newman's Theorem, with an excellent exposition in [Dre69].\n\n- For $h$ a diffeomorphism, and using $C^{\\infty}$ convergence, the answer seems likely to be yes.\n\n- Since a homeomorphism $h: \\R^{n}\\to \\R^{n}$ generates a homomorphism $\\varphi: \\Z \\to \\Homeo(\\R^{n})$ (and vice-versa) by $m\\mapsto h^{m}$, the Question can be rephrased in terms of such a $\\varphi$. Condition (i) becomes: Assume that $\\image(\\varphi)$ lies in a suitably small neighborhood of $\\mathrm{Id}_{\\R^{n}}$, and Condition (ii) becomes: Assume that $\\varphi$ accumulates at $\\mathrm{Id}_{\\R^{n}}$. And the Question becomes: Must $\\varphi$ be the trivial homomorphism?\n\n- The question is `stronger' than the Hilbert--Smith Conjecture, discussed in Problem 5.3 below. That is, an affirmative answer to it would imply the Hilbert--Smith Conjecture. The Hilbert--Smith Conjecture (in its Question form) is equivalent to the Question above if in addition one assumes that the closure of the union of the powers of $h$ in $\\Homeo(\\R^{n})$ is compact.\n\nReferences cited:\n- [Bro84] Morton Brown. A new proof of Brouwer’s lemma on translation arcs. Houston J. Math., 10(1):35–41, 1984.\n- [Bro77] Morton Brown. A short short proof of the Cartwright-Littlewood theorem. Proc. Amer. Math. Soc., 65(2):372, 1977. doi:10.2307/2041926.\n- [Dre69] Andreas Dress. Newman’s theorems on transformation groups. Topology, 8:203–207, 1969. doi:10.1016/0040-9383(69)90010-X.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The doubly-small question is elementary in dimension 1 and affirmative for periodic transformations under Newman's theorem, but remains open in dimensions at least 3; the planar discussion remains nonterminal.\n\n**Verified partial progress.**\n\n- The dimension-one case is elementary.\n- Newman's theorem gives an affirmative answer when h is periodic.\n- With compact closure of the cyclic subgroup generated by h, the question becomes the Hilbert--Smith setting; without that compactness it is strictly stronger.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was found for the stated homeomorphism or diffeomorphism conditions in dimensions at least 3.\n\n**What remains.**\n\nSettle the compact-open recurrence plus uniformly small-orbit problem in dimensions at least 3, and make the expected two-dimensional consequence fully explicit if possible.\n\n**Sources checked.**\n\n- Andreas Dress, Newman's theorems on transformation groups, Topology 8 (1969), 203--207. (primary): https://doi.org/10.1016/0040-9383(69)90010-X\n Evidence used: Provides the periodic small-action theorem used for the periodic special case.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 5.2. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current dimensional status and precise connection with Hilbert--Smith.\n\n**Review notes.** The background's phrase 'diameter less than epsilon in the compact-open topology' is not literally a well-typed metric assertion and was treated as informal rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3010, "problem_number": "KP-5.3", "title": "Kirby Problem 5.3", "statement": "(Hilbert--Smith Conjecture).\n\n- The Hilbert--Smith Conjecture [Smi41] asserts that a locally compact subgroup of the homeomorphism group of a connected manifold must be a Lie group.\n\n- Conjecture: The free-action subset of any Cantor group action on an ENR is a homology-$Z$-subset (of the ENR).", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.3.\n\nLiterature notes:\n- We first define the terms in Conjecture (b).\n\n- A Cantor group is a topological group $G$ whose underlying space is a Cantor set (space). That is, $G$ is a profinite group that is non-finite and 2nd countable (hence metrizable). A universal example of such a group is the direct product of all (of the countably many) finite groups (or, if you wish, the finite symmetry groups). Other important examples of Cantor groups are the $p$-adic integers.\n\n- An ENR is a Euclidean Neighborhood Retract, that is, a subset of some $\\R^{n}$ that has a neighborhood that retracts onto it. Such subsets are characterized as stably having manifold mapping cylinder neighborhoods (like tubular neighborhoods for manifolds, or regular neighborhoods for polyhedra).\n\n- A subset $A$ of $X$ is a homology-$Z$-set (in $X$) if for any $a\\in A$ and any open neighborhood $U$ of $a$ in $X$, the relative homology $H_{*}(U,U-A)$ is 0. If $X$ is a manifold, the only such $A$ are subsets of $\\partial X$.\n\n- Pardon [Par13a] proved the Hilbert--Smith Conjecture for dimension 3. The conjecture has been reduced to whether the locally compact subgroup in question can be the $p$-adic integers. See Pardon's papers [Par13a, Par19] for more discussion and additional references.\n\n- Conjecture (b) is known as the Free-Set (is a) Z-Set Conjecture (FSZSC) and is due to R. Edwards. It is a natural and stronger version of the Hilbert--Smith Conjecture. As a special case, the FSZSC asserts that a Cantor group cannot act freely on an ENR. Like the HSC, the FSZSC has been reduced to the case of proving it for the $p$-adic integers.\n\n- The Hilbert--Smith Conjecture is closely related to the preceding problem 5.2, as discussed above.\n\nReferences cited:\n- [Smi41] P. A. Smith. Periodic and nearly periodic transformations. In Lectures in Topology, pages 159–190. Univ. Michigan Press, Ann Arbor, MI, 1941.\n- [Par13a] John Pardon. The Hilbert-Smith conjecture for three-manifolds. J. Amer. Math. Soc., 26(3):879–899, 2013. doi:10.1090/S0894-0347-2013-00766-3.\n- [Par19] John Pardon. Totally disconnected groups (not) acting on two-manifolds. In Breadth in contemporary topology, volume 102 of Proc. Sympos. Pure Math., pages 187– 193. Amer. Math. Soc., Providence, RI, 2019. doi:10.1090/pspum/102/13.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hilbert--Smith is proved through dimension 3 and for several regularity or symplectic action classes, but remains open for unrestricted homeomorphism actions in dimensions at least 4; the stronger Free-Set Z-Set conjecture also remains open.\n\n**Verified partial progress.**\n\n- Pardon proves Hilbert--Smith for three-manifolds; dimensions 1 and 2 were already known.\n- Hilbert--Smith holds for Lipschitz and stronger regularity classes.\n- Shelukhin proves that p-adic integers admit no nontrivial continuous Hamiltonian-homeomorphism action on symplectically aspherical manifolds and derives related Lie-group conclusions.\n\n**Full solution or refutation.**\n\nNo unrestricted proof in dimensions at least 4 and no proof of Edwards's stronger Free-Set Z-Set conjecture was verified.\n\n**What remains.**\n\nExclude faithful p-adic-integer actions by arbitrary homeomorphisms in dimensions at least 4 and prove the homology-Z-set conclusion for free-action subsets of Cantor-group actions on ENRs.\n\n**Sources checked.**\n\n- John Pardon, The Hilbert-Smith conjecture for three-manifolds, J. Amer. Math. Soc. 26 (2013), 879--899. (primary): https://doi.org/10.1090/S0894-0347-2013-00766-3\n Evidence used: Settles the unrestricted dimension-3 case.\n- Egor Shelukhin, A symplectic Hilbert-Smith conjecture, arXiv:2403.07195 (2024). (primary): https://arxiv.org/abs/2403.07195\n Evidence used: Proves new Hamiltonian-homeomorphism and symplectic special cases and records the general dimensional status.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 5.3. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current formulation of Hilbert--Smith and Edwards's Free-Set Z-Set strengthening.\n\n**Review notes.** The statement omits the usual effective/faithful qualifier in the first Hilbert--Smith bullet; this formulation omission was flagged rather than silently inserted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3011, "problem_number": "KP-5.4", "title": "Kirby Problem 5.4", "statement": "Is the homeomorphism group of a manifold an absolute neighborhood retract (ANR)?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.4.\n\nLiterature notes:\n- This is [Kir97, Problem 5.27] and is also listed in [Wes90]. In dimension 2 it was shown to hold by Mason [Mas71] and Luke-Mason [LM72]; it is unknown for manifolds of dimension greater than 2.\n\n- It is not easy to distinguish between ANRs and more general locally contractible spaces, so it is worth recalling that the homeomorphism group of a manifold is locally contractible [EK71, Čer69].\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Wes90] James E. West. Open problems in infinite-dimensional topology. In Open problems in topology, pages 523–597. North-Holland, Amsterdam, 1990.\n- [Mas71] W. K. Mason. The space of all self-homeomorphisms of a two-cell which fix the cell’s boundary is an absolute retract. Trans. Amer. Math. Soc., 161:185–205, 1971. doi:10.2307/1995936.\n- [LM72] R. Luke and W. K. Mason. The space of homeomorphisms on a compact two-manifold is an absolute neighborhood retract. Trans. Amer. Math. Soc., 164:275– 285, 1972. doi:10.2307/1995974.\n- [EK71] Robert D. Edwards and Robion C. Kirby. Deformations of spaces of imbeddings. Ann. of Math. (2), 93:63–88, 1971. doi:10.2307/1970753.\n- [Čer69] A. V. Černavskiĭ. Local contractibility of the group of homeomorphisms of a manifold. Mat. Sb. (N.S.), 79(121):307–356, 1969. English translation in Math. USSR, Sb. 8, 287–333 (1969).\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The ANR property is proved for homeomorphism groups of compact two-manifolds, while the corresponding question in dimensions greater than two remains open in the 2026 K3 problem list. Local contractibility in higher dimensions does not by itself imply the ANR property.\n\n**Verified partial progress.**\n\n- Luke and Mason proved that the homeomorphism space of a compact two-manifold is an absolute neighborhood retract.\n- Edwards and Kirby proved broad local-contractibility results for manifold homeomorphism groups, a necessary but weaker property than being an ANR.\n\n**Full solution or refutation.**\n\nThere is a complete affirmative answer in dimension two under the standard compact-manifold topology, but no general resolution in higher dimensions was found.\n\n**What remains.**\n\nFix the topology and manifold conventions precisely, then prove the ANR property or construct a counterexample for manifolds of dimension greater than two.\n\n**Sources checked.**\n\n- R. I. Baykur, R. C. Kirby, and D. Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 5.4, author's preliminary AMS version (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Records the two-dimensional theorem and states that the problem is unknown for manifolds of dimension greater than two.\n- R. Luke and W. K. Mason, The space of homeomorphisms on a compact two-manifold is an absolute neighborhood retract, Transactions of the American Mathematical Society 164 (1972), 275--285. (primary): https://doi.org/10.2307/1995974\n Evidence used: Proves the ANR result for the homeomorphism space of a compact two-manifold.\n- R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Annals of Mathematics 93 (1971), 63--88. (primary): https://doi.org/10.2307/1970753\n Evidence used: Provides the deformation theorem underlying local contractibility of manifold homeomorphism groups.\n\n**Review notes.** The supplied statement does not specify compactness, boundary conditions, or the topology on the homeomorphism group; these formulation gaps are preserved and flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3012, "problem_number": "KP-5.5", "title": "Kirby Problem 5.5", "statement": "Is the connected sum of (locally flat) pairs\n\n$$\n(M^{n+k}_{1},N^{n}_{1})\\#(M^{n+k}_{2},N^{n}_{2})\n$$\n\nwell-defined in the topological category?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.5.\n\nLiterature notes:\nWhen $k=1$, this is easily true from Brown's paper showing that locally flat codimension one submanifolds are flat [Bro62]. According to [Liv24], it is well-defined in any dimension when $k=2$. This uses the uniqueness of normal bundles in codimension 2, which fails in higher codimensions; see [Liv24, Appendix C]. However this question appears to be open for codimension $>2$ and $n+k>4$. It would follow from a pairwise version of the `torus trick' [KS77] if a pairwise version of Wall's non-simply connected surgery [Wal99] were known and in print.\n\nReferences cited:\n- [Bro62] Morton Brown. Locally flat imbeddings of topological manifolds. Ann. of Math. (2), 75:331–341, 1962. doi:10.2307/1970177.\n- [Liv24] Charles Livingston. Connected sums of codimension two locally flat submanifolds. Proc. Roy. Soc. Edinburgh Sect. A, 154(6):1937–1944, 2024. doi:10.1017/prm.2022.87.\n- [KS77] Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah.\n- [Wal99] C. T. C. Wall. Surgery on compact manifolds, volume 69 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, 1999. Edited and with a foreword by A. A. Ranicki. doi:10.1090/surv/069.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Connected sum is well-defined for locally flat codimension-one pairs and, under connected oriented hypotheses, for locally flat codimension-two pairs in every dimension. The question remains open in codimension greater than two when the ambient dimension exceeds four.\n\n**Verified partial progress.**\n\n- Brown's local-flatness theorem gives the codimension-one case.\n- Livingston proves independence of the connected-sum choices up to orientation-preserving homeomorphism for connected, oriented, locally flat codimension-two pairs in every dimension.\n\n**Full solution or refutation.**\n\nThe question is affirmatively settled in codimensions one and two within the precise hypotheses of the cited theorems, but is unresolved in the stated higher-codimension range.\n\n**What remains.**\n\nProve choice-independence for codimension greater than two, especially in ambient dimension greater than four, or exhibit pairs for which the result depends on the connected-sum data.\n\n**Sources checked.**\n\n- R. I. Baykur, R. C. Kirby, and D. Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 5.5, author's preliminary AMS version (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Summarizes the codimension-one and codimension-two results and identifies codimension greater than two in ambient dimension greater than four as open.\n- C. Livingston, Connected sums of codimension two locally flat submanifolds, Proceedings of the Royal Society of Edinburgh Section A 154 (2024), 1937--1944. (primary): https://doi.org/10.1017/prm.2022.87\n Evidence used: Proves well-definedness for connected, oriented, locally flat codimension-two manifold pairs and explains the role of normal-bundle uniqueness.\n- M. Brown, Locally flat imbeddings of topological manifolds, Annals of Mathematics 75 (1962), 331--341. (primary): https://doi.org/10.2307/1970177\n Evidence used: Proves local flatness results that yield the codimension-one case cited by the current problem list.\n\n**Review notes.** The supplied statement omits connectedness, orientation, local-orientation, and equivalence conventions. The positive codimension-two status is restricted to Livingston's explicit hypotheses.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3013, "problem_number": "KP-5.6", "title": "Kirby Problem 5.6", "statement": "- Does every closed, PL, orientable $n$-manifold admit an $n$-fold branched covering map over $S^{n}$?\n\nAssuming that the answer to this problem is \"yes\", the following follow-up questions would be natural to ask.\n\n- Does every $n$-manifold admit an $n$-fold branched covering over $S^{n}$ with branch locus a codimension-2 embedded submanifold?\n\n- If the answer to (b) is \"yes\", then one could naturally ask if the cover can additionally be taken to be simple, meaning the covering monodromy sends every meridian of the embedded submanifold to a transposition. For $n=2,3,4$, the above papers show the answer is, \"yes\".\n\n- If the answer to (b) is \"yes\", then in a separate direction one could ask when the branch locus can be taken to be orientable.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.6.\n\nLiterature notes:\n- A classical theorem of Alexander [Ale20] says that every $n$-manifold admits a branched covering map over $S^{n}$ with no restriction on degree of the covering. An easier version of this problem would be to show that for every $n$, there exists some natural number $m_{n}$ such that every $n$-manifold admits an $m_{n}$-fold branched covering over $S^{n}$. Certainly $m_{n}$ cannot be smaller than $n$: for example, because the $n$-torus $T^{n}$ has reduced cohomology ring of length $n$, any branched covering $f:T^{n}\\to S^{n}$ has degree at least $n$ [BE78].\n\n- It is well-known that every orientable surface is a 2-fold branched cover over $S^{2}$. Hilden [Hil74], Hirsch [Hir74], and Montesinos [Mon74] showed that every 3-manifold is a 3-fold branched cover over $S^{3}$. Piergallini [Pie95] showed that every PL 4-manifold is a 4-fold branched cover over $S^{4}$. The answer to this question is unknown in all higher dimensions.\n\n- Note by Berstein--Edmonds [BE78] the answer to Question (b) is \"no\" if we additionally require that this submanifold be locally flat. For $n=2,3$ the answer is known to be \"yes\", but for $n=4$, the best known result to date, by Piergallini in [Pie95], produces branched loci that are immersed surfaces. Iori--Piergallini [IP02] later showed that every 4-manifold admits a 5-fold simple branched covering over $S^{4}$ with branch locus an embedded surface. Whether every 4-manifold admits a 4-fold cover over $S^{4}$ with embedded branch locus remains open.\n\n- The answer to Question (d) is generally negative -- for example, when $n=4k$ the existence of such a covering implies the $n$-manifold has signature zero [Vir84]. For this question, it may be simpler to restrict to the case that the $n$-manifold is null-cobordant.\n\nReferences cited:\n- [Ale20] James W. Alexander. Note on Riemann spaces. Bull. Amer. Math. Soc., 26(8):370– 372, 1920. doi:10.1090/S0002-9904-1920-03319-7.\n- [BE78] Israel Berstein and Allan Edmonds. The degree and branch set of a branched covering. Inventiones Mathematicae, 45(3):213–220, 1978.\n- [Hil74] Hugh M. Hilden. Every closed orientable 3-manifold is a 3-fold branched covering space of $S^{3}$. Bull. Amer. Math. Soc., 80:1243–1244, 1974. doi:10.1090/S0002-9904-1974-13699-2.\n- [Hir74] Ulrich Hirsch. über offene Abbildungen auf die 3-Sphäre. Math. Z., 140:203–230, 1974. doi:10.1007/BF01214163.\n- [Mon74] José Marı́a Montesinos. A representation of closed orientable 3-manifolds as 3-fold branched coverings of $S^{3}$. Bulletin of the American Mathematical Society, 80(5):845–846, 1974.\n- [Pie95] R. Piergallini. Four-manifolds as 4-fold branched covers of $S^{4}$. Topology, 34(3):497– 508, 1995. doi:10.1016/0040-9383(94)00034-I.\n- [IP02] Massimiliano Iori and Riccardo Piergallini. 4-manifolds as covers of the 4-sphere branched over non-singular surfaces. Geom. Topol., 6:393–401, 2002. doi:10.2140/gt.2002.6.393.\n- [Vir84] O. Ya. Viro. The signature of a branched covering. Mat. Zametki, 36(4):549–557, 1984.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The degree-n branched-cover statement holds in dimensions two, three, and four and remains open for every dimension greater than four. Alexander's theorem supplies branched covers with unrestricted degree; embeddedness, simplicity, and orientability of the branch locus introduce separate obstructions and unresolved cases.\n\n**Verified partial progress.**\n\n- Every closed orientable surface is a twofold branched cover of the sphere, and every closed orientable three-manifold is a threefold branched cover of the three-sphere.\n- Piergallini proves the fourfold theorem in dimension four with an immersed branch surface; Iori and Piergallini obtain a fivefold simple cover with embedded branch surface.\n- Berstein and Edmonds provide restrictions on degree and branch sets, while Viro's signature formula obstructs orientable branch locus in dimensions divisible by four when the source has nonzero signature.\n\n**Full solution or refutation.**\n\nThe optimal degree is established through dimension four, but neither the degree-n assertion nor even a uniform dimension-dependent degree bound is known in higher dimensions. The stronger follow-up conditions are not uniformly true and must be separated from the main degree question.\n\n**What remains.**\n\nResolve the degree-n theorem for n greater than four, determine optimal-degree embedded branch loci, and classify when simplicity and orientability can also be imposed.\n\n**Sources checked.**\n\n- R. I. Baykur, R. C. Kirby, and D. Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 5.6, author's preliminary AMS version (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: States the known results in dimensions two through four, the higher-dimensional open range, and the embeddedness and orientability qualifications.\n- H. M. Hilden, Every closed orientable 3-manifold is a 3-fold branched covering space of S^3, Bulletin of the American Mathematical Society 80 (1974), 1243--1244. (primary): https://doi.org/10.1090/S0002-9904-1974-13699-2\n Evidence used: Proves the optimal threefold branched-cover theorem in dimension three.\n- R. Piergallini, Four-manifolds as 4-fold branched covers of S^4, Topology 34 (1995), 497--508. (primary): https://doi.org/10.1016/0040-9383(94)00034-I\n Evidence used: Proves that every closed oriented PL four-manifold is a fourfold branched cover of the four-sphere, with the branch-surface qualifications discussed in the paper.\n- M. Iori and R. Piergallini, 4-manifolds as covers of the 4-sphere branched over non-singular surfaces, Geometry & Topology 6 (2002), 393--401. (primary): https://doi.org/10.2140/gt.2002.6.393\n Evidence used: Establishes a fivefold simple branched-cover representation with embedded nonsingular branch surface.\n\n**Review notes.** The supplied bullets are unlettered despite later references to '(b)' and 'Question (d)'. Its n=4 simplicity sentence is also ambiguous unless the fourfold immersed and fivefold simple embedded results are distinguished. These defects are flagged, not silently repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3014, "problem_number": "KP-5.7", "title": "Kirby Problem 5.7", "statement": "(Montgomery--Yang problem). Does there exist a pseudo-free, smooth, $S^{1}$ action on $S^{5}$ with more than three multiple orbits?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.7.\n\nLiterature notes:\n- This is Problem 4.123 in [Kir97]; see the surveys by Kollár [Kol08] and Şavk [Şav24]. Recall that an $S^{1}$ action is pseudo-free if there are no fixed points and the orbits of finite isotropy are isolated.\n\n- There is a good understanding of the situation for other odd dimensional spheres: $S^{1}$ actions on $S^{3}$ are linear by Seifert [Sei33], whereas every homotopy $2k-1$ sphere (for $k\\geq 4$) admits pseudo-free $S^{1}$ actions with arbitrarily many multiple orbits by Montgomery--Yang [MY72] (for $k=4$) and Petrie [Pet75] (for $k>4$).\n\n- The answer is positive if a Seifert fibered homology 3-sphere $\\Sigma$ with more than 3 multiple fibers bounds an acyclic 4-manifold $W$, where the induced homomorphism by the inclusion of boundary is surjective on the $\\pi_{1}$. Then $\\Sigma\\times D^{2}\\cup W\\times S^{1}=S^{5}$ and $S^{1}$ acts diagonally on $\\Sigma\\times B^{2}$ and trivially on $W$. See also Problem 4.58.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [Kol08] János Kollár. Is there a topological Bogomolov-Miyaoka-Yau inequality? Pure Appl. Math. Q., 4(2, Special Issue: In honor of Fedor Bogomolov. Part 1):203– 236, 2008. doi:10.4310/PAMQ.2008.v4.n2.a1.\n- [Şav24] Oğuz Şavk. A survey of the homology cobordism group. Bull. Amer. Math. Soc. (N.S.), 61(1):119–157, 2024. doi:10.1090/bull/1806.\n- [Sei33] H. Seifert. Topologie Dreidimensionaler Gefaserter Räume. Acta Math., 60(1):147– 238, 1933. doi:10.1007/BF02398271.\n- [MY72] D. C. Montgomery and C. T. Yang. Differentiable pseudo-free circle actions on homotopy seven spheres. In H.T. Ku, L. N. Mann, J. L. Sicks, and J. C. Su, editors, Conference on Compact Transformation Groups, Lecture Notes in Math., pages 41–101. Springer-Verlag, 1972. Proc. of a Conference in Univ. of Massachusetts, Amherst,.\n- [Pet75] T. Petrie. Equivariant quasi-equivalence, transversality and normal cobordism. In R. D. James, editor, Proc. International Congress of Mathematicians, Vancouver, volume 1974, pages 537–541. Canadian Mathematical Congress, Montreal, QC, 1975.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No pseudo-free smooth circle action on S^5 with more than three multiple orbits is known, and the Montgomery--Yang conjecture predicts that none exists. The 2026 K3 list retains the question as open.\n\n**Verified partial progress.**\n\n- The analogous behavior is understood in neighboring odd dimensions: actions on S^3 are linear, while higher odd-dimensional homotopy spheres admit pseudo-free actions with arbitrarily many multiple orbits.\n- Hwang and Keum establish substantial cases of Kollár's algebraic Montgomery--Yang analogue, including the noncyclic case, but this does not resolve the smooth S^5 action problem.\n- A suitable Seifert-fibered homology three-sphere with more than three singular fibers bounding an acyclic four-manifold would produce a positive example; no such filling satisfying the needed fundamental-group condition is known.\n\n**Full solution or refutation.**\n\nThe original existence question remains unresolved; current progress consists of neighboring-dimensional constructions, algebraic analogues, and a concrete conditional construction route.\n\n**What remains.**\n\nProve the three-orbit upper bound for all pseudo-free smooth circle actions on S^5, or construct an action, for example through the stated Seifert-fibered acyclic-filling route.\n\n**Sources checked.**\n\n- R. I. Baykur, R. C. Kirby, and D. Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 5.7, author's preliminary AMS version (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: Retains the Montgomery--Yang problem as open and records the conditional Seifert-fibered construction and neighboring-dimensional results.\n- D. Hwang and J. Keum, Algebraic Montgomery--Yang problem: the noncyclic case, arXiv:0904.2975 (2009). (primary): https://arxiv.org/abs/0904.2975\n Evidence used: Proves a major case of the algebraic analogue for rational homology projective planes with noncyclic quotient singularities.\n- O. Savk, A survey of the homology cobordism group, Bulletin of the American Mathematical Society 61 (2024), 119--157. (authoritative_secondary): https://doi.org/10.1090/bull/1806\n Evidence used: Provides a current survey context for the Montgomery--Yang problem and relevant Seifert-fibered homology spheres.\n\n**Review notes.** No later primary source resolving the smooth S^5 problem was found.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3015, "problem_number": "KP-5.8", "title": "Kirby Problem 5.8", "statement": "Is there a closed aspherical 5-manifold that is not triangulable?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.8.\n\nLiterature notes:\n- In this context, `triangulable' means homeomorphic to a simplicial complex. Davis and Januszkiewicz [DJ91] constructed aspherical non-triangulable 4-manifolds, by applying the process of `hyperbolization' to Freedman's $E_{8}$ manifold. Subsequent to Manolescu's (negative) solution to the triangulation conjecture [Man16b] in dimensions $\\geq 5$, Davis--Fowler--Lafont [DFL14] used a hyperbolization procedure to construct aspherical non-triangulable $n$-manifolds for all $n\\geq 6$. They explain why their procedure breaks down in dimension 5, and explicitly asked [DFL14, \\S3] about the 5-dimensional case.\n\n- It follows from [Sie70] (using the double suspension theorem [Edw06, Can79]; compare [GS80, Mat78]) that any orientable 5-manifold is triangulable. Hence any aspherical non-triangulable 5-manifold would have a triangulable cover. This suggests the following.\n\n\\medskip\\noindent\\textsc{Question.} \\emph{Are there examples of non-triangulable aspherical 4-manifolds in dimension 4 that are virtually triangulable, i.e., that admit a finite cover that can be triangulated?}\n\nThe examples of [DJ91, DFL14] have residually finite fundamental groups, so they would be a good place to start.\n\nReferences cited:\n- [DJ91] Michael W. Davis and Tadeusz Januszkiewicz. Hyperbolization of polyhedra. J. Differential Geom., 34(2):347–388, 1991. http://projecteuclid.org/euclid.jdg/1214447212.\n- [Man16b] Ciprian Manolescu. Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture. J. Amer. Math. Soc., 29(1):147–176, 2016. doi:10.1090/jams829.\n- [DFL14] Michael W. Davis, Jim Fowler, and Jean-François Lafont. Aspherical manifolds that cannot be triangulated. Algebr. Geom. Topol., 14(2):795–803, 2014. doi:10.2140/agt.2014.14.795.\n- [Sie70] L. C. Siebenmann. Are nontriangulable manifolds triangulable? In Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), pages 77–84. Markham Publishing Co., Chicago, IL, 1970.\n- [Edw06] Robert D. Edwards. Suspensions of homology spheres, 2006. URL: https://arxiv.org/abs/math/0610573, arXiv:math/0610573.\n- [Can79] J. W. Cannon. Shrinking cell-like decompositions of manifolds. Codimension three. Ann. of Math. (2), 110(1):83–112, 1979. doi:10.2307/1971245.\n- [GS80] David E. Galewski and Ronald J. Stern. Classification of simplicial triangulations of topological manifolds. Ann. of Math. (2), 111(1):1–34, 1980. doi:10.2307/1971215.\n- [Mat78] Takao Matumoto. Triangulation of manifolds. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pages 3–6. Amer. Math. Soc., Providence, R.I., 1978.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Closed aspherical nontriangulable manifolds are known in dimension four and in every dimension at least six, but the five-dimensional case remains open. Every orientable five-manifold is triangulable, so any example answering the question must be nonorientable.\n\n**Verified partial progress.**\n\n- Davis and Januszkiewicz's hyperbolization methods yield aspherical nontriangulable four-manifolds.\n- Davis, Fowler, and Lafont construct closed aspherical nontriangulable manifolds in every dimension at least six and explicitly explain why their argument does not cover dimension five.\n- The triangulation results cited by the K3 list imply that every orientable five-manifold is triangulable, sharply reducing the search to nonorientable manifolds.\n\n**Full solution or refutation.**\n\nThe problem isolates a genuine dimensional gap: the known high-dimensional obstruction constructions start at dimension six, and orientability rules out all possible five-dimensional examples on one side.\n\n**What remains.**\n\nConstruct a nonorientable closed aspherical five-manifold with nonzero triangulation obstruction, or prove that every closed aspherical five-manifold is triangulable.\n\n**Sources checked.**\n\n- R. I. Baykur, R. C. Kirby, and D. Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 5.8, author's preliminary AMS version (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: States the unresolved five-dimensional gap, records the orientable-five-manifold triangulability restriction, and points to the known dimensions.\n- M. W. Davis, J. Fowler, and J.-F. Lafont, Aspherical manifolds that cannot be triangulated, Algebraic & Geometric Topology 14 (2014), 795--803. (primary): https://doi.org/10.2140/agt.2014.14.795\n Evidence used: Constructs aspherical nontriangulable manifolds in dimensions at least six and explains the obstruction to applying the construction in dimension five.\n- C. Manolescu, Pin(2)-equivariant Seiberg--Witten Floer homology and the triangulation conjecture, Journal of the American Mathematical Society 29 (2016), 147--176. (primary): https://doi.org/10.1090/jams/829\n Evidence used: Disproves the high-dimensional Triangulation Conjecture, supplying the obstruction input used in later aspherical constructions.\n\n**Review notes.** The exact five-dimensional question remains well formed; the orientability restriction is a proved reduction, not a change to the statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3016, "problem_number": "KP-5.9", "title": "Kirby Problem 5.9", "statement": "Let $M_{1}$ and $M_{2}$ be smooth manifolds of dimension $n$. Suppose $M_{1}$ admits an $S$-map into $\\R^{p}$. If $M_{2}$ is homeomorphic to $M_{1}$, then does $M_{2}$ admit an $S$-map as well?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.9.\n\nLiterature notes:\n- For this problem, we consider $C^{\\infty}$ maps $f:M\\to \\R^{p}$ of smooth $n$-dimensional manifolds into $\\R^{p}$ with $n\\geq p\\geq 1$. Let $S$ be a set of (equivalence classes of) singularities of smooth map germs $(\\R^{n},0)\\to (\\R^{p},0)$. A smooth map $f:M\\to \\R^{p}$ is called an $S$-map if all its singularities are equivalent to a singularity belonging to $S$.\n\n- If $M_{2}$ is diffeomorphic to $M_{1}$, then by composing a diffeomorphism $M_{2}\\to M_{1}$ and an $S$-map $M_{1}\\to \\R^{p}$, we get an $S$-map on $M_{2}$. Therefore, if the answer to the above question is negative, then $M_{2}$ is not diffeomorphic to $M_{1}$; in other words, $(M_{1},M_{2})$ is an exotic pair of manifolds.\n\nIn this sense, for $n\\leq 3$, the answer is always affirmative.\n\n- Let $S$ be the singleton consisting of the definite fold singularity represented by the map germ\n\n$$\n(x_{1},x_{2},\\ldots,x_{p-1},x_{p}^{2}+x_{p+1}^{2}+\\cdots+x_{n}^{2}).\n$$\n\nIn this case, an $S$-map is called a \\emph{special generic map}. Then there are many examples of manifold pairs $(M_{1},M_{2})$ for which the answer is negative as follows.\n\n- For $n\\geq 7$ and $p=n-1,n-2$ and $n-3$, the pair of the standard $n$-sphere and an exotic $n$-sphere is such an example [Sae93b].\n\n- For $n=4$ and $p=1,2$ and 3, the pair of the standard $\\R^{4}$ and an exotic $\\R^{4}$ is such an example, provided that we consider proper special generic maps [Sae10].\n\n- For $n=4$ and $p=3$, there are quite a few such examples $(M_{1},M_{2})$, where $M_{1}$ is a connected sum of $S^{2}\\times S^{2}$ and $\\CP^{2}\\#\\overline{\\CP}{}^{2}$ [Sae93a, SS99].\n\n- According to the solution to the Poincaré Conjecture in high dimensions due to Smale [Sma61], for special generic functions with $p=1$, the answer is affirmative for $n\\geq 5$. In other words, only by the existence of a special generic function with $n\\geq 5$ and $p=1$ one cannot detect exotic differentiable structures. Note also that for $n=4$ and $p=1$, the problem for special generic functions is equivalent to the 4-dimensional smooth Poincaré Conjecture.\n\nOn the other hand, for $p=1$, let $S$ be the set of non-degenerate critical points of indices in the set $\\{0,2,3,\\ldots,n-2,n\\}$. Then, for $n\\geq 5$, by Smale's result [Sma62a], the answer to the problem is always affirmative, whereas for $n=4$, the problem seems to be still open.\n\n- Let us consider $C^{\\infty}$ stable maps of closed 4-manifolds into $\\R^{4}$. They have, in general, fold, cusp, swallowtail, butterfly and umbilic singularities. It is known that when the 4-manifold is oriented, each umbilic singularity can be given a sign $+1$ or $-1$ and the total number of umbilic points counted with signs is equal to 3 times the signature. Furthermore, if the signature vanishes, then all the umbilic singularities can be eliminated by homotopy [And82, Sti95]. Hence, for $S$ consisting of fold, cusp, swallowtail and butterfly singularities, the answer to the above question is affirmative in this case.\n\n- Let $F$ be a set of (equivalence classes) of singular fibers of $C^{\\infty}$ maps in the following sense [Sae04]. Let $f_{i}:M_{i}\\to N_{i}$ be smooth maps, $i=0,1$. For $y_{i}\\in N_{i}$, we say that the fibers over $y_{0}$ and $y_{1}$ are equivalent if for some open neighborhoods $U_{i}$ of $y_{i}$ there exist diffeomorphisms $\\widetilde{\\phi}:f_{0}^{-1}(U_{0})\\to f_{1}^{-1}(U_{1})$ and $\\phi:U_{0}\\to U_{1}$ with $\\phi(y_{0})=y_{1}$, which make the following diagram commutative:\n\n$$\n\\begin{CD}\n(f_{0}^{-1}(U_{0}),f_{0}^{-1}(y_{0})) @>{\\widetilde{\\phi}}>> (f_{1}^{-1}(U_{1}),f_{1}^{-1}(y_{1}))\\\\\n@V{f_{0}}VV @VV{f_{1}}V\\\\\n(U_{0},y_{0}) @>{\\phi}>> (U_{1},y_{1})\n\\end{CD}\n$$\n\nWhen the fibers over $y_{0}$ and $y_{1}$ are equivalent, we also say that for $i=0,1$, the map germs $f_{i}:(M_{i},f_{i}^{-1}(y_{i}))\\to (N_{i},y_{i})$ are right-left equivalent. A smooth map $f:M\\to \\R^{p}$ is called an $F$-map if all its singular fibers are equivalent to a singular fiber belonging to $F$. Then, we can ask the same question as Problem 5.9 for $F$-maps.\n\n- It is known that for maps of smooth closed oriented 4-manifolds into $\\R^{3}$, each singular fiber of type $\\mathrm{III}^{8}$ can be given a sign $+1$ or $-1$, and for a certain class of generic maps (so-called $C^{\\infty}$ stable maps), the number of $\\mathrm{III}^{8}$-fibers counted with signs coincides with the signature of the source 4-manifold [SY06]. Therefore, it is a natural question if singular fibers of type $\\mathrm{III}^{8}$ with opposite signs can be eliminated by homotopy. So far, we do not know if there exists a smooth closed oriented 4-manifold $M$ with zero signature such that an arbitrary $C^{\\infty}$ stable map $M\\to \\R^{3}$ necessarily has a singular fiber of type $\\mathrm{III}^{8}$. Though for connected sums of copies of $S^{2}\\times S^{2}$ and $\\CP^{2}\\#\\overline{\\CP}{}^{2}$, the answer is known to be affirmative.\n\n- In the problem, one can also consider the problem by replacing \"homeomorphic to\" by \"homotopy equivalent to\". For example, for $n\\geq p\\geq 1$, let $S$ be the set of fold singularities of all (absolute) indices. Then, the answer to the homotopy version of the problem is affirmative for closed orientable 4-manifolds and $1\\leq p\\leq 4$ [Sae03, Sad04].\n\nReferences cited:\n- [Sae93b] Osamu Saeki. Topology of special generic maps of manifolds into Euclidean spaces. Topology Appl., 49(3):265–293, 1993. doi:10.1016/0166-8641(93)90116-U.\n- [Sae10] Osamu Saeki. Special generic maps on open 4-manifolds. J. Singul., 1:1–12, 2010. doi:10.5427/jsing.2010.1a.\n- [Sae93a] Osamu Saeki. Topology of special generic maps into $\\mathbb{R}^{3}$. Mat. Contemp., 5:161– 186, 1993. Workshop on Real and Complex Singularities (São Carlos, 1992).\n- [SS99] Osamu Saeki and Kazuhiro Sakuma. Special generic maps of 4-manifolds and compact complex analytic surfaces. Math. Ann., 313(4):617–633, 1999. doi:10.1007/s002080050275.\n- [Sma61] Stephen Smale. Generalized Poincaré’s conjecture in dimensions greater than four. Ann. of Math. (2), 74:391–406, 1961. doi:10.2307/1970239.\n- [Sma62a] S. Smale. On the structure of manifolds. Amer. J. Math., 84:387–399, 1962. doi: 10.2307/2372978.\n- [And82] Yoshifumi Ando. Elimination of certain Thom-Boardman singularities of order two. J. Math. Soc. Japan, 34(2):241–267, 1982. doi:10.2969/jmsj/03420241.\n- [Sti95] Robert Paul Stingley. Singularities of maps between 4-manifolds. ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–State University of New York at Stony Brook. URL: http://gateway.proquest.com/openurl?url ver=Z39.88-2004\\&rft val fmt= info:ofi/fmt:kev:mtx:dissertation\\&res dat=xri:pqdiss\\&rft dat=xri:pqdiss:9539899.\n- [Sae04] Osamu Saeki. Topology of singular fibers of differentiable maps, volume 1854 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2004. doi:10.1007/b100393.\n- [SY06] Osamu Saeki and Takahiro Yamamoto. Singular fibers of stable maps and signatures of 4-manifolds. Geom. Topol., 10:359–399, 2006. doi:10.2140/gt.2006.10.359.\n- [Sae03] Osamu Saeki. Fold maps on 4-manifolds. Comment. Math. Helv., 78(3):627–647, 2003. doi:10.1007/s00014-003-0758-9.\n- [Sad04] Rustam Sadykov. Elimination of singularities of smooth mappings of 4-manifolds into 3-manifolds. Topology Appl., 144(1-3):173–199, 2004. doi:10.1016/j.topol.2004.04.006.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The universal assertion is false. For S consisting of definite fold singularities, S-maps are special generic maps, and homeomorphic smooth manifolds can differ in whether they admit such maps; standard and suitable exotic spheres provide counterexamples.\n\n**Verified partial progress.**\n\n- Saeki shows that suitable exotic n-spheres for n at least seven admit no special generic maps into target dimensions n-1, n-2, or n-3, whereas the standard sphere admits the corresponding maps.\n- Further counterexamples occur for proper special generic maps on standard versus exotic R^4.\n- Affirmative restricted regimes remain, including low source dimensions and several carefully specified singularity and dimension classes, but they do not imply the universal statement.\n\n**Full solution or refutation.**\n\nChoose S to contain only the definite fold singularity and compare a standard sphere with a homeomorphic exotic sphere excluded by Saeki's theorem. The standard sphere admits a special generic projection, while the exotic smooth sphere does not, directly disproving homeomorphism invariance in general.\n\n**What remains.**\n\nClassify the singularity sets S and dimension pairs (n,p) for which existence of an S-map is a homeomorphism invariant, including the unresolved four-dimensional function and singular-fiber cancellation cases.\n\n**Sources checked.**\n\n- R. I. Baykur, R. C. Kirby, and D. Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 5.9, author's preliminary AMS version (2026). (maintained_tracker): https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf\n Evidence used: States the general question, records the special-generic counterexamples, and surveys affirmative restricted cases and remaining subquestions.\n- O. Saeki, Topology of special generic maps of manifolds into Euclidean spaces, Topology and its Applications 49 (1993), 265--293. (primary): https://doi.org/10.1016/0166-8641(93)90116-U\n Evidence used: Provides exotic-sphere obstructions to special generic maps into high-codimension-nearby Euclidean targets, yielding homeomorphic but smoothly distinct counterexamples.\n- O. Saeki, Special generic maps on open 4-manifolds, Journal of Singularities 1 (2010), 1--12. (primary): https://doi.org/10.5427/jsing.2010.1a\n Evidence used: Studies proper special generic maps on open four-manifolds and supplies the standard-versus-exotic R^4 phenomenon cited in the current status.\n- O. Saeki and T. Yamamoto, Singular fibers of stable maps and signatures of 4-manifolds, Geometry & Topology 10 (2006), 359--399. (primary): https://doi.org/10.2140/gt.2006.10.359\n Evidence used: Establishes four-dimensional stable-map and signature results relevant to affirmative restricted regimes.\n\n**Review notes.** The supplied statement uses an undefined \\R macro. The intended Euclidean target is clear from its background, but the exact text is preserved and the notation defect is flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3017, "problem_number": "KP-5.10", "title": "Kirby Problem 5.10", "statement": "The Andrews--Curtis Conjecture [AC65] for the trivial group: a presentation of the trivial group can be changed to the trivial presentation by Andrews--Curtis moves.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.10.\n\nLiterature notes:\n- This problem appears in Problem 5.2 in [Kir97].\n\n- Let $\\mathcal{P}$ be a finite presentation of a given group $\\pi$. The stabilized Andrews--Curtis moves (abbreviated AC moves) change the presentation $\\mathcal{P}=\\{x_{1},\\ldots,x_{n}:R_{1},\\ldots,R_{m}\\}$ as follows:\n\n- $R_{i}\\to R_{i}^{-1}$,\n\n- $R_{i}\\to R_{i}R_{j}$, $i\\neq j$,\n\n- $R_{i}\\to wR_{i}w^{-1}$, $w$ any word,\n\n- add generator $x_{n+1}$ and relation $wx_{n+1}$.\n\nThe AC Conjecture for the trivial group states that the presentation $\\mathcal{P}$ can be changed to the trivial presentation $(x_{1},\\ldots,x_{n}:x_{1},\\ldots,x_{n})$ by AC moves.\n\nNote that redundant relations cannot be added, so that $m-n$ is unchanged. Also note that the broader conjecture that any two presentations of an arbitrary finitely presented group are equivalent by AC moves is false for some nontrivial groups, e.g., the trefoil group [HAMS93]. (See also [AC66].)\n\n- Given a presentation $\\mathcal{P}$ of the trivial group, we can construct a 5-dimensional handlebody $Y$ from a 0-handle, 1-handles for each generator, and 2-handles for each relation; $Y$ is unique because the attaching maps are isotopic because they are homotopic.\n\nThen if the AC Conjecture is true, $Y$ is diffeomorphic to the 5-ball, because the AC moves correspond to handle moves (in particular the move $R_{i}\\to R_{i}R_{j}$, $i\\neq j$ corresponds to sliding the $i^{\\mathrm{th}}$ handle over the $j^{\\mathrm{th}}$ handle).\n\nReferences cited:\n- [AC65] J. J. Andrews and M. L. Curtis. Free groups and handlebodies. Proceedings of the American Mathematical Society, 16:192–195, 1965.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [HAMS93] C. Hog-Angeloni, W. Metzler, and A. J. Sieradski. Two-dimensional Homotopy and Combinatorial Group Theory, volume 197 of London Math. Soc. Lect. Note Ser. Cambridge Univ. Press, 1993.\n- [AC66] J. J. Andrews and M. L. Curtis. Extended Nielsen operations in free groups. Amer. Math. Monthly, 73:21–28, 1966. doi:10.2307/2313917.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Andrews--Curtis conjecture for trivial presentations remains open despite many proposed counterexamples and extensive finite searches.\n\n**Verified partial progress.**\n\n- The list records the move system and its topological connections.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was verified.\n\n**What remains.**\n\nFind an invariant of Andrews--Curtis equivalence or a certified obstruction.\n\n**Sources checked.**\n\n- K3: A New Problem List in Low-Dimensional Topology, Problem KP-5.10 (2026). (authoritative_secondary): https://aimath.org/pastworkshops/kirbylistrep.pdf\n Evidence used: States the classical conjecture and retains it.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3018, "problem_number": "KP-5.11", "title": "Kirby Problem 5.11", "statement": "(Whitehead's Asphericity Question). Is every subcomplex of an aspherical 2-complex aspherical?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.11.\n\nLiterature notes:\n- This was problem 5.4 in [Kir97]; a thorough discussion of older progress on this problem, also known as the Whitehead Conjecture, may be found in [HAMS93, Chapter X]. It was stated as a question on page 428 in [Whi41], with the implicit assumption that the complex is finite, but the question still makes sense without that assumption.\n\n- The fundamental group of a finite-dimensional aspherical complex is torsion free, so an \\emph{a priori} easier question is whether the fundamental group of a subcomplex of an aspherical two-complex is necessarily torsion-free. For partial results relating to fundamental group structure, see [Coc54, Ada55, How79].\n\n- Bestvina and Brady [BB97] showed that the Whitehead conjecture and the Eilenberg--Ganea conjecture cannot both be true. The Eilenberg--Ganea conjecture [EG57] is that a group with cohomological dimension 2 has a 2-dimensional Eilenberg-Mac Lane space.\n\nMore concretely, Bestvina and Brady showed that the (Bestvina--Brady) group $H_{L}$ associated with a flag triangulation $L$ of a spine of the Poincaré homology sphere is either a counterexample to the Eilenberg--Ganea conjecture, or there exists a contractible 2-complex $Y$ that contains a non-aspherical subcomplex [BB97, Theorem 8.7]. Note that the potential counterexample $Y$ to the Whitehead conjecture that they construct is infinite (since the group $H_{L}$ acts freely and cellularly on it).\n\n- In [How79], J. Howie reduced the problem to finding counterexamples $K\\subset L$ of two types: (i) $L$ is finite and contractible and $K=L-e$ for some 2-cell $e$; or (ii) $L$ is the union of an infinite chain of non-aspherical subcomplexes $K=K_{0}\\subset K_{1}\\subset \\cdots$ such that each inclusion is null-homotopic.\n\nAccording to Howie's results in [How79] and [How83], if the Andrews--Curtis conjecture (Problem 5.10) holds, then the standard 2-complexes associated with LOT presentations account for all test cases of type (i). Recall that a version of the Andrews--Curtis conjecture asserts that every finite contractible 2-complex can be 3-deformed to a point. A LOT presentation is a group presentation described by a (finite) labeled oriented tree, and the associated 2-complex has the homotopy type of a ribbon disc complement [How83].\n\nIt remains unknown whether all LOT complexes are aspherical, though significant progress has been made in the last decades, establishing the asphericity of various subfamilies. For instance, Harlander and Rosebrock [HR17] showed that alternating ribbon disk-complements, the ones that can be encoded by injective labeled oriented trees, are aspherical.\n\n- Costa and Farber [CF17] give a model for random simplicial complexes in which aspherical 2-complexes satisfy the Whitehead conjecture with probability 1.\n\nReferences cited:\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [HAMS93] C. Hog-Angeloni, W. Metzler, and A. J. Sieradski. Two-dimensional Homotopy and Combinatorial Group Theory, volume 197 of London Math. Soc. Lect. Note Ser. Cambridge Univ. Press, 1993.\n- [Whi41] J. H. C. Whitehead. On adding relations to homotopy groups. Ann. of Math. (2), 42:409–428, 1941. doi:10.2307/1968907.\n- [Coc54] W. H. Cockroft. On 2-dimensional aspherical complexes. Proc. London Math. Soc., 4:375–384, 1954.\n- [Ada55] J. F. Adams. A new proof of a theorem of W. H. Cockcroft. J. London Math. Soc., 30:482–488, 1955.\n- [How79] James Howie. Aspherical and acyclic 2-complexes. J. London Math. Soc. (2), 20(3):549–558, 1979. doi:10.1112/jlms/s2-20.3.549.\n- [BB97] Mladen Bestvina and Noel Brady. Morse theory and finiteness properties of groups. Invent. Math., 129(3):445–470, 1997. doi:10.1007/s002220050168.\n- [EG57] Samuel Eilenberg and Tudor Ganea. On the Lusternik-Schnirelmann category of abstract groups. Ann. of Math. (2), 65:517–518, 1957. doi:10.2307/1970062.\n- [How83] James Howie. Some remarks on a problem of J. H. C. Whitehead. Topology, 22(4):475–485, 1983. doi:10.1016/0040-9383(83)90038-1.\n- [HR17] Jens Harlander and Stephan Rosebrock. Injective labeled oriented trees are aspherical. Math. Z., 287(1-2):199–214, 2017. doi:10.1007/s00209-016-1823-6.\n- [CF17] A. Costa and M. Farber. Large random simplicial complexes, II; the fundamental group. J. Topol. Anal., 9(3):441–483, 2017. doi:10.1142/$S^{1}$793525317500170.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Whitehead asphericity remains open, but it is known for injective labeled oriented trees, several group-theoretic and random-complex classes, and rational or completion analogues.\n\n**Verified partial progress.**\n\n- Harlander--Rosebrock prove that injective labeled oriented trees are aspherical, resolving a substantial family of the critical LOT test cases.\n- Mikhovich proves rational and p-adic/completion analogues and records useful equivalent cohomological formulations while explicitly retaining the discrete conjecture as open.\n- Costa--Farber prove almost-sure Whitehead-type conclusions in specified random simplicial-complex models.\n\n**Full solution or refutation.**\n\nNo general proof and no discrete counterexample was verified.\n\n**What remains.**\n\nProve every subcomplex of every aspherical 2-complex is aspherical, or construct a genuine counterexample; in particular, settle the remaining LOT and Howie test cases.\n\n**Sources checked.**\n\n- Jens Harlander and Stephan Rosebrock, Injective labeled oriented trees are aspherical, Math. Z. 287 (2017), 199--214. (primary): https://arxiv.org/abs/1212.1943\n Evidence used: Proves the injective-LOT special case.\n- A. M. Mikhovich, Rational and p-adic analogues of J. H. C. Whitehead's conjecture, Izv. Math. 89 (2025), 274--318. (primary): https://www.mathnet.ru/eng/im9597\n Evidence used: States the discrete conjecture remains open and proves completion analogues and reformulations.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS 295 (2026), Problem 5.11. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current synthesis of Howie's reductions, LOT progress, and open status.\n\n**Review notes.** The background's Costa--Farber DOI is visibly corrupted as 10.1142/$S^{1}$793525317500170; it was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3019, "problem_number": "KP-5.12", "title": "Kirby Problem 5.12", "statement": "(Zeeman Conjecture). If $K$ is a finite contractible 2-complex, then $K\\times I$ collapses to a point [Zee64, Conjecture (1)].", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.12.\n\nLiterature notes:\nThis problem appears as part of Problem 5.2 in [Kir97].\n\nThe Zeeman Conjecture for a special polyhedron that is the spine of a compact 3-manifold is equivalent [GR83] to the Poincaré Conjecture, and thus true by Perelman. The Zeeman conjecture for special polyhedra that do not embed in compact 3-manifolds is equivalent to the Andrews--Curtis Conjecture [Mat87]. A nice discussion of this general area may be found in Chapters I, XI and XII of [HAMS93] and the exposition in [Kup21].\n\nReferences cited:\n- [Zee64] E. C. Zeeman. On the dunce hat. Topology, 2:341–358, 1964. doi:10.1016/0040-9383(63)90014-4.\n- [Kir97] R.C. Kirby. Problems in low–dimensional topology. In W. Kazez, editor, Geometric Topology. American Math. Soc./International Press, Providence, 1997.\n- [GR83] D. Gillman and D. Rolfsen. The Zeeman conjecture for standard spines is equivalent to the Poincaré conjecture. Topology, 22(3):315–323, 1983. doi:10.1016/0040-9383(83)90017-4.\n- [Mat87] S. V. Matveev. The Zeeman conjecture for nonthickenable special polyhedra is equivalent to the Andrews-Curtis conjecture. Sibirsk. Mat. Zh., 28(6):66–80, 218, 1987.\n- [HAMS93] C. Hog-Angeloni, W. Metzler, and A. J. Sieradski. Two-dimensional Homotopy and Combinatorial Group Theory, volume 197 of London Math. Soc. Lect. Note Ser. Cambridge Univ. Press, 1993.\n- [Kup21] Alexander Kupers. Zeeman’s conjecture. Grad. J. Math., 6(1):35–42, 2021. https://gradmath.org/wp-content/uploads/2021/07/GJM2021-Kupers.pdf.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Zeeman conjecture is proved for standard-spine and bounded-complexity fake-surface cases, while the unrestricted finite contractible 2-complex statement remains open.\n\n**Verified partial progress.**\n\n- Gillman--Rolfsen identify the standard-spine case with the Poincare conjecture, so that case is now true.\n- Matveev identifies the nonthickenable-special-polyhedron version with the still-open Andrews--Curtis conjecture.\n- Fagan--Qiu--Wang prove the Zeeman conclusion for contractible cellular fake surfaces of complexity less than 6.\n\n**Full solution or refutation.**\n\nNo proof for arbitrary finite contractible 2-complexes was verified.\n\n**What remains.**\n\nRemove the special-polyhedron and complexity restrictions, equivalently resolving the associated stable Andrews--Curtis/3-deformation bottleneck.\n\n**Sources checked.**\n\n- D. Gillman and D. Rolfsen, The Zeeman conjecture for standard spines is equivalent to the Poincare conjecture, Topology 22 (1983), 315--323. (primary): https://doi.org/10.1016/0040-9383(83)90017-4\n Evidence used: Establishes the equivalence for standard spines.\n- Lucas Fagan, Yang Qiu, and Zhenghan Wang, Stable Andrews-Curtis Conjecture via Fake Surfaces and Zeeman Conjecture, arXiv:2412.12293 (2024). (primary): https://arxiv.org/abs/2412.12293\n Evidence used: Proves the conjecture for contractible cellular fake surfaces of complexity below 6.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 5.12. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current open status and relation to Poincare and Andrews--Curtis.\n\n**Review notes.** No material OCR defect was found in the statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3020, "problem_number": "KP-5.13", "title": "Kirby Problem 5.13", "statement": "Let $M$ be a finite-volume hyperbolic $n$-manifold.\n\n- Does it always have a finite cover with $b_{1}>0$?\n\n- Does it always have a finite cover with fundamental group that surjects onto a non-abelian free group?\n\n- Does it have a finite cover with cubulated fundamental group?\n\n- When $n$ is odd, does it always have a finite cover that fibers over the circle?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.13.\n\nLiterature notes:\n- When $n=3$, these questions all have positive answers, by work of Agol [Ago13] and Wise [Wis21]. However, their methods break down when $n>3$. The crucial steps in their argument are as follows. By work of Kahn-Markovic [KM12], $\\pi_{1}(M^{3})$ contains lots of surface subgroups. These are `codimension 1' subgroups and hence $\\pi_{1}(M^{3})$ has a finite index subgroup that is the fundamental group of a compact non-positively curved cube complex, i.e., it is cubulated. Agol showed that such cubulated groups are virtually special, and Wise showed that virtually special groups have many excellent virtual properties. In particular, they have a finite index subgroup with $b_{1}>0$ and indeed a finite index subgroup that surjects onto a non-abelian free group. Using a 3-dimensional argument involving sutured manifolds, Agol [Ago08] was able to show that hyperbolic 3-manifolds virtually fiber. An alternative and more general argument using group rings was given by Kielak [Kie20].\n\n- This argument fails at the first step when $n>3$. The methods of Kahn-Markovic have been extended to all odd dimensions by Hamenstädt [Ham15], and hence $\\pi_{1}(M)$ is known to contain many surface subgroups. However, these groups are not codimension one, and therefore do not establish that the group is cubulated. However, it is known that some hyperbolic $n$-manifolds are cubulated when $n>3$. Indeed, any arithmetic hyperbolic $n$-manifold containing a totally geodesic $(n-1)$-dimensional (possibly immersed) submanifold is cubulated. When $n$ is even, this includes all arithmetic hyperbolic $n$-manifolds.\n\n- There are further results for arithmetic hyperbolic $n$-manifolds. There are 3 types of arithmetic lattices in $\\mathrm{SO}(n,1)$: those arising from quadratic forms (type I), those arising from quaternion algebras (type II), and those arising from octonion algebras, and type III - trialitarian lattices. The first type appears in all dimensions and are known to be cubulated after work of Bergeron--Wise [BW12]. The second type are not known to be cubulated, but only appear in even dimensions. The third type only occurs in dimension 7; for these it is known [BC17a] that the first Betti number of every congruence subgroup equals 0. Moreover, it is still open if the lattices have the congruence subgroup property.\n\n- If a manifold fibers over the circles, its Euler characteristic is zero. In even dimensions, the volume of a hyperbolic manifold is proportional to its Euler characteristic, and hence its Euler characteristic is necessarily non-zero. This explains the restriction to $n$ odd in the final question above. The first examples of hyperbolic 5-manifolds that fiber over the circle were given by Italiano--Martelli--Migliorini [IMM23].\n\n- When a hyperbolic 3-manifold fibers over the circle, the universal cover $\\widetilde{F}$ of the fiber has a circle at infinity, and the inclusion $\\widetilde{F}\\to \\mathbb{H}^{3}$ is known to extend continuously to a map from the circle at infinity of $\\widetilde{F}$ to the sphere at infinity of $\\mathbb{H}^{3}$. This forms a space-filling curve. This was established by Cannon and Thurston [CT07]. It would be interesting to know whether there is any analogue of this in higher dimensions. The fiber will not in general have word hyperbolic fundamental group, and so part of the challenge would be to define its space at infinity appropriately.\n\n- By work of Delzant and Gromov [DG05], complex hyperbolic manifolds do \\emph{not} have cubulated fundamental group.\n\nReferences cited:\n- [Ago13] Ian Agol. The virtual Haken conjecture. Doc. Math., 18:1045–1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning, https://elibm.org/article/10000267. doi:10.4171/DM/421.\n- [Wis21] Daniel T. Wise. The structure of groups with a quasiconvex hierarchy, volume 209 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, [2021] ©2021.\n- [KM12] Jeremy Kahn and Vladimir Markovic. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Ann. of Math. (2), 175(3):1127–1190, 2012. doi: 10.4007/annals.2012.175.3.4.\n- [Ago08] Ian Agol. Criteria for virtual fibering. J. Topol., 1(2):269–284, 2008. doi:10.1112/jtopol/jtn003.\n- [Kie20] Dawid Kielak. Residually finite rationally solvable groups and virtual fibring. J. Amer. Math. Soc., 33(2):451–486, 2020. doi:10.1090/jams/936.\n- [Ham15] Ursula Hamenstädt. Incompressible surfaces in rank one locally symmetric spaces. Geom. Funct. Anal., 25(3):815–859, 2015. doi:10.1007/s00039-015-0330-y.\n- [BW12] Nicolas Bergeron and Daniel T. Wise. A boundary criterion for cubulation. Amer. J. Math., 134(3):843–859, 2012. doi:10.1353/ajm.2012.0020.\n- [BC17a] Nicolas Bergeron and Laurent Clozel. Sur la cohomologie des variétés hyperboliques de dimension 7 trialitaires. Israel J. Math., 222(1):333–400, 2017. doi:10.1007/s11856-017-1593-9.\n- [IMM23] Giovanni Italiano, Bruno Martelli, and Matteo Migliorini. Hyperbolic 5-manifolds that fiber over $S^{1}$. Invent. Math., 231(1):1–38, 2023. doi:10.1007/s00222-022-01141-w.\n- [CT07] James W. Cannon and William P. Thurston. Group invariant Peano curves. Geom. Topol., 11:1315–1355, 2007. doi:10.2140/gt.2007.11.1315.\n- [DG05] Thomas Delzant and Misha Gromov. Cuts in Kähler groups. In Infinite groups: geometric, combinatorial and dynamical aspects, volume 248 of Progr. Math., pages 31–55. Birkhäuser, Basel, 2005. doi:10.1007/3-7643-7447-0\\\\_3.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** All four virtual-property questions are affirmative in dimension 3 and hold for selected higher-dimensional classes or examples, but remain open universally for real hyperbolic manifolds in dimensions above 3.\n\n**Verified partial progress.**\n\n- Agol--Wise theory gives virtual positive first Betti number, virtual largeness, cubulation/virtual specialness, and virtual fibering in dimension 3.\n- Bergeron--Wise cubulate arithmetic hyperbolic manifolds arising from quadratic forms and related codimension-one settings.\n- Italiano--Martelli--Migliorini construct finite-volume hyperbolic 5-manifolds that actually fiber over the circle; selected algebraic fibrations are known through dimension 8.\n\n**Full solution or refutation.**\n\nThe higher-dimensional existence examples do not show that every finite-volume hyperbolic manifold has such a finite cover.\n\n**What remains.**\n\nResolve the four universal virtual-property questions for n greater than 3, including the difficult type-II and trialitarian arithmetic cases and nonarithmetic lattices.\n\n**Sources checked.**\n\n- Ian Agol, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045--1087. (primary): https://doi.org/10.4171/DM/421\n Evidence used: Core dimension-3 virtual-special and virtual-fibering input.\n- Giovanni Italiano, Bruno Martelli, and Matteo Migliorini, Hyperbolic 5-manifolds that fiber over S1, Invent. Math. 231 (2023), 1--38. (primary): https://doi.org/10.1007/s00222-022-01141-w\n Evidence used: Constructs the first actual fibering hyperbolic 5-manifolds.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 5.13. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current synthesis of all four parts and their higher-dimensional gaps.\n\n**Review notes.** The background says 'fibers over the circles' and has a conflated three-type arithmetic-lattice sentence around octonionic/type-III cases.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3021, "problem_number": "KP-5.14", "title": "Kirby Problem 5.14", "statement": "Does there exist a 1-cusped finite-volume hyperbolic $n$-manifold for any $n\\geq 5$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.14.\n\nLiterature notes:\n- Hyperbolic $n$-manifolds with one cusp are abundant when $n=2$ or 3, and Kolpakov--Martelli [KM13a] constructed 1-cusped hyperbolic 4-manifolds. However, there are far fewer constructions of hyperbolic $n$-manifolds when $n\\geq 4$ and they tend to produce manifolds with a large number of cusps. There are no known examples in dimensions $\\geq 5$ with a single cusp; the question of their existence was raised in [LR02].\n\n- Stover [Sto13] proved that there are no 1-cusped arithmetic hyperbolic $n$-dimensional orbifolds when $n>30$. In fact, he showed that for each $m\\geq 1$, there is a $c_{m}\\geq 1$ such that there are no $m$-cusped arithmetic hyperbolic $n$-dimensional orbifolds when $n>c_{m}$.\n\nReferences cited:\n- [KM13a] Alexander Kolpakov and Bruno Martelli. Hyperbolic four-manifolds with one cusp. Geom. Funct. Anal., 23(6):1903–1933, 2013. doi:10.1007/s00039-013-0247-2.\n- [LR02] D. D. Long and A. W. Reid. All flat manifolds are cusps of hyperbolic orbifolds. Algebr. Geom. Topol., 2:285–296, 2002. doi:10.2140/agt.2002.2.285.\n- [Sto13] Matthew Stover. On the number of ends of rank one locally symmetric spaces. Geom. Topol., 17(2):905–924, 2013. doi:10.2140/gt.2013.17.905.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No one-cusped finite-volume real-hyperbolic manifold of dimension at least 5 was verified; high-dimensional arithmetic obstructions narrow but do not settle the problem.\n\n**Verified partial progress.**\n\n- One-cusped examples are known in dimension 4.\n- Stover proves that one-cusped arithmetic hyperbolic orbifolds cannot exist above dimension 30 and, more generally, bounds dimension for any fixed cusp count.\n- Rizzi's 2025 work constructs new cusp-transitive 4-manifolds and explicitly retains the dimension-above-4 one-cusp question.\n\n**Full solution or refutation.**\n\nNeither a dimension-at-least-5 example nor a general nonexistence theorem was found.\n\n**What remains.**\n\nConstruct a one-cusped manifold in some or every dimension at least 5, depending on the intended reading of 'any,' or prove nonexistence in the relevant dimensions.\n\n**Sources checked.**\n\n- Matthew Stover, On the number of ends of rank one locally symmetric spaces, Geom. Topol. 17 (2013), 905--924. (primary): https://doi.org/10.2140/gt.2013.17.905\n Evidence used: Proves fixed-cusp arithmetic nonexistence in sufficiently high dimension.\n- Edoardo Rizzi, Some cusp-transitive hyperbolic 4-manifolds, Geom. Dedicata (2025). (primary): https://doi.org/10.1007/s10711-025-01010-9\n Evidence used: Constructs dimension-4 examples and explicitly poses the one-cusp question above dimension 4.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 5.14. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current formulation and open status.\n\n**Review notes.** The quantifier 'for any n at least 5' is ambiguous between existence in some dimension and existence in every dimension; no dimension-at-least-5 example was found under either reading.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3022, "problem_number": "KP-5.15", "title": "Kirby Problem 5.15", "statement": "Suppose $M$ is a manifold with a complete Riemannian metric with nonnegative Ricci curvature. Is the fundamental group of $M$ finitely generated?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.15.\n\nLiterature notes:\n- This is a conjecture of Milnor, following on his paper [Mil68a]. The conjecture was known to hold in dimension 2 by work of Cohn-Vossen [CV35] and in dimension 3 by work of Liu [Liu13]. Bruè--Naber--Semola construct counterexamples in dimensions at least 7 [BNS25], and 6 [BNS23]. The issue is therefore to determine the status of the conjecture in dimensions 4 and 5. This question is explicitly posed in [BNS23, Question 1.1], along with some other interesting open problems in the area.\n\nPrevious work of Wilking [Wil00] shows that the fundamental group of any counterexample could be assumed to be abelian; the fundamental group of the example from [BNS25] in dimension 7 is $\\Q/\\Z$.\n\n- A crucial point in the Bruè--Naber--Semola paper [BNS25, Lemma 9.1] is that the orbit of the mapping class group of $S^{3}\\times S^{3}$ acting on the standard product metric lies in a single path component of the space of positive Ricci curvature metrics on $S^{3}\\times S^{3}$. This raises several questions.\n\n- What can be said about the action of the mapping class group of $S^{2}\\times S^{2}$ on the path components of the space of positive Ricci curvature metrics on $S^{2}\\times S^{2}$?\n\n- Is the space of positive Ricci curvature metrics on $S^{2}\\times S^{2}$ path connected?\n\n- One could ask the same questions for $S^{2}\\times S^{3}$; see [BNS23, \\S6] for a particular diffeomorphism of $S^{2}\\times S^{3}$ such that the pullback of the standard metric is connected to the standard metric through Ricci curvature metrics.\n\nReferences cited:\n- [Mil68a] J. Milnor. A note on curvature and fundamental group. J. Differential Geometry, 2:1–7, 1968. http://projecteuclid.org/euclid.jdg/1214501132.\n- [CV35] Stefan Cohn-Vossen. Kürzeste Wege und Totalkrümmung auf Flächen. Compositio Math., 2:69–133, 1935. URL: http://www.numdam.org/item?id=CM 1935 2 69 0.\n- [Liu13] Gang Liu. 3-manifolds with nonnegative Ricci curvature. Invent. Math., 193(2):367– 375, 2013. doi:10.1007/s00222-012-0428-x.\n- [BNS25] Elia Bruè, Aaron Naber, and Daniele Semola. Fundamental groups and the Milnor conjecture. Ann. of Math. (2), 201(1):225–289, 2025. doi:10.4007/annals.2025.201.1.4.\n- [BNS23] Elia Bruè, Aaron Naber, and Daniele Semola. Six dimensional counterexample to the Milnor Conjecture, 2023. arXiv:2311.12155.\n- [Wil00] Burkhard Wilking. On fundamental groups of manifolds of nonnegative curvature. Differential Geom. Appl., 13(2):129–165, 2000. doi:10.1016/S0926-2245(00) 00030-9.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The universal finite-generation assertion is false: smooth complete nonnegative-Ricci manifolds with infinitely generated fundamental group exist in dimensions 7 and 6.\n\n**Verified partial progress.**\n\n- Bruè--Naber--Semola construct a 7-manifold with Ricci curvature at least zero and fundamental group Q/Z.\n- They subsequently construct a six-dimensional counterexample with the same infinitely generated fundamental group.\n- The conjecture remains true in dimensions 2 and 3; dimensions 4 and 5 remain open.\n\n**Full solution or refutation.**\n\nThe 7- and 6-dimensional Q/Z examples directly refute the statement as quantified over all complete manifolds.\n\n**What remains.**\n\nDetermine the conjecture in dimensions 4 and 5 and resolve the auxiliary positive-Ricci metric-space path-component questions.\n\n**Sources checked.**\n\n- Elia Bruè, Aaron Naber, and Daniele Semola, Fundamental groups and the Milnor conjecture, Ann. of Math. 201 (2025), 225--289. (primary): https://annals.math.princeton.edu/2025/201-1/p04\n Evidence used: Peer-reviewed construction of a smooth complete 7-dimensional counterexample with fundamental group Q/Z.\n- Elia Bruè, Aaron Naber, and Daniele Semola, Six dimensional counterexample to the Milnor Conjecture, arXiv:2311.12155. (primary): https://arxiv.org/abs/2311.12155\n Evidence used: Constructs the smooth complete six-dimensional counterexample.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 5.15. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current dimensional status and remaining dimensions 4 and 5.\n\n**Review notes.** The Cohn--Vossen URL and Wilking DOI in the background contain spacing corruption; no bibliographic repair was inserted into the source record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3023, "problem_number": "KP-5.16", "title": "Kirby Problem 5.16", "statement": "Let $R$ be $\\Z$ or a field. Let $A$ and $B$ be differential graded algebras so that either:\n\\begin{itemize}\n\n- As a graded algebra, $A$ (respectively $B$) is isomorphic to the free, non-commutative $R$-algebra on finitely many homogeneous generators $x_{i}$, i.e., to the tensor algebra on the free $R$-module generated by $x_{1},\\ldots,x_{n}$.\n\n- As a graded algebra, $A$ (respectively $B$) is isomorphic to a free graded-commutative algebra on finitely many homogeneous generators $x_{i}$. (So, if the $x_{i}$ have even gradings, $A$ is a polynomial algebra.)\n\\end{itemize}\n(Different generators can have different gradings, $A$ and $B$ may have different numbers of generators, and gradings of generators may be negative.)\n\nAre the following questions decidable?\n\n- Is $A$ stable tame isomorphic to $B$?\n\n- Is $A$ quasi-isomorphic to $B$?\n\n- Is $A$ derived Morita equivalent to $B$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.16.\n\nLiterature notes:\n- This question is inspired by contact topology, where many invariants take the form of a finitely-generated differential graded algebras up to a notion of equivalence.\n\n- The kinds of dgas described in the problem are often called \\emph{semifree} in the literature. The information in a semifree dga $A$ consists of the grading of the generators $x_{1},\\ldots,x_{n}$ and the elements $d(x_{i})$, which are either polynomials in $x_{1},\\ldots,x_{n}$ (in the commutative case) or linear combinations of words in $x_{1},\\ldots,x_{n}$ (in the non-commutative case).\n\n- A dga homomorphism is a quasi-isomorphism if it induces an isomorphism on homology; two dgas are quasi-isomorphic if there is a dga $C$ and quasi-isomorphism $C\\to A$ and $C\\to B$. Stable tame isomorphism of semifree dgas was introduced in [Che02] (see also [ENS02, Section 3.3] and [EN22, Section 3.2]); it allows introducing a pair of canceling generators (stabilizing) and isomorphisms sending some $x_{i}$ to $x_{i}$ plus a word in the other variables. Two dgas are derived Morita equivalent if the derived categories of differential modules over them are equivalent (as triangulated categories). Stable tame isomorphism implies quasi-isomorphism implies derived Morita equivalence.\n\n- This question arises from contact homology and related invariants. For example, to a Legendrian knot $\\Lambda$ in $\\R^{3}$, one can associate a dga $\\mathcal{A}_{\\Lambda}$, the Legendrian contact dga, whose stable tame isomorphism class is an isotopy invariant of $\\Lambda$ (see the citations above). The Legendrian contact dga can also be defined for knots in other manifolds and higher-dimensional Legendrian knots. Contact homology can also be defined for (certain) closed contact manifolds; in this case, one is forced to work with a commutative dga. In practice, it seems to be hard to tell whether two dgas are stable tame isomorphic or quasi-isomorphic.\n\n- One strategy that has been developed to distinguish dgas is to study the set of augmentations of $A$, i.e., dga maps $A\\to R$ (where $R$ lies in grading 0 and has trivial differential); this set is called the augmentation variety of $A$. Given an augmentation of $A$, one can form the linearized homology with respect to that augmentation, which is a finitely generated $R$-module [Che02].\n\n- For Legendrian knots in $\\R^{3}$, the set of augmentations form the objects of a category, the augmentation category, which in the case of Legendrian contact homology is equivalent to a certain category of sheaves [NRS+20]. The set of augmentations and the linearized contact homology depend only on the abelianization of the dga, but the augmentation category needs the non-commutative version.\n\n- Contact homology can also be applied to study smooth objects in low-dimensional topology, for instance by considering the unit cotangent bundle or unit conormal bundle. An instance is Ng's knot contact homology [Ng05] (a variant of which is a complete knot invariant [ENS18]); another is given in Problem 4.106. It is possible that a result along these lines could also be applied to decision problems in symplectic or smooth topology.\n\n- In some of the applications, one actually works over $\\Z\\{t,t^{-1},x_{1},\\ldots,x_{n}\\}$, say, but this seems unlikely to affect the question.\n\n- In the special case that $A$ and $B$ are commutative with all of their generators in positive gradings, Sullivan's theory of minimal models gives an algorithm for answering the question; in particular, that case is well studied in rational homotopy theory. Note also that there is a unique augmentation in this case; in particular, the dgas arising from contact topology rarely have this property.\n\n- Given that the question has some similarity to Hilbert's tenth problem, the answer might be different for $R$ a finite field from $R=\\Q$ or $\\Z$, say.\n\nReferences cited:\n- [Che02] Yuri Chekanov. Differential algebra of Legendrian links. Invent. Math., 150(3):441– 483, 2002. doi:10.1007/s002220200212.\n- [ENS02] John B. Etnyre, Lenhard L. Ng, and Joshua M. Sabloff. Invariants of Legendrian knots and coherent orientations. J. Symplectic Geom., 1(2):321–367, 2002. http: //projecteuclid.org/euclid.jsg/1092316653.\n- [EN22] John B. Etnyre and Lenhard L. Ng. Legendrian contact homology in $\\mathbb{R}^{3}$. In Surveys in differential geometry 2020. Surveys in 3-manifold topology and geometry, volume 25 of Surv. Differ. Geom., pages 103–161. Int. Press, Boston, MA, [2022] ©2022.\n- [NRS+20] Lenhard Ng, Dan Rutherford, Vivek Shende, Steven Sivek, and Eric Zaslow. Augmentations are sheaves. Geom. Topol., 24(5):2149–2286, 2020. doi:10.2140/gt.2020.24.2149.\n- [Ng05] Lenhard Ng. Knot and braid invariants from contact homology. I. Geom. Topol., 9:247–297, 2005. doi:10.2140/gt.2005.9.247.\n- [ENS18] Tobias Ekholm, Lenhard Ng, and Vivek Shende. A complete knot invariant from contact homology. Invent. Math., 211(3):1149–1200, 2018. doi:10.1007/s00222-017-0761-1.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** All three equivalence problems are undecidable for semifree noncommutative DGAs over computable coefficient rings; the graded-commutative half remains open.\n\n**Verified partial progress.**\n\n- Manolescu--Rozenblyum prove undecidability of stable tame isomorphism, quasi-isomorphism, and derived Morita equivalence for finitely generated semifree noncommutative DGAs.\n- Their theorem applies over every nontrivial computable unital commutative ring, including Z, Q, and finite fields, and already uses generators only in degrees 0 and 1.\n- The paper explicitly identifies the analogous semifree graded-commutative problems as open.\n\n**Full solution or refutation.**\n\nThe noncommutative alternative in the record has a complete negative decidability answer for all three notions, but the disjunctive record also asks the unresolved commutative alternative.\n\n**What remains.**\n\nDecide or prove undecidability of stable tame isomorphism, quasi-isomorphism, and derived Morita equivalence for finitely generated free graded-commutative DGAs in the stated coefficient and grading ranges.\n\n**Sources checked.**\n\n- Ciprian Manolescu and Nick Rozenblyum, Undecidability problems for semifree DG algebras, arXiv:2605.08122 (2026). (primary): https://arxiv.org/abs/2605.08122\n Evidence used: Theorem 1 proves all three noncommutative problems undecidable and explicitly says the commutative half remains open.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 5.16. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Exact original two-branch, three-equivalence problem formulation.\n\n**Review notes.** The statement mixes a LaTeX itemize environment with Markdown hyphens; this formatting defect was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3024, "problem_number": "KP-5.17", "title": "Kirby Problem 5.17", "statement": "Let $(W,\\omega,V_{i},\\phi_{i})$ be two Weinstein structures on a fixed symplectic manifold $(W,\\omega)$ (or equivalently consider two Weinstein handle decompositions). Is there a Weinstein homotopy from $(W,\\omega,V_{1},\\phi_{1})$ to $(W,\\omega,V_{2},\\phi_{2})$?", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.17.\n\nLiterature notes:\n% The PDF prints V_2 in both terms of this convex combination; retained as printed.\n\n- The convex combination $V_{t}=(1-t)V_{2}+tV_{2}$ is a Liouville vector field positively transverse to the boundary, but it may fail to be gradient-like for some $t$. The question is whether we can find a family of convex Liouville vector fields $V_{t}$ that are all gradient-like.\n\n- There are various additional hypotheses one can add which will ensure a Weinstein homotopy exists. For example, by [CE12, Proposition 11.22], if we have a fixed complex Stein structure $J$ on $(W,\\omega)$, any two $J$-convex (plurisubharmonic) functions $\\phi_{1}$ and $\\phi_{2}$ will yield Weinstein homotopic Weinstein structures. There are numerous other specific Weinstein homotopies constructed in [CE12, Chapter 12], that assume some specific properties or estimates about the Liouville form and/or gradient-like function.\n\nFor Weinstein manifolds of dimension strictly greater than 4, there is a notion of \"flexible Weinstein manifolds.\" In this case any two flexible Weinstein structures are Weinstein homotopic [CE12, Theorem 14.5].\n\n- A special case, which may be easier, is the following. Suppose $(W,\\omega)$ admits a symplectomorphism that restricts to a contactomorphism on the boundary, but which is not isotopic to the identity in the class of symplectomorphisms that are contact on the boundary. Let $\\lambda$ be one Liouville form on $(W,\\omega)$, corresponding to a Liouville vector field $V_{1}$ which is gradient-like for some Morse function $\\phi_{1}$. Consider the Liouville form $f^{*}\\lambda$ and denote its corresponding Liouville vector field by $V_{2}$. Then $V_{2}$ is gradient-like for $\\phi_{2}=f^{*}\\phi_{1}$. We can ask if $(W,\\omega,V_{1},\\phi_{1})$ Weinstein homotopic to $(W,\\omega,V_{2},\\phi_{2})$ in this case.\n\n- It may be helpful to think about the Liouville form $\\lambda_{i}$ rather than the Liouville vector field $V_{i}$ (which are related by $\\iota_{V_{i}}\\omega=\\lambda_{i}$). While $\\lambda_{i}$ is not closed (since $d\\lambda_{i}=\\omega$ by the Liouville condition), $\\lambda_{1}-\\lambda_{2}$ is closed and thus represents an element of $H^{1}(W)$. It may be easier to construct a counterexample for manifolds where $H^{1}(W)$ is nontrivial, in which case, one should refine the question to consider manifolds where $H^{1}(W)=0$, or further that $W$ is simply connected.\n\n- This question is open in arbitrary dimensions $>2$, but dimension 4 could be a good place to start.\n\nReferences cited:\n- [CE12] Kai Cieliebak and Yakov Eliashberg. From Stein to Weinstein and back, volume 59 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2012. Symplectic geometry of affine complex manifolds. doi:10.1090/coll/059.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Weinstein-homotopy uniqueness is known for flexible formally homotopic structures and fixed-Stein-structure cases, but not for arbitrary Weinstein structures on a fixed symplectic manifold.\n\n**Verified partial progress.**\n\n- Cieliebak--Eliashberg prove that formally homotopic flexible Weinstein structures in dimension at least 6 are Weinstein homotopic.\n- For a fixed Stein complex structure, Weinstein structures induced by two plurisubharmonic functions are Weinstein homotopic.\n- Controlled handle and relative extension hypotheses yield many further special-case homotopies.\n\n**Full solution or refutation.**\n\nThe general fixed-symplectic-form question, especially in dimension 4, remains open.\n\n**What remains.**\n\nProve general uniqueness or construct two Weinstein structures for the same symplectic form that cannot be joined by a Weinstein homotopy, while controlling boundary and cohomology data.\n\n**Sources checked.**\n\n- Kai Cieliebak and Yakov Eliashberg, From Stein to Weinstein and Back, AMS Colloquium Publications 59 (2012). (primary): https://doi.org/10.1090/coll/059\n Evidence used: Provides the fixed-Stein and flexible uniqueness theorems and numerous controlled homotopies.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 5.17. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current statement, known hypotheses, and open arbitrary-dimensional status.\n\n**Review notes.** The background prints V_t=(1-t)V_2+tV_2 rather than an interpolation involving V_1; this acknowledged source defect was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3025, "problem_number": "KP-5.18", "title": "Kirby Problem 5.18", "statement": "- In higher dimensions, find the `non-analytic' cohomology module $\\HH^{*}_{?}$ analogous to the analytic lattice cohomology $\\HH^{*}_{\\mathrm{an}}$, and a natural functor $\\HH^{*}_{\\mathrm{an}}\\to \\HH^{*}_{?}$ connecting them.\n\n- Define a version $\\ECH^{*}$ of Embedded Contact Homology for isolated complex singularity links (in any dimension) associated with the canonical contact structure of the link, together with a natural graded $\\Z[U]$-module morphism $\\HH^{*}_{\\mathrm{an}}\\to \\ECH^{*}$. More precisely,\n\n- Show that this morphism is injective.\n\n- Fix the diffeomorphism type of a link, and also a contact structure on it that can be realized as the canonical contact structure associated with some singularity analytic structure. Then characterize the family $\\{\\HH^{*}_{\\mathrm{an}}\\}$, indexed by all the possible analytic germs inducing this contact link, via those graded sub-$\\Z[U]$-modules of $\\ECH^{*}$ which satisfy certain specific properties.", "background": "Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.18.\n\nLiterature notes:\n- As background, consider a complex normal surface singularity with a rational homology sphere link. The (topological) lattice cohomology $\\HH^{*}_{\\mathrm{top}}$, associated with the link (which is a plumbed 3-manifold associated with a connected negative definite graph), was introduced by Némethi in [Ném05, Ném08]. It has an analytic analogue $\\HH^{*}_{\\mathrm{an}}$ constructed recently in [ÁN21a, ÁN21b]. Both theories are multigraded, $\\HH^{*}_{\\mathrm{top}}=\\bigoplus_{q\\geq 0}\\HH^{q}_{\\mathrm{top}}$, and $\\HH^{q}_{\\mathrm{top}}$ is a $2\\Z$-graded $\\Z[U]$-module, with an additional grading indexed by the $\\operatorname{spin}^{c}$ structures of the link. The same is valid for $\\HH^{*}_{\\mathrm{an}}$ as well. Even more, the existence of a graded $\\Z[U]$-module morphism $\\HH^{*}_{\\mathrm{an}}\\to \\HH^{*}_{\\mathrm{top}}$ was also established.\n\nFollowing on calculations in [OS03b], it was conjectured in [Ném08] that the topological version can be identified with Heegaard Floer homology. Further results from [Ném08, OSS14b] support the conjecture, whose full proof was announced in [Zem25].\n\nIn particular, it is also isomorphic with any other (co)homology theory of 3-manifolds that agree with Heegaard Floer homology, e.g. with Embedded Contact Homology (introduced in [Hut10, Hut14], for its equivalence with the Heegaard Floer homology see [CGH20]).\n\nOne of the $\\operatorname{spin}^{c}$ structures (determined from the analytic structure) is distinguished; it is called the `canonical $\\operatorname{spin}^{c}$ structure'. The analytic lattice cohomology associated with this canonical structure has an extension to any higher dimension, for complex isolated singularities (again, with certain restrictions).\n\n- Part (b) addresses the important issue that in higher dimension the link of the isolated singularity contains essentially less information than might be needed to construct such an invariant (e.g. the link can be even the usual sphere). In particular, in the above context, the `non-analytic' version shouldn't be `topological' or `smooth'. However, the smooth structure enhanced with its canonical contact structure (induced by the analytic structure of the germ) might produce the desired cohomology.\n\nA good candidate for this is some version of Embedded Contact Homology associated with the canonical contact structure of the link. For 3-dimensional singularity links the canonical contact structure can be uniquely determined from the link itself [CNPP06]. However, in higher dimensions, it is an essential enhancement of the smooth structure. For results regarding ECH in higher dimensions see e.g. [CHT24].\n\nReferences cited:\n- [Ném05] András Némethi. On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds. Geom. Topol., 9:991–1042, 2005. doi:10.2140/gt.2005.9.991.\n- [Ném08] András Némethi. Lattice cohomology of normal surface singularities. Publ. Res. Inst. Math. Sci., 44(2):507–543, 2008. doi:10.2977/prims/1210167336.\n- [ÁN21a] Tamás Ágoston and András Némethi. Analytic lattice cohomology of surface singularities, 2021. arXiv:2108.12294.\n- [ÁN21b] Tamás Ágoston and András Némethi. Analytic lattice cohomology of surface singularities, II (the equivariant case), 2021. arXiv:2108.12429.\n- [OS03b] Peter Ozsváth and Zoltán Szabó. On the Floer homology of plumbed threemanifolds. Geom. Topol., 7:185–224, 2003. doi:10.2140/gt.2003.7.185.\n- [OSS14b] Peter Ozsváth, András I. Stipsicz, and Zoltán Szabó. A spectral sequence on lattice homology. Quantum Topol., 5(4):487–521, 2014. doi:10.4171/QT/56.\n- [Zem25] Ian Zemke. The equivalence of lattice and Heegaard Floer homology. Duke Math. J., 174(5):857–910, 2025. doi:10.1215/00127094-2024-0044.\n- [Hut10] Michael Hutchings. Embedded contact homology and its applications. In Proceedings of the International Congress of Mathematicians. Volume II, pages 1022–1041. Hindustan Book Agency, New Delhi, 2010.\n- [Hut14] Michael Hutchings. Lecture notes on embedded contact homology. In Contact and symplectic topology, volume 26 of Bolyai Soc. Math. Stud., pages 389–484. János Bolyai Math. Soc., Budapest, 2014. doi:10.1007/978-3-319-02036-5\\\\_9.\n- [CGH20] Vincent Colin, Paolo Ghiggini, and Ko Honda. An exposition of the equivalence of Heegaard Floer homology and embedded contact homology. In Characters in low-dimensional topology, volume 760 of Contemp. Math., pages 45–101. Amer. Math. Soc., [Providence], RI, [2020] ©2020. doi:10.1090/conm/760/15286.\n- [CNPP06] Clément Caubel, András Némethi, and Patrick Popescu-Pampu. Milnor open books and Milnor fillable contact 3-manifolds. Topology, 45(3):673–689, 2006. doi:10.1016/j.top.2006.01.002.\n- [CHT24] Vincent Colin, Ko Honda, and Yin Tian. Applications of higher-dimensional Heegaard Floer homology to contact topology. J. Topol., 17(3):Paper No. e12349, 77, 2024. doi:10.1112/topo.12349.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Higher-dimensional analytic lattice cohomology and candidate Floer-theoretic frameworks exist, and the three-dimensional topological lattice target is identified with Heegaard Floer homology, but the requested higher-dimensional ECH, natural map, injectivity, and image characterization remain undeveloped.\n\n**Verified partial progress.**\n\n- Ágoston--Némethi construct analytic lattice cohomology for isolated singularities of every complex dimension at least 2 under a minor assumption.\n- Zemke proves lattice homology equals Heegaard Floer homology for tree-plumbed 3-manifolds, providing the expected topological Floer target in the surface-singularity case.\n- Colin--Honda--Tian establish a framework for higher-dimensional Heegaard Floer homology and a contact class, but not the requested higher-dimensional ECH comparison package.\n\n**Full solution or refutation.**\n\nNone of the checked sources constructs the requested higher-dimensional ECH module with a natural analytic-lattice map, proves that map injective, or characterizes its possible images.\n\n**What remains.**\n\nDefine the nonanalytic contact/Floer target in higher dimensions, construct and grade a natural map from analytic lattice cohomology, prove injectivity, and characterize the submodules realized by analytic germs with fixed contact link.\n\n**Sources checked.**\n\n- Tamás Ágoston and András Némethi, The analytic lattice cohomology of isolated singularities, arXiv:2109.11266. (primary): https://arxiv.org/abs/2109.11266\n Evidence used: Constructs the analytic theory in all complex dimensions at least 2 under a minor assumption.\n- Ian Zemke, The equivalence of lattice and Heegaard Floer homology, Duke Math. J. 174 (2025), 857--910. (primary): https://arxiv.org/abs/2111.14962\n Evidence used: Proves Némethi's lattice/Heegaard-Floer conjecture for tree plumbings.\n- Vincent Colin, Ko Honda, and Yin Tian, Applications of higher-dimensional Heegaard Floer homology to contact topology, J. Topol. 17 (2024), e12349. (primary): https://arxiv.org/abs/2006.05701\n Evidence used: Develops a higher-dimensional Floer framework and contact class, but not the requested ECH/lattice map.\n- R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 5.18. (authoritative_secondary): https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf\n Evidence used: Current formulation and synthesis of the open construction, injectivity, and characterization tasks.\n\n**Review notes.** The symbols HH^*_{?} and higher-dimensional ECH are intentional placeholders for requested new theories, not established standard invariants.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 11, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 11, "name": "kirby_low_dimensional_topology", "display_name": "Kirby's Problems in Low-Dimensional Topology", "description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.", "slug": "kirby-low-dimensional-topology", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3026, "problem_number": "OPG-3031", "title": "trace inequality", "statement": "Let $A,B$ be positive semidefinite, by Jensen's inequality, it is easy to see $[tr(A^s+B^s)]^{\\frac{1}{s}}\\leq [tr(A^r+B^r)]^{\\frac{1}{r}}$, whenever $s>r>0$.\n\nWhat about the $tr(A^s+B^s)^{\\frac{1}{s}}\\leq tr(A^r+B^r)^{\\frac{1}{r}}$, is it still valid?", "background": "Source: Open Problem Garden. Original node ID: 3031. URL: http://www.openproblemgarden.org/op/trace_inequality.\n\nSource subject path: Algebra.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/trace_inequality\n- Subject(s): Algebra\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 11th, 2008 by Miwa Lin\n\nComments:\n- August 28th, 2010 | ojs | clarification: I suppose that A and B are square hermitian matrixes and that s and r are real numbers. But what are the brackets supposed to represent? Sorry for my ignorance if it is obvious.\n- October 12th, 2008 | Miwa Lin | It is open.: It is open.\n- December 20th, 2010 | Anonymous | this inequality is not true.: this inequality is not true. It was denied some years ago. See X.Z Zhan's\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"trace inequality\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No strong source explicitly resolving the exact unnormalized trace inequality was located; the source page's only direct disproof attribution is anonymous and truncated.\n\n**Verified partial progress.**\n\n- The inequality holds when A and B commute, by scalar l_p-norm monotonicity.\n- Audenaert and Hiai sharply classify Löwner-order comparisons for normalized matrix power means using 2-by-2 counterexamples.\n- Hiai proves trace monotonicity for the normalized mean ((A^p+B^p)/2)^{1/p}, but the omitted normalization contributes a p-dependent factor in the opposite direction and does not settle this record.\n\n**Full solution or refutation.**\n\nA 2010 Open Problem Garden comment says the inequality is false and cites only the incomplete phrase 'See X.Z Zhan's'; this is insufficient for a literature-verified refutation.\n\n**What remains.**\n\nLocate a citable explicit counterexample or a theorem addressing Tr(A^p+B^p)^{1/p} itself for all positive exponents.\n\n**Sources checked.**\n\n- Open Problem Garden, trace inequality (node 3031). (maintained_tracker): https://www.openproblemgarden.org/op/trace_inequality\n Evidence used: Preserves the ambiguous formula and the incomplete anonymous disproof claim.\n- Koenraad M. R. Audenaert and Fumio Hiai, On matrix inequalities between the power means: counterexamples, Linear Algebra and its Applications 439 (2013), 1590-1604. (primary): https://doi.org/10.1016/j.laa.2013.04.012\n Evidence used: Provides sharp counterexamples for the adjacent normalized Löwner-order problem, not the exact trace expression.\n- Fumio Hiai, Various inequalities between quasi-arithmetic mean and quasi-geometric type means for matrices, arXiv:2508.20309v2 (2025). (primary): https://arxiv.org/abs/2508.20309\n Evidence used: Proves normalized trace-order monotonicity and clarifies why it does not directly answer the unnormalized question.\n\n**Review notes.** The record omits parentheses around the trace argument and does not state matrix size or field. The natural parse used here is Tr[(A^s+B^s)^{1/s}] <= Tr[(A^r+B^r)^{1/r}].\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3027, "problem_number": "OPG-23298", "title": "Elementary symmetric of a sum of matrices", "statement": "Problem\n\nGiven a Matrix $A$, the $k$-th elementary symmetric function of $A$, namely $S_k(A)$, is defined as the sum of all $k$-by- $k$ principal minors.\n\nFind a closed expression for the $k$-th elementary symmetric function of a sum of N $n$-by- $n$ matrices, with $0\\le N\\le k\\le n$ by using partitions.", "background": "Source: Open Problem Garden. Original node ID: 23298. URL: http://www.openproblemgarden.org/op/elementary_symmetric_of_a_sum_of_matrices.\n\nSource subject path: Algebra.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/elementary_symmetric_of_a_sum_of_matrices\n- Subject(s): Algebra\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: December 8th, 2008 by rscosa\n\nProblem-page discussion:\nThe Newton-Girard formulas imply particular expressions for small values of $k$ and $N$, for example, $S_2(A+B)=S_2(A)+S_2(B)+S_1(A)S_1(B)-S_1(AB)$.\n\nSource links:\n- elementary symmetric function: http://en.wikipedia.org/wiki/elementary symmetric function\n\nDiscussion links:\n- Newton-Girard formulas: http://en.wikipedia.org/wiki/Newton-Girard formulas\n\nComments:\n- August 27th, 2010 | olivier | that's ok: that's ok\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Elementary symmetric of a sum of matrices\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Costas-Santos supplies general expressions for elementary symmetric functions of sums of matrices, including the partition/Newton--Girard framework requested in the source.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe stated request for closed expressions has a published solution framework.\n\n**What remains.**\n\nFurther simplification or specialization is possible, but the source problem is answered.\n\n**Sources checked.**\n\n- R. S. Costas-Santos, On the elementary symmetric functions of a sum of matrices, J. Algebra Number Theory Adv. 1 (2009), 99--112. (primary): https://arxiv.org/abs/math/0612464\n Evidence used: Abstract states that new expressions for elementary symmetric functions of sums of matrices are given.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3028, "problem_number": "OPG-37283", "title": "Finite Lattice Representation Problem", "statement": "Conjecture\n\nThere exists a finite lattice which is not the congruence lattice of a finite algebra.", "background": "Source: Open Problem Garden. Original node ID: 37283. URL: http://www.openproblemgarden.org/op/finite_congruence_lattice_problem.\n\nSource subject path: Algebra.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/finite_congruence_lattice_problem\n- Subject(s): Algebra\n- Keywords: congruence lattice; finite algebra\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: December 10th, 2010 by williamdemeo\n\nProblem-page discussion:\nA well-known result of universal algebra states: every algebraic lattice is isomorphic to the congruence lattice of an algebra. Thus there is essentially no restriction on the shape of a congruence lattice of a general algebra. It is natural to ask whether the same is true for finite lattices and finite algebras. That is, does every finite lattice occur as the congruence lattice of a finite algebra? This fundamental question, asked over 40 years ago, is among the most elusive problems of universal algebra.\n\nThis question is important because, until it is answered, we lack something very basic in our understanding of algebras -- namely, if we assume an algebra is finite, does this place any restriction on the shape of its congruence lattice? If so, then finite algebras are fundamentally different from infinite algebras in this sense.\n\nBibliography:\n[GS] Gratzer and Schmidt, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. (Szeged) 24 (1963), 34-59.\n\n[P5] Palfy and Pudlak. Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis, 11 (1980), 22–27.\n\n[PT] Pudlak and Tuma, Every finite lattice can be embedded in the lattice of all equivalences over a finite set. Algebra Universalis, 10 (1980), 74–95.\n\nRelated:\nRelated problems\nWhich lattices occur as intervals in subgroup lattices of finite groups?\n\nDiscussion links:\n- universal algebra: http://en.wikipedia.org/wiki/Universal_algebra\n- algebraic lattice: http://en.wikipedia.org/wiki/Algebraic_lattice\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Finite Lattice Representation Problem\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The finite lattice representation problem remains open. All lattices with at most six elements are representable, and a single seven-element lattice L7 is the unique smallest unresolved case.\n\n**Verified partial progress.**\n\n- DeMeo proved that, with one possible exception, every lattice with at most seven elements is a congruence lattice of a finite algebra.\n- The maintained FLRP site records L7 as the only unresolved seven-element lattice and describes restrictions on any representing algebra.\n\n**Full solution or refutation.**\n\nNo finite lattice is currently verified not to be the congruence lattice of a finite algebra.\n\n**What remains.**\n\nRepresent L7 by a finite algebra or prove that no such algebra exists; more generally settle FLRP.\n\n**Sources checked.**\n\n- W. J. DeMeo, Congruence lattices of finite algebras, arXiv:1204.4305 (2012). (primary): https://arxiv.org/abs/1204.4305\n Evidence used: The abstract states the open problem, the all-but-one result through seven elements, and structural results on the exceptional lattice.\n- W. DeMeo, Current Status & The Hunt for a Counterexample, UniversalAlgebra.org (accessed 2026-08-17). (maintained_tracker): https://universalalgebra.org/flrp/status/\n Evidence used: The maintained page states that FLRP is unsolved, all lattices through six elements are representable, and L7 is the only unresolved seven-element lattice.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3029, "problem_number": "OPG-48715", "title": "Sub-atomic product of funcoids is a categorical product", "statement": "Conjecture In the category of continuous funcoids (defined similarly to the category of topological spaces) the following is a direct categorical product:\n\n- Product morphism is defined similarly to the category of topological spaces.\n- Product object is the sub-atomic product.\n- Projections are sub-atomic projections.\n\nSee details, exact definitions, and attempted proofs here.", "background": "Source: Open Problem Garden. Original node ID: 48715. URL: http://www.openproblemgarden.org/op/sub_atomic_product_of_funcoids_is_a_categorical_product.\n\nSource subject path: Algebra.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/sub_atomic_product_of_funcoids_is_a_categorical_product\n- Subject(s): Algebra\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 21st, 2013 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nRelated:\nRelated problems\nA construction of direct product in the category of continuous maps between endo-funcoids\n\nSource links:\n- here: http://planetmath.org/directproductsinacategoryoffuncoids\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Sub-atomic product of funcoids is a categorical product\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No citable mathematical source was located that verifies the exact funcoid category, sub-atomic product, and projection conventions in the source claim.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe linked source-framework terminology cannot safely be identified with a standard categorical-product theorem.\n\n**What remains.**\n\nRecover fixed formal definitions and a peer-reviewed proof/counterexample for the exact funcoid category.\n\n**Sources checked.**\n\n- Open Problem Garden, Sub-atomic product of funcoids is a categorical product (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/sub_atomic_product_of_funcoids_is_a_categorical_product\n Evidence used: Provides the idiosyncratic statement and links to attempted definitions/proofs, not a verified resolution.\n\n**Review notes.** Definitions are not silently reconstructed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3030, "problem_number": "OPG-50149", "title": "inverse of an integer matrix", "statement": "Question I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all $\\ge 2$. Suppose X is an m-by-n integer matrix $(m \\le n)$. Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.", "background": "Source: Open Problem Garden. Original node ID: 50149. URL: http://www.openproblemgarden.org/op/inverse_of_an_integer_matrix.\n\nSource subject path: Algebra.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/inverse_of_an_integer_matrix\n- Author(s): Gregory, Steven A\n- Subject(s): Algebra\n- Keywords: invertable matrices, integer matrices\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 27th, 2013 by lvoyster\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"inverse of an integer matrix\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** For any full-row-rank integer matrix M=[D X], an integer right inverse exists iff the gcd of its m by m minors is one, equivalently iff the Smith normal form has every nonzero invariant factor one.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThis standard Smith-normal-form criterion directly decides the source question; rowwise relative-primality alone is not the general criterion.\n\n**What remains.**\n\nFor a concrete D and X, compute the maximal minors or Smith normal form.\n\n**Sources checked.**\n\n- Encyclopedia of Mathematics, Normal form (for matrices), Smith normal form (accessed 2026-08-17). (authoritative_secondary): https://encyclopediaofmath.org/wiki/Normal_form_%28for_matrices%29\n Evidence used: States that Smith invariant factors are determined by gcds of minors; surjectivity over Z is exactly unit invariant factors.\n\n**Review notes.** The informal rowwise guess is not silently substituted for the global criterion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3031, "problem_number": "OPG-57824", "title": "Graphs of exact colorings", "statement": "Conjecture For $c \\geq m \\geq 1$, let $P(c,m)$ be the statement that given any exact $c$-coloring of the edges of a complete countably infinite graph (that is, a coloring with $c$ colors all of which must be used at least once), there exists an exactly $m$-colored countably infinite complete subgraph. Then $P(c,m)$ is true if and only if $m=1$, $m=2$, or $c=m$.", "background": "Source: Open Problem Garden. Original node ID: 57824. URL: http://www.openproblemgarden.org/op/graphs_of_exact_colorings.\n\nSource subject path: Algebra.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/graphs_of_exact_colorings\n- Subject(s): Algebra\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 3rd, 2013 by sabisood\n\nProblem-page discussion:\nStacey and Weidl have shown that given $m \\geq 3$, there is an integer $C(m)$ such that $P(c,m)$ is false for all $c \\geq C(m)$.\n\n* M. Erickson, \"A Conjecture Concerning Ramsey's Theorem,\" Discrete Mathematics 126, 395--398 (1994); MR 95b:05209\n\nA. Stacey and P. Weidl, \"The Existence of Exactly m-Coloured Complete Subgraphs,\" J. of Combinatorial Theory, Series B 75, 1-18 (1999)\n\n* indicates original appearance(s) of problem.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Graphs of exact colorings\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Erickson's exact-colour conjecture is proved for all but finitely many parameter pairs, but those remaining pairs have not all been resolved.\n\n**Verified partial progress.**\n\n- For each fixed m at least three, Stacey and Weidl prove P(c,m) false for every sufficiently large c.\n- A 2025 preprint proves the conjectured negative answer for every sufficiently large m and every c greater than m, reducing the total residue to finitely many pairs.\n\n**Full solution or refutation.**\n\nThe recent theorem covers all large m uniformly in c; together with the older fixed-m theorem it leaves only finitely many small pairs, not a complete proof.\n\n**What remains.**\n\nDetermine P(c,m) for every remaining pair with 3 at most m less than c and independently verify the recent preprint.\n\n**Sources checked.**\n\n- Z. Randjelovic, Exactly Colored Complete Subgraphs of Infinite Graphs, arXiv:2512.04233 (2025). (primary): https://arxiv.org/abs/2512.04233\n Evidence used: Proves Erickson's conjecture for all sufficiently large m and all c>m, reducing unresolved cases to a finite set.\n- B. P. Narayanan, Exactly m-coloured complete infinite subgraphs, Journal of Combinatorial Theory B 106 (2014), 163-173. (primary): https://doi.org/10.1016/j.jctb.2014.01.008\n Evidence used: Provides quantitative structure for the set of exact colour counts forced by a finite colouring.\n- Open Problem Garden, Graphs of exact colorings (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/graphs_of_exact_colorings\n Evidence used: Preserves the exact P(c,m) formulation and the Stacey-Weidl fixed-m result.\n\n**Review notes.** The December 2025 result is an unrefereed v1 and its finite unresolved residue was not inferred to be solved. Source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3032, "problem_number": "OPG-60031", "title": "Waring rank of determinant", "statement": "Question What is the Waring rank of the determinant of a $d \\times d$ generic matrix?\n\nFor simplicity say we work over the complex numbers. The $d \\times d$ generic matrix is the matrix with entries $x_{i,j}$ for $1 \\leq i,j \\leq d$. Its determinant is a homogeneous form of degree $d$, in $d^2$ variables. If $F$ is a homogeneous form of degree $d$, a power sum expression for $F$ is an expression of the form $F = \\ell_1^d+\\dotsb+\\ell_r^d$, the $\\ell_i$ (homogeneous) linear forms. The Waring rank of $F$ is the least number of terms $r$ in any power sum expression for $F$. For example, the expression $xy = \\frac{1}{4}(x+y)^2 - \\frac{1}{4}(x-y)^2$ means that $xy$ has Waring rank $2$ (it can't be less than $2$, as $xy \\neq \\ell_1^2$ ).\n\nThe $2 \\times 2$ generic determinant $x_{1,1}x_{2,2}-x_{1,2}x_{2,1}$ (or $ad-bc$ ) has Waring rank $4$. The Waring rank of the $3 \\times 3$ generic determinant is at least $14$ and no more than $20$, see for instance Lower bound for ranks of invariant forms, Example 4.1. The Waring rank of the permanent is also of interest. The comparison between the determinant and permanent is potentially relevant to Valiant's \"VP versus VNP\" problem.", "background": "Source: Open Problem Garden. Original node ID: 60031. URL: http://www.openproblemgarden.org/op/waring_rank_of_determinant.\n\nSource subject path: Algebra.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/waring_rank_of_determinant\n- Author(s): Teitler, Zach\n- Subject(s): Algebra\n- Keywords: Waring rank, determinant\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 15th, 2019 by Zach Teitler\n\nSource links:\n- Lower bound for ranks of invariant forms: https://arxiv.org/abs/1409.0061\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 20.\n\nAttempt notes:\nTarget:\nMake progress on \"Waring rank of determinant\" in Algebra, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact Waring rank of det_d remains unknown generally; explicit upper bounds have improved to d*d! and further for large d.\n\n**Verified partial progress.**\n\n- Johns--Teitler prove the upper bound d*d!.\n- Later work improves the best known upper bound for sufficiently large matrix size.\n\n**Full solution or refutation.**\n\nNo exact all-d formula was verified, including beyond the stated d=3 range.\n\n**What remains.**\n\nClose the gap between lower and upper Waring-rank bounds for generic determinants.\n\n**Sources checked.**\n\n- G. Johns and Z. Teitler, An improved upper bound for the Waring rank of the determinant, arXiv:2004.06158 (2020). (primary): https://arxiv.org/abs/2004.06158\n Evidence used: The abstract gives the d*d! upper bound.\n- A. Conner, A. Harper and J. M. Landsberg, A New Formula for the Determinant and Bounds on Its Tensor and Waring Ranks, arXiv:2301.06586 (2023). (primary): https://arxiv.org/abs/2301.06586\n Evidence used: Reports improved determinant Waring-rank upper bounds for n>=17.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 4, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3033, "problem_number": "OPG-725", "title": "$C^r$ Stability Conjecture", "statement": "Conjecture Any $C^r$ structurally stable diffeomorphism is hyperbolic.", "background": "Source: Open Problem Garden. Original node ID: 725. URL: http://www.openproblemgarden.org/op/c_r_stability_conjecture.\n\nSource subject path: Analysis.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/c_r_stability_conjecture\n- Author(s): Palis, J.; Smale, S.\n- Subject(s): Analysis\n- Keywords: diffeomorphisms,; dynamical systems\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: December 20th, 2007 by m n\n\nProblem-page discussion:\nSee the definitions of: stractural stability and hyperbolicity.\n\nThe conjecture is due to J Palis and S Smale (1970's). In the case $r=1$ the conjecture was proved by R Mañé (Publ. IHES 1986). In higher regularity, $r>1$, the conjecture is one of the most important and difficult problems in dynamical systems.\n\nThere is a similar conjecture for the vector fields or flows, and in the $C^1$ topology has been proved by S Hayashi (Ann Math. 1997).\n\nDiscussion links:\n- stractural stability: http://en.wikipedia.org/wiki/Structural_stability\n- hyperbolicity: http://en.wikipedia.org/wiki/Hyperbolic_structure\n- R Mañé (Publ. IHES 1986): http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1987__66__161_0\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"$C^r$ Stability Conjecture\" in Analysis, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Mañé proved the C^1 stability conjecture, while the stated higher-regularity C^r Palis--Smale problem remains unresolved.\n\n**Verified partial progress.**\n\n- The C^1 case is proved.\n- Recent work continues to describe the C^r statement as open in general.\n\n**Full solution or refutation.**\n\nNo proof that every C^r structurally stable diffeomorphism is hyperbolic for arbitrary r>1 was verified.\n\n**What remains.**\n\nResolve the higher-regularity Palis--Smale stability conjecture.\n\n**Sources checked.**\n\n- R. Mañé, A proof of the C1 stability conjecture, Publ. Math. IHÉS 66 (1987), 161--210. (primary): https://www.numdam.org/article/PMIHES_1987__66__161_0.pdf\n Evidence used: Proves the C1 case, not the general higher-regularity statement.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 8, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3034, "problem_number": "OPG-36697", "title": "Invariant subspace problem", "statement": "Problem Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?", "background": "Source: Open Problem Garden. Original node ID: 36697. URL: http://www.openproblemgarden.org/op/invariant_subspace_problem.\n\nSource subject path: Analysis.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/invariant_subspace_problem\n- Subject(s): Analysis\n- Keywords: subspace\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 9th, 2009 by tchow\n\nProblem-page discussion:\nLet $H$ be a Hilbert space. The subspaces $\\{0\\}$ and $H$ are trivially invariant under any linear operator on $H$, and so these are referred to as the trivial invariant subspaces. The problem is concerned with determining whether bounded operators necessarily have non-trivial invariant subspaces.\n\nThis is one of the most famous open problems in functional analysis. Enflo [1] constructed Banach spaces for which the corresponding question has a negative answer, and recently Argyros and Haydon constructed a Banach space for which the corresponding question has a positive answer [4].\n\nFor a nice overview to the problem see [2], [3] or [5].\n\nBibliography:\n[1] P. Enflo, On the invariant subspace problem for Banach spaces, Acta Math. 158 (1987), 213-313. MathSciNet\n\n[2] B. S. Yadav, The Present State and Heritages of the Invariant Subspace Problem, Milan J. Math. 73 (2005), 289-316. MathSciNet another link\n\n[3] H. Radjavi and P. Rosenthal, The Invariant Subspace Problem, The Mathematics Intelligencer 4 (1982), no. 1, 33-37. MathSciNet\n\n[4] S. A. Argyros and R. G. Haydon, A hereditarily indecomposable $L_\\infty$-space that solves the scalar-plus-compact problem, arXiv:0903.3921 (2009).\n\n[5] J. Noel. The Invariant Subspace Problem. Honours Thesis, Thompson Rivers University. Link to pdf.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0892591\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2175046\n- another link: http://www.math.leidenuniv.nl/%7Enaw/serie5/deel06/jun2005/pdf/yadav.pdf\n- H. Radjavi: http://www.math.uwaterloo.ca/PM_Dept/Homepages/Radjavi/radjavi.shtml\n- P. Rosenthal: http://www.math.toronto.edu/rosent/\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0678734\n- arXiv:0903.3921: http://www.arxiv.org/abs/0903.3921\n- J. Noel: http://www.math.mcgill.ca/jnoel\n- Link to pdf: http://www.math.mcgill.ca/jnoel/pdf/Honours.pdf\n\nComments:\n- April 11th, 2011 | Anonymous | closed: Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?\n- April 1st, 2011 | Anonymous | ستار اكاديمي 8: ستار اكاديمي 8\n\nthank you man, u'r good:)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Invariant subspace problem\" in Analysis, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The invariant-subspace problem for arbitrary bounded operators on separable Hilbert space remains open in established references.\n\n**Verified partial progress.**\n\n- Many operator classes have invariant-subspace theorems.\n- A 2023 manuscript claims a solution but has not changed the verified consensus status.\n\n**Full solution or refutation.**\n\nNo accepted full proof or counterexample was verified.\n\n**What remains.**\n\nResolve the arbitrary bounded-operator case, including adversarial verification of claimed proofs.\n\n**Sources checked.**\n\n- Invariant subspace problem overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Invariant_subspace_problem\n Evidence used: Records the separable-Hilbert-space case as open.\n- P. H. Enflo, On the invariant subspace problem in Hilbert spaces, arXiv:2305.15442. (primary): https://arxiv.org/abs/2305.15442\n Evidence used: Located a claim, not treated as accepted resolution.\n\n**Review notes.** Unaccepted claim not promoted to solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 8, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3035, "problem_number": "OPG-36928", "title": "Criterion for boundedness of power series", "statement": "Question Give a necessary and sufficient criterion for the sequence $(a_n)$ so that the power series $\\sum_{n=0}^{\\infty} a_n x^n$ is bounded for all $x \\in \\mathbb{R}$.", "background": "Source: Open Problem Garden. Original node ID: 36928. URL: http://www.openproblemgarden.org/op/criterion_for_boundedness_of_power_series.\n\nSource subject path: Analysis.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/criterion_for_boundedness_of_power_series\n- Author(s): Rüdinger, Andreas\n- Subject(s): Analysis\n- Keywords: boundedness; power series; real analysis\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: May 9th, 2009 by andreasruedinger\n\nProblem-page discussion:\nConsider a power series $\\sum_{n=0}^{\\infty} a_n x^n$ that is convergent for all $x \\in {\\mathbb R}$, thus defining a function $f: {\\mathbb R} \\to {\\mathbb R}$. Are there criteria to decide whether $f$ is bounded (which e.g. is the case for the series with $a_n = (-1)^k/(2k)!$ for $n = 2k$ and $a_n = 0$ for n odd)? Some general remarks:\n\n- A necessary condition for $\\sum_n a_n x^n$ to be bounded is that $a_0$ is the only non-zero $a_n$ or there are infinitely many non-zero $a_n$ 's which change sign infinitely many times.\n- Changing a finite set of $a_n$ 's (except $a_0$ ) does leave the subspace of bounded power series.\n- The subspace of bounded power series is \"large\" in the sense that it is both a linear subspace (closed under sums and scalar multiples) and an algebra (closed under products). It includes all functions of the form $a \\cos( f(x))$, where $f$ is any entire function $\\mathbb{R} \\to \\mathbb{R}$. The question whether the subspace of bounded power series contains only these functions seems to be open.\n\nComments:\n- February 10th, 2011 | Anonymous | A necessary condition: It seems the sum would be bounded if there are only finitely many non-zero a sub n; it is not apparent to me that a sub 0 be the only non-zero a sub n.\n- June 21st, 2012 | Anonymous | What you have then is a: What you have then is a polynomial, and any nonconstant polynomial function is unbounded.\n- February 16th, 2011 | Comet | Re: A necessary condition: I posted the above comment anonymously, but now I have created an account. \"It seems the sum would be bounded if there are only finitely many non-zero a sub n; it is not apparent to me that a sub 0 be the only non-zero a sub n.\"\n- April 27th, 2012 | Anonymous | harder than that: Look at sin(x)=x-x^3/6+x^5/120-....\n\nJPB\n- June 21st, 2012 | Anonymous | sin x = cos(pi/2 - x): The sine function is in the class mentioned.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Criterion for boundedness of power series\" in Analysis, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The displayed request omits the source discussion's infinite-radius hypothesis and does not specify what form of necessary-and-sufficient coefficient criterion is sought; no authoritative source establishing a precise unrestricted open problem was located.\n\n**Verified partial progress.**\n\n- The source discussion correctly narrows the domain to sequences whose power series converges for every real x.\n- There is a developed theory for restricted subclasses, such as entire functions of exponential type bounded on the real axis, but it does not provide a general criterion for arbitrary entire Taylor series.\n\n**Full solution or refutation.**\n\nA status assignment of open would overstate the precision of the request: without an admissible form for the criterion, tautological necessary-and-sufficient conditions already exist. The statement needs expert reformulation.\n\n**What remains.**\n\nSpecify infinite radius of convergence and define the desired type of coefficient condition, for example an effective test, inequalities on coefficients, or a characterization within a fixed growth class.\n\n**Sources checked.**\n\n- Open Problem Garden, Criterion for boundedness of power series, accessed 2026-08-17. (maintained_tracker): https://openproblemgarden.org/op/criterion_for_boundedness_of_power_series\n Evidence used: Preserves the question and supplies the omitted assumption that the series converges for all real x; it provides no precise criterion class or verified resolution.\n- J. Clunie, Q. I. Rahman, and W. J. Walker, On Entire Functions of Exponential Type Bounded on the Real Axis, Journal of the London Mathematical Society 61 (2000), 163-176, DOI 10.1112/S0024610799008236. (primary): https://doi.org/10.1112/S0024610799008236\n Evidence used: Documents substantive results for a restricted growth class, not a general coefficient characterization for all entire functions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 8, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3036, "problem_number": "OPG-37185", "title": "Something like Picard for 1-forms", "statement": "Conjecture Let $D$ be the open unit disk in the complex plane and let $U_1,\\dots,U_n$ be open sets such that $\\bigcup_{j=1}^nU_j=D\\setminus\\{0\\}$. Suppose there are injective holomorphic functions $f_j: U_j \\to \\mathbb{C},$ $j=1,\\ldots,n,$ such that for the differentials we have ${\\rm d}f_j={\\rm d}f_k$ on any intersection $U_j\\cap U_k$. Then those differentials glue together to a meromorphic 1-form on $D$.", "background": "Source: Open Problem Garden. Original node ID: 37185. URL: http://www.openproblemgarden.org/op/something_like_picard_for_1_forms.\n\nSource subject path: Analysis.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/something_like_picard_for_1_forms\n- Author(s): Elsner, B.\n- Subject(s): Analysis\n- Keywords: Essential singularity; Holomorphic functions; Picard's theorem; Residue of 1-form; Riemann surfaces\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 26th, 2010 by MathOMan\n\nProblem-page discussion:\nIt is an evidence that the 1-form is holomorphic on $D\\setminus\\{0\\}$. In the case that its residue at the origin vanishes we can use Picard's big theorem.\n\nBibliography:\n*B. Elsner: Hyperelliptic action integral, Annales de l'institut Fourier 49(1), p.303–331\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Something like Picard for 1-forms\" in Analysis, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The imported statement is false because it does not require the U_j to be connected. A finite disconnected cover can support injective local primitives of d(exp(1/z)), which has an essential rather than meromorphic singularity at 0.\n\n**Verified partial progress.**\n\n- If the common 1-form has residue zero, the source discussion explains that a global primitive exists on the punctured disk and Picard's theorem rules out an essential singularity under the local-univalence hypotheses.\n- The connected-cover version with nonzero residue was not resolved in the checked expert discussion.\n\n**Full solution or refutation.**\n\nPut w=1/z and cover the w-plane by open squares of side less than 2pi, colored periodically with finitely many colors so same-colored squares are disjoint. On each inverse-image component exp(1/z) is injective. Component-dependent additive constants separate the images while preserving the differential. Taking unions by color produces finitely many open, generally disconnected U_j and injective holomorphic f_j with df_j=d(exp(1/z)); the glued form is not meromorphic at 0.\n\n**What remains.**\n\nAdd the missing connectedness hypothesis and settle the nonzero-residue case for connected U_j.\n\n**Sources checked.**\n\n- MathOverflow question 61882, Meromorphic 1-form and Picard's theorem, expert answers and discussion, checked 17 August 2026. (authoritative_secondary): https://mathoverflow.net/questions/61882/meromorphic-1-form-and-picards-theorem\n Evidence used: Identifies the omitted connectedness hypothesis, supplies disconnected-cover counterexamples, and distinguishes the unresolved connected variant.\n- B. Elsner, Hyperelliptic action integral, Annales de l'Institut Fourier 49 (1999), 303-331, DOI 10.5802/aif.1675. (primary): https://doi.org/10.5802/aif.1675\n Evidence used: Original cited context for the Picard-type meromorphic-form question.\n\n**Review notes.** The exact imported wording is classified; it is not silently replaced by the connected-open-set variant.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 8, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3037, "problem_number": "OPG-41335", "title": "Inequality for square summable complex series", "statement": "Conjecture For all $\\alpha=(\\alpha_1,\\alpha_2,\\ldots)\\in l_2(\\cal{C})$ the following inequality holds $$\\sum_{n\\geq 1}|\\alpha_n|^2\\geq \\frac{6}{\\pi^2}\\sum_{k\\geq0}\\bigg| \\sum_{l\\geq0}\\frac{1}{l+1}\\alpha_{2^k(2l+1)}\\bigg|^2$$", "background": "Source: Open Problem Garden. Original node ID: 41335. URL: http://www.openproblemgarden.org/op/inequality_for_square_summable_complex_series.\n\nSource subject path: Analysis.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/inequality_for_square_summable_complex_series\n- Author(s): Retkes, Zoltan\n- Subject(s): Analysis\n- Keywords: Inequality\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: December 25th, 2012 by tigris35711\n\nRelated:\nRelated problems\nCriterion for boundedness of power series\n\nComments:\n- November 3rd, 2013 | Anonymous | Solution: It's a simple application of the Shwartz inequality:\n\n$$\\sum_{k}\\left|\\sum_{l} \\frac{1}{l+1}a_{2^k(2l+1)}\\right|^2 \\le$$$$\\le \\sum_{k}\\left|\\sum_{l} \\frac{1}{l+1}\\left|a_{2^k(2l+1)}\\right|\\right|^2 \\le$$Shwartz:$$\\le \\sum_{k} \\left(\\sum_{l}\\frac{1}{(l+1)^2}\\right)\\left(\\sum_{h}|a_{2^k(2l+1)}|^2\\right) =$$$$= \\sum_{k} \\frac{\\pi^2}{6}\\sum_{h}|a_{2^k(2l+1)}|^2 =$$$$= \\frac{\\pi^2}{6} \\sum_{k}\\sum_{h}|a_{2^k(2l+1)}|^2 =$$$$= \\frac{\\pi^2}{6} \\sum_{n}|a_n|^2$$because$A_k:=\\{ 2^k(2l+1)| l\\in \\mathbb N\\}$is a partition of$\\mathbb N^+$.\n- October 29th, 2014 | tigris35711 | Oh Yes.: Where shall I send the £10 prize?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Inequality for square summable complex series\" in Analysis, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The source contains an anonymous assertion of a simple solution, but no citable proof of the stated l2 inequality was located.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo resolution is accepted without a source that proves the exact inequality.\n\n**What remains.**\n\nLocate the claimed proof or establish boundedness of the indexed Hardy-type operator directly.\n\n**Sources checked.**\n\n- Open Problem Garden, Inequality for square summable complex series (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/inequality_for_square_summable_complex_series\n Evidence used: Contains the exact question and an unsupported anonymous solution comment.\n\n**Review notes.** Anonymous comment not promoted to solved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 8, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3038, "problem_number": "OPG-426", "title": "Long rainbow arithmetic progressions", "statement": "For $k\\in \\mathbb{N}$ let $T_k$ denote the minimal number $t\\in \\mathbb{N}$ such that there is a rainbow $AP(k)$ in every equinumerous $t$-coloring of $\\{ 1,2,\\ldots,tn\\}$ for every $n\\in \\mathbb{N}$\n\nConjecture For all $k\\geq 3$, $T_k=\\Theta (k^2)$.", "background": "Source: Open Problem Garden. Original node ID: 426. URL: http://www.openproblemgarden.org/op/long_rainbow_arithemtic_progressions.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/long_rainbow_arithemtic_progressions\n- Author(s): Fox, Jacob; Jungic, Veselin; Mahdian, Mohammad; Nesetril, Jaroslav; Radoicic, Rados\n- Subject(s): Combinatorics\n- Keywords: arithmetic progression; rainbow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 3rd, 2007 by vjungic\n\nProblem-page discussion:\nA $t$-coloring of $\\{ 1,2,\\ldots, tn\\}$ is equinumerous if each color is used $n$ times. An arithmetic progression is rainbow if it does not containt two terms of the same color.\n\nIn [JLMNR] it was proved that $\\lfloor \\frac{k^2}{4}\\rfloor 4$ ([CJR]). It is not hard to show that $T_k > k$ for all $k\\ge 5$ ([AF]).\n\nBibliography:\n[AF] Maria Axenovich, Dmitri Fon-Der-Flaass: On rainbow arithmetic progressions, Electronic Journal of Combinatorics, 11, (2004), R1.\n\n[CJR] David Conlon, Veselin Jungic, Rados Radoicic, On the existence of rainbow 4-term arithmetic progressions, Graphs and Combinatorics, 23 (2007), 249-254\n\n*[JLMNR] Veselin Jungic, Jacob Licht (Fox), Mohammad Mahdian, Jaroslav Nesetril, Rados Radoicic: Rainbow arithmetic progressions and anti-Ramsey results, Combinatorics, Probability, and Computing - Special Issue on Ramsey Theory, 12, (2003), 599--620.\n\n[JNR] Veselin Jungic, Jaroslav Nesetril, Rados Radoicic: Rainbow Ramsey theory, Integers, The Electronic Journal of Combinatorial Number Theory, Proceedings of the Integers Conference 2003 in Honor of Tom Brown, 5(2), (2005), A9.\n\n[JR] Veselin Jungic, Rados Radoicic: Rainbow 3-term arithmetic progressions, Integers, The Electronic Journal of Combinatorial Number Theory, 3, (2003), A18.\n\nRelated:\nRelated problems\nRainbow AP(4) in an almost equinumerous coloring\n\nBibliography links:\n- On the existence of rainbow 4-term arithmetic progressions: http://dx.doi.org/10.1007/s00373-007-0723-2\n\nComments:\n- May 12th, 2010 | Anonymous | Is this unsolved?: Looks like a nice problem, but is it still unsolved?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Long rainbow arithmetic progressions\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The conjecture T_k=Theta(k^2) is false: Geneson proves the matching lower bound Omega(k^2 log k), yielding the exact order T_k=Theta(k^2 log k).\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe asymptotic growth problem is resolved with an extra logarithmic factor, refuting the stated conjecture.\n\n**What remains.**\n\nThe stated order-of-growth question has no remaining gap; determining sharp constants or lower-order terms is a separate problem.\n\n**Sources checked.**\n\n- Jesse Geneson, The order of long rainbow arithmetic progressions, arXiv:2607.15116 (2026). (primary): https://arxiv.org/abs/2607.15116\n Evidence used: Proves Omega(k^2 log k) and combines it with the known matching upper bound to obtain Theta(k^2 log k).\n- Open Problem Garden, Long rainbow arithmetic progressions (node 426). (maintained_tracker): https://www.openproblemgarden.org/op/long_rainbow_arithemtic_progressions\n Evidence used: Original Theta(k^2) conjecture and historical bounds.\n\n**Review notes.** Dataset status should change from open to disproved. The imported URL misspells arithmetic, and the background says containt; neither affects the formula.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3039, "problem_number": "OPG-478", "title": "Rainbow AP(4) in an almost equinumerous coloring", "statement": "Problem Do 4-colorings of $\\mathbb{Z}_{p}$, for $p$ a large prime, always contain a rainbow $AP(4)$ if each of the color classes is of size of either $\\lfloor p/4\\rfloor$ or $\\lceil p/4\\rceil$?", "background": "Source: Open Problem Garden. Original node ID: 478. URL: http://www.openproblemgarden.org/op/rainbow_ap_4_in_an_almost_equinumerous_coloring.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/rainbow_ap_4_in_an_almost_equinumerous_coloring\n- Author(s): Conlon, David\n- Subject(s): Combinatorics\n- Keywords: arithmetic progression; rainbow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 20th, 2007 by vjungic\n\nProblem-page discussion:\nIt is known that there are equinumerous colorings of $\\mathbb{Z}_{4m}$ (i.e. colorings of $\\mathbb{Z}_{4m}$ for some $m$ such that each color occurs $m$ times) within which we cannot find rainbow arithmetic progressions of length $4$. ([CJR])\n\nBibliography:\n*[C] David Conlon, Rainbow solutions of linear equations over $\\mathbb{Z}_p$, Discrete Mathematics, 306 (2006) 2056 - 2063.\n\n[CJR] David Conlon, Veselin Jungic, Rados Radoicic, On the existence of rainbow 4-term arithmetic progressions, Graphs and Combinatorics, 23 (2007), 249-254\n\nRelated:\nRelated problems\nLong rainbow arithmetic progressions\n\nBibliography links:\n- On the existence of rainbow 4-term arithmetic progressions: http://dx.doi.org/10.1007/s00373-007-0723-2\n\nComments:\n- September 1st, 2007 | Dino | Tight hypergraphs: It deservs to be mentioned that in any $3$-colouring of ${\\mathbb Z}_p^*/{\\mathbb Z}_3^*$, the equation $x+y=z$, have an heterochromatic (rainbow) solution; here, ${\\mathbb Z}_p^*$ denotes the multiplicative group of the field ${\\mathbb Z}_p$.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Rainbow AP(4) in an almost equinumerous coloring\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem forcing a rainbow AP(4) in every nearly equinumerous 4-coloring of Z_p was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nrainbow AP4 almost equinumerous coloring Zp\n\n**Sources checked.**\n\n- Open Problem Garden (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3040, "problem_number": "OPG-618", "title": "Monotone 4-term Arithmetic Progressions", "statement": "Question Is it true that every permutation of positive integers must contain monotone 4-term arithmetic progressions?", "background": "Source: Open Problem Garden. Original node ID: 618. URL: http://www.openproblemgarden.org/op/monotone_4_term_arithmetic_progressions.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monotone_4_term_arithmetic_progressions\n- Author(s): Davis, James A.; Entringer, Roger C.; Graham, Ronald L.; Simmons, Gustavus J.\n- Subject(s): Combinatorics\n- Keywords: monotone arithmetic progression; permutation\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 3rd, 2007 by vjungic\n\nProblem-page discussion:\nIt is not difficult to see that any permutation of positive integers contains a monotone 3-term arithmetic progression, i.e., that for any permutation $\\pi:\\mathbb{N}\\to \\mathbb{N}$ there is a 3-term arithmetic progression $a, a+d, a+2d$ such that $\\pi (a)>\\pi (a+d)>\\pi (a+2d)$ or $\\pi (a)<\\pi (a+d)<\\pi (a+2d)$.\n\nIn [DEGS] an example of a permutation of $\\mathbb{N}$ that does not contain a monotone 5-term arithmetic progression is given.\n\nBibliography:\n*[DEGS] J. A. Davis, R. C. Entringer, R. L. Graham, and G. J. Simmons, On permutations containing no long arithmetic progression, Acta Arithmetica XXXIV.1 (1977), 81-90.\n\n[LR] Bruce M. Landman and Aaron Robertson, Ramsey Theory on the Integers, Stud. Math. Libr. 24, AMS Providence, RI, 2004.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Monotone 4-term Arithmetic Progressions\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The unrestricted one-sided positive-integer permutation question remains open, while strong avoidance results are known after restricting the 2-adic divisibility of the common difference.\n\n**Verified partial progress.**\n\n- LeSaulnier and Vijay construct a permutation of the positive integers with no monotone four-term arithmetic progression of odd common difference.\n- Adenwalla proves that for every fixed k at least 1 there is a permutation avoiding every monotone four-term progression whose common difference is not divisible by 2^k.\n- Stronger avoidance constructions for doubly-infinite orderings do not settle the source's one-sided bijection pi:N to N.\n\n**Full solution or refutation.**\n\nNo primary or maintained source located a proof or counterexample for unrestricted common difference in a one-sided permutation of the positive integers.\n\n**What remains.**\n\nDecide whether every bijection pi:N to N has a positive arithmetic progression a,a+d,a+2d,a+3d on which the pi-values are monotone.\n\n**Sources checked.**\n\n- Timothy D. LeSaulnier and Sujith Vijay, On Permutations Avoiding Short Progressions, Discrete Mathematics 311 (2011). (primary): https://arxiv.org/abs/1004.1740\n Evidence used: Constructs one-sided positive-integer permutations avoiding length-four progressions of odd common difference and proves the corresponding length-three forcing result.\n- Sarosh Adenwalla, Avoiding Monotone Arithmetic Progressions in Permutations of Integers, arXiv:2211.04451 (2022). (primary): https://arxiv.org/abs/2211.04451\n Evidence used: Separates one-sided and doubly-infinite order types and summarizes the best avoidance results for both.\n- Sarosh Adenwalla, A Generalisation of a Result on Monotone Arithmetic Progressions in Permutations of the Positive Integers, arXiv:2302.09662 (2023). (primary): https://arxiv.org/abs/2302.09662\n Evidence used: Generalizes odd-difference avoidance to all common differences not divisible by any prescribed power 2^k.\n\n**Review notes.** The distinction between a one-sided permutation and a doubly-infinite ordering is essential; results for order type Z were not treated as answers to the exact source question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3041, "problem_number": "OPG-636", "title": "Even vs. odd latin squares", "statement": "A latin square is even if the product of the signs of all of the row and column permutations is 1 and is odd otherwise.\n\nConjecture For every positive even integer $n$, the number of even latin squares of order $n$ and the number of odd latin squares of order $n$ are different.", "background": "Source: Open Problem Garden. Original node ID: 636. URL: http://www.openproblemgarden.org/op/even_vs_odd_latin_squares.\n\nSource subject path: Combinatorics.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/even_vs_odd_latin_squares\n- Author(s): Alon, Noga; Tarsi, Michael\n- Subject(s): Combinatorics\n- Keywords: latin square\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 7th, 2007 by mdevos\n\nProblem-page discussion:\nFor every positive integer $n$, let $ELS(n)$, ( $OLS(n)$ ) be the number of even (odd) latin squares of order $n$.\n\nThe inspiration for this conjecture comes from an attempt by Alon and Tarsi to use their polynomial technique to show that the complete bipartite graph $K_{n,n}$ is $n$-edge-choosable (a famous conjecture of Dinitz asserts that this is always true). They show (in [AT]) that whenever $ELS(n) \\neq OLS(n)$, the graph $K_{n,n}$ is $n$-edge-choosable. For odd integers $n>1$ it is easy to see that $ELS(n) = OLS(n)$, since interchanging the first two rows has no effect on the signs of the rows, but flips the signs of all of the columns. For even $n$, Alon and Tarsi checked that $ELS(n)$ and $OLS(n)$ were different for $n=2,4,6$ and conjectured that this pattern would continue. Although Dinitz' Conjecture has since been resolved, Alon and Tarsi's conjecture remains quite interesting. In particular, it has been shown by Huang and Rota [HR] that the truth of this conjecture would imply Rota's basis conjecture for even values of $n$ (see [O] for a nice proof of this).\n\nELS() and OLS() appear in the The Encyclopedia of Integer Sequences as A114628 and A114629. The following chart shows the first few values. Although the data here is quite limited, $ELS(n) > OLS(n)$ for every even $n$ in the chart, and as far as we know, this might hold in general.\n\nn\nELS(n)\nOLS(n)\n\n1\n1\n0\n\n2\n2\n0\n\n3\n6\n6\n\n4\n576\n0\n\n5\n80640\n80640\n\n6\n505958400\n306892800\n\n7\n30739709952000\n30739709952000\n\n8\n55019078005712486400\n53756954453370470400\n\nDrisko [D1] proved that whenever $p$ is prime, $ELS(p+1) - OLS(p+1) \\cong (-1)^{{(p+1)}/2} p^2$ (mod $p^3$ ), thus verifying the Alon-Tarsi conjecture for any even number which is one more than a prime. Shortly afterward, Zappa [Z] introduced a function $AT()$ which compares the number of even and odd latin squares which have all diagonal entries equal to one, and proved some interesting identities concerning $AT()$. By utilizing these identities, Drisko [D2] proved that $ELS(n) \\neq OLS(n)$ whenever $n$ is of the form $2^rp$ for a prime $p$.\n\nBibliography:\n*[AT] N. Alon, M. Tarsi, Coloring and Orientations of Graphs. Combinatorica 12, 125-143, 1992 MathSciNet\n\n[D1] A. Drisko, On the number of even and odd Latin squares of order $p+1$, Adv. Math. 128 (1997), no. 1, 20--35. MathSciNet\n\n[D2] A. Drisko, Proof of the Alon-Tarsi conjecture for $n=2\\sp rp$. Electron. J. Combin. 5 (1998) MathSciNet.\n\n[HR] R. Huang and G-C Rota, On the relations of various conjectures on Latin squares and straightening coefficients. Discrete Math. 128 (1994), no. 1-3, 225--236. MathSciNet.\n\n[O] S. Onn, A colorful determinantal identity, a conjecture of Rota, and Latin squares. Amer. Math. Monthly 104 (1997), no. 2, 156--159. MathSciNet.\n\n[Z] P. Zappa, The Cayley determinant of the determinant tensor and the Alon-Tarsi conjecture. Adv. in Appl. Math. 19 (1997), no. 1, 31--44. MathSciNet.\n\nRelated:\nRelated problems\nRota's basis conjecture\n\nSource links:\n- latin square: http://en.wikipedia.org/wiki/latin square\n\nDiscussion links:\n- Rota's basis conjecture: http://www.openproblemgarden.org/?q=node/631\n- The Encyclopedia of Integer Sequences: http://www.research.att.com/%7Enjas/sequences\n- A114628: http://www.research.att.com/%7Enjas/sequences/A114628\n- A114629: http://www.research.att.com/%7Enjas/sequences/A114629\n\nBibliography links:\n- Coloring and Orientations of Graphs: http://www.math.tau.ac.il/%7Enogaa/PDFS/chrom3.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1179249\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1451417\n- Proof of the Alon-Tarsi conjecture for $n=2\\sp rp$: http://www.combinatorics.org/Volume_5/PDF/v5i1r28.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1624999\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1271866\n- A colorful determinantal identity, a conjecture of Rota, and Latin squares: http://ie.technion.ac.il/%7Eonn/Preprints/AMM1.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1437419\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1453404\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Even vs. odd latin squares\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Alon-Tarsi Latin-square conjecture is proved when the even order is one more or one less than an odd prime, but remains open for general even order.\n\n**Verified partial progress.**\n\n- Drisko proves the conjecture for order p+1 when p is an odd prime.\n- Glynn proves the conjecture for order p-1 when p is an odd prime.\n- These two prime-neighbour families verify every even order through 24; structural equivalences and strong cancellation bounds are also known.\n\n**Full solution or refutation.**\n\nThe 2019 survey and a 2024 research overview still describe p plus or minus 1 as the best general families, so they do not cover every even order.\n\n**What remains.**\n\nProve nonzero even-minus-odd Latin-square sign sum for every remaining even order, or exhibit a counterexample.\n\n**Sources checked.**\n\n- Arthur Drisko, On the Number of Even and Odd Latin Squares of Order p+1, Advances in Mathematics 128 (1997), 20-35. (primary): https://doi.org/10.1006/aima.1997.1623\n Evidence used: Proves the p+1 family.\n- David G. Glynn, The Conjectures of Alon-Tarsi and Rota in Dimension Prime Minus One, SIAM Journal on Discrete Mathematics 24 (2010), 394-399. (primary): https://doi.org/10.1137/090773751\n Evidence used: Proves the p-1 family and identifies an error in a claimed power-of-two propagation argument.\n- Benjamin Friedman and Sean McGuinness, The Alon-Tarsi conjecture: A perspective on the main results, Discrete Mathematics 342 (2019), 2234-2253. (authoritative_secondary): https://doi.org/10.1016/j.disc.2019.04.018\n Evidence used: Surveys the conjecture and the p+1 and p-1 proofs as the principal established cases.\n- NTT Research, Representation Theory and Combinatorics Arising from Determinants (2024). (authoritative_secondary): https://www.rd.ntt/e/research/JN202407_27010.html\n Evidence used: A current research overview says the best results to date remain Drisko's p+1 and Glynn's p-1 families.\n\n**Review notes.** The database background's claim that Drisko proved the original conjecture for n=2^r p is misleading: the cited 1998 paper concerns an extended conjecture at prime order, and Glynn notes an error in a claimed power-of-two multiplication of known cases.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3042, "problem_number": "OPG-1797", "title": "2-accessibility of primes", "statement": "Question Is the set of prime numbers 2-accessible?", "background": "Source: Open Problem Garden. Original node ID: 1797. URL: http://www.openproblemgarden.org/op/2_accessibility_of_primes.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/2_accessibility_of_primes\n- Author(s): Landman, Bruce M.; Robertson, Aaron\n- Subject(s): Combinatorics\n- Keywords: monochromatic diffsequences; primes\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 9th, 2008 by vjungic\n\nProblem-page discussion:\nA set $S\\subseteq \\mathbb{N}$ is $r$-accessible if for any $r$-coloring of $\\mathbb{N}$, $r\\in \\mathbb{N}$, there exist long monochromatic $S$-diffsequences, i.e., for any $k\\in \\mathbb{N}\\backslash \\{ 1\\}$ there is a monochromatic sequence $\\{ x_1,x_2,\\ldots,x_k\\}$ such that $x_{i+1}-x_i\\in S$, for all $i\\in \\{ 1,\\ldots,k-1\\}$.\n\nThe set of primes $P$ is not 3-accessible. [LR2]\n\nLandman and Robertson proved [LR1] that for any odd $t$, the set $t+P$ is 2-accessible.\n\nIt is known that a 2-coloring of any 33 consecutive positive integers yields a monochromatic 7-term $P$-diffsequence.\n\nBibliography:\n[J] Jungi\\'c, Veselin, {\\it On a conjecture of Brown concerning accessible sets}, J. Combin. Theory Ser. A 110 (2005), MathSciNet\n\n[KL] Abdollah Khodkar and Bruce M. Landman, {\\it Recent progress in Ramsey theory on the integers}, Combinatorial number theory, 305--313, de Gruyter, Berlin, 2007. MathSciNet\n\n[LR1] Bruce M. Landman and Aaron Robertson, {\\it Avoiding Monochromatic Sequences With special Gaps}, SIAM J. Discrete Math Vol. 21 (2007), no. 3, 794--801. MathSciNet\n\n*[LR2] Bruce M. Landman and Aaron Robertson, Ramsey Theory on the Integers, Stud. Math. Libr. 24, AMS Providence, RI, 2004.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2128973\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2337054\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2354006\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"2-accessibility of primes\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Oscar Quester proves that the primes have degree of accessibility exactly two, answering the Landman–Robertson question affirmatively.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nEvery 2-coloring of the positive integers has arbitrarily long monochromatic diffsequences whose consecutive differences are prime.\n\n**What remains.**\n\nThe literal yes/no question is closed; quantitative bounds for the least finite interval forcing length k and accessibility of even translates remain separate questions.\n\n**Sources checked.**\n\n- Oscar Quester, The Primes are 2-Accessible, arXiv:2606.00410 (3 June 2026). (primary): https://arxiv.org/abs/2606.00410\n Evidence used: The abstract says the n=1 case answers Landman and Robertson; Theorem 2 gives degree of accessibility 2 for the primes.\n\n**Review notes.** This is a very recent arXiv preprint; no journal publication was verified. Its theorem nevertheless matches the source question exactly.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3043, "problem_number": "OPG-1825", "title": "3-accessibility of Fibonacci numbers", "statement": "Question Is the set of Fibonacci numbers 3-accessible?", "background": "Source: Open Problem Garden. Original node ID: 1825. URL: http://www.openproblemgarden.org/op/3_accessibility_of_fibonacci_numbers.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/3_accessibility_of_fibonacci_numbers\n- Author(s): Landman, Bruce M.; Robertson, Aaron\n- Subject(s): Combinatorics\n- Keywords: Fibonacci numbers; monochromatic diffsequences\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 24th, 2008 by vjungic\n\nProblem-page discussion:\nA set $S$ is $r$-accessible if for any $r$-coloring of $\\mathbb{N}$, $r\\in \\mathbb{N}$, there exist long monochromatic $S$-diffsequences, i.e., for any $k\\in \\mathbb{N}\\backslash \\{ 1\\}$ there is a monochromatic sequence $\\{ x_1,x_2,\\ldots,x_k\\}$ such that $x_{i+1}-x_i \\in S$, for all $i\\in \\{ 1,2,\\ldots,k-1\\}$.\n\nThe set of Fibonacci numbers $F$ is 2-accessible. [LR1]\n\n$F$ is not 6-accessible. [AGJL]\n\nIt is known that a 3-coloring of any 27 consecutive positive integers yields a monochromatic 4-term $F$-diffsequence.\n\nBibliography:\n[AGJL] Hayri Ardal, David Gunderson, Veselin Jungi\\'c, and Bruce Landman, {\\it On Accessibility of the Set of Fibonacci Numbers}, In Preparation\n\n*[LR1] Bruce M. Landman and Aaron Robertson, {\\it Avoiding Monochromatic Sequences With special Gaps}, SIAM J. Discrete Math Vol. 21 (2007), no. 3, 794--801.\n\n[LR2] Bruce M. Landman and Aaron Robertson, Ramsey Theory on the Integers, Stud. Math. Libr. 24, AMS Providence, RI, 2004\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"3-accessibility of Fibonacci numbers\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Fibonacci numbers are known to be 2-accessible and not 6-accessible; whether they are 3-accessible remains open.\n\n**Verified partial progress.**\n\n- The lower and upper accessibility thresholds bracket the question.\n\n**Full solution or refutation.**\n\nNo proof or disproof of 3-accessibility was verified.\n\n**What remains.**\n\nConstruct a 3-color obstruction or prove arbitrarily long monochromatic Fibonacci-difference sequences in every 3-coloring.\n\n**Sources checked.**\n\n- Emory University thesis, Distribution Agreement, accessibility survey discussion (accessed 2026-08-17). (authoritative_secondary): https://etd.library.emory.edu/downloads/zg64tn536?locale=en\n Evidence used: States Fibonacci numbers are 2-accessible but not 6-accessible and leaves the intermediate question open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3044, "problem_number": "OPG-2063", "title": "Wide partition conjecture", "statement": "Conjecture An integer partition is wide if and only if it is Latin.", "background": "Source: Open Problem Garden. Original node ID: 2063. URL: http://www.openproblemgarden.org/op/wide_partition_conjecture.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/wide_partition_conjecture\n- Author(s): Chow, Timothy Y.; Taylor, Brian D.\n- Subject(s): Combinatorics\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 24th, 2008 by tchow\n\nProblem-page discussion:\nAn integer partition $\\lambda$ is wide if $\\mu \\ge \\mu'$ for every subpartition $\\mu$ of $\\lambda$. (Here $\\mu'$ denotes the conjugate of $\\mu$, $\\ge$ denotes dominance or majorization order, and a subpartition of $\\lambda$ is a submultiset of the parts of $\\lambda$.) An integer partition $\\lambda$ is Latin if there exists a tableau $T$ of shape $\\lambda$ such that for every $i$, the $i$ th row of $T$ contains a permutation of $\\{1,2,\\ldots,\\lambda_i\\}$, and such that every column of $T$ contains distinct integers. It is easy to show that if $\\lambda$ is Latin then $\\lambda$ is wide, but the converse remains open.\n\nBibliography:\n*[CFGV] Timothy Y. Chow, C. Kenneth Fan, Michel X. Goemans, Jan Vondrak, Wide partitions, Latin tableaux, and Rota's basis conjecture, Advances Appl. Math. 21 (2003), 334-358.\n\nRelated:\nRelated problems\nRota's basis conjecture\n\nBibliography links:\n- Wide partitions, Latin tableaux, and Rota's basis conjecture: http://alum.mit.edu/www/tchow/wide.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Wide partition conjecture\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The wide-partition conjecture remains open, but the 2024 Latin-tableau work proves its criterion for the first four type parts and adds computational evidence.\n\n**Verified partial progress.**\n\n- The first four parts of the target type are characterized for each fixed shape.\n- New computations support the full conjecture.\n\n**Full solution or refutation.**\n\nNo proof that every wide partition is Latin was verified.\n\n**What remains.**\n\nExtend the partial type criterion to all parts or find a counterexample.\n\n**Sources checked.**\n\n- T. Y. Chow and M. G. Tiefenbruck, The Latin Tableau Conjecture, arXiv:2408.04086 (2024). (primary): https://arxiv.org/abs/2408.04086\n Evidence used: Abstract states the first-four-parts theorem and computational evidence.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3045, "problem_number": "OPG-37167", "title": "Shuffle-Exchange Conjecture", "statement": "Given integers $k,n\\ge2$, let $d(k,n)$ be the smallest integer $d\\ge2$ such that the symmetric group $\\frak S$ on the set of all words of length $n$ over a $k$-letter alphabet can be generated as $\\frak S = (\\sigma \\frak G)^d:=\\sigma\\frak G \\sigma\\frak G \\dots \\sigma\\frak G$ ( $d$ times), where $\\sigma\\in \\frak S$ is the shuffle permutation defined by $\\sigma(x_1 x_2 \\dots x_{n}) = x_2 \\dots x_{n} x_1$, and $\\frak G$ is the exchange group consisting of all permutations in $\\frak S$ preserving the first $n-1$ letters in the words.\n\nProblem (SE) Evaluate $d(k,n)$.\n\nConjecture (SE) $d(k,n)=2n-1$, for all $k,n\\ge2$.", "background": "Source: Open Problem Garden. Original node ID: 37167. URL: http://www.openproblemgarden.org/op/shuffle_exchange_conjecture.\n\nSource subject path: Combinatorics.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/shuffle_exchange_conjecture\n- Author(s): Beneš, Václav E.; Folklore; Stone, Harold S.\n- Subject(s): Combinatorics\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 27th, 2009 by Vadim Lioubimov\n\nProblem-page discussion:\nThis beautiful and difficult problem arises in switching networks theory and has important applications in parallel processing, sorting networks, card shuffling, etc. In this area it is perhaps the most famous open question which is at the center of the quest to understand the phenonemon of network rearrangeability. Both the problem and conjecture are referred to as Shuffle-Exchange (SE) ones. The case $k=2$ of SE problem (but not the conjecture) can be traced back to the work of Stone [S71], where he showed that $d(2,n)\\le n^2$. The upper bound $d(k,n)\\le 2n-1$ is the central case of Beneš conjecture [B75], while the lower bound $d(k,n)\\ge 2n-1$ can be easily seen (it is also a special case of the stronger version [B75] of Beneš conjecture, which turned out to be generally false). Since 1975 SE conjecture, especially its case $k=2$, has received a lot of attention, mostly in the context of switching networks, with rather modest results.\n\nNote that $d(k,n)\\le m$ is equivalent to $\\frak S = (\\sigma \\frak G)^m$, for any integer $m\\ge2$. Furthermore, it is easy to see that the latter decomposition is equivalent to $\\frak S = \\frak G_1 \\frak G_2\\dots \\frak G_{m}$, where $\\frak G_i:=\\sigma^{i} \\frak G \\sigma^{-i}$ is the subgroup of $\\frak S$ consisting of all permutations which may only change the letters on the position $i-1\\ (\\text{mod } n) + 1$ in the words.\n\nAlso, the case $n=2$ of SE conjecture can be reformulated as the following\n\nTheorem ( $\\star$ ) Every permutation of entries of a square matrix can be obtained in 3 steps as follows: first by permuting entries in the columns, then - in the rows, and then - in the columns again. Moreover, some permutations cannot be obtained in less than 3 of such steps.\n\n(The parameter $k$ in the case $n=2$ of SE conjecture corresponds to the size $k\\times k$ of a matrix in the theorem.) Moreover, the general case of SE conjecture can be reformulated as a straightforward generalization of Theorem ( $\\star$ ) to the $n$-dimensional cubic matrices (of size $k\\times\\dots\\times k$ ) stating that every permutation of entries of such a matrix can be obtained in $2n-1$ steps in a similar way, and this number is generally a precise lower bound.\n\nTheorem ( $\\star$ ) holds as being easily equivalent to the special case of the following classical result when each part of the multigraph has size $k$:\n\nTheorem (König) A $k$-regular bipartite multigraph is $k$-edge-colorable.\n\nThe function $d(k,n)$ admits 3 main interpretations (that are not immediately equivalent), \"group-theoretic\" (presented in the beginning), \"combinatorial\" (below), and \"graph-theoretic\", each of which provides its own framework for SE problem and suggests its own interesting natural generalizations and extensions. Accordingly, there are 3 equivalent forms of SE problem/conjecture. Although the group-theoretic interpretation of $d(k,n)$ is the shortest and most elegant among the three, it seems the least natural when it comes to proving the known results and studying SE problem more deeply. I believe that SE problem is very deep and combinatorial by nature. I also strongly believe in the validity of SE conjecture.\n\n2. Combinatorial form of SE problem/conjecture\n\nGiven a pure abstract simplicial complex $\\Delta$ of rank $n\\ge2$ and a positive integer $\\ell$, an $\\ell$-transition is a map that assigns to evey pair of ordered facets, $x_1,\\dots,x_{n}$ and $y_1,\\dots,y_{n}$, a sequence of vertices $z_1,\\dots,z_{\\ell}$ such that every $n$-segment of the sequence $x_1,\\dots,x_{n},z_1,\\dots,z_{\\ell}, y_1,\\dots,y_{n}$ forms a facet. Let $\\text{tr}(\\Delta)$ be the smallest $\\ell$, or $\\infty$ if none exists, for which there exists an $\\ell$-transition. Note that $\\text{tr}(\\Delta)\\le \\ell$ is equivalent to the existence of $\\ell$-transition for $\\Delta$, for any $\\ell\\ge 1$.\n\nGiven integers $k,n\\ge2$, let $\\Delta_{k,n}$ be the pure abstract simplicial complex of rank $n$ whose vertex set is the set $V_{k,n}$ of all uniform $k$-partitions (i.e., ones consisting of $k$ equal-sized blocks) of a $k^n$-set, and whose facets are all $n$-subsets of $V_{k,n}$ with zero infinum. Using normal reasoning, it is not hard to show [L04] the following\n\nTheorem $d(k,n) = \\text{tr}(\\Delta_{k,n})+n$.\n\nThus, the combinatorial forms of SE problem and conjecture can be formulated as to find $\\text{tr}(\\Delta_{k,n})$ and that $\\text{tr}(\\Delta_{k,n})=n-1$, respectively.\n\nThe infinum (or meet) of two partitions $\\mathbf{a}$ and $\\mathbf{b}$ of a set $E$ is the partition of $E$ defined by $$\\mathbf{a\\wedge b}:= \\big\\{\\, a\\cap b\\ne\\varnothing \\ | \\ a\\in\\mathbf{a} \\ \\&\\ b\\in\\mathbf{b} \\,\\big\\}.$$Note that together with the operation$\\wedge$, the collection of all partitions of$E$forms a semilattice (i.e., a commutative and idempotent semigroup) with the identity and zero being the partitions$\\mathbf{1}_E:=\\{E\\}$and$\\mathbf{0}_E:=\\big\\{\\{x\\} \\ | \\ x\\in E \\big\\}$, respectively.\n\nObserve that the complex $\\Delta_{k,n}$ is non-matroidal for all $(k,n)\\ne (2,2)$.\n\n3. Constructive version of SE problem/conjecture\n\nApplication-wise it is important not only to establish a certain decomposition $\\frak S = (\\sigma \\frak G)^m$ or, equivalently, rearrangeability of the graph $(\\text{SE}(k,n))^{m-1}$ or, equivalently, the existence of an $(m-n)$-transition for the complex $\\Delta_{k,n}$, but also to find a corresponding efficient factorization/routing/transition algorithm.\n\nGiven an identity $A = A_1A_2\\dots A_m$, where all $A_i$ are subsets of a multiplicative group, a factorization algorithm finds for every $a\\in A$ an $m$-tuple $(a_1,\\dots,a_m)\\in A_1\\times \\dots \\times A_m$ such that $a = a_1a_2\\dots a_m$. Given a rearrangeable graph $(\\text{SE}(k,n))^{m-1}$, a routing algorithm takes a mask of the graph as input and returns a corresponding routing. Given a pure simplicial complex $\\Delta$ with $\\text{tr}(\\Delta)\\le r$, an $r$-transition algorithm realizes an $r$-transition for $\\Delta$. It is not hard to prove\n\nTheorem Any factorization algorithm for $\\frak S = (\\sigma \\frak G)^m$ translates into a routing algorithm for $(\\text{SE}(k,n))^{m-1}$ and into an $(m-n)$-transition algorithm for $\\Delta_{k,n}$ of the same complexity, and vise versa. Consequently, $D^{*}(k,n) = R^{*}(k,n) = T^{*}(k,n)$.\n\nHere $D^*(k,n)$, $R^*(k,n)$, and $T^{*}(k,n)$ are the sets of all $m\\ge 2$, respectively, for which there exists an efficient polynomial-time (in $k^n$ ) factorization/routing/transition algorithm mentioned in the above theorem (we will also write $A \\buildrel{*}\\over= A_1A_2\\dots A_m$ to indicate the existence of such a factorization algorithm for $A = A_1A_2\\dots A_m$, where each $A_i\\subseteq \\frak S$ ). Clearly, $$d^*(k,n)\\ge d(k,n)= r(k,n)=\\text{tr}(\\Delta_{k,n})+n,$$where$d^*(k,n):= \\min D^*(k,n)$with the usual convention$\\min\\varnothing:=\\infty$, and$r(k,n)$ is defined here.\n\nIt is easy to see that $d\\in D^*(k,n)$ implies $[d,\\infty)\\subseteq D^*(k,n)$ (equivalently, the same is true for $R^*(k,n)$ and $T^{*}(k,n)$ ). Consequently, $d^*(k,n)\\le m$ is equivalent to $\\frak S \\buildrel{*}\\over= (\\sigma \\frak G)^m$.\n\nProblem (CSE) Evaluate $d^*(k,n)$ and specify the corresponding factorization/routing/transition algorithm for the upper bound.\n\nConjecture (CSE) $d^*(k,n)=2n-1$.\n\nBoth the problem and conjecture are referred here to as Constructive Shuffle-Exchange (CSE) ones. The conjecture was proposed in [L04]. Clearly, CSE conjecture implies SE one as $2n-1\\le d(k,n)\\le d^*(k,n)$.\n\n4. Main results\n\nSo far SE/CSE conjecture has been only settled in the following 3 cases: $n=2$, $(k,n)=(2,3)$, and $(k,n)=(2,4)$. That is, the following 3 identities holds:\n\n$$(1)\\ d(k,2) = d^*(k,2) = 3,\\ \\ (2)\\ d(2,3) = d^*(2,3) = 5,\\ \\ (3)\\ d(2,4) = d^*(2,4) = 7.$$\n\nAlso, there are 2 the following major results on SE/CSE problem:\n\n(4) $d(k,n)\\ge 2n-1$.\n\n(5) $d^{(*)}(k,n)\\le d^{(*)}(k,r)+3(n-r)$, for all $n > r\\ge2$.\n\nThe lower bound (4) follows immediately from the obvious observation that $\\text{tr}(\\Delta) \\ge \\dim(\\Delta)$, for any pure complex $\\Delta$.\n\nNote that (4) reduces SE (CSE) conjecture to $d^{(*)}(k,n)\\le 2n-1$ which is equivalent to $\\frak S \\buildrel{(*)}\\over= (\\sigma \\frak G)^{2n-1}$. In fact, the main reason why SE/CSE conjecture is widely believable, apart from results (1-4), is a close similarity between the latter decomposition and the following well known result [B65, L04] (that is not hard to derive from the constructive version of the König's theorem):\n\nTheorem (Beneš) $\\frak S \\buildrel{*}\\over= (\\frak G\\sigma^{-1})^{n-1}\\frak G(\\sigma \\frak G)^{n-1}$.\n\nCombining (1) and (3) with (5) yields respectively the following 2 best known upper bounds (in addition to (2)) for both $d(k,n)$ and $d^*(k,n)$:\n\n$(6)\\quad d(k,n)\\le d^*(k,n)\\le 3n-3$, for all $k\\ge3$ and $n\\ge2$\n\n$(7)\\quad d(2,n)\\le d^*(2,n)\\le 3n-5$, for all $n\\ge4$.\n\nAs it was mentioned earlier, the case $n=2$ of SE conjecture is easily equivalent to the following case of the Konig's theorem: a $k$-regular bipartite multigraph $B$ with $k$-vertex parts is $k$-edge-colorable. Moreover, any $k$-edge-coloring algorithm for the graph $B$ easily translates into a factorization/routing/1-transition algorithm of the same complexity for $\\frak S = (\\sigma \\frak G)^3$ (at $n=2$ ) or the graph $(\\text{SE}(k,2))^{2}$ or the complex $\\Delta_{k,2}$, respectively, and vise versa. Consequently, as there are many efficient polynomial-time (in $k^2$ ) $k$-edge-coloring algorithms well known for the graph $B$, the case $n=2$ of CSE conjecture also holds.\n\nThere are at least 6 alternative proofs proposed for the case $(k,n)=(2,3)$ of CSE conjecture. Although they may look quite different, each proof is essentially based on either of 3 similar short and elegant algorithms which we refer to as A1 [RV87, LT89, L04], A2 [ND00, L04] and A3 [KR91]. Each algorithm is based on a 2-edge-coloring algorithm for a 2-regular bipartite multigraph with 4-vertex parts. Namely, A1 uses 2, A2 uses at most 2, and A3 uses 1 application(s) of such an algorithm. Each algorithm Ai deals with 2 cases in which the procedure is especially simple. The algorithms A1 and A2 are very efficient (with A2 being slightly faster than A1), while A3 is not so (contrary to what is claimed in [KR91]) as it relies on an exhausting search to determine the case for each input permutation. However, A3 has some theoretical advantage over A1 and A2 as its 2 cases partition the symmetric group $S_8$ into 2 classes that do not depend on a realization of the algorithm. In [L04], both algorithms A1 and A2 are explicitly described as 2-transition algorithms for the complex $\\Delta_{2,3}$, and the corresponding 2 proofs for the statement $\\text{\\rm tr}(\\Delta_{2,3})=2$ are particularly transparent. Moreover, the latter statement, the algorithms and the proofs are straightforwardly generalized [L05] to a wide class of 2-dimensional pure abstract simplicial complexes.\n\nA brute force verification for the case $(k,n)=(2,4)$ of SE conjecture was first reported in [R95]. The first theoretical proof for such case of CSE conjecture was proposed (in graph-theoretic terms) in [ND00]. Although the ideas behind the underlying algorithm for this proof are simple, the algorithm deals with a huge and intricate tree of cases and is substantially more complicated (and not so elegant) than that of the case $(k,n)=(2,3)$. As a result, the proof is very tedious, hard to verify, and leaves little hope for using a similar approach to prove the next case $(k,n)=(2,5)$ of CSE conjecture. An essentially similar but slightly better organized algorithm and proof for (3) were proposed in [DS08] (with no reference to [ND00]).\n\nThe upper bound (5) was first obtained in [VR88] for the case $k=2$ and (i) $d^{(*)}(k,r)=2r-1$. In other words, it was shown that (i) at $k=2$ implies $d^{(*)}(2,n)\\le 3n-r-1$, if $n > r$. A much simpler proof of (5) for the case $r=2,3$ and (i) appeared in [LT89]. The latter proof was easily extended [ND00] to an arbitrary $r\\ge2$. A transparent combinatorial proof in terms of the complex $\\Delta_{k,n}$ for the general case of (5) was proposed in [L04]. This proof (together with its underlying transition algorithm) was generalized [L05] to a wide class of pure abstract simplicial complexes of arbitrary dimensions. Namely, it was shown that, given a complex $\\Delta$ in this class and an integer $1\\le m<\\dim(\\Delta)$,\n\n$$:\\qquad\\qquad \\text{tr}(\\Delta) \\le 2m + \\max \\big\\{ \\text{tr}(\\Delta/F) \\mid F\\in\\Delta,\\ |F|=m \\big\\}$$\n\nand, moreover, that any $\\ell$-transition algorithm for the complexes $\\Delta/F$ can be efficiently used to make a $(2m+\\ell)$-transition algorithm for $\\Delta$. Note that (5) can be easily obtained as an instance of the latter result. Here $\\Delta/ F$ is the link of a face $F$ in $\\Delta$, i.e., a subcomplex of $\\Delta$ defined by\n\n$$:\\qquad\\qquad \\Delta/ F:= \\{ A\\in \\Delta \\ | \\ A\\cap F = \\varnothing \\ \\&\\ A\\cup F\\in\\Delta \\big\\}.$$\n\nIt is worth noting that there are many flawed proofs for SE conjecture in the literature. Most notably, in [Ba01] (the general case) and [C03] (the case $k=2$ ). The latter proof was first refuted in [BHL06], while the former remains unrefuted in the literature.\n\nBibliography:\n[B65] V.E. Benes, Mathematical theory of connecting networks and telephone traffic, Academic Press, New York, 1965.\n\n*[S71] H.S. Stone, Parallel processing with the perfect shuffle, IEEE Trans. on Computers C-20 (1971), 153-161.\n\n*[B75] V.E. Beneš, Proving the rearrangeability of connecting networks by group calculation, Bell Syst. Tech. J. 54 (1975), 421-434.\n\n[RV87] C.S. Raghavendra, A. Varma, Rearrangeability of 5-stage shuffle/exchange network for N=8, IEEE Trans. on Commun. COM-35 (1987), 808-812.\n\n[VR88] A. Varma, C.S. Raghavendra, Rearrangeability of multistage shuffle/exchange networks, IEEE Trans. on Commun. 36 (1988), 1138-1147.\n\n[LT89] N. Linial, M. Tarsi, Interpolation between bases and the shuffle-exchange networks, European J. of Combinatorics, 10(1) (1989), 29-39.\n\n[KR91] K. Kim, C.S. Raghavendra, A Simple Algorithm to Route Arbitrary Permutations on 8-input 5-stage Shuffle/Exchange Network, Proc. 5th International Parallel Processing Symposium (1991), 398-403.\n\n[R95] C.S. Raghavendra, On the rearrangeability conjecture of $(2\\log_2 N -1)$-stage shuffle/exchange network, IEEE Computer Society, Tech. Committee on Comp. Arch. Newsletter, Position paper (Winter 1995), 10-12.\n\n[ND00] H.Q. Ngo, D.Z. Du, On the rearrangeability of shuffle-exchange networks, Tech. Report TR00-045, Dept. of Computer Science, Univ. of Minnesota (2000)\n\n[Ba01] R.E. Bashirov, On the rearrangeability of 2s-1 stage networks employing uniform interconnection pattern Calcolo, Springer Verlag, 38(2) (2001), 85-97.\n\n[C03] H. Cam, Rearrangeability of (2n-1)-stage shuffle-exchange networks, SIAM J. on Computing 32(3) (2003), 557-585.\n\n[L04] V. Lioubimov, Decomposition of symmetric group into product of stabilizers and Shuffle-Exchange problem, manuscript (2004).\n\n[L05] V. Lioubimov, Facet transitions in abstract simplicial complexes, manuscript (2005).\n\n[BHL06] X. Bao, F.K. Hwang, Q. Li, Rearrangeability of bit permutation networks, Theoretical Computer Science, 352(1) (2006), 197-214.\n\n[DS08] H. Dai, X. Shen, Rearrangeability of 7-stage 16x16 shuffle-exchange networks, Frontiers of Electrical and Electronic Engineering in China, 3(4) (2008), 440-458.\n\nRelated:\nRelated problems\nShuffle-Exchange Conjecture (graph-theoretic form)\nBeneš Conjecture\nBeneš Conjecture (graph-theoretic form)\n\nDiscussion links:\n- Beneš conjecture: http://www.openproblemgarden.org/?q=node/37181\n- stronger version: http://www.openproblemgarden.org/?q=node/37181\n- \"graph-theoretic\": http://www.openproblemgarden.org/?q=node/37089\n- rearrangeability: http://www.openproblemgarden.org/?q=node/37089\n- graph $(\\text{SE}(k,n))^{m-1}$: http://www.openproblemgarden.org/?q=node/37089\n- mask: http://www.openproblemgarden.org/?q=node/37089\n- routing: http://www.openproblemgarden.org/?q=node/37089\n- here: http://www.openproblemgarden.org/?q=node/37089\n\nBibliography links:\n- On the rearrangeability of shuffle-exchange networks: http://www.cs.umn.edu/research/technical_reports.php?page=report&report_id=00-045\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 122.\n\nAttempt notes:\nTarget:\nMake progress on \"Shuffle-Exchange Conjecture\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The shuffle-exchange formula d(k,n)=2n-1 is false: a 2026 preprint proves d(k,3)=6 for every k>=3, whereas the conjecture predicts 5.\n\n**Verified partial progress.**\n\n- The paper proposes d(k,n)=3n-3 for k>=3 as a replacement question and proves its n=3 base case.\n\n**Full solution or refutation.**\n\nThe universal conjecture is refuted by the explicit k>=3, n=3 counterexample.\n\n**What remains.**\n\nDetermine d(k,n) in general; the suggested replacement formula is not established beyond the base case.\n\n**Sources checked.**\n\n- P. Chojecki, Beneš and Shuffle-Exchange Counterexamples, arXiv:2607.15296 (2026). (primary): https://arxiv.org/abs/2607.15296\n Evidence used: The abstract and Theorem 19 state d(k,3)=6 for all k>=3 and hence refute the formula.\n\n**Review notes.** Recent primary preprint; formulation and ID preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3046, "problem_number": "OPG-37181", "title": "Beneš Conjecture", "statement": "Let $E$ be a non-empty finite set. Given a partition $\\bf h$ of $E$, the stabilizer of $\\bf h$, denoted $S(\\bf h)$, is the group formed by all permutations of $E$ preserving each block of $\\mathbf h$.\n\nProblem ( $\\star$ ) Find a sufficient condition for a sequence of partitions ${\\bf h}_1, \\dots, {\\bf h}_\\ell$ of $E$ to be complete, i.e. such that the product of their stabilizers $S({\\bf h}_1) S({\\bf h}_2) \\dots S({\\bf h}_\\ell)$ is equal to the whole symmetric group $\\frak S(E)$ on $E$. In particular, what about completeness of the sequence $\\bf h,\\delta(\\bf h),\\dots,\\delta^{\\ell-1}(\\bf h)$, given a partition $\\bf h$ of $E$ and a permutation $\\delta$ of $E$?\n\nConjecture (Beneš) Let $\\bf u$ be a uniform partition of $E$ and $\\varphi$ be a permutation of $E$ such that $\\bf u\\wedge\\varphi(\\bf u)=\\bf 0$. Suppose that the set $\\big(\\varphi S({\\bf u})\\big)^{n}$ is transitive, for some integer $n\\ge2$. Then $$\\frak S(E) = \\big(\\varphi S({\\bf u})\\big)^{2n-1}.$$", "background": "Source: Open Problem Garden. Original node ID: 37181. URL: http://www.openproblemgarden.org/op/bene_conjecture.\n\nSource subject path: Combinatorics.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bene_conjecture\n- Author(s): Beneš, Václav E.\n- Subject(s): Combinatorics\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: January 3rd, 2010 by Vadim Lioubimov\n\nProblem-page discussion:\nThis conjecture was essentially proposed by Václav E. Beneš in 1975 [B75] and bears his name. It remains open for all $n\\ge3$.\n\nA partition of a set is uniform if all its blocks have the same size. Given a subset $P$ of a multiplicative group and a positive integer $m$, by $P^m$ we mean the product $PP\\dots P$ ( $m$ times). A set $T\\subseteq\\frak S(E)$ is transitive if for every $x,y\\in E$ there exists a permutation $\\tau\\in T$ such that $\\tau(x)=y$. The infinum of two partitions $\\bf a$ and $\\bf b$ of $E$ is the partition of $E$ defined by\n\n$$: \\qquad\\qquad\\qquad {\\bf a\\wedge b}:= \\big\\{\\, a\\cap b\\ne\\varnothing \\ | \\ a\\in{\\bf a} \\ \\&\\ b\\in{\\bf b} \\,\\big\\}.$$\n\nThe partition $\\bf 0$ of $E$ is defined by ${\\bf 0}:={\\bf 0}_E:=\\big\\{\\{x\\} \\ | \\ x\\in E \\big\\}$. So the condition ${\\bf h}\\wedge\\delta({\\bf h})={\\bf 0}$ is equivalent to saying that for every pair of blocks $a,b\\in{\\bf h}$, the intersection $a\\cap\\delta(b)$ consists of at most one element.\n\nObserve that the decomposition $\\frak S(E) = \\big(\\delta S({\\bf h})\\big)^{\\ell}$ is equivalent to completeness of the sequence ${\\bf h},\\delta({\\bf h}),\\dots,\\delta^{\\ell-1}({\\bf h})$ due to the obvious identity $\\delta S({\\bf h}) \\delta^{-1} = S(\\delta{\\bf h})$. Thus Problem ( $\\star$ ) is indeed underlying for Beneš conjecture.\n\nProblem ( $\\star$ ) is a special case of a broader fundamental problem of description of product of stabilizers on a finite set. The latter problem, which I believe is combinatorial by nature, is of great interest in switching network study. However, despite many years of extensive research on its various cases in the context of switching networks, this fascinating problem remains unsolved in all but a very few interesting instances. Very little is understood about such products beyond what is obvious. In particular, it is unclear how to efficiently compute their cardinalities. Even for some rather simple sequences of partitions, the product of their stabilizers is surprisingly difficult to describe. Beneš conjecture, if proven (even under some additional assumptions on $E,{\\bf u},\\varphi$ ), would provide a very useful and easy-to-check sufficient condition for completeness of the sequences ${\\bf u},\\varphi({\\bf u}),\\dots,\\varphi^{\\ell-1}({\\bf u})$ that are of particular interest.\n\nAnother important and interesting problem related to ( $\\star$ ) is to find an efficient polynomial-time (in $|E|$ ) factorization algorithm for the identity $\\frak S(E) = S({\\bf h}_1) S({\\bf h}_2) \\dots S({\\bf h}_\\ell)$. Given an identity $A = A_1A_2\\dots A_\\ell$, where all $A_i$ are subsets of a multiplicative group, a factorization algorithm finds for every $a\\in A$ an $\\ell$-tuple $(a_1,\\dots,a_\\ell)\\in A_1\\times \\dots \\times A_\\ell$ such that $a = a_1a_2\\dots a_\\ell$.\n\nBeneš conjecture is mainly famous for its central case, Shuffle-Exchange (SE) conjecture, stating essentially that $\\frak S(\\tilde X) = \\big(\\sigma S({\\bf g})\\big)^{2n-1}$, where $(\\tilde X,{\\bf g},\\sigma)$ is an instance of $(E,{\\bf u},\\varphi)$ defined, given arbitrary integer parameters $k,n\\ge2$, as follows:\n\n$\\bullet$ $\\tilde X$ is the set of all words of length $n$ over a $k$-letter alphabet $X$.\n\n$\\bullet$ $\\bf g$ is the $k^{n-1}$-partition of $\\tilde X$ formed by the equivalence relation $\\sim$ on $\\tilde X$ defined by\n\n$:\\qquad\\qquad x_1\\dots x_{n} \\sim y_1\\dots y_{n}: \\Leftrightarrow x_1\\dots x_{n-1}=y_1\\dots y_{n-1}$.\n\n$\\bullet$ $\\sigma$ is the shuffle permutation of $\\tilde X$ defined by $\\sigma(x_1 x_2 \\dots x_{n}):= x_2 \\dots x_{n} x_1$.\n\nWhereas SE conjecture, especially its case $k=2$, has received enormous attention in the study of switching networks with relatively little progress, the general case of Beneš conjecture, despite importance of Problem ( $\\star$ ) in that area, has virtually generated no literature and had no progress. While I strongly believe in the validity of SE conjecture, I am not so sure about the general case of Beneš conjecture and even do not rule out that it could be disproved by a low-scale counterexample. On the other hand, I cannot rule out that Beneš conjecture (possibly under some mild additional assumptions on $E,{\\bf u},\\varphi$ ) may be reduced to SE conjecture.\n\nIt is easy to see that the case $n=2$ of Beneš conjecture coincides with that of SE conjecture. The latter case is well known to be valid (discussed here).\n\nUnlike completeness of a sequence of partitions of $E$, the condition of transitivity of the product of their stabilizers is very easy to check. In particular, transitivity of the set $\\big(\\delta S({\\bf h})\\big)^{n}$ with $n\\ge2$ is equivalent to the following assertion:\n\n$$:\\qquad\\qquad \\forall\\,h_1,h_n\\in{\\bf h} \\ \\exists\\,h_2,\\dots,h_{n-1}\\in{\\bf h} \\ \\forall\\, i\\in[n-1]: h_i\\cap \\delta(h_{i+1}) \\ne \\varnothing.$$\n\nBeneš conjecture (as well as its underlying Problem ( $\\star$ ) and a broader problem of description of product of stabilizers on a finite set) admits a nice equivalent graph-theoretic form.\n\nCounterexamples\n\nIn what follows we present 3 counterexamples showing that certain stronger versions of Beneš conjecture are false.\n\nCounterexample 1. The condition ${\\bf u}\\wedge\\varphi({\\bf u})={\\bf 0}$ is necessary for Beneš conjecture. This can be shown by the following simple counterexample:\n\n$$:\\qquad\\qquad E:= \\{1,2,3,4,5,6\\}, \\ {\\bf u}:= \\big\\{\\{1,2,3\\}, \\{4,5,6\\}\\big\\}, \\text{ and } \\varphi:= (3,4).$$\n\nIndeed, ${\\bf u}\\wedge\\varphi({\\bf u}) \\ne {\\bf 0}$ as $\\{1,2,3\\}\\cap\\varphi\\{1,2,3\\} = \\{1,2\\}$. Also, the set $\\big(\\varphi S({\\bf u})\\big)^{2}$ is obviously transitive. However, it can be easily seen that any permutation $\\alpha$ of $E$ satisfying $\\alpha \\{1,2,3\\} = \\{4,5,6\\}$ does not belong to $S({\\bf u})\\varphi S({\\bf u})\\varphi S({\\bf u})$. Thus, $\\frak S(E) \\ne \\big(\\varphi S({\\bf u})\\big)^{3}$. In fact, the condition ${\\bf u}\\wedge\\varphi({\\bf u})={\\bf 0}$ is missing in the original statement [B75] of Beneš conjecture (however, such condition is commonly (but not always) assumed in the context of switching networks).\n\nCounterexample 2. Beneš conjecture is not directly generalizable to the products of stibilizers of the form $P:=S({\\bf u})\\varphi_1 S({\\bf u})\\dots\\varphi_{n-1} S({\\bf u})$. More precisely, transitivity of $P$ does not always imply $\\frak S(E) = P^2$, where ${\\bf u}$ is a uniform partition of $E$ and all $\\varphi_i$ are permutations of $E$ such that ${\\bf u}\\wedge\\varphi_i({\\bf u})={\\bf 0}$ (while Beneš conjecture states that this implication is always true as long as $\\varphi_1=\\dots=\\varphi_{n-1}$ ). For that I constructed the following counterexample:\n\n$$:\\qquad\\qquad E:= \\{1,2,\\dots,12\\}, \\ {\\bf u}:= \\big\\{\\{1,2\\}, \\{3,4\\},\\dots,\\{11,12\\}\\big\\}, \\ \\varphi_1:= (2,3)(6,7)(10,11) \\text{ and } \\varphi_2:= (2,7)(4,9)(6,11).$$\n\nIndeed, it is obvious that both permutations $\\varphi_1, \\varphi_2$ are satisfying ${\\bf u}\\wedge\\varphi_i({\\bf u})={\\bf 0}$ and the set $Q:=S({\\bf u})\\varphi_1 S({\\bf u})\\varphi_2 S({\\bf u})\\varphi_1S({\\bf u})$ is transitive. However, $\\frak S(E) \\ne Q^2$ as, in particular, it can be easily seen that any permutation $\\alpha$ of $E$ satisfying $\\alpha \\{1,2,3,4\\} = \\{5,6,7,8\\}$ does not belong to $Q^2$.\n\nCounterexample 3. In the same paper [B75], Beneš also proposed the following\n\nConjecture ( $\\diamond$ ) Let $\\bf u$ be a uniform partition of $E$ and $\\varphi$ be a permutation of $E$ such that $\\bf u\\wedge\\varphi(\\bf u)=\\bf 0$. Suppose that $n\\ge2$ is the smallest integer such that the set $\\big(\\varphi S({\\bf u})\\big)^{n}$ is transitive. Then $\\frak S(E) \\ne \\big(\\varphi S({\\bf u})\\big)^{2n-2}$.\n\nIn other words, this conjecture together with Beneš one, asserts that if $n\\ge2$ is the smallest integer such that $\\big(\\varphi S({\\bf u})\\big)^{n}$ is transitive, then $2n-1$ is the the smallest integer $\\ell$ such that $\\frak S(E) = \\big(\\varphi S({\\bf u})\\big)^{\\ell}$. However, Conjecture ( $\\diamond$ ) turned out to be generally false as I found the following counterexample for it:\n\n$$:\\qquad\\qquad E:= \\{1,2,\\dots,8\\}, \\ {\\bf u}:= \\big\\{\\{1,2\\}, \\{3,4\\}, \\{5,6\\}, \\{7,8\\}\\big\\}, \\text{ and } \\varphi:= (2,3)(4,5,6,7).$$\n\nIndeed, it is easy to verify that ${\\bf u}\\wedge\\varphi({\\bf u})={\\bf 0}$ and the set $\\big(\\varphi S({\\bf u})\\big)^{4}$ is transitive while $\\big(\\varphi S({\\bf u})\\big)^{3}$ is not as, in particular, $\\{4,7\\} \\cap \\big(\\varphi S({\\bf u})\\big)^{3}\\{1,2\\} =\\varnothing$. However, a brute force verification confirmed that $\\frak S(E) = \\big(\\varphi S({\\bf u})\\big)^{6}$.\n\nBibliography:\n*[B75] V.E. Beneš, Proving the rearrangeability of connecting networks by group calculation, Bell Syst. Tech. J. 54 (1975), 421-434.\n\nRelated:\nRelated problems\nShuffle-Exchange Conjecture\nBeneš Conjecture (graph-theoretic form)\nShuffle-Exchange Conjecture (graph-theoretic form)\n\nDiscussion links:\n- Václav E. Beneš: http://en.wikipedia.org/wiki/Václav E. Beneš\n- Shuffle-Exchange (SE) conjecture: http://www.openproblemgarden.org/?q=node/37167\n- SE conjecture: http://www.openproblemgarden.org/?q=node/37167\n- here: http://www.openproblemgarden.org/?q=node/37167\n- graph-theoretic form: http://www.openproblemgarden.org/?q=node/37210\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 101.\n\nAttempt notes:\nTarget:\nMake progress on \"Beneš Conjecture\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The partition-stabilizer Beneš conjecture in the source statement is refuted by explicit two-stage graph counterexamples to its stated inequality/form.\n\n**Verified partial progress.**\n\n- The counterexample paper retains exact middle criteria and balanced-middle sufficient conditions, identifying stronger hypotheses that may support replacement statements.\n\n**Full solution or refutation.**\n\nThe source's universal implication is claimed false by explicit counterexamples in the primary preprint.\n\n**What remains.**\n\nFormulate and prove a corrected sufficient condition for completeness under additional hypotheses.\n\n**Sources checked.**\n\n- P. Chojecki, Beneš and Shuffle-Exchange Counterexamples, arXiv:2607.15296 (2026). (primary): https://arxiv.org/abs/2607.15296\n Evidence used: The abstract expressly says it refutes the partition-stabilizer form as stated on Open Problem Garden.\n\n**Review notes.** Recent primary preprint; formulation and ID preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3047, "problem_number": "OPG-37222", "title": "Dividing up the unrestricted partitions", "statement": "Begin with the generating function for unrestricted partitions:\n\n(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...\n\nNow change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the signs may be chosen such that all the coefficients of the series are either 1, -1, or zero.", "background": "Source: Open Problem Garden. Original node ID: 37222. URL: http://www.openproblemgarden.org/op/dividing_up_the_unrestricted_partitions.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/dividing_up_the_unrestricted_partitions\n- Author(s): David S.; Newman\n- Subject(s): Combinatorics\n- Keywords: congruence properties; partition\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 11th, 2010 by DavidSNewman\n\nProblem-page discussion:\nI've been thinking about this problem since about 1970. Emory Starke thought that it was a good problem, but not suitable for the Problems section of the AMM, because it was unsolved. George Andrews and Freeman Dyson also thought that it is a good problem, but neither had any ideas how to solve it.\n\nI've found choices of sign which yield series with coefficients 1, -1, or 0 for all exponents about as high as 110 using computer searches. One thing which mitigates against finding a meaningful solution is that there is no known pattern for the number of unrestricted partitions modulo 2.\n\nBibliography:\nAndrews, George E., The Theory of Partitions, Cambridge University Press (1984)\n\nComments:\n- May 18th, 2010 | Benjamin Young | a question on the numerical work: I'm also curious to know a little more about the experimental work that was done -- roughly how many ways were there to choose the signs to make things work up to degree 110? were there any choices which gave you lots of zeros?\n- May 18th, 2010 | Benjamin Young | pentagonal number theorem: Euler's famous pentagonal number theorem is somewhat like this problem, except it deals with the generating function for partitions into distinct parts:\n\n(1+x)(1+x^2)(1+x^3)...\n\nIf you change *all* of the + signs in the above into minus signs, then the statement of your conjecture holds; indeed there is an explicit formula for the terms of the generating function involving the pentagonal numbers, hence the name of the theorem. This theorem has several pretty and well-publicized proofs (see Chapter 1 of the introduction to \"The Theory of Partitions\" by George Andrews, or Chapter 14.5 of \"Introduction to Analytic Number Theory\" by Tom Apostol, or \"Proofs from the book\" by Aigner-Ziegler, or Wikipedia).\n\nI would wager that this observation isn't terribly helpful, but still. Was this was the motivation of the problem?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Dividing up the unrestricted partitions\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The exact global sign-choice problem remains open in the checked primary source: it is not known whether signs can be assigned independently in the unrestricted-partition product so every coefficient is 0, 1, or -1.\n\n**Verified partial progress.**\n\n- Andrews and Newman construct a plus-or-minus-one power series f whose product over odd dilates f(q^(2n-1)) equals Euler's pentagonal-number series.\n- Their construction solves a structured binary-factor analogue, not the original unrestricted independent-sign assignment.\n- A weaker one-coefficient-at-a-time question has an affirmative greedy argument, but the signs may depend on the selected coefficient and do not solve the simultaneous problem.\n\n**Full solution or refutation.**\n\nThe 2017 Andrews-Newman manuscript opens by restating the exact unrestricted-partition sign question and explicitly says its answer is still unknown, then proves a related theta-function identity via binary representations.\n\n**What remains.**\n\nChoose one coherent infinite array of signs for all factors and powers with every product coefficient in {0,1,-1}, or prove that no such global choice exists.\n\n**Sources checked.**\n\n- G. E. Andrews and D. Newman, Binary Representations and Theta Functions, manuscript dated 19 January 2017. (primary): https://georgeandrews1.github.io/pdf/322.pdf\n Evidence used: Restates the exact problem, says it remains unknown, and proves a related structured binary-product result.\n- S. J. Miller (ed.), Combinatorial and Additive Number Theory Problem Sessions: 2009-2016, 2014.2.2 David Newman. (authoritative_secondary): https://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/CANTProblemSessions.pdf\n Evidence used: Records the exact sign question and Newman's finite verification through degree 101.\n\n**Review notes.** The 2017 paper's affirmative theorem concerns a different structured factorization; it is not misreported as solving this record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3048, "problem_number": "OPG-37226", "title": "Sequence defined on multisets", "statement": "Conjecture Define a $2 \\times n$ array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array $[1; 1]$, the sequence is: $[1; 1]$-> $[1; 2]$-> $[1, 2; 1, 1]$-> $[1, 2; 3, 1]$-> $[1, 2, 3; 2, 1, 1]$-> $[1, 2, 3; 3, 2, 1]$-> $[1, 2, 3; 2, 2, 2]$-> $[1, 2, 3; 1, 4, 1]$-> $[1, 2, 3, 4; 3, 1, 1, 1]$-> $[1, 2, 3, 4; 4, 1, 2, 1]$-> $[1, 2, 3, 4; 3, 2, 1, 2]$-> $[1, 2, 3, 4; 2, 3, 2, 1]$, and we now have a fixed point (loop of one array).\n\nThe process always results in a loop of 1, 2, or 3 arrays.", "background": "Source: Open Problem Garden. Original node ID: 37226. URL: http://www.openproblemgarden.org/op/a_sequence_defined_on_multisets.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_sequence_defined_on_multisets\n- Author(s): Erickson, Martin\n- Subject(s): Combinatorics\n- Keywords: multiset; sequence\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 29th, 2010 by Martin Erickson\n\nBibliography:\n* Erickson, Martin J., \"Introduction to Combinatorics,\" Wiley, 1996.\n\nComments:\n- July 18th, 2011 | Anonymous | Solution: This problem has recently been solved by the author.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Sequence defined on multisets\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The source page has an unlinked 2011 assertion that the author solved the multiset iteration problem, but no independently verifiable proof or publication was located.\n\n**Verified partial progress.**\n\n- The source comment provides a lead for further bibliographic verification.\n\n**Full solution or refutation.**\n\nThe claimed resolution is not accepted without a citable proof.\n\n**What remains.**\n\nLocate the author's proof/publication or independently verify the finite-state/loop claim.\n\n**Sources checked.**\n\n- Open Problem Garden, Sequence defined on multisets (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/a_sequence_defined_on_multisets\n Evidence used: The page contains a 2011 comment asserting a solution but supplies neither proof nor reference.\n\n**Review notes.** Unlinked solution claim retained as a lead only; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3049, "problem_number": "OPG-37228", "title": "Square achievement game on an n x n grid", "statement": "Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \\times n$ grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides is the winner. What is the outcome of the game given optimal play? Note: Roland Bacher and Shalom Eliahou proved that every 15 x 15 binary matrix contains four equal entries (all 0's or all 1's) at the vertices of a square with horizontal and vertical sides. So the game must result in a winner (the first player) when n=15.", "background": "Source: Open Problem Garden. Original node ID: 37228. URL: http://www.openproblemgarden.org/op/a_game_on_an_n_x_n_grid.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_game_on_an_n_x_n_grid\n- Author(s): Erickson, Martin\n- Subject(s): Combinatorics\n- Keywords: game\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 29th, 2010 by Martin Erickson\n\nBibliography:\nR. Bacher and S. Eliahou, \"Extremal binary matrices without constant 2-squares,\" J. of Combinatorics, Volume 1, Number 1, 77-100, 2010.\n\n* Erickson, Martin, \"Pearls of Discrete Mathematics,\" CRC Press, 2010\n\nComments:\n- September 13th, 2011 | Carolus | A new reference: My new article \"Guaranteed successful strategies for a square achievement game on an n by n grid\" concerning that problem is now accessible via http://arxiv.org/abs/1109.2341.\n- August 13th, 2011 | porton | This is not mathematics: I suspect this problem can be solved only with brute force for every particular n and is not a nice mathematical conjecture (just like the question who wins in chess, white or black, playing both with the optimal strategy).\n\n-- Victor Porton - http://www.mathematics21.org\n- August 14th, 2011 | Carolus | On drawn games for n up to 14: I should have read the referenced article by Bacher and Eliahou before trying to find drawn games. But I've read it now.\n\nAmong many other things, the authors describe parametric families of square-free configurations for n=14.\n\nNot all of those configurations do provide the required relation of the numbers of symbols of both sorts needed to be an end-configuration of the game.\n\nBut one can, for example, take the family A1, give the variables x1 up to x8 the value 0 (or O, resp.) and the variables x9 up to x16 the value 1 (or X, resp.).\n\nThen the numbers of symbols of both sorts are equal and that square-free configuration is also a correct end-configuration of the game for n=14.\n- August 13th, 2011 | Anonymous | An URL of the technical report version of the referenced article: http://www-lmpa.univ-littoral.fr/publications/articles/lmpa404.pdf\n- August 13th, 2011 | Carolus | Drawn game for n=12: I've had not much hope to find one in an acceptable time but after running another couple of hours my program delivered this square-free end-configuration for n=12:\n\noooooooxxxox\n\nxoxoxoxoxoxx\n\nxoxxooxooxxo\n\noooxxxooxxoo\n\noxooxoxxxoxo\n\nxxxooxoxoxox\n\nxoxxxxooooxx\n\noxoxoxxoxoox\n\nooxoooxxoxox\n\nxxooxooxoxxo\n\noooxoxoxooxx\n\nxoxxxoxxxoox\n- August 11th, 2011 | Carolus | Drawn games for grid sizes from 3 to 11: Running a self-written program I've found the square-less end-configurations to be listed (because of the 1000 characters limit) in a comment of this comment for the grid sizes (n) from 3 to 11.\n\nIf I understand the note in the problem text right, this proofs that there is no sure winning strategy for those sizes.\n\nOn my 1 GHz Celeron/PIII the search for n=11 took 65 seconds.\n\nBut I gave up the search for n=12 after circa 12 hours of calculation estimating the 130-fold duration for the complete search. On the other side: In the first 8 hours there was a square-less configuration just before the occupation of the last field.\n- August 13th, 2011 | Carolus | I withdraw the conclusion: The note in the problem text let me assume that it is proven that if the game for some grid size n must have a winner there is a sure winning strategy for the first player.\n\nTaking in account that implication only, one cannot, as I did, conclude that there is no sure winning strategy if a game could be drawn.\n- August 12th, 2011 | Carolus | Text corrections: It should stand 'proves' instead of 'proofs' and 'square-free' instead of 'square-less'.\n- August 13th, 2011 | Carolus | A list of square-free end-configurations for n from 3 to 11: 3\n\nooo\n\noxo\n\nxxx\n\n4\n\noooo\n\noxox\n\nxoxx\n\nxxxo\n\n5\n\nooooo\n\noxoxo\n\nooxxo\n\nxoxox\n\nxxxxx\n\n6\n\noooooo\n\noxoxox\n\nooxxoo\n\nxoxoxx\n\nxxxxox\n\nxooxxx\n\n7\n\nooooooo\n\noxoxoxo\n\nooxxoox\n\nxxooxxx\n\noxoxxox\n\nxxxooox\n\nxoxxxxo\n\n8\n\noooooooo\n\noxoxoxox\n\nooxxooxx\n\nxxooxxxo\n\noxoxxoxx\n\nxxxoxoox\n\nxoxoxxox\n\nxooooxxx\n\n9\n\nooooooooo\n\noxoxoxoxo\n\nooxxooxxo\n\nxoooxxxoo\n\noxoxxoxxx\n\nxxxoxooox\n\nxoxoxxoxx\n\nxxoxoxoox\n\noxxxoxxox\n\n10\n\nooooooooox\n\noxoxoxoxoo\n\noxxooxxxxo\n\nooxxxoxoxo\n\noxooxoxxoo\n\nxxxooooxxx\n\nxoxoxxoxox\n\nxooxoxxoxx\n\nxxoxxoxoox\n\nxoxxooxxox\n\n11\n\noooooooooxx\n\noxoxoxxoxox\n\noxxoxxooxox\n\nooxxxooxxoo\n\noxooxoxxoox\n\nxxxxoooooxx\n\nxoxooxoxxxo\n\nxooxoxxoxoo\n\nxxooxxoooxo\n\nxoxxxooxoxx\n\nxoxoxoxxxxo\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Square achievement game on an n x n grid\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Verified strategy analysis gives draws for boards of size 3 and 4 and a first-player win for size 5; the general optimal-play outcome is not established by this evidence.\n\n**Verified partial progress.**\n\n- Bacher--Eliahou's binary-matrix theorem ensures that a winner occurs by size 15.\n- Jenrich supplies certified small-board strategies.\n\n**Full solution or refutation.**\n\nKnown small cases do not determine every n.\n\n**What remains.**\n\nClassify the game outcome for all board sizes or give a general strategy.\n\n**Sources checked.**\n\n- T. Jenrich, Guaranteed successful strategies for a square achievement game on an n by n grid, arXiv:1109.2341 (2011). (primary): https://arxiv.org/abs/1109.2341\n Evidence used: The abstract states second-player non-loss for n=3,4 and first-player wins for n=5.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3050, "problem_number": "OPG-37230", "title": "Transversal achievement game on a square grid", "statement": "Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \\times n$ grid. The first player (if any) to occupy a set of $n$ cells having no two cells in the same row or column is the winner. What is the outcome of the game given optimal play?", "background": "Source: Open Problem Garden. Original node ID: 37230. URL: http://www.openproblemgarden.org/op/a_transversal_achievement_game_on_a_square_grid.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_transversal_achievement_game_on_a_square_grid\n- Author(s): Erickson, Martin\n- Subject(s): Combinatorics\n- Keywords: game\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 29th, 2010 by Martin Erickson\n\nComments:\n- January 6th, 2021 | Anonymous | Solution: The problem has been solved. Link:- https://arxiv.org/abs/2101.00770\n- February 13th, 2013 | Anonymous | Are there a simple solution?: I suspect, there are no simple answer and it can be solved only by heavy calculations, that is essentally there is no solution to this problem.\n- February 22nd, 2013 | mshj | history and application: i'm not sure but i think to solve this problem, i was wondering if any body gives me some information about the history and application of this problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Transversal achievement game on a square grid\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The transversal achievement game is a draw under optimal play for every n>1.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nKrishna proves the all-n draw statement; Dumitrescu gives an independent short analysis.\n\n**What remains.**\n\nThe source question is answered; further work could simplify strategies.\n\n**Sources checked.**\n\n- N. Krishna, Transversal achievement game on a square grid, arXiv:2101.00770 (2021). (primary): https://arxiv.org/abs/2101.00770\n Evidence used: The abstract states and proves that the game is a draw for all n greater than 1.\n- A. Dumitrescu, On a two-player transversal game on a square grid, arXiv:2101.05065 (2021). (primary): https://arxiv.org/abs/2101.05065\n Evidence used: The abstract gives a short analysis of the same Erickson game.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3051, "problem_number": "OPG-37416", "title": "Length of surreal product", "statement": "Conjecture Every surreal number has a unique sign expansion, i.e. function $f: o\\rightarrow \\{-, +\\}$, where $o$ is some ordinal. This $o$ is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of $s$ as $\\ell(s)$.\n\nIt is easy to prove that\n\n$$\\ell(s+t) \\leq \\ell(s)+\\ell(t)$$\n\nWhat about\n\n$$\\ell(s\\times t) \\leq \\ell(s)\\times\\ell(t)$$?", "background": "Source: Open Problem Garden. Original node ID: 37416. URL: http://www.openproblemgarden.org/op/length_of_surreal_product.\n\nSource subject path: Combinatorics.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/length_of_surreal_product\n- Author(s): Gonshor, Harry\n- Subject(s): Combinatorics\n- Keywords: surreal numbers\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: April 7th, 2012 by Lukáš Lánský\n\nProblem-page discussion:\nThis is strongly conjectured to be true by Gonshor in [Gon86]. There is an easy way to prove that\n\n$$\\ell(s\\times t) \\leq 3^{\\ell(s)+\\ell(t)}$$\n\nBibliography:\n*[Gon86] Harry Gonshor, An Introduction to the Theory of Surreal Numbers, Cambridge University Press, Cambridge, 1986.\n\nSource links:\n- surreal number: http://en.wikipedia.org/wiki/surreal number\n\nComments:\n- May 30th, 2012 | vprusso | Proof Already Exists?: I believe the proof for the conjectured statement was proven in the affirmative in the paper \"Fields of Surreal Numbers and Exponentiation\" by Dries and Ehrlich. Specifically, Lemma 3.3 on page 6: http://www.ohio.edu/people/ehrlich/EhrlichvandenDries.pdf\n\nIf this satisfies the conjecture adequately great, if not, let me know if you would like to work toward a solution together on something similar or related.\n\nThanks.\n\n-Vincent Russo\n- June 5th, 2012 | Lukáš Lánský | Maybe!: Thank you! I wasn't aware of this paper. At first sight I think that the part you refer to establish the required result just for surreals in the form $r\\cdot\\omega^x$, but I'll find time to go through it thoroughly as it is most relevant for the matter.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Length of surreal product\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Sign expansions and birthdays are standard, but no source proving or refuting the exact ordinal inequality for multiplication was verified.\n\n**Verified partial progress.**\n\n- Modern surreal-number sources develop sign-sequence lengths and product constructions.\n\n**Full solution or refutation.**\n\nNo status beyond uncertain is justified by the sources found.\n\n**What remains.**\n\nLocate a theorem in Gonshor/Conway theory or give a counterexample involving noncommutative ordinal multiplication.\n\n**Sources checked.**\n\n- J. van der Hoeven, Surreal substructures (accessed 2026-08-17). (authoritative_secondary): https://www.texmacs.org/joris/sss/sss.html\n Evidence used: Documents sign-sequence lengths and surreal substructure/product machinery without settling this stated bound.\n\n**Review notes.** Ordinal-multiplication convention requires expert check; source preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3052, "problem_number": "OPG-58213", "title": "Roller Coaster permutations", "statement": "Let $S_n$ denote the set of all permutations of $[n]=\\set{1,2,\\ldots,n}$. Let $i(\\pi)$ and $d(\\pi)$ denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in $\\pi$. Let $X(\\pi)$ denote the set of subsequences of $\\pi$ with length at least three. Let $t(\\pi)$ denote $\\sum_{\\tau\\in X(\\pi)}(i(\\tau)+d(\\tau))$.\n\nA permutation $\\pi\\in S_n$ is called a Roller Coaster permutation if $t(\\pi)=\\max_{\\tau\\in S_n}t(\\tau)$. Let $RC(n)$ be the set of all Roller Coaster permutations in $S_n$.\n\nConjecture For $n\\geq 3$,\n\n- If $n=2k$, then $|RC(n)|=4$.\n- If $n=2k+1$, then $|RC(n)|=2^j$ with $j\\leq k+1$.\n\nConjecture (Odd Sum conjecture) Given $\\pi\\in RC(n)$,\n\n- If $n=2k+1$, then $\\pi_j+\\pi_{n-j+1}$ is odd for $1\\leq j\\leq k$.\n- If $n=2k$, then $\\pi_j + \\pi_{n-j+1} = 2k+1$ for all $1\\leq j\\leq k$.", "background": "Source: Open Problem Garden. Original node ID: 58213. URL: http://www.openproblemgarden.org/op/roller_coaster_permutations.\n\nSource subject path: Combinatorics.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/roller_coaster_permutations\n- Author(s): Ahmed, Tanbir; Snevily, Hunter S.\n- Subject(s): Combinatorics\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 14th, 2013 by Tanbir Ahmed\n\nBibliography:\n*[AS] Tanbir Ahmed, Hunter Snevily, Some properties of Roller Coaster permutations. To appear in Bull. Institute of Combinatorics and its Applications, 2013.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"Roller Coaster permutations\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The enumeration and Odd Sum conjectures for Roller Coaster permutations remain open, with additional finite optimization data and structural work available.\n\n**Verified partial progress.**\n\n- An integer-linear-programming model yields exact optimum examples through n=17 and improves lower-bound/candidate data through n=40.\n- A later thesis identifies a counterexample to an exchange argument previously offered for the separate alternating-structure conjecture, so that claimed proof cannot be reused.\n\n**Full solution or refutation.**\n\nNo all-n proof or counterexample to either displayed conjecture was verified; finite computations extend evidence but do not determine every member of RC(n).\n\n**What remains.**\n\nProve the power-of-two/four enumeration and opposite-position sum laws for every n, or find the first counterexample.\n\n**Sources checked.**\n\n- F. Botler and B. L. Netto, New Bounds on Roller Coaster Permutations, Encontro de Teoria da Computacao (2021). (primary): https://doi.org/10.5753/etc.2021.16378\n Evidence used: Gives a cubic evaluation algorithm, an ILP model, exact additional examples, and improved finite lower bounds.\n- B. R. Lima Netto, Two Problems in Combinatorics: Roller Coaster Permutations & The Erdos-Sos Conjecture, master's dissertation, Universidade Federal do Rio de Janeiro (2022). (authoritative_secondary): https://www.cos.ufrj.br/index.php/pt-BR/publicacoes-pesquisa/details/15/3043\n Evidence used: Reviews the conjectures, extends finite optimization evidence, and documents the flaw in a proposed alternating-structure proof.\n- Open Problem Garden, Roller Coaster permutations (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/roller_coaster_permutations\n Evidence used: Retains both exact displayed conjectures as open problems.\n\n**Review notes.** The source LaTeX '[n]=\\set{1,2,...,n}' is malformed and should denote set braces; it is flagged without alteration. No local computation was run.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3053, "problem_number": "OPG-60000", "title": "The Double Cap Conjecture", "statement": "Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of $\\mathbb{R}^n$ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $\\pi/4$ around the north and south poles.", "background": "Source: Open Problem Garden. Original node ID: 60000. URL: http://www.openproblemgarden.org/op/the_double_cap_conjecture.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_double_cap_conjecture\n- Author(s): Kalai, Gil\n- Subject(s): Combinatorics\n- Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 15th, 2015 by Jon Noel\n\nProblem-page discussion:\nThe problem of determining the maximum was first considered by Witsenhausen [Wit] who proved that the measure of such a set is at most $\\frac{1}{n}$ times the surface measure of the sphere. In $\\mathbb{R}^3$, DeCorte and Pikhurko [DP] improved the multiplicative constant to $0.313< 1/3$. The conjecture above would imply that the measure is at most $1-1/\\sqrt{2} \\approx 0.2928$.\n\nBibliography:\n[DP] E. DeCorte and O. Pikhurko, Spherical sets avoiding a prescribed set of angles, arXiv:1502.05030v2.\n\n[Kalai] G. Kalai, How Large can a Spherical Set Without Two Orthogonal Vectors Be? https://gilkalai.wordpress.com/2009/05/22/how-large-can-a-spherical-set-without-two-orthogonal-vectors-be/\n\n[Wit] H. S. Witsenhausen. Spherical sets without orthogonal point pairs. American Mathematical Monthly, pages 1101–1102, 1974.\n\nRelated:\nRelated problems\nCircular colouring the orthogonality graph\nPartitioning the Projective Plane\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"The Double Cap Conjecture\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Double Cap Conjecture remains open. In dimension three the lower construction has density 0.29289..., while a May 2026 paper improves the upper bound to 0.2953.\n\n**Verified partial progress.**\n\n- Czifra--Dúcz--Matolcsi--Varga--Zsámboki prove alpha_3<=0.2953, improving the prior 0.2977 upper bound.\n- Mudgal proves that the dimension-three supremum can be approximated by orthogonal-pair-free sets that are finite unions of mutually disjoint convex sets.\n\n**Full solution or refutation.**\n\nNo proof that the two antipodal pi/4 caps are optimal in dimension three or in every dimension was verified.\n\n**What remains.**\n\nClose the dimension-three gap from 0.2953 to 1-1/sqrt(2), and establish or refute the double-cap extremizer in all dimensions.\n\n**Sources checked.**\n\n- Domonkos Czifra, Ákos Dúcz, Máté Matolcsi, Dániel Varga, and Pál Zsámboki, Improved bounds for the double cap conjecture, arXiv:2605.28709. (primary): https://arxiv.org/abs/2605.28709\n Evidence used: Calls the conjecture open and proves the current alpha_3 upper bound 0.2953.\n- Apurva Mudgal, Convexity of near-optimal orthogonal-pair-free sets on the unit sphere, arXiv:2403.18404. (primary): https://arxiv.org/abs/2403.18404\n Evidence used: Proves the finite-union-of-convex-sets approximation reduction for the dimension-three extremal problem.\n- Open Problem Garden, The Double Cap Conjecture. (maintained_tracker): https://www.openproblemgarden.org/op/the_double_cap_conjecture\n Evidence used: Preserves the exact all-dimensional extremal statement and its historical bounds.\n\n**Review notes.** The sphere is S^(n-1). Open versus closed cap boundaries have measure zero, so that convention does not change the extremal measure.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3054, "problem_number": "OPG-60002", "title": "Saturation in the Hypercube", "statement": "Question What is the saturation number of cycles of length $2\\ell$ in the $d$-dimensional hypercube?", "background": "Source: Open Problem Garden. Original node ID: 60002. URL: http://www.openproblemgarden.org/op/saturation_in_the_hypercube.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/saturation_in_the_hypercube\n- Author(s): Morrison, Natasha; Noel, Jonathan A.; Scott, Alex\n- Subject(s): Combinatorics\n- Keywords: cycles; hypercube; minimum saturation; saturation\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 20th, 2015 by Jon Noel\n\nProblem-page discussion:\nLet $G$ and $H$ be graphs. Say that a spanning subgraph $F$ of $G$ is $(G,H)$-saturated if $F$ contains no copy of $H$ but $F+e$ contains a copy of $H$ for every edge $e\\in E(G)\\setminus E(F)$. Let $\\text{sat}(G,H)$ denote the minimum number of edges in a $(G,H)$-saturated graph. Saturation was introduced by Erdős, Hajnal and Moon [EHM] who proved the following:\n\nTheorem (Erdős, Hajnal and Moon) For $n\\geq k\\geq2$ we have $\\text{sat}(K_n,K_k) = \\binom{n}{2} = \\binom{n-k+2}{2}$.\n\nLet $Q_d$ denote the $d$-dimensional hypercube. Saturation of $4$-cycles in the hypercube has been studied by Choi and Guan [CG] who proved that $\\text{sat}(Q_d,C_4)\\leq \\left(\\frac{1}{4} + o(1)\\right)|E(Q_d)|$. This was drastically improved by Johnson and Pinto [JP] to $\\text{sat}(Q_d,C_4) < 10\\cdot 2^d$. The saturation number for longer cycles in the hypercube is not known, though. The question above addresses this.\n\nAnother open problem is to determine the saturation number of sub-hypercubes in $Q_d$. This was first considered by Johnson and Pinto [JP] who proved that $\\text{sat}(Q_d,Q_m) = o\\left(|E(Q_d)|\\right)$ for fixed $m$ and $d\\to \\infty$. This upper bound was improved to $(1+o(1))72m^2 2^d$ by Morrison, Noel and Scott [MNS]. The best known lower bound on $\\text{sat}(Q_d,Q_m)$ for fixed $m$ and large $d$, also due to [MNS], is $(m-1-o(1))2^d$.\n\nProblem Improve the upper and lower bounds on $\\text{sat}(Q_d,Q_m)$ for fixed $m$ and large $d$.\n\nThe results of [MNS] show that $\\text{sat}(Q_d,Q_m) = \\Theta(2^d)$ for fixed $m$. Howver, the precise asymptotic behaviour of this quantity is unknown.\n\nQuestion (Morrison, Noel and Scott) For fixed $m\\geq 2$, is it true that $\\frac{\\text{sat}(Q_d,Q_m)}{2^d}$ converges as $d\\to \\infty$?\n\nBibliography:\n[CG] S. Choi and P. Guan, Minimum critical squarefree subgraph of a hypercube, Proceedings of the Thirty-Ninth Southeastern International Conference on Combinatorics, Graph Theory and Computing, vol. 189, 2008, pp. 57–64.\n\n[EHM] P. Erdős, A. Hajnal, and J. W. Moon, A problem in graph theory, Amer. Math. Monthly 71 (1964), 1107–1110.\n\n[JP] J. R. Johnson and T. Pinto, Saturated subgraphs of the hypercube, arXiv:1406.1766v1, preprint, June 2014.\n\n[MNS] N. Morrison, J. A. Noel and A. Scott, Saturation in the Hypercube and Bootstrap Percolation, arXiv:1408.5488v2, June 2015.\n\nRelated:\nRelated problems\nSaturated $k$-Sperner Systems of Minimum Size\nTurán Problem for $10$-Cycles in the Hypercube\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"Saturation in the Hypercube\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The saturation number sat(Q_d,C_2ell) remains undetermined in general. The C4 case has an O(2^d) construction, while the published hypercube-saturation literature continues to pose longer cycles as open.\n\n**Verified partial progress.**\n\n- Johnson--Pinto prove sat(Q_d,C4)<10*2^d, improving an earlier upper bound proportional to the full edge set.\n- Related work determines the order Theta(2^d) for fixed subcube saturation and exact weak-saturation values, but those results do not determine strong saturation by cycles.\n\n**Full solution or refutation.**\n\nNo exact formula or general fixed-ell asymptotic for strong C_2ell-saturation in Q_d was verified.\n\n**What remains.**\n\nFor each feasible ell>=2, determine exact values or at least matching asymptotic bounds for sat(Q_d,C_2ell), beginning beyond C4.\n\n**Sources checked.**\n\n- J. Robert Johnson and Trevor Pinto, Saturated subgraphs of the hypercube, arXiv:1406.1766. (primary): https://arxiv.org/abs/1406.1766\n Evidence used: Proves the less-than-10 times 2^d upper bound for C4 saturation and poses related hypercube saturation questions.\n- Natasha Morrison, Jonathan A. Noel, and Alex Scott, Saturation in the Hypercube and Bootstrap Percolation, Combinatorics, Probability and Computing 26 (2017), 78-98; arXiv:1408.5488. (primary): https://arxiv.org/abs/1408.5488\n Evidence used: Lists determination of sat(Q_d,C_2ell) as open and proves related fixed-subcube and weak-saturation results.\n- Open Problem Garden, Saturation in the Hypercube. (maintained_tracker): https://garden.irmacs.sfu.ca/op/saturation_in_the_hypercube\n Evidence used: Maintains the exact cycle-saturation question and historical C4 bound.\n\n**Review notes.** The question omits ell and d range conditions; the background indicates feasible cycles with ell>=2. The background also mangles the Erdős--Hajnal--Moon complete-host formula by omitting a subtraction.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3055, "problem_number": "OPG-60003", "title": "Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube", "statement": "Problem Determine the smallest percolating set for the $4$-neighbour bootstrap process in the hypercube.", "background": "Source: Open Problem Garden. Original node ID: 60003. URL: http://www.openproblemgarden.org/op/extremal_4_neighbour_bootstrap_percolation_in_the_hypercube.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/extremal_4_neighbour_bootstrap_percolation_in_the_hypercube\n- Author(s): Morrison, Natasha; Noel, Jonathan A.\n- Subject(s): Combinatorics\n- Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 20th, 2015 by Jon Noel\n\nProblem-page discussion:\nThe $r$-neighbour bootstrap process starts with an initial set of \"infected\" vertices in a graph and, at each step, a healthy vertex becomes infected if it has at least $r$ infected neighbours. Say that the initial set of infected vertices percolates if every vertex of $G$ is eventually infected. Let $m(G,r)$ denote the smallest percolating set in $G$ under the $r$-neighbour process.\n\nLet $Q_d$ denote the hypercube of dimension $d$. Balogh and Bollobás [BB] proved the following.\n\nTheorem (Balogh and Bollobás) $m(Q_d,2) = \\left\\lceil \\frac{d}{2}\\right\\rceil +1$ for all $d\\geq 2$.\n\nThey also conjectured that $m(Q_d,r) = \\frac{1+o(1)}{r}\\binom{d}{r-1}$ for fixed $r$ and $d\\to\\infty$. This conjecture was proved by Morrison and Noel [MN], who also showed the following.\n\nTheorem (Morrison and Noel) $m(Q_d,3) = \\left\\lceil \\frac{d(d+3)}{6} \\right\\rceil +1$ for all $d\\geq 3$.\n\nIt seems possible that one could obtain a general formula for $m(Q_d,r)$ for all $r$ and $d\\geq r$. However, the precise formula for $m(Q_d,r)$ (in terms of $d$ ) is not known for any fixed $r\\geq4$. A solution to this problem may have applications in proving probabilistic results for bootstrap percolation in the hypercube; see [BBM].\n\nBibliography:\n[BB] J. Balogh and B. Bollobás, Bootstrap percolation on the hypercube, Probab. Theory Related Fields 134 (2006), no. 4, 624–648.\n\n[BBM] J. Balogh, B. Bollobás and R. Morris, Bootstrap percolation in high dimensions, Combin. Probab. Comput. 19 (2010), no. 5-6, 643–692.\n\n[MN] N. Morrison and J. A. Noel, Extremal Bounds for Bootstrap Percolation in the Hypercube, preprint, arXiv:1506.04686v1.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Morrison--Noel lower-bound value for m(Q_d;4) is attained for infinitely many dimensions, and for every dimension the best new construction is only O(d) above that lower bound; an exact all-d formula remains open.\n\n**Verified partial progress.**\n\n- Noel proves m(Q_d;4)=d(d^2+3d+14)/24+1 for infinitely many d.\n- For arbitrary d, Noel gives an upper bound differing from the Morrison--Noel lower bound by an additive O(d).\n- Morrison--Noel had already determined the fixed-r asymptotic lower scale and the exact r=3 case.\n\n**Full solution or refutation.**\n\nInfinitely many exact values and an asymptotically near-exact bound are known, but the minimum is not determined for every dimension.\n\n**What remains.**\n\nDetermine m(Q_d;4) exactly in the dimensions not covered by the 2026 constructions, or close the residual additive O(d) gap.\n\n**Sources checked.**\n\n- Jonathan A. Noel, Optimal and Near-Optimal Constructions for Bootstrap Percolation in Hypercubes, arXiv:2604.15534 (2026). (primary): https://arxiv.org/abs/2604.15534\n Evidence used: States the exact formula for infinitely many d and an additive-O(d) gap for general d.\n- Natasha Morrison and Jonathan A. Noel, Extremal Bounds for Bootstrap Percolation in the Hypercube, Journal of Combinatorial Theory, Series A 156 (2018), 61--84. (primary): https://doi.org/10.1016/j.jcta.2017.11.018\n Evidence used: Provides the lower bound matched by Noel and the earlier fixed-r asymptotics.\n\n**Review notes.** The short statement omits d; the background makes the intended target the function d mapsto m(Q_d;4).\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3056, "problem_number": "OPG-60006", "title": "Turán Problem for $10$-Cycles in the Hypercube", "statement": "Problem Bound the extremal number of $C_{10}$ in the hypercube.", "background": "Source: Open Problem Garden. Original node ID: 60006. URL: http://www.openproblemgarden.org/op/turan_problem_for_10_cycles_in_the_hypercube.\n\nSource subject path: Combinatorics.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/turan_problem_for_10_cycles_in_the_hypercube\n- Author(s): Erdos, Paul\n- Subject(s): Combinatorics\n- Keywords: cycles; extremal combinatorics; hypercube\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 20th, 2015 by Jon Noel\n\nProblem-page discussion:\nThe problem of bounding the extremal number for cycles in the hypercube was first considered by Erdős [Erd1,Erd2] who conjectured that $\\text{ex}(Q_d,C_4) = (1/2 +o(1)) |E(Q_d)|$ and that $\\text{ex}(Q_d,C_{2t}) = o(|E(Q_d)|)$ for all $t\\geq3$. The first conjecture is still open, and the second is known to be false in the case $t=3$ (see [BDT, Chu, Cond]).\n\nChung [Chu] proved that $\\text{ex}(Q_d,C_{2t}) = o(|E(Q_d)|)$ for even $t\\geq 4$ and Füredi and Özkahya [FO1,FO2] proved the same for odd $t\\geq 7$. Conlon [Conl] gave a unified proof of these results, which also applies to more general subgraphs of the hypercube. However, the case of $C_{10}$ remains unsolved.\n\nBibliography:\n[BDT] A. E. Brouwer, I. J. Dejter and C. Thomassen, Highly symmetric subgraphs of hypercubes, J. Algebraic Combin. 2 (1993), 25–29.\n\n[Chu] F. Chung, Subgraphs of a hypercube containing no small even cycles, J. Graph Theory 16 (1992), 273–286.\n\n[Cond] M. Conder, Hexagon-free subgraphs of hypercubes, J. Graph Theory 17 (1993), 477–479.\n\n[Conl] D. Conlon, An extremal theorem in the hypercube, Electron. J. Combin. 17 (2010), no. 1, Research Paper 111, 7 pages\n\n[Erd1] P. Erdős, On some problems in graph theory, combinatorial analysis and combinatorial number theory, in: Graph Theory and Combinatorics (Cambridge, 1983), Academic Press, London, 1984, 1–17.\n\n[Erd2] P. Erdős, Some of my favourite unsolved problems, in: A tribute to Paul Erdős, Cambridge University Press, 1990, 467–478.\n\n[FO1] Z. Füredi and L. Özkahya, On 14-cycle-free subgraphs of the hypercube, Combin. Probab. Comput. 18 (2009), 725–729.\n\n[FO2] Z. Füredi and L. Özkahya, On even-cycle-free subgraphs of the hypercube, Electronic Notes in Discrete Mathematics 34 (2009), 515–517.\n\nRelated:\nRelated problems\nSaturation in the Hypercube\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Turán Problem for $10$-Cycles in the Hypercube\" in Combinatorics, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** C10 is now known to have positive hypercube Turán density: ex(Q_n,C10)>0.024|E(Q_n)|, while a 2026 preprint gives density at most 0.36577; the exact density and finite-n extremal numbers remain unknown.\n\n**Verified partial progress.**\n\n- Grebennikov--Marciano prove ex(Q_n,C10)>0.024|E(Q_n)| for every n, settling the old zero-versus-positive-density question.\n- Pejic proves pi_square(C10)<=pi_square(C6), which with Baber's bound yields pi_square(C10)<=0.36577.\n- Axenovich--Martin--Winter had previously established a superlinear-in-2^n lower bound before positive density was known.\n\n**Full solution or refutation.**\n\nThe historical qualitative problem is resolved in favor of positive density, but the open-ended request to bound or determine ex(Q_n,C10) is only partially answered.\n\n**What remains.**\n\nNarrow or determine the interval 0.024<=pi_square(C10)<=0.36577 and obtain sharper finite-dimensional bounds.\n\n**Sources checked.**\n\n- Alexandr Grebennikov and Joao Pedro Marciano, C10 Has Positive Turan Density in the Hypercube, Journal of Graph Theory 109 (2025), 31--34. (primary): https://doi.org/10.1002/jgt.23217\n Evidence used: Proves the explicit 0.024-density lower bound for every n.\n- Marko Pejic, On the Turan Density of C10 in the Hypercube, arXiv:2608.12237 (2026). (primary): https://arxiv.org/abs/2608.12237\n Evidence used: Proves pi_square(C10)<=pi_square(C6)<=0.36577; submitted 2026-08-12.\n- Maria Axenovich, Ryan R. Martin, and Christian Winter, On Graphs Embeddable in a Layer of a Hypercube and Their Extremal Numbers, Annals of Combinatorics (2024). (primary): https://doi.org/10.1007/s00026-024-00705-2\n Evidence used: Records C10 as the formerly unresolved cycle case and proves an earlier lower bound.\n\n**Review notes.** The word 'Bound' is open-ended; the old zero-density subquestion is solved, but the full extremal function is not.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3057, "problem_number": "OPG-37196", "title": "Perfect 2-error-correcting codes over arbitrary finite alphabets.", "statement": "Conjecture Does there exist a nontrivial perfect 2-error-correcting code over any finite alphabet, other than the ternary Golay code?", "background": "Source: Open Problem Garden. Original node ID: 37196. URL: http://www.openproblemgarden.org/op/perfect_2_error_correcting_codes_over_arbitrary_finite_alphabets.\n\nSource subject path: Combinatorics > Codes.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/perfect_2_error_correcting_codes_over_arbitrary_finite_alphabets\n- Subject(s): Combinatorics; Codes\n- Keywords: 2-error-correcting; code; existence; perfect; perfect code\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 3rd, 2010 by davidcullen\n\nProblem-page discussion:\nVery few perfect codes are known to exist over any alphabet. The trivial examples are codes with 1 or 2 codewords, or q-ary (n, M, d) codes with all of the q^n vectors being codewords. Other than this, we have an infinite family of perfect 1-error-correcting Hamming codes, and two unique Golay codes, the binary one which corrects 1 error, the ternary one which corrects 2 errors. Recent research activity has discovered a large number of previously unknown perfect 1-error correcting codes which are not isomorphic to the Hamming codes.\n\nIt is well known (see Van Lint) that the answer is negative for codes over alphabets of size equal to a power of a prime number. Further results (see Hong, Best) establish that there are no perfect t-error-correcting codes for any t > 2 over any finite alphabet, which establishes the fact that 2 is the largest number of errors which any new perfect code could possibly correct. Lloyd's theorem plays a key role in ruling out t > 2, but provides less information than needed in the t = 2 case. Establishing the result in the negative would likely require an ad-hoc combinatorial argument, while establishing it in the positive could be done by any clever construction.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Perfect 2-error-correcting codes over arbitrary finite alphabets.\" in Combinatorics; Codes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The ternary Golay code remains the only known nontrivial perfect 2-error-correcting code. Nonexistence is proved for all prime-power alphabet sizes and many composite sizes, but arbitrary non-prime-power alphabets are not classified.\n\n**Verified partial progress.**\n\n- Classical work excludes prime-power q, all q=2^r 3^s with r,s positive, and q in {6,10,15,21,22,26,30,35}.\n- Cazorla García excludes every non-prime-power q at most 200 except q=94 and 166, and further cases up to 600 whose prime divisors lie in {2,3,5,7,11}.\n- For every fixed non-prime-power alphabet size q at least 6, only finitely many perfect 2-error-correcting codes can exist.\n\n**Full solution or refutation.**\n\nThe 2026 revision reduces the perfect-code sphere-packing equation to generalized Ramanujan-Nagell equations, combines integral-point calculations with Lloyd's theorem, and greatly enlarges the excluded set of composite alphabet sizes. It explicitly retains the general q>3 conjecture as open.\n\n**What remains.**\n\nExclude q=94, q=166, and ultimately every composite q>3 not covered by existing families, or construct a new perfect 2-error-correcting code.\n\n**Sources checked.**\n\n- P.-J. Cazorla García, Perfect codes over non-prime power alphabets: an approach based on Diophantine equations, arXiv:2405.03347, revision dated 22 March 2026. (primary): https://arxiv.org/abs/2405.03347\n Evidence used: Explicitly says the non-prime-power classification is open and proves the stated 172 new exclusions and fixed-q finiteness theorem.\n\n**Review notes.** The current result is a 2026 arXiv revision; computational claims are reported from the paper and were not rerun.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3058, "problem_number": "OPG-762", "title": "Combinatorial covering designs", "statement": "A $(v, k, t)$ covering design, or covering, is a family of $k$-subsets, called blocks, chosen from a $v$-set, such that each $t$-subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by $C(v, k, t)$.\n\nProblem Find a closed form, recurrence, or better bounds for $C(v,k,t)$. Find a procedure for constructing minimal coverings.", "background": "Source: Open Problem Garden. Original node ID: 762. URL: http://www.openproblemgarden.org/op/covering_designs.\n\nSource subject path: Combinatorics > Designs.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/covering_designs\n- Author(s): Gordon, D.M.; Mills, W.H.; Rödl, V.; Schönheim, J.\n- Subject(s): Combinatorics; Designs\n- Keywords: recreational mathematics\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: April 11th, 2008 by Pseudonym\n\nProblem-page discussion:\nThe problem has applications in file design, but is also known at the \"lottery cover problem\", for its strategic application in playing lotteries.\n\nCurrent \"best\" covers have been collected by Dan Gordon.\n\nThe trivial lower bound is $C(v,k,t) \\geq \\dfrac{\\binom{v}{t}}{\\binom{k}{t}}$. When equality holds, the resulting design is called a Steiner system, and often denoted $S(t,k,v)$. If $S(t,k,v)$ exists, so does $S(t-1,k-1,v-1)$: just remove all occurrences of a point from the blocks containing it, and discard the blocks that didn't contain it before the deletion.\n\nBibliography:\nJ. Schönheim, On coverings, Pacific Journal of Mathematics, 14:1405–1411, 1964.\n\nDaniel M. Gordon, Oren Patashnik, Greg Kuperberg (1995) New constructions for covering designs, J. Combinatorial Designs 3(4), 269-284.\n\nD. T. Todorov. Combinatorial Coverings. PhD thesis, University of Sofia, 1985.\n\nDiscussion links:\n- Dan Gordon: http://www.ccrwest.org/cover.html\n\nBibliography links:\n- New constructions for covering designs: http://www.ccrwest.org/cover/cover.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Combinatorial covering designs\" in Combinatorics; Designs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exact covering numbers are known for many parameters and asymptotically optimal coverings exist for fixed k,t, but the broad all-parameter request is unsolved.\n\n**Verified partial progress.**\n\n- Rödl's theorem yields asymptotic optimality for fixed k,t as v grows.\n- The maintained La Jolla tables record current finite-parameter upper bounds and exact cases.\n\n**Full solution or refutation.**\n\nThere is no universal closed form or all-parameter construction of minimal coverings.\n\n**What remains.**\n\nImprove finite parameter bounds and determine exact C(v,k,t) beyond the known regimes.\n\n**Sources checked.**\n\n- D. Gordon, G. Kuperberg, O. Patashnik and J. Spencer, Asymptotically optimal covering designs, arXiv:math/9511224. (primary): https://arxiv.org/abs/math/9511224\n Evidence used: Records Rödl's asymptotic theorem and constructions meeting the bound.\n- La Jolla Covering Repository (accessed 2026-08-17). (maintained_tracker): https://ljcr.dmgordon.org/cover.html\n Evidence used: Maintained tables give current upper bounds and mark best-possible cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3059, "problem_number": "OPG-148", "title": "A nowhere-zero point in a linear mapping", "statement": "Conjecture If ${\\mathbb F}$ is a finite field with at least 4 elements and $A$ is an invertible $n \\times n$ matrix with entries in ${\\mathbb F}$, then there are column vectors $x,y \\in {\\mathbb F}^n$ which have no coordinates equal to zero such that $Ax=y$.", "background": "Source: Open Problem Garden. Original node ID: 148. URL: http://www.openproblemgarden.org/op/a_nowhere_zero_point_in_a_linear_mapping.\n\nSource subject path: Combinatorics > Matrices.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_nowhere_zero_point_in_a_linear_mapping\n- Author(s): Jaeger, Francois\n- Subject(s): Combinatorics; Matrices\n- Keywords: invertible; nowhere-zero flow\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2007 by mdevos\n\nProblem-page discussion:\nThe motivation for this problem comes from the study of nowhere-zero flows on graphs. If $A$ is the directed incidence matrix of a graph $G$, then a nowhere-zero ${\\mathbb F}$-flow on $G$ is precisely a vector $x$ so that $x$ has all entries nonzero, and $Ax=0$. The above conjecture is similar, but is for general (invertible) matrices. Alon and Tarsi have resolved this conjecture for all fields not of prime order using their polynomial technique.\n\nDefinition: Say that a $m \\times n$ matrix $A$ is $(a,b)$-choosable if for all $X_1,X_2,\\ldots,X_m \\subseteq {\\mathbb F}$ with $|X_i|=a$ and for all $Y_1,Y_2,\\ldots,Y_n \\subseteq {\\mathbb F}$ with $|Y_j|=b$, there exists a vector $x \\in X_1 \\times X_2 \\ldots \\times X_m$ and a vector $y \\in Y_1 \\times Y_2 \\ldots \\times Y_n$ so that $Ax=y$. Note that every matrix is $(1,|{\\mathbb F}|)$-choosable, but that an $n \\times n$ matrix is $(|{\\mathbb F}|,1)$-choosable if and only if it is invertible.\n\nAlon and Tarsi actually prove a stronger property than Jaeger conjectured for fields not of prime order. They prove that if ${\\mathbb F}$ has characteristic $p$, then every invertible matrix over ${\\mathbb F}$ is $(p,|{\\mathbb F}|-1)$-choosable. This result has been extended by DeVos [D] who showed that every such matrix is $(p,|{\\mathbb F}|-p+1)$-choosable. Yang Yu [Y] has verified that the conjecture holds for $n \\times n$ matrices with entries in ${\\mathbb Z}_p$ when $n < 2^{p-2}$.\n\nJaeger's conjecture is true in a very strong sense for fields of characteristic 2. DeVos [D] proved that every invertible matrix over such a field is $(k+1,|{\\mathbb F}|-k)$-choosable for every $k$. The following conjecture asserts that invertible matrices over fields of prime order have choosability properties nearly as strong.\n\nConjecture [The choosability in ${\\mathbb Z}_p$ conjecture (DeVos)] Every invertible matrix with entries in ${\\mathbb Z}_p$ for a prime $p$ is $(k+2,p-k)$-choosable for every $k$.\n\nThis is essentially the strongest choosability conjecture one might hope to be true over fields of prime order. I (M. DeVos) don't have any experimental evidence for this at all, so it could be false already for some small examples. However, I suspect that if The permanent conjecture is true, that this conjecture should also be true. In any case, I (M. DeVos) am offering a bottle of wine for this conjecture.\n\nBibliography:\n[A] N. Alon, Combinatorial Nullstellensatz, Combinatorics Probability and Computing 8 (1999) no. 1-2, 7-29. MathSciNet\n\n[AT] N. Alon, M. Tarsi, A Nowhere-Zero Point in Linear Mappings, Combinatorica 9 (1989), 393-395. MathSciNet\n\n[BBLS] R. Baker, J. Bonin, F. Lazebnik, and E. Shustin, On the number of nowhere-zero points in linear mappings, Combinatorica 14 (2) (1994), 149-157. MathSciNet\n\n[D] M. DeVos, Matrix Choosability, J. Combinatorial Theory, Ser. A 90 (2000), 197-209. MathSciNet\n\n[Y] Y. Yu, The Permanent Rank of a Matrix, J. Combinatorial Theory Ser. A 85 (1999), 237-242. MathSciNet\n\nBibliography links:\n- Combinatorial Nullstellensatz: http://www.math.tau.ac.il/%7Enogaa/PDFS/null2.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1684621\n- A Nowhere-Zero Point in Linear Mappings: http://www.springerlink.com/content/y9766rt81243188l/\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1054015\n- On the number of nowhere-zero points in linear mappings: http://www.springerlink.com/content/jx477824728366p0/\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1289069\n- Matrix Choosability: http://www.ams.org/leavingmsn?url=http://www.sciencedirect.com/science/journal/00973165\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1749430\n- The Permanent Rank of a Matrix: http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.1006/jcta.1998.2904\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1673948\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"A nowhere-zero point in a linear mapping\" in Combinatorics; Matrices, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The nowhere-zero matrix conjecture is proved for proper prime-power fields and for all sufficiently large prime fields, but current 2026 work still treats the remaining small prime-field cases as conjectural.\n\n**Verified partial progress.**\n\n- Alon-Tarsi prove every non-prime-order finite field case.\n- Nagy-Pach prove all sufficiently large prime cases; subsequent hyperplane-cover work strengthens the result.\n\n**Full solution or refutation.**\n\nNo proof for every finite field of size at least four was verified.\n\n**What remains.**\n\nResolve the remaining small prime fields, including the dimension-uniform problem over those fields.\n\n**Sources checked.**\n\n- J. Nagy and P. P. Pach, The Alon-Jaeger-Tarsi conjecture via group ring identities, arXiv:2107.03956. (primary): https://arxiv.org/abs/2107.03956\n Evidence used: Proves the conjecture for sufficiently large primes and records the earlier proper-prime-power theorem.\n- J. Nagy and P. P. Pach, On a group ring identity related to the Alon-Jaeger-Tarsi conjecture, arXiv:2604.26320. (primary): https://arxiv.org/abs/2604.26320\n Evidence used: Still formulates an implication route to the full conjecture in April 2026.\n- Open Problem Garden, A nowhere-zero point in a linear mapping (node 148). (maintained_tracker): https://www.openproblemgarden.org/op/a_nowhere_zero_point_in_a_linear_mapping\n Evidence used: Preserves the original statement and classical partial results.\n\n**Review notes.** Do not misread Honold-Schauz's ring extension as settling all prime fields.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3060, "problem_number": "OPG-150", "title": "The additive basis conjecture", "statement": "Conjecture For every prime $p$, there is a constant $c(p)$ (possibly $c(p)=p$ ) so that the union (as multisets) of any $c(p)$ bases of the vector space $({\\mathbb Z}_p)^n$ contains an additive basis.", "background": "Source: Open Problem Garden. Original node ID: 150. URL: http://www.openproblemgarden.org/op/the_additive_basis_conjecture.\n\nSource subject path: Combinatorics > Matrices.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_additive_basis_conjecture\n- Author(s): Jaeger, Francois; Linial, Nathan; Payan, Charles; Tarsi, Michael\n- Subject(s): Combinatorics; Matrices\n- Keywords: additive basis; matrix\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: Let $V$ be a finite dimensional vector space over the field ${\\mathbb Z}_p$. We call a multiset $B$ with elements in $V$ an additive basis if for every $v \\in V$, there is a subset of $B$ which sums to $v$.\n\nIt is worth noting that this conjecture would also imply that every $2c(3)$-edge-connected graph has a nowhere-zero 3-flow, thus resolving The weak 3-flow conjecture.\n\nDiscussion links:\n- The weak 3-flow conjecture: http://www.openproblemgarden.org/?q=op/3_flow_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"The additive basis conjecture\" in Combinatorics; Matrices, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjecture is now proved for p=3 with c(3)=4, while the dimension-independent assertion for every prime remains open.\n\n**Verified partial progress.**\n\n- Yang Yu proves that the union of any four bases over Z_3 is an additive basis.\n- General results previously required O_p(log n) bases; a strong weak-additive-basis variant is also known.\n\n**Full solution or refutation.**\n\nThe first nontrivial prime case is solved, not the all-primes conjecture.\n\n**What remains.**\n\nEstablish a dimension-independent c(p) for primes p>=5.\n\n**Sources checked.**\n\n- Yang Yu, Note on the Additive Basis Conjecture, arXiv:2510.01300 (2026 version). (primary): https://arxiv.org/abs/2510.01300\n Evidence used: Proves p=3 with c(3)=4 and explicitly retains the general conjecture.\n- J. Nagy, P. P. Pach and I. Tomon, Additive bases, coset covers, and non-vanishing linear maps, arXiv:2111.13658. (primary): https://arxiv.org/abs/2111.13658\n Evidence used: Proves the weak additive-basis conjecture in strong form and surveys the unresolved full conjecture.\n- Open Problem Garden, The additive basis conjecture (node 150). (maintained_tracker): https://www.openproblemgarden.org/op/the_additive_basis_conjecture\n Evidence used: Original tracked statement.\n\n**Review notes.** The source's implication to the weak 3-flow conjecture is historically superseded by independent flow results.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3061, "problem_number": "OPG-151", "title": "The permanent conjecture", "statement": "Conjecture If $A$ is an invertible $n \\times n$ matrix, then there is an $n \\times n$ submatrix $B$ of $[A A]$ so that $perm(B)$ is nonzero.", "background": "Source: Open Problem Garden. Original node ID: 151. URL: http://www.openproblemgarden.org/op/the_permanent_conjecture.\n\nSource subject path: Combinatorics > Matrices.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_permanent_conjecture\n- Author(s): Kahn, Jeff\n- Subject(s): Combinatorics; Matrices\n- Keywords: invertible; matrix; permanent\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2007 by mdevos\n\nProblem-page discussion:\nIf true, this conjecture would imply the nowhere-zero point in a linear mapping conjecture via the Alon-Tarsi polynomial technique. I believe Yang Yu was the first to suggest the following generalization of the permanent conjecture.\n\nConjecture (Yu) If $A,B$ are invertible $n \\times n$ matrices over the same field, then there is an $n \\times n$ submatrix $C$ of $[A B]$ so that $perm(C)$ is nonzero.\n\nThis conjecture when restricted to the field ${\\mathbb Z}_3$ is a consequence of the Alon-Tarsi basis conjecture. In addition to implying the above conjecture, the truth of this conjecture for matrices over the field ${\\mathbb Z}_3$ would imply that every 6-edge-connected graph has a nowhere-zero 3-flow, thus resolving The weak 3-flow conjecture.\n\nDiscussion links:\n- nowhere-zero point in a linear mapping conjecture: http://www.openproblemgarden.org/?q=op/a_nowhere_zero_point_in_a_linear_mapping\n- The weak 3-flow conjecture: http://www.openproblemgarden.org/?q=op/3_flow_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"The permanent conjecture\" in Combinatorics; Matrices, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No field-uniform proof or counterexample to Kahn's permanent-submatrix conjecture was found; current papers continue to cite it as a conjectural route to nowhere-zero vectors.\n\n**Verified partial progress.**\n\n- Computational algebra work verifies the associated permanental-ideal assertion through n<=5.\n\n**Full solution or refutation.**\n\nThe exact [A A] conjecture remains open.\n\n**What remains.**\n\nShow every invertible A has full permanent rank after duplication, or produce a counterexample over a specified field.\n\n**Sources checked.**\n\n- J. Nagy and P. P. Pach, The Alon-Jaeger-Tarsi conjecture via group ring identities, arXiv:2107.03956. (primary): https://arxiv.org/abs/2107.03956\n Evidence used: States the Permanent Conjecture as an unresolved stronger route to the main conjecture.\n- Open Problem Garden, The permanent conjecture (node 151). (maintained_tracker): https://www.openproblemgarden.org/op/the_permanent_conjecture\n Evidence used: Maintains the exact statement and Yu strengthening.\n\n**Review notes.** The statement omits the base field; the source discussion intends arbitrary fields.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3062, "problem_number": "OPG-152", "title": "The Alon-Tarsi basis conjecture", "statement": "Conjecture If $B_1,B_2,\\ldots B_p$ are invertible $n \\times n$ matrices with entries in ${\\mathbb Z}_p$ for a prime $p$, then there is a $n \\times (p-1)n$ submatrix $A$ of $[B_1 B_2 \\ldots B_p]$ so that $A$ is an AT-base.", "background": "Source: Open Problem Garden. Original node ID: 152. URL: http://www.openproblemgarden.org/op/the_alon_tarsi_basis_conjecture.\n\nSource subject path: Combinatorics > Matrices.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_alon_tarsi_basis_conjecture\n- Author(s): Alon, Noga; Linial, Nathan; Meshulam, Roy\n- Subject(s): Combinatorics; Matrices\n- Keywords: additive basis; matrix\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: If $A$ is an $n \\times (p-1)n$ matrix over a field of characteristic $p$, then we say that $A$ is an Alon-Tarsi basis (or AT-basis) if the permanent of the $(p-1)n \\times (p-1)n$ matrix obtained by stacking $p-1$ copies of $A$ is nonzero.\n\nIt follows from the Alon-Tarsi polynomial technique that if $A$ is an AT-base then for every $X_1,X_2,\\ldots,X_{(p-1)n} \\subseteq {\\mathbb Z}_p$ of size 2 and for every $y \\in {\\mathbb Z}_p^n$, there exists a vector $x \\in X_1 \\times X_2 \\ldots \\times X_{(p-1)n}$ so that $Ax=y$ (using the notation from A nowhere-zero point in a linear mapping, $A$ is (2,1)-choosable). It follows from this that every Alon-Tarsi base over ${\\mathbb Z}_p$ is also an additive basis. Thus, the above conjecture, if true, would imply The additive basis conjecture. The following strengthening of this conjecture was suggested in [D]\n\nConjecture (The strong Alon-Tarsi basis conjecture (DeVos)) If $B_1,B_2,\\ldots,B_p$ are invertible $n \\times n$ matrices with entries in a field of characteristic $p$, then we may partition the columns of $[B_1 B_2 \\ldots B_p]$ into an $n \\times (p-1)n$ matrix $A$ and an $n \\times n$ matrix $C$ so that $A$ is an AT-base and $C$ is invertible.\n\nIn addition to implying the conjecture, above, if true, this conjecture would imply both The permanent conjecture and The choosability in ${\\mathbb Z}_p$ conjecture.\n\nDiscussion links:\n- A nowhere-zero point in a linear mapping: http://www.openproblemgarden.org/?q=op/a_nowhere_zero_point_in_a_linear_mapping\n- The additive basis conjecture: http://www.openproblemgarden.org/?q=op/the_additive_basis_conjecture\n- The permanent conjecture: http://www.openproblemgarden.org/?q=op/the_permanent_conjecture\n- The choosability in ${\\mathbb Z}_p$ conjecture: http://www.openproblemgarden.org/?q=op/a_nowhere_zero_point_in_a_linear_mapping\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"The Alon-Tarsi basis conjecture\" in Combinatorics; Matrices, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No general proof of the required AT-base selection was verified; newer additive-basis results establish weaker consequences rather than the nonzero stacked permanent.\n\n**Verified partial progress.**\n\n- The p=2 case follows immediately because permanent equals determinant in characteristic two.\n- DeVos proves related matrix-choosability theorems and formulates a stronger partition conjecture.\n\n**Full solution or refutation.**\n\nThe nontrivial prime cases remain open in the literature checked.\n\n**What remains.**\n\nSelect (p-1)n columns with nonzero permanent after stacking p-1 copies, beginning with p=3.\n\n**Sources checked.**\n\n- M. DeVos, Matrix Choosability, Journal of Combinatorial Theory Series A 90 (2000), 197-209. (primary): https://www.sfu.ca/~mdevos/papers/pliant.pdf\n Evidence used: Defines AT-bases, states the conjectural selection/partition properties, and proves related choosability results.\n- Yang Yu, Note on the Additive Basis Conjecture, arXiv:2510.01300. (primary): https://arxiv.org/abs/2510.01300\n Evidence used: Proves a weaker additive-basis consequence for p=3 but does not claim the AT-base conjecture.\n- Open Problem Garden, The Alon-Tarsi basis conjecture (node 152). (maintained_tracker): https://www.openproblemgarden.org/op/the_alon_tarsi_basis_conjecture\n Evidence used: Maintains the exact statement and implications.\n\n**Review notes.** No OCR defect detected.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3063, "problem_number": "OPG-361", "title": "Rota's unimodal conjecture", "statement": "Let $M$ be a matroid of rank $r$, and for $0 \\le i \\le r$ let $w_i$ be the number of closed sets of rank $i$.\n\nConjecture $w_0,w_1,\\ldots,w_r$ is unimodal.\n\nConjecture $w_0,w_1,\\ldots,w_r$ is log-concave.", "background": "Source: Open Problem Garden. Original node ID: 361. URL: http://www.openproblemgarden.org/op/rotas_unimodal_conjecture.\n\nSource subject path: Combinatorics > Matroid Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/rotas_unimodal_conjecture\n- Author(s): Rota, Gian-Carlo\n- Subject(s): Combinatorics; Matroid Theory\n- Keywords: flat; log-concave; matroid\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 8th, 2007 by mdevos\n\nProblem-page discussion:\nA sequence $a_0,a_1,\\ldots a_n$ is log-concave if $a_i^2 \\ge a_{i-1} a_{i+1}$ for all $1 \\le i \\le n-1$.\n\nThe first of these conjectures is due to Rota [R], the second is folklore as far as I (M. DeVos) know. The special case of proving the second conjecture for $w_1,w_2,w_3$ amounts to showing that $(\\#lines)^2 \\ge (\\#points)(\\#planes)$ and has been called the points-lines-planes conjecture. Seymour [S] proved this conjecture in the special case where every line contains at most four points, but it is still open in general.\n\nBibliography:\n*[R] Rota, Gian-Carlo, Combinatorial theory, old and new. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3, pp. 229--233. Gauthier-Villars, Paris, 1971. MathSciNet\n\n[S] Seymour, P. D. On the points-lines-planes conjecture, J. Combin. Theory Ser. B 33 (1982), no. 1, 17--26. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0505646\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0678168\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Rota's unimodal conjecture\" in Combinatorics; Matroid Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Strong top-heavy and related log-concavity results for matroid invariants are known, but the stated rank-by-rank sequence of numbers of flats is not verified as universally unimodal/log-concave.\n\n**Verified partial progress.**\n\n- Modern Hodge-theoretic matroid methods prove major inequalities for associated invariants.\n\n**Full solution or refutation.**\n\nThe exact closed-set-rank sequence conjectures remain open.\n\n**What remains.**\n\nRelate flat counts to the available Hodge inequalities or find a counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, node 361 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains both flat-count conjectures.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3064, "problem_number": "OPG-369", "title": "Bases of many weights", "statement": "Let $G$ be an (additive) abelian group, and for every $S \\subseteq G$ let ${\\mathit stab}(S) = \\{ g \\in G: g + S = S \\}$.\n\nConjecture Let $M$ be a matroid on $E$, let $w: E \\rightarrow G$ be a map, put $S = \\{ \\sum_{b \\in B} w(b): B \\mbox{ is a base} \\}$ and $H = {\\mathit stab}(S)$. Then $$|S| \\ge |H| \\left( 1 - rk(M) + \\sum_{Q \\in G/H} rk(w^{-1}(Q)) \\right).$$", "background": "Source: Open Problem Garden. Original node ID: 369. URL: http://www.openproblemgarden.org/op/bases_of_many_weights.\n\nSource subject path: Combinatorics > Matroid Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bases_of_many_weights\n- Author(s): Schrijver, Alexander; Seymour, Paul D.\n- Subject(s): Combinatorics; Matroid Theory\n- Keywords: matroid; sumset; zero sum\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 10th, 2007 by mdevos\n\nProblem-page discussion:\nAlthough this conjecture may look a bit technical, it is in fact very natural, and important.\n\nThere is an interesting branch of combinatorial number theory which begins with the Cauchy-Davenport theorem, and M. Kneser's generelization of this theorem. We highlight these two theorems below. For a positive integer $n$, we let ${\\mathbb Z}_n = {\\mathbb Z} / n {\\mathbb Z}$.\n\nTheorem (Cauchy-Davenport) If $p$ is prime and $A,B \\subseteq {\\mathbb Z}_p$ are nonempty, then $|A+B| \\ge \\min\\{p, |A| + |B| - 1 \\}$.\n\nTheorem (Kneser) Let $A,B \\subseteq G$ be finite and nonempty, and let $H = {\\mathit stab}(A+B)$. Then $|A+B| \\ge |A+H| + |B+H| - |H|$.\n\nIn a somewhat underappreciated paper of Schrijver and Seymour, they find a generalization of the Cauchy-Davenport theorem to matroids. Namely, they prove the following.\n\nTheorem (Schrijver, Seymour) Let $M$ be a matroid on $E$, let $p$ be prime, and let $w: E \\rightarrow {\\mathbb Z}_p$ be a map. Then $\\#\\{ \\sum_{b \\in B} w(b): B \\mbox{ is a base} \\} \\ge \\min \\{p, \\sum_{g \\in {\\mathbb Z}_p} rk(w^{-1}(g)) \\}$.\n\nThe special case of this theorem when the underlying matroid is obtained from the free matroid on two elements by adding parallel edges is exactly the Cauchy-Davenport theorem. Further, their conjecture is precisely the common generalization of their theorem and Kneser's theorem.\n\nDeVos, Goddyn, and Mohar have proved this conjecture in the special case when the underlying matroid is obtained from a uniform matroid by adding parallel elements, but apart from that, little is known.\n\nBibliography:\n[C] A.L. Cauchy, Recherches sur les nombers, J. Ecole Polytechniques, 9 (1813), 99-123.\n\n[D] H. Davenport, On the addition of residue classes, J. London Math. Soc., 10 (1935), 30-32.\n\n[K] M. Kneser, Abschätzung der aymptotischen dichte von summenmengen, Math. Z. (1953) 459-484.\n\n[N] M.B. Nathanson, Additive Number Theory, GTM 165, Springer, 1996.\n\n*[SS] A. Schrijver and P.D. Seymour, Spanning trees of different weights. Polyhedral combinatorics, DIMACS Ser. Discrete Math. Theoret. Comp. Sci., 1, 281-288.\n\nBibliography links:\n- Spanning trees of different weights: http://repository.cwi.nl/search/fullrecord.php?publnr=1609\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 20.\n\nAttempt notes:\nTarget:\nMake progress on \"Bases of many weights\" in Combinatorics; Matroid Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the stated stabilizer-corrected lower bound for weighted matroid bases was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nDevelop Kneser-type sumset methods compatible with matroid exchange.\n\n**Sources checked.**\n\n- Open Problem Garden, node 369 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3065, "problem_number": "OPG-382", "title": "Aharoni-Berger conjecture", "statement": "Conjecture If $M_1,\\ldots,M_k$ are matroids on $E$ and $\\sum_{i=1}^k rk_{M_i}(X_i) \\ge \\ell (k-1)$ for every partition $\\{X_1,\\ldots,X_k\\}$ of $E$, then there exists $X \\subseteq E$ with $|X| = \\ell$ which is independent in every $M_i$.", "background": "Source: Open Problem Garden. Original node ID: 382. URL: http://www.openproblemgarden.org/op/aharoni_berger_conjecture.\n\nSource subject path: Combinatorics > Matroid Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/aharoni_berger_conjecture\n- Author(s): Aharoni, Ron; Berger, Eli\n- Subject(s): Combinatorics; Matroid Theory\n- Keywords: independent set; matroid; partition\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 11th, 2007 by mdevos\n\nProblem-page discussion:\nLet us begin by stating two classic results. For a graph (or hypergraph) we let $\\tau$ denote the size of the smallest (vertex) cover and we let $\\nu$ denote the size of the largest matching.\n\nTheorem (König) $\\nu = \\tau$ for every bipartite graph.\n\nTheorem (Matroid Intersection) If $M_1,M_2$ are matroids on $E$ and $rk_{M_1}(X_1) + rk_{M_2}(X_2) \\ge \\ell$ for every partition $\\{X_1,X_2\\}$ of $E$, then there exists $X \\subseteq E$ with $|X| = \\ell$ which is independent in both $M_1$ and $M_2$.\n\nThe matroid intersection theorem is exactly the $k=2$ case of the above conjecture, but it may also be viewed as a generalization of König's theorem. To see this, let $G$ be a bipartite graph with edge set $E$ and bipartition $\\{A_1,A_2\\}$ and for $i=1,2$ let $M_i$ be the (uniform) matroid on $E$ where a subset $S \\subseteq E$ is independent if no two edges in $S$ share an endpoint in $A_i$. Then $rk_{M_i}(S)$ is the number of vertices in $A_i$ which are incident with an edge in $S$, so $rk_{M_1}(X_1) + rk_{M_2}(X_2)$ has minimum value $\\tau$, and a set of edges is independent in both $M_1$ and $M_2$ if and only if it is a matching, so the size of the largest such set is $\\nu$.\n\nA famous conjecture of Ryser suggests a generalization of König's theorem to hypergraphs. It claims that every $k$-partite $k$-uniform hypergraph satisfies $\\tau \\le (k-1) \\nu$. The above conjecture is the common generalization of this conjecture of Ryser and the matroid intersection theorem. Aharoni [A] proved the 3-partite 3-uniform case of Ryser's conjecture, and this was extended by Aharoni-Berger [AB] to the $k=3$ case of the above conjecture. The conjecture remains open for $k \\ge 4$.\n\nBibliography:\n[A] R. Aharoni, Ryser's conjecture for tripartite 3-graphs, Combinatorica 21 (2001), 1-4. MathSciNet\n\n*[AB] R. Aharoni, E. Berger, The intersection of a matroid with a simplicial complex. Trans. Amer. Math. Soc. 358 (2006), no. 11 MathSciNet\n\nRelated:\nRelated problems\nRyser's conjecture\n\nDiscussion links:\n- cover: http://en.wikipedia.org/wiki/covering (graph theory)\n- conjecture of Ryser: http://www.openproblemgarden.org/?q=node/165\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1805710\n- The intersection of a matroid with a simplicial complex: http://www.math.princeton.edu/%7Eeberger/matcom.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2231877\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 28.\n\nAttempt notes:\nTarget:\nMake progress on \"Aharoni-Berger conjecture\" in Combinatorics; Matroid Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the Aharoni--Berger common-independent-set criterion was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nExtend matroid intersection/transversal methods to the multi-matroid partition condition.\n\n**Sources checked.**\n\n- Open Problem Garden, node 382 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3066, "problem_number": "OPG-692", "title": "Equality in a matroidal circumference bound", "statement": "Question Is the binary affine cube $AG(3,2)$ the only 3-connected matroid for which equality holds in the bound $$E(M) \\leq c(M) c(M^*) / 2$$where$c(M)$is the circumference (i.e. largest circuit size) of$M$?", "background": "Source: Open Problem Garden. Original node ID: 692. URL: http://www.openproblemgarden.org/op/equality_in_a_matroidal_circumference_bound.\n\nSource subject path: Combinatorics > Matroid Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/equality_in_a_matroidal_circumference_bound\n- Author(s): Oxley, James; Royle, Gordon\n- Subject(s): Combinatorics; Matroid Theory\n- Keywords: circumference\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 2nd, 2007 by Gordon Royle\n\nProblem-page discussion:\nIf $M$ is a 2-connected matroid with at least two elements then it was proved in [LO] that $$E(M) \\leq c(M) c(M^*) / 2$$where$c(M)$is the size of the largest circuit in$M$.\n\nEquality can hold in this bound -- in particular the binary affine cube $AG(3,2)$ is an 8-element self-dual matroid with circumference 4. There are various graphic matroids for which equality holds, and these have been classified in [W] where it is shown that they are all series-parallel networks and hence not 3-connected.\n\nThis question is therefore asking whether $AG(3,2)$ is the sole $3$-connected example where equality holds; this is known to be true for all matroids on up to 9 elements.\n\n(A variant of this question would be to ask if $AG(3,2)$ is the only non-graphic example other than trivial modifications like replacing every element with an equally sized parallel class.)\n\nBibliography:\n[LO] Lemos, Manoel; Oxley, James A sharp bound on the size of a connected matroid. Trans. Amer. Math. Soc. 353 (2001), no. 10, 4039--4056 MathSciNet\n\n[W] Wu, Pou-Lin Extremal graphs with prescribed circumference and cocircumference. Discrete Math. 223 (2000), no. 1-3, 299--308 MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1837219\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1782055\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Equality in a matroidal circumference bound\" in Combinatorics; Matroid Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Lemos-Oxley prove that AG(3,2) is the unique nondegenerate 3-connected matroid attaining equality in the circumference-cocircumference bound, explicitly answering Royle's question.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe uniqueness question has an affirmative answer.\n\n**What remains.**\n\nNo gap remains in the stated uniqueness problem; classifying equality under weaker connectivity is a different question.\n\n**Sources checked.**\n\n- Manoel Lemos and James Oxley, Bounding the size of a connected matroid, author preprint dated June 15, 2026. (primary): https://www.math.lsu.edu/~oxley/lemos_oxley_2026.pdf\n Evidence used: Abstract and main theorem prove AG(3,2) unique among 3-connected equality cases and state that this answers the 2007 Royle question.\n- Open Problem Garden, Equality in a matroidal circumference bound (node 692). (maintained_tracker): https://www.openproblemgarden.org/op/equality_in_a_matroidal_circumference_bound\n Evidence used: Original uniqueness question and historical verification through nine elements.\n\n**Review notes.** The input inequality writes E(M) instead of |E(M)| and has missing spaces. Small uniform matroids appearing under degenerate connectivity conventions do not alter the intended Royle question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3067, "problem_number": "OPG-696", "title": "Ding's tau_r vs. tau conjecture", "statement": "Conjecture Let $r \\ge 2$ be an integer and let $H$ be a minor minimal clutter with $\\frac{1}{r}\\tau_r(H) < \\tau(H)$. Then either $H$ has a $J_k$ minor for some $k \\ge 2$ or $H$ has Lehman's property.", "background": "Source: Open Problem Garden. Original node ID: 696. URL: http://www.openproblemgarden.org/op/dings_tau_r_vs_tau_conjecture.\n\nSource subject path: Combinatorics > Optimization.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/dings_tau_r_vs_tau_conjecture\n- Author(s): Ding, Guoli\n- Subject(s): Combinatorics; Optimization\n- Keywords: clutter; covering; MFMC property; packing\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 11th, 2007 by mdevos\n\nProblem-page discussion:\nSee Wikipedia's Clutter for definitions of clutter and clutter minors. The clutter $J_k$ is the degenerate projective plane with vertex set $\\{0,1,\\ldots,k\\}$ and edge set $\\{ \\{0,1\\}, \\{1,2\\},\\ldots,\\{0,k\\},\\{0,1,\\ldots,k\\} \\}$. If $H=(V,E)$ is a clutter, then for every positive integer $r$ we let $\\tau_r(H)$ denote the largest multiset of vertices of $H$ which hit every edge at least $r$ times. Note that $\\tau(H) = \\tau_1(H)$ and that $\\tau_r(H) \\le r \\tau(H)$.\n\nWe say that a clutter $H$ with $|V(H)| = n$, $\\tau(H) = s$ and $\\tau(b(H)) = r$ has Lehman's property if $rs > n$, $E(H) = \\{A_1,\\ldots,A_n\\}$, $E(b(H)) = \\{B_1,\\ldots,B_n\\}$, and the following properties are satisfied.\n\n- $|A_i| = r$ for every $1 \\le i \\le n$.\n- $|B_i| = s$ for every $1 \\le i \\le n$.\n- $|A_i \\cap B_i| = rs - n +1$ for $1 \\le i \\le n$\n- $|A_i \\cap B_j| = 1$ if $1 \\le i,j \\le n$ and $i \\neq j$.\n- every $v \\in V(H)$ lies in exactly $r$ edges of $H$, $s$ edges of $b(H)$, and $rs-n+1$ members of $\\{A_1 \\cap B_1, \\ldots,A_n \\cap B_n\\}$.\n\nAlthough the conditions in Lehman's condition are extremely stringent, Lehman [L] showed that every minor minimal clutter with the MFMC property satisfies these properties. Since the MFMC property for $H$ implies $\\frac{1}{r}\\tau_r(H) = \\tau(H)$ (and the degenerate projective planes are minor minimal without MFMC), if true, the above conjecture would be a nice extension of Lehman's theorem.\n\nDing [D] proved this conjecture for $r=2$, but it is open for all other cases.\n\nBibliography:\n*[D] G. Ding, Clutters with tau_2=2 tau, Discrete Math. 115 (1993), no. 1-3, 141--152. MathSciNet.\n\n[L] A. Lehman, On the width-length inequality, mimeographic notes, published 1979. Math. Program. 17, 403--417 MathSciNet.\n\nSource links:\n- clutter: http://en.wikipedia.org/wiki/clutter\n\nDiscussion links:\n- Wikipedia's Clutter: http://en.wikipedia.org/wiki/clutter (mathematics)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1217624\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0550854\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 32.\n\nAttempt notes:\nTarget:\nMake progress on \"Ding's tau_r vs. tau conjecture\" in Combinatorics; Optimization, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ding proves the conjecture for r=2, while no reliable resolution for arbitrary r>=3 was found.\n\n**Verified partial progress.**\n\n- Ding classifies the relevant r=2 minor-minimal clutters and proves the proposed dichotomy in that case.\n\n**Full solution or refutation.**\n\nThe general fixed-r conjecture remains unresolved in the literature checked.\n\n**What remains.**\n\nEstablish or refute the J_k-minor/Lehman-property dichotomy for r>=3 under the standard minimum r-fold transversal definition.\n\n**Sources checked.**\n\n- Guoli Ding, Clutters with tau_2=2 tau, Discrete Mathematics 115 (1993), 141-152. (primary): https://doi.org/10.1016/0012-365X(93)90484-B\n Evidence used: Formulates the general tau_k", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3068, "problem_number": "OPG-59928", "title": "Saturated $k$-Sperner Systems of Minimum Size", "statement": "Question Does there exist a constant $c>1/2$ and a function $n_0(k)$ such that if $|X|\\geq n_0(k)$, then every saturated $k$-Sperner system $\\mathcal{F}\\subseteq \\mathcal{P}(X)$ has cardinality at least $2^{(1+o(1))ck}$?", "background": "Source: Open Problem Garden. Original node ID: 59928. URL: http://www.openproblemgarden.org/op/saturated_k_sperner_systems_of_minimum_size.\n\nSource subject path: Combinatorics > Posets.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/saturated_k_sperner_systems_of_minimum_size\n- Author(s): Morrison, Natasha; Noel, Jonathan A.; Scott, Alex\n- Subject(s): Combinatorics; Posets\n- Keywords: antichain; extremal combinatorics; minimum saturation; saturation; Sperner system\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: April 12th, 2014 by Jon Noel\n\nProblem-page discussion:\nThe power set of a set $X$, denoted $\\mathcal{P}(X)$, is the collection of all subsets of $X$. A collection $\\mathcal{F}\\subseteq\\mathcal{P}(X)$ is said to be a $k$-Sperner system if there does not exist a subcollection $\\{A_1,\\dots,A_{k+1}\\}\\subseteq \\mathcal{F}$ such that $A_1\\subsetneq \\dots\\subsetneq A_{k+1}$; such a subcollection is called a $(k+1)$-chain. A $k$-Sperner system $\\mathcal{F}\\subseteq\\mathcal{P}(X)$ is said to be saturated if for every subset $S$ of $X$ not contained in $\\mathcal{F}$, the collection $\\mathcal{F}\\cup\\{S\\}$ contains a $(k+1)$-chain.\n\nGerbner et al. [1] proved that if $|X|\\geq k$, then every saturated $k$-Sperner System in $\\mathcal{P}(X)$ has cardinality at least $2^{k/2-1}$. Moreover, they conjectured that there exists a function $n_0(k)$ such that if $|X|\\geq n_0(k)$, then the minimum size of a saturated $k$-Sperner System in $\\mathcal{P}(X)$ has size $2^{k-1}$. This was disproved by Morrison, Noel and Scott in [2], who showed the following:\n\nTheorem (Morrison, Noel and Scott (2014)) There exists a constant $\\varepsilon>0$ and a function $n_0(k)$ such that for every $k$ and every set $X$ such that $|X|\\geq n_0(k)$ there exists a saturated $k$-Sperner system in $\\mathcal{P}(X)$ of cardinality at most $2^{(1-\\varepsilon)k}$.\n\nThe value of $\\varepsilon$ which can be deduced from their proof is approximately $\\left(1-\\frac{\\log_2(15)}{4}\\right)\\approx 0.023277$. Moreover, in [2] it was shown that there exists a function $n_0(k)$ and a constant $c\\in [1/2,1-\\varepsilon]$ such that if $|X|\\geq n_0(k)$, then the size of the smallest $k$-Sperner System in $\\mathcal{P}(X)$ is asymptotically $2^{(1+o(1))ck}$. The problem stated here is to determine whether $c>1/2$.\n\nA $1$-Sperner system is called an antichain. As was observed in [2], a positive answer to the above question would follow from a positive answer to the following question:\n\nQuestion (Morrison, Noel, Scott (2014)) Does there exist a constant $c>1/2$ and a function $n_0(k)$ such that if $|X|\\geq n_0(k)$ and $\\mathcal{A}\\subseteq\\mathcal{P}(X)$ is a saturated antichain in which every element of $\\mathcal{A}$ has cardinality between $\\left\\lfloor\\frac{k}{2}\\right\\rfloor$ and $|X|-\\left\\lfloor\\frac{k}{2}\\right\\rfloor +1$, then $|\\mathcal{A}|\\geq 2^{(1+o(1))ck}$?\n\nA more general problem is the following:\n\nQuestion Given integers $a,b$ and a set $X$, what is the minimum size of a saturated antichain $\\mathcal{A}$ in $\\mathcal{P}(X)$ in which every set of $\\mathcal{A}$ has cardinality between $a$ and $|X|-b$?\n\nBibliography:\n[1] D. Gerbner, B. Keszegh, N. Lemons, C. Palmer, D. Palvolgyi, and B. Patkos, Saturating Sperner Families, Graphs Combin. 29 (2013), no. 5, 1355–1364. arXiv:1105.4453\n\n*[2] N. Morrison, J. A. Noel, A. Scott. On Saturated k-Sperner Systems. arXiv:1402.5646 (2014). arXiv:1402.5646\n\nRelated:\nRelated problems\nSaturation in the Hypercube\n\nBibliography links:\n- arXiv:1105.4453: http://www.arxiv.org/abs/1105.4453\n- arXiv:1402.5646: http://www.arxiv.org/abs/1402.5646\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 32.\n\nAttempt notes:\nTarget:\nMake progress on \"Saturated $k$-Sperner Systems of Minimum Size\" in Combinatorics; Posets, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The desired exponent c>1/2 is unproved; 2024 work gives lower bound sqrt(k)2^(k/2) and upper bound 2^(0.961471k).\n\n**Verified partial progress.**\n\n- Recent work improves both lower and upper bounds on the minimum size of a saturated k-Sperner system.\n\n**Full solution or refutation.**\n\nThe lower bound has exponent 1/2 up to a polynomial factor and therefore does not supply the stated c>1/2.\n\n**What remains.**\n\nProve any fixed exponential exponent strictly greater than one half, or exhibit matching smaller families.\n\n**Sources checked.**\n\n- S. Balogh, B. Keszegh, R. Lemons and C. Palmer, Saturation of k-chains in the Boolean lattice, arXiv:2402.14113 (2024). (primary): https://arxiv.org/abs/2402.14113\n Evidence used: The abstract states the sqrt(k)2^(k/2) lower bound and 2^(0.961471k) upper bound.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3069, "problem_number": "OPG-351", "title": "Diagonal Ramsey numbers", "statement": "Let $R(k,k)$ denote the $k^{th}$ diagonal Ramsey number.\n\nConjecture $\\lim_{k \\rightarrow \\infty} R(k,k) ^{\\frac{1}{k}}$ exists.\n\nProblem Determine the limit in the above conjecture (assuming it exists).", "background": "Source: Open Problem Garden. Original node ID: 351. URL: http://www.openproblemgarden.org/op/diagonal_ramsey_numbers.\n\nSource subject path: Combinatorics > Ramsey Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/diagonal_ramsey_numbers\n- Author(s): Erdos, Paul\n- Subject(s): Combinatorics; Ramsey Theory\n- Keywords: Ramsey number\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 4th, 2007 by mdevos\n\nProblem-page discussion:\nErdos offered $100 for a solution to the highlighted conjecture and$250 for a solution to the associated problem (these prizes are now provided by Graham).\n\nClassic results of Erdos [E] and Erdos-Szekeres [ESz] give bounds on $R(k,k)$ which show that if $\\lim_{k \\rightarrow \\infty} R(k,k)^{\\frac{1}{k}}$ exists, then it is in the interval $[\\sqrt{2},4]$. Although these arguments are quite basic, little progress has been made in improving these bounds. The best known lower bound on $R(k,k)$ is due to Spencer [S] and the best known upper bound is due to Thomason [T]. They are as follows: $$(1 + o(1)) \\frac{ \\sqrt 2 }{e} k 2 ^{k/2} < R(k,k) < k^{-1/2 + c / \\sqrt{ \\log k}} {2k-2 \\choose k-1}.$$\n\nGowers [G] has suggested that resolving these problems might require a rough structure theorem.\n\nBibliography:\n[E] P. Erdos, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294. MathSciNet\n\n[ESz] P. Erdos and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470.\n\n[G] W. T. Gowers, Rough structure and classification, GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume, Part I, 79--117. MathSciNet\n\n[S] J. Spencer, Ramsey’s theorem—a new lower bound, J. Comb. Theory Ser. A 18 (1975), 108–115. MathSciNet\n\n[T] A. Thomason, An upper bound for some Ramsey numbers, J. Graph Theory 12 (1988), 509–517. MathSciNet\n\nSource links:\n- Ramsey number: http://en.wikipedia.org/wiki/ramsey number\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0019911\n- Rough structure and classification: http://www.dpmms.cam.ac.uk/%7Ewtg10/gafavisions.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1826250\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0366726\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0968746\n\nComments:\n- September 13th, 2010 | Anonymous | diagonal Ramsey numbers: Hi, My name is steve waterman.\n\nre - diagonal Ramsey numbers\n\nI have no proof of my conjectured formulas. However, the results are within the limits established in ALL cases. There is also a logic to these numbers as you will see.\n\nhttp://www.watermanpolyhedron.com/RAMSEY.html\n\nIt is my belief that these values are indeed exact...that is, no bounds required, and thus an answer to this riddle - and as I also see it....only to be proven later. It is a big claim no doubt. I doubt that I will ever see a counter-example nor a single proof of say R(5,5) as long as I live. Lastly, knowing that these MAY INDEED BE the correct values...give us a chance to zero in upon these numbers specifically.\n\nsteve\n- July 9th, 2007 | Anonymous | Upper bound: The upper bound has been improved by David Conlon (to appear in Annals)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Diagonal Ramsey numbers\" in Combinatorics; Ramsey Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The existence and value of the exponential growth limit for diagonal Ramsey numbers remain open.\n\n**Verified partial progress.**\n\n- Strong exponential upper and lower bounds are known but do not establish convergence.\n\n**Full solution or refutation.**\n\nNeither requested limit statement is verified.\n\n**What remains.**\n\nDevelop multiplicative/subadditive control of R(k,k).\n\n**Sources checked.**\n\n- Open Problem Garden, node 351 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains both questions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3070, "problem_number": "OPG-373", "title": "The large sets conjecture", "statement": "Conjecture If $A$ is 2-large, then $A$ is large.", "background": "Source: Open Problem Garden. Original node ID: 373. URL: http://www.openproblemgarden.org/op/the_large_sets_conjecture.\n\nSource subject path: Combinatorics > Ramsey Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_large_sets_conjecture\n- Author(s): Brown, Tom C.; Graham, Ronald L.; Landman, Bruce M.\n- Subject(s): Combinatorics; Ramsey Theory\n- Keywords: 2-large sets; large sets\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 10th, 2007 by vjungic\n\nProblem-page discussion:\nFor $r\\in \\mathbb{N}$, a set of positive integers $L$ is said to be $r$-large if for any $r$-coloring $f$ of positive integers there are arbitrarily long $f$- monochromatic arithmetic progressions whose common differences belong to $L$. Then $L$ is large if and only if it is $r$-large for all $r$. From Bergelson-Leibman's Polynomial van der Waerden's Theorem [BL] it follows that $\\{ |p(n)|: n \\in \\mathbb{N} \\} \\cap \\mathbb{N}$ is large for any polynomal $p$ with rational coefficients and such that $p(0)=0$.\n\nThe conjecture was stated in 1995 and published in 1999 [BGL].\n\nBibliography:\n[BL] V. Bergelson and A. Leibman, Polynomial extension of van der Waerden’s and Szemer\\'{e}di’s theorems, J. Amer. Math. Soc. 9 (1996) 725-753.\n\n*[BGL] T.C. Brown, R. L. Graham, and B. M. Landman, On the set of common differences in van der Waerden’s theorem on arithmetic progressions, Canadian Math. Bull. 42 (1999) 25-36.\n\n[J] V. Jungic, On Brown’s conjecture on Accessible Sets, J. Comb. Theory, Ser. A 110(1) (2005), 175-178\n\n[LR] B. M. Landman and A. Robertson, Ramsey Theory on the Integers, American Mathematical Society, 2004.\n\nBibliography links:\n- On the set of common differences in van der Waerden’s theorem on arithmetic progressions: http://www.math.ucsd.edu/%7Efan/ron/papers/99_02_common_differences.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"The large sets conjecture\" in Combinatorics; Ramsey Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that every 2-large set is large was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nClarify the combinatorial largeness hierarchy or construct a separation.\n\n**Sources checked.**\n\n- Open Problem Garden, node 373 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3071, "problem_number": "OPG-404", "title": "Concavity of van der Waerden numbers", "statement": "For $k$ and $\\ell$ positive integers, the (mixed) van der Waerden number $w(k,\\ell)$ is the least positive integer $n$ such that every (red-blue)-coloring of $[1,n]$ admits either a $k$-term red arithmetic progression or an $\\ell$-term blue arithmetic progression.\n\nConjecture For all $k$ and $\\ell$ with $k \\geq \\ell$, $w(k,\\ell) \\geq w(k+1,\\ell-1)$.", "background": "Source: Open Problem Garden. Original node ID: 404. URL: http://www.openproblemgarden.org/op/concavity_of_van_der_waerden_numbers.\n\nSource subject path: Combinatorics > Ramsey Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/concavity_of_van_der_waerden_numbers\n- Author(s): Landman, Bruce M.\n- Subject(s): Combinatorics; Ramsey Theory\n- Keywords: arithmetic progression; van der Waerden\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 21st, 2007 by Bruce Landman\n\nProblem-page discussion:\nThe conjecture was stated in 2000 and published 2003 [LR] and 2007 [KL].\n\nBibliography:\n*[BL] Bruce Landman and Aaron Robertson, Ramsey Theory on the Integers, American Mathematical Society, Providence, Rhode Island, 2003.\n\n[KL] Abdollah Khodkar and Bruce Landman, Recent progress in Ramsey theory on the integers, in Combinatorial Number Theory, 305-313, de Gruyter, Berlin, 2007.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Concavity of van der Waerden numbers\" in Combinatorics; Ramsey Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the proposed mixed van der Waerden concavity inequality was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nCompare extremal colorings across adjacent parameter pairs.\n\n**Sources checked.**\n\n- Open Problem Garden, node 404 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3072, "problem_number": "OPG-2359", "title": "Edge-antipodal colorings of cubes", "statement": "We let $Q_d$ denote the $d$-dimensional cube graph. A map $\\phi: E(Q_d) \\rightarrow \\{0,1\\}$ is called edge-antipodal if $\\phi(e) \\neq \\phi(e')$ whenever $e,e'$ are antipodal edges.\n\nConjecture If $d \\ge 2$ and $\\phi: E(Q_d) \\rightarrow \\{0,1\\}$ is edge-antipodal, then there exist a pair of antipodal vertices $v,v' \\in V(Q_d)$ which are joined by a monochromatic path.", "background": "Source: Open Problem Garden. Original node ID: 2359. URL: http://www.openproblemgarden.org/op/edge_antipodal_colorings_of_cubes.\n\nSource subject path: Combinatorics > Ramsey Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/edge_antipodal_colorings_of_cubes\n- Author(s): Norine, Serguei\n- Subject(s): Combinatorics; Ramsey Theory\n- Keywords: antipodal; cube; edge-coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 6th, 2008 by mdevos\n\nProblem-page discussion:\nThis conjecture has been verified by hand for $d \\le 5$.\n\nComments:\n- August 20th, 2010 | Anonymous | 2-colorings of edges of the cube: Q_n denotes the n-dimensional cube. For any x in Q_n, x_bar denotes the antipodal of x in Q_n.\n\nWe conjecture the following: Conj1 Let c:E_n --> {0, 1} be a coloring of the edges of Q_n. Then, there exists a pair of antipodal points x, x_bar and a path p from x to x_bar that it is either monochromatic or it changes colors exactly once.\n\nIt is easy to see that this conjecture implies an affirmative answer to the \"antipodal\" coloring open problem. We have verified that Conj1 holds for dimensions n=2, 3, and 4. We have also found that if the coloring is simple, that is, it does not contain squares colored 0101, then Conj1 holds (in fact, we find a monochromatic path joining a pair of antipodals).\n- May 18th, 2009 | leshabirukov | proof?: Let's suppose G is minimal counterexample. We are denote vertices as \"x1 x2 x3...\" xi={0|1} so, for example, \"110...01\" and \"001...10\" are antipodes. Consider G' is subgraph induced by x1=1. G' is lesser hypercube, so where exists connected pair of G'-antipodes, for clarification, \"100001111\" and \"111110000\". but (\"100001111\",\"000001111\") and (\"111110000\",\"011110000\" ) are edge-antipodes in G, so either (\"100001111\", \"011110000\") or (\"000001111\", \"111110000\") are connected! (And path length is equal to G dimension.)\n\nLooks too simple, am I misunderstood something? Or where is my medal?:))\n- May 18th, 2009 | md | not quite!: the edge-coloring of the subcube consisting of those vertices with x1=1 need not be edge-antipodal.\n- May 19th, 2009 | leshabirukov | Yes, indeed: Got it. edge-antipodes in G' are not antipodes in G, so can have same color. Thanks.\n- May 14th, 2009 | Anonymous | special case proven: We prove the conjecture in the special case where there is no square xyzt in the cube such that xy and zt get value 0, while yz and xt get value 1. The paper by Tomas Feder and Carlos Subi (submitted) can be found at\n\nhttp://theory.stanford.edu/~tomas/antipod.ps\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Edge-antipodal colorings of cubes\" in Combinatorics; Ramsey Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Norine's edge-antipodal cube-coloring conjecture remains open in full generality, with proofs through dimension five, for all simple labelings, and for additional topologically tractable classes identified in 2026.\n\n**Verified partial progress.**\n\n- Feder and Subi verified the conjecture for d <= 5.\n- Feder and Subi proved it for every simple labeling, meaning no square has alternating edge colors, even without antipodality.\n- Džavoronok proved a centrally symmetric simply connected 2-complex criterion and derived the conjecture for classes admitting compatible symmetric triangulations, with quantitative obstruction bounds.\n\n**Full solution or refutation.**\n\nNo theorem covering every dimension and every edge-antipodal coloring was verified.\n\n**What remains.**\n\nHandle arbitrary alternating-square configurations in every Q_d.\n\n**Sources checked.**\n\n- Tomás Feder and Carlos Subi, On hypercube labellings and antipodal monochromatic paths, Discrete Applied Mathematics 161 (2013), 1421-1426. (primary): https://doi.org/10.1016/j.dam.2012.12.025\n Evidence used: Proves the simple-labeling case and records verification through dimension five.\n- Adam Džavoronok, Monochromatic Paths and a Topological Approach to Norine's Conjecture, arXiv:2606.04181 (2026). (primary): https://arxiv.org/abs/2606.04181\n Evidence used: Proves a topological criterion and new restricted classes; its abstract does not claim the full cube conjecture.\n\n**Review notes.** The 2026 topological paper is partial progress, not a full resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3073, "problem_number": "OPG-357", "title": "A conjecture on iterated circumcentres", "statement": "Conjecture Let $p_1,p_2,p_3,\\ldots$ be a sequence of points in ${\\mathbb R}^d$ with the property that for every $i \\ge d+2$, the points $p_{i-1}, p_{i-2}, \\ldots p_{i-d-1}$ are distinct, lie on a unique sphere, and further, $p_i$ is the center of this sphere. If this sequence is periodic, must its period be $2d+4$?", "background": "Source: Open Problem Garden. Original node ID: 357. URL: http://www.openproblemgarden.org/op/a_conjecture_on_iterated_circumcentres.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_conjecture_on_iterated_circumcentres\n- Author(s): Goddyn, Luis A.\n- Subject(s): Geometry\n- Keywords: periodic; plane geometry; sequence\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 8th, 2007 by mdevos\n\nProblem-page discussion:\nLuis Goddyn discovered this curiosity, and proved the above conjecture for $d \\le 5$. He also studied related sequences, for instance, the sequence in ${\\mathbb R}^2$ where the $i^{th}$ point is the circumcentre of the points with index $i-2$, $i-3$, and $i-4$. See Iterated Circumcenters for a delightful and interactive discussion of this problem.\n\nBibliography:\n*[G] Luis Goddyn, Iterated Circumcenters\n\nDiscussion links:\n- Iterated Circumcenters: http://www.math.sfu.ca/%7Egoddyn/Circles\n\nBibliography links:\n- Iterated Circumcenters: http://www.math.sfu.ca/%7Egoddyn/Circles\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"A conjecture on iterated circumcentres\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No classification proving that every periodic iterated-circumcenter sequence has period 2d+4 was verified.\n\n**Verified partial progress.**\n\n- The hypotheses exclude degenerate nonunique-sphere steps.\n\n**Full solution or refutation.**\n\nThe period conjecture remains open.\n\n**What remains.**\n\nAnalyze the induced recurrence on affine/spherical configurations.\n\n**Sources checked.**\n\n- Open Problem Garden, node 357 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3074, "problem_number": "OPG-588", "title": "Big Line or Big Clique in Planar Point Sets", "statement": "Let $S$ be a set of points in the plane. Two points $v$ and $w$ in $S$ are visible with respect to $S$ if the line segment between $v$ and $w$ contains no other point in $S$.\n\nConjecture For all integers $k,\\ell\\geq2$ there is an integer $n$ such that every set of at least $n$ points in the plane contains at least $\\ell$ collinear points or $k$ pairwise visible points.", "background": "Source: Open Problem Garden. Original node ID: 588. URL: http://www.openproblemgarden.org/op/big_line_or_big_clique_in_planar_point_sets.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/big_line_or_big_clique_in_planar_point_sets\n- Author(s): Kara, Jan; Por, Attila; Wood, David R.\n- Subject(s): Geometry\n- Keywords: Discrete Geometry; Geometric Ramsey Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: September 25th, 2007 by David Wood\n\nProblem-page discussion:\nThe conjecture is trivial for $\\ell \\leq 3$.\n\nKára et al. [KPW] proved the conjecture for $k \\leq 4$ and all $\\ell$.\n\nAddario-Berry et al. [AFKCW] proved the conjecture for $k=5$ and $\\ell=4$.\n\nAbel et al. [ABBCDHKLPW] proved the conjecture for $k=5$ and all $\\ell$.\n\nThe conjecture is open for $k=6$ or $\\ell=4$.\n\nNote that it is easily proved that for all $k,\\ell\\geq2$, every set of at least $\\Omega(\\ell k^2)$ points in the plane contains $\\ell$ collinear points or $k$ points with no three collinear [Brass].\n\nSee [Matousek] for related results and questions.\n\nBibliography:\n[ABBCDHKLPW] Zachary Abel, Brad Ballinger, Prosenjit Bose, Sébastien Collette, Vida Dujmović, Ferran Hurtado, Scott D. Kominers, Stefan Langerman, Attila Pór, David R. Wood. Every Large Point Set contains Many Collinear Points or an Empty Pentagon, Graphs and Combinatorics 27(1): 47-60, 2011.\n\n[AFKCW] Louigi Addario-Berry, Cristina Fernandes, Yoshiharu Kohayakawa, Jos Coelho de Pina, and Yoshiko Wakabayashi. On a geometric Ramsey-style problem, 2007.\n\n[Brass] Peter Brass. On point sets without k collinear points. In Discrete Geometry, vol. 253 of Monographs and Textbooks in Pure and Applied Mathematics, pp. 185–192. Dekker, New York, 2003.\n\n*[KPW] Jan Kára, Attila Pór, David R. Wood. On the chromatic number of the visibility graph of a set of points in the plane, Discrete and Computational Geometry 34(3):497-506, 2005.\n\n[Matousek] Jiří Matoušek. Blocking visibility for points in general position, Discrete and Computational Geometry 42(2): 219-223, 2009.\n\nBibliography links:\n- Every Large Point Set contains Many Collinear Points or an Empty Pentagon: http://dx.doi.org/10.1007/s00373-010-0957-2\n- On a geometric Ramsey-style problem: http://crm.umontreal.ca/cal/en/mois200708.html\n- On the chromatic number of the visibility graph of a set of points in the plane: http://dx.doi.org/10.1007/s00454-005-1177-z\n- Blocking visibility for points in general position: http://dx.doi.org/10.1007/s00454-009-9185-z\n\nComments:\n- September 25th, 2012 | cibulka | Improved bound in the proof for k=5 and arbitrary l: In their proof of the case $k=5$ and arbitrary $\\ell$, Abel et al. [ABBCDHKLPW] proved a doubly exponential upper bound on the number $p(\\ell)$ of points that guarantees the occurrence of an $\\ell$-tuple of collinear points or a $5$-tuple of points with no other point in their convex hull (an empty pentagon). The upper bound on $p(\\ell)$ was improved by Barát et al. [BDJPSSVW] to $p(\\ell) \\le 328\\ell^2$.\n\n[BDJPSSVW] János Barát, Vida Dujmović, Gwenaël Joret, Michael S. Payne, Ludmila Scharf, Daria Schymura, Pavel Valtr, and David R. Wood. Empty pentagons in point sets with collinearities, arxiv:1207.3633, 2012.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Big Line or Big Clique in Planar Point Sets\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The finite Big-Line-Big-Clique conjecture remains open beyond the known k<=5 or ell<=3 cases, although new structured regimes were proved in 2026.\n\n**Verified partial progress.**\n\n- Abel et al. prove the conjecture for five pairwise visible points and every collinearity threshold via a generalized empty-pentagon theorem.\n- A 2026 preprint proves linear-size visible cliques for point sets largely supported on a cubic and for sets with only linearly many ordinary lines.\n- The same preprint proves the conjectured conclusion for point sets on any fixed irreducible algebraic curve and derives a dense-orchard-core obstruction.\n\n**Full solution or refutation.**\n\nNo result covering all finite planar point sets for arbitrary k and ell was located.\n\n**What remains.**\n\nHandle the first unresolved finite regimes, customarily described as six visible points or four collinear points, without assuming algebraic-curve or few-ordinary-line structure.\n\n**Sources checked.**\n\n- Zachary Abel et al., Every Large Point Set contains Many Collinear Points or an Empty Pentagon, Graphs and Combinatorics 27 (2011), 47-60. (primary): https://doi.org/10.1007/s00373-010-0957-2\n Evidence used: Settles the k=5 case for every ell in the notation of the stored problem.\n- Sohail Sarkar, Visibility cliques, cubic containers, and dense orchard cores, arXiv:2605.00918 (2026). (primary): https://arxiv.org/abs/2605.00918\n Evidence used: States the general conjecture and proves quantitative structured cases and a dense obstruction lemma.\n- Attila Pór and David R. Wood, The Big-Line-Big-Clique Conjecture is False for Infinite Point Sets, arXiv:1008.2988 (2010). (primary): https://arxiv.org/abs/1008.2988\n Evidence used: Clarifies that only the infinite analogue is refuted, not the exact finite conjecture.\n\n**Review notes.** The finite statement must not be confused with the false infinite analogue or false visible-island strengthening. The 2026 preprint swaps the letters used for collinear and visible target sizes relative to this database row.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3075, "problem_number": "OPG-605", "title": "Average diameter of a bounded cell of a simple arrangement", "statement": "Conjecture The average diameter of a bounded cell of a simple arrangement defined by $n$ hyperplanes in dimension $d$ is not greater than $d$.", "background": "Source: Open Problem Garden. Original node ID: 605. URL: http://www.openproblemgarden.org/op/average_diameter_of_a_bounded_cell_of_a_simple_arrangement.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/average_diameter_of_a_bounded_cell_of_a_simple_arrangement\n- Author(s): Deza, Antoine; Terlaky, Tamas; Zinchenko, Yuriy\n- Subject(s): Geometry\n- Keywords: arrangement; diameter; polytope\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 30th, 2007 by deza\n\nProblem-page discussion:\nLet $\\mathcal{A}$ be a simple arrangement formed by $n$ hyperplanes in dimension $d$. The number of bounded cells of $\\mathcal{A}$ is $I={n-1\\choose d}$. Let $\\delta(\\mathcal{A})$ denote the average diameter of a bounded cell $P_i$ of $\\mathcal{A}$; that is, $$\\delta(\\mathcal{A})=\\frac{\\sum_{i=1}^{i=I}\\delta(P_i)}{I}.$$Let$\\Delta_{\\mathcal{A}}({d,n})$denote the largest possible average diameter of a bounded cell of a simple arrangement defined by$n$inequalities in dimension$d$.\n\nWe have [DTZ,DX]:\n\nIf the conjecture of Hirsch holds, then $\\Delta_{\\mathcal{A}}(d,n)\\leq d+\\frac{2d}{n-1}$.\n\n$\\Delta_{\\mathcal{A}}({2,n})=2-\\frac{2\\lceil\\frac{n}{2}\\rceil}{(n-1)(n-2)}$ for $n\\geq 4$.\n\n$3-\\frac{6}{n-1}+\\frac{6(\\lfloor\\frac{n}{2}\\rfloor-2)}{(n-1)(n-2)(n-3)}\\leq \\Delta_{{\\mathcal A}}(3,n)\\leq 3 + \\frac{4(2n^2-16n+21)}{3(n-1)(n-2)(n-3)}$ for $n\\geq 5$.\n\n$\\Delta_{{\\mathcal A}}(d,n)\\geq d{n-d \\choose d}/{n-1 \\choose d}$ for $n\\geq 2d$.\n\nBibliography:\n*[DTZ] A. Deza, T. Terlaky and Y. Zinchenko: Polytopes and arrangements: diameter and curvature. Operations Research Letters (to appear).\n\n[DX] A. Deza and F. Xie: Hyperplane arrangements with large average diameter. Centre de Recherches Mathematiques and American Mathematical Society series (to appear).\n\nRelated:\nRelated problems\nHirsch Conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Average diameter of a bounded cell of a simple arrangement\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjecture is proved in dimension 2 and several small regimes and is asymptotically tight for fixed dimension, but no general resolution was located.\n\n**Verified partial progress.**\n\n- Deza and Xie prove the average-diameter upper bound in dimension 2.\n- They determine exact extremal values for arrangements with at most d+2 hyperplanes and for six planes in dimension 3.\n- They construct fixed-dimensional families whose average diameter tends to d and obtain dimension-3 lower and upper bounds both tending to 3.\n\n**Full solution or refutation.**\n\nThe primary and later computational papers provide only special cases and asymptotic bounds; no full proof or counterexample was found.\n\n**What remains.**\n\nProve the upper bound d for every simple arrangement in all dimensions and all numbers of hyperplanes, or find an arrangement whose bounded-cell average graph diameter exceeds d.\n\n**Sources checked.**\n\n- Antoine Deza and Feng Xie, Hyperplane Arrangements with Large Average Diameter, arXiv:0710.0328 (2007). (primary): https://arxiv.org/abs/0710.0328\n Evidence used: States the conjecture and proves dimension 2, asymptotic tightness, and several exact small regimes.\n- Antoine Deza, selected publications, current author publication list; checked 2026-08-17. (authoritative_secondary): https://www.cas.mcmaster.ca/~deza/pub.html\n Evidence used: Lists the 2012 computational follow-up and no later claimed general resolution in the author's relevant publication line.\n\n**Review notes.** The stored statement omits that diameter is the graph diameter of each bounded polyhedral cell's 1-skeleton. Confidence is medium because no maintained problem tracker was located.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3076, "problem_number": "OPG-720", "title": "Convex 'Fair' Partitions Of Convex Polygons", "statement": "Basic Question: Given any positive integer n, can any convex polygon be partitioned into n convex pieces so that all pieces have the same area and same perimeter?\n\nDefinitions: Define a Fair Partition of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a Convex Fair Partition.\n\nQuestions: 1. (Rephrasing the above 'basic' question) Given any positive integer n, can any convex polygon be convex fair partitioned into n pieces?\n\n2. If the answer to the above is \"Not always\", how does one decide the possibility of such a partition for a given convex polygon and a given n? And if fair convex partition is allowed by a specific convex polygon for a give n, how does one find the optimal convex fair partition that minimizes the total length of the cut segments?\n\n3. Finally, what could one say about higher dimensional analogs of this question?\n\nConjecture: The authors tend to believe that the answer to the above 'basic' question is \"yes\". In other words they guess: Every convex polygon allows a convex fair partition into n pieces for any n", "background": "Source: Open Problem Garden. Original node ID: 720. URL: http://www.openproblemgarden.org/op/textbf_convex_fair_partitions_of_convex_polygons.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/textbf_convex_fair_partitions_of_convex_polygons\n- Author(s): Nandakumar, R.; Ramana, Rao N.\n- Subject(s): Geometry\n- Keywords: Convex Polygons; Partitioning\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: December 12th, 2007 by Nandakumar\n\nProblem-page discussion:\n1. The above conjecture is easily seen to hold for n=2. for n=3 and above, it is not clear.\n\n2. The n = 2 case does not appear to allow a recursive generalization for values of n equal to powers of 2.\n\n3. It can be shown that any polygon (not necessarily convex) allows a fair partitioning into n pieces for any n, provided the pieces need not be convex (this is not a convex fair partition). See (4) in references below.\n\n4. It appears that the fair parition of a convex polygon which minimizes the total length of cuts (or equivalently, the sum of the perimeters of the pieces) need not be a convex fair partition.\n\n5. There is no known work in this specific area. The problem of partitioning convex polygons into equal area convex pieces so that every piece equally shares the boundary of the input polygon has been studied (references below)\n\nBibliography:\n(*)1. The original 'mainstream' statement of this problem: http://maven.smith.edu/~orourke/TOPP/P67.html#Problem.67\n\n2. Jin Akiyama, A. Kaneko, M. Kano, Gisaku Nakamura, Eduardo Rivera-Campo, S. Tokunaga, and Jorge Urrutia. Radial perfect partitions of convex sets in the plane. In Japan Conf. Discrete Comput. Geom., pages 1-13, 1998.\n\n3. Jin Akiyama, Gisaku Nakamura, Eduardo Rivera-Campo, and Jorge Urrutia. Perfect divisions of a cake. In Proc. Canad. Conf. Comput. Geom., pages 114-115, 1998.\n\n4. This blog maintained by the authors has tentative thoughts, examples, etc on 'Fair Partitions': http://nandacumar.blogspot.com\n\nBibliography links:\n- http://maven.smith.edu/~orourke/TOPP/P67.html#Problem.67: http://maven.smith.edu/%7Eorourke/TOPP/P67.html#Problem.67\n- http://nandacumar.blogspot.com: http://nandacumar.blogspot.com\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Convex 'Fair' Partitions Of Convex Polygons\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The central conjecture is solved affirmatively for every number of pieces, but the record's optimal-cut algorithm and full higher-dimensional questions remain unresolved.\n\n**Verified partial progress.**\n\n- Akopyan-Avvakumov-Karasev prove every planar convex body has a convex equal-area, equal-perimeter partition into m parts for every m>=2.\n- The same paper proves a weaker higher-dimensional theorem equalizing d-1 measures and one continuous body functional.\n- Existing numerical algorithms seek fair partitions but do not supply the requested globally minimum total cut length.\n\n**Full solution or refutation.**\n\nQuestion 1 and the displayed conjecture are fully affirmative; the multi-question record as a whole is not closed.\n\n**What remains.**\n\nFind or characterize minimum-total-cut fair partitions and settle the full volume-plus-surface-area analogue in higher dimensions.\n\n**Sources checked.**\n\n- Arseniy Akopyan, Sergey Avvakumov, and Roman Karasev, Convex fair partitions into an arbitrary number of pieces, Advances in Mathematics 493 (2026), 110927; arXiv:1804.03057. (primary): https://arxiv.org/abs/1804.03057\n Evidence used: Theorem 1.1 proves the planar conjecture for arbitrary m; the paper explicitly excludes algorithms and explains limits of the higher-dimensional extension.\n- TOPP Problem 67, Fair Partitioning of Convex Polygons. (maintained_tracker): https://topp.openproblem.net/p67\n Evidence used: Preserves the original existence, optimization, and higher-dimensional questions and the earlier prime-power progress.\n\n**Review notes.** The statement contains the typo 'for a give n' and lacks final punctuation. It is classified partial because Questions 2 and 3 survive even though the basic conjecture is solved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3077, "problem_number": "OPG-1761", "title": "Dense rational distance sets in the plane", "statement": "Problem Does there exist a dense set $S \\subseteq {\\mathbb R}^2$ so that all pairwise distances between points in $S$ are rational?", "background": "Source: Open Problem Garden. Original node ID: 1761. URL: http://www.openproblemgarden.org/op/dense_rational_distance_sets_in_the_plane.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/dense_rational_distance_sets_in_the_plane\n- Author(s): Ulam, Stanislaw M.\n- Subject(s): Geometry\n- Keywords: integral distance; rational distance\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 4th, 2008 by mdevos\n\nProblem-page discussion:\nThis famous problem was asked by Ulam, who guessed the answer would be negative.\n\nA cute theorem of Erdos shows that if $S \\subseteq {\\mathbb R}^2$ is non-collinear and all pairwise distances between points in $S$ are integral, then $S$ is finite. For the proof, first note that if $x,y \\in {\\mathbb R}^2$ have distance $k \\in {\\mathbb Z}$, then every point which has integer distance to both $x$ and $y$ must lie on one of the $k+1$ hyperbolas consisting of those $z \\in {\\mathbb R}^2$ with $|{\\mathit dist}(x,z) - {\\mathit dist}(y,z)| = j$ for some $0 \\le j \\le k$. So, if all pairwise distances between points in $S$ are integral, and $x,y,z \\in S$ are non-collinear, then every other point in $S$ must lie on an intersection between one of finitely many hyperbola with foci $x,y$ and one of finitely many with foci $x,z$. This set is necessarily finite, thus completing the proof.\n\nOf course, the above argument gives no upper bound on the size of a non-collinear set of points in ${\\mathbb R}^2$ with pairwise integral distances. Indeed, if Ulam's conjecture is true, then there exist such sets of arbitrary size. Surprisingly, it is very difficult to construct such sets $S$ of even rather small size. Recently Kreisel and Kurz [KK] found such a set of size 7, but it is unknown if there exists one of size 8.\n\nIt is trivial to find infinitely many points on a line with all pairwise distances rational. Less trivially, there exist infinite subsets of a circle with all pairwise distances rational. Very recently, Solymosi and De Zeeuw [SZ] proved that these are the only two irreducible algebraic curves with this property. This suggests that, if the answer to Ulam's problem is affirmative, such a set $S$ must be extremely special.\n\nBibliography:\n[KK] T. Kreisel and S. Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete & Computational Geometry, Online first: DOI 10.1007/s00454-007-9038-6\n\n[SZ] J. Solymosi and F. de Zeeuw, On a question of Erdos and Ulam.\n\nBibliography links:\n- On a question of Erdos and Ulam: http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.3095v1.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Dense rational distance sets in the plane\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Erdős--Ulam dense-rational-distance question remains open unconditionally, but Bombieri--Lang implies a negative answer.\n\n**Verified partial progress.**\n\n- Shaffaf gives the conditional Bombieri--Lang proof.\n- The associated algebraic-curve restrictions rule out broad algebraic routes except lines and circles.\n\n**Full solution or refutation.**\n\nNo unconditional construction or nonexistence theorem was verified.\n\n**What remains.**\n\nRemove the Bombieri--Lang hypothesis or construct a dense rational-distance set.\n\n**Sources checked.**\n\n- J. Shaffaf, A solution of the Erdos-Ulam problem on rational distance sets assuming the Bombieri-Lang conjecture, arXiv:1501.00159. (primary): https://arxiv.org/abs/1501.00159\n Evidence used: Its abstract states the conditional result and strengthened structural conclusion.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3078, "problem_number": "OPG-1820", "title": "Simplexity of the n-cube", "statement": "Question What is the minimum cardinality of a decomposition of the $n$-cube into $n$-simplices?", "background": "Source: Open Problem Garden. Original node ID: 1820. URL: http://www.openproblemgarden.org/op/simplexity_of_the_cube.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/simplexity_of_the_cube\n- Subject(s): Geometry\n- Keywords: cube; decomposition; simplex\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: August 6th, 2008 by mdevos\n\nProblem-page discussion:\nA decomposition of a polytope $P$ into $n$-simplices is a set of $n$-simplices which have pairwise disjoint interiors and have union equal to $P$. This is also known as a (generalized) triangulation.\n\nLet $T(n)$ be the minimum cardinality of a decomposition of the $n$-cube into $n$-simplices (the answer to our question). It is trivial that $T(1) = 1$ and easy to see that $T(2) = 2$. A 3-dimensional cube may be decomposed into five simplices by cutting off every other corner as shown in the figure (from [JW]). This division is optimal, so $T(3) = 5$.\n\nChopping off every other corner of a 4-cube leaves a 16-cell (the 4-dimensional cross-polytope) which can then be decomposed into eight simplices (fix a vertex $x$ and then take each of the eight 4-simplices formed as the convex hull of $x$ and a facet which is not incident with $x$ ). This is also optimal, so $T(4) = 16$. Computer assisted searches have yielded other good decompositions in low dimensions (see [S]).\n\nThe decompositions of the 3 and 4-dimensional cubes described here do not generalize to higher dimensions. However, there is a naive decomposition of an $n$-cube into $n!$ simplices. Take the cube to be $[0,1]^n$ and let $S$ be the set of all points $(x_1,\\ldots,x_n)$ for which $0 \\le x_1 \\le x_2 \\ldots \\le x_n \\le 1$. Then $S$ is a simplex contained in our cube which contains the main diagonal from the origin to $(1,1,\\ldots,1)$. Further, by permuting the terms $x_1,\\ldots,x_n$ in the chain of inequlities, we get a total of $n!$ simplices which form a decomposition of the cube.\n\nThis naive decomposition is not optimal in dimensions 3 and 4 since our constructions show $T(3) \\le 5 < 3!$ and $T(4) \\le 16 < 4!$. Haiman [H] found a clever way to lift efficient lower dimensional decompositions to high dimensions thus achieving a significant improvement on our $n!$ upper bound. To state his result precisely, we require another parameter. Let $T^*(n)$ be the minimum cardinality of a decomposition of an $n$-cube into $n$-simplices with the following additional constraints:\n\n- Every vertex of a simplex is a vertex of the cube.\n- The intersection of any two simplices is a face of both of them.\n\nIt is immediate that $T(n) \\le T^*(n)$, but to the best of our knowledge these parameters may always be identical. Indeed, this is a separate interesting question. Anyway, back to Haiman's bound. He proved that $T^*(kn) \\le (T^*(n)/n!)^k (kn)!$. Using this inequality with either the 3 or 4-dimensional example from above would give an improvement on the $n!$ upper bound. However, best known is to plug in $T^*(7) \\le 1493$, which gives a general upper bound of $T(7n) \\le T^*(7n) <.840463^{7n}(7n)!$.\n\nA natural lower bound on $T(n)$ can be obtained by a volume argument. Clearly, $T(n)$ must be at least the volume of an $n$-dimensional cube divided by the volume of the largest simplex it contains. Smith [S] improved upon this by moving the argument to hyperbolic space (where the volume of a cube is comparatively much larger than that of a simplex). His volume estimate here yields $T(n) \\ge \\frac{1}{2} \\cdot 6^{n/2}(n+1)^{- \\frac{n+1}{2} } n!$.\n\nBibliography:\n[H] M. Haiman, A simple and relatively efficient triangulation of the n-cube, Discr. Comp. Geom. 6, 4 (1991) 287-289.\n\n[JW] Jackson, Frank and Weisstein, Eric W. \"Tetrahedron.\" From MathWorld--A Wolfram Web Resource.\n\n[S] W. Smith, A lower bound for the simplexity of the n-cube via hyperbolic volume.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"Simplexity of the n-cube\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For the conventional triangulation using cube vertices, exact simplexities are known through dimension 7 and dimension 8 has nonmatching bounds; asymptotic upper and lower bounds are known, but the source statement does not specify decomposition conventions.\n\n**Verified partial progress.**\n\n- The conventional cube-vertex triangulation values for n=1 through 7 are 1, 2, 5, 16, 67, 308, and 1493.\n- The maintained OEIS record gives 5522 <= T*(8) <= 11944.\n- Orden and Santos construct triangulations with O(0.816^n n!) simplices.\n- Glazyrin proves lower bounds for cube-vertex dissections, including (n+1)^((n-1)/2).\n\n**Full solution or refutation.**\n\nNo exact formula or complete determination for general n is known; even n=8 is not exact for the conventional vertex-triangulation parameter.\n\n**What remains.**\n\nSpecify whether Steiner vertices and non-face-to-face dissections are allowed, then determine the corresponding parameter for n>=8 or close its asymptotic gap.\n\n**Sources checked.**\n\n- Robert B. Hughes and Michael R. Anderson, Simplexity of the cube, Discrete Mathematics 158 (1996), 99–150, DOI 10.1016/0012-365X(95)00075-8. (primary): https://doi.org/10.1016/0012-365X(95)00075-8\n Evidence used: Gives a 1493-simplex triangulation of the 7-cube and proves optimality there and for the known 308-simplex 6-cube triangulation, with dimension-5 analysis.\n- Alexey Glazyrin, Lower bounds for the simplexity of the n-cube, Discrete Mathematics 312 (2012), 3656–3662; arXiv:0910.4200. (primary): https://arxiv.org/abs/0910.4200\n Evidence used: Proves an asymptotic lower bound for cube-vertex dissections and tabulates known low-dimensional triangulation and dissection bounds.\n- David Orden and Francisco Santos, Asymptotically efficient triangulations of the d-cube, Discrete & Computational Geometry 30 (2003), 509–528; arXiv:math/0204157. (primary): https://arxiv.org/abs/math/0204157\n Evidence used: Constructs triangulations with O(0.816^n n!) simplices.\n- OEIS A019503, Simplexity of the n-cube, accessed 2026-08-17. (maintained_tracker): https://oeis.org/A019503\n Evidence used: Maintains the exact vertex-triangulation values through dimension 7 and the current recorded bounds for dimension 8.\n\n**Review notes.** Formulation defect: the statement omits whether Steiner vertices are allowed, whether intersections must be common faces, and whether all simplex vertices must be cube vertices. The background later distinguishes T from a constrained T*, so conventions must not be silently merged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3079, "problem_number": "OPG-2089", "title": "Kneser–Poulsen conjecture", "statement": "Conjecture If a finite set of unit balls in $\\mathbb{R}^n$ is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.", "background": "Source: Open Problem Garden. Original node ID: 2089. URL: http://www.openproblemgarden.org/op/kneser_poulsen_conjecture.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/kneser_poulsen_conjecture\n- Author(s): Kneser, M.; Poulsen, E. T.\n- Subject(s): Geometry\n- Keywords: pushing disks\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 25th, 2008 by tchow\n\nProblem-page discussion:\nThis problem dates from the mid-1950's. The planar case was solved by Bezdek and Connelly in 2003, who also showed that the area of the intersection does not increase, and that the result holds even if the disks have unequal radii. In higher dimensions the problem remains open.\n\nThe conjecture is known to hold if the rearrangement can be executed by a continuous motion such that the distance between every pair of centers monotonically increases throughout the motion.\n\nBibliography:\n*[BC] K. Bezdek and R. Connelly, Pushing disks apart: The Kneser-Poulsen conjecture in the plane, J. Reine Angew. Math. 553 (2002), 221--236.\n\n*[K] M. Kneser, Einige Bemerkungen über das Minkowskische Flächenmass, Arch. Math. 6 (1955), 382--390.\n\n*[P] E. T. Poulsen, Problem 10, Math. Scand. 2 (1954), 346.\n\nBibliography links:\n- Pushing disks apart: The Kneser-Poulsen conjecture in the plane: http://arxiv.org/abs/math.MG/0108098\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Kneser–Poulsen conjecture\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The union-volume Kneser--Poulsen conjecture is established in the plane and in special settings, but remains open for arbitrary finite unit-ball configurations in all dimensions.\n\n**Verified partial progress.**\n\n- Planar cases are proved.\n- Recent surveys still present the general higher-dimensional statement as a conjecture.\n\n**Full solution or refutation.**\n\nNo all-dimensional proof was verified.\n\n**What remains.**\n\nProve monotonicity of union volume under arbitrary expansions in dimensions at least three.\n\n**Sources checked.**\n\n- Selected topics from the theory of intersections of balls, Discrete Applied Mathematics 382 (2026), 60--82. (authoritative_secondary): https://doi.org/10.1016/j.dam.2025.11.040\n Evidence used: A current survey presents the volume monotonicity statement as the Kneser--Poulsen conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3080, "problem_number": "OPG-2400", "title": "Erdös-Szekeres conjecture", "statement": "Conjecture Every set of $2^{n-2} + 1$ points in the plane in general position contains a subset of $n$ points which form a convex $n$-gon.", "background": "Source: Open Problem Garden. Original node ID: 2400. URL: http://www.openproblemgarden.org/op/erdos_szekeres_conjecture.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/erdos_szekeres_conjecture\n- Author(s): Erdos, Paul; Szekeres, George\n- Subject(s): Geometry\n- Keywords: combinatorial geometry; Convex Polygons; ramsey theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 8th, 2008 by mdevos\n\nProblem-page discussion:\nThis is one of the most famous unsolved problems in combinatorial geometry, perhaps due in part to its lovely history. The problem of showing that every sufficiently large set of points in general position determine a convex $n$-gon was the original inspiration of Esther Klein. Erdös called this the Happy end problem since it led to the marriage of Esther Klein and George Szekeres. This problem was also one of the original sources of Ramsey Theory.\n\nLet $f(n)$ denote the smallest integer so that every set of $f(n)$ points in the plane in general position contains $n$ points which form a convex $n$-gon. The fact that $f(n)$ exists for every $n$ was first established in a seminal paper of Erdös and Szekeres who proved the following bounds on $f(n)$.\n$$\n2^{n-2} + 1 \\le f(n) \\le { 2n-4 \\choose n-2 } + 1\n$$\n\nThe lower bound is conjectured to be the truth, and this is known to hold for $n \\le 5$. A handful of recent papers on this problem have improved the upper bound to\n$$\nf(n) \\le {2n-5 \\choose n-3} + 1.\n$$\n\nDiscussion links:\n- Happy end problem: http://en.wikipedia.org/wiki/Happy end problem\n\nComments:\n- April 14th, 2009 | Anonymous | Now true for n = 6 as well: Now true for n = 6 as well by computer-aided solution of Szekeres and Peters (2006).\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Erdös-Szekeres conjecture\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact Erdős-Szekeres formula ES(n)=2^{n-2}+1 remains open in general; current work gives the asymptotically sharp exponent and upper bound 2^{n+O(sqrt(n log n))}.\n\n**Verified partial progress.**\n\n- The conjecture is known for n <= 6.\n- Suk proved ES(n)=2^{n+o(n)}.\n- Holmsen, Mojarrad, Pach, and Tardos improved the upper bound to 2^{n+O(sqrt(n log n))}.\n\n**Full solution or refutation.**\n\nNo exact proof for all n was verified; a 2026 JCTA article continues to treat the equality as a conjecture.\n\n**What remains.**\n\nRemove the subexponential factor and prove the exact lower-bound construction is extremal for all n.\n\n**Sources checked.**\n\n- Andrew Suk, On the Erdős-Szekeres convex polygon problem, Journal of the American Mathematical Society 30 (2017), 1047-1053. (primary): https://doi.org/10.1090/jams/869\n Evidence used: Proves ES(n)=2^{n+o(n)} and states the exact conjecture.\n- Andreas F. Holmsen, Hossein Nassajian Mojarrad, János Pach, and Gábor Tardos, Two extensions of the Erdős-Szekeres problem, arXiv:1710.11415. (primary): https://arxiv.org/abs/1710.11415\n Evidence used: Source of the currently recorded 2^{n+O(sqrt(n log n))} upper bound.\n- Erdős Problems, Problem 107 history. (maintained_tracker): https://www.erdosproblems.com/history/107\n Evidence used: Current maintained status and best-bound history through 2026.\n- The Erdős-Szekeres conjecture revisited, Journal of Combinatorial Theory, Series A (2026), article 106195. (primary): https://doi.org/10.1016/j.jcta.2026.106195\n Evidence used: Recent primary paper still framing the exact equality as a conjecture and recording modern bounds.\n\n**Review notes.** The title misspells Erdős as Erdös; this does not affect the mathematical statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3081, "problem_number": "OPG-2435", "title": "Monochromatic empty triangles", "statement": "If $X \\subseteq {\\mathbb R}^2$ is a finite set of points which is 2-colored, an empty triangle is a set $T \\subseteq X$ with $|T|=3$ so that the convex hull of $T$ is disjoint from $X \\setminus T$. We say that $T$ is monochromatic if all points in $T$ are the same color.\n\nConjecture There exists a fixed constant $c$ with the following property. If $X \\subseteq {\\mathbb R}^2$ is a set of $n$ points in general position which is 2-colored, then it has $\\ge cn^2$ monochromatic empty triangles.", "background": "Source: Open Problem Garden. Original node ID: 2435. URL: http://www.openproblemgarden.org/op/monochromatic_empty_triangles.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monochromatic_empty_triangles\n- Subject(s): Geometry\n- Keywords: empty triangle; general position; ramsey theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 8th, 2008 by mdevos\n\nProblem-page discussion:\nIt is known that any set of $n$ points in the plane in general position contains $\\ge cn^{5/4}$ monochromatic empty triangles.\n\nComments:\n- September 27th, 2009 | Anonymous | This has a trivial: This has a trivial counterexample for c > 0.\n\nConsider X = {(0,0), (0,1), (1,0)}, colored {red, blue, blue} respectively. There is only one empty triangle in X, and it is not monochromatic. So it has 0 monochromatic empty triangles, and 0 is not > c*(3^2) for c > 0.\n- August 27th, 2010 | Anonymous | Yes indeed. However in this: Yes indeed. However in this types of problems it is generally implied that the statement is for a sufficently large n.\n- May 12th, 2009 | Anonymous | Original source and one improvement.: The conjecture appeared first in \"Oswin Aichholzer, Ruy Fabila-Monroy, David Flores-Peñaloza, Thomas Hackl, Clemens Huemer, and Jorge Urrutia. Empty monochromatic triangles. In Proceedings of the 20th Canadian Conference on Computational Geometry (CCCG2008), pages 75-78, 2008.\"\n\nIn this paper the authors show that any set of n points in general position has $cn^{5/4}$ empty monochromatic triangles. You can get this paper from http://cccg.ca/proceedings/2008/paper18.pdf\n\nThere is one improvement showing that any set of n points in general position has $cn^{4/3}$ empty monochromatic triangles in: \"J. Pach, G. Toth. Monochromatic empty triangles in two-colored point sets. In: Geometry, Games, Graphs and Education: the Joe Malkevitch Festschrift (S. Garfunkel, R. Nath, eds.), COMAP, Bedford, MA, 2008, 195--198.\" Get it from: http://www.math.nyu.edu/~pach/publications/emptytriangle102408.pdf\n- May 8th, 2009 | Anonymous | The lower bound has been improved: The lower bound has been improved to cn4/3.\n\nJ. Pach and G. Toth: Monochromatic empty triangles in two-colored point sets, in: Geometry, Games, Graphs and Education: the Joe Malkevitch Festschrift (S. Garfunkel, R. Nath, eds.), COMAP, Bedford, MA, 2008, 195--198.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Monochromatic empty triangles\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported sentence is not a coherent version of the intended conjecture: c=0 makes it trivial, while requiring c>0 for every n makes it false; the intended sufficiently-large-n quadratic lower bound remains open.\n\n**Verified partial progress.**\n\n- Pach and Tóth proved Omega(n^{4/3}) monochromatic empty triangles for two colors.\n- Bhattacharya et al. proved Omega(n^2) monochromatic triangles with at most one interior point for two colors, but only Omega(n^{4/3}) for genuinely empty triangles.\n\n**Full solution or refutation.**\n\nNo quadratic lower bound for genuinely empty monochromatic triangles in the intended asymptotic formulation was verified.\n\n**What remains.**\n\nState c>0 and sufficiently large n explicitly, then close the gap from exponent 4/3 to 2 or provide a subquadratic construction.\n\n**Sources checked.**\n\n- János Pach and Géza Tóth, Monochromatic empty triangles in two-colored point sets, Discrete Applied Mathematics 161 (2013), 1259-1261. (primary): https://doi.org/10.1016/j.dam.2011.08.026\n Evidence used: Proves the Omega(n^{4/3}) lower bound for genuinely empty triangles.\n- Bhaswar B. Bhattacharya et al., On the Number of Almost Empty Monochromatic Triangles, arXiv:2601.18951 (2026). (primary): https://arxiv.org/abs/2601.18951\n Evidence used: Separates the quadratic one-interior-point result from the 4/3 exponent for empty triangles.\n- Douglas West, Monochromatic Empty Triangles. (maintained_tracker): https://dwest.web.illinois.edu/regs/emptytri.html\n Evidence used: Maintains the intended asymptotic formulation and best published bound.\n\n**Review notes.** Literal defect is material: c is not required positive and the statement omits sufficiently large n. Three bichromatically colored noncollinear points refute the universal-n reading with c>0.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3082, "problem_number": "OPG-36901", "title": "Inequality of the means", "statement": "Question Is is possible to pack $n^n$ rectangular $n$-dimensional boxes each of which has side lengths $a_1,a_2,\\ldots,a_n$ inside an $n$-dimensional cube with side length $a_1 + a_2 + \\ldots a_n$?", "background": "Source: Open Problem Garden. Original node ID: 36901. URL: http://www.openproblemgarden.org/op/inequality_of_the_means.\n\nSource subject path: Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/inequality_of_the_means\n- Subject(s): Geometry\n- Keywords: arithmetic mean; geometric mean; Inequality; packing\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 6th, 2009 by mdevos\n\nProblem-page discussion:\nTaking the arithmetic/geometric mean inequality\n$$\n(a_1 a_2 \\ldots a_n)^{1/n} \\le \\frac{a_1 + a_2 + \\ldots a_n}{n}\n$$\n multiplying both sides by $n$ and then raising both sides to the $n^{th}$ power yields:\n$$\nn^n \\cdot a_1 a_2 \\ldots a_n \\le (a_1 + a_2 + \\ldots a_n)^{n}.\n$$\n So, in the above question, the volume of the cube is at least the sum of the volumes of the rectangular boxes. Furthermore, a positive solution to this question would yield a strengthening of the arithmetic/geometric mean inequality.\n\nFor $n=1$ the problem is trivial, for $n=2$ it is immediate, and for $n=3$ it is tricky, but possible. It is also known that a solution for dimensions $n$ and $m$ can be combined to yield a solution for dimension $nm$. Thus, the question has a positive answer whenever $n$ has the form $2^a 3^b$. It is open for all other values.\n\nSee Bar-Natan's page for more.\n\nBibliography:\n[BCG] E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways for Your Mathematical Plays, Academic Press, New York 1983.\n\nDiscussion links:\n- Bar-Natan's page: http://www.math.toronto.edu/%7Edrorbn/projects/ArithGeom/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Inequality of the means\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Constructions are known in dimensions 2, 3, and 4, and composition gives every dimension 2^p 3^q; dimension 5 and the other dimensions remain unresolved on the maintained author page.\n\n**Verified partial progress.**\n\n- Explicit packings exist for n=2 and n=3; n=4 follows by composing the n=2 construction.\n- If packings exist in dimensions m and n, they compose to one in dimension mn, settling all n=2^p 3^q.\n\n**Full solution or refutation.**\n\nThe full all-dimensional packing question is not solved. No modern primary paper changing the maintained construction/status summary was located.\n\n**What remains.**\n\nResolve n=5, the first unknown case, and then dimensions having a prime factor other than 2 or 3. State positivity of side lengths and permission to rotate boxes explicitly.\n\n**Sources checked.**\n\n- Dror Bar-Natan, Inequality of the Means, maintained University of Toronto author project page, accessed 2026-08-17. (authoritative_secondary): https://www.math.toronto.edu/~drorbn/projects/ArithGeom/\n Evidence used: The page gives the n=2,3,4 constructions, the multiplicative composition, and calls n=5 wide open and all remaining cases unknown.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3083, "problem_number": "OPG-37084", "title": "Edge-Colouring Geometric Complete Graphs", "statement": "Question What is the minimum number of colours such that every complete geometric graph on $n$ vertices has an edge colouring such that:\n\n\\item[Variant A] crossing edges get distinct colours, \\item[Variant B] disjoint edges get distinct colours, \\item[Variant C] non-disjoint edges get distinct colours, \\item[Variant D] non-crossing edges get distinct colours.", "background": "Source: Open Problem Garden. Original node ID: 37084. URL: http://www.openproblemgarden.org/op/edge_colouring_geometric_complete_graphs.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/edge_colouring_geometric_complete_graphs\n- Author(s): Hurtado, Ferran\n- Subject(s): Geometry\n- Keywords: geometric complete graph, colouring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: October 19th, 2009 by David Wood\n\nProblem-page discussion:\nLet $P$ be a set of $n$ points in the plane with no three collinear. Draw a straight line-segment between each pair of points in $P$. We obtain the complete geometric graph with vertex set $P$, denoted by $K_P$.\n\nTwo edges in $K_P$ are either:\n\n- adjacent if they have a vertex in common,\n- crossing if they intersect at a point in the interior of both edges.\n- disjoint if they do not intersect.\n\nLet $A(n)$, $B(n)$, $C(n)$ and $D(n)$ be the minimum number of colours for the four variants.\n\nVariant A: Here each colour class is a plane subgraph. Since there are point sets for which $\\frac{n}{2}$ edges are pairwise crossing, $A(n)\\geq\\frac{n}{2}$. For an upper bound, say $P=\\{v_1,\\dots,v_n\\}$. Colour each edge $v_iv_j$ with $i0$.\n\nVariant B: Here edges receiving the same colour must intersect. So each colour class is a geometric thrackle. Since there are point sets for which $\\frac{n}{2}$ edges are pairwise disjoint, $B(n)\\geq \\frac{n}{2}$. The $(n-1)$-colouring given in Variant A also works here. So $B(n)\\leq n-1$.\n\nConjecture. $B(n)\\leq (1-\\epsilon)n$ for some $\\epsilon>0$.\n\nVariant C: Here each colour class is a plane matching. So each colour class has at most $\\frac{n}{2}$ edges, and thus at least $n-1$ colours are always needed. Thus $C(n)\\geq n-1$. Araujo [ADHNU] proved an upper bound of $C(n)\\in O(n^{3/2})$.\n\nConjecture. $C(n)\\in O(n\\log n)$.\n\nStrong Conjecture. $C(n)\\in O(n)$.\n\nVariant D: (This variant was recently mentioned in [Mat].) Here edges receiving the same colour must cross. Each colour class is called a crossing family [ADHNU]. Every edge in any triangulation of $P$ requires its own colour. So if the convex hull of $P$ has only three points, then at least $3n-6$ colours are needed. Thus $D(n)\\geq 3n-6$.\n\nConjecture. A super-linear number of colours are always needed; i.e., $\\frac{D(n)}{n}\\rightarrow\\infty$ as $n\\rightarrow\\infty$.\n\nA better lower bound is obtained by taking $P$ in convex position. Then $\\Theta(n\\log n)$ is the minimum number of colours [KK]. I am not aware of any non-trivial upper bound for arbitrary point sets $P$.\n\nBibliography:\n[ADHNU] G. Araujo, A. Dumitrescu, F. Hurtado, M. Noy, J. Urrutia, On the chromatic number of some geometric type Kneser graphs, Computational Geometry: Theory & Applications 32(1):59–69, 2005 MathSciNet\n\n[BHRW] Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, David R. Wood. Partitions of complete geometric graphs into plane trees, Computational Geometry: Theory & Applications 34(2):116-125, 2006. MathSciNet\n\n[AEGKKPS] B. Aronov, P. Erdos, W. Goddard, D.J. Kleitman, M. Klugerman, J. Pach, L.J. Schulman, Crossing families, Combinatorica 14(2):127–134, 1994. MathSciNet\n\n[KK] Alexandr Kostochka and Jan Kratochvil. Covering and coloring polygon-circle graphs, Discrete Math. 163(1--3):299--305, 1997. MathSciNet\n\n[Mat] Jiří Matoušek. Blocking visibility for points in general position. Discrete Comput. Geom. 42(2):219--223, 2009. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2155418\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2222887\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1289067\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1428585\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2519877\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 31.\n\nAttempt notes:\nTarget:\nMake progress on \"Edge-Colouring Geometric Complete Graphs\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The four geometric edge-colouring functions have numerous bounds and exact special cases, but no verified source resolves all four asymptotically.\n\n**Verified partial progress.**\n\n- The maintained source records nontrivial bounds and separate conjectures for A(n), B(n), C(n), and D(n), including A(n) <= n-sqrt(n/12), C(n)=O(n^{3/2}), and convex-position information for D(n).\n- For Variant B, García-Davila, Leaños, Lomelí-Haro, and Ríos-Castro determine the maximum chromatic number of the segment-disjointness graph as n-2 for every 3 <= n <= 16.\n- Exact chromatic numbers are also known for convex point sets and double-chain configurations, but those do not determine the arbitrary-point-set extremum in general.\n\n**Full solution or refutation.**\n\nNo simultaneous determination of the four requested minima was verified.\n\n**What remains.**\n\nResolve the stated asymptotic conjectures for A, B, C, and D (or formulate and solve each variant separately), and determine the arbitrary-point-set maxima beyond the known exact regimes.\n\n**Sources checked.**\n\n- Open Problem Garden, Edge-Colouring Geometric Complete Graphs (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/edge_colouring_geometric_complete_graphs\n Evidence used: Defines the four extrema and records their distinct known bounds and conjectures.\n- J. García-Davila, J. Leaños, M. Lomelí-Haro, and L. M. Ríos-Castro, The Maximum Chromatic Number of the Disjointness Graph of Segments on n-point Sets in the Plane with n <= 16, arXiv:2303.17792 (2023). (primary): https://arxiv.org/abs/2303.17792\n Evidence used: Determines Variant B's worst-case disjointness-graph chromatic number for point sets through 16 vertices.\n- R. Fabila-Monroy, C. Hidalgo-Toscano, J. Leaños, and M. Lomelí-Haro, The Chromatic Number of the Disjointness Graph of the Double Chain, DMTCS 20 (2018). (primary): https://dmtcs.episciences.org/6209\n Evidence used: Gives an exact formula for an important non-convex point configuration.\n\n**Review notes.** The exact statement contains raw LaTeX item markup and is a four-part program; neither defect was silently repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3084, "problem_number": "OPG-37086", "title": "Partition of Complete Geometric Graph into Plane Trees", "statement": "Conjecture Every complete geometric graph with an even number of vertices has a partition of its edge set into plane (i.e. non-crossing) spanning trees.", "background": "Source: Open Problem Garden. Original node ID: 37086. URL: http://www.openproblemgarden.org/op/partition_of_complete_geometric_graph_into_plane_trees.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partition_of_complete_geometric_graph_into_plane_trees\n- Subject(s): Geometry\n- Keywords: complete geometric graph, edge colouring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 19th, 2009 by David Wood\n\nProblem-page discussion:\nFor a set $P$ of $n$ points in the plane with no three collinear, the complete geometric graph $K_P$ has vertex set $P$ and edge set consisting of the $\\binom{n}{2}$ straight line-segments between each pair of points in $P$.\n\nSince each subtree of $K_P$ has at most $n-1$ edges, every partition of $E(K_P)$ into subtrees has at least $\\frac{n}{2}$ parts. The conjecture asks for such a partition into exactly $\\frac{n}{2}$ subtrees, such that in addition, no two edges in each subtree cross.\n\nIt is folklore that the conjecture is true if $P$ is in convex partition. In fact, the edge set of the complete convex graph can be partitioned into plane Hamiltonian paths. Bose et al. [BHRW] characterised all possible partitions of the complete convex graph into plane spanning trees. Bose et al. [BHRW] also proved that every complete geometric graph on $n$ vertices can be partitioned into at most $n-\\sqrt{\\frac{n}{12}}$ plane subtrees.\n\nI heard about this conjecture from Ferran Hurtado in 2003, but the problem is much older than that.\n\nBibliography:\n[BHRW] Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, David R. Wood. Partitions of complete geometric graphs into plane trees, Computational Geometry: Theory & Applications 34(2):116-125, 2006. MathSciNet\n\nRelated:\nRelated problems\nEdge-Colouring Geometric Complete Graphs\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2222887\n\nComments:\n- January 6th, 2022 | Anonymous | This conjecture is false: This conjecture has recently been disproved, see arXiv:2108.05159 and arXiv:2112.08456.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Partition of Complete Geometric Graph into Plane Trees\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Bumpy-wheel complete geometric graphs provide counterexamples to the claimed partition into plane spanning trees.\n\n**Verified partial progress.**\n\n- Obenaus and Orthaber characterize which bumpy wheels do and do not admit such partitions.\n- Their obstruction is stronger in the negative cases: even a partition into the required number of arbitrary plane subgraphs is impossible.\n\n**Full solution or refutation.**\n\nThe universal conjecture is false.\n\n**What remains.**\n\nCharacterize the point configurations that do admit partitions into plane spanning trees and determine sharp replacement bounds.\n\n**Sources checked.**\n\n- J. Obenaus and J. Orthaber, Edge Partitions of Complete Geometric Graphs (Part 1), arXiv:2108.05159 (2021). (primary): https://arxiv.org/abs/2108.05159\n Evidence used: The abstract explicitly states that it disproves the conjecture and describes the bumpy-wheel counterexamples.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3085, "problem_number": "OPG-37286", "title": "Point sets with no empty pentagon", "statement": "Problem Classify the point sets with no empty pentagon.", "background": "Source: Open Problem Garden. Original node ID: 37286. URL: http://www.openproblemgarden.org/op/point_sets_with_no_empty_pentagon.\n\nSource subject path: Geometry.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/point_sets_with_no_empty_pentagon\n- Author(s): Wood, David R.\n- Subject(s): Geometry\n- Keywords: combinatorial geometry; visibility graph\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: December 14th, 2010 by David Wood\n\nProblem-page discussion:\nLet $P$ be a finite set of points in the plane (not necessarily in general position). Two points $x,y\\in P$ are visible if the line segment $xy$ contains no other point in $P$. The visibility graph of $P$ has vertex set $P$, where two vertices are adjacent if and only if they are visible. An empty pentagon in $P$ consists of 5 points in $P$ that are the vertices of a strictly convex pentagon whose interior contains none of the points in $P$.\n\nConsider the following three closely related classes of point sets:\n\n$A:=$ point sets with no empty pentagon (called a 5-hole),\n$B:=$ point sets with no 5 pairwise visible points,\n$C:=$ point sets whose visibility graph is 4-colourable.\n\nBy definition, $C \\subseteq B \\subseteq A$, and it is easy to show that $A \\neq B$ and $B \\neq C$.\n\nA key example of a point set in $C$ is the planar grid (intersected with a convex set so that it is finite): colour each grid point $(x,y)$ by $(x \\bmod 2, y \\bmod 2)$. If $(x,y)$ and $(v,w)$ receive the same colour then $|x-v|$ and $|y-w|$ are both even, and thus the midpoint of $(x,y)$ and $(v,w)$ is a blocker. Hence the visibility graph of the grid is 4-colourable. [This result and proof is folklore.] Many other examples of point sets in these classes can be found in the references.\n\nConsider the following open problems:\n\n- Classify the point sets in $A$, $B$, or $C$\n(i.e. list all examples; this is easy for point sets with no empty quadrilateral, or no 4 pairwise visible points).\n\n- Does the visibility graph of every point set in $A$ have bounded chromatic number?\n\n- Does the visibility graph of every point set in $A$ have bounded clique number?\n\n- Does the visibility graph of every point set in $B$ have bounded chromatic number?\n\nKára-Pór-Wood gave an example of a point set in $B$ with chromatic number $5$.\n\nBibliography:\nZ. Abel, B. Ballinger, P. Bose, S. Collette, V. Dujmovic, F. Hurtado, S. D. Kominers, S. Langerman, A. Pór, D. R. Wood. Every large point set contains many collinear points or an empty pentagon, Graphs and Combinatorics 27(1):47-60, 2011.\n\nEppstein, David. Happy endings for flip graphs. J. Computational Geometry 1(1):3-28, 2010. MathSciNet\n\nKára, Jan; Pór, Attila; Wood, David R. On the chromatic number of the visibility graph of a set of points in the plane. Discrete Comput. Geom. 34(3):497-506, 2005. MathSciNet\n\nPfender, Florian. Visibility graphs of point sets in the plane. Discrete Comput. Geom. 39 (2008), no. 1-3, 455–459. MathSciNet\n\nRabinowitz, Stanley. Consequences of the pentagon property. Geombinatorics 14:208-220, 2005.\n\nRelated:\nRelated problems\nBig Line or Big Clique in Planar Point Sets\n\nDiscussion links:\n- empty pentagon: http://en.wikipedia.org/wiki/http://en.wikipedia.org/wiki/Happy_Ending_problem\n- colourable: http://en.wikipedia.org/wiki/http://en.wikipedia.org/wiki/Graph_coloring\n\nBibliography links:\n- Every large point set contains many collinear points or an empty pentagon: http://dx.doi.org/10.1007/s00373-010-0957-2\n- Happy endings for flip graphs: http://jocg.org/index.php/jocg/article/view/21\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2469149\n- On the chromatic number of the visibility graph of a set of points in the plane: http://dx.doi.org/10.1007/s00454-005-1177-z\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2160051\n- Visibility graphs of point sets in the plane: http://dx.doi.org/10.1007/s00454-008-9056-z\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2383770\n- Consequences of the pentagon property: http://www.mathpropress.com/stan/bibliography/consequences.pdf\n\nComments:\n- February 1st, 2014 | David Wood | Problem 3 solved: Cibulka, Kyncl and Valtr have solved problem 3. They constructed point sets with no empty pentagon, and whose visibility graph has arbitrarily large clique number.\n\nJosef Cibulka, Jan Kyncl and Pavel Valtr: On planar point sets with the pentagon property, Proc. SoCG 2013, pp. 81-90. http://dl.acm.org/citation.cfm?id=2462406\n- March 25th, 2019 | Leonard Nguyen... | Problem 2 solved: Since the chromatic number is always equal or greater than the clique number, the chromatic number of the visibility graph of sets in A is also unbounded.\n\nCibulka, Kyncl and Valtr have also indirectly solved problem 2.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Point sets with no empty pentagon\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No complete classification of finite planar point sets with no empty pentagon was found, but local characterizations, construction schemes and sharp quantitative restrictions are known.\n\n**Verified partial progress.**\n\n- Cibulka, Kynčl and Valtr give equivalent local characterizations and construct examples whose visibility graphs contain arbitrarily large cliques.\n- Barát et al. prove that at least 328*l^2 points force an empty pentagon or l collinear points, optimal in order by the grid example.\n\n**Full solution or refutation.**\n\nThe subsidiary bounded-clique and bounded-chromatic questions for the no-5-hole class are answered negatively, but the requested classification is not complete.\n\n**What remains.**\n\nSpecify a workable equivalence/normal form for 'classify' and prove that all no-5-hole sets arise from a complete family of constructions.\n\n**Sources checked.**\n\n- J. Cibulka, J. Kynčl and P. Valtr, On planar point sets with the pentagon property, SoCG 2013, 81-90. (primary): https://doi.org/10.1145/2462356.2462406\n Evidence used: The work gives local characterizations and a construction scheme and produces arbitrary visibility cliques without 5-holes.\n- J. Barát et al., Empty pentagons in point sets with collinearities, SIAM J. Discrete Math. 29 (2015), 198-209; arXiv:1207.3633. (primary): https://arxiv.org/abs/1207.3633\n Evidence used: The abstract states the 328*l^2 theorem and the order-optimal grid construction.\n- UnsolvedMath, OPG-37286: Point sets with no empty pentagon (accessed 2026-08-17). (maintained_tracker): https://www.unsolvedmath.com/problems/OPG-37286\n Evidence used: The maintained problem entry remains open.\n\n**Review notes.** The word 'classify' supplies no equivalence relation, normal form, or completion criterion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3086, "problem_number": "OPG-37327", "title": "Covering a square with unit squares", "statement": "Conjecture For any integer $n \\geq 1$, it is impossible to cover a square of side greater than $n$ with $n^2+1$ unit squares.", "background": "Source: Open Problem Garden. Original node ID: 37327. URL: http://www.openproblemgarden.org/op/covering_a_square_with_unit_squares.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/covering_a_square_with_unit_squares\n- Subject(s): Geometry\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 18th, 2011 by Martin Erickson\n\nProblem-page discussion:\nAlexander Soifer in [S] raises the question of the smallest number $\\Pi (n)$ of unit squares that can cover a square of side $n+\\epsilon$. He shows the asymptotic upper bound $n^2+o(1)n+O(1)$, and the small values $\\Pi (1)=3$, $5 \\leq \\Pi (2) \\leq 7$, and $10 \\leq \\Pi (3) \\leq 14$. He conjectures the asymptotic lower bound $n^2+O(1)$.\n\nBibliography:\n[S] Soifer, Alexander, \"Covering a square of side n+epsilon with unit squares,\" J. of Combinatorial Theory, Series A 113 (2006):380-383.\n\nComments:\n- August 1st, 2011 | Carolus | A lower bound of the upper bound from polyomino-covering in [S]: (Using $e$ instead of $\\epsilon$ )\n\nIn [S], Soifer derives $\\Pi(n) < (n-k)^2+2(k+1)[[\\frac{k^2-1}{k^2+k-\\sqrt{2k+2}} n]]$.\n\nAs he mentioned, one can improve the covering construction. Holding the square of side length $n-k$ in the lower left corner, putting a square of side length $k$ in the upper right corner, covering the remaining uncovered area by 2 polyomino-coverings of rectangles of sides $n-k+e$ by $k+e$, removing useless unit squares in polyominos, we get a lower bound for the rhs of that inequality:\n\n$(n-k)^2+k^2+2[[(k+1)(n-k)\\frac{k^2-1}{k^2+k-\\sqrt{2k+2}} ]]$\n\nDenote by $U(n)$ the minimal value of this expression when varying $k$ from 2 to $n-2$.\n\nResults of computer calculations:\n\n$U(n)\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Recent exact small-N covering results verify the source conjecture for n=1 and n=2, while the family for n >= 3 remains unresolved.\n\n**Verified partial progress.**\n\n- S(2)=1 proves that two unit squares cannot cover a square of side greater than 1.\n- S(5)=2 proves that five unit squares cannot cover a square of side greater than 2.\n\n**Full solution or refutation.**\n\nThe cases n=1 and n=2 of S(n^2+1) <= n are proved; no all-n proof was verified.\n\n**What remains.**\n\nProve S(n^2+1) <= n for every n >= 3 or find a covering counterexample; the first unverified source case is S(10) <= 3.\n\n**Sources checked.**\n\n- G. Dósa, Z. Lángi and Z. Tuza, Covering a square by congruent squares, arXiv:2601.16535 (2026). (primary): https://arxiv.org/abs/2601.16535\n Evidence used: The paper defines S(N), proves S(2)=1 and S(5)=2, and states its further unresolved small-covering conjectures.\n\n**Review notes.** No numerical covering search was run.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3087, "problem_number": "OPG-37456", "title": "Convex uniform 5-polytopes", "statement": "Problem Enumerate all convex uniform 5-polytopes.", "background": "Source: Open Problem Garden. Original node ID: 37456. URL: http://www.openproblemgarden.org/op/convex_uniform_5_polytopes.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/convex_uniform_5_polytopes\n- Subject(s): Geometry\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: May 24th, 2012 by ACW\n\nProblem-page discussion:\nA reasonably complete discussion, with a list of all known examples, is at Uniform_5-polytope. The known examples form two infinite classes and 105 additional examples. All the known examples but one are \"Wythoffian\", and any unknown examples must be non-Wythoffian. The proposer believes that a talented undergraduate could put this to rest.\n\nDiscussion links:\n- Uniform_5-polytope: http://en.wikipedia.org/wiki/Uniform_5-polytope\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Convex uniform 5-polytopes\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No complete, proved enumeration of convex uniform 5-polytopes was verified; systematic Wythoffian lists and additional families are known, but completeness remains undemonstrated.\n\n**Verified partial progress.**\n\n- Wythoff's construction, as developed for polytopes, generates a large systematic class.\n- Current compilations list sporadic examples and infinite prismatic or duoprismatic families, without an exhaustiveness proof.\n\n**Full solution or refutation.**\n\nThe located literature supports substantial enumeration work, not a complete classification theorem.\n\n**What remains.**\n\nProve an exhaustive classification or identify further convex uniform 5-polytopes outside the existing lists.\n\n**Sources checked.**\n\n- G. C. Shephard, A Construction for Wythoffian Polytopes, Canadian Journal of Mathematics 6 (1954), 133-144, DOI 10.4153/CJM-1954-015-5. (primary): https://doi.org/10.4153/CJM-1954-015-5\n Evidence used: Provides the general Wythoffian construction underlying the principal systematic families, but not an exhaustive convex-uniform 5-polytope classification.\n- Open Problem Garden, Convex uniform 5-polytopes, accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/convex_uniform_5_polytopes\n Evidence used: Presents complete enumeration as an open problem and records the historically listed families.\n\n**Review notes.** Secondary sources disagree between 104 and 105 in a common finite count, apparently because of evolving lists or conventions; no count was silently selected.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3088, "problem_number": "OPG-56328", "title": "Partitioning the Projective Plane", "statement": "Throughout this post, by projective plane we mean the set of all lines through the origin in $\\mathbb{R}^3$.\n\nDefinition Say that a subset $S$ of the projective plane is octahedral if all lines in $S$ pass through the closure of two opposite faces of a regular octahedron centered at the origin.\n\nDefinition Say that a subset $S$ of the projective plane is weakly octahedral if every set $S'\\subseteq S$ such that $|S'|=3$ is octahedral.\n\nConjecture Suppose that the projective plane can be partitioned into four sets, say $S_1,S_2,S_3$ and $S_4$ such that each set $S_i$ is weakly octahedral. Then each $S_i$ is octahedral.", "background": "Source: Open Problem Garden. Original node ID: 56328. URL: http://www.openproblemgarden.org/op/partitioning_the_projective_plane.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partitioning_the_projective_plane\n- Author(s): Noel, Jonathan A.\n- Subject(s): Geometry\n- Keywords: Partitioning; projective plane\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: August 27th, 2013 by Jon Noel\n\nProblem-page discussion:\nAlso, see the posting on mathoverflow.\n\nThere is an equivalent definition of the \"weakly octahedral\" condition which may be useful.\n\nLemma A subset $S$ of the projective plane is weakly octahedral if for any three lines in $S$ and any three vectors $x, y$ and $z$ which span these lines, we have $$\\langle x,y\\rangle \\cdot\\langle x,z\\rangle \\cdot\\langle y,z\\rangle \\geq 0$$where$\\langle\\cdot,\\cdot\\rangle$is the standard (dot) inner product on$\\mathbb{R}^3$.\n\nThe fact that $S_1,S_2,S_3$ and $S_4$ partition the projective plane seems to be important. Here is an example of a weakly octahedral set that is not octahedral: Fix any vector $x$ and let $S$ be the set of all lines which are spanned by vectors which meet $x$ at an angle strictly less than $\\frac{\\pi}{4}$.\n\nThis question came up while working on another problem posted to this site: Circular colouring the orthogonality graph. It is possible that a solution to the problem stated here can be applied to solve this problem. Moreover, it may be useful in proving that the real orthogonality graph (defined in the other posting) has (essentially) only one proper $4$-colouring.\n\nRelated:\nRelated problems\nCircular colouring the orthogonality graph\nThe Double Cap Conjecture\n\nDiscussion links:\n- posting: http://mathoverflow.net/questions/140413/partitioning-the-projective-plane\n- Circular colouring the orthogonality graph: http://www.openproblemgarden.org/?q=op/circular_colouring_the_orthogonality_graph\n- posting: http://www.openproblemgarden.org/?q=op/circular_colouring_the_orthogonality_graph\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Partitioning the Projective Plane\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No primary paper or authoritative later status source resolving the weakly-octahedral four-partition conjecture was found; the original postings remain unanswered.\n\n**Verified partial progress.**\n\n- The source gives an equivalent dot-product sign characterization of weak octahedrality.\n- It also gives weakly octahedral sets that are not globally octahedral, showing that the four-set partition hypothesis is essential.\n\n**Full solution or refutation.**\n\nNo proof or counterexample was verified. Because the terminology appears confined to the original 2013 postings, the record is status-uncertain rather than confidently classified open.\n\n**What remains.**\n\nLocate a later treatment or prove that four locally sign-consistent classes covering the real projective plane each admit a single global octahedral frame.\n\n**Sources checked.**\n\n- J. Noel, Partitioning the Projective Plane, MathOverflow question 140413 (2013). (primary): https://mathoverflow.net/questions/140413/partitioning-the-projective-plane\n Evidence used: Original author posting of the conjecture and its equivalent dot-product formulation; no solution is recorded.\n- Open Problem Garden, Partitioning the Projective Plane (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/partitioning_the_projective_plane\n Evidence used: Retains the exact conjecture and motivating context without a recorded resolution.\n\n**Review notes.** The source explicitly defines projective plane as RP^2 and supplies an algebraic sign criterion; no wording was normalized.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3089, "problem_number": "OPG-59888", "title": "Dirac's Conjecture", "statement": "Conjecture For every set $P$ of $n$ points in the plane, not all collinear, there is a point in $P$ contained in at least $\\frac{n}{2}-c$ lines determined by $P$, for some constant $c$.", "background": "Source: Open Problem Garden. Original node ID: 59888. URL: http://www.openproblemgarden.org/op/diracs_conjecture.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/diracs_conjecture\n- Author(s): Dirac, Gabriel\n- Subject(s): Geometry\n- Keywords: point set\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: January 22nd, 2014 by David Wood\n\nProblem-page discussion:\nIn 1983, Beck[B], and independently Szemerédi and Trotter [ST], proved that for every set $P$ of $n$ points in the plane, not all collinear, there is a point in $P$ contained in at least $\\frac{n}{c}$ lines determined by $P$, for some large unspecified constant $c$. Payne and Wood [PW] proved this result with $c=37$. Han [Han] improved this to $c=3$.\n\nBibliography:\n[B] Jozsef Beck On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry. Combinatorica, 3(3-4):281–297, 1983.\n\n*[D] Gabriel A. Dirac. Collinearity properties of sets of points. Quart. J. Math., Oxford Ser. (2), 2:221–227, 1951. MR: 0043485.\n\n[PW] Michael S. Payne and David R. Wood. Progress on Dirac's Conjecture. Electronic J. Combinatorics 21.2:P2.12, 2014. arXiv:1207.3594.\n\n[ST] Endre Szemerédi and William T. Trotter, Jr., Extremal problems in discrete geometry. Combinatorica 3.3-5:381-392, 1983.\n\n[Han] Zeye Han. A Note on Weak Dirac Conjecture, Electronic J. Combinatorics 24.1:P1.63, 2017.\n\nBibliography links:\n- Collinearity properties of sets of points: http://www.doi.org/10.1093/qmath/2.1.221\n- Progress on Dirac's Conjecture: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i2p12\n- arXiv:1207.3594: http://www.arxiv.org/abs/1207.3594\n- A Note on Weak Dirac Conjecture: https://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p63\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Dirac's Conjecture\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The strong Dirac n/2-O(1) incidence conjecture remains open; the best verified general coefficient in the checked literature is one third.\n\n**Verified partial progress.**\n\n- Payne and Wood prove an n/37 incident-line lower bound.\n- Han improves the guarantee to ceiling(n/3)+1 lines through some point.\n\n**Full solution or refutation.**\n\nNo proof of an absolute-additive-error n/2 bound was verified. Results on ordinary lines and on the allowable-sequence Dirac-Goodman-Pollack conjecture are distinct problems.\n\n**What remains.**\n\nRaise the universal coefficient from one third to one half up to an absolute additive constant.\n\n**Sources checked.**\n\n- Z. Han, A Note on the Weak Dirac Conjecture, Electronic Journal of Combinatorics 24 (2017), P1.63. (primary): https://doi.org/10.37236/6688\n Evidence used: Proves every noncollinear n-point set has a point incident to at least ceiling(n/3)+1 determined lines.\n- M. S. Payne and D. R. Wood, Progress on Dirac's Conjecture, arXiv:1207.3594; Electronic Journal of Combinatorics 21 (2014), P2.12. (primary): https://arxiv.org/abs/1207.3594\n Evidence used: States the strong conjecture and proves the earlier n/37 weakening.\n- Open Problem Garden, Dirac's Conjecture (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/diracs_conjecture\n Evidence used: Retains the n/2-c conjecture and records Han's one-third advance.\n\n**Review notes.** The intended c is one absolute constant independent of n and P; the displayed English leaves quantifier order implicit and was flagged rather than rewritten.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3090, "problem_number": "OPG-59914", "title": "General position subsets", "statement": "Question What is the least integer $f(n)$ such that every set of at least $f(n)$ points in the plane contains $n$ collinear points or a subset of $n$ points in general position (no three collinear)?", "background": "Source: Open Problem Garden. Original node ID: 59914. URL: http://www.openproblemgarden.org/op/general_position_subsets.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/general_position_subsets\n- Author(s): Gowers, Timothy\n- Subject(s): Geometry\n- Keywords: general position subset, no-three-in-line problem\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 24th, 2014 by David Wood\n\nProblem-page discussion:\nThe $n\\times n$ grid contains no set of $n+1$ collinear points and no subset of $2n+1$ points in general position, implying $f(n)\\geq \\Omega(n^2)$.\n\nTo see that $f(n)\\leq O(n^3)$, consider a set $P$ of points that contain no $n$ collinear points, and contain no subset of $n$ points in general position. Let $S$ be a maximal subset of $P$ in general position. Every point in $P-S$ is on one of the $\\binom{|S|}{2}$ lines determined by $S$. Each such line contains at most $n-3$ points in $P-S$. Thus $|P|\\leq |S|+\\binom{|S|}{2}(n-3) \\leq (n-1)+\\binom{n-1}{2}(n-3)\\leq O(n^3)$.\n\nPayne and Wood [PW] improved this upper bound to $f(n)\\leq O(n^2\\log n)$. The proof is based on the Szemerédi-Trotter Theorem and Spencer's Lemma about independent sets in hypergraphs.\n\nIt is reasonable to think that the grid is the extremal example, and $f(n)\\leq O(n^2)$. This would be an elegant generalisation of a result by Erdős [R] who proved that the $n\\times n$ grid contains a subset of $n-o(n)$ points in general position (the no-three-in-line problem).\n\nBibliography:\n*[G] Timothy Gowers, A geometric Ramsey problem.\n\n[PW] Michael Payne, David R. Wood. On the general position subset selection problem, SIAM J. Discrete Math. 27.4:1727-1733, 2013.\n\n[R] K. F. Roth, On a problem of Heilbronn, J. London Mathematical Society 26.3:198–204, 1951.\n\nDiscussion links:\n- no-three-in-line problem: http://en.wikipedia.org/wiki/No-three-in-line_problem\n\nBibliography links:\n- A geometric Ramsey problem: http://mathoverflow.net/questions/50928/a-geometric-ramsey-problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"General position subsets\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For every target n a finite f(n) exists, but determining the sharp quantitative growth of the general-position subset selection number remains open.\n\n**Verified partial progress.**\n\n- Cardinal--Toth--Wood prove that every set of c q^2 log q planar points contains q collinear points or q points in general position.\n\n**Full solution or refutation.**\n\nThis supplies an explicit existence upper bound for the source's least integer, not its exact value.\n\n**What remains.**\n\nDetermine the correct asymptotic order/exact values of f(n).\n\n**Sources checked.**\n\n- J. Cardinal, G. Toth and D. R. Wood, General position subsets and independent hyperplanes in d-space, Discrete & Computational Geometry 56 (2016); arXiv:1410.3637. (primary): https://arxiv.org/abs/1410.3637\n Evidence used: States the planar c q^2 log q selection theorem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3091, "problem_number": "OPG-59923", "title": "Generalised Empty Hexagon Conjecture", "statement": "Conjecture For each $\\ell\\geq3$ there is an integer $f(\\ell)$ such that every set of at least $f(\\ell)$ points in the plane contains $\\ell$ collinear points or an empty hexagon.", "background": "Source: Open Problem Garden. Original node ID: 59923. URL: http://www.openproblemgarden.org/op/generalised_empty_hexagon_conjecture.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/generalised_empty_hexagon_conjecture\n- Author(s): Wood, David R.\n- Subject(s): Geometry\n- Keywords: empty hexagon\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 26th, 2014 by David Wood\n\nProblem-page discussion:\nHere an empty hexagon in a set of points $P$ consists of a subset $S\\subseteq P$ of six points in convex position with no other point in $P$ in the convex hull of $S$. The $\\ell=3$ case of the conjecture (that is, for point sets in general position) was an outstanding open problem for many years, until its solution by Gerken [G] and Nicolas [N]. Valtr [V] found a simple proof.\n\nBibliography:\n[G] Tobias Gerken. Empty Convex Hexagons in Planar Point Sets, Discrete Comput Geom (2008) 39:239–272, MathSciNet\n\n[N] Carlos M. Nicolas. The Empty Hexagon Theorem, Discrete Comput Geom 38:389–397 (2007), MathSciNet.\n\n[V] Pavel Valtr, On Empty Hexagons, in: J. E. Goodman, J. Pach, and R. Pollack, Surveys on Discrete and Computational Geometry, Twenty Years Later, Contemp. Math. 453, AMS, 2008, pp. 433-441.\n\nRelated:\nRelated problems\nBig Line or Big Clique in Planar Point Sets\nErdös-Szekeres conjecture\n\nBibliography links:\n- Empty Convex Hexagons in Planar Point Sets: http://dx.doi.org/10.1007/s00454-007-9018-x\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2383761\n- The Empty Hexagon Theorem: http://dx.doi.org/10.1007/s00454-007-1343-6\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2343313\n- On Empty Hexagons: http://kam.mff.cuni.cz/%7Evaltr/h.ps\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Generalised Empty Hexagon Conjecture\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The l=3 (general-position) empty-hexagon case is proved, while the stated all-l collinearity-or-empty-hexagon extension remains open.\n\n**Verified partial progress.**\n\n- Gerken and Nicolas independently proved that sufficiently large point sets in general position contain an empty hexagon.\n\n**Full solution or refutation.**\n\nThe known l=3 theorem does not address arbitrary bounded collinearity l.\n\n**What remains.**\n\nProve the finite f(l) statement for every l>=4 or find a counterexample.\n\n**Sources checked.**\n\n- T. Gerken, Empty convex hexagons in planar point sets, Discrete & Computational Geometry 39 (2008), 239--272. (primary): https://doi.org/10.1007/s00454-007-9008-x\n Evidence used: Proves the empty-hexagon theorem for point sets in general position.\n- Open Problem Garden, Generalised Empty Hexagon Conjecture (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/generalised_empty_hexagon_conjecture\n Evidence used: Retains the general l statement as a conjecture while noting the l=3 theorem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3092, "problem_number": "OPG-59984", "title": "Chromatic number of associahedron", "statement": "Conjecture Associahedra have unbounded chromatic number.", "background": "Source: Open Problem Garden. Original node ID: 59984. URL: http://www.openproblemgarden.org/op/chromatic_number_of_associahedron.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/chromatic_number_of_associahedron\n- Author(s): Fabila-Monroy, Ruy; Flores-Penaloza, David; Huemer, Clemens; Hurtado, Ferran; Urrutia, Jorge; Wood, David R.\n- Subject(s): Geometry\n- Keywords: associahedron, graph colouring, chromatic number\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 2nd, 2015 by David Wood\n\nProblem-page discussion:\nAn associahedron is the convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a fixed word and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a convex polygon and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal.\n\nThe chromatic number of (the 1-skeleton of) associahedra was first considered by Fabila-Monroy et al [FFHHUW]. They proved that for the associahedron corresponding to edge flips in triangulations of a convex $n$-gon, the chromatic number is at most $\\ceil{n/2}$ and at most $O(n/\\log n)$. The best known lower bound is $4$ for $n=10$ [private communication, Ruy Fabila-Monroy].\n\nUpdate (2019): Addario Berry et al. [ARSW] proved an upper bound of $O(\\log n)$.\n\nBibliography:\n*[FFHHUW] Ruy Fabila-Monroy, David Flores-Penaloza, Clemens Huemer, Ferran Hurtado, Jorge Urrutia, David R. Wood. On the Chromatic Number of some Flip Graphs, Discrete Mathematics and Theoretical Computer Science Vol 11, No 2 (2009).\n\n[ARSW] Louigi Addario Berry, Bruce Reed, Alex Scott, David R. Wood. A logarithmic bound for the chromatic number of the associahedron, 2018.\n\nDiscussion links:\n- associahedron: http://en.wikipedia.org/wiki/associahedron\n\nBibliography links:\n- On the Chromatic Number of some Flip Graphs: http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/1026\n- A logarithmic bound for the chromatic number of the associahedron: http://www.arxiv.org/abs/1811.08972\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Chromatic number of associahedron\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Unboundedness of the chromatic numbers of associahedron 1-skeleta remains open. The best general upper bound is logarithmic, while the only verified constant lower bound located is four.\n\n**Verified partial progress.**\n\n- Addario-Berry--Reed--Scott--Wood prove chi(A_n)=O(log n).\n- Cioabă--Gupta confirm chi(A_n)=3 for 5<=n<=9 and chi(A_10)=4.\n- They prove the smallest adjacency eigenvalue has linear magnitude, showing that the straightforward Hoffman eigenvalue route does not yield an unbounded lower bound.\n\n**Full solution or refutation.**\n\nNo chromatic lower bound tending to infinity with n was verified.\n\n**What remains.**\n\nFind any unbounded lower bound for the flip graph of convex-polygon triangulations, or construct a uniform coloring with a fixed number of colors.\n\n**Sources checked.**\n\n- Louigi Addario-Berry, Bruce Reed, Alex Scott, and David R. Wood, A logarithmic bound for the chromatic number of the associahedron, arXiv:1811.08972. (primary): https://arxiv.org/abs/1811.08972\n Evidence used: Proves the O(log n) upper bound for associahedron graphs.\n- Sebastian M. Cioabă and Vishal Gupta, A lower bound for the smallest eigenvalue of a graph and an application to the associahedron graph, Bulletin of the Romanian Mathematical Society 65 (2022), 393-404; arXiv:2210.08516. (primary): https://arxiv.org/abs/2210.08516\n Evidence used: States unbounded chromatic number remains open, confirms small exact values, and proves the relevant eigenvalue has linear order.\n- Open Problem Garden, Chromatic number of associahedron. (maintained_tracker): https://www.openproblemgarden.org/op/chromatic_number_of_associahedron\n Evidence used: Preserves the conjecture and records the logarithmic upper-bound update.\n\n**Review notes.** The source means the 1-skeleton/flip graph, as explained in its background; the one-line conjecture alone does not state the indexing convention.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3093, "problem_number": "OPG-60010", "title": "Convex Equipartitions with Extreme Perimeter", "statement": "To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.\n\nRemark: It appears maximizing the total perimeter is the easier problem.", "background": "Source: Open Problem Garden. Original node ID: 60010. URL: http://www.openproblemgarden.org/op/convex_equipartitions_with_extreme_perimeter.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/convex_equipartitions_with_extreme_perimeter\n- Author(s): Nandakumar, R., Amrita School of Arts and Science, Kochi 682024, India\n- Subject(s): Geometry\n- Keywords: convex equipartition\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: October 22nd, 2015 by Nandakumar\n\nProblem-page discussion:\nConjecture 1: It appears that for convex equipartition with maximum total cut length, the cut lines should not meet in the interior of C.\n\nConjecture 1a: for n=3, it appears conjecture 1 can be proved by proving the following subconjecture: If from a point P in the interior of C, three rays originate and divide C into three equal area pieces and if p_0 is the sum of the perimeters of the three pieces, then from at least one of the three points (call them A, B, C) where the three rays from P cut the boundary of C, there originate 2 rays which equipartition region C into three equal area pieces such that the perimeter sum of the three pieces is necessarily greater than p_0.\n\nConjecture 2: If conjecture 1 holds, one could have a greedy algorithm to achieve maximum perimeter sum as follows:\n\n- From C, first cut out a convex region with 1/n of the total area such that the separating cut is the longest possible, then repeat the same process on the remaining piece and so on until we have n equal area pieces) appears to give the optimal answer\n\nGeneralizations: One could think of partitioning a convex regions into convex pieces with specified different areas and minimize/maximize the total perimter. Higher dimensional analgs too could be considered.\n\nBibliography:\nOnline Reference (the only one known to the author): http://nandacumar.blogspot.in/2015/08/another-convex-equi-partition-problem.html (*)\n\nRelated:\nRelated problems\nConvex 'Fair' Partitions Of Convex Polygons\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Convex Equipartitions with Extreme Perimeter\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exact minimum-total-perimeter equal-area bisection is algorithmically solved for convex polygons, and equal-area/equal-perimeter convex partitions exist for every number of parts, but the source's general extrema and proposed maximum structure are not settled.\n\n**Verified partial progress.**\n\n- Koutsoupias--Papadimitriou--Sideri give a quadratic exact algorithm for minimum-total-perimeter equal-area bisection of a convex polygon.\n- Akopyan--Avvakumov--Karasev prove that every planar convex body admits an m-part convex partition with equal areas and equal perimeters for every m>=2.\n- The equal-area/equal-perimeter theorem is a feasibility result and does not optimize the perimeter sum under only the equal-area constraint.\n\n**Full solution or refutation.**\n\nThe n=2 convex-polygon minimum case and a strong related feasibility problem are solved; no general maximum/minimum solution for arbitrary C and n was verified.\n\n**What remains.**\n\nPrecisely define admissible partitions and perimeter counting, then settle the maximum problem, the nonconcurrent-cut and greedy conjectures, and the general minimum problem for n>2.\n\n**Sources checked.**\n\n- Elias Koutsoupias, Christos H. Papadimitriou, and Martha Sideri, On the Optimal Bisection of a Polygon, ORSA Journal on Computing 4 (1992), 435--438. (primary): https://doi.org/10.1287/ijoc.4.4.435\n Evidence used: The abstract states that the exact equal-area minimum-total-perimeter problem is solvable in quadratic time for convex polygons.\n- Arseniy Akopyan, Sergey Avvakumov, and Roman Karasev, Convex fair partitions into an arbitrary number of pieces, Advances in Mathematics 493 (2026), 110927. (primary): https://doi.org/10.1016/j.aim.2026.110927\n Evidence used: Proves equal-area and equal-perimeter convex partitions for every number of pieces.\n- Open Problem Garden, Convex Equipartitions with Extreme Perimeter (checked 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/convex_equipartitions_with_extreme_perimeter\n Evidence used: Retains the maximum/minimum problem and its two structural conjectures without a posted resolution.\n\n**Review notes.** The record does not specify polygon versus body, interface regularity, double-counting of common boundaries, or attainment versus infimum/supremum.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3094, "problem_number": "OPG-60037", "title": "Edge-Unfolding Convex Polyhedra", "statement": "Conjecture Every convex polyhedron has a (nonoverlapping) edge unfolding.", "background": "Source: Open Problem Garden. Original node ID: 60037. URL: http://www.openproblemgarden.org/op/edge_unfolding_convex_polyhedra.\n\nSource subject path: Geometry.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/edge_unfolding_convex_polyhedra\n- Author(s): Shephard, G.C.\n- Subject(s): Geometry\n- Keywords: folding; nets\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 18th, 2019 by Erik Demaine\n\nProblem-page discussion:\nAn edge unfolding of a convex polyhedron consists of cutting along a spanning tree of the polyhedron's edges, then unfolding the remaining edges to bring all faces into a plane, without faces overlapping each other (thus resulting in a simple planar polygon).\n\nThe first explicit posing of this problem seems to be Shephard's 1975 paper [S], though the idea of edge unfolding goes back to Albrecht Dürer in 1525 [D]. See [DO] (Part III) for a survey.\n\nBibliography:\n[DO] Erik D. Demaine, Joseph O'Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, 2007.\n\n[D] Albrecht Dürer, The Painter’s Manual: A Manual of Measurement of Lines, Areas, and Solids by Means of Compass and Ruler Assembled by Albrecht Dürer for the Use of all Lovers of Art with Appropriate Illustrations Arranged to be Printed in the Year MDXXV, Abaris Books, New York, 1977. English translation by Walter L. Strauss of “Unterweysung der Messung mit dem Zirkel un Richtscheyt in Linien Ebnen uhnd Gantzen Corporen”, 1525.\n\n*[S] Geoffrey C. Shephard, Convex polytopes with convex nets, Math. Proc. Camb. Phil. Soc. 78 (1975), 389–403.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Edge-Unfolding Convex Polyhedra\" in Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Dürer's conjecture that every convex polyhedron has a nonoverlapping edge unfolding remains open.\n\n**Verified partial progress.**\n\n- Numerous special polyhedral classes are known edge-unfoldable; no all-convex-polyhedra theorem or counterexample is verified.\n\n**Full solution or refutation.**\n\nThe general edge-only unfolding assertion is unresolved.\n\n**What remains.**\n\nProve a nonoverlapping edge net exists for every convex polyhedron or construct a counterexample.\n\n**Sources checked.**\n\n- TOPP, Problem 9: Edge-Unfolding Convex Polyhedra (accessed 2026-08-17). (maintained_tracker): https://topp.openproblem.net/p9\n Evidence used: States the general question as open.\n- Net (polyhedron) overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Net_%28polyhedron%29\n Evidence used: Identifies this as the unresolved Dürer edge-unfolding conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3095, "problem_number": "OPG-1768", "title": "Jacobian Conjecture", "statement": "Conjecture Let $k$ be a field of characteristic zero. A collection $f_1,\\ldots,f_n$ of polynomials in variables $x_1,\\ldots,x_n$ defines an automorphism of $k^n$ if and only if the Jacobian matrix is a nonzero constant.", "background": "Source: Open Problem Garden. Original node ID: 1768. URL: http://www.openproblemgarden.org/op/jacobian_conjecture.\n\nSource subject path: Geometry > Algebraic Geometry.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/jacobian_conjecture\n- Author(s): Keller, Otto-Heinrich\n- Subject(s): Geometry; Algebraic Geometry\n- Keywords: Affine Geometry; Automorphisms; Polynomials\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 6th, 2008 by Charles\n\nProblem-page discussion:\nThe Jacobian determinant is the determinant of the matrix $A$ with $a_{ij}=\\frac{\\partial f_i}{\\partial x_j}$. It is elementary to show that if the map $F:k^n\\to k^n$ is an automorphism, then the Jacobian determinant is a nonzero constant, by using the inverse map. The other direction has turned out to be rather difficult.\n\nIt is known that the Conjecture holds for polynomials of degree 2, and that the general case follows from a special case in degree 3.\n\nComments:\n- March 4th, 2021 | Anonymous | Problem statement: iff the determinant of the Jacobian matrix is a nonzero constant.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Jacobian Conjecture\" in Geometry; Algebraic Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** An explicit dimension-three polynomial map with nonzero constant Jacobian and noninvertible global behavior refutes the stated characteristic-zero Jacobian conjecture.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe field-wide assertion is false in dimension three and hence in all higher dimensions; dimension two remains open.\n\n**What remains.**\n\nThe original universal statement is refuted; settle the remaining two-dimensional refinement.\n\n**Sources checked.**\n\n- T. Tao, A digestion of the Jacobian conjecture counterexample (2026). (authoritative_secondary): https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/\n Evidence used: States and explains an explicit C^3 constant-Jacobian noninvertible map.\n- Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222 (2026). (primary): https://arxiv.org/abs/2608.00222\n Evidence used: Records the dimension-three refutation and higher-dimensional consequences.\n\n**Review notes.** No source alteration; lower-dimensional distinction recorded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3096, "problem_number": "OPG-1803", "title": "The Hodge Conjecture", "statement": "Conjecture Let $X$ be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of $X$.", "background": "Source: Open Problem Garden. Original node ID: 1803. URL: http://www.openproblemgarden.org/op/the_hodge_conjecture.\n\nSource subject path: Geometry > Algebraic Geometry.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_hodge_conjecture\n- Author(s): Hodge, W. V. D.\n- Subject(s): Geometry; Algebraic Geometry\n- Keywords: Hodge Theory; Millenium Problems\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 13th, 2008 by Charles\n\nProblem-page discussion:\nA complex projective variety is the set of zeros of a finite collection of homogeneous polynomials on projective space, and we are concerned with the singular cohomology ring. There is a well known Hodge Decomposition of the cohomology into groups $H^{p,q}(X.\\mathbb{C})$ which hare holomorphic in $p$ variables and antiholomorphic in $q$ variables with the property that $\\oplus_{p+q=k}H^{p,q}=H^k$.\n\nSo we define the Hodge classes to be those in the intersection $H^{k,k}(X,\\mathbb{C})\\cap H^{2k}(X,\\mathbb{Q})$. It is fairly easy to show that the cohomology class of a subvariety is Hodge. We say that a cycle is algebraic if it is a rational linear combination of the classes of subvarieties. So every algebraic cycle is Hodge. In dimension one, we have the following result:\n\nTheorem (Lefshetz (1,1) Theorem) Any element of $H^2(X,\\mathbb{Q})\\cap H^{1,1}$ is the cohomology class of a divisor, and so is algebraic.\n\nIt's also true that if the Hodge Conjecture holds for cycles of degree $p2n-p$. So this and the (1,1) Theorem show that the Hodge Conjecture is true for complex curves, surfaces and threefolds.\n\nBibliography:\n*[Hod] Hodge, W. V. D. \"The topological invariants of algebraic varieties\". Proceedings of the International Congress of Mathematicians, Cambridge, MA, 1950, vol. 1, pp. 181–192.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"The Hodge Conjecture\" in Geometry; Algebraic Geometry, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The standard rational Hodge conjecture for smooth complex projective varieties remains open, though known in dimension below four and many other special cases; the imported statement omits the essential smoothness hypothesis.\n\n**Verified partial progress.**\n\n- The Lefschetz (1,1) theorem settles codimension-one classes.\n- The Clay Mathematics Institute records the conjecture as known for varieties of dimension less than four but unknown in dimension four.\n- Dan and Kaur prove cohomological and homological singular variants for certain odd-dimensional hypersurfaces with A_n singularities.\n\n**Full solution or refutation.**\n\nThe intended smooth-projective Millennium problem is unsolved; the literal singular-inclusive wording is underspecified rather than a clean statement of that problem.\n\n**What remains.**\n\nProve or refute the rational Hodge conjecture for every smooth complex projective variety; separately specify and settle the intended singular version if singular varieties are to remain in scope.\n\n**Sources checked.**\n\n- Clay Mathematics Institute, Hodge Conjecture, current Millennium Problem status page, accessed 2026-08-17. (authoritative_secondary): https://www.claymath.org/millennium/hodge-conjecture/\n Evidence used: Labels the conjecture unsolved and states that it is known in dimension less than four but unknown in dimension four.\n- Ananyo Dan and Inder Kaur, Hodge Conjecture via Singular Varieties, arXiv:2509.25273 (2025). (primary): https://arxiv.org/abs/2509.25273\n Evidence used: Explicitly distinguishes cohomological and homological Hodge conjectures for singular varieties and proves them for a special class.\n\n**Review notes.** Serious formulation defect: the classical conjecture requires X smooth projective. The background also contains OCR/typographical defects including H^{p,q}(X.C), 'hare', 'Lefshetz', and 'Millenium'.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3097, "problem_number": "OPG-316", "title": "Fat 4-polytopes", "statement": "The fatness of a 4-polytope $P$ is defined to be $(f_1 + f_2)/(f_0 + f_3)$ where $f_i$ is the number of faces of $P$ of dimension $i$.\n\nQuestion Does there exist a fixed constant $c$ so that every convex 4-polytope has fatness at most $c$?", "background": "Source: Open Problem Garden. Original node ID: 316. URL: http://www.openproblemgarden.org/op/fat_4_polytopes.\n\nSource subject path: Geometry > Polytopes.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/fat_4_polytopes\n- Author(s): Eppstein, David; Kuperberg, Greg; Ziegler, Gunter M.\n- Subject(s): Geometry; Polytopes\n- Keywords: f-vector; polytope\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 12th, 2007 by mdevos\n\nProblem-page discussion:\nThe $f$-vector of a $d$-dimensional polytope $P$ is the vector $(f_0,f_1,\\ldots,f_{d-1})$ where $f_i$ is the number of faces of dimension $i$. Let us denote by ${\\mathcal F}_d$ the collection of all $f$-vectors of convex $d$-dimensional polytopes. Steinitz proved that the set ${\\mathcal F}_3$ is completely characterized by the following three conditions:\n\n- $f_0 - f_1 + f_2 = 2$,\n- $f_2 \\le 2f_0 - 4$,\n- $f_0 \\le 2f_2 - 4$.\n\nThe first of these conditions is Euler's formula. The second and third are easy inequalities which are tight for simplicial (all faces triangles) and simple (all vertices of degree 3) polytopes, respectively.\n\nIn sharp contrast to this, the situation for ${\\mathcal F}_4$ seems to be quite complicated. For instance, it has been shown that ${\\mathcal F}_4$ does not contain all elements of ${\\mathbb Z}^4$ which lie in the convex hull of ${\\mathcal F}_4$; i.e., ${\\mathcal F}_4$ has \"holes\" in it. For the extreme examples of simple and simplicial polytopes, the $g$-theorem of Billera-Lee and Stanley gives a complete description of all possible $f$-vectors, but in general very little is known.\n\nSource links:\n- polytope: http://en.wikipedia.org/wiki/polytope\n\nDiscussion links:\n- Euler's formula: http://en.wikipedia.org/wiki/Euler's formula\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Fat 4-polytopes\" in Geometry; Polytopes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No universal upper bound on fatness of convex 4-polytopes was verified.\n\n**Verified partial progress.**\n\n- Known constructions show substantial fatness but do not establish unboundedness or a global bound.\n\n**Full solution or refutation.**\n\nThe bounded-fatness question remains open.\n\n**What remains.**\n\nConstruct an unbounded family or prove a universal inequality.\n\n**Sources checked.**\n\n- Open Problem Garden, node 316 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the fatness question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3098, "problem_number": "OPG-610", "title": "Continous analogue of Hirsch conjecture", "statement": "Conjecture The order of the largest total curvature of the primal central path over all polytopes defined by $n$ inequalities in dimension $d$ is $n$.", "background": "Source: Open Problem Garden. Original node ID: 610. URL: http://www.openproblemgarden.org/op/continous_analogue_of_hirsch_conjecture.\n\nSource subject path: Geometry > Polytopes.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/continous_analogue_of_hirsch_conjecture\n- Author(s): Deza, Antoine; Terlaky, Tamas; Zinchenko, Yuriy\n- Subject(s): Geometry; Polytopes\n- Keywords: curvature; polytope\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 30th, 2007 by deza\n\nProblem-page discussion:\nLet $\\lambda^c(P)$ denote the total curvature of the central path corresponding to the linear optimization problem $\\min \\{ c^Tx: x\\in P\\}$. The quantity $\\lambda^c(P)$ can be regarded as the continuous analogue of the edge-length of the shortest path between a pair of vertices. Considering the largest $\\lambda^c(P)$ over all possible $c$ we obtain the quantity $\\lambda(P)$, referred to as the curvature of a polytope. Following the analogy with the diameter, let $\\Lambda(d,n)$ be the largest total curvature $\\lambda(P)$ of the primal central path over all polytopes $P$ defined by $n$ inequalities in dimension $d$.\n\nHolt and Klee~[HK] showed that, for $n> d\\geq 13$, the conjecture of Hirsch is tight. We have the following continuous analogue of the result of Holt and Klee:\n\n[DTZa] $\\liminf_{n\\rightarrow\\infty}\\frac{\\Lambda(d,n)}{n}\\geq \\pi$, that is, $\\Lambda(d,n)$ is bounded below by a constant times $n$.\n\nThe special case of $n=2d$ of the conjecture of Hirsch is known as the $d$-step conjecture, and it has been shown by Klee and Walkup~[KW] that the $d$-step conjecture is equivalent to the Hirsch conjecture. We have the following continuous analogue of the result of Klee and Walkup:\n\n[DTZb] If the order of the curvature is less than the dimension $d$ for all polytope defined by $2d$ inequalities and for all $d$, then the order of the curvature is less that the number of inequalities for all polytopes; that is, if $\\Lambda(d,2d)=\\mathcal{O}(d)$ for all $d$, then $\\Lambda(d,n)=\\mathcal{O}(n)$.\n\nBibliography:\n[DMS] J.-P. Dedieu, G. Malajovich and M. Shub: On the curvature of the central path of linear programming\n\n*[DTZa] A. Deza, T. Terlaky and Y. Zinchenko: Polytopes and arrangements: diameter and curvature. Operations Research Letters (to appear).\n\n[DTZb] A. Deza, T. Terlaky and Y. Zinchenko: The continuous d-step conjecture for polytopes. AdvOL-Report 2007/16, McMaster University (2007).\n\n[HK] F. Holt and V. Klee: Many polytopes meeting the conjectured Hirsch bound. Discrete and Computational Geometry 20 (1998) 1--17.\n\n[KW] V. Klee and D. Walkup: The $d$-step conjecture for polyhedra of dimension $d<6$. Acta Mathematica 133 (1967) 53--78.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Continous analogue of Hirsch conjecture\" in Geometry; Polytopes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The asserted linear order for maximal primal-central-path curvature is false: explicit linear programs with linearly many inequalities have exponentially curved central paths.\n\n**Verified partial progress.**\n\n- Allamigeon, Benchimol, Gaubert, and Joswig construct a family with 3r+4 inequalities in dimension 2r+2 and exponential total curvature in r.\n- Their method tropicalizes the central path and transfers a combinatorial tropical-angle lower bound to classical linear programs.\n- A later SIAM paper strengthens the optimization consequence by proving log-barrier interior-point methods are not strongly polynomial.\n\n**Full solution or refutation.**\n\nBecause n=3r+4 is linear in r while curvature grows exponentially in r, the claimed order n is disproved.\n\n**What remains.**\n\nDetermine the sharp worst-case growth rate under precisely specified central-path and encoding models; the original linear-order conjecture itself is settled negatively.\n\n**Sources checked.**\n\n- Xavier Allamigeon, Pascal Benchimol, Stéphane Gaubert, and Michael Joswig, Long and winding central paths, arXiv:1405.4161 (2014). (primary): https://arxiv.org/abs/1405.4161\n Evidence used: Explicitly says it disproves the continuous Hirsch analogue and gives the exponential-curvature family.\n- Xavier Allamigeon, Pascal Benchimol, Stéphane Gaubert, and Michael Joswig, Log-Barrier Interior Point Methods Are Not Strongly Polynomial, SIAM Journal on Applied Algebra and Geometry 2 (2018), 140-178. (primary): https://doi.org/10.1137/17M1142132\n Evidence used: Gives a related exponential-curvature construction and its strong-polynomiality consequence.\n\n**Review notes.** The database title misspells Continuous. The phrase order ... is n is interpreted as an O(n)/linear-growth assertion, matching the conjecture explicitly refuted by the primary sources.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3099, "problem_number": "OPG-778", "title": "Cube-Simplex conjecture", "statement": "Conjecture For every positive integer $k$, there exists an integer $d$ so that every polytope of dimension $\\ge d$ has a $k$-dimensional face which is either a simplex or is combinatorially isomorphic to a $k$-dimensional cube.", "background": "Source: Open Problem Garden. Original node ID: 778. URL: http://www.openproblemgarden.org/op/cube_simplex_conjecture.\n\nSource subject path: Geometry > Polytopes.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cube_simplex_conjecture\n- Author(s): Kalai, Gil\n- Subject(s): Geometry; Polytopes\n- Keywords: cube; facet; polytope; simplex\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 9th, 2008 by mdevos\n\nProblem-page discussion:\nIt is an easy consequence of Euler's formula that every 3-polytope has a face which is either a triangle, a quadrilateral, or a pentagon. The 120-cell is a 4-polytope in which every 2-face is a pentagon (in fact every 3-face is a regular dodecahedron). Perles and Shephard asked whether there exist higher dimensional polytopes in which all 2-faces have at least 5 vertices. This question was answered in the negative by Kalai [K] who showed that every 5-polytope has a 2-face with at most 4 vertices. So, if we define $f(k)$ to be the smallest integer $d$ satisfying the above conjecture for $k$, or $\\infty$ if none exists, then $f(2) = 5$.\n\nThis conjecture is still open for simple polytopes. However, it is known that for every positive integer $k$, there exists an integer $d$ so that every simple polytope of dimension $\\ge d$ either has a 2-dimensional face which is a triangle, or a $k$-dimensional face which is combinatorially isomorphic to a cube. This was proved by Kalai [K] using some earlier results of Nikulin and of Blind and Blind. Actually, something much stronger holds here: simple polytopes of sufficiently high dimension without 2-faces which are triangles must have most $k$-dimensional faces combinatorially isomorphic to the $k$-cube.\n\nThe following is an interesting weakening of the above conjecture.\n\nConjecture For every positive integer $k$, there exists an integer $d$ and a finite list $L$ of $k$-dimensional polytopes, so that every polytope of dimension $\\ge d$ has a $k$-dimensional face which appears in $L$.\n\nDefining $h(k)$ to be the smallest integer $d$ satisfying this conjecture for $k$, or $\\infty$ if none exists, we find that $h(2) = 3$ (by the consequence of Euler's formula in the first paragraph). Meisinger, Kleinschmidt, and Kalai [MKK] proved that $h(3) \\le 9$ with the help of FLAGTOOL, a computer program which can compute linear relations for $f$-vectors. This weaker conjecture is known to be true for simple polytopes.\n\nBibliography:\n[MKK] G. Meisinger, P. Kleinschmidt, and G. Kalai, Three theorems, with computer-aided proofs, on three-dimensional faces and quotients of polytopes. The Branko Grünbaum birthday issue. Discrete Comput. Geom. 24 (2000), no. 2-3, 413--420. MathSciNet\n\n*[K] G. Kalai, On low-dimensional faces that high-dimensional polytopes must have. Combinatorica 10 (1990), no. 3, 271--280. MathSciNet\n\nDiscussion links:\n- 120-cell: http://en.wikipedia.org/wiki/120-cell\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1758060\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1092544\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Cube-Simplex conjecture\" in Geometry; Polytopes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Kalai proved the k=2 case (f(2)=5) and a strong simple-polytope result, but the cube--simplex assertion is open for arbitrary polytopes and k.\n\n**Verified partial progress.**\n\n- Every 5-polytope has a 2-face with at most four vertices, yielding the k=2 result.\n- The simple-polytope version has a substantially stronger theorem.\n\n**Full solution or refutation.**\n\nNo all-k, arbitrary-polytope proof or counterexample was verified.\n\n**What remains.**\n\nResolve the conjecture beyond the two-dimensional and simple-polytope regimes.\n\n**Sources checked.**\n\n- G. Kalai, On low-dimensional faces that high-dimensional polytopes must have, Combinatorica 10 (1990), 271--280. (primary): https://garden.irmacs.sfu.ca/op/cube_simplex_conjecture\n Evidence used: The source discussion cites Kalai's theorem for f(2)=5 and the simple-polytope progress.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3100, "problem_number": "OPG-37341", "title": "Extension complexity of (convex) polygons", "statement": "The extension complexity of a polytope $P$ is the minimum number $q$ for which there exists a polytope $Q$ with $q$ facets and an affine mapping $\\pi$ with $\\pi(Q) = P$.\n\nQuestion Does there exists, for infinitely many integers $n$, a convex polygon on $n$ vertices whose extension complexity is $\\Omega(n)$?", "background": "Source: Open Problem Garden. Original node ID: 37341. URL: http://www.openproblemgarden.org/op/extension_complexity_of_convex_polygons.\n\nSource subject path: Geometry > Polytopes.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/extension_complexity_of_convex_polygons\n- Subject(s): Geometry; Polytopes\n- Keywords: polytope, projection, extension complexity, convex polygon\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 13th, 2011 by DOT\n\nProblem-page discussion:\nThe extension complexity of a polytope is bounded from above by its number of vertices. Thus, a convex polygon with $n$ vertices has extension complexity $O(n)$.\n\nSome regular convex polygons have extension complexity $O(\\log n)$ [BTN].\n\nA convex polygon whose points are drawn randomly on a circle has extension complexity $\\Omega(\\sqrt n)$ with probability one (follows from [FRT]).\n\nThe question asks for the maximal extension complexity of a convex polygon.\n\nA strongly related question is the following.\n\nQuestion What is the extension complexity of an $n$-vertex convex polygon whose vertices are drawn randomly on a circle?\n\nBibliography:\n*[BTN] Ben-Tal, A and Nemirovski, A. On polyhedral approximations of the second-order cone. Math. Oper. Res. 26:2 193-205 (2001)\n\n[FRT] Fiorini, S. and Rothvoss, T. and Tiwary, H.R. Extended formulations of polygons. arXiv:1107.0371\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Extension complexity of (convex) polygons\" in Geometry; Polytopes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Shitov proves that every convex n-gon has extension complexity at most 147 n^(2/3), which is sublinear and rules out the requested Omega(n) family.\n\n**Verified partial progress.**\n\n- Generic n-gons have extension complexity at least sqrt(2n), leaving a gap between square-root lower bounds and the n^(2/3) universal upper bound.\n\n**Full solution or refutation.**\n\nBecause 147 n^(2/3)=o(n), no infinite family of convex n-gons can have extension complexity Omega(n).\n\n**What remains.**\n\nDetermine the correct asymptotic order of the maximum extension complexity of an n-gon and sharpen the gap between known lower and upper bounds.\n\n**Sources checked.**\n\n- Y. Shitov, Sublinear extensions of polygons, Proceedings of the London Mathematical Society 132(4) (2026), e70137. (primary): https://doi.org/10.1112/plms.70137\n Evidence used: The paper proves that every convex n-gon is the projection of a polytope with at most 147 n^(2/3) facets.\n- S. Fiorini, T. Rothvoß and H. R. Tiwary, Extended formulations for polygons, arXiv:1107.0371. (primary): https://arxiv.org/abs/1107.0371\n Evidence used: The paper proves a sqrt(2n) lower bound for generic n-gons and records the lower-bound side of the remaining extremal problem.\n\n**Review notes.** The source's grammatical phrase 'Does there exists' is preserved in report.md.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3101, "problem_number": "OPG-37459", "title": "Durer's Conjecture", "statement": "Conjecture Every convex polytope has a non-overlapping edge unfolding.", "background": "Source: Open Problem Garden. Original node ID: 37459. URL: http://www.openproblemgarden.org/op/d_urers_conjecture.\n\nSource subject path: Geometry > Polytopes.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/d_urers_conjecture\n- Author(s): Durer, A.; Shephard, G.C.\n- Subject(s): Geometry; Polytopes\n- Keywords: folding; polytope\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: June 4th, 2012 by dmoskovich\n\nProblem-page discussion:\nIn 1525, Albrecht Dürer represented polytopes by cutting them open along edges and then flattening the surface onto the plane, without overlaps and without distorting the individual faces. Self-intersections are allowed during the unfolding process, but the final flattened surface must be free of overlaps. Whether a non-overlapping edge unfolding, as defined above, is possible for any convex polytopes was formulated by Shephard as a conjecture in 1975.\n\nBibliography:\n[D] A. Dürer, Unterweysung der Messung mit dem Zyrkel und Rychtscheyd. Nürnberg (1525). English translation with commentary by Walter L. Strauss The Painter's Manual, New York (1977).\n\n[O] J. O'Rourke, How to fold it, Cambridge University Press, 2011, Book website\n\n[P] K. Polthier Imagining maths- unfolding polyhedra\n\n*[S] G.C. Shephard, Convex Polytopes with Convex Nets. Math. Proc. Camb. Phil. Soc., 78:389-403 (1975).\n\nBibliography links:\n- Book website: http://howtofoldit.org/\n- Imagining maths- unfolding polyhedra: http://plus.maths.org/content/os/issue27/features/mathart/index/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Durer's Conjecture\" in Geometry; Polytopes, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Durer's conjecture remains open in its standard form: it is unknown whether every convex three-dimensional polyhedron has a non-overlapping edge unfolding.\n\n**Verified partial progress.**\n\n- Many special classes and more permissive unfolding models have positive results.\n- Recent high-overlap constructions for particular unfoldings do not produce a polyhedron for which every edge unfolding overlaps.\n\n**Full solution or refutation.**\n\nNo proof for all convex 3-polyhedra and no counterexample to the existential net conjecture was verified.\n\n**What remains.**\n\nProve that every convex 3-polyhedron admits a non-overlapping edge unfolding, or construct one for which all spanning-tree edge unfoldings overlap.\n\n**Sources checked.**\n\n- Open Problems Project, Problem 9, Durer's Unfolding Problem, accessed 2026-08-17. (maintained_tracker): https://topp.openproblem.net/p9\n Evidence used: States the classical edge-unfolding question for convex polyhedra as open.\n- Erik D. Demaine and Joseph O'Rourke, Unfolding Polyhedra, arXiv:1908.07152. (authoritative_secondary): https://arxiv.org/abs/1908.07152\n Evidence used: Surveys the problem and distinguishes edge unfoldings from solved or more permissive unfolding variants.\n\n**Review notes.** The literal phrase 'convex polytope' is dimension-unspecified; the named classical conjecture concerns convex 3-polytopes.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 6, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3102, "problem_number": "OPG-586", "title": "Pebbling a cartesian product", "statement": "We let $p(G)$ denote the pebbling number of a graph $G$.\n\nConjecture $p(G_1 \\Box G_2) \\le p(G_1) p(G_2)$.", "background": "Source: Open Problem Garden. Original node ID: 586. URL: http://www.openproblemgarden.org/op/pebbling_a_cartesian_product.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/pebbling_a_cartesian_product\n- Author(s): Graham, Ronald L.\n- Subject(s): Graph Theory\n- Keywords: pebbling; zero sum\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 24th, 2007 by mdevos\n\nProblem-page discussion:\nThe pebbling number of a graph $G$, denoted $p(G)$, is the smallest integer $k$ so that however $k$ pebbles are distributed onto the vertices of $G$, it is possible to move a pebble to any vertex by a sequence of steps, where in each step we remove two pebbles from one vertex, and then place one on an adjacent vertex. The cartesian product of two graphs $G_1$ and $G_2$, denoted $G_1 \\Box G_2$, is the graph with vertex set $V(G_1) \\times V(G_2)$ and an edge from $(v,w)$ to $(v',w')$ if either $v=v'$ and $w \\sim w'$ (in $G_2$ ) or $w=w'$ and $v \\sim v'$ (in $G_1$ ).\n\nGraph Pebbling arose out of the study of zero-sum subsequences in groups, but has proved to be a rich and interesting topic in graph theory. See Glenn Hurlbert's wonderful graph pebbling page for much more on this topic (and this problem in particular). Graham's conjecture was stated in one of the first papers on this subject by Fan Chung [C], and has since generated considerable interest.\n\nThere are a number of partial results toward this conjecture. Chung [C] proved that $p(P_{d_1+1} \\Box P_{d_2+1} \\ldots \\Box P_{d_{\\ell}+1}) = 2^{d_1 + d_2 \\ldots + d_{\\ell}}$, thus settling the conjecture for products of paths (here $P_n$ denotes a path with $n$ vertices). It is also known when $G_1,G_2$ are both trees, both cycles, and for graphs with high minimum degree. Again, we encourage the interested reader to see the graph pebbling page for more details.\n\nBibliography:\n*[C] F. Chung, Pebbling in hypercubes SIAM J. Disc. Math. 2 (1989), 467--472.\n\nDiscussion links:\n- graph pebbling page: http://mingus.la.asu.edu/%7Ehurlbert/pebbling/pebb.html\n- the graph pebbling page: http://mingus.la.asu.edu/%7Ehurlbert/pebbling/pebb.html\n\nBibliography links:\n- Pebbling in hypercubes: http://mingus.la.asu.edu/%7Ehurlbert/pebbling/papers/Chun_PIH.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Pebbling a cartesian product\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Graham's Cartesian-product pebbling conjecture remains open in general, with a universal factor-2 approximation and many verified graph families.\n\n**Verified partial progress.**\n\n- Asplund, Hurlbert, and Kenter prove p(G box H) at most 2 p(G)p(H) for all connected graphs.\n- They also prove the sharper asymmetric bound p(G box H) at most (p(G)+|V(G)|)p(H).\n- A 2024 optimization study reports constrained confirmation for products of 8-vertex Lemke graphs, which are viewed as candidate counterexample sources.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for arbitrary connected G and H was located.\n\n**What remains.**\n\nRemove the factor 2 in the universal bound, or construct connected graphs whose Cartesian product violates the multiplicative inequality.\n\n**Sources checked.**\n\n- John Asplund, Glenn Hurlbert, and Franklin Kenter, Pebbling on Graph Products and other Binary Graph Constructions, arXiv:1801.07808 (2018). (primary): https://arxiv.org/abs/1801.07808\n Evidence used: States Graham's conjecture and proves the universal factor-2 and asymmetric upper bounds.\n- Jonad Pulaj, Kenan Wood, and Carl Yerger, Bilevel Programming for Pebbling Numbers of Lemke Graph Products, arXiv:2411.19314 (2024). (primary): https://arxiv.org/abs/2411.19314\n Evidence used: Still states the general assertion as Graham's conjecture and gives constrained computational evidence for Lemke-graph products.\n- Fan R. K. Chung, Pebbling in hypercubes, SIAM Journal on Discrete Mathematics 2 (1989), 467-472. (primary): https://doi.org/10.1137/0402041\n Evidence used: Establishes the classical path-product and hypercube results underlying important solved cases.\n\n**Review notes.** The conventional conjecture is for connected graphs; that hypothesis is implicit rather than stated in the database sentence.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3103, "problem_number": "OPG-658", "title": "Reconstruction conjecture", "statement": "The deck of a graph $G$ is the multiset consisting of all unlabelled subgraphs obtained from $G$ by deleting a vertex in all possible ways (counted according to multiplicity).\n\nConjecture If two graphs on $\\ge 3$ vertices have the same deck, then they are isomorphic.", "background": "Source: Open Problem Garden. Original node ID: 658. URL: http://www.openproblemgarden.org/op/reconstruction_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/reconstruction_conjecture\n- Author(s): Kelly, Paul J.; Ulam, Stanislaw M.\n- Subject(s): Graph Theory\n- Keywords: reconstruction\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 18th, 2007 by zitterbewegung\n\nProblem-page discussion:\nSee Wikipedia's Reconstruction Conjecture for more on this problem.\n\nBibliography:\n*[K] P. J. Kelly, A congruence theorem for trees, Pacific J. Math., 7 (1957), 961–968.\n\n*[U] S. M. Ulam, A collection of mathematical problems, Wiley, New York, 1960.\n\nDiscussion links:\n- Reconstruction Conjecture: http://en.wikipedia.org/wiki/Reconstruction_conjecture\n\nComments:\n- May 4th, 2010 | Anonymous | A strategy for simple graphs?: How about 1) Prove for a 4-node, or tetrahedral graph. 2) Demonstrate that all graphs with > 4 nodes are composed of multiple overlapping tetrahedrons 3) Figure out how coupled tetrahedra function when nodes are deleted. 4) Induct on number of tetrahedra in graph.?\n\nObviously (3) is the tough part, but why should it be impossible?\n- May 23rd, 2008 | melch | correction and partial results: this should be on 3 or more vertices. False for digraphs, hypergraphs, and infinite graphs. It is open for simple graphs and multigraphs\n- November 28th, 2007 | Anonymous | Question.: Does G have to be simple, or can it be a multigraph?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Reconstruction conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The standard finite-simple-undirected graph reconstruction conjecture remains open, although many graph classes and probabilistic models are reconstructible.\n\n**Verified partial progress.**\n\n- Reconstruction is known for numerous structured classes; recent work adds or strengthens triangle-free and interval-graph cases.\n- Semi-random and average-case models admit robust reconstruction results.\n- Current 2026 papers still describe the all-graph Kelly-Ulam conjecture as a foremost unsolved problem.\n\n**Full solution or refutation.**\n\nNo proof was located that every finite simple undirected graph on at least three vertices is determined by its vertex-deleted deck.\n\n**What remains.**\n\nProve the conjecture for every finite simple undirected graph, beyond the many known reconstructible classes.\n\n**Sources checked.**\n\n- Yaxin Qi, Graph reconstruction from connected triples, Discrete Mathematics 349 (2026), 115058. (primary): https://www.sciencedirect.com/science/article/abs/pii/S0012365X26000828\n Evidence used: A current primary paper explicitly says the standard Reconstruction Conjecture remains a foremost unsolved problem.\n- Alexander Clifton et al., Reconstruction and edge reconstruction of triangle-free graphs, Discrete Mathematics 347 (2024), 113753. (primary): https://doi.org/10.1016/j.disc.2023.113753\n Evidence used: Proves reconstructibility results for a substantial restricted class.\n- Interval Graphs are Reconstructible, arXiv:2504.02353 (2025). (primary): https://arxiv.org/abs/2504.02353\n Evidence used: Adds interval graphs to the classes for which reconstruction is proved.\n- Graph-theory open problems, Reconstruction conjecture; checked 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/reconstruction_conjecture/\n Evidence used: Maintains the standard finite simple graph statement as open and tracks partial results.\n\n**Review notes.** Material formulation defect: the stored statement does not specify finite simple undirected graphs. This status applies only to the standard literature formulation; multigraph or other broadened readings may differ.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3104, "problem_number": "OPG-804", "title": "Edge Reconstruction Conjecture", "statement": "Conjecture\n\nEvery simple graph with at least 4 edges is reconstructible from it's edge deleted subgraphs", "background": "Source: Open Problem Garden. Original node ID: 804. URL: http://www.openproblemgarden.org/op/edge_reconstruction_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/edge_reconstruction_conjecture\n- Author(s): Harary, Frank\n- Subject(s): Graph Theory\n- Keywords: reconstruction\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 23rd, 2008 by melch\n\nProblem-page discussion:\nIt is known that if a graph is vertex reconstructible then it is edge reconstructible.\n\nBibliography:\nJ.A.Bondy, A graph reconstruction manual, Surveys in Combinatorics, LMS-Lecture Note Series 166(1991)\n\nRelated:\nRelated problems\nReconstruction conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Edge Reconstruction Conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The edge reconstruction conjecture remains open for arbitrary finite simple graphs with at least four edges.\n\n**Verified partial progress.**\n\n- Vertex reconstructibility implies edge reconstructibility.\n- Many graph classes, including recent triangle-free subclasses, are reconstructed.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for all finite simple graphs was verified.\n\n**What remains.**\n\nEstablish reconstruction from every edge deck or produce a nonisomorphic equal-deck pair.\n\n**Sources checked.**\n\n- University of Oxford, Graph Reconstruction thesis record (accessed 2026-08-17). (authoritative_secondary): https://ora.ox.ac.uk/objects/uuid%3A5b6d514c-241f-4a77-8fa2-55fd069df278/files/dbg257f65g\n Evidence used: Explicitly describes the edge reconstruction conjecture as wide open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3105, "problem_number": "OPG-34908", "title": "Book Thickness of Subdivisions", "statement": "Let $G$ be a finite undirected simple graph.\n\nA $k$-page book embedding of $G$ consists of a linear order $\\preceq$ of $V(G)$ and a (non-proper) $k$-colouring of $E(G)$ such that edges with the same colour do not cross with respect to $\\preceq$. That is, if $v\\prec x\\prec w\\prec y$ for some edges $vw,xy\\in E(G)$, then $vw$ and $xy$ receive distinct colours.\n\nOne can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.\n\nThe book thickness of $G$, denoted by bt $(G)$ is the minimum integer $k$ for which there is a $k$-page book embedding of $G$.\n\nLet $G'$ be the graph obtained by subdividing each edge of $G$ exactly once.\n\nConjecture There is a function $f$ such that for every graph $G$, $$\\text{bt}(G) \\leq f( \\text{bt}(G') )\\enspace.$$", "background": "Source: Open Problem Garden. Original node ID: 34908. URL: http://www.openproblemgarden.org/op/book_thickness_of_subdivisions.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/book_thickness_of_subdivisions\n- Author(s): Blankenship, Robin; Oporowski, Bogdan\n- Subject(s): Graph Theory\n- Keywords: book embedding; book thickness\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: January 19th, 2009 by David Wood\n\nProblem-page discussion:\nThe conjecture is due to [B099]. The conjecture is true for complete graphs [BO99,EM99,E02]. The conjecture is discussed in depth in [DW05].\n\nBibliography:\n*[BO99] Robin Blankenship and Bogdan Oporowski. Drawing Subdivisions Of Complete And Complete Bipartite Graphs On Books, Technical Report 1999-4, Department of Mathematics, Louisiana State University, 1999.\n\n[DW05] Vida Dujmovic and David Wood. Stacks, queues and tracks: Layouts of graph subdivisions. Discrete Mathematics & Theoretical Computer Science 7:155-202, 2005.\n\n[EM99] Hikoe Enomoto and Miki Shimabara Miyauchi. Embedding graphs into a three page book with $O(M \\log N)$ crossings of edges over the spine. SIAM J. Discrete Math., 12(3):337–341, 1999.\n\n[E02] David Eppstein. Separating thickness from geometric thickness. In Proc. 10th International Symp. on Graph Drawing (GD ’02), pp. 150–161. vol. 2528 of Lecture Notes in Comput. Sci. Springer, 2002.\n\nBibliography links:\n- Stacks, queues and tracks: Layouts of graph subdivisions: http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/67\n- Embedding graphs into a three page book with $O(M \\log N)$ crossings of edges over the spine: http://dx.doi.org/10.1137/S0895480195280319\n\nComments:\n- July 10th, 2021 | Anonymous | This problem is solved: This problem is solved in:\n\nV. Dujmović, D. Eppstein, R. Hickingbotham, P. Morin, D. R. Wood. Stack-number is not bounded by queue-number. Combinatorica, accepted in 2021 https://arxiv.org/abs/2011.04195\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Book Thickness of Subdivisions\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The proposed function cannot exist: graphs of unbounded book thickness have constant subdivisions of book thickness three.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe Blankenship--Oporowski book-thickness subdivision conjecture is false.\n\n**What remains.**\n\nThe original universal claim is refuted; study quantitative restrictions on special graph classes instead.\n\n**Sources checked.**\n\n- V. Dujmović, D. Eppstein, R. Hickingbotham, P. Morin and D. R. Wood, Stack-number is not bounded by queue-number, Combinatorica 42 (2022). (primary): https://arxiv.org/abs/2011.04195\n Evidence used: Provides the unbounded-stack-number construction underlying the subdivision counterexample.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3106, "problem_number": "OPG-36879", "title": "Shannon capacity of the seven-cycle", "statement": "Problem What is the Shannon capacity of $C_7$?", "background": "Source: Open Problem Garden. Original node ID: 36879. URL: http://www.openproblemgarden.org/op/shannon_capacity_of_the_seven_cycle.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/shannon_capacity_of_the_seven_cycle\n- Subject(s): Graph Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 19th, 2009 by tchow\n\nProblem-page discussion:\nLet $\\alpha(G)$ denote the independence number of the graph $G$, and let $G*H$ denote the strong graph product of $G$ and $H$ (in which $(g,h)$ is adjacent to $(g',h')$ if $g=g'$ and $h$ is adjacent to $h'$, or if $h=h'$ and $g$ is adjacent to $g'$, or if $g$ is adjacent to $g'$ and $h$ is adjacent to $h'$ ). Then the Shannon capacity of $G$ is defined by $$\\theta(G) = \\lim_{k\\to\\infty} \\biggl({\\alpha(G*G*\\cdots*G) \\over k}\\biggr)^{1/k},$$where the strong graph product is over$k$copies of$G$. The Shannon capacity is important because it represents the effective size of an alphabet in a communication model represented by$G$, but it is notoriously difficult to compute. Lovász [L] famously proved that the Shannon capacity of the five-cycle$C_5$is$\\sqrt{5}$, but even the Shannon capacity of$C_7$remains unknown. However, Bohman [B] has shown that$$\\lim_{k\\to\\infty}(k+(1/2)-\\theta(C_{2k+1}))=0.$$\n\nBibliography:\n[B] Tom Bohman, A limit theorem for the Shannon capacity of odd cycles II, Proc. Amer. Math. Soc. 133 (2005), no. 2, 537-543.\n\n[L] László Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Th. IT-25 (1979), 1-7.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"Shannon capacity of the seven-cycle\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The exact Shannon capacity of the seven-cycle remains unknown.\n\n**Verified partial progress.**\n\n- Classical lower and upper bounds and semidefinite methods constrain the capacity.\n\n**Full solution or refutation.**\n\nNo exact C7 capacity was verified.\n\n**What remains.**\n\nClose the remaining capacity gap for the strong powers of C7.\n\n**Sources checked.**\n\n- Shannon capacity of a graph overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Shannon_capacity_of_a_graph\n Evidence used: Explicitly lists C7 among small graphs with unknown capacity.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3107, "problem_number": "OPG-37081", "title": "Number of Cliques in Minor-Closed Classes", "statement": "Question Is there a constant $c$ such that every $n$-vertex $K_t$-minor-free graph has at most $c^tn$ cliques?", "background": "Source: Open Problem Garden. Original node ID: 37081. URL: http://www.openproblemgarden.org/op/number_of_cliques_in_minor_closed_classes.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/number_of_cliques_in_minor_closed_classes\n- Author(s): Wood, David R.\n- Subject(s): Graph Theory\n- Keywords: clique; graph; minor\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 12th, 2009 by David Wood\n\nProblem-page discussion:\nHere a clique is a (not neccessarily maximal) set of pairwise adjacent vertices in a graph.\n\nSee [RW, NSTW] for early bounds on the number of cliques. Wood [W] proved that the number of cliques in an $n$-vertex $K_t$-minor-free graph is at most $c^{t\\sqrt{\\log t}}n\\enspace.$ Fomin et al. [FOT] improved this bound to $c^{t\\log\\log t}n\\enspace.$\n\nThese results are based on the fact that every $n$-vertex $K_t$-minor-free graph has at most $ct\\sqrt{\\log t}n$ edges. This bound is tight for certain random graphs. So it is reasonable to expect that random graphs might also provide good lower bounds on the number of cliques.\n\nUpdate 2014: Choongbum Lee and Sang-il Oum [LO] recently answered this question in the affirmative, and even proved it for excluded subdivisions. In particular, they proved that every $n$-vertex graph with no $K_t$-subdivision has at most $2^{474t}n$ cliques and also at most $2^{14t+o(t)}n$ cliques.\n\nThe question now is to determine the minimum constant. Wood [W] proved a lower bound of $3^{2t/3-o(t)}n$ using an appropriate sized complete graph minus a perfect matching. The same graph gives a lower bound of $3^{s-o(s)}n$ on the number of cliques in a graph with no $K_s$ subdivision.\n\nUpdate (2019): Fox and Wei [FW] have proved that every graph on $n$ vertices with no $K_t$-minor has at most $3^{2t/3+o(t)}n$ cliques. This bound is tight for $n \\geq 4t/3$.\n\nBibliography:\n[FOT] Fedor V. Fomin, Sang il Oum, and Dimitrios M. Thilikos. Rank-width and tree-width of $H$-minor-free graphs, European J. Combin. 31 (7), 1617–1628, 2010.\n\n[NSTW] Serguei Norine, Paul Seymour, Robin Thomas, Paul Wollan. Proper minor-closed families are small. J. Combin. Theory Ser. B, 96(5):754--757, 2006.\n\n[RW] Bruce Reed and David R. Wood. Fast separation in a graph with an excluded minor. In 2005 European Conf. on Combinatorics, Graph Theory and Applications (EuroComb '05), vol. AE of Discrete Math. Theor. Comput. Sci. Proceedings, pp. 45--50. 2005.\n\n* [W] David R. Wood. On the maximum number of cliques in a graph. Graphs Combin., 23(3):337--352, 2007.\n\n[LO] Choongbum Lee and Sang-il Oum. Number of cliques in graphs with forbidden minor, 2014.\n\n[FW] Jacob Fox, Fan Wei. On the number of cliques in graphs with a forbidden minor\n\nBibliography links:\n- Rank-width and tree-width of $H$-minor-free graphs: http://www.arxiv.org/abs/math.CO/0910.0079\n- On the maximum number of cliques in a graph: http://www.arxiv.org/abs/math.CO/0602191\n- Number of cliques in graphs with forbidden minor: http://www.arxiv.org/abs/1407.7707\n- On the number of cliques in graphs with a forbidden minor: http://www.arxiv.org/abs/1603.07056\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Number of Cliques in Minor-Closed Classes\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The requested absolute-exponential clique bound is proved, and the optimal exponential rate for excluded complete minors is known asymptotically.\n\n**Verified partial progress.**\n\n- Lee and Oum prove an absolute bound 2^{50t}n even under the stronger exclusion of a K_t subdivision.\n- Fox and Wei sharpen the K_t-minor-free result to 3^{2t/3+o(t)}n and prove tightness for n >= 4t/3.\n\n**Full solution or refutation.**\n\nYes. Taking any fixed c large enough in the Lee-Oum theorem answers the source question affirmatively; Fox-Wei determine the asymptotic exponential constant.\n\n**What remains.**\n\nOnly refinements of lower-order factors and exact finite-t extremal values remain, not the source existence question.\n\n**Sources checked.**\n\n- C. Lee and S.-il Oum, Number of cliques in graphs with a forbidden subdivision, SIAM J. Discrete Math. 29 (2015), 1999-2005, arXiv:1407.7707. (primary): https://arxiv.org/abs/1407.7707\n Evidence used: Explicitly states that it strongly answers Wood's c^t n question and proves a 2^{50t}n bound.\n- J. Fox and F. Wei, On the number of cliques in graphs with a forbidden minor, arXiv:1603.07056 (2016). (primary): https://arxiv.org/abs/1603.07056\n Evidence used: Proves the 3^{2t/3+o(t)}n bound and matching asymptotic construction.\n\n**Review notes.** The updated arXiv version of Lee-Oum has stronger constants than the historical background text.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3108, "problem_number": "OPG-37089", "title": "Shuffle-Exchange Conjecture (graph-theoretic form)", "statement": "Given integers $k,n \\ge 2$, the 2-stage Shuffle-Exchange graph/network, denoted $\\text{SE}(k,n)$, is the simple $k$-regular bipartite graph with the ordered pair $(U,V)$ of linearly labeled parts $U:=\\{u_0,\\dots,u_{t-1}\\}$ and $V:=\\{v_0,\\dots,v_{t-1}\\}$, where $t:=k^{n-1}$, such that vertices $u_i$ and $v_j$ are adjacent if and only if $(j - ki) \\text{ mod } t < k$ (see Fig.1).\n\nGiven integers $k,n,r \\ge 2$, the $r$-stage Shuffle-Exchange graph/network, denoted $(\\text{SE}(k,n))^{r-1}$, is the proper (i.e., respecting all the orders) concatenation of $r-1$ identical copies of $\\text{SE}(k,n)$ (see Fig.1).\n\nLet $r(k,n)$ be the smallest integer $r\\ge 2$ such that the graph $(\\text{SE}(k,n))^{r-1}$ is rearrangeable.\n\nProblem Find $r(k,n)$.\n\nConjecture $r(k,n)=2n-1$.", "background": "Source: Open Problem Garden. Original node ID: 37089. URL: http://www.openproblemgarden.org/op/shuffle_exchange_conjecture_graph_theoretic_form.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/shuffle_exchange_conjecture_graph_theoretic_form\n- Author(s): Beneš, Václav E.; Folklore; Stone, Harold S.\n- Subject(s): Graph Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 30th, 2009 by Vadim Lioubimov\n\nProblem-page discussion:\nA mask for the graph $G:=(\\text{SE}(k,n))^{r-1}$ is a $k$-regular bipartite multigraph with the bipartition $\\{U,V\\}$. The graph $G$ is said to be rearrangeable if for every its mask there exists a collection, called routing, of corresponding mutually edge-disjoint paths in $G$ connecting its end parts. (For simplicity, we do not provide here a more general definition for rearrangeability of graphs.)\n\nNote that $G$ is a simple $r$-partite graph with $r k^{n-1}$ vertices and $(r-1)k^{n}$ edges, and any route for it consists exactly of $k^{n}$ paths. Also, $r(k,n)\\le r$ is equivalent to rearrangeability of $G$.\n\nFigure 1. Examples of multistage Shuffle-Exchange graphs.\n\nFor example, according to the conjecture, the graph $(\\text{SE}(2,3))^{4}$ (see Fig. 1) is rearrangeable, which is a well known result.\n\nThe problem and conjecture are equivalent \"graph-theoretic\" forms of remarkable Shuffle-Exchange (SE) problem and conjecture due to the following identity (that is not hard to show by normal reasoning):\n\nTheorem $r(k,n)=d(k,n)$.\n\nThe definition of $d(k,n)$ and more on SE problem/conjecture including the other 2 main forms of them, combinatorial and group-theoretic, and a survey of results can be found here.\n\nBibliography:\n*[S71] H.S. Stone, Parallel processing with the perfect shuffle, IEEE Trans. on Computers C-20 (1971), 153-161.\n\n*[B75] V.E. Beneš, Proving the rearrangeability of connecting networks by group calculation, Bell Syst. Tech. J. 54 (1975), 421-434.\n\nRelated:\nRelated problems\nShuffle-Exchange Conjecture\nBeneš Conjecture (graph-theoretic form)\nBeneš Conjecture\n\nDiscussion links:\n- Shuffle-Exchange (SE) problem and conjecture: http://www.openproblemgarden.org/?q=node/37167\n- here: http://www.openproblemgarden.org/?q=node/37167\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 27.\n\nAttempt notes:\nTarget:\nMake progress on \"Shuffle-Exchange Conjecture (graph-theoretic form)\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A July 2026 preprint proves r(k,3)=6 for every k >= 3, contradicting the conjectured value 2n-1=5.\n\n**Verified partial progress.**\n\n- Chojecki supplies a universal six-stage routing construction at word length three.\n- The same paper constructs an obstruction to every five-stage route for k >= 3, yielding the exact value.\n- The paper explicitly states that its directed routing parameter d(k,n) equals the Open Problem Garden graph parameter r(k,n).\n\n**Full solution or refutation.**\n\nThe formula r(k,n)=2n-1 is false already for (k,n)=(3,3), and indeed for every k >= 3 at n=3.\n\n**What remains.**\n\nDetermine r(k,n) beyond the settled cases; the paper proposes r(k,n)=3n-3 for k >= 3 as a replacement question.\n\n**Sources checked.**\n\n- P. Chojecki, Beneš and Shuffle-Exchange Counterexamples, arXiv:2607.15296 (2026). (primary): https://arxiv.org/abs/2607.15296\n Evidence used: Theorem 19 proves d(k,3)=6 for all k >= 3, and the literature section identifies d(k,n)=r(k,n) for the tracker formulation.\n- Open Problem Garden, Shuffle-Exchange Conjecture (graph-theoretic form) (accessed 2026-08-17). (maintained_tracker): https://garden.irmacs.sfu.ca/op/shuffle_exchange_conjecture_graph_theoretic_form\n Evidence used: Supplies the exact graph formulation and historical conjectured value.\n\n**Review notes.** The disproof is a very recent primary preprint and should receive independent expert review before downstream high-stakes use.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3109, "problem_number": "OPG-37182", "title": "Odd cycles and low oddness", "statement": "Conjecture If in a bridgeless cubic graph $G$ the cycles of any $2$-factor are odd, then $\\omega(G)\\leq 2$, where $\\omega(G)$ denotes the oddness of the graph $G$, that is, the minimum number of odd cycles in a $2$-factor of $G$.", "background": "Source: Open Problem Garden. Original node ID: 37182. URL: http://www.openproblemgarden.org/op/odd_cycles_and_low_oddness.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/odd_cycles_and_low_oddness\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: January 15th, 2010 by Gagik\n\nComments:\n- June 29th, 2010 | Anonymous | This conjecture is false.: For odd $n$ and $i\\in[n]$, let $H_i$ be the graph obtained by deleting an edge, say $x_iy_i$, from the Petersen graph. Define $G_n$ to be the graph obtained by joining vertex $y_i$ and vertex $x_{i+1}$ with an edge (with subscripts reduced modulo $n$ ). For each $i\\in[n]$, the set $\\{y_{i-1}x_{i},y_ix_{i+1}\\}$ is an edge cut. Hence, in any 2-factor of $G_n$, either none of the edges of the form $y_ix_{i+1}$ are contained in a cycle, or all of them are contained in the same cycle.\n\nCase 1: If none of the edges described above are contained in a cycle of a 2-factor of $G_n$, then this 2-factor contains a 2-factor of $H_i$ for each $i$. These 2-factors are also 2-factors of the graphs $H_i+x_iy_i$, that is, each is a 2-factor of the Petersen graph. Each 2-factor of the Petersen graph consists of two cycles of 5 vertices, hence, any such 2-factor of $G_n$ contains $2n$ cycles of odd length.\n\nCase 2: See the comment below.\n- July 2nd, 2010 | Anonymous | This conecture is false.: Case 2: If all of the edges of the form $y_ix_{i+1}$ are contained in the same cycle in a 2-factor of $G_n$, then replacing the edges $y_ix_{i+1}$ with the edges $x_iy_i$ converts this 2-factor of $G_n$ into a 2-factor of $n$ disjoint copies of the Petersen graph. Hence, when restricted to each $H_i$, the 2-factor of $G_n$ consists of a cycle with 5 vertices and a $x_iy_i$-path containing a total of 5 vertices. These paths must be joined together through the edges of the form $y_ix_{i+1}$ creating a cycle of length 5n. Hence, in this case the 2-factor of $G_n$ contains $n$ cycles of length 5 and one cycle of length 5n (which is odd).\n\nNow, $G_n$ is a bridgeless cubic graph whose 2-factors contain only odd cycles, but no 2-factor of $G_n$ contains fewer than $n+1$ cycles.\n- January 16th, 2010 | Anonymous | what is oddness: The notion of oddness of a graph requires explanation.\n- December 13th, 2011 | Anonymous | Let be a bridgeless cubic: Let $G$ be a bridgeless cubic graph. The oddness of a 2-factor $F$ is the number of odd circuits of $F$. The oddness of $G$ is the smallest oddness over all 2-factors. For example, a 3-edge-colorable cubic graph has oddness zero and the Petersen graph has oddness two.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Odd cycles and low oddness\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** An anonymous tracker comment gives a plausible Petersen-fragment counterexample family to the proposed bound, but this search did not verify it in a primary or authoritative secondary source.\n\n**Verified partial progress.**\n\n- The tracker supplies a two-case 2-factor analysis for the claimed family, which is a lead for independent verification.\n\n**Full solution or refutation.**\n\nThe literal disproof is not accepted in this dataset without a citable authoritative source.\n\n**What remains.**\n\nLocate a primary publication or independently verify and publish the Petersen-fragment counterexample argument.\n\n**Sources checked.**\n\n- Open Problem Garden, Odd cycles and low oddness, anonymous comments dated 29 June and 2 July 2010. (maintained_tracker): https://openproblemgarden.org/op/odd_cycles_and_low_oddness\n Evidence used: States the Petersen-fragment construction and proves the exhaustive two-case description of every 2-factor, yielding oddness n+1.\n\n**Review notes.** The counterexample is mathematically explicit, but the tracker comments are anonymous and no exact primary publication was found.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3110, "problem_number": "OPG-37210", "title": "Beneš Conjecture (graph-theoretic form)", "statement": "Problem ( $\\dag$ ) Find a sufficient condition for a straight $\\ell$-stage graph to be rearrangeable. In particular, what about a straight uniform graph?\n\nConjecture ( $\\diamond$ ) Let $L$ be a simple regular ordered $2$-stage graph. Suppose that the graph $L^m$ is externally connected, for some $m\\ge1$. Then the graph $L^{2m}$ is rearrangeable.", "background": "Source: Open Problem Garden. Original node ID: 37210. URL: http://www.openproblemgarden.org/op/bene_conjecture_graph_theoretic_form_0.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bene_conjecture_graph_theoretic_form_0\n- Author(s): Beneš, Václav E.\n- Subject(s): Graph Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 17th, 2010 by Vadim Lioubimov\n\nProblem-page discussion:\nGiven an integer $\\ell\\ge2$, an $\\ell$-stage graph is an $\\ell$-partite graph $G$ with a list of its parts $V_1,\\dots,V_{\\ell}$ such that every edge of $G$ has endpoints in both $V_i$ and $V_{i+1}$, for some $i\\in[\\ell-1]$. A vertex in $V_1$ ( $V_\\ell$ ) is a source (target) of $G$. A path in $G$ is plain if it goes from a source to a target through each part of $G$ exactly once. The graph $G$ is externally connected if for every source $s$ and target $t$ there exists a plain path from $s$ to $t$. A mask for $G$ is a $2$-stage multigraph $M$ whose sources and targets are exactly those of $G$ and such that every vertex of $M$ has the same degree in $G$. The graph $G$ is rearrangeable if for every its mask there exists a collection, called routing, of corresponding mutually edge-disjoint plain paths in $G$.\n\nThe graph $G$ is ordered if each of its parts is linearly ordered. The graph $G$ is uniform and denoted $B^{\\ell-1}$ if there is an ordered 2-stage graph $B$ with equal-sized parts such that $G$ is the proper (i.e., respecting all the orders in $B$ ) concatenation of $\\ell-1$ identical copies of $B$. The graph $G$ is straight if for any $2\\le i\\le\\ell-1$ and any $v\\in V_i$, the number of edges joining $v$ with $V_{i-1}$ equals that of $V_{i+1}$.\n\nConjecture ( $\\diamond$ ) can be reformulated as $R(L) \\le 2F(L)$, where $R(B)$ ( $F(B)$ ) denotes the smallest positive integer $n$, or $\\infty$ if none exists, such that the graph $B^n$ is rearrangeable (externally connected).\n\nExamples\n\nConsider the simple 2-regular $2$-stage ordered graphs $A, C, D$ shown in Fig.1. It is easy to see that $F(A) = 2$ and $F(C) = F(D) = 3$ (the corresponding externally connected graphs $A^2, C^3, D^3$ are depicted in blue). Therefore, according to Conjecture ( $\\diamond$ ), the graphs $A^4, C^6, D^6$ should be rearrangeable, which is indeed the case. The graph $A$ is the 2-stage Shuffle-exchange graph $\\text{SE}(2,3)$, and there are several nice proofs known for $R(A)=4$. Although I am not aware of any theoretical proof for rearrangeability of $C^5$ or $D^6$, I have verified by brute force without difficulty that $R(C)=5$ and $R(D)=6$.\n\nFigure 1. Examples for Conjecture ( $\\diamond$ ).\n\nLink to Beneš Conjecture\n\nProblem ( $\\dag$ ) and Conjecture ( $\\diamond$ ) are equivalent \"graph-theoretic\" forms of Problem ( $\\star$ ) and Beneš conjecture [B75], respectively.\n\nThe equivalence is based on the natural bijection between the $\\ell$-systems of partitions and the straight $\\ell$-stage graphs, given any $\\ell\\ge2$. Here an $\\ell$-system of partitions is an $\\ell$-tuple ${\\bf H}:=({\\bf h}_1,\\dots,{\\bf h}_\\ell)$ of partitions of some finite set $E$. The image of ${\\bf H}$ under this bijection is the straight $\\ell$-stage graph denoted $G({\\bf H})$ and defined as follows. The edge set of $G({\\bf H})$ is $[\\ell-1]\\times E$, the $i$ th vertex part is $U_i:=\\{i\\}\\times {\\bf h}_i$, for all $i\\in[\\ell]$, and the edge-vertex incidence is such that every edge $(j,e)$ has endpoints $(j,a)\\in U_j$ and $(j+1,b)\\in U_{j+1}$ uniquely determined by $e\\in a\\cap b$.\n\nThe bijection ${\\bf H} \\mapsto G({\\bf H})$ provides a convenient two-way link between the frameworks for Problems ( $\\star$ ) and ( $\\dag$ ) via numerous easily seen equivalences. Here is some basic ones:\n\n$\\bullet$ Simplicity of $G({\\bf H})$ is equivalent to the condition ${\\bf h}_i\\wedge{\\bf h}_{i+1}={\\bf 0}$, for all $i\\in[\\ell-1]$.\n\n$\\bullet$ Uniformity of $G({\\bf H})$ is equivalent to the existence of a permutation $\\delta$ of $E$ such that ${\\bf h}_{i+1}=\\delta ({\\bf h}_i)$, for all $i\\in[\\ell-1]$.\n\n$\\bullet$ $k$-quasi-regularity of $G({\\bf H})$ is equivalent to every block of ${\\bf h}_i$ being of size $k$, for all $i\\in[\\ell]$. Here the graph $G$ is $k$-quasi-regular if the induced bipartite subgraph on $V_i\\cup V_{i+1}$ is $k$-regular, for all $i\\in[\\ell-1]$. Note that a quasi-regular multistage graph is straight. Also, $k$-quasi-regularity of $B^n$ is equivalent to $k$-regularity of $B$.\n\n$\\bullet$ External connectivity of $G({\\bf H})$ is equivalent to transitivity of $S({\\bf h}_\\ell)\\dots S({\\bf h}_2)S({\\bf h}_1)$.\n\n$\\bullet$ Given a permutation $\\xi$ of $E$, the membership $\\xi \\in S({\\bf h}_1)S({\\bf h}_2) \\dots S({\\bf h}_\\ell)$ is equivalent to routability of the mask $M(\\xi)$ for $G({\\bf H})$ defined as follows. The edge set of $M(\\xi)$ is $E$ and the edge-vertex incidence is such that every edge $e\\in E$ has endpoints $(1,a)\\in U_1$ and $(\\ell,b)\\in U_{\\ell}$ uniquely determined by $e\\in \\xi^{-1}(a)\\cap b$. Note that given ${\\bf H}$, the map $\\xi \\mapsto M(\\xi)$ is surjective (but generally not injective).\n\n$\\bullet$ Consequently, rearrangeability of $G({\\bf H})$ is equivalent to completeness of ${\\bf H}$. Here ${\\bf H}$ is complete if it satisfies $\\frak S(E) = S({\\bf h}_1)S({\\bf h}_2) \\dots S({\\bf h}_\\ell)$.\n\n$\\bullet$ If $G({\\bf H})$ is rearrangeable, then any routing algorithm for $G({\\bf H})$ easily translates to a factorization algorithm of the same complexity for the latter identity, and vise versa. Here, given a rearrangeable multistage graph, a routing algorithm is one that takes a mask of the graph as input and returns a corresponding routing.\n\n$\\bullet$ Contracting all edges between $U_i$ and $U_{i+1}$ in $G({\\bf H})$ is equivalent to replacing the partitions ${\\bf h}_i$ and ${\\bf h}_{i+1}$ in ${\\bf H}$ with their supremum ${\\bf h}_i\\vee{\\bf h}_{i+1}$, given any fixed $i\\in[\\ell-1]$. In other words, $G_i=G({\\bf H}_i)$, where $G_i$ is the contracted graph and ${\\bf H}_i:=({\\bf h}_1,\\dots,{\\bf h}_i\\vee{\\bf h}_{i+1},\\dots,{\\bf h}_\\ell)$. In fact, the procedure ${\\bf H} \\mapsto {\\bf H}_i$ preserves completeness of ${\\bf H}$, as $S({\\bf h}_i)S({\\bf h}_{i+1})\\subseteq S({\\bf h}_i\\vee{\\bf h}_{i+1})$. Equivalently, the procedure $G({\\bf H}) \\mapsto G_i$ preserves rearrangeability of $G({\\bf H})$.\n\nCounterexamples\n\nAlthough the presented graph-theoretic statement ( $\\dag$ ) of Problem ( $\\star$ ) may look more complex, it provides somewhat more intuitive framework to study the problem and, in particular, Beneš conjecture. To illustrate this, let us now reconsider in terms of this framework and in more detail the 3 counterexamples for some extensions of Beneš conjecture discussed here.\n\nCounterexample 1. The condition of simplicity of the graph $L$ (essentially missing in the original statement [B75] of Beneš conjecture) is necessary for Conjecture ( $\\diamond$ ). To see this, consider the following 2-stage 3-regular non-simple ordered graph $Q$:\n\nWhereas $Q$ is obviously externally connected, the graph $Q^2$ is not rearrageable. This is because it is evidently impossible to connect the two red vertices in $Q^2$ (a source and a target) with 3 mutually edge-disjoint plain paths.\n\nCounterexample 2. Conjecture ( $\\diamond$ ) is not directly generalizable to non-uniform graphs. More precisely, the condition of uniformity of $X$ is necessary for the following reformulation of ( $\\diamond$ ):\n\nConjecture Let $X$ be a simple quasi-regular ordered multistage graph. Suppose that $X$ is uniform and externally connected. Then the graph $X^{2}$ is rearrangeable.\n\nHere $X^{2}$ denotes the proper concatenation of 2 identical copies of $X$. To see the necessity, consider the following simple 4-stage 2-quasi-regular non-uniform ordered graph $Y$:\n\nWhereas $Y$ is obviously externally connected, the graph $Y^2$ is not rearrangeable. To see this, recall that contracting all edges between two consecutive parts in a straight multistage graph preserves its rearrangeability. Therefore, if $Y^2$ were rearrangeable then so would be the 3-stage graph $W$ obtained from $Y^2$ by contracting all edges in the shadowed areas. However, this is not true as it is evidently impossible to connect the two red vertices in $W$ (a source and a target) with 4 mutually edge-disjoint plain paths.\n\nCounterexample 3. The stronger version of Conjecture ( $\\diamond$ ) (proposed essentially in the same paper [B75]), claiming that $R(L) = 2F(L)$, is false. The graph $C$ shown in Fig.1 is a counterexample as $R(C) = 2F(C)-1$.\n\nMore information on Problem ( $\\dag$ ) and Conjecture ( $\\diamond$ ) can be found here (via Problem ( $\\star$ ) and Beneš conjecture).\n\nBibliography:\n*[B75] V.E. Beneš, Proving the rearrangeability of connecting networks by group calculation, Bell Syst. Tech. J. 54 (1975), 421-434.\n\nRelated:\nRelated problems\nBeneš Conjecture\nShuffle-Exchange Conjecture\nShuffle-Exchange Conjecture (graph-theoretic form)\n\nDiscussion links:\n- 2-stage Shuffle-exchange graph $\\text{SE}(2,3)$: http://www.openproblemgarden.org/?q=node/37089\n- proofs known for $R(A)=4$: http://www.openproblemgarden.org/?q=node/37167\n- Problem ( $\\star$ ) and Beneš conjecture: http://www.openproblemgarden.org/?q=node/37181\n- ( $\\star$ ): http://www.openproblemgarden.org/?q=node/37181\n- complete: http://www.openproblemgarden.org/?q=node/37181\n- factorization algorithm: http://www.openproblemgarden.org/?q=node/37181\n- Problem ( $\\star$ ): http://www.openproblemgarden.org/?q=node/37181\n- Beneš conjecture: http://www.openproblemgarden.org/?q=node/37181\n- here: http://www.openproblemgarden.org/?q=node/37181\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 95.\n\nAttempt notes:\nTarget:\nMake progress on \"Beneš Conjecture (graph-theoretic form)\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No proof or counterexample to the precise general inequality R(L) <= 2F(L) for simple regular ordered two-stage graphs was located. Important shuffle-exchange instances and small cases are known, while several stronger or hypothesis-weakened variants are false.\n\n**Verified partial progress.**\n\n- For the classical shuffle-exchange formulation, Ngo and Du verified the Beneš conjecture for n=4 and improved the general upper bound to 3n-5 stages.\n- The Open Problem Garden formulation records examples with R(A)=4, R(C)=5, and R(D)=6, and explicit counterexamples showing simplicity and uniformity cannot simply be dropped.\n- The stronger equality R(L)=2F(L) is false; the surviving conjecture is the inequality.\n\n**Full solution or refutation.**\n\nBeneš's 1975 group-theoretic formulation initiated the conjecture. The checked literature supports special networks and small parameter values but does not supply a general theorem for the graph-theoretic statement in this record.\n\n**What remains.**\n\nProve R(L) <= 2F(L) for every simple regular ordered L, or give a counterexample satisfying all three hypotheses; more broadly, characterize rearrangeable straight uniform multistage graphs.\n\n**Sources checked.**\n\n- V. E. Beneš, Proving the Rearrangeability of Connecting Networks by Group Calculations, Bell System Technical Journal 54 (1975), 421-434, DOI 10.1002/j.1538-7305.1975.tb02845.x. (primary): https://doi.org/10.1002/j.1538-7305.1975.tb02845.x\n Evidence used: Original group-calculation formulation and network context.\n- H. Q. Ngo and D.-Z. Du, Remarks on Beneš Conjecture, in Switching Networks: Recent Advances (2001), 257-258, DOI 10.1007/978-1-4613-0281-0_11. (primary): https://doi.org/10.1007/978-1-4613-0281-0_11\n Evidence used: Reports verification for n=4, the 3n-5 upper bound, and gaps in earlier claimed proofs.\n- Open Problem Garden, Beneš Conjecture (graph-theoretic form), checked 17 August 2026. (maintained_tracker): https://garden.irmacs.sfu.ca/op/bene_conjecture_graph_theoretic_form_0\n Evidence used: Gives the exact R(L), F(L) reformulation, tested examples, and counterexamples to stronger variants.\n\n**Review notes.** The record bundles an open-ended sufficient-condition problem with the precise inequality; no claim is made that progress on the classical shuffle-exchange subcase settles the full graph form.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3111, "problem_number": "OPG-37211", "title": "Approximation Ratio for Maximum Edge Disjoint Paths problem", "statement": "Conjecture Can the approximation ratio $O(\\sqrt{n})$ be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than $\\mathcal{APX}$-hardness?", "background": "Source: Open Problem Garden. Original node ID: 37211. URL: http://www.openproblemgarden.org/op/approximation_ratio_for_maximum_edge_disjoint_paths_problem.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/approximation_ratio_for_maximum_edge_disjoint_paths_problem\n- Author(s): Bentz, Cedric\n- Subject(s): Graph Theory\n- Keywords: approximation algorithms; Disjoint paths; planar graph; polynomial algorithm\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 18th, 2010 by jcmeyer\n\nProblem-page discussion:\nAssume a flow graph $G = (V, E)$ with $n$ vertices and $m$ edges. Each edge has a capacity function $c: E \\rightarrow \\mathbb{Z}^+$ (Flow network). The graph contains a list $\\mathcal{N}$ of terminal vertices called sources ( $s_i$ ) and sinks ( $s_i'$ ). Each pair ( $s_i, s_i'$ ) defines a net or commodity.\n\nA Multiflow is a way of routing commodities from their sources to the respective sinks while ensuring that the flow of each commodity is conserved at each non-terminal vertex and that the sum of the flows of all commodities through an edge does not exceed the capacity of the edge.\n\nThe Maximum Integer Multiflow problem (MaxIMF) seeks to maximize the number of flow units routed between the nets in the graph. The Maximum Edge Disjoint Paths (MaxEDP) problem seeks to find the maximum number of disjoint paths between the sources and sinks. When the capacities for all edges are set to one, MaxIMF simplifies into the MaxEDP problem.\n\nBentz provides an algorithm to find the MaxEDP with a proven approximation ratio (Approximation and integrality gap) of $O(\\sqrt{n})$. Can the approximation ratio be improved for MaxEDP in planar graphs, or can an inapproximability result stronger than $\\mathcal{APX}$-hardness be proved for this problem? And what about the general graphs?\n\nBibliography:\nCédric Bentz, Edge disjoint paths and max integral muliflow/min multicut theorems in planar graphs, Electronic Notes in Discrete Mathematics 22 (2005), 55–60\n\nDiscussion links:\n- Flow network: http://en.wikipedia.org/wiki/Flow network\n- Approximation and integrality gap: http://en.wikipedia.org/wiki/Linear_programming_relaxation#Approximation_and_integrality_gap\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Approximation Ratio for Maximum Edge Disjoint Paths problem\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For general planar MaxEDP with no congestion, the O(sqrt(n)) approximation barrier and the gap above APX-hardness remain unresolved. Stronger ratios are known under additional planarity or connectivity assumptions.\n\n**Verified partial progress.**\n\n- Eulerian planar and 4-edge-connected planar instances admit an O(log n)-approximation without congestion.\n- Fully planar instances, where adding all demand edges preserves planarity, admit a constant-factor approximation.\n- Allowing congestion 2 yields constant or polylogarithmic approximations, but that is not the literal edge-disjoint problem.\n\n**Full solution or refutation.**\n\nThe known improvements do not apply to arbitrary planar supply graphs with arbitrary demand pairs and congestion one. The natural multicommodity-flow relaxation also has an Omega(sqrt(n)) integrality gap on planar instances, so beating the algorithmic ratio requires a different relaxation or approach.\n\n**What remains.**\n\nImprove O(sqrt(n)) for all planar MaxEDP instances without congestion, or prove a quantitatively stronger inapproximability threshold than APX-hardness for that exact class.\n\n**Sources checked.**\n\n- C. Chekuri, S. Khanna, and F. B. Shepherd, An O(sqrt(n)) Approximation and Integrality Gap for Disjoint Paths and Unsplittable Flow, Theory of Computing 2 (2006), 137-146. (primary): https://theoryofcomputing.org/articles/v002a007/\n Evidence used: Establishes the baseline O(sqrt(n)) approximation and matching-scale LP integrality gap.\n- K. Kawarabayashi and Y. Kobayashi, An O(log n)-Approximation Algorithm for the Edge-Disjoint Paths Problem in Eulerian Planar Graphs, ACM Transactions on Algorithms 9 (2013), DOI 10.1145/2438645.2438648. (primary): https://doi.org/10.1145/2438645.2438648\n Evidence used: Proves O(log n) for Eulerian or 4-edge-connected planar supply graphs.\n- C.-C. Huang, M. Mari, C. Mathieu, K. Schewior, and J. Vygen, An Approximation Algorithm for Fully Planar Edge-Disjoint Paths, SIAM Journal on Discrete Mathematics 35 (2021), DOI 10.1137/20M1319401. (primary): https://doi.org/10.1137/20M1319401\n Evidence used: Proves a constant-factor approximation when the supply graph together with demand edges is planar.\n\n**Review notes.** Results allowing congestion greater than one are recorded only as nearby progress and not treated as resolutions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3112, "problem_number": "OPG-37217", "title": "Approximation ratio for k-outerplanar graphs", "statement": "Conjecture Is the approximation ratio for the Maximum Edge Disjoint Paths (MaxEDP) or the Maximum Integer Multiflow problem (MaxIMF) bounded by a constant in $k$-outerplanar graphs or tree-width graphs?", "background": "Source: Open Problem Garden. Original node ID: 37217. URL: http://www.openproblemgarden.org/op/approximation_ratio_for_k_outerplanar_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/approximation_ratio_for_k_outerplanar_graphs\n- Author(s): Bentz, Cedric\n- Subject(s): Graph Theory\n- Keywords: approximation algorithms; planar graph; polynomial algorithm\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 18th, 2010 by jcmeyer\n\nProblem-page discussion:\nAssume a flow graph $G = (V, E)$ with $n$ vertices and $m$ edges (Flow network). Each edge has a capacity function $c: E \\rightarrow \\mathbb{Z}^+$. The graph contains a list $\\mathcal{N}$ of terminal vertices called sources ( $s_i$ ) and sinks ( $s_i'$ ). Each pair ( $s_i, s_i'$ ) defines a net or commodity.\n\nA Multiflow is a way of routing commodities from their sources to the respective sinks while ensuring that the flow of each commodity is conserved at each non-terminal vertex and that the sum of the flows of all commodities through an edge does not exceed the capacity of the edge.\n\nThe Maximum Integer Multiflow problem (MaxIMF) seeks to maximize the number of flow units routed between the nets in the graph. The Maximum Edge Disjoint Paths (MaxEDP) problem seeks to find the maximum number of disjoint paths between the sources and sinks. When the capacities for all edges are set to one, MaxIMF simplifies into the MaxEDP problem.\n\nIs the approximation ratio (Approximation and integrality gap) for MaxEDP or MaxIMF bounded by a constant in $k$-outerplanar graphs (Outerplanar graphs) or tree-width graphs?\n\nBibliography:\nC. Bentz, “Disjoint paths in sparse graphs,” Discrete Appl. Math., vol. 157, no. 17, pp. 3558–3568, 2009\n\nDiscussion links:\n- Flow network: http://en.wikipedia.org/wiki/Flow network\n- Approximation and integrality gap: http://en.wikipedia.org/wiki/Linear_programming_relaxation#Approximation_and_integrality_gap\n- Outerplanar graphs: http://en.wikipedia.org/wiki/Planar_graph#Outerplanar_graphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Approximation ratio for k-outerplanar graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The MaxEDP part has an affirmative answer on bounded-treewidth graphs: treewidth at most r admits an O(r^3)-approximation, constant for fixed r and hence for fixed k-outerplanar classes. The bundled MaxIMF question is not settled by this theorem.\n\n**Verified partial progress.**\n\n- Ene, Mnich, Pilipczuk, and Risteski prove an O(r^3)-approximation and O(r^3) multicommodity-flow integrality gap for edge-capacitated MaxEDP on treewidth-r graphs.\n- Because k-outerplanar graphs have treewidth bounded as a function of k, the theorem supplies a constant depending on k for MaxEDP.\n- Outerplanar graphs also have constant-factor approximation results for more general weighted edge-disjoint-path formulations.\n\n**Full solution or refutation.**\n\nThe 2016 bounded-treewidth result removes dependence on the number of vertices and demands for MaxEDP, exactly the central approximation-ratio issue for fixed treewidth. It does not by itself establish the same guarantee for the record's separately named Maximum Integer Multiflow objective.\n\n**What remains.**\n\nClarify whether the original 'or' asks for both MaxEDP and MaxIMF. For the unresolved reading, establish a constant depending only on treewidth or k for MaxIMF, or prove a contrary hardness result.\n\n**Sources checked.**\n\n- A. Ene, M. Mnich, M. Pilipczuk, and A. Risteski, On Routing Disjoint Paths in Bounded Treewidth Graphs, SWAT 2016, LIPIcs 53, Article 15, DOI 10.4230/LIPIcs.SWAT.2016.15. (primary): https://doi.org/10.4230/LIPIcs.SWAT.2016.15\n Evidence used: Theorem gives an O(r^3)-approximation for MaxEDP on graphs of treewidth at most r.\n- G. Naves, B. Shepherd, and H. Xia, Maximum Weight Disjoint Paths in Outerplanar Graphs via Single-Tree Cut Approximators, arXiv:2007.10537 (2020). (primary): https://arxiv.org/abs/2007.10537\n Evidence used: Provides a constant-factor approximation for a general weighted no-congestion EDP formulation on outerplanar graphs.\n\n**Review notes.** Formulation defect: 'tree-width graphs' must mean a class of bounded treewidth, and the scope of the connective 'or' is ambiguous.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3113, "problem_number": "OPG-37218", "title": "Finding k-edge-outerplanar graph embeddings", "statement": "Conjecture It has been shown that a $k$-outerplanar embedding for which $k$ is minimal can be found in polynomial time. Does a similar result hold for $k$-edge-outerplanar graphs?", "background": "Source: Open Problem Garden. Original node ID: 37218. URL: http://www.openproblemgarden.org/op/finding_k_edge_outerplanar_graph_embeddings.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/finding_k_edge_outerplanar_graph_embeddings\n- Author(s): Bentz, Cedric\n- Subject(s): Graph Theory\n- Keywords: planar graph; polynomial algorithm\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 18th, 2010 by jcmeyer\n\nProblem-page discussion:\nA $k$-outerplanar graph [Baker] with $k > 0$ is a planar graph having an embedding with at most $k$ layers of vertices such that after removing iteratively the vertices (and their adjacent edges) lying on the outer face $k$ times, we obtain the empty graph.\n\nA $k$-edge-outerplanar graph [Bentz] is defined to be a planar graph having an embedding with at most $k$ layers of edges such that after removing iteratively the edges lying on the outer face $k$ times, we obtain a graph with no edge. All $k$-edge-outerplanar graphs are $k$-outerplanar graphs.\n\nGiven a planar graph, Bienstock and Monma have shown that a $k$-outerplanar embedding for which $k$ is minimal can be found in polynomial time. Does a similar result hold for $k$-edge-outerplanar graphs?\n\nBibliography:\nC. Bentz, “Disjoint paths in sparse graphs,” Discrete Appl. Math., vol. 157, no. 17, pp. 3558–3568, 2009\n\nD. Bienstock and C. L. Monma, “On the complexity of embedding planar graphs to minimize certain distance measures,” Algorithmica, vol. 5, no. 1–4, pp. 93–109, 1990\n\nB. S. Baker, “Approximation algorithms for np-complete problems on planar graphs,” J. ACM, vol. 41, no. 1, pp. 153–180, 1994\n\nDiscussion links:\n- $k$-outerplanar graph: http://en.wikipedia.org/wiki/Planar_graph#Outerplanar_graphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Finding k-edge-outerplanar graph embeddings\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A July 2026 preprint gives an affirmative answer: a planar embedding of minimum possible edge-outerplanarity can be found in polynomial time for every finite loopless planar graph.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nYu proves a fixed-embedding formula equating an edge's peeling round with one plus the minimum dual distance of its incident faces. Marker-triangle gadgets reduce optimization of edge-outerplanarity to the known polynomial-time minimum-face-depth embedding problem. Projecting an optimal embedding of the auxiliary graph back to the original graph preserves the optimum, yielding an O(n^4)-type polynomial algorithm.\n\n**What remains.**\n\nIndependent expert verification and conventional peer review are advisable because the result is a very recent preprint. Extensions to graphs with loops would require a definition and separate treatment.\n\n**Sources checked.**\n\n- H. Yu, Minimum Edge-Outerplanar Embeddings are Polynomial-Time Computable, arXiv:2607.08110 (2026). (primary): https://arxiv.org/abs/2607.08110\n Evidence used: States and proves the polynomial-time theorem, explicitly identifying it as a resolution of Bentz's 2009 question.\n- C. Bentz, Disjoint paths in sparse graphs, Discrete Applied Mathematics 157 (2009), 3558-3568, DOI 10.1016/j.dam.2009.03.009. (primary): https://doi.org/10.1016/j.dam.2009.03.009\n Evidence used: Original source of the edge-outerplanar embedding question.\n\n**Review notes.** The solving source is unrefereed and reports that its initial proof was AI-generated before manual verification and polishing; confidence is therefore medium rather than high.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3114, "problem_number": "OPG-37229", "title": "Exact colorings of graphs", "statement": "Conjecture For $c \\geq m \\geq 1$, let $P(c,m)$ be the statement that given any exact $c$-coloring of the edges of a complete countably infinite graph (that is, a coloring with $c$ colors all of which must be used at least once), there exists an exactly $m$-colored countably infinite complete subgraph. Then $P(c,m)$ is true if and only if $m=1$, $m=2$, or $c=m$.", "background": "Source: Open Problem Garden. Original node ID: 37229. URL: http://www.openproblemgarden.org/op/exact_colorings_of_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/exact_colorings_of_graphs\n- Author(s): Erickson, Martin\n- Subject(s): Graph Theory\n- Keywords: graph coloring; ramsey theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 29th, 2010 by Martin Erickson\n\nProblem-page discussion:\nStacey and Weidl have shown that given $m \\geq 3$, there is an integer $C(m)$ such that $P(c,m)$ is false for all $c \\geq C(m)$.\n\nBibliography:\n* M. Erickson, \"A Conjecture Concerning Ramsey's Theorem,\" Discrete Mathematics 126, 395--398 (1994); MR 95b:05209\n\nA. Stacey and P. Weidl, \"The Existence of Exactly m-Coloured Complete Subgraphs,\" J. of Combinatorial Theory, Series B 75, 1-18 (1999)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Exact colorings of graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A 2025 paper proves Erickson's proposed classification for all sufficiently large m and every c>m, reducing the unresolved work to finitely many parameter cases.\n\n**Verified partial progress.**\n\n- Stacey--Weidl had already proved failure for c sufficiently large relative to each fixed m>=3.\n- Ranđelović proves the conjecture for all sufficiently large m and all c>m.\n\n**Full solution or refutation.**\n\nA finite set of exceptional parameter pairs remains to be resolved.\n\n**What remains.**\n\nSettle the remaining finite set of pairs or produce a counterexample to the conjectured classification.\n\n**Sources checked.**\n\n- Ž. Ranđelović, Exactly Colored Complete Subgraphs of Infinite Graphs, arXiv:2512.04233 (2025). (primary): https://arxiv.org/abs/2512.04233\n Evidence used: The abstract says the result holds for all sufficiently large m and every c>m, reducing verification to finitely many cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3115, "problem_number": "OPG-37271", "title": "Star chromatic index of cubic graphs", "statement": "The star chromatic index $\\chi_s'(G)$ of a graph $G$ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.\n\nQuestion Is it true that for every (sub)cubic graph $G$, we have $\\chi_s'(G) \\le 6$?", "background": "Source: Open Problem Garden. Original node ID: 37271. URL: http://www.openproblemgarden.org/op/star_chromatic_index_of_cubic_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/star_chromatic_index_of_cubic_graphs\n- Author(s): Dvorak, Zdenek; Mohar, Bojan; Samal, Robert\n- Subject(s): Graph Theory\n- Keywords: edge coloring; star coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 16th, 2010 by Robert Samal\n\nProblem-page discussion:\nThe star chromatic number is the more usual concept [ACKKR,FRR]. Star chromatic index of a graph $G$ is simply the star chromatic number of the line graph $L(G)$; the definition given above is easily seen to be equivalent.\n\nDvořák, Mohar, and Šámal [DMS] show that every (sub)cubic graph $G$, satisfies $\\chi_s'(G) \\le 7$? On the other hand, it is simple to check that $\\chi_s'(K_{3,3])=6$, so the conjecture, if true, is tight.\n\nBibliography:\n[ACKKR] Albertson, Michael O.; Chappell, Glenn G.; Kierstead, Hal A.; Kündgen, André; Ramamurthi, Radhika: Coloring with no 2-Colored P4's, The Electronic Journal of Combinatorics 11 (1).\n\n[FRR] Fertin, Guillaume; Raspaud, André; Reed, Bruce, Star coloring of graphs, Journal of Graph Theory 47 (3): 163-182, doi:10.1002/jgt.20029.\n\n*[DMS] Dvořák, Zdeněk; Mohar, Bojan; Šámal, Robert: Star chromatic index, arXiv:1011.3376.\n\nDiscussion links:\n- star chromatic number: http://en.wikipedia.org/wiki/Star coloring\n- line graph: http://en.wikipedia.org/wiki/line graph\n\nBibliography links:\n- doi:10.1002/jgt.20029: http://dx.doi.org/10.1002/jgt.20029\n- arXiv:1011.3376: http://www.arxiv.org/abs/1011.3376\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Star chromatic index of cubic graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjectured 6-color bound for all subcubic graphs remains open, while it is established for cubic Halin graphs and other restricted classes.\n\n**Verified partial progress.**\n\n- Casselgren--Granholm--Raspaud settle the bound for cubic Halin graphs.\n\n**Full solution or refutation.**\n\nNo proof for every (sub)cubic graph was verified.\n\n**What remains.**\n\nProve the 6-star-edge-color bound generally or find a subcubic counterexample.\n\n**Sources checked.**\n\n- C. J. Casselgren, J. B. Granholm and A. Raspaud, On star edge colorings of bipartite and subcubic graphs, arXiv:1912.02467 (2019). (primary): https://arxiv.org/abs/1912.02467\n Evidence used: The abstract calls the subcubic 6-color statement a well-known conjecture and proves the cubic-Halin case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3116, "problem_number": "OPG-37275", "title": "Star chromatic index of complete graphs", "statement": "Conjecture Is it possible to color edges of the complete graph $K_n$ using $O(n)$ colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?\n\nEquivalently: is the star chromatic index of $K_n$ linear in $n$?", "background": "Source: Open Problem Garden. Original node ID: 37275. URL: http://www.openproblemgarden.org/op/star_chromatic_index_of_complete_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/star_chromatic_index_of_complete_graphs\n- Author(s): Dvorak, Zdenek; Mohar, Bojan; Samal, Robert\n- Subject(s): Graph Theory\n- Keywords: complete graph; edge coloring; star coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: November 16th, 2010 by Robert Samal\n\nProblem-page discussion:\nThe star chromatic index $\\chi_s'(G)$ of a graph $G$ is the minimum number of colors needed to properly color the edges of $G$ so that no path or cycle of length four is bi-colored. An equivalent definition is that $\\chi_s'(G)$ is the star chromatic number of the line graph $L(G)$.\n\nDvořák, Mohar, and Šámal [DMS] show that $\\chi_s'(G) \\ge (2+o(1))n$. On the other hand, the best known upper bound (also in \\cite{DMS]) is superlinear: $$\\chi_s'(K_n) \\le n \\cdot \\frac{ 2^{ 2\\sqrt2(1+o(1)) \\sqrt{\\log n} } }{(\\log n)^{1/4}} \\,.$$\n\nIt may be possible to decrease the upper bound by elementary methods.\n\nBibliography:\n*[DMS] Dvořák, Zdeněk; Mohar, Bojan; Šámal, Robert: Star chromatic index, arXiv:1011.3376.\n\nRelated:\nRelated problems\nStar chromatic index of cubic graphs\n\nDiscussion links:\n- line graph: http://en.wikipedia.org/wiki/line graph\n\nBibliography links:\n- arXiv:1011.3376: http://www.arxiv.org/abs/1011.3376\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Star chromatic index of complete graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The question whether the star chromatic index of K_n is O(n) remains open; a 2026-reviewed tracker reports that the original near-linear upper bound and asymptotic 2n lower bound remain the state of the art.\n\n**Verified partial progress.**\n\n- Dvořák, Mohar and Šámal proved (2+o(1))n as a lower bound for complete graphs.\n- They proved a near-linear upper bound n*2^{2sqrt(2)(1+o(1))sqrt(log n)}/(log n)^{1/4}.\n\n**Full solution or refutation.**\n\nNo linear upper bound or superlinear obstruction has been verified.\n\n**What remains.**\n\nProve chi'_s(K_n)=O(n), or prove a superlinear lower bound.\n\n**Sources checked.**\n\n- Z. Dvořák, B. Mohar and R. Šámal, Star chromatic index, arXiv:1011.3376 (2010; rev. 2011). (primary): https://arxiv.org/abs/1011.3376\n Evidence used: The abstract defines the invariant and states the near-linear upper bound and asymptotic complete-graph lower bound.\n- Graph-theory open problems, Star chromatic index of complete graphs (reviewed 2026-05-08). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/star_chromatic_index_of_complete_graphs/\n Evidence used: The maintained page labels the problem open with high confidence and says the original bounds remain the state of the art.\n\n**Review notes.** The statement is clear; '4-edge path' is read as a path of length four.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3117, "problem_number": "OPG-37316", "title": "Vertex Coloring of graph fractional powers", "statement": "Conjecture Let $G$ be a graph and $k$ be a positive integer. The $k-$ power of $G$, denoted by $G^k$, is defined on the vertex set $V(G)$, by connecting any two distinct vertices $x$ and $y$ with distance at most $k$. In other words, $E(G^k)=\\{xy:1\\leq d_G(x,y)\\leq k\\}$. Also $k-$ subdivision of $G$, denoted by $G^\\frac{1}{k}$, is constructed by replacing each edge $ij$ of $G$ with a path of length $k$. Note that for $k=1$, we have $G^\\frac{1}{1}=G^1=G$.\nNow we can define the fractional power of a graph as follows:\nLet $G$ be a graph and $m,n\\in \\mathbb{N}$. The graph $G^{\\frac{m}{n}}$ is defined by the $m-$ power of the $n-$ subdivision of $G$. In other words $G^{\\frac{m}{n}}\\isdef (G^{\\frac{1}{n}})^m$.\nConjecture. Let $G$ be a connected graph with $\\Delta(G)\\geq3$ and $m$ be a positive integer greater than 1. Then for any positive integer $n>m$, we have $\\chi(G^{\\frac{m}{n}})=\\omega(G^\\frac{m}{n})$.\nIn [1], it was shown that this conjecture is true in some special cases.", "background": "Source: Open Problem Garden. Original node ID: 37316. URL: http://www.openproblemgarden.org/op/vertex_coloring_of_graph_fractional_powers.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/vertex_coloring_of_graph_fractional_powers\n- Author(s): Iradmusa, Moharram\n- Subject(s): Graph Theory\n- Keywords: chromatic number, fractional power of graph, clique number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: April 23rd, 2011 by Iradmusa\n\nBibliography:\n[1] Iradmusa, Moharram N., On colorings of graph fractional powers. Discrete Math. 310 (2010), no. 10-11, 1551–1556.\n\nComments:\n- July 13th, 2011 | Anonymous | Needs revision: Note that if K_t is the complete graph on t vertices with t even, then the 2-power of the 2-subdivision of K_t is isomorphic to the total graph of K_t. That is the graph T(K_t) whose vertex set is V(K_t) union E(K_t) and two vertices are adjacent in T(K_t) if their either adjacent or incident in K_t.\n\nclique number of T(K_t) is t + 1 and the chromatic number of T(K_t) is >= t+2.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"Vertex Coloring of graph fractional powers\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Hartke, Liu, and Petříčková explicitly disprove Iradmusa's conjecture with a graph H satisfying chi(H^(3/5)) > omega(H^(3/5)).\n\n**Verified partial progress.**\n\n- The conjectured equality holds whenever m is even.\n- For odd m and maximum degree at least 4, chi(G^(m/n)) <= omega(G^(m/n)) + 2.\n\n**Full solution or refutation.**\n\nThe universal equality is false; the 3/5-power counterexample satisfies all of the conjecture's parameter inequalities.\n\n**What remains.**\n\nClassify the odd-m cases attaining a gap and determine when equality still holds.\n\n**Sources checked.**\n\n- S. Hartke, H. Liu and Š. Petříčková, On coloring of fractional powers of graphs, arXiv:1212.3898 (2012; revised 2014). (primary): https://arxiv.org/abs/1212.3898\n Evidence used: The abstract identifies the source conjecture, supplies an H with chi(H^(3/5)) greater than omega(H^(3/5)), proves the even-m case, and gives the additive-two odd-m bound.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3118, "problem_number": "OPG-37325", "title": "Covering powers of cycles with equivalence subgraphs", "statement": "Conjecture Given $k$ and $n$, the graph $C_{n}^k$ has equivalence covering number $\\Omega(k)$.", "background": "Source: Open Problem Garden. Original node ID: 37325. URL: http://www.openproblemgarden.org/op/covering_powers_of_cycles_with_equivalence_subgraphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/covering_powers_of_cycles_with_equivalence_subgraphs\n- Subject(s): Graph Theory\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: July 7th, 2011 by Andrew King\n\nProblem-page discussion:\nGiven a graph $G$, a subgraph $H$ of $G$ is an equivalence subgraph of $G$ if $H$ a disjoint union of cliques. The quivalence covering number of $G$, denoted $eq(G)$, is the least number of equivalence subgraphs needed to cover the edges of $G$.\n\nThis problem has been studied by various people since the 80s [A]. For line graphs, the equivalence covering number is known to within a constant factor [EGK]. It is therefore tempting to examine the situation for quasi-line graphs and claw-free graphs. Powers of cycles are perhaps the simplest interesting class of claw-free graphs that are not necessarily line graphs. However, even for $n$ very large compared to $k$, no upper bound is known beyond trivial linear bounds of order $\\Theta(k)$. Furthermore, it is not even certain that a nontrivial lower bound (i.e. going to infinity as $k$ goes to infinity) is known. It is possible that this can be related somehow to a known result, but for now it seems at least superficially that this problem is wide open.\n\nBibliography:\n[A] N. Alon, Covering graphs with the minimum number of equivalence relations, Combinatorica 6 (1986) 201–206.\n\n[EGK] L. Esperet, J. Gimbel, A. King, Covering line graphs with equivalence relations, Discrete Applied Mathematics Volume 158, Issue 17, 28 October 2010, Pages 1902-1907.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Covering powers of cycles with equivalence subgraphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The displayed Omega(k) claim omits the required asymptotic regime. Uniformly over all n and k it fails in the complete-graph range, while the intended regime n very large relative to k remains an open lower-bound problem on the maintained source page.\n\n**Verified partial progress.**\n\n- eq(C_n^k) = k+1 when k+1 divides n.\n- eq(C_n^k) <= 2k for all parameters in the intended nondegenerate cycle-power setting.\n- If n <= 2k+1, then C_n^k is complete and eq(C_n^k)=1, showing why a uniform reading cannot be intended.\n\n**Full solution or refutation.**\n\nNo resolution is claimed because the literal wording and the intended large-n regime are different mathematical statements.\n\n**What remains.**\n\nState a precise limiting regime, such as a uniform lower bound for n >= f(k), and then prove an unbounded or linear lower bound or construct a sublinear equivalence covering.\n\n**Sources checked.**\n\n- D. West, Equivalence covering of cycle-powers, REGS problem page (accessed 2026-08-17). (maintained_tracker): https://dwest.web.illinois.edu/regs/eqcov.html\n Evidence used: The page defines eq, notes eq(K_n)=1, gives eq(C_n^k)=k+1 when k+1 divides n and the universal 2k upper bound, and says no good general lower bound is known.\n\n**Review notes.** The missing asymptotic quantifier is explicitly flagged rather than silently supplied from the background.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3119, "problem_number": "OPG-37357", "title": "Obstacle number of planar graphs", "statement": "Does there exist a planar graph with obstacle number greater than 1? Is there some $k$ such that every planar graph has obstacle number at most $k$?", "background": "Source: Open Problem Garden. Original node ID: 37357. URL: http://www.openproblemgarden.org/op/obstacle_number_of_planar_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/obstacle_number_of_planar_graphs\n- Author(s): Alpert, Hanna; Koch, Christina; Laison, Joshua D.\n- Subject(s): Graph Theory\n- Keywords: graph drawing; obstacle number; planar graph; visibility graph\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: November 23rd, 2011 by Andrew King\n\nProblem-page discussion:\nA $k$-obstacle drawing of a graph $G$ is a mapping of the vertices of $G$ to points in the plane, along with a set of polygonal obstacles $P_1,\\ldots, P_k$, such that two vertices are adjacent precisely if the line segment connecting their corresponding points in $\\mathbb R^2$ does not intersect any obstacle. The {\\em obstacle number} of a graph $G$ is the minimum $k$ such that $G$ has a $k$-obstacle drawing.\n\nThis invariant was recently introduced by Alpert, Koch, and Laison [AKL], who proved that every outerplanar graph has obstacle number 1. The next question, then, follows naturally: what is the obstacle number of a planar graph? So far no planar graph has been proved to have obstacle number greater than 1. Alpert, Koch, and Laison specifically ask what the obstacle numbers of the icosahedron and dodecahedron are [AKL].\n\nBibliography:\n[AKL] Hannah Alpert, Christina Koch, and Joshua D. Laison: Obstacle numbers of graphs. Discrete Comput. Geom. (2010) 44:223-244.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Obstacle number of planar graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The first question is answered affirmatively by the icosahedron, whose standard obstacle number is 2; the existence of a universal constant for all planar graphs remains open in the verified sources.\n\n**Verified partial progress.**\n\n- The icosahedron and a related polyhedral graph have standard obstacle number 2.\n- Every bipartite planar graph of order at least 3 has standard obstacle number 1.\n- The authors who found the icosahedral example conjecture that every planar graph has standard obstacle number at most 2.\n\n**Full solution or refutation.**\n\nA planar graph with obstacle number greater than 1 exists, but no universal constant bound for all planar graphs has been proved.\n\n**What remains.**\n\nProve the conjectured bound obs(G) <= 2 for every planar graph or construct planar graphs with larger or unbounded standard obstacle number.\n\n**Sources checked.**\n\n- L. W. Berman, G. G. Chappell, J. R. Faudree, J. Gimbel, C. Hartman and G. I. Williams, Graphs with Obstacle Number Greater than One, Journal of Graph Algorithms and Applications 21(6) (2017), 1107-1119. (primary): https://doi.org/10.7155/jgaa.00452\n Evidence used: The paper proves that the icosahedron has obstacle number 2 and explicitly poses/conjectures a universal bound of 2 for planar graphs.\n- J. Gimbel, P. Ossona de Mendez and P. Valtr, Obstacle Numbers of Planar Graphs, arXiv:1706.06992. (primary): https://arxiv.org/abs/1706.06992\n Evidence used: The paper proves standard obstacle number 1 for bipartite planar graphs and distinguishes this parameter from planar obstacle number, whose extremal value is n-3.\n- Graph-theory open problems index, Obstacle number of planar graphs (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/index.html\n Evidence used: The maintained index marks the two-part source problem partial rather than solved.\n\n**Review notes.** The standard obstacle number is kept distinct from planar obstacle number throughout.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3120, "problem_number": "OPG-37364", "title": "Matching cut and girth", "statement": "Question For every $d$ does there exists a $g$ such that every graph with average degree smaller than $d$ and girth at least $g$ has a matching-cut?", "background": "Source: Open Problem Garden. Original node ID: 37364. URL: http://www.openproblemgarden.org/op/matching_cut_and_girth.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/matching_cut_and_girth\n- Subject(s): Graph Theory\n- Keywords: matching cut, matching, cut\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 30th, 2011 by w\n\nProblem-page discussion:\nLet $G=(V,E)$ be a graph. A matching $M$ is a matching-cut if there exists a set $S\\subset V$ such that $M = E(S:V\\setminus S)$. Graphs having a matching-cut are called decomposable.\n\nIt is known that every graph with $|E| < 3(|V|-1)/2$ is decomposable [BFP11].\n\nBibliography:\n[C84] V. Chvátal, Recognizing decomposable graphs, J Graph Theory 8 (1984), 51–53\n\n[BFP11] P. Bonsma, A. Farley, A. Proskurowski, Extremal graphs having no matching cuts, J Graph Theory (2011)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Matching cut and girth\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The matching-cut/girth question has a negative answer: high-girth bounded-degree graphs without a matching cut are constructed for every prescribed girth.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe cited ISAAC paper explicitly identifies and answers the Open Problem Garden question negatively.\n\n**What remains.**\n\nStudy sharp average-degree thresholds or restricted graph classes, rather than the refuted universal assertion.\n\n**Sources checked.**\n\n- C. Feghali, F. Lucke, D. Paulusma and B. Ries, Matching Cuts in Graphs of High Girth and H-Free Graphs, ISAAC 2023, arXiv:2212.12317. (primary): https://arxiv.org/abs/2212.12317\n Evidence used: The paper's introduction states that its high-girth NO-instances answer the quoted Open Problem Garden question in the negative.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3121, "problem_number": "OPG-37420", "title": "Minimal graphs with a prescribed number of spanning trees", "statement": "Conjecture Let $n \\geq 3$ be an integer and let $\\alpha(n)$ denote the least integer $k$ such that there exists a simple graph on $k$ vertices having precisely $n$ spanning trees. Then $\\alpha(n) = o(\\log{n}).$", "background": "Source: Open Problem Garden. Original node ID: 37420. URL: http://www.openproblemgarden.org/op/minimal_graphs_with_a_prescribed_number_of_spanning_trees.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/minimal_graphs_with_a_prescribed_number_of_spanning_trees\n- Author(s): Azarija, Jernej; Skrekovski, Riste\n- Subject(s): Graph Theory\n- Keywords: number of spanning trees, asymptotics\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: April 22nd, 2012 by azi\n\nProblem-page discussion:\nObserve that $\\alpha(n)$ is well defined for $n \\geq 3$ since $C_n$ has $n$ spanning trees.\n\nThe function was introduced by Sedlacek [S] who has shown that for large enough $n$ $\\alpha(n) \\leq \\frac{n+6}{3} \\mbox{if } n \\equiv 0 \\pmod{3}$ and $\\alpha(n) \\leq \\frac{n+4}{3} \\mbox{if } n \\equiv 2 \\pmod{3}.$\n\nUsing the fact that almost all positive integers $n$ are expressible as $n = ab+ac+bc$ for integers $0 < a < b < c$ it can be shown [A] that for large enough $n$\n\n$\\alpha(n) \\leq \\frac{n+4}{3} \\mbox{if } n \\equiv 2 \\pmod{3}$ and $\\alpha(n) \\leq \\frac{n+9}{4}$ otherwise.\n\nMoreover, the only fixed points of $\\alpha$ are 3, 4, 5, 6, 7, 10, 13 and 22.\n\nThe conjecture is motivated by the following graph (ploted for a very small sample of vertices)\n\nThe conjecture [C] is justifiable for highly composite numbers $n$ since in this case one can construct the graph obtained after taking cycles $C_{p_1}, \\ldots,C_{p_k}$ for every odd prime factor $p_i$ of $n$.\n\nBibliography:\n[S] J. Sedlacek, On the minimal graph with a given number of spanning trees, Canad. Math. Bull. 13 (1970) 515-517.\n\n[A] J. Azarija, R. Skrekovski, Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees, IMFM preprints 49 (2011) Link to paper\n\n* [C] Minimal graphs with a prescribed number of spanning trees\n\nBibliography links:\n- Link to paper: http://www.imfm.si/preprinti/PDF/01157.pdf\n- Minimal graphs with a prescribed number of spanning trees: http://mathoverflow.net/questions/93656/minimal-graphs-with-a-prescribed-number-of-spanning-trees\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Minimal graphs with a prescribed number of spanning trees\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The simple-undirected graph question remains open, while a directed analogue achieves O(log n/log log n) vertices for every required number of rooted spanning arborescences.\n\n**Verified partial progress.**\n\n- A base-k construction realizes every integer up to k^k by a simple directed graph on O(k) vertices.\n\n**Full solution or refutation.**\n\nThe directed result does not prove alpha(n)=o(log n) for undirected simple graphs.\n\n**What remains.**\n\nTransfer the construction to undirected graphs or prove a lower obstruction.\n\n**Sources checked.**\n\n- MathOverflow, Minimal graphs with a prescribed number of spanning trees (accessed 2026-08-17). (authoritative_secondary): https://mathoverflow.net/questions/93656/minimal-graphs-with-a-prescribed-number-of-spanning-trees\n Evidence used: The expert discussion gives the directed O(log n/log log n) construction and identifies the undirected challenge.\n\n**Review notes.** Directed/undirected distinction explicit; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3122, "problem_number": "OPG-37670", "title": "The Borodin-Kostochka Conjecture", "statement": "Conjecture Every graph with maximum degree $\\Delta \\geq 9$ has chromatic number at most $\\max\\{\\Delta-1, \\omega\\}$.", "background": "Source: Open Problem Garden. Original node ID: 37670. URL: http://www.openproblemgarden.org/op/the_borodin_kostochka_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_borodin_kostochka_conjecture\n- Author(s): Borodin, Oleg V.; Kostochka, Alexandr V.\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 10th, 2012 by Andrew King\n\nProblem-page discussion:\nThe Borodin-Kostochka conjecture proposes that for any graph $G$ with maximum degree $\\Delta$ and clique number $\\omega < \\Delta$, $G$ is $\\Delta-1$ colourable so long as $\\Delta$ is sufficiently large (specifically, $\\Delta\\geq 9$ ). The requirement that $\\Delta \\geq 9$ is necessary, as one can see by looking at the strong product of $C_5$ and $K_3$.\n\nReed [R] proved that there exists a $\\Delta_0$ for which the conjecture holds whenever $\\Delta \\geq \\Delta_0$. Specifically he proved that $\\Delta_0 \\leq 10^{14}$, but claims that more careful analysis could reduce $\\Delta_0$ to 1000.\n\nThe conjecture was recently proven by Cranston and Rabern for claw-free graphs [CR]. In their paper they mention an unpublished strengthening proposed by Borodin and Kostochka, namely that one can replace the chromatic number in the statement of the conjecture with the list chromatic number.\n\nBibliography:\n[BK] O. V. Borodin and A. V. Kostochka. On an upper bound of a graph's chromatic number, depending on the graph's degree and density. JCTB 23 (1977), 247--250.\n\n[CR] D. W. Cranston and L. Rabern. Coloring claw-free graphs with $\\Delta-1$ colors, arXiv 1206.1269, 2012.\n\n[R] B. A. Reed. A strengthening of Brooks’ Theorem. J. Comb. Theory Ser. B, 76:136–149, 1999.\n\nBibliography links:\n- Coloring claw-free graphs with $\\Delta-1$ colors: http://www.arxiv.org/abs/1206.1269\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"The Borodin-Kostochka Conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Borodin-Kostochka conjecture remains open in general; large maximum degree and many graph classes are known, and the unresolved core has been reduced to Delta = 9.\n\n**Verified partial progress.**\n\n- Reed proves the conjecture for sufficiently large maximum degree.\n- Modern reductions show that proving the maximum-degree-nine case suffices, and recent work handles additional forbidden-substructure cases.\n\n**Full solution or refutation.**\n\nNo proof covering every graph with maximum degree at least 9 was verified.\n\n**What remains.**\n\nSettle the general Delta = 9 case, which would complete the conjecture after the known reductions.\n\n**Sources checked.**\n\n- B. Reed, A strengthening of Brooks' theorem, Journal of Combinatorial Theory, Series B 76 (1999), 136-149, DOI 10.1006/jctb.1998.1891. (primary): https://doi.org/10.1006/jctb.1998.1891\n Evidence used: Proves the conjectured inequality for sufficiently large maximum degree.\n- Rachel Galindo and Jessica McDonald, On graphs with chromatic number and maximum degree both equal to nine, arXiv:2408.12693. (primary): https://arxiv.org/abs/2408.12693\n Evidence used: States the current reduction to Delta = 9 and proves new structural and forbidden-subgraph cases rather than the full conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3123, "problem_number": "OPG-46443", "title": "Stable set meeting all longest directed paths.", "statement": "Conjecture Every digraph has a stable set meeting all longest directed paths", "background": "Source: Open Problem Garden. Original node ID: 46443. URL: http://www.openproblemgarden.org/op/stable_set_meeting_all_longest_directed_paths.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/stable_set_meeting_all_longest_directed_paths\n- Author(s): Laborde, Jean-Marie; Payan, Charles; Xuong N.H.\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 1st, 2013 by fhavet\n\nProblem-page discussion:\nIf the stability number is 1, that is if the digraph is a tournament, it follows Redei's Theorem stating that every tournament has a directed hamiltonian path. The conjecture has been proved by Havet [H] for digraphs having stability number 2.\n\nThe conjecture would give an easy inductive proof of Gallai-Roy Theorem: every digraph with chromatic number $k$ contains a directed path on $k$ vertices.\n\nHahn and Jackson [HJ] conjectured that in contrast there is no directed path meeting every maximum stable set. In fact, they conjectured the following: For each positive integer $k$, there is a digraph $D$ with stability number $k$ such that deleting the vertices of any $k-1$ directed paths in $D$ leaves a digraph with stability number $k$. This was proved by Fox and Sudakov [FS] by a probabilistic argument.\n\nBibliography:\n[FS] J. Fox and B. Sudakov, Paths and stability number in digraphs, Electronic Journal of Combiantorics, 16 (2009), no.1, N23.\n\n[HJ] G. Hahn and B. Jackson, A note concerning paths and independence number in digraphs, Discrete Math. 82 (1990), 327–329.\n\n[H] F. Havet. Stable set meeting every longest path. Discrete Mathematics, 289 (2004), no. 1-3, 169-173.\n\n*[LPX] J.M. Laborde, C. Payan, and N.H. Xuong, Independent sets and longest directed paths in digraphs. In Graphs and other Combinatorial Topics (Prague, 1982)}, Teubner-Texte Math., Vol. 59 (1983), 173-177, Teubner, Leipzig.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Stable set meeting all longest directed paths.\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The stable-set transversal conjecture for longest directed paths is open generally but proved for several structured digraph classes.\n\n**Verified partial progress.**\n\n- The statement holds for tournaments by Rédei's theorem and for further locally semicomplete/special classes.\n\n**Full solution or refutation.**\n\nNo proof for arbitrary digraphs was verified.\n\n**What remains.**\n\nProve a stable transversal for longest directed paths in every digraph or find a counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, Stable set meeting all longest directed paths (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/stable_set_meeting_all_longest_directed_paths\n Evidence used: The page identifies tournament and special-class results while retaining the general conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3124, "problem_number": "OPG-46496", "title": "Arc-disjoint strongly connected spanning subdigraphs", "statement": "Conjecture There exists an ineteger $k$ so that every $k$-arc-connected digraph contains a pair of arc-disjoint strongly connected spanning subdigraphs?", "background": "Source: Open Problem Garden. Original node ID: 46496. URL: http://www.openproblemgarden.org/op/arc_disjoint_strongly_connected_spanning_subdigraphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/arc_disjoint_strongly_connected_spanning_subdigraphs\n- Author(s): Bang-Jensen, Joergen; Yeo, Anders\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 2nd, 2013 by fhavet\n\nProblem-page discussion:\nBang-Jensen and Yeo [BY] proved the conjecture for several classes like tournaments. There is stronger conjecture for tournaments. Yeo (See [BG, Theorem 13.10.1]) showed that it is NP-complete to decide whether a 2-regular digraph has two arc-disjoint strongly connected spanning subdigraphs.\n\nA similar question can be asked about arc-disjoint out-branching and in-branching. Several related problems are mentioned in the survey of Bang-Jensen and Kriesell [BK].\n\nBibliography:\n[BG] J. Bang-Jensen, G. Gutin, Digraphs: Theory, Algorithms and Applications, 2nd. ed., Springer Verlag (2009).\n\n[BK] J. Bang-Jensen, M. Kriesell, Disjoint sub(di)graphs in digraphs, Electronic Notes in Discrete Mathematics 34 (2009), 179-183.\n\n*[BY] J. Bang-Jensen, A. Yeo, Decomposing k-arc-strong tournaments into strong spanning subdigraphs, Combinatorica 24 (2004), 331-349.\n\nRelated:\nRelated problems\nArc-disjoint out-branching and in-branching\nDecomposing k-arc-strong tournament into k spanning strong digraphs\n\nDiscussion links:\n- stronger conjecture for tournaments: http://www.openproblemgarden.org/?q=node/4765\n- arc-disjoint out-branching and in-branching: http://www.openproblemgarden.org/?q=node/46495\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Arc-disjoint strongly connected spanning subdigraphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The strong arc decomposition conjecture remains open for general digraphs. It holds for every 3-arc-strong split digraph, while infinite 2-arc-strong split counterexample families show that two does not suffice even in that class.\n\n**Verified partial progress.**\n\n- Bang-Jensen--Wang prove every 3-arc-strong split digraph has a strong arc decomposition, constructible in polynomial time.\n- They provide infinite classes of 2-arc-strong split digraphs without a strong arc decomposition.\n- For semicomplete digraphs, 2-arc-strongness suffices apart from one four-vertex exception.\n\n**Full solution or refutation.**\n\nNo universal arc-connectivity constant for arbitrary digraphs was verified.\n\n**What remains.**\n\nExtend the split/semicomplete methods to arbitrary highly arc-strong digraphs or construct counterexamples for every connectivity level.\n\n**Sources checked.**\n\n- Jørgen Bang-Jensen and Yun Wang, Strong arc decompositions of split digraphs, Journal of Graph Theory 108 (2025), 76-109; arXiv:2309.06904. (primary): https://arxiv.org/abs/2309.06904\n Evidence used: Explicitly restates the general conjecture, proves the 3-arc-strong split case, and gives infinite 2-arc-strong split counterexamples.\n- Open Problem Garden, Arc-disjoint strongly connected spanning subdigraphs. (maintained_tracker): https://www.openproblemgarden.org/op/arc_disjoint_strongly_connected_spanning_subdigraphs\n Evidence used: Preserves the source formulation and classical semicomplete results.\n\n**Review notes.** The input misspells integer as ineteger, uses k-arc-connected rather than the modern k-arc-strong terminology, and ends the conjecture with a question mark. The status match uses the standard directed meaning.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3125, "problem_number": "OPG-46538", "title": "Do any three longest paths in a connected graph have a vertex in common?", "statement": "Conjecture Do any three longest paths in a connected graph have a vertex in common?", "background": "Source: Open Problem Garden. Original node ID: 46538. URL: http://www.openproblemgarden.org/op/do_any_three_longest_paths_in_a_connected_graph_have_a_vertex_in_common.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/do_any_three_longest_paths_in_a_connected_graph_have_a_vertex_in_common\n- Author(s): Gallai, Tibor\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 3rd, 2013 by fhavet\n\nProblem-page discussion:\nIt is a well-known exercise that every two longest paths in a connected graph have a common vertex. Skupien [S] showed connected graphs where 7 longest paths do not share a common vertex.\n\nBibliography:\n*[G] T. Gallai, Problem 6. In Theory of Graphs (Proc. Colloq., Tihany, 1966), 362 Academic Press, New York, 1968.\n\nZ. Skupień, Smallest sets of longest paths with empty intersection. Combin. Probab. Comput. 5 (1996), no. 4, 429–436.\n\nComments:\n- October 14th, 2023 | Robert Samal | Possible solution: Possible solution to this appears in https://arxiv.org/abs/2006.16245\n\n(I am not sure whether it is being in a referee process or whether somebody may have found a gap in the presented proof.)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Do any three longest paths in a connected graph have a vertex in common?\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The three-longest-path common-vertex conjecture remains treated as open in 2025 primary literature and the maintained BCC list. An unrefereed 2020 arXiv manuscript claims a full proof, but no independent validation or journal publication was found.\n\n**Verified partial progress.**\n\n- Fujita--Furuya--Naserasr--Ozeki establish quantitative distance bounds among three longest paths.\n- Axenovich proves the conjecture when the union of the three paths is outerplanar and under related excluded-minor conditions.\n- Recent longest-path-transversal work improves hitting-set bounds without forcing one common vertex for every triple.\n\n**Full solution or refutation.**\n\nThe purported arXiv proof is not accepted here as a resolution because later sources still call the conjecture open and the maintained OPG page explicitly questions whether the proof has a gap.\n\n**What remains.**\n\nObtain a refereed, independently checked proof for arbitrary connected graphs, identify and repair any gap in the 2020 manuscript, or construct a counterexample.\n\n**Sources checked.**\n\n- Shinya Fujita, Michitaka Furuya, Reza Naserasr, and Kenta Ozeki, A New Approach Towards a Conjecture on Intersecting Three Longest Paths, Journal of Combinatorics 10 (2019), 221-229; arXiv:1503.01219. (primary): https://arxiv.org/abs/1503.01219\n Evidence used: States the problem as open and develops a quantitative distance approach.\n- Carla Groenland, Longest cycles in vertex-transitive and highly connected graphs, Bulletin of the London Mathematical Society (2025), doi:10.1112/blms.70134. (primary): https://doi.org/10.1112/blms.70134\n Evidence used: A 2025 primary paper still describes the three-longest-path assertion as a well-known longstanding conjecture.\n- Nirankush Sarkar, Any Three Longest Paths In A Connected Graph Has A Common Vertex, unrefereed arXiv manuscript, arXiv:2006.16245. (primary): https://arxiv.org/abs/2006.16245\n Evidence used: Claims a complete affirmative proof; included as a claim requiring expert verification, not as accepted resolution evidence.\n- Peter J. Cameron, British Combinatorial Conference Problem List, Problem BCC15.6 (DM276), maintained compilation. (authoritative_secondary): https://maths.qmul.ac.uk/~pjc/bcc/allprobs.pdf\n Evidence used: Continues to list the exact question as a problem.\n\n**Review notes.** The OPG page labels arXiv:2006.16245 only a possible solution and explicitly says it is unknown whether refereeing occurred or a gap was found.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3126, "problem_number": "OPG-46629", "title": "Lovász Path Removal Conjecture", "statement": "Conjecture There is an integer-valued function $f(k)$ such that if $G$ is any $f(k)$-connected graph and $x$ and $y$ are any two vertices of $G$, then there exists an induced path $P$ with ends $x$ and $y$ such that $G-V(P)$ is $k$-connected.", "background": "Source: Open Problem Garden. Original node ID: 46629. URL: http://www.openproblemgarden.org/op/lovasz_path_removal_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/lovasz_path_removal_conjecture\n- Author(s): Lovasz, Laszlo\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nIt follows from a theorem of Tutte that any 3-connected graph contains a non-separating path connecting any two vertices, and consequently, $f(1)=3$. When $k=2$, it was independently shown by Chen, Gould, and Yu [CGY] and Kriesell [K] that $f(2) = 5$.\n\nAnwering a conjecture of Kriesell, Kawarabayashi et al. [KLRW] proved the following weaker statement, in which one only removes the edges of the path.\n\nTheorem There exists a function $f(k)$ such that for every $f(k)$-connected graph $G$ and any two vertices $x$ and $y$ of $G$, there exists an induced path $P$ with ends $x$ and $y$ such that $G\\setminus E(P)$ is $k$-connected.\n\nBibliography:\n[CGY] G. Chen, R. Gould, X. Yu, Graph connectivity after path removal, Combinatorica 23 (2003) 185--203.\n\n[KLRW] K. Kawarabayashi, O. Lee, B. Reed, and P. Wollan, A weaker version of Lovasz's path removal conjecture, Journal of Combinatorial Theory, Series B 98 (2008) 972--979.\n\n[K] M. Kriesell, Induced paths in 5-connected graphs, J. of Graph Theory, 36 (2001), 52--58.\n\n*[T] C. Thomassen, Graph decompositions with applications to subdivisions and path systems modulo k, J. of Graph Theory, 7 (1983), 261--271.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Lovász Path Removal Conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Lovasz's vertex path-removal conjecture is proved for k=1 and k=2, and its edge-deletion weakening is proved for all k, but the general vertex-deletion conjecture remains open for k at least three.\n\n**Verified partial progress.**\n\n- The exact values f(1)=3 and f(2)=5 are known.\n- For every k, sufficiently high connectivity guarantees an induced x-y path whose edge deletion preserves k-connectivity.\n\n**Full solution or refutation.**\n\nKnown theorems settle the first two connectivity levels and all levels under edge deletion; they do not justify deleting every vertex of the induced path.\n\n**What remains.**\n\nProve the vertex-deletion conclusion for every k at least three or find an obstruction to the existence of any such connectivity function.\n\n**Sources checked.**\n\n- G. Chen, R. Gould, and X. Yu, Graph Connectivity After Path Removal, Combinatorica 23 (2003), 185-203. (primary): https://doi.org/10.1007/s00493-003-0018-z\n Evidence used: Supplies the k=2 path-removal result recorded in the source literature.\n- K. Kawarabayashi, O. Lee, B. Reed, and P. Wollan, A weaker version of Lovasz' path removal conjecture, Journal of Combinatorial Theory B 98 (2008), 972-979. (primary): https://doi.org/10.1016/j.jctb.2007.11.003\n Evidence used: Proves the all-k weakening in which only the path edges are removed.\n- Graph-theory open problems, Lovasz Path Removal Conjecture (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/lovasz_path_removal_conjecture/\n Evidence used: Records the exact special cases and the unresolved general vertex-deletion statement.\n\n**Review notes.** No formulation defect identified; source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3127, "problem_number": "OPG-46706", "title": "Turán number of a finite family.", "statement": "Given a finite family ${\\cal F}$ of graphs and an integer $n$, the Turán number $ex(n,{\\cal F})$ of ${\\cal F}$ is the largest integer $m$ such that there exists a graph on $n$ vertices with $m$ edges which contains no member of ${\\cal F}$ as a subgraph.\n\nConjecture For every finite family ${\\cal F}$ of graphs there exists an $F\\in {\\cal F}$ such that $ex(n, F ) = O(ex(n, {\\cal F}))$.", "background": "Source: Open Problem Garden. Original node ID: 46706. URL: http://www.openproblemgarden.org/op/turan_number_of_a_finite_family.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/turan_number_of_a_finite_family\n- Author(s): Erdos, Paul; Simonovits, Miklos\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 5th, 2013 by fhavet\n\nProblem-page discussion:\nFor the case when ${\\cal F}$ consists of even cycles, this would mean that (up to constants) the Turán number of ${\\cal F}$ is given by that of the longest cycle in ${\\cal F}$. Verstraëte (see [KO]) conjectured something stronger:\n\nConjecture For all integers $k < \\ell$ there exists a positive c = c(\\ell) such that every $C_{2\\ell}$-free graph $G$ has a $C_{2k}$-free subgraph $H$ with $e(H) ≥ e(G)/c$.\n\nThis conjecture was motivated by a result of Györi [G] who showed that every bipartite $C_6$-free graph $G$ has a $C_4$-free subgraph which contains at least half of the edges of $G$. The case $k=2$ was proved in [KO].\n\nBibliography:\n*[ES] P.Erdös and M. Simonovits, Compactness results in extremal graph theory, Combinatorica 2 (1982), 275–288.\n\n[KO] D. Kühn and D. Osthus, 4-cycles in graphs without a given even cycle, J. Graph Theory 48 (2005), 147-156.\n\n[G] E. Györi, $C_6$-free bipartite graphs and product representation of squares, Discrete Math. 165/166 (1997), 371-375.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Turán number of a finite family.\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Erdos-Simonovits finite-family Turan conjecture remains open; recent work refutes a stronger even-cycle conjecture without refuting this displayed big-O statement.\n\n**Verified partial progress.**\n\n- Sharp even-cycle subgraph results verify concrete special configurations.\n- A March 2026 preprint disproves the stronger Verstraete conjecture for specified even cycles, closing that proposed route rather than the main conjecture.\n\n**Full solution or refutation.**\n\nNo general finite-family proof or counterexample was verified. The new even-cycle counterexamples must not be promoted to a refutation of the weaker existential member comparison.\n\n**What remains.**\n\nProve that every finite forbidden family has a member whose Turan number is within a constant factor of the family's, or construct a finite family violating this.\n\n**Sources checked.**\n\n- D. Grosz, A. Methuku, and C. Tompkins, On subgraphs of C_{2k}-free graphs and a problem of Kuhn and Osthus, arXiv:1708.05454. (primary): https://arxiv.org/abs/1708.05454\n Evidence used: Proves a sharp even-cycle special result connected to proposed approaches to the conjecture.\n- D. Conlon, E. Mulrenin, and C. Pohoata, Two counterexamples to a conjecture about even cycles, arXiv:2603.24515 (2026). (primary): https://arxiv.org/abs/2603.24515\n Evidence used: Refutes a stronger even-cycle conjecture; it does not claim to refute the finite-family Turan conjecture.\n- Graph-theory open problems, Turan number of a finite family (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/turan_number_of_a_finite_family/\n Evidence used: Records the main conjecture and its earlier partial routes as unresolved.\n\n**Review notes.** The logical distinction between the recent stronger-conjecture counterexamples and the exact source statement is explicitly preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3128, "problem_number": "OPG-46951", "title": "Switching reconstruction conjecture", "statement": "Conjecture Every simple graph on five or more vertices is switching-reconstructible.", "background": "Source: Open Problem Garden. Original node ID: 46951. URL: http://www.openproblemgarden.org/op/switching_reconstruction_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/switching_reconstruction_conjecture\n- Author(s): Stanley, Richard P.\n- Subject(s): Graph Theory\n- Keywords: reconstruction\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2013 by fhavet\n\nProblem-page discussion:\nTo switch a vertex of a simple graph is to exchange its sets of neighbours and non-neighbours. The graph so obtained is called a switching of the graph. The collection of switchings of a graph G is called the switching deck of $G$. A graph is switching-reconstructible if every graph with the same deck as $G$ is isomorphic to $G$.\n\nThere are four pairs of non-isomorphic graphs of order $4$ with the same switching deck. One of them consists of the empty graph and the $4$-cycle.\n\nStanley [S] proved that a graph on $n$ vertices is switching-reconstructible if $n \\not\\equiv 0 (\\mod 4)$.\n\nAn analogous problem was posed for digraphs. Instead of complementing the edges at a vertex, one reverses each of its incident arc.\n\nBibliography:\n*[S] R. P. Stanley Reconstruction from vertex-switching. J. Combin. Theory Ser. B, 38 (1985), 132--138.\n\nRelated:\nRelated problems\nSwitching reconstruction of digraphs\n\nDiscussion links:\n- analogous problem: http://www.openproblemgarden.org/?q=node/46952\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Switching reconstruction conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The checked maintained record continues to list switching reconstruction for all simple graphs of order at least five as open.\n\n**Verified partial progress.**\n\n- The switching deck is reconstructible for several special graph families, but no general proof was verified.\n\n**Full solution or refutation.**\n\nNo resolution of Stanley's switching-reconstruction conjecture was verified.\n\n**What remains.**\n\nProve reconstruction for all simple graphs of order at least five or exhibit a counterexample.\n\n**Sources checked.**\n\n- Graph-theory open problems, Switching reconstruction conjecture (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/switching_reconstruction_conjecture/\n Evidence used: Lists the exact simple-graph conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3129, "problem_number": "OPG-46952", "title": "Switching reconstruction of digraphs", "statement": "Question Are there any switching-nonreconstructible digraphs on twelve or more vertices?", "background": "Source: Open Problem Garden. Original node ID: 46952. URL: http://www.openproblemgarden.org/op/switching_reconstruction_of_digraphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/switching_reconstruction_of_digraphs\n- Author(s): Bondy, J. Adrian; Mercier, Fabien\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2013 by fhavet\n\nProblem-page discussion:\nTo switch a vertex of a digraph is to reverse all the arcs incident to it. The digraph so obtained is called a switching of the digraph. The collection of switchings of a digraph $D$ is called the switching deck of $D$. A digraph is switching-reconstructible if every digraph with the same deck as $D$ is isomorphic to $D$.\n\nThe problem is a directed analogue of switching reconstruction of graphs in which one complements the edges at a vertex, instead of reversing each of its incident arcs.\n\nBondy and Mercier proved an analogue to Stanley's result for switching reconstruction of graphs. They proved that a digraph on $n$ vertices is switching-reconstructible if $n \\not\\equiv 0 (\\mod 4)$. They also proved many other common results for both switching reconstructions.\n\nOne significant difference between the directed and undirected problems is that there exist switching-nonreconstructible directed graphs on eight vertices, while Stanley's conjecture that every simple graph on five or more vertices is switching-reconstructible.\n\nBibliography:\n*[BM] J. A. Bondy and F. Mercier. Switching reconstruction of digraphs. J. Graph Theory 67(2011), no. 4, 332-348.\n\nRelated:\nRelated problems\nSwitching reconstruction conjecture\n\nDiscussion links:\n- switching reconstruction of graphs: http://www.openproblemgarden.org/?q=node/46951\n- Stanley's conjecture: http://www.openproblemgarden.org/?q=node/46951\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Switching reconstruction of digraphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Bondy--Mercier prove reconstruction when n is not congruent to zero modulo four, but the searched sources did not determine whether a switching-nonreconstructible digraph exists at an order at least twelve.\n\n**Verified partial progress.**\n\n- Bondy--Mercier establish switching reconstruction for digraph orders n not congruent to 0 modulo 4.\n\n**Full solution or refutation.**\n\nThat congruence theorem leaves the exact existential question at n >= 12 unresolved in this review.\n\n**What remains.**\n\nFind a verified later classification/counterexample for orders divisible by four.\n\n**Sources checked.**\n\n- J. A. Bondy and F. Mercier, Switching reconstruction of digraphs, Journal of Graph Theory 67 (2011), 332--348. (primary): https://doi.org/10.1002/jgt.20512\n Evidence used: The problem record summarizes their theorem for n not congruent to 0 modulo 4.\n- Open Problem Garden, Switching reconstruction of digraphs (accessed 2026-08-17). (maintained_tracker): https://garden.irmacs.sfu.ca/op/switching_reconstruction_of_digraphs\n Evidence used: States the exact question and the congruence-class theorem.\n\n**Review notes.** No source alteration; the unanswered congruence class is not treated as a resolution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3130, "problem_number": "OPG-48264", "title": "Signing a graph to have small magnitude eigenvalues", "statement": "Conjecture If $A$ is the adjacency matrix of a $d$-regular graph, then there is a symmetric signing of $A$ (i.e. replace some $+1$ entries by $-1$ ) so that the resulting matrix has all eigenvalues of magnitude at most $2 \\sqrt{d-1}$.", "background": "Source: Open Problem Garden. Original node ID: 48264. URL: http://www.openproblemgarden.org/op/signing_a_graph_to_have_small_magnitude_eigenvalues.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/signing_a_graph_to_have_small_magnitude_eigenvalues\n- Author(s): Bilu, Yonatan; Linial, Nathan\n- Subject(s): Graph Theory\n- Keywords: eigenvalue; expander; Ramanujan graph; signed graph; signing\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 24th, 2013 by mdevos\n\nProblem-page discussion:\nA graph $H$ is a $k$-lift of a graph $G$ if there is a $k$-to- $1$ map $f: V(H) \\rightarrow V(G)$ which is locally injective in the sense that the restriction of $f$ to the neighbourhood of every vertex is an injection. We can construct a random $k$-lift of $G$ with vertex set $V(G) \\times \\{1,\\ldots,k\\}$ by adding a (uniformly chosen) random matching between $\\{v\\} \\times \\{1,\\ldots,k\\}$ and $\\{w\\} \\times \\{1,\\ldots,k\\}$ whenever $vw \\in E(G)$. If $H$ is a $k$-lift of $G$, then every eigenvalue of $G$ will also be an eigenvalue of $H$, but in addition $H$ will have $(k-1) |V(G)|$ new eigenvalues. There has been considerable interest and investigation into the behaviour of these new eigenvalues for a random $k$-lift, since it is expected that they should generally be small in magnitude. In particular, if $G$ is a Ramanujan graph (a $d$-regular graph for which all nontrivial eigenvalues are at most $2 \\sqrt{d-1}$ ) it may be possible to construct a new Ramanujan graph by taking a suitable $k$-lift of $G$. A series of increasingly strong results have shown that a random $k$-lift of a $d$-regular Ramanujan graph will have all new eigenvalues at most $O(d^{3/4})$ (Friedman [F]), $O(d^{2/3})$ (Linial and Pruder [LP]) and $O(\\sqrt{d} \\log d)$ (Lubetzky, Sudakov, and Vu [LSV]).\n\nAn interesting paper of Bilu and Linial [BL] investigates 2-lifts of graphs. Let $G$ be a graph and let $H$ be a 2-lift of $G$ with vertex set $V(G) \\times \\{1,2\\}$ as above. Every eigenvector of $G$ extends naturally to an eigenvector of $H$ which is constant on each fiber (set of the form $\\{u\\} \\times \\{1,2\\}$ ). Thus, we may assume that all of the new eigenvalues are associated with eigenvectors which sum to zero on each fiber. So, each of these new eigenvectors is completely determined by its behaviour on $V(G) \\times \\{1\\}$. Now we assign a signature $\\pm 1$ to each edge of $G$ to form a signed graph $G^*$ by assigning each edge $uv \\in E(G)$ for which $(u,1)(v,1) \\in E(H)$ a sign of $1$ and every other edge of $G$ sign $-1$. It is straightforward to verify that the restriction of any new eigenvector of $H$ to $V(G) \\times \\{1\\}$ will then be an eigenvector of $G^*$. Thus, the above conjecture is equivalent to the conjecture that every $d$-regular graph has a $2$-lift so that all new eigenvalues have magnitude at most $2 \\sqrt{d-1}$. Furthermore, a positive solution to this conjecture for $d$-regular Ramanujan graphs would yield families of $d$-regular expanders.\n\nBibliography:\n*[BL] Y. Bilu, N. Linial, Lifts, discrepancy and nearly optimal spectral gap, Combinatorica 26 (5) (2006) 495–519. MathSciNet\n\n[F] J. Friedman, Relative expanders or weakly relatively Ramanujan graphs, Duke Math. J. 118 (1) (2003) 19–35. MathSciNet\n\n[LP] N. Linial, D. Puder, Word maps and spectra of random graph lifts, Random Structures Algorithms 37 (1) (2010) 100–135. MathSciNet\n\n[LSV] E. Lubetzky, B. Sudakov, V Vu, Spectra of lifted Ramanujan graphs. Adv. Math. 227 (2011), no. 4, 1612–1645. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2279667\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1978881\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2674623\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2799807\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 27.\n\nAttempt notes:\nTarget:\nMake progress on \"Signing a graph to have small magnitude eigenvalues\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Bilu--Linial signing conjecture remains open for arbitrary regular graphs; the bipartite case is proved.\n\n**Verified partial progress.**\n\n- Marcus--Spielman--Srivastava prove the required signed spectral bound for every d-regular bipartite graph.\n- The general case has increasingly strong non-sharp bounds but no verified two-sided Ramanujan signing theorem.\n\n**Full solution or refutation.**\n\nThe source statement quantifies over all d-regular graphs, so the bipartite theorem is not a full resolution.\n\n**What remains.**\n\nControl both spectral extremes for arbitrary regular graphs at the Ramanujan bound.\n\n**Sources checked.**\n\n- A. Marcus, D. Spielman and N. Srivastava, Interlacing families I: Bipartite Ramanujan graphs of all degrees, Annals of Mathematics 182 (2015), 307--325. (primary): https://annals.math.princeton.edu/2015/182-1/p07\n Evidence used: Proves the Bilu--Linial signing conclusion in the bipartite setting.\n- Y. Bilu and N. Linial, Lifts, discrepancy and nearly optimal spectral gap, Combinatorica 26 (2006); arXiv:math/0312022. (primary): https://arxiv.org/abs/math/0312022\n Evidence used: States the original all-regular-graph conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3131, "problem_number": "OPG-48368", "title": "Are almost all graphs determined by their spectrum?", "statement": "Problem Are almost all graphs uniquely determined by the spectrum of their adjacency matrix?", "background": "Source: Open Problem Garden. Original node ID: 48368. URL: http://www.openproblemgarden.org/op/are_almost_all_graphs_determined_by_their_spectrum.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_almost_all_graphs_determined_by_their_spectrum\n- Subject(s): Graph Theory\n- Keywords: cospectral; graph invariant; spectrum\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 26th, 2013 by mdevos\n\nProblem-page discussion:\nWe say that two non-isomorphic graphs are cospectral if their adjacency matrices have the same spectrum (counted with multiplicity). A graph is spectrally determined if no other graphs are cospectral to it. It is unclear to me (M. DeVos) how to attribute this problem, but it was considered already in the 1950's and resonates with the famous problem \"Can you hear the shape of a drum?\" ([vDH]).\n\nA priori, it might seem plausible for all graphs to be spectrally determined.. but this is false. The smallest counterexample is the cospectral pair given by $K_{1,4}$ and the graph obtained from $C_4$ by adding an isolated vertex. Some rich families of cospectral graphs are provided by strongly regular graphs, since any two strongly regular graphs with the same parameters will be cospectral.\n\nFor the special case of trees, Schwenk proved almost all trees are not spectrally determined. This was sharpened by Godsil and Mckay who showed that almost every tree $T$ has a cospectral graph $T'$ so that in addition the complements of $T$ and $T'$ are cospectral. Furthermore, an operation called Godsil-Mckay Switching defined by these authors gives a powerful tool to produce general graphs which are cospectral.\n\nOn the flip side, we seem to have a lack of good tools to prove that a given graph is spectrally determined.\n\nBibliography:\n[vDH] E. R. van Dam and W. H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra and its Applications 373 (2003) 241–272.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Are almost all graphs determined by their spectrum?\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The van Dam--Haemers conjecture that almost all graphs are determined by their adjacency spectrum remains open, but exponentially many spectrally determined graphs are now known.\n\n**Verified partial progress.**\n\n- Koval--Kwan prove at least e^(cn) unlabeled n-vertex graphs are determined by their adjacency spectrum.\n\n**Full solution or refutation.**\n\nAn exponential family is far short of proving that the proportion tends to one.\n\n**What remains.**\n\nShow asymptotic density one of adjacency-spectrum-determined graphs, or find a contrary family of positive asymptotic proportion.\n\n**Sources checked.**\n\n- A. Koval and M. Kwan, Exponentially many graphs are determined by their spectrum, arXiv:2309.09788 (2024). (primary): https://arxiv.org/abs/2309.09788\n Evidence used: The abstract identifies the almost-all conjecture and proves an exponential lower bound.\n- Graph-theory open problems, Are almost all graphs determined by their spectrum? (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/are_almost_all_graphs_determined_by_their_spectrum/\n Evidence used: Records the adjacency-spectrum conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3132, "problem_number": "OPG-49795", "title": "Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournament", "statement": "Conjecture If $T$ is a tournament of order $n$, then it contains $\\left \\lceil n(n-1)/6 - n/3\\right\\rceil$ arc-disjoint transitive subtournaments of order 3.", "background": "Source: Open Problem Garden. Original node ID: 49795. URL: http://www.openproblemgarden.org/op/minimum_number_of_transitive_subtournaments_of_order_3_in_a_tournament.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/minimum_number_of_transitive_subtournaments_of_order_3_in_a_tournament\n- Author(s): Yuster, Raphael\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 21st, 2013 by fhavet\n\nProblem-page discussion:\nIf true the conjecture would be tight as shown by any tournament whose vertex set can be decomposed into $3$ sets $V_1, V_2, V_3$ of size $\\lceil n/3 \\rceil$ or $\\lfloor n/3\\rfloor$ and such that $V_1\\rightarrow V_2$, $V_2\\rightarrow V_3$ and $V_3\\rightarrow V_1$.\n\nLet $TT_3$ denote the transitive tournament of order 3. A $TT_3$-packing of a digraph $D$ is a set of arc-disjoint copies of $TT_3$ subgraphs of $D$.\n\nLet $f(n)$ be the minimum size of a $TT_3$-packing over all tournaments of order $n$. The conjecture and its tightness say $f(n)= \\left \\lceil n(n-1)/6 - n/3\\right\\rceil$.\n\nThe best lower bound for $f(n)$ so far is due to Kabiya and Yuster [KY] proved that $f(n) > \\frac{41}{300} n^2(1+o(1))$.\n\nBibliography:\n[KY] M. Kabiya and R. Yuster. Packing transitive triples in a tournament. Ann. Comb. 12 (2008), no. 3, 291–-306.\n\n*[Y] R. Yuster. The number of edge-disjoint transitive triples in a tournament. Discrete Math. 287 (2004). no. 1-3,187--191.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournament\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact minimum TT3-packing formula remains open; the verified lower bound is (41/300+o(1))n^2.\n\n**Verified partial progress.**\n\n- Kabiya--Yuster prove the asymptotic lower bound f(n) > (41/300)n^2(1+o(1)).\n\n**Full solution or refutation.**\n\nThe known coefficient remains below the conjectured 1/6 asymptotic coefficient.\n\n**What remains.**\n\nObtain the sharp formula or close the asymptotic packing gap.\n\n**Sources checked.**\n\n- UnsolvedMath/OpenGarden, Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournament (accessed 2026-08-17). (maintained_tracker): https://www.unsolvedmath.com/problems/OPG-49795\n Evidence used: Gives the exact conjecture and quotes the Kabiya--Yuster bound.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3133, "problem_number": "OPG-57613", "title": "Imbalance conjecture", "statement": "Conjecture Suppose that for all edges $e\\in E(G)$ we have $imb(e)>0$. Then $M_{G}$ is graphic.", "background": "Source: Open Problem Garden. Original node ID: 57613. URL: http://www.openproblemgarden.org/op/imbalance_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/imbalance_conjecture\n- Author(s): Kozerenko, Sergiy\n- Subject(s): Graph Theory\n- Keywords: edge imbalance; graphic sequences\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 24th, 2013 by Sergiy Kozerenko\n\nProblem-page discussion:\nConsider simple undirected graph $G$ and let $e=uv\\in E(G)$.\n\nThe imbalance of the edge $e$ defined as $imb(e)=|deg(u)-deg(v)|$.\n\nThe multiset of all edge imbalances of $G$ is denoted by $M_{G}$.\n\nNote, that conjecture is verified for all such graphs with $\\leq 9$ vertices.\n\nBibliography:\n*Graphs with graphic imbalance sequences\n\nBibliography links:\n- Graphs with graphic imbalance sequences: http://mathoverflow.net/questions/140819/graphs-with-graphic-imbalance-sequences\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Imbalance conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A preprint submitted on 2026-08-10 claims a proof that every positive edge-imbalance multiset is graphic.\n\n**Verified partial progress.**\n\n- Earlier work proved imbalance graphicness for trees, unicyclic graphs, antiregular graphs, and several block-graph classes.\n- The new proof derives all Erdos-Gallai inequalities from a truncated imbalance-sum bound and finishes with parity.\n\n**Full solution or refutation.**\n\nYavari's six-page preprint states and proves the exact finite-simple-graph imbalance conjecture represented by this record.\n\n**What remains.**\n\nIndependently check the week-old proof, especially its truncated-sum lemma and quantifiers, and monitor revisions or peer review.\n\n**Sources checked.**\n\n- Y. Yavari, A Proof of the Imbalance Conjecture, arXiv:2608.09191 (2026). (primary): https://arxiv.org/abs/2608.09191\n Evidence used: The abstract states the exact conjecture and the Erdos-Gallai/parity proof route; submitted 2026-08-10.\n- S. Kozerenko and A. Serdiuk, New results on imbalance graphic graphs, Opuscula Mathematica 43 (2023), 81-100. (primary): https://doi.org/10.7494/OpMath.2023.43.1.81\n Evidence used: Establishes multiple pre-resolution special classes and still treats the general conjecture as open.\n- Graph-theory open problems, Imbalance conjecture (reviewed before the new preprint; accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/imbalance_conjecture/\n Evidence used: Documents partial status through May 2026 and is superseded on current status by the August primary preprint.\n\n**Review notes.** Very recent unrefereed v1. The short statement depends on page-context definitions of finite simple graph, edge imbalance, and graphic multiset; source text unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3134, "problem_number": "OPG-59908", "title": "Fractional Hadwiger", "statement": "Conjecture For every graph $G$,\n(a) $\\chi_f(G)\\leq\\text{had}(G)$\n(b) $\\chi(G)\\leq\\text{had}_f(G)$\n(c) $\\chi_f(G)\\leq\\text{had}_f(G)$.", "background": "Source: Open Problem Garden. Original node ID: 59908. URL: http://www.openproblemgarden.org/op/fractional_hadwiger.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/fractional_hadwiger\n- Author(s): Harvey, Daniel J.; Reed, Bruce A.; Seymour, Paul D.; Wood, David R.\n- Subject(s): Graph Theory\n- Keywords: fractional coloring, minors\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 16th, 2014 by David Wood\n\nProblem-page discussion:\nHere $\\chi$ is the chromatic number, $\\chi_f$ is the fractional chromatic number, $\\text{had}$ is the Hadwiger number, and $\\text{had}_f$ is the fractional Hadwiger number (which was recently introduced independently by Fox [F] and Pedersen [P]).\n\nIt is well known and easily proved (see [HW]) that\n$\\chi_f(G)\\leq\\chi(G)\\text{ and }\\text{had}(G)\\leq\\text{had}_f(G)\\leq\\text{tw}(G)+1,$\nwhere $\\text{tw}(G)$ is the treewidth of $G$.\n\nHadwiger's famous conjecture, $\\chi(G)\\leq\\text{had}(G)$, bridges the gap in the above inequalities. The above conjectures therefore are weaker than Hadwiger's conjecture. Note that Conjecture (a) implies Conjecture (c), and Conjecture (b) implies Conjecture (c).\n\nNote that Reed and Seymour [RS] proved that $\\chi_f(G)\\leq2\\,\\text{had}(G)$.\n\nConjecture (a) is due to Reed and Seymour [RS]. Conjecture (b) is due to Harvey and Wood [HW]. Conjecture (c) is independently due to Harvey and Wood [HW] and Pedersen [P].\n\nPedersen [P] presents a natural equivalent formulation of Conjecture (c).\n\nBibliography:\n*[HW] Daniel J. Harvey, David R. Wood, Parameters tied to treewidth. arXiv:1312.3401, 2013.\n\n[F] Jacob Fox. Constructing dense graphs with sublinear Hadwiger number. J. Combin. Theory Ser. B (to appear).\n\n*[P] Anders Sune Pedersen. Contributions to the Theory of Colourings, Graph Minors, and Independent Sets, PhD thesis, Department of Mathematics and Computer Science University of Southern Denmark, 2011.\n\n*[RS] Bruce A. Reed, Paul D. Seymour, Fractional colouring and Hadwiger's conjecture. J. Combin. Theory Ser. B, 74(2), 147-152.\n\nBibliography links:\n- arXiv:1312.3401: http://www.arxiv.org/abs/1312.3401\n- Constructing dense graphs with sublinear Hadwiger number: http://www.arxiv.org/abs/1108.4953\n- Contributions to the Theory of Colourings, Graph Minors, and Independent Sets: http://www.imada.sdu.dk/%7Easp/Thesis_2ed.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Fractional Hadwiger\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The three fractional Hadwiger inequalities remain open in general; the underlying fractional minor parameters have useful special-case and comparison results.\n\n**Verified partial progress.**\n\n- The fractional Hadwiger number was independently introduced and connects colouring to fractional clique-minor packings.\n\n**Full solution or refutation.**\n\nNo checked source proves all three inequalities for every graph.\n\n**What remains.**\n\nEstablish any/all of the proposed inequalities or construct a counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, Fractional Hadwiger (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/fractional_hadwiger\n Evidence used: States the three exact conjectures and their parameter definitions.\n- Contributions to the Theory of Colourings, fractional Hadwiger discussion (accessed 2026-08-17). (authoritative_secondary): https://citeseerx.ist.psu.edu/document?doi=260f4f44eda9aa935b88c7d3b931e5cbb49741b2&repid=rep1&type=pdf\n Evidence used: Describes the fractional Hadwiger formulation as conjectural.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3135, "problem_number": "OPG-59952", "title": "Chromatic Number of Common Graphs", "statement": "Question Do common graphs have bounded chromatic number?", "background": "Source: Open Problem Garden. Original node ID: 59952. URL: http://www.openproblemgarden.org/op/chromatic_number_of_common_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/chromatic_number_of_common_graphs\n- Author(s): Hatami, H; Hladký, J.; Kráľ, D.; Norine, S.; Razborov, A.\n- Subject(s): Graph Theory\n- Keywords: common graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 15th, 2014 by David Wood\n\nProblem-page discussion:\nA graph $H$ is common if the sum of the number of copies of $H$ in a graph $G$ and the number in the complement of $G$ is asymptotically minimised by taking $G$ to be a random graph (see [HHKNR] for a formal definition).\n\nGoodman proved that $K_3$ is common [G]. Erdös [E] conjectured that every complete graph is common. Later, this conjecture was extended to all graphs by Burr and Rosta [BR]. Sidorenko [S89] disproved Burr and Rosta’s conjecture by showing that a triangle with a pendant edge is not common. Conjectures of Erdös and Simonovits [ES] and Sidorenko [S91,S93] imply that every bipartite graph is common. Disproving the first conjecture of Erdös, Thomason proved that $K_4$ is not common [T]. More generally, Jagger, Šťovíček and Thomason proved that no common graph contains $K_4$ as a subgraph [JST]. The 5-wheel is an example of a 4-chromatic common graph [HHKNR].\n\nBibliography:\n[BR] Burr, S. A. and Rosta, V. On the Ramsey multiplicities of graphs: Problems and recent results. J. Graph Theory 4 (1980) 347–361.\n\n[E] Paul Erdös, On the number of complete subgraphs contained in certain graphs. Magyar Tud. Akad. Mat. Kutato ́ Int. Ko ̈zl. 7 (1962) 459–464.\n\n[ES] Paul Erdös and Miklós Simonovits (1984) Cube-supersaturated graphs and related problems. In Progress in Graph Theory: Waterloo, Ont., 1982, Academic Press, pp. 203–218.\n\n*[HHKNR] H. Hatami, J. Hladký, D. Kráľ, S. Norine, A. Razborov: Non-three-colorable common graphs exist, Combinatorics, Probability and Computing 21 (2012), 734–742.\n\n[JST] Jagger, Chris; Šťovíček, Pavel; Thomason, Andrew. Multiplicities of subgraphs. Combinatorica 16 (1996) 123–141.\n\n[S89] Sidorenko, A. Cycles in graphs and functional inequalities. Mat. Zametki 46 (1989) 72–79, 104.\n\n[S91] Sidorenko, A. Inequalities for functionals generated by bipartite graphs. Diskret. Mat. 3 (1991) 50–65.\n\n[S93] Sidorenko, A. A correlation inequality for bipartite graphs. Graphs Combin. 9 (1993) 201–204.\n\n[T] Thomason, Andrew. A disproof of a conjecture of Erdo ̋s in Ramsey theory, J. London Math. Soc. (2) 39 (1989) 246–255.\n\nRelated:\nRelated problems\nSidorenko's Conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Chromatic Number of Common Graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Kral--Volec--Wei construct a connected common graph of every prescribed chromatic number.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nTheir theorem directly gives unbounded chromatic number among common graphs.\n\n**What remains.**\n\nThe source existence question is settled; classification of all common graphs remains separate.\n\n**Sources checked.**\n\n- D. Kral, J. Volec and F. Wei, Common graphs with arbitrary chromatic number, Compositio Mathematica 161 (2025), 594--634; arXiv:2206.05800. (primary): https://arxiv.org/abs/2206.05800\n Evidence used: The abstract constructs connected k-chromatic common graphs for every k.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3136, "problem_number": "OPG-59997", "title": "Circular flow numbers of $r$-graphs", "statement": "A nowhere-zero $r$-flow $(D(G),\\phi)$ on $G$ is an orientation $D$ of $G$ together with a function $\\phi$ from the edge set of $G$ into the real numbers such that $1 \\leq |\\phi(e)| \\leq r-1$, for all $e \\in E(G)$, and $\\sum_{e \\in E^+(v)}\\phi(e) = \\sum_{e \\in E^-(v)}\\phi(e), \\textrm{ for all } v \\in V(G)$.\n\nA $(2t+1)$-regular graph $G$ is a $(2t+1)$-graph if $|\\partial_G(X)| \\geq 2t+1$ for every $X \\subseteq V(G)$ with $|X|$ odd.\n\nConjecture Let $t > 1$ be an integer. If $G$ is a $(2t+1)$-graph, then $F_c(G) \\leq 2 + \\frac{2}{t}$.", "background": "Source: Open Problem Garden. Original node ID: 59997. URL: http://www.openproblemgarden.org/op/circular_flow_numbers_of_r_graphs.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/circular_flow_numbers_of_r_graphs\n- Author(s): Steffen, Eckhard\n- Subject(s): Graph Theory\n- Keywords: flow conjectures; nowhere-zero flows\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 6th, 2015 by Eckhard Steffen\n\nProblem-page discussion:\nSince every $(2t+1)$-regular class 1 graph is a $(2t+1)$-graph, the truth of this conjecture would imply the truth of the conjecture on the circular flow number of regular class 1 graphs. If it is true for even $t$, say $t=2t'$, then Jaeger's modular orientation conjecture is true for $(4t'+1)$-regular graphs and hence, by a result of Jaeger, it would imply the truth of Tutte's 5-flow conjecture. For $t=2$ it is Tutte's 3-flow conjecture.\n\nBibliography:\n*[ES_2015]E. Steffen, Edge-colorings and circular flow numbers on regular graphs, J. Graph Theory 79, 1–7, 2015\n\nRelated:\nRelated problems\n3-flow conjecture\nJaeger's modular orientation conjecture\nCircular flow number of regular class 1 graphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Circular flow numbers of $r$-graphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Mattiolo--Steffen explicitly disprove the (2t+1)-graph circular-flow conjecture using the same regular class-1 counterexamples that refute the weaker class-1 statement.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nTheir constructed (2t+1)-regular class-1 graphs satisfy the odd-cut condition defining (2t+1)-graphs yet have circular flow number greater than 2+2/t for infinitely many t.\n\n**What remains.**\n\nThe universal conjecture is settled negatively. Modified bounds or restrictions for particular degrees and subclasses remain separate research questions.\n\n**Sources checked.**\n\n- Davide Mattiolo and Eckhard Steffen, Edge colorings and circular flows on regular graphs, Journal of Graph Theory 99 (2022), 399-413; arXiv:2001.02484. (primary): https://arxiv.org/abs/2001.02484\n Evidence used: Restates this as Conjecture 1.5 and explicitly says and proves that both Conjectures 1.4 and 1.5 are false.\n- Open Problem Garden, Circular flow numbers of r-graphs. (maintained_tracker): https://www.openproblemgarden.org/op/circular_flow_numbers_of_r_graphs\n Evidence used: Preserves the source definition and implication to the class-1 conjecture.\n\n**Review notes.** The title uses r-graphs while the body specializes to r=2t+1. The disproof is to the exact odd-regular formulation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3137, "problem_number": "OPG-60027", "title": "3-Decomposition Conjecture", "statement": "Conjecture (3-Decomposition Conjecture) Every connected cubic graph $G$ has a decomposition into a spanning tree, a family of cycles and a matching.", "background": "Source: Open Problem Garden. Original node ID: 60027. URL: http://www.openproblemgarden.org/op/3_decomposition_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/3_decomposition_conjecture\n- Author(s): Arthur; Hoffmann-Ostenhof\n- Subject(s): Graph Theory\n- Keywords: cubic graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: January 24th, 2017 by arthur\n\nProblem-page discussion:\nWe state the conjecture in a more precise manner:\n\nLet $G$ be a connected cubic graph. Then $G$ contains a spanning tree $H_1$, a $2$-regular subgraph $H_2$ and a matching $H_3$ (where only $H_3$ and not $H_1$ or $H_2$ may be empty) such that $E(H_1) \\cup E(H_2) \\cup E(H_3) = E(G)$ and $E(H_i) \\cap E(H_j) =\\emptyset$ for every $\\{i,j\\} \\subseteq \\{1,2,3\\}$ with $i\\not=j$.\n\nThe conjecture holds for all hamiltionian cubic graphs and for all connected planar cubic graphs, see [1] and see also [7].\n\nEvery cubic graph G which has a spanning tree T such that every vertex of T has degree three or one (such spanning tree T is called a HIST) obviously satisfies this conjecture. But not every connected cubic graph has a HIST, see [2].\n\nThe 3-Decomposition Conjecture has been shown to be equivalent to the following conjecture:\n\nConjecture (2-Decomposition Conjecture) Let $G$ be connected graph where every vertex has degree two or three. Suppose that for every cycle $C$ of $G$, $G-E(C)$ is disconnected, then $G$ has a decomposition into a spanning tree $T$ and a matching $M$, i.e $G-M=T$.\n\nNote that every cycle $C$ which passes through a vertex of degree two satisfies the condition that G-E(C) is disconnected.\n\nRemark: The 3-Decomposition Conjecture has also been shown to hold for other classes of cubic graphs, see for instance [3,4]. A survey on the 3-Decompostion conjecture has been given by the author 2015 in Pilsen (at that time the planar case was still open) see iti.zcu.cz/plzen15/talks/1-2a-Arthur-Survey_decomposition.ppt (and press play if you find the play button). Note that there are several papers on the problem whether a planar graph $G$ has a matching $M$ such that $G-M$ is acyclic, see for instance [6].\n\nBibliography:\n[1] Arthur Hoffmann-Ostenhof, Tomáš Kaiser, Kenta Ozeki, \\arXiv[Decomposing planar cubic graphs] 1609.05059 [math.CO]\n[2] Arthur Hoffmann-Ostenhof, Kenta Ozeki, \\arXiv[On HISTs in Cubic Graphs] 1507.07689 [math.CO]\n[3] F. Abdolhosseini, S. Akbari, H. Hashemi, M.S. Moradian, \\arXiv[Hoffmann-Ostenhof's conjecture for traceable cubic graphs] 1607.04768[math.CO]\n[4] Anna Bachstein, Dong Ye (talk): www.rwoodroofe.math.msstate.edu/workshop2014/bachstein_slides.pdf\n[5] Arthur Hoffmann-Ostenhof (talk): www.iti.zcu.cz/plzen15/talks/1-2a-Arthur-Survey_decomposition.ppt\n[6] Yingqian Wang, Qijun Zhang, Discrete Mathematics 311 (2011) 844–849, Decomposing a planar graph with girth at least 8 into a forest and a matching\n[7] Kenta Ozeki, Dong Ye, Decomposing plane cubic graphs, European Journal of Combinatorics 52 (2016) 40-46.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"3-Decomposition Conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The 3-Decomposition Conjecture remains open for general connected cubic graphs; every such graph has a quantitative near-decomposition, and exhaustive verification now covers all connected cubic graphs through 28 vertices.\n\n**Verified partial progress.**\n\n- Fan--Zhou prove a decomposition into a spanning tree, cycles, and vertex-disjoint paths of length at most two, with at most (n-4)/6 two-edge paths.\n- Goedgebeur--Jooken--Renders verify the conjecture for all connected cubic graphs on at most 28 vertices.\n- Known positive classes include connected planar, traceable, claw-free, selected 2-factor, and bounded tree/path-width cubic graphs.\n\n**Full solution or refutation.**\n\nNo counterexample or proof for all connected cubic graphs is known in the checked literature; the matching requirement is the remaining gap in the near-decomposition theorem.\n\n**What remains.**\n\nEliminate the two-edge paths from the general near-decomposition or construct a connected cubic counterexample beyond the verified range.\n\n**Sources checked.**\n\n- Genghua Fan and Chuixiang Zhou, Hoffmann-Ostenhof's 3-Decomposition Conjecture, Discrete Mathematics 348 (2025), 114454. (primary): https://doi.org/10.1016/j.disc.2025.114454\n Evidence used: States the general conjecture and proves the quantitative path near-decomposition with at most (n-4)/6 two-edge paths.\n- Jan Goedgebeur, Jorik Jooken, and Matthew Renders, The 3-Decomposition Conjecture: A SAT-Based Approach with Specialized Propagators, LIPIcs CP 2025, Article 39. (primary): https://doi.org/10.4230/LIPIcs.CP.2025.39\n Evidence used: Verifies the conjecture exhaustively for all connected cubic graphs up to 28 vertices.\n- Graph-theory open problems, 3-Decomposition Conjecture (auto-reviewed 2026-05-07). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/3_decomposition_conjecture/\n Evidence used: Records the status as partial/high-confidence and summarizes known graph classes and the 28-vertex verification.\n\n**Review notes.** No source-statement alteration; 'family of cycles' is interpreted as the edge-disjoint 2-regular part specified in the background.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3138, "problem_number": "OPG-60029", "title": "Cycle Double Covers Containing Predefined 2-Regular Subgraphs", "statement": "Conjecture Let $G$ be a $2$-connected cubic graph and let $S$ be a $2$-regular subgraph such that $G-E(S)$ is connected. Then $G$ has a cycle double cover which contains $S$ (i.e all cycles of $S$ ).", "background": "Source: Open Problem Garden. Original node ID: 60029. URL: http://www.openproblemgarden.org/op/cycle_double_covers_containing_predefined_2_regular_subgraphs.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cycle_double_covers_containing_predefined_2_regular_subgraphs\n- Author(s): Arthur; Hoffmann-Ostenhof\n- Subject(s): Graph Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 21st, 2017 by arthur\n\nProblem-page discussion:\nUsed definitions in the above conjecture: a \"cycle\" is a connected 2-regular subgraph, a \"cycle double cover\" of a graph $G$ is a set of cycles of $G$ such that every edge of $G$ is contained in precisely two cycles of the set. This conjecture has been motivated by Theorem 3, respectively, Theorem 4 in www.arxiv.org/abs/1711.10614. A weaker conjecture (Conjecture 14) has been stated in \"Snarks with special spanning trees\" (see www.arxiv.org/abs/1706.05595).\n\nRelated:\nRelated problems\nCycle double cover conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Cycle Double Covers Containing Predefined 2-Regular Subgraphs\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The full prescribed 2-regular-subgraph cycle-double-cover conjecture remains open, with tree-complement/at-most-three-circuits and single-cycle special cases proved.\n\n**Verified partial progress.**\n\n- Hoffmann-Ostenhof--Zhang--Zhang prove the case of at most three circuits when the complement is a spanning tree.\n- Recent work strengthens the non-separating single-cycle case to 5-CDC/6-CDC conclusions.\n\n**Full solution or refutation.**\n\nNo theorem covers arbitrary 2-regular S with merely connected complement.\n\n**What remains.**\n\nExtend the prescribed CDC construction to arbitrary S or find a counterexample.\n\n**Sources checked.**\n\n- Graph-theory open problems, Cycle Double Covers Containing Predefined 2-Regular Subgraphs (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/cycle_double_covers_containing_predefined_2_regular_subgraphs/\n Evidence used: Records the full conjecture as open and the named special cases.\n- A. Hoffmann-Ostenhof, C.-Q. Zhang and Z. Zhang, Cycle double covers and non-separating cycles, European Journal of Combinatorics 76 (2019); arXiv:1711.10614. (primary): https://arxiv.org/abs/1711.10614\n Evidence used: Proves the motivating restricted prescribed-subgraph case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3139, "problem_number": "OPG-60030", "title": "Monochromatic vertex colorings inherited from Perfect Matchings", "statement": "Conjecture For which values of $n$ and $d$ are there bi-colored graphs on $n$ vertices and $d$ different colors with the property that all the $d$ monochromatic colorings have unit weight, and every other coloring cancels out?", "background": "Source: Open Problem Garden. Original node ID: 60030. URL: http://www.openproblemgarden.org/op/monochromatic_vertex_colorings_inherited_from_perfect_matchings.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monochromatic_vertex_colorings_inherited_from_perfect_matchings\n- Subject(s): Graph Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: March 4th, 2019 by Mario Krenn\n\nProblem-page discussion:\nBackground: This and many related questions are directly inspired from quantum physics, and their solutions would directly contribute to new understanding in quantum physics.\n\nBi-Colored Graph: A bi-colored weighted graph $G=(V(G),E(G))$, on $n$ vertices with $d$ colors is an undirected, not necessarily simple graph where there is a fixed ordering of the vertices $V(G)=v_1, \\ldots, v_n$ and to each edge $e \\in E(G)$ there is a complex weight $w_e$ and an ordered pair of (not necessarily different) colors $(c_1(e),c_2(e))$ associated with it from the $d$ possible colors. We say that an edge is monochromatic if the associated pair of colors are not different, otherwise the edge is bi-chromatic. Moreover, if $e$ is an edge incident to the vertices $v_i,v_j \\in V(G)$ with $i\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The source does not specify the weight, cancellation rule, or bi-coloured-graph formalism sufficiently to identify a standard theorem or a verified resolution.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo exact mathematical status can be assigned safely.\n\n**What remains.**\n\nRecover definitions and locate the intended source/normalisation before a literature claim can be evaluated.\n\n**Sources checked.**\n\n- Open Problem Garden, Monochromatic vertex colorings inherited from Perfect Matchings (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/monochromatic_vertex_colorings_inherited_from_perfect_matchings\n Evidence used: Provides the same underspecified source wording.\n\n**Review notes.** Formulation defect flagged; no reconstruction.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3140, "problem_number": "OPG-60039", "title": "Sidorenko's Conjecture", "statement": "Conjecture For any bipartite graph $H$ and graph $G$, the number of homomorphisms from $H$ to $G$ is at least $\\left(\\frac{2|E(G)|}{|V(G)|^2}\\right)^{|E(H)|}|V(G)|^{|V(H)|}$.", "background": "Source: Open Problem Garden. Original node ID: 60039. URL: http://www.openproblemgarden.org/op/sidorenkos_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/sidorenkos_conjecture\n- Author(s): Sidorenko, A.\n- Subject(s): Graph Theory\n- Keywords: density problems; extremal combinatorics; homomorphism\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 10th, 2019 by Jon Noel\n\nProblem-page discussion:\nA homomorphism from a graph $H$ to a graph $G$ is a mapping $f:V(H)\\to V(G)$ which preserves edges. Given graphs $H$ and $G$, the homomorphism density of $H$ in $G$, denoted $t(H,G)$, is the probability that a random function $f:V(H)\\to V(G)$ is a homomorphism. That is,\n\n$$t(H,G)=\\frac{\\left|\\left\\{f: V(H)\\to V(G): f\\text{ is a homomorphism from }H\\text{ to }G\\right\\}\\right|}{|V(G)|^{|V(H)|}}.$$\n\nIn this language, Sidorenko's Conjecture says that, if $H$ is bipartite, then every graph $G$ satisfies\n\n$$t(H,G)\\geq t(K_2,G)^{|E(H)|}.$$\n\nThere are lots of results on Sidorenko's Conjecture; rather than listing them all here, we encourage the reader to see the references of the recent paper [CL].\n\nBibliography:\n[CL] David Conlon and Joonkyung Lee: Sidorenko's conjecture for blow-ups, submitted.\n\nRelated:\nRelated problems\nChromatic Number of Common Graphs\n\nBibliography links:\n- Sidorenko's conjecture for blow-ups: http://www.arxiv.org/abs/math.CO/1809.01259\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Sidorenko's Conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Sidorenko's conjecture remains open for arbitrary bipartite H but is proved for many large graph families.\n\n**Verified partial progress.**\n\n- All trees, even cycles, complete bipartite graphs, and numerous further constructions satisfy the conjecture.\n\n**Full solution or refutation.**\n\nNo universal proof for every bipartite graph was verified.\n\n**What remains.**\n\nProve the homomorphism lower bound for all bipartite H or find a counterexample.\n\n**Sources checked.**\n\n- Sidorenko conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Sidorenko%27s_conjecture\n Evidence used: Records the general conjecture as open and summarizes verified classes.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3141, "problem_number": "OPG-60046", "title": "3-Edge-Coloring Conjecture", "statement": "Conjecture Suppose $G$ with $|V(G)|>2$ is a connected cubic graph admitting a $3$-edge coloring. Then there is an edge $e \\in E(G)$ such that the cubic graph homeomorphic to $G-e$ has a $3$-edge coloring.", "background": "Source: Open Problem Garden. Original node ID: 60046. URL: http://www.openproblemgarden.org/op/3_edge_coloring_conjecture.\n\nSource subject path: Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/3_edge_coloring_conjecture\n- Author(s): Arthur; Hoffmann-Ostenhof\n- Subject(s): Graph Theory\n- Keywords: 3-edge coloring; 4-flow; removable edge\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Prize listed: no\n- Posted: April 28th, 2020 by arthur\n\nProblem-page discussion:\nReformulation via 4-flows:\n\nConjecture Suppose $G$ is a cubic graph with a nowhere-zero $4$-flow, then there is an edge $e \\in E(G)$ such that $G-e$ has a nowhere-zero $4$-flow.\n\nComments:\n- June 23rd, 2022 | Anonymous | Context: Is this conjecture missing some greater context? It seems obviously false on its own\n- June 22nd, 2022 | Anonymous | question: wouldn't removing any edge from a cubic graph make the graph not cubic?\n- August 3rd, 2021 | Anonymous | A counterexample?: What would be the cubic graph homeomorphic to K4-e? I think I can show there does not exist a cubic graph homeomorphic to K4-e. If so, this would seem to contradict the conjecture's claim.\n- November 13th, 2020 | Anonymous | Is there yet any progress on this problem?: Hello, I would like to know whether anybody made any progress on this. I tried to google and found nothing. Also, why is there nothing in Bibliography of this problem? Is there any paper involving or proposing it? Thanks in advance\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"3-Edge-Coloring Conjecture\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The removable-edge 3-edge-colouring conjecture is proved for cubic graphs of girth at most seven, but remains open generally.\n\n**Verified partial progress.**\n\n- A 2026 paper proves the conjecture for every 3-edge-colourable cubic graph of girth at most seven.\n\n**Full solution or refutation.**\n\nThe girth-restricted theorem does not cover arbitrary connected cubic graphs.\n\n**What remains.**\n\nRemove the girth bound or find a 3-edge-colourable cubic counterexample with no removable edge.\n\n**Sources checked.**\n\n- Removable edges and stability of 4-flow, Discrete Mathematics (2026). (authoritative_secondary): https://www.sciencedirect.com/science/article/abs/pii/S0012365X26002578\n Evidence used: The article snippet states the girth-at-most-seven theorem for the exact conjecture.\n- Open Problem Garden, 3-Edge-Coloring Conjecture (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/3_edge_coloring_conjecture\n Evidence used: Gives the exact general conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3142, "problem_number": "OPG-60055", "title": "Chromatic number of $\\frac{3}{3}$-power of graph", "statement": "Let $G$ be a graph and $m,n\\in \\mathbb{N}$. The graph $G^{\\frac{m}{n}}$ is defined to be the $m$-power of the $n$-subdivision of $G$. In other words, $G^{\\frac{m}{n}}=(G^{\\frac{1}{n}})^m$.\n\nConjecture Let $G$ be a graph with $\\Delta(G)\\geq 2$. Then $\\chi(G^{\\frac{3}{3}})\\leq 2\\Delta(G)+1$.", "background": "Source: Open Problem Garden. Original node ID: 60055. URL: http://www.openproblemgarden.org/op/chromatic_number_of_frac_3_3_power_of_graph.\n\nSource subject path: Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/chromatic_number_of_frac_3_3_power_of_graph\n- Subject(s): Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 20th, 2023 by Iradmusa\n\nBibliography:\n[1] Mahsa Mozafari-Nia and M. N. Iradmusa, Simultaneous coloring of vertices and incidences of graphs, Australasian Journal of Combinatorics, Vol. 85, Mo. 3, pp. 287-307, 2023.\n\n[2] Mahsa Mozafari-Nia and M. N. Iradmusa, Simultaneous coloring of vertices and incidences of outerplanar graphs, Electronic Journal of Graph Theory and Applications, Vol.11, No.1, pp.245-262, 2023.\n\n[3] Mahsa Mozafari-Nia and M. N. Iradmusa, A note on coloring of 3/3-power of subquartic graphs, Australasian Journal of Combinatorics, Vol. 79, No. 3, pp. 454-460, 2021.\n\n[4] M. N. Iradmusa, A short proof of 7-colorability of 3/3-power of subcubic graphs, Iranian Journal of Science and Technology, Transactions A: Science, Vol. 44, No. 1, pp. 225-226, 2020.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Chromatic number of $\\frac{3}{3}$-power of graph\" in Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general chi(G^(3/3)) <= 2Delta(G)+1 conjecture remains open, while it has been reformulated as simultaneous vertex-incidence colouring and established for several graph classes.\n\n**Verified partial progress.**\n\n- The 2023 simultaneous-colouring paper identifies chi(G^(3/3)) with the vi-simultaneous chromatic number and treats outerplanar graphs.\n- Earlier work treats subquartic and other special graph classes.\n\n**Full solution or refutation.**\n\nNo checked theorem proves the proposed 2Delta+1 bound for every graph of maximum degree at least two.\n\n**What remains.**\n\nProve the simultaneous vertex-incidence colouring bound for arbitrary graphs or find a counterexample.\n\n**Sources checked.**\n\n- M. Mozafari-Nia and M. N. Iradmusa, Simultaneous coloring of vertices and incidences of graphs, Australasian Journal of Combinatorics 85 (2023), 287--307. (primary): https://ajc.maths.uq.edu.au/pdf/85/ajc_v85_p287.pdf\n Evidence used: States the 3/3-power conjecture and its equivalence with the simultaneous-colouring parameter.\n- M. Mozafari-Nia and M. N. Iradmusa, Simultaneous coloring of vertices and incidences of outerplanar graphs, Electronic Journal of Graph Theory and Applications 11 (2023), 245--262. (primary): https://www.ejgta.org/index.php/ejgta/article/download/1629/pdf_267\n Evidence used: Proves exact outerplanar bounds, a special-case advance.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3143, "problem_number": "OPG-160", "title": "57-regular Moore graph?", "statement": "Question Does there exist a 57-regular graph with diameter 2 and girth 5?", "background": "Source: Open Problem Garden. Original node ID: 160. URL: http://www.openproblemgarden.org/op/57_regular_moore_graph.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/57_regular_moore_graph\n- Author(s): Hoffman, Alan J.; Singleton, Robert R.\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: cage; Moore graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 18th, 2007 by mdevos\n\nProblem-page discussion:\nA Moore graph is a graph with diameter $d$ and girth $2d+1$. It is known that every Moore graph is regular [S] (even distance-regular), and two (rather trivial) families of such graphs are provided by complete graphs and odd cycles. In one of the founding papers in the subject of algebraic graph theory, Hoffman and Singleton [HS] proved that every $r$-regular Moore graph with diameter 2 must have $r \\in \\{2,3,7,57\\}$. For $r=2$ and $r=3$ such graphs exist, are unique, and are familiar: the pentagon, and the Petersen graph. For $r=7$ Hoffman and Singleton constructed such a graph - now known as the Hoffman-Singleton graph, but for $r=57$ we are still uncertain whether such a graph exists.\n\nThe pentagon, the Petersen graph, and the Hoffman-Singleton graph are all very highly symmetric graphs, and much of the interest in these objects is related to exceptional phenomena in small finite groups. In contrast to this, Higman proved that a 57-regular Moore graph cannot be vertex transitive (see [C]). In some sense, this is indication that even if a 57-regular Moore graph exists, it will be of less interest than its younger siblings. Nevertheless, as a lingering problem left by one of the first papers in algebraic graph theory, this is viewed as an important question.\n\nThere are some easily established properties of a 57-regular Moore graph. For instance it must have 3250 vertices and independence number at most 400. However it seems not nearly enough is known to narrow the search sufficiently.\n\nBibliography:\n[C] P. J. Cameron, Automorphisms of graphs in: Selected topics in graph theory, Volume 2, eds. L. W. Beineke and R. J. Wilson (Academic Press, London) 1983, pp. 89-127. MathSciNet\n\n* [HS] A. J. Hoffman and R. R. Singleton, On Moore graphs with diameters 2 and 3. IBM J. Res. Develop. 4 (1960) 497--504. MathSciNet\n\n[S] R. R. Singleton, There is no irregular Moore graph. American Mathematical Monthly 75, vol 1 (1968) 42–43. MathSciNet\n\nSource links:\n- regular: http://en.wikipedia.org/wiki/regular graph\n- diameter: http://en.wikipedia.org/wiki/diameter (graph theory)\n- girth: http://en.wikipedia.org/wiki/girth\n\nDiscussion links:\n- Moore graph: http://en.wikipedia.org/wiki/Moore graph\n- Petersen graph: http://en.wikipedia.org/wiki/Petersen graph\n- Hoffman-Singleton graph: http://en.wikipedia.org/wiki/Hoffman-Singleton graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0797250\n- On Moore graphs with diameters 2 and 3: http://www.research.ibm.com/journal/rd/045/ibmrd0405H.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0140437\n- There is no irregular Moore graph: http://www.jstor.org/view/00029890/di991528/99p1853x/0\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0225679\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"57-regular Moore graph?\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Existence of the 57-regular Moore graph remains open; a claimed 2020 nonexistence proof has a critical error, and a 2026 result shows any hypothetical graph has no involutory automorphism.\n\n**Verified partial progress.**\n\n- Any such graph must be strongly regular on 3250 vertices with parameters (3250,57,0,1).\n- Ishida proves in 2026 that its automorphism group would contain no involutions.\n\n**Full solution or refutation.**\n\nNeither a construction nor a valid nonexistence proof is known.\n\n**What remains.**\n\nConstruct the graph or derive a contradiction from its strongly regular parameters and increasingly restricted automorphism structure.\n\n**Sources checked.**\n\n- V. Faber and J. Keegan, Existence of a Moore graph of degree 57 is still open, arXiv:2210.09577. (primary): https://arxiv.org/abs/2210.09577\n Evidence used: Identifies the error in Makhnev's claimed nonexistence proof and restores open status.\n- Yawara Ishida, No involutions in the missing Moore graph, arXiv:2606.29183. (primary): https://arxiv.org/abs/2606.29183\n Evidence used: Proves the new no-involutory-automorphisms restriction and explicitly says existence remains open.\n- Open Problem Garden, 57-regular Moore graph? (node 160). (maintained_tracker): https://www.openproblemgarden.org/op/57_regular_moore_graph\n Evidence used: Original question and classical Hoffman-Singleton constraints.\n\n**Review notes.** The invalid Makhnev claim is not treated as a solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3144, "problem_number": "OPG-161", "title": "Hamiltonian paths and cycles in vertex transitive graphs", "statement": "Problem Does every connected vertex-transitive graph have a Hamiltonian path?", "background": "Source: Open Problem Garden. Original node ID: 161. URL: http://www.openproblemgarden.org/op/hamiltonian_paths_and_cycles_in_vertex_transitive_graphs.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamiltonian_paths_and_cycles_in_vertex_transitive_graphs\n- Author(s): Lovasz, Laszlo\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: cycle; hamiltonian; path; vertex-transitive\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 18th, 2007 by mdevos\n\nProblem-page discussion:\nThe question posed here is due to Lovasz [L], but the general problem of finding Hamiltonian paths and cycles in highly symmetric graphs is much older. Knuth has traced it back to bell ringing, and it appears again in gray codes and in the knight's tour of a chessboard.\n\nVertex-transitive graphs are, of course, very special, very well-behaved graphs, and it seems unsurprising that many of them have Hamiltonian cycles. What is surprising is that there are only five connected ones known which do not have Hamiltonian cycles. This list consists of the complete graph on 2 vertices, the Petersen graph, Coxeter's graph, and the graphs obtained from Petersen and Coxeter by truncating every vertex (inflate each vertex to a triangle). In particular, we do not know of a vertex transitive graph without a Hamiltonian path.\n\nInterestingly, there seems to be considerable disagreement among experts as to what the answer will be. On one hand, there does not appear to be any particular reason why vertex-transitive graphs should almost always have Hamiltonian cycles. On the other hand, such graphs have been studied and searched for at great length, and so far every one investigated with the exception of the five listed above has proved to have a Hamiltonian cycle. Babai formulated the following conjecture which is in quite sharp contrast to the problem above.\n\nConjecture (Babai [B96]) There exists $\\epsilon > 0$ so that there are infinitely many connected vertex-transitive graphs $G$ with longest cycle of length $<(1-\\epsilon)|V(G)|$.\n\nFor general vertex-transitive graphs, very little is known. Babai [B79] has shown that a vertex-transitive graph on $n$ vertices has a cycle of length $\\ge \\sqrt{3n}$, but (though a very clever arguement) this is obviously quite far from the conjecture. Considerable attention has been given to the special case of Cayley graphs. Here we have the following conjecture.\n\nConjecture Every connected Cayley graph (apart from $K_2$ ) has a Hamiltonian cycle.\n\nThe above conjecture is not difficult to prove for abelian groups. Witte [W] proved it for $p$-groups, and it has also been established for certain special types of generating sets. Two other results of note are a theorem of Pak-Radocic [PR] showing that every group $G$ has a generating set of size $\\le \\log_2(|G|)$ for which the corresponding Cayley graph is Hamiltonian, and a theorem of Krivelevich-Sudakov [KS] showing that almost surely taking a random set of $\\log^5(|G|)$ elements of $G$ as generators yields a Hamiltonian graph.\n\nBibliography:\n[B79] L. Babai, Long cycles in vertex-transitive graphs. J. Graph Theory 3 (1979), no. 3, 301--304. MathSciNet\n\n[B96] L. Babai, Automorphism groups, isomorphism, reconstruction, in Handbook of Combinatorics, Vol. 2, Elsevier, 1996, 1447-1540. MathSciNet\n\n[KS] M. Krivelevich and B. Sudakov, Sparse pseudo-random graphs are Hamiltonian. J. Graph Theory 42 (2003), no. 1, 17--33. MathSciNet\n\n[L] L. Lov\\'{a}sz, \"Combinatorial structures and their applications\", (Proc. Calgary Internat. Conf., Calgary, Alberta, 1969), pp. 243-246, Problem 11, Gordon and Breach, New York, 1970.\n\n[PR] I. Pak and R. Radocic, Hamiltonian paths in Cayley graphs, preprint\n\n[W] D. Witte, Cayley digraphs of prime-power order are Hamiltonian. J. Combin. Theory Ser. B 40 (1986), no. 1, 107--112. MathSciNet\n\n[WG] D. Witte and J.A. Gallian, A survey: Hamiltonian cycles in Cayley graphs. Discrete Math. 51 (1984), no. 3, 293--304. MathSciNet\n\nSource links:\n- vertex-transitive graph: http://en.wikipedia.org/wiki/vertex-transitive graph\n- Hamiltonian path: http://en.wikipedia.org/wiki/Hamiltonian path\n\nDiscussion links:\n- bell ringing: http://en.wikipedia.org/wiki/bell ringing\n- gray codes: http://en.wikipedia.org/wiki/gray codes\n- knight's tour: http://en.wikipedia.org/wiki/knight's tour\n- Petersen graph: http://en.wikipedia.org/wiki/Petersen graph\n- Coxeter's graph: http://mathworld.wolfram.com/CoxeterGraph.html\n- Cayley graphs: http://en.wikipedia.org/wiki/cayley graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0542553\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1373683\n- Sparse pseudo-random graphs are Hamiltonian: http://www.math.princeton.edu/%7Ebsudakov/pseudo-hamiltonian.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1943104\n- Hamiltonian paths in Cayley graphs: http://www-math.mit.edu/%7Epak/hamcayley8.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0830597\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0762322\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamiltonian paths and cycles in vertex transitive graphs\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Lovász Hamilton-path problem for connected vertex-transitive graphs remains open, but strong long-cycle bounds and many infinite families are known.\n\n**Verified partial progress.**\n\n- Every connected vertex-transitive graph on n at least 3 vertices has a cycle of length Omega(n^(13/21)).\n- Every Kneser graph has a Hamilton cycle except the Petersen graph; the same work covers connected generalized Johnson graphs with that exception.\n\n**Full solution or refutation.**\n\nThe 2025 long-cycle theorem improves the general guaranteed cycle length, while the 2023 Kneser result settles a broad special family. Neither supplies a spanning path for every connected vertex-transitive graph.\n\n**What remains.**\n\nProve that every finite connected vertex-transitive graph has a Hamiltonian path or construct a counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, Hamiltonian paths and cycles in vertex transitive graphs (OPG-161), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/hamiltonian_paths_and_cycles_in_vertex_transitive_graphs\n Evidence used: Maintains the exact Lovász Hamilton-path question and its classical context.\n- C. Groenland, S. Longbrake, R. Steiner, J. Turcotte, and L. Yepremyan, Longest cycles in vertex-transitive and highly connected graphs, Bulletin of the London Mathematical Society 57 (2025), 2975-2990. (primary): https://doi.org/10.1112/blms.70134\n Evidence used: Proves the Omega(n^(13/21)) cycle-length lower bound for connected vertex-transitive graphs.\n- A. Merino, T. Mütze, and Namrata, Kneser graphs are Hamiltonian, STOC 2023, 963-970. (primary): https://doi.org/10.1145/3564246.3585137\n Evidence used: Proves Hamiltonicity for Kneser and connected generalized Johnson graphs, apart from the Petersen exception.\n\n**Review notes.** The supplied statement is unambiguous and was not altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3145, "problem_number": "OPG-345", "title": "Triangle free strongly regular graphs", "statement": "Problem Is there an eighth triangle free strongly regular graph?", "background": "Source: Open Problem Garden. Original node ID: 345. URL: http://www.openproblemgarden.org/op/triangle_free_strongly_regular_graphs.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/triangle_free_strongly_regular_graphs\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: strongly regular; triangle free\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 28th, 2007 by mdevos\n\nProblem-page discussion:\nA regular graph $G$ is strongly regular if there exist integers $\\lambda, \\mu$ so that every pair of adjacent vertices have exactly $\\lambda$ common neighbors, and every pair of nonadjacent vertices have exactly $\\mu$ common neighbors. To eliminate degeneracies, we shall further assume that $\\mu \\ge 1$. If $G$ is $k$-regular and $|V(G)| = n$, then we say that $G$ is a $(n,k,\\lambda,\\mu)$ strongly regular graph.\n\nThere are exactly seven triangle-free strongly regular graphs known: The five cycle, the Petersen Graph, The Clebsch Graph, the Hoffman-Singleton Graph, The Gewirtz Graph, the Higman-Sims Graph, and a $(77,16,0,4)$ strongly regular subgraph of the Higman-Sims graph. Every Moore Graph of diameter 2 is a triangle-free strongly regular graph, so if there is a 57-regular Moore Graph of diameter 2, this would add another to the list.\n\nSee Andries Brouwer's graph descriptions for more on these graphs.\n\nBibliography:\n[G] C. D. Godsil, Problems in Algebraic Combinatorics, Electronic Journal of Combinatorics, Volume 2, F1\n\nDiscussion links:\n- strongly regular: http://en.wikipedia.org/wiki/strongly regular graph\n- Petersen Graph: http://en.wikipedia.org/wiki/Petersen Graph\n- Hoffman-Singleton Graph: http://en.wikipedia.org/wiki/Hoffman-Singleton Graph\n- Moore Graph: http://en.wikipedia.org/wiki/Moore graph\n- 57-regular Moore Graph: http://www.openproblemgarden.org/?q=op/57_regular_moore_graph\n- Andries Brouwer's graph descriptions: http://www.win.tue.nl/%7Eaeb/drg/graphs/index.html\n\nBibliography links:\n- Problems in Algebraic Combinatorics: http://www.combinatorics.org/Volume_2/PDFFiles/v2i1f1.pdf\n\nComments:\n- May 30th, 2020 | Anonymous | Higman-Sims Graph: In fact, all (seven) known primitive triangle-free strongly regular graphs are actual *subgraphs* of the Higman-Sims graph (which btw was first constructed by Dale Mesner). A Moore graph of degree 57 would of course break this mold.\n- February 23rd, 2012 | Anonymous | Reply..: I hate math..Lol _________________________________________________________________________________________ toll free number\n- February 8th, 2012 | Anonymous | The complement of $2 K_n$: The complement of $2 K_n$ is triangle free srg with parameters $(2n,n,0,n)$. Probably this infinite case should be excluded from the conjecture.\n- April 21st, 2011 | Anonymous | If the number of the: If the number of the vertices are even, we can determine regular triangle free graphs up to half the number of vertices of any degree. However, this does not hold true if the vertices numbers are odd. Some of the examples of even vertices graphs are Petersen graph (10 vertices), Heawood graph (14 vertices), Clebsch graph (16 vertices), Pappus graph (18 vertices) and odd vertices graph includes Schläfli graph (27 vertices), Perkel graph (57 vertices). 800 numbers\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Triangle free strongly regular graphs\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No eighth triangle-free strongly regular graph was verified.\n\n**Verified partial progress.**\n\n- Only a small finite catalog of such graphs is currently known.\n\n**Full solution or refutation.**\n\nThe existence question remains open.\n\n**What remains.**\n\nUse feasibility parameter constraints and spectral constructions.\n\n**Sources checked.**\n\n- Open Problem Garden, node 345 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3146, "problem_number": "OPG-348", "title": "Half-integral flow polynomial values", "statement": "Let $\\Phi(G,x)$ be the flow polynomial of a graph $G$. So for every positive integer $k$, the value $\\Phi(G,k)$ equals the number of nowhere-zero $k$-flows in $G$.\n\nConjecture $\\Phi(G,5.5) > 0$ for every 2-edge-connected graph $G$.", "background": "Source: Open Problem Garden. Original node ID: 348. URL: http://www.openproblemgarden.org/op/half_integral_flow_polynomial_values.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/half_integral_flow_polynomial_values\n- Author(s): Mohar, Bojan\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: nowhere-zero flow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 31st, 2007 by mohar\n\nProblem-page discussion:\nBy Seymour's 6-flow theorem, $\\Phi(G,k) > 0$ for every 2-edge-connected graph $G$ and every integer $k\\ge6$.\n\nIt would be interesting to find any non-integer rational number $x>5$ so that $\\Phi(G,x) > 0$ for every 2-edge-connected graph $G$. It is known that zeros of flow polynomials are dense in the complex plane.\n\nSource links:\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Half-integral flow polynomial values\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that every 2-edge-connected graph has positive flow polynomial at 5.5 was verified.\n\n**Verified partial progress.**\n\n- The assertion refines nowhere-zero-flow behavior through real flow roots.\n\n**Full solution or refutation.**\n\nThe half-integral positivity conjecture remains open.\n\n**What remains.**\n\nControl real flow roots for 2-edge-connected graphs.\n\n**Sources checked.**\n\n- Open Problem Garden, node 348 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3147, "problem_number": "OPG-372", "title": "Ramsey properties of Cayley graphs", "statement": "Conjecture There exists a fixed constant $c$ so that every abelian group $G$ has a subset $S \\subseteq G$ with $-S = S$ so that the Cayley graph ${\\mathit Cayley}(G,S)$ has no clique or independent set of size $> c \\log |G|$.", "background": "Source: Open Problem Garden. Original node ID: 372. URL: http://www.openproblemgarden.org/op/ramsey_properties_of_cayley_graphs.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/ramsey_properties_of_cayley_graphs\n- Author(s): Alon, Noga\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: Cayley graph; Ramsey number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 10th, 2007 by mdevos\n\nProblem-page discussion:\nThe classic bounds from Ramsey theory show that every $n$ vertex graph must have either a clique or an independent set of size $c \\log n$ and further random graphs almost surely have this property (using different values of $c$ ). The above conjecture asserts that every group has a Cayley graph with similar behavior.\n\nImproving upon some earlier results of Agarwal et. al. [AAAS], Green [G] proved that there exists a constant $c$ so that whenever a set $S \\subseteq {\\mathbb Z}_n$ is chosen at random, and we form the graph with vertex set ${\\mathbb Z}_n$ and two vertices $i$, $j$ joined if $i+j \\in S$, then this graph almost surely has both maximum clique size and maximum independent size $O(\\log n)$. The reader should note that such graphs are not generally Cayley graphs - although the definition is similar.\n\nAs a word of caution, Green [G] also shows that a randomly chosen subset of the group ${\\mathbb Z}_2^n$ almost surely has both max. clique and max. independent set of size $\\Theta( \\log N \\log \\log N )$ where $N = 2^n$.\n\nBibliography:\n[AAAS] P. K. Agarwal, N. Alon, B. Aronov, S. Suri, Can visibility graphs be represented compactly? Discrete Comput. Geom. 12 (1994), no. 3, 347--365. MathSciNet\n\n*[C] Problem BCC14.6 from the BCC Problem List (edited by Peter Cameron)\n\n[G] B. Green, Counting sets with small sumset, and the clique number of random Cayley graphs, Combinatorica 25 (2005), no. 3, 307--326. MathSciNet\n\nSource links:\n- Cayley graph: http://en.wikipedia.org/wiki/cayley graph\n\nBibliography links:\n- Can visibility graphs be represented compactly?: http://www.math.tau.ac.il/%7Enogaa/PDFS/main.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1298916\n- BCC Problem List: http://www.maths.qmul.ac.uk/%7Epjc/bcc/allprobs.pdf\n- Counting sets with small sumset, and the clique number of random Cayley graphs: http://arxiv.org/PS_cache/math/pdf/0304/0304183v2.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2141661\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 17.\n\nAttempt notes:\nTarget:\nMake progress on \"Ramsey properties of Cayley graphs\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No uniform logarithmic Ramsey Cayley-graph construction for every abelian group was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nConstruct pseudorandom symmetric subsets uniformly across abelian groups.\n\n**Sources checked.**\n\n- Open Problem Garden, node 372 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3148, "problem_number": "OPG-407", "title": "Laplacian Degrees of a Graph", "statement": "Conjecture If $G$ is a connected graph on $n$ vertices, then $c_k(G) \\ge d_k(G)$ for $k = 1, 2, \\dots, n-1$.", "background": "Source: Open Problem Garden. Original node ID: 407. URL: http://www.openproblemgarden.org/op/laplacian_degrees_of_a_graph.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/laplacian_degrees_of_a_graph\n- Author(s): Guo, Ji-Ming\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: degree sequence; Laplacian matrix\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 22nd, 2007 by Robert Samal\n\nProblem-page discussion:\n(Reproduced from [M].)\n\nLet $L = D - A$ be the Laplacian matrix of a graph $G$ of order $n$. Let $t_k$ be the $k$-th largest eigenvalue of $L$ ( $k = 1,\\dots,n$ ). For the purpose of this problem, we call the number $$c_k = c_k(G) = t_k + k - 2$$the$k$-th Laplacian degree of$G$. In addition to that, let$d_k(G)$be the$k$-th largest (usual) degree in$G$. It is known that every connected graph satisfies$c_k(G) \\ge d_k(G)$for$k = 1$[GM],$k = 2$[LP] and for$k = 3$ [G].\n\nBibliography:\n[GM] R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math.7 (1994) 221-229. MathSciNet\n\n[LP] J.S. Li, Y.L. Pan, A note on the second largest eigenvalue of the Laplacian matrix of a graph, Linear Multilin. Algebra 48 (2000) 117-121. MathSciNet\n\n*[G] J.-M. Guo, On the third largest Laplacian eigenvalue of a graph, Linear Multilin. Algebra 55 (2007) 93-102. MathSciNet\n\n[M] B. Mohar, Problem of the Month\n\nDiscussion links:\n- Laplacian matrix: http://en.wikipedia.org/wiki/Laplacian matrix\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1271994\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1813439\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2281876\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0612_LaplacianDegrees.html\n\nComments:\n- February 29th, 2008 | Anonymous | Proved: Proved by Willem Haemers and Andries Brouwer, see guo.pdf.\n- September 13th, 2007 | Gordon Royle | Equality?: Any conjectures about the structure of the graphs for which equality holds (for any particular value of k)?\n\nGordon Royle\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 20.\n\nAttempt notes:\nTarget:\nMake progress on \"Laplacian Degrees of a Graph\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the Laplacian-degree majorization inequality was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nRelate Laplacian coefficients to degree-sequence inequalities.\n\n**Sources checked.**\n\n- Open Problem Garden, node 407 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3149, "problem_number": "OPG-824", "title": "Cores of strongly regular graphs", "statement": "Question Does every strongly regular graph have either itself or a complete graph as a core?", "background": "Source: Open Problem Garden. Original node ID: 824. URL: http://www.openproblemgarden.org/op/cores_of_strongly_regular_graphs.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cores_of_strongly_regular_graphs\n- Author(s): Cameron, Peter J.; Kazanidis, Priscila A.\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: core; strongly regular\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 16th, 2008 by mdevos\n\nProblem-page discussion:\nIf true, this curious question indicates a very interesting property of strongly regular graphs. While on the surface, there would appear to be no particular reason for it to hold, it has already been verified for a number of interesting classes of graphs. Cameron and Kazanidis [CK] showed that it holds for rank-3 graphs, while Godsil and Royle [GR] have showed that it holds for point graphs of generalized quadrangles, block graphs of Steiner systems and orthogonal arrays with sufficiently many points, and for all strongly regular graphs on at most 36 vertices.\n\nBibliography:\n*[CK] P. J. Cameron and P. A. Kazanidis, Cores of symmetric graphs, J. Australian Math. Soc., to appear.\n\n[GR] C. Godsil and G.F. Royle, Cores of Geometric Graphs\n\nSource links:\n- strongly regular graph: http://en.wikipedia.org/wiki/strongly regular graph\n- core: http://en.wikipedia.org/wiki/core (graph theory)\n\nBibliography links:\n- Cores of Geometric Graphs: http://xxx.tau.ac.il/pdf/0806.1300v1\n\nComments:\n- October 18th, 2011 | Anonymous | Correction: I believe you mean \"Godsil and Royle\", not \"Gordon and Royle\".\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Cores of strongly regular graphs\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Roberson proved the Cameron--Kazanidis core-complete conjecture for all strongly regular graphs.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nEvery strongly regular graph is either a core or has a complete graph as core.\n\n**What remains.**\n\nNo open part of the stated question remains.\n\n**Sources checked.**\n\n- D. E. Roberson, Homomorphisms of strongly regular graphs, Algebraic Combinatorics 2 (2019), 481--510. (primary): https://www.numdam.org/article/ALCO_2019__2_4_481_0.pdf\n Evidence used: The abstract says it confirms and strengthens the Cameron--Kazanidis conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3150, "problem_number": "OPG-36880", "title": "Does the chromatic symmetric function distinguish between trees?", "statement": "Problem Do there exist non-isomorphic trees which have the same chromatic symmetric function?", "background": "Source: Open Problem Garden. Original node ID: 36880. URL: http://www.openproblemgarden.org/op/does_the_symmetric_chromatic_function_distinguish_trees.\n\nSource subject path: Graph Theory > Algebraic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/does_the_symmetric_chromatic_function_distinguish_trees\n- Author(s): Stanley, Richard P.\n- Subject(s): Graph Theory; Algebraic Graph Theory\n- Keywords: chromatic polynomial; symmetric function; tree\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 25th, 2009 by mdevos\n\nProblem-page discussion:\nStanley [S] introduced the following symmetric function associated with a graph. Let $x_1,x_2,\\ldots$ be commuting indeterminates, and for every graph $G=(V,E)$ let ${\\mathcal C}_G$ be the set of all proper colorings $f: V \\rightarrow {\\mathbb N}$. Then the chromatic symmetric function is defined to be\n$$\nX_G = \\sum_{f \\in {\\mathcal C}_G} \\prod_{v \\in V} x_{f(v)}.\n$$\n So, the coefficient of a term $x_1^{d_1} x_2^{d_2} \\ldots$ in $X_G$ is precisely the number of proper colorings of $G$ where color $i$ appears exactly $d_i$ times. It is immediate that $X_G$ is homogeneous of degree $|V|$ and is symmetric.\n\nIf we set $x_1,x_2,\\ldots,x_k = 1$ and $x_{k+1}, x_{k+2} \\ldots = 0$ and evaluate, we get the number of proper colorings of $G$ using the colors $1,2,\\ldots,k$. Therefore, the chromatic symmetric function contains all of the information of the chromatic polynomial. In fact, the chromatic symmetric function contains strictly more information about the graph, since there exist examples of graphs which have distinct chromatic symmetric functions but have the same chromatic polynomial.\n\nThis natural problem of Stanley remains wide open. It has recently been established for some special classes of trees, namely caterpillars and spiders [MMW].\n\nBibliography:\n[MMW] J. Martin, M. Morin, and J. D. Wagner, On distinguishing trees by their chromatic symmetric functions. J. Combin. Theory Ser. A 115 (2008), no. 2, 237–253. MathSciNet\n\n*[S] R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166–194.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2382514\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Does the chromatic symmetric function distinguish between trees?\" in Graph Theory; Algebraic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Stanley's tree-distinguishability question is still open, but the chromatic symmetric function is proved to recover several new tree invariants and many tree classes.\n\n**Verified partial progress.**\n\n- The function determines the generalized degree sequence of a tree.\n- It distinguishes many named classes and has been verified for finite orders.\n\n**Full solution or refutation.**\n\nNo nonisomorphic trees with equal chromatic symmetric function, nor a universal proof of distinction, was verified.\n\n**What remains.**\n\nEither find a collision or prove the function reconstructs all trees.\n\n**Sources checked.**\n\n- J. Aliste-Prieto, J. Martin, J. Wagner and J. Zamora, Chromatic symmetric functions and polynomial invariants of trees, arXiv:2402.10333. (primary): https://arxiv.org/abs/2402.10333\n Evidence used: Establishes the generalized-degree-sequence result while the main question remains open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3151, "problem_number": "OPG-164", "title": "Graham's conjecture on tree reconstruction", "statement": "Problem for every graph $G$, we let $L(G)$ denote the line graph of $G$. Given that $G$ is a tree, can we determine it from the integer sequence $|V(G)|, |V(L(G))|, |V(L(L(G)))|, \\ldots$?", "background": "Source: Open Problem Garden. Original node ID: 164. URL: http://www.openproblemgarden.org/op/grahams_conjecture_on_tree_reconstruction.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/grahams_conjecture_on_tree_reconstruction\n- Author(s): Graham, Ronald L.\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: reconstruction; tree\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 18th, 2007 by mdevos\n\nProblem-page discussion:\nGraph reconstruction is a notoriously difficult subject. This conjecture is an unusual type of reconstruction problem where our class of graphs is very limited - just trees, but we are also given relatively little information - just a sequence of integers.\n\nBibliography:\n[GR] C. Godsil and G. Royle, Algebraic graph theory. Graduate Texts in Mathematics, 207. Springer-Verlag, New York, 2001 (page 18).\n\nSource links:\n- line graph: http://en.wikipedia.org/wiki/line graph\n\nComments:\n- April 14th, 2009 | Anonymous | Reference: Could someone pleas give a proper reference? If I'm not mistaken, the problem is just *mentioned* in G&R, without references (anyway, I didn't find any).\n- December 22nd, 2008 | Anonymous | Graph theory: Does for any tree T there exist n that L^n(T) is a regular graph? Or perhaps for all graph?\n- January 16th, 2013 | leshabirukov | No: Consider L^i(T) is a graph with \"even triangle\"(triangle with even degrees of vertices) subgraph. Edges of even triangle produce new even triangle in L^(i+1)(T). And if there is an odd degree vertex adjacent to parent triangle, there would be another one adjacent to child. So, irregular subgraph remains.\n- June 10th, 2010 | Anonymous | No: Let G be a star graph of order 5. Then L(G) = C_4. Note that L(C_4) = C_4.\n- January 8th, 2013 | Anonymous | If G is a star graph of: If G is a star graph of order 5, then L(G) = K_5.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Graham's conjecture on tree reconstruction\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Graham's conjecture that the full iterated line-graph order sequence determines a tree remains open, with both structural counting progress and finite verification.\n\n**Verified partial progress.**\n\n- The number of Graham-equivalence classes of trees on n vertices is at least exp(Omega((log n)^(3/2))).\n- A 2025 experimental study reports complete verification for all trees through 11 vertices and partial verification from 12 through 16 vertices.\n\n**Full solution or refutation.**\n\nCooper, Kay, and Swifton construct superpolynomially many distinguishable Graham classes using caterpillars and Prouhet-Tarry-Escott partitions. This remains far below the exponential number of unlabeled trees.\n\n**What remains.**\n\nProve injectivity of the Graham sequence on all finite trees or find two nonisomorphic trees having identical full sequences.\n\n**Sources checked.**\n\n- Open Problem Garden, Graham's conjecture on tree reconstruction (OPG-164), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/grahams_conjecture_on_tree_reconstruction\n Evidence used: States the tree reconstruction question from iterated line-graph orders.\n- J. Cooper, B. Kay, and A. Swifton, Graham's Tree Reconstruction Conjecture and a Waring-Type Problem on Partitions, Journal of Combinatorics 9 (2018), 469-488. (primary): https://doi.org/10.4310/JOC.2018.v9.n3.a3\n Evidence used: Proves the exp(Omega((log n)^(3/2))) lower bound on the number of distinguishable Graham classes.\n- K. Weatherspoon and D. Zeilberger, An Experimental Note on Graham's Tree Reconstruction Conjecture (2025). (primary): https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/treeline.html\n Evidence used: Author-hosted computational paper and code reporting finite verification through 11 vertices and partial checks beyond.\n\n**Review notes.** The awkward opening 'for every graph G' followed by the tree restriction is preserved and flagged in report.md. No computation was run for this triage.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3152, "problem_number": "OPG-801", "title": "Nearly spanning regular subgraphs", "statement": "Conjecture For every $\\epsilon > 0$ and every positive integer $k$, there exists $r_0 = r_0(\\epsilon,k)$ so that every simple $r$-regular graph $G$ with $r \\ge r_0$ has a $k$-regular subgraph $H$ with $|V(H)| \\ge (1- \\epsilon) |V(G)|$.", "background": "Source: Open Problem Garden. Original node ID: 801. URL: http://www.openproblemgarden.org/op/nearly_spanning_regular_subgraphs.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/nearly_spanning_regular_subgraphs\n- Author(s): Alon, Noga; Mubayi, Dhruv\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: regular; subgraph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 22nd, 2008 by mdevos\n\nProblem-page discussion:\nPetersen's theorem asserts that every regular graph of even degree contains a 2-factor (i.e. a spanning 2-regular subgraph). Iterating this easy result we find that for any pair of positive even integers $k,r$, every $r$-regular graph has a spanning $k$-regular subgraph. The cases when either $k$ or $r$ is odd are considerably more complicated. There are some nice general results (see [AFK]) which show that every regular graph of sufficiently high degree contains a $k$-regular subgraph. However these theorems give no bound on the size of this subgraph.\n\nFor $k=1$ this conjecture is an easy consequence of Vizing's Theorem. Indeed, this theorem implies that every $d$-regular graph $G$ has a 1-regular subgraph $H$ with $|V(H)| \\ge (1 - \\frac{1}{d+1}) |V(G)|$ (just choose a largest color class from a $(d+1)$-edge coloring). Alon [A] proved the conjecture for $k=2$ with the help of two famous results on permanents: the Minc Conjecture (proved by Bregman), and the van der Waerden conjecture (proved by Falikman and Egorichev). It is open for all $k \\ge 3$.\n\nBibliography:\n*[A] N. Alon, Problems and results in extremal combinatorics, J, Discrete Math. 273 (2003), 31-53.\n\n[AFK] N. Alon, S. Friedland and G. Kalai, Regular subgraphs of almost regular graphs, J. Combinatorial Theory, Ser. B 37(1984), 79-91.\n\nBibliography links:\n- Problems and results in extremal combinatorics: http://www.math.tau.ac.il/%7Enogaa/PDFS/extremal1.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Nearly spanning regular subgraphs\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The nearly-spanning conclusion is proved for k=1 and k=2 and in parity-forced even cases, but the conjecture remains open for general k>=3.\n\n**Verified partial progress.**\n\n- Vizing's theorem gives the k=1 case.\n- Alon proves the k=2 case; even r and k give spanning even factors by Petersen-type factorization.\n\n**Full solution or refutation.**\n\nNo theorem covering all fixed k and sufficiently large r was verified.\n\n**What remains.**\n\nHandle odd or general k>=3 with a near-spanning size bound.\n\n**Sources checked.**\n\n- N. Alon, Problems and results in extremal combinatorics, Discrete Math. 273 (2003), 31--53. (primary): https://www.math.kit.edu/iag6/lehre/extprobcomb2020w/media/alon-regular-subgraphs-section-3.pdf\n Evidence used: States the conjecture and proves its k=2 case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3153, "problem_number": "OPG-826", "title": "Complete bipartite subgraphs of perfect graphs", "statement": "Problem Let $G$ be a perfect graph on $n$ vertices. Is it true that either $G$ or $\\bar{G}$ contains a complete bipartite subgraph with bipartition $(A,B)$ so that $|A|, |B| \\ge n^{1 - o(1)}$?", "background": "Source: Open Problem Garden. Original node ID: 826. URL: http://www.openproblemgarden.org/op/complete_bipartite_subgraphs_of_perfect_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/complete_bipartite_subgraphs_of_perfect_graphs\n- Author(s): Fox, Jacob\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: perfect graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 17th, 2008 by mdevos\n\nProblem-page discussion:\nEvery perfect graph on $n$ vertices either has a clique or an independent set of size $\\ge n^{1/2}$, so weakening the bound on $|A|$, $|B|$ to $\\lfloor \\frac{1}{2} n^{1/2} \\rfloor$ gives a true statement. Jacob Fox [F] has proved that every comparability graph $G$ on $n$ vertices has a complete bipartite subgraph of size $\\ge c \\frac{n}{\\log n}$, and (up to the constant) this is best possible.\n\nBibliography:\n[F] J. Fox, A Bipartite Analogue of Dilworth’s Theorem, Order 23 (2006), 197-209.\n\nBibliography links:\n- A Bipartite Analogue of Dilworth’s Theorem: http://www.princeton.edu/%7Ejacobfox/papers/bipdil.pdf\n\nComments:\n- January 19th, 2021 | Anonymous | Claimed to be solved: In this recent preprint on Paul Seymour's webpage: http://web.math.princeton.edu/~pds/papers/pure5/paper.pdf (not on ArXiv nor published yet)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Complete bipartite subgraphs of perfect graphs\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No n^(1-o(1)) balanced complete-bipartite conclusion for every perfect graph or complement was verified.\n\n**Verified partial progress.**\n\n- The source records the elementary n^(1/2)-scale weakening from the clique-or-stable-set property of perfect graphs.\n\n**Full solution or refutation.**\n\nNo proof or counterexample at the proposed near-linear exponent was verified.\n\n**What remains.**\n\nImprove the exponent substantially or construct a perfect-graph obstruction.\n\n**Sources checked.**\n\n- Open Problem Garden, node 826 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/complete_bipartite_subgraphs_of_perfect_graphs\n Evidence used: Retains the exact formulation and the n^(1/2)-scale baseline.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3154, "problem_number": "OPG-36933", "title": "Asymptotic Distribution of Form of Polyhedra", "statement": "Problem Consider the set of all topologically inequivalent polyhedra with $k$ edges. Define a form parameter for a polyhedron as $\\beta:= v/(k+2)$ where $v$ is the number of vertices. What is the distribution of $\\beta$ for $k \\to \\infty$?", "background": "Source: Open Problem Garden. Original node ID: 36933. URL: http://www.openproblemgarden.org/op/asymptotic_distribution_of_form_of_polyhedra.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/asymptotic_distribution_of_form_of_polyhedra\n- Author(s): Rüdinger, Andreas\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: polyhedral graphs, distribution\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 9th, 2009 by andreasruedinger\n\nProblem-page discussion:\nConsider the set of all topologically inequivalent polyhedra on a sphere with k edges (i.e. polyhedral graphs, Sloan Sequence A002840 ). Due to duality the distribution of the form parameter $\\beta:= v/(k+2)$ is symmetric about $\\beta=1/2$. Now a natural question is whether the distribution of beta tends to a limiting distribution when the number of edges tends to infinity. Is there any nontrivial limit theorem by means of rescaling? Some numerical values can be found on Counting Polyhedra suggesting that the distribution concentrates around $\\beta=1/2$.\n\nDiscussion links:\n- Sloan Sequence A002840: http://www.research.att.com/%7Enjas/sequences/A002840\n- Counting Polyhedra: http://home.att.net/%7Enumericana/data/polycount.htm\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Asymptotic Distribution of Form of Polyhedra\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No checked source proves a limiting or rescaled distribution for the exact uniform-unlabelled polyhedral-graph model conditioned on edge count. Closely related rooted-map and labelled-graph results use different probability measures.\n\n**Verified partial progress.**\n\n- Duijvestijn and Federico enumerated polyhedral graphs jointly by vertices and edges through small edge counts and discussed conjectured asymptotics.\n- Euler duality sends v to k+2-v, making beta's finite distribution symmetric about 1/2 under a duality-invariant uniform model.\n- Bivariate enumeration and limit laws are available for related rooted-map or labelled-planar-graph models, but a transfer to uniform unlabelled isomorphism classes was not verified.\n\n**Full solution or refutation.**\n\nThe exact problem appears unresolved but is classified uncertain rather than open because the probability model and the requested rescaling are implicit, and relevant enumeration literature may not use the same weighting.\n\n**What remains.**\n\nSpecify uniform sampling over unlabelled 3-connected planar graphs with exactly k edges and a centering/scaling, then prove concentration or a limit law for the vertex count under that measure.\n\n**Sources checked.**\n\n- A. J. W. Duijvestijn and P. J. Federico, The number of polyhedral (3-connected planar) graphs, Mathematics of Computation 37 (1981), 523-532, DOI 10.1090/S0025-5718-1981-0628713-3. (primary): https://doi.org/10.1090/S0025-5718-1981-0628713-3\n Evidence used: Provides finite joint counts by vertices and edges and discusses conjectured enumeration asymptotics, but not the requested limit distribution.\n- Graph-theory open problems, Asymptotic Distribution of Form of Polyhedra, checked 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/asymptotic_distribution_of_form_of_polyhedra/\n Evidence used: Current status review finds related labelled/rooted results but no result for the exact fixed-edge uniform-unlabelled distribution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3155, "problem_number": "OPG-37038", "title": "Domination in cubic graphs", "statement": "Problem Does every 3-connected cubic graph $G$ satisfy $\\gamma(G) \\le \\lceil |G|/3 \\rceil$?", "background": "Source: Open Problem Garden. Original node ID: 37038. URL: http://www.openproblemgarden.org/op/domination_in_cubic_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/domination_in_cubic_graphs\n- Author(s): Reed, Bruce A.\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: cubic graph; domination\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 19th, 2009 by mdevos\n\nProblem-page discussion:\nIf the girth of $G$ is sufficiently large, then $\\gamma(G)\\leq 0.2999|G|$ [KSV].\n\nBibliography:\n[KSV] Daniel Kral, Petr Skoda, Jan Volec: Domination number of cubic graphs with large girth\n\nBibliography links:\n- Domination number of cubic graphs with large girth: http://arxiv.org/abs/0907.1166\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Domination in cubic graphs\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The 3-connected cubic-graph domination bound remains open, with stronger bounds proved for large-girth and other cycle-restricted subclasses.\n\n**Verified partial progress.**\n\n- Král', Škoda, and Volec prove gamma(G) <= 0.299871|G| + O(|G|/g) for cubic graphs of girth g, hence a coefficient below 1/3 for sufficiently large girth.\n- Dorbec and Henning prove the related girth-at-least-six 1/3 conjecture when 7- and 8-cycles are absent, plus a bipartite variant under explicit cycle exclusions.\n\n**Full solution or refutation.**\n\nNo proof or counterexample for all 3-connected cubic graphs was verified.\n\n**What remains.**\n\nProve gamma(G) <= ceil(|G|/3) for every 3-connected cubic graph or exhibit a 3-connected counterexample.\n\n**Sources checked.**\n\n- D. Král', P. Škoda, and J. Volec, Domination number of cubic graphs with large girth, J. Graph Theory 69 (2012), arXiv:0907.1166. (primary): https://arxiv.org/abs/0907.1166\n Evidence used: Proves the 0.299871n + O(n/g) large-girth bound.\n- P. Dorbec and M. A. Henning, The 1/3-conjectures for domination in cubic graphs, arXiv:2401.17820 (2024). (primary): https://arxiv.org/abs/2401.17820\n Evidence used: Proves cycle-restricted 1/3 results for related cubic and bipartite conjectures, not the full 3-connected problem.\n- Graph-theory open problems, Domination in cubic graphs (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/domination_in_cubic_graphs/\n Evidence used: Explicitly records the original 3-connected conjecture as open and distinguishes the newer related results.\n\n**Review notes.** The exact source statement was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3156, "problem_number": "OPG-37163", "title": "Friendly partitions", "statement": "A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.\n\nProblem Is it true that for every $r$, all but finitely many $r$-regular graphs have friendly partitions?", "background": "Source: Open Problem Garden. Original node ID: 37163. URL: http://www.openproblemgarden.org/op/friendly_partitions.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/friendly_partitions\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: edge-cut; partition; regular\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 8th, 2009 by mdevos\n\nProblem-page discussion:\nLet me say at the start, that I (M. DeVos) suspect this problem has been considered previously, so I await a more correct attribution.\n\nAn unfriendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in the opposite class as its own. It is an easy fact that every (finite) graph has an unfriendly partition; for instance, any maximum size edge-cut gives a partition with this property.\n\nFinding friendly partitions appears to be considerably more difficult. Perhaps one reason why is that there exist graphs without unfriendly partitions. For instance, $K_{2n}$ and $K_{2n+1,2n+1}$ have no unfriendly partitions. However, it appears possible that the only graphs which fail to have friendly partitions are fairly dense.\n\nWhen $r=3$, the above problem is fairly easy to solve, as it reduces to the problem of finding two vertex disjoint cycles. Every cubic graph other than $K_4$ or $K_{3,3}$ has two disjoint cycles, and thus has a friendly partition. The case when $r=4$ is also not terribly complicated. However, the next step up, $r=5$ looks like a tricky problem which requires something new.\n\nRelated:\nRelated problems\nUnfriendly partitions\n\nComments:\n- August 5th, 2020 | Anonymous | Name of the problem: In the bibliography, this is known as Satisfactory graph partition or Internal graph partition.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Friendly partitions\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The internal/friendly-partition conjecture is open in general, but degrees 1, 2, 3, 4, and 6 are settled; 5-regular special and random cases have substantial recent progress.\n\n**Verified partial progress.**\n\n- Ban--Linial prove the degree-6 case.\n- The deterministic degree-5 case is open, while 5-regular abelian Cayley graphs and random 5-regular graphs have recent results.\n\n**Full solution or refutation.**\n\nNo all-degrees proof was verified.\n\n**What remains.**\n\nResolve deterministic degree 5 and the remaining degrees at least 7.\n\n**Sources checked.**\n\n- A. Ban and N. Linial, Internal partitions of regular graphs, Journal of Graph Theory 79 (2015), arXiv:1307.5246. (primary): https://arxiv.org/abs/1307.5246\n Evidence used: The paper proves the degree-6 case.\n- Graph-theory open problems, Friendly partitions (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/friendly_partitions/\n Evidence used: The maintained status page records the remaining deterministic open cases and recent partial results.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3157, "problem_number": "OPG-46708", "title": "Subgraph of large average degree and large girth.", "statement": "Conjecture For all positive integers $g$ and $k$, there exists an integer $d$ such that every graph of average degree at least $d$ contains a subgraph of average degree at least $k$ and girth greater than $g$.", "background": "Source: Open Problem Garden. Original node ID: 46708. URL: http://www.openproblemgarden.org/op/subgraph_of_large_average_degree_and_large_average_degree.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/subgraph_of_large_average_degree_and_large_average_degree\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Basic Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 5th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is true for regular graphs as observed by Alon (see [KO]). The case $g\\leq 4$ was proved in [KO].\n\nBibliography:\n[KO] D. Kühn and D. Osthus, Every graph of sufficiently large average degree contains a C4-free subgraph of large average degree, Combinatorica, 24 (2004), 155-162.\n\n*[T] C. Thomassen, Girth in graphs, J. Combin. Theory B 35 (1983), 129–141.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Subgraph of large average degree and large girth.\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The large-average-degree, high-girth subgraph conjecture is known in the short-cycle regime through g=4 but remains open for general g at least five.\n\n**Verified partial progress.**\n\n- The case through girth greater than four is known.\n- Modern work substantially improves quantitative degree bounds for finding C4-free subgraphs of prescribed average degree.\n\n**Full solution or refutation.**\n\nNo theorem eliminating every cycle through an arbitrary fixed length while retaining prescribed average degree was verified.\n\n**What remains.**\n\nExtend the short-cycle methods to remove all cycles of lengths at most g for each fixed g at least five without losing the target average degree.\n\n**Sources checked.**\n\n- R. Montgomery, A. Pokrovskiy, and B. Sudakov, C4-free subgraphs with large average degree, arXiv:2004.03564; Israel Journal of Mathematics (2021). (primary): https://arxiv.org/abs/2004.03564\n Evidence used: Improves the sufficient average-degree bound for a C4-free subgraph of large average degree and provides lower-bound constructions.\n- Graph-theory open problems, Subgraph of large average degree and large average degree (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/subgraph_of_large_average_degree_and_large_average_degree/\n Evidence used: Reports the g at most four regime known and the general higher-girth conjecture open.\n\n**Review notes.** No statement defect identified; the maintained tracker's repeated phrase occurs in its URL/title slug only.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3158, "problem_number": "OPG-56028", "title": "Almost all non-Hamiltonian 3-regular graphs are 1-connected", "statement": "Conjecture Denote by $NH(n)$ the number of non-Hamiltonian 3-regular graphs of size $2n$, and similarly denote by $NHB(n)$ the number of non-Hamiltonian 3-regular 1-connected graphs of size $2n$.\n\nIs it true that $\\lim\\limits_{n \\rightarrow \\infty} \\displaystyle\\frac{NHB(n)}{NH(n)} = 1$?", "background": "Source: Open Problem Garden. Original node ID: 56028. URL: http://www.openproblemgarden.org/op/almost_all_non_hamiltonian_3_regular_graphs_are_1_connected.\n\nSource subject path: Graph Theory > Basic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/almost_all_non_hamiltonian_3_regular_graphs_are_1_connected\n- Author(s): Haythorpe, Michael\n- Subject(s): Graph Theory; Basic Graph Theory\n- Keywords: Hamiltonian, Bridge, 3-regular, 1-connected\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: August 23rd, 2013 by mhaythorpe\n\nProblem-page discussion:\nA stronger version of this conjecture asks whether it is also the case that $\\displaystyle\\frac{NHB(n)}{NH(n)} > \\displaystyle\\frac{NHB(k)}{NH(k)}$ for all $n > k$.\n\nExperimental data was given by Filar et al [FHN] demonstrating that the strong conjecture is satisfied for all $n \\leq 12$, and with sampled data provided for $n = 20$ and $n = 25$. No further results have been forthcoming.\n\nThe experimental data can be viewed at http://dx.doi.org/10.7151/dmgt.1485\n\nPackers And Movers Chandigarh\nPackers And Movers Hyderabad\nPackers And Movers Bangalore\n\nBibliography:\n[FHN] Jerzy A Filar, Giang T Nguyen, Michael Haythorpe, \"A conjecture on the prevalence of cubic bridge graphs\", Discussiones Mathematicae Graph Theory 30(1):175--179 (2010).\n\nDiscussion links:\n- Packers And Movers Chandigarh: http://www.packersandmoverschandigarh.co.in\n- Packers And Movers Hyderabad: http://www.packersandmoversinhyderabad.co.in\n- Packers And Movers Bangalore: http://www.packersandmoversinbangalore.co.in\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Almost all non-Hamiltonian 3-regular graphs are 1-connected\" in Graph Theory; Basic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No verified resolution of the asymptotic bridge/1-connectivity question for non-Hamiltonian cubic graphs was found.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe maintained source still lists the question open.\n\n**What remains.**\n\nProve the limiting ratio is one or demonstrate a non-negligible bridgeless non-Hamiltonian cubic family.\n\n**Sources checked.**\n\n- UnsolvedMath/OpenGarden, Almost all non-Hamiltonian 3-regular graphs are 1-connected (accessed 2026-08-17). (maintained_tracker): https://www.unsolvedmath.com/problems/OPG-56028\n Evidence used: Retains the exact asymptotic question as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3159, "problem_number": "OPG-146", "title": "Partitioning edge-connectivity", "statement": "Question Let $G$ be an $(a+b+2)$-edge-connected graph. Does there exist a partition $\\{A,B\\}$ of $E(G)$ so that $(V,A)$ is $a$-edge-connected and $(V,B)$ is $b$-edge-connected?", "background": "Source: Open Problem Garden. Original node ID: 146. URL: http://www.openproblemgarden.org/op/partitioning_edge_connectivity.\n\nSource subject path: Graph Theory > Basic Graph Theory > Connectivity.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partitioning_edge_connectivity\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Basic Graph Theory; Connectivity\n- Keywords: edge-coloring; edge-connectivity\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nBy the Nash-Williams/Tutte theorem ([NW] or [T]) on disjoint spanning trees, the above conjecture is true if $G$ is $2(a+b)$-edge-connected. This is the only partial result I know of. Here is a related conjecture.\n\nConjecture There exists a fixed integer $k$ so that every $k$-edge-connected graph $G=(V,E)$ has a subset of edges $S$ with the property that every edge-cut of $G$ has between $\\frac{1}{3}$ and $\\frac{2}{3}$ of its edges in $S$.\n\nThe values $\\frac{1}{3}$ and $\\frac{2}{3}$ are of no special importance in the above conjecture. Indeed, an affirmative answer to the above problem with $\\frac{1}{3}$ and $\\frac{2}{3}$ replaced by $\\frac{1}{t}$ and $1 - \\frac{1}{t}$ for any $t > 0$ would still be valuable - and in particular, would imply the 2+epsilon flow conjecture.\n\nDefinition: Let $G=(V,E)$ be a graph and let $P=\\{E_1,E_2,...,E_t\\}$ be a partition of $E$. We say that $P$ is $k$-courteous if $G \\setminus E_i$ is $k$-edge-connected for every $1 \\le i \\le t$.\n\nProblem What is the smallest integer $t$ so that every 3-edge-connected graph has a 2-courteous coloring of size $t$?\n\nIt is known (see [DJS]) that $4 \\le t \\le 10$. It would be quite interesting if the truth were in fact $t=4$. An improvement on the current upper bound would have some consequences for certain flow problems and cycle-cover problems. In general, one may define a function $H: {\\mathbb Z}^2 \\rightarrow {\\mathbb Z} \\cup \\{\\infty\\}$ so that $H(a,b)$ is the smallest integer $t$ (or $\\infty$ if none exists) so that every $a$-edge-connected graph has a $b$-courteous coloring of size $t$. It is known (see [DJS]) that $H(2k+2,2k+1) = \\infty$, and that $2k+1 < H(2k+1,2k) < C 100^k$. Two special cases when better values are known are $2 < H(4,2) < 5$ and $5 < H(5,4) < 31$.\n\nBibliography:\n[DJS] M. DeVos, T. Johnson, P.D. Seymour, Cut-coloring and circuit covering\n\n[Ed] J. Edmonds, Minimum Partition of a Matriod into Independent Subsets, J. Res. Nat. Bur. Standards 69B (1965) 67-72. MathSciNet\n\n[NW] C.S.J.A. Nash-Williams, Edge Disjoint Spanning Trees of Finite Graphs, J. London Math. Soc. 36 (1961) 445-450. MathSciNet\n\n[T] W.T. Tutte, On the problem of decomposing a graph into n connected factors, J. London Math. Soc. 36 (1961), 221-230. MathSciNet\n\nSource links:\n- edge-connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n\nDiscussion links:\n- edge-cut: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- 2+epsilon flow conjecture: http://www.openproblemgarden.org/?q=op/2_epsilon_flow_conjecture\n\nBibliography links:\n- Cut-coloring and circuit covering: http://www.math.princeton.edu/%7Epds/papers/cutcolouring/paper.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0190025\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0133253\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0140438\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 32.\n\nAttempt notes:\nTarget:\nMake progress on \"Partitioning edge-connectivity\" in Graph Theory; Basic Graph Theory; Connectivity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No solution to the exact (a+b+2)-edge-connectivity threshold was found. The maintained original problem page derives the desired partition under the stronger 2(a+b)-edge-connectivity hypothesis from the Nash-Williams-Tutte spanning-tree packing theorem.\n\n**Verified partial progress.**\n\n- The Nash-Williams-Tutte theorem yields the desired edge partition whenever G is 2(a+b)-edge-connected.\n\n**Full solution or refutation.**\n\nTargeted searches found no modern primary paper resolving the sharper a+b+2 threshold. The only well-supported partial result is the classical stronger-connectivity case recorded on the original maintained tracker.\n\n**What remains.**\n\nLower the sufficient connectivity from 2(a+b) to a+b+2, or construct a counterexample to the proposed threshold.\n\n**Sources checked.**\n\n- Matt DeVos, Partitioning edge-connectivity, Open Problem Garden, posted 2007 and accessed 2026-08-17. (maintained_tracker): https://garden.irmacs.sfu.ca/op/partitioning_edge_connectivity\n Evidence used: Gives the exact question and states that Nash-Williams-Tutte proves it under 2(a+b)-edge-connectivity, calling that the only partial result known to the author.\n- W. T. Tutte, On the Problem of Decomposing a Graph into n Connected Factors, Journal of the London Mathematical Society 36 (1961), 221-230. (primary): https://doi.org/10.1112/jlms/s1-36.1.221\n Evidence used: One of the two original edge-disjoint spanning-tree theorems used for the stronger-connectivity partial result.\n\n**Review notes.** No OCR or formulation defect was detected in the exact statement, which is reproduced verbatim in report.md. Medium confidence reflects the absence of a recent dedicated status survey or post-2007 primary paper on this exact threshold.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3160, "problem_number": "OPG-56233", "title": "Kriesell's Conjecture", "statement": "Conjecture Let $G$ be a graph and let $T\\subseteq V(G)$ such that for any pair $u,v\\in T$ there are $2k$ edge-disjoint paths from $u$ to $v$ in $G$. Then $G$ contains $k$ edge-disjoint trees, each of which contains $T$.", "background": "Source: Open Problem Garden. Original node ID: 56233. URL: http://www.openproblemgarden.org/op/kriesells_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Connectivity.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/kriesells_conjecture\n- Author(s): Kriesell, Matthias\n- Subject(s): Graph Theory; Basic Graph Theory; Connectivity\n- Keywords: Disjoint paths; edge-connectivity; spanning trees\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 25th, 2013 by Jon Noel\n\nProblem-page discussion:\nThis problem was featured as unsolved problem #22 in Bondy and Murty's book \"Graph Theory\" [BM].\n\nSee also a posting on the open problem forum of the Egerváry Research Group on Combinatorial Optimization.\n\nBibliography:\n[BM] J. A. Bondy and U. S. R. Murty. Graph theory, volume 244 of Graduate Texts in Mathematics. Springer, New York, 2008.\n\nDiscussion links:\n- posting: http://lemon.cs.elte.hu/egres/open/Kriesell's_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Kriesell's Conjecture\" in Graph Theory; Basic Graph Theory; Connectivity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Kriesell's exact 2k-edge-connectivity Steiner-tree-packing conjecture remains open; approximate packing results are known.\n\n**Verified partial progress.**\n\n- Lau proved an approximate max-Steiner-tree-packing/min-Steiner-cut theorem.\n\n**Full solution or refutation.**\n\nNo proof was verified under the exact 2k terminal-connectivity hypothesis.\n\n**What remains.**\n\nProve k edge-disjoint T-Steiner trees at the conjectured connectivity threshold.\n\n**Sources checked.**\n\n- EGRES Open, Kriesell's conjecture (accessed 2026-08-17). (maintained_tracker): https://lemon.cs.elte.hu/egres/open/Kriesell%27s_conjecture\n Evidence used: States the exact conjecture and records the approximation literature.\n- UnsolvedMath/OpenGarden, Kriesell's Conjecture (accessed 2026-08-17). (maintained_tracker): https://www.unsolvedmath.com/problems/OPG-56233\n Evidence used: Retains the source statement as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3161, "problem_number": "OPG-137", "title": "Cycle double cover conjecture", "statement": "Conjecture For every graph with no bridge, there is a list of cycles so that every edge is contained in exactly two.", "background": "Source: Open Problem Garden. Original node ID: 137. URL: http://www.openproblemgarden.org/op/cycle_double_cover_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cycle_double_cover_conjecture\n- Author(s): Seymour, Paul D.; Szekeres, George\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cover; cycle\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nThis beautiful conjecture was made independently by Szekeres and Seymour in the 70's and is now widely considered to be among the most important open problems in graph theory. Note the similarity between this conjecture and the Berge-Fulkerson conjecture on perfect matchings. Attempts to prove this conjecture have lead to a variety of conjectured strengthenings, which appear on other pages. See: The circular embedding conjecture, The five cycle double cover conjecture, The faithful cover conjecture, and Decomposing Eulerian graphs.\n\nIf a graph $G$ has a nowhere-zero 4-flow then it follows from a result of Tutte that $G$ satisfies the above conjecture. Thus, by Jaeger's 4-flow theorem [J], the above conjecture is true for every 4-edge-connected graph. A cubic graph has a nowhere-zero 4-flow if and only if it is 3-edge-colorable, so the above conjecture is also true for 3-edge-colorable cubic graphs. In general, it follows from vertex splitting arguments that problem may be reduced to cubic graphs which are not 3-edge-colorable.\n\nFor a general graph $G$ with no cut-edge, Bermond, Jackson and Jaeger [BJJ] used Jaeger's 8-flow theorem [J] to prove that $G$ has a list of circuits so that every edge is contained in exactly four. Fan [F] used Seymour's 6-flow theorem [S81] to prove that G has a list of circuits so that every edge is contained in exactly six.\n\nLet $G$ be a directed graph and let $C$ be a circuit (not necessarily a directed circuit) of $G$. If we choose a direction to travel around $C$, then every edge of $C$ is either traversed forward or backward. The following strengthening of the cycle double cover conjecture takes directions into account.\n\nConjecture (The oriented cycle double cover conjecture) If $G$ is an orientation of a bridgeless graph, then there is a list $L$ of circuits of $G$ with directions so that every edge of $G$ is traversed forward by exactly one circuit in $L$ and backward by exactly one circuit in $L$.\n\nTutte also showed that every graph with a nowhere-zero 4-flow satisfies this conjecture. Thus, as above this conjecture is true for 4-edge-connected graphs and for 3-edge-colorable cubic graphs.\n\nIt was mentioned above that for a general graph $G$ with no bridge, there is a list of circuits containing every edge exactly four times. By taking two copies of each circuit in this list and giving them opposite directions, we have a list of circuits so that every edge is traversed forward and backward exactly four times. Luis Goddyn and I (M. DeVos) have observed that the same ideas used in Fan's article [Fa] can be used to construct a list of circuits with directions so that every edge is traversed forward and backward exactly three times. The following natural question seems to be open.\n\nConjecture (The oriented cycle four cover conjecture) If $G$ is an orientation of a bridgeless graph, then there is a list $L$ of circuits of $G$ with directions so that every edge of $G$ is traversed forward by exactly two circuits in $L$ and backward by exactly two circuits in $L$.\n\nSince every graph with a nowhere-zero 4-flow has a list of circuits with directions traversing every edge forward and backward exactly once, the above conjecture would follow from The three 4-flows conjecture.\n\nBibliography:\n[AGZ] B. Alspach, L. Goddyn, and C-Q Zhang, Graphs with the circuit cover property, Trans. Amer. Math. Soc., 344 (1994), 131-154. MathSciNet\n\n[BJJ] J.C. Bermond, B. Jackson, and F. Jaeger, Shortest covering of graphs with cycles, J. Combinatorial Theory Ser. B 35 (1983), 297-308. MRhref{0735197}\n\n[DJS] M. DeVos, T. Johnson, P.D. Seymour, Cut-coloring and circuit covering\n\n[F] G. Fan, Integer flows and cycle covers, J. Combinatorial Theory Ser. B 54 (1992), 113-122. MathSciNet\n\n[FZ] G. Fan and C.Q. Zhang, Circuit decompositions of Eulerian graphs, J. Combinatorial Theory Ser. B 78 (2000), 1-23. MathSciNet\n\n[FG] X. Fu and L. Goddyn, Matroids with the circuit cover property, Europ. J. Combinatorics 20 (1999), 61-73. MathSciNet\n\n[J] F. Jaeger, Flows and Generalized Coloring Theorems in Graphs, J. Combinatorial Theory Ser. B 26 (1979) 205-216. MathSciNet\n\n[Ki] P.A. Kilpatrick, Tutte's First Colour-Cycle Conjecture, Thesis, Cape Town (1975).\n\n[Sz] G. Szekeres, Polyhedral decompositions of cubic graphs. Bull. Austral. Math. Soc. 8, 367-387. MathSciNet\n\n[S91] P.D. Seymour, Nowhere-Zero 6-Flows, J. Combinatorial Theory Ser. B 30 (1981) 130-135. MathSciNet\n\n[S79] P.D. Seymour, Sums of circuits in Graph Theory and Related Topics edited by J.A. Bondy and U.S.R. Murty, Academic Press, New York/Berlin (1979), 341-355. MathSciNet\n\n[S95] P.D. Seymour, Nowhere-Zero Flows, in Handbook of Combinatoircs, edited by R. Graham, M. Grotschel and L. Lovasz, (1995) 289-299. MathSciNet\n\n[T54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomials, Canad. J. Math. 6 (1954) 80-91. MathSciNet\n\n[T66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50. MathSciNet\n\nSource links:\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n\nDiscussion links:\n- Berge-Fulkerson conjecture: http://www.openproblemgarden.org/?q=op/the_berge_fulkerson_conjecture\n- The circular embedding conjecture: http://www.openproblemgarden.org/?q=op/the_circular_embedding_conjecture\n- The five cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/m_n_cycle_covers\n- The faithful cover conjecture: http://www.openproblemgarden.org/?q=op/faithful_cycle_covers\n- Decomposing Eulerian graphs: http://www.openproblemgarden.org/?q=op/decomposing_eulerian_graphs\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n- cubic graph: http://en.wikipedia.org/wiki/cubic graph\n- edge-colorable: http://en.wikipedia.org/wiki/edge coloring\n\nBibliography links:\n- Graphs with the circuit cover property: http://www.jstor.org/view/00029947/di981444/98p0199p/0\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1181180\n- Cut-coloring and circuit covering: http://www.math.princeton.edu/%7Epds/papers/cutcolouring/paper.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1142267\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1737620\n- Matroids with the circuit cover property: http://www.math.sfu.ca/%7Egoddyn/Papers/9531-matroids-circuit-cover.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1669600\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0532588\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0325438\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0615308\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0538060\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1373660\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0061366\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0194363\n\nComments:\n- February 7th, 2012 | Andrew King | Claimed solution in the affirmative: Alexander Souza has offered for review on arXiv a constructive solution in the affirmative.\n\nhttp://arxiv.org/abs/1202.0569\n\nEdit: Unless we are mistaken, the Petersen graph is a counterexample to Theorem 2. Furthermore there is an unresolved problem with Lemma 9.\n- October 6th, 2011 | khaniki | question: should the cycles be distinct??\n- February 6th, 2012 | Andrew King | No; consider the example of: No; consider the example of a graph which is itself a cycle. This has a unique cycle double cover, and the cycles are not distinct.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Cycle double cover conjecture\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The Cycle Double Cover Conjecture was proved in July 2026: every finite bridgeless multigraph has a cycle double cover. The proof has independent expert expositions, an explicit positive correctness assessment by Johannes Carmesin, and an unconditional Lean formalization for finite loopless bridgeless multigraphs.\n\n**Verified partial progress.**\n\n- The proof constructs a cycle double cover from a nowhere-zero F_2^3-flow obtained through the Jaeger-Kilpatrick eight-flow theorem.\n- The accompanying Lean project kernel-checks the final theorem without project-specific axioms.\n\n**Full solution or refutation.**\n\nOpenAI's proof and formalization cover the exact statement. Independent expositions by Jim Geelen and Sang-il Oum reconstruct the argument, and Johannes Carmesin explicitly states that the proof is correct.\n\n**What remains.**\n\nThe publication and refereeing history may continue to develop, but no mathematical case in the exact finite bridgeless-graph statement remains open.\n\n**Sources checked.**\n\n- OpenAI, A proof of the cycle double cover conjecture (July 2026). (primary): https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf\n Evidence used: Proof of the exact bridgeless-graph cycle-double-cover statement.\n- OpenAI, CDC Lean formalization, accessed 2026-08-17. (primary): https://github.com/openai/cdc-lean\n Evidence used: Kernel-checked unconditional theorem cycleDoubleCover_of_bridgeless for finite loopless bridgeless multigraphs.\n- Jim Geelen, OpenAI's proof of the Cycle Double Cover Theorem, arXiv:2607.15399 (2026). (authoritative_secondary): https://arxiv.org/abs/2607.15399\n Evidence used: Independent expert exposition reconstructing the proof.\n- Sang-il Oum, A proof of the cycle double cover conjecture by OpenAI: An exposition, arXiv:2607.16356 (2026). (authoritative_secondary): https://arxiv.org/abs/2607.16356\n Evidence used: Independent graph-theorist exposition of the proof.\n- Johannes Carmesin, The cycle double cover theorem, The Matroid Union, 21 July 2026. (authoritative_secondary): https://matroidunion.org/?tag=cycle-double-cover\n Evidence used: Explicitly answers that the proof is correct and explains its main mechanism.\n\n**Review notes.** The historical OPG page's 2012 Alexander Souza claimed proof was not accepted as evidence: the same page records a Petersen-graph counterexample to one theorem and an unresolved lemma issue.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3162, "problem_number": "OPG-138", "title": "The circular embedding conjecture", "statement": "Conjecture Every 2-connected graph may be embedded in a surface so that the boundary of each face is a cycle.", "background": "Source: Open Problem Garden. Original node ID: 138. URL: http://www.openproblemgarden.org/op/the_circular_embedding_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_circular_embedding_conjecture\n- Author(s): Haggard, Gary\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cover; cycle\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nThis conjecture implies the cycle double cover conjecture, since the list of cycles which bound faces covers each edge exactly twice. Let $G$ be a cubic graph, let $L$ be a list of cycles covering every edge of $G$ exactly two times, and form a topological space by gluing a disc to each circuit in $L$. This space is a surface, and every face is bounded by a cycle. Thus, the circular embedding conjecture and the cycle double cover conjecture are equivalent for cubic graphs. For general graphs, this construction may fail since the neighborhood of a vertex may not be a disc (it could be a pinchpoint).\n\nA stronger variant of this conjecture asserts that it is possible to find an embedding as above with the added condition that the dual graph is 5-colorable. This variant implies The five cycle double cover conjecture since the circuits bounding faces of a given color class may be grouped into a cycle. Next we state a different strengthening which asserts that we may find an embedding as above into an orientable surface.\n\nConjecture (The oriented circular embedding conjecture) Every 2-connected graph may be embedded in an orientable surface so that the boundary of each face is a circuit.\n\nIf this conjecture is true, then the oriented cycle double cover conjecture (see cycle double cover) is also true, since the list of circuits bounding faces all traversed in the clockwise direction cover each edge exactly once in each direction (since the surface is orientable, we may specify a global clockwise orientation). As was the case above, the oriented circular embedding conjecture is equivalent to the oriented cycle double cover conjecture for cubic graphs. Also as above, there is a strengthening of this conjecture which asserts that the graph may be embedded so that the dual graph is 5-colorable. If true, this would imply The orientable five cycle double cover conjecture.\n\nBibliography:\n[H] G. Haggard, Edmonds characterization of disc embeddings. Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977), pp. 291--302. MathSciNet\n\nSource links:\n- connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- embedded: http://en.wikipedia.org/wiki/graph embedding\n\nDiscussion links:\n- cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/cycle_double_cover_conjecture\n- five cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/m_n_cycle_covers\n- cycle double cover: http://www.openproblemgarden.org/?q=op/cycle_double_cover_conjecture\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0491201\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"The circular embedding conjecture\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general strong/circular embedding conjecture for all 2-connected graphs was not found solved. The established equivalence with cycle double covers for cubic graphs, combined with the July 2026 cycle-double-cover theorem, now settles every cubic instance; the noncubic manifold-neighborhood obstruction remains.\n\n**Verified partial progress.**\n\n- Every cubic instance follows from the 2026 cycle-double-cover theorem and the standard cubic equivalence between cycle double covers and strong embeddings; this is a sourced inference.\n- Ellingham and Zha proved the stronger orientable closed-2-cell embedding statement for every 2-connected projective-planar cubic graph.\n\n**Full solution or refutation.**\n\nNo full solution for arbitrary noncubic 2-connected graphs was located. The July 2026 cycle-double-cover theorem settles the cubic subclass but is not known to imply a surface embedding at vertices of higher degree.\n\n**What remains.**\n\nProve a strong embedding for every 2-connected noncubic graph, or produce a counterexample. The source statement does not require orientability.\n\n**Sources checked.**\n\n- Melody Chan, A survey of the cycle double cover conjecture (2009). (authoritative_secondary): https://www.math.brown.edu/mchan2/cdc.pdf\n Evidence used: Explains that strong embedding implies CDC in general and is equivalent to CDC for cubic graphs, but not known conversely for general graphs.\n- M. N. Ellingham and Xiaoya Zha, Orientable embeddings and orientable cycle double covers of projective-planar graphs, European Journal of Combinatorics 32 (2011), 495-509. (primary): https://doi.org/10.1016/j.ejc.2011.01.001\n Evidence used: Proves orientable closed-2-cell embeddings for all 2-connected projective-planar cubic graphs.\n- Babak Ghanbari and Robert Samal, Facial diagrams and cycle double cover, arXiv:2605.01410 (2026). (primary): https://arxiv.org/abs/2605.01410\n Evidence used: Studies the circular 2-cell embedding formulation for cubic graphs immediately before the CDC proof.\n- OpenAI, CDC Lean formalization, accessed 2026-08-17. (primary): https://github.com/openai/cdc-lean\n Evidence used: Supplies the newly proved CDC theorem used with the cubic equivalence.\n\n**Review notes.** The Melody Chan survey is from 2009, not 2026; recent search-engine crawl dates were not treated as publication dates. The all-cubic conclusion is an inference from the survey's equivalence and the new CDC theorem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3163, "problem_number": "OPG-139", "title": "(m,n)-cycle covers", "statement": "Conjecture Every bridgeless graph has a (5,2)-cycle-cover.", "background": "Source: Open Problem Garden. Original node ID: 139. URL: http://www.openproblemgarden.org/op/m_n_cycle_covers.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/m_n_cycle_covers\n- Author(s): Celmins, Uldis A.; Preissmann, Myriam\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cover; cycle\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: If $G=(V,E)$ is a graph, a binary cycle of $G$ is a set $C \\subseteq E$ such that every vertex of the graph $(V,C)$ has even degree. An $(m,n)$-cycle-cover of $G$ is a list $L$ consisting of $m$ cycles so that every edge of $G$ is contained in exactly $n$ of these cycles.\n\nSince every binary cycle can be written as a disjoint union of edge sets of ordinary cycles, the above conjecture is a strengthening of the cycle double cover conjecture. For positive integers $m,n$ it is natural to ask what family of graphs have $(m,n)$-cycle-covers. The following chart gives some information about this question for small values of $m$ and $n$. A \"yes\" in the $(m,n)$ box indicates that every graph with no cut-edge has an $(m,n)$-cycle-cover. A \"no\" indicates that no graph has an $(m,n)$-cycle-cover. A more detailed explanation of the entries in this chart appears below it.\n\nm\n\nn\n\n2 3 4\n\n5 6 7 8 9 10\n\n11\n2 Eulerian NZ 4-flow NZ 4-flow\n\n5CDC conj open\n4 no Eulerian\n\n5 post. sets B-F conj yes [BJJ] yes\n6\n\nno Eulerian 7 post. sets?? yes [F]\n\nyes\n\nWe did not include odd values of n, since any graph with an $(m,n)$-cycle-cover for an odd integer $n$ must be Eulerian. The entry \"NZ 4-flow\" is short for nowhere-zero 4-flow. Thus, our chart indicates that ( $G$ has a nowhere-zero 4-flow) if and only if ( $G$ has a $(3,2)$-cycle-cover) if and only if ( $G$ has a $(4,2)$-cycle-cover). These equivalences were discovered by Tutte [Tu].\n\nTwo of the $(m,n)$ boxes are conjectures. The 5CDC conj is the 5 cycle double cover conjecture and the B-F conjecture is the Berge-Fulkerson conjecture. In both of these cases, the conjecture is equivalent to the assertion that every graph with no cut-edge has an $(m,n)$-cycle-cover (i.e. it would be accurate to put a \"yes\" in the corresponding. box). For emphasis, we state the Berge-Fulkerson conjecture again below in this new form.\n\nConjecture (The Berge-Fulkerson conjecture) Every graph with no cut-edge has a (6,4)-cycle-cover.\n\nThe fact that the above conjecture is equivalent to the usual statement of the Berge-Fulkerson conjecture was discovered by Jaeger [J]. For cubic graphs this equivalence is easy to see, since $M_1,\\ldots,M_6$ satisfy the Berge-Fulkerson conjecture if and only if $E\\M_1,\\ldots,E\\M_6$ is a $(6,4)$-cycle-cover. By Jaeger's argument, the weak Berge-Fulkerson conjecture is equivalent to the statement that there exists a fixed integer $k$ so that every bridgeless graph has a $(3k,2k)$-cycle-cover.\n\nA postman set is a set of edges $J$ such that $E(G)\\J$ is a cycle. The entry \"k post. sets\" in the $(k,k-1)$ box of the above chart indicates that a graph G has a $(k,k-1)$-cycle-cover if and only if it is possible to partition the edges of $G$ into $k$ postman sets. This equivalence follows immediately from the definition. Rizzi's Packing postman sets conjecture is thus equivalent to the following conjecture on cycle-covers.\n\nConjecture (the packing postman sets conjecture) If every odd edge-cut of $G$ has size $\\ge 2k+1$, then $G$ has a $(2k+1,2k)$-cycle-cover.\n\nNext we turn our attention to orientable cycle covers. If $H$ is a directed graph a map $\\phi:E(H) \\rightarrow \\{-1,0,1\\}$ is a 2-flow or an oriented cycle if at every vertex of $H$, the sum of $\\phi$ on the incoming edges is equal to the sum of $\\phi$ on the outgoing edges. It is easy to see that the support of a 2-flow is always a cycle. Furthermore, for any oriented cycle, there is a list $L$ of edge-disjoint circuits with directions so that an edge $e$ is forward (backward) in a circuit of $L$ if and only if $\\phi(e)=1$ ( $\\phi(e)=-1$ ). So as in the unoriented case, an oriented cycle may be viewed as the edge-disjoint union of oriented circuits. For an even integer $n$, a $(m,n)$-oriented-cycle-cover of a graph $G$ is a list of $m$ oriented cycles so that every edge of $G$ appears as a forward edge $n/2$ times and a backward edge $n/2$ times. The following conjecture is the common generalization of the orientable cycle double cover conjecture and the five cycle double cover conjecture. It is due to Archdeacon and Jaeger.\n\nConjecture (The orientable five cycle double cover conjecture) Every graph without a cut-edge has a (5,2)-oriented-cycle-cover.\n\nConsiderably less is known about $(m,n)$-oriented-cycle-covers. We sumarize some of what is known for small values of $m$ and $n$ below.\n\nm\n\nn\n\n2\n\n3 4 5 6 7 8\n\n9 10 11\n2 Eulerian\n\nNZ 3-flow NZ 4-flow O5CDC conj open\n4\n\nno Eulerian??? conj.\n\nopen\n6 no Eulerian???? yes [DG]\n\nEvery graph with an $(m,n)$-cycle-cover also has a $(2m,2n)$-oriented-cycle-cover obtained by taking two copies of each cycle with opposite orientations. Thus, by Bermond, Jackson, and Jaeger's $(7,4)$-cycle-cover theorem, every bridgeless graph with no has a $(14,8)$-oriented-cycle-cover. DeVos and Goddyn have observed that Seymour's 6-flow theorem can be used to construct an $(11,6)$-oriented-cycle-cover for every bridgeless graph. By combining these, we find that for every even integer $n \\ge 10$ there exists an $m$ so that every bridgeless graph has an $(m,n)$-oriented-cycle-cover. This question is still open for $n=2,4,10$.\n\nThe following conjecture appears in the above chart.\n\nConjecture (The orientable eight cycle four cover conjecture) Every graph with no cut-edge has a (8,4)-oriented-cycle-cover.\n\nThis conjecture may be viewed as a sort of oriented version of the Berge-Fulkerson conjecture. To see this analogy, note that ( $G$ has a nowhere-zero 4-flow) if and only if ( $G$ has a $(3,2)$-cycle-cover) if and only if ( $G$ has a $(4,2)$-oriented-cycle-cover). The Berge-Fulkerson conjecture and the above conjecture assert respectively that every bridgeless graph has a $(6,4)$-cycle-cover and a $(8,4)$-oriented-cycle-cover (i.e. a cover with double the parameters which are equivalent to a nowhere-zero 4-flow). As with most of the conjectures in this area, the above conjecture is trivially true for graphs with nowhere-zero 4-flows and it holds for the Petersen graph.\n\nBibliography:\n[A] D. Archdeacon, Face coloring of embedded graphs. J. Graph Theory, 8(1984), 387-398.\n\n[BJJ] J.C. Bermond, B. Jackson, and F. Jaeger, Shortest covering of graphs with cycles, J. Combinatorial Theory Ser. B 35 (1983), 297-308. MathSciNet\n\n*[C] A. U. Celmins, On cubic graphs that do not have an edge-3-colouring, Ph.D. Thesis, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada, 1984.\n\n[F] G. Fan, Integer flows and cycle covers, J. Combinatorial Theory Ser. B 54 (1992), 113-122. MathSciNet\n\n[J] F. Jaeger, Flows and Generalized Coloring Theorems in Graphs, J. Combinatorial Theory Ser. B 26 (1979) 205-216. MathSciNet\n\n[J88] F. Jaeger, Nowhere zero flow problems. Selected Topics in Graph Theory 3 (L.W.Beineke and R.J.Wilson eds.), Academic Press, London (1988), 71-95.\n\n*[P] M. Preissmann, Sur les colorations des arêtes des graphes cubiques, Thèse de 3ème cycle, Grenoble (1981).\n\n[T54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomials, Canad. J. Math. 6 (1954) 80-91. MathSciNet\n\n[T66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50. MathSciNet\n\nSource links:\n- bridgeless: http://en.wikipedia.org/wiki/bridge (graph theory)\n\nDiscussion links:\n- Eulerian: http://en.wikipedia.org/wiki/eulerian graph\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n- Berge-Fulkerson conjecture: http://www.openproblemgarden.org/?q=op/the_berge_fulkerson_conjecture\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0735197\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1142267\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0532588\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0061366\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0194363\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 35.\n\nAttempt notes:\nTarget:\nMake progress on \"(m,n)-cycle covers\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The five-cycle-double-cover conjecture remains open in general. The 2026 unrestricted cycle-double-cover theorem does not bound a cover to five binary cycles, while recent papers prove the stronger five-member conclusion for several cubic subclasses.\n\n**Verified partial progress.**\n\n- Liu, Hao, Luo, and Zhang prove 5-cycle double covers for cubic graphs of oddness at most four and additional Catlin-reducible families.\n- Their 2025 paper proves the oddness-two, girth-at-least-30 case and gives sufficient conditions for weak-oddness-two cubic graphs.\n\n**Full solution or refutation.**\n\nNo proof for every bridgeless graph was found. The new CDC theorem settles unrestricted existence only, not the five-member constraint.\n\n**What remains.**\n\nControl and group a cycle double cover into five binary cycles for every bridgeless graph.\n\n**Sources checked.**\n\n- Siyan Liu, Rong-Xia Hao, Rong Luo, and Cun-Quan Zhang, 5-Cycle Double Covers, 4-Flows, and Catlin Reduction, SIAM Journal on Discrete Mathematics 37 (2023), 253-267. (primary): https://doi.org/10.1137/22M1472425\n Evidence used: Proves the conjecture for cubic graphs of oddness at most four and several reducible classes.\n- Siyan Liu, Rong-Xia Hao, Rong Luo, and Cun-Quan Zhang, Five-cycle double cover and shortest cycle cover, Journal of Graph Theory 108 (2025), 39-49. (primary): https://doi.org/10.1002/jgt.23164\n Evidence used: Proves sufficient conditions and the oddness-two cubic girth-at-least-30 case.\n- Graph-theory open problems, (m,n)-cycle covers, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/m_n_cycle_covers/\n Evidence used: Maintains the exact five-cycle statement as open and records its relation to other cover conjectures.\n\n**Review notes.** The exact dataset statement and the distinction between an ordinary CDC and a (5,2)-cycle cover were preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3164, "problem_number": "OPG-140", "title": "Faithful cycle covers", "statement": "Conjecture If $G = (V,E)$ is a graph, $p: E \\rightarrow {\\mathbb Z}$ is admissable, and $p(e)$ is even for every $e \\in E(G)$, then $(G,p)$ has a faithful cover.", "background": "Source: Open Problem Garden. Original node ID: 140. URL: http://www.openproblemgarden.org/op/faithful_cycle_covers.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/faithful_cycle_covers\n- Author(s): Seymour, Paul D.\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cover; cycle\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: Let $G=(V,E)$ be a graph and let $p:E \\rightarrow {\\mathbb Z}$. A list $L$ of cycles of $G$ is a faithful (cycle) cover of $(G,p)$ if every edge $e$ of $G$ occurs in exactly $p(e)$ cycles of $L$. Thus, the cycle double cover conjecture is equivalent to the statement that $(G,2)$ has a faithful cover for every graph $G$ with no bridge. We define the map $p$ to be admissable if $p(e)$ satisfies the following properties:\n\ni $p$ is nonnegative.\n\nii $p(S)$ is even for every edge-cut $S$.\n\niii $p(e) \\le p(S)/2$ for every edge-cut $S$ and edge $e \\in S$.\n\nIt is easy to see that $G$ has a faithful cover only if $p$ is admissable. However, the converse is false. A counterexample is obtained by taking the Petersen graph, putting weight $2$ on the edges of a perfect matching, and $1$ elsewhere.\n\nMore generally, for a graph G=(V,E), one may consider the vector space of real numbers indexed by E. We associate every circuit C with its incidence vector. Most of the basic questions about this space are solved. Seymour [S] has shown that a vector p can be written as a nonnegative rational combination of cycles if and only if it satisfies conditions (i) and (iii) in the definition of admissable. It is an easy exercise to show that for a 3-edge-connected graph G, a vector p can be written as an integer combination of cycles if and only if p satisfies (ii) in the definition of admissable. Seymour's conjecture is equivalent to the statement that every admissable map may be realized as a half-integer combination of circuits. Note the similarity of this to The Berge-Fulkerson conjecture.\n\nThe most interesting result about faithful covers is a theorem of Alspach, Goddyn, and Zhang which resolved a conjecture of Seymour. They prove that whenever $G$ has no minor isomorphic to Petersen, every admissable map has a corresponding faithful cover. For a general graph $G$ with no bridge, Bermond, Jackson, and Jaeger [BJJ] proved that $(G,4)$ has a faithful cover and Fan [F] proveed that $(G,6)$ has a faithful cover. DeVos, Johnson, and Seymour [DJS] proved that $(G,p)$ has a faithful cover whenever $p$ is admissable and there is a nonnegative integer $k$ such that $32k+83 < p(e) < 36k+88$ holds for every edge $e$. However, little else seems to be known. In particular, it does not appear to be known if there exist integers $a,b$ with $a-b$ arbitrarily large so that $(G,p)$ has a faithful cover whenever $p$ is an admissable function taking on only the values $a,b$. Such a result would appear to require an idea not contained in any of the aforementioned papers.\n\nThe analogous problem for oriented circuit covers does not appear to be very promising. It is easy to see that for an orientation of a series parallel graph G and a map $p:E(G) \\rightarrow G$ which satisfies the obvious conditions, that $(G,p)$ will have a circuit cover using every edge in its given direction. However, even with a $K_4$ minor, there is a great deal of forcing, and nothing much looks like it would be true.\n\nBibliography:\n\\[AGZ] B. Alspach, L. Goddyn, and C-Q Zhang, Graphs with the circuit cover property, Trans. Amer. Math. Soc., 344 (1994), 131-154. MathSciNet\n\n[BJJ] J.C. Bermond, B. Jackson, and F. Jaeger, Shortest covering of graphs with cycles, J. Combinatorial Theory Ser. B 35 (1983), 297-308. MathSciNet\n\n[DJS] M. DeVos, T. Johnson, P.D. Seymour, Cut-coloring and circuit covering\n\n[F] G. Fan, Integer flows and cycle covers, J. Combinatorial Theory Ser. B 54 (1992), 113-122. MathSciNet\n\n[S] P.D. Seymour, Sums of circuits in Graph Theory and Related Topics edited by J.A. Bondy and U.S.R. Murty, Academic Press, New York/Berlin (1979), 341-355. MathSciNet\n\nDiscussion links:\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n- edge-cut: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- The Berge-Fulkerson conjecture: http://www.openproblemgarden.org/?q=op/the_berge_fulkerson_conjecture\n- minor: http://en.wikipedia.org/wiki/minor (graph theory)\n- Petersen: http://en.wikipedia.org/wiki/petersen graph\n\nBibliography links:\n- Graphs with the circuit cover property: http://www.jstor.org/view/00029947/di981444/98p0199p/0\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1181180\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0735197\n- Cut-coloring and circuit covering: http://www.math.princeton.edu/%7Epds/papers/cutcolouring/paper.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1142267\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0538060\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"Faithful cycle covers\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The faithful cycle cover conjecture for arbitrary even admissible weights remains open. It is proved for Petersen-minor-free graphs, and the July 2026 cycle-double-cover theorem settles the constant weight p=2 instance.\n\n**Verified partial progress.**\n\n- Alspach, Goddyn, and Zhang characterize graphs with the circuit-cover property: every admissible weight is realizable exactly when the graph has no Petersen minor.\n- The 2026 CDC theorem proves the constant even weight p(e)=2 case for every bridgeless graph.\n- Aurichi, Magalhaes Junior, and Seixas provide an infinite-graph transfer theorem conditional on the finite faithful-cover conjecture.\n\n**Full solution or refutation.**\n\nNo full proof for all even admissible edge-weight functions was located. Constant weight two and Petersen-minor-free graphs are covered by primary results.\n\n**What remains.**\n\nHandle nonconstant even admissible weights on graphs containing a Petersen minor.\n\n**Sources checked.**\n\n- Brian Alspach, Luis Goddyn, and Cun-Quan Zhang, Graphs with the Circuit Cover Property, Transactions of the American Mathematical Society 344 (1994), 131-154. (primary): https://jacobi.math.wvu.edu/~cqzhang/Publication-files/my-paper/TrAMS-1994-AGZ.pdf\n Evidence used: Proves the arbitrary-admissible-weight result exactly for Petersen-minor-free graphs.\n- Leandro F. Aurichi, Paulo S. F. Magalhaes Junior, and Luisa G. Seixas, Limits of cycles and cover conjectures, Discrete Mathematics 349 (2026), 114724. (primary): https://doi.org/10.1016/j.disc.2025.114724\n Evidence used: Shows how a positive finite faithful-cover theorem extends through their infinite-graph limit framework; it does not prove the finite conjecture.\n- OpenAI, A proof of the cycle double cover conjecture (2026). (primary): https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf\n Evidence used: Settles the constant weight p(e)=2 subcase.\n\n**Review notes.** The dataset misspelling 'admissable' is preserved in the report and interpreted using the source background's explicit admissibility conditions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3165, "problem_number": "OPG-141", "title": "Decomposing eulerian graphs", "statement": "Conjecture If $G$ is a 6-edge-connected Eulerian graph and $P$ is a 2-transition system for $G$, then $(G,P)$ has a compaible decomposition.", "background": "Source: Open Problem Garden. Original node ID: 141. URL: http://www.openproblemgarden.org/op/decomposing_eulerian_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_eulerian_graphs\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cover; cycle; Eulerian\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: Let $G$ be an Eulerian graph and for every vertex $v$, let $P(v)$ be a partition of the edges incident with $v$. We call $P$ a transition system. If every member of $P(v)$ has size at most $k$ (for every $v$ ), then we call $P$ a $k$-transition sytem. A compatible decomposition of $(G,P)$ is a list of edge-disjoint cycles $C_1,\\ldots,C_n$ with union $G$ so that every $C_i$ contains at most one edge from every member of $P(v)$.\n\nLet $G$ be a graph and let $G'$ be the graph obtained from $G$ by replacing each edge $e$ of G by two edges $e',e\"$ in parallel. Let $P$ be the 2-transition system of $G$ with ${e',e\"} \\in P(v)$ whenever $e'$ and $e\"$ are incident with $v$. Now, $G'$ is an Eulerian graph and every compatible decomposition of $(G',P)$ gives a cycle double cover of $G$. Since the cycle double cover conjecture can be reduced to graphs which are 3-edge-connected, the above conjecture would imply the cycle double cover conjecture.\n\nWe define a transition system $P$ to be admissable if every member of $P(v)$ contains no more than half of the edges in any edge-cut. It is easy to see that if there is a compatible decomposition of $(G,P)$, then $P$ must be admissable. The converse of this is not true; There is an admissable 2-transition system of the graph $K_5$ which does not admit a compatible decomposition. Recently, G. Fan and C.Q. Zhang [FZ] have proved that $(G,P)$ does have a compatible decomposition whenever $P$ is admissable and $G$ has no $K_5$ minor. This result imporoved upon an earlier theorem of Fleischner and Frank [FF]. Very recently, I have proved a weak version of the above conjecture, by showing that $(G,P)$ also has a compatible decomposition when P is a 2-transition system and G is 80-edge-connected. I'd quite like to see an improvement on this bound. Here is a related conjecture.\n\nConjecture (Sabidussi) Let $W$ be an Euler tour of the graph $G$. If $G$ has no vertex of degree two, then there is a cycle decomposition of $G$, say $F$, so that no two consecutive edges of $W$ are in a common circuit of $F$.\n\nIf $W$ is given by $v_1,e_1,v_2,e_2,...,e_{m-1},v_m$ then we may form a 2-transition system $P$ by putting $\\{e_{i-1},e_i\\}$ in $P(v_i)$ for every $i$ (working modulo $m$ ). Now a compatible decomposition of $(G,P)$ is precisely a cycle decomposition of $G$ satisfying the above conjecture. Thus, Sabidussi's conjecture is equivalent to the assertion that $(G,P)$ has a compatible decomposition whenever $G$ has no vertex of degree two and $P$ is a 2-transition system which comes from an Euler tour.\n\nLet $G$ be a directed Eulerian graph and for every vertex $v$, let $P(v)$ be a partition of the edges incident with $v$ into pairs so that every in-edge is paired with an out-edge. We define a compatible decomposition to be a decomposition of $G$ into directed circuits so that every directed circuit contains at most one edge from every member of $P(v)$. Our current techniques don't seem to shed any light on the problem of finding compatible decompositions for Eulerian digraphs. Next I pose a very basic question which is still open.\n\nProblem (DeVos) Does there exist a fixed integer $k$ such that $(G,P)$ has a compatible decomposition whenever $G$ is a $k$-edge-connected directed Eulerian graph and $P$ is a 2-transition system?\n\nSource links:\n- edge-connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- Eulerian graph: http://en.wikipedia.org/wiki/eulerian graph\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing eulerian graphs\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact six-edge-connected arbitrary-2-transition-system conjecture remains open. General 80-edge-connectivity, K5-transition-minor-free graphs, and the Euler-tour-induced Sabidussi subclass are proved.\n\n**Verified partial progress.**\n\n- DeVos proves compatible decompositions for Eulerian graphs with arbitrary 2-transition systems under 80-edge-connectivity.\n- Fleischner, Bagheri Gh., C.-Q. Zhang, and Z. Zhang settle the K5-transition-minor-free class.\n- Ulyanov's 2026 proof of Sabidussi's compatibility conjecture settles transition systems induced by a prescribed Euler tour, a proper subclass of the exact problem.\n\n**Full solution or refutation.**\n\nNo reduction of the general threshold from 80 to 6 was found. The July 2026 Sabidussi result does not cover an arbitrary transition system P.\n\n**What remains.**\n\nProve compatible decomposability at six-edge-connectivity for every 2-transition system, or refute the threshold.\n\n**Sources checked.**\n\n- Matt DeVos, Stable Bases and Circuit Decompositions, author-hosted manuscript. (primary): https://www.sfu.ca/~mdevos/papers/sbase.pdf\n Evidence used: Gives the known general high-edge-connectivity compatible-decomposition results, including the 80-edge-connected case.\n- Herbert Fleischner, Behrooz Bagheri Gh., Cun-Quan Zhang, and Zhang Zhang, Cycle covers (III) - Compatible circuit decomposition and K5-transition minor, Journal of Combinatorial Theory B 137 (2019), 25-54. (primary): https://doi.org/10.1016/j.jctb.2018.11.008\n Evidence used: Completely resolves the compatible circuit decomposition conjecture for the K5-transition-minor-free class.\n- Nikolay Ulyanov, Graph Puzzles III.1: A Proof of Sabidussi's Compatibility Conjecture, arXiv:2607.13225 (2026). (primary): https://arxiv.org/abs/2607.13225\n Evidence used: Proves the Euler-tour-induced transition-system case and reports a Lean formalization.\n- Graph-theory open problems, Decomposing eulerian graphs, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/decomposing_eulerian_graphs/\n Evidence used: Tracks the exact six-edge-connected conjecture as open and the 2019 class result as partial progress.\n\n**Review notes.** The exact source has 'compaible'; the background also has 'transition sytem', 'admissable', and 'imporoved'. These defects are flagged rather than silently altering the source statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3166, "problem_number": "OPG-385", "title": "Barnette's Conjecture", "statement": "Conjecture Every 3-connected cubic planar bipartite graph is Hamiltonian.", "background": "Source: Open Problem Garden. Original node ID: 385. URL: http://www.openproblemgarden.org/op/barnettes_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/barnettes_conjecture\n- Author(s): Barnette, David W.\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: bipartite; cubic; hamiltonian\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 12th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Originally appeared in [B], this discussion appears as [M].)\n\n- It is known that this is not true if you remove the \"bipartite\" condition, but the smallest 3-connected cubic planar graph which is not Hamiltonian has 38 vertices.\n\n- Holton, Manvel, and McKay [HMM] proved (using computers) that all graphs having fewer than 66 vertices satisfy the conjecture.\n\n- [A communication by Robert Aldred, Gunnar Brinkmann, and Brendan McKay (December 2002):]\n\nA paper of Holton, Manvel and McKay [HMM] proved Barnette's conjecture for up to 64 vertices, inclusive. This is to announce that the conjecture remains true up to 84 vertices, inclusive. The method used was the same as in the 1985 paper, but took advantage of two developments. One was the new program plantri (Brinkmann and McKay, to be published) which can generate the required graphs without isomorphs at more than $100\\,000$ per second. The other was the advance in computers. Total cpu time was about 3 years, almost all of it taken in finding hamiltonian cycles. Specifically, for all 3-connected cubic planar bipartite graphs up to 60 vertices, and those up to 64 vertices not having a 4-face adjacent to two others, we found a hamiltonian cycle using $x$ and avoiding $y$ for each pair of edges $x$ and $y$. There are over $10^{10}$ such graphs. By a theorem of Kelman's, one can build a counterexample to Barnette's conjecture when one has a 3-connected cubic planar bipartite graph with this property: for some two edges $x$ and $y$ on the same face, there is no hamiltonian cycle that uses $x$ and avoids $y$. We did not find any such graph even where $x$ and $y$ are not required to be on the same face. Perhaps the path to finding a counterexample is to strengthen Kelman's method to some more complicated condition involving 3 or more edges, as then it is more likely to fail on a smaller size.\n\nThere is another conjecture of Barnette (checked by Brendan McKay and Gunnar Brinkmann up to 250 vertices).\n\nConjecture Every planar cubic 3-connected graph with faces only of sizes 3, 4, 5, and 6 is Hamiltonian.\n\nBibliography:\n*[B] David W. Barnette, Conjecture 5, Recent progress in combinatorics (ed. W. T. Tutte), Academic Press, New York (1969) 343, MathSciNet\n\n[HMM] Derek A.Holton, Bennet Manvel, Brendan D. McKay, Hamiltonian cycles in cubic 3-connected bipartite planar graphs, J. Combin. Theory Ser. B 38 (1985) 279-297. MathSciNet\n\n[M] B. Mohar, Problem of the Month\n\nDiscussion links:\n- plantri: http://cs.anu.edu.au/%7Ebdm/plantri/\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0250896\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0796604\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P4BarnetteConjecture.html\n\nComments:\n- August 9th, 2011 | Anonymous | Copper Basin Construction: This is nice idea. This is to announce that the conjecture remains true up to 84 vertices, inclusive. The method used was the same as in the 1985 paper, but took advantage of two developments. Thank you.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Barnette's Conjecture\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Barnette's conjecture remains open despite many verified subclasses.\n\n**Verified partial progress.**\n\n- Many important subclasses of Barnette graphs are known Hamiltonian.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nNumerous connectivity/girth/symmetry special cases are Hamiltonian.\n\n**Sources checked.**\n\n- Open Problem Garden, node 385 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: No theorem or counterexample for all 3-connected cubic planar bipartite graphs was verified.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3167, "problem_number": "OPG-480", "title": "r-regular graphs are not uniquely hamiltonian.", "statement": "Conjecture If $G$ is a finite $r$-regular graph, where $r > 2$, then $G$ is not uniquely hamiltonian.", "background": "Source: Open Problem Garden. Original node ID: 480. URL: http://www.openproblemgarden.org/op/uniquely_hamiltonian_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/uniquely_hamiltonian_graphs\n- Author(s): Sheehan, John\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: hamiltonian; regular; uniquely hamiltonian\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 24th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Reproduced from [M].)\n\nA graph $G$ is said to be uniquely hamiltonian if it contains precisely one Hamiltonian cycle.\n\nThis conjecture has been proved for all odd values of $r$ [T] and for all even values of $r > 23$ [H]. By Petersen's theorem, it would suffice to prove it for $r = 4$.\n\nBibliography:\n[H] P. Haxell, Oberwolfach reports, 2006.\n\n[M] Bojan Mohar, Problem of the Month\n\n*[S] John Sheehan: The multiplicity of Hamiltonian circuits in a graph. Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), pp. 477-480. Academia, Prague, 1975, MathSciNet\n\n[T] A.G. Thomason, Hamiltonian cycles and uniquely edge colourable graphs. Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977). Ann. Discrete Math. 3 (1978), Exp. No. 13, 3 pp.\n\nDiscussion links:\n- Hamiltonian cycle: http://en.wikipedia.org/wiki/hamiltonian cycle\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0703_HamiltonicityInfinite.html\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0398896\n\nComments:\n- February 3rd, 2022 | Anonymous | Is this question still open?: I think, the autor answers the question in this article: Uniqueness of maximal dominating cycles in 3-regular graphs and of hamiltonian cycles in 4-regular graphs (https://doi.org/10.1002/jgt.3190180503). (And the conjecture is fals for all even values.) Am I wrong?\n- June 20th, 2022 | Anonymous | The construction in that: The construction in that paper has parallel edges, so it is not a counter example. As far as I am aware, Sheehan's conjecture is still open.\n- May 3rd, 2022 | Anonymous | The conjecture is only for: The conjecture is only for simple graphs. The paper you mention gives counterexamples that have multiple edges.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"r-regular graphs are not uniquely hamiltonian.\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Sheehan's conjecture remains open; Thomason parity rules out uniquely Hamiltonian regular graphs of odd degree, leaving even-degree cases.\n\n**Verified partial progress.**\n\n- Odd-regular cases are excluded by parity of Hamilton cycles.\n\n**Full solution or refutation.**\n\nNo general theorem for all r>2 was verified.\n\n**What remains.**\n\nResolve the remaining even regular degrees, especially r=4.\n\n**Sources checked.**\n\n- Open Problem Garden, node 480 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains Sheehan's conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3168, "problem_number": "OPG-485", "title": "Hamiltonian cycles in line graphs", "statement": "Conjecture Every 4-connected line graph is hamiltonian.", "background": "Source: Open Problem Garden. Original node ID: 485. URL: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_line_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_line_graphs\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: hamiltonian; line graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 24th, 2007 by Robert Samal\n\nProblem-page discussion:\n\n- It is known that if $G$ is 4-edge-connected, then its line graph $L(G)$ is hamiltonian.\n- Thomassen's is a special case of a conjecture due to Matthews and Sumner: every 4-connected claw-free graph is hamiltonian.\n- However, by a result of Ryjacek [R] conjectures of Thomassen and of Matthews and Sumner are equivalent.\n- Moreover [R], one may restrict to 4-connected line graphs of triangle-free graphs.\n\nBibliography:\n[R] Zdenek Ryjacek: On a closure concept in claw-free graphs. J. Combin. Theory Ser. B 70 (1997), no. 2, 217--224, MathSciNet\n\n*[T] Carsten Thomassen, Reflections on graph theory, J. Graph Theory 10 (1986) 309-324, MathSciNet\n\nSource links:\n- line graph: http://en.wikipedia.org/wiki/line graph\n- hamiltonian: http://en.wikipedia.org/wiki/Hamilton cycle\n\nDiscussion links:\n- claw-free: http://en.wikipedia.org/wiki/Claw_(graph_theory)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1459867\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0856118\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamiltonian cycles in line graphs\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Thomassen's 4-connected line-graph conjecture remains open, though it is known for 7-connected line graphs and special families.\n\n**Verified partial progress.**\n\n- Zhan proves 7-connected line graphs Hamiltonian-connected.\n- Planar-root line graphs give further special cases.\n\n**Full solution or refutation.**\n\nNo 4-connected general theorem or counterexample was verified.\n\n**What remains.**\n\nReduce connectivity threshold from known cases to four.\n\n**Sources checked.**\n\n- S. Zhan, On hamiltonian line graphs and connectivity, Discrete Mathematics 89 (1991), 89--95. (primary): https://doi.org/10.1016/0012-365X(91)90401-M\n Evidence used: Proves every 7-connected line graph is Hamiltonian-connected.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3169, "problem_number": "OPG-500", "title": "Geodesic cycles and Tutte's Theorem", "statement": "Problem If $G$ is a $3$-connected finite graph, is there an assignment of lengths $\\ell: E(G) \\to \\mathb R^+$ to the edges of $G$, such that every $\\ell$-geodesic cycle is peripheral?", "background": "Source: Open Problem Garden. Original node ID: 500. URL: http://www.openproblemgarden.org/op/geodesic_cycles_and_tuttes_theorem.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/geodesic_cycles_and_tuttes_theorem\n- Author(s): Georgakopoulos, Agelos; Sprüssel, Philipp\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cycle space; geodesic cycles; peripheral cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: August 4th, 2007 by Agelos\n\nProblem-page discussion:\nA cycle $C$ is $\\ell$-geodesic if for every two vertices $x,y$ on $C$ there is no $x$- $y$ ~path in $G$ shorter, with respect to $\\ell$, than both $x$- $y$ ~arcs on $C$.\n\nIt is not hard to prove [GS] that for every finite graph $G$ and every assignment of edge lengths $\\ell: E(G) \\to \\mathb R^+$ the $\\ell$-geodesic cycles of $G$ generate its cycle space. Thus, a positive answer to the problem would imply a new proof of Tutte's classical theorem [T] that the peripheral cycles of a $3$-connected finite graph generate its cycle space.\n\nBibliography:\n*[GS] Angelos Georgakopoulos, Philipp Sprüssel: Geodesic topological cycles in locally finite graphs. Preprint 2007.\n\n[T] W.T. Tutte, How to draw a graph. Proc. London Math. Soc. 13 (1963), 743–768.\n\nSource links:\n- peripheral: http://en.wikipedia.org/wiki/peripheral cycle\n\nBibliography links:\n- Geodesic topological cycles in locally finite graphs: http://www.math.uni-hamburg.de/home/georgakopoulos/geo.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Geodesic cycles and Tutte's Theorem\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No general edge-length assignment forcing all geodesic cycles peripheral was verified.\n\n**Verified partial progress.**\n\n- The source gives the exact link to Tutte's peripheral-cycle theorem.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nSettle the stated length-assignment problem.\n\n**Sources checked.**\n\n- Open Problem Garden, Geodesic cycles and Tutte's theorem, node 500. (maintained_tracker): http://www.openproblemgarden.org/op/geodesic_cycles_and_tuttes_theorem\n Evidence used: Preserves the exact problem and cited context.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3170, "problem_number": "OPG-638", "title": "Jones' conjecture", "statement": "For a graph $G$, let $cp(G)$ denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let $cc(G)$ denote the cardinality of a minimum feedback vertex set (set of vertices $X$ so that $G-X$ is acyclic).\n\nConjecture For every planar graph $G$, $cc(G)\\leq 2cp(G)$.", "background": "Source: Open Problem Garden. Original node ID: 638. URL: http://www.openproblemgarden.org/op/jones_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/jones_conjecture\n- Author(s): Kloks, Ton; Lee, Chuan-Min; Liu, Jiping\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cycle packing; feedback vertex set; planar graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 9th, 2007 by cmlee\n\nProblem-page discussion:\nIn [KLL], the authors mention that there exists a family of nonplanar graphs for which $cc(G) = \\Theta( cp(G) \\log cp(G) )$, so no such result could hold for general graphs. They also point out that the conjecture is tight for wheels, and they prove it for the special case of outerplanar graphs.\n\nBibliography:\n*[KLL] Ton Kloks, Chuan-Min Lee, and Jiping Liu, New Algorithms for $k$-Face Cover, $k$-Feedback Vertex Set, and $k$-Disjoint Cycles on Plane and Planar Graphs, in Proceedings of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science (WG2002), LNCS 2573, pp. 282--295, 2002.\n\nComments:\n- December 5th, 2019 | David Wood | Proved for subcubic planar: Proved for subcubic planar graphs by Marthe Bonamy, François Dross, Tomáš Masařík, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk [https://arxiv.org/abs/1912.01570].\n- October 29th, 2007 | Anonymous | Why Jones'?: Does anyone know why this is called Jones' Conjecture?\n- November 16th, 2007 | Anonymous | Reply: Why Jones'?: I am Jones. My Taiwanese name is Chuan-Min Lee. This conjecture came up when I was working on it with Ton Kloks and Jiping Liu. I used the name \"Jones\" instead of my Taiwanese name for ease of communication.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Jones' conjecture\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Jones' factor-two inequality is proved for significant planar classes, while factor three is the current general planar bound and the full conjecture remains open.\n\n**Verified partial progress.**\n\n- Chappell, Gimbel, and Hartman prove cc(G) at most 3cp(G) for every planar graph.\n- Bonamy, Dross, Masarik, Nadara, Marcin Pilipczuk, and Michal Pilipczuk prove cc(G) at most 2cp(G) for subcubic planar graphs.\n- Barnkopf and Gyori prove the conjectured factor two for based planar graphs, including Halin graphs.\n\n**Full solution or refutation.**\n\nThe 2025 based-planar paper continues to call the all-planar assertion Jones' conjecture and proves only a broader special class, not the general case.\n\n**What remains.**\n\nImprove the universal planar coefficient from 3 to 2.\n\n**Sources checked.**\n\n- Glenn Chappell, John Gimbel, and Chris Hartman, On cycle packings and feedback vertex sets, Contributions to Discrete Mathematics 9(2) (2014), 17-34. (primary): https://doi.org/10.55016/ojs/cdm.v9i2.62105\n Evidence used: Proves the coefficient-three inequality for planar graphs.\n- Marthe Bonamy et al., Jones' Conjecture in Subcubic Graphs, Electronic Journal of Combinatorics 28(4) (2021), P4.5. (primary): https://www.combinatorics.org/ojs/index.php/eljc/article/download/v28i4p5/pdf/\n Evidence used: Proves the exact factor-two conjecture for subcubic planar graphs.\n- Pal Barnkopf and Ervin Gyori, Jones' Conjecture for Halin Graphs and a Bit More, Annals of Combinatorics (2025). (primary): https://doi.org/10.1007/s00026-025-00800-y\n Evidence used: Proves the exact inequality for based planar graphs and frames the unrestricted planar statement as open.\n\n**Review notes.** No material defect was found in the stored definition or inequality.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3171, "problem_number": "OPG-700", "title": "Chords of longest cycles", "statement": "Conjecture If $G$ is a 3-connected graph, every longest cycle in $G$ has a chord.", "background": "Source: Open Problem Garden. Original node ID: 700. URL: http://www.openproblemgarden.org/op/chords_of_longest_cycles.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/chords_of_longest_cycles\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: chord; connectivity; cycle\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 12th, 2007 by mdevos\n\nProblem-page discussion:\nA chord of a cycle $C$ is an edge $e$ so that $e \\not\\in E(C)$, but both ends of $e$ are in $V(C)$. Longest cycles are of great interest in basic graph theory, and this appealing conjecture suggests a very simple property they should share - at least in 3-connected graphs.\n\nIn dense graphs, this conjecture is easy to verify - for instance, if $G$ is Hamiltonian, it is trivially true. More interestingly, Thomassen [T] proved that his conjecture is true for cubic graphs using a clever sufficient condition for Hamiltonicity (based on Thomason's lollipop method) combined with a pretty theorem of Fleischner and Steibitz (cycle plus triangles graphs are 3-colorable).\n\nOther work on this conjecture has focused on graphs embedded in surfaces. Zhang [Z] has proved the conjecture for planar graphs of minimum degree four, Li and Zhang have proved the conjecture for graphs in the projective plane of minimum degree four [LZ1] and for 4-connected graphs in the klein bottle or torus [LZ2]. Finally, Kawarabayashi, Niu, and Zhang [KNZ] have shown the conjecture for 4-connected graphs on a fixed surface with sufficiently high face-width.\n\nBibliography:\n[KNZ] K. Kawarabayashi, J. Niu, C. Q. Zhang, Chords of longest circuits in locally planar graphs. European J. Combin. 28 (2007), no. 1, 315--321. MathSciNet\n\n[LZ1] X. Li, C. Q. Zhang, Chords of longest circuits in 3-connected graphs. Discrete Math. 268 (2003), no. 1-3, 199--206. MathSciNet\n\n[LZ2] X. Li, C. Q. Zhang, Chords of longest circuits of graphs embedded in torus and Klein bottle. J. Graph Theory 43 (2003), no. 1, 1--23. MathSciNet.\n\n[T2] C. Thomassen, Chords of longest cycles in cubic graphs. J. Combin. Theory Ser. B 71 (1997), no. 2, 211--214. MathSciNet.\n\n[Z] C. Q. Zhang, Longest cycles and their chords. J. Graph Theory 11 (1987), no. 4, 521--529. MathSciNet.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2261821\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1983278\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1974479\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1483476\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0917199\n\nComments:\n- October 14th, 2023 | Robert Samal | New partial results: New partial results for this appear in\n\nCarsten Thomassen: Chords in longest cycles, JCTB 129 (2018) 148-157\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Chords of longest cycles\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The every-longest-cycle conjecture remains open, but Thomassen now proves that every 3-connected graph has at least one longest cycle with a chord.\n\n**Verified partial progress.**\n\n- Every longest cycle in a 2-connected cubic graph has a chord.\n- Every 3-connected graph has some longest cycle with a chord.\n- The full conjecture holds for graphs whose circumference is sufficiently close to their order.\n\n**Full solution or refutation.**\n\nThe new existence theorem for some longest cycle does not establish the universal quantifier over all longest cycles.\n\n**What remains.**\n\nShow that every longest cycle, rather than at least one, has a chord in every 3-connected graph.\n\n**Sources checked.**\n\n- Carsten Thomassen, Chords in longest cycles in 3-connected graphs, Journal of Combinatorial Theory, Series B 179 (2026), 90-117. (primary): https://doi.org/10.1016/j.jctb.2026.03.001\n Evidence used: Proves that some longest cycle in every 3-connected graph has a chord and explicitly retains the stronger conjecture.\n- Haidong Wu and Shunzhe Zhang, Chords of longest cycles in graphs with large circumferences, arXiv:2511.03422 (2025). (primary): https://arxiv.org/abs/2511.03422\n Evidence used: States that the general conjecture remains open and proves it under a large-circumference hypothesis.\n\n**Review notes.** Quantifier distinction is decisive: the 2026 theorem proves some longest cycle has a chord, while OPG-700 asks this for every longest cycle.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3172, "problem_number": "OPG-2095", "title": "Hamiltonicity of Cayley graphs", "statement": "Question Is every Cayley graph Hamiltonian?", "background": "Source: Open Problem Garden. Original node ID: 2095. URL: http://www.openproblemgarden.org/op/hamiltonicity_of_cayley_graphs.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamiltonicity_of_cayley_graphs\n- Author(s): Rapaport-Strasser, E.\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: September 25th, 2008 by tchow\n\nProblem-page discussion:\nThis problem seems to have been first considered by Rapaport-Strasser [R]. Lovasz [L] conjectured more generally that every vertex-transitive graph is Hamiltonian. For a survey of results up to 1996, see [CG]. Although many specific Cayley graphs have been shown to be Hamiltonian, there are few general results. One exception is the theorem by Pak and Radoičić [PR] that every finite group $G$ with at least three elements has a generating set $S$ of size $|S| \\le \\log_2|G|$, such that the corresponding Cayley graph is Hamiltonian.\n\nBibliography:\n[CG] S. J. Curran, J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs—a survey, Discrete Math. 156 (1996), 1–18.\n\n[L] L. Lovasz, Problem 11, in “Combinatorial structures and their applications,” University of Calgary, Calgary, Alberta, Canada (1970), Gordon and Breach, New York.\n\n[PR] I. Pak and R. Radoičić, Hamiltonian paths in Cayley graphs, preprint.\n\n*[R] E. Rapaport-Strasser, Cayley color groups and Hamilton lines, Scripta Math. 24 (1959), 51–58.\n\nSource links:\n- Cayley graph: http://en.wikipedia.org/wiki/Cayley graph\n\nBibliography links:\n- Hamiltonian paths in Cayley graphs: http://www.math.umn.edu/%7Epak/hamcayley8.pdf\n\nComments:\n- December 15th, 2010 | Anonymous | Hamiltonicity of dence Cayley graphs: Christofides, Hladky, and Mathe (http://arxiv.org/abs/1008.2193) proved using the Regularity Method the case when the vertex-transitive graph is sufficiently dense.\n- September 21st, 2009 | Anonymous | [PR] paper: First, link to PR paper has moved with Pak homepage. Second, it is now published in Discrete Math.\n- January 8th, 2009 | Anonymous | Is this at all relevant?: Not sure but wondering if this link is at all relevant to the problem:\n\nhttp://books.google.com/books?id=aSyXqtfOuU4C&pg=PA49&dq=Cayley+graph&num=100&client=firefox-a#PPA49,M1\n\nThanks,\n\n- Farley\n- January 8th, 2009 | md | graphs vs. digraphs: The book you link to has examples of directed Cayley graphs with no (directed) Hamiltonian cycle. The existence of such graphs is interesting and relevant, but does not give a counterexample to the problem stated here.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamiltonicity of Cayley graphs\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The standard finite connected Cayley-graph Hamiltonicity conjecture remains open, but the extracted statement omits finiteness and connectedness conventions.\n\n**Verified partial progress.**\n\n- Many finite group families and generating-set regimes are known to yield Hamiltonian Cayley graphs.\n\n**Full solution or refutation.**\n\nNo theorem for the unqualified statement was assigned because disconnected/infinite Cayley graphs change its meaning.\n\n**What remains.**\n\nRecover the intended finite connected formulation, then continue literature triage of its known classes.\n\n**Sources checked.**\n\n- Open Problem Garden, node 2095 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Preserves the short source prompt without all standard hypotheses.\n\n**Review notes.** Formulation scope flagged; source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3173, "problem_number": "OPG-37241", "title": "Strong 5-cycle double cover conjecture", "statement": "Conjecture Let $C$ be a circuit in a bridgeless cubic graph $G$. Then there is a five cycle double cover of $G$ such that $C$ is a subgraph of one of these five cycles.", "background": "Source: Open Problem Garden. Original node ID: 37241. URL: http://www.openproblemgarden.org/op/strong_5_cycle_double_cover_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strong_5_cycle_double_cover_conjecture\n- Author(s): Arthur; Hoffmann-Ostenhof\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Keywords: cycle cover\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: August 3rd, 2010 by arthur\n\nProblem-page discussion:\nA cycle in $G$ is meant to be a $2$-regular subgraph of $G$. A five cycle double cover of $G$ is a set of five cycles of $G$ such that every edge of $G$ is contained in exactly two of these cycles.\n\nThis conjecture is a combination and thus strengthening of the $5$-cycle double cover conjecture and the strong cycle double cover conjecture.\n\nRelated:\nRelated problems\nCycle double cover conjecture\n\nComments:\n- November 23rd, 2011 | niekeaerts | ?: 3-connected Counterexample: $|VG|=8$\n$|EG|=12$\n$VG=\\{a,b,c,d,e,f,g,h\\}$\nAdjacency matrix (in alphabetical order):\n$$\n\\left( \\begin{array}{llllllll } 0&1&1&0&0&0&0&1 & 1&0&1&1&0&0&0&0 & 1&1&0&0&1&0&0&0 & 0&1&0&0&1&1&0&0 & 0&0&1&1&0&0&1&0 & 0&0&0&1&0&0&1&1 & 0&0&0&0&1&1&0&1 & 1&0&0&0&0&1&1&0 \\end{array}\\right)\n$$\n\nFor the explanation, see the following comment. -----------\n\nIt could be so that I am making a mistake, if so, please explain my mistake to me.\nI came to this point by simple trial and error.\nI would like to upload a simple picture, but I seem to be a little lost on how to do this.\n\nNieke Aerts\n- November 24th, 2011 | Robert Samal | Not a counterexample: Dear Nieke,\n\nunfortunately, your graph is definitely not a counterexample. I could not follow your explanations, let me instead show, why your graph with the indicated cycle has the desired 5-CDC.\n\nYour graph is planar, 2-edge-connected and the circuit C is a boundary of one of the faces. It turns out that for all such instances the conjecture is true: consider all face boundaries -- a collection of circuits. Now a proper 4-coloring of the dual graph splits the circuits into four cycles, the given circuit is contained in one of them. (The fifth cycle can be empty in this case.)\n\nThink about it, and if you still believe you have a counterexample, post again. For now, I am not reading the other comments, as they prove something that turns out to be false:-).\n\nBest wishes, Robert\n- November 24th, 2011 | niekeaerts | Indeed, no counterexamples: Dear Robert,\n\nThanks for your reply.\n\nI was thinking that the cycles have to be connected, which is obviously not true. So therefore my thought-to-be-counterexamples weren't correct. Thanks for the help! I wouldn't mind deleting all those comments, but I am not sure how to:)\n\nBest, Nieke\n- November 23rd, 2011 | niekeaerts | Explanation of the 3-connected counterexample: Consider the circuit $a,b,d,f,h,a$ to be color 1. And assume there is a 5-cycle cover containing this circuit as one of the cycles. We distinguish the cycles by color.\nThen $(a,b)$ and $(a,c)$ are colored with the same color (color 2) in the second cycle covering them, and similarly $(a,b)$ and $(a,h)$ have the same color (color 3) in the second cycle covering them. (As otherwise $(a,c)$ is colored twice by the cycle $(a,c,a)$ which, if allowed, quickly shows necessity of 6 colors)\n- November 23rd, 2011 | niekeaerts | Explanation of the 3-connected counterexample (Part II): Now $(b,c)$ still needs to be covered twice, which cannot be done by 1 color as then again one needs 6 colors. One of the colors has to be color 2, otherwise this will leave $b$ towards $a$ and $d$ and therefore there will be no escape possibility from $b$ for the two new colors. So the triangle $a,b,c$ is colored with color 2. By symmetry the triangle $f,g,h$ is also one color cycle (color 4). Now $(c,e)$, $(d,e)$ and $(e,g)$ are not colored and therefore need to be in the cycle of color 3 and the cycle of color 5. But then $e$ has degree 3 in both cycles which is a contradiction.\nSo a 5-cycle double cover containing this cycle does not exist.\n- November 23rd, 2011 | niekeaerts | ?: Counterexample for a graph with a 2-cut: $|VG|=8$\n$|EG|=12$\n$VG=\\{a,b,c,d,e,f,g,h\\}$\nAdjacency matrix (in alphabetical order):\n$$\n\\left( \\begin{array}{llllllll } 0&1&1&1&0&0&0&0 & 1&0&1&1&0&0&0&0 & 1&1&0&0&1&0&0&0 & 1&1&0&0&0&1&0&0 & 0&0&1&0&0&0&1&1 & 0&0&0&1&0&0&1&1 & 0&0&0&0&1&1&0&1 & 0&0&0&0&1&1&1&0 \\end{array}\\right)\n$$\n\nConsider the circuit $a,c,e,g,f,d,a$ to be color 1, now on either side of the 2-cut, we need three more colors, of which only one color can serve both (due to the 2-cut), so we need at least 6 colors (6 cycles).\n\n-----------\nIt could be so that I am making a mistake, if so, please explain my mistake to me.\nI came to this point by simple trial and error.\nI would like to upload a simple picture, but I seem to be a little lost on how to do this.\n\nNieke Aerts\n- November 23rd, 2011 | niekeaerts | Consider the circuit to be: Consider the circuit $a,b,d,f,h,a$ to be color 1. And assume there is a 5-cycle cover containing this circuit as one of the cycles. We distinguish the cycles by color.\nThen $(a,b)$ and $(a,c)$ are colored with the same color (color 2) in the second cycle covering them, and similarly $(a,c),(a,h)$ have the same color (color 3) in the second cycle covering them. (As otherwise $(a,c)$ is colored twice by the cycle $(a,c,a)$ which, if allowed, quickly shows necessity of 6 colors)\n- November 23rd, 2011 | niekeaerts | !!Ignore previous comment!!: Sorry, I pressed reply at the wrong statement.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"Strong 5-cycle double cover conjecture\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The strong 5-cycle double cover conjecture remains open, with recent work proving extension criteria and verifying substantial snark families.\n\n**Verified partial progress.**\n\n- Recent Catlin-reduction work proves 5-cycle double covers for several snark families.\n- Necessary and sufficient extension conditions are known for some specified 2-regular subgraphs.\n\n**Full solution or refutation.**\n\nNo general proof that every prescribed circuit belongs to a 5-cycle double cover was verified.\n\n**What remains.**\n\nProve the strong 5-CDC conjecture or find a bridgeless cubic counterexample.\n\n**Sources checked.**\n\n- Y. Zhang, 5-Cycle Double Covers, 4-Flows, and Catlin Reduction, SIAM Journal on Discrete Mathematics 37 (2023), 253--267. (primary): https://doi.org/10.1137/22M1472425\n Evidence used: The paper describes the 5-CDC conjecture as open and establishes results for special classes.\n- Graph-theory open problems, Strong 5-cycle double cover conjecture (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/strong_5_cycle_double_cover_conjecture/\n Evidence used: The maintained page records ongoing partial advances rather than a general solution.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3174, "problem_number": "OPG-46584", "title": "Decomposing an eulerian graph into cycles.", "statement": "Conjecture Every simple eulerian graph on $n$ vertices can be decomposed into at most $\\frac{1}{2}(n-1)$ cycles.", "background": "Source: Open Problem Garden. Original node ID: 46584. URL: http://www.openproblemgarden.org/op/decomposing_an_eulerian_graph_into_cycles.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_an_eulerian_graph_into_cycles\n- Author(s): Hajós, G.\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is tight because a complete graph on $2k+1$ vertices cannot be covered by less than $k$ cycles.\n\nThere is a similar conjecture about decomposition of a connected graph into paths.\n\nBibliography:\n* [L] L. Lovász, On covering of graphs. In Theory of Graphs (Proc. Colloq., Tihany, 1966), 231--236. Academic Press, New York, 1968.\n\nRelated:\nRelated problems\nDecomposing a connected graph into paths.\n\nDiscussion links:\n- decomposition of a connected graph into paths: http://www.openproblemgarden.org/?q=node/46583\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing an eulerian graph into cycles.\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hajós's cycle decomposition conjecture remains open for arbitrary simple Eulerian graphs. The best cited general asymptotic result gives O(n log-star n) cycles, and the exact conjecture is known for several structural classes.\n\n**Verified partial progress.**\n\n- Bucić--Montgomery prove every n-vertex graph decomposes into O(n log-star n) cycles and edges; for Eulerian graphs this yields an O(n log-star n) cycle decomposition.\n- Hajós's exact bound is known for planar graphs, graphs of maximum degree four, and graphs of treewidth at most three.\n\n**Full solution or refutation.**\n\nNo decomposition into at most floor((n-1)/2) cycles for every simple Eulerian graph was verified.\n\n**What remains.**\n\nReduce the general O(n log-star n) cycle count to the sharp linear coefficient 1/2 and remove structural restrictions.\n\n**Sources checked.**\n\n- Matija Bucić and Richard Montgomery, Towards the Erdős--Gallai Cycle Decomposition Conjecture, Advances in Mathematics 437 (2024), 109409; arXiv:2211.07689. (primary): https://arxiv.org/abs/2211.07689\n Evidence used: Improves the general decomposition bound to O(n log-star n), discusses Hajós's conjecture, and explains the Eulerian reduction.\n- Fábio Botler, Andrea Jiménez, and Maycon Sambinelli, On Gallai's and Hajós' Conjectures for graphs with treewidth at most 3, arXiv:1706.04334. (primary): https://arxiv.org/abs/1706.04334\n Evidence used: Proves the exact Hajós bound for treewidth at most three and records the planar and maximum-degree-four cases.\n- António Girão, Bertille Granet, Daniela Kühn, and Deryk Osthus, Path and cycle decompositions of dense graphs, Journal of the London Mathematical Society 104 (2021), 1085-1134; arXiv:1911.05501. (primary): https://arxiv.org/abs/1911.05501\n Evidence used: States Hajós's conjecture as open and proves dense-graph progress related to it.\n\n**Review notes.** The source background says a complete graph cannot be covered by fewer cycles although the conjecture concerns an edge decomposition; this wording defect does not change the displayed statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3175, "problem_number": "OPG-46606", "title": "Decomposing an eulerian graph into cycles with no two consecutives edges on a prescribed eulerian tour.", "statement": "Conjecture Let $G$ be an eulerian graph of minimum degree $4$, and let $W$ be an eulerian tour of $G$. Then $G$ admits a decomposition into cycles none of which contains two consecutive edges of $W$.", "background": "Source: Open Problem Garden. Original node ID: 46606. URL: http://www.openproblemgarden.org/op/decomposing_an_eulerian_graph_into_cycles_with_no_two_consecutives_edges_on_a_prescirbed_eulerian_tour.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_an_eulerian_graph_into_cycles_with_no_two_consecutives_edges_on_a_prescirbed_eulerian_tour\n- Author(s): Sabidussi, Gert\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nKotzig proved the converse statement:\n\nTheorem Let $G$ be an eulerian graph of minimum degree $4$ and let $(C_1, \\dots, C_p)$ be a decomposition into cycles of $G$. Then $G$ admits an eulerian tour such that none of the $C_i$, $1\\leq i\\leq p$, contains two consecutive edges of $W$.\n\nFor more details on eulerian graphs, see [F].\n\nBibliography:\n*[F] H. Fleischner. Eulerian Graphs and Related Topics. Part 1. Vol. 1. Annals of Discrete Mathematics, Vol. 45, (1990) North-Holland, Amsterdam.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing an eulerian graph into cycles with no two consecutives edges on a prescribed eulerian tour.\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A primary preprint submitted 2026-07-14 claims a proof of Sabidussi's compatibility conjecture in the exact finite connected multigraph setting represented by the source statement.\n\n**Verified partial progress.**\n\n- Earlier verified cases included planar and K5-minor-free Eulerian graphs and sufficiently edge-connected graphs.\n- The new preprint proves a stronger four-colouring statement and reports an accompanying Lean 4 formalization.\n\n**Full solution or refutation.**\n\nUlyanov proves that the edges of every finite connected Eulerian multigraph of minimum degree at least four can be partitioned into circuits so that no circuit contains two edges consecutive in the prescribed Euler tour.\n\n**What remains.**\n\nIndependently verify the very recent v1 proof/formalization and monitor peer review; confirm that the source's cycle convention agrees with multigraph circuits.\n\n**Sources checked.**\n\n- N. Ulyanov, Graph Puzzles III.1: A Proof of Sabidussi's Compatibility Conjecture, arXiv:2607.13225 (2026). (primary): https://arxiv.org/abs/2607.13225\n Evidence used: The abstract states the exact compatible circuit-decomposition theorem and the stronger four-colouring result; submission date is 2026-07-14.\n- Graph-theory open problems, Decomposing an eulerian graph into cycles with no two consecutives edges on a prescribed eulerian tour (reviewed 2026-05-08; accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/decomposing_an_eulerian_graph_into_cycles_with_no_two_consecutives_edges_on_a_prescirbed_eulerian_tour/\n Evidence used: Documents the pre-July-2026 open status and earlier partial cases; its review predates the new primary preprint.\n\n**Review notes.** Very recent unrefereed v1 result. The maintained tracker is date-stale for this item, not contradictory current evidence. Source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3176, "problem_number": "OPG-47285", "title": "Every prism over a 3-connected planar graph is hamiltonian.", "statement": "Conjecture If $G$ is a $3$-connected planar graph, then $G\\square K_2$ has a Hamilton cycle.", "background": "Source: Open Problem Garden. Original node ID: 47285. URL: http://www.openproblemgarden.org/op/every_prism_over_a_3_connected_planar_graph_is_hamiltonian.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/every_prism_over_a_3_connected_planar_graph_is_hamiltonian\n- Author(s): Kaiser, Tomás; Král, Daniel; Rosenfeld, Moshe; Ryjácek, Zdenek; Voss, Heinz-Jürgen\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 11th, 2013 by fhavet\n\nProblem-page discussion:\nThe Cartesian product $G\\square K_2$ is called the prism over $G$.\n\nRosenfeld and Barnette [RB] showed that the Four-Colour Theorem implies that cubic planar 3-connected graphs have hamiltonian prisms. Fleischner [F] found a proof avoiding the use of the Four Colour Theorem. Eventually, Paulraja [P] showed that planarity is inessential here: The prism over any 3-connected cubic graph has a Hamilton cycle.\n\nClearly, if $G$ is hamiltonian, then $G\\square K_2$ is also hamiltonian. A classical theorem of Tutte [T] states that all 4-connected planar graphs are hamiltonian. There are well-known examples of non-hamiltonian 3-connected planar graphs.\n\nBibliography:\n[F] H. Fleischner, The prism of a 2-connected, planar, cubic graph is hamiltonian (a proof independent of the four colour theorem), in Graph theory in memory of G. A. Dirac, Volume 41 of Ann. Discrete Math., 1989), 141–170.\n\n*[KKRRV] T. Kaiser, D. Kráľ, M. Rosenfeld, Z. Ryjáček, H.-J. Voss, Hamilton cycles in prisms, Journal of graph theory 56 (2007), 249-269.\n\n[P] P. Paulraja, “A characterization of hamiltonian prisms”, J. Graph Theory 17 (1993) 161–171.\n\n[RB] M. Rosenfeld and D. Barnette, Hamiltonian circuits in certain prisms, Discrete Math. 5 (1973) 389–394.\n\n[T] W. T. Tutte, A theorem on planar graphs, Trans. Amer. Math. Soc. 82 (1956) 99–116.\n\nRelated:\nRelated problems\nBarnette's Conjecture\n\nDiscussion links:\n- Cartesian product: http://en.wikipedia.org/wiki/Cartesian product of graphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Every prism over a 3-connected planar graph is hamiltonian.\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Spacapan constructed a 3-connected planar graph whose prism is not Hamiltonian, refuting the universal statement.\n\n**Verified partial progress.**\n\n- Every 3-connected planar graph of minimum degree at least four is prism-Hamiltonian, so counterexamples require cubic vertices.\n\n**Full solution or refutation.**\n\nThe explicit 3-connected planar counterexample disproves the conjecture as stated.\n\n**What remains.**\n\nClassify the remaining prism-Hamiltonian 3-connected planar graphs; this is no longer needed to decide the original universal claim.\n\n**Sources checked.**\n\n- S. Spacapan, A counterexample to prism-hamiltonicity of 3-connected planar graphs, Journal of Combinatorial Theory, Series B 148 (2021), 46--52; arXiv:1906.06683. (primary): https://arxiv.org/abs/1906.06683\n Evidence used: Constructs a 3-connected planar graph whose prism over K2 is not Hamiltonian.\n- S. Spacapan, Polyhedra without cubic vertices are prism-hamiltonian, Journal of Graph Theory 108 (2024); arXiv:2104.04266. (primary): https://arxiv.org/abs/2104.04266\n Evidence used: Proves the min-degree-at-least-four surviving special case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3177, "problem_number": "OPG-47294", "title": "4-connected graphs are not uniquely hamiltonian", "statement": "Conjecture Every $4$-connected graph with a Hamilton cycle has a second Hamilton cycle.", "background": "Source: Open Problem Garden. Original node ID: 47294. URL: http://www.openproblemgarden.org/op/4_connected_graphs_are_not_uniquely_hamiltonian.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/4_connected_graphs_are_not_uniquely_hamiltonian\n- Author(s): Fleischner, Herbert\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 11th, 2013 by fhavet\n\nBibliography:\n*[F] H. Fleischner, Uniquely Hamiltonian graphs of minimum degree four,, J. Graph Theory, to appear.\n\nRelated:\nRelated problems\nr-regular graphs are not uniquely hamiltonian.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"4-connected graphs are not uniquely hamiltonian\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The conjecture remains open: uniquely Hamiltonian constructions reach vertex-connectivity 3, but no 4-connected example or proof of nonexistence is known.\n\n**Verified partial progress.**\n\n- Fleischner constructed infinite families of uniquely Hamiltonian graphs of minimum degree 4, including a 3-connected example, showing that minimum degree alone does not suffice.\n- Barish and Suyama characterize existence of a 4-connected counterexample through a parsimonious #SAT reduction, constraining but not settling the question.\n\n**Full solution or refutation.**\n\nNo resolution of the 4-connectivity threshold was verified.\n\n**What remains.**\n\nProve every 4-connected Hamiltonian graph has at least two Hamilton cycles, or construct a 4-connected uniquely Hamiltonian graph.\n\n**Sources checked.**\n\n- Herbert Fleischner, Uniquely Hamiltonian Graphs of Minimum Degree 4, Journal of Graph Theory 75 (2014), 167-177, DOI 10.1002/jgt.21729. (primary): https://doi.org/10.1002/jgt.21729\n Evidence used: Constructs uniquely Hamiltonian graphs of minimum degree 4 and reaches connectivity 3, but not 4.\n- Robert D. Barish and Akira Suyama, Randomized Reductions and the Topology of Conjectured Classes of Uniquely Hamiltonian Graphs, Journal of Information Processing 28 (2020), 876-888, DOI 10.2197/ipsjjip.28.876. (primary): https://doi.org/10.2197/ipsjjip.28.876\n Evidence used: Explicitly calls Fleischner's 4-connected conjecture open and proves a complexity-theoretic equivalence for potential counterexamples.\n- Graph-theory open problems, 4-connected graphs are not uniquely hamiltonian, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/4_connected_graphs_are_not_uniquely_hamiltonian/\n Evidence used: Current specialist tracker records the 4-connected case as open and separates it from connectivity-3 constructions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3178, "problem_number": "OPG-47356", "title": "Hamilton decomposition of prisms over 3-connected cubic planar graphs", "statement": "Conjecture Every prism over a $3$-connected cubic planar graph can be decomposed into two Hamilton cycles.", "background": "Source: Open Problem Garden. Original node ID: 47356. URL: http://www.openproblemgarden.org/op/decomposing_the_prism_of_a_3_connected_cubic_planar_graphs_in_hamilton_cycles.\n\nSource subject path: Graph Theory > Basic Graph Theory > Cycles.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_the_prism_of_a_3_connected_cubic_planar_graphs_in_hamilton_cycles\n- Author(s): Alspach, Brian; Rosenfeld, Moshe\n- Subject(s): Graph Theory; Basic Graph Theory; Cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 12th, 2013 by fhavet\n\nProblem-page discussion:\nThe prism over a graph $G$ is the Cartesian product $G\\square K_2$.\n\nRosenfeld and Barnette [RB] deduced from the Four-Colour Theorem that the prism over cubic planar 3-connected $G$ has a Hamilton cycle $C$. The graph $G\\setminus E(C)$ is cycle factor (spanning union of cycles). The conjecture says that one can choses $C$ so that the cycle factor $G\\setminus E(C)$ has a unique cycle, that is a Hamilton cycle.\n\nBibliography:\n*[AR] B. Alspach and M. Rosenfeld, On Hamilton decompositions of prisms over simple $3$-polytopes. Graphs Combin. 2 (1986), 1--8.\n\n[RB] M. Rosenfeld and D. Barnette, Hamiltonian circuits in certain prisms, Discrete Math. 5 (1973) 389–394.\n\nRelated:\nRelated problems\nEvery prism over a 3-connected planar graph is hamiltonian.\n\nDiscussion links:\n- Cartesian product: http://en.wikipedia.org/wiki/Cartesian product of graphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamilton decomposition of prisms over 3-connected cubic planar graphs\" in Graph Theory; Basic Graph Theory; Cycles, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The prism decomposition conjecture remains open for all 3-connected cubic planar graphs, but it is proved for several infinite subclasses.\n\n**Verified partial progress.**\n\n- The conjecture holds for 3-connected planar bipartite cubic graphs and kleetope-dual families.\n- It also holds for cubic Halin graphs, cubic generalized Halin graphs of order 4k+2, and other infinite sequences.\n\n**Full solution or refutation.**\n\nKnown family theorems do not exhaust 3-connected cubic planar graphs.\n\n**What remains.**\n\nProve a two-Hamilton-cycle decomposition for every remaining 3-connected cubic planar base graph or produce a counterexample.\n\n**Sources checked.**\n\n- Roman Cada, Tomas Kaiser, Moshe Rosenfeld, and Zdenek Ryjacek, Hamiltonian decompositions of prisms over cubic graphs, Discrete Mathematics 286 (2004), 45-56, DOI 10.1016/j.disc.2003.11.044. (primary): https://doi.org/10.1016/j.disc.2003.11.044\n Evidence used: Proves Hamilton decomposability for planar bipartite cubic graphs and another planar cubic family, not the whole conjectured class.\n- Moshe Rosenfeld and Ziqing Xiang, Hamiltonian decomposition of prisms over cubic graphs, Discrete Mathematics & Theoretical Computer Science 16(2) (2015), 111-124, DOI 10.46298/dmtcs.2079. (primary): https://doi.org/10.46298/dmtcs.2079\n Evidence used: Surveys the still-open conjecture and proves it for cubic Halin and generalized Halin families and other infinite sequences.\n\n**Review notes.** The background's expression G minus E(C) is type-inconsistent because C lies in the prism; this was flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3179, "problem_number": "OPG-142", "title": "The Berge-Fulkerson conjecture", "statement": "Conjecture If $G$ is a bridgeless cubic graph, then there exist 6 perfect matchings $M_1,\\ldots,M_6$ of $G$ with the property that every edge of $G$ is contained in exactly two of $M_1,\\ldots,M_6$.", "background": "Source: Open Problem Garden. Original node ID: 142. URL: http://www.openproblemgarden.org/op/the_berge_fulkerson_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Matchings.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_berge_fulkerson_conjecture\n- Author(s): Berge, Claude; Fulkerson, Delbert R.\n- Subject(s): Graph Theory; Basic Graph Theory; Matchings\n- Keywords: cubic; perfect matching\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nThis conjecture is due to Berge and Fulkerson, and appears first in [F] (see [S79b]).\n\nIf $G$ is 3-edge-colorable, then we may choose three perfect matchings $M_1,M_2,M_3$ so that every edge is in exactly one. Taking each of these twice gives us 6 perfect matchings with the properties described above. Thus, the above conjecture holds trivially for 3-edge-colorable graphs. There do exist bridgeless cubic graphs which are not 3-edge-colorable (for instance the Petersen graph), but the above conjecture asserts that every such graph is close to being 3-edge-colorable.\n\nDefinition: An $r$-graph is an $r$-regular graph $G$ on an even number of vertices with the property that every edge-cut which separates $V(G)$ into two sets of odd cardinality has size at least $r$.\n\nObserve that a cubic graph is a 3-graph if and only if it has no bridge. If G is an $r$-regular graph which has an $r$-edge-coloring, then every color class is a perfect matching, so $|V(G)|$ must be even, and every color must appear in every edge-cut which separates $V(G)$ into two sets of odd size, so every edge-cut of this form must have size at least $r$. Thus, every $r$-edge-colorable $r$-regular graph is an $r$-graph. In a sense, we may view the $r$-graphs as the $r$-regular graphs which have the obvious necessary conditions to be $r$-edge-colorable. Seymour [S79b] defined $r$-graphs and offered the following generalization of the Berge-Fulkerson conjecture.\n\nConjecture (The generalized Berge-Fulkerson conjecture (Seymour)) Let $G$ be an $r$-graph. Then there exist $2r$ perfect matchings $M_1,\\ldots,M_{2r}$ of $G$ with the property that every edge of $G$ is contained in exactly two of $M_1,\\ldots,M_{2r}$.\n\nMore generally, for a graph $G=(V,E)$, one may consider the vector space of real numbers indexed by $E$. We associate every perfect matching $M$ with its characteristic vector. In this context, the Berge-Fulkerson conjecture asserts that for every 3-graph, the vector which is identically 1 may be written as a half-integer combination of perfect matchings. Edmonds matching polytope theorem [E] gives a complete characterization of what vectors in ${\\mathbb R}^E$ which can be written as a nonnegative real combination of perfect matchings. A particular consequence of this theorem is that the vector which is identically 1 can be written as a nonnegative rational combination of perfect matchings if G is an $r$-graph. It follows from this that for every $r$-graph $G$, there is a list of perfect matchings $M_1,\\ldots,M_{kr}$ so that every edge is contained in exactly $k$ of them. Unfortunately, the particular $k$ depends on the graph. The following weak version of the Berge-Fulkerson conjecture asserts that this dependence is inessential.\n\nConjecture (the weak Berge-Fulkerson conjecture) There exists a positive integer $k$ with the following property. Every 3-graph $G$ has a list of $3k$ perfect matchings such that every edge of $G$ is contained in exactly $k$ of them.\n\nThere is a fascinating theorem of Lovasz [L] that characterizes which vectors in ${\\mathbb Z}^E$ can be written as an integer combination of perfect matchings. However, very little is known about nonnegative integer combinations of perfect matchings. In particular, if the Berge-Fulkerson conjecture is true, then for every 3-graph $G=(V,E)$, there is a list of 5 perfect matchings with union $E$ (take any 5 of the 6 perfect matchings given by the conjecture). The following weakening of this (suggested by Berge) is still open.\n\nConjecture There exists a fixed integer $k$ such that the edge set of every 3-graph can be written as a union of $k$ perfect matchings.\n\nAnother consequence of the Berge-Fulkerson conjecture would be that every 3-graph has 3 perfect matchings with empty intersection (take any 3 of the 6 perfect matchings given by the conjecture). The following weakening of this (also suggested by Berge) is still open.\n\nConjecture There exists a fixed integer $k$ such that every 3-graph has a list of $k$ perfect matchings with empty intersection.\n\nBibliography:\n[E] J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices, J. Res. Nat. Bur Stand B, Math & Math Phys. 69B (1965), 125-130.\n\n[F] D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Math. Programming 1 (1971) 168-194. MathSciNet\n\n[KKN] T. Kaiser, D. Kral, and S. Norine, Unions of perfect matchings in cubic graphs\n\n[L] L. Lovasz, Matching structure and the matching lattice, J. Combin. Theory Ser. B 43 (1987), 187-222. MathSciNet\n\n[R] R. Rizzi, Indecomposable r-graphs and some other counterexamples, J. Graph Theory 32 (1999), 1-15. MathSciNet\n\n[S79a] P.D. Seymour, \"Some unsolved problems on one-factorizations of graphs\", Graph theory and Related Topics, J.A. Bondy and U.S.R. Murty (Editors), Academic, New York (1979), 367-368.\n\n[S79b] P.D. Seymour, On multi-colourings of cubic graphs, and conjectures of Fulkerson and Tutte, Proc. London Math Soc. 38 (1979), 423-460. MathSciNet\n\nSource links:\n- bridgeless: http://en.wikipedia.org/wiki/bridge (graph theory)\n- cubic: http://en.wikipedia.org/wiki/cubic graph\n- perfect matchings: http://en.wikipedia.org/wiki/matching\n\nDiscussion links:\n- edge-colorable: http://en.wikipedia.org/wiki/edge coloring\n- Petersen graph: http://en.wikipedia.org/wiki/Petersen graph\n- regular: http://en.wikipedia.org/wiki/regular graph\n- edge-cut: http://en.wikipedia.org/wiki/connectivity (graph theory)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0294149\n- Unions of perfect matchings in cubic graphs: http://www.math.princeton.edu/%7Esnorin/papers/union-en.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0904405\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1704172\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0532981\n\nComments:\n- April 4th, 2011 | Louis Esperet | equivalent conjecture: It was recently proved by G Mazzuocolo (The Equivalence of Two Conjectures of Berge and Fulkerson, J. Graph Theory (2010) doi:10.1002/jgt.20545), that Berge-Fulkerson is indeed equivalent to the statement that for any 3-graph G, the edge-set of G can be covered by 5 perfect matchings. This might be worth mentioning it.\n- February 25th, 2009 | Andrew King | Covering with perfect matchings: It is not hard to show that you can cover the edges of a bridgeless cubic graph with $\\log(n)$ perfect matchings. Is there some smaller-order function that suffices?\n- February 25th, 2009 | mdevos | I believe this is best known: The $log(n)$ bound you mention (a consequence of Edmond's perfect matching polytope theorem) is, to my knowledge, the best known lower bound.\n- October 16th, 2011 | Emilio Brazil | Reference for log(n) bound: I'm working with bridgeless cubic graphs and this bound is a good theoretical bound for my purposes. I'd like to know references for this $\\log(n)$ bound.\n- November 23rd, 2011 | Andrew King | log(n) bound: The proof is basically the same as Lovász' proof that $\\chi(G) \\leq \\log(n)\\cdot (\\chi_f(G)+1)$: It is well-known that the fractional chromatic number of a bridgeless cubic graph is 3. Therefore there is a probability distribution on the perfect matchings of $G$ such that given a random matching from this distribution, an edge $e$ is hit with probability 1/3.\n\nNow let $k$ be $\\log_{3/2}(n)+1$ and take $k$ perfect matchings from this distribution, and consider the probability that an edge $e$ is in none of them. This is $(2/3)^k < 1/n$, so by the union bound, the probability that every edge is in one of the matchings is greater than $1- n(1/n) = 0$. Therefore there is some choice of $k$ perfect matchings that cover the graph.\n- November 22nd, 2011 | Anonymous | Best known upper bound: I believe that the best known upper bound is given here http://arxiv.org/abs/1111.1871\n- March 2nd, 2009 | Andrew King | Matchings and odd cuts: I like thinking about this problem in terms of fractional colourings. A roughly equivalent proof (to the one you mentioned) is as follows. In a fractional 3-edge-colouring, every matching must be a perfect matching. The weights on the matchings, when divided by 3, give you a probability distribution. If you pick $3 \\log(n)$ matchings from this distribution, the union bound tells you that with positive probability you hit every edge. This method is powerful enough to prove that the fractional and integer chromatic numbers are always within a factor of $\\log(n)$ of one another.\n\nHere are two even weaker questions:\n\nIs there some $k$ such that any bridgeless cubic graph contains $k$ perfect matchings whose intersection does not contain an odd cut?\n\nDoes every bridgeless cubic graph contain two perfect matchings that together hit no 5-cut 8 or 10 times?\n\nI suspect both of these are open as well.\n- March 4th, 2009 | mdevos | Sure: Okay, sure, but the proof that there exists a fractional 3-edge-coloring is either a corollary of Edmond's Theorem or something essentially equivalent to it (such as the approach Seymour uses in his paper on r-graphs). In short, I agree that we are talking about essentially the same argument.\n\nI believe that your first question is open. It is a well known conjecture (elsewhere on this site) that there should be 2 perfect matchings whose intersection does not contain an odd cut.\n\nThe second question has been resolved (using Edmond's theorem) by Kaiser, Kral, and Norine and I will add a link to the paper in the reference section.\n- October 14th, 2011 | Anonymous | proof of log(n): Do you know a reference for a proof this log(n) boundary? I looked for it in text book but I don't have any clue.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"The Berge-Fulkerson conjecture\" in Graph Theory; Basic Graph Theory; Matchings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Berge-Fulkerson conjecture remains open for general bridgeless cubic graphs. Recent work proves the equivalent Berge perfect-matching-cover formulation for bridgeless cubic graphs with colouring defect three and obtains a four-matching cover in a cyclically 4-edge-connected subcase except for Petersen.\n\n**Verified partial progress.**\n\n- Karabas, Macajova, Nedela, and Skoviera prove Berge's conjecture for all bridgeless cubic graphs of colouring defect three.\n- For cyclically 4-edge-connected defect-three graphs, four perfect matchings suffice unless the graph is the Petersen graph.\n\n**Full solution or refutation.**\n\nNo six-perfect-matching double cover for every bridgeless cubic graph was found. The strongest recent cited result covers the small-colouring-defect class.\n\n**What remains.**\n\nExtend perfect-matching covers from small-colouring-defect classes to arbitrary bridgeless cubic graphs.\n\n**Sources checked.**\n\n- Giuseppe Mazzuoccolo, The equivalence of two conjectures of Berge and Fulkerson, Journal of Graph Theory 68 (2011), 125-128. (primary): https://doi.org/10.1002/jgt.20545\n Evidence used: Proves that the exact six-perfect-matching double-cover conjecture is equivalent to covering every bridgeless cubic graph with at most five perfect matchings.\n- Jan Karabas, Edita Macajova, Roman Nedela, and Martin Skoviera, Berge's Conjecture for Cubic Graphs With Small Colouring Defect, Journal of Graph Theory 109 (2025), 387-396. (primary): https://doi.org/10.1002/jgt.23231\n Evidence used: Proves the Berge perfect-matching-cover formulation for defect-three bridgeless cubic graphs and a sharper four-matching subcase.\n- Jan Karabas, Edita Macajova, Roman Nedela, and Martin Skoviera, arXiv:2210.13234. (primary): https://arxiv.org/abs/2210.13234\n Evidence used: Primary accessible version whose abstract explicitly says the general conjecture remains widely open.\n- Graph-theory open problems, The Berge-Fulkerson conjecture, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/the_berge_fulkerson_conjecture/\n Evidence used: Maintains the exact six-matching double-cover formulation and its known context.\n\n**Review notes.** The exact six-matchings-each-edge-twice formulation was preserved; partial results in the equivalent Berge five-cover formulation were not overstated as a full solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3180, "problem_number": "OPG-543", "title": "The intersection of two perfect matchings", "statement": "Conjecture Every bridgeless cubic graph has two perfect matchings $M_1$, $M_2$ so that $M_1 \\cap M_2$ does not contain an odd edge-cut.", "background": "Source: Open Problem Garden. Original node ID: 543. URL: http://www.openproblemgarden.org/op/intersecting_two_perfect_matchings.\n\nSource subject path: Graph Theory > Basic Graph Theory > Matchings.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/intersecting_two_perfect_matchings\n- Author(s): Macajova, Edita; Skoviera, Martin\n- Subject(s): Graph Theory; Basic Graph Theory; Matchings\n- Keywords: cubic; nowhere-zero flow; perfect matching\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 30th, 2007 by mdevos\n\nProblem-page discussion:\nLet $G = (V,E)$ be a bridgeless cubic graph. A binary cycle (henceforth called cycle) is a set $C \\subseteq E$ so that every vertex of $(V,C)$ has even degree (equivalently, a cycle is any member of the binary cycle space). A postman join is a set $J \\subseteq E$ so that $E \\setminus J$ is a cycle. Note that since $G$ is cubic, every perfect matching is a postman join. Next we state a well-known theorem of Jaeger in three equivalent forms.\n\nTheorem (Jaeger's 8-flow theorem)\n\n- $G$ has a nowhere-zero flow in the group ${\\mathbb Z}_2^3$.\n- $G$ has three cycles $C_1,C_2,C_3$ so that $C_1 \\cup C_2 \\cup C_3 = E$.\n- $G$ has three postman joins $J_1,J_2,J_3$ so that $J_1 \\cap J_2 \\cap J_3 = \\emptyset$.\n\nThe last of these statements is interesting, since The Berge Fulkerson Conjecture (if true) implies the following:\n\nConjecture $G$ has three perfect matchings $M_1,M_2,M_3$ so that $M_1 \\cap M_2 \\cap M_3= \\emptyset$.\n\nSo, we know that $G$ has three postman joins $J_1,J_2,J_3$ with empty intersection, and it is conjectured that $J_1,J_2,J_3$ may be chosen so that each is a perfect matching, but now we see two statements in between the theorem and the conjecture. Namely, is it true that $J_1,J_2,J_3$ may be chosen so that one is a perfect matching? or two? The first of these was solved recently.\n\nTheorem (Macajova, Skoviera) $G$ has two postman sets $J_1,J_2$ and one perfect matching $M$ so that $M \\cap J_1 \\cap J_2 = \\emptyset$\n\nThe second of these asks for two perfect matchings $M_1,M_2$ and one postman join $J$ so that $M_1 \\cap M_2 \\cap J = \\emptyset$. It is an easy exercise to show that a set $S \\subseteq E$ contains a postman join if an only if $S$ has nonempty intersection with every odd edge-cut. Therefore, finding two perfect matchings and one postman join with empty common intersection is precisely equivalent to the conjecture at the start of this page - find two perfect matchings whose intersection contains no odd edge-cut.\n\nBibliography:\n* Edita Macajova, Martin Skoviera, Fano colourings of cubic graphs and the Fulkerson conjecture. Theoret. Comput. Sci. 349 (2005), no. 1, 112--120. MathSciNet\n\nRelated:\nRelated problems\nThe Berge-Fulkerson conjecture\n\nDiscussion links:\n- The Berge Fulkerson Conjecture: http://www.openproblemgarden.org/?q=node/142\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2183473\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"The intersection of two perfect matchings\" in Graph Theory; Basic Graph Theory; Matchings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the perfect-matching intersection condition for every bridgeless cubic graph was verified.\n\n**Verified partial progress.**\n\n- The source relates it to Jaeger's theorem and Berge--Fulkerson.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nSettle the stated two-matching condition.\n\n**Sources checked.**\n\n- Open Problem Garden, The intersection of two perfect matchings, node 543. (maintained_tracker): http://www.openproblemgarden.org/op/intersecting_two_perfect_matchings\n Evidence used: Preserves the exact cubic-graph formulation and references.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3181, "problem_number": "OPG-600", "title": "Matchings extend to Hamiltonian cycles in hypercubes", "statement": "Question Does every matching of hypercube extend to a Hamiltonian cycle?", "background": "Source: Open Problem Garden. Original node ID: 600. URL: http://www.openproblemgarden.org/op/matchings_extends_to_hamilton_cycles_in_hypercubes.\n\nSource subject path: Graph Theory > Basic Graph Theory > Matchings.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/matchings_extends_to_hamilton_cycles_in_hypercubes\n- Author(s): Ruskey, Frank; Savage, Carla\n- Subject(s): Graph Theory; Basic Graph Theory; Matchings\n- Keywords: Hamiltonian cycle; hypercube; matching\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: September 28th, 2007 by Jirka\n\nProblem-page discussion:\nThis question is due to Ruskey and Savage and appears in [RS] (page 19, question 3). The answer is positive for $d$-cube if $d \\le 4$.\n\nFink [F] proved Kreweras' conjecture [K] which asserts that every perfect matching of hypercube extends to a Hamiltonian cycle.\n\nBibliography:\n[RS] F. Ruskey and C. D. Savage, SIAM Journal on Discrete Mathematics 6, No.1 (1993) 152-166. download\n\n[F] J. Fink. Perfect matchings extend to Hamilton cycles in hypercubes. J. Comb. Theory, Ser. B, 97(6):1074-1076, 2007. download\n\n[K] G. Kreweras, Matchings and Hamiltonian cycles on hypercubes, Bull. Inst. Combin. Appl. 16 (1996), 87--91.\n\nSource links:\n- matching: http://en.wikipedia.org/wiki/matching\n- hypercube: http://en.wikipedia.org/wiki/hypercube\n- Hamiltonian cycle: http://en.wikipedia.org/wiki/Hamiltonian path\n\nDiscussion links:\n- perfect matching: http://en.wikipedia.org/wiki/matching\n\nBibliography links:\n- download: http://www4.ncsu.edu/%7Esavage/AVAILABLE_FOR_MAILING/transposition_matching.ps\n- download: http://kam.mff.cuni.cz/%7Efink/publications/kreweras1.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Matchings extend to Hamiltonian cycles in hypercubes\" in Graph Theory; Basic Graph Theory; Matchings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Ruskey-Savage conjecture remains open for arbitrary matchings of Q_n, while long-cycle and several Hamiltonian-extension regimes are now proved.\n\n**Verified partial progress.**\n\n- Fink and Mütze prove that every matching extends to a cycle visiting at least two thirds of all hypercube vertices.\n- Wang and Zhang prove Hamiltonian extension for every matching with at most 2n-1 edges.\n- Fink and Hotmar prove Hamiltonian extension when the matching uses at most five coordinate directions; the perfect-matching case was already known.\n\n**Full solution or refutation.**\n\nNo Hamiltonian-extension theorem for every matching of every Q_n was located.\n\n**What remains.**\n\nUpgrade the universal two-thirds cycle to a spanning cycle for unrestricted matchings, or find a counterexample.\n\n**Sources checked.**\n\n- Jiří Fink and Torsten Mütze, Matchings in hypercubes extend to long cycles, arXiv:2401.01769 (2024). (primary): https://arxiv.org/abs/2401.01769\n Evidence used: States the Ruskey-Savage conjecture and proves the universal two-thirds long-cycle theorem.\n- Fan Wang and Heping Zhang, Prescribed matchings extend to Hamiltonian cycles in hypercubes with faulty edges, arXiv:1301.2931 (2013). (primary): https://arxiv.org/abs/1301.2931\n Evidence used: Proves Hamiltonian extension for matchings of size at most 2n-1, with additional faulty-edge robustness.\n- Jiří Fink and Vojtěch Hotmar, Matchings of five directions in hypercube extend to Hamilton cycles and paths with prescribed ends, arXiv:2501.19029 (2025). (primary): https://arxiv.org/abs/2501.19029\n Evidence used: Proves the conjecture for matchings whose edges span at most five coordinate directions.\n\n**Review notes.** The stored wording omits Q_n and n>=2; the literature reconstruction as the Ruskey-Savage conjecture is unambiguous.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3182, "problem_number": "OPG-743", "title": "Random stable roommates", "statement": "Conjecture The probability that a random instance of the stable roommates problem on $n \\in 2{\\mathbb N}$ people admits a solution is $\\Theta( n ^{-1/4} )$.", "background": "Source: Open Problem Garden. Original node ID: 743. URL: http://www.openproblemgarden.org/op/random_stable_roommates.\n\nSource subject path: Graph Theory > Basic Graph Theory > Matchings.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/random_stable_roommates\n- Author(s): Mertens, Stephan\n- Subject(s): Graph Theory; Basic Graph Theory; Matchings\n- Keywords: stable marriage; stable roommates\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 26th, 2008 by mdevos\n\nProblem-page discussion:\nA system of preferences for a graph $G$ is a family $\\{ >_v \\}_{v \\in V(G)}$ so that every $>_v$ is a linear ordering of the neighbors of the vertex $v$. We say that $v$ prefers $u$ to $u'$ if $u >_v u'$. A perfect matching $M$ in $G$ is stable if there do not exist $uv,u'v' \\in M$ so that $u$ prefers $v'$ to $v$ and $v'$ prefers $u$ to $u'$.\n\nA famous theorem of Gale-Shapley [GS] proves that every system of preferences on a complete bipartite graph $K_{n,n}$ admits a stable perfect matching. Indeed, they provide an amusing algorithm to construct one. On complete graphs, this problem is known as either the homosexual stable marriage problem, or more commonly, the stable roommate problem. Here there does not always exist a solution (that is, a stable perfect matching), but Irving [I] constructed an algorithm which runs in polynomial time, and outputs a solution if one exists.\n\nLet $P_n$ denote the probability that a random instance of the stable roommates problem has a solution (so the above conjecture asserts that $P_n = \\Theta( n^{-1/4}$ ). The following are the best known asymptotic bounds for $P_n$ (with $n$ even) and hold for $n$ sufficiently large. The lower bound is due to Pittel [P] and the upper bound to Pittel and Irving [IP]\n\n$$\n\\frac{2 e ^{3/2} }{ \\sqrt{\\pi n}} \\le P_n \\le \\frac{\\sqrt{e}}{2}\n$$\n\nMertens [M] did an extensive Monte-Carlo simulation to obtain the above conjecture. Indeed, by guessing at the constant he even offers the stronger conjecture $P_n \\simeq e \\sqrt{ \\frac{2}{\\pi} } n ^{-1/4}$.\n\nBibliography:\n[GS] D. Gale D and L. S. Shapley, College admissions and the stability of marriage, Am. Math. Mon. 69 9-15.\n\n[I] R. W. Irving, An efficient algorithm for the stable roommates problem, J. Algorithms 6 577-95.\n\n[IP] B. Pittel and R. W. Irving, An upper bound for the solvability of a random stable roommates instance, Random Struct. Algorithms 5 465-87.\n\n*[M] S. Mertens, Random stable matchings, J. Stat. Mech. Theory Exp. 2005, no. 10 MathSciNet\n\n[P] B. Pittel, The 'stable roommates' problem with random preferences, Ann. Probab. 21 1441-77\n\nBibliography links:\n- Random stable matchings: http://www.iop.org/EJ/article/1742-5468/2005/10/P10008/jstat5_10_p10008.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2185394\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Random stable roommates\" in Graph Theory; Basic Graph Theory; Matchings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Rigorous asymptotic work on random stable partitions and bounds exists, but the conjectured Theta(n^{-1/4}) solvability probability is not verified.\n\n**Verified partial progress.**\n\n- Pittel analyzes random stable partitions and derives asymptotic information about stable matchings.\n- Published and surveyed bounds leave a gap around exponent 1/4.\n\n**Full solution or refutation.**\n\nNo proof of the proposed two-sided Theta(n^{-1/4}) asymptotic was verified.\n\n**What remains.**\n\nProve matching upper and lower bounds at exponent 1/4, or identify a different asymptotic regime.\n\n**Sources checked.**\n\n- B. Pittel, On random stable partitions, arXiv:1705.08340 (2017). (primary): https://arxiv.org/abs/1705.08340\n Evidence used: Studies random stable partitions and relevant stable-roommates asymptotics.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3183, "problem_number": "OPG-153", "title": "Highly connected graphs with no K_n minor", "statement": "Problem Is it true for all $n \\ge 0$, that every sufficiently large $n$-connected graph without a $K_n$ minor has a set of $n-5$ vertices whose deletion results in a planar graph?", "background": "Source: Open Problem Garden. Original node ID: 153. URL: http://www.openproblemgarden.org/op/high_connectivity_no_k_n.\n\nSource subject path: Graph Theory > Basic Graph Theory > Minors.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/high_connectivity_no_k_n\n- Author(s): Thomas, Robin\n- Subject(s): Graph Theory; Basic Graph Theory; Minors\n- Keywords: connectivity; minor\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 10th, 2007 by mdevos\n\nProblem-page discussion:\nA famous conjecture of Jorgensen asserts that every 6-connected graph without a $K_6$-minor is apex (planar plus one vertex). If true, Jorgensen's conjecture does not generalize (naively) to higher connectivities, since for sufficiently large $n$, there do exist $n$-connected graphs which are not close to planar in the sense we are considering (many more than $n-5$ vertices must be deleted to leave a planar graph). This conjecture of Thomas asserts that all such graphs are small in size.\n\nFor $n \\le 6$ this conjecture is true. For $n \\le 4$ this conjecture is trivial, since any graph without a $K_4$-minor is planar. The $n=5$ case follows from a theorem of Wagner which gives a construction for all graphs without $K_5$-minors (and from which it follows that every 4-connected graph with no $K_5$ minor is planar). The $n=6$ case was recently resolved by DeVos, Hegde, Kawarabayashi, Norine, Thomas, and Wollan. The difficulties associated with finding $K_n$ minors in graphs make this conjecture appear daunting, but if true, it would yield powerful insight into the structure of graphs.\n\nSource links:\n- connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- $K_n$: http://en.wikipedia.org/wiki/complete graph\n- minor: http://en.wikipedia.org/wiki/minor (graph theory)\n- planar graph: http://en.wikipedia.org/wiki/planar graph\n\nDiscussion links:\n- conjecture of Jorgensen: http://www.openproblemgarden.org/?q=op/jorgensens_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Highly connected graphs with no K_n minor\" in Graph Theory; Basic Graph Theory; Minors, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The n=6 sufficiently-large case is proved, while no all-n theorem for n>=7 was verified; the literal n>=0 formulation is undefined below n=5.\n\n**Verified partial progress.**\n\n- Every sufficiently large 6-connected K_6-minor-free graph is apex.\n- Classical structure theorems cover the intended smaller cases.\n\n**Full solution or refutation.**\n\nThe general high-connectivity assertion remains open after correcting the intended parameter range.\n\n**What remains.**\n\nProve the (n-5)-apex conclusion for all fixed n>=7 and sufficiently large order.\n\n**Sources checked.**\n\n- K. Kawarabayashi, S. Norine, R. Thomas and P. Wollan, K_6 minors in large 6-connected graphs, Journal of Combinatorial Theory Series B 129 (2018), 158-203. (primary): https://arxiv.org/abs/1203.2192\n Evidence used: Proves the n=6 sufficiently-large case.\n- Open Problem Garden, Highly connected graphs with no K_n minor (node 153). (maintained_tracker): https://garden.irmacs.sfu.ca/op/high_connectivity_no_k_n\n Evidence used: Provides the exact statement and intended low-n discussion.\n\n**Review notes.** Formulation defect preserved: for n<5, a set of n-5 vertices has negative cardinality, so the statement as written is undefined.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3184, "problem_number": "OPG-154", "title": "Jorgensen's Conjecture", "statement": "Conjecture Every 6-connected graph without a $K_6$ minor is apex (planar plus one vertex).", "background": "Source: Open Problem Garden. Original node ID: 154. URL: http://www.openproblemgarden.org/op/jorgensens_conjecture.\n\nSource subject path: Graph Theory > Basic Graph Theory > Minors.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/jorgensens_conjecture\n- Author(s): Jorgensen, Leif K.\n- Subject(s): Graph Theory; Basic Graph Theory; Minors\n- Keywords: connectivity; minor\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 10th, 2007 by mdevos\n\nProblem-page discussion:\nFor $n \\le 5$, the class of graphs with no $K_n$ minor is very well understood. Simple graphs without $K_3$ minors are forests. Graphs without $K_4$ minors are called series-parallel graphs, and have a simple construction. Finally, Wagner [W] obtained a construction for all graphs without $K_5$ minors. For $n \\ge 6$, an explicit characterization of those graphs without $K_n$ minors appears hopeless. The graph minors project of Robertson and Seymour give a rough structure theorem for such classes, but much remains unknown. In particular, this conjecture and Thomas' conjecture highly connected graphs with no $K_n$-minor suggest interesting properties of highly connected graphs without $K_n$ minors which appear quite difficult to resolve.\n\nPart of the interest in graphs without $K_n$ minors stems from Hadwiger's conjecture (every loopless graph without a $K_{n+1}$ minor is $n$-colorable). Indeed, Wagner's work on graphs with no $K_5$ minor was done while studying the $n=4$ case of Hadwiger. More recently, Robertson, Seymour, and Thomas [RST] proved Hadwiger's conjecture for $n=5$, and in doing so came somewhat close to proving Jorgensoen's conjecture. The thrust of their argument is to prove that any minimal counterexample to Hadwiger for $n=5$ is apex. However, in doing so, they exploit both connectivity and coloring properties of a minimal counterexample. It would appear to be difficult to modify their argument to prove Jorgensen's conjecture.\n\nDeVos, Hegde, Kawarabayashi, Norine, Thomas, and Wollan proved this conjecture true for all sufficiently large graphs [KNTWa,KNTWb].\n\nBibliography:\n[RST] N. Robertson, P. D. Seymour, R. Thomas, Hadwiger's conjecture for $K\\sb 6$-free graphs. Combinatorica 13 (1993), no. 3, 279-361. MathSciNet\n\n[W] K. Wagner Uber eine Eigenschaft der ebenen Komplexe, Math. Ann 114 (1937) 570-590. MathSciNet\n\n[KNTWa] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. $K_6$ minors in 6-connected graphs of bounded tree-width. J. Combinatorial Theory, Series B, 136:1--32, 2019\n\n[KNTWb] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. $K_6$ minors in large 6-connected graphs. J. Combinatorial Theory, Series B, 129:158-203, 2019.\n\nSource links:\n- connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- $K_6$: http://en.wikipedia.org/wiki/complete graph\n- minor: http://en.wikipedia.org/wiki/minor (graph theory)\n\nDiscussion links:\n- series-parallel graphs: http://en.wikipedia.org/wiki/series-parallel graph\n- highly connected graphs with no $K_n$-minor: http://www.openproblemgarden.org/?q=op/high_connectivity_no_k_n\n\nBibliography links:\n- Hadwiger's conjecture for $K\\sb 6$-free graphs: http://www.math.gatech.edu/%7Ethomas/PAP/hadwiger.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1238823\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1513158\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Jorgensen's Conjecture\" in Graph Theory; Basic Graph Theory; Minors, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Jorgensen's conjecture is proved for all sufficiently large graphs and for several structural subclasses, but remains open without a size or subclass restriction.\n\n**Verified partial progress.**\n\n- Kawarabayashi-Norine-Thomas-Wollan prove every sufficiently large 6-connected K_6-minor-free graph is apex.\n- Aigner-Horev-Krakovski settle the conjecture for girth at least six.\n\n**Full solution or refutation.**\n\nOnly a finite-order exceptional range remains, but no full theorem was verified.\n\n**What remains.**\n\nEliminate all remaining bounded-order counterexamples to the apex conclusion.\n\n**Sources checked.**\n\n- K. Kawarabayashi, S. Norine, R. Thomas and P. Wollan, K_6 minors in large 6-connected graphs, JCTB 129 (2018), 158-203. (primary): https://arxiv.org/abs/1203.2192\n Evidence used: Proves the conjecture for sufficiently large graphs.\n- E. Aigner-Horev and R. Krakovski, Extremal results regarding K_6-minors in graphs of girth at least 5, Journal of Combinatorics 2 (2011), 463-479. (primary): https://arxiv.org/abs/1012.5795\n Evidence used: Proves the conjecture for girth at least six.\n- Open Problem Garden, Jorgensen's Conjecture (node 154). (maintained_tracker): https://www.openproblemgarden.org/op/jorgensens_conjecture\n Evidence used: Maintains the unrestricted conjecture and records large-graph progress.\n\n**Review notes.** The source misspells Jorgensen once as Jorgensoen in discussion only.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3185, "problem_number": "OPG-729", "title": "Seagull problem", "statement": "Conjecture Every $n$ vertex graph with no independent set of size $3$ has a complete graph on $\\ge \\frac{n}{2}$ vertices as a minor.", "background": "Source: Open Problem Garden. Original node ID: 729. URL: http://www.openproblemgarden.org/op/seagull_problem.\n\nSource subject path: Graph Theory > Basic Graph Theory > Minors.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/seagull_problem\n- Author(s): Seymour, Paul D.\n- Subject(s): Graph Theory; Basic Graph Theory; Minors\n- Keywords: coloring; complete graph; minor\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: January 6th, 2008 by mdevos\n\nProblem-page discussion:\nThis conjecture is significant because it is an interesting unproved consequence of Hadwiger's conjecture (this implication is proved next). In fact, some experts have suggested that this problem might hold the key to finding a counterexample to Hadwiger. I (M. DeVos) have attributed this conjecture to Seymour, but I believe that it may have been independently suggested by Mader and by others. Its curious title will be explained later in this discussion.\n\nHadwiger's conjecture (every loopless graph with chromatic number $\\ge n$ has $K_n$ as a minor) is one of the outstanding problems in graph theory. This conjecture has been resolved for small values of $n$; when $n \\le 4$ it is relatively easy, for $n=5,6$ it has been proven to be equivalent to the Four color theorem. The Seagull problem concerns the other extreme - when the size of the chromatic number is close to the order of the graph. If $G$ is an $n$ vertex graph with no independent set of size 3, then $\\chi(G) \\ge \\frac{n}{2}$ since each color class has size $\\le 2$. If Hadwiger's conjecture holds for $G$, it must then have a minor which is a complete graph on $\\ge \\frac{n}{2}$ vertices. This is precisely the statement of the Seagull problem.\n\nThe (essentially) best known bound for the conjecture is that every $n$ vertex graph $G$ with no independent set of size 3 has a complete graph on $\\ge \\frac{n}{3}$ vertices as a minor. This argument is where the name of this conjecture arises. Let us call a seagull of $G$ an induced subgraph which is a 2-edge path (such a subgraph may be drawn suggestively as a seagull). Then, for every seagull $S$ and every vertex $v$ not in $S$, there must be an edge between $v$ and one of the two endpoints of $S$ (this follows from the assumption that $G$ has no independent set of size 3). This feature makes seagulls especially useful for constructing complete graphs as minors - as we now demonstrate. Choose a maximal collection ${\\mathcal S}$ of pairwise disjoint seagulls of $G$. The graph $G'$ obtained from $G$ by deleting every vertex which appears in a seagull in ${\\mathcal S}$ cannot have any two vertices at distance 2 from one another (since this would yield a seagull), so $G'$ must be a disjoint union of complete graphs. Since $G'$ has no independent set of size 3, it is in fact a disjoint union of at most two complete graphs. By deleting every vertex in the smaller complete subgraph of $G'$ from $G$ and then contracting both edges in every seagull in ${\\mathcal S}$, we obtain a complete graph minor of $G$ with size $\\ge \\frac{n}{3}$.\n\nKawarabayashi, Plummer, and Toft have improved this bound slightly by showing that $G$ must have a complete graph minor of size $\\ge \\frac{n + \\omega(G)}{3}$, but it looks very difficult to get much more out of this type of argument.\n\nBibliography:\n[KPT] K. Kawarabayashi, M. Plummer, and B. Toft, Improvements of the theorem of Duchet and Meynial's theorem on Hadwiger's Conjecture, J. Combin. Theory Ser. B. 95 (2005) 152-167.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Seagull problem\" in Graph Theory; Basic Graph Theory; Minors, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The complete-minor assertion remains open, but exact seagull-packing structure and dense-minor results have substantially advanced the alpha(G)<=2 setting.\n\n**Verified partial progress.**\n\n- Chudnovsky--Seymour characterize k disjoint seagulls in this graph class.\n- Norin--Seymour construct a dense minor with the conjectured number of vertices.\n\n**Full solution or refutation.**\n\nNo general K_{ceil(n/2)} minor theorem was verified.\n\n**What remains.**\n\nUpgrade dense or specially structured minors to a complete minor, or find a counterexample.\n\n**Sources checked.**\n\n- M. Chudnovsky and P. Seymour, Packing seagulls, Combinatorica 32 (2012), 251--282. (primary): https://doi.org/10.1007/s00493-012-2594-2\n Evidence used: Establishes the relevant seagull-packing theorem.\n- S. Norin and P. Seymour, Dense minors of graphs with independence number two. (primary): https://web.math.princeton.edu/~pds/papers/densehad/paper.pdf\n Evidence used: Gives a dense minor on n/2 branch sets, not a complete minor.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3186, "problem_number": "OPG-37379", "title": "Forcing a $K_6$-minor", "statement": "Conjecture Every graph with minimum degree at least 7 contains a $K_6$-minor.\n\nConjecture Every 7-connected graph contains a $K_6$-minor.", "background": "Source: Open Problem Garden. Original node ID: 37379. URL: http://www.openproblemgarden.org/op/forcing_a_k_6_minor.\n\nSource subject path: Graph Theory > Basic Graph Theory > Minors.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/forcing_a_k_6_minor\n- Author(s): Barát,János; Joret, Gwenaël; Wood, David R.\n- Subject(s): Graph Theory; Basic Graph Theory; Minors\n- Keywords: connectivity; graph minors\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: January 16th, 2012 by David Wood\n\nProblem-page discussion:\nThe first conjecture implies the second.\n\nWhether the second conjecture is true was first asked in [KT]. Both conjectures were stated in [BJW].\n\nThe second conjecture is implied by Jørgensen’s conjecture, which asserts that every $6$-connected $K_6$-minor-free graph is apex (which have minimum degree at most $6$ and are thus not $7$-connected). Since Jørgensen’s conjecture is true for sufficiently large graphs [KNTWa,KNTWb], the second conjecture is true for sufficiently large graphs.\n\nBibliography:\n*[BJW] János Barát, Gwenaël Joret, David R. Wood. Disproof of the List Hadwiger Conjecture, Electronic J. Combinatorics 18:P232, 2011.\n\n*[KT] Ken-ichi Kawarabayashi and Bjarne Toft. Any 7-chromatic graph has $K_7$ or $K_{4,4}$ as a minor. Combinatorica 25 (3), 327–353, 2005.\n\n[KNTWa] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. $K_6$ minors in $6$-connected graphs of bounded tree-width. http://arxiv.org/abs/1203.2171\n\n[KNTWb] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. $K_6$ minors in large $6$-connected graphs. http://arxiv.org/abs/1203.2192\n\nRelated:\nRelated problems\nJorgensen's Conjecture\n\nDiscussion links:\n- Jørgensen’s conjecture: http://www.openproblemgarden.org/?q=node/154\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Forcing a $K_6$-minor\" in Graph Theory; Basic Graph Theory; Minors, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Both the minimum-degree-seven and 7-connected K6-minor conjectures remain open, with strong structural reductions and many special cases known.\n\n**Verified partial progress.**\n\n- The 7-connected claim follows from Jørgensen's stronger apex conjecture if that conjecture is proved.\n- Bipartite and other structured high-degree cases have K6-minor theorems.\n\n**Full solution or refutation.**\n\nNo general K6-minor forcing theorem matching either source statement was verified.\n\n**What remains.**\n\nProve either conjecture or give a counterexample.\n\n**Sources checked.**\n\n- Graph-theory open problems, Forcing a K6-minor (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/forcing_a_k_6_minor/\n Evidence used: The maintained entry explicitly records both stated conjectures as open and summarizes reductions.\n- P. D. Seymour, Bipartite graphs with no K6 minor, Journal of Combinatorial Theory B 118 (2016), 214--236. (primary): https://web.math.princeton.edu/~pds/papers/bipK6/paper.pdf\n Evidence used: The paper proves a substantive structured high-degree K6-minor theorem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3187, "problem_number": "OPG-59911", "title": "Forcing a 2-regular minor", "statement": "Conjecture Every graph with average degree at least $\\frac{4}{3}t-2$ contains every 2-regular graph on $t$ vertices as a minor.", "background": "Source: Open Problem Garden. Original node ID: 59911. URL: http://www.openproblemgarden.org/op/forcing_a_2_regular_minor.\n\nSource subject path: Graph Theory > Basic Graph Theory > Minors.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/forcing_a_2_regular_minor\n- Author(s): Reed, Bruce A.; Wood, David R.\n- Subject(s): Graph Theory; Basic Graph Theory; Minors\n- Keywords: minors\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 16th, 2014 by David Wood\n\nProblem-page discussion:\nReed and Wood [RW] explained that a result of Corradi and Hajnal [CH] implies that if $H$ is the graph consisting of $k$ disjoint triangles, then every graph with average degree at least $4k-2$ contains $H$ as a minor. Moreover, the bound of $4k-2$ is best possible since the complete bipartite graph $K_{2k-1,n}$ contains no $H$-minor, but has average degree tending to $4k-2$ (as $n\\rightarrow\\infty$ ). Thus the conjecture would generalise this result.\n\nUpdate: There has been a lot of recent progress on this conjecture [HW,CNLWY].\n\nBibliography:\n[CH] Keresztely Corradi and Andras Hajnal. On the maximal number of independent circuits of a graph. Acta Math. Acad. Sci. Hungar., 14:423–443, 1963.\n\n*[RW] Bruce Reed and David R. Wood. Forcing a sparse minor, arXiv:1402.0272, 2013.\n\n[HW] Daniel J. Harvey and David R. Wood. Cycles of given size in a dense graph. SIAM J. Discrete Math. 29.4:2336–2349, 2015.\n\n[CNLWY] E. Csóka, S. Norin, I. Lo, H. Wu and L. Yepremyan. The extremal function for disconnected minors. J. Comb. Theory B 126 (2017), 162-174.\n\nBibliography links:\n- 1402.0272: http://www.arxiv.org/abs/1402.0272\n- Cycles of given size in a dense graph: http://dx.doi.org/10.1137/15M100852X\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Forcing a 2-regular minor\" in Graph Theory; Basic Graph Theory; Minors, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Csoka--Lo--Norin--Wu--Yepremyan prove that average degree at least 4t/3-2 forces every 2-regular t-vertex graph as a minor.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThis is exactly the quantified assertion in the source record.\n\n**What remains.**\n\nThe theorem is sharp; no further work is needed to decide the source conjecture.\n\n**Sources checked.**\n\n- E. Csoka, I. Lo, S. Norin, H. Wu and L. Yepremyan, The extremal function for disconnected minors, Journal of Combinatorial Theory, Series B 126 (2017), 162--174; arXiv:1509.01185. (primary): https://arxiv.org/abs/1509.01185\n Evidence used: Theorem 1 proves the exact 4t/3-2 bound for every 2-regular t-vertex minor.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3188, "problem_number": "OPG-46583", "title": "Decomposing a connected graph into paths.", "statement": "Conjecture Every simple connected graph on $n$ vertices can be decomposed into at most $\\frac{1}{2}(n+1)$ paths.", "background": "Source: Open Problem Garden. Original node ID: 46583. URL: http://www.openproblemgarden.org/op/decomposing_a_connected_graph_into_paths.\n\nSource subject path: Graph Theory > Basic Graph Theory > Paths.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_a_connected_graph_into_paths\n- Author(s): Gallai, Tibor\n- Subject(s): Graph Theory; Basic Graph Theory; Paths\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is tight because a complete graph on $n$ vertices cannot be covered by less than $(n+1)/2$ cycles.\n\nThere is a similar conjecture about decomposition of an eulerian graph into cycles.\n\nBibliography:\n* [L] L. Lovász, On covering of graphs. In Theory of Graphs (Proc. Colloq., Tihany, 1966), 231--236. Academic Press, New York, 1968.\n\nRelated:\nRelated problems\nDecomposing an eulerian graph into cycles.\n\nDiscussion links:\n- decomposition of an eulerian graph into cycles: http://www.openproblemgarden.org/?q=node/46584\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing a connected graph into paths.\" in Graph Theory; Basic Graph Theory; Paths, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Gallai's path decomposition conjecture remains open for general connected graphs. The best general upper bound is floor(2n/3), while the conjecture is proved for all connected planar graphs and several other broad classes.\n\n**Verified partial progress.**\n\n- Dean--Kouider and independently Yan proved that every n-vertex graph can be decomposed into at most floor(2n/3) paths.\n- Blanché--Bonamy--Bonichon prove the conjecture for every connected planar graph, with floor(n/2) paths except for K3 and K5 minus one edge.\n- The conjecture is also known for maximum degree at most five and for bounded-treewidth classes including treewidth at most four.\n\n**Full solution or refutation.**\n\nNo path-decomposition theorem attaining ceil(n/2) for every connected graph was verified.\n\n**What remains.**\n\nClose the general gap from floor(2n/3) to ceil(n/2), beyond the many solved structural classes.\n\n**Sources checked.**\n\n- Alexandre Blanché, Marthe Bonamy, and Nicolas Bonichon, Gallai's path decomposition in planar graphs, arXiv:2110.08870. (primary): https://arxiv.org/abs/2110.08870\n Evidence used: Proves Gallai's conjecture for all connected planar graphs and states the stronger bound with two exceptions.\n- Yanan Chu and Yan Wang, Path decompositions of Eulerian graphs, Discrete Mathematics 349 (2026), 114830; arXiv:2510.12806. (primary): https://arxiv.org/abs/2510.12806\n Evidence used: A recent primary paper states the conjecture remains open, records the best general 2n/3 bound, and adds an Eulerian special case.\n- Matija Bucić and Richard Montgomery, Towards the Erdős--Gallai Cycle Decomposition Conjecture, Advances in Mathematics 437 (2024), 109409; arXiv:2211.07689. (primary): https://arxiv.org/abs/2211.07689\n Evidence used: Surveys the current general path bound and identifies connected planar graphs as the latest major exact class.\n\n**Review notes.** Because a path count is integral, the input's at most (n+1)/2 convention is equivalent to at most floor((n+1)/2)=ceil(n/2).\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3189, "problem_number": "OPG-46613", "title": "Partition of a cubic 3-connected graphs into paths of length 2.", "statement": "Problem Does every $3$-connected cubic graph on $3k$ vertices admit a partition into $k$ paths of length $2$?", "background": "Source: Open Problem Garden. Original node ID: 46613. URL: http://www.openproblemgarden.org/op/partition_of_a_cubic_3_connected_graphs_into_paths_of_length_2.\n\nSource subject path: Graph Theory > Basic Graph Theory > Paths.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partition_of_a_cubic_3_connected_graphs_into_paths_of_length_2\n- Author(s): Kelmans, Alexander K.\n- Subject(s): Graph Theory; Basic Graph Theory; Paths\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nMore generally, the following question is posed.\n\nProblem Does every $3$-connected cubic graph on at least $3k$ vertices contain $k$ pairwise vertex-disjoint paths of length $2$?\n\nIn [K1], Kelmans gave a construction that provided infi nitely many 2-connected graphs for which the above statement is false.\n\nBibliography:\n[K1] Alexander K. Kelmans, Packing 3-vertex paths in 2-connected graphs\n\n*[K2] Alexander K. Kelmans, On $\\Lambda$--Packing in 3--connected Graphs, RUTCOR Research Report 23--2005, Rutgers University. See also Packing 3-vertex Paths In Cubic 3-connected Graphs\n\nBibliography links:\n- Packing 3-vertex paths in 2-connected graphs: http://www.arxiv.org/abs/0712.4151\n- Packing 3-vertex Paths In Cubic 3-connected Graphs: http://www.arxiv.org/abs/0910.2766v2\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Partition of a cubic 3-connected graphs into paths of length 2.\" in Graph Theory; Basic Graph Theory; Paths, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The conjecture that every cubic 3-connected graph of order divisible by three has a P3-factor remains open.\n\n**Verified partial progress.**\n\n- Kelmans proved equivalence to several stronger-looking edge-avoidance and deletion formulations.\n- Cyclically 6-connected constructions show that natural two-P3 deletion strengthenings fail.\n\n**Full solution or refutation.**\n\nNo proof or counterexample to the exact P3-factor assertion was verified; the strongest primary source supplies equivalences and sharp obstructions to strengthenings.\n\n**What remains.**\n\nProve a P3-factor exists in every cubic 3-connected graph whose order is divisible by three, or construct a counterexample.\n\n**Sources checked.**\n\n- A. Kelmans, Packing 3-vertex Paths In Cubic 3-connected Graphs, arXiv:0910.2766; accepted in Discrete Mathematics. (primary): https://arxiv.org/abs/0910.2766\n Evidence used: Defines Lambda-factors, states the conjecture, proves equivalent formulations, and gives limitations to stronger variants.\n- Graph-theory open problems, Partition of a cubic 3-connected graphs into paths of length 2 (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/partition_of_a_cubic_3_connected_graphs_into_paths_of_length_2/\n Evidence used: Reports the exact conjecture open after searches through 2026.\n\n**Review notes.** The source statement is clear. A grammatical error occurs only in the source title.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3190, "problem_number": "OPG-170", "title": "Linial-Berge path partition duality", "statement": "Conjecture The minimum $k$-norm of a path partition on a directed graph $D$ is no more than the maximal size of an induced $k$-colorable subgraph.", "background": "Source: Open Problem Garden. Original node ID: 170. URL: http://www.openproblemgarden.org/op/linial_berge_path_partition_duality.\n\nSource subject path: Graph Theory > Coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/linial_berge_path_partition_duality\n- Author(s): Berge, Claude; Linial, Nathan\n- Subject(s): Graph Theory; Coloring\n- Keywords: coloring; directed path; partition\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 27th, 2007 by berger\n\nProblem-page discussion:\nDefinitions: Let $D$ be a directed graph. A path partition of $D$ is a set of vertex disjoint paths in it (some might be singletons), covering all vertices. Let $k$ be a positive integer. The $k$ norm of a path partition is the sum of $\\min\\{|V(P)|,k\\}$ for all paths $P$ in it.\n\nThis conjecture is known for acyclic graphs and for $k =1,2$.\n\nComments:\n- September 19th, 2007 | Anonymous | Berge-Linial conjecture: There is a typo in the formulation:\n\nReplace \"maximum k-norm\" by \"minimum k-norm\".\n\nThanks! Best, Andras Sebo\n- September 19th, 2007 | mdevos | Thanks Andras!: Thanks much for the correction, I've updated the problem.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Linial-Berge path partition duality\" in Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Linial's weak path-partition conjecture remains open for general digraphs and k, with proofs for acyclic digraphs, small k, spine classes, and ranges near the longest-path length.\n\n**Verified partial progress.**\n\n- The conjecture holds for acyclic digraphs and for k=1,2.\n- It holds for arc-spine digraphs.\n- Berge's stronger orthogonality conjecture is proved for k at least lambda-3, where lambda is the maximum directed-path order.\n\n**Full solution or refutation.**\n\nThe arc-spine theorem verifies Linial's exact inequality for a nontrivial structural class, while results on Berge's stronger conjecture imply additional parameter ranges.\n\n**What remains.**\n\nProve pi_k(D) <= alpha_k(D) for every finite digraph D and positive integer k, or find a counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, Linial-Berge path partition duality (OPG-170), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/linial_berge_path_partition_duality\n Evidence used: States the exact weak path-partition inequality and definitions.\n- L. R. Yoshimura, M. Sambinelli, C. N. da Silva, and O. Lee, Linial's Conjecture for Arc-spine Digraphs, Electronic Notes in Theoretical Computer Science 346 (2019), 735-746. (primary): https://doi.org/10.1016/j.entcs.2019.08.064\n Evidence used: Proves Linial's conjecture for arc-spine digraphs and records the general conjecture as open.\n- D. Herskovics, Proof of Berge's path partition conjecture for k>=lambda-3, Discrete Applied Mathematics 209 (2016), 137-143. (primary): https://doi.org/10.1016/j.dam.2015.07.039\n Evidence used: Proves the stronger Berge conjecture for k=lambda-2 and lambda-3, supplementing the previously known nearby cases.\n\n**Review notes.** The displayed statement depends on definitions in the source background; it was preserved without inserting those definitions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3191, "problem_number": "OPG-562", "title": "Three-chromatic (0,2)-graphs", "statement": "Question Are there any (0,2)-graphs with chromatic number exactly three?", "background": "Source: Open Problem Garden. Original node ID: 562. URL: http://www.openproblemgarden.org/op/three_chromatic_0_2_graphs.\n\nSource subject path: Graph Theory > Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/three_chromatic_0_2_graphs\n- Author(s): Payan, Charles\n- Subject(s): Graph Theory; Coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 7th, 2007 by Gordon Royle\n\nProblem-page discussion:\nA (0,2)-graph is a graph such that every pair of distinct vertices has either 0 or 2 common neighbours. It is fairly easy to see that a (0,2)-graph is necessarily regular and a variety of other properties can be shown to hold. Although (0,2)-graphs with chromatic number 2, 4 and 5 are known it is open as to whether there can be any (0,2)-graphs with chromatic number exactly three.\n\nBibliography:\n*[P] Payan, Charles: On the chromatic number of cube-like graphs, Discrete Math. 103 (1992), no. 3, 271--277.\n\nComments:\n- February 6th, 2013 | Anonymous | Finite Three-Chromatic (0,2)-graphs: An infinite three-chromatic (0, 2)-graph is easy to construct. See Payan.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Three-chromatic (0,2)-graphs\" in Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No (0,2)-graph of chromatic number three, nor exclusion proof, was verified.\n\n**Verified partial progress.**\n\n- The source supplies the exact formulation.\n\n**Full solution or refutation.**\n\nNo resolution was verified.\n\n**What remains.**\n\nConstruct or exclude such a graph.\n\n**Sources checked.**\n\n- Open Problem Garden, (0,2)-graphs, node 562. (maintained_tracker): http://www.openproblemgarden.org/op/0_2_graphs\n Evidence used: Preserves the source question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3192, "problem_number": "OPG-815", "title": "Total Colouring Conjecture", "statement": "Conjecture A total coloring of a graph $G = (V,E)$ is an assignment of colors to the vertices and the edges of $G$ such that every pair of adjacent vertices, every pair of adjacent edges and every vertex and incident edge pair, receive different colors. The total chromatic number of a graph $G$, $\\chi\"(G)$, equals the minimum number of colors needed in a total coloring of $G$. It is an old conjecture of Behzad that for every graph $G$, the total chromatic number equals the maximum degree of a vertex in $G$, $\\Delta(G)$ plus one or two. In other words,\n$$\n\\chi\"(G)=\\Delta(G)+1\\ \\ or \\ \\ \\Delta(G)+2.\n$$", "background": "Source: Open Problem Garden. Original node ID: 815. URL: http://www.openproblemgarden.org/op/behzads_conjecture.\n\nSource subject path: Graph Theory > Coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/behzads_conjecture\n- Author(s): Behzad, M.\n- Subject(s): Graph Theory; Coloring\n- Keywords: Total coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 4th, 2008 by Iradmusa\n\nProblem-page discussion:\nThe lower bound $\\Delta(G)+1$ is trivial by looking at the number of colours required on a vertex of maximum degree and its incident edges. It is easy to prove $\\chi\"(G)\\leq 2\\Delta(G)+2$. Molloy and Reed [MR] showed that there exists a constant $C$ such that $\\chi\"(G)\\leq Delta(G)+C$ for every graph $G$.\n\nThe Edge list coloring conjecture would imply that $\\chi\"(G)\\leq \\Delta(G)+3$.\n\nThe Total Colouring Conjecture was proved for $\\Delta(G)=3$ by Rosenfeld [R] and also by Vijayaditya [V], and for $\\Delta(G)\\in\\{4,5\\}$ by Kostochka [K1,K2,K3]; in fact the proof for $\\Delta(G)=5$ holds for multigraphs.\n\nThe Conjecture has also been established for many graph classes. For every planar graph G with $\\Delta(G) \\geq 7$, the following clever argument proves it. By the 4 Color Theorem, we can color the vertices with the colors 1, 2, 3, 4. By a result of Sanders and Zhao [SZ], we can color the edges of the graph with the colors $3, 4, \\ldots, \\Delta(G) + 1, \\Delta(G) + 2$. Uncolor each edge that was colored 3 or 4. Note that each uncolored edge has exactly two colors from $\\{1,2,3,4\\}$ forbidden. Hence, each uncolored edge has at least two colors available. Note that the uncolored edges induce a disjoint union of paths and even cycles. Thus, by a special case of a theorem of Erdos, Rubin, and Taylor [ERT], we can color the edges from their lists of two available colors each.\n\nBibliography:\n*[B] M. Behzad, Graphs and their chromatic numbers, Ph.D. Thesis, Michigan State University, 1965.\n\n[ERT] P. Erdos, A.L. Rubin, and H. Taylor, Choosability in graph, Cong. Numer. 26, 125-157, 1979.\n\n[K1] A.V. Kostochka, The total coloring of a multigraph with maximal degree 4. Discrete Math. 17, 161-163, 1977.\n\n[K2] A.V. Kostochka, Upper bounds of chromatic functions of graph (in Russian). Ph.D. Thesis, Novosibirsk, 1978.\n\n[K3] A.V. Kostochka, Exact upper bound for the total chromatic number of a graph (in Russian). In: Proc. 24th Int. Wiss. Koll., Tech Hochsch. Ilmenau, 1979, pages 33-36, 1979.\n\n[MR] M. Molloy and B.Reed. A bound on the total chromatic number. Combinatorica, 18(2), 241-280, 1998.\n\n[R] M. Rosenfeld, On the total coloring of certain graphs. Israel J. Math. 9, 396-402, 1971.\n\n[SZ] D.P. Sanders and Y. Zhao, Planar Graphs of Maximum Degree Seven are Class 1. J. Comb. Theory B. 83, 201-212, 2001.\n\n[V] N. Vijayaditya, On total chormatic number of a graph. J. London Math. Soc. (2) 3, 405-408, 1971.\n\nRelated:\nRelated problems\nEdge list coloring conjecture\nList Total Colouring Conjecture\n\nDiscussion links:\n- Edge list coloring conjecture: http://www.openproblemgarden.org/?q=node/2110\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Total Colouring Conjecture\" in Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Total Colouring Conjecture remains open in general, although it is proved for numerous degree ranges and graph classes.\n\n**Verified partial progress.**\n\n- The conjecture is known for subcubic graphs and several small maximum degrees.\n- A constant-additive general upper bound is known, but not the desired Delta+2 in full generality.\n\n**Full solution or refutation.**\n\nAn arXiv claim of a general proof was located but not accepted as verified literature; no general resolution was confirmed.\n\n**What remains.**\n\nProve chi''(G)<=Delta(G)+2 for all simple graphs or find a counterexample.\n\n**Sources checked.**\n\n- Total colorings--a survey, AKCE International Journal of Graphs and Combinatorics (2023). (authoritative_secondary): https://doi.org/10.1080/09728600.2023.2187960\n Evidence used: Survey presents the Delta+2 statement as the Total Coloring Conjecture and records partial class results.\n\n**Review notes.** No source alteration; unverified preprint claim not treated as a solution.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3193, "problem_number": "OPG-830", "title": "4-regular 4-chromatic graphs of high girth", "statement": "Problem Do there exist 4-regular 4-chromatic graphs of arbitrarily high girth?", "background": "Source: Open Problem Garden. Original node ID: 830. URL: http://www.openproblemgarden.org/op/high_girth_low_degree_4_chromatic_graphs.\n\nSource subject path: Graph Theory > Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/high_girth_low_degree_4_chromatic_graphs\n- Author(s): Grunbaum, Branko\n- Subject(s): Graph Theory; Coloring\n- Keywords: coloring; girth\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 18th, 2008 by mdevos\n\nProblem-page discussion:\nGrunbaum conjectured that for every $m$, there exist $m$-regular $m$-chromatic graphs of arbitrarily high girth. However, this was shown dramatically false by Johansson, who proved that every triangle free graph $G$ with maximum degree $\\Delta$ satisfies $\\chi(G) \\le C \\frac{\\Delta}{\\log \\Delta}$ for some fixed constant $C$. Neverless, some interesting smaller cases of Grunbaum's conjecture, such as the one highlighted above, might still be true.\n\nThere are only a few 4-regular 4-chromatic graphs of girth $\\ge 4$ which are known. These include the Chvatal graph, Brinkmann graph (discovered independently by Kostochka), and Grunbaum graph. To the best of my (M. DeVos') knowledge, this might be the full list of such graphs.\n\nThere do exist 4-chromatic graphs of minimum degree $\\le 6$ and arbitrarily high girth, but it is open wether there exist 4-chromatic graphs of minimum degree 5 and arbitrary girth.\n\nDiscussion links:\n- Chvatal graph: http://mathworld.wolfram.com/ChvatalGraph.html\n- Brinkmann graph: http://mathworld.wolfram.com/BrinkmannGraph.html\n- Grunbaum graph: http://mathworld.wolfram.com/GruenbaumGraphs.html\n\nComments:\n- January 29th, 2012 | Anonymous | The list of known graphs is not full: Your list of 4 regular 4 chromatic graphs with girth >= 4 is not complete. Check \"A note on 4-regular 4 chromatic graphs with girth 4\" http://math.nju.edu.cn/~zkmfl/kmzhang/41-60/Zhang44.pdf. Girth 5 is known too.\n- November 13th, 2008 | Anonymous | What about higher degree?: What maximum degree is needed in a triangle-free graph (or higher girth) with chromatic number 5? What about chromatic number 6 and 7? Are good bounds known on these maximum degrees?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"4-regular 4-chromatic graphs of high girth\" in Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No construction of 4-regular 4-chromatic graphs of arbitrarily high girth was verified.\n\n**Verified partial progress.**\n\n- There are 4-chromatic graphs of arbitrarily high girth without the exact regularity requirement.\n- Several finite 4-regular 4-chromatic examples of small girth are known.\n\n**Full solution or refutation.**\n\nThe combined degree-four, chromatic-four, arbitrary-girth request remains open.\n\n**What remains.**\n\nConstruct an unbounded-girth family or prove a structural obstruction.\n\n**Sources checked.**\n\n- Open Problem Garden, node 830 (accessed 2026-08-17). (maintained_tracker): https://www.unsolvedmath.com/problems/OPG-830\n Evidence used: Retains the exact problem and distinguishes the known non-regular high-girth results.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3194, "problem_number": "OPG-46533", "title": "Coloring the union of degenerate graphs", "statement": "Conjecture The union of a $1$-degenerate graph (a forest) and a $2$-degenerate graph is $5$-colourable.", "background": "Source: Open Problem Garden. Original node ID: 46533. URL: http://www.openproblemgarden.org/op/coloring_the_union_of_degenerate_graphs.\n\nSource subject path: Graph Theory > Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/coloring_the_union_of_degenerate_graphs\n- Author(s): Tarsi, Michael\n- Subject(s): Graph Theory; Coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 3rd, 2013 by fhavet\n\nProblem-page discussion:\nA graph is $k$-degenerate if it can be reduced to $K_1$ (the graph with a unique vertex) by repeatedly deleting vertices of degree at most $k$. A $1$-degenerate graph $G_1$ admits a proper $2$-colouring $c_1$, and a $2$-degenerate graph $G_2$ admits a proper $3$-colouring $c_2$. Thus, $(c_1,c_2)$ is a proper $6$-colouring of $G_1$ and $G_2$.\n\nThe conjecture is tigth because $K_5$ is the union of a $1$-degenerate graph and a $2$-degenerate graph.\n\nBased on a decompostion of the complete graph, Klein and Schönheim [KlSc93] generalised this conjecture to $(m_1, \\dots, m_s)$-composed graphs, which are unions of $s$ graphs $G_1, \\dots, G_s$ such that $G_i$ is $m_i$-degenerate, $1\\leq i\\leq s$.\n\nConjecture Every $(m_1, \\dots, m_s)$-composed graph is $\\left(\\sum_{i=1}^s m_i+\\bigg\\lfloor\\frac{1}{2}\\bigg(1+\\sqrt{1+8\\sum_{1\\leq i\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Tarsi's (1,2)-degenerate union colouring problem remains listed as open in the maintained British Combinatorial Conference problem list. The classical general theorem gives six colours, while the conjecture asks for five.\n\n**Verified partial progress.**\n\n- Klein's general m-composed-graph theorem gives a 2m+2 colour upper bound; for a forest plus a 2-degenerate graph this is six.\n- Klein--Schönheim identify Tarsi's case and record the conjectural sharp value five.\n\n**Full solution or refutation.**\n\nNo proof of five-colourability or 6-chromatic counterexample was located in the targeted literature search.\n\n**What remains.**\n\nEither improve the general six-colour bound to five for every (1,2)-composed graph or construct a 6-chromatic example.\n\n**Sources checked.**\n\n- R. Klein, On the colorability of m-composed graphs, Discrete Mathematics 133 (1994), 181-190. (primary): https://doi.org/10.1016/0012-365X(94)90025-6\n Evidence used: Proves the 2m+2 upper bound and identifies the sharper conjectural problem.\n- R. Klein and J. Schönheim, On the colorability of graphs decomposable into degenerate graphs with specified degeneracy, Australasian Journal of Combinatorics 12 (1995), 201-208. (primary): https://ajc.maths.uq.edu.au/pdf/12/ocr-ajc-v12-p201.pdf\n Evidence used: States that existence of a 6-chromatic (1,2)-composed graph was unknown and conjectures value five.\n- Peter J. Cameron, British Combinatorial Conference Problem List, Problem BCC15.5 (DM275), maintained compilation. (authoritative_secondary): https://maths.qmul.ac.uk/~pjc/bcc/allprobs.pdf\n Evidence used: Continues to present the generalization and explicitly identifies the (1,2) case as Tarsi's problem.\n\n**Review notes.** The open conclusion is conservative because the strongest direct primary sources are old; the maintained BCC list is the current-status evidence.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3195, "problem_number": "OPG-56449", "title": "List Total Colouring Conjecture", "statement": "Conjecture If $G$ is the total graph of a multigraph, then $\\chi_\\ell(G)=\\chi(G)$.", "background": "Source: Open Problem Garden. Original node ID: 56449. URL: http://www.openproblemgarden.org/op/list_total_colouring_conjecture.\n\nSource subject path: Graph Theory > Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/list_total_colouring_conjecture\n- Author(s): Borodin, Oleg V.; Kostochka, Alexandr V.; Woodall, Douglas R.\n- Subject(s): Graph Theory; Coloring\n- Keywords: list coloring; Total coloring; total graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 29th, 2013 by Jon Noel\n\nProblem-page discussion:\nThe list chromatic number of a graph $G$, denoted $\\chi_\\ell(G)$, is defined here. Given a multigraph $H$, the total graph $T(H)$ of $H$ is a graph on vertex set $V(T(H)):=V(H)\\cup E(H)$ where\n\n- two elements of $V(H)$ are adjacent in $T(H)$ if and only if they are adjacent in $H$;\n- two elements of $E(H)$ are adjacent in $T(H)$ if and only if they share an endpoint;\n- an element of $V(H)$ is adjacent to an element of $E(H)$ in $T(H)$ if it is incident with it.\n\nThis problem is related to the List (Edge) Colouring Conjecture as well as the Total Colouring Conjecture.\n\nKostochka and Woodall [KW] conjectured that $\\chi_\\ell(G^2)=\\chi(G^2)$ for every graph $G$; this was known as the List Square Colouring Conjecture. It is stronger than the List Total Colouring Conjecture since, given a multigraph $H$, the total graph of $H$ can be obtained by subdividing each edge of $H$ and taking the square. Moreover, the graph obtained from $H$ by subdividing each edge is bipartite and one part of the bipartition consists of vertices of degree $2$. Thus, the List Total Colouring Conjecture corresponds to this (very) special case of the List Square Colouring Conjecture.\n\nHowever, the List Square Colouring Conjecture is not true in general. For a family of counterexamples, see the paper of Kim and Park [KP].\n\nBibliography:\n*[BKW] O. V. Borodin, A. V. Kostochka, and D. R. Woodall. List edge and list totalcolourings of multigraphs. J. Combin. Theory Ser. B, 71(2):184–204, 1997.\n\n[KW] A. V. Kostochka and D. R. Woodall. Choosability conjectures and multicircuits. Discrete Math., 240(1-3):123–143, 2001.\n\n[KP] Seog-Jin Kim and Boram Park: Counterexamples to the List Square Coloring Conjecture, submitted.\n\nRelated:\nRelated problems\nEdge list coloring conjecture\nTotal Colouring Conjecture\nChoosability of Graph Powers\n\nDiscussion links:\n- here: http://en.wikipedia.org/wiki/List coloring\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"List Total Colouring Conjecture\" in Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The List Total Colouring Conjecture remains open for total graphs of arbitrary multigraphs.\n\n**Verified partial progress.**\n\n- List-total-colouring equality or sharp bounds are known for several restricted graph classes, including high-degree planar regimes.\n- Counterexamples to the stronger List Square Colouring Conjecture do not specialize to total graphs and hence do not refute this conjecture.\n\n**Full solution or refutation.**\n\nNo proof that every total graph is chromatic-choosable and no total-graph counterexample was verified.\n\n**What remains.**\n\nProve chi_l(T(H))=chi(T(H)) for every multigraph H or construct a multigraph whose total graph violates equality.\n\n**Sources checked.**\n\n- O. V. Borodin, A. V. Kostochka, and D. R. Woodall, List Edge and List Total Colourings of Multigraphs, Journal of Combinatorial Theory B 71 (1997), 184-204. (primary): https://doi.org/10.1006/jctb.1997.1780\n Evidence used: Develops list-total-colouring bounds and proves restricted-class results while formulating the broader programme.\n- Graph-theory open problems, List Total Colouring Conjecture (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/list_total_colouring_conjecture/\n Evidence used: Reports the exact conjecture open with high confidence after a post-2013 search.\n\n**Review notes.** No formulation defect identified; source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3196, "problem_number": "OPG-143", "title": "Petersen coloring conjecture", "statement": "Conjecture Let $G$ be a cubic graph with no bridge. Then there is a coloring of the edges of $G$ using the edges of the Petersen graph so that any three mutually adjacent edges of $G$ map to three mutually adjancent edges in the Petersen graph.", "background": "Source: Open Problem Garden. Original node ID: 143. URL: http://www.openproblemgarden.org/op/petersen_coloring_conjecture.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/petersen_coloring_conjecture\n- Author(s): Jaeger, Francois\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: cubic; edge-coloring; Petersen graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nThis extrordainary conjecture asserts that in a very strong sense, every bridgeless cubic graph has all of the cycle-space properties posessed by the Petersen graph. If true, this conjecture would imply both The Berge-Fulkerson conjecture and The five cycle double cover conjecture.\n\nIf $G$ is a graph and $C \\subseteq E(G)$ we say that $C$ is a binary cycle if every vertex in the graph $(V(G),C)$ has even degree. If $H$ is a graph and $f: E(G) \\rightarrow E(H)$ is a map, we say that $f$ is cycle-continuous if the pre-image of every binary cycle is a binary cycle. The following conjecture is an equivalent reformulation of the Petersen coloring conjecture.\n\nConjecture (Petersen coloring conjecture (2)) Every bridgeless graph has a cycle-continuous mapping to the Petersen graph.\n\nSource links:\n- cubic: http://en.wikipedia.org/wiki/cubic graph\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n- Petersen: http://en.wikipedia.org/wiki/petersen graph\n\nDiscussion links:\n- The Berge-Fulkerson conjecture: http://www.openproblemgarden.org/?q=op/the_berge_fulkerson_conjecture\n- The five cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/m_n_cycle_covers\n\nComments:\n- November 8th, 2011 | Anonymous | Question: For which bridgeless cubic graphs has this been checked for?\n- November 24th, 2011 | Robert Samal | Re:: It is trivially true for all that are 3-edge-colorable -- which is the vast majority. Among the rest, I checked it using computer and lists of snarks for all graphs upto 34 vertices. (And some more -- e.g. all flower-snarks.)\n\nBest wishes, Robert\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Petersen coloring conjecture\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Jaeger's Petersen colouring conjecture remains open. Sharp approximate normal-colouring results and several family-specific weaker normal six-colourings are known, but no primary source establishes a Petersen colouring for every bridgeless cubic graph.\n\n**Verified partial progress.**\n\n- Pirot, Sereni, and Skrekovski prove that every bridgeless cubic graph has a four-edge-colouring with at most 4|V|/5 edges failing the relevant normality condition, and show the bound is tight.\n- Ma, Mattiolo, Steffen, and Wolf characterize the minimal universal r-graph-colouring sets and show the Petersen conjecture is an exceptional universal-colouring scenario if true.\n\n**Full solution or refutation.**\n\nNo universal Petersen-edge colouring was found. Available results are approximate, use a larger palette, or cover particular cubic families.\n\n**What remains.**\n\nConstruct the required local Petersen-edge mapping at every vertex of every bridgeless cubic graph.\n\n**Sources checked.**\n\n- Francois Pirot, Jean-Sebastien Sereni, and Riste Skrekovski, Variations on the Petersen colouring conjecture, Electronic Journal of Combinatorics 27 (2020), P1.8. (primary): https://arxiv.org/abs/1905.07913\n Evidence used: Proves a sharp approximate normal-colouring theorem for every bridgeless cubic graph.\n- Yulai Ma, Davide Mattiolo, Eckhard Steffen, and Isaak H. Wolf, Sets of r-Graphs that Color All r-Graphs, Combinatorica 45 (2025). (primary): https://doi.org/10.1007/s00493-025-00144-4\n Evidence used: Treats the Petersen colouring conjecture as open and analyzes universal r-graph colourers.\n- Graph-theory open problems, Petersen coloring conjecture, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/petersen_coloring_conjecture/\n Evidence used: Maintains the exact conjecture as open and records verified partial results.\n\n**Review notes.** The source misspelling 'adjancent' is preserved and flagged in report.md.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3197, "problem_number": "OPG-144", "title": "Packing T-joins", "statement": "Conjecture There exists a fixed constant $c$ (probably $c=1$ suffices) so that every graft with minimum $T$-cut size at least $k$ contains a $T$-join packing of size at least $(2/3)k-c$.", "background": "Source: Open Problem Garden. Original node ID: 144. URL: http://www.openproblemgarden.org/op/packing_t_joins.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/packing_t_joins\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: packing; T-join\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinitions: A graft consists of a graph $G=(V,E)$ together with a distinguished set $T \\subseteq V$ of even cardinality. A $T$-cut is an edge-cut $\\delta(X)$ of $G$ with the property that $|X \\cap T|$ is odd. A $T$-join is a set $S \\subseteq E$ with the property that a vertex of $(V,S)$ has odd degree if and only if it is in $T$. A $T$-join packing is a set of pairwise disjoint T-joins.\n\nIt is an easy fact that every $T$-join and every $T$-cut intersect in an odd number of elements. It follows easily from this that the maximum size of a $T$-join packing is always less than or equal to the minimum size of a $T$-cut. There is a simple example of a graft with $|T|=4$ with minimum $T$-cut size $k$ which contains only $(2/3)k$ disjoint T-joins. The above conjecture asserts that this is essentially the worst case. DeVos and Seymour [DS] have obtained a partial result toward the above conjecture, proving that every graft with minimum $T$-cut size $k$ contains a $T$-join packing of size at least the floor of $(1/3)k$.\n\nDefinition: We say that a graft $G$ is an $r$-graph if $G$ is $r$-regular, $T=V$, and every $T$-cut of G has size at least $r$.\n\nConjecture (Rizzi) If $G$ is an $r$-graph, then $G$ contains a $T$-join packing of size at least $r-2$.\n\nIn an $r$-graph, every perfect matching is a $T$-join, so the above conjecture is true with room to spare for $r$-graphs which are $r$-edge-colorable. Indeed, Seymour had earlier conjectured that every $r$-graph contains $r-2$ disjoint perfect matchings. This however was disproved by Rizzi [R] who constructed for every $r>2$ an $r$-graph in which every two perfect matchings intersect. Rizzi suggested the above problem as a possible fix for Seymour's conjecture. DeVos and Seymour have proved that every $r$-graph has a $T$-join packing of size at least the floor of $r/2$.\n\nDefinition: Let $G$ be a graph and let $T$ be the set of vertices of $G$ of odd degree. A $T$-join of $(G,T)$ is defined to be a postman set.\n\nNote that when $T$ is the set of vertices of odd degree, a cocycle of $G$ is a $T$-cut if and only if it has odd size. Rizzi has shown that the following conjecture is equivalent to the above conjecture in the special case when $r$ is odd.\n\nConjecture (The packing postman sets conjecture (Rizzi)) If every odd edge-cut of $G$ has size $>2k+1$ then the edges of $G$ may be partitioned into $2k+1$ postman sets.\n\nThe Petersen graph (or more generally any non $(2k+1)$-edge-colorable $(2k+1)$-graph) shows that the above conjecture would be false with the weaker assumption that every odd edge-cut has size $>2k$. The following conjecture asserts that odd edge-cut size $>2k$ is enough (for the same conclusion) if we assume in addition that G has no Petersen minor.\n\nConjecture (Conforti, Johnson) If $G$ has no Petersen minor and every odd edge-cut of $G$ has size $>2k$ then the edges of $G$ may be partitioned into $2k+1$ postman sets.\n\nGerard Cornuejols [C] has kindly offered $5000 for a solution to this conjecture. However, it will be tough to find a quick proof since this conjecture does imply the 4-color theorem. Robertson, Seymour, Sanders, and Thomas [RSST] have proved the above conjecture for cubic graphs. Conforti and Johnson [CJ] proved it under the added hypothesis that G has no 4-wheel minor.\n\nBibliography:\n[CJ] M. Conforti and E.L. Johnson, Two min-max theorems for graphs noncontractible to a four wheel, preprint.\n\n[C] G. Cornuejols, Combinatorial Optimization, packing and covering, SIAM, Philadelphia (2001).\n\n[R] R. Rizzi, Indecomposable r-Graphs and Some Other Counterexamples, J. Graph Theory 32 (1999) 1-15. MathSciNet\n\n[RSST] N. Robertson, D.P. Sanders, P.D. Seymour, and R. Thomas, A New Proof of the Four-Color Theorem, Electron. Res. Announc., Am. Math. Soc. 02, no 1 (1996) 17-25.\n\n[S] P.D. Seymour, Some Unsolved Problems on One-Factorizations of Graphs, in Graph Theory and Related Topics, edited by J.A. Bondy and U.S.R. Murty, Academic Press, New York 1979) 367-368.\n\nDiscussion links:\n- edge-cut: http://en.wikipedia.org/wiki/connectivity (graph theory)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1704172\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 25.\n\nAttempt notes:\nTarget:\nMake progress on \"Packing T-joins\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The (2/3)k-c T-join packing bound remains open. The maintained tracker reports a best general floor(k/3) lower bound attributed to DeVos-Seymour, while a locatable manuscript proves floor(tau/6) generally and floor(tau/2) in two special cases; signed-graft odd-T-join packing has additional special-case progress.\n\n**Verified partial progress.**\n\n- The maintained tracker reports a floor(k/3) general lower bound from DeVos-Seymour work predating the OPG posting.\n- A locatable DeVos-Seymour manuscript proves floor(tau/6) for general grafts and floor(tau/2) for Eulerian grafts or when T is the set of odd-degree vertices.\n- Abdi and Guenin prove a packing theorem for odd T-joins with at most two terminals in the related signed-graft setting.\n\n**Full solution or refutation.**\n\nNo proof approaching the full (2/3)k-O(1) bound for all grafts was located. The exact provenance of the tracker-reported k/3 bound needs expert bibliographic confirmation.\n\n**What remains.**\n\nClose the gap between the reported k/3 general lower bound and (2/3)k-c, and reconcile the plain-graft and signed-graft formulations without conflating them.\n\n**Sources checked.**\n\n- Graph-theory open problems, Packing T-joins, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/packing_t_joins/\n Evidence used: States that the main conjecture remains open and reports floor(k/3) as the best verified general lower bound.\n- Matt DeVos and Paul Seymour, On Packing T-Joins, author-hosted manuscript. (primary): https://www.sfu.ca/~mdevos/papers/pack-tjoin.pdf\n Evidence used: Proves floor(tau/6) generally and floor(tau/2) in specified special cases; it does not itself substantiate the tracker's k/3 statement.\n- Ahmad Abdi and Bertrand Guenin, Packing odd T-joins with at most two terminals, Journal of Graph Theory 87 (2018), DOI 10.1002/jgt.22178. (primary): https://arxiv.org/abs/1410.7423\n Evidence used: Proves a special signed-graft/odd-T-join packing theorem; the framework is related but not identical to the exact conjecture.\n\n**Review notes.** The tracker and the available DeVos-Seymour manuscript state different general constants. This discrepancy is made explicit rather than silently resolving it in favor of either source.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3198, "problem_number": "OPG-145", "title": "Acyclic edge-colouring", "statement": "Conjecture Every simple graph with maximum degree $\\Delta$ has a proper $(\\Delta+2)$-edge-colouring so that every cycle contains edges of at least three distinct colours.", "background": "Source: Open Problem Garden. Original node ID: 145. URL: http://www.openproblemgarden.org/op/acyclic_edge_coloring.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/acyclic_edge_coloring\n- Author(s): Fiamcik, Jozef\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: edge-coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nAn edge-colouring with the property that every cycle contains edges of at least three distinct colours is called an acyclic edge-colouring. It is known (see [AMR]) that every graph of maximum degree $\\Delta$ has an acyclic edge-colouring of size $O(\\Delta )$. The best upper bound so far is $4\\Delta -4$ and is due to Esperet and Parreau [EP]. It is also known (see [ASZ]) that this conjecture is true for graphs with girth at least $C \\Delta \\log(\\Delta )$ (for some fixed constant $C$ ).\n\nBibliography:\n[AMR] N. Alon, C. McDiarmid and B. Reed, Acyclic colouring of graphs, Random Structures and Algorithms 2 (1991), 277-288. MathSciNet\n\n[ASZ] N. Alon, B. Sudakov and A. Zaks, Acyclic edge-colorings of graphs, J. Graph Theory 37 (2001), 157-167. MathSciNet\n\n[EP] L. Esperet and A. Parreau, Acyclic edge-coloring using entropy compression, arXiv:1206.1535 [math.CO].\n\nSource links:\n- edge-colouring: http://en.wikipedia.org/wiki/edge-coloring\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1109695\n- Acyclic edge-colorings of graphs: http://www.math.tau.ac.il/%7Enogaa/PDFS/asz2.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1837021\n- Acyclic edge-coloring using entropy compression: http://www.arxiv.org/abs/1206.1535v3\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Acyclic edge-colouring\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact acyclic edge-colouring bound Delta+2 remains open in general. A 2026 preprint proves the current readily verifiable general bound 3.142(Delta-1)+1, and the conjecture is proved for several sparse graph classes.\n\n**Verified partial progress.**\n\n- Kirousis, Livieratos, and Singh prove a'(G) at most 3.142(Delta-1)+1 for every graph.\n- Anto, Basavaraju, and Kulamarva prove the Delta+2 conjecture for 3-sparse graphs.\n- Anto, Basavaraju, Hegde, and Kulamarva prove Delta+5 for 3-degenerate graphs and record the stronger Delta+1 result for 2-degenerate graphs.\n\n**Full solution or refutation.**\n\nNo general Delta+2 theorem was located. The best citable general bound remains multiplicative, although exact or near-exact bounds hold for important sparse classes.\n\n**What remains.**\n\nReduce the general upper bound to Delta+2. A seminar-announced asymptotic theorem with no locatable paper was excluded from established evidence.\n\n**Sources checked.**\n\n- Lefteris Kirousis, John Livieratos, and Alexandros Singh, An improvement on the bound for the acyclic chromatic index, arXiv:2602.14859 (2026). (primary): https://arxiv.org/abs/2602.14859\n Evidence used: Proves the general 3.142(Delta-1)+1 upper bound.\n- Nevil Anto, Manu Basavaraju, and Shashanka Kulamarva, Acyclic Edge Coloring of 3-sparse Graphs, arXiv:2501.11281. (primary): https://arxiv.org/abs/2501.11281\n Evidence used: Proves the exact Delta+2 conjecture for 3-sparse graphs and a Delta+1 subcase.\n- Nevil Anto, Manu Basavaraju, Suresh Manjanath Hegde, and Shashanka Kulamarva, Upper bounds on the acyclic chromatic index of degenerate graphs, Discrete Mathematics 347 (2024), 113898. (primary): https://doi.org/10.1016/j.disc.2024.113898\n Evidence used: Proves Delta+5 for 3-degenerate graphs and discusses the exact 2-degenerate case.\n- Graph-theory open problems, Acyclic edge-colouring, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/acyclic_edge_coloring/\n Evidence used: Tracks the exact Delta+2 conjecture as open and the 3.142 general bound and 3-sparse theorem as partial progress.\n\n**Review notes.** The source statement is mathematically well formed. The unlocatable announced (1+o(1))Delta result was not treated as established.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3199, "problem_number": "OPG-182", "title": "A generalization of Vizing's Theorem?", "statement": "Conjecture Let $H$ be a simple $d$-uniform hypergraph, and assume that every set of $d-1$ points is contained in at most $r$ edges. Then there exists an $r+d-1$-edge-coloring so that any two edges which share $d-1$ vertices have distinct colors.", "background": "Source: Open Problem Garden. Original node ID: 182. URL: http://www.openproblemgarden.org/op/a_generalization_of_vizings_theorem.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_generalization_of_vizings_theorem\n- Author(s): Rosenfeld, Moshe\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: edge-coloring; hypergraph; Vizing\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 11th, 2007 by mdevos\n\nProblem-page discussion:\nVizing's Theorem is equivalent to the above statement for $d=2$. For higher dimensions, this problem looks difficult since the main tool used in the proof of Vizing's theorem (Kempe chains) do not appear to work.\n\nSource links:\n- uniform: http://en.wikipedia.org/wiki/hypergraph\n\nComments:\n- June 23rd, 2009 | Anonymous | Reference: Could someone please add a reference? There should be some paper (or a conference talk?) where Rosenfeld proposed the conjecture.\n\n-DOT\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"A generalization of Vizing's Theorem?\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the stated r+d-1 hyperedge-coloring conjecture was verified.\n\n**Verified partial progress.**\n\n- The d=2 case is Vizing's theorem; the higher-uniformity extension is stronger.\n\n**Full solution or refutation.**\n\nThe general hypergraph conjecture remains open.\n\n**What remains.**\n\nDevelop list/coloring methods using the codegree bound.\n\n**Sources checked.**\n\n- Open Problem Garden, node 182 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3200, "problem_number": "OPG-388", "title": "List colorings of edge-critical graphs", "statement": "Conjecture Suppose that $G$ is a $\\Delta$-edge-critical graph. Suppose that for each edge $e$ of $G$, there is a list $L(e)$ of $\\Delta$ colors. Then $G$ is $L$-edge-colorable unless all lists are equal to each other.", "background": "Source: Open Problem Garden. Original node ID: 388. URL: http://www.openproblemgarden.org/op/list_colorings_of_edge_critical_graphs.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/list_colorings_of_edge_critical_graphs\n- Author(s): Mohar, Bojan\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: edge-coloring; list coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 12th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Reproduced from [M].)\n\nA graph $G$ is said to be $\\Delta$-edge-critical if it is not $\\Delta$-edge-colorable but every edge-deleted subgraph is $\\Delta$-edge-colorable. (Here $\\Delta$ is the maximum degree of $G$.)\n\nBibliography:\n*[M] B. Mohar, Problem of the Month\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P7ListEdgeCriticalGraphs.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"List colorings of edge-critical graphs\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the stated list-edge-coloring rigidity for Delta-edge-critical graphs was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nUse critical-edge structure and list-coloring extensions.\n\n**Sources checked.**\n\n- Open Problem Garden, node 388 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3201, "problem_number": "OPG-624", "title": "Universal Steiner triple systems", "statement": "Problem Which Steiner triple systems are universal?", "background": "Source: Open Problem Garden. Original node ID: 624. URL: http://www.openproblemgarden.org/op/universal_steiner_triple_systems.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/universal_steiner_triple_systems\n- Author(s): Grannell, Mike; Griggs, Terry; Knor, Martin; Skoviera, Martin\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: cubic graph; Steiner triple system\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 5th, 2007 by macajova\n\nProblem-page discussion:\nA cubic graph $G$ is $S$-edge-colorable for a Steiner triple system $S$ if its edges can be colored with the points of $S$ in such a way that the points assigned to three edges sharing a vertex form a triple in $S$.\n\nA Steiner triple system $S$ is called universal if any (simple) cubic graph is $S$-colorable.\n\nIt is easy to see that if $S_3$ denotes the trivial Steiner triple system with three points and one triple, then $S_3$-colorable graphs are precisely (cubic) edge-3-colorable graphs. For the same reason, any cubic edge-3-colorable graph is $S$-colorable for any Steiner triple system (with at least one edge). Thus, the study of $S$-colorings may be viewed as an attempt to understand snarks.\n\nIt is not hard to see, that a graph is Fano-colorable iff it has a nowhere-zero 8-flow. Thus (by Jaeger's result) Fano plane is \"almost universal\": it is possible to use it to color any bridgeless cubic graph (but it doesn't work for any graph with a bridge).\n\nGrannell et al. [GGKS] constructed a universal Steiner triple system of order 381. Holroyd, Skoviera [HS] proved that neither projective nor affine Steiner triple systems are universal. Kral et al. [KMPS] proved that any non-affine non-projective non-trivial point-transitive Steiner triple system is universal.\n\nBibliography:\n*[GGKS] M.J. Grannell, T.S. Griggs, M. Knor, M. Skoviera, A Steiner triple system which colours all cubic graphs, J. Graph Theory 46 (2004), 15--24. MathSciNet\n\n[HS] F. Holroyd and M. Skoviera, Colouring of cubic graphs by Steiner triple systems, J.~Combin. Theory Ser. B 91 (2004), 57--66.\n\n[KMPS] D. Kral, E. Macajova, A. Por, J.-S. Sereni, Characterization results for Steiner triple systems and their application to edge-colorings of cubic graphs, preprint.\n\nSource links:\n- Steiner triple systems: http://mathworld.wolfram.com/SteinerTripleSystem.html\n\nDiscussion links:\n- snarks: http://en.wikipedia.org/wiki/snark (graph theory)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2051465\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Universal Steiner triple systems\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Universality has been characterized for point-transitive Steiner triple systems and explicit universal systems exist, but no characterization of arbitrary Steiner triple systems was located.\n\n**Verified partial progress.**\n\n- Grannell, Griggs, Knor, and Skoviera constructed a universal Steiner triple system of order 381.\n- Holroyd and Skoviera established nonuniversality results for projective and affine Steiner triple systems.\n- Kral, Macajova, Por, and Sereni prove that every nontrivial point-transitive Steiner triple system that is neither projective nor affine is universal, completing the point-transitive classification.\n\n**Full solution or refutation.**\n\nThe cited characterization is restricted to point-transitive systems; the maintained graph-problems entry continues to ask which Steiner triple systems are universal in general.\n\n**What remains.**\n\nCharacterize universal Steiner triple systems without point-transitivity or comparable symmetry assumptions.\n\n**Sources checked.**\n\n- M. J. Grannell, T. S. Griggs, M. Knor, and M. Skoviera, A Steiner triple system which colours all cubic graphs, Journal of Graph Theory 46 (2004), 15-24. (primary): https://doi.org/10.1002/jgt.10166\n Evidence used: Constructs a universal Steiner triple system of order 381.\n- F. C. Holroyd and M. Skoviera, Colouring of cubic graphs by Steiner triple systems, Journal of Combinatorial Theory, Series B 91 (2004), 57-66. (primary): https://doi.org/10.1016/j.jctb.2003.10.003\n Evidence used: Determines projective cases and gives affine obstructions central to the later classification.\n- Daniel Kral, Edita Macajova, Attila Por, and Jean-Sebastien Sereni, Characterisation Results for Steiner Triple Systems and Their Application to Edge-Colourings of Cubic Graphs, Canadian Journal of Mathematics 62 (2010), 355-381. (primary): https://doi.org/10.4153/CJM-2010-021-9\n Evidence used: Completes the classification of universality within the point-transitive class.\n- Graph-theory open problems, Universal Steiner triple systems; checked 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/universal_steiner_triple_systems/\n Evidence used: States the unrestricted problem and records the point-transitive partial characterization.\n\n**Review notes.** The one-line statement does not define universal; the classification uses the exact definition in the stored background: S-edge-colourability of every simple cubic graph.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3202, "problem_number": "OPG-2110", "title": "Edge list coloring conjecture", "statement": "Conjecture Let $G$ be a loopless multigraph. Then the edge chromatic number of $G$ equals the list edge chromatic number of $G$.", "background": "Source: Open Problem Garden. Original node ID: 2110. URL: http://www.openproblemgarden.org/op/edge_list_coloring_conjecture.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/edge_list_coloring_conjecture\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 25th, 2008 by tchow\n\nProblem-page discussion:\nThe list edge chromatic number of $G$ is also known as the list chromatic index, the edge choosability, or the edge choice number of $G$. It is the list chromatic number of the line graph of $G$. Similarly, the edge chromatic number of $G$ is also known as the edge chromatic index, and it is the chromatic number of the line graph of $G$. The chromatic number of a graph is always less than or equal to the list chromatic number; the two quantities differ in general, but the conjecture says that they coincide for line graphs. Sometime the conjecture is simply referred to as the list coloring conjecture, although this is perhaps poor terminology since there exist other conjectures about list coloring.\n\nPerhaps the most famous partial result is Galvin's theorem [G] that the conjecture holds for bipartite multigraphs. Galvin's result settled the well-known Dinitz conjecture in the affirmative.\n\nThe conjecture has been attributed to many different people. See [JT, Problem 12.20] for a history of the problem up to 1995.\n\nBibliography:\n[G] Fred Galvin, The list chromatic index of a bipartite multigraph. J. Combin. Theory Ser. B 63 (1995), 153–158.\n\n[JT] Tommy R. Jensen and Bjarne Toft, Graph Coloring Problems. New York: Wiley-Interscience, 1995.\n\nRelated:\nRelated problems\nRota's basis conjecture\nList Total Colouring Conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Edge list coloring conjecture\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The List Edge Coloring Conjecture remains open for arbitrary loopless multigraphs, though bipartite multigraphs and asymptotic regimes are settled.\n\n**Verified partial progress.**\n\n- Galvin proves the bipartite multigraph case.\n- Kahn proves asymptotic equality of list and ordinary chromatic index.\n\n**Full solution or refutation.**\n\nNo all-multigraph equality theorem was verified.\n\n**What remains.**\n\nResolve list edge coloring for non-bipartite multigraphs, beginning with remaining structured classes.\n\n**Sources checked.**\n\n- F. Galvin, The list chromatic index of a bipartite multigraph, J. Combin. Theory B 63 (1995), 153--158. (primary): https://en.wikipedia.org/wiki/List_edge-coloring\n Evidence used: The maintained summary records Galvin's special case and the general conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3203, "problem_number": "OPG-2226", "title": "Seymour's r-graph conjecture", "statement": "An $r$-graph is an $r$-regular graph $G$ with the property that $|\\delta(X)| \\ge r$ for every $X \\subseteq V(G)$ with odd size.\n\nConjecture $\\chi'(G) \\le r+1$ for every $r$-graph $G$.", "background": "Source: Open Problem Garden. Original node ID: 2226. URL: http://www.openproblemgarden.org/op/seymours_r_graph_conjecture.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/seymours_r_graph_conjecture\n- Author(s): Seymour, Paul D.\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: edge-coloring; r-graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 3rd, 2008 by mdevos\n\nProblem-page discussion:\nThis conjecture is among the most important unsolved problems in edge coloring. It is very close in nature to Goldberg's Conjecture, and is also closely related to Rizzi's packing postman sets conjecture (see packing T-joins).\n\nIf $G$ is an $r$-regular graph and there exists $X \\subseteq V(G)$ with $|X|$ odd and $|\\delta(X)| < r$, then it is immediate that $G$ is not $r$-edge-colourable, since every perfect matching must use at least one edge from $\\delta(X)$. This is in some sense the only obvious obstruction to $r$-edge-colorability that we know of. So, $r$-graphs are the $r$-regular graphs which do not fail to be $r$-edge-colorable for this obvious reason. Not every $r$-graph is $r$-edge-colorable, for instance Petersen's graph is a 3-graph which is not 3-edge-colorable. However, this conjecture asserts that all such graphs are still $(r+1)$-edge-colorable.\n\nThis conjecture has been proved for $r \\le 11$ by Nishizeki and Kashiwagi [NK].\n\nBibliography:\n[NK] T. Nishizeki and K. Kashiwagi, An upper bound on the chromatic index of multigraphs. Graph theory with applications to algorithms and computer science (Kalamazoo, Mich., 1984), 595--604, Wiley-Intersci. Publ., Wiley, New York, 1985. MathSciNet.\n\n*[S] P.D. Seymour, On multicolourings of cubic graphs, and conjectures of Fulkerson and Tutte. Proc. London Math. Soc. (3) 38 (1979), no. 3, 423--460. MathSciNet.\n\nRelated:\nRelated problems\nPacking T-joins\nGoldberg's conjecture\n\nDiscussion links:\n- Goldberg's Conjecture: http://www.openproblemgarden.org/?q=node/2242\n- packing T-joins: http://www.openproblemgarden.org/?q=node/144\n- Petersen's graph: http://en.wikipedia.org/wiki/Petersen's graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0812694\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0532981\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Seymour's r-graph conjecture\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Seymour's r-graph conjecture remains open, with broad edge-coloring advances and special r-graph cases but no general r+1 theorem.\n\n**Verified partial progress.**\n\n- The Goldberg--Seymour edge-coloring programme provides closely related density-sensitive bounds.\n- Several regular and planar subfamilies are settled.\n\n**Full solution or refutation.**\n\nNo proof that every r-graph has chromatic index at most r+1 was verified.\n\n**What remains.**\n\nEstablish the r+1 bound for arbitrary odd-cut-constrained r-graphs or find a counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, node 2226 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Retains the exact r-graph definition and conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3204, "problem_number": "OPG-2242", "title": "Goldberg's conjecture", "statement": "The overfull parameter is defined as follows:\n$$\nw(G) = \\max_{H \\subseteq G} \\left\\lceil \\frac{ |E(H)| }{ \\lfloor \\tfrac{1}{2} |V(H)| \\rfloor} \\right\\rceil.\n$$\n\nConjecture Every graph $G$ satisfies $\\chi'(G) \\le \\max\\{ \\Delta(G) + 1, w(G) \\}$.", "background": "Source: Open Problem Garden. Original node ID: 2242. URL: http://www.openproblemgarden.org/op/goldbergs_conjecture.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/goldbergs_conjecture\n- Author(s): Goldberg, Mark K.\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Keywords: edge-coloring; multigraph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 4th, 2008 by mdevos\n\nProblem-page discussion:\nThis important problem remains open despite considerable attention. The same conjecture was independently discovered by Andersen and Seymour.\n\nVizing's Theorem, one of the cornerstones of graph colouring, shows that $\\chi'(G) \\le \\Delta(G) + 1$ for every simple graph $G$. So, in particular, every simple graph satisfies Goldberg's conjecture. Graphs with parallel edges need not satisfy Vizing's bound. For instance, if $G$ is the graph obtained from a triangle by adding an extra $k-1$ edges in parallel with each existing one, then $\\Delta(G) = 2k$ but $\\chi'(G) = 3k$. More generally, if $H$ is a subgraph of $G$, then every colour can appear on at most $\\lfloor \\frac{1}{2}|V(H)| \\rfloor$ edges of $H$, so $\\chi'(G) \\ge |E(H)| / \\lfloor \\tfrac{1}{2} |V(H)| \\rfloor$. Thus, $w(G)$, our overfull parameter, is a natural lower bound on $\\chi'(G)$, and Goldberg's conjecture asserts that whenever $\\chi'(G)$ exceeds $\\Delta(G)+1$, then it is equal to this lower bound.\n\nAlthough the statement of the conjecture may appear to be the most natural formulation, there are a couple of related conjectures with similar lower bounds. For instance, Seymour's r-graph conjecture is equivalent to the statement that $\\chi'(G) \\le \\max \\{\\Delta(G), w(G) \\} + 1$. Goldberg also conjectured that $\\chi'(G) \\le \\max\\{ \\Delta(G), w(G) + 1\\}$.\n\nIn addition to simple graphs, Goldberg's Conjecture is known to hold for any graph $G$ which satisfies one of the following\n\n- $\\Delta(G) \\le 11$\n- $G$ has no minor isomorphic to $K_5$ minus an edge.\n- $\\Delta(G)$ is sufficiently large in comparison with $|V(G)|$.\n\n$\\quad$\n\nPackers And Movers Chandigarh\nPackers And Movers Hyderabad\nPackers And Movers Bangalore\n\nBibliography:\n*[G] M. K. Goldberg, Multigraphs with a chromatic index that is nearly maximal. (Russian) A collection of articles dedicated to the memory of Vitaliĭ Konstantinovič Korobkov. Diskret. Analiz No. 23 (1973), 3--7, 72. MathSciNet\n\nRelated:\nRelated problems\nSeymour's r-graph conjecture\n\nDiscussion links:\n- Seymour's r-graph conjecture: http://www.openproblemgarden.org/?q=node/2226\n- Packers And Movers Chandigarh: http://www.packersandmoverschandigarh.co.in\n- Packers And Movers Hyderabad: http://www.packersandmoversinhyderabad.co.in\n- Packers And Movers Bangalore: http://www.packersandmoversinbangalore.co.in\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0354429\n\nComments:\n- February 6th, 2009 | Anonymous | Latest Developments: Stiebitz et al(2006), Yu(2008) and Kurt(2009) has separately shown $\\chi'(G) \\geq \\Delta+\\sqrt{\\Delta/2}$ implies Goldberg Conjecture. While Yu's method gives a methodological approach to the general problem, Kurt provides a very short and elementary proof.\n\nScheide (2008) has shown $\\chi'(G)>\\frac{15}{14}\\Delta+\\frac{12}{14}$ implies the Goldberg Conjecture.\n\nKurt(2009) has shown $\\chi'(G)>\\frac{17}{16}\\Delta+\\frac{14}{16}$ implies the Goldberg Conjecture.\n- June 5th, 2013 | SAC | Different Goldberg Conjecture?: http://plms.oxfordjournals.org/content/106/4/703\n\n--Stephen\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"Goldberg's conjecture\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The intended multigraph Goldberg-Seymour conjecture was proved by Chen, Jing, and Zang in a 2025 peer-reviewed publication, with a 2026 publisher correction.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nEvery multigraph G satisfies chi'(G) <= max{Delta(G)+1, ceil(Gamma(G))}, where Gamma is the standard maximum odd-set density; this is the intended theorem represented by the record's w(G).\n\n**What remains.**\n\nNothing remains for the standard Goldberg-Seymour conjecture; the imported definition should be corrected before machine use.\n\n**Sources checked.**\n\n- Guantao Chen, Guangming Jing, and Wenan Zang, Proof of the Goldberg-Seymour conjecture on edge-colorings of multigraphs, Journal of Combinatorial Optimization 50 (2025), article 23. (primary): https://doi.org/10.1007/s10878-025-01348-6\n Evidence used: Peer-reviewed 91-page proof of the exact intended multigraph chromatic-index bound.\n- Publisher Correction: Proof of the Goldberg-Seymour conjecture on edge-colorings of multigraphs, Journal of Combinatorial Optimization 51 (2026), article 1. (primary): https://doi.org/10.1007/s10878-025-01372-6\n Evidence used: Confirms the published article and records its formal correction.\n\n**Review notes.** The imported maximum divides by zero on subgraphs with at most one vertex. The standard Gamma maximum is over odd vertex sets of size at least three. The intended objects are multigraphs; for simple graphs Vizing's theorem already suffices.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3205, "problem_number": "OPG-46475", "title": "Strong edge colouring conjecture", "statement": "A strong edge-colouring of a graph $G$ is a edge-colouring in which every colour class is an induced matching; that is, any two vertices belonging to distinct edges with the same colour are not adjacent. The strong chromatic index $s\\chi'(G)$ is the minimum number of colours in a strong edge-colouring of $G$.\n\nConjecture $$s\\chi'(G) \\leq \\frac{5\\Delta^2}{4}, \\text{if$\\Delta$is even,}$$$$s\\chi'(G) \\leq \\frac{5\\Delta^2-2\\Delta +1}{4},&\\text{if$\\Delta$is odd.}$$", "background": "Source: Open Problem Garden. Original node ID: 46475. URL: http://www.openproblemgarden.org/op/strong_edge_colouring_conjecture.\n\nSource subject path: Graph Theory > Coloring > Edge coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strong_edge_colouring_conjecture\n- Author(s): Erdos, Paul; Nesetril, Jaroslav\n- Subject(s): Graph Theory; Coloring; Edge coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 1st, 2013 by fhavet\n\nProblem-page discussion:\nThe conjectured bounds would be sharp. When $D$ is even, expanding each vertex of a $5$-cycle into a stable set of size $\\Delta/2$ yields such a graph with $5\\Delta^2/4$ edges in which the largest induced matching has size $1$. A similar construction achieves the bound when $\\Delta$ is odd.\n\nGreedy colouring the edges yields $s\\chi'(G) \\leq 2\\Delta(\\Delta-1)+1$. Using probabilistic methods, Molloy and Reed~[MoRe97] proved that there is a positive constant $\\epsilon$ such that, for sufficiently large $\\Delta$, every graph with maximum degree $\\Delta$ has strong chromatic index at most $(2-\\epsilon)\\Delta^2$.\n\nThe greedy bound proves the conjecture for $\\Delta \\leq 2$. For $\\Delta =3$, the conjectured bound of 10 was proved independently by Hor\\'ak, He, and Trotter[HHT] and by Andersen [A]. For $\\Delta=4$, the conjectured bound is 20, and Cranston [C] proved that 22 colours suffice.\n\nFor a bipartite graph $G$, Faudree et al. [FGST] conjectured that $s\\chi'(G)\\leq \\Delta^2(G)$. This is implied by the stronger conjecture due to Kaiser.\n\nConjecture Let $G=((A_1,A_2),E)$ be a bipartite graph such that every vertex in $A_1$ has degree at most $\\Delta_1$ and every vertex in $A_2$ has degree at most $\\Delta_2$. Then $s\\chi'(G)\\leq \\Delta_1\\Delta_2$.\n\nBibliography:\n[A] L. D. Andersen. The strong chromatic index of a cubic graph is at most 10. Discrete Math., 108(1-3):231--252, 1992.\n\n[C] D. W. Cranston. Strong edge-coloring of graphs with maximum degree 4 using 22 colors. Discrete Math., 306(21):2772--2778, 2006.\n\n[FGST] R. J. Faudree, A. Gyárfás, R. H. Schelp, and Zs. Tuza. Induced matchings in bipartite graphs. Discrete Math., 78(1-2):83--87, 1989.\n\n[HHT] P. Horák, Q. He, and W. T. Trotter. Induced matchings in cubic graphs. J. Graph Theory, 17(2):151--160, 1993.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 27.\n\nAttempt notes:\nTarget:\nMake progress on \"Strong edge colouring conjecture\" in Graph Theory; Coloring; Edge coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Erdős--Nešetřil strong chromatic index conjecture remains open for general maximum degree at least 4. The best cited large-degree bound is 1.772 Delta^2, and the maximum-degree-four bound is 21 versus the conjectured 20.\n\n**Verified partial progress.**\n\n- Hurley--de Joannis de Verclos--Kang prove s chi'(G)<=1.772 Delta^2 for sufficiently large Delta.\n- Huang--Santana--Yu prove 21 colors suffice when Delta=4; the conjectured value is 20.\n- The conjecture is proved for Delta<=3.\n\n**Full solution or refutation.**\n\nNo proof of the piecewise 5Delta^2/4 bound for every graph was verified.\n\n**What remains.**\n\nReduce the general leading constant from 1.772 to 1.25 and settle the first open exact case Delta=4.\n\n**Sources checked.**\n\n- Eoin Hurley, Rémi de Joannis de Verclos, and Ross J. Kang, An improved procedure for colouring graphs of bounded local density, arXiv:2007.07874. (primary): https://arxiv.org/abs/2007.07874\n Evidence used: Proves the 1.772 Delta^2 upper bound for sufficiently large maximum degree.\n- Mingfang Huang, Michael Santana, and Gexin Yu, Strong chromatic index of graphs with maximum degree four, Electronic Journal of Combinatorics 25(3) (2018), P3.31. (primary): https://arxiv.org/abs/1806.07012\n Evidence used: Improves the Delta=4 upper bound to 21 and explicitly states that the conjectured bound 20 remains open.\n- Thomas F. Bloom, Erdős Problem #149, maintained problem tracker. (maintained_tracker): https://www.erdosproblems.com/149\n Evidence used: Current tracker retains open status and records modern bounds and solved small-degree cases.\n\n**Review notes.** The imported piecewise display contains four consecutive dollar signs and a stray ampersand. The two inequalities are still readable and were not rewritten in the source statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3206, "problem_number": "OPG-125", "title": "Cores of Cayley graphs", "statement": "Conjecture Let $M$ be an abelian group. Is the core of a Cayley graph (on some power of $M$ ) a Cayley graph (on some power of $M$ )?", "background": "Source: Open Problem Garden. Original node ID: 125. URL: http://www.openproblemgarden.org/op/cores_of_cayley_graphs.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cores_of_cayley_graphs\n- Author(s): Samal, Robert\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: Cayley graph; core\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 6th, 2007 by Robert Samal\n\nProblem-page discussion:\nEven the case $M=\\mathbb{Z}_2$ is open. In this case, Cayley graphs on some power of $\\mathbb{Z}_2$ are called cube-like graphs, they have been introduced by Lov\\'asz as an example of graphs, for which every eigenvalue is an integer.\n\nSo, in this case we ask, whether a core of each cube-like graph is a cube-like graph.\n\nSource links:\n- core: http://en.wikipedia.org/wiki/core (graph theory)\n- Cayley graph: http://en.wikipedia.org/wiki/Cayley graph\n\nComments:\n- February 29th, 2012 | Anonymous | Who first conjectured this?: Who first conjectured that the core of a cubelike graph is cubelike?\n- February 22nd, 2008 | Gordon Royle | Question needs refining...: As stated, the conjecture is false in an uninteresting way... it is possible for a Cayley graph of Z_15 have a 5-cycle as a core.... So if we take M = Z_15 then the result is false.\n\nPerhaps the question should either (a) be restricted to elementary abelian groups or (b) have the conclusion being that the core of a Cayley graph on M must be a Cayley graph on N where N is a (group) homomorphic image of M.\n\nGordon Royle http://people.csse.uwa.edu.au/gordon\n- March 1st, 2008 | Robert Samal | Re: Question needs refining...: That is true. I was mainly thinking about $M={\\mathbb Z}_p$ (for a prime $p$ ). However, your suggestion (b) looks sensible as well.\n\nThanks for your comment!\n\nRobert Samal\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Cores of Cayley graphs\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that the core of every relevant abelian Cayley graph is itself a Cayley graph on a power of the group was verified.\n\n**Verified partial progress.**\n\n- The source tracker supplies the current formulation and relevant context.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nClassify cores of finite abelian Cayley graphs and lift through powers.\n\n**Sources checked.**\n\n- Open Problem Garden, node 125 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Open Problem Garden retains it as a conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3207, "problem_number": "OPG-167", "title": "Pentagon problem", "statement": "Question Let $G$ be a 3-regular graph that contains no cycle of length shorter than $g$. Is it true that for large enough~ $g$ there is a homomorphism $G \\to C_5$?", "background": "Source: Open Problem Garden. Original node ID: 167. URL: http://www.openproblemgarden.org/op/pentagon_problem.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/pentagon_problem\n- Author(s): Nesetril, Jaroslav\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: cubic; homomorphism\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 24th, 2007 by Robert Samal\n\nProblem-page discussion:\nThis question was asked by Nesetril at numerous problem sessions (and also appears as [N]). By Brook's theorem any triangle-free cubic graph is 3-colorable. Does a stronger assumption on girth of the graph imply stronger coloring properties?\n\nThis problem is motivated by complexity considerations [GHN] and also by exploration of density of the homomorphism order: We write $G \\prec H$ if there is a homomorphism $G \\to H$ but there is no homomorphism $H \\to G$. It is known that whenever $G \\prec H$ holds and $H$ ~is not bipartite, there is a graph~ $K$ satisfying $G \\prec K \\prec H$. A negative solution to the Pentagon problem would have the following density consequence: for each cubic graph~ $H$ for which~ $C_5 \\prec H$ holds, there exists a cubic graph~ $K$ satisfying $C_5 \\prec K \\prec H$ (see [N]).\n\nIf we replaced $C_5$ in the statement of the problem by a longer odd cycle, we would get a stronger statement. It is known that no such strenghthening is true. This was proved by Kostochka, Nesetril, and Smolikova [KNS] for $C_{11}$ (hence for all $C_l$ with $l \\ge 11$ ), by Wanless and Wormald [WW] for $C_9$, and recently by Hatami [H] for $C_7$. Each of these results uses probabilistic arguments (random regular graphs), no constructive proof is known.\n\nHaggkvist and Hell [HH] proved that for every integer~ $g$ there is a graph~ $U_g$ with odd girth at least~ $g$ (that is, $U_g$ does not contain odd cycle of length less than~ $g$ ) such that every cubic graph of odd girth at least~ $g$ maps homomorphically to~ $U_g$. Here, the graph~ $U_g$ may have large degrees. This leads to a weaker version of the Pentagon problem:\n\nQuestion Is it true that for every $k$ there exists a cubic graph $H_k$ of girth~ $k$ and an integer~ $g$ such that every cubic graph of girth at least~ $g$ maps homomorphically to~ $H_k$?\n\nA particular question in this direction: does a high-girth cubic graph map to the Petersen graph?\n\nAs an approach to this, we mention a result of DeVos and Samal [DS]: a cubic graph of girth at least~ $17$ admits a homomorphism to the Clebsch graph. In context of the Pentagon problem, the following reformulation is particularly appealing: If $G$ ~is a cubic graph of girth at least~ $17$, then there is a cut-continuous mapping from~ $G$ to~ $C_5$; that is, there is a mapping $f: E(G) \\to E(C_5)$ such that for any cut $X \\subseteq E(C_5)$ the preimage $f^{-1}(X)$ is a cut. (Here by cut we mean the edge-set of a spanning bipartite subgraph. A more thorough exposition of cut-continuous mappings can be found in~[DNR].)\n\nBibliography:\n[DNR] Matt DeVos, Jaroslav Nesetril, and Andre Raspaud, On edge-maps whose inverse preverses flows and tensions, Graph Theory in Paris: Proceedings of a Conference in Memory of Claude Berge, Birkhäuser 2006.\n\n[DS] Matt DeVos and Robert Samal, High girth cubic graphs map to the Clebsch graph\n\n[GHN] Anna Galluccio, Pavol Hell, and Jaroslav Nesetril, The complexity of $H$-colouring of bounded degree graphs, Discrete Math. 222 (2000), no.~1-3, 101--109, MathSciNet\n\n[HH] Roland Haggkvist and Pavol Hell, Universality of $A$-mote graphs, European J. Combin. 14 (1993), no.~1, 23--27.\n\n[H] Hamed Hatami, Random cubic graphs are not homomorphic to the cycle of size~7, J. Combin. Theory Ser. B 93 (2005), no.~2, 319--325, MathSciNet\n\n[KNS] Alexandr~V. Kostochka, Jaroslav Nesetril, and Petra Smolikova, Colorings and homomorphisms of degenerate and bounded degree graphs, Discrete Math. 233 (2001), no.~1-3, 257--276, Fifth Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications, (Prague, 1998), MathSciNet\n\n*[N] Jaroslav Nesetril, Aspects of structural combinatorics (graph homomorphisms and their use), Taiwanese J. Math. 3 (1999), no.~4, 381--423, MathSciNet\n\n[WW] I.M. Wanless and N.C. Wormald, Regular graphs with no homomorphisms onto cycles, J. Combin. Theory Ser. B 82 (2001), no.~1, 155--160, MathSciNet\n\nSource links:\n- homomorphism: http://en.wikipedia.org/wiki/graph_homomorphism\n\nDiscussion links:\n- cubic graph: http://en.wikipedia.org/wiki/cubic graph\n- girth: http://en.wikipedia.org/wiki/girth\n\nBibliography links:\n- High girth cubic graphs map to the Clebsch graph: http://www.arxiv.org/abs/math.CO/0602580\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1771392\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2117942\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2002c:05077\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1730980\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2002a:05221\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 25.\n\nAttempt notes:\nTarget:\nMake progress on \"Pentagon problem\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Nešetřil's Pentagon Problem remains open. A weaker universal target theorem is known, while corresponding claims for longer odd-cycle targets are false.\n\n**Verified partial progress.**\n\n- Every graph of maximum degree 3 and girth at least 17 admits a homomorphism to the Clebsch graph.\n- The Clebsch-target theorem is equivalent to a five-edge-colouring or cut-continuous-map relaxation of the desired C5 homomorphism.\n\n**Full solution or refutation.**\n\nDeVos and Šámal establish a uniform girth-17 theorem with the Clebsch graph as target, which is a strict relaxation of a homomorphism to C5.\n\n**What remains.**\n\nProve an absolute girth threshold forcing a homomorphism to C5 or construct cubic graphs of arbitrarily large girth with no such homomorphism.\n\n**Sources checked.**\n\n- Open Problem Garden, Pentagon problem (OPG-167), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/pentagon_problem\n Evidence used: Maintains the exact C5-homomorphism question and its known relaxations.\n- M. DeVos and R. Šámal, High-girth cubic graphs are homomorphic to the Clebsch graph, Journal of Graph Theory 66 (2011), 241-259. (primary): https://doi.org/10.1002/jgt.20580\n Evidence used: Proves the girth-17 Clebsch-target theorem and explicitly describes it as a weak version of the Pentagon Problem.\n\n**Review notes.** The nonbreaking-space marker after 'enough' is a typography artifact; the intended existential girth threshold is clear and the statement was not changed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3208, "problem_number": "OPG-412", "title": "Mapping planar graphs to odd cycles", "statement": "Conjecture Every planar graph of girth $\\ge 4k$ has a homomorphism to $C_{2k+1}$.", "background": "Source: Open Problem Garden. Original node ID: 412. URL: http://www.openproblemgarden.org/op/mapping_planar_graphs_to_odd_cycles.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/mapping_planar_graphs_to_odd_cycles\n- Author(s): Jaeger, Francois\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: girth; homomorphism; planar graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 24th, 2007 by mdevos\n\nProblem-page discussion:\nThis conjecture is Jaeger's modular orientation conjecture restricted to planar graphs and then dualized. To see this duality, first note that circular coloring and circular flows are dual for planar graphs, and then observe that $(2k+1)$-orientations are equivalent to $2 + \\frac{1}{k}$-flows and $2 + \\frac{1}{k}$-colorings are equivalent to homomorphisms to $C_{2k+1}$. So if $G$ and $G^*$ are dual planar graphs, then we have the following equivalences.\n\n- $G$ has a $(2k+1)$-orientation.\n- $G$ has a $2 + \\frac{1}{k}$-flow.\n- $G^*$ has a $2 + \\frac{1}{k}$-coloring.\n- $G^*$ has a homomorphism to $C_{2k+1}$.\n\nThere is an easy family of graphs which show that the above conjecture (if true) is best possible. Let $H_k$ be the graph obtained from an odd circuit of length $4k-1$ by adding a new vertex $u$ joined to every existing vertex by a path of length $2k-1$. Now, $H_k$ is a planar graph of girth $4k-1$, but there is no homomorphism from $H_k$ to $C_{2k+1}$. To see the latter claim, suppose (for a contradiction) that such a homomorphsim $f$ exists, let $C$ be the unique circuit of $H_k \\setminus u$ and let $a=f(u)$. Now, no vertex in $C$ can map to $a$ since every such vertex is distance $2k-1$ from $u$. However we must then have a homomorphism from $C$ to $C_{2k+1} \\setminus a$, which is impossible since $C$ is an odd circuit and $C_{2k+1} \\setminus u$ is bipartite.\n\nThe k=1 case of the above conjecture asserts that every (loopless) triangle free planar graph has a homomorphism to the triangle. In other words, every (loopless) triangle free planar graph is 3-colorable. This is a well known theorem of Grotszch. For every k>1, the above conjecture is still open. Actually, I think this conjecture is already quite interesting for k=2. One reason is that this case of the conjecture implies the 5-color theorem for planar graphs. To see this implication, suppose that the above conjecture is true for k=2, let G be a simple loopless planar graph, and let G' be the graph obtained from G by subdividing each edge two times. Now, G' has girth at least 9, so by our assumption there is a homomorphism from G' to C_5. It is easy to see that adjacent vertices of G must map to different vertices of C_5 under this homomorphism. Thus, we have a proper 5-coloring of G as desired.\n\nLet us call a homomorphism to $C_{2k+1}$ a $C_{2k+1}$-coloring. It is quite easy to show that every planar graph of girth > 10k has a $C_{2k+1}$-coloring. This follows from a simple degeneracy argument: Every such (nonempty) graph must have a either a vertex of degree $\\le 1$, or a path $P$ of length $2k-1$ all of whose internal vertices have degree two. Both of these configurations are reducible, in the sense that we may delete either a vertex of degree $\\le 1$ or the interior vertices of $P$ and then extend any $C_{2k+1}$-coloring of the resulting graph to a $C_{2k+1}$-coloring of the original. By more complicated, but similar degeneracy arguments, we can approach this conjecture. To my knowledge, the best result to date is as follows.\n\nTheorem (Borodin, Kim, Kostochka, West) Every planar graph of girth $\\ge \\frac{20k-2}{3}$ has a homomorphism to $C_{2k+1}$.\n\nFor the special case of the conjecture when $k=2$, Matt DeVos and Adam Deckelbaum have an unpublished improvement showing that every planar graph with odd girth $\\ge 11$ has a homomorphism to $C_5$.\n\nBibliography:\n[BKKW] O. V. Borodin, S. J. Kim, A. V. Kostochka, D. B. West, Homomorphisms from sparse graphs with large girth. Dedicated to Adrian Bondy and U. S. R. Murty. J. Combin. Theory Ser. B 90 (2004), no. 1, 147--159. MathSciNet\n\n[Ja] F. Jaeger, On circular flows in graphs in Finite and Infinite Sets, volume 37 of Colloquia Mathematica Societatis Janos Bolyai, edited by A. Hajnal, L. Lovasz, and V.T. Sos. North-Holland (1981) 391-402.\n\n[Zh] X. Zhu, Circular chromatic number of planar graphs of large odd girth, Electronic Journal of Combinatorics Vol. 8 no. 1 (2001).\n\nRelated:\nRelated problems\nJaeger's modular orientation conjecture\n\nDiscussion links:\n- Jaeger's modular orientation conjecture: http://www.openproblemgarden.org/?q=node/130\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2041323\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"Mapping planar graphs to odd cycles\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Jaeger-Zhang odd-cycle homomorphism conjecture remains open for general k>1; a general odd-girth 6k+1 theorem and sharper target-specific results approach but do not reach the conjectured 4k+1 threshold.\n\n**Verified partial progress.**\n\n- The k=1 case is Grötzsch's theorem.\n- Lovász-Thomassen-Wu-Zhang proved that odd girth at least 6k+1 suffices in general.\n- Cranston-Li-Wang-Wei proved that odd girth 23 suffices for a homomorphism to C_9.\n\n**Full solution or refutation.**\n\nNo proof at the conjectured threshold for every k was verified.\n\n**What remains.**\n\nClose the gap between odd-girth 6k+1 (with target-specific improvements) and 4k+1 for all k>1.\n\n**Sources checked.**\n\n- Daniel W. Cranston, Jiaao Li, Zhouningxin Wang, and Chunyan Wei, Planar Graphs with Homomorphisms to the 9-cycle, arXiv:2402.02689 (2024). (primary): https://arxiv.org/abs/2402.02689\n Evidence used: States the modern Jaeger-Zhang conjecture, records the general 6k+1 bound, and proves the C_9 odd-girth-23 special case.\n- Open Problem Garden, Mapping planar graphs to odd cycles (node 412). (maintained_tracker): https://www.openproblemgarden.org/op/mapping_planar_graphs_to_odd_cycles\n Evidence used: Original formulation, duality, sharpness construction, and classical bounds.\n\n**Review notes.** Current papers usually state odd girth >=4k+1. This is equivalent to the input's girth >=4k formulation after observing that bipartite graphs map to an edge of C_{2k+1}.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3209, "problem_number": "OPG-434", "title": "Weak pentagon problem", "statement": "Conjecture If $G$ is a cubic graph not containing a triangle, then it is possible to color the edges of $G$ by five colors, so that the complement of every color class is a bipartite graph.", "background": "Source: Open Problem Garden. Original node ID: 434. URL: http://www.openproblemgarden.org/op/weak_pentagon_problem.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/weak_pentagon_problem\n- Author(s): Samal, Robert\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: Clebsch graph; cut-continuous mapping; edge-coloring; homomorphism; pentagon\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 13th, 2007 by Robert Samal\n\nProblem-page discussion:\nThis conjecture has several reformulations: the conclusion of the conjecture can be replaced by either of the following:\n\n- $G$ has a homomorphism to the Clebsch graph.\n- there is a cut-continuous mapping from $G$ to $C_5$.\n\nFor the latter variant, few definitions are in place. A cut-continuous mapping from a graph~ $G$ to a graph~ $H$ is a mapping $f: E(G) \\to E(H)$ such that the preimage of every cut in~ $H$ is a cut in~ $G$. Here, by a cut in~ $H$ we mean the edge-set of a spanning bipartite subgraph of~ $H$---less succinctly, it is the set of all edges leaving some subset of vertices of~ $H$.\n\nCut-continuous mappings are closely related with graph homomorphisms (see [DNR], [S]). In particular, every homomorphism from~ $G$ to~ $H$ naturally induces a cut-continuous mapping from~ $G$ to~ $H$; thus, the presented conjecture can be thought of as a weaker version of Nesetril's Pentagon problem.\n\nWe mention a generalization of the conjecture, that deals with longer cycles/larger number of colors. The $n$-dimensional projective cube, denoted $PQ_n$, is the simple graph obtained from the $(n+1)$-dimensional cube~ $Q_{n+1}$ by identifying pairs of antipodal vertices (vertices that differ in all coordinates). Note that $PQ_4$ is the Clebsch graph.\n\nQuestion What is the largest integer $k$ with the property that all cubic graphs of sufficiently high girth have a homomorphism to $PQ_{2k}$?\n\nAgain, the question has several reformulations due to the following simple proposition.\n\nProposition For every graph $G$ and nonnegative integer $k$, the following properties are equivalent.\n\n- There exists a coloring of~ $E(G)$ by $2k+1$ colors so that the complement of every color class is a bipartite graph.\n- $G$ has a homomorphism to $PQ_{2k}$\n- $G$ has a cut-continuous mapping to~ $C_{2k+1}$\n\nThere are high-girth cubic graphs with the largest cut of size less then $0.94\\cdot |E|$. Such graphs do not admit a homomorphism to $PQ_{2k}$ for any $k \\ge 8$, so there is indeed some largest integer~ $k$ in the above question. To bound this largest~ $k$ from below, recall that every cubic graph maps homomorphically to $K_4 = PQ_2$. Moreover, it is known [DS] that cubic graphs of girth at least 17 admit a homomorphism to $PQ_4$ (the Clebsch graph). This shows $k\\ge 2$ (and also provides a support for the main conjecture).\n\nBibliography:\n[DNR] Matt DeVos, Jaroslav Nesetril and Andre Raspaud: On edge-maps whose inverse preserves flows and tensions, \\MRref{MR2279171}\n\n*[DS] Matt Devos, Robert Samal: \\arXiv[High Girth Cubic Graphs Map to the Clebsch Graph}{math.CO/0602580}\n\n[S] Robert Samal, On XY mappings, PhD thesis, Charles University 2006, tech. report\n\nRelated:\nRelated problems\nPentagon problem\n\nDiscussion links:\n- Clebsch graph: http://en.wikipedia.org/wiki/Gallery_of_named_graphs\n- graph homomorphisms: http://en.wikipedia.org/wiki/graph homomorphism\n- Pentagon problem: http://www.openproblemgarden.org/?q=node/167\n- girth: http://en.wikipedia.org/wiki/girth\n\nBibliography links:\n- tech. report: http://kam.mff.cuni.cz/%7Ekamserie/serie/clanky/2006/s772.ps\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Weak pentagon problem\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The weak pentagon conjecture remains open for arbitrary triangle-free cubic graphs, but is proved for maximum-degree-three graphs of girth at least 17 and for all triangle-free K_5-minor-free graphs.\n\n**Verified partial progress.**\n\n- DeVos-Šámal prove the Clebsch homomorphism for maximum degree at most three and girth at least 17.\n- Naserasr-Nigussie-Škrekovski prove it for every triangle-free graph with no K_5 minor.\n\n**Full solution or refutation.**\n\nNo result covering all triangle-free cubic graphs was verified.\n\n**What remains.**\n\nHandle triangle-free cubic graphs that have short cycles and K_5 minors, or construct a counterexample.\n\n**Sources checked.**\n\n- Matt DeVos and Robert Šámal, High-girth cubic graphs are homomorphic to the Clebsch graph, Journal of Graph Theory 66 (2011), 241-259. (primary): https://doi.org/10.1002/jgt.20580\n Evidence used: Proves the equivalent five-edge-coloring/Clebsch-homomorphism assertion at girth at least 17.\n- Reza Naserasr, Yared Nigussie, and Riste Škrekovski, Homomorphisms of triangle-free graphs without a K5-minor, Discrete Mathematics 309 (2009), 5789-5798. (primary): https://doi.org/10.1016/j.disc.2009.04.032\n Evidence used: Extends the Clebsch-homomorphism theorem to all triangle-free K_5-minor-free graphs.\n- Graph-theory open problems, Weak pentagon problem. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/weak_pentagon_problem/\n Evidence used: Current tracker explicitly retains open status in full generality.\n\n**Review notes.** No formulation defect detected; the equivalence with homomorphism to the Clebsch graph is standard and documented in the cited papers.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3210, "problem_number": "OPG-37232", "title": "Algorithm for graph homomorphisms", "statement": "Question\n\nIs there an algorithm that decides, for input graphs $G$ and $H$, whether there exists a homomorphism from $G$ to $H$ in time $O(c^{|V(G)|+|V(H)|})$ for some constant $c$?", "background": "Source: Open Problem Garden. Original node ID: 37232. URL: http://www.openproblemgarden.org/op/algorithm_for_graph_homomorphisms.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/algorithm_for_graph_homomorphisms\n- Author(s): Fomin, Fedor V.; Heggernes, Pinar; Kratsch, Dieter\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: algorithm; Exponential-time algorithm; homomorphism\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 8th, 2010 by jfoniok\n\nProblem-page discussion:\nAn affirmative answer is known in several cases: if $H=K_k$ (graph coloring) [L], [BH], [K]; if $H$ has bounded treewidth [FHK]; if $H$ has bounded cliquewidth [W].\n\nBibliography:\n[BH] Andreas Björklund, Thore Husfeldt: Inclusion--Exclusion Algorithms for Counting Set Partitions, Proc. FOCS'06 (2006).\n\n*[FHK] Fedor V. Fomin, Pinar Heggernes, Dieter Kratsch: Exact Algorithms for Graph Homomorphisms, Theory Comput. Syst. 41 (2007), no. 2, 381--393. MathSciNet\n\n[K] Mikko Koivisto: An $O^\\ast(2^n)$ Algorithm for Graph Coloring and Other Partitioning Problems via Inclusion--Exclusion, Proc. FOCS'06 (2006).\n\n[L] Eugene L. Lawler: A note on the complexity of the chromatic number problem, Information Processing Lett. 5 (1976), no. 3, 66--67. MathSciNet\n\n[W] Magnus Wahlström: New Plain-Exponential Time Classs for Graph Homomorphism, CSR2009, LNCS5675 (2009), 346--355.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2329330\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0464675\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Algorithm for graph homomorphisms\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Under ETH, no uniform algorithm of the requested c^(|V(G)|+|V(H)|) form exists: setting |V(G)| and |V(H)| comparable would contradict the established 2^(Omega(n log h)) lower bound.\n\n**Verified partial progress.**\n\n- Plain-exponential algorithms exist for important restricted target classes, such as bounded-treewidth targets.\n\n**Full solution or refutation.**\n\nThis is a conditional negative answer, not an unconditional impossibility theorem.\n\n**What remains.**\n\nEither refute ETH or obtain an unconditional lower bound matching the requested uniform runtime barrier.\n\n**Sources checked.**\n\n- F. V. Fomin, A. Golovnev, A. S. Kulikov and I. Mihajlin, Tight Bounds for Subgraph Isomorphism and Graph Homomorphism, SODA 2016, arXiv:1507.03738. (primary): https://arxiv.org/abs/1507.03738\n Evidence used: The abstract proves an ETH lower bound h^(Omega(n/log log h)) for HOM, ruling out single-exponential dependence when h grows.\n\n**Review notes.** Conditional ETH interpretation explicitly flagged; source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3211, "problem_number": "OPG-37619", "title": "Circular choosability of planar graphs", "statement": "Let $G = (V, E)$ be a graph. If $p$ and $q$ are two integers, a $(p,q)$-colouring of $G$ is a function $c$ from $V$ to $\\{0,\\dots,p-1\\}$ such that $q \\le |c(u)-c(v)| \\le p-q$ for each edge $uv\\in E$. Given a list assignment $L$ of $G$, i.e.~a mapping that assigns to every vertex $v$ a set of non-negative integers, an $L$-colouring of $G$ is a mapping $c: V \\to N$ such that $c(v)\\in L(v)$ for every $v\\in V$. A list assignment $L$ is a $t$- $(p,q)$-list-assignment if $L(v) \\subseteq \\{0,\\dots,p-1\\}$ and $|L(v)| \\ge tq$ for each vertex $v \\in V$. Given such a list assignment $L$, the graph G is $(p,q)$- $L$-colourable if there exists a $(p,q)$- $L$-colouring $c$, i.e. $c$ is both a $(p,q)$-colouring and an $L$-colouring. For any real number $t \\ge 1$, the graph $G$ is $t$- $(p,q)$-choosable if it is $(p,q)$- $L$-colourable for every $t$- $(p,q)$-list-assignment $L$. Last, $G$ is circularly $t$-choosable if it is $t$- $(p,q)$-choosable for any $p$, $q$. The circular choosability (or circular list chromatic number or circular choice number) of G is $$cch(G):= \\inf\\{t \\ge 1: G \\text{ is circularly$t$-choosable}\\}.$$\n\nProblem What is the best upper bound on circular choosability for planar graphs?", "background": "Source: Open Problem Garden. Original node ID: 37619. URL: http://www.openproblemgarden.org/op/circular_choosability_of_planar_graphs.\n\nSource subject path: Graph Theory > Coloring > Homomorphisms.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/circular_choosability_of_planar_graphs\n- Author(s): Mohar, Bojan\n- Subject(s): Graph Theory; Coloring; Homomorphisms\n- Keywords: choosability; circular colouring; planar graphs\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: August 23rd, 2012 by rosskang\n\nProblem-page discussion:\nThe problem was first posed in 2003 by Mohar (Problem 4 of link*) who suggested the answer should be between 4 and 5.\n\nSome time later, Havet, Kang, Müller, and Sereni [HKMS] showed that in fact the answer is somewhere between 6 and 8. The upper bound extends a celebrated planar choosability proof due to Thomassen [T]. The lower bound is by way of an elementary, though rather large, construction.\n\nBibliography:\n[HKMS] F. Havet, R. J. Kang, T. Müller, and J.-S. Sereni. Circular choosability. J. Graph Theory 61 (2009), no. 4, 241--270.\n\n[T] C. Thomassen. Every planar graph is 5-choosable. J. Combinatorial Theory B 62 (1994) 180--181\n\nDiscussion links:\n- link: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0201ChoosabilityCircular.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"Circular choosability of planar graphs\" in Graph Theory; Coloring; Homomorphisms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For the supremum tau of circular choice number over planar graphs, the best verified bounds remain 6 <= tau <= 8; the exact best upper bound is unknown.\n\n**Verified partial progress.**\n\n- Havet, Kang, Muller, and Sereni prove the lower bound 6 and universal planar upper bound 8.\n\n**Full solution or refutation.**\n\nThe literature narrows the planar circular-choosability problem to the interval from 6 to 8 but does not determine the optimum.\n\n**What remains.**\n\nLower the universal upper bound below 8, raise the lower bound above 6, or determine the exact supremum.\n\n**Sources checked.**\n\n- F. Havet, R. J. Kang, T. Muller, and J.-S. Sereni, Circular choosability, Journal of Graph Theory 61 (2009), 241-270, DOI 10.1002/jgt.20375. (primary): https://doi.org/10.1002/jgt.20375\n Evidence used: Proves the stated bounds 6 <= tau <= 8 for planar graphs.\n- Ross J. Kang, research page, accessed 2026-08-17. (authoritative_secondary): https://staff.fnwi.uva.nl/j.r.kang/\n Evidence used: The coauthor's current research summary continues to list planar circular choosability as lying between 6 and 8.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3212, "problem_number": "OPG-439", "title": "Graceful Tree Conjecture", "statement": "Conjecture All trees are graceful", "background": "Source: Open Problem Garden. Original node ID: 439. URL: http://www.openproblemgarden.org/op/graceful_tree_conjecture.\n\nSource subject path: Graph Theory > Coloring > Labeling.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/graceful_tree_conjecture\n- Subject(s): Graph Theory; Coloring; Labeling\n- Keywords: combinatorics; graceful labeling\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 13th, 2007 by kintali\n\nProblem-page discussion:\nLabel the vertices of a simple undirected graph $G(V,E)$ (where $|V| = n$ and $|E| = m$ ) with integers from $0$ to $m$. Now label each edge with absolute difference of the labels of its incident vertices. The labeling is said to be graceful if the edges are labelled $1$ through $m$ inclusive (with no number repeated).\n\nA graph is called graceful if it has at least one such labeling. This labeling was originally introduced in 1967 by Rosa. The name graceful labeling was coined later by Golomb.\n\nGracefully labeled graphs serve as models in a wide range of applications including coding theory and communication network addressing.\n\nThe graceful labeling problem is to determine which graphs are graceful. It is conjectured (by Kotzig, Ringel and Rosa) that all trees are graceful.\n\nDespite numerous (more than 200) publications on graceful labeling for over three decades, only a very restricted classes of trees (and also of some other graphs) have been shown to be graceful. These restricted classes include paths, stars, complete bipartite graphs, prism graphs, wheel graphs, caterpillar graphs, olive trees, and symmetrical trees.\n\nComments:\n- November 29th, 2020 | Anonymous | Bibliography: Many, many references on the topic appear in Joseph Gallian's \"A Dynamic Survey of Graph Labeling\", section 2 ( https://www.combinatorics.org/ds6 ).\n- September 29th, 2014 | Anonymous | Graceful Tree examples: If you want to develop intuition for a proof, feel free to use this program I wrote (http://bl.ocks.org/NPashaP/7683252). Good luck.\n- November 30th, 2009 | Anonymous | applications of graceful graphs: can you explain some applications of graceful graphs\n- June 12th, 2010 | Anonymous | Apllication of graceful labeling: I want the application of graceful label\n- November 30th, 2009 | Anonymous | graph theory: Can you pls send some applications of gracefulgraphs\n- May 5th, 2009 | Anonymous | Tentative proof: I just saw this paper on arxiv, entitled \"A complete proof of The Graceful Tree Conjecture using the concept of Edge Degree\".\n\nI'm surprised to see such a short proof for such a long-standing open problem, but surely people who are a lot more into the subject than I will be able to provide more constructive comments on the paper.\n- May 6th, 2009 | Robert Samal | Re: Tentative proof: It's a bit worrisome that this is already the eight version on the arXiv... this probably means that the previous seven versions had some sort of flaw in it... I haven't read the paper though...\n- April 14th, 2009 | Anonymous | injective labeling: Rosa (1967) required the labeling to be injective.\n- August 1st, 2007 | Anonymous | Err..: The complete bipartite graph isn't a tree.\n- August 2nd, 2007 | Anonymous | not a problem: graceful labellings are defined for arbitrary graphs, not just trees.\n- August 2nd, 2007 | Anonymous | no duh..: I was just pointing out the statement\n\n\"...only a very restricted classes of *trees* have been shown to be graceful. *These* restricted classes include...the complete bipartite graph\".\n\ndoesn't make sense.\n- August 4th, 2007 | rs | Re: wording: Truly, this was unfortunate choice of wording; I corrected it. (Btw to provide a graceful labeling for complete bipartite graphs is quite an easy exercise.)\n- April 12th, 2008 | Anonymous | graceful labeling of complete bipartite graphs: how to give a graceful labeling to complete bipartite graph. Can u suggest some efficient algorithm or scheme for that Plz email it to rocky_blt@yahoo.co.in if u have any...\n\nthanx\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Graceful Tree Conjecture\" in Graph Theory; Coloring; Labeling, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Graceful Tree Conjecture remains open, but every sufficiently large tree now has gracesize at least (1-epsilon)n, and earlier work gives almost-graceful labelings for broad bounded-degree families.\n\n**Verified partial progress.**\n\n- Letzter-Pokrovskiy-Williams prove every sufficiently large n-vertex tree has gracesize at least (1-epsilon)n.\n- Adamaszek-Allen-Grosu-Hladký prove approximate graceful labelings for controlled-maximum-degree trees, hence asymptotically almost all trees.\n\n**Full solution or refutation.**\n\nThe exact requirement of n-1 distinct edge differences for every n-vertex tree remains unproved.\n\n**What remains.**\n\nRemove the epsilon loss in the uniform gracesize bound, including all finite exceptional trees.\n\n**Sources checked.**\n\n- Shoham Letzter, Alexey Pokrovskiy, and Ella Williams, On the gracesize of trees, arXiv:2511.11331 (2025). (primary): https://arxiv.org/abs/2511.11331\n Evidence used: Proves the uniform asymptotic lower bound on gracesize and states the exact conjecture as longstanding.\n- Anna Adamaszek, Peter Allen, Codrut Grosu, and Jan Hladký, Almost all trees are almost graceful, arXiv:1608.01577. (primary): https://arxiv.org/abs/1608.01577\n Evidence used: Proves an approximate labeling theorem for trees of controlled maximum degree.\n\n**Review notes.** The statement is terse and relies on the standard definition of graceful supplied by the source background; no accepted claimed full proof was found.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3213, "problem_number": "OPG-37323", "title": "Good Edge Labelings", "statement": "Question What is the maximum edge density of a graph which has a good edge labeling?\n\nWe say that a graph is good-edge-labeling critical, if it has no good edge labeling, but every proper subgraph has a good edge labeling.\n\nConjecture For every $c<4$, there is only a finite number of good-edge-labeling critical graphs with average degree less than $c$.", "background": "Source: Open Problem Garden. Original node ID: 37323. URL: http://www.openproblemgarden.org/op/good_edge_labelings.\n\nSource subject path: Graph Theory > Coloring > Labeling.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/good_edge_labelings\n- Author(s): Araújo, Julio; Cohen, Nathann; Giroire, Frédéric; Havet, Frédéric\n- Subject(s): Graph Theory; Coloring; Labeling\n- Keywords: good edge labeling, edge labeling\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 30th, 2011 by DOT\n\nProblem-page discussion:\nLet $G$ be a finite undirected simple graph. A good edge labeling of $G$ is an assignment of distinct numbers to the edges such that every cycle has at least two local maxima. (The distinctness of the labels is required only to make the term `local maximum' unambiguous.)\n\nEquivalently, a labeling of the edges is good, if for every pair of distinct vertices $u,v$, there is at most one increasing path from $u$ to $v$.\n\nHaving a good edge labeling is inherited by subgraphs.\n\nIt is easy to verify that the graphs $K_3$ and $K_{2,3}$ have no good edge labeling. In [ACGH2] an infinite class of graphs without good edge labelings is given, none of whom is a subgraph of the other. In [BFT] contains an example of a minimal graph without good edge labeling which as average degree < 3 (thus refuting an earlier conjecture saying that a good-edge-labeling critical graph with average degree less than three is either $K_3$ or $K_{2,3}$ ). In that same paper it is shown that every such graph must have girth at most 4.\n\nGood edge labeling of graphs was introduced in [BCP] in the context of the so-called Routing and Wavelength Assignment (RWA) problem. The problems above are proposed in [ACGH1] and [ACGH2]. There the algorithmic problem of determining whether a graph has a good edge labeling is shown to be NP-hard. Moreover, the authors also prove that every planar graph with girth at least six has a good edge labeling.\n\nBibliography:\n[BCP] J-C. Bermond, M. Cosnard, and S. Pérennes. Directed acyclic graphs with unique path property. Technical report 6932, INRIA, May 2009\n\n[ACGH1] J. Araújo, N. Cohen, F. Giroire, F. Havet. Good edge-labelling of graphs. (English summary) LAGOS'09—V Latin-American Algorithms, Graphs and Optimization Symposium, 275–280, Electron. Notes Discrete Math., 35, Elsevier Sci. B. V., Amsterdam, 2009. MathSciNet\n\n[ACGH2*] J. Araujo, N. Cohen, F. Giroire, and F. Havet. Good edge-labelling of graphs. Research Report 6934, INRIA, 2009.\n\n[BFT] M. Bode, B. Farzad, D.O. Theis. Good edge-labelings and graphs of girth at least 5. (arXiv:1109.1125)\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2579442\n\nComments:\n- February 16th, 2013 | Anonymous | First question (almost) solved: Mehrabian, Mitsche, and Pralat showed that any $n$-vertex graph with a good edge-labelling has at most $n \\log_2 n$ edges, and that for each $n$ there is an $n$-vertex graphs with a good edge-labelling having $n \\log_2 n - O(n)$ edges.\n\nhttp://arxiv.org/abs/1211.2641\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Good Edge Labelings\" in Graph Theory; Coloring; Labeling, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The density question is asymptotically settled and exact for infinitely many orders, while the distinct conjecture about critical graphs of average degree below 4 remains unresolved.\n\n**Verified partial progress.**\n\n- Every n-vertex graph admitting a good edge-labeling has at most n log_2(n)/2 edges.\n- The upper bound is attained for infinitely many n, and every n admits a construction with n log_2(n)/2 - O(n) edges.\n\n**Full solution or refutation.**\n\nThe first question has a sharp asymptotic answer, but no proof of the stated finiteness conjecture for every c < 4 was verified.\n\n**What remains.**\n\nDetermine the exact maximum for all n and prove or refute finiteness of good-edge-labeling critical graphs below every average-degree threshold c < 4.\n\n**Sources checked.**\n\n- A. Mehrabian, D. Mitsche and P. Prałat, On the Maximum Density of Graphs with Good Edge-Labellings, arXiv:1211.2641 (2012). (primary): https://arxiv.org/abs/1211.2641\n Evidence used: The abstract proves the n log_2(n)/2 upper bound, equality for infinitely many n, and n log_2(n)/2 - O(n) constructions for every n.\n- Graph-theory open problems, Good Edge Labelings (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/good_edge_labelings/\n Evidence used: The maintained page records the critical-graph conjecture as unresolved and separates recent algorithmic work from the extremal questions.\n\n**Review notes.** The Open Problem Garden comment misstated the factor 1/2; the primary paper's bound is used.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3214, "problem_number": "OPG-126", "title": "5-flow conjecture", "statement": "Conjecture Every bridgeless graph has a nowhere-zero 5-flow.", "background": "Source: Open Problem Garden. Original node ID: 126. URL: http://www.openproblemgarden.org/op/5_flow_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/5_flow_conjecture\n- Author(s): Tutte, William T.\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: cubic; nowhere-zero flow\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nFor planar graphs, this theorem follows from flow/coloring duality, and the Five color theorem (every loopless planar graph is 5-colorable). In light of this, we may view this conjecture as a widesweeping generalization of the 5-color-theorem. The Petersen graph does not have a nowhere-zero 4-flow, which shows that this conjecture (if true) is best possible.\n\nIt is far from obvious that there should exist a fixed number $k$ so that every bridgeless graph has a nowhere-zero $k$-flow. Indeed, this weaker conjecture was also made by Tutte, but was resolved by Kilpatrick [K] and independently Jaeger [J], who both proved that bridgeless graphs have nowhere-zero 8-flows. Seymour [S] improved upon this result by showing that bridgeless graphs have nowhere-zero 6-flows.\n\nBibliography:\n[J] F. Jaeger, Flows and Generalized Coloring Theorems in Graphs, J. Combinatorial Theory Ser. B 26 (1979) 205-216. MathSciNet\n\n[K] P.A. Kilpatrick, Tutte's First Colour-Cycle Conjecture, Thesis, Cape Town (1975).\n\n[S] P.D. Seymour, Nowhere-Zero 6-Flows, J. Combinatorial Theory Ser. B 30 (1981) 130-135. MathSciNet\n\n[T54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomials, Canad. J. Math. 6 (1954) 80-91. MathSciNet\n\n[Tt66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50. MathSciNet\n\nSource links:\n- bridgeless: http://en.wikipedia.org/wiki/bridge (graph theory)\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nDiscussion links:\n- planar graphs: http://en.wikipedia.org/wiki/planar graphs\n- flow/coloring duality: http://en.wikipedia.org/wiki/nowhere-zero flows\n- Five color theorem: http://en.wikipedia.org/wiki/Five color theorem\n- Petersen graph: http://en.wikipedia.org/wiki/Petersen graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0532588\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0615308\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0061366\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0194363\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"5-flow conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Tutte's 5-flow conjecture remains open; Seymour's 6-flow theorem and many special cases do not settle it.\n\n**Verified partial progress.**\n\n- Seymour's 6-flow theorem is a weaker general result.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nImprove flow reduction methods from six to five.\n\n**Sources checked.**\n\n- Open Problem Garden, node 126 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Open Problem Garden retains the 5-flow conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3215, "problem_number": "OPG-127", "title": "4-flow conjecture", "statement": "Conjecture Every bridgeless graph with no Petersen minor has a nowhere-zero 4-flow.", "background": "Source: Open Problem Garden. Original node ID: 127. URL: http://www.openproblemgarden.org/op/4_flow_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/4_flow_conjecture\n- Author(s): Tutte, William T.\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: minor; nowhere-zero flow; Petersen graph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nIt is a consequence of a theorem of Tutte that a cubic graph has a nowhere-zero 4-flow if and only if it is 3-edge-colorable. Thus, the 4-flow conjecture implies that every bridgeless cubic graph with no Petersen minor is 3-edge-colorable (another conjecture of Tutte). Note that the Four Color Theorem is equivalent to the assertion that planar cubic graphs without bridges are 3-edge-colorable, so even this weaker conjecture is a strengthening of the Four Color Theorem. This weaker conjecture was recently proved by Robertson, Seymour, and Thomas [RST]. Their proof involves a reduction to the case of nearly planar graphs, and then an application of 4-color-theorem type techniques (computer assisted) to color these graphs.\n\nMost conjectures about flows can be easily reduced to the case of cubic graphs by splitting arguments. The idea is to take a vertex $v$ incident with edges $e_1,\\ldots,e_k$ and \"split\" $v$, that is, replace $v$ by two new vertices $v_1$ and $v_2$, and for every edge $e_i$ join it to either $v_1$ or $v_2$ (sometimes the edge $v_1 v_2$ is also added). For instance, this technique can be used to reduce the general 5-flow conjecture down to the special case of cubic graphs. Unfortunately, that technique does not apply here, since splitting a vertex may introduce a Petersen minor.\n\nPetersen's graph is not an apex graph (deleting any vertex still leaves a nonplanar graph). It follows that no apex graph can have a Petersen minor, so the above conjecture implies that every bridgeless apex graph has a nowhere-zero 4-flow. By splitting the vertices which lie in the plane this can be reduced to the special case where all vertices which lie in the plane have degree 3. This is then equivalent to the following old conjecture of Gr\\\"{o}tzsch.\n\nConjecture (Gr\\\"{o}tzsch) If $G$ is a 2-connected connected planar graph of maximum degree 3, then $G$ is 3-edge-colorable unless it has exactly one vertex of degree 2.\n\nBibliography:\n[AH] K. Appel, W. Haken, Every Planar Map is Four Colorable, Bull. Amer. Math. Soc. 82 (1976) 711-712. MathSciNet\n\n[RSST] N. Robertson, D.P. Sanders, P.D. Seymour, and R. Thomas, A New Proof of the Four-Color Theorem, Electron. Res. Announc., Am. Math. Soc. 02, no 1 (1996) 17-25. MathSciNet\n\n[RST] N. Robertson, P.D. Seymour, and R. Thomas, Tutte's edge-colouring conjecture. J. Combin. Theory Ser. B 70 (1997), no. 1, 166--183. MathSciNet\n\n[Tut54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomials, Canad. J. Math. 6 (1954) 80-91. MathSciNet\n\n[Tut66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50. MathSciNet\n\nSource links:\n- bridgeless: http://en.wikipedia.org/wiki/bridge (graph theory)\n- Petersen: http://en.wikipedia.org/wiki/petersen graph\n- minor: http://en.wikipedia.org/wiki/minor (graph theory)\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nDiscussion links:\n- cubic graph: http://en.wikipedia.org/wiki/cubic graph\n- edge-colorable: http://en.wikipedia.org/wiki/edge coloring\n- Four Color Theorem: http://en.wikipedia.org/wiki/Four Color Theorem\n- planar: http://en.wikipedia.org/wiki/planar graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0424602\n- A New Proof of the Four-Color Theorem: http://www.ams.org/era/1996-02-01/S1079-6762-96-00003-0/S1079-6762-96-00003-0.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1405965\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1441265\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0061366\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0194363\n\nComments:\n- April 11th, 2009 | Anonymous | fix the problm: fix the problm\n- April 12th, 2009 | Anonymous | What is wrong?: Perhaps you could elaborate.. what needs to be fixed?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"4-flow conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the Petersen-minor 4-flow conjecture was verified.\n\n**Verified partial progress.**\n\n- The source tracker supplies the current formulation and relevant context.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nUse Petersen-minor structure theory to construct a 4-flow.\n\n**Sources checked.**\n\n- Open Problem Garden, node 127 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Open Problem Garden retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3216, "problem_number": "OPG-128", "title": "3-flow conjecture", "statement": "Conjecture Every 4-edge-connected graph has a nowhere-zero 3-flow.", "background": "Source: Open Problem Garden. Original node ID: 128. URL: http://www.openproblemgarden.org/op/3_flow_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/3_flow_conjecture\n- Author(s): Tutte, William T.\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: nowhere-zero flow\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nGrotzsch proved that every triangle free (and loopless) planar graph is 3-colorable. By flow/coloring duality, this is equivalent to the statement that every 4-edge-connected planar graph has a nowhere-zero 3-flow. The 3-flow conjecture asserts that this is still true without the assumption of planarity.\n\nJaeger proved that 4-edge-connected graphs have nowhere-zero 4-flows, but very little is known about nowhere-zero 3-flows. In particular, the following weak version of the 3-flow conjecture is still wide open.\n\nConjecture (The weak 3-flow conjecture (Jaeger)) There exists a fixed integer $k$ so that every $k$-edge-connected graph has a nowhere-zero 3-flow.\n\nLai and Zhang [LZ] have proved that if $G$ has $n$ vertices and edge-connectivity at least $4 \\log_2(n)$ then $G$ has a nowhere-zero 3-flow. A similar result (edge connectivity at least $4 \\log(n) + 2$ ) also follows from a theorem of Alon, Linial, and Meshulam [ALM] on additive bases of vector spaces.\n\nBibliography:\n[ALM] N. Alon, N. Linial, and R. Meshulam Additive Bases of Vector Spaces over Prime Fields J. Combinatorial Theory Ser. A 57 (1991), 203-210.\n\n[J] F. Jaeger, Flows and Generalized Coloring Theorems in Graphs, J. Combinatorial Theory Ser. B 26 (1979) 205-216.\n\n[LZ] H.J. Lai and C.Q. Zhang, Nowhere-Zero 3-Flows of Highly Connected Graphs, Discrete Math 110 (1992) 179-183.\n\n[T54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomial, Canad. J. Math. 6 (1954) 80-91.\n\n[T66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50.\n\nSource links:\n- edge-connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nDiscussion links:\n- planar graph: http://en.wikipedia.org/wiki/planar graph\n- colorable: http://en.wikipedia.org/wiki/graph coloring\n- flow/coloring duality: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nComments:\n- April 12th, 2011 | Flo Pfender | weak 3-flow conjecture proved: Carsten Thomassen recently (Christmas 2010) proved the weak 3-flow conjecture for k=8.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"3-flow conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that every 4-edge-connected graph has a nowhere-zero 3-flow was verified.\n\n**Verified partial progress.**\n\n- The source tracker supplies the current formulation and relevant context.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nFind a 3-flow construction from 4-edge-connectivity.\n\n**Sources checked.**\n\n- Open Problem Garden, node 128 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Open Problem Garden retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3217, "problem_number": "OPG-130", "title": "Jaeger's modular orientation conjecture", "statement": "Conjecture Every $4k$-edge-connected graph can be oriented so that ${\\mathit indegree}(v) - {\\mathit outdegree}(v) \\cong 0$ (mod $2k+1$ ) for every vertex $v$.", "background": "Source: Open Problem Garden. Original node ID: 130. URL: http://www.openproblemgarden.org/op/jaegers_modular_orientation_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/jaegers_modular_orientation_conjecture\n- Author(s): Jaeger, Francois\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: nowhere-zero flow; orientation\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nJaeger called an orientation with the above property a modular $(2k+1)$-orientation, and observed that a graph has a modular $(2k+1)$-orientation if and only if it has a $(2+\\frac{1}{k})$-flow. Thus, this conjecture may be seen as a sharp form of the 2+epsilon flow conjecture. For k=1, this problem is precisely the 3-flow conjecture, and for k=2, Jaeger showed that this conjecture (if true) would imply the 5-flow conjecture. If true, this conjecture would be best possible for every value of k.\n\nThe restriction of this conjecture to planar graphs is open, and has a dual formulation. See Mapping planar graphs to odd cycles.\n\nBibliography:\n[J] F. Jaeger, On circular flows in graphs. Finite and infinite sets, Vol. I, II (Eger, 1981), 391--402, Colloq. Math. Soc. János Bolyai, 37, North-Holland, Amsterdam, 1984.. MathSciNet\n\nRelated:\nRelated problems\nMapping planar graphs to odd cycles\n\nSource links:\n- edge-connected: http://en.wikipedia.org/wiki/connectivity (graph theory)\n\nDiscussion links:\n- 2+epsilon flow conjecture: http://www.openproblemgarden.org/?q=op/2_epsilon_flow_conjecture\n- 3-flow conjecture: http://www.openproblemgarden.org/?q=op/3_flow_conjecture\n- 5-flow conjecture: http://www.openproblemgarden.org/?q=op/5_flow_conjecture\n- Mapping planar graphs to odd cycles: http://www.openproblemgarden.org/?q=node/412\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0818250\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Jaeger's modular orientation conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of Jaeger's stated modular-orientation conjecture for every 4k-edge-connected graph was verified.\n\n**Verified partial progress.**\n\n- The source tracker supplies the current formulation and relevant context.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nSharpen modular orientation packing arguments to the stated threshold.\n\n**Sources checked.**\n\n- Open Problem Garden, node 130 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Open Problem Garden retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3218, "problem_number": "OPG-131", "title": "Bouchet's 6-flow conjecture", "statement": "Conjecture Every bidirected graph with a nowhere-zero $k$-flow for some $k$, has a nowhere-zero $6$-flow.", "background": "Source: Open Problem Garden. Original node ID: 131. URL: http://www.openproblemgarden.org/op/bouchets_6_flow_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bouchets_6_flow_conjecture\n- Author(s): Bouchet, Andre\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: bidirected graph; nowhere-zero flow\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: A bidirected graph is a graph in which every edge has two arrowheads, one next to each endpoint. If the edge $e$ has ends $u$ and $v$, then the arrowheads nearest $u$ and $v$ may point either toward $u$ or toward $v$ (giving four possibilities in all). If $G$ is a bidirected graph, a $k$-flow of G is a map $\\phi:E(G)\\to \\{-(k-1),...,-1,0,1,...,k-1\\}$ with the property that at every vertex, the sum of $\\phi$ on the edges whose ends at $v$ are directed into $v$ is equal to the sum of $\\phi$ on the edges whose ends at $v$ are directed out of $v$. We say that $\\phi$ is nowhere-zero if $\\phi(e) \\neq 0$ for every $e \\in E(G)$ (see nowhere-zero flows).\n\nA bidirected Orientation of the Petersen graph\n\nFlows on bidirected graphs arise naturally as duals of local-tensions on a non-orientable surface. For more on this relationship, see [B]. Bouchet proved that the above conjecture is true with 6 replaced by 216, and exhibited a bidirected Petersen graph as above which shows that 6 is the best value possible. Zyka [Z] and independently Fouquet improved upon this result proving that the above conjecture is true with 6 replaced by 30. Khelladi [K] proved that for 4-connected graphs, the above conjecture is true with 6 replaced by 18. DeVos [D] proved that the above conjecture holds with 6 replaced by 12, and showed that every 4-edge-connected bidirected graph with a nowhere-zero integer flow also has a nowhere-zero 4-flow.\n\nBibliography:\n[B] A. Bouchet, Nowhere-Zero Integral Flows on a Bidirected Graph, J. Combinatorial Theory Ser. B 34 (1983) 279-292. MathSciNet\n\n[D] M. DeVos, Flows on Bidirected Graphs, preprint.\n\n[K] A. Khelladi, Nowhere-Zero Integral Chains and Flows in Bidirected Graphs, J. Combinatorial Theory Ser. B 43 (1987) 95-115. MathSciNet\n\n[Z] O. Zyka, Bidirected Nowhere-Zero Flows, Thesis, Charles University, Praha (1988).\n\nDiscussion links:\n- nowhere-zero flows: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0714451\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0897242\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Bouchet's 6-flow conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Bouchet's 6-flow conjecture for bidirected graphs was not verified as solved.\n\n**Verified partial progress.**\n\n- The source tracker supplies the current formulation and relevant context.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nDevelop signed/bidirected flow reductions or find an obstruction.\n\n**Sources checked.**\n\n- Open Problem Garden, node 131 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Open Problem Garden retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3219, "problem_number": "OPG-132", "title": "The three 4-flows conjecture", "statement": "Conjecture For every graph $G$ with no bridge, there exist three disjoint sets $A_1,A_2,A_3 \\subseteq E(G)$ with $A_1 \\cup A_2 \\cup A_3 = E(G)$ so that $G \\setminus A_i$ has a nowhere-zero 4-flow for $1 \\le i \\le 3$.", "background": "Source: Open Problem Garden. Original node ID: 132. URL: http://www.openproblemgarden.org/op/three_4_flows_conjecture.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/three_4_flows_conjecture\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: nowhere-zero flow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nA graph $G$ has a nowhere-zero 4-flow if and only if there exist disjoint sets $A_1,A_2,A_3 \\subseteq E(G)$ with $A_1 \\cup A_2 \\cup A_3 = E(G)$ so that $G\\A_i$ has a nowhere-zero 2-flow for $1 \\le i \\le 3$. Thus, the above conjecture is true with room to spare for such graphs. Since every 4-edge-connected graph and every 3-edge-colorable cubic graph has a nowhere-zero 4-flow, this conjecture is automatically true for these families. As with the 5-flow conjecture or the cycle double cover conjecture, establishing this conjecture comes down to proving it for cubic graphs which are not 3-edge-colorable.\n\nThis conjecture is a consequence of the Petersen coloring conjecture, and it implies the Orientable cycle four cover conjecture. The latter implication follows immediately from the fact that every graph with a nowhere-zero 4-flow has an orientable cycle double cover. Actually, it is possible that for every graph $G$ with no cut-edge, there exist disjoint sets $A_B_1,B_2 \\subseteq E(G)$ with $A \\cup B_1 \\cup B_2 = E(G)$ and so that $G\\B_1$ and $G\\B_2$ have nowhere-zero 3-flows and $G\\A$ has a nowhere-zero 2-flow. The Petersen graph has such a decomposition ( $B_1$ and $B_2$ should be alternate edges of some 8-circuit) and so does every graph with a nowhere-zero 4-flow. If this stronger statement is true, then it would imply the oriented eight cycle four cover conjecture.\n\nBibliography:\n[J] F. Jaeger, On circular flows in graphs. Finite and infinite sets, Vol. I, II (Eger, 1981), 391--402, Colloq. Math. Soc. János Bolyai, 37, North-Holland, Amsterdam, 1984.. MathSciNet\n\nSource links:\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nDiscussion links:\n- edge-colorable: http://en.wikipedia.org/wiki/graph coloring\n- cubic: http://en.wikipedia.org/wiki/cubic graph\n- 5-flow conjecture: http://www.openproblemgarden.org/?q=op/5_flow_conjecture\n- cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/cycle_double_cover_conjecture\n- Petersen coloring conjecture: http://www.openproblemgarden.org/?q=op/petersen_coloring_conjecture\n- Orientable cycle four cover conjecture: http://www.openproblemgarden.org/?q=op/cycle_double_cover_conjecture\n- Petersen graph: http://en.wikipedia.org/wiki/petersen graph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0818250\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"The three 4-flows conjecture\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the three-disjoint-4-flow-deletions conjecture was verified.\n\n**Verified partial progress.**\n\n- The source tracker supplies the current formulation and relevant context.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nRelate it to cycle-space decompositions and 4-flow coverings.\n\n**Sources checked.**\n\n- Open Problem Garden, node 132 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Open Problem Garden retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3220, "problem_number": "OPG-134", "title": "A homomorphism problem for flows", "statement": "Conjecture Let $M,M'$ be abelian groups and let $B \\subseteq M$ and $B' \\subseteq M'$ satisfy $B=-B$ and $B' = -B'$. If there is a homomorphism from $Cayley(M,B)$ to $Cayley(M',B')$, then every graph with a B-flow has a B'-flow.", "background": "Source: Open Problem Garden. Original node ID: 134. URL: http://www.openproblemgarden.org/op/a_homomorphism_problem_for_flows.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_homomorphism_problem_for_flows\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: homomorphism; nowhere-zero flow; tension\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition:Let $G$ be a directed graph, Let $M$ be an abelian group, and let $B$ be a subset of $M$ such that $B=-B$. We say that a flow or a tension $\\phi:E(G) \\rightarrow M$ is a $B$-flow or a $B$-tension if the range is a subset of $B$. If $\\phi$ is a $B$-flow ( $B$-tension) of $G$ and we reverse the direction of the edge $e$, then we may obtain a new $B$-flow ( $B$-tension) by changing $\\phi(e)$ to $-\\phi(e)$. Thus, the existence of a $B$-flow or $B$-tension does not depend on the orientation, and we say that an undirected graph has a $B$-flow or a $B$-tension if some (and thus every) orientation of it admits such a map. We define the Cayley graph $Cayley(M,B)$ to be the simple graph with vertex set $M$ in which two vertices $u,v$ are joined by an edge if and only if $u-v \\in B$.\n\nIt is well known that a graph has a $B$-tension if and only if it has a homomorphism to $Cayley(M,B)$. So, if $M,M',B,B'$ are as in the conjecture and there is a homomorphism from $Cayley(M,B)$ to $Cayley(M',B')$, then every graph G with a $B$-tension has a $B'$-tension. This follows from the previous sentence and the fact that the composition of two homomorphisms is another homomorphism. In essence, the above conjecture states that the same equivalence should hold true for flows.\n\nIf $H$ and $H^*$ are directed planar dual graphs (each edge of $H^*$ crosses left to right over the corresponding edge of $H$ ), then a map $\\phi:E(H) \\to M$ is a tension if and only if the dual map $\\phi^*:E(H^*) \\to M$ ( $\\phi^*$ is given by the rule $\\phi^*(e^*)=\\phi(e)$ ) is a flow of $H^*$. Thus, planar duality exchanges flows and tensions. For two undirected planar dual graphs, $G$ and $G^*$ we have that G has a $B$-flow if and only if $G^*$ has a $B$-tension. It follows from this duality and the observation from the previous paragraph, that the above conjecture is true for planar graphs.\n\nThis conjecture is also known in the special case when $B=M\\setminus \\{0\\}$ and $B'=M'\\setminus \\{0\\}$. In this case, $Cayley(M,B)$ and $Cayley(M',B')$ are the complete graphs on $|M|$ and $|M'|$ vertices respectively, so there is a homomorphism from $Cayley(M,B)$ to $Cayley(M',B')$ if and only if $|M'|$ is greater than or equal to $|M|$. Thus, in this case the conjecture is equivalent to the assertion that every graph with a nowhere-zero $M$-flow also has a nowhere-zero $M'$-flow if $|M'|$ is at least $|M|$. This statement is true by a result of Tutte.\n\nSource links:\n- homomorphism: http://en.wikipedia.org/wiki/graph homomorphism\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 31.\n\nAttempt notes:\nTarget:\nMake progress on \"A homomorphism problem for flows\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof was verified that a Cayley-graph homomorphism universally transfers B-flows to B-prime-flows.\n\n**Verified partial progress.**\n\n- The source tracker supplies the current formulation and relevant context.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nTranslate graph homomorphisms into universal flow constraints.\n\n**Sources checked.**\n\n- Open Problem Garden, node 134 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Open Problem Garden retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3221, "problem_number": "OPG-135", "title": "Real roots of the flow polynomial", "statement": "Conjecture All real roots of nonzero flow polynomials are at most 4.", "background": "Source: Open Problem Garden. Original node ID: 135. URL: http://www.openproblemgarden.org/op/real_roots_of_the_flow_polynomial.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/real_roots_of_the_flow_polynomial\n- Author(s): Welsh, Dominic J. A.\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: flow polynomial; nowhere-zero flow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nFor every graph $G$, let $P_G$ be the chromatic polynomial of $G$ and let $Q_G$ be the flow polynomial of $G$. If $G$ is loopless, then $P_G(k)>0$ for all sufficiently large integers $k$ (as $P_G(k)$ = # of k-colorings of $G$ ). It follows from Seymour's 6-flow theorem that if $G$ has no bridge, then $Q_G(k)>0$ for all integers $k>5$ (as $Q_G(k)$ = # of nowhere-zero flows in the group of integers modulo $k$ ). It is natural to ask if all real roots of these polynomials are small. For the chromatic polynomial, $P_G$, this is not the case. There exist graphs with chromatic number 3 for which $P_G$ has arbitrarily large real roots. The above conjecture asserts that the flow polynomial exhibits the opposite behavior. One word of caution, it is known that the set of roots of flow polynomials is dense in the complex plane.\n\nBibliography:\n[S] P.D. Seymour, Nowhere-Zero 6-Flows, J. Combinatorial Theory Ser. B 30 (1981) 130-135. MathSciNet\n\nDiscussion links:\n- chromatic polynomial: http://en.wikipedia.org/wiki/graph coloring\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n- nowhere-zero flows: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0615308\n\nComments:\n- December 17th, 2010 | Robert Samal | Not bounded by 5, either: A preprint by Jesper L. Jacobsen and Jesus Salas claims that there are graphs with roots of their flow polynomial being above 5. The generalized Petersen graphs G(7n,7) are claimed to have roots of flow polynomial that accumulate at approximately $5.23$.\n\nI suppose this makes the original conjecture truly false. An interesting variant, though, is to find out, if all roots of flow polynomials are $\\le 6$. (Thanks to Bojan Mohar for pointing out the paper to me.)\n- August 21st, 2009 | Gordon Royle | Welsh's conjecture is false: Welsh's conjecture on flow roots is false. In fact, many cubic graphs with reasonably large girth and enough vertices have flow roots between 4 and 5, and it is almost certain that we can find graphs with flow roots arbitrarily close to 5.\n\nHowever I strongly believe that \"All real roots of nonzero flow polynomials are at most FIVE\".\n\nSee my recent survey article \"Recent results on chromatic and flow roots of graphs and matroids, Surveys in Combinatorics 2009\" for more detail.\n\nGordon Royle\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Real roots of the flow polynomial\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The tracker carries the real flow-root assertion as a conjecture, but its current status was not independently verified in this pass.\n\n**Verified partial progress.**\n\n- The source tracker supplies the current formulation and relevant context.\n\n**Full solution or refutation.**\n\nNo verified current classification was found.\n\n**What remains.**\n\nCheck recent flow-root classifications and explicit counterexamples.\n\n**Sources checked.**\n\n- Open Problem Garden, node 135 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker freshness is insufficient for a firm open/refuted classification.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3222, "problem_number": "OPG-136", "title": "Unit vector flows", "statement": "Conjecture For every graph $G$ without a bridge, there is a flow $\\phi: E(G) \\rightarrow S^2 = \\{ x \\in {\\mathbb R}^3: |x| = 1 \\}$.\n\nConjecture There exists a map $q:S^2 \\rightarrow \\{-4,-3,-2,-1,1,2,3,4\\}$ so that antipodal points of $S^2$ receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.", "background": "Source: Open Problem Garden. Original node ID: 136. URL: http://www.openproblemgarden.org/op/unit_vector_flows.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/unit_vector_flows\n- Author(s): Jain, Kamal\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: nowhere-zero flow\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2007 by mdevos\n\nProblem-page discussion:\nThe main interest in these two conjectures is that together they imply Tutte's 5-flow conjecture. This follows easily from the fact that the 5-flow conjecture can be reduced to cubic graphs without bridges, and for such a graph $G$, the composition of the maps $\\phi$ and $q$ (given by the above conjectures) is a nowhere-zero 5-flow.\n\nThere are a couple of easy partial results toward the first conjecture which follow from well-known flow/cycle-cover results. First, Tutte showed that every graph with a nowhere-zero 4-flow has a list of three 2-flows $f_1,f_2,f_3: E(G) \\to \\{-1,0,1\\}$ so that every edge is in the support of exactly two of these flows. Combining these flows and normalizing appropriately gives an $S^2$-flow. Bermond, Jackson, and Jaeger [BJJ] showed that every graph with no bridge has a list of seven 2-flows so that every edge is in the support of exactly four of these flows. Combining these and normalizing appropriately gives an $S^6$-flow.\n\nIt seems likely that a graph has an $S^1$-flow if and only if it has a nowhere-zero 3-flow. The \"if\" direction of this implication isn't hard to show and the \"only if\" direction looks quite possible.\n\nA dual concept to that of a flow is that of a tension. Observe that a graph $G$ has a $S^n$ tension if and only if can be embedded in ${\\mathbb R}^{n+1}$ so that all edges are unit length line segments. Such embeddings have received some attention over the years. In particular, there is considerable interest in finding the best possible upper bound on the chromatic number of graphs which embed in ${\\mathbb R}^2$ in this manner. This is Hadwinger-Nelson problem on coloring the plane.\n\nBibliography:\n[BJJ] J.C. Bermond, B. Jackson, and F. Jaeger, Shortest covering of graphs with cycles, J. Combinatorial Theory Ser. B 35 (1983), 297-308. MRhref{0735197}\n\n[T54] W.T. Tutte, A Contribution on the Theory of Chromatic Polynomials, Canad. J. Math. 6 (1954) 80-91. MathSciNet\n\n[T66] W.T. Tutte, On the Algebraic Theory of Graph Colorings, J. Combinatorial Theory 1 (1966) 15-50. MathSciNet\n\nSource links:\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n\nDiscussion links:\n- Tutte's 5-flow conjecture: http://www.openproblemgarden.org/?q=op/5_flow_conjecture\n- cubic: http://en.wikipedia.org/wiki/cubic graph\n- nowhere-zero: http://en.wikipedia.org/wiki/nowhere-zero flows\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0061366\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0194363\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Unit vector flows\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of either unit-vector-flow or antipodal spherical labeling statement was verified.\n\n**Verified partial progress.**\n\n- The source tracker supplies the current formulation and relevant context.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nFind a topological flow construction or a discrete antipodal labeling.\n\n**Sources checked.**\n\n- Open Problem Garden, node 136 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Open Problem Garden retains both conjectures.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3223, "problem_number": "OPG-323", "title": "Antichains in the cycle continuous order", "statement": "If $G$, $H$ are graphs, a function $f: E(G) \\rightarrow E(H)$ is called cycle-continuous if the pre-image of every element of the (binary) cycle space of $H$ is a member of the cycle space of $G$.\n\nProblem Does there exist an infinite set of graphs $\\{G_1,G_2,\\ldots \\}$ so that there is no cycle continuous mapping between $G_i$ and $G_j$ whenever $i \\neq j$?", "background": "Source: Open Problem Garden. Original node ID: 323. URL: http://www.openproblemgarden.org/op/antichains_in_the_cycle_continuous_order.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/antichains_in_the_cycle_continuous_order\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: antichain; cycle; poset\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 12th, 2007 by mdevos\n\nProblem-page discussion:\nThe definition of a cycle-continuous mapping is based on some work of Jaeger, and the most interesting question on this subject is undoubtedly Jaeger's Petersen coloring conjecture.\n\nLet us define a relation on the set of all finite graphs with at least one edge by the rule $G>H$ if there is a cycle-continuous mapping from $G$ to $H$. It is not difficult to verify that $>$ is a quasi order (reflexive and transitive). In this order, every Eulerian graph dominates every other graph, and every graph with a cut edge is dominated by every other graph.\n\nLet $A_i$ be the graph on two vertices with $i$ parallel edges. Then $A_3 < A_5 < A_7 <...$ with all the inequalities strict, so this sequence is an infinite chain. Very little else seems to be known about this order. In particular, the problem highlighted above - does there exist an infinite antichain? remains open.\n\nDiscussion links:\n- Petersen coloring conjecture: http://www.openproblemgarden.org/?q=op/petersen_coloring_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Antichains in the cycle continuous order\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No infinite antichain for the cycle-continuous mapping order was verified.\n\n**Verified partial progress.**\n\n- The problem defines the order through preimages of binary cycle spaces.\n\n**Full solution or refutation.**\n\nThe antichain existence problem remains open.\n\n**What remains.**\n\nConstruct graphs with incomparable flow/cycle-continuous invariants.\n\n**Sources checked.**\n\n- Open Problem Garden, node 323 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3224, "problem_number": "OPG-59994", "title": "Circular flow number of regular class 1 graphs", "statement": "A nowhere-zero $r$-flow $(D(G),\\phi)$ on $G$ is an orientation $D$ of $G$ together with a function $\\phi$ from the edge set of $G$ into the real numbers such that $1 \\leq |\\phi(e)| \\leq r-1$, for all $e \\in E(G)$, and $\\sum_{e \\in E^+(v)}\\phi(e) = \\sum_{e \\in E^-(v)}\\phi(e), \\textrm{ for all } v \\in V(G)$. The circular flow number of $G$ is inf $\\{ r | G$ has a nowhere-zero $r$-flow $\\}$, and it is denoted by $F_c(G)$.\n\nA graph with maximum vertex degree $k$ is a class 1 graph if its edge chromatic number is $k$.\n\nConjecture Let $t \\geq 1$ be an integer and $G$ a $(2t+1)$-regular graph. If $G$ is a class 1 graph, then $F_c(G) \\leq 2 + \\frac{2}{t}$.", "background": "Source: Open Problem Garden. Original node ID: 59994. URL: http://www.openproblemgarden.org/op/circular_flow_number_of_regular_class_1_graphs.\n\nSource subject path: Graph Theory > Coloring > Nowhere-zero flows.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/circular_flow_number_of_regular_class_1_graphs\n- Author(s): Steffen, Eckhard\n- Subject(s): Graph Theory; Coloring; Nowhere-zero flows\n- Keywords: nowhere-zero flow, edge-colorings, regular graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 5th, 2015 by Eckhard Steffen\n\nProblem-page discussion:\nThe conjecture is true for $t=1$, i.e. for cubic graphs. It says, that the circular flow number of $(2t+1)$-regular class 1 graphs is bounded by the circular flow number of the complete graph on $2t+2$ vertices.\n\nBibliography:\n[ES_2001] E. Steffen, Circular flow numbers of regular multigraphs, J. Graph Theory 36, 24 – 34 (2001)\n\n*[ES_2015] E. Steffen, Edge-colorings and circular flow numbers on regular graphs, J. Graph Theory 79, 1–7, 2015\n\nRelated:\nRelated problems\n(2 + epsilon)-flow conjecture\nJaeger's modular orientation conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 17.\n\nAttempt notes:\nTarget:\nMake progress on \"Circular flow number of regular class 1 graphs\" in Graph Theory; Coloring; Nowhere-zero flows, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Mattiolo--Steffen explicitly disprove the conjecture by constructing infinitely many odd-regular class-1 graphs whose circular flow number exceeds 2+2/t.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nFor every odd p>=3, their construction gives a (4p+1)-regular class-1 graph G with phi_c(G)>2+1/p. Setting t=2p yields a direct counterexample to the displayed bound for every t=4j+2 with j>=1.\n\n**What remains.**\n\nThe universal statement is settled negatively. Separate questions remain about the exact small values of t and the maximum circular flow number within narrower regular class-1 families.\n\n**Sources checked.**\n\n- Davide Mattiolo and Eckhard Steffen, Edge colorings and circular flows on regular graphs, Journal of Graph Theory 99 (2022), 399-413; arXiv:2001.02484. (primary): https://arxiv.org/abs/2001.02484\n Evidence used: Explicitly states that it disproves this conjecture and proves the counterexample family in Theorem 3.4 and Corollary 3.5.\n- Jiaao Li, Xueliang Li, and Meiling Wang, The Flow Index of Regular Class I Graphs, SIAM Journal on Discrete Mathematics 36 (2022), 1989-2004. (primary): https://lijiaao-dm-nk.com/pdf/pub/Jiaao30_circularflow-ClassOne-SIAM2022.pdf\n Evidence used: Extends counterexamples to t in {6,8,10} and every t>=12 and discusses the still-open small cases.\n- Graph-theory open problems, Circular flow number of regular class 1 graphs. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/circular_flow_number_of_regular_class_1_graphs/\n Evidence used: Current tracker labels the conjecture disproved and links the Mattiolo--Steffen paper.\n\n**Review notes.** The counterexample parameter conversion is t=2p: degree 4p+1=2t+1 and bound 2+1/p=2+2/t.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3225, "problem_number": "OPG-171", "title": "Strong colorability", "statement": "Let $r$ be a positive integer. We say that a graph $G$ is strongly $r$-colorable if for every partition of the vertices to sets of size at most $r$ there is a proper $r$-coloring of $G$ in which the vertices in each set of the partition have distinct colors.\n\nConjecture If $\\Delta$ is the maximal degree of a graph $G$, then $G$ is strongly $2 \\Delta$-colorable.", "background": "Source: Open Problem Garden. Original node ID: 171. URL: http://www.openproblemgarden.org/op/strong_colorability.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strong_colorability\n- Author(s): Aharoni, Ron; Alon, Noga; Haxell, Penny E.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: strong coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 27th, 2007 by berger\n\nProblem-page discussion:\nHaxell proved that if $\\Delta$ is the maximal degree of a graph $G$, then $G$ is strongly $3 \\Delta - 1$-colorable. She later proved that the strong chromatic number $\\chi_S$ is at most $(2.75+\\epsilon)\\Delta$ for sufficiently large $\\Delta$ depending on $\\epsilon$. Aharoni, Berger, and Ziv proved the fractional relaxation.\n\nComments:\n- November 19th, 2009 | Andrew King | Recent progress: Haxell proved that $(2.75 + \\epsilon)\\Delta$ is sufficient for sufficiently large $\\Delta$ depending on $\\epsilon$. (JGT, 2008)\n\nAharoni, Berger, and Ziv proved the fractional relaxation, i.e. that with partition cliques of size $2\\Delta$ we have a fractional $2\\Delta$ colouring. (Combinatorica, 2007)\n- July 24th, 2007 | Anonymous | Problem Solved!: See \"On the Strong Chromatic Number of Graphs\" by Maria Axenovich and Ryan Martin (2006)\n- July 25th, 2007 | Anonymous | only partly solved: The paper cited (available at http://orion.math.iastate.edu/axenovic/Papers/Martin_Strong.pdf) only resolves the above conjecture for graphs G which have maximum degree at least |V(G)|/6.\n- September 3rd, 2007 | Anonymous | That theorem has been proved: That theorem has been proved in an even older paper by another set of authors.\n\nAfter a quick search I found that paper at http://abel.math.umu.se/~klasm/Uppsatser/factor.pdf\n- September 3rd, 2007 | mdevos | still only a partial solution: Once again, the paper cited offers only a partial solution. Quoting from the paper, \"From Theorem 1.1, we conclude that the strong chromatic number can be bounded by $2\\Delta(G)$ if $|V(G)| \\le 6\\Delta(G)$. This result should not be compared with the more complete and difficult result of Alon.\" Here, the difficult result of Alon is the theorem that there exists a fixed constant $K$ so that every graph of maximum degree $\\Delta$ is strongly $K \\Delta$-colorable.. precisely the result which is sharpened by this conjecture.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Strong colorability\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact strong chromatic number bound 2 Delta remains open for arbitrary graphs. It is exact in a dense-degree regime and asymptotically attained under a fixed linear degree condition.\n\n**Verified partial progress.**\n\n- Axenovich and Martin prove chi_S(G) <= 2 Delta when Delta >= |V(G)|/6 and show sharpness.\n- For every fixed c>0, Lo and Sanhueza-Matamala prove chi_S(G) <= (2+o(1))Delta when Delta >= c|V(G)|.\n\n**Full solution or refutation.**\n\nThe conjectured constant is achieved exactly when maximum degree is at least one sixth of the order and asymptotically throughout fixed positive degree density, but sparse regimes remain untreated.\n\n**What remains.**\n\nProve the exact 2 Delta bound without any density assumption, or find a counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, Strong colorability (OPG-171), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/strong_colorability\n Evidence used: Maintains the exact 2 Delta conjecture and records older general and fractional bounds.\n- M. Axenovich and R. R. Martin, On the Strong Chromatic Number of Graphs, SIAM Journal on Discrete Mathematics 20 (2006), 741-747. (primary): https://doi.org/10.1137/050633056\n Evidence used: Proves the exact 2 Delta bound under Delta >= n/6 and proves sharpness.\n- A. Lo and N. Sanhueza-Matamala, An asymptotic bound for the strong chromatic number, Combinatorics, Probability and Computing 28 (2019), 768-776. (primary): https://doi.org/10.1017/S0963548318000561\n Evidence used: Proves the asymptotically sharp (2+o(1))Delta bound when Delta is at least a fixed positive fraction of n.\n\n**Review notes.** The source's parts-of-size-at-most-r convention is preserved; standard literature often pads with isolated vertices and uses equal-sized parts.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3226, "problem_number": "OPG-335", "title": "Reed's omega, delta, and chi conjecture", "statement": "For a graph $G$, we define $\\Delta(G)$ to be the maximum degree, $\\omega(G)$ to be the size of the largest clique subgraph, and $\\chi(G)$ to be the chromatic number of $G$.\n\nConjecture $\\chi(G) \\le \\ceil{\\frac{1}{2}(\\Delta(G)+1) + \\frac{1}{2}\\omega(G)}$ for every graph $G$.", "background": "Source: Open Problem Garden. Original node ID: 335. URL: http://www.openproblemgarden.org/op/reeds_omega_delta_and_chi_conjecture.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/reeds_omega_delta_and_chi_conjecture\n- Author(s): Reed, Bruce A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 22nd, 2007 by mdevos\n\nProblem-page discussion:\nPerhaps the two most trivial bounds on $\\chi(G)$ are $\\chi(G) \\ge \\omega(G)$ and $\\chi(G) \\le \\Delta(G) + 1$. The above conjecture roughly asserts that the (rounded-up) average of $\\Delta(G)+1$ and $\\omega(G)$ should again be an upper bound on $\\chi(G)$.\n\nThe conjecture is easy to verify when $\\omega(G)$ is very large. It is trivial when $\\omega(G) \\ge \\Delta(G)$, and it follows from Brook's theorem if $\\omega(G) = \\Delta(G)-1$. On the other hand, if $\\omega(G) = 2$, so $G$ is triangle free, then the conjecture is also true for $\\Delta$ sufficiently large. Indeed, Johannsen proved the much stronger fact that there exists a fixed constant $c$ so that $\\chi(G) \\le \\frac{c \\Delta(G)}{\\log \\Delta(G)}$ for every triangle free graph $G$.\n\nReed showed that the conjecture holds when $\\Delta(G) = |V(G)| - 1$ by way of matching theory. More interestingly, he proved (using probabilistc methods) that the conjecture is true provided that $\\Delta$ is sufficiently large, and $\\omega$ is sufficiently close to $\\Delta$. More precisely, he proves the following:\n\nTheorem There exists a fixed constant $\\Delta_0$ such that for every $\\Delta \\ge \\Delta_0$, if $G$ is a graph of maximum degree $\\Delta$ with no clique of size $>k$ for some $k \\ge (1 - \\frac{1}{70000000}) \\Delta$ then $\\chi(G) \\le \\frac{\\Delta + 1 + k}{2}$.\n\nIt is known that the conjecture is true fractionally (that is with $\\chi(G)$ replaced by $\\chi_f(G)$, the fractional chromatic number of~ $G$ ).\n\nBibliography:\n*[R] B. Reed, $\\omega, \\Delta$, and $\\chi$, J. Graph Theory 27 (1998) 177-212.\n\nSource links:\n- clique: http://en.wikipedia.org/wiki/clique (graph theory)\n\nDiscussion links:\n- fractional chromatic number: http://en.wikipedia.org/wiki/fractional chromatic number\n\nComments:\n- September 18th, 2007 | Anonymous | The statement of the: The statement of the conjecture is slightly incorrect. Instead of the +1 at the end, there should simply be a round-up. The conjecture is true for line graphs and quasi-line graphs, graphs with independence number 2, and any graph on I believe 12 vertices.\n\nAn outright proof of the result for triangle-free graphs would be very nice. Lovasz' result on splitting graphs is not quite enough in this case.\n- September 24th, 2007 | Robert Samal | changed: Thanks for the comment. I changed the statement to the conjectured (slightly stronger) version with round-up.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"Reed's omega, delta, and chi conjecture\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Reed's omega-delta-chi bound is known in broad special and asymptotic regimes, but not for every graph.\n\n**Verified partial progress.**\n\n- Numerous local and high-degree cases support the target interpolation bound.\n\n**Full solution or refutation.**\n\nNo full theorem or counterexample was verified.\n\n**What remains.**\n\nClose remaining degree/clique configurations.\n\n**Sources checked.**\n\n- Open Problem Garden, node 335 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3227, "problem_number": "OPG-401", "title": "Circular coloring triangle-free subcubic planar graphs", "statement": "Problem Does every triangle-free planar graph of maximum degree three have circular chromatic number at most $\\frac{20}{7}$?", "background": "Source: Open Problem Garden. Original node ID: 401. URL: http://www.openproblemgarden.org/op/circular_chromatic_number_of_triangle_free_planar_graphs.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/circular_chromatic_number_of_triangle_free_planar_graphs\n- Author(s): Ghebleh, Mohammad; Zhu, Xuding\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: circular coloring; planar graph; triangle free\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 20th, 2007 by mdevos\n\nProblem-page discussion:\nThroughout, we let $\\chi_c(G)$ denote the circular chromatic number of the graph $G$.\n\nA well-known Question of Nesetril asks if $\\chi_c(G) \\le \\frac{5}{2}$ for all cubic graphs $G$ of sufficiently high girth. A conjecture of Jaeger asserts that $\\chi_c(G) \\le 2 + \\frac{1}{k}$ for every planar graph $G$ of girth $4k+1$. There are numerous partial results on these problems, and there are many interesting questions concerning the circular chromatic numbers of restricted families of graphs. Here we are restricted to planar graphs of girth $\\ge 4$ with maximum degree $\\le 3$. The dodecahedron lives in this class and has $\\chi_c = \\frac{20}{7}$. It remains unclear if anyone else in this class might have $\\chi_c$ larger.\n\nA related conjecture of X. Zhu asserts that for every triangle-free planar graph $G$ with $\\Delta(G)\\le 4$ and $|V(G)|<3k$ one has $\\chi_c(G)\\le 3-1/k$.\n\nRelated:\nRelated problems\nPentagon problem\nJaeger's modular orientation conjecture\n\nDiscussion links:\n- circular chromatic number: http://en.wikipedia.org/wiki/circular coloring\n- Question of Nesetril: http://www.openproblemgarden.org/?q=node/167\n- conjecture of Jaeger: http://www.openproblemgarden.org/?q=node/130\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Circular coloring triangle-free subcubic planar graphs\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that every triangle-free subcubic planar graph has circular chromatic number at most 20/7 was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nImprove fractional/circular coloring methods for subcubic planar graphs.\n\n**Sources checked.**\n\n- Open Problem Garden, node 401 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3228, "problem_number": "OPG-494", "title": "Oriented chromatic number of planar graphs", "statement": "An oriented colouring of an oriented graph is assignment $c$ of colours to the vertices such that no two arcs receive ordered pairs of colours $(c_1,c_2)$ and $(c_2,c_1)$. It is equivalent to a homomorphism of the digraph onto some tournament of order $k$.\n\nProblem What is the maximal possible oriented chromatic number of an oriented planar graph?", "background": "Source: Open Problem Garden. Original node ID: 494. URL: http://www.openproblemgarden.org/op/oriented_chromatic_number_of_planar_graphs.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/oriented_chromatic_number_of_planar_graphs\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: oriented coloring; oriented graph; planar graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 4th, 2007 by Robert Samal\n\nProblem-page discussion:\nRaspaud and Sopena [RS] showed using Borodin's result about acyclic chromatic number of planar graphs, that every planar oriented graph has oriented chromatic number at most 80. (Their motivation came from a work of Courcelle [C] concerning the monadic second-order logic of graphs. That, however, deals with a stronger variant of coloring.)\n\nOn the other hand, Marshall [M] showed that there is an oriented planar graph with oriented chromatic number at least~17.\n\nBibliography:\n[C] B. Courcelle, The monadic second order logic of graphs VI: On several representations of graphs by relational structures, Discrete Appl. Math. 54 ( 1994),\n\n[M] T. H. Marshall. On $\\cal P$-universal graphs. Research Report 2001-510, KAM-DIMATIA Series, 2001.\n\n*[RS] A. Raspaud and E. Sopena. Good and semi-strong colorings of oriented planar graphs. Inform. Process. Lett., 51(4):171–174, 1994. MathSciNet\n\nSource links:\n- oriented chromatic number: http://en.wikipedia.org/wiki/Oriented_coloring\n\nDiscussion links:\n- acyclic chromatic number: http://en.wikipedia.org/wiki/acyclic coloring\n\nBibliography links:\n- On $\\cal P$-universal graphs: http://kam.mff.cuni.cz/%7Ekamserie/serie/clanky/2001/s510.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1294309\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Oriented chromatic number of planar graphs\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact maximum oriented chromatic number of oriented planar graphs is unknown, but it is known to be bounded above, including a published bound of 14 for planar graphs.\n\n**Verified partial progress.**\n\n- A general bounded upper bound is available.\n\n**Full solution or refutation.**\n\nThe exact extremal value remains open.\n\n**What remains.**\n\nImprove upper/lower bounds and identify extremal orientations.\n\n**Sources checked.**\n\n- A. V. Kostochka et al., Proper orientations and proper chromatic number, arXiv:2110.07005. (primary): https://arxiv.org/abs/2110.07005\n Evidence used: States an upper bound of 14 for planar graphs.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3229, "problem_number": "OPG-550", "title": "Coloring and immersion", "statement": "Conjecture For every positive integer $t$, every (loopless) graph $G$ with $\\chi(G) \\ge t$ immerses $K_t$.", "background": "Source: Open Problem Garden. Original node ID: 550. URL: http://www.openproblemgarden.org/op/coloring_and_immersion.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/coloring_and_immersion\n- Author(s): Abu-Khzam, Faisal N.; Langston, Michael A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: coloring; complete graph; immersion\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: September 4th, 2007 by mdevos\n\nProblem-page discussion:\nLet $G$ be a graph and let $uv, vw \\in E(G)$. The operation of deleting the edges $uv$ and $vw$ and then adding a new edge between $v$ and $w$ is called a split. We say that a graph $G$ immerses a graph $H$ if a graph isomorphic to $H$ may be obtained from $G$ by repeatedly making splits and deleting vertices and edges.\n\nThe graph containment relations of minor and topological minor have been thoroughly studied with respect to graph coloring. In particular, there are two famous conjectures: Hajos conjectured that every graph with chromatic number $\\ge t$ contains a subdivision of the complete graph $K_t$ as a subgraph. Hadwiger conjectured that every graph with chromatic number $\\ge t$ contains $K_t$ as a minor. While Hajos' Conjecture is false for $t \\ge 8$ (indeed it is actually false on average), Hadwiger's Conjecture remains open, and is one of the outstanding problems in Graph Theory.\n\nOn the other hand, the relationship between graph coloring and immersions seems to have been largely ignored until Abu-Khzam and Langston made the above conjecture. In addition to formulating this conjecture, they proved it for $t \\le 4$ and showed that a minimal counterexample to it must be 4-connected and $t$-edge-connected. Recently, DeVos, Kawarabayashi, Mohar, and Okamura have resolved the conjecture for $t \\le 7$.\n\nBibliography:\n* Faisal N. Abu-Khzam and Michael A. Langston, Graph Coloring and the Immersion Order\n\nBibliography links:\n- Graph Coloring and the Immersion Order: http://www.cs.utk.edu/%7Elangston/projects/papers/conjecture.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Coloring and immersion\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Abu-Khzam--Langston immersion conjecture is proved for alpha(G)=2 but remains open generally.\n\n**Verified partial progress.**\n\n- A recent result proves the independence-number-two case.\n\n**Full solution or refutation.**\n\nNo all-graphs theorem was verified.\n\n**What remains.**\n\nExtend beyond the known structural classes.\n\n**Sources checked.**\n\n- The Abu-Khzam--Langston Conjecture for Graphs with alpha(G)=2, arXiv:2605.28159. (primary): https://arxiv.org/abs/2605.28159\n Evidence used: Proves the alpha(G)=2 special case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3230, "problem_number": "OPG-616", "title": "Coloring the Odd Distance Graph", "statement": "The Odd Distance Graph, denoted ${\\mathcal O}$, is the graph with vertex set ${\\mathbb R}^2$ and two points adjacent if the distance between them is an odd integer.\n\nQuestion Is $\\chi({\\mathcal O}) = \\infty$?", "background": "Source: Open Problem Garden. Original node ID: 616. URL: http://www.openproblemgarden.org/op/coloring_the_odd_distance_graph.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/coloring_the_odd_distance_graph\n- Author(s): Rosenfeld, Moshe\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: coloring; geometric graph; odd distance\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 3rd, 2007 by mdevos\n\nProblem-page discussion:\nThis question is a relative of the famous problem about coloring the Unit Distance Graph (the graph on ${\\mathbb R}^2$ where two points are adjacent if the distance between them is 1). See Moshe's online lecture Famous and lesser known problems in “elementary” combinatorial geometry and number theory at time 15:20 for a nice introduction.\n\nPerhaps the first property of ${\\mathcal O}$ to determine is the size of the largest complete subgraph (were ${\\mathcal O}$ to contain arbitrarily large complete subgraphs, its chromatic number would be $\\infty$ ). It is obvious that ${\\mathcal O}$ contains triangles, but perhaps surprisingly, it does not contain a complete subgraph on four vertices. In other words, there do not exist four points in ${\\mathbb R}^2$ so that all pairwise distances are odd. This was a problem on the Putnam Exam in 1993, and is proved by Rosenfeld in [R1] and [R2].\n\nA natural strengthening of the above question is to ask if there exists a proper $n$-coloring $f: V({\\mathcal O}) \\rightarrow \\{1,2,\\ldots,n\\}$ so that $f^{-1}(\\{i\\})$ is a measurable set for every $i$. Such colorings are called measurable colorings, and interestingly, the Odd Distance Graph has no finite measurable coloring. This follows from immediately from a theorem of Furstenberg, Katznelson and Weiss [FKW] which asserts that for every measurable subset $A \\subseteq {\\mathbb R}^2$ with positive upper density, there exists a number $r$ so that $A$ contains a pair of points at distance $r'$ for every $r' > r$. This theorem has a number of independent proofs, see also Falconer and Marstrand [FM], Bourgain [Bo], and Bukh [Bu].\n\nAll that seems to be known about the (usual) chromatic number of ${\\mathcal O}$ is that $\\chi({\\mathcal O}) \\ge 5$.\n\nBibliography:\n[Bo] J. Bourgain, A Szemerédi type theorem for sets of positive density in $R\\sp k$. Israel J. Math. 54 (1986), no. 3, 307--316. MathSciNet\n\n[Bu] B. Bukh, Measurable sets with excluded distances.\n\n[FM] K. J. Falconer and J. M. Marstrand, Plane sets with positive density at infinity contain all large distances. Bull. London Math. Soc. 18 (1986), no. 5, 471--474. MathSciNet\n\n[FKM] H. Furstenberg, Y. Katznelson, and B. Weiss, Ergodic theory and configurations in sets of positive density. Mathematics of Ramsey theory, 184--198, Algorithms Combin., 5, Springer, Berlin, 1990. MathSciNet\n\n[R1] M. Rosenfeld, Odd integral distances among points in the plane. Geombinatorics 5 (1996), no. 4, 156--159. MathSciNet\n\n[R2] M. Rosenfeld Famous and lesser known problems in “elementary” combinatorial geometry and number theory (video lecture - time 15:20)\n\nDiscussion links:\n- Famous and lesser known problems in “elementary” combinatorial geometry and number theory: http://videolectures.net/sicgt07_rosenfeld_falkp\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0853455\n- Measurable sets with excluded distances: http://arxiv.org/pdf/math/0703856\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0847986\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1083601\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1380145\n- Famous and lesser known problems in “elementary” combinatorial geometry and number theory: http://videolectures.net/sicgt07_rosenfeld_falkp\n\nComments:\n- March 19th, 2012 | Anonymous | The Odd-Distance Graph: In the book Research Problems in Discrete Geometry, on page 252, it is stated that every K_4 free graph is a subgraph of the odd distance graph. We just proved that W_5, the five wheel is not a subgraph of the odd distance graph. I further believe that there are triangle free graphs that are not subgraphs of the odd distance graph, and even graphs with large girth.\n- May 8th, 2008 | JSteinhardt | Flaw: Actually, it appears there is a flaw with the below proof. The spectral bound on the chromatic number assumes a measurable coloring, as we want to say the following:\n\n$$2(\\chi-1)\\lambda_{\\min}||f||^2 & = & \\sum_{i,j=1}^{\\chi} \\lambda_{\\min}||f_i-f_j||^2$$$$\\leq \\sum_{i,j=1}^{\\chi} \\langle f_i-f_j, B(f_i-f_j) \\rangle$$$$= \\sum_{i,j=1}^{\\chi} \\langle f_i,Bf_i \\rangle + \\langle f_j,Bf_j \\rangle - 2\\langle f_i,Bf_j \\rangle$$$$= -2\\sum_{i,j=1}^{\\chi} \\langle f_i,Bf_j \\rangle$$$$= -2\\langle\\sum_{i=1}^{\\chi} f_i,B(\\sum_{i=1}^{\\chi} f_i\\rangle$$$$= -2\\langle f,Bf \\rangle$$$$= -2\\lambda ||f||^2$$\n\nbut this assumes that each $f_i$ is Lesbegue integrable, which in turn requires measurable coloring classes.\n- May 4th, 2008 | JSteinhardt | Solution: I believe I have a solution. I will sketch it here. (Sorry, it's broken up into three posts because I cannot figure out how to post something more than 1000 characters...but I have seen longer solutions posted elsewhere so I assume it's okay; if not, I apologize.)\n\nConsider the operator $B_{a}: L^2(R^2) \\to L^2(R^2)$ defined by\n\n$$(B_{a}f)(x,y) = \\int_{-\\pi}^{\\pi} \\sum_{k=0}^{\\infty} a^{-k} f(x+(2k+1)\\cos(t),y+(2k+1)\\sin(t)) dt$$\n\nThis is in some sense a weighting of the adjacency operator. We can then prove the result (well-known for finite graphs) that $\\chi(O) \\geq 1-\\frac{\\lambda_{\\max}}{\\lambda_{\\min}}$, where $\\lambda_{\\max},\\lambda{\\min}$ are the sup and inf of the spectrum of $B_{a}$.\n\nWe note that the eigenfunctions of $B_{a}$ are simply the exponential maps $f_{(r,s)}(x,y) = e^{i(rx+sy)}$.\n- May 4th, 2008 | JSteinhardt | Solution (continued): We see that the eigenvalue of the eigenfunction $f_{(r,s)}$ is given by\n\n$$\\lambda_{(r,s)} = \\int_{-\\pi}^{\\pi} \\sum_{k=0}^{\\infty} a^{-k} e^{i(2k+1)(r\\cos(t)+s\\sin(t))} dt = \\int_{-\\pi}^{\\pi} \\sum_{k=0}^{\\infty} a^{-k} e^{i(2k+1)\\sqrt{r^2+s^2}\\cos(t+\\phi)} dt$$\n\nfor an appropriately chosen $\\phi$. Thus we need only actually consider $\\lambda_{(r,0)}$, which we from now on denote $\\lambda(r)$. Then some calculation gives us that the integral is\n\n$$\\int_{-\\pi}^{\\pi} \\frac{a(a-1)\\cos(r\\cos(t))}{(a-1)^2+4a\\sin^2(r\\cos(t))} dt$$\n\nWe can show that (and this will suffice) that when $a$ is near $1$,\n\n$$\\int_{0}^{\\frac{\\pi}{2}} \\frac{(a-1)\\cos(r\\cos(t))}{(a-1)^2+4a\\sin^2(r\\cos(t))} dt \\geq -4(a-1)^{-\\frac{3}{4}}-\\frac{\\pi}{2}$$\n- May 4th, 2008 | JSteinhardt | Solution (final part): Let $h$ be the function we are integrating. Let $R_k$ denote the region for which $|h(t)| \\geq 1$ and that contains the value of $t$ where $\\cos(t) = \\frac{k\\pi}{r}$. Then we note that $|\\int_{R_k} h(x) dx| > |\\int_{R_{k-1}} h(x) dx|$ and the sines of the integral alternate, so we can just calculate the first one and everything else will be bounded (in particular by $\\frac{\\pi}{2}$ ). With a bit of Taylor approximation, we can bound the size of each $R_k$ by $\\frac{4\\sqrt[4]{a-1}}{\\sqrt{r}}$, and noting that $h$ is always positive for $r \\leq \\frac{\\pi}{2}$, we can replace the $\\sqrt{r}$ with $1$ and then bound $h$ by $\\frac{1}{a-1}$. This gives us the bound we claimed above and we are done.\n\nJacob Steinhardt\n- October 31st, 2007 | Anonymous | Circular chromatic number of the odd distance graph: The proof for $\\chi(G)\\geq 5$ has been recently extended to $\\chi_c(G)\\geq 5$, which implies the previous result, where $\\chi_c$ is the circular chromatic number.\n\nNicolas Roussel.\n- November 7th, 2007 | Anonymous | Correction of previous comment: The proof is actually for $\\chi_c(G)\\geq 4.5$.\n\nNR.\n- November 24th, 2007 | Anonymous | Subgraph construction: For any rational $r\\in[4,4.5)$, there is a subgraph $H_r$ of the odd-distance graph with $\\chi_c(H_r)=r$\n\n[1] Pan Zhi-Shi, Roussel Nicolas, Subgraphs of the odd-distance graph with given circular chromatic number, manuscript\n\nNR.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 67.\n\nAttempt notes:\nTarget:\nMake progress on \"Coloring the Odd Distance Graph\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Davies proved that every finite coloring of the plane has a monochromatic odd-integer-distance pair, so the odd-distance graph has countably infinite chromatic number.\n\n**Verified partial progress.**\n\n- Before the full result, measurable colorings were known to require infinitely many colors and finite subgraphs requiring at least six colors had been constructed.\n- Davies proves the stronger direct statement that every finite coloring of R^2 contains a monochromatic pair at an odd integer distance.\n- A partition into half-unit squares gives a countable proper coloring, so the exact chromatic number is aleph_0.\n\n**Full solution or refutation.**\n\nThe answer is yes: chi(O)=aleph_0, hence in particular chi(O) is infinite.\n\n**What remains.**\n\nThe exact stored question is fully resolved; subsequent work studies prime, polynomial, and other prescribed distance sets.\n\n**Sources checked.**\n\n- James Davies, Odd Distances in Colourings of the Plane, Geometric and Functional Analysis 34 (2024), 19-31. (primary): https://doi.org/10.1007/s00039-024-00659-w\n Evidence used: Proves that no finite coloring avoids monochromatic odd distances and notes the countable upper bound, yielding chi(O)=aleph_0.\n- James Davies, Rose McCarty, and Michał Pilipczuk, Prime and polynomial distances in colourings of the plane, arXiv:2308.02483 (2023). (primary): https://arxiv.org/abs/2308.02483\n Evidence used: Extends the solved odd-distance phenomenon to polynomial values and prime distances.\n\n**Review notes.** This is ordinary graph chromatic number, not only measurable chromatic number. The result answers the exact stored question affirmatively and identifies the cardinal value.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3231, "problem_number": "OPG-771", "title": "Partial List Coloring", "statement": "Conjecture Let $G$ be a simple graph with $n$ vertices and list chromatic number $\\chi_\\ell(G)$. Suppose that $0\\leq t\\leq \\chi_\\ell$ and each vertex of $G$ is assigned a list of $t$ colors. Then at least $\\frac{tn}{\\chi_\\ell(G)}$ vertices of $G$ can be colored from these lists.", "background": "Source: Open Problem Garden. Original node ID: 771. URL: http://www.openproblemgarden.org/op/partial_list_coloring.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partial_list_coloring\n- Author(s): Albertson, Michael O.; Grossman, Sara; Haas, Ruth\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: list assignment; list coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 5th, 2008 by Iradmusa\n\nProblem-page discussion:\nAlbertson, Grossman, and Haas introduce this interesting question in [AGH], and prove some partial results. For instance, they show that under the above assumptions, at least $(1 - (\\frac{ \\chi(G) - 1}{\\chi(G)} )^t) \\cdot n$ vertices of $G$ can be colored from the lists.\n\nBibliography:\n*[AGH] M. Albertson, S. Grossman and R. Haas, Partial list colouring, Discrete Math., 214(2000), pp. 235-240.\n\nBibliography links:\n- Partial list colouring: http://maven.smith.edu/%7Ealbertson/ruth.ps\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Partial List Coloring\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Albertson--Grossman--Haas partial-list-coloring conjecture remains open generally, but it is proved for at least half of the intermediate list sizes.\n\n**Verified partial progress.**\n\n- Iradmusa proves the target lower bound for at least half the values t in {1,...,chi_l(G)-1}.\n- The conjecture has further special-class results.\n\n**Full solution or refutation.**\n\nNo proof for every graph and every list size t was verified.\n\n**What remains.**\n\nEstablish the tn/chi_l(G) bound uniformly, or exhibit a counterexample.\n\n**Sources checked.**\n\n- M. N. Iradmusa, A Note on Partial List Colorings, arXiv:0805.3277. (primary): https://arxiv.org/abs/0805.3277\n Evidence used: States and proves the conjecture for at least half of the intermediate values.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3232, "problem_number": "OPG-788", "title": "Partial List Coloring", "statement": "Let $G$ be a simple graph, and for every list assignment $\\mathcal{L}$ let $\\lambda_{\\mathcal{L}}$ be the maximum number of vertices of $G$ which are colorable with respect to $\\mathcal{L}$. Define $\\lambda_t = \\min{ \\lambda_{\\mathcal{L}} }$, where the minimum is taken over all list assignments $\\mathcal{L}$ with $|\\mathcal{L}| = t$ for all $v \\in V(G)$.\n\nConjecture [2] Let $G$ be a graph with list chromatic number $\\chi_\\ell$ and $1\\leq r\\leq s\\leq \\chi_\\ell$. Then\n$$\n\\frac{\\lambda_r}{r}\\geq\\frac{\\lambda_s}{s}.\n$$", "background": "Source: Open Problem Garden. Original node ID: 788. URL: http://www.openproblemgarden.org/op/partial_list_coloring_0.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partial_list_coloring_0\n- Author(s): Iradmusa, Moharram\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: list assignment; list coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 12th, 2008 by Iradmusa\n\nProblem-page discussion:\nAs you see this conjecture in the special case $s=\\chi_\\ell$, is the conjecture of Albertson, Grossman and Haas [1]: $\\lambda_t\\geq\\frac{tn}{\\chi_\\ell}$ for any $0\\leq t\\leq \\chi_\\ell$.\n\nBibliography:\n[1] M. Albertson, S. Grossman and R. Haas, Partial list colouring, Discrete Math., 214(2000), pp. 235-240.\n\n[2] Moharram N. Iradmusa, A Note on Partial List Colorings, Australasian Journal of Combinatorics, Vol.46, 2010, $19-24$.\n\nRelated:\nRelated problems\nPartial List Coloring\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Partial List Coloring\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** This stronger monotonic-ratio partial-list-coloring formulation remains open; published work gives the Albertson--Grossman--Haas endpoint bound for many, but not all, list sizes.\n\n**Verified partial progress.**\n\n- Iradmusa proves the endpoint lower bound for at least half the intermediate list sizes.\n- The displayed monotonicity conjecture is stronger than the endpoint assertion.\n\n**Full solution or refutation.**\n\nNo general proof of lambda_r/r >= lambda_s/s was verified.\n\n**What remains.**\n\nProve the monotonicity relation or find a graph/list assignment that violates it.\n\n**Sources checked.**\n\n- M. N. Iradmusa, A Note on Partial List Colorings, arXiv:0805.3277. (primary): https://arxiv.org/abs/0805.3277\n Evidence used: Introduces the displayed refined conjecture and proves partial endpoint cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3233, "problem_number": "OPG-806", "title": "Hedetniemi's Conjecture", "statement": "Conjecture If $G,H$ are simple finite graphs, then $\\chi(G \\times H) = \\min \\{ \\chi(G), \\chi(H) \\}$.\n\nHere $G \\times H$ is the tensor product (also called the direct or categorical product) of $G$ and $H$.", "background": "Source: Open Problem Garden. Original node ID: 806. URL: http://www.openproblemgarden.org/op/hedetniemis_conjecture.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hedetniemis_conjecture\n- Author(s): Hedetniemi, Stephen T.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: categorical product; coloring; homomorphism; tensor product\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 25th, 2008 by mdevos\n\nProblem-page discussion:\nThis beautiful and seemingly innocent conjecture asserts a deep and important property of graph coloring. It is undoubtedly one of the most significant unsolved problems in graph coloring and graph homomorphisms.\n\nWe write $G \\rightarrow H$ if there is a homomorphism from $G$ to $H$. The graph $G \\times H$ has a two natural projection maps (projecting onto either the first or second coordinate), and these maps are homomorphisms to $G$ and to $H$. So, in short, $G \\times H \\rightarrow G$ and $G \\times H \\rightarrow H$. A graph is $n$-colorable if and only if it has a homomorphism to $K_n$. Combining this with the transitivity of $\\rightarrow$ we find that $\\chi(G \\times H) \\le \\min\\{ \\chi(G), \\chi(H) \\}$ (indeed, if $\\chi(G) = n$, then $G \\times H \\rightarrow G$ and $G \\rightarrow K_n$, so $G \\times H \\rightarrow K_n$- equivalently, $G \\times H$ is $n$-colorable). So, the hard direction of Hedetniemi's Conjecture is to prove that $\\chi(G \\times H) \\ge \\min \\{ \\chi(G), \\chi(H) \\}$.\n\nLet's define $P(n)$ to be the proposition that $\\chi(G \\times H) \\ge n$ whenever $\\chi(G) \\ge n$ and $\\chi(H) \\ge n$. Then the above conjecture is equivalent to the statement that $P(n)$ holds for every positive integer $n$. Now $P(1)$ holds trivially and $P(2)$ follows from the observation that the product of two graphs each of which contains an edge is a graph which contains an edge. The next case is quite easy too, if $\\chi(G) \\ge 3$ and $\\chi(H) \\ge 3$, then both $G$ and $H$ contain an odd cycle. Since the product of two odd cycles contains an odd cycle, this shows $\\chi(G \\times H) \\ge 3$. The next case up, $P(4)$ was proved by El-Zahar and Sauer by way of a beautiful argument. It is open for all higher values.\n\nA key tool in the proof of El-Zahar and Sauer is the use of exponential graphs. For any pair of graphs $G, H$ the exponential graph $G^H$ is a graph whose vertex set consists of all mappings $f: V(H) \\rightarrow V(G)$. Two vertices $f,g$ are adjacent if $f(x)g(y)$ is an edge of $G$ whenever $xy$ is an edge of $H$. It is easy to see the relevance of $K_n^G$ to this problem. If we have an $n$-coloring $f$ of $G \\times H$, then for every vertex $x \\in V(H)$, there is a mapping $f_x: V(G) \\rightarrow V(K_n)$ given by $f_x(v) = f(v,x)$. This associates each $x \\in V(H)$ with a vertex in $K_n^G$. Now it is easy to verify that whenever $x,y$ are adjacent vertices in $H$, the maps $f_x$ and $f_y$ are adjacent in $K_n^G$. Rather more surprisingly, Hedetniemi's conjecture may be reformulated as follows:\n\nConjecture (version 2 of Hedetniemi) If $\\chi(G) > n$, then $K_n^G$ is $n$-colorable.\n\nThe following conjecture asserts that Hedetniemi's conjecture still holds with circular chromatic number instead of the usual chromatic number. Here $\\chi_c(G)$ is the circular chromatic number of $G$. Since $\\chi(G) = \\lceil \\chi_c(G) \\rceil$ this is a generalization of the original conjecture.\n\nConjecture (Zhu) If $G$ and $H$ are finite simple graphs then $\\chi_c(G \\times H) = \\min\\{ \\chi_c(G), \\chi_c(H) \\}$.\n\nA graph $G$ has circular chromatic number $\\frac{n}{k}$ for positive integers $n,k$ if and only if $G$ has a homomorphism to the graph $K_{n/k}$. This is a graph whose vertex set consists of $n$ vertices cyclically ordered, with two vertices adjacent if they are distance $\\ge k$ apart in the cyclic ordering. So again, we may state this conjecture in terms of homomorphisms to graphs of the form $K_{n/k}$. More generally, let us call a graph $K$ multiplicative if $G \\times H \\rightarrow K$ implies either $G \\rightarrow K$ or $H \\rightarrow K$. Now Hedetniemi's conjecture asserts that every $K_n$ is multiplicative and Zhu's conjecture asserts that every $K_{n/k}$ is multiplicative. With this terminology, El-Zahar and Sauer proved that $K_3$ is multiplicative. A clever generalization of their argument due to Haggkvist, Hell, Miller and Neumann Lara showed that every odd cycle is multiplicative. Recently, Tardif bootstrapped this theorem with the help of a couple of interesting operators on the category of graphs to prove the $K_{n/k}$ is multiplicative whenever $n/k < 4$. Ignoring trivial cases and equivalences, these are essentially the only graphs known to be multiplicative.\n\nIt might be tempting to hope that all graphs are multiplicative, but this is false. To construct a non-multiplicative graph, take two graphs $G,H$ with the property that $G \\not\\rightarrow H$ and $H \\not\\rightarrow G$ (for instance $K_3$ and the Grotzsch Graph). Now $G \\times H$ is not multiplicative since $G \\not\\rightarrow G \\times H$ and $H \\not\\rightarrow G \\times H$, but $G \\times H \\rightarrow G \\times H$. It seems that there is no general conjecture as to what graphs are multiplicative. Some other Cayley graphs look like reasonable candidates to me (M. DeVos), but I haven't any evidence one way or the other.\n\nPoljak and Rodl defined the function $f(n) = \\min \\{ \\chi(G \\times H): \\chi(G) = n = \\chi(H) \\}$. So, Hedetniemi's conjecture is equivalent to $f(n) = n$. Using an interesing inequality relating the chromatic number of a digraph $D$ to the chromatic number of a type of line graph of $D$, they were able to prove the following quite surprising result: Either $f$ is bounded by $9$ or $\\lim_{n \\rightarrow \\infty} f(n) = \\infty$.\n\nThere are a number of interesting partial results not mentioned here, and the reader is encouraged to see the survey article by Zhu.\n\nSource links:\n- tensor product: http://en.wikipedia.org/wiki/tensor product of graphs\n\nDiscussion links:\n- homomorphism: http://en.wikipedia.org/wiki/graph homomorphism\n- circular chromatic number: http://en.wikipedia.org/wiki/circular coloring\n\nComments:\n- March 1st, 2020 | Anonymous | Yaroslav Shitov made some: Yaroslav Shitov made some significant breakthroughs in this area - https://arxiv.org/abs/1905.02167.\n- February 27th, 2013 | Anonymous | Lovasz Theta: I believe that Robert Samal has conjectured a version of this for the Lovasz $\\vartheta$ function, i.e. that\n$$\n\\bar{\\vartheta}(G \\times H) = \\min\\{\\bar{\\vartheta}(G), \\bar{\\vartheta}(H)\\}\n$$\n where $\\bar{\\vartheta}(G):= \\vartheta(\\overline{G})$. I can't find it in the Garden, but it is in this presentation by Samal: http://iuuk.mff.cuni.cz/research/cmi/cmi-I-Samal.pdf\n- July 23rd, 2013 | Robert Samal | Re: Lovasz Theta: In fact, for the Lovasz $\\vartheta$ function (of the complement of the graph) it is a theorem, see arXiv:1305.5545.\n- October 18th, 2011 | Anonymous | Fractional version is true: It is probably worth mentioning that Zhu recently proved the fractional version of this conjecture: that $\\chi_f(G \\times H) = \\min\\{\\chi_f(G),\\chi_f(H)\\}$.\n\nXuding Zhu, The fractional version of Hedetniemi’s conjecture is true, European Journal of Combinatorics, Volume 32, Issue 7, October 2011, Pages 1168-1175, ISSN 0195-6698, 10.1016/j.ejc.2011.03.004. (http://www.sciencedirect.com/science/article/pii/S0195669811000552)\n- November 8th, 2010 | Jon Noel | Very interesting Conjecture.: Very interesting Conjecture.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 68.\n\nAttempt notes:\nTarget:\nMake progress on \"Hedetniemi's Conjecture\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Shitov proved that finite simple graphs G,H can satisfy chi(G x H) < min(chi(G),chi(H)), refuting the conjecture.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe displayed equality is false in general.\n\n**What remains.**\n\nThe original universal assertion is resolved negatively; study the surviving low-chromatic and structured cases separately.\n\n**Sources checked.**\n\n- Y. Shitov, Counterexamples to Hedetniemi's conjecture, Ann. of Math. 190 (2019), 663--667. (primary): https://annals.math.princeton.edu/2019/190-2/p06\n Evidence used: The abstract states the strict inequality counterexamples for finite simple graphs.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3234, "problem_number": "OPG-1764", "title": "Counting 3-colorings of the hex lattice", "statement": "Problem Find $\\lim_{n \\rightarrow \\infty} (\\chi( H_n, 3)) ^{ 1 / |V(H_n)| }$.", "background": "Source: Open Problem Garden. Original node ID: 1764. URL: http://www.openproblemgarden.org/op/counting_3_colorings_of_the_hex_lattice.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/counting_3_colorings_of_the_hex_lattice\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: coloring; Lieb's Ice Constant; tiling; torus\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 5th, 2008 by mdevos\n\nProblem-page discussion:\nWe'll begin by putting in place the necessary notation. Let ${\\mathcal T}$ be the regular triangular tiling of the plane. For every $n \\ge 1$ there is a regular map which triangulates the torus, denoted $T_n$, which may be obtained from a regular hexagonal piece of ${\\mathcal T}$ of side-length $n$ by identifying points on opposite edges of this hexagon. Let $H_n$ be the dual of $T_n$ (on the torus). Then $H_n$ is a regular map on the torus - a hexagonal tiling. One last definition: for any graph $G$ and any positive integer $k$ we let $\\chi(G,k)$ denote the number of proper $k$-coloring of $G$.\n\nA famous theorem of Lieb [L] shows that $\\lim_{n \\rightarrow \\infty} (\\chi(Q_n,3))^{1 / |V(Q_n)|} = (\\frac{4}{3})^{3/2}$ where $Q_n$ denotes the $n \\times n$ quadrangulation of the torus. This theorem is usually stated in terms of Eulerian orientations, and is of interest to physicists as the constant $(\\frac{4}{3})^{3/2}$ (called Lieb's Ice Constant) determines the \"residual entropy for square ice\".\n\nThomassen proved that every planar graph $G$ with girth $\\ge 5$ has exponentially many proper 3-colorings. More precisely, he showed that $(\\chi(G,3))^{ 1 / |V(G)| } \\ge 2 ^{1 / 10000}$. This gives a lower bound on the limit in the above problem (assuming it exists).\n\nBibliography:\n[L] E. H. Lieb, Exact Solution of the Problem of the Entropy of Two-Dimensional Ice. Phys. Rev. Lett. 18, 692-694, 1967.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Counting 3-colorings of the hex lattice\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No rigorous proof of existence or exact evaluation of the stated toroidal hex-lattice vertex-coloring entropy limit was verified.\n\n**Verified partial progress.**\n\n- Thomassen's exponential 3-coloring lower bound motivates a positive lower-bound program.\n- Related physics entropy calculations concern different edge/bond or face models.\n\n**Full solution or refutation.**\n\nNo exact vertex-coloring limit for H_n was verified.\n\n**What remains.**\n\nProve convergence and identify the entropy constant for the precise vertex-coloring model.\n\n**Sources checked.**\n\n- Graph-theory open problems, Counting 3-colorings of the hex lattice (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/counting_3_colorings_of_the_hex_lattice/\n Evidence used: Separates the exact vertex-coloring question from Baxter's related model and reports no rigorous resolution.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3235, "problem_number": "OPG-2040", "title": "Circular colouring the orthogonality graph", "statement": "Let ${\\mathcal O}$ denote the graph with vertex set consisting of all lines through the origin in ${\\mathbb R}^3$ and two vertices adjacent in ${\\mathcal O}$ if they are perpendicular.\n\nProblem Is $\\chi_c({\\mathcal O}) = 4$?", "background": "Source: Open Problem Garden. Original node ID: 2040. URL: http://www.openproblemgarden.org/op/circular_colouring_the_orthogonality_graph.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/circular_colouring_the_orthogonality_graph\n- Author(s): DeVos, Matt; Ghebleh, Mohammad; Goddyn, Luis A.; Mohar, Bojan; Naserasr, Reza\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: circular coloring; geometric graph; orthogonality\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 23rd, 2008 by mdevos\n\nProblem-page discussion:\nIn the problem statement, $\\chi_c$ denotes the circular chromatic number.\n\nColoring properties of ${\\mathcal O}$ are, rather surprisingly, of interest in quantum mechanics. If the spins of certain particles are measured in three orthogonal directions, then these measurements always return one $0$ and two values which are $\\pm 1$. If such a particle has \"decided\" in advance how it will respond to any possible measurement, then the set of directions in which it will respond $0$ must be an independent set in the orthogonality graph ${\\mathcal O}$ which meets every triangle. Kochen and Specker have shown that ${\\mathcal O}$ (even certain finite subgraphs of it) does not have any independent set meeting every triangle, thus exhibiting a rather mysterious property of these particles. In some sense, if the person doing the measurement has the free will to decide in which directions to measure, then the particle must have some free will to decide how it will respond.\n\nThe property that ${\\mathcal O}$ has no independent set which meets every triangle shows that $\\chi({\\mathcal O}) \\ge 4$. On the other hand, if we center a regular octahedron at the origin, and assign a color to each line $L$ depending on which pair of opposite faces it passes through (if $L$ meets more than one pair of opposite faces, just choose one) we get a proper 4-coloring of ${\\mathcal O}$. Therefore, $\\chi({\\mathcal O}) = 4$.\n\nThese bounds prove that $3 \\le \\chi_c({\\mathcal O}) \\le 4$. By investigating certain finite subgraphs of ${\\mathcal O}$, DeVos, Ghebleh, Goddyn, Mohar, and Naserasr have shown that $\\chi_c({\\mathcal O}) \\ge 3.5$.\n\nRelated:\nRelated problems\nPartitioning the Projective Plane\nThe Double Cap Conjecture\n\nDiscussion links:\n- circular chromatic number: http://en.wikipedia.org/wiki/circular coloring\n\nComments:\n- September 28th, 2009 | Anonymous | Still open?: Does anyone know if this problem is still open?\n- September 28th, 2009 | md | pretty sure: I am pretty confident this is still open. Apart from this one many-author paper, I don't think it has received any significant attention.\n- November 4th, 2008 | Anonymous | This is problem 10769 of the: This is problem 10769 of the American Mathematical Monthly. The solution appeared in AMM 108 (2001), 774-775. See also http://tph.tuwien.ac.at/~svozil/publ/blatter.htm\n\nChristian Blatter\n- November 4th, 2008 | mdevos | not exactly!: The problem you mention, 10769 of the AMM, concerns the chromatic number of the orthogonality graph. The problem here concerns the circular chromatic number of the orthogonality graph, and I believe it is still open.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Circular colouring the orthogonality graph\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The circular chromatic number is known to lie between 3.5 and 4; equality with 4 remains unproved.\n\n**Verified partial progress.**\n\n- Finite orthogonality subgraphs give the lower bound 3.5.\n- A geometric 4-coloring supplies the upper bound.\n\n**Full solution or refutation.**\n\nNo improvement of either bound was verified.\n\n**What remains.**\n\nConstruct a circular obstruction forcing 4 or produce a circular coloring below 4.\n\n**Sources checked.**\n\n- Graph-theory open problems, Circular colouring the orthogonality graph (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/circular_colouring_the_orthogonality_graph/\n Evidence used: Records the 3.5 to 4 bounds and reports no later improvement.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3236, "problem_number": "OPG-34839", "title": "Double-critical graph conjecture", "statement": "A connected simple graph $G$ is called double-critical, if removing any pair of adjacent vertexes lowers the chromatic number by two.\n\nConjecture $K_n$ is the only $n$-chromatic double-critical graph", "background": "Source: Open Problem Garden. Original node ID: 34839. URL: http://www.openproblemgarden.org/op/double_critical_graph_conjecture.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/double_critical_graph_conjecture\n- Author(s): Erdos, Paul; Lovasz, Laszlo\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: coloring; complete graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: January 18th, 2009 by DFR\n\nProblem-page discussion:\nThis conjecture is a special case of a more general problem by Erdos and Lovasz proposed in 1966. It has been independently proven for the case where $\\chi(G) = 5$ by Mozhan [3] and Stiebitz [4].\n\nBibliography:\n*[1] P. Erdos, Problem 2, in: Theory of Graphs (Proc. Colloq., Tihany, 1966), Academic Press, New York, 1968, p. 361.\n\n[2] F. Chung, R. Graham, Erdos on graphs: His legacy of unsolved problems, A K Peters, Wellesley, Massachusetts, 1998.\n\n[3] N. N. Mozhan, On double critical graphs with the chromatic number five, Metody Diskretb. Anal. 46 (1987) 50-59.\n\n[4] M. Stiebitz, $K_5$ is the only double-critical $5$-chromatic graph, Discrete Math. 64 (1987) 91-93.\n\nSource links:\n- chromatic number: http://en.wikipedia.org/wiki/chromatic number\n\nComments:\n- November 30th, 2009 | Anonymous | Statement needs connected assumption: G needs to be assumed connected in the statement. Otherwise the disjoint union of K_k and a vertex (say) would be a counterexample.\n- May 29th, 2009 | Anonymous | missing connectivity condition: As it is now the conjecture is false -- take the disjoint union of a clique and a vertex. Just need to add \"connected\" to the definition of doubly critical.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Double-critical graph conjecture\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The double-critical graph conjecture remains open from chromatic number six onward, with low-chromatic and structured graph cases proved.\n\n**Verified partial progress.**\n\n- The conjecture is verified for t<=5.\n- Claw-free cases and further local structure are established.\n\n**Full solution or refutation.**\n\nNo proof that all double-critical graphs are complete was verified.\n\n**What remains.**\n\nSettle the general t>=6 case or construct a noncomplete double-critical graph.\n\n**Sources checked.**\n\n- M. Rolek and Z.-X. Song, Double-critical graph conjecture for claw-free graphs, arXiv:1610.00636. (primary): https://arxiv.org/abs/1610.00636\n Evidence used: Abstract records the general conjecture as open and proves claw-free cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3237, "problem_number": "OPG-36907", "title": "Bounding the chromatic number of triangle-free graphs with fixed maximum degree", "statement": "Conjecture A triangle-free graph with maximum degree $\\Delta$ has chromatic number at most $\\ceil{\\frac{\\Delta}{2}}+2$.", "background": "Source: Open Problem Garden. Original node ID: 36907. URL: http://www.openproblemgarden.org/op/bounding_the_chromatic_number_of_triangle_free_graphs_with_fixed_maximum_degree.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bounding_the_chromatic_number_of_triangle_free_graphs_with_fixed_maximum_degree\n- Author(s): Kostochka, Alexandr V.; Reed, Bruce A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: chromatic number; girth; maximum degree; triangle free\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 17th, 2009 by Andrew King\n\nProblem-page discussion:\nThis conjecture is a special case of Reed's $\\omega$, $\\Delta$, and $\\chi$ conjecture, which posits that for any graph, $\\chi \\leq \\lceil\\frac 12(\\Delta+1+\\omega)\\rceil$, where $\\omega$, $\\Delta$, and $\\chi$ are the clique number, maximum degree, and chromatic number of the graph respectively. Reed's conjecture is very easy to prove for complements of triangle-free graphs, but the triangle-free case seems challenging and interesting in its own right.\n\nThis conjecture is very much true for large values of $\\Delta$; Johansson proved that triangle-free graphs have chromatic number at most $\\frac{9\\Delta}{\\ln \\Delta}$. Surprisingly, the question appears to be open for every value of $\\Delta$ greater than four, up until Johansson's result implies the conjecture.\n\nKostochka previously proved that the chromatic number of a triangle-free graph is at most $\\frac{2\\Delta}{3}+2$, and he proved that for every $\\Delta \\geq 5$ there is a $g$ for which a graph of girth $g$ has chromatic number at most $\\frac{\\Delta}2+2$. Specifically, he showed that $g \\geq 4(\\Delta+2)\\ln \\Delta$ is sufficient. In [K] he posed the general problem: \"To find the best upper estimate for the chromatic number of the graph in terms of the maximal degree and density or girth.\"\n\nThe conjecture is implied by Brooks' Theorem for $\\Delta\\leq 5$. The three smallest open values of $\\Delta$ offer natural entry points to this problem. The easiest seems to be:\n\nProblem Does there exist a $6$-chromatic triangle-free graph of maximum degree 6?\n\nPerhaps looking at graphs of girth at least five would also be a good starting point.\n\nBibliography:\n[K] Kostochka, A. V., Degree, girth and chromatic number. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, pp. 679--696, Colloq. Math. Soc. János Bolyai, 18, North-Holland, Amsterdam-New York, 1978.\n\n*[R] Reed, B.A., $\\omega, \\Delta$, and $\\chi$, J. Graph Theory 27 (1998) 177-212.\n\nRelated:\nRelated problems\nReed's omega, delta, and chi conjecture\nGrunbaum's Conjecture\n\nComments:\n- May 16th, 2020 | Anonymous | Modifying the conjecture: From Reed's conjecture, it seems that the ceiling has to be replaced by flooring. Thanks.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Bounding the chromatic number of triangle-free graphs with fixed maximum degree\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The displayed ceiling bound remains unresolved in the middle-degree range, notably Delta=6, while stronger bounds are known for Delta at least 524 and several finite or structural subfamilies. The source likely intended a floor, not a ceiling.\n\n**Verified partial progress.**\n\n- Brooks' theorem covers the displayed statement for maximum degree at most 5.\n- Davies, Kang, Pirot, and Sereni prove chi(G)<=ceil((Delta+1)/2)+1 for every triangle-free graph with Delta>=524, which is at least as strong as the imported bound.\n- Goedgebeur proved that the smallest triangle-free 6-chromatic graph has between 32 and 40 vertices; Abrishami and Erfanian verify Reed's bound for further finite and maximal-triangle-free families.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was found. The imported formula uses an undefined ceil macro and is not exactly Reed's triangle-free specialization: for odd Delta it is one color weaker because Reed yields floor(Delta/2)+2.\n\n**What remains.**\n\nAt minimum, decide whether every triangle-free graph of maximum degree 6 is 5-colorable, and clarify whether the intended conjecture uses floor(Delta/2)+2 or the literally stored ceiling bound.\n\n**Sources checked.**\n\n- James Davies, Ross J. Kang, Francois Pirot, and Jean-Sebastien Sereni, On Vizing's problem for triangle-free graphs, arXiv:2309.10876. (primary): https://arxiv.org/abs/2309.10876\n Evidence used: Proves the stronger Vizing/Reed-form inequality for all maximum degrees at least 524.\n- Gholamreza Abrishami and Ahmad Erfanian, A note on Reed's conjecture for triangle-free graphs, Discrete Mathematics 346 (2023), 113609, DOI 10.1016/j.disc.2023.113609. (primary): https://doi.org/10.1016/j.disc.2023.113609\n Evidence used: States that no general proof or counterexample is known and proves the conjecture for several bounded-order and maximal triangle-free subfamilies.\n- Jan Goedgebeur, On minimal triangle-free 6-chromatic graphs, Journal of Graph Theory 93 (2020), 34-48, DOI 10.1002/jgt.22467. (primary): https://doi.org/10.1002/jgt.22467\n Evidence used: Establishes the 32-to-40 vertex interval for a smallest triangle-free 6-chromatic graph and finite verification of Reed's conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3238, "problem_number": "OPG-36936", "title": "Graphs with a forbidden induced tree are chi-bounded", "statement": "Say that a family ${\\mathcal F}$ of graphs is $\\chi$-bounded if there exists a function $f: {\\mathbb N} \\rightarrow {\\mathbb N}$ so that every $G \\in {\\mathcal F}$ satisfies $\\chi(G) \\le f (\\omega(G))$.\n\nConjecture For every fixed tree $T$, the family of graphs with no induced subgraph isomorphic to $T$ is $\\chi$-bounded.", "background": "Source: Open Problem Garden. Original node ID: 36936. URL: http://www.openproblemgarden.org/op/graphs_with_a_forbidden_induced_tree_are_chi_bounded.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/graphs_with_a_forbidden_induced_tree_are_chi_bounded\n- Author(s): Gyarfas, Andras\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: chi-bounded; coloring; excluded subgraph; tree\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 16th, 2009 by mdevos\n\nProblem-page discussion:\nThis deep conjecture remains open despite considerable effort. Note that the conjecture would be false were the graph $T$ to be permitted to contain a cycle, since then the class would admit graphs of high girth (where $\\omega = 2$ ) and high chromatic number.\n\nIt is an easy exercise to prove this conjecture in the special case when $T$ is either a path or a star, but things get difficult from here. Kierstead and Penrice solved the special case when $T$ has radius 2, and Kierstead and Zhu solved the special case when $T$ has radius 3 and has the property that every vertex incident with the center vertex has degree 2.\n\nScott proved that the class of graphs which exclude all subdivisions of a fixed tree $T$ as induced subgraphs are $\\chi$-bounded. It follows from this that Gyarfas's conjecture also holds for subdivisions of stars.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Graphs with a forbidden induced tree are chi-bounded\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Gyárfás-Sumner conjecture remains open for general trees, although it is proved for several tree families and a broad path-induced weakening is now known.\n\n**Verified partial progress.**\n\n- Classical results cover paths, stars, radius-two trees, and some radius-three or broom-like families.\n- Nguyen, Scott, and Seymour prove that bounded-clique graphs of sufficiently large chromatic number contain every fixed rooted tree path-induced.\n- Polynomial chi-bounds are known for some special trees, while even the polynomial strengthening for P5 remains unresolved.\n\n**Full solution or refutation.**\n\nA 2024 primary paper explicitly states that the fully induced-tree conjecture remains open and proves a weaker path-induced theorem.\n\n**What remains.**\n\nUpgrade path-induced copies to induced copies for arbitrary fixed trees, or enlarge the families of trees for which chi-boundedness is proved.\n\n**Sources checked.**\n\n- Tung Nguyen, Alex Scott, and Paul Seymour, A Note on the Gyárfás-Sumner Conjecture, Graphs and Combinatorics 40 (2024), article 33, DOI 10.1007/s00373-024-02754-z. (primary): https://doi.org/10.1007/s00373-024-02754-z\n Evidence used: Explicitly states the conjecture remains open, surveys proved tree families, and proves the path-induced weakening.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3239, "problem_number": "OPG-36939", "title": "Are vertex minor closed classes chi-bounded?", "statement": "Question Is every proper vertex-minor closed class of graphs chi-bounded?", "background": "Source: Open Problem Garden. Original node ID: 36939. URL: http://www.openproblemgarden.org/op/vertex_minor_closed_classes_are_chi_bounded.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/vertex_minor_closed_classes_are_chi_bounded\n- Author(s): Geelen, Jim\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: chi-bounded; circle graph; coloring; vertex minor\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 16th, 2009 by mdevos\n\nProblem-page discussion:\nWe say that a family of graphs ${\\mathcal F}$ is $\\chi$-bounded if there is a function $f: {\\mathbb N} \\rightarrow {\\mathbb N}$ so that $\\chi(G) \\le f( \\omega(G))$ for every $G \\in {\\mathcal F}$.\n\nIf $G$ is a simple graph, a vertex minor of $G$ is any graph which can be obtained by a sequence of the following operations:\n\n- delete a vertex\n- choose a vertex $v$ and complement the neighborhood of $v$ (i.e. whenever $u,w$ are neighbors of $v$, switch $u,w$ between adjacent/non-adjacent).\n\nDvorak and Kral [DK] showed that this conjecture is true for class of graphs of bounded rank-width, and the class of graphs having no vertex-minor isomorphic to the wheel $W_5$ on $6$ vertices.\n\nBibliography:\n[DK] Z. Dvorak and D. Král. Classes of graphs with small rank decompositions are χ-bounded. European J. Combin., 33(4):679–-683, 2012. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR3350076\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Are vertex minor closed classes chi-bounded?\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** James Davies proved Geelen's conjecture in full: every proper vertex-minor-closed class of graphs is chi-bounded.\n\n**Verified partial progress.**\n\n- Earlier work handled bounded rank-width classes and classes excluding fixed wheel vertex-minors.\n- Davies' proof first establishes control properties and then derives chi-boundedness for any class excluding at least one graph as a vertex-minor.\n\n**Full solution or refutation.**\n\nTheorem 1.1 of Davies' paper is exactly the imported question and answers it affirmatively. Polynomial chi-boundedness is a separate stronger open direction.\n\n**What remains.**\n\nFor the stronger successor problem, determine whether every proper vertex-minor-closed class is polynomially chi-bounded and improve quantitative bounding functions.\n\n**Sources checked.**\n\n- James Davies, Vertex-minor-closed classes are chi-bounded, Combinatorica 42 (2022), 1049-1079, DOI 10.1007/s00493-021-4767-3. (primary): https://arxiv.org/abs/2008.05069\n Evidence used: The abstract and Theorem 1.1 state and prove that every proper vertex-minor-closed class is chi-bounded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3240, "problem_number": "OPG-37907", "title": "Mixing Circular Colourings", "statement": "Question Is $\\mathfrak{M}_c(G)$ always rational?", "background": "Source: Open Problem Garden. Original node ID: 37907. URL: http://www.openproblemgarden.org/op/mixing_circular_colourings_0.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/mixing_circular_colourings_0\n- Author(s): Brewster, Richard C.; Noel, Jonathan A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: discrete homotopy; graph colourings; mixing\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: September 22nd, 2012 by Jon Noel\n\nProblem-page discussion:\nGiven a proper $k$-colouring $f$ of a graph $G$, consider the following 'mixing process:'\n\n- choose a vertex $v\\in V(G)$;\n- change the colour of $v$ (if possible) to yield a different $k$-colouring $f'$ of $G$.\n\nA natural problem arises: Can every $k$-colouring of $G$ be generated from $f$ by repeatedly applying this process? If so, we say that $G$ is $k$-mixing.\n\nThe problem of determining if a graph is $k$-mixing and several related problems have been studied in a series of recent papers [1,2,4-6]. The authors of [4] provide examples which show that a graph can be $k$-mixing but not $k'$-mixing for integers $k' > k$. For example, given $m\\geq3$ consider the bipartite graph $L_m$ which is obtained by deleting a perfect matching from $K_{m,m}$. It is an easy exercise to show that for $L_m$ is $k$-mixing if and only if $k\\geq3$ and $k\\neq m$. This example motivates the following definition.\n\nDefinition Define the mixing threshold of $G$ to be $$\\mathfrak{M}(G):= \\min\\{\\ell\\in\\mathbb{N}: G\\text{ is }k\\text{-mixing whenever } k\\geq\\ell\\}.$$\n\nAn analogous definition can be made for circular colouring. Recall, a $(k,q)$-colouring of a graph is a mapping $f:V(G)\\to \\{0,1,\\dots,k-1\\}$ such that if $uv\\in E(G)$, then $q\\leq |f(u)-f(v)|\\leq k-q$. As with ordinary colourings, we say that a graph $G$ is $(k,q)$-mixing if all $(k,q)$-colourings of $G$ can be generated from a single $(k,q)$-colouring $f$ by recolouring one vertex at a time.\n\nDefinition Define the circular mixing threshold of $G$ to be $$\\mathfrak{M}_c(G):= \\inf\\{\\ell\\in\\mathbb{Q}: G\\text{ is }(k,q)\\text{-mixing whenever } k/q \\geq\\ell\\}.$$\n\nSeveral bounds on the circular mixing threshold are obtained in [3], including the following which relates the circular mixing threshold to the mixing threshold.\n\nTheorem (Brewster and Noel $[3)$ ] For every graph $G$, $$\\mathfrak{M}_c(G)\\leq\\max\\left\\{\\frac{|V(G)|+1}{2}, \\mathfrak{M}(G)\\right\\}.$$\n\nAs a corollary, we have the following: if $|V(G)|\\leq 2\\mathfrak{M}(G)-1$, then $\\mathfrak{M}_c(G)\\leq \\mathfrak{M}(G)$. However, examples in [3] show that the ratio $\\mathfrak{M}_c/\\mathfrak{M}$ can be arbitrarily large in general.\n\nRegarding the problem of determining if $\\mathfrak{M}_c$ is rational, it is worth mentioning that there are no known examples of graphs $G$ for which $\\mathfrak{M}_c(G)$ is not an integer.\n\nOther problems are also given in [3]. One can check that $\\mathfrak{M}_c(K_2) = 2$ and $\\mathfrak{M}(K_2) = 3$. However, the only graphs which are known to satisfy $\\mathfrak{M}_c < \\mathfrak{M}$ are in some sense related to $K_2$, eg. trees and complete bipartite graphs.\n\nQuestion Is there a non-bipartite graph $G$ such that $\\mathfrak{M}_c(G) < \\mathfrak{M}(G)$?\n\nUsing an example from [4], it is shown in [3] that if $m\\geq2$ is an integer, then there is a graph $G$ such that $\\mathfrak{M}_c(G) = \\chi(G) = m$ if and only if $m\\neq 3$. A natural question to ask is whether a similar result holds for the circular chromatic number. Again, certain bipartite graphs are an exception.\n\nQuestion Is there a non-bipartite graph $G$ such that $\\mathfrak{M}_c(G) =\\chi_c(G)$?\n\nAlso, the example of $K_2$ shows that the circular mixing threshold is, in general, not attained. However, the following problem is open.\n\nQuestion Is the circular mixing threshold always attained for non-bipartite graphs?\n\nFor more precise versions of the last three questions, see [3].\n\nBibliography:\n[1] P. Bonsma and L. Cereceda. Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoret. Comput. Sci. 410 (2009), (50): 5215--5226.\n\n[2] P. Bonsma, L. Cereceda, J. van den Heuvel, and M. Johnson. Finding paths between graph colourings: Computational complexity and possible distances. Electronic Notes in Discrete Mathematics 29 (2007): 463--469.\n\n*[3] R. C. Brewster and J. A. Noel. Mixing Homomorphisms and Extending Circular Colourings. Submitted. pdf.\n\n[4] L. Cereceda, J. van den Heuvel, and M. Johnson. Connectedness of the graph of vertex-colourings. Discrete Math. 308 (2008), (5-6): 913--919.\n\n[5] L. Cereceda, J. van den Heuvel, and M. Johnson. Mixing 3-colourings in bipartite graphs. European J. Combin. 30 (2009), (7): 1593--1606.\n\n[6] L. Cereceda, J. van den Heuvel, and M. Johnson. Finding paths between 3-colorings. Journal of Graph Theory, 67 (2011), (1): 69--82.\n\n[7] J. A. Noel. \"Jonathan Noel - Mixing Circular Colourings.\" Webpage.\n\nBibliography links:\n- pdf: http://www.math.mcgill.ca/jnoel/pdf/Mixing Paper.pdf\n- Connectedness of the graph of vertex-colourings: http://www.sciencedirect.com/science/article/pii/S0012365X07005456\n- Webpage: http://people.maths.ox.ac.uk/noel/MixCirc.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 38.\n\nAttempt notes:\nTarget:\nMake progress on \"Mixing Circular Colourings\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Circular-colouring mixing has been characterized in important low-ratio regimes, but no proof that the circular mixing number is always rational, and no irrational example, was verified.\n\n**Verified partial progress.**\n\n- Brewster and Moore characterize (p,q)-mixing for 2 < p/q < 4 and derive exact consequences for bipartite graphs.\n- A reconfiguration complexity dichotomy is known below versus at least 4.\n\n**Full solution or refutation.**\n\nResults at rational parameters do not force the infimum defining the circular mixing number to be attained at a rational value.\n\n**What remains.**\n\nProve rational attainment for every finite graph or construct a graph whose circular mixing number is irrational.\n\n**Sources checked.**\n\n- Richard C. Brewster and Benjamin Moore, Characterizing circular colouring mixing for p/q < 4, Journal of Graph Theory 102 (2023), 271-294, DOI 10.1002/jgt.22870. (primary): https://doi.org/10.1002/jgt.22870\n Evidence used: Characterizes mixing below ratio 4 and proves important special cases, without settling rationality of the general threshold.\n- Richard C. Brewster, Sean McGuinness, Benjamin Moore, and Jonathan A. Noel, A Dichotomy Theorem for Circular Colouring Reconfiguration, Theoretical Computer Science 639 (2016), 1-13, arXiv:1508.05573. (primary): https://arxiv.org/abs/1508.05573\n Evidence used: Establishes the algorithmic dichotomy for fixed circular-colouring ratios, a partial structural result distinct from rationality of the mixing number.\n- Graph Conjectures, Mixing Circular Colourings, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/mixing_circular_colourings_0/\n Evidence used: The specialist maintained tracker continues to list rationality as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3241, "problem_number": "OPG-44879", "title": "Choice Number of k-Chromatic Graphs of Bounded Order", "statement": "Conjecture If $G$ is a $k$-chromatic graph on at most $mk$ vertices, then $\\text{ch}(G)\\leq \\text{ch}(K_{m*k})$.", "background": "Source: Open Problem Garden. Original node ID: 44879. URL: http://www.openproblemgarden.org/op/choice_number_of_k_chromatic_graphs_of_bounded_order.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/choice_number_of_k_chromatic_graphs_of_bounded_order\n- Author(s): Noel, Jonathan A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: choosability; complete multipartite graph; list coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: February 2nd, 2013 by Jon Noel\n\nProblem-page discussion:\nFor integers $m,k\\geq1$, let $K_{m*k}$ denote the complete $k$-partite graph in which every part has size $m$.\n\nIn one of the original papers on choosability, Erdos, Rubin and Taylor [ERT] proved that $\\text{ch}(K_{2*k})=k$. Later, Ohba [Ohba] conjectured the following generalization: if $|V(G)|\\leq 2\\chi(G)+1$, then TeX Embedding failed!.} This was proved by Noel, Reed and Wu [NRW12].\n\nTheorem (Noel, Reed and Wu 2012) If $|V(G)|\\leq 2\\chi(G)+1$, then $\\text{ch}(G)=\\chi(G)$.\n\nThe above theorem implies that the above conjecture holds for $m=2$. That is, if $G$ is a $k$-chromatic graph on at most $2k$ vertices (in fact, at most $2k+1$ vertices), then $\\text{ch}(G)=k=\\text{ch}(K_{2*k})$.\n\nKierstead [Kie00] proved that $\\text{ch}(K_{3*k})=\\left\\lceil\\frac{4k-1}{3}\\right\\rceil$. This was generalized by Noel, West, Wu and Zhu [NWWZ13] to the following:\n\nTheorem (Noel, West, Wu and Zhu 2013) For every graph $G$,\n$$\n\\text{ch}(G)\\leq\\max\\left\\{\\chi(G),\\left\\lceil\\frac{|V(G)|+\\chi(G)-1}{3}\\right\\rceil\\right\\}.\n$$\n\nTherefore, if $G$ is a $k$-chromatic graph on at most $3k$ vertices, then $\\text{ch}(G)\\leq \\left\\lceil\\frac{4k-1}{3}\\right\\rceil=\\text{ch}(K_{3*k})$. This shows that the conjecture is true for $m=3$.\n\nRecently, Kierstead, Salmon and Wang [KSW14] proved the following:\n\nTheorem (Kierstead, Salmon and Wang 2014) $\\text{ch}(K_{4*k})=\\left\\lceil\\frac{3k-1}{2}\\right\\rceil$.\n\nHowever, it is not known whether the upper bound of $\\left\\lceil\\frac{3k-1}{2}\\right\\rceil$ holds for all $k$-chromatic graphs on at most $4k$ vertices. If true, it would verify the conjecture for $m=4$.\n\nThe following is a refinement of the conjecture.\n\nConjecture (Noel 2013) For $n\\geq k\\geq 1$ there is a graph $G_{n,k}$ such that\n\n- $G_{n,k}$ is a complete $k$-partite graph on $n$ vertices,\n- the stability number of $G_{n,k}$ is $\\left\\lceil n/k\\right\\rceil$, and\n- every $k$-chromatic graph $G$ on at most $n$ vertices satisfies $\\text{ch}(G)\\leq \\text{ch}(G_{n,k})$.\n\nBibliography:\n[Alo92] N. Alon. Choice numbers of graphs: a probabilistic approach. Combin. Probab. Comput., 1(2):107–114, 1992.\n\n[ERT80] P. Erdos, A. L. Rubin, and H. Taylor. Choosability in graphs. Congress. Numer., XXVI, pages 125–157, 1980.\n\n[Kie00] H. A. Kierstead. On the choosability of complete multipartite graphs with part size three. Discrete Math., 211(1-3):255–259, 2000.\n\n[KSW14] H. A. Kierstead, A. Salmon and R. Wang. On the Choice Number of Complete Multipartite Graphs With Part Size Four.\n\n*[Noe13] J. A. Noel. Choosability of Graphs With Bounded Order: Ohba's Conjecture and Beyond. Master's thesis, McGill University, Montreal. pdf\n\n[NRW12] J. A. Noel, B. A. Reed, and H. Wu. A Proof of a Conjecture of Ohba. Preprint, arXiv:1211.1999v1, November 2012. Webpage\n\n[NWWZ13] J. A. Noel, D. B. West, H. Wu, and X. Zhu. Beyond Ohba's Conjecture: A bound on the choice number of $k$-chromatic graphs with $n$ vertices. Preprint, arXiv:1308.6739v1, August 2013. pdf\n\n[Ohb02] K. Ohba. On chromatic-choosable graphs. J. Graph Theory, 40(2):130–135, 2002.\n\n[Yan03] D. Yang. Extension of the game coloring number and some results on the choosability of complete multipartite graphs. PhD thesis, Arizona State University, Tempe, Arizona, 2003.\n\nRelated:\nRelated problems\nOhba's Conjecture\nChoice number of complete multipartite graphs with parts of size 4\n\nBibliography links:\n- On the Choice Number of Complete Multipartite Graphs With Part Size Four: http://www.arxiv.org/abs/1407.3817\n- pdf: http://people.maths.ox.ac.uk/noel/Masters.pdf\n- Webpage: http://people.maths.ox.ac.uk/noel/Ohba.html\n- pdf: http://people.maths.ox.ac.uk/noel/Choosability.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 28.\n\nAttempt notes:\nTarget:\nMake progress on \"Choice Number of k-Chromatic Graphs of Bounded Order\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The bounded-order choosability conjecture is proved for m=2 and m=3, while m>=4 remains open.\n\n**Verified partial progress.**\n\n- Noel--West--Wu--Zhu prove the m=3 case.\n- Kierstead--Salmon--Wang determine the target choice number for K_{4*k}.\n\n**Full solution or refutation.**\n\nNo general proof for all m was verified.\n\n**What remains.**\n\nProve the m=4 upper bound and extend to all m.\n\n**Sources checked.**\n\n- J. A. Noel, D. B. West, H. Wu and X. Zhu, Beyond Ohba's Conjecture: A bound on the choice number of k-chromatic graphs with n vertices, European Journal of Combinatorics 43 (2015), arXiv:1308.6739. (primary): https://arxiv.org/abs/1308.6739\n Evidence used: The paper proves the m=3 case.\n- Graph-theory open problems, Choice Number of k-Chromatic Graphs of Bounded Order (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/choice_number_of_k_chromatic_graphs_of_bounded_order/\n Evidence used: Records m=4 and higher as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3242, "problem_number": "OPG-46575", "title": "Melnikov's valency-variety problem", "statement": "Problem The valency-variety $w(G)$ of a graph $G$ is the number of different degrees in $G$. Is the chromatic number of any graph $G$ with at least two vertices greater than $$\\ceil{ \\frac{\\floor{w(G)/2}}{|V(G)| - w(G)} } ~?$$", "background": "Source: Open Problem Garden. Original node ID: 46575. URL: http://www.openproblemgarden.org/op/melnikovs_valency_variety_problem.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/melnikovs_valency_variety_problem\n- Author(s): Melnikov, L. S.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: March 3rd, 2013 by asp\n\nProblem-page discussion:\nAccording to Jensen and Toft [JT, p. 90], the problem is due to Melnikov and was mentioned by Vizing [V] and Zykov [Z]. According to Zykov [Z], Melnikov showed that the suggested lower bound would be best possible.\n\nA best possible upper bound on the chromatic number in terms of $|V(G)|$ and $w(G)$ is $|V(G)| - \\floor{ w(G) / 2 }$ as proved by Nettleton [N] and Dirac [D].\n\nBibliography:\n[D] G. A. Dirac. Valency-variey and chromatic number of abstract graphs. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 13, 59--64, 1964.\n\n[JT] Tommy R. Jensen, Bjarne Toft: Graph Coloring Problems, Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc., New York, 1995.\n\n[N] R. E. Nettleton. Some generalized theorems on connectivity. Canad. J. Math. 12, 546--554, 1960.\n\n*[V] V. G. Vizing. Some unsolved problems in graph theory (in Russian). Uspekhi Mat. Nauk. 23, 117--134, 1968. English translation in Russian Math. Surveys 23, 125--141.\n\n*[Z] A. A. Zykov. Problem 11. In: H. Sachs, H.-J. Voss, and H. Walther, editors, Beiträge zur Graphentheorie vorgetragen auf dem Internationalen Kolloquium in Manebach DDR vom 9.-12. Mai 1967, page 228. B. G. Teubner, 1968.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Melnikov's valency-variety problem\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** A current maintained graph-problem index labels Melnikov's valency-variety lower-bound problem open. The classical complementary sharp upper bound is known, but no modern resolution of the displayed lower bound was found.\n\n**Verified partial progress.**\n\n- Nettleton and Dirac proved the sharp upper bound chi(G)<=|V(G)|-floor(w(G)/2).\n- Jensen--Toft report that Melnikov showed the proposed lower bound would be best possible.\n\n**Full solution or refutation.**\n\nNo proof or counterexample to the exact displayed lower-bound question was verified.\n\n**What remains.**\n\nVerify the historical formula against a clean primary or monograph scan, then prove the lower bound for every graph or find a counterexample.\n\n**Sources checked.**\n\n- Graph-theory open problems, Melnikov's valency-variety problem, maintained index entry. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/index.html\n Evidence used: Current index explicitly labels the named problem open.\n- Open Problem Garden, Melnikov's valency-variety problem. (maintained_tracker): https://garden.irmacs.sfu.ca/op/melnikovs_valency_variety_problem\n Evidence used: Preserves the displayed inequality, historical attributions, and the Nettleton--Dirac upper bound.\n- Tommy R. Jensen and Bjarne Toft, Graph Coloring Problems, Wiley, 1995, p. 90. (authoritative_secondary): https://www.gbv.de/dms/goettingen/152233997.pdf\n Evidence used: Authoritative monograph source for the problem, attribution, sharpness statement, and complementary upper bound.\n\n**Review notes.** The old formula OCR is fragile and the source uses nonstandard ceil/floor LaTeX commands. No typographical correction was inferred.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3243, "problem_number": "OPG-46900", "title": "Earth-Moon Problem", "statement": "Problem What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon?", "background": "Source: Open Problem Garden. Original node ID: 46900. URL: http://www.openproblemgarden.org/op/earth_moon_problem.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/earth_moon_problem\n- Author(s): Ringel, G.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 6th, 2013 by fhavet\n\nProblem-page discussion:\nIn term of graphs, it can be rephrased as follows. What is the maximum chromatic number of a graph $G$ which is the union of two planar graphs (on the same vertex set)?\n\nIf a graph $G$ on $n$ vertices is the union of two planar graphs, then it has at most $2(3n-6)$ edges, and so it is has a vertex of degree at most $11$. An easy induction shows that $G$ is 12-colourable, as observed by Heawood [He]. Gardner [G] reported an example requiring 9 colours (the join of $C_5$ and $K_6$ ). It is not known if configurations exist requiring 10, 11, or 12 colours.\n\nMore generally, one may ask for the maximum chromatic number of the union of $k$ planar graphs.\n\nProblem What is the maximum chromatic number of a graph $G$ which is the union of $k$ planar graphs?\n\nThe same reasoning as above shows that $6k$ colours are always sufficient. The minimum number of planar graphs to decompose a complete graph [BW] gives a lower bound of $6k-2$ for $k \\ne 2$.\n\nBibliography:\n[BW] L. W. Beineke and A. T. White, Topological Graph Theory, Selected Topics in Graph Theory, L. W. Beineke and R. J. Wilson, eds., Academic Press, 15-50, 1978.\n\n[G] M. Gardner, Mathematical Recreations: The Coloring of Unusual Maps Leads Into Uncharted Territory. Sci. Amer. 242, 14-22, 1980.\n\n[He] P.J. Heawood, Map Colour Theorems. Quart. J. Pure Appl. Math. 24, 332-338, 1890.\n\n*[R] G. Ringel, Färbungsprobleme auf Flachen und Graphen. Berlin: Deutsche Verlag der Wissenschaften, 1959.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Earth-Moon Problem\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The biplanar chromatic maximum remains open with the classical bounds 9 <= chi <= 12; recent work rules out some proposed routes rather than settling it.\n\n**Verified partial progress.**\n\n- Eppstein disproved the conjecture that 2-blowups of planar graphs are always biplanar.\n- SAT-based work excludes at least one proposed 10-chromatic candidate as biplanar.\n\n**Full solution or refutation.**\n\nNo verified 10-, 11-, or 12-chromatic biplanar graph, nor improved universal upper bound, resolves the question.\n\n**What remains.**\n\nDetermine whether a biplanar graph needs 10, 11, or 12 colours, or prove an improved upper bound.\n\n**Sources checked.**\n\n- D. Eppstein, On the Biplanarity of Blowups, Graph Drawing and Network Visualization 2023; arXiv:2301.09246. (primary): https://arxiv.org/abs/2301.09246\n Evidence used: Constructs iterated Kleetopes whose 2-blowups are not biplanar and gives class-specific positive results.\n- Graph-theory open problems, Earth--Moon problem (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/earth_moon_problem/\n Evidence used: Reports the 9--12 bounds unchanged and the recent partial investigations.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3244, "problem_number": "OPG-46940", "title": "Acyclic list colouring of planar graphs.", "statement": "Conjecture Every planar graph is acyclically 5-choosable.", "background": "Source: Open Problem Garden. Original node ID: 46940. URL: http://www.openproblemgarden.org/op/acyclic_list_colouring_of_planar_graphs.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/acyclic_list_colouring_of_planar_graphs\n- Author(s): Borodin, Oleg V.; Fon-Der-Flasss, D. G.; Kostochka, Alexandr V.; Raspaud, André; Sopena, Eric\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture would imply to celebrated 5-colour theorems: one due toBorodin [B] stating that every planar graph is acyclically 5-colourable, and one due to Thomassen [T] stating that every plaanar graph is 5-choosable. This two theorems are best possible, because there are planar graphs which are not acyclically 4-colourable and others which are not 4-choosable.\n\nBorodin et al. [BFKRS] showed that every planar graph is acyclically 7-choosable.\n\nBibliography:\n[B] O.V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979) 211-236.\n\n*[BFKRS] O.V. Borodin, D.G. Fon-Der Flaass, A.V. Kostochka, A. Raspaud, and E. Sopena, Acyclic list 7-coloring of planar graphs, J. of Graph Theory 40-2 (2002) 83-90.\n\n[T] C. Thomassen. Every planar graph is 5-choosable. J. Combin. Theory Ser. B, 62(1994), no. 1, 180–181.\n\n[V] M. Voigt. List colourings of planar graphs. Discrete Math., 120(1993):no. 1-3, 215–219.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Acyclic list colouring of planar graphs.\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every planar graph remains known acyclically 7-choosable, and restricted classes attain smaller bounds, but universal acyclic 5-choosability is unresolved.\n\n**Verified partial progress.**\n\n- Borodin--Ivanova prove the 5-list assertion for planar graphs without 4-cycles.\n- Sun--Chen obtain acyclic 4-choosability for a more restricted forbidden-cycle class.\n\n**Full solution or refutation.**\n\nNo source establishes the all-planar-graphs 5-list statement.\n\n**What remains.**\n\nRemove all cycle restrictions from a 5-list-colouring proof, or find a counterexample.\n\n**Sources checked.**\n\n- Graph-theory open problems, Acyclic list colouring of planar graphs (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/acyclic_list_colouring_of_planar_graphs/\n Evidence used: Reports the conjecture open and documents the restricted-class advances.\n- Y. Sun and M. Chen, Acyclic 4-choosability of planar graphs without 4-cycles, Czechoslovak Mathematical Journal 70 (2020). (primary): https://doi.org/10.21136/CMJ.2019.0197-18\n Evidence used: Provides a verified restricted-class advance.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3245, "problem_number": "OPG-47358", "title": "List chromatic number and maximum degree of bipartite graphs", "statement": "Conjecture There is a constant $c$ such that the list chromatic number of any bipartite graph $G$ of maximum degree $\\Delta$ is at most $c \\log \\Delta$.", "background": "Source: Open Problem Garden. Original node ID: 47358. URL: http://www.openproblemgarden.org/op/list_chromatic_number_and_maximum_degree_of_bipartite_graphs.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/list_chromatic_number_and_maximum_degree_of_bipartite_graphs\n- Author(s): Alon, Noga\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 12th, 2013 by fhavet\n\nProblem-page discussion:\nFor definitions and an introduction to list colouring, see the related Wikipedia page.\n\nAlon [A] showed that the list chromatic number of a graph (not necessarily bipartite) of maximum degree $\\Delta$ is at least $\\frac{1}{2}(1-o(1))\\log_2\\Delta$. Random bipartite graphs show that this is tight up to a multiplicative factor $(2+o(1))$.\n\nIt is not diffcult to see that the list chromatic number of any bipartite graph $G$ of maximum degree $\\Delta$ is at most $O(\\Delta/\\log \\Delta)$. It also follows a more general result of Johansson [J] on triangle-free graphs.\n\nBibliography:\n*[A] N. Alon, Degrees and choice numbers, Random Structures Algorithms, 16 (2000), 364--368.\n\n[AK] N. Alon and M. Krivelevich, The choice number of random bipartite graphs, Annals of Combi- natorics 2 (1998), 291-297.\n\n[J] A. Johansson. Asymptotic choice number for triangle free graphs. Technical Report 91–95, DIMACS, 1996.\n\nDiscussion links:\n- Wikipedia page: http://en.wikipedia.org/wiki/list coloring\n\nBibliography links:\n- Degrees and choice numbers: http://www.cs.tau.ac.il/%7Enogaa/PDFS/degch1.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"List chromatic number and maximum degree of bipartite graphs\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjectured O(log Delta) list-chromatic bound for all bipartite graphs remains open; the best verified general upper bound is still of order Delta/log Delta.\n\n**Verified partial progress.**\n\n- Bradshaw, Mohar, and Stacho prove chi_l(G) < (4/5-10^-3) Delta/log Delta for all sufficiently large Delta, improving the previous leading constant 1 for bipartite graphs.\n\n**Full solution or refutation.**\n\nThe new upper bound is quantitatively stronger but remains far above logarithmic order.\n\n**What remains.**\n\nBridge the gap from Theta(Delta/log Delta) upper bounds to O(log Delta), or disprove the conjecture with bipartite graphs of larger choice number.\n\n**Sources checked.**\n\n- Peter Bradshaw, Bojan Mohar, and Ladislav Stacho, Bipartite graphs are (4/5-epsilon) Delta/log Delta-choosable, arXiv:2409.01513 (2024). (primary): https://arxiv.org/abs/2409.01513\n Evidence used: Explicitly states the Alon-Krivelevich conjecture remains open and proves the improved upper bound with epsilon=10^-3.\n- Graph-theory open problems, List chromatic number and maximum degree of bipartite graphs, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/list_chromatic_number_and_maximum_degree_of_bipartite_graphs/\n Evidence used: Current specialist tracker records the conjecture as open and the 2024 bound as partial progress.\n\n**Review notes.** The logarithm base is absorbed into the unspecified absolute constant.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3246, "problem_number": "OPG-47470", "title": "Colouring the square of a planar graph", "statement": "Conjecture Let $G$ be a planar graph of maximum degree $\\Delta$. The chromatic number of its square is\n\n- at most $7$ if $\\Delta =3$,\n- at most $\\Delta+5$ if $4\\leq\\Delta\\leq 7$,\n- at most $\\left\\lfloor\\frac32\\,\\Delta\\right\\rfloor+1$ if $\\Delta\\ge8$.", "background": "Source: Open Problem Garden. Original node ID: 47470. URL: http://www.openproblemgarden.org/op/colouring_the_square_of_a_planar_graph.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/colouring_the_square_of_a_planar_graph\n- Author(s): Wegner\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 13th, 2013 by fhavet\n\nProblem-page discussion:\nThe square of a graph $G$ is the graph $G^2$ on the same set of vertices, in which two vertices are adjacent when their distance in $G$ is at most 2.\n\nWegner [W] also gave examples showing that these bounds would be tight. For $\\Delta\\geq 8$, they are the following.\n\nTight examples for Wegner's conjecture\n\nFor $4\\leq \\Delta \\leq 9$, the examples are planar graphs on $\\Delta+5$ with maximum degree $\\Delta$ whose square is a complete graph.\n\nThis conjecture has also been generalized to the list chromatic number.\n\nConjecture Let $G$ be a planar graph of maximum degree $\\Delta$. The list chromatic number of its square is\n\n- at most $7$ if $\\Delta =3$,\n- at most $\\Delta+5$ if $4\\leq\\Delta\\leq 7$,\n- at most $\\left\\lfloor\\frac32\\,\\Delta\\right\\rfloor+1$ if $\\Delta\\ge8$.\n\nCranston and Kim [CK] showed that the square of every connected graph (non necessarily planar) which is subcubic (i.e., with $\\Delta\\le3$ ) is 8-choosable, except for the Petersen graph. However, the 7-choosability of the square of subcubic planar graphs is still open.\n\nHavet et al. [HHMR] proved the conjecture asymptotically:\n\nTheorem The square of every planar graph $G$ of maximum degree $\\Delta$ has list chromatic number at most $(1+o(1))\\,\\frac32\\,\\Delta$.\n\nIn fact, they proved this results for more general classes of graph. This led them to pose the following problem.\n\nProblem Is it true that for every minor-closed family ${\\cal F}$ of graphs (with ${\\cal F}$ not the set of all graphs), we have $\\chi(G^2)\\le \\bigl(\\frac32+o(1)\\bigr) \\Delta(G)$ for all $G\\in{\\cal F}$?\n\nBibliography:\n[HHMR] F. Havet, J. van den Heuvel, C. McDiarmid, and B. Reed. List Colouring Squares of Planar Graphs. Research Report RR-6586, INRIA, July 2008.\n\n[CK] D. W. Cranston and S.-J. Kim. List-coloring the square of a subcubic graph, J. Graph Theory, 57(1):65--87, 2008.\n\n*[W] G. Wegner. Graphs with given diameter and a coloring problem. Technical report, 1977.\n\nDiscussion links:\n- square: http://en.wikipedia.org/wiki/Graph power\n- list chromatic number: http://en.wikipedia.org/wiki/list coloring\n\nBibliography links:\n- List Colouring Squares of Planar Graphs: http://hal.inria.fr/inria-00303303/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Colouring the square of a planar graph\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Wegner's conjecture is proved and sharp for maximum degree 3, while all supplied clauses for maximum degree at least 4 remain open.\n\n**Verified partial progress.**\n\n- Thomassen proves every planar cubic graph has a 7-colorable square, and 7 cannot be lowered.\n- For large maximum degree, the conjectured leading term is known asymptotically, even for list coloring: (3/2+o(1)) Delta colors suffice.\n\n**Full solution or refutation.**\n\nOnly the first of the three degree regimes in the supplied statement is fully settled.\n\n**What remains.**\n\nProve the exact Delta+5 bounds for 4 <= Delta <= 7 and floor(3 Delta/2)+1 for Delta >= 8, or find counterexamples.\n\n**Sources checked.**\n\n- Carsten Thomassen, The square of a planar cubic graph is 7-colorable, Journal of Combinatorial Theory, Series B 128 (2018), 192-218, DOI 10.1016/j.jctb.2017.08.010. (primary): https://doi.org/10.1016/j.jctb.2017.08.010\n Evidence used: Proves the Delta=3 clause and its sharpness.\n- Frederic Havet, Jan van den Heuvel, Colin McDiarmid, and Bruce Reed, List Colouring Squares of Planar Graphs, arXiv:0807.3233. (primary): https://arxiv.org/abs/0807.3233\n Evidence used: Proves the asymptotic (3/2+o(1)) Delta list-colouring upper bound for squares of planar graphs.\n- Graph-theory open problems, Colouring the square of a planar graph, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/colouring_the_square_of_a_planar_graph/\n Evidence used: Current specialist tracker distinguishes the solved Delta=3 case from the open Delta>=4 regimes.\n\n**Review notes.** The stronger list-colouring analogue in the background is not part of the supplied top-level conjecture and was kept separate.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3247, "problem_number": "OPG-47511", "title": "Weighted colouring of hexagonal graphs.", "statement": "Conjecture There is an absolute constant $c$ such that for every hexagonal graph $G$ and vertex weighting $p:V(G)\\rightarrow \\mathbb{N}$, $$\\chi(G,p) \\leq \\frac{9}{8}\\omega(G,p) + c$$", "background": "Source: Open Problem Garden. Original node ID: 47511. URL: http://www.openproblemgarden.org/op/weighted_colouring_of_hexagonal_graphs.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/weighted_colouring_of_hexagonal_graphs\n- Author(s): McDiarmid, Colin; Reed, Bruce A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 13th, 2013 by fhavet\n\nProblem-page discussion:\nA hexagonal graph is an induced subgraph of the triangular lattice. The triangular lattice $TL$ may be described as follows. The vertices are all integer linear combinations $a\\mathbf{e_1} + b\\mathbf{e_2}$ of the two vectors $\\mathbf{e_1}=(1,0)$ and $\\mathbf{e_2}=(\\frac{1}{2}, \\frac{\\sqrt{3}}{2})$. Two vertices are adjacent when the Euclidean distance between them is 1.\n\nLet $G$ be a graph and $p$ a vertex weighting $p:V(G)\\rightarrow \\mathbb{N}$. The weighted clique number of $(G,p)$, denoted by $\\omega(G,p)$, is the maximum weight of a clique, that is $\\max \\{p(C) \\tq C \\mbox{ clique of } G\\}$, where $p(C)=\\sum_{v\\in C} p(v)$. A $k$-colouring of a $(G,p)$ is a mapping $C:V(G)\\ra {\\cal P}(\\{1, \\dots, k\\})$ such that for every vertex $v\\in V(G)$, $|C(v)|=p(v)$ and for all edge $uv\\in E(G)$, $C(u)\\cap C(v)=\\emptyset$. The chromatic number of $(G,p)$, denoted by $\\chi(G,p)$, is the least integer $t$ such that $(G,p)$ admits a $t$-colouring.\n\nThe conjecture would be tight because of $C_9$ the cycle of length 9. The maximum size of stable set in $C_9$ is $4$. Thus $\\chi(C_9,\\mathbf{k})\\geq 9k/4$ and $\\omega(G,\\mathbf{k})=2k$, where $\\mathbf{k}$ is the all $k$ function.\n\nMcDiarmid and Reed [MR] proved that $\\chi(G,p)\\leq \\frac{4\\omega(G,p)+1}{3}$ for any hexagonal graph $G$ and vertex weighting $p$. Havet [H] proved that if a hexagonal graph $G$ is triangle-free, then $\\chi(G,p)\\leq\\frac{7}{6}\\omega(G,p) + 5$ (See also [SV]).\n\nThe conjecture would be implied by the following one, where $\\mathbf{4}$ is the all $4$ function.\n\nConjecture $\\chi(G,\\mathbf{4})\\leq 9$ for every hexagonal graph.\n\nSince $\\chi(G,\\mathbf{4}) \\geq 4|V(G)|/\\alpha(G)$, where $\\alpha(G)$ is the stability number (the maximum size of a stable set). A first step to this later conjecture would be to prove the following conjecture of McDiarmid.\n\nConjecture Let $G$ be a triangle-free hexagonal graph. $$\\alpha(G)\\geq \\frac{4}{9}|V(G)|$$\n\nBibliography:\n[H] F.Havet. Channel assignment and multicolouring of the induced subgraphs of the triangular lattice. Discrete Mathematics 233:219--231, 2001.\n\n*[MR] C. McDiarmid and B. Reed. Channel assignment and weighted coloring, Networks, 36:114--117, 2000.\n\n[SV] K. S. Sudeep and S. Vishwanathan. A technique for multicoloring triangle-free hexagonal graphs. Discrete Mathematics, 300(1-3), 256--259, 2005.\n\nDiscussion links:\n- stable set: http://en.wikipedia.org/wiki/Independent set (graph theory)\n\nBibliography links:\n- Channel assignment and multicolouring of the induced subgraphs of the triangular lattice: http://www.sciencedirect.com/science/article/pii/S0012365X00002417\n- A technique for multicoloring triangle-free hexagonal graphs: http://www.sciencedirect.com/science/article/pii/S0012365X05002694\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 35.\n\nAttempt notes:\nTarget:\nMake progress on \"Weighted colouring of hexagonal graphs.\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The 9/8 weighted-colouring bound remains open; the best verified general asymptotic ratio located is 5/4 with an additive constant.\n\n**Verified partial progress.**\n\n- Zerovnik proves a bound of the form chi(G,p) <= 15 floor(omega(G,p)/12)+O(1), improving the asymptotic ratio from 4/3 to 5/4.\n- For triangle-free hexagonal graphs, an earlier 7/6 ratio plus an additive constant is known.\n\n**Full solution or refutation.**\n\nNeither the general 9/8 ratio nor an equivalent bounded additive-gap theorem was verified.\n\n**What remains.**\n\nImprove the general 5/4 ratio to 9/8 with a uniform additive constant, or construct a family exceeding that bound.\n\n**Sources checked.**\n\n- Janez Zerovnik, Improved approximative multicoloring of hexagonal graphs, arXiv:1606.01328 (2016). (primary): https://arxiv.org/abs/1606.01328\n Evidence used: Explicitly states the 9/8 conjecture and proves the improved 5/4 asymptotic ratio.\n- Colin McDiarmid and Bruce Reed, Channel assignment and weighted coloring, Networks 36 (2000), 114-117, DOI 10.1002/1097-0037(200009)36:2<114::AID-NET5>3.0.CO;2-G. (primary): https://doi.org/10.1002/1097-0037(200009)36:2%3C114::AID-NET5%3E3.0.CO;2-G\n Evidence used: Original source for the weighted hexagonal-graph bound and the 9/8 conjecture.\n- Graph-theory open problems, Weighted colouring of hexagonal graphs, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/weighted_colouring_of_hexagonal_graphs/\n Evidence used: Current specialist tracker records the conjecture as open and the 5/4 theorem as partial progress.\n\n**Review notes.** Versioned preprints differ in the additive constant (18 versus 27); only the stable 5/4 asymptotic conclusion is reported. Background OCR macros were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3248, "problem_number": "OPG-48583", "title": "Bounding the on-line choice number in terms of the choice number", "statement": "Question Are there graphs for which $\\text{ch}^{\\text{OL}}-\\text{ch}$ is arbitrarily large?", "background": "Source: Open Problem Garden. Original node ID: 48583. URL: http://www.openproblemgarden.org/op/bounding_the_on_line_choice_number_in_terms_of_the_choice_number.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/bounding_the_on_line_choice_number_in_terms_of_the_choice_number\n- Author(s): Zhu, Xuding\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: choosability; list coloring; on-line choosability\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: April 11th, 2013 by Jon Noel\n\nProblem-page discussion:\nWe let $\\text{ch}$ denote the (classical) choice number. For a definition of the on-line choice number of $G$ (denoted $\\text{ch}^{\\text{OL}}(G)$ ), see the following posting: On-Line Ohba's Conjecture.\n\nA result of Alon [Alo93] says that the choice number of a graph is bounded above and below by a function of the colouring number, defined as follows: $\\text{col}(G):=\\max\\{\\delta(H):H\\subseteq G\\}$.\n\nZhu [Zhu09] demonstrated that the on-line choice number is bounded above by the colouring number. By combining this with Alon's result, we have that there is a function $g$ such that $\\text{ch}^{\\text{OL}}(G)\\leq g(\\text{ch}(G))$ for every graph $G$. However, the function $g$ from Alon's result is exponential. In [Zhu09], Zhu asked if we can do better (polynomial? linear? etc).\n\nIt is known that there are graphs for which $\\text{ch}^{\\text{OL}}(G) = \\text{ch}(G) + 1$. Interestingly, as is mentioned in [CLM+13], it is not even known whether there is a graph $G$ such that $\\text{ch}^{\\text{OL}}(G)>\\text{ch}(G)+1$.\n\nThere are not many graphs for which the choice number (let alone the on-line choice number) is known exactly. For this reason, it seems that a natural starting point for this problem is to study the complete $k$-partite graph in which every part has size $3$, denoted $K_{3*k}$. Kierstead [Kie00] proved that $\\text{ch}(K_{3*k}) = \\left\\lceil\\frac{4k-1}{3}\\right\\rceil$. Kozik, Micek and Zhu proved that the $\\text{ch}^{\\text{OL}}(K_{3*k})\\leq\\frac{3k}{2}$.\n\nIt may be the case that $\\text{ch}^{\\text{OL}}(K_{3*k})>\\left\\lceil\\frac{4k-1}{3}\\right\\rceil$. Is it larger than $\\left\\lceil\\frac{4k-1}{3}\\right\\rceil+1$?\n\nBibliography:\n[Alo93] N. Alon. Restricted colorings of graphs. In Surveys in combinatorics, 1993 (Keele), volume 187 of London Math. Soc. Lecture Note Ser., pages 1–33. Cambridge Univ. Press, Cambridge, 1993.\n\n[CLM+13] J. Carraher, S. Loeb, T. Mahoney, G. Puleo, M.-T. Tsai, and D. West. Three Topics in Online List Coloring. Preprint, February 2013.\n\n[HWZ12] P. Huang, T. Wong, and X. Zhu. Application of polynomial method to on-line list colouring of graphs. European J. Combin., 33(5):872–883, 2012.\n\n[Kie00] H. A. Kierstead. On the choosability of complete multipartite graphs with part size three. Discrete Math., 211(1-3):255–259, 2000.\n\n[KKLZ12] S.-J. Kim, Y. S. Kwon, D. D.-F. Liu, and X. Zhu. On-line list colouring of complete multipartite graphs. Electron. J. Combin., 19(1):Paper 41, 13, 2012.\n\n[KMZ12] J. Kozik, P. Micek, and X. Zhu. Towards on-line Ohba’s conjecture. Preprint, arXiv:1111.5458v2, December 2012.\n\n[Sch09] U. Schauz. Mr. Paint and Mrs. Correct. Electron. J. Combin., 16(1):Research Paper 77, 18, 2009.\n\n[Sch10] U. Schauz. A paintability version of the combinatorial Nullstellensatz, and list colorings of k-partite k-uniform hypergraphs. Electron. J. Combin., 17(1):Research Paper 176, 13, 2010.\n\n*[Zhu09] X. Zhu. On-line list colouring of graphs. Electron. J. Combin., 16(1):Research Paper 127, 16, 2009.\n\nRelated:\nRelated problems\nOn-Line Ohba's Conjecture\nOhba's Conjecture\nChoice number of complete multipartite graphs with parts of size 4\n\nDiscussion links:\n- On-Line Ohba's Conjecture: http://www.openproblemgarden.org/op/on_line_ohbas_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Bounding the on-line choice number in terms of the choice number\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The checked literature still describes arbitrarily large online-minus-offline choice-number gaps as conjectural; no proof or counterexample to this exact question was verified.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo current exact resolution was found in the available sources.\n\n**What remains.**\n\nLocate a later theorem resolving unbounded ch^OL-ch, or prove/unprove the asserted gap family.\n\n**Sources checked.**\n\n- G. Gutowski, online graph colouring thesis/discussion (accessed 2026-08-17). (authoritative_secondary): https://grzegorzgutowski.staff.tcs.uj.edu.pl/content/phd.pdf\n Evidence used: States the unbounded-difference assertion as a conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3249, "problem_number": "OPG-53019", "title": "Choosability of Graph Powers", "statement": "Question (Noel, 2013) Does there exist a function $f(k)=o(k^2)$ such that for every graph $G$,\n$$\n\\text{ch}\\left(G^2\\right)\\leq f\\left(\\chi\\left(G^2\\right)\\right)?\n$$", "background": "Source: Open Problem Garden. Original node ID: 53019. URL: http://www.openproblemgarden.org/op/choosability_of_graph_powers.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/choosability_of_graph_powers\n- Author(s): Noel, Jonathan A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: choosability; chromatic number; list coloring; square of a graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: July 13th, 2013 by Jon Noel\n\nProblem-page discussion:\nFor a survey of choosability, including relevant definitions, see [Noe] or click here.\n\nThe List Square Colouring Conjecture, due to Kostochka and Woodall [KW], states that $\\text{ch}\\left(G^2\\right) = \\chi\\left(G^2\\right)$ for every graph $G$. This was disproved by Kim and Park [KP], who proved that there is a sequence $\\{G_n\\}_n$ of graphs and a constant $c_1$ such that $\\chi\\left(G^2_n\\right)\\to\\infty$ and $\\text{ch}\\left(G_n^2\\right) \\geq c_1 \\chi\\left(G_n^2\\right)\\log\\left(\\chi\\left(G_n^2\\right)\\right)$ for all $n$. To obtain this lower bound from the construction of Kim and Park, one can apply the well-known result of Alon [Alo].\n\nIt may be the case that the correct upper bound for all graphs is of the same order of magnitude as the example in the result of Kim and Park.\n\nQuestion (Noel, 2013) Does there exist a positive constant $c_2$ such that every graph $G$ satisfies $\\text{ch}\\left(G^2\\right) \\leq c_2\\chi\\left(G^2\\right)\\log{\\chi\\left(G^2\\right)}$?\n\nBy calculating the clique number and maximum degree of $G^2$, one can easily show that $\\text{ch}\\left(G^2\\right)\\leq\\chi\\left(G^2\\right)^2$ (this observation is due to Young Soo Kwon), but it seems that no significantly better bound is known.\n\nProposition If $G$ contains an edge, then\n$$\n\\text{ch}\\left(G^2\\right)< \\chi\\left(G^2\\right)^2.\n$$\n\nProof We observe the following bounds:\n$$\n\\chi\\left(G^2\\right) \\geq \\omega\\left(G^2\\right) \\geq \\Delta(G)+1,\n$$\n\n$$\n\\text{ch}\\left(G^2\\right) \\leq \\Delta\\left(G^2\\right)+1 \\leq \\Delta(G)\\left(\\Delta(G)-1\\right) + \\Delta(G)+1 = \\Delta(G)^2+1.\n$$\n Therefore, since $\\Delta(G)>0$, we have\n$$\n\\text{ch}\\left(G^2\\right)\\leq \\Delta(G)^2+1 < \\left(\\Delta(G)+1\\right)^2 \\leq \\chi\\left(G^2\\right)^2.\n$$\n This completes the proof.\n\nThese questions are related to a problem of Zhu (see Doug West's webpage for more info) who asked whether there exists an integer $k$ such that for every graph $G$, we have that $G^k$ has choice number equal to chromatic number. This conjecture has been disproved independently by Kim, Kwon and Park [KKP] and Kosar, Petrickova, Reigniger and Yeager [KPRY]. The example of [KPRY] also yields, for every $k$, a sequence $\\{G_n\\}_n$ of graphs and a constant $c$ such that $\\chi\\left(G^k_n\\right)\\to\\infty$ and $\\text{ch}\\left(G_n^k\\right) \\geq c \\chi\\left(G_n^k\\right)\\log\\left(\\chi\\left(G_n^k\\right)\\right)$ for all $n$. They ask the following, more general, questions:\n\nQuestion (Kosar et al., 2013) Given $k\\geq2$, does there exist a function $f_k(x)=o(x^2)$ such that for every graph $G$,\n$$\n\\text{ch}\\left(G^k\\right)\\leq f_k\\left(\\chi\\left(G^k\\right)\\right)?\n$$\n\nTo our knowledge, it is not known whether there exists a function $f_k(x) = o(x^k)$ such that the same conclusion holds. (Intuitively, it seems that higher values of $k$ should yield a smaller separation between $\\text{ch}(G^k)$ and $\\chi(G^k)$; however, there seems to be no hard evidence to support this.)\n\nQuestion (Kosar et al., 2013) Given $k\\geq2$, does there exist a positive constant $c_k$ such that every graph $G$ satisfies $\\text{ch}\\left(G^k\\right) \\leq c_k\\chi\\left(G^k\\right)\\log{\\chi\\left(G^k\\right)}$? Moreover, can the constant $c_k$ be made independent of $k$?\n\nThese questions are also related to the so-called List Total Colouring Conjecture of Borodin, Kostochka and Woodall [BKW], which says that the total graph of a multigraph always satisfies $\\text{ch}=\\chi$. Given a multigraph $G$, the total graph of $G$ can be obtained by subdividing every edge of $G$ and then taking the square of the resulting graph.\n\nBibliography:\n[Alo] Noga Alon. Choice numbers of graphs: a probabilistic approach. Combin. Probab. Comput., 1(2):107–114, 1992.\n\n[BKW] Oleg V. Borodin, Alexandr V. Kostochka, and Douglas R. Woodall. List edge and list total colourings of multigraphs. J. Combin. Theory Ser. B, 71(2):184–204, 1997.\n\n[KP] Seog-Jin Kim and Boram Park: Counterexamples to the List Square Coloring Conjecture, submitted.\n\n[KKP] Seog-Jin Kim, Young Soo Kwon and Boram Park: Chromatic-choosability of the power of graphs.\n\n[KPRY] Nicholas Kosar, Sarka Petrickova, Benjamin Reiniger, Elyse Yeager: A note on list-coloring powers of graphs.\n\n[KW] Alexandr V. Kostochka and Douglas R. Woodall. Choosability conjectures and multicircuits, Discrete Math., 240 (2001), 123--143.\n\n[Noe] Jonathan A. Noel. Choosability of Graphs with Bounded Order: Ohba's Conjecture and Beyond, Master's thesis. McGill University (2013). pdf.\n\nRelated:\nRelated problems\nChoice number of complete multipartite graphs with parts of size 4\nChoice Number of k-Chromatic Graphs of Bounded Order\nList Total Colouring Conjecture\n\nDiscussion links:\n- click here: http://www.openproblemgarden.org/ http:/www.math.mcgill.ca/jnoel/pdf/Masters.pdf\n- Doug West's webpage: http://www.openproblemgarden.org/ http:/www.math.uiuc.edu/%7Ewest/regs/listkpow.html\n\nBibliography links:\n- Counterexamples to the List Square Coloring Conjecture: http://www.arxiv.org/abs/1305.2566\n- Chromatic-choosability of the power of graphs: http://www.arxiv.org/abs/1309.0888\n- A note on list-coloring powers of graphs: http://www.arxiv.org/abs/1309.7705\n- pdf: http://www.openproblemgarden.org/ http:/www.math.mcgill.ca/jnoel/pdf/Masters.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 32.\n\nAttempt notes:\nTarget:\nMake progress on \"Choosability of Graph Powers\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The subquadratic bound remains open; the stronger List Square Colouring equality conjecture is false and logarithmic separation examples are known.\n\n**Verified partial progress.**\n\n- Kim--Park construct graph squares with choice number at least c chi log chi, disproving equality ch(G^2)=chi(G^2).\n\n**Full solution or refutation.**\n\nA logarithmic lower bound does not rule out an o(k^2) upper bound.\n\n**What remains.**\n\nFind a subquadratic universal upper bound or construct near-quadratic separations.\n\n**Sources checked.**\n\n- Open Problem Garden, Choosability of Graph Powers (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/choosability_of_graph_powers\n Evidence used: Gives the exact open question and the Kim--Park logarithmic lower bound.\n- S.-J. Kim, Y. S. Kwon and B. Park, Chromatic-choosability of the power of graphs (accessed 2026-08-17). (authoritative_secondary): https://citeseerx.ist.psu.edu/document?doi=b1de8dd815b758ba58c85171930eb805e5a69f8c&repid=rep1&type=pdf\n Evidence used: Restates Noel's subquadratic question and discusses lower-bound constructions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3250, "problem_number": "OPG-55984", "title": "Erdős–Faber–Lovász conjecture", "statement": "Conjecture If $G$ is a simple graph which is the union of $k$ pairwise edge-disjoint complete graphs, each of which has $k$ vertices, then the chromatic number of $G$ is $k$.", "background": "Source: Open Problem Garden. Original node ID: 55984. URL: http://www.openproblemgarden.org/op/erdos_faber_lovasz_conjecture.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/erdos_faber_lovasz_conjecture\n- Author(s): Erdos, Paul; Faber, Vance; Lovasz, Laszlo\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: chromatic number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: August 22nd, 2013 by Jon Noel\n\nProblem-page discussion:\nFrom [Erd81]: \"Faber, Lovász and I made this harmless looking conjecture at a party in Boulder Colorado in September 1972. Its diff\u001eculty was realised only slowly. I now off\u001ber 500 dollars for a proof or disproof. (Not long ago I only offered 50; the increase is not due to inflation but to the fact that I now think the problem is very diff\u001ecult. Perhaps I am wrong.)\"\n\nThe conjecture can be equivalently formulated in terms of seating assignments or hypergraph colouring; see Wikipedia or Doug West's Webpage.\n\nBibliography:\n[Erd81] P. Erdős. On the combinatorial problems which I would most like to see solved. Combinatorica, 1(1):25–42, 1981.\n\nRelated:\nRelated problems\nAlon-Saks-Seymour Conjecture\n\nDiscussion links:\n- Wikipedia: http://en.wikipedia.org/wiki/Erdős–Faber–Lovász_conjecture\n- Doug West's Webpage: http://www.math.uiuc.edu/%7Ewest/regs/efl.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Erdős–Faber–Lovász conjecture\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Erdős--Faber--Lovász conjecture is proved for every sufficiently large k, but the source's unrestricted all-k formulation has not been verified as fully closed.\n\n**Verified partial progress.**\n\n- Kang--Kelly--Kuehn--Methuku--Osthus prove EFL for every sufficiently large order and give stability results.\n\n**Full solution or refutation.**\n\nThe theorem is a decisive asymptotic resolution but retains a finite all-order qualification.\n\n**What remains.**\n\nResolve or explicitly discharge the remaining finite range for the exact universal source statement.\n\n**Sources checked.**\n\n- D. Y. Kang, T. Kelly, D. Kuehn, A. Methuku and D. Osthus, A proof of the Erdős--Faber--Lovász conjecture, arXiv:2101.04698 (2021). (primary): https://arxiv.org/abs/2101.04698\n Evidence used: The abstract proves the conjecture for every sufficiently large n.\n\n**Review notes.** Large-order theorem not silently upgraded to all orders.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3251, "problem_number": "OPG-56230", "title": "2-colouring a graph without a monochromatic maximum clique", "statement": "Conjecture If $G$ is a non-empty graph containing no induced odd cycle of length at least $5$, then there is a $2$-vertex colouring of $G$ in which no maximum clique is monochromatic.", "background": "Source: Open Problem Garden. Original node ID: 56230. URL: http://www.openproblemgarden.org/op/2_colouring_a_graph_without_a_monochromatic_maximum_clique.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/2_colouring_a_graph_without_a_monochromatic_maximum_clique\n- Author(s): Hoang, Chinh T.; McDiarmid, Colin\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: maximum clique; Partitioning\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 25th, 2013 by Jon Noel\n\nProblem-page discussion:\nA $2$-division of a graph $G$ is a partitioning of $G$ into two subgraphs, neither of which contains a maximum clique. It is known that every perfect graph admits a $2$-division. Thus, by the Strong Perfect Graph Theorem [CRS], a graph which does not contain an induced copy of an odd cycle of length at least $5$ or its complement has a $2$-division. Hoàng and McDiarmid [HMcD] also prove that a claw-free graph admits a 2-division if and only if it does not contain an induced odd cycle of length at least $5$. The conjecture says that this holds for all graphs.\n\nThis problem was featured as unsolved problem #49 in Bondy and Murty's book \"Graph Theory\" [BM].\n\nSee also a posting on the American Institute of Mathematics website, contributed by Bruce Reed.\n\nBibliography:\n[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: The strong perfect graph theorem, Ann. of Math. (2) 164 (2006), no. 1, 51--229. MathSciNet\n\n[HMcD] C.T. Hoàng, C. McDiarmid, On the divisibility of graphs, Discrete Math. 242 (1–3) (2002) 145–156.\n\n[BM] J. A. Bondy and U. S. R. Murty. Graph theory, volume 244 of Graduate Texts in Mathematics. Springer, New York, 2008.\n\nDiscussion links:\n- posting: http://www.aimath.org/WWN/perfectgraph/articles/html/19a/\n\nBibliography links:\n- The strong perfect graph theorem: http://www.arxiv.org/abs/math.CO/0212070\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2233847\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"2-colouring a graph without a monochromatic maximum clique\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Hoang--McDiarmid odd-hole-free 2-division conjecture remains open generally, with multiple hereditary subclasses now proved.\n\n**Verified partial progress.**\n\n- Hoang proves a stronger perfect-divisibility statement for (banner, odd-hole)-free graphs.\n- Chudnovsky--Sivaraman prove 2-divisibility for (P5,C5)-free graphs and further cases.\n\n**Full solution or refutation.**\n\nNo proof covers every graph with no induced odd cycle of length at least five.\n\n**What remains.**\n\nExtend 2-divisibility to all odd-hole-free graphs or find a counterexample.\n\n**Sources checked.**\n\n- Graph-theory open problems, 2-colouring a graph without a monochromatic maximum clique (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/2_colouring_a_graph_without_a_monochromatic_maximum_clique/\n Evidence used: Reports the full conjecture open and records the restricted-class theorems.\n- M. Chudnovsky and V. Sivaraman, Perfect divisibility and 2-divisibility, Journal of Graph Theory 90 (2019); arXiv:1704.06667. (primary): https://arxiv.org/abs/1704.06667\n Evidence used: Proves several named restricted classes are 2-divisible.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3252, "problem_number": "OPG-59927", "title": "List Colourings of Complete Multipartite Graphs with 2 Big Parts", "statement": "Question Given $a,b\\geq2$, what is the smallest integer $t\\geq0$ such that $\\chi_\\ell(K_{a,b}+K_t)= \\chi(K_{a,b}+K_t)$?", "background": "Source: Open Problem Garden. Original node ID: 59927. URL: http://www.openproblemgarden.org/op/list_colourings_of_complete_multipartite_graphs_with_2_big_parts.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/list_colourings_of_complete_multipartite_graphs_with_2_big_parts\n- Author(s): Allagan, Julian\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: complete bipartite graph; complete multipartite graph; list coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: April 12th, 2014 by Jon Noel\n\nProblem-page discussion:\nThe list chromatic number of a graph $G$, denoted $\\chi_\\ell(G)$, is the minimum $k$ such that for every assignment of lists of size $k$ to the vertices of $G$ there is a proper colouring in which every vertex is mapped to a colour in its own list. For more background on the list chromatic number, see [3].\n\nGiven graphs $G$ and $H$, the join of $G$ and $H$, denoted $G+H$, is obtained by taking disjoint copies of $G$ and $H$ and adding all edges between them. Ohba [1] proved that for every graph $G$ there exists $t\\geq0$ such that $\\chi_\\ell(G+K_t)= \\chi(G+K_t)$. The question above asks to determine the minimum value of $t$ in the case that $G$ is a complete bipartite graph. It seems that it was first studied in [4], although this is unclear; for the time being, we have chosen to attribute this problem to J. Allagan.\n\nDefine $\\phi(a,b)$ to be the minimum $t$ such that $\\chi_\\ell(K_{a,b}+K_t)= \\chi(K_{a,b}+K_t)$. Note that, if $G$ is a complete multipartite graph with at most one non-singleton part, then we see that $\\chi_\\ell(G)=\\chi(G)$ by colouring the vertices of the non-singleton part last. Thus, if $a$ or $b$ is equal to 1, then $\\phi(a,b)=0$. As it turns out, $\\phi(2,2)=\\phi(2,3)=0$ and $\\phi(3,3)=\\phi(2,4)=1$. This can be deduced from the following result of [2] and the fact that $\\chi_\\ell(K_{3,3})=\\chi_\\ell(K_{4,2})=3$:\n\nTheorem (Noel, Reed, Wu (2012)) If $|V(G)|\\leq 2\\chi(G)+1$, then $\\chi_\\ell(G)=\\chi(G)$.\n\nThe above result of [2] implies that if $a+b\\geq 5$, then $\\phi(a,b)\\leq a+b-5$. However it seems that, for most values of $a,b$, this bound is far from tight.\n\nA simple observation is that, since $\\chi_\\ell(K_{a,b}+K_t)\\geq \\chi_\\ell(K_{a,b})$ for all $t$, we must have\n$$\n\\phi(a,b)\\geq \\chi_\\ell(K_{a,b}) - \\chi(K_{a,b}) = \\chi_\\ell(K_{a,b}) -2.\n$$\n\nThe following is a result of Allagan [4]:\n\nTheorem (Allagan (2009)) If $a\\geq5$, then\n$$\n\\lfloor \\sqrt{a}\\rfloor - 1 \\leq \\phi(a,2)\\leq \\left\\lceil\\frac{-7+\\sqrt{8a+17}}{2}\\right\\rceil.\n$$\n\nThis implies that $\\phi(a,2)=1$ for $4\\leq a\\leq 8$ and that $\\phi(a,2)=2$ for $9\\leq a\\leq 13$.\n\nBibliography:\n[1] K. Ohba. On chromatic-choosable graphs, J. Graph Theory. 40 (2002) 130--135. MathSciNet.\n\n[2] J. A. Noel, B. A. Reed, H. Wu. A Proof of a Conjecture of Ohba. Submitted. pdf.\n\n[3] J. A. Noel. Choosability of Graphs with Bounded Order: Ohba's Conjecture and Beyond. Master's Thesis, McGill University. pdf.\n\n[4] J. A. D. Allagan. Choice Numbers, Ohba Numbers and Hall Numbers of some complete $k$-partite graphs. PhD Thesis. Auburn University. 2009.\n\nRelated:\nRelated problems\nOhba's Conjecture\nChoice Number of k-Chromatic Graphs of Bounded Order\nChoice number of complete multipartite graphs with parts of size 4\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1899118\n- pdf: http://www.openproblemgarden.org/people.maths.ox.ac.uk/noel/Ohba Paper.pdf\n- pdf: http://www.openproblemgarden.org/people.maths.ox.ac.uk/noel/Masters.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 30.\n\nAttempt notes:\nTarget:\nMake progress on \"List Colourings of Complete Multipartite Graphs with 2 Big Parts\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Exact phi(a,b) values are known in small cases and upper/lower bounds are known generally, but the minimum t for arbitrary a,b remains unresolved.\n\n**Verified partial progress.**\n\n- Ohba's theorem guarantees a finite t for every fixed graph.\n- The maintained source records exact values phi(2,2)=phi(2,3)=0 and phi(3,3)=phi(2,4)=1, plus Allagan bounds for phi(a,2).\n\n**Full solution or refutation.**\n\nNo general formula for phi(a,b) was verified.\n\n**What remains.**\n\nDetermine phi(a,b) for all a,b>=2.\n\n**Sources checked.**\n\n- Open Problem Garden, List Colourings of Complete Multipartite Graphs with 2 Big Parts (accessed 2026-08-17). (maintained_tracker): https://garden.irmacs.sfu.ca/op/list_colourings_of_complete_multipartite_graphs_with_2_big_parts\n Evidence used: Gives the exact parameter, small values, and known bounds.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3253, "problem_number": "OPG-59943", "title": "List Hadwiger Conjecture", "statement": "Conjecture Every $K_t$-minor-free graph is $c t$-list-colourable for some constant $c\\geq1$.", "background": "Source: Open Problem Garden. Original node ID: 59943. URL: http://www.openproblemgarden.org/op/list_hadwiger_conjecture.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/list_hadwiger_conjecture\n- Author(s): Kawarabayashi, Ken-ichi; Mohar, Bojan\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: Hadwiger conjecture; list colouring; minors\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 7th, 2014 by David Wood\n\nProblem-page discussion:\nHadwiger's conjecture asserts that every $K_t$-minor-free graph is $(t − 1)$-colourable. Robertson, Seymour and Thomas [RST] proved Hadwiger's conjecture for $t \\leq 6$. It remains open for $t \\geq 7$. In fact, it is open whether every $K_t$-minor-free graph is $ct$-colourable for some constant $c\\geq 1$. It is natural to consider analogous problems for list colourings.\n\nFirst, consider planar graphs. While every planar graph is 4-colourable, Erdös, Rubin and Taylor. [ERT] conjectured that some planar graph is not 4-list-colourable, and that every planar graph is 5-list-colourable. The first conjecture was verified by Voigt [V] and the second by Thomassen [T].\n\nMore generally, Borowiecki [B] asked whether every $K_t$-minor-free graph is $(t − 1)$-list-colourable, which is true for $t \\leq 4$ but false for $t = 5$ by Voigt’s example. Kawarabayashi and Mohar [KM] proposed the stated conjecture, and suggested it might be true with $c=\\frac{3}{2}$. Barát, Joret and Wood [BJW] proved that $c\\geq\\frac{4}{3}$. In particular, they constructed a $K_{3t+2}$-minor-free graph that is not $4t$-list-colourable.\n\nBibliography:\n[B] Mieczyslaw Borowiecki. Research problem 172. Discrete Math., 121:235–236, 1993..\n\n[BJW] Janos Barát, Gwenael Joret, David R. Wood. Disproof of the List Hadwiger Conjecture, Electronic J. Combinatorics 18:P232, 2011.\n\n[ERT] Paul Erdo ̋s, Arthur L. Rubin, and Herbert Taylor. Choosability in graphs. In Proc. West Coast Conference on Combinatorics, Graph Theory and Computing, vol. XXVI of Congress. Numer., pp. 125–157. Utilitas Math., 1980. MathSciNet.\n\n*[KM] Ken-ichi Kawarabayashi and Bojan Mohar. A relaxed Hadwiger’s conjecture for list colorings. J. Combin. Theory Ser. B, 97(4):647–651, 2007. MathSciNet.\n\n[RST] Neil Robertson, Paul D. Seymour, and Robin Thomas. Hadwiger’s conjecture for $K_6$-free graphs. Combinatorica, 13(3):279–361, 1993. MathSciNet.\n\n[T] Carsten Thomassen. Every planar graph is 5-choosable. J. Combin. Theory Ser. B, 62(1):180–181, 1994. MathSciNet.\n\nDiscussion links:\n- minor: http://en.wikipedia.org/wiki/Graph_minor\n- list colourings: http://en.wikipedia.org/wiki/List_coloring\n\nBibliography links:\n- Research problem 172: http://dx.doi.org/10.1016/0012-365X(93)90557-A\n- Disproof of the List Hadwiger Conjecture: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p232\n- Choosability in graphs: http://www.renyi.hu/∼p erdos/1980-07.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=593902\n- A relaxed Hadwiger’s conjecture for list colorings: http://dx.doi.org/10.1016/j.jctb.2006.11.002\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2325803\n- Hadwiger’s conjecture for $K_6$-free graphs: http://dx.doi.org/10.1007/BF01202354\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1238823\n- Every planar graph is 5-choosable: http://dx.doi.org/10.1006/jctb.1994.1062\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1290638\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"List Hadwiger Conjecture\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The linear weak list-Hadwiger conjecture remains open; the best known universal upper bound is O(t(log t)^beta) for every beta>1/4, with lower bound (2-o(1))t.\n\n**Verified partial progress.**\n\n- Norin--Postle establish the almost-linear universal upper bound.\n- Steiner's construction rules out the proposed c=3/2 constant.\n\n**Full solution or refutation.**\n\nNeither result proves the existence of an absolute linear list-colouring constant.\n\n**What remains.**\n\nRemove the logarithmic factor or prove a superlinear lower bound.\n\n**Sources checked.**\n\n- Graph-theory open problems, List Hadwiger Conjecture (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/list_hadwiger_conjecture/\n Evidence used: Reports the current upper/lower bounds and the remaining linear conjecture.\n- S. Norin and L. Postle, Connectivity and choosability of graphs with no K_t minor, arXiv:2004.10367 (2020). (primary): https://arxiv.org/abs/2004.10367\n Evidence used: Proves the O(t(log t)^beta) list-colouring bound.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3254, "problem_number": "OPG-60001", "title": "Cycles in Graphs of Large Chromatic Number", "statement": "Conjecture If $\\chi(G)>k$, then $G$ contains at least $\\frac{(k+1)(k-1)!}{2}$ cycles of length $0\\bmod k$.", "background": "Source: Open Problem Garden. Original node ID: 60001. URL: http://www.openproblemgarden.org/op/cycles_in_graphs_of_large_chromatic_number.\n\nSource subject path: Graph Theory > Coloring > Vertex coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cycles_in_graphs_of_large_chromatic_number\n- Author(s): Brewster, Richard C.; McGuinness, Sean; Moore, Benjamin; Noel, Jonathan A.\n- Subject(s): Graph Theory; Coloring; Vertex coloring\n- Keywords: chromatic number; cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 20th, 2015 by Jon Noel\n\nProblem-page discussion:\nChudnovsky, Plumettaz, Scott and Seymour [CPSS] proved that every graph with chromatic number at least $4$ contains a cycle of length $0\\bmod 3$. A simpler proof was found by Wrochna [W]. Wrochna's argument was generalised by Brewster, McGuinness, Moore and Noel [BMMN] to the following: if $\\chi(G)>k$, then $G$ contains at least TeX Embedding failed! cycles of length $0\\bmod k$.} The compete graph on $k+1$ vertices has exactly $\\frac{(k+1)(k-1)!}{2}$ cycles of length $0\\bmod k$ and so the conjecture above, if true, would be best possible.\n\nBibliography:\n[BMMN] R. C. Brewster, S. McGuinness, B. Moore, J. A. Noel, A Dichotomy Theorem for Circular Colouring Reconfiguration, submitted, arXiv:1508.05573v1.\n\n[CPSS] M. Chudnovsky, M. Plumettaz, A. Scott, and P. Seymour, The Structure of Graphs with no Cycles of Length Divisible by Three, in preparation.\n\n[Wro] M. Wrochna, unpublished.\n\nRelated:\nRelated problems\nBounding the chromatic number of graphs with no odd hole\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Cycles in Graphs of Large Chromatic Number\" in Graph Theory; Coloring; Vertex coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The record omits the essential published range k>=3. Read without that restriction it is false for k=1 and k=2; the intended k>=3 conjecture remains open in general, with the full k=3 case now proved.\n\n**Verified partial progress.**\n\n- Kim--Picollelli prove that every 4-chromatic graph contains at least four cycles of length divisible by three and that K4 uniquely attains equality.\n- They also prove a broader critical-graph special case for cycles in specified residue classes.\n\n**Full solution or refutation.**\n\nNo status can be assigned to the unrestricted imported sentence as the named literature conjecture: it is trivially false at small k, whereas the published conjecture starts at k=3.\n\n**What remains.**\n\nConfirm and restore the intended hypothesis k>=3 at the source level; under that formulation, prove the claimed count for every k>=4.\n\n**Sources checked.**\n\n- Sean Kim and Michael E. Picollelli, 4-Chromatic Graphs Have At Least 4 Cycles of Length 0 mod 3, arXiv:2312.05945. (primary): https://arxiv.org/abs/2312.05945\n Evidence used: States the literature conjecture with the missing condition k>=3, proves its k=3 case, and gives further critical-graph progress.\n- Open Problem Garden, Cycles in Graphs of Large Chromatic Number. (maintained_tracker): https://www.openproblemgarden.org/op/cycles_in_graphs_of_large_chromatic_number\n Evidence used: Preserves the imported statement and background but does not display the required k range.\n\n**Review notes.** Literal counterexamples: K3 at k=2 has chromatic number 3 and no even cycle; any nontrivial tree at k=1 has chromatic number greater than 1 and no cycle. These expose the missing hypothesis rather than repair it.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3255, "problem_number": "OPG-169", "title": "The Two Color Conjecture", "statement": "Conjecture If $G$ is an orientation of a simple planar graph, then there is a partition of $V(G)$ into $\\{X_1,X_2\\}$ so that the graph induced by $X_i$ is acyclic for $i=1,2$.", "background": "Source: Open Problem Garden. Original node ID: 169. URL: http://www.openproblemgarden.org/op/partitioning_planar_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partitioning_planar_digraphs\n- Author(s): Neumann-Lara, Victor\n- Subject(s): Graph Theory; Directed Graphs\n- Keywords: acyclic; digraph; planar\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 26th, 2007 by mdevos\n\nProblem-page discussion:\nThis is a type of coloring digraphs introduced by V. Neumann-Lara. More generally, if $G$ is a digraph, we wish to partition the vertex set of $G$ into as few parts as possible so that each induces an acyclic subgraph.\n\nBibliography:\n* [N] V. Neumann-Lara (1985). Vertex colourings in digraphs. Some problems. Technical report, University of Waterloo.\n\nComments:\n- May 31st, 2007 | Anonymous | Riste Skrekovski conjectured: Riste Skrekovski conjectured at a graph theory meeting in Budmerice, Slovakia, in 2005 that this conjecture is false.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"The Two Color Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The planar digraph two-colour conjecture remains open, but it is proved for digirth at least four and reduced equivalently to oriented K5-minor-free graphs.\n\n**Verified partial progress.**\n\n- Every planar digraph of digirth at least 4 can be partitioned into two acyclic colour classes.\n- The full conjecture is equivalent to 2-colourability of every oriented K5-minor-free graph.\n\n**Full solution or refutation.**\n\nLi and Mohar settle all planar orientations without directed triangles. Steiner's equivalence expands the structural setting but leaves the same directed-triangle obstruction unresolved.\n\n**What remains.**\n\nHandle planar orientations containing directed triangles, or find a counterexample to two acyclic colour classes.\n\n**Sources checked.**\n\n- Open Problem Garden, The Two Color Conjecture (OPG-169), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/partitioning_planar_digraphs\n Evidence used: Maintains Neumann-Lara's exact planar orientation statement.\n- Z. Li and B. Mohar, Planar Digraphs of Digirth Four are 2-Colorable, SIAM Journal on Discrete Mathematics 31 (2017), 2201-2205. (primary): https://doi.org/10.1137/16M108080X\n Evidence used: Proves the conjecture for planar digraphs of digirth at least four.\n- R. Steiner, A Note on Graphs of Dichromatic Number 2, Discrete Mathematics & Theoretical Computer Science 22:4 (2020), article 11. (primary): https://arxiv.org/abs/1907.00351\n Evidence used: Proves equivalence with 2-colourability of all oriented K5-minor-free graphs.\n\n**Review notes.** For orientations, the digirth-at-least-four theorem leaves directed triangles as the unresolved regime.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3256, "problem_number": "OPG-179", "title": "Woodall's Conjecture", "statement": "Conjecture If $G$ is a directed graph with smallest directed cut of size $k$, then $G$ has $k$ disjoint dijoins.", "background": "Source: Open Problem Garden. Original node ID: 179. URL: http://www.openproblemgarden.org/op/woodalls_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/woodalls_conjecture\n- Author(s): Woodall, Douglas R.\n- Subject(s): Graph Theory; Directed Graphs\n- Keywords: digraph; packing\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 5th, 2007 by mdevos\n\nProblem-page discussion:\nDefinitions: Let $G$ be a directed graph. A directed cut of $G$ is an edge-cut in which all edges are directed the same way. A dijoin is a set of edges which intersect every directed cut.\n\nThere is an important theorem of Lucchesi and Younger [LY] which asserts that the dual problem has an optimum integer packing. That is, for every digraph $G$ in which the smallest dijoin has size $k$, there exist $k$ pairwise disjoint directed cuts. This result implies more generally (in the terminology of Corneujols [C]) that the clutter of (minimal) directed cuts has the max-flow min-cut property (MFMC). It follows from this and Lehman's theorem [L] that the clutter of (minimal) dijoins is ideal. Therefore, if the smallest directed cut of $G$ has size $k$, then there exist nonnegative rational numbers $x_1,x_2,\\ldots,x_m$ summing to $k$ and dijoins $J_1,J_2,\\ldots,J_m$ so that if each dijoin $J_i$ is taken with weight $x_i$, the total weight of the dijoins containing any edge is at most 1. The above conjecture asserts that such a combination exists with $x_1,x_2,\\ldots,x_m$ integral.\n\nSchrijver [Sc80] has found a digraph in which the clutter of (minimal) dijoins does not have the MFMC property. Thus, the weighted version of Woodall's conjecture is not true in general. However, this clutter does have the MFMC property in a number of interesting special cases. One such example (due to Schrijver [Sc82] and Feofiloff, Younger [FY]) is when the digraph is acyclic and every source is joined to every sink by a directed path. Since every such digraph has the MFMC property, every such digraph must satisfy Woodall's conjecture.\n\nThere seem to be few positive results towards Woodall's conjecture for general digraphs. Although this was probably already known, Seymour and I (M. DeVos) observed that the conjecture is true for $k=2$. To see this, note that the underlying graph is 2-edge-connected, so it may be oriented to give a strongly connected digraph, call it $H$. Now partition the edges into two sets $\\{X,Y\\}$ where $X$ consists of those edges which have the same orientation in both $H$ and $G$, and $Y$ consists of those edges with different orientations in the two graphs. It is immediate that both $X$ and $Y$ meet every directed cut, so each is a dijoin. Extending this to $k=3$ appears difficult. Indeed, I believe the following weak version of this is still open.\n\nConjecture Prove that there exists a fixed integer $k$ so that every digraph with all directed cuts of size $\\ge k$ contains three pairwise disjoint dijoins.\n\nThe restriction of this conjecture to the special case of planar graphs is also open. Here we can use duality to restate the conjecture. Call a set of edges $A$ a feedback arc-set if $A$ intersects every directed cycle (or equivalently, $G-A$ is acyclic).\n\nConjecture If $G$ is a planar digraph with all directed cycles of length $\\ge k$, then $G$ contains $k$ pairwise disjoint feedback arc-sets.\n\nThe above conjecture is known to fail when the assumption of planarity is removed. On the other hand, it does hold for digraphs which are orientations of series-parallel graphs [LW]. It is also still open for $k=3$.\n\nBibliography:\n[C] G. Cornuejols, Combinatorial Optimization, Packing and Covering SIAM, Philadelphia (2001). MathSciNet\n\n[FY] P. Feofiloff and D. H. Younger, Directed cut transversal packing for source-sink connected graphs. Combinatorica 7 (1987), no. 3, 255--263. MathSciNet\n\n[LW] O. Lee, Y. Wakabayashi, Note on a min-max conjecture of Woodall. J. Graph Theory 38 (2001), no. 1, 36--41. MathSciNet\n\n[LY] C.L. Lucchesi and D. H. Younger A minimax theorem for directed graphs, Journal of the London Math. Soc. (2) 17 (1978) 369-374. MathSciNet\n\n[Sc80] A. Schrijver, A counterexample to a conjecture of Edmonds and Giles, Discrete Math. 32 (1980) 213-214. MathSciNet\n\n[Sc82] A. Schrijver, Min-max relations for directed graphs, Annals of Discrete Math. 16 (1982) 261-280. MathSciNet\n\n[W] D. R. Woodall, Menger and König systems. Theory and applications of graphs (Proc. Internat. Conf., Western Mich. Univ., Kalamazoo, Mich., 1976), pp. 620--635, Lecture Notes in Math., 642, Springer, Berlin, 1978. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1828452\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0918396\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1849557\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0500618\n- A counterexample to a conjecture of Edmonds and Giles: http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.1016/0012-365X(80)90057-6\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0592858\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0686312\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0499529\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Woodall's Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Woodall's conjecture on packing k disjoint dijoins remains open.\n\n**Verified partial progress.**\n\n- There are special cases and weakening/approximation results.\n\n**Full solution or refutation.**\n\nNo general equality between minimum directed-cut size and maximum disjoint dijoin packing was verified.\n\n**What remains.**\n\nProve the packing theorem or construct a counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, node 179 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains Woodall's conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3257, "problem_number": "OPG-611", "title": "The Bermond-Thomassen Conjecture", "statement": "Conjecture For every positive integer $k$, every digraph with minimum out-degree at least $2k-1$ contains $k$ disjoint cycles.", "background": "Source: Open Problem Garden. Original node ID: 611. URL: http://www.openproblemgarden.org/op/the_bermond_thomassen_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_bermond_thomassen_conjecture\n- Author(s): Bermond, Jean-Claude; Thomassen, Carsten\n- Subject(s): Graph Theory; Directed Graphs\n- Keywords: cycles\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 1st, 2007 by JS\n\nProblem-page discussion:\nThis conjecture is a simple observation when $k=1$. It was proved by Thomassen~[Tho83] in 1983 when $k=2$, and more recently the case $k=3$ was settled~[LPS07].\n\nThe bound offered would be optimal — just consider a symmetric complete graph on $2k-1$ vertices. In 1996, Alon~[Alo96] proved that the statement is true with $2k-1$ replaced by $64k$. The conjecture was also verified for tournaments of minimum in-degree at least $2k-1$ ~[BLS07].\n\nBang-Jensen et al. [BBT] made a stronger conjecture for digraph with sufficiently large girth.\n\nConjecture For every integer $g >1$, every digraph $D$ with girth at least $g$ and with minimum out-degree at least $\\frac{g}{g-1}k$ contains $k$ disjoint cycles.\n\nThe constant $\\frac{g}{g-1}$ is best possible. Indeed, for every integers $p$ and $g$, consider the digraph $D(g,p)$ on $n = p(g − 1) + 1$ vertices with vertex set $\\{x_1, \\dots, x_n\\}$ and arc set $\\{x_ix_j: j − i \\mod n \\in \\{1,\\dots p\\}\\}$. It has girth $g$ and out-degree $p = \\left \\lfloor \\frac{g}{g−1} k \\right \\rfloor$. Moreover, for $n = 0 \\mod g$, the digraph $D(g,p)$ admits a partition into $k$ vertex disjoint 3-cycles and no more. For g = 3, the first case of this conjecture which differs from Bermond-Thomassen Conjecture and which is not already known corresponds to the following question:\n\nQuestion Does every digraph D without 2-cycles and out-degree at least 6 admit four vertex disjoint cycles?\n\nBibliography:\n[Alo96] N. Alon: Disjoint directed cycles, J. Combin. Theory Ser. B, 68(2):167--178, 1996. PDF\n\n[BBT] J. Bang-Jensen, S. Bessy and S. Thomassé, Disjoint 3-cycles in tournaments: a proof of the Bermond-Thomassen conjecture for tournaments, J. Graph Theory, to appear.\n\n*[BeTh81] J.-C. Bermond and C.~Thomassen: Cycles in digraphs---a survey, J. Graph Theory, 5(1):1--43, 1981. MathSciNet\n\n[BLS07] S.~Bessy, N.~Lichiardopol, and J.-S. Sereni: Two proofs of the {B}ermond-{T}homassen conjecture for tournaments with bounded minimum in-degree, Discrete Math., Special Issue dedicated to CS06, to appear.\n\n[LPS07] N.~Lichiardopol, A.~ P\\'or, and J.-S. Sereni: A step towards the Bermond-Thomassen conjecture about disjoint cycles in digraphs, Submitted, 2007.\n\n[Tho83] C.~Thomassen, Disjoint cycles in digraphs, Combinatorica, 3(3-4):393--396, 1983. MathSciNet\n\nBibliography links:\n- PDF: http://www.math.tau.ac.il/%7Enogaa/PDFS/dicycles3.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR604304\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR729792\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"The Bermond-Thomassen Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Bermond-Thomassen conjecture remains open for general digraphs from k=4 onward, but is proved for k<=3, all tournaments, and several other classes.\n\n**Verified partial progress.**\n\n- The conjecture is known for k=1,2,3; k=4 is the first unresolved general case.\n- Bang-Jensen, Bessy, and Thomassé prove the conjecture for every tournament, in fact with k disjoint directed triangles.\n- Alon proves the general linear sufficient bound f(k)<=64k, and 2023 work verifies additional multipartite-tournament classes.\n\n**Full solution or refutation.**\n\nNo proof of the sharp 2k-1 threshold for arbitrary digraphs and all k was located.\n\n**What remains.**\n\nProve the k=4 case and then all k for arbitrary simple digraphs, or construct a digraph of minimum outdegree 2k-1 without k vertex-disjoint directed cycles.\n\n**Sources checked.**\n\n- Yandong Bai and Yannis Manoussakis, On the number of vertex-disjoint cycles in digraphs, arXiv:1805.02999 (2018). (primary): https://arxiv.org/abs/1805.02999\n Evidence used: Treats the general statement as open and gives a shorter proof of the k=3 case.\n- Jørgen Bang-Jensen, Stéphane Bessy, and Stéphan Thomassé, Disjoint 3-cycles in tournaments: a proof of the Bermond-Thomassen conjecture for tournaments, Journal of Combinatorial Theory, Series B 102 (2012), 1178-1194. (primary): https://www.lirmm.fr/~bessy/publis/bt.pdf\n Evidence used: Proves the conjecture for all tournaments.\n- Gregory Gutin, Wei Li, Shujing Wang, Anders Yeo, and Yacong Zhou, Note on Disjoint Cycles in Multipartite Tournaments, arXiv:2311.13369 (2023). (primary): https://arxiv.org/abs/2311.13369\n Evidence used: Still states the general conjecture and verifies triangle-free multipartite and 3-partite tournament cases.\n\n**Review notes.** Disjoint cycles means vertex-disjoint directed cycles. Tournament results are special cases and do not settle arbitrary digraphs.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3258, "problem_number": "OPG-646", "title": "Seymour's Second Neighbourhood Conjecture", "statement": "Conjecture Any oriented graph has a vertex whose outdegree is at most its second outdegree.", "background": "Source: Open Problem Garden. Original node ID: 646. URL: http://www.openproblemgarden.org/op/seymours_second_neighbourhood_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/seymours_second_neighbourhood_conjecture\n- Author(s): Seymour, Paul D.\n- Subject(s): Graph Theory; Directed Graphs\n- Keywords: Caccetta-Häggkvist; neighbourhood; second; Seymour\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: October 9th, 2007 by nkorppi\n\nProblem-page discussion:\nBy the $n$ th outdegree of $v$, we mean the number of vertices for which the minimal outward-directed path from $v$ to them is of length $n$.\n\nChen, Shen, and Yuster [CSY] proved that in any oriented graph there is a vertex whose second outdegree is at least $\\gamma$ times its outdegree, where $\\gamma=0.657298...$ is the unique real root of $2x^3+x^2 -1=0$.\n\nThis conjecture implies a special case of the \\Oprefnum[Caccetta-Häggkvist Conjecture]{46385}.\n\nBibliography:\n[ASY] Chen, G.; Shen, J.; Yuster, R. Second neighborhood via first neighborhood in digraphs, Annals of Combinatorics, 7 (2003), 15--20.\n\n[F] Fisher, David C. Squaring a tournament: a proof of Dean's conjecture. J. Graph Theory 23 (1996), no. 1, 43--48.\n\n[KL] Kaneko, Yoshihiro; Locke, Stephen C. The minimum degree approach for Paul Seymour's distance 2 conjecture. Proceedings of the Thirty-second Southeastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 2001). Congr. Numer. 148 (2001), 201--206.\n\nRelated:\nRelated problems\nCaccetta-Häggkvist Conjecture\n\nComments:\n- December 12th, 2009 | Anonymous | It is proved: This conjecture has been proved. You can find the proof here.\n- December 12th, 2009 | Robert Samal | Re: It is proved: Thanks for the reference. The proof, however, seems flawed. The basic outline of the proof (if I understood it correctly) is as follows:\n\n- If $G$ has a sink, then this sink satisfies the conditions.\n- An oriented graph without directed cycles has a sink.\n- Then one proves directly that a graph containing a directed cycle has a vertex (even on that cycle) that satisfies the conditions.\n\nThe last part (Lemma 2 of the manuscript), is not true -- take a directed cycle $C$, add many independent points $X$, and add all the arcs from $C$ to $X$. Now the conjecture is true for this graph (one can take any of the sinks -- vertices in $X$ ), but it is not true that one can choose a vertex of $C$.\n\nThe omission in the proof of Lemma 2 is that it's tacitly assumed, that adjacent vertices of the cycle have no common out-neighbours.\n- March 14th, 2010 | Anonymous | So is this proof valid or: So is this proof valid or not? I am currently working on a proof and am wondering whether or not I should be bothering\n- August 8th, 2010 | rs | Re: So is this proof valid or: I believe the mentioned prof is not valid. Is your proof working? And sorry for the late reply, our system falsely recognized your comment as a spam.\n- March 18th, 2011 | sjcjoosten | Counterexample for the proof: As was indicated, Lemma 2 is false. Here is a counterexample to lemma 2. Note that it is not a counterexample to the conjecture, since it has two sinks.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Seymour's Second Neighbourhood Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Seymour's Second Neighborhood Conjecture remains open, with a current universal ratio of 0.715538 and exact proofs for many structured or probabilistic classes.\n\n**Verified partial progress.**\n\n- Huang and Peng prove that some vertex u always satisfies |N++(u)| at least 0.715538 times |N+(u)|.\n- Botler, Moura, and Naia verify the conjecture for several random and pseudorandom regimes and show that almost every oriented graph satisfies it.\n- Current 2026 papers extend dense and small-order cases while continuing to state the general assertion as a conjecture.\n\n**Full solution or refutation.**\n\nNo valid proof of the universal factor-one assertion was located; the proof linked in an old Garden comment has a documented false lemma.\n\n**What remains.**\n\nRaise the universal 0.715538 ratio to 1 for every oriented graph.\n\n**Sources checked.**\n\n- Hao Huang and Fei Peng, An improved bound on Seymour's second neighborhood conjecture, arXiv:2412.20234 (2024). (primary): https://arxiv.org/abs/2412.20234\n Evidence used: Proves the current 0.715538 general bound and explicitly describes the factor-one statement as open.\n- Fabio Botler, Phablo Moura, and Tassio Naia, Seymour's Second Neighborhood Conjecture for orientations of (pseudo)random graphs, Discrete Mathematics 346 (2023), 113583. (primary): https://doi.org/10.1016/j.disc.2023.113583\n Evidence used: Proves exact special cases for random and pseudorandom orientations.\n- A dense-case theorem for Seymour's second neighborhood conjecture, arXiv:2608.11530 (2026). (primary): https://arxiv.org/abs/2608.11530\n Evidence used: Provides a current dense-case theorem and still treats the unrestricted assertion as conjectural.\n\n**Review notes.** The stored background itself records the counterexample to Lemma 2 of a purported 2009 proof, so that claim was excluded.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3259, "problem_number": "OPG-1793", "title": "Non-edges vs. feedback edge sets in digraphs", "statement": "For any simple digraph $G$, we let $\\gamma(G)$ be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and $\\beta(G)$ be the size of the smallest feedback edge set.\n\nConjecture If $G$ is a simple digraph without directed cycles of length $\\le 3$, then $\\beta(G) \\le \\frac{1}{2} \\gamma(G)$.", "background": "Source: Open Problem Garden. Original node ID: 1793. URL: http://www.openproblemgarden.org/op/non_edges_vs_feedback_edge_sets_in_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/non_edges_vs_feedback_edge_sets_in_digraphs\n- Author(s): Chudnovsky, Maria; Seymour, Paul D.; Sullivan, Blair\n- Subject(s): Graph Theory; Directed Graphs\n- Keywords: acyclic; digraph; feedback edge set; triangle free\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 8th, 2008 by mdevos\n\nProblem-page discussion:\nIf $G$ satisfies $\\gamma(G) = 0$, then $G$ is a tournament, and it is easy to check that $G$ will have a directed cycle of length three unless it is acyclic, in which case $\\beta(G) = 0$. So in this case, the conjecture holds. More generally, it is natural to suspect that a digraph with few non-edges and no directed triangles should be close to acyclic. Indeed, this conjecture asserts a precise relationship of this form.\n\nIf true, the above conjecture is essentially tight for a number of examples. We noted above that it is tight for transitive tournaments. Here is another basic class: let $G_k$ be the circulant digraph obtained by placing $3k+1$ vertices in a circle, and adding an edge directed from $u$ to $v$ whenever $v$ is distance $\\le k$ from $u$ in the clockwise order. Such examples may be nested to obtain new ones.\n\nChudnovsky, Seymour, and Sullivan [CSS] utilized a clever double counting argument to prove that $\\beta(G) \\le \\gamma(G)$ always holds. They also proved their conjecture in the case when $V(G)$ is the union of two cliques, and when $G$ is a circular interval digraph.\n\nBibliography:\n*[CSS] M. Chudnovsky, P.D. Seymour, and B. Sullivan, Cycles in dense digraphs.\n\nSource links:\n- feedback edge set: http://en.wikipedia.org/wiki/feedback arc set\n\nBibliography links:\n- Cycles in dense digraphs: http://www.math.princeton.edu/%7Epds/papers/triangles/paper.ps\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Non-edges vs. feedback edge sets in digraphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The conjectured beta(G) <= gamma(G)/2 bound remains open in general; beta(G) <= 0.8616 gamma(G) is proved universally, and the half-bound holds for several special classes.\n\n**Verified partial progress.**\n\n- Chudnovsky, Seymour, and Sullivan proved beta(G) <= gamma(G) and the half-bound for unions of two cliques and circular interval digraphs.\n- Chen, Karson, Liu, and Shen proved beta(G) <= 0.8616 gamma(G).\n- The half-bound is verified for relevant directed Cayley graphs over Z/NZ.\n\n**Full solution or refutation.**\n\nNo general proof of the coefficient 1/2 and no counterexample were verified.\n\n**What remains.**\n\nImprove the universal coefficient from 0.8616 to 0.5 or construct a 3-free digraph violating the half-bound.\n\n**Sources checked.**\n\n- Maria Chudnovsky, Paul Seymour, and Blair D. Sullivan, Cycles in dense digraphs, Combinatorica 28 (2008), 1–18, DOI 10.1007/s00493-008-2331-z; arXiv:math/0702147. (primary): https://arxiv.org/abs/math/0702147\n Evidence used: The abstract states the general beta <= gamma theorem, formulates the half-bound conjecture, and proves it in two special cases.\n- Kevin Chen, Sean Karson, Dan Liu, and Jian Shen, On the Chudnovsky-Seymour-Sullivan Conjecture on Cycles in Triangle-free Digraphs, Electronic Journal of Linear Algebra 28 (2015), 117–123; arXiv:0909.2468. (primary): https://arxiv.org/abs/0909.2468\n Evidence used: The abstract states and proves the improved universal coefficient 0.8616.\n- Open Problem Garden, Non-edges vs. feedback edge sets in digraphs (node 1793), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/non_edges_vs_feedback_edge_sets_in_digraphs\n Evidence used: Preserves the exact half-bound and the original special cases.\n\n**Review notes.** The no-cycles-of-length-at-most-three hypothesis matches the standard 3-free condition: no loops, digons, or directed triangles.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3260, "problem_number": "OPG-46167", "title": "Oriented trees in n-chromatic digraphs", "statement": "Conjecture Every digraph with chromatic number at least $2k-2$ contains every oriented tree of order $k$ as a subdigraph.", "background": "Source: Open Problem Garden. Original node ID: 46167. URL: http://www.openproblemgarden.org/op/oriented_trees_in_n_chromatic_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/oriented_trees_in_n_chromatic_digraphs\n- Author(s): Burr, S. A.\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 25th, 2013 by fhavet\n\nProblem-page discussion:\nThe conjectured bound is best possible, because a regular tournament of order $2k-3$ does not contain the oriented tree consisting of a vertex dominating $k-1$ leaves.\n\nLet $f$ be the function $f$ such that every oriented tree of order $k$ is $f(k)$-universal, that is contained in every digraph with chromatic number at least $f(k)$. Burr proved that $f(k) \\leq (k-1)^2$. This was slightly improved by Addario-Berry et al. [AHL+] who proved $f(k)\\leq k^2/2-k/2+1$.\n\nBurr's conjecture has been proved only in few particular cases of digraphs: tournaments, and acyclic digraphs. Kühn, Mycroft, and Osthus [KMS] showed that every oriented tree of order $k$ is contained in every tournament of order $2k-2$ for all sufficiently large $k$ (so proving a Conjecture of Sumner); Addario-Berry et al. [AHL+] proved that every acyclic digraph with chromatic number $k$ contains every oriented tree of order $k$.\n\nBurr's conjecture or some approximation have been also proved for special classes of trees. Gallai-Roy's celebrated theorem states that every directed path of order $k$ is $k$-universal; El-Sahili [E] proved that every oriented path of order $4$ is $4$-universal and that the antidirected path of order $5$ is $5$-universal; Addario-Berry, Havet, and Thomassé [AHT] showed that every oriented path of order $k$ whose vertex set can be partioned into two directed paths is $k$-universal; Addario-Berry et al. [AHL+] showed that antidirected trees (oriented trees in which every vertex has in-degree $0$ or out-degree $0$ ) are $5k$-universal.\n\nHavet, generalizing a conjecture of Havet and Thomassé (see [H]) on tournaments, conjectured that the following could also be true.\n\nConjecture Every digraph with chromatic number at least $k+\\ell+1$ contains every oriented tree of order $k$ with $k$ leaves.\n\nBibliography:\n[AHL+] L. Addario-Berry, F. Havet, C. Linhares Sales, B. Reed, and S. Thomassé. Oriented trees in digraphs. Discrete Mathematics, 313(8):967-974, 2013.\n\n[AHT] L. Addario-Berry, F. Havet, and S. Thomassé, Paths with two blocks in $n$-chromatic digraphs, J. of Combinatorial Theory Ser. B, 97 (2007), 620--626.\n\n* [B] A. Burr, Subtrees of directed graphs and hypergraphs, Proceedings of the Eleventh Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, Congr. Numer., 28 (1980), 227--239.\n\n[H] F. Havet, Trees in tournaments. Discrete Mathematics 243 (2002), no. 1-3, 121--134.\n\n[KOM] D. Kühn, D. Osthus, and R. Mycroft, A proof of Sumner's universal tournament conjecture for large tournaments, Proceedings of the London Mathematical Society 102 (2011), 731--766.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Oriented trees in n-chromatic digraphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Burr's 2k-2 chromatic bound remains open, but every oriented tree of order k is now known in every O(k sqrt(k))-chromatic digraph.\n\n**Verified partial progress.**\n\n- The first subquadratic general upper bound was proved in 2024.\n- Tournaments and several special tree/digraph classes satisfy sharp or better bounds.\n\n**Full solution or refutation.**\n\nThe best possible linear threshold is unproved.\n\n**What remains.**\n\nReduce the general bound to 2k-2 or find a counterexample.\n\n**Sources checked.**\n\n- S. Bessy, D. Gonçalves and A. Reinald, Oriented trees in O(k sqrt(k))-chromatic digraphs, arXiv:2402.19351 (2024). (primary): https://arxiv.org/abs/2402.19351\n Evidence used: The abstract states the subquadratic universal bound.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3261, "problem_number": "OPG-46279", "title": "Antidirected trees in digraphs", "statement": "An antidirected tree is an orientation of a tree in which every vertex has either indegree 0 or outdergree 0.\n\nConjecture Let $D$ be a digraph. If $|A(D)| > (k-2) |V(D)|$, then $D$ contains every antidirected tree of order $k$.", "background": "Source: Open Problem Garden. Original node ID: 46279. URL: http://www.openproblemgarden.org/op/antidirected_trees_in_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/antidirected_trees_in_digraphs\n- Author(s): Addario-Berry, Louigi; Havet, Frédéric; Linhares Sales, Claudia; Reed, Bruce A.; Thomassé, Stéphan\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 26th, 2013 by fhavet\n\nProblem-page discussion:\nThe value $k-2$ would be best possible, since the oriented tree consisting of a vertex dominating $k-1$ other vertices is not contained in any digraph in which every vertex has outdegree $k-2$. The condition on the trees be antidirected cannot be suppressed. In a bipartite digraph $D$ with bipartition $(A,B)$ such that all arcs are directed from $A$ to $B$, all the trees contained in $D$ are antidirected.\n\nThis conjecture for symmetric digraphs is equivalent to the celebrated Erdös-Sos conjecture for undirected graphs. (see [E]).\n\nConjecture Let $G$ be a graph. If $|E(G)| > \\frac{1}{2} (k-2) |V(G)|$, then $G$ contains every tree of order $k$.\n\nAddario-Berry et al. Conjecture also implies Burr's conjecture (see Oriented trees in n-chromatic digraphs) for antidirected trees, since every digraph with chromatic number $2k-2$ contains a colour-critical digraph has minimum degree at least $2k-3$, and so whose number of vertices is at least $\\frac{2k-3}{2}|V(D)|$, which exceeds $(k-2) |V(D)|$.\n\nThis conjecture has only been proved [AHL+] for antidirected trees of diameter at most $3$.\n\nBibliography:\n*[AHL+] L. Addario-Berry, F. Havet, C. Linhares Sales, B. Reed, and S. Thomassé. Oriented trees in digraphs. Discrete Mathematics, 313(8):967-974, 2013.\n\n[E] P. Erdös, Some problems in graph theory, Theory of Graphs and Its Applications, M. Fielder, Editor, Academic Press, New York, 1965, pp. 29--36.\n\nRelated:\nRelated problems\nOriented trees in n-chromatic digraphs\n\nDiscussion links:\n- Oriented trees in n-chromatic digraphs: http://www.openproblemgarden.org/?q=node/46167]\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Antidirected trees in digraphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The sharp arc-density threshold k-2 remains open, while broad dense-oriented-graph cases for balanced bounded-degree antidirected trees are now known.\n\n**Verified partial progress.**\n\n- A 2022 result asymptotically confirms the conjecture in dense oriented graphs for balanced bounded-degree trees.\n\n**Full solution or refutation.**\n\nNo proof for every digraph at the exact density threshold was verified.\n\n**What remains.**\n\nProve the sharp general density theorem.\n\n**Sources checked.**\n\n- M. Stein and C. Zárate-Guerén, Antidirected subgraphs of oriented graphs, arXiv:2212.00769 (2022). (primary): https://arxiv.org/abs/2212.00769\n Evidence used: The abstract gives the asymptotic dense-oriented-graph result and explicitly addresses the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3262, "problem_number": "OPG-46359", "title": "Directed path of length twice the minimum outdegree", "statement": "Conjecture Every oriented graph with minimum outdegree $k$ contains a directed path of length $2k$.", "background": "Source: Open Problem Garden. Original node ID: 46359. URL: http://www.openproblemgarden.org/op/directed_cycle_of_length_twice_the_minimum_outdegree.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/directed_cycle_of_length_twice_the_minimum_outdegree\n- Author(s): Thomassé, Stéphan\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 28th, 2013 by fhavet\n\nProblem-page discussion:\nIn fact, Thomassé made the following stronger conjecture which implies the celebrated Cacetta-Häggkvist Conjecture.\n\nConjecture Every digraph with minimum outdegree $k$ and directed girth $g$, contains a directed path of length $(g-1)k$.\n\nThis conjecture holds easily when $g=2$. For $g=3$ it is the above conjecture which is still open.\n\nBibliography:\n*[S] Blair D. Sullivan: A Summary of Problems and Results related to the Caccetta-Haggkvist Conjecture\n\nRelated:\nRelated problems\nCaccetta-Häggkvist Conjecture\n\nDiscussion links:\n- Cacetta-Häggkvist Conjecture: http://www.openproblemgarden.org/?q=node/46385\n\nBibliography links:\n- A Summary of Problems and Results related to the Caccetta-Haggkvist Conjecture: http://www.arxiv.org/abs/math.CO/0605646\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Directed path of length twice the minimum outdegree\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Thomassé's directed-path bound remains open; the stronger directed-girth formulation is open already for girth three.\n\n**Verified partial progress.**\n\n- The girth-two case is elementary.\n\n**Full solution or refutation.**\n\nNo general 2k directed-path theorem was verified.\n\n**What remains.**\n\nProve the girth-three case or produce a counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, Directed path of length twice the minimum outdegree (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/directed_cycle_of_length_twice_the_minimum_outdegree\n Evidence used: The current page explicitly states the girth-three/base conjecture remains open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3263, "problem_number": "OPG-46385", "title": "Caccetta-Häggkvist Conjecture", "statement": "Conjecture Every simple digraph of order $n$ with minimum outdegree at least $r$ has a cycle with length at most $\\lceil n/r\\rceil$", "background": "Source: Open Problem Garden. Original node ID: 46385. URL: http://www.openproblemgarden.org/op/caccetta_haggkvist_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/caccetta_haggkvist_conjecture\n- Author(s): Caccetta, L.; Häggkvist, Roland\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 28th, 2013 by fhavet\n\nProblem-page discussion:\nIt is one of the most famous conjectures in graph theory. It has many alternative formulations and lots of work have been done around it. Many interesting conjectures are related to it. See [Sul]. It is in particular implied by a conjecture of Thomassé and Hoàng-Reed Conjecture.\n\nThe Caccetta-Häggkvist Conjecture is a generalization of an earlier conjecture of Behzad, Chartrand, and Wall, who conjectured it only for diregular digraphs. Caccetta-H äggkvist Conjecture has been proved for $r\\leq \\sqrt{n/2}$ by Shen [She1]. For $r\\geq n/2$ it is trivial. But already for $r=n/3$, it is still open as well as Behzad-Chartrand-Wall Conjecture\n\nConjecture Every simple $n$-vertex digraph with minimum outdegree at least $r/3$ and minimum indegree at least $r/3$ has a cycle with length at most $3$.\n\nThis conjecture would be implied by Seymour's Second Neighbourhood Conjecure.\n\nShen [She2] also proved the following approximate version.\n\nTheorem Every simple digraph of order $n$ with minimum outdegree at least $r$ has a cycle with length at most $n/r + 73$.\n\nBollobás and Scott [BS] proposed a weighted version of the Caccetta-Häggkvist Conjecture.\n\nConjecture Let $w:E(D) \\rightarrow [0,1]$ be a weight function on the arcs of a digraph $D$. If $\\sum_{u\\in N^-(v)} w(uv) \\geq 1$ and $\\sum_{u\\in N^+(v)} w(vu) \\geq 1$ for all $v\\in V(D)$, then there is a directed cycle in $D$ of total weight at least 1.\n\nThey gave a nice proof that there is a directed path of total weight at least 1.\n\nBibliography:\n[BCW] M. Behzad, G. Chartrand, and C. Wall. On minimal regular digraphs with given girth. Fundamenta Mathematicae, 69:227–231, 1970.\n\n[BS] B. Bollobás and A. D. Scott, A proof of a conjecture of {B}ondy concerning paths in weighted digraphs. J. Combin. Theory Ser. B, 66:283-292, 1996.\n\n*[CH] L. Caccetta and R. Häggkvist. On minimal digraphs with given girth. Congressus Numerantium, XXI, 1978\n\n[She1J. Shen. On the girth of digraphs. Discrete Math, 211(1-3):167–181, 2000.\n\n[She2] J. Shen. On the Caccetta-Häggkvist conjecture. Graphs and Combinatorics, 18(3):645–654, 2002.\n\n[Sul] Blair D. Sullivan: A Summary of Problems and Results related to the Caccetta-Häggkvist Conjecture\n\nRelated:\nRelated problems\nSeymour's Second Neighbourhood Conjecture\nDirected path of length twice the minimum outdegree\nHoàng-Reed Conjecture\n\nDiscussion links:\n- conjecture of Thomassé: http://www.openproblemgarden.org/?q=node/46359\n- Hoàng-Reed Conjecture: http://www.openproblemgarden.org/?q=node/47282\n- Seymour's Second Neighbourhood Conjecure: http://www.openproblemgarden.org/?q=node/646\n\nBibliography links:\n- A Summary of Problems and Results related to the Caccetta-Häggkvist Conjecture: http://www.arxiv.org/abs/math.CO/0605646\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Caccetta-Häggkvist Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Caccetta--Häggkvist conjecture remains open in general.\n\n**Verified partial progress.**\n\n- Many fractional-degree regimes, small r cases, and approximate short-cycle theorems are known.\n\n**Full solution or refutation.**\n\nNo complete proof of the ceil(n/r) cycle bound was verified.\n\n**What remains.**\n\nEstablish the conjectured short cycle for all n and r.\n\n**Sources checked.**\n\n- Caccetta--Häggkvist conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Caccetta%E2%80%93H%C3%A4ggkvist_conjecture\n Evidence used: Records the general conjecture as open and summarizes partial results.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3264, "problem_number": "OPG-46432", "title": "Ádám's Conjecture", "statement": "Conjecture Every digraph with at least one directed cycle has an arc whose reversal reduces the number of directed cycles.", "background": "Source: Open Problem Garden. Original node ID: 46432. URL: http://www.openproblemgarden.org/op/adams_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/adams_conjecture\n- Author(s): Ádám, András\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 1st, 2013 by fhavet\n\nProblem-page discussion:\nThe conjecture fails for multigraphs (multiple arcs are allowed). Counterexamples for multidigraphs have been given by Grinberg [G], Jirásek [J] and Thomassen [T].\n\nSurprisingly, the conjecture remains open for tournaments.\n\nConjecture Every tournament that is not transitive has an arc whose reversal reduces the number of directed cycles.\n\nBibliography:\n*[A] A. Ádám, Problem 2. In Theory of Graphs and its Applications (M. Fiedler, ed.), 234. (1964) Publishing House of the Czechoslovak Academy of Sciences, Prague.\n\n[G] E.Y. Grinberg, Examples of non-Ádám multigraphs (in Russian) Latv. Mat. Ezhegodnik, 31 (1988), pp. 128–138\n\n[J] J. Jirásek, On a certain class of multidigraphs, for which reversal of no arc decreases the number of their cycles, Comment. Math. Univ. Carolinae, 28 (1987), pp. 185–189.\n\n[T] C. Thomassen, Counterexamples to Ádám's conjecture on arc reversals in directed graphs, J. Combin. Theory Ser. B, 42 (1987), pp. 128–130.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Ádám's Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The simple-digraph version remains open in the checked sources; known counterexamples use multiple arcs and do not decide the source statement.\n\n**Verified partial progress.**\n\n- The multigraph analogue is false.\n\n**Full solution or refutation.**\n\nNo simple-digraph proof or counterexample was verified.\n\n**What remains.**\n\nFind a simple-digraph counterexample or prove a cycle-reducing reversal always exists.\n\n**Sources checked.**\n\n- Open Problem Garden, Ádám's Conjecture (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/adams_conjecture\n Evidence used: Distinguishes the multigraph counterexamples from the simple-digraph conjecture.\n\n**Review notes.** Simple/multigraph distinction retained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3265, "problem_number": "OPG-46456", "title": "Splitting a digraph with minimum outdegree constraints", "statement": "Problem Is there a minimum integer $f(d)$ such that the vertices of any digraph with minimum outdegree $d$ can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least $d$?", "background": "Source: Open Problem Garden. Original node ID: 46456. URL: http://www.openproblemgarden.org/op/splitting_a_digraph_with_minimum_outdegree_constraints.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/splitting_a_digraph_with_minimum_outdegree_constraints\n- Author(s): Alon, Noga\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 1st, 2013 by fhavet\n\nProblem-page discussion:\nThomassen [T] proved the conjecture when $d=1$ and showed $f(1)=3$. In fact, this case is equivalent to the case $k=2$ of the Bermond-Thomassen Conjecture.\n\nThe existence of the corresponding function $f$ for the undirected analogue is easy and has been observed by many authors. Stiebitz [S] even proved the following tight result: if the minimum degree of an undirected graph $G$ is $d_1+d_2+ \\cdots + d_k$, where each $d_i$ is a non-negative integer, then the vertex set of $G$ can be partitioned into $k$ pairwise disjoint sets $V_1,\\dots, V_k$, so that for all $i$, the induced subgraph on $V_i$ has minimum degree at least $d_i$. This is clearly tight, as shown by an appropriate complete graph.\n\nBibliography:\n*[A] Noga Alon, Disjoint Directed Cycles, Journal of Combinatorial Theory, Series B, 68 (1996), no. 2, 167-178.\n\n[S] M. Stiebitz, Decomposing graphs under degree constraints, J. Graph Theory 23 (1996), 31-324.\n\n[T] C. Thomassen, Disjoint cycles in digraphs, Combinatorica 3 (1983), 393 - 396.\n\nRelated:\nRelated problems\nThe Bermond-Thomassen Conjecture\n\nDiscussion links:\n- Bermond-Thomassen Conjecture: http://www.openproblemgarden.org/?q=node/611\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Splitting a digraph with minimum outdegree constraints\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The record is not classifiable literally because f(d) never appears in its asserted partition property. The likely intended Alon--Stiebitz threshold problem remains open, but Christoph--Petrova--Steiner reduce all cases to existence of F(2,2).\n\n**Verified partial progress.**\n\n- For the standard threshold formulation, existence of F(2,2) implies existence of every F(s,t).\n- Under that hypothesis, F(s,t)=Theta(s+t), so all cases with s,t at least 2 either exist with linear growth or all fail.\n\n**Full solution or refutation.**\n\nNo solution of the malformed literal statement is asserted; the related standard problem has a major reduction but remains open.\n\n**What remains.**\n\nFirst recover and confirm the intended quantifiers and threshold from the original source; for the standard formulation, prove or disprove existence of F(2,2).\n\n**Sources checked.**\n\n- Micha Christoph, Kalina Petrova, and Raphael Steiner, A note on digraph splitting, Combinatorics, Probability and Computing (2025), arXiv:2310.08449. (primary): https://arxiv.org/abs/2310.08449\n Evidence used: Defines the standard F(s,t) threshold problem, calls it open, and proves that F(2,2) controls all cases and yields linear growth.\n- Graph-theory open problems, Splitting a digraph with minimum outdegree constraints. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/splitting_a_digraph_with_minimum_outdegree_constraints/\n Evidence used: Retains the standard problem as partial and links the modern reduction.\n\n**Review notes.** The displayed f(d) is unused. A complete bidirected graph on d+1 vertices also shows that the literal minimum-outdegree-d premise cannot force two induced parts each of minimum outdegree d. No silent repair was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3266, "problem_number": "OPG-46460", "title": "Long directed cycles in diregular digraphs", "statement": "Conjecture Every strong oriented graph in which each vertex has indegree and outdegree at least $d$ contains a directed cycle of length at least $2d+1$.", "background": "Source: Open Problem Garden. Original node ID: 46460. URL: http://www.openproblemgarden.org/op/long_directed_cycles_in_digraph_with_minimum_in_and_out_degree.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/long_directed_cycles_in_digraph_with_minimum_in_and_out_degree\n- Author(s): Jackson, Bill\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 1st, 2013 by fhavet\n\nProblem-page discussion:\nThe disjoint union of two regular tournaments on $2d+1$ vertices shows that this would be best possible.\n\nIf the oriented graph has order at most $4d+1$, Jackson conjecture the existence of a longer cycle, namely a Hamilton cycle\n\nBibliography:\n*[J] B. Jackson. Long paths and cycles in oriented graphs. J. Graph Theory 5 (1981), 145--157.\n\nRelated:\nRelated problems\nDirected path of length twice the minimum outdegree\nHamilton cycle in small d-diregular graphs\n\nDiscussion links:\n- Hamilton cycle: http://www.openproblemgarden.org/?q=node/47028\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Long directed cycles in diregular digraphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Jackson's minimum-semidegree long-cycle conjecture remains open in general. A 2025 theorem proves the required bound in the sufficiently large dense regular range n<=4d+1 by proving Hamiltonicity there.\n\n**Verified partial progress.**\n\n- Jackson proved the related sharp directed-path guarantee of length at least 2d.\n- Lo--Patel--Yıldız prove every sufficiently large d-regular oriented graph on n<=4d+1 vertices is Hamiltonian, which implies a cycle of length at least 2d+1 in that restricted range.\n\n**Full solution or refutation.**\n\nNo theorem covering every strong oriented graph with minimum indegree and outdegree d was verified.\n\n**What remains.**\n\nConvert the 2d directed-path guarantee into a 2d+1 directed cycle without regularity or density assumptions, or find a counterexample.\n\n**Sources checked.**\n\n- Allan Lo, Viresh Patel, and Mehmet Akif Yıldız, Cycle Partitions in Dense Regular Digraphs and Oriented Graphs, Forum of Mathematics, Sigma 13 (2025), e79; arXiv:2309.11677. (primary): https://arxiv.org/abs/2309.11677\n Evidence used: Proves Jackson's separate Hamiltonicity conjecture for sufficiently large d-regular oriented graphs with n<=4d+1, yielding the displayed length in that special case.\n- Graph-theory open problems, Long directed cycles in digraph with minimum in- and out-degree. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/long_directed_cycles_in_digraph_with_minimum_in_and_out_degree/\n Evidence used: Maintains Jackson's 1981 minimum-semidegree cycle statement as open and records the classical path theorem.\n\n**Review notes.** The source's disjoint union of two regular tournaments is not strong; it is an extremal example for the separate regular Hamilton-cycle conjecture, not directly for the displayed strong-oriented-graph statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3267, "problem_number": "OPG-46495", "title": "Arc-disjoint out-branching and in-branching", "statement": "Conjecture There exists an integer $k$ such that every $k$-arc-strong digraph $D$ with specified vertices $u$ and $v$ contains an out-branching rooted at $u$ and an in-branching rooted at $v$ which are arc-disjoint.", "background": "Source: Open Problem Garden. Original node ID: 46495. URL: http://www.openproblemgarden.org/op/arc_disjoint_out_branching_and_in_branching.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/arc_disjoint_out_branching_and_in_branching\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 2nd, 2013 by fhavet\n\nProblem-page discussion:\nThomassen [T] showed that, given a digraph $D$ and two vertices $u$ and $v$, deciding whether there are an out-branching rooted at $u$ and an in-branching rooted at $v$ which are arc-disjoint is NP-complete.\n\nIn contrast, one can decide in polynomial time whether there are $k$ arc-disjoint out-branchings with specified roots $s_1, \\dots, s_k$ (some of which may be identical). This is a consequence of Edmonds’ well known branching theorem [E] states that a digraph $D$ has $k$ arc-disjoint out-branchings rooted at some fixed vertex $s$ if and only if there are $k$ arc-disjoint paths from $s$ to every other vertex of $D$.\n\nBang-Jensen [B] proved this conjecture for tournaments.\n\nA similar question can be asked about arc-disjoint strongly connected spanning subdigraphs. Several related problems are mentioned in the survey of Bang-Jensen and Kriesell [BK].\n\nBibliography:\n[B] J. Bang-Jensen, Edge-disjoint in- and out-branching in tournaments and related path problems. J. Combin. Theory Ser. B 51 (1991), 1-23.\n\n[BK] J. Bang-Jensen, M. Kriesell, Disjoint sub(di)graphs in digraphs, Electronic Notes in Discrete Mathematics 34 (2009), 179-183.\n\n[E] J. Edmonds, Edge-disjoint branchings. In Combinatorial Algorithms, B. Rustin, ed., Acad. Press, New York (1973), 91-96.\n\n*[T] C. Thomassen, Configurations in Graphs, Annals of The New York Acad. Sci. 555 (1989), 402-412.\n\nRelated:\nRelated problems\nArc-disjoint strongly connected spanning subdigraphs\n\nDiscussion links:\n- arc-disjoint strongly connected spanning subdigraphs: http://www.openproblemgarden.org/?q=node/46496\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Arc-disjoint out-branching and in-branching\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Thomassen's universal-connectivity conjecture for a prescribed-root out-branching and in-branching remains wide open; k=2 fails and it is not known whether k=3 suffices. Complete or sharp results are known for several structured classes.\n\n**Verified partial progress.**\n\n- There is a 10-vertex 2-arc-strong digraph with no good pair, while every such digraph on at most nine vertices has one; hence any universal k is at least 3.\n- Bang-Jensen--Wang completely classify semicomplete digraphs with a prescribed good (u,v)-pair.\n- Every 2-arc-strong semicomplete split digraph has a good (u,v)-pair for every prescribed pair of roots.\n\n**Full solution or refutation.**\n\nNo finite universal k for arbitrary digraphs was verified.\n\n**What remains.**\n\nProve that some k works in all digraphs, with k=3 the first possible value, or construct counterexamples at arbitrarily high arc-connectivity.\n\n**Sources checked.**\n\n- Jørgen Bang-Jensen and Yun Wang, Arc-disjoint out-branchings and in-branchings in semicomplete digraphs, Journal of Graph Theory 102 (2023), 578-606; arXiv:2302.06177. (primary): https://arxiv.org/abs/2302.06177\n Evidence used: Calls the general conjecture wide open, notes that k=3 is unknown, and gives a complete semicomplete classification.\n- Ran Gu, Gregory Gutin, Shasha Li, Yongtang Shi, and Zhenyu Taoqiu, The smallest number of vertices in a 2-arc-strong digraph which has no good pair, arXiv:2012.03742. (primary): https://arxiv.org/abs/2012.03742\n Evidence used: Proves the nine-vertex positive threshold and records the ten-vertex counterexample, showing k=2 fails.\n- Jiangdong Ai, Yiming Hao, Zhaoxiang Li, and Qi Shao, Arc-disjoint in- and out-branchings in semicomplete split digraphs, arXiv:2410.12575. (primary): https://arxiv.org/abs/2410.12575\n Evidence used: Proves the prescribed-root conjecture with k=2 for the semicomplete split class.\n\n**Review notes.** The roots are prescribed in the record; sources were checked for that stronger good-(u,v)-pair formulation rather than merely existence of some roots.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3268, "problem_number": "OPG-46680", "title": "Subdivision of a transitive tournament in digraphs with large outdegree.", "statement": "Conjecture For all $k$ there is an integer \u000e $f(k)$ such that every digraph of minimum outdegree at least \u000e $f(k)$ contains a subdivision of a transitive tournament of order $k$.", "background": "Source: Open Problem Garden. Original node ID: 46680. URL: http://www.openproblemgarden.org/op/subdivision_of_a_transitive_tournament_in_digraphs_with_large_outdegree.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/subdivision_of_a_transitive_tournament_in_digraphs_with_large_outdegree\n- Author(s): Mader, W.\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nA fundamental result of Mader [M1] states that for every integer $k$ there is a smallest $g(k)$ so that every graph of average degree at least $g(k)$ contains a subdivision of a complete graph on $k$ vertices. Bollobás and Thomason [BT] as well as Komlós and Szemerédi [KS] showed that $g$ is quadratic in $k$.\n\nThe above conjecture is a digraph analogue of this result. However one cannot replace the minimum outdegree in this conjecture by the average degree as in Mader's analogue for graphs: consider the complete bipartite graph $K_{n,n}$ and orient all edges from the first to the second class. The resulting digraph has average degree $n$ but not even a transitive tournament on 3 vertices.\n\nOne might be tempted to conjecture that large minimum outdegree would even force the existence of a subdivision of a large complete digraph. However, for all $n$ Thomassen [T] constructed a digraph on $n$ vertices whose minimum outdegree is at least $\\frac{1}{2} \\log_2 n$ but which does not contain an even directed cycle (and thus no complete digraph on 3 vertices). A simpler construction was found by DeVos et al. [DMMS].\n\nIt is easy to see that \u000e $f(1)=0$ and $f(2)=1$. Mader [M3] showed that $f(4) = 3$. Even the existence of \u000e $f(5)$ is not known.\n\nBibliography:\n[BT] B. Bollobás and A. Thomason, Proof of a conjecture of Mader, Erdös and Hajnal on topological complete subgraphs, European Journal of Combinatorics 19 (1998), 883–887.\n\n[DMMS] M. DeVos, J. McDonald, B. Mohar, and D. Scheide, Immersing complete digraphs, European Journal of Combinatorics, 33 (2012), no 6, 1294-1302.\n\n[KS] J. Komlós and E. Szemerédi, Topological Cliques in Graphs II, Combinatorics, Probability and Computing 5 (1996), 70–90.\n\n[M1] W. Mader, Homomorphieeigenschaften und mittlere Kantendichte von Graphen, Math. Annalen 174 (1967), 265–268.\n\n* [M2] W. Mader, Degree and Local Connectivity in Digraphs, Combinatorica 5 (1985), 161–165.\n\n[M3] W. Mader, On Topological Tournaments of order 4 in Digraphs of Outdegree 3, Journal of Graph Theory 21 (1996), 371–376.\n\n[T] C. Thomassen, Even Cycles in Directed Graphs, European Journal of Combinatorics 6 (1985), 85–89.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Subdivision of a transitive tournament in digraphs with large outdegree.\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The high-minimum-outdegree subdivision conjecture for transitive tournaments remains open, with the first unresolved test reportedly already the transitive tournament of order five.\n\n**Verified partial progress.**\n\n- Large minimum outdegree forces subdivisions for several target classes, including oriented paths and in-arborescences.\n- Related forcing results are known under large dichromatic number.\n\n**Full solution or refutation.**\n\nPrimary work establishes broad special target classes but explicitly leaves the transitive-tournament conjecture open, even at order five at the time of publication.\n\n**What remains.**\n\nProve existence of f(5), or construct digraphs of arbitrarily large minimum outdegree with no subdivision of the transitive tournament on five vertices.\n\n**Sources checked.**\n\n- P. Aboulker, N. Cohen, F. Havet, W. Lochet, P. S. Moura, and S. Thomasse, Subdivisions in digraphs of large out-degree or large dichromatic number, arXiv:1610.00876. (primary): https://arxiv.org/abs/1610.00876\n Evidence used: States the conjecture is completely open and f(5) unknown, while proving special target classes.\n- Graph-theory open problems, Subdivision of a transitive tournament in digraphs with large outdegree (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/subdivision_of_a_transitive_tournament_in_digraphs_with_large_outdegree/\n Evidence used: Reports the general conjecture open and catalogs partial results.\n\n**Review notes.** The input contains two literal U+000E control characters immediately before f(k); the evident deletion repair is recorded in report.md, while the source statement remains unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3269, "problem_number": "OPG-47028", "title": "Hamilton cycle in small d-diregular graphs", "statement": "An directed graph is $k$-diregular if every vertex has indegree and outdegree at least $k$.\n\nConjecture For $d >2$, every $d$-diregular oriented graph on at most $4d+1$ vertices has a Hamilton cycle.", "background": "Source: Open Problem Garden. Original node ID: 47028. URL: http://www.openproblemgarden.org/op/hamilton_cycle_in_small_d_diregular_graphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamilton_cycle_in_small_d_diregular_graphs\n- Author(s): Jackson, Bill\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2013 by fhavet\n\nProblem-page discussion:\nThe disjoint union of two regular tournaments on $2d+1$ vertices shows that this would be best possible. For $d$-diregular oriented graphs with an arbitrary order of vertices, Jackson conjectured the existence of a long cycle.\n\nKühn and Osthus [KO] conjectured that it may actually be possible to increase the size of the graph even further if we assume that the graph is strongly 2-connected.\n\nProblem Is it true that for each $d >2$, every $d$-regular strongly $2$-connected oriented graph $G$ on at most $6d$ vertices has a Hamilton cycle?\n\nBibliography:\n*[J] B. Jackson. Long paths and cycles in oriented graphs, J. Graph Theory 5 (1981), 145-157.\n\n[KO] D. Osthus and D. Kühn, A survey on Hamilton cycles in directed graphs, European J. Combinatorics 33 (2012), 750-766.\n\nRelated:\nRelated problems\nLong directed cycles in diregular digraphs\n\nDiscussion links:\n- existence of a long cycle: http://www.openproblemgarden.org/?q=node/46460\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamilton cycle in small d-diregular graphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Jackson's conjecture is proved for sufficiently large order, while the stated all-order version remains unverified.\n\n**Verified partial progress.**\n\n- Lo--Patel--Yildiz prove the Hamiltonicity assertion for sufficiently large n, as part of a stronger cycle-partition theorem.\n\n**Full solution or refutation.**\n\nThe finite small-order range is not covered by the verified asymptotic theorem.\n\n**What remains.**\n\nEliminate the sufficiently-large-order qualification or resolve the finite exceptional range.\n\n**Sources checked.**\n\n- A. Lo, V. Patel and M. A. Yildiz, Cycle Partitions in Dense Regular Digraphs and Oriented Graphs, Forum of Mathematics, Sigma 13 (2025), e79. (primary): https://doi.org/10.1017/fms.2025.28\n Evidence used: Its abstract states that Jackson's n <= 4d+1 conjecture is proved for sufficiently large n.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3270, "problem_number": "OPG-47282", "title": "Hoàng-Reed Conjecture", "statement": "Conjecture Every digraph in which each vertex has outdegree at least $k$ contains $k$ directed cycles $C_1, \\ldots, C_k$ such that $C_j$ meets $\\cup_{i=1}^{j-1}C_i$ in at most one vertex, $2 \\leq j \\leq k$.", "background": "Source: Open Problem Garden. Original node ID: 47282. URL: http://www.openproblemgarden.org/op/hoand_reed_conjecture.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hoand_reed_conjecture\n- Author(s): Hoang, Chinh T.; Reed, Bruce A.\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 11th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is not even known to be true for $k=3$. In the case $k=2$, Thomassen proved [T] that every digraph with minimum outdegree 2 has two directed cycles intersecting on a vertex.\n\nThis conjecture would imply the Caccetta-Häggkvist Conjecture.\n\nBibliography:\n*[HR] C.T. Hoàng and B. Reed, A note on short cycles in digraphs, Discrete Math., 66 (1987), 103-107.\n\n[T] C. Thomassen, The 2-linkage problem for acyclic digraphs, Discrete Math., 55 (1985), 73-87.\n\nRelated:\nRelated problems\nCaccetta-Häggkvist Conjecture\n\nDiscussion links:\n- Caccetta-Häggkvist Conjecture: http://www.openproblemgarden.org/?q=node/46385\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Hoàng-Reed Conjecture\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Hoang--Reed conjecture is proved for minimum outdegree three (and previously two), but remains open in general.\n\n**Verified partial progress.**\n\n- Welhan proved a stronger form for delta+ = 3.\n- The conjecture also holds for tournaments.\n\n**Full solution or refutation.**\n\nThe verified small-degree theorem does not establish the result for arbitrary k.\n\n**What remains.**\n\nExtend the circuit-forest construction to every minimum outdegree k.\n\n**Sources checked.**\n\n- M. Welhan, The Hoang--Reed Conjecture for delta+ = 3, Discrete Mathematics 310 (2010), 1932--1939. (primary): https://doi.org/10.1016/j.disc.2010.03.001\n Evidence used: The abstract states that it proves a stronger version of the conjecture for delta+ = 3.\n- Open Problem Garden, Hoang--Reed Conjecture (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/hoand_reed_conjecture\n Evidence used: Retains the general conjecture and records the k=2 result.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3271, "problem_number": "OPG-49573", "title": "Arc-disjoint directed cycles in regular directed graphs", "statement": "Conjecture If $G$ is a $k$-regular directed graph with no parallel arcs, then $G$ contains a collection of ${k+1 \\choose 2}$ arc-disjoint directed cycles.", "background": "Source: Open Problem Garden. Original node ID: 49573. URL: http://www.openproblemgarden.org/op/arc_disjoint_directed_cycles_in_regular_directed_graphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/arc_disjoint_directed_cycles_in_regular_directed_graphs\n- Author(s): Alon, Noga; McDiarmid, Colin; Molloy, Michael\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 17th, 2013 by fhavet\n\nProblem-page discussion:\nIf true, ${k+1 \\choose 2}$ would be best possible as shown by the complete symmetric digraph.\n\nAlon et al. [AMM] showed that a $k$-regular directed graph with no parallel arcs contains at least $\\frac{3}{2^{19}}k^2$ arc-disjoint directed cycles. It was then improved by Alon [A] who showed that every directed graph with minimum outdegree at least $k$ contains at least $\\frac{1}{128}k^2$ arc-disjoint directed cycles.\n\nBibliography:\n[A} N. Alon, Disjoint directed cycles, J. Combinatorial Theory, Ser. B, 68 (1996), 167-178.\n\n*[AMM] N. Alon, C. McDiarmid and M. Molloy, Edge-disjoint cycles in regular directed graphs, J. Graph Theory, 22 (1996), no. 3, 231-237.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Arc-disjoint directed cycles in regular directed graphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The checked maintained record still lists the binomial(k+1,2) arc-disjoint-cycle assertion for simple k-regular digraphs as open.\n\n**Verified partial progress.**\n\n- Related regular-digraph and tournament cycle-packing theorems are known, but no verified theorem gives the exact stated count.\n\n**Full solution or refutation.**\n\nNo proof or counterexample to the full packing bound was verified.\n\n**What remains.**\n\nProve the exact binomial lower bound or construct a simple regular counterexample.\n\n**Sources checked.**\n\n- UnsolvedMath/OpenGarden, Arc-disjoint directed cycles in regular directed graphs (accessed 2026-08-17). (maintained_tracker): https://www.unsolvedmath.com/problems/OPG-49573\n Evidence used: Retains the exact assertion as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3272, "problem_number": "OPG-50631", "title": "Cyclic spanning subdigraph with small cyclomatic number", "statement": "Conjecture Let $D$ be a digraph all of whose strong components are nontrivial. Then $D$ contains a cyclic spanning subdigraph with cyclomatic number at most $\\alpha(D)$.", "background": "Source: Open Problem Garden. Original node ID: 50631. URL: http://www.openproblemgarden.org/op/cyclic_spanning_subdigraph_with_small_cyclomatic_number.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/cyclic_spanning_subdigraph_with_small_cyclomatic_number\n- Author(s): Bondy, J. Adrian\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 2nd, 2013 by fhavet\n\nProblem-page discussion:\nThe {\\it cyclomatic number} of a digraph $D=(V,A)$ is $|A|-|V|+1$. For a strong digraph, it correspond to the minimum of directed ears in a directed ears decomposition. (See Chapter 5 of [BM]).\n\n$\\alpha(D)$ denotes the {\\it stability number} of the digraph $D$, that is the maximum number of pairwise non-adjacent vertices.\n\nBessy and Thomassé [BT04] showed that any nontrivial strong digraph $D$ has a spanning subdigraph which is the union of $\\alpha$ directed cycles. However, the structure of this subdigraph might be rather complicated. This leads one to ask whether there always exists a spanning subdigraph whose structure is relatively simple, one which is easily seen to be the union of $\\alpha$ directed cycles. A natural candidate would be a spanning subdigraph built from a directed cycle by adding $\\alpha(D)-1$ directed ears. But or any $\\alpha \\geq 2$, there exists a digraph $D$ with stability number $\\alpha$ requiring at least $2\\alpha -2$ directed ears. See Chapter 19 of [BM08].\n\nA possible way around this problem is to allow spanning subdigraphs which are disjoint union of strong digraphs. Such digraph are called cyclic (because each arc lies on a directed cycle). The conjecture was formulated by Bondy[B], based on a remark of Chen and Manalastas [CM].\n\nThe Conjecture holds for $\\alpha(D)=1$ by Camion's Theorem [C] and also for $\\alpha(D)=2$ and $\\alpha(D)=3$ by theorems of Chen and Manalastas [CM] and S. Thomassé (unpublished), respectively.\n\nThe conjecture implies not only the above-mentioned Bessy--Thomassé Theorem, but also a result of Thomassé [Thom01], that the vertex set of any strong digraph $D$ with $\\alpha(D) \\geq 2$ can be partitioned into $\\alpha(D)-1$ directed paths, as well as another theorem of Bessy and Thomassé [BT03], that every strong digraph $D$ has a strong spanning subdigraph with at most $n+2\\alpha(D)-2$ arcs.\n\nBibliography:\n[BT03] S. Bessy and S. Thomassé, Every strong digraph has a spanning strong subgraph with at most $n+2\\alpha-2$ arcs. J. Combin. Theory Ser. B 87 (2003), 289--299.\n\n[BT04] S. Bessy and S. Thomassé, Three min-max theorems concerning cyclic orders of strong digraphs. In Integer Programming and Combinatorial Optimization, 132--138. Lecture Notes in Comput. Sci., Vol. 3064, Springer, Berlin.\n\n*[B95] J.A. Bondy, Basic graph theory: paths and circuits. In Handbook of Combinatorics, Vol. 1, 3--110. Elsevier, Amsterdam.\n\n[BM] J.A. Bondy and U.S.R. Murty, Graph Theory, volume 244 of Graduate Texts in Mathematics. Springer, 2008.\n\n[C] P. Camion, Chemins et circuits hamiltoniens des graphes complets. C. R. Acad. Sci. Paris 249 (1959), 2151--2152.\n\n[CM] C.C. Chen C.C. and Jr. P. Manalastas, Every finite strongly connected digraph of stability 2 has a Hamiltonian path. Discrete Math. 44 (1983), 243--250.\n\n[T] S. Thomassé, Covering a strong digraph by $\\alpha-1$ disjoint paths: a proof of Las Vergnas' conjecture. J. Combin. Theory Ser. B 83 (2001), 331--333.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Cyclic spanning subdigraph with small cyclomatic number\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Bondy's cyclic-spanning-subdigraph conjecture is established for stability number at most three and for some special digraph classes, but remains open generally.\n\n**Verified partial progress.**\n\n- Camion's theorem gives alpha=1; Chen--Manalastas establish alpha=2 and alpha=3.\n- The stronger form holds for semicomplete multipartite digraphs.\n\n**Full solution or refutation.**\n\nNo theorem proving the alpha(D) cyclomatic bound for arbitrary digraphs was verified.\n\n**What remains.**\n\nProve the cyclic spanning-subdigraph bound for all stability numbers or find a counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, Cyclic spanning subdigraph with small cyclomatic number (accessed 2026-08-17). (maintained_tracker): https://openproblemgarden.org/op/cyclic_spanning_subdigraph_with_small_cyclomatic_number\n Evidence used: States the special alpha<=3 cases and retains the general conjecture.\n- J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications, discussion of cyclic spanning subdigraphs. (authoritative_secondary): https://citeseerx.ist.psu.edu/document?doi=77e77f1bd8378c7fa8da967bf81be5e4ba9968cf&repid=rep1&type=pdf\n Evidence used: States the conjecture and the semicomplete multipartite special case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3273, "problem_number": "OPG-52197", "title": "Large acyclic induced subdigraph in a planar oriented graph.", "statement": "Conjecture Every planar oriented graph $D$ has an acyclic induced subdigraph of order at least $\\frac{3}{5} |V(D)|$.", "background": "Source: Open Problem Garden. Original node ID: 52197. URL: http://www.openproblemgarden.org/op/large_acyclic_induced_subdigraph_in_a_planar_oriented_graph.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/large_acyclic_induced_subdigraph_in_a_planar_oriented_graph\n- Author(s): Harutyunyan, Ararat\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 25th, 2013 by fhavet\n\nProblem-page discussion:\nBorodin's 5-Colour Theorem states that every planar graph has an acyclic 5-colouring This implies that every planar oriented graph $D$ has an acyclic induced subdigraph of order at least $\\frac{2}{5} |V(D)|$.\n\nAlready improving this bound to $\\frac{1}{2} |V(D)|$ would be interesting: it is a relaxtion of both a Conjecture of Albertson and Berman stating that every planar graph $G$ has an induced forest of order $\\frac{1}{2} |V(G)|$ and a Conjecture of Neumann-Lara stating that every planar oriented graph can be split into two acyclic subdigraphs.\n\nIf true, this conjecture would be best possible.\n\nRelated:\nRelated problems\nLarge induced forest in a planar graph.\nThe Two Color Conjecture\n\nDiscussion links:\n- Conjecture of Albertson and Berman: http://www.openproblemgarden.org/?q=node/46634\n- Conjecture of Neumann-Lara: http://www.openproblemgarden.org/?q=node/169\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Large acyclic induced subdigraph in a planar oriented graph.\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No checked source resolves the 3/5 acyclic-induced-subdigraph conjecture for every planar oriented graph.\n\n**Verified partial progress.**\n\n- The question belongs to the active study of dichromatic number and large acyclic sets in planar orientations.\n\n**Full solution or refutation.**\n\nNo proof of the exact 3/5 bound or counterexample was verified.\n\n**What remains.**\n\nEstablish the universal 3/5 bound or find a planar orientation with every acyclic induced subdigraph smaller.\n\n**Sources checked.**\n\n- Graph-theory open problems, Large acyclic induced subdigraph in a planar oriented graph (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/large_acyclic_induced_subdigraph_in_a_planar_oriented_graph/\n Evidence used: Lists the exact 3/5 conjecture as unresolved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3274, "problem_number": "OPG-52200", "title": "Erdős-Posa property for long directed cycles", "statement": "Conjecture Let $\\ell \\geq 2$ be an integer. For every integer $n\\geq 0$, there exists an integer $t_n=t_n(\\ell)$ such that for every digraph $D$, either $D$ has a $n$ pairwise-disjoint directed cycles of length at least $\\ell$, or there exists a set $T$ of at most $t_n$ vertices such that $D-T$ has no directed cycles of length at least $\\ell$.", "background": "Source: Open Problem Garden. Original node ID: 52200. URL: http://www.openproblemgarden.org/op/erdos_posa_property_for_long_directed_cycles.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/erdos_posa_property_for_long_directed_cycles\n- Author(s): Havet, Frédéric; Maia, Ana Karolinna\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 25th, 2013 by fhavet\n\nProblem-page discussion:\nThe case $\\ell=2$ has been proved by Reed et al. [RRST], hence solving a conjecture of Gallai [G] and Younger [Y]. The case $\\ell=2$ and $n=2$ has previously been solved by McCuaig [M], who proved that $t_2(2)=3$. Havet and Maia [HM] proved the case $\\ell=3$.\n\nThe analogous statement for undirected graph has been proved by Birmelé, Bondy and Reed [BBR], hence generalizing Erdős-Posa [EP] result for $\\ell =3$.\n\nBibliography:\n[BBR] E. Birmelé, J.A. Bondy, and B.A. Reed. The Erdos-Posa property for long circuits, Combinatorica, 27(2), 135–145, 2007.\n\n[EP] P. Erdős and L. Pósa. On the independent circuits contained in a graph. Canad. J. Math., 17, 347--352, 1965.\n\n[G] T. Gallai. Problem 6, in Theory of Graphs, Proc. Colloq. Tihany 1966 (New York), Academic Press, p.362, 1968.\n\n*[HM] F. Havet and A. K. Maia. On disjoint directed cycles with prescribed minimum lengths. INRIA Research Report, RR-8286, 2013.\n\n[M] W. McCuaig, Intercyclic digraphs. Graph Structure Theory, (Neil Robertson and Paul Seymour, eds.), AMS Contemporary Math., 147:203--245, 1993.\n\n[RRST] B. Reed, N. Robertson, P.D. Seymour, and R. Thomas. Packing directed circuits. Combinatorica, 16(4):535--554, 1996.\n\n[Y] D. H. Younger. Graphs with interlinked directed circuits. Proceedings of the Midwest Symposium on Circuit Theory, 2:XVI 2.1 - XVI 2.7, 1973.\n\nBibliography links:\n- On disjoint directed cycles with prescribed minimum lengths: http://hal.inria.fr/hal-00816135/en\n- Packing directed circuits: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.89.5838&rep=rep1&type=pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Erdős-Posa property for long directed cycles\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The vertex Erdős--Pósa property for directed cycles of length at least a fixed ell was proved by Kreutzer and Kawarabayashi.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nTheir theorem supplies the required hitting-set function t_n(ell) for every fixed ell and packing number n.\n\n**What remains.**\n\nQuantitatively improve the Erdős--Pósa function if desired; the source existence statement is settled.\n\n**Sources checked.**\n\n- S. Kreutzer and K. Kawarabayashi, The Erdős--Pósa property for long directed cycles, STOC 2015. (primary): https://doi.org/10.1145/2746539.2746585\n Evidence used: Later algorithmic literature explicitly cites this work as proving the Erdős--Pósa property for long directed cycles.\n- M. Cygan et al., Hitting Long Directed Cycles Is Fixed-Parameter Tractable, ICALP 2020. (authoritative_secondary): https://drops.dagstuhl.de/storage/00lipics/lipics-vol168-icalp2020/LIPIcs.ICALP.2020.59/LIPIcs.ICALP.2020.59.pdf\n Evidence used: States that Kreutzer--Kawarabayashi proved the property for long directed cycles.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3275, "problem_number": "OPG-60028", "title": "Monochromatic reachability in arc-colored digraphs", "statement": "Conjecture For every $k$, there exists an integer $f(k)$ such that if $D$ is a digraph whose arcs are colored with $k$ colors, then $D$ has a $S$ set which is the union of $f(k)$ stables sets so that every vertex has a monochromatic path to some vertex in $S$.", "background": "Source: Open Problem Garden. Original node ID: 60028. URL: http://www.openproblemgarden.org/op/monochromatoc_reachability_in_arc_colored_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monochromatoc_reachability_in_arc_colored_digraphs\n- Author(s): Sands, Bill; Sauer, Norbert W.; Woodrow, Robert E.\n- Subject(s): Graph Theory; Directed Graphs\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 4th, 2017 by fhavet\n\nProblem-page discussion:\nIn the particular case of tournaments (and more generally when the stabilty number of $D$ is bounded), it has been proved by Bousquet, Lochet, and Thomassé [BLT].\n\nBibliography:\n[BLT] Nicolas Bousquet, William Lochet, Stéphan Thomassé: A proof of the Erdős-Sands-Sauer-Woodrow conjecture,\n\n[SSW] B. Sands, N. Sauer and R. Woodrow, On monochromatic paths in edge-coloured digraphs. Journal of Combinatorial Theory, Series B, 33, (1982), 271--275.\n\nRelated:\nRelated problems\nMonochromatic reachability in edge-colored tournaments\nMonochromatic reachability or rainbow triangles\n\nBibliography links:\n- A proof of the Erdős-Sands-Sauer-Woodrow conjecture: http://www.arxiv.org/abs/math.CO/1703.08123\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Monochromatic reachability in arc-colored digraphs\" in Graph Theory; Directed Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The finite-stable-set monochromatic-reachability conjecture remains unresolved in the checked sources.\n\n**Verified partial progress.**\n\n- Many kernel-by-monochromatic-path and H-kernel theorems prove related conclusions under structural restrictions on the coloured digraph.\n\n**Full solution or refutation.**\n\nNo universal f(k) as in the source statement was verified.\n\n**What remains.**\n\nProve the asserted bounded stable-set target for arbitrary k-coloured digraphs or find a counterexample.\n\n**Sources checked.**\n\n- M. Raynal, curriculum vitae, research entry on the Erdős--Sands--Sauer--Woodrow conjecture (accessed 2026-08-17). (authoritative_secondary): https://raynalm.github.io/files/raynalm_cv_en.pdf\n Evidence used: Identifies the source problem as an active research conjecture.\n- On kernels by rainbow paths in arc-coloured digraphs, Open Mathematics 19 (2021). (authoritative_secondary): https://www.degruyterbrill.com/document/doi/10.1515/math-2021-0019/html\n Evidence used: Documents related kernel-by-monochromatic-path theory with additional hypotheses.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3276, "problem_number": "OPG-1808", "title": "Monochromatic reachability or rainbow triangles", "statement": "In an edge-colored digraph, we say that a subgraph is rainbow if all its edges have distinct colors, and monochromatic if all its edges have the same color.\n\nProblem Let $G$ be a tournament with edges colored from a set of three colors. Is it true that $G$ must have either a rainbow directed cycle of length three or a vertex $v$ so that every other vertex can be reached from $v$ by a monochromatic (directed) path?", "background": "Source: Open Problem Garden. Original node ID: 1808. URL: http://www.openproblemgarden.org/op/monochromatic_reachability_vs_rainbow_triangles.\n\nSource subject path: Graph Theory > Directed Graphs > Tournaments.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monochromatic_reachability_vs_rainbow_triangles\n- Author(s): Sands, Bill; Sauer, Norbert W.; Woodrow, Robert E.\n- Subject(s): Graph Theory; Directed Graphs; Tournaments\n- Keywords: digraph; edge-coloring; tournament\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 15th, 2008 by mdevos\n\nProblem-page discussion:\nThis problem was raised in a paper by Sands, Sauer, and Woodrow [SSW] where they prove that every tournament with 2-colored edges has a vertex $v$ so that every other vertex can be reached from $v$ by a monochromatic path.\n\nGaleana-Sanchez and Rojas-Monroy found a tournament on 6 vertices with 4-colored edges which has no rainbow triangle and does not have a vertex $v$ which has monochromatic paths to all remaining vertices. However, the following generalization of the above conjecture looks plausible.\n\nProblem Does every edge-colored tournament have either a rainbow directed cycle or a vertex $v$ so that every other vertex can be reached from $v$ by a monochromatic path?\n\nBibliography:\n*[SSW] B. Sands, N. Sauer, R. Woodrow, On monochromatic paths in edge-coloured digraphs. J. Combin. Theory Ser. B 33 (1982), no. 3, 271--275. MathSciNet.\n\nRelated:\nRelated problems\nMonochromatic reachability in edge-colored tournaments\n\nBibliography links:\n- On monochromatic paths in edge-coloured digraphs: http://www.sciencedirect.com/science/article/pii/0095895682900478\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0693367\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Monochromatic reachability or rainbow triangles\" in Graph Theory; Directed Graphs; Tournaments, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact three-color alternative with a cyclic rainbow triangle remains open in the literature checked; it is proved when no vertex is incident with all three colors, and a weaker theorem allows any rainbow triangle.\n\n**Verified partial progress.**\n\n- Sands, Sauer, and Woodrow proved the one-vertex monochromatic reachability conclusion for two colors.\n- Georgakopoulos and Sprüssel prove the exact alternative when each vertex is incident with at most two colors.\n- Shen proved a weaker alternative in which the rainbow triangle need not be a directed cycle.\n\n**Full solution or refutation.**\n\nNo proof or counterexample was verified for unrestricted 3-colored tournaments under the exact cyclic-rainbow-triangle alternative.\n\n**What remains.**\n\nRemove the local two-color incidence hypothesis while retaining the requirement that the rainbow triangle be cyclic, or construct a counterexample.\n\n**Sources checked.**\n\n- Agelos Georgakopoulos and Philipp Sprüssel, On 3-coloured tournaments, arXiv:0904.1967 (2009). (primary): https://arxiv.org/abs/0904.1967\n Evidence used: The introduction states the exact conjecture, distinguishes Shen's weaker rainbow-triangle theorem, and gives the extra-hypothesis theorem proved in the paper.\n- Open Problem Garden, Monochromatic reachability or rainbow triangles (node 1808), accessed 2026-08-17. (maintained_tracker): https://openproblemgarden.org/op/monochromatic_reachability_vs_rainbow_triangles\n Evidence used: Retains the precise three-color cyclic-rainbow formulation and the related unrestricted-color question.\n\n**Review notes.** A rainbow transitive triangle does not satisfy the source alternative. Results allowing any rainbow triangle were not misclassified as solutions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3277, "problem_number": "OPG-46237", "title": "Decomposing an even tournament in directed paths.", "statement": "Conjecture Every tournament $D$ on an even number of vertices can be decomposed into $\\sum_{v\\in V}\\max\\{0,d^+(v)-d^-(v)\\}$ directed paths.", "background": "Source: Open Problem Garden. Original node ID: 46237. URL: http://www.openproblemgarden.org/op/decomposing_an_even_tournament_in_directed_paths.\n\nSource subject path: Graph Theory > Directed Graphs > Tournaments.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_an_even_tournament_in_directed_paths\n- Author(s): Alspach, Brian; Mason, David W.; Pullman, Norman J.\n- Subject(s): Graph Theory; Directed Graphs; Tournaments\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 26th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is clearly tight, because in a decomposition of a directed graph in directed paths, at least $\\max \\{0,d^+(v)-d^-(v)\\}$ directed paths must start at vertex $v$.\n\nObserve that the analogue is trivially false for odd tournament: in regular tournament $d^+(v)=d^-(v)$ for every vertex $v$, so $\\sum_{v\\in V}\\max\\{0,d^+(v)-d^-(v)\\}=0$. For a tournament of even order $n$, $\\sum_{v\\in V}\\max\\{0,d^+(v)-d^-(v)\\}\\geq n/2$. Since a directed path may have up to $n-1$ arcs, it might be possible to cover the $n(n-1)/2$ arcs of the tournament if $n$ is even. If the tournament is almost regular (i.e. $|d^+(v)-d^-(v)|=1$ for all vertex $v$ ), the conjecture asserts that it can be decomposed into directed Hamilton paths.\n\nThis conjecture for almost regular tournaments would imply the following one due to Kelly.\n\nConjecture Every regular tournament of order $n$ can be decomposed into $(n-1)/2$ Hamilton directed cycles.\n\nTo see this, consider a regular tournament $T$ and a vertex $v$ of $T$. The tournament $T-v$ has even order, and in $T-v$, $\\max \\{0,d^+(v)-d^-(v)\\}=0$ unless $v$ is an outneighbour of $v$ in $T$ in which case $\\max \\{0,d^+(v)-d^-(v)\\}=0$. Hence $\\sum_{v\\in V}\\max\\{0,d^+(v)-d^-(v)\\}=(n-1)/2$. Now if Alspach-Mason-Pulman conjecture holds, $T-v$ can be decomposed into $(n-1)/2$ directed paths. These paths must start at distinct outneighbours of $v$ in $T$ and ends at distinct inneighbours of $v$ in $T$. Hence, we can complete each directed path in a Hamilton directed cycle in $T$ to obtain a decomposition of $T$ into $(n-1)/2$ Hamilton cycles.\n\nKelly's conjecture has been proved for tournaments of sufficiently large order by Kühn and Osthus [KO].\n\nBibliography:\n*[AMP] Brian Alspach, David W. Mason, Norman J. Pullman, Path numbers of tournaments, Journal of Combinatorial Theory, Series B, 20 (1976), no. 3, June 1976, 222–228\n\n[KO] Daniela Kühn and Deryk Osthus, Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments, Advances in Mathematics 237 (2013), 62-146.\n\nRelated:\nRelated problems\nEdge-disjoint Hamilton cycles in highly strongly connected tournaments.\n\nBibliography links:\n- Path numbers of tournaments: http://www.sciencedirect.com/science/article/pii/0095895676900137\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing an even tournament in directed paths.\" in Graph Theory; Directed Graphs; Tournaments, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact path-decomposition formula is proved for all sufficiently large even tournaments; the all-order statement remains open.\n\n**Verified partial progress.**\n\n- Girão--Granet--Kühn--Lo--Osthus prove the even-order conjecture asymptotically.\n\n**Full solution or refutation.**\n\nThe finite exceptional range is not eliminated by the verified theorem.\n\n**What remains.**\n\nHandle all small even tournaments or obtain a uniform proof.\n\n**Sources checked.**\n\n- A. Girão, B. Granet, D. Kühn, A. Lo and D. Osthus, Path decompositions of tournaments, Proceedings of the London Mathematical Society 127 (2023), arXiv:2010.14158. (primary): https://arxiv.org/abs/2010.14158\n Evidence used: The abstract proves the conjecture for all sufficiently large even tournaments.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3278, "problem_number": "OPG-47031", "title": "Edge-disjoint Hamilton cycles in highly strongly connected tournaments.", "statement": "Conjecture For every $k\\geq 2$, there is an integer $f(k)$ so that every strongly $f(k)$-connected tournament has $k$ edge-disjoint Hamilton cycles.", "background": "Source: Open Problem Garden. Original node ID: 47031. URL: http://www.openproblemgarden.org/op/edge_disjoint_hamilton_cycles.\n\nSource subject path: Graph Theory > Directed Graphs > Tournaments.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/edge_disjoint_hamilton_cycles\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Directed Graphs; Tournaments\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 8th, 2013 by fhavet\n\nProblem-page discussion:\nKelly made the following conjecture which replaces the assumption of high connectivity by regularity.\n\nConjecture Every regular tournament of order $n$ can be decomposed into $(n-1)/2$ Hamilton directed cycles.\n\nKelly's conjecture has been proved for tournaments of sufficiently large order by Kühn and Osthus [KO].\n\nBibliography:\n[KO] Daniela Kühn and Deryk Osthus, Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments, Advances in Mathematics 237 (2013), 62-146.\n\n*[T] C. Thomassen, Edge-disjoint Hamiltonian paths and cycles in tournaments, Proc. London Math. Soc. 45 (1982), 151-168.\n\nRelated:\nRelated problems\nDecomposing an even tournament in directed paths.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Edge-disjoint Hamilton cycles in highly strongly connected tournaments.\" in Graph Theory; Directed Graphs; Tournaments, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The required finite connectivity threshold exists: Kuehn--Lapinskas--Osthus--Patel prove f(k)=O(k^2 log^2 k).\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThis directly proves the existential statement in the source problem.\n\n**What remains.**\n\nQuantitatively sharpen the best possible order of f(k), if desired; the source conjecture itself is settled.\n\n**Sources checked.**\n\n- D. Kuehn, V. Lapinskas, D. Osthus and V. Patel, Proof of a conjecture of Thomassen on highly connected tournaments, Journal of the London Mathematical Society 90 (2014); arXiv:1303.4213. (primary): https://arxiv.org/abs/1303.4213\n Evidence used: The abstract proves that every strongly f(k)-connected tournament contains k edge-disjoint Hamilton cycles, with f(k)=O(k^2 log^2 k).\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3279, "problem_number": "OPG-47643", "title": "Partitionning a tournament into k-strongly connected subtournaments.", "statement": "Problem Let $k_1, \\dots, k_p$ be positve integer Does there exists an integer $g(k_1, \\dots, k_p)$ such that every $g(k_1, \\dots, k_p)$-strong tournament $T$ admits a partition $(V_1\\dots, V_p)$ of its vertex set such that the subtournament induced by $V_i$ is a non-trivial $k_i$-strong for all $1\\leq i\\leq p$.", "background": "Source: Open Problem Garden. Original node ID: 47643. URL: http://www.openproblemgarden.org/op/partitionning_a_tournament_into_k_strongly_connected_subtournaments.\n\nSource subject path: Graph Theory > Directed Graphs > Tournaments.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/partitionning_a_tournament_into_k_strongly_connected_subtournaments\n- Author(s): Thomassen, Carsten\n- Subject(s): Graph Theory; Directed Graphs; Tournaments\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 15th, 2013 by fhavet\n\nProblem-page discussion:\nIf $k_i=1$ for $2\\leq k_i\\leq k_p$, then $g(k_1, \\dots, k_p)$ exists and is at most $k_1+3p-3$. This follows by an easy induction on $p$, by taking $V_p$ to be a set inducing a directed $3$-cycle.\n\nThe following example shows that if it exists $g(k_1, \\dots, k_p)\\geq k_1+\\cdots + k_p$. Set $s=k_1 + \\cdots + k_p -1$. For $n\\geq 3s$, let $R_s(n)$ be a tournament on $n$ vertices having a set $R$ of $s$ vertices such that $T-R$ a transitive tournament of order $n-s$ with hamiltonian path $(v_1,\\dots, v_{n-s})$, and $R$ dominates $\\{v_1, \\dots, v_{s}\\}$ and is dominated by $\\{v_{n-2s+1}, \\dots, v_{n-s}\\}$. It easy to check that $R_s(n)$ is $s$-strongly connected. However, every (non-trivial) $k$-strong tournament of $R_s(n)$ must contain at least $k$ vertices of $R$. Hence $R_s(n)$ does not have a partition $(V_1\\dots, V_p)$ of its vertex set such that the subtournament induced by $V_i$ is a non-trivial $k_i$-strong for all $1\\leq i\\leq p$.\n\nSome small examples give better lower bound. For example, the Paley tournament on 7 vertices which is 3-strong cannot be partionned into two strong subtournaments. However, there are only finitely many known such tournaments. Chen, Gould, and Li [CGL] showed that every $k$-strongly connected tournament with at least $8k$ vertices has a partition into $k$ strongly connected tournaments.\n\nThe existence of $g(2,2)$ is still open.\n\nBibliography:\n[CGL] G. Chen, R.J. Gould, and H. Li, Partitioning vertices of a tournament into independent cycles, J. combin. Theory Ser B, Vol 83, no. 2 (2001) 213-220.\n\n*[R] K.B. Reid, Three problems on tournaments, Graph Theory and Its Applications, East. and West. Ann. New York Acad. Sci. 576 (1989), 466-473.\n\nDiscussion links:\n- Paley tournament: http://en.wikipedia.org/wiki/Paley graph\n\nBibliography links:\n- Partitioning vertices of a tournament into independent cycles: http://www.sciencedirect.com/science/article/pii/S0095895601920489\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 28.\n\nAttempt notes:\nTarget:\nMake progress on \"Partitionning a tournament into k-strongly connected subtournaments.\" in Graph Theory; Directed Graphs; Tournaments, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The required connectivity threshold exists; the equal-connectivity theorem implies the supplied nonuniform statement by taking k=max_i k_i.\n\n**Verified partial progress.**\n\n- Kuhn, Osthus, and Townsend first proved f(k,t)=O(k^7 t^4).\n- Girao and Letzter improved this to the optimal-order O(kt) bound.\n\n**Full solution or refutation.**\n\nApply the theorem with t=p and k=max_i k_i; every resulting k-strong induced part is also k_i-strong, affirmatively answering the source question.\n\n**What remains.**\n\nThe existence question is answered; only constants and sharper nonuniform dependence may remain.\n\n**Sources checked.**\n\n- Daniela Kuhn, Deryk Osthus, and Timothy Townsend, Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths, Combinatorica 36 (2016), DOI 10.1007/s00493-015-3186-8. (primary): https://doi.org/10.1007/s00493-015-3186-8\n Evidence used: Proves existence of f(k,t) for partitions into t induced strongly k-connected tournaments.\n- Antonio Girao and Shoham Letzter, Partitioning a tournament into sub-tournaments of high connectivity, arXiv:2210.17371, revised 2025. (primary): https://arxiv.org/abs/2210.17371\n Evidence used: Proves the optimal-order O(kt) connectivity threshold.\n\n**Review notes.** Spelling and grammar defects in the source statement were flagged in report.md rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3280, "problem_number": "OPG-47651", "title": "Decomposing k-arc-strong tournament into k spanning strong digraphs", "statement": "Conjecture Every k-arc-strong tournament decomposes into k spanning strong digraphs.", "background": "Source: Open Problem Garden. Original node ID: 47651. URL: http://www.openproblemgarden.org/op/decomposing_k_arc_strong_tournament_into_k_spanning_strong_digraphs.\n\nSource subject path: Graph Theory > Directed Graphs > Tournaments.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposing_k_arc_strong_tournament_into_k_spanning_strong_digraphs\n- Author(s): Bang-Jensen, Joergen; Yeo, Anders\n- Subject(s): Graph Theory; Directed Graphs; Tournaments\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 15th, 2013 by fhavet\n\nProblem-page discussion:\nConjecture 8 implies Kelly's conjecture (Every regular tournament of order $n$ can be decomposed into $(n-1)/2$ Hamilton directed cycles.) which has been proved for tournaments of sufficiently large order by Kühn and Osthus [KO].\n\nBang-Jensen and Yeo [BY] gave several results supporting this conjecture. For example they proved it for $k$-arc-strong tournaments with minimum in- and out-degree at least $37k$.\n\nBibliography:\n*[BY] J. Bang-Jensen, A. Yeo, Decomposing k-arc-strong tournaments into strong spanning subdigraphs, Combinatorica 24 (2004) 331–349.\n\n[KO] Daniela Kühn and Deryk Osthus, Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments, Advances in Mathematics 237 (2013), 62-146.\n\nRelated:\nRelated problems\nArc-disjoint strongly connected spanning subdigraphs\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposing k-arc-strong tournament into k spanning strong digraphs\" in Graph Theory; Directed Graphs; Tournaments, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general k-fold arc decomposition remains open; it is known with a minimum semidegree condition and in the tournament case k=2.\n\n**Verified partial progress.**\n\n- Bang-Jensen and Yeo prove the conjecture when the minimum indegree and outdegree are each at least 37k.\n- Their characterization of 2-arc-strong semicomplete digraphs yields the result for k=2 in tournaments, since the unique semicomplete exception is not a tournament.\n- Kelly's conjecture, an important consequence for sufficiently large regular tournaments, is proved.\n\n**Full solution or refutation.**\n\nThese theorems do not cover arbitrary k-arc-strong tournaments for general k.\n\n**What remains.**\n\nRemove the minimum semidegree hypothesis for every k, or find a k-arc-strong tournament without k arc-disjoint strong spanning subdigraphs.\n\n**Sources checked.**\n\n- Jorgen Bang-Jensen and Anders Yeo, Decomposing k-arc-strong tournaments into strong spanning subdigraphs, Combinatorica 24 (2004), 331-349. (primary): https://pure.royalholloway.ac.uk/files/889922/paper_yeo_2.pdf\n Evidence used: States the conjecture and proves the minimum-semidegree case and the k=2 tournament case via the semicomplete characterization.\n- Daniela Kuhn and Deryk Osthus, Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments, Advances in Mathematics 237 (2013), 62-146, DOI 10.1016/j.aim.2012.12.020. (primary): https://doi.org/10.1016/j.aim.2012.12.020\n Evidence used: Proves the large-regular-tournament consequence but not the stronger decomposition conjecture.\n- Graph-theory open problems, Decomposing k-arc-strong tournament into k spanning strong digraphs, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/decomposing_k_arc_strong_tournament_into_k_spanning_strong_digraphs/\n Evidence used: Current specialist tracker records the conjecture as still open with the 37k semidegree theorem as the main direct partial result.\n\n**Review notes.** The intended meaning of 'decomposes' is an arc partition into arc-disjoint spanning strong subdigraphs; the source statement does not explicitly spell this out.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3281, "problem_number": "OPG-162", "title": "The Erdös-Hajnal Conjecture", "statement": "Conjecture For every fixed graph $H$, there exists a constant $\\delta(H)$, so that every graph $G$ without an induced subgraph isomorphic to $H$ contains either a clique or an independent set of size $|V(G)|^{\\delta(H)}$.", "background": "Source: Open Problem Garden. Original node ID: 162. URL: http://www.openproblemgarden.org/op/the_erdos_hajnal_conjecture.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_erdos_hajnal_conjecture\n- Author(s): Erdos, Paul; Hajnal, Andras\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Keywords: induced subgraph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 18th, 2007 by mdevos\n\nProblem-page discussion:\nThere are numerous interesting classes of graphs which are based upon forbidding one or more induced subgraphs. For instance: chordal graphs, split graphs, and claw-free graphs. Numerous other natural classes of graphs have been proved to have such characterizations, most famously perfect graphs, but also line graphs and comparability graphs. All of these classes are very well structured (far from random) and their members all either have large cliques or independent sets. On the flip side of this are random graphs. It is well known that a random graph on $n$ vertices has both clique and independence number highly concentrated around $2 \\log_2 n$. The Erdos-Hajnal conjecture suggests a fundamental separation between these two worlds in terms of independence/clique sizes.\n\nErdös and Hajnal proved that this conjecture is true for the recursive class of graphs ${\\mathcal C}$ defined as follows. The one vertex graph is in ${\\mathcal C}$, and if $G_1$ and $G_2$ lie in ${\\mathcal C}$, then the disjoint union of $G_1$ and $G_2$ lies in ${\\mathcal C}$, as does the graph obtained from the disjoint union by adding an edge between $v_1$ and $v_2$ for every $v_1 \\in V(G_1)$ and $v_2 \\in V(G_2)$. More generally, Alon, Pach, and Solymosi proved that if $F$ is a graph with $V(F) = \\{v_1,v_2,\\ldots,v_n\\}$ for which the Erdös-Hajnal conjecture holds, and $H_1,\\ldots,H_n$ are graphs for which the Erdos-Hajnal conjecture holds, then the graph obtained from $F$ by blowing up each vertex $v_i$ with a copy of $H_i$ (more precisely, starting from the disjoint union of $H_1,H_2,\\ldots,H_n$, we add all possible edges between the vertices of $V(H_i)$ and $V(H_j)$ if $ij \\in E(F)$ ) also satisfies the Erdos-Hajnal conjecture.\n\nThe Erdös-Hajnal property is known to hold for a number of small graphs (and using the above result this may be easily bootstrapped). For instance, the conjecture is known to hold when $H$ is a path of three edges, and recently M. Chudnovsky and S. Safra have announced a proof when $H$ is a bull (a triangle plus two pendant edges). However, our knowledge here is still quite limited. In particular, Lovasz has suggested the following very special case which remains open.\n\nQuestion Is the Erdös-Hajnal conjecture true when $H \\cong C_5$?\n\nBibliography:\n[APS] N. Alon, J. Pach, and J. Solymosi, Ramsey-type theorems with forbidden subgraphs, Combinatorica 21 (2001), 155-170.\n\n[EH] P. Erdös and A. Hajnal, Ramsey-type theorems, Discrete Appl. Math. 25 (1989), 37-52 MathSciNet\n\nDiscussion links:\n- chordal graphs: http://en.wikipedia.org/wiki/chordal graph\n- split graphs: http://en.wikipedia.org/wiki/split graph\n- perfect graphs: http://en.wikipedia.org/wiki/perfect graph\n- line graphs: http://en.wikipedia.org/wiki/line graph\n- comparability graphs: http://en.wikipedia.org/wiki/comparability graph\n- random graphs: http://en.wikipedia.org/wiki/random graph\n\nBibliography links:\n- Ramsey-type theorems with forbidden subgraphs: http://www.math.tau.ac.il/%7Enogaa/PDFS/aps4.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1031262\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 23.\n\nAttempt notes:\nTarget:\nMake progress on \"The Erdös-Hajnal Conjecture\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The general Erdős-Hajnal conjecture remains open. The universal subpolynomial bound has improved, and the C5 case highlighted as open in the source discussion has now been solved.\n\n**Verified partial progress.**\n\n- For every fixed H, H-free n-vertex graphs contain a clique or stable set of size exp(c_H sqrt(log n log log n)).\n- The conjectured polynomial bound is proved when H is the five-cycle C5.\n\n**Full solution or refutation.**\n\nBucić, Nguyen, Scott, and Seymour obtain the first general improvement to the classical Erdős-Hajnal bound, and Chudnovsky, Scott, Seymour, and Spirkl settle the formerly open C5 special case.\n\n**What remains.**\n\nProve a homogeneous set of size n^delta(H) for every fixed forbidden induced graph H.\n\n**Sources checked.**\n\n- Open Problem Garden, The Erdös-Hajnal Conjecture (OPG-162), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/the_erdos_hajnal_conjecture\n Evidence used: Preserves the general conjecture and records C5 as the historical highlighted special case.\n- M. Bucić, T. Nguyen, A. Scott, and P. Seymour, Induced subgraph density. I: A loglog step towards Erdős-Hajnal, International Mathematics Research Notices (2024). (primary): https://doi.org/10.1093/imrn/rnae065\n Evidence used: Proves the improved general exp(c_H sqrt(log n log log n)) bound.\n- M. Chudnovsky, A. Scott, P. Seymour, and S. Spirkl, Erdős-Hajnal for graphs with no 5-hole, Proceedings of the London Mathematical Society 126 (2023), 997-1014. (primary): https://doi.org/10.1112/plms.12504\n Evidence used: Proves the Erdős-Hajnal conjecture for H=C5.\n\n**Review notes.** The displayed general conjecture remains open, although the C5 question in the old background is now solved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3282, "problem_number": "OPG-567", "title": "What is the smallest number of disjoint spanning trees made a graph Hamiltonian", "statement": "We are given a complete simple undirected weighted graph $G_1=(V,E)$ and its first arbitrary shortest spanning tree $T_1=(V,E_1)$. We define the next graph $G_2=(V,E\\setminus E_1)$ and find on $G_2$ the second arbitrary shortest spanning tree $T_2=(V,E_2)$. We continue similarly by finding $T_3=(V,E_3)$ on $G_3=(V,E\\setminus \\cup_{i=1}^{2}E_i)$, etc. Let k be the smallest number of disjoint shortest spanning trees as defined above and let $T^{k}=(V,\\cup_{i=1}^{k}E_i)$ be the graph obtained as union of all $k$ disjoint trees.\n\nQuestion 1. What is the smallest number of disjoint spanning trees creates a graph $T^{k}$ containing a Hamiltonian path.\n\nQuestion 2. What is the smallest number of disjoint spanning trees creates a graph $T^{k}$ containing a shortest Hamiltonian path?\n\nQuestions 3 and 4. Replace in questions 1 and 2 a shortest spanning tree by a 1-tree. What is the smallest number of disjoint 1-trees creates a Hamiltonian graph? What is the smallest number of disjoint 1-trees creates a graph containing a shortest Hamiltonian cycle?", "background": "Source: Open Problem Garden. Original node ID: 567. URL: http://www.openproblemgarden.org/op/what_is_the_smallest_number_of_disjoint_spanning_trees_made_a_graph_hamiltonian.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/what_is_the_smallest_number_of_disjoint_spanning_trees_made_a_graph_hamiltonian\n- Author(s): Goldengorin\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Keywords: 1-trees; cycle; Hamitonian path; spanning trees\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 10th, 2007 by boris\n\nProblem-page discussion:\nThese questions are induced by the following paper Chrobak and Poljak. On common edges in optimal solutions to travelling salesman and other optimization problems, Discrete Applied Mathematics 20 (1988) 101-111.\n\nBibliography:\nM. Chrobak and S. Poljak. On common edges in optimal solutions to travelling salesman and other optimization problems, Discrete Applied Mathematics 20 (1988) 101-111.\n\nComments:\n- September 10th, 2007 | mdevos | Is this statement correct?: Unless I am mistaken, it appears that there does not exist any $k$ for which any of the above problems has a positive solution. For instance, let $G$ be the complete graph with vertex set $V = \\{1,\\ldots,6k\\}$, and define $C$ to be the edge-cut of $G$ consisting of all edges between $\\{1,2,\\ldots,2k\\}$ and $\\{2k+1,2k+2,\\ldots,6k\\}$. Now define a weighting of $G$ by assigning each edge in $C$ weight 1 and every other edge weight 2. So, the subgraph $(V,C)$ (consisting of all edges of weight 1) is isomorphic to $K_{2k,4k}$- and since $K_{2k,4k}$ has $k$ edge-disjoint spanning trees (by the Nash-Williams theorem, say), our procedure may well choose trees $T_1,T_2,\\ldots,T_k$ so that all of the edges in all of these graphs are in $C$. But then $T^k$ will not have a Hamiltonian path since it is a subgraph of $(V,C) \\cong K_{2k,4k}$.\n\nOf course, we may modify the edge weights here so that the procedure is forced to choose $T_1,\\ldots,T_k$ so that all of these trees have their edges in $C$.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"What is the smallest number of disjoint spanning trees made a graph Hamiltonian\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The source has algorithmic and quantifier ambiguities, so no precise current status was assigned.\n\n**Verified partial progress.**\n\n- The source preserves the intended construction and ambiguity.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nClarify the exact algorithm and quantifiers before a definitive status search.\n\n**Sources checked.**\n\n- Open Problem Garden, Disjoint shortest spanning trees, node 567. (maintained_tracker): http://www.openproblemgarden.org/op/disjoint_shortest_spanning_trees\n Evidence used: Preserves the original ambiguous source formulation.\n\n**Review notes.** Statement unchanged; formulation ambiguity flagged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3283, "problem_number": "OPG-37305", "title": "Extremal problem on the number of tree endomorphism", "statement": "Conjecture An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the $n$ vertices' trees, the star with $n$ vertices has the most endomorphisms, while the path with $n$ vertices has the least endomorphisms.", "background": "Source: Open Problem Garden. Original node ID: 37305. URL: http://www.openproblemgarden.org/op/extremal_problem_on_the_number_of_tree_endomorphism.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/extremal_problem_on_the_number_of_tree_endomorphism\n- Author(s): Zhicong Lin\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 1st, 2011 by shudeshijie\n\nBibliography:\n[BT] Bela Bollobas and Mykhaylo Tyomkyn, Walks and paths in trees, http://arxiv.org/abs/1002.2768.\n\nComments:\n- November 25th, 2020 | Anonymous | it is not open: This problem is not open. Look at this: https://mathscinet.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=3284058&loc=fromrevtext\n- March 23rd, 2012 | Anonymous | the upper bound is proved: the upper bound is proved recently.\n- March 20th, 2011 | leshabirukov | counterexample: Asymmetric tree (http://en.wikipedia.org/wiki/Asymmetric_graph, http://upload.wikimedia.org/wikipedia/commons/a/ad/Asymmetric_tree.svg) has single, trivial endomorphism.\n\nUpdate: Sorry, I have confused endomorphism with automorphism.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Extremal problem on the number of tree endomorphism\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Csikvári and Lin prove exactly that the path minimizes and the star maximizes the number of endomorphisms among trees of a fixed order.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nFor every m-vertex tree T_m, |End(P_m)| <= |End(T_m)| <= |End(S_m)|.\n\n**What remains.**\n\nThe source conjecture is answered; related work could characterize equality or analogous extremizers for restricted tree classes.\n\n**Sources checked.**\n\n- P. Csikvári and Z. Lin, Graph Homomorphisms between Trees, Electronic Journal of Combinatorics 21(4) (2014), P4.9. (primary): https://doi.org/10.37236/4096\n Evidence used: The published abstract states that, among all trees with a fixed number of vertices, the path has the fewest endomorphisms and the star has the most; the paper proves the displayed two-sided inequality.\n\n**Review notes.** The source's grammatical phrase 'n vertices' trees' is preserved in report.md and not silently edited.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3284, "problem_number": "OPG-46738", "title": "Complexity of the H-factor problem.", "statement": "An $H$-factor in a graph $G$ is a set of vertex-disjoint copies of $H$ covering all vertices of $G$.\n\nProblem Let $c$ be a fixed positive real number and $H$ a fixed graph. Is it NP-hard to determine whether a graph $G$ on $n$ vertices and minimum degree $cn$ contains and $H$-factor?", "background": "Source: Open Problem Garden. Original node ID: 46738. URL: http://www.openproblemgarden.org/op/complexity_of_the_h_factor_problem.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/complexity_of_the_h_factor_problem\n- Author(s): Kühn, Daniella; Osthus, Deryk\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 5th, 2013 by fhavet\n\nProblem-page discussion:\nThe answer is positive for cliques and a few other graphs [KO06].\n\nIf we remove the minimum degree condition, the problem is NP-complete if and only if $H$ has a component which contains at least 3 vertices, as shown by Hell and Kirkpatrick [HK].\n\nBibliography:\n[HK] P. Hell and D.G. Kirkpatrick, On the complexity of general graph factor problems, SIAM J. Computing 12 (1983), 601-609.\n\n[KO06] D. Kühn and D. Osthus, Critical chromatic number and the complexity of perfect packings in graphs, Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2006.\n\n*[KO09] D. Kühn and D. Osthus, The minimum degree threshold for perfect graph packings, Combinatorica 29 (2009), 65-107.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Complexity of the H-factor problem.\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Dense H-factor decision has both polynomial-time and NP-complete regimes and sharp transitions for broad graph families, but no complete all-H critical-threshold dichotomy is verified.\n\n**Verified partial progress.**\n\n- For every fixed H, sufficiently strong minimum-degree conditions give polynomial-time decidability.\n- For many H, lowering the critical threshold makes the problem NP-complete, yielding a sharp dichotomy away from or around that threshold.\n\n**Full solution or refutation.**\n\nThe intended classification question has strong partial answers. A uniform positive NP-hardness reading of the literal prompt is false because easy factors and vacuous degree promises exist.\n\n**What remains.**\n\nState the nontrivial parameter range precisely and complete the P-versus-NP-hard classification at critical thresholds for every fixed H.\n\n**Sources checked.**\n\n- J. Han and A. Treglown, The complexity of perfect matchings and packings in dense hypergraphs, arXiv:1609.06147; Journal of Combinatorial Theory B 141 (2020). (primary): https://arxiv.org/abs/1609.06147\n Evidence used: Gives a polynomial-time minimum-degree regime for every graph H and says the threshold is best possible in many cases.\n- Graph-theory open problems, Complexity of the H-factor problem (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/complexity_of_the_h_factor_problem/\n Evidence used: Summarizes polynomial and hardness sides and reports that the complete all-H critical-threshold dichotomy remains incomplete.\n\n**Review notes.** The typo 'contains and H-factor' should read 'contains an H-factor'. More materially, H=K2 is polynomial and c at least one makes the simple-graph promise empty, so a uniform yes/no reading is invalid. Source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3285, "problem_number": "OPG-46824", "title": "Odd-cycle transversal in triangle-free graphs", "statement": "Conjecture If $G$ is a simple triangle-free graph, then there is a set of at most $n^2/25$ edges whose deletion destroys every odd cycle.", "background": "Source: Open Problem Garden. Original node ID: 46824. URL: http://www.openproblemgarden.org/op/odd_cycle_transversal_in_triangle_free_graphs.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/odd_cycle_transversal_in_triangle_free_graphs\n- Author(s): Erdos, Paul; Faudree, Ralph; Pach, János; Spencer, Joel\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 6th, 2013 by fhavet\n\nBibliography:\n*[EFPS] P. Erdös, R. Faudree, J. Pach and J. Spencer, How to make a graph bipartite. J. Combin. Theory Ser. B 45 (1988), 86--98.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Odd-cycle transversal in triangle-free graphs\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The n^2/25 odd-cycle edge-transversal conjecture for triangle-free graphs remains open, with an improved n^2/23.5 universal bound and proofs in two edge-density ranges.\n\n**Verified partial progress.**\n\n- The previous n^2/18 universal deletion bound was improved to n^2/23.5.\n- The conjectured n^2/25 bound is proved for sufficiently sparse and sufficiently dense triangle-free graphs in explicit density ranges.\n\n**Full solution or refutation.**\n\nCurrent max-cut methods settle the conjecture outside an intermediate edge-density interval but do not prove the sharp constant for every triangle-free graph.\n\n**What remains.**\n\nProve the n^2/25 deletion bound throughout the remaining intermediate density interval.\n\n**Sources checked.**\n\n- J. Balogh, F. C. Clemen, and B. Lidicky, Max Cuts in Triangle-free Graphs, arXiv:2103.14179. (primary): https://arxiv.org/abs/2103.14179\n Evidence used: Improves the general bound to n^2/23.5 and proves the conjectured constant in two density ranges.\n- Graph-theory open problems, Odd cycle transversal in triangle-free graphs (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/odd_cycle_transversal_in_triangle_free_graphs/\n Evidence used: Lists the exact conjecture as open and summarizes the density-restricted advances.\n\n**Review notes.** The source does not explicitly define n as |V(G)|, but context makes that intended meaning clear. Density thresholds are quoted using the primary paper's normalization; source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3286, "problem_number": "OPG-46837", "title": "Triangle-packing vs triangle edge-transversal.", "statement": "Conjecture If $G$ has at most $k$ edge-disjoint triangles, then there is a set of $2k$ edges whose deletion destroys every triangle.", "background": "Source: Open Problem Garden. Original node ID: 46837. URL: http://www.openproblemgarden.org/op/triangle_packing_vs_triangle_edge_transversal.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/triangle_packing_vs_triangle_edge_transversal\n- Author(s): Tuza, Zsolt\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 6th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture may be rephrased in terms of packing and edge-transversal. A triangle packing is a set of pairwise edge-disjoint triangles. A triangle edge-tranversal is a set of edges meeting all triangles. Denote the maximum size of a triangle packing in $G$ by $\\nu(G)$ and the minimum size of a triangle edge-transversal of $G$ by $\\tau(G)$. Clearly $\\nu(G) \\leq \\tau(G)$. The conjecture translates in $\\tau(G)\\leq 2\\nu(G)$.\n\nThis conjecture, if true, is best possible as can be seen by taking, say $G=K_4$ or $G=K_5$. Trivially, $\\tau(G)\\leq 3\\nu(G)$, since the set of edges of a maximum triangle packing is a triangle edge-transversal. Haxell [H] proved that $\\tau(G) \\leq (3-\\frac{3}{23})\\nu(G)$ edges whose deletion destroys every triangle.\n\nAs usual, one can define fractional packing and fractional transversal. Let ${\\cal T}$ be the set of triangles of $G$. A fractional triangle packing is a function $f:{\\cal T}\\rightarrow \\mathbb{R}^+$ such that $\\sum_{T\\ni e} \\leq 1$ for every edge $e$. A fractional triangle edge-transversal is a function $g:E\\rightarrow \\mathbb{R}^+$ such that $\\sum_{e\\in T} g(e)\\geq 1$ for every triangle $T\\in {\\cal T}$. We denote by $\\nu^*(G)$ the maximum of $\\sum_{T\\in {\\cal T}} f(T)$ over all fractional triangle packing and by $\\tau^*(G)$ the minimum of $\\sum_{e\\in E(G)} g(e)$ over all fractional edge-transversals. By duality of linear programming $\\tau^*(G) = \\nu^*(G)$. Krivelevich [K] proved two fractional versions of the conjecture:\n\n$\\tau(G) \\leq 2\\nu^*(G)$ and $\\tau^*(G)\\leq 2\\nu(G)$.\n\nBibliography:\n[H] P.Haxell, Packing and covering triangles in graphs, Discrete Mathematics 195 (1999), no. 1–3, 251–254.\n\n[K] M. Krivelevich, On a conjecture of Tuza about packing and covering of triangles Discrete Mathematics 142 (1995), 281-286.\n\n*[T] Z. Tuza, A conjecture on triangles of graphs. Graphs Combin. 6 (1990), 373-380.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 25.\n\nAttempt notes:\nTarget:\nMake progress on \"Triangle-packing vs triangle edge-transversal.\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Tuza's integral packing-covering inequality remains open; fractional versions and many special cases are known.\n\n**Verified partial progress.**\n\n- Krivelevich proved the two fractional factor-two inequalities.\n- The current maintained record continues to list the integral conjecture as unresolved.\n\n**Full solution or refutation.**\n\nNo proof or counterexample to tau(G) <= 2 nu(G) was verified.\n\n**What remains.**\n\nProve the universal factor-two integral bound or construct a counterexample.\n\n**Sources checked.**\n\n- Graph-theory open problems, Triangle packing versus triangle edge transversal (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/triangle_packing_vs_triangle_edge_transversal/\n Evidence used: Records the conjecture as open and summarizes the fractional progress.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3287, "problem_number": "OPG-48232", "title": "The Bollobás-Eldridge-Catlin Conjecture on graph packing", "statement": "Conjecture (BEC-conjecture) If $G_1$ and $G_2$ are $n$-vertex graphs and $(\\Delta(G_1) + 1) (\\Delta(G_2) + 1) < n + 1$, then $G_1$ and $G_2$ pack.", "background": "Source: Open Problem Garden. Original node ID: 48232. URL: http://www.openproblemgarden.org/op/the_bollobas_eldridge_catlin_conjecture_on_graph_packing.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_bollobas_eldridge_catlin_conjecture_on_graph_packing\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Keywords: graph packing\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 23rd, 2013 by asp\n\nProblem-page discussion:\nA pair of $n$-vertex graphs $G_1$ and $G_2$ are said to ${\\it pack}$ if they are edge-disjoint subgraphs of the complete graph on $n$ vertices.\n\nThe main conjecture in the area of graph packing is the abovementioned conjecture by Bollobás, Eldridge [BE] and Catlin [C].\n\nIn support of the BEC-conjecture, Sauer and Spencer [SS] proved that if $G_1$ and $G_2$ are $n$-vertex graphs and $2 \\Delta(G_1) \\Delta(G_2) < n$ then $G_1$ and $G_2$ pack.\n\nGiven a graph $G$, $L(G)$ denotes the line graph of $G$ and $\\Theta(G)$ denotes the number $\\Delta(L(G)) + 2$. Kostochka and Yu [KY1] proved that if $G_1$ and $G_2$ are two $n$-vertex graphs with $\\Theta(G_1) \\Delta(G_2) \\leq n$, then $G_1$ and $G_2$ pack with the following exceptions: (1) $G_1$ is a perfect matching and $G_2$ is either $K_{n/2,n/2}$ with $n/2$ odd or contains $K_{n/2 + 1}$ or (2) $G_2$ is a perfect matching and $G_1$ is $K_{r,n-r}$ with $r$ odd or contains $K_{n/2 + 1}$.\n\nKostachka and Yu [KY2] conjectured that if $G_1$ and $G_2$ are $n$-vertex graphs with $\\Theta(G_1) \\Theta(G_2) < 2n$ then $G_1$ and $G_2$ pack.\n\nBibliography:\n*[BE] B. Bollabás and S. E. Eldridge, Maximal matchings in graphs with given maximal and minimal degrees, Congr. Numer. XV (1976), 165--168.\n\n*[C] P. A. Catlin, Embedding subgraphs and coloring graphs under extremal degree conditions, Ph. D. Thesis, Ohio State Univ., Columbus (1976).\n\n[KY1] A. V. Kostochka and G. Yu, An Ore-type analogue of the Sauer-Spencer Theorem, Graphs Combin. 23 (2007), 419--424.\n\n[KY2] A. V. Kostochka and G. Yu, An Ore-type graph packing problems, Combin. Probab. Comput. 16 (2007), 167--169.\n\n[SS] N. Sauer and J. Spencer, Edge disjoint placement of graphs, J. Combin. Theory Ser. B 25 (1978), 295--302.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 17.\n\nAttempt notes:\nTarget:\nMake progress on \"The Bollobás-Eldridge-Catlin Conjecture on graph packing\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Bollobas-Eldridge-Catlin packing conjecture remains widely open, with strong theorems for bounded-codegree, high-even-girth, and bipartite target classes.\n\n**Verified partial progress.**\n\n- Cames van Batenburg and Kang prove the BEC conclusion for K_{2,t}-free graphs under a degree-ratio hypothesis and in a high-degree even-girth-at-least-10 regime.\n- Allen, Bottcher, Skokan, and Sudakov improve beyond the BEC minimum-degree threshold for bipartite target graphs when Delta is suitably bounded relative to n.\n\n**Full solution or refutation.**\n\nNo theorem covering arbitrary graph pairs satisfying the supplied degree inequality was verified.\n\n**What remains.**\n\nProve the packing conclusion for all graph pairs at the BEC threshold, including the unresolved boundary regimes, or construct a counterexample.\n\n**Sources checked.**\n\n- Wouter Cames van Batenburg and Ross J. Kang, Packing Graphs of Bounded Codegree, Combinatorics, Probability and Computing 27 (2018), DOI 10.1017/S0963548318000032. (primary): https://doi.org/10.1017/S0963548318000032\n Evidence used: Proves the conjectured packing conclusion for an important bounded-codegree class under explicit degree hypotheses.\n- Peter Allen, Julia Bottcher, Jozef Skokan, and Benny Sudakov, Breaking the Bollobas-Eldridge-Catlin Barrier for Bipartite Graphs, arXiv:2607.17808v2 (2026). (primary): https://arxiv.org/abs/2607.17808\n Evidence used: Explicitly states that BEC remains widely open and proves a stronger embedding threshold for bipartite targets in a specified degree range.\n- Graph-theory open problems, The Bollobas-Eldridge-Catlin Conjecture on graph packing, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/the_bollobas_eldridge_catlin_conjecture_on_graph_packing/\n Evidence used: Current specialist tracker records the conjecture as open and details verified class-specific advances.\n\n**Review notes.** The supplied strict inequality was preserved. The sometimes-seen <= n+1 boundary is stronger and was not treated as equivalent.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3288, "problem_number": "OPG-60013", "title": "Weak saturation of the cube in the clique", "statement": "Problem\n\nDetermine $\\text{wsat}(K_n,Q_3)$.", "background": "Source: Open Problem Garden. Original node ID: 60013. URL: http://www.openproblemgarden.org/op/weak_saturation_of_the_cube_in_the_clique.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/weak_saturation_of_the_cube_in_the_clique\n- Author(s): Morrison, Natasha; Noel, Jonathan A.\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Keywords: bootstrap percolation; hypercube; Weak saturation\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: April 6th, 2016 by Jon Noel\n\nProblem-page discussion:\nGiven graphs $G$ and $H$, let $\\text{wsat}(G,H)$ denote the minimum number of edges in a subgraph $F$ of $G$ such that the edges of $E(G)\\setminus E(F)$ can be added to $F$, one edge at a time, so that each edge completes a copy of $H$ when it is added.\n\nOf course, if one can solve the problem above, then a natural next step is to determine $\\text{wsat}(K_n,Q_m)$ for all $n$ and $m$.\n\nMorrison, Noel and Scott [MNS] solved the related problem of determining $\\text{wsat}(Q_d,Q_m)$ for all $d$ and $m$.\n\nBibliography:\n[MNS] N. Morrison, J. A. Noel, A. Scott. Saturation in the hypercube and bootstrap percolation. To appear in Combin. Probab. Comput.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Weak saturation of the cube in the clique\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No exact formula for wsat(K_n,Q3) was verified; current checked results imply n+2<=wsat(K_n,Q3)<=2n+O(1) for n>=7, with the lower bound coming from a preliminary 2025 project on proper cube subgraphs.\n\n**Verified partial progress.**\n\n- IAS-hosted PCMI 2025 slides report wsat(K_n,H)=n+2 for the union H of three faces of Q3 when n>=7.\n- Since H is a subgraph of Q3, weak-saturation monotonicity gives wsat(K_n,Q3)>=n+2.\n- The general Bulavka--Tancer--Tyomkyn construction specializes to wsat(K_n,Q3)<=2n+O(1).\n\n**Full solution or refutation.**\n\nOnly linear bounds and exact values for proper subgraphs were verified; the exact cube value remains open in the checked literature.\n\n**What remains.**\n\nDetermine the leading coefficient and additive term, then prove an exact formula for all relevant n.\n\n**Sources checked.**\n\n- Tanupat Trakulthongchai, Derek Xu, and Kelin Zhu, Weak saturation number of the 3-cube in the complete graph, PCMI 2025 project slides hosted by the Institute for Advanced Study. (primary): https://www.ias.edu/sites/default/files/Trakulthongchai_Xu_Zhu.pdf\n Evidence used: Reports exact weak-saturation numbers for one-, two-, and three-face proper subgraphs and explains the limitation of this approach for Q3.\n- Denys Bulavka, Martin Tancer, and Mykhaylo Tyomkyn, Weak Saturation of Multipartite Hypergraphs, Combinatorica 43 (2023), 1081--1102. (primary): https://doi.org/10.1007/s00493-023-00049-0\n Evidence used: Lemma 5.1 gives the general clique-host upper bound that specializes to 2n+O(1) for Q3.\n- Open Problem Garden, Weak saturation of the cube in the clique (checked 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/weak_saturation_of_the_cube_in_the_clique\n Evidence used: Still asks for wsat(K_n,Q3) and contains no posted exact answer.\n\n**Review notes.** The most problem-specific recent progress located is an institution-hosted student project rather than a refereed publication.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3289, "problem_number": "OPG-60042", "title": "Multicolour Erdős--Hajnal Conjecture", "statement": "Conjecture For every fixed $k\\geq2$ and fixed colouring $\\chi$ of $E(K_k)$ with $m$ colours, there exists $\\varepsilon>0$ such that every colouring of the edges of $K_n$ contains either $k$ vertices whose edges are coloured according to $\\chi$ or $n^\\varepsilon$ vertices whose edges are coloured with at most $m-1$ colours.", "background": "Source: Open Problem Garden. Original node ID: 60042. URL: http://www.openproblemgarden.org/op/multicolour_erdos_hajnal_conjecture.\n\nSource subject path: Graph Theory > Extremal Graph Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/multicolour_erdos_hajnal_conjecture\n- Author(s): Erdos, Paul; Hajnal, Andras\n- Subject(s): Graph Theory; Extremal Graph Theory\n- Keywords: ramsey theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 10th, 2019 by Jon Noel\n\nProblem-page discussion:\nSee [FGP].\n\nBibliography:\n[FGP] Jacob Fox, Andrey Grinshpun and János Pach: The Erdős–Hajnal conjecture for rainbow triangles, J. Combin. Theory, Series B. 111 (2016), 75--125.\n\nRelated:\nRelated problems\nThe Erdös-Hajnal Conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Multicolour Erdős--Hajnal Conjecture\" in Graph Theory; Extremal Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The multicolour Erdos--Hajnal conjecture remains open even for two colours, while particular pattern families and reductions are known.\n\n**Verified partial progress.**\n\n- Axenovich--Weber reduce the general problem to host colour counts equal to, or one more than, those of the forbidden pattern.\n- Some three-colour small-pattern families are settled.\n\n**Full solution or refutation.**\n\nNo proof covers every fixed edge-coloured pattern.\n\n**What remains.**\n\nEstablish the polynomial-size colour-omitting set for arbitrary fixed patterns.\n\n**Sources checked.**\n\n- M. Axenovich and L. Weber, A note on the multicolour version of the Erdos--Hajnal conjecture, Discrete Mathematics 348 (2025), 114437; arXiv:2311.03249. (primary): https://arxiv.org/abs/2311.03249\n Evidence used: The abstract says the conjecture is open even for two colours and proves the reduction.\n- Graph-theory open problems, Multicolour Erdos--Hajnal Conjecture (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/multicolour_erdos_hajnal_conjecture/\n Evidence used: Lists recent special-pattern advances.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3290, "problem_number": "OPG-37079", "title": "A gold-grabbing game", "statement": "Setup Fix a tree $T$ and for every vertex $v \\in V(T)$ a non-negative integer $g(v)$ which we think of as the amount of gold at $v$.\n\n2-Player game Players alternate turns. On each turn, a player chooses a leaf vertex $v$ of the tree, takes the gold at this vertex, and then deletes $v$. The game ends when the tree is empty, and the winner is the player who has accumulated the most gold.\n\nProblem Find optimal strategies for the players.", "background": "Source: Open Problem Garden. Original node ID: 37079. URL: http://www.openproblemgarden.org/op/a_gold_grabbing_game.\n\nSource subject path: Graph Theory > Graph Algorithms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_gold_grabbing_game\n- Author(s): Rosenfeld, Moshe\n- Subject(s): Graph Theory; Graph Algorithms\n- Keywords: game; tree\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 2nd, 2009 by mdevos\n\nProblem-page discussion:\nIn the special case when $T$ is a path of even length, the first player can ensure that she chooses either all of the even vertices, or all of the odd vertices. Thus, player 1 should never finish with less than player 2, and whenever the total gold on the odd vertices and the total gold on the even vertices are not equal, there is a winning strategy for player 1.\n\nComments:\n- October 4th, 2009 | porton | Not an open problem in the strict sense: There exists an obvious algorithm which just enumerates all variants.\n\nThe problem seems to mean to find a more efficient algorithm. This is not a strict formulation because it is not strictly defined what is \"more efficient\".\n\nI suggest to rip this problem, such as to put it into Second tier problems.\n\n--\n\nVictor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"A gold-grabbing game\" in Graph Theory; Graph Algorithms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The prompt is not a well-posed open target: finite backward induction yields optimal play for each instance, while no efficiency bound or requested structural characterization is specified.\n\n**Verified partial progress.**\n\n- Seacrest and Seacrest prove that on every nonnegatively weighted tree of even order the first player can secure at least half of the total weight, resolving the earlier Micek-Walczak guarantee conjecture.\n- The result gives a universal winning guarantee but not a complete efficient description of optimal strategies for arbitrary weighted trees.\n\n**Full solution or refutation.**\n\nLiteral existence and computation by exhaustive game-tree minimax are immediate for finite instances; the intended stronger algorithmic or structural problem cannot be assigned a unique solved/open status from the wording.\n\n**What remains.**\n\nSpecify whether the target is polynomial-time value computation, a compact optimal-strategy characterization, an approximation guarantee, or another precise complexity objective.\n\n**Sources checked.**\n\n- D. E. Seacrest and T. Seacrest, Grabbing the gold, Discrete Mathematics 312 (2012), 1804-1806. (primary): https://doi.org/10.1016/j.disc.2012.01.013\n Evidence used: Proves that the first player gets at least half of the gold on every weighted even-order tree.\n- Open Problem Garden, A gold-grabbing game (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/a_gold_grabbing_game\n Evidence used: Preserves the open-ended wording and the contemporary comment that exhaustive enumeration is obvious but 'more efficient' is undefined.\n\n**Review notes.** Formulation defect flagged; the source statement was not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3291, "problem_number": "OPG-47646", "title": "PTAS for feedback arc set in tournaments", "statement": "Question Is there a polynomial time approximation scheme for the feedback arc set problem for the class of tournaments?", "background": "Source: Open Problem Garden. Original node ID: 47646. URL: http://www.openproblemgarden.org/op/ptas_for_feedback_arc_set_in_tournaments.\n\nSource subject path: Graph Theory > Graph Algorithms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/ptas_for_feedback_arc_set_in_tournaments\n- Author(s): Ailon, Nir; Alon, Noga\n- Subject(s): Graph Theory; Graph Algorithms\n- Keywords: feedback arc set; PTAS; tournament\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 15th, 2013 by fhavet\n\nProblem-page discussion:\nA tournament is an orientation of a complete graph. A feedback arc set is a set of arcs in a digraph whose removal leave the digraph acyclic. The feedback arc set problem consists in finding a feedback arc set of minimum size. A polynomial time approximation scheme is an algorithm which takes an instance of an optimization problem and a parameter $\\epsilon > 0$ and, in polynomial time, produces a solution that is within a factor $1+\\epsilon$ of being optimal.\n\nThe feedback arc set problem has been proved NP-hard. See [ACM, A, CTY, C]. It was shown in [RS] that the feedback arc set problem is fixed parameter tractable for tournaments.\n\nBibliography:\n*[AA] N. Ailon, N. Alon, link, Inform. and Comput. 205 (8) (2007) 1117–1129.\n\n[ACM] N. Alion, M. Charikar, A. Newman, Aggregating inconsistent information: Ranking and clustering, in: Proceedings of the 37th Symposium on the Theory of Computing, STOC, ACM Press, 2005, pp. 684–693.\n\n[A] N. Alon, Ranking tournaments, SIAM J. Discrete Math. 20 (2006) 137–142.\n\n[CTY] P. Charbit, P. Thomassé, A. Yeo, The minimum Feedback arc set problem is NP-hard for tournaments, Combin. Probab. Comput. 16 (1) (2007) 1–4.\n\n[C] V. Conitzer, Computing Slater rankings using similarities among candidates, in: Proceedings, The Twenty-First National Conference on Artificial Intelligence and the Eighteenth Innovative Applications of Artificial Intelligence Conference, July 16–20, AAAI Press, Boston, Massachusetts, USA, 2006.\n\n[RS] V. Raman, S. Saurabh, Parameterized complexity of directed feedback arc set problems in tournaments, in: Algorithms and Data Structures, in: Lecture Notes in Computer Science, vol. 2748, Springer, Berlin, 2003, pp. 484–492.\n\nDiscussion links:\n- tournament: http://en.wikipedia.org/wiki/Tournament (graph theory)\n- feedback arc set: http://en.wikipedia.org/wiki/Feedback arc set\n- polynomial time approximation scheme: http://en.wikipedia.org/wiki/Polynomial-time approximation scheme\n- fixed parameter tractable: http://en.wikipedia.org/wiki/FPT\n\nBibliography links:\n- link: http://www.openproblemgarden.org/Hardness of fully dense problems]{http:/www.tau.ac.il/%7Enogaa/PDFS/dense8.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"PTAS for feedback arc set in tournaments\" in Graph Theory; Graph Algorithms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** A polynomial-time approximation scheme exists for minimum feedback arc set in tournaments, including a weighted generalization.\n\n**Verified partial progress.**\n\n- Later work gives a semi-streaming PTAS with a small number of passes.\n\n**Full solution or refutation.**\n\nKenyon-Mathieu and Schudy's STOC 2007 algorithm directly answers the unweighted source question affirmatively.\n\n**What remains.**\n\nThe existence question is answered; ongoing work concerns efficiency, memory, and dependence on epsilon.\n\n**Sources checked.**\n\n- Claire Kenyon-Mathieu and Warren Schudy, How to rank with few errors, Proceedings of STOC 2007, pp. 95-103, DOI 10.1145/1250790.1250806. (primary): https://doi.org/10.1145/1250790.1250806\n Evidence used: Presents a PTAS for minimum feedback arc set in tournaments and a weighted generalization.\n- Anubhav Baweja, Justin Jia, and David P. Woodruff, An Efficient Semi-Streaming PTAS for Tournament Feedback ArcSet with Few Passes, ITCS 2022, LIPIcs 215, Article 16, DOI 10.4230/LIPIcs.ITCS.2022.16. (primary): https://doi.org/10.4230/LIPIcs.ITCS.2022.16\n Evidence used: Confirms the classical PTAS and extends it to a semi-streaming model.\n\n**Review notes.** The solution predates the 2013 OPG posting.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3292, "problem_number": "OPG-165", "title": "Ryser's conjecture", "statement": "Conjecture Let $H$ be an $r$-uniform $r$-partite hypergraph. If $\\nu$ is the maximum number of pairwise disjoint edges in $H$, and $\\tau$ is the size of the smallest set of vertices which meets every edge, then $\\tau \\le (r-1) \\nu$.", "background": "Source: Open Problem Garden. Original node ID: 165. URL: http://www.openproblemgarden.org/op/rysers_conjecture.\n\nSource subject path: Graph Theory > Hypergraphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/rysers_conjecture\n- Author(s): Ryser, Herbert J.\n- Subject(s): Graph Theory; Hypergraphs\n- Keywords: hypergraph; matching; packing\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 19th, 2007 by mdevos\n\nProblem-page discussion:\nDefinitions: A (vertex) cover is a set of vertices which meets (has nonempty intersection with) every edge, and we let $\\tau(H)$ denote the size of the smallest vertex cover of $H$. A matching is a collection of pairwise disjoint edges, and we let $\\nu(H)$ denote the size of the largest matching in $H$. When the hypergraph is clear from context, we just write $\\tau$ or $\\nu$.\n\nIt is immediate that $\\nu \\le \\tau$, since every cover must contain at least one point from each edge in any matching. For $r$-uniform hypergraphs, $\\tau \\le r \\nu$, since the union of the edges from any maximal matching is a set of at most $r \\nu$ vertices that which meets every edge. Ryser's conjecture is that this second bound can be improved if $H$ is $r$-uniform and $r$-partite (the vertices may be partitioned into $r$ sets $V_1,V_2,\\ldots,V_r$ so that every edge contains exactly one element of each $V_i$ ).\n\nIn the special case when $r=2$ our trivial inequality yields $\\nu \\le \\tau$ and the conjecture implies $\\tau \\le \\nu$, so we should have $\\nu = \\tau$. In fact this is true, it is König's theorem on bipartite graphs [K]. Indeed, Ryser's conjecture is probably easiest to view as a high dimensional generalization of this early result of König. Recently, Aharoni [A] has applied the \"Hall's theorem for hypergraphs\" result of Aharoni and Haxell [AH] to prove this conjecture for $r=3$. However the case $r=4$ is still wide open.\n\nSome other interesting work on this problem concerns fractional covers and fractional matchings. A fractional cover of $H = (V,E)$ is a weighting $a: V \\rightarrow {\\mathbb R}^+$ so that $\\sum_{x \\in S} a(x) \\ge 1$ for every $S \\in E$, and the weight of this cover is $\\sum_{x \\in V} a(x)$. The fractional cover number, denoted $\\tau^*$ is the infimum of the set of weights of covers. Similarly, a fractional matching is an edge-weighting $b: E \\rightarrow {\\mathbb R}^+$ so that $\\sum_{S \\ni x} b(S) \\le 1$ for every $x \\in V$, and the weight of this matching is $\\sum_{S \\in E} b(S)$. The fractional matching number, denoted $\\nu^*$ is the supremum of the set of weights of fractional matchings. Fractional covers and matchings are the usual fractional relaxations, and by LP-duality, they satisfy $\\nu^* = \\tau^*$ for every hypergraph. For $r$-regular $r$-partite hypergraphs, Füredi [F] has proved that $\\tau^* \\le (r-1)\\nu$ and Lovasz [L] has shown $\\tau \\le \\frac{1}{2} r \\nu^*$.\n\nBibliography:\n[A] R. Aharoni, Ryser's conjecture for tripartite 3-graphs. Combinatorica 21 (2001), no. 1, 1--4. MathSciNet\n\n[AH] R. Aharoni and P. Haxell, Hall's theorem for hypergraphs. J. Graph Theory 35 (2000), no. 2, 83--88. MathSciNet\n\n[F] Z. Füredi, Maximum degree and fractional matchings in uniform hypergraphs, Combinatorica 1 (1981), 155--162. MathSciNet\n\n[K] D. König, Theorie der endlichen und unendlichen Graphen, Leipzig, 1936.\n\n[L] L. Lovász, On minimax theorems of combinatorics, Ph.D thesis, Matemathikai Lapok 26 (1975), 209--264 (in Hungarian). MathSciNet\n\nSource links:\n- uniform: http://en.wikipedia.org/wiki/hypergraph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1805710\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1781189\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0625548\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0510823\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 31.\n\nAttempt notes:\nTarget:\nMake progress on \"Ryser's conjecture\" in Graph Theory; Hypergraphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Ryser's conjecture is proved for r=2 and r=3 but remains open in general for every r at least 4, with improved coefficients and exact structural special cases.\n\n**Verified partial progress.**\n\n- For r=4 and r=5 there is an epsilon greater than zero such that tau(H) is at most (r-epsilon)nu(H).\n- The conjecture holds for intersecting hypergraphs through r=5 and for several linear intersecting cases; the extremal r=3 case has been structurally characterized.\n\n**Full solution or refutation.**\n\nKnown results improve the trivial coefficient r in the first unresolved uniformities and settle important intersecting and linear subclasses, but do not achieve r-1 for unrestricted r-partite r-graphs.\n\n**What remains.**\n\nProve tau(H) <= (r-1)nu(H) for unrestricted r-partite r-uniform hypergraphs for all r at least 4, beginning with r=4.\n\n**Sources checked.**\n\n- Open Problem Garden, Ryser's conjecture (OPG-165), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/rysers_conjecture\n Evidence used: States the exact conjecture and the solved r=2,3 cases.\n- P. E. Haxell and A. D. Scott, On Ryser's conjecture, Electronic Journal of Combinatorics 19 (2012), P23. (primary): https://doi.org/10.37236/1175\n Evidence used: Proves tau <= (r-epsilon)nu for r=4 and r=5.\n- P. Haxell, L. Narins, and T. Szabó, Extremal hypergraphs for Ryser's Conjecture, Journal of Combinatorial Theory, Series A 158 (2018), 492-547. (primary): https://doi.org/10.1016/j.jcta.2018.04.004\n Evidence used: Characterizes the tight 3-partite case and explicitly surveys the remaining open r>=4 landscape.\n\n**Review notes.** No conflict was found between the supplied definitions and modern formulations.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3293, "problem_number": "OPG-547", "title": "¿Are critical k-forests tight?", "statement": "Conjecture\n\nLet $H$ be a $k$-uniform hypergraph. If $H$ is a critical $k$-forest, then it is a $k$-tree.", "background": "Source: Open Problem Garden. Original node ID: 547. URL: http://www.openproblemgarden.org/op/are_critical_k_forests_tight.\n\nSource subject path: Graph Theory > Hypergraphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_critical_k_forests_tight\n- Author(s): Strausz, Ricardo\n- Subject(s): Graph Theory; Hypergraphs\n- Keywords: heterochromatic number\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 1st, 2007 by Dino\n\nProblem-page discussion:\nWe say that a hypergraph $H=(V,E)$ is a $k$-graph if it is $k$-uniform, and denote its order by $n=|V|$ and its size by $m=|E|$.\n\nLaszlo Lovasz introduced the following concept: a $k$-graph $H=(V,E)$ is said to be a $k$-forest if for every edge $e\\in E$ there exists a $k$-colouing $\\varsigma\\colon V\\to[k]$ such that $\\varsigma(e')=[k]\\Leftrightarrow e'=e$; that is, such that only the edge $e$ receives the $k$ colours in its vertices. Clearly a $2$-forest is simply a forest in the usual sense (i.e., an acyclic graph). Lovasz proved that\n\nTheorem A $k$-forest has size at most $m\\leq{n-1\\choose k-1}$.\n\nOn the other hand, Victor Neumann-Lara introduced the following invariant: the heterochromatic number of a $k$-graph $H=(V,G)$ is the minimum number of colours $c$ such that, in every colouring $\\varsigma\\colon V\\to[c]$ there is an edge wich receives different colours in each of its vertices; that is, there exists $e\\in E$ such that $|\\varsigma(e)|=k$. If the heterochromatic number and the rank are equal, the hypergraph is said to be tight. Clearly a $2$-graph is tight if and only if it is connected. A tight $k$-forest is called a $k$-tree.\n\nI can prove the following\n\nTheorem If a $k$-forest has size $m={n-1\\choose k-1}$ then it is tight — and therefore a $k$-tree.\n\nFinally, we say that a $k$-forest is critical if no edge can be added to it without loosing the property of being a $k$-forest; it is maximal (in size) with such a property. Observe that there are critical $k$-forests of size $m<{n-1\\choose k-1}$, whenever $k>2$.\n\nSo, the conjecture is to motivate the question: ¿are critical $k$-forests tight?\n\nSource links:\n- uniform: http://en.wikipedia.org/wiki/hypergraph\n\nDiscussion links:\n- acyclic: http://en.wikipedia.org/wiki/Tree (graph theory)\n- connected: http://en.wikipedia.org/wiki/Connectivity (graph theory)\n\nComments:\n- May 5th, 2014 | Anonymous | This has been solved.: See http://arxiv.org/abs/1109.3390\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"¿Are critical k-forests tight?\" in Graph Theory; Hypergraphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem that every critical k-forest is a k-tree was verified.\n\n**Verified partial progress.**\n\n- The source supplies the specialized formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nSettle the critical-k-forest assertion.\n\n**Sources checked.**\n\n- Open Problem Garden, Critical k-forests and k-trees, node 547. (maintained_tracker): http://www.openproblemgarden.org/op/critical_k_forests_and_k_trees\n Evidence used: Preserves the exact source formulation.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3294, "problem_number": "OPG-2108", "title": "Frankl's union-closed sets conjecture", "statement": "Conjecture Let $F$ be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists $x$ such that $x$ is an element of at least half the members of $F$.", "background": "Source: Open Problem Garden. Original node ID: 2108. URL: http://www.openproblemgarden.org/op/frankls_union_closed_sets_conjecture.\n\nSource subject path: Graph Theory > Hypergraphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/frankls_union_closed_sets_conjecture\n- Author(s): Frankl, Peter\n- Subject(s): Graph Theory; Hypergraphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 25th, 2008 by tchow\n\nProblem-page discussion:\nThis conjecture is notoriously difficult, even though (or should we say `because'?) it involves almost no mathematical structure whatsoever. It was posed by Frankl in the late 1970's. The recent paper of Morris [M] provides a good illustration of the kind of partial results known: Morris extends earlier work to show that the conjecture holds for families containing three 3-subsets of a 5-set, four 3-subsets of a 6-set, or eight 4-subsets of a 6-set. In a different direction, Czédli [C] has proved the conjecture in the case when $|F| \\ge 2^n - 2^{n/2}$ where $n = |\\bigcup F| \\ge 3$.\n\nBibliography:\n[C] G. Czédli, On averaging Frankl's conjecture for large union-closed-sets, J. Combin. Theory Ser. A, to appear.\n\n[M] R. Morris, FC-families and improved bounds for Frankl's conjecture, European J. Combin. 27 (2006), no. 2, 269–282.\n\n[P] B. Poonen, Union-closed families, J. Combin. Theory Ser. A 59 (1992), no. 2, 253–268.\n\n[V] T. P. Vaughan, Three-sets in a union-closed family, J. Combin. Math. Combin. Comput. 49 (2004), 73–84.\n\n[W] P. Wójcik, Union-closed families of sets, Discrete Math. 199 (1999), no. 1–3, 173–182.\n\nComments:\n- March 7th, 2009 | Anonymous | Frankl's conjecture: For mor einformation and many references see also West's account in: http://www.math.uiuc.edu/~west/openp/unionclos.html\n- December 17th, 2009 | Anonymous | question: Does anyone know if there is any linear bound known instead of 1/2?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Frankl's union-closed sets conjecture\" in Graph Theory; Hypergraphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Frankl's union-closed sets conjecture remains open, with numerous verified special cases and quantitative lower bounds below one half.\n\n**Verified partial progress.**\n\n- Recent optimization and extremal work continues to establish special sufficient conditions rather than the full half-frequency conclusion.\n\n**Full solution or refutation.**\n\nNo general proof or counterexample was verified.\n\n**What remains.**\n\nShow some element belongs to at least half the member sets for every nontrivial finite union-closed family.\n\n**Sources checked.**\n\n- Union-closed sets conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Union-closed_sets_conjecture\n Evidence used: Records the problem as open and summarizes partial cases.\n\n**Review notes.** No source alteration; unverified claimed proofs were not accepted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3295, "problem_number": "OPG-46817", "title": "Simultaneous partition of hypergraphs", "statement": "Problem Let $H_1$ and $H_2$ be two $r$-uniform hypergraph on the same vertex set $V$. Does there always exist a partition of $V$ into $r$ classes $V_1, \\dots, V_r$ such that for both $i=1,2$, at least $r!m_i/r^r -o(m_i)$ hyperedges of $H_i$ meet each of the classes $V_1, \\dots, V_r$?", "background": "Source: Open Problem Garden. Original node ID: 46817. URL: http://www.openproblemgarden.org/op/simultaneous_partition_of_hypergraphs.\n\nSource subject path: Graph Theory > Hypergraphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/simultaneous_partition_of_hypergraphs\n- Author(s): Kühn, Daniella; Osthus, Deryk\n- Subject(s): Graph Theory; Hypergraphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 6th, 2013 by fhavet\n\nProblem-page discussion:\nThe bound on the number of hyperedges is what one would expect for a random partition. For graphs, the question was answered in the a\u000effirmative in [KO]. Keevash and Sudakov observed that the answer is negative if we consider many hypergraphs instead of just 2 (see [KO] for the example).\n\nBibliography:\n*[KO] D. Kühn and D. Osthus, Maximizing several cuts simultaneously, Combinatorics, Probability and Computing 16 (2007), 277-283.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Simultaneous partition of hypergraphs\" in Graph Theory; Hypergraphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The simultaneous partition problem is solved for two graphs and for two uniform hypergraphs under a small-codegree condition, but remains open without that condition.\n\n**Verified partial progress.**\n\n- Kuhn and Osthus prove the simultaneous asymptotically optimal cut result for two graphs.\n- The hypergraph conclusion holds when each maximum pair-codegree is o(m_i).\n\n**Full solution or refutation.**\n\nNo unconditional theorem for two arbitrary r-uniform hypergraphs was verified; counterexamples for many simultaneous hypergraphs do not settle the case of exactly two.\n\n**What remains.**\n\nAfter specifying the asymptotic regime, remove the maximum-codegree hypothesis for exactly two r-uniform hypergraphs.\n\n**Sources checked.**\n\n- D. Kuehn and D. Osthus, Maximizing several cuts simultaneously, arXiv:math/0503403; Combinatorics, Probability and Computing 16 (2007), 277-283. (primary): https://arxiv.org/abs/math/0503403\n Evidence used: The primary abstract proves the two-graph case and records results for more classes; the full paper is the source of the conditional hypergraph result.\n- Graph-theory open problems, Simultaneous partition of hypergraphs (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/simultaneous_partition_of_hypergraphs/\n Evidence used: Reports the Delta_2(H_i)=o(m_i) conditional theorem and the unrestricted two-hypergraph case open.\n\n**Review notes.** The source omits the definition m_i=|E(H_i)| and does not specify the asymptotic/uniformity regime for o(m_i); these were flagged without altering the statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3296, "problem_number": "OPG-47343", "title": "Turán's problem for hypergraphs", "statement": "Conjecture Every simple $3$-uniform hypergraph on $3n$ vertices which contains no complete $3$-uniform hypergraph on four vertices has at most $\\frac12 n^2(5n-3)$ hyperedges.\n\nConjecture Every simple $3$-uniform hypergraph on $2n$ vertices which contains no complete $3$-uniform hypergraph on five vertices has at most $n^2(n-1)$ hyperedges.", "background": "Source: Open Problem Garden. Original node ID: 47343. URL: http://www.openproblemgarden.org/op/turans_problem_for_hypergraphs.\n\nSource subject path: Graph Theory > Hypergraphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/turans_problem_for_hypergraphs\n- Author(s): Turan, Paul\n- Subject(s): Graph Theory; Hypergraphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 12th, 2013 by fhavet\n\nProblem-page discussion:\nLet $V$ be an $n$-set. A $k$-uniform hypergraph $(V,{\\cal F})$ is complete if ${\\cal F}={V \\choose k}$, the set of all ${n\\choose{k}}$ $k$-subsets of $V$.\n\nLet $\\{X,Y,Z\\}$ be a partition of $V$ into three sets which are as nearly equal in size as possible, and let ${\\cal F}$ be the union of $\\{\\{x,y,z\\}:x\\in X, y\\in Y, z\\in Z\\}$, $\\{\\{x_1,x_2,y\\}:x_1\\in X, x_2\\in X, y\\in Y\\}$, $\\{\\{y_1,y_2,z\\}:y_1\\in Y, y_2\\in Y, z\\in Z\\}$, and $\\{\\{z_1,z_2,x\\}:z_1\\in Z, z_2\\in Z, x\\in X\\}$. This $3$-uniform hypergraph has $\\frac12 n^2(5n-3)$ hyperedges and contains no complete $3$-uniform hypergraph on four vertices. Hence the first conjecture asserts that this hypergraph is extremal with this prpoerty.\n\nLet $\\{X,Y\\}$ be a partition of $V$ into two sets which are as nearly equal in size as possible, and let ${\\cal F}$ be the set of all $3$-subsets of $V$ which intersect both $X$ and $Y$. This $3$-uniform hypergraph has $n^2(n-1)$ hyperedges and contains no complete $3$-uniform hypergraph on five vertices. Hence the second conjecture asserts that this hypergraph is extremal with this property.\n\nBibliography:\n*[T] P. Turán, Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48 (1941), 436--452.\n\nDiscussion links:\n- hypergraph: http://en.wikipedia.org/wiki/hypergraph\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 20.\n\nAttempt notes:\nTarget:\nMake progress on \"Turán's problem for hypergraphs\" in Graph Theory; Hypergraphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Both exact 3-uniform Turan conjectures remain open; even the predicted asymptotic densities 5/9 for K4^(3) and 3/4 for K5^(3) are unproved.\n\n**Verified partial progress.**\n\n- Turan's construction gives the 5/9 lower bound for the K4^(3)-free density, while Razborov's flag-algebra method gives the upper bound 0.561666.\n- The balanced bipartite construction gives the conjectured 3/4 density lower bound for the K5^(3)-free problem.\n\n**Full solution or refutation.**\n\nNeither of the two finite exact extremal formulas in the source has been proved or refuted.\n\n**What remains.**\n\nClose the asymptotic density gaps and then establish the exact finite formulas, or find denser counterexamples.\n\n**Sources checked.**\n\n- Alexander A. Razborov, On 3-Hypergraphs with Forbidden 4-Vertex Configurations, SIAM Journal on Discrete Mathematics 24 (2010), 946-963, DOI 10.1137/090747476. (primary): https://doi.org/10.1137/090747476\n Evidence used: Provides the flag-algebra upper bound for the tetrahedron Turan density, leaving a gap above 5/9.\n- Peter Keevash, Hypergraph Turan Problems, Surveys in Combinatorics 2011, pp. 83-140, DOI 10.1017/CBO9781139004114.004. (authoritative_secondary): https://doi.org/10.1017/CBO9781139004114.004\n Evidence used: Authoritative survey explains that the tetrahedron problem remains open even asymptotically and surveys the complete-hypergraph context.\n- Graph-theory open problems, Turan's problem for hypergraphs, accessed 2026-08-17. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/turans_problem_for_hypergraphs/\n Evidence used: Current specialist tracker explicitly records both supplied conjectures as unresolved.\n\n**Review notes.** The source discussion's spelling defect and reused parameter n were flagged; the two exact statements were preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3297, "problem_number": "OPG-333", "title": "Seymour's self-minor conjecture", "statement": "Conjecture Every infinite graph is a proper minor of itself.", "background": "Source: Open Problem Garden. Original node ID: 333. URL: http://www.openproblemgarden.org/op/seymours_self_minor_conjecture.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/seymours_self_minor_conjecture\n- Author(s): Seymour, Paul D.\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: infinite graph; minor\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 22nd, 2007 by mdevos\n\nProblem-page discussion:\nRobertson and Seymour famously proved that the set of all finite graphs is well-quasi-ordered by the minor relation. More precisely, if we let $G_1,G_2,\\ldots$ be an infinite sequence of graphs, then there exist $i < j$ so that $G_i$ is a minor of $G_j$. Their theory also gives us a rough structure theorem for the family of graphs without a fixed minor - a powerful tool for studying minors in finite graphs.\n\nOn the other hand, there are still large gaps in our understanding of minors of infinite graphs. For instance, while it is known that Wagner's conjecture does not hold in general for infinite graphs [T], it is possible that countably infinite graphs are well quasi-ordered.\n\nThe conjecture highlighted above is especially interesting, because (if true) it would imply the well quasi-ordering of finite graphs. Indeed, the well quasi-ordering of finite graphs is equivalent to the statement that every infinite set $\\Omega$ of finite graphs contains two distinct members, one of which is a minor of the other. This latter statement follows from the above conjecture, since we may form a single infinite graph $G$ from the disjoint union of the graphs in $\\Omega$, and the proper self-minor of $G$ gives us a pair of graphs in $\\Omega$ with one a minor of the other.\n\nBibliography:\n[T] R. Thomas, A counterexample to \"Wagner's conjecture\" for infinite graphs. Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 1, 55--57. MathSciNet\n\nSource links:\n- minor: http://en.wikipedia.org/wiki/minor (graph theory)\n\nDiscussion links:\n- well-quasi-ordered: http://en.wikipedia.org/wiki/well quasi-order\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0913450\n\nComments:\n- April 4th, 2010 | Anonymous | Counterexample to Seymour's self-minor conjecture: This conjecture has been shown to be false:\n\nB. Oporowski, 'A counterexample to Seymour's self-minor conjecture.' Journal of Graph Theory, (14) 5 (521--524)\n- April 15th, 2010 | Anonymous | Real Problem: The real problem, of course, is to prove Seymour's Conjectre for countable graphs (which suffices for the Graph Minor Theorem). Note that Oporowski's example involves uncountable many vertices.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Seymour's self-minor conjecture\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof that every infinite graph has a proper self-minor was verified.\n\n**Verified partial progress.**\n\n- The assertion belongs to infinite graph minor theory.\n\n**Full solution or refutation.**\n\nThe self-minor conjecture remains open.\n\n**What remains.**\n\nDevelop well-quasi-order or end-structure methods.\n\n**Sources checked.**\n\n- Open Problem Garden, node 333 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3298, "problem_number": "OPG-349", "title": "Unions of triangle free graphs", "statement": "Problem Does there exist a graph with no subgraph isomorphic to $K_4$ which cannot be expressed as a union of $\\aleph_0$ triangle free graphs?", "background": "Source: Open Problem Garden. Original node ID: 349. URL: http://www.openproblemgarden.org/op/unions_of_triangle_free_graphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/unions_of_triangle_free_graphs\n- Author(s): Erdos, Paul; Hajnal, Andras\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: forbidden subgraph; infinite graph; triangle free\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 4th, 2007 by mdevos\n\nProblem-page discussion:\nShelah [S] has proved that the existence of such a graph is consistent with ZFC.\n\nBibliography:\n*[EH] P. Erdos and A. Hajnal, On decomposition of graphs, Acta Math. Acad. Sci. Hungar. 18 (1967), 359–377.\n\n[S] S. Shelah, Consistency of positive partition theorems for graphs and models, in Set Theory and Applications, Springer Lecture Notes 1401, (Toronto, ON, 1987), 167–193.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Unions of triangle free graphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No K4-free graph requiring more than countably many triangle-free subgraphs was verified, nor a universal decomposition theorem.\n\n**Verified partial progress.**\n\n- The question is inherently infinite-graph theoretic.\n\n**Full solution or refutation.**\n\nThe existence/decomposition question remains open.\n\n**What remains.**\n\nConstruct high-chromatic infinite graphs with persistent triangles or prove a countable cover.\n\n**Sources checked.**\n\n- Open Problem Garden, node 349 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3299, "problem_number": "OPG-484", "title": "Infinite uniquely hamiltonian graphs", "statement": "Problem Are there any uniquely hamiltonian locally finite 1-ended graphs which are regular of degree $r > 2$?", "background": "Source: Open Problem Garden. Original node ID: 484. URL: http://www.openproblemgarden.org/op/infinite_uniquely_hamiltonian_graphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/infinite_uniquely_hamiltonian_graphs\n- Author(s): Mohar, Bojan\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: hamiltonian; infinite graph; uniquely hamiltonian\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 24th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Originally appeared as [M].)\n\nLet $G$ be a locally finite infinite graph and let $I(G)$ be the set of ends of~ $G$. The Freudenthal compactification of $G$ is the topological space $|G|$ which is obtained from the usual topological space of the graph, when viewed as a 1-dimensional cell complex, by adding all points of $I(G)$ and setting, for each end $t \\in I(G)$, the basic set of neighborhoods of $t$ to consist of sets of the form $C(S, t) \\cup I(S,t) \\cup E'(S,t)$, where $S$ ranges over the finite subsets of $V(G)$, $C(S, t)$ is the component of $G - S$ containing all rays in $t$, the set $I(S,t)$ contains all ends in $I(G)$ having rays in $C(S, t)$, and $E'(S,t)$ is the union of half-edges $(z,y]$, one for every edge $xy$ joining $S$ and $C(S,t)$. We define a hamilton circle in $|G|$ as a homeomorphic image $C$ of the unit circle $S^1$ into $|G|$ such that every vertex (and hence every end) of $G$ appears in $C$. More details about these notions can be found in [D].\n\nA graph $G$ (finite or infinite) is said to be uniquely hamiltonian if it contains precisely one hamilton circle.\n\nFor finite graphs, the celebrated Sheehan's conjecture states that there are no $r$-regular uniquely hamiltonian graphs for $r>2$; this is known for all odd $r$ and even $r > 23$. For infinite graphs this is false even for odd $r$ (e.g. for the two-way infinite ladder), but each of the known counterexamples has at least 2 ends, leading to the problem stated.\n\nAnother way to extend Sheehan's conjecture to infinite graphs is to define degree of an end $t \\in I(G)$ to be the maximal number of disjoint rays in $t$ and ask the following:\n\nProblem Are there any uniquely hamiltonian locally finite graphs where every vertex and every end has the same degree $r > 2$?\n\nBibliography:\n[D] R. Diestel, Graph Theory, Third Edition, Springer, 2005.\n\n*[M] Bojan Mohar, Problem of the Month\n\nRelated:\nRelated problems\nr-regular graphs are not uniquely hamiltonian.\n\nDiscussion links:\n- Freudenthal compactification: http://en.wikipedia.org/wiki/End_(topology)\n- Sheehan's conjecture: http://www.openproblemgarden.org/?q=node/480\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0703_HamiltonicityInfinite.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"Infinite uniquely hamiltonian graphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No uniquely Hamiltonian locally finite one-ended regular graph of degree greater than two was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\ninfinite uniquely Hamiltonian locally finite one ended regular graph\n\n**Sources checked.**\n\n- Open Problem Garden (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3300, "problem_number": "OPG-488", "title": "Hamiltonian cycles in line graphs of infinite graphs", "statement": "Conjecture\n\n- If $G$ is a 4-edge-connected locally finite graph, then its line graph is hamiltonian.\n- If the line graph $L(G)$ of a locally finite graph $G$ is 4-connected, then $L(G)$ is hamiltonian.", "background": "Source: Open Problem Garden. Original node ID: 488. URL: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_line_graphs_of_infinite_graphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_line_graphs_of_infinite_graphs\n- Author(s): Georgakopoulos, Agelos\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: hamiltonian; infinite graph; line graphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 24th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Reproduced from [M].)\n\nA locally finite graph is hamiltonian, if its Freudenthal compactification (also called the end compactification, see [D]) contains a hamilton circle, i.e. a homeomorphic copy of $S^1$ containing all vertices.\n\nThe first part is known for finite graphs. The proof uses the existence of two edge-disjoint spanning trees in 4-edge-connected graphs. In the infinite case, it would be enough to prove that a 4-edge-connected locally finite graph $G$ has two edge-disjoint topological spanning trees (see [D]), one of which is connected as a subgraph of $G$. The problem is open even for the 1-ended case (where hamilton circles correspond to 2-way-infinite paths).\n\nThe second part is widely open even in the finite case, where it was proposed by Thomassen [T].\n\nBibliography:\n[D] Reinhard Diestel, Graph Theory, Third Edition, Springer, 2005.\n\n*[G] A. Georgakopoulos, Oberwolfach reports, 2007.\n\n[M] Bojan Mohar, Problem of the Month\n\n[T] Carsten Thomassen, Reflections on graph theory, J. Graph Theory 10 (1986) 309-324, MathSciNet\n\nRelated:\nRelated problems\nHamiltonian cycles in line graphs\n\nSource links:\n- line graph: http://en.wikipedia.org/wiki/line graph\n\nDiscussion links:\n- Thomassen: http://www.openproblemgarden.org/?q=node/485\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0703_HamiltonicityInfinite.html\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0856118\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamiltonian cycles in line graphs of infinite graphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of either infinite line-graph Hamiltonicity conjecture was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\ninfinite graph line graph 4 edge connected Hamiltonian\n\n**Sources checked.**\n\n- Open Problem Garden (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3301, "problem_number": "OPG-490", "title": "Hamiltonian cycles in powers of infinite graphs", "statement": "Conjecture\n\n- If $G$ is a countable connected graph then its third power is hamiltonian.\n- If $G$ is a 2-connected countable graph then its square is hamiltonian.", "background": "Source: Open Problem Garden. Original node ID: 490. URL: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_powers_of_infinite_graphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hamiltonian_cycles_in_powers_of_infinite_graphs\n- Author(s): Georgakopoulos, Agelos\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: hamiltonian; infinite graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 24th, 2007 by Robert Samal\n\nProblem-page discussion:\n(Reproduced from [M].)\n\nBoth results are known to be true for finite graphs (the second part is the celebrated result of Fleischner) and also for locally finite graphs [G].\n\nBibliography:\n[G] A. Georgakopoulos, Oberwolfach reports, 2007.\n\n[M] Bojan Mohar, Problem of the Month\n\nRelated:\nRelated problems\nHamiltonian cycles in line graphs of infinite graphs\n\nSource links:\n- power: http://en.wikipedia.org/wiki/Glossary_of_graph_theory#Distance\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P0703_HamiltonicityInfinite.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Hamiltonian cycles in powers of infinite graphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the two countable-graph power Hamiltonicity assertions was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\ncountable graph third power Hamiltonian square 2 connected\n\n**Sources checked.**\n\n- Open Problem Garden (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3302, "problem_number": "OPG-677", "title": "Universal highly arc transitive digraphs", "statement": "An alternating walk in a digraph is a walk $v_0,e_1,v_1,\\ldots,v_m$ so that the vertex $v_i$ is either the head of both $e_i$ and $e_{i+1}$ or the tail of both $e_i$ and $e_{i+1}$ for every $1 \\le i \\le m-1$. A digraph is universal if for every pair of edges $e,f$, there is an alternating walk containing both $e$ and $f$\n\nQuestion Does there exist a locally finite highly arc transitive digraph which is universal?", "background": "Source: Open Problem Garden. Original node ID: 677. URL: http://www.openproblemgarden.org/op/universal_highly_arc_transitive_digraphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/universal_highly_arc_transitive_digraphs\n- Author(s): Cameron, Peter J.; Praeger, Cheryl E.; Wormald, Nicholas C.\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: arc transitive; digraph\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 21st, 2007 by mdevos\n\nProblem-page discussion:\nLet $D$ be a digraph. For a nonnegative integer $s$, a $s$-arc in $D$ is a sequence $(x_0,x_1,\\ldots,x_s)$ of vertices so that $(x_i,x_{i+1})$ is an edge for every $0 \\le i \\le s-1$ and $x_{i-1} \\neq x_{i+1}$ for every $1 \\le i \\le s-1$. We say that $D$ is $s$-arc transitive if its automorphism group acts transitively on the set of $s$ arcs, and we say that $D$ is highly arc transitive if it is $s$-arc transitive for every $s$. Note that the condition $0$-arc transitive is precisely equivalent to vertex transitive.\n\nIt is an easy exercise to show that the only finite digraphs which are highly arc transitive are directed cycles. Since such graphs have only trivial alternating walks (only one edge can be used), they are not universal. Thus, any graph satisfying the criteria of the conjecture must be infinite.\n\nLet $P$ be a two way infinite directed path (i.e. the Cayley graph on ${\\mathbb Z}$ with generating set $\\{1\\}$ ). The digraph $P$ is not universal, but moreover, any digraph with a homomorphism onto $P$ cannot be universal. In the same article where the above question was posed, the authors asked wether there exist infinite highly transitive digraphs with no homomorphism onto $P$. This question has since been resolved in the affirmative: Evans [E] constructed such a digraph with infinite indegree, and Malnic et. al. [MMSZ] have constructed a locally finite one.\n\nIn a vertex transitive digraph, every vertex must have the same indegree and the same outdegree, and we shall denote these by $d^-$ and $d^+$ respectively. A theorem of Praeger [P] shows that every locally finite highly transitive digraph for which $d^- \\neq d^+$ has a homomorphism onto $P$ and thus is not universal. More recently, Malnic et. al. [MMMSTZ] have established a condition on edge stabilizers in arc transitive digraphs which implies that any such digraph with $d^- = d^+$ a prime is not universal. It follows that any digraph satisfying the conditions of the highlighted question must have $d^+ = d^-$ a composite number.\n\nBibliography:\n*[CPW] P. J. Cameron, C. E. Praeger, and N. C. Wormald, Infinite highly arc transitive digraphs and universal covering digraphs. Combinatorica 13 (1993), no. 4, 377--396. MathSciNet.\n\n[E] D. M. Evans, An infinite highly arc-transitive digraph, European J. Combin., 18 (1997) 281--286. MathSciNet.\n\n[MMMSTZ] A. Malnic, D. Marusic, R. G. Moller, N. Seifter, V. Trofimov, and B. Zgrablic, Highly arc transitive digraphs: reachability, topological groups. European J. Combin. 26 (2005), no. 1, 19--28. MathSciNet.\n\n[MMSZ] A. Malnic, D. Marusic, N. Seifter, and B. Zgrablic, Highly arc-transitive digraphs with no homomorphism onto Z. Combinatorica 22 (2002), no. 3, 435--443. MathSciNet\n\n[P] C. E. Praeger, On homomorphic images of edge transitive directed graphs, Australas. J. Combin., 3 (1991), 207--210. MathSciNet.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1262915\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1437003\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2101032\n- Highly arc-transitive digraphs with no homomorphism onto Z: http://www.ijp.si/ftp/pub/preprints/ps/99/pp659.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1932063\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1122225\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"Universal highly arc transitive digraphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** DeVos-Mohar-Šámal construct a locally finite highly arc-transitive digraph with universal reachability relation, answering the existence question affirmatively.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nSuch a digraph exists; universal reachability is exactly the condition that every pair of edges belongs to a common alternating walk.\n\n**What remains.**\n\nThe existence question is closed; classification and minimum-valency questions are separate refinements.\n\n**Sources checked.**\n\n- Matt DeVos, Bojan Mohar, and Robert Šámal, Highly arc-transitive digraphs -- structure and counterexamples, Combinatorica 37 (2017); arXiv:1110.2945. (primary): https://arxiv.org/abs/1110.2945\n Evidence used: Abstract explicitly constructs a locally finite highly arc-transitive digraph with universal reachability and says the Cameron-Praeger-Wormald problem is resolved.\n- Open Problem Garden, Universal highly arc transitive digraphs (node 677). (maintained_tracker): https://www.openproblemgarden.org/op/universal_highly_arc_transitive_digraphs\n Evidence used: Original definition and existence question.\n\n**Review notes.** Dataset status should change from open to solved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3303, "problem_number": "OPG-687", "title": "Unfriendly partitions", "statement": "If $G$ is a graph, we say that a partition of $V(G)$ is unfriendly if every vertex has at least as many neighbors in the other classes as in its own.\n\nProblem Does every countably infinite graph have an unfriendly partition into two sets?", "background": "Source: Open Problem Garden. Original node ID: 687. URL: http://www.openproblemgarden.org/op/unfriendly_partitions.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/unfriendly_partitions\n- Author(s): Cowan, Robert H.; Emerson, William R.\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: coloring; infinite graph; partition\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 22nd, 2007 by mdevos\n\nProblem-page discussion:\nIt is a simple property that every finite graph $G$ has an unfriendly partition into two sets - just choose a partition of $V(G)$ into two sets so that the number of edges with one end in each is maximum. Cowan and Emerson [CE] conjectured that the same property should hold true of infinite graphs. A counterexample to this was constructed by Milner and Shelah [MS], but their construction uses uncountably many vertices, leaving the countable case (highlighted above) still open. In the same article by Milner and Shelah [MS], they show that every graph does have an unfriendly partition into three sets.\n\nCuriously, it is quite easy to see that the answer to the above question is yes in the case when all vertices have finite degree, and also in the case when all vertices have infinite degree. The former follows from the unfriendly partition property for finite graphs together with a standard compactness argument. The latter can be achieved with a \"back and forth\" construction. Thus, the difficult case is the mixed one. Aharoni, Milner, and Prikry [AMP] showed that every graph with only finitely many vertices of infinite degree has an unfriendly partition into two sets, but this seems the extent of our knowledge.\n\nIt does not appear that there is any consensus among experts as to whether this conjecture should be true or false.\n\nBibliography:\n*[CE] R. Cowan and W. Emerson, Proportional colorings of graphs, unpublished.\n\n[MS] E. C. Milner and S. Shelah, Graphs with no unfriendly partitions. A tribute to Paul Erdös, 373--384, Cambridge Univ. Press, Cambridge, 1990. MathSciNet.\n\n[AMP] R. Aharoni, E. C. Milner, K. Prikry, Unfriendly partitions of a graph. J. Combin. Theory Ser. B 50 (1990), no. 1, 1--10. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1117030\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1070461\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Unfriendly partitions\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Whether every countable graph has an unfriendly bipartition remains open, but it is now proved for every countable graph without an alternating ray and for several other broad classes.\n\n**Verified partial progress.**\n\n- Aurichi-Real prove the conjecture for countable graphs whose rays do not meet infinitely many finite-degree and infinitely many infinite-degree vertices.\n- The conjecture is known for rayless graphs, graphs without a subdivision of the infinite clique, and line graphs of any cardinality.\n\n**Full solution or refutation.**\n\nNo proof or countable counterexample for the unrestricted problem was verified.\n\n**What remains.**\n\nResolve countable graphs containing alternating rays, the structural case not covered by the latest theorem.\n\n**Sources checked.**\n\n- Leandro Fiorini Aurichi and Lucas Real, Remarks on the countable case of the Unfriendly Partition Problem, arXiv:2412.14151 (2024). (primary): https://arxiv.org/abs/2412.14151\n Evidence used: States that the countable problem remains unsolved and proves the no-alternating-ray case.\n- Graph-theory open problems, Unfriendly partitions. (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/unfriendly_partitions/\n Evidence used: Maintains current open status and collates verified positive subclasses.\n\n**Review notes.** Milner-Shelah counterexamples are uncountable and do not refute the literal countable problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3304, "problem_number": "OPG-690", "title": "Strong matchings and covers", "statement": "Let $H$ be a hypergraph. A strongly maximal matching is a matching $F \\subseteq E(H)$ so that $|F' \\setminus F| \\le |F \\setminus F'|$ for every matching $F'$. A strongly minimal cover is a (vertex) cover $X \\subseteq V(H)$ so that $|X' \\setminus X| \\ge |X \\setminus X'|$ for every cover $X'$.\n\nConjecture If $H$ is a (possibly infinite) hypergraph in which all edges have size $\\le k$ for some integer $k$, then $H$ has a strongly maximal matching and a strongly minimal cover.", "background": "Source: Open Problem Garden. Original node ID: 690. URL: http://www.openproblemgarden.org/op/strong_matchings_and_covers.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strong_matchings_and_covers\n- Author(s): Aharoni, Ron\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: cover; infinite graph; matching\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 23rd, 2007 by mdevos\n\nProblem-page discussion:\nThe theory of matching in finite graphs is quite well understood. Now, thanks to the work of Aharoni and others, much of this theory has been extended to infinite graphs. On the other hand, matching in hypergraphs - both finite and infinite - is a subject where our knowledge apears to be lacking. The above conjecture asserts a rather basic property of hypergraphs which would be nice to verify.\n\nThis conjecture is (of course) trivial for finite hypergraphs, but it looks very difficult for infinite ones. It has been proved by Aharoni [A2] for the case when $k=2$, that is, for infinite graphs. Here the key tool is an infinite version of the Tutte-Edmonds-Gallai decomposition theorem [A1].\n\nNext we offer another interesting conjecture of Aharoni on minimal covers.\n\nConjecture If $G$ is a (possibly infinite) graph and $H$ is the hypergraph of independent sets in $G$, then $H$ has a strongly minimal cover.\n\nBibliography:\n[A1] R. Aharoni, Matchings in infinite graphs. J. Combin. Theory Ser. B 44 (1988), no. 1, 87--125. MathSciNet.\n\n*[A2] R. Aharoni, Infinite matching theory. Directions in infinite graph theory and combinatorics (Cambridge, 1989). Discrete Math. 95 (1991), no. 1-3, 5--22. MathSciNet.\n\nSource links:\n- hypergraph: http://en.wikipedia.org/wiki/hypergraph\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0923268\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1141929\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Strong matchings and covers\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Both literal existence assertions are false: a 3-uniform hypergraph can lack a strongly maximal matching, and even a graph can lack a strongly minimal vertex-cover.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nHollom-Randall Shaw supply bounded-edge counterexamples to the matching assertion and to the explicitly defined vertex-cover assertion.\n\n**What remains.**\n\nCharacterize restricted hypergraph classes where either strong object exists; the universal conjecture itself is refuted.\n\n**Sources checked.**\n\n- Lawrence Hollom and Benedict Randall Shaw, Counterexamples to conjectures on strong maximality and minimality, Electronic Journal of Combinatorics 33(2) (2026), P2.11; arXiv:2511.13709. (primary): https://arxiv.org/abs/2511.13709\n Evidence used: Theorem 1.2 gives a 3-uniform hypergraph without a strongly maximal matching; Section 3.4 gives a graph without a strongly minimal vertex-cover.\n- Open Problem Garden, Strong matchings and covers (node 690). (maintained_tracker): https://www.openproblemgarden.org/op/strong_matchings_and_covers\n Evidence used: Preserves the literal vertex-cover definition and conjecture.\n\n**Review notes.** Later literature distinguishes vertex-covers from edge-covers. The input explicitly asks for vertex-covers; the 2026 paper refutes that version too. The imported claim that the full k=2 case was proved appears overbroad.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3305, "problem_number": "OPG-691", "title": "Highly arc transitive two ended digraphs", "statement": "Conjecture If $G$ is a highly arc transitive digraph with two ends, then every tile of $G$ is a disjoint union of complete bipartite graphs.", "background": "Source: Open Problem Garden. Original node ID: 691. URL: http://www.openproblemgarden.org/op/highly_arc_transitive_two_ended_digraphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/highly_arc_transitive_two_ended_digraphs\n- Author(s): Cameron, Peter J.; Praeger, Cheryl E.; Wormald, Nicholas C.\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: arc transitive; digraph; infinite graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 29th, 2007 by mdevos\n\nProblem-page discussion:\nIt follows from a theorem of Dunwoody [D] that every vertex transitive graph $G$ with two ends has a system of imprimitivity $\\{ X_i: i \\in {\\mathbb Z} \\}$ with finite blocks so that the cyclic order $\\ldots X_{-2}, X_{-1},X_0,X_1,X_2,\\ldots$ is preserved by the automorphism group (of $G$ ). If $G$ is edge-transitive, then every edge of $G$ must have its ends in two consecutive blocks, so in this case $G$ is an edge-disjoint union of the (isomorphic) bipartite graphs $G[X_i,X_{i+1}]$ for $i \\in {\\mathbb Z}$- which we shall call tiles. Note that the tiles are edge-transitive.\n\nThis gives us a good description of edge-transitive graphs with two ends; each is made up by gluing together copies of a tile in a linear order. If $G$ is a 2-arc transitive digraph with two ends, then all edges in each tile must be oriented consistently, so by possibly reordering, we may assume that every edge in $G[X_i,X_{i+1}]$ is oriented from $X_i$ to $X_{i+1}$. The above conjecture asserts that under the added symmetry condition of high arc transitivity, each tile has a simple structure - namely it is a union of (consistently oriented) complete bipartite graphs.\n\nIt is easy to construct a highly arc transitive two ended graph by simply using the complete bipartite graph $K_{n,n}$ (with all edges oriented consistently) as a tile. Mckay and Praeger found the following pretty construction of a highly arc transitive digraph with tiles isomorphic to a disjoint union of complete bipartite graphs: Let $S$ be a finite set, let $n$ be a positive integer, and define $G$ to be the digraph with vertex set ${\\mathbb Z} \\times S^n$ and an edge from $(i, \\mathbf{x}, y)$ to $(i+1, z, \\mathbf{x})$ if $i \\in {\\mathbb Z}$, $\\mathbf{x} \\in S^{n-1}$, and $y,z \\in S$. A generalized (twisted) version of this construction was introduced by Cameron, Praeger, and Wormald [CPW], but again, every tile in this construction is a disjoint unions of bipartite graphs, and it looks hard to do anything else.\n\nBibliography:\n*[CPW] P. J. Cameron, C. E. Praeger, and N. C. Wormald, Infinite highly arc transitive digraphs and universal covering digraphs. Combinatorica 13 (1993), no. 4, 377--396. MathSciNet.\n\n[D] M. J. Dunwoody, Cutting up graphs. Combinatorica 2 (1982), no. 1, 15--23. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1262915\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0671142\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Highly arc transitive two ended digraphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The tile conjecture is false: two-ended highly arc-transitive digraphs exist whose finite bipartite building blocks are not disjoint unions of complete bipartite graphs.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nDeVos-Mohar-Šámal and independently Neumann construct counterexamples.\n\n**What remains.**\n\nA complete structure theorem for all two-ended highly arc-transitive digraphs remains separate from the refuted tile assertion.\n\n**Sources checked.**\n\n- Matt DeVos, Bojan Mohar, and Robert Šámal, Highly arc-transitive digraphs -- structure and counterexamples, Combinatorica 37 (2017); arXiv:1110.2945. (primary): https://arxiv.org/abs/1110.2945\n Evidence used: Abstract explicitly constructs two-ended counterexamples with building blocks not of the conjectured form.\n- Christoph Neumann, Constructing highly arc transitive digraphs using a direct fibre product, Discrete Mathematics 313 (2013), 2816-2829. (primary): https://doi.org/10.1016/j.disc.2013.08.019\n Evidence used: Independently obtains counterexamples to the Cameron-Praeger-Wormald structural conjecture.\n\n**Review notes.** The one-line statement relies on the background definition of tile. Dataset status should change from open to disproved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3306, "problem_number": "OPG-735", "title": "End-Devouring Rays", "statement": "Problem Let $G$ be a graph, $\\omega$ a countable end of $G$, and $K$ an infinite set of pairwise disjoint $\\omega$-rays in $G$. Prove that there is a set $K'$ of pairwise disjoint $\\omega$-rays that devours $\\omega$ such that the set of starting vertices of rays in $K'$ equals the set of starting vertices of rays in $K$.", "background": "Source: Open Problem Garden. Original node ID: 735. URL: http://www.openproblemgarden.org/op/end_devouring_rays.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/end_devouring_rays\n- Author(s): Georgakopoulos, Agelos\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: end; ray\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: February 3rd, 2008 by Agelos\n\nProblem-page discussion:\nWe say that a set of rays $K$ devours the end $\\omega$ if every ray in $\\omega$ meets some ray in $K$. An end is countable if there is a countable set of rays devouring it.\n\nIf $K$ is a finite set of rays then it is not hard to prove (see [G]) that this problem has a positive answer:\n\nTheorem For every graph $G$ and every countable end $\\omega$ of $G$, if $G$ has a set $K$ of $k\\in \\mathcal N$ pairwise disjoint $\\omega$-rays, then it also has a set $K'$ of $k$ pairwise disjoint $\\omega$-rays that devours $\\omega$. Moreover, $K'$ can be chosen so that its rays have the same starting vertices as the rays in~ $K$.\n\nBibliography:\n*[G] A. Georgakopoulos, Infinite Hamilton Cycles in Squares of Locally Finite Graphs, Preprint.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"End-Devouring Rays\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Georgakopoulos and Pitz prove exactly the prescribed-start-vertices end-devouring-rays conclusion for every countable end.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe stated theorem is proved, in a result whose abstract explicitly says it confirms Georgakopoulos's conjecture.\n\n**What remains.**\n\nNo open part of the stated countable-end problem remains.\n\n**Sources checked.**\n\n- A. Georgakopoulos and M. Pitz, Infinite end-devouring sets of rays with prescribed start vertices, Discrete Mathematics 341 (2018), 2117--2120. (primary): https://arxiv.org/abs/1704.06577\n Evidence used: The theorem gives disjoint devouring rays with the same prescribed starting vertices.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3307, "problem_number": "OPG-1750", "title": "Characterizing (aleph_0,aleph_1)-graphs", "statement": "Call a graph an $(\\aleph_0,\\aleph_1)$-graph if it has a bipartition $(A,B)$ so that every vertex in $A$ has degree $\\aleph_0$ and every vertex in $B$ has degree $\\aleph_1$.\n\nProblem Characterize the $(\\aleph_0,\\aleph_1)$-graphs.", "background": "Source: Open Problem Garden. Original node ID: 1750. URL: http://www.openproblemgarden.org/op/characterizing_aleph_0_aleph_1_graphs.\n\nSource subject path: Graph Theory > Infinite Graphs.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/characterizing_aleph_0_aleph_1_graphs\n- Author(s): Diestel, Reinhard; Leader, Imre\n- Subject(s): Graph Theory; Infinite Graphs\n- Keywords: binary tree; infinite graph; normal spanning tree; set theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 26th, 2008 by mdevos\n\nProblem-page discussion:\nThe motivation for this problem comes from a lovely paper of Diestel and Leader [DL] where they prove that an infinite graph has a normal spanning tree (the natural infinite analogue of a depth-first search tree) if and only if it has no minor isomorphic to either an $(\\aleph_0,\\aleph_1)$-graph or an Aronszajn tree. (An earlier conjecture of Halin asserted that only the first of these excluded minors was needed.) So, $(\\aleph_0,\\aleph_1)$-graphs appear as a forbidden minor obstruction to the existence of a kind of depth-first search tree for infinite graphs.\n\nThe obvious example of an $(\\aleph_0,\\aleph_1)$-graph is $K_{\\aleph_0,\\aleph_1}$, but there are other natural families of such graphs. For instance, let $T$ be an infinite binary tree with root $r$, and let $X$ be the set of all rays (one way infinite paths) with endpoint $r$. Now, form a bipartite graph with vertex bipartition $(X,V(T))$ and adjacency given by the rule that $v \\in V(T)$ adjacent to $x \\in X$ if and only if $v$ lies on the ray $x$ (in $G$ ). Any $(\\aleph_0,\\aleph_1)$-graph which is isomorphic to a subgraph of this graph is said to be of binary type.\n\nSay that a $(\\aleph_0,\\aleph_1)$-graph is divisible if there exist disjoint subsets $A',A\" \\subseteq A$ and disjoint subsets $B',B\" \\subseteq B$ so that the graphs induced by both $A' \\cup B'$ and $A\" \\cup B\"$ are $(\\aleph_0,\\aleph_1)$-graphs. It is not difficult to show that every binary type graph is divisible. Curiously, the existence of non-divisible $(\\aleph_0,\\aleph_1)$-graphs depends on the Continuum Hypothesis (see [DL]).\n\nAlthough it is not clear wether or not there is a nice characterization of $(\\aleph_0,\\aleph_1)$-graphs, it would certainly be interesting to find more natural families of these graphs. The following rather more concrete question is posed by Diestel and Leader who suspect the answer is 'no'.\n\nProblem Does every $(\\aleph_0,\\aleph_1)$-graph have an $(\\aleph_0,\\aleph_1)$-graph as a minor which is either indivisible or of binary type?\n\nBibliography:\n[DL] R. Diestel and I. Leader, Normal spanning trees, Aronszajn trees and excluded minors, J. London Math. Soc. 63 (2001), 16-32;\n\nBibliography links:\n- Normal spanning trees, Aronszajn trees and excluded minors: http://www.math.uni-hamburg.de/home/diestel/papers/Aronszajn.pdf\n\nComments:\n- November 21st, 2011 | Anonymous | problem claimed to be solved by scott streit texas professor: Has his solution ever been viewed/reviewed? He claims to have solved it with his students\n- November 24th, 2011 | Robert Samal | Re: problem claimed to be solved by scott streit texas professor: Please be more specific: who claims it a where? If I googled the right Scott Streit, he doesn't seem to be the kind of person interested in this kind of problems.\n\nBest wishes, Robert\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 20.\n\nAttempt notes:\nTarget:\nMake progress on \"Characterizing (aleph_0,aleph_1)-graphs\" in Graph Theory; Infinite Graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A full characterization is not known, but normal-spanning-tree theory and binary-type/indivisibility structure give substantial information about (aleph_0,aleph_1)-graphs.\n\n**Verified partial progress.**\n\n- Diestel--Leader identify these graphs as excluded-minor obstructions to normal spanning trees.\n- Binary-type examples and CH-dependent nondivisibility behavior are documented.\n\n**Full solution or refutation.**\n\nNo complete classification or definitive finite family of canonical types was verified.\n\n**What remains.**\n\nCharacterize all types, in particular the proposed binary-type versus indivisible-minor alternative.\n\n**Sources checked.**\n\n- Open Problem Garden, node 1750 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/characterizing_aleph_0_aleph_1_graphs\n Evidence used: Provides the exact definitions, Diestel--Leader theorem, and the remaining concrete question.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3308, "problem_number": "OPG-827", "title": "Coloring random subgraphs", "statement": "If $G$ is a graph and $p \\in [0,1]$, we let $G_p$ denote a subgraph of $G$ where each edge of $G$ appears in $G_p$ with independently with probability $p$.\n\nProblem Does there exist a constant $c$ so that ${\\mathbb E}(\\chi(G_{1/2})) > c \\frac{\\chi(G)}{\\log \\chi(G)}$?", "background": "Source: Open Problem Garden. Original node ID: 827. URL: http://www.openproblemgarden.org/op/coloring_random_subgraphs.\n\nSource subject path: Graph Theory > Probabilistic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/coloring_random_subgraphs\n- Author(s): Bukh, Boris\n- Subject(s): Graph Theory; Probabilistic Graph Theory\n- Keywords: coloring; random graph\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 18th, 2008 by mdevos\n\nProblem-page discussion:\nIt is a classical result that the above problem has a positive answer when $G$ is the complete graph. More generally, the lower bound ${\\mathbb E}(\\chi(G_{1/2})) \\ge c \\frac{\\chi(G)}{\\log |V(G)|}$ is known.\n\nIt is easy to obtain the bound ${\\mathbb E}(\\chi(G_{1/2})) \\ge (\\chi(G))^{1/2}$, since we may imagine forming two random subgraphs $H,H'$ of $G$ by putting each edge of $G$ in either $H$ or $H'$ independently with probability $1/2$. Then $\\chi(H) \\chi(H') \\ge \\chi(G)$ and this gives the desired bound. A similar argument with three subgraphs shows that ${\\mathbb E}(\\chi(G_{1/3})) \\ge (\\chi(G))^{1/3}$, however these arguments all seem to require integer multiples, so the best known lower bound on ${\\mathbb E}(\\chi(G_{49/100}))$ of this form is $(\\chi(G))^{1/3}$.\n\nBibliography:\n*[B] Boris Bukh's problem page.\n\nBibliography links:\n- Boris Bukh's problem page: http://www.math.princeton.edu/%7Ebbukh/problems.html\n\nComments:\n- June 4th, 2009 | chrisrudy502 | No?: I haven't worked through the details yet, but what if $G$ is the union of $K_n$ and a sufficiently huge bipartite graph $H$? Then $\\chi(G) = n$, and by taking $H$ huge enough, you can get $\\mathbb{E}(\\chi(G_{1/2}))$ as close to 2 as you like, forcing $c$ as small as you like.\n\nEDIT: Whoo boy. Nevermind.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 19.\n\nAttempt notes:\nTarget:\nMake progress on \"Coloring random subgraphs\" in Graph Theory; Probabilistic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Bukh's bound is proved for graphs with suitably small independence number, but remains open for arbitrary graphs.\n\n**Verified partial progress.**\n\n- Shinkar proves E chi(G_1/2)=Omega(chi(G)/log chi(G)) when alpha(G)=O(n/chi(G)).\n- General lower and tail bounds are also obtained.\n\n**Full solution or refutation.**\n\nNo universal constant c for every graph G was verified.\n\n**What remains.**\n\nRemove the independence-number hypothesis or find a counterexample family.\n\n**Sources checked.**\n\n- I. Shinkar, On Coloring Random Subgraphs of a Fixed Graph, arXiv:1612.04319. (primary): https://arxiv.org/abs/1612.04319\n Evidence used: Its abstract states the Bukh bound for the stated large graph family.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3309, "problem_number": "OPG-1757", "title": "Negative association in uniform forests", "statement": "Conjecture Let $G$ be a finite graph, let $e,f \\in E(G)$, and let $F$ be the edge set of a forest chosen uniformly at random from all forests of $G$. Then\n$$\n{\\mathbb P}(e \\in F \\mid f \\in F}) \\le {\\mathbb P}(e \\in F)\n$$", "background": "Source: Open Problem Garden. Original node ID: 1757. URL: http://www.openproblemgarden.org/op/negative_association_in_uniform_forests.\n\nSource subject path: Graph Theory > Probabilistic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/negative_association_in_uniform_forests\n- Author(s): Pemantle, Robin\n- Subject(s): Graph Theory; Probabilistic Graph Theory\n- Keywords: forest; negative association\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 30th, 2008 by mdevos\n\nProblem-page discussion:\nThe FKG inequality is the cornerstone of a respectable theory of positive association; If a natural lattice condition holds, we can use it to deduce positive association. On the other hand, the theory of negative associations is still lacking good techniques. See Pemantle's lovely paper [P] for an excellent description of this situation. The conjecture highlighted above seems to be almost obviously true, but we have no tools to prove it.\n\nModifying the conjecture by replacing \"forest\" by \"spanning tree\" gives a true statement which was proved by Feder and Mihail [FM]. Actually, they prove that this holds more generally for uniform bases of balanced matroids. Perhaps surprisingly, this is false for general matroids, see [SW].\n\nBibliography:\n[FM] T. Feder and M. Mihail, Balanced Matroids. Proc 24th Annual STOC 26 - 38 (1992).\n\n*[P] R. Pemantle, Towards a theory of negative dependence, Journal of Mathematical Physics 41 (2000), 1371–1390.\n\n[SW] P. D. Seymour and D. J. A. Welsh, Combinatorial applications of an inequality from statistical mechanics. Math. Proc. Camb. Phil. Soc. 77 485 - 495 (1975).\n\nBibliography links:\n- Towards a theory of negative dependence: http://arxiv.org/abs/math/0404095\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Negative association in uniform forests\" in Graph Theory; Probabilistic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Pairwise negative correlation for a uniform forest is unproved for every finite graph, although it has computational and special-family evidence.\n\n**Verified partial progress.**\n\n- Grimmett--Winkler verify the conjecture computationally for small graph ranges.\n- The uniform spanning-tree analogue is known, but is not the same measure.\n\n**Full solution or refutation.**\n\nNo general proof of the displayed conditional-probability inequality was verified.\n\n**What remains.**\n\nProve pairwise negative association for arbitrary uniform forests or find a finite counterexample.\n\n**Sources checked.**\n\n- G. R. Grimmett and S. N. Winkler, Negative association in uniform forests and connected graphs, arXiv:math/0302185. (primary): https://arxiv.org/abs/math/0302185\n Evidence used: States the uniform-forest conjecture, records small-graph verification, and distinguishes the known spanning-tree case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3310, "problem_number": "OPG-37656", "title": "Chromatic number of random lifts of complete graphs", "statement": "Question Is the chromatic number of a random lift of $K_5$ concentrated on a single value?", "background": "Source: Open Problem Garden. Original node ID: 37656. URL: http://www.openproblemgarden.org/op/chromatic_number_of_random_lifts_of_complete_graphs.\n\nSource subject path: Graph Theory > Probabilistic Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/chromatic_number_of_random_lifts_of_complete_graphs\n- Author(s): Amit, Linial, Matousek\n- Subject(s): Graph Theory; Probabilistic Graph Theory\n- Keywords: random lifts, coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 3rd, 2012 by DOT\n\nProblem-page discussion:\nLet $G$ be a graph with vertex set $V$ and edge set $E$. An $h$-lift $H$ is a graph with vertex set $V\\times\\{1,\\dots,h\\}$, such that $(u,k)$ and $(v,\\ell)$ may only be adjacent in $H$ if $uv \\in E$, and for each $uv\\in E$, the edges between $\\{u\\}\\times\\{1,\\dots,h\\}$ and $\\{v\\}\\times\\{1,\\dots,h\\}$ form a perfect matching.\n\nA random $h$-lift of $G$ is a graph drawn uniformly at random from the set of all $h$-lifts of $G$. This amounts to choosing, independently at random, a perfect matching for each edge of $G$. One is generally interested in properties of random $h$-lifts when $h\\to\\infty$.\n\nAmit, Linial, and Matousek [ALM02] have studied the chromatic number of random lifts. They ask whether a the chromatic number of a random $h$-lift of $K_5$ is asymptotically almost surely a single number.\n\nIt is easy to see that this number may be either 3 or 4. Farzad and Theis [FT12] have shown that random lifts of $K_5\\setminus e$ are asymptotically almost surely 3-colorable.\n\nA more general question is this.\n\nQuestion Is the chromatic number of a random lift of $K_n$ concentrated on a single value?\n\nAmit, Linial, and Matousek [ALM02] have shown that the chromatic number of a random lift of $K_n$ is in $\\Theta(n/\\log n)$.\n\nBibliography:\n*[ALM02] Random Lifts of Graphs III: Independence and Chromatic Number, A. Amit, N. Linial and J. Matousek, Random Structures and Algorithms, 20(2002) 1-22.\n\n[FT12] Random lifts of $K_5\\setminus e$ are 3-colourable. B. Farzad and D.O. Theis. SIAM J. Discrete Math. 26:1 (2012), 169–179.\n\nDiscussion links:\n- $h$-lift: http://en.wikipedia.org/wiki/Covering_graph\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 17.\n\nAttempt notes:\nTarget:\nMake progress on \"Chromatic number of random lifts of complete graphs\" in Graph Theory; Probabilistic Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The chromatic number of a random lift of K5 is asymptotically almost surely in {3,4}, but concentration on one of those values is still unresolved.\n\n**Verified partial progress.**\n\n- Nir and Perez-Gimenez obtain a general two-value window for random lifts of complete graphs and exact concentration for many other base degrees.\n- Random lifts of K5 minus one edge are asymptotically almost surely 3-colourable.\n\n**Full solution or refutation.**\n\nThe source's single-value concentration question for K5 itself remains open despite the sharp two-value window.\n\n**What remains.**\n\nDetermine whether the probability of 3-colourability tends to 0 or 1, or otherwise show that single-value concentration fails.\n\n**Sources checked.**\n\n- J. D. Nir and Xavier Perez-Gimenez, The chromatic number of random lifts of complete graphs, arXiv:2109.13347. (primary): https://arxiv.org/abs/2109.13347\n Evidence used: Gives the two-point concentration framework and leaves the K5 case at the values 3 and 4.\n- B. Farzad and D. O. Theis, Random lifts of K5 minus e are 3-colourable, SIAM Journal on Discrete Mathematics 26 (2012), 169-179, arXiv:1003.1527. (primary): https://arxiv.org/abs/1003.1527\n Evidence used: Proves asymptotic 3-colourability for the nearby base graph K5 minus an edge, not for K5.\n\n**Review notes.** The nearby K5-minus-edge theorem is explicitly kept separate from the K5 question.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3311, "problem_number": "OPG-36922", "title": "Domination in plane triangulations", "statement": "Conjecture Every sufficiently large plane triangulation $G$ has a dominating set of size $\\le \\frac{1}{4} |V(G)|$.", "background": "Source: Open Problem Garden. Original node ID: 36922. URL: http://www.openproblemgarden.org/op/domination_in_plane_triangulations.\n\nSource subject path: Graph Theory > Topological Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/domination_in_plane_triangulations\n- Author(s): Matheson, Lesley R.; Tarjan, Robert E.\n- Subject(s): Graph Theory; Topological Graph Theory\n- Keywords: coloring; domination; multigrid; planar graph; triangulation\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 4th, 2009 by mdevos\n\nProblem-page discussion:\nMotivated by some problems in multigrid computations, Matheson and Tarjan [MT] considered the problem of finding small dominating sets in plane triangulations. They proved that every such graph $G$ has a dominating set of size $\\le \\frac{1}{3} |V(G)|$ and posed the above question.\n\nThe Octahedron is a triangulation with 6 vertices for which every dominating set has size $\\ge 2$, so no constant better than $\\frac{1}{3}$ can be achieved in general. However, it appears that one can do better for larger graphs. The most extreme examples here (also from [MT]) are constructed as follows: Start with $n$ disjoint copies of $K_4$ embedded in the plane, and then add edges to complete this graph to a triangulation (with $4n$ vertices). Now each of the original copies of $K_4$ has an inner vertex which has degree 3 in the final graph, and in order to cover it, one must take at least one vertex from this $K_4$. It follows that every dominating set has size $\\ge n$.\n\nSince the Matheson-Tarjan proof is short and instructive, we sketch it here. In fact, we shall prove (as they did) the stronger statement that every near-triangulation (a graph embedded in the plane with all finite faces of size three) has a (possibly improper) 3-coloring so that each color class is a dominating set and so that the subgraph induced by those vertices incident with the infinite face is properly colored. This stronger fact we prove by induction. If the infinite face is not bounded by a cycle or the infinite face is bounded by a cycle which has a chord, then the graph may be written as the union of two near-triangulations $G_1,G_2$ where $G_1$ and $G_2$ either share one vertex or two adjacent vertices and one edge. In either case, the result follows by applying induction to $G_1$ and $G_2$. Otherwise, choose a vertex $v$ on the infinite face, delete $v$ and apply induction. Since the neighbors of $v$ are all on the infinite face, and do not form an independent set, there are at least two colors, say $1$ and $2$, which appear on the neighbors of $v$. Now giving $v$ the color $3$ gives a solution.\n\nBibliography:\n[MT] L. R. Matheson, R. E. Tarjan, Dominating sets in planar graphs. European J. Combin. 17 (1996), no. 6, 565--568. MathSciNet\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1401911\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 16.\n\nAttempt notes:\nTarget:\nMake progress on \"Domination in plane triangulations\" in Graph Theory; Topological Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Matheson-Tarjan n/4 conjecture remains open; the best checked general bound is 2n/7 for n>10, with n/4 proved for maximum degree at most 6 and other special classes.\n\n**Verified partial progress.**\n\n- Christiansen, Rotenberg, and Rutschmann prove every n-vertex plane triangulation with n>10 has domination number at most 2n/7.\n- King and Pelsmajer prove the conjectured n/4 bound for plane triangulations of maximum degree at most 6.\n- Francis, Illickan, Jose, and Rajendraprasad give the bound (1-alpha)n/2 in terms of the maximum independent-set ratio alpha, improving 2n/7 in stated extreme alpha ranges.\n\n**Full solution or refutation.**\n\nRecent primary work continues to call the n/4 bound open, so the conjecture is not solved in general.\n\n**What remains.**\n\nClose the gap between the general upper bound 2n/7 and the sharp lower-bound scale n/4, or prove the n/4 bound for additional structural classes.\n\n**Sources checked.**\n\n- Aleksander B. G. Christiansen, Eva Rotenberg, and Daniel Rutschmann, Triangulations Admit Dominating Sets of Size 2n/7, SODA 2024, 1194-1240, DOI 10.1137/1.9781611977912.47. (primary): https://doi.org/10.1137/1.9781611977912.47\n Evidence used: Proves the current checked general 2n/7 upper bound for n>10.\n- P. Francis, Abraham M. Illickan, Lijo M. Jose, and Deepak Rajendraprasad, Face-hitting Dominating Sets in Planar Graphs, arXiv:2403.02808. (primary): https://arxiv.org/abs/2403.02808\n Evidence used: Explicitly says the Matheson-Tarjan conjecture remains open, identifies 2n/7 as the best general bound, and gives an independence-number refinement.\n- Erika L. C. King and Michael J. Pelsmajer, Dominating Sets in Plane Triangulations, arXiv:0806.2421. (primary): https://arxiv.org/abs/0806.2421\n Evidence used: Proves the n/4 conjecture for maximum degree at most 6.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3312, "problem_number": "OPG-46634", "title": "Large induced forest in a planar graph.", "statement": "Conjecture Every planar graph on $n$ verices has an induced forest with at least $n/2$ vertices.", "background": "Source: Open Problem Garden. Original node ID: 46634. URL: http://www.openproblemgarden.org/op/large_induced_forest_in_a_planar_graph.\n\nSource subject path: Graph Theory > Topological Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/large_induced_forest_in_a_planar_graph\n- Author(s): Abertson, Michael O.; Berman, David M.\n- Subject(s): Graph Theory; Topological Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 4th, 2013 by fhavet\n\nProblem-page discussion:\nThis conjecture is best possible. (See [AW]). It follows from Borodin's theorem stating that every planar graph has an acyclic $5$-colouring that every planar graph on $n$ verices has an induced forest with at least $2n/5$ vertices. The conjecture holds for planar graph with girth at least $5$, because they can be partitionned into a stable set and a forest [BG] (see also [KT]).\n\nAkiyama-Watanabe [AW] conjectured an even larger induced forest for bipartite planar graphs.\n\nConjecture Every bipartite planar graph on $n$ verices has an induced forest with at least $5n/8$ vertices.\n\nThis conjecture is also best possible. (See [AW]).\n\nBibliography:\n*[AB] M. O. Albertson and D. M. Berman. A conjecture on planar graphs. Graph Theory and Related Topics (J. A. Bondy and U. S. R. Murty, eds.), (Academic Press, 1979), 357.\n\n[AW] J. Akiyama and M. Watanabe. Maximum induced forests of planar graphs. Graphs and Combinatorics 3 (1987), 201--202.\n\n[B] O. V. Borodin. A proof of B. Grünbaum's conjecture on the acyclic 5-colorability of planar graphs. (Russian) Dokl. Akad. Nauk SSSR 231 (1976), no. 1, 18--20.\n\n[BG] O. V. Borodin and A. N. Glebov. On the partition of a planar graph of girth 5 into an empty graph and an acyclic subgraph. Diskretn. Anal. Issled. Oper. Ser. 1 8:34–53, 2001\n\n[KT] K. Kawarabayashi and C. Thomassen. Decomposing a planar graph of girth 5 into an independent set and a forest. Journal of Combinatorial Theory, Series B 99(4):674–684, 2009.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Large induced forest in a planar graph.\" in Graph Theory; Topological Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Albertson-Berman n/2 induced-forest conjecture remains open for general planar graphs, with a 2n/5 universal bound and stronger results for restricted planar classes.\n\n**Verified partial progress.**\n\n- Acyclic five-colourability gives an induced forest on at least 2n/5 vertices.\n- Triangle-free, girth-at-least-five, and bipartite planar graphs have stronger published lower bounds.\n\n**Full solution or refutation.**\n\nNo general n/2 theorem or counterexample was verified; modern primary papers improve restricted classes rather than settling arbitrary planar graphs.\n\n**What remains.**\n\nRaise the universal induced-forest lower bound from 2n/5 to n/2, or find a planar counterexample.\n\n**Sources checked.**\n\n- F. Dross, M. Montassier, and A. Pinlou, Large induced forests in planar graphs with girth 4 or 5, arXiv:1409.1348. (primary): https://arxiv.org/abs/1409.1348\n Evidence used: Gives the stated triangle-free and girth-five lower bounds.\n- Y. Wang, Q. Xie, and X. Yu, Induced forests in bipartite planar graphs, arXiv:1605.00047. (primary): https://arxiv.org/abs/1605.00047\n Evidence used: Proves the ceiling of (4n+3)/7 lower bound for bipartite planar graphs.\n- Graph-theory open problems, Large induced forest in a planar graph (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/large_induced_forest_in_a_planar_graph/\n Evidence used: Continues to list the n/2 conjecture as open and summarizes classical bounds.\n\n**Review notes.** The literal source typo 'verices' means 'vertices'; it was flagged and not silently altered.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3313, "problem_number": "OPG-46943", "title": "Every 4-connected toroidal graph has a Hamilton cycle", "statement": "Conjecture Every 4-connected toroidal graph has a Hamilton cycle.", "background": "Source: Open Problem Garden. Original node ID: 46943. URL: http://www.openproblemgarden.org/op/every_4_connected_toroidal_graph_has_a_hamilton_cycle.\n\nSource subject path: Graph Theory > Topological Graph Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/every_4_connected_toroidal_graph_has_a_hamilton_cycle\n- Author(s): Grunbaum, Branko; Nash-Williams, Crispin, St. J. A.\n- Subject(s): Graph Theory; Topological Graph Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 7th, 2013 by fhavet\n\nProblem-page discussion:\nTutte [Tu] proved that every 4-connected planar graph has a Hamilton cycle. (See also [Th]). Thomas and Yu [TY] proved that every 4-connected projective-planar graph has a Hamilton cycle.\n\nThomas and Yu [TY] also proved that every 5-connected toroidal graph has a Hamilton cycle. In fact, they show something stronger: every edge in a 5-connected toroidal graph is contained in a Hamilton cycle. This stronger result cannot be extended to 4-connected toroidal graphs: Thomassen [Th] showed 4-connected toroidal graphs in which certain edges are not contained in any Hamilton cycle.\n\nBibliography:\n*[G] B. Grünbaum, Polytopes, graphs, and complexes. Bull. Amer. Math. Soc. 76 (1970) 1131-1201.\n\n*[N] C. St. J. A. Nash-Williams, Unexplored and semi-explored territories in graph theory. New Directions in Graph Theory, Academic Press, New York (1973) 169-176.\n\n[TY] R. Thomas and X. Yu, 5-connected toroidal graphs are hamiltonian. J. Combinat. Theory Ser. B 69 (1997), no.1, 79-96.\n\n[Th] C. Thomassen, A theorem on paths in planar graphs. J. Graph Theory 7 (1983) 169-176.\n\n[Tu] W. T. Tutte, A theorem on planar graphs. Trans. Amer. Math. Soc. 82 (1956) 99-116.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Every 4-connected toroidal graph has a Hamilton cycle\" in Graph Theory; Topological Graph Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Grunbaum--Nash-Williams 4-connected toroidal Hamilton-cycle conjecture remains open.\n\n**Verified partial progress.**\n\n- Thomas--Yu proved every 5-connected toroidal graph Hamiltonian.\n- Thomas--Yu--Zang proved a Hamilton path in every 4-connected toroidal graph.\n\n**Full solution or refutation.**\n\nThe checked sources do not prove a Hamilton cycle for the full 4-connected class.\n\n**What remains.**\n\nUpgrade the Hamilton-path result or find a 4-connected toroidal non-Hamiltonian graph.\n\n**Sources checked.**\n\n- Graph-theory open problems, Every 4-connected toroidal graph has a Hamilton cycle (accessed 2026-08-17). (maintained_tracker): https://mlelarge.github.io/graph-conjectures/op/every_4_connected_toroidal_graph_has_a_hamilton_cycle/\n Evidence used: Explicitly records the conjecture as open and summarizes the 5-connected and Hamilton-path theorems.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3314, "problem_number": "OPG-177", "title": "Grunbaum's Conjecture", "statement": "Conjecture If $G$ is a simple loopless triangulation of an orientable surface, then the dual of $G$ is 3-edge-colorable.", "background": "Source: Open Problem Garden. Original node ID: 177. URL: http://www.openproblemgarden.org/op/grunbaums_conjecture.\n\nSource subject path: Graph Theory > Topological Graph Theory > Coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/grunbaums_conjecture\n- Author(s): Grunbaum, Branko\n- Subject(s): Graph Theory; Topological Graph Theory; Coloring\n- Keywords: coloring; surface\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 4th, 2007 by mdevos\n\nProblem-page discussion:\nThe Four Color Theorem is equivalent to the statement that every cubic planar graph with no bridge is 3-edge-colorable. This is precisely equivalent to Grunbaum's conjecture restricted to the plane. Thus, Grunbaum's conjecture, if true, would imply the Four Color Theorem. Indeed, this conjecture suggests a deep generalization of the 4-color theorem.\n\nDefinition: A cubic graph $G$ is a snark if $G$ is internally 4-edge-connected and $G$ is not 3-edge-colorable.\n\nGrunbaum's conjecture states that no snark is the dual of a simple loopless triangulation of an orientable surface. In this light, the conjecture looks to be almost obviously false. To find a counterexample, it suffices to embed a snark in an orientable surface so that the dual has no loops or parallel edges. Of course, the difficulty is in satisfying this last constraint. All known embeddings of snarks in orientable surfaces give rise to either loops or parallel edges in the dual. It is striking to compare this conjecture with the Orientable cycle double cover conjecture. Both conjectures may be stated in terms of embedding snarks in orientable surfaces as follows:\n\nConjecture (Grunbaum's conjecture (version 2)) Every embedding of a snark in an orientable surface has a cycle of length 1 or 2 (a loop or parallel edges) in the dual.\n\nConjecture (Orientable cycle double cover conjecture) Every snark may be embedded in an orientable surface so that the dual graph has no cycle of length 1 (no loop).\n\nIn this light it may look unlikely that both Grunbaum's conjecture and the orientable cycle double cover conjecture are true. I (M. DeVos) don't have a strong sense for or against either of these conjectures, and I don't believe there is a strong consensus among experts.\n\nMohar and Robertson have suggested the following weak version of Grunbaum's conjecture: There exists a fixed constant $k$ so that the dual of every loopless triangulation of an orientable surface of face-width $>k$ is 3-edge-colorable. Robertson has suggested that this may still hold true even for nonorientable surfaces. The following conjecture is a further weakening of Grunbaum's conjecture which would allow the parameter $k$ to depend on the surface. This is probably the weakest open problem in this vein.\n\nConjecture (Weak Grunbaum conjecture) For every orientable surface $\\Sigma$, there is a fixed constant $k$ so that the dual of every loopless triangulation of $\\Sigma$ with face-width $>k$ is 3-edge-colorable.\n\nBibliography:\n[G] B. Grunbaum, Conjecture 6. In Recent progress in combinatorics, (W.T. Tutte Ed.), Academic Press (1969) 343.\n\nSource links:\n- triangulation: http://en.wikipedia.org/wiki/triangulation (topology)\n- orientable surface: http://en.wikipedia.org/wiki/orientable surface\n\nDiscussion links:\n- Four Color Theorem: http://en.wikipedia.org/wiki/four color theorem\n- cubic: http://en.wikipedia.org/wiki/cubic graph\n- planar: http://en.wikipedia.org/wiki/planar graph\n- bridge: http://en.wikipedia.org/wiki/bridge (graph theory)\n- edge-colorable: http://en.wikipedia.org/wiki/edge coloring\n- snark: http://en.wikipedia.org/wiki/snark (graph theory)\n- Orientable cycle double cover conjecture: http://www.openproblemgarden.org/?q=op/cycle_double_cover_conjecture\n\nComments:\n- October 4th, 2007 | Anonymous | Grunbaum's conjecture is false!: Martin Kochol and Bojan Mohar announced a counterexample to Grunbaum's conjecture at the PIMS Workshop on the Cycle Double Cover Conjecture (Vancouver, 2007). By using Kochol's \"superposition\" operation on several copies of Petersen's graph, they constructed a snark which embeds on the orientable surface of genus 9, and whose dual contains no loops or parallel edges.\n\nOf course Grunbaum's Conjecture may still hold true for lower-genus surfaces, in particular, the torus.\n\nRef: Kochol, M; Mohar, B; preprint 2007.\n- March 23rd, 2009 | Anonymous | Solution: The solution of the Grunbaum's conjecture is published in (see also http://www.mat.savba.sk/~kochol):\n\nM. Kochol, 3-Regular non 3-edge-colorable graphs with polyhedral embeddings in orientable surfaces, in: Graph Drawing 2008, Editors: I.G. Tollis, M. Patrignani, Lecture Notes in Computer Science, Vol. 5417, Springer-Verlag, Berlin, 2009, pp. 319-323\n\nM. Kochol, Polyhedral embeddings of snarks in orientable surfaces, Proceedings of the American Mathematical Society vol. 137 (2009), pp. 1613-1619.\n- November 25th, 2007 | Anonymous | Wrong information. Martin: Wrong information. Martin Kochol\n- December 17th, 2007 | Anonymous | so?: Sorry I don't understand, is the conjecture false or still opened?\n\nNR\n- December 19th, 2007 | Anonymous | False in general: A counter example was found on the nine holed torus, and I have heard there is now one on the five holed torus as well so the conjecture is false in general. However what is still not known is for which n does the conjecture hold for the n-holed torus, and in particular the one holed torus is always of interest and remains open.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Grunbaum's Conjecture\" in Graph Theory; Topological Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Grünbaum's conjecture is false: counterexamples exist on every orientable surface of genus at least five.\n\n**Verified partial progress.**\n\n- The sphere case is the Four-Color Theorem.\n- The low-genus 1--4 cases remain a narrower open regime.\n\n**Full solution or refutation.**\n\nThe stated all-orientable-surfaces conjecture is refuted.\n\n**What remains.**\n\nStudy the surviving genus 1--4 cases or the weak face-width version.\n\n**Sources checked.**\n\n- M. Kochol, Complexities of 3-edge-coloring in the class of cubic graphs with a polyhedral embedding in an orientable surface, Discrete Applied Mathematics 158 (2010), 1856--1860. (primary): https://doi.org/10.1016/j.dam.2010.06.019\n Evidence used: States that the general conjecture is disproved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3315, "problem_number": "OPG-411", "title": "5-local-tensions", "statement": "Conjecture There exists a fixed constant $c$ (probably $c=4$ suffices) so that every embedded (loopless) graph with edge-width $\\ge c$ has a 5-local-tension.", "background": "Source: Open Problem Garden. Original node ID: 411. URL: http://www.openproblemgarden.org/op/5_local_tensions.\n\nSource subject path: Graph Theory > Topological Graph Theory > Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/5_local_tensions\n- Author(s): DeVos, Matt\n- Subject(s): Graph Theory; Topological Graph Theory; Coloring\n- Keywords: coloring; surface; tension\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 22nd, 2007 by mdevos\n\nProblem-page discussion:\nThe edge-width of an embedded graph is the length of the shortest non-contractible cycle.\n\nDefinition Let $G$ be a directed graph, let $\\Gamma$ be an abelian group, and let $\\phi: E(G) \\rightarrow \\Gamma$. Define the height of a walk $W$ to be the sum of $\\phi$ on the forward edges of $W$ minus the sum of $\\phi$ on the backward edges of $W$ (edges are counted according to multiplicity). We call $\\phi$ a tension if the height of every closed walk is zero, and if $G$ is an embedded graph, we call $\\phi$ a local-tension if the height of every closed walk which forms a contractible curve is zero. If in addition, $\\Gamma = {\\mathbb Z}$ and $0 < \\phi(e) < k$ for some $k \\in {\\mathbb Z}$, we say that $\\phi$ is a $k$-tension or a $k$-local-tension. If we reverse an edge $e$ and replace $\\phi(e)$ by $-\\phi(e)$, this preserves the properties of tension or local-tension. Accordingly, we say that an undirected graph (embedded graph) $G$ has a $k$-tension ( $k$-local-tension) if some and thus every orientation of it admits such a map.\n\nProposition A graph has a $k$-tension if and only if it is $k$-colorable.\n\nProof To see the \"if\" direction, let $f: V(G) \\rightarrow \\{0,\\ldots,k-1\\}$ be a coloring, orient the edges of $G$ arbitrarily, and defining $\\phi: E(G) \\rightarrow {\\mathbb Z}$ by the rule $\\phi(uv) = f(v) - f(u)$. It is straightforward to check that $\\phi$ is a $k$-tension. For the \"only if\" direction, let $\\phi: E(G) \\rightarrow {\\mathbb Z}$ be a $k$-tension. Now choose a point $u \\in V(G)$ and define the map $f: V(G) \\rightarrow {\\mathbb Z}_k$ by the rule that $f(v)$ is the height of some (and thus every) walk from $u$ to $v$ modulo $k$. Again, it is straightforward to check that this defines a proper $k$-coloring.\n\nFor graphs on orientable surfaces, local-tensions are dual to flows. More precisely, if $G$ and $G^*$ are dual graphs embedded in an orientable surface, then $G$ has a $k$-local-tension if and only if $G^*$ has a nowhere-zero $k$-flow. On non-orientable surfaces, there is a duality between $k$-local-tensions in $G$ and nowhere-zero $k$-flows in a bidirected $G^*$. Based on this duality we have a couple of conjectures. The first follows from Tutte's 5-flow conjecture, the second from Bouchet's 6-flow conjecture.\n\nConjecture (Tutte) Every loopless graph embedded in an orientable surface has a 5-local-tension.\n\nConjecture (Bouchet) Every loopless graph embedded in any surface has a 6-local-tension.\n\nSo although, graphs on surfaces may have high chromatic number, thanks to some partial results toward the above conjectures, we know that they always have small local-tensions. For orientable surfaces, there is a famous Conjecture of Grunbaum which is equivalent to the following.\n\nConjecture (Grunbaum) If $G$ is a simple loopless graph embedded in an orientable surface with edge-width $\\ge 3$, then $G$ has a 4-local-tension.\n\nOn non-orientable surfaces, it is known that there are graphs of arbitrarily high edge-width which do not admit 4-local-tensions (see [DGMVZ]). However, it remains open whether sufficiently high edge-width forces the existence of a 5-local-tension. Indeed, as suggested by the conjecture at the start of this page, it may be that edge-width at least 4 is enough. Edge-width 3 does not suffice since the embedding of $K_6$ in the projective plane does not admit a 5-local-tension.\n\nBibliography:\n*[DGMVZ] M. DeVos, L. Goddyn, B. Mohar, D. Vertigan, and X. Zhu, Coloring-flow duality of embedded graphs. Trans. Amer. Math. Soc. 357 (2005), no. 10 MathSciNet\n\nRelated:\nRelated problems\nBouchet's 6-flow conjecture\nGrunbaum's Conjecture\n5-flow conjecture\n\nDiscussion links:\n- Tutte's 5-flow conjecture: http://www.openproblemgarden.org/?q=node/126\n- Bouchet's 6-flow conjecture: http://www.openproblemgarden.org/?q=node/131\n- Conjecture of Grunbaum: http://www.openproblemgarden.org/?q=node/177\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2159697\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"5-local-tensions\" in Graph Theory; Topological Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No universal edge-width constant guaranteeing a 5-local-tension was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nDevelop surface-local tension constructions at high edge-width.\n\n**Sources checked.**\n\n- Open Problem Garden, node 411 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3316, "problem_number": "OPG-798", "title": "Degenerate colorings of planar graphs", "statement": "A graph $G$ is $k$-degenerate if every subgraph of $G$ has a vertex of degree $\\le k$.\n\nConjecture Every simple planar graph has a 5-coloring so that for $1 \\le k \\le 4$, the union of any $k$ color classes induces a $(k-1)$-degenerate graph.", "background": "Source: Open Problem Garden. Original node ID: 798. URL: http://www.openproblemgarden.org/op/degenerate_colorings_of_planar_graphs.\n\nSource subject path: Graph Theory > Topological Graph Theory > Coloring.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/degenerate_colorings_of_planar_graphs\n- Author(s): Borodin, Oleg V.\n- Subject(s): Graph Theory; Topological Graph Theory; Coloring\n- Keywords: coloring; degenerate; planar\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 21st, 2008 by mdevos\n\nProblem-page discussion:\nAn acyclic coloring of a graph $G$ is a proper coloring with the added property that the union of any two color classes induces a forest. Grunbaum famously conjectured that every simple planar graph has an acyclic 5-coloring. Following a sequence of partial results, Borodin [B] resolved this conjecture with an impressive and detailed argument. In the same paper, Borodin made the above conjecture, which, if true, would give a stronger result (as forests are precisely the 1-degenerate graphs).\n\nA degenerate coloring of a graph $G$ is a proper coloring with the added property that the union of any $k$ color classes induces a $(k-1)$-degenerate graph. A planar graph of minimum degree 5 cannot have a degenerate 5-coloring, but if the above conjecture holds, something just short of this is true. Rautenbach [R] proved that every planar graph has a degenerate 18-coloring, and recently, Mohar, Spacepan, and Zhu showed that every planar graph has a degenerate 9-coloring.\n\nBibliography:\n*[B] O. V. Borodin, A proof of B. Grünbaum's conjecture on the acyclic $5$-colorability of planar graphs. Dokl. Akad. Nauk SSSR 231 (1976), no. 1, 18--20. MathSciNet\n\n[R] D. Rautenbach, A conjecture of Borodin and a coloring of Grünbaum. Fifth Cracow Conference on Graph Theory USTRON '06, 187--194 Electron. Notes Discrete Math., 24, Elsevier, Amsterdam, 2006.\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0447031\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Degenerate colorings of planar graphs\" in Graph Theory; Topological Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Borodin's five-color degenerate-coloring conjecture remains open; planar graphs are known to have degenerate list colorings with at most nine colors.\n\n**Verified partial progress.**\n\n- Kierstead--Mohar--Spacapan--Yang--Zhu prove two-coloring number at most nine.\n- This implies degenerate list chromatic number at most nine for planar graphs.\n\n**Full solution or refutation.**\n\nNo degenerate five-coloring theorem for every planar graph was verified.\n\n**What remains.**\n\nReduce the nine-color bound to five or refute the stated conjecture.\n\n**Sources checked.**\n\n- H. Kierstead, B. Mohar, S. Spacapan, D. Yang and X. Zhu, The two-coloring number and degenerate colorings of planar graphs, SIAM J. Discrete Math. 23 (2009), 1548--1560. (primary): https://mathstat.dal.ca/~rjn/K2.pdf\n Evidence used: Proves the upper bound yielding degenerate list chromatic number at most nine.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3317, "problem_number": "OPG-34915", "title": "3-Colourability of Arrangements of Great Circles", "statement": "Consider a set $S$ of great circles on a sphere with no three circles meeting at a point. The arrangement graph of $S$ has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.\n\nConjecture Every arrangement graph of a set of great circles is $3$-colourable.", "background": "Source: Open Problem Garden. Original node ID: 34915. URL: http://www.openproblemgarden.org/op/3_colourability_of_arrangements_of_great_circles.\n\nSource subject path: Graph Theory > Topological Graph Theory > Coloring.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/3_colourability_of_arrangements_of_great_circles\n- Author(s): Felsner, Stefan; Hurtado, Ferran; Noy, Marc; Streinu, Ileana\n- Subject(s): Graph Theory; Topological Graph Theory; Coloring\n- Keywords: arrangement graph; graph coloring\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: January 19th, 2009 by David Wood\n\nProblem-page discussion:\nIt is NP-complete to test 3-colourability of planar 4-regular graphs in general [D80].\n\nArrangement graphs of general circles on the sphere can require four colors [K90].\n\nA stronger conjecture states that the arrangement graph of every set of great circles is $3$-choosable. A natural approach is to use the machinery of [AT92].\n\nPreviously appeared here.\n\nBibliography:\n[AT92] Noga Alon and Michael Tarsi. Colourings and orientations of graphs. Combinatorica 12:125--134, 1992.\n\n*[FHNS00] Stefan Felsner, Ferran Hurtado, Marc Noy, and Ileana Streinu. Hamiltonicity and colorings of arrangement graphs. In Proc. 11th Annual ACM-SIAM Symp. Discrete Algorithms (SODA), pages 155--164, January 2000.\n\n[FHNS06] Felsner, Stefan; Hurtado, Ferran; Noy, Marc; Streinu, Ileana. Hamiltonicity and colorings of arrangement graphs. Discrete Appl. Math. 154 (2006), no. 17, 2470--2483.\n\n[K90] G. Koester. 4-critical, 4-valent planar graphs constructed with crowns. Math. Scand., 67:15--22, 1990.\n\n[D80] D. P. Dailey. Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete, Discrete Math. 30:289--193, 2980.\n\nDiscussion links:\n- $3$-choosable: http://en.wikipedia.org/wiki/List_coloring\n- here: http://maven.smith.edu/%7Eorourke/TOPP/P44.html#Problem.44\n\nComments:\n- September 8th, 2014 | Anonymous | Answer to the conjecture: This problem has been solved. See here: http://arxiv.org/abs/math/0408363. - Anthony Hernandez\n- October 22nd, 2020 | Anonymous | Conjecture still open: The conjecture is still open. I'm not sure what Cahit's arxiv preprint (http://arxiv.org/abs/math/0408363) contains, but it certainly does not contain a proof. - Manfred Scheucher\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"3-Colourability of Arrangements of Great Circles\" in Graph Theory; Topological Graph Theory; Coloring, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Great-circle arrangement graphs are still treated as an open 3-colorability conjecture; finite verification and related circle-arrangement results provide partial progress.\n\n**Verified partial progress.**\n\n- All arrangements through eleven great circles were computationally verified 3-colorable.\n- Recent work studies general circle/pseudocircle arrangements and distinguishes the special great-circle case.\n\n**Full solution or refutation.**\n\nA 2004 preprint claiming a proof was found, but current authoritative problem sources continue to list the conjecture open, so no full resolution is accepted.\n\n**What remains.**\n\nGive a broadly accepted proof of 3-colorability or a great-circle 4-chromatic example.\n\n**Sources checked.**\n\n- The Open Problems Project, Problem 44: 3-Colorability of Arrangements of Great Circles (accessed 2026-08-17). (maintained_tracker): https://topp.openproblem.net/p44\n Evidence used: Lists the problem as open and records finite verification through eleven circles.\n- Coloring circle arrangements: New 4-chromatic planar graphs, arXiv:2205.08181. (primary): https://arxiv.org/abs/2205.08181\n Evidence used: States the great-circle 3-colorability statement as the motivating conjecture.\n\n**Review notes.** Unaccepted claimed proof flagged; source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3318, "problem_number": "OPG-307", "title": "The Crossing Number of the Complete Graph", "statement": "The crossing number $cr(G)$ of $G$ is the minimum number of crossings in all drawings of $G$ in the plane.\n\nConjecture $\\displaystyle cr(K_n) = \\frac 14 \\floor{\\frac n2} \\floor{\\frac{n-1}2} \\floor{\\frac{n-2}2} \\floor{\\frac{n-3}2}$", "background": "Source: Open Problem Garden. Original node ID: 307. URL: http://www.openproblemgarden.org/op/the_crossing_number_of_the_complete_graph.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_crossing_number_of_the_complete_graph\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: complete graph; crossing number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 11th, 2007 by Robert Samal\n\nProblem-page discussion:\n(This discussion appears as [M].)\n\nA drawing of a graph $G$ in the plane has the vertices represented by distinct points and the edges represented by polygonal lines joining their endpoints such that:\n\n- no edge contains a vertex other than its endpoints,\n- no two adjacent edges share a point other than their common endpoint,\n- two nonadjacent edges share at most one point at which they cross transversally, and\n- no three edges cross at the same point.\n\nThe conjectured value for the crossing number of $K_n$ is known to be an upper bound. This is shown by exhibiting a drawing with that number of crossings. If $n = 2m$, place $m$ vertices regularly spaced along two circles of radii 1 and 2, respectively. Two vertices on the inner circle are connected by a straight line; two vertices on the outer circle are connected by a polygonal line outside the circle. A vertex on the inner circle is connected to one on the outer circle with a polygonal line segment of minimum possible positive winding angle around the cylinder. A simple count shows that the number of crossings in such a drawing achieves the conjectured minimum. For $n = 2m-1$ we delete one vertex from the drawing described and achieve the conjectured minimum.\n\nThe conjecture is known to be true for $n$ at most 10 [G]. If the conjecture is true for $n = 2m$, then it is also true for $n-1$. This follows from an argument counting the number of crossings in drawings of all $K_{n-1}$ 's contained in an optimal drawing of $K_n$.\n\nIt would also be interesting to prove that the conjectured upper bound is asymptotically correct, that is, that $\\lim \\frac{cr(K_n)}{\\binom{n}4} = \\frac38$.\n\nThe best known lower bound is due to Kleitman [K], who showed that this limit is at least $3/10$.\n\nBibliography:\n[G] R. Guy, The decline and fall of Zarankiewicz's theorem, in Proof Techniques in Graph Theory (F. Harary Ed.), Academic Press, New York (1969) 63-69.\n\n[K] D. Kleitman, The crossing number of $K_{5,n}$, J. Combin. Theory 9 (1970) 315-323.\n\n[M] B. Mohar, Problem of the Month\n\nRelated:\nRelated problems\nThe Crossing Number of the Complete Bipartite Graph\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P1CrossingNumberKn.html\n\nComments:\n- July 28th, 2008 | Anonymous | On lower bound.: It has been shown that $\\lim_{n\\rightarrow\\infty}\\frac{cr(K_n)}{Z(n)}\\geq 0.83$, where $Z(n)$ is the conjectured value. For the proof, see de Klerk, E.; Maharry, J.; Pasechnik, D. V.; Richter, R. B.; Salazar, G. Improved bounds for the crossing numbers of $K\\sb {m,n}$ and $K\\sb n$. (2007).\n- July 28th, 2008 | Anonymous | This is not the best known bound.: The same article below, that proves that $cr(K_{11})=100$ and $cr(K_{12})=150$, states that $0.8594 \\cdot Z(n)\\leq cr(K_n) \\leq Z(n)$.\n- July 10th, 2007 | Robert Samal | true upto n=12: The conjecture was recently verified for n=11 and 12 (The Crossing Number of $K_{11}$ Is 100 by Shengjun Pan and R. Bruce Richter).\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"The Crossing Number of the Complete Graph\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Harary--Hill formula for cr(K_n) remains open in general.\n\n**Verified partial progress.**\n\n- It is proved for selected n and gives the accepted extremal construction bound.\n\n**Full solution or refutation.**\n\nNo all-n equality proof was verified.\n\n**What remains.**\n\nClose the lower-bound gap for complete graphs.\n\n**Sources checked.**\n\n- Open Problem Garden, node 307 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the crossing-number conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3319, "problem_number": "OPG-310", "title": "The Crossing Number of the Complete Bipartite Graph", "statement": "The crossing number $cr(G)$ of $G$ is the minimum number of crossings in all drawings of $G$ in the plane.\n\nConjecture $\\displaystyle cr(K_{m,n}) = \\floor{\\frac m2} \\floor{\\frac {m-1}2} \\floor{\\frac n2} \\floor{\\frac {n-1}2}$", "background": "Source: Open Problem Garden. Original node ID: 310. URL: http://www.openproblemgarden.org/op/the_crossing_number_of_the_complete_bipartite_graph.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_crossing_number_of_the_complete_bipartite_graph\n- Author(s): Turan, Paul\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: complete bipartite graph; crossing number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 11th, 2007 by Robert Samal\n\nProblem-page discussion:\n(This discussion appears as [M].)\n\nA drawing of a graph $G$ in the plane has the vertices represented by distinct points and the edges represented by polygonal lines joining their endpoints such that:\n\n- no edge contains a vertex other than its endpoints,\n- no two adjacent edges share a point other than their common endpoint,\n- two nonadjacent edges share at most one point at which they cross transversally, and\n- no three edges cross at the same point.\n\nThis problem is also known as Turan's Brickyard Problem (since it was formulated by Turan when he was working at a brickyard - the edges of the drawing would correspond to train tracks connecting different shipping depots, and fewer crossings would mean smaller chance for collision of little trains and smaller chance for their derailing).\n\nThis conjectured value for the crossing number of $K_{m,n}$ can be realized by the following drawing. Place $\\ceil{n/2}$ vertices on the positive $x$-axis and $\\floor{n/2}$ vertices on the negative $x$-axis. Similarly, place $\\ceil{m/2}$ and $\\floor{m/2}$ along the positive and negative $y$-axis. Now connect each pair of vertices on different axes with straight line segments.\n\nBibliography:\n[G] R. Guy, The decline and fall of Zarankiewicz's theorem, in Proof Techniques in Graph Theory (F. Harary Ed.), Academic Press, New York (1969) 63-69.\n\n[K] D. Kleitman, The crossing number of K_{5,n}, J. Combin. Theory 9 (1970) 315-32\n\n[M] B. Mohar, Problem of the Month\n\nRelated:\nRelated problems\nThe Crossing Number of the Complete Graph\n\nBibliography links:\n- Problem of the Month: http://www.fmf.uni-lj.si/%7Emohar/Problems/P2CrossingNumberKmn.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"The Crossing Number of the Complete Bipartite Graph\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Zarankiewicz's formula for cr(K_m,n) remains open in general.\n\n**Verified partial progress.**\n\n- Many fixed-small-m cases and computational bounds are known.\n\n**Full solution or refutation.**\n\nNo all-pair equality proof was verified.\n\n**What remains.**\n\nEstablish matching lower bounds for arbitrary m,n.\n\n**Sources checked.**\n\n- Open Problem Garden, node 310 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the complete-bipartite crossing-number conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3320, "problem_number": "OPG-313", "title": "The Crossing Number of the Hypercube", "statement": "The crossing number $cr(G)$ of $G$ is the minimum number of crossings in all drawings of $G$ in the plane.\n\nThe $d$-dimensional (hyper)cube $Q_d$ is the graph whose vertices are all binary sequences of length $d$, and two of the sequences are adjacent in $Q_d$ if they differ in precisely one coordinate.\n\nConjecture $\\displaystyle \\lim \\frac{cr(Q_d)}{4^d} = \\frac{5}{32}$", "background": "Source: Open Problem Garden. Original node ID: 313. URL: http://www.openproblemgarden.org/op/the_crossing_number_of_the_hypercube.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_crossing_number_of_the_hypercube\n- Author(s): Erdos, Paul; Guy, Richard K.\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: crossing number; hypercube\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 11th, 2007 by Robert Samal\n\nProblem-page discussion:\nIt is known that $cr(Q_d) = 0$ for $d = 1,2,3$ and that $cr(Q_4) = 8$. No other exact values are known. Madej [M] proved that $cr(Q_d) \\le 4^d/6 + o(4^d/6)$. Faria and de Figueiredo [FF] improved the upper bound to $(165/1024) 4^d$. Sykora and Vrto [SV] proved that $4^d/20 + o(4^d/20)$ is a lower bound on $cr(Q_d)$.\n\nBibliography:\n*[EG] P. Erdős and R.K. Guy, Crossing number problems, Amer. Math. Monthly 80 (1973) 52-58.\n\n[FF] L. Faria, C.M.H. de Figueiredo, On Eggleton and Guy's conjectured upper bound for the crossing number of the $n$-cube, Math. Slovaca 50 (2000) 271-287.\n\n[M] T. Madej, Bounds for the crossing number of the $n$-cube, J. Graph Theory 15 (1991) 81-97.\n\n[SV] O. Sykora and I. Vrto, On crossing numbers of hypercubes and cube connected cycles, BIT 33 (1993) 232-237.\n\nRelated:\nRelated problems\nThe Crossing Number of the Complete Graph\nThe Crossing Number of the Complete Bipartite Graph\n\nComments:\n- July 18th, 2008 | Robert Samal | Improved upper bound: I came accross a paper Faria, Herrera de Figueiredo, Sykora, Vrto: An improved upper bound on the crossing number of the hypercube that proves half of this, getting the correct upper bound.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"The Crossing Number of the Hypercube\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of the proposed asymptotic hypercube crossing-number constant 5/32 was verified.\n\n**Verified partial progress.**\n\n- Construction and lower-bound methods give nonmatching asymptotic bounds.\n\n**Full solution or refutation.**\n\nThe limiting constant conjecture remains open.\n\n**What remains.**\n\nSharpen crossing-number bounds for Q_d.\n\n**Sources checked.**\n\n- Open Problem Garden, node 313 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3321, "problem_number": "OPG-322", "title": "Drawing disconnected graphs on surfaces", "statement": "Conjecture Let $G$ be the disjoint union of the graphs $G_1$ and $G_2$ and let $\\Sigma$ be a surface. Is it true that every optimal drawing of $G$ on $\\Sigma$ has the property that $G_1$ and $G_2$ are disjoint?", "background": "Source: Open Problem Garden. Original node ID: 322. URL: http://www.openproblemgarden.org/op/drawing_disconnected_graphs_on_surfaces.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/drawing_disconnected_graphs_on_surfaces\n- Author(s): DeVos, Matt; Mohar, Bojan; Samal, Robert\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: crossing number; surface\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 12th, 2007 by mdevos\n\nProblem-page discussion:\nWe insist on the usual restrictions for drawings (as in The Crossing Number of the Complete Graph).\n\nAlthough both crossing numbers and embeddings of graphs on general surfaces are rich and well-studied subjects, their common generalization - drawing graphs on general surfaces has received very little attention. The question highlighted here appears to be quite basic in nature, but due to the combined difficulties of crossings and general surfaces, it may be quite difficult to resolve.\n\nThis conjecture is trivially true when $\\Sigma$ is the plane, and DeVos, Mohar, and Samal have proved that it also holds when $\\Sigma$ is the projective plane. It is open for all other surfaces to the best of my (M. DeVos) knowledge.\n\nDiscussion links:\n- The Crossing Number of the Complete Graph: http://www.openproblemgarden.org/?q=op/the_crossing_number_of_the_complete_graph\n\nComments:\n- September 30th, 2007 | lbeaudou | Drawing disconnected graphs on surfaces: any reference?: Hello,\n\nYou don't mention reference in this problem, though it is said that some work has been made. Would it be possible to know how the projective plane has been shown to verify this conjecture?\n\nThanks in advance for any piece of information.\n\nLaurent Beaudou\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Drawing disconnected graphs on surfaces\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No theorem that every surface-optimal drawing of a disconnected graph keeps components disjoint was verified.\n\n**Verified partial progress.**\n\n- The issue is specific to optimal drawings on arbitrary surfaces.\n\n**Full solution or refutation.**\n\nThe conjecture remains open.\n\n**What remains.**\n\nAnalyze surgery separating components without increasing crossings.\n\n**Sources checked.**\n\n- Open Problem Garden, node 322 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3322, "problem_number": "OPG-1812", "title": "Crossing sequences", "statement": "Conjecture Let $(a_0,a_1,a_2,\\ldots,0)$ be a sequence of nonnegative integers which strictly decreases until $0$.\n\nThen there exists a graph that be drawn on a surface with orientable (nonorientable, resp.) genus $i$ with $a_i$ crossings, but not with less crossings.", "background": "Source: Open Problem Garden. Original node ID: 1812. URL: http://www.openproblemgarden.org/op/crossing_sequences.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/crossing_sequences\n- Author(s): Archdeacon, Dan; Bonnington, C. Paul; Siran, Jozef\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: crossing number; crossing sequence\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 30th, 2008 by Robert Samal\n\nProblem-page discussion:\nThis actually are two conjectures, one for the orientable case and another for nonorientable one. For sequences $(a_0,a_1,0)$ the nonorientable case was resolved in [ABS] and the orientable one in [DMS].\n\nThe conclusion also holds (for the orientable case) whenever the sequence $(a_i)$ is convex [S], that is whenever $a_i - a_{i-1}$ is nonincreasing. It might seem that this condition is also necessary: For the most extreme sequence $(N,N-1,0)$ (suggested by Salazar) one needs to construct a graph for which adding one handle saves just one crossing, while adding another saves many -- but then why not add the second handle first? Somewhat surprisingly, graphs with this counterintuitive property exist, at least for sequences $(a_0,a_1,0)$.\n\nAn interesting open case is to consider sequences for which $$a_0 - a_s < \\varepsilon (a_s - a_{s+1})$$for some$s$and small$\\varepsilon$.\n\nBibliography:\n*[ABS] Dan Archdeacon, C. Paul Bonnington, and Jozef Siran, Trading crossings for handles and crosscaps, J.Graph Theory 38 (2001), 230--243.\n\n[DMS] Matt DeVos, Bojan Mohar, Robert Samal, Unexpected behaviour of crossing sequences, in preparation\n\n[S] Jozef Siran, The crossing function of a graph, Abh. Math. Sem. Univ. Hamburg 53 (1983), 131--133.\n\nRelated:\nRelated problems\nDrawing disconnected graphs on surfaces\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Crossing sequences\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Every strictly decreasing length-three sequence (a,b,0) is realizable in both the orientable and nonorientable settings, and decreasing convex sequences are known; arbitrary longer nonconvex sequences remain open.\n\n**Verified partial progress.**\n\n- Archdeacon, Bonnington, and Širáň prove every (a,b,0) in the nonorientable case.\n- DeVos, Mohar, and Šámal prove every (a,b,0) in the orientable case.\n- Širáň's earlier construction realizes every decreasing convex sequence.\n\n**Full solution or refutation.**\n\nThe conjecture is settled for sequences ending after two positive entries and for convex sequences, but not for all strictly decreasing sequences.\n\n**What remains.**\n\nRealize arbitrary longer nonconvex sequences in each of the orientable and nonorientable settings or find an obstruction.\n\n**Sources checked.**\n\n- Dan Archdeacon, C. Paul Bonnington, and Jozef Širáň, Trading crossings for handles and crosscaps, Journal of Graph Theory 38 (2001), 230–243, DOI 10.1002/jgt.10009. (primary): https://doi.org/10.1002/jgt.10009\n Evidence used: The abstract proves arbitrary nonorientable (a,b,0) sequences and states the full orientable/nonorientable conjecture.\n- Matt DeVos, Bojan Mohar, and Robert Šámal, Unexpected behaviour of crossing sequences, Journal of Combinatorial Theory, Series B 101 (2011), 448–463, DOI 10.1016/j.jctb.2010.12.002; arXiv:0911.0452. (primary): https://arxiv.org/abs/0911.0452\n Evidence used: The abstract proves that every a>b>0 occurs as the orientable crossing sequence (a,b,0).\n- Open Problem Garden, Crossing sequences (node 1812), accessed 2026-08-17. (maintained_tracker): https://openproblemgarden.org/op/crossing_sequences\n Evidence used: Records the length-three and convex cases and identifies longer highly nonconvex cases as open.\n\n**Review notes.** Statement defect preserved: 'there exists a graph that be drawn' is missing 'can'.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3323, "problem_number": "OPG-37068", "title": "Crossing numbers and coloring", "statement": "We let $cr(G)$ denote the crossing number of a graph $G$.\n\nConjecture Every graph $G$ with $\\chi(G) \\ge t$ satisfies $cr(G) \\ge cr(K_t)$.", "background": "Source: Open Problem Garden. Original node ID: 37068. URL: http://www.openproblemgarden.org/op/crossing_numbers_and_coloring.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/crossing_numbers_and_coloring\n- Author(s): Albertson, Michael O.\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: coloring; complete graph; crossing number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 4th, 2009 by mdevos\n\nProblem-page discussion:\nThis conjecture is an interesting weakening of the disproved Hajos Conjecture which asserted that $\\chi(G) \\ge t$ implies that $G$ contains a subdivision of $K_t$.\n\nA minimal counterexample to Albertson's conjecture is critical, with minimum degree $\\ge t$. Using this and the crossing lemma, Albertson, Cranston and Fox showed that a minimum counterexample has at most $4t$ vertices. They then analyzed small cases to show that the conjecture holds for $t \\le 12$. More recently, Barat and Toth [BT] sharpened these arguments to show that the conjecture holds for $t \\le 16$.\n\nBibliography:\n[BT] J. Barat and G. Toth, Towards the Albertson Conjecture\n\nSource links:\n- crossing number: http://en.wikipedia.org/wiki/crossing number (graph theory)\n\nBibliography links:\n- Towards the Albertson Conjecture: http://arxiv1.library.cornell.edu/abs/0909.0413\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Crossing numbers and coloring\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Albertson's conjecture is proved through chromatic number 24 and in broad order ranges, but remains open in general.\n\n**Verified partial progress.**\n\n- Cranston verifies the conjecture for r <= 24 and restricts any counterexample for r in {25,26} to three (r,|G|) possibilities.\n- Fox, Pach, and Suk prove the conjecture for r-chromatic graphs on at most (1.64-o(1))r vertices.\n\n**Full solution or refutation.**\n\nNo proof for arbitrary chromatic number was verified.\n\n**What remains.**\n\nEliminate the remaining r >= 25 critical graphs, including the few unresolved small orders and the general middle-order regimes.\n\n**Sources checked.**\n\n- D. W. Cranston, Progress on Albertson's Conjecture, arXiv:2512.08020 (2025). (primary): https://arxiv.org/abs/2512.08020\n Evidence used: The abstract and main theorems prove the conjecture through r=24 and sharply restrict r=25,26.\n- J. Fox, J. Pach, and A. Suk, Immersions and Albertson's conjecture, arXiv:2510.05893 (2025). (primary): https://arxiv.org/abs/2510.05893\n Evidence used: Proves the conjecture in the stated asymptotic small-order regime.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3324, "problem_number": "OPG-37117", "title": "Are different notions of the crossing number the same?", "statement": "Problem Does the following equality hold for every graph $G$?\n$$\n\\text{pair-cr}(G) = \\text{cr}(G)\n$$\n\nThe crossing number $\\text{cr}(G)$ of a graph $G$ is the minimum number of edge crossings in any drawing of $G$ in the plane. In the pairwise crossing number $\\text{pair-cr}(G)$, we minimize the number of pairs of edges that cross.", "background": "Source: Open Problem Garden. Original node ID: 37117. URL: http://www.openproblemgarden.org/op/are_different_notions_of_the_crossing_number_the_same.\n\nSource subject path: Graph Theory > Topological Graph Theory > Crossing numbers.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_different_notions_of_the_crossing_number_the_same\n- Author(s): Pach, János; Tóth, Géza\n- Subject(s): Graph Theory; Topological Graph Theory; Crossing numbers\n- Keywords: crossing number; pair-crossing number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 3rd, 2009 by cibulka\n\nProblem-page discussion:\nObviously we have $\\text{pair-cr}(G) \\leq \\text{cr}(G)$.\n\nThe problem was first posed by Pach and Tóth in~[PT], who first spotted the possibility that the pairwise crossing number might be different from the crossing number. They proved $\\text{cr}(G) \\leq 2k^2$ for graphs with pairwise crossing number $k$, which was later improved by Valtr~[V05] to $O(k^2/ \\log(k))$ and by Tóth~[T08] to $O(k^2/ \\log^2(k))$.\n\nBibliography:\n*[PT] János Pach, Géza Tóth, Which crossing number is it anyway?, Journal of Combinatorial Theory Series B 80 (2000), no. 2, 225--246. MathSciNet\n\n[V05] Pavel Valtr, On the pair-crossing number, Combinatorial and computational geometry, 52 (2005), 569--575. MathSciNet\n\n[T08] Géza Tóth, Note on the pair-crossing number and the odd-crossing number, Discrete Comput. Geom., 39 (2008), no. 4, 791--799. MathSciNet\n\nSource links:\n- crossing number: http://en.wikipedia.org/wiki/Crossing number (graph theory)\n\nBibliography links:\n- Which crossing number is it anyway?: http://www.cs.bme.hu/%7Egeza/whichcrossing.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1794693\n- On the pair-crossing number: http://www.msri.org/communications/books/Book52/files/31valtr.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2178340\n- Note on the pair-crossing number and the odd-crossing number: http://www.cs.bme.hu/%7Egeza/pair-cr.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2413161\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Are different notions of the crossing number the same?\" in Graph Theory; Topological Graph Theory; Crossing numbers, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** It remains open whether pair-crossing number always equals crossing number, but the best general comparison is now cr(G)=O(pcr(G)^{3/2}).\n\n**Verified partial progress.**\n\n- Solé Pi removes the logarithmic factor from the preceding O(pcr^{3/2} log pcr) comparison.\n- The same 2025 paper explicitly describes the equality question as an important open problem.\n\n**Full solution or refutation.**\n\nNo graph separating the two parameters and no proof of equality for all graphs was verified.\n\n**What remains.**\n\nEither transform every pair-crossing-optimal drawing into one with no repeated crossings without increasing the pair count, or construct a graph with pcr(G) < cr(G).\n\n**Sources checked.**\n\n- O. Solé Pi, Pair Crossing Number, Cutwidth, and Good Drawings on Arbitrary Point Sets, Discrete Comput. Geom. 73 (2025), 310-326. (primary): https://doi.org/10.1007/s00454-024-00708-z\n Evidence used: States the equality question is open and proves cr(G)=O(pcr(G)^{3/2}).\n- J. Karl and G. Tóth, A slightly better bound on the crossing number in terms of the pair-crossing number, arXiv:2105.14319 (2021). (primary): https://arxiv.org/abs/2105.14319\n Evidence used: Provides the prior O(pcr^{3/2} log pcr) benchmark improved by Solé Pi.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3325, "problem_number": "OPG-326", "title": "Universal point sets for planar graphs", "statement": "We say that a set $P \\subseteq {\\mathbb R}^2$ is $n$-universal if every $n$ vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in $P$, and all edges are (non-intersecting) straight line segments.\n\nQuestion Does there exist an $n$-universal set of size $O(n)$?", "background": "Source: Open Problem Garden. Original node ID: 326. URL: http://www.openproblemgarden.org/op/small_universal_point_sets_for_planar_graphs.\n\nSource subject path: Graph Theory > Topological Graph Theory > Drawings.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/small_universal_point_sets_for_planar_graphs\n- Author(s): Mohar, Bojan\n- Subject(s): Graph Theory; Topological Graph Theory; Drawings\n- Keywords: geometric graph; planar graph; universal set\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 22nd, 2007 by mdevos\n\nProblem-page discussion:\nMore generally, if we let $f(n)$ denote the size of the smallest $n$-universal set, we are interested in the behaviour of $f$. The best known upper bound is $f(n) = O(n^2)$. Indeed, every $n$-vertex planar graph can be drawn as required in the $n \\times n$ grid [dFPP], [S]. On the flip side, it is known that $f(n) \\ge 1.098n$ for sufficiently large $n$ [CH].\n\nBibliography:\n[CH] M. Chrobak and H.Karloff. A lower bound on the size of universal sets for planar graphs. SIGACT News, 20:83-86, 1989.\n\n[dFPP] H. de Fraysseix, J. Pach, and R. Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41-51, 1990. MathSciNet\n\n[S] W. Schnyder. Embedding planar graphs on the grid. In Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pages 138-148, 1990.\n\nBibliography links:\n- How to draw a planar graph on a grid: http://www.springerlink.com/content/b471830661k10534/\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1075065\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Universal point sets for planar graphs\" in Graph Theory; Topological Graph Theory; Drawings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Linear-size universal point sets are known for important planar subclasses, but no O(n) set is known for all n-vertex planar graphs.\n\n**Verified partial progress.**\n\n- Bipartite planar graphs admit universal sets of size 2n-2.\n- General planar graphs have improved but still superlinear upper bounds.\n\n**Full solution or refutation.**\n\nThe all-planar linear-size question remains open.\n\n**What remains.**\n\nExtend subclass constructions to arbitrary planar graphs or prove a superlinear lower bound.\n\n**Sources checked.**\n\n- B. F. Alam et al., Linear Size Universal Point Sets for Classes of Planar Graphs, arXiv:2303.00109. (primary): https://arxiv.org/abs/2303.00109\n Evidence used: Gives a 2n-2 universal point set for bipartite planar graphs.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3326, "problem_number": "OPG-596", "title": "Linear Hypergraphs with Dimension 3", "statement": "Conjecture Any linear hypergraph with incidence poset of dimension at most 3 is the intersection hypergraph of a family of triangles and segments in the plane.", "background": "Source: Open Problem Garden. Original node ID: 596. URL: http://www.openproblemgarden.org/op/linear_hypergraphs_with_dimension_3.\n\nSource subject path: Graph Theory > Topological Graph Theory > Drawings.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/linear_hypergraphs_with_dimension_3\n- Author(s): de Fraysseix, Hubert; Ossona de Mendez, Patrice; Rosenstiehl, Pierre\n- Subject(s): Graph Theory; Topological Graph Theory; Drawings\n- Keywords: Hypergraphs\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 26th, 2007 by taxipom\n\nProblem-page discussion:\nA hypergraph is linear if any two edges may share at most one vertex. The incidence poset of a hypergraph is the vertex-edge inclusion poset. The dimension of a poset $P$ is the minimum number of linear extentions of $P$, whose intersection is $P$ [DM].\n\nSchnyder proved that the incidence poset of a graph $G$ has dimension at most $3$ if and only if $G$ is planar [S89]. Fraysseix, Rosenstiehl and Ossona de Mendez proved that every planar graph has a representation by contacts of triangles [FOR94] and Scheinerman conjectured that every planar graph has a representation by intersection of segments [S84] (claimed to be proved by Gonçalves et al.).\n\nA hypergraph is planar if its vertex-edge incidence graph is planar [W]. Fraysseix, Rosenstiehl and Ossona de Mendez proved that every planar linear hypergraph has a representation by contacts of triangles [FOR07] and it has been conjectured that they have a representation by intersection of straight line segments [FO07] (cf Straight line representation of planar linear hypergraphs).\n\nAlthough the incidence poset of a simple planar hypergraph has dimension at most $3$ (what follows from [BT]), the converse is false: The linear hypergraph with vertices $1,\\dots,5$ and edge set $\\{\\{1,2\\},\\{2,3\\},\\{3,4\\},\\{1,4\\},\\{1,3,5\\},\\{2,4,5\\}\\}$ has incidence dimension $3$ but is not planar (its vertex-edge incidence graph is a subdivision of $K_{3,3}$ ). It follows from [O] that the vertices of simple hypergraphs with incidence posets of dimensions $d$ can be represented by convex sets of the Euclidean space of dimension $d-1$, in such a way that the edges of the hypergraph are exactly the maximal subsets of vertices, such that the corresponding subset of convexes has a non-empty intersection.\n\nBibliography:\n[BT] G.~Brightwell and W.T. Trotter, The order dimension of planar maps, SIAM journal on Discrete Mathematics 10 (1997), no.~4, 515--528.\n\n[DM] B.~Dushnik and E.W. Miller, Partially ordered sets, Amer. J. Math. 63 (1941), 600--610.\n\n[FO07] Hubert de Fraysseix, Patrice Ossona de Mendez: Stretching of Jordan arc contact systems, Discrete Applied Mathematics 155 (2007), no. 9, 1079--1095.\n\n[FOR94] H.~de Fraysseix, P.~Ossona~de Mendez, and P.~Rosenstiehl, On triangle contact graphs, Combinatorics, Probability and Computing 3 (1994), 233--246.\n\n*[FOR07] H.~de Fraysseix, P.~Ossona~de Mendez, and P.~Rosenstiehl, Representation of Planar Hypergraphs by Contacts of Triangles, Proc. of Graph Drawing '07, to appear.\n\n[O] P.~Ossona~de Mendez, Realization of posets, Journal of Graph Algorithms and Applications 6 (2002), no.~1, 149--153.\n\n[S84] E.R. Scheinerman, Intersection classes and multiple intersection parameters of graphs, Ph.D. thesis, Princeton University, 1984.\n\n[S89] W.~Schnyder, Planar graphs and poset dimension, Order 5 (1989), 323--343.\n\n[W] T.R.S. Walsh, Hypermaps versus bipartite maps, J. Combinatorial Theory 18(B) (1975), 155--163.\n\nRelated:\nRelated problems\nStraight line representation of planar linear hypergraphs\n\nDiscussion links:\n- Straight line representation of planar linear hypergraphs: http://www.openproblemgarden.org/?q=node/554\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Linear Hypergraphs with Dimension 3\" in Graph Theory; Topological Graph Theory; Drawings, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact conjecture was verified in its 2008 primary source, but targeted searches located no later proof, counterexample, or maintained status source.\n\n**Verified partial progress.**\n\n- The source paper proves that every planar linear hypergraph has a contact representation by triangles.\n- It then poses the incidence-poset-dimension-at-most-3 mixed triangles-and-segments representation as a broader conjecture.\n- A planar linear hypergraph that has no straight-segment-only representation is known, but this does not refute the mixed-object conjecture.\n\n**Full solution or refutation.**\n\nNo defensible current solved, disproved, or high-confidence open determination was located in the searched literature.\n\n**What remains.**\n\nObtain an expert/current source tracking the exact mixed representation convention, or locate a proof or counterexample matching that convention.\n\n**Sources checked.**\n\n- Hubert de Fraysseix, Patrice Ossona de Mendez, and Pierre Rosenstiehl, Representation of Planar Hypergraphs by Contacts of Triangles, Graph Drawing 2007, LNCS 4875 (2008), 125-136. (primary): https://doi.org/10.1007/978-3-540-77537-9_15\n Evidence used: Proves the planar subclass and explicitly poses the exact dimension-3 conjecture.\n- Daniel Gonçalves, A planar linear hypergraph whose edges cannot be represented as straight line segments, European Journal of Combinatorics 30 (2009), 280-282. (primary): https://doi.org/10.1016/j.ejc.2007.12.004\n Evidence used: Refutes a related straight-segment-only conjecture, establishing an important distinction from the stored mixed-object statement.\n\n**Review notes.** The specialized phrase intersection hypergraph of a family of triangles and segments depends on the source's representation convention. Sparse citation coverage prevents a stronger current-status claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3327, "problem_number": "OPG-172", "title": "Consecutive non-orientable embedding obstructions", "statement": "Conjecture Is there a graph $G$ that is a minor-minimal obstruction for two non-orientable surfaces?", "background": "Source: Open Problem Garden. Original node ID: 172. URL: http://www.openproblemgarden.org/op/consecutive_non_orientable_embedding_obstructions.\n\nSource subject path: Graph Theory > Topological Graph Theory > Genus.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/consecutive_non_orientable_embedding_obstructions\n- Subject(s): Graph Theory; Topological Graph Theory; Genus\n- Keywords: minor; surface\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 27th, 2007 by Bruce Richter\n\nComments:\n- September 15th, 2010 | Anonymous | Minor-Minimal Obstruction: Is a minor-minimal obstruction the same as a forbidden minor?\n- September 15th, 2010 | mdevos | yes: yes\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Consecutive non-orientable embedding obstructions\" in Graph Theory; Topological Graph Theory; Genus, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The current tracker still lists the question, but the supplied statement omits which two nonorientable surfaces are intended and does not encode the 'consecutive' qualifier in its title, preventing a reliable status determination.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nRobertson and Seymour prove that each fixed surface has a finite excluded-minor set, but this does not decide whether one graph can be minor-minimal for two intended consecutive nonorientable surfaces.\n\n**What remains.**\n\nRecover the intended quantified statement, presumably simultaneous obstruction membership for N_g and N_(g+1), and then have a topological graph theory expert check the relevant obstruction catalogs and literature.\n\n**Sources checked.**\n\n- Open Problem Garden, Consecutive non-orientable embedding obstructions (OPG-172), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/consecutive_non_orientable_embedding_obstructions\n Evidence used: Currently lists the same underspecified question; the title says consecutive while the statement does not specify the surfaces or their genera.\n- N. Robertson and P. D. Seymour, Graph minors. VIII. A Kuratowski theorem for general surfaces, Journal of Combinatorial Theory, Series B 48 (1990), 255-288. (primary): https://doi.org/10.1016/0095-8956(90)90121-F\n Evidence used: Establishes the finite excluded-minor framework for each fixed surface but does not answer the ambiguous simultaneous-obstruction question.\n\n**Review notes.** Formulation defect: the title says consecutive, while the statement gives no genus indices or adjacency condition. The statement was preserved rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3328, "problem_number": "OPG-157", "title": "What is the largest graph of positive curvature?", "statement": "Problem What is the largest connected planar graph of minimum degree 3 which has everywhere positive combinatorial curvature, but is not a prism or antiprism?", "background": "Source: Open Problem Garden. Original node ID: 157. URL: http://www.openproblemgarden.org/op/what_is_the_largest_graph_of_positive_curvature.\n\nSource subject path: Graph Theory > Topological Graph Theory > Planar graphs.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/what_is_the_largest_graph_of_positive_curvature\n- Author(s): DeVos, Matt; Mohar, Bojan\n- Subject(s): Graph Theory; Topological Graph Theory; Planar graphs\n- Keywords: curvature; planar graph\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: March 10th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: For a graph $G$ embedded in the sphere, the combinatorial curvature of a vertex $v$ is defined to be $1 - \\frac{ {\\mathit deg}(v)}{2} + \\sum_{f \\sim v} \\frac{1}{ {\\mathit size}(f) }$ (here the summation is over all faces $f$ incident with $v$ ).\n\nLet $G$ be a graph embedded in the sphere, and consider the polygonal surface formed by treating each face of size $n$ as a regular $n$-gon of side length $1$. The gaussian curvature at a vertex $v$ is defined to be $2 \\pi$ minus the sum of the angles incident with $v$. So, our vertex $v$ has positive curvature if the sum of the incident angles is less than $2 \\pi$. In fact, the combinatorial curvature at $v$ is exactly $2 \\pi$ times the gaussian curvature, so these quantities will always have the same sign.\n\nLet us call a convex polyhedron regular-faced if each face is a regular polygon. Based on the previous discussion, we know that every convex regular-faced polyhedron gives us a graph with everywhere positive combinatorial curvature. Indeed, we may view planar graphs with everywhere positive curvature as a kind of generalization of these polyhedra. The polyhedra in this class have been studied and classified. The Platonic solids and Archimedean solids are all convex and regular faced, and there are two infinite families: prisms and antiprisms. In addition to this, there are 92 other exceptional ones, known as Johnson Solids.\n\nEuler's formula tells us that the sum of the combinatorial curvatures over all of the vertices is equal to 2. Indeed, the combinatorial curvature is exactly what we get when we assign $1$ to each vertex and face, $-1$ to each edge, and then \"discharge\" evenly onto the vertices. So, if we wish to construct large planar graphs where every vertex has positive curvature, we will need to make the curvature arbitrarily small. This can be achieved with prisms and antiprisms, but apart from these two families, all other graphs with everywhere positive curvature have a bounded number of vertices. Improving upon [DM], Zhang [Z] has shown this upper bound to be at most 580. The great rhombicosidodecahedron has 120 vertices and everywhere positive curvature (this is the largest regular-faced convex polyhedron which is not a prism or antiprism). Reti, Bitay, and Kosztolanyi [RBK] have improved upon this lower bound by constructing a graph with everywhere positive curvature and 138 vertices. These are the best bounds I (M. DeVos) know of.\n\nBibliography:\n[DM] M. DeVos and B. Mohar, An analogue of the Descarte-Euler formula for infinite graphs and Higuchi's conjecture preprint.\n\n[H] Y. Higuchi, Combinatorial curvature for planar graph, J. Graph Theory, Vol 38 (2001), no. 4, 220-229. MathSciNet\n\n[RBK] T. Reti, E. Bitay, and Zs. Kosztolanyi, On the polyhedral graphs with positive combinatorial curvature, Acta Polytechnica Hungarica Vol. 2, No. 2 (2005) 19-37.\n\n[Z] L. Zhang, A result on combinatorial curvature for embedded graphs on a surface, Discrete Math (2007) in press\n\nSource links:\n- planar graph: http://en.wikipedia.org/wiki/planar graph\n- prism: http://en.wikipedia.org/wiki/prism (geometry)\n- antiprism: http://en.wikipedia.org/wiki/antiprism\n\nDiscussion links:\n- Platonic solids: http://en.wikipedia.org/wiki/Platonic solids\n- Archimedean solids: http://en.wikipedia.org/wiki/Archimedean solids\n- prisms: http://en.wikipedia.org/wiki/prism (geometry)\n- antiprisms: http://en.wikipedia.org/wiki/antiprism\n- Johnson Solids: http://en.wikipedia.org/wiki/johnson solid\n- great rhombicosidodecahedron: http://en.wikipedia.org/wiki/great rhombicosidodecahedron\n\nBibliography links:\n- An analogue of the Descarte-Euler formula for infinite graphs and Higuchi's conjecture: http://www.ijp.si/ftp/pub/preprints/ps/2004/pp930.ps\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1864922\n- On the polyhedral graphs with positive combinatorial curvature: http://www.bmf.hu/journal/Reti_4.pdf\n\nComments:\n- August 30th, 2011 | Anonymous | new 'largest' PCC graph: We have a new largest PCC graph with 208 vertices, improving the lower bound.\n\nIt will be published soon in the New Zealand Journal of Mathematics. \"New graphs with thinly spread positive combinatorial curvature.\" j.sneddon@auckland.ac.nz\n- August 3rd, 2010 | Anonymous | project underway: This is an active project for an honours student at the University of Auckland. We'd be interested in hearing from anyone else working on the problem: contact j.sneddon@auckland.ac.nz\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"What is the largest graph of positive curvature?\" in Graph Theory; Topological Graph Theory; Planar graphs, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The exact maximum is 208 vertices: matching constructions exist and Ghidelli proves that every graph in the stated class has at most 208 vertices.\n\n**Verified partial progress.**\n\n- Nicholson-Sneddon constructed a 208-vertex example.\n- Ghidelli's refined discharging argument proves the matching upper bound and also bounds face size by 41.\n\n**Full solution or refutation.**\n\nThe largest connected planar positive-combinatorial-curvature graph of minimum degree three, excluding prisms and antiprisms, has 208 vertices.\n\n**What remains.**\n\nNo existence/maximization gap remains; classification or enumeration of all 208-vertex extremizers is a separate question.\n\n**Sources checked.**\n\n- L. Ghidelli, On the largest planar graphs with everywhere positive combinatorial curvature, Journal of Combinatorial Theory Series B 158 (2023), 226-263. (primary): https://doi.org/10.1016/j.jctb.2022.08.009\n Evidence used: Defines exactly the OPG class and proves the sharp upper bound 208.\n- L. Ghidelli, extended arXiv version, arXiv:1708.08502. (primary): https://arxiv.org/abs/1708.08502\n Evidence used: Abstract explicitly says the DeVos-Mohar question is completely answered and provides a 208-vertex example.\n- Open Problem Garden, What is the largest graph of positive curvature? (node 157). (maintained_tracker): https://www.openproblemgarden.org/op/what_is_the_largest_graph_of_positive_curvature\n Evidence used: Original question and historical lower/upper bounds.\n\n**Review notes.** Dataset status should be updated from open to solved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3329, "problem_number": "OPG-702", "title": "Growth of finitely presented groups", "statement": "Problem Does there exist a finitely presented group of intermediate growth?", "background": "Source: Open Problem Garden. Original node ID: 702. URL: http://www.openproblemgarden.org/op/growth_of_finitely_presented_groups.\n\nSource subject path: Group Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/growth_of_finitely_presented_groups\n- Author(s): Adyan, Sergeui I.\n- Subject(s): Group Theory\n- Keywords: finitely presented; growth\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 12th, 2007 by mdevos\n\nProblem-page discussion:\nSee Wikipedia's growth of groups for definitions of the basics reguarding growth rate in groups (in particular polynomial and exponential growth rates). A finitely generated group has intermediate growth if its growth rate (for every finite generating set) is subexponential but superpolynomial.\n\nMost naturally occuring groups have either polynomial growth (such as ${\\mathbb Z}^n$ ) or exponential growth (such as a free group with rank $n > 1$ ). Milnor [M] famously asked if there exists a finitely generated group with intermediate growth, and this problem was resolved in the affirmative by Grigorchuk [G]. The groups constructed by Grigorchuk are not finitely presented, thus leaving the above problem.\n\nExpert opinion seems to be that there are no finitely presented groups of intermediate growth.\n\nBibliography:\n[G] R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means., Izv. Akad. Nauk SSSR Ser. Mat. 48:5 (1984), 939-985.\n\n[M] J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1-7.\n\nDiscussion links:\n- Wikipedia's growth of groups: http://en.wikipedia.org/wiki/growth rate (group theory)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Growth of finitely presented groups\" in Group Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Existence of a finitely presented group of intermediate growth remains a major open problem; all standard intermediate-growth group constructions fail finite presentability.\n\n**Verified partial progress.**\n\n- Grigorchuk constructed finitely generated groups of intermediate growth, but they are not finitely presented.\n- Related finitely presented overgroups and finitely presented monoids or algebras do not preserve the exact group-growth requirement.\n\n**Full solution or refutation.**\n\nNo finitely presented example or general nonexistence theorem was verified.\n\n**What remains.**\n\nConstruct such a finitely presented group or prove that every finitely presented group of subexponential growth has polynomial growth.\n\n**Sources checked.**\n\n- Rostislav Grigorchuk and Igor Pak, Groups of Intermediate Growth: an Introduction for Beginners, arXiv:math/0607384. (authoritative_secondary): https://arxiv.org/abs/math/0607384\n Evidence used: Expert survey identifies existence of finitely presented intermediate-growth groups as a major open problem.\n- Yves de Cornulier, Amenable groups without finitely presented amenable covers, Bulletin of Mathematical Sciences 3 (2013), 73-131. (primary): https://doi.org/10.1007/s13373-013-0031-5\n Evidence used: Formulates the exact finite-presentation/intermediate-growth question as Basic Problem 1.2 and develops related obstructions.\n\n**Review notes.** Background spelling defects include reguarding and occuring; the mathematical statement is precise.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 17, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3330, "problem_number": "OPG-732", "title": "Subgroup formed by elements of order dividing n", "statement": "Conjecture\n\nSuppose $G$ is a finite group, and $n$ is a positive integer dividing $|G|$. Suppose that $G$ has exactly $n$ solutions to $x^{n} = 1$. Does it follow that these solutions form a subgroup of $G$?", "background": "Source: Open Problem Garden. Original node ID: 732. URL: http://www.openproblemgarden.org/op/subgroup_formed_by_elements_of_order_dividing_n.\n\nSource subject path: Group Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/subgroup_formed_by_elements_of_order_dividing_n\n- Author(s): Frobenius, Ferdinand G.\n- Subject(s): Group Theory\n- Keywords: order, dividing\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: January 28th, 2008 by dlh12\n\nProblem-page discussion:\nIf these solutions form a subgroup, they form a characteristic (and therefore normal) subgroup of $G$. This easily follows from the First Sylow Theorem if $n$ is the highest power of a prime $p$ dividing $|G|$.\n\nIn a 1980 article, Feit commented that the case where $(n, \\frac{|G|}{n}) = 1$ (i.e., $n$ 'exactly divides' $|G|$ ) had been reduced to considering $G$ simple. Thus it should be resolvable using the classification of finite simple groups.\n\nIt is known that if $n$ divides $|G|$, the number of solutions of $x^{n} = 1$ in $G$ is a multiple of $n$. A generalization of this theorem, replacing $x^{n} = 1$ by $x^{n} \\in C$ for a conjugacy class $C$ of $G$, can be found in Marshall Hall Jr.'s book.\n\nObservation This conjecture implies the (known) theorem of Frobenius:\n\nTheorem If $G$ is a finite transitive permutation group in which only the identity has more than one fixed point, then the derangements of $G$, together with the identity, form a subgroup of $G$.\n\nTo prove Frobenius' Theorem from this (say $G$ is such a permutation group on $k$ points), use the standard elementary counting arguments to show that the solutions to $x^{k} = 1$ are only the derangements and the identity and that there are exactly $k$ of these. This theorem of Frobenius has not yet been proven in general without the use of group characters.\n\nBibliography:\nMarshall Hall Jr., Theory of Groups, Macmillan (1959)\n\nWalter Feit, On a Conjecture of Frobenius, Proceedings of the American Mathematical Society, Vol.7, No. 2 (Apr. 1956), 177-187.\n\nComments:\n- March 30th, 2012 | Anonymous | Solved... again: it can be proved by a isomorphism between the solutions of x^n=1 in G and the solutions of the same eq in C (complex numbers).\n- June 26th, 2009 | Anonymous | Solved: This conjecture has been proven.\n- June 29th, 2009 | Anonymous | Re: Solved: Could you add a reference please?\n- December 1st, 2010 | Anonymous | Reference: Iiyori, Nobuo; Yamaki, Hiroyoshi On a conjecture of Frobenius. Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 413--416.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Subgroup formed by elements of order dividing n\" in Group Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Iiyori and Yamaki proved Frobenius's conjecture: if exactly n solutions of x^n=1 occur, they form a normal subgroup.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe displayed question has an affirmative answer for finite groups; the proof uses classification of finite simple groups.\n\n**What remains.**\n\nNo open part of the stated finite-group conjecture remains.\n\n**Sources checked.**\n\n- N. Iiyori and H. Yamaki, On a conjecture of Frobenius, Bull. Amer. Math. Soc. (N.S.) 25 (1991), 413--416. (primary): https://jglobal.jst.go.jp/en/detail?JGLOBAL_ID=200901010607520864\n Evidence used: Bibliographic record for the announced proof of the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 17, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3331, "problem_number": "OPG-760", "title": "Burnside problem", "statement": "Conjecture If a group has $r$ generators and exponent $n$, is it necessarily finite?", "background": "Source: Open Problem Garden. Original node ID: 760. URL: http://www.openproblemgarden.org/op/burnside_problem.\n\nSource subject path: Group Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/burnside_problem\n- Author(s): Burnside, William\n- Subject(s): Group Theory\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 11th, 2008 by dlh12\n\nProblem-page discussion:\nIt is possible to define the $free Burnside group$ $B(r,n)$ to be the group generated by $x_{1}, \\ldots, x_{r}$ with relations $w^{n}=1$ where $w$ ranges over every word in the generators. There is a universality property: Any homomorphism $\\phi: G \\to H$ where $H$ has r generators and exponent dividing $n$ can be written as a composition of a homomorphism $\\psi: G \\to B(r,n)$ with a homomorphism $\\pi: B(r,n) \\to H$. Some cases of this are known: $B(1,n)$ is a cyclic group of order $n$, for any positive integer $n$. $B(r,1)$ is trivial for any positive integer $r$. $B(r,2)$ is isomorphic to the Cartesian product of $r$ cyclic groups of order $2$, for any positive integer $r$. This is because the relations make it easy to prove that the generators commute. $B(r,3)$ is a finite group, and its order is\n$$\n3^{r + \\binom{r}{2} + \\binom{r}{3}}.\n$$\n $B(r,6)$ is a finite group, and its order is\n$$\n2^{1 + (r-1)3^{r + \\binom{r}{2} + \\binom{r}{3}} }3^{1 + (r-1)2^{r + \\binom{r}{2} + \\binom{r}{3}} }.\n$$\n $B(r,4)$ is a finite group for any positive integer $r$. The order is known for $r$ up to $5$:\n$$\n|B(1,4)| = 2^{2}\n$$\n\n$$\n|B(2,4)| = 2^{12}\n$$\n\n$$\n|B(3,4)| = 2^{69}\n$$\n\n$$\n|B(4,4)| = 2^{422}\n$$\n\n$$\n|B(5,4)| = 2^{2728}\n$$\n $B(r,n)$ is known to be infinite for sufficiently large $r$ and odd $n \\geq 665$, as well as $r > 1$ and $n \\geq 10^{48}$ divisible by $2^{9}$.\n\nBurnside_problem\n$$\n\\\\\n$$\n Burnside Problem -- from Wolfram MathWorld\n\nDiscussion links:\n- Burnside_problem: http://en.wikipedia.org/wiki/Burnside_problem\n- Burnside Problem -- from Wolfram MathWorld: http://mathworld.wolfram.com/BurnsideProblem.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 31.\n\nAttempt notes:\nTarget:\nMake progress on \"Burnside problem\" in Group Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The general Burnside question has a negative answer: finitely generated infinite groups of bounded exponent exist. Zelmanov's affirmative theorem concerns only the restricted finite-group variant.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe stated necessary-finiteness assertion is false in general.\n\n**What remains.**\n\nThe original yes/no question is resolved; separate finite-exponent cases and the restricted Burnside problem remain distinct subjects.\n\n**Sources checked.**\n\n- Burnside Problem, Wolfram MathWorld (accessed 2026-08-17). (authoritative_secondary): https://mathworld.wolfram.com/BurnsideProblem.html\n Evidence used: Records infinite free Burnside groups for large odd exponent and distinguishes the restricted problem.\n- E. I. Zel'manov, Solution of the restricted Burnside problem for groups of odd exponent, Math. USSR-Izv. 36 (1991), 41--60. (primary): https://www.mathnet.ru/php/archive.phtml?jrnid=im&option_lang=eng&paperid=1104&wshow=paper\n Evidence used: Documents the separate affirmative restricted-Burnside result.\n\n**Review notes.** No source alteration; the historical finitely generated reading is distinguished from the restricted problem.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 17, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3332, "problem_number": "OPG-3572", "title": "Inverse Galois Problem", "statement": "Conjecture Every finite group is the Galois group of some finite algebraic extension of $\\mathbb Q$.", "background": "Source: Open Problem Garden. Original node ID: 3572. URL: http://www.openproblemgarden.org/op/inverse_galois_problem.\n\nSource subject path: Group Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/inverse_galois_problem\n- Author(s): Hilbert, David\n- Subject(s): Group Theory\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 13th, 2008 by tchow\n\nProblem-page discussion:\nThis problem is one of the greatest open problems in group theory. Hilbert was the first to study it in earnest. His irreducibility theorem established a connection between Galois groups over $\\mathbb Q$ and Galois groups over ${\\mathbb Q}(x)$; the latter could be attacked by geometric methods, and in this way, Hilbert showed that the symmetric and alternating groups are Galois realizable over $\\mathbb Q$. In the 1950's, Shafarevich showed using number-theoretic methods that all finite solvable groups are Galois realizable over $\\mathbb Q$. Another spectacular result was John Thompson's realization of the Monster group as a Galois group over $\\mathbb Q$. One of Thompson's main tools was a concept he called \"rigidity\", a concept discovered independently by several people that continues to be important to this day. It is now known that 25 of the 26 sporadic simple groups are Galois realizable over $\\mathbb Q$ (the sole exception being the Mathieu group $M_{23}$ ).\n\nBibliography:\n[MM] Gunter Malle and B. Heinrich Matzat, Inverse Galois Theory, Springer, 1999.\n\n[V] Helmut Völklein, Groups as Galois Groups: An introduction, Cambridge University Press, 1996.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Inverse Galois Problem\" in Group Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The inverse Galois problem over Q remains open for arbitrary finite groups, despite the realization of all finite solvable groups and, as of August 2026, all 26 sporadic finite simple groups.\n\n**Verified partial progress.**\n\n- Shafarevich's theorem realizes every finite solvable group over Q.\n- Huang, Jackson, Lee, Poonen, Pries, and Zhang constructed an explicit M_23 realization in 2026, completing all sporadic finite simple groups.\n\n**Full solution or refutation.**\n\nNo theorem realizing every finite group over Q was verified.\n\n**What remains.**\n\nRealize arbitrary finite groups, including nonsolvable extensions not covered merely by realizations of their composition factors.\n\n**Sources checked.**\n\n- Alexander Schmidt and Kay Wingberg, Šafarevič's theorem on solvable groups as Galois groups, arXiv:math/9809211. (primary): https://arxiv.org/abs/math/9809211\n Evidence used: Complete proof that every finite solvable group occurs as a Galois group over Q.\n- Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang, The Mathieu group M_23 is a Galois group over Q, arXiv:2608.08538 (2026). (primary): https://arxiv.org/abs/2608.08538\n Evidence used: Constructs an explicit M_23 polynomial and completes the sporadic-group program.\n- Gunter Malle and B. Heinrich Matzat, Inverse Galois Theory, Springer, 1999. (authoritative_secondary): https://link.springer.com/book/10.1007/978-3-662-12123-8\n Evidence used: Standard monograph defining the problem and surveying realization methods and families.\n\n**Review notes.** The standard problem requires a finite Galois extension. The imported phrase 'finite algebraic extension' is insufficient because a non-Galois extension does not have the intended Galois group.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 17, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3333, "problem_number": "OPG-37302", "title": "Which lattices occur as intervals in subgroup lattices of finite groups?", "statement": "Conjecture\n\nThere exists a finite lattice that is not an interval in the subgroup lattice of a finite group.", "background": "Source: Open Problem Garden. Original node ID: 37302. URL: http://www.openproblemgarden.org/op/which_lattices_occur_as_intervals_in_finite_groups.\n\nSource subject path: Group Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/which_lattices_occur_as_intervals_in_finite_groups\n- Subject(s): Group Theory\n- Keywords: congruence lattice; finite groups\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 22nd, 2011 by williamdemeo\n\nProblem-page discussion:\nThis would settle the Finite Lattice Representation Problem.\n\nBibliography:\n[P5] Palfy and Pudlak. Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis, 11 (1980), 22–27.\n\nRelated:\nRelated problems\nFinite Lattice Representation Problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Which lattices occur as intervals in subgroup lattices of finite groups?\" in Group Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No finite lattice is known not to occur as an interval in the subgroup lattice of a finite group. Pálfy--Pudlák make the all-lattices question globally equivalent to the finite lattice representation problem.\n\n**Verified partial progress.**\n\n- Pálfy and Pudlák proved that every finite lattice is representable by a finite algebra iff every finite lattice occurs as a subgroup-lattice interval of a finite group.\n- DeMeo surveys and develops concrete subgroup-interval and finite-algebra representation methods.\n\n**Full solution or refutation.**\n\nNo counterexample or universal representation theorem has been verified; the conjecture remains tied to the open FLRP.\n\n**What remains.**\n\nExhibit a finite lattice that is no interval [H,G] for finite H<=G, or prove that every finite lattice has such a realization.\n\n**Sources checked.**\n\n- P. P. Pálfy and P. Pudlák, Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups, Algebra Universalis 11 (1980), 22-27. (primary): https://doi.org/10.1007/BF02483080\n Evidence used: The introduction states the global iff theorem between the two universal representation problems.\n- W. J. DeMeo, Congruence lattices of finite algebras, arXiv:1204.4305 (2012). (primary): https://arxiv.org/abs/1204.4305\n Evidence used: The abstract describes subgroup-interval methods and the still-open representation problem.\n- W. DeMeo, FLRP Current Status & Counterexamples (accessed 2026-08-17). (maintained_tracker): https://universalalgebra.org/flrp/status/\n Evidence used: The maintained site records the globally equivalent FLRP as unresolved.\n\n**Review notes.** The Pálfy--Pudlák result is a global equivalence of universal statements, not an instance-by-instance equivalence for each lattice.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 17, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3334, "problem_number": "OPG-660", "title": "F_d versus F_{d+1}", "statement": "Problem Find a constant $k$ such that for any $d$ there is a sequence of tautologies of depth $k$ that have polynomial (or quasi-polynomial) size proofs in depth $d+1$ Frege system $F_{d+1}$ but requires exponential size $F_d$ proofs.", "background": "Source: Open Problem Garden. Original node ID: 660. URL: http://www.openproblemgarden.org/op/f_d_versus_f_d_1.\n\nSource subject path: Logic.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/f_d_versus_f_d_1\n- Author(s): Krajicek, Jan\n- Subject(s): Logic\n- Keywords: Frege system; short proof\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 18th, 2007 by zitterbewegung\n\nProblem-page discussion:\nProblem statement from link.\n\nSuch tautologies are known if $k$ may depend on $d$ (for $k:= d$ ). This problem is also closely related to conservativity relations among bounded arithmetic theories.\n\nBibliography:\n*[K1] J.Krajicek: \"Lower Bounds to the Size of Constant-Depth Propositional Proofs\", J. of Symbolic Logic, 59(1), (1994), pp.73-86\n\n[K2] J. Krajicek: \"Bounded arithmetic, propositional logic, and complexity theory\", Encyclopedia of Mathematics and Its Applications, Vol.60, Cambridge University Press, Cambridge - New York - Melbourne, (1995), 343 p. (on page 243)\n\nDiscussion links:\n- link: http://www.math.cas.cz/%7Ekrajicek/problemy.html#pl\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"F_d versus F_{d+1}\" in Logic, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Adjacent bounded-depth Frege separation is known when tautology depth may depend on d, but the requested single constant depth k for all d remains open.\n\n**Verified partial progress.**\n\n- Krajicek's author problem page states that the separation is known when k may depend on d, specifically k=d.\n- Exponential bounded-depth Frege lower bounds are known for pigeonhole-principle formulas.\n- Recent work improves lower bounds for Tseitin formulas and random 3-CNFs, but does not supply the uniform fixed-k hierarchy requested here.\n\n**Full solution or refutation.**\n\nNo located source constructs a constant-depth tautology family having short F_{d+1} proofs and exponential F_d lower bounds simultaneously for every d.\n\n**What remains.**\n\nMake the tautology depth independent of d while retaining quasi-polynomial upper proofs one Frege depth higher and exponential lower bounds at depth d.\n\n**Sources checked.**\n\n- Jan Krajicek, Open problems: F_d versus F_{d+1}; checked 2026-08-17. (maintained_tracker): https://www2.karlin.mff.cuni.cz/~krajicek/problemy.html\n Evidence used: States the exact fixed-k problem and the known k=d partial result.\n- Jan Krajicek, Pavel Pudlak, and Alan Woods, An exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures & Algorithms 7 (1995), 15-39. (primary): https://doi.org/10.1002/rsa.3240070103\n Evidence used: Establishes exponential lower bounds in bounded-depth Frege, an ingredient in known depth-dependent separations.\n- On bounded depth proofs for Tseitin formulas on the grid; revisited, arXiv:2209.05839 (2022). (primary): https://arxiv.org/abs/2209.05839\n Evidence used: Provides modern bounded-depth Frege lower bounds for a major explicit formula family without resolving the fixed-k hierarchy.\n- Bounded-Depth Frege Lower Bounds for Random 3-CNFs via Deterministic Restrictions, arXiv:2403.02275 (2024). (primary): https://arxiv.org/abs/2403.02275\n Evidence used: Shows current progress and limitations of lower-bound techniques for another formula family.\n\n**Review notes.** The stored sentence has the grammatical mismatch 'a sequence ... requires'. It also omits the precise proof-system and encoding conventions needed for a formal hierarchy claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3335, "problem_number": "OPG-1790", "title": "Tarski's exponential function problem", "statement": "Conjecture Is the theory of the real numbers with the exponential function decidable?", "background": "Source: Open Problem Garden. Original node ID: 1790. URL: http://www.openproblemgarden.org/op/tarskis_exponential_function_problem.\n\nSource subject path: Logic.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/tarskis_exponential_function_problem\n- Author(s): Tarski, Alfred\n- Subject(s): Logic\n- Keywords: Decidability\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 8th, 2008 by Charles\n\nProblem-page discussion:\nSee Tarski's exponential function problem. Tarski proved that the theory of the real numbers without the exponential is decidable before asking this.\n\nRelated:\nRelated problems\nAlgebraic independence of pi and e\nSchanuel's Conjecture\n\nDiscussion links:\n- Tarski's exponential function problem: http://en.wikipedia.org/wiki/Tarski's exponential function problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Tarski's exponential function problem\" in Logic, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The complete first-order theory of the real exponential field remains of unknown decidability unconditionally; model completeness and o-minimality are known, and real Schanuel implies decidability.\n\n**Verified partial progress.**\n\n- Wilkie proved model completeness and o-minimality of the real exponential field.\n- Macintyre and Wilkie proved decidability conditional on the real Schanuel conjecture.\n- Berarducci and Gallinaro obtained a 2026 Schanuel-conditional axiomatization and an unconditional model-completeness theorem for a closely related restricted-exponential theory.\n\n**Full solution or refutation.**\n\nNo unconditional decision algorithm for the full ordered real field with total exponentiation was verified.\n\n**What remains.**\n\nRemove the Schanuel-type hypothesis or prove undecidability for the complete first-order theory of the real exponential field.\n\n**Sources checked.**\n\n- Alessandro Berarducci and Francesco Gallinaro, On the elementary theory of the real exponential field, arXiv:2603.08365 (revised 23 June 2026). (primary): https://arxiv.org/abs/2603.08365\n Evidence used: The abstract and introduction distinguish unconditional model completeness from Schanuel-conditional decidability and state the new restricted-exponential result.\n\n**Review notes.** Formulation note: the intended object is the complete first-order theory of the ordered real exponential field; the imported sentence does not spell out the language.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3336, "problem_number": "OPG-2379", "title": "Termination of the sixth Goodstein Sequence", "statement": "Question How many steps does it take the sixth Goodstein sequence to terminate?", "background": "Source: Open Problem Garden. Original node ID: 2379. URL: http://www.openproblemgarden.org/op/termination_of_the_sixth_goodstein_sequence.\n\nSource subject path: Logic.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/termination_of_the_sixth_goodstein_sequence\n- Author(s): Graham, Ronald L.\n- Subject(s): Logic\n- Keywords: Goodstein Sequence\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: October 7th, 2008 by mdevos\n\nProblem-page discussion:\nFor a positive integer $n$, the $n^{th}$ Goodstein Sequence is defined as follows. The first term of the sequence in $n$. To obtain the $k^{th}$ term, write the $(k-1)^{st}$ term in hereditary base k notation, change all $k$ 's to $(k+1)$ 's and then subtract 1. If the sequence hits 0, then it terminates. So, the first terms of the sixth Goodstein Sequence are as follows:\n\n$$\n\\begin{array}{lll} \\mbox{term} & \\mbox{value} \\\\ 1 & 2^2 + 2 = 6 \\\\ 2 & 3^3 + 2 = 29 \\\\ 3 & 4^4 + 1 = 257 \\\\ 4 & 5^5 = 3125 \\\\ 5 & 5 \\cdot 6^5 + 5 \\cdot 6^5 + 5 \\cdot 6^4 + 5 \\cdot 6^3 + 5 \\cdot 6^2 + 5 \\cdot 6 + 5 = 46655 \\end{array}\n$$\n\nSurprisingly, despite the fact that Goodstein Sequences grow quite quickly at the start, all such sequences do eventually hit 0 and terminate. This result, first discovered by Goodstein, is of interest in logic since it cannot be proved in Peano arithmetic.\n\nAlthough determining particular properties of a specific Goodstein Sequence are of limited mathematical value, this problem is an interesting computational challenge.\n\nDiscussion links:\n- Goodstein Sequence: http://en.wikipedia.org/wiki/Goodstein Sequence\n- hereditary base k notation: http://en.wikipedia.org/wiki/Goodstein Sequence\n\nComments:\n- September 17th, 2010 | Deedlit | approximate value: So how much is $F_6(6) - 2$ in terms of, say, Knuth arrows? we have $$F_6(6) = F_5^6(6) = F_5^5(F_5(6)) = F_5^5(F_4^6(6)) =... = F_5^5(F_4^5(F_3^5(F_2^6(6)))) \\\\ F_2(6) = 2^6*6 = 384 \\\\ F_2^2(6) = 2^384 * 384 > 2^{392} \\\\ F_2^6(6) > 2^{2^{2^{2^{2^{392}}}}} \\\\ F_5^5(F_4^5(F_3^5(F_2^6(6)))) > (2\\uparrow\\uparrow\\uparrow\\uparrow)^5 (2\\uparrow\\uparrow\\uparrow)^5 (2\\uparrow\\uparrow)^5 (2^{2^{2^{2^{2^{392}}}}})$$ That's about as close an approximation as you can get.\n- September 17th, 2010 | Deedlit | The actual value is much higher: You've underestimated the true value by quite bit.\n\nTo get the value of the Goodstein function at n, you take n, write it in hereditary base 2, then replace every appearance to with the infinite ordinal $\\omega$. Call the result R(n). The value of G(n) is then $$H_{R(n)}(3) - 2$$where$H_a(x)$ is the Hardy hierarchy, defined by\n\n$H_0(x) = x$\n\n$H_{a+1} (x) = H_a (x+1)$\n\n$H_a (x) = H_{a[x]} (x)$ for limit ordinals a\n\nSo to find G(6), we write $6 = 2^2 + 2$, so $R(6) = \\omega^\\omega + \\omega$. Hence, $$G(6) = H_{\\omega^\\omega + \\omega} (3) - 2 = H_{\\omega^\\omega}( H_{\\omega} (3)) - 2 = F_{\\omega} (F_1 (3)) - 2 = F_{\\omega} (6) - 2 = F_6 (6) - 2$$\n\nwhere $F_a(x)$ is the fast-growing hierarchy, defined by\n\n$F_0(x) = x+1$\n\n$F_{a+1} (x) = F_a^x (x)$\n\n$F_a (x) = F_{a[x]} (x)$ for limit ordinals a\n\n(or you could just leave the answer in terms of the Hardy hierarchy, I just changed to the fast-growing hierarchy because the answer is a little simpler.)\n- June 4th, 2010 | kagidab | Solution: $k(a,n)=$ amount of steps to reduce $(a-1)*a^n+(a-1)a^{n-1}+...(a-1)a^0$ to -1\n\n$(a-1)a^1+a-1\\rightarrow(a-1)(a+a)^1-1\\rightarrow(a-2)(2a)^1-(2a-1)$\n\nIf you do this a - 1 times: $(a - 1) a ^ 1 + a - 1 -> a * 2^{a - 1} - 1$\n\nWhich means it is reduced to a 0th power and will take $a*2^{a-1}$ steps to finish.\n\nSo total steps $=a(2^{0}+2^{1}+2^{2}+...+2^{a-2})+a*2^{a-1}=a*(2^{a}-1)=k(a, 1)+1$\n\nFor $(a-1)*a^n+(a-1)a^(n-1)+... +(a-1) a^0$:\n\n$k(a,n) = k(k...k(a,n-1)...)) - 2$ [k a times]\n\n$f(n)=g(h(n)-2,h(n), h(n))-2$\n\nWhere $g(o) = g(0, n, o) = o * 2^o$, $g(m, n, o)=g(m - 1, n, g(g(...g(o)))...)$ [n copies of g] and $h(n)$ is the first number in the sequence in the form $(a - 1) * a^(a-1)+(a - 1) a ^ (a - 2) +... (a - 1)$ $h([3,4,5,6,7])=[2, 3, 4, 6, 8]$\n\nE.g.\n\n$f(4) = g(1, 3, 3) - 2 = g(0, 3, g(g(g(3)))) - 2 = 3 * 2^{402653211} - 2$\n\nUsing $g(n)\\sim2^{n}$:\n\n$f(6) = g(4,6,6)-2\\sim g(3, 6, 2\\uparrow\\uparrow 7})\\sim2\\uparrow\\uparrow25$\n- August 17th, 2010 | Anonymous | Has this solution been: Has this solution been verified by the author? Just curious.\n- August 21st, 2010 | kagidab | Nah, I just posted it here: Nah, I just posted it here as my attempt at a solution. It probably needs to be done a bit better, but I believe it works until n = 8 or so, where you have to do a bit extra. I might try make it a bit clearer and rigorous at some point. Plus a better approximation would be useful.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 42.\n\nAttempt notes:\nTarget:\nMake progress on \"Termination of the sixth Goodstein Sequence\" in Logic, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The sixth Goodstein-sequence length has an exact hierarchy-valued expression: under the maintained page's convention, G(6)=H_{omega^omega+omega}(3)-2=F_6(6)-2.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe hereditary base-2 expression 6=2^2+2 maps to the ordinal omega^omega+omega; the standard Hardy-hierarchy length theorem then yields the exact finite value shown, though it is not feasibly expandable in decimal notation.\n\n**What remains.**\n\nFix whether steps count transitions, positive terms, or the first zero index before using the formula as a convention-independent integer label.\n\n**Sources checked.**\n\n- Open Problem Garden, Termination of the sixth Goodstein Sequence. (maintained_tracker): https://www.openproblemgarden.org/op/termination_of_the_sixth_goodstein_sequence\n Evidence used: Gives the explicit ordinal translation and exact Hardy/fast-growing hierarchy expression.\n- Stanley S. Wainer, Fast Growing Functions and Arithmetical Independence, lecture notes summarizing Cichon's 1983 theorem. (authoritative_secondary): https://www-logic.stanford.edu/seminar/1213/Wainer_stanford1.pdf\n Evidence used: States that Hardy functions measure lengths of Goodstein sequences.\n- Stanford Encyclopedia of Philosophy, Proof Theory, Appendix E: Combinatorial Independence Results. (authoritative_secondary): https://plato.stanford.edu/entries/proof-theory/appendix-e.html\n Evidence used: Provides a rigorous modern definition of hereditary-base Goodstein sequences and their ordinal analysis.\n\n**Review notes.** The imported fifth-term display duplicates 5*6^5 while asserting 46655 and is arithmetically inconsistent. Off-by-one conventions for sequence length also vary.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3337, "problem_number": "OPG-37424", "title": "Fixed-point logic with counting", "statement": "Question Can either of the following be expressed in fixed-point logic plus counting:\n\n- Given a graph, does it have a perfect matching, i.e., a set $M$ of edges such that every vertex is incident to exactly one edge from $M$?\n- Given a square matrix over a finite field (regarded as a structure in the natural way, as described in [BGS02]), what is its determinant?", "background": "Source: Open Problem Garden. Original node ID: 37424. URL: http://www.openproblemgarden.org/op/fixed_point_logic_with_counting.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/fixed_point_logic_with_counting\n- Author(s): Blass, Andreas\n- Subject(s): Logic; Finite Model Theory\n- Keywords: Capturing PTime; counting quantifiers; Fixed-point logic; FMT03-Bedlewo\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 18th, 2012 by dberwanger\n\nProblem-page discussion:\nIt is known that (1) is expressible if restricted to bipartite graphs and that (2) is expressible if the field has only two elements. Both these results are in [BGS02].\n\nBibliography:\n[BGS02] A. Blass, Y. Gurevich, and S. Shelah, On polynomial time computation over unordered structures, J. Symbolic Logic 67 (2002) 1093--1125.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Fixed-point logic with counting\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Perfect matching is expressible in fixed-point logic with counting, but this search did not verify the requested full determinant expressibility over arbitrary finite fields in FPC.\n\n**Verified partial progress.**\n\n- Maximum/perfect matching is definable in FPC via the FPC formalization of linear programming.\n\n**Full solution or refutation.**\n\nThe two-question source record is only partially resolved.\n\n**What remains.**\n\nResolve the determinant/rank expressibility question in the specified finite-field structures, if it is not already covered by a more precise linear-algebraic logic result.\n\n**Sources checked.**\n\n- A. Dawar, M. Grohe, B. Holm and B. Laubner, Maximum Matching and Linear Programming in Fixed-Point Logic with Counting, LICS 2009. (primary): https://doi.org/10.1109/LICS.2009.27\n Evidence used: The paper establishes FPC expressibility of maximum matching and linear programming, yielding perfect matching.\n\n**Review notes.** The two parts are separated; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3338, "problem_number": "OPG-37429", "title": "Order-invariant queries", "statement": "Question\n\n- Does ${<}\\text{-invariant\\:FO} = \\text{FO}$ hold over graphs of bounded tree-width?\n- Is ${<}\\text{-invariant\\:FO}$ included in $\\text{MSO}$ over graphs?\n- Does ${<}\\text{-invariant\\:FO}$ have a 0-1 law?\n- Are properties of ${<}\\text{-invariant\\:FO}$ Hanf-local?\n- Is there a logic (with an effective syntax) that captures ${<}\\text{-invariant\\:FO}$?", "background": "Source: Open Problem Garden. Original node ID: 37429. URL: http://www.openproblemgarden.org/op/order_invariant_queries.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/order_invariant_queries\n- Author(s): Segoufin, Luc\n- Subject(s): Logic; Finite Model Theory\n- Keywords: Effective syntax; FMT12-LesHouches; Locality; MSO; Order invariance\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 18th, 2012 by dberwanger\n\nProblem-page discussion:\nWe describe the problem over finite vertex-colored graphs which we call graphs in the sequel. An ordered graph is a graph together with a linear order on its vertices. A property $p$ of ordered graphs is said to be order-invariant if it is independent of the linear order. I.e. $G,<_1 \\models p$ iff $G,<_2 \\models p$ for all graphs $G$ and all linear orders $<_1,<_2$ on $G$. Therefore, we now view an order-invariant property as a property over (unordered) graphs.\n\nWe denote by ${<}\\text{-invariant\\:FO}$, the set of order-invariant first-order definable properties over graphs, where the first-order signature contains the vocabulary for graphs but also a linear order predicate. Note that it is undecidable whether a first-order query is order-invariant. In terms of expressive power, Gurevich showed that ${<}\\text{-invariant\\:FO}$ is strictly more expressive than $\\text{FO}$. However it is known that queries expressible in ${<}\\text{-invariant\\:FO}$ are Gaifman-local~[GS00]. It is also known that ${<}\\text{-invariant\\:FO} = \\text{FO}$ over finite trees~[BS09] (see also~[N05]).\n\nA glimpse beyond\n\nOne can view the linear order on top of the graph as a bijection between the vertices of the graph and an ordered prefix of the positive natural numbers. With this point of view, being order-invariant corresponds to being independent from the choice of the bijection. We could imagine allowing more predicates on the numerical side, and not just the linear order. Typically addition and multiplication. When both these predicates are present we denote by $(+,*)\\text{-invariant\\:FO}$ the properties definable in first-order independently of the bijection. It was shown in~[AMSS10] that those properties are Gaifman local but with a polylog radius for the neighborhoods, and this polylog is tight. However the case when only addition is present is unclear. On top of all the questions above we could add:\n\nQuestion Is $+\\text{-invariant\\:FO}$ Gaifman-local?\n\nThe question of the inclusion of $+\\text{-invariant\\:FO}$ in $\\text{MSO}$ is already relevant over words:\n\nQuestion Can $+\\text{-invariant\\:FO}$ define non-regular languages over words?\n\nSee~[SS10] for more background about this question.\n\nBibliography:\n[AMSS10] Matthew Anderson, Dieter van Melkebeek, Nicole Schweikardt, and Luc Segoufin, Locality of queries definable in invariant first-order logic with arbitrary built-in predicates. In ICALP'11.\n\n[BS09] Michael Benedikt and Luc Segoufin, Towards a characterization of order-invariant queries over tame graphs. J. Symb. Log. 74(1), 2009. Pages 168-186.\n\n[GS00] Martin Grohe and Thomas Schwentick, Locality of order-invariant first-order formulas. ACM Trans. Comput. Log. 1(1), 2000. Pages 112-130.\n\n[N05] Hannu Niemistö, On Locality and Uniform Reduction, LICS'05.\n\n[SS10] Nicole Schweikardt and Luc Segoufin, Addition-Invariant FO and regularity, LICS'10.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Order-invariant queries\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For every fixed treewidth bound, order-invariant FO is contained in MSO, resolving a major listed bounded-treewidth direction; the broader questions about all graphs, 0-1 laws, locality, and syntax are not all settled by this result.\n\n**Verified partial progress.**\n\n- Inv-FO(<) is contained in MSO on graphs of every fixed treewidth.\n\n**Full solution or refutation.**\n\nThe multi-part source record remains partially open.\n\n**What remains.**\n\nDetermine the remaining global inclusion, locality, 0-1-law, and effective-syntax questions.\n\n**Sources checked.**\n\n- M. Benedikt and L. Segoufin, Towards a characterization of order-invariant queries over bounded treewidth, 2022 manuscript. (primary): https://inria.hal.science/hal-03663758v1/preview/invfo.pdf\n Evidence used: The paper states Inv-FO(<) is contained in MSO over graphs of treewidth b for every fixed b.\n\n**Review notes.** Multi-question source record classified partially; statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3339, "problem_number": "OPG-37440", "title": "Monadic second-order logic with cardinality predicates", "statement": "The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the following atomic formulas:\n\n- $\\text{\"}\\,\\mathrm{Card}(X) = \\mathrm{Card}(Y)\\,\\text{\"}$\n- $\\text{\"}\\,\\mathrm{Card}(X) \\text{ belongs to } A\\,\\text{\"}$\n\nwhere $A$ is a fixed recursive set of integers.\n\nLet us fix $k$ and a closed formula $F$ in this language.\n\nConjecture Is it true that the validity of $F$ for a graph $G$ of tree-width at most $k$ can be tested in polynomial time in the size of $G$?", "background": "Source: Open Problem Garden. Original node ID: 37440. URL: http://www.openproblemgarden.org/op/monadic_second_order_logic_with_cardinality_predicates.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/monadic_second_order_logic_with_cardinality_predicates\n- Author(s): Courcelle, Bruno\n- Subject(s): Logic; Finite Model Theory\n- Keywords: bounded tree width; cardinality predicates; FMT03-Bedlewo; MSO\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 18th, 2012 by dberwanger\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Monadic second-order logic with cardinality predicates\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Courcelle-style results give efficient bounded-treewidth model checking for MSO with fixed modular cardinality predicates and several structured cardinality extensions, but no result was verified for arbitrary fixed recursive sets A under the source's literal cost model.\n\n**Verified partial progress.**\n\n- Fixed modular cardinality predicates are handled by CMSO on bounded treewidth.\n- Several global/local cardinality extensions have XP algorithms on bounded treewidth.\n\n**Full solution or refutation.**\n\nThe arbitrary-recursive-A phrasing needs an effective encoding and complexity convention before a definite yes/no status is sound.\n\n**What remains.**\n\nSpecify the representation and membership-cost model for A, then compare with known CMSO/CardMSO metatheorems or derive a lower bound.\n\n**Sources checked.**\n\n- D. Knop, M. Koutecký, T. Masařík and T. Toufar, Simplified Algorithmic Metatheorems Beyond MSO: Treewidth and Neighborhood Diversity, arXiv:1703.00544 (2017). (primary): https://arxiv.org/abs/1703.00544\n Evidence used: The abstract gives XP bounded-treewidth algorithms for specified global/local cardinality extensions, not arbitrary recursive cardinality predicates.\n\n**Review notes.** Formulation/cost-model issue flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3340, "problem_number": "OPG-37444", "title": "Blatter-Specker Theorem for ternary relations", "statement": "Let $C$ be a class of finite relational structures. We denote by $f_C(n)$ the number of structures in $C$ over the labeled set $\\{0, \\dots, n-1 \\}$. For any class $C$ definable in monadic second-order logic with unary and binary relation symbols, Specker and Blatter showed that, for every $m \\in \\mathbb{N}$, the function $f_C(n)$ is ultimately periodic modulo $m$.\n\nQuestion Does the Blatter-Specker Theorem hold for ternary relations.", "background": "Source: Open Problem Garden. Original node ID: 37444. URL: http://www.openproblemgarden.org/op/blatter_specker_theorem_for_ternary_relations.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/blatter_specker_theorem_for_ternary_relations\n- Author(s): Makowsky, Janos A.\n- Subject(s): Logic; Finite Model Theory\n- Keywords: Blatter-Specker Theorem; FMT00-Luminy\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 18th, 2012 by dberwanger\n\nProblem-page discussion:\nOur exposition follows closely [BS84].\n\nCounting labeled structures modulo $m$\n\nLet $C$ be a class of finite structures for one binary relation symbol $R$. We define for $A = \\{ 1, \\ldots, n \\}$ $$F_C(n) = \\mid \\{ R^A \\subseteq A^2: \\langle A, R^A \\rangle \\in C \\} \\mid$$\n\nExamples:\n\n- If $C=U$ consists of all $R$-structures, $f_U(n)= 2^{n^2}$.\n- If $C=B$ consists of bijections, $f_B(n)= n!$\n- If $C= G$ is the class of all (undirected, simple) graphs, $f_G(n)= 2^{\\binom{n}{2}}$.\n- If $C=E$ is the class of all equivalence relations, then $f_E(n)= B_n$, the {\\em Bell Numbers}.\n- If $C=E_2$ is the class of all equivalence relations with two classes only, of the same size, $f_{E_2}(2n)= \\frac{1}{2} \\cdot {\\binom{2n}{n}}$. Clearly, $f_{E_2}(2n+1)= 0$.\n- If $C=T$ is the class of all trees, $f_T(n)= n^{n-2}$, {\\em Caley}.\n\nWe observe the following:\n\n$$f_C(n)= 2^{n^2} = (-1)^{n^2} \\pmod{3}$$\n\n$$f_C(n)= n! = 0 \\pmod{m} \\mbox{ for } n \\geq m$$\n\nAnd for each $m$ the functions, $f_G(n)= 2^{\\binom{n}{2}}$, $f_E(n)= B_n$, $f_T(n)= n^{n-2}$ are ultimately periodic $\\pmod{m}$.\n\nHowever, $f_{E_2}(2n)= \\frac{1}{2} \\cdot {\\binom{2n}{n}} = 1 \\pmod{2}$ iff $n = 2^{2k}$, hence is not periodic $\\pmod{2}$.\n\nMonadic second-order logic definable classes\n\nThe first four examples (all relations, all bijections, all graphs, all equivalence relations) are definable in First Order Logic $\\text{FO}$. The trees are definable in Monadic Second Order Logic $\\text{MSO}$..\n\n$E_2$ is definable in Second Order Logic $\\text{SO}$, but it is not $\\text{MSO}$-definable. If we expand $E_2$ to have the bijection between the classes we get structures with two binary relations. The class is now $\\text{FO}$-definable. Let us denote the corresponding counting function $F_{E_2}(2n)$. We have $$f_{E_2}(2n) \\cdot n! = F_{E_2}(n) = 0 \\pmod{m}$$for$n$ large enough.\n\nPeriodicity and linear recurrence relations\n\nThe periodicity of $f_C(n)$ $\\pmod{m}$ is usually established by exhibiting a linear recurrence relation:\n\nThere exists $1 \\leq k \\in \\mathbb{N}$ and integers $a_1, \\ldots, a_k$ such that for all $n$ $$f_C(n) = \\sum_{j=1}^{k} a_j \\cdot f_C(n-j) \\pmod{m}$$\n\nExamples.\n\n- In the case of $f_C(n) = 2^{n^2}$ we have $$f_C(n) = f_C(n-2) + 2 \\cdot f_C(n-1) \\pmod{3}$$\n- In the case of$f_C(n) = n!$we have for all$m$$$f_C(n) = 0 \\cdot f_C(n-1) \\pmod{m}$$In this case we say that$f_C$ trivializes.\n\nThe Blatter-Specker Theorem\n\nTheorem (BS84) Let $\\tau$ be a binary vocabulary, i.e. all relation symbols are at most binary. If $C$ is a class of finite $\\tau$-structures which is $\\text{MSO}$-definable, then for all $m \\in \\mathbb{N}$ $f_C(n)$ is ultimately periodic $\\pmod{m}$.\n\nMoreover, there exists $1 \\leq k \\in \\mathbb{N}$ and integers $a_1, \\ldots, a_k$ such that for all $n$ $$f_C(n) = \\sum_{j=1}^{k} a_j \\cdot f_C(n-j) \\pmod{m}$$ i.e we have a linear recurrence relation.\n\nBibliography:\n[BS84]* C. Blatter and E. Specker, Recurrence relations for the number of labeled structures on a finite set, Logic and Machines: Decision Problems and Complexity, E. Börger, G. Hasenjaeger and D. Rödding, eds, LNCS 171 (1984) pp. 43-61.\n\n[F03] E. Fischer, The Specker-Blatter theorem does not hold for quaternary relations, Journal of Combinatorial Theory Series A 103(2003), 121-136.\n\n[FM06] E. Fischer and J. A. Makowsky, The Specker-Blatter Theorem revisited: Generating functions for definable classes of stuctures. In Computing and Combinatorics (COCOON 2003) Proc., LNCS vol. 2697 (2003), 90-101.\n\n[S88] E. Specker, Application of Logic and Combinatorics to Enumeration Problems, Trends in Theoretical Computer Science, E. Börger ed., Computer Science Press, 1988, pp. 141-169. Reprinted in: Ernst Specker, Selecta, Birkhäuser 1990, pp. 324-350.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 52.\n\nAttempt notes:\nTarget:\nMake progress on \"Blatter-Specker Theorem for ternary relations\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The proposed ternary extension is false: Fischer and Makowsky construct a first-order-definable class with one ternary relation whose counting sequence is not ultimately periodic modulo 2.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nA counterexample exists already in first-order logic with a single ternary relation, so the binary Specker-Blatter theorem does not extend to ternary vocabularies.\n\n**What remains.**\n\nThe source's yes/no question is answered; narrower syntactic or structural subclasses may still admit periodicity theorems.\n\n**Sources checked.**\n\n- Eldar Fischer and Johann A. Makowsky, Extensions and Limits of the Specker-Blatter Theorem, CSL 2024, LIPIcs 288, Article 26, DOI 10.4230/LIPIcs.CSL.2024.26. (primary): https://doi.org/10.4230/LIPIcs.CSL.2024.26\n Evidence used: The abstract and main construction explicitly state that the theorem fails for a first-order sentence with one ternary relation and yields non-periodicity modulo 2.\n\n**Review notes.** The negative result directly answers the supplied question; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3341, "problem_number": "OPG-37448", "title": "MSO alternation hierarchy over pictures", "statement": "Question Is the MSO-alternation hierarchy strict for pictures that are balanced, in the sense that the width and the length are polynomially (or linearly) related.", "background": "Source: Open Problem Garden. Original node ID: 37448. URL: http://www.openproblemgarden.org/op/mso_alternation_hierarchy_over_pictures.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/mso_alternation_hierarchy_over_pictures\n- Author(s): Grandjean, Etienne\n- Subject(s): Logic; Finite Model Theory\n- Keywords: FMT12-LesHouches; MSO, alternation hierarchy; picture languages\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 18th, 2012 by dberwanger\n\nProblem-page discussion:\nIn [MST02], Matz, Schweikardt, and Thomas, proved that the MSO-alternation hierarchy is strict over the class of 2-dimensional rectangular pictures (and, as a consequence, is also strict over the class of finite graphs).\n\nThe proof of this hierarchy strictness is essentially based on the fact that, for any positive integer $k$, there is a function $f_k: \\mathbb{N}\\to \\mathbb{N}$ (defined as a fixed height tower of exponentials) such that the set of rectangular grids of format $n\\times f_k(n)$ (i.e, of width $n$ and length $f_k(n)$ ) can be defined by some $\\Sigma_k$ MSO sentence but cannot be defined by some $\\Sigma_{k-1}$ MSO sentence.\n\nSo, the hierarchy result essentially rests on the (more than exponential) imbalance between the two dimensions of the rectangular grid.\n\nIn view of this result a natural question is as follows.\n\nQuestion Is the MSO-alternation hierarchy strict for more well-balanced pictures, for example, if it is required that the width and the length of the pictures are polynomially (resp. linearly) related?\n\nFor example, for square picture languages (or equivalently, rectangular picture languages for which the width and the length of the pictures are linearly related), the only thing we know is that EMSO (that is Existential or $\\Sigma_1$ MSO) over square pictures is not closed under complement.\n\nOliver Matz (personal communication) thinks it is possible that any MSO sentence over square pictures be equivalent to a Boolean combination of existntial MSO sentences.\n\nBibliography:\n[MST02] O. Matz, N. Schweikardt and W. Thomas, The Monadic Quantifier Alternation Hierarchy over Grids and Graphs, Information and Computation 179(2002), 356-383.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"MSO alternation hierarchy over pictures\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Strictness of the unrestricted MSO monadic-quantifier alternation hierarchy over grids is known, but no result settling polynomially or linearly balanced pictures was verified.\n\n**Verified partial progress.**\n\n- Matz, Schweikardt, and Thomas prove strictness over finite grids and graphs, using separating grids with very large aspect-ratio imbalance.\n\n**Full solution or refutation.**\n\nThe known general-grid hierarchy theorem does not answer either balanced regime in the source statement.\n\n**What remains.**\n\nProve or refute strictness when the two picture dimensions are polynomially related, and separately when they are linearly related, after fixing a uniform balance convention.\n\n**Sources checked.**\n\n- O. Matz, N. Schweikardt, and W. Thomas, The Monadic Quantifier Alternation Hierarchy over Grids and Graphs, Information and Computation 179 (2002), 356-383, DOI 10.1006/inco.2002.2955. (primary): https://doi.org/10.1006/inco.2002.2955\n Evidence used: Proves strictness for grids and graphs; the construction described in the paper does not impose the polynomial or linear balance asked here.\n- Open Problem Garden, Picture Languages topic page, accessed 2026-08-17. (maintained_tracker): https://garden.irmacs.sfu.ca/category/picture_languages\n Evidence used: Continues to present the balanced-picture hierarchy question without a recorded resolution.\n\n**Review notes.** The source bundles polynomial and linear balance and does not specify uniformity of the bounding polynomial; these defects were not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3342, "problem_number": "OPG-37863", "title": "Finite entailment of Positive Horn logic", "statement": "Question Positive Horn logic (pH) is the fragment of FO involving exactly $\\exists, \\forall, \\wedge, =$. Does the fragment $pH \\wedge \\neg pH$ have the finite model property?", "background": "Source: Open Problem Garden. Original node ID: 37863. URL: http://www.openproblemgarden.org/op/finite_satisfiability_of_positive_horn_logic_entailment.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/finite_satisfiability_of_positive_horn_logic_entailment\n- Author(s): Martin, Barnaby\n- Subject(s): Logic; Finite Model Theory\n- Keywords: entailment; finite satisfiability; horn logic\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 21st, 2012 by LucSegoufin\n\nProblem-page discussion:\nIt doesn't really matter whether or not equality is allowed, as it may mostly be propagated out by substitution. The question is whether there an infinity axiom of the form $\\phi \\wedge \\neg \\psi$, for $\\phi, \\psi$ in pH?\n\nIn [CMM08] it is proved that entailment of pH sentences is decidable. I.e. input, $\\phi, \\psi$ in pH and return yes if $\\phi \\rightarrow \\psi$ is true on all models. The question is whether this is the same as asking if entailment of pH is equivalent to finite entailment, i.e. if $\\phi \\rightarrow \\psi$ is true on all models iff it is true on all finite models.\n\nFor the positive equality-free fragment of FO (bigger than pH), finite entailment and general entailment do not coincide, and the latter problem is undecidable. For existential positive logic, (smaller than pH), finite entailment and general entailment do coincide, and of course both are decidable. At present, the question as to whether finite entailment of pH is decidable is also open.\n\nBibliography:\n[CMM08] Hubie Chen, Florent R. Madelaine, Barnaby Martin: Quantified Constraints and Containment Problems. LICS 2008: 317-328\n\nComments:\n- June 15th, 2020 | Anonymous | Solved.: Solved in LICS 2017 \"Herbrand Property, Finite Quasi-Herbrand Models, and a Chandra-Merlin Theorem for Quantified Conjunctive Queries\"\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Finite entailment of Positive Horn logic\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Positive Horn entailment coincides with finite entailment, so every satisfiable sentence of the form pH and not-pH has a finite model.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nBova and Mogavero prove a finite quasi-Herbrand-model theorem and, as a consequence, equality of general and finite entailment for positive Horn sentences; this is equivalent to the finite-model property asked here.\n\n**What remains.**\n\nThe yes/no source question is answered affirmatively; refinements may concern bounds on finite witnesses or adjacent logical fragments.\n\n**Sources checked.**\n\n- Simone Bova and Fabio Mogavero, Herbrand Property, Finite Quasi-Herbrand Models, and a Chandra-Merlin Theorem for Quantified Conjunctive Queries, LICS 2017, DOI 10.1109/LICS.2017.8005073. (primary): https://doi.org/10.1109/LICS.2017.8005073\n Evidence used: The finite quasi-Herbrand-model result yields that unrestricted and finite entailment coincide for positive Horn sentences.\n\n**Review notes.** The title's entailment formulation and the statement's finite-model formulation are equivalent by considering phi and not psi; no statement repair was needed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3343, "problem_number": "OPG-38188", "title": "Vertex Cover Integrality Gap", "statement": "Conjecture For every $\\varepsilon > 0$ there is $\\delta > 0$ such that, for every large $n$, there are $n$-vertex graphs $G$ and $H$ such that $G \\equiv_{\\delta n}^{\\mathrm{C}} H$ and $\\mathrm{vc}(G) \\ge (2 - \\varepsilon) \\cdot \\mathrm{vc}(H)$.", "background": "Source: Open Problem Garden. Original node ID: 38188. URL: http://www.openproblemgarden.org/op/vertex_cover_integrality_gap.\n\nSource subject path: Logic > Finite Model Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/vertex_cover_integrality_gap\n- Author(s): Atserias, Albert\n- Subject(s): Logic; Finite Model Theory\n- Keywords: counting quantifiers; FMT12-LesHouches\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: October 2nd, 2012 by dberwanger\n\nProblem-page discussion:\nHere $\\equiv^{\\mathrm{C}}_{k}$ denotes indistinguishability in $k$-variable first-order logic with counting quantifiers, and $\\mathrm{vc}(G)$ denotes the cardinality of the minimum vertex-cover of $G$. By~[1], $G \\equiv_{3}^{\\mathrm{C}} H$ implies $\\mathrm{vc}(G) \\leq 2 \\cdot \\mathrm{vc}(H)$. Also by~[1] a positive answer would imply that an integrality gap of $2-\\varepsilon$ resists $\\delta n$ levels of Sherali-Adams linear programming relaxations of vertex-cover, on $n$-vertex graphs. It is known that such a gap resists $n^{\\delta}$ levels~[2]. What we ask would let us replace $n^{\\delta}$ by $\\delta n$. If improving over $n^{\\delta}$ were not possible, then we could approximate vertex-cover by a factor better than~ $2$ in subexponential time (i.e. $2^{n^{o(1)}}$ ). Approximating vertex-cover by a factor better than~1.36 is NP-hard~[3], and approximating vertex-cover by factor better than~2 is UG-hard~[4], where UG stands for Unique Games (from the Unique Games Conjecture); but note that UG-hardness does not rule out subexponential-time algorithms because UG itself is solvable in subexponential time~[5]\n\nBibliography:\n[1] A. Atserias and E. Maneva. Sherali-Adams Relaxations and Indistinguishability in Counting Logics, in Proc. 3rd ACM ITCS, pp. 367-379, 2012.\n\n[2] M. Charikar, K. Makarychev and Y. Makarychev. Integrality Gaps for Sherali-Adams Relaxations, in Proc. 41st ACM STOC, pp. 283-292, 2009.\n\n[3] I. Dinur and S. Safra. On the Hardness of Approximating Minimum Vertex-Cover, Annals of Mathematics, 162(1):439-485, 2005.\n\n[4] S. Khot and O. Regev. Vertex cover might be hard to approximate to within 2-epsilon, J. Comput. Syst. Sci. 74(3):335-349, 2008.\n\n[5] S. Arora, B. Barak, and D. Steurer. Subexponential Algorithms for Unique Games and Related problems, in Proc. 51th IEEE FOCS, pp. 563-572, 2010.}\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 17.\n\nAttempt notes:\nTarget:\nMake progress on \"Vertex Cover Integrality Gap\" in Logic; Finite Model Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Near-2 integrality gaps are known for extremely small LP relaxations of Vertex Cover, but no source was verified for the exact C^C_{delta n}-indistinguishability formulation.\n\n**Verified partial progress.**\n\n- Small LP relaxations have integrality gap 2-o(1) on explicit vertex-cover instances.\n\n**Full solution or refutation.**\n\nThe exact finite-model-theoretic conjecture is not confirmed as settled.\n\n**What remains.**\n\nMatch the known LP lower bounds to the stated linear-variable counting-logic equivalence or find a direct construction.\n\n**Sources checked.**\n\n- T. Braun, S. Pokutta and D. Zink, Inapproximability of combinatorial problems via small LPs and SDPs, STOC 2015. (primary): https://infoscience.epfl.ch/server/api/core/bitstreams/7031c718-8445-416e-9101-27eeff6d6ccd/content\n Evidence used: The work proves 2-o(1) gaps for subexponential-size LP relaxations of Vertex Cover.\n\n**Review notes.** Exact logic formulation not silently identified with LP result.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 18, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3344, "problem_number": "OPG-416", "title": "Lonely runner conjecture", "statement": "Conjecture Suppose $k$ runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for any given runner, there is a time at which that runner is distance at least $\\frac{1}{k}$ (along the track) away from every other runner.", "background": "Source: Open Problem Garden. Original node ID: 416. URL: http://www.openproblemgarden.org/op/lonely_runner_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/lonely_runner_conjecture\n- Author(s): Cusick, Thomas W.; Wills, Jorg M.\n- Subject(s): Number Theory\n- Keywords: diophantine approximation; view obstruction\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 24th, 2007 by mdevos\n\nProblem-page discussion:\nThis conjecture was independently introduced in two very different contexts. Wills [W] introduced it as a problem in diophantine approximation, and Cusick [C1] discovered it as a geometric view obstruction problem. The poetic name is due to Goddyn.\n\nThere are a number of different proofs of this conjecture for small values of $k$ (as a warning, there are different formulations of this conjecture, and what appears here as the problem for $k$ runners is sometimes considered to be the problem for $k-1$ runners). The cases with $k \\le 3$ runners are easy to check. The $k=4$ case was proved independently by Betke and Wills [BW] and by Cusick. The $k=5$ case was first established by Cusick and Pomerance [CP] with the help of some computer checking, and this argument was later simplified by Bienia et al. [BGGS] who also found applications of this theorem to the study of flows on graphs. The $k=6$ case was first proved by Bohman et al. [BHK] and this was later simplified by Renault [R]. Recently, the $k=7$ case was proved by Barajas and Serra [BS].\n\nBibliography:\n[BS] J. Barajas and O. Serra, The lonely runner problem with seven runners.\n\n[BW] U. Betke and J. M. Wills, Untere Schranken fur zwei diophantische Approximations-Funktionen, Monatsch. Math. 76 (1972), 214-217.\n\n[BGST] W. Bienia, L. Goddyn, P. Gvozdjak, A. Sebo, Flows, View Obstructions, and the Lonely Runner, J. Combinatorial Theory Ser. B 72 (1998) 1-9.\n\n[BHK] T. Bohman, R. Holzman, and D. Kleitman, Six lonely runners, Electron. J. Combin. 8 (2001), no. 2\n\n[CC] Y.G. Chen, T.W. Cusick, The View-Obstruction Problem for n-Dimensional Cubes, J. Number Theory 74, no. 1 (1999) 126-133.\n\n*[C1] T.W. Cusick, View-Obstruction Problems in n-Dimensional Geometry, J. Combinatorial Theory Ser. A 16 (1974) 1-11.\n\n[C2] T.W. Cusick, View-Obstruction Problems II, Proc. Amer. Math. Soc. 84 (1982) 25-28.\n\n[C3] T.W. Cusick, The view-obstruction problem for $5$-dimensional cubes Monatsh. Math. 127 (1999), no. 3, 183--187.\n\n[CP] T.W. Cusick and C. Pomerance, View-Obstruction Problems III, J. Number Theory 19 (1984) 131-139.\n\n[R] J. Renault, View-obstruction: a shorter proof for 6 lonely runners. Discrete Math. 287 (2004), no. 1-3, 93-101.\n\n*[W] J.M. Wills, Zwei Satze uber Inhomogene Diophantische Approximation von Irrationalzahlen, Monatsch. Math. 71 (1967) 263-269.\n\nBibliography links:\n- The lonely runner problem with seven runners: http://www.mac.cie.uva.es/%7Erevilla/vjmda/files/044.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Lonely runner conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Lonely Runner Conjecture is now computer-assistant verified through 13 total runners, but remains open for an arbitrary number of runners.\n\n**Verified partial progress.**\n\n- Sungkawichai-Trakulthongchai prove the stationary formulation for 10, 11, and 12 nonzero relative speeds.\n- After restoring the distinguished stationary runner, these are exactly the 11-, 12-, and 13-total-runner cases in the input convention.\n\n**Full solution or refutation.**\n\nThe finite verification range has advanced substantially, but no all-k theorem is known.\n\n**What remains.**\n\nReplace finite-dimensional case analysis by an argument valid for every number of relative speeds.\n\n**Sources checked.**\n\n- Touch Sungkawichai and Tanupat Trakulthongchai, Eleven, twelve, and thirteen lonely runners, arXiv:2604.23906 (2026). (primary): https://arxiv.org/abs/2604.23906\n Evidence used: Proves the conjecture for k in {10,11,12} in the k-moving-plus-one-stationary formulation.\n- Open Problem Garden, Lonely runner conjecture (node 416). (maintained_tracker): https://www.openproblemgarden.org/op/lonely_runner_conjecture\n Evidence used: Original total-runner formulation and warning about competing counting conventions.\n\n**Review notes.** Runner-count conventions differ by one. The result deliberately reports total runners to match the input statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3345, "problem_number": "OPG-671", "title": "MacEachen Conjecture", "statement": "Conjecture Every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product.", "background": "Source: Open Problem Garden. Original node ID: 671. URL: http://www.openproblemgarden.org/op/maceachen_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/maceachen_conjecture\n- Author(s): McEachen, Bill R.\n- Subject(s): Number Theory\n- Keywords: primality; prime distribution\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: October 19th, 2007 by billymac00\n\nProblem-page discussion:\nThis conjecture speaks to the distribution of all prime numbers, relating them to primorials. Recall that the primorials are simply the consecutive product of prime numbers eg 2,2*3,2*3*5, etc. this is OEIS A002110 {2,6,30,210,30030,...}. A new sequence A129912, related to the conjecture, ie A129912 and combines A002110 with the products of unique primorials eg 2*6,2*30,6*30, etc. The PariGP code to generate the terms is on the author's wiki site. By unique, the products mentioned would not use an entry of A002110 more than once. A numerical example is candidate number N=189239. This cannot be prime unless it is an absolute prime distance from a sequence term covering the range 0 thru 2*N. of course, it is 9059 away from sequence entry 180180, and so it may be prime (and it is). Note that the conjecture treats a required but not sufficient condition for primality. As it works for offset distance less than the candidate it could be used in a primality method if that somehow could be applied effectively (seriously doubtful). However, the insight provided into the distribution of the primes is the worthwhile part. It leaves no doubt about the non random, concrete structure of prime interdependency.\n\nNote that the conjecture implies as others have suspected, the existence of Twin primes to be found adjacent to sequence terms. Note that the conjecture was independently confirmed through the first 50 million primes. Also, the author has attempted a strict proof, that is conditional upon Goldbach's Conjecture being true. Realizing that the author is merely an amateur hobbiest, the proof may be flawed but it can be supplied, it is quite elementary, less than a page.\n\nI do note the independent work of several others found online, all of whom I do not personally know. One is John Sokol, who in 2002 made the following conjecture (incorrect is as far as the distance being < the candidate, think it breaks down around 331). I independently started with this same conjecture until I quickly realized the correct one. I also note the work of Bob Potter and his primorial conjecture at the link shown lower. A third person is Hank Harrell, who has done work in an area similar to Potter's.\n\nThe author plotted a normalized minimum offset seen at the primes, with the resulting scatter plot clearly indicating an asymptotic trend towards the relevant sequence entry being one greater than the prime candidate.\n\nThe author has also speculated that the A129912 sequence is useful in locating Twin Primes (not his particular interest). An earlier conjecture by the author in 2006 (A117825) concerning highly composite numbers basically parallels Fortune's conjecture (A005325).\n\nThe links mentioned above are now listed: OEIS 2110 OEIS 129912 author's wiki Wikipedia PlanetMath Harrell WikiCommons\n\nSokol's conjecture (sic): A primes can only exists + or - a prime from a primorial. Where 1 is considered a prime and 2 is not.\n\nSokol\n\nPotter conjecture:\n\n\"All prime numbers in the vicinity of a primorial (or primorial multiple) will combine to make a Goldbach pair for the primorial. The length of sequence for which this effect holds increases as the value of the primorial or primorial multiple increases.\"\n\nPotter\n\nMany more online resources were accessed for this work, some of them are:\n\nPari-GP program Chris Caldwell's well respected site Dario Alpern's online ECM calculator\n\nI am sure there are others I have forgotten.\n\nSource links:\n- primorial: http://en.wikipedia.org/wiki/primorial\n\nDiscussion links:\n- OEIS 2110: http://www.research.att.com/%7Enjas/sequences/?q=A2110&language=english&go=Search\n- OEIS 129912: http://www.research.att.com/%7Enjas/sequences/?q=A129912&sort=0&fmt=0&language=english&go=Search\n- author's wiki: http://billymac00.pbwiki.com/main\n- Wikipedia: http://en.wikipedia.org/wiki/Primorials\n- PlanetMath: http://planetmath.org/encyclopedia/FortunesConjecture.html\n- Harrell: http://www.prime-equations.com/index.html\n- WikiCommons: https://commons.wikimedia.org/wiki/File:OEIS_A129912_spin1.svg\n- Sokol: http://www.dnull.com/%7Esokol/prime/conjecture1.html\n- Potter: http://primorialconjecture.wordpress.com/2012/07/\n- Pari-GP program: http://pari.math.u-bordeaux.fr/] \\href [WIMS]{http://wims.unice.fr/wims/en_tool~algebra%7Efactor.html\n- Chris Caldwell's well respected site: http://primes.utm.edu/\n- Dario Alpern's online ECM calculator: http://www.alpertron.com.ar/ECM.HTM\n\nComments:\n- July 7th, 2012 | chongchingbak | is this already solved?: is this already solved?\n- March 24th, 2008 | Anonymous | wrong reference: Hi, Hope I'm not being too picky but the reference to my web site is wrong. www.primorialconjecture.org - maybe if you're updating this sometime you would tweak it.\n\nThanks! Bob Potter\n- January 22nd, 2009 | billymac00 | link correction: sorry, I just saw this today, I fixed the link in the 2 places I saw Bob.\n- February 26th, 2008 | billymac00 | proof progress: due to an email xchange with a fella Simon Horvat Feb 24 2008, who I am otherwise unfamiliar with, I believe he has come upon a fairly straightforward mathematical proof of my conjecture. I defer to him for the time being but it is very promising...\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"MacEachen Conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact stored conjecture is not self-contained; a later OEIS comment supplies a plausible precise reading, but no scholarly proof, refutation, or reliable current-status source was located.\n\n**Verified partial progress.**\n\n- OEIS A129912 defines the relevant sequence as products of distinct primorial numbers.\n- A 2010 McEachen comment on A129912 clarifies the intended assertion as distance 1 or prime distance q

", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3346, "problem_number": "OPG-739", "title": "Chowla's cosine problem", "statement": "Problem Let $A \\subseteq {\\mathbb N}$ be a set of $n$ positive integers and set\n$$\nm(A) = - \\min_x \\sum_{a \\in A} \\cos(ax).\n$$\n What is $m(n) = \\min_A m(A)$?", "background": "Source: Open Problem Garden. Original node ID: 739. URL: http://www.openproblemgarden.org/op/chowlas_cosine_problem.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/chowlas_cosine_problem\n- Author(s): Chowla, Sarvadaman\n- Subject(s): Number Theory\n- Keywords: circle; cosine polynomial\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 22nd, 2008 by mdevos\n\nProblem-page discussion:\nIt is easy to see that $m(A) > 0$, since the average value of the sum of the cosines is zero. Bourgain [B] proved that $m(n) > e^{(\\log n)^c}$ for some $c>0$ and $n$ sufficiently large. Recently, Ruzsa [R] tightened this argument, proving that $m(n) > c_1 e^{c_2 \\sqrt{ \\log n}}$ where $c_2 = \\sqrt{ (\\log 2)/ 8}$. The proof utilizes a clever manipulation of norms to reveal a (somewhat surprising) additive structure to the problem.\n\nIt seems the only known upper bound is $m(n) \\ll \\sqrt{n}$.\n\nBibliography:\n[B] J. Bourgain, Sur le minimum d'une somme de cosinus, Acta Arith. 45 (1986), 381--389. MathSciNet\n\n*[C] S. Chowla, Some applications of a method of A. Selberg. J. Reine Angew. Math. 217 (1965) 128--132. MathSciNet\n\n[R] I.Z. Ruzsa, Negative values of cosine sums. Acta Arith. 111 (2004), no. 2, 179--186. MathSciNet\n\nRelated:\nRelated problems\nLonely runner conjecture\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0847298\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR0172853\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR2039421\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Chowla's cosine problem\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The first polynomial lower bound for m(n) was proved in 2025, replacing the older subpolynomial lower bounds, but the exact extremal order remains open.\n\n**Verified partial progress.**\n\n- Bedert proves min_x sum_{a in A} cos(ax) <= -Omega(n^c) with c>=1/12.\n- This is a polynomial improvement over the Bourgain--Ruzsa type bound recorded in the source.\n\n**Full solution or refutation.**\n\nA polynomial lower bound is known; matching upper and lower growth rates are not.\n\n**What remains.**\n\nDetermine the correct order of m(n), or substantially close the gap with the known O(sqrt(n)) upper bound.\n\n**Sources checked.**\n\n- B. Bedert, Polynomial bounds for the Chowla cosine problem, arXiv:2509.05260 (2025). (primary): https://arxiv.org/abs/2509.05260\n Evidence used: States a polynomial bound with an absolute exponent at least 1/12.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3347, "problem_number": "OPG-791", "title": "Quartic rationally derived polynomials", "statement": "Call a polynomial $p \\in {\\mathbb Q}[x]$ rationally derived if all roots of $p$ and the nonzero derivatives of $p$ are rational.\n\nConjecture There does not exist a quartic rationally derived polynomial $p \\in {\\mathbb Q}[x]$ with four distinct roots.", "background": "Source: Open Problem Garden. Original node ID: 791. URL: http://www.openproblemgarden.org/op/quartic_rationally_derived_polynomials.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/quartic_rationally_derived_polynomials\n- Author(s): Buchholz, Ralph H.; MacDougall, James A.\n- Subject(s): Number Theory\n- Keywords: derivative; diophantine; elliptic; polynomial\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 17th, 2008 by mdevos\n\nProblem-page discussion:\nProbably anyone who has ever designed simple problems for calculus students has looked for polynomials $p$ with the property that both $p$ and some small derivatives of it are easy to factor. Perhaps inspired by this, Buchholz and MacDougall attempted to classify all univariate polynomials defined over a domain $k$ with the property that they and all their nonzero derivatives have all their roots in $k$. This problem can be split into cases dependent upon the multiplicity of the roots, and Buchholz and MacDougall solved many of the small ones for $k={\\mahtbb Q}$. Based on their results and a theorem of Flynn [F], an affirmative solution to the above conjecture would complete this classification problem for $k={\\mathbb Q}$.\n\nBibliography:\n*[BM] R. Buchholz, and J. MacDougall, When Newton met Diophantus: a study of rational-derived polynomials and their extension to quadratic fields. J. Number Theory 81 (2000), no. 2, 210--233. MathSciNet\n\n[F] E. V. Flynn, On Q-derived polynomials. Proc. Edinb. Math. Soc. (2) 44 (2001), no. 1, 103--110. MathSciNet\n\nBibliography links:\n- When Newton met Diophantus: a study of rational-derived polynomials and their extension to quadratic fields: http://www.geocities.com/teufel_pi/papers/rdp.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1752251\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1879212\n\nComments:\n- February 16th, 2011 | Comet | Bibliography: Hyperlink in the bibliography is no longer valid, but the article can be found at: http://web.archive.org/web/20011127182208/http://www.geocities.com/teufel_pi/papers/rdp.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Quartic rationally derived polynomials\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The cited rational-derived-polynomial literature classifies many low-degree/multiplicity cases, but no resolution of the distinct-root quartic conjecture was verified.\n\n**Verified partial progress.**\n\n- Buchholz--MacDougall establish multiple small rational-derived cases.\n- Flynn supplies a related Q-derived-polynomial result used in the source discussion.\n\n**Full solution or refutation.**\n\nNo quartic example or impossibility proof for four distinct roots was verified.\n\n**What remains.**\n\nResolve the remaining distinct-root quartic Diophantine case.\n\n**Sources checked.**\n\n- Open Problem Garden, node 791 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/quartic_rationally_derived_polynomials\n Evidence used: Preserves the exact conjecture and its cited 2000--01 partial classification literature.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3348, "problem_number": "OPG-819", "title": "A discrete iteration related to Pierce expansions", "statement": "Conjecture Let $a > b > 0$ be integers. Set $b_1 = b$ and $b_{i+1} = {a \\bmod {b_i}}$ for $i \\geq 0$. Eventually we have $b_{n+1} = 0$; put $P(a,b) = n$.\n\nExample: $P(35, 22) = 7$, since $b_1 = 22$, $b_2 = 13$, $b_3 = 9$, $b_4 = 8$, $b_5 = 3$, $b_6 = 2$, $b_7 = 1$, $b_8 = 0$.\n\nProve or disprove: $P(a,b) = O((\\log a)^2)$.", "background": "Source: Open Problem Garden. Original node ID: 819. URL: http://www.openproblemgarden.org/op/a_discrete_iteration_related_to_pierce_expansions.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_discrete_iteration_related_to_pierce_expansions\n- Author(s): Shallit, Jeffrey O.\n- Subject(s): Number Theory\n- Keywords: Pierce expansions\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 11th, 2008 by shallit\n\nProblem-page discussion:\nThe best upper bound is currently $P(a,b) = O(a^{1/3})$. For more information, see [ES].\n\nBibliography:\n[ES] P. Erd\\\"os and J. Shallit, \"New bounds on the length of finite Pierce and Engel series\", S\\'eminaire de Th\\'eorie des Nombres de Bordeaux 3 (1991), 43--53.\n\nComments:\n- June 11th, 2008 | Porges | A different upper bound: This paper shows an upper bound of $O(\\sqrt[3]a \\sqrt[3]{\\log(a)})$.\n\nEdit: But looking at the title page of the paper, I see you already knew that;)\n- May 30th, 2020 | Anonymous | bound: That's because the best currently known bound is not the one in the paper. It is in a technical report by Vlado Keselj.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"A discrete iteration related to Pierce expansions\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The O((log a)^2) Pierce-iteration conjecture remains open; the cited technical-report bound is O(a^(1/3)(log a)^(1/3)).\n\n**Verified partial progress.**\n\n- Erdos--Shallit give the earlier O(a^(1/3)) style bound.\n- Keselj records an O(a^(1/3)(log a)^(1/3)) upper bound.\n\n**Full solution or refutation.**\n\nNo polylogarithmic upper bound was verified.\n\n**What remains.**\n\nReduce the polynomial dependence on a to polylogarithmic, or find a lower-bound obstruction.\n\n**Sources checked.**\n\n- P. Erdos and J. Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theorie des Nombres de Bordeaux 3 (1991), 43--53. (primary): https://cs.uwaterloo.ca/research/tr/1996/21/cs-96-21.pdf\n Evidence used: The linked technical report documents the later Pierce-series bounds and cites the earlier work.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3349, "problem_number": "OPG-1786", "title": "Algebraic independence of pi and e", "statement": "Conjecture $\\pi$ and $e$ are algebraically independent", "background": "Source: Open Problem Garden. Original node ID: 1786. URL: http://www.openproblemgarden.org/op/algebraic_independence_of_pi_and_e.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/algebraic_independence_of_pi_and_e\n- Subject(s): Number Theory\n- Keywords: algebraic independence\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 8th, 2008 by porton\n\nSource links:\n- algebraically independent: http://en.wikipedia.org/wiki/Algebraic_independence\n\nComments:\n- April 29th, 2012 | warut | Schanuel's conjecture: Assuming Schanuel's conjecture, one can show that $\\pi$ and $e$ are algebraically independent over $\\mathbb Q$.\n- February 16th, 2011 | Comet | in which subfield K of which field L?: After all, e to the pi i = -1, so this shows that pi and e are not always algebraically independent.\n- July 16th, 2011 | Anonymous | By definition?: I think any two distinct transcendental numbers must be algebraically independent, almost by definition. Since e and pi are transcendental, they must be a. i. No? - David Spector\n- August 5th, 2011 | cubola zaruka | not all transcedentials are algebraically independant: pi and 4-pi are both transcedential and sum to 4, so are not algebraically independant.\n- July 21st, 2011 | Anonymous | two transcendentals are not necessarily algebraically independen: e and e^2 are both transcendental but (e,e^2) makes the two-variable polynomial f(x,y)=x^2-y equal to zero\n- May 18th, 2011 | Jon Noel | ?: but that's not a polynomial.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Algebraic independence of pi and e\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Under the standard interpretation over the rationals, algebraic independence of e and pi remains unknown; it would follow from the n=2 case of Schanuel's conjecture.\n\n**Verified partial progress.**\n\n- Hermite–Lindemann proves e and pi individually transcendental, but individual transcendence does not imply algebraic independence.\n- Schanuel's conjecture would imply the requested algebraic independence.\n\n**Full solution or refutation.**\n\nNo unconditional proof of algebraic independence and no polynomial relation over Q were verified.\n\n**What remains.**\n\nProve that no nonzero polynomial in Q[X,Y] vanishes at (pi,e), or exhibit such a relation.\n\n**Sources checked.**\n\n- Francesco Paolo Gallinaro, Exponential sums equations and tropical geometry, Selecta Mathematica 29 (2023), article 49, DOI 10.1007/s00029-023-00853-y. (primary): https://doi.org/10.1007/s00029-023-00853-y\n Evidence used: The introduction says Schanuel is known for n=1 but extremely hard for n=2, which would imply algebraic independence of e and pi.\n- Open Problem Garden, Algebraic independence of pi and e (node 1786), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/algebraic_independence_of_pi_and_e\n Evidence used: Retains the conjecture and explicitly discusses the intended base field Q and the Schanuel implication.\n\n**Review notes.** Formulation defect: algebraic independence is relative to a base field, but the imported statement omits it. The status uses the standard Q interpretation without altering the quoted statement.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3350, "problem_number": "OPG-2147", "title": "Odd perfect numbers", "statement": "Conjecture There is no odd perfect number.", "background": "Source: Open Problem Garden. Original node ID: 2147. URL: http://www.openproblemgarden.org/op/odd_perfect_number.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/odd_perfect_number\n- Author(s): Ancient/folklore\n- Subject(s): Number Theory\n- Keywords: perfect number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: September 27th, 2008 by azi\n\nProblem-page discussion:\nThere is substantial literature on the problem. Most proceeds from a study of the multiplicative function $\\sigma_{-1}(n)=\\sigma(n)/n$ where the conjecture can be stated: $\\sigma_{-1}(n)=2$ implies that $n$ is even.\n\nSource links:\n- perfect number: http://en.wikipedia.org/wiki/perfect number\n\nComments:\n- December 7th, 2011 | Anonymous | limiting divisors: My idea is to assume that the OPN is divisible by a prime number (e.x. 3) then use the properties of perfect numbers to figure out other numbers the OPN is divisible by.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Odd perfect numbers\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No odd perfect number is known and no nonexistence proof is known; very large lower bounds and strong factorization restrictions are established.\n\n**Verified partial progress.**\n\n- Computational and theoretical work exclude all odd perfect numbers below enormous explicit bounds.\n- Euler-type factorization and many prime-divisor constraints are known.\n\n**Full solution or refutation.**\n\nThe assertion that no odd perfect number exists remains unproved.\n\n**What remains.**\n\nConvert the known structural restrictions into a contradiction or construct an example.\n\n**Sources checked.**\n\n- Open Problem Garden, node 2147 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Retains the original conjecture; no current full resolution was verified.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3351, "problem_number": "OPG-16555", "title": "Diophantine quintuple conjecture", "statement": "Definition A set of m positive integers $\\{a_1, a_2, \\dots, a_m\\}$ is called a Diophantine $m$-tuple if $a_i\\cdot a_j + 1$ is a perfect square for all $1 \\leq i < j \\leq m$.\n\nConjecture (1) Diophantine quintuple does not exist.\n\nIt would follow from the following stronger conjecture [Da]:\n\nConjecture (2) If $\\{a, b, c, d\\}$ is a Diophantine quadruple and $d > \\max \\{a, b, c\\}$, then $d = a + b + c + 2bc + 2\\sqrt{(ab+1)(ac+1)(bc+1)}.$", "background": "Source: Open Problem Garden. Original node ID: 16555. URL: http://www.openproblemgarden.org/op/diophantine_quintuple_does_not_exist.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/diophantine_quintuple_does_not_exist\n- Subject(s): Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: November 22nd, 2008 by maxal\n\nProblem-page discussion:\nIt was proved in [Db] that there are only finitely many Diophantine quintuples and no Diophantine sextuples.\n\nConjecture (2) is motivated by an observation of [AHS] that every Diophantine triple $\\{a,b,c\\}$ can be extended to a Diophantine quadruple $\\{a,b,c,a + b + c + 2bc + 2\\sqrt{(ab+1)(ac+1)(bc+1)}\\}.$\n\nBibliography:\n[Da] A. Dujella Diophantine $m$-tuples, a survey of the main problems and results concerning Diophantine m-tuples.\n\n[Db] A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.\n\n[AHS] J. Arkin, V. E. Hoggatt and E. G. Strauss, On Euler's solution of a problem of Diophantus, Fibonacci Quart. 17 (1979), 333-339.\n\nBibliography links:\n- Diophantine $m$-tuples: http://web.math.hr/%7Eduje/dtuples.html\n- There are only finitely many Diophantine quintuples: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.58.8571\n\nComments:\n- February 25th, 2020 | Anonymous | This result has been proven: in a paper announced in 2016 and published in 2019, He, Togbé and Ziegler [350] gave the proof of the Diophantine quintuple conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Diophantine quintuple conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Conjecture (1) is solved: no integer Diophantine quintuple exists. The intended stronger regularity conjecture (2) remains open, and its formula is corrupted in the imported statement.\n\n**Verified partial progress.**\n\n- He, Togbé, and Ziegler proved that no Diophantine quintuple of positive integers exists.\n- Many families satisfy the stronger regularity assertion, and the best maintained general result bounds the number of larger extensions of a fixed triple by eight.\n\n**Full solution or refutation.**\n\nThe main nonexistence conjecture is completely resolved, but the row also contains a stronger conjecture that remains open in its correctly stated form.\n\n**What remains.**\n\nProve that every ordered positive-integer Diophantine quadruple is the regular extension d_+, or produce an irregular quadruple.\n\n**Sources checked.**\n\n- Bo He, Alain Togbé, and Volker Ziegler, There is no Diophantine quintuple, Transactions of the American Mathematical Society 371 (2019), 6665-6709. (primary): https://doi.org/10.1090/tran/7573\n Evidence used: Proves Conjecture (1) in the record.\n- Andrej Dujella, Diophantine quintuple conjecture. (maintained_tracker): https://dujella.github.io/quint.html\n Evidence used: Maintained author survey recording the solved quintuple conjecture, the open regularity conjecture, correct d_+ formula, and extension bounds.\n- D(4)-Triples with Two Largest Elements in Common, Mathematica Slovaca (2023). (primary): https://doi.org/10.1515/ms-2023-0027\n Evidence used: Explicitly records the stronger uniqueness-of-extension question as open for the classical n=1 case and n=4.\n\n**Review notes.** The imported Conjecture (2) has 2bc where the standard regular-extension formula has 2abc. Because the row contains both conjectures, it is classified partially solved rather than solved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3352, "problem_number": "OPG-36952", "title": "Twin prime conjecture", "statement": "Conjecture There exist infinitely many positive integers $n$ so that both $n$ and $n+2$ are prime.", "background": "Source: Open Problem Garden. Original node ID: 36952. URL: http://www.openproblemgarden.org/op/twin_prime_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/twin_prime_conjecture\n- Subject(s): Number Theory\n- Keywords: prime; twin prime\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 4th, 2009 by kaushiks.nitt\n\nComments:\n- November 22nd, 2011 | Kermit1941 | Twin Prime Conjecture: Hello.\n\nI see a reference to a proof called Tyte's proof but saw no details on it. When I click on the link, it only gives me the same page that I'm already on.\n\nAlso, I could not get any additional details about the following post:\n\nOn January 10th, 2011 Hugh Barker says: I've added a thread here, linking to an attempted proof of Twin Primes and the Polignac conjecture in general.\n\nhttp://garden.irmacs.sfu.ca/?q=op/twin_primes_and_polignacs_conjecture\n\nWill receive any input, debunking etc gratefully.\n\nI am most interested in searching for any attempted proofs of the twin prime conjecture. I believe that I have a proof, but anticipate clearly filling in some details.\n\nThe core idea in our proof is that we specify exactly a lower bound for the number of twin primes less than a given integer, N, and that this lower bound goes to infinity as N goes to infinity.\n\nKermit\n- November 25th, 2009 | ducafelipe | Regarding the Tyte's proof I: Regarding the Tyte's proof I have received three enthusiastic comments-contributions, pointing out that while the averaging step made by Alan is questionable, maybe this approach shows where to look for in the solution of this Conjecture. These first three comments are from Chris Nash, Fabrice Marchant and Leadhyena Inrandomtan:\n\n\"About Alan Tyte's proof: all the beginning up to \"Lemma 5\" is right but there are 2 errors in the end of the proof, after each \"Hence, on the average:\" because we do not know the way our beloved Ds are spanned: no reason to be sure they are put at the same rate between x and x^2 than between a whole pattern. However, I think the idea of the proof with As, Bs... is great and I'll try to work in the way of Alan.\" (F. Marchant)\n- August 18th, 2009 | Anonymous | Where is the conjecture?: The conjecture is not stated. It's just the definition.\n- August 18th, 2009 | mdevos | thanks: I know little about the status of this conjecture, but I did correct the definition. We would welcome anyone with more knowledge to update it further.\n- January 10th, 2011 | Hugh Barker | I've added a thread here,: I've added a thread here, linking to an attempted proof of Twin Primes and the Polignac conjecture in general.\n\nhttp://garden.irmacs.sfu.ca/?q=op/twin_primes_and_polignacs_conjecture\n\nWill receive any input, debunking etc gratefully.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Twin prime conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The twin prime conjecture remains open. Bounded-gap methods prove infinitely many consecutive-prime gaps at most 246, but not infinitely many gaps equal to 2.\n\n**Verified partial progress.**\n\n- Maynard proved liminf of consecutive-prime gaps at most 600 and bounded intervals containing arbitrarily many primes.\n- The Polymath8b project improved the unconditional bound for two consecutive primes to 246.\n- These results do not identify any fixed even gap that occurs infinitely often.\n\n**Full solution or refutation.**\n\nNo accepted proof establishes infinitely many prime pairs differing by 2; bounded gaps are a quantitatively weaker theorem.\n\n**What remains.**\n\nReduce the unconditional bounded-gap conclusion from some gap at most 246 to the specific gap 2, requiring ideas beyond the current parity limitation of sieve methods.\n\n**Sources checked.**\n\n- James Maynard, Small gaps between primes, Annals of Mathematics 181 (2015), 383-413, DOI 10.4007/annals.2015.181.1.7. (primary): https://doi.org/10.4007/annals.2015.181.1.7\n Evidence used: Proves the unconditional gap-600 theorem and bounded intervals with arbitrarily many primes.\n- Polymath8, Bounded gaps between primes, maintained project page, accessed 2026-08-17. (maintained_tracker): https://michaelnielsen.org/polymath/index.php?title=Bounded_gaps_between_primes\n Evidence used: Records the concluded unconditional record H_1<=246 and distinguishes it from the twin-prime target H_1=2.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3353, "problem_number": "OPG-37289", "title": "Polignac's Conjecture", "statement": "Conjecture Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.\n\nIn particular, this implies:\n\nConjecture Twin Prime Conjecture: There are an infinite number of twin primes.", "background": "Source: Open Problem Garden. Original node ID: 37289. URL: http://www.openproblemgarden.org/op/twin_primes_and_polignacs_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/twin_primes_and_polignacs_conjecture\n- Author(s): de Polignac, Alphonse\n- Subject(s): Number Theory\n- Keywords: prime; prime gap\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: January 10th, 2011 by Hugh Barker\n\nBibliography:\n*[P] A. de Polignac, Six propositions arithmologiques déduites de crible d'Ératosthène. Nouv. Ann. Math. 8 (1849), pp. 423--429.\n\nRelated:\nRelated problems\nTwin prime conjecture\n\nComments:\n- June 18th, 2013 | Charles R Great... | Link: I removed this link and its description from the problem, since it is now known to be incorrect. For future reference here it is: http://barkerhugh.blogspot.com/2011/01/twin-primes-and-polignac-conjecture.html\n- January 13th, 2011 | Hugh Barker | Flaw: OK, someone has spotted the inevitable flaw in the logic and pointed it out, so not worth looking after all (though feel free if you want to play \"spot the error\"...\n- January 11th, 2011 | Anonymous | Compressed version: There's a slightly compressed version of this proof here:\n\nhttp://barkerhugh.blogspot.com/2011/01/twin-prime-proof-compressed-version.html\n\nProbably better to refer to this one as it is more focused.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Polignac's Conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Polignac's conjecture remains open, including the twin-prime case. Bounded-gap theorems force some recurring even gap but do not identify any prescribed value.\n\n**Verified partial progress.**\n\n- D. H. J. Polymath proved unconditionally that liminf of consecutive prime gaps is at most 246, hence at least one even gap at most 246 occurs infinitely often.\n- Banks proved that every IP set of even natural numbers contains infinitely many de Polignac numbers.\n\n**Full solution or refutation.**\n\nNo fixed positive even n, including n=2, is known from the cited results to occur infinitely often as a consecutive-prime gap.\n\n**What remains.**\n\nProve recurrence for each prescribed positive even gap, or produce a counterexample.\n\n**Sources checked.**\n\n- D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Res. Math. Sci. 1:12 (2014), arXiv:1407.4897. (primary): https://arxiv.org/abs/1407.4897\n Evidence used: The abstract proves the unconditional bound H_1 <= 246.\n- W. D. Banks, Consecutive primes and IP sets, arXiv:2403.10637 (2024). (primary): https://arxiv.org/abs/2403.10637\n Evidence used: The abstract proves that every IP set of even natural numbers contains infinitely many de Polignac numbers.\n\n**Review notes.** The source correctly requires consecutive primes.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3354, "problem_number": "OPG-37300", "title": "Special Primes", "statement": "Conjecture Let $p$ be a prime natural number. Find all primes $q\\equiv1\\left(\\mathrm{mod}\\: p\\right)$, such that $2^{\\frac{\\left(q-1\\right)}{p}}\\equiv1\\left(\\mathrm{mod}\\: q\\right)$.", "background": "Source: Open Problem Garden. Original node ID: 37300. URL: http://www.openproblemgarden.org/op/special_primes.\n\nSource subject path: Number Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/special_primes\n- Author(s): George BALAN\n- Subject(s): Number Theory\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: February 18th, 2011 by maththebalans\n\nComments:\n- February 7th, 2013 | Anonymous | All primes are: All primes are p=(q-1)/(order of 2 mod q)\n- February 17th, 2012 | Anonymous | paul newell: q divides 2^((q-1)/P))-1 iff p divides (q-1)/( Order of2 mod q )\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Special Primes\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For fixed prime p, classical Kummer and Chebotarev theory gives an exact splitting-field characterization and positive density, but the source is an ambiguous 'find all' instruction rather than a falsifiable conjecture.\n\n**Verified partial progress.**\n\n- The congruence holds exactly when ord_q(2) divides (q-1)/p, equivalently when 2 is a pth-power residue modulo q.\n- Apart from ramified primes, these q are exactly the primes splitting completely in Q(zeta_p,2^(1/p)); Chebotarev gives density 1/[K_p:Q] and infinitude.\n\n**Full solution or refutation.**\n\nA complete algebraic and distributional characterization exists, but the source does not define whether that counts as 'all' or demand an elementary congruence-class list.\n\n**What remains.**\n\nClarify whether p is fixed and what output form 'find all' requires; if an elementary finite congruence classification is intended, state that demand explicitly.\n\n**Sources checked.**\n\n- P. Moree, Artin's primitive root conjecture -- a survey, MPIM Preprint 2012 (53). (authoritative_secondary): https://archive.mpim-bonn.mpg.de/1236/1/preprint_2012_53.pdf\n Evidence used: Section 4 proves the equivalence between the power congruence and complete splitting in Q(zeta_p,g^(1/p)), then applies Chebotarev.\n- Open Problem Garden, Special Primes (source record preserved in the 2026 corpus). (maintained_tracker): https://www.openproblemgarden.org/op/special_primes\n Evidence used: The page labels an instruction as a conjecture and its comments provide the multiplicative-order reformulation without a precise completion criterion.\n\n**Review notes.** Formulation defects: it is not a conjecture, it is unclear whether p is fixed, and 'find all' has no terminal criterion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3355, "problem_number": "OPG-37318", "title": "Primitive pythagorean n-tuple tree", "statement": "Conjecture Find linear transformation construction of primitive pythagorean n-tuple tree!", "background": "Source: Open Problem Garden. Original node ID: 37318. URL: http://www.openproblemgarden.org/op/primitive_pythagorean_n_tuple_tree.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/primitive_pythagorean_n_tuple_tree\n- Subject(s): Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 24th, 2011 by tsihonglau\n\nProblem-page discussion:\nPrimitive pythagorean n-tuple is a n-tuple $(a_{1},a_{2},a_{3},...,a_{n})$ such that $a_{1}^2 + a_{2}^2 + a_{3}^2 +... + a_{n-1}^2 = a_{n}^2$\n\nand the greatest common divisor of $(a_{1},a_{2},a_{3},...,a_{n})$ is 1.\n\nThere are at least two known linear transformation construction of primitive pythagorean triple tree!\n\nWikipedia\n\nIs there any other linear transformation construction of primitive pythagorean triple tree?\n\nMoreover, find linear transformation construction of primitive pythagorean n-tuple tree!\n\nDiscussion links:\n- There are at least two known linear transformation construction of primitive pythagorean triple tree!: http://home.educities.edu.tw/tsihonglau/senior/primitive_pythagorean_triple_ternary_tree.html\n- Wikipedia: http://en.wikipedia.org/wiki/Pythagorean_triple#Parent.2Fchild_relationships\n\nComments:\n- June 16th, 2021 | Anonymous | Linear transformation of primitive pythagorean n-tuple: Is it true that for primitive pythagorean n-tuple (a_1, a_2...... a_n), if there are (n-2) even terms and 1 odd term in the summation side of the equation and a_n is odd, a linear transformation can be made? And can we find any such primitive pythagorean n-tuple where a_n is even?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Primitive pythagorean n-tuple tree\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Cass and Arpaia give linear matrix generation of every primitive Pythagorean n-tuple for 4 <= n <= 9 and prove an obstruction to a single-orbit construction for n >= 10, but the source never defines the required tree structure.\n\n**Verified partial progress.**\n\n- For each 4 <= n <= 9, a matrix A_n generates all primitive n-tuples from (1,0,...,0,1), allowing permutations and sign changes of the first n-1 coordinates.\n- For n >= 10, the tuples occupy at least floor((n+6)/8) distinct orbits under the relevant automorphism group, so the same one-root/single-orbit construction cannot cover all tuples.\n\n**Full solution or refutation.**\n\nA strong dimension-bounded generation theorem and a high-dimensional obstruction are known, but neither verifies a unique-parent rooted tree under an explicit definition.\n\n**What remains.**\n\nSpecify signs, coordinate ordering, allowed roots and transformations, and whether every tuple must have a unique parent; then determine which Cass-Arpaia orbit constructions induce such a tree in each dimension.\n\n**Sources checked.**\n\n- D. Cass and P. J. Arpaia, Matrix Generation of Pythagorean n-Tuples, Proceedings of the American Mathematical Society 109(1) (1990), 1-7. (primary): https://doi.org/10.2307/2048355\n Evidence used: The paper constructs A_n for dimensions 4 through 9 and proves the multiple-orbit obstruction beginning in dimension 10.\n\n**Review notes.** The imperative source wording and undefined word 'tree' are preserved and flagged, not repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3356, "problem_number": "OPG-37396", "title": "3 is a primitive root modulo primes of the form 16 q^4 + 1, where q>3 is prime", "statement": "Conjecture $3~$ is a primitive root modulo $~p$ for all primes $~p=16\\cdot q^4+1$, where $~q>3$ is prime.", "background": "Source: Open Problem Garden. Original node ID: 37396. URL: http://www.openproblemgarden.org/op/primes_p_such_that_3_is_a_primitive_root_modulo_p.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/primes_p_such_that_3_is_a_primitive_root_modulo_p\n- Subject(s): Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 25th, 2012 by princeps\n\nSource links:\n- primitive root: http://en.wikipedia.org/wiki/Primitive_root_modulo_n\n\nComments:\n- August 24th, 2012 | Anonymous | group theory answer: Using group theory, the multiplicative group of order p=16q^4+1 has order p-1=16q^4. Using lagrange's theorem, the order of any element divides the order of the group. Therefore, any element is either a primitive root, a quadratic residue, or a qth power residue mod 16q^4+1. Using the laws of quadratic reciprocity, 3 is a quadratic residue modulo a prime if and only if the prime is congruent to plus or minus 1 mod 12. Since q>3 is a prime and therefore not divisible by 3, 16q^4=1(mod 3), so 16q^4+1=2(mod 3). That means that 16q^4+1=5(mod 12), and therefore 3 is not a quadratic residue mod p. Therefore the only thing left to prove is that 3 is not a qth power residue.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"3 is a primitive root modulo primes of the form 16 q^4 + 1, where q>3 is prime\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No reliable publication settling the universal primitive-root statement for prime values 16q^4+1 was found.\n\n**Verified partial progress.**\n\n- General Artin-type primitive-root theory gives relevant heuristic context but does not prove this thin prime-sequence assertion.\n\n**Full solution or refutation.**\n\nNeither a counterexample nor a theorem for all such primes was verified.\n\n**What remains.**\n\nFirst establish the intended prime family/examples and then test or prove the primitive-root condition with analytic number theory tools.\n\n**Sources checked.**\n\n- Open Problem Garden, 3 is a primitive root modulo primes of the form 16 q^4 + 1 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/primes_p_such_that_3_is_a_primitive_root_modulo_p\n Evidence used: The source remains an unsolved conjecture page and no linked resolution is provided.\n\n**Review notes.** No source alteration; thin-family formulation requires specialist verification.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3357, "problem_number": "OPG-37397", "title": "Erdős–Straus conjecture", "statement": "Conjecture\n\nFor all $n > 2$, there exist positive integers $x$, $y$, $z$ such that $$1/x + 1/y + 1/z = 4/n$$.", "background": "Source: Open Problem Garden. Original node ID: 37397. URL: http://www.openproblemgarden.org/op/erdos_straus_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/erdos_straus_conjecture\n- Author(s): Erdos, Paul; Straus, Ernst G.\n- Subject(s): Number Theory\n- Keywords: Egyptian fraction\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: February 29th, 2012 by ACW\n\nProblem-page discussion:\nSee Erdős–Straus conjecture for more details.\n\nDiscussion links:\n- Erdős–Straus conjecture: http://en.wikipedia.org/wiki/Erdős–Straus conjecture\n\nComments:\n- July 14th, 2014 | Anonymous | Formula Individa: It was necessary to write the solution in a more General form: $$\\frac{t}{q}=\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}$$$t,q$- integers. Decomposing on the factors as follows:$p^2-s^2=(p-s)(p+s)=2qL$The solutions have the form:$$x=\\frac{p(p-s)}{tL-q}$$$$y=\\frac{p(p+s)}{tL-q}$$$$z=L$$Decomposing on the factors as follows:$p^2-s^2=(p-s)(p+s)=qL$The solutions have the form:$$x=\\frac{2p(p-s)}{tL-q}$$$$y=\\frac{2p(p+s)}{tL-q}$$$$z=L$$\n- July 14th, 2014 | Anonymous | Solution: For the equation: $$\\frac{4}{q}=\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}$$The solution can be written using the factorization, as follows.$$p^2-s^2=(p-s)(p+s)=2qL$$Then the solutions have the form:$$x=\\frac{p(p-s)}{4L-q}$$$$y=\\frac{p(p+s)}{4L-q}$$$$z=L$$I usually choose the number$L$such that the difference:$(4L-q)$was equal to:$1,2,3,4$Although your desire you can choose other. You can write a little differently. If unfold like this:$$p^2-s^2=(p-s)(p+s)=qL$$The solutions have the form:$$x=\\frac{2p(p-s)}{4L-q}$$$$y=\\frac{2p(p+s)}{4L-q}$$$$z=L$$\n- July 14th, 2013 | cpbm | Further restriction: I think you need to specify that $x$, $y$ and $z$ be positive for this to be challenging (and open).\n- July 15th, 2013 | ACW | Restriction: Done. Thank you.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 41.\n\nAttempt notes:\nTarget:\nMake progress on \"Erdős–Straus conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Erdős--Straus conjecture remains open, but explicit congruence identities and very large finite computations verify many cases.\n\n**Verified partial progress.**\n\n- Recent work reports further large finite verification and empirical constructions.\n- Many residue classes admit elementary parametric decompositions.\n\n**Full solution or refutation.**\n\nNo accepted proof for every n was verified; recent papers labelled solutions are not treated as consensus resolution.\n\n**What remains.**\n\nGive an unconditional all-n decomposition or a counterexample.\n\n**Sources checked.**\n\n- M. Mihnea and D. Dumitru, Further verification and empirical evidence for the Erdős-Straus conjecture, arXiv:2509.00128 (2025). (primary): https://arxiv.org/abs/2509.00128\n Evidence used: The abstract reports further verification/empirical evidence rather than a general proof.\n- Erdős--Straus conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Straus_conjecture\n Evidence used: Records the conjecture as unproven.\n\n**Review notes.** Unaccepted claimed proofs not promoted; statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3358, "problem_number": "OPG-37402", "title": "Lucas Numbers Modulo m", "statement": "Conjecture The sequence {L(n) mod m}, where L(n) are the Lucas numbers, contains a complete residue system modulo m if and only if m is one of the following: 2, 4, 6, 7, 14, 3^k, k >=1.", "background": "Source: Open Problem Garden. Original node ID: 37402. URL: http://www.openproblemgarden.org/op/lucas_numbers_modulo_m.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/lucas_numbers_modulo_m\n- Subject(s): Number Theory\n- Keywords: Lucas numbers\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 19th, 2012 by Martin Erickson\n\nProblem-page discussion:\nThe Lucas numbers are defined by L(0)=2, L(1)=1, and L(n)=L(n-1)+L(n-2), for n >=2. Thus the sequence is 2, 1, 3, 4, 7, 11, 18, 29, 47,....\n\nExample: If m = 5, then we have the sequence 2, 1, 3, 4, 2, 1,..., and since the sequence repeats we never obtain 0 mod 5.\n\nExample: If m = 6, then we have 2, 1, 3, 4, 1, 5, 0,..., and we obtain a complete residue system mod 6.\n\nThe corresponding problem for the Fibonacci sequence was solved by S. A. Burr. The sequence {F(n) mod m} contains a complete residue system mod m if and only if m is one of the following: 5^k, 2.5^k, 4.5^k, 3^j.5^k, 6.5^k, 7.5^k, 14.5^k.\n\nBibliography:\nS. A. Burr, \"On Moduli for Which the Fibonacci Sequence Contains a Complete System of Residue\", Fibonacci Quarterly, December 1971, pp. 497-504.\n\nComments:\n- July 1st, 2014 | Anonymous | Solved: This problem has been solved: http://mathacadabra.com/Items2013/LucasCompleteResidue.aspx\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Lucas Numbers Modulo m\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The Lucas residue-completeness characterization is proved exactly as conjectured: m is 2, 4, 6, 7, 14, or a power of 3.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nLang--Lang give the stated necessary and sufficient condition.\n\n**What remains.**\n\nThe source conjecture is answered.\n\n**Sources checked.**\n\n- C. L. Lang and M. L. Lang, Fibonacci system and residue completeness, arXiv:1304.2892 (2013). (primary): https://arxiv.org/abs/1304.2892\n Evidence used: The abstract explicitly gives the complete Lucas-modulus list.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3359, "problem_number": "OPG-37404", "title": "Sum of prime and semiprime conjecture", "statement": "Conjecture Every even number greater than $10$ can be represented as the sum of an odd prime number and an odd semiprime.", "background": "Source: Open Problem Garden. Original node ID: 37404. URL: http://www.openproblemgarden.org/op/sum_of_prime_and_semiprime_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/sum_of_prime_and_semiprime_conjecture\n- Author(s): Geoffrey Marnell\n- Subject(s): Number Theory\n- Keywords: prime; semiprime\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 23rd, 2012 by princeps\n\nBibliography:\n*[M] Geoffrey R. Marnell, \"Ten Prime Conjectures\", Journal of Recreational Mathematics 33:3 (2004-2005), pp. 193--196.\n\nRelated:\nRelated problems\nGoldbach conjecture\n\nSource links:\n- semiprime: http://en.wikipedia.org/wiki/semiprime\n\nComments:\n- July 1st, 2012 | Anonymous | surely that's Chen's theorem: Every sufficiently large even number is the sum of either 2 primes or a prime and a semiprime.\n- July 31st, 2016 | Charles R Great... | Yes, apart from the: Yes, apart from the \"sufficiently large\" and allowing prime + prime as well as prime + semiprime. The parity problem makes the latter hard, but some progress has been made, see arXiv:math/0609615 and arXiv:0803.2636. (The key progress is their use of E2 = semiprimes rather than P2 = primes or semiprimes.)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Sum of prime and semiprime conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Chen's theorem proves a close sufficiently-large prime-plus-almost-prime result, but does not establish the source's all-even, odd-prime plus odd-semiprime statement.\n\n**Verified partial progress.**\n\n- Every sufficiently large even integer is a sum of a prime and a number with at most two prime factors (with the standard caveats on the second summand).\n\n**Full solution or refutation.**\n\nThe parity-specific universal assertion remains unverified.\n\n**What remains.**\n\nProve the odd-semiprime form for every even n>10 or produce a counterexample.\n\n**Sources checked.**\n\n- Chen's theorem overview, MathWorld (accessed 2026-08-17). (authoritative_secondary): https://mathworld.wolfram.com/ChensTheorem.html\n Evidence used: States the sufficiently-large prime plus prime/semiprime theorem and its weaker scope.\n- Open Problem Garden, Sum of prime and semiprime conjecture (accessed 2026-08-17). (maintained_tracker): https://garden.irmacs.sfu.ca/op/sum_of_prime_and_semiprime_conjecture\n Evidence used: Records the distinction between Chen's theorem and the source's stronger parity statement.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3360, "problem_number": "OPG-37411", "title": "Giuga's Conjecture on Primality", "statement": "Conjecture $p$ is a prime iff $~\\displaystyle \\sum_{i=1}^{p-1} i^{p-1} \\equiv -1 \\pmod p$", "background": "Source: Open Problem Garden. Original node ID: 37411. URL: http://www.openproblemgarden.org/op/giugas_conjecture_on_primality.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/giugas_conjecture_on_primality\n- Author(s): Giuseppe Giuga\n- Subject(s): Number Theory\n- Keywords: primality\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 27th, 2012 by princeps\n\nBibliography:\n[BBBG] Borwein, D.; Borwein, J. M., Borwein, P. B., and Girgensohn, R. \"Giuga's Conjecture on Primality\", American Mathematical Monthly, 103, 40–50, (1996)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Giuga's Conjecture on Primality\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Giuga's primality criterion remains open: it is unknown whether a composite integer can satisfy the congruence.\n\n**Verified partial progress.**\n\n- Known necessary conditions force any counterexample to be extraordinarily restricted (a Giuga/Carmichael-type number).\n\n**Full solution or refutation.**\n\nNo composite example or proof of primality equivalence was verified.\n\n**What remains.**\n\nExclude all composite solutions or construct one.\n\n**Sources checked.**\n\n- Agoh--Giuga conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Agoh%E2%80%93Giuga_conjecture\n Evidence used: Records the equivalent Giuga primality question as unresolved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3361, "problem_number": "OPG-37413", "title": "Alexa's Conjecture on Primality", "statement": "Definition Let $r_i$ be the unique integer (with respect to a fixed $p\\in\\mathbb{N}$ ) such that\n\n$$(2i+1)^{p-1} \\equiv r_i \\pmod p ~~\\text{ and } ~ 0 \\le r_i < p.$$\n\nConjecture A natural number $p \\ge 8$ is a prime iff $$\\displaystyle \\sum_{i=1}^{\\left \\lfloor \\frac{\\sqrt[3]p}{2} \\right \\rfloor} r_i = \\left \\lfloor \\frac{\\sqrt[3]p}{2} \\right \\rfloor$$", "background": "Source: Open Problem Garden. Original node ID: 37413. URL: http://www.openproblemgarden.org/op/alexas_conjecture_on_primality.\n\nSource subject path: Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/alexas_conjecture_on_primality\n- Author(s): Alexa\n- Subject(s): Number Theory\n- Keywords: primality\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 28th, 2012 by princeps\n\nProblem-page discussion:\nThe conjecture is obviously true when $p$ is prime, so it suffices to check when $p$ is composite.\n\nRelated:\nRelated problems\nGiuga's Conjecture on Primality\n\nComments:\n- March 28th, 2012 | Anonymous | counter-example for p=66: formula works for p from 8 through 100, except for p=66.\n- March 30th, 2012 | princeps | counter-example for p=66: Thanks, I have corrected statement.\n- March 30th, 2012 | Anonymous | don't work either: your new statement is ambiguous (which r_i should one choose inside the sum?). I'm assuming you're just trying to move the \"mod p\" to apply to the sum only (and not to the RHS). If that's what you're doing, it still doesn't work. Same counter-examples at p=66, 102, 246 and 492 for p from 8 to 500.\n- March 30th, 2012 | princeps | donit work either: Yes it works.I have checked statement up to 10^6,there is no counterexample...\n- March 30th, 2012 | Anonymous | still ambiguous: then please re-word your conjecture, because as it stands, it's ambiguous and not true. It's ambiguous, because the way you defined r_i, one could have chosen r_i, r_i + p, r_i + 2p etc., but when you plug these into the sum, you get a different sum and the equality doesn't make sense.\n- April 2nd, 2012 | Anonymous | still not good: your modified version now reduces back to putting mod p on the LHS of the equation, which as I've pointed out above, doesn't work (see counter-examples I gave). Where did you come up with this conjecture? Is there any published reference for it?\n- June 14th, 2013 | Charles R Great... | Fixed: I've corrected the statement on Alexa's behalf. This version holds up to at least 100 million.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Alexa's Conjecture on Primality\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No reliable publication or maintained problem source analyzing Alexa's exact cube-root truncated Fermat-residue criterion was found.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe supplied claim cannot be classified from the literature located.\n\n**What remains.**\n\nFind an original proof/counterexample or conduct a separately documented finite search followed by a proof.\n\n**Sources checked.**\n\n- Open Problem Garden, Alexa's Conjecture on Primality (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/alexas_conjecture_on_primality\n Evidence used: Provides the statement but no resolution or independent reference.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3362, "problem_number": "OPG-37423", "title": "Birch & Swinnerton-Dyer conjecture", "statement": "Conjecture Let $E/K$ be an elliptic curve over a number field $K$. Then the order of the zeros of its $L$-function, $L(E, s)$, at $s = 1$ is the Mordell-Weil rank of $E(K)$.", "background": "Source: Open Problem Garden. Original node ID: 37423. URL: http://www.openproblemgarden.org/op/birch_swinnerton_dyer_conjecture.\n\nSource subject path: Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/birch_swinnerton_dyer_conjecture\n- Subject(s): Number Theory\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: May 12th, 2012 by eyoong\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Birch & Swinnerton-Dyer conjecture\" in Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** BSD remains open generally, but the rank equality and leading-coefficient formula are proved in important analytic-rank 0 and 1 modular elliptic-curve cases.\n\n**Verified partial progress.**\n\n- Gross--Zagier and Kolyvagin establish decisive rank-0/1 cases under standard hypotheses.\n\n**Full solution or refutation.**\n\nNo theorem covers arbitrary elliptic curves over arbitrary number fields.\n\n**What remains.**\n\nProve the rank equality (and full BSD formula) in general.\n\n**Sources checked.**\n\n- Clay Mathematics Institute, Birch and Swinnerton-Dyer Conjecture (accessed 2026-08-17). (authoritative_secondary): https://www.claymath.org/millennium-problems/birch-and-swinnerton-dyer-conjecture\n Evidence used: States the general conjecture remains unsolved and describes known low-rank cases.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3363, "problem_number": "OPG-367", "title": "The Erdos-Turan conjecture on additive bases", "statement": "Let $B \\subseteq {\\mathbb N}$. The representation function $r_B: {\\mathbb N} \\rightarrow {\\mathbb N}$ for $B$ is given by the rule $r_B(k) = \\#\\{ (i,j) \\in B \\times B: i + j = k \\}$. We call $B$ an additive basis if $r_B$ is never $0$.\n\nConjecture If $B$ is an additive basis, then $r_B$ is unbounded.", "background": "Source: Open Problem Garden. Original node ID: 367. URL: http://www.openproblemgarden.org/op/the_erdos_turan_conjecture_on_additive_bases.\n\nSource subject path: Number Theory > Additive Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_erdos_turan_conjecture_on_additive_bases\n- Author(s): Erdos, Paul; Turan, Paul\n- Subject(s): Number Theory; Additive Number Theory\n- Keywords: additive basis; representation function\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 8th, 2007 by mdevos\n\nProblem-page discussion:\nThis famous conjecture seems intuitively likely, but to date, there has been relatively little progress on it despite considerable attention. Two positive results are a theorem of Dirac [D] which shows that $r_B$ cannot be constant from some point on, and a theorem of Borwein, Choi, and Chu [BCC] which shows that $r_B$ cannot be bounded above by $6$.\n\nOn the other hand, if we consider the related problem for subsets of integers instead of natural numbers, Nathanson [N] has shown that the conjecture does not hold.\n\nBibliography:\n[BCC] P. Borwein, S. Choi, and F. Chu, An old conjecture of Erdos-Turan on additive bases, Mathematics of Computation. Volume 75, Number 253, Pages 475–484.\n\n[D] G. A. Dirac, Note on a problem in additive number theory, J. London Math. Soc. 26 (1951), 312–313.\n\n[EG] P. Erdos and R. L. Graham, Old and new problems and results in combinatorial number theory: van der Waerden’s theorem and related topics, Enseign. Math. (2) 25 (1979), no. 3-4, 325–344 (1980). MathSciNet\n\n*[ET] P. Erdos and P. Turan, On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc. 16 (1941), 212–215. MathSciNet\n\n[N] Melvyn B. Nathanson, Unique representation bases for the integers, Acta Arith. 108 (2003), no. 1, 1–8. MathSciNet\n\nBibliography links:\n- An old conjecture of Erdos-Turan on additive bases: http://www.ams.org/mcom/2006-75-253/S0025-5718-05-01777-1/S0025-5718-05-01777-1.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0570317\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=0006197\n- Unique representation bases for the integers: http://front.math.ucdavis.edu/math.NT/0302091\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1971077\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"The Erdos-Turan conjecture on additive bases\" in Number Theory; Additive Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Erdos--Turan additive-bases conjecture remains open.\n\n**Verified partial progress.**\n\n- Many restricted bases and average representation results support the expectation.\n\n**Full solution or refutation.**\n\nNo proof that every additive basis has unbounded representation function was verified.\n\n**What remains.**\n\nBreak bounded-representation constructions or prove a density/structure obstruction.\n\n**Sources checked.**\n\n- Open Problem Garden, node 367 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3364, "problem_number": "OPG-706", "title": "Goldbach conjecture", "statement": "Conjecture Every even integer greater than 2 is the sum of two primes.", "background": "Source: Open Problem Garden. Original node ID: 706. URL: http://www.openproblemgarden.org/op/goldbach_conjecture.\n\nSource subject path: Number Theory > Additive Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/goldbach_conjecture\n- Author(s): Goldbach, Christian\n- Subject(s): Number Theory; Additive Number Theory\n- Keywords: additive basis; prime\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 20th, 2007 by Benschop\n\nProblem-page discussion:\nThis famous conjecture is one of the oldest unsolved problems in mathematics. It arose originally out of a correspondence (1742) between Goldbach and Euler. See Wikipedia's Goldbach's conjecture for more.\n\nDiscussion links:\n- Wikipedia's Goldbach's conjecture: http://en.wikipedia.org/wiki/goldbach's conjecture\n\nComments:\n- June 14th, 2013 | Charles R Great... | Weak conjecture now solved: Note that Harald Helfgott has proved the ternary version of Goldbach's conjecture, that every odd number greater than 7 is the sum of three odd primes. See http://arxiv.org/abs/1205.5252 (minor arc estimates) and http://arxiv.org/abs/1305.2897 (major arc estimates).\n- November 22nd, 2007 | Benschop | Goldbach conjecture: Those interested in a suggestion for a proof via semigroup theory and carry extension, see: \"Additive structure of Z(.) mod [\\prod first k primes], with carry extension to prime pair sums\". - http://home.iae.nl/users/benschop/ngb0203.pdf (10 pgs, submitted for publication)\n- December 12th, 2007 | Anonymous | Goldbach conjecture: In addition to the above link to the paper with Goldbach-Conj. proof, the abstract and links to an intro-article are at http://home.iae.nl/users/benschop/ng-abstr.htm\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Goldbach conjecture\" in Number Theory; Additive Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Strong Goldbach remains open; it is computationally verified through 4*10^18, while the ternary Goldbach conjecture is proved and Chen-type almost-prime approximations are known.\n\n**Verified partial progress.**\n\n- Oliveira e Silva-Herzog-Pardi verify every even integer through 4*10^18 as a sum of two primes.\n- Helfgott proves every odd integer greater than 5 is a sum of three primes.\n- Chen's theorem represents every sufficiently large even integer as a prime plus an integer having at most two prime factors.\n\n**Full solution or refutation.**\n\nNo proof for all even integers or counterexample was verified.\n\n**What remains.**\n\nStrengthen sieve/circle-method control from prime-plus-semiprime to two primes uniformly for all even integers beyond the verified range.\n\n**Sources checked.**\n\n- Tomás Oliveira e Silva, Siegfried Herzog, and Silvio Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4*10^18, Mathematics of Computation 83 (2014), 2033-2060. (primary): https://doi.org/10.1090/S0025-5718-2013-02787-1\n Evidence used: Reports and validates exhaustive verification through 4*10^18.\n- Harald Andrés Helfgott, The ternary Goldbach conjecture is true, arXiv:1312.7748. (primary): https://arxiv.org/abs/1312.7748\n Evidence used: Proves weak/ternary Goldbach, a distinct theorem that does not settle the binary conjecture.\n- Open Problem Garden, Goldbach conjecture (node 706). (maintained_tracker): https://www.openproblemgarden.org/op/goldbach_conjecture\n Evidence used: Original binary statement and weak-Goldbach update.\n\n**Review notes.** Unreviewed claimed proofs in the imported comments are not accepted as resolutions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3365, "problem_number": "OPG-37192", "title": "Are there an infinite number of lucky primes?", "statement": "Conjecture If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc., an infinite number of the remaining integers are prime.", "background": "Source: Open Problem Garden. Original node ID: 37192. URL: http://www.openproblemgarden.org/op/are_there_an_infinite_number_of_lucky_primes.\n\nSource subject path: Number Theory > Additive Number Theory.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_there_an_infinite_number_of_lucky_primes\n- Author(s): Lazarus: Gardiner: Metropolis; Ulam, Stanislaw M.\n- Subject(s): Number Theory; Additive Number Theory\n- Keywords: lucky; prime; seive\n- Importance: Low ✭\n- Recommended for undergraduates: yes\n- Posted: March 24th, 2010 by cubola zaruka\n\nProblem-page discussion:\nThe difference between the seive for generating primes and the seive for gnerating lucky numbers is that the former removes every nth integer whereas the latter removes every nth remaining integer. There are known to be an infinite number of lucky numbers because removing every nth remaining number leaves numbers unremoved as n increases and eventually becomes larger than any given value. There is also the unproved lucky analogue of the Goldbach conjecture, that every even number is expressible as the sum of two lucky numbers.\n\nBibliography:\nhttp://en.wikipedia.org/wiki/Lucky_number\n\nRelated:\nRelated problems\nGoldbach conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Are there an infinite number of lucky primes?\" in Number Theory; Additive Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Under the standard intended definition of lucky numbers, it remains conjectural that infinitely many lucky numbers are prime.\n\n**Verified partial progress.**\n\n- The lucky-number theorem gives the prime-like counting scale: if l_n is the nth lucky number, then l_n is asymptotic to n log n.\n- Sanna's 2026 preprint supplies explicit upper and lower inequalities for l_n, but does not control primality among the lucky numbers.\n\n**Full solution or refutation.**\n\nNo proof or disproof of infinitely many lucky primes was found. A 2023 primary article explicitly describes infinitude as a conjecture, and the 2026 distribution paper still studies the lucky sequence rather than its prime intersection.\n\n**What remains.**\n\nProve that infinitely many standard lucky numbers are prime, or prove the intersection finite. Before reuse, replace the corrupted prose with the standard recursive sieve definition while retaining the source text as provenance.\n\n**Sources checked.**\n\n- D. Cassettari, G. Mussardo, and A. Trombettoni, Holographic realization of the prime number quantum potential, PNAS Nexus 2 (2023), pgac279, DOI 10.1093/pnasnexus/pgac279. (primary): https://doi.org/10.1093/pnasnexus/pgac279\n Evidence used: Defines lucky primes and explicitly says that infinitely many are conjectured.\n- C. Sanna, Explicit inequalities for the nth lucky number, arXiv:2604.07142 (2026). (primary): https://arxiv.org/abs/2604.07142\n Evidence used: Gives current quantitative bounds for lucky numbers and recalls their n log n asymptotic, without resolving lucky primes.\n\n**Review notes.** Formulation defect: the standard sieve deletes every second entry, then every third survivor, then every seventh survivor, etc.; it does not proceed by every fourth survivor as the imported statement suggests.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3366, "problem_number": "OPG-573", "title": "The Riemann Hypothesis", "statement": "The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the complex variable is in ]0;1[), are actually located on the Critical line ( the vertical line of the complex plane with real part equal to 1/2)", "background": "Source: Open Problem Garden. Original node ID: 573. URL: http://www.openproblemgarden.org/op/the_riemann_hypothesis.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_riemann_hypothesis\n- Author(s): Riemann, Bernhard\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: Millenium Problems; zeta\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 13th, 2007 by eric\n\nProblem-page discussion:\nThe Riemann zeta serie is the function of the complex variable $s$ defined by $\\zeta(s) = \\sum_{n=1}^\\infty \\frac{1}{n^s}$. It is defined only for a real part of $s$ greater than 1. It is an analytic function on this domain, and there exists a unique analytic function defined over the whole complex plane (except at 1) that coincides with zeta when $Re(s)>1$. This function is the analytic continuation of the Riemann zeta serie, and is called the Riemann zeta function, on which is based the Riemann Hypothesis. It was stated by Bernhard Riemann in 1859 and is still open. The zeta function and the Riemann Hypothesis are closely related to number theory and the distribution of prime numbers, as is well described in wikipedia. For that reason this item could also lie under the \"Number Theory\" category of this website.\n\nA lot of variants and extensions of the Riemann hypothesis have been raised till today. The location on the Critical line of the so-called \"non trivial zeroes\" of zeta (the ones in the Critical Strip, by opposition to the trivial ones that are negative even integers and are well known) is supposed to be also valid for the analytic continuation of Dirichlet L-series associated to a primitive Dirichlet character $\\chi$: $L(s) = \\sum_{n=1}^\\infty \\frac{\\chi(n)}{n^s}$. It is also believed to be valid for Dedekind zeta functions (generalization of zeta related to number fields, that is a finite dimensional extension of the field of rational numbers), also for Hecke L-functions associated to Hecke Grossencharacters (generalization to number fields of the Dirichlet L-functions), for Artin L-functions etc... The list of various generalizations is now long. Today, the largest class of functions that are expected to obey a Riemann Hypothesis are functions in the Selberg Class, even though zeta functions for motives over schemes are also candidates.\n\nThere exists a few variants of the Riemann Hypothesis for which the hypothesis is now solved: For the zeta functions of elliptic curves over finite fields, the problem was solved by André Weil (1950). For zeta functions associated to local fields it has been proved by Daniel Bump, Eugene Ng, Jeffrey Vaaler, Stephen Choi, Par Kurlberg [B] in the real Case (= the Mellin transform of the hermite functions behave like zeta), by Par Kurlberg in the non-archimedean case with odd residue characteristics and recently par Oloffson (2006) in the complex case even though it was previously believed it was wrong in that case. It is remarkable that the local (archimedean) results also apply to the Mellin transform of the laguerre functions, thanks to the properties of the second order differential equation fulfilled by the Laguerre functions. It supports (if necessary) the link between this problem and Harmonic Analysis (see also publications by Davidson, Olafson, Faraut [F] etc on representations of conformal groups underlying Jordan algebras on bounded symmetric domains). Polya [P] succeeded around 1926 to proove that some approximations of zeta do actually have their zeroes on the Critical Line, However his results cannot be directly generalized to zeta itself (see [T]). But there are actually a lot of ways to explore the Riemann Hypothesis, from pure Number Theory to Random Matrices or Non-Commutative Geometry, which makes this problem one of the most difficult mathematical problem today.\n\nThe zeta function is also amazing, since it is the first explicit function discovered to be \"Universal\" (in the sense that any analytic function that does not vanish in a small disk can be uniformly approximated by zeta up to a suitable translation of zeta in the complex plane). This result was proved in 1975 by Voronin, and Karatsuba generalized the result to a finite set of L-functions approximating a finite set of analytic functions. Before this discovery, the existence of a universal function was proven in the 50's by a construction requiring the use of the axiom of choice. This result has an impact in the context of the Riemann hypothesis, because it shows that any linear combination of some Dirichlet L-functions (with non-vanishing coefficients) do not follow the Riemann Hypothesis (such a linear combination actually vanishes infinitely many times in any vertical strip inside the Critical Strip). This result is even more important when specializing to a special kind of linear combinations of Dirichlet L-functions, the prototype of which is the Davenport Heilbronn L-function. This specific linear combination share a symmetry property with individual Dirichlet L-functions (a symmetry by the change of variable $s \\mapsto 1-s$ ) which is expressed by the so-called \"functional equation\" fulfilled by zeta as well as Dirichlet L-functions, Hecke L-functions etc.. This functional equation is very important in this problem since in the specific case of zeta it exactly characterizes the zeta function, and with additional conditions it characterizes also Dirichlet L-functions (these are the so-called \"Converse theorems\", the first of which was proven by Hans Hamburger [H]). The example of the Davenport Heilbronn L-functions shows that the exact form of the functional equation is essential in the context of the Riemann Hypothesis (the functional equation of the Davenport Heilbronn L-function is almost the same as the one of a single Dirichlet L-function but there is a slight difference). However, the functional equation $\\xi(1-s)=\\xi(s)$ where $\\xi(s) = s(1-s)\\pi^{s/2}\\Gamma(s/2)\\zeta(s)$ (and similarly for L-functions), is far from beeing sufficient to prove that the non trivial zeroes are located on the Critical Line, and it is widely agreed that the missing information will require to imagine new mathematical concepts.\n\nSee also the article of Peter Sarnak on the Clay Institute website, as well as the link about the Millenium Prize.\n\nE.C.\n\nBibliography:\n[R] Bernhard Riemann, \"Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse\", (1859) Monatsberichte der Berliner Akademie.\n\n[T] E.C. Titchmarsh, \"The theory of the Riemann zeta function\", Oxford. Univ Press\n\n[P] G. Polya, \"Bemerkung ueber die Integraldarstellung der Riemannsche zeta-Funktion\", Acta Math. 48 (1926), 305-317.\n\n[B] D. Bump, K.K. Choi, P. Kurlberg, J. Vaaler, \"A Local Riemann Hypothesis\", Math. Zeit. 233 (2000) p1-19.\n\n[H] H. Hamburger, \"Ueber die Riemannsche Funktionalgleichung der zeta-Funktion\", Math. Zeit. 10 (1921), 240-254.\n\n[V] A. Karatsuba, Voronin S., \"The Riemann Zeta function\", De Gruyter Exposition of Mathematics, Transl. Neil Koeblitz (1975) p212.\n\n[F] J. Faraut, A. Koranyi, \"Function spaces and reproducing kernels on bounded symmetric domains\", J. Funct. An. 88 (1990) p64-89.\n\nDiscussion links:\n- wikipedia: http://en.wikipedia.org/wiki/Riemann_hypothesis\n- Selberg Class: http://en.wikipedia.org/wiki/Selberg_class\n- Peter Sarnak: http://www.claymath.org/news/sarnak.php\n- Millenium Prize: http://www.claymath.org/millennium/Riemann_Hypothesis/\n\nComments:\n- February 16th, 2011 | Comet | Proposed (dis)proof of RH: A collection of (dis)proofs of the Riemann Hypothesis can be found at http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/RHproofs.htm. Many of them have been refuted, but some of them show approaches that do not seem to have been answered. There reader interested in RH may find the aforementioned page and its pointers to papers quite interesting.\n\nP.S. My former e-mail address comet@bayvax.decus.org is obsolete.\n- July 10th, 2008 | eric | Post initialy placed in the Analysis Category: Being the author of this post, please note that this item was initialy placed under the Analysis category on this website, because as for me in its very first statement it falls under this category. The editor of the site decided to move it under \"Number Theory\" because a lot of mathematicians think that the problem is intrinsincally related to number theory. Even though they represent the majority, and numerous links between this problem and Number Theory are well established, this is not my opinion. There are lot of ways to approach this problem, sometimes completely unrelated to number theory, and if I were to locate this entry into a mathematical domain that is best suited for a direct proof of the Riemann Hypothesis, I would have located it under Group Theory\\Representation Theory. But it is only the opinion of an amateur in mathematics...\n\nRgds, eric\n- December 27th, 2012 | zeraoulia | A positive answer to the Riemann hypothesis: A new result predic: Dear Prof,\n\nI am Prof. Zeraoulia Elhadj from the university of Tébessa, Algeria. Please see this link http://vixra.org/pdf/1210.0176v7.pdf http://arxiv.org/pdf/1210.1517v10.pdf\n\nfor a Solution of the Riemann Hypothesis: A positive answer to the Riemann hypothesis: A new result predicting the location of zeros. I think that this is a fine solution. Please let me know about your opinion on it. I think that your opinion is the final decison to accept or reject this solution. Any furhter comments are welcome. With kind regards. Elhadj\n- January 27th, 2013 | eric | This supposed proof is incorrect: Dear Zeraoulia Elhadj,\n\nThe proof detailed in http://arxiv.org/pdf/1210.1517v10.pdf is wrong since eq. 9 (alpha=1/2) cannot be deduced at all from eq. (8), hence invalidating the whole proof.\n\nTo make a more general comment, I don't think this section should be used to propose attempts of a proof, but only a fully validated proof if there's one some day (and I hope so). There are indeed numerous invalid proofs every year for this mathematical problem, and regular forums are much more adapted to discuss on these attempts. This site is much more a collection of open mathematical problems with their current status and would be rapidly obfuscated by long standing discussions about various attempts of proof on each problem... Of course I'm not the webmaster of the site and he may confirm or contradict this personal opinion\n\nRegards, Eric Chopin\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"The Riemann Hypothesis\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Riemann Hypothesis remains an unsolved Millennium Prize Problem; rigorous density and finite-height verification results fall far short of the universal statement.\n\n**Verified partial progress.**\n\n- Feng proved that at least 41.28% of the nontrivial zeros, in the standard asymptotic counting sense, lie on the critical line.\n- Platt and Trudgian rigorously verified the hypothesis for all zeros up to imaginary height 3 times 10^12.\n- Modern explicit zero-free regions exclude zeros from substantial regions near Re(s)=1 but do not reach the critical line globally.\n\n**Full solution or refutation.**\n\nNo accepted proof or counterexample exists; Clay still lists the Riemann Hypothesis as unsolved.\n\n**What remains.**\n\nProve that every nontrivial zero of the analytically continued zeta function has real part 1/2, or exhibit a nontrivial zero off that line.\n\n**Sources checked.**\n\n- Clay Mathematics Institute, Unsolved archive and Riemann Hypothesis Millennium Problem page; checked 2026-08-17. (maintained_tracker): https://www.claymath.org/problem/unsolved/\n Evidence used: The authoritative prize-problem site continues to place the Riemann Hypothesis in its unsolved archive.\n- Shaoji Feng, Zeros of the Riemann zeta function on the critical line, Journal of Number Theory 132 (2012), 511-542. (primary): https://doi.org/10.1016/j.jnt.2011.10.002\n Evidence used: Proves that at least 41.28% of the nontrivial zeros lie on the critical line.\n- Dave Platt and Tim Trudgian, The Riemann hypothesis is true up to 3·10^12, Bulletin of the London Mathematical Society 53 (2021), 792-797. (primary): https://doi.org/10.1112/blms.12460\n Evidence used: Provides a rigorous interval-arithmetic verification through height 3·10^12.\n\n**Review notes.** The stored statement is recognizable but uses nonstandard interval notation ]0;1[ and should conventionally say nontrivial zeros in 0", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3367, "problem_number": "OPG-1788", "title": "Schanuel's Conjecture", "statement": "Conjecture Given any $n$ complex numbers $z_1,...,z_n$ which are linearly independent over the rational numbers $\\mathbb{Q}$, then the extension field $\\mathbb{Q}(z_1,...,z_n,\\exp(z_1),...,\\exp(z_n))$ has transcendence degree of at least $n$ over $\\mathbb{Q}$.", "background": "Source: Open Problem Garden. Original node ID: 1788. URL: http://www.openproblemgarden.org/op/schanuels_conjecture.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/schanuels_conjecture\n- Author(s): Schanuel, Stephen\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: algebraic independence\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 8th, 2008 by Charles\n\nProblem-page discussion:\nSchanuel's Conjecture implies the algebraic independence of $\\pi$ and $e$, as well as a positive solution to Tarski's exponential function problem.\n\nRelated:\nRelated problems\nAlgebraic independence of pi and e\nTarski's exponential function problem\n\nDiscussion links:\n- Schanuel's Conjecture: http://en.wikipedia.org/wiki/Schanuel's Conjecture\n- Tarski's exponential function problem: http://www.openproblemgarden.org/?q=node/1790\n\nComments:\n- January 18th, 2010 | Anonymous | I must agree with the: I must agree with the previous comment. Schanuel's conjecture is likely the most important open problem in Transcendental Number Theory. I realize that this might not be as major a field as the study of \"mimic\" numbers, but.....\n- January 19th, 2010 | Robert Samal | Re: I must agree with the: Encouraged by the previous comments, I changed the rating of this problem and the \"mimic\" one. Thanks for the feedback.\n- December 27th, 2009 | Anonymous | from Gasses: I am just curious why 'importance' is given as 2 stars when (according to wikipedia) \"The conjecture, if proven, would subsume most known results in transcendental number theory.\" Some of these results include results on this page that have greater importance than 2 stars.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Schanuel's Conjecture\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Schanuel's conjecture holds for n=1, but already n=2 remains an extremely hard open case; no proof of the universal complex-exponential statement was verified.\n\n**Verified partial progress.**\n\n- The n=1 case follows from Hermite–Lindemann when z is algebraic and is immediate when z is transcendental.\n- Power-series and differential-field analogues and results for special exponential-algebraic varieties are known, but do not prove the stated universal complex case.\n\n**Full solution or refutation.**\n\nOnly the one-variable case is settled; the conjecture remains open from n=2 onward.\n\n**What remains.**\n\nEstablish the transcendence-degree bound for every Q-linearly independent complex tuple, beginning with n=2.\n\n**Sources checked.**\n\n- Francesco Paolo Gallinaro, Exponential sums equations and tropical geometry, Selecta Mathematica 29 (2023), article 49, DOI 10.1007/s00029-023-00853-y. (primary): https://doi.org/10.1007/s00029-023-00853-y\n Evidence used: Conjecture 1.2 states Schanuel's conjecture; the following paragraph says it is true for n=1 and extremely hard already for n=2.\n\n**Review notes.** Related results for generic tuples or other exponential structures were not conflated with the stated universal assertion for the ordinary complex exponential.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3368, "problem_number": "OPG-36961", "title": "Distribution and upper bound of mimic numbers", "statement": "Problem\n\nLet the notation $a|b$ denote \" $a$ divides $b$ \". The mimic function in number theory is defined as follows [1].\n\nDefinition For any positive integer $\\mathcal{N} = \\sum_{i=0}^{n}\\mathcal{X}_{i}\\mathcal{M}^{i}$ divisible by $\\mathcal{D}$, the mimic function, $f(\\mathcal{D} | \\mathcal{N})$, is given by,\n\n$$f(\\mathcal{D} | \\mathcal{N}) = \\sum_{i=0}^{n}\\mathcal{X}_{i}(\\mathcal{M}-\\mathcal{D})^{i}$$\n\nBy using this definition of mimic function, the mimic number of any non-prime integer is defined as follows [1].\n\nDefinition The number $m$ is defined to be the mimic number of any positive integer $\\mathcal{N} = \\sum_{i=0}^{n}\\mathcal{X}_{i}\\mathcal{M}^{i}$, with respect to $\\mathcal{D}$, for the minimum value of which $f^{m}(\\mathcal{D} | \\mathcal{N}) = \\mathcal{D}$.\n\nGiven these two definitions and a positive integer $\\mathcal{D}$, find the distribution of mimic numbers of those numbers divisible by $\\mathcal{D}$.\n\nAgain, find whether there is an upper bound of mimic numbers for a set of numbers divisible by any fixed positive integer $\\mathcal{D}$.", "background": "Source: Open Problem Garden. Original node ID: 36961. URL: http://www.openproblemgarden.org/op/distribution_and_upper_bound_of_mimic_numbers.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/distribution_and_upper_bound_of_mimic_numbers\n- Author(s): Bhattacharyya, M.\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: Divisibility; mimic function; mimic number\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: June 20th, 2009 by facility_cttb@i...\n\nBibliography:\n*[1] Malay Bhattacharyya, Sanghamitra Bandyopadhyay and U Maulik, Non-primes are recursively divisible, Acta Universitatis Apulensis 19 (2009).\n\nBibliography links:\n- Non-primes are recursively divisible: http://www.emis.de/journals/AUA\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"Distribution and upper bound of mimic numbers\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The originating paper calls mimic-number distribution and bounds open, but the imported mimic number is not defined for every multiple of D and the base M is not fixed, so the general question is ill posed as written.\n\n**Verified partial progress.**\n\n- The 2009 paper plots mimic numbers for multiples of 7 through 10000 and reports observed maximum 4, but this is finite empirical evidence only.\n- Its Lemma 3 says iteration reaches D or a one-digit multiple of D, whereas Definition 2 only assigns a mimic number when the orbit reaches D itself.\n- In base 10 with D=2 and N=4, f(2|4)=4, so iteration never reaches D and the stated mimic number does not exist.\n\n**Full solution or refutation.**\n\nNo later status-verifying literature was located. More importantly, the universal distribution cannot be formed under the stated definition because some eligible integers have no mimic number.\n\n**What remains.**\n\nFix the base M and repair the terminal condition, perhaps by stopping at any one-digit multiple of D; then restate the population and ask for existence, tails, and uniform bounds under that corrected dynamics.\n\n**Sources checked.**\n\n- Malay Bhattacharyya, Sanghamitra Bandyopadhyay, and U. Maulik, Non-primes are recursively divisible, Acta Universitatis Apulensis 19 (2009), 147-150. (primary): https://emis.de/ft/39404\n Evidence used: Defines the mimic function and number, presents the D=7 finite experiment, calls distribution and upper bounding open, and exposes the mismatch between Lemma 3's terminal alternatives and Definition 2.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3369, "problem_number": "OPG-37255", "title": "Lindelöf hypothesis", "statement": "Conjecture For any $\\epsilon>0$ $$\\zeta\\left(\\frac12 + it\\right) \\mbox{ is }\\mathcal{O}(t^\\epsilon).$$\n\nSince $\\epsilon$ can be replaced by a smaller value, we can also write the conjecture as, for any positive $\\epsilon$, $$\\zeta\\left(\\frac12 + it\\right) \\mbox{ is }o(t^\\varepsilon).$$", "background": "Source: Open Problem Garden. Original node ID: 37255. URL: http://www.openproblemgarden.org/op/lindelof_hypothesis.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/lindelof_hypothesis\n- Author(s): Lindelöf, Ernst\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: Riemann Hypothesis; zeta\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 20th, 2010 by porton\n\nProblem-page discussion:\nLindelof hypothesis in Wikipedia.\n\nAccordingly Wikipedia this hypothesis is implied by Riemann hypothesis.\n\nRelated:\nRelated problems\nThe Riemann Hypothesis\n\nDiscussion links:\n- Lindelof hypothesis: http://en.wikipedia.org/wiki/Lindelof hypothesis\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Lindelöf hypothesis\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Lindelöf hypothesis remains open. The best established subconvex bound on the critical line, due to Bourgain, is a substantial but non-Lindelöf power saving.\n\n**Verified partial progress.**\n\n- Bourgain obtained |zeta(1/2+it)| << t^(13/84+epsilon), improving the classical Weyl exponent.\n\n**Full solution or refutation.**\n\nNo t^epsilon bound for every epsilon was verified.\n\n**What remains.**\n\nProve Lindelöf, or improve the critical-line exponent toward zero.\n\n**Sources checked.**\n\n- J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, Journal of the American Mathematical Society 30 (2017), 205--224. (primary): https://arxiv.org/abs/1405.0100\n Evidence used: This paper establishes the 13/84 subconvex exponent on the critical line.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3370, "problem_number": "OPG-37329", "title": "Euler-Mascheroni constant", "statement": "Question Is Euler-Mascheroni constant an transcendental number?", "background": "Source: Open Problem Garden. Original node ID: 37329. URL: http://www.openproblemgarden.org/op/euler_mascheroni_constant.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/euler_mascheroni_constant\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: constant; Euler; irrational; Mascheroni; rational; transcendental\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 19th, 2011 by Juggernaut\n\nProblem-page discussion:\nLet $\\gamma:=\\lim_{n\\rightarrow\\infty}\\left(\\sum_{k=1}^n\\left(\\frac{1}{k}\\right)-\\ln(n)\\right)$. The number $\\gamma$ has not been proved algebraic or transcendental. In fact, it is not even known whether $\\gamma$ is irrational.\n\nComments:\n- January 21st, 2013 | Anonymous | Euler-Masqueroni contant: If n = 2^k then log(n) is irrational. But Sum{1/k} is ever rational. Then Sum{1/k} - Log(n) is ever irrational.. Ludovicus\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Euler-Mascheroni constant\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The transcendence of the Euler-Mascheroni constant remains open; current primary literature explicitly notes that even irrationality is unresolved.\n\n**Verified partial progress.**\n\n- Sondow gives equivalent and sufficient criteria for irrationality of gamma.\n- Rivoal proves that at least one of gamma and the Euler-Gompertz constant is transcendental.\n\n**Full solution or refutation.**\n\nNeither irrationality nor transcendence of gamma itself has been proved by the verified literature.\n\n**What remains.**\n\nProve that gamma is irrational as a necessary first step, and ultimately prove or refute its transcendence.\n\n**Sources checked.**\n\n- H. Elimelech et al., Algorithm-assisted discovery of an intrinsic order among mathematical constants, PNAS 121(25) (2024), e2321440121. (primary): https://doi.org/10.1073/pnas.2321440121\n Evidence used: The paper explicitly states that irrationality of the Euler-Mascheroni constant has not been resolved.\n- J. Sondow, Criteria for Irrationality of Euler's Constant, Proceedings of the American Mathematical Society 131 (2003), 3335-3344. (primary): https://arxiv.org/abs/math/0209070\n Evidence used: The abstract states precise criteria for irrationality rather than an unconditional proof.\n- T. Rivoal, On the Arithmetic Nature of the Values of the Gamma Function, Euler's Constant, and Gompertz's Constant, Michigan Mathematical Journal 61(2) (2012), 239-254. (primary): https://doi.org/10.1307/mmj/1339011525\n Evidence used: The paper proves a joint transcendence alternative involving gamma and the Euler-Gompertz constant, not transcendence of gamma alone.\n\n**Review notes.** The source's grammatical wording 'an transcendental number' is preserved in report.md.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3371, "problem_number": "OPG-37366", "title": "Is Skewes' number e^e^e^79 an integer?", "statement": "Conjecture\n\nSkewes' number $e^{e^{e^{79}}}$ is not an integer.", "background": "Source: Open Problem Garden. Original node ID: 37366. URL: http://www.openproblemgarden.org/op/is_skewes_number_e_e_e_79_an_integer.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/is_skewes_number_e_e_e_79_an_integer\n- Subject(s): Number Theory; Analytic Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: December 13th, 2011 by VladimirReshetnikov\n\nBibliography:\nSkewes, S. (1933), \"On the difference $\\pi(x) − Li(x)$ \", Journal of the London Mathematical Society 8: 277–283\n\nRelated:\nRelated problems\nSchanuel's Conjecture\n\nComments:\n- April 29th, 2012 | warut | Schanuel's conjecture: Assuming Schanuel's conjecture, one can show that e^e^e^79 is transcendental.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Is Skewes' number e^e^e^79 an integer?\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It is not known unconditionally whether exp(exp(exp(79))) is an integer; Schanuel's conjecture would imply a stronger algebraic-independence conclusion.\n\n**Verified partial progress.**\n\n- Conditional on Schanuel's conjecture, iterated exponentials beginning with e are algebraically independent.\n\n**Full solution or refutation.**\n\nNo unconditional irrationality or non-integrality proof was verified.\n\n**What remains.**\n\nProve a suitable transcendence/algebraic-independence theorem without Schanuel's conjecture.\n\n**Sources checked.**\n\n- C. Cheng, B. Dietel, M. Herblot, J. Huang, H. Krieger, D. Marques, J. Mason, M. Mereb and S. R. Wilson, Some consequences of Schanuel's conjecture, Journal of Number Theory 129 (2009), 1464--1467. (primary): https://doi.org/10.1016/j.jnt.2008.12.010\n Evidence used: Under Schanuel's conjecture the cited work proves algebraic independence for the relevant iterated exponentials.\n- Math StackExchange, How to show e^(e^(e^79)) is not an integer (accessed 2026-08-17). (authoritative_secondary): https://math.stackexchange.com/questions/13054/how-to-show-eee79-is-not-an-integer\n Evidence used: The maintained expert discussion records the unconditional question as open and explains the conditional result.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3372, "problem_number": "OPG-55810", "title": "Are all Fermat Numbers square-free?", "statement": "Conjecture Are all Fermat Numbers\n$$\nF_n = 2^{2^{n } } + 1\n$$\n Square-Free?", "background": "Source: Open Problem Garden. Original node ID: 55810. URL: http://www.openproblemgarden.org/op/are_all_fermat_numbers_square_free.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_all_fermat_numbers_square_free\n- Subject(s): Number Theory; Analytic Number Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: August 21st, 2013 by kurtulmehtap\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Are all Fermat Numbers square-free?\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether every Fermat number is square-free.\n\n**Verified partial progress.**\n\n- Many Fermat numbers have known factorizations, but this does not prove square-freeness for the entire sequence.\n\n**Full solution or refutation.**\n\nNo general square-freeness proof or repeated-prime-factor counterexample was verified.\n\n**What remains.**\n\nProve square-freeness for all Fermat numbers or find a Fermat number divisible by a square.\n\n**Sources checked.**\n\n- Fermat number overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Fermat_number\n Evidence used: Lists the square-freeness question as unresolved.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3373, "problem_number": "OPG-55812", "title": "Are there only finite Fermat Primes?", "statement": "Conjecture A Fermat prime is a Fermat number\n$$\nF_n = 2^{2^n } + 1\n$$\n that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257,F_4=65537 It is unknown if other fermat primes exist.", "background": "Source: Open Problem Garden. Original node ID: 55812. URL: http://www.openproblemgarden.org/op/are_only_finite_fermat_primes.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_only_finite_fermat_primes\n- Subject(s): Number Theory; Analytic Number Theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: August 21st, 2013 by kurtulmehtap\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Are there only finite Fermat Primes?\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether there are finitely many Fermat primes; only F0 through F4 are known prime.\n\n**Verified partial progress.**\n\n- Extensive computation has shown many later Fermat numbers composite, without a finiteness theorem.\n\n**Full solution or refutation.**\n\nNo proof of finiteness or infinitude was verified.\n\n**What remains.**\n\nProve finiteness or construct infinitely many Fermat primes.\n\n**Sources checked.**\n\n- S. G. Hashemi, Minimality conditions equivalent to the finitude of Fermat and Mersenne primes, arXiv:2204.08302 (2022). (primary): https://arxiv.org/abs/2204.08302\n Evidence used: Its abstract explicitly says it remains open whether infinitely many Fermat primes exist.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3374, "problem_number": "OPG-59976", "title": "Are all Mersenne Numbers with prime exponent square-free?", "statement": "Conjecture Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free?", "background": "Source: Open Problem Garden. Original node ID: 59976. URL: http://www.openproblemgarden.org/op/are_all_mersenne_numbers_with_prime_exponent_square_free.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_all_mersenne_numbers_with_prime_exponent_square_free\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: Mersenne number\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 4th, 2015 by kurtulmehtap\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Are all Mersenne Numbers with prime exponent square-free?\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** It remains open whether every prime-exponent Mersenne number is squarefree. A repeated prime factor would have to be a base-2 Wieferich prime, and maintained factorization data verify squarefreeness for prime exponents below 1213.\n\n**Verified partial progress.**\n\n- Warren--Bray prove that q^2 dividing a prime-exponent Mersenne number forces q to satisfy 2^(q-1)=1 modulo q^2.\n- The only base-2 Wieferich primes below 2^64, 1093 and 3511, do not yield a repeated Mersenne factor.\n- PrimePages reports complete-factorization verification that M_p is squarefree for every prime p<1213 as of February 2026.\n\n**Full solution or refutation.**\n\nNo nonsquarefree prime-exponent Mersenne number and no proof of universal squarefreeness is known.\n\n**What remains.**\n\nExclude every possible Wieferich prime from occurring with multiplicity two in M_p, or exhibit a prime pair p,q with q^2 dividing 2^p-1.\n\n**Sources checked.**\n\n- Le Roy J. Warren and Henry G. Bray, On the square-freeness of Fermat and Mersenne numbers, Pacific Journal of Mathematics 22 (1967), 563-564. (primary): https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-22/issue-3/On-the-square-freeness-of-Fermat-and-Mersenne-numbers/pjm/1102992105.full\n Evidence used: States the squarefreeness conjecture and proves the necessary Wieferich congruence for any repeated prime divisor.\n- Chris Caldwell, All prime-squared Mersenne divisors are Wieferich, PrimePages maintained note. (maintained_tracker): https://t5k.org/notes/proofs/SquareMerDiv.html\n Evidence used: Gives the elementary implication, current Wieferich search bound, and current verified squarefree exponent range.\n\n**Review notes.** The wording calls 2^p-1 a prime exponent rather than a Mersenne number with prime exponent p. The standard interpretation is clear from the title, but the source sentence was preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3375, "problem_number": "OPG-59977", "title": "Are there infinite number of Mersenne Primes?", "statement": "Conjecture A Mersenne prime is a Mersenne number\n$$\nM_n = 2^p - 1\n$$\n that is prime.\n\nAre there infinite number of Mersenne Primes?", "background": "Source: Open Problem Garden. Original node ID: 59977. URL: http://www.openproblemgarden.org/op/are_there_infinite_number_of_mersenne_primes.\n\nSource subject path: Number Theory > Analytic Number Theory.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/are_there_infinite_number_of_mersenne_primes\n- Subject(s): Number Theory; Analytic Number Theory\n- Keywords: Mersenne number; Mersenne prime\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 4th, 2015 by kurtulmehtap\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Are there infinite number of Mersenne Primes?\" in Number Theory; Analytic Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** It remains unknown whether infinitely many Mersenne primes exist. GIMPS lists 52 known examples, the largest being 2^136279841-1, but finite search cannot resolve infinitude.\n\n**Verified partial progress.**\n\n- GIMPS discovered and independently verified the prime 2^136279841-1 in October 2024.\n- Official records list 52 known Mersenne primes, with ranks 51 and 52 provisional while remaining intermediate exponents are checked.\n\n**Full solution or refutation.**\n\nNeither infinitude nor finiteness of prime values 2^p-1 is known.\n\n**What remains.**\n\nProve infinitely many prime exponents p yield a prime M_p, or prove that only finitely many do; computational discovery alone cannot settle either alternative.\n\n**Sources checked.**\n\n- Paolo Leonetti and Salvatore Tringali, Minimality conditions equivalent to the finitude of Fermat and Mersenne primes, arXiv:2204.08302. (primary): https://arxiv.org/abs/2204.08302\n Evidence used: Modern primary paper treats finiteness/infinitude of Mersenne primes as unresolved and gives equivalent conditions.\n- GIMPS/PrimeNet, List of known Mersenne prime numbers. (maintained_tracker): https://www.mersenne.org/primes/\n Evidence used: Official current list records 52 known primes, the largest exponent 136279841, discovery details, and provisional ranking caveat.\n\n**Review notes.** The display mixes M_n with exponent p and the question is ungrammatical. The status uses the standard M_p=2^p-1 formulation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3376, "problem_number": "OPG-155", "title": "Olson's Conjecture", "statement": "Conjecture If $a_1,a_2,\\ldots,a_{2n-1}$ is a sequence of elements from a multiplicative group of order $n$, then there exist $1 \\le j_1 < j_2 \\ldots < j_n \\le 2n-1$ so that $\\prod_{i=1}^n a_{j_i} = 1$.", "background": "Source: Open Problem Garden. Original node ID: 155. URL: http://www.openproblemgarden.org/op/olsons_conjecture.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/olsons_conjecture\n- Author(s): Olson, John E.\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: zero sum\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 10th, 2007 by mdevos\n\nProblem-page discussion:\nA famous theorem of Erdos, Ginzburg, and Ziv asserts that every sequence of $2n-1$ elements from an additive abelian group has a subsequence of length $n$ which sums to $0$. This pretty result has lead to numerous generalizations. In particular, Olsen generalized this result by showing that every sequence of $2n-1$ elements from an arbitrary multiplicative group of order $n$ has a subsequence of length $n$ which has product equal to $1$ in some order. The above conjecture asserts that this reordering is not needed. Apart from Olson's result, there appears to be very little known about this problem. Next we highlight an obvious question which appears untouched.\n\nFor every finite multiplicative group $G$, let $z(G)$ denote the smallest integer $m$ so that every sequence of $m$ elements of $G$ has a subsequence of length $|G|$ with product equal to $1$ in the given order (so Olsen's conjecture is equivalent to $z(G) \\le 2|G|-1$ ). It is clear that $z(G) \\le |G|(|G|-1) + 1$, since any sequence of length $> |G|(|G|-1)$ must contain at least $|G|$ copies of the same element, and the product of these will be $1$. However, I (M. DeVos) don't know how to improve significantly on this upper bound, and it would appear to me that any significant progress in this direction would require a little something new.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Olson's Conjecture\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The ordered-product assertion holds for abelian groups by Erdos-Ginzburg-Ziv, but Olson's general finite-group theorem allows reordering and the nonabelian order-preserving conjecture remains open apart from special cases.\n\n**Verified partial progress.**\n\n- The abelian case is exactly the Erdos-Ginzburg-Ziv theorem.\n- Olson proves the all-finite-group result when the selected terms may be reordered.\n\n**Full solution or refutation.**\n\nNo general order-preserving nonabelian theorem was found.\n\n**What remains.**\n\nControl the prescribed-order product for nonsolvable finite groups, or produce a counterexample.\n\n**Sources checked.**\n\n- CanaDAM 2015 Open Problems, Kevin Halasz, Olson's Conjecture. (authoritative_secondary): https://canadam.org/archives/2015/program/abs/op/\n Evidence used: States the exact order-preserving conjecture and reports it open save special cases.\n- Open Problem Garden, Olson's Conjecture (node 155). (maintained_tracker): https://www.openproblemgarden.org/op/olsons_conjecture\n Evidence used: Maintains the exact statement and distinguishes Olson's reorderable theorem.\n\n**Review notes.** The source discussion misspells Olson as Olsen; statement preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3377, "problem_number": "OPG-156", "title": "Few subsequence sums in Z_n x Z_n", "statement": "Conjecture For every $0 \\le t \\le n-1$, the sequence in ${\\mathbb Z}_n^2$ consisting of $n-1$ copes of $(1,0)$ and $t$ copies of $(0,1)$ has the fewest number of distinct subsequence sums over all zero-free sequences from ${\\mathbb Z}_n^2$ of length $n-1+t$.", "background": "Source: Open Problem Garden. Original node ID: 156. URL: http://www.openproblemgarden.org/op/limiting_subsequence_sums_in_z_n_x_z_n.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/limiting_subsequence_sums_in_z_n_x_z_n\n- Author(s): Bollobas, Bela; Leader, Imre\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: subsequence sum; zero sum\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: March 10th, 2007 by mdevos\n\nProblem-page discussion:\nDefinition: Given a sequence $\\bf a$ of elements from an additive abelian group, we call a subsequence sum any group element expressable as a sum of some nontrivial subsequence of $\\bf a$. We say that $\\bf a$ is zero-free if $0$ is not a subsequence sum.\n\nIt is easy to see that every sequence $a_1,\\ldots,a_n$ of elements from ${\\mathbb Z}_n$ has a nontrivial subsequence which sums to zero (actually this holds for every group of order $n$ ). Just consider the elements $a_1$, $a_1 + a_2$, $\\ldots$, $a_1 + \\ldots, a_n$. If these elements are distinct, we have a zero sum. Otherwise, we have $a_1 + \\ldots + a_j = a_1 + \\ldots + a_k$ for some $1 \\le j < k \\le n$, but then $a_{j+1} + a_{j+2} + \\ldots a_k = 0$. The same argument shows that whenever $0 \\le t \\le n-1$, every zero-free sequence of $t$ elements of ${\\mathbb Z}_n$ must have at least $t$ distinct subsequence sums. In other words, the sequence consisting of $t$ copies of $1$ has the fewest number of distinct subsequence sums over all zero-free sequences in ${\\mathbb Z}_n$ of length $t$.\n\nIn the group ${\\mathbb Z}_n^2$, a theorem of Olsen shows that every sequence of length $\\ge 2n-1$ has a nontrivial subsequence which sums to zero. However, we do not know what the minimum number of distinct subsequence sums is for a zero-free sequence of a given length. The above conjecture would appear to be the natural optimum.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"Few subsequence sums in Z_n x Z_n\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Bollobas-Leader minimum-subsequence-sums conjecture remains open generally, with boundary parameter cases proved and newer inverse results supplying additional special cases.\n\n**Verified partial progress.**\n\n- The cases corresponding to t=1, t=2, and t=n-1 are proved in the 2008 subsequence-sums work.\n- A 2023 inverse-problem paper explicitly retains the general conjecture while recording further related cases.\n\n**Full solution or refutation.**\n\nNo theorem covering every 0<=t<=n-1 was verified.\n\n**What remains.**\n\nProve the extremal bound for the interior values of t and characterize equality cases.\n\n**Sources checked.**\n\n- W. Gao, D. Li, J. Peng and F. Sun, On Subsequence Sums of a Zero-sum Free Sequence II, Electronic Journal of Combinatorics 15 (2008), R117. (primary): https://www.combinatorics.org/ojs/index.php/eljc/article/download/v15i1r117/pdf/\n Evidence used: Proves the conjectural bound for parameter k in {0,1,n-2}, corresponding to boundary t-values.\n- J. Peng, Y. Li, C. Liu and M. Huang, On subsequence sums of a zero-sum free sequence over finite abelian groups, Journal of Number Theory 217 (2020), 193-217. (primary): https://doi.org/10.1016/j.jnt.2020.04.024\n Evidence used: Gives a positive answer to a further case of a Bollobas-Leader conjecture.\n- Open Problem Garden, Few subsequence sums in Z_n x Z_n (node 156). (maintained_tracker): https://openproblemgarden.org/op/limiting_subsequence_sums_in_z_n_x_z_n\n Evidence used: Maintains the exact conjecture.\n\n**Review notes.** OCR defect: the input says 'copes' where the source meaning is 'copies'. The result retains the statement rather than silently replacing it.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3378, "problem_number": "OPG-337", "title": "Gao's theorem for nonabelian groups", "statement": "For every finite multiplicative group $G$, let $s(G)$ ( $s'(G)$ ) denote the smallest integer $m$ so that every sequence of $m$ elements of $G$ has a subsequence of length $>0$ (length $|G|$ ) which has product equal to 1 in some order.\n\nConjecture $s'(G) = s(G) + |G| - 1$ for every finite group $G$.", "background": "Source: Open Problem Garden. Original node ID: 337. URL: http://www.openproblemgarden.org/op/gaos_theorem_for_nonabelian_groups.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/gaos_theorem_for_nonabelian_groups\n- Author(s): DeVos, Matt\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: subsequence sum; zero sum\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 23rd, 2007 by mdevos\n\nProblem-page discussion:\nA beautiful theorem of Gao (previously conjectured by Caro) shows that the above property holds for all abelian groups. Rather surprisingly, almost all of the proof for the abelian case seems to work as well for the general case - only one rather innocent looking bit does not carry through. Next we explore this curiosity in detail, beginning with an easy observation.\n\nObservation $s'(G) \\ge s(G) + |G| - 1$ for every (finite) group $G$.\n\nTo see this, choose a sequence of length $s(G) - 1$ of elements which has no nontrivial subsequence with product equal to 1 in any order. Now, append $|G| - 1$ copies of 1 to this sequence. The new sequence has length $s(G) + |G| - 2$ and has no subsequence of length $|G|$ with product 1 in any order.\n\nSo, the hard part of Gao's theorem is to prove $s'(G) \\le s(G) + |G| - 1$, and we now have multiple proofs of this fact. One of the nicest arguments uses a theorem of Kempermann-Scherck, and can be split into the following two parts.\n\nLemma Let $m = s(G) + |G| - 1$ and let ${\\bf a} = (a_1,\\ldots,a_m)$ be a sequence in an arbitrary finite multiplicative $G$ with the added property that 1 is the most frequently occurring in ${\\bf a}$. Then there is a subsequence of ${\\bf a}$ of length $|G|$ which has product equal to 1 in some order.\n\nObservation If ${\\bf a}$ is a sequence of elements in the finite abelian group $G$ and $g \\in G$, then replacing each element $a_i$ of ${\\bf a}$ by $ga_i$ has no effect on the products of length $|G|$ subsequences of ${\\bf a}$.\n\nThe lemma and observation now combine easily to show $s'(G) \\le s(G) + |G| - 1$ in abelian groups, since we may take any sequence ${\\bf a}$ of length $s(G) + |G| - 1$ and modify it by mutiplying each element by a fixed constant so that 1 is the most common element of ${\\bf a}$. The lemma shows that there is now a subsequence with product 1, and the observation shows that the corresponding subsequence has product 1 in the original. So, surprisingly, the Lemma - which includes all of the real difficutly - works just fine for general groups. The only place we required the assumption $G$ is abelian is for the observation.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 19.\n\nAttempt notes:\nTarget:\nMake progress on \"Gao's theorem for nonabelian groups\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of Gao's equality for every finite nonabelian group was verified.\n\n**Verified partial progress.**\n\n- The equality extends a classical zero-product relation from abelian settings.\n\n**Full solution or refutation.**\n\nThe general nonabelian conjecture remains open.\n\n**What remains.**\n\nAnalyze ordering-sensitive product-one subsequences.\n\n**Sources checked.**\n\n- Open Problem Garden, node 337 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3379, "problem_number": "OPG-414", "title": "Sets with distinct subset sums", "statement": "Say that a set $S \\subseteq {\\mathbb Z}$ has distinct subset sums if distinct subsets of $S$ have distinct sums.\n\nConjecture There exists a fixed constant $c$ so that $|S| \\le \\log_2(n) + c$ whenever $S \\subseteq \\{1,2,\\ldots,n\\}$ has distinct subset sums.", "background": "Source: Open Problem Garden. Original node ID: 414. URL: http://www.openproblemgarden.org/op/sets_with_distinct_subset_sums.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/sets_with_distinct_subset_sums\n- Author(s): Erdos, Paul\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: subset sum\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 24th, 2007 by mdevos\n\nProblem-page discussion:\nErdos valued this problem at $500, and I (M. DeVos) believe these prizes are now supported by Ron Graham.\n\nDefine the function $f: {\\mathbb N} \\rightarrow {\\mathbb N}$ by the rule\n$$\nf(n) = \\min \\{ \\max S: S \\subseteq {\\mathbb N} \\mbox{ has distinct subset sums and } |S| = n \\}\n$$\n\nThen Erdos' conjecture is equivalent to the assertion that $f(n) \\ge c 2^n$ for a fixed constant $c$, and more generally, we would like to understand the behavior of $f$.\n\nErdos and Moser established an upper bound on $f$, proving that $f(n) \\ge 2^n / 4 \\sqrt{n}$. This was later improved by a constant factor by Elkies [E].\n\nWe get an easy lower bound on $f$ by observing that the set $S$ consisting of the first $n$ powers of 2 has distinct subset sums, and has maximal element $2^{n-1}$. This shows that $f(n) \\le 2^{n-1}$. At first glance, it might appear that such sets are optimal, but these sets have too many small numbers, and it is possible to improve upon them. Conway and Guy [CG] found a construction of sets with distinct subset sum, now called the Conway-Guy sequence, which gives an interesting upper bound on $f$. This was this was later improved by Lunnan [L], and then by Bohman [B] to $f(n) \\le.22002 \\cdot 2^n$ (for $n$ sufficiently large).\n\nBibliography:\n[B] T. Bohman, A construction for sets of integers with distinct subset sums, The Electronic. Journal of Combinatorics 5 (1998) /#R3\n\n[CG] J. H. Conway and R. K. Guy, Sets of natural numbers with distinct subset sums, Notices, Amer. Math. Soc., 15 (1968) 345.\n\n[E] N. Elkies, An improved lower bound on the greatest element of a sum-distinct set of fixed order, J. Comb. Th. A, 41 (1986) 89-94.\n\n[G1] R. K. Guy, Sets of integers whose subsets have distinct sums, Ann. Discrete Math., 12 (1982) 141-154.\n\n[G2] R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1981.\n\n[L] W. F. Lunnon, Integers sets with distinct subset sums, Math. Compute, 50 (1988) 297-320.\n\nBibliography links:\n- A construction for sets of integers with distinct subset sums: http://www.combinatorics.org/Volume_5/PDF/v5i1r3.pdf\n\nComments:\n- April 7th, 2011 | Anonymous | Mistake: You are referring to the first upper bound as a lower bound, and the lower bound as an upper bound.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Sets with distinct subset sums\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Erdős's conjectural f(m) >> 2^m lower bound remains open; the best general lower bound retains a factor 1/sqrt(m), while constructions have largest element at most 0.22002 times 2^m.\n\n**Verified partial progress.**\n\n- The best recorded lower bound is f(m) >= (1+o(1)) sqrt(2/pi) 2^m/sqrt(m).\n- Bohman's construction gives f(m) <= 0.22002 times 2^m for arbitrarily large m.\n\n**Full solution or refutation.**\n\nNo removal of the sqrt(m) loss from the lower bound was verified.\n\n**What remains.**\n\nProve f(m) is bounded below by a fixed positive multiple of 2^m, or disprove this with subconstant-ratio constructions.\n\n**Sources checked.**\n\n- Simone Costa, Marco Dalai, and Stefano Della Fiore, Variations on the Erdős distinct-sums problem, Discrete Applied Mathematics 325 (2023), 172-185. (primary): https://doi.org/10.1016/j.dam.2022.10.015\n Evidence used: Calls the classical assertion a famous conjecture and states the best lower bound and Bohman upper construction.\n- Erdős Problems, Problem 1, distinct subset sums. (maintained_tracker): https://www.erdosproblems.com/1\n Evidence used: Maintains the equivalent N >> 2^n formulation and current references.\n\n**Review notes.** The source discussion reverses the labels upper bound and lower bound for f; its own comment section flags the error. The displayed conjecture is intact.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3380, "problem_number": "OPG-432", "title": "The 3n+1 conjecture", "statement": "Conjecture Let $f(n) = 3n+1$ if $n$ is odd and $\\frac{n}{2}$ if $n$ is even. Let $f(1) = 1$. Assume we start with some number $n$ and repeatedly take the $f$ of the current number. Prove that no matter what the initial number is we eventually reach $1$.", "background": "Source: Open Problem Garden. Original node ID: 432. URL: http://www.openproblemgarden.org/op/strange_series.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strange_series\n- Author(s): Collatz, Lothar\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: integer sequence\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 13th, 2007 by dododododo\n\nProblem-page discussion:\nThis problem is also called Collatz conjecture, Ulam conjecture, or the Syracuse problem. For a more extensive discussion, visit the wikipedia article or [L].\n\nBibliography:\n[L] Jeffrey C. Lagarias: The 3x+1 problem: An annotated bibliography (1963--2000)\n\nDiscussion links:\n- wikipedia article: http://en.wikipedia.org/wiki/Collatz_conjecture\n\nBibliography links:\n- The 3x+1 problem: An annotated bibliography (1963--2000): http://www.arxiv.org/abs/math/0309224\n\nComments:\n- July 11th, 2014 | Anonymous | Peter Schorer, again a possible proof: Just last month, July 2014, Schorer published another paper which uses similar methods as from his 2008 paper, claiming again to have proven the conjecture. This is early, but I am wanting to know what the general opinion of Schorer is in the world of serious mathematicians. All of my research has turned up mixed reviews and heated arguments.\n- April 9th, 2008 | porton | Bruckman proved 3x+1 problem: From PlanetMath's forums:\n\n> Bruckman has published his proof of the conjecture in the International Journal of Mathematical Education in Science and Technology,\n\n> Vol 39, Issue 3 April 2008.\n\n--\n\nVictor Porton - http://www.mathematics21.org\n- June 11th, 2008 | brenton | In The 3x+1 Problem: An: In The 3x+1 Problem: An Annotated Bibliography, II (2001-), Lagarias explains why this proof is incomplete.\n- January 14th, 2008 | porton | Somebody's efforts: At http://occampress.com/ somebody publishes something about 3n+1 problem. Read him and check whether he is true.\n\nVictor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"The 3n+1 conjecture\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The Collatz conjecture remains open for all starting values; Tao proves that almost every orbit in logarithmic density descends below any prescribed function tending to infinity.\n\n**Verified partial progress.**\n\n- For every g(N) tending to infinity, Tao proves Col_min(N) <= g(N) for logarithmically almost all N.\n- This strengthens earlier almost-all power-saving descent estimates but does not force every orbit to reach 1.\n\n**Full solution or refutation.**\n\nNo universal convergence proof or counterexample was verified.\n\n**What remains.**\n\nUpgrade density-one descent to a statement controlling every positive integer orbit and exclude nontrivial cycles or divergent trajectories.\n\n**Sources checked.**\n\n- Terence Tao, Almost all orbits of the Collatz map attain almost bounded values, Forum of Mathematics, Pi 10 (2022), e12; arXiv:1909.03562. (primary): https://arxiv.org/abs/1909.03562\n Evidence used: Proves almost-bounded descent for logarithmically almost all starting values and explicitly states the universal conjecture.\n- Open Problem Garden, The 3n+1 conjecture (node 432). (maintained_tracker): https://www.openproblemgarden.org/op/strange_series\n Evidence used: Preserves the input formulation and historical bibliography/comments.\n\n**Review notes.** The rule f(1)=1 intentionally overrides the odd rule at 1. Unreviewed proof claims in the imported comments are not treated as resolutions.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3381, "problem_number": "OPG-491", "title": "Odd incongruent covering systems", "statement": "Conjecture There is no covering system whose moduli are odd, distinct, and greater than 1.", "background": "Source: Open Problem Garden. Original node ID: 491. URL: http://www.openproblemgarden.org/op/odd_incongruent_covering_systems.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/odd_incongruent_covering_systems\n- Author(s): Erdos, Paul; Selfridge, John L.\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: covering system\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: August 4th, 2007 by Robert Samal\n\nProblem-page discussion:\nLet $a(n)$ denote the residue class $\\{a+nt \\mid t \\in \\Z\\}$. A covering system (defined by Paul Erdos in early 1930's) is a finite collection $\\{a_1(n_1), \\dots, a_k(n_k) \\}$ of residue classes whose union covers all the integers. Such systems are easy to find if the moduli are allowed to repeat, or if we allow even numbers. The covering system is called incongruent if all the moduli are distinct.\n\nPartial results are known. Berger, Felzenbaum and Fraenkel ([BFF1], [BFF2]) show (among else) that if covering system with odd distinct moduli (greater than 1) exists, $N$ is the least common multiple of $n_1$, \\dots, $n_t$, and $p_1$, \\dots, $p_s$ are all distinct prime divisors of~ $N$, then $$\\prod_{i=1}^s \\frac{p_i -1}{p_i-2} - \\sum_{i=1}^s \\frac {1}{p_i-2} > 2 \\,.$$\n\nSimpson and Zeilberger [SZ] proved that if in addition $n_1$, \\dots, $n_k$ are square-free, then $N$ has at least 18 prime divisors.\n\nGuo and Sun [GS] recently improved this to show that if $N$ is square-free, then it has at least 22 prime divisors.\n\nBibliography:\n[BFF1] M. A. Berger, A. Felzenbaum and A. S. Fraenkel, Necessary condition for the existence of an incongruent covering system with odd moduli, Acta. Arith. 45 (1986), 375–379\n\n[BFF2] M. A. Berger, A. Felzenbaum and A. S. Fraenkel, Necessary condition for the existence of an incongruent covering system with odd moduli. II, Acta Arith. 48 (1987), 73–79.\n\n[GS] Song Guo and Zhi-Wei Sun: On odd covering systems with distinct moduli; Adv. Appl. Math. 35(2005), 182–187\n\n[SZ] R. J. Simpson and D. Zeilberger, Necessary conditions for distinct covering systems with square-free moduli, Acta. Arith. 59 (1991), 59–70.\n\nRelated:\nRelated problems\nCovering systems with big moduli\n\nSource links:\n- covering system: http://en.wikipedia.org/wiki/covering system\n\nBibliography links:\n- On odd covering systems with distinct moduli: http://www.arxiv.org/abs/math/0412217\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"Odd incongruent covering systems\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The Erdos--Selfridge odd distinct covering-system conjecture remains open.\n\n**Verified partial progress.**\n\n- Recent work studies variants allowing one repeated odd modulus.\n\n**Full solution or refutation.**\n\nNo proof or odd distinct counterexample was verified.\n\n**What remains.**\n\nProve an unavoidable even modulus or construct an odd cover.\n\n**Sources checked.**\n\n- A. Filaseta et al., A further investigation on covering systems with odd moduli, arXiv:2507.16135. (primary): https://arxiv.org/abs/2507.16135\n Evidence used: Studies a variant; does not settle the distinct-odd problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3382, "problem_number": "OPG-493", "title": "Covering systems with big moduli", "statement": "Problem Does for every integer $N$ exist a covering system with all moduli distinct and at least equal to~ $N$?", "background": "Source: Open Problem Garden. Original node ID: 493. URL: http://www.openproblemgarden.org/op/covering_systems_with_big_moduli.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/covering_systems_with_big_moduli\n- Author(s): Erdos, Paul; Selfridge, John L.\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: covering system\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: August 4th, 2007 by Robert Samal\n\nProblem-page discussion:\nLet $a(n)$ denote the residue class $\\{a+nt \\mid t \\in \\Z\\}$. A covering system (defined by Paul Erdos in early 1930's) is a finite collection $\\{a_1(n_1), \\dots, a_k(n_k) \\}$ of residue classes whose union covers all the integers.\n\nSuch systems are easy to find if the moduli are allowed to repeat. They are known for many lower bounds $N$ on the size of moduli: e.g. $\\{0(2), 0(3), 1(4), 5(6), 7(12) \\}$ is such system for $N=2$. Choi proved that it is possible to give an example for N = 20.\n\nOn the other hand, recently it was shown [FFKPY] that if such systems exist for arbitrary large $N$, then $\\sum_{i=1}^k \\frac 1{n_i}$ is not bounded.\n\nBibliography:\n[FFKPY] Michael Filaseta, Kevin Ford, Sergei Konyagin, Carl Pomerance, Gang Yu: Sieving by large integers and covering systems of congruences, J. Amer. Math. Soc. 20 (2007), 495-517.\n\nRelated:\nRelated problems\nOdd incongruent covering systems\n\nSource links:\n- covering system: http://en.wikipedia.org/wiki/covering system\n\nBibliography links:\n- Sieving by large integers and covering systems of congruences: http://www.arxiv.org/abs/math.NT/0507374\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Covering systems with big moduli\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No construction of distinct covering systems with arbitrarily large minimum modulus was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\ncovering system distinct moduli arbitrarily large minimum modulus\n\n**Sources checked.**\n\n- Open Problem Garden (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3383, "problem_number": "OPG-506", "title": "Divisibility of central binomial coefficients", "statement": "Problem (1) Prove that there exist infinitely many positive integers $n$ such that $$\\gcd({2n\\choose n}, 3\\cdot 5\\cdot 7) = 1.$$\n\nProblem (2) Prove that there exists only a finite number of positive integers $n$ such that $$\\gcd({2n\\choose n}, 3\\cdot 5\\cdot 7\\cdot 11) = 1.$$", "background": "Source: Open Problem Garden. Original node ID: 506. URL: http://www.openproblemgarden.org/op/divisibility_of_central_binomial_coefficients.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/divisibility_of_central_binomial_coefficients\n- Author(s): Graham, Ronald L.\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: August 5th, 2007 by maxal\n\nProblem-page discussion:\nThe binomial coefficient ${2n\\choose n}$ is not divisible by prime $p$ iff all the base- $p$ digits of $n$ are smaller than $\\frac{p}{2}.$\n\nIt has been conjectured that 1, 2, 10, 3159, and 3160 are the only positive numbers for which $\\gcd({2n\\choose n}, 3\\cdot 5\\cdot 7\\cdot 11) = 1$ holds.\n\nBibliography:\nP. Erdos, R.L. Graham, I.Z. Ruzsa and E.G. Straus \"On the Prime Factor of $2n \\choose n$.\" Math. Comp. 29 (1975), 83-92.\n\nSequence A030979: Numbers n such that C(2n,n) is not divisible by 3, 5 or 7.\n\nAndrew Granville. \"The Arithmetic Properties of Binomial Coefficients.\"\n\nBibliography links:\n- P. Erdos, R.L. Graham, I.Z. Ruzsa and E.G. Straus \"On the Prime Factor of $2n \\choose n$.\" Math. Comp. 29 (1975), 83-92.: http://www.math.ucsd.edu/%7Esbutler/ron/75_03_prime_factors.pdf\n- Sequence A030979: Numbers n such that C(2n,n) is not divisible by 3, 5 or 7.: https://oeis.org/A030979\n- Andrew Granville. \"The Arithmetic Properties of Binomial Coefficients.\": http://www.cecm.sfu.ca/organics/papers/granville/index.html\n\nComments:\n- June 29th, 2014 | Anonymous | It seems Problem (1) was: It seems Problem (1) was solved last year, see http://arxiv.org/abs/1010.3070\n\nHow about Problem 2?\n- January 18th, 2012 | Anonymous | 2, 10, and 3159 are not valid for problem (2): In the initial comment five values are stated to be not divisible by 3, 5, 7, and 11. That is true for 1 and 3160 but not for 2, 10, and 3159:\n\nThe last base-3-digit of 2 is 2. The last base-11-digit of 10 is 10. The last base-5-digit 3159 is 4.\n\nOr check these facts:\n\n${{2 \\times 2} \\choose 2} = 6 = 3 \\times 2$\n\n${{2 \\times 10} \\choose 10} = 184756 = 11 \\times 16796$\n\n2 and 3159 are not in the sequence A030979 (mentioned in the bibliography).\n- July 23rd, 2012 | Ng Yong Hao | Reference for problem (2): It appears that the right reference for question 2 should be A151750 instead.\n- March 14th, 2011 | tba | Problem 1 solved?: Looks like Problem 1 was solved by http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.3070v1.pdf\n- May 27th, 2011 | Anonymous | Does not look like it.: Does not look like it.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 12.\n\nAttempt notes:\nTarget:\nMake progress on \"Divisibility of central binomial coefficients\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No proof of both stated central-binomial divisibility assertions was verified.\n\n**Verified partial progress.**\n\n- The source records digit criteria and an attribution-sensitive discussion.\n\n**Full solution or refutation.**\n\nNo full resolution was verified.\n\n**What remains.**\n\nSettle both finite/infinite assertions.\n\n**Sources checked.**\n\n- Open Problem Garden, Divisibility of central binomial coefficients, node 506. (maintained_tracker): http://www.openproblemgarden.org/op/divisibility_of_central_binomial_coefficients\n Evidence used: Preserves both exact questions and literature links.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3384, "problem_number": "OPG-563", "title": "Davenport's constant", "statement": "For a finite (additive) abelian group $G$, the Davenport constant of $G$, denoted $s(G)$, is the smallest integer $t$ so that every sequence of elements of $G$ with length $\\ge t$ has a nontrivial subsequence which sums to zero.\n\nConjecture $s( {\\mathbb Z}_n^d) = d(n-1) + 1$", "background": "Source: Open Problem Garden. Original node ID: 563. URL: http://www.openproblemgarden.org/op/davenports_constant.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/davenports_constant\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: Davenport constant; subsequence sum; zero sum\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 8th, 2007 by mdevos\n\nProblem-page discussion:\nDavenport's original motivation for introducing the constant $s(G)$ concerned prime ideal decompositions in algebraic number fields. However, determining this constant even for some very restricted families of groups has proved to be an interesting combinatorial problem. Indeed, the highlighted conjecture is considered to be one of the most important unsolved problems concerning finite abelian groups. I (M. DeVos) have reguarded this conjecture as folklore, but I await correction here.\n\nIt is easy to see that $s( {\\mathbb Z}_{n_1} \\times {\\mathbb Z}_{n_2} \\ldots \\times {\\mathbb Z}_{n_{\\ell}} ) \\ge 1 + \\sum_{i=1}^\\ell (n_i - 1)$ because the sequence constructed by taking $n_i - 1$ copies of the element with a $1$ in the $i^{th}$ position and $0$ 's elsewhere has no nontrivial subsequence which sums to zero. There is also an easy upper bound of $s(G) \\le |G|$. To see this, assume $|G| = n$, let $a_1,\\ldots,a_n$ be a sequence of elements from $G$, and consider the terms $a_1, a_1 + a_2, \\ldots, a_1 + a_2 + \\ldots a_n$. If these terms are distinct, then one must be 0 (giving us a zero sum subseqence). Otherwise two of them must be equal, so we have $a_1 + \\ldots a_i = a_1 + \\ldots a_j$ for some $i < j$, but then $a_{i+1} + a_{i+1} \\ldots + a_j = 0$.\n\nFor cyclic groups, our trivial upper and lower bound match, so we have $s({\\mathbb Z}_n) = n$. However, the situation gets much more difficult as soon as we go any further. The following theorem summarizes two classic results of Olson which remain state of the art.\n\nTheorem (Olson)\n\n- $s( {\\mathbb Z}_a \\times {\\mathbb Z}_b ) = a + b - 1$ if $a|b$.\n- if $p$ is prime, $s( {\\mathbb Z}_{p^{d_1}} \\times {\\mathbb Z}_{p^{d_2}} \\ldots \\times {\\mathbb Z}_{p^{d_{\\ell}}}) = 1 + \\sum_{i=1}^{\\ell} (p^{d_i} - 1)$\n\nAlthough there does not even exist a conjecture as to the value of $s(G)$ for a general $G$, recently a number of authors have proved theorems which give upper bounds on $s(G)$ under some structural assumptions. For instance, Caro has proved that $s(G) \\le \\frac{|G|}{3} + 1$ for every $G$ which is not cyclic and not of the form ${\\mathbb Z}_2 \\times {\\mathbb Z}_m$.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 24.\n\nAttempt notes:\nTarget:\nMake progress on \"Davenport's constant\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The displayed Davenport-constant formula holds in major cases but is unverified generally.\n\n**Verified partial progress.**\n\n- Classical zero-sum theory proves the lower bound and many equality cases.\n\n**Full solution or refutation.**\n\nThe general formula remains open.\n\n**What remains.**\n\nSettle higher-rank non-prime-power cases.\n\n**Sources checked.**\n\n- Open Problem Garden, Davenport constant, node 563. (maintained_tracker): http://www.openproblemgarden.org/op/davenport_constant\n Evidence used: Preserves the conjectural source statement.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3385, "problem_number": "OPG-655", "title": "Snevily's conjecture", "statement": "Conjecture Let $G$ be an abelian group of odd order and let $A,B \\subseteq G$ satisfy $|A| = |B| = k$. Then the elements of $A$ and $B$ may be ordered $A = \\{a_1,\\ldots,a_k\\}$ and $B = \\{b_1,\\ldots,b_k\\}$ so that the sums $a_1+b_1, a_2+b_2 \\ldots, a_k + b_k$ are pairwise distinct.", "background": "Source: Open Problem Garden. Original node ID: 655. URL: http://www.openproblemgarden.org/op/snevilys_conjecture.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/snevilys_conjecture\n- Author(s): Snevily, Hunter S.\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: addition table; latin square; transversal\n- Importance: High ✭✭✭\n- Recommended for undergraduates: yes\n- Posted: October 13th, 2007 by mdevos\n\nProblem-page discussion:\nThe motivation for this question comes from the study of latin squares. The addition table of every (additive) group forms a latin square, and this gives us a rich source of interesting squares. To explain further, we require a couple of definitions. A transversal of a $k \\times k$ matrix is a collection of $k$ cells, no two of which are in the same row or column, and we say that a transversal is latin if no two of its cells contain the same element. Latin transversals are nice structures to find in latin squares. In particular, note that the cells of a $k \\times k$ latin square $L$ may be partitioned into $k$ latin transversals if and only if there is a latin square orthogonal to $L$ (see this for a definition of orthogonal latin squares). The above conjecture is perhaps most naturally phrased in terms of latin transversals as follows.\n\nConjecture (Snevily's conjecture - version 2) Every $k \\times k$ submatrix of the addition table of every abelian group of odd order has a latin transversal.\n\nSnevily's conjecture was proved by Alon [A] for abelian groups of prime order using a fairly standard application of the Alon-Tarsi polynomial technique. Later, Dasgupta, Karolyi, Serra, and Szegedy [DKSSz] used a sneaky application of the same technique to prove the conjecture for cyclic groups of odd order (the key to their approach is the fact that for $n$ odd, ${\\mathbb Z}_n$ is a subgroup of the multiplicative group of the field of order $2^{\\phi(n)}$ where $\\phi$ is Euler's totient function). The conjecture is still open for non-cyclic groups.\n\nThe full addition table of ${\\mathbb Z}_{2n}$ does not have a latin transversal. To see this, note that the sum of the elements in this group is equal to $n$ (here we identify $\\{0,1,\\ldots,2n-1\\}$ with ${\\mathbb Z}_{2n}$ in the usual manner). So, if $a_1,\\ldots,a_{2n}$ and $b_1,\\ldots,b_{2n}$ are two orderings of ${\\mathbb Z}_{2n}$, then $\\sum_{i=1}^{2n} (a_i + b_i) = 0$, and therefore $a_1 + b_1,\\ldots,a_{2n} + b_{2n}$ cannot be an ordering of ${\\mathbb Z}_{2n}$. This parity problem is the only obstruction known, and the following conjecture asserts that apart from it, the above conjectures holds for cyclic groups of even order.\n\nConjecture (Snevily) Every $k \\times k$ submatrix of the addition table of ${\\mathbb Z}_{2n}$ has a latin transversal, unless it is a translate of a cyclic subgroup of ${\\mathbb Z}_{2n}$ of even order.\n\nIn fact, it appears that the above conjecture might hold with ${\\mathbb Z}_{2n}$ replaced by any abelian group.\n\nBibliography:\n[A] N. Alon, Additive Latin transversals. Israel J. Math. 117 (2000), 125--130. MathSciNet\n\n[DKSSz] S. Dasgupta, Gy. Károlyi, O. Serra, B. Szegedy, Transversals of additive Latin squares. Israel J. Math. 126 (2001), 17--28. MathSciNet\n\n*[S] H. S. Snevily, Unsolved Problems: The Cayley Addition Table of Z $\\sb n$. Amer. Math. Monthly 106 (1999), no. 6, 584--585. MathSciNet.\n\nDiscussion links:\n- latin squares: http://en.wikipedia.org/wiki/latin square\n- this: http://www.cut-the-knot.org/arithmetic/latin3.shtml\n\nBibliography links:\n- Additive Latin transversals: http://www.tau.ac.il/%7Enogaa/PDFS/alt2.pdf\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1760589\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1882032\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=MR1543489\n\nComments:\n- May 27th, 2011 | G. Eric Moorhouse | Proof of Snevily's Conjecture: Snevily's Conjecture was in fact proved in 2009. See: Bodan Arsovski, 'A proof of Snevily's Conjecture', Israel Journal of Mathematics, vol. 182 (2011), pp. 505-508. See also Gergely Harcos, Gyula Károlyi and Géza Kós, 'Remarks to Arsovski's proof of Snevily's Conjecture', Annales Univ. Sci. Budapest., vol. 54 (2011), pp. 57-61.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 22.\n\nAttempt notes:\nTarget:\nMake progress on \"Snevily's conjecture\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Arsovski proved Snevily's conjecture for all finite abelian groups of odd order, exactly matching the stored claim.\n\n**Verified partial progress.**\n\n- Earlier work established prime-order and cyclic odd-order cases.\n- Harcos, Karolyi, and Kos gave a shorter exposition of Arsovski's full proof.\n\n**Full solution or refutation.**\n\nFor any finite abelian group G of odd order and equal-size subsets A and B, the elements can be paired so that all sums are distinct.\n\n**What remains.**\n\nNothing remains for the exact odd-order abelian-group statement; related even-order variants are separate conjectures.\n\n**Sources checked.**\n\n- Bodan Arsovski, A proof of Snevily's conjecture, Israel Journal of Mathematics 182 (2011), 505-508. (primary): https://doi.org/10.1007/s11856-011-0040-6\n Evidence used: Proves the full odd-order abelian-group conjecture.\n- Gergely Harcos, Gyula Karolyi, and Geza Kos, Remarks to Arsovski's proof of Snevily's conjecture, arXiv:1004.0253 (2010). (primary): https://arxiv.org/abs/1004.0253\n Evidence used: Confirms Arsovski's result and supplies a shortened proof.\n- Open Problem Garden, Snevily's conjecture; checked 2026-08-17. (maintained_tracker): https://openproblemgarden.org/op/snevilys_conjecture\n Evidence used: Records the 2011 notice that Arsovski proved the conjecture.\n\n**Review notes.** The exact stored statement has a missing comma between a_2+b_2 and the ellipsis; this typographical defect does not change the recoverable claim.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3386, "problem_number": "OPG-17958", "title": "Frobenius number of four or more integers", "statement": "Problem Find an explicit formula for Frobenius number $g(a_1, a_2, \\dots, a_n)$ of co-prime positive integers $a_1, a_2, \\dots, a_n$ for $n\\geq 4$.", "background": "Source: Open Problem Garden. Original node ID: 17958. URL: http://www.openproblemgarden.org/op/frobenius_number_of_four_or_more_integers.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/frobenius_number_of_four_or_more_integers\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: November 26th, 2008 by maxal\n\nProblem-page discussion:\nFor $n=2$, the formula $g(a_1,a_2) = a_1 a_2 − a_1 − a_2$ was discovered by Sylvester discovered in 1884 [S]. For $n=3$, an explicit solution is also known [G,R,SB]. No explicit solution is known for $n\\geq 4$.\n\nBibliography:\n[G] Greenberg, H. \"Solution to a Linear Diophantine Equation for Nonnegative Integers.\" J. Algorithms 9, 343-353, 1988.\n\n[R] Rødseth, Ø. J. \"On a Linear Diophantine Problem of Frobenius.\" J. reine angew. Math. 301, 171-178, 1978.\n\n[SB] Selmer, E. S. and Beyer, Ö. \"On the Linear Diophantine Problem of Frobenius in Three Variables.\" J. reine angew. Math. 301, 161-170, 1978.\n\n[S] Sylvester, J. J. \"Question 7382.\" Mathematical Questions from the Educational Times 41, 21, 1884.\n\nSource links:\n- Frobenius number: http://en.wikipedia.org/wiki/Frobenius number\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Frobenius number of four or more integers\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No universal closed formula for arbitrary n>=4 is known, but the Frobenius number is polynomial-time computable for every fixed n, NP-hard when n varies, and explicit formulas exist for many structured families.\n\n**Verified partial progress.**\n\n- Kannan gave a polynomial-time algorithm for each fixed number n of generators.\n- Ramírez Alfonsín proved NP-hardness when n is part of the input.\n- Roune and Woods proved eventual quasi-polynomial formulas for linear parametric generators and for at most three quasi-polynomial generators.\n\n**Full solution or refutation.**\n\nAlgorithmic computation and broad special-family formulas are available, but no single general closed expression matching the vague request was verified.\n\n**What remains.**\n\nSpecify an accepted class of 'explicit formulas' and either derive one for arbitrary collectively coprime inputs or prove an appropriate impossibility statement.\n\n**Sources checked.**\n\n- Ravi Kannan, Lattice translates of a polytope and the Frobenius problem, Combinatorica 12 (1992), 161-177. (primary): https://doi.org/10.1007/BF01204720\n Evidence used: Gives a polynomial-time algorithm for every fixed n.\n- Jorge L. Ramírez Alfonsín, Complexity of the Frobenius problem, Combinatorica 16 (1996), 143-147. (primary): https://doi.org/10.1007/BF01300131\n Evidence used: Establishes NP-hardness for variable n.\n- Bjarke Hammersholt Roune and Kevin Woods, The Parametric Frobenius Problem, arXiv:1502.06009. (primary): https://arxiv.org/abs/1502.06009\n Evidence used: Proves eventual quasi-polynomial behavior for major parametric special cases.\n- Jorge L. Ramírez Alfonsín, The Diophantine Frobenius Problem, Oxford University Press, 2005. (authoritative_secondary): https://doi.org/10.1093/acprof:oso/9780198568209.001.0001\n Evidence used: Authoritative monograph surveying algorithms and formula families.\n\n**Review notes.** Collective gcd 1, not pairwise coprimality, is the condition for finiteness. 'Explicit formula' is undefined, so the record is a research program rather than a precise conjecture.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3387, "problem_number": "OPG-60034", "title": "Singmaster's conjecture", "statement": "Conjecture There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number $1$.\n\nThe number $2$ appears once in Pascal's triangle, $3$ appears twice, $6$ appears three times, and $10$ appears $4$ times. There are infinite families of numbers known to appear $6$ times. The only number known to appear $8$ times is $3003$. It is not known whether any number appears more than $8$ times. The conjectured upper bound could be $8$; Singmaster thought it might be $10$ or $12$. See Singmaster's conjecture.", "background": "Source: Open Problem Garden. Original node ID: 60034. URL: http://www.openproblemgarden.org/op/singmasters_conjecture.\n\nSource subject path: Number Theory > Combinatorial Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/singmasters_conjecture\n- Author(s): Singmaster, David\n- Subject(s): Number Theory; Combinatorial Number Theory\n- Keywords: Pascal's triangle\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: March 15th, 2019 by Zach Teitler\n\nSource links:\n- Singmaster's conjecture: http://en.wikipedia.org/wiki/Singmaster's conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Singmaster's conjecture\" in Number Theory; Combinatorial Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Singmaster's conjecture that nonunit Pascal-triangle multiplicities are uniformly bounded remains open.\n\n**Verified partial progress.**\n\n- Known infinite families have multiplicity six; 3003 is the only currently known entry of multiplicity eight.\n\n**Full solution or refutation.**\n\nNo finite universal bound or unbounded-multiplicity construction was verified.\n\n**What remains.**\n\nProve a uniform multiplicity bound or find entries of arbitrarily large multiplicity.\n\n**Sources checked.**\n\n- Singmaster's conjecture overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/Singmaster%27s_conjecture\n Evidence used: Records the conjecture as unresolved and the known multiplicity data.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3388, "problem_number": "OPG-508", "title": "A sextic counterexample to Euler's sum of powers conjecture", "statement": "Problem Find six positive integers $x_1, x_2, \\dots, x_6$ such that $$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$ or prove that such integers do not exist.", "background": "Source: Open Problem Garden. Original node ID: 508. URL: http://www.openproblemgarden.org/op/a_sextic_counterexample_to_eulers_sum_of_powers_conjecture.\n\nSource subject path: Number Theory > Computational Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_sextic_counterexample_to_eulers_sum_of_powers_conjecture\n- Author(s): Euler, Leonhard P.\n- Subject(s): Number Theory; Computational Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: August 5th, 2007 by maxal\n\nProblem-page discussion:\nEuler's sum of powers conjecture states that for $k\\geq 3$ the Diophantine equation $\\sum_{i=1}^{n} a_i^k = b^k$ does not have solutions in positive integers as soon as $n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No positive solution of five sixth powers equal to a sixth power, nor impossibility proof, was verified.\n\n**Verified partial progress.**\n\n- Euler counterexamples at other exponents do not settle this sextic equation.\n\n**Full solution or refutation.**\n\nThe literal sextic problem remains open.\n\n**What remains.**\n\nFind a solution or prove none exists.\n\n**Sources checked.**\n\n- Open Problem Garden, Sextic counterexample to Euler's sum of powers conjecture, node 508. (maintained_tracker): http://www.openproblemgarden.org/op/a_sextic_counterexample_to_eulers_sum_of_powers_conjecture\n Evidence used: Preserves the exact sextic form and source context.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3389, "problem_number": "OPG-511", "title": "Counterexamples to the Baillie-PSW primality test", "statement": "Problem (1) Find a counterexample to Baillie-PSW primality test or prove that there is no one.\n\nProblem (2) Find a composite $n\\equiv 3$ or $7\\pmod{10}$ which divides both $2^{n-1} - 1$ (see Fermat pseudoprime) and the Fibonacci number $F_{n+1}$ (see Lucas pseudoprime), or prove that there is no such $n$.", "background": "Source: Open Problem Garden. Original node ID: 511. URL: http://www.openproblemgarden.org/op/counterexamples_to_the_baillie_psw_primality_test.\n\nSource subject path: Number Theory > Computational Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/counterexamples_to_the_baillie_psw_primality_test\n- Subject(s): Number Theory; Computational Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: August 7th, 2007 by maxal\n\nProblem-page discussion:\nSelfridge, Wagstaff, and Pomerance offered $500 +$100 + $20 for$n$satisfying Problem 2, and$20 + $100 +$500 for a proof that there is no such $n$ (R. Guy, 1994).\n\nBibliography:\nCarl Pomerance. \"Are There Counterexamples to the Baillie-PSW Primality Test?\"\n\nThomas R. Nicely. \" The Baillie-PSW primality test.\"\n\nR. K. Guy. \"Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes\". §A12 in \"Unsolved Problems in Number Theory\", 2nd ed. New York: Springer-Verlag, pp. 27-30, 1994.\n\nSource links:\n- Baillie-PSW primality test: http://en.wikipedia.org/wiki/Baillie-PSW primality test\n- Fermat pseudoprime: http://en.wikipedia.org/wiki/Fermat pseudoprime\n- Lucas pseudoprime: http://en.wikipedia.org/wiki/Lucas pseudoprime\n\nBibliography links:\n- Carl Pomerance. \"Are There Counterexamples to the Baillie-PSW Primality Test?\": http://www.pseudoprime.com/dopo.pdf\n- Thomas R. Nicely. \" The Baillie-PSW primality test.\": http://www.trnicely.net/misc/bpsw.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Counterexamples to the Baillie-PSW primality test\" in Number Theory; Computational Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No counterexample to Baillie--PSW, or proof of its correctness, was verified.\n\n**Verified partial progress.**\n\n- Known computation and pseudoprime restrictions do not establish universality.\n\n**Full solution or refutation.**\n\nBoth exact requests remain open.\n\n**What remains.**\n\nClassify the specified simultaneous pseudoprimes.\n\n**Sources checked.**\n\n- Open Problem Garden, Counterexamples to Baillie-PSW primality test, node 511. (maintained_tracker): http://www.openproblemgarden.org/op/counterexamples_to_the_baillie_psw_primality_test\n Evidence used: Preserves both stated forms and source literature.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3390, "problem_number": "OPG-822", "title": "Wall-Sun-Sun primes and Fibonacci divisibility", "statement": "Conjecture For any prime $p$, there exists a Fibonacci number divisible by $p$ exactly once.\n\nEquivalently:\n\nConjecture For any prime $p>5$, $p^2$ does not divide $F_{p-\\left(\\frac p5\\right)}$ where $\\left(\\frac mn\\right)$ is the Legendre symbol.", "background": "Source: Open Problem Garden. Original node ID: 822. URL: http://www.openproblemgarden.org/op/fibonacci_divisibility.\n\nSource subject path: Number Theory > Computational Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/fibonacci_divisibility\n- Subject(s): Number Theory; Computational Number Theory\n- Keywords: Fibonacci; prime\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: June 14th, 2008 by adudzik\n\nProblem-page discussion:\nLet $p$ be an odd prime, and let $\\nu_p(n)$ denote the $p$-adic valuation of $n$. Let $F_{k(p)}$ be the smallest Fibonacci number that is divisible by $p$ (which must exist by a simple counting argument). A well-known result says that $\\nu_p(F_n)=0$ unless $k(p)$ divides $n$, and $\\nu_p(F_{k(p)m}) = \\nu_p(F_{k(p)}) + \\nu_p(m)$. This conjecture asserts that $\\nu_p(F_{k(p)})=1$ for all $p$. This has been verified up to at least $p<10^{14}$. [EJ]\n\nThis conjecture is equivalent to non-existence of Wall-Sun-Sun primes.\n\nBibliography:\n[EJ] Andreas-Stephan Elsenhansand and Jörg Jahnel, The Fibonacci sequence modulo p^2\n\n[R] Marc Renault, Properties of the Fibonacci Sequence Under Various Moduli\n\n*[W] D. D. Wall, Fibonacci Series Modulo m, American Mathematical Monthly, 67 (1960), pp. 525-532.\n\nDiscussion links:\n- Wall-Sun-Sun primes: http://en.wikipedia.org/wiki/Wall-Sun-Sun_prime\n\nBibliography links:\n- The Fibonacci sequence modulo p^2: http://www.uni-math.gwdg.de/tschinkel/gauss/Fibon.pdf\n- Properties of the Fibonacci Sequence Under Various Moduli: http://www.math.temple.edu/%7Erenault/fibonacci/thesis.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Wall-Sun-Sun primes and Fibonacci divisibility\" in Number Theory; Computational Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No Wall--Sun--Sun prime is known, but no theorem excludes them all; hence the stated universal Fibonacci nondivisibility remains open.\n\n**Verified partial progress.**\n\n- Large finite searches have excluded many primes from being Wall--Sun--Sun primes.\n\n**Full solution or refutation.**\n\nNeither a counterexample prime nor a proof of the universal assertion was verified.\n\n**What remains.**\n\nFind a Wall--Sun--Sun prime or prove p^2 never divides the stated Fibonacci number.\n\n**Sources checked.**\n\n- Wall-Sun-Sun Prime, Wolfram MathWorld (accessed 2026-08-17). (authoritative_secondary): https://mathworld.wolfram.com/Wall-Sun-SunPrime.html\n Evidence used: Records the equivalent condition and certified finite search bounds, while no example is known.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3391, "problem_number": "OPG-16570", "title": "Magic square of squares", "statement": "Question Does there exist a $3\\times 3$ magic square composed of distinct perfect squares?", "background": "Source: Open Problem Garden. Original node ID: 16570. URL: http://www.openproblemgarden.org/op/magic_square_of_squares.\n\nSource subject path: Number Theory > Computational Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/magic_square_of_squares\n- Author(s): LaBar, Martin\n- Subject(s): Number Theory; Computational Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: November 23rd, 2008 by maxal\n\nProblem-page discussion:\nThis question was first asked in 1984 by Martin LaBar and popularized in 1996 by Martin Gardner, who offered $100 to the first person to construct such a square. In 2005 Christian Boyer offered €1,000 and a bottle of champagne for a solution to a somewhat easier problem [Bc]. For a review of the history of research, see [Ba, Bb, Bc]. For basic facts about the anticipated$3\\times 3$ magic square of squares, see [Br, Mo].\n\nBibliography:\n[Ba] Christian Boyer. Some notes on the magic squares of squares problem. The Mathematical Intelligencer 27 (2005), 2, 52-64.\n\n[Bb] Christian Boyer. Magic squares of squares, \"Multimagic Squares\" website.\n\n[Bc] Christian Boyer. Latest research on the \"3x3 magic square of squares\" problem, \"Multimagic Squares\" website.\n\n[Br] Kevin Brown. Magic Square of Squares, \"Math Pages\" website.\n\n[Mo] Lee Morgenstern. 3x3 Magic Square of Squares Formulations\n\nSource links:\n- magic square: http://en.wikipedia.org/wiki/magic square\n\nBibliography links:\n- Magic squares of squares: http://multimagie.com/English/SquaresOfSquares.htm\n- Latest research on the \"3x3 magic square of squares\" problem: http://multimagie.com/English/SquaresOfSquaresSearch.htm\n- Magic Square of Squares: http://www.mathpages.com/HOME/kmath417.htm\n- 3x3 Magic Square of Squares Formulations: http://home.earthlink.net/%7Emorgenstern/magic/sq3.htm\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Magic square of squares\" in Number Theory; Computational Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No 3-by-3 magic square with nine distinct positive square entries and no impossibility proof is known; the strongest classical construction has seven square entries.\n\n**Verified partial progress.**\n\n- Bremner reduced substantial subproblems to rational points on elliptic and hyperelliptic curves.\n- Bremner exhibited a genuine 3-by-3 magic square with seven square entries; no eight- or nine-square example is recorded on Boyer's maintained page.\n\n**Full solution or refutation.**\n\nThe traditional distinct-positive-square problem remains open.\n\n**What remains.**\n\nConstruct nine distinct square entries satisfying all rows, columns, and diagonals, or prove nonexistence.\n\n**Sources checked.**\n\n- Andrew Bremner, On squares of squares, Acta Arithmetica 88 (1999), 289-297. (primary): https://eudml.org/doc/207247\n Evidence used: Primary curve-based study and seven-square construction.\n- Christian Boyer, Magic squares of squares, maintained Multimagie page accompanying Some notes on the magic squares of squares problem, Mathematical Intelligencer 27(2) (2005), 52-64. (maintained_tracker): https://www.multimagie.com/English/SquaresOfSquares.htm\n Evidence used: Maintains open status, known constructions, and primary bibliography.\n\n**Review notes.** The exact statement does not say positive squares or whether zero is allowed and relies on the standard eight-line definition of a magic square. The literature's traditional problem uses distinct positive integer squares.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3392, "problem_number": "OPG-37221", "title": "Perfect cuboid", "statement": "Conjecture Does a perfect cuboid exist?", "background": "Source: Open Problem Garden. Original node ID: 37221. URL: http://www.openproblemgarden.org/op/perfect_cuboid.\n\nSource subject path: Number Theory > Computational Number Theory.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/perfect_cuboid\n- Subject(s): Number Theory; Computational Number Theory\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: May 4th, 2010 by tsihonglau\n\nProblem-page discussion:\nPerfect cuboid is a cuboid whose edges and face and body diagonals are all integers. In other words, is there any solution to the following system of Diophantine equations:\n\n$a^2 + b^2 = d^2$\n\n$b^2 + c^2 = e^2$\n\n$a^2 + c^2 = f^2$\n\n$a^2 + b^2 + c^2 = g^2$\n\nDiscussion links:\n- Perfect cuboid: http://en.wikipedia.org/wiki/Perfect cuboid\n\nComments:\n- August 24th, 2010 | Anonymous | Primitive perfect cuboid (Primitive perfect Euler brick): I think that I have a simple proof that there cannot be any primitive perfect cuboid (primitive perfect Euler brick). I am willing to provide it if anyone requests it. T Herndon\n- January 27th, 2012 | Anonymous | the proof: send to tomk@globalserve.net\n- December 28th, 2010 | Anonymous | well, I'll bite...: If you still think you have a proof, I'd love to take a look - my email is timro21@gmail.com, or you could have it published on the Unsolved Problems web site at http://www.unsolvedproblems.org/\n\nTim\n- May 4th, 2010 | tsihonglau | Is there any 4D Euler brick?: Perfect cuboid is related to Euler brick whose edges and face diagonals are all integers. It is know that there are infinite Euler bricks. But is there any 4D Euler brick? In other words, is there any solution to the following system of Diophantine equations:\n\n$a^2 + b^2 = e^2$\n\n$a^2 + c^2 = f^2$\n\n$b^2 + c^2 = g^2$\n\n$a^2 + d^2 = h^2$\n\n$b^2 + d^2 = i^2$\n\n$c^2 + d^2 = j^2$\n\nI computed a, b, c, d up to 1 million with brute force and found no solution. Any idea?\n- April 24th, 2011 | tsihonglau | Divisibility: I found the divisibility conditions of four sides a, b, c and d in a primitive 4d euler brick (if exists):\n\n1. One is divided by 64, another by 16, another by 4, another odd.\n\n2. One is divided by 27, another by 9, another by 3, another not by 3.\n\n3. Two is divided by 5.\n\n4. Two is divided by 11.\n\n5. One is divided by 13.\n\n6. One is divided by 19.\n- October 1st, 2010 | tsihonglau | Without loss of generality,: Without loss of generality, we can suppose a > b > c > d and remove a Diophantine equation from the system. I found some solutions, for example: without the last equation, the following quadruple is a solution. a=6325,b=5796,c=5520,d=528. They are so small, so i guess 4D euler bricks should exist.\n- May 19th, 2010 | Anonymous | 4d brick: Well, it's easy to show that for any primitive 4D brick (that is, one where a, b, c, and d have no common factor), then exactly one of a, b, c, and d must be odd, and the rest even....\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Perfect cuboid\" in Number Theory; Computational Number Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The existence of a perfect cuboid remains open: no accepted integer cuboid with all three face diagonals and the space diagonal integral is known, and no accepted nonexistence proof was located.\n\n**Verified partial progress.**\n\n- De Grey, Gibbs, and Helm give two more efficient search algorithms and new two-parameter families of edge cuboids.\n- They exclude large classes of possible face and internal-rectangle aspect ratios for any perfect cuboid.\n\n**Full solution or refutation.**\n\nRecent primary work continues to frame existence as an open age-old question and develops necessary conditions and searches. Several online manuscripts claim nonexistence, but no authoritative verification of those claims was found, so they are not treated as solutions.\n\n**What remains.**\n\nConstruct positive integers satisfying all seven Pythagorean conditions, or prove the Diophantine system has no nontrivial positive solution.\n\n**Sources checked.**\n\n- A. de Grey, P. Gibbs, and L. Helm, Novel required properties of, and efficient algorithms to seek, perfect cuboids, Geombinatorics Quarterly XXXIII (2024), 107ff.; arXiv:2401.06784. (primary): https://arxiv.org/abs/2401.06784\n Evidence used: Treats existence as an unresolved question and proves new necessary restrictions and search improvements.\n\n**Review notes.** Unrefereed claimed proofs of nonexistence were excluded because no authoritative validation was found.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3393, "problem_number": "OPG-60052", "title": "KPZ Universality Conjecture", "statement": "Conjecture Formulate a central limit theorem for the KPZ universality class.", "background": "Source: Open Problem Garden. Original node ID: 60052. URL: http://www.openproblemgarden.org/op/kpz_universality_conjecture.\n\nSource subject path: Probability.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/kpz_universality_conjecture\n- Subject(s): Probability\n- Keywords: KPZ equation, central limit theorem\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 3rd, 2020 by Tomas Kojar\n\nProblem-page discussion:\nThe KPZ equation is given by\n\n$$\\partial_{t}h(x,t)=\\partial_{x}^{2}h(x,t)+\\lambda(\\partial_{x}h(x,t))^{2}+\\xi,$$\n\nwhere $\\xi$ denotes space-time white noise and $\\lambda\\in \\mathbb{R}$ is a parameter describing the strength of its \"asymmetry\". It has been conjectured (see [BPRS93, BG97, Cor 12,GJ14,HQ18] for a number of results in this direction) that the KPZ equation has a “universal” character in the sense that any one-dimensional model of surface growth should converge to it provided that it has the following features:\n\n• There is a microscopic smoothing mechanism. Pictorially this means that large valleys are quickly filled.\n\n• The system has microscopic fluctuations with short-range correlations. Pictorially this means that height function change depends only on neighboring heights.\n\n• The system has some “lateral growth” mechanism in the sense that the growth speed depends in a nontrivial way on the slope. The vertical effective growth rate depends non-linearly on local slope.\n\n• At the microscopic scale, the strengths of the growth and fluctuation mechanisms are well separated: either the growth mechanism dominates (intermediate disorder) or the fluctuations dominate (weak asymmetry). Growth is drive by noise which quickly decorrelates in space / time and is not heavy tailed.\n\nHere is a concrete surface growth mathematical model to give a sense of the above features. The random deposition model is one of the simplest (and least realistic) models for a randomly growing one-dimensional interface. Unit blocks fall independently and in parallel from the sky above each site of $\\mathbb{Z}$ according to exponentially distributed waiting times. Recall that a random variable X has exponential distribution of rate $\\lambda>0$ (or mean $1/\\lambda$ ) if $P(X > x) = e^{-\\lambda x}$. Such random variables are characterized by the memoryless property – conditioned on the event that $X > x$, $X - x$ still has the exponential distribution of the same rate. Consequently, the random deposition model is Markov – its future evolution only depends on the present state (and not on its history). The ballistic deposition (or sticky block) model was introduced by Vold [V59] in 1959 and, as one expects in real growing interfaces, displays spatial correlation. As before, blocks fall according to iid exponential waiting times, however, now a block will stick to the first edge against which it becomes incident. This creates overhangs and we define the height function h(t, x) as the maximal height above x which is occupied by a box.\n\nBibliography:\n[BG97] L. BERTINI and G. GIACOMIN. Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183, no. 3, (1997), 571–607.\n\n[BPRS93] L. BERTINI, E. PRESUTTI, B. RUDIGER ¨, and E. SAADA. Dynamical fluctuations at the critical point: convergence to a nonlinear stochastic PDE. Teor. Veroyatnost. i Primenen. 38, no. 4, (1993), 689–741\n\n[Cor 12] I. Corwin, The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), 1130001, 76. MR 2930377. Zbl 1247.82040. http://dx.doi.org\n\n[GJ14] P. GONC¸ ALVES and M. JARA. Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212, no. 2, (2014), 597–644\n\n[HQ18] HAIRER, M. and QUASTEL, J. (2018). A class of growth models rescaling to KPZ. Forum Math. Pi 6 e3\n\n[V59] M. J. Vold. A numerical approach to the problem of sediment volume. J. Colloid Sci., 14:168 (1959).\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 10.\n\nAttempt notes:\nTarget:\nMake progress on \"KPZ Universality Conjecture\" in Probability, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Precise KPZ limit theorems and weak-universality theorems are known for substantial model classes, but the source's request for a CLT for the whole KPZ universality class is not a single theorem and remains unfulfilled in that breadth.\n\n**Verified partial progress.**\n\n- Hairer--Shen prove a KPZ-equation central-limit/weak-universality theorem for mixing non-Gaussian noise in a weakly asymmetric regime.\n- The KPZ fixed point gives a precise universal candidate for several integrable models.\n\n**Full solution or refutation.**\n\nNo verified theorem covers every one-dimensional growth model satisfying an informal KPZ-membership criterion.\n\n**What remains.**\n\nSpecify a robust model class and hypotheses, then prove convergence to the KPZ fixed point/universal field for that class.\n\n**Sources checked.**\n\n- M. Hairer and H. Shen, A central limit theorem for the KPZ equation, Annals of Probability 45 (2017), 4167--4221; arXiv:1507.01237. (primary): https://arxiv.org/abs/1507.01237\n Evidence used: Proves a weakly asymmetric convergence theorem for stationary mixing noise.\n- J. Baik, KPZ limit theorems, ICM 2022 survey. (authoritative_secondary): https://ems.press/books/standalone/278/5554\n Evidence used: States the broad universality conjecture and surveys model-specific limit theorems.\n\n**Review notes.** Open-ended formulation retained exactly.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 19, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3394, "problem_number": "OPG-36887", "title": "Sums of independent random variables with unbounded variance", "statement": "Conjecture If $X_1, \\dotsc, X_n \\geq 0$ are independent random variables with $\\mathbb{E}[X_i] \\leq \\mu$, then $$\\mathrm{Pr} \\left( \\sum X_i - \\mathbb{E} \\left[ \\sum X_i \\right ] < \\delta \\mu \\right) \\geq \\min \\left ( (1 + \\delta)^{-1} \\delta, e^{-1} \\right).$$", "background": "Source: Open Problem Garden. Original node ID: 36887. URL: http://www.openproblemgarden.org/op/sums_of_independent_random_variables_with_unbounded_variance.\n\nSource subject path: Theoretical Computer Science.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/sums_of_independent_random_variables_with_unbounded_variance\n- Author(s): Feige, Uriel\n- Subject(s): Theoretical Computer Science\n- Keywords: Inequality; Probability Theory; randomness in TCS\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 2nd, 2009 by cwenner\n\nProblem-page discussion:\nIn comparison to most probabilistic inequalities (like Hoeffding's), Feige's inequality does not deteriorate as $n$ goes to infinity, something that is useful for computer scientists.\n\nLet $T = \\mathbb{E}\\left [ \\sum X_i \\right ] + \\delta$. Feige argued that to prove the conjecture, one only needs to prove it for the case when $\\mu = 1$ and each variable $X_i$ has the entire probability mass distributed on 0 and $t_i$ for some $\\mathbb{E}[X_i] \\leq t_i \\leq T$. He proved that $\\mathrm{Pr} \\left( \\sum X_i - \\mathbb{E} \\left[ \\sum X_i \\right ] < \\delta \\right) \\geq \\min \\left ( (1 + \\delta)^{-1} \\delta, 1/13 \\right),$ and conjectured that the constant 1/13 may be replaced with $e^{-1}$. It was further conjectured that \"the worst case\" would be one of\n\n- one variable has $1 + \\delta$ as maximum value and the remaining $n-1$ random variables are always 1 (hence the probability that the sum is less than $T$ is $(1 + \\delta)^{-1} \\delta$ ),\n- each variable has $T = n + \\delta$ as maximum (hence the probability that the sum is less than $T$ is $\\left(1 - \\frac{1}{T}\\right)^n \\stackrel{n \\rightarrow \\infty}{\\longrightarrow} e^{-1}$ ).\n\nOne way to initiate an attack on this problem is to assume $\\delta = \\mathbb{E}[X_i] = 1$ and argue that the case when each variable assumes $n + 1$ with probability $(n+1)^{-1}$ and otherwise 0 is indeed the worst.\n\nBibliography:\n*[F04] Uriel Feige: On sums of independent random variables with unbounded variance, and estimating the average degree in a graph, STOC '04: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing (2004), pp. 594 - 603. ACM\n\n*[F05] Uriel Feige: On sums of independent random variables with unbounded variance, and estimating the average degree in a graph, Manuscript, 2005, [pdf]\n\nThe problem was also referenced at population algorithms, the blog.\n\nBibliography links:\n- ACM: http://doi.acm.org/10.1145/1007352.1007443\n- [pdf]: http://www.wisdom.weizmann.ac.il/%7Efeige/Others/newmarkov.pdf\n- population algorithms, the blog: http://petar.blog.lcs.mit.edu/?p=66\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 21.\n\nAttempt notes:\nTarget:\nMake progress on \"Sums of independent random variables with unbounded variance\" in Theoretical Computer Science, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Feige proves the lower-tail inequality with constant 1/13; replacement by the conjectured optimal e^-1 constant remains unverified.\n\n**Verified partial progress.**\n\n- Reduction to two-point distributions is part of Feige's approach.\n- The 1/13 universal constant is rigorous.\n\n**Full solution or refutation.**\n\nNo proof of the displayed e^-1 bound or counterexample was verified.\n\n**What remains.**\n\nEstablish the sharp constant or identify a worse extremal independent distribution.\n\n**Sources checked.**\n\n- Open Problem Garden, node 36887 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/sums_of_independent_random_variables_with_unbounded_variance\n Evidence used: Records Feige's 1/13 theorem and the e^-1 conjecture.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3395, "problem_number": "OPG-661", "title": "P vs. NP", "statement": "Problem Is P = NP?", "background": "Source: Open Problem Garden. Original node ID: 661. URL: http://www.openproblemgarden.org/op/p_vs_np.\n\nSource subject path: Theoretical Computer Science > Algorithms.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/p_vs_np\n- Author(s): Cook, Stephen; Levin, Leonid A.\n- Subject(s): Theoretical Computer Science; Algorithms\n- Keywords: Complexity Class; Computational Complexity; Millenium Problems; NP; P; polynomial algorithm\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 18th, 2007 by zitterbewegung\n\nProblem-page discussion:\nThis problem is the central open problem in Theoretical Computer Science and is regarded as one of the most outstanding unsolved problems in mathematics. See Wikipedia's P versus NP problem and The Clay Mathematical Institute's P vs. NP problem for more.\n\nRelated:\nRelated problems\nOne-way functions exist\n\nDiscussion links:\n- P versus NP problem: http://en.wikipedia.org/wiki/P versus NP problem\n- P vs. NP problem: http://www.claymath.org/millenium-problems/p-vs-np-problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"P vs. NP\" in Theoretical Computer Science; Algorithms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** P versus NP remains an officially unsolved Millennium Prize Problem, with no generally accepted proof of either equality or inequality.\n\n**Verified partial progress.**\n\n- The Cook-Levin theory identifies NP-complete problems, so a polynomial-time algorithm for one would prove P=NP.\n- Large bodies of conditional lower bounds, restricted-model separations, algorithms, and proof barriers shape the problem but do not decide the exact classes.\n\n**Full solution or refutation.**\n\nNo solution exists in the accepted literature as of the check date; the Clay Mathematics Institute lists the problem in its unsolved section.\n\n**What remains.**\n\nProve P=NP by a deterministic polynomial-time algorithm for an NP-complete problem, or prove an unconditional separation P not equal to NP.\n\n**Sources checked.**\n\n- Clay Mathematics Institute, The Millennium Prize Problems; checked 2026-08-17. (maintained_tracker): https://www.claymath.org/millennium-problems/\n Evidence used: The official current page lists P versus NP under Unsolved problems.\n- Stephen Cook, The P versus NP Problem, official Clay Mathematics Institute problem description. (authoritative_secondary): https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf\n Evidence used: Gives the official technical statement, equivalent formulations, and foundational context.\n\n**Review notes.** The exact stored statement is concise but mathematically standard. The background's misspelling of Millennium does not affect it.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3396, "problem_number": "OPG-36311", "title": "Exponential Algorithms for Knapsack", "statement": "Conjecture\n\nThe famous 0-1 Knapsack problem is: Given $a_{1},a_{2},\\dots,a_{n}$ and $b$ integers, determine whether or not there are $0-1$ values $x_{1},x_{2},\\dots,x_{n}$ so that $$\\sum_{i=1}^{n} a_{i}x_{i} = b.$$The best known worst-case algorithm runs in time$2^{n/2}$times a polynomial in$n$. Is there an algorithm that runs in time$2^{n/3}$?", "background": "Source: Open Problem Garden. Original node ID: 36311. URL: http://www.openproblemgarden.org/op/exponential_algorithms_for_knapsack.\n\nSource subject path: Theoretical Computer Science > Algorithms.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/exponential_algorithms_for_knapsack\n- Author(s): Lipton, Dick\n- Subject(s): Theoretical Computer Science; Algorithms\n- Keywords: Algorithm construction; Exponential-time algorithm; Knapsack\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: February 4th, 2009 by dick lipton\n\nBibliography:\n[SS] Richard Schroeppel and Adi Shamir. A T=O(2n/2), S=O(2n/4) algorithm for certain NP-complete problems, SIAM J. Comput. 10 (1981), no. 3, 456--464.\n\nComments:\n- August 6th, 2010 | rzwaan | References: Solving this problem in pseudopolynomial time with polynomial space: Daniel Lokshtanov and Jesper Nederlof: \"Saving Space by Algebraization\". To appear in the proceedings of ACM Symposium on Theory of Computing (STOC 2010).\n\nThe next reference solves this problem (even general knapsack, I believe). An 2^{0.311 n} Algorithm: Nick Howgrave-Graham and Antoine Joux: \"A new generic algorithm for hard knapsacks\". To appear in the proceedings of EUROCRYPT 2010.\n\nHowever, the problem might be easily adjusted to \"find an algorithm that takes O(2^{c n}) time, where c < 0.311\", etc etc.\n- August 3rd, 2010 | Anonymous | Updates: Publications from 2010 on this topic: Pseudopolynomial algorithm with only polynomial space: Daniel Lokshtanov and Jesper Nederlof: \"Saving Space by Algebraization\". To appear in the proceedings of ACM Symposium on Theory of Computing (STOC 2010).\n\nFast knapsack: Nick Howgrave-Graham and Antoine Joux: \"New Generic Algorithms for Hard Knapsacks\". To appear in the proceedings of EUROCRYPT 2010 The running time is O^*(2^0.311 n), which is faster than the question posted here.\n\nObviously, the question could become: find a lower constant than 0.311.\n- March 29th, 2010 | Stephen Le Guen | 0-1 knapsack: I do not know how to determine the run time but I have an algorithm based on dynamic programming which solves the problem more efficiently than a simple search but not in polynomial time!\n\nI have implemented the procedure as a Java program and tried to express it as a flow chart. Files for both of these are available at; www.cybase.co.uk/wlcs/Software.html\n\nalthough at the moment the website is down.\n\nI have contacted the ISP to try and restore it.\n- February 7th, 2010 | Anonymous | This appears to be answered: see paper at http://www.joux.biz/publications/Knapsacks.pdf and rjlipton's blog post about it at http://rjlipton.wordpress.com/2010/02/05/a-2010-algorithm-for-the-knapsack-problem/\n\nof course, this is an open ended question, there is still room for better algorithms.\n- January 13th, 2010 | Anonymous | A solution has been announced: A (randomized, probably heuristic) algorithm has been announced to solve this problem in time close to 2^(0.311 n), at the ESC 2010 seminar.\n\nFor details, see: https://cryptolux.org/ESC/Antoine_Joux\n- September 8th, 2009 | Anonymous | Subset sum or 0-1?: When your values - your a_n - can be negative, and your b (the goal) is zero, then it's called \"subset sum\". If the a_n are non-negative (i.e., some of the a's may be zero), the b is positive, and the choice is to either exclude (0) or include (1) one \"copy\" of each value, then it's a \"0-1 knapsack\" problem. Usually, but not always, a knapsack problem has components with multiple values, and the goal is a minimax problem: maximize the a's, while minimizing the c's.\n\nIt's always darkest, just after the lights go out.\n- April 4th, 2009 | cwenner | 0-1 Knapsack?: I know this problem as subset sum and not as 0-1 knapsack.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Exponential Algorithms for Knapsack\" in Theoretical Computer Science; Algorithms, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A 2010 algorithm reaches about 2^(0.3113n) for almost all density-one knapsacks, but this does not verify the requested worst-case 2^(n/3) algorithm.\n\n**Verified partial progress.**\n\n- Howgrave-Graham--Joux improve the meet-in-the-middle exponent on almost all density-one instances.\n\n**Full solution or refutation.**\n\nThe historical tracker called the target solved, but the primary paper's scope is average-case/density-one; no matching worst-case theorem was verified.\n\n**What remains.**\n\nEstablish a true worst-case exponent below 1/3 for the stated decision problem, or clarify an intended random-instance variant.\n\n**Sources checked.**\n\n- N. Howgrave-Graham and A. Joux, New Generic Algorithms for Hard Knapsacks, EUROCRYPT 2010. (primary): https://eprint.iacr.org/2010/189.pdf\n Evidence used: Abstract gives about 2^(0.3113n) for almost all density-one knapsacks, not arbitrary worst-case inputs.\n\n**Review notes.** Historical tracker overstates scope; exact source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3397, "problem_number": "OPG-445", "title": "The robustness of the tensor product", "statement": "Problem Given two codes $R,C$, their Tensor Product $R \\otimes C$ is the code that consists of the matrices whose rows are codewords of $R$ and whose columns are codewords of $C$. The product $R \\otimes C$ is said to be robust if whenever a matrix $M$ is far from $R \\otimes C$, the rows (columns) of $M$ are far from $R$ ( $C$, respectively).\n\nThe problem is to give a characterization of the pairs $R,C$ whose tensor product is robust.", "background": "Source: Open Problem Garden. Original node ID: 445. URL: http://www.openproblemgarden.org/op/the_robustness_of_the_tensor_product_0.\n\nSource subject path: Theoretical Computer Science > Coding Theory.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_robustness_of_the_tensor_product_0\n- Author(s): Ben-Sasson, Eli; Sudan, Madhu\n- Subject(s): Theoretical Computer Science; Coding Theory\n- Keywords: codes; coding; locally testable; robustness\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 13th, 2007 by ormeir\n\nProblem-page discussion:\nThe question is studied in the context of Locally Testable Codes.\n\nBibliography:\n*[BS] Eli Ben-Sasson, Madhu Sudan, Robust locally testable codes and products of codes, APPROX-RANDOM 2004, pp. 286-297 (See ECCC TR04-046).\n\n[CR] D. Coppersmith and A. Rudra, On the robust testability of tensor products of codes, ECCC TR07-061.\n\n[DSW] Irit Dinur, Madhu Sudan and Avi Wigderson, Robust local testability of tensor products of LDPC codes, APPROX-RANDOM 2006, pp. 304-315 (See ECCC TR06-118).\n\n[GM] Oded Goldreich, Or Meir, The Tensor Product of Two Good Codes Is Not Necessarily Robustly Testable, ECCC TR07-062.\n\n[M] Or Meir, On the Rectangle Method in proofs of Robustness of Tensor Products, ECCC TR07-061.\n\n[V] Paul Valiant, The Tensor Product of Two Codes Is Not Necessarily Robustly Testable, APPROX-RANDOM 2005, pp. 472-481.\n\nBibliography links:\n- ECCC TR04-046: http://eccc.hpi-web.de/eccc-reports/2004/TR04-046/index.html\n- ECCC TR07-061: http://eccc.hpi-web.de/eccc-reports/2005/TR05-104/index.html\n- ECCC TR06-118: http://eccc.hpi-web.de/eccc-reports/2006/TR06-118/index.html\n- ECCC TR07-062: http://eccc.hpi-web.de/eccc-reports/2007/TR07-062/index.html\n- ECCC TR07-061: http://eccc.hpi-web.de/eccc-reports/2007/TR07-061/index.html\n\nComments:\n- February 17th, 2010 | Anonymous | Results in: Eli Ben-Sasson and Michael Viderman. \"Composition of semi-LTCs by two-wise Tensor Products\" (RANDOM 09)\n\nEli Ben-Sasson and Michael Viderman. \"Tensor Products of Weakly Smooth Codes are Robust\" (RANDOM 08)\n- July 16th, 2007 | ormeir | The formal definition of robustness, and of the problem: In all of the following definitions, the term \"distance\" refers to \"relative Hamming distance\".\n\nGiven a matrix $M$, let $\\delta_{R \\otimes C}(M)$ denote the distance from $M$ to the nearest codeword of $R \\otimes C$. Let $\\delta_{\\rm{row}}(M)$ denote the average distance of a row of $M$ to $R$, and let $\\delta_{\\rm{col}}(M)$ denote the average distance of a column of $M$ to $C$. Finally, let $\\rho(M)$ denote the average of $\\delta_{\\rm{row}}(M)$ and $\\delta_{\\rm{col}}(M)$.\n\nThe tensor product $R \\otimes C$ is said to be $\\alpha$-robust iff for every matrix $M$ we have that $\\rho(M) \\ge \\alpha \\cdot \\delta_{R \\otimes C}(M)$.\n\nThe question is, under what conditions the tensor product $R \\otimes C$ is $\\alpha$-robust for some constant $\\alpha$.\n- July 14th, 2007 | Robert Samal | To be more precise...: 1) When you say codes, do you mean linear codes?\n\n2) What distance you are using when you're saying \"far from\"?\n- July 16th, 2007 | ormeir | To be more precise: 1) The question is most interesting for linear codes, but it can also be defined for non-linear codes.\n\n2) The distance is (relative or absolute) Hamming Distance.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 11.\n\nAttempt notes:\nTarget:\nMake progress on \"The robustness of the tensor product\" in Theoretical Computer Science; Coding Theory, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No general necessary-and-sufficient characterization of robust tensor-code pairs was verified; substantial sufficient conditions, good-code counterexamples, and duality/product-expansion obstructions are known.\n\n**Verified partial progress.**\n\n- Weakly smooth codes tensored with good-distance codes give robust products.\n- Constant-rate, constant-relative-distance component codes need not have a robust product.\n- Random linear-code pairs are robust with high probability, while a code paired with its dual is never robust.\n\n**Full solution or refutation.**\n\nThe literature supplies broad positive and negative criteria but not the requested full characterization.\n\n**What remains.**\n\nFix a quantitative asymptotic definition and identify necessary and sufficient structural conditions, plausibly through product expansion/agreement testability.\n\n**Sources checked.**\n\n- Eli Ben-Sasson and Michael Viderman, Tensor Products of Weakly Smooth Codes are Robust, Theory of Computing 5 (2009), 239-255. (primary): https://theoryofcomputing.org/articles/v005a012/\n Evidence used: Gives a broad sufficient condition for robust tensor products.\n- Oded Goldreich and Or Meir, The tensor product of two good codes is not necessarily robustly testable, Information Processing Letters 112 (2012), 351-355. (primary): https://doi.org/10.1016/j.ipl.2012.01.007\n Evidence used: Constructs nonrobust products whose two component codes both have constant rate and distance.\n- Gleb Kalachev and Pavel Panteleev, Two-sided Robustly Testable Codes, arXiv:2206.09973. (primary): https://arxiv.org/abs/2206.09973\n Evidence used: Proves random-pair robustness, the code-dual obstruction, and links to product expansion and agreement testability.\n\n**Review notes.** The statement omits alphabet/field, distance normalization, robustness constant, and family quantifiers. These choices must be fixed before a unique characterization question is well-posed.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3398, "problem_number": "OPG-163", "title": "Subset-sums equality (pigeonhole version)", "statement": "Problem Let $a_1,a_2,\\ldots,a_n$ be natural numbers with $\\sum_{i=1}^n a_i < 2^n - 1$. It follows from the pigeon-hole principle that there exist distinct subsets $I,J \\subseteq \\{1,\\ldots,n\\}$ with $\\sum_{i \\in I} a_i = \\sum_{j \\in J} a_j$. Is it possible to find such a pair $I,J$ in polynomial time?", "background": "Source: Open Problem Garden. Original node ID: 163. URL: http://www.openproblemgarden.org/op/theoretical_computer_science/subset_sums_equality.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/theoretical_computer_science/subset_sums_equality\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: polynomial algorithm; search problem\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: March 18th, 2007 by mdevos\n\nProblem-page discussion:\nThis is one of a class of search problems for which a positive solution is garaunteed (so the corresponding decision problem is trivial) based on a theoretical property of the problem. Another such problem is given a Hamiltonian cycle in a cubic graph, find a second Hamiltonian cycle (here a theorem of Smith guarantee's a positive solution). The above problem is particularly attractive, since the proof that a pair $I,J$ must exist is quite simple, but it gives no insight into how to find the pair $I,J$.\n\nIt seems to be consensus among the cryptography community that this problem is hard.\n\nComments:\n- July 13th, 2007 | Anonymous | The problems are in PPP and PPA, respectively: Both this problem and the Smith/2nd Hamiltonian Path problems were suggested by Papadimitriou in his 1991 paper 'On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence\". (See http://www.cs.berkeley.edu/~christos/papers/ for a copy of the journal version.)\n\nThese problems are in natural complexity classes related to though apparently somewhat larger than PPAD. The subset sum problem is in the class PPP and the Smith problem is in the class PPA. Neither is known to be complete for the respective complexity class as far as I know.\n- July 9th, 2007 | kvanetten | You're right: I finally located a copy of the paper on the author's web page at http://www.lri.fr/~santha/ (why didn't that occur to me before?), and it's much clearer to me now--you're absolutely right. I guess the one suggestion I would make is to adopt their terminology and include the term 'pigeonhole' in the name of the problem to distinguish this variation from the more general case.\n- July 9th, 2007 | mdevos | How's this?: I am quite certian I have seen this problem called simply \"subset-sums equality\" before, so I have just appended \"(pigeonhole version)\" to the title. I hope that this is increasing the clarity.\n- July 9th, 2007 | mdevos | I think it is still open: First a disclaimer: I am no expert on this problem.. I heard it awhile ago, liked it, wrote it down, and that's what you see here. However, I do believe it is still open. If my understanding is correct, the problem considered in the paper you mention (\"Efficient approximation algorithms for the subset-sums equality problem\" by Bazgan, Santha and Tuza, JCSS Vol.64 Issue 2, March 2002) is a relaxed version of the one stated here where the restriction $\\sum_{i=1}^n a_i < 2^n - 1$ is not present, and the goal is to find a pair of nonempty disjoint subsets $I,J \\subseteq \\{1,\\ldots,n\\}$ so that $\\sum_{i \\in I} a_i$ and $\\sum_{j \\in J} a_j$ are as close as possible. I believe the only hardness results they obtain are for this problem.\n\nThe thing still missing is what to do with the funny assumption $\\sum_{i=1}^n a_i < 2^n - 1$. Although the problem is a search problem which looks roughly like a knapsack-type problem (and thus should be NP-hard), the associated decision problem is trivially \"true\", yet there is no obvious way to use this information.\n\nPS: thanks for the post!\n- July 8th, 2007 | kvanetten | A little more: Just an afterthought to my previous comment... Even if this problem is known not to be computable in polynomial time, it would still be interesting to know if it is NP-hard or not.\n- July 15th, 2007 | Anonymous | Re: A little more: Problems with guaranteed-to-exist solutions (like this one, & others in PPA, PPAD, etc.) are not NP-complete unless NP = coNP and the Polynomial Hierarchy collapses. Now, we don't yet know that P!= NP, or even that P!= PH.\n- July 16th, 2007 | kvanetten | Re: A little more: Yes, I made that comment before I read the paper by Bazgan, Santha and Tuza. They mentioned in passing that the problem isn't NP-hard unless NP = coNP, and apparently felt that it was obvious enough to not require any further comment. I'm probably missing something obvious, but I can't quite follow this. With factorization (for example), a number's prime decomposition is unique and can serve as a certificate for both 'yes' and 'no' instances. For pigeonhole subset-sums equality, the solutions are not necessarily unique. It would depend on exactly how one crafted the corresponding decision problem, but it's not clear to me what the certificate would be for a 'no' instance.\n- July 8th, 2007 | kvanetten | Should this problem still be considered open?: I was doing a quick search for background on this question and came across this article: http://dx.doi.org/10.1006/jcss.2001.1784, \"Efficient approximation algorithms for the subset-sums equality problem\" by Bazgan, Santha and Tuza, JCSS Vol.64 Issue 2 (March 2002). I only have access to the abstract, so I might be misinterpreting this, but they say, \"... On the other hand, we show that in the case where the value of a solution is the positive difference between the two partial sums, the problem is not 2nk -approximable in polynomial time unless P = NP, for any constant k.\" I take this to mean that there are no exact polynomial time algorithms unless P=NP. If so, this problem is only open in the sense that any question reducible to P vs. NP can be considered open.\n- June 30th, 2007 | Anonymous | Problem formulation: It seems to me that there are some typos in the problem formulation. The sum should be less than $2^n - 1$. The subsets should be nonempty.\n\n- JS\n- July 2nd, 2007 | mdevos | Thanks!: Thanks for your comment, I will edit the problerm and correct it.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Subset-sums equality (pigeonhole version)\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Pigeonhole Equal Sums remains without a polynomial-time algorithm. The best verified general exact algorithm is exponential but improves substantially over meet-in-the-middle.\n\n**Verified partial progress.**\n\n- The problem is a natural total-search problem in PPP.\n- Jin and Wu give an O*(2^(0.4n))-time algorithm and an O*(2^(0.75n))-time polynomial-space algorithm.\n\n**Full solution or refutation.**\n\nThe 2024 ICALP algorithm improves the O*(2^(n/2)) barrier for this promised equal-sums problem, but remains exponential in n and therefore does not answer the polynomial-time question.\n\n**What remains.**\n\nFind an algorithm polynomial in the binary input length, or establish an appropriate conditional hardness or PPP-completeness result.\n\n**Sources checked.**\n\n- Open Problem Garden, Subset-sums equality (pigeonhole version) (OPG-163), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/theoretical_computer_science/subset_sums_equality\n Evidence used: Gives the exact promise problem and historical PPP discussion.\n- C. Jin and H. Wu, A Faster Algorithm for Pigeonhole Equal Sums, 51st ICALP, LIPIcs 297 (2024), Article 94. (primary): https://doi.org/10.4230/LIPIcs.ICALP.2024.94\n Evidence used: States the exact same promise problem, gives the O*(2^(0.4n)) algorithm, and identifies polynomial time and PPP-completeness as unresolved.\n\n**Review notes.** Natural numbers are treated as positive integers encoded in binary, matching the modern Pigeonhole Equal Sums formulation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3399, "problem_number": "OPG-467", "title": "Complexity of square-root sum", "statement": "Question What is the complexity of the following problem?\n\nGiven $a_1,\\dots,a_n; k$, determine whether or not $\\sum_i \\sqrt{a_i} \\leq k.$", "background": "Source: Open Problem Garden. Original node ID: 467. URL: http://www.openproblemgarden.org/op/complexity_of_square_root_sum.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/complexity_of_square_root_sum\n- Author(s): Goemans, Michel (?)\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: semi-definite programming\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 18th, 2007 by abie\n\nProblem-page discussion:\nAs of a 1998 survey, the complexity of this problem was unknown. I'm not sure if that's still the case. But I wanted to see how easy it was to make a page about it.\n\nThis is the key to determining if semi-definite programming is truly solvable in polynomial time (it can be approximated to within $\\varepsilon$ using the interior point method or the ellipsoid algorithm in time polynomial in the size of the instance and $\\log 1/\\varepsilon$.\n\nBibliography:\n[G] Michal Goemans, Semidefinite Programming and Combinatorial Optimization\n\nDiscussion links:\n- a 1998 survey: http://www.emis.ams.org/journals/DMJDMV/xvol-icm/17/Goemans.MAN.ps.gz\n\nBibliography links:\n- Semidefinite Programming and Combinatorial Optimization: http://www.emis.ams.org/journals/DMJDMV/xvol-icm/17/Goemans.MAN.ps.gz\n\nComments:\n- December 29th, 2009 | Anonymous | Complexity: The complexity of the problem as stated is worst case O(n^3) in the magnitude of the greatest value a[i]. The reason being that it is a simple sum, as stated, of square roots. The square root operation is O(n^2) in the number of bits, which has a 3.xxx:1 relation to the magnitude of each a[i]. Either the problem is mistated or it has not been significant enough for anyone to waste time on stating the obvious.\n- December 30th, 2009 | Anonymous | Filching 10:00(2): I think the problem is that you may need to compute square roots to high precision. If you need to know the first m digits in the decimal expansion of sqrt(5), it will probably take time linear in m to find them, even though 5 took only constantly many bits to encode.\n- June 23rd, 2008 | Anonymous | A slightly more general: A slightly more general problem (in which the square roots may be subtracted as well as added) is very important in the context of computational geometry, both for actual algorithms and for proving complexity-theoretic bounds on problems, as it encapsulates the difficulty of comparing lengths of polygonal chains; see http://maven.smith.edu/~orourke/TOPP/P33.html.\n\n—David Eppstein\n- July 19th, 2007 | Robert Samal | Some update: Quotation from Etessami and Yannakakis, 2007:\n\n- [Sqrt-sum problem] is known to be solvable in PSPACE but it has been a major open problem ([GareyGrahamJohnson’76]) whether it is solvable even in NP.\n- [Allender et. al.,’06] Showed that Sqrt-Sum reduces to a more general problem, which they showed lies in the 4th level of the Counting Hierarchy ( $P^{PP^{PP^{PP}}}$ ).\n- July 1st, 2011 | Anonymous | From Allender et al.: square-root sum is in CH: http://ftp.cs.rutgers.edu/pub/allender/slp.pdf\n\nCorollary 1.5 The Sum-of-square-roots problem and the Euclidean Traveling Salesman Problem are in CH.\n\n(CH = Counting hierarchy.)\n\nDo we also know that it is in P^(PP^(PP^PP))?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Complexity of square-root sum\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** For the standard binary-integer formulation, Square-Root-Sum is in the counting hierarchy, specifically P with three nested PP oracles, but it remains unknown whether it is in P or even NP.\n\n**Verified partial progress.**\n\n- Allender-Bürgisser-Kjeldgaard-Pedersen-Miltersen place Square-Root-Sum in the counting hierarchy via PosSLP.\n- Special input regimes, such as unary or structured radicands, have stronger results but do not settle binary Square-Root-Sum.\n\n**Full solution or refutation.**\n\nA nontrivial classical upper bound is known, but the basic P/NP membership questions remain unresolved.\n\n**What remains.**\n\nGive polynomially checkable separation certificates or a polynomial-time comparison algorithm for binary inputs, or prove a meaningful hardness/lower-bound result.\n\n**Sources checked.**\n\n- Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, and Peter Bro Miltersen, On the Complexity of Numerical Analysis, SIAM Journal on Computing 38 (2009), 1987-2006. (primary): https://doi.org/10.1137/070697926\n Evidence used: Develops the PosSLP framework and places Sum-of-Square-Roots in the counting hierarchy.\n- Open Problem Garden, Complexity of square-root sum (node 467). (maintained_tracker): https://www.openproblemgarden.org/op/complexity_of_square_root_sum\n Evidence used: Preserves the original complexity question and its classical context.\n\n**Review notes.** The literal input omits the domains and encodings of a_i and k, so complexity is undefined without repair. This result explicitly uses nonnegative binary integers and a binary integer threshold, the standard formulation.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3400, "problem_number": "OPG-474", "title": "Linear-size circuits for stable $0,1 < 2$ sorting?", "statement": "Problem Can $O(n)$-size circuits compute the function $f$ on $\\{0,1,2\\}^*$ defined inductively by $f(\\lambda) = \\lambda$, $f(0x) = 0f(x)$, $f(1x) = 1f(x)$, and $f(2x) = f(x)2$?", "background": "Source: Open Problem Garden. Original node ID: 474. URL: http://www.openproblemgarden.org/op/linear_size_circuits_for_stable_0_1_2_sorting.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/linear_size_circuits_for_stable_0_1_2_sorting\n- Author(s): Regan, Kenneth\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: Circuits; sorting\n- Importance: Medium ✭✭\n- Recommended for undergraduates: yes\n- Posted: July 19th, 2007 by KWRegan\n\nProblem-page discussion:\nThis function moves all 2s in $x$ flush-right, leaving the sequence of 0s and 1s the same, and represents stable topological sort of the partial order $0,1 < 2$. It is linear-time computable in any model that supports the operations of a double-ended queue in $O(1)$ time, including multi-tape Turing machines, but is to me the \"easiest\" function for which I do not know linear-size circuits. By contrast sorting $0 < 1 < 2$, called the \"Dutch National Flag Problem\", has $O(n)$-size circuits by counting. It suffices to compute $f(x)$ when $|x|$ is a power of $2$ and exactly half the entries are $2$. For this and more see my Computational Complexity blog item, PDF file here.\n\nDiscussion links:\n- Computational Complexity blog item: http://weblog.fortnow.com/2007/07/concrete-open-problem.html\n- here: http://www.cse.buffalo.edu/%7Eregan/InfoFlow.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Linear-size circuits for stable $0,1 < 2$ sorting?\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No linear-size circuit construction for the stated stable 0,1,2 sorting function was verified.\n\n**Verified partial progress.**\n\n- The source tracker provides the exact current formulation.\n\n**Full solution or refutation.**\n\nNo full solution or counterexample was verified.\n\n**What remains.**\n\nstable 0 1 2 sorting linear size circuits\n\n**Sources checked.**\n\n- Open Problem Garden (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/\n Evidence used: Tracker retains the problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3401, "problem_number": "OPG-2150", "title": "Discrete Logarithm Problem", "statement": "If $p$ is prime and $g,h \\in {\\mathbb Z}_p^*$, we write $\\log_g(h) = n$ if $n \\in {\\mathbb Z}$ satisfies $g^n = h$. The problem of finding such an integer $n$ for a given $g,h \\in {\\mathbb Z}^*_p$ (with $g \\neq 1$ ) is the Discrete Log Problem.\n\nConjecture There does not exist a polynomial time algorithm to solve the Discrete Log Problem.", "background": "Source: Open Problem Garden. Original node ID: 2150. URL: http://www.openproblemgarden.org/op/discrete_logarithm_problem.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/discrete_logarithm_problem\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: discrete log; NP\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 27th, 2008 by cplxphil\n\nProblem-page discussion:\nThe Discrete Logarithm Problem is a critical problem in number theory, and is similar in many ways to the integer factorization problem. If it were possible to compute discrete logs efficiently, it would be possible to break numerous thought-to-be unbreakable cryptographic schemes. However, although most mathematicians and computer scientists believe that the DLP is unsolvable, this conjecture is difficult to establish, because such a proof would imply that P!= NP...which is the most difficult open problem in theoretical computer science.\n\nAvi Wigderson has shown that there is no \"natural proof\" (in the sense of [RR]) that the DLP requires circuits of greater than half-exponential size. The key idea is that a natural proof that the DLP is hard would yield a method for breaking discrete-log-based cryptosystems, and this is a contradiction. Of course, it could still be that DLP is provably hard, but by a proof that is not \"natural.\"\n\nBibliography:\n[RR] Alexander A. Razborov and Steven Rudich, Natural proofs, Journal of Computer and System Sciences 55 (1997), 24–35.\n\nComments:\n- December 29th, 2009 | Anonymous | Discrete Logrithm is polynomial: The discrete logarithm problem is little more than an integer analog to a rotor-code cipher problem. The later is a problem in finding concurrent zeros for two periodic functions where the periodicity of one tracks the value of x and the other the value of y. The value of x being x, x^2, x^3,... x^n. The value of y being y, y+p, y+2p,..., y+np. These forms of problems are amenable to solution using a multi dimensional difference (recurrence) expression (as is the Elliptic Curve Cryptographic problem which is a direct application of DE over algebraic fields). The Discrete Logarithm problem is solvable by a deterministic polynomial time algorithm in O(n^3). Google a paper titled \"Computing a Discrete Logarithm in O(n^3)\", which can be found at Cornell's arXiv website. Example code for the algorithm is also provided by the author of that paper.\n- February 25th, 2013 | Anonymous | N!= n: The above paper solves the discrete logarithm in time O(N^3) not O(n^3), two very different things. N being the size of the modulus, n being log_2 of N (the binary length). There are many algorithms that will solve the discrete log problem much faster than this method, brute force search runs at a worst case of O(N), or in other words O(2^n). The provided algorithm in the above paper runs in (2^(3*n)).\n- December 29th, 2009 | Anonymous | A polynomial algorithm: See the paper at http://arxiv1.library.cornell.edu/abs/0912.2269.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Discrete Logarithm Problem\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No classical polynomial-time discrete-log algorithm is known, but Shor's quantum algorithm is polynomial time; the source does not specify the computation model.\n\n**Verified partial progress.**\n\n- Classical subexponential algorithms are known in many group settings.\n- Quantum polynomial-time computation changes the answer.\n\n**Full solution or refutation.**\n\nNo unconditional classical lower bound rules out polynomial time, and the unqualified wording is false under standard quantum computation.\n\n**What remains.**\n\nState the intended classical or quantum model and the group/input representation before assigning a single formal status.\n\n**Sources checked.**\n\n- Open Problem Garden, node 2150 (accessed 2026-08-17). (authoritative_secondary): https://www.openproblemgarden.org/\n Evidence used: The historical prompt omits a classical-versus-quantum model qualification.\n\n**Review notes.** Computational-model ambiguity flagged; source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3402, "problem_number": "OPG-36892", "title": "P vs. PSPACE", "statement": "Problem Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, does P = PSPACE?", "background": "Source: Open Problem Garden. Original node ID: 36892. URL: http://www.openproblemgarden.org/op/p_vs_pspace.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/p_vs_pspace\n- Author(s): Folklore\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: P; PSPACE; separation; unconditional\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 4th, 2009 by cwenner\n\nProblem-page discussion:\nIf $P \\neq NP$, then $P \\neq NP, NP^{NP}, \\dotsc, PH, P^{\\#P}, PSPACE = NPSPACE$, and a whole bunch more separations can be shown. In the light of this, if one believes that $P \\neq NP$, then it is naive to try to directly separate P from NP. In the words of the great George Polya, \"If there is a problem you can’t solve, then there is an easier problem you can solve: find it.\" The P versus PSPACE question is one of the easiest such questions and would constitute an astonishing discovery in its own right. In particular, it would be the first separation result of this kind.\n\nProblem Approaches\n\nHow do we approach this problem? I don't know, readers please contribute. My personal take would be circuit complexity, i.e. functions that are known not to be computable with circuits of polynomial size. Another would be descriptive complexity: show that there is a property that can be expressed in second-order logic with a transitive closure operation which cannot be recognized in polynomial time.[N87] Fenner offered an interesting characterization of the P versus PSPACE question in terms of reductions.[FKR89] P versus PSPACE was also as an intermediate step towards the P versus NP prize by the Clay institute.[F05]\n\nBibliography:\n[F05] Harvey M. Friedman: Clay Millenium Problem: P = NP, manuscript, 2005. [pdf]\n\n[FKR89] Fenner,, S. A. and Kurtz,, S. A. and Royer,, J. A.: Every polynomial-time 1-degree collapses iff P=PSPACE, SFCS '89: Proceedings of the 30th Annual Symposium on Foundations of Computer Science, 1988, pp. 624-629, citeseer acm [pdf]\n\n[N87] Neil Immerman: Languages that Capture Complexity Classes, SIAM Journal of Computation 16:4, 1987. citeseer [pdf]\n\nRelated:\nRelated problems\nP vs. NP\n\nBibliography links:\n- [pdf]: http://www.math.ohio-state.edu/%7Efriedman/pdf/P=NP10290512pt.pdf\n- citeseer: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.3854\n- acm: http://dx.doi.org/10.1109/SFCS.1989.63545\n- [pdf]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.29.3854&rep=rep1&type=pdf\n- citeseer: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.9176\n- [pdf]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.54.9176&rep=rep1&type=pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"P vs. PSPACE\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether P equals PSPACE remains open. The imported first sentence asks for P != PSPACE, while its purported formal restatement asks whether P = PSPACE, so the yes/no polarity is reversed.\n\n**Verified partial progress.**\n\n- Williams proved that every time-t multitape Turing computation can be simulated in O(sqrt(t log t)) space.\n- The same work gives explicit problems in linear space requiring n^(2-epsilon) time on multitape Turing machines for every epsilon>0, but not a polynomial-time versus polynomial-space separation.\n\n**Full solution or refutation.**\n\nNo unconditional equality or separation is known. The intended task is to decide P versus PSPACE, but the two sentences in the stored statement phrase opposite answers.\n\n**What remains.**\n\nProve either P=PSPACE or exhibit a language in PSPACE outside P; repair the statement so its informal and formal questions have consistent polarity.\n\n**Sources checked.**\n\n- R. Ryan Williams, Simulating Time With Square-Root Space, Proceedings of STOC 2025, 13-23, DOI 10.1145/3717823.3718225. (primary): https://arxiv.org/abs/2502.17779\n Evidence used: The abstract states the square-root-space simulation and explicitly describes the derived lower bound as only a little progress on P versus PSPACE.\n- Albert-Ludwigs-Universitat Freiburg, Theoretical Computer Science II lecture notes, accessed 2026-08-17. (authoritative_secondary): https://ac.informatik.uni-freiburg.de/teaching/ss_18/tcs/lecturenotes.pdf\n Evidence used: The notes state P subset NP subset PSPACE and explicitly identify strictness/equality as open.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3403, "problem_number": "OPG-59968", "title": "One-way functions exist", "statement": "Conjecture One-way functions exist.", "background": "Source: Open Problem Garden. Original node ID: 59968. URL: http://www.openproblemgarden.org/op/one_way_functions_exist.\n\nSource subject path: Theoretical Computer Science > Complexity.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/one_way_functions_exist\n- Subject(s): Theoretical Computer Science; Complexity\n- Keywords: one way function\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: October 12th, 2014 by porton\n\nProblem-page discussion:\nIn fact, their existence would prove that the complexity classes P and NP are not equal.\n\nBibliography:\nOne-way functions (Wikipedia)\n\nRelated:\nRelated problems\nP vs. NP\n\nSource links:\n- One-way functions: http://en.wikipedia.org/wiki/One-way_function\n\nBibliography links:\n- One-way functions: http://en.wikipedia.org/wiki/One-way_function\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"One-way functions exist\" in Theoretical Computer Science; Complexity, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The existence of cryptographic one-way functions remains unproved.\n\n**Verified partial progress.**\n\n- One-way functions have many conditional equivalents and implications, including standard pseudorandomness and private-key cryptographic primitives.\n\n**Full solution or refutation.**\n\nNo unconditional construction with a proof of average-case one-wayness was verified.\n\n**What remains.**\n\nProve one-way functions exist or establish a contrary complexity-theoretic result.\n\n**Sources checked.**\n\n- One-way function overview (accessed 2026-08-17). (authoritative_secondary): https://en.wikipedia.org/wiki/One-way_function\n Evidence used: Records existence as an open conjecture and summarizes its complexity consequences.\n- E. Hirahara, O. Ilango and R. A. Oliveira, On One-Way Functions and Kolmogorov Complexity, arXiv:2009.11514 (2020). (primary): https://arxiv.org/abs/2009.11514\n Evidence used: States equivalences conditional on the existence of one-way functions.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3404, "problem_number": "OPG-454", "title": "Unconditional derandomization of Arthur-Merlin games", "statement": "Problem Prove unconditionally that $\\mathcal{AM}$ $\\subseteq$ $\\Sigma_2$.", "background": "Source: Open Problem Garden. Original node ID: 454. URL: http://www.openproblemgarden.org/op/unconditional_derandomization_of_mathcal_am.\n\nSource subject path: Theoretical Computer Science > Complexity > Derandomization.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/unconditional_derandomization_of_mathcal_am\n- Author(s): Shaltiel, Ronen; Umans, Christopher\n- Subject(s): Theoretical Computer Science; Complexity; Derandomization\n- Keywords: Arthur-Merlin; Hitting Sets; unconditional\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 16th, 2007 by ormeir\n\nProblem-page discussion:\nIt is trivial to show that $\\mathcal{AM} \\subseteq \\Pi_2$. It is also known that under hardness assumptions $\\mathcal{AM}= \\mathcal{NP}$ (See [MV99]). The question is, can we prove unconditionally that $\\mathcal{AM} \\subseteq \\Sigma_2$.\n\nBibliography:\n*[GSTS03] Danny Gutfreund, Ronen Shaltiel and Amnon Ta-Shma, Uniform hardness vs. randomness for Arthur-Merlin games, Proc. of CCC 2003. Can be downloaded from Ronen Shaltiel's web site.\n\n[MV99] Peter Bro Miltersen and N. Variyam Vinodchandran, \"Derandomizing Arthur-Merlin games using hitting sets\", STOC 1999, pages 71-80. Can be downloaded from N. Variyam Vinodchandran's web site.\n\n[SU01] Ronen Shaltiel and Christopher Umans, Simple extractor for all min-entropies and new pseudo-random generator, Proc. of FOCS 2001, pages 648-657. Can be downloaded from Ronen Shaltiel's web site.\n\n[SU07] Ronen Shaltiel and Christopher Umans, Low-end uniform hardness vs. randomness tradeoffs for AM, STOC 2007. Can be downloaded from Ronen Shaltiel's web site.\n\nSource links:\n- $\\mathcal{AM}$: http://en.wikipedia.org/wiki/Arthur-Merlin_protocol\n- $\\Sigma_2$: http://en.wikipedia.org/wiki/Polynomial_hierarchy\n\nBibliography links:\n- web site: http://cs.haifa.ac.il/%7Eronen/online_papers/online_papers.html\n- web site: http://www.cse.unl.edu/%7Evinod/papers/index.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Unconditional derandomization of Arthur-Merlin games\" in Theoretical Computer Science; Complexity; Derandomization, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No unconditional proof that AM is contained in Sigma_2^P was verified; the standard unconditional upper bound remains Pi_2^P, with only weaker or conditional derandomizations known.\n\n**Verified partial progress.**\n\n- Stull proves that Sigma_2-promise-AM problems can be decided in Sigma_2SUBEXP with polynomial advice.\n- Hardness assumptions yield stronger derandomization, but do not answer the unconditional question.\n\n**Full solution or refutation.**\n\nThe exact unconditional polynomial-hierarchy containment remains open.\n\n**What remains.**\n\nConstruct unconditional polynomial-size hitting sets sufficient to swap the AM randomness into an existential-first second-level simulation, or establish a barrier/counterrelativization.\n\n**Sources checked.**\n\n- D. M. Stull, Some Results on Circuit Lower Bounds and Derandomization of Arthur-Merlin Problems, arXiv:1701.04428. (primary): https://arxiv.org/abs/1701.04428\n Evidence used: Proves a weak unconditional promise-AM derandomization and does not claim AM subset Sigma_2^P.\n- Open Problem Garden, Unconditional derandomization of Arthur-Merlin games (node 454). (maintained_tracker): https://www.openproblemgarden.org/op/unconditional_derandomization_of_mathcal_am\n Evidence used: States the exact open inclusion and records Pi_2 containment and conditional derandomization context.\n\n**Review notes.** The source omits the superscript P on Sigma_2; the intended class is Sigma_2^P. The promise/subexponential/advice result is not a solution to the displayed inclusion.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3405, "problem_number": "OPG-51618", "title": "P vs. BPP", "statement": "Conjecture Can all problems that can be computed by a probabilistic Turing machine (with error probability < 1/3) in polynomial time be solved by a deterministic Turing machine in polynomial time? That is, does P = BPP?", "background": "Source: Open Problem Garden. Original node ID: 51618. URL: http://www.openproblemgarden.org/op/p_vs_bpp.\n\nSource subject path: Theoretical Computer Science > Complexity > Derandomization.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/p_vs_bpp\n- Author(s): Folklore\n- Subject(s): Theoretical Computer Science; Complexity; Derandomization\n- Keywords: BPP; circuit complexity; pseudorandom generators\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: June 14th, 2013 by Charles R Great...\n\nProblem-page discussion:\nBPP has long been considered tractable. Many problems in BPP have been derandomized, showing that they are in fact in P. Is this true for all problems in BPP? All that is known at the moment is $P\\subseteq BPP\\subseteq NEXP.$\n\nThis problem has been shown to have deep connections to circuit complexity (see for example Impagliazzo & Wigderson). It is folklore that the existence of appropriate pseudorandom generators suffices to give P = BPP; Goldreich shows that their existence also follows from P = BPP.\n\nBibliography:\nAndrea E.\u0002 F.\u0002 Clement, Jos\u0003e D.\u0002 P.\u0002 Rolim,\u0002 and Luca Trevisan, Recent Advances Towards Proving P \u0002= BPP (1998).\n\nOded Goldreich, In a World of BPP=P, Studies in complexity and cryptography, Lecture Notes in Comput. Sci., 6650, Springer, Heidelberg, 2011, pp. 191-–232. See also the presentation.\n\nRussell Impagliazzo and Avi Wigderson, P=BPP unless E has sub-exponential circuits: derandomizing the XOR Lemma, following STOC '97.\n\nRyan Williams, Towards NEXP versus BPP?, Computer Science---Theory and Applications, Lecture Notes in Computer Science Volume 7913 (2013), pp. 174--182.\n\nRelated:\nRelated problems\nP vs. NP\n\nBibliography links:\n- Recent Advances Towards Proving P \u0002= BPP: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.5568\n- In a World of BPP=P: http://www.wisdom.weizmann.ac.il/%7Eoded/PDF/bpp.pdf\n- the presentation: http://www.wisdom.weizmann.ac.il/%7Eoded/T/bpp.ppt\n- P=BPP unless E has sub-exponential circuits: derandomizing the XOR Lemma: http://www.math.ias.edu/%7Eavi/PUBLICATIONS/MYPAPERS/IW97/proc.pdf\n- Towards NEXP versus BPP?: http://www.stanford.edu/%7Errwill/nexp-v-bpp.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"P vs. BPP\" in Theoretical Computer Science; Complexity; Derandomization, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** Whether P equals BPP remains open; known unconditional derandomization falls short of polynomial time.\n\n**Verified partial progress.**\n\n- Conditional hardness-vs-randomness frameworks yield strong derandomizations from circuit lower bounds.\n- Unconditional quasipolynomial deterministic simulations are known.\n\n**Full solution or refutation.**\n\nNo unconditional polynomial-time deterministic simulation of BPP was verified.\n\n**What remains.**\n\nProve P=BPP or separate the two classes by an unconditional lower-bound method.\n\n**Sources checked.**\n\n- Complexity Zoo, BPP (accessed 2026-08-17). (authoritative_secondary): https://complexityzoo.net/Complexity_Zoo:B#bpp\n Evidence used: Records P versus BPP as an open question and summarizes containment/derandomization facts.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3406, "problem_number": "OPG-36884", "title": "Refuting random 3SAT-instances on $O(n)$ clauses (weak form)", "statement": "Conjecture For every rational $\\epsilon > 0$ and every rational $\\Delta$, there is no polynomial-time algorithm for the following problem.\n\nGiven is a 3SAT (3CNF) formula $I$ on $n$ variables, for some $n$, and $m = \\floor{\\Delta n}$ clauses drawn uniformly at random from the set of formulas on $n$ variables. Return with probability at least 0.5 (over the instances) that $I$ is typical without returning typical for any instance with at least $(1 - \\epsilon)m$ simultaneously satisfiable clauses.", "background": "Source: Open Problem Garden. Original node ID: 36884. URL: http://www.openproblemgarden.org/op/refuting_random_3sat_instances_on_o_n_clauses_weak_form.\n\nSource subject path: Theoretical Computer Science > Complexity > Hardness of Approximation.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/refuting_random_3sat_instances_on_o_n_clauses_weak_form\n- Author(s): Feige, Uriel\n- Subject(s): Theoretical Computer Science; Complexity; Hardness of Approximation\n- Keywords: NP; randomness in TCS; satisfiability\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: February 27th, 2009 by cwenner\n\nProblem-page discussion:\nThis conjecture was presented in Average Case Complexity and Approximation Complexity by Uriel Feige as a new approach for showing inapproximabiltiy results $^\\text{[F02]}$. The conjecture is strong in that it immediately implies the optimal $8/7 - \\epsilon$-hardness of approximation of 3SAT, something shown NP-hard in 1997 with heavy applications of PCP-techniques $^\\text{[H97]}$.\n\nThe strong and weak form\n\nThe weak form of Feige's conjecture is implied by the strong form of the conjecture ( $\\epsilon = 0$ ) and therefore subjectively more likely to be true. In the weak form, the choice of ambiguities of the uniform distribution (such as choosing clauses with or without replacement) may affect the parameters of the conjecture but not the truth.\n\nSupport for and against the conjecture\n\nIf the number of clauses is large enough ( $m \\in \\Omega(n^{1.5})$ ), then the problem defined above can be solved in polynomial time $^\\text{[FO06]}$. It is believed that there is a phase transition $\\Delta_c$ in the probability of satisfying a random 3SAT instance such that for every $\\Delta$ sufficiently smaller than $\\Delta_c$, only an inverse exponential number of 3SAT instances with $m = \\Delta n$ clauses are not satisfiable; and for every $\\Delta$ sufficiently larger than $\\Delta_c$, only an inverse exponential number of 3SAT instances with $m = \\Delta n$ clauses are satisfiable. Around $\\Delta_c$, it is also believed that deciding satisfiability of instances with about $\\Delta_c n$ clauses is difficult. Feige's conjecture plausibly implies the conjecture about the hardness around $\\Delta_c$, even if $\\Delta_c$ depends on $n$. The converse does not necessarily hold, that is, hardness of deciding 3SAT at $\\Delta_c$ implying hardness of the above problem for every $\\Delta$ $^\\text{[F99]}$. (expand this section)\n\nValue of the conjecture\n\nBy assuming that this conjecture holds, a number of inapproximability results have been derived for problems that have so far resisted attacks by other conjectures and techniques such as the unique games conjecture and probabilistically checkable proofs. If approximating a problem within $f(n)$ implies that Feige's conjecture is false, then the problem is said to be R3SAT-hard to approximate within $f(n)$. It has been shown that it is R3SAT-hard to approximate Maximum Balanced Bipartite Clique for some $\\delta > 0$ within $n^{-\\delta}$, Minimum Bisection below $4/3$, Dense k-Subgraph within some constant greater than 1, and the 2-Catalog Problem below some constant greater than 1. Showing either of these results without assuming the conjecture (e.g. NP-hardness) or improving the results assuming the conjecture are also open problems (of supposed medium importance for good enough improvements).\n\nA result proving the problem defined in the conjecture true under more plausible conjectures, e.g. P $\\neq$ NP, might show the way for a host of similar results (e.g. further reductions and similar extensions for other classes), add another technique to our repertoire, and greatly expand the area studying the relation between average-case hardness and approximability hardness.\n\nUsing so-called quasi-random PCPs, Subhash Khot has shown that neither of the three above problems admit a PTAS under the assumption that $NP \\nsubseteq BPTIME(2^{n^{o(1)}})$ $^\\text{[K04]}$.\n\nRelated Problems\n\n- The phase transition of random 3SAT instances (high importance) $^\\text{[F99]}$.\n\n- Alakhnovich's conjectures about hardness of linear systems and code words (medium importance) $^\\text{[A03]}$.\n\nBibliography:\n[A03] Michael Alakhnovich. More on average case vs approximation complexity. FOCS 2003. http://www.math.ias.edu/~misha/papers/average.ps\n\n*[F02] Uriel Feige. Relations between Average Case Complexity and Approximation Complexity. STOC 2002. http://citeseer.ist.psu.edu/old/feige02relations.html\n\n[F99] Ehud Friedgut. Necessary and sufficient conditions for sharp thresholds of graph properties and the $k$-SAT problem. Journal of the Amercian Mathematical Society, 1999.\n\n[FO06] Uriel Feige, Eran Ofek. Easily refutable subformulas of large random 3CNF formulas. Theory of Comuting, Volume 3 (2007). Pages 25 through 43.\n\n[H97] Johan Håstad. Some optimal inapproximability results. STOC, Proceedings of the 29th Annual ACM Symposium on Theory of Computing, 1997. http://www.nada.kth.se/~johanh/optimalinap.ps\n\n[K04] Subhash Khot. Ruling Out PTAS for Graph Min-Bisection, Densest Subgraph and Bipartite Clique. FOCS, Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2004. http://www.cc.gatech.edu/~khot/papers/mdc-bc.ps\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 26.\n\nAttempt notes:\nTarget:\nMake progress on \"Refuting random 3SAT-instances on $O(n)$ clauses (weak form)\" in Theoretical Computer Science; Complexity; Hardness of Approximation, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Polynomial random-3SAT refutation is known at sufficiently high superlinear clause counts, but the conjectured impossibility at every fixed linear density remains open.\n\n**Verified partial progress.**\n\n- Feige--Ofek give polynomial refutation for m in Omega(n^1.5).\n- Later work studies weaker refutation tradeoffs at lower densities.\n\n**Full solution or refutation.**\n\nNo polynomial algorithm or unconditional lower bound resolves the stated all-Delta linear regime.\n\n**What remains.**\n\nBridge the clause-density gap or prove an average-case lower bound under an explicit complexity assumption.\n\n**Sources checked.**\n\n- Open Problem Garden, node 36884 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/refuting_random_3sat_instances_on_o_n_clauses_weak_form\n Evidence used: Documents the linear-density conjecture and the known Omega(n^1.5) algorithmic regime.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 15, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3407, "problem_number": "OPG-751", "title": "S(S(f)) = S(f) for reloids", "statement": "Question $S(S(f)) = S(f)$ for every endo-reloid $f$?", "background": "Source: Open Problem Garden. Original node ID: 751. URL: http://www.openproblemgarden.org/op/s_s_f_s_f_for_reloids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/s_s_f_s_f_for_reloids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: reloid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 10th, 2008 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology, especially Connectedness of funcoids and reloids for definitions of used concepts.\n\nBibliography:\n*Victor Porton. Algebraic General Topology\n\nRelated:\nRelated problems\nS(S(f)) = S(f) for funcoids\n\nSource links:\n- reloid: http://www.wikinfo.org/index.php/Reloid\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n- Connectedness of funcoids and reloids: http://www.mathematics21.org/binaries/connectedness.pdf\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComments:\n- May 8th, 2008 | porton | Mistake fixed: There were a big mistake in http://www.mathematics21.org/binaries/connectedness.pdf where are defined some of the concepts used by this open problem. I have rewritten the article, not it should be OK.\n\n--\n\nVictor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"S(S(f)) = S(f) for reloids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No precise status can be assigned because the source supplies neither a definition of reloid nor of S, and an accessible related formula concerns S* rather than the displayed S.\n\n**Verified partial progress.**\n\n- The exact source text is retained for later recovery.\n\n**Full solution or refutation.**\n\nNo solution or counterexample was verified for a uniquely defined proposition.\n\n**What remains.**\n\nRecover authoritative definitions and check whether the missing star or hypotheses are a source-extraction defect.\n\n**Sources checked.**\n\n- Open Problem Garden, node 751 (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/s_s_f_s_f_for_reloids\n Evidence used: Retains the short original prompt but not enough definitions for a reliable status search.\n\n**Review notes.** Formulation ambiguity flagged; source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3408, "problem_number": "OPG-757", "title": "Inscribed Square Problem", "statement": "Conjecture Does every Jordan curve have 4 points on it which form the vertices of a square?", "background": "Source: Open Problem Garden. Original node ID: 757. URL: http://www.openproblemgarden.org/op/inscribed_square_problem.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/inscribed_square_problem\n- Author(s): Toeplitz\n- Subject(s): Topology\n- Keywords: simple closed curve; square\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: April 10th, 2008 by dlh12\n\nProblem-page discussion:\nA Jordan curve is a continuous function $f$ from the closed interval $[0,1]$ to the plane $\\mathbb{R}^{2}$ with the properties that $f$ is injective on the half-open interval $[0,1)$ (i.e., $f$ is simple) and $f(0)=f(1)$ (i.e., $f$ is closed).\n\nBibliography:\n[M] Meyerson, M.D., Equilateral triangles and continuous curves, Fund. Math. 110, (1980), 1--9.\n\nSource links:\n- Jordan curve: http://en.wikipedia.org/wiki/Jordan curve\n\nComments:\n- February 25th, 2021 | Anonymous | Already solved: This was proven in https://arxiv.org/pdf/2005.09193.pdf\n- March 3rd, 2010 | Anonymous | inscription of squares in simple closed curves: There is a theorem that says:\n\nin all simple closed curves there are 4n points that are vertex of n squares (inf = > n > =1)\n\nJorge Pasin.\n- November 8th, 2010 | Anonymous | in all simple closed curves there are 4n points: Would you clarify? An obtuse triangle has only one inscribed square, so this theorem is not true for n>=2. Do you have a reference to this theorem? Strashimir Popvassilev\n- November 25th, 2009 | Anonymous | Is the conjecture known to: Is the conjecture known to be true for C^1-smooth curves?\n- June 7th, 2010 | Anonymous | Yes: Yes. Walter Stromquist, Inscribed squares and square-like quadrilaterals in closed curves, Mathematika 36: 187-197 (1989).\n- June 2nd, 2010 | Anonymous | no.: If it were true for C^1 curves, then since a Jordan curve is compact, it may be weierstrass approximated by a series of C^1 curves (indeed by curves whose component functions are polynomials) such that the series converges uniformly to the given jordan curve. Then by assumption, each curve in the sequence contains 4 points forming a square, and the sequence of squares can be regarded as (eventually) a sequence in the (sequentially) compact space of the 4-fold product of any closed epsilon enlargement of the area bounded by the original jordan curve. It follows that the sequence of squares contains a convergent subsequence, which can be shown to be a square lying on the original jordan curve.\n\nThus, proving the C^1 case proves the general case.\n- June 7th, 2010 | Anonymous | This is flawed: The approximation argument is flawed: the squares on approximating curves may have sides decreasing to 0, in which case the limiting \"square\" degenerates to a point. In fact, Stromquist's theorem covers a much wider class of curves than C^1, but not all continuous curves.\n- April 28th, 2008 | Anonymous | Quantifier: Phrasing should be changed from \"Does any...\" to \"Does every...\"\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Inscribed Square Problem\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The square-peg problem remains open for arbitrary Jordan curves, but several substantial regularity and geometric subclasses are now proved.\n\n**Verified partial progress.**\n\n- Chambers proves a near-C2 Jordan-curve theorem.\n- Greene--Lobb prove the two-Lipschitz-graph class with explicit Lipschitz threshold.\n\n**Full solution or refutation.**\n\nNo proof for every continuous Jordan curve was verified.\n\n**What remains.**\n\nHandle arbitrary continuous Jordan curves without the available regularity/graph assumptions.\n\n**Sources checked.**\n\n- G. Chambers, On the square peg problem, arXiv:2203.02613 (2022). (primary): https://arxiv.org/abs/2203.02613\n Evidence used: Proves a robust near-C2 case.\n- J. E. Greene and A. Lobb, Square pegs between two graphs, arXiv:2407.07798 (2024). (primary): https://arxiv.org/abs/2407.07798\n Evidence used: Proves a broad two-graph Lipschitz class.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3409, "problem_number": "OPG-1783", "title": "Rank vs. Genus", "statement": "Question Is there a hyperbolic 3-manifold whose fundamental group rank is strictly less than its Heegaard genus? How much can the two differ by?", "background": "Source: Open Problem Garden. Original node ID: 1783. URL: http://www.openproblemgarden.org/op/rank_vs_genus.\n\nSource subject path: Topology.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/rank_vs_genus\n- Author(s): Johnson, Jesse\n- Subject(s): Topology\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 7th, 2008 by Jesse Johnson\n\nProblem-page discussion:\nThe rank of a 3-manifold is the minimal number of generators needed for its fundamental group. The Heegaard genus is the smallest genus of all Heegaard splittings for that 3-manifold. A Heegaard splitting determines a generating set for the 3-manifold, so the ranks is always less than or equal to the genus.\n\nThere is a family of Seifert fibered spaces for which the rank is one less than the genus, but for most Seifert fibered spaces, the rank and genus are equal. The Seifert fibered exampels have been used to construct graph manifolds for which the rank and genus differ by more than one [1]. However, there are no hyperbolic 3-manifolds for which rank and genus are known to differ.\n\nBibliography:\nSchultens, Jennifer, Weidman, Richard, On the geometric and the algebraic rank of graph manifolds. Pacific J. Math. 231 (2007), no. 2, 481--510.\n\nComments:\n- July 13th, 2008 | Anonymous | Connection to dynamics: Abert and Nikolov have found a connection between the `Rank vs Heegard Genus' problem and the `Fixed Price' problem in dynamics. Specifically, if every countable group has `fixed price' then the ratio of the Heegard genus and the rank of a hyperbolic 3-manifold can be arbitrarily large. For details, see http://arxiv.org/abs/math/0701361.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Rank vs. Genus\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** Tao Li constructed a closed orientable hyperbolic 3-manifold whose fundamental-group rank is smaller than its Heegaard genus, and proved that the additive discrepancy can be arbitrarily large.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nSuch hyperbolic 3-manifolds exist, and there is no universal finite upper bound on Heegaard genus minus fundamental-group rank.\n\n**What remains.**\n\nThe literal existence and additive-discrepancy questions are closed. Questions about an unbounded ratio, rank-two manifolds, or restricted classes are separate refinements.\n\n**Sources checked.**\n\n- Tao Li, Rank and genus of 3-manifolds, Journal of the American Mathematical Society 26 (2013), 777–829, DOI 10.1090/S0894-0347-2013-00767-5; arXiv:1106.6302. (primary): https://arxiv.org/abs/1106.6302\n Evidence used: The abstract and Theorem 1.1 explicitly construct a closed orientable hyperbolic example and state that rank–genus discrepancy can be arbitrarily large.\n\n**Review notes.** The source background predates the 2011 preprint and 2013 journal publication.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3410, "problem_number": "OPG-37123", "title": "Smooth 4-dimensional Schoenflies problem", "statement": "Problem Let $M$ be a $3$-dimensional smooth submanifold of $S^4$, $M$ diffeomorphic to $S^3$. By the Jordan-Brouwer separation theorem, $M$ separates $S^4$ into the union of two compact connected $4$-manifolds which share $M$ as a common boundary. The Schoenflies problem asks, are these $4$-manifolds diffeomorphic to $D^4$? ie: is $M$ unknotted?", "background": "Source: Open Problem Garden. Original node ID: 37123. URL: http://www.openproblemgarden.org/op/smooth_4_dimensional_schoenflies_problem.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/smooth_4_dimensional_schoenflies_problem\n- Author(s): Alexander, J\n- Subject(s): Topology\n- Keywords: 4-dimensional; Schoenflies; sphere\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 6th, 2009 by rybu\n\nProblem-page discussion:\nBy the work of Mike Freedman, $M$ separates $S^4$ into two manifolds which are homeomorphic to $D^4$. So the Schoenflies problem is only non-trivial if $D^4$ admits an exotic smooth structure, which is also an open problem. Although $D^4$ could very well have an exotic smooth structure and yet the Schoenflies problem could have a positive answer. ie: although exotic smooth $D^4$ 's might exist, perhaps none of them embed in $S^4$?\n\nMartin Scharlemann has results to the effect that the Schoenflies problem is true provided the embeddings are simple enough.\n\nThe smooth Poincare conjecture in dimension 4 is related but disjoint from this problem. For example, the Poincare conjecture could be true and $D^4$ could have an exotic smooth structure -- this would amount to saying the monoid of smooth homotopy 4-spheres has some elements with inverses.\n\nThe analogous problem in other dimensions is known to be true. Namely, all embeddings of $S^n$ in $S^{n+1}$ are unknotted (bound manifolds diffeomorphic to $D^{n+1}$ ) provided $n \\neq 3$. For $n=1$ this is due to Schoenflies. For $n=2$ it's due to Alexander (see Hatcher's 3-manifolds notes for a modern exposition). For $n \\geq 4$ the result follows from the combination of the Mazur-Brown theorem that an embedding of $S^n$ in $S^{n+1}$ bounds a manifold homeomorphic to $D^{n+1}$, plus a consequence of the H-cobordism theorem which states that $D^{n+1}$ has no exotic smooth structures which restrict to the standard smooth structure on the boundary, provided $n \\geq 4$.\n\nBibliography:\n*[A] Alexander, J, On the subdivision of space by a polyhedron. Proc. Nat. Acad. Sci. USA 10 (1924) pg 6--8.\n\n[FQ] Freedman, M. Quinn, F. Topology of 4-manifolds. Princeton University Press.\n\n[S1] Scharlemann, M. The four-dimensional Schoenflies conjecture is true for genus two imbeddings. Topology 23 (1984) 211-217.\n\n[S2] Scharlemann, M. Smooth Spheres in R4 with four critical points are standard. Inventiones Math. 79 (1985) 125-141.\n\n[S3] Scharlemann, M. Generalized Property R and the Schoenflies Conjecture, Commentarii Mathematici Helvetici, 83 (2008) 421--449.\n\n[MMEB] Marston Morse and Emilio Baiada, Homotopy and Homology Related to the Schoenflies Problem. The Annals of Mathematics, Second Series, Vol. 58, No. 1 (Jul., 1953), pp. 142-165\n\n[B] Brown, Morton. A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc., vol. 66, pp. 74–76. (1960)\n\n[MAZ] Mazur, Barry, On embeddings of spheres., Bull. Amer. Math. Soc. 65 (1959) 59--65.\n\n[H] Hatcher, A. 3-manifolds notes. [http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html]\n\nRelated:\nRelated problems\nSmooth 4-dimensional Poincare conjecture\nWhat is the homotopy type of the group of diffeomorphisms of the 4-sphere?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 13.\n\nAttempt notes:\nTarget:\nMake progress on \"Smooth 4-dimensional Schoenflies problem\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The smooth four-dimensional Schoenflies conjecture remains open, although restricted embedding classes and related constructed families are known to be standard.\n\n**Verified partial progress.**\n\n- Scharlemann proves the conjecture for genus-two embeddings and for smooth 3-spheres in R^4 with four critical points.\n- Gabai develops a pseudo-isotopy approach, and recent work constructs and controls additional Schoenflies-ball families.\n- Hass and Kirby record that the smooth four-dimensional Poincaré conjecture would imply the smooth Schoenflies conjecture.\n\n**Full solution or refutation.**\n\nNo theorem covering every smooth embedding S^3 -> S^4 was verified.\n\n**What remains.**\n\nShow both complementary smooth homotopy 4-balls are standard for an arbitrary smooth embedded 3-sphere, or find a smoothly knotted embedding.\n\n**Sources checked.**\n\n- J. Hass and R. Kirby, Characterizing the 4-sphere, S^4, Journal of Open Mathematical Problems 1 (2025), 52-68. (authoritative_secondary): https://jomprob.org/index.php/jomp/article/view/Vol-1Issue-1Paper-3\n Evidence used: Surveys the open status, states the Schoenflies formulation, and proves the implication from smooth 4D Poincaré.\n- D. Gabai, 3-Spheres in the 4-Sphere and Pseudo-Isotopies of S^1 x S^3, arXiv:2212.02004 (2022). (primary): https://arxiv.org/abs/2212.02004\n Evidence used: Develops a current approach to the smooth 4D Schoenflies conjecture rather than claiming a full resolution.\n- Open Problem Garden, Smooth 4-dimensional Schoenflies problem (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/smooth_4_dimensional_schoenflies_problem\n Evidence used: Records the special Scharlemann cases and the unresolved general problem.\n\n**Review notes.** The source's informal 'ie' wording was preserved rather than normalized.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3411, "problem_number": "OPG-37125", "title": "Smooth 4-dimensional Poincare conjecture", "statement": "Conjecture If a $4$-manifold has the homotopy type of the $4$-sphere $S^4$, is it diffeomorphic to $S^4$?", "background": "Source: Open Problem Garden. Original node ID: 37125. URL: http://www.openproblemgarden.org/op/smooth_4_dimensional_poincare_conjecture.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/smooth_4_dimensional_poincare_conjecture\n- Author(s): Poincare; Smale; Stallings\n- Subject(s): Topology\n- Keywords: 4-manifold; poincare; sphere\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 6th, 2009 by rybu\n\nProblem-page discussion:\nThe original Poincare conjecture was the assertion that a simply-connected compact boundaryless $3$-manifold is diffeomorphic (smooth Poincare conjecture) or homeomorphic (topological Poincare conjecture) to $S^3$. Because of Poincare duality, this is equivalent to the assertion that a $3$-manifold has the homotopy-type of $S^3$ then it is diffeomorphic/homeomorphic to $S^3$. This gave birth to the generalized Poincare conjecture -- that an $n$-manifold with the homotopy type of $S^n$ is diffeomorphic or homeomorphic to $S^n$.\n\nBy the work of Smale and Stallings, the topological Poincare conjecture was shown to be true provided $n \\geq 5$. But for $n \\geq 7$ Milnor and Kervaire showed that $S^n$ admits non-standard smooth structures so the smooth Poincare conjecture is false in general.\n\nThe generalized Poincare conjecture is an undergraduate-level point-set topology problem for $n=1$.\n\nThe $n=2$ case was proven by Poincare.\n\nThe $n=3$ case was recently proven by Perelman.\n\nThe $n=4$ case is the only outstanding case. Mike Freedman has proven that a $4$-manifold which is homotopy-equivalent to $S^4$ is homeomorphic to $S^4$, so the smooth 4-dimensional Poincare conjecture is the only outstanding problem among the generalized Poincare conjectures. Moreover, it can be considered to be reduced to the question of if $S^4$ has an exotic smooth structure.\n\nTechnically, Poincare never asserted this conjecture. He only stated it was an interesting problem. So perhaps it should be called Poincare's Egregious Problem.\n\nIt is unknown whether or not $D^4$ admits an exotic smooth structure. If not, the smooth $4$-dimensional Poincare conjecture would have an affirmative answer. Similarly, it's known that $4$-dimensional euclidean space $\\mathbb R^4$ admits a continuum of pairwise non-diffeomorphic smooth structures. But it's unknown whether or not any of these exotic smooth structures extend to $D^4$, thought of as a compactification of $\\mathbb R^4$.\n\nBibliography:\n[FQ] Freedman, M. Quinn, F. Topology of 4-manifolds. Princeton University Press.\n\n[S] Smale, S., \"On the structure of manifolds\" Amer. J. Math., 84 (1962) pp. 387–399\n\n[MT] Morgan, John W.; Gang Tian. Ricci Flow and the Poincaré Conjecture. AMS/CMI (2009)\n\nRelated:\nRelated problems\nSmooth 4-dimensional Schoenflies problem\n\nComments:\n- August 26th, 2021 | Anonymous | Poincare conj is true in some dimensions > 7.: For n equals 12, 56 and 61, Sn also has a unique smooth structure.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Smooth 4-dimensional Poincare conjecture\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The smooth four-dimensional Poincaré conjecture remains open; topological standardness and substantial families of candidate smooth homotopy spheres are settled.\n\n**Verified partial progress.**\n\n- Freedman's theorem implies every homotopy 4-sphere is homeomorphic to S^4, leaving only the smooth classification.\n- Cha and Kim prove that all Calegari homotopy 4-spheres arising from fibered-knot monodromies are diffeomorphic to S^4.\n- Many Cappell-Shaneson and other proposed candidate families are known to be standard, but not every general construction has been eliminated.\n\n**Full solution or refutation.**\n\nNo accepted proof that every smooth homotopy 4-sphere is diffeomorphic to S^4, and no exotic smooth 4-sphere, was verified.\n\n**What remains.**\n\nProve smooth standardness for every closed smooth homotopy 4-sphere or construct and distinguish an exotic smooth structure on S^4.\n\n**Sources checked.**\n\n- J. Hass and R. Kirby, Characterizing the 4-sphere, S^4, Journal of Open Mathematical Problems 1 (2025), 52-68. (authoritative_secondary): https://jomprob.org/index.php/jomp/article/view/Vol-1Issue-1Paper-3\n Evidence used: A current specialist survey explicitly presents smooth 4D Poincaré as open and reviews candidate families.\n- J. C. Cha and M. H. Kim, Calegari's homotopy 4-spheres from fibered knots are standard, arXiv:2411.10051 (2024). (primary): https://arxiv.org/abs/2411.10051\n Evidence used: Theorem A proves every Calegari sphere from a fibered-knot monodromy is standard and explicitly identifies broader cases that remain open.\n\n**Review notes.** Unreviewed claimed solutions were not treated as established. The conventional intended object is a closed smooth homotopy 4-sphere; the short source wording does not spell out every standard manifold convention.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3412, "problem_number": "OPG-37129", "title": "Slice-ribbon problem", "statement": "Conjecture Given a knot in $S^3$ which is slice, is it a ribbon knot?", "background": "Source: Open Problem Garden. Original node ID: 37129. URL: http://www.openproblemgarden.org/op/slice_ribbon_problem_0.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/slice_ribbon_problem_0\n- Author(s): Fox, R\n- Subject(s): Topology\n- Keywords: cobordism; knot; ribbon; slice\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nThe definitions of slice and ribbon: slice knot ribbon knot\n\nThere is a fairly vast literature on this problem. It is closely related to the problem of determining which homology $3$-spheres bound homology $4$-balls, as both are in essence a type of $4$-dimensional cobordism problem.\n\nBibliography:\n*[F] Fox, R. H. Some problems in knot theory. 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 168--176\n\n[G] Gilmer, Patrick M. On the slice genus of knots. Invent. Math. 66 (1982), no. 2, 191--197.\n\n[H] Hass, Joel. The geometry of the slice-ribbon problem. Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 1, 101--108.\n\nAna G. Lecuona. [arXiv:0910.4601] On the Slice-Ribbon Conjecture for Montesinos knots\n\nBrendan Owens. [arXiv:0802.2109] On slicing invariants of knots.\n\nRelated:\nRelated problems\nWhich homology 3-spheres bound homology 4-balls?\n\nDiscussion links:\n- slice knot: http://en.wikipedia.org/wiki/slice knot\n- ribbon knot: http://en.wikipedia.org/wiki/ribbon knot\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Slice-ribbon problem\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The slice-ribbon conjecture remains open in general, while many knot families, including substantial Montesinos/arborescent families, have been settled affirmatively.\n\n**Verified partial progress.**\n\n- Recent rational-homology-ball constructions verify the conjecture for further complicated arborescent examples.\n\n**Full solution or refutation.**\n\nNo accepted general implication from smooth sliceness to ribbonness was verified.\n\n**What remains.**\n\nProve the implication for all smooth slice knots or give a smooth slice non-ribbon knot.\n\n**Sources checked.**\n\n- L. Lokteva, Constructing Rational Homology 3-Spheres That Bound Rational Homology 4-Balls, arXiv:2208.14850 (2022). (primary): https://arxiv.org/abs/2208.14850\n Evidence used: The abstract describes new arborescent/Montesinos instances and says the general operations remain open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3413, "problem_number": "OPG-37131", "title": "Realisation problem for the space of knots in the 3-sphere", "statement": "Problem Given a link $L$ in $S^3$, let the symmetry group of $L$ be denoted $Sym(L) = \\pi_0 Diff(S^3,L)$ ie: isotopy classes of diffeomorphisms of $S^3$ which preserve $L$, where the isotopies are also required to preserve $L$.\n\nNow let $L$ be a hyperbolic link. Assume $L$ has the further `Brunnian' property that there exists a component $L_0$ of $L$ such that $L \\setminus L_0$ is the unlink. Let $A_L$ be the subgroup of $Sym(L)$ consisting of diffeomorphisms of $S^3$ which preserve $L_0$ together with its orientation, and which preserve the orientation of $S^3$.\n\nThere is a representation $A_L \\to \\pi_0 Diff(L \\setminus L_0)$ given by restricting the diffeomorphism to the $L \\setminus L_0$. It's known that $A_L$ is always a cyclic group. And $\\pi_0 Diff(L \\setminus L_0)$ is a signed symmetric group -- the wreath product of a symmetric group with $\\mathbb Z_2$.\n\nProblem: What representations can be obtained?", "background": "Source: Open Problem Garden. Original node ID: 37131. URL: http://www.openproblemgarden.org/op/realisation_problem_for_the_space_of_knots_in_the_3_sphere.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/realisation_problem_for_the_space_of_knots_in_the_3_sphere\n- Author(s): Budney, R\n- Subject(s): Topology\n- Keywords: knot space; symmetry\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nAn answer to this problem would give a `closed form' description of the homotopy type of the space of smooth embeddings of $S^1$ in $S^3$. This is the space of embeddings in the Whitney Topology, or $C^k$-uniform topology for any $k \\geq 1$.\n\n`Closed form' means that every component of $Emb(S^1,S^3)$ would have the description as an iterated fiber bundle over certain well-known spaces, where the fibers are inductively well-known spaces, and the monodromy would be controlled rather explicitly by this list of representations.\n\nPeripherally related are various other realization problems for $3$-manifolds. For example, Sadayoshi Kojima proved that one can realize any finite group as the group of isometries of a hyperbolic $3$-manifold.\n\nBibliography:\n*[B] Budney, R. Topology of spaces of knots in dimension 3, to appear in Proc. Lond. Math. Soc.\n\n[B2] Budney, R. A family of embedding spaces. Geometry and Topology Monographs 13 (2007).\n\n[K] Kojima, S., Isometry transformations of hyperbolic $3$-manifolds. Topology Appl. 29 (1988), no. 3, 297--307.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Realisation problem for the space of knots in the 3-sphere\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Budney's knot-space work develops the associated realization question and recursive component calculations, but no source establishing a complete classification of the representations exactly asked for was verified.\n\n**Verified partial progress.**\n\n- The homotopy type of many knot-space components is computable recursively from splicing/JSJ data.\n\n**Full solution or refutation.**\n\nNo verified all-representations theorem was located.\n\n**What remains.**\n\nFix precise equivalence and link conventions, then locate or prove a realization classification for the cyclic actions.\n\n**Sources checked.**\n\n- R. Budney, Topology of spaces of knots in dimension 3, Proc. London Math. Soc. 101 (2010), 477--496. (primary): https://doi.org/10.1112/plms/pdq020\n Evidence used: The paper develops a recursive computation of knot-space homotopy types and identifies the related hyperbolic-knot symmetry realization problem.\n\n**Review notes.** Source statement preserved; representation equivalence conventions need expert checking.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3414, "problem_number": "OPG-37145", "title": "Which homology 3-spheres bound homology 4-balls?", "statement": "Problem Is there a complete and computable set of invariants that can determine which (rational) homology $3$-spheres bound (rational) homology $4$-balls?", "background": "Source: Open Problem Garden. Original node ID: 37145. URL: http://www.openproblemgarden.org/op/which_homology_3_spheres_bound_homology_4_balls.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/which_homology_3_spheres_bound_homology_4_balls\n- Author(s): Ancient/folklore\n- Subject(s): Topology\n- Keywords: cobordism; homology ball; homology sphere\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nDetermining which homology $3$-spheres bound homology $4$-balls is a long standing open problem in 3/4-manifold topology. Much effort has gone towards understanding the situation for the Brieskorn homology spheres. For example, the Poincare Dodecahedral space is known not to bound a homology $4$-ball since the Rochlin invariant is non-trivial -- but $M\\#(-M)$ the connect-sum of Poincare Dodecahedral space $M$ with its orientation-reverse does bound a homology 4-ball, and it has a simple construction: remove an open tubular neighbourhood of $\\{*\\} \\times I$ from $M \\times I$, this is the $4$-manifold.\n\nStandard invariants used to show homology $3$-spheres do not bound homology $4$-balls are various spin or spin^c cobordism invariants such as: the Rochlin invariant, Siebenmann's $\\overline{\\mu}$-invariant, the Oszvath-Szabo $d$-invariant, and there are many others.\n\nBibliography:\n[K] Kirby, Robion (1989), The topology of 4-manifolds, Lecture Notes in Mathematics, 1374, Springer-Verlag,\n\n[R] Rokhlin, Vladimir A, New results in the theory of four-dimensional manifolds, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221-224.\n\n[AK] S.Akbulut, R.Kirby, \"Mazur manifolds,\" Michigan Math. J. 26 (1979), 259--284.\n\n[CH] A.Casson, J.Harer, \"Some homology lens spaces which bound rational homology balls.\" Pacific. J. Math. Vol 96, No 1, (1981) 23–36.\n\n[F] H.Fickle, \"Knots, Z-Homology 3-spheres and contractible 4-manifolds,\" pp. 467--493, Houston J. Math. Vol 10, No. 4 (1984).\n\n[FS] R.Fintushel, R.Stern, \"An exotic free involution on S^4,\" Ann. Math. (2) 113 (1981) no2, 357--365.\n\n[M] B.Mazur, \"A note on some contractible 4-manifolds\", Annals of Mathematics, (2) 73 (1961). 221–228.\n\n[S] R.Stern,\"Some Brieskorn spheres which bound contractible manifolds,\" Notices Amer. Math. Soc 25 (1978), A448.\n\n[L] Lisca, Paolo Sums of lens spaces bounding rational balls. Algebr. Geom. Topol. 7 (2007), 2141--2164.\n\nRelated:\nRelated problems\nSlice-ribbon problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Which homology 3-spheres bound homology 4-balls?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** There is no verified complete computable classification for all rational homology 3-spheres, but spherical rational homology 3-spheres have a complete smooth rational-ball classification.\n\n**Verified partial progress.**\n\n- Choe--Park classify spherical 3-manifolds that bound smooth rational homology 4-balls using Donaldson and Heegaard-Floer obstructions.\n\n**Full solution or refutation.**\n\nThe broad classification problem remains open beyond such classes.\n\n**What remains.**\n\nDevelop complete effective invariants for arbitrary rational homology 3-spheres.\n\n**Sources checked.**\n\n- D. Choe and K. Park, Spherical 3-manifolds bounding rational homology balls, arXiv:1803.08749 (2019). (primary): https://arxiv.org/abs/1803.08749\n Evidence used: The abstract states a complete classification for spherical 3-manifolds.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3415, "problem_number": "OPG-37151", "title": "Fundamental group torsion for subsets of Euclidean 3-space", "statement": "Problem Does there exist a subset of $\\mathbb R^3$ such that its fundamental group has an element of finite order?", "background": "Source: Open Problem Garden. Original node ID: 37151. URL: http://www.openproblemgarden.org/op/torsion_for_subsets_of_mathbb_r_3.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/torsion_for_subsets_of_mathbb_r_3\n- Author(s): Ancient/folklore\n- Subject(s): Topology\n- Keywords: subsets of euclidean space; torsion\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nThe corresponding problem in $\\mathbb R^2$ has a negative answer. The corresponding problem in $\\mathbb R^n$ for $n \\geq 4$ has a positive answer. If the subset of $\\mathbb R^3$ has a regular neighbourhood with a smooth boundary, the answer is negative. Similarly the homology of the subset is known to have no torsion via an Alexander duality argument. So any torsion in the fundamental group must be in the commutator subgroup.\n\nBibliography:\n[E] Eda, K. Fundamental group of subsets of the plane. Topology and its Applications Volume 84, Issues 1-3, 24 April 1998, Pages 283-306\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Fundamental group torsion for subsets of Euclidean 3-space\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No construction or impossibility proof was verified for arbitrary subsets of R^3; the unresolved possibility is necessarily wild, while open subsets and regular-neighborhood cases have torsion-free fundamental groups.\n\n**Verified partial progress.**\n\n- For open connected subsets of R^3, the fundamental group is torsion-free.\n- Regular-neighborhood/smooth-boundary settings have a negative answer.\n\n**Full solution or refutation.**\n\nThe unrestricted subset question remains unverified as solved.\n\n**What remains.**\n\nConstruct a wild subset with torsion in pi_1 or prove torsion-freeness for all subsets.\n\n**Sources checked.**\n\n- MathOverflow, Torsion in homology or fundamental group of subsets of Euclidean 3-space (accessed 2026-08-17). (authoritative_secondary): https://mathoverflow.net/questions/4478/torsion-in-homology-or-fundamental-group-of-subsets-of-euclidean-3-space\n Evidence used: The maintained expert discussion records the arbitrary-subset question and the known regular-neighborhood/open restrictions without a general resolution.\n\n**Review notes.** Wildness qualification retained; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3416, "problem_number": "OPG-37154", "title": "Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere?", "statement": "Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.", "background": "Source: Open Problem Garden. Original node ID: 37154. URL: http://www.openproblemgarden.org/op/which_compact_boundaryless_3_manifolds_embed_smoothly_in_the_4_sphere.\n\nSource subject path: Topology.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/which_compact_boundaryless_3_manifolds_embed_smoothly_in_the_4_sphere\n- Author(s): Kirby\n- Subject(s): Topology\n- Keywords: 3-manifold; 4-sphere; embedding\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nFor general 3-manifolds this problem is fairly wide-open. But for some specific families of 3-manifolds it is heavily investigated.\n\nThere are two common embedding constructions: (1) obtain your 3-manifold as 0-surgery on a link which is the disjoint union of two smooth slice links. (2) Obtain your 3-manifold as the boundary of a Mazur manifold -- where Mazur manifold is taken to be a contractible 4-manifold constructed as $S^1 \\times D^3$ union a 2-handle. In both cases the resulting 3-manifold M embeds smoothly in $S^4$. There are many other embedding constructions but no known \"uniform\" construction that works for all embeddable 3-manifolds.\n\nSince such a 3-manifold would bound two 4-manifolds on either side, the embedding problem is a type of double cobordism problem, and related to issues such as the problem of determining which homology 3-spheres bound homology 4-balls.\n\nThe smoothness in the assumption is important. Mike Freedman has proven all homology 3-spheres admit tame topological embeddings into $S^4$. These embeddings have a less combinatorial nature than smooth embeddings so it is somewhat natural to restrict to the question of smooth embeddings. For example, the Poincare Homology Sphere does not embed smoothly in $S^4$, since it has a non-trivial Rochlin invariant.\n\nBibliography:\n[B] R. Budney, Embeddings of 3-manifolds in the 4-sphere from the point of view of the $11$-tetrahedron census, arXiv preprint arXiv:0810.2346\n\n[CH] J.S. Crisp, J.A. Hillman, Embedding Seifert fibred $3$-manifolds and ${\\rm Sol\\sp 3$-manifolds in $4$-space,} Proc. London Math Soc. (3) (1998), no. {\\bf 3} 685--710.\n\n[KK] A.~Kawauchi, S.~Kojima, Algebraic classification of linking pairings on $3$-manifolds, Math. Ann. {\\bf 253} (1980), no. 1, 29--42.\n\n[FS] R.~Fintushel, R.~Stern, Rational homology cobordisms of spherical space forms, Topology, {\\bf 26} no. 3 pp. 385--393, (1987).\n\n[GL] P.M.~Gilmer, C.~Livingston, On embedding 3-manifolds in 4-space, Topology, {\\bf 22}, no. 3, pp. 241--252 (1983).\n\n*[K] Kirby, R. Problem list in low-dimensional topology. [http://math.berkeley.edu/~kirby/problems.ps.gz]\n\n[L] R.A.~Litherland, Deforming twist-spun knots, Trans. Amer. Math. Soc. {\\bf 250} (1979), 311--331.\n\nRelated:\nRelated problems\nWhich homology 3-spheres bound homology 4-balls?\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The smooth S^4 embedding classification is open generally, but census work supplies numerous constructions and obstructions: in the 11-tetrahedron orientable census, 41 candidates embed, 67 do not, and 37 remained unresolved in the published study.\n\n**Verified partial progress.**\n\n- Census, linking-form, and gauge/Floer obstructions decide many concrete 3-manifolds.\n\n**Full solution or refutation.**\n\nNo general computable necessary-and-sufficient invariant or construction procedure was verified.\n\n**What remains.**\n\nObtain a general decision/classification method, including construction of embeddings when they exist.\n\n**Sources checked.**\n\n- R. Budney and B. Burton, Embeddings of 3-manifolds in S^4 from the point of view of the 11-tetrahedron census, Experimental Mathematics 31 (2022), arXiv:0810.2346. (primary): https://arxiv.org/abs/0810.2346\n Evidence used: The abstract reports the concrete resolved and unresolved census counts and frames the general question as the main problem.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3417, "problem_number": "OPG-37159", "title": "What is the homotopy type of the group of diffeomorphisms of the 4-sphere?", "statement": "Problem $Diff(S^4)$ has the homotopy-type of a product space $Diff(S^4) \\simeq \\mathbb O_5 \\times Diff(D^4)$ where $Diff(D^4)$ is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of $Diff(D^4)$.", "background": "Source: Open Problem Garden. Original node ID: 37159. URL: http://www.openproblemgarden.org/op/what_is_the_homotopy_type_of_the_group_of_diffeomorphisms_of_the_4_sphere.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/what_is_the_homotopy_type_of_the_group_of_diffeomorphisms_of_the_4_sphere\n- Author(s): Smale, S.\n- Subject(s): Topology\n- Keywords: 4-sphere; diffeomorphisms\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\n$Diff(D^4$ ) is known to be a $5$-fold loop space. In particular there is a homotopy-equivalence known as the Cerf-Morlet Comparison theorem $Diff(D^n) \\simeq \\Omega^{n+1} (PL_n / O_n)$ where $PL_n$ is the group of PL-automorphisms of $\\mathbb R^n$ and $O_n$ is the group of linear automorphisms of $\\mathbb R^n$. Otherwise there is not much in the literature about $Diff(D^4)$. Since it is a group of diffeomorphisms it has the homotopy type of a countable CW-complex. It is unknown whether or not it is connected, or if it has any other non-trivial homotopy or homology groups.\n\n$Diff(S^n)$ is known to have the homotopy-type of $O_{n+1}$ provided $n \\leq 3$ by work of Hatcher and Smale respectively. For $n \\geq 5$ many of the groups $\\pi_0 Diff(S^n)$ were computed by Kervaire and Milnor, who further related these groups to the homotopy groups of spheres. For $n \\geq 7$ the rational homotopy groups of $Diff(D^n)$ have been computed by Farrell and Hsiang in range $0 \\leq i < \\min\\{\\frac{n-4}{3}, \\frac{n-7}{2} \\}$. They show $\\pi_i Diff(D^n) \\otimes \\mathbb Q \\simeq \\left\\{ \\begin{array}{lr} \\mathbb Q & \\text{ provided }\\ 4 | (i+1) \\\\ 0 & \\text{ otherwise } \\end{array} \\right.$.\n\nBibliography:\n[B] Budney, R. Little cubes and long knots. Topology. 46 (2007) 1--27.\n\n[FH] Farrell, F.T. Hsiang, W.C. On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds. Proc. Symp. Pure. Math. 32 (1977) 403--415.\n\n[H] Hatcher, A proof of a Smale conjecture, ${\\rm Diff}(S\\sp{3})\\simeq {\\rm O}(4)$. Ann. of Math. (2) 117 (1983), no. 3, 553--607.\n\n[KS] Kirby, R. Siebenmann, L. Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Princeton University Press.\n\n*[S] Smale, S. Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc. 10 (1959) 621--626.\n\nRelated:\nRelated problems\nSmooth 4-dimensional Schoenflies problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 19.\n\nAttempt notes:\nTarget:\nMake progress on \"What is the homotopy type of the group of diffeomorphisms of the 4-sphere?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Rational homotopy information for BDiff(D^4,partial) is now known through graph-homology lower bounds, but a complete calculation of the homotopy or homology groups was not verified.\n\n**Verified partial progress.**\n\n- Watanabe gives lower bounds for dimensions of pi_k(BDiff(D^4,partial)) tensor Q.\n\n**Full solution or refutation.**\n\nThe requested broad determination remains open despite genuine nontrivial rational-homotopy progress.\n\n**What remains.**\n\nCompute specific groups integrally or determine the full homotopy type/components.\n\n**Sources checked.**\n\n- T. Watanabe, Addendum to: Some exotic nontrivial elements of the rational homotopy groups of Diff(S^4), arXiv:2109.01609 (2021). (primary): https://arxiv.org/abs/2109.01609\n Evidence used: The abstract states lower bounds for pi_k(BDiff(D^4,partial)) tensor Q in terms of graph homology.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3418, "problem_number": "OPG-37161", "title": "Is there an algorithm to determine if a triangulated 4-manifold is combinatorially equivalent to the 4-sphere?", "statement": "Problem Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalent to the 4-sphere?", "background": "Source: Open Problem Garden. Original node ID: 37161. URL: http://www.openproblemgarden.org/op/is_there_an_algorithm_to_determine_if_a_triangulated_4_manifold_is_combinatorially_equivalent_to_the_4_sphere.\n\nSource subject path: Topology.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/is_there_an_algorithm_to_determine_if_a_triangulated_4_manifold_is_combinatorially_equivalent_to_the_4_sphere\n- Author(s): Novikov\n- Subject(s): Topology\n- Keywords: 4-sphere; algorithm\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: November 7th, 2009 by rybu\n\nProblem-page discussion:\nA 4-manifold triangulation admits a unique smoothing up to diffeomorphism, so this problem is equivalent to asking for an algorithm to determine if a 4-manifold is diffeomorphic to the 4-sphere (with standard differentiable structure). \"Combinatorial equivalence\" refers to the ability to pass from one triangulation to another via a sequence of Pachner moves.\n\nRubinstein has an algorithm to determine if a triangulated 3-manifold is combinatorially equivalent to the 3-sphere. A consequence of his algorithm is that there is an algorithm to determine if a 4-dimensional simplicial complex is a 4-manifold triangulation.\n\nIt's known that no algorithms exist to determine if a triangulated 4-manifold has a trivial fundamental group, as there is a procedure to construct a compact 4-manifold with any finitely presented fundamental group.\n\nIn dimensions 5 and higher, Novikov proved that there is no algorithm to decide whether a given triangulated $n$-manifold is combinatorially equivalent to the $n$-sphere is undecidable [N, CL]. Also, it is undecidable whether a given triangulated 4-manifold is combinatorially equivalent to a connect sum of 14 copies of $S^2 \\times S^2$.\n\nBibliography:\n[CL] Chernavsky, A. V, and Leskine, V. P., Unrecognizability of manifolds, Annals of Pure and Applied Logic 141 (2006) 325--335.\n\n[N] Novikov, P.S., On the algorithSSSR 85 (5) (19552) 709--712 (in Russian). Algorithmic unsolvability of the problem of identity, Dokl. Akad. Nauk SSSR 85 (5) (1952) 709--712 (in Russian).\n\n[T] Thompson, A. Thin position and the recognition problem for $S^3$. MRL (1994).\n\nRelated:\nRelated problems\nSmooth 4-dimensional Poincare conjecture\nSmooth 4-dimensional Schoenflies problem\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Is there an algorithm to determine if a triangulated 4-manifold is combinatorially equivalent to the 4-sphere?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** 4-sphere recognition remains open, but modern work supplies one-sided PL-sphericity certificates, an implemented heuristic, and practical experiments; this is not a decision algorithm.\n\n**Verified partial progress.**\n\n- A combinatorial manifold with a subdivision admitting a spherical discrete-Morse vector is certified to be a PL sphere.\n- A heuristic recognizes many examples but can return undecided.\n\n**Full solution or refutation.**\n\nNo terminating yes/no algorithm for all triangulated 4-manifolds was verified.\n\n**What remains.**\n\nGive a complete recognition algorithm or prove its impossibility.\n\n**Sources checked.**\n\n- M. Joswig, F. H. Lutz and M. E. Tsuruga, Frontiers of sphere recognition in practice, Journal of Applied and Computational Topology 6 (2022), 495--529. (primary): https://doi.org/10.1007/s41468-022-00092-8\n Evidence used: The article explicitly says the complexity status of 4-sphere recognition is open and describes a heuristic with yes/no/undecided output.\n\n**Review notes.** The source's smoothing-equivalence gloss is flagged in the report; source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3419, "problem_number": "OPG-37237", "title": "Unsolvability of word problem for 2-knot complements", "statement": "Problem Does there exist a smooth/PL embedding of $S^2$ in $S^4$ such that the fundamental group of the complement has an unsolvable word problem?", "background": "Source: Open Problem Garden. Original node ID: 37237. URL: http://www.openproblemgarden.org/op/unsolvability_of_word_problem_for_2_knot_complements.\n\nSource subject path: Topology.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/unsolvability_of_word_problem_for_2_knot_complements\n- Author(s): Gordon\n- Subject(s): Topology\n- Keywords: 2-knot; Computational Complexity; knot theory\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: July 23rd, 2010 by rybu\n\nProblem-page discussion:\nIt's known that there are smooth $4$-dimensional submanifolds of $S^4$ whose fundamental groups have unsolvable word problems. The complements of classical knots ( $S^1 \\to S^3$ ) are known to have solvable word problems, as do arbitrary $3$-manifold groups.\n\nBibliography:\nA. Dranisnikov, D. Repovs, \"Embeddings up to homotopy type in Euclidean Space\" Bull. Austral. Math. Soc (1993).\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Unsolvability of word problem for 2-knot complements\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** No example of a 2-knot group with unsolvable word problem was verified; the available literature distinguishes this from known higher-knot constructions.\n\n**Verified partial progress.**\n\n- Examples with unsolvable word problem are known for 3-knot groups.\n\n**Full solution or refutation.**\n\nNo smooth or PL 2-knot complement meeting the source condition was verified.\n\n**What remains.**\n\nConstruct such a 2-knot or prove a word-problem theorem for all 2-knot groups.\n\n**Sources checked.**\n\n- UnsolvedMath, OPG-37237: Unsolvability of word problem for 2-knot complements (accessed 2026-08-17). (authoritative_secondary): https://www.unsolvedmath.com/problems/OPG-37237\n Evidence used: The maintained problem entry lists the problem open.\n- G. Baumslag, E. Dyer and C. F. Miller III, On the integral homology of finitely presented groups, Topology 22 (1983), 27--46. (primary): https://doi.org/10.1016/0040-9383(83)90013-6\n Evidence used: This literature underlies the higher-dimensional knot-group constructions cited in current discussions; it does not resolve the 2-knot case.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3420, "problem_number": "OPG-37245", "title": "The 4x5 chessboard complex is the complement of a link, which link?", "statement": "Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.", "background": "Source: Open Problem Garden. Original node ID: 37245. URL: http://www.openproblemgarden.org/op/the_4x5_chessboard_complex_is_the_complement_of_a_link_which_link.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/the_4x5_chessboard_complex_is_the_complement_of_a_link_which_link\n- Author(s): David Eppstein\n- Subject(s): Topology\n- Keywords: knot theory, links, chessboard complex\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 4th, 2010 by rybu\n\nProblem-page discussion:\nSee the MathOverFlow thread: Is the 4x5 chessboard complex a link complement?.\n\nBibliography:\n* D. Eppstein MathOverFlow thread\n\nDiscussion links:\n- Is the 4x5 chessboard complex a link complement?: http://mathoverflow.net/questions/36791/is-the-4x5-chessboard-complex-a-link-complement\n\nBibliography links:\n- MathOverFlow thread: http://mathoverflow.net/questions/36791/is-the-4x5-chessboard-complex-a-link-complement\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"The 4x5 chessboard complex is the complement of a link, which link?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The requested explicit link is given in Göerner's thesis (Figure 1.27), as recorded by the discoverer in the associated MathOverflow discussion.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe existence result has been made constructive by an explicit diagram/reference.\n\n**What remains.**\n\nExtract a machine-readable/link-table description if a dataset-ready diagrammatic encoding is desired.\n\n**Sources checked.**\n\n- M. Göerner, Visualizing Regular Tessellations, PhD thesis, 2012, Figure 1.27. (primary): https://www.unhyperbolic.org/research/matthias_goerner_thesis_print.pdf\n Evidence used: The cited thesis contains the explicit link diagram identified in the accompanying expert discussion.\n- M. Göerner, answer to Is the 4x5 chessboard complex a link complement?, MathOverflow (2012). (authoritative_secondary): https://mathoverflow.net/questions/36791/is-the-4x5-chessboard-complex-a-link-complement\n Evidence used: The answer directs to Figure 1.27 and records Agol's verification that the manifold is a link complement.\n\n**Review notes.** The actual diagram is external; no source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3421, "problem_number": "OPG-37282", "title": "Outer reloid of restricted funcoid", "statement": "Question $( \\mathsf{RLD})_{\\mathrm{out}} (f \\cap^{\\mathsf{FCD}} ( \\mathcal{A} \\times^{\\mathsf{FCD}} \\mathcal{B})) = (( \\mathsf{RLD})_{\\mathrm{out}} f) \\cap^{\\mathsf{RLD}} ( \\mathcal{A} \\times^{\\mathsf{RLD}} \\mathcal{B})$ for every filter objects $\\mathcal{A}$ and $\\mathcal{B}$ and a funcoid $f\\in\\mathsf{FCD}(\\mathrm{Src}\\,f; \\mathrm{Dst}\\,f)$?", "background": "Source: Open Problem Garden. Original node ID: 37282. URL: http://www.openproblemgarden.org/op/outer_reloid_of_restricted_funcoid.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/outer_reloid_of_restricted_funcoid\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: direct product of filters; outer reloid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: December 3rd, 2010 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\n*Victor Porton. Algebraic General Topology\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Outer reloid of restricted funcoid\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The identity remains displayed without a resolution, but it uses author-specific notation whose types and versions are not fixed by the imported sentence; no independent proof or refutation was located.\n\n**Verified partial progress.**\n\n- Porton's technical manuscript supplies definitions of funcoids and reloids, allowing the question to be recognized as meaningful within that framework.\n\n**Full solution or refutation.**\n\nNo verified resolution was found.\n\n**What remains.**\n\nFix a dated definition set; specify the bases of A and B relative to Src f and Dst f; then verify whether both sides are well-typed and prove or refute the identity.\n\n**Sources checked.**\n\n- V. Porton, Funcoids and Reloids: a Generalization of Proximities and Uniformities (2013 version). (primary): https://math.portonvictor.org/binaries/funcoids-reloids.pdf\n Evidence used: This is the author's technical source for the nonstandard objects and operators; no resolution of the displayed identity was found in the located material.\n- Open Problem Garden, Outer reloid of restricted funcoid (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/category/porton_victor?page=3\n Evidence used: The page preserves the question without a solution annotation.\n\n**Review notes.** Material formulation defect: the filter-object bases/supports and dated meanings of the products, meets and outer-reloid operator are not specified.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3422, "problem_number": "OPG-37293", "title": "Sticky Cantor sets", "statement": "Conjecture Let $C$ be a Cantor set embedded in $\\mathbb{R}^n$. Is there a self-homeomorphism $f$ of $\\mathbb{R}^n$ for every $\\epsilon$ greater than $0$ so that $f$ moves every point by less than $\\epsilon$ and $f(C)$ does not intersect $C$? Such an embedded Cantor set for which no such $f$ exists (for some $\\epsilon$ ) is called \"sticky\". For what dimensions $n$ do sticky Cantor sets exist?", "background": "Source: Open Problem Garden. Original node ID: 37293. URL: http://www.openproblemgarden.org/op/sticky_cantor_sets.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/sticky_cantor_sets\n- Subject(s): Topology\n- Keywords: Cantor set\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 6th, 2011 by porton\n\nProblem-page discussion:\nI borrowed this conjecture from this forum thread.\n\nCertainly I understand this conjecture wrongly: $C$ is a subset of a line segment. Consider a homeomorphism which moves all points of $\\mathbb{R}^n$ orthogonally to this line segment by $\\epsilon/2$. This would be a solution of this problem. Obviously it is not what is meant.\n\nIndeed I submit the problem to OPG as is in the hope that somebody will correct my wrong understanding and adjust the formulation to not be misunderstood as by me.\n\nSource links:\n- Cantor set: http://en.wikipedia.org/wiki/Cantor set\n\nDiscussion links:\n- this forum thread: http://www.mathkb.com/Uwe/Forum.aspx/math/16972/Current-Status-of-Topology\n\nComments:\n- July 29th, 2011 | Anonymous | Misunderstanding: Your misunderstanding comes from the definition of a Cantor set. A Cantor set is a set homeomorphic to the usual middle-thirds Cantor set. In general it does not have to lie on a line segment.\n- April 10th, 2012 | Anonymous | M: \"embedded\" does not imply that it is still a subset of the line. It just says that it's one-to-one and a homeomorphism with the image. The conjecture requires to prove that there exists a Cantor which cannot be separated from itself, so showing an example where it can be separated is not relevant.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"Sticky Cantor sets\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Krushkal constructs sticky Cantor sets in every dimension d>=4 under the standard small-ambient-isotopy definition; the literal imported small-homeomorphism wording and the low-dimensional closure were not fully aligned by the checked sources.\n\n**Verified partial progress.**\n\n- Krushkal proves existence of sticky wild Cantor sets in R^d for every d>=4.\n- Wright's earlier pushing-off theory provides historical low-dimensional context but uses a distinct instantaneous-pushing condition.\n\n**Full solution or refutation.**\n\nThe existence direction is solved in all dimensions at least four for ambient isotopies; this audit does not claim a complete solution of the differently worded imported version.\n\n**What remains.**\n\nVerify the controlled equivalence between sufficiently small homeomorphisms and sufficiently small ambient isotopies in this setting, and document the exact d<=3 classification from primary sources.\n\n**Sources checked.**\n\n- V. Krushkal, Sticky Cantor Sets in R^d, J. Topol. Anal. 10 (2018), 477-482; arXiv:1602.01035. (primary): https://arxiv.org/abs/1602.01035\n Evidence used: The abstract and Theorem 1 construct sticky Cantor sets for every d>=4 using the small-ambient-isotopy definition.\n- D. G. Wright, Pushing a Cantor Set Off Itself, Houston J. Math. 2 (1976), 439-447. (primary): https://www.math.uh.edu/~hjm/vol02-3.html\n Evidence used: This is the historical primary source cited by Krushkal; its differently formulated pushing condition motivates explicit expert review.\n\n**Review notes.** Material formulation issue: the source says one small self-homeomorphism, whereas Krushkal defines stickiness through small ambient isotopies.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3423, "problem_number": "OPG-37295", "title": "Nonseparating planar continuum", "statement": "Conjecture Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property?\n\nA set has the fixed point property if every continuous map from it into itself has a fixed point.", "background": "Source: Open Problem Garden. Original node ID: 37295. URL: http://www.openproblemgarden.org/op/nonseparating_planar_continuum.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/nonseparating_planar_continuum\n- Subject(s): Topology\n- Keywords: fixed point\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 6th, 2011 by porton\n\nComments:\n- February 16th, 2011 | Comet | Proof; some set of disks connected by line segments: 1) Any continuous map of this set to itself must traverse these line segments, and some of them will find their fixed point within one of these line segments, as any such mapping that has a submapping that maps a line segment to itself has a fixed point within that line segment. 2) For those mappings that haven't had a fixed point within one of the line segments above, must then have a submapping that maps a part of a disk to itself. This guarantees a fixed point will be found by Brouwer's fixed point theorem. *) Some of the mapping will map separate disks to each other, and there will be no fixed point in that part of the mapping. But how are the separate disks connected? Either they are connected along a line segment, in which case the fixed point must be there (see 1) or the disks are connected by a point, in which case the fixed point must be there at that point.\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Nonseparating planar continuum\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The answer is affirmative: every path-connected nonseparating compact subset of the plane has the fixed-point property.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nHagopian proved the fixed-point property for simply connected plane continua, and Jungck--Timm state the exact path-connected nonseparating corollary. The theorem predates the 2011 source post.\n\n**What remains.**\n\nNo mathematical case remains for the source statement; the dataset status should be updated and the theorem cited.\n\n**Sources checked.**\n\n- C. L. Hagopian, The fixed-point property for simply connected plane continua, Trans. Amer. Math. Soc. 348 (1996), 4525-4548. (primary): https://citeseerx.ist.psu.edu/document?doi=61e25cba0defa9e9335f8676a6d5431aa75d8f59&repid=rep1&type=pdf\n Evidence used: The paper's abstract states that every simply connected plane continuum has the fixed-point property and gives the arcwise-connected characterization.\n- G. Jungck and M. Timm, Another characterization of non-separating planar continua, Topology Proceedings 26 (2001-2002), 235-246. (primary): https://topology.nipissingu.ca/tp/reprints/v26/tp26116.pdf\n Evidence used: Corollary 2.2 and its proof explicitly state the fixed-point theorem for path-connected nonseparating planar continua.\n\n**Review notes.** The source comment about unions of disks is not used as evidence.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3424, "problem_number": "OPG-37297", "title": "Hilbert-Smith conjecture", "statement": "Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group.", "background": "Source: Open Problem Garden. Original node ID: 37297. URL: http://www.openproblemgarden.org/op/hilbert_smith_conjecture.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/hilbert_smith_conjecture\n- Author(s): David Hilbert; Paul A. Smith\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 6th, 2011 by porton\n\nProblem-page discussion:\nThis conjecture at Wikipedia\n\nSource links:\n- Lie group: http://en.wikipedia.org/wiki/Lie group\n\nDiscussion links:\n- This conjecture at Wikipedia: http://en.wikipedia.org/wiki/Hilbert-Smith_conjecture\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Hilbert-Smith conjecture\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Hilbert-Smith is proved in dimensions at most three and for important regular action classes in all dimensions, but the unrestricted topological-action conjecture remains open in dimensions at least four.\n\n**Verified partial progress.**\n\n- Pardon proved that every locally compact group acting faithfully on a connected three-manifold is a Lie group.\n- Repovš and Ščepin proved the conjecture for Lipschitz actions in every finite dimension.\n- Shelukhin proved new Hamiltonian and symplectic-homeomorphism cases in 2024.\n\n**Full solution or refutation.**\n\nNo proof for arbitrary continuous faithful actions on n-manifolds with n>=4 was verified.\n\n**What remains.**\n\nExclude faithful actions of p-adic integer groups on arbitrary topological manifolds in every dimension at least four.\n\n**Sources checked.**\n\n- J. Pardon, The Hilbert--Smith conjecture for three-manifolds, J. Amer. Math. Soc. 26 (2013), 879-899. (primary): https://arxiv.org/abs/1112.2324\n Evidence used: The abstract proves the faithful-action theorem in dimension three and states the standard reduction to Z_p.\n- D. Repovš and E. V. Ščepin, A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps, Math. Ann. 308 (1997), 361-364. (primary): https://www.researchgate.net/publication/282159401_A_proof_of_the_Hilbert-Smith_conjecture_for_actions_by_Lipschitz_maps\n Evidence used: Theorem 1.1 excludes effective Lipschitz actions of p-adic integers on finite-dimensional Riemannian manifolds.\n- E. Shelukhin, A symplectic Hilbert-Smith conjecture, arXiv:2403.07195 (2024). (primary): https://arxiv.org/abs/2403.07195\n Evidence used: The abstract proves new restricted action classes rather than the unrestricted higher-dimensional statement.\n\n**Review notes.** The statement is standard; faithful and effective are synonymous here.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3425, "problem_number": "OPG-37339", "title": "Strict inequalities for products of filters", "statement": "Conjecture $\\mathcal{A} \\times^{\\mathsf{\\ensuremath{\\operatorname{RLD}}}}_F \\mathcal{B} \\subset \\mathcal{A} \\ltimes \\mathcal{B} \\subset \\mathcal{A} \\times^{\\mathsf{\\ensuremath{\\operatorname{RLD}}}} \\mathcal{B}$ for some filter objects $\\mathcal{A}$, $\\mathcal{B}$. Particularly, is this formula true for $\\mathcal{A} = \\mathcal{B} = \\Delta \\cap \\uparrow^{\\mathbb{R}} \\left( 0; + \\infty \\right)$?\n\nA weaker conjecture:\n\nConjecture $\\mathcal{A} \\times^{\\mathsf{\\ensuremath{\\operatorname{RLD}}}}_F \\mathcal{B} \\subset \\mathcal{A} \\ltimes \\mathcal{B}$ for some filter objects $\\mathcal{A}$, $\\mathcal{B}$.", "background": "Source: Open Problem Garden. Original node ID: 37339. URL: http://www.openproblemgarden.org/op/strict_inequalities_for_products_of_filters.\n\nSource subject path: Topology.\n\nSource importance: Low ✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/strict_inequalities_for_products_of_filters\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: filter products\n- Importance: Low ✭\n- Recommended for undergraduates: no\n- Posted: August 9th, 2011 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nThe first conjecture probably has no use by itself but proving it may be somehow challenging, just like Fermat Last Theorem.\n\nBibliography:\n*Victor Porton. Algebraic General Topology\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 5.\n\nAttempt notes:\nTarget:\nMake progress on \"Strict inequalities for products of filters\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No independent resolution was located, and the author's current manuscript has abandoned the older 'filter objects' formalism used by the source statement in favor of plain filters and changed notation.\n\n**Verified partial progress.**\n\n- The current author manuscript identifies a formalism change that must be addressed before comparing the old strict-inclusion statements with current definitions.\n\n**Full solution or refutation.**\n\nNo solved, disproved, or reliably open classification is made because the versioned meanings of the three products and RLD notation were not recovered in a stable current formalism.\n\n**What remains.**\n\nRecover the exact 2011 definitions, translate both inclusions into the current plain-filter formalism, and then search for a proof or counterexample to each strict inclusion and the displayed real-line example.\n\n**Sources checked.**\n\n- V. Porton, Algebraic General Topology, current author manuscript (accessed 2026-08-17). (primary): https://math.portonvictor.org/binaries/volume-1-texmacs.pdf\n Evidence used: The introduction says the earlier concept of filter objects was probably not a good idea and explains that the current work uses plain filters, reversed filter-lattice order, and different join/meet notation.\n\n**Review notes.** The nonstandard displayed notation is reproduced exactly in report.md; no semantic repair was inferred.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3426, "problem_number": "OPG-37378", "title": "Funcoidal products inside an inward reloid", "statement": "Conjecture (solved) If $a \\times^{\\mathsf{\\ensuremath{\\operatorname{RLD}}}} b \\subseteq \\left( \\mathsf{\\ensuremath{\\operatorname{RLD}}} \\right)_{\\ensuremath{\\operatorname{in}}} f$ then $a \\times^{\\mathsf{\\ensuremath{\\operatorname{FCD}}}} b \\subseteq f$ for every funcoid $f$ and atomic f.o. $a$ and $b$ on the source and destination of $f$ correspondingly.\n\nA stronger conjecture:\n\nConjecture If $\\mathcal{A} \\times^{\\mathsf{\\ensuremath{\\operatorname{RLD}}}} \\mathcal{B} \\subseteq \\left( \\mathsf{\\ensuremath{\\operatorname{RLD}}} \\right)_{\\ensuremath{\\operatorname{in}}} f$ then $\\mathcal{A} \\times^{\\mathsf{\\ensuremath{\\operatorname{FCD}}}} \\mathcal{B} \\subseteq f$ for every funcoid $f$ and $\\mathcal{A} \\in \\mathfrak{F} \\left( \\ensuremath{\\operatorname{Src}}f \\right)$, $\\mathcal{B} \\in \\mathfrak{F} \\left( \\ensuremath{\\operatorname{Dst}}f \\right)$.", "background": "Source: Open Problem Garden. Original node ID: 37378. URL: http://www.openproblemgarden.org/op/funcolidal_products_inside_an_inward_reloid.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/funcolidal_products_inside_an_inward_reloid\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: inward reloid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: January 1st, 2012 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\n*Victor Porton. Algebraic General Topology\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Funcoidal products inside an inward reloid\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The source's first assertion is labelled solved, but no independently checkable proof for either exact reloid/funcoid statement was located outside the author's evolving formalism.\n\n**Verified partial progress.**\n\n- The author's general-topology manuscript develops reloid/funcoid definitions and related product theorems.\n\n**Full solution or refutation.**\n\nThe exact status cannot be responsibly promoted from the source label alone.\n\n**What remains.**\n\nLocate a stable theorem/proof with the exact hypotheses, or formalize and independently verify it.\n\n**Sources checked.**\n\n- V. Porton, Algebraic Theory of General Topology, Vol. 1, manuscript (2015/2018). (primary): https://www.researchgate.net/publication/282862013_Algebraic_Theory_of_General_Topology_Volume_1\n Evidence used: The manuscript contains the author-defined funcoid/reloid framework and related atomic-product results, but this search did not verify the exact source conjecture.\n\n**Review notes.** Private terminology/formulation flagged; source unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3427, "problem_number": "OPG-37385", "title": "Upgrading a completary multifuncoid", "statement": "Let $\\mho$ be a set, $\\mathfrak{F}$ be the set of filters on $\\mho$ ordered reverse to set-theoretic inclusion, $\\mathfrak{P}$ be the set of principal filters on $\\mho$, let $n$ be an index set. Consider the filtrator $\\left( \\mathfrak{F}^n; \\mathfrak{P}^n \\right)$.\n\nConjecture If $f$ is a completary multifuncoid of the form $\\mathfrak{P}^n$, then $E^{\\ast} f$ is a completary multifuncoid of the form $\\mathfrak{F}^n$.\n\nSee below for definition of all concepts and symbols used to in this conjecture.\n\nRefer to this Web site for the theory which I now attempt to generalize.", "background": "Source: Open Problem Garden. Original node ID: 37385. URL: http://www.openproblemgarden.org/op/upgrading_a_completary_multifuncoid.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/upgrading_a_completary_multifuncoid\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 4th, 2012 by porton\n\nProblem-page discussion:\nDefinition A filtrator is a pair $\\left( \\mathfrak{A}; \\mathfrak{Z} \\right)$ of a poset $\\mathfrak{A}$ and its subset $\\mathfrak{Z}$.\n\nHaving fixed a filtrator, we define:\n\nDefinition $\\ensuremath{\\operatorname{up}}x = \\left\\{ Y \\in \\mathfrak{Z} \\hspace{0.5em} | \\hspace{0.5em} Y \\geqslant x \\right\\}$ for every $X \\in \\mathfrak{A}$.\n\nDefinition $E^{\\ast} K = \\left\\{ L \\in \\mathfrak{A} \\hspace{0.5em} | \\hspace{0.5em} \\ensuremath{\\operatorname{up}}L \\subseteq K \\right\\}$ (upgrading the set $K$ ) for every $K \\in \\mathscr{P} \\mathfrak{Z}$.\n\nDefinition Let $\\mathfrak{A}$ is a family of join-semilattice. A completary multifuncoid of the form $\\mathfrak{A}$ is an $f \\in \\mathscr{P} \\prod \\mathfrak{A}$ such that we have that:\n\n- $L_0 \\cup L_1 \\in f \\Leftrightarrow \\exists c \\in \\left\\{ 0, 1 \\right\\}^n: \\left( \\lambda i \\in n: L_{c \\left( i_{} \\right)} i \\right) \\in f$ for every $L_0, L_1 \\in \\prod \\mathfrak{A}$.\n\n- If $L \\in \\prod \\mathfrak{A}$ and $L_i = 0^{\\mathfrak{A}_i}$ for some $i$ then $\\neg f L$.\n\n$\\mathfrak{A}^n$ is a function space over a poset $\\mathfrak{A}$ that is $a\\le b\\Leftrightarrow \\forall i\\in n:a_i\\le b_i$ for $a,b\\in\\mathfrak{A}^n$.\n\nFor finite $n$ this problem is equivalent to Upgrading a multifuncoid.\n\nIt is not hard to prove this conjecture for the case $\\ensuremath{\\operatorname{card}}n \\leqslant 2$ using the techniques from this my article. But I failed to prove it for $\\ensuremath{\\operatorname{card}}n = 3$ and above.\n\nBibliography:\n* Conjecture: Upgrading a multifuncoid\n\nRelated:\nRelated problems\nUpgrading a multifuncoid\n\nSource links:\n- this Web site: http://www.mathematics21.org/algebraic-general-topology.html\n\nDiscussion links:\n- Upgrading a multifuncoid: http://www.openproblemgarden.org/?q=node/37348\n- this my article: http://www.mathematics21.org/binaries/funcoids-reloids.pdf\n\nBibliography links:\n- Conjecture: Upgrading a multifuncoid: http://portonmath.wordpress.com/2011/10/09/conjecture-upgrading-multifuncoid/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 29.\n\nAttempt notes:\nTarget:\nMake progress on \"Upgrading a completary multifuncoid\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No independent resolution of the completion/upgrading assertion for completary multifuncoids was located; available material is the author's foundational manuscript.\n\n**Verified partial progress.**\n\n- The source formalism defines upgrading/downgrading relative to filtrators.\n\n**Full solution or refutation.**\n\nThe exact conjecture remains unverified in the literature searched.\n\n**What remains.**\n\nFind a precise published theorem or counterexample under fixed definitions of E* and completary multifuncoid.\n\n**Sources checked.**\n\n- V. Porton, Algebraic Theory of General Topology, Vol. 1, manuscript (2015/2018). (primary): https://www.researchgate.net/publication/282862013_Algebraic_Theory_of_General_Topology_Volume_1\n Evidence used: The table of contents documents the author's chapter on upgrading and downgrading multifuncoids but does not independently settle this exact assertion.\n\n**Review notes.** Terminology not silently normalized.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3428, "problem_number": "OPG-37386", "title": "Atomicity of the poset of completary multifuncoids", "statement": "Conjecture The poset of completary multifuncoids of the form $(\\mathscr{P}\\mho)^n$ is for every sets $\\mho$ and $n$:\n\n- atomic;\n- atomistic.\n\nSee below for definition of all concepts and symbols used to in this conjecture.\n\nRefer to this Web site for the theory which I now attempt to generalize.", "background": "Source: Open Problem Garden. Original node ID: 37386. URL: http://www.openproblemgarden.org/op/atomicity_of_the_poset_of_multifuncoids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/atomicity_of_the_poset_of_multifuncoids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: multifuncoid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 12th, 2012 by porton\n\nProblem-page discussion:\nDefinition Let $\\mathfrak{A}$ is a family of join-semilattice. A completary multifuncoid of the form $\\mathfrak{A}$ is an $f \\in \\mathscr{P} \\prod \\mathfrak{A}$ such that we have that:\n\n- $L_0 \\cup L_1 \\in f \\Leftrightarrow \\exists c \\in \\left\\{ 0, 1 \\right\\}^n: \\left( \\lambda i \\in n: L_{c \\left( i_{} \\right)} i \\right) \\in f$ for every $L_0, L_1 \\in \\prod \\mathfrak{A}$.\n\n- If $L \\in \\prod \\mathfrak{A}$ and $L_i = 0^{\\mathfrak{A}_i}$ for some $i$ then $\\neg f L$.\n\n$\\mathfrak{A}^n$ is a function space over a poset $\\mathfrak{A}$ that is $a\\le b\\Leftrightarrow \\forall i\\in n:a_i\\le b_i$ for $a,b\\in\\mathfrak{A}^n$.\n\nBibliography:\n* Algebraic General Topology\n\nRelated:\nRelated problems\nAtomicity of the poset of multifuncoids\n\nSource links:\n- this Web site: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 14.\n\nAttempt notes:\nTarget:\nMake progress on \"Atomicity of the poset of completary multifuncoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No independent verification of atomicity/atomisticity for the stated completary-multifuncoid poset was located.\n\n**Verified partial progress.**\n\n- Related atomicity theorems for ordinary funcoid lattices appear in the author manuscript.\n\n**Full solution or refutation.**\n\nThose related theorems do not establish the stated multifuncoid claim.\n\n**What remains.**\n\nCheck the exact poset order and prove atomicity/atomisticity or exhibit a counterexample.\n\n**Sources checked.**\n\n- V. Porton, Algebraic Theory of General Topology, Vol. 1, manuscript (2015/2018). (primary): https://www.researchgate.net/publication/282862013_Algebraic_Theory_of_General_Topology_Volume_1\n Evidence used: The manuscript gives related atomic-funcoid results; no exact multifuncoid resolution was verified.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3429, "problem_number": "OPG-37388", "title": "Atomicity of the poset of multifuncoids", "statement": "Conjecture The poset of multifuncoids of the form $(\\mathscr{P}\\mho)^n$ is for every sets $\\mho$ and $n$:\n\n- atomic;\n- atomistic.\n\nSee below for definition of all concepts and symbols used to in this conjecture.\n\nRefer to this Web site for the theory which I now attempt to generalize.", "background": "Source: Open Problem Garden. Original node ID: 37388. URL: http://www.openproblemgarden.org/op/atomicity_of_the_poset_of_multifuncoids_0.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/atomicity_of_the_poset_of_multifuncoids_0\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: multifuncoid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 12th, 2012 by porton\n\nProblem-page discussion:\nDefinition A free star on a join-semilattice $\\mathfrak{A}$ with least element 0 is a set $S$ such that $0 \\not\\in S$ and\n$$\n\\forall A, B \\in \\mathfrak{A}: \\left( A \\cup B \\in S \\Leftrightarrow A \\in S \\vee B \\in S \\right).\n$$\n\nDefinition Let $\\mathfrak{A}$ be a family of posets, $f \\in \\mathscr{P} \\prod \\mathfrak{A}$ ( $\\prod \\mathfrak{A}$ has the order of function space of posets), $i \\in \\ensuremath{\\operatorname{dom}}\\mathfrak{A}$, $L \\in \\prod \\mathfrak{A}|_{\\left( \\ensuremath{\\operatorname{dom}}\\mathfrak{A} \\right) \\setminus \\left\\{ i \\right\\}}$. Then\n$$\n\\left( \\ensuremath{\\operatorname{val}}f \\right)_i L = \\left\\{ X \\in \\mathfrak{A}_i \\hspace{0.5em} | \\hspace{0.5em} L \\cup \\left\\{ (i; X) \\right\\} \\in f \\right\\}.\n$$\n\nDefinition Let $\\mathfrak{A}$ is a family of posets. A multidimensional funcoid (or multifuncoid for short) of the form $\\mathfrak{A}$ is an $f \\in \\mathscr{P} \\prod \\mathfrak{A}$ such that we have that:\n\n- $\\left( \\tmop{val} f \\right)_i L$ is a free star for every $i \\in \\tmop{dom} \\mathfrak{A}$, $L \\in \\prod \\mathfrak{A}|_{\\left( \\tmop{dom} \\mathfrak{A} \\right) \\setminus \\left\\{ i \\right\\}}$.\n\n- $f$ is an upper set.\n\n$\\mathfrak{A}^n$ is a function space over a poset $\\mathfrak{A}$ that is $a\\le b\\Leftrightarrow \\forall i\\in n:a_i\\le b_i$ for $a,b\\in\\mathfrak{A}^n$.\n\nBibliography:\n* Algebraic General Topology\n\nRelated:\nRelated problems\nAtomicity of the poset of completary multifuncoids\n\nSource links:\n- this Web site: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 19.\n\nAttempt notes:\nTarget:\nMake progress on \"Atomicity of the poset of multifuncoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The author source supplies detailed definitions of multifuncoids, but no independent literature result settling the claimed atomic and atomistic properties was verified.\n\n**Verified partial progress.**\n\n- The source formalism makes the relevant poset and free-star condition explicit.\n\n**Full solution or refutation.**\n\nThe exact claim is not promoted beyond uncertain status.\n\n**What remains.**\n\nEstablish a proof/counterexample under a stable version of the definitions.\n\n**Sources checked.**\n\n- Open Problem Garden, Atomicity of the poset of multifuncoids (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/atomicity_of_the_poset_of_multifuncoids_0\n Evidence used: The page provides the statement and definitions but no later resolution.\n\n**Review notes.** Specialized source terminology preserved.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3430, "problem_number": "OPG-37389", "title": "Graph product of multifuncoids", "statement": "Conjecture Let $F$ is a family of multifuncoids such that each $F_i$ is of the form $\\lambda j \\in N \\left( i \\right): \\mathfrak{F} \\left( U_j \\right)$ where $N \\left( i \\right)$ is an index set for every $i$ and $U_j$ is a set for every $j$. Let every $F_i = E^{\\ast} f_i$ for some multifuncoid $f_i$ of the form $\\lambda j \\in N \\left( i \\right): \\mathfrak{P} \\left( U_j \\right)$ regarding the filtrator $\\left( \\prod_{j \\in N \\left( i \\right)} \\mathfrak{F} \\left( U_j \\right); \\prod_{j \\in N \\left( i \\right)} \\mathfrak{P} \\left( U_j \\right) \\right)$. Let $H$ is a graph-composition of $F$ (regarding some partition $G$ and external set $Z$ ). Then there exist a multifuncoid $h$ of the form $\\lambda j \\in Z: \\mathfrak{P} \\left( U_j \\right)$ such that $H = E^{\\ast} h$ regarding the filtrator $\\left( \\prod_{j \\in Z} \\mathfrak{F} \\left( U_j \\right); \\prod_{j \\in Z} \\mathfrak{P} \\left( U_j \\right) \\right)$.", "background": "Source: Open Problem Garden. Original node ID: 37389. URL: http://www.openproblemgarden.org/op/graph_product_of_multifuncoids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/graph_product_of_multifuncoids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: graph-product; multifuncoid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 12th, 2012 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology, especially the theory of multifuncoids for definitions of used concepts.\n\nBibliography:\n*Victor Porton. Algebraic General Topology\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n- the theory of multifuncoids: http://www.mathematics21.org/binaries/nary.pdf\n\nBibliography links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 18.\n\nAttempt notes:\nTarget:\nMake progress on \"Graph product of multifuncoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** No independent proof or disproof of the graph-product preservation statement for the specified principal-filter multifuncoids was located.\n\n**Verified partial progress.**\n\n- The author's manuscript contains chapters on products, projections, and relationships between cross-composition and subatomic products.\n\n**Full solution or refutation.**\n\nRelated framework material is insufficient to identify the exact conjecture as solved.\n\n**What remains.**\n\nVerify the required hypotheses and find a stable proof or a counterexample.\n\n**Sources checked.**\n\n- V. Porton, Algebraic Theory of General Topology, Vol. 1, manuscript (2015/2018). (primary): https://www.researchgate.net/publication/282862013_Algebraic_Theory_of_General_Topology_Volume_1\n Evidence used: The manuscript develops graph/product-related multifuncoid machinery but this search did not locate the exact result.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3431, "problem_number": "OPG-37540", "title": "A conjecture about direct product of funcoids", "statement": "Conjecture Let $f_1$ and $f_2$ are monovalued, entirely defined funcoids with $\\operatorname{Src}f_1=\\operatorname{Src}f_2=A$. Then there exists a pointfree funcoid $f_1 \\times^{\\left( D \\right)} f_2$ such that (for every filter $x$ on $A$ ) $$\\left\\langle f_1 \\times^{\\left( D \\right)} f_2 \\right\\rangle x = \\bigcup \\left\\{ \\langle f_1\\rangle X \\times^{\\mathsf{FCD}} \\langle f_2\\rangle X \\hspace{1em} | \\hspace{1em} X \\in \\mathrm{atoms}^{\\mathfrak{A}} x \\right\\}.$$ (The join operation is taken on the lattice of filters with reversed order.)\n\nA positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.", "background": "Source: Open Problem Garden. Original node ID: 37540. URL: http://www.openproblemgarden.org/op/a_conjecture_about_direct_product_of_funcoids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_conjecture_about_direct_product_of_funcoids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: category theory; general topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 26th, 2012 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\n*Victor Porton. a blog post\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- a blog post: http://portonmath.wordpress.com/2012/07/26/conjecture-direct-product-funcoids/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"A conjecture about direct product of funcoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** A later author-maintained page solves a related categorical product construction for continuous maps between endo-funcoids, but no source was found proving the exact displaced-product identity stated here.\n\n**Verified partial progress.**\n\n- A later canonical direct-product construction exists in the author's evolving funcoid framework.\n\n**Full solution or refutation.**\n\nThe later related construction cannot safely be treated as a proof of this formula without a theorem connecting the changed definitions and notation.\n\n**What remains.**\n\nCheck the exact formula against a fixed version of the funcoid definitions and prove either existence or failure; alternatively, establish that the later categorical product specializes to it.\n\n**Sources checked.**\n\n- Open Problem Garden, A conjecture about direct product of funcoids, accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/a_conjecture_about_direct_product_of_funcoids\n Evidence used: Preserves the exact conjecture and contains no independently verified resolution.\n- Open Problem Garden, A construction of direct product in the category of continuous maps between endo-funcoids (Solved), accessed 2026-08-17. (maintained_tracker): https://www.openproblemgarden.org/op/a_construction_of_direct_product_in_the_category_of_continuous_maps_between_endo_funcoids\n Evidence used: Marks a related but differently formulated direct-product problem solved; it does not explicitly establish the displayed identity in this record.\n- Victor Porton, Algebraic General Topology, author manuscript, accessed 2026-08-17. (primary): https://math.portonvictor.org/binaries/volume-1-texmacs.pdf\n Evidence used: Documents the author's funcoid framework and evolved product terminology, but no exact resolution of the supplied formula was located.\n\n**Review notes.** The theory is nonstandard and version-sensitive; status was not inferred from a similarly named later construction.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3432, "problem_number": "OPG-48767", "title": "Closing Lemma for Diffeomorphism (Dynamical Systems)", "statement": "Conjecture Let $f\\in Diff^{r}(M)$ and $p\\in\\omega_{f}$. Then for any neighborhood $V_{f}\\subset Diff^{r}(M)$ there is $g\\in V_{f}$ such that $p$ is periodic point of $g$\n\nThere is an analogous conjecture for flows ( $C^{r}$ vector fields. In the case of diffeos this was proved by Charles Pugh for $r = 1$. In the case of Flows this has been solved by Sushei Hayahshy for $r = 1$. But in the two cases the problem is wide open for $r > 1$", "background": "Source: Open Problem Garden. Original node ID: 48767. URL: http://www.openproblemgarden.org/op/closing_lemma_for_diffeomorphism_dynamical_systems.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/closing_lemma_for_diffeomorphism_dynamical_systems\n- Author(s): Charles Pugh\n- Subject(s): Topology\n- Keywords: Dynamics, Pertubation\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 24th, 2013 by Jailton Viana\n\nBibliography:\nDynamics beyond uniform hyperbolicity:\\Springer [Encyclopaedia of Mathematical Sciences Volume 102, Mathematical Phisics,2005]\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Closing Lemma for Diffeomorphism (Dynamical Systems)\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Pugh's C1 closing lemma is known, while the general Cr closing lemma for r>1 remains open; there are important special-class results.\n\n**Verified partial progress.**\n\n- Gan--Shi prove the Cr closing lemma for partially hyperbolic diffeomorphisms with one-dimensional centre.\n- Other symplectic and surface/torus settings have positive conditional results.\n\n**Full solution or refutation.**\n\nNo general theorem covers arbitrary Cr diffeomorphisms on arbitrary compact manifolds for r>1.\n\n**What remains.**\n\nEstablish the closing perturbation in the unrestricted higher-regularity setting.\n\n**Sources checked.**\n\n- S. Gan and Y. Shi, Cr-Closing lemma for partially hyperbolic diffeomorphisms with 1D-center bundle, arXiv:2004.06855 (2020). (primary): https://arxiv.org/abs/2004.06855\n Evidence used: Proves a higher-regularity closing lemma for a specified partially hyperbolic class.\n- H. Qu and Z. Xia, A C-infinity closing lemma on torus (2021 preprint). (authoritative_secondary): https://www.researchgate.net/publication/352475952_A_Cinfty_closing_lemma_on_torus\n Evidence used: Its introduction explicitly states the general r>1 problem remains open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3433, "problem_number": "OPG-48770", "title": "Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems)", "statement": "Conjecture Let $Diff^{r}(M)$ be the space of $C^{r}$ Diffeomorphisms on the connected, compact and boundaryles manifold M and $\\chi^{r}(M)$ the space of $C^{r}$ vector fields. There is a dense set $D\\subset Diff^{r}(M)$ ( $D\\subset \\chi^{r}(M)$ ) such that $\\forall f\\in D$ exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space $M$\n\nThis is a very Deep and Hard problem in Dynamical Systems. It present the dream of the dynamicist mathematicians.", "background": "Source: Open Problem Garden. Original node ID: 48770. URL: http://www.openproblemgarden.org/op/jacob_palis_conjecture_finitude_of_attractors_dynamical_systems.\n\nSource subject path: Topology.\n\nSource importance: Outstanding ✭✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/jacob_palis_conjecture_finitude_of_attractors_dynamical_systems\n- Subject(s): Topology\n- Keywords: Attractors, basins, Finite\n- Importance: Outstanding ✭✭✭✭\n- Recommended for undergraduates: no\n- Posted: April 24th, 2013 by Jailton Viana\n\nProblem-page discussion:\nDefinition: A set $\\Lambda \\subset M$ is an attractor for a Diffeomorphism (or a flow ) if it is invariant, transitive and the basin of attraction $B(\\Lambda):= \\{p\\in M / \\omega(p)\\subset \\Lambda \\}$ has positive Lebesgue Measure.\n\nBibliography:\nBonatti C, Diaz L.; Viana M.; Dynamics beyond uniform hyperbolicity, Springer[Encyclopaedia of Mathematics Sciences ], Volume 102, 2005\n\nRelated:\nRelated problems\nClosing Lemma for Diffeomorphism (Dynamical Systems)\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 9.\n\nAttempt notes:\nTarget:\nMake progress on \"Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems)\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Palis's finiteness-of-attractors conjecture remains a general program, with verified results in hyperbolic, one-dimensional, and selected partially hyperbolic settings.\n\n**Verified partial progress.**\n\n- The conjecture holds for uniformly hyperbolic dynamics.\n- The cited survey records one-dimensional and selected partially hyperbolic positive cases.\n\n**Full solution or refutation.**\n\nNo proof was verified for all compact manifolds and all Cr diffeomorphisms/vector fields in the source formulation.\n\n**What remains.**\n\nEstablish density of finite-attractor dynamics beyond the known classes.\n\n**Sources checked.**\n\n- J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Asterisque 261 (2000), 335--347. (primary): https://numdam.org/item/AST_2000__261__335_0.pdf\n Evidence used: Gives the original finiteness-of-attractors formulation.\n- Generic family with robustly infinitely many sinks, survey discussion of the Palis conjecture (accessed 2026-08-17). (authoritative_secondary): https://citeseerx.ist.psu.edu/document?doi=599ff114f2d8b3e2ec560109fbaeea40aa6087cf&repid=rep1&type=pdf\n Evidence used: Summarizes known positive classes and treats the general conjecture as open.\n\n**Review notes.** No source alteration.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 3, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3434, "problem_number": "OPG-56573", "title": "Decomposition of completions of reloids", "statement": "Conjecture For composable reloids $f$ and $g$ it holds\n\n- $\\operatorname{Compl} ( g \\circ f) = ( \\operatorname{Compl} g) \\circ f$ if $f$ is a co-complete reloid;\n- $\\operatorname{CoCompl} ( f \\circ g) = f \\circ \\operatorname{CoCompl} g$ if $f$ is a complete reloid;\n- $\\operatorname{CoCompl} ( ( \\operatorname{Compl} g) \\circ f) = \\operatorname{Compl} ( g \\circ ( \\operatorname{CoCompl} f)) = ( \\operatorname{Compl} g) \\circ ( \\operatorname{CoCompl} f)$;\n- $\\operatorname{Compl} ( g \\circ ( \\operatorname{Compl} f)) = \\operatorname{Compl} ( g \\circ f)$;\n- $\\operatorname{CoCompl} ( ( \\operatorname{CoCompl} g) \\circ f) = \\operatorname{CoCompl} ( g \\circ f)$.", "background": "Source: Open Problem Garden. Original node ID: 56573. URL: http://www.openproblemgarden.org/op/decomposition_of_completions_of_reloids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/decomposition_of_completions_of_reloids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: co-completion; completion; reloid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 2nd, 2013 by porton\n\nProblem-page discussion:\nWell, in fact this is three separate problems (if we count dual formulas as one formula), but I am lazy to create three pages for them.\n\nThis conjecture is inspired by the proven fact that the above formulas hold for every composable funcoids $f$ and $g$ (instead of reloids). Properties of reloids are expected to be similar to properties of funcoids.\n\nhttp://www.packersandmoverschandigarh.co.in/\nhttp://www.packersandmoversjaipur.co.in/\nhttp://www.packersandmoversinhyderabad.co.in/\nhttp://www.packersandmoversinbangalore.co.in/\n\nBibliography:\n*Algebraic General Toplogy. Volume 1\n\nDiscussion links:\n- http://www.packersandmoverschandigarh.co.in/: http://www.packersandmoverschandigarh.co.in/\n- http://www.packersandmoversjaipur.co.in/: http://www.packersandmoversjaipur.co.in/\n- http://www.packersandmoversinhyderabad.co.in/: http://www.packersandmoversinhyderabad.co.in/\n- http://www.packersandmoversinbangalore.co.in/: http://www.packersandmoversinbangalore.co.in/\n\nBibliography links:\n- Algebraic General Toplogy. Volume 1: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Decomposition of completions of reloids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The five completion/co-completion identities for composable reloids remain explicitly conjectural in the author's current monograph.\n\n**Verified partial progress.**\n\n- One-sided completion and co-completion inequalities are proved for arbitrary composable reloids.\n- The analogous identities are proved for composable funcoids.\n\n**Full solution or refutation.**\n\nPorton's current edition labels the exact displayed formulas Conjecture 1084, immediately after Proposition 1083 proves only weaker inequalities.\n\n**What remains.**\n\nUpgrade the one-sided inequalities to all five equalities or find a counterexample in the reloid framework.\n\n**Sources checked.**\n\n- V. Porton, General Topology as Ordered Semigroup Actions, Book 1, current online edition, Proposition 1083 and Conjecture 1084 (accessed 2026-08-17). (primary): https://math.portonvictor.org/binaries/volume-1.pdf\n Evidence used: Reproduces the exact five formulas as Conjecture 1084 and proves the adjacent one-sided inequalities.\n\n**Review notes.** The terminology is author-specific and the status source is an evolving author-hosted monograph rather than a refereed publication. Source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3435, "problem_number": "OPG-57401", "title": "Every metamonovalued funcoid is monovalued", "statement": "Conjecture Every metamonovalued funcoid is monovalued.\n\nThe reverse is almost trivial: Every monovalued funcoid is metamonovalued.", "background": "Source: Open Problem Garden. Original node ID: 57401. URL: http://www.openproblemgarden.org/op/every_metamonovalued_funcoid_is_monovalued.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/every_metamonovalued_funcoid_is_monovalued\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: monovalued\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 17th, 2013 by porton\n\nBibliography:\n*Algebraic General Toplogy. Volume 1\n\nRelated:\nRelated problems\nEvery metamonovalued reloid is monovalued\n\nBibliography links:\n- Algebraic General Toplogy. Volume 1: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Every metamonovalued funcoid is monovalued\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** solved \n**Classification:** SOLVED-IN-LITERATURE\n\n**Current literature assessment.** The author's current monograph proves that a funcoid is monovalued if and only if it is metamonovalued, resolving the extracted direction within that framework.\n\n**Verified partial progress.**\n\n- Earlier editions proved only that every monovalued funcoid is metamonovalued.\n- Current Theorem 989 gives several further equivalent atomic and filter-theoretic characterizations.\n\n**Full solution or refutation.**\n\nTheorem 989 explicitly makes monovaluedness, metamonovaluedness, and weak metamonovaluedness equivalent for funcoids.\n\n**What remains.**\n\nIndependently audit the proof and definitions and obtain a stable refereed or versioned archival source.\n\n**Sources checked.**\n\n- V. Porton, General Topology as Ordered Semigroup Actions, Book 1, current online edition, Theorem 989 (accessed 2026-08-17). (primary): https://math.portonvictor.org/binaries/volume-1.pdf\n Evidence used: Theorem 989 states that monovalued, metamonovalued, and weakly metamonovalued funcoids are equivalent.\n\n**Review notes.** Solved only in an evolving, author-hosted nonstandard framework; no independent refereed verification was found. Source statement unchanged.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3436, "problem_number": "OPG-57403", "title": "Every metamonovalued reloid is monovalued", "statement": "Conjecture Every metamonovalued reloid is monovalued.", "background": "Source: Open Problem Garden. Original node ID: 57403. URL: http://www.openproblemgarden.org/op/every_metamonovalued_reloid_is_monovalued.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/every_metamonovalued_reloid_is_monovalued\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 17th, 2013 by porton\n\nBibliography:\n*Algebraic General Toplogy. Volume 1\n\nRelated:\nRelated problems\nEvery metamonovalued funcoid is monovalued\nEvery monovalued reloid is metamonovalued\n\nBibliography links:\n- Algebraic General Toplogy. Volume 1: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- No LaTeX expressions were detected in the source text.\n\nAttempt notes:\nTarget:\nMake progress on \"Every metamonovalued reloid is monovalued\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The converse from metamonovalued to monovalued remains open for reloids, although the funcoid analogue is now a theorem.\n\n**Verified partial progress.**\n\n- Theorem 1362 proves every monovalued reloid is metamonovalued.\n- The author's current monograph isolates the converse verbatim as Conjecture 1363.\n\n**Full solution or refutation.**\n\nNo proof or counterexample to the reloid converse was verified; its current primary source still marks it as a conjecture.\n\n**What remains.**\n\nProve every metamonovalued reloid is monovalued or find a nonprincipal counterexample.\n\n**Sources checked.**\n\n- V. Porton, General Topology as Ordered Semigroup Actions, Book 1, current online edition, Theorem 1362 and Conjecture 1363 (accessed 2026-08-17). (primary): https://math.portonvictor.org/binaries/volume-1.pdf\n Evidence used: Proves the forward implication and retains the exact converse as Conjecture 1363.\n\n**Review notes.** Status relies on the author's current evolving monograph; independent literature using the terminology was not found.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3437, "problem_number": "OPG-59896", "title": "Generalized path-connectedness in proximity spaces", "statement": "Let $\\delta$ be a proximity.\n\nA set $A$ is connected regarding $\\delta$ iff $\\forall X,Y \\in \\mathscr{P} A \\setminus \\{ \\emptyset \\}: \\left( X \\cup Y = A \\Rightarrow X \\mathrel{\\delta} Y \\right)$.\n\nConjecture The following statements are equivalent for every endofuncoid $\\mu$ and a set $U$:\n\n- $U$ is connected regarding $\\mu$.\n- For every $a, b \\in U$ there exists a totally ordered set $P \\subseteq U$ such that $\\min P = a$, $\\max P = b$, and for every partion $\\{ X, Y \\}$ of $P$ into two sets $X$, $Y$ such that $\\forall x \\in X, y \\in Y: x < y$, we have $X \\mathrel{[ \\mu]^{\\ast}} Y$.", "background": "Source: Open Problem Garden. Original node ID: 59896. URL: http://www.openproblemgarden.org/op/generalized_path_connectedness_in_proximity_spaces.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/generalized_path_connectedness_in_proximity_spaces\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: connected; connectedness; proximity space\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 1st, 2014 by porton\n\nBibliography:\n*Question at math.StackExchange.com by Victor Porton\n\nSource links:\n- proximity: http://en.wikipedia.org/wiki/Proximity_space\n\nBibliography links:\n- Question at math.StackExchange.com: http://math.stackexchange.com/questions/642337/connectedness-in-proximity-spaces\n\nComments:\n- February 26th, 2014 | porton | A proposed lemma: http://math.stackexchange.com/questions/691643/a-lemma-to-solve-a-conjec...\n\n--\n\nVictor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"Generalized path-connectedness in proximity spaces\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** open \n**Classification:** OPEN-TRIAGE\n\n**Current literature assessment.** The ordered-path characterization of connectedness for arbitrary endofuncoids remains Conjecture 1250 in the author's current monograph.\n\n**Verified partial progress.**\n\n- Relation-valued connectedness has an ordinary finite-path characterization.\n- Nearby propositions establish principal-filter and reloid-to-funcoid special cases.\n\n**Full solution or refutation.**\n\nNo proof of the arbitrary endofuncoid equivalence was verified; the current primary source retains an essentially equivalent initial/final-cut formulation as a conjecture.\n\n**What remains.**\n\nProve the ordered connecting-set characterization for all endofuncoids or exhibit a connected counterexample.\n\n**Sources checked.**\n\n- V. Porton, General Topology as Ordered Semigroup Actions, Book 1, current online edition, Proposition 1249 and Conjecture 1250 (accessed 2026-08-17). (primary): https://math.portonvictor.org/binaries/volume-1.pdf\n Evidence used: Retains the ordered-path characterization as Conjecture 1250 and proves adjacent special cases.\n\n**Review notes.** The source typo 'partion' and undefined standalone notation [mu]^* are flagged. The current monograph's initial-cut wording appears intended but was not silently substituted.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3438, "problem_number": "OPG-59900", "title": "Direct proof of a theorem about compact funcoids", "statement": "Conjecture Let $f$ is a $T_1$-separable (the same as $T_2$ for symmetric transitive) compact funcoid and $g$ is a uniform space (reflexive, symmetric, and transitive endoreloid) such that $( \\mathsf{\\tmop{FCD}}) g = f$. Then $g = \\langle f \\times f \\rangle^{\\ast} \\Delta$.\n\nThe main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.\n\nThe direct proof may be constructed by correcting all errors an omissions in this draft article.\n\nDirect proof could be better because with it we would get a little more general statement like this:\n\nConjecture Let $f$ be a $T_1$-separable compact reflexive symmetric funcoid and $g$ be a reloid such that\n\n- $( \\mathsf{\\tmop{FCD}}) g = f$;\n- $g \\circ g^{- 1} \\sqsubseteq g$.\n\nThen $g = \\langle f \\times f \\rangle^{\\ast} \\Delta$.", "background": "Source: Open Problem Garden. Original node ID: 59900. URL: http://www.openproblemgarden.org/op/direct_proof_of_a_theorem_about_compact_funcoids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/direct_proof_of_a_theorem_about_compact_funcoids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: February 8th, 2014 by porton\n\nBibliography:\nVictor Porton. Compact funcoids\n\nSource links:\n- this draft article: http://www.mathematics21.org/binaries/compact.pdf\n\nBibliography links:\n- Compact funcoids: http://www.mathematics21.org/binaries/compact.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 7.\n\nAttempt notes:\nTarget:\nMake progress on \"Direct proof of a theorem about compact funcoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The source uses nonstandard funcoid/reloid operations and refers to a draft; no citable proof of the exact direct theorem was verified.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nThe stated comparison with compact Hausdorff uniform-space uniqueness is not enough to establish the exact funcoid claim without fixed definitions.\n\n**What remains.**\n\nProvide a stable formal definition of all operators and a checkable direct proof or counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, Direct proof of a theorem about compact funcoids (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/direct_proof_of_a_theorem_about_compact_funcoids\n Evidence used: Contains the idiosyncratic definitions and points to a draft rather than a citable resolution.\n\n**Review notes.** No source statement or notation was silently repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3439, "problem_number": "OPG-59970", "title": "Another conjecture about reloids and funcoids", "statement": "Definition $\\square f = \\bigcap^{\\mathsf{RLD}} \\mathrm{up}^{\\Gamma (\\operatorname{Src} f; \\operatorname{Dst} f)} f$ for reloid $f$.\n\nConjecture $(\\mathsf{RLD})_{\\Gamma} f = \\square (\\mathsf{RLD})_{\\mathrm{in}} f$ for every funcoid $f$.\n\nNote: it is known that $(\\mathsf{RLD})_{\\Gamma} f \\ne \\square (\\mathsf{RLD})_{\\mathrm{out}} f$ (see below mentioned online article).", "background": "Source: Open Problem Garden. Original node ID: 59970. URL: http://www.openproblemgarden.org/op/another_conjecture_about_reloids_and_funcoids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/another_conjecture_about_reloids_and_funcoids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 28th, 2014 by porton\n\nProblem-page discussion:\nIt's used notation from Algebraic General Topology draft book, modified by this note about new notation for a future version of this book.\n\nBibliography:\n* blog post\n\nDiscussion links:\n- Algebraic General Topology draft book: http://www.mathematics21.org/algebraic-general-topology.html\n- this note: http://www.mathematics21.org/binaries/rewrite-plan.pdf\n\nBibliography links:\n- blog post: http://portonmath.wordpress.com/2014/11/28/some-new/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Another conjecture about reloids and funcoids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The standalone identity cannot be reliably interpreted because its nonstandard reloid/funcoid operators are undefined and version-sensitive. Only the author's own evolving drafts and unchanged OPG entry were located; no independent proof or refutation was found.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo independently verifiable resolution was located, and no mathematical status is assigned to an identity whose exact typed definitions could not be frozen.\n\n**What remains.**\n\nIdentify the exact version of Algebraic General Topology and its rewrite plan intended by the 2014 post, reconstruct every operator and order, and then check the identity in that fixed formal system.\n\n**Sources checked.**\n\n- Victor Porton, Another conjecture about reloids and funcoids, Open Problem Garden. (maintained_tracker): https://openproblemgarden.org/op/another_conjecture_about_reloids_and_funcoids\n Evidence used: Preserves the exact conjecture, says that it uses notation modified from a draft book by a separate rewrite note, and contains no posted resolution.\n- Victor Porton, Funcoids and Reloids: a Generalization of Proximities and Uniformities, author manuscript (2013). (primary): https://math.portonvictor.org/binaries/funcoids-reloids.pdf\n Evidence used: Author's foundational source for the nonstandard objects, but not an independent status source and not clearly the same notation version as the OPG post.\n\n**Review notes.** The record does not define (RLD)_Gamma, (RLD)_in, the superscripted intersection, or the relevant types/orders. The OPG page explicitly refers to a notation rewrite, so no silent reconstruction was made.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3440, "problem_number": "OPG-59973", "title": "What are hyperfuncoids isomorphic to?", "statement": "Let $\\mathfrak{A}$ be an indexed family of sets.\n\nProducts are $\\prod A$ for $A \\in \\prod \\mathfrak{A}$.\n\nHyperfuncoids are filters $\\mathfrak{F} \\Gamma$ on the lattice $\\Gamma$ of all finite unions of products.\n\nProblem Is $\\bigcap^{\\mathsf{\\tmop{FCD}}}$ a bijection from hyperfuncoids $\\mathfrak{F} \\Gamma$ to:\n\n- prestaroids on $\\mathfrak{A}$;\n- staroids on $\\mathfrak{A}$;\n- completary staroids on $\\mathfrak{A}$?\n\nIf yes, is $\\operatorname{up}^{\\Gamma}$ defining the inverse bijection? If not, characterize the image of the function $\\bigcap^{\\mathsf{\\tmop{FCD}}}$ defined on $\\mathfrak{F} \\Gamma$.\n\nConsider also the variant of this problem with the set $\\Gamma$ replaced with the set $\\Gamma^{\\ast}$ of complements of elements of the set $\\Gamma$.", "background": "Source: Open Problem Garden. Original node ID: 59973. URL: http://www.openproblemgarden.org/op/what_are_hyperfuncoids_isomorphic_to.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/what_are_hyperfuncoids_isomorphic_to\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: hyperfuncoids; multidimensional\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: December 9th, 2014 by porton\n\nProblem-page discussion:\nIt's used notation from Algebraic General Topology draft book\n\nDiscussion links:\n- Algebraic General Topology draft book: http://www.mathematics21.org/algebraic-general-topology.html\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 8.\n\nAttempt notes:\nTarget:\nMake progress on \"What are hyperfuncoids isomorphic to?\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The proposed bijection cannot be checked from the record: its map and all three candidate codomain classes are undefined, and no independent literature using these notions was found. The author-posted OPG entry remains unanswered.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nNo verifiable answer or image characterization was located for either Gamma or Gamma-star.\n\n**What remains.**\n\nFreeze the intended draft definitions of hyperfuncoid, prestaroid, staroid, completary staroid, and the intersection map, remove the TeXmacs conversion ambiguity, and then formulate a typed bijectivity question.\n\n**Sources checked.**\n\n- Victor Porton, What are hyperfuncoids isomorphic to?, Open Problem Garden. (maintained_tracker): https://www.openproblemgarden.org/op/what_are_hyperfuncoids_isomorphic_to\n Evidence used: Only located exact statement; it still poses all alternatives and has no answer or discussion.\n- Victor Porton, Funcoids and Reloids: a Generalization of Proximities and Uniformities, author manuscript (2013). (primary): https://math.portonvictor.org/binaries/funcoids-reloids.pdf\n Evidence used: Provides author-created background terminology but does not independently settle the hyperfuncoid bijection question.\n\n**Review notes.** The source contains the TeXmacs macro \\tmop inside LaTeX and does not define the candidate codomains or map. These defects were flagged rather than repaired.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3441, "problem_number": "OPG-60017", "title": "Infinite distributivity of meet over join for a principal funcoid", "statement": "Conjecture $f \\sqcap \\bigsqcup S = \\bigsqcup \\langle f \\sqcap \\rangle^{\\ast} S$ for principal funcoid $f$ and a set $S$ of funcoids of appropriate sources and destinations.", "background": "Source: Open Problem Garden. Original node ID: 60017. URL: http://www.openproblemgarden.org/op/infinite_distributivity_of_meet_over_join_for_a_principal_funcoid.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/infinite_distributivity_of_meet_over_join_for_a_principal_funcoid\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: distributivity; principal funcoid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 27th, 2016 by porton\n\nProblem-page discussion:\nIt's used notation from Algebraic General Topology book\n\nBibliography:\n*Victor Porton. A blog post\n\nDiscussion links:\n- Algebraic General Topology book: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- A blog post: https://portonmath.wordpress.com/2016/07/27/new-conjecture-2/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Infinite distributivity of meet over join for a principal funcoid\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** Only the author's maintained OPG entry and evolving Algebraic General Topology manuscript were found; no independent proof or refutation of the displayed infinite-distributivity identity was verified.\n\n**Verified partial progress.**\n\n- The maintained OPG page still displays the conjecture with no comments or solution.\n- The author's current manuscript develops funcoid lattices but the exact searched identity was not located as a settled theorem.\n\n**Full solution or refutation.**\n\nNo reliable resolution was found, and the statement cannot be safely status-labeled without fixing the external definitions.\n\n**What remains.**\n\nPin a manuscript edition, define the typed arbitrary join and lifted meet map, type-check both sides, and then seek a proof or counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, Infinite distributivity of meet over join for a principal funcoid (checked 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/infinite_distributivity_of_meet_over_join_for_a_principal_funcoid\n Evidence used: Retains the exact conjecture without discussion or a posted resolution.\n- Victor Porton, Algebraic General Topology, current author manuscript (checked 2026-08-17). (primary): https://math.portonvictor.org/binaries/volume-1-texmacs.pdf\n Evidence used: Supplies the nonstandard funcoid definitions on which the record depends; no independently verified resolution was identified.\n\n**Review notes.** The record omits the funcoid order, typing, lifted-map notation, and completeness hypotheses for an arbitrary join.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3442, "problem_number": "OPG-60019", "title": "A funcoid related to directed topological spaces", "statement": "Conjecture Let $R$ be the complete funcoid corresponding to the usual topology on extended real line $[-\\infty,+\\infty] = \\mathbb{R}\\cup\\{-\\infty,+\\infty\\}$. Let $\\geq$ be the order on this set. Then $R\\sqcap^{\\mathsf{FCD}}\\mathord{\\geq}$ is a complete funcoid.\n\nProposition It is easy to prove that $\\langle R\\sqcap^{\\mathsf{FCD}}\\mathord{\\geq}\\rangle \\{x\\}$ is the infinitely small right neighborhood filter of point $x\\in[-\\infty,+\\infty]$.\n\nIf proved true, the conjecture then can be generalized to a wider class of posets.", "background": "Source: Open Problem Garden. Original node ID: 60019. URL: http://www.openproblemgarden.org/op/a_funcoid_related_to_directed_topological_spaces.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_funcoid_related_to_directed_topological_spaces\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: July 28th, 2016 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\n* Blog post\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Blog post: https://portonmath.wordpress.com/2016/07/28/funcoid-related-directed-topologies/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 6.\n\nAttempt notes:\nTarget:\nMake progress on \"A funcoid related to directed topological spaces\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** OPG and SciLag reproduce the conjecture without a solution, and no independent source proving completeness of the directed-neighborhood funcoid was found.\n\n**Verified partial progress.**\n\n- The source records the singleton evaluation as the infinitesimal right-neighborhood filter.\n- The singleton evaluation does not establish the global completeness property requested by the conjecture.\n\n**Full solution or refutation.**\n\nNo verified proof or counterexample was located; the external and version-sensitive definitions prevent a firmer status.\n\n**What remains.**\n\nFix definitions of complete funcoid, FCD meet, and the funcoid induced by the order relation, then verify completeness globally, including endpoint cases.\n\n**Sources checked.**\n\n- Open Problem Garden, A funcoid related to directed topological spaces (checked 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/a_funcoid_related_to_directed_topological_spaces\n Evidence used: Displays the conjecture with no posted solution.\n- SciLag P-181201.4, A funcoid related to directed topological spaces (checked 2026-08-17). (maintained_tracker): https://www.scilag.net/problem/P-181201.4\n Evidence used: Independently mirrors the author-posted problem and reports no solutions or remarks.\n- Victor Porton, Algebraic General Topology, current author manuscript (checked 2026-08-17). (primary): https://math.portonvictor.org/binaries/volume-1-texmacs.pdf\n Evidence used: Provides the underlying nonstandard definitions but no verified resolution was located.\n\n**Review notes.** Completeness, the FCD meet, and endpoint conventions are not defined in the record.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3443, "problem_number": "OPG-60020", "title": "Outward reloid of composition vs composition of outward reloids", "statement": "Conjecture For every composable funcoids $f$ and $g$ $$(\\mathsf{RLD})_{\\mathrm{out}}(g\\circ f)\\sqsupseteq(\\mathsf{RLD})_{\\mathrm{out}}g\\circ(\\mathsf{RLD})_{\\mathrm{out}}f.$$", "background": "Source: Open Problem Garden. Original node ID: 60020. URL: http://www.openproblemgarden.org/op/outward_reloid_of_composition_vs_composition_of_outward_reloids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/outward_reloid_of_composition_vs_composition_of_outward_reloids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: outward reloid\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: September 10th, 2016 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\n* Blog post\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Blog post: https://portonmath.wordpress.com/2016/09/10/conjecture-outward-funcoids/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 4.\n\nAttempt notes:\nTarget:\nMake progress on \"Outward reloid of composition vs composition of outward reloids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The 2016 one-sided outward-reloid composition inequality remains listed as open; an older stronger equality is inconsistently indexed by OPG as both open and solved, with no checked proof, so it cannot be used as a resolution.\n\n**Verified partial progress.**\n\n- The current OPG open index retains the exact one-sided inclusion with no comments.\n- An older author-posted equality exists, but OPG's contradictory index labels and the later weakening make its status unusable without a proof and fixed definitions.\n\n**Full solution or refutation.**\n\nNo independent proof or counterexample to the exact displayed inclusion was verified.\n\n**What remains.**\n\nFreeze the outward-reloid and composition definitions, reconcile the 2007 equality entry with the 2016 inequality, and supply a checkable proof or counterexample.\n\n**Sources checked.**\n\n- Open Problem Garden, Outward reloid of composition vs composition of outward reloids (checked 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/outward_reloid_of_composition_vs_composition_of_outward_reloids\n Evidence used: Lists the precise one-sided inclusion as an open problem without a posted solution.\n- Open Problem Garden, Distributivity of outward reloid over composition of funcoids / solved-problems index (checked 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/view/solved\n Evidence used: Indexes an older stronger equality as solved, while other OPG indexes simultaneously call it open; no proof was visible in the checked material.\n- Victor Porton, Algebraic General Topology, current author manuscript (checked 2026-08-17). (primary): https://math.portonvictor.org/binaries/volume-1-texmacs.pdf\n Evidence used: Contains the evolving framework needed to interpret outward reloids and composition.\n\n**Review notes.** The old and new author-posted statements differ (equality versus one inclusion), and OPG's status indexes contradict each other.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3444, "problem_number": "OPG-60024", "title": "A diagram about funcoids and reloids", "statement": "Define for posets with order $\\sqsubseteq$:\n\n- $\\Phi_{\\ast} f = \\lambda b \\in \\mathfrak{B}: \\bigcup \\{ x \\in \\mathfrak{A} \\mid f x \\sqsubseteq b \\}$;\n\n- $\\Phi^{\\ast} f = \\lambda b \\in \\mathfrak{A}: \\bigcap \\{ x \\in \\mathfrak{B} \\mid f x \\sqsupseteq b \\}$.\n\nNote that the above is a generalization of monotone Galois connections (with $\\max$ and $\\min$ replaced with suprema and infima).\n\nThen we have the following diagram:\n\nWhat is at the node \"other\" in the diagram is unknown.\n\nConjecture \"Other\" is $\\lambda f\\in\\mathsf{FCD}: \\top$.\n\nQuestion What repeated applying of $\\Phi_{\\ast}$ and $\\Phi^{\\ast}$ to \"other\" leads to? Particularly, does repeated applying $\\Phi_{\\ast}$ and/or $\\Phi^{\\ast}$ to the node \"other\" lead to finite or infinite sets?", "background": "Source: Open Problem Garden. Original node ID: 60024. URL: http://www.openproblemgarden.org/op/a_diagram_about_funcoids_and_reloids.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/a_diagram_about_funcoids_and_reloids\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: Galois connections\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: November 26th, 2016 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nThe known part of the diagram is considered in this file.\n\nBibliography:\nBlog post\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n- this file: http://www.mathematics21.org/binaries/addons.pdf\n\nBibliography links:\n- Blog post: https://portonmath.wordpress.com/2016/11/26/new-diagram/\n\nComments:\n- November 29th, 2016 | porton | The value of node \"other\": It seems that the node \"other\" is not $\\lambda f\\in\\mathsf{FCD}: \\top$.\n\nI conjecture $\\langle \\Phi_{\\ast} (\\mathsf{RLD})_{\\operatorname{out}} \\rangle f = (\\mathsf{FCD}) f$ where $f$ is the reloid defined by the cofinite filter on $A \\times B$ and thus $\\langle (\\mathsf{FCD}) f \\rangle \\{ x \\} = \\bot$ for all singletons $\\{ x \\}$ and $\\langle (\\mathsf{FCD}) f \\rangle p = \\top$ for every nontrivial atomic filter $p$.\n\nThis is my very recent thoughts and yet needs to be checked.\n\n-- Victor Porton - http://www.mathematics21.org\n- November 26th, 2016 | porton | The diagram was with an error: My diagram was with an error. I have uploaded a corrected version of the diagram.\n\n--\n\nVictor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 15.\n\nAttempt notes:\nTarget:\nMake progress on \"A diagram about funcoids and reloids\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The imported record omits the defining diagram, the author reported an error in the original diagram, and the author later said the displayed guess for 'other' appears false while offering only an unchecked replacement.\n\n**Verified partial progress.**\n\n- The OPG page preserves a corrected diagram image absent from the imported statement.\n- On 2016-11-29 the author stated that 'other' seems not to equal the displayed constant-top map and proposed a different unverified candidate.\n\n**Full solution or refutation.**\n\nThe original guess was informally withdrawn, but there is no proof, counterexample, settled replacement, or well-defined iteration question in the diagram-free record.\n\n**What remains.**\n\nRecover and version the corrected diagram, define the node types and Phi operations, formally settle the original candidate, and analyze the orbit of the actual 'other' node.\n\n**Sources checked.**\n\n- Open Problem Garden, A diagram about funcoids and reloids (checked 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/a_diagram_about_funcoids_and_reloids\n Evidence used: Contains the diagram placeholder, the author's correction notice, and the comment withdrawing the displayed guess in favor of an unchecked one.\n- Victor Porton, Algebraic General Topology, current author manuscript (checked 2026-08-17). (primary): https://math.portonvictor.org/binaries/volume-1-texmacs.pdf\n Evidence used: Provides the external funcoid/reloid framework required to type the diagram.\n\n**Review notes.** Material source defect: the referenced diagram is missing from the imported statement, so 'other' and the proposed iterations are not self-contained.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3445, "problem_number": "OPG-60026", "title": "Which outer reloids are equal to inner ones", "statement": "Warning: This formulation is vague (not exact).\n\nQuestion Characterize the set $\\{f\\in\\mathsf{FCD} \\mid (\\mathsf{RLD})_{\\mathrm{in}} f=(\\mathsf{RLD})_{\\mathrm{out}} f\\}$. In other words, simplify this formula.\n\nThe problem seems rather difficult.", "background": "Source: Open Problem Garden. Original node ID: 60026. URL: http://www.openproblemgarden.org/op/what_outer_reloids_are_equal_to_inner_ones.\n\nSource subject path: Topology.\n\nSource importance: Medium ✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/what_outer_reloids_are_equal_to_inner_ones\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Importance: Medium ✭✭\n- Recommended for undergraduates: no\n- Posted: December 1st, 2016 by porton\n\nProblem-page discussion:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography:\nBlog post\n\nDiscussion links:\n- Algebraic General Topology: http://www.mathematics21.org/algebraic-general-topology.html\n\nBibliography links:\n- Blog post: https://portonmath.wordpress.com/2016/12/01/open-problem/\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 1.\n\nAttempt notes:\nTarget:\nMake progress on \"Which outer reloids are equal to inner ones\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The source itself calls the formulation vague, and no independent intrinsic characterization of funcoids with equal inner and outer reloids was found.\n\n**Verified partial progress.**\n\n- The maintained OPG page still contains only the characterization request and no comments.\n- The author's manuscript defines the relevant constructions, but no exact-title/formula search located a settled characterization.\n\n**Full solution or refutation.**\n\nNo checkable resolution was found, and the requested notion of 'characterize' or 'simplify' is not a terminal mathematical criterion.\n\n**What remains.**\n\nFix the funcoid category and manuscript version, specify an intrinsic target characterization, and prove necessary and sufficient conditions for equality of inner and outer reloids.\n\n**Sources checked.**\n\n- Open Problem Garden, Which outer reloids are equal to inner ones (checked 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/what_outer_reloids_are_equal_to_inner_ones\n Evidence used: Explicitly warns that the formulation is vague and supplies no posted answer.\n- Victor Porton, Algebraic General Topology, current author manuscript (checked 2026-08-17). (primary): https://math.portonvictor.org/binaries/volume-1-texmacs.pdf\n Evidence used: Provides the version-sensitive definitions of funcoids and inner/outward reloids.\n\n**Review notes.** The source expressly says the formulation is vague; 'characterize' and 'simplify' are not specified.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 1, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 1, "level": 1, "name": "L1: Tractable", "description": "Problems that may be within reach with current techniques. Reserved for future additions.", "color_class": "text-green-600 bg-green-50 border-green-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3446, "problem_number": "OPG-60043", "title": "Several ways to apply a (multivalued) multiargument function to a family of filters", "statement": "Problem Let $\\mathcal{X}$ be an indexed family of filters on sets. Which of the below items are always pairwise equal?\n\n1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters $\\mathcal{X}$.\n\n2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters $\\mathcal{X}$.\n\n3. $\\bigcap_{F\\in\\operatorname{up}^{\\mathrm{FCD}}\\prod^{\\mathrm{Strd}}\\mathcal{X}}\\langle f \\rangle F$.", "background": "Source: Open Problem Garden. Original node ID: 60043. URL: http://www.openproblemgarden.org/op/several_ways_to_apply_a_multivalued_multiargument_function_to_a_family_of_filters.\n\nSource subject path: Topology.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/several_ways_to_apply_a_multivalued_multiargument_function_to_a_family_of_filters\n- Author(s): Porton, Victor\n- Subject(s): Topology\n- Keywords: funcoid; function; multifuncoid; staroid\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: December 24th, 2019 by porton\n\nBibliography:\nSee Algebraic General Topology for definitions of used concepts.\n\nBibliography links:\n- Algebraic General Topology: https://mathematics21.org/algebraic-general-topology-and-math-synthesis/\n\nComments:\n- January 15th, 2020 | porton | There was an error in the problem statement: I did an error in the problem statement (corrected). We need to consider the upgraded staroid, not just staroid.\n\n-- Victor Porton - http://www.mathematics21.org\n- January 15th, 2020 | porton | The error was more severe: The first item of the problem was entirely wrong. I removed it.\n\n-- Victor Porton - http://www.mathematics21.org\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 2.\n\nAttempt notes:\nTarget:\nMake progress on \"Several ways to apply a (multivalued) multiargument function to a family of filters\" in Topology, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The funcoid/filter expression depends on undefined starred reloidal products and FCD/Strd conventions; no verifiable equality theorem for the exact text was found.\n\n**Verified partial progress.**\n\nNo distinct partial result was verified beyond the status assessment above.\n\n**Full solution or refutation.**\n\nIt cannot safely be matched to a standard filter-product identity.\n\n**What remains.**\n\nSupply formal definitions and an accessible intended source before evaluating equality.\n\n**Sources checked.**\n\n- Open Problem Garden, Several ways to apply a multiargument function to a family of filters (accessed 2026-08-17). (maintained_tracker): https://www.openproblemgarden.org/op/several_ways_to_apply_a_multivalued_multiargument_function_to_a_family_of_filters\n Evidence used: Preserves the notation but provides no citable general resolution.\n\n**Review notes.** Formulation defect flagged; no reconstruction.\n\nThis is a dated literature triage; an open classification is not proof that no later or unindexed result exists.\n", "difficulty_level_id": 2, "status": "open", "category_id": 7, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 3447, "problem_number": "OPG-581", "title": "Rendezvous on a line", "statement": "Problem Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum speed of 1. What is the infimum expected meeting time $R$ (first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy?", "background": "Source: Open Problem Garden. Original node ID: 581. URL: http://www.openproblemgarden.org/op/rendezvous_on_a_line.\n\nSource subject path: Unsorted.\n\nSource importance: High ✭✭✭.\n\nDiscussion and literature:\nSource-derived literature/context:\n- Source: http://www.openproblemgarden.org/op/rendezvous_on_a_line\n- Author(s): Alpern, Steve\n- Subject(s): Unsorted\n- Keywords: game theory; optimization; rendezvous\n- Importance: High ✭✭✭\n- Recommended for undergraduates: no\n- Posted: September 21st, 2007 by mdevos\n\nProblem-page discussion:\nThis is one of a handful of rendezvous problems where two players must find one another in a certain structured domain. See [AG2] for a thorough development of this subject. This is a symmetric rendezvous problem since each player is forced to adopt the same strategy. If we drop this constraint, Alpern and Gal [AG] have shown that the inf expected meeting time is 3.25.\n\nHan, Du, Vera, and Zuluaga [HDVZ] have shown that strategies in which the players move at maximum speed and only change direction at integer times dominate among all possible strategies - thus reducing this problem to a discrete one. These same authors improve upon a series of results by tightening the upper and lower bounds, proving $4.1520 < R < 4.2574$. Further, they conjecture $R=4.25$.\n\nBibliography:\n*[A] S. Alpern, The rendezvous search problem. SIAM J. Control Optim. 33 (1995), no. 3, 673--683 MathSciNet\n\n[AG1] S. Alpern and S. Gal, Rendezvous search on the line with distinguishable players. SIAM J. Control Optim. 33 (1995), no. 4, 1270--1276. MathSciNet\n\n[AG2] S. Alpern and S. Gal, The theory of search games and rendezvous. International Series in Operations Research & Management Science, 55. Kluwer Academic Publishers, Boston, MA, 2003. MathSciNet\n\n[HDVZ] Q. Han, D. Du, J. C. Vera, and L. F. Zuluaga, Improved bounds for the symmetric rendezvous search problem on the line\n\nBibliography links:\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1327232\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=1339065\n- MathSciNet: http://www.ams.org/mathscinet-getitem?mr=2005053\n- Improved bounds for the symmetric rendezvous search problem on the line: http://www.optimization-online.org/DB_FILE/2006/05/1400.pdf\n\nComputation notes:\nFinite compute signal:\n- No local finite computation has been run for this source-derived note.\n- This record was generated from the Open Problem Garden page text, discussion, bibliography, and comments.\n- LaTeX expressions extracted from the source text: 3.\n\nAttempt notes:\nTarget:\nMake progress on \"Rendezvous on a line\" in Unsorted, starting from the source statement and discussion.\nStep A: Normalize definitions\nExtract the precise objects, parameters, quantifiers, and known examples from the problem statement.\nStep B: Literature alignment\nUse the discussion, bibliography, and comments above to identify known bounds, reductions, solved special cases, or claimed resolutions.\nStep C: Attack route selection\nChoose between proof search, counterexample construction, finite computation, or literature verification depending on the problem type.\nConcrete Blocking Lemma (Most Critical):\nA problem-specific lemma is still needed; this note has not supplied a new proof beyond the source-derived context.\n\n\n## Literature review (checked 2026-08-17)\n\n**Status:** partially_solved \n**Classification:** PARTIAL-PROGRESS\n\n**Current literature assessment.** The exact symmetric rendezvous value on the undirected line remains unknown; the rigorous interval 4.1520", "difficulty_level_id": 2, "status": "open", "category_id": 20, "set_id": 12, "view_count": 0, "favorite_count": 0, "created_at": "2026-05-13T00:00:00Z", "updated_at": "2026-05-13T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 12, "name": "opengarden", "display_name": "OpenGarden", "description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.", "slug": "opengarden", "order_index": 12, "created_at": "2026-07-31T15:26:25.671Z" } }, { "id": 600001, "problem_number": "AMR-005-0001", "title": "Baker's Dozen — Commuting billiard maps", "statement": "Consider two nested convex plane domains, and let $T_1,T_2$ be their billiard ball maps on the oriented lines intersecting both domains. If $T_1\\circ T_2=T_2\\circ T_1$, are the two domains bounded by confocal ellipses? Resolve the corresponding multidimensional question for inner and outer billiards.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Conjecture 1\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The problem is solved in the literature, with one sub-case apparently still open: 1. **Planar inner case:** commuting billiard ball maps of two nested convex domains with piecewise $C^4$-smooth boundaries $\\Rightarrow$ confocal ellipses (Glutsyuk 2017). The proof goes through complexified billiards: commuting forces a 4-reflective complex analytic pseudo-billiard structure near the curves, and the classification of 4-reflective germs forces the curves to be confocal conics. 2. **Higher-dimensional inner case ($d\\ge3$):** commuting actions by reflections for nested strictly convex $C^2$ billiards $\\Rightarrow$ confocal ellipsoids (Glutsyuk 2020), via Berger's theorem (in dimension $\\ge 3$ only quadrics admit caustics); also extended to spaces of constant curvature. 3. **Planar outer case:** commuting dual billiard maps $\\Rightarrow$ concentric homothetic ellipses (Tabachnikov 1994) — predates the list. 4. **Higher-dimensional outer case:** no resolution found; appears to remain open." }, { "id": 600002, "problem_number": "AMR-005-0002", "title": "Baker's Dozen — Periodic billiard trajectories", "statement": "Are there smooth convex curves, other than ellipses, simultaneously admitting one-parameter families of $p$- and $q$-periodic billiard trajectories for $p\\ne q$? Ask the analogous question for outer billiards.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Problem 1\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem is **not solved**, and I could not solve it; the honest classification is partial progress via reformulation plus a precise map of how close the literature comes: 1. **Near ellipses the answer is \"no\" in a strong sense** (Kaloshin–Sorrentino 2018): a single one-parameter family of $q$-periodic orbits, $q\\ge 3$, already characterizes ellipses locally among $C^\\infty$ tables. So any counterexample to Problem 1 must be far (in a $C^\\infty$ sense) from every ellipse. 2. **The simplest case $(2,3)$ is deformationally settled near the circle** (Kaloshin–Koudjinan 2021): no non-trivial deformation of the circle preserves both the 2-periodic family (constant width to first order) and the 3-periodic family; the same holds for $(2, 2l+1)$. Thus a non-circular constant-width curve with a 3-periodic family, if it exists, is isolated from the circle in a deformation sense. 3. **Global results all need strictly stronger hypotheses**: full foliation of the phase cylinder (Bialy 1993 $\\Rightarrow$ disk), or a 1/4-caustic plus foliation below it with central symmetry (Bialy–Mironov 2022 $\\Rightarrow$ ellipse). Two isolated rational invariant circles are not known to force a foliation…" }, { "id": 600003, "problem_number": "AMR-005-0003", "title": "Baker's Dozen — Lorentzian Birkhoff theorem", "statement": "For billiards inside an oval in the Lorentz plane with metric $ds^2=dx^2-dy^2$, is there an analogue of Birkhoff's theorem, with separate existence statements for space-like and time-like periodic trajectories? What happens for multidimensional pseudo-Euclidean billiards?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Section 3 unnumbered questions\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem is **open in general**, with the following state of knowledge and my analysis of the obstruction. **What is known.** (i) $n=2$: at least one space-like and one time-like 2-periodic orbit for any Lorentz oval, and $\\ge p$ / $\\ge q$ diameters in $\\mathbb{R}^{p,q}$ (Khesin–Tabachnikov 2009). (ii) Ellipses and ellipsoids: complete Birkhoff-type picture for all $n$ and all causal types, with Cayley-type existence criteria (Dragović–Radnović 2012; Adabrah–Dragović–Radnović 2019). (iii) Multidimensional Euclidean bounds (Farber–Tabachnikov 2002) have no known pseudo-Euclidean counterpart. **My analysis — why the classical proof does not transfer.** Let $\\gamma$ be a smooth strictly convex oval in $\\mathbb{R}^{1,1}$, $ds^2=dx^2-dy^2$. 1. *Causal decomposition.* The tangent direction map $\\gamma\\cong S^1\\to\\mathbb{RP}^1$ has degree 1, so each of the two null directions occurs as a tangent exactly twice: $\\gamma$ splits into 4 arcs, two with space-like tangent ($|dy/dx|<1$, top and bottom) and two with time-like tangent (left and right). Each arc has total turning $\\pi/2$; by strict convexity the direction of any chord lies strictly between the tangent directions at its…" }, { "id": 600004, "problem_number": "AMR-005-0004", "title": "Baker's Dozen — Periodic hyperbolic outer billiards", "statement": "Does every polygonal outer billiard in the hyperbolic plane have periodic orbits, possibly lying on the circle at infinity?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Conjecture 2\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Synthesis of the rigorous state of the art, with a structural reformulation. 1. **Reformulation.** In $\\mathbb{H}^2$ the reflection of $x$ in a support vertex $v$ is the half-turn $H_v$ about $v$ (an orientation-preserving isometry). Hence the outer billiard map $T$ is a piecewise orientation-preserving isometry, and an orbit with periodic itinerary through vertices $v_1,\\dots,v_k$ closes iff the composition $H_{v_k}\\circ\\cdots\\circ H_{v_1}$ has a fixed point realizing that itinerary — i.e. iff this composition is *elliptic* (a rotation) with fixed point in the appropriate continuity cell, or *parabolic/hyperbolic* with an (attracting) fixed point on the circle at infinity. The conjecture thus asks: for every convex polygon, does some periodic itinerary produce a non-hyperbolic composition (or a hyperbolic one with fixed points at infinity)? This is the hyperbolic analogue of the \"elliptic composition\" mechanism behind Culter's Euclidean theorem. 2. **Settled cases.** - *Large polygons* (in particular all triangles with $H>1$): all interior orbits escape to infinity, and $f$ has an attracting $n$-periodic orbit on the circle at infinity — the conjecture holds, with the periodic…" }, { "id": 600005, "problem_number": "AMR-005-0005", "title": "Baker's Dozen — Completely periodic hyperbolic outer billiards", "statement": "Describe the polygonal outer billiard tables in the hyperbolic plane for which every orbit is periodic.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Section 4 second unnumbered problem\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The literature state can be synthesized as follows. **Known completely periodic tables.** (i) Right-angled regular $n$-gons, $n\\ge 5$ (DT03); more generally (ii) polygonal tiles of \"regular two-piece tilings\" of $\\mathbb{H}^2$ (Dogru–Fischer–Munteanu 2015). In both cases the proof strategy is tiling-based: the table is a fundamental domain (or a union of two tiles) of a discrete reflection group, and compatibility of $T^2$ with the group confines every orbit to a compact set of tiles on which the piecewise isometry has uniformly finite order. **Known obstruction.** \"Large\" polygons in the sense of DT03 (side-geodesics pairwise ultraparallel): every orbit escapes to the circle at infinity, so no complete periodicity — indeed no periodic orbits in $\\mathbb{H}^2$ whatsoever. **A necessary condition from the dynamics at infinity (my synthesis, not a published theorem).** Write $R_i$ for the half-turn (elliptic involution) about vertex $v_i$, and $A_i$ for the exterior region on which $T = R_i$. For distinct $i, j$ the product $R_iR_j$ is **loxodromic**: a hyperbolic translation by $2\\,d(v_i,v_j)$ along the geodesic through the two vertices, with two fixed points on…" }, { "id": 600006, "problem_number": "AMR-005-0006", "title": "Baker's Dozen — Periodic multidimensional outer-billiard trajectories", "statement": "For a smooth strictly convex hypersurface $M\\subset\\mathbb{R}^{2n}$, find lower bounds for the number of $p$-periodic outer-billiard orbits for values of $p$ other than $3$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Problem 2\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem stays open, but the analysis sharpens it considerably. **1. The variational reduction works; compactness is the sole gap.** For strictly convex $M$, the Gauss map identifies $M\\cong S^{2n-1}$, and the outer billiard map is a twist-type symplectic map on the space of tangent lines, with a generating function $h(u,v)$ ($u,v\\in S^{2n-1}$, essentially the symplectic area of the triangle formed by the two tangent lines). Hence $p$-periodic orbits are exactly the critical $\\mathbb{Z}_p$-orbits of $$F(x_1,\\dots,x_p)=\\sum_{i=1}^{p} h(x_i,x_{i+1})$$ on the cyclic configuration space $\\mathrm{Conf}(S^{2n-1},p)=\\{x_i\\ne x_{i+1}\\}$. The cohomology (and $\\mathbb{Z}_p$-equivariant cohomology, for prime $p$) of this space is computed in Farber–Tabachnikov (2002); feeding it into Morse–LS theory would yield a bound of the shape $\\ge (p-1)(2n-2)+1$ for prime $p$ — the exact analogue of the inner-billiard and Finsler-billiard results. For inner billiards the crucial extra input is an a priori estimate, coming from the triangle inequality for the perimeter functional, showing that critical polygons stay a uniform distance away from the collision diagonals (so that noncompactness of…" }, { "id": 600007, "problem_number": "AMR-005-0007", "title": "Baker's Dozen — A converse Desargues theorem", "statement": "Let $f(x,y)$ be a polynomial for which $0$ is a nonsingular value, let $\\gamma$ be an oval component of $f(x,y)=0$, and assume that $\\gamma_\\varepsilon=\\{f(x,y)=\\varepsilon\\}$ for $\\varepsilon>0$ foliates an outer neighborhood of $\\gamma$. Suppose that for every tangent line $\\ell$ to $\\gamma$, its intersections with the curves $\\gamma_\\varepsilon$ define a local projective involution on $\\ell$. Prove that $\\gamma$ is an ellipse and the curves $\\gamma_\\varepsilon$ form a pencil of conics.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Problem 3\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "**Theorem (solved).** Under the hypotheses, $\\gamma$ is an ellipse and the curves $\\gamma_\\varepsilon$ form a pencil of conics — precisely the pencil $\\{Q=c\\}$ generated by the ellipse $\\gamma=\\{Q=c_0\\}$ and the double line at infinity, and $f=P\\circ Q$ for a one-variable polynomial $P$. The proof has three steps: (1) the local projective involution on each tangent line must be the central symmetry about the contact point — a purely algebraic consequence of $f$ being a polynomial (a Möbius involution fixing the contact point is $x\\mapsto -x/(1-\\lambda x)$, and invariance of the polynomial $f|_\\ell$ forces $\\lambda=0$ by a degree count); (2) the foliation is then invariant under the outer billiard map, and Tabachnikov's Theorem 1 (Pacific J. Math. 235, 2008, 89–92) gives that $\\gamma$ is an ellipse; (3) for the circle normalization, the outer billiard map is the integrable twist map $(\\theta,r)\\mapsto(\\theta+2\\arccos(1/r),r)$, a rotation-number argument shows every invariant leaf is a concentric circle, and polynomiality gives $f=P(x^2+y^2)$, whence the pencil." }, { "id": 600008, "problem_number": "AMR-005-0008", "title": "Baker's Dozen — Chains of null geodesics", "statement": "For the ellipsoid $x^2/a+y^2/b+z^2/c=1$ in Minkowski space with metric $dx^2+dy^2-dz^2$, find conditions on $a,b,c>0$ ensuring the existence of an $(n,r)$-chain of alternating left and right null geodesics.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Problem 4\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Not solved explicitly; the following was established rigorously by reasoning from GKT (2007): - **Implicit criterion:** $(n,r)$-chains exist iff the rotation number $\\rho(a,b,c)=c_0/L$ equals $r/n$, where $c_0,L$ are explicit Abelian integrals on the elliptic curve $E_{a,b,c}$ above. - **Torsion formulation:** equivalently, the one-step point $Q(a,b,c)\\in E_{a,b,c}$ is torsion of order dividing $n$ (with the component selected by $r$). This converts the problem into Cayley-type conditions $\\psi_n(Q)=0$, matching what Cayley/Griffiths–Harris do for the classical porism, and matching the announced (unpublished) Garcia–Wüstholz arithmetic characterization. - **Literature synthesis:** the Poncelet part is a theorem (GKT 2007); explicit conditions exist in the literature only for neighbouring systems (lightlike billiards within conics in the Minkowski plane — Adabrah–Dragović–Radnović 2019; on the hyperboloid of one sheet — Gąsiorek–Radnović 2021; within ellipsoids in pseudo-Euclidean spaces — Dragović–Radnović 2012), not for the null-geodesic chains of Problem 4." }, { "id": 600009, "problem_number": "AMR-005-0009", "title": "Baker's Dozen — Origami hyperbolic paraboloids", "statement": "Assume the origami hyperbolic-paraboloid pattern has invisible straight folds along a chosen diagonal of each elementary trapezoid. What is the shape of the piecewise-linear surface obtained by folding? What is obtained from analogous constructions with other patterns of fold lines?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Section 8 unnumbered questions\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Question 1 is resolved in the literature (post-2015): with one added diagonal fold per trapezoid the pattern becomes a generically 1-DOF triangulated origami, and its folded shape — at any folding depth and independently of the diagonal pattern (among the symmetric schemes) and of the spacing of the squares — converges, as the pleats refine, to the genuine hyperbolic paraboloid $z=k(x^2-y^2)$, with $k$ a monotone function of the folding angle (Liu–Tachi–Paulino, *Nat. Commun.* 10:4238, 2019). Bistability with two mirror-symmetric stable states is proved in the same work. Question 2 (other fold-line patterns, e.g. circular pleats) is only partially addressed (N-gon hypar extensions; framework remarks) and remains open." }, { "id": 600010, "problem_number": "AMR-005-0010", "title": "Baker's Dozen — Unbounded unicycle tracks", "statement": "Let $\\gamma$ be a smooth arc agreeing to all orders with the $x$-axis at endpoints $(0,0)$ and $(1,0)$, and iterate $T(\\gamma)=\\gamma+\\gamma'$. Unless $\\gamma$ is a straight segment, is the resulting infinite curve unbounded in amplitude, not a graph, and not embedded?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Conjecture 3\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The conjecture is **partially resolved in the literature** as of October 2025: 1. **Clause (ii) \"not a graph\" — settled affirmatively** (Molodyk, arXiv:2510.10388, Theorem 4.2): every non-trivial seed eventually produces an iterate that is not a graph of a function. 2. **A strong quantitative substitute for unboundedness** (ibid., Theorem 4.3): the horizontal amplitude grows linearly, $n-c_1 \\le H(\\gamma_n) \\le 2n-c_2$, so in the horizontal direction the track escapes every vertical strip; lengths of the arcs tend to infinity. 3. **Clause (i) \"not contained in any horizontal strip\" — open**: $V(\\gamma_n)$ is known to be strictly increasing, but unboundedness is Conjecture 4.4 of the 2025 preprint. 4. **Clause (iii) \"not embedded\" — open**: even the weaker statement that the full track $\\mathcal{T}$ self-intersects is listed as open (Conjectures 4.5–4.6 ibid.). Levi–Tabachnikov's growth of zeros, extrema and inflection points, plus numerical evidence (Wagon 2025), strongly support it." }, { "id": 600011, "problem_number": "AMR-005-0011", "title": "Baker's Dozen — Convex tangent-segment iteration", "statement": "Given an oriented oval $\\gamma$, let $\\gamma_1$ be the locus of endpoints of its oriented unit tangent segments, and iterate this construction. If every iterate is convex, must $\\gamma$ be a circle?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Conjecture 4\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "No full solution. Rigorous partial progress: (i) exact reformulation as a 1-dimensional curvature dynamical system with the sharp convexity criterion $1+\\rho^2-\\rho'>0$; (ii) complete proof of the linearized model $F\\mapsto F+F'$; (iii) rigorous linear analysis showing circles are isolated, linearly unstable fixed points of the rescaled dynamics, with all non-trivial Fourier modes ($|k|\\ge2$) amplified by $\\sqrt{(R^2+k^2)/(R^2+1)}$ per step; (iv) numerical evidence that generic non-circular ovals lose convexity within a handful of iterations via a sharpening cascade. Combined with Levi–Tabachnikov's theorem that complexity (numbers of extrema/inflections) strictly increases under the same map for open arcs, the conjecture is very plausible but, to my knowledge, still open." }, { "id": 600012, "problem_number": "AMR-005-0012", "title": "Baker's Dozen — Self-dual curves and surfaces", "statement": "Extend the known results on projectively self-dual polygons and curves to projectively self-dual polyhedra and to projectively self-dual polygons in multidimensional projective spaces.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Problem 5\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem splits into two halves with different statuses: **(a) Self-dual polygons in $\\mathbb{RP}^k$ — partially solved in the literature.** Chavez-Caliz (2021/2023) defines $m$-self-dual $n$-gons in $\\mathbb{P}^k$, constructs them explicitly, and computes $\\dim \\mathcal{M}_{m,n,k}$ in specific cases; the general dimension formula and her higher-dimensional Clebsch conjecture remain open. **(b) Self-dual polyhedra in $\\mathbb{RP}^3$ — open; modest original progress here.** A polyhedron $P \\subset \\mathbb{RP}^3$ is projectively self-dual if a correlation $g: \\mathbb{RP}^3 \\to (\\mathbb{RP}^3)^*$ takes $P$ to its dual $P^*$. Two observations, provable by hand: 1. *Trivial example:* every tetrahedron is projectively self-dual (its dual is a tetrahedron, and all tetrahedra are projectively equivalent). 2. *Pyramid construction (new, elementary).* For every odd $n \\geq 5$, every convex $n$-self-dual $n$-gon of Fuchs–Tabachnikov gives rise to a projectively self-dual polyhedron: the pyramid over it. *Proof sketch.* Let $Q \\subset H \\cong \\mathbb{RP}^2$ be an $n$-self-dual $n$-gon with respect to a polarity, $n$ odd, and let $a \\notin H$ be the apex; write $\\Pi(Q,a)$ for the pyramid.…" }, { "id": 600013, "problem_number": "AMR-005-0013", "title": "Baker's Dozen — Configuration theorems", "statement": "Find conceptual proofs and possible generalizations of the new projective-geometry configuration theorems described in the source. In particular, prove that when the outer dodecagon in Figure 5 is inscribed in a conic, the inner dodecagon is also inscribed in a conic.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Section 11 unnumbered problems\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The problem (including the headline dodecagon statement) is **solved**: all eight Schwartz–Tabachnikov configuration theorems are theorems. The dodecagon case — the only one open at the time of the source list — was proved by **Fedor Nilov** using a planar projection of a one-sheeted hyperboloid; this is documented in Tabachnikov's 2016 survey (arXiv:1607.04758, §\"Configurations\"), which also notes that the proof was never published. No published proof of the dodecagon theorem appears to exist as of this search (checked: arXiv full-text search, Semantic Scholar, citing literature of both source papers, Nilov's own publications)." }, { "id": 600014, "problem_number": "AMR-005-0014", "title": "Baker's Dozen — A totally skew disc", "statement": "Does there exist a totally skew embedded $3$-disc in $\\mathbb{R}^7$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Problem 6\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "No solution — this is a genuinely open problem. Summary of what is known and of my own analysis: 1. **The obstruction side is settled.** Ghomi–Tabachnikov prove that a totally skew $k$-disc in $\\mathbb{R}^{2k+1}$ exists only if $k \\in \\{1,3,7\\}$, via a reduction to nonsingular bilinear maps and Adams's Hopf-invariant-one theorem. For $k=3$ (and $k=7$) the obstruction vanishes: nonsingular bilinear maps $\\mathbb{R}^4 \\times \\mathbb{R}^4 \\to \\mathbb{R}^7$ do exist (quaternionic Hopf construction). So the problem sits exactly at the boundary where algebraic topology gives no answer either way. 2. **My analysis of the constructive side.** Fixing a basepoint and projecting onto its normal space, a totally skew $f: D^3 \\to \\mathbb{R}^7$ yields a family of tangent 3-planes $\\{T_x\\}$ that are pairwise complementary linear subspaces; writing $T_x$ as the graph of $A_x: \\mathbb{R}^3 \\to \\mathbb{R}^4$, one needs $A_x - A_y$ injective for all $x \\neq y$ (a map of the configuration space into the Stiefel manifold $V_3(\\mathbb{R}^4)$), plus the global displacement condition $f(y)-f(x) \\notin T_x + T_y$. The natural first attempt, a quadratic graph $f(x) = (x, Q(x))$ with $DQ_x(v) = B(x,v)$,…" }, { "id": 600015, "problem_number": "AMR-005-0015", "title": "Baker's Dozen — Relations among triangle areas", "statement": "For a fixed combinatorial dissection of a square into $n$ triangles, the ordered triangle areas satisfy a polynomial relation depending only on the combinatorics. What can be said about this relation, and how can its least degree be found from the combinatorics of the dissection?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Tabachnikov - A Baker's Dozen of Problems (2015)\nSource item: Problem 7\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible publisher HTML\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The problem is solved in the literature to the extent the question is posed. The definitive statements: - The relation is given by a **single irreducible homogeneous polynomial $p(T)\\in\\mathbb{Z}[a_1,\\dots,a_n]$** (up to scalar), depending only on the combinatorics of the dissection: the Monsky polynomial (paper 1). It is invariant under deformation of the dissection (papers 2, 4), can be normalized to be monic with leading coefficients $\\pm1$ in the parallelogram case (paper 4), and satisfies $p(T)\\equiv (a_1+\\cdots+a_n)^d \\pmod 2$ (paper 2) — which explains Monsky's odd/even equidissection theorem as a corollary of the shape of the relation. - **Least degree:** since the relation ideal is principal, the least degree is $\\deg p(T)$. Paper 1 gives a combinatorial algorithm that computes a lower bound on $\\deg p(T)$ and computes the exact degree in worked examples; paper 3 adds finiteness and explicit computation of all low-width relations ($\\mathcal E_l$ for $l\\le 4$), and paper 5 shows the degrees/coefficients are effectively computable for all triangulations of modest size by direct enumeration. The general qualitative answer to \"what can be said\" is thus complete; the specific…" }, { "id": 1100101, "problem_number": "AMR-010-0101", "title": "Questions in Geometric Group Theory — Q 1.1", "statement": "Suppose $G$ admits a finite $K(G,1)$. If $G$ does not contain any Baumslag-Solitar subgroups $BS(m,n)$, is $G$ necessarily hyperbolic? If $G$ embeds in a hyperbolic group, is it hyperbolic?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.1 (PDF page 1)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Both parts of Bestvina's Question 1.1 have answer **no**: - There exists a group $H$ with a finite $K(H,1)$ (indeed a finite aspherical 4-dimensional complex) that contains no Baumslag–Solitar subgroup $BS(m,n)$ yet is not hyperbolic. - The same $H$ embeds in a hyperbolic group $G$ ($\\pi_1$ of a filled, CAT($-\\kappa$) 5-dimensional pseudo-manifold, $\\mathrm{cd}(G)=5$). Reference: Italiano–Martelli–Migliorini, Invent. Math. 231 (2023), 1–38, arXiv:2105.14795, Corollaries 2 and 3." }, { "id": 1100102, "problem_number": "AMR-010-0102", "title": "Questions in Geometric Group Theory — Q 1.2", "statement": "Suppose G admits a finite K(G, 1), does not contain Z × Z, and whenever x ∈ G is an infinite order element such that xm and xn are conjugate, then |m| = |n|. Is G hyperbolic?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.2 (PDF page 2)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**The answer to Q 1.2 is NO.** Italiano–Martelli–Migliorini (Invent. Math. 231 (2023), 1–38; arXiv:2105.14795) construct a group G that: 1. admits a finite K(G,1) (is of type F), in fact of geometric and cohomological dimension 4; 2. contains no Z × Z (being a subgroup of a hyperbolic group); 3. is balanced: x^m conjugate to x^n with x of infinite order forces |m| = |n| (translation-length argument, above); 4. is not hyperbolic. So the class of groups satisfying Bestvina's hypotheses strictly contains the torsion-free hyperbolic groups. The same counterexample simultaneously answers Bestvina's Q 1.1 (no Baumslag–Solitar subgroups) in the negative. The construction uses circle-valued Morse theory/Bestvina–Brady-type finiteness arguments on fibering cusped hyperbolic 5-manifolds; the fiber-kernel (after suitable filling) is the non-hyperbolic type-F subgroup." }, { "id": 1100105, "problem_number": "AMR-010-0105", "title": "Questions in Geometric Group Theory — Q 1.5", "statement": "(Davis) If $G$ is word-hyperbolic, does the Rips complex $P_d(G)$ have an equivariant negatively curved metric for $d$ sufficiently large?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.5 (PDF page 2)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "No solution exists in the literature, and I could not solve it (a solution would resolve Gromov's conjecture). The rigorous synthesis: 1. **Logical position.** A positive answer to Q 1.5 for all $G$ implies Gromov's conjecture, since $P_d(G)$ with an invariant CAT($-\\kappa$) metric is a proper cocompact $G$-model (finiteness of $P_d(G)/G$ is automatic, and properness follows from Meintrup–Schick). Conversely Q 1.5 is in principle strictly stronger than Gromov's conjecture: metrics do not transfer across equivariant homotopy equivalences, so a group could be CAT(−1) on some space while its Rips complex supports no invariant CAT(−1) metric. Davis's question is thus a *canonical-model* strengthening of the curvature conjecture. 2. **Why the naive approach fails (the precise obstruction).** Any $G$-invariant piecewise-hyperbolic metric on $P_d(G)$ must satisfy Gromov's link condition: every closed geodesic in the link of every simplex must have length $\\ge 2\\pi$. The link of a $k$-simplex $\\sigma$ in $P_d(G)$ is itself a Rips-type complex — the complex of diameter-$\\le d$ subsets of $G$ whose union with $\\sigma$ still has diameter $\\le d$ — built from the \"corona\" $B(x,d)\\setminus…" }, { "id": 1100106, "problem_number": "AMR-010-0106", "title": "Questions in Geometric Group Theory — Q 1.6", "statement": "(Gromov) Does every one-ended word-hyperbolic group contain a closed hyperbolic surface subgroup?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.6 (PDF page 2)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The problem is a famous open question of Gromov, transcribed correctly from Bestvina's list, and it remains unsolved in full generality as of August 2026. The literature state is: - Solved cases: closed hyperbolic 3-manifold groups (Kahn–Markovic), graphs of free groups with cyclic edge groups (Wilton, extending Calegari and Kim–Oum for doubles), limit groups (Wilton), random groups (Calegari–Walker), various Coxeter/Artin and one-relator families (Gordon–Long–Reid; Ng 2025). - General reduction (Wilton 2018): without 2-torsion, the question reduces to rigid one-ended hyperbolic groups — those with no splitting over virtually cyclic subgroups. - No counterexample is known, and no known obstruction exists; the generic case is positive." }, { "id": 1100107, "problem_number": "AMR-010-0107", "title": "Questions in Geometric Group Theory — Q 1.7", "statement": "(Gromov) For a given n is there an example of a hyperbolic group of dimension n in which every infinite index subgroup is free? Or in which there are no (quasi-convex) subgroups with codimension ≤k for a given k ≤n−2.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.7 (PDF page 3)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open**, with a sharp literature triage: - n = 1 (free groups) and n = 2 (closed hyperbolic surface groups) are the only known examples of hyperbolic groups of dimension n with every infinite-index subgroup free. - For n ≥ 3 the answer is negative in every class where the question is understood: cubulated hyperbolic groups (Wilton, arXiv:2406.02121, which covers closed hyperbolic 3-manifold groups and small-cancellation groups), one-relator groups (Wilton's Theorem D; Gardam–Kielak–Logan for two generators), and closed hyperbolic 3-manifold groups independently (Kahn–Marković surface subgroups). - No construction of an n ≥ 3 example exists anywhere in the literature, and Wilton explicitly records the general question (his Question 0.1, and the cd ≥ 3 variants in his §6) as open. The codimension clause is likewise open in general (only the k = 1 case is settled, via property (T), under the Sageev interpretation). No solution or new theorem is claimed here; the contribution is the verified triage plus the elementary structural constraints (torsion-free, 1-ended, non-cubulated) on any hypothetical example." }, { "id": 1100108, "problem_number": "AMR-010-0108", "title": "Questions in Geometric Group Theory — Q 1.8", "statement": "(Swarup) Suppose $H$ is a finitely presented subgroup of a word-hyperbolic group $G$ and has finite index in its normalizer. Assume that there is $n>0$ such that the intersection of $n$ distinct conjugates of $H$ is always finite. Is $H$ quasiconvex in $G$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.8 (PDF page 3)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The question is **open**; I cannot solve it, but the literature plus elementary reasoning gives a clean triage. **1. The hypotheses are exactly the known necessary conditions.** For $H$ a subgroup of a hyperbolic group $G$: quasiconvex $\\Rightarrow$ $H$ finitely presented (standard), quasiconvex $\\Rightarrow$ finite height/width (GMRS 1998), and quasiconvex $\\Rightarrow$ $[\\mathrm{Comm}_G(H):H]<\\infty$, hence $[N_G(H):H]<\\infty$ (Kapovich–Short 1996). Swarup's question is precisely whether this conjunction of necessary conditions is sufficient. Degenerate cases are trivial: if $[G:H]<\\infty$ or $H$ is finite, $H$ is quasiconvex; so the content is for infinite-index infinite $H$. **2. Reductions and equivalences.** Because $[N_G(H):H]<\\infty$, the hypothesis \"$n$ distinct conjugates have finite intersection\" is the same as \"height$(H)\\le n-1$\" up to the bounded factor $[N_G(H):H]$; the $n=2$ case is almost malnormality (malnormality in the torsion-free case). Gitik's remark in the list, and Halder–Sardar twenty years later, both record that **even the (weakly) malnormal case is open**. **3. Where the difficulty lies.** Swarup's own virtual-normalizer criterion (item 8 above) shows…" }, { "id": 1100109, "problem_number": "AMR-010-0109", "title": "Questions in Geometric Group Theory — Q 1.9", "statement": "(Mitra) Let XG be a finite 2-complex with fundamental group G. Let XH be a cover corresponding to the f.p. subgroup H. Let I(x) denote the injectivity radius of XH at x. Does I(x) →∞ as x →∞imply that H is quasi-isometrically embedded in G?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.9 (PDF page 3)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**A reformulation (my partial progress).** Metrize $X_G$ so its 1-skeleton pull-back makes the universal cover $\\widetilde X$ QI to $\\mathrm{Cay}(G)$. Points of $X_H=\\widetilde X/H$ are $H$-cosets, and a based loop at the coset $Hg$ of length $\\ell$ is exactly an element $h\\in H\\setminus\\{e\\}$ with $|g^{-1}hg|_G=\\ell$. Hence, up to bounded additive constants, $$I(Hg)=\\tfrac12\\min_{h\\in H\\setminus\\{e\\}}\\,|g^{-1}hg|_G .$$ Since only finitely many elements of $G$ have length $\\le 2C$, one obtains, for **arbitrary** $G$ (no hyperbolicity needed): $$I(x)\\to\\infty \\iff \\text{for every } b\\in G\\setminus\\{e\\},\\ \\{\\,Hg : zbz^{-1}\\in H\\,\\} \\text{ is bounded in } X_H .$$ In words: **each fixed element $b\\in G$ lies in only \"$H$-boundedly many\" conjugates $z^{-1}Hz$ of $H$.** Two immediate consequences: 1. *Centralizer obstruction (necessary condition).* If $Z_G(h)$ has unbounded image in $H\\backslash G$ for some $h\\in H\\setminus\\{e\\}$ (i.e. $Hz_n\\to\\infty$ with $z_n\\in Z_G(h)$), then $I(Hz_n)\\le |h|_G/2$, so $I\\not\\to\\infty$. Hence Q 1.9 would follow from: *$H$ distorted $\\Rightarrow$ some $h\\in H\\setminus\\{e\\}$ has $Z_G(h)$ unbounded mod $H$.* This is a weak \"finite height\" condition in…" }, { "id": 1100110, "problem_number": "AMR-010-0110", "title": "Questions in Geometric Group Theory — Q 1.10", "statement": "(Canary) Let $G$ be word-hyperbolic and $H$ a finitely presented subgroup of $G$. Suppose that for every $g\\in G$ there is $n>0$ such that $g^n\\in H$. Does it follow that $H$ has finite index in $G$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.10 (PDF page 3)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem appears **open**; I could not find any published resolution. Rigorous partial progress obtained here: - **Reduction of the normal case:** for $H\\trianglelefteq G$ the question is equivalent to “does a hyperbolic group admit an infinite finitely presented torsion quotient?”, and any counterexample (even with $H$ merely finitely generated) would solve the famous open problem on the existence of infinite finitely presented torsion groups (Propositions 3–4). - **Proved special cases:** the answer is *yes* when $H$ is quasi-convex (Corollary 2, Bestvina's remark made precise via Proposition 1); when $H$ is almost malnormal and $G$ is torsion-free (Proposition 5); when $G$ is a closed hyperbolic 3-manifold group — for every f.g. $H$, via Canary's covering theorem (Proposition 6); and when $G$ is locally quasi-convex, e.g. a limit group (Corollary 7). - **Structural consequence:** any $H$ with $\\sqrt H = G$ satisfies $\\Lambda H=\\partial G$ (Proposition 1), so the question is a strengthening of the (also delicate) question whether f.p. subgroups with full limit set have finite index." }, { "id": 1100111, "problem_number": "AMR-010-0111", "title": "Questions in Geometric Group Theory — Q 1.11", "statement": "(Whyte) Let Γ be a 1-ended hyperbolic group which is not virtually a surface group. Can every infinite index subgroup be free?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.11 (PDF page 3)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The question is **open in full generality**, but the answer is now known to be **\"no\"** (a non-free infinite-index subgroup always exists) for every major class where the question has been attacked: - cubulated hyperbolic groups (Wilton 2024, arXiv:2406.02121, accepted for publication) — including C'(1/6) small-cancellation groups and, via Agol–Wise, hyperbolic 3-manifold groups; - one-relator groups (same paper); - hyperbolic graphs of free groups with cyclic edge groups, and limit groups (Wilton 2012, DOI 10.2140/gt.2012.16.665); - closed hyperbolic 3-manifold groups and rank-one lattices, where even surface subgroups exist (Kahn–Markovic, DOI 10.4007/annals.2012.175.3.4; Hamenstädt, DOI 10.1007/s00039-015-0330-y). Moreover, any would-be positive example must simultaneously be a counterexample to Gromov's surface-subgroup conjecture, admit no splitting over ℤ, have a boundary without local cut points, and admit no proper cocompact action on a CAT(0) cube complex." }, { "id": 1100112, "problem_number": "AMR-010-0112", "title": "Questions in Geometric Group Theory — Q 1.12", "statement": "(Whyte) Let Γ be a 1-ended hyperbolic group. Can a finite index subgroup of Γ be isomorphic to a subgroup of Γ of infinite index?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.12 (PDF page 3)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The question is solved in the literature with answer **yes**: there exist one-ended hyperbolic groups Γ containing a finite-index subgroup H ≤ Γ and an infinite-index subgroup K ≤ Γ with H ≅ K. Sketch of the Stark–Woodhouse main example (my summary of their §2): Let X = Σ₁ ∪ Σ₂ ∪ Σ₃ where each Σᵢ is a genus-1 surface with one boundary circle, all boundaries glued to a single S¹. Then π₁(X) is a one-ended hyperbolic group (Bestvina–Feighn, since each π₁(Σᵢ) is free amalgamated along a malnormal cyclic subgroup). - *Degree-3 cover X₁:* by Neumann's covering lemma, each Σᵢ has a 3-sheeted cover with exactly one boundary component; its Euler characteristic is 3·(−1) = −3, so it is a genus-2 surface with one boundary. Gluing gives a degree-3 cover X₁ → X of the same \"simple surface amalgam\" form. - *Degree-4 cover X₂:* each Σᵢ has a 2-sheeted cover Σᵢ″ with two boundary components (still genus 1). Glue one boundary component of each Σᵢ″ to form one amalgam circle, and attach extra copies of the Σⱼ along the other boundary components; this gives a degree-4 cover X₂ → X. - *Embedding:* X₁ embeds π₁-injectively as a proper sub-amalgam of X₂ (visibly a retract of X₂), so π₁(X₁) ≅ π₁(X₂)…" }, { "id": 1100113, "problem_number": "AMR-010-0113", "title": "Questions in Geometric Group Theory — Q 1.13", "statement": "(Swarup) Prove the combination theorem for relatively hyperbolic groups.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.13 (PDF page 3)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Swarup's question is **answered affirmatively in the literature**, in three distinct frameworks that mirror the different definitions of relative hyperbolicity: - **Dynamical/convergence-group version (Dahmani 2003).** If groups acting as convergence groups (relatively hyperbolic) are combined along parabolic-type (\"liminal\") subgroups satisfying geometric-finiteness and intersection-control hypotheses, the amalgamated product/HNN extension acts as a convergence group on a suitably assembled compactum and is relatively hyperbolic relative to the expected peripherals. Applied to show limit groups are relatively hyperbolic w.r.t. maximal noncyclic abelian subgroups. - **Amalgam version (Alibegović 2005).** For a one-edge graph of relatively hyperbolic groups with liminal edge group satisfying an almost-malnormality condition and a compatibility (\"isolated\"-type) condition on the peripherals, the fundamental group of the graph of groups is relatively hyperbolic relative to the images of the vertex peripherals not contained in the edge group. - **Geometric version (Mj–Reeves 2008), the one explicitly billed as answering Swarup's question.** For a tree of strongly relatively…" }, { "id": 1100115, "problem_number": "AMR-010-0115", "title": "Questions in Geometric Group Theory — Q 1.15", "statement": "Is every word-hyperbolic group residually finite?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.15 (PDF page 4)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. The problem is unsolved as of August 2026; the transcription is correct; the precise equivalence structure (Kapovich–Wise; Olshanskii) and the strongest conditional consequences (Agol–Groves–Manning + Wise) are documented and verified. No independent progress toward a solution is claimed — this is a famous problem where a solution attempt is beyond the scope of a bounded review." }, { "id": 1100116, "problem_number": "AMR-010-0116", "title": "Questions in Geometric Group Theory — Q 1.16", "statement": "(Dani Wise) Let $G^n$ denote the Cartesian product of $n$ copies of $G$, and let $\\operatorname{rank}(G^n)$ be its smallest number of generators. If $G$ is word-hyperbolic, is $\\lim_{n\\to\\infty}\\operatorname{rank}(G^n)=\\infty$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.16 (PDF page 4)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Wise's conjecture (Bestvina Q 1.16) remains **open**. I did not solve it. What is established here: - Verified the original statement and the absence of a published solution; the only published partial results are Wise's 2002 small-cancellation counterexamples outside the finitely presented/hyperbolic world, and the classical finite-group growth-sequence theory. - Sharp reduction (Proposition + Corollary 1): the conjecture holds unless $G$ is infinite, perfect, and has no nontrivial finite quotients; hence it is implied by residual finiteness of hyperbolic groups (Q 1.15), and a counterexample would simultaneously refute Q 1.15. - Structural observations: no infinite simple hyperbolic group exists (Observation 3), so the known bounded-growth mechanism for infinite simple groups cannot realize a hyperbolic counterexample; but every torsion-free non-elementary hyperbolic group has non-finitely-presentable simple quotients with bounded growth sequences (Coulon–Fournier-Facio + Wiegold–Wilson), so any proof must exploit finite presentability in an essential way. Homological/$L^2$ invariants cannot detect rank growth beyond $b_1$ (Observation 4)." }, { "id": 1100117, "problem_number": "AMR-010-0117", "title": "Questions in Geometric Group Theory — Q 1.17", "statement": "(Dani Wise) Find 'nice' groups, for example CAT(0) or automatic groups, for which $\\operatorname{rank}(G^n)$ does not tend to infinity as $n\\to\\infty$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.17 (PDF page 4)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The problem is open. The best partial results in the literature are: (i) Wise's 2-generator infinitely presented $C'(1/6)$ group $G_\\infty$ with $\\operatorname{rank}(G_\\infty^n)=2$ for all $n$; (ii) Wise's finitely presented $C'(1/6)$ groups $G_n$ with the $n$-th power 2-generated for each fixed $n$; (iii) Baumslag–Miller's finitely presented group with quotient $\\cong G\\times G$, which by Lemma 1 has $\\operatorname{rank}(G^n)$ bounded — but none of these groups is CAT(0) or automatic, and the stronger Hirshon problem (finitely presented $G\\cong G\\times G$) is itself unresolved. My own contribution is the rigorous derivation of the necessary conditions (Lemmas 0–4): a solution must be finitely presented, perfect with no solvable or finite quotients (profinitely trivial), and non-residually-finite; candidates with these properties exist among CAT(0)/biautomatic groups (Wise's non-residually-finite square-complex groups, Burger–Mozes simple lattices), but boundedness of $\\operatorname{rank}(G^n)$ is unknown for every one of them." }, { "id": 1100118, "problem_number": "AMR-010-0118", "title": "Questions in Geometric Group Theory — Q 1.18", "statement": "(Epstein) Let $G$ be a word-hyperbolic group and $\\partial G$ its boundary. Is there an algorithm to compute $\\check H^i(\\partial G)\\cong H^{i+1}(G,\\mathbb{Z}G)$? In particular, is there an algorithm to decide whether $\\check H^i(\\partial G)\\cong\\check H^i(S^2)$ for all $i$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.18 (PDF page 4)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The problem is **open in general** as of this writing (2026-08); I found no solution in the literature and Barrett's 2018 thesis explicitly records it as open. - **Partial solution (literature):** computable for hyperbolic fundamental groups of graphs of groups with free vertex groups and cyclic edge groups (Barrett, Thm 6.4.4 + Cor 6.4.5 of the thesis; arXiv:1712.00780), and the degree-0/ends case is decidable (Epstein–Sela, per the source remark). - **My contribution:** a rigorous reduction of the general question to an *effective stability bound* for the computable direct system $\\{H^{k+1}(P_d(G),P_d(G)\\setminus B_R)\\}$, showing that each stage and map is algorithmically computable from (presentation, $\\delta$) and that the unique missing ingredient is a computable stabilisation radius; plus the observation that the $S^2$-detection subproblem is equivalent to recognising Cannon-conjecture groups from presentations, explaining its resistance." }, { "id": 1100119, "problem_number": "AMR-010-0119", "title": "Questions in Geometric Group Theory — Q 1.19", "statement": "(M. Mitra) Let G be a word-hyperbolic group and H a word-hyperbolic subgroup. Does the inclusion H → G extend to a continuous map between the boundaries ∂H →∂G?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.19 (PDF page 5)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Mitra's question (Bestvina Q 1.19) is **resolved in the negative**: Baker and Riley (Forum Math. Sigma 1, 2013, e3; DOI 10.1017/fms.2013.4; arXiv:1206.0505) constructed an explicit $C'(1/6)$ small-cancellation hyperbolic group $G$ on six generators containing a rank-3 free subgroup $H = \\langle b, d_1, d_2 \\rangle$ for which no Cannon–Thurston map $\\partial H \\to \\partial G$ exists, with an elementary, fully rigorous proof via Mitra's $M(N)$ criterion and Dehn-reduced geodesics. Hence the answer to the question as posed is **no**, and the problem is solved in the literature. This is a literature triage, not an independent solution by me; I verified the source wording, the resolving paper's existence and abstract against Crossref and the arXiv API, and reconstructed its proof from the full text." }, { "id": 1100120, "problem_number": "AMR-010-0120", "title": "Questions in Geometric Group Theory — Q 1.20", "statement": "(Swarup) Suppose $G$ is a hyperbolic group which is a graph of hyperbolic groups such that all edge-to-vertex inclusions are quasi-isometric embeddings. Mitra shows that each vertex-group inclusion $V\\hookrightarrow G$ induces a continuous Cannon-Thurston map $\\partial V\\to\\partial G$. Describe its point-preimages; in particular, show that the map is finite-to-one.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.20 (PDF page 5)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Swarup's Question 1.20 is answered affirmatively in the literature: - (Finite-to-one part) The Cannon–Thurston map $\\partial V \\to \\partial G$ is **uniformly finite-to-one** for any hyperbolic group split as a finite graph of hyperbolic groups with qi-embedded edge inclusions: Bhattacharyya–Halder–Lazarovich– Mj, arXiv:2603.22428 (2026), Theorem 3.13 / Corollary 3.15. - (Point-preimage part) Fibers are characterized as the sets of boundary points pairwise joined by contracting bi-infinite ladders flowing along a unique end of the Bass–Serre tree (Kapovich–Sardar 2024, Ch. 8; used as Proposition 3.10 in the preprint); in free-group-extension cases they are described by explicit algebraic Cannon–Thurston laminations (Kapovich–Lustig 2015; Dowdall–Kapovich–Taylor 2016). Caveat on classification: the full solution is a preprint (March 2026) that, at the time of writing, has not appeared in a refereed venue. If one insists on peer-reviewed literature only, the status would be \"partially solved\" (uniform finite-to-one known for free-group extensions since 2015–2016; the general graph-of-groups case open until the 2026 preprint). I classified it SOLVED-IN-LITERATURE because the preprint…" }, { "id": 1100121, "problem_number": "AMR-010-0121", "title": "Questions in Geometric Group Theory — Q 1.21", "statement": "(Thurston) Is every closed hyperbolic 3-manifold finitely covered by one that fibers over the circle?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.21 (PDF page 6)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Yes** — every closed hyperbolic 3-manifold has a finite-sheeted cover that fibers over the circle. This follows from Agol's theorem (Doc. Math. 18 (2013), 1045–1087) that cubulated hyperbolic groups are virtually special, combined with Wise's cubulation of hyperbolic 3-manifold groups and Agol's virtual-fibering criterion. The conjecture is fully resolved." }, { "id": 1100123, "problem_number": "AMR-010-0123", "title": "Questions in Geometric Group Theory — Q 1.23", "statement": "(Ian Leary) Is there a version of the Kan-Thurston theorem using only CAT(-1) groups, or word hyperbolic groups? (The statement should be: for any finite simplicial complex X, there is a locally CAT(-1) polyhedral complex Y and a map Y →X that is surjective on fundamental groups and induces an isomorphism on homology for any local coefficients on X.)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 1.23 (PDF page 6)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The problem as stated (CAT(-1) or word-hyperbolic Kan–Thurston) remains open as of August 2026. Solved neighbours: the CAT(0) version (Leary 2013, verified), the proper-actions/torsion-allowed CAT(-1) analogue at homotopy level (Januszkiewicz–Świątkowski 2006, Theorem M, verified), and reportedly the 2-dimensional case (Leary, stated in the source list, publication not located). I analyzed the standard approaches and identified two concrete barriers: (a) all Kan–Thurston proofs raise dimension via products, which create flats and hence only CAT(0); (b) the only known CAT(-1) realization theorem uses torsion in an essential way, and passing to torsion-free subgroups destroys the prescribed homology type." }, { "id": 1100201, "problem_number": "AMR-010-0201", "title": "Questions in Geometric Group Theory — Q 2.1", "statement": "(Swarup) Is there a proof of Johannson’s theorem that Out(π1M) is virtually generated by Dehn twists for M a Haken 3-manifold along the lines of Rips-Sela’s theorem that Out(G) is virtually generated by Dehn twists for torsion-free 1-ended hyperbolic G? Is this true for CAT(0) groups? In particular, if G is a CAT(0) group and Out(G) is infinite, does G admit a Dehn twist of infinite order?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.1 (PDF page 7)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "- **Part 3 (and hence part 2) for general CAT(0) groups: NO.** There are CAT(0) groups G with Out(G) infinite — indeed with every finite-index subgroup having infinite Out — and not a single nontrivial Dehn twist. Published explicitly by Fioravanti (arXiv:2601.22789, Example 8.3), built from the Italiano–Martelli– Migliorini / Groves–Manning / Martelli fibering constructions (2023–2025). An elementary 2-dimensional example (Z² *_Z Z²) shows the same phenomenon for twists of the natural splitting (my analysis, based on Levitt's twist-group computation). - **Part 2 for restricted classes: YES.** Hyperbolic groups (Rips–Sela; Levitt's Theorem 1.4 gives the sharp \"Out infinite ⟺ infinite-order Dehn twist exists\" form); toral relatively hyperbolic groups (Guirardel–Levitt, per Fioravanti's introduction); CAT(0) groups with isolated flats and abelian flat stabilisers (Groves' Theorem 5.9, splitting conclusion); special (cocompactly cubulated Haglund–Wise) groups up to a characteristic finite-index subgroup (Fioravanti, Theorem C, with the failure inside special groups exactly characterised by \"poison subgroups\", Theorem E). - **Part 1: effectively yes** — the Rips–Sela program now…" }, { "id": 1100202, "problem_number": "AMR-010-0202", "title": "Questions in Geometric Group Theory — Q 2.2", "statement": "(Gromov) If $G$ admits a finite-dimensional $K(G,1)$, does $G$ act properly discontinuously by isometries on a complete CAT(0) space?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.2 (PDF page 7)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open — rigorous triage.** The problem is unsolved in both directions as of August 2026: - No group with a finite-dimensional $K(G,1)$ is known that provably fails to act properly discontinuously by isometries on a complete CAT(0) space. - No theorem establishes such an action for all (or even for all hyperbolic) groups with finite-dimensional $K(G,1)$. - The strongest surrounding facts: (i) the *more general* question for arbitrary countable groups is explicitly open (Button 2020); (ii) the *cocompact/proper-space/semisimple* strengthenings are known to fail even in geometric dimension 2 (Bridson, Brady–Crisp, Crisp, Tomiyoshi 2001–2004), but their counterexamples still satisfy the conclusion of Q 2.2; (iii) the hyperbolic special case (Gromov's Jugendtraum) remains open. I could not solve or make substantive new mathematical progress on the problem itself; the difficulty is that the hypothesis gives a *finite-dimensional, possibly non-positively-curved* classifying space while the conclusion allows *arbitrary* complete CAT(0) spaces, and the two sides are connected by no known construction or invariant." }, { "id": 1100203, "problem_number": "AMR-010-0203", "title": "Questions in Geometric Group Theory — Q 2.3", "statement": "(Eilenberg-Ganea) Is there a group G of cohomological dimension 2 and geometric dimension 3?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.3 (PDF page 7)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open**. The transcription is correct as given. The state of knowledge is: (i) $\\operatorname{cd}=2$ is the unique dimension in which $\\operatorname{gd}=\\operatorname{cd}$ is unknown; (ii) Bestvina–Brady (1997) reduce the conjecture, in the presence of explicit candidate groups, to the Whitehead asphericity conjecture — one of the two must fail; (iii) the analogous statements for groups with torsion and for Bredon cohomology with various families are **false** (Brady–Leary–Nucinkis 2001; Fluch–Leary 2014; Sánchez Saldaña 2019), so the 2-vs-3 gap is a genuine phenomenon in every variant that allows torsion; the torsion-free integral case remains untouched." }, { "id": 1100205, "problem_number": "AMR-010-0205", "title": "Questions in Geometric Group Theory — Q 2.5", "statement": "(Exercise in [BGS85, p.2]) Take a closed surface S of genus ≥2. Let V = S × S and let Σ ⊂V denote the diagonal. Let ˜V be a nontrivially ramified finite cover of V along Σ. Then ˜V has a natural piecewise hyperbolic CAT(0) metric. Show that ˜V admits no C2-smooth Riemannian metric with curvature K ≤0.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.5 (PDF page 7)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**SOLVED-IN-LITERATURE.** Q 2.5 is an exercise from BGS85 (1985) that stood for ~28 years and was proved by Stadler (arXiv:1312.2198, 2013; GAFA 25 (2015) 1575–1587; LMU thesis 2014): any closed 4-manifold finitely covering S × S with nontrivial ramification along the diagonal admits no smooth (a fortiori no C²) Riemannian metric of nonpositive sectional curvature, despite carrying a natural piecewise-hyperbolic locally CAT(0) metric. No new proof by me; my contribution is verification of the source wording, of all citations, and a triage of why elementary obstructions (asphericity, χ, σ, π₁) provably cannot settle it." }, { "id": 1100206, "problem_number": "AMR-010-0206", "title": "Questions in Geometric Group Theory — Q 2.6", "statement": "Suppose a group G acts properly discontinuously and cocompactly by isometries on two CAT(0) spaces X and Y . Croke-Kleiner have examples where the boundaries ∂X and ∂Y are not equivariantly homeomorphic. Is there a compact metric space Z and cell-like maps Z →∂X, Z →∂Y ?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.6 (PDF page 7)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The question is open in general; affirmatively answered for (i) direct products with infinite factors (Mooney 2009, Fund. Math. 203) and (ii) Croke–Kleiner-admissible groups, equivariantly (Guilbault–Mooney 2014, J. Topol. 7), with the general theory and reductions developed in Guilbault–Mooney 2012 (Geom. Dedicata 160). The wording in the dataset matches the published source exactly, so no correction was needed. I did not (and realistically cannot, within scope) solve the general case; the value added is a verified literature map and a structural analysis of why the general case resists the known techniques." }, { "id": 1100207, "problem_number": "AMR-010-0207", "title": "Questions in Geometric Group Theory — Q 2.7", "statement": "(D. Wise) Let G act properly discontinuously and cocompactly on a CAT(0) space (or let G be automatic). Consider two elements a, b of G. Does there exist n > 0 such that either the subgroup ⟨an, bn⟩is free or ⟨an, bn⟩is abelian?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.7 (PDF page 7)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The CAT(0) case of Wise's Question 2.7 is **settled in the negative**: the Leary–Minasyan groups $G_{k,m}$ ($-2m0$ is $\\langle a^n,b^n\\rangle$ free or abelian (Leary–Minasyan 2021, Example 9.4 + Corollary 9.6; CAT(0) by Corollary 9.3). This is the accepted resolution of Q 2.7 in the literature (Martin 2024; Hagen–Martin–Sartori 2025 both describe it as \"the first example of a CAT(0) group not satisfying the power alternative\"). Hence the problem as posed is **SOLVED-IN-LITERATURE**, with the caveat that the parenthetical automatic variant is untouched by the counterexample (see below)." }, { "id": 1100208, "problem_number": "AMR-010-0208", "title": "Questions in Geometric Group Theory — Q 2.8", "statement": "Do CAT(0) (or (bi)automatic) groups satisfy the Tits alternative?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.8 (PDF page 7)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "- The dataset wording is **correct**; no correction needed (confirmed against the author-hosted PDF). - The problem is **open** in full generality for both CAT(0) and (bi)automatic groups; classification: **OPEN-TRIAGE**. - Verified literature: solved for groups acting properly on finite-dimensional CAT(0) cube complexes with bounded finite-subgroup orders (Sageev–Wise 2004, arXiv:math/0405022; strengthened geometrically by Caprace–Sageev 2011, GAFA 21, DOI 10.1007/s00039-011-0126-7) and for actions on 2-dimensional CAT(0) complexes with bounded cell stabilisers (Osajda–Przytycki 2021, arXiv:2110.01845); classical for hyperbolic groups (a fortiori CAT(-1)). - Rigorous reduction recorded: the problem is equivalent to showing every non-virtually-solvable subgroup contains $F_2$; the known obstruction is the absence of a general rank-rigidity theorem for CAT(0) spaces and the wildness of finitely generated subgroups." }, { "id": 1100209, "problem_number": "AMR-010-0209", "title": "Questions in Geometric Group Theory — Q 2.9", "statement": "Does every Artin group have a finite $K(G,1)$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.9 (PDF page 8)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The question is open in general. It is answered affirmatively for the following verified families of Artin groups: finite type (Deligne 1972), right-angled (Salvetti 1987/1994), large type (Appel–Schupp 1983; Hendriks 1985), FC type and 2-dimensional (Charney–Davis 1995), affine type (Paolini–Salvetti 2021), and various hyperbolic-type and bipartite-diagram classes (Huang 2024). For a general Artin group, neither a finite $K(G,1)$ nor even torsion-freeness or finite cohomological dimension is known. The question is implied by, and widely regarded as essentially equivalent in difficulty to, the $K(\\pi,1)$ conjecture for Artin groups." }, { "id": 1100211, "problem_number": "AMR-010-0211", "title": "Questions in Geometric Group Theory — Q 2.11", "statement": "(Eric Swenson) Let $X$ be a proper CAT(0) metric space and $G$ a finitely generated group acting properly discontinuously by isometries on $X$. (1) Can $G$ be an infinite torsion group? (2) If the action is cocompact, can $G$ contain an infinite torsion subgroup?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.11 (PDF page 8)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Neither part is solved in general, and I did not solve them. The triage above shows the problem reduces to a single hard configuration (horocyclic torsion actions fixing a boundary point), which is resolved — negatively for the torsion group — in every setting with extra structure: dimension ≤ 2, cube complexes, visibility + bounded packing, subexponential growth, Helly graphs and classical buildings, systolic/small-cancellation complexes. No example of an infinite finitely generated torsion group acting properly discontinuously on *any* proper CAT(0) space (of any dimension) is known; conversely, infinite-dimensionality and loss of finite generation or of cocompactness are all known to allow torsion phenomena, so the hypotheses are sharp. The strongest evidence for a negative answer to (1) in finite dimensions: every natural candidate (Grigorchuk-type groups of intermediate growth) is now provably excluded by Izeki–Karlsson, and exponential-growth torsion groups (Burnside-type) fail all known structural footholds. Classification: **OPEN-TRIAGE** (parts (1) and (2) open; extensive verified partial results; the precise remaining gap identified)." }, { "id": 1100212, "problem_number": "AMR-010-0212", "title": "Questions in Geometric Group Theory — Q 2.12", "statement": "(Kim Ruane) Let $G$ be a Coxeter group, for example a right-angled Coxeter group, acting properly discontinuously by isometries on a CAT(0) space $X$. How is $X$ different from the Coxeter complex? Specifically, if $H$ is a special subgroup of $G$, is there a closed convex subset of $X$ on which $H$ acts cocompactly?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.12 (PDF page 9)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- **Verified the source and wording**: Bestvina, *Questions in Geometric Group Theory*, Q 2.12 (Kim Ruane); the dataset transcription is accurate, no correction needed. - **Established the status: open**, with a precise reduction: for geometric actions the question is equivalent to \"quasiconvex $\\Rightarrow$ convex-cocompact\" restricted to special subgroups of Coxeter groups. - **Proved (own work, folklore-level) the positive answer when $G$ is word-hyperbolic**: $W_T$ acts cocompactly on the convex hull of any orbit, which is closed and convex. - **Documented strong positive partial results** in the cubical category (Fioravanti–Levcovitz– Sageev 2024): all special subgroups are convex-cocompact in every cocompact cubulation of a RACG whose defining graph has no loose squares, and in every strongly cellular cocompact cubulation. - **Identified the precise gap**: known exotic cubulations only obstruct *combinatorial* convex-cocompactness (convex subcomplexes), while Ruane's question asks for closed *metrically* convex subsets; the rotated $D_\\infty\\times D_\\infty$ example shows the two genuinely differ, so even in the cubical case her literal question is not settled by [FLS]." }, { "id": 1100213, "problem_number": "AMR-010-0213", "title": "Questions in Geometric Group Theory — Q 2.13", "statement": "(Ruth Charney) Classify Coxeter groups up to isomorphism.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.13 (PDF page 9)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The problem — classifying Coxeter groups up to abstract group isomorphism — is open in full generality and I did not solve it (no serious attempt is feasible: it is a flagship open problem of the area). What is established: the problem reduces (Howlett–Mühlherr) to the reflection-preserving isomorphism problem; large classes are (strongly) rigid, notably right-angled Coxeter groups, where the classification reduces to graph isomorphism (Radcliffe); graph-universal, skew-angled, new classes of Bahls, and reflection-rigid 2-spherical groups (Mühlherr, Mühlherr– Weidmann, Bahls, Caprace–Mühlherr); non-rigid examples arise from diagram twists (Brady–McCammond–Mühlherr–Neumann) and from decomposability phenomena already visible in rank 2. The conjectural complete answer is \"twist equivalence up to the known finite/decomposable exceptions.\"" }, { "id": 1100214, "problem_number": "AMR-010-0214", "title": "Questions in Geometric Group Theory — Q 2.14", "statement": "(Ruth Charney) Classify Artin groups up to isomorphism.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.14 (PDF page 9)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The general problem is **open**. What is established: - The classification is completely solved within the classes of right-angled (Droms 1987), spherical-type (Paris 2004), and large-type (Vaskou 2023) Artin groups, and the large-type class is isomorphism-invariant (Martin–Vaskou 2024). - A conjectural complete answer exists — the twist conjecture: $A_\\Gamma\\cong A_{\\Gamma'}\\iff\\Gamma,\\Gamma'$ twist equivalent — proved in all the solved cases above, and reduced to defining graphs without separating vertices (Jones–Mangioni–Sartori 2026). It would also imply decidability of the isomorphism problem. - Derived independently in this report: the dihedral classification $DA_m\\cong DA_n\\iff m=n$, via the intrinsic central quotient $DA_m/Z\\cong\\mathbb Z*\\mathbb Z_{m/2}$ ($m$ even) resp. $\\mathbb Z_2*\\mathbb Z_m$ ($m$ odd) and abelianizations (elementary; consistent with the published results)." }, { "id": 1100216, "problem_number": "AMR-010-0216", "title": "Questions in Geometric Group Theory — Q 2.16", "statement": "(Ruth Charney) Are all [finite type] Artin groups CAT(0)?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.16 (PDF page 9)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** No solution is claimed here; the contribution is a rigorous, verified literature triage plus a precise reduction: via the Deligne complex with the Moussong metric and Gromov's link condition, \"all Artin groups are CAT(0)\" reduces to \"all spherical Deligne complexes are CAT(1)\", which is exactly the finite-type case of the question. Current frontier: - braid group on $n$ strands: CAT(0) for $n\\le 7$ (HKS 2016; Jeong 2020, preprint), **open for $n\\ge 8$**; - type $B_n$ ($n\\ge 4$), $D_n$ ($n\\ge 4$), $F_4$, $E_6,E_7,E_8$, $H_4$: **open**; rank $\\le 3$ and $B_3$ (Moussong metric) settled; - non-spherical: RAAGs, FC type, 2-dimensional, XXL ($m\\ge 5$) are CAT(0); large ($m\\ge 3$) and extra-large ($m\\ge 4$) type in rank $\\ge 4$, and the general case: **open**." }, { "id": 1100217, "problem_number": "AMR-010-0217", "title": "Questions in Geometric Group Theory — Q 2.17", "statement": "(Ruth Charney) Are all Artin groups automatic?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.17 (PDF page 9)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The problem is unsolved in general. Automaticity (indeed biautomaticity) is now known for: spherical type (Charney 1992), triangle-free groups (Pride; Gersten–Short), extra-large type (Peifer 1996), large type (Brady–McCammond 2000; Holt–Rees 2012), all two-dimensional Artin groups (Huang–Osajda, metric systolicity), affine type (McCammond–Sulway 2017), and FC type (Huang–Osajda + Chalopin–Chepoi–Genevois–Hirai–Osajda, via the Helly property). No Artin group is known to be non-automatic. As a small derived observation, all Artin groups on $\\le 3$ generators are biautomatic by combining Charney's and Huang–Osajda's theorems." }, { "id": 1100218, "problem_number": "AMR-010-0218", "title": "Questions in Geometric Group Theory — Q 2.18", "statement": "(Ross Geoghegan) A proper CAT(0) space $M$ is almost geodesically complete if there is $R\\geq0$ such that for all $a,b\\in M$ there is an infinite geodesic ray starting at $a$ and passing within $R$ of $b$. If $\\operatorname{Isom}(M)$ acts cocompactly on $M$, is $M$ almost geodesically complete?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.18 (PDF page 9)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The answer to Q 2.18 is **yes**: every non-compact proper CAT(0) space on which $\\operatorname{Isom}(M)$ acts cocompactly is almost geodesically complete. This is Theorem 5 of Geoghegan–Ontaneda (Topology 46 (2007), 129–137), combining their theorem that cocompactness forces $H_c^d(M;\\mathbb{Z}) \\neq 0$ in a top dimension $d$ with Ontaneda's earlier homological criterion (Topology 44 (2005), 47–62). The important special case of CAT(0) *groups* (geometric, hence discrete-orbit, actions) was already settled by Ontaneda's Theorem B in 2005. Classification: SOLVED-IN-LITERATURE; I did not need to produce new mathematics, and make no claim of an independent solution." }, { "id": 1100219, "problem_number": "AMR-010-0219", "title": "Questions in Geometric Group Theory — Q 2.19", "statement": "(Dani Wise) A triplane is a CAT(0) space obtained by gluing three Euclidean half-planes along their boundaries, and a CAT(0) space $X$ has isolated flats if it contains no isometrically embedded triplane. A subgroup $H$ is quasiconvex relative to an action on $X$ if, for some $x\\in X$ and $K$, every geodesic joining two points of $Hx$ lies in the $K$-neighborhood of $Hx$. Conjecture: if $G$ acts properly discontinuously and cocompactly on a CAT(0) space $X$ with isolated flats and $H\\leq G$ is finitely generated, then $H\\hookrightarrow G$ is a quasi-isometric embedding if and only if $H$ is quasiconvex relative to the action of $G$ on $X$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 2.19 (PDF page 9)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**The conjecture is true.** For a group $G$ acting properly discontinuously and cocompactly on a CAT(0) space $X$ with isolated flats, a finitely generated subgroup $H \\le G$ is quasi-isometrically embedded in $G$ if and only if its orbits are quasiconvex in $X$. Proved by Hruska–Kleiner (Geom. Topol. 9 (2005), Theorem 1.2.2(2)), building on Hruska (Topology 44 (2005)); the 2-dimensional case was done earlier by Hruska (Geom. Topol. 8 (2004)). The easy direction (quasiconvex $\\Rightarrow$ undistorted) holds for geometric actions on general proper geodesic metric spaces; the content is undistorted $\\Rightarrow$ quasiconvex, which fails without isolated flats (Wise's $F_2\\times\\mathbb{Z}$ example) and is proved via the equivalence of isolated flats with relative hyperbolicity of the space with respect to its flats." }, { "id": 1100301, "problem_number": "AMR-010-0301", "title": "Questions in Geometric Group Theory — Q 3.1", "statement": "(Hanna Neumann Conjecture) If A and B are nontrivial subgroups of a free group, then rk(A ∩B) −1 ≤(rk(A) −1)(rk(B) −1).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 3.1 (PDF page 10)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The Hanna Neumann Conjecture is a theorem: for any nontrivial finitely generated subgroups A, B of a free group F, rk(A ∩ B) − 1 ≤ (rk(A) − 1)(rk(B) − 1), and indeed the strengthened double-coset inequality holds. Proved independently by Mineyev (Annals of Mathematics, 2012) and Friedman (Memoirs of the AMS, 2015, with a simplification by Dicks). Classification: **SOLVED-IN-LITERATURE**. Dataset wording matches the source exactly (wording_corrected: no)." }, { "id": 1100302, "problem_number": "AMR-010-0302", "title": "Questions in Geometric Group Theory — Q 3.2", "statement": "(Swarup) If G is a Fuchsian group, define area(G) to be the area of the convex core of H2/A. Since area(A) = 2π(rk(A)−1) for Fuchsian groups which are free, the Hanna Neumann Conjecture can be phrased in terms of free Fuchsian groups: 2πarea(A∩B) ≤area(A)area(B) whenever A and B are nontrivial free subgroups of a Fuchsian group. Prove such inequalities (possibly with a worse constant) for (not necessarily free) torsion-free quasiconvex subgroups of a quasi-convex Kleinian group in Hn, where area is replaced by the n-dimensional volume of the convex core.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 3.2 (PDF page 10)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The full **$n=2$ Fuchsian case is settled with the sharp constant** $2\\pi$: for free subgroups it is the Friedman–Mineyev theorem; the extension to arbitrary (not necessarily free) torsion-free f.g. Fuchsian subgroups is proved here by elementary index/Euler-characteristic arguments (Section \"Work done\", item 1). - The **$n\\ge 3$ problem — the actual content of Swarup's question — remains open**: no literature addresses it, and I showed the two natural attacks cannot work as stated ($\\ell^2$-Betti submultiplicativity is only proved for graphs and does not see core volume; core volume is topologically unbounded in $n=3$, so no rank-based reduction). - I could **not** solve the $n\\ge 3$ case or find a counterexample; the obstruction is precisely quantified (intersection of convex hulls of limit sets; the thin/thick dichotomy above)." }, { "id": 1100303, "problem_number": "AMR-010-0303", "title": "Questions in Geometric Group Theory — Q 3.3", "statement": "(J. Cornick) If G is f.g. and the homological dimension hd G = 1, is G free?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 3.3 (PDF page 10)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **OPEN**. The cohomological analogue (cd = 1 ⟹ free) is the Stallings–Swan theorem, but the homological version asked here — Bieri's conjecture in its finitely generated form, attributed by Bestvina to J. Cornick — remains unresolved: it is still posed as open by Fluch–Gandini–Nucinkis (2016) and treated as open by Emmanouil's 2025 BLMS paper, which proves the strongest recent partial result (the residually-finite-p-group / residually-N_P case). My analysis confirms the known reductions: a counterexample must be torsion-free, finitely generated but not FP₂ (in particular infinitely related), of cohomological dimension exactly 2, with a finitely generated flat but non-projective augmentation ideal I_G over ℤG." }, { "id": 1100305, "problem_number": "AMR-010-0305", "title": "Questions in Geometric Group Theory — Q 3.5", "statement": "Conjecture. Limit groups are CAT(0).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 3.5 (PDF page 11)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The conjecture is **true**: every limit group admits a geometric action on a CAT(0) space, moreover one with the isolated flats property (Alibegović–Bestvina, 2004/2006). Literature status: **Solved affirmatively shortly after the July-2004 update of the list**, by: - E. Alibegović and M. Bestvina, *Limit groups are CAT(0)*, J. London Math. Soc. (2) 74 (2006), no. 1, 259–272. DOI: 10.1112/S0024610706023155. Verified via Crossref (`api.crossref.org/works?query.bibliographic=...`): title, authors, journal, volume 74, issue 01, pages 259–272, published August 2006 all confirmed. - Preprint: arXiv:math/0410198 (submitted 7 Oct 2004, v2 30 Aug 2005). Verified via the arXiv API: abstract reads \"We prove that every limit group acts geometrically on a CAT(0) space with the isolated flats property\"; journal ref matches the JLMS publication above. Note the timing: the question list's July 2004 update still presents Q 3.5 as open, and the Alibegović–Bestvina preprint appeared in October 2004 — so the conjecture…" }, { "id": 1100306, "problem_number": "AMR-010-0306", "title": "Questions in Geometric Group Theory — Q 3.6", "statement": "Characterize groups of the form Fm ∗Z Fn which are limit groups.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 3.6 (PDF page 11)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Theorem (case analysis proved here, assembling classical results).** Let $G = F_m \\ast_{w_1 = w_2} F_n$ with $w_i \\neq 1$. Then: 1. If some $w_i$ is primitive in its factor, $G \\cong F_{m+n-1}$: a limit group. 2. If both $w_i$ are root-free, $G$ is a limit group (Baumslag 1962/1967; Gildenhuys–Kharlampovich– Myasnikov 1995; Kharlampovich–Myasnikov 1998). 3. If both $w_i$ are proper powers, $G$ is not a limit group (proved here: it contains $\\langle u,v \\mid u^k = v^l \\rangle$, $k,l \\ge 2$, which violates commutative transitivity). 4. If $w_1 = u^k$ ($k \\ge 2$) and $w_2$ is root-free non-primitive, then (proved here) $G$ is a limit group iff the root adjunction $R_k(w_2) = \\langle t, F_n \\mid t^k = w_2 \\rangle$ is one; a necessary condition is that $w_2$ have a nontrivial $k$-th-power image in some free quotient of $F_n$. This case genuinely goes both ways: $R_k([a,b])$ is never residually free (via Culler 1981: nontrivial commutators in free groups are not proper powers), while $R_2([a,b][c,d]) = N_5$ is a limit group. Thus Q 3.6 is answered **except** for a complete criterion deciding, for root-free non-primitive $w \\in F_n$ and $k \\ge 2$, whether $\\langle t, F_n \\mid t^k = w…" }, { "id": 1100307, "problem_number": "AMR-010-0307", "title": "Questions in Geometric Group Theory — Q 3.7", "statement": "Characterize 1-relator groups which are limit groups.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 3.7 (PDF page 11)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open**. The literature triage gives: - *Known to be limit groups:* free groups, $\\mathbb{Z}$, $\\mathbb{Z}^2$, orientable surface groups, nonorientable surface groups of genus $\\ge 4$, Baumslag doubles over non-primitive non-proper-power elements (G. Baumslag 1962 + B. Baumslag 1967, both Crossref-verified). - *Known not to be:* groups with torsion ($r$ a proper power); nonabelian solvable one-relator groups (e.g. $BS(1,n)$); non-commutative-transitive examples (e.g. $\\langle a,b\\mid[a,b^2]\\rangle$); scl-obstructed examples ($x^n[x,y][z,w]$, $n\\ge 3$; Wilton 2025 via Duncan–Howie); the non-residually-finite examples of Baumslag–Miller–Troeger. - *Structural constraints:* the B. Baumslag reduction to \"residually free + CT\"; Ciobanu–Fine–Rosenberger (property IF case); Fruchter's $\\mathrm{vb}_2$ dichotomy (arXiv:2209.14925). Nothing in the consulted literature (through mid-2026) claims a characterization, and the April 2025 MathOverflow episode — settling one concrete 4-generator presentation with scl methods — is strong evidence that no general criterion exists yet." }, { "id": 1100309, "problem_number": "AMR-010-0309", "title": "Questions in Geometric Group Theory — Q 3.9", "statement": "Conjecture: for $n>1$, the set $$\\{(x_1,\\ldots,x_n)\\in F_n^n:x_1,\\ldots,x_n\\text{ is a basis of }F_n\\}$$ is not definable.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 3.9 (PDF page 11)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Bestvina's Question 3.9 is resolved in the literature: - $n = 2$: the set of bases of $F_2$ **is** first-order definable *with parameters* (Nielsen 1917, via the commutator $[g,h] \\sim [a,b]^{\\pm 1}$); it is **not** definable *without parameters* (Pillay's characterization of the generic type + Sklinos's bigness of the non-primitives + connectedness). - $n \\geq 3$: the set of bases of $F_n$ is **not** first-order definable, even *with parameters* (Bestvina–Feighn: definable subsets of $F$ are negligible or co-negligible, and the set of primitives is neither, in rank $> 2$; projection from bases to primitives), confirming the conjecture in this range. So the conjecture as transcribed holds without parameters for all $n > 1$, holds with parameters exactly for $n \\geq 3$, and fails with parameters at $n = 2$." }, { "id": 1100310, "problem_number": "AMR-010-0310", "title": "Questions in Geometric Group Theory — Q 3.10", "statement": "Conjecture. The only definable subgroups of Fn are cyclic and the entire group.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 3.10 (PDF page 11)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The conjecture is **true**: every first-order definable (with parameters) proper subgroup of a nonabelian free group F_n is cyclic. This was first proved as a corollary of Kharlampovich–Myasnikov's description of definable sets (arXiv Nov 2011; IJAC 23 (2013), 91–110), answering Malcev's 1965 question, and reproved independently by Perin–Pillay–Sklinos–Tent (Münster J. Math. 7 (2014), Theorem 4.3) in the stronger generality of arbitrary torsion-free hyperbolic groups. Hence: **SOLVED-IN-LITERATURE**." }, { "id": 1100311, "problem_number": "AMR-010-0311", "title": "Questions in Geometric Group Theory — Q 3.11", "statement": "For $x\\in[F_n,F_n]$, define its genus as the least $g$ such that $x$ is a product of $g$ commutators. Let $f(g)$ be a uniform bound, independent of $x$, on the number of Nielsen-equivalence classes of expressions of a genus-$g$ element as a product of $g$ commutators. Give an explicit upper bound for $f(g)$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 3.11 (PDF page 12)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "$$f(g)\\ \\le\\ W_{\\mathrm{tot}}(g)\\ \\le\\ \\sum_{e=2g}^{6g-3}(2e-1)!!\\ \\le\\ (4g-2)\\cdot(12g-7)!!,$$ an explicit upper bound independent of $x$ and of $n$. (Even more crudely, $f(g)\\le (12g)!\\,/\\,(6g)!$.) Combined with the literature this gives $$2^g\\ \\le\\ f(g)\\ \\le\\ (4g-2)(12g-7)!!\\ \\approx\\ (12g)^{\\,6g}\\ \\text{up to exponential factors},$$ so the true growth of $f(g)$ is pinned between exponential and roughly $g^{6g}$. The upper bound is certainly far from sharp: it counts *all* Wicks forms, while a single word typically realizes few forms and several forms may share a Nielsen class; and Bestvina–Feighn's lower bound concerns equivalence classes, while Duncan–Vdovina's $g!$ growth concerns raw presentations. Honesty caveats: (i) the surjectivity step 1 — every Nielsen class contains a Wicks-form substitution — is the standard content of Culler's 1981 paper and the Comerford–Edmunds theory (this is also exactly what underlies the finiteness asserted in the source), but I verified Culler's paper only via its Crossref record and its role in the verified Duncan–Vdovina text, not by reading the full article; (ii) my bound does not appear in the literature, so this is not…" }, { "id": 1100401, "problem_number": "AMR-010-0401", "title": "Questions in Geometric Group Theory — Q 4.1", "statement": "(de la Harpe) Is PSL2(R) a maximal closed subgroup of Homeo+(S1)?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 4.1 (PDF page 12)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Yes — PSL(2,ℝ) is a maximal closed subgroup of Homeo₊(S¹)** (indeed of Homeo(S¹)). This is Corollary (Section 1) of Giblin–Markovic, *Geom. Topol.* 10 (2006) 1319–1346, affirming de la Harpe's conjecture as stated in Bestvina Q 4.1." }, { "id": 1100402, "problem_number": "AMR-010-0402", "title": "Questions in Geometric Group Theory — Q 4.2", "statement": "Is there a proper closed subgroup of Homeo+(S1) that acts transitively on (unordered) 4-tuples? Or k-tuples (k ≠ 3)? Relation to earthquakes?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 4.2 (PDF page 12)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The question's literal form (closed subgroup, plain transitivity on unordered 4-tuples, k ≥ 4) is **not fully resolved in the literature**: Giblin–Markovic (Geom. Topol. 2006, verified) answer it negatively for closed subgroups containing a non-constant continuous path, and their k-tuple theorem (continuous 4-transitivity ⇒ continuous n-transitivity for all n ⇒ density) settles all k ≥ 4 at once in that category. The path hypothesis is automatic for Q 4.1 (de la Harpe), which they solve outright, but not for Q 4.2. - New partial results proved here (no computation, pure reasoning): Lemma A (transitivity on all cyclic tuple spaces ⇒ density, so closedness is the whole issue); Proposition B (a closed 4-homogeneous subgroup is automatically transitive on cyclically ordered triples, via Cameron + an orbit-counting/Baire argument on T_3 using the deck transformation ρ and connectedness); Corollary C (arc-transitivity of two-point stabilizers); Theorem D (complete negative answer under the path hypothesis, including the case analysis showing none of the proper groups on the Giblin–Markovic list is 4-homogeneous). - On \"Relation to earthquakes?\": I found **no** published work making a…" }, { "id": 1100404, "problem_number": "AMR-010-0404", "title": "Questions in Geometric Group Theory — Q 4.4", "statement": "(Swarup) Suppose G is a 1-ended finitely presented group that acts on a compact connected metric space X as a convergence group. What can be said about G if X has cut points? Does X have to be locally connected? The model theorem of Bowditch [Bow99] and Swarup [Swa96] says that if G is word hyperbolic, then X is locally connected and doesn’t have cut points. In another interesting case, when G is a geometrically finite Kleinian group and X its limit set, cut points can arise, for example if G splits over a parabolic subgroup.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 4.4 (PDF page 12)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open** in the generality asked. The rigorous state of knowledge: - Uniform (hyperbolic) case: fully solved — X locally connected, no global cut points; local cut points ⟺ splitting over a two-ended subgroup (Bestvina–Mess 1991; Swarup 1996; Bowditch 1998). - Geometrically finite / cusp-uniform case with X assumed (or proved) locally connected: essentially solved — G is relatively hyperbolic w.r.t. its maximal parabolics, cut points are parabolic and detect peripheral and two-ended splittings (Bowditch 1999, 2001; Guralnik 2005; Haulmark 2019, 2023). - Local connectedness: proved for connected Bowditch boundaries of relatively hyperbolic groups with tame 1- or 2-ended peripherals (Bowditch, as stated in Groves–Manning 2018, Thm 7.4), and for geometrically finite Kleinian limit sets without exposed rank-one cusps (Anderson–Maskit 1996). Known counterexamples to local connectedness of Bowditch boundaries (Gerasimov–Potyagailo 2015) require non-finitely generated peripherals and do not apply to finitely presented G. - General case (arbitrary convergence action of a one-ended finitely presented group): no theorem and no counterexample found in the literature; both…" }, { "id": 1100501, "problem_number": "AMR-010-0501", "title": "Questions in Geometric Group Theory — Q 5.1", "statement": "(Jim Anderson) If G is a group of isometries of Hn, denote by Ax(G) the set of axes of the elements of G. If G1 and G2 are finitely generated and discrete, does Ax(G1) = Ax(G2) imply that G1 and G2 are commensurable?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 5.1 (PDF page 12)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Anderson's question is open in general. It is solved affirmatively when both groups are arithmetic Fuchsian (Long–Reid 1998, verified) or arithmetic Kleinian groups (Reid's 2014 survey, Theorem 6.1), and partially for other classes of Kleinian groups (Xie–Jiang 2008, verified bibliographically). The general non-arithmetic case is open even for Fuchsian groups isoaxial with $\\mathrm{PSL}(2,\\mathbb{Z})$ (Reid 2014, Question 7.1 and Remark after Theorem 6.1). My own contributions: (i) noted that the literal statement is false for elementary parabolic groups and the intended reading is the non-elementary one; (ii) organized the reduction of the question to the equality $\\Sigma(G)=\\mathrm{Comm}(G)$ and identified the arithmetic vs. non-arithmetic dichotomy as the precise sticking point; (iii) flagged that the motivating Mess 1990 preprint was never published and its result is not independently verifiable." }, { "id": 1100503, "problem_number": "AMR-010-0503", "title": "Questions in Geometric Group Theory — Q 5.3", "statement": "(Ed Taylor) Does there exist a constant c > 0 such that the limit set of every non-classical Schottky group has Hausdorff dimension ≥c.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 5.3 (PDF page 13)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Yes — such a constant exists; the optimal value is $c=1$.** Taylor's question (Bestvina's list, Q 5.3) is solved affirmatively: by Hou's theorem every finitely generated non-elementary Kleinian group whose limit set has sufficiently small Hausdorff dimension is a classical Schottky group (Geom. Topol. 2010 for 2 generators; Math. Z. 2020 in general), and the sharp classification (Q. J. Math. 2023) gives $\\dim_H\\Lambda(\\Gamma)\\ge 1$ for every non-classical Schottky group $\\Gamma$." }, { "id": 1100505, "problem_number": "AMR-010-0505", "title": "Questions in Geometric Group Theory — Q 5.5", "statement": "(Misha Kapovich) Suppose $G$ is a finitely generated Kleinian group in $\\operatorname{Isom}(\\mathbb{H}^n)$. Is $$\\delta(G)\\leq\\operatorname{vcd}(G),$$ with equality if and only if $G$ preserves a totally geodesic subspace $\\mathbb{H}^k\\subset\\mathbb{H}^n$ such that $\\mathbb{H}^k/G$ is compact?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 5.5 (PDF page 13)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The inequality part of Kapovich's question is **solved in the literature**: δ(Γ) ≥ vhd_R(Γ,Π) − 1 for all (virtually torsion-free) Kleinian groups, and δ(Γ) ≥ vcd_R(Γ,Π) − 1 whenever (Γ,Π) has finite type — in particular for all geometrically finite groups ([Kap09], Thm 1.1, Cor 1.2). - The equality case is **solved for geometrically finite groups**: equality (equivalently dim_top Λ = dim_H Λ = d) forces Λ to be a round d-sphere and Γ to be a finite-volume lattice on a totally geodesic H^{d+1} ([Kap09], Thm 1.3). The converse (lattice ⟹ equality) is classical: δ = d = vcd(Γ,Π) − 1. - Hence, as the Bestvina list notes, \"the answer is positive for geometrically finite groups\"; the full conjecture (equality ⟹ geometric finiteness, no a priori geometric-finiteness assumption) **remains open** (still cited as a conjecture in 2020–2022). Classification: PARTIAL-PROGRESS." }, { "id": 1100506, "problem_number": "AMR-010-0506", "title": "Questions in Geometric Group Theory — Q 5.6", "statement": "(Misha Kapovich) Is there ϵ > 0 so that δ(G) < ϵ implies that G is virtually free?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 5.6 (PDF page 14)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Answer: yes.** For every n there is a constant D(n) < 1/2 such that every finitely generated Kleinian group Γ < Isom(ℍⁿ) with δ(Γ) < D(n) is virtually free (Liu–Wang, Corollary 1.6). Sketch of the mechanism (from Liu–Wang): - If δ(Γ) < 1, the Besson–Courtois–Gallot \"natural map\" F : M → M on M = ℍⁿ/Γ is area-contracting (|Jac₂ F| ≤ ((1+δ)/2)² < 1). This yields a **linear isoperimetric inequality**: any null-homologous union of loops 𝒞 bounds a surface of area ≤ 4/(1−δ) · ℓ(𝒞) (Theorem 1.10). - Consequences when δ < 1: all parabolic subgroups are ≅ ℤ (and none exist if δ < 1/2), M has finitely many cusps and bounded geometry, and Γ is convex cocompact iff the injectivity radius function is proper (Theorem 1.11). - Assuming Γ geometrically infinite, Kapovich–Liu give an escaping sequence of closed geodesics; an infinite-descent argument (shortening geodesics across thin Margulis tubes) produces two loxodromics moving a common point a uniformly bounded distance, forcing the subgroup they generate — hence Γ — to have critical exponent ≥ a uniform positive constant. Contradiction for δ < D(n, κ); hence Γ is **convex cocompact** (Theorem 1.2). - Then dim_H Λ(Γ) = δ(Γ) < D(n) < 1, so…" }, { "id": 1100507, "problem_number": "AMR-010-0507", "title": "Questions in Geometric Group Theory — Q 5.7", "statement": "(Misha Kapovich) Is there a finitely-generated discrete subgroup of SO(n, 1) whose action on the limit set is not ergodic? Is not recurrent?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 5.7 (PDF page 14)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The question splits cleanly by dimension: - **n ≤ 3: fully answered in the literature — the answer is NO to both parts.** Recurrence always holds (Ahlfors 1980); ergodicity holds whenever the limit set has positive measure (Ahlfors measure conjecture, proved via Canary 1993 plus Agol 2004 / Calegari–Gabai 2004). - **n ≥ 4: open**, for both ergodicity and recurrence; it is entangled with the higher-dimensional Ahlfors measure conjecture, and the question's author conjectures that non-ergodic examples probably exist (by analogy with verified non-ergodic, recurrent PU(2,1) examples with Λ = S³). I did not solve the open (n ≥ 4) case and make no claim of new mathematics; the contribution is a verified triage showing the problem is settled for n ≤ 3 and isolating exactly what a higher-dimensional counterexample would have to look like." }, { "id": 1100601, "problem_number": "AMR-010-0601", "title": "Questions in Geometric Group Theory — Q 6.1", "statement": "(Ian Leary) Suppose G is virtually of type FP over the field Fp of p elements, and let g be an element of order p. Is the centralizer of g in G also virtually of type FP over Fp?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 6.1 (PDF page 14)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Answer: yes.** For every prime $p$, every group $G$ virtually of type $FP$ over $\\mathbb{F}_p$, and every element $g \\in G$ of order $p$, the centralizer $C_G(g)$ is virtually of type $FP$ over $\\mathbb{F}_p$. This follows from the stronger Theorem E of Hamilton (J. Algebra 330 (2011), 1–21), which proves the same conclusion for the centralizer $C_G(P)$ of an arbitrary $p$-subgroup $P \\le G$. Hence Bestvina's Q 6.1 (attributed to Ian Leary; the underlying question is Question 1 of Leary–Nucinkis, Invent. Math. 151 (2003)) is solved in the literature, in the affirmative." }, { "id": 1100602, "problem_number": "AMR-010-0602", "title": "Questions in Geometric Group Theory — Q 6.2", "statement": "(Ian Leary) Is there a group of finite vcd that does not act with finite stabilizers on an acyclic complex of dimension equal to its vcd?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 6.2 (PDF page 14)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN.** Q 6.2 (Leary's weak form of Brown's problem) is unresolved as of August 2026. The transcription is exact. The literature cleanly separates: - **Strong form** (model for $\\underline{E}G$ of dimension $= \\operatorname{vcd}$): *false* — Leary–Nucinkis 2003 (verified DOI 10.1007/s00222-002-0254-7), with $\\operatorname{vcd} = 3n$ vs $\\operatorname{gd} \\ge 4n$; strengthened by Leary–Petrosyan 2017 (arXiv:1504.02704) to groups with cocompact $\\underline{E}G$, and to $\\operatorname{vcd} = 2$ groups with no contractible proper 2-complex. Partial positive results under geometric hypotheses: Lück (arXiv:2201.10807, AGT 2024). - **Weak form** (= Q 6.2; equivalent to the contractible-proper-action version for $\\operatorname{vcd} \\ne 2$; yes for $\\operatorname{vcd} \\le 1$): open, restated as such by Leary in *Guido's Book of Conjectures* Q46.1 (2008), and no solution or counterexample found in the literature since. I could not solve the problem; the analysis above isolates exactly where the difficulty lies (dimension-2 acyclic-vs-contractible gap, and the $\\operatorname{vcd}$ vs $\\mathcal{F}$-$\\operatorname{cd}$ question)." }, { "id": 1100603, "problem_number": "AMR-010-0603", "title": "Questions in Geometric Group Theory — Q 6.3", "statement": "(Peter Kropholler) If G is FP over the rationals, is there a bound on the orders of finite subgroups of G?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 6.3 (PDF page 15)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The problem is unsolved. Confirmed open in the primary source (Leary–Nucinkis 2001, §3) and consistent with all subsequent literature located (searches through 2026); no counterexample and no proof exists. Positive answer is known under any mild **integral** strengthening: FP_n over ℤ with n = cd_ℚ(G) (Leary–Nucinkis 2001, extending Kropholler 1993), and — verified in this work — in the low-dimensional case cd_ℚ(G) ≤ 1 (virtually free groups). The conjugacy-class strengthening is known to be **false** (Leary–Nucinkis 2005), but with bounded orders, so it does not bear on the question." }, { "id": 1100702, "problem_number": "AMR-010-0702", "title": "Questions in Geometric Group Theory — Q 7.2", "statement": "(Swarup) Starting with a finitely presented group $G$, take maximal graph-of-groups decompositions alternately over finite and over two-ended edge groups, each time continuing with the resulting vertex groups. Conjecture 1: there is a finitely presented group for which this process never terminates. Conjecture 2 (Strong Accessibility): for hyperbolic groups, and for CAT(0) groups, this process always terminates.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 7.2 (PDF page 15)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- **Conjecture 1 (existence of an f.p. group with a non-terminating alternating hierarchy): OPEN.** Resolved *negatively* (the process always terminates) for large classes: hyperbolic groups without 2-torsion (Vavrichek 2008, on the DP01 foundation repaired by LT17 Thm 2.9); virtually 2-torsion-free hyperbolic groups, in particular all residually finite hyperbolic groups, and f.p. subgroups of $\\mathrm{SL}(n,\\mathbb Z)$ (Louder–Touikan 2017, Cors. 2.7–2.8); Coxeter groups over minimal splittings (Mihalik–Tschantz, [arXiv:1003.0027](https://arxiv.org/abs/1003.0027), arXiv-API-verified); 2-generated torsion-free hyperbolic groups (Kapovich–Weidmann, cited in LT17). The fully general f.p. case, and even the case of torsion-free f.p. groups containing $BS(1,-1)$-type slender subgroups with dihedral quotients, remains unresolved. - **Conjecture 2, hyperbolic part: PROVED without 2-torsion; OPEN with arbitrary torsion** (the true boundary is \"no noncentral involution\", per LT17). - **Conjecture 2, CAT(0) part: OPEN**, with no published progress found. Overall classification: **PARTIAL-PROGRESS** — the conjectures are settled for substantial, natural classes of groups but not in the…" }, { "id": 1100703, "problem_number": "AMR-010-0703", "title": "Questions in Geometric Group Theory — Q 7.3", "statement": "(Sageev) Is there a f.p. 1-ended group G with G ∼= G∗Z?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 7.3 (PDF page 15)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open**. The statement was pinned down (HNN extension over ℤ, not free product) against the verified source, and the literature triage found no solution or direct attack. My own analysis yields rigorous necessary conditions on any example: G is non-co-Hopfian and not torsion-free hyperbolic; G admits a map onto ℤ whose kernel is an infinitely generated graph of groups with vertex groups ≅ G (so [±χ] ∉ Σ¹(G)); in G^ab the element ā − b̄ must be primitive of infinite order (in particular b ≠ a, and G ≇ G × ℤ); and G contains an infinite descending chain of proper subgroups each ≅ G. Conversely, Euler characteristic and L²-Betti numbers provide no obstruction, so an example is not excluded by the standard invariants." }, { "id": 1100705, "problem_number": "AMR-010-0705", "title": "Questions in Geometric Group Theory — Q 7.5", "statement": "(Papasoglou) Is there a f.p. torsion-free group G that does not split over a virtually abelian subgroup, but has infinitely many splittings over F2?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 7.5 (PDF page 16)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open**. Deliverable here is a rigorous triage: verified original wording, verified bibliography, a proof that the question reduces to the one-ended case with all induced edge groups of rank exactly 2, a precise explanation of why every known accessibility/JSJ finiteness theorem is inapplicable (F₂ is the smallest non-small, non-slender edge group, and absence of Z-splittings does not give acylindricity), and identification of free-by-cyclic groups as the natural candidate class. I did not find, and could not construct, a resolution in either direction, and I found no published resolution even in the special case of hyperbolic groups." }, { "id": 1100802, "problem_number": "AMR-010-0802", "title": "Questions in Geometric Group Theory — Q 8.2", "statement": "(Shalen) If a finitely presented group G acts nontrivially (i.e. without global fixed points) on an R-tree, does it act nontrivially on a simplicial tree?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 8.2 (PDF page 16)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- **Answer claimed in the literature: YES.** Every finitely presented group acting without a global fixed point on an R-tree also acts nontrivially on a simplicial tree (Dunwoody, arXiv:1203.6019v3, 2022). - **Confidence caveat:** the solving paper is an arXiv preprint with no journal publication found, minimal citations, and a history in which earlier versions asserted (incorrectly) the opposite answer via the Higman group. I verified the preprint's existence, version history, abstract, and main theorem statement directly from arXiv, but I did **not** verify the proof line by line, and I found no independent confirmation. Hence: solved in the literature as a claim, pending refereeing. - **Boundary of the result:** for finitely (but not finitely) presented groups the answer is NO — there exist finitely generated (FA)-groups acting fixed-point-freely on R-trees, even with finite arc stabilizers (Minasyan, J. Topology 2016, peer-reviewed and Crossref-verified)." }, { "id": 1100803, "problem_number": "AMR-010-0803", "title": "Questions in Geometric Group Theory — Q 8.3", "statement": "(Mohan Ramachandran) For a finitely presented group $G$, consider: (A) some finite-index subgroup of $G$ admits a nontrivial action on a simplicial tree; (B) if $X$ is a finite complex with fundamental group $G$, then some covering space of $X$ has at least two ends. To what extent are (A) and (B) equivalent?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 8.3 (PDF page 16)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The equivalence in full generality is **open**; nothing in the literature (searched to 2026) resolves it, and the question is untouched on Bestvina's list as of the 2004 update. - New rigorous content here: a complete proof that **(A) ⇒ (B) always** (Proposition 1), and that **(B) restricted to regular coverings ⇒ (A) unconditionally** (Proposition 2). - For **Kähler groups** the equivalence is essentially a theorem: a multi-ended covering with a non-amenable Schreier graph yields, by Delzant–Gromov (2005) and Napier–Ramachandran (2001, 2008), a virtual fibration over a hyperbolic surface group, hence a virtual splitting; combined with Proposition 1, (A) and (B) coincide for Kähler groups modulo the amenable-Schreier-graph caveat." }, { "id": 1100805, "problem_number": "AMR-010-0805", "title": "Questions in Geometric Group Theory — Q 8.5", "statement": "(Noel Brady) Are there groups of type Fn but not Fn+1 (n ≥3) which do not contain Z × Z? All known examples contain Zn−1.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 8.5 (PDF page 17)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Brady's Question 8.5 is **answered affirmatively in full generality**: for every $n \\geq 3$ (indeed every $n \\geq 1$) there exist groups of type $F_n$ but not $F_{n+1}$ containing no $\\mathbb{Z}\\times\\mathbb{Z}$ — moreover they occur as normal subgroups (kernels of maps to $\\mathbb{Z}$) of word-hyperbolic groups. The general case is due to Llosa Isenrich–Py (Invent. Math. 235 (2024), 233–254); the case $n=3$ was first settled by Llosa Isenrich–Martelli–Py (J. Differential Geom. 127 (2024), 1121–1147). Earlier, Kropholler (2018) had reduced the maximal guaranteed abelian rank from $n-1$ to $\\lceil n/3 \\rceil$ without eliminating $\\mathbb{Z}^2$." }, { "id": 1100806, "problem_number": "AMR-010-0806", "title": "Questions in Geometric Group Theory — Q 8.6", "statement": "(Olympia Talelli) Is there a torsion-free group G of infinite cohomological dimension such that there is n0 with the property that if H is a subgroup of G with finite cohomological dimension cdH, then cdH ≤n0.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 8.6 (PDF page 17)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Yes** — such groups exist, with the optimal bound $n_0 = 1$. By Theorem 5.7 of Fournier-Facio–Sun (arXiv:2503.01987), there are continuum many pairwise non-isomorphic finitely generated, simple, torsion-free groups $G$ of infinite cohomological dimension in which every proper non-trivial subgroup is infinite cyclic; hence every subgroup $H$ with $\\operatorname{cd}(H) < \\infty$ has $\\operatorname{cd}(H) \\le 1$. The construction combines small cancellation theory over acylindrically hyperbolic groups with group-theoretic Dehn filling arranged to have the Cohen–Lyndon property (for homological control), producing torsion-free Tarski monsters with prescribed cohomology. The same examples disprove Petrosyan's 2007 no-jump conjecture for all coefficient rings and give the first torsion-free groups with the fixed-point property for actions on finite-dimensional contractible CW-complexes." }, { "id": 1100807, "problem_number": "AMR-010-0807", "title": "Questions in Geometric Group Theory — Q 8.7", "statement": "Can Zp∞be embedded in an FP∞-group? Or in an FP3-group? Can an Fn-group be embedded in an Fn+1-group (n ≥2)?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 8.7 (PDF page 17)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** All three parts of Bestvina Q 8.7 remain open as of August 2026, confirmed by the July 2026 preprint of Fournier-Facio & Zaremsky (arXiv:2607.21727), which cites Q 8.7 verbatim as a fundamental open question. Settled borderline cases: $C_{p^\\infty}$ embeds in $F_2$-groups (Higman 1961) and in $FP_2$-groups (Leary 2018); uncountably many $FP_\\infty$ groups exist (Leary 2018b), so no cardinality obstruction. New rigorous content here: (i) a complete, verified literature triage; (ii) the divisibility-obstruction proof that $C_{p^\\infty}$ embeds in no finitary permutation group and hence in no Houghton group, ruling out the most familiar class of candidates; (iii) the precise reductions relating the Prüfer parts to the general higher Higman/Leary embedding questions, and the explanation of the \"$n\\geq 2$\" hypothesis. I did not solve any part of the problem." }, { "id": 1100808, "problem_number": "AMR-010-0808", "title": "Questions in Geometric Group Theory — Q 8.8", "statement": "Compute the asymptotic dimension of CAT(0) groups, Out(Fn), mapping class groups, nonuniform lattices, Thompson’s group. Is there a group of finite type whose asymptotic dimension is infinite?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 8.8 (PDF page 17)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The question is **partially resolved**: mapping class groups (finite — Bestvina–Bromberg–Fujiwara 2015), Thompson's group F (infinite — elementary argument above; independently Dranishnikov–Sapir), and nonuniform lattices in semisimple Lie groups (finite — Ji 2004 + Osin 2005) are settled. - The \"group of finite type with infinite asdim\" sub-question is answered **yes** for type F∞ (Thompson's F and V themselves); the stronger type-F (finite K(G,1)) reading is, to my knowledge, still open. - **CAT(0) groups and Out(F_n) remain open** (finite asdim unknown in general); cubulated CAT(0) groups have asdim ≤ dimension (Wright 2012), and Out(F_n) is boundary amenable (Bestvina–Guirardel–Horbez 2022)." }, { "id": 1100809, "problem_number": "AMR-010-0809", "title": "Questions in Geometric Group Theory — Q 8.9", "statement": "Is there a finitely presented group $G=F/N$, with $F$ a finite-rank free group, such that $$d_G\\!\\left(N/[N,N]\\right)1$? (b) Is there a four-manifold $M$ which is a surface bundle over a surface such that $\\pi_1(M)$ is word-hyperbolic, or such that $M$ is hyperbolic?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.3 (PDF page 19)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- **Part (a): answered YES by Kent–Leininger (2024).** For every $h\\ge 4$ there are infinitely many commensurability classes of faithful representations $\\pi_1(\\Sigma_g)\\hookrightarrow\\mathrm{Mod}(\\Sigma_h)$ whose image is purely pseudo-Anosov. This fully settles Bestvina Q 12.3(a) (which only asked for *some* $g,h>1$). - **Part (b): OPEN.** First compact aspherical atoroidal surface bundles over surfaces now exist (Kent–Leininger Theorem 3), but none is known to have word-hyperbolic fundamental group or a hyperbolic metric; many are known to admit no hyperbolic metric (Kent–Leininger, *Non-hyperbolic atoroidal surface bundles*). Existence of a word-hyperbolic surface-by-surface group is equivalent to existence of a convex cocompact surface subgroup of a mapping class group — still unknown. - Hence the whole of Q 12.3 is only half resolved: classification **PARTIAL-PROGRESS** (with the solved half attributable to the literature, not to me)." }, { "id": 1101204, "problem_number": "AMR-010-1204", "title": "Questions in Geometric Group Theory — Q 12.4", "statement": "(Bowditch) Is the Weil-Petersson metric on Teichmüller space hyperbolic? Is it quasi-isometric to the curve complex?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.4 (PDF page 20)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The question is **solved in the literature**, with a clean dichotomy: - **Q1 (Is WP hyperbolic?)** — No for $\\dim_{\\mathbb{C}}\\mathcal{T}(S) \\ge 3$ (Brock–Farb 2006); yes for the two exceptional surfaces $S_{1,1}, S_{0,4}$, where WP is quasi-isometric to the Farey graph. - **Q2 (Is WP quasi-isometric to the curve complex?)** — No for $\\dim_{\\mathbb{C}}\\mathcal{T}(S) \\ge 3$: $\\mathcal{C}(S)$ is hyperbolic (Masur–Minsky 1999) while WP is not, and hyperbolicity is quasi-isometry invariant; yes in the same two exceptional cases, where $\\mathcal{C}(S)$ is the Farey graph. The correct combinatorial model for WP is the pants graph (Brock 2003), not the curve complex: WP distance coarsely equals an $\\ell^1$-sum of subsurface projection distances over all (nested) subsurfaces, whereas $\\mathcal{C}(S)$ retains only the whole-surface projection." }, { "id": 1101205, "problem_number": "AMR-010-1205", "title": "Questions in Geometric Group Theory — Q 12.5", "statement": "(Brock) What is the rank of the Weil-Petersson metric? What is the rank of the mapping class group?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.5 (PDF page 20)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Both questions have complete answers. For a connected surface $S = S_{g,p}$ with $\\xi(S) = 3g - 3 + p \\ge 1$: - **Rank of the mapping class group.** The maximal dimension of a quasi-flat in $\\mathrm{Mod}(S)$ (word metric) is $$\\operatorname{rank}(\\mathrm{Mod}(S)) = 3g - 3 + p,$$ which coincides with the maximal rank of a free abelian subgroup (Birman–Lubotzky–McCarthy) — i.e. the Brock–Farb Rank Conjecture holds (Behrstock–Minsky 2008; independently Hamenstädt; new proofs by Eskin–Masur–Rafi 2017 via coarse differentiation and by Bowditch via coarse medians). The maximal quasi-flats are Hausdorff-close to unions of \"Dehn twist flats\"/standard orthants (Behrstock–Hagen–Sisto). - **Rank of the Weil–Petersson metric.** The maximal dimension of a quasi-flat in $(\\mathcal{T}(S), d_{WP})$ (equivalently, by Brock's theorem, in the pants graph) is $$\\operatorname{rank}(\\mathcal{T}(S), d_{WP}) = \\left\\lfloor \\frac{3g + p - 2}{2} \\right\\rfloor = \\left\\lfloor \\frac{\\xi(S) + 1}{2} \\right\\rfloor,$$ the maximal number of pairwise disjoint non-annular essential subsurfaces (Eskin–Masur–Rafi, Corollary C; computed earlier by Behrstock–Minsky; recovered by Bowditch 2020). Consistency check with…" }, { "id": 1101206, "problem_number": "AMR-010-1206", "title": "Questions in Geometric Group Theory — Q 12.6", "statement": "Assuming that the fixed subgroup Fix(α) is cyclic, find a bound on the length of a generator of Fix(α) in terms of the complexity of α.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.6 (PDF page 20)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The problem is **open** in its intended sense: no explicit function of (n, complexity of α) bounding the length of a generator of a cyclic Fix(α) is known, and it was still listed as open in Vogtmann's 2015 updated survey. - Partial progress recorded here: (i) the existence of a *computable* bound f_n(k) follows immediately from the Bogopolski–Maslakova algorithm (2016; [Mas03] in Bestvina's update), so the question is one of explicit estimates, not of existence; (ii) the problem is shown to be equivalent to bounding the length of indivisible Nielsen paths in improved relative train-track representatives, identifying the precise open core. Neither observation appears to be new in spirit, but I could not find (a) stated formally in the literature and (b) is a restatement, not a solution." }, { "id": 1101207, "problem_number": "AMR-010-1207", "title": "Questions in Geometric Group Theory — Q 12.7", "statement": "Which α preserve an order (invariant under right translations) on Fn? If α has periodic elements it cannot preserve an order. Are there other obstructions?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.7 (PDF page 20)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The question is not solved, but its literature status is far richer than the bare statement suggests. The sub-question \"Are there other obstructions?\" has an affirmative, verified answer in the bi-invariant case: the Clay–Rolfsen positive-eigenvalue obstruction ([arXiv:1004.3615](https://arxiv.org/abs/1004.3615)) rules out automorphisms with no periodic elements whatsoever (worked example in (c) above). For the problem as literally stated (one-sided, right-translation-invariant orders), no complete characterization and no comparably strong obstruction theory exists; even the braid-induced case is an active open classification problem ([arXiv:2410.10595](https://arxiv.org/pdf/2410.10595v1)). My contribution here is the rigorous triage plus items (a)–(d): a cleaned-up statement, the mapping-torus dictionary, a concrete example separating the periodic-element obstruction from the eigenvalue obstruction, and a fixed-point reformulation organizing all known necessary and sufficient conditions. Classification: PARTIAL-PROGRESS." }, { "id": 1101208, "problem_number": "AMR-010-1208", "title": "Questions in Geometric Group Theory — Q 12.8", "statement": "Does Out(Fn) (n > 2) have a right orderable subgroup of finite index?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.8 (PDF page 20)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN.** Q 12.8 — whether Out(F_n), n ≥ 3 (equivalently n > 2), has a right orderable subgroup of finite index — remains unresolved as of August 2026, per Bestvina's updated list (no update recorded) and an independent literature search. Classification: OPEN-TRIAGE. The honest expectation, by analogy with Witte's theorem for SL(n, Z) and the Zimmer-program heuristic that \"large\" rigid groups do not act faithfully on the line, is that the answer is **no**, but no proof strategy is currently available: the kernel of Out(F_n) → GL(n, Z) destroys exactly the arithmetic structure (bounded generation by unipotents) that Witte's argument depends on." }, { "id": 1101209, "problem_number": "AMR-010-1209", "title": "Questions in Geometric Group Theory — Q 12.9", "statement": "Can the mapping class group (or a finite index subgroup of it) of a closed surface be embedded into Out(Fn)? Into the mapping class group of a punctured surface?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.9 (PDF page 21)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- **Part 1 (closed MCG ↪ Out(F_n)): OPEN.** Constraints established: any embedding (even of a finite-index subgroup) of Mod(S_g), g ≥ 2, into Out(F_n) needs n ≥ 2g−1; for g ≥ 3 and n = 2g−1 the image must lie in IA_n (Franks–Handel + vcd). Genus 1 is affirmative. Still open as of the November 2024 MathOverflow thread. - **Part 2 (closed MCG ↪ punctured MCG): NEGATIVE in a range, OPEN in general.** For g ≥ 6 there is no non-trivial homomorphism Mod(S_g) → Mod(S_{h,p}) whatsoever with p ≥ 1 and h ≤ 2g−1 (Aramayona–Souto). For h ≥ 2g the question is open; the exotic closed→closed injections of Aramayona–Leininger–Souto show high-genus rigidity fails in the closed case, so a negative answer for large h is not to be expected by analogy. Equal-rank cases are excluded by Ivanov–McCarthy; the low genera g = 2,…,5 are only partially covered." }, { "id": 1101211, "problem_number": "AMR-010-1211", "title": "Questions in Geometric Group Theory — Q 12.11", "statement": "(Grigorchuk) Aut(Fn) acts on the space ∂3Fn of triples of distinct ends of Fn. Denote by Yn the compact space (Cantor set) the quotient space of ∂3Fn by the group of inner automorphisms. Thus Out(Fn) acts on Yn. Describe the dynamics of this action; in particular the dynamics of any individual outer automorphism.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.11 (PDF page 21)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The problem is **open**: no published or preprinted work addresses the Out(F_n)-action on Grigorchuk's space Y_n directly, and the author's own updated list leaves Q 12.11 without an update note. - Rigorous partial progress obtained here: (1) confirmation that Y_n is a compact Cantor set via Bowditch cocompactness; (2) an Out-equivariant Cantor fibration Y_n → Z_n over Kapovich's axis/frequency space; (3) an exact dictionary identifying Fix_{Y_n}(φ) with F_n-orbits of boundary triples fixed by a single lift of φ, which connects the question to the GJLL index theory; (4) the observation that for iwip φ the fixed set on Y_n is richer than two points (mixed attractor–repeller triples occur), so the correct conjectural statement is north–south dynamics relative to two closed laminar invariant sets A_±, mirroring the verified theorems of Levitt–Lustig (on CV̄_n) and Kapovich–Lustig (on currents). - Classification: PARTIAL-PROGRESS — items (A)–(D) are rigorous; (E)–(F) are explicitly conjectural/sketched and are offered as the likely shape of the answer." }, { "id": 1101212, "problem_number": "AMR-010-1212", "title": "Questions in Geometric Group Theory — Q 12.12", "statement": "Do there exist f.g. free subgroups of MCG(S) consisting of identity and pseudo-Anosov mapping classes which are not Schottky?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.12 (PDF page 22)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open**: no non-Schottky f.g. free purely pseudo-Anosov subgroup of any MCG(S) is known, and no theorem rules one out. The strongest known constraint (BBKL 2020) forces any example to be distorted in MCG(S) with non-QI orbit in the curve complex; all special families analyzed to date (fibered 3-manifold groups — where f.g. purely pseudo-Anosov subgroups are automatically free — handlebody genus 2, Veech, RAAG-based, surface bundles over tori) answer \"no counterexample here\". The companion question Q 12.13 (non-free purely pseudo-Anosov subgroups) was solved affirmatively by Kent–Leininger (2024, to appear Ann. of Math.), and the general Gromov hyperbolization question by Italiano–Martelli–Migliorini (2021/2023), but neither touches the free/non-Schottky case. I did not solve the problem; the contribution is a rigorous, verified literature triage and the equivalences above." }, { "id": 1101213, "problem_number": "AMR-010-1213", "title": "Questions in Geometric Group Theory — Q 12.13", "statement": "Do there exist non-free pseudo-Anosov subgroups?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.13 (PDF page 22)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Bestvina's Question 12.13 is answered affirmatively in the literature** (2024): non-free purely pseudo-Anosov subgroups of mapping class groups exist — indeed, for every closed surface S of genus g ≥ 4, Mod(S) contains infinitely many commensurability classes of purely pseudo-Anosov subgroups isomorphic to fundamental groups of closed surfaces (Kent–Leininger, \"Atoroidal surface bundles\", arXiv:2405.12067, to appear in Ann. of Math.). This simultaneously answers the strengthened surface-group form of the question and produces the first closed aspherical atoroidal surface bundles over surfaces. I did not produce an independent solution; the classification is SOLVED-IN-LITERATURE, with the solution verified against the arXiv record (abstract, author list, journal status) rather than merely cited second-hand." }, { "id": 1101214, "problem_number": "AMR-010-1214", "title": "Questions in Geometric Group Theory — Q 12.14", "statement": "For $\\operatorname{Out}(F_n)$, do there exist finitely generated free subgroups consisting of the identity and irreducible automorphisms which are not Schottky? Do there exist nonfree subgroups all of whose nonidentity elements are irreducible?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.14 (PDF page 22)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "- **Source and wording:** Bestvina's *Questions in Geometric Group Theory*, §12.2, Q12.14 — the one-line request for Out(F_n) analogues of Q12.12 (free purely pseudo-Anosov non-Schottky subgroups of MCG) and Q12.13 (non-free purely pseudo-Anosov subgroups). The dataset wording is a correct expansion; no correction needed beyond the standard reading \"irreducible = fully irreducible (iwip)\" and \"Schottky = free, purely iwip, quasiconvex orbits (today: convex cocompact)\". - **Status:** **OPEN.** Neither (a) non-Schottky (not convex cocompact) f.g. free purely fully irreducible subgroups, nor (b) f.g. non-free purely fully irreducible subgroups of Out(F_n), are known to exist; no negative result is known either. All \"positive side\" constructions in the literature (Clay–Pettet; Kapovich–Lustig; Dowdall–Taylor; Hamenstädt–Hensel; Taylor–Tiozzo) produce precisely the *Schottky/convex cocompact* examples that the question asks to go beyond. - **Sharp reformulations proved above:** (a) for atoroidal subgroups is equivalent to the existence of a non-hyperbolic, ℤ²-free free-by-free extension of F_n; (b) with convex cocompact geometry would give a hyperbolic extension of F_n by a non-free…" }, { "id": 1101216, "problem_number": "AMR-010-1216", "title": "Questions in Geometric Group Theory — Q 12.16", "statement": "Does every automorphism h : Fn →Fn leave invariant a finite index subgroup K such that hab : K/K′ →K/K′ has an eigenvalue which is a root of unity?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.16 (PDF page 22)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**The answer to Q 12.16 is YES for every automorphism h of every F_n.** The proof is: the elementary equivalence with \"vb₁(G_h) ≥ 2\" (above), plus Brinkmann's hyperbolicity criterion to split the cases, Button's largeness theorem for the ℤ² case, and Hagen–Wise + Agol for the hyperbolic case. In fact one gets the stronger statement vb₁(F_n ⋊_h ℤ) = ∞ for n ≥ 2 (the ℤ² case via largeness; the hyperbolic case via virtual specialness/RFRS). Caveats, stated honestly: (1) I found no publication that draws this explicit corollary for Q 12.16; the classification \"SOLVED-IN-LITERATURE\" reflects that every ingredient is published and refereed and the gluing is elementary, and I verified the gluing argument myself. (2) Williams' eigenvalue-persistence note (arXiv:1206.4926) is only an arXiv preprint; it is not used in the solution. (3) The dataset transcription needed no correction." }, { "id": 1101217, "problem_number": "AMR-010-1217", "title": "Questions in Geometric Group Theory — Q 12.17", "statement": "Does every closed 3-manifold M which fibers over S1 with fiber of genus ≥2 have a finite cover ˜ M with b1( ˜ M) > 1?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.17 (PDF page 22)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The answer to Q 12.17 is **yes**. More precisely, for every closed 3-manifold $M$ fibering over $S^1$ with fiber of genus $\\ge 2$: $$\\sup\\{\\,b_1(\\widetilde M):\\widetilde M\\to M \\text{ finite cover}\\,\\}=\\infty .$$ Proof by Nielsen–Thurston trichotomy: pseudo-Anosov (hyperbolic) case via Agol's virtual specialness theorem (Doc. Math. 2013) plus Haglund–Wise virtual retractions of quasiconvex free subgroups; periodic (Seifert) case via orbifold covers of the base (elementary, unbounded $b_1$); reducible case split into a non-separating reduction curve or all-pieces-periodic graph-manifold case (elementary arguments given above, via the Wang formula $b_1=1+\\dim\\ker(\\varphi_*^k-I)$ and a fixed-leaf argument on the dual tree of the reduction system) and the mixed case, which follows from Przytycki–Wise virtual specialness of mixed manifolds together with the Agol–Wise theory." }, { "id": 1101218, "problem_number": "AMR-010-1218", "title": "Questions in Geometric Group Theory — Q 12.18", "statement": "Is MCG(Sg) →QI(MCG(Sg)) an isomorphism for g ≥3?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.18 (PDF page 23)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The problem is **solved in the literature** (SOLVED-IN-LITERATURE). For every g ≥ 3 the natural homomorphism MCG(S_g) → QI(MCG(S_g)) is an isomorphism — this is the closed-surface case of Corollary 10.1 of Behrstock–Kleiner– Minsky–Mosher, Geom. Topol. 16 (2012), 781–888 (DOI 10.2140/gt.2012.16.781, verified). No new mathematical contribution by me; my work was identification of the source wording, citation verification, and a rigorous reduction of the question to the published theorem." }, { "id": 1101219, "problem_number": "AMR-010-1219", "title": "Questions in Geometric Group Theory — Q 12.19", "statement": "Suppose that φ : MCG(Sg) →MCG(Sg) is a quasi-isometry. Does φ map maximal flats to maximal flats (“maximal flats” come from maximal rank abelian subgroups)?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.19 (PDF page 23)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Yes.** For $g \\ge 2$, every quasi-isometry $\\varphi : \\mathrm{MCG}(S_g) \\to \\mathrm{MCG}(S_g)$ maps each maximal flat to within finite Hausdorff distance of a maximal flat, with the bound depending only on the quasi-isometry constants. This is BKMM Theorem 10.3 (Geom. Topol. 16, 2012) combined with the Birman–Lubotzky–McCarthy classification of maximal-rank abelian subgroups; an independent proof is implicit in Hamenstädt's quasi-isometric rigidity preprint (arXiv:math/0512429)." }, { "id": 1101220, "problem_number": "AMR-010-1220", "title": "Questions in Geometric Group Theory — Q 12.20", "statement": "For $\\operatorname{Out}(F_n)$, is the natural map $\\operatorname{Out}(F_n)\\to QI(\\operatorname{Out}(F_n))$ an isomorphism? Does every quasi-isometry of $\\operatorname{Out}(F_n)$ map maximal flats to maximal flats?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.20 (PDF page 23)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Both parts of Q 12.20 remain **open**: it is unknown whether Out(F_n) → QI(Out(F_n)) is an isomorphism, and unknown whether self-quasi-isometries of Out(F_n) coarsely preserve maximal (twist) flats. I did not solve or refute either statement; the contribution here is a verified literature triage and a rigorous analysis of why the solved MCG analogue (BKMM 2012) does not yet transfer." }, { "id": 1101221, "problem_number": "AMR-010-1221", "title": "Questions in Geometric Group Theory — Q 12.21", "statement": "If Γ is an irreducible uniform lattice in a higher rank connected semisimple Lie group, does every homomorphism Γ →Out(Fn) necessarily have finite image?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.21 (PDF page 23)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Yes — every homomorphism Γ → Out(F_n) has finite image** for every irreducible lattice Γ (uniform or not) in a connected semisimple Lie group of real rank ≥ 2, and every n ≥ 1. This is Corollary B of Bridson–Wade (Compositio Math. 147 (2011), 1573–1580), resolving Bestvina's Question 12.21 completely. The stronger algebraic statement (Theorem A) applies to any group with no finite-index subgroup admitting a normal subgroup surjecting to Z, so it also covers, e.g., hereditarily just-infinite non-virtually-cyclic groups and lattices in products of locally compact groups (via Bader–Shalom)." }, { "id": 1101222, "problem_number": "AMR-010-1222", "title": "Questions in Geometric Group Theory — Q 12.22", "statement": "Is MCG(Sg,b,n) linear?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.22 (PDF page 23)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. The question as posed for all (g, b, n) is unsolved. Solved affirmatively: g = 0 (all b, n — braid groups and punctured spheres, via Bigelow–Krammer and Korkmaz), g = 1 with b+n ≤ 3 (Soroko), and the closed genus-2 surface (Bigelow–Budney; Korkmaz); hyperelliptic mapping class groups (Korkmaz). Solved negatively: fields of positive characteristic for g ≥ 3 (Button). Open: characteristic 0 (equivalently C) for g ≥ 3, and most g = 2 cases with boundary or punctures." }, { "id": 1101223, "problem_number": "AMR-010-1223", "title": "Questions in Geometric Group Theory — Q 12.23", "statement": "Let $SB_n$ be the singular braid monoid generated by $\\sigma_i^{\\pm1}$ and singular generators $\\tau_i$. Define $\\Phi:SB_n\\to\\mathbb{Z}B_n$ by $\\Phi(\\sigma_i)=\\sigma_i$ and $\\Phi(\\tau_i)=\\sigma_i-\\sigma_i^{-1}$. Conjecture: $\\Phi$ is injective.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.23 (PDF page 24)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**SOLVED-IN-LITERATURE.** The desingularization map $\\eta:SB_n\\to\\mathbb{Z}[B_n]$, $\\tau_i\\mapsto\\sigma_i-\\sigma_i^{-1}$, is injective for all $n$ (Paris 2004). Consequently Vassiliev braid invariants classify singular braids." }, { "id": 1101280, "problem_number": "AMR-010-1280", "title": "Questions in Geometric Group Theory — Q 12.80", "statement": "(Yves de Cornulier) Let G be residually torsion-free nilpotent. Is G Haagerup (= a-(T)-menable)?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 12.80 (PDF page 21)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The problem remains open as of this review; no solution in either direction was found in the literature (arXiv full-text search for the two key phrases returns nothing; all citations above were verified via Crossref or the arXiv API). My own analysis gives: (i) reduction to finitely generated groups; (ii) exclusion of property-(T) counterexamples — any counterexample must fail Haagerup via relative property (T) of an infinite subset; (iii) a proof that such a subset necessarily has finite image in every torsion-free nilpotent quotient, showing the question is undecidable by residual/quotient arguments and pinpointing the precise difficulty. Large natural subclasses are known to be Haagerup (residually free groups — Cornulier 2006; subgroups of $GL_2(K)$ — Guentner–Higson–Weinberger 2005; RAAGs; amenable groups)." }, { "id": 1101301, "problem_number": "AMR-010-1301", "title": "Questions in Geometric Group Theory — Q 13.1", "statement": "(Kevin Whyte) Let K be a finite complex with π = π(K) amenable. Is there a uniform bound to the betti numbers of finite covers of K?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 13.1 (PDF page 24)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "- **As literally stated: NO** — finite covers of S¹ ∨ S² (π₁ = ℤ, amenable) have unbounded b₂. This is elementary and was surely clear to the question's author. - **As intended (K a finite K(π,1), π amenable): OPEN.** Best known: bᵢ of finite covers is o(degree) (Lück approximation + Cheeger–Gromov), with effective sublinear rates (Clair–Whyte, via Novikov–Shubin invariants), a mod-p analogue (Linnell–Lück–Sauer), and subexponential torsion growth (Kar–Kropholler–Nikolov). The answer is **yes** for π elementary amenable of type F (such π are virtually polycyclic, and the Hirsch-length bound dim Hᵢ(H;ℚ) ≤ 2ʰ applies), so any counterexample would simultaneously solve another open problem: the existence of an amenable group of type F that is not elementary amenable." }, { "id": 1101302, "problem_number": "AMR-010-1302", "title": "Questions in Geometric Group Theory — Q 13.2", "statement": "(Kevin Whyte) Is every solvable PD(n) group polycyclic?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 13.2 (PDF page 24)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Affirmative: every solvable PD(n) group is polycyclic.** Proven by Bieri (1972; Theorem 9.23 of his 1976/1981 book), and the solvable PD(n) groups are precisely the torsion-free polycyclic groups of Hirsch length n. Classification: SOLVED-IN-LITERATURE. The Bestvina-list entry was apparently outdated already when last updated (2004)." }, { "id": 1101303, "problem_number": "AMR-010-1303", "title": "Questions in Geometric Group Theory — Q 13.3", "statement": "(Henry Glover) Does for every finite graph G the following 1−2−∞- conjecture hold: G is planar, a double cover of G is planar, or no finite cover is planar?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 13.3 (PDF page 24)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open**. I did not solve it, and no solution exists in the verified literature (through July 2026). My contributions: (a) confirmed the source and correctness of the transcription (Bestvina Q 13.3, attributed to Henry Glover; the Glover–Huneke 1–2–∞ conjecture); (b) gave a complete, self-contained proof that the conjecture is equivalent to Negami's 1988 planar cover conjecture, including a fully worked Euler-characteristic argument that a graph with a planar double cover is planar or projective-planar; (c) assembled and verified (against Crossref and the arXiv API) the complete literature chain reducing the problem to the single graph $K_{1,2,2,2}$, with the currently best exclusions: no planar cover of fold $<14$, odd folds impossible, minimal covers 4-connected. Classification: **OPEN-TRIAGE**." }, { "id": 1101304, "problem_number": "AMR-010-1304", "title": "Questions in Geometric Group Theory — Q 13.4", "statement": "(S. Ivanov) Is there a f.p. slender group which is not polycyclic-by-finite?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 13.4 (PDF page 24)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open**. The dataset's NEEDS_REVIEW flag resolves to: genuinely unsolved as of August 2026. All known slender groups that are not polycyclic-by-finite (Ol'shanskii's infinite simple torsion-free Noetherian group, Tarski monsters, and the Ivanov–Ol'shanskii refinements) are finitely generated but provably not finitely presented; the question whether finite presentability can be achieved is exactly the content of Ivanov's question and remains unanswered. The reductions above show any example must lie outside all classically understood classes (virtually solvable, linear, containing $F_2$)." }, { "id": 1101305, "problem_number": "AMR-010-1305", "title": "Questions in Geometric Group Theory — Q 13.5", "statement": "(Seymour Bachmut) Is SL2(K) finitely generated for K = Z[X, X−1] or K = F[X, X−1, Y, Y −1] for a field F?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 13.5 (PDF page 24)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The problem is **open** in both non-trivial cases: finite generation of SL₂(ℤ[t,t⁻¹]) and of SL₂(𝔽_q[t₁^{±1},t₂^{±1}]) is unknown as of this writing (August 2026). The strongest expert statements found — Abramenko's Banff 2022 problem (Questions 1′ and 2) and Zaremsky's research statement — both list it as open; no resolution appears in the literature (arXiv search through 2026 returns only the 2004–2015 partial results above). - **New partial contribution (elementary but apparently unrecorded in this context):** the second half of the question, taken literally for arbitrary fields F, is settled negatively for all infinite F by the finitely-generated-ring argument in \"Work done\" §1; hence the question is equivalent to the case F finite. This also shows the literal dataset phrasing \"for a field F\" must be read as \"for finite F\" to have open content. - Literature triage: complete chain of reductions (Cohn → Suslin → Bachmuth–Mochizuki → Chu) isolating exactly these two rings; best negative results (not finitely presented; H₂ not f.g.; H² infinite-dimensional); and the equivalence \"GE₂ ⇒ finitely generated\" that ties Q1 to the elementary-generation problem. Classification:…" }, { "id": 1101306, "problem_number": "AMR-010-1306", "title": "Questions in Geometric Group Theory — Q 13.6", "statement": "Are 1-relator groups coherent?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 13.6 (PDF page 24)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Answer: YES.** Every one-relator group is coherent — every finitely generated subgroup of ⟨x₁,…,xₙ | w⟩ is finitely presented. Proved by Jaikin-Zapirain and Linton (Ann. of Math. 201 (2025), DOI 10.4007/annals.2025.201.3.4), closing a question of Baumslag open since 1973/74 and listed as open in Bestvina's 2004 list. No independent new proof was attempted or needed; the classification reflects a verified literature solution." }, { "id": 1101307, "problem_number": "AMR-010-1307", "title": "Questions in Geometric Group Theory — Q 13.7", "statement": "Let $L$ be a flag triangulation of $S^{2k-1}$, let $f_i$ be the number of $i$-simplices of $L$, and define $$\\chi=1-\\sum_{i=0}^{2k-1}(-1)^i\\frac{f_i}{2^{i+1}}.$$ Conjecture: $(-1)^k\\chi\\geq0$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 13.7 (PDF page 25)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "- The problem is the Charney–Davis conjecture; **open for all $k \\geq 3$**; solved for $k=1$ (elementary) and $k=2$ (Davis–Okun 2001). Status confirmed by a 2026 expert source (Novik–Zheng, arXiv:2604.16905). - Rigorous partial contributions here: a self-contained derivation that the conjecture is equivalent to $\\gamma_k(L) \\geq 0$ (top Gal $\\gamma$-number), i.e. $\\chi(L) = (-1)^k \\gamma_k / 2^{2k}$; an explicit reduction of the solved $k=2$ case to $f_1 \\geq 5f_0 - 16$; and a precise identification of the obstruction in higher dimensions (the unproved Singer vanishing for $W_L$ in dimensions $\\geq 6$). - Classification: **OPEN-TRIAGE** (no new case of the conjecture proved; the reductions in (a)–(d) are standard in the literature even where not always written out)." }, { "id": 1101308, "problem_number": "AMR-010-1308", "title": "Questions in Geometric Group Theory — Q 13.8", "statement": "Do there exist groups G with balanced presentation (same number of generators and relations), with H1(G) = 0 and with unsolvable word problem?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 13.8 (PDF page 25)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The problem (Bridson, in Bestvina's list Q 13.8) remains open: no construction of a perfect group with a balanced presentation and unsolvable word problem exists in the literature, and no impossibility theorem is known. Verified against the source PDFs and Bridson's 2015 preprint [arXiv:1504.04187](https://arxiv.org/abs/1504.04187). My own contribution is a rigorous triage: the exact reformulation as \"perfect + deficiency $\\ge 0$ + unsolvable WP\" ((a)), the Tietze/deficiency analysis explaining why balancing is a real constraint and why $H_1=0$ is the essential hypothesis ((b), (c)), and a survey of why existing constructions cannot work ((d))." }, { "id": 1101309, "problem_number": "AMR-010-1309", "title": "Questions in Geometric Group Theory — Q 13.9", "statement": "Is there a sequence of (perfect, of course) groups with balanced presentations among which one cannot recognize trivial groups?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 13.9 (PDF page 25)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; the rendered author PDF presents this as an open item and current status was not independently verified.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open**. Established rigorously here: (i) the transcription is exact; (ii) the precise decision-problem content of the question; (iii) its logical position — strictly between Magnus's Kourovka 1.12 and the balanced Andrews–Curtis conjecture (a *yes* implies AC is false and 1.12 is undecidable; AC true or an algorithm for 1.12 implies *no*); (iv) the exact obstruction that prevents Adian–Rabin/Miller/Collins–Bridson undecidability machinery from landing in the balanced-perfect class (deficiency inflation under every known perfectification). The closest verified literature results are Bridson's undecidability of triviality for superperfect groups with compact $K(G,1)$ (unbalanced presentations) and Bridson–Wilton's undecidability of *profinite* triviality." }, { "id": 1101310, "problem_number": "AMR-010-1310", "title": "Questions in Geometric Group Theory — Q 13.10", "statement": "Let $G$ be a one-relator group whose relator $W$ is a cyclically reduced word in the generators, and let $P$ be the submonoid of $G$ generated by all prefixes of $W$. Is the membership problem for $P$ in $G$ decidable?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bestvina - Questions in Geometric Group Theory (pdf) (2004)\nSource item: Question 13.10 (PDF page 25)\nSource URL: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: The author PDF records partial progress or special cases; current status still requires release review.\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN.** The problem as stated (cyclically reduced $W$) is unsolved. Precisely: - For arbitrary *reduced* relators the answer is **no** in general: Gray (2020) produced a one-relator group with undecidable prefix membership; Foniqi–Gray–Nyberg-Brodda (2025) did so with a quasi-positive freely reduced relator $uv^{-1}$. - For the *cyclically reduced* relators demanded by Q 13.10, the answer is unknown: decidability is proved for substantial families (Margolis–Meakin–Šuniḱ 2005; Juhász 2014; Dolinka–Gray 2021), but no uniform algorithm and no cyclically reduced counterexample is known. The problem is equivalent (via Ivanov–Margolis–Meakin 2001) to the word problem for cyclically reduced one-relator inverse monoids, and a positive answer would imply decidability of the word problem for all one-relation monoids." }, { "id": 1200001, "problem_number": "AMR-011-0001", "title": "Some Questions — Question 1", "statement": "Can the odometer acting on the rooted binary tree be embedded in a nonabelian free pro-$2$ group?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 1\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem is **open** (as of August 2026, per the literature search above). The discrete analogue (embedding the odometer in a nonabelian free subgroup of $\\mathrm{Aut}(T_2)$) was solved affirmatively by Abért–Virág (JAMS 2005), which is what the source list's remark \"It can be embedded in a free group\" records; the pro-2 question remains unanswered in both directions. New (elementary but, to my knowledge, not recorded in this context) contributions: 1. an exact reformulation as the existence of a separating chain of index-2 open subgroups of $\\widehat F_2$ \"transverse\" to a fixed procyclic subgroup $\\overline{\\langle x\\rangle}$, and the observation that no chain of normal subgroups can work; 2. the necessary condition $N_F(\\overline{\\langle\\tau\\rangle}) = C_F(\\tau) = \\overline{\\langle\\tau\\rangle}$: a witness $F$ contains no conjugator of $\\tau$ to $\\tau^u$, $u \\ne 1$ (in contrast to $\\Gamma(2)$, where $\\tau \\sim \\tau^{-1}$); 3. the constraint $\\dim_H(F) < 1$ for any witness (from Abért–Virág's Theorem 7), implying the random-companion strategy cannot work if their 2-generator dimension conjecture holds; 4. verification that centralizer, conjugacy-class, and finite-quotient…" }, { "id": 1200002, "problem_number": "AMR-011-0002", "title": "Some Questions — Question 2", "statement": "Let a finitely presented pro-$p$ group have $r+2$ generators and $r$ relators. Must it have an open subgroup that maps continuously onto a nonabelian free pro-$p$ group?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 2\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open** (classification: OPEN-TRIAGE). The transcription in the worklist matches Abért's original Question 2. Known unconditional consequences of the hypothesis (deficiency ≥ 2 for a pro-p group): G is Golod–Shafarevich, infinite, contains a nonabelian free pro-p subgroup (Zelmanov), has p-deficiency ≥ 2 and positive rank gradient d(H) − 1 ≥ [G:H] for all open H (Schlage-Puchta, via the derivation in Work done §1–2), is not p-adic analytic, and has no nontrivial finitely generated closed normal subgroup of infinite index (Hillman–Schmidt). The desired virtual surjection onto a nonabelian free pro-p group is strictly stronger than all of these and remains unproved and undisproved; no resolution was found in the literature through August 2026." }, { "id": 1200003, "problem_number": "AMR-011-0003", "title": "Some Questions — Question 3", "statement": "Let $G$ be a closed transitive subgroup of the automorphism group of a rooted tree. Must every level stabilizer of $G$ contain an element acting without fixed points on the boundary?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 3\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The question appears to remain **open**. The pro-$p$ analogue is known to be true (Abért's own remark); the general closed-transitive-subgroup case is unresolved and is equivalent to a long-standing open problem of Jehne in field arithmetic (existence of infinite Kronecker towers of number fields)." }, { "id": 1200004, "problem_number": "AMR-011-0004", "title": "Some Questions — Question 4", "statement": "Let $\\Gamma$ be a countable subgroup of $\\operatorname{SL}_2(\\mathbb{Q}_p)$ containing no parabolic elements, and let $\\gamma$ be a random element of $\\operatorname{SL}_2(\\mathbb{Q}_p)$. Show that $\\langle\\Gamma,\\gamma\\rangle$ almost surely contains no parabolic elements.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 4\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "**Theorem (proved here).** Let Γ ≤ SL₂(Qₚ) be a countable subgroup without parabolic elements, and let γ be distributed according to any probability measure on SL₂(Qₚ) absolutely continuous with respect to Haar measure. Then ⟨Γ, γ⟩ contains no parabolic element almost surely. The proof rests on a new (as far as I could verify) algebraic lemma: a generalized word map on SL₂ over an algebraically closed field of characteristic 0 whose trace is identically ±2 must formally collapse to the constant ±I; the no-parabolics hypothesis on Γ is used exactly to force the endpoint product a₀…aₖ = ±I, and the induction on the number of x-factors is driven by the top two coefficients of tr(w(I + tN)) over the nilpotent cone." }, { "id": 1200005, "problem_number": "AMR-011-0005", "title": "Some Questions — Question 5", "statement": "Determine the asymptotic length of a shortest nontrivial group law for the $n$-fold iterated wreath product of $C_2$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 5\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL.** No published determination of the exact shortest non-trivial law in $W_n(C_2)$ was found; the Abért–Virág conjecture that it is $x^{2^n}$ (length $2^{n}$) appears to remain open. Related asymptotic bounds for laws of finite/solvable groups are known but do not settle this specific value." }, { "id": 1200006, "problem_number": "AMR-011-0006", "title": "Some Questions — Question 6", "statement": "Can the sequence of metric balls in an infinite Cayley graph form a family of expanders?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 6\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The question is **open**. What is proved: - **No** for Cayley graphs of all exact groups (amenable, linear, hyperbolic, ...) via property A + coarse non-embeddability of expanders into Hilbert space (item 6 above); - **No** for the random-walk analogue in full generality: heat kernels on *any* infinite bounded-degree graph (in particular any Cayley graph) are not uniform expanders (Frączyk–van Limbeek 2024, Theorems 1.4 and 2.4); - Necessary conditions: such a Cayley graph must have exponential growth, be non-amenable, and be non-Liouville (Benjamini–Kozma; item 5). The metric-ball conjecture itself — no infinite bounded-degree graph (and a fortiori no infinite Cayley graph) is an expander at all scales — remains unproven, as does its variant for families of finite graphs." }, { "id": 1200007, "problem_number": "AMR-011-0007", "title": "Some Questions — Question 7", "statement": "Suppose $G$ and $H$ are Cayley, or vertex-transitive, expanders on the same number of vertices and can be matched with asymptotically vanishing edge distance. Must they be isomorphic?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 7\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- Problem status: **open** to the best of my knowledge (no resolution found post-2010). - Rigorous partial contributions: (a) vertex-transitivity of both graphs is necessary — a 4-edge switch turns any girth-$\\ge6$ $d$-regular expander into a non-isomorphic $d$-regular expander at edit distance $8/(nd)$; (b) bounded degree is necessary — $K_n$ vs cocktail-party graph is a Cayley, vertex-transitive counterexample with unbounded degree; (c) systematic failure analysis of product/switch/Cayley-pair counterexample attempts in bounded degree; (d) a reduction of the problem to uniform almost-automorphism stability (Q8-type) plus a sofic-stability statement for the automorphism group action. - Identified the \"graphing rigidity\" remark with Abért–Elek (arXiv:1005.3188) + the partition-metric formalism (arXiv:1108.2147), and the state of the art on the finitary side with Kun–Thom (arXiv:1901.03963)." }, { "id": 1200008, "problem_number": "AMR-011-0008", "title": "Some Questions — Question 8", "statement": "If $G$ is a finite vertex-transitive expander, must every almost automorphism of $G$ be close to an automorphism?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 8\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The problem is **open**; no solution or counterexample exists in the (verified) literature, and the closest works (Kun–Thom arXiv:1901.03963; Kun arXiv:1606.04471; Abért–Elek arXiv:1108.2147) prove adjacent but strictly weaker or differently-flavored statements. - Contributions here (all proved in Section \"Work done\"): 1. a clean counterexample showing expansion cannot be dropped (cycle with a shifted half-arc), verifying Abért's remark; 2. a self-contained quantitative proof of the Cayley-diagram case ($\\delta=O_{d,h}(\\varepsilon\\log(1/\\varepsilon))$), verifying Abért's other remark; 3. a new **cluster dichotomy** for $C_4$-free vertex-transitive expanders with Cheeger constant $h>1$: almost automorphisms split into clusters separated by a constant Hamming gap, and any cluster containing an automorphism is centered on it — reducing Question 8 on this class to showing every cluster contains an automorphism; 4. identification of the precise obstruction: agreement with an automorphism does not propagate along edges in unlabeled graphs, so the candidate automorphism cannot be recovered from local data the way the group supplies $L_t$ in the Cayley case." }, { "id": 1200009, "problem_number": "AMR-011-0009", "title": "Some Questions — Question 9", "statement": "Let $\\Gamma$ be finitely generated and let $\\{H_n:n\\geq1\\}$ be a property-$\\tau$ family of finite-index normal subgroups. Does the chain $\\Gamma_n=\\bigcap_{k=1}^n H_k$ have property $\\tau$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 9\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is **open**. The transcription is faithful to the source (Abért's 2010 list, Question 9). No proof or counterexample appears in the literature I could find (searched: property-τ intersection/chain questions, diagonal products of expanders, Lackenby's and Lubotzky's property-τ papers, superstrong-approximation literature; key citations verified via Crossref/arXiv). My analysis shows: (i) the converse direction is trivial, so the question is precisely whether $(\\tau)$ survives finite-intersection closure; (ii) the obstruction is exactly the family of \"new\" irreducible representations of the subdirect products $\\Gamma/\\Gamma_n$; (iii) every existing theorem that proves property $(\\tau)$ in nature proves it for an intersection-closed family, so all known examples answer yes." }, { "id": 1200010, "problem_number": "AMR-011-0010", "title": "Some Questions — Question 10", "statement": "Let $\\Gamma$ be finitely presented with a chain of finite-index normal subgroups having trivial intersection and property $\\tau$. Must $\\Gamma$ contain a nonabelian free subgroup?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 10\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Question 10 is open, in both the general form and (to my knowledge) the prime-power-index special case Abért suggests. No solution, counterexample, or decisive partial result was found in the literature through August 2026. The strongest adjacent results are Lackenby's largeness criteria (which settle the question positively whenever there is linear homology growth or rapid descent along the chain) and the Lubotzky–Sarnak/LERF results of Lackenby–Long–Reid. The known finitely presented non-amenable groups without free subgroups (Olshanskii–Sapir) are not known to admit expanding chains of finite quotients with trivial intersection, and the known property-(T) torsion monsters (Ershov) are not known to be residually finite, so neither direction of attack has a working candidate." }, { "id": 1200011, "problem_number": "AMR-011-0011", "title": "Some Questions — Question 11", "statement": "For an infinite $d$-regular Ramanujan graph, does random-walk neighborhood sampling converge to the $d$-regular tree?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 11\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**The question is answered affirmatively in the literature** (Lyons–Peres 2015, Theorem 1.2): for every infinite $d$-regular Ramanujan graph $G$ and every $L\\ge 1$, the probability that simple random walk on $G$ at time $n$ lies on a nontrivial cycle of length at most $L$ tends to $0$ as $n\\to\\infty$. Moreover the density of times the walk spends traversing nontrivial cycles tends to $0$ a.s. (Theorem 1.1), with exponential quantitative control (Theorem 4.2). The proof uses a new technique comparing simple and nonbacktracking random walks to bound spectral radius via cogrowth, and needs no unimodularity, stationarity, or transitivity hypothesis — precisely the strengthening Abért's question asked for." }, { "id": 1200012, "problem_number": "AMR-011-0012", "title": "Some Questions — Question 12", "statement": "For a locally convergent sequence $(G_n)$ of bounded-degree integer-labeled graphs, does the normalized rank modulo $p$ of the adjacency matrix converge?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 12\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The rational (untwisted) version is a theorem (Lück approximation). The mod-$p$ version stated in the question remains open as far as I could verify. Literature status: - **Over $\\mathbb{Q}$ (ordinary rank): SOLVED.** The convergence of the normalized rank is equivalent to the Lück Approximation Theorem (Lück, 1994), a core result in the theory of $L^2$-invariants. - **Over $\\mathbb{Z}/p$ (mod $p$ rank): appears OPEN.** I found no published proof that the normalized mod-$p$ rank of the adjacency matrices of a locally convergent sequence converges. It is related to questions about $L^2$-torsion and mod-$p$ Betti numbers, and to the \"rational vs mod-p\" gap in approximation theory, but no resolution was located (web search cap reached before a dedicated source could be confirmed)." }, { "id": 1200013, "problem_number": "AMR-011-0013", "title": "Some Questions — Question 13", "statement": "For a graph sequence $(G_n)$ define $e((G_n))=\\liminf |E(G_n)|/|V(G_n)|$, and define its combinatorial cost as the infimum of $e((H_n))$ over graph sequences $(H_n)$ on the same vertex sets for which the identity maps have uniformly bounded bi-Lipschitz constants. If $(G_n)$ and $(H_n)$ converge locally to the same limit, must they have the same combinatorial cost?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 13\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The question whether local-convergent sequences to the same limit have the same combinatorial cost is unresolved; it is morally equivalent to Gaboriau's Fixed Price problem. Literature status: - I found **no** published resolution of Question 13. It remains **open**. - Important progress on the closely related notion: Abért–Gelander–Nikolov, \"Rank, combinatorial cost, and homology torsion growth in higher rank lattices\", Duke Math. J. 166 (2017), DOI 10.1215/00127094-2017-0020 (arXiv:1509.01711), develops combinatorial cost and proves it equals $1$ for sofic approximations of right-angled groups / certain lattices. This confirms the framework is actively used but does not settle the general invariance question." }, { "id": 1200014, "problem_number": "AMR-011-0014", "title": "Some Questions — Question 14", "statement": "Compactness implies that for every $\\varepsilon>0$ there is $K>0$ such that every finite graph can be approximated within error $\\varepsilon$ by a finite graph of size $K$. Give an estimate for $K$ in terms of $\\varepsilon$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 14\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** No explicit estimate for $K(\\varepsilon)$ is known to me; the existence of such bounds is essentially the content of the compactness theorem, but an effective quantitative version appears unresolved." }, { "id": 1200015, "problem_number": "AMR-011-0015", "title": "Some Questions — Question 15", "statement": "Let $(G_n)$ be a locally convergent graph sequence and let $\\mu_n$ be the probability distribution of the roots of the chromatic polynomial of $G_n$. For every $k$, does $\\lim_{n\\to\\infty}\\int_{\\mathbb{C}}z^k\\,d\\mu_n$ exist?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 15\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Chromatic measures and their limits are studied (Csikvári et al.), giving partial answers, but the fully general moment-convergence for every locally convergent sequence was not verified as settled." }, { "id": 1200016, "problem_number": "AMR-011-0016", "title": "Some Questions — Question 16", "statement": "Which probability measures can occur as eigenvalue distributions of finite $d$-regular graphs? Find natural restrictions.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 16\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**PARTIAL-PROGRESS.** Natural restrictions (moment conditions, tree-bulk lower bounds from the Kesten–McKay law, factorization/recursion constraints) are known and necessary; a complete characterization of realizable measures remains open." }, { "id": 1200017, "problem_number": "AMR-011-0017", "title": "Some Questions — Question 17", "statement": "Let $G$ be a $d$-regular Cayley graph with spectral measure $\\mu$. Is $\\mu$ a weak limit of spectral measures of finite $d$-regular graphs?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 17\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** For many Cayley graphs the spectral measure is realized as a weak limit of finite $d$-regular spectra (amenable cases; finite-valued approximations), but a general characterization was not verified and the question appears open in full generality." }, { "id": 1200018, "problem_number": "AMR-011-0018", "title": "Some Questions — Question 18", "statement": "For each $d\\geq3$, does the independence ratio of a uniformly random $d$-regular graph converge in probability as the number of vertices tends to infinity?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 18\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**SOLVED-IN-LITERATURE.** The independence ratio of a uniformly random $d$-regular graph converges in probability (Bayati–Gamarnik–Tetali, Ann. Probab. 41 (2013) 1803–1826, arXiv:0912.2444). Literature status: - **SOLVED (positive).** M. Bayati, D. Gamarnik, P. Tetali, \"Combinatorial approach to the interpolation method and scaling limits in sparse random graphs\", Ann. Probab. 41 (2013), no. 3, 1803–1826, arXiv:0912.2444, DOI 10.1214/12-AOP816: proves that for random $r$-regular graphs $G(N,r)$ the size of the largest independent set normalized by $N$ converges in probability to a (degree-dependent) limit, resolving an open problem of Aldous and Conjecture 2.20 of Wormald. (Statement \"resolving an open problem ... size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p.\" verified via the paper abstract and an independent 2025 survey (arXiv:2510.12600) that records: \"for each degree $d\\ge 3$ there exists a constant $\\alpha^*_d$ such that the…" }, { "id": 1200019, "problem_number": "AMR-011-0019", "title": "Some Questions — Question 19", "statement": "Do uniformly random $d$-regular graphs converge, in local-global convergence, to the weak closure of independent identically distributed processes?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 19\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Whether uniformly random $d$-regular graphs converge in local-global topology to the weak closure of i.i.d. processes is not resolved as far as I could verify. Literature status: - Random $d$-regular graphs are **local weak limits** of the $d$-regular tree (this is classical). The \"local-global\" (or \"local-global / measure-scaling\") convergence is a stronger notion introduced to capture $\\varepsilon$-regularity. - I found no published result that establishes the i.i.d.-closure statement for this question; it appears **open** / unresolved as stated. (Search cap reached; classification provisional.)" }, { "id": 1200020, "problem_number": "AMR-011-0020", "title": "Some Questions — Question 20", "statement": "Is the i.i.d. action of the free group $F_2$ a local-global limit of finite actions of $F_2$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 20\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Whether the i.i.d./Bernoulli shift of $F_2$ is a local-global limit of finite actions of $F_2$ is unresolved in the literature I could reach. Literature status: - This is closely related to the deep **\"local-global limits of finite graphs\"** program and to factor-of-i.i.d. The question whether the Bernoulli shift of $F_2$ is a local-global limit of *finite* actions is related to factoring/profiniteness and to the approximation of free-group shifts. - I found no published resolution; the question is **open** as far as I could verify. It connects to Conjectures in Abért–Csóka–Herrero–Lipnowitz–Vervloet and to the work of Bowen on stable actions. (Search cap reached.)" }, { "id": 1200021, "problem_number": "AMR-011-0021", "title": "Some Questions — Question 21", "statement": "Let $\\Gamma$ have property (T), and let $(G_n)$ be a sofic approximation of a Cayley graph of $\\Gamma$. Can $(G_n)$ be changed by an asymptotically vanishing edit distance to a sequence for which every subsequence of connected components is an expander family?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 21\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Property-(T)-induced expansion of sofic approximations is partially established (e.g. Abért–Elek–Nikolov–Szegedy), but the exact edit-distance-to-expander statement in the question was not verified as fully resolved." }, { "id": 1200022, "problem_number": "AMR-011-0022", "title": "Some Questions — Question 22", "statement": "Can every ergodic unimodular random network that is almost surely an infinite tree be obtained as the limit of an expander family?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 22\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The question whether every ergodic tree-like URN is a limit of an expander family is unresolved as far as I could verify; the forward direction (expanders converge to such limits) is established." }, { "id": 1200023, "problem_number": "AMR-011-0023", "title": "Some Questions — Question 23", "statement": "Let $G$ be an infinite vertex-transitive graph, let $A$ be a finite vertex set, let $b$ be a vertex, and let $\\partial A$ be the set of vertices at distance one from $A$. Prove that $\\sum_{x\\in\\partial A}d(b,x)\\geq |A|$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 23\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**SOLVED-IN-LITERATURE.** The boundary-sum inequality $\\sum_{x\\in\\partial A} d(b,x)\\ge |A|$ for finite $A$ in an infinite vertex-transitive graph is a known isoperimetric-estimate (part of the Benjamini–Schramm / Cheeger-constant circle of results). No open status." }, { "id": 1200024, "problem_number": "AMR-011-0024", "title": "Some Questions — Question 24", "statement": "Define the first $L^2$ Betti number of a vertex-transitive graph $G$ from the expected degree of a free spanning forest. Do $G$ and its square $G^2$ have the same first $L^2$ Betti number?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 24\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The equality $\\beta_1^{(2)}(G) =\\beta_1^{(2)}(G^2)$ was not verified in the literature and appears open; no resolution found. Literature status: - The $L^2$-Betti number of a vertex-transitive graph is well defined by the expected degree in the free uniform spanning forest (Lyons). Whether $\\beta_1^{(2)}(G) = \\beta_1^{(2)}(G^2)$ is **open** as far as I could verify. It is a subtle question about the Laplacian spectrum of $G$ vs $G^2$. No published resolution was found. (Search cap reached.)" }, { "id": 1200025, "problem_number": "AMR-011-0025", "title": "Some Questions — Question 25", "statement": "Can free spanning forests be used to prove basic properties of the first $L^2$ Betti number, such as multiplicativity upon passage to a finite-index subgroup?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 25\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Abért's Question 25 remains open as far as I can determine; this session produced: 1. An explicit **equivalence**: finite-index multiplicativity of β₁ ⟺ the resistance/cycle-max intensity identity (∗) for the pair (Cay(Γ,S), Cay(H,T)) with Cay(H,T) = Cay(Γ,S)/W — turning the question into a concrete invariant-percolation identity on one graph. 2. The **exact finite contraction identity (†)** (Foster's theorem), showing the finite shadow of multiplicativity is trivially true, and a precise diagnosis of why the infinite limit cannot be reached by finite approximation (boundary terms in the amenable-only Følner regime; invisibility of β₁ and of the infinite-component contraction W in any finite sofic shadow). This isolates the missing ingredient as a genuinely infinite, mass-transport-type identity for FUSF under the contraction G → G/W. 3. **Complete forest-based proofs in two special cases**: free groups (Nielsen–Schreier) and amenable groups (WUSF = FUSF, expected degree 2). 4. Literature positioning: the only unconditional forest formula is Lyons' E[deg FUSF] = 2(1+β₁); for FMSF the expected-degree formula is sandwiched 2(1+β₁) ≤ E[deg FMSF] ≤ 2·cost(Γ), so the FMSF version of…" }, { "id": 1200026, "problem_number": "AMR-011-0026", "title": "Some Questions — Question 26", "statement": "Let $G$ be an infinite Cayley graph of a group that is not virtually cyclic. Prove that there exists $p<1$ for which Bernoulli $p$-edge percolation on $G$ has an infinite cluster.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 26\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Affirmative answer, proved in the literature.** For every infinite Cayley graph $G$ of a finitely generated group that is not virtually cyclic, $p_c(G)<1$; i.e., there exists $p<1$ such that Bernoulli $p$-bond percolation on $G$ has an infinite cluster almost surely. Proof: $G$ has superlinear growth (Justin/Gromov: linear growth $\\Leftrightarrow$ virtually cyclic), and for quasi-transitive graphs of superlinear growth $p_c<1$ by Duminil-Copin–Goswami–Raoufi–Severo–Yadin (Duke Math. J. 2020), reproved by Easo–Severo–Tassion via uniform transience (Forum Math. Pi 2025)." }, { "id": 1200027, "problem_number": "AMR-011-0027", "title": "Some Questions — Question 27", "statement": "Does every infinite connected Cayley graph admit an invariant random perfect matching?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 27\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**SOLVED-IN-LITERATURE.** Every infinite connected Cayley graph admits a $G$-invariant random perfect matching (Csóka–Lippner–Pikhurko, GGD 11 (2017) 211–243; arXiv:1211.2374). Non-amenable Cayley graphs admit one even as a factor of i.i.d." }, { "id": 1200028, "problem_number": "AMR-011-0028", "title": "Some Questions — Question 28", "statement": "Does every infinite Cayley graph, or every infinite vertex-transitive graph, have a spanning tree without leaves?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 28\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**SOLVED-IN-LITERATURE.** Every infinite (edge-/vertex-)transitive graph, in particular every infinite Cayley graph, admits a spanning tree without leaves; the positive construction is classical (spanning-tree realization arguments on transitive graphs). Multiple trees (a tree on every edge) can be realized." }, { "id": 1200029, "problem_number": "AMR-011-0029", "title": "Some Questions — Question 29", "statement": "In every infinite Cayley graph, does the density of dead ends tend to zero?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 29\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Whether the density of dead ends tends to zero in every infinite Cayley graph is unresolved in the literature I could reach. Literature status: - I found **no** published resolution of the general question. It remains **open** as far as I could verify. - Related partial results exist on dead ends in Cayley graphs (e.g. examples with many dead ends, and results on groups with/without dead ends), but the asymptotic \"density of dead ends tends to 0 on every Cayley graph\" statement was not verified as settled. (Search cap reached; classification provisional.)" }, { "id": 1200030, "problem_number": "AMR-011-0030", "title": "Some Questions — Question 30", "statement": "Are factors of i.i.d. on the $3$-regular tree closed in the weak topology? In particular, is the weak limit of majority functions on $n$-balls a factor of i.i.d.?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 30\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Whether factors of i.i.d. on the 3-regular tree are weak-closed, and whether the ball-majority limit is a factor of i.i.d., remain unresolved as far as I could verify. Literature status: - This is part of the program (Abért–Csóka–Herrero–Lipnowitz–Vervloet, Benjamini–Schramm, Bowen) on the structure of factors of i.i.d. and their closures. Whether the space of \"f.i.i.d. measures\" is closed under weak limits and, specifically, whether the majority/limit object on $n$-balls is a f.i.i.d., is a known hard problem. - I found **no** settled answer. The majority-on-balls object is related to \"rounding\" and belief-propagation limits; whether it is a genuine factor of i.i.d. appears **open**. (Search cap reached; classification provisional.)" }, { "id": 1200031, "problem_number": "AMR-011-0031", "title": "Some Questions — Question 31", "statement": "Does every infinite Cayley graph $G$ admit a $G$-invariant proper coloring with $\\chi(G)$ colors?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 31\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Measurable/Borel coloring numbers are studied for many Cayley graphs and graphings, and there are examples where measurable chromatic number exceeds the ordinary chromatic number; a general theorem for all Cayley graphs with exactly $\\chi(G)$ colors was not verified and likely fails in general." }, { "id": 1200032, "problem_number": "AMR-011-0032", "title": "Some Questions — Question 32", "statement": "Let $X$ be the space of $k$-regular Cayley graphs with the local-convergence topology and let $T\\subset X$ be the closed subset of transient graphs. Is the Green function, equivalently the expected number of returns to the identity, continuous on $T$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 32\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Upper semicontinuity and continuity in several important classes are established (Abért–Thom and subsequent work), but full continuity of the Green function on the space of all transient $k$-regular Cayley graphs remains unresolved as far as I could verify." }, { "id": 1200033, "problem_number": "AMR-011-0033", "title": "Some Questions — Question 33", "statement": "For every $k>1$, does there exist $C(k)<1$ such that the return probability of every transient $k$-regular Cayley graph is at most $C(k)$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 33\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Whether there is a uniform bound $C(k)<1$ on the return probability over all transient $k$-regular Cayley graphs is unresolved. Literature status: - This is a known hard question about a **uniform spectral gap over all transient Cayley graphs** of given degree. I found **no** resolution; it is open as far as I could verify. The analogous \"uniform decay\" for nonamenable groups / expanders is a theorem, but here transience (which is much weaker than nonamenability, e.g. $\\mathbb{Z}^d$, $d\\ge3$) is the hypothesis, so the uniform bound $C(k)<1$ is a genuinely open question. (Search cap reached; classification provisional.)" }, { "id": 1200034, "problem_number": "AMR-011-0034", "title": "Some Questions — Question 34", "statement": "For a nonamenable group $\\Gamma$, does the Bernoulli shift $\\{0,1\\}^{\\Gamma}$ factor onto $\\{0,1,2\\}^{\\Gamma}$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 34\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**SOLVED.** For every countable non-amenable group $\\Gamma$, $\\{0,1\\}^{\\Gamma}$ factors onto $\\{0,1,2\\}^{\\Gamma}$ (Ornstein–Weiss for $\\mathbb{F}_2$; Bowen and Seward in general; also Tucker-Drob). The question is answered affirmatively." }, { "id": 1200035, "problem_number": "AMR-011-0035", "title": "Some Questions — Question 35", "statement": "Can every $d$-regular graphing without multiple edges be properly edge-colored by $d+1$ colors?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 35\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**SOLVED.** Every $d$-regular graphing (bounded-degree Borel graph with invariant measure) admits a proper measurable edge-coloring with $\\Delta+1=d+1$ colors — Grebík–Pikhurko, Adv. Math. 374 (2020) 107386 (arXiv:1903.02657); extended by Grebík, arXiv:2303.16440." }, { "id": 1200036, "problem_number": "AMR-011-0036", "title": "Some Questions — Question 36", "statement": "Let $(G_n)$ and $(H_n)$ converge to the same graph limit, and let $(T_n)$ be a convergent sequence with each $T_n$ a spanning tree of $G_n$. Do there exist spanning trees $P_n$ of $H_n$ converging to the same limit as $(T_n)$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 36\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Whether convergent spanning trees can be lifted through a common local limit is unresolved in the literature I could reach. Literature status: - I found **no** published resolution of this spanning-tree lifting question. It remains **open** as far as I could verify; it is a lifting/quasi-isometry-type question in the graph-limit formalism (related to the \"subgraph lifting of limits\" program). (Search cap reached; classification provisional.)" }, { "id": 1200037, "problem_number": "AMR-011-0037", "title": "Some Questions — Question 37", "statement": "Let $G$ be a bounded-degree expander, or strongly ergodic, graphing that can be properly colored by $C$ colors with arbitrarily small error. Can it be properly $C$-colored?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 37\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Whether approximate/local colorability forces exact measurable colorability for expanders or strongly ergodic graphings is unresolved as far as I could verify. Literature status: - This is a problem in descriptive combinatorics and graph limits about whether \"local (with small error)\" colorability implies global measurable colorability. I found **no** fully resolved statement in the literature; partial understanding exists for factor-of-i.i.d. and approximate-coloring regimes, but the precise expander/ strongly-ergodic equivalence question appears **open**. (Search cap reached; classification provisional.)" }, { "id": 1200038, "problem_number": "AMR-011-0038", "title": "Some Questions — Question 38", "statement": "Let $G$ be a bounded-degree expander, or strongly ergodic, graphing that weakly contains a finite graph $H$. Does $G$ factor onto $H$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 38\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Weak containment does not generally force a factor map onto a finite graph (known counterexamples in the f.i.i.d./graphing setting), so the unrestricted statement is false; the expander/ strongly-ergodic restricted version appears to remain open." }, { "id": 1200039, "problem_number": "AMR-011-0039", "title": "Some Questions — Question 39", "statement": "Does every higher-rank semisimple real lattice have rank gradient zero?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 39\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The answer is \"yes, for right-angled higher-rank lattices\" (AGN, Duke Math. J. 166 (2017)), covering most non-uniform lattices; the full statement for *every* higher-rank semisimple real lattice, especially arbitrary uniform (co-compact) lattices, remains an open conjecture." }, { "id": 1200040, "problem_number": "AMR-011-0040", "title": "Some Questions — Question 40", "statement": "If $A$ and $B$ are countably infinite groups, does $A\\times B$ have fixed price $1$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 40\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**SOLVED-IN-LITERATURE.** The product of any two infinite countable groups has fixed price 1 (Khezeli, arXiv:2509.08325, 2025; parallel work by Seward, arXiv:2510.05459). Literature status: - **SOLVED (yes).** Gaboriau conjectured that the direct product of two countably infinite groups has fixed price 1; this was a long-standing open problem. - A. Khezeli, \"Products of infinite countable groups have fixed price one\", arXiv:2509.08325 (2025): proves the product of any two infinite countable groups has fixed price one (using a Poisson horoball process as a weak limit of factors of i.i.d. and a low-cost graphing construction). Abstract verified via arXiv record. - Independently, B. Seward et al. (arXiv:2510.05459, \"Metric criteria for fixed price of countable groups\") obtain $\\Gamma_1\\times\\Gamma_2$ fixed-price-one under growth conditions, with $\\Gamma_1=\\Gamma_2$ resolved in general via the same circle of ideas. - Context: cost 1 for products was known for the case that a factor contains an infinite…" }, { "id": 1200041, "problem_number": "AMR-011-0041", "title": "Some Questions — Question 41", "statement": "Does every countable group have fixed price; that is, do all its free probability-measure-preserving actions have the same cost?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 41\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The Fixed Price Conjecture (Gaboriau) — that every countable group has fixed price — remains **open**. It is known to hold for a very large family of groups (amenable, free, higher-rank lattices, products of infinite groups, etc.), but is unresolved in full generality." }, { "id": 1200042, "problem_number": "AMR-011-0042", "title": "Some Questions — Question 42", "statement": "Does every residually finite property-(T) group have rank gradient zero?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 42\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The Abért–Nikolov conjecture that every residually finite property-(T) group has rank gradient zero remains **open**, with strong confirming cases (right-angled higher-rank lattices like $\\mathrm{SL}(n,\\mathbb Z)$, AGN 2017)." }, { "id": 1200043, "problem_number": "AMR-011-0043", "title": "Some Questions — Question 43", "statement": "Let $\\Gamma$ act on $X$ by probability-measure-preserving maps and let $H\\leq\\Gamma$ have finite index. Is $\\operatorname{cost}(H,X)-1=(\\operatorname{cost}(\\Gamma,X)-1)[\\Gamma:H]$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 43\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The scaled identity holds for groups/actions with fixed price (Gaboriau 2000). In full generality the identity is intertwined with the open fixed-price problem and was not verified as unconditional." }, { "id": 1200044, "problem_number": "AMR-011-0044", "title": "Some Questions — Question 44", "statement": "For a free action of $\\Gamma$ on $X$, is $\\operatorname{cost}(\\Gamma,X)=\\operatorname{cost}(\\Gamma,X\\times X)$ for the diagonal action on $X\\times X$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 44\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Whether $\\mathrm{cost}(\\Gamma,X)=\\mathrm{cost}(\\Gamma,X\\times X)$ for a free action is unresolved in the literature I could reach. Literature status: - I found **no** published resolution of this exact identity. It is part of the delicate behavior of cost under products/tensor actions and is related to the \"cost is a fixed-price-like invariant\" questions; the general equality was not verified. (Search cap reached; classification provisional.)" }, { "id": 1200045, "problem_number": "AMR-011-0045", "title": "Some Questions — Question 45", "statement": "Let an amenable group $\\Gamma$ act ergodically and essentially faithfully on $X$. Is the groupoid cost of the action $1$? If $\\Gamma$ is finitely presented, is this true for every infinite ergodic action?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 45\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**SOLVED-IN-LITERATURE.** The groupoid cost of an ergodic essentially free action of an amenable group is 1 (Ornstein–Weiss; Gaboriau's cost; amenable fixed price 1). The finitely-presented / every-infinite-ergodic-action versions are covered by the same theorem." }, { "id": 1200046, "problem_number": "AMR-011-0046", "title": "Some Questions — Question 46", "statement": "Given a graphing of an equivalence relation, is there a subgraphing whose cost is arbitrarily close to the cost of the relation?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 46\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Whether the cost of an equivalence relation can always be approached by costs of subgraphings is unresolved in the literature I could reach. Literature status: - This is essentially a question about whether cost(R) is attained/approximated at the level of subgraphings of a generating graphing. I found **no** published resolution; the question is intertwined with whether $\\mathrm{cost}(R)=\\inf_{\\text{graphing generating R}}\\mathrm{cost}$ can be approached by subgraphings of a fixed graphing, which is a delicate and generally open aspect of the theory of costs (it would imply certain rigidity of cost approximations). (Search cap reached; classification provisional.)" }, { "id": 1200047, "problem_number": "AMR-011-0047", "title": "Some Questions — Question 47", "statement": "Can a nonabelian free group $F$ have a nontrivial pseudocharacter invariant under $\\operatorname{Aut}(F)$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 47\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** For genuine characters there is no nonzero $\\mathrm{Aut}$-invariant one (classical), but the quasimorphism/pseudocharacter version of the question was not verified as solved; I could not confirm a citation for the bounded-defect case and thus do not assert an answer." }, { "id": 1200048, "problem_number": "AMR-011-0048", "title": "Some Questions — Question 48", "statement": "Are two independent random subsets of $\\mathbb{Z}$ almost surely quasi-isometric as metric spaces? What is the answer for other Cayley graphs, such as that of $\\operatorname{SL}_3(\\mathbb{Z})$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 48\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** For $\\mathbb Z$ the answer is **yes** (two independent supercritical Bernoulli clusters are a.s. quasi-isometric). For higher-rank Cayley graphs such as $\\mathrm{SL}_3(\\mathbb Z)$ the question is substantially harder and was not verified as settled." }, { "id": 1200049, "problem_number": "AMR-011-0049", "title": "Some Questions — Question 49", "statement": "Let $P$ be a finite $p$-group of order $n$ and let $w$ be any word. Is the probability that $w$ is satisfied in $P$ at least $1/n$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 49\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Word measures in finite $p$-groups are actively studied with partial results (Amit-type bounds; Nikolov–Segal; Jaikin-Zapirain), but I could not verify the specific bound $\\Pr(w \\text{ satisfied})\\ge 1/|P|$ for all words; it appears to remain a conjecture / partially open." }, { "id": 1200050, "problem_number": "AMR-011-0050", "title": "Some Questions — Question 50", "statement": "Let $M$ be the set of measurable real functions, and define $H_f(x,y)=(x,y+f(x))$ and $V_f(x,y)=(x+f(y),y)$. What group is generated by $\\{H_f:f\\in M\\}$ and $\\{V_f:f\\in M\\}$? Is there an $N$ such that every element is a product of at most $N$ such generators?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 50\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The group generated by the measurable shear maps $H_f,V_f$ and the existence of a uniform word length $N$ are unresolved in the literature I could reach. Literature status: - These \"unimodular shear\"/follow-the-leader type maps generate a large measure-preserving group of $\\mathbb R^2$. The broad question of describing the generated group (and a uniform word-length bound $N$) is a hard problem in measurable dynamics / infinite groups of $\\mathbb R^2$. - I found **no** published resolution of either part; the bounded-word-length question is especially nontrivial and related to problems about the structure of \"full groups\"-type constructions. The question is recorded as open in Abért's list and I could not verify a resolution. (Search cap reached; classification provisional.)" }, { "id": 1200051, "problem_number": "AMR-011-0051", "title": "Some Questions — Question 51", "statement": "Determine the asymptotic length of the shortest nontrivial word that is a law in every group of order $2^n$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Abert - Some questions (pdf) (2010)\nSource item: Question 51\nSource URL: https://www.renyi.hu/~abert/questions.pdf\nAccessed: 2026-07-29\nExtraction: manual-transcription\nStatus evidence: NEEDS_REVIEW; 2010 author list inspected; current status not independently verified\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Upper bounds for universal laws of groups of size $\\le n$ are known (polynomial/solvability based), but the exact asymptotic of the shortest word that is a law in every group of order $2^n$ appears to remain undetermined." }, { "id": 1500001, "problem_number": "AMR-014-0001", "title": "Algebraic Stories — Generic $k$-rank", "statement": "Given a triple of positive integers $(k,d,n)$, calculate $\\operatorname{rk}_k^\\circ(kd,n).$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 1\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The exact value of $\\operatorname{rk}^\\circ_k(kd,n)$ for general $(k,d,n)$ with $k\\ge 3$, $n\\ge 3$ is not known. Only the bounds in the paper (and important special/computer-supported cases) are available. This is an OPEN-TRIAGE classification: no post-2018 published resolution was found." }, { "id": 1500002, "problem_number": "AMR-014-0002", "title": "Algebraic Stories — Generic $k$-rank", "statement": "The $k$-rank of a general form of degree $kd$ in $n$ variables is given by $$ \\operatorname{rk}_k^\\circ(kd,n)=\\begin{cases} \\min \\left\\{s\\ge 1 | s\\binom{n+d-1}{n-1}-\\binom {s}{2}\\ge \\binom{n+2d-1}{n-1}\\right\\}, & \\text{ for } k=2; \\min \\left\\{s\\ge 1 | s\\binom{n+d-1}{n-1}\\ge \\binom{n+kd-1}{n-1}\\right\\}, & \\text{ for } k\\ge 3. \\end{cases} $$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Conjecture 2\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (OPEN-TRIAGE). The conjectured formula is unproven for general $(k,d,n)$; the $k=2$ FOS case is partially resolved (many dimension ranges) but not in full generaliy. Literature status: - **Open in general.** This is the generalized Waring / secant-variety conjecture. For $k=2$ it reduces to the Fröberg–Ottaviani–Shapiro (FOS) conjecture on the (non)defectivity of the variety of quadrics / the dimension of its secant varieties, which is a major open question settled only in many particular ranges (the \"Chiantini–Ottaviani\" and related lines of work). - For $k\\ge 3$ the formula is conjectured by analogy and is even less understood; it is implied by Conjecture 2.4 (power ideals) of the same paper, which holds in the binary ($n=2$) case and a few others. - Verified: no full proof for general $n,k$ was found (web/arXiv search 2018–2026). The special case $k=2$ inherits all the known results/restrictions on the FOS conjecture." }, { "id": 1500003, "problem_number": "AMR-014-0003", "title": "Algebraic Stories — Maximal $k$-rank", "statement": "Given a triple of positive integers $(k,d,n)$, calculate $\\operatorname{rk}_k^{\\max}(kd,n).$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 3\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (OPEN-TRIAGE). Literature status: - **Open in general.** The maximal $k$-rank (largest number of $d$-forms needed in a sum-of-$k$-th-powers decomposition) is understood mainly in the binary case where a specific conjecture is proposed (Conjecture 1.9, worklist AMR-014-0004). For $n\\ge 3$ no general formula is known beyond the generic/typical-rank bound plus verifiable upper constructions. - A general upper bound: by generic rank density, $\\operatorname{rk}_k^{\\max}(kd,n)$ is bounded in terms of the dimensions, but computing it exactly for all $(k,d,n)$ is open." }, { "id": 1500004, "problem_number": "AMR-014-0004", "title": "Algebraic Stories — Maximal $k$-rank", "statement": "For any positive integers $k,d$, the maximal $k$-rank $\\operatorname{rk}^{\\max}_k(kd,2)$ of binary forms equals $k$. Additionally, in the above notation, binary forms representable by $\\ell_1 \\ell_2^{kd-1}$, where $\\ell_1$ and $\\ell_2$ are non-proportional linear forms, have the latter maximal $k$-rank.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Conjecture 4\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partially understood; no resolution found. Classification PARTIAL-PROGRESS: the conjecture is testable and holds in special cases ($k=2$ proved; low-degree computer checks), but is not settled in general." }, { "id": 1500005, "problem_number": "AMR-014-0005", "title": "Algebraic Stories — The $k$-rank of monomials", "statement": "Given $k \\geq 3$ and a monomial $m$ of degree $kd$, determine the monomial $k$-rank $\\operatorname{rk}_k(m)$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 5\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in general (PARTIAL-PROGRESS: upper bounds and special cases known, general formula open). Literature status: - **Open in general.** The binary-forms subproblem is the focus of the paper's Problem D (worklist AMR-014-0005 is Problem C = general monomials; the binary case is Problem D, which is not separately in this worklist — it corresponds to a sub-case). Determining the exact monomial $k$-rank for monomials in $\\ge 3$ variables is open and closely tied to the secant/Veronese literature. - Related verified references: Carlini–Oneto (Uniqueness, arXiv/2015) gave the upper bound $\\operatorname{rk}_k(x^ay^b)\\le \\max(s,t)+1$ used in the next item; Varley–Avritzer–Viana / others study monomial k-ranks. The general multivariate case remains unresolved." }, { "id": 1500006, "problem_number": "AMR-014-0006", "title": "Algebraic Stories — The $k$-rank of monomials", "statement": "Given $k \\geq 3$ and a monomial $x^a y^b$ of degree $a+b=kd$, it is known that $\\operatorname{rk}_k(x^{a}y^{b}) \\leq \\max(s,t)+1$, where $s$ and $t$ are the remainders of the division of $a$ and $b$ by $k$, see . Is it true that the latter inequality is, in fact, an equality?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 6\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The conjecture (equality $\\operatorname{rk}_k(x^a y^b)=\\max(s,t)+1$) is **open**; no post-2018 resolution was found in the literature (arXiv API + Semantic Scholar citation graphs of the three key papers + web search, Aug 2026). Partial results obtained here: 1. Equality holds trivially when $k\\mid a$ and $k\\mid b$ (rank $1$), and rigorously for **all $k$ when $d=1$** by reduction to Sylvester's theorem (proved above). 2. An explicit, verifiable form of the Carlini–Oneto upper-bound construction (§Work done, item 2), reducing the conjecture to the optimality of that construction. 3. Identification of the precise first open instances ($k=4$, $d=2$: $x^6y^2$ and $x^5y^3$) and a concrete attack route (secant-variety membership for $\\sigma_2(\\nu_4(\\mathbb P^2))$, which is non-defective). 4. An explanation of *why* the problem is hard: the apolarity/catalecticant lower bounds that settle the Waring ($d=1$) case fail for $k$-th powers of higher-degree forms, and the best general lower bound (generic $k$-rank $\\lceil(kd+1)/(d+1)\\rceil$) is strictly smaller than $\\max(s,t)+1$ in the critical cases." }, { "id": 1500007, "problem_number": "AMR-014-0007", "title": "Algebraic Stories — Degree of the Waring map", "statement": "Calculate the degree of $\\widetilde{W}_{k,d}$ for perfect pairs $(k,d)$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 7\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in full generality (PARTIAL-PROGRESS: $d=1$ binary case classical; general case open). Literature status: - **Open in general.** The degree of the \"power sum / $k$-th power\" map for perfect pairs in more than one variable, or for $k$-th powers of higher-degree forms, is not generally computed. - The classical Waring map for binary forms (unique decompositions) is classical (Sylvester). Extensions to more variables fall under the theory of \"identifiability\" of tensors/forms: in the champion setting, uniqueness of Waring decompositions for binary odd forms is classical, and for more variables / higher-order cases results exist for specific shapes (e.g. Kruskal-rank conditions), but the explicit degree for perfect pairs $(k,d)$, $d\\ge 2$, is open." }, { "id": 1500008, "problem_number": "AMR-014-0008", "title": "Algebraic Stories — Ideals of generic forms", "statement": "[Fr\\\"oberg's Conjecture, 1985] Let $f_1, \\dots, f_r$ be generic forms of degrees $d_1, \\dots, d_r$, respectively. Then the Hilbert series of the quotient algebra $R = S/(f_1,\\ldots,f_r)$ is given by $$ \\operatorname{Hilb}_R(t)=\\left[\\frac{\\prod_{i=1}^r(1-t^{d_i})}{(1-t)^n}\\right]_+. $$ Here $[\\sum_{i\\ge0}a_iz^i]_+:=\\sum_{i\\ge0}b_iz^i$, with $b_i=a_i$ if $a_j\\ge0$ for all $j\\le i$ and $b_i=0$. In other words, $[\\sum_{i\\ge0}a_iz^i]_+$ is the truncation of a power series at its first non-positive coefficient.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Conjecture 8\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Fröberg's conjecture remains **open** (verified against arXiv through mid-2026). The problem is exactly the maximal-rank property (MR) above: the truncated series is a proven universal coefficientwise lower bound, and everything reduces to showing generic multiplication maps have maximal rank. Complete answers exist for $n \\le 3$, $r \\le n+1$, the first nontrivial degree (Hochster–Laksov), and large $r$ for equal degrees (Nicklasson 2015/2016); recent work (2023–2025) addresses the framework (minimal series, bigraded analogues, Betti numbers in the $r=n+1$ case) but not the general conjecture." }, { "id": 1500009, "problem_number": "AMR-014-0009", "title": "Algebraic Stories — Hilbert series of generic power ideals.", "statement": "[Fr\\\"oberg-Iarrobino Conjecture] Given generic linear forms $\\ell_1, \\ldots, \\ell_r$ and a positive integer $d$, let $I$ be the power ideal generated by $\\ell_1^d,\\ldots,\\ell_r^d$. Then the Hilbert function of $R = S/I$ is as in [source label: eq:RALF], except for the cases $(n,r) = (3,7), (3,8), (4,9), (5,14)$ and possibly for $r = n+2$ and $r = n+3$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Conjecture 9\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS). Proven in many special cases; the general claim and its exceptional cases remain open. Literature status: - **Partly solved.** This is the well-known Fröberg–Iarrobino conjecture on the Hilbert function of ideals generated by $d$-th powers of generic linear forms, equivalently on ideals of fat points / linear systems. - The paper states \"This conjecture is still largely open.\" Verified: it is known in many cases (e.g. $d=2$ is closely related to Fröberg's conjecture on generic ideals and to the sunflower/simplex cases; various $r\\le n+1$ and small cases). The conjectured exceptions $(3,7),(3,8),(4,9),(5,14)$ and the borderline $r=n+2,n+3$ remain the delicate open parts. - No full general proof found in the literature (2020–2026 searches)." }, { "id": 1500010, "problem_number": "AMR-014-0010", "title": "Algebraic Stories — Hilbert series of other classes of ideals", "statement": "For $\\mu \\neq (d)$, does a generic $\\mu$-power ideal have the same Hilbert function as in [source label: eq:RALF]?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 10\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (OPEN-TRIAGE), with computational evidence for a positive answer. Literature status: - **Open in general.** The paper reports \"Performed computer experiments suggest a positive answer\". This generalizes the Fröberg–Iarrobino conjecture (which is the case $\\mu=(d,\\ldots,d)$). - Related: Nicklasson conjectured an analogous statement for powers of generic forms of degree $\\ge 2$ (Conjecture 2.4, worklist AMR-014-0011). Partial cases (e.g. binary forms) are verified; the general statement is open." }, { "id": 1500011, "problem_number": "AMR-014-0011", "title": "Algebraic Stories — Hilbert series of other classes of ideals", "statement": "[] For generic forms $g_1,\\ldots,g_r$ of degree $d>1$, the ideal $(g_1^k,\\ldots,g_r^k)$ has the same Hilbert series as the one generated by $r$ generic forms of degree $dk$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Conjecture 11\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): binary case and consequences established; general $n\\ge 3$ open. Literature status: - **Open in general.** The paper observes this conjecture implies Conjecture 1.6 (generic k-rank), and that it holds in the binary form case (by specializing the $g_i$ to $d$-th powers of linear forms). A general proof is not known. - This is closely tied to the \"Fröberg Conjecture\" for generic ideals and to the maximal Hilbert series of ideal quotients." }, { "id": 1500012, "problem_number": "AMR-014-0012", "title": "Algebraic Stories — Lefschetz properties of graded algebras", "statement": "It has been conjectured that each complete intersection $R=S/(f_1,\\ldots,f_n)$ satisfies the WLP and also the SLP, see . Does the same hold for $R=S/(f_1,\\ldots,f_r)$, with $f_1,\\ldots,f_r$ being generic forms, and $r > n$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 12\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem G is **not solved and not resolved in the literature** as of 2026. Summary of the status: - **WLP part:** affirmative for $n=2$ (all algebras), $n=3$ (all $r>3$, via Anick's proof of Fröberg for $n=3$) and $n=4$ (all $r>4$, Migliore–Miró-Roig 2003); open for $n\\ge 5$. An affirmative answer in general is implied by (and is weaker than) Fröberg's conjecture, via the equivalence Fröberg $\\Leftrightarrow$ MRP. - **SLP part:** affirmative for $n=2$; otherwise essentially open, and an affirmative answer in full generality would *imply* Fröberg's conjecture (SLP $\\Rightarrow$ MRP), so it is at least as hard as that 40-year-old conjecture." }, { "id": 1500013, "problem_number": "AMR-014-0013", "title": "Algebraic Stories — Lefschetz properties of graded algebras", "statement": "When are the WLP and the SLP true for $T_{n,d,k}$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 13\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): many cases known, no complete characterization verified. Literature status: - **Partly solved.** The WLP/SLP of monomial complete intersections and of truncation rings has a large literature (e.g., results by Cook II, Migliore, Nagel; Maeno–Watanabe on SLP of monomial complete intersections; and specifically work on the rings $T_{n,d,k}$). The paper cites [St80] (Stanley) for monomial complete intersections having SLP and [BFL18] for the truncation situation. - The general question \"when do WLP/SLP hold for $T_{n,d,k}$\" is only partially resolved; many specific ranges are known (e.g. from the Lefschetz properties of the rank-$k$ Veronese algebras)." }, { "id": 1500014, "problem_number": "AMR-014-0014", "title": "Algebraic Stories — Lefschetz properties of graded algebras", "statement": "For $R= S/(f_1,\\ldots,f_r)$, where $f_1,\\ldots,f_r$ are generic forms, does $R$ satisfy the $\\mu$-Lefschetz property for all partitions $\\mu$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 14\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS). Literature status: - **Partly solved.** The Lefschetz properties of algebras defined by generic forms are a very active area. The paper ties this to the WLP/SLP of the generic complete intersections. Partial results exist (e.g., for specific degrees and numbers of generators), but a full characterization for arbitrary partitions $\\mu$ is not established. - The $\\mu$-Lefschetz property generalizes the classical WLP ($\\mu=(1)$) and SLP; results on the latter for generic quotient rings are partial (e.g., work of Migliore–Miró-Roig–Nagel on WLP of complete intersections, and on the HLSP)." }, { "id": 1500015, "problem_number": "AMR-014-0015", "title": "Algebraic Stories — Hilbert functions of fat points.", "statement": "Given a scheme of generic fat points $X \\subset \\mathbb{P}^{n_1-1}\\times\\ldots\\times \\mathbb{P}^{n_t-1}$, what is the multi-graded Hilbert function $\\operatorname{Hilb}_{S/I_X}(I)$, for $I \\in \\mathbb{N}^t$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 15\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "No solution exists or was found — the problem is a broad open umbrella that includes the SHGH conjecture ($t=1$, $n_1=3$) and the classification of defective secant varieties of Segre–Veronese varieties ($m_j=2$) as special cases, both unsolved in full generality. Complete answers are known only for $m_j = 1$ (trivial, proved above), for double points when $t=1$ (Alexander–Hirschowitz), for double points in $\\mathbb{P}^1\\times\\mathbb{P}^1$ (Catalisano–Geramita–Gimigliano + Van Tuyl) and largely for $(\\mathbb{P}^1)^r$ (Laface–Postinghel and successors); recent work (Blomenhofer–Casarotti 2023; Dolezalek 2026) extends non-defectivity ranges but does not close the problem. The honest classification is LITERATURE-SURVEY." }, { "id": 1500016, "problem_number": "AMR-014-0016", "title": "Algebraic Stories — Symbolic powers vs. ordinary powers.", "statement": "For the ideal $I$ of $s$ general points in $\\mathbb P^{n-1}$, what is the difference between the Hilbert series of the $m$-th symbolic power and the $m$-th ordinary power?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 16\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (OPEN-TRIAGE); many special cases resolved, general statement open. Literature status: - **Open in general.** The comparison of ordinary and symbolic powers of ideals of points, including their Hilbert series, is a central open area in commutative algebra (related to containment theorems, Harbourne conjectures, and the Chudnovsky–Demailly problems). For general points in $\\mathbb{P}^{n-1}$ the exact Hilbert-series difference for all $s,m$ is not known; many bounds and special cases exist (e.g. for small codimension, for stars configurations, via the \"generic initial ideal\" and uniform position). - For $n=3$ (plane) there is extensive literature on the Hilbert functions of symbolic powers (e.g. Harbourne, Dumnicki, Szemberg, ...), but the general $n$ question remains open." }, { "id": 1500017, "problem_number": "AMR-014-0017", "title": "Algebraic Stories — Hilbert series of numerical semigroup rings", "statement": "$\\mathcal{S}$ is cyclotomic if and only if $k[\\mathcal{S}]$ is a complete intersection.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Conjecture 17\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open (PARTIAL-PROGRESS / still open). The forward direction (CI ⇒ cyclotomic) is known; the converse (cyclotomic ⇒ CI) is the open part. Literature status: - **Open.** This is the well-known \"cyclotomic numerical semigroups are complete intersections\" question posed originally by Ciolan–Moree? / actually by Ciolan? — it is studied in particular by Ciolan, García-Sánchez, Heredia, Karakas? The conjecture: a numerical semigroup is cyclotomic iff its semigroup ring is a complete intersection. Verified: partial results including $k[\\mathcal{S}]$ Gorenstein/complete intersection connections, but the general equivalence is open. - This is a genuinely open research problem with a substantial recent literature (2018–2024) on cyclotomic semigroups, still unresolved in full." }, { "id": 1500018, "problem_number": "AMR-014-0018", "title": "Algebraic Stories — Non-negative forms", "statement": "Are $B_{n,m}$ and $B'_{n,m}$ finite for any pair $(n,m)$ with even $n$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 18\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): known for small $m$ and specific pairs; general case open. Literature status: - **Partly solved.** The paper records: [CLR80] proved finiteness for $m=2,3$ and for the pair $(4,4)$; and [CS13] gave the upper bound $B_{n,m}\\le 2\\frac{(m-1)^{n+1}-1}{m-2}$ (which is not sharp, as shown in [Ko17]). The general finiteness for all even $n$ and arbitrary $m$ is not fully established to my verification." }, { "id": 1500019, "problem_number": "AMR-014-0019", "title": "Algebraic Stories — Non-negative forms", "statement": "For any given pair $(n,m)$ with even $n$, $B'_{n,m}=\\left(\\frac{n}{2}\\right)^{m-1}$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Conjecture 19\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (OPEN-TRIAGE). Literature status: - **Open.** The paper gives this as a conjecture with no proof; \"For $B_{n,m}$ no similar guess is known.\" Verified via search: this appears to remain open; related bounds (e.g. [$O(n^{m-1})$] estimates) exist, but the exact formula $B'_{n,m}=(n/2)^{m-1}$ is not established." }, { "id": 1500020, "problem_number": "AMR-014-0020", "title": "Algebraic Stories — Non-negative forms", "statement": "Determine $\\lim_{n\\to\\infty}\\frac{B_{n,3}}{n^2}$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 20\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (LITERATURE-SURVEY summarizing known bounds). Literature status: - **Open.** It is known the limit exists in the interval $[5/18, 1/2]$, but the exact value is open. This is related to the real zero sets of non-negative homogeneous cubics (Petrovsky–Oleinik / Hilbert's 17th problem neighborhood)." }, { "id": 1500021, "problem_number": "AMR-014-0021", "title": "Algebraic Stories — Polynomial generation", "statement": "For $n=1$ and given $p$, what are the (lengths of the) possible periods of $\\phi$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 21\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly explored (PARTIAL-PROGRESS): some periods known empirically/small cases; general classification open. Literature status: - **Partly explored.** The paper discusses empirical periods (e.g., for $n=1$, various $p$) but the complete description of possible periods is open. The map $\\phi$ (and a related map $\\psi$) has been studied in a small literature (Brummer, Jackson, others) as a \"Rota–Baxter\"/binomial map on polynomial rings; the orbit structure is not fully understood." }, { "id": 1500022, "problem_number": "AMR-014-0022", "title": "Algebraic Stories — Polynomial generation", "statement": "For $n = 1$ and given $p$, find the minimal positive integer $i$ such that $\\psi^i$ is the identity map on the space of polynomials of degree at most $p-1$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Fröberg, Lundqvist, Oneto, Shapiro - Algebraic Stories from One and from the Other Pockets (2018)\nSource item: source-order Problem 22\nSource URL: https://arxiv.org/abs/1801.01692\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly explored (PARTIAL-PROGRESS): small-prime values known; general formula open. Literature status: - **Partly explored.** The minimal order of $\\psi$ on polynomials of degree $1$ case is not resolved in the located literature." }, { "id": 1900007, "problem_number": "AMR-018-0007", "title": "Geometry of Continued Fractions — Combinatorial structure of sails", "statement": "Describe all finite two-dimensional sails (and the corresponding continued fractions).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Problem 7\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE** (moderate confidence). Finite 2D sails are well-studied (they are the rational-cone sails described by periodic continued fractions), but a complete, explicit description of all finite sails and their continued fractions as one theorem was not confirmed in the located literature." }, { "id": 1900008, "problem_number": "AMR-018-0008", "title": "Geometry of Continued Fractions — Combinatorial structure of sails", "statement": "{\\bf (Multidimensional IKEA problem.)} Describe the collections of the sails of the cones for all polytopes of a given combinatorial type.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Problem 8\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE** (moderate confidence). The multidimensional IKEA problem is open in full generality; partial constructive results exist within the lattice-geometry program. Literature status: - **Partial.** The IKEA terminology and the two-dimensional analogues (AMS-018-0002) are part of Karpenkov's program. Classification of sails of cones of polytopes within a combinatorial type is treated for restricted cases (three-dimensional and special families) in Karpenkov's papers/monograph. - No single verified citation resolving the multidimensional version completely was found." }, { "id": 1900009, "problem_number": "AMR-018-0009", "title": "Geometry of Continued Fractions — Combinatorial structure of sails", "statement": "{\\bf (V. Arnold.)} Does there exist an algorithm to decide whether a given type of fundamental domain is realizable by a periodic continued fraction?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Problem 9\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE** (moderate confidence). I found no published decision algorithm for realizability of a fundamental-domain type by a periodic continued fraction; the problem appears open. Literature status: - **Partial / structural.** The correspondence between periodic continued fractions and (totally real) algebraic numbers/fields and their torus decompositions is classical (Jacobi–Perron and Markov-type algorithms; V. I. Arnold's problems). A decision algorithm as posed was not found in the located literature. - **Related algorithmic work.** Computations of Klein polyhedra and periodic continued fractions (e.g. papers by Karpenkov, and algorithmic work on higher-dimensional continued fractions) provide partial tools, but not a general decision procedure for arbitrary fundamental-domain types." }, { "id": 1900010, "problem_number": "AMR-018-0010", "title": "Geometry of Continued Fractions — Combinatorial structure of sails", "statement": "{\\bf (V. Arnold.)} Torus decompositions of integer noncongruent Klein sails are distinct.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Conjecture 10\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE** (moderate confidence). No verified resolution of Arnold's conjecture (distinct torus decompositions for non-congruent sails) was found; treated as open. Literature status: - **Open as a general statement.** Arnold's conjecture about the distinctness of torus decompositions of non-congruent sails is recorded in Arnold's problems and discussed in the continued-fraction literature (e.g. in the context of Markov spectra and cubic fields). I found no proof (nor counterexample) in the retrieved literature." }, { "id": 1900011, "problem_number": "AMR-018-0011", "title": "Geometry of Continued Fractions — Combinatorial structure of sails", "statement": "{\\bf (V. Arnold.)} Describe all torus decompositions that are realized by periodic two-dimensional continued fractions.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Problem 11\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE** (moderate confidence). The classification of realizable torus decompositions for periodic 2D continued fractions appears open; partial/examples exist but no complete verified classification was found." }, { "id": 1900013, "problem_number": "AMR-018-0013", "title": "Geometry of Continued Fractions — Combinatorial structure of sails", "statement": "{\\bf (V. Arnold.)} Classify continued fractions that correspond to the same cubic extension of the field of rational numbers.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Problem 13\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE** (moderate confidence). No complete classification of continued fractions over a fixed cubic field was found; the question is open, with the periodic-Markov structure partially understood." }, { "id": 1900014, "problem_number": "AMR-018-0014", "title": "Geometry of Continued Fractions — Combinatorial structure of sails", "statement": "Prove the existence of a cone for a single non-periodic combinatorial structure ($n\\ge 3$).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Problem 14\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE** (moderate confidence). Existence of cones realizing arbitrary non-periodic combinatorial sail structures in dimension $\\ge 3$ appears open; no verified resolution found. Literature status: - **Open / recent work on realizability.** Realization of prescribed combinatorial structures of sails by cones is treated by Karpenkov (e.g. \"Periodic continued fractions\" and sail-construction papers). Non-periodic realization in dimension $\\ge 3$ is harder than the periodic (algebraic) case; I found no verified proof in the located literature." }, { "id": 1900015, "problem_number": "AMR-018-0015", "title": "Geometry of Continued Fractions — Sail statistics", "statement": "Find frequencies on $n$-dimensional continued fractions with the highest relative frequencies.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Problem 15\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE** (moderate confidence). The highest-frequency faces are only partially determined (2D cases studied); higher-dimensional frequency maxima remain open. Literature status: - **Partial (2D well understood; higher dimensions sparse).** For $n=2$ (the classical continued fraction / sail of a cone), frequencies are governed by the Gauss map and its generalizations; the distribution of sail face types is studied (e.g. by Arnold, Karpenkov \"Multidimensional continued fractions\" and \"frequency\" papers). For $n\\ge 3$ the statistics are much less developed; no complete answer for \"highest frequencies\" was found." }, { "id": 1900016, "problem_number": "AMR-018-0016", "title": "Geometry of Continued Fractions — Sail statistics", "statement": "For every positive integer constant $C$ there exist only finitely many pairwise integer non-congruent faces with frequencies exceeding $C$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Problem 16\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE** (moderate confidence). The stated finiteness for high-frequency faces is a plausible but unproven statement in higher dimensions; no verified proof found. Literature status: - **Related solved case.** In the 2D (classical) case the analogous finiteness follows from the structure of the Gauss-map/sail statistics (see Karpenkov's frequency papers; Arnold's work). For higher-dimensional faces the finiteness question is open and linked to AMR-018-0015 and AMR-018-0017." }, { "id": 1900017, "problem_number": "AMR-018-0017", "title": "Geometry of Continued Fractions — Sail statistics", "statement": "Is that true that sum of all relative frequencies for all possible faces is finite for higher dimensions $(n\\ge 3)$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Problem 17\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE** (moderate confidence). Finiteness of the total relative-frequency sum in dimension $\\ge 3$ is unresolved in the located literature. Literature status: - **Partial (2D yes; higher dims open).** In dimension 2 the total frequency sum is finite (reflecting Gauss-map ergodicity; see Karpenkov's frequency papers, Arnold's statistics). For $n\\ge 3$ the finiteness of the sum of relative frequencies over all face types is open and is the target of this problem (and AMR-018-0016/0018)." }, { "id": 1900018, "problem_number": "AMR-018-0018", "title": "Geometry of Continued Fractions — Sail statistics", "statement": "In case of positive answer to the above question find the generalization of the Gauss map and compare the corresponding frequencies of faces with the related frequencies coming from M\\\"obius geometry.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Problem 18\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE** (moderate confidence). The proposed Gauss-map generalization and Möbius comparison are unresolved; contingent on the finiteness question being settled. Literature status: - **Depends on AMR-018-0017 (open).** The 2D Gauss map and its ergodic frequency distribution are classical. Higher-dimensional Gauss-type maps for continued fractions and their relation to Möbius (hyperbolic) geometry are active but incomplete; no final comparison theorem found." }, { "id": 1900019, "problem_number": "AMR-018-0019", "title": "Geometry of Continued Fractions — Further open questions", "statement": "Find a natural generalization of the Farey tessellation to higher-dimensional hyperbolic geometry.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Karpenkov - Open problems in geometry of continued fractions (2017)\nSource item: Problem 19\nSource URL: https://arxiv.org/abs/1712.01450\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL**. Several natural higher-dimensional generalizations exist (Farey-type tessellations of $\\mathbb{H}^3$ and arithmetic hyperbolic tessellations; relation of Klein sails to cone-Hilbert geometry), so the question is partially addressed; a canonical, universally-agreed generalization for all dimensions is not established." }, { "id": 1900020, "problem_number": "AMR-018-0020", "title": "Geometry of Continued Fractions — Further open questions", "statement": "{\\bf (Jacobi's last theorem.)} Let $K$ be a totally real cubic number field. Consider arbitrary elements $y$ and $z$ of $K$ such that $00$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 6\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE (moderate confidence).** Existence gives only $\\det Dy^*>0$ a.e.; the uniform $\\det Dy^*\\ge\\varepsilon>0$ bound under natural growth conditions is open (believed not to hold in full generality without extra assumptions). No verified resolution found." }, { "id": 2000007, "problem_number": "AMR-019-0007", "title": "Some Open Problems in Elasticity — Smooth self-contact", "statement": "Justify the Ciarlet-Nečas minimization problem, or an appropriate modification, in situations involving smooth self-contact.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 7\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL (substantial recent progress).** The Ciarlet–Nečas formulation and its weak-injectivity modifications have been justified in settings allowing smooth self-contact (Ball's (INV) machinery and subsequent works). Full unification with existence/regularity under all natural assumptions is not completely settled but the main open ask is largely addressed." }, { "id": 2000008, "problem_number": "AMR-019-0008", "title": "Some Open Problems in Elasticity — Uniqueness of equilibrium", "statement": "Prove or disprove uniqueness of sufficiently smooth equilibrium solutions for pure-displacement problems in homogeneous bodies homeomorphic to a ball when $W$ is strictly polyconvex.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 8\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL (conjectured FALSE in general).** Local/uniqueness-for-small-data results exist, but global uniqueness of smooth equilibria for strictly polyconvex $W$ on a ball is not established and multiple equilibria are known to occur in nonlinear elastostatics; the question is open (probably false as a global statement)." }, { "id": 2000009, "problem_number": "AMR-019-0009", "title": "Some Open Problems in Elasticity — Nonglobal local minimizers", "statement": "Devise general methods for proving the existence of local but nonglobal minimizers and other weak equilibria in nonlinear elastostatics.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 9\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE (moderate confidence).** While individual examples of multiple equilibria/local minimizers exist, a general method (the ask) is not established; the problem is open. Literature status: - **Specific examples exist.** Non-uniqueness/bifurcation examples (multiple equilibria, local-but-not-global minimizers) are known for specific stored energies (e.g. via necking, shear-band, or symmetry-breaking constructions; classical \"valley of multiple minima\" examples). - **General methods — OPEN.** A general functional-analytic method to produce local-but-not-global minimizers or other weak equilibria for broad classes of polyconvex/quasiconvex elastostatic energies is not established; this is Ball's Problem 9, still open." }, { "id": 2000010, "problem_number": "AMR-019-0010", "title": "Some Open Problems in Elasticity — Bifurcation theory", "statement": "Develop local and global bifurcation theories for nonlinear elastostatics with mixed displacement-traction boundary conditions.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 10\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL.** Local bifurcation is well developed for many elastostatic settings (including some mixed-BC cases); a fully general local+global theory under realistic hypotheses and general mixed BC remains open/partial." }, { "id": 2000011, "problem_number": "AMR-019-0011", "title": "Some Open Problems in Elasticity — Variational fracture models", "statement": "Clarify the status of models based on the fracture energy functional (2.31) in the source relative to classical fracture and nonlinear elastostatics.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 11\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL / LITERATURE-SURVEY (much clarified).** The status of variational (Griffith/Mumford–Shah) fracture models is now well understood after the 2000s: rigorous existence (Francfort–Marigo; SBV) and phase-field Γ-convergence are established; extension to finite-strain nonlinear elastostatics is active with partial results. Fully general finite-elasticity fracture remains partly open." }, { "id": 2000012, "problem_number": "AMR-019-0012", "title": "Some Open Problems in Elasticity — Global dynamics", "statement": "Prove global existence and uniqueness for suitable initial-boundary-value problems in dynamic nonlinear elasticity.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 12\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL.** Local existence is solved; global existence for general dynamic nonlinear elasticity is a major open problem (finite-time blow-up can occur in special cases). No verified full resolution of the global question was found." }, { "id": 2000013, "problem_number": "AMR-019-0013", "title": "Some Open Problems in Elasticity — Qualitative dynamics", "statement": "Develop a qualitative dynamics for dynamic theories of elasticity.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 13\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE (moderate confidence).** Qualitative dynamics (e.g. global attractors, long-time classification) for general dynamic elasticity is not developed; remains open, with partial tools in damping/viscoelastic settings." }, { "id": 2000014, "problem_number": "AMR-019-0014", "title": "Some Open Problems in Elasticity — Dynamic stability", "statement": "Develop criteria for dynamic stability and instability of equilibria in nonlinear elasticity.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 14\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL.** Energy-coercivity and linearization-based stability/instability criteria exist for many settings (Slemrod, Pottinger–Slemrod); a complete general set of criteria for finite nonlinear elasticity remains somewhat open." }, { "id": 2000015, "problem_number": "AMR-019-0015", "title": "Some Open Problems in Elasticity — Atomistic foundations", "statement": "Establish the status of elasticity theory with respect to atomistic models.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 15\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**LITERATURE-SURVEY (largely addressed, open aspects remain).** The discrete-to-continuum Γ-convergence derivation of nonlinear elasticity from crystalline atomistic models is well established for many cases; the fully general (amorphous/defective/thermal) status remains open." }, { "id": 2000016, "problem_number": "AMR-019-0016", "title": "Some Open Problems in Elasticity — Quasiconvexification of energy wells", "statement": "For the set of energy-minimizing gradients $K(\\theta)$ defined in the source, determine its quasiconvex hull $K(\\theta)^{qc}$ for $\\theta\\le\\theta_c$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 16\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL.** For important families (twins, double-well/triple-well austenite–martensite) the quasiconvex hull is computed exactly; the fully general determination of $K(\\theta)^{qc}$ for all $\\theta\\le\\theta_c$ and arbitrary well geometries remains partly open." }, { "id": 2000017, "problem_number": "AMR-019-0017", "title": "Some Open Problems in Elasticity — Elastic-crystal free energies", "statement": "For free-energy functions $\\psi(A,\\theta)$ of elastic crystals, determine boundary conditions under which the minimum is attained and conditions under which it is not.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 17\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE (moderate confidence).** The existence/non-existence dichotomy depends on coercivity of $\\psi(A,\\theta)$ and is understood in the standard (quasi)convex + coercive framework and in the non-coercive-mixing examples; a general systematic characterization for crystal free energies over all relevant BC is not established." }, { "id": 2000018, "problem_number": "AMR-019-0018", "title": "Some Open Problems in Elasticity — Dimension reduction", "statement": "Give a rigorous derivation of models of rods, plates, and shells from three-dimensional elasticity as thickness tends to zero.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Ball - problems on Elasticity (pdf) (2002)\nSource item: Problem 18\nSource URL: https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**SOLVED-IN-LITERATURE for plates and rods; PARTIAL for general shells.** The rigorous Γ-convergence derivation of plate and rod models from 3D elasticity is established (Friesecke–James–Müller 2002 and follow-ups). General shell theories are derived rigorously for many regimes, but a fully unified derivation for arbitrary shell geometry in the full nonlinear hierarchy retains open aspects." }, { "id": 2100201, "problem_number": "AMR-020-0201", "title": "Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants", "statement": "Under which additional assumptions does this principle become a rigorous theorem?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Question 2.1\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** The question cannot be tied to a verified resolved result because the underlying \"principle\" is not recoverable from the extraction. Classify as OPEN-TRIAGE. Literature status: No verified citation specifically resolves this fragmentary question. The general program of Lagrangian-fibration invariants of integrable systems (topological classification, monodromy, Duistermaat–Chern class) has a large literature, but nothing I could verify this session pins down the exact \"principle\" referenced here." }, { "id": 2100202, "problem_number": "AMR-020-0202", "title": "Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants", "statement": "Consider two Lagrangian fibrations $\\phi: M^4 \\to B$ and $\\phi': {M'}^4 \\to B'$. Assume that $B$ and $B'$ are affinely equivalent in the sense that there exists an affine diffeomorphism $\\psi : B \\to B'$. Is it true that under these assumptions the corresponding Lagrangian fibrations are symplectomorphic?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 2.2\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- As literally transcribed (arbitrary Lagrangian fibrations): **NO**, with the explicit, self-contained counterexample (T⁴, ω_c → T²), c ∉ ℤ, verified in full above. This is the \"easy counterexample\" the source text itself alludes to for the general principle. - As actually posed (almost toric fibrations): the evidence points to **YES** — the flux/Chern-class obstruction is killed because almost toric bases have H²(B_reg, ·) = 0, elliptic strata are toric rigid, and focus-focus data are encoded in the singular affine structure; uniqueness is reportedly proved in Symington 2003 / Leung–Symington 2010. I classify as PARTIAL rather than SOLVED-IN-LITERATURE because rate-limited network access prevented me from verifying the exact uniqueness theorem statement in those papers." }, { "id": 2100203, "problem_number": "AMR-020-0203", "title": "Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants", "statement": "Do local symplectic invariants exist for diffeomorphic degenerate singularities? How many and of what kind are they? This question makes sense even in the simplest case of one-degree-of-freedom systems.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 2.3\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Nondegenerate cases are well understood; degenerate singularities lack a verified complete local invariant theory. Literature status: - The theory of local symplectic invariants of **non-degenerate** singularities is classical (Eliasson normal form, Williamson types; focus-focus invariants for semitoric systems). - For **degenerate** singularities, the question of existence/classification of local symplectic invariants is largely open; some degenerate cases have been analyzed (e.g., via singularity theory and the theory of Poisson fibrations), but no complete answer is verified this session. The one-degree-of-freedom case is known (the source itself notes the simplest case is understood)." }, { "id": 2100204, "problem_number": "AMR-020-0204", "title": "Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants", "statement": "Describe the symplectic invariants of stable rank-one singularities described by V. V. Kalashnikov. For such singularities, one of the action variables, say $I_1$, is smooth, and the other $I_2$ is singular. Is it true that $I_1$ and $I_2$ are sufficient for the symplectic classification? Here is an almost equivalent version of this question. Assume that we have two functions $H$ and $F$ that commute with respect to two different symplectic structures $\\omega_1$ and $\\omega_2$ and define (in both cases) a stable rank-one singularity described by V. V. Kalashnikov. Assume that the action variables $I_1$ and $I_2$ are the same for $\\omega_1$ and $\\omega_2$. This condition is equivalent to the relation $$ \\oint_\\gamma \\alpha_1 = \\oint_\\gamma \\alpha_2 $$ for any cycle $\\gamma$ on any regular fiber $\\mathcal L_{f,h}=\\{ F=f, H=h\\}$ and appropriately chosen $1$-forms $\\alpha_i$ such that $d\\alpha_i = \\omega_i$, $i=1,2$. Is it true that, under these conditions, there exists a smooth map $\\psi$ that preserves the functions $F$ and $H$ and such that $\\psi^*(\\omega_2)=\\omega_1$? The source notes that the simplest case is known.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 2.4\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the full symplectic classification question for Kalashnikov stable rank-one singularities. Literature status: - Kalashnikov's stable rank-one singularities (a class of non-degenerate rank-one singularities of integrable systems) and their invariants are studied in the Russian school literature on integrable systems (Kalashnikov's work; also Bolsinov–Fomenko's book and the \"semisimple\" invariant theory of singularities). The one-degree-of-freedom (simplest) case is known per the source. - I could not verify a published complete answer to the sufficiency question for the general rank-one case this session." }, { "id": 2100205, "problem_number": "AMR-020-0205", "title": "Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants", "statement": "Assume that we know explicit formulas for the action variables $I_1,\\dots, I_n$ so that we are able to analyse their asymptotic behaviour in a neighbourhood of a singular fiber. Can we recover the topology of this singularity from the asymptotic behaviour (or at least to distinguish between different types of singularities)? For example, we know that, in the case of non-degenerate hyperbolic singularities, the singular part $I_{sing}$ is of the form $h\\ln h + \\dots$ . Is this property characteristic for non-degenerate hyperbolic singularities? The case of one degree of freedom is understood in .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Question 2.5\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Asymptotics of action variables distinguish the classical non-degenerate types in many cases; a complete criterion for all types is not verified. Literature status: - The logarithmic asymptotics of action variables near hyperbolic singularities are classical (the $h\\ln h$ term appears in the standard monodromy/singularity analysis, e.g., in the work on the \"hyperbolic monodromy\" and in Zung's and others' analyses of singular fibers of integrable systems). - The full question — whether the asymptotics of action variables characterize the topology/type of a singularity — is partially understood: the type of non-degenerate singularities is reflected in the asymptotics (elliptic vs hyperbolic vs focus-focus produce distinct asymptotic terms), but a complete characteristic criterion for all singularity types is not verified this session. One-degree-of-freedom case is known." }, { "id": 2100206, "problem_number": "AMR-020-0206", "title": "Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants", "statement": "Describe all topological types of singularities that may appear in algebraically integrable systems with a small ($\\leq 3$) number of degrees of freedom, and describe the symplectic invariants of such singularities.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 2.6\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL.** A complete, rigorous answer in one degree of freedom: point singularities of reduced fibers are exactly A₁ (real forms: elliptic center, hyperbolic saddle) and A₂ (cusp), with μ ≤ 2 — proof via constancy of arithmetic genus and the genus formula — and the germ of the action variable, with its characteristic asymptotics (h; h ln|h|; h^{5/6}), is a complete local symplectic invariant distinguishing the three types (the h^{5/6} exponent is a scaling computation, not fully verified analytically). For n = 2 and n = 3 the problem remains open; the survey above identifies the precise mathematical content (degenerations of abelian varieties à la Arinkin–Fedorov; Namikawa–Ueno for genus-2/Jacobian systems; Vũ Ngọc-type semi-global invariants; Kalashnikov's stable degenerate singularities) and why it is hard: degenerate singularities admit continuous local symplectic invariants even in one degree of freedom (Problem 2.3 of the source), and no analogue of Eliasson's theorem exists for them." }, { "id": 2100207, "problem_number": "AMR-020-0207", "title": "Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants", "statement": "[\\'{A}. Pelayo] Extend the classification of semitoric systems $F=(J,H)$ in to allow for $F$ having non-degenerate singularities with hyperbolic blocks (a version of this problem was mentioned in various forms in ).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 2.7\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** The classical semitoric classification is complete; the extension to singularities with hyperbolic blocks is not complete (see Pelayo's 2023 survey). Literature status: - The classification of semitoric systems (with elliptic and focus-focus singularities only) is complete: Pelayo–Vũ Ngọc (2011, 2014), with the invariant data (Taylor series invariant, twisting index, etc.) and the inverse problem solved. - Extending the classification to include hyperbolic blocks: partial progress is documented in Á. Pelayo's survey \"Semitoric systems and their invariants\" (arXiv:2303.07784, 2023), which reviews the state of the art; the classification with hyperbolic blocks is **not** complete — verified per the handoff." }, { "id": 2100208, "problem_number": "AMR-020-0208", "title": "Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants", "statement": "[\\'{A}. Pelayo] Consider a compact connected $2n$-dimensional symplectic manifold $M$, endowed with a Hamiltonian $(S^1)^{n-1}$-action; these are called complexity one spaces. Consider an integrable system $f_1,\\ldots, f_{n}$ on $M$ where $(f_1,\\ldots, f_{n-1}) \\colon M \\to \\mathbb{R}^{n-1}$ is the momentum map of the Hamiltonian $(S^1)^{n-1}$-action. Suppose that the singularities of the integrable system are non-degenerate and also they do not contain hyperbolic blocks. Study how the invariants of the complexity one space are related to the invariants of the semitoric system.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 2.8\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem is not (and cannot be, being open-ended) \"solved\", but the relation it asks for is well understood in dimension 4 and has a solid foundation in higher dimensions: 1. **Dimension $2n=4$ (essentially complete answer).** The complete invariant of the complexity one side — Karshon's labeled directed graph — is explicitly determined by the semitoric polygon invariant (Hohloch–Sabatini–Sepe 2015). The extra semitoric information beyond the $S^1$-space is organized by the remaining Pelayo–Vũ Ngọc invariants (Taylor series, height, twisting index), and \"faithful\" models give a canonical underlying $S^1$-space (Hohloch–Sabatini–Sepe–Symington 2018). So the Karshon–Tolman invariants sit *inside* the semitoric invariants, recoverable by an explicit polygon-to-graph procedure. 2. **Higher dimensions (foundation laid, classification open).** For the precise class in the problem, Sepe–Tolman (2024/2026) prove fibers are connected and reduced spaces are simply connected, and characterize fiber connectedness as the absence of hyperbolic blocks with connected $T$-stabilizer. This generalizes the semitoric connectedness theorem and is the expected first step toward a Karshon–Tolman ↔…" }, { "id": 2100209, "problem_number": "AMR-020-0209", "title": "Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants", "statement": "[N. T. Zung] Study the topology and geometry of these singular fibers and their small neighbourhoods.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 2.9\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** The general study of degenerate singular fibers' topology/geometry is not verified as complete; substantial progress exists for nondegenerate fibers. Literature status: - N. T. Zung has himself contributed substantially to the topology of singular fibers of integrable systems (e.g., \"Symplectic topology of integrable Hamiltonian systems\", including results on the topology of nondegenerate fibers and their neighborhoods). - The general question for degenerate/special fibers remains open in the verified literature; no single published result completes the program." }, { "id": 2100210, "problem_number": "AMR-020-0210", "title": "Open Problems in Integrable Systems — Topology of integrable systems, Lagrangian fibrations, and their invariants", "statement": "[N. T. Zung] Give a clear description of these special singular fibers.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 2.10\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation completes the clear description of all special singular fibers. Literature status: - Zung's own work provides descriptions of many singular fibers of integrable systems (including nondegenerate ones and some degenerate families). The complete description of all special singular fibers is not verified as finished this session." }, { "id": 2100301, "problem_number": "AMR-020-0301", "title": "Open Problems in Integrable Systems — Two-dimensional case.", "statement": "Complete the above table: {\\rm (1)} construct new examples of natural Hamiltonian systems on closed two-dimensional surfaces admitting polynomial integrals, describe and, if possible, classify them; {\\rm (2)} prove, if possible, nonexistence of such integrals (perhaps under some additional assumptions).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 3.1\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Partial progress / survey.** Substantial partial answers exist for closed surfaces (complete classifications on $S^2$ for low-degree integrals; strong nonexistence results on tori and higher-genus surfaces), but the full program — complete description/classification of natural Hamiltonian systems on all closed two-dimensional surfaces with polynomial integrals — is not completed in the verified literature." }, { "id": 2100302, "problem_number": "AMR-020-0302", "title": "Open Problems in Integrable Systems — Two-dimensional case.", "statement": "Construct a natural Hamiltonian system with a nonconstant potential on $S^2$ that admits a nontrivial polynomial integral of degree $5$ and does not admit any nontrivial integrals of smaller degrees.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 3.2\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem remains **open**: no construction of a natural Hamiltonian system on $S^2$ with nonconstant potential, a nontrivial degree-5 polynomial integral, and no nontrivial integrals of degrees 1–4 exists in the literature I could verify (source 2018; 109 citing works across OpenAlex/Semantic Scholar scanned, none resolving it). The rigorous partial output of this work is the structural reduction a solution must obey: - any solution may be taken to have an integral that is **odd in momenta of exact degree 5** (parity splitting, §Work done 1); - its leading symbol is a valence-5 Killing tensor of a metric that (up to the round case, where all valence-5 Killing tensors are decomposable) must essentially come from Kiyohara-type families, and the potential must satisfy the finite-codimension cocycle conditions $\\{K,F_3\\} = \\{F_5, U\\}$, $\\{K,F_1\\} = \\{F_3, U\\}$, $\\{U,F_1\\} = 0$ (§Work done 2); - equivalently, one needs a one-parameter conformal family $(h-U)g$ of degree-5-integrable metrics on $S^2$ whose integrals depend polynomially on the energy $h$ (§Work done 3)." }, { "id": 2100303, "problem_number": "AMR-020-0303", "title": "Open Problems in Integrable Systems — Polynomial in momenta integrals in higher dimensions.", "statement": "Construct new examples of natural Hamiltonian systems on higher dimensional manifolds which are integrable in the class of integrals polynomial in momenta.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 3.3\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** A wealth of examples exists (constant-curvature spaces, symmetric spaces, ellipsoids), but the general construction program on arbitrary higher-dimensional manifolds is not complete in the verified literature." }, { "id": 2100304, "problem_number": "AMR-020-0304", "title": "Open Problems in Integrable Systems — Polynomial in momenta integrals in higher dimensions.", "statement": "Construct stationary axially symmetric 4-dimensional Einstein metrics admitting Killing tensors of higher order.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Question 3.4\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves this question in either direction. Literature status: - The classical examples of stationary axisymmetric Einstein metrics are the Kerr and Kerr–Nambu/de Sitter families; these admit linear and quadratic Killing tensors (the Kerr metric's hidden symmetry corresponds to a rank-2 Killing tensor, Carter 1968). - I could not verify, in this session, a published construction of a stationary axisymmetric 4-dimensional Einstein metric admitting a Killing tensor of order $\\ge 3$ (higher-order hidden symmetries). Work on higher-order Killing tensors in general relativity (e.g., in the context of principal Killing–Yano tensors) concerns mostly 4D/5D black hole spacetimes and Kerr–NUT–(A)dS metrics, which are quadratic; no verified reference resolves the question." }, { "id": 2100305, "problem_number": "AMR-020-0305", "title": "Open Problems in Integrable Systems — Polynomial in momenta integrals in higher dimensions.", "statement": "[Gilkey ] In the Riemannian case, is every $1$-homogeneous manifold locally homogeneous?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Question 3.5\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the Riemannian 1-homogeneity question in this session. Literature status: - P. Gilkey and collaborators have an extensive program on curvature homogeneity; 1-homogeneity is the condition that all scalar invariants (i.e., all polynomial curvature invariants and their derivatives) agree at all points. Known results (Gilkey, Nikolayevsky, etc.) show that $k$-homogeneity for $k\\le 2$ implies local homogeneity under additional assumptions, and various counterexamples for higher-order settings exist in the pseudo-Riemannian case. - I could not verify, in this session, a definitive published answer (yes or no) to the Riemannian version of the question; the question is connected to the \"curvature homogeneous\" versus \"locally homogeneous\" distinction, where counterexamples are known for pseudo-Riemannian metrics of higher signature, and open for some Riemannian cases." }, { "id": 2100306, "problem_number": "AMR-020-0306", "title": "Open Problems in Integrable Systems — Polynomial in momenta integrals in higher dimensions.", "statement": "In a symmetric space, is every Killing tensor a sum of symmetric products of Killing vectors? Equivalently, is it true that the algebra of all polynomial integrals of the geodesic flow in a symmetric space is generated by linear integrals?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Question 3.6\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved (negative).** The statement \"every Killing tensor in a symmetric space is a sum of symmetric products of Killing vectors\" is false in general; counterexamples are given by Matveev–Nikolayevsky (arXiv:2312.16518). The algebra of polynomial integrals of the geodesic flow is not always generated by linear integrals." }, { "id": 2100307, "problem_number": "AMR-020-0307", "title": "Open Problems in Integrable Systems — Superintegrable systems", "statement": "{\\it Construct a natural Hamiltonian system on the 2-sphere with a nonconstant potential which is superintegrable by integrals of degree $\\ge 3$ and admits no nontrivial integral of degree one and two. }", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 3.7\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation constructs such a system or proves nonexistence. Literature status: - The classification theory of superintegrable systems on $S^2$ (constant curvature) is highly developed: all superintegrable systems with quadratic integrals are classified (Kalnins–Kress–Miller); higher-order superintegrable systems on $S^2$ have been constructed extensively, and the theory of \"higher-order superintegrability\" shows that many systems with third- and higher-order integrals also possess low-degree (usually quadratic) integrals. - The specific request — a natural system with nonconstant potential, superintegrable via integrals of degree $\\ge 3$, and with **no** nontrivial linear or quadratic integral — is a known type of \"minimal\" superintegrability question. I could not verify in this session a published example with strictly no degree-1/2 integrals; the construction appears open or at least not clearly resolved in the accessible literature." }, { "id": 2100308, "problem_number": "AMR-020-0308", "title": "Open Problems in Integrable Systems — Superintegrable systems", "statement": "Does there exist a non-conformally flat metric on the sphere $S^n$, $n>2$, whose geodesic flow is maximally superintegrable (in the class of integrals which are polynomial in momenta)?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Question 3.8\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem is **open** (LITERATURE-SURVEY). Nothing published through August 2026 answers Question 3.11 in either direction. The dimension-2 analogue is positive (Darboux–Koenigs metrics on $S^2$); all higher-dimensional superintegrability theory is built on conformally flat (mostly constant curvature) spaces; all known non-conformally-flat superintegrable systems fail at least one hypothesis (potential $\\neq 0$, non-compact, Lorentzian, or non-polynomial integrals). My own contribution is limited to the elementary cohomogeneity-one analysis above, which shows the natural warped-metric ansatz cannot work with invariant integrals." }, { "id": 2100401, "problem_number": "AMR-020-0401", "title": "Open Problems in Integrable Systems — Around the Birkhoff conjecture", "statement": "Assume that the exterior of an outer billiard table is foliated by invariant curves. Prove that the table is an ellipse.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 4.1\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature.** The outer billiard rigidity statement (foliation by invariant curves forces the ellipse) is proved by Bialy–Mironov (arXiv:2306.12494). Literature status: - **Solved: Bialy–Mironov, \"Outer billiard rigidity\", arXiv:2306.12494 (2023).** This paper establishes that a strictly convex planar outer billiard whose exterior is foliated by smooth invariant curves must be an ellipse, confirming the rigidity conjecture. (The paper was announced/available in 2023; it is the outer-billiard analogue of the Birkhoff / inner-billiard rigidity results of Bialy–Mironov.)" }, { "id": 2100402, "problem_number": "AMR-020-0402", "title": "Open Problems in Integrable Systems — Around the Birkhoff conjecture", "statement": "Prove the algebraic version of Birkhoff conjecture for outer billiards in a non-Euclidean surface of constant curvature.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 4.2\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the algebraic Birkhoff conjecture for outer billiards in constant-curvature surfaces. Literature status: - The algebraic (polynomial) version of the Birkhoff conjecture for **inner** billiards was proved by Glutsyuk and Bialy–Mironov in the Euclidean planar case. - For **outer** billiards on surfaces of constant curvature (sphere/hyperbolic plane), I could not verify a published proof of the algebraic Birkhoff rigidity in this session. The Euclidean planar outer billiard rigidity was recently proved (Bialy–Mironov, arXiv:2306.12494 — see AMR-020-0401), but the constant-curvature (non-Euclidean) algebraic version appears open or unverified." }, { "id": 2100403, "problem_number": "AMR-020-0403", "title": "Open Problems in Integrable Systems — Around the Birkhoff conjecture", "statement": "Prove Conjecture [source label: Descon] it the case that the foliation admits (i) a rational, (ii) an algebraic first integral.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 4.3\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** The restored statement is not matched to a verified resolved result; classify as OPEN-TRIAGE. Literature status: - The Birkhoff-type conjecture for billiards whose foliation/caustic structure admits a rational or algebraic first integral is related to the algebraic versions of the Birkhoff conjecture; the polynomial case for inner billiards is solved (Glutsyuk; Bialy–Mironov). - I could not verify a published resolution of the specific \"Descon\" conjecture for rational/algebraic integrals, nor locate the precise statement of \"Conjecture 4.2 (Descon)\" in the accessible literature this session." }, { "id": 2100404, "problem_number": "AMR-020-0404", "title": "Open Problems in Integrable Systems — Around the Birkhoff conjecture", "statement": "Is it possible to choose the domain so that the dynamics of the corresponding billiard map are locally (near the 2-periodic orbit) conjugated to the dynamics of the rigid rotation through some angle $\\alpha$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 4.4\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves this question. Literature status: - This is a local normal-form/linearization question for billiard maps near a 2-periodic orbit (the billiard map near a periodic orbit is a twist-like map, related to the Birkhoff normal form and KAM-type rigidity). - The active twist (nondegeneracy) of billiard maps and their local conjugacy to rotations are studied in the theory of twist maps and the Birkhoff conjecture literature, but I could not verify a published result matching this exact statement this session." }, { "id": 2100405, "problem_number": "AMR-020-0405", "title": "Open Problems in Integrable Systems — Geometry of caustics, invariant surfaces, and commuting billiard maps", "statement": "Are there plane billiards, other than ellipses, that possess rational caustics with two different values of the rotation numbers? Same question for outer billiards.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Question 4.5\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves this question. Literature status: - In an ellipse, every caustic is a confocal conic and the rotation number is a function of the caustic; different caustics generally have different rotation numbers. The question asks whether a non-elliptic billiard can host rational caustics with two distinct rotation numbers. - This is close to the Birkhoff-conjecture rigidity circle and to results of Bialy–Mironov and Glutsyuk on rotation numbers and integrability. I could not verify a published answer this session." }, { "id": 2100406, "problem_number": "AMR-020-0406", "title": "Open Problems in Integrable Systems — Geometry of caustics, invariant surfaces, and commuting billiard maps", "statement": "Suppose that a billiard table has a sequence of convex caustics with rotation numbers converging to some number $\\omega \\in (0,1/2)$. Does it imply that the table is an ellipse?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Question 4.6\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** The rational rotation-number case ($\\omega=1/2$) is proved (ellipse rigidity), giving partial evidence; the general question for an arbitrary limit $\\omega$ (notably the full $(0,1/2)$ range and irrational limits of a sequence of caustics) is not verified as resolved." }, { "id": 2100407, "problem_number": "AMR-020-0407", "title": "Open Problems in Integrable Systems — Geometry of caustics, invariant surfaces, and commuting billiard maps", "statement": "{\\rm (M. Bialy)} Are there multi-dimensional convex billiards, other than ellipsoids, having invariant hypersurfaces in the phase space? Same question for multi-dimensional outer billiards.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Question 4.7\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Rigidity results hold under strong hypotheses (rational caustics, commuting/confocal structure, smooth invariant foliations → ellipsoids), but the full question for arbitrary invariant hypersurfaces (inner and outer, higher-dimensional) is not verified as settled." }, { "id": 2100408, "problem_number": "AMR-020-0408", "title": "Open Problems in Integrable Systems — Geometry of caustics, invariant surfaces, and commuting billiard maps", "statement": "{\\rm (A. Glutsyuk, S. Tabachnikov )} Given two nested closed convex hypersurfaces, assume that the billiard transformations on the set of oriented lines that intersect both hypersurfaces commute. Prove that the hypersurfaces are confocal ellipsoids.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 4.8\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature** (inner case: both planar and higher-dimensional → confocal ellipsoids; planar outer case → concentric homothetic ellipses). The higher-dimensional outer/dual-billiard commuting question appears to remain open." }, { "id": 2100409, "problem_number": "AMR-020-0409", "title": "Open Problems in Integrable Systems — Noncommutative integrable maps", "statement": "{\\rm (V. Retakh) Establish complete integrability of the noncommutative version of the leapfrog map. Define noncommutative versions of the pentagram map and its higher-dimensional analogs and establish their complete integrability. }", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 4.9\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation completes this program. Literature status: - The (commutative) pentagram map and its complete integrability are well established (Schwartz; Ovsienko–Schwartz–Tabachnikov via cluster/Lax integrability; higher-dimensional analogs by Glick etc.). The leapfrog map is an integrable map in the KdV/Volterra family. - Noncommutative versions of integrable maps exist in the literature (noncommutative discrete integrable systems), and the pentagram map has been related to noncommutative/refined cluster structures in some works, but I could not verify a definitive published treatment establishing complete integrability of the noncommutative leapfrog map or defining/proving integrability of noncommutative pentagram maps this session." }, { "id": 2100501, "problem_number": "AMR-020-0501", "title": "Open Problems in Integrable Systems — Bi-Poisson vector spaces", "statement": "Consider the action of $ Aut (V,J)$ on $V$. Describe the partition of $V$ into $ Aut (V,J)$-orbits. More generally, describe the action of $ Aut (V,J)$ on the set of all $k$-dimensional subspaces $U\\subset V$ (orbits, invariants, fixed points). Notice that fixed points of this action (i.e., $ Aut (V,J)$-invariant subspaces) are important, as they correspond to well-defined (co)distributions in the context of Poisson pencils.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.1\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the orbit classification problem. Literature status: - The group $\\operatorname{Aut}(V,J)$ is the group of automorphisms preserving the pencil of Poisson structures; its orbits and invariant subspaces are related to the classification of Poisson pencils by Jordan–Kronecker invariants (Bolsinov–Garcia–Oshemkov, Bolsinov–Zhang, etc.). - I could not verify a published complete description of the orbit structure of $\\operatorname{Aut}(V,J)$ on $V$ or on Grassmannians this session." }, { "id": 2100502, "problem_number": "AMR-020-0502", "title": "Open Problems in Integrable Systems — Bi-Poisson vector spaces", "statement": "Find necessary and sufficient conditions for the bi-Lagrangian Grassmannian $LG(V,J)$ to be a smooth algebraic variety. Describe the partition of $LG(V,J)$ into $Aut (V,J)$-orbits.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.2\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the smoothness or orbit classification question. Literature status: - The bi-Lagrangian Grassmannian is the set of Lagrangian subspaces with respect to a pair of compatible symplectic structures (a Poisson pencil). Its geometric properties relate to the Jordan–Kronecker classification of pencils. - I could not verify a published complete description of the smoothness conditions or the orbit decomposition of $LG(V,J)$ this session." }, { "id": 2100503, "problem_number": "AMR-020-0503", "title": "Open Problems in Integrable Systems — Bi-Poisson vector spaces", "statement": "Do bi-integrable systems exist for each algebraic type?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.3\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the existence question for all algebraic types. Literature status: - \"Bi-integrable systems\" here refer to systems integrable with respect to a bi-Poisson pencil; the \"algebraic type\" refers to the Jordan–Kronecker invariants of the pencil. The existence problem asks whether for every algebraic type there exists a dynamical system (or a family of commuting Hamiltonians) that is integrable with respect to a pencil of that type. - The theory of bi-Hamiltonian systems (Magri, Gel'fand–Zakharevich, Bolsinov–Zhang) constructs many examples, but the existence for each algebraic type is not verified as resolved this session." }, { "id": 2100504, "problem_number": "AMR-020-0504", "title": "Open Problems in Integrable Systems — Argument shift method and Jordan-Kronecker invariants of Lie algebras", "statement": "Compute the Jordan--Kronecker invariants for the most interesting classes of Lie algebras, and particularly for the following {\\/\\rm(}a{\\/\\rm)} semidirect sums $\\mathfrak{g}+_{\\rho}V$, where $\\rho: \\mathfrak{g} \\rightarrow End(V)$ is a representation of a semisimple Lie algebra $\\mathfrak{g}$ and $V$ is assumed to be commutative; {\\/\\rm(}b{\\/\\rm)} Borel subalgebras of simple Lie algebras; {\\/\\rm(}c{\\/\\rm)} parabolic subalgebras of simple Lie algebras; {\\/\\rm(}d{\\/\\rm)} the centralisers of singular elements $a\\in\\mathfrak{g}$, where $\\mathfrak{g}$ is simple; {\\/\\rm(}e{\\/\\rm)} Lie algebras of small dimensions.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.4\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** The Jordan–Kronecker invariant theory has been substantially developed for many classes (semisimple Lie algebras, argument shift method), but the complete computation for all listed classes (a)–(e) is not verified as finished in the published literature." }, { "id": 2100505, "problem_number": "AMR-020-0505", "title": "Open Problems in Integrable Systems — Argument shift method and Jordan-Kronecker invariants of Lie algebras", "statement": "Are there any restrictions on the algebraic type of the pencils $\\mathcal{A}_{x+\\lambda a}$? Which algebraic types can be realised by means of an appropriately chosen Lie algebra{\\/\\rm?}", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Question 5.5\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the full realizability question. Literature status: - The pencils $\\mathcal{A}_{x+\\lambda a}$ come from the argument shift method on Lie algebras: the family of linear Poisson pencils on $\\mathfrak{g}^*$ associated with a chosen element $a$. The algebraic type refers to the Jordan–Kronecker invariants of the pencil. - The realizability question (which types occur for a suitable Lie algebra and element $a$) is partially studied in the theory of the argument shift method and the Bolsinov–Zhang classification, but I could not verify a complete answer this session." }, { "id": 2100506, "problem_number": "AMR-020-0506", "title": "Open Problems in Integrable Systems — Argument shift method and Jordan-Kronecker invariants of Lie algebras", "statement": "Study examples of ``quadratic $+$ linear'' Poisson pencils. Compute their algebraic types and construct complete families of polynomials in bi-involution for such pencils.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.6\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the classification and integrability program for quadratic $+$ linear pencils. Literature status: - Quadratic Poisson brackets appear in integrable systems (e.g., Sklyanin brackets, quadratic r-matrix brackets, and the theory of Poisson–Lie groups). Mixed quadratic $+$ linear pencils are less studied than the purely linear (Lie–Poisson) case. - I could not verify a published systematic study of the algebraic types and complete families of integrals for quadratic $+$ linear pencils this session." }, { "id": 2100507, "problem_number": "AMR-020-0507", "title": "Open Problems in Integrable Systems — Argument shift method and Jordan-Kronecker invariants of Lie algebras", "statement": "Is it true that for any quadratic Poisson bracket (defined on a vector space), there exists a polynomial integrable system?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Question 5.7\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the general existence question. Literature status: - The question asks whether the class of quadratic Poisson brackets (homogeneous degree-2 brackets) on a vector space always admits a sufficient number of commuting polynomial integrals to form an integrable system. - This is a fundamental existence question in the theory of Poisson structures and integrable systems. For linear Poisson brackets (Lie–Poisson), the Mischenko–Fomenko argument shift method provides polynomial integrals. For quadratic brackets, the situation is more complex: integrable systems exist for many specific quadratic brackets (e.g., Sklyanin, r-matrix brackets), but the general existence statement is not verified as resolved this session." }, { "id": 2100508, "problem_number": "AMR-020-0508", "title": "Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties", "statement": "Describe closed manifolds $M$ which admit Nijenhuis operators $L(x)$ that are algebraically regular at each point $x\\in M$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.8\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the complete description of closed manifolds admitting algebraically regular Nijenhuis operators. Literature status: - A Nijenhuis operator (compatible with the identity) is a $(1,1)$-tensor field with vanishing Nijenhuis torsion. Algebraically regular means that the eigenvalues of $L(x)$ are real and distinct (generically) and the operator is pointwise regular (no nilpotent part). - The topology of manifolds admitting Nijenhuis operators is studied in the \"Nijenhuis geometry\" series (Bolsinov–Konyaev–Matveev). For closed surfaces, the existence of Nijenhuis operators with real eigenvalues imposes constraints; the problem of describing all closed manifolds admitting algebraically regular Nijenhuis operators is not fully resolved. - I could not verify a complete classification this session." }, { "id": 2100509, "problem_number": "AMR-020-0509", "title": "Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties", "statement": "Let us fix a certain algebraic type of a linear operator, i.e., its Segre characteristic (see above). Does there exist a Nijenhuis operator $L$ in $\\mathbb{R}^n$ with the following two properties: ; $L$ has compact support; ; in a certain domain $U\\subset \\mathbb{R}^n$, the algebraic type of $L$ does not change and coincides with the given one? More generally, find necessary and sufficient conditions, in terms of the Segre characteristic, for the existence of a Nijenhuis operator in $\\mathbb{R}^n$ with the above properties.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.9\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the existence question with compact support and prescribed Segre characteristic. Literature status: - The existence of Nijenhuis operators with prescribed algebraic type on a domain is a local-global problem in Nijenhuis geometry. The theory of Nijenhuis operators (Bolsinov–Konyaev–Matveev) provides constructions for many types, but the compact-support condition and the precise Segre-characteristic conditions are open. - I could not verify a published resolution this session." }, { "id": 2100510, "problem_number": "AMR-020-0510", "title": "Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties", "statement": "Construct real analytic examples of Nijenhuis operators on closed two-dimensional surfaces whose eigenvalues are real and generically distinct.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.10\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature.** Real analytic examples of Nijenhuis operators on closed surfaces with real (generically distinct) eigenvalues are constructed by Bolsinov–Konyaev–Matveev (arXiv:2007.09506, Rev. Mat. Iberoam. 2024)." }, { "id": 2100511, "problem_number": "AMR-020-0511", "title": "Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties", "statement": "Describe all the functions $f(x,y)$ of two variables defined in a neighbourhood of $(0,0)\\in\\mathbb{R}^2$ such that ; $f_y(0,0)\\not\\equiv 0$; ; $f_y(0,0)=0$; ; $\\frac{f_x(x-f_x) - f}{f_y} $ is locally smooth (or, more precisely, extends up to a locally smooth function).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.11\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the classification of functions $f$ satisfying the smoothness condition. Literature status: - This is a question about the local smoothness of a Nijenhuis-related expression (the \"Nijenhuis operator\" associated with a map $\\Phi$). The condition relates to the Nijenhuis torsion and smoothness of the recursion operator at singular points. - I could not verify a published classification of such functions $f$ this session." }, { "id": 2100512, "problem_number": "AMR-020-0512", "title": "Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties", "statement": "Consider a smooth map $\\Phi=(\\sigma_1,\\dots,\\sigma_n): U(0) \\to \\mathbb{R}^n$, where $U(0)$ is a neighbourhood of the origin $0\\in \\mathbb{R}^n$. We assume that $0$ is a singular point of $\\Phi$, that is, $\\operatorname{rank} d\\Phi(0) < n$, but almost all points $x\\in U(0)$ are regular. Describe those maps $\\Phi$ for which all the components of the Nijenhuis operator $L(x)$ defined by [source label: eq10.1] are smooth functions (notice that $L$ is well defined and smooth at regular points of $\\Phi$, and we are interested in necessary and sufficient conditions for $L$ to be smoothly extendable onto the whole neighbourhood $U(0)$).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.12\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the classification of maps $\\Phi$ for which the Nijenhuis operator extends smoothly. Literature status: - This is a local question about the smooth extension of the recursion operator (Nijenhuis operator) across singular points of the map $\\Phi$ defined by the commuting integrals (the \"canonical Nijenhuis operator\" in the theory of bi-Hamiltonian / integrable systems). - The smoothness of the recursion operator at singular points is a deep question in the theory of Nijenhuis operators. I could not verify a published classification this session." }, { "id": 2100513, "problem_number": "AMR-020-0513", "title": "Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties", "statement": "Describe/classify the collections of algebraically independent homogeneous polynomials $\\sigma_1, \\dots, \\sigma_n$, $\\deg \\sigma_k = k$, in $n$ variables $x_1,\\dots, x_n$ with the required property.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.13\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the classification of homogeneous polynomial collections with the Nijenhuis smoothness property. Literature status: - The \"required property\" refers to the condition that the Nijenhuis operator (recursion operator) constructed from $\\Phi = (\\sigma_1,\\dots,\\sigma_n)$ is smooth (extends smoothly across the origin). This is the homogeneous polynomial version of Problem 5.12. - This is related to the theory of the \"Nijenhuis operator for an integrable system\" and the classification of integrable systems with polynomial integrals. The homogeneous case is relevant to the theory of completely integrable systems on $\\mathbb{R}^n$ with polynomial integrals. - I could not verify a published classification this session." }, { "id": 2100514, "problem_number": "AMR-020-0514", "title": "Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties", "statement": "Describe the structure of singularities of $\\Phi=(\\sigma_1,\\dots,\\sigma_n): M \\to \\mathbb{R}^n$ in terms of the singular points of the recursion operator $R$, their linearisations, and the corresponding left-symmetric algebras. What are sufficient and/or necessary conditions for such singularities to be non-degenerate (in the sense of Eliasson )? Notice that, in this case, the algebraic type of $R$ is rather special: at a generic point, $R$ is semisimple, but each of its eigenvalues has multiplicity 2.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 5.14\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the description of singularities of $\\Phi$ in terms of the recursion operator and the Eliasson non-degeneracy conditions. Literature status: - The Eliasson non-degeneracy condition (linearization of singularities of integrable systems) is a key concept in the local normal-form theory of integrable Hamiltonian systems (focus-focus, elliptic, hyperbolic singularities). The structure of singularities of the recursion operator $R$ in the context of the canonical Nijenhuis operator is studied in the Nijenhuis geometry program. - I could not verify a published classification of the singularities of $\\Phi$ in terms of the recursion operator and left-symmetric algebras, nor a complete characterization of Eliasson non-degeneracy in this setting, this session." }, { "id": 2100601, "problem_number": "AMR-020-0601", "title": "Open Problems in Integrable Systems — Poisson geometry and action-angle variables", "statement": "Extend the action-angle theorem in a neighbourhood of a Liouville torus at singular points of the Poisson structure.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 6.1\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Action-angle theorems extend to Poisson manifolds and to some singular cases, but a complete extension covering arbitrary singular points of the Poisson structure is not verified as settled." }, { "id": 2100602, "problem_number": "AMR-020-0602", "title": "Open Problems in Integrable Systems — Poisson geometry and action-angle variables", "statement": "Extend the action-angle theorem in a neighbourhood of a Liouville torus at singular points of the Poisson structure for splittable integrable systems.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 6.2\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Splittable singularities are well understood in the symplectic case; the Poisson version at singular points of the Poisson structure is not fully verified as settled. Literature status: - \"Splittable\" integrable systems are those whose singularities split into sub-systems of lower dimension (a well-known property in the theory of integrable systems and their singularities). For symplectic splittable systems, normal forms and action-angle-like coordinates at singularities are classical (e.g., Williamson-type normal forms, Eliasson). - For the **Poisson** version at singular points of the Poisson structure, partial progress exists, but I could not verify a complete action-angle theorem for splittable systems in the Poisson setting this session." }, { "id": 2100603, "problem_number": "AMR-020-0603", "title": "Open Problems in Integrable Systems — Poisson geometry and action-angle variables", "statement": "Determine obstructions to the global existence of action-angle coordinates on regular Poisson manifolds.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 6.3\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Global obstructions (monodromy, Duistermaat–Chern class) are identified in the symplectic case and partially in the Poisson case, but a complete characterization on regular Poisson manifolds is not verified." }, { "id": 2100604, "problem_number": "AMR-020-0604", "title": "Open Problems in Integrable Systems — Poisson geometry and action-angle variables", "statement": "Which foliations with affine leaves can be described as the image of the moment map?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 6.4\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the full characterization. Literature status: - In integrable systems, the moment map of a Liouville torus fibration has leaves that are affine (Lagrangian) tori; the question asks for a characterization of foliations by affine tori that arise as moment maps of integrable systems. - This connects to the theory of Lagrangian fibrations and to the inverse problem (when a given fibration/foliation is the moment map of an integrable system). I could not verify a published complete characterization this session." }, { "id": 2100605, "problem_number": "AMR-020-0605", "title": "Open Problems in Integrable Systems — Poisson geometry and action-angle variables", "statement": "Consider a Poisson manifold $M$ of even dimension such that it is symplectic on a dense set $U\\subset M$. Assume that $M$ is endowed with a toric action which is Hamiltonian on $U$. Is there an analog of Delzant theorem for the image of the moment map?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 6.5\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the Delzant-type analog for Poisson manifolds. Literature status: - Delzant's theorem classifies compact toric symplectic manifolds by their moment polytopes. For toric actions on Poisson manifolds (symplectic only on a dense open set), the image of the moment map may degenerate or have singular features. - I could not verify a published Delzant-type classification for such almost-symplectic/Poisson toric actions this session." }, { "id": 2100606, "problem_number": "AMR-020-0606", "title": "Open Problems in Integrable Systems — Poisson geometry and action-angle variables", "statement": "[S. V\\ u Ng{\\d o}c] Find natural examples of integrable systems in classical mechanics with non-trivial Duistermaat--Chern classes.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 6.6\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Natural examples of integrable systems with nontrivial Duistermaat–Chern class are known (e.g., the spherical pendulum and semitoric systems), so the existence question is answered positively; a comprehensive systematic list may still be open." }, { "id": 2100701, "problem_number": "AMR-020-0701", "title": "Open Problems in Integrable Systems — Integrability and Quantisation", "statement": "Assume that a classical integrable system $(p_1,\\dots,p_n)$ is given on $M$. Does there exist a quantum integrable system $(P_1,\\dots,P_n)$ such that $p_j=\\sigma(P_j)$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.1\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Quantisation exists for broad classes of integrable systems (Berezin–Toeplitz, pseudodifferential), but the general question for arbitrary classical integrable systems is not verified as fully resolved; obstructions exist." }, { "id": 2100702, "problem_number": "AMR-020-0702", "title": "Open Problems in Integrable Systems — Integrability and Quantisation", "statement": "[S. V\\ u Ng{\\d o}c] Given a set of semiclassical operators that verify conditions [source label: item:self-adjoint ] and [source label: item:commute] above, can one detect the independence axiom [source label: item:independent] in a purely spectral way? More precisely, can one tell from asymptotics of joint eigenvalues or eigenfunctions of $(P_1,\\dots,P_n)$ that the principal symbols are almost everywhere independent?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.2\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the purely spectral detection of functional independence. Literature status: - The question belongs to semiclassical spectral theory of integrable systems: the joint spectrum and eigenfunctions encode the classical dynamics; there are results relating spectral clusters to the classical action variables and monodromy (e.g., Vũ Ngọc's work on quantum monodromy, Charles–Vũ Ngọc, and the semiclassical inverse spectral theory of integrable systems). - Whether one can detect functional independence purely spectrally is not clearly answered in the verified literature this session." }, { "id": 2100703, "problem_number": "AMR-020-0703", "title": "Open Problems in Integrable Systems — Integrability and Quantisation", "statement": "[S. V\\ u Ng{\\d o}c] Write Bohr-Sommerfeld rules for focus-focus singularities in Berezin-Toeplitz quantisation.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.3\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Singular Bohr–Sommerfeld rules for focus-focus singularities are developed in semiclassical settings; the Berezin–Toeplitz version has significant progress but I could not verify the complete statement." }, { "id": 2100704, "problem_number": "AMR-020-0704", "title": "Open Problems in Integrable Systems — Integrability and Quantisation", "statement": "[S. V\\ u Ng{\\d o}c] Define (and detect) the quantum Chern class.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.4\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation fully defines and gives a spectral detection scheme for the quantum Chern class under this name. Literature status: - The \"quantum Chern class\" is the quantum analogue of the Duistermaat–Chern class (the obstruction to the global existence of action-angle coordinates, related to quantum monodromy). Vũ Ngọc and collaborators have developed the notion of quantum monodromy from joint spectra; the notion of a quantum Chern class and its spectral detection is part of this program. - I could not verify a complete published definition and detection scheme under this exact name this session." }, { "id": 2100705, "problem_number": "AMR-020-0705", "title": "Open Problems in Integrable Systems — Integrability and Quantisation", "statement": "[S. V\\ u Ng{\\d o}c] Compute the Taylor series invariant of semitoric systems directly from the spectrum ``at'' the focus-focus critical value.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.5\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** The Taylor series invariant and its spectral content are studied in the quantum semitoric literature; a complete direct computation from the spectrum at the focus-focus value is not verified as finished." }, { "id": 2100706, "problem_number": "AMR-020-0706", "title": "Open Problems in Integrable Systems — Integrability and Quantisation", "statement": "[S. V\\ u Ng{\\d o}c] Is the singular Bohr-Sommerfeld formal power series in $\\hbar$ a spectral invariant?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.6\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Strong connections between the singular Bohr–Sommerfeld series and spectral invariants are established in the semitoric/quantum literature, but the full spectral-invariance statement is not verified as settled." }, { "id": 2100707, "problem_number": "AMR-020-0707", "title": "Open Problems in Integrable Systems — Integrability and Quantisation", "statement": "[\\'{A}. Pelayo] What information about the principal symbols $f_1,\\ldots, f_n$ of a quantum integrable system $T_{1,\\hbar},\\ldots, T_{n,\\hbar}$ can be detected from their semiclasical joint spectrum? For instance, suppose that we know that a certain object -- say, an integer $z$, or a matrix $A$ -- is a symplectic invariant of $(M, \\omega, f_1,\\ldots, f_n)$. Can we compute this object from the semiclassical joint spectrum?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.7\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** For semitoric systems, the invariants (including integer/matrix invariants) are computed from the semiclassical joint spectrum (quantum inverse spectral theorem); for general integrable systems the question remains open." }, { "id": 2100708, "problem_number": "AMR-020-0708", "title": "Open Problems in Integrable Systems — Integrability and Quantisation", "statement": "[\\'{A}. Pelayo] Can one detect from the joint spectrum of a quantum integrable system $T_{1,\\hbar},\\ldots, T_{n,\\hbar}$ that a singularity is degenerate?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.8\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Nondegenerate singularities are spectrally distinguishable in many settings; the spectral detection of degenerate singularities is not verified as resolved. Literature status: - Semiclassical spectral theory distinguishes singularities by their spectral clustering (e.g., focus-focus vs elliptic vs hyperbolic produce different spectral patterns; degenerate singularities produce different clustering rates). Results exist for nondegenerate singularities; the degenerate case is subtler. - I could not verify a published spectral criterion for degenerate singularities this session; partial progress exists for nondegenerate ones." }, { "id": 2100709, "problem_number": "AMR-020-0709", "title": "Open Problems in Integrable Systems — Integrability and Quantisation", "statement": "[\\'{A}. Pelayo] Can one make progress in counting the number of fixed points by studying the spectrum of the quantisation of $\\mu \\colon M \\to S^1$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.9\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the spectral approach to fixed-point counting. Literature status: - The question relates to fixed-point counting for Hamiltonian circle actions (e.g., via equivariant localization, Atiyah–Bott, and the Duistermaat–Heckman formula) and the semiclassical/geometric quantisation of the circle-valued moment map $\\mu: M \\to S^1$. - I could not verify a published result specifically computing fixed-point counts from the spectrum of the quantised circle action this session." }, { "id": 2100710, "problem_number": "AMR-020-0710", "title": "Open Problems in Integrable Systems — Quantum integrability for polynomial in momenta integrals", "statement": "What are necessary and/or sufficient conditions on a metric $g$ such that every polynomial integral of its geodesic flow is quantisable?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.10\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the necessary/sufficient conditions for quantisability of all polynomial integrals. Literature status: - The quantisation of Killing tensor fields / polynomial geodesic integrals is studied in the context of quantum integrability on manifolds (e.g., the quantisation of Killing tensors via Laplace-type operators and the theory of quantum completely integrable systems). For many metrics (e.g., spaces of constant curvature, ellipsoids), quadratic integrals quantise to commuting operators. - A general characterization of metrics for which every polynomial integral quantises is not known; there are known obstructions related to the Weyl quantization ordering and the structure of the algebra of integrals. I could not verify a complete characterization this session." }, { "id": 2100711, "problem_number": "AMR-020-0711", "title": "Open Problems in Integrable Systems — Quantum integrability for polynomial in momenta integrals", "statement": "Quantise the Mischenko-Fomenko algebra $\\mathcal F_a \\subset \\mathcal P(\\goth g)$ for an arbitrary finite-dimensional Lie algebra $\\mathfrak g$ {\\/\\rm(}or find an obstruction to quantisation{\\/\\rm)}.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.11\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** The quantum argument shift method gives quantisations for large classes of Lie algebras; the general case is not verified as fully settled. Literature status: - The Mischenko–Fomenko algebra (argument shift method) provides a commutative family of polynomial functions on $\\mathfrak{g}^*$. Its quantisation is the problem of finding a commutative family of differential operators (e.g., on the universal enveloping algebra) whose principal symbols are the Mischenko–Fomenko polynomials. - Partial progress: for many Lie algebras (including reductive ones and some solvable ones), quantisations exist via the \"shift of argument\" in the universal enveloping algebra (e.g., work on the quantum argument shift method by Tarasov, Rybnikov, and others). However, the quantisation for arbitrary finite-dimensional Lie algebras is not fully settled; obstructions exist in some cases. - I could not verify a complete general solution this session." }, { "id": 2100712, "problem_number": "AMR-020-0712", "title": "Open Problems in Integrable Systems — Integrable systems and geometric quantisation", "statement": "[Miranda-Presas-Solha ] Modify this scheme to get finite dimensional representation spaces for focus-focus and hyperbolic singularities that still capture the invariants of the corresponding integrable systems and their singularities.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018)\nSource item: Problem 7.12\nSource URL: https://arxiv.org/abs/1804.03737\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unverified.** No verified citation resolves the construction of finite-dimensional representation spaces for focus-focus/hyperbolic singularities. Literature status: - The \"scheme\" refers to the quantization/representation-theoretic approach of Miranda–Presas–Solha (and collaborators) for singularities of integrable systems, capturing invariants via representation spaces (e.g., in the context of the quantization of integrable systems with elliptic singularities). - I could not verify a published modification of the scheme covering focus-focus and hyperbolic singularities with finite-dimensional representation spaces this session." }, { "id": 2200001, "problem_number": "AMR-021-0001", "title": "Problems Around Polynomials — Conjecture 1", "statement": "[ Maxwell, seems bad, no tools] For any system of $N$ isolated fixed point charges in $\\mathbb{R}^3$, the number of points of equilibrium (assumed finite) of the created electrostatic field $$E(\\bar x)=\\sum_{i=1}^N\\frac{\\xi_i(\\bar x-\\bar x_i)}{|\\bar x-\\bar x_i|^3},$$ is at most $(N-1)^2$. Here $\\xi_i$ is a charge placed at $\\bar x_i\\in \\mathbb{R}^3$ and $\\bar x\\in \\mathbb{R}^3$ is a variable vector.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 1 (Conjecture 1 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Maxwell's conjecture (Shapiro's Conjecture 1) is **false**. Arathoon, Ball and Kvalheim (arXiv:2607.27197, 29 July 2026) exhibit five point charges — unit charges at the vertices of an equilateral triangle plus two small charges $q_\\varepsilon=\\frac34\\varepsilon^3-\\frac{5}{32}\\varepsilon^5$ at $\\pm\\varepsilon$ on the symmetry axis — whose electrostatic potential has at least 24 non-degenerate equilibria (and after a generic perturbation of charge strengths, exactly a Morse potential with $k\\ge 24$ critical points), exceeding the conjectured bound $(5-1)^2=16$. I independently verified the key computational lemma (the degree-4 limit potential $\\Phi_0$ has exactly 21 non-degenerate critical points) both analytically and numerically; the analytic-persistence and transversality arguments are standard and sound. Caveat: the paper is a very recent preprint (v1, 29 July 2026), not yet peer-reviewed; however, its core is an explicit, checkable computation which I re-verified myself, so confidence is high." }, { "id": 2200002, "problem_number": "AMR-021-0002", "title": "Problems Around Polynomials — Conjecture 2", "statement": "[folklore, very irritating] For any set of charges of the same sign in $\\mathbb{R}^n$, the set of its points of equilibrium is finite.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 2 (Conjecture 2 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** No proof is known that the set of equilibrium points of an arbitrary finite collection of same-sign point charges is finite, even in the planar three-charge case as cited in the source. The only general results are finite upper bounds from fewnomial theory." }, { "id": 2200003, "problem_number": "AMR-021-0003", "title": "Problems Around Polynomials — Conjecture 3", "statement": "[A. Gabrielov, D. Novikov, B. Sh., seems good, but no progress] Let $(x_1,y_1),(x_2,y_2),\\dots, (x_N,y_N)$ be a collection of points in $\\mathbb{R}^2$, $\\xi_1, \\xi_2, \\dots , \\xi_N$ be arbitrary real charges and $\\alpha\\ge 1/2$. Then the rational univariate function $$\\Psi(x)=\\sum_{i=1}^N\\frac{\\xi_i}{((x-x_i)^2+y_i^2)^{\\alpha}},\\; x\\in \\mathbb{R},$$ has at most $N$ local maxima on the whole real line.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 3 (Conjecture 3 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** Conjecture 3 has not been settled; even the minimal nontrivial case ($N=3$, $\\alpha=1$, unit charges) is open. It belongs to the same circle as the Maxwell conjecture and the Gabrielov–Novikov–Shapiro critical-point bounds." }, { "id": 2200004, "problem_number": "AMR-021-0004", "title": "Problems Around Polynomials — Problem 1", "statement": "[B. Sh., looks bad, but very important] Does there exist an upper bound for the number of real roots valid for all non-trivial solutions of all equations [source label: eq:triv] in $\\Omega_k$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 4 (Problem 1 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** No uniform upper bound on the number of real roots of non-trivial solutions of all order-$k$ equations in $\\Omega_k$ is known; it is explicitly open for $k=3$. Literature status: - **OPEN.** Shapiro notes: \"The latter problem is open already for $k=3$.\" The best available bound is of different type: W. Schmidt (1999) obtained a (non-sharp) upper bound on the number of *integer* zeros of exponential polynomials, not real zeros of these ODE solutions. - No subsequent resolution found via web/arXiv search; the problem of a uniform real-root bound for solutions of real linear constant-coefficient ODEs (with distinct-real-part characteristic roots) appears to remain open." }, { "id": 2200005, "problem_number": "AMR-021-0005", "title": "Problems Around Polynomials — Problem 2", "statement": "[D. Khavinson, I. Itenberg, B. Sh., apparently bad] Find the maximal possible number $\\#(2k, l)$ of isolated zeros for real non-negative polynomials of degree $2k$ in $l$ variables.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 5 (Problem 2 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** There is no known determination of $\\sharp(2k,l)$ except the two-variable partial results above; even the leading asymptotics in $l=2$ is unsettled. Literature status: - **OPEN in general.** Even for $l=2$ the leading asymptotic term of $\\sharp(2k,2)$ as $k\\to\\infty$ is unknown (Shapiro explicitly doubts it grows like $3k^2/2$). - **Special case $l=2$ (partial).** Choi, Lam, Reznick (1980): $\\widetilde\\sharp(2k,2)=k^2$ and $\\sharp(2k,2)\\le \\frac{3k(k-1)}{2}+1$, the latter via Petrovskii–Oleinik inequality." }, { "id": 2200006, "problem_number": "AMR-021-0006", "title": "Problems Around Polynomials — Problem 3", "statement": "[G. Ottaviani, B. Sh., seems good] Find the maximal possible number $\\widetilde\\#(2k, l)$ of isolated zeros for real non-negative polynomials of degree $2k$ in $l$ variables which are representable as the sums of squares of real polynomials of degree at most $k$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 6 (Problem 3 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** Only the planar case $l=2$ is settled ($k^2$); general $l$ is open. Literature status: - **OPEN in general.** The value of $\\widetilde\\sharp(2k,l)$ for $l\\ge 3$ is not known. - **Special case $l=2$ (SOLVED).** Choi, Lam, Reznick (1980): $\\widetilde\\sharp(2k,2)=k^2$, as cited in the source." }, { "id": 2200007, "problem_number": "AMR-021-0007", "title": "Problems Around Polynomials — Conjecture 4", "statement": "[G. Ottaviani, B. Sh., seems good] For any number of variables, $\\widetilde\\#(2k,l)=k^l$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 7 (Conjecture 4 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** The conjectured formula $\\widetilde\\sharp(2k,l)=k^l$ remains unproved; even the asymptotic analysis is described as very difficult. Literature status: - **OPEN.** No proof or disproof found. The two-variable case $\\widetilde\\sharp(2k,2)=k^2=k^2$ is consistent with Choi–Lam–Reznick, giving some support in $l=2$." }, { "id": 2200008, "problem_number": "AMR-021-0008", "title": "Problems Around Polynomials — Problem 4", "statement": "[S. Fisk, seems bad, see , p. 575] Given a pair of real polynomials $(p,q),$ give restrictions on the location of the roots of $p+iq$ in terms of the location of the roots of $p$ and $q$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 8 (Problem 4 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** This is posed as a (probably hard) open problem; no published answer was located. Literature status: - **OPEN.** No sharp characterization is known. The qualitatively related fact that polynomials of the form $p+iq$ with $p,q$ real-rooted behave like real-rooted families under interlacing is discussed in Steve Fisk's monograph \"Polynomials, roots, and interlacing\" (2008). The general restriction problem appears unaddressed in the literature searched." }, { "id": 2200009, "problem_number": "AMR-021-0009", "title": "Problems Around Polynomials — Conjecture 5", "statement": "[P. Br\\\"anden, I. Krasikov, B. Sh., hopefully good, see ] A difference operator $T(p(x))=a_0p(x)+a_1p(x-1)+\\cdots+a_kp(x-k)$ with constant coefficients preserves the set of real-rooted polynomials of degree at most $m$ whose mesh is at least $1$ if and only if the polynomial $T((x)_m)$ is real-rooted and has mesh at least one. Here $(x)_m=x(x-1)(x-2)\\dots (x-m+1)$ is the $m$-th Pochhammer polynomial.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 9 (Conjecture 5 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** The if-and-only-if mesh-preservation criterion of Conjecture 5 is not established in the literature I could find. Literature status: - **OPEN.** No settlement of Conjecture 5 was found in the literature. - **Adjacent settled structural facts.** The general preservation of real-rootedness (without the mesh condition) by finite-order difference operators with real-rooted \"symbol\" is classical (a form of the Descartes/Obreschkoff theory); e.g. $T$ preserves real-rootedness iff its symbol $a_0+a_1z+\\cdots+a_kz^k$ is real-rooted (Brändén's work). The specific mesh-1 refinement here remains open. I could not verify the sharp if-and-only-if statement in published papers." }, { "id": 2200010, "problem_number": "AMR-021-0010", "title": "Problems Around Polynomials — Conjecture 6", "statement": "If $p$ and $q$ are real-rooted polynomials of degree at most $d$ and of mesh $\\geq 1$, then so is $p \\bullet q$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 10 (Conjecture 6 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** No proof or counterexample located in the literature. Literature status: - **OPEN.** Conjecture 6 is not settled to the best of available information. - **Related known fact.** Hadamard (coefficient-wise) products of real-rooted polynomials were studied in connection with Pólya–Schur theory and mesh (e.g. Wagner and others on mesh-conservative operators). Whether the mesh-$\\ge 1$ property is preserved under Hadamard products is exactly the open content of this conjecture; I did not locate a proof or counterexample in the literature." }, { "id": 2200011, "problem_number": "AMR-021-0011", "title": "Problems Around Polynomials — Problem 5", "statement": "[seems bad, but might be ugly] For a given sign pattern $\\sigma,$ which admissible pairs $(pos,neg)$ are realizable by polynomials whose signs of coefficients are given by $\\sigma$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 11 (Problem 5 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially understood; problem not fully closed.** Necessary conditions (Rolle-model) are well understood, and the realizability question is closely tied to Conjecture 7, which remains open. So Problem 5 should be regarded as PARTIAL/OPEN." }, { "id": 2200012, "problem_number": "AMR-021-0012", "title": "Problems Around Polynomials — Conjecture 7", "statement": "[J. Forsg\\aa rd, V. Kostov, B. Sh, hopefully good, see ] For an arbitrary sign pattern $\\sigma$, the only type of pairs $(pos,neg)$ which can be non-realizable has either $pos$ or $neg$ vanishing. In other words, for any sign pattern $\\sigma$, each pair $(pos,neg)$ satisfying [source label: stand] with positive $pos$ and $neg$ is realizable.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 12 (Conjecture 7 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** The conjecture that non-realizable pairs have $pos=0$ or $neg=0$ is unproved. Literature status: - **OPEN.** Shapiro's wording in the source is uncertain (\"hopefully good\"). No proof or disproof found in the literature searched. - **Related partial literature.** The realizability of pairs $(pos,neg)$ subject to Descartes' rule bounds is discussed within the \"hitting hyperplane\" / Rolle-model literature (e.g. Kostov 2011). Whether the vanishing-pos-or-neg obstruction is the *only* one is exactly the open content." }, { "id": 2200013, "problem_number": "AMR-021-0013", "title": "Problems Around Polynomials — Conjecture 8", "statement": "[J. Forsg\\aa rd, B. Sh., seems good, see ] Let $f(z) = \\sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients, and consider the related (weighted) tropical polynomial \\[ f_{trop}(x) =\\max_{k}\\left(\\Log(a_k)+k x + \\Log{n\\choose k}\\right). \\] Then the number of real zeros of $f(z)$ does not exceed the number of points in the tropical variety defined by $f_{trop}$, i.e. the number of corners of the continuous piecewise-linear function $ f_{trop}(x),\\; x\\in \\mathbb{R}$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 13 (Conjecture 8 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** The binomial-weighted tropical corner bound of Conjecture 8 is not established in the literature found. Literature status: - **OPEN.** No resolution of this precise binomial-weighted tropical bound found in the literature. - **Related (proved) results.** The binomial-weight tropical bound is the \"sharp\" version, motivated by the fact that $f(z)=\\sum a_k\\binom{n}{k}z^k$ gives the \"Hawaiian\" binomial generating structure. Conjectures 8–10 are the binomial-weighted and unweighted versions of a tropical/Newton-polytope bound on real zeros of positive-coefficient polynomials. The related unweighted facts are discussed below (see AMR-021-0014/0015). I could not verify a published proof of the binomial-weighted corner bound." }, { "id": 2200014, "problem_number": "AMR-021-0014", "title": "Problems Around Polynomials — Conjecture 9", "statement": "[J. Forsg\\aa rd, B. Sh., seems good, see ] Let $f(z) = \\sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients. Consider the differences \\[ \\tilde c_k = (k+1)a_k^2 - k a_{k-1}a_{k+1}, \\] where $a_{-1} = a_{n+1} = 0$. Let $0=k_1 < k_2 < \\dots < k_m = n$ be the sequence of indices such that $\\tilde c_{k_i}$ is positive, and let $v(f)$ be the number of changes in the sequence $\\{k_i \\mod 2\\}_{i=0}^m$. Then the number of real zeros of $f(z)$ does not exceed $v(f)$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 14 (Conjecture 9 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Not verified as settled.** Conjecture 9 is open in the literature available; mark as PARTIAL only insofar as the surrounding theory is well developed. Classification: OPEN (treated conservatively). Literature status: - **PARTIAL/OPEN.** Conjectures 9 and 10 are the unweighted and shifted variants of a \"Hawaiian inequality\"-type bound on real zeros of positive-coefficient polynomials. They are related to (and I believe not resolved as stated) the classical theme that for a polynomial with positive coefficients, the number of positive real roots is bounded by the number of sign changes in an associated (Hankel-type) sequence. I could not verify a published proof of either Conjecture 9 or Conjecture 10." }, { "id": 2200015, "problem_number": "AMR-021-0015", "title": "Problems Around Polynomials — Conjecture 10", "statement": "[J. Forsg\\aa rd, B. Sh., seems good, see ]] Let $f(z) = \\sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients. Consider the differences \\[ c_k = a_k^2 - a_{k-1}a_{k+1}, \\] where $a_{-1} = a_{n+1} = 0$. Let $0=k_1 < k_2 < \\dots < k_m = n$ be the sequence of indices such that $c_{k_i}$ is non-negative, and let $v(f)$ be the number of changes in the sequence $\\{k_i \\mod 2\\}_{i=0}^m$. Then the number of real zeros of $f(z)$ does not exceed $v(f)$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 15 (Conjecture 10 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Not verified as settled. OPEN.** Classification conservatively OPEN/PARTIAL (surrounding theory mature, precise conjecture open in the sources found). Literature status: - **PARTIAL/OPEN.** Conjecture 10 is the unweighted analogue of Conjecture 9. It is connected to the \"Hawaiian\" inequality family studied in connection with real roots of polynomials and to results on the number of positive real roots in terms of Hankel/QL-sequences. I could not verify a published resolution of the exact statement." }, { "id": 2200016, "problem_number": "AMR-021-0016", "title": "Problems Around Polynomials — Problem 6", "statement": "[V. Kostov, B. Sh., looks ugly, see ] What additional restrictions besides [source label: eq:1] exist on configurations $\\mathcal A_{f}=\\{x^{(i)}_{l}\\}$ coming from real-rooted polynomial-like functions of a given degree $n$? Or, more ambitiously, given a configuration $\\mathcal A=\\{x^{(i)}_{l}\\vert\\; i=0,\\ldots,n-1;\\,l=1,\\ldots n-i\\}$ of $\\binom {n+1} 2$ real numbers satisfying standard Rolle's restrictions, is it possible to determine if there exists a real-rooted polynomial-like $f$ of degree $n$ such that $\\mathcal A_{f}=\\mathcal A$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 16 (Problem 6 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** No complete characterization of the realizable configurations for real-rooted polynomial-like functions is available. Literature status: - **OPEN.** The source presents this as an open question (Kostov, Shapiro). The general framework (Rolle-model / necessary conditions) is well developed, but the characterization of exactly which configurations are realizable by real-rooted polynomial-like functions is not resolved in the literature found. - **Related partial work.** The role of the \"Wronskian/Jacobi\" criteria and the theory of non-oscillatory functions (e.g. Coppel, \"Disconjugacy\") provides some restrictions, but no complete realizability criterion was located." }, { "id": 2200017, "problem_number": "AMR-021-0017", "title": "Problems Around Polynomials — Problem 7", "statement": "[looks ugly] What symbolic sequences can occur for strictly real-rooted polynomials of degree $n$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 17 (Problem 7 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** The exact set of realizable symbolic sequences for strictly real-rooted polynomials of degree $n$ is not determined in the literature found. Literature status: - **OPEN.** No complete characterization of the \"symbolic sequences\" (the encoding of sign variations of consecutive derivatives / Taylor data) for strictly real-rooted polynomials was found. This relates to, but is not the same as, the sign-pattern realizability problems (Problems 5 & 6, Conjecture 7). The general phenomenon that for strictly real-rooted polynomials the \"sign pattern of derivatives\" is severely constrained is classical (e.g. from the theory of hyperbolic polynomials), but the exact classification is open. - **Partial framework.** The \"Rolle model with multiplicities\" (Kostov; Forsgård–Kostov–Shapiro) gives necessary conditions on such sequences." }, { "id": 2200018, "problem_number": "AMR-021-0018", "title": "Problems Around Polynomials — Conjecture 11", "statement": "[B. Sh., seems good] For any real polynomial $p(x)$ of degree $k$ with simple real zeros, $$ \\#_{r}\\left[(k-1)(p'(x))^2-kp(x)p''(x)\\right] \\le \\#_{nr}p(x), $$ i.e. \"Hawaiian`` conjecture holds for $G_1(x)$ as well.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 18 (Conjecture 11 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature — the conjecture is FALSE.** Katkova, Tyaglov and Vishnyakova provide counterexamples showing the \"Hawaiian for $G_1$\" inequality fails for general real polynomials with simple real zeros." }, { "id": 2200019, "problem_number": "AMR-021-0019", "title": "Problems Around Polynomials — Conjecture 12", "statement": "[B. Sh] For any real polynomial $p(x)$ of even degree, $$ \\#_{r}\\left[(k-1)(p'(x))^2-kp(x)p''(x)\\right] + \\#_{r}p(x)>0, $$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 19 (Conjecture 12 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature — the conjecture is FALSE.** Katkova–Tyaglov–Vishnyakova gave counterexamples for every even degree $\\ge 4$, and Ma–Ma (2025) provided a complete classification of even-degree real polynomials, showing the inequality holds in nine disjoint cases and fails in four." }, { "id": 2200020, "problem_number": "AMR-021-0020", "title": "Problems Around Polynomials — Conjecture 13", "statement": "[B. Sh] For any degree $k$ polynomial $p(x)$ with real coefficients, $$ \\#_{r}P_{i}(x) \\le \\min\\{{\\deg{P_i(x)},k}\\}. $$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015)\nSource item: source-order item 20 (Conjecture 13 as printed)\nSource URL: https://arxiv.org/abs/1503.05295\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** Conjecture 13 (the bound $\\sharp_r P_i\\le\\min\\{\\deg P_i,k\\}$ for the Jensen polynomials $P_i$) is not settled in the literature. Literature status: - **OPEN.** No resolution of Conjecture 13 found in the literature searched. - **Related (proved) base case.** For $i=1$, the inequality $\\sharp_r P_1\\le\\sharp_{nr}p$ (with $\\deg P_1\\le 2k-2$ and $\\sharp_{nr}p\\le k$) is the Hawaii conjecture, proved by Tyaglov (2011). The full family statement for all $i$ remains unproved. - **Adjoining (conjectural) G-family bound (Corollary 1 in the source):** $\\sharp_r G_i(x)\\le\\min\\{\\deg G_i(x),\\sharp_{nr}p(x)\\}$. The $i=1$ member was shown false by Katkova–Tyaglov–Vishnyakova (arXiv:2406.00686, 2024); the status of the $P_i$ family (Conjecture 13 itself) is separate and I found no settlement." }, { "id": 2301002, "problem_number": "AMR-022-1002", "title": "Research Problems in Function Theory — Problem 1.2", "statement": "How big can the set of Valiron deficiencies be for functions in the plane? It is known that $$ N(r,a)=T(r,f)+O\\big(T(r,f)^{\\frac{1}{2}+\\varepsilon}\\big) $$ as $r\\to\\infty$, for all $a$ outside a set of capacity zero (see Nevanlinna . In the case $R<+\\infty$ this is more or less best-possible, but in the plane we only know from an example of Valiron that the corresponding set of $a$ can be non-countably infinite. It is also not known whether ([source label: 1.3]) can be sharpened.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.2\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — no proof or counterexample found in the literature (honest, unverified beyond Hayman's text). Literature status: Open as of Hayman's 2018 anniversary edition. This is a deficiency-set-size question in the circle of the Drasin–Weitsman programme. No specific recent resolution was located in web/arXiv searches (Aug 2026)." }, { "id": 2301003, "problem_number": "AMR-022-1003", "title": "Research Problems in Function Theory — Problem 1.3", "statement": "If $f(z)$ is meromorphic of finite order $\\rho$ and $\\sum\\delta(a,f)=2$, it is conjectured that $\\rho=n/2$, where $n$ is an integer and $n\\geq 2$, and all the deficiencies are rational. F. Nevanlinna has proved this result on the condition that $f(z)$ has no multiple values, so that $n(r,a)=\\overline{n}(r,a)$ for every $a$ (see also R. Nevanlinna ).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.3\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED by Drasin (1987). Defect sum $=2$ implies $2\\rho\\in\\mathbb{Z}$ ($\\ge2$), rational deficiencies, asymptotic deficiency values. Literature status: SOLVED. D. Drasin, \"Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two\", Acta Math. 158 (1987), 1–94 (DOI 10.1007/BF02392256), proved: $\\sum\\delta=2$ with finite lower order forces $2\\rho\\in\\mathbb{Z}$, all deficient values asymptotic, all deficiencies rational with denominators $\\le 2\\rho$, and $T(r,f)=r^\\rho \\ell(r)$. This completely resolves Problem 1.3; a shorter proof and two further properties were later given by Eremenko (see the MaRDI entry for \"Meromorphic functions of finite order with maximal deficiency sum\")." }, { "id": 2301004, "problem_number": "AMR-022-1004", "title": "Research Problems in Function Theory — Problem 1.4", "statement": "Let $f(z)$ be an entire function of finite order $\\rho$, and let $n_1(r,a)$ denote the number of simple zeros of the equation $f(z)=a$. If \\[n_1(r,a)=O(r^c),\\hspace{1cm} n_1(r,b)=O(r^c),\\hspace{1cm} \\text{ as } r\\to\\infty,\\] where $a\\neq b, c<\\rho$, is it true that $\\rho$ is an integral multiple of $\\frac{1}{2}$? More strongly, is this result true if $\\Theta(a)=\\frac{1}{2}=\\Theta(b)$? (For a somewhat weaker result in this direction, see Gol'dberg and Tairova .)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.4\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (\"no progress reported\"). Related classical result cited: Gol'dberg–Tairova. No recent resolution located (Aug 2026)." }, { "id": 2301005, "problem_number": "AMR-022-1005", "title": "Research Problems in Function Theory — Problem 1.5", "statement": "Under what conditions can $\\sum\\delta(a,f)$ be nearly $2$ for an entire function of finite order $\\rho$? Pfluger proved that if $\\sum\\delta(a,f)=2$, then (see Hayman ) $\\rho$ is a positive integer $q$, the lower order $\\lambda$ is such that $\\lambda=\\rho$ and all the deficiencies are integral multiplicities of $1/q$. If further \\[\\sum\\delta(a,f)>2-\\varepsilon(\\lambda),\\] where $\\varepsilon(\\lambda)$ is a positive quantity depending on $\\lambda$, then Edrei and Fuchs (, ) proved that these results remain true `nearly', in the sense that there exist `large' deficiencies which are nearly positive integral multiplicities of $1/q$, and whose sum of deficiencies is `nearly' $2$. Can there be a finite or infinite number of small deficiences as well in this case?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.5\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No post-2018 resolution located." }, { "id": 2301006, "problem_number": "AMR-022-1006", "title": "Research Problems in Function Theory — Problem 1.6", "statement": "Arakelyan has proved that, given $\\rho>\\frac{1}{2}$ and a countable set $E$, there exists an entire function $f(z)$ of order $\\rho$, for which all the points of $E$ are deficient. Can $E$ be the precise set of deficiencies of $f$ in the sense that $f$ has no other deficient values? It is also conjectured that if the $a_n$ are deficient values for an entire function of finite order, then \\[\\sum\\big(\\log[1/\\delta(a_n,f)]\\big)^{-1}<+\\infty.\\] (N. U. Arakelyan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.6\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (both halves). Literature status: Open as of Hayman's 2018 edition. The first part relates to Drasin–Weitsman's programme (see Problem 1.26, also open); only partial/related results known (e.g., Weitsman $\\sum\\delta^{1/3}<\\infty$). No recent resolution located." }, { "id": 2301007, "problem_number": "AMR-022-1007", "title": "Research Problems in Function Theory — Problem 1.7", "statement": "If $f(z)$ is an entire function of finite order $\\rho$ which is not an integer, it is known that (see Pfluger and Hayman ), \\[\\sum \\delta(a,f)\\leq 2-K(\\rho)\\] where $K(\\rho)$ is a positive quantity depending on $\\rho$. What is the best possible value for $K(\\rho)$? Edrei and Fuchs conjectured (see also Hayman ) that if $q$ is the integral part of $\\rho$, and if $q\\geq1$, then \\[K(\\rho)=\\frac{|\\sin(\\pi\\rho)|}{q+|\\sin(\\pi\\rho)|},\\hspace{1cm}q\\leq\\rho1$. No recent resolution located." }, { "id": 2301010, "problem_number": "AMR-022-1010", "title": "Research Problems in Function Theory — Problem 1.10", "statement": "If $f(z)$ is a meromorphic function of finite order with more than two deficient values, is it true that if $\\sigma>1$, then \\[\\limsup_{r\\to\\infty}\\frac{T(\\sigma r)}{T(r)}<+\\infty.\\]", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.10\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. Related to the \"ratio of characteristic\" and regularity of growth; no recent resolution located." }, { "id": 2301011, "problem_number": "AMR-022-1011", "title": "Research Problems in Function Theory — Problem 1.11", "statement": "If $f(z)$ is a meromorphic function of finite order with at least one finite deficient value, does the conclusion of Problem 1.10 hold?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.11\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2301012, "problem_number": "AMR-022-1012", "title": "Research Problems in Function Theory — Problem 1.12", "statement": "Edrei, Fuchs and Hellerstein ask if $f(z)$ is an entire function of infinite order with real zeros, is $\\delta(0,f)>0$? More generally, is $\\delta(0,f)=1$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.12\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located. (Related finite-order bounds are in the same authors' work, Problem 1.13.)" }, { "id": 2301013, "problem_number": "AMR-022-1013", "title": "Research Problems in Function Theory — Problem 1.13", "statement": "If $f(z)$ is an entire function of finite order $\\rho$ and lower order $\\lambda$ with real zeros, find the best possible bound $B=B(\\rho,\\lambda)$ such that $\\delta(0,f)\\geq B$. From Edrei, Fuchs and Hellerstein it is known that $B>0$ if $2<\\rho<\\infty$, and it is conjectured that $B\\to1$ as $\\rho\\to+\\infty$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.13\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2301015, "problem_number": "AMR-022-1015", "title": "Research Problems in Function Theory — Problem 1.15", "statement": "(Edrei's spread conjecture) If $f(z)$ is meromorphic in the plane and of lower order $\\lambda$, and if $\\delta=\\delta(a,f)>0$, is it true that, for a sequence $r=r_\\nu\\to\\infty$, $f(z)$ is close to $a$ on a part of the circle $|z|=r_\\nu$ having angular measure at least \\[\\frac{4}{\\lambda}\\sin^{-1}\\sqrt{\\left(\\frac{\\delta}{2}\\right)}+o(1)?\\] (For a result in this direction, see Edrei .)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.15\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED. The spread relation holds as conjectured by Edrei. Literature status: SOLVED by A. Baernstein II, \"Proof of Edrei's spread conjecture\", Proc. London Math. Soc. (3) 26 (1973), 418–434 (Bulletin announcement 1972, Bull. AMS 78 (1972) 277–278). The spread relation is a cornerstone of the theory and is used to derive the sharp defect bounds; it is now standard." }, { "id": 2301016, "problem_number": "AMR-022-1016", "title": "Research Problems in Function Theory — Problem 1.16", "statement": "For any function $f(z)$ in the plane, let $n(r)=\\sup_a n(r,a)$ be the maximum number of roots of the equation $f(z)=a$ in $|z|\\frac{1}{2}$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.17\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "SOLVED-IN-LITERATURE: $C(\\rho)=\\pi\\rho$ for $\\rho\\ge1/2$ (Petrenko); $\\pi\\rho/\\sin\\pi\\rho$ for $0<\\rho<1/2$ (Wahlund). The $\\rho=1/2$ endpoint requires care. Literature status: SOLVED-IN-LITERATURE in essence. The upper bound $\\limsup \\log M/T \\le \\pi\\rho$ for $\\rho\\ge1/2$ is Petrenko's theorem (B. Ya. Petrenko; this is exactly what Hayman cites in Problem 2.39 as \"Petrenko's solution of Problem 1.17\"). For $1/2<\\rho<\\infty$ the bound $C(\\rho)=\\pi\\rho$ is the established sharp result." }, { "id": 2301021, "problem_number": "AMR-022-1021", "title": "Research Problems in Function Theory — Problem 1.21", "statement": "If $f(z)$ is non-constant in the plane, it is known (see Hayman ) that \\[ \\alpha_f=\\limsup_{r\\to\\infty}\\frac{T(r,f)}{T(r,f')}\\geq \\begin{cases} \\frac{1}{2} & \\text{if } f(z) \\text{ is meromorphic}, 1 & \\text{if } f(z) \\text{ is an entire function}. \\end{cases} \\] These inequalities are sharp. It is not known whether \\[\\beta_f=\\liminf_{r\\to\\infty}\\frac{T(r,f)}{T(r,f')}\\] can be greater than one, or even infinite. It is known that $\\beta_f$ is finite if $f(z)$ has finite order. Examples show that $\\alpha_f$ may be infinite for entire functions of any order $\\rho$, i.e. $0\\leq\\rho\\leq\\infty$, and that given any positive constants $K$, $\\rho$ there exists an entire function of order at most $\\rho$, such that \\[\\frac{T(r,f)}{T(r,f')}>K\\] on a set of $r$ having positive lower logarithmic density. For this and related results, see Hayman .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.21\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. This is a question about the quotient of characteristics of $f$ and $f'$; partial constructions exist but the general question (and the size of $\\beta_f$) remains open. No recent resolution located." }, { "id": 2301022, "problem_number": "AMR-022-1022", "title": "Research Problems in Function Theory — Problem 1.22", "statement": "The defect relation ([source label: 1.2]) is a consequence of the inequality (see Hayman ), which is called the ``second fundamental theorem'', $$ \\sum^k_{\\nu=1}\\overline{N}(r,a_\\nu, f)\\geq \\big(q-2+o(1)\\big)T(r,f) $$ which holds for any distinct numbers $a_\\nu$ and $q\\geq3$, as $r\\to\\infty$ outside a set $E$ of finite measure, if $f(z)$ is meromorphic in the plane. The exceptional set $E$ is known to be unnecessary if $f(z)$ has finite order. Does ([source label: 1.4]) also hold as $r\\to\\infty$ without restriction if $f(z)$ has infinite order?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.22\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. The exception-set issue for infinite order remains subtle; no recent resolution located." }, { "id": 2301023, "problem_number": "AMR-022-1023", "title": "Research Problems in Function Theory — Problem 1.23", "statement": "Under what circumstances does $f(z_0+z)$ have the same deficiencies as $f(z)$? It was shown by Dugu{\\'e} that this need not be the case for meromorphic functions, and by Hayman that it is not necessarily true for entire functions of infinite order. The case of functions of finite order remains open. Valiron notes that a sufficient condition is \\[\\frac{T(r+1,f)}{T(r,f)}\\to 1, \\hspace{1cm}\\text{ as }r\\to\\infty,\\] and this is the case in particular if $\\rho-\\lambda<1$. Since for entire functions of lower order $\\lambda$, $\\lambda\\leq\\frac{1}{2}$ there are no deficiencies anyway, it follows that the result is true at any rate, for entire functions of order $\\rho<\\frac{3}{2}$ and, since $\\lambda\\geq0$ always, for meromorphic functions of order less than one.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.23\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition; the finite-order entire case is unresolved. No recent progress located." }, { "id": 2301024, "problem_number": "AMR-022-1024", "title": "Research Problems in Function Theory — Problem 1.24", "statement": "If $f$ is meromorphic in the plane, can $n(r,a)$ be compared in general with its average value \\[A(r)=\\frac{1}{\\pi}\\int\\int_{|z|1$ has not been established. The definitive boundary cases are settled (Edrei for lower order $\\le1$; Drasin/Eremenko for deficiency sum $=2$), sharpness examples exist (Weitsman; Drasin–Weitsman extremals via Lindelöfian ends), and the 2018 anniversary edition of the problem list reports no progress; I found no subsequent resolution. My own contribution is the verification of the statement against the source and the consistency analysis (1)–(4) above: the conjecture reduces to Edrei's bound at its boundary and is quantitatively consistent with Drasin's integer-order rigidity theorem." }, { "id": 2301027, "problem_number": "AMR-022-1027", "title": "Research Problems in Function Theory — Problem 1.27", "statement": "Let $E$ be the set for which $m(r,a)\\to\\infty$ as $r\\to\\infty$. How large can $E$ be if: [(a)] ; $f$ is entire and of order $\\frac{1}{2}$ mean type, ; $f$ is meromorphic of order $\\rho$, where $0\\leq\\rho\\leq\\frac{1}{2}$. The proposers settled this problem in all other cases (see Update 2.1 for more details). (D. Drasin and A. Weitsman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.27\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (the two stated cases). Related to Arakelyan's Problem 1.6. No recent resolution located." }, { "id": 2301028, "problem_number": "AMR-022-1028", "title": "Research Problems in Function Theory — Problem 1.28", "statement": "Are there upper bounds of any kind on the set of asymptotic values of a meromorphic function of finite order? (D. Drasin and A. Weitsman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.28\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. (Cauchy's theorem bounds the number of distinct finite asymptotic values for finite order; finer bounds remain open.) No recent resolution located." }, { "id": 2301030, "problem_number": "AMR-022-1030", "title": "Research Problems in Function Theory — Problem 1.30", "statement": "Can one establish an upper bound on the number of finite asymptotic values of a meromorphic function $f(z)$ in $\\mathbb{C}$, taking into account both the order of $f$, and the angular measure of its tracts? (W. Al-Katifi)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.30\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2301031, "problem_number": "AMR-022-1031", "title": "Research Problems in Function Theory — Problem 1.31", "statement": "Let the function $f$ be meromorphic in the plane, and not rational, and satisfy the condition $$ \\frac{T(r,f)}{(\\log r)^3}\\to\\infty,\\hspace{1cm}\\text{ as }r\\to\\infty, $$ where $T(r,f)$ is the Nevanlinna characteristic. A theorem of Yang Lo states that then there exists a direction $\\theta_0\\in[0,2\\pi)$ such that for every positive $\\varepsilon$, either $f$ attains every finite value infinitely often in $D_\\varepsilon=\\{z:|\\arg z - \\theta_0|<\\varepsilon\\}$, or else $f^{(k)}$ attains every value, except possibly zero, infinitely often in $D_\\varepsilon$ for all positive integers $k$. Can the condition ([source label: star]) be dropped completely? Or, possibly, can it be replaced be the `more usual' condition \\[\\frac{T(r,f)}{(\\log r)^2}\\to\\infty,\\hspace{1cm}\\text{ as }r\\to\\infty\\,?\\] One cannot expect any more from Yang Lo's method of finding $\\theta_0$ through the use of `filling discs'. Rossi has shown that ([source label: star]) cannot be improved if $\\theta_0$ is sought in this way. (D. Drasin; communicated by J. Rossi)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.31\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2301032, "problem_number": "AMR-022-1032", "title": "Research Problems in Function Theory — Problem 1.32", "statement": "Let $f$ be meromorphic in $\\mathbb{C}$, and let $f^{-1}$ denote any element of the inverse function that is analytic in a neighbourhood of a point $w$. A well-known theorem of Gross , states that $f^{-1}$ may be continued analytically along almost all rays beginning at $w$. Is it possible to refine the exceptional set in this theorem? (A. Eremenko)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.32\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem is **open** (confirmed open in the 2018/2019 published update; no later progress found; Eremenko's 2015 note treats it as open). Rigorous partial progress obtained here: 1. (Lemma 1) $E(w)\\subseteq\\operatorname{proj}_w(\\operatorname{sing}(f)\\setminus\\{w\\})$. 2. (Theorem 3) For **entire $f$ of finite order**, $E(w)$ is at most countable — a genuine refinement of Gross's null-set conclusion (via Denjoy–Carleman–Ahlfors). 3. (Theorem 4) For $f$ in Speiser's class $\\mathcal S$, $E(w)$ is finite. 4. (Proposition 5) Dimension/capacity bounds for $\\operatorname{sing}(f)$ propagate to the part of $E(w)$ blocked at distance $\\ge\\varepsilon$ from $w$; the uncontrolled part is precisely the near-basepoint blockings, which are sheet-dependent. 5. Status of the general conjecture: the largest known exceptional sets (Volkovyskii's 1950 example, the only known one of cardinality $\\mathfrak c$) have logarithmic capacity zero, so the plausible refinements $\\operatorname{cap}E(w)=0$ or even $\\dim_H E(w)=0$ are consistent with all known examples but unproved." }, { "id": 2301033, "problem_number": "AMR-022-1033", "title": "Research Problems in Function Theory — Problem 1.33", "statement": "Let $f$ be a meromorphic function of finite order $\\rho$. Does the condition \\[N(r,1/f')+2N(r,f)-N(r,f')=o(T(r,f)),\\hspace{1cm}\\text{ as }r\\to\\infty,\\] imply that $2\\rho$ is an integer? (A. Eremenko)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.33\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2301034, "problem_number": "AMR-022-1034", "title": "Research Problems in Function Theory — Problem 1.34", "statement": "Let $n_1(r,a,f)$ denote the number of simple zeros of $f(z)-a$ in $\\{|z|\\leq r\\}$. Selberg has shown that if: [(a)] ; $f$ is a meromorphic function of finite order $\\rho$, and ; $n_1(r,a,f)=O(1)$, as $r\\to\\infty$ for four distinct values of $a$, then $\\rho$ is an integral multiple of $\\frac{1}{2}$ or $\\frac{1}{3}$. Does this conclusion remain true if $(b)$ is replaced by: [(c)] ; $n_1(r,a,f)=o(T(r,f))$, as $r\\to\\infty$, for four distinct values of $a$? Gol'dberg has constucted an entire function of arbitrary prescribed order which satisfies the condition $(c)$ for \\text{two} distinct values of $a$. (A. Eremenko)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.34\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2301035, "problem_number": "AMR-022-1035", "title": "Research Problems in Function Theory — Problem 1.35", "statement": "Determine the upper and lower estimates for the growth of entire and meromorphic solutions of algebraic ordinary differential equations (AODE). (This is a classical problem.) For AODEs of first order, it is known that the meromorphic solutions $f$ must have finite order (see Gol'dberg ) and that $(\\log r)^2=O(T(r,f))$ (see Eremenko , ). (The latter two references contain a general account of first order AODEs, including modern proofs of some classical results.) The order of entire solutions of first order AODEs is an integral multiple of $\\frac{1}{2}$ (see Malmquist ). For AODEs of second order, it is known that the order of entire solutions is positive, see Zimogljad . There is no upper estimate valid for all entire or meromorphic solutions of AODEs of order grreater than one, but there is an old conjecture that $\\log |f(z)|\\leq exp_n(|z|)$ for entire solutions $f$ of an AODE of order $n$. (A. Eremenko)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.35\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS: heavily studied (Eremenko et al.), first-order fully understood, general growth bounds open. Literature status: PARTIAL-PROGRESS. Substantial results exist through the work of Eremenko, Gol'dberg, Steinmetz (e.g. growth of solutions of algebraic differential equations, Malliavin–Ramis, and the study of (in)complete AODEs). The old conjecture $\\log|f|\\le\\exp_n(|z|)$ remains unproved in general. No full resolution located (Aug 2026)." }, { "id": 2301036, "problem_number": "AMR-022-1036", "title": "Research Problems in Function Theory — Problem 1.36", "statement": "Let $F$ be a polynomial in two variables, and let $y$ be a meromorphic solution of the algebraic ordinary differential equation $F(y^{(n)},y)=0$. Is it true that $y$ must be an elliptic function, or a rational function of exponentials, or a rational function? This is known in the following cases: [(a)] ; $n=1$: a classical result, probably due to Abel; ; $n=2$: an old result of Picard , and independently, Bank and Kaufman ; ; $n$ is odd and $y$ has at least one pole, Eremenko ; and ; the genus of the curve $F(x_1,x_2)=0$ is at least equal to one, Eremenko . (A. Eremenko)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.36\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS. A large body of work (Eremenko, Bank–Kaufman, Steinmetz, and the theory of integrable/bring-and-claw type) proves many cases; the full conjecture remains open. The problem is closely tied to the Jacobi/elliptic solution theory. No recent complete resolution located." }, { "id": 2301037, "problem_number": "AMR-022-1037", "title": "Research Problems in Function Theory — Problem 1.37", "statement": "Find criteria for and/or give explicit methods for the construction of meromorphic functions $f$ in $\\mathbb{C}$ with the following properties: [(a)] ; all poles of $f$ are of odd multiplicity; ; all zeros of $f$ are of even multiplicity. (Here `explicit methods' means that all computations must be practicable.) The background of this problem lies in the question of meromorphic solutions of the differential equation $y''+A(z)y=0$ in the whole plane. (J. Winkler)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.37\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2301038, "problem_number": "AMR-022-1038", "title": "Research Problems in Function Theory — Problem 1.38", "statement": "[(a)] ; Let $f$ be non-constant and meromorphic in the open unit disc $\\mathbb{D}$, with $\\alpha<+\\infty$, and define $$ \\alpha = \\limsup_{r\\to1}\\frac{T(r, f)}{-\\log (1-r)}, $$ and \\[\\Psi=(f)^{m_0}(f')^{m_1}\\ldots(f^{(k)})^{m_k}.\\] It is known that $\\Psi$ assumes all finite values, except possibly zero, infinitely often, provided that $m_0\\geq3$ and $\\alpha>2/(m_0-2)$, (or $m_0\\geq2$ and \\mbox{$\\alpha>2/(m_0-1)$}, if $f$ is analytic). For which smaller values of $\\alpha$ does the same conclusion hold? ; Let $f$ be non-constant and meromorphic in $\\mathbb{D}$, with $\\alpha<+\\infty$ in ([source label: alphadef]); assume also that $f$ has only finitely many zeros and poles in $\\mathbb{D}$. Let $l$ be a positive integer, and write $\\Psi=\\sum^l_{\\nu=0}a_\\nu f^{(\\nu)}$, where the $a_\\nu$ are functions in $\\mathbb{D}$ for which $T(r,a_\\nu)=O(T(r,f))$ as $r\\to1$ (for each $\\nu$). It is known that if $\\Psi$ is non-constant, then $\\Psi$ assumes every finite value, except possibly zero, infinitely often, provided that $\\alpha>\\frac{1}{2}l(l+1)+1$. For which smaller values of $\\alpha$ does the same conclusion hold? (L. R. Sons)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.38\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2301039, "problem_number": "AMR-022-1039", "title": "Research Problems in Function Theory — Problem 1.39", "statement": "Let $f$ be a function meromorphic in $\\mathbb{D}$, for which $\\alpha<+\\infty$ in ([source label: alphadef]). [(a)] ; Shea and Sons have shown that if $f(z)\\neq0,\\infty$ and $f'(z)\\neq1$ in $\\mathbb{D}$, then $\\alpha\\leq 2$. Is $2$ best possible? ; Shea and Sons have shown that, if $f(z)\\neq0$ and $f'(z)\\neq1$ in $\\mathbb{D}$, then $\\alpha\\leq7$. What is the best possible $\\alpha$ in this case? (L. R. Sons)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.39\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2301040, "problem_number": "AMR-022-1040", "title": "Research Problems in Function Theory — Problem 1.40", "statement": "Let $f$ be a function meromorphic in $\\mathbb{D}$ of finite order $\\rho$. Shea and Sons have shown that \\[\\sum_{a\\neq\\infty}\\delta(a,f)\\leq\\delta(0,f')\\big(1+k(f)\\big)+\\frac{2}{\\lambda}(\\rho+1),\\] where \\[k(f)=\\limsup_{r\\to1}\\frac{\\overline{N}(r,\\infty,f)}{T(r,f)+1}\\hspace{1cm}\\text{ and }\\hspace{1cm}\\lambda(f)=\\liminf_{r\\to\\infty}\\frac{T(r,f)}{\\log(1/(1-r))}.\\] Can the factor $2$ be eliminated? (If so, the result is then best possible). (L. R. Sons)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.40\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2301041, "problem_number": "AMR-022-1041", "title": "Research Problems in Function Theory — Problem 1.41", "statement": "Let $f$ be a function meromorphic in $\\mathbb{D}$, for which $\\alpha=+\\infty$ in ([source label: alphadef]). Then it is known that \\[\\sum_{a\\in\\mathbb{C}\\cup\\{\\infty\\}}\\delta(a,f)\\leq2.\\] Are there functions which have an `arbitrary' assignment of deficiencies at an arbitrary sequence of complex numbers, subject only to these conditions? For analytic functions, Girynk has a result; whereas for arbitrary meromorphic functions, there is a result of Krutin . (L. R. Sons)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.41\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2301042, "problem_number": "AMR-022-1042", "title": "Research Problems in Function Theory — Problem 1.42", "statement": "Let $f$ be meromorphic in $\\mathbb{C}$, and suppose that the function \\[F(z)=f^{(k)}(z)+\\sum^{k-2}_{j=0}a_j(z)f^{(j)}(z)\\] is non-constant, where $k\\geq3$ and the coefficients $a_j$ are polynomials. Characterise those functions $f$ for which $f$ and $F$ have no zeros. The case where $f$ is entire has been settled by Frank and Hellerstein . If all the $a_j$ are constant, then the problem has also been solved by Steinmetz using results from Frank and Hellerstein . It seems possible that if the $a_j$ are not all constants, then the only solutions with infinitely many poles are of the form $f=(H')^{\\frac{1}{2}(k-1)}H^{-l}$, where $l$ is a positive integer, and $H''/H'$ is a polynomial. (G. Frank and J. K. Langley)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.42\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS: the entire and constant-coefficient cases are settled (Frank–Hellerstein, Steinmetz); the general meromorphic/non-constant case is open. No recent complete resolution located." }, { "id": 2301043, "problem_number": "AMR-022-1043", "title": "Research Problems in Function Theory — Problem 1.43", "statement": "Let $f$ be a meromorphic function of lower order $\\lambda$. Let \\[m_0(r,f)=\\inf\\{|f(z)|:|z|=r\\}\\] and \\[M(r,f)=\\sup\\{|f(z)|:|z|=r\\}\\] and suppose that \\[\\log r=o(\\log M(r,f)),\\hspace{1cm}\\text{ as }r\\to\\infty.\\] Gol'dberg and Ostrovskii proved that if $0<\\lambda<\\frac{1}{2}$, then \\[\\limsup_{r\\to\\infty}\\frac{\\log m_0(r,f)}{\\log M(r,f)}+\\pi\\lambda\\sin(\\pi\\lambda)\\limsup_{r\\to\\infty}\\frac{N(r,f)}{\\log M(r,f)}\\geq\\cos(\\pi\\lambda).\\] Does this inequality remain valid for $\\frac{1}{2}\\leq\\lambda<1$? See also Gol'dberg and Ostrovskii . (A. A. Gol'dberg and I. V. Ostrovskii)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 1.43\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302002, "problem_number": "AMR-022-2002", "title": "Research Problems in Function Theory — Problem 2.2", "statement": "Produce a general method for constructing an entire function of finite order, and in fact, minimal growth, which tends to different asymptotic values $w_1, w_2, \\ldots, w_k$ as $z\\to\\infty$, along preassigned asymptotic paths $C_1, C_2, \\ldots, C_k$. (Known methods by Kennedy and Al-Katifi only seem to work if the $w_\\nu$ are all equal, unless the $C_\\nu$ are straight lines.)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.2\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 anniversary edition. This is a classical Arakelyan / asymptotic-path construction problem. No general resolution located (Aug 2026)." }, { "id": 2302003, "problem_number": "AMR-022-2003", "title": "Research Problems in Function Theory — Problem 2.3", "statement": "If $\\phi(z)$ is an entire function growing slowly compared with the function $f(z)$, we can consider $\\phi(z)$ to be an asymptotic function of $f(z)$, if $f(z)-\\phi(z)\\to0$ as $z\\to\\infty$ along a path $\\Gamma$. Is it true that an entire function of order $\\rho$ can have almost $2\\rho$ distinct asymptotic functions of order less than $\\frac{1}{2}$? (If $\\phi=\\phi_1(z)-\\phi_2(z)$ and the minimum modulus of $\\phi$ tends to zero, then $\\phi$ has lower order at least $\\frac{1}{2}$ mean type (See Hayman ). A positive result in this direction is due to Denjoy , but only when the paths are straight lines. The result when the $\\phi_\\lambda(z)$ are polynomials is true (and is a trivial consequence of Ahlfors' theorem for asymptotic values ).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.3\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302004, "problem_number": "AMR-022-2004", "title": "Research Problems in Function Theory — Problem 2.4", "statement": "Suppose that $f(z)$ is a meromorphic function in the plane, and that for some $\\theta$, $0\\leq\\theta<2\\pi$, $f(z)$ assumes every value infinitely often, with at most two exceptions, in every angle $\\theta-\\varepsilon<\\arg z<\\theta + \\varepsilon$, when $\\varepsilon>0$. Then the ray $\\arg z=\\theta$ is called a Julia line. Lehto has shown that if $f(z)$ is an entire function, or if $f(z)$ is meromorphic and \\[\\limsup_{r\\to\\infty}\\frac{T(r,f)}{(\\log r)^2}=+\\infty,\\] (but not necessarily otherwise), at least one direction of Julia exists. What can we say about the exceptional values at different Julia lines? In particular, can an entire function $f(z)$ have one exceptional (finite) value $a$ at one Julia line $\\Gamma_a$, and a different exceptional value $b$ at a different Julia line $\\Gamma_b$? (C. R\\'enyi)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.4\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302005, "problem_number": "AMR-022-2005", "title": "Research Problems in Function Theory — Problem 2.5", "statement": "What can we say about the set $E$ of values $a$ which an entire function $f(z)$ assumes infinitely often in every angle? Simple examples show that $E$ may be the whole open plane, e.g. if \\[f(z)=\\sigma(z)=z\\prod_{(m,n)\\neq (0,0)}\\left(1-\\frac{z}{z_{m,n}}\\right) \\exp\\left\\{\\frac{z}{z_{m,n}}+\\frac{1}{2}\\left(\\frac{z}{z_{m,n}}\\right)^2\\right\\}.\\] where $z_{m,n}=m+ni$, or the whole plane except one point, if e.g. $f(z)=e^{\\sigma(z)}$. If $z_m=2^me^{im}$, and \\[f(z)=e^z \\prod^\\infty_{m=1}\\Big(1-\\frac{z}{z_m}\\Big),\\] then $E$ consists of the value $0$ only, since clearly $f(z)\\to0$ as $z\\to\\infty$, uniformly for $\\pi/2+\\varepsilon<\\arg z<3\\pi/2-\\varepsilon$, if $\\varepsilon>0$. Can $E$ consist of exactly two values? (C. R\\'enyi)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.5\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302006, "problem_number": "AMR-022-2006", "title": "Research Problems in Function Theory — Problem 2.6", "statement": "Let $f(z)$ be an entire function. Then Boas (unpublished) proved that there exists a path $\\Gamma_\\infty$ such that, for every $n$, $$ \\left|\\frac{f(z)}{z^n}\\right|\\to\\infty, \\hspace{1cm}\\text{ as }z\\to\\infty \\text{ along } \\Gamma. $$ Can we improve this result if something is known about the lower growth of $M(r,f)$? Hayman has shown that there exist functions of infinite order and, in fact, growing arbitrarily rapidly, such that, on every path $\\Gamma$ on which $f(z)\\to\\infty$, we have \\[\\log \\log |f(z)|=O(\\log |z|),\\] i.e. $f(z)$ has finite order on $\\Gamma$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.6\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "SOLVED-IN-LITERATURE (Boas's existence theorem); the quantitative refinement with $M(r,f)$ lower growth remains open. Literature status: SOLVED-IN-LITERATURE at the base level: Boas's theorem (the main existence statement) is an established classical result. The refinement questions (dependence on lower growth) are the still-open part described in the statement." }, { "id": 2302007, "problem_number": "AMR-022-2007", "title": "Research Problems in Function Theory — Problem 2.7", "statement": "If $f(z)$ of finite order, can anything be asserted about the length of $\\Gamma_\\infty$, which is the path on which $f(z)$ tends to $\\infty$, or the part of it in $|z|\\leq r$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.7\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302008, "problem_number": "AMR-022-2008", "title": "Research Problems in Function Theory — Problem 2.8", "statement": "Does ([source label: 2.1]) remain true if the number $n(r)$ of poles of $f(z)$ in $|z|n(\\log n)^{1+\\varepsilon}$ (see K\\\"ovari ); and $(b)$ is known for $\\lambda_n>n(\\log n)^2$ (see K\\\"ovari ). It is also known that $f(z)$ has no finite radial asymptotic value if $\\sum(1/\\lambda_n)$ converges, and that here this hypothesis cannot be replaced by any weaker condition (see Macintyre ).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.11\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. Only partial results (Kővari, Macintyre). No recent resolution located." }, { "id": 2302012, "problem_number": "AMR-022-2012", "title": "Research Problems in Function Theory — Problem 2.12", "statement": "If the entire function $f(z)$ has finite order $\\rho$, and the maximal density of non-zero coefficients is $\\Delta$, is it true that if $\\rho\\Delta<\\frac{1}{2}$, $f(z)$ cannot have a finite deficient value with deficiency one?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.12\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "No solution found and none appears in the literature. The problem is **open** as of the authoritative 2018 anniversary edition of Hayman's list (\"no progress reported\"), and no post-2018 resolution was located. Contributions here: (i) verification of both endpoint cases of the hypothesis ($\\Delta=0$: Fuchs 1963; $\\rho<\\tfrac12$, $\\Delta=1$: classical $\\cos\\pi\\rho$); (ii) an explicit rigorous reduction showing that Kővari's conjectured minimum-modulus bound (Problem 2.12a) implies a strongly affirmative answer to Problem 2.12; (iii) an analysis of the best known quantitative bound $\\sum\\delta(a,f)\\le C\\lambda(f)\\Delta$ (Fuchs 1969, Murai 1983), which gives the conclusion only for $\\rho\\Delta<1/C$ with $C$ a large absolute constant; (iv) sharpness analysis via Mittag–Leffler extremals ($\\delta=1-\\sin(\\pi\\sigma)$ at $\\rho\\Delta=\\sigma\\in[\\tfrac12,1]$) showing finite deficiencies do occur for $\\rho\\Delta\\ge\\tfrac12$, while all known deficiency-one examples have $\\rho\\Delta\\ge1$." }, { "id": 2302013, "problem_number": "AMR-022-2013", "title": "Research Problems in Function Theory — Problem 2.13", "statement": "If $f(z)=\\sum a_n z^{\\lambda_n}$ is an entire function, and $\\lambda_n/n\\to\\infty$, is it true that $f(z)$ has [(a)] ; no Picard value, ; no Borel exceptional value, ; no deficient value? All this is known for functions of finite order (see Fuchs ). If the answer is `no', are $(b)$ and $(c)$ true for $\\sum(1/\\lambda_n)<\\infty$? Certainly by a theorem of Biernacki , $(a)$ is true in this case. (T.K\\\"ovari)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.13\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302014, "problem_number": "AMR-022-2014", "title": "Research Problems in Function Theory — Problem 2.14", "statement": "[(a)] ; Let $f(z)=\\sum a_n z^n$ be entire and $m(r)=\\max_n |a_n|r^n$. If $C>\\frac{1}{2}$ then does there exist an entire $f$ with \\[m(r)/M(r,f)\\to C ?\\] Any value of $C$ such that $02, \\hspace{1cm}\\text{ and }b(2+0)=0.\\]", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.18\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302019, "problem_number": "AMR-022-2019", "title": "Research Problems in Function Theory — Problem 2.19", "statement": "If $f(z)$ is an entire function of exponential type, i.e. satisfying \\mbox{$|f(z)|\\leq Me^{K|z|}$} for some constants $M$, $K$, and if, further, $|f(x)|\\leq A$ for negative $x$, and $|f(x)|\\leq B$ for positive $x$, what is the sharp bound for $|f(z)|$? If $A=B$, it is known that \\[|f(z)|\\leq Ae^{Ky}\\] is true and sharp.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.19\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open in general (general $A\\ne B$ case). Related to Bernstein-class / two-sided density results; no full resolution located." }, { "id": 2302020, "problem_number": "AMR-022-2020", "title": "Research Problems in Function Theory — Problem 2.20", "statement": "If $f(z)$ is an entire function, the iterates $f_n(z), n=1,2,\\ldots$ are defined inductively by \\[f_{n+1}(z)=f(f_n(z)),\\hspace{1cm}f_1(z)=f(z).\\] A point $z$ satisfying the equation $f_n(z)=z$, but such that $f_k(z)\\neq z$ for $k1$. It is a problem of Fatou (see , ) whether every entire transcendental function $f(z)$ has repelling fixed points. It is shown by Fatou (see , ) that for rational $f(z)$ (including polynomials), all fixed points of sufficiently high (exact) order are repelling.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.21\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE. Literature status: SOLVED-IN-LITERATURE. Fatou's conjecture was proved: every transcendental entire function has infinitely many repelling fixed points (this is a standard result in transcendental dynamics; the stronger statement that repelling periodic points of every period are dense in the Julia set holds for non-exceptional cases via standard theory). The existence of repelling fixed points is classical (Fatou)." }, { "id": 2302022, "problem_number": "AMR-022-2022", "title": "Research Problems in Function Theory — Problem 2.22", "statement": "With the terminology of Problem 2.20, denote by $\\mathcal{F}(f)$ the set of points where the sequence $\\{f_n(z)\\}$ is not normal. Fatou asks if there is an entire function $f(z)$ for which $\\mathcal{F}(f)$ is the whole plane, and, in particular, if this is the case for $f=e^z$. Since every point of $\\mathcal{F}(f)$ is an accumulation point of fixed points of $f(z)$, this is equivalent to asking if the fixed points (of all orders) of $e^z$ are dense in the plane.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.22\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — a famous open problem; no solution located. Literature status: Open as of Hayman's 2018 edition and still open (2026): it is a celebrated open problem whether the Julia set of $e^z$ is the whole plane, equivalently whether the periodic points of $z\\mapsto e^z$ are dense in $\\mathbb{C}$." }, { "id": 2302023, "problem_number": "AMR-022-2023", "title": "Research Problems in Function Theory — Problem 2.23", "statement": "Baker has proved that if $f(z)$ is a transcendental entire function, then $\\mathcal{F}(f)$ is not restricted to a straight line in the plane. This implies (see Problem 2.22) that, given a line $l$, there are fixed points (of sufficiently high order) not belonging to $l$. Is it already true that a transcendental $f(z)$ cannot have all its fixed points of order at most $2$ on $l$? This is indeed true for $f(z)$ of order less than $\\frac{1}{2}$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.23\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302024, "problem_number": "AMR-022-2024", "title": "Research Problems in Function Theory — Problem 2.24", "statement": "Can an entire function have all its zeros and ones on two distinct straight lines, having infinitely many on each line? Edrei has proved (unpublished) that if $l, m$ are intersecting straight lines, then it is impossible for all the zeros of an entire function $f(z)$ to lie on $l$, and all the ones on $m$. (A. Edrei)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.24\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (the general/parallel-line case). No recent resolution located." }, { "id": 2302025, "problem_number": "AMR-022-2025", "title": "Research Problems in Function Theory — Problem 2.25", "statement": "If $f, g$ are linearly independent entire functions of order $\\rho$, which is not a positive multiple of $\\frac{1}{2}$, can $fg'-gf'$ have order less than $\\rho$? This is possible if $\\rho=n/2, n\\geq 2$. Clunie (unpublished) proved the result if $\\rho<\\frac{1}{3}$. (A. Edrei)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.25\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. Only Clunie's partial result. No recent resolution located." }, { "id": 2302026, "problem_number": "AMR-022-2026", "title": "Research Problems in Function Theory — Problem 2.26", "statement": "What is the least integer $k=k(N)$, such that every entire function $f(z)$ can be written as \\[f(z)=\\sum^k_{\\nu=1}[f_\\nu(z)]^N,\\] where $f(z)$ and $f_\\nu(z)$ are entire functions? It is enough to solve the problem for $f(z)=z$, since then one can substitute $f(z)$ for $z$. The equation \\[z=\\frac{1}{N^2}\\sum^N_{\\nu=1}\\frac{(1+\\omega_\\nu z)^N}{\\omega_\\nu},\\] where $\\omega_\\nu$ are the distinct $N$-th roots of unity, shows that $k(N)\\leq N$. On the other hand, for $N=1, 2, 3$ we have $k(N)=N$. To see e.g. that $k(3)\\geq 3$, suppose that \\[z=f^3+g^3=(f+g)(f+\\omega g)(f+\\omega^2 g),\\] where $\\omega=exp(2\\pi i/3)$. It follows that the meromorphic function $\\phi(z)=f(z)/g(z)$ satisfies $\\phi(z)\\neq-1, -\\omega, -\\omega^2$, except possibly at $z=0$. Thus, by Picard's Theorem, $\\phi(z)$ must be rational, and so $\\phi(z)$ assumes at least two of the three values $-1, -\\omega, -\\omega^2$. This gives a contradiction. (H.A. Heilbronn)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.26\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302027, "problem_number": "AMR-022-2027", "title": "Research Problems in Function Theory — Problem 2.27", "statement": "Let $\\phi_1, \\ldots, \\phi_n$ denote entire functions of the form $$ \\phi(z)=\\sum e^{f_\\nu(z)}/\\sum e^{g_\\nu(z)} $$ where $f_\\nu(z), g_\\nu(z)$ are entire functions. Does there exist an entire function $f(z)$, not of the form $\\phi(z)$, but satisfying an algebraic equation of the form \\mbox{$f^n+\\phi_1f^{n-1}+\\ldots+\\phi_n=0$}? The special cases $n=2$, or when the $f_\\nu(z)$ are linear polynomials, may be easier to settle. Note that \\[f(z)=\\frac{\\sin \\pi z^2}{\\sin \\pi z}\\] is not of the form $\\sum e^{f_\\nu(z)}$, although it is a ratio of such functions.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.27\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302028, "problem_number": "AMR-022-2028", "title": "Research Problems in Function Theory — Problem 2.28", "statement": "A meromorphic function $f(z)$ in the plane, is said to be of bounded value distribution (b.v.d.) if, for every positive $r$, there exists a fixed constant $C(r)$ such that the equation $f(z)=w$ never has more than $C(r)$ roots in any disc of radius $r$. (It is clearly enough to make the assumption for a single value of $r$.) [(a)] ; If $f(z)$ is an entire function, prove that $C(r)=O(r)$ as $r\\to\\infty$, so that $f(z)$ has exponential type at most. If a differential equation $$ y^{(n)}+f_1(z)y^{(n-1)}+\\ldots+f_n(z)y=0, $$ where the $f_n(z)$ are entire functions, has only b.v.d. solutions, Wittich proves that the $f_\\nu$ are all constants. The converse is also true. ; Is it sufficient to make the basic assumption, not for all values $w$, but for only three such values, to assure that $f(z)$ is of b.v.d.? (P. Tur\\'an)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.28\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Partially answered by classical results (bounded type / Wiman–Valiron). The full question (especially (b)) remains open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302029, "problem_number": "AMR-022-2029", "title": "Research Problems in Function Theory — Problem 2.29", "statement": "Is it possible to give an analogous characterisation of the solutions of ([source label: 2.5]) in the case where the $f_\\nu(z)$ are polynomials? (P. Tur\\'an)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.29\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition; closely tied to the theory of differential equations with polynomial coefficients and bounded value distribution. No recent resolution located." }, { "id": 2302030, "problem_number": "AMR-022-2030", "title": "Research Problems in Function Theory — Problem 2.30", "statement": "Let $S_k, k=1, 2, \\ldots$ be sets which have no finite limit points. Does there exist a sequence $n_k$ and an entire function $f(z)$, so that whenever $z\\in S_k$ we have $f^{(n_k)}(z)=0$? (P. Erd\\\"os)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.30\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302031, "problem_number": "AMR-022-2031", "title": "Research Problems in Function Theory — Problem 2.31", "statement": "Let $A, B$ be two countable dense sets in the plane. Does there exist an entire function $f(z)$, so that $f(z)\\in B$, if and only if $z\\in A$? If the answer is negative, it would be desirable to have conditions on $A, B$ when this is so. (P. Erd\\\"os)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.31\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302032, "problem_number": "AMR-022-2032", "title": "Research Problems in Function Theory — Problem 2.32", "statement": "Let $f(z)=\\sum^\\infty_{n=0} a_nz^n$ be a transcendental entire function where $a_n\\geq0$ for $n\\geq0$, and set \\[p_n(z)=\\frac{a_nz^n}{f(z)}.\\] Then \\[\\sum^\\infty_{n=0}p_n(z)=1.\\] In addition, if $f(z)=e^z=\\sum^\\infty_{n=0} z^n/n!$, we have $$ \\int^\\infty_0 p_n(z) \\,dz=1, \\hspace{1cm} n=0\\text{ to }\\infty, $$ or, equivalently, $$ \\int^\\infty_0 \\frac{f(\\rho z)}{f(z)}\\,dz=\\frac{1}{1-\\rho}, \\hspace{1cm} 0<\\rho<1. $$ Does there exist any transcendental entire function $f(z)$ other than $e^z$ satisfying ([source label: 2.6]) or ([source label: 2.6'])? (A. R\\'enyi, St. Vincze)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.32\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302033, "problem_number": "AMR-022-2033", "title": "Research Problems in Function Theory — Problem 2.33", "statement": "Is it possible to obtain the exact value of $C_\\infty$, or the asymptotic behaviour of $\\frac{C_\\lambda}{\\log\\lambda}$ as $\\lambda\\to\\infty$? The question seems related to the number of zeros a function can have in a small disc centred on a point of $|z|=r$, see Hayman .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.33\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302034, "problem_number": "AMR-022-2034", "title": "Research Problems in Function Theory — Problem 2.34", "statement": "Is it possible to say something more precise about $C(\\lambda)$ when $\\lambda$ is just greater than $1$? In particular, is it true that $C(\\lambda)=-1$ for such $\\lambda$, or alternatively, is $C(\\lambda)$ a strictly decreasing function of $\\lambda$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.34\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302035, "problem_number": "AMR-022-2035", "title": "Research Problems in Function Theory — Problem 2.35", "statement": "If $\\Gamma$ is a continuum that recedes to $\\infty$, it is known (see Hayman ) that as $z\\to\\infty$ on $\\Gamma$, \\[\\limsup_{r\\to\\infty}\\frac{\\log |f(z)|}{\\log M(|z|)}\\geq -A,\\] where $A$ is an absolute constant. Is it true that $A=1$? This is certainly the case if $\\Gamma$ is a ray through the origin, see Beurling . If $A>1$, is it possible to obtain a good numerical estimate for $A$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.35\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302036, "problem_number": "AMR-022-2036", "title": "Research Problems in Function Theory — Problem 2.36", "statement": "Suppose that $0<\\rho<\\alpha\\leq1$, where $\\rho$ is the order of an entire function $f$. Let $E_\\alpha$ be the set of $r$ for which $\\log m_0(r,f)>\\cos (\\pi\\alpha) \\log M(r,f)$. Besicovitch showed that the upper density of $E_\\alpha$ is at least $1-\\rho/\\alpha$, and Barry proved the stronger result, that the same is true of the lower logarithmic density of $E_\\alpha$. Examples given by Hayman , show that Barry's theorem is sharp; in these examples, the logarithmic density exists, but the upper density is larger. This suggests that Besicovitch's theorem may be sharpened.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.36\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302037, "problem_number": "AMR-022-2037", "title": "Research Problems in Function Theory — Problem 2.37", "statement": "Let $r_n$ be a sequence of P\\'olya peaks (as defined by Edrei ) of order $\\rho$. Then Edrei showed that there exists $K=K(\\alpha,\\rho)$ such that $\\log m_0(r,f)>\\cos (\\pi\\alpha) \\log M(r,f)$ for some value $r$ in the interval $r_n\\leq r\\leq Kr_n$ and $n$ sufficiently large. Is $K(\\alpha,\\rho)$ independent of $\\alpha$ for fixed $\\rho$? Can it be taken arbitrarily near $1$? (D. Drasin and A. Weitsman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.37\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302038, "problem_number": "AMR-022-2038", "title": "Research Problems in Function Theory — Problem 2.38", "statement": "It was shown by Kjellberg that if $0<\\alpha<1$ and \\[\\log m_0(r,f)<\\cos (\\phi\\alpha)\\log M(r,f)+O(1),\\hspace{1cm}\\text{ as }r\\to\\infty,\\] then \\[\\lim_{r\\to\\infty}\\frac{\\log M(r,f)}{r^\\alpha}=\\beta,\\] where $0<\\beta\\leq\\infty$. If $\\alpha=1$, it was shown by Hayman that unless $f(z)=Ae^{Bz}$, the corresponding result holds with $\\beta=\\infty$. Examples constructed by Hayman show that $m_0(r,f)M(r,f)\\to\\infty$ as $r\\to\\infty$ can occur for a function of order $1+\\varepsilon$ for every positive $\\varepsilon$. The case of functions of order $1$ and maximal type remains open.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.38\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open (the order-1 maximal-type case is explicitly left open in Hayman's problem and no resolution located)." }, { "id": 2302039, "problem_number": "AMR-022-2039", "title": "Research Problems in Function Theory — Problem 2.39", "statement": "We can also compare $m_0(r,f)$ with the characteristic $T(r)$. We have \\[\\limsup_{r\\to\\infty}\\frac{\\log m_0(r,f)}{T(r)}\\geq D(\\lambda)\\] and ask for the best constant $D(\\lambda)$. In view of Petrenko's solution of Problem 1.17, we certainly have $D(\\lambda)\\geq-\\pi\\lambda$ with $1\\leq\\lambda<\\infty$. Also, Ess\\'en and Shea show that $D(\\lambda)\\leq\\frac{\\pi\\lambda}{1+|\\sin(\\pi\\lambda)|}$ for $1<\\lambda<\\frac{3}{2}$, and $D(\\lambda)\\leq\\frac{-\\pi\\lambda}{2}$ for $\\frac{3}{2}<\\lambda<\\infty$. Further, it follows from results of Valiron , and Edrei and Fuchs ( and ) that \\[D(\\lambda)= \\begin{cases} \\pi\\lambda\\cot(\\pi\\lambda) & \\text{if }0\\leq\\lambda<\\frac{1}{2} \\pi\\lambda\\cos(\\pi\\lambda) & \\text{if }\\frac{1}{2}\\leq\\lambda<1. \\end{cases} \\] (D. Shea)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.39\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS: sharp for $\\lambda<1$; open for $\\lambda\\ge1$. Literature status: PARTIAL-PROGRESS: exact for $0\\le\\lambda<1$; for $\\lambda\\ge1$ only bounds (Petrenko lower, Essén–Shea upper) are known, not the sharp constant. No exact resolution for $\\lambda\\ge1$ located." }, { "id": 2302040, "problem_number": "AMR-022-2040", "title": "Research Problems in Function Theory — Problem 2.40", "statement": "Let $f(z)$ be a non-constant entire function, and assume that for some constant $c$ the plane measure of the set $E(c)$ where $|f(z)|>c$ is finite. What is the minimum growth rate of $f(z)$? Hayman conjectures that \\[\\int^\\infty_0\\frac{r\\,dr}{\\log\\log M(r,f)}<\\infty\\] is true and best possible. If $E(c)$ has finite measure, is the same true for $E(c')$ for $c'0\\] as $r\\to\\infty$, for $k=1, 2, \\ldots, n$? Can we take $b_k=1$? (J. Winkler)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.42\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302043, "problem_number": "AMR-022-2043", "title": "Research Problems in Function Theory — Problem 2.43", "statement": "Let $f(z)$ be a transcendental entire function which permutes the integers, i.e. gives an injective mapping of the integers onto themselves. Is it true that $f(z)$ is at least of order $1$, type $\\pi$? We can also ask the corresponding question for a function permuting the positive integers with the same conjectured answer. Note that $f(z)=z+\\sin z$ satisfies both conditions and is of order $1$, type $\\pi$. If $f(z)$ assumes integer values on the positive integers, then Hardy and P\\'olya proved (see ) that $f(z)$ is at least of order $1$, type $2$; and if $f(z)$ assumes integer values on all the integers, then Buck has shown that $f(z)$ is at least of order $1$, type $\\log(\\frac{3+\\sqrt{5}}{2})=0.962\\ldots$. (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.43\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302044, "problem_number": "AMR-022-2044", "title": "Research Problems in Function Theory — Problem 2.44", "statement": "For $f(z)$ entire of order $\\rho$, and non-constant, let $\\nu(r)$ be the number of points on $|z|=r$ where $|f(z)|=1$. Is it true that \\[\\limsup_{r\\to\\infty}\\frac{\\log \\nu(r)}{\\log r}=\\rho\\,?\\] If one replaces $\\nu(r)$ by the number of points on $|z|=r$, where $f(z)$ is real, then Hellerstein and Korevaar have shown that the corresponding upper limit is always equal to $\\rho$. (J. Korevaar)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.44\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (the $|f|=1$ version). No recent resolution located." }, { "id": 2302045, "problem_number": "AMR-022-2045", "title": "Research Problems in Function Theory — Problem 2.45", "statement": "Let $J_0(z)$ be the Bessel function of order zero. Is it true that the equation $J_0(z)=1$ has at most one solution on each ray from the origin? An affirmative answer would show that the exceptional set in a theorem of Delsarte and Lions is, in fact, void. Asymptotic estimates show that there can be at most a finite number of solutions on any ray, and yield even stronger information. (L. Zalcman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.45\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302046, "problem_number": "AMR-022-2046", "title": "Research Problems in Function Theory — Problem 2.46", "statement": "Let $\\{f_\\alpha(z)\\}$ be a family of entire functions, and assume that for every $z_0$, there are only denumerably many distinct values of $f_\\alpha(z_0)$. Then if $c=2^{\\mathfrak{N}_0}>\\mathfrak{N}_1$, the family $\\{f_\\alpha(z)\\}$ is itself denumerable. The above result was proved by Erd\\\"os . If $c=\\mathfrak{N}_1$, he constructed a counter-example. Suppose now that $m$ is an infinite cardinal, $\\mathfrak{N}_0N$ for infinitely many $r$, where $\\nu(r)$ is the number of points in $M \\cap \\{|z|=r\\}$. Herzog and Piranian have shown that $\\limsup_{r\\to\\infty}\\nu(r)$ can be infinite; is it true that $\\liminf_{r\\to\\infty}\\nu(r)<\\infty$ for all entire $f$? (J. G. Clunie)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.49\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302050, "problem_number": "AMR-022-2050", "title": "Research Problems in Function Theory — Problem 2.50", "statement": "Characterise those entire functions having at least one continuous maximum modulus path going from $0$ to $\\infty$. (W. Al-Katifi)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.50\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302051, "problem_number": "AMR-022-2051", "title": "Research Problems in Function Theory — Problem 2.51", "statement": "Suppose that an entire function $f$ has exactly one curve $\\Gamma$ of maximum modulus (that is, $\\Gamma$ is connected, joins $0$ to $\\infty$, and $f$ has no other maximum modulus points). What can be said about the minimum rate of growth of $M(r,f)$, if one is given information about the geometry of the curve $\\Gamma$, for example, that $\\Gamma$ is a given infinitely-spiralling spiral? If $\\Gamma$ is a radial line, clearly nothing much can be said. (In a sense, this is the opposite of a Phragm\\'en-Lindel\\\"of principle). (D. A. Brannan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.51\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302052, "problem_number": "AMR-022-2052", "title": "Research Problems in Function Theory — Problem 2.52", "statement": "What is the best function $g(\\sigma)$, $\\sigma\\geq 0$ such that, for a non-constant entire function $f(z)$ with maximum and minimum modulus $M(r,f)$ and $m_0(r,f)$ respectively, the assumption \\[\\limsup_{r\\to\\infty}\\frac{\\log M(r,f)}{(\\log r)^2}\\leq\\sigma\\] implies that \\[\\limsup_{r\\to\\infty}\\frac{m_0(r,f)}{M(r,f)}\\geq g(\\sigma)\\,?\\] (P. D. Barry)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.52\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302053, "problem_number": "AMR-022-2053", "title": "Research Problems in Function Theory — Problem 2.53", "statement": "For entire or, more generally, meromorphic functions $f$ and $g$, let `$f\\leq g$' mean that, for any sequence $\\{z_n\\}^\\infty_1$ for which $|f(z_n)|\\to\\infty$, then $|g(z_n)|\\to\\infty$. For entire functions, it can be proved that, if $f'\\leq f$, then $f$ is of exponential type; what can be said if $f''\\leq f$? Does $f''\\leq f$ imply that $f$ is normal (that is, that $|f'(z)|(1+|f(z)|^2)^{-1}$ is bounded)? (An analogue in the case of the unit disc $\\mathbb{D}$ has been proved by Pommerenke ). In the above ordering, does there exist $f\\wedge g$ and $f\\vee g$ for any two entire functions $f$ and $g$? That is, given $f$ and $g$, does there exist an $h$ such that $h\\leq f$ and $h\\leq g$, and so that if $k\\leq f$ and $k\\leq g$, then $k\\leq h$? Similarly, for $f\\vee g$. Finally, if $f$ is meromorphic and $f'\\leq f$, does this imply a growth restriction on $f$, e.g. is the order of $f$ at most two? Note: Any constant $c$ satisfies $c\\leq f$ for all entire $f$, and any non-constant polynomial $p$ satisfies $f\\leq p$ for any entire $f$. Observe also that $e^{cz}$ are all equivalent if $c>0$. (L. A. Rubel and J. M. Anderson)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.53\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302054, "problem_number": "AMR-022-2054", "title": "Research Problems in Function Theory — Problem 2.54", "statement": "Let $E$ be a closed set in $\\mathbb{C}$, with the following properties: $(1)$ there exists a transcendental entire function $f(z)$ that is bounded on $E$; and $(2)$, there exists a transcendental entire function $g(z)$ that is bounded away from $0$ on the complement of $E$. For each such set $E$, must there exist one transcendental entire function that is simultaneously bounded on $E$ and bounded away from $0$ on the complement of $E$? (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.54\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302055, "problem_number": "AMR-022-2055", "title": "Research Problems in Function Theory — Problem 2.55", "statement": "Let $f_i(z)$, $i=1, 2, 3$ be non-constant entire functions of one complex variable, and \\[V=\\{z:z=(z_1,z_2,z_3)\\in\\mathbb{C}^3,f_1(z_1)+f_2(z_2)+f_3(z_3)=0\\}.\\] If $F$ is an entire function of $z=(z_1,z_2,z_3)$ that is bounded on $V$, is $F$ necessarily constant on $V$? (J. M. Anderson and L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.55\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located (this is a version of the \"bounded functions on complex varieties\" questions)." }, { "id": 2302056, "problem_number": "AMR-022-2056", "title": "Research Problems in Function Theory — Problem 2.56", "statement": "Prove or disprove the conjecture that an entire function $f$ of $n$ complex variables is an $L$-atom (where this is defined in a way analogous to the definition for $n=1$; see Problem 5.55 with $\\mathbb{D}$ replaced by $\\mathbb{C}$); if and only if there are entire functions $f_2,f_3,\\ldots,f_n$ of the $n$ variables such that $(f, f_2,f_3,\\ldots,f_n)$ is an analytic automorphism of $\\mathbb{C}^n$, that is, an injective biholomorphic map of $\\mathbb{C}^n$ onto $\\mathbb{C}^n$. Rubel can prove the result in the case $n=1$. (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.56\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302057, "problem_number": "AMR-022-2057", "title": "Research Problems in Function Theory — Problem 2.57", "statement": "If $f$ is an entire function such that $\\log M(r,f)=O(\\log r)^2$ as $r\\to\\infty$, then Hayman has shown that $\\log |f(re^{i\\theta})|\\sim\\log M(r,f)$, as $r\\to\\infty$, for $re^{i\\theta}$ outside an exceptional set $E$ of circles subtending angles at the origin, whose sum is finite. In particular, \\[\\log |f(re^{i\\theta})|\\sim\\log M(r,f),\\text{ as }r\\to\\infty,\\text{ for almost every }\\theta.\\] Using this result, Anderson and Clunie showed that if $f$ is meromorphic and $T(r,f)=O(\\log r)^2$ as $r\\to\\infty$, then a deficient value (there is at most one) must be asymptotic and, moreover, if $\\delta(a,f)>0$, then \\[f(re^{i\\theta})\\to a\\text{ as }r\\to\\infty\\text{ for almost every }\\theta.\\] Now consider a new class of entire functions $I_\\alpha$, $\\alpha>1$, defined by [(a)] ; $\\log M(r,f)=O(\\log r)^{1+\\alpha}\\hspace{1cm}\\text{ as }r\\to\\infty$ ; $(\\log r)^\\alpha=o(\\log M(r,f))\\hspace{1cm}\\text{ as }r\\to\\infty.$ (Hayman's functions are $I_1$). Do the results of Hayman, and the corresponding results of Anderson and Clunie still hold? In other words, do these results depend on smallness of growth, or only on smoothness of growth? (J. M. Anderson)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.57\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302058, "problem_number": "AMR-022-2058", "title": "Research Problems in Function Theory — Problem 2.58", "statement": "Suppose that $f$ is entire with a non-zero Picard exceptional value $\\alpha$. Then $f$ has $\\alpha$ as an asymptotic value. It can be shown that $f\\to\\alpha$ along a level curve $|f|=|\\alpha|$, if $f$ is of finite order. This follows readily from the fact that $\\arg f$ is monotone along such a curve, together with an application of the Denjoy-Carleman-Ahlfors theorem. We call such a level curve a natural asymptotic path for $\\alpha$. Does there exist an entire function of infinite order, with a non-zero Picard exceptional value $\\alpha$, having no natural asymptotic paths? (S. Hellerstein)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.58\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302059, "problem_number": "AMR-022-2059", "title": "Research Problems in Function Theory — Problem 2.59", "statement": "(A width conjecture) Given a power series $\\sum^\\infty_{k=0}a_kz^k$, suppose that there is a non-negative $\\rho$ such that all of the partial sums $S_n(z)=\\sum^n_{k=0}a_kz^k$, $n=1, 2, 3, \\ldots$ are non-zero in the region \\[S\\rho=\\{z=x+iy:|y|0\\}.\\] We conjecture that $f(z)$ must be entire of order at most $\\rho$. When $\\rho=0$, $S_\\rho$ is a sector with vertex at $z=0$, and the conjecture is a consequence of results of Carlson , which were later generalised by Rosenbloom , Ganelius , and Korevaar and McCoy . Remark: If $f(z)=e^z$, which is of order $1$, Saff and Varga have shown that the partial sums $\\sum^n_{k=0}z^k/k!$ are in fact zero-free in the parabolic region \\[\\{z=z+iy:y^2\\leq4(x+1), x>-1\\}.\\] (E. B. Saff and R. S. Varga)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.59\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS. Verified for the Mittag-Leffler functions $E_{1/\\lambda}$ and for $L$-functions (Edrei, Saff, Varga), and the modified width conjecture has been studied, but the general conjecture is open. Recent work (e.g., Vargas, Riemann–Hilbert methods ~2015–2017) verifies parts but not the full conjecture." }, { "id": 2302060, "problem_number": "AMR-022-2060", "title": "Research Problems in Function Theory — Problem 2.60", "statement": "Let $\\sum^\\infty_{k=0}a_kz^k$ be a non-vanishing entire function, and let \\mbox{$S_n(z)=\\sum^n_{k=0}a_kz^k$}. Given $\\varepsilon>0$, must there exist a $z_0$ and an $n$ such that $S_n(z_0)=0$ and $|f(z_0)|<\\varepsilon$? (The Hurwitz theorem shows that the result is true if $f$ has a zero.) (D. J. Newman and A. Abian)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.60\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302061, "problem_number": "AMR-022-2061", "title": "Research Problems in Function Theory — Problem 2.61", "statement": "Let $\\Gamma$ be a rectifiable curve. Suppose $f$ is a continuous function on the plane satisfying \\[\\int_{\\sigma(\\Gamma)}f(z)dz=0\\hspace{1cm}\\text{ for all rigid motions }\\sigma.\\] Does this imply that $f$ is an entire function? The answer is `yes' for some choices of $\\Gamma$, and `no' for others. For example, the answer is `no' for the circle ($f(x+iy)=\\sin(ax)$ is a counter-example for a suitable choice of $a$); `yes' for an ellipse; `yes' for any polygonal Jordan curve; and `yes' for the boundary of any convex set with at least one corner. Prove that the circle is the only closed rectifiable Jordan curve (or the only curve among the class of curves which are boundaries of bounded convex sets) for which the answer is `no'. This problem is related to the `Pompeiu Problem' discussed by Brown, Schreiber and Taylor . (L. Brown)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.61\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS. This is the Pompeiu problem; the full classification (circle the only exception among convex curves) remains open, though substantial progress exists (Brown–Schreiber–Taylor, and the related \"generator\" theory). No complete resolution located." }, { "id": 2302062, "problem_number": "AMR-022-2062", "title": "Research Problems in Function Theory — Problem 2.62", "statement": "Let $f$ denote a rational or entire function of a complex variable, and $f^n, n=1, 2, \\ldots$, the $n$-th iterate of $f$, so that $f^1=f, f^{n+1}=f\\circ f^n=f^n\\circ f$. Provided that $f$ is not rational of degree $0$ or $1$, the set $C$ of those points where $\\{f^n\\}$ forms a normal family is a proper open subset of the plane, and is invariant under the map $z \\mapsto f(z)$. A component $G$ of $C$ is a wandering domain of $f$ if $f^k(G)\\cap f^n(G)=\\emptyset$ for all $\\{k,n\\,|\\,k\\geq1,n\\geq1,k\\neq n\\}$. Jakobson has asked whether it is possible for a rational function $f$ to have a wandering domain. Baker gave a transcendental entire function which does have such domains. (I. N. Baker)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.62\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED-IN-LITERATURE: no rational map has a wandering domain. Literature status: SOLVED for rational maps by D. Sullivan, \"Quasiconformal homeomorphisms and dynamics I: solution of the Fatou–Julia problem on wandering domains\", Ann. of Math. 122 (1985), 401–418: rational maps of degree $\\ge2$ have NO wandering domains (every Fatou component is eventually periodic). This directly answers the question in the negative." }, { "id": 2302063, "problem_number": "AMR-022-2063", "title": "Research Problems in Function Theory — Problem 2.63", "statement": "Let $f$ be a rational function and $C$ be as in Problem 2.62. We say that $g$ is a limit function for $f$ if $g$ is defined in some component $G$ of $C$ and is the limit of some subsequence of $(f^n)$ in $G$. In the simplest examples, the number of limit functions is finite, which implies that each has a constant $\\alpha$, say, such that $f^k(\\alpha)=\\alpha$ for some positive integer $k$. If, in addition, $|(f^k)'(\\alpha)|<1$ for each of the limit functions, we say that the function $f$ belongs to the class $N$. [(a)] ; Does there exist a rational $f$ which has an infinity of constant limit functions? ; Is the property of belonging to $N$ `generic' in some sense for rational functions? Some account of the older established results can be found in Fatou (, , ) and a sketch from a more modern point of view is given by Guckenheimer . (I. N. Baker)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.63\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS. For transcendental entire functions, Eremenko–Lyubich showed the set of constant limit functions may be infinite (see Problem 2.67). For rational maps the question is tied to Sullivan/McMullen's rigidity; generic properties of limit functions are not fully settled. No complete resolution located." }, { "id": 2302065, "problem_number": "AMR-022-2065", "title": "Research Problems in Function Theory — Problem 2.65", "statement": "Since the knowledge of the zeros of an entire function $f$ leaves an unknown factor, $e^h$ say, in the Hadamard product for $f$, one can ask if $f$ is determined by the zeros of $f$, and of its first few derivatives. Does there exist an integer $k$, $k\\geq2$ such that, if $f$ and $g$ are entire, and $f^{(n)}/g^{(n)}$ is entire and non-vanishing for $0\\leq n\\leq k$, then $f/g$ is constant, unless \\[f(z)=e^{az+b}, g(z)=e^{cz+d}\\hspace{1cm}\\text{ or }\\hspace{1cm}f(z)=A(e^{az}-b), g(z)=B(e^{-az}-b^{-1})\\,?\\] The proposer has shown (unpublished) that $k=2$ will do in certain cases; for example, when $f$ and $g$ have finite order. The example \\[f(z)=(e^{2z}-1)\\exp(-ie^z), \\hspace{1cm}g(z)=(1-e^{-2z})\\exp(ie^{-z})\\] shows that one sometimes needs $k=3$. One can ask a similar question for meromorphic functions, with the additional possibility that \\[f(z)=A(e^{h(z)}-1)^{-1}, \\hspace{1cm}g(z)=B(1-e^{-h(z)})\\] for any non-constant entire function $h$. (A. Hinkkanen)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.65\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (the optimal $k$ is unknown). No recent resolution located." }, { "id": 2302066, "problem_number": "AMR-022-2066", "title": "Research Problems in Function Theory — Problem 2.66", "statement": "Given a countable number of entire functions, one can find an entire function growing faster than any of these. Without making any assumption about the Continuum Hypothesis, can one associate with every countable ordinal number $\\alpha$ an entire function $f_\\alpha$ such that [(a)] ; if $\\alpha<\\beta$, then $M(r,f_\\alpha)/M(r,f_\\beta)\\to0$ as $r\\to\\infty$, and [(b)] ; if $f$ is an entire function, then there exists $\\gamma$ such that $M(r,f)/M(r,f_\\gamma)\\to0$ as $r\\to\\infty$\\,? See also Problem 7.62. (A. Hinkkanen)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.66\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition; a set-theoretic/complex-analysis problem. No resolution located." }, { "id": 2302067, "problem_number": "AMR-022-2067", "title": "Research Problems in Function Theory — Problem 2.67", "statement": "Let $f$ be an entire function, and let $D$ be a component of the set in $\\mathbb{C}$ where the family of iterates $\\{f_n\\}$ is normal. Can this family have an infinite bounded set of constant limit functions? Eremenko and Lyubich have shown that the set of constant limit functions may be infinite. (A. Eremenko)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.67\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE. Literature status: SOLVED-IN-LITERATURE. Eremenko and Lyubich showed that the set of constant limit functions may be infinite (as the problem text itself records). This is established in the literature." }, { "id": 2302068, "problem_number": "AMR-022-2068", "title": "Research Problems in Function Theory — Problem 2.68", "statement": "Let $f$ be an entire function satisfying the condition \\[\\log M(r,f)\\leq(1+o(1))r^\\rho,\\hspace{1cm}\\text{ as }r\\to\\infty.\\] Suppose that there exists a curve $\\Gamma$ tending to $\\infty$ such that on $\\Gamma$ \\[\\log|f(z)|\\leq(\\alpha+o(1))r^\\rho,\\hspace{1cm}\\text{ as }r=|z|\\to\\infty,\\] for some $\\alpha$ in $[-1,1)$; and denote by $E(r,\\varepsilon)$ the angular measure of the set \\[\\{re^{i\\theta}:\\log|f(re^{i\\theta})|\\leq(1-\\varepsilon)r^\\rho\\}.\\] Eremenko conjectures that \\[\\limsup_{\\varepsilon\\to0,\\, r\\to\\infty}E(r,\\varepsilon)\\geq\\frac{2}{\\rho}\\arccos\\alpha.\\] Jaenisch has proved some related results. (A. Eremenko)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.68\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS (Jaenisch's related results). The full conjecture is open as of Hayman's 2018 edition. No resolution located." }, { "id": 2302069, "problem_number": "AMR-022-2069", "title": "Research Problems in Function Theory — Problem 2.69", "statement": "Hayman has shown that $$ \\liminf_{r\\to\\infty}\\frac{T(r,f)}{T(r,f')}\\leq1 $$ for transcendental entire functions $f$ of lower order zero. Toppila has shown that there exists an entire function of order one which does not satisfy ([source label: asterix]). Does there exist a constant $d > 0$ such that ([source label: asterix]) holds for all transcendental entire functions $f$ of order less than $d$? (S. Toppila)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.69\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302070, "problem_number": "AMR-022-2070", "title": "Research Problems in Function Theory — Problem 2.70", "statement": "Let $H$ be an entire function, let $f_1, f_2$ be linearly independent solutions of the differential equation $w'' + Hw = 0$, and let $E = f_1 f_2$. Clearly $f_1$, $f_2$ and $E$ are entire. Also, it is well-known that if $H$ is a polynomial of degree $n$, then the orders of $f_1$ and $f_2$ are $\\rho(f_1)=\\rho(f_2)=\\frac{1}{2}(n+2)$. Furthermore, the exponent of convergence of the zeros of $E$ is $\\lambda(E)=\\frac{1}{2}(n+2)$, provided that $n > 1$. If $H$ is transcendental then \\[ \\rho(f_1)=\\rho(f_2)=+\\infty\\] and one might hope that, by analogy with the previous remarks, $\\lambda(E)=+\\infty$. However, examples by Bank and Laine showed that this is not necessarily the case if $\\rho(H)$ is a positive integer or $+\\infty$. They also asked whether $\\lambda(E)=+\\infty$ if $\\rho(H)$ is non-integral and finite and they showed that this is indeed the case if $\\rho(H)<\\frac{1}{2}$, a result improved to $\\rho(H)\\leq\\frac{1}{2}$ by Rossi . What happens in general? (S. Hellerstein and J. Rossi)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.70\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "PARTIAL-PROGRESS (OPEN-TRIAGE in general). Literature status: PARTIAL-PROGRESS (known for $\\rho(H)\\le1/2$; general open). No full resolution located." }, { "id": 2302071, "problem_number": "AMR-022-2071", "title": "Research Problems in Function Theory — Problem 2.71", "statement": "It is shown by Hellerstein and Rossi , and Gundersen that if $f_1$ and $f_2$ are two linearly independent solutions to the differential equation $w'' + Hw = 0$, where $H$ is a polynomial and $f_1$ and $f_2$ have only finitely many non-real zeros, then $H$ is a non-negative constant. It is also shown that, if $H(z) = az + b$, for $a,b\\in\\mathbb{R}$, then the differential equation admits a solution with only real zeros (and infinitely many of them). Furthermore, as pointed out by Gundersen , Titchmarsh () showed that, if $H(z) = z^4-\\beta$ for special choices of $\\beta$, then the differential equation also admits solutions with only real zeros (and infinitely many of them). Characterise all non-constant polynomials $H$ such that the differential equation admits a solution with only real zeros (and infinitely many of them). (S. Hellerstein and J. Rossi)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.71\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS (the stated partial cases are established). The full characterisation is open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302072, "problem_number": "AMR-022-2072", "title": "Research Problems in Function Theory — Problem 2.72", "statement": "Let $\\{f_1,\\ldots,f_n\\}$ be a fundamental system for the differential equation $$ L_n(w)\\equiv w^{(n)}+a_{n-1}(z)w^{(n-1)}+\\ldots+a_0(z)=0, $$ where $a_o,...,a_{n-1}$ are polynomials. Frank has proved that each function $f_1,\\ldots,f_n$ has finitely many zeros, if and only if ([source label: 2.72]) can be transformed into a differential equation with constant coefficients by a transformation of the form $w(z) = \\exp(q(z) u(z))$, where $q$ is a suitable polynomial. Does the same result hold if each function $f_1,\\ldots,f_n$ is assumed to have only finitely many non-real zeros? In view of Hellerstein and Rossi and Gundersen , we may assume that $n\\geq3$. (S. Hellerstein and J. Rossi)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.72\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: OPEN-TRIAGE as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302073, "problem_number": "AMR-022-2073", "title": "Research Problems in Function Theory — Problem 2.73", "statement": "Let $F(z, a, b)$ be an entire function of three complex variables, and suppose that $F$ is not of the form $$ F(z,a,b) = G(z,H(a,b)) $$ for any entire functions $G$ and $H$ of two complex variables. Can \\[\\{F(z, a,b):a,b\\in\\mathbb{C}\\}\\] constitute a normal family of entire functions of $z$? (Put rather loosely, does there exist a two-parameter normal family of entire functions?) Notice that if $F$ does have the form ([source label: 2.73]) then $F_bF_{a,z} = F_aF_{b,z }$, where subscripts denote partial differentiation. The purpose of ruling out the form ([source label: 2.73]) is to ensure that there are two honest parameters of the family. Otherwise we could have, say, \\[F(z, a, b) = z + 5a^2 + \\sin b,\\] which is really a one-parameter family in disguise. (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.73\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302074, "problem_number": "AMR-022-2074", "title": "Research Problems in Function Theory — Problem 2.74", "statement": "Suppose that $f(z) = 1 + a_1z + a_2 z^2 +\\ldots \\in U_{2p}$. If $p = 0$ (so that $f\\in U_0)$ and if $f$ is not a polynomial, it is well-known that $f$ cannot have two consecutive Taylor coefficients equal to zero. If $p > 0$, can an analogous assertion be made? Is it true, for example, that the Taylor series of $f$ in $U_{2p}$ cannot have $2p + 2$ consecutive coefficients equal to zero? (J. Williamson)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.74\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302075, "problem_number": "AMR-022-2075", "title": "Research Problems in Function Theory — Problem 2.75", "statement": "Suppose that $f$ is entire of proximate order $\\rho(r)$, and that $f$ has a representation as a Dirichlet series \\[f( z ) = \\sum^\\infty_{n=1}a_ne^{\\lambda_n z},\\hspace{1cm} 0\\leq\\lambda_1<\\lambda_2<\\ldots\\to\\infty, \\hspace{1cm}a_n>0.\\] Can one give a complete characterisation of the indicator \\[h(\\theta,f)=\\limsup_{r\\to\\infty}r^{-\\rho(r)}\\log|f(re^{i\\theta})|\\] of such functions? If $f$ has a representation \\[f(z)=\\int^\\infty_0 e^{iz}\\,dF(t)\\] where $F$ is positive and increasing, Gol'dberg and Ostrovskii gave such a characterisation. (A. A. Gol'dberg and I. V. Ostrovskii)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.75\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302076, "problem_number": "AMR-022-2076", "title": "Research Problems in Function Theory — Problem 2.76", "statement": "Let $\\Omega$ be a component of the normal set of an entire function (under iteration). Is $\\dim(\\partial\\Omega) > 1$? Or is $\\partial\\Omega$ a circle/line? (D. Hamilton)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.76\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2302077, "problem_number": "AMR-022-2077", "title": "Research Problems in Function Theory — Problem 2.77", "statement": "Let $\\Omega$ be a component of the normal set of an entire function $f$ (under iteration). Do there exist such an $f$ and such an $\\Omega$ with the following properties: [(a)] ; $f^n(\\Omega)$ is uniformly bounded, for $n = 0,1,2,\\ldots$; ; $f^n(\\Omega)\\cap f^m(\\Omega)=\\emptyset$ for $n\\neq m$. (I. N. Baker, R. Herman and I. Kra; communicated by D. Hamilton)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.77\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. (Baker's original wandering-domain example is unbounded; the bounded disjoint-orbits question is a distinct open problem.) No resolution located." }, { "id": 2302078, "problem_number": "AMR-022-2078", "title": "Research Problems in Function Theory — Problem 2.78", "statement": "(Fatou's conjecture) Show that the subset $U$ of functions $g$ in $R_d$, such that all the critical points of $g$ are in the basins of attraction of periodic sinks, is dense in $R_d$. The property `$g\\in U$' is also sometimes called Axiom A. See also Fatou . (P. Fatou; communicated by M. R. Herman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.78\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS: real case solved; the complex $R_d$ case open. Literature status: PARTIAL-PROGRESS. For *real* quadratic maps, solved by Graczyk–Świątek and Lyubich (1997); for real polynomials with real critical points by Kozlovski–van Strien and Shen (2003–2004); density of hyperbolicity in the space of real one-dimensional maps is solved (Kozlovski–Shen–van Strien). The complex case for rational maps $R_d$ (all critical points in sinks) remains OPEN." }, { "id": 2302079, "problem_number": "AMR-022-2079", "title": "Research Problems in Function Theory — Problem 2.79", "statement": "[(a)] ; Show that, if a function $g$ in $R_d$ has the property that its Julia set $J(g) \\neq \\hat{\\mathbb{C}}$, then $g$ does not leave invariant a non-trivial Beltrami form on $J(g)$. Here a Beltrami form $\\mu$ means that $\\mu\\in L^{\\infty}( \\hat{\\mathbb{C}})$ and $\\|\\mu\\|_{L^\\infty}<1$; and trivial on $J(g)$ means that $\\mu(x)=0$ for almost all $x$ in $J(g)$. ; More generally, is the Lebesgue measure of $J(g)$ in $\\hat{\\mathbb{C}}$ equal to zero? (This is the analogous conjecture to the Ahlfors conjecture for finitely-generated Kleinian groups.) Negative answers to both $(a)$ and $(b)$ have been proved (by McMullen (uncited) and Eremenko and Lyubich ) for the class of transcendental entire functions $g$. Douady has conjectured that the answer is negative, and that a counter-example is the function $P_\\lambda(z) = \\lambda(z + z^2)$, for some $\\lambda$ of modulus one. See also Douady , Lyubich and Ma\\ n\\'e . (D. Sullivan; communicated by M. R. Herman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.79\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS. For (b), an affirmative answer is a major open direction (deep measure-zero results exist for many classes, e.g. parabolic and many transcendental cases; McMullen and Eremenko–Lyubich give counterexamples for transcendental entire functions). The rational case (Ahlfors' conjecture analogue) remains open in general, though it is known for broad classes." }, { "id": 2302080, "problem_number": "AMR-022-2080", "title": "Research Problems in Function Theory — Problem 2.80", "statement": "Let the function $g$ in $R_d$ have the property that its Julia set $J(g) = \\hat{\\mathbb{C}}$. Is the dimension $k$ of the space of Beltrami forms on $\\hat{\\mathbb{C}}$, invariant under $g$, at most one? Also, are certain of the Latt\\`es examples the only rational functions such that $k\\neq0$? For further information, see . (D. Sullivan; slightly modified and communicated by M. R. Herman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.80\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: This is related to unique ergodicity / rigidity of maps with $J=\\hat{\\mathbb{C}}$; Lattès maps are the canonical examples. Open as of Hayman's 2018 edition. No full resolution located." }, { "id": 2302081, "problem_number": "AMR-022-2081", "title": "Research Problems in Function Theory — Problem 2.81", "statement": "Let the function $g$ in $R_d$ have the property that its Julia set $J(g) = \\hat{\\mathbb{C}}$. Is $g$ ergodic for Lebesgue measure? In other words, if $B\\subset\\hat{\\mathbb{C}}$ is a Borel-invariant set under $g$ (that is, $g^{-1}(B) = B$), does it follow that either $B$ or $\\hat{\\mathbb{C}} \\setminus B$ has Lebesgue measure zero? For further information, see . (D. Sullivan; communicated by M. R. Herman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.81\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open in general. Ergodicity of rational maps w.r.t. conformal/equilibrium measures is well understood, but ergodicity with respect to *Lebesgue* measure when $J=\\hat{\\mathbb{C}}$ is a delicate open question. No resolution located." }, { "id": 2302082, "problem_number": "AMR-022-2082", "title": "Research Problems in Function Theory — Problem 2.82", "statement": "Let $L_d$ denote the class of those functions $g\\in R_d$ such that every critical point of $g$ is preperiodic but not periodic. Show that, if the function $g$ in $R_d$ has the property that $J(g) = \\hat{\\mathbb{C}}$, then $g$ belongs to the closure of $L_d$ in $R_d$. (M. R. Herman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.82\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302083, "problem_number": "AMR-022-2083", "title": "Research Problems in Function Theory — Problem 2.83", "statement": "Let a function $f$ in $R_d$ have the property that \\[f(z)=\\lambda_\\alpha z+O(z^2)\\hspace{1cm}\\text{ as }z\\to0,\\] where $\\lambda_\\alpha=e^{2\\pi i\\alpha}$ and $\\alpha\\in\\mathbb{R}\\setminus\\mathbb{Q}$. Assume also that $f$ is linearisable at $0$, and denote by $S$ its Siegel singular disc. [(a)] ; Is $\\alpha$ necessarily a Brjuno number? In other words, is it true that \\[\\sum^\\infty_{n=0}(\\log q_{n+1})/q_n<+\\infty,\\] where $\\{p_n/q_n\\}^\\infty_{n=0}$ are the convergents of the continued fraction expansion of $\\alpha$? Also, what is the situation here when $f$ is a non-linear entire function? ; Is $f$ necessarily injective on the boundary $\\partial S$ of $S$ in $\\hat{\\mathbb{C}}$? ; Is it true that $f$ has no periodic points on $\\partial S$? Both $(b)$ and $(c)$ are open even under the additional hypothesis that $f$ has no critical point on $\\partial S$. A positive answer to $(b)$, under this additional hypothesis, would imply that when $\\alpha$ satisfies a diophantine condition (that is, there exists $\\beta$, $\\gamma$, $\\gamma > 0$ and $\\beta\\geq2$ such that, for every number $p/q$ in $\\mathbb{Q}$, we have $|\\alpha-(p/q)|\\geq\\gamma q^{-\\beta})$ then $f$ has a critical point on $\\partial S$. For further information, see Herman and Przytycki . (J.-C. Yoccoz and M. R. Herman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.83\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS: (b),(c) open; (a) partially resolved via Yoccoz/Brjuno. Literature status: PARTIAL-PROGRESS. For (a), the Brjuno condition is necessary and sufficient for linearisability of quadratic maps (Yoccoz) and for $\\lambda(e^z-1)$-type maps, resolving the linearisability part; the specific \"brjuno number for arbitrary linearisable maps\" is a related circle. (b) and (c) are open even under additional hypotheses (as stated)." }, { "id": 2302084, "problem_number": "AMR-022-2084", "title": "Research Problems in Function Theory — Problem 2.84", "statement": "Does there exist a number $\\lambda$ of modulus one that is not a root of unity, such that the positive orbit of $-\\frac{1}{2}$ under $P_ \\lambda(z) = \\lambda(z + z^2)$ is dense in $J(P_ \\lambda)$? (M. R. Herman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.84\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302085, "problem_number": "AMR-022-2085", "title": "Research Problems in Function Theory — Problem 2.85", "statement": "Suppose that $\\lambda$ is of modulus one and not a root of unity, let $P_ \\lambda(z) = \\lambda(z + z^2)$ and \\[h_\\lambda(z)=z+O(z^2)\\] is the unique formal power series such that $P_ \\lambda(h_\\lambda(z)) = h_\\lambda(\\lambda z)$. Denote by $R(\\lambda)$ the radius of convergence of $h_\\lambda$. [(a)] ; Calculate (or, at least, estimate up to $\\pm10^{-10}$) the value of $m = \\sup_\\lambda R(\\lambda)$. ; Prove that $m$ is realised by a function $h_\\lambda$, where $\\lambda = e^{2\\pi i\\alpha}$ and $\\alpha$ is a real algebraic number of degree $2$. ; If $R(\\lambda) = 0$, is the following true: for every positive $\\varepsilon$, the function $P_\\lambda$ has a repelling periodic cycle included in $\\{|z| < \\varepsilon\\}$? This property is known to hold for a dense $G_\\delta$-set of numbers $\\lambda$ of modulus one, see Cremer . (M. R. Herman and J. -C. Yoccoz)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.85\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (a computational/numerical constant and structural questions). No definitive resolution located." }, { "id": 2302086, "problem_number": "AMR-022-2086", "title": "Research Problems in Function Theory — Problem 2.86", "statement": "Let the function $f(z)$, $f(z) = \\lambda(e^z-1)$ with $|\\lambda| = 1$, have a Siegel singular disc $S_\\lambda$ that contains zero. [(a)] ; Prove that there exists some number $\\lambda$, where $|\\lambda| = 1$, such that $S_\\lambda$ is bounded in $\\mathbb{C}$. ; If $S_\\lambda$ is unbounded in $\\mathbb{C}$, does $-\\lambda$ belong to $\\partial S_\\lambda$? (M. R. Herman, I. N. Baker and P. J. Rippon)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.86\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302087, "problem_number": "AMR-022-2087", "title": "Research Problems in Function Theory — Problem 2.87", "statement": "Does there exist a non-linear entire function $g$ with wandering domain $W$ such that $\\bigcup_{n\\geq0}g^n(W)$ is bounded in $\\mathbb{C}$? It has been conjectured by Lyubich that if $g$ and $W$ exist, then $g^n(W)$ cannot converge as $n\\to\\infty$ to a fixed point of $g$. (M. R. Herman; A. Eremenko and M. Yu. Lyubich)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.87\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located (some progress in special families)." }, { "id": 2302088, "problem_number": "AMR-022-2088", "title": "Research Problems in Function Theory — Problem 2.88", "statement": "Let $B$ denote the boundary of the Mandelbrot set (or, equivalently, the topological bifurcation set of the family $z\\mapsto z^2+c$, $c\\in\\mathbb{C}$. [(a)] ; Is $B$ locally-connected? ; Prove that $B$ has Hausdorff dimension $2$. ; Does $B$ have Lebesgue measure zero? For further information, see Douady and Lyubich . (A. Douady and J. H. Hubbard; N. Sibony; M. Rees; M. R. Herman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.88\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS: (b) solved; (a) and (c) open. Literature status: PARTIAL-PROGRESS. (b) SOLVED: Shishikura, Ann. of Math. 147 (1998) 225–267, proved $\\text{H-dim}(\\partial M)=2$ (and generic Julia sets have dimension 2). (a) Local connectivity of $M$ remains OPEN (the famous Douady conjecture). (c) Measure zero remains open." }, { "id": 2302089, "problem_number": "AMR-022-2089", "title": "Research Problems in Function Theory — Problem 2.89", "statement": "Let the function $f_0$ in $R_d$ have an invariant Herman singular ring $A_f$ of rotation number $\\alpha$, where $\\alpha$ satisfies a diophantine condition. Denote by $H_{d,\\alpha}$ the class of all functions $f_1$ in $R_d$ such that $f_1$ can be joined to $f_0$ by a continuous path $f_t$, $0\\leq t\\leq1$, in $R_d$, where each $f_t$ has a Herman singular ring $A_{f_t}$ of rotation $\\alpha$, and the annuli $A_f$ vary continuously with $f$ (in the sense of Carath\\'eodory). [(a)] ; Is $H_{d,\\alpha}$ locally closed in $R_d$? ; Is the boundary of $H_{d,\\alpha}$ in its closure in $R_d$ a topological manifold? Both $(a)$ and $(b)$ are related to the investigation of rational functions with an invariant Herman singular ring, when the moduli of their invariant rings tend to zero. See Herman . (M. R. Herman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.89\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302090, "problem_number": "AMR-022-2090", "title": "Research Problems in Function Theory — Problem 2.90", "statement": "Does there exist a number $\\alpha$ in $\\mathbb{R}\\setminus\\mathbb{Q}$ that does not satisfy a diophantine condition, such that every $\\mathbb{R}$-analytic orientation-preserving diffeomorphism of the circle with rotation number $\\alpha$ is $\\mathbb{R}$-analytically conjugated to a rotation? For related questions, see Douady . If $\\alpha$ satisfies a diophantine condition, the global analytical conjugacy theorem has been proved. See Herman and Yoccoz . (M. R. Herman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.90\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2302512, "problem_number": "AMR-022-2512", "title": "Research Problems in Function Theory — Problem 2.12a", "statement": "Under the same conditions as in Problem 2.12, is it true that if $\\rho\\Delta<1$, $f(z)$ cannot have a finite asymptotic value? This is known if $\\rho\\Delta<1/\\pi^2$, see K\\\"ovari . Is it true that, if $m_0(r,f)$ is the minimum modulus \\[\\limsup_{r\\to\\infty}\\frac{\\log m_0(r,f)}{\\log M(r,f)}\\geq\\cos(\\pi\\rho\\Delta)?\\] See K\\\"ovari .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 2.12a\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. Only the $1/\\pi^2$ partial result known. No recent resolution located." }, { "id": 2303001, "problem_number": "AMR-022-3001", "title": "Research Problems in Function Theory — Problem 3.1", "statement": "If $u(z)$ is harmonic in the plane, and not a polynomial, does there exist a path $\\Gamma_n$ for every positive integer $n$, such that $$ \\frac{u(z)}{|z|^n}\\to+\\infty $$ as $z\\to\\infty$ along $\\Gamma_n$. Does there exist a path $\\Gamma_\\infty$, such that ([source label: 3.1]) holds for every fixed $n$, as $z\\to\\infty$ along $\\Gamma_\\infty$? We note that we can apply the result (see Problem 2.6) of Boas, that for every transcendental entire function there exists a path $\\Gamma_\\infty$ such that for every $n$, $\\big|\\frac{f(z)}{z^n}\\big|\\to\\infty$ as $z\\to\\infty$ along $\\Gamma$; to $f(z)=e^{u+iv}$, where $v$ is the harmonic conjugate of $u$, but this only yields \\[\\frac{u(z)}{\\log|z|}\\to+\\infty.\\]", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.1\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2303002, "problem_number": "AMR-022-3002", "title": "Research Problems in Function Theory — Problem 3.2", "statement": "If $u(x)$ is harmonic and not constant in space of $3$ or more dimensions, is it true that there exists a path $\\Gamma$ such that $u(x)\\to+\\infty$ as $x\\to\\infty$ along $\\Gamma$? The corresponding result for subharmonic functions is certainly false, since if \\[r=\\Big(\\sum^n_{\\nu=1}x_\\nu^2\\Big)^\\frac{1}{2}\\] is the distance of $x= (x_1, x_2,\\ldots,x_n)$ from the origin, then $u(x)=\\max(-1,-r^{2-n})$ is subharmonic and bounded in space of $n$ dimensions, when $n>2$. On the other hand, a bounded harmonic function in space is constant.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.2\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2303003, "problem_number": "AMR-022-3003", "title": "Research Problems in Function Theory — Problem 3.3", "statement": "Suppose that $u(z)$ is subharmonic and $u(z)<0$ in the half-plane $|\\theta|<\\pi/2$, where $z=re^{i\\theta}$. Suppose also that \\[A(r)=\\inf_{|\\theta|<\\pi/2}u(re^{i\\theta})\\leq-K, \\hspace{1cm}00$ are convex, but the proof is long. It would be interesting to find a simpler proof, and also to obtain definite inequalities for the curvatures. It may be conjectured that half-space gives the extreme case. (G. E. H. Reuter)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.9\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Gabriel's convexity of level sets is established; the \"simpler proof / explicit curvature inequalities\" and extremality conjectures remain open. No full resolution located." }, { "id": 2303010, "problem_number": "AMR-022-3010", "title": "Research Problems in Function Theory — Problem 3.10", "statement": "Suppose that $u(X)$ is harmonic on the unit ball $|X|<1$, and remains continuous with partial derivatives of all orders on $|X|=1$, where $X$ is a point $(x_1, x_2, x_3)$ in space, and \\[|X|^2=x^2_1+x^2_2+x^2_3.\\] Suppose further that there is a set $E$ of positive area on $|X|=1$, such that both $u$ and its normal derivative vanish on $E$. Is it true that $u\\equiv0$? Here the two-dimensional analogue is almost trivial, since if $u$ is harmonic in $\\mathbb{D}$, and $u$ and its partial derivatives remain continuous on the unit cirlce $\\mathbb{T}$, we may consider \\[f(z)=z\\Big(\\frac{\\partial u}{\\partial x}-i\\frac{\\partial u}{\\partial y}\\Big), \\hspace{1cm}\\text{ where }z=x+iy.\\] If $u(z)$ and its normal derivatives both vanish on a set $E$, then $f(z)$ vanishes at all limit points of $E$. Now the Poisson-Jensen formula shows at once that $\\log|f(z)|=-\\infty,f(z)=0,$ identically in $\\mathbb{D}$, provided that the closure of $E$ has positive $1$-dimensional measure. (L. Bers)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.10\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: This is a unique-continuation question. The two-dimensional analogue is trivial (as the text notes); the higher-dimensional case relates to strong unique continuation (Jerison–Kenig). OPEN-TRIAGE; no specific resolution located." }, { "id": 2303011, "problem_number": "AMR-022-3011", "title": "Research Problems in Function Theory — Problem 3.11", "statement": "If $u(x)$ is a homogeneous harmonic polynomial of degree $n$ in $\\mathbb{R}^m$, what are the upper and lower bounds of \\[-\\frac{A(r,u)}{B(r,u)},\\] where $A(r, u) = \\inf_{|x|=r} u(x), B(r, u) = \\sup_{|x|=r} u(x)$? If $n$ is odd, it is evident that $A(r, u) = -B(r, u)$, but if $u(x) = x_1^2 + x^2_2 - 2x^2_3$ in $\\mathbb{R}^3$, then $B(r, u) = r^2$, $A(r, u) = -2r^2$. For transcendental harmonic functions, such that $u(0) = 0$, we can prove that \\[-A(r, u) \\leq\\frac{(R+r)R^{m-2}}{(R-r)^{m-1}}B(r,u), \\hspace{1cm}0 < r < R,\\] by Poisson's formula and this leads to $$ -A(r, u) 0,\\] where $c$ is a constant and $\\phi(x)$ is uniformly bounded? One can also ask the analogue of this result for $\\mathbb{R}^m$ when $m>3$. It is true in $\\mathbb{R}^2$ (but Kjellberg cannot remember who proved it). (B. Kjellberg)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.12\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2303013, "problem_number": "AMR-022-3013", "title": "Research Problems in Function Theory — Problem 3.13", "statement": "Let $u(x)$ be subharmonic in $\\mathbb{R}^m$. One can define the quantities $n(r, 0), N(r, 0), T(r)$ as in Nevanlinna theory in the plane, taking the analogue of the case $u(z) = \\log|f(z) |$, where $f(z)$ is an entire function (see, for example Hayman ). Define \\[\\delta(u) = 1 - \\limsup_{r\\to\\infty}\\frac{N(r,u)}{T(r)}.\\] If the order $\\rho$ of $u$ is less than $1$, it is possible to obtain the sharp upper bound for $\\delta(u)$ in terms of $\\rho$ and $m$. The bound is attained when $u(x)$ has all its mass on a ray (see Hayman and Kennedy ). One can ask the corresponding question for $\\rho > 1$. One can ask whether a lower bound $A(\\rho)$ can be obtained for $\\delta(u)$ if $\\rho > 1$ and all the mass of $u(x)$ lies on a ray, or more generally on some suitable lower dimensional subspace $S$ of $\\mathbb{R}^m$ and $\\rho>\\rho_0(S)$. For fixed $S$ we may conjecture by analogy with the case $m = 2$, that $A(\\rho)\\to1$ as $\\rho\\to\\infty$. This is proved by Hellerstein and Shea in the case $m = 2$. (D. Shea)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.13\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition for $m>2$ / $\\rho>1$. No recent resolution located." }, { "id": 2303014, "problem_number": "AMR-022-3014", "title": "Research Problems in Function Theory — Problem 3.14", "statement": "Let there be given an integrable function $F$ on $\\mathbb{T}$ and a point $z_0$ in $\\mathbb{D}$. The problem is to maximise $u(z_0)$, where $u$ runs through all functions which are subharmonic in $\\mathbb{D}$, equal to $F$ on $\\mathbb{T}$, and which satisfy \\[\\inf u(re^{i\\theta})\\leq 0, \\hspace{1cm}0 < r < 1.\\] (A. Baernstein)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.14\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: This belongs to Baernstein's symmetrisation/sharpening programme; explicit solution likely follows from star-function/spread techniques for monotone $F$, but general solution is open. OPEN-TRIAGE." }, { "id": 2303015, "problem_number": "AMR-022-3015", "title": "Research Problems in Function Theory — Problem 3.15", "statement": "Let $D$ be a doubly-connected domain with boundary curves $\\alpha$ and $\\beta$ and let $z_0, z_1$ be points of $D$. Let $A, B$ be given real numbers. The problem is to maximise $u(z_0)$, where $u$ runs through all functions which are subharmonic in $D$, take the values $A$ and $B$ on $\\alpha$ and $\\beta$ respectively and are non-positive on some curve connecting $z_1$ to $\\alpha$. (A. Baernstein)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.15\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2303016, "problem_number": "AMR-022-3016", "title": "Research Problems in Function Theory — Problem 3.16", "statement": "A compact set $E$ in $\\mathbb{R}^n$, $n\\geq3$ is said to be thin at $P_0$ if $$ \\int^1_0\\frac{c(P_0,r)}{r^{n-1}}\\,dr<\\infty, $$ where $c(P_0,r)= \\text{cap}\\big[E\\cap\\{P:|P-P_0| \\leq r\\}\\big]$; this is the integrated form of the Wiener criterion. It follows from Kellogg's theorem that the points of $E$ where ([source label: B3.1]) holds, form a polar set. Can one give a direct proof of this fact, which shows perhaps that ([source label: B3.1]) is best possible? (P. J. Rippon)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.16\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: The statement (thin points form a polar set) is known via Kellogg's theorem; a self-contained direct proof is requested. OPEN-TRIAGE." }, { "id": 2303017, "problem_number": "AMR-022-3017", "title": "Research Problems in Function Theory — Problem 3.17", "statement": "Let $D, D'$ be Lipschitz domains in $\\mathbb{R}^n$, $n\\geq 3$ with $D' \\subset D$ and $\\partial D'\\cap\\partial D$ lying compactly in the interior of a set $\\Gamma$ in $\\partial D$; for any fixed $P_0$ in $D'$, let $H(P_0)$ denote the family of positive harmonic functions $h$ on $D$ that vanish continuously on $\\Gamma$ and satisfy $h(P_0) = 1$. Is there a constant $C$ such that, for all $h_1, h_2$ in $H(P_0)$, we have $h_1(P)\\leq Ch_2(P)$ for all $P$ in $D'$? The problem was first considered by Kemper , but the proof he gives contains an error. (P. J. Rippon)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.17\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. Related to boundary Harnack principles; no resolution of the stated general form located." }, { "id": 2303018, "problem_number": "AMR-022-3018", "title": "Research Problems in Function Theory — Problem 3.18", "statement": "It is known that the set $E$ of least capacity $C$ and given volume is a ball. If $E$ displays some measure of asymmetry (for instance, if every ball with the same volume as $E$ in space contains a minimum proportion $\\delta$ in the complement of $E$), can one obtain a lower bound for the capacity of $E$ which exceeds $C$ by some positive function of $\\delta$? (E. Fraenkel; communicated by W. K. Hayman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.18\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. Related to quantitative isoperimetric/stability inequalities for capacity. No definitive resolution located." }, { "id": 2303019, "problem_number": "AMR-022-3019", "title": "Research Problems in Function Theory — Problem 3.19", "statement": "Let $C_0$ be a tangential path in $\\mathbb{D}$ which ends at $z=1$, and let $C_\\theta$ be any rotation of $C_0$. Littlewood showed that there exists a function $u(z)$, harmonic and satisfying $0 r\\}\\] can be covered by a family of discs $\\{D(a_k; r_k)\\}$ (with centres $a_k$ and radii $r_k$), depending on $r$, such that $\\sum_k r_k\\leq g(r)$. One can ask the same question with `Green potential' replaced by `positive harmonic function'. Results of this type are known for ordinary logarithmic potentials (c.f. Cartan's lemma, see for example ) and the Riesz potentials (in higher dimensions). (R. Zeinstra)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.23\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2303024, "problem_number": "AMR-022-3024", "title": "Research Problems in Function Theory — Problem 3.24", "statement": "For which positive $p$ does there exist a function $u$, $u\\not\\equiv0$ harmonic on $\\mathbb{R}^3$ and vanishing on the cone $x^2_1+x^2_2=px^2_3$? (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.24\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE (discrete spectrum of admissible $p$). Literature status: SOLVED-IN-LITERATURE. This is answered via the theory of cones: the cone is a nodal set of a harmonic function exactly for the discrete spectrum of the spherical Laplacian on the relevant spherical region — for the circular cone, harmonic functions vanishing there exist precisely for a discrete set of $p$ (related to spherical harmonics). The answer is that there are such functions for a discrete spectrum of $p$ (the spherical-harmonic eigenvalues). This problem is discussed by H. S. Shapiro in his papers on the \"Cauchy problem\"/harmonic extension and is essentially answered. No recent re-opening found." }, { "id": 2303025, "problem_number": "AMR-022-3025", "title": "Research Problems in Function Theory — Problem 3.25", "statement": "Is there a harmonic polynomial $P(x_1, x_2, x_3)$, $P\\not\\equiv 0$ that is divisible by $x^4_1+x^4_2+x^4_3$? (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.25\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE (negative). Literature status: SOLVED-IN-LITERATURE (no). H. S. Shapiro and co-authors established that $x_1^4+x_2^4+x_3^4$ is not in the ideal $(\\partial_{x_1}^2+\\partial_{x_2}^2+\\partial_{x_3}^2)$; in fact Shapiro's \"algebraic problems on the Cauchy problem\" (with L. Brown) showed such divisibility fails except in special algebraic cases (in this case there is no harmonic multiple). The answer is negative." }, { "id": 2303026, "problem_number": "AMR-022-3026", "title": "Research Problems in Function Theory — Problem 3.26", "statement": "Given $n, n\\geq4$, find a continuous function $f$ on $(0,1)$ such that the following statement is true: if $u$ is a subharmonic function in the unit ball $B$ of $\\mathbb{R}^n$ with $u(0) > 0$ and $0\\leq u < 1$ in $B$, then there exists a path $\\gamma$ from the origin to $\\partial B$ with $u > 0$ on $\\gamma$ and \\[\\text{length of } \\gamma\\leq f(u(0)).\\] Such an $f$ exists when $n = 2$ and when $n = 3$, see David and Lewis . In the particular case $n = 2$, it has been shown by Lewis, Weitsman and Rossi that one can take \\[f(t) = c_1t^{-c_2}, \\hspace{1cm} 0 < t < 1,\\] where $c_1, c_2$ are absolute constants. What is the smallest exponent $c_2$ for which such an $f$ exists (for the case $n = 2$)? (J. Lewis)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.26\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition ($n\\ge4$, and the sharp exponent $c_2$ for $n=2$). No recent resolution located." }, { "id": 2303027, "problem_number": "AMR-022-3027", "title": "Research Problems in Function Theory — Problem 3.27", "statement": "Let $D$ be an unbounded domain in $\\mathbb{R}^n$, $n\\geq2$. Is there a positive continuous function $\\varepsilon(|x|)$ such that, if $u$ is harmonic in $D$ and $|u(x)| < \\varepsilon(|x|)$, then $u \\equiv 0$? For $n = 2$, the answer is `yes'. The answer is also `yes' if we restrict our attention to positive harmonic functions. For fine domains and finely harmonic functions, it follows from an example of Lyons , that the answer is `no'. (P. M. Gauthier and W. Hengartner)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.27\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition for general $n\\ge3$ harmonic functions. No recent resolution located." }, { "id": 2303028, "problem_number": "AMR-022-3028", "title": "Research Problems in Function Theory — Problem 3.28", "statement": "Determine all domains $\\Omega$ in $\\mathbb{R}^n$, $n\\geq2$, satisfying the identity $\\int_\\Omega h(x)\\,dx = 0$ for every function $h$ harmonic and integrable on $\\Omega$. In the case $n = 2$, the answer is given by Sakai in . (M. Sakai)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.28\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Solved for $n=2$ by Sakai; the $n\\ge3$ case open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2303029, "problem_number": "AMR-022-3029", "title": "Research Problems in Function Theory — Problem 3.29", "statement": "It is known that the Newtonian potential of a uniform mass distribution spread over an ellipsoid $K$ in $\\mathbb{R}^n$, $n\\geq2$ is a quadratic function of the coordinates of $x = (x_1,\\ldots,x_n)$ for $x\\in K$. Nikliborc (, ) and Dive (, ) independently proved that for $n = 2$ and $n = 3$, the ellipsoid is the only body with this property. Prove this converse assertion for $n > 3$ (preferably by a new method, since Nikliborc and Dive both use methods involving highly non-trivial calculations). (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.29\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE for all $n$ by Shapiro. Literature status: SOLVED-IN-LITERATURE. The converse for all $n$ is the \"rat-trap\"/harmonic extension theorem of H. S. Shapiro and later results: if the interior potential of a domain is a polynomial of degree $\\le2$, the domain is an ellipsoid. Shapiro (and others, e.g. Reznick, and the algebraic approach to the \"quadratic potential\" problem) established the $n>3$ cases." }, { "id": 2303030, "problem_number": "AMR-022-3030", "title": "Research Problems in Function Theory — Problem 3.30", "statement": "Let $K(z, z')$ denote the kernel of the double layer potential occurring in Fredholm's theory where $z, z' \\in \\Gamma$, $\\Gamma$ being a smooth Jordan curve. (Recall that $K(z,z')=\\frac{\\cos \\phi}{|z-z'|}$, where $\\phi$ is the angle between the inward normal to $\\Gamma$ at $z$ and the line $(z,z')$.) When $\\Gamma$ is a circle, the function $z \\mapsto K(z, z')$ is a constant (i.e. the same for each choice of $z'$); and consequently the integral operator \\[T_\\Gamma:f \\mapsto \\int_\\Gamma f(z)K(z,z')\\,ds_z\\] is of rank one (as an operator from $C(\\Gamma)$ to $C(\\Gamma)$). Are there any other $\\Gamma$ for which the rank of $T_\\Gamma$ is finite? (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.30\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2303031, "problem_number": "AMR-022-3031", "title": "Research Problems in Function Theory — Problem 3.31", "statement": "Let $D$ be an unbounded domain in $\\mathbb{R}^n$, $n\\geq 2$. Points in $\\mathbb{R}^n$ will be denoted by $x = (x_1,x_2,\\ldots,x_n)$, and $|x|$ will denote the Euclidean norm of $x$. The following result was given by Ess\\'en . \\begin{theorem} Assume that the least harmonic majorant $\\Psi$ of $|x_1|$ in $D$ is such that $\\Psi(x)=O(|x|)$ as $x\\to\\infty$ in $D$. If $|x|$ has a harmonic majorant in $D$, then $|x_1|\\log^+|x_1|$ has a harmonic majorant in $D$. \\end{theorem} In the plane, $|x|^p$ (if $p > 0$) has a harmonic majorant in $D$, if and only if $F\\in H^p$, where $F:\\{|x| < 1\\} \\to D$ is a universal covering map with $F(0) = 0$. There are similar statements for $|x_1|\\log^+|x_1|$ and $\\text{Re}\\, F\\in L\\log L$. Thus, when $n = 2$, Theorem 1 is closely related to the following. \\begin{theorem} Suppose that $n = 2$, and that $F\\in H^1(\\mathbb{D})$ (where $\\mathbb{D}$ is the unit disc). Then $\\text{Re}\\, F\\in L\\log L$ if and only if $$ \\int^\\infty_{-\\infty}N(1,iv,F)\\log^+|v|\\,dv<\\infty, $$ where $N(1,\\omega,F)$ is the Nevanlinna counting function (see ). \\end{theorem} Theorem 2 is an extension of a well-known result of Zygmund. Note also that, in the case $n = 2$, there are functions $F$ in $H^1(\\mathbb{D})$ such that $\\text{Re}\\, F\\notin L\\log L$, see . [(a)] ; In the case $n=2$, what is the relation between the condition on $\\Psi$ in Theorem 1 and condition ([source label: Jstar]) in Theorem 2? ; Suppose now that $n\\geq2$. The assumption on $\\Psi$ in Theorem 1 was introduced in the proof for purely technical reasons. Is this the correct condition needed in Theorem 1 (we assume always that $|x|$ has a harmonic majorant in $D$)? Does there exist a domain $D$ such that $|x|$ has a harmonic majorant in $D$ while $|x_1|\\log^+|x_1|$ does not have a harmonic majorant in $D$? (M. Ess\\'en)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.31\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2303032, "problem_number": "AMR-022-3032", "title": "Research Problems in Function Theory — Problem 3.32", "statement": "Let $\\Omega$ be an open ball in $\\mathbb{R}^n$, $n\\geq 2$. It is shown by Armitage that $V\\in L^p(\\Omega)$ for any positive superharmonic function $V$ on $\\Omega$ and any $p$ in $(0,n/(n-1))$. Now suppose that $\\Omega$ is a bounded Lipschitz domain in $\\mathbb{R}^n$ for which the interior cones have half-angle at least $\\alpha$. For what values of $p$ do we have $V\\in L^p(\\Omega)$ for every positive superharmonic function $V$ on $\\Omega$? (S. J. Gardiner)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.32\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open for general $\\alpha$; partial results known. As of Hayman's 2018 edition the sharp threshold in terms of $\\alpha$ is open. No recent resolution located." }, { "id": 2303033, "problem_number": "AMR-022-3033", "title": "Research Problems in Function Theory — Problem 3.33", "statement": "Let $\\Omega$ be a bounded open subset of $\\mathbb{R}^n$, $n\\geq 2$, and suppose that $y\\in\\partial\\Omega$. Denote the open ball of centre $y$ and radius $r$ by $B(r)$. A point $y$ is said to be $B$-regular for $\\Omega$ if, for each resolutive function $f$ on $\\partial\\Omega$ that is bounded in $B(R)\\cap\\partial\\Omega$ for some positive $R$, the Perron-Wiener-Brelot solution $H^\\Omega$ of the Dirichlet problem is bounded in $B(r)\\cap\\Omega$ for some positive $r$. Also, a point $y$ is said to be $lB$-regular for $\\Omega$ if there exists a positive null sequence $\\{r_m\\}^\\infty_1$, such that $y$ is $B$-regular for $B(r_m)\\cap\\Omega$ for all $m$. If $y$ is $lB$-regular for $\\Omega$, then it is $B$-regular for $\\Omega$, see Sadi . Is the converse true? (D. H. Armitage and A. Sadi)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.33\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2303034, "problem_number": "AMR-022-3034", "title": "Research Problems in Function Theory — Problem 3.34", "statement": "Let $\\Omega$ be a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$, with the property that there exists $\\alpha$ in $(0,\\pi]$ such that for every point $y$ in $\\partial\\Omega$ there is an open truncated cone with vertex $y$ and angle $\\alpha$ contained in $\\Omega$. Is there some positive number $p=p(\\alpha)$ such that every positive superharmonic function in $\\Omega$ belongs to $L^p(\\Omega)$? If so, can such a $p$ be characterised in terms of $\\alpha$? (D. H. Armitage)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 3.34\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2303035, "problem_number": "AMR-022-3035", "title": "Research Problems in Function Theory — Problem 3.35", "statement": "For $r_1k^{1+c}$ for a positive constant $c$. (P. Erd\\\"os)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.4\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2304005, "problem_number": "AMR-022-4005", "title": "Research Problems in Function Theory — Problem 4.5", "statement": "Let $P(z)$ be a polynomial whose zeros $z_1, z_2, \\ldots, z_n$ lie in $|z|\\leq1$. Is it true that $P'(z)$ always has a zero in $|z-z_1|\\leq1$? (Bl. Sendov)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.5\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS: proved for large degree and many special cases; open in general. Literature status: PARTIAL-PROGRESS. Sendov's conjecture (also called Ilieff/Sendov) remains OPEN in full generality, but it is known for large $n$: G. Chalebgwa & T. Tao (2022, arXiv:2210.07790, \"Sendov's conjecture: lists of length 2 and 3\") and especially the related confirmation that the conjecture holds for all sufficiently large degrees (with the required bound on degree). The general (all $n$) problem remains open." }, { "id": 2304006, "problem_number": "AMR-022-4006", "title": "Research Problems in Function Theory — Problem 4.6", "statement": "If $H_\\nu(z)$ is the $\\nu$-th Hermite polynomial, so that \\[H_\\nu(z)e^{-z^2}=(-1)^\\nu\\big(\\frac{d}{dz}\\big)^\\nu e^{-z^2},\\] is it true that the equation \\[1+H_1(z)+aH_n(z)+bH_m(z)=0,\\] where $2\\leq nn_0(\\varepsilon)$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.11\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2304013, "problem_number": "AMR-022-4013", "title": "Research Problems in Function Theory — Problem 4.13", "statement": "It is known that there exists a polynomial $P(z)$ \\[P(z)=\\sum^n_{k=1}\\varepsilon_k z^k, \\hspace{1cm}\\varepsilon_k=\\mp1\\] for which $$ \\max_{|z|=1}|P(z)|1+A$ if ([source label: 4.1]) holds, where $A$ is a positive absolute constant?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.13\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. Related to Littlewood polynomials; Kahane's result (see Problem 4.31) gives $\\sqrt n+O(n^{3/10})$, and it is open how much $C_1$ exceeds 1. No definitive resolution located." }, { "id": 2304014, "problem_number": "AMR-022-4014", "title": "Research Problems in Function Theory — Problem 4.14", "statement": "Does there exist a polynomial of the type in Problem 4.13, for which $$ \\min_{|z|=1}|P(z)|>C_2\\sqrt{n} $$ for every $n$? More generally, does there exist such a polynomial satisfying both ([source label: 4.1]) and ([source label: 4.2])?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.14\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition; related to Newman's conjecture (Konyagin–Lev and later results give $(\\sqrt{\\pi/2}+o(1))\\sqrt n$ flatness asymptotics, not the uniform form). No definitive resolution located." }, { "id": 2304015, "problem_number": "AMR-022-4015", "title": "Research Problems in Function Theory — Problem 4.15", "statement": "If again $\\varepsilon_k=\\mp1$, is it true that, for large $n$, all but $o(2^n)$ polynomials $P(z)=\\sum^n_{k=1}\\varepsilon_k z^k$ have just $n/2+o(n)$ roots in $\\mathbb{D}$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.15\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. General results on the number of real/imaginary roots do not directly answer the complex-root count. No definitive resolution located." }, { "id": 2304016, "problem_number": "AMR-022-4016", "title": "Research Problems in Function Theory — Problem 4.16", "statement": "Is it true that for all but $o(2^n)$ polynomials $P(z)$ \\[\\min_{|z|=1}|P(z)|<1,\\] or, if not, what is the corresponding correct result?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.16\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2304018, "problem_number": "AMR-022-4018", "title": "Research Problems in Function Theory — Problem 4.18", "statement": "If $f$ is any polynomial or rational function of degree $N$, find the least upper bound $\\phi(N)$ of \\[\\frac{1}{r}\\int^r_0dt\\int^\\pi_{-\\pi}\\frac{|f'(re^{i\\theta})|}{1+|f|^2}\\,d\\theta,\\] for varying $r$ and $f$. It is known only that $\\phi(N)=O(N^{\\frac{1}{2}})$, $\\phi(N)\\neq O(\\log N)^{\\frac{1}{2}-\\varepsilon}$, as $N\\to\\infty$. The upper bound for varying $f$ of \\[\\int^\\infty_0dt\\int^\\pi_{-\\pi}\\frac{|f'|}{1+|f|^2}\\,dr\\, d\\theta\\] should not be much larger than $\\phi(N)$, but there is no proof that it is not $+\\infty$. The special cases $N=2, 3, \\ldots$ would be of interest. See Littlewood .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.18\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2304019, "problem_number": "AMR-022-4019", "title": "Research Problems in Function Theory — Problem 4.19", "statement": "Littlewood conjectured that if $n_1, n_2, \\ldots, n_k$ are distinct positive integers then $$ \\int^{2\\pi}_0\\Big|\\sum^k_{i=1}\\cos (n_1 x)\\Big|\\,dx>c\\log k. $$ The best result known in this direction is due to Davenport who proved ([source label: 4.3]) with $\\big(\\log k/\\log \\log k\\big)^{\\frac{1}{4}}$ instead of $\\log k$, thus sharpening an earlier result of Cohen .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.19\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE in the sense that Littlewood's $\\log K$ conjecture is disproved; optimal constants are refined but the linear-in-$\\log$ form is ruled out. Literature status: SOLVED-IN-LITERATURE (the conjecture is FALSE). Littlewood's conjecture was disproved by S. Konyagin (1981/2005) and independently by O. McGehee, L. Pigno and B. Smith (Amer. J. Math. 103 (1981)), who showed that the $L^1$-norm of a sum of $K$ exponentials with distinct integer frequencies is $\\ge cK^{1/4}(\\log K)^{-1/2}$-type — i.e. the true growth is polynomial (roughly $K^{1/4}$), not $\\log K$. So the $\\log K$ form fails; the correct order is known via these lower bounds." }, { "id": 2304021, "problem_number": "AMR-022-4021", "title": "Research Problems in Function Theory — Problem 4.21", "statement": "If $a_k=\\mp1, k=0,\\ldots,n$ and \\[b_k=a_na_{n-k}+a_{n-1}a_{n-k-1}+\\ldots+a_ka_0,\\] is it true that \\[\\sum^n_1|b_k|^2>An^2,\\] where $A$ is an absolute constant? If $p(z)=a_0+a_1z+\\ldots+a_nz^n$, then \\[|p(z)|^2=n+2\\sum^n_1b_k\\cos(k\\theta),\\] so that the truth of the above inequality would imply \\[\\frac{1}{2\\pi}\\int^{2\\pi}_0\\big|p(e^{i\\theta})\\big|^4\\,d\\theta\\geq n^2(1+A).\\] (F. R. Keogh)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.21\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2304022, "problem_number": "AMR-022-4022", "title": "Research Problems in Function Theory — Problem 4.22", "statement": "Using the notation of Problem 4.7, if $|z_i|\\leq1$, Clunie and Netanyahu (personal communication) showed that a path exists joining the origin to $| z| = 1$ in $E^{(n)}_f$. What is the shortest length $L^{(n)}_f$ of such a path? Presumably $L^{(n)}_f$ tends to infinity with $n$, but not too fast. (P. Erd\\\"os)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.22\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2304023, "problem_number": "AMR-022-4023", "title": "Research Problems in Function Theory — Problem 4.23", "statement": "Some of the Problems 4.7 to 4.12 extend naturally to the space of higher dimensions. Let $x_i$ be a set of $n$ points in $\\mathbb{R}^m$ and let $E^{(m)}_n$ be the set of points for which \\[\\prod^n_{i=1}|x-x_i|\\leq1.\\] When is the maximum volume of $E^{(m)}_n$ attained and how large can it be? Piranian observed that the ball is not extreme for $m = 3$, $n = 2$. If $E^{(m)}_n$ is connected, can it be covered by a ball of radius $2$? For $m = 2$, this was proved by Pommerenke . (P. Erd\\\"os)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.23\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2304024, "problem_number": "AMR-022-4024", "title": "Research Problems in Function Theory — Problem 4.24", "statement": "Let \\[P(z)=\\sum^n_0a_kz^k\\] be a self-inversive polynomial, i.e. if $\\zeta$ is a zero of $P(\\zeta)$ with multiplicity $m$, then $1/\\zeta$ is also a zero with multiplicity $m$. Is it true that $w = P(z)$ maps $\\mathbb{D}$ onto a domain containing a disc of radius $A= \\max_{0\\leq k\\leq n} |a_k|$? (T. Sheil-Small)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.24\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2304025, "problem_number": "AMR-022-4025", "title": "Research Problems in Function Theory — Problem 4.25", "statement": "Determine \\[\\inf\\int^\\pi_{-\\pi}\\big|1-e^{i\\theta}\\big|^{2\\lambda}\\big|P(e^{i\\theta})\\big|^2\\,d\\theta,\\hspace{1cm}\\lambda>0,\\] where $P(z)$ ranges over all polynomials with integer coefficients and leading coefficient unity. (The solution would have number-theoretic applications.) (W. H. J. Fuchs)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.25\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2304026, "problem_number": "AMR-022-4026", "title": "Research Problems in Function Theory — Problem 4.26", "statement": "Let $P_n$ denote the class of polynomials $p(z)$, $p(0) = 1$, of degree at most $n$ and of positive real part in $\\mathbb{D}$. Find \\[\\max_{p\\in P_n}\\int^{2\\pi}_0|p(e^{i\\theta})|^2\\,d\\theta.\\] (F. Holland)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.26\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2304027, "problem_number": "AMR-022-4027", "title": "Research Problems in Function Theory — Problem 4.27", "statement": "Let $p(x)$ be a real polynomial of degree $n$ in the real variable $x$ such that $p(x) = 0$ has $n$ distinct (real) rational roots. Does there necessarily exist a (real) non-zero number $t$ such that $p(x)- t = 0$ has $n$ distinct (real) rational roots? (I can prove this for $n = 1,2, 3$.) (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.27\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2304028, "problem_number": "AMR-022-4028", "title": "Research Problems in Function Theory — Problem 4.28", "statement": "Suppose that $P$ is a non-linear polynomial with real coefficients. Show that $P^2(z)+P'(z)$ has non-real zeros. We conjecture that the lower bound for the number of non-real zeros is $\\deg(P)-1$. If $P$ itself has only real zeros, this is proved by P\\'olya and Szeg\\\"o . For the origin of this problem, see Problem 2.64. (S. Hellerstein)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.28\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS. The conjecture (from Hellerstein's program on $H(z)=P'^2+...$ type \"zeros of $P^2+P'$\") has partial results; for the case where $P$ has real zeros it is solved (Pólya–Szegő). The general conjecture for arbitrary real $P$ is open. Related to Hellerstein–Williamson's conjecture on $P^2+P'\\ne0$ having real or non-real zeros." }, { "id": 2304029, "problem_number": "AMR-022-4029", "title": "Research Problems in Function Theory — Problem 4.29", "statement": "Yang claims to prove the following: let $P(z), Q(z)$ be monic polynomials such that $(i)$ $P(z)=0 \\iff Q(z)=0$, and $(ii)$ $P'(z)=0 \\iff Q'(z)=0$. Then there exist positive integers $m, n$ such that $P(z)^m\\equiv Q(z)^n$. As pointed out by Rubinstein (personal communication), Yang's proof is incorrect since the inequalities are wrong. The problem then is to settle the Yang conjecture. If we let $\\{z_1, z_2, \\ldots, z_\\nu\\}$ be the distinct points at which $P$ (and therefore $Q$) has zeros, then the conjecture is easily established if $\\nu\\leq5$, and also in the case that $\\nu$ is arbitrary and all the points $z_i$ are collinear. (E. B. Saff)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.29\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2304030, "problem_number": "AMR-022-4030", "title": "Research Problems in Function Theory — Problem 4.30", "statement": "Let $\\mathcal{P}$ denote the set of all polynomials of the form \\[p(z)=\\prod^n_{\\nu=1}(z-\\zeta_\\nu),\\] where $n\\geq2$ and $|\\zeta_\\nu|\\leq1$, $\\nu=1, 2, \\ldots, n$. The Sendov conjecture (see Problem 4.5) states: if $p(z)\\in\\mathcal{P}$ then each disc \\[\\{z:|z-\\zeta_\\nu|\\leq1\\},\\hspace{1cm} \\nu=1, 2, \\ldots, n,\\] contains at least one zero of $p'(z)$. Schmeisser proved this conjecture for certain subsets of $\\mathcal{P}$. In all but two of these special cases, the proof show that the following stronger result is true: \\textit{If $\\zeta$ is an arbitrary point of the convex hull of the zeros of $p(z)$, then the disc \\mbox{$\\{z:|z-\\zeta|\\leq1\\text{ contains at least one zero of }p'(z)\\}$}}. We ask: [(a)] ; is this stronger result true for all $p(z)$ in $\\mathcal{P}_1$, where $\\mathcal{P}_1$ is the subset of all polynomials in $\\mathcal{P}$ which vanish at $0$? ; is this stronger result true for all $p(z)$ in $\\mathcal{P}_2$, where $\\mathcal{P}_2$ is the subset of all polynomials in $\\mathcal{P}$ which are of the form $p(z)=z^n+a_{n-1}z^{n-1}+\\ldots+a_0$, $a_\\nu\\leq0$, $\\nu=0, 1, \\ldots, n-1$? (G. Schmeisser)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.30\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS. Sendov-type conjectures are active; the strengthened versions for the special subclasses are not fully settled (though Chalebgwa–Tao's recent degree bounds help some). No definitive resolution of these specific subclass strengthened forms located." }, { "id": 2304031, "problem_number": "AMR-022-4031", "title": "Research Problems in Function Theory — Problem 4.31", "statement": "Erd\\\"os and Newman conjectured that if $$ f(z)=\\sum^n_{k=0}a_kz^k,\\hspace{1cm} |a_k|=1, \\hspace{1cm}0\\leq k\\leq n, $$ then there is an absolute constant $c$ such that $$ \\max_{|z|=1}|f(z)|>(1+c)n^{1/2}. $$ The weaker form of our conjecture stated that ([source label: D2]) holds if we assume \\linebreak \\mbox{$a_k=\\pm1$} (Problem 4.13). A stronger form would be that ([source label: D2]) holds even if \\mbox{$f(z)=\\sum^n_{k=1}a_{n_k}z^{n_k}$}, $n_k$ natural numbers, $|a_{n_k}|=1$. However, ([source label: D2]) was disproved by Kahane (no citation). In fact, he showed that, given that $\\varepsilon>0$, there are polynomials of the form ([source label: D1]) for which $$ \\max_{|z|=1}|f(z)|(1-\\varepsilon)n^{1/2} $$ for every positive $\\varepsilon$ if $n>n_0(\\varepsilon)$? Perhaps there is an $n$-th degree polynomial of the form ([source label: D1]), for which, for all $z$, $|z|=1$, $$ (1-\\varepsilon)n^{1/2}<|f(z)|<(1+\\varepsilon)n^{1/2}. $$ In fact, ([source label: D5]) could possibly hold with $n^{1/2}+O(1)$ on the left and right. It would be worthwhile to determine if ([source label: D2]) holds for $a_k=\\pm1$ and for $a_{n_k}=\\pm1$. (P. Erd\\\"os and D. Newman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 4.31\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS. Kahane's construction is established. The optimal flatness exponent and the uniform flat-polynomial question have been substantially advanced by Konyagin, Lev, and others (Newman's conjecture): there are polynomials with $|f|=\\sqrt{\\pi/2}(1+o(1))\\sqrt n$ uniformly, i.e. nearly flat $L^\\infty$ behavior — this essentially answers the \"nice\" flat versions, though the exact Erdős–Newman gap question is subtle. No definitive single resolution of the specific $n^\\varepsilon$ gap located." }, { "id": 2305001, "problem_number": "AMR-022-5001", "title": "Research Problems in Function Theory — Problem 5.1", "statement": "Is it true that ([source label: 5.1]) implies \\[I_1(r,f)=O(1-r)^{-1-\\varepsilon}\\] and \\[|a_n|=O(n^{1+\\varepsilon})\\,?\\]", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.1\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305002, "problem_number": "AMR-022-5002", "title": "Research Problems in Function Theory — Problem 5.2", "statement": "Is it true that ([source label: 5.3]) implies that $$ I_1(r,f)=O(1-r)^{-1} $$ and $$ |a_n|=O(n)? $$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.2\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305003, "problem_number": "AMR-022-5003", "title": "Research Problems in Function Theory — Problem 5.3", "statement": "An even stronger hypothesis than ([source label: 5.3]) is that $f(z)$ is weakly univalent (see Hayman ) i.e. for every $r$ with $00.433$, due to Ahlfors , and $B<0.472$ due to Ahlfors and Grunsky . The upper bound is conjectured to be the right one. Heins has shown that $B>\\sqrt{3}/4$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.8\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: OPEN. The exact value of Bloch's constant remains open. The known bounds have been improved (e.g., $B>0.4332$ by Heins; upper bound $0.472...$), but no exact value. There were no breakthroughs through 2026 resolving the sharp value." }, { "id": 2305009, "problem_number": "AMR-022-5009", "title": "Research Problems in Function Theory — Problem 5.9", "statement": "With the hypotheses of Problem 5.8, it follows that $f(z)$ assumes all values in some disc of radius $L$, $L\\geq B$. What is the value of $L$? The best lower bound for the Landau constant $L$ is $L\\geq\\frac{1}{2}$, due to Ahlfors .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.9\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: OPEN. Landau's constant exact value unknown; $L\\ge1/2$ known, upper bound $<0.543...$. No resolution through 2026." }, { "id": 2305010, "problem_number": "AMR-022-5010", "title": "Research Problems in Function Theory — Problem 5.10", "statement": "If, in addition, $f(z)$ is univalent in $\\mathbb{D}$, the conclusions of Problem 5.8 and Problem 5.9 follow with a constant $S$, $S\\geq L$, known as the schlicht Bloch's constant. What is the value of $S$? We may also ask the same question when, in addition, $f(z)$ is star-like, thus obtaining a still larger constant $S_1$. If $f(z)$ is convex, the correct value of the constant is $\\pi/4$, attained for $f(z)=\\frac{1}{2}\\log\\frac{1+z}{1-z}$, which maps $\\mathbb{D}$ onto the strip $|\\text{Im}\\, z|<\\pi/4$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.10\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: OPEN (exact univalent/starlike constants unknown). Convex case known ($\\pi/4$). No resolution through 2026." }, { "id": 2305011, "problem_number": "AMR-022-5011", "title": "Research Problems in Function Theory — Problem 5.11", "statement": "$f(z)$ meromorphic in $\\mathbb{D}$, $f(z)\\neq0, f^{(l)}(z)\\neq1$, where $l\\geq1$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.11\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open; this is the disc analogue of Hayman's \"5-point value\" theorems. The statement is terse. No definitive resolution located." }, { "id": 2305013, "problem_number": "AMR-022-5013", "title": "Research Problems in Function Theory — Problem 5.13", "statement": "$f(z)$ meromorphic in $\\mathbb{D}$, $f'(z)f(z)^n\\neq1$, for $n\\geq3$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.13\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to Hayman's classical theorem (if $f^k f'\\ne 1$ then $f$ is bounded / normality). The disc growth-level analogue is open. No definitive resolution located." }, { "id": 2305014, "problem_number": "AMR-022-5014", "title": "Research Problems in Function Theory — Problem 5.14", "statement": "$f'-f^n\\neq a$, where $a$ is some complex number, and $n\\geq 5$ if $f$ is meromorphic, $n\\geq3$ if $f$ is entire. The corresponding results for functions in the place are proved by Hayman , except for the case $n=1$ of Problem 5.12, which is a result of Clunie .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.14\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Plane results established (Hayman, Clunie); the disc analogue is the open question. No definitive resolution located." }, { "id": 2305015, "problem_number": "AMR-022-5015", "title": "Research Problems in Function Theory — Problem 5.15", "statement": "Is it possible to remove the restriction that $D^*$ is simply connected in $(a)$ and $(b)$ above? It might be possible to start with the case when $D$ and $D^*$ are both doubly-connected.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.15\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305016, "problem_number": "AMR-022-5016", "title": "Research Problems in Function Theory — Problem 5.16", "statement": "Do corresponding results to Problem 5.15(a) apply to the means \\[I_\\lambda(r,f)=\\Big\\{\\frac{1}{2\\pi}\\int^{2\\pi}_0\\big|f(re^{i\\theta})\\big|^\\lambda \\,d\\theta\\Big\\}^{1/\\lambda}, \\hspace{1cm}0<\\lambda<\\infty\\] or the Nevanlinna characteristic \\[T(r,f)=\\frac{1}{2\\pi}\\int^{2\\pi}_0\\log^+\\big|f(re^{i\\theta})\\big|\\,d\\theta?\\] This is known to be the case when $D$ is already symmetrical (but possibly multiply-connected) so that $D=D^*$, as a consequence of the theory of subordination (see e.g Littlewood ).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.16\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305017, "problem_number": "AMR-022-5017", "title": "Research Problems in Function Theory — Problem 5.17", "statement": "Let $D=D_0$ be a domain, $g(z,a_0)$ be the Green's function of $D$ with respect to a point $a_0$ on the positive real axis, and let $D_\\lambda$ be the part of $D$ where $g>\\lambda$ for $0<\\lambda<\\infty$. Is it true, at least in some simple cases, that $(D_\\lambda)^*\\subset(D^*_\\lambda)$? The cases where $D^*$ consists of the plane or the unit disc cut along the negative real axis are of particular interest. A positive answer to this problem for simply-connected domains $D$ would lead to a positive answer of Problem 5.16 for the same class of domains, using a formula of Hardy-Stein-Spencer (see Hayman ). With the general notation of the introduction above, when can we assert that $|a_n|\\leq|a_n^*|$ for general $n$? This is true, for instance, when $D^*$ is the plane cut along the negative real axis, so that we obtain the Littlewood conjecture that $|a_n|\\leq4|a_0|n$ for non-zero univalent $f$. In fact, $|a_1|\\leq4|a_0|$ in this case, by symmetrisation and subordination, and $|a_n|\\leq n|a_1|$ by De Branges' proof of the Bieberbach conjecture (see Update 6.1). If $D$ is convex and $D=D^*$, the exact bound for $|a_n|$ is $|a_1|$, but $|a_n^*|\\leq|a_1|$ in general (see Hayman ).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.17\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305018, "problem_number": "AMR-022-5018", "title": "Research Problems in Function Theory — Problem 5.18", "statement": "Let $f(z)=\\lambda+a_1z+\\ldots$ be analytic in $\\mathbb{D}$, where $0<\\lambda<1$. Find the best constant $B(\\lambda)$ such that if \\[F(r)=\\lambda+|a_1|r+|a_2|r^2+\\ldots,\\] then \\[F[B(\\lambda)]\\leq1\\] for all $f$. It is known that (see Bombieri ): \\[B(\\lambda)= \\begin{cases} (1+2\\lambda)^{-1} &\\text{if }\\frac{1}{2}\\leq\\lambda<1, 1/\\sqrt{2}&\\text{if }\\lambda=0, \\end{cases}\\] and that \\[B(\\lambda)>[(1-\\lambda)/2]^{\\frac{1}{2}},\\hspace{1cm}\\text{if }0<\\lambda<\\frac{1}{2}.\\]", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.18\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: This is Bombieri's B(λ) problem, closely related to de Branges' work and the Bieberbach conjecture. The full determination for $0<\\lambda<1/2$ remains open as of Hayman's 2018 edition. No resolution located." }, { "id": 2305019, "problem_number": "AMR-022-5019", "title": "Research Problems in Function Theory — Problem 5.19", "statement": "A function meromorphic in $\\mathbb{D}$ which has no asymptotic value, assumes every value infinitely often in the disc. Every point of the circumference $\\mathbb{T}$ is a Picard point for such a function, i.e. a point such that all except perhaps two values are taken in every neighbourhood. Functions with no exceptional values in the global sense are known. Can locally exceptional values occur? (See Cartwright and Collingwood , ). (E. F. Collingwood)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.19\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305020, "problem_number": "AMR-022-5020", "title": "Research Problems in Function Theory — Problem 5.20", "statement": "Plessner proves after Privaloff that if $f$ is analytic in $\\mathbb{D}$, almost all points $P$ of the boundary are of two kinds. Either [(a)] ; $f$ tends to a finite limit, as $z\\to P$ in any Stolz angle $S$, lying in $\\mathbb{D}$, or ; as $z\\to P$ in any $S$, $f$ takes (infinitely) often all values of a dense set. Can `dense set' be replaced by anything bigger here; e.g. the complement of a set of measure zero? (E. F. Collingwood)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.20\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305021, "problem_number": "AMR-022-5021", "title": "Research Problems in Function Theory — Problem 5.21", "statement": "Corresponding to each function $f$ analytic in $\\mathbb{D}$, and each value $w$, with $(|w|<1)$, write \\[f_w(z)=f\\Big(\\frac{z-w}{1-wz}\\Big)=\\sum^\\infty_{n=0}a_n^{(w)}z^n,\\] \\[\\|f_w\\|=\\sum^\\infty_{n=0}|a^{(w)}_n|\\] and let $W_f$ be the set of all values $w$, with $(|w|<1)$ for which \\[\\|f_w\\|<\\infty,\\] Since $a^{(w)}_n$ is a continuous function of $w$, it follows that $W_f$ is a set of type $F_\\sigma$. What more can be said? For example, if $W_f$ is everywhere dense in $D$ (or uncountable, or of positive measure), is $W_f$ the unit disc? It is known that $W_f$ may be a proper non-empty subset of $D$. Is $W_f$ either empty, or all of $D$ if $f$ is univalent?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.21\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305022, "problem_number": "AMR-022-5022", "title": "Research Problems in Function Theory — Problem 5.22", "statement": "Let $H^p$ be the space of functions $f(z)=\\sum^\\infty_{n=0}a_nz^n$ analytic in $\\mathbb{D}$, and such that \\[\\int^{2\\pi}_0\\big|f(re^{i\\theta})\\big|^p\\,d\\theta\\] remains bounded as $r\\to1$. We define $H^\\infty$ to be the class of bounded functions in $\\mathbb{D}$. For $0 < p < 1$ describe the coefficient multipliers from $H^p$ to $H^p$. That is, for each such $p$ describe the sequences $\\lambda_n$ such that \\[\\sum\\lambda_n a_n z^n\\in H^p\\hspace{1cm}\\text{whenever }\\sum a_nz^n\\in H^p.\\] (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.22\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: The multipliers of $H^p$ ($00\\] in $D$. Can one find bounded analytic functions $g_\\nu(z)$ in $D$ such that \\[\\sum^n_{\\nu=1}f_\\nu(z)g_\\nu(z)\\equiv1\\] in $D$? When $D$ is a disc, Carleson proved that the answer is `yes', and the result extends to finitely connected domains. The result is also known to be true for certain infinitely connected domains (see Behrens , Gamelin ), but false for general Riemann surfaces of infinite genus (Cole, unpublished). Presumably the answer for the general plane domain is negative. Proofs of all positive results depend on Carleson's theorem. (L. Zalcman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.27\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS. Carleson proved the corona theorem for the disc (1962); extended to finitely connected and some infinitely connected domains; Cole gave counterexamples on Riemann surfaces. The general-plane-domain corona problem is open/expected negative. No full resolution located." }, { "id": 2305028, "problem_number": "AMR-022-5028", "title": "Research Problems in Function Theory — Problem 5.28", "statement": "Let $f$ be continuous in $\\overline{\\mathbb{D}}$ and analytic in $\\mathbb{D}$. Let \\[\\omega(f,\\delta)=\\sup|f(z)-f(w)|,\\hspace{1cm}\\text{for }|z-w|\\leq\\delta\\text{ and }z, w\\in \\mathbb{D},\\] \\[\\tilde{\\omega}(f,\\delta)=\\sup|f(z)-f(w)|,\\hspace{1cm}\\text{for }|z-w|\\leq\\delta\\text{ and }|z|=|w|=1.\\] Is it true that \\[\\lim_{\\delta\\to0}\\frac{\\omega(f,\\delta)}{\\tilde{\\omega}(f,\\delta)}=1?\\] The Ruben, Shields and Taylor (unpublished) have shown that there is an absolute constant $C$ such that \\[\\omega(f,\\delta)\\leq C\\tilde{\\omega}(f,\\delta),\\] but that one may not take $C = 1$. (L. A. Rubel, A. Shields)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.28\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305029, "problem_number": "AMR-022-5029", "title": "Research Problems in Function Theory — Problem 5.29", "statement": "A $G_\\delta$ set is a subset of a topological space that is a countable intersection of open sets. Let $E$ be a $G_\\delta$ set of measure zero on $|z|=1$. Then does there exist an $f$ in $H^{\\infty}$, $f\\neq0$ such that $f = 0$ on $E$, and every point of the unit circle is a Fatou point of $f$? (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.29\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305030, "problem_number": "AMR-022-5030", "title": "Research Problems in Function Theory — Problem 5.30", "statement": "Let $\\mathcal{B}$ be the space of Bloch functions, that is the space of functions analytic in $\\mathbb{D}$ with \\[\\|f\\|_\\mathcal{B}=|f(0)|+\\sup_\\mathbb{D}(1-|z|^2)|f'(z)|<\\infty.\\] Let $\\mathcal{B}_S$ be the space of functions of the form $$ f(z) = \\log g'(z),\\hspace{1cm} g \\in S $$ where $S$ is the class of functions as in Problem 5.23(a). Let $\\mathcal{B}_Q$ be the space of functions $g$ in $S$ that have a quasi-conformal extension to the closed plan, see Anderson, Clunie and Pommerenke . [(a)] ; Is $\\mathcal{B}_S$ connected in the norm topology? ; Is $\\mathcal{B}_Q$ dense in $\\mathcal{B}_S$ in the norm topology? (L. Bers)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.30\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305031, "problem_number": "AMR-022-5031", "title": "Research Problems in Function Theory — Problem 5.31", "statement": "It was shown by Becker that \\[\\big\\{f:\\|f\\|_\\mathcal{B}<1\\big\\}\\subset \\mathcal{B}_Q.\\] Is the radius 1 best possible? Is it true that for $f \\in \\mathcal{B}_S$, \\[ \\limsup_{|z|\\to1}\\,(1 - |z|^2)|f'(z)| < 1\\hspace{0.5cm}\\implies\\hspace{0.5cm} f \\in\\mathcal{B}.\\] (Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.31\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305032, "problem_number": "AMR-022-5032", "title": "Research Problems in Function Theory — Problem 5.32", "statement": "Suppose that $f_n\\in\\mathcal{B}_S$. What does $\\|f_n-f\\|_\\mathcal{B}\\to0$ as $n\\to\\infty$ mean geometrically for the functions $g_n$ related to $f_n$ by ([source label: B5.11])? (Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.32\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305033, "problem_number": "AMR-022-5033", "title": "Research Problems in Function Theory — Problem 5.33", "statement": "Let $L$ be a regular triangular lattice in the plane. Let $f(z)$ map $\\mathbb{D}$ onto the universal covering surface over the complement of $L$. Is it true that the coefficients $a_n$ of $f$ tend to $0$ as $n\\to\\infty$? (Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.33\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305034, "problem_number": "AMR-022-5034", "title": "Research Problems in Function Theory — Problem 5.34", "statement": "It was proved by Hall that every Bloch function has (possibly infinite) angular limits on an uncountably dense subset of $|z | = 1$. Do there always exist angular limits on a set of positive measure relative to some fixed Hausdorff measure, such as logarithmic measure for example? (J. E. McMillan, Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.34\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305035, "problem_number": "AMR-022-5035", "title": "Research Problems in Function Theory — Problem 5.35", "statement": "Let $F$ be any discontinuous group of M\\\"obius transformations of $\\mathbb{D}$. Does there always exist a meromorphic function automorphic with respect to $\\Gamma$ and normal, i. e. such that \\[(1-|z|^2)\\frac{|f'(z)|}{1+|f(z)|^2}\\] is bounded in $\\mathbb{D}$? (Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.35\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305036, "problem_number": "AMR-022-5036", "title": "Research Problems in Function Theory — Problem 5.36", "statement": "Let $(n_k)$ be a sequence of positive integers such that $$ n_{k+1}>\\lambda n_k,\\text{ where }\\lambda>1, $$ and suppose that $$ f(z)=\\sum^\\infty_{k=0}a_kz^{n_k} $$ is analytic in $\\mathbb{D}$. Is it true that if $\\sum^\\infty_{k=0}|a_k|=\\infty$, then $f(z)$ assumes every finite value [(a)] ; at least once; ; infinitely often; ; in every angle $\\alpha<\\arg z<\\beta$ of $| z | < 1$? See Weiss and Weiss . (J. P. Kahane)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.36\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305037, "problem_number": "AMR-022-5037", "title": "Research Problems in Function Theory — Problem 5.37", "statement": "Suppose that $f(z)$ is a function as in ([source label: B5.13]) and define \\[\\mu=\\limsup_{r\\to1}\\frac{\\log\\log M(r,f)}{-\\log(1-r)},\\] where $M(r,f)$ is the maximum modulus of $f(z)$ on $|z| = r$. We do not now assume ([source label: B5.12]), but let $N^0(t)$ be the number of $n_k$ not greater than $t$. If $N(r, a)$ is the function of Nevanlinna theory (see Chapter 1) it is known that \\[\\limsup_{r\\to1}\\frac{N(r,0)}{\\log M(r,f)}=1\\] provided that either [(a)] ; $\\mu > 0$ and \\[\\liminf_{k\\to\\infty}\\frac{\\log(n_{k+1}-n_k)}{\\log n_k}>\\frac{1}{2}\\Big(\\frac{2+\\mu}{1+\\mu}\\Big),\\] (this is implicit in Wiman, see Sunyer and Balaguer ), or ; $\\mu>\\frac{1-\\beta}{\\beta}$ and $N^0(t)=O(t^{1-\\beta})$ as $t\\to\\infty$, where $0<\\beta<1$, see Sons . If $0<\\mu<\\frac{1-\\beta}{\\beta}$ with $N^0(t)=O(t^{1-\\beta})$ as $t\\to\\infty$ we ask $(a), (b)$ and $(c)$ of the preceding problem, at least for those cases not covered above. In particular we may consider the cases $n_k = [k^\\alpha]$, where $1 < \\alpha < \\frac{3}{2}$. (L. R. Sons)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.37\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305038, "problem_number": "AMR-022-5038", "title": "Research Problems in Function Theory — Problem 5.38", "statement": "Tao-Shing Shah has shown that if $g(z)\\prec f(z)$ in $\\mathbb{D}$, $g'(0)/f'(0)$ is real, and $f$ in $S$ then $$ |g(z)|\\leq|f(z)|\\text{ for }|z|\\leq\\frac{1}{2}(3-\\sqrt{5}), $$ $$ |g'(z)|\\leq|f'(z)|\\text{ for }|z|\\leq3-\\sqrt{8}. $$ Both constants are `best-possible'. Shah's proofs are technically very involved and it would be nice to have simpler proofs. Goluzin gave simpler proofs but with worse constants in each case. His methods appear to be incapable of yielding ([source label: C5.1]) and ([source label: C5.2]). (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.38\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: The results are established (Shah); the request is for simpler proofs. Open as a \"proof simplification\" problem. No simpler proof located." }, { "id": 2305039, "problem_number": "AMR-022-5039", "title": "Research Problems in Function Theory — Problem 5.39", "statement": "Goluzin has shown that, if $g(z)\\prec f(z)$ in $\\mathbb{D}$, then \\[M_2(r,g')\\leq M_2(r,f'),\\hspace{1cm}0\\leq r\\leq\\frac{1}{2}.\\] Here, for $p > 0$, \\[M_p(r,h)=\\Big(\\frac{1}{2\\pi}\\int^{2\\pi}_0|h(re^{i\\theta})|^p\\,d\\theta\\Big)^{1/p}.\\] The result is not necessarily true if $r > \\frac{1}{2}$, as $f(z) = z, g(z) = z^2$ shows, though it follows from a theorem of Littlewood that, for all $p$, \\[M_p(r,g)\\leq M_p(r,f),\\hspace{1cm} 0 < r < 1.\\] Find the largest number $r_p, 0 < r_p < 1$, independent of $f$ and $g$ so that \\[M_p(r,g')\\leq M_p(r,f'),\\hspace{1cm} 0 0, r \\leq\\sqrt{2}-1$. This improves what one gets if one applied ([source label: C5.2]) to the $p$-th means since $\\sqrt{2}-1>3-\\sqrt{8}$; but it may not be best possible. (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.39\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305040, "problem_number": "AMR-022-5040", "title": "Research Problems in Function Theory — Problem 5.40", "statement": "Suppose that $f(z) = \\sum^\\infty_0 a_nz^n$ and that $F(z)$ is analytic in $\\mathbb{D}$, with $f\\prec F$. What non-trivial conditions on $F$ imply that $$ a_n\\to0\\hspace{1cm}\\text{ as }n\\to\\infty? $$ In particular, is ([source label: C5.3]) implied by $$ F\\text{ is a Bloch function, }and $$ $$ \\int^{2\\pi}_0|f'(re^{i\\theta})|^2\\,d\\theta=o\\Big(\\frac{1}{1-r}\\Big)^2,\\hspace{1cm}\\text{ as }r\\to1^-\\,? $$ Is it true that ([source label: C5.5]) by itself is preserved under subordination? It is known that ([source label: C5.3]) holds if \\[(1-r)|F'(re^{i\\theta})|\\to0\\] uniformly in $\\theta$ as $r\\to1$ and that ([source label: C5.3]) holds if both ([source label: C5.4]) is satisfied, and given any positive $\\varepsilon$, $F$ can be written in the form \\[F(z)=F_1(z)+F_2(z)\\] where $(1-|z|^2)|F_1'(z)|\\leq\\varepsilon$ and $F_2$ has bounded characteristic in $\\mathbb{D}$. (W. K. Hayman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.40\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305042, "problem_number": "AMR-022-5042", "title": "Research Problems in Function Theory — Problem 5.42", "statement": "Suppose that \\[f(z)=\\sum^\\infty_{n=0}a_nz^n\\] is analytic in $\\mathbb{D}$, with \\[\\sum^\\infty_{n=0}|a_n|=1,\\hspace{1cm} |f(z)|\\geq\\delta>0\\text{ in }\\mathbb{D},\\hspace{1cm}\\text{ and}\\hspace{1cm}\\frac{1}{f(z)}=\\sum^\\infty_{n=0}b_nz^n.\\] The following facts are known about $M=\\sum^\\infty_{n=0}|b_n|$: [(a)] ; $M < +\\infty$; ; if $\\delta<\\frac{1}{2}, M$ cannot be bounded in terms of $\\delta$ (Katznelson (no citation)). ; if $\\delta>2^{-\\frac{1}{2}}, M$ can be bounded in terms of $\\delta$ (Katznelson (no citation), Newman (no citation)). What is the infimum of those $\\delta$ such that $M$ can be bounded in terms of $\\delta$? (A likely guess is $\\frac{1}{2}$). (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.42\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305043, "problem_number": "AMR-022-5043", "title": "Research Problems in Function Theory — Problem 5.43", "statement": "Determine the Laurent coefficient bodies for analytic functions taking values of modulus at most unity in a given annulus \\[A_r = \\{z : r < |z| < 1\\}.\\] Determine the extremal functions. (M. Heins)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.43\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305044, "problem_number": "AMR-022-5044", "title": "Research Problems in Function Theory — Problem 5.44", "statement": "Suppose that $0 < \\alpha < 1$ and \\[\\frac{(1+xz)^\\alpha}{1-z}=\\sum^\\infty_{n=0}A_n(x)z^n,\\hspace{1cm} A_0(x)=1,\\hspace{1cm} |x|=1.\\] Is it true that \\[|A_{2n+1}(x)|\\leq|A_{2n+1}(1)|,\\hspace{1cm}n\\geq2,\\hspace{1cm}|x|=1?\\] The above is true for $n = 1$; the corresponding result is false for all \\mbox{$A_{2m}(x)$} with $m \\geq 1$, see Brannan . More generally, one can ask the same question for the coefficients of \\[(1+xz)^\\alpha(1-z)^{-\\beta}, \\hspace{1cm}|x|=1, \\hspace{1cm}\\alpha>0,\\hspace{1cm}\\beta>0.\\] Here it is not even known if $|A_3(x)| \\leq A_3(1)$. (D. A. Brannan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.44\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition; related to the \"Close-to-convex coefficient\" and Rogosinski-type problems. No definitive resolution located." }, { "id": 2305045, "problem_number": "AMR-022-5045", "title": "Research Problems in Function Theory — Problem 5.45", "statement": "If $A$ is any analytic subset of the Riemann sphere it was shown by Kierst that $A$ is (exactly) the set of asymptotic values of a function meromorphic in $\\mathbb{D}$. If $\\infty\\in A$, Kierst also proved that $A$ is the set of asymptotic values of a function $f$ analytic in $\\mathbb{D}$. However, there exist analytic sets which are not the set of asymptotic values of a function analytic in $\\mathbb{D}$. Ryan has characterised those subsets of the Riemann sphere which are the set of asymptotic values of a function analytic in $\\mathbb{D}$. Can one find a simpler characterisation? (K. F. Barth)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.45\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (the request is for a simpler characterisation). No resolution located." }, { "id": 2305046, "problem_number": "AMR-022-5046", "title": "Research Problems in Function Theory — Problem 5.46", "statement": "A non-constant function $f$ analytic in $\\mathbb{D}$, is said to be in the MacLane class $\\mathcal{A}$ if the set of points of the unit circle $\\mathbb{T}$ at which $f$ has an asymptotic value is a dense subset of $\\mathbb{T}$ (see MacLane ). A function $f$ is said to have an arc tract if there exists a sequence $\\{\\gamma_n\\}$ of arcs, $\\gamma_n\\subset \\mathbb{D}$, and a non-degenerate subarc $\\gamma$ of $\\mathbb{T}$ such that $\\gamma_n\\to\\gamma$ (in the obvious fashion) and $\\min\\{|f(z)| :z\\in\\gamma_n\\}\\to\\infty$ as $n\\to\\infty$. Does there exist a function $f\\in\\mathcal{A}$ with an arc tract and with non-zero derivative (see MacLane )? (K. F. Barth)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.46\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305047, "problem_number": "AMR-022-5047", "title": "Research Problems in Function Theory — Problem 5.47", "statement": "Let \\[f(z)=\\sum^\\infty_{n=0}a_nz^{\\lambda_n}\\] be analytic in $\\mathbb{D}$ and have Hadamard gaps, i.e. $\\lambda_{n+1}/\\lambda_n\\geq q>1$. Need $f$ have any radial limits (finite or infinite)? If not, need $f$ have any asymptotic value on a path ending at a single point? For the case $q\\geq3$, see MacLane . (J. M. Anderson and R. Hornblower)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.47\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open for $1 N(\\varepsilon)$, $J_n$ lies in the domain $\\{z: 1 - \\varepsilon < |z| < 1\\}$, ; $\\min\\{|f(z)| : z \\in J_n\\}\\ \\to\\infty$ as $n\\to\\infty$. Let $\\mathbb{T}$ denote the unit circle and let $Z'(f)$ denote the set of limit points of the zeros of an annular function $f$. Write \\[S(f) = \\{a:a\\in\\mathbb{C}\\text{ and }Z'(f-a) \\neq T\\},\\] and let $|S(f)|$ denote the cardinality of this set. Can $|S(f)| = \\mathfrak{N}_0$? The cases $|S(f)| = 1$ (Barth and Schneider ) and $|S(f)| = 2$ (Osada (no citation)) are known, however neither construction can be easily adapted to the general case. (K. F. Barth and D. D. Bonar)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.50\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305051, "problem_number": "AMR-022-5051", "title": "Research Problems in Function Theory — Problem 5.51", "statement": "It can be shown that there exists a Blaschke product $B(z)$ with \\mbox{$B(0) = 0$} such that \\[\\frac{1+B(z)}{1-B(z)}\\] is a Bloch function. Give an explicit construction for such a product. (F. Holland)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.51\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Existence is shown (as stated); an explicit construction is requested. Open as of Hayman's 2018 edition. No explicit construction located." }, { "id": 2305052, "problem_number": "AMR-022-5052", "title": "Research Problems in Function Theory — Problem 5.52", "statement": "Let F be a Fuchsian group in $\\mathbb{D}$, and let $B$ be a set on $\\mathbb{T}$ such that $B\\cap \\gamma(B)=\\emptyset$ for $\\gamma\\in\\Gamma,\\gamma\\neq I$. (This is the case, for instance, if \\[B=(\\partial F\\cap\\mathbb{T})\\setminus C\\hspace{1cm}(C\\text{ countable})\\] where $F$ is a normal fundamental domain.) If cap $B > 0$, does it follow that $F$ is of convergence type? (Conversely, it is known that, for every group $F$ of convergence type, there exists such a set $B$ with cap $B > 0$). (Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.52\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305053, "problem_number": "AMR-022-5053", "title": "Research Problems in Function Theory — Problem 5.53", "statement": "The ratio $R$ of two Blaschke products $B(z,a_n), B(z,b_n)$ is of bounded characteristic, but need not be a normal meromorphic function. That is, if $a_n$ is `close' to $b_n$ for infinitely many $n$ we can arrange that the spherical derivative of $R$ is too large for $R$ to be normal. When is $R$ a normal function? Cima and Colwell have shown that if $\\{a_n\\}$ and $\\{b_n\\}$ are both interpolating sequences, then $R$ is normal if and only if $\\{a_n\\} \\cup \\{b_n\\}$ is also an interpolating sequence. What happens if the sequences are not interpolating sequences? (J. M. Anderson)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.53\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305054, "problem_number": "AMR-022-5054", "title": "Research Problems in Function Theory — Problem 5.54", "statement": "Suppose that \\[f(z)=\\sum^\\infty_{n=0}a_nz^n\\] is convergent in $\\mathbb{D}$, that $|z_0| = 1$, and that neither of the series \\[\\sum^\\infty_{n=0}(\\text{Re}\\, a_n)z_0^n,\\hspace{1cm}\\sum^\\infty_{n=0}(\\text{Im}\\, a_n)z_0^n,\\] is absolutely convergent. Let $S$ be the set of complex values assumed by the series \\[\\sum^\\infty_{n=0}\\varepsilon_n a_n z_0^n,\\] where $\\varepsilon=\\pm1$, and is allowed to vary over all possible choices. Is it true that $S = \\mathbb{C}$? (A. C. Offord; communicated by J. G. Clunie)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.54\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305055, "problem_number": "AMR-022-5055", "title": "Research Problems in Function Theory — Problem 5.55", "statement": "Let the function $h(z)$ be analytic in $\\mathbb{D}$, $G_h = \\{h(\\mathbb{D})\\}$, $K_h$ a compact subset of $G_h$. The function $h(z)$ will be said to have the $L$-property (`$h(z_n)$ leaves the range of $h$') on the sequence $\\{z_n\\}^\\infty_1$ with $\\lim |z_n|=1$, if only finitely many points of $\\{h(z_n)\\}^\\infty_1$ lie in $K_h$ for every $K_h$. We will define two functions $f,g$ analytic in $\\mathbb{D}$ to be an ordered $L$-pair if, on any sequence for which $f$ has the $L$-property, $g$ also has the $L$-property. A non-constant function $\\alpha(z)$ analytic in $\\mathbb{D}$ will be called an $L$-atom if, corresponding to any function $f$ such that $f, \\alpha$ is an ordered $L$-pair, there exists a function $\\phi_f$ analytic on $G_\\alpha$ such that $f=\\phi_f\\circ\\alpha$. Prove or disprove the conjecture that $\\alpha$ is an $L$-atom if and only if it is univalent. What happens if $\\mathbb{D}$ is replaced by more general domains? (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.55\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305056, "problem_number": "AMR-022-5056", "title": "Research Problems in Function Theory — Problem 5.56", "statement": "By a first-order property we shall mean a ring-theoretic property in the first-order predicate calculus. For functions $f$ analytic in $\\mathbb{D}$, are the following first-order properties: [(a)] ; $f$ is constant? ; $f$ is bounded ? ; $f$ is admissible, i.e. $T(r,f) \\neq O(\\log(1-r)^{-1})?$ What is the situation in more general domains? (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.56\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305057, "problem_number": "AMR-022-5057", "title": "Research Problems in Function Theory — Problem 5.57", "statement": "Suppose that $f$ is analytic in $\\mathbb{D}$. Plessner's theorem asserts that at almost all points $e^{i\\theta}$ on the unit circle, either $f$ has an angular limit, or else the image $f(S)$ of every Stolz angle $S$ with vertex at $e^{i\\theta}$ is dense in $\\mathbb{C}$. How much can `dense' be improved? In particular, is it true that at almost all $e^{i\\theta}$ either $f$ has an angular limit, or else $f(S)$ is all of $\\mathbb{C}$, except perhaps for a set of zero logarithmic capacity? This result would be best possible, since if $E$ is any closed set of capacity zero then the universal covering map of the disc onto $\\mathbb{C} \\setminus E$ has angular limits almost nowhere. (A. Baernstein II)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.57\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305058, "problem_number": "AMR-022-5058", "title": "Research Problems in Function Theory — Problem 5.58", "statement": "Suppose that $f$ is univalent and zero-free in $\\mathbb{D}$. It has been shown by Baernstein that, for each $p\\in(0,\\frac{1}{2})$, $f$ admits a factorisation $f=B_p(F_p)^{1/p}$, where $B_p\\in H^\\infty, 1/B_p\\in H^\\infty$, and $\\text{Re}\\, F_p > 0$. Is it possible to pass to the limit $p = \\frac{1}{2}$, and thus factor $f$ into a bounded function times a function subordinate to a map onto a slit plane? (A. Baernstein II)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.58\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. Related to Baernstein–Solynin work. No definitive resolution located." }, { "id": 2305059, "problem_number": "AMR-022-5059", "title": "Research Problems in Function Theory — Problem 5.59", "statement": "(Subordination and extreme point problem) Let $g(z)=\\sum^\\infty_{n=0}B_nz^n$ be analytic in $\\mathbb{D}$. Denote by $S_g$ the family of functions $f(z)$ subordinate to $g$. Find general conditions on $g$ so that the only extreme points of the closed convex hull of $S_g$ are the functions $g(ze^{it})(0\\leq t<2\\pi)$. This is known for certain functions $g$, see Clunie , for example: [(a)] ; $g(z)=[(1+cz)/(1-z)]^\\alpha$ where $|c|\\leq1,\\alpha\\geq1$; ; $g(z)=\\exp[(1+z)/(1-z)]$. Sheil-Small can prove it for a functions of the form \\linebreak\\mbox{$g(z)=h((1+z)/(1-z))$}, where $h(w)$ is a univalent quadratic polynomial in $\\text{Re}\\, w>0$. One might expect that the conclusion would hold for a functions $g$ with positive coefficients increasing in a suitably regular manner. (T. Sheil-Small)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.59\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305060, "problem_number": "AMR-022-5060", "title": "Research Problems in Function Theory — Problem 5.60", "statement": "(Hadamard convolutions) Suppose that $\\alpha\\geq1, \\beta\\geq1$ and that $\\phi$ is analytic in $\\mathbb{D}$, and satisfies \\[\\phi(z)\\ast\\frac{(1+xz)^\\alpha}{(1-z)^\\beta}\\neq0,\\hspace{1cm}|x|=1, |z|<1.\\] Is it true that \\[\\phi(z)\\ast\\frac{(1+xz)^{\\alpha-1}}{(1-z)^\\beta}\\neq0,\\hspace{1cm}|x|=1, |z|<1\\,?\\] This is true when $\\alpha$ is a natural number. For the proof and further background, see Sheil-Small . (T. Sheil-Small)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.60\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305061, "problem_number": "AMR-022-5061", "title": "Research Problems in Function Theory — Problem 5.61", "statement": "Let $w(z)$ be analytic in $\\mathbb{D}$ with $w(0)=0$. If \\mbox{$|w(z)+zw'(z)|<1$}, for $|z|<1$, then a simple application of Schwarz's lemma shows that $|w(z)|<1$, for $|z|<1$. Miller and Mocanu showed that \\[|w(z)+zw'(z)+z^2w''(z)|<1,\\hspace{1cm}|z|<1\\] implies that $|w(z)|<1$, for $|z|<1$. Is it true that \\[|w+zw'+z^2w''+\\ldots+z^nw^{(n)}|<1\\hspace{0.5cm}\\implies\\hspace{0.5cm}|w(z)|<1,\\] for $n=1, 2, 3, \\ldots$? (S. Miller)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.61\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No final resolution located (the question is about higher-order differential subordination)." }, { "id": 2305062, "problem_number": "AMR-022-5062", "title": "Research Problems in Function Theory — Problem 5.62", "statement": "Let $u$ be a continuous real-valued function on the unit circle $\\mathbb{T}$. Give a necessary and sufficient condition on $u$ such that $u$ is the real part of a function $f$ in the disc algebra $A(\\overline{\\mathbb{D}})$. Remarks: [(a)] ; A solution would have applications in the algebraic ideal theory of $A(\\overline{\\mathbb{D}})$. ; An answer to the analogous problem for $L^p(T), H^p(\\mathbb{D})$ is the Burkholder-Gundy-Silverstein Theorem (see Peterson ). (M. von Renteln)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.62\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305063, "problem_number": "AMR-022-5063", "title": "Research Problems in Function Theory — Problem 5.63", "statement": "One of the many equivalent norms on BMOA on $\\mathbb{D}$ is defined by \\[\\|f\\|_h=\\inf_q\\sup_{z\\in\\mathbb{D}}|f(z)+\\overline{q(z)}|,\\] the infimum being taken over all functions analytic in $\\mathbb{D}$. Given $f$ in BMOA, does there exist a $q$ such that \\[|f(z)+\\overline{q(z)}|\\equiv\\|f\\|_h\\text{ a.e. on }|z|=1?\\] The answer is `yes' when $f$ is a rational function. (J. A. Hempel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.63\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305064, "problem_number": "AMR-022-5064", "title": "Research Problems in Function Theory — Problem 5.64", "statement": "Let $f$ be analytic in $\\mathbb{D}$ with $$ |f(z)|=O((1-|z|)^{-k}),\\hspace{1cm}k\\geq0. $$ Then $f$ induces a distribution on $C^\\infty(T)$, as follows. For $\\phi\\in C^\\infty(T)$ \\[\\lim_{r\\to1}\\Big(\\frac{1}{2\\pi}\\int f(re^{i\\theta})\\phi(e^{-i\\theta})\\,d\\theta\\Big)=\\Lambda_f (\\phi).\\] What can be said about the order of the distributions satisfying ([source label: D5.64.1]) and \\[\\lim_{|z|\\to1}|f(z)|(1-|z|)^k=0\\] (J. A. Cima)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.64\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305065, "problem_number": "AMR-022-5065", "title": "Research Problems in Function Theory — Problem 5.65", "statement": "Does there exist a non-constant function $f$ in the disc algebra such that $f(e^{i\\theta})\\in f(\\mathbb{D})$ for almost all $\\theta$? Caution: the Rudin-Carleson theorem (see Bishop ) allows construction of a good candidate, but it is not immediately clear whether it does work. (K. Stephenson)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.65\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305066, "problem_number": "AMR-022-5066", "title": "Research Problems in Function Theory — Problem 5.66", "statement": "Let $B$ be an infinite Blaschke product in $\\mathbb{D}$. Does there exist a positive $\\delta$, depending on $B$, such that, for every $w$, $|w|<\\delta$, the set $B^{-1}(\\{w\\})$ is infinite? Stephenson has obtained some related results. (K. Stephenson)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.66\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305067, "problem_number": "AMR-022-5067", "title": "Research Problems in Function Theory — Problem 5.67", "statement": "Let the function $f$ in $\\mathbb{D}$ be given by \\[f(z)=\\sum^\\infty_{k=0}a_kz^{n_k},\\hspace{1cm}\\frac{n_{k+1}}{n_k}\\geq\\lambda>1,\\hspace{1cm} k\\geq0,\\] with \\[m_0(r,f) \\equiv \\max_{k\\geq0} |a_k|r^{n_k}\\to\\infty\\hspace{1cm}\\text{ as }r\\to 1^-.\\] Define \\[E=\\Big\\{\\theta:\\liminf_{r\\to1^-}\\frac{|f(re^{i\\theta})|}{m_0(r,f)}>0\\Big\\}.\\] Is it true that $E$ has measure $0$? This has been proved for the case that $\\mu(\\frac{1}{2}(1+r))/m_0(r,f)\\leq$ constant. (D. Gnuschke and Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.67\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305068, "problem_number": "AMR-022-5068", "title": "Research Problems in Function Theory — Problem 5.68", "statement": "Let the function $f$ where \\[f(z)=1+\\sum^\\infty_{n=1}a_nz^n,\\hspace{1cm}|z|\\leq1,\\] be a Bloch function with positive real part in $\\mathbb{D}$. Determine the rate of growth (as $N\\to\\infty$) of the sequence $\\big\\{\\sum^N_{n=1}|a_n|^2\\big\\}$. (F. Holland)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.68\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305069, "problem_number": "AMR-022-5069", "title": "Research Problems in Function Theory — Problem 5.69", "statement": "Let the function $f$ where \\[f(z)=1+\\sum^\\infty_{n=1}a_nz^n,\\hspace{1cm}|z|\\leq1,\\] be a Bloch function with positive real part in $\\mathbb{D}$ and such that each $a_n\\geq0$. Does it follow that $\\sum^\\infty_{n=1}a^2_n<\\infty$? An equivalent formulation of the problem is the following: let $\\mu$ be a probability measure in Zygmund's class $\\Lambda_*$ on the circle, and let \\[\\hat{\\mu}(n)=\\int^{2\\pi}_0e^{-inx}\\,d\\mu(x)\\geq0,\\hspace{1cm}n\\in\\mathbb{Z}.\\] Is it true that $\\hat{\\mu}\\in l_2$? A counter-example, if one exists, cannot be constructed using Riesz products (see Duren , and Holland and Twomey ). An affirmative answer would mean that, if $f(z) = 1 + \\sum^\\infty_{n=1}a_nz^n$ is a Bloch function in $\\mathbb{D}$ with positive real part, then $\\sum^\\infty_{n=1}|a_n|^4<\\infty$. Even if this last inequality is false, perhaps it is still true in the general case that there exists $p > 4$, such that $\\sum^\\infty_{n=1}|a_n|^p<\\infty$. (F. Holland)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.69\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305070, "problem_number": "AMR-022-5070", "title": "Research Problems in Function Theory — Problem 5.70", "statement": "Barth and Clunie have constructed a bounded analytic function in $\\mathbb{D}$ with a level set component of infinite length; this component is highly branched. Can one construct a bounded analytic function with an unbranched level set component of infinite length? (K. F. Barth and J. G. Clunie)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.70\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305071, "problem_number": "AMR-022-5071", "title": "Research Problems in Function Theory — Problem 5.71", "statement": "Suppose that \\[f(z) = \\sum^\\infty_{k=1}a_kz^{n_k}, \\hspace{1cm}n_{k+1}/n_k\\geq q>1,\\] is an analytic function in $\\mathbb{D}$ with Hadamard gaps, such that $T(r,f)\\to\\infty$ as $r\\to1$. Does $\\delta(w,f)=0$ hold for every (finite) complex number $w$? (T. Murai)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.71\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305072, "problem_number": "AMR-022-5072", "title": "Research Problems in Function Theory — Problem 5.72", "statement": "Let the function $f$ have the power series $f(z)=\\sum^\\infty_{n=0}a_nz^n$ of radius of convergence $1$; let $E$ be the singular set on $\\mathbb{T}$, and suppose that \\[\\sup_{\\xi\\in E}\\, \\sup_{N\\geq0}\\Big|\\sum^N_{n=0}a_n\\xi\\Big|<\\infty.\\] It has been shown by Allan, O'Farrell and Ransford that, if $E$ has measure zero, then $\\sum^\\infty_{n=0}a_nz^n$ converges to $f(z)$ at each point $z$ in $\\mathbb{T}\\setminus E$. Does the conclusion remain true if $E$ has positive measure? (Nothing appears to be known either way.) (T. J. Ransford)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.72\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (\"nothing appears known either way\"). No recent resolution located." }, { "id": 2305073, "problem_number": "AMR-022-5073", "title": "Research Problems in Function Theory — Problem 5.73", "statement": "Let $0<\\alpha<1$ and let $R_\\alpha$ denote the set of all Riesz potentials $p(x)$ of finite positive Borel measures $\\mu$ on $\\mathbb{R}$: \\[p(x)=\\int^\\infty_{-\\infty}|x-t|^{-\\alpha}\\,d\\mu(t).\\] Characterise those non-negative measurable functions $f(x)$ on $\\mathbb{R}$ that are dominated by some function $p$ in $R_\\alpha$. (B. Korenblum)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.73\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305074, "problem_number": "AMR-022-5074", "title": "Research Problems in Function Theory — Problem 5.74", "statement": "Characterise those non-negative measurable functions $f$ on the unit circle that are dominated almost everywhere by moduli of the boundary values of an analytic function in the unit disc with positive real part. (Problem 5.73 might be considered a step towards solving this problem.) (B. Korenblum)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.74\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305075, "problem_number": "AMR-022-5075", "title": "Research Problems in Function Theory — Problem 5.75", "statement": "Does there exist a bounded analytic function in $\\mathbb{D}$ such that the image of every radius has infinite length? See, for example, Anderson , and Rudin . (W. Rudin; communicated by K. F. Barth)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.75\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located (related work by Anderson, Rudin)." }, { "id": 2305076, "problem_number": "AMR-022-5076", "title": "Research Problems in Function Theory — Problem 5.76", "statement": "Let $\\gamma$ be a non-tangential arc that lies in $\\mathbb{D}$ except for one endpoint at $z = 1$, and define \\[\\gamma_\\theta=e^{i\\theta\\gamma},\\hspace{1cm}\\theta\\in[0,2\\pi).\\] Does there exist a function $g\\in H^\\infty$ such that \\[\\lim_{z\\to e^{i\\theta},\\,z\\in\\gamma_\\theta}g(z)\\] exists for no value of $\\theta$? (See Rudin .) If $\\gamma$ is tangential, the answer is `yes', see Collingwood and Lohwater . (W. Rudin; communicated by K. F. Barth)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.76\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No recent resolution located." }, { "id": 2305077, "problem_number": "AMR-022-5077", "title": "Research Problems in Function Theory — Problem 5.77", "statement": "In general, the radial behaviour of the derivative of a bounded analytic function in $\\mathbb{D}$ can be pretty arbitrary; in fact, even under much stronger restrictions than just bounded, it can still be quite arbitrary. Is it true that, given any measurable function $m(\\theta)$ on $[0,2\\pi)$, there exists a function $f(z)$, continuous on $\\overline{\\mathbb{D}}$ and univalent on $\\mathbb{D}$, such that \\[\\lim_{r\\to1}f'(re^{i\\theta})=m(\\theta)\\] for almost all $\\theta$? (In terms of known results, the conjecture seems quite plausible. Ortel and Schneider showed that the conjecture is true under slightly strengthened hypotheses; and Lohwater, Piranian and Rudin showed that the conjecture is true with a slightly weakened conclusion.) (W. J. Schneider)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.77\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305078, "problem_number": "AMR-022-5078", "title": "Research Problems in Function Theory — Problem 5.78", "statement": "Let $H^1$ denote Hausdorff one-dimensional measure on $\\mathbb{C}$, and $\\mathbb{T}$; let $g:\\mathbb{T}\\to[-\\infty,\\infty]$ denote an arbitrary Borel function. Does there exist a corresponding function $f$, analytic in $\\mathbb{D}$ and with bounded Taylor coefficients, such that \\[\\lim_{r\\to1^-}f(rz)=g(z)\\] for $H^1$-almost all $z$ in $\\mathbb{T}$? For work on related questions, see Ortel and Schneider . (M. Ortel and W. J. Schneider)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.78\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2305079, "problem_number": "AMR-022-5079", "title": "Research Problems in Function Theory — Problem 5.79", "statement": "[(a)] ; Let $f$ be a non-constant analytic function in $\\mathbb{D}$, $m$ be a positive integer, and define $\\psi = (f)^mf'$. Then it is shown by Sons () that, when $f$ and $\\psi$ both belong to MacLane's class $A$ , either [(i)] ; $f$ has finite asymptotic values on a dense subset of $\\mathbb{T}$, or ; $\\psi$ assumes every finite value infinitely often. What replacements can be found for $(i)$ (to give another correct theorem)? Can $\\psi$ be taken to be of the form \\[\\psi=(f)^{m_0}(f')^{m_1}\\ldots(f^{(k)})^{m_k},\\] whenever $k,m_0,m_1,\\ldots,m_k\\in\\mathbb{N}\\cup\\{0\\}$? ; It is known that the conclusion in part $(a)$ is true when $\\psi$ is replaced by \\[\\psi=f^{(l)}+\\sum^{i-1}_{\\nu=0}a_\\nu f^{(\\nu)},\\] where $l\\in\\mathbb{N}$ and the $a_\\nu$ are analytic functions in $\\{|z| < 1 +r\\}$, for some positive $r$. What replacements can be found in this case for $(i)$ to give another correct theorem? Can the $a_\\nu$ be taken as functions analytic in $\\mathbb{D}$ with $T(r,a_\\nu) = o(T(r,f))$ as $r\\to1$? (For both parts, compare Problem 1.38.) (L. R. Sons)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 5.79\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306001, "problem_number": "AMR-022-6001", "title": "Research Problems in Function Theory — Problem 6.1", "statement": "The Bieberbach conjecture Is it true that $|a_n|\\leq n$ for $f$ in $S$ with equality only for $f(z)\\equiv f_\\theta(z)$? The result is known to be true for $n=2$ (see Bieberbach ), $n=3$ (see L\\\"owner ), and $n=4$ (see Garabedian and Schiffer , Charzynski and Schiffer ).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.1\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED. $|a_n|\\le n$ for all $n\\in S$, equality only for Koebe rotations. Literature status: SOLVED-IN-LITERATURE. L. de Branges, \"A proof of the Bieberbach conjecture\", Acta Math. 154 (1985), 137–152, proved $|a_n|\\le n$ for all $n$, with equality iff $f$ is a rotation of the Koebe function $z/(1-z)^2$. This is one of the most celebrated results in geometric function theory." }, { "id": 2306002, "problem_number": "AMR-022-6002", "title": "Research Problems in Function Theory — Problem 6.2", "statement": "Define $A_n=\\sup_{f\\in S}|a_n|$. It is shown by Hayman that \\[\\frac{A_n}{n}\\to K_0,\\hspace{1cm}\\text{ as }n\\to\\infty.\\] Is it true that $K_0=1$? The best known result so far is $K_0<1.243$ due to Milin .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.2\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED: $K_0=1$. Literature status: SOLVED-IN-LITERATURE. Yes, $K_0=1$, as a consequence of de Branges' proof of the Bieberbach conjecture (which gives $A_n=n$, so $K_0=1$)." }, { "id": 2306003, "problem_number": "AMR-022-6003", "title": "Research Problems in Function Theory — Problem 6.3", "statement": "If $f(z)$ in $S$ it is shown by Bombieri , that there exist constants $c_n$ such that for $f(z)$ in $S$ \\[|\\text{Re}\\,(n-a_n)|\\leq c_n\\text{Re}\\,(2-a_2).\\] What is it the exact size of the constants $c_n$? Is it true that there exists $d_n$ such that \\[\\big|n-|a_n|\\big|\\leq d_n\\big(2-|a_2|\\big)\\,?\\] (E. Bombieri)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.3\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: The de Branges proof makes the $a_2$-refinement questions (Bombieri-type) partially tractable, but the exact constants $c_n,d_n$ are not fully settled. OPEN-TRIAGE." }, { "id": 2306005, "problem_number": "AMR-022-6005", "title": "Research Problems in Function Theory — Problem 6.5", "statement": "If it proves too difficult to obtain sharp bounds for all of the coefficients in Problem 6.4, we ask for the orders of magnitude. An area principle shows that \\[\\sum^\\infty_{n=1}n|b_n|^2\\leq1\\] and hence \\[b_n=o(n^{-\\frac{1}{2}}).\\] Clunie and Pommerenke have shown that \\[|b_n|=O(n^{-\\frac{1}{2}-\\frac{1}{300}}).\\] In the opposite direction, examples due to Clunie show that \\[|b_n|>n^{-1+\\delta}\\] is possible for indefinitely many $n$ and a fixed $F(z)$ in $\\Sigma$, where $\\delta$ is an absolute constant.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.5\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (the exact rates remain open). No definitive resolution located." }, { "id": 2306006, "problem_number": "AMR-022-6006", "title": "Research Problems in Function Theory — Problem 6.6", "statement": "What are the orders of magnitude of the $c_n$ in Problem 6.4? Springer obtained the estimate \\[|c_n|\\leq\\frac{2^n}{n}\\] and also showed that, given $\\varepsilon>0$, \\[|c_{2n-1}|>(1-\\varepsilon)2^{2n-2}e/(\\pi n^3)^{\\frac{1}{2}}\\] is possible for all sufficiently large $n$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.6\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306007, "problem_number": "AMR-022-6007", "title": "Research Problems in Function Theory — Problem 6.7", "statement": "If $f(z)$ in $S$ and is bounded, i.e. satisfies $|f(z)|n^{-1+\\delta}\\] for infinitely many $n$ and a fixed $f(z)$. The problems for this class of functions seem very analogous to the corresponding problems for $\\Sigma$, see Problem 6.5.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.7\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306008, "problem_number": "AMR-022-6008", "title": "Research Problems in Function Theory — Problem 6.8", "statement": "We write \\[I_\\lambda(r,f)=\\Big\\{\\frac{1}{2\\pi}\\int^{2\\pi}_0|f(re^{i\\theta}|^\\lambda\\,d\\theta\\Big\\}^{1/\\lambda}.\\] What are the exact bounds for $I_\\lambda(r,f)$ and $I_\\lambda(r,f')$ when $f$ in $S$ or $f$ in $\\Sigma$? If $f(z)$ in $S$, it is known that for fixed $\\lambda$, the orders of magnitude of $I_\\lambda(r,f)$ and $I_\\lambda(r,f')$ are maximal when $f(z)$ is the Koebe function. For the best results in this direction, see Bazilevi\\v c . If $f$ in $S$ and $|f|1$ by $F(z)$ is convex.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.8\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (exact bounds not all determined). No definitive resolution located." }, { "id": 2306009, "problem_number": "AMR-022-6009", "title": "Research Problems in Function Theory — Problem 6.9", "statement": "(Schoenberg's conjecture) If $f(z)=\\sum^\\infty_{n=1}a_nz^n$ and $g(z)=\\sum^\\infty_{n=1}b_nz^n$ are convex, and $f$, $g$ belong to $S$, is it true that \\[f\\ast g=\\sum^\\infty_{n=1}a_nb_nz^n\\] is also a convex function in $S$? See P\\'olya and Schoenberg .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.9\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED: convolution of convex functions in $S$ is convex. Literature status: SOLVED-IN-LITERATURE. This is the Pólya–Schoenberg conjecture, proved by St. Ruscheweyh and T. Sheil-Small, \"Hadamard products of Schlicht functions and the Pólya–Schoenberg conjecture\", Comment. Math. Helv. 48 (1973), 119–135. The conclusion holds: the convolution of two convex functions in $S$ is convex." }, { "id": 2306010, "problem_number": "AMR-022-6010", "title": "Research Problems in Function Theory — Problem 6.10", "statement": "If $F(z)$, $G(z)$ are convex functions in $\\Sigma$, it is known that for $0<\\lambda<1$, \\[H(z)=\\lambda F(z)+(1-\\lambda)G(z)\\in\\Sigma,\\] see Pommerenke . Is it true that $H(z)$ is also convex? (Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.10\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306011, "problem_number": "AMR-022-6011", "title": "Research Problems in Function Theory — Problem 6.11", "statement": "If $f(z)$, $g(z)$ are convex functions in $S$, is it true that for $0<\\lambda<1$, $\\lambda f+(1-\\lambda)g$ is star-like and univalent? A function $w=f(z)$ in $S$ is said to be star-like if the image domain $D$ is star-like with respect to the origin $O$, that is, if for any point $P$ in $D$ the straight line segment $OP$ lies in $D$. It is known that $f(z)$ is convex if and only if $zf'(z)$ is star-like, see e.g. Hayman .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.11\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE (affirmative via known theory). Literature status: The starlike-univalent conclusion for convex combinations of convex functions is a known result (follows from close-to-convexity / the theory of convex combinations of convex mappings — this is a positive theorem of Pommerenke and others). It is essentially answered affirmatively. Mark SOLVED-IN-LITERATURE with the note that univalence+starlikeness of convex combinations of convex functions holds." }, { "id": 2306012, "problem_number": "AMR-022-6012", "title": "Research Problems in Function Theory — Problem 6.12", "statement": "If $f(z)=z+\\sum^\\infty_{k=2}a_{n_k}z^{n_k}\\in S$, and \\[\\liminf_{k\\to\\infty}\\frac{n_{k+1}}{n_k}>1,\\] then Pommerenke has proved that $$ a_n=o\\Big(\\frac{1}{n}\\Big) $$ and this is sharp. If \\[\\liminf_{k\\to\\infty}(n_{k+1}-n_k)>4,\\] then Hayman has shown that $$ a_n=o(n^{-\\frac{1}{2}}). $$ Are there intermediate gap conditions which allow us to interpolate between ([source label: 6.3]) and ([source label: 6.4])?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.12\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive interpolation resolution located." }, { "id": 2306013, "problem_number": "AMR-022-6013", "title": "Research Problems in Function Theory — Problem 6.13", "statement": "Suppose that $f(z)$ in $S$, and that positive integers $k, m, n,$ are given. It is known that there exist complex numbers $c_0, c_1,\\ldots,c_m,$ depending on $k, m, n$ and $f$, such that $c_0=1$, $|c_m|\\geq4^{-m}$ and $$ |c_0a_{n+j}+c_1a_{n+j+1}+\\ldots+c_ma_{n+j+m}|\\leq Kn^{\\alpha_m},\\hspace{1cm}0\\leq j\\leq k, $$ where $K$ is a constant depending on $k, m$ and $f$, and $\\alpha_m=8/\\sqrt{m-\\frac{1}{2}}$. What are the best possible values for the $\\alpha_m$? See Pommerenke . It is known that $\\alpha_1=0$, and it may be conjectured that $\\alpha_2=-\\frac{1}{3}$, $\\alpha_3=-\\frac{1}{2}+\\varepsilon$, and $\\alpha_m<-\\frac{1}{2}$ for $m>4$. If, in addition, $f(z)$ is star-like, then Pommerenke has shown that ([source label: 6.5]) holds with $\\alpha_m=-1+\\frac{2}{m+1}$. The result is sharp.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.13\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (best $\\alpha_m$ unknown). No definitive resolution located." }, { "id": 2306014, "problem_number": "AMR-022-6014", "title": "Research Problems in Function Theory — Problem 6.14", "statement": "If $f(z)$ in $S$, set \\[A^{(k)}_n= \\begin{vmatrix} a_n,&a_{n+1},&\\ldots,&a_{n+k-1} \\hdotsfor{4} a_{n+k-1},&a_{n+k},&\\ldots,&a_{n+2k-2} \\end{vmatrix}\\] Using ([source label: 6.5]), Pommerenke proved that if $k$ is fixed $$ |A^{(k)}_n|^{1/k}=O(n^{j_k}),\\hspace{1cm}\\text{ as }n\\to\\infty, $$ with $j_k=-\\frac{1}{2}+16/\\sqrt{k}$. Here the conjecture is that $j_k<-\\frac{1}{2}$ for large $k$. It can be shown that for $k=2$, ([source label: 6.6]) is false with $j_2=0$, and holds with $j_2=\\frac{1}{4}$, see Hayman . If $f(z)$ is in addition star-like, then Pommerenke showed that the best possible values of $j_k$ are $j_k=-1+2/k$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.14\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306015, "problem_number": "AMR-022-6015", "title": "Research Problems in Function Theory — Problem 6.15", "statement": "If $f(z)$ in $S$, write \\[f_\\alpha(z)=\\int^z_0f'(\\zeta)^\\alpha\\, d\\zeta.\\] For what values of $\\alpha$, is it true that $f_\\alpha(z)\\in S$? The result is known to hold for $\\alpha\\leq(\\sqrt{5}-2)/3$ (see Duren, Shapiro and Shields ), but not for $\\alpha>\\frac{1}{3}$, see Royster . If for $z$ in $\\mathbb{D}$, $f(z)$ is analytic and $|f''(z)/f'(z)|\\leq c/(1-|z|^2)$, then $f(z)$ is univalent if $$ c\\leq2(\\sqrt{5}-2), $$ see Duren, Shapiro and Shields ; but not if $c>2$, see Hille . What is the best value of $c$? (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.15\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (the sharp $\\alpha$ range and constant $c$ are not determined). No definitive resolution located." }, { "id": 2306016, "problem_number": "AMR-022-6016", "title": "Research Problems in Function Theory — Problem 6.16", "statement": "Let $S^*$ be the class of all star-like functions $f(z)$ in $S$. Marx conjectured that for each fixed $z_0$, $|z_0|<1$, the set of all numbers $f'(z_0)$ for $f\\in S^*$ coincides with the set of all number $k'(z)$, $|z|\\leq |z_0|$, where \\[k(z)=\\frac{z}{(1-z)^2}\\] is the Koebe function. This is known to be true for $|z|\\leq 0.736$, see Duren . (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.16\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306017, "problem_number": "AMR-022-6017", "title": "Research Problems in Function Theory — Problem 6.17", "statement": "If $f(z)=z+\\sum^\\infty_{n=2}a_nz^n$ in $S$, then \\[ A=\\pi\\sum^\\infty_{n=1}n|a_n|^2\\] is the area of the image domain. What is the minimum value of $A$ when $a_2$ is given? Clearly $A\\geq\\pi(1+2|a_2|^2)$ always, but this bound is not sharp if $|a_2|>\\frac{1}{2}$, since in this case $f(z)=z+a_2z^2$ is not univalent in $\\mathbb{D}$. (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.17\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306018, "problem_number": "AMR-022-6018", "title": "Research Problems in Function Theory — Problem 6.18", "statement": "If $F(z)=z+\\sum^\\infty_{n=1}b_nz^{-n}$ in $\\Sigma$, then \\[A(F)=\\pi-\\pi\\sum^\\infty_{n=1}n|b_n|^2\\] is the area of the set of values not assumed by $F(z)$ in $\\mathbb{D}$. If $F_n(z)\\in\\Sigma$, and \\[F_n(z)\\to F(z),\\hspace{1cm}\\text{ as }n\\to\\infty,\\] for $|z|>1$, under what additional hypotheses is it true that $$ A(F_n)\\to A(F),\\hspace{1cm}\\text{ as }n\\to\\infty\\,? $$ It is suggested that ([source label: 6.7]) might be true under some hypotheses on $(1-|z|^2)^2\\{F(z),z\\}$ where \\[\\{F(z),z\\}=\\Big(\\frac{F''}{F'}\\Big)^\\prime-\\frac{1}{2}\\Big(\\frac{F''}{F'}\\Big)^2\\] is the Schwarzian derivative of $F(z)$. (L. Bers)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.18\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306019, "problem_number": "AMR-022-6019", "title": "Research Problems in Function Theory — Problem 6.19", "statement": "If $f(z)=\\sum^\\infty_{n=1}a_nz^n$ is analytic in $\\mathbb{D}$ and $\\sum^\\infty_{n=1}|a_n|<+\\infty$, can $f(z)$ map the unit circle $\\mathbb{T}$ onto a curve of positive two-dimensional measure if [(a)] ; $f(z)$ in $S$, ; more generally, $f'(z)\\neq0$ in $\\mathbb{D}$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.19\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306020, "problem_number": "AMR-022-6020", "title": "Research Problems in Function Theory — Problem 6.20", "statement": "Let $C$ be a closed curve inside the unit circle $\\mathbb{T}$. Under what conditions on $C$ does there exist a univalent function $f$ in $\\mathbb{D}$ such that $f(C)$ and $f(\\mathbb{T})$ are both convex?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.20\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306021, "problem_number": "AMR-022-6021", "title": "Research Problems in Function Theory — Problem 6.21", "statement": "A function $f(z)$ analytic in $\\mathbb{D}$ is said to be typically real if $f(z)$ is real, when and only when $z$ is real, see Rogosinski . If $f(z)=z+\\sum^\\infty_{n=2}a_nz^n$ is typically real in $\\mathbb{D}$, then \\[f(z)=\\frac{z}{1-z^2}P(z),\\] where $P(0)=0$, $\\text{Re}\\, P(z)>0$ in $\\mathbb{D}$. What other conditions must $P(z)$ satisfy to make $f(z)$ univalent in $\\mathbb{D}$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.21\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306022, "problem_number": "AMR-022-6022", "title": "Research Problems in Function Theory — Problem 6.22", "statement": "If $f(z)=z+\\sum^\\infty_{n=2}a_nz^n$ is univalent and star-like of order $\\frac{1}{2}$ in $\\mathbb{D}$, i.e. \\[\\text{Re}\\,\\frac{zf'(z)}{f(z)}\\geq\\frac{1}{2},\\] find the radius of the largest disc $|z|0,\\] where $\\text{Re}\\, P(z)>0$ in $\\mathbb{D}$ and $P(0)=1$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.22\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: The radius of convexity of starlike functions of order $\\alpha$ has known values for various $\\alpha$; open as of Hayman's 2018 edition for the exact general problem. No definitive resolution located." }, { "id": 2306023, "problem_number": "AMR-022-6023", "title": "Research Problems in Function Theory — Problem 6.23", "statement": "A related problem concerns upper bounds for $|a_{n+1}|-|a_n|$ when $f(z)$ is mean $p$-valent. Lucas has proved that \\[\\big||a_{n+1}|-|a_n|\\big|=O(n^{j_d}),\\] where $j_d=2p-2$, if $p\\geq1$; $j_d\\leq2p-2\\sqrt{p}$ if $\\frac{1}{4}0$, then if $K$ is a sufficiently large positive constant $g(z)+K$ is circumferentially mean $\\varepsilon$-valent. On the other hand, if $p>\\frac{1}{4}$ and $f(z)$ is fixed, ([source label: 6.10]) holds for all sufficiently large $n$, see Hayman . In the special case when $p=1$ and $f(z)$ is univalent, ([source label: 6.10]) reduces to the Littlewood conjecture $|a_n|\\leq4|a_0|n$. This conjecture is somewhat weaker than the Bieberbach conjecture. It was shown by Nehari (see also Bombieri ) that we have at any rate $|a_n|\\leq 4K_0|a_0|n$ if $f(z)$ is univalent, where $K_0$ is the constant of Problem 6.2. Thus, Littlewood's conjecture holds since $K_0=1$, as was pointed out in Update 6.2. It also seems likely that for $\\lambda p>1$ \\[I_\\lambda(r,f)\\leq I_\\lambda(r,F)\\] if $f(z)$ satisfies the above hypotheses, and $F(z)$ is given by ([source label: 6.11]).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.26\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS. The $p=1$ univalent case (Littlewood's conjecture) is SOLVED via de Branges (since $K_0=1$). The general mean-$p$-valent form remains open. No full resolution located." }, { "id": 2306027, "problem_number": "AMR-022-6027", "title": "Research Problems in Function Theory — Problem 6.27", "statement": "Suppose that \\[g(z) = z + b_0 + b_1z^{-1} + \\ldots\\] is univalent in $|z|>1$. Is it true that for each positive $\\varepsilon$ we have \\[n|b_n|=O(n^\\varepsilon)\\big\\{\\max_{0<|\\nu-n|<\\frac{n}{2}}(\\nu(|b_\\nu|+1)\\big\\},\\hspace{1cm}\\text{ as }n\\to\\infty\\,?\\] This is suggested by some results of Clunie and Pommerenke. (J. Clunie, Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.27\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306028, "problem_number": "AMR-022-6028", "title": "Research Problems in Function Theory — Problem 6.28", "statement": "Suppose that $f(z) = z+\\sum^\\infty_{n=2}a_nz^n$ in $S$ and that \\mbox{$P(z) = \\sum^n_{k=0}b_kz^k$} is a polynomial of degree at most $n$. Is it true that \\[\\max_{|z|=1} |P(z)\\ast f(z)| \\leq n \\max_{|z|=1} |P(z )|?\\] Here $P \\ast f = \\sum^n_{k=0}a_kb_kz^k$. The above result would imply Rogosinski's generalised Bieberbach conjecture but is weaker than Robertson's conjecture, see Sheil-Small . (T. Sheil-Small)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.28\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306029, "problem_number": "AMR-022-6029", "title": "Research Problems in Function Theory — Problem 6.29", "statement": "With the above notation $f (z)$ in $S$ if and only if for each pair of numbers $\\xi_1, \\xi_2$ satisfying $|\\xi_1|\\leq1$, $|\\xi_2|\\leq1$, we have \\[f(z)\\ast\\frac{z}{(1-\\xi_1z)(1-\\xi_2z)}\\neq0,\\hspace{1cm}0<|z|<1.\\] On the other hand it is true that if $F(z) \\ast f(z) \\neq 0$, $0 <|z| < 1$ whenever $f$ in $S$, then $F(z)$ is star-like. What is the complete class of star-like functions having this property? $F(z) = z + z^n/n$ has the property since the Bieberbach conjecture holds. (T. Sheil-Small)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.29\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306030, "problem_number": "AMR-022-6030", "title": "Research Problems in Function Theory — Problem 6.30", "statement": "If $f$ in $S$, Baernstein has shown that \\[\\int^{2\\pi}_0|f(re^{i\\theta})|^p\\,d\\theta\\leq\\int^{2\\pi}_0|k(re^{i\\theta})|^p\\,d\\theta,\\hspace{1cm}00$, then \\[\\frac{a_n}{n}=\\lambda+O\\Big(\\frac{1}{\\log n}\\Big), \\hspace{1cm}\\text{ as }n\\to\\infty.\\] To what extent can this estimate be improved? (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.31\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306032, "problem_number": "AMR-022-6032", "title": "Research Problems in Function Theory — Problem 6.32", "statement": "Let $S_\\alpha$, $0 < \\alpha \\leq 1$ be the subclass of $S$ of functions $f$ such that $\\mathbb{C}\\setminus f(\\mathbb{D})$ is a single piecewise analytic slit from some finite point $\\omega_0$ to $\\infty$ that makes an angle at most $\\alpha \\pi/2$ with the radii vectors. What can be said about the Taylor coefficients of functions in $S_\\alpha$? If $f(-1) = \\infty$ and $f(e^{i\\phi_f}) = \\omega_0$, $-\\pi< \\phi < \\pi$, find $\\sup_{f\\in S_\\alpha} |\\phi_f|$. (K. W. Lucas)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.32\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306033, "problem_number": "AMR-022-6033", "title": "Research Problems in Function Theory — Problem 6.33", "statement": "The same questions as in Problem 6.32 can be asked under the alternative hypothesis that $\\mathbb{C}\\setminus\\{f(\\mathbb{D})\\}$ is a single piecewise analytic slit from some finite point ($\\omega_0$, say) to $\\infty$ lying in an infinite sector with opening $\\alpha\\pi$ $(0<\\alpha\\leq2)$ and vertex $\\omega_0$. (K. W. Lucas and D. A. Brannan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.33\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306034, "problem_number": "AMR-022-6034", "title": "Research Problems in Function Theory — Problem 6.34", "statement": "A function $f(z) = z + a_2z^2 +\\ldots$ analytic in $\\mathbb{D}$ is said to belong to Ruscheweyh's class $M$ if the $*$ (i.e. Hadamard) convolution of $f$ with every (normalised) convex function is univalent. All close-to-convex functions lie in $M$. Suppose that $g(z) = z + b_2z^2 +\\ldots$ is analytic in $\\mathbb{D}$ and satisfies the condition $$ \\text{Re}\\,\\Big\\{\\frac{\\phi*(gF)}{\\phi*g}\\Big\\}>0\\hspace{1cm}|z|<1, $$ for all normalised convex functions $\\phi$, and all normalised functions $F$ of positive real part in $\\mathbb{D}$. If $g$ is star-like, then ([source label: C6.1]) certainly holds; is ([source label: C6.1]) true for any other $g$, or (maybe) for some significant larger family of $g$? If $g$ satisfies ([source label: 6.2]) and the condition \\[\\text{Re}\\, \\Big(\\frac{zf'}{g}\\Big) > 0,\\hspace{1cm} |z| < 1,\\] then $f\\in M$; does this classify $M$? (T. Sheil-Small)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.34\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306035, "problem_number": "AMR-022-6035", "title": "Research Problems in Function Theory — Problem 6.35", "statement": "Let $\\mathbb{O}$ be a subset of $\\mathbb{D}=\\{|\\omega|< 1\\}$. Find a characterisation of those $\\mathbb{O}$ that are of the form $(\\mathbb{C}\\setminus f(\\mathbb{D}))\\cap\\mathbb{D}$ for some $f$ in $S$. What is the maximum area of $\\mathbb{O}$ ? Given that $\\omega_1, \\omega_2\\in\\{|\\omega|<1\\}$, how does one tell whether there exists such an $f$ with $\\omega_1, \\omega_2 \\in\\mathbb{O}_f$? The same question could be asked for $\\omega_1, \\omega_2, \\omega_3$ etc. (A. W. Goodman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.35\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306036, "problem_number": "AMR-022-6036", "title": "Research Problems in Function Theory — Problem 6.36", "statement": "Suppose that $f$ in $S$ and define \\[f_p(z)=[f(z)]^p=z^p+\\sum^\\infty_{n=p+1}a_{n, p}z^n.\\] What can be said about bounds for $a_{n,p}$? If $|a_{n,1}|\\leq Kn$ for all $n$ and fixed $K$, then, for integral $p$, \\[|a_{n,\\,p}|\\leq K^p\\frac{2p(2p+1)\\ldots(n+p-1)}{(n-p)!},\\] but it might be easier to obtain bounds for $f_p$ than for $f$. (W. K. Hayman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.36\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306037, "problem_number": "AMR-022-6037", "title": "Research Problems in Function Theory — Problem 6.37", "statement": "Suppose that $f(z)=z + c_3z^3 + c_5z^5 +\\ldots$ is an odd univalent function in $\\mathbb{D}$, and let $d_n = |c_{2n+1}|-|c_{2n-1}|$. It is known that $d_n\\to0$, the best known estimate being $d_n =O(n^{1-\\sqrt{2}})$; can this be improved to $$ d_n=O(n^{-\\frac{1}{2}})\\,? $$ (Nothing better is possible, as is shown by the fourth-root transform of the Koebe-function). Milin (see ; ; and ) proves that $d_n\\leq K(\\alpha)n^{-\\frac{1}{2}}$ for functions $f$ such that $g(z) = [f(z^{\\frac{1}{2}})]^2 = z + a_2z^2 +\\ldots$ is univalent in $\\mathbb{D}$ and has positive Hayman number $\\alpha = \\lim_{n\\to\\infty} n^{-1}|a_n|$; but $K(\\alpha)\\to\\infty$ as $\\alpha\\to0$. Levin showed that $d_n=O(n^{-\\frac{1}{2}}\\log n)$ if $c_n$ vanishes for $n\\not\\equiv1\\pmod 4$. (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.37\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306038, "problem_number": "AMR-022-6038", "title": "Research Problems in Function Theory — Problem 6.38", "statement": "With the notation of Problem 6.37, is it true that \\[\\sum^\\infty_{n=1}n^{-\\beta}d_n^2<\\infty\\] where $\\beta=(\\sqrt{2}-1)^2$? (K. W. Lucas)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.38\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306039, "problem_number": "AMR-022-6039", "title": "Research Problems in Function Theory — Problem 6.39", "statement": "Suppose $f$ in $S$ and define $h(z) = \\{f(z^2)\\}^{\\frac{1}{2}} = z + c_3z^3 + c_5z^5 +\\ldots$ Robertson's conjecture (see Sheil-Small ) asserts that \\[1+|c_3|^2+|c_5|^2+\\ldots+|c_{2n-1}|^2\\leq n.\\] This is known to be true if $f$ is star-like; is it true if $f$ has real coefficients, or if $f$ is close-to-convex? (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.39\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED: Robertson's conjecture holds (de Branges). Literature status: SOLVED-IN-LITERATURE. Robertson's conjecture was proved by de Branges as part of his proof of the Bieberbach conjecture (the Robertson conjecture for all $n$ is a consequence of de Branges' inequalities). So $1+\\sum|c_{2k-1}|^2\\le n$ holds in general." }, { "id": 2306040, "problem_number": "AMR-022-6040", "title": "Research Problems in Function Theory — Problem 6.40", "statement": "If $f(z)$ in $S$ and if the $a_n$ are real, then $$ 1+a_3+\\ldots+a_{2n-1}\\geq a_n^2,\\hspace{1cm}n\\geq1. $$ The Bieberbach conjecture for such functions follows easily, see e.g. Fitzgerald . It was pointed out by Clunie and Robertson that ([source label: C6.2]) holds for normalised typically-real functions; the inequality is clear from the representation formula for these functions. The Bieberbach conjecture for $S$ would follow analogously if we could prove that if $f$ in $S$, $$ 1+|a_3|+\\ldots+|a_{2n-1}|\\geq|a_n|^2,\\hspace{1cm}n\\geq1. $$ Bshouty (unpublished) has shown that if $f$ in $S$, then there exists an $N(f)$ such that ([source label: C6.3]) holds for $n > N(f)$. (C. Fitzgerald)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.40\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (the general form for all $n$). No definitive resolution located." }, { "id": 2306041, "problem_number": "AMR-022-6041", "title": "Research Problems in Function Theory — Problem 6.41", "statement": "Let $K(\\alpha)$ and $S^*(\\alpha)$ be those subsets of $S$ consisting of the class of functions convex in $\\mathbb{D}$ of order $\\alpha$ i.e. \\[\\text{Re}\\,\\Big[1+\\frac{zf''(z)}{f'(z)}\\Big]\\geq\\alpha,\\hspace{1cm}|z|<1,\\] and star-like of order $\\alpha$ in $\\mathbb{D}$ i.e. \\[\\text{Re}\\,\\Big[z\\frac{f'z)}{f(z)}\\Big]\\geq\\alpha,\\hspace{1cm}|z|<1,\\] respectively. [(a)] ; Prove that (see Goel ) \\[\\min_{f\\in K(\\alpha)}\\,\\min_{|z|=r}\\Big|\\frac{zf'(z)}{f(z)}\\Big|=\\min_{f\\in K(\\alpha)}\\,\\min_{|z|=r}\\Big[\\text{Re}\\,\\frac{zf'(z)}{f(z)}\\Big].\\] ; Show that the functions \\[(2\\alpha-1)^{-1}[1-(1-z)^{2\\alpha-1}],\\hspace{1cm}\\alpha\\neq\\frac{1}{2}\\hspace{1cm}\\text{ and }\\hspace{1cm}-\\log(1-z)\\] are star-like of order \\[4^\\alpha(2\\alpha-1)[4-4.2^\\alpha]^{-1}\\hspace{1cm}\\text{ and }\\hspace{1cm}(\\log 4)^{-1}\\] respectively, see MacGregor . Either $(a)$ or $(b)$, combined with the work of I. S. Jack would solve the following problem of F. R. Keogh: find \\[\\max_{f\\in K(\\alpha)}\\{\\beta: f\\in S^*(\\beta)\\}.\\] (D. Benjamin)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.41\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306042, "problem_number": "AMR-022-6042", "title": "Research Problems in Function Theory — Problem 6.42", "statement": "If $f$ in $S$, write \\[\\log[f(z)/z]=2\\sum^\\infty_{k=1}\\gamma_kz^k.\\] If $f$ is star-like then $|\\gamma_k| \\leq1/k$; this is false in general, even in order of magnitude. Milin (see ) has shown that \\[\\sum^n_{k=1}k|\\gamma_k|^2\\leq\\sum^n_{k=1}\\frac{1}{k}+\\delta\\] where $\\delta<0.312$, and that $\\delta$ cannot be reduced to $0$; Milin conjectured that $$ \\sum^N_{n=1}\\sum^n_{k=1}k|\\gamma_k|^2\\leq\\sum^N_{n=1}\\sum^n_{k=1}\\frac{1}{k}, $$ which would imply Robertson's conjecture (see Problem 6.39). Inequality ([source label: C6.4]) is known to be true for $N = 1, 2, 3$ (see Grin\\v span ). Is it true in general if $f$ has real coefficients, or if $f$ is close-to-convex? (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.42\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED-IN-LITERATURE (the Milin conjecture underlying this problem). Literature status: The strong Milin/Lebedev–Milin conjecture is SOLVED (de Branges, as part of Bieberbach). The specific real-coefficient/close-to-convex variant questions are thus partly answered." }, { "id": 2306043, "problem_number": "AMR-022-6043", "title": "Research Problems in Function Theory — Problem 6.43", "statement": "Using the notation of Problem 6.42, it is well-known that \\[\\Big|\\sum^\\infty_{k=1}k\\gamma_kz^k\\Big|=O\\Big(\\frac{1}{1-r}\\Big),\\hspace{1cm}r\\to1-,\\] for $|z| = r < 1$; is it true that \\[\\sum^\\infty_{k=1}k|\\gamma_k|r^k=O\\Big(\\frac{1}{1-r}\\Big),\\hspace{1cm}r\\to1-\\,?\\] (D. Aharonov)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.43\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306044, "problem_number": "AMR-022-6044", "title": "Research Problems in Function Theory — Problem 6.44", "statement": "Let $f$, $g$ be formal power series \\[\\sum^\\infty_{n=0}a_nz^n,\\hspace{1cm} \\sum^\\infty_{n=0}b_nz^n\\] respectively, and define \\[(f\\otimes g)(z)=\\sum^\\infty_{n=1}a_nb_nn^{-1}z^n.\\] Let $S_R$ denote the class of functions in $S$ with real coefficients. Prove (or disprove) that $f,g\\in S_R$ implies that $f\\otimes g\\in S_R$. (Robertson (uncited) has proved the corresponding result for typically-real functions. (J. G. Krzyz)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.44\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306045, "problem_number": "AMR-022-6045", "title": "Research Problems in Function Theory — Problem 6.45", "statement": "Let $S^*(\\alpha)$ be the class of $\\alpha$-strongly-star-like functions $f$, that is, those $f$ in $S$ for which \\[\\Big|\\arg\\Big(\\frac{zf'(z)}{f(z)}\\Big)\\Big|<\\frac{\\alpha\\pi}{2}\\hspace{1cm}\\text{ for } |z|<1,\\] where $0 < \\alpha < 1$. [(a)] ; Prove (or disprove) that $S^*(\\alpha)$ is closed under $\\otimes$ (see Problem 6.44). ; Prove (or disprove) that, if $f\\in S^*(\\alpha)$ and $g \\in S^*(\\beta)$, then $f\\otimes g\\in S^*(\\gamma)$ where $\\gamma=\\gamma(\\alpha, \\beta)<1$. One could ask the same question for different convolutions in place of $\\otimes$. (J. G. Krzyz)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.45\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306046, "problem_number": "AMR-022-6046", "title": "Research Problems in Function Theory — Problem 6.46", "statement": "Suppose that $f$ in $S$ and is star-like. Is it true that $$ \\big||a_{n+1}|-|a_n|\\big|\\leq1? $$ This is certainly true if $\\lim_{r\\to1} (1-r)M(r,f) > 0$, (D. A. Brannan, unpublished). T. Sheil-Small (uncited) has obtained an upper bound $2$ in ([source label: C6.5]). Notice that in addition to the `obvious' extremal functions $z(1-z^2)^{-1}$ and $z(1-z)^{-2}$ we have $z(1+z+z^2)^{-1}$. (J. G. Clunie)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.46\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306047, "problem_number": "AMR-022-6047", "title": "Research Problems in Function Theory — Problem 6.47", "statement": "If $f$ in $S$ and $f'$ is also univalent in $\\mathbb{D}$, what can be said about $\\max|a_n|$, $n \\geq 2$? The function $z(1-z)^{-1}$ shows that $\\max|a_n|\\geq1$ whenever $n\\geq2$. (J. G. Clunie)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.47\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306048, "problem_number": "AMR-022-6048", "title": "Research Problems in Function Theory — Problem 6.48", "statement": "Suppose that $f$ in $S$. The coefficient problem, except in certain cases, remains open for each of the following subclasses of univalent functions. (We limit ourselves to one-parameter families). [(a)] ; $B(\\alpha)$, the class of Basilevi\\^c functions comprising functions $f$ such that \\[f(z)=\\Big[\\int^z_0 p(t)s^\\alpha(t)t^{-1}\\,dt\\Big]^{1/\\alpha},\\] where $\\alpha> 0$, $p(t) = 1+p_1t+\\ldots$ is analytic and of positive real part in $|t|<1$ and $s(t) = t + s_2t^2 +\\ldots$ is star-like in $|t| < 1$. ; $M(\\alpha)$, the class of Mocanu-Reade functions comprising functions $f$ such that \\[\\text{Re}\\,\\Big[\\alpha\\Big(1+\\frac{zf''}{f'}\\Big)+(1-\\alpha)\\frac{zf'}{f}\\Big]>0,\\] where $0 <\\alpha < 1$. ; $S^*(\\alpha)$, the class of strongly-star-like functions, comprising functions $f$ such that \\[\\Big|\\arg\\Big(\\frac{zf'}{f}\\Big)\\Big|<\\frac{\\alpha\\pi}{2},\\] where $0 < \\alpha < 1$. (D. A. Brannan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.48\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (partial sharp results for special $\\alpha$ exist). No definitive complete resolution located." }, { "id": 2306049, "problem_number": "AMR-022-6049", "title": "Research Problems in Function Theory — Problem 6.49", "statement": "What are the extreme points of the following classes of functions? [(a)] ; Basilevi\\^c functions (see Problem 6.48). ; $S^*(\\alpha)$ (see Problem 6.48). ; Close-to-convex functions of order $\\alpha$, $0 < \\alpha < 1$. ; Functions of boundary rotation $k\\pi$, $2 < k < 4$. (D. A. Brannan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.49\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (some partial results). No definitive complete resolution located." }, { "id": 2306050, "problem_number": "AMR-022-6050", "title": "Research Problems in Function Theory — Problem 6.50", "statement": "If $0 \\le \\alpha \\le 1$, and $f(z)$, $g(z)\\in \\Sigma$, and if we define $F(z)$ by \\begin{eqnarray} F(z)&=&f(z)^{1-\\alpha}g(z)^\\alpha, \\hspace{1cm}|z|>1; &=&z+\\sum^\\infty_{n=0}A_nz^{-n}, \\hspace{1cm} |z|\\text{ sufficiently large}; \\end{eqnarray} is it true that \\[\\sum^\\infty_{n=1}n|A_n|^2\\le 1?\\] See Thomas . (D. K. Thomas)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.50\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306051, "problem_number": "AMR-022-6051", "title": "Research Problems in Function Theory — Problem 6.51", "statement": "Let $D$ be a domain in $\\mathbb{C}$ (containing the origin) of connectivity $n$, and let $S(D)$ be the class of analytic univalent functions in $D$ with $f(0) = 0$, $f'(0) = 1$. Find the functions $f$ in $S(D)$ that minimise \\[\\int_D\\int|f'(z)|^2\\,d\\sigma_z.\\] (D. Aharonov)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.51\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306052, "problem_number": "AMR-022-6052", "title": "Research Problems in Function Theory — Problem 6.52", "statement": "Suppose that $f(z)$ is analytic in $\\mathbb{D}$, and has the whole complex plane as its range. Does there necessarily exist a bounded univalent function $g(z)$ in $\\mathbb{D}$ such that $f(z)+g(z)$ has the whole complex plane as its range? (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.52\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306053, "problem_number": "AMR-022-6053", "title": "Research Problems in Function Theory — Problem 6.53", "statement": "Hentgartner and Schobe and Goodman and Saff have shown that if $f(z) = z + a_2z^2 +\\ldots$ maps $\\mathbb{D}$ univalently onto a domain $G_1$ that is convex in the direction of the imaginary axis (CIA), then $G_r = \\{f(|z| < r)\\}$ is not necessarily CIA for all $r$, $r < 1$, or even for $r$ bigger than some constant. [(a)] ; Find $\\sup\\{r : G_r \\text{ is necessarily CIA}\\}$. (A result of Goodman and Saff suggests that this is $\\sqrt{2}-1$.) ; Find reasonable sufficient conditions on $G_1$ (or, equivalently, on $f$) that imply that $G_r$ is in CIA for all $r$ in $(0,1)$. ; Suppose that we slit $G_1$ along the real axis and let $G'$, $G''$ be those two components of the resulting family of domains that have $0$ on their boundaries. If $G_1 = G' \\cup G''$, does it necessarily follow that $G_r$ is CIA for all $r$ in $(0,1)$? ; If $G_1$ is CIA but $G_{r_0}$ is not CIA, is it true that $G_r$ is not CIA for $r_0 < r < 1$? (D. A. Brannan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.53\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306054, "problem_number": "AMR-022-6054", "title": "Research Problems in Function Theory — Problem 6.54", "statement": "Let $D$ be a Jordan domain with boundary $C$, $\\{F_n(z)\\}^\\infty_1$ the sequence of Faber polynomials for $D$, and $S(D)$ the class of univalent functions \\[f(z)=F_1(z)+\\sum^\\infty_{n=2}a_nF_n(z)\\] in $D$. Does the coefficient region of at least one $a_n$ have the same shape as $C$, or is it at least in the subclass of star-like functions? Royster has shown that, if $f$ is star-like in an ellipse, then there exists a direction $\\theta_f$ and sequences \\[\\{\\lambda_n\\}^\\infty_2,\\hspace{1cm}\\{\\mu_n\\}^\\infty_2,\\hspace{1cm}\\lambda_n>\\mu_n>0,\\] such that each coefficient $a_n$ lies in an ellipse of centre $0$, major axis $\\lambda_n$, minor axis $\\mu_n$, inclined at an angle $\\theta_f$ to the real axis. (The $\\lambda_n, \\mu_n$ are known.) (K. W. Lucas)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.54\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306055, "problem_number": "AMR-022-6055", "title": "Research Problems in Function Theory — Problem 6.55", "statement": "Let $f(z)$ be a normalised bounded star-like function in $\\mathbb{D}$, and set \\[f(\\xi)=\\lim_{r\\to1-}f(r\\xi),\\] where $|\\xi|=1$, $\\xi\\in E$, $E\\subset\\{|z|=1\\}$. Is it true that $f(E) = \\{f(\\xi):\\xi\\in E\\}$ has zero linear (= one-dimensional Hausdorff) measure if $\\text{cap } E = 0$, or at least if $E$ has zero logarithmic measure? (It is known that $\\text{meas} \\{\\arg f(\\xi):\\xi\\in E\\}=0$ if $\\text{cap } E=0$.) (Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.55\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306056, "problem_number": "AMR-022-6056", "title": "Research Problems in Function Theory — Problem 6.56", "statement": "Let $S_R(q)$ be the class of normalised univalent functions in $\\mathbb{D}$ with real coefficients that admit a quasi-conformal extension to the whole plane with complex dilatation bounded (in modulus) almost everywhere by $q$, $q < 1$. Following the notation in Problem 6.44, prove (or disprove) that, if $f\\in S_R(p)$ and $g\\in S_R(q)$, then $f\\otimes g\\in S_R(r)$. (J. G. Krzyz)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.56\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306058, "problem_number": "AMR-022-6058", "title": "Research Problems in Function Theory — Problem 6.58", "statement": "Following the notation in Problem 6.57, the well-known Golusin inequality for functions $f$ in $\\Sigma(q)$ (defined in Problem 6.57) is: $$ \\Big|\\log\\frac{f'(z)f'(\\zeta)(z-\\zeta)^2}{[f(z)-f(\\zeta)]^2}\\Big|\\leq q\\log\\frac{|z\\overline{\\xi}-1|^2}{(|z|^2-1)(|\\zeta|^2-1)}. $$ Prove (or disprove) that ([source label: C6.6]) is also sufficient for $f$ to have a quasi-conformal extension to the whole plane, possibly for $q$ sufficiently small. (J. G. Krzyz)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.58\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306059, "problem_number": "AMR-022-6059", "title": "Research Problems in Function Theory — Problem 6.59", "statement": "Let $D$ be a plane domain containing $\\infty$. Let there be given a continuous assignment of numbers (thought of as angles) to the components of $\\mathbb{C}\\setminus D$. Consider conformal mappings of $D$ onto the complement in the extended plane of straight line segments. Show that one of these mappings is such that the straight line segments make angles with the positive real axis equal to the corresponding preassigned angles. In other words, under the mapping, each component of $\\mathbb{C}\\setminus D$ is associated with a slit in its preassigned direction. (B. Rodin; communicated by C. FitzGerald)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.59\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306060, "problem_number": "AMR-022-6060", "title": "Research Problems in Function Theory — Problem 6.60", "statement": "Let $C$ be a closed Jordan curve. Then if $f(z) = z + a_2z^2 +\\ldots$, $g(z) =z^{-1}+b_0 +b_1z + \\ldots$ map $\\mathbb{D}$ onto the inside and outside of $C$ respectively, the area principle shows that $C$ is the unit circumference $\\mathbb{T}$. If we remove the normalisation on $g$ and replace $g$ by $g_1(z) = b_{-1}z^{-1} + b_0 + b_1z + \\ldots$, what can be said about the connection between $f$ and $g_1$? For example, what about the asymptotic behaviour of their coefficients, or their Lipschitz behaviour on $\\mathbb{T}$, or about prime ends (if we make $C$ slightly non-Jordan)? (D. A. Brannan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.60\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306061, "problem_number": "AMR-022-6061", "title": "Research Problems in Function Theory — Problem 6.61", "statement": "Let $D_1, D_2$ be Jordan domains bounded by rectifiable curves $C_1, C_2$ of equal length. Suppose that an isometric sewing of $C_1$ and $C_2$ is everywhere conformally admissible, thus generating a Riemann surface equivalent to a sphere $S$. Is the curve $C$ on $S$ which corresponds to $C_1$ (and $C_2$) necessarily rectifiable? (A. Huber)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.61\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306062, "problem_number": "AMR-022-6062", "title": "Research Problems in Function Theory — Problem 6.62", "statement": "Let $D_1$ and $D_2$ be bounded Jordan domains, bounded by curves $C_1$ and $C_2$ of bounded boundary rotation (in the sense of Paatero, see e.g. Noonan ); then $C_1$ and $C_2$ are rectifiable, and we shall assume that they have the same length. It is known that in this case, every isometric sewing is conformally admissible and generates a Riemann surface which is equivalent to a sphere $S$. It follows from results of Aleksandrov (no citation) and Reshetnjak (no citation) that the curve $C$ on $S$ which corresponds to $C_1$ (and $C_2$) is of bounded boundary rotation. Can one find a function-theoretic proof of this? (A. Huber)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.62\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: The statement is known (Aleksandrov–Reshetnjak); the request is for a function-theoretic proof. Open as of Hayman's 2018 edition. No function-theoretic proof located." }, { "id": 2306063, "problem_number": "AMR-022-6063", "title": "Research Problems in Function Theory — Problem 6.63", "statement": "Let $\\alpha$ be a homeomorphic mapping of $(0, \\infty)$ onto $(\\alpha(0), \\infty)$, $\\alpha(0)\\geq 0$, such that $x\\to\\alpha(x)+i$ defines a conformal sewing of the half-strip \\[H = \\big\\{z:\\text{Re}\\, z > 0, 0 \\leq \\text{Im}\\,(z) \\leq 1\\big\\}.\\] If $\\alpha$ is hyperbolic (that is, if the generated Riemann surface has a hyperbolic end at $\\infty$), does there exist a positive continuous function $\\varepsilon: (0,\\infty)\\to\\mathbb{R}$ with the property that each conformal sewing $x\\to\\beta(x)+i$ of $H$ satisfying the inequality \\[|\\beta(x)-\\alpha(x)|<\\varepsilon(x),\\hspace{1cm}00\\] for $|z|<1$. Note that $M_0=S^*$, the class of star-like functions and $M_1=K$, the class of convex functions. It is known that $M_\\alpha\\subset S^*$ for all $\\alpha$. Clunie and Keogh have shown the following: [(a)] ; If $\\sum^\\infty_{n=2}n|a_n|<1$ then $f\\in M_0$ ; If $\\sum^\\infty_{n=2}n^2|a_n|<1$ then $f\\in M_1$. What generalisation of these conditions implies that $f$ belongs to $M_\\alpha$? (S. Miller)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.64\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306065, "problem_number": "AMR-022-6065", "title": "Research Problems in Function Theory — Problem 6.65", "statement": "Given $M$, $11\\}$ with the usual normalisation $f(z)=z+\\sum^\\infty_{n=0}b_nz^{-n}$. Let $S_f$ denote the Schwarzian derivative of $f$. Does \\[\\sup_{|z|>1}(|z|^2-1)^2\\big|S_{f_n}(z)-S_f(z)\\big|\\to0\\hspace{1cm}\\text{ as }n\\to\\infty\\] imply that \\[\\sup_{|z|>1}(|z|^2-1)\\Big|\\frac{f_n''(z)}{f_n'(z)}-\\frac{f''(z)}{f'(z)}\\Big|\\to0\\hspace{1cm}\\text{ as }n\\to\\infty,\\] if $f, f_n\\in\\Sigma$ $(n=1, 2, \\ldots)$? This is known to be true if $f$, $f_n$ have quasiconformal extensions onto the plane. (J. Becker)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.68\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306069, "problem_number": "AMR-022-6069", "title": "Research Problems in Function Theory — Problem 6.69", "statement": "Let $B$ be the Banach space of analytic functions $\\phi$ in $\\{|z|>1\\}$ with finite norm \\[\\|\\phi\\|:=\\sup_{|z|>1}(|z|^2-1)|z\\phi(z)|.\\] Let $S$ and $T$ be the subsets defined by $S:=\\{f''/f':f\\in \\sum\\}$ and $T:=\\{f''/f':f\\in\\sum, f\\text{ has a quasiconformal extension to }\\mathbb{C}\\}$. It is known that $T$ is topologically equivalent to the universal Teichm\\\"uller space. From results of Ahlfors and Gehring (no citations), it follows that $T$ is a subdomain in $B$ and that $S\\setminus\\overline{T}\\neq\\emptyset$, where $\\overline{T}$ denotes the closure of $T$. Is it also true (analogous to another result of Gehring (no citation)) that $T=S^0$, where $S^0$ denotes the interior of $S$? This would follow if the answer to Problem 6.68 were affirmative. The problem is closely related to a characterisation of quasicircles given by Gehring. (J. Becker)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.69\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306070, "problem_number": "AMR-022-6070", "title": "Research Problems in Function Theory — Problem 6.70", "statement": "Is every extreme point of $S$ a support point? Is every support point an extreme point? (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.70\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (partial results exist). No definitive resolution located." }, { "id": 2306071, "problem_number": "AMR-022-6071", "title": "Research Problems in Function Theory — Problem 6.71", "statement": "For each $f$ in $S$, it can be shown that \\[\\int^{2\\pi}_0\\Big|\\frac{f'(Re^{i\\theta})}{f(Re^{i\\theta})}\\Big|^2d\\theta=O\\Big(\\frac{1}{1-R}\\log\\frac{1}{1-R}\\Big)\\] as $R\\to1$. Hayman has constructed an example showing that `$O$' cannot be replaced by `$o$'. For certain subclasses of $S$, such as the star-like functions and the functions with positive Hayman index (including all support points of $S$), the estimate can be improved to $O(1/(1-R))$. Can the same improvement be made for the extreme points of $S$?; for close-to-convex functions? (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.71\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306072, "problem_number": "AMR-022-6072", "title": "Research Problems in Function Theory — Problem 6.72", "statement": "Let $\\Gamma$ be the analytic arc omitted by a support point of $S$. Must $\\Gamma$ have monotonic argument? Must the angle between the radius and tangent vectors be monotonic on $\\Gamma$? (The second property implies the first. Brown has shown that the support points associated with point-evaluation functions have both properties.) (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.72\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306073, "problem_number": "AMR-022-6073", "title": "Research Problems in Function Theory — Problem 6.73", "statement": "Let $f(z)=z+\\sum^\\infty_{n=2}a_nz^n$ be in $S$. Is it true that \\[\\limsup_{n\\to\\infty}\\big||a_{n+1}|-|a_n|\\big|\\leq1?\\] Hamilton has proved that this is true for odd functions in $S$, functions of maximal growth in $S$, and spiral-like functions. (D. H. Hamilton)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.73\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306074, "problem_number": "AMR-022-6074", "title": "Research Problems in Function Theory — Problem 6.74", "statement": "Suppose $f(z)=z+\\sum^\\infty_{n=2}a_nz^n$ is univalent and bounded by $M$ in $\\mathbb{D}$. Find \\[\\sup_t \\max_{0\\leq t\\leq 2\\pi}|s_n(e^{it})|,\\] where \\[s_n(z)=z+\\sum^n_{k=2}a_kz^k.\\] (F. Holland)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.74\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (related to partial-sum problems, some resolved by Pommerenke/others). No definitive resolution located." }, { "id": 2306075, "problem_number": "AMR-022-6075", "title": "Research Problems in Function Theory — Problem 6.75", "statement": "Let $\\mathcal{P}_n$ be the class of polynomials \\[P_n(z)=z+a_2z^2+\\ldots+a_nz^n\\] univalent in $\\mathbb{D}$, and let \\[A_m(n)=\\max_{\\mathcal{P}_n}|a_m|.\\] If $n$ is fixed, is it true that as $m$ increases, the quantity $A_m(n)$ increases strictly up to some index $n_0$, and then decreases strictly? Again, for fixed (but arbitrary) $n$ determine the least $n_1$ such that \\[A_m(n)\\leq1,\\hspace{1cm}n_1\\leq m\\leq n.\\] (D. A. Brannan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.75\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306076, "problem_number": "AMR-022-6076", "title": "Research Problems in Function Theory — Problem 6.76", "statement": "Let $\\mathcal{V}_n$ denote the class of polynomials \\[P_n(z)=z+a_2z^2+\\ldots+a_nz^n\\] analytic and bi-univalent in $\\mathbb{D}$ (that is, $P_n$ and $P^{-1}_n$ are both univalent in the unit disc). Determine $\\max_{\\mathcal{V}_n}|a_2|$ and $\\max_{\\mathcal{V}_n}|a_n|$. (D. A. Brannan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.76\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306077, "problem_number": "AMR-022-6077", "title": "Research Problems in Function Theory — Problem 6.77", "statement": "Let $\\mathcal{P}_n$ be the class of polynomials \\[p_n(z)=z+a_2z^2+\\ldots+a_nz^n\\] univalent in $\\mathbb{D}$. Determine \\[\\max_{p\\in\\mathcal{P}_n}\\int^{2\\pi}_0|p(e^{it})|^q\\,dt,\\hspace{1cm}0\\frac{4}{3}+0.02$ (see Styer and Wright ). Prove or disprove the statement that, if $f\\in\\sigma$, then \\[d_n\\equiv\\big||a_{n+1}|-|a_n|\\big|\\leq A\\] for some absolute constant $A$ with $0 \\frac{1}{2}$ in $\\mathbb{D}$. Is it true that, if $f\\in S^*(\\frac{1}{2})$, $r\\in (0,1)$ and $\\theta\\in[0,2\\pi)$, then $$ \\frac{1}{|f(re^{-i\\theta})|}\\int^r_0|f'(te^{i\\theta})|\\,dt\\leq\\frac{\\arcsin r}{r}? $$ (The left-hand-side of ([source label: J6.89]) is the ratio of the length of the image of a radius and the distance between the endpoints of that image (as the crow flies).) Inequality ([source label: J6.89]) is true when $f(z) = z/(1-z)$; in addition, the left-hand-side of ([source label: J6.89]) never exceeds $\\pi/2$ for any $r$ or $f$. Let $z_1,z_2$ lie in $\\mathbb{D}$. Then, in the smaller class of convex functions $f$, what can be said about \\[\\frac{1}{|f(z_1)-f(z_2)|}\\int^{z_2}_{z_1}|f'(t)|\\,|dt|?\\] (R. R. Hall)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.89\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306090, "problem_number": "AMR-022-6090", "title": "Research Problems in Function Theory — Problem 6.90", "statement": "Let $E$ be a set of positive logarithmic capacity on the unit circle $\\mathbb{T}$. Is $E$ necessarily a set of uniqueness for functions univalent in the unit disc $\\mathbb{D}$? Carleson has shown that this is false for functions $f$ analytic in $\\mathbb{D}$ for which ${\\int\\int}_{|z|<1}|f'(z)|^2\\,dx\\,dy<\\infty$. Beurling has shown that univalent functions cannot have constant boundary values on a set of positive capacity. (D. H. Hamilton)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.90\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306091, "problem_number": "AMR-022-6091", "title": "Research Problems in Function Theory — Problem 6.91", "statement": "Let $\\Omega$ be an arbitrary domain in $\\mathbb{C}$. Does there necessarily exist a set $E$ in $\\partial\\Omega$, of full harmonic measure, with the following property: for each $z$ in $E$, there exist circular arcs $C_r$ in $\\Omega$, of radius $r$ (where $r$ is small) and centred at $z$, for which \\[\\lim_{R\\to0}\\Big(\\frac{1}{\\pi R^2}\\int^R_0\\theta(C_r)r\\,dr\\Big)=\\frac{1}{2},\\] where $\\theta(\\,\\cdot\\,)$ denotes angular measure? For simply-connected domains $\\Omega$, this is a theorem of McMillan . For general domains $\\Omega$, at least there is a sequence $\\{r_n\\}$ decreasing to $0$ for which $\\theta(C_{r_n})$ approaches $\\pi/4$. (D. Stegenga and K. Stephenson)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.91\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306092, "problem_number": "AMR-022-6092", "title": "Research Problems in Function Theory — Problem 6.92", "statement": "If $\\mathbb{R}^2_+=\\{(x,y)\\in\\mathbb{R}^2:y>0\\}$, suppose that $E\\subset\\mathbb{R}^2_+$, and let $f:\\mathbb{R}^2_+\\to B^2$ be analytic and conformal where $B^2 = \\{(x, y)\\in\\mathbb{R}^2_+: x^2 + y^2 < 1\\}$; assume also that \\[\\lim_{x\\to0,\\,x\\in E}f(x)=\\alpha.\\] Lindel\\\"of's theorem shows that $\\alpha$ is an angular limit of $f$ at $0$ if $E$ is a curve terminating at $0$. Vuorinen has shown that the same conclusion holds under much weaker hypotheses on $E$; for instance, the condition that \\[\\liminf_{r\\to0}m(A\\cap(0,r))/r>0,\\] where $A = \\{|x|: x \\in E\\}$, is sufficient (see Vuorinen ). This problem concerns a converse result. Denote by $\\mathcal{F}$ the class of all analytic and conformal maps of $\\mathbb{R}^2_+$ into $B^2$. Let $K$ be a subset of $\\mathbb{R}^2_+$ with the property that, whenever \\[\\lim_{x\\to0,\\,x\\in E}f(x)=\\alpha\\hspace{1cm}\\text{ and }\\hspace{1cm}f\\in\\mathcal{F},\\] then necessarily $f$ has an angular limit $\\alpha$ at $0$. Since $f$ is conformal it follows (see Vuorinen ) that \\[\\lim_{x\\to0,\\,x\\in K_1}f(x)=\\alpha,\\] where $K_1 = \\{(x,y)\\in\\mathbb{R}^2_+ : \\rho((x,y), K) < l\\}$ and $\\rho$ is the Poincar\\'e metric of $\\mathbb{R}^2_+$. What can be said about the thickness of $K_1$ at $0$? Is it true that $$ \\limsup_{r\\to0}m\\big(B^\\prime\\cap(0,r)\\big)/r>0, $$ where $B^\\prime = \\{(x^2 + y^2)^{1/2} : (x,y)\\in K_1\\}$? It is easy to see that ([source label: J6.92]) must hold if $K$ is contained in an angle with vertex at $0$ whose closure lies inside $\\mathbb{R}^2_+$. (M. Vuorinen)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.92\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306093, "problem_number": "AMR-022-6093", "title": "Research Problems in Function Theory — Problem 6.93", "statement": "Let the function $f(z) = z + a_2z^2 + \\ldots$ map $\\mathbb{D}$ univalently onto a domain $\\Omega$, and let $F: \\Omega\\to\\mathbb{D}$ denote the inverse of $f$. Is it true that \\[\\int_{\\Omega\\cap \\mathbb{R}}|F'(x)|^p\\,dx<\\infty\\hspace{1cm}\\text{ for }1\\leq p<2?\\] Hayman and Wu and Garnett, Gehring and Jones have shown that the answer is `yes' for $p = 1$. For $p = 2$, it is possible that $\\int_{\\Omega\\cap\\mathbb{R}}|F'(x)|^2\\,dx = \\infty$; for example, when $\\Omega = \\{w: |w| < R, R > 1\\}\\setminus L$ where, for a suitably chosen $R_1$, $L = \\{(u,0): -R\\leq u \\leq-R_1 < 0\\}$. (A. Baernstein II)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.93\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open for $p\\in(1,2)$ as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306094, "problem_number": "AMR-022-6094", "title": "Research Problems in Function Theory — Problem 6.94", "statement": "Let $\\Omega$ be a simply-connected domain in $\\mathbb{C}$ with at least two boundary points, and let the function $\\phi$ map $\\Omega$ analytically and conformally onto $\\mathbb{D}$. For which values of $p$ is it true that $$ {\\int\\int}_\\Omega|\\phi'|^p\\,dx\\,dy<\\infty? $$ If $\\Omega$ is star-like or close-to-convex, ([source label: J6.94]) holds for $\\frac{4}{3} < p < 4$ and this is sharp. More generally, it is known that there is a universal constant $\\tau$, independent of $\\Omega$, with $0 < \\tau < 1$, such that ([source label: J6.94]) holds whenever $\\frac{4}{3} < p < 3 + \\tau$. Is $\\frac{4}{3}< p < 4$ the correct range for general types of $\\Omega$? (For background material and additional information, see .) (J. E. Brennan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.94\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: This is the Brennan conjecture circle. The sharp $4/32$. Based on extensive computer experiments Kraetzer conjectured that $B(p)=p^2/4$ for $|p|\\leq2$. See Garnett and Marshall . (Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.96\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: OPEN in general. Brennan's conjecture (the case $p\\ge2$ giving $B(p)=p-1$) is a celebrated open problem; proved for special domains. The $-2 p$, we have \\[|a_n|\\leq\\sum^p_{k=1}\\frac{2k(n+p)!}{(p+k)!(p-k)!(n-p-1)!(n^2-k^2)}|a_k|.\\] If the bound is true, then it is sharp in all of the variables $p$, $n$, $a_1,\\ldots, a_p$. The conjecture has been proved for large subclasses (, , ), but is still open in general. The simplest case, namely for all $2$-valent functions, is still open; but Watson has made important contributions towards proving this inequality. (A. W. Goodman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.97\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (proved for many subclasses; the general and $p=2$ cases open; Watson contributed). No definitive resolution located." }, { "id": 2306098, "problem_number": "AMR-022-6098", "title": "Research Problems in Function Theory — Problem 6.98", "statement": "The coefficients of a $p$-valent function are bounded by some function of its zeros. In particular, let the function \\[f(z)=z^q+\\sum^\\infty_{n=q+1}a_nz^n\\] be $p$-valent in $\\mathbb{D}$ and have $s$ zeros $\\beta_k$, $k= 1,2, \\ldots, s$, where $0 < |\\beta_k|<1$. Goodman conjectured that, under these hypotheses, $|a_n|\\leq |A_n|$ where $A_n$ is defined by the identity \\[F(z)=\\frac{z^q}{(1-z)^{2q+2s}}\\Big(\\frac{1+z}{1-z}\\Big)^{2t}\\prod^s_{k=1}\\Big(1+\\frac{z}{|\\beta_k|}\\Big)\\big(1+|\\beta_k|z\\big)=z+\\sum^\\infty_{n=2}A_nz^n,\\] where $t = p-q-s\\geq0$. The conjecture has been proved if $t = 0$ and $f(z)$ is $p$-valent star-like with respect to the origin . However, it is still open in general. (A. W. Goodman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.98\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306099, "problem_number": "AMR-022-6099", "title": "Research Problems in Function Theory — Problem 6.99", "statement": "A function $f(z) = z + a_2 z^2 +\\ldots$ is said to belong to the class $CV(R_1,R_2)$ if it is univalent and convex in $\\mathbb{D}$, and if on $f(\\{|z|=1\\})$ the curvature $\\rho$ satisfies the inequalities $R_1\\leq\\rho\\leq R_2$. So far, little progress has been made on the study of this class of functions (see Goodman and ); we do not even know the sharp bound for $|a_2|$ in $CV(R_1,R_2)$. (A. W. Goodman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.99\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306100, "problem_number": "AMR-022-6100", "title": "Research Problems in Function Theory — Problem 6.100", "statement": "Given two functions $f$, $g$ in the (usual) class $S$, we can form the new functions (arithmetic and geometric mean functions) \\[F(z)=\\alpha f(z)+\\beta g(z)\\hspace{1cm}\\text{ and }\\hspace{1cm} G(z)=z\\Big(\\frac{f(z)}{z}\\Big)^\\alpha\\Big(\\frac{g(z)}{z}\\Big)^\\beta,\\] where $\\alpha$, $\\beta\\in(0,1)$ and $\\alpha+\\beta=1$. It is known that, if \\[0.042\\simeq\\frac{1}{1+e^\\pi}<\\alpha,\\hspace{1cm}\\beta<\\frac{e^\\pi}{1+e^\\pi}\\simeq0.988,\\] then there are functions $f$, $g$ in $S$ such that $F$ and $G$ have valence infinity in $\\mathbb{D}$. What can be said about the `fringes' of the interval $(0,1)$? Is there some bound on the valence of $F$ and $G$ that is a function of $\\alpha$ for $0 < \\alpha\\leq 1/(1+e^\\pi)$? (A. W. Goodman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.100\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306101, "problem_number": "AMR-022-6101", "title": "Research Problems in Function Theory — Problem 6.101", "statement": "Let $K$ be a closed set of points in $\\mathbb{C}$, and let $F(K)$ denote the family of functions $f$ of the form \\[f(z)=\\sum^n_{k=1}\\frac{A_k}{z-a_k},\\] where $A_k > 0$ and $a_k\\in K$, $k=1, 2,\\ldots, n$. Find a maximal domain of $p$-valence for the class $F(K)$. For $p=1$, this problem was completely solved by Distler ; however, for $p> 1$, we do not even have a good conjecture. (See also Goodman .) (A. W. Goodman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.101\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open for $p>1$ as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306102, "problem_number": "AMR-022-6102", "title": "Research Problems in Function Theory — Problem 6.102", "statement": "Let $\\{v_n\\}^\\infty_1$ be a sequence of positive integers (which may include $\\infty$); the sequence is called a valence sequence if there is a function $f(z)$, analytic in $\\mathbb{D}$, such that $f^{(n)}(z)$ has valence $v_n$ in $\\mathbb{D}$, for $n = 1,2,\\ldots$. Find interesting necessary conditions for $\\{v_n\\}$ to be a valence sequence. Also, find sufficient conditions for $\\{v_n\\}$ to be a valence sequence. (For some results of this type, see Goodman .) (A. W. Goodman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.102\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306103, "problem_number": "AMR-022-6103", "title": "Research Problems in Function Theory — Problem 6.103", "statement": "The function \\[k(z)=2\\text{Re }\\Big(\\frac{z+\\frac{1}{3}z^3}{(1-z)^3}\\Big)=\\sum^\\infty_{n=1}\\frac{1}{3}(2n^2+1)r^n(e^{in\\theta}+e^{-in\\theta}),\\] where $z = re^{i\\theta}$, lies in the closure of $S_H$. Prove that $k$ is extremal for the coefficient bounds in $S_H$. It is known by Clunie and Sheil-Small that $|a_n| <\\frac{1}{3}(2n^2+1)$ for functions in $S_H$ with real coefficients, and that $|a_n| \\leq\\frac{1}{3}(2n^2+1)$ for functions in $S_H$ for which the domain $f(\\mathbb{D})$ is close-to-convex. (T. Sheil-Small)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.103\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306104, "problem_number": "AMR-022-6104", "title": "Research Problems in Function Theory — Problem 6.104", "statement": "It is known that, for functions $f$ in $S^0_H$, $\\{|w| < \\frac{1}{16}\\}\\subset f(\\mathbb{D})$. Prove that the correct value $d$, of the Koebe constant for the class $S^0_H$ is $\\frac{1}{6}$. Note that the function \\[k_0(z)=\\text{Re }\\Big(\\frac{z+\\frac{1}{3}z^3}{(1-z)^3}\\Big)+i\\text{Im }\\Big(\\frac{z}{(1-z)^2}\\Big)\\] belongs to $S^0_H$ and maps $\\mathbb{D}$ onto the plane cut along the real axis from $-\\frac{1}{6}$ to $-\\infty$, and that $\\frac{1}{6}$ is the correct constant for those functions $f$ in $S^0_H$ for which $f(\\mathbb{D})$ is close-to-convex. Also, determine the number \\[\\alpha=\\sup\\{|a_2|:f\\in S_H\\}.\\] The best known estimate for $\\alpha$ is $\\alpha < 57.05$, see Sheil-Small . It is also known that $d\\geq 1/(2\\alpha)$. (T. Sheil-Small)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.104\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306105, "problem_number": "AMR-022-6105", "title": "Research Problems in Function Theory — Problem 6.105", "statement": "What are the convolution multipliers $\\phi^*:K_H\\to K_H$, where $K_H$ is the subclass of functions $f$ in $S_H$ with convex images $f(\\mathbb{D})$? A particularly interesting case is the radius of convexity problem: for which values of $r$ in $(0,1)$ is the function $z\\mapsto f(rz)$ convex in $\\mathbb{D}$, when $f$ is convex in $\\mathbb{D}$? (It is known that $r\\leq\\sqrt(2)-1$, see Clunie and Sheil-Small .) (T. Sheil-Small)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.105\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306106, "problem_number": "AMR-022-6106", "title": "Research Problems in Function Theory — Problem 6.106", "statement": "Let $J$ be a Jordan curve in $\\mathbb{C}$ bounding a domain $D$. Suppose that $f:e^{it}\\mapsto f(e^{it})$ is a sense-preserving homeomorphism of the unit cirlce $\\mathbb{T}$ onto $J$, and that the harmonic extension of $f$ to $\\mathbb{D}$ satisfies the relation $f(\\mathbb{D})\\subset D$. Prove that $f$ is a homeomorphism of $\\mathbb{D}$ onto $D$. This is known to be true if $J$ is convex, when the hypothesis $f(\\mathbb{D})\\subset D$ is automatically satisfied because of the positivity of the Poisson kernel by the Kneser-Rado-Choquet theorem, see Duren . The result is also known to be true when $\\partial f/\\partial t$ is continuous and non-zero (on $\\mathbb{T}$), and when the co-analytic and analytic parts of $f$ have continuous derivatives on $\\overline{\\mathbb{D}}$. (T. Sheil-Small)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.106\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306107, "problem_number": "AMR-022-6107", "title": "Research Problems in Function Theory — Problem 6.107", "statement": "Prove that, for $f\\in S^0_H$, $$ \\big||a_n|-|a_{-n}|\\big|\\leq n,\\hspace{1cm} n=2,3,4,\\ldots. $$ (This is a generalisation of the Bieberbach conjecture for $S$.) It is known that ([source label: J6.107]) holds in the following cases: [(a)] ; when $f$ has real coefficients, see Clunie and Sheil-Small ; ; when $f(\\mathbb{D})$ is star-like with respect to the origin, see Sheil-Small ; ; when $f(\\mathbb{D})$ is convex in one direction, see Sheil-Small . (T. Sheil-Small)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.107\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306108, "problem_number": "AMR-022-6108", "title": "Research Problems in Function Theory — Problem 6.108", "statement": "Let $f$ be analytic univalent in $\\mathbb{D}$, and consider \\[I_\\lambda(r,f')=\\Big(\\frac{1}{2\\pi}\\int^{2\\pi}_0\\big|f'(re^{i\\theta})\\big|^\\lambda\\,d\\theta\\Big)^{1/\\lambda}\\] where $\\lambda > 0$. ; What is the maximal order of magnitude of $I_\\lambda(r,f')$ as $r\\to1$, where $0\\leq \\lambda\\leq\\frac{2}{5}$? If $\\lambda>\\frac{2}{5}$, it is known that $I_\\lambda(r,f') = O\\big(1/(1-r)^{3-1/\\lambda}\\big)$ with equality when $f$ is the Koebe function. The case $\\lambda >\\frac{1}{2}$ follows easily from classical facts, while the case $\\frac{2}{5}\\leq \\lambda \\leq\\frac{1}{2}$ is due to Feng and MacGregor . (It was formerly conjectured that the Koebe function would still be extremal for $\\lambda>\\frac{1}{3}$; but an example of Makarov has shown that this is not the case for $\\lambda\\leq\\frac{1}{3}+\\varepsilon$, for some positive $\\varepsilon$.) ; Now normalise the function $f$ to belong to the usual class $S$. For $\\lambda>\\frac{2}{5}$, we know that \\[I_\\lambda(r,f')\\leq C_\\lambda I_\\lambda(r,k'),\\] where $k$ is the Koebe function. For which $\\lambda$ is the best constant $C_\\lambda$ equal to $1$? (It follows from de Branges' theorem that $C_\\lambda = 1$ for \\mbox{$\\lambda = 2,4,6,\\ldots$}. Presumably $C_\\lambda=1$ for $\\lambda\\geq2$, but the proposer knows of no proof of this.) (A. Baernstein)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.108\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS (case $\\lambda>2/5$ known; $0\\le\\lambda\\le2/5$ and the $C_\\lambda=1$ question open). No full resolution located." }, { "id": 2306109, "problem_number": "AMR-022-6109", "title": "Research Problems in Function Theory — Problem 6.109", "statement": "Let $f$ be analytic univalent in $\\mathbb{D}$, and consider \\[I_{-\\lambda}(r,f')=\\Big(\\frac{1}{2\\pi}\\int^{2\\pi}_0\\big|f'(re^{i\\theta})\\big|^{-\\lambda}\\,d\\theta\\Big)^{1/\\lambda},\\] where $\\lambda>0$. Except for the elementary case $\\lambda=\\infty$, the maximal order of magnitude as $r\\to1$ is not known for any positive $\\lambda$. A particularly interesting case is when $\\lambda=2$. It seems possible that $$ I_{-2}(r,f')=O\\big(I_{-2}(r,k')\\big)=O\\big((1-r)^{-1/2}\\big) $$ might be true, where $k$ is the Koebe function. This equation ([source label: J6.109]) is slightly stronger than an earlier conjecture of Brennan (see Problem 6.94). The best known bounds are due to Pommerenke . Related information appears in . (A. Baernstein)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.109\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306110, "problem_number": "AMR-022-6110", "title": "Research Problems in Function Theory — Problem 6.110", "statement": "Let $\\Omega$ be a simply-connected domain in the finite plane whose complement contains $n$ disjoint closed balls with centres on the interval $[0,1]$ and common radius $\\varepsilon$, $\\varepsilon<1/n$, (such domains are the so-called `ball and chain domains'). Let $z_0$ in $\\Omega$ be a point at a distance at least $1$ from each ball, and let $w(z_0,\\Omega)$ denote the harmonic measure at $z_0$ of the union of the balls, relative to $\\Omega$. Is it true that, for every positive $\\delta$, there is an estimate $$ w(z_0,\\Omega)\\leq C_\\delta(n\\varepsilon)^{\\frac{1}{2}-\\delta}, $$ where $C_\\delta$ depends only on $\\delta$. An affirmative answer would imply the $L^p$ extension, for $1 < p < 2$, of the Hayman-Wu theorem mentioned in Problem 6.93. In the case $n = 1$, the Beurling-Nevanlinna projection theorem shows that ([source label: J1.110]) is true with exponent $\\frac{1}{2}$; however, for large $n$ there are examples that show that this is then false. (A. Baernstein)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.110\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "SOLVED-IN-LITERATURE (the $L^p$ Hayman–Wu extension for $1 0$, define a $\\Sigma_\\delta(f)$ neighbourhood of $f$ to be \\[\\Big\\{g:g\\in A,\\big|(g'(z)-f'(z))-\\frac{1}{z}(g(z)-f(z))\\big|+\\big|(g'(z)-f'(z))+\\frac{1}{z}|g(z)-f(z))\\big|<2\\delta\\Big\\}.\\] Clearly, $N_\\delta(f)\\subset\\sum_\\delta(f)$. Then it is known by Sheil-Small and Silvia that $\\sum_1(z)\\subset St$, and \\mbox{$\\sum_\\frac{1}{4}(f)\\subset St$} for convex functions $f$, with the notation in Problem 6.111. Given a normal family $\\mathcal{F}$ in $A$, the dual, $\\mathcal{F}^*$, of $\\mathcal{F}$ is the set \\[\\big\\{f:f\\in A,f\\ast g\\neq0\\text{ for all }g\\in\\mathcal{F},0<|z|<1\\big\\},\\] where $\\ast$ denotes the Hadamard product, see Ruscheweyh . The Bieberbach conjecture (see de Branges ) is equivalent to the statement that $N_1(z)\\subset S^*$ (where $S^*$ is the dual of $S$). It seems reasonable therefore to ask whether $\\sum_1(z)\\subset S^*$. (Recall that $S^*\\subset St$, see Ruscheweyh .) (T. Sheil-Small and E. M. Silvia)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.112\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306113, "problem_number": "AMR-022-6113", "title": "Research Problems in Function Theory — Problem 6.113", "statement": "Following the notation of Problem 6.111 and 6.112, it is known that, if $|x|\\leq\\rho\\leq1$ and $\\gamma=1/(1+\\rho)^2$, then \\[N_\\gamma\\Big(\\frac{z}{1-xz}\\Big)\\subset S^*.\\] Is it true that \\[\\Sigma_\\gamma\\Big(\\frac{z}{1-xz}\\Big)\\subset S^*?\\] Notice that, since not all convex functions belong to $S^*$ (see Sheil-Small and Silvia ), it follows that we cannot replace in the above, $N_\\gamma(z/(1-xz))$ by $N_\\gamma(g)$, where $g$ is an arbitrary convex function. (T. Sheil-Small and E. M. Silvia)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.113\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306114, "problem_number": "AMR-022-6114", "title": "Research Problems in Function Theory — Problem 6.114", "statement": "Let $\\Gamma$ be a regular curve and $f$ an analytic and conformal function in the open unit disc. Does $f^{-1}(\\Gamma)$ necessarily have finite length? (A curve $\\Gamma$ is said to be regular if the intersection of $\\Gamma$ with a disc of radius $r$ has one-dimensional measure at most $Cr$, where $C$ is a constant independent of $r$.) (J. L. Fernandez and D. Hamilton)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.114\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306115, "problem_number": "AMR-022-6115", "title": "Research Problems in Function Theory — Problem 6.115", "statement": "Let $\\Gamma$ be a rectifiable curve, and let $E$ be a subset of $\\Gamma$ having zero length. If $\\Omega$ is any simply-connected domain and $z\\in\\Omega$, is it true that the harmonic measure satisfies the equation $\\omega(z,E,\\Omega)=0$? \\textit{(B. \\O{}ksendal; R. Kaufman and J.-M. G. Wu; communicated by D. Hamilton)}", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.115\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306116, "problem_number": "AMR-022-6116", "title": "Research Problems in Function Theory — Problem 6.116", "statement": "Let $D$ be a domain in $\\mathbb{C}$ containing the origin $0$; for $t>0$, let $\\Omega_t$ be the component of $D \\cap\\{|z|\\leq t\\}$ containing $0$. In the usual notation for harmonic measure, define the function $\\omega = \\omega_D:[0, \\infty)\\to[0,1]$ by the formula \\[\\omega(t)=\\omega\\big(0,\\partial\\Omega_t\\cap\\{|z|=t\\},\\Omega_t\\big).\\] What can be said about $\\omega$ and about its relations with $D$? For instance: [(a)] ; What are necessary and sufficient conditions on a function to be $\\omega_D$ for some $D$? ; If $\\omega_{D_1 }\\equiv \\omega_{D_2}$, are $D_1$ and $D_2$ essentially the same? (That is, will $D_1$ be a rotation or reflection of $D_2$? Or will $D_1$ and $D_2$ differ only on sets of capacity zero?) What happens if $\\omega_{D_1 }\\equiv \\omega_{D_2}$ only on some subinterval of $[0,1]$? ; Given a function $\\omega_D$, can one `reconstruct' $D$? ; Can one infer such properties as connectivity of $D$ fromthe behaviour of $\\omega_D$? One might start by dealing with domains $D$ that are circularly symmetric. (K. Stephenson)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.116\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306117, "problem_number": "AMR-022-6117", "title": "Research Problems in Function Theory — Problem 6.117", "statement": "Let $G$ be a domain in $\\mathbb{C}$ that contains the origin $0$ and is axially-symmetric with respect to the real axis, that is, if a point $z\\in G$ then the line segment with endpoints $z$ and $\\bar{z}$ also lies in $G$. (In particular it follows that $G$ is simply-connected.) For $t>0$, let $G_t$ and $G_t'$ be the components of the domains \\[\\{z:z\\in G,\\text{Im }z< t\\}\\hspace{1cm}\\text{ and }\\hspace{1cm}\\{z:z\\in G,\\text{Re }z< t\\}\\] containing $0$. Let $\\omega_t$ and $\\omega_t'$ denote the harmonic measure at $0$ of the sets \\[\\partial G_t\\cap\\{\\text{Im }z=t\\}\\hspace{1cm}\\text{ and }\\hspace{1cm}\\partial G_t'\\cap\\{\\text{Re }z=t\\}\\] with respect to the domains $G_t$ and $G_t'$ respectively. Now let $D_1$ and $D_2$ be two domains in $\\mathbb{C}$ of this type, and use for each the notation just described. Then: [(a)] ; If $\\omega_t(D_1)\\equiv \\omega_t(D_2)$ for each $t>0$, is it true that $D_1 = D_2$? ; If $\\omega_t'(D_1)\\equiv \\omega_t'(D_2)$ for each $t>0$, is it true that $D_1 = D_2$? ; If both $(a)$ and $(b)$ fail, is it true that $D_1 = D_2$ if $\\omega_t(D_1)\\equiv \\omega_t(D_2)$ for each $t>0$ and $\\omega_t'(D_1)\\equiv \\omega_t'(D_2)$ for each $t>0$? If the answer to $(a)$ or $(b)$ is `yes', in that case, how few $t$'s does one need for the conclusion to hold? (For example, infinitely many $t$'s such that $\\{\\text{Im }z = t\\}$ or $\\{\\text{Re }z = t\\}$ meet the domains? Or do the $t$'s need to be dense?) If the answer to $(a)$, $(b)$ or $(c)$ is `yes', in that case can one replace `axially-symmetric' by `simply-connected', or perhaps drop this requirement completely? (D. A. Brannan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.117\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306802, "problem_number": "AMR-022-6802", "title": "Research Problems in Function Theory — Problem 6.2′", "statement": "If $A^{(p)}_n=\\sup_{f\\in S_p}|a_n|$ is it true that \\[\\frac{A^{(p)}_n}{n^{2p-1}}\\to K_p,\\hspace{1cm}\\text{ as }n\\to\\infty,\\] and if so, what is $K_p$? It is known that for a fixed $f$ in $S(p)$, the limit \\[\\alpha_f=\\lim_{n\\to\\infty}\\frac{|a_n|}{n^{2p-1}}\\] exists if $p>\\frac{1}{4}$, see Hayman and Eke .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.2′\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306807, "problem_number": "AMR-022-6807", "title": "Research Problems in Function Theory — Problem 6.7′", "statement": "Here our counter-example shows that $|a_n|=o(n^{-\\frac{1}{2}})$ is best possible for bounded $f(z)$ in $S(p)$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.7′\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE (sharpness of $o(n^{-1/2})$). Literature status: The sharpness statement is established (it is the counterexample described). Mark SOLVED-IN-LITERATURE for the sharpness fact." }, { "id": 2306808, "problem_number": "AMR-022-6808", "title": "Research Problems in Function Theory — Problem 6.8′", "statement": "If we ask the analogous problems to those of Problem 6.8 for the class $S(p)$, the correct orders of magnitude are again known in many cases, but not the exact bounds. If $f$ in $S(p)$ and $f$ is bounded, then \\[I_1(r,f')=o(1-r)^{-\\frac{1}{2}}\\] and this is sharp for the class, by our introductory example.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.8′\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (exact bounds not all determined). No definitive resolution located." }, { "id": 2306813, "problem_number": "AMR-022-6813", "title": "Research Problems in Function Theory — Problem 6.13′", "statement": "The results of Pommerenke were proved in fact for mean $p$-valent functions, and if $f(z)=\\sum^\\infty_{n=0}a_nz^n$ is mean $p$-valent with $p>\\frac{1}{4}$, then ([source label: 6.5]) holds with $\\alpha_m=-\\frac{1}{2}+8p^\\frac{3}{2}/\\sqrt{m}$. This result is probably far from the best possible, though clearly $\\alpha_m\\geq-\\frac{1}{2}$ in all cases. Lucas has shown that $\\alpha_1=2p-2$ if $p\\geq1$, and $\\alpha_1\\leq 2p-2\\sqrt{p}$ for $\\frac{1}{4}4p$. The functions \\[f(z)=\\Big(\\frac{1+z^{m+1}}{1-z^{m+1}}\\Big)^{2p/(m+1)}=1+\\sum^\\infty_{n=2}a_nz^n\\] provide counter-examples. These functions are mean $p$-valent, and $a_n=0$ except when $(m+1)$ divides $n$, and for the remaining values of $n$ we have, as $n\\to\\infty$, \\[|a_n|\\sim Cn^{2p/(m+1)-1},\\hspace{1cm}\\text{ where }C\\text{ is a constant}.\\] Thus ([source label: 6.5]) cannot hold with $\\alpha_m<2p/(m+1)-1$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.13′\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2306814, "problem_number": "AMR-022-6814", "title": "Research Problems in Function Theory — Problem 6.14′", "statement": "Here again the main conclusions extend to mean $p$-valent functions. In this case ([source label: 6.6]) holds with $j_k=-\\frac{1}{2}+16(p^3/k)^{\\frac{1}{2}}$. This is still unlikely to be best possible.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 6.14′\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307001, "problem_number": "AMR-022-7001", "title": "Research Problems in Function Theory — Problem 7.1", "statement": "Let $E$ be the compact plane set of transfinite diameter ($=$capacity) $d(E)=1$ and let \\[d_n(E)^{n(n-1)/2}=\\max_{w_\\nu\\in E}\\prod_{1\\leq\\mu<\\nu\\leq n}|w_\\mu-w_\\nu|.\\] It is known that $d_n(E)$ decreases with $n$ and $d_n(E)\\to d(E)$ as $n\\to\\infty$, so that $d_n(E)\\geq1$ is trivial. It is known by Pommerenke , that $d_n(E)\\geq n^{2/(n-1)}$ if $E$ is connected. Is this inequality true in general? Further, is it true that \\[d_n(E)^{(n-1)/2}\\leq Kn\\] if $E$ is connected, where $K$ is some absolute constant? It is known that \\[d_n(E)^{(n-1)/2}\\leq\\Big(\\frac{4}{e}\\log n +4\\Big)n,\\] in this case. (Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.1\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307002, "problem_number": "AMR-022-7002", "title": "Research Problems in Function Theory — Problem 7.2", "statement": "Let $f(z)$ be analytic in a simply-connected domain $D$. It is known that $f(z)$ can be expanded in a series of Faber polynomials \\[f(z)=\\sum^\\infty_{n=0}a_nP_n(z).\\] Find the domain of variability $V$ of $a_n$ as $f(z)$ runs through all functions analytic in $D$, and having positive real part there. It is known that if $D$ is a circle, $V$ is a circle; and if $D$ is an ellipse, then $V$ is an ellipse, see Royster . (W. C. Royster)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.2\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307003, "problem_number": "AMR-022-7003", "title": "Research Problems in Function Theory — Problem 7.3", "statement": "Let $z_i$, $1\\leq i\\leq n$ be a finite sequence of complex numbers such that $|z_i|\\leq1$. Set \\[S_k=\\sum^n_{i=1}z^k_i.\\] Can we have $$ \\max_{2\\leq k\\leq n+1}|S_k|c\\] with $c=\\frac{1}{3}$. What is the best value for the constant $c$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.4\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307006, "problem_number": "AMR-022-7006", "title": "Research Problems in Function Theory — Problem 7.6", "statement": "We consider the range of the random function \\[F(z)=\\sum^\\infty_{n=0}\\pm a_nz^n\\] ($F$ chosen at random in the natural way) defined in $\\mathbb{D}$, where $\\sum|a_n|^2=\\infty$. Is the image of $w=F(z)$ with probability one [(a)] ; everywhere dense in the plane? ; the whole plane? ; does it contain any given point with probability one? If $a_n=n^\\lambda$, $(b)$ holds if $\\lambda>\\frac{1}{2}$, and $(a)$ holds if $-\\frac{1}{2}<\\lambda<+\\frac{1}{2}$. (J. P. Kahane)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.6\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307008, "problem_number": "AMR-022-7008", "title": "Research Problems in Function Theory — Problem 7.8", "statement": "Is it possible to express each $K$-quasiconformal map in $3$-space as the composition of two quasiconformal maps with maximal dilatation less than $K$? The corresponding plane result is true. (F. W. Gehring)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.8\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307009, "problem_number": "AMR-022-7009", "title": "Research Problems in Function Theory — Problem 7.9", "statement": "Suppose that $f$ is a plane $K$-quasiconformal mapping of the unit disc $\\mathbb{D}$ onto itself. Show that there exists a finite constant $b = b(K)$ such that \\[m(f(E)) \\leq b \\{ m(E)\\}^{1/K}\\] for each measurable set $E \\subset D$. Here $m$ denotes plane Lebesgue measure. Such an inequality is known (see ) with the exponent $\\frac{1}{K}$ replaced by a constant $a=a(K)$. (F. W. Gehring)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.9\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307010, "problem_number": "AMR-022-7010", "title": "Research Problems in Function Theory — Problem 7.10", "statement": "It was proved by Boyarski\\u\\i\\, that the partial derivatives of a plane $K$-quasiconformal mapping are locally $L$-integrable for $2\\leq p < 2+c$, where $c = c(K) > 0$. Show that this is true with $c =\\frac{2}{K-1}$. The example $f(z) = |z|^{\\frac{1}{K}-1}.z$ shows that such a result would be sharp. (F. W. Gehring)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.10\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "PARTIAL-PROGRESS ($n=2$ resolved via Astala; higher dimension sharp constants open). Literature status: PARTIAL-PROGRESS. The optimal $L^p$-integrability constants $p 0$. This was shown by Gehring to be true if $f$ is injective. (F. W. Gehring)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.12\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition for general quasi-analytic (non-injective) functions. No definitive resolution located." }, { "id": 2307013, "problem_number": "AMR-022-7013", "title": "Research Problems in Function Theory — Problem 7.13", "statement": "One part of Nevanlinna theory is devoted to the following problem. How does the geometric structure of a simply connected Riemann covering surface of the sphere influence the value distribution of the meromorphic function generating the surface? It is suggested that one also consider halfsheets among the constituent pieces of such surfaces, which have an infinite number of branch points on their boundary. A typical example is the covering surface generated by $e^z - z$. It contains a right half-plane with a second order branch point at $2\\pi in$ for each integer $n$. (F. Huckeman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.13\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307014, "problem_number": "AMR-022-7014", "title": "Research Problems in Function Theory — Problem 7.14", "statement": "(Boundary values of Cauchy integrals). Let $\\gamma$ be a $C^1$ curve in the plane and let $f$ be a continuous function on $\\gamma$. Put \\[F(z)=\\int_\\gamma\\frac{f(t)}{t-z}\\,dt\\hspace{1cm}(z\\notin\\gamma).\\] Does $F$ have non-tangential boundary values almost everywhere on $\\gamma$? This is a very old question which has been studied quite extensively, especially by Russian mathematicians. It is known that the answer is `yes' if slightly greater smoothness is assumed for $\\gamma$ or $f$ (see e.g. Havin for an outstanding contribution to this subject). Suppose now that $\\gamma$ is a Jordan curve and let $\\phi$ be a conformal map from the unit disc onto the inside of $\\gamma$. Does $F\\circ\\phi$ belong to $H^p$ for some $p < 1$, or perhaps to the class $N$ of functions with bounded Nevanlinna characteristic? (A. Baernstein)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.14\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307015, "problem_number": "AMR-022-7015", "title": "Research Problems in Function Theory — Problem 7.15", "statement": "Let $D$ be a domain in the extended complex plane. A finite point $z$ on the boundary $\\partial D$ of $D$ is called angular (relative to $D$) if there exists $\\varepsilon > 0$ such that every component domain of $D\\cap\\{|z - z_0| < \\varepsilon\\}$ which has $z_0$ as a boundary point, is contained in an angle less than $\\pi$ with vertex at $z_0$. Angularity at $\\infty$ is similarly defined. Let $A = A(D)$ be the set of angular points of $\\partial D$ relative to $D$. Obviously $A$ not empty implies that $\\partial D$ has positive capacity. The set $A(D)$ can have positive linear measure. E.g. let $C$ be a Cantor set on $|z|=1$ and let $D$ consist of the open unit disc from which have been deleted all points $rz$ with $z \\in C$ and $\\frac{1}{2}\\leq r < 1$. Then $A(D) = C$ which can, of course, have positive linear measure. Yet the following holds for arbitrary domains: $A(D)$ is either empty or its harmonic measure, relative to any point of $D$, is zero. This result follows easily from an unpublished theorem on Brownian paths $\\omega(t)$ in the complex plane. This says that almost all such paths have the property that for every real $t_0$ and every $\\varepsilon > 0$ the set of numbers $\\Big|\\frac{\\omega(t)-\\omega(t_0)}{\\omega(t)-\\omega(t_0)}\\Big|$ with $t_0 0$. Suppose that $|\\alpha|<\\frac{1}{2}\\pi$ and that $(r_n)$ is a sequence of positive integers with $\\sum\\frac{1}{r_n}=\\infty$. Show that the exponential type of $f$ on the sequence $z_n = r_ne^{i\\alpha}$ is equal to the type of $f$ on the ray $z = re^{i\\alpha}$. (Proofs by Boas and by Levinson for $\\alpha = 0$ do not seem to work for $|\\alpha|>\\frac{1}{4}\\pi$.) (J. Korevaar)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.17\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307018, "problem_number": "AMR-022-7018", "title": "Research Problems in Function Theory — Problem 7.18", "statement": "Let $\\Gamma$ be a Jordan curve and suppose that $z = 0$ lies inside it. Wermer showed that when $\\Gamma$ has infinite length, the powers $z^n$, $n\\neq0$ span all of $C(\\Gamma)$. One can indicate conditions on $\\Gamma$ under which the powers $z^n$, $n\\neq n_1,\\ldots, n_k$, form a spanning set (see Korevaar and Pfluger, ). Under what conditions on $\\Gamma$ can one omit an infinite set of powers, and still have a spanning set? (J. Korevaar)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.18\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open (related to Müntz-type/rational approximation on curves). No definitive resolution located." }, { "id": 2307019, "problem_number": "AMR-022-7019", "title": "Research Problems in Function Theory — Problem 7.19", "statement": "For what sets $\\Omega$ of lattice points $(m_k, n_k)$ do the monomials $x^{m_k}y^{n_k}$ span $L^2$ or $C_0$ on the unit square $0\\leq x\\leq1$, $0\\leq y\\leq1$? It is conjectured that the condition $\\sum\\frac{1}{m_kn_k}=\\infty$ is sufficient for sets $\\Omega$ in an angle $\\varepsilon x \\leq y \\leq x/\\varepsilon$, $\\varepsilon>0$. Hellerstein has shown that the condition is not necessary. (J. Korevaar)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.19\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307020, "problem_number": "AMR-022-7020", "title": "Research Problems in Function Theory — Problem 7.20", "statement": "(Two constant theorems for the polydisc) Let $F(z_1, z_2)$ be defined for $|z_1\\leq1, |z_2|\\leq2$ except when $z_1 = z_2$ and $|z_1|=|z_2|=1$. Suppose that $F$ is plurisubharmonic in $|z_1|<1, |z_2|<1$ and that $F(z_1,z_2)\\leq\\log\\frac{1}{|z_1-z_2|}$, whenever the left hand side is defined. Further, suppose that $F(z_1, z_2) \\leq 0$ for $\\{|z_1| = |z_2| = 1, z_1 \\neq z_2\\}$. Does it follow that $F(z_1,z_2)\\leq0$ for $|z_1|<1$, $|z_2|<1$? (L. A. Rubel, A. Shields)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.20\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307021, "problem_number": "AMR-022-7021", "title": "Research Problems in Function Theory — Problem 7.21", "statement": "Suppose that $|z_k | = 1$ $(1 \\leq k < \\infty)$. Put \\[A_l=\\limsup_{m\\to\\infty}\\big|\\sum^m_{k=1}z_k^l\\big|.\\] It is easy to see that there is a sequence $z_k,$ for which $A_lcl^{\\frac{1}{2}}$ for infinitely many $l$. Is there a sequence for which $A_l = o(l)$ as $l\\to\\infty$? (P. Erd\\\"os)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.21\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307022, "problem_number": "AMR-022-7022", "title": "Research Problems in Function Theory — Problem 7.22", "statement": "Suppose that $A$ and $B$ are disjoint linked Jordan curves in $\\mathbb{R}^3$ which lie at a distance $1$ from each other. Show that the length of $A$ is at least $2\\pi$. The corresponding result with a positive absolute constant instead of $2\\pi$ is due to Gehring . (F. W. Gehring)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.22\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "SOLVED-IN-LITERATURE (length of a linked curve $\\ge 2\\pi$). Literature status: SOLVED-IN-LITERATURE. The sharp bound (length at least $2\\pi$ up to the constant) was established in the theory of linking and the \"length of linked curves\" — this is essentially tied to a theorem from differential geometry / the work on the length of nontrivially linked curves giving length $\\ge 2\\pi$ (up to exact constant resolved). Mark SOLVED-IN-LITERATURE with moderate confidence." }, { "id": 2307023, "problem_number": "AMR-022-7023", "title": "Research Problems in Function Theory — Problem 7.23", "statement": "The expression $u(z, \\zeta)$ of Problem 6.57 is closely related to the Schwarzian derivative $\\{f(z), z\\}$, e.g. it is invariant under compositions with M\\\"obius transformations and \\[\\lim_{\\zeta\\to z}u(z,\\zeta)=\\frac{1}{6}\\{f(z),z\\}\\] K\\\"uhnau (no citation) showed that if $f(z)$ is analytic in $\\mathbb{D}$ and has a quasiconformal extension to the whole plane with \\[|f_{\\overline{z}}/f_z|\\leq q<1\\hspace{1cm}\\text{ a.e.}\\] then $$ |u(z,\\zeta)|\\leq q(1-|z|^2)^{-1}(1-|\\zeta|^2)^{-1}. $$ Show that ([source label: C7.1]) is also sufficient for $f$ to have a quasi-conformal extension to the whole plane (possibly with $\\frac{1}{3}\\leq q < 1$). (J. G. Krzyz)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.23\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307024, "problem_number": "AMR-022-7024", "title": "Research Problems in Function Theory — Problem 7.24", "statement": "Let $E = \\{|z|< 1\\}$, let $0$, $a > 0$, $b = |b|e^{i\\beta}(-\\pi < \\beta\\leq\\pi)$ be distinct points of $E$; and let $\\mathcal{K}$ be the family of continua $K$ with the properties that $\\{a, b\\} \\subset K \\subset E\\setminus\\{0\\}$ and $E\\setminus K$ is connected. On any $K\\in\\mathcal{K}$ there is a continuous function $\\arg z$; we thus subdivide $\\mathcal{K}$ into homotopy classes $\\{\\mathcal{K}_n\\}$ according to the value of \\[V(K)=\\arg b-\\arg a\\in\\{\\beta+2n\\pi,n\\in\\mathbb{Z}\\}.\\] Let us call $K\\in\\mathcal{K}$ a natural continuum if $K$ is a trajectory of a quadratic differential $a$ with the following properties: [(a)] ; $0, a, b$ are simple poles of $\\sigma$, and there is no other pole of $\\sigma$ in $E$; ; $\\sigma$ is real on $\\partial E$. There are many problems that then arise, for example: [(i)] ; Do all homotopy classes contain natural continua? (There are many in $\\mathcal{K}_n$ when $|\\beta+2n\\pi| < 2\\pi$.) ; Find all natural continua in $\\mathcal{K}$. ; How does the modulus of $E\\setminus K$ vary when $K$ runs through the natural continua in $\\mathcal{K}_n$? (F. Huckemann)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.24\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307025, "problem_number": "AMR-022-7025", "title": "Research Problems in Function Theory — Problem 7.25", "statement": "Let $K$ be a compact set of positive measure in $\\mathbb{C}$. Does there necessarily exist a non-constant analytic function in $\\mathbb{C}\\setminus K$ with $f(\\infty) = 0$ such that $[f(z)-f(\\zeta)]/[z-\\zeta]\\neq\\pm1$ for any $z, \\zeta\\in\\mathbb{C}\\setminus K$? Conceivably, the hypotheses even imply the existence of non-linear analytic functions $f$ with $|f(z) —f(\\zeta)| / |z — \\zeta|$ bounded, but this is a well-known unsolved problem. Of course one can pose more general problems again such as requiring that the difference quotient omit all values in some pre-assigned plane set. (These problems arise in the use of variational methods.) (D. Aharonov and H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.25\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307026, "problem_number": "AMR-022-7026", "title": "Research Problems in Function Theory — Problem 7.26", "statement": "Is there a homeomorphism of the open unit ball in $\\mathbb{R}^3$ onto $\\mathbb{R}^3$, whose coordinate functions are harmonic? In other words, do there exist $u_1,u_2,u_3$ harmonic in $|x|<1$, $x = (x_1, x_2, x_3)$, such that \\[ (x_1, x_2, x_3)\\to (u_1, u_2, u_3)\\] is a homeomorphism of $|x| < 1$ onto all of $\\mathbb{R}^3$? The analogous problem in $\\mathbb{R}^2$ is answered negatively; the result is due to T. Rado (no citation), and is an important lemma in the theory of minimal surfaces. (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.26\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (the $\\mathbb{R}^3$ harmonic homeomorphism question). No definitive resolution located." }, { "id": 2307027, "problem_number": "AMR-022-7027", "title": "Research Problems in Function Theory — Problem 7.27", "statement": "For a domain $D$ in $\\mathbb{C}$, define \\[\\rho(x,y) = \\sup\\{|f(x)-f(y)| : x,y\\in D; f \\text{ analytic in }D; |f'|\\leq1\\text{ in }D\\}.\\] If $D$ is convex, then $\\rho(x,y) = |x-y|$, but not otherwise. Clearly $\\rho(x,y)\\leq L(x,y)$, the infimum of the lengths of paths in $G$ that join $x$ to $y$. What can be said about $\\rho$ for general $D$, in terms of the geometry of $D$? (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.27\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307028, "problem_number": "AMR-022-7028", "title": "Research Problems in Function Theory — Problem 7.28", "statement": "Suppose that $f(z)$ is continuous in a domain $D$ and that either [(i)] ; $\\int_{|\\zeta-z|=r}f(\\zeta)\\,d\\zeta=0$ for all $z\\in D$ and $00,\\hspace{1cm}0\\leq a_n\\leq n,\\hspace{1cm} n\\geq1,\\hspace{1cm} b_n=\\sum^n_{\\nu=1}a_\\nu,\\hspace{1cm} c_n=\\sum^n_{\\nu=1}b_\\nu.\\] Then $\\sum(a_n/c_n)^\\alpha<\\infty$ if $\\alpha > \\frac{1}{2}$. For what other functions $f(t)$ is it true that $\\sum f(c_n/a_n)<\\infty$? Is it true, for instance, that (under some smoothness condition on $f$) $\\sum f(c_n/a_n)$ converges with $\\sum f(n^2)$? The analogous result for $c_n/b_n$ was obtained by Borwein , that if $xf(x)$ is positive and non-increasing for $x\\geq a>0$ and $\\sum f(n)<\\infty$, then $\\sum f(c_n/b_n)<\\infty$. (W. K. Hayman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.31\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307032, "problem_number": "AMR-022-7032", "title": "Research Problems in Function Theory — Problem 7.32", "statement": "Let $\\mu(t)$ be a continuous monotonic increasing function of $t$ for $t\\in[0,1]$ and let $\\omega_1(h,\\mu)$, $\\omega_2(h,\\mu)$ denote its modulus of continuity and modulus of smoothness respectively. It is known that, if \\[\\omega_1(h)=O(h)\\hspace{1cm}(h\\to0),\\] or \\[\\omega_2(h)=O(h(\\log 1/h)^{-c})\\hspace{1cm}(h\\to0),\\] where $c > \\frac{1}{2}$, then $\\mu(t)$ is absolutely continuous (w.r.t. Lebesgue measure). It is also known that each of these conditions is essentially best-possible. Are they simultaneously best possible? More precisely, is it true that given any function $\\phi(t)\\uparrow\\infty$ as $t\\downarrow0$, there is a continuous, monotonic increasing singular function $\\mu(t)$ such that \\[\\omega_1(h)=O(h\\phi(h))\\hspace{1cm}(h\\to0)\\] and \\[\\omega_2(h)=O(h(\\log 1/h)^{-\\frac{1}{2}})\\hspace{1cm}(h\\to0)?\\] (J. M. Anderson)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.32\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307033, "problem_number": "AMR-022-7033", "title": "Research Problems in Function Theory — Problem 7.33", "statement": "Let $P(\\theta)=\\sum^N_{n=1}e^{i\\lambda_n\\theta}$ be a finite Dirichlet series with exponents $\\gamma_m\\neq\\gamma_n$ for $m\\neq n$. What can be said about $\\mu\\equiv\\inf|P(\\theta)|$? A trivial argument shows that $\\mu\\leq (N-1)^{1/2}$. In fact, $|P|^2=N+2\\sum_{m\\neq n}\\cos(\\lambda_m-\\lambda_n)\\theta$. If $w(\\theta)=|P|^2-N\\geq-c$, then let $h(\\theta)=c+w(\\theta)=2\\sum b_n\\cos(\\delta_n\\theta)$, where the $b_n$ are positive integers. Then \\[b_n=\\lim_{T\\to\\infty}\\frac{1}{2T}\\int^T_{-T}h(t)\\cos(\\delta_nt)\\,dt\\leq\\lim_{T\\to\\infty}\\frac{1}{2T}\\int^T_{-T}h(t)\\,dt=c.\\] Thus $c\\geq1$. The problem arises in prediction theory where $\\mu\\leq(N-1)^{1/2}$ is adequate. If the $\\lambda_n$ are rational, then the problem reduces to a problem on polynomials with coefficients $0$ and $1$. (S. Rudolfer and W. K. Hayman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.33\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307034, "problem_number": "AMR-022-7034", "title": "Research Problems in Function Theory — Problem 7.34", "statement": "Let $\\beta_j\\in\\mathbb{R}^+$ and $\\zeta_j\\in\\mathbb{C}$, and for suitable small $z$ define $$ f(z)=\\prod^n_{j=1}(1-\\zeta_jz)^{\\beta_j}=1+a_1z+a_2z^2+\\ldots $$ Suppose further that the $a_k$ are all real. Then there exists $N=N(\\beta_1, \\beta_2, \\beta_3, \\ldots, \\beta_n)<\\infty$, independent of the $\\zeta_j$, such that $\\min(a_1, a_2, \\ldots, a_N)\\leq0$. Find a sharp, or good, upper bound for $N$ as a function of the $\\beta$'s. This would be significant for Tur\\'an's power sum method. To see that $N<\\infty$: if the $\\beta_j$'s are all integers, then $f$ is a polynomial and $N=\\beta_1+\\beta_2+\\beta_3+\\ldots+\\beta_n+1$. (The sort of estimate wanted in the general case.) If not, assume that $|\\zeta_1|\\leq|\\zeta_2|\\leq\\ldots\\leq|\\zeta_n|\\leq1$, and let $m=\\max\\{j:\\beta_j\\in\\mathbb{Z}\\}$. The radius of convergence of ([source label: 7.341]) is $|\\zeta_m|^{-1}$ and if the $a_k$ were all positive (or even greater than or equal to $0$). then by Pringshein's theorem (see Titchmarsh ), $|\\zeta_m|^{-1}$ would be a singularity of $f$. Since $f$ only has the singularities $\\{\\zeta^{-1}_j:\\beta_j\\notin\\mathbb{Z}\\}$ we deduce that $\\zeta_m>0$. We multiply through by $(1-\\zeta_m)^{-\\beta_m}$ which has positive coefficients, and repeat the arguments. After at most $n$ steps, we obtain a contradiction. This gives a finite $N$ for any particular set of $\\zeta_j$ and the uniformity is straightforward. (R. Hall)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.34\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (sharp $N$ unknown). No definitive resolution located." }, { "id": 2307035, "problem_number": "AMR-022-7035", "title": "Research Problems in Function Theory — Problem 7.35", "statement": "According to Fefferman's theorem (no citation), a real function $u$ on the unit circle which has bounded mean oscillation, can be decomposed as $u=b_1+\\tilde{b_2}$ where $b_1$ and $b_2$ are bounded functions and $\\tilde{b_2}$ denotes the conjugate of $b_2$. Given $u$, what is the smallest possible $\\|b_2\\|_\\infty$? This problem is discussed by Baernstein (Section 10). An affirmative answer to would prove the conjecture about factoring non-zero univalent functions that Baernstein made (see Problem 5.58). (A. Baernstein)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.35\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307036, "problem_number": "AMR-022-7036", "title": "Research Problems in Function Theory — Problem 7.36", "statement": "This problem is equivalent to Problem 7.9 due to Gehring and Reich about best bounds for area distribution under quasiconformal mapping. Let $E$ denote a measurable subset of the unit disc $\\mathbb{D}$, $m$ denote $2$-dimensional measure, and define \\[f_E(z)=-\\frac{1}{\\pi}\\int\\int_E\\frac{dm(w)}{(w-z)^2},\\hspace{1cm}z\\in\\mathbb{C}\\setminus E.\\] Thus, $f_E$ is the $2$-dimensional Hilbert transform of the characteristic function of $E$. It follows from the Calder\\'on-Zygmund theory of singular integrals that there are constants $a$ and $b$ such that \\[\\int\\int_{\\mathbb{D}\\setminus E}|f_E|\\,dm\\leq am(E)\\log\\frac{\\pi}{m(E)}+bm(E)\\] for every $E$. The problem is to find the smallest possible $a$ (for which there exists some $b$ such that the inequality holds for every $E\\subset\\mathbb{D}$). Consideration of $E=\\{z:|z|<\\delta\\}$ for small $\\delta$ shows that $a=1$ would be best possible, and this is conjectured by Gehring and Reich. An analogous sharp inequality for sets $E\\subset[-1,1]$ and $1$-dimensional Hilbert transforms \\[f_E(x)=-\\frac{1}{\\pi}\\int_E\\frac{dt}{x-t}\\] is known, and can be proved either by a subordination argument or by use of a theorem of Stein and Weiss. (A. Baernstein)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.36\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307037, "problem_number": "AMR-022-7037", "title": "Research Problems in Function Theory — Problem 7.37", "statement": "Let $T$ denote the class of all rational functions $g$ of the form \\[g(z)=\\sum^n_{j=1}\\frac{\\lambda_j}{(z-z_j)^2},\\] where the constants $\\lambda_j$ satisfy $\\lambda_j>0$ and $\\sum^n_{j=1}\\lambda_j=1$. Prove (or disprove): There is a constant $C$ with the property that for each $g$ in $T$ we can find a set $S=S(g)$ with $m(S)=\\pi$ such that \\[\\int\\int_{\\Delta(R)\\setminus S}|g|\\,dx\\,dy\\leq2\\pi\\log R+C\\] for every $R$ in $(1,\\infty)$. Here $\\Delta(R)=\\{z:|z|0$ such that whenever $r\\leq\\delta$ and $N\\geq[1/r]^{2-\\delta}$, then $\\int^{2\\pi}_0h(z_j+2re^{i\\theta})\\,d\\theta\\leq r^{2+\\delta}$, holds for at least one $D(z_j,r)$? Here $[1/r]$ denotes the greatest integer less than or equal to $1/r$. An affirmative answer should imply a weak version of Arakelian's conjecture for entire functions (Problem 1.6). (J. Lewis)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.40\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307041, "problem_number": "AMR-022-7041", "title": "Research Problems in Function Theory — Problem 7.41", "statement": "Let $\\{z_\\nu\\}$ be a sequence of distinct points in the unit disc $\\mathbb{D}$ such that $\\sum(1-|z_\\nu|)<\\infty$. Let $B$ be the Blaschke product corresponding to this sequence. If $00$ such that if $f$ is a conformal self-map of $D$ satisfying $|f(z)-z|<\\varepsilon$, then $f(z)\\equiv z$. Clearly, if each prime end of $D$ is a singleton, then $D$ is not conformally rigid. Show the converse. (P. M. Gauthier)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.43\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307044, "problem_number": "AMR-022-7044", "title": "Research Problems in Function Theory — Problem 7.44", "statement": "Let $U$ be an open set in the plane and $\\lambda^U_a$ be the harmonic measure at the point $a$ with respect to $U$. Then \\O{}ksendal showed that $\\lambda^U_a$ is singular with respect to area measure. Is it also true that $\\lambda^U_a$ is singular with respect to $\\beta$-dimensional Hausdorff measure for all $\\beta>1$? The same question can be asked for the Keldysh measure $\\mu^U_a$ at $a\\in K$ with respect to a compact set $K$. \\O{}ksendal showed that $\\mu^U_a$ is singular with respect to area measure. (B. \\O ksendal)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.44\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307045, "problem_number": "AMR-022-7045", "title": "Research Problems in Function Theory — Problem 7.45", "statement": "Let $D$ be the unit disc cut along $p$ radial slits from the outer boundary, all of the same given length. Let $u$ be the harmonic measure of $\\{|z|=1\\}$ in $D$. Find the configuration of slits which makes $u(0)$ minimal, when $p$ is fixed. (A. A. Gonchar, communicated by M. Ess\\'en)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.45\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307046, "problem_number": "AMR-022-7046", "title": "Research Problems in Function Theory — Problem 7.46", "statement": "(A `Universal' Phr\\'agmen-Lindel\\\"of Theorem) Let $D$ be an arbitrary unbounded plane domain. Suppose that $f(z)$ is analytic on $D$ and continuous on $\\overline{D}$. If $|f(z)|\\leq1$ on $\\partial D$ and $f(z)=o(|z|)$ at $\\infty$, show that $|f(z)|\\leq1$ throughout $D$. The $o(|z|)$ would then be the `right' condition since that is what is needed for the case $D=\\{z:|z|>1\\}$. (D. J. Newman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.46\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307047, "problem_number": "AMR-022-7047", "title": "Research Problems in Function Theory — Problem 7.47", "statement": "Let $K\\subset\\mathbb{C}$ be compact and let $x_0\\in K$ be a non-peak point for $R(K)$, the uniform limits on $K$ of rational functions with poles outside $K$. Will there always exist a continuous curve $\\Gamma$ in $K$ terminating at $x_0$? It is easy to see that if $\\sum^\\infty_{n=1}2^n M_1(A_n(x_0)\\setminus K)<\\infty$, where $A_n(x_0)=\\{z:2^{-n-1}\\leq|z-x_0|\\leq2^{-n}\\}$, and $M_1$ denotes $1$-dimensional Hausdorff content, then $\\Gamma$ can be chosen to be a straight line segment. In this case, \\O ksendal showed that the integrated Brownian motion starting at $x_0$ stays inside $K$ for a positive period of time, almost surely. There is an example of a compact set $K$ and a non-peak point $x_0\\in K$ such that no straight line segment terminating at $x_0$ is included in $K$. (B. \\O ksendal)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.47\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307048, "problem_number": "AMR-022-7048", "title": "Research Problems in Function Theory — Problem 7.48", "statement": "A domain $D\\subseteq \\mathbb{R}^n$ is said to be linearly accessible, if each point in the complement of $D$ can be joined to $\\infty$ by a ray which does not meet $D$. Let $g(\\,\\cdot\\,,x_0)$ be the Green's function for $D$ with pole at $x_0$ in $D$. Is $\\{x:g(x,x_0)>t\\}$ linearly accessible for $00$. If $u$ is differentiable, put $\\triangledown u=(u_{x_1}, u_{x_2},\\ldots, u_{x_n})$. Prove there are no homogeneous polynomials $u$ with real coefficients, $m\\geq2$, such that $$ \\triangledown u\\cdot(|\\triangledown u|^{p-2}\\triangledown u)\\equiv0,\\hspace{1cm}p\\text{ fixed},\\hspace{1cm}p\\neq2,\\hspace{1cm}11$, with smooth boundary. Denote by $A^\\infty(D)$ the class of functions analytic in $D$, continuous on $\\overline{D}$, all of whose derivatives are continuous on $\\overline{D}$. Let $E$ be a closed subset of the boundary of $D$ which is not a set of uniqueness for $A^\\infty(D)$, i.e. there exists a function $f\\not\\equiv0$, which belongs to $A^\\infty(D)$ such that $f$ vanishes exactly on $E$ and all of the derivatives of $f$ vanish on $E$. Is every closed subset of $E$ a set of non-uniqueness for $A^\\infty(D)$? This is true in the case of the unit disc in $\\mathbb{C}$. (A. -M. Chollet)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.51\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307052, "problem_number": "AMR-022-7052", "title": "Research Problems in Function Theory — Problem 7.52", "statement": "Let $K\\subset\\mathbb{C}^n$ be a compact set and let $P_0(K)$ be the set of all polynomials on $K$. The $P$-hull of $K$, the polynomial convex hull of $K$, is defined by \\[\\text{$P$-hull } K=\\{z\\in\\mathbb{C}^n:|p(z)|\\leq\\sup_k|P(z)|\\text{ for all }p\\in P_0(K)\\}.\\] Let $P(K)$ be the uniform closure of $P_0(K)$ in $C(K)$, the continuous function on $K$. Let \\v{S}ilov Bd $(P(K))$ denote the \\v{S}ilov boundary of the uniform algebra $P(K)$. Determine all compact sets $K\\subset\\mathbb{C}^n (n>1)$ such that \\v{S}ilov Bd $(P(K))=\\text{Boundary }(\\text{$P$-hull }K)$. For $n=1$, every compact set $K\\subset\\mathbb{C}$ has this property. For $n>2$, examples of $K$ are compact sets and closed spheres. (S. Kilambi)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.52\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307053, "problem_number": "AMR-022-7053", "title": "Research Problems in Function Theory — Problem 7.53", "statement": "[(i)] ; (One-dimensional version.) Let $E$ be a compact set in $\\mathbb{R}$; and, for each $x \\in E$, let $\\delta_x > 0$ be given. Let $I_x = (x-\\delta_x, x + \\delta_x)$. For what values of $c$ can one always find a disjoint collection of such intervals, $\\{I_{x_j}\\}$ say, such that $\\sum_j|I_{x_j}|\\geq c|E|$? It is known that this is possible for $c = \\frac{1}{2}$, but is impossible in general for $c > \\frac{2}{3}$. ; ($n$-dimensional version.) Let $E$ be a compact set in $\\mathbb{R}^n$; and let $K$ be an open bounded symmetric convex set in $\\mathbb{R}^n$. For each $x \\in E$, let $\\delta_x > 0$ be given; and let $K_x = x + \\delta_xK$, the dilation of $K$ by a factor $\\delta_x$, centred at $x$. For what values of $c$ can one always find a disjoint collection of such sets, $\\{K_x\\}$ say, such that $\\sum_j|K_{x_j}|\\geq c|E|$? If $c(K)$ denotes the best value, it is known that $2^{-n}\\leq c(K)<1$. The facts in $(i)$ and $(ii)$, together with some sketchy information on $c(Q_n)$ (where $Q_n$ is the $n$-cube) and $c(S_n)$ (where $S_n$ is the $n$-sphere), are given in , but there is no information on the correct asymptotic behaviour of $c(Q_n)$ or $c(S_n)$. The problem has applications to the best constants in results concerning the Hardy-Littlewood maximal function. (P. L. Walker)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.53\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (exact/asymptotic constants unknown). No definitive resolution located." }, { "id": 2307054, "problem_number": "AMR-022-7054", "title": "Research Problems in Function Theory — Problem 7.54", "statement": "Let $\\phi_t(z)= e^{tz}-1$, $\\phi^1_t=\\phi_t$ and $\\phi^{n+1}_t=\\phi_t\\circ\\phi^n_t$ for $n\\geq1$; it follows that \\[\\phi^1_t(-1)=e^{-t}-1 = -t+\\frac{t}{2!}-\\ldots,\\] \\[\\phi^2_t(-1)=e^{t(e^{-t}-1)}-1 = -t^2+\\ldots\\text{ etc.}\\] Are the coefficients in these formal power series for $\\{\\phi^n_t(-1)\\}^\\infty_{n=1}$ uniformly bounded by $1$ in modulus? (P. J. Rippon)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.54\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307055, "problem_number": "AMR-022-7055", "title": "Research Problems in Function Theory — Problem 7.55", "statement": "It is known that any quasi-conformal homeomorphism of $B^n = \\{x : x \\in\\mathbb{R}^n,|x|<1\\}$ onto a Jordan domain $D$ in $\\mathbb{R}^n$ can be extended to a homeomorphism of $B^n$ onto $D$. If $\\partial D$ is rectifiable (in the sense that $\\Lambda^{n-1}(\\partial D)<\\infty$), is $f|_{\\partial B^n}$ absolutely continuous (in the sense that $\\Lambda^{n-1}\\big(f(E)\\big)=0$ for every set $E$ in $\\partial B^n$ with $|E|=0$)? One can also ask the analogous question about $f^{-1}$. When $n = 2$, the answer to both questions is `yes' for conformal mappings, but `no' for quasi-conformal mappings. When $n = 3$, Gehring has proved that, if in addition the function $f$ has a quasi-conformal extension to $\\mathbb{R}^n$, then $f|_{\\partial B^n}$ is absolutely continuous; but, even in this special case, it is not known if $f^{-1}|_{\\partial D}$ is absolutely continuous. (A. Baernstein II)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.55\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307056, "problem_number": "AMR-022-7056", "title": "Research Problems in Function Theory — Problem 7.56", "statement": "Let $\\Gamma$ be a closed Jordan curve in the extended plane, and suppose that $\\infty\\in\\Gamma$. Let $f_1, f_2$ map the upper and lower half-planes, respectively, onto the two different domains in $\\mathbb{C}$ bounded by $\\Gamma$, with $f_1(\\infty)=f_2(\\infty)=\\infty$. Then, if $h=f^{-1}_2\\circ f_2$, $h$ is a homeomorphism of $\\mathbb{R}$ onto $\\mathbb{R}$. It is known that $\\Gamma$ is a quasi-circle if and only if $h$ is quasi-symmetric (that is, there exists a constant $c$ such that \\[\\frac{1}{c}\\leq\\frac{h(x+t)-h(x)}{h(x)-h(x-t)}\\leq c\\] for all $x,t \\in\\mathbb{R}$). Can one characterise the function $h$ for general Jordan curves $\\Gamma$? In particular, can every function $h:\\mathbb{R}\\to\\mathbb{R}$ be generated in this fashion? (L. Bers; communicated by A. Baernstein II)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.56\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307057, "problem_number": "AMR-022-7057", "title": "Research Problems in Function Theory — Problem 7.57", "statement": "For $n\\geq2$, let \\[\\mathbb{R}^n_+=\\big\\{(x_1,\\ldots,x_n)\\in\\mathbb{R}^n:x_n>0\\big\\}\\] and \\[\\mathbb{R}^n_-=\\big\\{(x_1,\\ldots,x_n)\\in\\mathbb{R}^n:x_n<0\\big\\};\\] let $E$, $F$ be non-empty compact subsets of $\\mathbb{R}^n_+$, $\\mathbb{R}^n_-$ (respectively), and let $F^*$ denote the symmetric image of $F$ in $\\partial \\mathbb{R}^n_+$. Denote by $\\Delta(E,F)$, $\\Delta(E,F^*)$ the families of all curves in $\\overline{\\mathbb{R}^n}$ joining $E$ and $F$, $E$ and $F^*$ (respectively). Is it true that $$ M\\big(\\Delta(E,F)\\big)\\leq M\\big(\\Delta(E,F^*)\\big), $$ where $M$ denotes the $n$-modulus of a curve family? It is easy to show that strict inequality holds in ([source label: J7.57]) if $E$ and $F$ are balls. Also, if $\\Delta(E,F)$ is obtained from $\\Delta(E, F^*)$ as a result of symmetrisation, then ([source label: J7.57]) holds. Note too that it follows from the symmetry principle for the modulus that $\\frac{1}{2}M\\big(\\Delta(E,F)\\big)\\leq M\\big(\\Delta(E,F^*)\\big)$, at least if $E\\cap f^*=\\emptyset$. (M. Vuorinen)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.57\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307058, "problem_number": "AMR-022-7058", "title": "Research Problems in Function Theory — Problem 7.58", "statement": "Let $E\\subset[0,1]$ be a compact set on the positive real axis in $\\mathbb{R}^2$; and let $E$ have positive conformal $2$-capacity, that is $M\\big(\\Delta(E, \\partial B^2(2);\\mathbb{R}^2)\\big)> 0$ where $\\Delta(E,\\partial B^2(2);\\mathbb{R}^2)$ is the family of all curves joining $E$ to $\\partial B^2(2)$ and $M(\\Delta)$ is the $2$-modulus of $\\Delta$. Is it true that $M\\big(\\Delta(E, F; \\mathbb{R}^2)\\big) = \\infty$, where $F = \\mathbb{R}\\setminus E$? (A. A. Gon\\^car; communicated by M. Vuorinen)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.58\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307059, "problem_number": "AMR-022-7059", "title": "Research Problems in Function Theory — Problem 7.59", "statement": "Let $(P, Q)$ denote a pair of polynomials with the following property: $$ \\text{the map }f\\mapsto P(D)(Qf)\\text{ carries }\\mathcal{E}\\text{ bijectively onto }\\mathcal{E}. $$ If $(P, Q)$ has the property ([source label: J7.59]), is it necessarily true that $(Q, P)$ also has the property ([source label: J7.59])? (There is little theoretical ground so far to support such a conjecture, but in all examples the proposer has been able to check it is true.) Note that it is fairly easy to show that, under the hypotheses above, the map $F\\mapsto Q(D)(PF)$ carries $\\tilde{\\mathcal{E}}$ bijectively onto $\\tilde{\\mathcal{E}}$, where $\\tilde{\\mathcal{E}} = \\{F : F \\in\\mathcal{E}, F \\text{ is of exponential type}\\}$. (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.59\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307060, "problem_number": "AMR-022-7060", "title": "Research Problems in Function Theory — Problem 7.60", "statement": "Let $P$ be a polynomial of degree $m$ in which the coefficient of $z^m_1$ is non-zero, and let $Q(z) = z^m_1$. [(a)] ; Does the pair $(P, Q)$ have the property ([source label: J7.59])? ; Does the pair $(Q, P)$ have the property ([source label: J7.59])? The proposer can prove the conjectures in the case that \\[Q(z) = z^m_1 + \\big(\\text{polynomial in }(z_2,\\ldots, z_n)\\big).\\] Note that if $(a)$ were true, then it would follow that the non-characteristic Cauchy problem with entire data on a hyperplane has a unique entire solution. (The uniqueness follows from classical results; only the entirety is in question.) (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.60\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (the general $P$; proposer proved a special case). No definitive resolution located." }, { "id": 2307061, "problem_number": "AMR-022-7061", "title": "Research Problems in Function Theory — Problem 7.61", "statement": "Is it true that, for any polynomial $P$ with complex coefficients, the mapping $f\\mapsto P^*(D)(Pf)$, where $P^*(z) = \\overline{P(\\overline{z})}$, is a bijection of $\\mathcal{E}$ onto $\\mathcal{E}$? The proposer can prove that this is true when $P$ is a homogeneous polynomial; and D. J. Newman told the proposer that he could prove the injectivity half of the conjecture, but the proposer has seen no details of the proof. The conjecture would follow if one could show that the partial differential equation $P^*(D)(Pf) = z^\\alpha$ where $P^*(z) = \\overline{P(\\overline{z})}$, has a solution $f$ that is entire and of exponential type for every multi-index $\\alpha$. (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.61\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307062, "problem_number": "AMR-022-7062", "title": "Research Problems in Function Theory — Problem 7.62", "statement": "Given a countable number of convergent series with positive decreasing terms, one can find such a series converging more slowly than any of these. Without making any assumption about the Continuum Hypothesis, can one associate with every countable ordinal number $\\alpha$ a convergent series $\\sum^\\infty_{n=1}x_{n,\\alpha}$ with $0\\leq x_{n+1,\\alpha}\\leq x_{n,\\alpha}$ such that [(a)] ; if $\\alpha<\\beta$, then $x_{n,\\alpha}/x_{n,\\beta}\\to0$ as $n\\to\\infty$, and ; if $x_n > 0$ and $\\sum^\\infty_{n=1}x_n<\\infty$, then there exists $\\alpha$ such that $x_n/x_{n,\\alpha}\\to0$ as $n\\to\\infty$. [See also Problem 2.66.] (A. Hinkkanen)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.62\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open; set-theoretic. No definitive resolution located." }, { "id": 2307063, "problem_number": "AMR-022-7063", "title": "Research Problems in Function Theory — Problem 7.63", "statement": "For $n\\geq 2$ and $\\alpha > 0$, let \\[\\widehat{T^\\alpha_Rf}(\\xi)=\\Big(1-\\frac{|\\xi|^2}{R^2}\\Big)^\\alpha_+\\hat{f}(\\xi)\\] where $R > 0$ and $f\\in\\mathcal{S}(\\mathbb{R}^n)$. Is it true, for all $f\\in L^{2n/(n+1)}(\\mathbb{R}^n)$ and all $\\alpha > 0$, that $T^\\alpha_Rf(x)\\to f(x)$ a.e. as $R\\to\\infty$? When $n = 2$ and $\\{R_j\\}$ is a lacunary sequence tending to $\\infty$, Carbery , C\\'ordoba and L\\'opez-Melero and also Igari have shown that the answer is `yes'. When $n\\geq3$, the more `elementary' problem of norm convergence remains unsolved. (See also , , .) (A. Carbery)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.63\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition; a-te. convergence of Bochner–Riesz means at the critical index remains a major open problem in harmonic analysis for $n\\ge3$. No resolution through 2026." }, { "id": 2307064, "problem_number": "AMR-022-7064", "title": "Research Problems in Function Theory — Problem 7.64", "statement": "Let $\\Gamma$ be a Fuchsian group in $\\mathbb{D}$, and let $i(z) \\equiv z$. Is it true that \\[\\sum_{\\gamma\\in\\Gamma}|\\gamma'(0)|\\geq\\prod_{\\gamma\\in\\Gamma,\\,\\gamma\\neq i}|\\gamma(0)|^2\\,?\\] Note that this is an equivalent formulation of a problem connecting the Bergman kernel function and the capacity function of a Riemann surface. (N. Suita; communicated by Ch. Pommerenke)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.64\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "PARTIAL-PROGRESS (the underlying Suita conjecture is solved; the exact Fuchsian-group form may need checking). Literature status: This is a version of the Suita conjecture / Suita's problem. The Suita conjecture (relating the logarithmic capacity to the Bergman kernel at a boundary point) was proved; this specific normalized Fuchsian-group form is closely related. Mark SOLVED-IN-LITERATURE for the Suita-conjecture essence with caution." }, { "id": 2307065, "problem_number": "AMR-022-7065", "title": "Research Problems in Function Theory — Problem 7.65", "statement": "Let $\\mathcal{W}$ be a hyperbolic Riemann surface, $G_\\omega$ the Green's function with pole $\\omega\\in\\mathcal{W}$, and $\\Gamma=\\{[\\gamma_n]\\}$ the fundamental group. Let $\\tilde{\\Gamma}$ be the subgroup of equivalence classes $[\\gamma]$ such that, for every $\\omega\\in\\mathcal{W}$, the harmonic conjugate $G^*_\\omega$ of $G_\\omega$ changes by an integral multiple of $2\\pi$ on a representative path in $[\\gamma]$. Certainly $[\\Gamma,\\Gamma]\\triangleleft\\tilde{\\Gamma}\\triangleleft\\Gamma$. Is it true that, if $[\\Gamma, \\Gamma] \\neq\\tilde{\\Gamma}$, then $\\mathcal{W}$ is necessarily of the form $\\mathcal{V}\\setminus A$ where $\\mathcal{V}$ is a hyperbolic surface and $A\\neq\\emptyset$ is a relatively closed subset of zero logarithmic capacity? The conjecture is true if $\\tilde{\\Gamma}=\\Gamma$ and $\\mathcal{V}$ is the unit disc; and it is false if $[\\Gamma,\\Gamma] = \\tilde{\\Gamma}$. (The conjecture arises from work in .) (K. Stephenson)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.65\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "SOLVED-IN-LITERATURE. Literature status: This is a known theorem in the theory of harmonic functions on Riemann surfaces / the subgroup $\\tilde\\Gamma$ related to multivalued harmonic conjugates. The conjecture as stated was resolved in the literature (see the theory of the subgroup of integral-period harmonic conjugates; Stephenson's work). Mark SOLVED-IN-LITERATURE with moderate confidence." }, { "id": 2307066, "problem_number": "AMR-022-7066", "title": "Research Problems in Function Theory — Problem 7.66", "statement": "A continuous mapping $f:B^n\\to\\mathbb{R}^n$, where $B^n = \\{x : x\\in\\mathbb{R}^n,|x|<1\\}$, and $n\\geq2$, is said to be proper if $f^{-1}(K)$ is a compact subset of $B^n$ whenever $K$ is a proper subset of $f(B^n)$. Let \\[B_f = \\{z : z \\in B^n, f\\text{ is not a local homeomorphism at }z\\}.\\] Is it true that, if [(i)] ; $n\\geq3$, ; $f:B^n\\to\\mathbb{R}^n$ is proper and quasi-analytic, and [(iii)] ; $B_f$ is compact then $f$ is necessarily injective? Note that the mapping $f(z) = z^2$, where $z \\in B^2$, shows that the conjecture is false when $n = 2$. The conjecture is known to be true in the special case that $f(B^n) = B^n$, $n\\geq3$. The conjecture is a special case of a more general open problem in . (M. Vuorinen)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.66\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307067, "problem_number": "AMR-022-7067", "title": "Research Problems in Function Theory — Problem 7.67", "statement": "Let $V$ be the zero set of some analytic function in a strictly pseudo-convex domain $\\Omega$ in $\\mathbb{C}^2$. If $V$ has finite area inside $\\Omega$, is it necessarily true that $V$ is the zero set of some bounded analytic function on $\\Omega$? Bo Berndttson (no citation) has shown that the answer is `yes' when $\\Omega$ is the ball; and easy examples show that the answer is `no' for strictly pseudo-convex domains in $\\mathbb{C}$ when $n > 2$. Skoda (, ) and independently Henkin have shown (under some cohomology condition on $\\Omega$) for all $n$ that $V$ is the zero set of a function of Nevanlinna class only if $V$ satisfies the Blaschke condition. (R. Zeinstra)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.67\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: SOLVED-IN-LITERATURE for the ball (Berndtsson); general case involves the corona-type questions. The finite-area implication for the ball is established. Mark PARTIAL-PROGRESS." }, { "id": 2307068, "problem_number": "AMR-022-7068", "title": "Research Problems in Function Theory — Problem 7.68", "statement": "For a Hadamard gap sequence $\\{n_k\\}^\\infty_{k=1}$, $n_{k+1}/n_k\\geq q>1$, is it true that the measure in $[0,2\\pi[$ of the set of those points $x$ for which \\[\\liminf_{m\\to\\infty}\\Big|\\sum^m_{k=1}\\cos(n_kx)-\\xi\\Big|=0,\\hspace{1cm}\\text{ for all }\\xi\\in\\mathbb{R},\\] equals $2\\pi$? (T. Murai)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.68\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307069, "problem_number": "AMR-022-7069", "title": "Research Problems in Function Theory — Problem 7.69", "statement": "Given an associative algebra $A$, with identity $1$ and countable basis, then for a finite-dimensional subspace $V$ spanned by the vectors $\\{e_j\\}^k_{j=1}$ we have the differential operator \\[\\sum^k_{j=1}e_j\\frac{\\partial}{\\partial x_j}\\] acting on differentiable functions defined over domains in $V$ and taking their values in $A$. We call a function $f: U\\subseteq V\\to A$ a left analytic function if \\[\\sum^k_{j=1}e_j\\frac{\\partial f}{\\partial x_j}(x)=0\\] for all $x\\in U$. Ryan has shown that there is a generalised Cauchy integral formula \\[f(x_0)=\\int_{\\partial M}G(x,x_0)\\sum^k_{j=1}(-1)^je_j\\,d\\hat{x}_jf(x),\\] with real-analytic kernel $G(x,x_0)$, where $M$ is an arbitrary, compact real $n$-dimensional manifold lying in $U$; and $x_0\\in \\mathring{M}$ if and only if there are elements $\\{p_j\\}^k_{j=1}\\subseteq A$ satisfying the relation \\[p_je_l+p_le_j=2\\delta_{jl}.\\] [(a)] ; Is the result still valid if we only assume $G(x,x_0)$ to be a $C^1$ function? ; What analogous result holds if we assume the algebra to be non-associative? (J. Ryan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.69\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307070, "problem_number": "AMR-022-7070", "title": "Research Problems in Function Theory — Problem 7.70", "statement": "Using arguments due to Ahlfors (see, for example ) any M\\\"obius transformation in $\\mathbb{R}^n$ can be written in the form $(ax + b)(cx+d)^{-1}$, where $x\\in\\mathbb{R}^n$ and $a, b, c, d$ are elements of a Clifford algebra $A_n$ that satisfies certain constraints. It can be shown that the linear differential equations whose solution spaces are conformally invariant are of the type \\[D^kf_k((ax+b)(cx+d)^{-1})=0,\\hspace{1cm} k\\in\\mathbb{N},\\] where $D$ is the Euclidean Dirac operator, and the associated conformal weight is \\[J_k(cx+d)= \\begin{cases} (cx+d)\\ast |cx+d|^{-n-1+k}&\\text{for }k=2p-1, |cx+d|^{-n+k}&\\text{for }k=2p, \\end{cases}\\] where $\\ast$ is the involution described in . ; What are the non-linear differential equations whose solution spaces are conformally invariant? ; Can their conformal weights also be expressed in terms of $cx + d$, and what relationship do these solutions have to the linear conformally-invariant differential equations? (J. Ryan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.70\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307071, "problem_number": "AMR-022-7071", "title": "Research Problems in Function Theory — Problem 7.71", "statement": "Given a domain of holomorphy $\\Omega\\subseteq\\mathbb{C}^n$, $n\\geq2$, what conditions are required on $\\Omega$ to admit an analytic function $p:\\Omega\\to\\mathbb{C}$ which cannot be analytically extended beyond the boundary of $\\Omega$, and satisfies the complex version \\[\\sum^n_{k=1}\\frac{\\partial^2}{\\partial z^2_j}p(z)=0\\] of Laplace's equation? (J. Ryan)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.71\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307072, "problem_number": "AMR-022-7072", "title": "Research Problems in Function Theory — Problem 7.72", "statement": "Let $N$ denote the class of complex-valued $L^\\infty$-functions $v$ on the unit disc $U$ such that $\\int_U v\\phi\\,dx\\,dy=0$ whenever $\\phi$ is analytic in $U$ with $\\int_U|\\phi(x+iy)|\\,dx\\,dy<\\infty$. The Cauchy principal value of \\[(Bv)(z)=\\frac{-1}{\\pi}\\int_U\\frac{v(\\zeta)}{(z-\\zeta)^2}\\,d\\xi\\,d\\eta,\\hspace{1cm}\\zeta=\\xi+i\\eta,\\] defines the Beurling transform $Bv$ of $v$. Is it true that $Bv\\in L^\\infty$ and, furthermore, that $$ \\|Bv\\|_\\infty\\leq C\\|v\\|_\\infty, $$ for some $C < \\infty$, whenever $v\\in N$. A weaker question is whether this holds for all $v\\in N\\cap P$, where $P$ is the class of all polynomials in $z$ and $\\overline{z}$. (The inequality ([source label: K7.72]) is true at least for certain subclasses of $N\\cap P$.) (A. Hinkkanen)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.72\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307073, "problem_number": "AMR-022-7073", "title": "Research Problems in Function Theory — Problem 7.73", "statement": "Let $D_1$, $D_2$ be domains in $\\{|z|< R\\}$, and let $\\lambda_1(z)\\,|dz|$ and $\\lambda_2(z)\\,|dz|$ be their hyperbolic metrics. What is the least number $A = A(R)$ such that the hyperbolic metric $\\lambda(z)\\,|dz|$ of $D_1 \\cap D_2$ satisfies the inequality \\[\\lambda(z)< A(\\lambda_1(z)+\\lambda_2(z))?\\] (W. H. J. Fuchs)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.73\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307074, "problem_number": "AMR-022-7074", "title": "Research Problems in Function Theory — Problem 7.74", "statement": "In their famous Acta paper , Hardy and Littlewood introduced the celebrated Hardy-Littlewood maximal function in connection with complex function theory. Since then it has proved an invaluable tool in real analysis. Here we ask some questions about the dependence of constants on dimension. Let $B$ be a convex compact symmetric body in $\\mathbb{R}^n$, normalised to have Euclidean volume $1$. Let the Hardy-Littlewood maximal functions be \\[Mf(x)=\\sup_{t>0}\\big(\\frac{1}{t^n}\\int_{tB}|f(x+y)|\\,dy\\Big)\\] and \\[\\tilde{M}f(x)=\\sup_{k\\in Z}\\big(\\frac{1}{2^{kn}}\\int_{2^kB}|f(x+y)|\\,dy\\Big).\\] [(a)] ; If $B$ is the Euclidean ball in $\\mathbb{R}^n$, does there exist a constant $C$ such that $\\text{meas}\\{x:\\tilde{M}f(x)>\\lambda\\}\\leq C\\lambda^{-1}\\|f\\|_1$ for all $\\lambda> 0$, with $C$ independent of $n$? ; If so, what is the answer to the same question for $Mf$? ; In the case that $n = 1$, a conjecture of F. Sonia and the proposer is that the best constant in the inequality $\\text{meas}\\{x:Mf(x)>\\lambda\\}\\leq C\\lambda^{-1}\\|f\\|_1$ is $C = \\frac{3}{2}$. Prove this. ; Let $1 0$ (independent of the choice of $E$ and $F$) such that \\[\\gamma(E\\cup F)\\leq K(\\gamma(E)+\\gamma(F)).\\] Perhaps one can take $K = 1$? See for related results. ; If $K$ is a compact subset of $\\mathbb{C}$ with $\\gamma(K) = 0$, then \\[\\gamma(E\\setminus K) = \\gamma(E)\\] for all compact subsets $E$ of $\\mathbb{C}$. Here $\\gamma(E\\setminus K)$ means the inner capacity \\[\\sup\\{\\gamma(L):L\\subseteq E\\setminus K, L\\text{ compact}\\}.\\] An interesting special case would be that when $K$ is the `corner quarters square Cantor set' (the so-called Garnett set). (A. G. O'Farrell)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.75\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS. The semiadditivity (c) is proved by Tolsa with some constant, and the problem whether $K=1$ (or the exact constant) is open. (d) relates to Tolsa's results on $\\gamma$ invariance. No definitive single resolution of all parts located." }, { "id": 2307076, "problem_number": "AMR-022-7076", "title": "Research Problems in Function Theory — Problem 7.76", "statement": "Prove or disprove the following statement: If $K$ is a compact subset of $\\mathbb{C}$ with continuous analytic capacity $\\alpha(K)= 0$, then \\[\\alpha(E\\setminus K)=\\alpha(E)\\] for all compact subsets $E$ of $\\mathbb{C}$. An interesting special case is when $K$ is a $C^1$-arc. The case when $K$ is a $C^{1+\\varepsilon}$-arc has already been settled. (A. G. O'Farrell)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.76\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307077, "problem_number": "AMR-022-7077", "title": "Research Problems in Function Theory — Problem 7.77", "statement": "We will say that $g(x)$ is a rearrangement of $f(x)$ if \\[m\\{x:g(x) < y\\} = m\\{x:f(x) < y\\}\\hspace{1cm}\\text{ for all } y\\in\\mathbb{R},\\] where $m$ is Lebesgue measure and $f$ and $g$ are defined on some finite interval $I$. What are those functions $f(x)$ for which $f'(x)$ is a rearrangement $f(x)$? Obvious examples of such functions are $f(x)=ke^x$ on any interval and $f(x) = k\\sin x$ on $[0,\\frac{1}{2}\\pi]$. What others are there? How about other `differential rearrangements' than $f'(z)\\sim f(x)$? (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.77\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307078, "problem_number": "AMR-022-7078", "title": "Research Problems in Function Theory — Problem 7.78", "statement": "Does there exist a sequence $\\{z_n\\}^\\infty_1$ of distinct complex numbers such that \\[\\sum\\frac{1}{|z_n|}<\\infty\\hspace{1cm}\\text{and}\\hspace{1cm}\\sum\\frac{1}{z-z_n}\\neq0,\\] for all $z\\in\\mathbb{C}$? This has the following physical interpretation. If we imagine electrons (really unit-charged wires perpendicular to the complex plane) placed at each point $z_n$, then these generate a logarithmic potential given by $\\sum\\log|z-z_j|$. The gradient of this potential is $\\sum1/(z-z_n)$. Thus the question is whether such a field must always have an equilibrium point - that is, a point where a free electron (or wire), once placed there, would remain there. Of course, the corresponding problem could be asked for $\\mathbb{R}^n$, $n\\geq3$ also. (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.78\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307079, "problem_number": "AMR-022-7079", "title": "Research Problems in Function Theory — Problem 7.79", "statement": "Let $f$ be analytic on a domain $G$ in $\\mathbb{C}$. We will say that a point $z_0\\in G$ is a MacLane point of $f$ if there exists some neighbourhood $N$ of $z_0$ such that the restrictions to $N$ of the successive derivatives of $f$, \\[\\{f^{(n)}|_N:n\\in\\mathbb{N}\\},\\] form a normal family of functions on $N$. Let $M(f)$ denote the set of MacLane points of $f$. What can be said about the set $M(f)$, besides the fact that it is open? Must $M(f)$ be connected? If $G$ is simply-connected, must $M(f)$ be simply-connected? Or can $M(f)$ be an arbitrarily prescribed open subset of $G$? Similar questions can be asked about functions meromorphic in $G$. Perhaps it is more natural, also, to ask such questions about$\\{f^{(n)}(z)/n!\\}$ rather than simply $\\{f^{(n)}(z)\\}$? Some relevant facts can be found in and . (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.79\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307080, "problem_number": "AMR-022-7080", "title": "Research Problems in Function Theory — Problem 7.80", "statement": "Let $f$ be an inner function, with $f(0) = 0$; then $f$ induces an ergodic (Lebesgue-) measure-preserving map of the circle onto itself. What is the entropy, $h(f)$, of $f$? It is conjectured that $h(f) < \\infty$ if and only if $f'$ belongs to the Nevanlinna class; and that, in that case, then \\[h(f)=\\frac{1}{2\\pi}\\int^{2\\pi}_0\\log\\big|f'(e^{i\\theta})\\big|\\,d\\theta.\\] (See and .) (J. L. Fernandez)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.80\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: PARTIAL-PROGRESS. The entropy of inner functions has been studied; the exact characterization is intricate. No definitive resolution of the conjecture located through 2026." }, { "id": 2307081, "problem_number": "AMR-022-7081", "title": "Research Problems in Function Theory — Problem 7.81", "statement": "Let $I$ denote the class of all inner functions. Then $I$, as a subset of $H^\\infty$, enjoys some of the properties that the collection of unimodular functions have as a subset of $L^\\infty_\\mathbb{R}$ (the real-valued functions in $L^\\infty(\\Pi)$ ). Is it true that if $\\{\\Lambda_n\\} \\subset (H^\\infty)^*$ and if for each function $\\phi\\in I$ one has $|\\Lambda_n(\\phi)|\\leq C(\\phi)$, then \\[\\sup_n\\|\\Lambda_n\\|_{(H^\\infty)^*}<\\infty?\\] This is known to hold if $\\Lambda_n\\in L^1\\setminus H^1_0(\\subset(H^\\infty)^*)$ (, , ); the corresponding real-variable result is also known . (J. L. Fernandez)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.81\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307082, "problem_number": "AMR-022-7082", "title": "Research Problems in Function Theory — Problem 7.82", "statement": "Let $E\\subset \\mathbb{C}$, and the function $F:E\\times \\mathbb{D}\\to\\mathbb{C}$ satisfy the following conditions: [(a)] ; $F$ is injective on $E$, for all $z\\in \\mathbb{D}$; ; $F$ is analytic in $z\\in \\mathbb{D}$, for each $w\\in E$; ; $F(z) = z$, when $w = 0$. Does there exist a function $G: \\mathbb{C}\\times \\mathbb{D} \\to \\mathbb{C}$ that satisfies conditions $(a)$, $(b)$ and $(c)$, and for which $G = F$ on $E$? (D. Sullivan, W. Thurston, H. Royden; communicated by D. Hamilton)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.82\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2307083, "problem_number": "AMR-022-7083", "title": "Research Problems in Function Theory — Problem 7.83", "statement": "Let $G$ be a finitely-generated Fuchsian group of the first kind. [(a)] ; If $F:\\Pi \\to \\Pi$ is a $G$-invariant quasi-symmetric function, is $F$ totally singular? ; Is the Teichmuller space $T_G$ dense in the space $S_G$ (Schwarzians of $G$-invariant univalent functions)? (O. Lehto; L. Bers; I. Kra; communicated by D. Hamilton)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 7.83\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308001, "problem_number": "AMR-022-8001", "title": "Research Problems in Function Theory — Problem 8.1", "statement": "Let $L = \\{L\\}^\\infty_0$ be a non-negative increasing sequence such that \\[\\sum^\\infty_{k=0}L_kr^k<\\infty\\hspace{1cm}(00$ such that, if $g\\in H^\\infty$ and \\[|g(z)|^\\kappa\\leq C[|f_1(z)|+\\ldots+|f_N(z)|],\\hspace{1cm}|z|<1,\\] then necessarily $g\\in I$? (If so, we must have $\\kappa\\geq2$.) As a special case, is it true that $J^2\\subset I$? This is true in appropriate algebras of functions defined in terms of faster rates of growth as $|z|\\to1$. (J. J. Kelleher)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.2\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition; related to the higher-dimensional corona-type questions. No definitive resolution located." }, { "id": 2308003, "problem_number": "AMR-022-8003", "title": "Research Problems in Function Theory — Problem 8.3", "statement": "For a bounded plane domain $D$, denote by $N(D)$ the class of functions analytic on $D$ of bounded characteristic (i.e., all quotients of functions in $H^\\infty(D)$ with nonvanishing denominator); let $f_1,\\ldots,f_N\\in N(D)$ have no common zeros in $D$. Find necessary and sufficient conditions on $f_1,\\ldots,f_N$ in order that they generate the full ring $N(D)$. Equivalently, if $g_1,\\ldots,g_N\\in H^\\infty(D)$, when does the ideal generated by $g_1,\\ldots,g_N$ in $H^\\infty(D)$, or in $N(D)$, contain a non-vanishing function? For $D = \\mathbb{D}$, for example, the zeros of $g_1,\\ldots,g_N$ should not get too close together as one approaches $\\partial D$, and one would like to obtain a corona-type theorem for this problem - i.e., a lower estimate for \\[|g_1(z)|+\\ldots|g_N(z)|\\] in $\\mathbb{D}$. (J. J. Kelleher)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.3\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308004, "problem_number": "AMR-022-8004", "title": "Research Problems in Function Theory — Problem 8.4", "statement": "Let $D$ be a simply-connected domain in the complex plane $\\mathbb{C}$ and $A(D)$ the ring of functions $f: D\\to\\mathbb{C}$ analytic in $D$. Bers (no citation) has shown that (for domains of arbitrary connectivity) the algebraic structure of $A(D)$ determines the conformal structure of $D$. Can $A(D)$ be the direct sum of two non-trivial subrings of itself? (This would represent a generalisation of Taylor's Theorem.) Is there a more general result for multiply-connected domains $D$? (J. J. Kelleher)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.4\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308005, "problem_number": "AMR-022-8005", "title": "Research Problems in Function Theory — Problem 8.5", "statement": "Let $G$ be a domain in $\\mathbb{C}$, and $H(G)$ the ring of functions analytic on $G$. It is known that, for two domains $G_1$ and $G_2$, $H(G_1)$ is isomorphic to $H(G_2)$ if and only if $G_1$ and $G_2$ are conformally (or anticonformally) equivalent. What can be said under only the hypothesis that $H(G_1)$ and $H(G_2)$ are elementarily equivalent in the sense of model theory? For a large class of domains the corresponding problem has been solved for $H_\\mathbb{C}(G)$, the algebra of functions analytic on $G$, by Henson and Rubel (see and ). (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.5\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308006, "problem_number": "AMR-022-8006", "title": "Research Problems in Function Theory — Problem 8.6", "statement": "Let $A^p$, $p > 0$ be the Bergmann space of functions $f(z)$ analytic in $|z|< 1$ such that \\[\\|f\\|_p=\\Huge({\\int\\int}_{|z|<1}|f(re^{i\\theta})|^p\\,r\\,dr\\,d\\theta\\Huge)^{1/p}<\\infty;\\] clearly $H^p \\subset A^p$. Horowitz has shown that if $f\\in A^p$ and $f$ has zeros $\\{z_k\\}$ in $|z|< 1$ then $$ \\prod^n_{k=1}|z_k|^{-1}=O(n^{(1/p)+\\varepsilon})\\hspace{1cm}\\text{ as }n\\to\\infty, $$ for all $\\varepsilon>0$; in ([source label: C8.1]) $\\varepsilon$ cannot be replaced by $0$. Recall that $\\{z_k\\}$ is a zero set for $H^p$ if and only if $\\sum(1-|z_k|)<\\infty$. Characterise the zero sets for $A^p$, or at least find some non-trivial converse to ([source label: C8.1]). (P. L. Duren)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.6\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308007, "problem_number": "AMR-022-8007", "title": "Research Problems in Function Theory — Problem 8.7", "statement": "We use the notation of Problem 8.6. Let $\\{z_n\\}^\\infty_1$ be a sequence of points in $\\mathbb{D}$ such that the kernel functions $k_n(z) = (1-z_nz)^{-2}$ do not span the space $A^2$, and let $\\{f_j\\}^\\infty_1$ be finite linear combinations of the $k_n$ which converge (in norm) to some function $f$ in $A^2$. Prove that, if the sequence $\\{f_j\\}$ converges uniformly to $0$ on some disc $\\Delta$ in $\\{|z| > 1\\}$, then $f\\equiv0$. Similar problems can of course be stated for the functions $(1- z_nz)^{-1}$, and for spaces other than $A^2$. This slightly `weird' problem arises in the theory of generalised analytic continuation; it is known to be true if the closure of $\\{z_n\\}^\\infty_1$ does not contain all of $\\{|z| = 1\\}$; it is also known that the analogous result for $H^2$ is true. (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.7\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308008, "problem_number": "AMR-022-8008", "title": "Research Problems in Function Theory — Problem 8.8", "statement": "Suppose that $F$ is a relatively-closed subset of $\\mathbb{D}$, and let \\[\\|f\\|_F=\\sup\\{|f(z)|;z\\in F\\}\\] for functions $f$ in the Bergman space $A^2$. Describe geometrically the set \\[\\{z: |z|< 1, |f(z)|\\leq\\|f\\|_F\\text{ for all }f\\in A^2\\}.\\] (A. Stray)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.8\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308009, "problem_number": "AMR-022-8009", "title": "Research Problems in Function Theory — Problem 8.9", "statement": "Determine \\[\\|\\Lambda\\|=\\sup\\Big|{\\int\\int}_{|z|<1}f(z)\\phi(z)\\,d\\sigma_z\\Big|\\] over those $f\\in A^1$ with $\\|f\\|_1\\leq1$, where \\[\\phi(z)=\\text{sgn}(\\text{Re } z).\\] It is shown by Reich and Strebel that $\\|\\Lambda\\|<1$, and that there exists an extremal function for the problem. See also Harrington and Ortel . Problems of this type are of interest in connection with the theorem of Hamilton in quasi-conformal mapping. (K. Strebel; communicated by M. Ortel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.9\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308010, "problem_number": "AMR-022-8010", "title": "Research Problems in Function Theory — Problem 8.10", "statement": "If $f$ and $1/f$ belong to the Bergman space $A^2$, does it follow that $\\mathcal{P}f$ is dense in $A^2$? Here $\\mathcal{P}f$ denotes the set of all polynomial multiples of $f$, i.e. \\[\\mathcal{P}f = \\{pf: p\\text{ a polynomial}\\}.\\] More generally, if $f\\in A^2$ and $|f(z)|\\geq c(1-|z|)^a$ for some $a$, $c > 0$, then does it follow that $\\mathcal{P}f$ is dense in $A^2$? (For partial results in this direction, see ). (A. L. Shields)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.10\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308011, "problem_number": "AMR-022-8011", "title": "Research Problems in Function Theory — Problem 8.11", "statement": "Let $g$ be a function in the space $D$ of functions analytic in $|z|< 1$ with finite Dirichlet integral, i.e., \\[{\\int\\int}_{|z|<1}|h'(z)|^2\\,dx\\,dy<\\infty.\\] If $\\mathcal{P}g$, as defined in Problem 8.10, is dense in $D$ and if, for some $f\\in D$, $|f(z)|\\geq|g(z)|$ for all $|z|<1$, is it necessarily true that $\\mathcal{P}f$ is dense in $D$? The analogous result is true in $H^2$ and in $A^2$. Shields solved the special case $g(z)\\equiv1$. (A. L. Shields)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.11\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308012, "problem_number": "AMR-022-8012", "title": "Research Problems in Function Theory — Problem 8.12", "statement": "Let $A$ denote the set of functions continuous on $|z|= 1$ which extend continuously to analytic functions on $|z|< 1$ (the disc algebra). Let $\\tilde{A}$ denote the set of functions of the form $f\\circ\\phi$ where $f$ ranges over $A$ and $\\phi$ ranges over the set of sense-preserving homeomorphisms of $|z| = 1$. Find a `good' characterisation of $\\tilde{A}$. Is there a function in $\\tilde{A}$ which coincides with \\[\\sum^\\infty_{n=1}2^{-n}\\exp(-i2^n\\theta)\\] on some subset of $|z|= 1$ having positive measure? The proposer conjectures not. (This problem is related to generalised analytic continuation.) (H. S. Shapiro)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.12\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308013, "problem_number": "AMR-022-8013", "title": "Research Problems in Function Theory — Problem 8.13", "statement": "Does there exist a singular measure in Zygmund's class $A^*$ all of whose Fourier-Stieltjes coefficients are non-negative? (F. Holland)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.13\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308014, "problem_number": "AMR-022-8014", "title": "Research Problems in Function Theory — Problem 8.14", "statement": "Which functions in $L^\\infty$ on the unit circle generate positive Hankel operators? (F. Holland)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.14\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open; characterisation of positive-Hankel-generating symbols is intricate (Hankel operator positivity). No definitive single resolution located." }, { "id": 2308015, "problem_number": "AMR-022-8015", "title": "Research Problems in Function Theory — Problem 8.15", "statement": "Characterise the Hankel operators on the Hardy space $H^2$ on the circle that are of trace class. (F. Holland)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.15\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED-IN-LITERATURE. Literature status: The trace-class Hankel operators on $H^2$ are classically described (a Hankel operator of trace class corresponds to a symbol in a Besov-type space); this is essentially known (Peller). Mark SOLVED-IN-LITERATURE with moderate confidence." }, { "id": 2308016, "problem_number": "AMR-022-8016", "title": "Research Problems in Function Theory — Problem 8.16", "statement": "Miles and Rudin have shown that in $\\mathbb{C}^n$ a function analytic in the unit polydisc and in the Hardy class $H^1$ may not be expressible as the product of two functions in $H^2$, if $n\\geq3$. Is this result also true for $n = 2$? (J. G. Clunie)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.16\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308017, "problem_number": "AMR-022-8017", "title": "Research Problems in Function Theory — Problem 8.17", "statement": "In the ring of bounded analytic functions on the unit ball or the polydisc in $n$ variables, is the intersection of two finitely-generated ideals again finitely generated? This was proved for $n = 1$ by McVoy and Rubel . (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.17\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open for $n>1$. No definitive resolution located." }, { "id": 2308018, "problem_number": "AMR-022-8018", "title": "Research Problems in Function Theory — Problem 8.18", "statement": "For which simply-connected domains $D$ (with $0 \\in D$) is it true that there is a constant $K = K(D)$ such that $$ {\\int\\int}_D|f|^2\\,dx\\,dy\\leq K{\\int\\int}_D|f'|^2\\,dx\\,dy $$ for all functions $f$ analytic on $D$ with $f(0) = 0$? This inequality ([source label: J8.18]) is known as the analytic Poincar\\'e inequality. Courant and Hilbert have given a Jordan domain for which ([source label: J8.18]) is false; and Hummel has given an example of a spiral domain $D$ for which ${\\int\\int}_D|f|^2\\,dx\\,dy = \\infty$ and ${\\int\\int}_D|f'|^2\\,dx\\,dy < \\infty$. (D. H. Hamilton)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.18\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308019, "problem_number": "AMR-022-8019", "title": "Research Problems in Function Theory — Problem 8.19", "statement": "Let $\\mu\\geq1$ be a singular measure on the boundary of the unit disc $\\mathbb{D}$; and let $S_\\mu$ be the corresponding singular inner function \\[S_\\mu(z)=\\exp\\Big\\{\\int^{2\\pi}_0\\frac{z+e^{i\\theta}}{z-e^{i\\theta}}\\,d\\mu(\\theta)\\Big\\},\\hspace{1cm}z\\in \\mathbb{D}.\\] The function $S_\\mu$ is said to be discrete if $\\mu$ is discrete, and continuous if it has no discrete part. If $S_\\mu$ is discrete, does there exist some $\\delta > 0$ such that $\\|S_\\mu-s_\\nu\\|_\\infty>\\delta$ for all continuous functions $S_\\nu$? The answer is `no' if `discrete' and `continuous' are interchanged . (K. Stephenson)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.19\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308020, "problem_number": "AMR-022-8020", "title": "Research Problems in Function Theory — Problem 8.20", "statement": "Let $f_1,\\ldots,f_n$ and $g$ be $H^\\infty$ functions on $\\mathbb{D}$. If $|g(z)| \\leq |f_1(z)| + \\ldots + |f_n(z)|$, are there necessarily functions $g_1,\\ldots,g_n\\in H^\\infty$ such that $$ g^2=f_1g_1+\\ldots+f_ng_n? $$ Wolff (see , page 329) has shown that ([source label: J8.20]) holds with $g^2$ replaced by $g^3$, and Rao (see ; and , Exercise 3, page 369) has shown that ([source label: J8.20]) is false with $g^2$ replaced by $g$. For general background on the problem see . (J. B. Garnett)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.20\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (the square case). No definitive resolution located." }, { "id": 2308021, "problem_number": "AMR-022-8021", "title": "Research Problems in Function Theory — Problem 8.21", "statement": "Let $w\\geq0$ be an integrable function on the circle or on the line. Then $w$ is said to belong to the class $A_2$ if \\[\\sup_l\\Big(\\frac{1}{|l|}\\int_lw\\,dx\\Big)\\Big(\\frac{1}{|l|}\\int_l\\frac{1}{w}\\,dx\\Big)<\\infty\\] over all (appropriate) intervals $I$ ; and $w$ is said to satisfy the Helson-Szeg\\\"o condition if $\\log w = u + \\tilde{v}$ where $u\\in L^\\infty,\\|v\\|_\\infty<\\frac{1}{2}\\pi$ and $\\tilde{v}$ is the conjugate function of $v$. It is known that the Helson-Szeg\\\"o condition is equivalent to $A_2$, since both conditions are necessary and sufficient for $$ \\int_l|\\tilde{f}|^2w\\,dx\\leq\\text{ const.}\\int_l|f|^2w\\,dx. $$ The problem is to prove that the Helson-Szeg\\\"o condition and $A_2$ are equivalent without using ([source label: J8.21]). For information on the case $\\|v\\|_\\infty<\\pi$, see . (J. B. Garnett and P. W. Jones)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.21\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE (the equivalence is established). Literature status: Both directions are classical; the request is for a direct proof. Open as a \"proof\" problem. No alternative proof located." }, { "id": 2308022, "problem_number": "AMR-022-8022", "title": "Research Problems in Function Theory — Problem 8.22", "statement": "Do there exist inner functions in any strictly pseudo-convex domain of $\\mathbb{C}^n$, $n\\geq2$? Alexandrov (no citation) and independently L{\\o}w (no citation) and Hakim and Sibony have shown the existence of inner functions in the case of the ball. (R. Zeinstra)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.22\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS (ball solved; general strictly pseudoconvex open). Literature status: SOLVED-IN-LITERATURE. The existence of non-trivial inner functions in the unit ball (and, more generally, in strictly pseudoconvex/smoothly bounded domains with some conditions) is established — Alexandrov proved the existence of inner functions in the ball for $n\\ge1$; the general strictly pseudoconvex case has positive results (e.g. via Alexandrov's method adapted to convex domains). Mark SOLVED for the ball; open in the fully general case." }, { "id": 2308023, "problem_number": "AMR-022-8023", "title": "Research Problems in Function Theory — Problem 8.23", "statement": "In Pompeiu's formula, $f(z)$ and $\\int_\\Gamma f(\\zeta)/(\\zeta-z)\\,d\\zeta$ make sense for merely continuous functions, but \\[\\int\\int_\\Omega\\frac{\\partial f}{\\partial\\overline{\\zeta}}\\frac{1}{\\zeta-z}\\,d\\zeta\\,d\\eta\\] needs at least some weak differentiability of $f$. It would be useful to extend the validity of the formula, for example to cover the case of functions $f\\in \\text{Lip }\\alpha$, $\\alpha> 0$. This prompts the following question. For which Borel sets $\\Omega$ is the inequality \\[\\Big|\\left<\\frac{\\partial\\chi_\\Omega}{\\partial\\overline{z}},f\\right>\\Big|\\leq C_\\Omega\\|f\\|_{\\text{Lip }\\alpha}\\] valid? Putting it another way, for which Borel sets $\\Omega$ does $\\chi_\\Omega$ act on the Besov space $B^{\\alpha-1}_{\\infty,\\infty}$? An interesting special case would be that when $\\Omega$ is a `Swiss cheese'. (A. G. O'Farrell)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.23\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308024, "problem_number": "AMR-022-8024", "title": "Research Problems in Function Theory — Problem 8.24", "statement": "Let the sequence $\\{M_k\\}^\\infty_0$ of positive numbers be such that \\[M_0=1\\hspace{1cm}\\text{ and }\\hspace{1cm}\\frac{M_{k+j}}{M_kM_j}\\geq\\begin{pmatrix}k+j j\\end{pmatrix}.\\] Assume also the non-quasianalyticity condition that $\\sum_k\\big(M_k/M_{k+1}\\big)<\\infty$. Consider those functions $f$ on a compact set $X \\subset\\mathbb{C}$ that are limits of rational functions with poles of $X$ in the norm \\[g\\mapsto\\sum^\\infty_{k=0}\\frac{1}{M_k}\\sup_{X}|g^{(k)}|;\\] these functions form a Banach algebra. Is $X$ its maximal ideal space? (The answer is `yes' if $X$ is the unit disc or the unit interval.) (A. G. O'Farrell)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.24\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308025, "problem_number": "AMR-022-8025", "title": "Research Problems in Function Theory — Problem 8.25", "statement": "Let $\\psi:S^1\\to S^1$ be a direction-reversing homeomorphism, and let $A_\\psi$ denote the set of functions $f:S^1\\to\\mathbb{C}$ such that both $f$ and $f\\circ\\psi$ belong to the disc algebra. When does $A_\\psi$ contain only constant functions? (A. G. O'Farrell)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.25\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2308026, "problem_number": "AMR-022-8026", "title": "Research Problems in Function Theory — Problem 8.26", "statement": "Let $\\psi:S^1\\to S^1$ be a homeomorphism. When is it true that $\\text{Re }A = \\text{Re }A \\circ \\psi$? That is, when is each function $f$ in the real part of the disc algebra also of the form $g\\circ\\psi$, for some function $g$ in the disc algebra? O'Connell has shown that it is necessary that $\\psi$ is absolutely continuous, and sufficient that $\\psi$ is $C^{1+\\varepsilon}$. (A. G. O'Farrell)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 8.26\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Literature status: PARTIAL-PROGRESS: necessary (a.c.) and sufficient ($C^{1+\\varepsilon}$) conditions known; the gap between them is the open part. No definitive resolution located." }, { "id": 2309001, "problem_number": "AMR-022-9001", "title": "Research Problems in Function Theory — Problem 9.1", "statement": "A sequence $\\{z_n\\}^\\infty_1$ in $|z|< 1$ is interpolating for bounded analytic (harmonic) functions if, for each bounded sequence $\\{\\alpha_n\\}^\\infty_1$ there exists a function $u(z)$ bounded and analytic (harmonic) in $|z|< 1$ with $u(z_n) = \\alpha_n (n\\geq 1)$. It is known that a sequence is interpolating for bounded harmonic functions if and only if it is interpolating for bounded analytic functions (, , ); however all such proofs require knowledge of conditions implying that a sequence interpolates for bounded analytic functions. Find a simple proof of this equivalence which does not rely on knowledge of such conditions. (L. Zalcman)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.1\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE (equivalence known); finding a simpler proof is open. Literature status: The equivalence is known (as stated). The request is for a simpler proof. No simpler proof located." }, { "id": 2309002, "problem_number": "AMR-022-9002", "title": "Research Problems in Function Theory — Problem 9.2", "statement": "Let $f(z)\\in H^\\infty$, and let $\\{z_n\\}$ be a Blaschke sequence \\[\\sum^\\infty_{n=1}(\\-|z_n|)<\\infty.\\] [(a)] ; Does there always exist a Blaschke product $B(z)$ of norm not necessarily equal to $1$, such that $$ B(z_n)=f(z_n) $$ for $n\\geq1$? This is certainly the case if the Blaschke sequence is uniformly separated, that is \\[\\inf_n\\prod_{m\\neq n}\\big|\\frac{z_m-z_n}{1-\\overline{z_m}z_n}\\big|>0.\\] See, for example, Earl . ; Is the unique $H^\\infty$ function of minimal norm assuming the values $\\{f(z_n)\\}$ at $\\{z_n\\}$ a constant multiple of a Blaschke product? The answer is `yes' if the sequence $\\{z_n\\}$ is finite, see Earl . What happens if we know, in addition, that $f(z_n)\\to0$ as $n\\to\\infty$? The answer is again `yes' if $z_n\\to1$ non-tangentially as $n\\to\\infty$. ; One can ask questions $(a)$ and $(b)$ for sequences $\\{z_n\\}$ that are weakly separated, viz \\[\\inf_{m\\neq n}\\big|\\frac{z_m-z_n}{1-\\overline{z_m}z_n}\\big|>0.\\] ; Also, one can ask questions $(a)$ and $(c)$ for inner functions instead of for Blaschke products. (J. P. Earl and A. Stray)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.2\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2309003, "problem_number": "AMR-022-9003", "title": "Research Problems in Function Theory — Problem 9.3", "statement": "[(a)] ; Suppose that $f, f_1, f_2,\\ldots,f_n\\in H^\\infty$, and \\[|f|\\leq|f_1|+|f_2|+\\ldots+|f_n|.\\] Do there necessarily exist $h_1,h_2,\\ldots,h_n\\in H^\\infty$ such that \\[f=f_1h_1+f_2h_2+\\ldots+f_nh_n?\\] (If $|f_1|+|f_2|+\\ldots+|f_n|\\geq\\delta>0$, this is the corona theorem.) ; Suppose that $f_1,f_2\\in H^\\infty$. Do there necessarily exist $f\\in H^\\infty$ and $\\delta>0$ such that \\[\\delta(|f_1|+|f_2|)\\leq|f|\\leq|f_1|+|f_2|?\\] If the answer is `yes' is $f$ necessarily or possibly of the form $h_1f_1+h_2f_2$ for some $h_1,h_2\\in H^\\infty$? (Notice that $(b)$ would imply $(a)$). (J. P. Earl)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.3\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2309004, "problem_number": "AMR-022-9004", "title": "Research Problems in Function Theory — Problem 9.4", "statement": "For each pair of $f,g\\in H^\\infty$, does there necessarily exist another pair of functions $a,b\\in H^\\infty$ such that $$ af+gb\\neq0,\\hspace{1cm}|z|<1\\,? $$ It is easy to see that a necessary condition for this is that $\\log(|f|-|g|)$ have a harmonic minorant. Is this condition also sufficient for ([source label: C9.1])? This problem is closely related to Problem 9.3. (B. A. Taylor; communicated by L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.4\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2309005, "problem_number": "AMR-022-9005", "title": "Research Problems in Function Theory — Problem 9.5", "statement": "Let $K_1, K_2, K_3$ be disjoint closed sets in the extended complex plane, and $C_1, C_2, C_3$ constants. Let $\\rho_n(f)$ be the best rational approximation to the function $f$ which equals $C_i$ on $K_i$ $(i = 1, 2, 3)$; i.e. \\[\\rho_n(f)=\\inf_{g\\in R_n}\\max_{z\\in\\cup_i K_i}|f(z)-g(z)|,\\] where $R_n$ is the class of rational functions $f$ order at most $n$. Find a geometric characterisation of $\\lim_{n\\to\\infty}\\rho_n^{1/n}$. For the case of two sets, see Gonchar . (T. Ganelius)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.5\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2309006, "problem_number": "AMR-022-9006", "title": "Research Problems in Function Theory — Problem 9.6", "statement": "Let $D$ be an open subset of the extended complex plane with non-empty boundary $\\partial D$, and let $F$ be a relatively-closed subset of $D$. Let $f$ be a function given on $F$, and $\\{f_n\\}^\\infty_1$ a sequence of functions analytic on $D$ such that $f_n\\to f$ uniformly on $F$. If $E\\subset(\\partial F_n\\cap\\partial D)$ and if $f$ extends continuously to $F\\cup E$, can each $f_n$ be extended continuously to $F\\cup E$? The answer is `yes' if $D$ is the unit disc, or if $E$ is compact. (A. Stray)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.6\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2309007, "problem_number": "AMR-022-9007", "title": "Research Problems in Function Theory — Problem 9.7", "statement": "Let us call a closed set $E$ in $\\mathbb{C}$ a weak Arakelian set if, corresponding to each function $g(z)$ continuous on $E$ and analytic in the interior of $E$, there exists an entire function $g(z)$ such that, for any sequence $\\{z_n\\}^\\infty_1$ in $E$, $|f(z_n)|\\to\\infty$ if and only if $|g(z_n)|\\to\\infty$. Find a geometric characterisation of the weak Arakelian sets. (L. A. Rubel)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.7\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2309008, "problem_number": "AMR-022-9008", "title": "Research Problems in Function Theory — Problem 9.8", "statement": "Let $\\gamma$ be a Jordan arc in $\\mathbb{C}^n$, $n\\geq2$ such that the projections $\\gamma_j$ on the complex coordinate planes $j=1,\\ldots,n$ have area zero. Then $R(\\gamma)=C(\\gamma)$. Is it true that $P(\\gamma)=C(\\gamma)$? See Korevaar and Wermer . (J. Korevaar)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.8\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2309009, "problem_number": "AMR-022-9009", "title": "Research Problems in Function Theory — Problem 9.9", "statement": "Does the condition $\\sum 1/p_n<\\infty$ for positive integers $p_n$ guarantee that the sequences of powers $\\{z^{p_n}\\}$ fails to span $C(\\gamma)$ for every Jordan arc $\\gamma$? Korevaar and Dixon have shown that for arcs of locally limited rotation (for example $C^1$ arcs), the condition \\[p_n\\geq nL(n),\\hspace{1cm}0 2)$. (A. G. O'Farrell)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.14\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition. No definitive resolution located." }, { "id": 2309015, "problem_number": "AMR-022-9015", "title": "Research Problems in Function Theory — Problem 9.15", "statement": "Let $f_1$ and $f_2\\in H^\\infty(\\text{unit disc}) = H^\\infty$; and let the function $g\\in H^\\infty$ satisfy the inequality \\[|g(z)|\\leq|f_1(z)|+|f_2(z)|.\\] Do there necessarily exist functions $g_1$ and $g_2$ in $H^\\infty$ such that \\[g^2=f_1g_1+f_2g_2\\,?\\] In other words, is it true that $g^2\\in I(f_1, f_2)$ (the ideal generated by $f_1$ and $f_2$)? Wolff has proved that $g^3\\in I(f_1,f_2)$. Also Rao has given an example of a function $g\\notin I(f_1,f_2)$. For related results by Tolokonnikov, see . (J. Garnett)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.15\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open as of Hayman's 2018 edition (same as Problem 8.20). No definitive resolution located." }, { "id": 2309016, "problem_number": "AMR-022-9016", "title": "Research Problems in Function Theory — Problem 9.16", "statement": "Let $\\Gamma$ be a curve of the form \\[\\{x+iA(x):-\\infty 0$, and let \\[\\Omega=\\mathbb{C}^*\\setminus E,\\hspace{1cm}\\text{ where }\\mathbb{C}^*=\\mathbb{C}\\cup\\{\\infty\\}.\\] Prove the corona theorem for $\\Omega$. (J. Garnett)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.16\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: The corona theorem for planar domains has been proved for many domains (Carleson for the disc; extensively for plane domains by a line of authors; corona theorems for infinitely connected domains). The specific Lipschitz-graph class is tied to the general \"corona problem\" which for planar domains with certain regularity is largely settled though the general problem remains subtle. Mark OPEN-TRIAGE." }, { "id": 2309017, "problem_number": "AMR-022-9017", "title": "Research Problems in Function Theory — Problem 9.17", "statement": "Let $K$ denote the $\\frac{1}{3}$-Cantor set on $\\mathbb{R}$; let $E = K\\times K$, and let $\\Omega=\\mathbb{C}^*\\setminus E$. Prove the corona theorem for $\\Omega$. (J. Garnett)", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Hayman - Research Problems in Function Theory (2018)\nSource item: Problem 9.17\nSource URL: https://arxiv.org/abs/1809.07200\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the 2018 source update was retained for status triage\nRights note: NEEDS_REVIEW; public arXiv source TeX inspected\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open (corona problem for infinitely connected/self-similar complement domains is subtle; not resolved for this exact set to my knowledge). No definitive resolution located." }, { "id": 2400002, "problem_number": "AMR-023-0002", "title": "Fuglede's conjecture in dimensions one and two", "statement": "For nonconvex subsets of $\\mathbb{R}$ and $\\mathbb{R}^2$, is a set spectral if and only if it tiles by translations?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Analysis problems from Wikipedia\nSource item: Wikipedia analysis item 2\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767\nAccessed: 2026-07-29\nExtraction: wiki-pinned-revision\nStatus evidence: Pinned Wikipedia revision 1366636767 lists this item under Unsolved problems; primary-source/current-status release review is still required\nRights note: CC BY-SA 4.0; adapted from the pinned Wikipedia revision with source attribution\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress / open. Fuglede's conjecture is true in the convex cases (dims $\\le2$), false in dimension $\\ge3$ (counterexamples in all directions), and **open** for nonconvex sets in $\\mathbb{R}$ and $\\mathbb{R}^2$." }, { "id": 2400005, "problem_number": "AMR-023-0005", "title": "Kung–Traub conjecture", "statement": "For an iteration without memory that uses $n$ evaluations of a function or its derivatives per step, is its convergence order always at most $2^{n-1}$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Analysis problems from Wikipedia\nSource item: Wikipedia analysis item 5\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767\nAccessed: 2026-07-29\nExtraction: wiki-pinned-revision\nStatus evidence: Pinned Wikipedia revision 1366636767 lists this item under Unsolved problems; primary-source/current-status release review is still required\nRights note: CC BY-SA 4.0; adapted from the pinned Wikipedia revision with source attribution\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. The Kung–Traub conjecture (order $\\le 2^{n-1}$ for $n$-evaluation memoryless iterations) is established only for small $n$; the general statement is unresolved. Literature status: - The conjecture, by Kung & Traub (1974), states that any $n$-evaluation multipoint method without memory has convergence order $\\le 2^{n-1}$. It is known to be **true for low $n$** (e.g., $n=1,2,3$: Newton, and the optimal 2-point/3-point methods) and is a central open conjecture for general $n$. - The conjecture is **open**; it is a well-known problem in numerical analysis/computer arithmetic. Some recent work gives partial results or confirms optimal orders for particular families, but no general proof or counterexample is known. - Difficulty above default L3 (it is a long-standing open conjecture; some formulations relate to computational complexity of root-finding)." }, { "id": 2400007, "problem_number": "AMR-023-0007", "title": "Mean value problem for polynomial critical points", "statement": "Given a complex polynomial $f$ of degree $d\\geq2$ and $z\\in\\mathbb{C}$, must there be a critical point $c$ of $f$ such that $|f(z)-f(c)|\\leq |f'(z)|\\,|z-c|$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Analysis problems from Wikipedia\nSource item: Wikipedia analysis item 7\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767\nAccessed: 2026-07-29\nExtraction: wiki-pinned-revision\nStatus evidence: Pinned Wikipedia revision 1366636767 lists this item under Unsolved problems; primary-source/current-status release review is still required\nRights note: CC BY-SA 4.0; adapted from the pinned Wikipedia revision with source attribution\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. The sharp mean-value inequality for polynomial critical points is proven only for low degrees and special families; the general conjecture is unresolved. Literature status: - This is a problem from Smale's mean value conjecture circle (Smale's mean value conjecture: for polynomial $f$ and $z$ not a critical point, there exists $c$ with $|f(z)-f(c)|/|z-c|\\le |f'(z)| \\cdot 4$; the constant is conjectured to be 1 in the \"mean-value\" form — related to and supporting the present formulation). - Smale's mean value conjecture (with the factor 4, or the sharper conjecture) remains **open** for general polynomials, though proved for degrees $\\le 8$ and for special families. The present \"must there be a critical point $c$ with $|f(z)-f(c)|\\le |f'(z)||z-c|$\" is the sharp/known-form variation, also open in general. - The literature (e.g., Dubinin, and surveys on Smale's mean value conjecture) confirms it is open; verified for low degrees and special cases only. - Difficulty above default L3." }, { "id": 2400008, "problem_number": "AMR-023-0008", "title": "Pompeiu problem", "statement": "Characterize the domains for which there exists a nonzero function whose integral vanishes over every congruent copy of the domain.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Analysis problems from Wikipedia\nSource item: Wikipedia analysis item 8\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767\nAccessed: 2026-07-29\nExtraction: wiki-pinned-revision\nStatus evidence: Pinned Wikipedia revision 1366636767 lists this item under Unsolved problems; primary-source/current-status release review is still required\nRights note: CC BY-SA 4.0; adapted from the pinned Wikipedia revision with source attribution\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress / open. The ball and many special domains are fully understood as Pompeiu sets, and necessary conditions (boundary regularity) are known, but a complete characterization of all Pompeiu domains (especially beyond moments/Fourier conditions) is open." }, { "id": 2400013, "problem_number": "AMR-023-0013", "title": "Flint Hills series", "statement": "Does the Flint Hills series $\\sum_{n=1}^{\\infty} 1/(n^3\\sin^2 n)$ converge?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Analysis problems from Wikipedia\nSource item: Wikipedia analysis item 13\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767\nAccessed: 2026-07-29\nExtraction: wiki-pinned-revision\nStatus evidence: Pinned Wikipedia revision 1366636767 lists this item under Unsolved problems; primary-source/current-status release review is still required\nRights note: CC BY-SA 4.0; adapted from the pinned Wikipedia revision with source attribution\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. It is unknown whether the Flint Hills series converges; convergence would follow from a sufficiently strong irrationality-measure bound for $\\pi$ (not currently available). Literature status: - The Flint Hills series is a well-known open problem in analysis/number theory. Its behavior is governed by the irrationality-measure properties of $\\pi$: the series converges if the irrationality measure $\\mu(\\pi)$ is small enough (roughly $\\mu(\\pi)<4$ loosely speaking), which is not known. Specifically, convergence is implied if $|\\pi - p/q|$ is not too well approximated; a sufficiently strong irrationality measure for $\\pi$ would settle it, but current bounds (e.g., Zeilberger/Zudilin-type results giving $\\mu(\\pi)\\le7.10...$) are too weak to decide convergence. - The problem is **open**: numerical evidence is inconclusive (oscillatory partial sums), and it is not known whether the series converges or diverges. - Difficulty above default L3 in the sense that it is a hard, well-known open problem (connected…" }, { "id": 2400014, "problem_number": "AMR-023-0014", "title": "Vlasov–Maxwell regularity", "statement": "Establish global regularity, or exhibit breakdown, for solutions of the Vlasov–Maxwell equations from appropriate smooth initial data.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Analysis problems from Wikipedia\nSource item: Wikipedia analysis item 14\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767\nAccessed: 2026-07-29\nExtraction: wiki-pinned-revision\nStatus evidence: Pinned Wikipedia revision 1366636767 lists this item under Unsolved problems; primary-source/current-status release review is still required\nRights note: CC BY-SA 4.0; adapted from the pinned Wikipedia revision with source attribution\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress. Global weak solutions and local/de-regularity results are established, and there is active recent progress for special data regimes, but the general global regularity (or breakdown) of smooth solutions to 3D Vlasov–Maxwell is open." }, { "id": 2400015, "problem_number": "AMR-023-0015", "title": "Infinitely many Lehmer pairs", "statement": "Are there infinitely many Lehmer pairs of zeros in the sense used in the theory of the de Bruijn–Newman constant?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Analysis problems from Wikipedia\nSource item: Wikipedia analysis item 15\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767\nAccessed: 2026-07-29\nExtraction: wiki-pinned-revision\nStatus evidence: Pinned Wikipedia revision 1366636767 lists this item under Unsolved problems; primary-source/current-status release review is still required\nRights note: CC BY-SA 4.0; adapted from the pinned Wikipedia revision with source attribution\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. It is believed there are infinitely many Lehmer pairs (consistent with the GUE pair-correlation heuristic, growth ~ $cT$), but infinitude is not proven unconditionally. Literature status: - Lehmer pairs (in the context of the de Bruijn–Newman constant and the search for zeros of $\\Xi_\\lambda(t)$) are connected to the work of de Bruijn, Newman, and the de Bruijn–Newman constant $\\Lambda\\le0$ (the 2018 result of Rodgers–Tao). The number of Lehmer pairs up to height $T$ is believed (unconditionally conjectured, linked to the Montgomery pair-correlation / GUE heuristic) to grow like $cT$, so infinitely many are expected. - However, proving that there are **infinitely many** Lehmer pairs (for a precise definition) is **open**; it is not established unconditionally. Numerical evidence strongly supports it (many discovered Lehmer pairs), but no proof of infinitude exists. - Difficulty above default L3 (it relates to the fine structure of the Riemann zeta zeros and pair-correlation)." }, { "id": 2500003, "problem_number": "AMR-024-0003", "title": "Finite-dimensional dynamics for two-dimensional Navier–Stokes", "statement": "Is the global attractor of the periodically forced two-dimensional Navier–Stokes equations conjugate to a smooth finite-dimensional dynamical system? Can its transient dynamics be described or controlled in finite-dimensional terms?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Constantin - Problems (pdf) (2001)\nSource item: source-order Question 3\nSource URL: https://web.math.princeton.edu/~const/2k.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The question of whether the global attractor of 2D Navier–Stokes is smoothly conjugate to a finite-dimensional system, and whether transient dynamics are finitely reducible, remains **open** (OPEN-TRIAGE)." }, { "id": 2500004, "problem_number": "AMR-024-0004", "title": "Dissipation bounds for flow past an obstacle", "statement": "For viscous incompressible flow in $\\mathbb{R}^3\\setminus B$ past a fixed obstacle $B$, with velocity approaching a nonzero constant vector at infinity, obtain realistic rigorous upper bounds on the energy-dissipation rate.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Constantin - Problems (pdf) (2001)\nSource item: source-order Question 4\nSource URL: https://web.math.princeton.edu/~const/2k.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; public author-hosted PDF\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Rigorous dissipation-rate/drag bounds are known in special geometries, but fully general and physically sharp (\"realistic\") bounds for 3D flow past an arbitrary obstacle remain **partially open** (PARTIAL-PROGRESS)." }, { "id": 2700001, "problem_number": "AMR-026-0001", "title": "Five Open Problems — Global Cauchy theory for one-dimensional Euler–Fourier flow", "statement": "Develop a global-in-time theory of the Cauchy problem for the one-dimensional Euler–Fourier system for initial data constrained only by finite energy and entropy, possibly with local boundedness assumptions on density, reciprocal density, velocity, temperature, and reciprocal temperature.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Serre - Problems (pdf) (2012)\nSource item: Open Problem 1\nSource URL: https://perso.ens-lyon.fr/denis.serre/DPF/Ouverts.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: The corrected author PDF and open journal article present this as one of five open problems; current status requires release review\nRights note: NEEDS_REVIEW; public author-hosted PDF and open journal version inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Denis Serre", "proposed_year": 2012, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: 1D Euler–Fourier global existence is settled under additional hypotheses (bounded density bounds, entropy control); the fully general finite-energy-and-entropy-only Cauchy theory (allowing vacuum up to the data) remains open." }, { "id": 2700002, "problem_number": "AMR-026-0002", "title": "Five Open Problems — Compensated compactness for symmetric matrices", "statement": "Develop a compensated-compactness calculus for symmetric matrices when compensated compactness yields only inequalities. As a first application, prove complete continuity of the semigroup for the multidimensional scalar-conservation-law Cauchy problem.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Serre - Problems (pdf) (2012)\nSource item: Open Problem 2\nSource URL: https://perso.ens-lyon.fr/denis.serre/DPF/Ouverts.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: The corrected author PDF and open journal article present this as one of five open problems; current status requires release review\nRights note: NEEDS_REVIEW; public author-hosted PDF and open journal version inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Denis Serre", "proposed_year": 2012, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: a compensated-compactness principle for symmetric matrices is known in limited forms; the proposed \"only inequalities\" calculus and its use to prove complete continuity of the multidimensional scalar conservation law semigroup remains open." }, { "id": 2700003, "problem_number": "AMR-026-0003", "title": "Five Open Problems — Compressible Navier–Stokes near vacuum", "statement": "For compressible Navier–Stokes equations with constant viscosities near vacuum, does the unphysical one-dimensional consequence identified by Hoff and Serre have a multidimensional counterpart?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Serre - Problems (pdf) (2012)\nSource item: Open Problem 3\nSource URL: https://perso.ens-lyon.fr/denis.serre/DPF/Ouverts.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: The corrected author PDF and open journal article present this as one of five open problems; current status requires release review\nRights note: NEEDS_REVIEW; public author-hosted PDF and open journal version inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Denis Serre", "proposed_year": 2012, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open/partial: the 1D Hoff–Serre near-vacuum degeneracy is established; whether a multidimensional analogue (same unphysical consequence) exists is an unresolved question. Literature status: - Partial/qualitative progress. The \"Hoff–Serre\" phenomenon (the one-dimensional incompatibility between constant viscosity and vacuum) is documented; its precise multidimensional analogue is subtle and not fully resolved either way. Some more recent works (e.g., by Hoff–Serre and followers) analyze the multidimensional vacuum interface for compressible Navier–Stokes. Whether the same \"unphysical\" one-dimensional mechanism (which for 1D forces solutions into only recursively weaker classes) has a true multidimensional counterpart is an open question. - I could not verify an explicit paper settling the multidimensional analogue (either an example showing the same failure in $d\\ge2$, or a proof that it does not occur). The problem retains the character of a genuinely open research question." }, { "id": 2700004, "problem_number": "AMR-026-0004", "title": "Five Open Problems — Eternal finite-energy compressible Euler flow", "statement": "Does the compressible Euler system in odd spatial dimension admit a nontrivial smooth eternal solution having finite nonzero mass and energy?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Serre - Problems (pdf) (2012)\nSource item: Open Problem 4\nSource URL: https://perso.ens-lyon.fr/denis.serre/DPF/Ouverts.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: The corrected author PDF and open journal article present this as one of five open problems; current status requires release review\nRights note: NEEDS_REVIEW; public author-hosted PDF and open journal version inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Denis Serre", "proposed_year": 2012, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open: existence of nontrivial smooth eternal finite-energy compressible Euler flows in odd dimensions is unresolved; no construction and no impossibility proof is available. Literature status: - Open. The existence of nontrivial smooth eternal solutions to compressible Euler with finite nonzero mass/energy is open. Classical intuition (dispersive/decay of acoustic waves) suggests such solutions may fail to exist (the total mass of a nontrivial compressible wave is generically not conserved to a static state), but no proof of non-existence nor construction is known. - Related known results: temporary smooth solutions exist locally (local well-posedness via energy methods); global smooth small-data solutions around constant state spread out and decay (Klainerman–Majda-type; Christodoulou for isentropic 2D), so they are not \"eternal nontrivial\" in the momentum-transport sense. There is also the fact that in 1D, entropy/rarefaction structures prevent compactly-supported eternal waves. The odd-dimension…" }, { "id": 2700005, "problem_number": "AMR-026-0005", "title": "Five Open Problems — Regular reflection without irrotationality", "statement": "Prove existence of a regular reflection for compressible flow against a wedge without assuming that the flow is irrotational.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Serre - Problems (pdf) (2012)\nSource item: Open Problem 5\nSource URL: https://perso.ens-lyon.fr/denis.serre/DPF/Ouverts.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: The corrected author PDF and open journal article present this as one of five open problems; current status requires release review\nRights note: NEEDS_REVIEW; public author-hosted PDF and open journal version inspected\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Denis Serre", "proposed_year": 2012, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: regular reflection existence is rigorously established for potential (irrotational) flows (Chen–Feldman), with recent 2024–25 work extending to cases with vorticity/non-potential flows. A fully general existence without any irrotationality hypothesis, in the exact Serre formulation, remains open." }, { "id": 2800101, "problem_number": "AMR-027-0101", "title": "10 Lectures and 42 Open Problems — Mallat-Zeitouni Gaussian-basis problem", "statement": "Let $X$ be a centered Gaussian random vector in $\\mathbb{R}^n$ with known covariance matrix, and for an orthonormal basis $B=(b_1,\\ldots,b_n)$ let $N_B=\\| (\\langle X,b_1\\rangle,\\ldots,\\langle X,b_n\\rangle)\\|_\\infty$. Is the Karhunen-Loève basis always optimal for minimizing $\\mathbb{E}N_B$ over all orthonormal bases?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 1.1\nSource URL: https://afonsobandeira.wordpress.com/2014/07/02/an-interesting-problem-by-mallat-and-zeitouni/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- Corrected the statement: the conjectured extremum is a **maximum**, not a minimum (dataset had it backwards; the minimum version is disproved already in dimension 2). - Proved the conjecture exactly in dimension $n=2$ for the sup-norm functional $\\mathbb{E}N_B$, with the closed form above (verified numerically). The minimizer in dimension 2 is the variance-equalizing basis. - Documented the literature status: exact conjecture open in general; best results are constant-factor (Litvak–Tikhomirov 2018) and $1+O(1/\\sqrt d)$-factor (Liu 2026) approximate optimality of the KL basis." }, { "id": 2800102, "problem_number": "AMR-027-0102", "title": "10 Lectures and 42 Open Problems — Gaussian singular-value monotonicity", "statement": "For a $d\\times d$ real Gaussian matrix $G_{\\mathbb{R}}$ and complex Gaussian matrix $G_{\\mathbb{C}}$, both normalized to entry variance $1/d$, define $\\alpha_{\\mathbb{K}}(d)=d^{-1}\\mathbb{E}\\sum_{k=1}^d\\sigma_k(G_{\\mathbb{K}})$. Is $\\alpha_{\\mathbb{R}}(d)$ monotonically increasing in $d$, and is $\\alpha_{\\mathbb{C}}(d)$ monotonically decreasing in $d$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 1.2\nSource URL: https://afonsobandeira.wordpress.com/2013/11/01/a-conjecture-on-the-singular-values-of-a-gaussian-matrix/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "- $\\alpha_{\\mathbb C}(d)$ is monotonically **decreasing** in $d$ — proven in the literature (arXiv:1606.00494). - $\\alpha_{\\mathbb R}(d)$ is conjectured to be monotonically **increasing** in $d$ — **still open** as far as verified." }, { "id": 2800103, "problem_number": "AMR-027-0103", "title": "10 Lectures and 42 Open Problems — Open Problem 1.3", "statement": "Let ${W}$ denote a symmetric Wigner matrix with i.i.d. entries ${W_{ij}\\sim \\mathcal{N}(0,1)}$ . Also, given ${B\\in\\mathbb{R}^{n\\times n}}$ symmetric, define: $Q(B) = \\max\\left\\{tr(BX): X\\succeq 0, X_{ii}=1 \\right\\}. $ Define ${q(\\xi)}$ as $q(\\xi) = \\lim_{n\\rightarrow\\infty} \\frac1n\\mathbb{E} Q\\left( \\frac{\\xi}n\\mathbf{1}\\mathbf{1}^T + \\frac1{\\sqrt{n}}W \\right). $ What is the value of ${\\xi_\\ast}$ , defined as $\\displaystyle \\xi_\\ast = \\inf\\{ \\xi\\geq 0: q(\\xi)>2\\}. $", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 1.3\nSource URL: https://afonsobandeira.wordpress.com/2015/09/18/10l42ppca/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "- The exact value of $\\xi_\\ast$ in Open Problem 1.3 is **not confirmed resolved** as of the current search (2026-08-05). Heuristically/qualitatively the transition should occur at a constant order-1 signal (consistent with the BBP / spiked-Wigner SDP phase transition), but no verifiable exact-value or sharp-theorem citation was found." }, { "id": 2800203, "problem_number": "AMR-027-0203", "title": "10 Lectures and 42 Open Problems — The planted clique problem", "statement": "Is there a polynomial time algorithm that is able to find the largest clique of $G$ (with high probability) for $\\omega \\ll \\sqrt{n}$ ? For example, for $\\omega \\approx \\frac{\\sqrt{n}}{\\log n}$ ? Is there a polynomial time algorithm that is able to distinguish, with high probability, $G$ from a draw of $G\\left( n, \\frac12\\right)$ for $\\omega \\ll \\sqrt{n}$ ? For example, for $\\omega \\approx \\frac{\\sqrt{n}}{\\log n}$ ? Is there a quasi-linear time algorithm able to find the largest clique of $G$ (with high probability) for $\\omega\\leq \\left(\\frac1{\\sqrt{e}}-\\epsilon \\right)\\sqrt{n}$ , for some $\\epsilon>0$ ? (This question was posed here )", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 2.3\nSource URL: https://afonsobandeira.wordpress.com/2015/09/30/10l42pgraphs/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "- The three sub-questions of Open Problem 2.3 (polynomial-time recovery, detection, and quasi-linear-time recovery at $\\omega=o(\\sqrt n)$ / $(\\tfrac1{\\sqrt e}-\\varepsilon)\\sqrt n$) remain **open**. - The **Planted Clique Conjecture** continues to be a central open conjecture in average-case complexity." }, { "id": 2800301, "problem_number": "AMR-027-0301", "title": "10 Lectures and 42 Open Problems — Hardness at the Cheeger square-root gap", "statement": "Does there exist $c>0$ such that it is NP-hard, given a graph $G$ and $\\phi>0$, to distinguish $h_G\\le\\phi$ from $h_G\\ge c\\sqrt{\\phi}$?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 3.1\nSource URL: https://afonsobandeira.wordpress.com/2015/10/05/10l42pcheeger/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The question of whether distinguishing $h_G\\le\\phi$ from $h_G\\ge c\\sqrt\\phi$ is $\\mathsf{NP}$-hard remains **open as posed**; the closest known results are UGC/SSE-based hardness for approximation factors and spectral-hardness for small sets, which do not settle the stated unconditional square-root-gap hardness. Recorded as partial-progress/open-triage." }, { "id": 2800302, "problem_number": "AMR-027-0302", "title": "10 Lectures and 42 Open Problems — Certifying that matrices are PSD", "statement": "Given a symmetric matrix ${M}$ with small condition number, is there a quasi-linear time (on ${n}$ and the number of non-zero entries of ${M}$ ) procedure that certifies that ${M\\succeq 0}$ . More specifically, the procedure can be randomized in the sense that it may, with some probably not certify that ${M\\succeq 0}$ even if that is the case, what is important is that it never produces erroneous certificates (and that it has a bounded-away-from-zero probably of succeeding, provided that ${M\\succeq 0}$ ).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 3.2\nSource URL: https://afonsobandeira.wordpress.com/2015/10/05/10l42pcheeger/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem appears to remain **open**. Partial progress established here: - A rigorous explanation of *why* the problem is hard: Krylov/residual information gives one-sided bounds in the wrong direction (item 1 contains an explicit counterexample to the naive Lanczos certificate); certificates from $O(1)$ quadratic forms are impossible (item 2); zero-error forces global spectral information. - An exact zero-error spectral certificate $\\mathrm{tr}((I-\\mu^{-1}M)^{2k})<1$ with a precise success condition $\\lambda_{\\min} > \\mu(1-n^{-1/2k})$, showing the obstruction is runtime, not existence of algebraic certificates (item 3). - Two provably correct quasi-linear zero-error certifiers for nontrivial special cases: diagonal dominance ($O(m)$) and the exact second-moment test ($O(m)$, succeeds when the spectrum is near-isotropic around the Gershgorin bound) (item 5). - A reduction of the general problem to certified one-sided spectral-norm estimation with a spectral gap (item 6), and the observation that the Kyng–Sachdeva nearly-linear approximate Cholesky (arXiv:1605.02353, verified) misses zero-error PSD certification even for Laplacians because its spectral guarantee is itself only…" }, { "id": 2800303, "problem_number": "AMR-027-0303", "title": "10 Lectures and 42 Open Problems — Open Problem 3.3", "statement": "Let ${G=(V,E,W)}$ be a graph and ${k}$ a positive integer, is the following true? $\\rho_G(k) \\leq \\mathrm{polylog}(k) \\sqrt{\\lambda_k}. \\ \\ \\ \\ \\ (2)$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 3.3\nSource URL: https://afonsobandeira.wordpress.com/2015/10/05/10l42pcheeger/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4 (higher-order Cheeger; known positive partial answers with refined polylogs)", "research_summary": "- The higher-order Cheeger inequality $\\rho_G(k)\\lesssim \\operatorname{polylog}(k)\\sqrt{\\lambda_k}$ is **open**; the current best general bound is $\\rho_G(k)\\le Ck^2\\sqrt{\\lambda_k}$ (Lee–Oveis Gharan–Trevisan), with near-linear-time $k^6$-type variants. No proof or counterexample for the polylog form was found in the search." }, { "id": 2800401, "problem_number": "AMR-027-0401", "title": "10 Lectures and 42 Open Problems — Improvement over Non-commutative Khintchine inequality", "statement": "Let $A_1,\\dots,A_n\\in \\mathbb{R}^{d\\times d}$ be symmetric matrices and $g_1,\\dots,g_n\\sim\\mathcal{N}(0,1)$ i.i.d.. Does the following hold? $\\mathbb{E} \\left\\| \\sum_{k=1}^{n} g_k A_k \\right\\| \\lesssim \\sigma + \\left(\\log d\\right)^{\\frac12}\\sigma_\\ast.$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 4.1\nSource URL: https://afonsobandeira.wordpress.com/2015/10/25/10l42concentration/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The conjectured improvement over the noncommutative Khintchine inequality is **false**, even with $(\\log d)^{1/2}$ replaced by any $(\\log d)^\\beta$: tensor sums of Wigner matrices $X_{n,N}$ have $\\sigma=\\sqrt n$ and $\\sigma_*=O(\\sqrt{n/N})$ but $\\mathbb{E}\\|X_{n,N}\\|\\ge(2-o(1))n$. This is exactly Proposition 8.2 of Bandeira–Boedihardjo–van Handel (arXiv:2108.06312; Invent. Math. 234, 2023), which explicitly identifies itself as disproving the conjecture from Bandeira's open-problem list. Intuitively: the summands $\\mathbf 1\\otimes\\cdots\\otimes G_k^N\\otimes\\cdots\\otimes\\mathbf 1$ are classically (not freely) independent, and norms of sums of classically independent copies add ($\\sim 2n$) rather than combine in $\\ell^2$ ($\\sim\\sqrt n$); any small parameter that is \"natural\" (subadditive, unitarily and tensor invariant, vanishing on Wigner) is blind to this, so the $\\log$-factor cannot be confined to such a parameter." }, { "id": 2800402, "problem_number": "AMR-027-0402", "title": "10 Lectures and 42 Open Problems — Lata $\\l$ a-Riemer-Schutt", "statement": "Let $X\\in\\mathbb{R}^{d\\times d}$ be a symmetric matrix with (otherwise) independent gaussian entries. Prove (or disprove): $\\mathbb{E} \\|X\\| \\lesssim\\mathbb{E} \\max_k \\|Xe_k\\|_2$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 4.2\nSource URL: https://afonsobandeira.wordpress.com/2015/10/25/10l42concentration/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "- The inequality $\\mathbb E\\|X\\|\\lesssim\\mathbb E\\max_k\\|Xe_k\\|_2$ **(and its reverse) holds** for arbitrary-variance symmetric Gaussian matrices; the Latała conjecture is **settled** (Latała–van Handel–Youssef 2018)." }, { "id": 2800403, "problem_number": "AMR-027-0403", "title": "10 Lectures and 42 Open Problems — Matrix version of 6 deviations suffice", "statement": "Prove or disprove: there exists a universal constant $C$ such that, for any choice of $n$ symmetric matrices $H_1,\\dots,H_n\\in\\mathbb{R}^{n\\times n}$ satisfying $\\|H_k\\|\\leq 1$ (for all $k=1,\\dots,n$ ), there exists $\\varepsilon_1,\\dots,\\varepsilon_n \\in \\{ \\pm1\\}$ such that $\\left\\| \\sum_{k=1}^n \\varepsilon_k H_k \\right\\| \\leq C \\sqrt{n}.$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 4.3\nSource URL: https://afonsobandeira.wordpress.com/2015/10/25/10l42concentration/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem (Matrix Spencer conjecture) is **open as of August 2026**, with major recent progress: the conjecture is proved for rank $\\le n/\\log^3 n$ (Bansal–Jiang–Meka, STOC 2023), for matrices in $C^*$-algebras of dimension $O(n)$ and hence for the Group Spencer variant (Sra 2026; Bandeira–Bölcskei 2026), while the natural variance-sensitive strengthening has been *disproved* (Sra 2026). Best general bound remains $O(\\sqrt{n\\log n})$ from random signing; the lower bound is $\\Omega(\\sqrt n)$. Elementary verifications and the bottleneck analysis above are my own; all substantial results are from the cited literature (each verified to exist via arXiv)." }, { "id": 2800404, "problem_number": "AMR-027-0404", "title": "10 Lectures and 42 Open Problems — OSNAP", "statement": "Part (3) of the problem: Let $s\\leq d\\leq m$ and $z_1,\\dots,z_m\\in \\mathbb{R}^d$ i.i.d. random vectors with i.i.d. entries $\\left( z_k\\right)_j = \\left\\{ \\begin{array}{rcc} -\\frac1{\\sqrt{s}} & \\text{ with probability } & \\frac{s}{2m} \\\\ 0 & \\text{ with probability } & 1-\\frac{s}{m} \\\\ \\frac1{\\sqrt{s}} & \\text{ with probability } & \\frac{s}{2m} \\end{array} \\right.$ Note that $\\mathbb{E} z_kz_k^T = \\frac1m I_{d\\times d}.$ The conjecture is that, there exists $c_1$ and $c_2$ positive universal constants such that $\\mathrm{Prob}\\left\\{ \\left\\| \\sum_{k=1}^m \\left[z_k z_k^T - \\mathbb{E} z_k z_k^T\\right] \\right\\| \\geq \\varepsilon \\right\\} < \\delta,$ for $m\\geq c_1 \\frac{d+\\log\\left( \\frac1{\\delta} \\right)}{\\varepsilon^2}$ and $s\\geq c_2 \\frac{\\log\\left( \\frac{d}{\\delta} \\right)}{\\varepsilon^2}$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 4.4\nSource URL: https://afonsobandeira.wordpress.com/2015/10/25/10l42concentration/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The sparse-sign ($z_k$) concentration of $\\sum_k z_kz_k^T$ holds up to logarithmic factors with $m=\\widetilde O(d/\\varepsilon^2)$, $s=\\widetilde O(\\log(d/\\delta)/\\varepsilon)$, matching the conjecture's form (Nelson–Nguyên OSNAP). The precise universal constants $c_1,c_2$ in the exact statement are not pinned as a theorem as written." }, { "id": 2800405, "problem_number": "AMR-027-0405", "title": "10 Lectures and 42 Open Problems — Random k-lifts of graphs", "statement": "Give a tight upperbound to $\\mathbb{E}\\left\\| A^{\\otimes k} -\\mathbb{E} A^{\\otimes k} \\right\\|.$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 4.5\nSource URL: https://afonsobandeira.wordpress.com/2015/10/25/10l42concentration/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- Random-lift spectra (including $2\\sqrt{\\Delta-1}$ new-eigenvalue bounds and general lift spectral-norm bounds) are well understood, and Bordenave–Collins extend to tensor products / quantum expanders — a strong partial answer. A crisply stated tight bound on $\\mathbb E\\|A^{\\otimes k}-\\mathbb E A^{\\otimes k}\\|$ as literally posed is not certified in the material found, so it remains partially open." }, { "id": 2800406, "problem_number": "AMR-027-0406", "title": "10 Lectures and 42 Open Problems — Feige's conjecture", "statement": "Prove or disprove the following conjecture by Feige : Given $n$ independent random variables $X_1,\\dots,X_n$ s.t., for all $i$ , $X_i \\geq 0$ and $\\mathbb{E} X_i = 1$ we have $\\mathrm{Prob}\\left( \\sum_{i=1}^n X_i \\geq n+1 \\right) \\leq 1 - e^{-1}$ .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 4.6\nSource URL: https://afonsobandeira.wordpress.com/2015/11/29/10l42panextraopenproblem/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Feige's conjecture as posed in AMR-027-0406 is **proved** (July 2026): the sharp bound $\\Pr(\\sum_{i=1}^n X_i \\ge n+1) \\le 1 - (\\frac{n}{n+1})^n \\le 1 - e^{-1}$ holds for all $n$. Primary references: arXiv:2607.23980 (Fu–Han–Wang–Yan–Zhang–Zhou; sharp for all $\\delta\\ge 1$; Lean-verified) and arXiv:2607.24528 (Nie–Wei; sharp for $\\delta\\le 1$), both built on arXiv:2607.08415 (Vlassis–Thomas, Gaffke's conjecture) and arXiv:2410.04741 (Letwin–Yaskin). Status caveat: preprints are one week old, not yet peer-reviewed, AI-assisted (disclosed); the existence of two independent proofs plus a Lean formalization gives high confidence." }, { "id": 2800501, "problem_number": "AMR-027-0501", "title": "10 Lectures and 42 Open Problems — Deterministic Restricted Isometry Property matrices", "statement": "Construct deterministic matrices $A\\in\\mathbb{C}^{M\\times N}$ (or $A\\in\\mathbb{R}^{M\\times N}$ ) satisfying the $(s,\\frac13)$ -RIP for $s\\approx\\frac{M^{0.6}}{\\mathrm{polylog}(N)}$", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 5.1\nSource URL: https://afonsobandeira.wordpress.com/2015/11/22/10l42gordon/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL.** The problem is open. My contribution: 1. A complete self-contained proof that the coherence/Gershgorin route yields deterministic $(s,\\tfrac13)$-RIP for $s\\asymp\\sqrt M$ (Lemma 1) and provably cannot exceed $s=O(\\sqrt M)$ (Lemma 2 + Corollary) — i.e., a rigorous special case (exponent $1/2$ instead of $0.6$) and a rigorous explanation of where every pre-2011 technique stalls. 2. A verified literature status: the only unconditional result beyond $\\sqrt M$ is Bourgain–Dilworth–Ford–Konyagin–Kutzarova (Duke Math. J. 2011) with effective exponent $1/2+\\varepsilon_0$, $\\varepsilon_0\\approx 10^{-24}$; the target exponent $0.6$ remains wide open, with conditional constructions (Paley-type, Satake–Gu 2020) and formal barriers (Ramsey-hardness of Gamarnik–Zadik; certification hardness of Ding–Kunisky–Wein–Bandeira) indicating genuine difficulty." }, { "id": 2800502, "problem_number": "AMR-027-0502", "title": "10 Lectures and 42 Open Problems — Certifying the Restricted Isometry Property", "statement": "Let $N = 2M$ . For which $s$ is there a polynomial time algorithm that is guaranteed to, with high probability, certify that a gaussian matrix $A$ is $\\left(s,\\frac13\\right)$ -RIP? In particular, a $\\left(s,\\frac13\\right)$ -RIP matrix has to not have $s$ sparse vectors in its nullspace. This motivates a second question: Let $N = 2M$ . for which $s$ is there a polynomial time algorithm that is guaranteed to, with high probability, certify that a gaussian matrix $A$ does not have $s$ -sparse vectors in its nullspace?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 5.2\nSource URL: https://afonsobandeira.wordpress.com/2015/11/22/10l42gordon/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Both questions are answered in the literature, post-2016: - **Q1 (RIP certification).** For $N = 2M$ Gaussian $A$: polytime certification of $(s,\\tfrac13)$-RIP exists for $s \\lesssim \\sqrt{M/\\log N}$ (entrywise/coherence thresholding; Wang–Berthet–Plan 2016), and the optimal average-case runtime for certification in the regime $\\sqrt M \\ll s \\lesssim M/\\log N$ is exactly $N^{\\tilde\\Theta(s^2/M)}$ — upper bound by the Koiran–Zouzias lazy algorithm (2014), lower bound by Ding–Kunisky–Wein–Bandeira (IEEE TIT 2021, arXiv:2005.11270) via a rigorous low-degree-likelihood-ratio bound. So polynomial time is achievable exactly at $s = \\tilde O(\\sqrt M)$, i.e. the famous \"square-root bottleneck\" is inherent to certification, not just to deterministic constructions. - **Q2 (nullspace certification).** Same threshold: polytime up to $s \\sim \\sqrt N$ (up to polylogs) via convex ($\\ell^1/\\ell^\\infty$) certificates; beyond that, the same low-degree lower bound $N^{\\tilde\\Omega(s^2/M)}$ applies. **Caveat (what \"solved\" means here).** The hardness direction is a rigorous unconditional lower bound against the class of low-degree polynomial algorithms (which captures all known polytime techniques…" }, { "id": 2800601, "problem_number": "AMR-027-0601", "title": "10 Lectures and 42 Open Problems — Random Partial Discrete Fourier Transform", "statement": "Consider a $A\\in\\mathbb{C}^{M\\times N}$ obtained by sampling random rows of a Discrete Fourier Tranform. How large does $M$ need to be in order for, with high probability, $A$ to satisfy the $(s,\\frac13)$ -RIP?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 6.1\nSource URL: https://afonsobandeira.wordpress.com/2015/12/07/10l42pcompressedsensing/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Best known answer: $M = O(s\\log^2 s\\,\\log N)$ uniformly random rows suffice for the $(s,\\tfrac13)$-RIP with high probability (Haviv–Regev 2015, improving Bourgain 2014), while $\\Omega(s\\log N)$ rows are necessary in general and $\\Omega(s\\log s\\,\\log N/\\log p)$ are necessary in the finite-field Fourier model of Rao (2019). The conjectured truth is $M\\asymp s\\log N$; the gap is the factor $\\log^2 s$, which is provably unremovable for the analogous Walsh–Hadamard system (Blasiok 2023) but unresolved for the DFT." }, { "id": 2800602, "problem_number": "AMR-027-0602", "title": "10 Lectures and 42 Open Problems — Mutually Unbiased Bases", "statement": "How many mutually unbiased bases are there in 6 dimensions?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 6.2\nSource URL: https://afonsobandeira.wordpress.com/2015/12/07/10l42pcompressedsensing/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "- The number of MUBs in dimension 6 is **unknown**; the problem is **open**. Three MUBs are known to exist; whether a fourth exists (and hence whether the maximum is 3 or more, up to 7) is unresolved." }, { "id": 2800604, "problem_number": "AMR-027-0604", "title": "10 Lectures and 42 Open Problems — The Paley ETF Conjecture", "statement": "Does the Paley Equiangular tight frame satisfy the Restricted Isometry Property pass the square root bottleneck? (even by logarithmic factors?).", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 6.4\nSource URL: https://afonsobandeira.wordpress.com/2015/12/07/10l42pcompressedsensing/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- Whether the Paley ETF satisfies RIP past the square-root bottleneck is **still open**. It is known **conditionally** (under a Legendre-symbol pseudorandomness conjecture) that it breaks the bottleneck (Bandeira–Mixon–Moreira), and 2024 work links it to Paley-graph extractors, but no unconditional proof is known." }, { "id": 2800605, "problem_number": "AMR-027-0605", "title": "10 Lectures and 42 Open Problems — Constructive Kadison-Singer", "statement": "Give a (polynomial time) construction of the tight frame partition satisfying the properties required in the Kadison-Singer problem (or the related Weaver’s conjecture). These partitions were proven to exist (with a non-constructive proof) in the recent breakthrough of Marcus, Spielman, and Srivastava .", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 6.5\nSource URL: https://afonsobandeira.wordpress.com/2015/12/07/10l42pcompressedsensing/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The existence side is **solved** non-constructively (MSS 2015). The **constructive (polynomial-time) side is open in general**. Partial progress: deterministic poly time in the dense regime $m\\ge49d^2$ (2024), and quasi-polynomial time for low dimensions (2023), alongside $\\mathsf{FNP}$-hardness of the optimization version." }, { "id": 2800701, "problem_number": "AMR-027-0701", "title": "10 Lectures and 42 Open Problems — Gilbert-Varshamov bound", "statement": "Explicit deterministic constructions of codes achieving the GV bound Is the GV bound tight?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 7.1\nSource URL: https://afonsobandeira.wordpress.com/2015/11/27/10l42grouptesting/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- **GV is not tight** — it can (and has been) improved (Jiang–Vardy binary; 2024 $q$-ary). - **Explicit codes achieving the GV bound are still open.** The current state of the art gives explicit codes *approaching* the GV tradeoff (Ta-Shma 2017) and even efficiently decodable ones near the large-distance curve (2023), but achieving the exact GV bound deterministically remains an outstanding challenge." }, { "id": 2800702, "problem_number": "AMR-027-0702", "title": "10 Lectures and 42 Open Problems — Boolean classification and annulus conjecture", "statement": "Prove or disprove: $R_A(\\alpha n,\\beta n,n)=\\alpha+(1-\\alpha)R_A(1,\\beta n,(1-\\alpha)n)+o(1)$.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 7.2\nSource URL: https://afonsobandeira.wordpress.com/2015/11/27/10l42grouptesting/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The annulus conjecture $R_A(\\alpha n,\\beta n,n)=\\alpha+(1-\\alpha)R_A(1,\\beta n,(1-\\alpha)n)+o(1)$ is **proven for $\\beta\\ge 2\\alpha$** and remains **open in general** (for $\\beta<2\\alpha$). Recent structural work corroborates but does not fully resolve it." }, { "id": 2800704, "problem_number": "AMR-027-0704", "title": "10 Lectures and 42 Open Problems — The Deletion Channel", "statement": "What are the asymptotics of $\\mathcal{D}\\left(n;\\frac12\\right)$ ? \\item An interesting aspect of the Deletion Channel is that different messages may have different difficulties of decoding. This motivates the following question: What are the two (distinct) binary sequences $x^{(1)}$ and $x^{(2)}$ that are more difficult to distinguish (let’s say that the receiver knows that either $x^{(1)}$ or $x^{(2)}$ was sent but not which)?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 7.4\nSource URL: https://afonsobandeira.wordpress.com/2015/11/27/10l42grouptesting/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The problem (both parts) remains **open** as of August 2026, but the landscape is transformed since 2015: worst-case trace reconstruction is now known to be quasipolynomial (Burudgunte–Valiant–Wang, arXiv:2607.04073, July 2026), and the average case $\\mathcal{D}(n;\\tfrac12)$ is sandwiched between $\\widetilde{\\Omega}(\\log^{5/2} n)$ (Chase 2021) and $\\exp(\\widetilde{O}(\\log^{1/5} n))$ (Rubinstein 2023). - Part (b) is the \"separating words\" problem; I showed (with proof) that the obvious sparse pair $(0^n, 0^{n-1}1)$ is distinguished by a single trace (TV $=\\tfrac12$ exactly), proved the equivalence of part (b) to worst-case reconstruction up to a factor $O(n)$, and recorded why the natural alternating-string candidate defeats elementary analysis. The extremal pair is uncharacterized; its trace complexity lies between $\\widetilde{\\Omega}(n^{3/2})$ and quasipolynomial. ### References (all verified this session) 1. T. Batu, S. Kannan, S. Khanna, A. McGregor, *Reconstructing strings from random traces*, SODA 2004, 910–918. 2. T. Holenstein, M. Mitzenmacher, R. Panigrahy, U. Wieder, *Trace reconstruction with constant deletion probability and related results*, SODA 2008, 389–398. 3.…" }, { "id": 2800802, "problem_number": "AMR-027-0802", "title": "10 Lectures and 42 Open Problems — Sum of Squares approximation ratio for Max-Cut", "statement": "What is the approximation ratio (or integrality gap) for the Sum-of-Squares (SOS) relaxation of degree 4 for the Max-Cut problem? What about other constant degrees?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 8.2\nSource URL: https://afonsobandeira.wordpress.com/2015/12/04/10l42pmaxcut/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "- The exact degree-4 SOS integrality gap / approximation ratio for Max-Cut is **unknown** (open). Degree-2 is settled at $0.878$; degree-4 (and other constant degrees $>2$) are undetermined. Literature status: - **Status: OPEN/PARTIAL.** The degree-2 SOS (= Goemans–Williamson) ratio $0.878567\\ldots$ is known. It is believed that constant-degree SOS relaxations can strictly improve on $0.878$ for Max-Cut, but **no exact ratio or integrality gap for degree 4 (or any degree $>2$) has been determined** as far as verified. - Known limitations: for **polylog-degree** SOS, the guarantee degrades to $0.878$ (Khot–Moshkovitz 2016, *Candidate hard unique games*; and the \"SOS for MaxCut\" results showing that degree-$\\Omega(\\sqrt n)$ is needed to beat UGC) — but these concern degree scaling, not the exact constant-degree-4 gap. - The exact degree-4 gap depends on solving an optimization over symmetric multilinear forms / the fourth-moment tensor, which is computationally hard to determine; no closed form is known.…" }, { "id": 2800803, "problem_number": "AMR-027-0803", "title": "10 Lectures and 42 Open Problems — The Grothendieck Constant", "statement": "What is the value of the (real) Grothendieck constant?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 8.3\nSource URL: https://afonsobandeira.wordpress.com/2015/12/04/10l42pmaxcut/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4 (celebrated, long-open constant)", "research_summary": "- The exact value of $K_G$ is **unknown**; the problem remains **open**. Known bounds are approximately $[1.676, 1.783]$. Literature status: - **Status: OPEN.** The exact value of the real Grothendieck constant $K_G$ is still unknown as of 2026. The best-known bounds improved over time but remain strictly separated: - **Lower bound:** the classical Krivine / Reeds *lower bound* $K_G \\ge \\tfrac{\\pi}{2\\ln(1+\\sqrt2)}=\\tfrac{\\pi/\\ln(1+\\sqrt2)}{2}\\approx 1.676\\ldots$ (this value coincides with $\\frac{1}{\\ln(1+\\sqrt2)}\\cdot\\frac{\\pi}{2}$ numerically). Krivine conjectured this value is exact; it is not. - **Upper bound:** Braverman, Makarychev, Makarychev, Naor (*The Grothendieck constant is strictly smaller than Krivine's bound*, Forum of Mathematics Pi 2013) proved $K_G<\\tfrac{\\pi}{2\\ln(1+\\sqrt2)}$, i.e. strictly below the Krivine value, establishing $K_G\\approx 1.782\\ldots$ as an upper bound. Together $K_G\\in[\\,1.676\\ldots,\\;1.782\\ldots\\,]$. - Since Bandeira posed this problem (2015), no exact value has…" }, { "id": 2800804, "problem_number": "AMR-027-0804", "title": "10 Lectures and 42 Open Problems — The Paley Clique Problem", "statement": "What is the clique number of the Paley graph? Can the the SOS degree 4 analogue of the theta number help upper bound it?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 8.4\nSource URL: https://afonsobandeira.wordpress.com/2015/12/04/10l42pmaxcut/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "- The clique/independence number of Paley graphs (in the sharp polylog/$\\sqrt q$-constant sense) is **open**, as is the question whether SOS degree-4 theta improves the upper bound. Both parts remain **open**." }, { "id": 2800805, "problem_number": "AMR-027-0805", "title": "10 Lectures and 42 Open Problems — Maximum and minimum bisections on random regular graphs", "statement": "Given a $d$ -regular graph on $n$ nodes $G$ . Let $MaxBis(G)$ and $MinBis(G)$ denote, respectively, the size of its largest and smallest bisection. Is it true that for every $d$ $MaxBis(G)$ $+ MinBis(G)$ $=$ $\\frac{d}{2} + o(1)$ , where $o(1)$ is a term that goes to zero as $n$ grows?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 8.5\nSource URL: https://afonsobandeira.wordpress.com/2015/12/04/10l42pmaxcut/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- $\\mathrm{MaxBis}$ and $\\mathrm{MinBis}$ of random $d$-regular graphs are each asymptotic-linear in $n$ with known (degree-dependent) constants; the literal exact identity $\\mathrm{MaxBis}+\\mathrm{MinBis}=\\tfrac{d}{4}n+o(n)$ **is not established as such** for all $d$. **Partial.**" }, { "id": 2800901, "problem_number": "AMR-027-0901", "title": "10 Lectures and 42 Open Problems — Detection Threshold for SBM for three of more communities", "statement": "What is the partial recovery threshold for the Stochastic Block Model on $k\\geq 3$ communities.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 9.1\nSource URL: https://afonsobandeira.wordpress.com/2015/11/28/10l42psbm/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "- The partial-recovery threshold for $k\\ge3$ communities is **sharp and known**: partial recovery is achievable exactly above a well-defined signal-to-noise threshold (the Kesten–Stieltjes/spectral regime), a *gap* exists between the spectral and Bayes thresholds for $k\\ge3$, and optimal misclassification rates are characterized (notably via the CH-divergence). **Solved.**" }, { "id": 2800902, "problem_number": "AMR-027-0902", "title": "10 Lectures and 42 Open Problems — Recovery Threshold for SBM for logarithmic many communities", "statement": "What is the exact recovery threshold for the Stochastic Block Model with a logarithm number of communities? Both computational and information theoretical.", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 9.2\nSource URL: https://afonsobandeira.wordpress.com/2015/11/28/10l42psbm/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "- The exact-recovery threshold for $k=O(\\log n)$ communities is **known**, both information-theoretically and computationally: exact recovery is possible iff the signal exceeds $\\frac{\\log n}{\\log k}\\cdot$ (constant depending on the model), achieved by polynomial-time (spectral/SDP) algorithms in the sharp regime. **Solved** (Abbe–Bandeira–Hall; Abbe–Sandon)." }, { "id": 2800903, "problem_number": "AMR-027-0903", "title": "10 Lectures and 42 Open Problems — Tightness of k-median LP", "statement": "Is the k-medians Linear Programming relaxation tight even for point clouds coming from generative models that do not have a community structure?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 9.3\nSource URL: https://afonsobandeira.wordpress.com/2015/11/28/10l42psbm/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- Worst-case $k$-median LP is not tight (constant gap). For **community/cluster generative models** it is tight w.h.p. under separation conditions (solved). For the literal \"generative models **without** community structure\" case, tightness is **open/general** — no full characterization found." }, { "id": 2800904, "problem_number": "AMR-027-0904", "title": "10 Lectures and 42 Open Problems — Stability conditions for tightness of k-median LP and k-means SDP", "statement": "Can one give conditions for integrality of the k-medians LP or the k-means SDP based on stability type properties (on the fact that the data is “well-explained” by a certain number of clusters)?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 9.4\nSource URL: https://afonsobandeira.wordpress.com/2015/11/28/10l42psbm/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- Tightness of $k$-median LP / $k$-means SDP is established under **strong separation (well-separated mixture / community) conditions**; under the more permissive notion of **stability (perturbation-stability)**, near-optimal algorithms exist, but **exact integrality of the $k$-median LP / $k$-means SDP is not generally proven** — the problem remains partially open as posed." }, { "id": 2800905, "problem_number": "AMR-027-0905", "title": "10 Lectures and 42 Open Problems — Positive PCA tightness", "statement": "Is the Semidefinite programming relaxation for the positive Principal Component Analysis problem tight with high probability for Wigner matrices?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 9.5\nSource URL: https://afonsobandeira.wordpress.com/2015/11/28/10l42psbm/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The Positive-PCA SDP (with Wigner input and a planted positive/nonneg spike) is known to be tight above a constant SNR in several regimes (via the spiked/nonnegative PCA and angular-sync SDP-tightness literature), but a sharp, clean characterization of the tightness threshold for Wigner matrices is **not fully certified** — partial progress recorded." }, { "id": 2801001, "problem_number": "AMR-027-1001", "title": "10 Lectures and 42 Open Problems — Angular Synchronization via Projected Power Method", "statement": "Does the projected power method converge (with high probability) to the optimal solution of the angular synchronization problem with (small enough) gaussian noise?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 10.1\nSource URL: https://afonsobandeira.wordpress.com/2015/12/09/42l10psynchronization/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "- With small enough Gaussian noise, projected power iteration **converges w.h.p. to the optimal (ground-truth) solution** of angular synchronization — **solved** by the non-convex optimization / PPM analyses (Boumal, Voroninski, Bandeira; Bandeira–Boumal–Singer)." }, { "id": 2801002, "problem_number": "AMR-027-1002", "title": "10 Lectures and 42 Open Problems — Sharp tightness of the Angular Synchronization SDP", "statement": "Is the SDP for angular synchronization tight (with high probability) for noise levels $\\sigma$ essentially until the solution of angular synchronization no longer correlates with the ground truth?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 10.2\nSource URL: https://afonsobandeira.wordpress.com/2015/12/09/42l10psynchronization/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The angular-sync SDP is **tight with high probability exactly up to the sharp (spectral/BBP-type) threshold** where the solution stops correlating with the ground truth, and **not** tight beyond it — **solved** (Bandeira–Khoo–Singer 2015/2016, arXiv:1410.1353)." }, { "id": 2801003, "problem_number": "AMR-027-1003", "title": "10 Lectures and 42 Open Problems — Tightness of the Multireference Alignment SDP", "statement": "For which levels of noise is the SDP for Multireference Alignment tight?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 10.3\nSource URL: https://afonsobandeira.wordpress.com/2015/12/09/42l10psynchronization/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- MRA SDP is tight in the **high-SNR** regime (above the Bandeira-Rigollet-Singer / Perry-Wein-Bandeira-Moitra threshold), but for intermediate noise levels SDP tightness is not established (some regimes are computationally hard for SDP-type methods while estimation remains possible IT-wise). **Partial** resolution." }, { "id": 2801004, "problem_number": "AMR-027-1004", "title": "10 Lectures and 42 Open Problems — Consistency and sample complexity of Multireference Alignment", "statement": "Is the Maximum likelihood for Multireference Alignment consistent? (after fixing the power spectrum) What is the sample complexity of the Multireference Alignment problem?", "background": "Collected from an AMR-indexed open-problem list; current status requires release review.\nSource list: Bandeira - 42 Open Problems in Data Science (2016)\nSource item: Open Problem 10.4\nSource URL: https://afonsobandeira.wordpress.com/2015/12/09/42l10psynchronization/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source presents this as an open item\nRights note: NEEDS_REVIEW; publicly accessible author blog\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Afonso S. Bandeira", "proposed_year": 2015, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- **Sample complexity** of MRA is characterized: $N=\\Theta(\\sigma^4 L/\\mathrm{SNR}^4)$ (high-SNR) with blow-up $\\sigma^{2L}$ at very low SNR (guaranteed-consistency threshold); **consistency** of ML/spectral methods holds in the feasible regime after fixing the power spectrum. Sharp all-regime ML-consistency statement remains partially open. **Partial.**" }, { "id": 2900001, "problem_number": "AMR-028-0001", "title": "Betti Posets and the Stanley Depth", "statement": "The Betti poset of a monomial ideal $I$ determines the Stanley projective dimension of $S/I$ and $I$. More precisely, if $I\\subseteq S$ and $I'\\subseteq S'$ are monomial ideals in polynomial rings $S$ and $S'$ with $\\mathcal{B}(I)\\cong\\mathcal{B}(I')$, then $\\operatorname{spdim}_{S}(S/I)=\\operatorname{spdim}_{S'}(S'/I')$ and $\\operatorname{spdim}_{S}I=\\operatorname{spdim}_{S'}I'$.", "background": "Difficulty assignment: default L3\nSource list: Katthän - Betti Posets and the Stanley Depth (2016)\nSource item: Conjecture 2.4\nSource URL: https://arxiv.org/abs/1509.08275\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1509.08275 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Lukas Katthän", "proposed_year": 2016, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Katthän's Conjecture 2.4 (Betti poset determines Stanley projective dimension) remains **open** as of 2026 (OPEN-TRIAGE). Literature status: - **Source.** L. Katthän, \"Betti posets and the Stanley depth\", arXiv:1509.08275 (2016; published version 2016), **Conjecture 2.4**: the Betti poset $\\mathcal B(I)$ determines the Stanley projective dimension of $S/I$ and of $I$. - **Status — OPEN as of source; no resolution found.** The paper states the conjecture as open. The Stanley depth (Stanley's conjecture and its context, disproved by Ichim–Katthän–Moyano-Fernández's 2022 counterexamples for the Stanley depth) is a subtle invariant; I found no published proof or disproof of Conjecture 2.4 specifically through 2026. - Classification **OPEN-TRIAGE**: the open status is sourced to the paper itself; a deeper 2024–2026 audit of the Stanley-depth literature is warranted." }, { "id": 3000001, "problem_number": "AMR-029-0001", "title": "Achieve global rigidity by pinning nodes", "statement": "Given a graph $G(V,E)$, find a minimum cardinality set $S \\subset V$ of nodes such that adding a complete graph on $S$ renders the graph $G+K_S$ globally rigid in 2-dimensional space.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Achieve global rigidity by pinning nodes\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (no verified resolution found). The 2D global-rigidity structural theory is well developed, suggesting the problem is likely polynomial, but I did not find a written algorithm. Literature status: The Egres Open list still classes this as open. Global rigidity in $\\mathbb{R}^2$ has a clean matroidal characterization (Hendrickson–Jordan: a graph is generically globally rigid iff it is 3-connected and redundantly rigid), and \"adding a clique on a set\" is a standard way to force global rigidity. No published polynomial-time algorithm or characterization for the *minimum* pinning set $S$ was verified in this study; the difficulty appears to lie in the interaction between the matroid base and the planted clique." }, { "id": 3000002, "problem_number": "AMR-029-0002", "title": "Acyclic orientation with connectivity prescriptions", "statement": "Problem 1. Given an undirected graph $\\displaystyle G=(V,E)$ and $\\displaystyle s,t\\in V,\\;\\; k\\in N$, decide whether the graph has an acyclic orientation, so that for every vertex $\\displaystyle v\\in V\\setminus\\{s,t\\}$, there are $\\displaystyle k$ pairwise edge-disjoint directed paths from $\\displaystyle s$ to $\\displaystyle v$, and also $\\displaystyle k$ pairwise edge-disjoint directed paths from $\\displaystyle v$ to $\\displaystyle t$. Problem 2. Given an undirected graph $\\displaystyle G=(V,E)$ and $\\displaystyle s, t_1, t_2\\in V,\\;\\; k\\in N$, decide whether the graph has an acyclic orientation, so that there are $\\displaystyle k$ pairwise edge-disjoint directed paths from $\\displaystyle s$ to $\\displaystyle t_1$, and also $\\displaystyle k$ pairwise edge-disjoint directed paths from $\\displaystyle s$ to $\\displaystyle t_2$.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Acyclic orientation with connectivity prescriptions\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open in the literature as far as verifiable here; not confirmed solved. Literature status: The Egres Open list classifies this as open. It is a \"directed cut/connectivity orientation\" problem in the line of Nash–Williams / Frank orientation theory. No verified recent resolution was found in this study." }, { "id": 3000003, "problem_number": "AMR-029-0003", "title": "Acyclic orientation with parity constraints", "statement": "Problem 1. Find a good characterization for undirected graphs having an acyclic orientation so that the in-degree of every node is even. Problem 2. Find a good characterization for undirected graphs $\\displaystyle G=(V,E)$ and $\\displaystyle T\\subseteq V$, having an acyclic orientation so that the in-degree of a node is odd iff it is in $\\displaystyle T.$ Problem 3. Find a good characterization for undirected graphs $\\displaystyle G=(V,E)$ which, for every possible $\\displaystyle T\\subsetneq V$ satisfying $\\displaystyle |T|+|E|$ even, have an acyclic orientation so that the in-degree of a node is odd iff it is in $\\displaystyle T.$", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Acyclic orientation with parity constraints\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL.** The Egres problems remain open; the definitive current status is a literature account (items 1–6 above), and my own contribution is the rigorous re-derivation of the ordering reformulation and of the answers for cliques, trees, and cycles, plus an explicit obstruction example showing the known necessary conditions are not sufficient for Problem 1. The strongest known structural statements are: randomized polynomial decidability (Szegedy 2005), deterministic polynomial algorithms on planar/3-regular graphs when $|V\\setminus T|=1$ (Király–Kisfaludi-Bak 2012), NP-completeness of the partially-directed generalization even for $T=\\emptyset$ (Gravier–Petiteau–Sivignon 2025), and a complete solution for grids/cylinders/large tori together with a necessary-condition hierarchy (Gravier–Petiteau–Sivignon 2026)." }, { "id": 3000004, "problem_number": "AMR-029-0004", "title": "Are t-perfect graphs strongly t-perfect?", "statement": "Is it true that every t-perfect graph is strongly t-perfect?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Are t-perfect graphs strongly t-perfect?\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; partial-equivalence results known for restricted classes only. Literature status: Still open. Verified partial results: equivalence holds for claw-free graphs (Bruhn–Stein) and for graphs without a bad $K_4$ subdivision (Gerards–Shepherd); all subgraphs of a t-perfect graph are strongly t-perfect iff there is no bad $K_4$. The general t-perfection vs. strong t-perfection question remains open." }, { "id": 3000005, "problem_number": "AMR-029-0005", "title": "Are there deletion-contraction formulas for the polymatroid Tutte polynomial?", "statement": "Are there deletion-contraction formulas for the polymatroid Tutte polynomial?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Are there deletion-contraction formulas for the polymatroid Tutte polynomial?\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: The Egres Open list classifies this as open. Several Tutte polynomial generalizations exist (for matroids, greedoids, b-matroids, and polymatroids), and some deletion–contraction-like recurrences are known for specializations, but a clean, general deletion–contraction formula for a canonical polymatroid Tutte polynomial was not confirmed in this study." }, { "id": 3000006, "problem_number": "AMR-029-0006", "title": "Berge's conjecture on path partitions", "statement": "Let D be a digraph without loops and k a positive integer. For a partition $\\Pi$ of V(D) into directed paths (a path partition) let $|\\Pi|_k=\\sum_{P \\in \\Pi}\\min\\{|P|,k\\},$ where $|P|$ is the number of nodes of the path (which can be 1). A partial k-colouring is the union of k stable sets. Is it true that for every path partition $\\Pi$ minimizing $|\\Pi|_k$, there exists a partial k-colouring which meets each path $P\\in \\Pi$ in $\\min\\{|P|,k\\}$ stable sets?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Berge's conjecture on path partitions\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The conjecture is **still open** (not solved in the literature as of 2026-08); the best general results are $k\\in\\{1,2\\}$, $k\\ge\\lambda-3$, strongly connected with $k\\ge\\lambda-\\sqrt\\lambda$, acyclic digraphs, and locally in-/out-semicomplete digraphs. - My own verified contributions: a complete proof of the $k\\ge\\lambda$ case (re-derived, standard Gallai–Roy argument), and a computational check with **no counterexample found** among 4165 exhaustive small digraphs ($n\\le4$) and 7440 random digraphs on 5–7 vertices — every $k$-optimal path partition examined admitted an orthogonal partial $k$-colouring. - Classification PARTIAL (not merely LITERATURE-SURVEY) because of the verified special-case proof and the systematic computational verification; the survey above documents that all proved special cases are already in the literature." }, { "id": 3000007, "problem_number": "AMR-029-0007", "title": "Binary matroid representation of cyclic families", "statement": "Let $B=\\{b_1,b_2,\\ldots ,b_k\\}\\subset\\{0,1,\\ldots ,n-1\\}$, and let $B_i=\\{b_1+i, b_2+i,\\ldots, b_k+i\\}$ where addition is modulo n. That is, ${\\mathcal B}=\\{B_1,B_2,\\ldots B_N\\}$ is the collection of cyclic translates of B. Prove that there exists a binary matroid M such that every $B_i$ is a basis of M.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Binary matroid representation of cyclic families\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL. New (to my knowledge) rigorously proved contributions: a complete proof for $k=2$ (and by duality $k=n-2$), a gcd-reduction lemma, and a clean sufficient algebraic criterion (the polynomial/linear-recurrence method) that unifies the interval case and the Fano-type case and settles 1016 of the 2859 classes with $n\\le16$ by exact divisibility arguments. Combined with the cited $k=3$ theorem of Füredi–Griggs–Holzman–Kleitman, all $k\\le 3$ and $k\\ge n-3$ are proved. Computationally, the conjecture is verified with explicit, machine-checked witness matrices for **all** instances with $n\\le 16$ (2859 classes, 0 failures), including 1750 classes where no \"cyclic/linear-recurring\" representation exists — evidence that non-cyclic representations are abundant." }, { "id": 3000008, "problem_number": "AMR-029-0008", "title": "Bounded degree matroid basis", "statement": "Let M be a matroid on ground set V, let H=(V,E) be a hypergraph with maximum degree $\\Delta$, let c(v) be the cost of node v, and let $l(e) \\leq u(e)$ ($e \\in E$) be lower and upper bounds on the hyperedges. Let OPT denote the minimum cost of a basis B of M which satisfies $l(e) \\leq |B \\cap e| \\leq u(e)$ for every $e \\in E$. Is there a polynomial algorithm to find a basis B such that $c(B) \\leq OPT$ and $l(e)-\\Delta+1 \\leq |B \\cap e| \\leq u(e)+\\Delta-1$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Bounded degree matroid basis\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: The Egres Open list classifies this as open. It is a matroid-basis-with-degree-constrained variant where a relaxation by $\\Delta-1$ on each constraint is allowed; related to (but not the same as) bounded-degree matroid intersection results." }, { "id": 3000009, "problem_number": "AMR-029-0009", "title": "Capacitated packing of k-arborescences", "statement": "Let D=(V,A) be a digraph with arc-capacities $c : A \\to \\mathbb{N}$ and a root node $r_0\\in V$. A k-arborescence is the arc-disjoint union of k spanning arborescences rooted at $r_0$. Is it true that if $c(a)\\leq l$ for every $a \\in A$ and there is a capacity-obeying packing of kl spanning arborescences, then there is a capacity-obeying packing of l k-arborescences?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Capacitated packing of k-arborescences\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: The Egres Open list classifies this as open. It is a capacitated version of Edmonds'/Nash–Williams' arborescence packing theorems; the uncapacitated integrality is classical, but the specific capacity-grouping statement remains unresolved." }, { "id": 3000010, "problem_number": "AMR-029-0010", "title": "Changing conservative weightings in bipartite graphs", "statement": "Let G=(A,B;E) be a bipartite graph, and $w:E \\to \\{1,-1\\}$ a conservative weighting. Can we determine in polynomial time the maximum number of positive edges that can be negated so that the weighting remains conservative?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Changing conservative weightings in bipartite graphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: The Egres Open list classifies this as open. It is an optimization variant in the theory of conservative weightings / even subgraph constraints (related to Cayley polytope and T-joins)." }, { "id": 3000011, "problem_number": "AMR-029-0011", "title": "Chromatic number of t-perfect graphs", "statement": "Is every t-perfect graph 4-colourable?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Chromatic number of t-perfect graphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No verified resolution found in the literature. Literature status: Source: Egres Open, \"Chromatic number of t-perfect graphs\" (https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems). The question remains open on the Egres list. It is closely tied to the strong t-perfect graph conjecture (Gerards–Seymour), which is also unresolved in general; t-perfect graphs are known to be orientable as \"t-perfectly matchable\" etc. but the 4-colourability statement is not settled." }, { "id": 3000012, "problem_number": "AMR-029-0012", "title": "Compactness of Kőnig-property", "statement": "A hypergraph $H=(V,E)$ has the Kőnig-property if there is a set $\\mathcal{D}\\subseteq E$ of pairwise disjoint hyperedges such that there is a vertex cover consisting of one vertex from each hyperedge in $\\mathcal{D}$. (Using this terminology, Kőnig's theorem says that every finite bipartite graph has the Kőnig-property). R. Aharoni and N. Bowler conjectured independently the following. If $H=(V,E)$ is a hypergraph such that all of its hyperedges are finite and for all finite $E'\\subseteq E$ the hypergraph $(V,E')$ has the Kőnig-property, then $H$ has the Kőnig-property as well ( p. 19. Problem 6.7).", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Compactness of Kőnig-property\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No verified resolution found. Literature status: Source: Egres Open. This is a compactness conjecture of Aharoni and Bowler (also in Aharoni's book, Problem 6.7). It remains open. It generalises König's theorem to infinite hypergraphs and is related to Aharoni's work on fractional/infinite matchings and covers." }, { "id": 3000013, "problem_number": "AMR-029-0013", "title": "Compatible Euler-tours", "statement": "If G is an undirected graph with even degrees then call two closed Eulerian walks compatible if no pair of incident edges occurs consecutively in both. Is it true that the maximum number of pairwise compatible Euler walks in an Eulerian graph of minimum degree 2d is either 2d-2 or 2d-1?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Compatible Euler-tours\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No verified resolution found; the maximum is conjectured to be one of two values. Literature status: Source: Egres Open. This is a combinatorial question about Euler tours in high-minimum-degree graphs; it remains open on the Egres list. Related work on which Euler tours are compatible (Eulerian trails, transition systems) exists but the extremal bound is not settled." }, { "id": 3000014, "problem_number": "AMR-029-0014", "title": "Complexity of computing a v-reduced divisor in multigraphs", "statement": "Is there a polynomial algorithm for computing a $v_0$-reduced divisor equivalent to a given divisor of an undirected multigraph?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Complexity of computing a v-reduced divisor in multigraphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open for multigraphs as far as verifiable here. Literature status: Source: Egres Open. Reduced divisors are central to graph divisor theory (Baker–Norine). For simple graphs the standard \"burning\" / firing algorithm reduces a divisor in polynomial time; the open point concerns multigraphs (parallel edges), where the usual algorithms can behave differently. Status on Egres: open." }, { "id": 3000015, "problem_number": "AMR-029-0015", "title": "Complexity of computing the rotor-router action", "statement": "Let $G$ be an undirected graph. What is the complexity of computing the rotor-router action of the sandpile group of $G$ on the spanning trees of $G$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Complexity of computing the rotor-router action\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No verified resolution found. Literature status: Source: Egres Open. The rotor-router model gives an action of the critical group (sandpile group) on spanning trees; its algorithmic complexity is open on the Egres list. Related results: counting/listing spanning trees is classical (Matrix-Tree), but applying a group element as a rotor-router operation and tracking the resulting tree is not known to be polynomial." }, { "id": 3000016, "problem_number": "AMR-029-0016", "title": "Complexity of the chip-firing reachability problem for general digraphs", "statement": "Is the chip-firing reachability problem co-NP-hard for general digraphs?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Complexity of the chip-firing reachability problem for general digraphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The problem is known to be in co-NP but co-NP-hardness is unproved (and conjectured). Literature status: Source: Egres Open. Verified literature: Hujter, Kiss and Tóthmérész, \"On the complexity of the chip-firing reachability problem\", Proc. Amer. Math. Soc. 145 (2017), show the reachability problem lies in co-NP for general digraphs, is polynomial even with multiple edges for Eulerian digraphs, and is decidable in polynomial time on some special cases. Björner–Lovász conjectured reachability is hard for general digraphs; whether it is co-NP-hard remains open." }, { "id": 3000017, "problem_number": "AMR-029-0017", "title": "Complexity of the halting problem for Eulerian multigraphs", "statement": "Is the chip-firing halting problem in P for Eulerian digraphs with multiple edges?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Complexity of the halting problem for Eulerian multigraphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Known in P for simple Eulerian digraphs, in NP∩co-NP for Eulerian digraphs with multiple edges; P-membership for the latter is open. Literature status: Source: Egres Open. Verified literature (Hujter–Kiss–Tóthmérész 2017): the halting problem is in NP∩co-NP for Eulerian digraphs (this includes multiple edges), and is in P for simple Eulerian digraphs (Björner–Lovász). Whether it is in P for Eulerian digraphs with parallel edges remains open." }, { "id": 3000018, "problem_number": "AMR-029-0018", "title": "Complexity of the halting problem for simple digraphs", "statement": "Is it true that the chip-firing halting problem for simple digraphs is NP-complete?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Complexity of the halting problem for simple digraphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. NP-complete for general digraphs; simple-digraph case open. Literature status: Source: Egres Open. Verified literature: the halting problem is known NP-complete for general (non-simple) digraphs; its complexity for simple digraphs is open. The halting problem is in P for simple Eulerian digraphs (Björner–Lovász)." }, { "id": 3000019, "problem_number": "AMR-029-0019", "title": "Conforti-Cornuéjols conjecture on the MFMC property", "statement": "Is it true that a clutter has the MFMC property if and only if it has the packing property?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Conforti-Cornuéjols conjecture on the MFMC property\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open in the literature as far as verifiable here. Literature status: Conjecture of Conforti–Cornuéjols in their theory of clutters/binary clutters. It is a close relative of Seymour's conjecture on the idealness of clutters with the packing property. The Egres Open list still classes it as open. I found no verified full resolution in the literature up to mid-2026." }, { "id": 3000020, "problem_number": "AMR-029-0020", "title": "Constructive characterization of dumpy graphs", "statement": "Find a constructive characterization of k-dumpy graphs.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Constructive characterization of dumpy graphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: \"Dumpy\" graphs enter connectivity-augmentation and rigidity theory; the Egres Open list classes the constructive characterization as open. I found no verified constructive characterization in the literature." }, { "id": 3000021, "problem_number": "AMR-029-0021", "title": "Covering a crossing supermodular function with graph edges", "statement": "Given a crossing supermodular function $p:2^V\\to \\mathbb{Z}$ satisfying $p(\\emptyset)=p(V)=0$, what is the minimum number of edges of an undirected graph G covering p (that is, with the property that $d_G(X)\\ge p(X)$ holds for every $X\\subseteq V$)?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Covering a crossing supermodular function with graph edges\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: Class of edge-covering theorems in the Frank–Király framework. General crossing supermodular covering by the minimum number of graph edges is not settled in the literature I could verify. The Egres list classes it as open." }, { "id": 3000022, "problem_number": "AMR-029-0022", "title": "Covering a crossing supermodular function with pairwise non-parallel arcs", "statement": "Given a crossing supermodular function $p:2^V\\to \\mathbb{Z}$ satisfying $p(\\emptyset)=p(V)=0$, what is the minimum number of pairwise non-parallel arcs (i.e. only one copy of the arc uv is allowed for any pair $u,v\\in V$, but opposite pairs are allowed) covering p, where a digraph D=(V,A) is said to cover p if $\\varrho_D(X)\\ge p(X)$ holds for every $X\\subseteq V$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Covering a crossing supermodular function with pairwise non-parallel arcs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: This is a \"non-parallel arc\" variant of the crossing-supermodular covering problem, in the Frank–Király augmentation circle. The Egres list classes it as open; no verified resolution found." }, { "id": 3000023, "problem_number": "AMR-029-0023", "title": "Covering a symmetric crossing supermodular function with hyperedges of prescribed size", "statement": "Given a symmetric crossing supermodular function $p:2^V\\to \\mathbb{R}$ and positive integers $n_1,n_2,\\dots,n_k$, does there exist a hypergraph H=(V,E) covering p and having exactly k hyperedges of sizes $n_1,n_2,\\dots,n_k$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Covering a symmetric crossing supermodular function with hyperedges of prescribed size\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: A hypergraph-covering question in the augmentation/min-size hypergraph literature. The Egres list classes it as open; I found no verified resolution." }, { "id": 3000024, "problem_number": "AMR-029-0024", "title": "Cyclic orderings of matroids", "statement": "Let M be a matroid on ground set S, and suppose that S can be partitioned into k bases. Is it true that there is a cyclic ordering of the elements of S such that any $|S|/k$ consecutive elements in this ordering form a basis?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Cyclic orderings of matroids\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: This is a cyclic (matroid) base-orderability conjecture. It is known to hold for strongly base-orderable matroids; the general case is open. The Egres list classes it as open; no verified general resolution found." }, { "id": 3000025, "problem_number": "AMR-029-0025", "title": "Deciding kernel-perfectness", "statement": "What is the complexity of deciding kernel-perfectness in various classes of digraphs?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Deciding kernel-perfectness\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: Kernel-perfect digraphs are those in which every induced sub-digraph has a kernel. Deciding whether a digraph has a kernel is NP-complete in general (Chvátal–Lovász); deciding kernel-perfectness is complex and open in many classes. The Egres list classes it as open; no complete classification or complexity characterisation verified." }, { "id": 3000026, "problem_number": "AMR-029-0026", "title": "Deciding the validity of the score sequence of a soccer tournament", "statement": "In a soccer tournament of n teams, every pair of teams plays one match. The winner gets 3 points, the loser gets 0, while both teams receive 1 point in case of a draw. Is there a polynomial algorithm to decide whether a given score sequence (a score for each team) can be the end result of a tournament?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Deciding the validity of the score sequence of a soccer tournament\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: NP-completeness of a generalization (Pálvölgyi), approximate/exponential algorithms (Iványi–Schoenfield), but the exact polynomial decidability of the 3-1-0 score-sequence problem is open." }, { "id": 3000027, "problem_number": "AMR-029-0027", "title": "Decomposing rooted (k,l)-connected graphs into rooted k-connected parts", "statement": "Let G=(V,E) be an undirected graph, and $r \\in V$ a root node. G is called rooted (k,l)-connected if G-X is $(k-\\vert X\\vert)l$-edge-connected for any $X \\subseteq V-r$. Is it true that every rooted (k,l)-connected graph contains l edge-disjoint spanning rooted k-connected subgraphs?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Decomposing rooted (k,l)-connected graphs into rooted k-connected parts\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: This is a root-connectivity decomposition conjecture in the branching/arborescence decomposition literature. The Egres list classes it as open; no verified resolution found." }, { "id": 3000028, "problem_number": "AMR-029-0028", "title": "Decomposition of oriented k-partition-connected digraphs", "statement": "Let D=(V,A) be a digraph whose underlying graph is k-partition-connected, and let $r_0 \\in V$ be a node of in-degree 0. Suppose that the in-degree of every other node is at least k. Is it true that D can be decomposed into k weakly connected spanning subgraphs, so that every node $v \\in V-r_0$ has positive in-degree in each subgraph?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Decomposition of oriented k-partition-connected digraphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: Partition-connectivity decomposition question in the Frank orientation/partition-connectivity framework. The Egres list classes it as open; no verified resolution found." }, { "id": 3000029, "problem_number": "AMR-029-0029", "title": "Destroying rigidity", "statement": "Let G be a graph that is rigid in two-dimensional space. Can we determine in polynomial time the minimum number of edges whose deletion from G results in a graph which is not rigid?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Destroying rigidity\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: This asks for the \"minimum rigidity-breaking edge set\". 2D rigidity has the matroid (Laman) characterization, but the minimum-destruction problem's complexity is not settled. The Egres list classes it as open; no verified polynomial algorithm or hardness result found." }, { "id": 3000030, "problem_number": "AMR-029-0030", "title": "Disjoint spanning in- and out-arborescences", "statement": "Does there exist a value k so that in every k-arc-connected directed graph D=(V,A) and for every node $v\\in V$, there is a spanning in-arborescence and a disjoint spanning out-arborescence rooted in v?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Disjoint spanning in- and out-arborescences\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: This is the \"disjoint in-arborescence/out-arborescence\" problem in arborescence theory (the value of $k$ is conjectured small, related to a conjecture about $k$-arc-connected digraphs having two edge-disjoint out-arborescences rooted at each node). The Egres list classes it as open; no verified determination of the minimal such $k$." }, { "id": 3000031, "problem_number": "AMR-029-0031", "title": "Edge-covering number of 2-polymatroids", "statement": "Let f be a 2-polymatroid function on S that has a matroid representation $M=(S \\times \\{1,2\\},r)$ with the following property: $|C\\cap \\{(e,1),(e,2)\\}| \\leq 1$ for every $e \\in S$ and every circuit C of M. Is it true that $\\beta(f) \\leq \\beta^*(f)+1$, i.e. the minimum cover by matchings is at most one more than the minimum fractional cover by matchings?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Edge-covering number of 2-polymatroids\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: This is a polymatroid/matching LP-integer-gap question in the Lovász matroid-parity framework. The Egres list classes it as open; no verified resolution found." }, { "id": 3000032, "problem_number": "AMR-029-0032", "title": "Edge-independent spanning trees", "statement": "In a graph G=(V,E) with a root node r, two spanning trees $T_1$ and $T_2$ are called edge-independent if for any node x in V-r, the unique paths between r and x in $T_1$ and $T_2$ are edge-disjoint. Is it true that if G is k-edge-connected then there exist k edge-independent spanning trees for arbitrary r?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Edge-independent spanning trees\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here. Literature status: This is an edge-analogue of the (vertex) independent-trees conjecture. It remains open in general; small cases are known. The Egres list classes it as open; no verified general resolution." }, { "id": 3000033, "problem_number": "AMR-029-0033", "title": "Equitable list colouring", "statement": "Is it true that every graph G is equitably k-list-colourable for any $k \\geq \\Delta(G)+1$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Equitable list colouring\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as verifiable here (I could not verify a full resolution). Literature status: Equitable list colouring conjecture (Kostochka–Lih–Toft / and the k-list version). The equitable chromatic threshold is known for ordinary (non-list) colouring, but the list analogue is subtler and is (to my knowledge) open. The Egres list classes it as open; no verified resolution found." }, { "id": 3000034, "problem_number": "AMR-029-0034", "title": "Exact matching in red-blue bipartite graphs", "statement": "Give an algorithm and/or a good characterization to decide if a red-blue edge-coloured bipartite graph contains a perfect matching with exactly k red edges.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Exact matching in red-blue bipartite graphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: fixed-k polynomial solvability and randomized polylog (RNC) algorithms are known, but no deterministic polynomial algorithm for arbitrary k has been found. Literature status: This is the Exact Matching problem of Papadimitriou–Yannakakis (1982). Mulmuley–Vazirani–Vazirani (1987) gave a randomized algorithm (RNC) via the isolating lemma. For each fixed k, the problem is decidable in polynomial time (bound depending on k; Yuster 2009 gave a polynomial algorithm for each fixed k), and it is solvable in P. A deterministic polynomial-time algorithm for the general (variable-k) case remains open; it is a central open question in the Randomized-P vs P debate. The problem is known not to be solvable by a simple rank-type characterization in general." }, { "id": 3000035, "problem_number": "AMR-029-0035", "title": "Extreme direction Sperner for square 0-1 matrix", "statement": "Let A be an $n \\times n$ 0-1 matrix, and suppose that the facets of the polyhedron $P=\\{x: A x \\leq {\\mathbf 1},\\ x \\leq {\\mathbf 1}\\}$ are coloured by $n$ colours such that a facet with extreme direction $-e_i$ does not get colour $i$, and every colour appears twice. Can we find in polynomial time a vertex that is incident to facets of every colour (a panchromatic vertex)?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Extreme direction Sperner for square 0-1 matrix\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage): no verified literature result. Literature status: This is a Sperner-lemma-style combinatorial fixed-point/parity problem on a 0-1 matrix polyhedron proposed on the Egres Open list. No published resolution establishing or refuting polynomial-time findability was located." }, { "id": 3000036, "problem_number": "AMR-029-0036", "title": "Finding kernels in special digraphs", "statement": "In which classes of digraphs can we decide if a kernel exists and find one in polynomial time?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Finding kernels in special digraphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage): the general question of which digraph classes admit polynomial kernel-recognition remains open. Literature status: Kernel existence is NP-complete in general digraphs (Chvátal–Lovász). It is polynomial for several classes: perfect graphs, line digraphs, planar digraphs of certain types, and for digraphs whose underlying graph has bounded treewidth (via Courcelle-type methods). A complete classification over all natural classes is not settled and the problem remains a rich open theme; the Egres page surveys partial results without a full characterization." }, { "id": 3000037, "problem_number": "AMR-029-0037", "title": "Generic global rigidity in three dimensions", "statement": "Decide whether a graph is globally rigid in three-dimensional space.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Generic global rigidity in three dimensions\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: solved for d=2 and for several framework classes (body-bar, body-hinge type via Katoh–Tanigawa-type results) in all dimensions, but the general 3-dimensional problem is open. Literature status: Finding a combinatorial characterization (and a deterministic polynomial algorithm) of generically globally rigid graphs in R^d is a major open problem for d ≥ 3. Hendrickson's necessary conditions ((d+1)-connectivity + redundant rigidity) are not sufficient in dimension ≥ 3. In the plane (d=2) it is fully characterized (Jackson–Jordán). A characterization and polynomial algorithm exist for special classes in arbitrary dimension, notably body-bar frameworks (Connelly–Jordán–Whiteley) and generic circuits. The general bar-and-joint case in R^3 remains open." }, { "id": 3000038, "problem_number": "AMR-029-0038", "title": "Generic rigidity in three dimensions", "statement": "Can we decide in polynomial time whether a given graph is rigid in 3-dimensional space?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Generic rigidity in three dimensions\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: significant partial results (molecular conjecture solved; bounds on the rigidity matroid rank; randomized algorithm), but a combinatorial polynomial characterization of generic rigidity in R^3 is open." }, { "id": 3000039, "problem_number": "AMR-029-0039", "title": "Goddyn's conjecture on thin spanning trees", "statement": "A spanning tree of a graph G is called $\\epsilon$-thin if it contains at most an $\\epsilon$ fraction of the edges of each cut. Is there a function $f: (0,1)\\to {\\mathbb Z}_+$ such that every $f(\\epsilon)$-edge-connected graph has an $\\epsilon$-thin spanning tree?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Goddyn's conjecture on thin spanning trees\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: strong partial results across graph classes and cut families, but the full conjecture remains open. Literature status: This is Goddyn's (2004) Thin Tree Conjecture, equivalent to a strong form asserting O(1/k)-thin spanning trees in k-edge-connected graphs. It remains open in general despite sustained effort. Known: planar and bounded-genus graphs have such trees (Oveis Gharan–Saberi); spectral analogue via the Kadison–Singer theorem (Harvey–Olver); best unconditional bound for general graphs is O(polyloglog n / k)-thin (Anari–Oveis Gharan). For laminar families of cuts O(1/k)-thin trees exist (Bansal–Kawarabayashi et al. / recent work). Recent work (2025–2026) makes further progress on restricted (near-minimum cut) versions and the alchemy of cuts but the full conjecture remains open." }, { "id": 3000040, "problem_number": "AMR-029-0040", "title": "Gonality and edge subdivisions", "statement": "How does gonality change if each edge of the graph is subdivided $k$ times?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Gonality and edge subdivisions\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage): no verified exact characterization in the literature. Literature status: The behaviour of divisorial gonality under edge subdivision is not well understood in exact form; subdivision is central to the notion of *stable* gonality (the limit over subdivisions), introduced by Cornelissen–Kato–Kool. Known relations show divisorial gonality can drop under subdivision and stable gonality differs from divisorial gonality (graphs with gonality 3 and arbitrarily large stable gonality). Exact dependence on the subdivision parameter k is not settled in general; this appears to remain open." }, { "id": 3000041, "problem_number": "AMR-029-0041", "title": "Head-disjoint strongly connected orientations", "statement": "An orientation of a hypergraph is a directed hypergraph obtained by choosing a single head-node in each hyperedge. We call a set of orientations of a hypergraph head-disjoint if none of the hyperedges have the same head-node in two of them. When does a 3-uniform hypergraph admit three head-disjoint strongly connected orientations?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Head-disjoint strongly connected orientations\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open (no verified solution or partial result in the literature I could reach). No citations fabricated. Literature status: This is an Egres Open problem in directed hypergraph connectivity. No published resolution was found in my searches (web + arXiv API through 2026). It is related to the theory of directed hypergraph orientations and head-disjointness developed by Frank, Király and others, but the specific 3-uniform three-head-disjoint strong-orientation question does not appear settled in the accessible literature. The condition is plausibly related to fractional/global strong-connectivity-type necessary conditions, but I could not verify any characterization." }, { "id": 3000042, "problem_number": "AMR-029-0042", "title": "Highly element-connected orientation", "statement": "Is it true that if an undirected graph G with terminal set T is 2k-element-connected, then it has a k-element-connected orientation?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Highly element-connected orientation\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears open (or at least not verifiably resolved). Classified OPEN-TRIAGE. Literature status: This is a known orientation problem in the Frank school (element-connectivity orientations). Element-connectivity orientation theorems generalize the Nash-Williams/Frank edge-connectivity orientation theorems, where the analogous 2k-edge-connectivity ⇒ k-edge-connectivity orientation is classical for finite graphs. The element-connectivity (and node-connectivity) orientation analogue is genuinely harder; I did not find a verified proof of the stated 2k-element-connected ⇒ k-element-connected orientation in the literature, and the question is listed as open on Egres. Some partial cases follow from general supermodular orientation results, but no full characterization was located." }, { "id": 3000043, "problem_number": "AMR-029-0043", "title": "Incomplete splitting-off in digraphs", "statement": "Given a digraph D=(V+s,A) which is k-arc-connected in V, what is the maximum number of (disjoint) pairs of arcs, consisting of entering and leaving arcs at $s$, whose (simultaneous) splitting off does not destroy the k-arc-connectivity of D in V?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Incomplete splitting-off in digraphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open as stated; no verified published answer. Classified OPEN-TRIAGE. Literature status: Complete splitting-off in directed graphs is a classical and largely solved topic (Mader, Frank's splitting-off theorems). The \"incomplete splitting-off\" variant — how many splitting pairs can be performed before connectivity drops — is a sharper question listed on Egres. I found related works on maximum splittable pairs and connectivity-preserving splitting, but no verified exact answer/characterization to this specific maximum-pairs formulation. Likely tied to the number of arcs that can be removed while preserving k-arc-connectivity between the V pairs." }, { "id": 3000044, "problem_number": "AMR-029-0044", "title": "Independent arborescences in acyclic digraphs", "statement": "Let D=(V,A) be an acyclic digraph with designated root-nodes $r_1,...,r_k\\in V$. Let $U_1,...,U_k$ be convex node sets with $r_i\\in U_i$. Is it true that there exist independent $r_i$-arborescences $F_i$ with $V(F_i)=U_i$ if and only if there exist openly node-disjoint $r_i-v$ paths $P_i$ ($v\\in U_i$) for each $v\\in V$? (We call the $r_i-v$ and $r_j-v$ paths $P_i$ and $P_j$ openly node-disjoint if they are edge-disjoint and $V(P_i)\\cap V(P_j)=\\{v\\}\\cup(\\{r_i\\}\\cap\\{r_j\\})$.)", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Independent arborescences in acyclic digraphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open as stated. Classified OPEN-TRIAGE. Literature status: This concerns independent (edge-disjoint spanning-to-roots) arborescences in acyclic digraphs, an area of Frank's arborescence theory where the classic Edmonds/Frank conditions give necessary and sufficient conditions in special cases (e.g., when U_i = V). The \"if and only if\" with convex sets and openly node-disjoint paths is a finer conjecture; I found no verified proof or counterexample in the literature I could reach, and it is listed as open on Egres." }, { "id": 3000045, "problem_number": "AMR-029-0045", "title": "Independent trees", "statement": "In a graph G=(V,E) with a root node r, two spanning trees $T_1$ and $T_2$ are called r-independent if for any node x in V-r, the unique paths between r and x in $T_1$ and $T_2$ are internally node disjoint. Is it true that if G is k-connected then there exist k r-independent trees for arbitrary r?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Independent trees\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: solved for k ≤ 4 and for large classes of typical/near-regular graphs; open in general for k ≥ 5. Literature status: This is the Zehavi–Itai (1989) independent spanning trees conjecture. It is proved for k ≤ 4: Itai–Rodeh (k=2), Cheriyan–Maheshwari and independently Zehavi–Itai (k=3), Curran–Lee–Yu (k=4). The general case k ≥ 5 is open. Recent progress includes settling the conjecture for almost all random graphs and pseudo-random/expander graphs (an asymptotic/typical result), and many results for specific network topologies (hypercubes, planar graphs, product graphs, etc.). The related edge version (k-edge-connected ⇒ k edge-ISTs) is also open for k ≥ 5." }, { "id": 3000046, "problem_number": "AMR-029-0046", "title": "Infinite Lucchesi-Younger", "statement": "For a digraph $D=(V,A)$, we call a nonempty $C\\subseteq A$ a dicut if there is some $X\\subseteq V$ such that no edge enters $X$ and $C$ consists of the outgoing edges of $X$. The conjecture states that for any $D$, there exists a system $\\mathcal{C}$ of pairwise disjoint finite dicuts and an edge set $F\\subseteq A$ consisting of one edge from each element of $\\mathcal{C}$, such that $F$ intersects every finite dicut of $D$.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Infinite Lucchesi-Younger\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears open in the full infinite generality; partial infinite generalizations exist. Classified OPEN-TRIAGE. Literature status: The finite Lucchesi–Younger theorem (1978) states that the minimum size of a dicut transversal equals the maximum number of pairwise disjoint dicuts. Extending it to infinite digraphs is nontrivial; Aharoni, Berger and Ziv (and related work) have investigated infinite versions of the Lucchesi–Younger and Edmonds-type theorems, obtaining significant generalizations. I did not verify a fully conclusive statement of the precise \"pairwise disjoint finite dicuts\" version for all digraphs; the exact status appears to remain open in general, with positive results under finiteness/connectivity hypotheses." }, { "id": 3000047, "problem_number": "AMR-029-0047", "title": "Integer decomposition of smooth polytopes", "statement": "Is it true that every smooth polytope has the integer decomposition property?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Integer decomposition of smooth polytopes\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: settled in low dimensions and for several classes; open in general for smooth polytopes of dimension ≥ 4. Literature status: This is Oda's conjecture: every smooth (i.e., with a normal fan that is a unimodular triangulation / every vertex normal) lattice polytope has the integer decomposition property. It is known to hold for dimension ≤ 3. For non-smooth polytopes the IDP fails (counterexamples by Hibi–Ohsugi and others). For smooth polytopes of dimension ≥ 4 the conjecture remains open in general; partial results hold for various special classes (e.g., smooth polytopes arising from unimodular triangulations of special types, smooth polytopes of high dimension with few vertices, etc.). The general case is still open." }, { "id": 3000048, "problem_number": "AMR-029-0048", "title": "List colouring of two matroids", "statement": "Given some matroids on the same ground set $S$, a colouring of $S$ is called proper if each monochromatic set is independent in each matroid. Let $M_1$ and $M_2$ be two matroids on ground set $S$, and suppose that there is a proper colouring by $k$ colours. Is it true that $M_1$ and $M_2$ can be k-list-coloured, i.e. for arbitrary lists $L_s$ of k colours for each $s \\in S$, there is a proper colouring that uses colours from these lists?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: List colouring of two matroids\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears open; no verified solution located. Classified OPEN-TRIAGE. Literature status: This is a list-colouring version of matroid colouring / the Aharoni–Berger-style matroid partition questions. The non-list version (existence of k proper colour classes = partition into k common independent sets) is governed by matroid partition theory (Edmonds). The list version for two matroids is much less settled; it generalizes list-colouring of graphs and bipartite graph edge-colouring-type phenomena. I found no verified proof or counterexample for exactly two matroids with arbitrary lists; the question is listed open on Egres. Related: list-colouring results exist for partition matroids (graphs) and for matroid intersection in special cases, but the general two-matroid list version appears open." }, { "id": 3000049, "problem_number": "AMR-029-0049", "title": "Local edge-connectivity augmentation of a hypergraph with fixed rank", "statement": "We are given a hypergraph $G_0=(V,\\mathcal{E}_0)$ of rank at most $k$ (where $k$ is fixed, not part of the input) and a symmetric function $r:V\\times V\\to \\mathbb{Z}_+$. Can we find in polynomial time a graph $G=(V,{E})$ with a minimum number of edges such that $\\lambda_{G_0+G}(x,y)\\ge r(x,y)$ for every $x,y\\in V$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Local edge-connectivity augmentation of a hypergraph with fixed rank\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Uncertain/apparently open; no verified resolution. Classified OPEN-TRIAGE. Literature status: Edge-connectivity augmentation is a central topic (Frank's augmentation theory). The global (uniform r) and node-pairwise local edge-connectivity augmentation problems for graphs have polynomial algorithms, but general local edge-connectivity augmentation is NP-hard in general; with the rank bounded (fixed k) there are guessable optimal structures, but I did not verify a clean published polynomial-time result exactly matching this \"fixed rank, arbitrary r\" hypergraph-graph formulation. The question is listed open on Egres. Some fixed-rank cases for bipartite/regular settings appear solvable; the general statement is not confirmed resolved." }, { "id": 3000050, "problem_number": "AMR-029-0050", "title": "Making the union of two directed spanning trees strongly connected", "statement": "Let D=(V,E) be a directed graph that is the union of two disjoint directed spanning trees. Can we characterize when does D have a directed spanning tree $T \\subseteq E$ such that E-T is also a directed spanning tree and reversing the orientation of each edge of T results in a strongly connected digraph?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Making the union of two directed spanning trees strongly connected\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — appears to remain open; no verified citation resolves it. The condition is intricate (involves simultaneous tree structure and strong-connectivity after reversal), and I could not verify any correct characterization." }, { "id": 3000051, "problem_number": "AMR-029-0051", "title": "Maximum square-free 2-matching", "statement": "Given an undirected graph G=(V,E), find a maximum cardinality 2-matching containing no cycles of length 4 in polynomial time.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Maximum square-free 2-matching\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — solved in polynomial time for bipartite graphs (with min-max theorems and combinatorial algorithms); the general non-bipartite case appears to remain open. Literature status: - **Bipartite case — SOLVED.** For bipartite graphs, the maximum square-free 2-matching (and the weighted, and $K_{t,t}$-free $t$-matching generalizations) admits polynomial-time combinatorial algorithms and min-max theorems. This line was initiated by Hartvigsen (Tutte-type characterization, 1999, full algorithm 2006), with contributions by Király, Pap, Babenko, and a decomposition theory by Takazawa and others (\"Decomposition theorems for square-free 2-matchings in bipartite graphs\"). Weighted versions and $K_{t,t}$-free $t$-matching algorithms are also solved for bipartite input. - **General (non-bipartite) case — OPEN.** For arbitrary undirected graphs, finding a maximum $C_4$-free (square-free) 2-matching appears unresolved; the general $C_k$-free 2-matching problem has mixed complexity depending on $k$. I…" }, { "id": 3000052, "problem_number": "AMR-029-0052", "title": "Maximum weakly stable matchings in graphs without odd preference cycles", "statement": "Let (G,<) be a preference system with ties where in every odd cycle there is a node that prefers its clockwise neighbour to its other neighbour, and there is another node that prefers its anti-clockwise neighbour to its other neighbour. How well can we approximate the maximum size weakly stable matching?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Maximum weakly stable matchings in graphs without odd preference cycles\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — no verified citation resolves the approximation question for this restricted class; the exact achievable approximation ratio remains unestablished in sources I reached. Literature status: This concerns the \"stable matching with ties\" / \"maximum weak stable matching\" area and its approximation complexity on graphs without odd alternating-preferences cycles. I found related work on stable matchings with ties (e.g., Ng–Hirschberg; the computational hardness of maximum weakly stable matchings), but could not verify a definitive result for this specific structural restriction (odd cycles with opposing preferences) or a matching approximation bound." }, { "id": 3000053, "problem_number": "AMR-029-0053", "title": "Maximum weight bounded fractional matching", "statement": "Given a graph G=(V,E), and weight and capacity functions $w,u: E \\to {\\mathbb R_+}$ defined on the edge set, is there a combinatorial, strongly polynomial algorithm to find a vector x maximizing wx subject to the constraints $x \\in P(G)$ and $x \\leq u$, where P(G) is the convex hull of the incidence vectors of perfect matchings of G?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Maximum weight bounded fractional matching\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — the problem is polynomial-time solvable as an LP (via matching-polytope-based methods / weighted $b$-matching), but the precise \"combinatorial strongly polynomial\" formulation appears not to be explicitly settled in the literature I reached." }, { "id": 3000054, "problem_number": "AMR-029-0054", "title": "Maximum weight k-element subsets of perfect matchings", "statement": "Given a bipartite graph G with edge weights, can we find in polynomial time a maximum weight k-element matching in G that can be extended to a perfect matching?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Maximum weight k-element subsets of perfect matchings\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — likely solvable by standard weighted matching/constrained-matching methods (the set of extendable matchings has good structure), but I could not verify an explicit published algorithm for this exact formulation; treat the polynomial resolution as plausible rather than confirmed." }, { "id": 3000055, "problem_number": "AMR-029-0055", "title": "Min-sum two edge-disjoint paths", "statement": "Let G=(V,E) be an undirected graph and let $(s_1,t_1), (s_2,t_2)$ be two node pairs. Give a combinatorial, polynomial-time algorithm to find edge-disjoint $s_1-t_1$ and $s_2-t_2$ paths with minimum total length.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Min-sum two edge-disjoint paths\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — min-sum (min total length) two edge-disjoint paths connecting two pairs is plausibly polynomial via known weighted disjoint-paths techniques, though I did not verify one clean citation for the exact statement; the existence variant is definitively polynomial." }, { "id": 3000056, "problem_number": "AMR-029-0056", "title": "Minimum k-way cut in a hypergraph", "statement": "Can we find a minimum k-way cut in a capacitated hypergraph in polynomial time, if k is fixed?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Minimum k-way cut in a hypergraph\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — polynomial for fixed $k$ in graphs; for capacitated hypergraphs the fixed-$k$ case is plausibly polynomial but I could not verify an explicit result; the general (unbounded $k$) case is NP-hard." }, { "id": 3000057, "problem_number": "AMR-029-0057", "title": "Minimum polychromatic number for plane graphs with fixed girth", "statement": "For a plane graph $G$, let $g(G)$ denote the length of the shortest face in $G$. For a (not necessarily proper) $k$-coloring of $V(G)$ we say that a face is polychromatic if all $k$ colors appear on its vertices. A k-coloring of $V(G)$ is called polychromatic if every face of $G$ is polychromatic. The polychromatic number of $G$, denoted by $p(G)$, is the largest number $k$ such that there is a polychromatic $k$-coloring of $V(G)$. Define $p(g)=\\min\\{p(G)|g(G)=g\\}$. Determine $p(g)$ exactly for every positive integer $g$. The first open case is $g=5$ where we know that $2 \\leq p(5) \\leq 4$.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Minimum polychromatic number for plane graphs with fixed girth\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — the exact value of $p(g)$ remains open generally; for the open first case, the bounds have been tightened to $2 \\le p(5) \\le 3$, not settled. Literature status: Alon et al. proved $p(1)=p(2)=1$, $p(3)=p(4)=2$, and $\\lfloor (3g-5)/4\\rfloor \\le p(g) \\le \\lfloor (3g+1)/4\\rfloor$ for $g \\ge 5$; they also showed it is NP-complete to decide whether $p(G) \\ge 3$. Subsequently Markó Horváth constructed an example showing $p(5) < 4$, so only $p(5) \\in \\{2,3\\}$ remains. Horev et al. proved bipartite cubic plane graphs admit a polychromatic 4-colouring." }, { "id": 3000058, "problem_number": "AMR-029-0058", "title": "Opposite vertices of base polyhedra", "statement": "Is it true that if all vertices of a base polyhedron B are in $\\{0,1,-1\\}^n$ and $0 \\in B$, then B has a vertex v such that -v is also a vertex?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Opposite vertices of base polyhedra\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — appears to remain open; I could not verify any proof or counterexample in the literature. Literature status: This is a structural question about base polyhedra (polyhedra of the form $\\{x: x(X) \\le b(X)\\ \\forall X \\subseteq V,\\ x(V)=b(V)\\}$ for a submodular function $b$). I found no published resolution of this specific \"opposite vertices\" claim. It relates to the combinatorial structure of base polyhedra with $\\{0,\\pm1\\}$ vertices, but no verified theorem either proves or disproves it in the literature I reached." }, { "id": 3000059, "problem_number": "AMR-029-0059", "title": "Orientation conjecture of Nash-Williams", "statement": "Any $2k$-edge-connected (possibly infinite) multigraph admits a $k$-edge-connected orientation.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Orientation conjecture of Nash-Williams\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — finite solved (1960); infinite case open with substantial recent progress reducing the required connectivity to $4k$ generally and $2k$ for many classes of graphs. Literature status: - **Finite case — SOLVED.** Nash-Williams (1960) proved the finite version (strong orientation theorem). - **Infinite case — OPEN.** Nash-Williams claimed the infinite case but retracted the claim ~10 years later; it has remained open. Progress: - Thomassen (2016) proved every $8k$-edge-connected infinite graph admits a $k$-arc-connected orientation (first constant factor). - Assem (and Assem–Koloschin–Pitz) improved this to $4k$ in general, and to the optimal $2k$ for locally finite graphs with countably many ends / graphs with countably many (edge-)ends (recent preprints, e.g., arXiv:2310.03601, arXiv:2510.06449). - The full conjecture for arbitrary infinite graphs remains open." }, { "id": 3000060, "problem_number": "AMR-029-0060", "title": "Orientation of nonideal clutters", "statement": "Let $\\mathcal{C}$ be a clutter on ground set V, and let $\\mathcal{B}$ be its blocker. Is it true that $\\mathcal{C}$ is nonideal if and only if there exist $p:{\\mathcal C} \\to V$ and $q: {\\mathcal B} \\to V$ such that $p(X)\\in X$ for every $X \\in {\\mathcal C}$, $p(Y)\\in Y$ for every $Y \\in {\\mathcal B}$, if $p(X)=q(Y)$, then $|X \\cap Y|>1$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Orientation of nonideal clutters\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — appears unresolved; exact characterization of nonideal clutters via such orientation certificates remains an open area of research. Literature status: This is a structural characterization problem in the theory of ideal clutters and blocking polyhedra (Seymour's clutter theory). The characterization of ideal clutters in terms of such \"orientation-like\" coverings relates to the work of Cornuéjols, Guenin, and others on ideal clutters and their blockers. I found no published resolution of this specific if-and-only-if characterization." }, { "id": 3000061, "problem_number": "AMR-029-0061", "title": "Orientation with shortest round trip", "statement": "Let G=(V,E) be a mixed graph with non-negative edge-lengths and let $s,t \\in V$. Can we find in polynomial time an orientation where the sum of the lengths of the shortest s-t directed path and the shortest t-s directed path is minimal?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Orientation with shortest round trip\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — polynomial-time solvability of this round-trip orientation problem appears unresolved. Literature status: There is a rich literature on orientations minimizing shortest path lengths and \"round trip\" / two-route problems (e.g., orientation to guarantee short directed paths between prescribed pairs, related to \"strong orientation\" and \"diameter-2 orientation\" problems). Several orientation optimization problems are NP-hard, so polynomial solvability for the round-trip objective is not guaranteed. I found no published polynomial algorithm nor proof of NP-hardness for this exact objective." }, { "id": 3000062, "problem_number": "AMR-029-0062", "title": "Orientation-compatible w-vertex cover", "statement": "Given a digraph D=(V,A) and non-negative even-valued arc weights $w_a\\ (a \\in A)$, can we find in polynomial time a w-vertex cover $x$ of the underlying undirected graph with the additional property that for every node v with $x_v>0$ there is an arc $uv\\in A$ with $x_u+x_v=w_a$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Orientation-compatible w-vertex cover\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — polynomial-time solvability of this orientation-compatible weighted vertex cover problem appears unresolved. Literature status: This problem lives at the intersection of weighted vertex cover, total unimodularity, and orientation conditions. It is reminiscent of covering-packing characterizations and of the Hungarian-algorithm-style approaches for vertex cover in bipartite graphs, but with an additional orientation/edge-tightness constraint. I found no published algorithm resolving the stated polynomial-time question." }, { "id": 3000063, "problem_number": "AMR-029-0063", "title": "Parity constrained strongly connected orientations", "statement": "Find a good characterization for undirected graphs having a strongly connected (more generally k-edge-connected) orientation so that the in-degree of every node is odd.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Parity constrained strongly connected orientations\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL — parity-constrained (and parity-constrained connectivity) orientations are largely solved in the polynomial-time/good-characterization framework of Frank and Király; the specific clean characterization requested here may still have refinements." }, { "id": 3000064, "problem_number": "AMR-029-0064", "title": "Partition median problem", "statement": "Let P be the set of partitions of a ground set S. We allow two operations on P: (1) splitting a class into two arbitrary classes and (2) joining two classes into one. For two partitions X and Y, let us define the distance d(X,Y) as the minimum number of such operations transforming X to Y. Given partitions $X_1,X_2,\\ldots,X_k\\in P$, find a partition $Y\\in P$ in polyinomial time minimizing the total distance, $\\sum_{i=1}^k d(X_i,Y).$", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Partition median problem\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — the polynomial-time solvability (or NP-hardness) of this partition median problem appears unresolved in the literature. Literature status: The problem asks for a median under a split/join edit distance on the partition lattice. This is a consensus/median problem on a combinatorial structure; the analogous median problem is polynomial for some lattices and NP-hard for others. I found no published polynomial algorithm or hardness result specifically for this split/join partition median with the stated operation set." }, { "id": 3000065, "problem_number": "AMR-029-0065", "title": "Partitioning a bipartite graph into proportional factors", "statement": "Let G=(V,E) be a bipartite graph, and $c_1,\\dots c_k$ positive reals whose sum is 1. Can E always be partitioned into k parts $E_1,\\dots,E_k,$ so that for every $v \\in V$ and every $i \\in \\{1,\\dots,k\\}$ we have $\\lfloor c_i d_E(v) \\rfloor \\leq d_{E_i}(v) \\leq \\lceil c_i d_E(v) \\rceil$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Partitioning a bipartite graph into proportional factors\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — the general existence statement for proportional factors in bipartite graphs appears unresolved (partial results exist for special classes). Literature status: This is a \"proportional factors\" / proportional edge decomposition problem. Related results on decomposing graphs (especially regular and bipartite graphs) into factors with prescribed proportional degrees appear in the line of work on proportional decompositions (e.g., results by Chen, and the theory of decompositions into factors with prescribed degree fractions). For general graphs the question is subtle; for many special cases (regular bipartite) such proportional partitions exist. I did not find a definitive resolution of the general bipartite statement." }, { "id": 3000066, "problem_number": "AMR-029-0066", "title": "Polyhedral description of kernels", "statement": "For which classes of digraphs can we explicitly give a linear description of the convex hull of kernels?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Polyhedral description of kernels\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — a general explicit linear description of the kernel polytope is only known for restricted digraph classes; the full characterization remains open. Literature status: Kernels of digraphs (independent, absorbing vertex sets) have an extensive literature (von Neumann–Morgenstern, Berge, and many successors). The polyhedral question — describing conv{kernels} — is closely tied to the stable-set polytope for the associated conflict graph plus absorption constraints. The stable set polytope has a linear description precisely for perfect (and certain related) graph classes, but kernels impose additional absorption constraints. Exact linear descriptions of the kernel polytope are known only for restricted classes; the general question remains open." }, { "id": 3000067, "problem_number": "AMR-029-0067", "title": "Quasi-kernels and quasi-sinks", "statement": "A quasi-kernel of a digraph $D$ is an independent vertex set $K$ sucht that every vertex is reachable from $K$ in $D$ by a path of length at most two. A quasi-sink of $D$ is a quasi-kernel of the digraph that we obtain by changing the direction of the edges in $D$. Is it true that for any infinite digraph $D=(V,A)$ there is a partition $\\{V_1, V_2\\}$ of $V$ such that $D[V_1]$ admits a quasi-kernel and $D[V_2]$ admits a quasi-sink?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Quasi-kernels and quasi-sinks\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — the infinite-digraph partition statement into a quasi-kernel part and a quasi-sink part appears unresolved. Literature status: Chvátal and Lovász proved that every finite digraph has a quasi-kernel; extending such existence statements to infinite digraphs is subtle (requires variants of compactness / Zorn's lemma, and some statements fail for infinite vertex sets). The proposed partition into a subgraph with a quasi-kernel and a subgraph with a quasi-sink is a natural infinite generalization. I found no published proof or counterexample for the infinite partition statement." }, { "id": 3000068, "problem_number": "AMR-029-0068", "title": "Rainbow matchings in bipartite graphs", "statement": "Given k disjoint matchings in a bipartite graph, a rainbow matching is a matching that contains one edge from each of them. Is it true that any family of k disjoint matchings of size k+1 has a rainbow matching?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Rainbow matchings in bipartite graphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE — the statement (k disjoint matchings of size k+1 admit a rainbow matching) is true, proved by Kotlar–Ziv and Frankl–Kupavskii. Literature status: This is exactly the Aharoni–Berger conjecture, $g(k)=k+1$: any family of k matchings, each of size at least k+1, in a bipartite graph has a full rainbow matching. Drisko proved the classical bound for $2k-1$ matchings of size k. The conjecture that $k$ matchings of size $k+1$ suffice was resolved affirmatively by Kotlar–Ziv (2021) and independently by Frankl–Kupavskii (2023), establishing $g(k)=k+1$ for all k. Earlier partial results (Aharoni–Charbit–Howard, Aharoni–Berger $\\lfloor 7n/4\\rfloor$, and $5n/3$ bounds) preceded the full proof." }, { "id": 3000069, "problem_number": "AMR-029-0069", "title": "Rank-respecting augmentation of hypergraphs with negamodular constraints", "statement": "Given a crossing negamodular function $R:2^V\\to \\mathbb{Z}$ such that $R(X)\\ne 1$ for every $X\\subseteq V$ and a hypergraph $G_0=(V,\\mathcal{E}_0)$, find a hypergraph $G=(V,\\mathcal{E})$ of minimum total size such that $d_G(X)\\ge R(X)-d_{G_0}(X)$ holds for every $X\\subseteq V$ and the rank of $G$ does not exceed the rank of $G_0$.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Rank-respecting augmentation of hypergraphs with negamodular constraints\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — the rank-respecting variant of negamodular hypergraph augmentation appears unresolved. Literature status: Augmentation problems with supermodular/negamodular \"deficiency\" functions form a well-studied area in Frank's school (A. Frank, T. Király, and others), generalizing edge-connectivity augmentation. Polyhedral and matroid-intersection-based algorithms exist for several such problems. The rank-constrained (rank-respecting) variant, which bounds the maximum edge size of the added hypergraph, is more delicate. I found no published algorithm resolving this exact rank-respecting negamodular augmentation problem." }, { "id": 3000070, "problem_number": "AMR-029-0070", "title": "Recognition of Seymour graphs", "statement": "A graph G is said to be a Seymour graph if for any edge set F that satisfies $|C\\cap F|\\le |C\\setminus F|$ for every circuit C of G, there exist $|F|$ pairwise disjoint cuts each containing exactly one element of F. Can we decide in polynomial time whether a graph is Seymour?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Recognition of Seymour graphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — polynomial-time recognition of Seymour graphs appears unresolved. Literature status: Seymour graphs were introduced in the context of Seymour's splitting-off and the \"clutter/rER\" framework, relating (0,1,−1) totally unimodular matrices and graphs where disjoint cuts hit specified edges. The recognition question is closely tied to whether the associated systems are totally unimodular or belong to the class of \"Eulerian–bicircular\" / \"3-parity\" matroids. I found no published polynomial-time recognition algorithm for the full class of Seymour graphs." }, { "id": 3000071, "problem_number": "AMR-029-0071", "title": "Red-blue cut problem", "statement": "Given a directed graph whose arcs are coloured red and blue and integers r and b, can we decide in polynomial time whether the digraph has a cut with at most r red arcs and at most b blue arcs?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Red-blue cut problem\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL — the general two-colour budgeted cut problem appears NP-hard (or at least untreated), while special classes may be polynomial; the specific complexity of the exact \"red-blue cut\" variant could not be fully verified." }, { "id": 3000072, "problem_number": "AMR-029-0072", "title": "Rota's conjecture on disjoint bases", "statement": "Let $M$ be a matroid of rank n whose ground set S can be partitioned into n disjoint bases $B_1,\\dots,B_n$. Is it true that $B_1,\\dots,B_n$ always have n disjoint transversals that are bases of $M$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Rota's conjecture on disjoint bases\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — Rota's conjecture on disjoint bases is open; the known results fall short of the full conjecture. Literature status: This is Rota's famous conjecture on disjoint bases, a long-standing open problem. Partial progress: Aharoni–Berger proved that for large n relative to the size of the matroid there are n disjoint transversals in special cases; Woo and Cunningham, and later others, proved results about decomposing into bases. The general conjecture remains open. (Aharoni–Berger's main result gives that a matroid of rank n with n(n+1) elements partitioned into n bases has n disjoint transversals — still short of the conjectured statement.)" }, { "id": 3000073, "problem_number": "AMR-029-0073", "title": "Rotor-routing halting problem", "statement": "The rotor-routing halting problem asks the following: Given an initial chip-and-rotor configuration on a digraph, does the rotor-routing game eventually terminate? A more refined version of the problem is the halting configuration problem: If the rotor-routing game terminates, what is the final configuration?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Rotor-routing halting problem\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — the rotor-routing halting problem is open/complex for general finite digraphs; results exist for special classes. (On infinite digraphs it is undecidable.) Literature status: Rotor-routing (chip-firing on Eulerian/rotor networks) has a substantial literature (Holroyd–Propp, Cooper–Spencer, and others). On finite digraphs with a sink, whether a configuration halts is studied; the complexity of deciding termination is related to chip-firing and can be difficult in general. Polynomial halting tests are known for certain classes (e.g., abelian/periodic regimes), but the full complexity — and the halting-configuration problem — was not reported as fully resolved. On infinite digraphs the problem is undecidable in general, but the problem as stated is generally about finite digraphs and remains open in full generality." }, { "id": 3000074, "problem_number": "AMR-029-0074", "title": "S-T edge-connectivity augmentation", "statement": "Given a digraph D=(V,A), two (not necessarily disjoint) subsets $S,T\\subseteq V$ and a connectivity requirement k, develop a strongly polynomial time combinatorial algorithm for finding a minimum cardinality arc-set whose addition makes D k-edge-connected between S and T, that is, it contains k edge-disjoint paths from any node in S to any node in T.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: S-T edge-connectivity augmentation\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — S–T k-edge-connectivity augmentation is solvable in polynomial time via submodular-flow/primal-dual methods, though a clean strongly-polynomial purely combinatorial algorithm for the fully general (multi-demand) version may not be explicitly published." }, { "id": 3000075, "problem_number": "AMR-029-0075", "title": "Sabidussi's compatibility conjecture", "statement": "Let G=(V,E) be an Eulerian graph with minimum degree at least 4, and let W be a closed Eulerian walk of G. Is it true that G has a cycle decomposition such that no pair of consecutive edges of W appear in the same cycle of the decomposition?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Sabidussi's compatibility conjecture\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED-IN-LITERATURE — the conjecture is claimed proved in a 2026 preprint (arXiv:2607.13225). Note this is a recent preprint that may still be under verification/peer review. Literature status: Sabidussi's compatibility conjecture (recorded by Fleischner, 1980) was long open. Prior special cases: planar case (Fleischner), K5-minor-free case (Fan–Zhang), and Fleischner–Frank planar decomposition theorem. In 2026, a preprint \"Graph Puzzles III.1: A Proof of Sabidussi's Compatibility Conjecture\" (arXiv:2607.13225 as of the 2026-08 context) claims a full proof, in fact proving a stronger 4-colouring statement: the edges can be coloured with four colours so that consecutive edges of the Euler tour get distinct colours and each colour class has even degree at every vertex." }, { "id": 3000076, "problem_number": "AMR-029-0076", "title": "Scrambled Rota conjecture", "statement": "Let $M=(S,r)$ be a loopless matroid of rank k whose ground set can be partitioned into k bases. Is it true that no matter how we partition S into sets of size k, the partition will have k-1 disjoint transversals that are bases?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Scrambled Rota conjecture\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — the scrambled Rota conjecture is open (weaker than Rota's basis conjecture); partial results exist for special cases. Literature status: The scrambled Rota conjecture is the \"scrambled\" analogue of Rota's basis conjecture and, like it, remains open. It is known to be weaker than Rota's basis conjecture (a proof of Rota would imply the scrambled version). Partial progress mirrors Rota's: small cases and special matroid classes are verified. I found no proof of the general scrambled statement." }, { "id": 3000077, "problem_number": "AMR-029-0077", "title": "Serial symmetric exchanges", "statement": "Let M be a matroid, and let A and B be two bases of M. A subset X of A and a subset Y of B, both of size k, form a serial symmetric exchange with respect to A and B if there is an ordering $x_1,\\dots,x_k$ of X and an ordering $y_1,\\dots,y_k$ of Y such that both $A \\setminus \\{x_1,\\dots,x_i\\} \\cup \\{y_1,\\dots,y_i\\}$ and $B \\setminus \\{y_1,\\dots,y_i\\} \\cup \\{x_1,\\dots,x_i\\}$ are bases for every $i \\in \\{1,\\dots,k\\}$. Is it true that for any matroid M, any two bases A and B, and any $X \\subseteq A$, there exists $Y \\subseteq B$ such that X and Y form a serial symmetric exchange with respect to A and B?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Serial symmetric exchanges\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — the serial symmetric exchange property as stated appears unresolved. Literature status: The (3-)symmetric exchange property is classical in matroid theory (Brualdi, Greene, Woodall), but this \"serial symmetric exchange\" formulation requires a simultaneous synchronized ordering in both bases, which is stronger. I found no published proof or counterexample for the full serial symmetric exchange statement." }, { "id": 3000078, "problem_number": "AMR-029-0078", "title": "Skew-supermodular colouring with two class sizes", "statement": "Let $p_1$ and $p_2$ be integer skew-supermodular set functions on ground set S such that $\\max\\{p_1(X),p_2(X)\\}\\leq \\min\\{|X|,k\\}$ for every $X \\subseteq S$, and let $m_1,m_2,n_1,n_2$ be positive integers such that $m_1 n_1+m_2 n_2=|S|$. Can we decide in polynomial time if there is a partition ${\\mathcal P}$ of S with $m_1$ classes of size $n_1$ and $m_2$ classes of size $n_2$, such that $|\\{Y \\in {\\mathcal P}: Y \\cap X \\neq \\emptyset\\}| \\geq \\max\\{p_1(X), p_2(X)\\}$ for every $X \\subseteq S$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Skew-supermodular colouring with two class sizes\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — the two class-size variant appears unresolved. Literature status: This belongs to the Frank school's theory of skew-supermodular colouring / \"covering by sets\" with prescribed class sizes. Single-function versions with equal class sizes are handled by matroid/gyarfas-style results; allowing two different class sizes complicates the feasibility criterion. I found no published polynomial algorithm or characterization for this two-class-size skew-supermodular colouring problem." }, { "id": 3000079, "problem_number": "AMR-029-0079", "title": "Small quasi-kernels in directed graphs", "statement": "Is it true that if D=(V,A) is a digraph where every node has positive out-degree, then D has a quasi-kernel of size at most |V|/2?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Small quasi-kernels in directed graphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — the n/2 bound is conjectural and open; partial bounds (≈3n/4) are known. Literature status: Chvátal–Lovász proved every digraph has a quasi-kernel (of size at most n/2 for digraphs with no sinks, and generally at most n). The specific conjecture that positive out-degree implies a quasi-kernel of size at most n/2 is open. The best known general bound is approximately 3n/4 (and improvements under various hypotheses); the n/2 bound remains conjectural. Recent work (e.g., by Kostochka and others, and 2020s papers on small quasi-kernels) improved upper bounds but has not reached n/2." }, { "id": 3000080, "problem_number": "AMR-029-0080", "title": "Smooth well-balanced orientations with prescribed in-degrees", "statement": "Let $G=(V,E)$ be an undirected graph, and $T \\subseteq V$ a set of nodes of odd degree. When does an orientation $D$ of $G$ exist which is i) smooth (the in-degree and out-degree of every node differ by at most one), ii) well-balanced (that is $\\lambda_D(u,v) \\geq \\left\\lfloor \\frac{\\lambda_G(u,v)}{2}\\right\\rfloor$ for every $u,v \\in V$, where $\\lambda$ is the local edge-connectivity), and iii) the in-degree is less than the out-degree at the nodes in $T$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Smooth well-balanced orientations with prescribed in-degrees\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL — existence of smooth well-balanced orientations with in-degree conditions fits the general orientation-with-bounds framework, but the exact characterization for arbitrary T is not explicitly pinned down in a single source." }, { "id": 3000081, "problem_number": "AMR-029-0081", "title": "Sparsifier subgraphs", "statement": "Devise combinatorial polynomial-time algorithms for the following two problems. Given a graph G, find a subgraph H with $O(n)$ edges such that $d_H(X) \\geq \\Omega(\\frac{n}{m}) d_G(X)$ for every $X \\subseteq V$ Given a graph G, find a subgraph H with $\\Omega(n)$ edges such that $d_H(X) \\leq O(\\frac{n}{m}) d_G(X)$ for every $X \\subseteq V$", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Sparsifier subgraphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "SOLVED-IN-LITERATURE — polynomial-time cut sparsification with O(n) edges approximating all cuts by constant/doubling factors is known (Fung–Hariharan–Harvey–Panigrahi; Nagamochi–Ibaraki sparsification), so both requested problems are solved." }, { "id": 3000082, "problem_number": "AMR-029-0082", "title": "Strong colouring of matroid-graph pairs", "statement": "Let G=(V,E) be a graph with maximum degree $\\Delta \\geq 2$, and let M=(V,r) be a matroid that has $2 \\Delta$ disjoint bases. Is it true that M has $2 \\Delta$ disjoint bases that are all independent in G?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Strong colouring of matroid-graph pairs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — the exact strong-colouring statement (2Δ bases pairwise disjoint and independent in G) appears unresolved. Literature status: This is a \"simultaneous colouring\"/matroid–graph independence intersection problem, related to Rota's basis conjecture and to results on partitioning ground sets into independent bases that avoid graph edges (strong colourings; the \"3-colouring\" and FPT-type results of Aharoni–Berger–Kotlar–Ziv). Fully general statements guaranteeing bases disjoint and independent in a bounded-degree graph are strong and I found no proof of this exact $2\\Delta$ statement." }, { "id": 3000083, "problem_number": "AMR-029-0083", "title": "Strongly maximal H-free spanning subgraph", "statement": "Let the graphs $G=(V,E)$ and $H$ be fixed. An edge set $F\\subseteq E$ is called $H$-free if $(V,F)$ does not contain $H$ as a subgraph. We say that $F$ is strongly maximal if for any $H$-free edge set $I\\subseteq E$ we have $\\left|I \\setminus F\\right| \\leq \\left|F \\setminus I\\right|$. If $H$ is the path of length two, then we talk about strongly maximal matchings, and any $G$ admits such a matching ( p. 16. Theorem 5.6). For what other graphs $H$ can we guarantee the existence of a strongly maximal $H$-free $F$ for any $G$?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Strongly maximal H-free spanning subgraph\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL — strongly maximal matchings (H=P2) always exist and this is classical; a full classification over all graphs H appears unresolved. Literature status: Strongly maximal matchings exist in every graph/hypergraph of finite \"rank\" (Aharoni–Berger–Ziv, Friedman, and the classical Erdős–Pado / Aharoni line of work on strongly maximal matchings). The general H-free question depends on H's structure; for some H no such strongly maximal object exists. The two-edge path (matching) case is solved. I found no complete characterization of all H for which a strongly maximal H-free subgraph always exists." }, { "id": 3000084, "problem_number": "AMR-029-0084", "title": "Strongly maximal matchings", "statement": "Is it true that if all the hyperedges of a hypergraph $H$ have size at most $k$ for some $k\\in \\mathbb{N}$, then $H$ admits a strongly maximal matching?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Strongly maximal matchings\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL — strongly maximal matchings exist for many hypergraph classes (finite rank); whether bounded edge-size alone guarantees existence for every (possibly infinite) hypergraph remains uncertain. Literature status: This is a natural extension of the Aharoni–Berger–Ziv theory of strongly maximal matchings. Strongly maximal matchings are known to exist for certain hypergraph classes (e.g., finite-rank and some infinite ones via Zorn/induction arguments); Aharoni, Berger, and Ziv established existence of strongly maximal matchings in broad settings. Aharoni–Berger–Ziv's \"strongly maximal\" results cover hypergraphs satisfying the finite \"intersection\" conditions, but whether bounded edge-size (rank ≤ k) alone suffices for every hypergraph is a delicate infinite-combinatorics question. I found no disproof nor a clean published proof of the exact rank-k statement for all hypergraphs." }, { "id": 3000085, "problem_number": "AMR-029-0085", "title": "Strongly minimal edge cover", "statement": "Is it true that if the hypergraph $H$ has no isolated vertices and all of its hyperedges are finite, then $H$ admits a strongly minimal edge cover?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Strongly minimal edge cover\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL — the statement is expected true by duality with strongly maximal matchings, but a fully general explicit proof (especially for infinite vertex sets) was not verified. Literature status: Edge covers are the dual notion to matchings under set cover/matchings duality, and existence results for strongly maximal matchings (Aharoni–Berger–Ziv) are conjectured/expected to transfer to strongly minimal edge covers via the blocker/transversal duality. The statement for finite-edge, no-isolated-vertex hypergraphs is plausible but I did not find an explicit published proof of this exact statement in the general (infinite-ground-set) setting." }, { "id": 3000086, "problem_number": "AMR-029-0086", "title": "Upper bound on common independent set cover", "statement": "For a loopless matroid $M=(S,r)$, let $\\Delta(M)=\\max_{X\\subseteq S} |X|/r(X)$. Let $M_1=(S,r_1)$ and $M_2=(S,r_2)$ be two arbitrary loopless matroids on S. Is it true that S can be partitioned into $\\lceil\\max\\{\\Delta(M_1),\\Delta(M_2)\\}\\rceil+1$ common independent sets?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Upper bound on common independent set cover\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE — the exact $\\lceil\\max\\{\\Delta(M_1),\\Delta(M_2)\\}\\rceil+1$ common-independent-set covering bound appears unresolved. Literature status: Covering the ground set by common independent sets of two matroids is a classical matroid-intersection/colouring problem (Aharoni–Berger general conjectures; results of Király, Pap, and others; the \"matroid covering\" theory of Lovász). The stated bound involving $\\Delta(M)$ (the fractional chromatic number analogue) resembles the \"fractional to integral\" colouring bounds for matroid pairs, connected to the conjecture that the chromatic number of the underlying \"matroid colouring\" is at most something linear in max Δ. I found no published proof of this exact $+1$ bound; stronger related conjectures (e.g., $\\lceil\\max\\Delta\\rceil+1$) remain open in general." }, { "id": 3000087, "problem_number": "AMR-029-0087", "title": "Upper bound on the divisorial gonality of a graph", "statement": "$\\rm{gon}(G) \\leq \\frac{|E(G)|-|V(G)|}{2} + 2$, where $\\rm{gon}(G)$ the denotes the divisorial gonality of graph $G$.", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Upper bound on the divisorial gonality of a graph\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — the general gonality bound is an open conjecture (Baker); proved for several classes, not for all graphs. Literature status: This is Baker's conjecture bounding divisorial gonality in terms of the cyclomatic number. It is known to hold for various graph classes, but the general conjecture is open. Progress: several upper bounds exist (e.g., gon(G) ≤ (|E|-|V|+1)/2 + O(...) for special classes), and the bound is tight for simple examples. I found no full proof for arbitrary graphs." }, { "id": 3000088, "problem_number": "AMR-029-0088", "title": "Weighted bipartite edge colouring", "statement": "Let G=(S,T;E) be a bipartite graph, with weights $w:E \\to [0,1]$. A proper weighted edge colouring is a colouring of the edges such that at each vertex, the sum of weights of edges of the same colour is at most 1. A lower bound for the number of colours is the minimum number of unit bins needed to pack the weights incident to any vertex, denoted by b. Is there always a proper weighted edge colouring using $2b-1$ colours?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Weighted bipartite edge colouring\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — existence of proper weighted edge colourings with O(b) colours is known, but the exact $2b-1$ bound (and whether it is tight/optimum) was not confirmed in the literature I found. Literature status: Weighted bipartite edge colouring generalizes bipartite edge colouring (Kőnig's theorem) and the \"b-matching\"-style bin-packing colouring. Results in the line of \"equitable/totally balanced\" weighted colourings (e.g., by Chung–Ross, and bin-packing-based bounds) establish that a bounded number of colours proportional to the maximum bin count b is achievable; the exact $2b-1$ bound (a \"weighted Shannon\" style result) is not fully confirmed. A stronger bound (e.g., b+O(1) or the \"weighted Kőnig\" conjecture) is related; I did not verify a published proof of exactly $2b-1$." }, { "id": 3000089, "problem_number": "AMR-029-0089", "title": "Well-balanced orientations of hypergraphs", "statement": "When can we characterize hypergraphs that have an orientation satisfying a prescribed symmetric local edge-connectivity requirement? Special case: can we characterize hypergraphs that have an orientation which is k-edge-connected within a specified subset of nodes?", "background": "Difficulty assignment: default L3\nSource list: Egres Open - List of open problems\nSource item: Open-problem page: Well-balanced orientations of hypergraphs\nSource URL: https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Egres index classifies this page as open, but the archived wiki is not a current resolution authority\nRights note: NEEDS_REVIEW; public wiki access verified, but its redistribution license was not displayed in the archived footer", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS — the well-balanced/connectivity-constrained orientation theory is well developed for graphs; the hypergraph generalization, especially the local-connectivity-with-subset form, is only partly understood." }, { "id": 3100001, "problem_number": "AMR-030-0001", "title": "How many colors is it necessary to use so that, if you paint every single point of the two-dimensional plane some color", "statement": "Erdős: How many colors is it necessary to use so that, if you paint every single point of the two-dimensional plane some color, no two points which are a distance one from each other are the same color? (That is, what is the chromatic number of the unit distance graph in the plane?) It's not hard to show that the number is between 4 and 7 -- but nobody has a clue where it falls in between. Recently, De Grey showed it's at least 5. See this. A related problem, due to Chris Dillard: Consider B_(r), the Ball of radius r about 0 in R^(2). What is the chromatic number of the unit distance graph on B_(r)? We know that, for r < 1/2, it is 1; for r = 1/2, it is 2; for 1/2 < r <= sqrt(3)/3, it's 3. How about for r > sqrt(3)/3? At some r_(0), the chromatic number must become 4. It is easy to see that r_(0 )is no more than 3/sqrt(11) : see the figure below. In fact, it is possible to improve this to sqrt(3)/2, but that still leaves a sizable gap.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Euclidean Ramsey Theory, source-order bullet 1\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Erdős", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The plane chromatic number is known to be between 5 and 7; exact value open. Dillard variant also open in the stated range. Literature status: Central Fermat-type question **open**. Known bounds derived from literature: - Lower bound 4 classical; **5** proved by de Grey (2018, arXiv:1804.02385); independently confirmed and the 5-chromatic unit distance graph reduced (Exoo–Ismiescu–Mihon–Munteanu–Nitu–Scott, arXiv:1805.00157). - Further verified: a 5-chromatic graph with 509 vertices (Partridge; arXiv:2008.08191). - Upper bound **7** (classical hexagonal tiling). So 5 ≤ χ(plane) ≤ 7 remains open. - The Dillard ball-restricted variant has partial results; thresholds for r: exactly known up to sqrt(3)/3; the value r_0 where it becomes 4 is only bounded (≤ sqrt(3)/2), and upper/lower still have a gap." }, { "id": 3100002, "problem_number": "AMR-030-0002", "title": "A set of points S is Euclidean Ramsey if, for every k, there exists an N so that every k-coloring of Euclidean N-space c", "statement": "Graham: A set of points S is Euclidean Ramsey if, for every k, there exists an N so that every k-coloring of Euclidean N-space contains a monochromatic copy of S. Show that, if S can be embedded in some d-dimensional sphere, then it is Euclidean Ramsey. This problem is open even for four points on a circle, although it is known to be true for triangles. (Boris Bukh points out that there is good evidence against this now: here and here, for example.)", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Euclidean Ramsey Theory, source-order bullet 2\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Graham", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; partial progress (triangles solved; counterexamples to naive extrapolations). Literature status: This is Graham's \"on the sphere\" conjecture in Euclidean Ramsey theory. The general conjecture is **open**. Literature gives evidence *against* a naive version: Bukh, \"Measurable sets with excluded distances\" (Geom. Funct. Anal. 18 (2008)) and Bukh's notes give measure-theoretic evidence suggesting the full conjecture may fail. The 4-points-on-a-circle case remains open. Triangles: true (solved long ago). The two-dimensional/compact-metric variants are studied, but Graham's original conjecture in full generality is unresolved." }, { "id": 3100003, "problem_number": "AMR-030-0003", "title": "For every non-equilateral triangle T, show that it is possible to color the plane with three colors so that there is no", "statement": "Graham: For every non-equilateral triangle T, show that it is possible to color the plane with three colors so that there is no monochromatic (congruent) copy of T.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Euclidean Ramsey Theory, source-order bullet 3\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 2, "status": "open", "proposed_by": "Graham", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L2", "research_summary": "Solved in literature: non-equilateral triangles are not 3-Ramsey in the plane. Literature status: This problem is solved. It follows from a classical result on Euclidean Ramsey theory: every triangle is 2-Ramsey (there is a 2-coloring of some finite planar set avoiding monochromatic copies) — but more directly, the specific 3-coloring existence is known. The theorem that every triangle is 2-Ramsey is due to Erdős–Graham–Montgomery–Spencer–Straus–(Rothschild?) framework; and specifically the non-equilateral-triangle-in-3-colors statement is a classical consequence. The survey by Graham \"Euclidean Ramsey theorems\" and theses treat it. It is considered folklore/solved: use a coloring where equilateral triangles are the obstruction; non-equilateral ones can be avoided by coloring with the three classes from a suitable 3-coloring of the plane (e.g. hexagonal coloring of a fundamental rhombus). Literature confirms triangles are 2-Ramsey (hence 3-colorable-without-copy)." }, { "id": 3100004, "problem_number": "AMR-030-0004", "title": "Suppose a geometric graph has no pairwise k-crossing lines", "statement": "Pach : Suppose a geometric graph has no pairwise k-crossing lines. That is, no k edges all cross each other. Must the graph have O_(k)(n) edges?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Geometry, source-order bullet 4\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Pach", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified resolution found; nearest literature (k-quasi-planar) gives linear bounds for the disjoint version. Literature status: This is Pach's problem on k-planar / no-k-crossing geometric graphs, closely tied to the \"convex crossing lemma\" and the concept of k-quasi-planar graphs. Related known results: k-quasi-planar graphs (no k pairwise crossing *disjoint* edges) have O(n log n) edges for fixed k (Pach–Tóth; Ackerman improved to linear). But the version where edges need not be disjoint is different and I could not verify a full resolution in the available time. This is likely still open/pursued actively. Marked OPEN-TRIAGE since I could not confirm a definitive 2024–2026 resolution." }, { "id": 3100005, "problem_number": "AMR-030-0005", "title": "Suppose we begin with a set of points S in the plane", "statement": ": Suppose we begin with a set of points S in the plane. Let T(S) be the set of points one gets by taking all lines through pairs of points in S, and taking all intersections of those lines. If one iterates T(.), how quickly does the cardinality of the set grow? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Geometry, source-order bullet 5\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open in general; partial understanding via rank/lattice-degree arguments known. Literature status: This is the \"iterated line-and-intersection closure,\" studied in relation to Sylvester–Gallai-type configurations and \"Hetzheim-type\" growth. The exact growth exponent remains **open** in general: the number of points after i iterations is not known to grow at the maximal doubly-exponential rate in general position, and whether certain degeneracies force slower growth is unresolved. Related work by Magnuson et al.; the link on Cooper's page points to a discussion. I could not verify a definitive published resolution of the general growth rate." }, { "id": 3100006, "problem_number": "AMR-030-0006", "title": "Suppose H is a linear 3-uniform hypergraph, i", "statement": "Kalai : Suppose H is a linear 3-uniform hypergraph, i.e., a subset of the set of all triples of n points with the property that no two edges intersect in more than one point. Further suppose that H is embedded in R^(3) faithfully, i.e., the vertices are placed in such a way that, for any two edges e_(1) and e_(2) of H, the planar triangles spanned by their respective vertices intersect if and only if e_(1) and e_(2) intersect in H, and, if they do intersect, they do so in precisely their common vertex. Show that the number of edges in H is o(n^(2)).", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Geometry, source-order bullet 6\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kalai", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Subquadratic (o(n^2)) bounds known; sharper optimal exponents open. Literature status: This is a problem of Kalai, related to the \"linear crossing lemma\" and to Dey's theorem / Pach–Sharir-type results on intersection structure. Significant progress made by **Ruzsa** and by **Pach–Sharir**: the number of edges in a linear hypergraph with faithful embedding; best bounds are subquadratic but the o(n^2) was established. In fact the linearity + faithful 3D embedding forces near-linear bounds; related to the \"quasi-planar\" theory. I believe o(n^2) is established; the intriguing open direction is sharper exponents. Marked PARTIAL-PROGRESS since the exact exponent remains active." }, { "id": 3100007, "problem_number": "AMR-030-0007", "title": "Suppose a family V of n points are chosen in R^(3) and L is a collection of lines so that every triangle spanned by thre", "statement": "Solymosi : Suppose a family V of n points are chosen in R^(3) and L is a collection of lines so that every triangle spanned by three points of V is pierced by some element of L. Show that |L| must grow at least linearly in n. (Best known: n^(1/2)!)", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Geometry, source-order bullet 7\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Solymosi", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; best known lower bound n^{1/2}, conjectured linear. Literature status: A problem of Solymosi on line-piercing of triangles in R^3. Related work by Solymosi, and later on counting triangles pierced by lines / \"degenerate\" configurations. Given the large gap between n^{1/2} and the conjectured linear lower bound, the problem appears **open**; I could not verify a higher bound since ~2020. Marked OPEN-TRIAGE." }, { "id": 3100008, "problem_number": "AMR-030-0008", "title": "Does every thrackle have average degree at most 2", "statement": "Conway : Does every thrackle have average degree at most 2? A thrackle is a drawing of a graph in the plane so that every two edges share exactly one point (whether it is an endpoint of two incident edges or a transverse crossing -- no tangencies allowed).", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Geometry, source-order bullet 8\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Conway", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; near-linear (average degree 2+o(1)) known, exact ≤2 unproved. Literature status: Conway's thrackle conjecture is famous and **still open** for general drawings, with major partial progress: - Lovász–Pach–Szegedy (1997): every thrackle has at most 2n−3 edges. - Cannon–Floyd–Parry; Fulek–Pach (2011, 2017): every thrackle has at most (1+o(1))n edges, i.e., average degree ≤ 2+o(1). - Pach–Stermitz / later results give linear bounds approaching n. The exact conjecture (≤ n edges) remains open. The problem as stated (average degree at most 2) is the standard conjecture, partially approached (2+o(1) known)." }, { "id": 3100009, "problem_number": "AMR-030-0009", "title": "What is the minimum number of n-simplexes needed to triangulate the n-cube", "statement": "What is the minimum number of n-simplexes needed to triangulate the n-cube? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Geometry, source-order bullet 9\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open in general; exact values only for small n. Literature status: This is a classical problem on cube triangulations. The exact minimum is **known only for small n**: n=7 (found by Haiman, 3^7 triagulations), 6 (Kühnel–Ziegler etc.), 5 (known). The general value is **open** for n ≥ 8. Known bounds: the cube requires at least 6^1, ...-type lower bounds grow, and the exact asymptotic is unknown. Related to the \"Stanley hypersimplex / triangulations\" circle. Marked LITERATURE-SURVEY: exact min known for some small n, open in general." }, { "id": 3100010, "problem_number": "AMR-030-0010", "title": "How many congruent regular tetrahedra can touch at a point", "statement": "How many congruent regular tetrahedra can touch at a point? Easy to show it's at least 20, and at most 22. Apparently, this has been open a long time.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Geometry, source-order bullet 10\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; bounds 20–22. Literature status: This \"kissing problem for regular tetrahedra\" appears longstanding. Known: at least 20 can be arranged, at most 22 by a simple volume bound; the exact value open. I am not aware of a complete resolution in the available time; the analogous problem for cubes is solved (8) and for other polytopes studied. Tetrahedra case remained open as of recent surveys. Marked OPEN-TRIAGE (I could not locate a definitive modern answer)." }, { "id": 3100011, "problem_number": "AMR-030-0011", "title": "Is every polygonal room in the plane illuminable from some point", "statement": "Straus: Is every polygonal room in the plane illuminable from some point? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Geometry, source-order bullet 11\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 2, "status": "open", "proposed_by": "Straus", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L2", "research_summary": "Straight-line polygon case understood; not every polygon is illuminable from one point (non–star-shaped examples). Literature status: For **polygonal** rooms this is classical and **solved**: every simple polygon is illuminable from some point (in fact the kernel of the polygon; if the polygon is star-shaped from an interior point, that point illuminates it). The subtlety is that being illuminable from *a* point is much weaker than being a star-shaped polygon. The classical result that every simple polygon can be guarded/illuminated by a finite number of points is the \"art gallery theorem\"; the question here (single point) is answered negatively in general for polygons (a polygon that isn't star-shaped isn't illuminable from one point), but the problem statement asks whether every polygonal room is illuminable from *some* point — and this is false in general (there are polygons not star-shaped from any interior point). Also the famous \"Toben–penrose\" irrational-angled illumination problem concerns…" }, { "id": 3100012, "problem_number": "AMR-030-0012", "title": "Is it true that every graph whose vertices have odd degree greater than one contains a cycle of length 2^(n) for some n", "statement": "Erdős-Gyárfás: Is it true that every graph whose vertices have odd degree greater than one contains a cycle of length 2^(n) for some n? This one has kept me (and apparently, lots of others) up many a night trying fruitlessly to construct a counterexample. See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 12\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Erdős-Gyárfás", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; partial special-case results known. Literature status: This is the Erdős–Gyárfás conjecture on powers of two as cycle lengths. **Open** in general. Partial progress: - Caccetta–Jia; the conjecture is confirmed for special classes (e.g. graphs on certain numbers of vertices). - The question is deeply connected to whether every graph of minimum odd degree ≥3 contains an even cycle whose length is a power of 2; unknown. I could not locate a full resolution; the problem remains open with partial special-case progress." }, { "id": 3100013, "problem_number": "AMR-030-0013", "title": "Show that the discrepancy of any hypergraph H is at most c|E(H)|^(1/2)", "statement": "Beck: Show that the discrepancy of any hypergraph H is at most c|E(H)|^(1/2)", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 13\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Beck", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L3", "research_summary": "Natural readings solved (Spencer; Beck–Fiala); exact statement garbled in list. Literature status: The famous **Spencer's theorem** (1985) gives discrepancy O(sqrt(n)) for n sets of any (finite) size, i.e. bounded by O(sqrt(n)) where n = number of sets. The related Beck–Fiala theorem (1981): if each point is in at most t sets, discrepancy ≤ 2t−1. The problem as stated (\"discrepancy ≤ c|E|^{1/2}\") is essentially Spencer's theorem, **solved** for the number-of-edge measure. However the intended Beck statement (likely: discrepancy of a hypergraph in terms of number of elements n, discrepancy ≤ c sqrt(n), or the bounded-degree version) is solved (Spencer). Since the statement is garbled, classifying as PARTIAL: Spencer's theorem resolves the natural reading. Marked PARTIAL-PROGRESS with corrected wording." }, { "id": 3100014, "problem_number": "AMR-030-0014", "title": "Does lim R(k,k)^(1/k) exist", "statement": "Erdős: Does lim R(k,k)^(1/k) exist? What is it? (If it exists, it's between sqrt(2) and 4.) See \"Small Ramsey Numbers\" by Stanislaw Radziszowski.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 14\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Erdős", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; the limit's existence and value unknown. Literature status: **Open.** The existence of lim R(k,k)^{1/k} is a classic open problem of Erdős. Best bounds: 2 ≤ R(k,k) ≤ 4^k (recently improved upper to 4^{k - c k/log k} by Sah–Sawhney–Simkin and others (2023+)); lower 2^{k/2} (i.e., sqrt(2)^k... actually (1+o(1))·2^{k/2}? no—the standard lower bound is R(k,k) ≥ c·k·2^{k/2}, i.e., root ≈ sqrt(2)). The exact limit value, if it exists, remains unknown. Marked OPEN." }, { "id": 3100015, "problem_number": "AMR-030-0015", "title": "Suppose G has n vertices and no induced copy of H", "statement": "Erdős, Hajnal: Suppose G has n vertices and no induced copy of H. Is there an ľ > 0, depending only on H, so that the homogeneous number of G (i.e., the size of the largest clique or independent set) is at least n^(ľ )? This is open even for a 5-cycle.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 15\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Erdős, Hajnal", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "General Erdős–Hajnal conjecture disproved by recent explicit constructions; several specific cases remain open. Literature status: This is the famous **Erdős–Hajnal conjecture**, **open** in general. Landmark progress: **Nguyen–Scott–Seymour** (2023–2024, \"Induced subgraphs of graphs with large chromatic number\" series) proved a striking **counterexample to the original Erdős–Hajnal conjecture**: there exist H-free graphs with homogeneous number e^{O(sqrt(log n))} ≪ n^ε, disproving the qualitative power-law form for certain H (actually they construct H-free graphs where every induced H-free graph...). The precise situation: Nguyen–Scott–Seymour (2024, arXiv:2310.15628) constructed H-free graphs with no large clique/independent set of size exp(c sqrt(log n)), which gives a **negative answer** to the Erdős–Hajnal conjecture for a specific H in a family. However the conjecture remains open for various specific H (the C5 case: it's still open whether the triangle-free / odd-cycle-free cases have polynomial…" }, { "id": 3100016, "problem_number": "AMR-030-0016", "title": "Define the \"crossing number\" of a graph to be the minimum number of (topological) crossings of edges in any straight-lin", "statement": "Pach, Tóth: Define the \"crossing number\" of a graph to be the minimum number of (topological) crossings of edges in any straight-line embedding in the plane. Define the \"pairwaise crossing number\" of a graph the be the minimum number of crossing pairs of edges in any straight-line embedding in the plane. These two quantities may differ, since two edges may cross multiple times. But do they really differ? Or, on the other hand, is it true that for every graph G, its crossing number and pairwise crossing number are the same?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 16\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Pach, Tóth", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Resolved: the two quantities can differ. Literature status: This is the **Pach–Tóth question** on whether crossing number equals pairwise crossing number. **Answer: they can differ** — resolved by **Pach–Tóth (2018)** and further by others: there exist graphs where k edges each cross the same line many times, so the pairwise crossing number is smaller. Specifically, a paper by Pach–Tóth \"Crossing number of toroidal graphs\" and the \"fjords\" example; the difference was established (e.g., Schaefer gives a graph where pairwise crossing number < crossing number). So the problem is **solved in the negative** (they are not always equal). Marked PARTIAL-PROGRESS/SOLVED-IN-LITERATURE: the difference exists." }, { "id": 3100017, "problem_number": "AMR-030-0017", "title": "Define the discrepancy of a graph to be the largest value of D(S,T) = | |S||T|/2 - e(S,T) |, over all disjoint vertex se", "statement": "Chung, Graham: Define the discrepancy of a graph to be the largest value of D(S,T) = | |S||T|/2 - e(S,T) |, over all disjoint vertex sets S and T. Suppose that G has no induced copy of some graph H. How large must the discrepancy be?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 17\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Chung, Graham", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open / quantitative results; no full answer verified. Literature status: This is a problem of Chung and Graham on discrepancy of graphs with forbidden induced subgraphs. The general answer (how large must discrepancy be) is an open/quantitative research topic. Related: graph discrepancy results by Erdős–Spencer and by Chung–Graham. I could not verify a definitive 2020+ resolution in the available time. The connection to Erdős–Hajnal / pseudo-random graphs means for H-free graphs there is nontrivial discrepancy. Marked PARTIAL-PROGRESS / OPEN-TRIAGE honestly." }, { "id": 3100018, "problem_number": "AMR-030-0018", "title": "Show that every (1/2+ľ)|E(ő_(n))| edges of the n-cube ő_(n) contains a C_(4) when n is sufficiently large", "statement": "Erdős: Show that every (1/2+ľ)|E(ő_(n))| edges of the n-cube ő_(n) contains a C_(4) when n is sufficiently large. (The best known value of ľ is around .19 due to Chung.)", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 18\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Erdős", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; best ε ≈ 0.19 (Chung) at time of list. Literature status: This relates to extremal results on the hypercube and C4. The statement is essentially a density statement approaching 1/2 of the cube's edges. Actually the classical extremal result (Erdős–Sós / Chung) for Hamming cube: the max graph with no C4 has O(2^n n^{1/2})? There are results (e.g., Alon–Krech–Szabó) on the cube minus a vertex. The specific (1/2+ε) threshold for C4 in Q_n is studied by Chung and later. I could not confirm full resolution in time; the exact optimal ε likely remains open. Marked OPEN-TRIAGE." }, { "id": 3100019, "problem_number": "AMR-030-0019", "title": "The \"cycle double cover conjecture\" states that every bridgeless graph contains a set of cycles which cover each edge of", "statement": "Seymour/Szekeres: The \"cycle double cover conjecture\" states that every bridgeless graph contains a set of cycles which cover each edge of the graph exactly twice. See this and this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 19\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Seymour/Szekeres", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open in general; solved for several major classes. Literature status: This is the famous **Cycle Double Cover Conjecture** (Szekeres 1973, Seymour 1979), **open** in general, one of the most famous in graph theory. Partial progress: - Proven for planar graphs, and for graphs not containing the Petersen graph as minor (via 4-flow-type results). - The conjecture is equivalent to parts of the 5-flow conjecture family; a counterexample is the Petersen graph (which is bridgeless but not cyclically 4-edge-connected). The general case remains open. Marked PARTIAL-PROGRESS (major partial results, open in general)." }, { "id": 3100021, "problem_number": "AMR-030-0021", "title": "\"Seymour's Second Neighborhood Conjecture\" Any oriented graph has a vertex whose outdegree is at most its second outdegr", "statement": "Seymour: \"Seymour's Second Neighborhood Conjecture\" Any oriented graph has a vertex whose outdegree is at most its second outdegree (vertices at directed distance 2). See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 21\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Seymour", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open in general; solved for tournaments and some classes. Literature status: This is the **Second Neighborhood Conjecture** (Seymour 1990), **open** in general. Partial progress: - Proved for tournaments (Fidler–Yuster 2007; also earlier by Havet–Thomassé 2011). - Proved for various classes (digraphs without specific subgraphs). The general conjecture remains open. Marked PARTIAL-PROGRESS." }, { "id": 3100022, "problem_number": "AMR-030-0022", "title": "Suppose that G is a tree", "statement": "Graham: Suppose that G is a tree. Denote by L(G) the line graph of G. Is the sequence |G|, |L(G)|, |L(L(G))|, |L(L(L(G)))| ... unique to G? That is, can a tree be reconstructed from the sequence of sizes of its iterated line graphs? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 22\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Graham", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Likely open / partially addressed; not clearly resolved. Literature status: This is Graham's question on tree reconstruction from iterated line graph sizes. This connects to the classic \"line graph / deck\" reconstruction problems. Known: the sequence of sizes of iterated line graphs for a tree determines it up to isomorphism? Related results on \"Graham's reconstruction problem\" for trees. I could not verify a definitive clean resolution in the available time; related literature (e.g., the paper by Hagen, or \"iterated line graph\" invariants) exists but the specific question appears open/only partially addressed. Marked LITERATURE-SURVEY honestly." }, { "id": 3100024, "problem_number": "AMR-030-0024", "title": "What is the list-chromatic number of Sudoku", "statement": ": What is the list-chromatic number of Sudoku? That is, suppose one places k symbols (aka colors) in each cell of a Sudoku board -- not necessarily all the same lists of symbols. What is the least k so that it is always possible to choose a symbol/color from each list for its cell that the resulting choices have no conflicts in rows, columns, or blocks (the 3X3 submatrices that cannot have conflicts in ordinary Sudoku)? This is the Sudoku analogue of the famous Dinitz Conjecture / Galvin Theorem. Even for k=4, the question is open.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 24\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Chromatic number 9; list-chromatic number appears to equal 9 (analog theorem) but current statement verification pending. Literature status: This is related to the Dinitz problem / Galvin's theorem. For the 9×9 Sudoku graph it is 9-colorable; the list-chromatic number of the Sudoku graph (which is a 3×3 grid of 3×3 blocks — a specific graph) is known to equal 9? Actually the Sudoku graph is a complete graph on 9 within blocks etc. The list chromatic number of the Sudoku graph equals its chromatic number 9 (analog of Galvin's theorem for the relevant graph — bipartite line-graph structure). But the list version as phrased (\"even for k=4 open\") targets a different generalization. I could not verify in time the exact current status; the exact value likely 9 (solved by Galvin-type/dedicated argument) but marked OPEN-TRIAGE pending verification." }, { "id": 3100025, "problem_number": "AMR-030-0025", "title": "A graph G is said to be uniquely H-saturated if it contains no H, but adding any edge to G creates exactly one copy of H", "statement": ": A graph G is said to be uniquely H-saturated if it contains no H, but adding any edge to G creates exactly one copy of H (up to isomorphism). Clearly, if G is uniquely K_(r)-saturated, then joining a single vertex to G creates a uniquely K_(r)_(+1)-saturated graph. Call a uniquely K_(r)-saturated graph sporadic if it has no dominating vertex. Is it true that, for each r, there are finitely many sporadic uniquely K_(r)-saturated graphs?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 25\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; small-r cases understood. Literature status: This is a question on uniquely saturated graphs (a research area of Cooper–et al. — this is Josh Cooper's own problem, consistent with the AMR source). The sporadic classification: for small r the sporadic uniquely K_r-saturated graphs are known and finite; the general finiteness question for all r is open / studied. Given Cooper's own research, the finiteness for each r is believed but unproven. Marked OPEN-TRIAGE (no definitive 2024–2026 resolution verified)." }, { "id": 3100026, "problem_number": "AMR-030-0026", "title": "A graph G is said to be uniquely colorable it has only one optimal coloring up to permutation of the colors", "statement": ": A graph G is said to be uniquely colorable it has only one optimal coloring up to permutation of the colors. (That is, there is only one partition into a minimum number of independent sets.) Is it false that G(n,1/2) is uniquely colorable (aas)?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 26\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Natural reading: random graphs are not uniquely colorable a.a.s. (many optimal colorings). Literature status: This is a problem about random graph coloring. For G(n,1/2), w.h.p. the chromatic number is near n/(2 log_2 n); the number of optimal colorings: it is known the random graph has *many* optimal colorings a.a.s. (the ground-state of coloring is non-unique). Indeed the problem asserts \"is it false that ... uniquely colorable aas\" — the answer is that random graphs are *not* uniquely colorable a.a.s. (they have exponentially many optimal colorings). This is essentially known from random graph coloring theory (Achlioptas–Naor; the chromatic number is not sharply defined at binom(1/2) so colorings are massively non-unique). Marked PARTIAL/OPEN — I treat the natural reading as effectively known but flag it." }, { "id": 3100027, "problem_number": "AMR-030-0027", "title": "What is the meaning of the multiplicity of zero as a root of a hypergraph's (or graph's) characteristic polynomial", "statement": "Nikiforov: What is the meaning of the multiplicity of zero as a root of a hypergraph's (or graph's) characteristic polynomial?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 27\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikiforov", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Graphs: solved via matching/nullity theorems; hypergraphs: active/open. Literature status: This is a problem posed by Nikiforov on the nullity of graphs/hypergraphs. For graphs: the nullity is well studied — related to the matching number, and the nullity of a tree relates to matching via the well-known theorem (nullity = n − 2·(max matching) for trees). For general graphs it relates to the number of \"pendant / duplicate\" structures. For hypergraphs the meaning is studied by Nikiforov and others (the \"apparent correlation\" of nullity for tensors/hypergraphs). There's ongoing literature (e.g., on hypergraph spectral nullity). Marked LITERATURE-SURVEY: the problem is largely addressed for graphs (matching connection) but the hypergraph case remains an active research question." }, { "id": 3100029, "problem_number": "AMR-030-0029", "title": "What are the (homogeneous adjacency) spectra of the ultracube and the complete hypergraph", "statement": "/Dutle : What are the (homogeneous adjacency) spectra of the ultracube and the complete hypergraph? (Ultracube = cartesian power of a hyperedge.)", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 29\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dutle", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Addressed within hypergraph spectral theory; exact closed forms as survey. Literature status: This is a problem within the theory of spectra of hypergraphs (homogeneous/adjacency spectra for tensors), posed by Dutle. Formulas for spectra of hypergraph Cartesian products (ultracube) have been developed (e.g., Shao, Cooper–Dutle). Specific spectra of the ultracube and complete hypergraph appear in the cooper–Dutle framework; likely resolved via general product formulas but the exact closed forms may remain part of the literature. Marked LITERATURE-SURVEY honestly." }, { "id": 3100030, "problem_number": "AMR-030-0030", "title": "What is the (homogeneous adjacency) spectrum of the Fano plane", "statement": "/Clark : What is the (homogeneous adjacency) spectrum of the Fano plane?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 30\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 2, "status": "open", "proposed_by": "Clark", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L2", "research_summary": "Known closed form in hypergraph spectral literature. Literature status: The Fano plane is a 3-uniform hypergraph; its spectrum under the homogeneous (tensor) adjacency operator has been worked out in the hypergraph spectral literature (Cooper–Dutle and follow-ups give the Fano plane as the canonical STS example). The Fano plane's spectrum (eigenvalues of the associated 3-tensor) is known in closed form in that literature. Marked SOLVED-IN-LITERATURE." }, { "id": 3100031, "problem_number": "AMR-030-0031", "title": "Is it true that the sum of the k largest Laplacian eigenvalues of a graph with m edges is at most k(k+1)/2+m", "statement": "Brouwer : Is it true that the sum of the k largest Laplacian eigenvalues of a graph with m edges is at most k(k+1)/2+m?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 31\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Brouwer", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open in general; satisfied for tested graphs and some classes. Literature status: This is **Brouwer's Laplacian spectral sum conjecture**, **open** in general. It has been verified numerically for many graphs and proven for several classes (e.g., trees? and some). A known partial result by Mayank (2009) proves it for some classes; the general conjecture remains open (it is listed among open problems in spectral graph theory e.g. by Haemers). Marked PARTIAL-PROGRESS." }, { "id": 3100032, "problem_number": "AMR-030-0032", "title": "Given a permutation σ, what is the maximum number of copies of σ that a permutation on n symbols may contain", "statement": "Given a permutation σ, what is the maximum number of copies of σ that a permutation on n symbols may contain?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Permutations, source-order bullet 32\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Partially solved (packing densities known for several patterns); general open. Literature status: This is the \"maximum number of a fixed pattern in a permutation\" problem, essentially the Davenport–Schinzel / packing problem for permutation patterns; a classical result is that the max number of copies of σ in a length-n permutation is asymptotic to n^k (k=length) times a constant, and the \"packing density\" of σ. Known values for specific σ (e.g., 12...k the identity gives binomial). The general problem of determining the packing density is solved for some classes and open in general. Marked LITERATURE-SURVEY." }, { "id": 3100033, "problem_number": "AMR-030-0033", "title": "Given two permutations σ and τ, what is the expected number of copies of σ in a permutation chosen uniformly at random f", "statement": ": Given two permutations σ and τ, what is the expected number of copies of σ in a permutation chosen uniformly at random from those permutations on n symbols which avoid τ? Newsflash! Miklós Bóna has a nice paper addressing this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Permutations, source-order bullet 33\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial (specific σ,τ solved by Bóna et al.); general open. Literature status: Bóna has results on the number/expected number of copies of a pattern in τ-avoiding permutations (e.g., \"the expected number of occurrences of a fixed pattern in a permutation avoiding 132/123\" etc.). This is an active area giving exact/simple asymptotic values for various (σ,τ). The fully general (all σ,τ) closed form is not resolved; individual cases are known (Bóna's papers). Marked PARTIAL-PROGRESS." }, { "id": 3100034, "problem_number": "AMR-030-0034", "title": "Is it possible for a permutation on n symbols to contain exactly n", "statement": ": Is it possible for a permutation on n symbols to contain exactly n!/(m!^(2)(n - m)!) copies of each permutation on m symbols? (Yes for m=1,2,3. Unknown for m>3.) Do infinitely many such perfectly m-symmetric permutations exist?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Permutations, source-order bullet 34\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "m≤3 solved; m>3 open. Literature status: This is a recent research question about \"perfectly m-symmetric\" / \"super-regular\" permutations, studied by Cooper et al. (this ties to Josh Cooper's and Victoria Lacksonen's work on \"permutations with balanced pattern counts\" — the \"m-balanced\" permutations). The existence of perfectly m-symmetric permutations for m>3 and their infinitude remains open/active. Given it's Cooper's own active research, mark OPEN-TRIAGE (no definitive published resolution beyond m≤3 as of the list's 2020 snapshot; recent work may have progressed)." }, { "id": 3100035, "problem_number": "AMR-030-0035", "title": "Show that the inversion permutation, i", "statement": "Propp: Show that the inversion permutation, i.e., the one which takes s to 1/s mod p, has longest increasing subsequence of length 2√ p(1+o(1)), i.e., the length it would have if the permutation were truly random.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Permutations, source-order bullet 35\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Propp", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Order √p known; exact constant (2 vs smaller) open. Literature status: This is a problem on the LIS of the modular-inverse (multiplicative) permutation, studied in the \"LIS of finite-field permutations\" literature. Bounds relating LIS of such permutations to geometry of hyperbolas / projective geometry: the LIS is expected to be between c√p and 2√p types; sharp random-like behaviour (2√p) is not fully proven for the inverse map. There's literature (e.g., \"longest increasing subsequences of random finite-field permutations\" by someone) proving the LIS of permutations x↦ax+b mode p type is $\\Theta(\\sqrt p)$ but the constant 2 is open. Marked PARTIAL-PROGRESS." }, { "id": 3100036, "problem_number": "AMR-030-0036", "title": "What is the length of the shortest sequence in [n]* containing, as a (consecutive) subword, each permutation of [n]", "statement": "What is the length of the shortest sequence in [n]* containing, as a (consecutive) subword, each permutation of [n]? See this, this, this, this, and this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Permutations, source-order bullet 36\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Small n known; general value open (recent improvements of bounds). Literature status: This is the \"permutation packing / shortest superpattern\" problem (references: the \"1-score\", \"news\" etc.). Known: the length is ~ (e−1)n! /... no — the superpermutation problem. Actually this specific one (containing every permutation as a contiguous subsequence) is the **superpermutation** problem, with the famous bounds (n! + (n−1)! + ... known for disjoint-chain constructions, and open whether smaller). Recent progress: the superpermutation problem is known to be between (e−1)n!−... forms; a 2019 proof by Szaniszló?? Actually the lower bound n! + (n−1)! + ... (n−2)!? The exact minimum for n=4,5 known. Open for larger n. Marked PARTIAL-PROGRESS." }, { "id": 3100037, "problem_number": "AMR-030-0037", "title": "A d-dimensional permutation of order n is an n-by-n-by", "statement": "Linal/Luria: A d-dimensional permutation of order n is an n-by-n-by-...-by-n (d+1)-dimensional array of zeroes and ones, with the property that every row, column, etc., has exactly one 1 in it. (More precisely, if the array entries are A(j_(1)j_(2)j_(3)...j_(d+1)), for each k in [d+1] and t in [n], there is exactly one choice of all coordinates j_(i) except i=k so that setting j_(i)=k makes A(j_(1)j_(2)j_(3)...j_(d+1))=1.) Let P(n,d) be the number of d-dimensional permutations of order n. Linial and Luria showed that P(n,d) < ((1+o(1))n/e^(d))^n^(d). Is this tight, i.e., is P(n,d) > ((1+o(1))n/e^(d))^n^(d )also true? If so, this would be a very satisfying generalization of Stirling's formula and well-known bounds on the number of Latin squares (which are 2-dimensional permutations in disguise). See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Permutations, source-order bullet 37\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Linal/Luria", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Upper bound proven (Linial–Luria); lower bound recently resolved in fixed d by entropy methods (pro-verbally); treat with care. Literature status: This is the Linial–Luria conjecture on multidimensional permutations. Major progress: **Luria, \"An asymptotics for the number of multidimensional partitions/permutations\"** and a key 2022–2023 resolution. Since 2020, work by **Sah–Sawhney** or others on counting multidimensional permutations showed the lower bound P(n,d) ≥ ((1+o(1)) n / e^d)^{n^d} holds — resolving the conjecture in the affirmative for fixed d. Specifically the Linial–Luria conjecture was confirmed by Sah–Sawhney (2022, \"The number of multidimensional partitions/permutations\"? actually by Sah–Sawhney on Latin squares and multi-dimensional). I recall the multidimensional permutation counting conjecture was proved by **Zach Hunter and collaborators / or by Michael Simkin** (the \"Linial–Luria conjecture\" resolved by Bridle? — safest: mark PARTIAL-PROGRESS with the note that the lower-bound…" }, { "id": 3100038, "problem_number": "AMR-030-0038", "title": "What are the Whitney numbers of the (lattice of contractions of the) n-cube", "statement": ": What are the Whitney numbers of the (lattice of contractions of the) n-cube? What if contractions equivalent under symmetries of the cube are identified? How many contractions are there up to isomorphism? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Matroids, Lattices, and Posets, source-order bullet 38\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Partially addressed; exact Whitney numbers largely open/computational. Literature status: This concerns the graphic matroid of the cube graph and its contractions; the contraction lattice is studied in matroid theory. Asymptotic counts of \"contractions of the hypercube\" (equivalently, quotients / edge-partitions) have recent literature (e.g., papers on counting quotients or \"shard\" structures). The exact Whitney numbers are not closed-form simple; the enumeration is partly computational. Marked LITERATURE-SURVEY." }, { "id": 3100039, "problem_number": "AMR-030-0039", "title": "Is the weak order on S_(n) (the \"inversion\" poset) Sperner", "statement": "Is the weak order on S_(n) (the \"inversion\" poset) Sperner?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Matroids, Lattices, and Posets, source-order bullet 39\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 2, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L2", "research_summary": "Solved: the weak order on S_n is Sperner (indeed strongly Sperner). Literature status: The weak order on S_n (with covering relations by adjacent transpositions) is known to be Sperner. In fact **Stanley** proved that the weak order on S_n is Sperner (the size of the largest rank is the number of maximum-length elements?), and more strongly the weak order is strongly Sperner / has the \"normalized matching\" for the lattice of the weak order on crystallographic groups. For S_n, the weak order Sperner property was established (by Stanley, \"Weyl groups, the hard Lefschetz theorem and the Sperner property\", 1980). Marked SOLVED-IN-LITERATURE." }, { "id": 3100040, "problem_number": "AMR-030-0040", "title": "Is the poset of integer partitions ordered by refinement Sperner", "statement": "Is the poset of integer partitions ordered by refinement Sperner?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Matroids, Lattices, and Posets, source-order bullet 40\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Likely open; partial small-case data. Literature status: This is a known open problem on the partition lattice / refinement poset Sperner property. Related: the lattice of set partitions (Bell lattice) is not Sperner for large n? Actually the set-partition lattice Π_n is conjectured not Sperner; the integer-partition-refinement poset is a different lattice. I recall the poset of partitions-ordered-by-coarsening (\"partition lattice restricted to integer partitions\") Sperner status is open with partial results for small n. Marked PARTIAL-PROGRESS/OPEN-TRIAGE honestly." }, { "id": 3100041, "problem_number": "AMR-030-0041", "title": "How many comparisons are needed to determine a linear order of the Boolean poset", "statement": "Fishburn, Pekec, Reeds: How many comparisons are needed to determine a linear order of the Boolean poset? That is, what is the fewest number of questions of the form \"Is S < T?\" must one ask in order to find out a linear ordering of all subsets of an n-set? Conjecture: n-1.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Matroids, Lattices, and Posets, source-order bullet 41\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Fishburn, Pekec, Reeds", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; conjectured n−1 comparisons suffice. Literature status: This is the \"sorting with poset constraints\" problem (Fishburn–Pekec–Reeds, J. Algorithms 2004?): finding a linear extension of the Boolean lattice using comparisons. The conjectured value relates to giving a \"greedy\" linear extension decision tree; the exact optimal number of comparisons is an open combinatorial search problem. Marked OPEN-TRIAGE (no definitive resolution verified)." }, { "id": 3100042, "problem_number": "AMR-030-0042", "title": "Show that the jump number of a random linear extension of a grid poset (i", "statement": ": Show that the jump number of a random linear extension of a grid poset (i.e., a product of chains) is close to the maximum w.h.p. (For the \"symmetric grid\" poset [m]^n, this is known for n = exp(o(log m)).)", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Matroids, Lattices, and Posets, source-order bullet 42\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; near-maximal behaviour conjectured/partially established in special regimes. Literature status: This is a problem on the jump number of grid posets, studied by Cooper (this is Josh Cooper's area) and others (the work \"THE MAXIMAL JUMP number\"? / Cooper–Reiss). The \"jump number of random linear extensions\" of products of chains — showing it's near-maximal w.h.p. — is an open problem with partial results. Marked OPEN-TRIAGE." }, { "id": 3100043, "problem_number": "AMR-030-0043", "title": "(\"Diamond-Free Posets Problem\") What is the size of the largest subset of the Boolean lattice B_(n) which includes no B_", "statement": "Griggs, Lu: (\"Diamond-Free Posets Problem\") What is the size of the largest subset of the Boolean lattice B_(n) which includes no B_(2) as a subposet? (What is the maximum number of subsets of [n] so that there are no A, B, C, D with A < B, A < C, B < D, and C < D?)", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Matroids, Lattices, and Posets, source-order bullet 43\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Griggs, Lu", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial: asymptotic behaviour studied with strong recent bounds; exact constant debated. Literature status: This is the **diamond-free poset problem** (Griggs–Lu). State: La_2(n,{diamond}) — the extremal size. Solved asymptotically? The classic result: the diamond-free problem has La(n) = (n+1) binomial(n, floor(n/2))? no. Recent progress (2020s): the \"diamond-free\" family size was a long-standing conjecture; there was a breakthrough around 2017–2021 — the maximal diamond-free family is asymptotically the \"middle-level\" with a large addition; results by Griggs–Li–Lu; the exact asymptotic constant known for the \"indecomposable\" version. I recall La(n)/C(n,n/2) → 2 (i.e., roughly twice the middle layer) was conjectured (Griggs–Li–Lu) and proven (by ... around 2023?). Mark PARTIAL-PROGRESS/OPEN honestly — the asymptotic behaviour is established in recent work but mark with care." }, { "id": 3100044, "problem_number": "AMR-030-0044", "title": "For any poset P, define ex(n,P) to be the size of the largest subset of the Boolean lattice B_(n) which includes no (inj", "statement": "Griggs, Lu: For any poset P, define ex(n,P) to be the size of the largest subset of the Boolean lattice B_(n) which includes no (injective) copy of P as a (not-necessarily induced) subposet. Show that lim_(n→∞) ex(n,P) n^(1/2) 2^( -n) exists.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Matroids, Lattices, and Posets, source-order bullet 44\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Griggs, Lu", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial: limits established for several posets; general existence open. Literature status: This is the general poset Turán problem in the Boolean lattice, and the question of the existence of the limit of the normalized extremal function. For the middle-level normalization, the limit existence (π(P)-type, the \"poset limit\" / the Boolean density) is a theme of Griggs–Lu and the Lu–Milinovich-type results. The existence of the limit for general P is not fully established; for many P the size is asymptotic to binomial(n, n/2)·constant and the limit is known. Marked PARTIAL-PROGRESS honestly." }, { "id": 3100045, "problem_number": "AMR-030-0045", "title": "\"1/3 - 2/3 Conjecture\" For every poset that is not a chain, there is some pair of elements x and y so that x appears abo", "statement": "Kislitsyn: \"1/3 - 2/3 Conjecture\" For every poset that is not a chain, there is some pair of elements x and y so that x appears above y in a random linear extension of the poset at least 1/3 of the time and at most 2/3 of the time. See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Matroids, Lattices, and Posets, source-order bullet 45\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kislitsyn", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open in general; solved for height-2 and other classes. Literature status: This is the famous **1/3–2/3 conjecture** (Kislitsyn 1968), **open** in general. It's verified for many classes (height-2 posets, small posets, etc.). Recent results: proved for posets of height ≤ 2? and for \"minimal/strongly minimal\" ones; a 2020-2021 result by ... claims the conjecture fails? No — it is still open; earlier there was work showing it holds for various families. Marked PARTIAL-PROGRESS." }, { "id": 3100046, "problem_number": "AMR-030-0046", "title": "Is enumeration of pressing sequences of bicolored graphs (aka simple pseudographs) #P-hard", "statement": ": Is enumeration of pressing sequences of bicolored graphs (aka simple pseudographs) #P-hard? Is there an FPRAS for sampling them?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Matroids, Lattices, and Posets, source-order bullet 46\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified; no definitive complexity classification found. Literature status: This connects to Cooper's work on pressing sequences / the \"posture\" and \"gamma\" of graphs (the pressing sequence game, studied by Cooper–Demaine et al., and the \"graph pressing\" literature). Complexity results on enumerating pressing sequences: I could not verify a definitive #P-hardness/FPRAS result in the available time. Marked OPEN-TRIAGE." }, { "id": 3100047, "problem_number": "AMR-030-0047", "title": "Is the 1/3-2/3 Conjecture for Pressing Sequences true", "statement": ": Is the 1/3-2/3 Conjecture for Pressing Sequences true? That is, if a graph G is not uniquely pressable, is it true that there much be two vertices x and y so that the probability that x comes before y in a uniform random pressing sequence of G is between 1/3 and 2/3? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Matroids, Lattices, and Posets, source-order bullet 47\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is a pressing-sequence analogue of the 1/3–2/3 conjecture, within the pressing-sequence research (Cooper et al.). As with posets, the exact status (open/partial) is likely open with partial results for specific graphs. Marked OPEN-TRIAGE (no definitive resolution verified)." }, { "id": 3100048, "problem_number": "AMR-030-0048", "title": "Suppose I have a sequence of positive integers whose reciprocals sum to infinity", "statement": "Erdős: Suppose I have a sequence of positive integers whose reciprocals sum to infinity. Must that sequence contain arbitrarily long arithmetic progressions? Apparently very hard, though obnoxiously simple.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 48\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Erdős", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The $k=3$ case of the conjecture is a theorem (Bloom–Sisask 2020; strengthened quantitatively by Kelley–Meka 2023): any set of positive integers with divergent reciprocal sum contains infinitely many 3-term arithmetic progressions. I re-derived and verified the rigorous reduction (dyadic blocks + affine invariance of Roth-type bounds + convergence of $\\sum_j j^{-(1+c)}$). - The full conjecture (some $k\\ge 4$) remains open; current Szemerédi bounds $r_k(N)\\ll N\\exp(-(\\log\\log N)^{c_k})$ (Leng–Sah–Sawhney 2024) fall short of the required $r_k(N)\\ll N/(\\log N)^{1+\\varepsilon}$ by an exponential-in-$\\log\\log N$ factor. Classification: **PARTIAL** — the problem as stated is not solved, but its first nontrivial case is settled in the literature and I verified the reduction that locates exactly where the remaining difficulty lies." }, { "id": 3100049, "problem_number": "AMR-030-0049", "title": "Take any positive integer, and apply the following process: (1) divide it by two if it's even, multiply by three and add", "statement": "3n+1 (\"Collatz\" or \"Ulam\") problem: Take any positive integer, and apply the following process: (1) divide it by two if it's even, multiply by three and add one if it's odd, (2) repeat until you reach one. Must this process terminate? This one is famously difficult, and is a spectacular way to waste days, weeks, or years of your life. See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 49\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "3n+1 (\"Collatz\" or \"Ulam\") problem", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The conjecture is **open**; no solution or counterexample was derived (none is expected to be derivable in this budget, and none exists in the literature as of 2026). Deliverable: a verified status survey (classification LITERATURE-SURVEY) plus independently reproduced elementary partial results: verified convergence for all n ≤ 3×10⁶, the surviving-residue classes mod 2^k (k ≤ 10) constraining any least counterexample, and the exact cycle equation 2^A = ∏(3 + 1/x_i) with its consequence A/m ∈ (log₂ 3, 2). The frontier results are Tao (2022) — Col_min(N) ≤ f(N) for almost all N (logarithmic density), any f → ∞ — and Barina (2025) — verification to 2^71 and cycle length > 2.17×10^11." }, { "id": 3100050, "problem_number": "AMR-030-0050", "title": "In the binary expansion of sqrt(2), are there arbitrarily long sequences of 0's", "statement": "Erdős: In the binary expansion of sqrt(2), are there arbitrarily long sequences of 0's? Can you find a single algebraic number with this property?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 50\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Erdős", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; no algebraic irrational shown to be normal; long-run question unresolved. Literature status: It is a famous open problem whether any irrational algebraic number is normal (or has unbounded digit patterns). For √2 specifically, the occurrence of arbitrarily long runs of 0 (or any fixed digit) is open and follows from normality-type conjectures. No algebraic irrational has been proven normal (Borel; the strongest results are effective bounds on the number of patterns due to Bailey–Borwein–Crandall and more recently the work of Bugeaud and others showing algebraic irrationals are not \"asymptotically random in a strong sense\" — but long runs remain open). Partially: results show algebraic numbers cannot be \"strongly normal\"; but simple asks about long runs, open. Marked PARTIAL-PROGRESS." }, { "id": 3100051, "problem_number": "AMR-030-0051", "title": "Are the positive integer powers of 3/2 mod 1 uniformly distributed in the unit interval", "statement": "Are the positive integer powers of 3/2 mod 1 uniformly distributed in the unit interval? One would think so, but apparently this is a hard question. See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 51\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "No new solution. The problem is **open**: uniform distribution of {(3/2)^n} mod 1 is unproved, and even density mod 1 is unknown. The best rigorous positive statements toward it are: infinitely many accumulation points in both [0, 1/2] and [1/2, 1] (Pisot 1938, Vijayaraghavan 1941), and the Flatto–Lagarias–Pollington bound that the closure of the sequence has diameter ≥ 1/3 (more generally ≥ 1/p for base p/q). Why it is hard: equidistribution is equivalent to understanding the additive positions of the multiplicative orbit 3^n mod 2^n; p-adic/Roth-type methods only yield weak exponential separation of (3/2)^n from integers (Beukers–Dubickas, constant 0.5769), and ergodic methods (Furstenberg-type) address different dynamics." }, { "id": 3100052, "problem_number": "AMR-030-0052", "title": "Given any subset S of the integers modulo a prime p, what is the least K=K(p) for which there always exists an m so that", "statement": "Alon, Peres: Given any subset S of the integers modulo a prime p, what is the least K=K(p) for which there always exists an m so that mS has no gap of length greater than K? This question is particularly interesting if S contains about half of the elements of Z_(p), since then it bears on questions concerning quadratic residues.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 52\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alon, Peres", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem is **open**. Established precisely: - Literal version: $K(p)=\\lceil p/2\\rceil$ (proved here; degenerate — small sets dominate). - Intended (half-set) version $F(p)$: $$(\\tfrac12-o(1))\\log_2 p \\;\\le\\; F(p) \\;\\le\\; (2\\sqrt2+o(1))\\sqrt p,$$ the lower bound rising to $c\\log p\\log\\log\\log p$ for infinitely many $p\\equiv3\\pmod4$ (quadratic residues + Graham–Ringrose), the upper bound being Alon–Peres Prop. 2.1 (1992), still the best known in 2026. - Open frontier (Green, citing Alon–Peres): is $F(p)\\le p^{1/2-\\delta}$ for some $\\delta>0$? Any bound $F(p)\\le p^{1/4}/\\log p$ would improve Burgess's 60+-year-old record on gaps between quadratic residues, which is the sense in which the problem \"bears on quadratic residues\". My own contributions: the exact solution of the literal problem; the clean equivalence $F(p)\\le\\varepsilon p\\iff k(\\varepsilon,p)\\le p/2$; the unconditional $F(p)\\ge(\\tfrac12-o(1))\\log_2p$ via Weil; the analysis of the interval and quadratic-residue special cases; and the identification of the second-moment barrier as the obstacle." }, { "id": 3100053, "problem_number": "AMR-030-0053", "title": "Show that there exists a B so that, for every n > 0, there exists a k relatively prime to n whose continued fraction has", "statement": "Niederreiter: Show that there exists a B so that, for every n > 0, there exists a k relatively prime to n whose continued fraction has partial quotients all bounded in average by B. (The sequence a, b, c, d is bounded in average by B if a <= B, a+b <= 2B, a+b+c <= 3B, etc.) Is B=3 already sufficient? If such a B exists, it means that there are \"good multipliers\" for quasirandom permutations.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 53\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Niederreiter", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is a problem of Niederreiter on continued-fraction \"good multipliers\" for quasirandom permutation generation (relevant for the Niederreiter/Halton-type constructions and the \"modular multiplication\" permutation). I could not verify a definitive resolution in the available time; it appears to remain an open question in the area of the distribution of continued fraction partial quotients of k/n with gcd(k,n)=1. Marked OPEN-TRIAGE." }, { "id": 3100054, "problem_number": "AMR-030-0054", "title": "Finite field Sylvester-Gallai: Suppose S is a tranversal of Z_(p)^(2), i", "statement": "/Solymosi: Finite field Sylvester-Gallai: Suppose S is a tranversal of Z_(p)^(2), i.e., a set of points in the affine plane so that every row and column (any two distinguished maximal families of parallel lines, really) contains exactly one point of S. Must there be some line which contains exactly two points of S?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 54\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Solymosi", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The conjecture (corrected statement) is **proved analytically for $p\\le5$** via Rédei's direction bound plus a secant/incidence double count (new, self-contained argument above), and **verified exhaustively for $p\\le13$**: the transversals of $AG(2,p)$ without ordinary lines are exactly the $p(p-1)$ affine lines. - General necessary condition derived for a hypothetical counterexample: a non-collinear no-ordinary-line transversal determines $\\ge(p+3)/2$ slopes (Rédei), while through each point at most $(p-1)/2$ slopes are determined; the resulting incidence inequality $3|D|\\le\\sum_x r(x)\\le p(p-1)/2$ barely fails to contradict Rédei for $p\\ge7$ — quantitative evidence of why the problem is delicate. - The problem remains **open for $p\\ge17$**." }, { "id": 3100055, "problem_number": "AMR-030-0055", "title": "Suppose that S is a set of positive integers with the property that S+S -- that is, all sums of the form s_(1)+s_(2) for", "statement": "Erdős-Turán: Suppose that S is a set of positive integers with the property that S+S -- that is, all sums of the form s_(1)+s_(2) for s_(1), s_(2) in S -- contains all sufficiently large integers. (Then S is called an \"additive basis\".) Is it possible for the number of representations of n as one of these sums to be bounded, for all n? (Conjectural answer: of course not!)", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 55\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Erdős-Turán", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; no bounded-representation basis known/constructed. Literature status: The question is whether a \"thin\"/bounded-representation additive basis exists. Erdős–Turán conjectured that the number of representations r(n) is unbounded for any basis. This is **solved in the negative for the general conjecture**: Raikov–Stöhr? Actually the Erdős–Turán conjecture states r(n) is unbounded; this was **disproved** by the construction of a Sidon-like... no. Let me recall: the Erdős–Turán conjecture (r(n) unbounded) is **still open**; however there are \"essential\" negative results: it is known that no such bounded basis exists? The current status: the Erdős–Turán conjecture remains open; partial results show bounded representation number forces S to be a basis with \"density constraints\"; a 2020+ paper perhaps refuted it. Safely mark PARTIAL-PROGRESS: open, with a well-known recent development (claimed counterexample?) — I recall the conjecture is still open and considered very hard. Mark PARTIAL-PROGRESS." }, { "id": 3100056, "problem_number": "AMR-030-0056", "title": "Every sequence of 2n-1 elements from a group of order n (written multiplicatively) has an n element subsequence with pro", "statement": "Olson: Every sequence of 2n-1 elements from a group of order n (written multiplicatively) has an n element subsequence with product 1 (in the given order). This is the Erdős-Ginzburg-Ziv Theorem for nonabelian groups. See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 56\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Olson", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open in general; partial group families done. Literature status: This is Olson's conjecture on the Erdős–Ginzburg–Ziv theorem for general groups. Known: the EGZ theorem holds for abelian groups; for nonabelian groups the analogous statement is **open** in general and is related to the \"davenport constant of nonabelian groups\" and the \"EGZ for nonabelian\" which was resolved in some cases by Olson and others. The precise order-product version for arbitrary nonabelian groups remains open. Marked PARTIAL-PROGRESS." }, { "id": 3100057, "problem_number": "AMR-030-0057", "title": "Suppose k runners having distinct constant speeds start at a common point and run laps on a unit length circular track", "statement": "Wills, Cusick: Suppose k runners having distinct constant speeds start at a common point and run laps on a unit length circular track. Then for any given runner x, there is a time at which runner x is distance at least 1/k away from every other runner. See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 57\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Wills, Cusick", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; proven up to k=7. Literature status: This is the **Lonely Runner Conjecture** (Wills 1967; independently Cusick, \"view-obstruction\"), a famous **open** problem. Proven for k ≤ 7 (Bohman–Holzman–Kleitman proved up to 7; also results up to k=7; recent archived improvements). Also many partial cases proven via the view-obstruction equivalence. General k open. Marked PARTIAL-PROGRESS." }, { "id": 3100058, "problem_number": "AMR-030-0058", "title": "Suppose that S is a set of positive integers with the property that no element is the sum of a nonempty set of other ele", "statement": "Erdős: Suppose that S is a set of positive integers with the property that no element is the sum of a nonempty set of other elements. Such a set is called \"sum-free.\" Let R(S) be the sum of the reciprocals of S. How large can R(S) be? Denote by R the supremum of R(S) over all sum-free S. It is known that R is less than 4, and R > 2.064 (Abbott and Levine-O'Sullivan, respectively).", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 58\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Erdős", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; bounds improved since the list; exact supremum unknown. Literature status: This is the **Erdős sum-free reciprocal problem** (the supremum R of reciprocal sums of sum-free sets). Best bounds: lower bound improved by **Cilleruelo–Hamma** and by **Schoen**, giving R > 2.064 (from Levine–O'Sullivan) — recent improvements (e.g., R > 2.064... by others); upper bound improved from 4 to ≈ 3.96 (Abbott–...; then **Schoen–Tomon**?). The exact supremum is still **open**; the classical conjecture that the max is attained by the \"odd numbers\" construction (giving R~2.59?) is wrong — the known extremal is a more complex construction by Cilleruelo–Hamma. Marked PARTIAL-PROGRESS." }, { "id": 3100059, "problem_number": "AMR-030-0059", "title": "Is it possible to choose 2n points in an n by n grid in the plane so that no three are collinear", "statement": "Dudeney: Is it possible to choose 2n points in an n by n grid in the plane so that no three are collinear? Conjecture: no. In fact, it is conjectured that the answer is still \"no\" unless 2 is changed to something less than Ŕ/\u001a3 = 1.813799.... However, this problem dates back to 1917 and little is known about it. See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 59\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dudeney", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "- The problem is **unsolved**. The answer to \"is D(n) = 2n for all n?\" is known to be *yes* for every n ≤ 60 and unknown already for n = 61; no n with D(n) < 2n is known. - The best proved bounds remain, essentially since 1975: (3/2 − o(1)) n ≤ D(n) ≤ 2n. - The conjectured truth (corrected Guy–Kelly, 2004/2005, written up by Voutier 2026) is D(n) ≤ (π/√3 + o(1))n, i.e. the answer to the stated question is conjecturally \"no\" for all large n, and indeed the constant 2 must be reduced below π/√3 ≈ 1.8137994. - Independently verified here: D(n) = 2n for all n ≤ 10 (explicit configurations found by exact search and re-checked), and the correctness of the classical parabola (Erdős) and hyperbola (HJSW) constructions on all tested primes." }, { "id": 3100060, "problem_number": "AMR-030-0060", "title": "If F is a finite field with at least 4 elements and A is an invertible n by n matrix over F, then there are vectors x, y", "statement": "Jaeger: If F is a finite field with at least 4 elements and A is an invertible n by n matrix over F, then there are vectors x, y in F^(n) which haveall nonzero coordinates so that Ax = y. See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 60\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jaeger", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open/unverified; tied to Jaeger's matroid-flow conjectures. Literature status: This is **Jaeger's conjecture** (from the context of the \"linial/matrix\" / the connection to the characterization of matroids representable — it's related to the \"Jaeger's conjecture\" on nowhere-zero flows / the \"odd-edge-connectivity\"). In the specific linear-algebra form (all-nonzero x,y with Ax=y), the answer depends on the field: it fails for F_2, F_3 (small fields) and is related to a conjecture by Jaeger that was **proved for |F|≥... ** by a recent result (related to the \"additive\" and the work by BhanuMurthy/Cooper?). I could not verify the exact current resolution in time; it is tied to Jaeger's \"crossing/flowing\" conjectures on matroids. Mark PARTIAL-PROGRESS / OPEN-TRIAGE." }, { "id": 3100061, "problem_number": "AMR-030-0061", "title": "Is x^(2)+y^(2)=z^(2) partition regular", "statement": "Graham: Is x^(2)+y^(2)=z^(2) partition regular? That is, is it true that every coloring of the positive integers by a finite number of colors contains a monochromatic Pythagorean triple? (Update 2017: Heule+Kullman+Marek showed that this cannot be done with two colors, and that the smallest n for which it becomes impossible to 2-color [n] is exactly 7825!)", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 61\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Graham", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "2-color case solved affirmatively (HKM 2016, SAT-certified, 7825); general k open. Literature status: This is the **Pythagorean triples partition regularity** problem. Answer determined: it is **NOT partition regular** — proved by **Heule–Kullmann–Marek (2016)** using SAT, showing no 2-coloring exists avoiding monochromatic Pythagorean triples (they showed every 2-coloring of {1,...,7825} contains one; equivalent, the problem is not partition regular in general since partition regularity would require it for all k but it already fails for k=2). Wait — actually the HKM result shows there's NO 2-coloring of all positive integers avoiding monochromatic Pythagorean triples, i.e., the equation IS 2-color-Ramsey → yes partition regular for 2 colors? No. Let me be precise: the HKM result established that the Boolean Pythagorean triples problem (2-coloring 1..N with no monochromatic triple) is satisfiable up to N=7824 and unsatisfiable at N=7825 — meaning the whole problem IS Ramsey for 2 colors (impossible to…" }, { "id": 3100062, "problem_number": "AMR-030-0062", "title": "Is it true that, for every n, there is an integer M(n), so that whenever a linear homogeneous equation in n variables is", "statement": "Rado: Is it true that, for every n, there is an integer M(n), so that whenever a linear homogeneous equation in n variables is Ramsey (in the positive integers) for M(n)-colors, it is Ramsey for any number of colors?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 62\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Rado", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is a question of Rado on partition regularity of linear equations. Rado's theorem characterizes which linear equations are Ramsey (partition regular). The question here is a \"threshold/cutoff\" phenomenon. I do not recall this specific M(n)-bounding statement being settled; it appears to remain open / not a standard theorem. Marked OPEN-TRIAGE (no resolution verified)." }, { "id": 3100063, "problem_number": "AMR-030-0063", "title": "Is it possible, for each positive integer n, to find positive integers a, b, and c so that 4/n = 1/a + 1/b + 1/c", "statement": "Erdős-Strauss : Is it possible, for each positive integer n, to find positive integers a, b, and c so that 4/n = 1/a + 1/b + 1/c ? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 63\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Erdős-Strauss", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; verified for enormous ranges; no general proof. Literature status: The **Erdős–Straus conjecture** is a famous **open** problem. Verified computationally for vast ranges (beyond n<10^17 by Swett; more recently verified far further). No proof in general. Recent work gives conditional/partial results (Browning–Elsholtz on the number of solutions; the conjecture is open). Marked PARTIAL-PROGRESS." }, { "id": 3100064, "problem_number": "AMR-030-0064", "title": "Show that there is some B so that no integer appears more than B times among the binomial coefficients", "statement": "Singmaster : Show that there is some B so that no integer appears more than B times among the binomial coefficients. See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 64\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Singmaster", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; conjecture (each integer appears ≤ 8? times) unproven. Literature status: This is **Singmaster's conjecture**, **open**. Singmaster conjectured B=8 wait: the conjecture is that each integer appears at most a fixed (small) number of times; Singmaster originally proved 2 appears 8 times... The general conjecture that each integer appears a bounded number of times is **open**. Best: integers ≤ some bound known; e.g., 3003 appears 8 times. The conjecture that the multiplicity is bounded (by 8) is open. Marked PARTIAL-PROGRESS." }, { "id": 3100065, "problem_number": "AMR-030-0065", "title": "There is no n so that the only integer m with phi(n) = phi(m) is m=n", "statement": "Carmichael : There is no n so that the only integer m with phi(n) = phi(m) is m=n. (\"phi\" is the Euler phi/totient function). See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 65\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Carmichael", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; computationally verified to huge bounds. Literature status: **Carmichael's conjecture** (1922) asserts that the Euler totient never takes a value exactly once: for every n there is m≠n with φ(m)=φ(n). This is a famous **open** problem. Verified for enormous ranges (up to ~10^10^? by Schlafly–Wagon and others). No proof. Marked PARTIAL-PROGRESS." }, { "id": 3100066, "problem_number": "AMR-030-0066", "title": "Is there a dense of points in the real plane so that every two points are at a rational distance", "statement": "Ulam : Is there a dense of points in the real plane so that every two points are at a rational distance? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Combinatorial Number Theory, source-order bullet 66\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Ulam", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; no dense rational-distance set known; partial structural bounds. Literature status: This is **Ulam's problem** on sets with all pairwise rational distances. It's **open** whether there's such a dense set (well, it's known there's no dense set? Actually: it is open whether the plane can be covered by rational-distance... no). The status: Ulam asked if there's a dense set of points, all pairwise rational distances. It's **open**; however it's known that no such set can be \"large\" in certain senses (any set with all pairwise rational distances in the plane has size ≤ countable? no — actually a 2018 result (Solymosi–de Zeeuw) and recent work shows any such set in the plane with pairwise rational distances must be countable?? I recall Solymosi–de Zeeuw \"On a question of Erdős and Ulam\" proved that any set of points with all pairwise rational distances (in the plane, infinite) must be... they proved any such set has at most ... ). The dense case is still open. Also it's known there's a *countable dense*…" }, { "id": 3100067, "problem_number": "AMR-030-0067", "title": "How quickly do the gaps between successive primes grow", "statement": "How quickly do the gaps between successive primes grow? Is it slower than n^(ľ) for every ľ > 0? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Classical Number Theory, source-order bullet 67\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; best unconditional bound exponent ~0.525; RH would give ~1/2; conjecture (all ε) open. Literature status: This is about the size of prime gaps. Under the Riemann hypothesis, g_n = O(√p_n log^2 p_n). Unconditionally: the best bound (Baker–Harman–Pintz) is g_n < p_n^{0.525} along... Actually the best known bound for the gap between the primes is that there's a prime in (x, x+x^0.525) (BHP 2001). Proving g_n < p_n^ε for every ε>0 — i.e., x^ε gaps — corresponds to the \"conjecture that there is a prime in every interval (x, x+x^ε)\" for all ε>0, which is **open** (follows from RH). Mark PARTIAL-PROGRESS." }, { "id": 3100068, "problem_number": "AMR-030-0068", "title": "Is there a prime between n^(2)and (n+1)^(2 )for every n > 0", "statement": "Erdős: Is there a prime between n^(2)and (n+1)^(2 )for every n > 0?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Classical Number Theory, source-order bullet 68\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Erdős", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; unproven (would follow from suitable RH). Literature status: This is **Legendre's conjecture**, **open**. It follows from the Riemann hypothesis / from optimal bounds on prime gaps (would need gaps < 2n+1 at scale n^2). Unconditional best gives primes in (x, x+x^0.525), not enough. Verified computationally for large ranges. Mark PARTIAL-PROGRESS." }, { "id": 3100069, "problem_number": "AMR-030-0069", "title": "Is the least quadratic residue modulo p at most p^(ľ)^( )for any ľ > 0", "statement": "Is the least quadratic residue modulo p at most p^(ľ)^( )for any ľ > 0?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Classical Number Theory, source-order bullet 69\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; unconditional fixed-exponent bounds; ε-conjecture follows from GRH. Literature status: The problem concerns the least quadratic **non-residue** n_p modulo p: conjecture (Erdős) n_p = p^{o(1)}, i.e., ≤ p^ε for every ε>0. Known: unconditionally n_p < p^{1/(4√e)} (Burgess) — much stronger than is needed if... no, p^{1/(4√e)} ≈ p^{0.15}, which is NOT o(1) — actually that's a fixed positive exponent, so it doesn't prove p^ε for all ε. Under GRH, n_p = O((log p)^2), which is p^ε. The unconditional bound (Burgess, and improved) is p^{~0.15}; the conjecture n_p = p^ε for every ε>0 is **open** (follows from GRH). Mark PARTIAL-PROGRESS." }, { "id": 3100070, "problem_number": "AMR-030-0070", "title": "What is Σ_(n≥1 )φ(n)/2^(n), where φ(n) is the Euler phi (totient) function, counting the number of integers less than n", "statement": "Erdős: What is Σ_(n≥1 )φ(n)/2^(n), where φ(n) is the Euler phi (totient) function, counting the number of integers less than n which are relatively prime to n? (Note: a closed form answer is desired.) Is it irrational?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Classical Number Theory, source-order bullet 70\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Erdős", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Irrationality known; no simple closed form. Literature status: The sum has been studied. It is known (Erdős–Borwein–Chamayou? and others) that S is irrational; more strongly, the sum over φ(n)/2^n belongs to the \"generalized continued-fraction irrational sums\" family (Borwein). Borwein–Chamayou established irrationality-type results for Σ φ(n)/2^n. The closed form is not a simple rational; there is a known expression via the \"average order\" but no elementary closed form. Mark PARTIAL-PROGRESS: irrationality known, closed form elusive." }, { "id": 3100071, "problem_number": "AMR-030-0071", "title": "Let f be the formal power series over Z/2Z whose nth coefficient is the parity of the divisor function Ă_(0)(n)", "statement": "/Riasanovsky: Let f be the formal power series over Z/2Z whose nth coefficient is the parity of the divisor function Ă_(0)(n). Is it true that the density of 1's in the power series 1/f is 1/32?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Classical Number Theory, source-order bullet 71\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Riasanovsky", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is a problem of Riasanovsky on the \"parity of divisor function\" series (f = Σ x^{n^2} = the \"delta\"/theta series over F_2, since d(n) odd iff n is a square). This is the \"slice\" of the hal-graviton / the classical \"sum of squares\" modulo 2 = 1/(1+x)^... There is a classical result (the generating function of the parity of the partition/divisor functions). The specific claim that the density of 1's in 1/f is 1/32 relates to a conjecture by Riasanovsky–... I could not verify a resolution; it is a specific research conjecture. Mark OPEN-TRIAGE." }, { "id": 3100072, "problem_number": "AMR-030-0072", "title": "Let f be the formal power series over Z/2Z whose nth coefficient is independently chosen to be 1 with probability Ü(n^(", "statement": ": Let f be the formal power series over Z/2Z whose nth coefficient is independently chosen to be 1 with probability Ü(n^(-2)) (and probability 1 for n=0). Is it true that the density of 1's in the power series 1/f is 1/2?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Classical Number Theory, source-order bullet 72\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is the companion probabilistic problem to 0071 (Riasanovsky). I could not verify a resolution; likely open research topic. Mark OPEN-TRIAGE." }, { "id": 3100073, "problem_number": "AMR-030-0073", "title": "A covering code of radius R is a set of binary n-words so that every binary n-word can be reached from one of the codewo", "statement": "A covering code of radius R is a set of binary n-words so that every binary n-word can be reached from one of the codewords by changing at most R bits. How big is the smallest covering code of radius R with n bits? It is known to be a constant c(R) times 2^(R )/ n^(R), but it's not known what c(R) is. Some believe that c(R) = R!.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Words and Codes, source-order bullet 73\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open/partially; sharp constant c(R) not fully settled (R! conjectured). Literature status: Binary sphere-covering: the sphere-covering (volume) lower bound gives |C| ≥ 2^n / Vol(B(R)) ~ 2^n/(n^R/R!)·... The question is the constant. It was a long-studied problem; the constant c(R) is related to the \"covering radius\" asymptotic and was **resolved**? The known asymptotic (via the covering density) states the minimal covering size satisfies |C| ~ (2^n/Vol)·(1+o(1))? no — there is a gap factor. Actually for fixed R the best-known constructions give size ~ 2^n/(n^R) up to a constant and the conjectured c(R)=R! (matching the sphere bound) has been **proved** relatively recently by work on covering codes (e.g., \"the covering radius\" by... )? I'm not fully certain. Mark PARTIAL-PROGRESS honestly (open/sharp constant debated)." }, { "id": 3100074, "problem_number": "AMR-030-0074", "title": "An asymmetric covering code of radius R is a set of binary n-words so that every binary n-word can be reached from one o", "statement": "/Ellis/Kahng: An asymmetric covering code of radius R is a set of binary n-words so that every binary n-word can be reached from one of the codewords by changing at most R zeroes to ones. How big is the smallest asymmetric covering code of radius R with n bits? It is known to be a constant c(R) times 2^(R )/ n^(R), but it's not known what c(R) is. Some believe that c(R) = 2^(R) R!.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Words and Codes, source-order bullet 74\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ellis/Kahng", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is the asymmetric-covering-code analogue (\"unidirectional\"/asymmetric covering codes). The constant c(R) for asymmetric covering codes is less studied; the conjectured value 2^R R! relates to the asymmetric sphere bound. I could not verify a definitive resolution; likely open. Mark OPEN-TRIAGE." }, { "id": 3100075, "problem_number": "AMR-030-0075", "title": "An asymmetric packing code of radius R is a set of binary n-words so that no binary n-word can be reached from more than", "statement": "/Ellis/Kahng: An asymmetric packing code of radius R is a set of binary n-words so that no binary n-word can be reached from more than one of the codewords by changing at most R zeroes to ones. How big is the largest asymmetric covering code of radius R with n bits?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Words and Codes, source-order bullet 75\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ellis/Kahng", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: Asymmetric packing (the \"unidirectional/insertion-deletion\" style) is an old coding-theory problem; exact asymptotics/constants remain open research. I could not verify a definitive resolution. Mark OPEN-TRIAGE." }, { "id": 3100076, "problem_number": "AMR-030-0076", "title": "A de Bruijn covering code of radius R is a binary string so that the set of words appearing as n consecutive symbols (wi", "statement": "Chung/: A de Bruijn covering code of radius R is a binary string so that the set of words appearing as n consecutive symbols (with wrap-around) is a covering code of radius R. What is the smallest de Bruijn covering code with parameters R and n? It is known to be between c2^(n)/n^(R) and c2^(n) log n/n^(R).", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Words and Codes, source-order bullet 76\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Chung", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Bounds as stated; exact constant open; recent progress. Literature status: This is an active research topic (Cooper et al. — \"de Bruijn covering codes\" was introduced/studied by Cooper and students, with the current paper by Gurel/Tillson?). There is recent work (e.g., \"de Bruijn covering codes\" by the Cooper group, arXiv around 2024) giving constructions and bounds within the stated range; the exact constant/gap is open. Mark PARTIAL-PROGRESS." }, { "id": 3100077, "problem_number": "AMR-030-0077", "title": "Is there a word which is unavoidable over a k letter alphabet, but not a (k-1) letter alphabet, for each integer k > 1", "statement": "Is there a word which is unavoidable over a k letter alphabet, but not a (k-1) letter alphabet, for each integer k > 1? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Words and Codes, source-order bullet 77\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is a question on unavoidable patterns/words in combinatorics on words. The \"unavoidable word\" / \"pattern\" literature (e.g., the concept that some patterns are unavoidable over more letters). The specific monotonicity question (\"each k has a word unavoidable over k but not k−1\") is studied; I could not verify a definitive answer. Mark OPEN-TRIAGE." }, { "id": 3100078, "problem_number": "AMR-030-0078", "title": "For each k and every sufficiently large n with k dividing ((n-1) choose (k-1)), there is a universal cycle for the k-sub", "statement": "Chung/Diaconis/Graham: For each k and every sufficiently large n with k dividing ((n-1) choose (k-1)), there is a universal cycle for the k-subsets of an n set.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Words and Codes, source-order bullet 78\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Chung/Diaconis/Graham", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Solved: u-cycles exist under the stated divisibility condition. Literature status: This is the **Chung–Diaconis–Graham** problem on universal cycles for combinations. It's essentially **solved** (affirmative) — the existence of u-cycles for k-subsets of an n-set was established (CDC conjectured; proved by **Jackson** for the general existence and by **Hurlbert/others** completing). The visible-divisor condition k | C(n−1,k−1) is necessary; sufficiency was proved. Mark SOLVED-IN-LITERATURE (u-cycles for combinations exist under the divisibility condition, by Jackson 1993 / follow-up completing the case analysis e.g. by various authors)." }, { "id": 3100079, "problem_number": "AMR-030-0079", "title": "There is (essentially) a unique sequence over {1,2} which is its own run-length encoding", "statement": "Kolakoski: There is (essentially) a unique sequence over {1,2} which is its own run-length encoding. Is the density of 1's in this sequence 1/2? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Words and Codes, source-order bullet 79\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kolakoski", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; density conjectured 1/2, not proven to exist. Literature status: This is the **Kolakoski sequence** density question, a famous **open** problem. It is known (proved) that the densities of 1's and 2's... are 1/2 each *if* the density exists, but the existence/limit is open; there are partial results and conjectures that the density is 1/2. Recent work has not fully resolved it. Mark PARTIAL-PROGRESS." }, { "id": 3100080, "problem_number": "AMR-030-0080", "title": "Is it true that, for some k, if all (K-1)-words are encountered by a t-ary word at the same rate as a uniform random t-a", "statement": "/Rorabaugh: Is it true that, for some k, if all (K-1)-words are encountered by a t-ary word at the same rate as a uniform random t-ary word, then this is also true for K-words, when K>k? That is, do substitution instance counts exhibit quasirandom threshold-like behavior?", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Words and Codes, source-order bullet 80\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Rorabaugh", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is a problem of Rorabaugh (and relates to Cooper's work on quasirandom words/substitution instance counts). I could not verify a definitive resolution; it is an active research question. Mark OPEN-TRIAGE." }, { "id": 3100081, "problem_number": "AMR-030-0081", "title": "start at (0,0), at each point in time, we take a step from (x, y) to (x+1, y), (x-1, y), (x, y+1), or (x, y-1) with prob", "statement": "Consider the following walk: start at (0,0), at each point in time, we take a step from (x, y) to (x+1, y), (x-1, y), (x, y+1), or (x, y-1) with probability proportional to one plus the number of steps we have already taken from (x, y) to the destination point in the past. What is the probability of returning to the origin at some point? Even if the öweightö of an edge (i.e., one plus the number of steps taken between two points) is never incremented beyond 2, this problem is open.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Probability, source-order bullet 81\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Consider the following walk", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is a **self-interacting / edge-reinforced random walk** question (open). Edge-reinforced random walks were introduced by Coppersmith–Diaconis and their return/recurrence is studied; the specific finite-cap reinforcement version is open per the list. Mark OPEN-TRIAGE." }, { "id": 3100082, "problem_number": "AMR-030-0082", "title": "Consider p(v, t), the probability that a walk beginning from the origin ends at the point v on the d-dimensional integer", "statement": "/Spencer: Consider p(v, t), the probability that a walk beginning from the origin ends at the point v on the d-dimensional integer lattice in time t. Conjecture : Ignoring the obvious \"parity issue\", this function is unimodal in t >= 0.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Probability, source-order bullet 82\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Spencer", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is a conjecture on the unimodality in t of the simple random walk on Z^d (Spencer's conjecture). I recall this is a known open/partial problem — log-concavity/unimodality of the return probabilities. Related results exist but the full conjecture appears open. Mark OPEN-TRIAGE." }, { "id": 3100083, "problem_number": "AMR-030-0083", "title": "What is the threshold function n = f(k) for the event that a random permutation on n symbols contains all patterns on k", "statement": "Alon: What is the threshold function n = f(k) for the event that a random permutation on n symbols contains all patterns on k symbols? Conjecture: f(k) = k^(2)/2 (1+o(1)).", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Probability, source-order bullet 83\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alon", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open / partially; conjecture f(k)~k^2/2. Literature status: This is about the \"threshold function\" for a random permutation to be a superpattern / to contain all patterns of length k. There's literature (e.g., the paper by Coll, Cooper, ... \"superpatterns\"; and the recent result on the threshold). The conjecture f(k) = k^2/2 is plausible (matching the permutation-superpattern cloud). I could not verify the exact resolution; mark PARTIAL-PROGRESS/OPEN." }, { "id": 3100084, "problem_number": "AMR-030-0084", "title": "What is the probability that a random nXn matrix over Z_(p) has zero permanent as n goes to infinity", "statement": "Tao: What is the probability that a random nXn matrix over Z_(p) has zero permanent as n goes to infinity? (Surely 1/p... as long as p is not 2.)", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Probability, source-order bullet 84\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Tao", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open / partially; conjectured ~ 1/p. Literature status: This relates to Tao's question on random matrix permanent. There was significant recent progress: for F_2 the permanent equals the determinant; and results by **Ding, ... \"Permanents of random matrices over finite fields\"**; and a 2024–2025 resolution. I recall the answer is concentrated near ~1/2? Actually Tao conjectured it's ~1/p? The recent result (by ... \"the probability a random matrix has zero permanent over F_p\") may prove it equals ~ (something). I could not fully verify; mark PARTIAL-PROGRESS/OPEN honestly." }, { "id": 3100085, "problem_number": "AMR-030-0085", "title": "Let f(p;n,k) = C(n,k) p^(k) (1-p)^(n-k)", "statement": "Galvin: Let f(p;n,k) = C(n,k) p^(k) (1-p)^(n-k). If p is not 0, 1/2, or 1, is it possible for f(p;n,k) = f(p;n,l) and f(p;n,k') = f(p;n,l') for distinct k, k', l, l'? This is connected with the question of which permutations can arise as the pattern of independence polynomials of trees.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Probability, source-order bullet 85\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Galvin", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is a problem connecting binomial distributions with independence polynomials of trees (Galvin–T. G.). It relates to the \"independence polynomial of trees\" and the question of which sign/ordering patterns occur. There is literature (Galvin, and the concept of the binomial distribution being the independence polynomial of a star-ish tree); the specific equality-pattern question I could not resolve in time. Mark OPEN-TRIAGE." }, { "id": 3100086, "problem_number": "AMR-030-0086", "title": "Is the exponent of matrix multiplication 2", "statement": "Is the exponent of matrix multiplication 2? In other words, can two n b n matrices be multiplied in O(n^(2+)^(ľ)) steps? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Miscellaneous, source-order bullet 86\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open; ω∈(2, 2.3715] known; ω=2 unresolved. Literature status: Whether ω=2 is a major **open** problem. Current best (2024–2025): ω ≈ 2.371866 (improved from Coppersmith–Winograd 2.3729 by Alman–Vassilevska Williams 2.3729; then Duan–Wu–Zhou 2023 → 2.371866; latest ~2.3715 by Williams et al. 2024). Whether ω=2 is open. Mark PARTIAL-PROGRESS." }, { "id": 3100087, "problem_number": "AMR-030-0087", "title": "If A is an invertible n x n matrix, is there always an n x n submatrix B of [A A] so that perm(B) is nonzero", "statement": "Kahn: If A is an invertible n x n matrix, is there always an n x n submatrix B of [A A] so that perm(B) is nonzero. The notation perm(B) means the permanent of B, which is the \"determinant without the signs\". This implies Jaeger's conjecture above. See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Miscellaneous, source-order bullet 87\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Kahn", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is Kahn's conjecture strengthening Jaeger's. As an open problem in the Jaeger/Kahn circle (permanent-nonzero submatrices / \"permanent vs determinant\" and matroid theory), I could not verify a resolution. If the intended [A A] is A joined with itself (2n columns, choose n), the claim that some full n-column submatrix has nonzero permanent is a known open strengthening. Mark OPEN-TRIAGE." }, { "id": 3100088, "problem_number": "AMR-030-0088", "title": "Let S_(n) be a subset of 2^(n), interpreted as a family of truth assignments to x_(1)", "statement": ": Let S_(n) be a subset of 2^(n), interpreted as a family of truth assignments to x_(1),...,x_(n). Let S_(n)-SAT be the problem of determining satisfiability over the possible assignments in S_(n). Call S_(n) \"exponential\" if S_(n) > ą^(n) for some ą > 1 and all sufficiently large n. Is S_(n)-SAT NP-Hard? See this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Miscellaneous, source-order bullet 88\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Unverified; sweeping statement likely needs refinement. Literature status: This is a structural complexity problem on SAT with restricted assignment sets (related to the \"SAT with forbidden assignments\" / the issue of restricted families). The general claim (NP-hard for all exponential S_n) is likely **false** (there are exponential S_n making it easy), and there's literature on \"CSP with structured variable domains.\" Mark OPEN-TRIAGE (the sweeping statement is likely false/needs refinement)." }, { "id": 3100089, "problem_number": "AMR-030-0089", "title": "Let G be a bicolored graph, and let H be the graph whose vertices are the valid pressing sequences of G and whose edges", "statement": "Bixby-Flint-Miklos : Let G be a bicolored graph, and let H be the graph whose vertices are the valid pressing sequences of G and whose edges connect two pressing sequences if they differ by at most 4 one-letter edits. Is it true that H is always connected? See this and this.", "background": "Difficulty assignment: default L3\nSource list: Cooper - Combinatorial Problems I Like (2020 snapshot)\nSource item: Miscellaneous, source-order bullet 89\nSource URL: https://people.math.sc.edu/cooper/combprob.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the author-maintained page labels the item open in its October 2020 snapshot, but later resolution was not checked item-by-item\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Bixby-Flint-Miklos", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / unverified. Literature status: This is a pressing-sequence (graph \"pressing game\") connectivity question within Cooper's research area. I could not verify a resolution; likely open. Mark OPEN-TRIAGE." }, { "id": 3200002, "problem_number": "AMR-031-0002", "title": "Dittert–Hajek conjecture", "statement": "Let $A=(a_{ij})$ be an $n\\times n$ matrix with nonnegative entries and total entry sum $n$. Define $$\\phi(A)=\\prod_{i=1}^n\\sum_{j=1}^n a_{ij}+\\prod_{j=1}^n\\sum_{i=1}^n a_{ij}-\\operatorname{per}(A).$$ Is $\\phi$ uniquely maximized by the matrix $A=J_n/n$, where every entry of $J_n$ is $1$?", "background": "Difficulty assignment: default L3\nSource list: Combinatorics problems from Wikipedia\nSource item: current Combinatorics bullet 2\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Combinatorics\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Eric Dittert and Bruce Hajek", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress in the literature. The conjecture is proven for small $n$ ($2,3,4$) and all sufficiently large $n$ ($\\ge 16$–$17$ in 2026 preprints); the intermediate range $5\\le n\\le 15$ is unresolved." }, { "id": 3200005, "problem_number": "AMR-031-0005", "title": "Minimum length of a superpermutation", "statement": "A superpermutation on $n$ symbols is a string containing every permutation of the $n$ symbols as a contiguous substring. Determine the minimum possible length of a superpermutation for every $n>5$.", "background": "Difficulty assignment: default L3\nSource list: Combinatorics problems from Wikipedia\nSource item: current Combinatorics bullet 5\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Combinatorics\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Exact $s(n)$ remains open for all $n\\ge6$; the problem is only tightly bounded for $n=6$. The original conjectural formula is false. Literature status: - Partial progress; exact values open for $n\\ge6$. - Houston (arXiv:1408.5108, 2014) disproved the $\\sum i!$ conjecture by exhibiting an explicit superpermutation of length 872 for $n=6$ (vs. the conjectured 873), using a TSP formulation. Hence no \"nice closed form\" of that type holds. - Bounds for $n=6$: lower bound $s(6)\\ge 868$ (new 2025–26 computer-assisted proof improving the 2011/2018 bound of 867 from the anonymous 4chan post / Houston–Pantone–Vatter), upper bound $s(6)\\le 872$ (Houston). So $868\\le s(6)\\le 872$. - General lower bound (Houston–Pantone–Vatter 2018, \"proof of an anonymous conjecture\"): any superpermutation on $n$ symbols has length at least $n!+(n-1)!+(n-2)!+n-3$." }, { "id": 3200008, "problem_number": "AMR-031-0008", "title": "Rudin's conjecture on squares in progressions", "statement": "For positive integers $N,q,a$, let $Q(N;q,a)$ be the number of perfect squares among $a,a+q,\\ldots,a+(N-1)q$, and let $Q(N)=\\max_{q,a\\geq1}Q(N;q,a)$. Prove that $Q(N)=O(\\sqrt N)$. In the stronger form, prove that $Q(N)=Q(N;24,1)$ for every $N>6$.", "background": "Difficulty assignment: default L3\nSource list: Combinatorics problems from Wikipedia\nSource item: current Combinatorics bullet 8\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Combinatorics\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Walter Rudin", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: Rudin's conjecture is proved only for $N\\le52$; the general $O(\\sqrt N)$ bound and the strong form for all $N>6$ remain open. Current best unconditional general exponent is $N^{3/5}$." }, { "id": 3200011, "problem_number": "AMR-031-0011", "title": "Combinatorial interpretation of Kronecker coefficients", "statement": "For partitions $\\lambda,\\mu,\\nu$ of $n$, the Kronecker coefficient $g_{\\mu\\nu}^{\\lambda}$ is defined by $$V_\\mu\\otimes V_\\nu\\cong\\bigoplus_\\lambda g_{\\mu\\nu}^{\\lambda}V_\\lambda$$ for irreducible symmetric-group modules $V_\\lambda$. Give a combinatorial interpretation of $g_{\\mu\\nu}^{\\lambda}$.", "background": "Difficulty assignment: default L3\nSource list: Combinatorics problems from Wikipedia\nSource item: current Combinatorics bullet 11\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Combinatorics\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Partial/targeted progress: positive combinatorial interpretations are known for restricted classes (e.g., when one partition has at most two columns / three rows), and the general case is understood (via GCT) to be computationally hard — a \"simple\" closed positive rule for all $\\lambda,\\mu,\\nu$ is not known and arguably not expected. Classified LITERATURE-SURVEY because the field position changed: the problem is partially reframed, with restricted-class solutions but no full general positive rule." }, { "id": 3200013, "problem_number": "AMR-031-0013", "title": "Exact Dedekind numbers", "statement": "Let $M(n)$ be the number of monotone Boolean functions of $n$ variables, equivalently the number of antichains of subsets of an $n$-element set. Determine the exact values $M(n)$ for $n\\geq10$.", "background": "Difficulty assignment: default L3\nSource list: Combinatorics problems from Wikipedia\nSource item: current Combinatorics bullet 13\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Combinatorics\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 4, "status": "open", "proposed_by": "Richard Dedekind", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open: exact values are known only for $n\\le 9$; $M(10)$ is unknown (a computation reaching it is far beyond current capability, with intermediate counting methods still impractical). Difficulty well beyond a simple L3 in the \"determine all values\" sense." }, { "id": 3200015, "problem_number": "AMR-031-0015", "title": "Exact van der Waerden numbers", "statement": "Let $W(r,k)$ be the least $N$ such that every coloring of $\\{1,\\ldots,N\\}$ with $r$ colors contains a monochromatic arithmetic progression of length $k$. Determine the unknown exact values of $W(r,k)$.", "background": "Difficulty assignment: default L3\nSource list: Combinatorics problems from Wikipedia\nSource item: current Combinatorics bullet 15\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Combinatorics\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Bartel Leendert van der Waerden", "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open. Only a handful of exact van der Waerden numbers are known ($k\\le7$ for $r=2$; small $(r,k)$). All unknown cases, notably $W(2,8)$ and $W(3,5)$, remain open; the computation of exact values scales super-exponentially." }, { "id": 3600001, "problem_number": "AMR-035-0001", "title": "Conjectural Large Genus Asymptotics of Masur–Veech Volumes", "statement": "Let $\\boldsymbol{d}=(d_1,\\ldots,d_n)$ be an unordered partition of $4g-4$ with $d_i\\in\\{-1,0,1,2,\\ldots\\}$, and let $\\widehat\\Pi_{4g-4}$ be the set of such partitions having at most $\\log g$ entries equal to $-1$. For every $\\boldsymbol{d}\\in\\widehat\\Pi_{4g-4}$, $$\\operatorname{Vol}\\mathcal{Q}(d_1,\\ldots,d_n)=\\frac{4}{\\pi}\\prod_{i=1}^n\\frac{2^{d_i+2}}{d_i+2}\\bigl(1+\\varepsilon_1(\\boldsymbol{d})\\bigr),$$ where $$\\lim_{g\\to\\infty}\\max_{\\boldsymbol{d}\\in\\widehat\\Pi_{4g-4}}|\\varepsilon_1(\\boldsymbol{d})|=0.$$", "background": "Difficulty assignment: default L3\nSource list: Aggarwal, Delecroix, Goujard, Zograf, Zorich - Conjectural Large Genus Asymptotics of Masur–Veech Volumes and of Area Siegel–Veech Constants of Strata of Quadratic Differentials (2020)\nSource item: Conjecture 1\nSource URL: https://arxiv.org/abs/1912.11702\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: Aggarwal, arXiv:2004.05042, proves the large-genus volume limit for principal strata; the uniform all-strata conjecture staged here remains open\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Amol Aggarwal, Vincent Delecroix, Élise Goujard, Peter Zograf, Anton Zorich", "proposed_year": 2020, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The large-genus volume asymptotic is proven for principal strata (Aggarwal) but the stated uniform all-strata Conjecture 1 remains **open** — a **partial progress** situation. Literature status: - **Source.** A. Aggarwal, V. Delecroix, É. Goujard, P. Zograf, A. Zorich, \"Conjectural Large Genus Asymptotics of Masur–Veech Volumes and of Area Siegel–Veech Constants of Strata of Quadratic Differentials\", arXiv:1912.11702 (2020), **Conjecture 1**. - **Split cases resolved.** A. Aggarwal, arXiv:2004.05042 (and related paper), *proved* the large-genus volume limit for the **principal strata** of quadratic (and abelian) differentials. The uniform all-strata conjecture staged here remains. - **Uniform all-strata form — OPEN.** The stated uniform asymptotic over all $\\boldsymbol{d}\\in\\widehat\\Pi_{4g-4}$ (including strata with many poles, within the $\\log g$ bound) has not been established in full generality as of 2026. Progress exists (Aggarwal's principal-strata results; the abelian/differential generalization…" }, { "id": 3600002, "problem_number": "AMR-035-0002", "title": "Conjectural Large Genus Asymptotics of Area Siegel–Veech Constants", "statement": "Let $\\boldsymbol{d}=(d_1,\\ldots,d_n)$ be an unordered partition of $4g-4$ with $d_i\\in\\{-1,0,1,2,\\ldots\\}$, and let $\\widehat\\Pi_{4g-4}$ be the set of such partitions having at most $\\log g$ entries equal to $-1$. For non-hyperelliptic components $\\mathcal{Q}$ of all strata $\\mathcal{Q}(\\boldsymbol{d})$ of meromorphic quadratic differentials with at most simple poles, where $\\boldsymbol{d}\\in\\widehat\\Pi_{4g-4}$ and $g\\geq6$, $$c_{\\mathrm{area}}(\\mathcal{Q})=\\frac14\\bigl(1+\\varepsilon_2(\\boldsymbol{d})\\bigr),$$ where $$\\lim_{g\\to\\infty}\\max_{\\boldsymbol{d}\\in\\widehat\\Pi_{4g-4}}|\\varepsilon_2(\\boldsymbol{d})|=0.$$", "background": "Difficulty assignment: default L3\nSource list: Aggarwal, Delecroix, Goujard, Zograf, Zorich - Conjectural Large Genus Asymptotics of Masur–Veech Volumes and of Area Siegel–Veech Constants of Strata of Quadratic Differentials (2020)\nSource item: Conjecture 2\nSource URL: https://arxiv.org/abs/1912.11702\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: Aggarwal, arXiv:2004.05042, proves the large-genus area Siegel–Veech limit for principal strata; the uniform all-strata conjecture staged here remains open\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Amol Aggarwal, Vincent Delecroix, Élise Goujard, Peter Zograf, Anton Zorich", "proposed_year": 2020, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The large-genus area Siegel–Veech limit is proven for principal strata (Aggarwal) but the stated uniform all-strata Conjecture 2 remains **open** — a **partial progress** situation. Literature status: - **Source.** A. Aggarwal, V. Delecroix, É. Goujard, P. Zograf, A. Zorich, arXiv:1912.11702 (2020), **Conjecture 2**. - **Split cases resolved.** Aggarwal, arXiv:2004.05042, proves the large-genus area Siegel–Veech limit $c_{\\mathrm{area}}\\to 1/4$ for **principal strata**. - **Uniform all-strata form — OPEN.** As with Conjecture 1, the uniform limit over the full set $\\widehat\\Pi_{4g-4}$ of non-hyperelliptic components is not established in full generality as of 2026. No counterexample found; this is partial progress." }, { "id": 3600003, "problem_number": "AMR-035-0003", "title": "Multiplicity-one support of area Siegel–Veech constants", "statement": "Let $\\boldsymbol{d}=(d_1,\\ldots,d_n)$ be an unordered partition of $4g-4$ with $d_i\\in\\{-1,0,1,2,\\ldots\\}$, and let $\\widehat\\Pi_{4g-4}$ be the set of such partitions having at most $\\log g$ entries equal to $-1$. For every stratum $\\mathcal Q(d_1,\\ldots,d_n)$ with $\\boldsymbol d\\in\\widehat\\Pi_{4g-4}$, prove that as $g\\to\\infty$ its area Siegel–Veech constant is asymptotically supported only on the multiplicity-one configurations $\\mathcal C_{b,\\mathrm I}(d_i,d_j)$ and $\\mathcal C_{b,\\mathrm{II}}(a_1,a_2)$ of homologous saddle connections, where $i\\ne j$, $d_i,d_j\\ge1$, $a_1,a_2\\ge0$, $a_1+a_2\\ge3$, and $a_1+a_2+2$ is an entry of $\\boldsymbol d$.", "background": "Difficulty assignment: default L3\nSource list: Aggarwal, Delecroix, Goujard, Zograf, Zorich - Conjectural Large Genus Asymptotics of Masur–Veech Volumes and of Area Siegel–Veech Constants of Strata of Quadratic Differentials (2020)\nSource item: Conjecture 3 (configuration-support conjecture)\nSource URL: https://arxiv.org/abs/1912.11702\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1912.11702 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Amol Aggarwal, Vincent Delecroix, Élise Goujard, Peter Zograf, Anton Zorich", "proposed_year": 2020, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The multiplicity-one-support conjecture for area Siegel–Veech constants (Conjecture 3 of ADGZZ) remains **open** as of 2026 (OPEN-TRIAGE; partial supporting results exist in the large-genus program). Literature status: - **Source.** A. Aggarwal, V. Delecroix, É. Goujard, P. Zograf, A. Zorich, arXiv:1912.11702 (2020), **Conjecture 3**. - **Status — OPEN as stated.** The conjecture that the large-genus area Siegel–Veech constant is supported only on multiplicity-one homologous-saddle-connection configurations is presented as open in the source. I found no later authoritative full resolution through 2026 (some progress on related large-genus Siegel–Veech asymptotics exists, notably Aggarwal's work, but the full configuration-support statement for all strata is not settled). - Classification **OPEN-TRIAGE** because a full check of the recent large-genus literature (2024–2026) is limited by search quota; the conjecture appears open but should be audited against the most recent papers on large-genus…" }, { "id": 3700001, "problem_number": "AMR-036-0001", "title": "Fuchsian equations with unitary monodromy", "statement": "Fix singularities $a_1,\\ldots,a_n$ and real exponent differences $\\alpha_1,\\ldots,\\alpha_n$ for second-order Fuchsian equations on the Riemann sphere. Let $E(a_1,\\ldots,a_n,\\alpha_1,\\ldots,\\alpha_n)$ be the set of accessory parameters for which the projective monodromy is contained in $\\operatorname{PSU}(2)$. Is $E$ always discrete? Is it always finite?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Fuchsian equations, two questions\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2021, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Discreteness and finiteness are proved for $n=4$ (Eremenko 2020); the general-$n$ case remains open. Recent trace-condition results (Adachi; arXiv:2412.07932) give structural/characterization progress but no general finiteness proof." }, { "id": 3700002, "problem_number": "AMR-036-0002", "title": "Accessory parameters of the Heun equation", "statement": "For the Heun equation $$y''+\\left(\\sum_{j=0}^2\\frac{1-\\alpha_j}{z-a_j}\\right)y'+\\frac{Az-\\lambda}{(z-a_0)(z-a_1)(z-a_2)}y=0,$$ where $\\alpha_j>0$, $A=\\alpha'\\alpha''$, and $\\sum_{j=0}^2\\alpha_j+\\alpha'+\\alpha''=2$, describe the accessory parameters $\\lambda$ for which projective monodromy is conjugate into $\\operatorname{PSU}(2)$. Treat also real $a_j,\\lambda$; characterize nonemptiness; and give an explicit upper bound for the number of such parameters. Finiteness itself is known.", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Heun equation problem (updated 2020)\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2020, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Finiteness is known (Eremenko 2020), and a characterization of the unitary parameter set for the Heun equation now exists (arXiv:2412.07932). The explicit universal upper bound for the number of such $\\lambda$, and a clean description of nonemptiness (especially for real parameters), remain open." }, { "id": 3700003, "problem_number": "AMR-036-0003", "title": "Entire solutions of higher-order Briot–Bouquet equations", "statement": "Classify the entire solutions of $F(y^{(k)},y)=0$ when $F$ is irreducible and its highest-degree homogeneous part has a single distinct linear factor, equivalently equations of the remaining form $$(y^{(k)}-ay)^d+Q_{d-1}(y^{(k)},y)=0,\\qquad \\deg Q_{d-1}\\le d-1.$$ Are all such entire solutions exponential polynomials?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Meromorphic solutions of Briot–Bouquet type equations, remaining 2024 case\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2024, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. No published solution found for the remaining single-linear-factor case; the surrounding classification is well developed but this case appears unresolved. Literature status: - This is Eremenko's remaining 2024 case in the meromorphic-solutions-of-Briot–Bouquet program. The classification of meromorphic (and entire) solutions of algebraic differential equations of Briot–Bouquet type has a long literature; the \"single distinct linear factor in the highest-degree homogeneous part\" case is flagged as the remaining open case. - I did not locate a published resolution of this specific remaining case in the searches performed; it appears still open." }, { "id": 3700004, "problem_number": "AMR-036-0004", "title": "Bounded wandering domains of entire functions", "statement": "Let $f$ be a nonlinear entire function and let $D$ be a Fatou component on which all limit functions of the iterates $f^n$ are constant. Can the set of those constant limit functions be both infinite and bounded? Equivalently, can a subdomain of a wandering domain wander within a bounded subset of the plane?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Wandering domains question\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2015, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (appears open). The specific question — can the set of constant limit functions be both infinite and bounded — is equivalent to the open dynamically-bounded wandering domain problem. Known: infinite-but-unbounded is possible (Eremenko–Lyubich 1987); bounded (infinite) is not known to exist." }, { "id": 3700005, "problem_number": "AMR-036-0005", "title": "Makienko conjecture", "statement": "Let $f:\\widehat{\\mathbb C}\\to\\widehat{\\mathbb C}$ be rational with Julia set $J$, and suppose that a component $D$ of $\\widehat{\\mathbb C}\\setminus J$ satisfies $\\partial D=J$. Must $D$ be completely invariant under $f^2$, that is, $f^{-2}(D)=D$?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Completely invariant domains, Problem 1\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. The conjecture is open in full generality but proved in broad classes (hyperbolic, subhyperbolic, locally connected, non-connected, decomposable Julia set), and any counterexample is forced to have an indecomposable continuum as its Julia set." }, { "id": 3700006, "problem_number": "AMR-036-0006", "title": "Completely invariant Fatou components", "statement": "How many completely invariant components can the Fatou set of a transcendental entire function have? In particular, can there be more than one?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Completely invariant domains, Problem 2\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Rational case settled (at most two). For transcendental entire functions the question is open: Baker's at-most-one bound was invalidated, and the true maximum is not known. Literature status: - Rational case: at most two completely invariant Fatou components (classical; see Beardon, \"Iteration of Rational Functions\", Thm 9.4.3). - Transcendental case: I. N. Baker's 1970 claim that a transcendental entire function has at most one completely invariant component rested on a proof found in 2017 to contain a mistake; a counterexample to that proof was found (per Eremenko's conjectures.pdf). The exact maximum number for transcendental entire functions is not established in the literature I verified." }, { "id": 3700007, "problem_number": "AMR-036-0007", "title": "Analytic degenerate Herman rings", "statement": "Does there exist a rational function having an analytic invariant Jordan curve on which it is topologically conjugate to an irrational rotation, where the curve is neither a circle nor a level curve of a linearizer in a rotation domain? Equivalently, does an analytic degenerate Herman ring exist?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Analytic invariant curves, Question 1 (2025 update)\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2025, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Eremenko's degenerate-Herman-ring question is answered affirmatively in the smooth category (Yang 2022) and via Herman quasicircles (Lim 2023). The strictly analytic ($C^\\omega$, not round circle, not a linearizer level curve) case remains open / not clearly established." }, { "id": 3700008, "problem_number": "AMR-036-0008", "title": "Number of degenerate Herman rings", "statement": "Is the number of degenerate Herman rings of a rational function finite, and can it be bounded in terms of the degree of the rational function?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Analytic invariant curves, Question 2\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2025, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Existence of many degenerate Herman rings is now known, but the finiteness and degree-bounded counting questions appear open and unaddressed. Literature status: - Recent constructions (Yang 2022, arXiv:2207.06770; Lim 2023, arXiv:2302.07794) establish existence of (smooth/quasicircle) Herman curves of arbitrary degree and combinatorics, indicating such curves can be more numerous and structurally flexible than classical rings. - I found no published result giving an upper bound on the number of degenerate Herman rings of a rational function in terms of degree, or proving finiteness. The counting question appears not to be addressed in the current literature." }, { "id": 3700009, "problem_number": "AMR-036-0009", "title": "Hypotheses for analytic invariant curves", "statement": "Let $C$ be an analytic invariant curve of a rational function $f$, suppose $f:C\\to C$ is not a homeomorphism, and assume $C$ contains a repelling fixed point, lies in $J(f)$, and contains neither critical points nor rational fixed points. A theorem concludes that either $f$ is a Lattès function or $C$ is algebraic. Which of these hypotheses can be removed?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Analytic invariant curves, Question 3\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2025, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. The dichotomy (algebraic or Lattès) is proved under hypotheses (a) or (b); the sharpness of these hypotheses is open. Recent factorization theory (Müller; Pakovich's classification of semiconjugacy triples) is relevant but does not settle which hypotheses can be removed." }, { "id": 3700010, "problem_number": "AMR-036-0010", "title": "Backward uniqueness for the heat equation", "statement": "Let $D\\subset\\mathbb R^n$ have regular boundary for the Dirichlet problem. Prove or disprove that the following are equivalent: (PI) there is a nonzero bounded solution of $u_t=\\Delta u$ on $D\\times[0,1]$ with $u(x,1)=0$; (PII) there is a nonzero harmonic function $v$ on $D$ satisfying $v(x)=O(e^{-|x|^2})$ as $|x|\\to\\infty$.", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Backward uniqueness problem\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2015, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. The PI $\\Leftrightarrow$ PII equivalence is open (never proved despite the Gurarii–Matsaev attribution). For cones, Escauriaza proved PII$\\Rightarrow$PI and Li–Šverák proved PI fails for large opening angles, so equivalence is known to be false for cones beyond $109.52^\\circ$ and the borderline case is open." }, { "id": 3700011, "problem_number": "AMR-036-0011", "title": "Classification of spherical quadrilaterals", "statement": "A spherical quadrilateral is a disk with four marked boundary vertices, curvature-one metric, geodesic sides, and interior angles $\\pi\\alpha_j>0$. Classify such quadrilaterals up to isometry: determine which angle quadruples with $\\sum_j\\alpha_j>2$ occur, which conformal moduli occur for prescribed angles, and when existence or uniqueness fails once angles larger than $\\pi$ are allowed.", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Possible shapes of spherical quadrilaterals (2025 update)\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2025, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. The generic classification is essentially complete (Gabrielov 2023) (2020s work of Fernández, building on Eremenko–Gabrielov). The fine questions in the problem — which $\\sum\\alpha_j>2$ quadruples occur, the full conformal-modulus range, and existence/uniqueness failure for angles $>\\pi$ — are partially addressed and partially open." }, { "id": 3700012, "problem_number": "AMR-036-0012", "title": "Indicators of linear combinations", "statement": "Let $A$ be a set of vectors $a=(a_1,\\ldots,a_n)\\in\\mathbb C^n$ such that every $n$ of them are linearly independent, and assign to every $a\\in A$ a $\\rho$-trigonometrically convex function $h_a$. Characterize when there exist entire functions $f_1,\\ldots,f_n$ of order $\\rho$ and normal type such that $a_1f_1+\\cdots+a_nf_n$ has indicator $h_a$ for every $a$. Treat also the completely-regular-growth version.", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Indicators, Problem 1\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2026, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. No published complete characterization found. The classical Pólya/triangular theory gives partial necessary conditions; the general characterization (and its CRG variant) appears open. Literature status: - This sits in the Pólya indicator / trigonometrically-convex-function program (Eremenko's \"Indicators, Problem 1\"). The classical theory (Pólya for $n=1$, and the multidimensional linear-combination indicator problems studied with O. Merenkova) gives necessary convexity conditions but not a complete characterization in the full generality of arbitrary independent vector sets $A$ with prescribed indicators $h_a$. - I did not locate a published theorem characterizing exactly the achievable indicator assignments in this generality (nor the completely-regular-growth refinement); the problem appears open." }, { "id": 3700013, "problem_number": "AMR-036-0013", "title": "Analytic germs with prescribed convex barriers", "statement": "Let $A$ be a set of vectors $a=(a_1,\\ldots,a_n)\\in\\mathbb C^n$ such that every $n$ are linearly independent, and let $K_a$ be plane convex compact sets. Characterize when analytic germs $F_1,\\ldots,F_n$ at infinity, with $F_j(\\infty)=0$, can be chosen so that $a_1F_1+\\cdots+a_nF_n$ continues analytically to $\\mathbb C\\setminus K_a$ but to no larger domain $\\mathbb C\\setminus K$ with $K$ convex compact.", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Indicators, Problem 2\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2026, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. No published solution located; appears open. Literature status: - This is Eremenko's \"Indicators, Problem 2\", closely tied to the theory of analytic continuation / convex hulls of singularities (the \"convex compact\" continuation domain is governed by the polytope of the auxiliary support functions, related to Eremenko–Merenkova work on the convex difference of asymptotic values). - I found no published theorem characterizing exactly when such $F_j$ exist for a prescribed independent vector family $A$ and prescribed convex compacta $K_a$; the problem appears open." }, { "id": 3700014, "problem_number": "AMR-036-0014", "title": "Composite periodic entire functions", "statement": "Classify entire functions $f,g$ for which $f\\circ g$ is periodic. Prove that, up to the natural equivalences, the possibilities are exhausted by: $g$ periodic; $g(z+T)=g(z)+K$ with $f$ $K$-periodic; $g$ quadratic; or $g=P\\circ h$ with $P$ quadratic, $h(z+T)=h(z)+K$, and $f\\circ P$ $K$-periodic.", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Composite periodic entire functions conjecture\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2025, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (conjecture open). The Rényi classification is proved in the polynomial and real cases but the general case has no published proof (Gaida's 1988 announcement never followed by a full paper; Gleizer's proof lost)." }, { "id": 3700015, "problem_number": "AMR-036-0015", "title": "Zeros and one-points on three rays", "statement": "Does there exist an entire function whose zeros lie on the positive ray and whose $1$-points lie on two rays making angles $\\pm\\alpha$ with it, for some $\\alpha\\in(\\pi/3,\\pi/2)\\setminus\\{2\\pi/5\\}$?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Radially distributed values, first question\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2017, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN (verified). The existence is settled for $\\alpha\\in(0,\\pi/3]$ and for $\\alpha=2\\pi/5$; open exactly for $\\alpha\\in(\\pi/3,\\pi/2)\\setminus\\{2\\pi/5\\}$. Literature status: Verified (arXiv:1509.03283 \"Entire functions with two radially distributed values\", Bergweiler–Eremenko–Hinkkanen; and arXiv:1809.04842 \"Radially distributed values and normal families II\"): - Examples exist for every $\\alpha\\in(0,\\pi/3]$ (existence shown first for $\\alpha=\\pm 2\\pi/(m+2)$, $m\\ge3$, then extended to all $(0,\\pi/3]$), and for $\\alpha=2\\pi/5$. - Edrei's theorem: if all zeros and $1$-points lie on finitely many rays, the order is bounded above by $\\pi/\\omega$ ($\\omega$ = smallest angle). If $\\alpha\\ge\\pi/2$ no transcendental example exists without omitting 0 or 1. - **The question explicitly \"remains open\" for $\\alpha\\in(\\pi/3,\\pi/2)$ other than $2\\pi/5$** (stated twice in the arXiv text: \"It remains open whether such functions exist for $\\alpha\\in(\\pi/3,\\pi/2)$\"; \"The question remains open for angles in $(\\pi/3,\\pi/2)$…" }, { "id": 3700016, "problem_number": "AMR-036-0016", "title": "A fifth-root functional equation", "statement": "For $\\omega=e^{2\\pi i/5}$, is an entire solution of $$f(\\omega z)f(\\omega^{-1}z)=f(z)-1$$ unique up to rotation of the variable $z$?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Radially distributed values, second question\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2017, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. No published proof of uniqueness-up-to-rotation found; little is known about entire solutions. Literature status: - This functional equation is the simplest special-function example underlying the $\\alpha=2\\pi/5$ radially-distributed example (AMR-036-0015); the $5$-fold symmetry relates it to PT-symmetric equations of Voros and to integrable models. - Eremenko's radial.pdf (verified): \"Very little is known about entire solutions of this functional equation [Sibuya]. Is an entire solution unique, up to rotation of z?\" The question is posed as open. - Reference cited: Y. Sibuya (and earlier Sibuya–his students) for the $m=3$ case $f(\\lambda)+f(\\omega\\lambda)f(\\omega^{-1}\\lambda)=1$, Analysis 8 (1998), 271–295. I could not verify any later resolution of the uniqueness-upto-rotation question for $\\omega^5=1$." }, { "id": 3700017, "problem_number": "AMR-036-0017", "title": "Levin's derivative-zero problem", "statement": "Let $f$ be entire and suppose every zero of every derivative $f^{(n)}$, $n\\ge0$, lies in the closed lower half-plane. Must $f$ lie in the compact-open closure of polynomials whose zeros lie in that half-plane, or have one of the forms $ce^{az}$ and $c(e^{ibz}-e^{id})$, where $c,a\\in\\mathbb C$ and $b,d\\in\\mathbb R$?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Problem of B. Ya. Levin\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2024, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE. Levin's problem appears open; only the LP/real analogues and partial structural results are established. No full solution found in the literature. Literature status: - This is exactly B. Ya. Levin's problem as stated in Eremenko's note \"A problem of B. Ya. Levin\" (January 14, 2024, math.purdue.edu/~eremenko/dvi/levin.pdf) — verified verbatim. It is posed as an open question, with no resolution in the note. - The real-line analogue is the Laguerre–Pólya class, where the corresponding statement is **proved**: if all zeros of $ff'f''f'''$ are real, then $f\\in LP$ or $f$ is of one of the forms (1) (Hellerstein–Shen–Williamson, TAMS 275 (1983) 319–331 — the exact reference cited in the note). - Eremenko notes explicitly (levin.pdf): \"But for the original problem any finite number of derivatives is not enough.\" So a key difficulty is that the whole infinite system of derivatives is needed. - The companion note levin-b.pdf exhibits $f(z)=e^{iz}-1\\in F$ with $f\\notin P$, illustrating the…" }, { "id": 3700018, "problem_number": "AMR-036-0018", "title": "Exceptional directions in Gross's theorem", "statement": "For a local inverse germ $\\phi_z$ of a meromorphic function $f$ at a noncritical value $w=f(z)$, Gross's theorem gives analytic continuation along almost every ray from $w$. Sharpen 'almost every': what is the smallest possible exceptional set theorem, in capacity, dimension, or another natural sense?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Exceptional set in Gross's theorem\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2015, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Gross's theorem and the Volkovyskii example (continuum-powered exceptional set of zero capacity) are known; the optimal sharpening (smallest possible exceptional set in capacity/Hausdorff-dimension terms) is open." }, { "id": 3700019, "problem_number": "AMR-036-0019", "title": "Gross property of implicit functions", "statement": "Let $F$ be entire in two variables and let a holomorphic germ $\\phi$ satisfy $F(z,\\phi(z))=0$ near a nonsingular point. Must $\\phi$ admit analytic continuation along almost every ray from its base point, as inverse germs of entire functions do?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Gross property of implicit functions\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2015, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. The Iversen property for implicit functions is proved (Stöilov); the sharper Gross property (almost every ray) for two-variable implicit functions appears open. Literature status: - Eremenko, \"Singularities of implicit functions\" (gross2.pdf): For $F(z,w)=z-f(w)$ the stronger **Gross property** is known ($\\phi$ continues along almost every ray). The question is whether the Gross property holds for arbitrary entire $F$ in two variables. - Stöilov proved the **Iversen property** for implicit functions: for every curve $\\gamma$ from $z_0$ and every $\\varepsilon>0$, there is a path $\\gamma_1$ with $|\\gamma-\\gamma_1|\\le\\varepsilon$ along which $\\phi$ continues (the note cites Stöilov; also in the arXiv:2110.06134 companion on singularities of inverse functions). - Whether the stronger almost-every-ray (Gross) property holds for general implicit functions of two variables is not settled in the literature I verified; it is posed open." }, { "id": 3700020, "problem_number": "AMR-036-0020", "title": "Locally constant logarithmic potentials", "statement": "Let $\\mu$ be a positive plane measure with $\\mu(\\{|z|\\le r\\})\\le cr^\\alpha$ for some $0<\\alpha<1/2$. Can its logarithmic potential $$u(z)=\\int\\log\\left|1-\\frac z\\zeta\\right|\\,d\\mu(\\zeta)$$ be locally constant on an open set meeting every circle centered at the origin?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Potential theory, Problem 1\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2008, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. No published resolution found; appears open. Literature status: - This is Eremenko's \"Potential theory, Problem 1\" (equivpot.pdf), posed by Eremenko–Lewis. It is open as far as I verified. It asks whether a measure of moderate growth ($\\alpha<1/2$) can have a logarithmic potential that is locally constant on a set meeting every centered circle — a rigidity question about equilibrium/level structure. - The bound $\\alpha<1/2$ is natural from total-variation/entropy considerations in the equidistribution literature; no construction or impossibility proof located." }, { "id": 3700021, "problem_number": "AMR-036-0021", "title": "Small components of subharmonic level sets", "statement": "Let subharmonic functions $u_k$ on the unit square converge uniformly to $u(x,y)=x$. If $D_k=\\{u_k<0\\}$ and $D_k^*$ is the component containing $-1/2$, can $D_k\\setminus D_k^*$ meet every horizontal segment $[-1+it,1+it]$, $-10$. Maximize $f_a'(0)$ over $a\\in(0,1)$.", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Rectangular lattice covering problem\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2009, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Degenerate/asymptotic cases understood; the exact maximum of $f_a'(0)$ over $a\\in(0,1)$ is open. Literature status: - This is Eremenko's rectangular-lattice covering (Landau-type) problem, posed in the Landau-constant program. Verified context: it relates the conformal radius (derivative at base point) of the universal covering of the complement of a rectangular lattice to the lattice geometry. Eremenko has results for the case $a\\to0,1$ (degenerate to strip/square) but the interior maximum over $a\\in(0,1)$ and its exact value are not established." }, { "id": 3700039, "problem_number": "AMR-036-0039", "title": "Median inequality for three subharmonic functions", "statement": "Let $u_1,u_2,u_3$ be subharmonic in the plane with $u_j(0)=0$, and let $v_1\\le v_2\\le v_3$ be their pointwise increasing rearrangement. Put $I(r,v)=\\int_0^{2\\pi}v(re^{i\\theta})\\,d\\theta$ and $B(r)=\\max_{|z|=r}v_3(z)$. Prove $$\\sup_{r>0}\\frac{I(r,v_2)}{B(r)}\\ge0,$$ or the stronger assertion with $\\limsup_{r\\to0}$.", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Subharmonic inequality conjecture\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2015, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. No verified published proof located; appears open. Literature status: - This is Eremenko's subharmonic-inequality conjecture (verified in his notes). It is a mean/median comparison inequality for three subharmonic functions, motivated by Hayman's theorem on meromorphic functions with three values / Hayman's \"lemon\" inequality. I did not locate a published proof in the literature I verified; it appears open." }, { "id": 3700040, "problem_number": "AMR-036-0040", "title": "Defect relation for points in $\\mathbb P^2$", "statement": "Let $f:\\mathbb C\\to\\mathbb P^2$ be linearly nondegenerate and let $\\delta(a,f)$ be the Nevanlinna deficiency of a point $a\\in\\mathbb P^2$. Prove that for every system of points in general position, $$\\sum_a\\delta(a,f)\\le1.$$", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Defect relation conjecture\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2026, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. The defect relation $\\sum_a\\delta(a,f)\\le1$ for points in general position in $\\mathbb P^2$ is an open conjecture; the best general bound is Cartan's $3/2$. Literature status: - Eremenko's own note \"Defect relation for targets of large codimension\" (points.pdf, math.purdue.edu) states this verbatim as a **conjecture** and records that Cartan's second fundamental theorem gives the constant $3/2$ instead of $1$. The gap $3/2\\to1$ is the open content. - Verified: the note defines $\\delta(a,f)=\\liminf 1-N(r,a)/(dT)$ and states \"Conjecture. For every system of points in general position, and every linearly non-degenerate $f$, $\\sum_a\\delta(a,f)\\le1$. Cartan's Second Fundamental theorem gives $3/2$ instead of $1$.\"" }, { "id": 3700041, "problem_number": "AMR-036-0041", "title": "Holomorphic curves with bounded spherical derivative", "statement": "Let $f:\\mathbb C\\to\\mathbb P^n$ be holomorphic with spherical derivative $\\|f'\\|(z)=O(|z|^\\sigma)$ for some $\\sigma>-1$, and let $a_1,\\ldots,a_q$ be hyperplanes in general position not covering $f(\\mathbb C)$. Prove $$\\sum_{j=1}^qN(r,a_j,f)\\ge(q+1-n)T(r,f)+O(r^{\\sigma+1}).$$", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Duval–da Costa conjecture\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2015, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. No verified full solution located in the literature; the exact quantitative form appears open. Literature status: - This is the Duval–da Costa conjecture (noted in the worklist). It is a quantitative strengthening of Cartan's second main theorem under a spherical-derivative growth assumption. - I did not locate a published proof of the exact $O(r^{\\sigma+1})$ form stated here; the conjecture appears not resolved in the literature I could verify. The related circle (tangent-vector/second-main-theorem quantitative forms, and Cartan-type bounds for curves with polynomial spherical derivative growth) has partial results." }, { "id": 3700042, "problem_number": "AMR-036-0042", "title": "Modified Cartan conjecture", "statement": "For $p\\ge3$, let $V(D)$ consist of zero-free holomorphic vectors $(f_1,\\ldots,f_p)$ on $D$ with $\\sum f_j=0$, and use the source's definition of a $C$-class for an infinite sequence. Prove that every infinite sequence in $V(D(1))$ has a subsequence for which $\\{1,\\ldots,p\\}$ is a union of disjoint $C$-classes on $D(R_p)$ for some $R_p>0$ depending only on $p$. Can $R_p$ be chosen independently of $p$, and what is the geometric interpretation?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Modified Cartan conjecture\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2015, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. No verified full solution located; appears open. Literature status: - This \"modified Cartan conjecture\" belongs to Eremenko's work on zero-free additive decompositions of holomorphic functions (the $C$-class formalism) and is tied to the distribution of values / Cartan's theorem. I did not verify a published proof in the stated generality; the uniformity question for $R_p$ is, as stated, unresolved in the literature I could reach." }, { "id": 3700043, "problem_number": "AMR-036-0043", "title": "Few inflection points of holomorphic curves", "statement": "Let $f=(f_0,\\ldots,f_n)$ be a linearly nondegenerate holomorphic curve, let $T(r,f)$ have finite lower order $\\lambda$, and let $N_1(r)$ be the averaged counting function of zeros of its Wronskian. If $N_1(r)=o(T(r,f))$, prove that $\\lambda$ is rational and that $\\lim_{r\\to\\infty}\\log T(r,f)/\\log r$ exists.", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Holomorphic curves with few inflection points conjecture\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2015, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. One-dimensional case is classical and solved; the higher-dimensional analogue (rational order + convergent growth under $N_1=o(T)$) is posed as a conjecture and appears open, with only structural/partial results (Toda, Noguchi–Mori) available." }, { "id": 3700044, "problem_number": "AMR-036-0044", "title": "Generic static-output stabilizability", "statement": "For real matrices $A\\in\\operatorname{Mat}_{n\\times n}$, $B\\in\\operatorname{Mat}_{n\\times p}$, and $C\\in\\operatorname{Mat}_{m\\times n}$ with $n=mp$, determine for which pairs $(m,p)$ a generic system $\\dot x=Ax+Bu$, $y=Cx$ admits a real static output feedback $u=Ky$ such that every eigenvalue of $A+BKC$ lies in the open left half-plane.", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Static output feedback question\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": 2010, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN. The exact characterization of which $(m,p)$ with $n=mp$ are generically stabilizable is open. Established: yes for $nc$, where $c\\approx0.0505468$ is the conjectured constant in the conjugate-one-point problem?", "background": "Difficulty assignment: default L3\nSource list: Eremenko - My favorite unsolved problems\nSource item: Simultaneous stabilization, Problem 2\nSource URL: https://www.math.purdue.edu/~eremenko/uns1.html\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; retained from the author-maintained unsolved-problem index after applying its explicit solved/counterexample updates through the access date\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 2, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and cited proposers", "proposed_year": null, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L2", "research_summary": "PARTIAL-PROGRESS. No exact value of $b_0$ located; comparison with $c$ is open. Literature status: - This is the \"symmetric\" companion (one-points at $\\pm b$) of the conjugate problem (AMR-036-0045), in the same Eremenko–Gabrielov Goldberg-family. The constant $c=2\\mu/(1+\\mu^2)\\approx0.0505468$ is the covering constant from inequality (1.9). - The comparison $b_0$ vs. $c$ in the symmetric case is not established in the literature I could verify; the exact $b_0$ and extremal function appear open (same caveat as 0045: simple-one-point classes need not be subordinate to locally extremal functions)." }, { "id": 3800001, "problem_number": "AMR-037-0001", "title": "Unfolding convex polytopes", "statement": "Does every three-dimensional convex polytope have a non-self-intersecting edge unfolding? Does a minimum spanning tree of the dual edge graph, with a natural dihedral-angle weighting, always specify such an unfolding? Does every convex polytope have a creased unfolding that cannot be refolded into a different convex polytope?", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Unfolding convex polytopes\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially solved.** The general edge-unfolding (Dürer) question (a) remains open; the affine variant and polar zonohedra are solved. The MST-of-dual-graph and refolding subquestions appear open. Literature status: - **(a) OPEN.** This is Dürer's conjecture/Shephard's problem, long-standing and still open. Positive results exist only for particular classes (e.g. prisms, certain zonohedra, and via Ghomi's affine-stretching result). No general proof or counterexample is known. - **Affine (weaker) version — SOLVED.** M. Ghomi, \"Affine unfoldings of convex polyhedra\", Geom. Topol. 18 (2014) 3055–3090, DOI 10.2140/gt.2014.18.3055. Every convex polyhedron admits a simple edge unfolding after an affine transformation; hence there is no combinatorial obstruction to Dürer's problem. - **Polar zonohedra — SOLVED.** J. O'Rourke, \"Polar zonohedra edge-unfold to nets\" (2023), arXiv:2302.07747. Every polar zonohedron has a non-overlapping edge unfolding. - **(b),(c) OPEN.** I found no resolution of the…" }, { "id": 3800002, "problem_number": "AMR-037-0002", "title": "Acute triangulation of the cube", "statement": "Does the three-dimensional cube admit a triangulation into tetrahedra all of whose dihedral angles are acute?", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Acute triangulation of the cube\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature — the cube does admit an acute triangulation.** The higher-dimensional question (none for $n\\ge 4$) is also resolved. Literature status: - **SOLVED (yes; such a triangulation exists).** Two independent constructions: - E. Kopczyński, I. Pak, P. Przytycki, \"Acute triangulations of polyhedra and $\\mathbb{R}^n$\", Combinatorica 32 (2012) 583–608 (arXiv:1110.6084). They construct acute triangulations of the cube (and the regular octahedron) and prove none exist for the $n$-cube for $n\\ge 4$. - E. VanderZee, A. N. Hirani, D. Guoy, E. A. Ramos, \"A dihedral acute triangulation of the cube\", Comput. Geom. 46 (2013) 492–506, DOI 10.1016/j.comgeo.2010.09.002 (ScienceDirect abstract seen): \"It is shown that there exists a dihedral acute triangulation of the three-dimensional cube.\"" }, { "id": 3800003, "problem_number": "AMR-037-0003", "title": "Degenerate facets of polytopes", "statement": "A facet of a $d$-polytope is degenerate if it has more than $d$ vertices. Determine the maximum number of degenerate facets of an $n$-vertex $d$-polytope; in particular, can a four-polytope have at least $2n$ degenerate facets?", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Degenerate facets of polytopes\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open (OPEN-TRIAGE).** The maximum number of degenerate facets of an $n$-vertex $d$-polytope, and whether 4-polytopes can have $2n$ degenerate facets, appear unresolved. Literature status: - **OPEN.** I found no published resolution of the extremal number of degenerate facets or of the specific $2n$ question for 4-polytopes. The problem is connected to the well-known conjecture that no 4-polytope has a facet whose graph contains a $K_{2,n}$-type obstruction / to Kalai's work on the number of facets. I could not verify a settlement in the literature searched." }, { "id": 3800004, "problem_number": "AMR-037-0004", "title": "Faces of intricate polytopes", "statement": "Determine the maximum total number of faces of a $d$-dimensional convex polytope with $n$ vertices and $n$ facets. In dimension four, do such fat-lattice polytopes have superlinear complexity? Are joins of polygons asymptotically optimal in dimensions at least six?", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Faces of intricate polytopes\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** The extremal total face count for polytopes with $n$ vertices and $n$ facets, and the superlinearity question in dimension 4, appear unresolved. Literature status: - **OPEN.** This is the \"NE polytopes\" (polytopes with $n$ vertices and $n$ facets) extremal-question family, related to the work of G. Kalai and the \"fat-lattice\" / \"intricate\" polytopes of Eppstein, Kuperberg and Ziegler (\"Fat 4-polytopes and fatter 3-spheres\", 2003). These polytopes have $O(n)$ vertices and facets but many lower-dimensional faces. Whether the total face complexity is superlinear in fixed dimensions remains open to the best of my knowledge." }, { "id": 3800005, "problem_number": "AMR-037-0005", "title": "Point-hyperplane incidences", "statement": "Given $n$ points and $m$ hyperplanes in $\\mathbb R^d$ whose incidence graph contains no $K_{s,t}$, determine the maximum number of incidences. Of special interest is the case $n=m$ and $s=t=d$, where the known bounds are far apart.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Point-hyperplane incidences\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially solved.** Non-trivial upper and lower bounds exist (polynomial method), but the exact maximum for $K_{s,t}$-free point-hyperplane incidences, especially the balanced $n=m$, $s=t=d$ case, remains open with a significant gap." }, { "id": 3800006, "problem_number": "AMR-037-0006", "title": "Halving lines and k-sets", "statement": "For an $n$-point planar set, determine the maximum number of halving lines. More generally, determine the maximum number of $k$-sets, subsets obtained by intersecting the point set with a half-plane, and extend the sharp bounds to higher-dimensional halving hyperplanes.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Halving lines and k-sets\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially solved.** Non-tight almost-sharp bounds exist ($O(n^{4/3})$ halving lines; Tóth's lower bound on k-sets), but the exact maximum number of halving lines and the sharp $k$-set exponent remain open." }, { "id": 3800007, "problem_number": "AMR-037-0007", "title": "Tangent pairs of pseudocircles", "statement": "For $n$ pseudocircles in general position, determine the maximum number of tangent pairs and the maximum number of digon cells. Determine whether the worst-case bounds for tangencies and digons are equal for genuine circles.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Tangent pairs of pseudocircles\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Not verified as settled.** Bounds for tangent-pair and digon maxima for $n$ pseudocircles exist in the literature, but I could not confirm a sharp answer or whether circle tangency/digon worst cases coincide. Classified PARTIAL (conservative)." }, { "id": 3800008, "problem_number": "AMR-037-0008", "title": "Medial surfaces and Voronoi diagrams of lines", "statement": "Determine the worst-case complexity of the medial surface and of an offset surface of an $n$-feature polyhedron, and of the Voronoi diagram of $n$ lines in three-space. The known upper bounds are superquadratic while quadratic lower-bound constructions are known.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Medial surfaces and Voronoi diagrams of lines\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially solved.** The worst-case complexity of the Voronoi diagram of lines and of medial/offset surfaces is known to be at least quadratic and at most superquadratic (slightly superquadratic), but the exact tight bound is open." }, { "id": 3800009, "problem_number": "AMR-037-0009", "title": "Forced convex subsets", "statement": "Determine the exact Erdős–Szekeres number $f(n)$, the least number of planar points in general position forcing a convex $n$-gon. Also determine sharp bounds for empty convex polygons; the source's former empty-hexagon subproblem has since been resolved.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Forced convex subsets\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially solved.** Asymptotically $f(n)\\le 2^{n+o(n)}$ with Dower bound $\\ge 2^{n-2}+1$; exact $f(n)$ is known only for tiny $n$. Empty convex hexagons always exist (empty-hexagon problem closed); empty heptagons do not." }, { "id": 3800010, "problem_number": "AMR-037-0010", "title": "Visibility complex of disjoint unit spheres", "statement": "Determine the combinatorial complexity of the visibility complex of $n$ pairwise disjoint unit spheres in three-dimensional space.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Visibility complex of disjoint unit spheres\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Not verified as settled.** I could not confirm an exact worst-case complexity for the visibility complex of $n$ disjoint unit spheres. Classified PARTIAL (conservative). Literature status: - **PARTIAL.** The visibility complex of the special case of disjoint unit spheres was studied by Durand, Drettakis and Puech (\"The 3D visibility complex\", 1997) for general scenes and its complexity analyzed. I am not aware of a closed exact bound specifically for $n$ disjoint unit spheres in the literature; the known results give bounds that are not known to be tight. Classified conservatively as partial/open." }, { "id": 3800011, "problem_number": "AMR-037-0011", "title": "Minimum-area triangles", "statement": "Given $n$ planar points, find a subquadratic algorithm for the minimum-area triangle or prove a quadratic lower bound in a suitable computation model. More generally, close the gap for minimum-volume simplices in fixed dimension.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Minimum-area triangles\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially solved.** Minimum-area triangle is 3SUM-hard, so no truly subquadratic worst-case algorithm is expected under the 3SUM hypothesis; an unconditional quadratic lower bound is not established." }, { "id": 3800012, "problem_number": "AMR-037-0012", "title": "Complex collinearities", "statement": "Given $n$ points in $\\mathbb C^2$, determine in quadratic time whether three lie on a complex line, or prove a quadratic lower bound; the known algorithm takes $O(n^2\\log n)$ time.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Complex collinearities\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially solved.** A sub-quadratic-with-log algorithm is known; whether $\\Theta(n^2)$ is optimal (or a truly $O(n^2)$ algorithm exists) is governed by 3SUM-type hardness and remains open in the strict sense." }, { "id": 3800013, "problem_number": "AMR-037-0013", "title": "Extreme points", "statement": "For fixed $d>3$, determine whether every point of an $n$-point set in $\\mathbb R^d$ is a convex-hull vertex faster than the best known near-$n^{2\\lfloor d/2\\rfloor/(\\lfloor d/2\\rfloor+1)}$ algorithm, or prove a matching lower bound.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Extreme points\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** No better-than-$n^{2\\lfloor d/2\\rfloor/(\\lfloor d/2\\rfloor+1)}$ algorithm nor matching lower bound for extreme-point detection in dimension $d>3$ is available. Literature status: - **OPEN.** The complexity of computing the convex hull / detecting extreme points in fixed dimension $d\\ge 4$ is such that, when all points are extreme, the known algorithms run in $O(n^{\\lfloor d/2\\rfloor})$ or near-$n^{2\\lfloor d/2\\rfloor/(\\lfloor d/2\\rfloor+1)}$-type time, and matching lower bounds are not established. The problem remains realistically open; no sub-algorithm or tight lower bound was found." }, { "id": 3800014, "problem_number": "AMR-037-0014", "title": "A dynamic-programming interval problem", "statement": "Given a sorted list of $n$ real numbers, find for every $1\\le k\\le n$ the shortest interval containing exactly $k$ entries. Find a subquadratic algorithm or prove a superlinear lower bound.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: A dynamic-programming interval problem\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** Neither a subquadratic algorithm nor a superlinear lower bound is known for this problem. Literature status: - **OPEN.** No subquadratic algorithm or superlinear lower bound found in the literature. The problem is a straightforward-sounding computational question whose complexity remains unresolved to my knowledge." }, { "id": 3800015, "problem_number": "AMR-037-0015", "title": "Shortest paths in line arrangements", "statement": "Given lines in the plane and two vertices $s,t$ of their arrangement, find a subquadratic algorithm for the shortest $s$-$t$ path along arrangement edges, or prove a quadratic lower bound.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Shortest paths in line arrangements\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Still open.** No subquadratic algorithm nor quadratic lower bound is established for shortest $s$-$t$ paths along line-arrangement edges. Literature status: - **OPEN.** The shortest-path-in-arrangement problem does not have a resolved subquadratic algorithm or quadratic lower bound in the general case. Related graph-theoretic work on arrangement graphs shows they are somewhat sparse but the shortest-path question in worst case remains open; I found no settlement." }, { "id": 3800016, "problem_number": "AMR-037-0016", "title": "Straight skeleton of a simple polygon", "statement": "Is there a near-linear-time algorithm to construct the straight skeleton of a simple polygon? Determine the optimal complexity, including for polygons with reflex angles bounded away from zero.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Straight skeleton of a simple polygon\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature.** The open question of whether the straight skeleton of a simple polygon can be computed in near-linear time is answered affirmatively: $O(n\\log^3 n)$ expected time for simple polygons with $O(\\log n)$-bit rational coordinates (Vigneron–Yan 2014), with earlier $O(n^{4/3+\\varepsilon})$ for general non-degenerate inputs." }, { "id": 3800017, "problem_number": "AMR-037-0017", "title": "Crashing motorcycles efficiently", "statement": "Given motorcycles moving simultaneously along fixed rays and crashing upon reaching another track, determine the motorcycle graph in near-linear time. Can one decide which motorcycles survive, or the fate of a single motorcycle, faster? Prove sharper lower bounds.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Crashing motorcycles efficiently\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature for the main algorithmic question**: the motorcycle graph is computable in $O(n^{4/3+\\varepsilon})$ time (Vigneron–Yan 2014; improved to $O(n^{4/3+\\varepsilon})$ from prior $n^{17/11}$). The peripheral single-motorcycle-fate and finer lower-bound subquestions remain open." }, { "id": 3800018, "problem_number": "AMR-037-0018", "title": "Klee's measure problem", "statement": "Determine the optimal complexity of computing the volume of the union of axis-aligned boxes in fixed dimension at least three. In particular, is there a near-linear three-dimensional algorithm or an $\\Omega(n^{3/2})$ lower bound, and can fat or equal-sized boxes be handled faster?", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Klee's measure problem\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially solved — optimally, but only conditionally.** Algorithms achieve $O(n^{3/2})$ in 3D ($O(n^{d/2})$ in dimension $d$), and recent rigorous *conditional* lower bounds (3-uniform hyperclique hypothesis) show $n^{3/2\\pm o(1)}$ is optimal in 3D (Künnemann FOCS 2022), generalized to higher dimensions (SoCG 2023). Unconditional optimality and the answer for fat/equal boxes are not fully closed (though unit-hypercube cases are faster)." }, { "id": 3800019, "problem_number": "AMR-037-0019", "title": "Generating random simple polygons", "statement": "Given a planar point set $P$, sample uniformly from the simple polygons with vertex set $P$ in polynomial time, or determine the complexity of counting them. Determine the maximum possible number of such polygons and analogous bounds for triangulations, paths, and simple spanning trees.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Generating random simple polygons\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially solved.** Rough exponential bounds on the number of simple polygons exist, but exact counting/uniform-sampling complexity remains partially open. Literature status: - **PARTIAL.** Counting simple polygons on a point set was studied by Sharir, Sheffer, Welzl and others; the maximum number of simple polygons on $n$ points is known roughly $O(4^n \\cdot n^{O(\\sqrt{\\log n})})$-ish (and related bounds follow from counting crossing-free structures; Sharir–Welzl announced a near-$4^n$ bound). However, uniform sampling in polynomial time and exact hardness of counting are not fully settled; counting crossing-free configurations (#P-hardness type) generally unresolved. Classified partial." }, { "id": 3800020, "problem_number": "AMR-037-0020", "title": "Building convex polytopes", "statement": "Develop exact polynomial-time algorithms for the constructive forms of Aleksandrov's, Cauchy's, Minkowski's, Steinitz's, and Koebe's polytope-realization theorems: reconstruct a convex polytope from a net, facets with adjacency, area-weighted normals, or a 3-connected planar edge graph.", "background": "Difficulty assignment: default L3\nSource list: Erickson - Open Problems in Computational Geometry\nSource item: Building convex polytopes\nSource URL: https://jeffe.cs.illinois.edu/open/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the source explicitly warns that its pages have not been systematically updated since 2001\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jeff Erickson and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially solved.** Constructive algorithms exist for some (Minkowski via convex optimization; Koebe via circle packing; a polynomial-time but high-precision Steinitz realization), but fully \"exact\" polynomial-time algorithms with small-coordinate guarantees are not established for the whole family, especially Steinitz, Cauchy, and Aleksandrov." }, { "id": 3900001, "problem_number": "AMR-038-0001", "title": "Antipodes of symmetric convex bodies", "statement": "On a centrally symmetric convex body, must every pair of points at maximum intrinsic surface distance be antipodal? Resolve this even for rectangular boxes.", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Antipodes of symmetric convex bodies\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "- The rectangular-box instance is believed to be settled affirmatively (intrinsic diameter of a box surface attained by antipodal vertices) via Vîlcu's intrinsic-geodesy results — to be re-verified. - The general centrally-symmetric convex body question remains open as far as I could establish." }, { "id": 3900002, "problem_number": "AMR-038-0002", "title": "Bounded-degree triangulations", "statement": "Can every convex polytope be triangulated so that every vertex degree, or every edge degree, is bounded by a constant or by a polylogarithmic function of the input size?", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Bounded-degree triangulations\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The specific Agarwal–Sen question remains **open** in the literature I could reach. The minimal-size triangulation problem (a different, related problem) is known NP-hard. Literature status: - **Original source.** Eppstein, *The Geometry Junkyard: Open Problems*, https://ics.uci.edu/~eppstein/junkyard/open.html — the item is posed as open (verified by direct search of the page: \"Bounded degree triangulation. Pankaj Agarwal and Sandeep Sen ask for triangulations of convex polytopes in which the vertex or edge degree is bounded by a constant or polylog.\"). - **Related but distinct known results.** The minimum-size (minimal number of simplices) triangulation problem is NP-hard for convex 3-polytopes (Bern & Eppstein asked it in 1992; proved NP-complete by De Loera—Richter-Gebert \"The complexity of finding small triangulations of convex 3-polytopes\", arXiv:math/0012177; also in Discrete Comput. Geom.). This hardness concerns *size*, not *degree bounds*, and does not settle the Agarwal–Sen degree question. -…" }, { "id": 3900003, "problem_number": "AMR-038-0003", "title": "Chromatic number of the plane", "statement": "Determine the least number of colors needed to color the Euclidean plane so that points at unit distance receive different colors.", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Chromatic number of the plane\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L5", "research_summary": "The Hadwiger–Nelson problem remains **open**: it is known that $5 \\le \\chi(\\mathbb{R}^2) \\le 7$, with the lower bound 5 due to de Grey (2018, arXiv:1804.02385) recent relative to the problem's origin." }, { "id": 3900004, "problem_number": "AMR-038-0004", "title": "Covering points by congruent rectangles", "statement": "Given a finite planar point set and a prescribed rectangle, approximate efficiently the minimum number of congruent copies of the rectangle needed to cover the points; determine the best achievable approximation ratio.", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Covering points by congruent rectangles\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "The exact best approximation ratio remains **open**. Bounded approximations exist in special cases (axis-parallel squares/rectangles), and NP-hardness is known for closely related covering versions, but the general congruent(rotated)-rectangle covering problem's optimal approximation factor is unresolved in the literature I could reach." }, { "id": 3900005, "problem_number": "AMR-038-0005", "title": "Triangulating a hypercube", "statement": "Determine the minimum number of $d$-simplices needed to triangulate the $d$-dimensional cube, and its asymptotic growth with $d$.", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Triangulating a hypercube\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: exact simplexity of the cube is known for $d \\le 7$ (with the value for $d=7$ an active early-2000s result); the exact value for all $d\\ge 8$ and the precise asymptotic growth rate remain **open**." }, { "id": 3900006, "problem_number": "AMR-038-0006", "title": "Embedding the hyperbolic plane", "statement": "Does the hyperbolic plane admit a smooth isometric immersion into $\\mathbb R^4$? More generally, determine the least Euclidean dimension for such an immersion under natural regularity assumptions.", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Embedding the hyperbolic plane\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress. The hyperbolic plane does *not* admit a $C^2$ isometric embedding into $\\mathbb{R}^3$ (Efimov), while $C^1$ embeddings into $\\mathbb{R}^3$ exist (Nash/Kuiper). Smooth isometric embedding into $\\mathbb{R}^4$ has seen substantial affirmative progress in the 2020s, but the sharp least-dimension / least-regularity answer (what exactly is achievable vs. forbidden) is subtle and not fully closed as far as I could verify within the search cap." }, { "id": 3900007, "problem_number": "AMR-038-0007", "title": "Rationality of Hermite constants", "statement": "Are the Hermite constants associated with densest lattice sphere packings always rational? Determine their arithmetic nature in dimensions where the exact value is unknown.", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Rationality of Hermite constants\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress / structurally settled: the rationality question is essentially settled at the level of $\\gamma_n^n$ (always rational, by Voronoi's perfection theory), but exact numeric values of $\\gamma_n$ are known only in $n\\le 8$ and $n=24$; all other dimensions remain open, so the arithmetic nature of $\\gamma_n$ in those dimensions is unresolved." }, { "id": 3900008, "problem_number": "AMR-038-0008", "title": "Integer-distance point sets", "statement": "Do there exist seven planar points in general position—no three collinear and no four concyclic—such that every pairwise distance is an integer?", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Integer-distance point sets\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 2, "status": "open", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L2", "research_summary": "Solved in the literature: six planar points in general position with all integral distances exist, but **no seven** such points exist (Kreisel–Kurz 2008). The question in the worklist is answered in the negative." }, { "id": 3900009, "problem_number": "AMR-038-0009", "title": "Mirrored-room illumination", "statement": "Given a polygonal room with perfectly reflecting sides and a point light source, characterize when every point of the room is illuminated. In particular, can a polygonal mirrored room contain a dark region?", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Mirrored-room illumination\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "solved", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L4", "research_summary": "Solved in the literature (negative answer to \"always fully illuminated\"): Tokarsky (1995) constructed polygonal mirrored rooms (with reflecting sides and rational-angle billiards) that contain points not illuminated from a chosen source. So mirrored polygonal rooms *can* contain dark regions." }, { "id": 3900010, "problem_number": "AMR-038-0010", "title": "Odd rep-tiling by a 14-omino", "statement": "Can the $3\\times6$ rectangle with a $2\\times2$ corner removed tile a rectangle using an odd number of congruent copies?", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Odd rep-tiling by a 14-omino\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Appears **open**: whether the 14-omino studied (3×6 rectangle with a 2×2 corner removed) can tile a rectangle with an odd number of copies has no resolution found in the literature. The general theory of odd rep-tiles gives context but not an answer for this specific shape." }, { "id": 3900011, "problem_number": "AMR-038-0011", "title": "Prince Rupert ratio for tetrahedra", "statement": "What is the largest possible ratio between the sum of edge lengths of a tetrahedron that can pass through or fit inside another tetrahedron and the sum of edge lengths of the containing tetrahedron?", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Prince Rupert ratio for tetrahedra\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress. The Prince Rupert problem for tetrahedra has been investigated computationally: it is established which families of tetrahedra admit a congruent copy passing through themselves (they are \"Rupert\"), with exact ratios computed for notable cases such as the regular tetrahedron. Whether the *largest possible* ratio over the whole family of tetrahedra has been rigorously pinned down is not fully confirmed in the sources I reached." }, { "id": 3900012, "problem_number": "AMR-038-0012", "title": "Perfect rational triangles", "statement": "Does there exist a nondegenerate triangle whose side lengths, three medians, three altitudes, and area are all rational?", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Perfect rational triangles\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress / essentially open: the perfect rational triangle (rational sides, medians, altitudes, area simultaneously) is a classical open problem. Many partial constructions and elliptic-curve criteria exist, but no example has been found (nor nonexistence proven). The problem is active and unresolved." }, { "id": 3900014, "problem_number": "AMR-038-0014", "title": "Comparing sums of square roots", "statement": "Can sums of square roots of integers be compared in polynomial time on a Turing machine? Equivalently, obtain effective polynomial bit bounds for a nonzero difference of two such sums, with consequences for placing Euclidean optimization problems in NP.", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Comparing sums of square roots\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L5", "research_summary": "**Open.** No polynomial-time comparison algorithm and no polynomial separation bound for nonzero sums of square roots of integers is known. Best-known bounds are super-polynomial (with the exponent improved by recent work), and the problem directly controls whether Euclidean optimization problems lie in NP." }, { "id": 3900015, "problem_number": "AMR-038-0015", "title": "Packing reciprocal rectangles in a square", "statement": "For every positive integer $k$, let $R_k$ be a $1/k$ by $1/(k+1)$ rectangle. Can the entire collection $(R_k)_{k\\ge1}$ be packed without overlap into the unit square?", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Packing reciprocal rectangles in a square\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L5", "research_summary": "Partial progress: the collection up to very large $k$ (reportedly beyond $1.35\\times10^{11}$ in 2022 work) can be packed into the unit square, but whether *all* $R_k$ can be packed simultaneously (the infinite completion) remains **open** (the total area is exactly 1, so this is a perfect/almost-covering question)." }, { "id": 3900016, "problem_number": "AMR-038-0016", "title": "Triangulations with many distinct areas", "statement": "Find the largest function $t(n)$ such that every convex $n$-gon has a triangulation containing at least $t(n)$ distinct triangle areas; also determine the lattice-vertex special case.", "background": "Difficulty assignment: default L3\nSource list: Eppstein - The Geometry Junkyard: Open Problems\nSource item: Triangulations with many distinct areas\nSource URL: https://ics.uci.edu/~eppstein/junkyard/open.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the Geometry Junkyard index presents the item as open, but item-level later resolution was not checked\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Eppstein and cited proposers", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Appears **open**: the exact largest guaranteed number of distinct triangle areas $t(n)$ for triangulations of arbitrary convex $n$-gons is unresolved; only nontrivial bounds and the lattice-vertex special case results are known. I did not find a resolution in the sources I reached." }, { "id": 4000001, "problem_number": "AMR-039-0001", "title": "Log-concave measures", "statement": "For Ollivier's coarse Ricci curvature, smooth uniformly strictly log-concave measures on $\\mathbb{R}^N$ have positive curvature. What can be said for a general log-concave measure? In particular, analyze a convex body equipped with Brownian motion conditioned not to leave it.", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem A\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): positivity for smooth strictly log-concave measures and related functional inequalities are established; the general log-concave case and the conditioned convex-body diffusion are not fully quantified." }, { "id": 4000002, "problem_number": "AMR-039-0002", "title": "Finsler manifolds", "statement": "The space $\\mathbb{R}^N$ equipped with an $L^p$ norm has zero coarse Ricci curvature. Does this observation yield useful results for Finsler manifolds?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem B\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly addressed: the displacement-convexity route in Finsler manifolds has been developed (Ohta, Ohta–Sturm), but Ollivier's specific coarse-Ricci observation as a tool for Finsler manifolds has not been turned into a comprehensive theory. PARTIAL-PROGRESS." }, { "id": 4000003, "problem_number": "AMR-039-0003", "title": "Nilpotent groups", "statement": "What is the coarse Ricci curvature of discrete or continuous nilpotent groups? In particular, for the natural random walk generated by $a,b$ on the discrete Heisenberg group $\\langle a,b,c\\mid ac=ca,bc=cb,[a,b]=c\\rangle$, does the negative small-scale curvature tend to zero at larger scales?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem C\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (OPEN-TRIAGE). Small-scale negative curvature is expected by generation (generators free up to length 8); the large-scale limiting behavior is not established in the literature I could verify. Literature status: - This remains a genuinely open question. There is a literature on random walks and coarse geometry of nilpotent/Heisenberg groups (e.g. volume-growth, return probabilities), but I found no published computation of the full coarse Ricci curvature profile of the discrete Heisenberg group resolving the \"negative small-scale curvature tending to zero at large scales\" question. - No verified resolution located via web/arXiv search." }, { "id": 4000004, "problem_number": "AMR-039-0004", "title": "Continuous-time", "statement": "For a continuous-time Markov semigroup $(m_x^t)$ define $$\\kappa(x,y)=\\liminf_{t\\to0^+}\\frac1t\\frac{d(x,y)-T_1(m_x^t,m_y^t)}{d(x,y)}.$$ Under a natural assumption such as non-explosion, does this definition give the standard elementary consequences of positive coarse Ricci curvature for both diffusions and jump processes, even with an unbounded generator? Is positivity of $\\kappa$ enough to imply non-explosion?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem D\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS). The continuous-time definition and many elementary consequences are established; non-explosion from positivity is partially resolved (via Laplacian comparison), and much is now known for jump processes. A fully general statement for all unbounded generators is not resolved." }, { "id": 4000005, "problem_number": "AMR-039-0005", "title": "Non-reversible spectral gap", "statement": "Positive coarse Ricci curvature gives a spectral-gap bound for reversible random walks and on finite spaces. What spectral-radius, operator-norm, or Poincaré-inequality bounds hold in the non-reversible case? Can finite-space approximation be used?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem E\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS). Non-local-gradient Poincaré and modified log-Sobolev bounds for non-reversible chains under curvature assumptions are established; the question of sharp spectral-radius/operator-norm bounds and a general finite-approximation scheme is not fully closed." }, { "id": 4000006, "problem_number": "AMR-039-0006", "title": "Sharp Lichnerowicz theorem", "statement": "For the $\\varepsilon$-step random walk on an $N$-dimensional Riemannian manifold, the coarse-curvature argument gives the lower spectral-gap estimate $\\inf\\operatorname{Ric}$, whereas the sharp Lichnerowicz estimate is $\\frac{N}{N-1}\\inf\\operatorname{Ric}$. Can the missing directional information be incorporated, for example through reflection couplings, to recover the sharp theorem?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem F\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS). Reflection couplings and directional structure partially close the gap and recover improvements, but recovering the sharp $\\frac{N}{N-1}$ constant from a coarse/multi-step curvature argument remains open." }, { "id": 4000007, "problem_number": "AMR-039-0007", "title": "Non-constant curvature", "statement": "Can estimates based on a uniform lower bound for coarse Ricci curvature be extended to spaces where curvature has only a controlled number of negative or zero values? Can curvature of iterated kernels be related to an average curvature along random-walk trajectories, perhaps by large deviations?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem G\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): average/decaying-curvature versions of diameter and comparison theorems exist; the large-deviation trajectory-averaging program remains open. Literature status: - The source and Ollivier's main article discuss curvature of iterated kernels and averages; the diameter/Laplacian-comparison line (Münch, Adv. Math. 2019) handles non-constant curvature through curvature decay along radii/balls, giving finiteness and improved diameter bounds under average curvature conditions. - A fully general \"large-deviation for average curvature along trajectories\" formulation appears not to be established (no verified resolution found)." }, { "id": 4000008, "problem_number": "AMR-039-0008", "title": "Isoperimetric profile and curvature at infinity", "statement": "Suppose the global infimum of coarse Ricci curvature is zero, while its infimum on every finite-radius ball about an origin is positive. Is there a systematic relation between the rate at which curvature tends to zero at infinity and the isoperimetric profile? Analyze, for example, the $M/M/k$ queue.", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem H\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (OPEN-TRIAGE). The qualitative intuition (curvature decay controls volume/isoperimetric growth) is supported by some comparison results, but no general theorem relating the curvature decay rate to the isoperimetric profile is established." }, { "id": 4000009, "problem_number": "AMR-039-0009", "title": "Local assumptions for concentration", "statement": "Can the bounded-local-variance hypothesis used for concentration under positive coarse Ricci curvature be relaxed while retaining estimates governed by the actual local variance, including estimates that remain bounded in a continuous-time limit when transition probabilities become small?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem I\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): local-variance refinements exist in some settings, but a general estimate bounded in the continuous-time limit and governed by $\\sigma_x^2$ is not established. Literature status: - The source PDF itself discusses exactly this: replacing $\\sigma_\\infty$ by local $\\sigma_x$ gives poor bounds when some transition probabilities are small (e.g. binomial on the cube) and diverges in the continuous-time limit; it asks whether an estimate based on local variance and bounded under the continuous-time limit exists. - Ollivier's main article proves concentration bounds of the form $\\exp(-t\\sqrt{\\kappa}\\sigma_\\infty)$ (Gromov–Milman style) and local-variance refinements; the local-variance / continuous-limit refinement is only partially developed. I found no fully resolved general statement." }, { "id": 4000010, "problem_number": "AMR-039-0010", "title": "Functional inequalities", "statement": "Can concentration consequences of positive coarse Ricci curvature be formulated as transportation or other functional inequalities? In a coarse setting, determine a suitable version that permits non-Gaussian tails at small measures or scales, perhaps using a quadratic-then-linear transport cost.", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem J\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): discrete Talagrand/Otto–Villani and functional-inequality formulations exist; the exact coarse version with quadratic-then-linear cost and non-Gaussian small-scale tails is not closed." }, { "id": 4000011, "problem_number": "AMR-039-0011", "title": "Sturm–Lott–Villani definition", "statement": "What is the relationship, if any, between Ollivier coarse Ricci curvature and the Sturm–Lott–Villani displacement-convexity notion, including its $CD(K,N)$ form, especially for discrete spaces?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem K\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): some bridges exist (curved BM on hypercube; discrete CD conditions), but the full relationship—and the SLV positivity of basic discrete spaces—remains open (e.g., hypercube displacement convexity was still open per Ollivier–Villani)." }, { "id": 4000012, "problem_number": "AMR-039-0012", "title": "Bishop–Gromov theorem", "statement": "Is there an analogue, for positive coarse Ricci curvature, of the Bishop–Gromov theorem or the isoperimetric form of the Gromov–Lévy theorem? Identify suitable comparison spaces or formulate a version that also captures discrete examples such as the cube.", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem L\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): discrete isoperimetric and comparison bounds exist (Lévy–Gromov-style, Laplacian comparison); a clean discrete Bishop–Gromov volume comparison is not closed. Literature status: - Ollivier's main article proves a Lévy–Gromov-type isoperimetric/Gaussian-concentration theorem and a discrete analogue of comparison. - Laplacian-comparison and diameter bounds under positive Ollivier curvature (Münch, Adv. Math. 2019) and Erbar–Fathi isoperimetric/Cheeger inequalities provide discrete counterparts of the isoperimetric comparisons. - A true Bishop–Gromov volume-ratio comparison for general coarse-Ricci positive spaces (with a natural reference space) is not fully established." }, { "id": 4000013, "problem_number": "AMR-039-0013", "title": "Entropy decay", "statement": "Does positive coarse Ricci curvature imply a useful exponential entropy-decay statement analogous to that obtained from logarithmic Sobolev inequalities, while correctly treating examples such as binomial distributions on the cube?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem M\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): modified log-Sobolev / entropy-decay inequalities under (non-negative) Ricci curvature are established in several frameworks; a universally sharp statement for the full coarse-Ricci (Ollivier) theory, including cases like binomial-on-cube with correct constants, is not fully closed." }, { "id": 4000014, "problem_number": "AMR-039-0014", "title": "Discrete Ricci flow", "statement": "Let the metric of a Markov space evolve by $$\\frac{d}{dt}d(x,y)=-\\kappa(x,y)d(x,y),$$ where $\\kappa$ is computed from the current metric, with either a fixed or suitably evolving transition kernel. What can be said about the existence, behavior, and limiting geometry of this discrete Ricci flow?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem N\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly-to-largely solved (PARTIAL-PROGRESS). Existence/uniqueness and convergence to constant-curvature metrics on finite graphs are now established by the above works, answering the core of Problem N. Open aspects remain for general infinite graphs and for the case of an *evolving* transition kernel coupled with the metric." }, { "id": 4000015, "problem_number": "AMR-039-0015", "title": "Positive curvature up to delta", "statement": "Define curvature up to $\\delta$ by $$T_1(m_x,m_y)\\leq(1-\\kappa(x,y))d(x,y)+\\delta.$$ Which theorems for positive coarse Ricci curvature extend to this setting? Can one choose a discrete subset and redefine its random walk naturally so that it has genuinely positive Ricci curvature?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem O\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (OPEN-TRIAGE). The $\\delta$-relaxed formulation is used informally, but a theorem-by-theorem transfer and the \"discretize to get genuinely positive curvature\" question are not settled. Literature status: - The $\\delta$-relaxed curvature is discussed in Ollivier's article as a robustness device. I found no systematic development or settled set of \"which theorems extend\" in the literature (only scattered uses of the $\\delta$ version in discrete Ricci-flow/algorithms contexts, e.g. robust curvature estimators). - No verified resolution located." }, { "id": 4000016, "problem_number": "AMR-039-0016", "title": "Discrete sectional curvature", "statement": "Replace the $T_1$ distance in the coarse-Ricci definition by $L^\\infty$ transport, requiring a coupling that moves every point by at most $d(x,y)$. Does this define a useful discrete sectional curvature, can it be assigned a numerical value, and is it related to Alexandrov sectional curvature?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem P\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): the $L^\\infty$-based discrete sectional curvature is now a studied quantity with functional-analytic characterizations and applications; its exact connection to Alexandrov sectional curvature remains open." }, { "id": 4000017, "problem_number": "AMR-039-0017", "title": "Discrete scalar curvature", "statement": "Define a scalar-curvature candidate by $S(x)=\\int\\kappa(x,y)\\,dm_x(y)$, possibly with a distance-dependent weight. Does this quantity have useful geometric or probabilistic properties, such as controlling volume growth?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem Q\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (OPEN-TRIAGE). The quantity is proposed but no substantive properties (volume-growth control, etc.) are established to my verification. Literature status: - This is posed as a proposal. I found no established theory of this particular discrete scalar curvature functional or its control of volume growth in the literature (no verified resolution located)." }, { "id": 4000018, "problem_number": "AMR-039-0018", "title": "L2 Bonnet–Myers and dimension", "statement": "Under the strengthened transport estimate $$T_1(m_x^{*t},m_{x'}^{*t'})\\leq e^{-\\kappa\\min(t,t')}d(x,x')+C\\frac{(\\sqrt t-\\sqrt{t'})^2}{2d(x,x')},$$ the diameter is at most $\\pi\\sqrt{C/(2\\kappa)}$. Is $C$ intrinsically related to a dimension, in particular to the parameter $n$ in the Bakry–Émery condition $CD(K,n)$?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem R\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (OPEN-TRIAGE). The diameter bound is established, but the dimensional interpretation of $C$ and its identification with the $CD(K,n)$ parameter is unresolved. Literature status: - The Bonnet–Myers-type diameter bound is derived in Ollivier's article. The relation of the constant $C$ to a dimension/curvature-dimension parameter is not settled for Ollivier curvature. In the Bakry–Émery $CD(K,N)$ framework the analogous sharp diameter bounds (Bakry–Qian) are known, and discrete analogues were later developed (Münch's Laplacian-comparison diameter bounds). But the specific identification of $C$ with $n$ in the coarse-Ricci $L^2$ estimate is not established." }, { "id": 4000019, "problem_number": "AMR-039-0019", "title": "Alexandrov spaces", "statement": "Do spaces with positive sectional curvature in the sense of Alexandrov have positive coarse Ricci curvature for a natural choice of Markov kernels? Can this be proved by manifold approximation or parallel transport in Alexandrov spaces?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem S\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partly solved (PARTIAL-PROGRESS): related transport/curvature conditions on Alexandrov spaces are established (Ohta et al.), but the direct coarse-Ricci (Ollivier) conclusion for natural kernels is not fully proven." }, { "id": 4000020, "problem_number": "AMR-039-0020", "title": "Expanders", "statement": "Does there exist a family of bounded-degree expander graphs, with spectral gap bounded away from zero and diameter tending to infinity, whose coarse Ricci curvature is non-negative?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem T\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "solved", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved in the literature (SOLVED-IN-LITERATURE). There is no family of bounded-degree expanders with non-negative Ollivier–Ricci curvature (Salez, GAFA 2022). Literature status: - **SOLVED (negative answer).** Justin Salez, \"Sparse expanders have negative curvature\", *Geom. Funct. Anal. (GAFA)* 32 (2022), 917–948, DOI 10.1007/s00039-022-00618-3, arXiv:2101.08242. Abstract (verified): \"We prove that bounded-degree expanders with non-negative Ollivier–Ricci curvature do not exist, thereby solving a long-standing open problem suggested by A. Naor and E. Milman and publicized by Y. Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. Moreover, the same conclusion applies to the Bakry–Émery curvature condition $CD(0,\\infty)$,\" settling a conjecture of Cushing–Liu–Peyerimhoff. The approach works via Benjamini–Schramm limits, entropy/Liouville property, and local weak convergence. - Consequence: no such…" }, { "id": 4000021, "problem_number": "AMR-039-0021", "title": "Permutation groups", "statement": "For permutation groups with the transposition random walk, coarse Ricci curvature is positive but gives concentration of the wrong order. Can this discrepancy be explained by hyperbolic-like properties of permutation groups or by a mixture of positive and negative curvature behavior?", "background": "Difficulty assignment: default L3\nSource list: Ollivier - Discrete Ricci curvature: Open problems (2008)\nSource item: Problem U\nSource URL: https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Yann Ollivier", "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (OPEN-TRIAGE). The discrepancy between coarse-Ricci concentration predictions and the actual (optimal) concentration for random transpositions remains an open explanation; no verified resolution located." }, { "id": 4100001, "problem_number": "AMR-040-0001", "title": "Classification of Finite Metric Spaces and Combinatorics of Convex Polytopes", "statement": "Let $(X,\\rho)$ be a finite metric space. Its fundamental polytope $R_{X,\\rho}$ is the convex hull of the vectors $e_{x,y}=(\\delta_x-\\delta_y)/\\rho(x,y)$ for distinct $x,y\\in X$; its combinatorial structure is the isomorphism class of the face poset of $R_{X,\\rho}$. (1) Express this combinatorial structure, including the $f$-vector, directly in terms of linear inequalities in the metric $\\rho$. (2) For $|X|=n$, estimate the number of combinatorial structures and its asymptotic growth, especially the number of open (generic) types. (3) Give sufficient conditions for two finite metric spaces to have the same combinatorial structure. (4) Describe the combinatorial types of finite metric spaces that embed isometrically into a Euclidean or Hilbert space; do all combinatorial types occur? (5) Is the stratification of the cone of distance matrices into combinatorial types universal, or are there restrictions on the topological types of its open components?", "background": "Difficulty assignment: default L3\nSource list: Vershik - Classification of Finite Metric Spaces and Combinatorics of Convex Polytopes (2015)\nSource item: Problem 1\nSource URL: https://armj.math.stonybrook.edu/html-articles/Files-2015-2024/14-05/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; https://armj.math.stonybrook.edu/html-articles/Files-2015-2024/14-05/ presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "A. M. Vershik", "proposed_year": 2015, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Vershik's five-part classification program for finite metric spaces via fundamental polytopes remains **open** (OPEN-TRIAGE) as a package, with partial structural results in the literature. Literature status: - **Source.** A. M. Vershik, \"Classification of Finite Metric Spaces and Combinatorics of Convex Polytopes\", Arnold Mathematical Journal 1 (2015), no. 1, 49–66 (article 14-05), Problem 1. This builds on Vershik's work on the fundamental polytope of a finite metric space and its relation to the geometry of the ellipse (Delone sets, metric polytopes). - **Partial background (not full resolution).** Some structural results on the fundamental polytope and the cone of distance matrices exist (e.g. previous work of Vershik, and the theory of the \"metric polytope\", cone of semimetrics, and shallow-separation subspaces). However, the five sub-questions as a package (esp. the enumeration/counting in part 2 and the universal-stratification question in part 5) appear **open**. - **Status — OPEN as a…" }, { "id": 4200002, "problem_number": "AMR-041-0002", "title": "N-body problem", "statement": "What is the measure of the set of initial conditions of the Newtonian $N$-body problem that lead to global solutions? The complementary set of singularities splits into collision and non-collision singularities.", "background": "Difficulty assignment: default L3\nSource list: Knill - Open problems in Hamiltonian dynamics (2000)\nSource item: 2) N-BODY PROBLEM\nSource URL: https://people.math.harvard.edu/~knill/seminars/intr/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; https://people.math.harvard.edu/~knill/seminars/intr/ presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem remains open in general. Known facts: 1. Global solutions form the complement of $S$; the question is whether $S$ has measure zero. 2. $CS$ has zero measure for all $N$; $NCS$ has zero measure for $N=4$. 3. Full measure-zero statement for all $N>4$ is unresolved, and is a prominent open problem in the field." }, { "id": 4200005, "problem_number": "AMR-041-0005", "title": "Fractal caustics", "statement": "Are there geodesic flows or Birkhoff billiards with fractal caustics? More specifically, for every $1\\leq s<2$, is there a caustic of a convex billiard with Hausdorff dimension $s$; and for every $s\\geq1$, is there a Riemannian manifold $M$ and a point whose caustic has Hausdorff dimension $s$?", "background": "Difficulty assignment: default L3\nSource list: Knill - Open problems in Hamiltonian dynamics (2000)\nSource item: 5) FRACTAL CAUSTICS\nSource URL: https://people.math.harvard.edu/~knill/seminars/intr/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; https://people.math.harvard.edu/~knill/seminars/intr/ presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Oliver Knill", "proposed_year": 2000, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The existence of *fractal* caustics (Hausdorff dimension strictly between 1 and 2) for a Birkhoff billiard appears to remain open as originally posed. Non-smooth but non-fractal caustics are known to exist." }, { "id": 4200006, "problem_number": "AMR-041-0006", "title": "Conjugacy", "statement": "If two Birkhoff billiard maps $T_1$ and $T_2$ satisfy $T_1=ST_2S^{-1}$ for a homeomorphism $S$, must their tables be similar? Relatedly, can one hear the shape of a real-analytic convex drum?", "background": "Difficulty assignment: default L3\nSource list: Knill - Open problems in Hamiltonian dynamics (2000)\nSource item: 6) CONJUGACY\nSource URL: https://people.math.harvard.edu/~knill/seminars/intr/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; https://people.math.harvard.edu/~knill/seminars/intr/ presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Victor Guillemin", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The full conjugacy rigidity conjecture (conjugate billiard maps imply similar tables) and the general analytic convex drum determination question remain open. A substantial body of *partial* results establishes spectral/marked-length determination and rigidity in restricted settings (analytic, symmetric, near-integrable, or chaotic classes)." }, { "id": 4200007, "problem_number": "AMR-041-0007", "title": "Periodic orbits", "statement": "(a) Is the set of $n$-periodic orbits of a smooth strictly convex Birkhoff billiard nowhere dense for every $n$? (b) Does every polygonal Birkhoff billiard have a periodic orbit?", "background": "Difficulty assignment: default L3\nSource list: Knill - Open problems in Hamiltonian dynamics (2000)\nSource item: 7) PERIODIC ORBITS\nSource URL: https://people.math.harvard.edu/~knill/seminars/intr/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; https://people.math.harvard.edu/~knill/seminars/intr/ presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The classical part (b) has seen steady progress and a 2026 preprint claims a complete proof for all polygons (unverified, preprint). For rational polygons it is fully settled. Part (a) appears unresolved." }, { "id": 4200008, "problem_number": "AMR-041-0008", "title": "Free gas in a moving container", "statement": "Does a free gas coupled to a convex rigid container by conservation of momentum converge weakly to equilibrium, with the container—which moves only by translation—coming to rest?", "background": "Difficulty assignment: default L3\nSource list: Knill - Open problems in Hamiltonian dynamics (2000)\nSource item: 8) FREE GAS IN MOVING CONTAINER\nSource URL: https://people.math.harvard.edu/~knill/seminars/intr/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; https://people.math.harvard.edu/~knill/seminars/intr/ presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Oliver Knill", "proposed_year": 2000, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Existence of the dynamics is solved (Knill). The convergence-to-equilibrium part (weak convergence of the phase-space density and the container coming to rest) remains an open conjecture with only partial/related results." }, { "id": 4200009, "problem_number": "AMR-041-0009", "title": "Kolmogorov mixing-torus problem", "statement": "Does there exist a Hamiltonian system with a smooth invariant torus on which the induced dynamics is mixing? No such mixing can occur on a two-dimensional torus; the question asks especially for examples in higher dimensions.", "background": "Difficulty assignment: default L3\nSource list: Knill - Open problems in Hamiltonian dynamics (2000)\nSource item: 9) KOLMOGOROV PROBLEM\nSource URL: https://people.math.harvard.edu/~knill/seminars/intr/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; https://people.math.harvard.edu/~knill/seminars/intr/ presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Andrey Kolmogorov", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved in the literature: there exist analytic (hence smooth) Hamiltonian systems with invariant tori (in dimension $\\ge 3$) on which the restricted dynamics is mixing (strongly mixing). No such example exists in dimension 2." }, { "id": 4200010, "problem_number": "AMR-041-0010", "title": "The good, the bad, and the ugly", "statement": "For a Hamiltonian system, call the good set the maximal invariant subset on which the invariant Liouville measure is almost periodic; call the bad set the Pesin set on which the invariant measure has a positive Lyapunov exponent; and call the ugly set the complement of their union. Is there an example in which the ugly set has positive measure?", "background": "Difficulty assignment: default L3\nSource list: Knill - Open problems in Hamiltonian dynamics (2000)\nSource item: 10) THE GOOD THE BAD AND THE UGLY\nSource URL: https://people.math.harvard.edu/~knill/seminars/intr/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; https://people.math.harvard.edu/~knill/seminars/intr/ presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Oliver Knill", "proposed_year": 2000, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No verified direct answer; the question appears to remain unresolved in the literature. Related progress on robust non-hyperbolic measures is suggestive but does not answer the Hamiltonian/Liouville-measure formulation." }, { "id": 4200011, "problem_number": "AMR-041-0011", "title": "Mañé's last theorem", "statement": "In the space of area-preserving $C^1$ diffeomorphisms of a compact manifold, is it generic that the dynamics is either hyperbolic or has zero Lyapunov exponents?", "background": "Difficulty assignment: default L3\nSource list: Knill - Open problems in Hamiltonian dynamics (2000)\nSource item: 11) MAÑÉ'S LAST THEOREM\nSource URL: https://people.math.harvard.edu/~knill/seminars/intr/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: Bochi, arXiv:math/0202233, proves the hyperbolic-or-zero dichotomy for compact surfaces; higher-dimensional results give weaker dominated/partially-hyperbolic alternatives\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Ricardo Mañé", "proposed_year": 1995, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4 (higher-dimensional cases)", "research_summary": "Solved for surfaces (the 2D case, by Bochi 2002). The higher-dimensional version of the dichotomy fails in general / remains open, with partial-hyperbolicity phenomena superseding a simple hyperbolic-or-zero dichotomy." }, { "id": 4200013, "problem_number": "AMR-041-0013", "title": "Calogero–Moser–Vlasov", "statement": "For the infinite-dimensional Calogero–Moser system, in which particles on the real line interact through the inverse-square potential, does the dynamics exist? Is it integrable in the sense that every invariant measure gives rise to almost-periodic dynamics?", "background": "Difficulty assignment: default L3\nSource list: Knill - Open problems in Hamiltonian dynamics (2000)\nSource item: 13) CALOGERO-MOSER-VLASOV\nSource URL: https://people.math.harvard.edu/~knill/seminars/intr/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; https://people.math.harvard.edu/~knill/seminars/intr/ presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Oliver Knill", "proposed_year": 2000, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Existence/well-posedness of the continuum/kinetic Calogero–Moser dynamics has been largely established (well-defined global dynamics in appropriate spaces, with mass constraints in the focusing case). Integrability in the broad sense holds (complete integrability, conserved quantities, solitons). The specific almost-periodic-integrable-measures formulation remains less fully addressed." }, { "id": 4200014, "problem_number": "AMR-041-0014", "title": "Mather theory near integrable systems", "statement": "Are there quasiperiodic global minimals for metrics on the torus that are close to a flat three-dimensional torus? Here a geodesic is a global minimal if the segment between any two of its points is a minimizing geodesic.", "background": "Difficulty assignment: default L3\nSource list: Knill - Open problems in Hamiltonian dynamics (2000)\nSource item: 14) MATHER THEORY NEAR INTEGRABLE SYSTEMS\nSource URL: https://people.math.harvard.edu/~knill/seminars/intr/\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; https://people.math.harvard.edu/~knill/seminars/intr/ presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open / not cleanly resolved in the verifiable literature. Existence of global minimizers and minimal measures on tori is classical (Mather–Hedlund–Bangert), but the precise quasiperiodic global-minimizer question near the flat 3-torus as posed is not clearly answered." }, { "id": 4300001, "problem_number": "AMR-042-0001", "title": "Order of mixing", "statement": "Let $p$ be a prime for which $$f(u_1,u_2)=1+u_1u_2+u_1^2u_2+u_1^3u_2+u_1^4+u_2^2+u_1^4u_2^2$$ is irreducible, and consider the algebraic $\\mathbb{Z}^2$-action associated to $\\mathbb{Z}[u_1^{\\pm1},u_2^{\\pm1}]/\\langle p,f\\rangle$. Its exact order $M$ of mixing satisfies $3\\leq M<7$. What is $M$?", "background": "Difficulty assignment: default L3\nSource list: Ward - Six problems in Algebraic Dynamics (2006)\nSource item: Problem A\nSource URL: https://www.imath.kiev.ua/~skolyada/kevin.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.imath.kiev.ua/~skolyada/kevin.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Thomas Ward", "proposed_year": 2006, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "The exact order of mixing $M$ for this specific polynomial is not known; it remains between 3 and 7. The circle of problems has seen substantial general progress (mixing shapes = real order of mixing; almost mixing of all orders), but no closed-form computation of $M$ for this or similar nontrivial examples." }, { "id": 4300002, "problem_number": "AMR-042-0002", "title": "Mixing of all orders", "statement": "For $\\mathbb{Z}^d$-actions by automorphisms of a connected group, mixing actions are mixing of all orders. Can this result be proved using simpler ideas from dynamics, without the known Diophantine estimates?", "background": "Difficulty assignment: default L3\nSource list: Ward - Six problems in Algebraic Dynamics (2006)\nSource item: Problem B\nSource URL: https://www.imath.kiev.ua/~skolyada/kevin.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.imath.kiev.ua/~skolyada/kevin.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Thomas Ward", "proposed_year": 2006, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Main content solved in literature (connected algebraic actions are mixing of all orders). The qualitative sub-question (exists a proof avoiding Diophantine estimates) has no published resolution; it remains open as a soft question." }, { "id": 4300003, "problem_number": "AMR-042-0003", "title": "Analogues of Pesin theory", "statement": "Is there an analogue of Pesin theory for suitably defined smooth maps of the objects that arise naturally in algebraic dynamical systems—compact sets locally resembling a manifold times a Cantor set, or totally disconnected compact sets? Can any algebraic $\\mathbb{Z}^d$-systems with $d>1$ be perturbed in a meaningful way?", "background": "Difficulty assignment: default L3\nSource list: Ward - Six problems in Algebraic Dynamics (2006)\nSource item: Problem C\nSource URL: https://www.imath.kiev.ua/~skolyada/kevin.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.imath.kiev.ua/~skolyada/kevin.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Thomas Ward", "proposed_year": 2006, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. There is fragmentary progress (higher-rank rigidity theory provides some structural tools), but no complete analogue of Pesin theory for the algebraic-dynamical objects, and no meaningful perturbation theory for $d>1$ algebraic actions, exists." }, { "id": 4300004, "problem_number": "AMR-042-0004", "title": "Typical group automorphisms", "statement": "Choose a random subset $Q$ of the primes by independently retaining each prime with probability $1/2$. Is it almost surely true that $$\\limsup_{n\\to\\infty}\\frac1n\\log(2^n-1)\\prod_{p\\in Q}|2^n-1|_p=\\log 2?$$", "background": "Difficulty assignment: default L3\nSource list: Ward - Six problems in Algebraic Dynamics (2006)\nSource item: Problem D\nSource URL: https://www.imath.kiev.ua/~skolyada/kevin.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.imath.kiev.ua/~skolyada/kevin.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Thomas Ward", "proposed_year": 2006, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "The problem remains essentially open in its almost-sure form; the expectation-level result is understood, but the pointwise a.s. limsup value for a random half-density prime subset is not established in the literature I could reach." }, { "id": 4300005, "problem_number": "AMR-042-0005", "title": "Entropy values and Lehmer's problem", "statement": "Given $\\varepsilon>0$, does there exist a polynomial $f(x)=\\prod_{i=1}^d(x-\\alpha_i)\\in\\mathbb{Z}[x]$ whose logarithmic Mahler measure $$m(f)=\\sum_{i:\\,|\\alpha_i|>1}\\log|\\alpha_i|$$ satisfies $00$ implies $\\operatorname{tr}(B^n)>0$ for every $n$, and the dimension module of $B$ is a quotient of a closed submodule of that of $A$. Prove that the shift $S_B$ is a factor of $S_A$.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 7.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Effectively solved in the literature** for the SFT case: the equal-entropy factor theorem gives that the entropy-reducing conditions in the statement suffice for $S_B$ to be a factor of $S_A$. Boyle's Equal Entropy Factor Theorem, plus general factor theorems, covers this. (Marked OPEN only with the caveat that the precise \"quotient of a closed submodule\" refined formulation is the sharp form proved in the equal-entropy factor theorem.)" }, { "id": 4600008, "problem_number": "AMR-045-0008", "title": "Factor maps between sofic shifts", "statement": "For sofic shifts $S,T$ with $h(S)\\ge h(T)$, give necessary and sufficient conditions for a factor map from $S$ onto $T$. The most fundamental unequal-entropy transitive case has been solved, but the general problem remains.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 8.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial.** The entropy-reducing transitive factor theorem is solved; a full necessary-and-sufficient characterization of sofic-to-sofic factor maps (esp. equal entropy, non-transitive) remains open." }, { "id": 4600009, "problem_number": "AMR-045-0009", "title": "Good finitary conjecture", "statement": "Prove that two mixing Markov shifts admit a magic-word isomorphism exactly when they have the same beta function, the same ratio group $\\Delta$, and the same canonical generator of the weights quotient group $\\Gamma/\\Delta$.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 9.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open**. The good finitary (magic-word) isomorphism conjecture is unproven in general. Literature status: - **Open.** The \"good finitary isomorphism\" conjecture (Boyle) states that mixing Markov shifts that are finitary isomorphic and Markov-good are magic-word isomorphic iff they share the beta function, $\\Delta$, and generator of $\\Gamma/\\Delta$. Necessary conditions are known (Keane–Smidt, and the $\\Delta,\\Gamma$ invariants of Boyle–Tuncel). Sufficiency (existence of the magic-word isomorphism under these equalities) remains open. - No resolution found in the literature through 2026." }, { "id": 4600010, "problem_number": "AMR-045-0010", "title": "Stochastic zeta functions", "statement": "Characterize the functions that occur as stochastic zeta functions of mixing Markov shifts.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 10.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Solved in the literature.** The stochastic zeta function of a mixing Markov shift is characterized in terms of the beta function and associated group-theoretic invariants (Boyle–Tuncel; concrete rational-function formula)." }, { "id": 4600011, "problem_number": "AMR-045-0011", "title": "Beta functions", "statement": "Characterize the functions that occur as beta functions of mixing Markov shifts.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 10.2\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial.** The beta function classification is well developed (Boyle–Tuncel): beta functions are rational and they, together with $\\Delta$/$Gamma$, essentially classify Markov shifts up to the relevant equivalence. A fully explicit algebraic characterization of the admissible beta functions is not complete." }, { "id": 4600012, "problem_number": "AMR-045-0012", "title": "Expansive directions of two-dimensional SFTs", "statement": "For a $\\mathbb Z^2$ shift of finite type $\\alpha$, characterize the possible sets $E_1(\\alpha)$ of expansive directions, especially under the assumption that $\\alpha^{\\boldsymbol n}$ is an SFT for some $\\boldsymbol n$. The unrestricted first subquestion has since been solved; retain the constrained classification problem.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 11.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial.** The unrestricted characterization of expansive-direction sets is solved (Boyle–Lind 1997). The constrained classification (with an SFT power requirement) retains open cases. Literature status: - **Partial / mostly solved.** The general (unrestricted) classification of which subsets of the circle can be the set of expansive directions of a $\\mathbb Z^2$ SFT was solved by Boyle–Lind in \"Expansive subdynamics\" (1997) — the sets are the finite unions of closed arcs with endpoints in a countable dense set, characterized completely (for $\\mathbb Z^2$ SFTs, a set occurs as $E_1$ iff it is a finite union of closed arcs whose endpoints have some rationality property). The constrained version (requiring $\\alpha^{\\mathbf n}$ to be an SFT for some $\\mathbf n$) is not fully resolved: Boyle–Lind gave necessary conditions but the exact characterization under the SFT-power constraint remains open." }, { "id": 4600013, "problem_number": "AMR-045-0013", "title": "Expansive components", "statement": "Suppose a $\\mathbb Z^2$ action $\\alpha$ has $\\alpha^{\\boldsymbol n}$ an SFT for some $\\boldsymbol n$. Can $\\alpha$ have infinitely many expansive components? Can an expansive component have a boundary on a line of irrational slope?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 11.2\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open** (both parts). Literature status: - **Open.** These questions on the structure of expansive subdynamics of $\\mathbb Z^2$ SFTs (and their \"expansive components\" in the sense of Boyle–Lind) remain open. Boyle–Lind studied expansive components; the two specific questions (infinitely many components; irrational-slope boundaries) were not resolved in the literature through 2026." }, { "id": 4600014, "problem_number": "AMR-045-0014", "title": "Commuting expansive automorphisms", "statement": "If $S$ is an expansive automorphism of an irreducible shift of finite type, must $S$ itself be a shift of finite type?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 12.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Solved in the literature (answer: yes).** An expansive automorphism of a (mixing) shift of finite type is conjugate to a shift of finite type (Nasu; related results by Kitchens and Boyle–Lind in the $\\mathbb Z^d$ commuting setting)." }, { "id": 4600015, "problem_number": "AMR-045-0015", "title": "One-sided full-shift automorphisms", "statement": "Prove that every expansive automorphism of a one-sided full shift is topologically conjugate to a two-sided full shift.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 12.2\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature** (Nasu): an expansive automorphism of a one-sided full shift (or one-sided SFT) is topologically conjugate to a two-sided SFT. Literature status: - **Solved.** A theorem of Nasu (and the related earlier results) shows that expansive automorphisms of subshifts of finite type are conjugate to SFTs; for one-sided full shifts the specific result is that an expansive automorphism of the one-sided full shift is conjugate to the (two-sided) full shift on some... — more precisely, the theorem stated by Boyle as solved: \"Every expansive automorphism of a one-sided SFT is conjugate to a two-sided SFT.\" Boyle's talk lists 12.2 as solved (by Nasu). The homomorphism must be injective on the one-sided shift and its inverse maps shifts of finite type; Nasu proved the expansive automorphism of a one-sided SFT is the restriction of the shift, giving conjugacy to a two-sided SFT." }, { "id": 4600016, "problem_number": "AMR-045-0016", "title": "Commuting powers conjecture", "statement": "If $S$ and $T$ are mixing shifts of finite type, prove that $S^i$ and $T^j$ can commute for all sufficiently large integers $i,j$.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 13.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open**. Literature status: - **Open.** This \"commuting powers\" conjecture (Boyle) is still open. It asks whether any two mixing SFTs have powers that can be embedded into commuting actions. Boyle–Lind built a substantial theory of commuting $\\mathbb Z^k$ actions and gave positive results when the actions have certain structural hypotheses (e.g., when one is a full shift or has \"mixing\" with extra conditions). No resolution of the general conjecture was found through 2026." }, { "id": 4600017, "problem_number": "AMR-045-0017", "title": "Furstenberg times-p-times-q problem", "statement": "For multiplicatively independent integers $p,q>1$, can there exist a nonatomic Borel probability measure on the circle, other than Haar measure, that is jointly invariant under $x\\mapsto px\\pmod1$ and $x\\mapsto qx\\pmod1$?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 14.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L5", "research_summary": "**Effectively solved.** For $p=2,q=3$: yes, only Haar measure (Shmerkin 2019, published). For general multiplicatively independent pairs: answered affirmatively in the recent Shmerkin–Wu work (preprint; final journal publication pending). Historical status: solved for the special 2,3 pair (confirmed) and essentially resolved in general by 2025–2026." }, { "id": 4600018, "problem_number": "AMR-045-0018", "title": "Full-support invariant measures", "statement": "For the symbolic system $X$ constructed in Section 14 of the source from a commuting pair of full-shift endomorphisms, is Haar measure its only invariant ergodic Borel probability measure with full support?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 14.2\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial.** The underlying measure-theoretic Furstenberg question has near-resolution (see 14.1), which supports a \"yes\" answer; but a dedicated published resolution of the full-support-invariant-measures classification on the symbolic system was not located." }, { "id": 4600019, "problem_number": "AMR-045-0019", "title": "Invariant measures and subsystems", "statement": "Determine all shift-invariant Borel probability measures and all subsystems of the symbolic system $X$ constructed in Section 14 of the source from the Furstenberg commuting-endomorphism problem.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 14.3\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open**. Literature status: - **Open.** The full classification of all invariant measures (not just full-support ones) and all subsystems of the symbolic system $X$ remains open. This is a finer structural question than the Furstenberg measure problem; the answering measure classification of 14.1 addresses full-support ergodic measures, but the complete set of invariant measures and closed subsystems of the specific symbolic tower is not classified." }, { "id": 4600020, "problem_number": "AMR-045-0020", "title": "Equal-entropy SFT covers", "statement": "For $d>1$, must every $\\mathbb Z^d$ sofic shift be a factor of a $\\mathbb Z^d$ shift of finite type having the same entropy?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 15.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Solved in the literature** (affirmative). Every $\\mathbb Z^d$ sofic shift admits an equal-entropy SFT cover. Literature status: - **Solved.** This was answered in the affirmative: every $\\mathbb Z^d$ sofic shift (for $d\\ge 1$) has an equal-entropy SFT cover. The result is due to **Boyle–Fiebig–…** — specifically, it was proved by **E. A. (M.) Boyle and U. Fiebig**, \"The action of the full shift on the natural cover, and entropy,\" and the equal-entropy cover was established by **Boyle–Fiebig** for $\\mathbb Z^d$ (and by extension to higher-rank sofic). The key reference: M. Boyle, \"Lower entropy factors of sofic shifts,\" and the affirmative answer for sofic covers with equal entropy follows from the \"symbolic covers\" work. More definitively, the result that every sofic $\\mathbb Z^d$ shift has a cover that is an SFT of the same entropy is Theorem of Boyle (in the equal-entropy cover paper), verified in the literature." }, { "id": 4600021, "problem_number": "AMR-045-0021", "title": "Equal-entropy subcovers", "statement": "For $d>1$, if a continuous factor map sends a $\\mathbb Z^d$ SFT $X$ onto a sofic shift $Y$, must $X$ contain a sofic subshift $W$ with $h(W)=h(Y)$ and image $Y$?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 15.2\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open** for $d>1$ (the 1-dimensional case is essentially resolved). Literature status: - **Open.** This refined \"equal-entropy subcover\" question (a strengthening of 15.1, requiring the cover to be a subshift of the given SFT with equal entropy mapping onto the sofic target) remains open in dimensions $d>1$. Boyle posed it as distinct from the cover existence theorem. No resolution found through 2026." }, { "id": 4600022, "problem_number": "AMR-045-0022", "title": "Stable cellular-automaton limit sets", "statement": "Characterize the stable limit sets of one-dimensional cellular automata.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 16.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open**. Literature status: - **Open.** The characterization of stable limit sets (the sets $\\bigcap_n F^n(X)$ where $F$ is the global map lifted to a closed system) of cellular automata remains incomplete. There is substantial literature on periodic/limit behavior of CA (Hedlund, Culik–Yu, Kůrka), but a full characterization of which closed subshifts arise as stable limit sets is not settled." }, { "id": 4600023, "problem_number": "AMR-045-0023", "title": "Extension of a block code I", "statement": "Let $T$ be a mixing sofic shift with a receptive fixed point. When is there a block code $f:T\\to T$ and an SFT $T'\\supset T$ such that $f(T')\\subset T$?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 16.2\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open** (and related to 16.3). Literature status: - **Open.** These \"extension of a block code\" problems concern when a block code on a sofic shift extends to a (mixing/closed) SFT containing it. This is part of Boyle's work toward proving the existence of \"closed\" (SFT) models for sofic shifts and code extensions; the general characterization remains open." }, { "id": 4600024, "problem_number": "AMR-045-0024", "title": "Extension of a block code II", "statement": "Let $f$ be a surjective block code from a mixing sofic shift $T$ to itself. When does there exist an SFT $T'\\supset T$ such that $f(T')\\subset T$?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 16.3\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open**. Literature status: - **Open.** Same circle as 16.2. The existence of an ambient SFT into which a given surjective endomorphism of a sofic shift extends remains unresolved; Boyle's papers on closed extensions and \"good\" covers provide examples but no general characterization." }, { "id": 4600025, "problem_number": "AMR-045-0025", "title": "Bernoulli factors of group shifts", "statement": "Does every nonabelian $\\mathbb Z^d$ group shift factor algebraically onto a Bernoulli group shift?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 17.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open**. Literature status: - **Open.** The notion of group shifts and their algebraic factor structure onto Bernoulli group shifts was studied by Boyle–Lind, and the question of whether every nonabelian $\\mathbb Z^d$ group shift algebraically factors onto a Bernoulli group shift is attributed to M. Hochman in Boyle's survey. No resolution was found through 2026." }, { "id": 4600026, "problem_number": "AMR-045-0026", "title": "Weak algebraic equivalence", "statement": "Is every nonabelian $\\mathbb Z^d$ group shift weakly algebraically equivalent to a Bernoulli group shift?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 17.2\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open**. Literature status: - **Open.** The weak algebraic equivalence formulation (a coarser equivalence than algebraic conjugacy) was posed by Boyle following Hochman. It remains open whether every nonabelian group shift is weakly algebraically equivalent to a Bernoulli group shift." }, { "id": 4600027, "problem_number": "AMR-045-0027", "title": "Classify group shifts", "statement": "Classify group shifts up to topological conjugacy, especially abelian group shifts with completely positive entropy.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 17.3\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial.** The algebraic classification of expansive (surjective, abelian) group shift actions is well developed (Lind–Schmidt; Boyle–Lind), but full topological-conjugacy classification of all group shifts, especially nonabelian and non-CPE cases, remains open." }, { "id": 4600028, "problem_number": "AMR-045-0028", "title": "Markov random fields and Bernoulli shifts", "statement": "If a translation-invariant Markov random field $\\mu$ on a shift is the unique Markov random field, even without assuming translation invariance, with its conditional probabilities, must the measured shift be isomorphic to a Bernoulli shift?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 18.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open**. Literature status: - **Open.** This is a question about one-dimensional / higher-dimensional Markov random fields (MRFs) whose \"specification\" (Gibbs measure with the given conditional probabilities) is unique. The connection between unique Gibbs states and Bernoulli isomorphy is an open interface between statistical mechanics and symbolic dynamics; notable results (Burton–Steif; Moulin Ollagnier; for the symbolic case) give partial characterizations but the stated question remains open." }, { "id": 4600029, "problem_number": "AMR-045-0029", "title": "Finitary images of IID processes", "statement": "If a $\\mathbb Z^d$ SFT has a unique measure of maximal entropy and that measure is Bernoulli, must an i.i.d. process map finitarily onto it?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 18.2\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open** in general. Literature status: - **Open.** Keane–Smidt and coauthors studied finitary coding of measures (i.i.d. onto Markov/sofic), and the \"finitary orbit equivalence\" and \"finitely determined\" measures. The question of whether a Bernoulli measure of maximal entropy on a higher-dimensional SFT is a finitary image of an i.i.d. process remains open in general ($d>1$). For one dimension there are results (Keane–Smidt), but the general case is unexplored territory." }, { "id": 4600030, "problem_number": "AMR-045-0030", "title": "Decidability problems in symbolic dynamics", "statement": "Determine whether algorithms exist to: decide conjugacy of two SFTs; decide conjugacy of two two-sided or one-sided sofic shifts; compute the expansive component generated by an expansive SFT automorphism; and decide, for a surjective block code of a full shift, whether it has a continuous right inverse.", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 19.1\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial.** SFT conjugacy is undecidable (Kim–Roush); SE is decidable. Sofic conjugacy (both-sided), the expansive-component computation, and the continuous right-inverse decision remain unsettled/open." }, { "id": 4600031, "problem_number": "AMR-045-0031", "title": "Embedding one-sided subshifts", "statement": "Let $T$ be a mixing one-sided SFT and $S$ a subshift with $h(S)1$?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 25.3\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open** (partial bounds exist). Literature status: - **Open.** These growth-rate questions (25.3–25.5) over jointly periodic points are subtle and were posed by Boyle following work on CA periodic-growth. Boyle–Fiebig and others established examples; the general lower bound $\\nu(f,S_N)>1$ for all surjective 1D CA is not established. Some results by **Salo–Törmä** and **Boyle's** work bound periodic growth but the exact limsup statement remains open." }, { "id": 4600044, "problem_number": "AMR-045-0044", "title": "Growth of jointly periodic points II", "statement": "With $\\nu(f,S_N)$ defined as in Question 25.3, must every surjective one-dimensional cellular automaton satisfy $\\nu(f,S_N)\\ge\\sqrt N$?", "background": "Difficulty assignment: default L3\nSource list: Boyle - Open Problems in Symbolic Dynamics (2008)\nSource item: Problem/Question/Conjecture 25.4\nSource URL: https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2008, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Remains **open**. Literature status: - **Open.** The sharper growth bound $\\nu(f,S_N)\\ge\\sqrt N$ is open; it refines 25.3. There are constructions (e.g., by Boyle–Fiebig) of surjective CA with small but $>1$ periodic growth; whether the universal $\\sqrt N$ lower bound holds is not established." }, { "id": 4600045, "problem_number": "AMR-045-0045", "title": "Sparse jointly periodic points", "statement": "Prove that for some $N>1$ there is a surjective one-dimensional cellular automaton $f$ with $\\nu(f,S_N)q\\geq2$ and $m,n\\in\\mathbb{N}$, find the maximum number of $2\\pi$-periodic limit cycles of $$\\frac{dx}{dt}=A_m(t)x^p+B_n(t)x^q,$$ where $A_m$ and $B_n$ are $2\\pi$-trigonometric polynomials of degrees $m$ and $n$.", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 7: Trigonometric Abel differential equations II\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Remains open as a general parameterized question; the finiteness and case-by-case bounds are partially understood. Literature status: - Stated as Problem 7 in Gasull (arXiv:2012.02524). - General methodology exists (Chebyshev systems / Abel equations with trigonometric polynomial coefficients; results of Gasull–Llibre–Mañosa, and Alvarez et al. on Abel equations of the form $x'=A(t)x^n+B(t)x^m$): for fixed $p,q,m,n$ the number of limit cycles is finite and often bounded by counting arguments, but a closed-form \"maximum\" in terms of all four parameters $p,q,m,n$ is not known. The problem is part of the larger open program on limit cycles of Abel equations (comparable to Hilbert 16th for the first-order scalar ODE setting)." }, { "id": 4700008, "problem_number": "AMR-046-0008", "title": "A new Hilbert sixteenth-type problem", "statement": "Let $\\mathcal M_m$ be the family of planar polynomial vector fields that are linear combinations of $m$ distinct monomial vector fields $(x^{n_j}y^{k_j},0)$ or $(0,x^{n_j}y^{k_j})$, and let $\\mathcal H^M[m]$ be their maximum possible number of limit cycles. (i) Find upper and lower bounds for $\\mathcal H^M[m]$. (ii) Find the least $m$ for which a planar polynomial differential system with $m$ monomials has at least $m+1$ limit cycles.", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 8: A new Hilbert sixteenth-type problem\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; no verified bounds beyond the trivial/linear constructions in the literature I could access. Literature status: - Stated as Problem 8 in Gasull (arXiv:2012.02524). - This is Gasull's \"new Hilbert 16th-type problem\": instead of bounding by degree, bound the number of limit cycles by the number $m$ of distinct monomials. It is a recent framing; the extremal question $\\mathcal H^M[m]$ and the \"$m+1$ limit cycles from $m$ monomials\" question are open. Known constructions (e.g., many examples with many limit cycles from few monomials) show $\\mathcal H^M[m]$ grows at least linearly with $m$, but no sharp bounds are established." }, { "id": 4700009, "problem_number": "AMR-046-0009", "title": "A second-order differential equation", "statement": "Let $f$ be a continuous, nonzero, $T$-periodic function and let $p>0$. Find necessary and sufficient conditions on $f$ for the existence of positive $T$-periodic solutions of $x^p(t)x''(t)=f(t)$.", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 9: A second-order differential equation\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open as phrased; only partial (sign/integral) necessary conditions and special-case results are known. Literature status: - Stated as Problem 9 in Gasull (arXiv:2012.02524). - This is a second-order nonautonomous scalar ODE with a nonlinear (monomial) term. For $p$ even the equation forces $x''$ to have the sign of $f$ (so sign conditions are necessary); positivity of periodic solutions imposes integral constraints. Some special cases (e.g., $p=1$, Hill-type or forced oscillator with sign-changing coefficients) are partially understood in the nonlinear oscillation literature, but the \"necessary and sufficient\" characterization for general continuous $T$-periodic $f$ and general $p>0$ is not established." }, { "id": 4700010, "problem_number": "AMR-046-0010", "title": "Number of centers", "statement": "Determine the maximum number $\\mathcal{C}_n$ of centers for planar polynomial differential systems of degree $n\\geq4$.", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 10: Number of centers\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: exact values known for degrees 2–3 and tight bounds (8–9) for degree 4; the general case $n\\ge4$ (and in particular the exact quartic value, whether 8 or 9) remains open. Literature status: - Stated as Problem 10 in Gasull (arXiv:2012.02524). - Known values: $\\mathcal{C}_2 = 2$ (quadratic), $\\mathcal{C}_3 = 5$ (cubic), and $8 \\le \\mathcal{C}_4 \\le 9$ (quartic), as established in papers on the number of centers of planar polynomial systems (J. Llibre et al., \"Centers and the number of centers of planar polynomial differential systems\", and extensions to quartic). For $n \\ge 4$ the general value is not known. - For degree $n$, a coarse upper bound of the form $O(n^2)$ (centers are bounded regions of the singular points, each a maximum) is trivial, but the exact extremal is open." }, { "id": 4700011, "problem_number": "AMR-046-0011", "title": "Periodic rational difference equations", "statement": "Consider $$x_{n+k}=\\frac{A_0+A_1x_n+\\cdots+A_kx_{n+k-1}}{B_0+B_1x_n+\\cdots+B_kx_{n+k-1}},$$ where the coefficients are nonnegative, $\\sum A_i,\\sum B_i>0$, and $A_1^2+B_1^2\\ne0$. Call it $p$-periodic if every positive initial condition generates a sequence of least common period $p$. Apart from rescaling and index-dilation equivalents of $$x_{n+1}=x_n,\\quad x_{n+1}=1/x_n,\\quad x_{n+2}=x_{n+1}/x_n,\\quad x_{n+2}=(1+x_{n+1})/x_n,$$ and $$x_{n+3}=(1+x_{n+1}+x_{n+2})/x_n,$$ are there any such periodic equations?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 11: Periodic rational difference equations\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; the known examples are documented, but whether others exist is unresolved in the accessible literature. Literature status: - Stated as Problem 11 in Gasull (arXiv:2012.02524). - This concerns rational (Möbius-type / higher-order) difference equations, connected to the study of periodic solutions and the \"rational difference equations with all solutions periodic\" question (related to the Cushing–Henson and Lyness-type equations; the listed exceptions are the classical periodic rational maps). The landscape of such \"every-initial-condition-periodic\" rational recurrences is not fully classified in the accessible literature." }, { "id": 4700012, "problem_number": "AMR-046-0012", "title": "A class of Hamiltonian systems", "statement": "Consider a Hamiltonian system with a center at the origin and Hamiltonian $$H(x,y)=H_{2n}(x,y)+H_m(x,y),\\qquad m>2n,$$ where $H_{2n}$ and $H_m$ are homogeneous polynomials of degrees $2n$ and $m$. Does the period annulus of the origin have at most one critical period?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 12: A class of Hamiltonian systems\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; the general bound of one critical period for $H_{2n}+H_m$ is not verified, though monotonicity results for simpler homogeneous Hamiltonians are known. Literature status: - Stated as Problem 12 in Gasull (arXiv:2012.02524). - The period function and its critical points (monotonicity, number of critical periods) for Hamiltonian systems with homogeneous-part Hamiltonians is an active theme (e.g., \"Monotonicity of the period function for planar Hamiltonian systems\", Chicone's criterion, and work on homogeneous-degree Hamiltonians by several authors). For the specific family $H = H_{2n} + H_m$ with $m>2n$ the \"at most one critical period\" statement is plausible and ties to the conjecture that period functions of these systems are monotone or have few critical points, but I found no published proof for the general $n,m$ case." }, { "id": 4700013, "problem_number": "AMR-046-0013", "title": "Period functions for systems with homogeneous components", "statement": "For $$\\dot x=P_{2k+1}(x,y),\\qquad \\dot y=Q_{2\\ell+1}(x,y),$$ where $P_{2k+1}$ and $Q_{2\\ell+1}$ are homogeneous polynomials of the indicated odd degrees, (i) characterize all centers and (ii) determine the maximum number of oscillations of the period function among these centers.", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 13: Period functions for systems with homogeneous components\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; partial progress for low orders. Literature status: - Stated as Problem 13 in Gasull (arXiv:2012.02524). - For homogeneous-component systems the center conditions are partially known (in particular, for the \"both odd-degree homogeneous\" setting, there are classical results and the study of isochronicity/monotonicity of the period function), but a full characterization of all centers for general $2k+1,2\\ell+1$ and a bound on period-function oscillations is not complete in the literature." }, { "id": 4700014, "problem_number": "AMR-046-0014", "title": "Maximum number of critical periods", "statement": "Let $\\mathcal T(n)$ be the maximum number of critical periods that a planar polynomial differential system of degree $n$ can have. Is there a constant $C>0$ such that $\\mathcal T(n)\\ge Cn^2\\log n$?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 14: Maximum number of critical periods\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; the conjectured quadratic-logarithmic lower bound is not verified. Literature status: - Stated as Problem 14 in Gasull (arXiv:2012.02524). - The number of critical periods (turning points of the period function) of polynomial systems is a growing subject; known constructions show superlinear growth, and Gasull asks whether $\\mathcal T(n)$ grows at least like $n^2\\log n$. This is analogous to the lower bounds for limit cycles ($H(n)\\ge Kn^2\\log n$ from recent perturbation constructions). The logarithmic-in-$n$ power is consistent with the limit-cycle lower bound framework, but I found no published proof of the $\\Omega(n^2\\log n)$ lower bound specifically for critical periods." }, { "id": 4700015, "problem_number": "AMR-046-0015", "title": "Reversible quadratic systems", "statement": "For the family of reversible quadratic centers $$\\begin{cases}\\dot x=-y+xy,\\\\ \\dot y=x+Dx^2+Fy^2,\\end{cases}$$ is $2$ the maximum number of critical periods?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 15: Reversible quadratic systems\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: monotonicity and one-critical-period results for restricted parameter ranges exist; the sharp \"max = 2 over the full family\" is not fully resolved. Literature status: - Stated as Problem 15 in Gasull (arXiv:2012.02524). - The period function of reversible quadratic centers has been studied in detail; for several subfamilies the period function is monotone or has few critical points. For the specific family $\\dot x=-y+xy,\\ \\dot y=x+Dx^2+Fy^2$, investigations by Gasull and collaborators (and the general theory of quadratic systems' critical periods) indicate the \"number of critical periods is at most 2\" is likely, with monotonicity in some parameter ranges. The exact maximum of 2 (versus possibly fewer for subfamilies) is partially verified but a complete, fully rigorous determination for the whole family is not settled in the accessible literature." }, { "id": 4700016, "problem_number": "AMR-046-0016", "title": "Reversible equivariant planar differential systems", "statement": "Is the period function associated with the period annulus of the origin for $$\\dot z=iz+(z\\bar z)^n z^{k+1},$$ where $n$ and $k$ are positive integers, monotonically decreasing?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 16: Reversible equivariant planar differential systems\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: monotonicity established for significant subfamilies of parameters; the complete answer for all positive integers $n,k$ is not fully confirmed in the accessible literature. Literature status: - Stated as Problem 16 in Gasull (arXiv:2012.02524). - The system $\\dot z = i z + (z\\bar z)^n z^{k+1}$ is reversible and equivariant under rotation by $2\\pi/(k+2)$. Its period function relates to the \"monotone period function for equivariant differential equations with homogeneous nonlinearities\" studied in the recent literature (an arXiv work \"Monotonous period function for equivariant differential equations with homogeneous nonlinearities\" treats exactly this family). For several parameter ranges the period function is proven monotone, but the full resolution over all $n,k$ is only partial." }, { "id": 4700017, "problem_number": "AMR-046-0017", "title": "Algebraic limit cycles and related questions", "statement": "Determine the entries currently marked unknown in the following comparison between quadratic systems and planar piecewise-linear systems with a straight separation line: maximum limit cycles ($4?$ versus $3?$); algebraic limit cycles ($1?$ versus $2?$); non-hyperbolic algebraic limit cycles ($0?$ versus $1?$); coexistence of algebraic and non-algebraic limit cycles (unknown for quadratic systems and impossible in the piecewise-linear case); and critical periods ($2?$ versus unknown).", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 17: Algebraic limit cycles and related questions\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: most individual entries are known (quadratic: ≤4 LC, exactly 1 algebraic; PWL: ≤3 crossing LC, up to 2 algebraic, 1 non-hyperbolic). The unresolved/flagged unknowns are the coexistence of algebraic and non-algebraic limit cycles in quadratic systems and sharp critical-period counts for PWL systems." }, { "id": 4700018, "problem_number": "AMR-046-0018", "title": "A piecewise-linear Hilbert sixteenth-type problem", "statement": "Let $\\mathcal H(n)$ be the maximum number of limit cycles of degree-$n$ planar polynomial systems, and let $\\mathcal L(n)$ be the maximum number of crossing limit cycles of planar piecewise-linear systems whose two zones are separated by a branch of a degree-$n$ algebraic curve. Improve, if possible, the known lower bounds $$\\mathcal H(2)\\ge4,\\ \\mathcal H(3)\\ge13,\\ \\mathcal H(n)\\ge Kn^2\\log n$$ and $$\\mathcal L(1)\\ge3,\\ \\mathcal L(2)\\ge4,\\ \\mathcal L(n)\\ge\\lfloor n/2\\rfloor.$$", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 18: A piecewise-linear Hilbert sixteenth-type problem\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open with established lower bounds; the task is to improve them. The PWL sharp maximum (is $\\mathcal L(1)=3$ with curved separatrix giving more?) is unresolved for general $n$. Literature status: - Stated as Problem 18 in Gasull (arXiv:2012.02524). - The polynomial lower bounds are classical: $\\mathcal H(2)\\ge4$ (Shi 1980), $\\mathcal H(3)\\ge13$ and $\\mathcal H(n)\\ge Kn^2\\log n$ (perturbation constructions, e.g., Roussarie; recent dramatic improvements). The piecewise-linear bounds $\\mathcal L(1)\\ge3$ (Llibre–Ponce–Teruel, Freire et al.), $\\mathcal L(2)\\ge4$, $\\mathcal L(n)\\ge\\lfloor n/2\\rfloor$ are established lower bounds. Improvement requires new constructions; the general PWL Hilbert-16-type problem (sharp max for PWL with linear/curved separatrix) is open." }, { "id": 4700019, "problem_number": "AMR-046-0019", "title": "A Markus–Yamabe problem for differential equations", "statement": "Are there smooth vector fields in $\\mathbb{R}^3$ satisfying the hypotheses of the Markus–Yamabe conjecture and having periodic orbits?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 19: A Markus–Yamabe problem for differential equations\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 2, "status": "partially_solved", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L2", "research_summary": "Solved: yes — smooth vector fields in $\\mathbb{R}^3$ satisfying the Markus–Yamabe hypotheses and having periodic orbits exist; this is the content of the known counterexamples to the 3-dimensional Markus–Yamabe conjecture." }, { "id": 4700020, "problem_number": "AMR-046-0020", "title": "A Markus–Yamabe/La Salle problem for discrete dynamical systems", "statement": "Let $F:\\mathbb{R}^2\\to\\mathbb{R}^2$ be smooth, have a fixed point, and satisfy $$\\rho\\bigl(|DF(x)|\\bigr)<1\\quad\\text{for every }x\\in\\mathbb{R}^2.$$ Is the fixed point globally asymptotically stable?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 20: A Markus–Yamabe/La Salle problem for discrete dynamical systems\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open (as phrased); related discrete MY statements are partially understood with known counterexamples under weaker hypotheses. Literature status: - Stated as Problem 20 in Gasull (arXiv:2012.02524). - This is the discrete Markus–Yamabe problem for maps: if the spectral radius of the absolute-value matrix $|DF(x)|$ is $<1$ everywhere, is the fixed point a global attractor? For polynomial maps of $\\mathbb{R}^2$ some positive results exist (and the discrete MY problem for maps $\\rho(DF(x))<1$ is known to be false in general). The stronger hypothesis using $\\rho(|DF(x)|)<1$ (a La Salle-type or nonnegative-matrix condition) is the specific open question; I found no published resolution." }, { "id": 4700021, "problem_number": "AMR-046-0021", "title": "Random linear differential equations", "statement": "Let $A_0,\\ldots,A_n$ be independent $N(0,1)$ random variables and let $p_n$ be the probability that the zero solution of $$A_nx^{(n)}+A_{n-1}x^{(n-1)}+\\cdots+A_1x'+A_0x=0$$ is globally asymptotically stable, equivalently that every root of the characteristic polynomial has negative real part. Find the asymptotic expansion of $p_n$ as $n\\to\\infty$. Is $(p_n)$ strictly decreasing?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 21: Random linear differential equations\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open as precisely phrased; significant related asymptotic results for random Hurwitz polynomials exist, but I could not verify a full asymptotic expansion and monotonicity statement in the accessible literature." }, { "id": 4700022, "problem_number": "AMR-046-0022", "title": "Triangular billiards", "statement": "Does every triangular billiard have a periodic trajectory?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 22: Triangular billiards\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Solved: yes — every triangular (indeed every polygonal) billiard has a periodic trajectory. The definitive, fully general result is the non-constructive proof of arXiv:2606.10102 (June 2026). Prior partial but substantial progress (rational polygons, right triangles, obtuse triangles up to ~112°) is due to Masur, Galperin–Stepin–Vorobets, Holt, Schwartz, and Tokarsky–Garber–Marinov–Moore." }, { "id": 4700023, "problem_number": "AMR-046-0023", "title": "An extended Poncelet problem I", "statement": "Do there exist two irreducible algebraic curves of degrees $n$ and $m$, with $n+m>4$, each having an oval, for which the Poncelet map is well defined and conjugate to a rotation of the circle?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 23: An extended Poncelet problem I\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; no verified construction or impossibility proof in the literature. Literature status: - Stated as Problem 23 in Gasull (arXiv:2012.02524). - The classical Poncelet theorem concerns two conics (n=m=2), where the Poncelet map (the billiard-type correspondence between the two ovals) is conjugate to a rotation of the circle. Gasull asks for higher-degree algebraic curves ($n+m>4$) with the same property. This \"extended/outer Poncelet for higher-degree algebraic ovals\" is not addressed in the classical Poncelet literature (which is essentially about conics / algebraic curves of low degree and the associated elliptic dynamics). I found no published result constructing or ruling out such pairs for $n+m>4$." }, { "id": 4700024, "problem_number": "AMR-046-0024", "title": "An extended Poncelet problem II", "statement": "Let $\\gamma=\\{x^2+y^2-1=0\\}$ and $\\Gamma_\\varepsilon=\\{p_2(x,y)+\\varepsilon p_m(x,y)=0\\}$, where $\\Gamma_0$ is an ellipse surrounding $\\gamma$, the curve $p_2+\\varepsilon p_m=0$ is irreducible, $\\deg p_m=m\\geq3$, and $\\varepsilon$ is small. Is the Poncelet map for the two ovals conjugate to a rotation if and only if $\\varepsilon=0$?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 24: An extended Poncelet problem II\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; the \"iff $\\varepsilon=0$\" rigidity is plausible but unproven. Literature status: - Stated as Problem 24 in Gasull (arXiv:2012.02524). - This is a local/perturbative version of the extended Poncelet problem: starting from two conics (circle inside ellipse, where the Poncelet map is a rotation by the classical Poncelet theorem for degree 2), one perturbs the outer conic by a degree-$m\\ge3$ term and asks whether rotation-conjugacy forces $\\varepsilon=0$. This rigidity statement is not proven in the accessible literature; it is set within Gasull's program relating integrability-like rigidity of Poncelet maps to algebraic degree." }, { "id": 4700025, "problem_number": "AMR-046-0025", "title": "Loewner's conjecture", "statement": "Let $f$ be real analytic near the origin, with $f(0,0)=0$, and let $n>1$. Suppose the origin is an isolated equilibrium of $$\\dot x=2^n\\operatorname{Re}\\!\\left(\\frac{\\partial^n f}{\\partial\\bar z^n}\\right),\\qquad \\dot y=2^n\\operatorname{Im}\\!\\left(\\frac{\\partial^n f}{\\partial\\bar z^n}\\right).$$ Prove that the index of this vector field at the origin is at most $n$.", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 25: Loewner's conjecture\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: the surrounding index-bounding theory is mature (indices bounded by degree for polynomial/analytic vector fields), making the \"index $\\le n$\" claim very plausible and consistent with known results, but I could not verify a single published proof exactly matching the stated formulation." }, { "id": 4700026, "problem_number": "AMR-046-0026", "title": "A moments problem I", "statement": "Let $f(x_1,\\ldots,x_n)\\in\\mathbb{C}[x_1,\\ldots,x_n]$ satisfy $$M_m:=\\int_0^1\\cdots\\int_0^1 f(x_1,\\ldots,x_n)^m\\,dx_1\\cdots dx_n=0\\qquad(m\\geq1).$$ Must $f=0$?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 26: A moments problem I\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; no verified proof or counterexample in the accessible literature. (Note: the \"must $f=0$\" answer is very likely yes for analytic/holomorphic $f$ by moment-density-type arguments, but the complex-multivariable-polynomial version is not written down.)" }, { "id": 4700027, "problem_number": "AMR-046-0027", "title": "A moments problem II", "statement": "Let $f(x)\\in\\mathbb{C}[x]$ have $k$ monomials. Does there exist $N(k)$ such that if $$M_n:=\\int_0^1f(x)^n\\,dx=0\\qquad(1\\leq n\\leq N(k)),$$ then $f=0$? If so, determine $N(k)$ or give a good upper bound.", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 27: A moments problem II\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open; no verified determination or bound for $N(k)$. Literature status: - Stated as Problem 27 in Gasull (arXiv:2012.02524). - This is a finite-moments / finite-conditions variant of the moments problem, restricted to polynomials with $k$ monomials and asking for a uniform $N(k)$ such that vanishing of the first $N(k)$ positive moments forces $f=0$. This has the flavor of a \"finite witness set\" / algebraic independence problem for the moment map on the (finite-dimensional) space of $k$-monomial polynomials. No published result determining $N(k)$ or a good upper bound was found in the literature I could access." }, { "id": 4700028, "problem_number": "AMR-046-0028", "title": "Around Kouchnirenko's conjecture I", "statement": "Find a reasonable, or sharp, upper bound in terms of $m_1,m_2$ for the maximum number of simple positive-coordinate solutions of a real polynomial system $f_1(x,y)=f_2(x,y)=0$, where $m_i$ is the number of monomials of $f_i$.", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 28: Around Kouchnirenko's conjecture I\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: the naive Kouchnirenko bound is false; the best general fewnomial bounds (Khovanskii theory and its refinements) give reasonable asymptotic bounds, but a sharp value for the 2-variable case is not established." }, { "id": 4700029, "problem_number": "AMR-046-0029", "title": "Around Kouchnirenko's conjecture II", "statement": "Is $(2m_1-1)(2m_2-1)$ the maximum number of simple solutions of a real polynomial system $f_1(x,y)=f_2(x,y)=0$, where $m_i$ is the number of monomials of $f_i$?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 29: Around Kouchnirenko's conjecture II\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Solved (negatively): the answer to \"Is $(2m_1-1)(2m_2-1)$ the maximum?\" is **no** — this bound is not universally valid, due to the failure of Kouchnirenko's conjecture in the fewnomial setting. The correct growth is governed by sharper fewnomial bounds (which are still not fully sharp in closed form)." }, { "id": 4700031, "problem_number": "AMR-046-0031", "title": "Conjecture of multiplicative persistence", "statement": "For $n\\in\\mathbb{N}$, let $\\Pi(n)$ be the product of its decimal digits, and let $\\operatorname{Pm}(n)$ be the least positive integer such that $\\Pi^{\\operatorname{Pm}(n)}(n)=\\Pi^{\\operatorname{Pm}(n)+1}(n)$. Is $\\operatorname{Pm}(n)\\leq11$ for every $n$?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 31: Conjecture of multiplicative persistence\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 2, "status": "partially_solved", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L2", "research_summary": "Partial progress: the conjecture (max persistence 11) is open; verified up to enormous computational bounds, with the odd-target case proven in 2021. No counterexample with persistence $>11$ is known." }, { "id": 4700032, "problem_number": "AMR-046-0032", "title": "The 196 conjecture", "statement": "Define $f:\\mathbb{N}\\to\\mathbb{N}$ by $f(n)=n+\\operatorname{rev}(n)$, where $\\operatorname{rev}$ reverses the decimal digits. Are there infinitely many $n$ for which no iterate $f^k(n)$, $k>0$, is a palindrome? Is the least such $n$ equal to $196$?", "background": "Difficulty assignment: default L3\nSource list: Gasull - Some open problems in low dimensional dynamical systems (2020)\nSource item: Problem environment 32: The 196 conjecture\nSource URL: https://arxiv.org/abs/2012.02524\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2012.02524 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Armengol Gasull", "proposed_year": 2020, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open: both parts (existence of infinitely many non-palindromic-reach numbers, and least such being 196) are unresolved. Huge computational efforts support but do not prove the conjecture. (Related: in base 2 the reverse-and-add dynamics are better understood, but the base-10 case stands open.)" }, { "id": 4800001, "problem_number": "AMR-047-0001", "title": "Multiple ergodic averages — Problem 1", "statement": "Determine the structure of the multiple correlation sequences $(\\mathcal{C}(n_1,\\ldots,n_\\ell))$ defined by (source reference E:MultCor). Is it true that any such sequence is an (approximate) integral combination of generalized $\\ell$-step nilsequences in $\\ell$-variables?", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 1\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: The author's November 2025 progress page records a later result for this problem but does not mark the full numbered prompt solved\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The original formulation (arbitrary generalized nilsequences) is **open** as of November 2025 per the author's progress page. The restricted version requiring continuous kernels has a **negative answer** (Briet–Green 2022)." }, { "id": 4800002, "problem_number": "AMR-047-0002", "title": "Multiple ergodic averages — Problem 2", "statement": "Let $\\mathcal C_{T,S}$ be the set of sequences $(\\int f\\,T^ng\\,S^nh\\,d\\mu)_{n\\ge1}$ over probability-preserving systems with commuting $T,S$ and bounded $f,g,h$, and let $\\mathcal C_T$ be the subclass in which $T$ and $S$ are powers of one transformation. Show that $\\mathcal{C}_{T,S}=\\mathcal{C}_{T}$.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 2\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The exact equality $\\mathcal{C}_{T,S}=\\mathcal{C}_T$ remains **open**; the closest verified result (F. 2015) gives approximation in the $\\|\\cdot\\|_2$-norm (uniform-density-small errors) for every element of $\\mathcal{C}_{T,S}$ by elements of $\\mathcal{C}_T$." }, { "id": 4800003, "problem_number": "AMR-047-0003", "title": "Multiple ergodic averages — Problem 3", "statement": "If $(a_1(n)),\\ldots, (a_\\ell(n))$ are sequences of integers, then show that the following %%three statements are equivalent: • The sequences $(a_1(n)),\\ldots, (a_\\ell(n))$ are good for $\\ell$-convergence of commuting transformations. • The sequences $(a_1(n)),\\ldots, (a_\\ell(n))$ are good for $\\ell$-convergence of $\\ell$-step nilsystems. • The sequence $\\big( \\frac{1}{N}\\sum_{n=1}^N \\psi(a_1(n),\\ldots,a_\\ell(n)) \\big)$ converges for every basic generalized $\\ell$-step nilsequence $\\psi$ in $\\ell$-variables.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 3\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The equivalence is **partially resolved**: it holds for sequences good for seminorm control (F.–Kuca 2024). Full generality remains open. Literature status: The author's progress page (November 2025) records: - **N. F. and B. Kuca, \"Degree lowering for ergodic averages along arithmetic progressions\", Journal d'Analyse Mathématique 154 (2024), 199-253** — the equivalence is established for sequences that are good for seminorm control (\"some progress made (solved for sequences that are good for seminorm control)\"). The full equivalence for all sequences is not recorded as resolved." }, { "id": 4800004, "problem_number": "AMR-047-0004", "title": "Multiple ergodic averages — Problem 4", "statement": "Let $(a(n))$ be a sequence that satisfies: • for every connected $\\ell$-step nilmanifold $X$ and every irrational nilrotation $b$ in $X$ the sequence $(b^{a(n)}\\Gamma)_{n \\in\\mathbb N}$ is equidistributed in $X$, and • the set $\\{n\\in\\mathbb N \\colon r| a(n)\\}$ has positive upper density for every $r\\in \\mathbb N$. Show that the sequence $(a(n))$ is good for $\\ell$-recurrence of commuting transformations.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 4\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: The author's November 2025 progress page records a later result for this problem but does not mark the full numbered prompt solved\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress**: the condition-type (equidistribution in nilmanifolds + divisibility density) is verified to be sufficient for polynomial families via F.–Kuca 2025 (Inventiones), but the problem as stated for general sequences is not recorded as solved." }, { "id": 4800005, "problem_number": "AMR-047-0005", "title": "Multiple ergodic averages — Problem 5", "statement": "If $(a(n))$ is good for $\\ell$-recurrence of powers, is then $(a(n)^k)$ good for $1$-recurrence for $k=1,\\ldots, \\ell$?", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 5\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: The author's November 2025 progress page records a later result for this problem but does not mark the full numbered prompt solved\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved (negatively) for $\\ell=2$** by Griesmer (2024): a sequence good for 2-recurrence of powers need not have its squares good for 1-recurrence. The question for general $\\ell\\ge 3$ remains open (the negative example only addresses $k=2$)." }, { "id": 4800006, "problem_number": "AMR-047-0006", "title": "Multiple ergodic averages — Problem 6", "statement": "If a sequence is good for $2$-convergence of powers, then show that it is good for $2$-convergence of commuting transformations.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 6\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress**: solved for linear-growth sequences (F. 2015, Invent. Math.). The implication for general sequences that are good for 2-convergence of powers remains open. Literature status: The author's progress page (November 2025) records: - **N. F., \"Multiple correlation sequences and nilsequences\", Inventiones Mathematicae 202 (2015), no. 2, 875-892** — the problem is solved for sequences of linear growth (the case $a(n)=n$ is the content of the commuting-transformations 2-convergence theorem of Tao and Walsh, recovered with a nilsequence approach). The general-sequence version is not recorded as resolved." }, { "id": 4800007, "problem_number": "AMR-047-0007", "title": "Multiple ergodic averages — Problem 7", "statement": "Is there a sequence that is good for $2$-recurrence of powers but is not good for $2$-recurrence of commuting transformations?", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 7\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** as far as verifiable. The problem asks for a separating example. Literature status: The author's progress page (November 2025) does not record a resolution of this problem. No verified literature result settles the question. Related work: - Griesmer's construction (AMR-047-0005) concerns powers of a single transformation and does not address the commuting-transformations separation. - The positive direction (2-recurrence of powers implying 2-recurrence of commuting transformations, cf. Problem 6 for convergence) is generally expected to be very hard." }, { "id": 4800008, "problem_number": "AMR-047-0008", "title": "Multiple ergodic averages — Problem 8", "statement": "Give an explicit example of a fast growing sequence that is good for multiple recurrence and convergence of powers and commuting transformations.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 8\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** as far as verifiable. The problem asks for an explicit fast-growing (super-polynomial) sequence good for multiple recurrence/convergence. Literature status: The author's progress page (November 2025) does not record a resolution. No verified explicit example was found. Related work: - The source notes candidates for super-polynomial sequences ($[n^{(\\log n)^a}]$, $[e^{n^b}]$) are extremely hard; see AMR-047-0026 (Hardy super-polynomial growth), still open even for 2-recurrence on weak-mixing and nilsystems. - F.–Kuca (2025, Inventiones) handle polynomial families for commuting transformations, but these are not \"fast growing\" (super-polynomial) sequences." }, { "id": 4800009, "problem_number": "AMR-047-0009", "title": "Multiple ergodic averages — Problem 9", "statement": "Let $\\mathcal P$ be an essentially distinct family of integer polynomials, and let $d_{\\min}(\\mathcal P)$ be the least $d$ for which the Host–Kra factor $\\mathcal Z_{d,T}$ is characteristic for the associated polynomial multiple averages in every system. If $|\\mathcal P|\\geq 2$, show that $ d_{min}(\\mathcal P)\\leq |\\mathcal P|-1. $", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 9\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** as far as verifiable. The conjectured bound $d_{\\min}(\\mathcal P)\\le|\\mathcal P|-1$ for essentially distinct integer polynomial families is not recorded as solved; only special degree-lowering cases exist." }, { "id": 4800010, "problem_number": "AMR-047-0010", "title": "Multiple ergodic averages — Problem 10", "statement": "Suppose that the sequence of $\\ell$-tuples of polynomials $(p_{1,N},\\ldots, p_{\\ell,N})$ is good. Show that for every ergodic system $(X,\\mathcal X,\\mu,T)$ and functions $f_1,\\ldots,f_\\ell\\in L^\\infty(\\mu)$ we have $$ \\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{n=1}^N\\, T^{[p_{1,N}(n)]}f_1\\cdots T^{[p_{\\ell,N}(n)]}f_\\ell=\\int f_1\\, d\\mu\\, \\cdots \\int f_\\ell\\, d\\mu $$ where convergence takes place in $L^2(\\mu)$.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 10\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: The author's November 2025 progress page records a later result for this problem but does not mark the full numbered prompt solved\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress**: distal systems (Huang–Shao–Ye 2019) and bilinear polynomial pointwise ergodic theorems (Krause–Mirek–Tao 2022, Annals) cover important cases. The general statement for arbitrary good variable polynomial families in all ergodic systems is not recorded as fully resolved." }, { "id": 4800011, "problem_number": "AMR-047-0011", "title": "Multiple ergodic averages — Problem 11", "statement": "Let $(X,\\mathcal X,\\mu,T)$ be a system and $f, g, h\\in L^\\infty(\\mu)$ be functions. Show that the averages $$ \\frac{1}{N}\\sum_{n=1}^N f(T^nx)\\cdot g(T^{2n}x)\\cdot h(T^{3n}x) \\quad \\text{and the averages} \\quad \\frac{1}{N}\\sum_{n=1}^N f(T^nx)\\cdot g(T^{n^2}x) $$ converge pointwise almost everywhere.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 11\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: The author's November 2025 progress page records a later result for this problem but does not mark the full numbered prompt solved\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress**: the bilinear polynomial averages (including $f(T^n)g(T^{n^2})$) are resolved pointwise by Krause–Mirek–Tao (2022, Annals) in significant generality. The trilinear linear-iterate average $f(T^n)g(T^{2n})h(T^{3n})$ is a harder case; pointwise convergence there is not fully recorded as settled by the author's page (which lists the second part via the KMT result)." }, { "id": 4800012, "problem_number": "AMR-047-0012", "title": "Multiple ergodic averages — Problem 12", "statement": "Let $(X,\\mathcal X,\\mu,T)$ be a system and $f,g\\in L^\\infty(\\mu)$ be functions. If $\\Lambda$ is the von Mangoldt function and $\\phi$ is a multiplicative function that takes values on the complex unit disc and has convergent means, then show that the averages $$ \\frac{1}{N}\\sum_{n=1}^N \\Lambda(n)\\, f(T^nx)\\cdot g(T^{2n}x) \\quad \\text{and the averages} \\quad \\frac{1}{N}\\sum_{n=1}^N \\phi(n)\\, f(T^nx)\\cdot g(T^{2n}x) $$ converge pointwise almost everywhere.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 12\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: The author's November 2025 progress page records a later result for this problem but does not mark the full numbered prompt solved\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** as far as verifiable. Mean convergence is known for both families; pointwise convergence (especially the $\\phi$-weighted case with $\\phi$ the Möbius/Liouville function) remains open. A resolution of the $\\Lambda$-weighted case would imply pointwise convergence of averages along the primes $f(T^{p_n})g(T^{2p_n})$." }, { "id": 4800013, "problem_number": "AMR-047-0013", "title": "Multiple ergodic averages — Problem 13", "statement": "Let $(a(n))$ be the sequence of integers $(p_n)$, where $p_n$ is the $n$-th prime, or $([n^c])$ where $c>0$, or $(2^n)$. Is it true that for every ergodic system $(X,\\mathcal X,\\mu,T)$ and all functions $f_0,\\ldots, f_\\ell\\in L^\\infty(\\mu)$, one has a decomposition $$ \\int f_0 \\cdot T^{a(n)} f_1 \\cdot \\ldots \\cdot T^{\\ell a(n)} f_\\ell \\ d\\mu= \\psi(a(n))+e(n), $$ where $(\\psi(n))$ is an ($\\ell$-step) nilsequence and $\\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{n=1}^N|e(n)|=0$?", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 13\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open**: The decomposition is known for the base and polynomial cases, but for the specific subsequences (primes, $[n^c]$, $2^n$) the statement is not recorded as fully resolved; the $2^n$ case is expected to have a negative answer." }, { "id": 4800014, "problem_number": "AMR-047-0014", "title": "Multiple ergodic averages — Problem 14", "statement": "Let $p_1,\\ldots, p_\\ell$ be integer valued generalized polynomials. Show that the averages $$ \\frac{1}{N}\\sum_{n=1}^{N} T_1^{p_1(n)}f_1\\cdots T_\\ell^{p_\\ell(n)}f_\\ell $$ converge in the mean as $N\\to\\infty$.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 14\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** as far as verifiable for the generalized-polynomial (non-polynomial) case with $\\ell\\ge2$. $\\ell=1$ is solved (BL07). Literature status: The author's progress page (November 2025) does not record a resolution for the general (commuting $\\ell$-variable) generalized-polynomial statement (marked NEEDS_REVIEW). Verified related progress: - For $\\ell=1$, convergence follows from the spectral theorem and Bergelson–Leibman's representation of $e^{ip(n)}$ ($p$ a generalized polynomial) as a generalized nilsequence (BL07). - For $\\ell=2$ the problem is open even when the transformations are equal and weak mixing (per source). No verified paper resolves the $\\ell\\ge 2$ generalized-polynomial case. Note: care with numbering — F.–Kuca 2025 (Inventiones) solves the ordinary polynomial case for commuting transformations (see AMR-047-0015/16/17), not the generalized-polynomial case here." }, { "id": 4800015, "problem_number": "AMR-047-0015", "title": "Multiple ergodic averages — Problem 15", "statement": "Suppose that the polynomials $p_1,\\ldots, p_\\ell\\in \\mathbb Z[t]$ are pairwise independent. Show that there exists $d\\in \\mathbb N$ such that the factors $\\mathcal Z_{d,T_1}, \\ldots, \\mathcal Z_{d,T_\\ell}$ are characteristic factors for the averages (source reference E:Multies).", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 15\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature** by F.–Kuca (Inventiones 2025). Pairwise-independent integer polynomial families with commuting transformations admit a common characteristic Host–Kra factor, and the associated joint-ergodicity conclusion holds." }, { "id": 4800016, "problem_number": "AMR-047-0016", "title": "Multiple ergodic averages — Problem 16", "statement": "Suppose that the polynomials $p_1,\\ldots, p_\\ell\\in \\mathbb Z[t]$ are rationally independent. Show that $\\mathcal{K}_{rat}(T_1),\\ldots, \\mathcal{K}_{rat}(T_\\ell)$ are characteristic factors for the averages (source reference E:Multies).", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 16\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature** by F.–Kuca (Inventiones 2025): rational Kronecker factors $\\mathcal K_{rat}(T_i)$ are characteristic for commuting transformations with rationally independent polynomial iterates." }, { "id": 4800017, "problem_number": "AMR-047-0017", "title": "Multiple ergodic averages — Problem 17", "statement": "Suppose that the polynomials $p_1,\\ldots,p_\\ell\\in \\mathbb Z[t]$ are rationally independent and have zero constant term. Show that for every $A\\in \\mathcal X$ and every $\\varepsilon>0$, there exists $n\\in \\mathbb N$ such that $$ \\mu(A\\cap T_1^{p_1(n)}A\\cap \\cdots\\cap T_\\ell^{p_\\ell(n)}A)\\geq \\mu(A)^{\\ell+1}-\\varepsilon. $$", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 17\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open**: The characteristic-factor and joint-ergodicity structure (Walsh 2012, Austin 2015, F.–Kuca 2025) is established, giving the multiple-recurrence quantity and its limit, but the sharp lower bound $\\mu(A)^{\\ell+1}-\\varepsilon$ for rationally independent polynomials is not recorded as settled." }, { "id": 4800018, "problem_number": "AMR-047-0018", "title": "Multiple ergodic averages — Problem 18", "statement": "Let $(X,\\mathcal X,\\mu,T_1,\\ldots, T_\\ell)$ be a system and $\\{p_1,\\ldots, p_\\ell\\}$ be a family of intersective integer polynomials. Show that for every set $A\\in \\mathcal X$ with $\\mu(A)>0$ one has $$ \\mu(A\\cap T_1^{p_1(n)}A\\cap \\cdots\\cap T_\\ell^{p_\\ell(n)}A)>0 $$ for some $n\\in\\mathbb N$.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 18\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** as far as verifiable for general intersective polynomial families with commuting transformations. Solved special cases: zero constant term (Polynomial Szemerédi), equal transformations (BLL08), rationally independent (F.–Kuca 2025)." }, { "id": 4800019, "problem_number": "AMR-047-0019", "title": "Multiple ergodic averages — Problem 19", "statement": "Let $(X,\\mathcal X,\\mu, T,S)$ be a system and $f,g\\in L^\\infty(\\mu)$ be functions. Show that the averages $$ \\frac1N \\sum_{n=1}^N f(T^nx)\\cdot g(S^nx) $$ converge pointwise almost everywhere.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 19\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress**: pointwise convergence for commuting $T,S$ holds in distal systems (Donoso–Sun 2018) and for linear-power cases (Hua–Sun 2014). The general (non-distal) case remains open. Literature status: The author's progress page (November 2025) records: - **S. Donoso and W. Sun, \"Pointwise convergence of some multiple ergodic averages\", Advances in Mathematics 330 (2018), no. 3, 946-996** — the problem (distal case) is solved: pointwise convergence for commuting transformations in distal systems. The source also credits Huang–Shao–Ye and Hu–Sun–… for specific cases. - The source (2016, verified) notes pointwise convergence was previously known for $S=T^k$ (linear powers of one transformation, Hua–Sun 2014) and for distal systems (Donoso–Sun). The full non-distal case remains open." }, { "id": 4800020, "problem_number": "AMR-047-0020", "title": "Multiple ergodic averages — Problem 20", "statement": "Is it true that one always has a decomposition $$ \\int f_0 \\cdot T_1^n f_1 \\cdot \\ldots \\cdot T_\\ell^n f_\\ell \\ d\\mu= \\psi(n)+e(n) $$ where $(\\psi(n))$ is an $\\ell$-step nilsequence and $\\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{n=1}^N|e(n)|=0$?", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 20\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature** as of 2025 by J. Leng (\"Structured extensions and multi-correlation sequences\", preprint 2025): the commuting-transformation multi-correlation sequence decomposes as an $\\ell$-step nilsequence plus a mean-zero null sequence. (Verified via the author's November 2025 progress page; the preprint itself could not be fetched under network restrictions.)" }, { "id": 4800021, "problem_number": "AMR-047-0021", "title": "Multiple ergodic averages — Problem 21", "statement": "Let $(X,\\mathcal X,\\mu)$ be a probability space, $T_1,\\ldots, T_\\ell \\colon X\\to X$ be invertible measure preserving transformations, and $p_1,\\ldots,p_\\ell$ be distinct polynomials with zero constant term. Show that for every $A\\in \\mathcal X$ with $\\mu(A)>0$ we have $$ \\mu(A\\cap T_1^{m+p_1(n)}A\\cap\\cdots\\cap T_\\ell^{m+p_\\ell(n)}A)>0 $$ for some $m,n\\in \\mathbb N$.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 21\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** in general for distinct non-commuting transformations. Solved special cases: rationally independent polynomials (F.–Zorin-Kranich 2015), weak-mixing (CF11), and possibly linear/equal cases. Literature status: The author's progress page (November 2025) does not record a full resolution (marked NEEDS_REVIEW). Verified related results: - The linear-polynomial analogue with extra variable (the averages (E:Polynomial), Chu–Frantzikinakis 2011, CF11) gives pointwise convergence; the multiple recurrence result is the open problem here. - For rationally independent polynomials, multiple recurrence for non-commuting transformations is known: **N. F. and P. Zorin-Kranich, \"Multiple recurrence for non-commuting transformations along rationally independent polynomials\", Ergodic Theory & Dynamical Systems 35 (2015), no. 2, 403-411** (this resolves the $\\ell$-variable rationally-independent case). - Weak-mixing case: known (characteristic factors trivial, CF11). - The distinct-polynomial general case (e.g.…" }, { "id": 4800022, "problem_number": "AMR-047-0022", "title": "Multiple ergodic averages — Problem 22", "statement": "Here $\\mathcal F=\\{a_1,\\ldots,a_\\ell\\}$ is a family of functions of polynomial growth in one Hardy field, and $\\operatorname{span}^*(\\mathcal F)$ denotes its nonzero linear combinations. . Show that the family of sequences $\\{([a_1(n)]),\\ldots,$ $([a_\\ell(n)])\\}$ is good for $\\ell$-convergence of a single transformation if and only if every function $a\\in\\text{span}^*(\\mathcal{F})$ satisfies one of the following conditions: • $|a(t)-cp(t)|/ \\log t\\to \\infty$ for every $c\\in\\mathbb R$ and $p\\in \\mathbb Z[t]$; \\text{ or } • $a(t)-cp(t)\\to d$ for some $c,d\\in\\mathbb R$; \\text{ or } • $|a(t)-t/m|\\leq C \\log{t}$ for some non-zero $m\\in\\mathbb Z$ and $C>0$.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 22\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress**: sufficient and necessary conditions are understood for substantial classes via the equidistribution work of Richter (2023) and Tsinas (2024) on Hardy-field functions and nilmanifolds, but the full characterization for all Hardy-field families is not recorded as solved." }, { "id": 4800023, "problem_number": "AMR-047-0023", "title": "Multiple ergodic averages — Problem 23", "statement": "Here $\\mathcal F=\\{a_1,\\ldots,a_\\ell\\}$ is a family of functions of polynomial growth in one Hardy field, and $\\operatorname{span}^*(\\mathcal F)$ denotes its nonzero linear combinations. and suppose that for every function $a\\in \\text{span}^*(\\mathcal{F})$ we have $|a(t)-cp(t)|/ \\log{t}\\to\\infty$ for every $c\\in \\mathbb R$ and $p\\in \\mathbb Z[t]$. Show that for every ergodic system $(X,\\mathcal{B},\\mu,T)$ and functions $f_1,\\dots,f_\\ell\\in L^\\infty(\\mu)$ we have $$ \\lim_{N\\to\\infty} \\frac1N \\sum_{n=1}^N T^{[a_1(n)]}f_1\\cdots T^{ [a_\\ell(n)]}f_\\ell=\\int f_1\\, d\\mu\\, \\cdots \\, \\int f_\\ell\\, d\\mu $$ where the convergence takes place in $L^2(\\mu)$.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 23\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature** by Bergelson–Moreira–Richter (Advances in Mathematics 443, 2024, 109597): the Hardy-field joint convergence averages converge to the product of integrals under the hypothesis that every nonzero linear combination grows faster than any $cp(t)$ relative to $\\log t$." }, { "id": 4800024, "problem_number": "AMR-047-0024", "title": "Multiple ergodic averages — Problem 24", "statement": "Let $a,b$ be distinct positive non-integers. Show that for every ergodic system $(X,\\mathcal{X},\\mu,T)$ and functions $f, g \\in L^\\infty(\\mu)$, we have $$ \\lim_{N\\to\\infty} \\frac1N \\sum_{n=1}^N f(T^{[n^a]}x) \\cdot g(T^{[n^b]}x)=\\int f \\ d\\mu \\cdot \\int g\\ d\\mu $$ for almost every $x\\in X$.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 24\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress**: mean convergence is established; pointwise convergence remains open in general (the $a,b>1$ cases are open), with recent partial results for restricted exponents ($c\\in(1,23/22)$, Daskalakis 2025) and bilinear polynomial pointwise theorems (Krause–Mirek–Tao 2022)." }, { "id": 4800025, "problem_number": "AMR-047-0025", "title": "Multiple ergodic averages — Problem 25", "statement": "Here $\\mathcal F=\\{a_1,\\ldots,a_\\ell\\}$ is a family of functions of polynomial growth in one Hardy field, and $\\operatorname{span}^*(\\mathcal F)$ denotes its nonzero linear combinations. and suppose that for every function $a\\in \\text{span}^*(\\mathcal{F})$ we have $|a(t)-cp(t)|\\to \\infty$ for every $c\\in \\mathbb R$ and $p\\in \\mathbb Z[t]$. Show that the collection of sequences $\\{([a_1(n)]),\\ldots,([a_\\ell(n)])\\}$ is good for $\\ell$-recurrence of a single transformation.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 25\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature** by Bergelson–Moreira–Richter (Advances 368, 2020, 107-146): the Hardy-field family $\\{([a_i(n)])\\}$ is good for $\\ell$-recurrence of a single transformation under the non-polynomial-growth hypothesis on all nonzero linear combinations." }, { "id": 4800026, "problem_number": "AMR-047-0026", "title": "Multiple ergodic averages — Problem 26", "statement": "Find an example of a function $a\\in \\mathcal H$ that grows faster than polynomials, meaning, $a(t)/t^k \\to\\infty$ for every $k\\in \\mathbb N$, such that the sequence $[a(n)]$ is good for multiple recurrence and convergence of powers.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 26\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** as far as verifiable. This is directly tied to AMR-047-0008 (explicit fast-growing good sequence); a positive answer here would provide such an example. Literature status: The author's progress page (November 2025) does not record a resolution. Verified: - The source notes natural candidates ($[n^{(\\log n)^a}]$, $[e^{n^b}]$) are extremely hard; for $[n^{\\log\\log n}]$ even 2-recurrence/2-convergence on all weak-mixing or nilsystems is not known. - Exponential-sum estimates for the exponential function case are largely unavailable (only $a\\in(0,1/2)$ for the first candidate, per Karamata-type estimates cited in the source). No verified example of a super-polynomial Hardy sequence good for multiple recurrence/convergence was found." }, { "id": 4800027, "problem_number": "AMR-047-0027", "title": "Multiple ergodic averages — Problem 27", "statement": "Let $c$ be a positive non-integer. Show that the sequence $([p_n^c])$ is good for multiple recurrence and convergence of powers.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 27\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature**: fractional powers of primes $([p_n^c])$ are good for multiple recurrence and convergence of powers (Koutsogiannis–Tsinas, to appear J. Modern Dynamics; supported by N.F. Sigma 2022). Recurrence for $c<1$ was already known (finite-miss range)." }, { "id": 4800028, "problem_number": "AMR-047-0028", "title": "Multiple ergodic averages — Problem 28", "statement": "Show that the sequence $([n \\sin n])$ is good for multiple recurrence and convergence of powers.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 28\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Open** for $\\ell\\ge2$ as far as verifiable; known for $\\ell=1$. The multiple-recurrence/convergence of powers for $[n\\sin n]$ is not recorded as solved. Literature status: The author's progress page (November 2025) does not record a resolution for the general oscillatory problem. Verified: - Known for $\\ell=1$ (via equidistribution results in [BK90], per source's remark). - The problem has not been studied for $\\ell\\ge 2$ even for nilsystems or weak-mixing systems (per source). Related: the fractional-power/corners results (Daskalakis 2025, F.–Kuca sparse corners) concern non-oscillatory fractional-power sequences, not $[n\\sin n]$. No verified progress on the $\\ell\\ge2$ oscillatory case was found." }, { "id": 4800029, "problem_number": "AMR-047-0029", "title": "Multiple ergodic averages — Problem 29", "statement": "Show that if $c>1$ is not an integer, then the sequence $([n^c])$ is good for multiple recurrence and convergence of commuting transformations. Moreover, show that if $(X,\\mathcal X,\\mu, T_1,\\ldots,T_\\ell)$ is a system and $f_1,\\ldots,f_\\ell\\in L^\\infty(\\mu)$, then the $L^2(\\mu)$-limit $$ \\lim_{N\\to\\infty} \\frac1N \\sum_{n=1}^N T^{[n^c]}_1f_1 \\cdots T^{[n^c]}_\\ell f_\\ell $$ is equal to the $L^2(\\mu)$-limit $\\lim_{N\\to\\infty} \\frac1N \\sum_{n=1}^N T^{n}_1 f_1\\cdots T^{n}_\\ell f_\\ell $.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 29\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Solved in the literature** (main content) by N. F. (TAMS 367, 2015, 5653-5692): $([n^c])$, $c>1$ non-integer, is good for multiple recurrence and convergence of commuting transformations, extending the multidimensional/Hardy joint-ergodicity framework." }, { "id": 4800030, "problem_number": "AMR-047-0030", "title": "Multiple ergodic averages — Problem 30", "statement": "Let $\\ell \\in \\mathbb N$ and $c,c_1,\\ldots, c_\\ell$ be positive real numbers. Show that the prime numbers contain patterns of the form $$ \\{m,m+[n^{c}],m+2[n^{c}], \\ldots, m+\\ell[n^c]\\} \\quad \\text{ and } \\quad \\{m,m+[n^{c_1}],\\ldots,m+[n^{c_\\ell}]\\} $$ for infinitely many $n\\in\\mathbb N$.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 30\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress**: prime patterns with fractional-power differences are established in restricted cases ($\\ell=2$, $c\\in(1,23/22)$ by Daskalakis 2025; two-polynomial weak convergence by F.–Kuca 2025 sparse corners). Full generality remains open." }, { "id": 4800031, "problem_number": "AMR-047-0031", "title": "Multiple ergodic averages — Problem 31", "statement": "Suppose that $n\\sigma_n\\to\\infty$. Show that almost surely the sequence $(a_n(\\omega))$ is good for multiple recurrence and convergence of commuting transformations. Moreover, show that almost surely the following holds: For every system $(X,\\mathcal X,\\mu, T_1,\\ldots,T_\\ell)$ and functions $f_1,\\ldots, f_\\ell \\in L^\\infty(\\mu)$, the averages %%$$ $$ \\frac{1}{N}\\sum_{n=1}^N T_1^{a_n(\\omega)}f_1\\cdots T_\\ell^{ a_n(\\omega)}f_\\ell $$ converge in $L^2(\\mu)$ and their limit equals the limit of the averages $\\frac{1}{N}\\sum_{n=1}^N T_1^{n}f_1\\cdots T_\\ell^nf_\\ell$.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 31\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** as far as verifiable; partial progress exists (FLW 2016 random differences; random-set Szemerédi threshold works). The a.s. multiple recurrence/convergence for commuting transformations under $n\\sigma_n\\to\\infty$ is not recorded as solved." }, { "id": 4800032, "problem_number": "AMR-047-0032", "title": "Multiple ergodic averages — Problem 32", "statement": "Suppose that $n\\sigma_n\\to\\infty$. Show that almost surely the following holds: For every system $(X,\\mathcal X,\\mu, T,S)$ and functions $f, g \\in L^\\infty(\\mu)$, we have $$ \\lim_{N\\to\\infty} \\frac{1}{N}\\sum_{n=1}^N T^nf\\cdot S^{a_n(\\omega)}g= \\mathbb E(f|\\mathcal{I}_T)\\cdot \\mathbb E(g|\\mathcal{I}_S) $$ where the limit is taken in $L^2(\\mu)$. Furthermore, if $\\sigma_n=n^{-a}$ for some $a\\in (0,1)$, show that the convergence also holds pointwise almost everywhere.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 32\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress**: the bilinear commuting limit is established for restricted growth regimes ($a\\in(0,1/14)$ in general, $a\\in(0,1/2)$ when $T=S$; FLW). The full range $a\\in(0,1)$ and the general pointwise statement remain open." }, { "id": 4800033, "problem_number": "AMR-047-0033", "title": "Multiple ergodic averages — Problem 33", "statement": "Suppose that $a,b\\in(0,1)$ and $a\\neq b$. Show that almost surely the following holds: For every system $(X,\\mathcal X,\\mu,T,S)$ and functions $f, g \\in L^\\infty(\\mu)$ we have $$ \\lim_{N\\to\\infty} \\frac{1}{N}\\sum_{n=1}^N T^{a_n(\\omega)}f\\cdot S^{b_n(\\omega)}g= \\mathbb E(f|\\mathcal{I}_T)\\cdot \\mathbb E(g|\\mathcal{I}_S) $$ where the limit is taken in $L^2(\\mu)$ or pointwise.", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 33\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** as far as verifiable. No values of $a,b\\in(0,1)$ are known for which the sharp conclusion (in $L^2$ or pointwise) holds. Literature status: The author's progress page (November 2025) does not record a resolution. The source notes the problem seems non-trivial even when $T=S$ is weak mixing, and that no values of $a,b\\in(0,1)$ are known for which the conclusion holds. No verified progress found." }, { "id": 4800034, "problem_number": "AMR-047-0034", "title": "Multiple ergodic averages — Problem 34", "statement": "Let $(X,\\mathcal X,\\mu, T_n)$ be a measure preserving system with multiplicative structure and $A\\in \\mathcal X$ with $\\mu(A)>0$. Is it true that there exist $m,n\\in \\mathbb N$, $m> n$, such that $$ \\mu\\bigl(T_{2mn} A\\cap T_{(m-n)(m+n)} A \\bigr)>0\\, ? $$", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 34\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** as far as verifiable. A positive answer would establish density regularity for Pythagorean pairs; only the \"easier\" quadratic pattern ($9x^2+16y^2$) is known. Literature status: The author's progress page (November 2025) does not record a resolution (marked NEEDS_REVIEW). Verified context: - The pairing indices $2mn$ and $(m-n)(m+n)$ correspond to the Pythagorean-pair equation $x^2+y^2=\\lambda^2$ (partition regularity of Pythagorean pairs); a positive answer would prove density regularity for Pythagorean pairs. - The related two-factor result (E:part) — $\\mu(T_{m(m+n)}A\\cap T_{(m+2n)(m+3n)}A)>0$ — is established (FH15a, F.–Host 2015) and yields partition regularity of $9x^2+16y^2=\\lambda^2$. No verified resolution of the Pythagorean-pair variant (this problem) was found." }, { "id": 4800035, "problem_number": "AMR-047-0035", "title": "Multiple ergodic averages — Problem 35", "statement": "Let $(X,\\mathcal X,\\mu, T_n)$ be a measure preserving system with multiplicative structure and $A\\in \\mathcal X$ with $\\mu(A)>0$. Is it true that there exist $m,n\\in \\mathbb N$ such that $$ \\mu\\bigl(T_{m(m+n)} A\\cap T_{(m+2n)(m+3n)}\\cap T_{(m+4n)(m+5n)} A \\bigr)>0\\, ? $$", "background": "Difficulty assignment: default L3\nSource list: Frantzikinakis - Some open problems on multiple ergodic averages (2016)\nSource item: Problem 35\nSource URL: https://arxiv.org/abs/1103.3808\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the author's November 2025 progress page does not record a resolution of this numbered problem\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikos Frantzikinakis", "proposed_year": 2016, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** as far as verifiable. Higher-order (three-factor) multiple recurrence for systems with multiplicative structure is not recorded as solved; a positive answer would give partition/density regularity for a three-variable quadratic equation." }, { "id": 4900001, "problem_number": "AMR-048-0001", "title": "Arnold and Arnold–Givental conjectures", "statement": "For a Hamiltonian diffeomorphism of a closed symplectic manifold, prove the Arnold lower bound on its number of fixed points in terms of Morse-theoretic data. In the Lagrangian Arnold–Givental form, prove the corresponding lower bound for intersections of a Lagrangian submanifold with its Hamiltonian image under the standard hypotheses.", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 1\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Vladimir Arnold and Alexander Givental", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The Arnold and Arnold–Givental lower bounds are established in the main geometric settings (symplectically aspherical, weakly monotone, monotone, exact Lagrangian) using Floer homology. A general unconditional statement for arbitrary closed symplectic manifolds / arbitrary Lagrangians remains open. Classification: PARTIAL-PROGRESS." }, { "id": 4900002, "problem_number": "AMR-048-0002", "title": "Berry–Tabor conjecture", "statement": "For a generic quantum system whose classical counterpart is integrable, prove that the unfolded high-energy level spacings have Poisson statistics.", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 2\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Michael Berry and Michael Tabor", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS: the Berry–Tabor prediction is proved for specific integrable models (two-point/pair correlation of almost all flat tori, inhomogeneous quadratic forms, Aharonov–Bohm ring under Diophantine hypotheses) but the full level-spacing conjecture for generic integrable systems remains open." }, { "id": 4900003, "problem_number": "AMR-048-0003", "title": "Banach's simple Lebesgue spectrum problem", "statement": "Does there exist an ergodic measure-preserving transformation whose Koopman operator has simple Lebesgue spectrum?", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 3\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Stefan Banach", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL. The σ-finite version of Banach's problem is **solved** (el Abdalaoui 2015, arXiv:1508.06439): a conservative ergodic σ-finite transformation with simple Lebesgue spectrum exists. The finite-measure (probability-space) version remains **open**." }, { "id": 4900006, "problem_number": "AMR-048-0006", "title": "Eden's conjecture on local Lyapunov dimension", "statement": "For a smooth dissipative dynamical system with a global attractor, is the supremum of the local Lyapunov dimension on the attractor attained at an equilibrium or at an unstable periodic orbit contained in the attractor?", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 6\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alp Eden", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears open in full generality (OPEN-TRIAGE). The local Lyapunov dimension is known to give rigorous Hausdorff-dimension bounds, and the conjecture that its supremum is attained at an equilibrium/unstable periodic orbit is confirmed in special (mostly low-dimensional or structurally stable) cases, but no general proof could be verified." }, { "id": 4900009, "problem_number": "AMR-048-0009", "title": "Kaplan–Yorke dimension conjecture", "statement": "Under the hypotheses in which the Lyapunov (Kaplan–Yorke) dimension is defined from the ordered Lyapunov exponents, prove that it equals the appropriate dimension of the attractor or invariant measure.", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 9\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "James Kaplan and James Yorke", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. The Lyapunov (Kaplan–Yorke) dimension is a rigorous upper bound for the Hausdorff dimension of invariant sets/measures in broad classes, and equality with the actual dimension is established for SRB and sufficiently regular (absolutely continuous) invariant measures (Ledrappier–Young). The fully general Kaplan–Yorke equality for arbitrary dissipative attractors/invariant measures remains open, and equality can fail for non-SRB measures." }, { "id": 4900010, "problem_number": "AMR-048-0010", "title": "Margulis measure-classification conjecture", "statement": "Classify invariant ergodic probability measures for higher-rank diagonalizable group actions on homogeneous spaces; in particular, prove that the measures satisfying the usual nondegeneracy hypotheses are algebraic (homogeneous).", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 10\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Grigory Margulis", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. The main positive-entropy case of Margulis's measure-classification / measure-rigidity program for higher-rank diagonal actions is a theorem (Einsiedler–Katok–Lindenstrauss 2006; Lindenstrauss 2006): ergodic invariant measures of positive entropy for a higher-rank diagonal subgroup of a semisimple group are homogeneous. The fully general conjecture covering all ergodic invariant measures (notably the zero-entropy / non-algebraic-measures case) remains open. Consequently, the statement \"measures satisfying the usual nondegeneracy hypotheses are algebraic\" is, under the positive-entropy (nondegeneracy) interpretation, essentially a theorem; under the widest interpretation it remains open." }, { "id": 4900013, "problem_number": "AMR-048-0013", "title": "Unbounded outer-billiard orbits for almost every polygon", "statement": "Prove that the outer billiard about almost every convex polygon has an unbounded orbit.", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 13\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. The general conjecture (\"almost every convex polygon has an unbounded outer-billiard orbit\") remains **open**. Strong partial results exist: unbounded orbits for all irrational kites (Schwartz), for semi-disks/near-semi-disks (Dolgopyat–Fayyad), while all orbits are bounded for quasi-rational polygons (Vivaldi–Shaidenko / Kolodziej / Gutkin–Simanyi). The original Moser–Neumann question is answered positively, but the full measure-generic statement is unsettled." }, { "id": 4900014, "problem_number": "AMR-048-0014", "title": "Quantum unique ergodicity", "statement": "Let $M$ be a compact negatively curved Riemannian manifold. Do the probability measures $|\\varphi_j|^2\\,d\\operatorname{vol}$ associated with every orthonormal sequence of Laplace eigenfunctions equidistribute to normalized volume as the eigenvalues tend to infinity?", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 14\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Zeev Rudnick and Peter Sarnak", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. QUE is **proved** for compact arithmetic (congruence) negatively curved surfaces (Lindenstrauss 2006; Soundararajan for noncompact), where the microlocal limits along the full Hecke-compatible orthonormal basis equidistribute. For **general** (non-arithmetic) negatively curved manifolds, QUE remains **open**; the general theorem is only quantum ergodicity (equidistribution along a density-one subsequence), with counterexamples to full QUE in non-ergodic settings (flat torus, round sphere, specific modular-surface bases). Anantharaman's positive-entropy theorem is a substantial partial step." }, { "id": 4900015, "problem_number": "AMR-048-0015", "title": "Rokhlin multiple-mixing problem", "statement": "Is every strongly mixing measure-preserving transformation strongly mixing of order three?", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 15\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 4, "status": "open", "proposed_by": "Vladimir Rokhlin", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. The Rokhlin multiple-mixing problem (does strong mixing imply mixing of all orders?) remains **open** in general. Positive results are known for wide natural classes (rank-one: Kalikow 1984; singular-spectrum systems: Ageev, Ledrappier), but neither a general proof nor a counterexample is available; a counterexample is widely expected." }, { "id": 4900018, "problem_number": "AMR-048-0018", "title": "Termination of juggler sequences", "statement": "Starting from a positive integer $a_0$, define $a_{n+1}=\\lfloor a_n^{1/2}\\rfloor$ when $a_n$ is even and $a_{n+1}=\\lfloor a_n^{3/2}\\rfloor$ when $a_n$ is odd. Does every such juggler sequence eventually reach $1$?", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 18\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. The juggler-sequence termination conjecture (\"every juggler sequence eventually reaches 1\") is **open**. It has been verified empirically for all starting values up to very large bounds (records in OEIS), but no proof of termination (or even uniform boundedness for all $a_0$) is available." }, { "id": 4900019, "problem_number": "AMR-048-0019", "title": "Completeness of Lyapunov's second method", "statement": "For which classes of ordinary differential equations do the classical and canonically generalized forms of Lyapunov's second method give necessary as well as sufficient conditions for stability or asymptotic stability of motion?", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 19\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "open", "proposed_by": "Aleksandr Lyapunov", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "LITERATURE-SURVEY. Lyapunov's second method is **complete (necessary and sufficient)** for the principal classes of ODEs under standard regularity and uniformity hypotheses: uniform asymptotic stability implies the existence of a smooth converse Lyapunov function (Massera 1949; Kurzweil 1956; Kellett–Teel for switched/hybrid and input-to-state settings). The residual caveats concern non-uniform or low-regularity (non-Lipschitz, discontinuous, differential-inclusion) systems where only weaker converse theorems are available. The question as phrased is therefore answered affirmatively for the \"classical and canonically generalized\" classes, with technical caveats rather than an open problem." }, { "id": 4900020, "problem_number": "AMR-048-0020", "title": "Local reversibility of reversible cellular automata", "statement": "In every dimension at least three, is each reversible cellular automaton locally reversible?", "background": "Difficulty assignment: default L3\nSource list: Dynamical-systems problems from Wikipedia\nSource item: current Dynamical systems bullet 20\nSource URL: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: NEEDS_REVIEW; the current Wikipedia list classifies the item as unsolved, but Wikipedia is not an authoritative resolution source\nRights note: Wikipedia text is available under CC BY-SA 4.0; attribution and share-alike compatibility require release review", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. The local-reversibility question for reversible cellular automata in dimension $d\\ge3$ could not be verified with an authoritative citation in this session. The classical theory (Richardson 1972; Hedlund) shows that in dimension 1 injectivity ⟺ surjectivity ⟺ invertibility by a CA, and known 2-dimensional reversible CAs can fail to be locally reversible; the $d\\ge3$ claim is a specific open conjecture that requires further authoritative confirmation before classification beyond OPEN-TRIAGE." }, { "id": 5000001, "problem_number": "AMR-049-0001", "title": "Closed-versus-preclosed trajectory lengths", "statement": "In a regular $n$-gon, call a trajectory preclosed when its endpoints divide their boundary edges into equal-length parts and meet those oriented edges at equal angles. Closed and preclosed trajectories are strongly parallel when they have the same sequence of reflection edges. After canceling all common factors from the ratios of lengths of a family of parallel closed trajectories, prove that the remaining integer factors relating closed-trajectory lengths to the corresponding preclosed-trajectory lengths are never equal to $n$.", "background": "Difficulty assignment: default L3\nSource list: Fuchs - Billiard Trajectories in Regular Polygons and Foliations by Closed Geodesics on Surfaces (2020)\nSource item: Conjecture 1.7\nSource URL: https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dmitry Fuchs", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. The conjecture that the integer factor $n$ never appears in the reduced closed/preclosed length ratios is unresolved. Literature status: - This is a conjecture formulated from computer experiments in Fuchs' paper; the paper presents it as open. - Related theory: Fuchs' earlier work (with Tabachnikov) on periodic trajectories in regular polyhedra/polygons gives the framework of closed vs. preclosed (sometimes \"pseudo-closed\") trajectories and length ratio structure; the specific number-theoretic conjecture (avoidance of the factor $n$) is not resolved. - No later resolution was located via web search through 2026." }, { "id": 5000002, "problem_number": "AMR-049-0002", "title": "Types of vertices of reachable polygons", "statement": "Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and meets no vertex in between; a reachable point is an endpoint vertex occurring in such an unfolding. For an oriented short trajectory, let $\\alpha$ and $\\beta$ be its source and target endpoint angles as in Definition 2.1 of the source, and write $A_k$ for the resulting one of $n-2$ trajectory types (indices modulo $n-2$); in particular $A_0$ is characterized by $\\beta-\\alpha=2\\pi/n$. Call an $n$-gon reachable if it is an $SL(2,\\mathbb R)$ image of the original regular $n$-gon and all its vertices except the distinguished vertex $O$ are reachable points. If $n\\ge5$ and its clockwise vertices are $v_0=O,v_1,\\ldots,v_{n-1}$, prove that $v_1,\\ldots,v_{n-1}$ have types $A_0,A_1,\\ldots,A_{n-3},A_0$, respectively.", "background": "Difficulty assignment: default L3\nSource list: Fuchs - Billiard Trajectories in Regular Polygons and Foliations by Closed Geodesics on Surfaces (2020)\nSource item: Conjecture 2.3\nSource URL: https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dmitry Fuchs", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. The vertex-type pattern $A_0,A_1,\\ldots,A_{n-3},A_0$ for reachable $n$-gons is unresolved. Literature status: - This is one of a family of computational-geometry conjectures (Conjectures 2.3–2.7) about reachable points and reachable polygons in regular $n$-gons, presented as open in Fuchs' 2020 paper. - The author's later work and the billiards literature (Fuchs 2017/2020; Fuchs–Tabachnikov) develop the \"short trajectory / reachable point\" framework; the specific vertex-type pattern conjecture is not resolved. - No later resolution was located via web search through 2026." }, { "id": 5000003, "problem_number": "AMR-049-0003", "title": "Reachable points lie on reachable polygons", "statement": "Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and meets no vertex in between; a reachable point is an endpoint vertex occurring in such an unfolding. For an oriented short trajectory, let $\\alpha$ and $\\beta$ be its source and target endpoint angles as in Definition 2.1 of the source, and write $A_k$ for the resulting one of $n-2$ trajectory types (indices modulo $n-2$); in particular $A_0$ is characterized by $\\beta-\\alpha=2\\pi/n$. Prove that every reachable point is a vertex of infinitely many reachable $n$-gons.", "background": "Difficulty assignment: default L3\nSource list: Fuchs - Billiard Trajectories in Regular Polygons and Foliations by Closed Geodesics on Surfaces (2020)\nSource item: Conjecture 2.4\nSource URL: https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dmitry Fuchs", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. It is not proven that every reachable point lies on infinitely many reachable $n$-gons. Literature status: - Part of the same conjectural family (2.3–2.7) in Fuchs' paper, presented as open. - The paper develops these conjectures from extensive computational evidence; resolution would require a full classification of reachable points/polygons. - No later resolution was located via web search through 2026." }, { "id": 5000004, "problem_number": "AMR-049-0004", "title": "Reachable points on lines through a unitary pair", "statement": "Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and meets no vertex in between; a reachable point is an endpoint vertex occurring in such an unfolding. For an oriented short trajectory, let $\\alpha$ and $\\beta$ be its source and target endpoint angles as in Definition 2.1 of the source, and write $A_k$ for the resulting one of $n-2$ trajectory types (indices modulo $n-2$); in particular $A_0$ is characterized by $\\beta-\\alpha=2\\pi/n$. For $n\\ge5$, call reachable points $u,v$ of type $A_0$ a unitary pair if $\\det(u,v)=\\sin((n-2)\\pi/n)$, and put $\\lambda=2\\cos(\\pi/n)$. Prove: (a) $u_m=u+m(\\lambda+1)v$ is reachable of type $A_0$ for every integer $m$ for which it lies in the upper half-plane; (b) $w_m=u+(\\lambda+m(\\lambda+1))v$ is reachable of type $A_1$ whenever it lies there; (c) there are no other reachable points on $u+\\mathbb Rv$; and (d) the analogous description on $v+\\mathbb Ru$ yields only types $A_0$ and $A_{n-3}$.", "background": "Difficulty assignment: default L3\nSource list: Fuchs - Billiard Trajectories in Regular Polygons and Foliations by Closed Geodesics on Surfaces (2020)\nSource item: Conjecture 2.5\nSource URL: https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dmitry Fuchs", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. The explicit parametrization of reachable points on lines through a unitary pair (and the exhaustion claim) is unresolved. Literature status: - Part of the conjectural family (2.3–2.7) in Fuchs' paper, presented as open. Gives an explicit arithmetic parametrization ($\\lambda+1 = 2\\cos(\\pi/n)+1$) of reachable points on the line through a unitary pair. - No later resolution was located via web search through 2026." }, { "id": 5000005, "problem_number": "AMR-049-0005", "title": "Types of parallel short trajectories", "statement": "Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and meets no vertex in between; a reachable point is an endpoint vertex occurring in such an unfolding. For an oriented short trajectory, let $\\alpha$ and $\\beta$ be its source and target endpoint angles as in Definition 2.1 of the source, and write $A_k$ for the resulting one of $n-2$ trajectory types (indices modulo $n-2$); in particular $A_0$ is characterized by $\\beta-\\alpha=2\\pi/n$. If a short trajectory emanating from $O$ with slope angle $\\alpha$ has type $A_k$, prove that the parallel short trajectory from $O$ with slope angle $\\ell\\pi/n+\\varepsilon\\alpha$, where $\\varepsilon\\in\\{-1,1\\}$, has type $A_{\\varepsilon k-\\ell}$. When $n$ is even, restrict $\\ell$ to even integers.", "background": "Difficulty assignment: default L3\nSource list: Fuchs - Billiard Trajectories in Regular Polygons and Foliations by Closed Geodesics on Surfaces (2020)\nSource item: Conjecture 2.6\nSource URL: https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dmitry Fuchs", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. The type-transformation rule $A_k\\mapsto A_{\\varepsilon k-\\ell}$ under slope rotation/reflection is unresolved. Literature status: - Part of the conjectural family (2.3–2.7) in Fuchs' paper, presented as open. This gives a symmetry rule for the type of parallel short trajectories from the origin under rotation/reflection of the slope angle. - No later resolution was located via web search through 2026." }, { "id": 5000006, "problem_number": "AMR-049-0006", "title": "Length ratios of parallel short trajectories", "statement": "Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and meets no vertex in between; a reachable point is an endpoint vertex occurring in such an unfolding. For an oriented short trajectory, let $\\alpha$ and $\\beta$ be its source and target endpoint angles as in Definition 2.1 of the source, and write $A_k$ for the resulting one of $n-2$ trajectory types (indices modulo $n-2$); in particular $A_0$ is characterized by $\\beta-\\alpha=2\\pi/n$. Prove that the length ratio of parallel short trajectories of types $A_k$ and $A_\\ell$ is $$\\sin\\frac{(k+1)\\pi}{n}:\\sin\\frac{(\\ell+1)\\pi}{n}.$$", "background": "Difficulty assignment: default L3\nSource list: Fuchs - Billiard Trajectories in Regular Polygons and Foliations by Closed Geodesics on Surfaces (2020)\nSource item: Conjecture 2.7\nSource URL: https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dmitry Fuchs", "proposed_year": 2020, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. The sine-ratio formula for lengths of parallel short trajectories is unresolved. Literature status: - Part of the conjectural family (2.3–2.7) in Fuchs' paper, presented as open. This gives a clean trigonometric formula (ratios of sines at $(k+1)\\pi/n$) for lengths of parallel short trajectories of the different types. - No later resolution was located via web search through 2026." }, { "id": 5000007, "problem_number": "AMR-049-0007", "title": "Short geodesics on the regular dodecahedron", "statement": "On a regular dodecahedron, unfold a geodesic beginning at a vertex $v$ through successive faces. Call it short if it ends at a vertex and meets no vertex in between, and call it type $A_0$ when the endpoint-angle parameters satisfy $\\beta-\\alpha=2\\pi/5$. Prove that such a type-$A_0$ short geodesic beginning at $v$ never ends at a vertex at graph distance $2$ from $v$ in the dodecahedron's edge graph.", "background": "Difficulty assignment: default L3\nSource list: Fuchs - Billiard Trajectories in Regular Polygons and Foliations by Closed Geodesics on Surfaces (2020)\nSource item: Conjecture 3.2\nSource URL: https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dmitry Fuchs", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. The claim that no type-$A_0$ short geodesic from $v$ ends at a graph-distance-2 vertex is unresolved. Literature status: - This is a conjecture about periodic/short geodesics on the regular dodecahedron, part of Fuchs' broader program (related to his and Tabachnikov's work on billiards in the cube/dodecahedron; the dodecahedron's short geodesics were studied by Fuchs—Tabachnikov \"More on periodic billiard trajectories in the cube\" and the dodecahedron case). - Presented as open in Fuchs' 2020 paper; no later resolution of this specific $A_0$-not-at-graph-distance-2 claim was located via web search through 2026." }, { "id": 5100001, "problem_number": "AMR-050-0001", "title": "Elliptic-billiard invariant k_{107}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Here $\\theta_i$ is the angle of $P$ at $P_i$, $L$ is its perimeter, and $l_i=|P''_i-P_i|$, $r_i=|P_{i+1}-P''_i|$. Prove that, for $N\\equiv0\\pmod4$, the quantity $(A'/A)\\prod_i\\sin(\\theta_i/2)$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{107}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{107}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Not fully proven, but verified partial progress exists (component/sibling invariants proven, or the N=3 closed form), as detailed above. A general-N proof of the exact quantity is still missing. Literature status: The source paper (arXiv:2004.12497, Table 2) lists $k_{107}$ (resp. $k_{108}$) as $k_{103}k_{105}$ (resp. $k_{103}/k_{105}$) but only for the **even** parities $N\\equiv0\\pmod4$ (resp. $N\\equiv2\\pmod4$), with 'proven ?'. Each factor ($k_{103}=A'/A$ and $k_{105}=\\prod\\sin(\\theta_i/2)$) is a proven invariant but **only for odd $N$** (k103: refs [6,11]; k105: ref [2]). Since the demanded parity here is even, neither factor's proof applies, and no published proof of the product/ratio combination for these even parities was located in the 2021-2026 follow-up literature (bicentric paper arXiv:2103.11260, inversive triangle arXiv:2012.03020, self-intersected paper arXiv:2011.06640). The invariant remains an open conjecture; it is placed under OPEN-TRIAGE pending a further targeted check." }, { "id": 5100002, "problem_number": "AMR-050-0002", "title": "Elliptic-billiard invariant k_{108}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Here $\\theta_i$ is the angle of $P$ at $P_i$, $L$ is its perimeter, and $l_i=|P''_i-P_i|$, $r_i=|P_{i+1}-P''_i|$. Prove that, for $N\\equiv2\\pmod4$, the quantity $(A'/A)/\\prod_i\\sin(\\theta_i/2)$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{108}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{108}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Not fully proven, but verified partial progress exists (component/sibling invariants proven, or the N=3 closed form), as detailed above. A general-N proof of the exact quantity is still missing. Literature status: The source paper (arXiv:2004.12497, Table 2) lists $k_{107}$ (resp. $k_{108}$) as $k_{103}k_{105}$ (resp. $k_{103}/k_{105}$) but only for the **even** parities $N\\equiv0\\pmod4$ (resp. $N\\equiv2\\pmod4$), with 'proven ?'. Each factor ($k_{103}=A'/A$ and $k_{105}=\\prod\\sin(\\theta_i/2)$) is a proven invariant but **only for odd $N$** (k103: refs [6,11]; k105: ref [2]). Since the demanded parity here is even, neither factor's proof applies, and no published proof of the product/ratio combination for these even parities was located in the 2021-2026 follow-up literature (bicentric paper arXiv:2103.11260, inversive triangle arXiv:2012.03020, self-intersected paper arXiv:2011.06640). The invariant remains an open conjecture; it is placed under OPEN-TRIAGE pending a further targeted check." }, { "id": 5100003, "problem_number": "AMR-050-0003", "title": "Elliptic-billiard invariant k_{109}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Here $\\theta_i$ is the angle of $P$ at $P_i$, $L$ is its perimeter, and $l_i=|P''_i-P_i|$, $r_i=|P_{i+1}-P''_i|$. Prove that, for odd $N$, the quantity $A/A''$ (and prove it equals the invariant $A'/A$) is constant as $P$ varies through the family. The source labels this assertion invariant code k_{109}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{109}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L3", "research_summary": "The invariant is resolved: for odd $N$, $A/A''=A'/A$ is constant (equal to $k_{103}$), as a corollary of the proven invariant $k_{112}$ ($A'A''/A^2=1$, reference [3]) together with the proven $k_{103}$." }, { "id": 5100004, "problem_number": "AMR-050-0004", "title": "Elliptic-billiard invariant k_{110}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Here $\\theta_i$ is the angle of $P$ at $P_i$, $L$ is its perimeter, and $l_i=|P''_i-P_i|$, $r_i=|P_{i+1}-P''_i|$. Prove that, for even $N$, the quantity $AA''$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{110}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{110}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{110}, cluster 'Distances, area, angles, curvature'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100005, "problem_number": "AMR-050-0005", "title": "Elliptic-billiard invariant k_{111}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Here $\\theta_i$ is the angle of $P$ at $P_i$, $L$ is its perimeter, and $l_i=|P''_i-P_i|$, $r_i=|P_{i+1}-P''_i|$. Prove that, for even $N$, the quantity $A'A''$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{111}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{111}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{111}, cluster 'Distances, area, angles, curvature'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100006, "problem_number": "AMR-050-0006", "title": "Elliptic-billiard invariant k_{114}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Here $\\theta_i$ is the angle of $P$ at $P_i$, $L$ is its perimeter, and $l_i=|P''_i-P_i|$, $r_i=|P_{i+1}-P''_i|$. Prove that, for $N\\equiv2\\pmod4$, the quantity $\\prod_i|P_i-f_1|$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{114}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{114}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{114}, cluster 'Distances, area, angles, curvature'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100007, "problem_number": "AMR-050-0007", "title": "Elliptic-billiard invariant k_{115}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Here $\\theta_i$ is the angle of $P$ at $P_i$, $L$ is its perimeter, and $l_i=|P''_i-P_i|$, $r_i=|P_{i+1}-P''_i|$. Prove that, for $N\\equiv0\\pmod4$, the quantity $\\prod_i|P'_i-f_1|$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{115}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{115}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{115}, cluster 'Distances, area, angles, curvature'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100008, "problem_number": "AMR-050-0008", "title": "Elliptic-billiard invariant k_{117}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Here $\\theta_i$ is the angle of $P$ at $P_i$, $L$ is its perimeter, and $l_i=|P''_i-P_i|$, $r_i=|P_{i+1}-P''_i|$. Prove that, for even $N$, the quantity each of $\\prod_i l_i$ and $\\prod_i r_i$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{117}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{117}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{117}, cluster 'Distances, area, angles, curvature'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100009, "problem_number": "AMR-050-0009", "title": "Elliptic-billiard invariant k_{118}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Here $\\theta_i$ is the angle of $P$ at $P_i$, $L$ is its perimeter, and $l_i=|P''_i-P_i|$, $r_i=|P_{i+1}-P''_i|$. Prove that, for odd $N$, the quantity each of $\\sum_i l_i$ and $\\sum_i r_i$, with value $L/2$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{118}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{118}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{118}, cluster 'Distances, area, angles, curvature'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100010, "problem_number": "AMR-050-0010", "title": "Elliptic-billiard invariant k_{120}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Here $\\theta_i$ is the angle of $P$ at $P_i$, $L$ is its perimeter, and $l_i=|P''_i-P_i|$, $r_i=|P_{i+1}-P''_i|$. Prove that, for all $N$, the quantity $\\sum_i\\cos\\angle P_i f_1 P_{i+1}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{120}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{120}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{120}, cluster 'Distances, area, angles, curvature'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100011, "problem_number": "AMR-050-0011", "title": "Elliptic-billiard invariant k_{203,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Drop perpendiculars from $M$ to the sides of $P$ and let $A_M$ be the area of the resulting pedal polygon. Prove that, for $N\\equiv0\\pmod4$, every $M$, the quantity $AA_M$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{203,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{203,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{203,a}, cluster 'N-periodic pedal polygons'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100012, "problem_number": "AMR-050-0012", "title": "Elliptic-billiard invariant k_{203,b}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Drop perpendiculars from $M$ to the sides of $P$ and let $A_M$ be the area of the resulting pedal polygon. Prove that, for $N\\not\\equiv2\\pmod4$, $M=O$, the quantity $AA_M$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{203,b}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{203,b}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{203,b}, cluster 'N-periodic pedal polygons'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100013, "problem_number": "AMR-050-0013", "title": "Elliptic-billiard invariant k_{204}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Drop perpendiculars from $M$ to the sides of $P$ and let $A_M$ be the area of the resulting pedal polygon. Prove that, for $N\\equiv2\\pmod4$, every $M$, the quantity $A/A_M$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{204}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{204}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{204}, cluster 'N-periodic pedal polygons'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100014, "problem_number": "AMR-050-0014", "title": "Elliptic-billiard invariant k_{303,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Drop perpendiculars from $M$ to the sides of $P'$; let $A'_M$ and $C'_2$ be the area and signed area centroid of the resulting pedal polygon. Prove that, for $N\\equiv2\\pmod4$, every $M$, the quantity $A'A'_M$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{303,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{303,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{303,a}, cluster 'Outer pedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants…" }, { "id": 5100015, "problem_number": "AMR-050-0015", "title": "Elliptic-billiard invariant k_{303,b}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Drop perpendiculars from $M$ to the sides of $P'$; let $A'_M$ and $C'_2$ be the area and signed area centroid of the resulting pedal polygon. Prove that, for $N\\not\\equiv0\\pmod4$, $M=O$, the quantity $A'A'_M$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{303,b}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{303,b}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{303,b}, cluster 'Outer pedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants…" }, { "id": 5100016, "problem_number": "AMR-050-0016", "title": "Elliptic-billiard invariant k_{304}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Drop perpendiculars from $M$ to the sides of $P'$; let $A'_M$ and $C'_2$ be the area and signed area centroid of the resulting pedal polygon. Prove that, for $N\\equiv0\\pmod4$, every $M$, the quantity $A'/A'_M$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{304}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{304}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{304}, cluster 'Outer pedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100017, "problem_number": "AMR-050-0017", "title": "Elliptic-billiard invariant k_{307}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Drop perpendiculars from $M$ to the sides of $P'$; let $A'_M$ and $C'_2$ be the area and signed area centroid of the resulting pedal polygon. Prove that, for even $N$, every $M$, the quantity the signed area centroid $C'_2$ of the pedal polygon is constant as $P$ varies through the family. The source labels this assertion invariant code k_{307}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{307}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{307}, cluster 'Outer pedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100018, "problem_number": "AMR-050-0018", "title": "Elliptic-billiard invariant k_{401}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The antipedal polygon of a polygon $R$ with respect to $M$ has sides through its vertices perpendicular to the corresponding $R_i-M$; $A_M^*,C_0^*,C_2^*$ denote its area and centroids, with primes for $R=P'$. Prove that, for $N\\equiv2\\pmod4$, every $M$, the quantity $A'A_M^*$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{401}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{401}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{401}, cluster 'Antipedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100019, "problem_number": "AMR-050-0019", "title": "Elliptic-billiard invariant k_{402}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The antipedal polygon of a polygon $R$ with respect to $M$ has sides through its vertices perpendicular to the corresponding $R_i-M$; $A_M^*,C_0^*,C_2^*$ denote its area and centroids, with primes for $R=P'$. Prove that, for $N\\equiv0\\pmod4$, every $M$, the quantity $A'/A_M^*$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{402}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{402}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{402}, cluster 'Antipedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100020, "problem_number": "AMR-050-0020", "title": "Elliptic-billiard invariant k_{403,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The antipedal polygon of a polygon $R$ with respect to $M$ has sides through its vertices perpendicular to the corresponding $R_i-M$; $A_M^*,C_0^*,C_2^*$ denote its area and centroids, with primes for $R=P'$. Prove that, for odd $N$, $M=O$, the quantity $A_MA_M^*$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{403,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{403,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{403,a}, cluster 'Antipedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100021, "problem_number": "AMR-050-0021", "title": "Elliptic-billiard invariant k_{403,b}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The antipedal polygon of a polygon $R$ with respect to $M$ has sides through its vertices perpendicular to the corresponding $R_i-M$; $A_M^*,C_0^*,C_2^*$ denote its area and centroids, with primes for $R=P'$. Prove that, for $N\\equiv0\\pmod4$, $M=f_1$ or $f_2$, the quantity $A_MA_M^*$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{403,b}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{403,b}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{403,b}, cluster 'Antipedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100022, "problem_number": "AMR-050-0022", "title": "Elliptic-billiard invariant k_{404}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The antipedal polygon of a polygon $R$ with respect to $M$ has sides through its vertices perpendicular to the corresponding $R_i-M$; $A_M^*,C_0^*,C_2^*$ denote its area and centroids, with primes for $R=P'$. Prove that, for $N\\equiv2\\pmod4$, $M=f_1$ or $f_2$, the quantity $A_M^*/A_M$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{404}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{404}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{404}, cluster 'Antipedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100023, "problem_number": "AMR-050-0023", "title": "Elliptic-billiard invariant k_{405}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The antipedal polygon of a polygon $R$ with respect to $M$ has sides through its vertices perpendicular to the corresponding $R_i-M$; $A_M^*,C_0^*,C_2^*$ denote its area and centroids, with primes for $R=P'$. Prove that, for even $N$, $M=O,f_1,$ or $f_2$, the quantity the vertex centroid $C_0^*$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{405}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{405}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{405}, cluster 'Antipedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100024, "problem_number": "AMR-050-0024", "title": "Elliptic-billiard invariant k_{406,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The antipedal polygon of a polygon $R$ with respect to $M$ has sides through its vertices perpendicular to the corresponding $R_i-M$; $A_M^*,C_0^*,C_2^*$ denote its area and centroids, with primes for $R=P'$. Prove that, for even $N$, $M=O$, the quantity the vertex and signed area centroids ${C'_0}^*,{C'_2}^*$ (both equal $O$) is constant as $P$ varies through the family. The source labels this assertion invariant code k_{406,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{406,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{406,a}, cluster 'Antipedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100025, "problem_number": "AMR-050-0025", "title": "Elliptic-billiard invariant k_{406,b}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The antipedal polygon of a polygon $R$ with respect to $M$ has sides through its vertices perpendicular to the corresponding $R_i-M$; $A_M^*,C_0^*,C_2^*$ denote its area and centroids, with primes for $R=P'$. Prove that, for $N=4$, $M=f_1$ or $f_2$, the quantity the centroids ${C'_0}^*,{C'_2}^*$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{406,b}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{406,b}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{406,b}, cluster 'Antipedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100026, "problem_number": "AMR-050-0026", "title": "Elliptic-billiard invariant k_{407}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The antipedal polygon of a polygon $R$ with respect to $M$ has sides through its vertices perpendicular to the corresponding $R_i-M$; $A_M^*,C_0^*,C_2^*$ denote its area and centroids, with primes for $R=P'$. Prove that, for even $N$, $M=f_1$ or $f_2$, the quantity the vertex centroid ${C'_0}^*$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{407}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{407}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{407}, cluster 'Antipedal polygon'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100027, "problem_number": "AMR-050-0027", "title": "Elliptic-billiard invariant k_{501}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Let $K,K',K''$ be the Steiner centroids of curvature of $P,P',P''$, and let $A_K,A'_{K'},A''_{K''}$ be the areas of their pedal polygons with respect to those centroids. Prove that, for odd $N$, the quantity $A/A_K$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{501}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{501}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Not fully proven, but verified partial progress exists (component/sibling invariants proven, or the N=3 closed form), as detailed above. A general-N proof of the exact quantity is still missing. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{501}, cluster 'Steiner curvature centroid'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source (arXiv:2004.12497, Table 6) lists the Steiner-curvature-centroid pedal ratios ($A/A_K$, $A'/A'_{K'}$, $A''/A''_{K''}$, odd $N$) as 'proven ?'. The authors note (in 'Fifty New Invariants', Arnold Math. J. 7 (2021), DOI 10.1007/s40598-021-00174-y, §3.6) that combining the proven $k_{103}$ (odd) and $k_{106}$ (even) yields as corollaries the invariance of the Steiner-pedal-area **ratios** $A'_{K'}/A_K, A''_{K''}/A_K, A''_{K''}/A'_{K'}$ for odd $N$. The…" }, { "id": 5100028, "problem_number": "AMR-050-0028", "title": "Elliptic-billiard invariant k_{502}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Let $K,K',K''$ be the Steiner centroids of curvature of $P,P',P''$, and let $A_K,A'_{K'},A''_{K''}$ be the areas of their pedal polygons with respect to those centroids. Prove that, for odd $N$, the quantity $A'/A'_{K'}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{502}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{502}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Not fully proven, but verified partial progress exists (component/sibling invariants proven, or the N=3 closed form), as detailed above. A general-N proof of the exact quantity is still missing. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{502}, cluster 'Steiner curvature centroid'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source (arXiv:2004.12497, Table 6) lists the Steiner-curvature-centroid pedal ratios ($A/A_K$, $A'/A'_{K'}$, $A''/A''_{K''}$, odd $N$) as 'proven ?'. The authors note (in 'Fifty New Invariants', Arnold Math. J. 7 (2021), DOI 10.1007/s40598-021-00174-y, §3.6) that combining the proven $k_{103}$ (odd) and $k_{106}$ (even) yields as corollaries the invariance of the Steiner-pedal-area **ratios** $A'_{K'}/A_K, A''_{K''}/A_K, A''_{K''}/A'_{K'}$ for odd $N$. The…" }, { "id": 5100029, "problem_number": "AMR-050-0029", "title": "Elliptic-billiard invariant k_{503}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. Let $K,K',K''$ be the Steiner centroids of curvature of $P,P',P''$, and let $A_K,A'_{K'},A''_{K''}$ be the areas of their pedal polygons with respect to those centroids. Prove that, for odd $N$, the quantity $A''/A''_{K''}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{503}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{503}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Not fully proven, but verified partial progress exists (component/sibling invariants proven, or the N=3 closed form), as detailed above. A general-N proof of the exact quantity is still missing. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{503}, cluster 'Steiner curvature centroid'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source (arXiv:2004.12497, Table 6) lists the Steiner-curvature-centroid pedal ratios ($A/A_K$, $A'/A'_{K'}$, $A''/A''_{K''}$, odd $N$) as 'proven ?'. The authors note (in 'Fifty New Invariants', Arnold Math. J. 7 (2021), DOI 10.1007/s40598-021-00174-y, §3.6) that combining the proven $k_{103}$ (odd) and $k_{106}$ (even) yields as corollaries the invariance of the Steiner-pedal-area **ratios** $A'_{K'}/A_K, A''_{K''}/A_K, A''_{K''}/A'_{K'}$ for odd $N$. The…" }, { "id": 5100030, "problem_number": "AMR-050-0030", "title": "Elliptic-billiard invariant k_{601}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j=1,2$, let $q_{j,i}$ and $q^*_{j,i}$ be the distances from $f_j$ to the corresponding pedal and antipedal vertices. Let $\\bar A_j,\\bar A'_j,\\bar A''_j$ be pedal areas of $P,P',P''$ with respect to $f_j$, and use a star for antipedal areas. Prove that, for odd $N$, the quantity $(\\sum_iq_{1,i})(\\sum_iq_{2,i})$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{601}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{601}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{601}, cluster 'Pairs of pedal polygons wrt foci'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100031, "problem_number": "AMR-050-0031", "title": "Elliptic-billiard invariant k_{602}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j=1,2$, let $q_{j,i}$ and $q^*_{j,i}$ be the distances from $f_j$ to the corresponding pedal and antipedal vertices. Let $\\bar A_j,\\bar A'_j,\\bar A''_j$ be pedal areas of $P,P',P''$ with respect to $f_j$, and use a star for antipedal areas. Prove that, for all $N$, the quantity $(\\prod_iq_{1,i})(\\prod_iq_{2,i})$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{602}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{602}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{602}, cluster 'Pairs of pedal polygons wrt foci'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100032, "problem_number": "AMR-050-0032", "title": "Elliptic-billiard invariant k_{603}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j=1,2$, let $q_{j,i}$ and $q^*_{j,i}$ be the distances from $f_j$ to the corresponding pedal and antipedal vertices. Let $\\bar A_j,\\bar A'_j,\\bar A''_j$ be pedal areas of $P,P',P''$ with respect to $f_j$, and use a star for antipedal areas. Prove that, for all $N$, the quantity $(\\sum_iq^*_{1,i})/(\\sum_iq^*_{2,i})$ (equal to $1$) is constant as $P$ varies through the family. The source labels this assertion invariant code k_{603}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{603}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{603}, cluster 'Pairs of pedal polygons wrt foci'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100033, "problem_number": "AMR-050-0033", "title": "Elliptic-billiard invariant k_{605,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j=1,2$, let $q_{j,i}$ and $q^*_{j,i}$ be the distances from $f_j$ to the corresponding pedal and antipedal vertices. Let $\\bar A_j,\\bar A'_j,\\bar A''_j$ be pedal areas of $P,P',P''$ with respect to $f_j$, and use a star for antipedal areas. Prove that, for odd $N$, the quantity $\\bar A'_1\\bar A'_2$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{605,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{605,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{605,a}, cluster 'Pairs of pedal polygons wrt foci'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100034, "problem_number": "AMR-050-0034", "title": "Elliptic-billiard invariant k_{606}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j=1,2$, let $q_{j,i}$ and $q^*_{j,i}$ be the distances from $f_j$ to the corresponding pedal and antipedal vertices. Let $\\bar A_j,\\bar A'_j,\\bar A''_j$ be pedal areas of $P,P',P''$ with respect to $f_j$, and use a star for antipedal areas. Prove that, for all $N$, the quantity $\\bar A_1/\\bar A_2=\\bar A'_1/\\bar A'_2$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{606}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{606}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{606}, cluster 'Pairs of pedal polygons wrt foci'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100035, "problem_number": "AMR-050-0035", "title": "Elliptic-billiard invariant k_{607}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j=1,2$, let $q_{j,i}$ and $q^*_{j,i}$ be the distances from $f_j$ to the corresponding pedal and antipedal vertices. Let $\\bar A_j,\\bar A'_j,\\bar A''_j$ be pedal areas of $P,P',P''$ with respect to $f_j$, and use a star for antipedal areas. Prove that, for $N\\equiv0\\pmod4$, the quantity $\\bar A^*_1/\\bar A^*_2$ (equal to $1$) is constant as $P$ varies through the family. The source labels this assertion invariant code k_{607}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{607}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{607}, cluster 'Pairs of pedal polygons wrt foci'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100036, "problem_number": "AMR-050-0036", "title": "Elliptic-billiard invariant k_{608}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j=1,2$, let $q_{j,i}$ and $q^*_{j,i}$ be the distances from $f_j$ to the corresponding pedal and antipedal vertices. Let $\\bar A_j,\\bar A'_j,\\bar A''_j$ be pedal areas of $P,P',P''$ with respect to $f_j$, and use a star for antipedal areas. Prove that, for even $N$, the quantity ${\\bar A'_1}^*/{\\bar A'_2}^*$ (equal to $1$) is constant as $P$ varies through the family. The source labels this assertion invariant code k_{608}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{608}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{608}, cluster 'Pairs of pedal polygons wrt foci'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100037, "problem_number": "AMR-050-0037", "title": "Elliptic-billiard invariant k_{609}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j=1,2$, let $q_{j,i}$ and $q^*_{j,i}$ be the distances from $f_j$ to the corresponding pedal and antipedal vertices. Let $\\bar A_j,\\bar A'_j,\\bar A''_j$ be pedal areas of $P,P',P''$ with respect to $f_j$, and use a star for antipedal areas. Prove that, for even $N$, the quantity $\\bar A''_1/\\bar A''_2$ (equal to $1$) is constant as $P$ varies through the family. The source labels this assertion invariant code k_{609}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{609}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{609}, cluster 'Pairs of pedal polygons wrt foci'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100038, "problem_number": "AMR-050-0038", "title": "Elliptic-billiard invariant k_{610}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j=1,2$, let $q_{j,i}$ and $q^*_{j,i}$ be the distances from $f_j$ to the corresponding pedal and antipedal vertices. Let $\\bar A_j,\\bar A'_j,\\bar A''_j$ be pedal areas of $P,P',P''$ with respect to $f_j$, and use a star for antipedal areas. Prove that, for even $N$, the quantity ${\\bar A''_1}^*/{\\bar A''_2}^*$ (equal to $1$) is constant as $P$ varies through the family. The source labels this assertion invariant code k_{610}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{610}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{610}, cluster 'Pairs of pedal polygons wrt foci'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100039, "problem_number": "AMR-050-0039", "title": "Elliptic-billiard invariant k_{701}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The evolute polygon of $R$ has vertices at successive intersections of perpendicular bisectors of its sides; $A_{ev},A'_{ev},A''_{ev}$ are the areas for $P,P',P''$. Prove that, for $N>4$, the quantity $A/A_{ev}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{701}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{701}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Not fully proven, but verified partial progress exists (component/sibling invariants proven, or the N=3 closed form), as detailed above. A general-N proof of the exact quantity is still missing. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{701}, cluster 'Evolute polygons'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source (arXiv:2004.12497, Table 8) lists the evolute-area ratios $A/A_{ev}$, $A'/A'_{ev}$, $A''/A''_{ev}$ (valid $N>4$) as 'proven ?'. In 'Fifty New Invariants' (Arnold Math. J. 7 (2021), §3.8) the authors note that combining $k_{103}$ (odd) and $k_{106}$ (even) yields the invariance of the evolute-area **sibling ratios** $A_{ev}/A'_{ev}, A_{ev}/A''_{ev}, A'_{ev}/A''_{ev}$ for all $N>4$ as corollaries. The individual ratios $A/A_{ev}$ etc. (present invariants) remain…" }, { "id": 5100040, "problem_number": "AMR-050-0040", "title": "Elliptic-billiard invariant k_{702}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The evolute polygon of $R$ has vertices at successive intersections of perpendicular bisectors of its sides; $A_{ev},A'_{ev},A''_{ev}$ are the areas for $P,P',P''$. Prove that, for $N>4$, the quantity $A'/A'_{ev}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{702}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{702}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Not fully proven, but verified partial progress exists (component/sibling invariants proven, or the N=3 closed form), as detailed above. A general-N proof of the exact quantity is still missing. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{702}, cluster 'Evolute polygons'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source (arXiv:2004.12497, Table 8) lists the evolute-area ratios $A/A_{ev}$, $A'/A'_{ev}$, $A''/A''_{ev}$ (valid $N>4$) as 'proven ?'. In 'Fifty New Invariants' (Arnold Math. J. 7 (2021), §3.8) the authors note that combining $k_{103}$ (odd) and $k_{106}$ (even) yields the invariance of the evolute-area **sibling ratios** $A_{ev}/A'_{ev}, A_{ev}/A''_{ev}, A'_{ev}/A''_{ev}$ for all $N>4$ as corollaries. The individual ratios $A/A_{ev}$ etc. (present invariants) remain…" }, { "id": 5100041, "problem_number": "AMR-050-0041", "title": "Elliptic-billiard invariant k_{703}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. The evolute polygon of $R$ has vertices at successive intersections of perpendicular bisectors of its sides; $A_{ev},A'_{ev},A''_{ev}$ are the areas for $P,P',P''$. Prove that, for $N>4$, the quantity $A''/A''_{ev}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{703}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{703}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Not fully proven, but verified partial progress exists (component/sibling invariants proven, or the N=3 closed form), as detailed above. A general-N proof of the exact quantity is still missing. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{703}, cluster 'Evolute polygons'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source (arXiv:2004.12497, Table 8) lists the evolute-area ratios $A/A_{ev}$, $A'/A'_{ev}$, $A''/A''_{ev}$ (valid $N>4$) as 'proven ?'. In 'Fifty New Invariants' (Arnold Math. J. 7 (2021), §3.8) the authors note that combining $k_{103}$ (odd) and $k_{106}$ (even) yields the invariance of the evolute-area **sibling ratios** $A_{ev}/A'_{ev}, A_{ev}/A''_{ev}, A'_{ev}/A''_{ev}$ for all $N>4$ as corollaries. The individual ratios $A/A_{ev}$ etc. (present invariants) remain…" }, { "id": 5100042, "problem_number": "AMR-050-0042", "title": "Elliptic-billiard invariant k_{802}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for all $N$, the quantity $L_j^\\dagger$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{802}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{802}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L3", "research_summary": "Resolved in the literature (all N): focus-inversions of elliptic-billiard N-periodics have constant perimeter (Roitman–Garcia–Reznik, AMJ 2021, Corollary 1(iii); also proved for N=3 in arXiv:2012.03020)." }, { "id": 5100043, "problem_number": "AMR-050-0043", "title": "Elliptic-billiard invariant k_{803}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for $N\\ne4$, the quantity $\\sum_i\\cos\\theta_{j,i}^\\dagger$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{803}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{803}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Not fully proven, but verified partial progress exists (component/sibling invariants proven, or the N=3 closed form), as detailed above. A general-N proof of the exact quantity is still missing. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{803}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source lists the focus-inversive sum-of-cosines (valid $N\\ne4$) as unproven. Verified **partial progress**: (a) the $N=3$ case is fully solved — D. Reznik, R. Garcia, M. Helman, 'The Talented Mr. Inversive Triangle in the Elliptic Billiard', arXiv:2012.03020, Proposition 4, gives $\\sum\\cos\\theta_{1,i}^\\dagger=\\tfrac{\\delta(a^2+c^2-\\delta)}{a^2c^2}$; (b) for general $N$, the bicentric paper (arXiv:2103.11260, Conjecture 1) conjectures—but does not prove—that the…" }, { "id": 5100044, "problem_number": "AMR-050-0044", "title": "Elliptic-billiard invariant k_{804,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for $N\\equiv0\\pmod4$, the quantity $AA_j^\\dagger$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{804,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{804,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{804,a}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100045, "problem_number": "AMR-050-0045", "title": "Elliptic-billiard invariant k_{804,b}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for $N=4$, the quantity $AA_j^\\dagger=4$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{804,b}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{804,b}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{804,b}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100046, "problem_number": "AMR-050-0046", "title": "Elliptic-billiard invariant k_{805}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for $N\\equiv2\\pmod4$, the quantity $A/A_j^\\dagger$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{805}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{805}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{805}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100047, "problem_number": "AMR-050-0047", "title": "Elliptic-billiard invariant k_{806,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for all $N$, the quantity ${A'_j}^\\dagger/A_j^\\dagger$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{806,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{806,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{806,a}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100048, "problem_number": "AMR-050-0048", "title": "Elliptic-billiard invariant k_{806,b}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for $N=4$, the quantity ${A'_j}^\\dagger/A_j^\\dagger=2$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{806,b}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{806,b}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{806,b}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100049, "problem_number": "AMR-050-0049", "title": "Elliptic-billiard invariant k_{807}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for even $N$, the quantity $AA^\\otimes$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{807}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{807}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{807}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100050, "problem_number": "AMR-050-0050", "title": "Elliptic-billiard invariant k_{808}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for odd $N$, the quantity $A/A^\\otimes$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{808}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{808}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{808}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100051, "problem_number": "AMR-050-0051", "title": "Elliptic-billiard invariant k_{809}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for even $N$, the quantity $A'{A'}^\\ominus$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{809}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{809}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{809}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100052, "problem_number": "AMR-050-0052", "title": "Elliptic-billiard invariant k_{810}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for odd $N$, the quantity $A'/{A'}^\\ominus$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{810}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{810}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{810}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100053, "problem_number": "AMR-050-0053", "title": "Elliptic-billiard invariant k_{811}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for all $N$, the quantity $\\sum_iw_i^2$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{811}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{811}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{811}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100054, "problem_number": "AMR-050-0054", "title": "Elliptic-billiard invariant k_{812,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for all $N$, the quantity $\\sum_i\\cos\\psi_{1,i}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{812,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{812,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{812,a}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100055, "problem_number": "AMR-050-0055", "title": "Elliptic-billiard invariant k_{812,b}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for $N=4$, the quantity $\\sum_i\\cos\\psi_{1,i}=0$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{812,b}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{812,b}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{812,b}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100056, "problem_number": "AMR-050-0056", "title": "Elliptic-billiard invariant k_{813}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for all $N$, the quantity $A_{j,pol}/A_j^\\dagger$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{813}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{813}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{813}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100057, "problem_number": "AMR-050-0057", "title": "Elliptic-billiard invariant k_{814}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for all $N$, the quantity $A_{j,pol}/A_{j,dual}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{814}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{814}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{814}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100058, "problem_number": "AMR-050-0058", "title": "Elliptic-billiard invariant k_{815}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for odd $N$, the quantity $A_{j,ped}^\\dagger A_{j,dual}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{815}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{815}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{815}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100059, "problem_number": "AMR-050-0059", "title": "Elliptic-billiard invariant k_{816}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for even $N$, the quantity $A_{j,ped}^\\dagger/A_{j,dual}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{816}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{816}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{816}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100060, "problem_number": "AMR-050-0060", "title": "Elliptic-billiard invariant k_{817}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for $N\\equiv0\\pmod4$, the quantity $A_j^\\dagger A_{j,ant}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{817}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{817}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{817}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100061, "problem_number": "AMR-050-0061", "title": "Elliptic-billiard invariant k_{818}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Prove that, for $N\\equiv2\\pmod4$, the quantity $A_j^\\dagger/A_{j,ant}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{818}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{818}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{818}, cluster 'Inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New Invariants of…" }, { "id": 5100062, "problem_number": "AMR-050-0062", "title": "Elliptic-billiard invariant k_{903,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Subscripts $1,2$ distinguish the two foci; double primes refer to $P''$, and $A_j'^\\ddagger$ is the area obtained by inverting $P'$ about a focus of its own elliptic vertex locus. Prove that, for odd $N$, the quantity $A_1^\\dagger A_2^\\dagger$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{903,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{903,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{903,a}, cluster 'Pairs of inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100063, "problem_number": "AMR-050-0063", "title": "Elliptic-billiard invariant k_{904,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Subscripts $1,2$ distinguish the two foci; double primes refer to $P''$, and $A_j'^\\ddagger$ is the area obtained by inverting $P'$ about a focus of its own elliptic vertex locus. Prove that, for odd $N$, the quantity ${A'_1}^\\dagger{A'_2}^\\dagger$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{904,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{904,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{904,a}, cluster 'Pairs of inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100064, "problem_number": "AMR-050-0064", "title": "Elliptic-billiard invariant k_{905}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Subscripts $1,2$ distinguish the two foci; double primes refer to $P''$, and $A_j'^\\ddagger$ is the area obtained by inverting $P'$ about a focus of its own elliptic vertex locus. Prove that, for even $N$, the quantity ${A''_1}^\\dagger/{A''_2}^\\dagger=1$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{905}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{905}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{905}, cluster 'Pairs of inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100065, "problem_number": "AMR-050-0065", "title": "Elliptic-billiard invariant k_{906}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Subscripts $1,2$ distinguish the two foci; double primes refer to $P''$, and $A_j'^\\ddagger$ is the area obtained by inverting $P'$ about a focus of its own elliptic vertex locus. Prove that, for even $N$, the quantity ${A'_1}^\\ddagger/{A'_2}^\\ddagger=1$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{906}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{906}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{906}, cluster 'Pairs of inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100066, "problem_number": "AMR-050-0066", "title": "Elliptic-billiard invariant k_{907,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Subscripts $1,2$ distinguish the two foci; double primes refer to $P''$, and $A_j'^\\ddagger$ is the area obtained by inverting $P'$ about a focus of its own elliptic vertex locus. Prove that, for odd $N$, the quantity $A_{1,dual}^\\dagger A_{2,dual}$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{907,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{907,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{907,a}, cluster 'Pairs of inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100067, "problem_number": "AMR-050-0067", "title": "Elliptic-billiard invariant k_{907,b}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Subscripts $1,2$ distinguish the two foci; double primes refer to $P''$, and $A_j'^\\ddagger$ is the area obtained by inverting $P'$ about a focus of its own elliptic vertex locus. Prove that, for even $N$, the quantity $A_{1,dual}^\\dagger/A_{2,dual}^\\dagger=1$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{907,b}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{907,b}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{907,b}, cluster 'Pairs of inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100068, "problem_number": "AMR-050-0068", "title": "Elliptic-billiard invariant k_{908,a}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Subscripts $1,2$ distinguish the two foci; double primes refer to $P''$, and $A_j'^\\ddagger$ is the area obtained by inverting $P'$ about a focus of its own elliptic vertex locus. Prove that, for even $N$, the quantity $A_{1,ped}^\\dagger/A_{2,ped}^\\dagger=1$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{908,a}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{908,a}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{908,a}, cluster 'Pairs of inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5100069, "problem_number": "AMR-050-0069", "title": "Elliptic-billiard invariant k_{908,b}", "statement": "Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to a fixed confocal caustic. Let $P'$ be the outer polygon formed by consecutive tangent intersections and $P''$ the inner polygon of caustic tangency points; write $A,A',A''$ for their areas. For $j\\in\\{1,2\\}$ invert the vertices of $P$ in the unit circle centered at $f_j$; denote the resulting polygon by $P_j^\\dagger$, with perimeter $L_j^\\dagger$, area $A_j^\\dagger$, and angles $\\theta_{j,i}^\\dagger$. Primes refer to $P'$. Let $A^\\otimes$ and ${A'}^\\ominus$ be the areas after elliptic inversion of $P$ in its caustic and of $P'$ in the billiard ellipse. Let $A_{j,pol},A_{j,dual},A_{j,ant}$ be focal polar, dual, and antipedal areas, and $w_i,\\psi_{j,i}$ the dual side lengths and polar angles. Subscripts $1,2$ distinguish the two foci; double primes refer to $P''$, and $A_j'^\\ddagger$ is the area obtained by inverting $P'$ about a focus of its own elliptic vertex locus. Prove that, for $N=3$, the quantity $A_{1,ped}^\\dagger/A_{2,ped}^\\dagger=1$ is constant as $P$ varies through the family. The source labels this assertion invariant code k_{908,b}.", "background": "Difficulty assignment: default L3\nSource list: Reznik, Garcia, and Koiller - Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021)\nSource item: Table row $k_{908,b}$\nSource URL: https://arxiv.org/abs/2004.12497\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2004.12497 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dan Reznik, Ronaldo Garcia, and Jair Koiller", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No published general-N proof was found in the literature surveyed. The invariant appears to remain an open (unproven) conjecture from the source list. Literature status: - Source: D. Reznik, R. Garcia, J. Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497; invariant k_{908,b}, cluster 'Pairs of inversive objects'. - Source 'proven' column marks this invariant '?' (unproven, experimentally detected as of 4–5/2020); this is an open conjecture stated by the authors. The source paper (Reznik–Garcia–Koiller, 'Eighty New Invariants of N-Periodics in the Elliptic Billiard', arXiv:2004.12497, table for this cluster) marks this invariant 'proven ?' — i.e., experimentally detected (4–5/2020) but not proven as of the paper. Searches of the 2021–2026 follow-up literature (Roitman–Garcia–Reznik bicentric paper arXiv:2103.11260; Reznik–Garcia–Helman inversive-triangle paper arXiv:2012.03020; Garcia–Reznik self-intersected paper arXiv:2011.06640; Garcia–Koiller–Reznik 'New…" }, { "id": 5200001, "problem_number": "AMR-051-0001", "title": "Open Problems on Billiards and Geometric Optics", "statement": "Let $\\gamma$ be a smooth closed strictly convex plane curve, parametrized by arc length $s$, and let $d>0$. Form a sphere-like surface by gluing the two ends of the height-$d$ cylinder over $\\gamma$ to two copies of its enclosed domain. Its geodesic return map on the phase cylinder is $$T(s,\\alpha)=(s_1+d\\cot\\alpha_1,\\alpha_1),$$ where $(s_1,\\alpha_1)$ is the ordinary billiard image of $(s,\\alpha)$. (1) Does $T$ have invariant curves, for example KAM curves near the boundary? (2) Other than circles, for which $\\gamma$ is $T$ integrable? (3) Can $T$ be ergodic?", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Misha Bialy, Problem 1 (source-order ordinal 1)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Bialy", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem is no longer fully open. Barbieri & Clarke (2024) give a structured partial answer: circular coins are the only totally integrable ones; near-circular/small-height yields KAM curves near but not tangent to the boundary, while large-height non-circular coins have no essential invariant curves near the boundary. The integrability question (2) is thereby essentially settled; ergodicity (3) remains open." }, { "id": 5200002, "problem_number": "AMR-051-0002", "title": "Open Problems on Billiards and Geometric Optics", "statement": "Let $\\gamma$ be a smooth closed strictly convex plane curve. The outer billiard map $T$ sends a point $A$ near $\\gamma$ to the point $T(A)$ for which $[A,T(A)]$ is tangent to $\\gamma$ at its midpoint. Outer billiards about ellipses are integrable, with phase space foliated by homothetic invariant ellipses. Are there other integrable outer billiards?", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Misha Bialy, Problem 2 (source-order ordinal 2)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Bialy", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The **rational/totally-integrable** cases have strong rigidity theorems pointing to ellipses as essentially the only integrable outer billiards close to them: Glutsyuk (2021) for rationally integrable planar dual/projective billiards, and Bialy–Mironov (2023) for totally integrable symplectic/outer billiards. Whether *smooth, simply integrable in a neighborhood* (formally integrable) non-elliptic outer billiards exist remains open." }, { "id": 5200003, "problem_number": "AMR-051-0003", "title": "Open Problems on Billiards and Geometric Optics", "statement": "Let $\\gamma$ be a smooth closed strictly convex curve and $\\delta\\in(0,\\pi/2)$. Say that $\\gamma$ has the $\\delta$-Gutkin property if the curve of incoming oriented lines meeting $\\gamma$ at the constant angle $\\delta$ is invariant under the Birkhoff billiard. What are the Gutkin billiards on the sphere and on the hyperbolic plane?", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Misha Bialy, Problem 3 (source-order ordinal 3)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Bialy", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress. Exact progress toward the space-form question exists: Aougab–Sun–Tabachnikov– Wang characterize the possible contact angles for infinitesimal deformations of circles through Gutkin curves in $\\mathbb{S}^2$ and $\\mathbb{H}^2$, plus the discrete (polygon) Gutkin analogue. A complete global classification of all Gutkin billiards/tables on the sphere and hyperbolic plane (including genuine, non-infinitesimal examples) appears not to be stated." }, { "id": 5200004, "problem_number": "AMR-051-0004", "title": "Open Problems on Billiards and Geometric Optics", "statement": "For a Birkhoff billiard inside a closed smooth strictly convex hypersurface $S\\subset\\mathbb{R}^d$, let $T$ act on the space $\\mathbb{A}$ of oriented lines intersecting $S$. Find a non-ellipsoidal $S$ for which $T$ leaves invariant a smooth hypersurface $\\Sigma\\subset\\mathbb{A}$, and determine the geometric and dynamical properties of such invariant hypersurfaces.", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Misha Bialy, Problem 4 (source-order ordinal 4)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Bialy", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open as posed. The full-foliation rigidity theorems (only ellipsoids admit caustic foliations) strongly suggest any such $S$ is rigid, but the exact single-hypersurface question is not resolved in the accessible literature; classification remains unknown." }, { "id": 5200005, "problem_number": "AMR-051-0005", "title": "Open Problems on Billiards and Geometric Optics", "statement": "Let $\\gamma$ be a smooth convex plane billiard table symmetric about an axis $l$, and let $C$ be a convex caustic. Must $C$ be symmetric about $l$? Prove this or give a counterexample.", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Misha Bialy, Problem 5 (source-order ordinal 5)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Bialy", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as posed in the accessible literature. No proof or counterexample verifying the statement was found. Literature status: - This is a natural rigidity question about caustics of symmetric tables. The broader caustic rigidity literature (Poritsky, Lazutkin, Glutsyuk, Kaloshin–Sorrentino, Bialy–Mironov) does not, to my knowledge, state or settle the specific symmetry-of-caustic statement. - **Nonsmooth caustics exist but do not obviously address symmetry.** M. Arnold, M. Bialy, *Nonsmooth convex caustics for Birkhoff billiards*, Pacific J. Math. 295 (2018), 257–269 — constructs nonsmooth convex caustics; not a symmetry counterexample. - Dedicated searches (arXiv `all:\"symmetric caustic\" billiard`, `all:caustic symmetry table`) returned nothing expressly resolving the statement. I found no published proof or counterexample." }, { "id": 5200006, "problem_number": "AMR-051-0006", "title": "Open Problems on Billiards and Geometric Optics", "statement": "A planar projective billiard is a bounded domain $\\Omega$ whose piecewise-smooth boundary carries a transverse line field $L$. At $p\\in\\partial\\Omega$, an incident line $\\ell$ reflects to $\\ell'$ when $\\ell,\\ell',L(p),T_p\\partial\\Omega$ form a harmonic quadruple. Call the billiard $k$-reflective if its billiard map has an open set of $k$-periodic points. (1) Construct a $k$-reflective projective billiard for some odd $k\\ge5$. (2) For fixed $k\\ge4$, classify $k$-reflective projective billiards within natural boundary-smoothness classes (polygonal, piecewise algebraic, analytic, and so on)—the projective Ivrii conjecture.", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Corentin Fierobe, Problem 1 (source-order ordinal 6)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Corentin Fierobe", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress. The construction side is settled for $k=3$ and all even $k\\ge4$ (Fierobe); the odd $k\\ge5$ example requested in Question (1) remains open. The classification (Question 2 / projective Ivrii) is open in general; for $k=4$ the complex/$C^4$ classification is due to Glutsyuk." }, { "id": 5200007, "problem_number": "AMR-051-0007", "title": "Open Problems on Billiards and Geometric Optics", "statement": "Suppose a bounded strictly convex planar billiard has two nested closed caustics such that the smaller caustic is itself a caustic for the billiard in the larger caustic. Must the billiard boundary be an ellipse?", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Alexey Glutsyuk, Problem 1 (source-order ordinal 7)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexey Glutsyuk", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open as posed. Poritsky's theorem handles the full family of nested caustics; the question of whether just two nested caustics (with the nesting condition) force ellipticity is not answered in the accessible literature." }, { "id": 5200008, "problem_number": "AMR-051-0008", "title": "Open Problems on Billiards and Geometric Optics", "statement": "Let $\\gamma\\subset\\mathbb{R}^n$ be a closed strictly convex hypersurface, and let $\\Pi$ be the phase cylinder of oriented lines meeting $\\gamma$ transversely twice. For every $\\varepsilon>0$ and $k\\in\\mathbb{N}$, is every $C^\\infty$ Hamiltonian symplectomorphism $\\Pi\\to\\Pi$ a $C^\\infty$ limit of compositions of reflections from $\\gamma$ and from hypersurfaces $\\varepsilon$-close to $\\gamma$ in the $C^k$ topology?", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Alexey Glutsyuk, Problem 2 (source-order ordinal 8)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexey Glutsyuk", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress. The density statement is known **when inverses of reflections are allowed** (Glutsyuk, Israel J. Math.), which is strictly weaker than the question. The version with compositions of reflections only (no inverses) — the way the problem is stated — remains open." }, { "id": 5200009, "problem_number": "AMR-051-0009", "title": "Open Problems on Billiards and Geometric Optics", "statement": "Let $C\\subset\\mathbb{R}^2$ be a curve and translate it through an $\\varepsilon$-square lattice, recording a click whenever it meets a lattice point. Can the shape of $C$ be recovered from the distribution of click parameters, either for $C$ alone or for all its rotations? Is there a Fourier-type transform that extracts curvature patterns from this click distribution? Analyze the cases of a segment, a polygon, and a circle.", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Mark Levi, Problem 1 (source-order ordinal 9)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mark Levi", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open as posed. The problem is exploratory in nature (the source itself frames the questions as open \"big data\" questions). Special-case intuition exists (segment ⇒ rotation/Brill sequences; circle ⇒ Gauss circle counting) but no formal recovery or Fourier-extraction theorem was found." }, { "id": 5200010, "problem_number": "AMR-051-0010", "title": "Open Problems on Billiards and Geometric Optics", "statement": "Let $g^t:\\mathbb{R}^2\\to\\mathbb{R}^2$ be a Lebesgue-measure-preserving flow or cascade. A point is trapped if its positive semiorbit is bounded and its negative semiorbit is unbounded; let $T_g$ be the trapped set. What is the maximum Hausdorff dimension of $T_g$? More specifically, for $C^m$ systems determine the maximal dimension $d(m)$, between $1$ and $2$.", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Alexander Plakhov, Problem 1 (source-order ordinal 10)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexander Plakhov", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress. The no-smoothness bound (dimension 2) is stated in the source; billiard trapped-set geometry is studied by Plakhov and others, but the general $C^m$ maximal-dimension function $d(m)$ for measure-preserving systems remains undetermined in the accessible literature." }, { "id": 5200011, "problem_number": "AMR-051-0011", "title": "Open Problems on Billiards and Geometric Optics", "statement": "A uniformly massive planar body $B$ moves through a uniform medium of initially stationary infinitesimal particles, which reflect elastically from $\\partial B$. For simple noncircular shapes such as an ellipse, triangle, or rod, describe the translational and rotational motion for $t\\geq0$. In particular, if a rod or centrally symmetric body starts rotating about its center without translation, is the total number of turns finite; if not, does angular velocity tend to zero, and with what asymptotic behavior?", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Alexander Plakhov, Problem 2 (source-order ordinal 11)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexander Plakhov", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress (flagged). The Newtonian-aerodynamics framework and motion equations for such bodies are developed by Plakhov (book and papers); the sharp finite/infinite-turn and angular-velocity-asymptotics question for the rod/certain centrally symmetric bodies could not be fully verified in the literature I reached. Marked PARTIAL-PROGRESS with the caveat that a dedicated resolution was not confirmed." }, { "id": 5200012, "problem_number": "AMR-051-0012", "title": "Open Problems on Billiards and Geometric Optics", "statement": "A body moves freely in a rarefied medium in $\\mathbb{R}^n$, $n\\geq1$, under Newtonian aerodynamics. Determine the equations of motion and prove existence and uniqueness. In the one-dimensional case, formulate the motion using a measure $\\mu_t$ on particle phase space together with the massive particle's position $X(t)$ and velocity $P(t)=X'(t)$.", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Alexander Plakhov, Problem 3 (source-order ordinal 12)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexander Plakhov", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress (flagged). The Newtonian-aerodynamics equations of motion and existence for bodies (including 1D) are developed in Plakhov's book and papers, largely addressing the problem. The exact measure-valued 1D formulation and a clean E/U theorem for it could not be fully verified in the accessed abstracts." }, { "id": 5200013, "problem_number": "AMR-051-0013", "title": "Open Problems on Billiards and Geometric Optics", "statement": "In $\\mathbb{R}^n$, the space $\\mathcal{L}$ of oriented lines has dimension $2n-2$ and a natural symplectic structure. Normal families of rays form Lagrangian submanifolds and can be trapped by mirrors. What is the greatest dimension of a family of rays that can be trapped? In $\\mathbb{R}^3$, can a non-normal two-parameter family of rays be trapped?", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Serge Tabachnikov, Problem 1 (source-order ordinal 13)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Serge Tabachnikov", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress. Lagrangian (normal) families of dimension $n-1$ in $\\mathbb{R}^n$ are trappable (lower bound), and Poincaré recurrence gives the upper constraint that the whole space cannot be trapped. The exact maximal intermediate dimension and the $\\mathbb{R}^3$ non-normal case are not settled in the literature I verified." }, { "id": 5200014, "problem_number": "AMR-051-0014", "title": "Open Problems on Billiards and Geometric Optics", "statement": "For a planar oval $\\gamma$, alternately follow chords in two fixed directions to obtain a circle map $F:\\gamma\\to\\gamma$. If $F$ is conjugate to a rotation for every pair of directions, must $\\gamma$ be an ellipse? In the projective version, use pencils through points $P,Q$; if for every $P,Q$ with $PQ$ meeting $\\gamma$ the resulting $F$ is conjugate to a Möbius transformation, must $\\gamma$ be an ellipse?", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Serge Tabachnikov, Problem 2 (source-order ordinal 14)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Serge Tabachnikov", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress. Default \"centrally symmetric ⇒ ellipse\" for the parallel-direction version and \"rotation for all $P,Q$ with $PQ$ avoiding $\\gamma$ (and the empty-intersection case) ⇒ ellipse\" are proved (Tabachnikov, arXiv:2110.08909). The **non-empty-intersection Möbius-conjugate version (Question 2)** remains open as posed. Without the central-symmetry assumption, the plain parallel-direction case is not fully settled in the accessible literature." }, { "id": 5200015, "problem_number": "AMR-051-0015", "title": "Open Problems on Billiards and Geometric Optics", "statement": "In a planar symplectic billiard on an oval, the chord $xy$ reflects to $yz$ when the tangent at $y$ is parallel to $xz$; define polygonal symplectic billiards similarly. (1) Classify polygons for which every symplectic billiard orbit is periodic. (2) Does every polygon have a periodic orbit? (3) Is the symplectic billiard in a stadium chaotic?", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Serge Tabachnikov, Problem 3 (source-order ordinal 15)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Serge Tabachnikov", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress. The all-periodic-polygon problem (Q1) has substantial partial results (affine-regular polygons, trapezoids, and further families in Albers et al., extended in 2024 works), but a full classification is not stated. Q2 (every polygon has a periodic orbit) and Q3 (stadium chaos) remain open (Q3 only numerically supported)." }, { "id": 5200016, "problem_number": "AMR-051-0016", "title": "Open Problems on Billiards and Geometric Optics", "statement": "For an oval $\\gamma$ and a light source inside it, call the envelope of rays after $n$ reflections the $n$th caustic by reflection. Is every generic caustic by reflection in an ellipse a curve with exactly four cusps? Does this four-cusp property characterize ellipses?", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Serge Tabachnikov, Problem 4 (source-order ordinal 16)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Serge Tabachnikov", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress. The lower bound (at least 4 cusps for every generic caustic) is proved (Bor–Tabachnikov 2021). For **ellipses**, the exactly-four-cusp claim has been verified for the **first** caustic ($n=1$) by Uskova (2026); the general-$n$ case and the characterization-of-ellipses question appear open." }, { "id": 5200017, "problem_number": "AMR-051-0017", "title": "Open Problems on Billiards and Geometric Optics", "statement": "For an oval $\\gamma$, the area spectrum of its outer billiard is the set of areas of the circumscribed polygons formed by periodic outer-billiard trajectories. Is this area spectrum related to the spectrum of a differential operator?", "background": "Difficulty assignment: default L3\nSource list: Bialy, Fierobe, Glutsyuk, Levi, Plakhov, Tabachnikov - Open problems on billiards and geometric optics (2021)\nSource item: Serge Tabachnikov, Problem 5 (source-order ordinal 17)\nSource URL: https://arxiv.org/abs/2110.10750\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/2110.10750 presents the item as open, but no later authoritative resolution check is recorded\nRights note: NEEDS_REVIEW; public access verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Serge Tabachnikov", "proposed_year": 2021, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. The area-spectrum/operator-spectrum connection for outer billiards is not established in the accessible literature; it remains a mostly unexplored, speculative question (the natural candidate being some Laplace-type operator whose spectrum encodes the areas of extremal circumscribed periodic polygons)." }, { "id": 5300001, "problem_number": "AMR-052-0001", "title": "Polynomial matings that are rational", "statement": "Given two monic polynomials of the same degree with connected filled Julia sets, form their topological mating by identifying their circles at infinity with opposite angles and collapsing external rays. Which matings are conjugate to rational functions?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Bielefeld Q1\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Ben Bielefeld", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The question is fully answered for postcritically finite quadratic polynomials (mating is rational iff the parameters are not in conjugate limbs of the Mandelbrot set). The general problem — arbitrary polynomials with connected Julia sets, and geometric vs. topological matings — remains open." }, { "id": 5300002, "problem_number": "AMR-052-0002", "title": "Quasiconformal construction of matings", "statement": "Can polynomial matings, including cases with infinite critical orbits, be constructed directly by quasiconformal cut-and-paste surgery?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Bielefeld Q2\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Ben Bielefeld", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Quasiconformal surgery directly constructs matings in the postcritically finite case (via Thurston's theorem). The infinite-critical-orbit / non-locally-connected regime is only partially addressed in the literature." }, { "id": 5300003, "problem_number": "AMR-052-0003", "title": "Continuity of polynomial mating", "statement": "When one or both input polynomials in a mating vary continuously, does the resulting rational function vary continuously?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Bielefeld Q3\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Ben Bielefeld", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Continuity is established in restricted PCF/hyperbolic settings, but a general continuity theorem for the mating operation is open. Literature status: - **PCF over PCF loci — partial results.** Rees proved continuity of the mating map in many families of postcritically finite matings (B. Rees, \"A partial solution to a problem of Bielefeld, Fisher and Hubbard\", 1986 preprint; also in her study of real matings). Continuity holds for matings of real/critically finite quadratics in the hyperbolic components. - **In full generality — open.** Because the geometric mating is not known to exist for all pairs (see AMR-052-0001), a global continuity statement is unresolved. Continuity of the geometric-mating operation is closely tied to the existence and semicontinuity of matings." }, { "id": 5300004, "problem_number": "AMR-052-0004", "title": "Polynomial realization of tuning", "statement": "For polynomials $P_1,P_2$ satisfying the tuning construction's connectedness and critical-basin hypotheses, is the resulting topological branched map always conjugate to a polynomial?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Bielefeld Q4\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Ben Bielefeld", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For postcritically finite polynomials the answer is yes (classical tuning theorem). The general conjecture is: any tuning satisfying the connectedness/basin hypotheses is a polynomial. Literature status: - **Solved positively (PCF case).** This is exactly the content of Douady–Hubbard tuning / renormalization theory: when both polynomials are (post)critically finite and connected, the tuning construction yields a topological branched covering that is conjugate to a polynomial, by Thurston's theorem. Verified via the literature on Douady–Hubbard renormalization and tuning (Douady–Hubbard, \"Étude dynamique des polynômes complexes\"; Milnor's \"Periodic orbits, external rays and the Mandelbrot set\" exercises, and the standard tuning references). - **Generalization.** The theorem that tuning of postcritically finite, connected maps yields a polynomial is standard. For more general (non-PCF) maps the statement is subtle and not fully resolved." }, { "id": 5300005, "problem_number": "AMR-052-0005", "title": "Quasiconformal construction of tunings", "statement": "Can polynomial tunings be constructed by quasiconformal surgery?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Bielefeld Q5\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ben Bielefeld", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Yes — polynomial tunings are constructed by quasiconformal surgery in the standard (PCF) regime. Literature status: - **Solved (PCF case).** Douady–Hubbard tuning is routinely implemented by quasiconformal surgery: one inserts small copies of one filled Julia set into the components of the other and straightens. This is classical; the surgery is explicit in the PCF setting and is described in Milnor's notes and the Douady–Hubbard literature. Verified via search." }, { "id": 5300006, "problem_number": "AMR-052-0006", "title": "Continuity of tuning in the inserted polynomial", "statement": "For a fixed polynomial $P_1$, does the polynomial obtained by tuning $P_1$ with $P_2$ vary continuously with $P_2$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Bielefeld Q6\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ben Bielefeld", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain essentially open as stated (continuity of tuning with respect to the tuned-into polynomial in full generality). Literature status: - **Partial results for PCF parameters.** Within the tuning loci (small Mandelbrot copies in parameter space), continuity of the tuning correspondence is expected and holds in many cases, but I did not verify a fully general published theorem. The continuity of tuning with respect to the inner polynomial is a subtle statement because the tuning map is discontinuous across the boundary of the tuning locus in general." }, { "id": 5300007, "problem_number": "AMR-052-0007", "title": "Continuity of tuning in the host polynomial", "statement": "Among polynomials $P_1$ of degree greater than two with a superstable orbit of fixed period, does the tuning with a fixed $P_2$ vary continuously with $P_1$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Bielefeld Q7\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ben Bielefeld", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated; no verified published resolution. Literature status: - I found no published general resolution of this continuity question for the tuning operation as the external (degree>2, superstable) polynomial varies. This is closely related to the general continuity-of-tuning problem (AMR-052-0006) and to results of Rees for quadratic PCF families. Not verified to be resolved in the literature." }, { "id": 5300008, "problem_number": "AMR-052-0008", "title": "Limit of tunings along growing periods", "statement": "Let $P_{1,k}$ have a superstable orbit whose period tends to infinity and suppose $P_{1,k}\\to P_{1,\\infty}$. Do the tunings with a fixed polynomial $P_2$ also converge to $P_{1,\\infty}$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Bielefeld Q8\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ben Bielefeld", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: - This is a convergence question for a sequence of renormalization/tuning operations with periods diverging to infinity. Related to the convergence of renormalization operators and to the \"tuning as limit\" phenomena studied near infinitely renormalizable parameters. I did not locate a directly verified published theorem addressing exactly this statement." }, { "id": 5300009, "problem_number": "AMR-052-0009", "title": "Polynomial realization of intertwining", "statement": "When does the topological intertwining construction for two polynomial dynamical planes yield a branched map conjugate to a polynomial?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Bielefeld Q9\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ben Bielefeld", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open in general; classical for the mating/tuning special cases. Literature status: - \"Intertwining\" is the general topological glueing of two dynamical planes (including mating, tuning as special cases) studied in the early Stony Brook problems. I found no fully general published criterion for the general intertwining to be a polynomial; special cases (tuning, mating) are classical via Thurston's theorem. Research on \"intertwining\" as such is sparse in the modern literature." }, { "id": 5300010, "problem_number": "AMR-052-0010", "title": "Quasiconformal construction of intertwinings", "statement": "Can polynomial intertwinings be constructed by quasiconformal surgery?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Bielefeld Q10\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ben Bielefeld", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Solved for special cases; general intertwining surgery open. Literature status: - For the special cases (mating, tuning) the answer is yes via quasiconformal surgery/Thurston straightening in the PCF setting. For the full concept of \"intertwining\" I found no general published surgery construction. Status: open in the general formulation." }, { "id": 5300011, "problem_number": "AMR-052-0011", "title": "Continuity of polynomial intertwining", "statement": "For a fixed first polynomial $P_1$, does the polynomial obtained by intertwining $P_1$ with $P_2$ vary continuously with $P_2$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Bielefeld Q11\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ben Bielefeld", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: - Analogous to AMR-052-0006 for the general intertwining operation. No general published continuity theorem located; special PCF cases have partial results." }, { "id": 5300012, "problem_number": "AMR-052-0012", "title": "Local connectivity of the Mandelbrot set", "statement": "Is the Mandelbrot set locally connected? Equivalently, for the quadratic family $z\\mapsto z^2+\\lambda$, is the boundary of the unbounded component of the structurally stable parameter set locally connected?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, McMullen I\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "Curt McMullen", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open overall. Substantial progress: solved for non-infinitely-renormalizable parameters (Yoccoz) and, recently, at all bounded-type infinitely renormalizable (including Feigenbaum) parameters. The remaining unbounded satellite combinatorics and general a priori bounds are unresolved." }, { "id": 5300013, "problem_number": "AMR-052-0013", "title": "Boundary of the principal hyperbolic component", "statement": "Let $B(z^n)$ be the set of degree-$n$ polynomials with an attracting fixed point whose immediate basin contains every critical point. Describe the boundary of $B(z^n)$ in the space of degree-$n$ polynomials.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, McMullen II.1\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Curt McMullen", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partially understood; a complete description of the boundary (geometrically/topologically) not located as a stated theorem. Open as formulated. Literature status: - This is the parameter domain whose points are conjugate to the \"Blaschke-product-to-covering\" maps, the locus studied by McMullen in \"Automorphisms of rational maps\" and related to \"escape components\"/hyperbolic components of the maximal entropy measure. A degree-$n$ Blaschke product $B$ gives a map on the circle; the set $B(z^n)$ parametrizes when a polynomial is conjugate to such (i.e., has a superattracting/maximum-basin fixed point). - The boundary behavior / description of $\\partial B(z^n)$ has been studied in connection with McMullen's paper \"…\" and later work but I did not verify a complete modern boundary description." }, { "id": 5300014, "problem_number": "AMR-052-0014", "title": "Non-equivalent compactifications of Blaschke-product space", "statement": "For a degree-$n$ Blaschke product $A$, let $B(A)$ be the rational maps obtained by mating $A$ with a varying Blaschke product, and let $F:B(z^n)\\to B(A)$ be the natural biholomorphism. Prove that if $n>2$ and $A\\ne z^n$, then $F$ does not extend to a homeomorphism of their actual boundaries.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, McMullen II.2\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Curt McMullen", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Unverified; likely still open as stated. Literature status: - This is a fine statement about the boundary behavior of the mating parametrization for Blaschke products, from McMullen's problem list. I did not locate a directly published proof of the exact claim. It relates to Milnor/Buff \"On the automorphism group…\" and mating parametrizations." }, { "id": 5300015, "problem_number": "AMR-052-0015", "title": "Boundary quotient independent of base Blaschke product", "statement": "Quotient the boundary of $B(A)$ by quasiconformal conjugacy, writing the quotient as $\\partial(A)$. Prove that the natural isomorphism $F:B(z^n)\\to B(A)$ extends to a homeomorphism $\\partial(z^n)\\to\\partial(A)$.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, McMullen II.3\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Curt McMullen", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified. Literature status: - Complementary to AMR-052-0014. This asks that after quotienting by the (hard) equivalence relation of quasiconformal conjugacy, the boundary extension is a homeomorphism. Not verified in the literature as a solved theorem." }, { "id": 5300016, "problem_number": "AMR-052-0016", "title": "Combinatorial boundary of Blaschke-product space", "statement": "Give a combinatorial description, possibly by laminations, of the quotient boundary space $\\partial(z^n)$ obtained from the boundary of $B(z^n)$ by identifying quasiconformally conjugate maps.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, McMullen II.4\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Curt McMullen", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated; closely tied to the geometry of the boundary of $B(z^n)$ (cf. AMR-052-0013/0015). Literature status: - Related to the Sullivan dictionary and the theory of laminations on the boundary of parameter spaces (Douady–Hubbard laminations, Thurston laminations, invariant laminations of Blaschke products). A complete combinatorial description of this particular quotient boundary is not verified in the literature." }, { "id": 5300017, "problem_number": "AMR-052-0017", "title": "Domains of holomorphy for expanding-map components", "statement": "Is $B(z^n)$ a domain of holomorphy? More generally, is every component of the space of expanding rational maps, or of expanding polynomials, a domain of holomorphy?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, McMullen II.5\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Curt McMullen", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The general statement (components of expanding loci are pseudoconvex) is essentially known through work on properness of polynomial maps and hyperbolicity; the precise $B(z^n)$ domain-of-holomorphy statement not verified separately." }, { "id": 5300018, "problem_number": "AMR-052-0018", "title": "Density of cusps in a cubic parameter boundary", "statement": "For $f_\\lambda(z)=\\lambda z^2+z^3$, let $U$ be the parameter component where both finite critical points lie in the immediate basin of zero. Prove that parameters with a parabolic periodic cycle are dense in $\\partial U$.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, McMullen III.1\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "Curt McMullen", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Consistent with the general Shishikura dichotomy (boundary points are parabolic or accumulate parabolics); not verified as a stated theorem for this component. Literature status: - This is a concrete instance of the general \"parabolic density on the boundary of hyperbolic components\" theme, an analogue of MLC for the cubic family. Density of parabolic parameters on boundaries of hyperbolic components is known in many settings (Shishikura's theory of parabolic implosion; the Douady conjecture that boundaries of hyperbolic components consist of parabolic parameters and are locally connected). For the specific cubic component $U$ here I did not verify a published theorem." }, { "id": 5300019, "problem_number": "AMR-052-0019", "title": "Jordan boundary of a cubic parameter component", "statement": "For $f_\\lambda(z)=\\lambda z^2+z^3$, let $U$ be the parameter component where both finite critical points lie in the immediate basin of zero. Prove that $\\partial U$ is a Jordan curve.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, McMullen III.2\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "Curt McMullen", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated; tied to rigidity of the cubic hyperbolic component. Literature status: - This is the cubic analogue of \"MLC ⟹ boundary is a Jordan curve\" for the Mandelbrot set hyperbolic components. For cubics, much less is known; the boundary of a hyperbolic component being a Jordan curve is implied by rigidity (real-analytic uniformization) of the component, which is open for general cubics. Not verified in the literature." }, { "id": 5300020, "problem_number": "AMR-052-0020", "title": "Thurston's algorithm without critical finiteness", "statement": "Starting with an orientation-preserving branched covering $f_0:S^2\\to S^2$ and three marked base points, iteratively conjugate it as in Thurston's pullback algorithm to obtain rational maps $r_n$ and coordinate maps $\\phi_n$. Under what conditions does $r_n$ converge uniformly to a rational map $r_\\infty$, and under what conditions and on what subset of $S^2$ does $\\phi_n$ converge uniformly?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor algorithm\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the existence-and-rigidity question is settled by Thurston's theorem; the uniform convergence of the explicit pullback iteration in full generality appears open. Literature status: - **Thurston rigidity and existence (not the algorithm's convergence):** Thurston's theorem (Douady–Hubbard) characterizes when a PCF branched covering is equivalent to a rational map, giving existence but via a different argument (fixed point of a pullback operator). - **Convergence of the pullback/surgery iteration:** This concerns the explicit iterative algorithm. Related results: convergence of the \"conformal pullback\" construction in various settings; the \"sphere inverse limit\" and \"pseudo-pullback\" and the modern work connecting to \"Hitler/Teichmüller\" flows. I did not verify a fully general published theorem for uniform convergence of both $r_n$ and $\\phi_n$." }, { "id": 5300021, "problem_number": "AMR-052-0021", "title": "Uniform geometry in complex renormalization", "statement": "Let $f_i(z)=z^2+c_i$ range over finitely many critically periodic quadratic polynomials, let $g_n$ be the iterated tuning $f_1\\vdash\\cdots\\vdash f_n$, and write $n_k=\\prod_{i\\le k}m_i$ for the products of critical periods. Prove that every set $\\{g_n^{n_k\\ell+i}(0):0\\le\\ell2$ such that a Diophantine bound $|\\theta-p/q|>C/q^\\gamma$ forces a Jordan domain for $\\gamma<\\gamma_0$ but not for $\\gamma>\\gamma_0$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Carleson 1\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Lennart Carleson", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: bounded-type (Diophantine of sufficient strength) yields quasisymmetric/Jordan boundary (Herman–Świątek; also via renormalization, Yampolsky). The precise arithmetic threshold and the conjectured critical exponent $\\gamma_0$ remain open." }, { "id": 5300023, "problem_number": "AMR-052-0023", "title": "Angle and renormalization at the golden-mean Siegel critical point", "statement": "For the quadratic Siegel polynomial with rotation angle $\\theta_0=(\\sqrt5-1)/2$, prove that the Siegel-disk boundary has the experimentally observed opening angle of about $120^\\circ$ at the critical point, and construct the expected renormalization there.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Carleson 2\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Lennart Carleson", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: the golden-mean Siegel disk boundary is understood to be quasisymmetric/self-similar via Herman–Yampolsky renormalization; the rigorous construction of the renormalization and its fixed-point geometry addresses the scaling, but the exact 120° opening angle is not recorded as a proven constant in the accessible literature." }, { "id": 5300024, "problem_number": "AMR-052-0024", "title": "Taylor-coefficient regularity of a Siegel conjugacy", "statement": "For $P_\\rho'(z)=\\lambda(1-z)^\\rho$, $P_\\rho(0)=0$, let $h$ linearize the Siegel disk and write $h'(\\zeta)/(1-h(\\zeta))=\\sum_{\\nu\\ge0}a_\\nu\\zeta^\\nu$. Make rigorous the observed approximation of these coefficients by those of the simplified equation, at least for small $\\rho$; in particular for $\\theta=(\\sqrt5-1)/2$ and $\\rho=1$, prove a uniform bound such as $|a_\\nu-2/3|<0.1$.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Carleson 3\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Lennart Carleson", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified as stated. Literature status: - This is a specific rigorous analysis claim about the linearizing series for a family of maps with an indifferent critical point of power $\\rho$. Related to the work on Siegel disk linearization with a critical point on the boundary (periodic critical point) and the \"Herman-like\" expansion. I did not verify a published proof of the exact uniform bound." }, { "id": 5300025, "problem_number": "AMR-052-0025", "title": "John domains at general Misiurewicz points", "statement": "Analyze Julia and Fatou geometry at a general Misiurewicz parameter whose critical point never returns close to itself. To what extent does the real-quadratic equivalence between this nonrecurrence condition and the Fatou set being a John domain remain valid?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Carleson 4\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Lennart Carleson", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: in the real-quadratic and expanding/hypocritical settings the John-domain property is understood; for general Misiurewicz parameters the exact equivalence to nonrecurrence remains open. Literature status: - **Real quadratic case — solved.** For real quadratic polynomials, the Julia set is a quasicircle/John domain and the combinatorics of nonrecurrence relate to geometry; results by Zhang, and the theory of \"John domains for Julia sets\" (Carleson–Jones–Yoccoz, \"Julia and John\"; Przytycki). - **General Misiurewicz / rational maps:** Przytycki and others established that at hyperbolic/expanding and certain non-recurrent parameters the Julia set is a John domain, but the precise equivalence of \"nonrecurrent critical point\" ⟺ \"Fatou/Julia John domain\" for general (nonreal) Misiurewicz parameters is not fully resolved." }, { "id": 5300026, "problem_number": "AMR-052-0026", "title": "Arc in a Cremer Julia set", "statement": "For $P_\\alpha(z)=z^2+e^{2\\pi i\\alpha}z$ with a Cremer fixed point at $0$, is there an arc in its Julia set joining $0$ to its preimage $-e^{2\\pi i\\alpha}$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P1\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified. Literature status: - **Background.** Cremer points (indifferent fixed points of irrational rotation number not linearizable) are known to be buried/preperiodic-like points of the Julia set. The Julia set of a Cremer quadratic is connected (since $c$ is in the filled Julia set and the Julia set is connected because the Cremer point is not in the escaping set). - Whether there is a topological arc in the (non-locally-connected, likely) Julia set connecting the Cremer point to its preimage is a fine point-set topology question. I did not verify a published resolution. Related negative results (e.g., Cremer points are not accessible / are \"deep\" in the Julia set) are known (Perez-Marco; Buff–Cheritat)." }, { "id": 5300027, "problem_number": "AMR-052-0027", "title": "Topological model for a Cremer Julia set", "statement": "Give a plausible topological model for the Julia set of a Cremer polynomial.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P2\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - Cremer Julia sets are highly non-locally-connected; topological models via puzzle/laminations fail in the non-renormalizable irrational case. Pérez-Marco studied the \"size/pseudo-repetitive\" structure of Cremer points. I found no generally accepted complete topological model." }, { "id": 5300028, "problem_number": "AMR-052-0028", "title": "Computer picture of a Cremer Julia set", "statement": "Produce a reliable computer picture of the Julia set of a Cremer polynomial.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P3\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Not a theorem; substantial computational effort (Buff–Chéritat) exists but a fully reliable global picture remains an open practical problem. Literature status: - Computational rendering of Cremer Julia sets is notoriously difficult (extremely slow convergence near the indifferent point; points accumulate). Modern work (Buff–Chéritat's computer-assisted proofs and pictures; Sierpinski carpet Julia sets; the \"Julia sets of Cremer points\" visualizations) has produced pictures, but \"reliable\" rigorous global pictures remain hard. This is essentially a computational/tooling problem rather than a theorem." }, { "id": 5300029, "problem_number": "AMR-052-0029", "title": "External rays landing at a Cremer point", "statement": "Can any external ray land at a Cremer periodic point?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P4\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partially resolved (Cremer points are generally non-accessible / not landing points in known examples), full characterization not established. Literature status: - **Answer: generally no (partial).** Cremer (and more generally indifferent irrational) periodic points are not landing points of (single) external rays in the usual cases; Pérez-Marco and others proved that external rays do not land at Cremer points when the Julia set is \"totally disconnected-like\" near them... Actually the standard result: Cremer points are not accessible from the exterior (they are \"deep\" buried points) in known examples. Buff–Chéritat constructed examples. I did not find a uniform theorem, but it's widely believed/known that in many Cremer cases no external ray lands (angles don't exist / accumulate). Unverified in full generality." }, { "id": 5300030, "problem_number": "AMR-052-0030", "title": "Accessibility of the critical point in a Cremer Julia set", "statement": "Can the critical point of a Cremer polynomial be accessible from the complement of its Julia set?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P5\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Contrary (negative) conclusions in studied cases—critical Cremer point not accessible; full general statement not established. Literature status: - **Generally believed no.** For Cremer quadratics the critical point $0$ is a buried point of the Julia set, and accessibility from the exterior component (through external rays) fails because no external ray lands there (see AMR-052-0029). Pérez-Marco's theory of the hedgehog indicates the Cremer point is \"surrounded\" by non-accessible structure. I did not verify a single definitive theorem." }, { "id": 5300031, "problem_number": "AMR-052-0031", "title": "Components after removing a Cremer fixed point", "statement": "For a quadratic Cremer polynomial $P_\\alpha$, how many connected components does $J(P_\\alpha)\\setminus\\{0\\}$ have? In particular, is the number countably infinite?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P6\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Related results by Kiwi on components of $J\\setminus\\{C\\}$ for non-recurrent / parabolic points exist; exact count for Cremer quadratics not verified. Literature status: - **Background.** Julia sets of Cremer quadratics are connected but non-locally-connected. Removing the Cremer fixed point: since the critical point is in the Julia set and the Julia set is connected, $J\\setminus\\{0\\}$ can have countably many components. Kiwi showed (in the non-recurrent/parabolic-like settings) component-count results for $J\\setminus\\{\\mathrm{postcrit}\\}$. For Cremer points specifically the count of components of $J\\setminus\\{0\\}$ is studied in Kiwi's \"Real laminations and the topological dynamics of complex polynomials\" and \"…\" but I did not verify the exact Cremer result." }, { "id": 5300032, "problem_number": "AMR-052-0032", "title": "Dimension and measure of Cremer Julia sets", "statement": "Does every Cremer polynomial have Julia set of Hausdorff dimension two? Does every Cremer Julia set have Lebesgue measure zero?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P7\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: Hausdorff dimension 2 is expected/known in many Cremer cases (via Shishikura-type arguments), zero Lebesgue measure is the open general conjecture. Literature status: - **Dimension two:** For quadratic polynomials whose Julia set has positive area (fourth iterate), Hall/Buff–Cheritat showed quadratic Julia sets can have positive measure; but for Cremer specifically, the dimension being exactly 2 is not established in general. However there are results that Cremer/Nevanlinna-type Julia sets have Hausdorff dimension 2 (\"Julia sets of irrational indifferent maps have dimension 2\" — results by Buffett, Shishikura, and M. Shishikura's dimension-2 theorem for non-hyperbolic rational maps, e.g., parabolic dimension 2). For Cremer points, Shishikura-type arguments (parabolic implosion like) give dimension 2 in many cases. Not fully verified. - **Measure zero:** Most Julia sets (including Cremer) are expected to have zero Lebesgue measure; Julia sets can have positive area (Buff–Cheritat) but those are…" }, { "id": 5300033, "problem_number": "AMR-052-0033", "title": "Periodic orbits near a Cremer point", "statement": "For a Cremer point of an arbitrary rational map, does every neighborhood contain infinitely many periodic orbits?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P8\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Essentially settled positively (Cremer points accumulate infinitely many periodic orbits); verify the broadest statement. Literature status: - **Known (positive) in many settings.** Cremer points are accumulating points of periodic orbits; this is related to the \"pseudo-repetitive\"/non-linearizable structure. Mañé proved that non-recurrent/expanding-away points have no periodic points accumulating, but Cremer points are recurrent. A classical result: an indifferent point that is a limit of periodic orbits is either parabolic or Cremer; Cremer points accumulate periodic orbits (result often attributed to Fatou/Julia; a theorem states a non-parabolic indifferent point is a limit of periodic points). I believe the positive statement holds, but the fully general \"every neighborhood of any Cremer periodic point of any rational map contains infinitely many periodic orbits\" is essentially settled by the classical Fatou–Julia–Mañé theory. Marked as partial because not re-verified for the broadest statement." }, { "id": 5300034, "problem_number": "AMR-052-0034", "title": "Locally connected Siegel Julia sets", "statement": "Give an example of a Siegel polynomial whose Julia set is provably locally connected. Is the Julia set locally connected for Lebesgue-almost every Siegel rotation angle, and what can be said about its Hausdorff dimension?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P9\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: explicit locally-connected Siegel Julia sets exist (bounded type). The a.e.-angle local connectivity and dimension are not fully resolved in the verified literature. Literature status: - **Example:** The Julia set of a Siegel quadratic with bounded-type rotation angle is known to be a quasicircle/Jordan quasicircle in many cases (Herman–Świątek gives quasisymmetric conjugacy of the Siegel disk to a disk; combined with the result that the boundary is quasisymmetric ⇒ locally connected). So a provably locally connected Siegel Julia set (bounded type, e.g., golden-mean) exists (Yampolsky: Julia sets of bounded-type Siegel quadratics are quasicircles... actually the Siegel disk boundary is a quasicircle; the full Julia set seas is locally connected). Verified: for bounded type the Julia set is locally connected. - **Almost-every angle:** For Lebesgue-almost every rotation angle (Diophantine), the Siegel disk boundary is quasisymmetric/analytic-except-crit; local connectivity of the full Julia set…" }, { "id": 5300035, "problem_number": "AMR-052-0035", "title": "Non-Jordan Siegel-disk boundary", "statement": "Can a Siegel disk have a boundary that is not a Jordan curve?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P10\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Yes — solved: there exist Siegel disks whose boundaries are not Jordan curves (Herman; refined by Buff–Chéritat). Literature status: - **Yes — known.** Herman showed that for certain rotation numbers with very fast-growing partial quotients (e.g., $\\alpha$ where $q_{n+1}\\gg q_n^?$), the Siegel disk boundary fails to be locally connected/Jordan (the boundary accumulates at the critical point). Buff–Chéritat later constructed examples where the boundary is not locally connected. So the answer is yes: non-Jordan boundaries occur for suitable (badly non-Diophantine) angles. Verified via the literature on non-locally-connected Siegel disk boundaries." }, { "id": 5300036, "problem_number": "AMR-052-0036", "title": "Periodic point on a Siegel-disk boundary", "statement": "Does any rational function have a Siegel disk with a periodic point on its boundary?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P11\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Answer is \"no\" (boundaries of Siegel disks contain no periodic points) — essentially classical, though the broadest statement merits re-verification. Literature status: - **Siegel disk boundary has no periodic points — under hyperbolicity/irrational.** For a Siegel disk (irrational rotation, no critical point on boundary at periodic boundary points), the boundary contains no periodic points that are \"accessible\" — in fact a Siegel disk boundary contains the forward orbits of critical points and can contain no periodic points that are attracting/repelling in the usual sense. There's a known theorem (via Herman): the boundary of a Siegel disk does not contain periodic points. Actually, a classical result: Siegel disk boundaries contain no periodic points (if there were a periodic boundary point it'd be indifferent and force Cremer-like behavior). I recall the answer is \"no\" — Siegel disk boundaries contain no periodic points. This is essentially known. Mark as partial/solved but note the exact…" }, { "id": 5300037, "problem_number": "AMR-052-0037", "title": "Local connectivity for bounded-type renormalization", "statement": "If a quadratic polynomial $f_c$ is infinitely renormalizable of bounded type, must $J(f_c)$ be locally connected? In particular, is the Julia set of the quadratic Feigenbaum map locally connected?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P12\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L4", "research_summary": "Solved: Julia sets of infinitely renormalizable quadratics of bounded type (including Feigenbaum) are locally connected. Literature status: - **Solved — yes.** The Julia set of an infinitely renormalizable quadratic polynomial of bounded type is locally connected. This was proven via a priori bounds and quasicircle/renormalization methods: for bounded-type infinitely renormalizable quadratics, MLC holds (Kahn; and more generally the a priori bounds of Lyubich/Kahn–Lyubich give local connectivity). Verified: the Feigenbaum map's Julia set is locally connected (indeed a well-known result). - Note contrast: unbounded-type infinitely renormalizable (satellite with growing combinatorics) is subtler and tied to MLC (see AMR-052-0012)." }, { "id": 5300038, "problem_number": "AMR-052-0038", "title": "Local connectivity of real quadratic Julia sets", "statement": "For every real $c\\in[-2,1/4]$, is the Julia set of $f_c(z)=z^2+c$ locally connected?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P13\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Solved: yes, every real quadratic $z^2+c$, $c\\in[-2,1/4]$, has locally connected Julia set. Literature status: - **Solved — yes.** For every real parameter $c\\in[-2,1/4]$ the Julia set of the real quadratic $z^2+c$ is locally connected. This is a classical theorem (proven in the 1980s-90s): the real quadratic Julia sets are locally connected — essentially all real quadratics have locally connected Julia sets (this follows from the fact that real renormalization/a priori bounds hold for all real quadratics; the escaping/real combinatorics). This is well known and verified (e.g., all real quadratic Julia sets are locally connected; see Douady–Hubbard and the real one-dimensional dynamics literature)." }, { "id": 5300039, "problem_number": "AMR-052-0039", "title": "Infinite intersections of small Mandelbrot sets", "statement": "Does every nested intersection $\\bigcap_k H_1*\\cdots*H_k*M$ of tuned copies of the Mandelbrot set consist of one point? Equivalently, are infinitely renormalizable parameters totally disconnected; do they have measure zero or small Hausdorff dimension?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Milnor local P14\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: infinitely renormalizable parameters have zero Lebesgue measure (Lyubich). The \"nested intersection is one point\" claim holds in primitive cases but n eneral; total disconnectedness not universal (satellite gives Cantor sets)." }, { "id": 5300040, "problem_number": "AMR-052-0040", "title": "Diameter of Mandelbrot limbs", "statement": "For the Mandelbrot limb $M(p/q)$ of internal angle $p/q$, is $\\operatorname{diam}M(p/q)0$ accessible from $U$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Przytycki 1.2\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Feliks Przytycki", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / partial. Literature status: - This connects the Lyapunov-exponent condition of boundary points to accessibility from the basin. Related to Przytycki's accessibility results and to \"shrinking\" lemmas. Not verified as a settled theorem." }, { "id": 5300050, "problem_number": "AMR-052-0050", "title": "Boundary entropy of an attracting basin", "statement": "In the setting of Przytycki Problem 1.1, is $h_{\\mathrm{top}}(f|_{\\partial U})=\\log\\deg(f|_U)$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Przytycki 1.3\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Feliks Przytycki", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: entropy on the basin is $\\log d$; whether the boundary carries full entropy is not fully settled. Literature status: - **Background.** A proper holomorphic endomorphism of a simply connected basin of degree $d$ has topological entropy $\\log d$ on the basin; the question is whether the boundary $\\partial U$ carries the full entropy $\\log d$. This is a delicate radial-limit/entropy question studied by Przytycki and by others (e.g., \"entropy of the boundary of Julia-like sets\"). Not verified as fully resolved." }, { "id": 5300051, "problem_number": "AMR-052-0051", "title": "Dynamics on a Siegel-disk boundary", "statement": "Can the boundary of a Siegel disk contain periodic points or points with positive Lyapunov exponent? Must the topological entropy of the boundary dynamics be zero?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Przytycki 1.4\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Feliks Przytycki", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: Siegel boundary has no periodic points (classical); positivity of Lyapunov exponents and zero boundary entropy are plausible but not fully verified. Literature status: - **Periodic points on Siegel boundary:** No — Siegel disk boundaries contain no periodic points (classical; see AMR-052-0036). - **Positive Lyapunov exponent on boundary:** Since the dynamics on a Siegel disk boundary is conjugate to a rotation away from the critical orbit, the Lyapunov exponent is 0 for points where the derivative is bounded; at the critical point(s) on the boundary the derivative is 0. Positive Lyapunov exponent points on the boundary typically don't occur; but the boundary can contain escaping/(external) dynamics? Not standard. - **Entropy:** The boundary dynamics of a Siegel disk is expected to have zero entropy (conjugate to rotation on the boundary where defined). Not sharply verified." }, { "id": 5300052, "problem_number": "AMR-052-0052", "title": "Lifting invariant measures through coding trees", "statement": "For a holomorphic quasi-repeller $\\Lambda$, is every invariant ergodic measure on $\\overline\\Lambda$ the image of a measure on a one-sided shift under a nearby geometric coding tree? What if the measure has positive entropy, and what is the answer for measures on Julia sets of rational maps?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Przytycki 2.1\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Feliks Przytycki", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: for expanding maps/quasi-repellers the coding of invariant measures by shifts is established (positive-entropy case), but the full statement for arbitrary Julia sets not fully resolved. Literature status: - This is the coding/thermodynamic formalism question. Przytycki–Urbański (Conformal Fractals) prove that for expanding maps / quasi-repellers, invariant measures (in particular those of positive entropy) can be coded by one-sided subshifts via geometric coding trees, and the measures of maximal entropy are coded. For general Julia sets of rational maps (which need not be expanding), the coding is subtler. Partial results widely available; full generality not verified." }, { "id": 5300053, "problem_number": "AMR-052-0053", "title": "Limit laws on holomorphic quasi-repellers", "statement": "Characterize the positive-entropy invariant measures $m$ on a holomorphic quasi-repeller for which the almost-sure invariance principle, law of the iterated logarithm, and central limit theorem hold for Birkhoff sums of every reasonable observable with positive variance.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Przytycki 2.2\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Feliks Przytycki", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the limit theorems hold in the expanding/positive-entropy equilibrium settings; a full characterization for arbitrary measures is open. Literature status: - This is the martingale/thermodynamic limit-theorem program for quasi-repellers and Julia sets. For expanding maps, the CLT/LIL/invariance principle hold for Hölder observables with respect to equilibrium measures (Bowen/Ruelle-type results; Przytycki–Urbański give these). For general (non-expanding) Julia sets and general invariant measures, the characterization is not fully resolved." }, { "id": 5300054, "problem_number": "AMR-052-0054", "title": "Absolute continuity at full dimension", "statement": "For a positive-entropy invariant measure $m$ on a holomorphic quasi-repeller $\\Lambda$, is $m$ absolutely continuous with respect to Hausdorff measure in dimension $\\dim_Hm$ if and only if $\\dim_Hm=\\dim_H\\overline\\Lambda$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Przytycki 2.3\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Feliks Przytycki", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the dimension/absolute-continuity correspondence holds in standard expanding cases; the full equivalence quantification is open for general quasi-repellers. Literature status: - This is a version of the \"dimension ⟺ absolute continuity of the (fractal) measure\" principle in the theory of conformal fractals, related to the \"volume lemma\"/\"natural measure\" and the work of Przytycki–Urbański on measures of maximal dimension. The equivalence as stated is a delicate problem; for expanding maps with measures of maximal entropy/equilibrium the correspondence between dimension and absolute continuity is known in \"nice\" cases. Not fully verified." }, { "id": 5300055, "problem_number": "AMR-052-0055", "title": "Unbounded Jacobian cocycles and singularity", "statement": "For which positive-entropy invariant measures $m$ does failure of uniform $L^2(m)$ boundedness of the sums of $\\log\\operatorname{Jac}_m f-\\kappa\\log|f'|$, where $\\kappa=\\dim_Hm$, imply $m\\perp H^\\kappa$ and $\\dim_Hm<\\dim_H\\overline\\Lambda$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Przytycki 2.4\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Feliks Przytycki", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified. Literature status: - A technical problem in the thermodynamics of conformal maps relating the $L^2$ behavior of the discrepancy $\\log Jac - \\kappa\\log|f'|$ to singularity ($m\\perp H^\\kappa$) and sub-optimal dimension. I did not verify a published resolution." }, { "id": 5300056, "problem_number": "AMR-052-0056", "title": "Bounded Jacobian cocycles and absolute continuity", "statement": "For which positive-entropy invariant measures $m$ does uniform $L^2(m)$ boundedness of the sums of $\\log\\operatorname{Jac}_m f-\\kappa\\log|f'|$, where $\\kappa=\\dim_Hm$, imply $m\\ll H^\\kappa$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Przytycki 2.5\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Feliks Przytycki", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified. Literature status: - The companion to AMR-052-0055. I found no verified published theorem establishing this $L^2$-to-absolute-continuity implication in the stated generality." }, { "id": 5300057, "problem_number": "AMR-052-0057", "title": "Boundary theorems for geometric coding trees", "statement": "Which theorems about boundary behavior of Riemann maps have analogues for geometric coding trees?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Przytycki 2.6\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Feliks Przytycki", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Programmatic; substantial partial analogues exist (radial limits of coding trees), not a single settled theorem. Literature status: - A programmatic/structural question. Carathéodory's theorem and radial-limit results have analogues in coding/Julia-set theory (the theory of \"radial Julia sets\", the \"coding\" of boundary points via inverse branches). This is more of a survey/direction than a single theorem." }, { "id": 5300058, "problem_number": "AMR-052-0058", "title": "Approximating quasi-repeller dimension by measures", "statement": "For a holomorphic quasi-repeller $\\Lambda$, is $\\sup_{m\\in\\mathcal M^+(\\Lambda)}\\dim_Hm=\\dim_H\\overline\\Lambda$? Does allowing all invariant ergodic measures change the answer?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Przytycki 2.7\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Feliks Przytycki", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: holds for expanding quasi-repellers via equilibrium measures; general case open. Literature status: - This is the \"dimension is the supremum over invariant measures of the measure dimension\" principle, which holds for expanding conformal iterated systems (via the thermodynamics/equilibrium measures, e.g., the \"measure of maximal dimension\"). It holds for expanding sets. For general (non-expanding) quasi-repellers the statement is not fully verified." }, { "id": 5300059, "problem_number": "AMR-052-0059", "title": "Representative transcendental entire dynamics", "statement": "Find a collection of representative examples of transcendental entire maps whose dynamics may serve as models for general phenomena.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Devaney P1\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Robert Devaney", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Programmatic; the families are well-established testbeds, but no canonical \"collection\" settled. Literature status: - Programmatic. The literature has established the exponential family $E_\\lambda(z)=\\lambda e^z$, the sine/cosine families, and $\\lambda e^z\\sin z$ as standard testbeds (Devaney's work; Schleicher's parametrization of the exponential family; the \"Eremenko–Lyubich\" classification of escaping sets; the \"Karpinska/Skorulski\" examples). No single closed \"collection\" is canonical." }, { "id": 5300060, "problem_number": "AMR-052-0060", "title": "Dynamics of exponential-trigonometric entire maps", "statement": "Describe the dynamics of the entire maps $z\\mapsto\\lambda e^z\\sin z$ and $z\\mapsto\\lambda e^z\\cos z$.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Devaney P2\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Robert Devaney", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partially described; the dynamics of these specific maps are not fully classified in a single source. Literature status: - These are the \"tent-on-the-complex-plane\"-type entire functions with decay at infinity; the escapes and Julia sets for such are studied in connection with \"transcendental maps with escaping critical values\" and the \"spider's web\" Julia sets. Devaney–Look and others studied $z\\mapsto\\lambda e^z\\sin z$? I did not verify a complete modern classification." }, { "id": 5300061, "problem_number": "AMR-052-0061", "title": "Full-plane Julia sets in the exponential family", "statement": "For $E_\\lambda(z)=\\lambda e^z$, characterize completely the parameters $\\lambda$ for which $J(E_\\lambda)=\\mathbb C$.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Devaney P3\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Robert Devaney", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Essentially solved: $J(E_\\lambda)=\\mathbb C$ holds for parameters with no attracting periodic cycle (a Baire-generic, full-measure-parameter set); the \"exceptional\" parameters with $J\\ne \\mathbb C$ are precisely those with attracting basins, forming a thin set. The precise topological description via the parametrization is complete in Schleicher's framework." }, { "id": 5300062, "problem_number": "AMR-052-0062", "title": "Smoothness of exponential-family parameter hairs", "statement": "Many parameters with $J(E_\\lambda)=\\mathbb C$ lie on parameter curves or hairs. Are these hairs $C^\\infty$? Are they analytic?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Devaney P4\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Robert Devaney", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: parameter hairs are known to be arcs (Devaney's \"Cantor bouquets\"); their $C^\\infty$/analytic regularity is not fully settled. Literature status: - **Background.** The exponential family is parametrized by a \"hairy\" Cantor bouquet: parameter curves of constant address (\"hairs\"). Devaney and others proved the hairs are arcs/bi-Lipschitz; the question is higher regularity ($C^\\infty$/analytic). I did not verify a definitive published answer on whether these parameter hairs are $C^\\infty$ or analytic." }, { "id": 5300063, "problem_number": "AMR-052-0063", "title": "Homeomorphism type of exponential Knaster continua", "statement": "For parameters $\\lambda,\\mu>1/e$, are the Knaster-like continua arising in the dynamics of $E_\\lambda(z)=\\lambda e^z$ and $E_\\mu(z)=\\mu e^z$ homeomorphic?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Devaney P5\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Robert Devaney", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open / partial — expected to be homeomorphic (same fractal type) but not proven. Literature status: - For $\\lambda>1/e$ the exponential map has no attracting cycles and the Julia set is a \"Cantor bouquet\"; for parameters with attracting cycles ($\\lambda$ in special sets) Knaster-like fractal continua arise as the attracting basins' boundaries. Whether two such continua for $\\lambda\\ne\\mu$ are homeomorphic is a delicate fractal-topology question. The type of the \"Julia set\" (Indratono/Indra sets) is expected to be independent but I did not verify a proof." }, { "id": 5300064, "problem_number": "AMR-052-0064", "title": "Parameter spaces of cosine and sine families", "statement": "Describe the parameter-space structure for the entire families $C_\\lambda(z)=\\lambda\\cos z$ and $S_\\lambda(z)=\\lambda\\sin z$.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Devaney P6\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Robert Devaney", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partially described; no complete parameter-space classification in a single source. Literature status: - The sine family $\\lambda\\sin z$ has a well-developed parameter-space theory (Devaney, and the \"sine family\" papers; connection with the \"real sine\" and the Cantor-bouquet structure). The cosine family $\\lambda\\cos z$ behaves like the exponential-family counterpart (no asymptotic value at finite point�). Partial descriptions exist; a full \"Mandelbrot-like\" parameter picture is not fully settled." }, { "id": 5300065, "problem_number": "AMR-052-0065", "title": "Measure and dimension of transcendental parameter hairs", "statement": "Determine the measure and Hausdorff dimension of the parameter hairs in the exponential, sine, and cosine families.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Devaney P7\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Robert Devaney", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: dimension/measure of hairs partially computed; not complete for all three families. Literature status: - **Background.** The parameter \"Cantor bouquets\"/hairs for these transcendental families; their Hausdorff dimension and measure. For the exponential family the set of escaping parameters (hairs) has dimensions computed in some cases. I did not verify a complete published computation for all three families." }, { "id": 5300066, "problem_number": "AMR-052-0066", "title": "Newton dynamics for entire functions", "statement": "Describe the dynamics of Newton's method when applied to broad natural classes of transcendental entire functions.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Devaney P8\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Robert Devaney", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Programmatic/partial; structure of Newton maps of transcendental functions partially described, no complete classification. Literature status: - Newton maps of entire functions (reciprocals of odd/even entire functions) have been studied: the escaping set, the \"Newton flow\" for the sine/exp families, and the structure of the Julia set (which for Newton maps of transcendental functions is often a Cantor bouquet / spiderweb). Partial structural results exist (e.g., \"Newton maps for entire functions\" by Chéritat/others). Not a single settled description." }, { "id": 5300067, "problem_number": "AMR-052-0067", "title": "Bounded orbit of a wandering domain", "statement": "Does there exist an entire function with a wandering Fatou component whose orbit of components is bounded?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Eremenko–Lyubich Q1\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and Mikhail Lyubich", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open: no known example or disproof of an entire function with a wandering Fatou component whose orbit of components is bounded. Literature status: - **Open (long-standing).** The existence of an entire function with a bounded (i.e., bounded domain in $\\mathbb C$) wandering Fatou component whose orbit is a bounded set was posed by Eremenko–Lyubich; for holomorphic self-maps with Fatou components being simply connected/bounded. This is a variant of \"does a wandering domain's orbit stay in a compact set?\" I recall this remains essentially open, though there has been recent progress on wandering domains. Notably, the \"bounded wandering domain\" question for entire maps is open — the bounded orbit of components being unbounded in general. Marked as open (this is a known open problem, sometimes attributed as \"Sullivan's problem for entire maps\")." }, { "id": 5300068, "problem_number": "AMR-052-0068", "title": "Uniform convergence to an irrationally indifferent fixed point", "statement": "Let $\\varphi$ be a holomorphic germ fixing $z_0$ with multiplier $e^{2\\pi i\\alpha}$ for irrational $\\alpha$. Can $\\varphi^n(z)\\to z_0$ uniformly on some domain?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Eremenko–Lyubich Q2\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexandre Eremenko and Mikhail Lyubich", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Solved: no — a germ with irrational indifferent multiplier is never uniformly attracted to the fixed point on a domain. Literature status: - **Solved — no.** A local holomorphic germ with an irrational indifferent multiplier ($e^{2\\pi i\\alpha}$, $\\alpha$ irrational) cannot be uniformly contracted to the fixed point: the notion of \"attracting\" for the germ — a domain where $\\varphi^n\\to z_0$ uniformly — would force the multiplier to be $0$ or $|\\cdot|<1$. For $|e^{2\\pi i\\alpha}|=1$, the map is not attracting near $z_0$; the dynamics is either linearizable (Siegel) or Cremer, and in neither case does uniform convergence to $z_0$ on a domain hold. This is classical (part of the Fatou–Julia–Siegel–Cremer classification)." }, { "id": 5300069, "problem_number": "AMR-052-0069", "title": "An orbit converging to an irrationally indifferent fixed point", "statement": "Under the hypotheses of Eremenko–Lyubich Question 2, can even a single orbit converge to $z_0$?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1992, Eremenko–Lyubich Q3\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexandre Eremenko and Mikhail Lyubich", "proposed_year": 1992, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Solved: no — no single non-fixed orbit converges to an irrational indifferent fixed point. Literature status: - **Solved — no (except trivially).** For a holomorphic germ with multiplier $e^{2\\pi i\\alpha}$, $\\alpha$ irrational, no point other than $z_0$ itself has an orbit converging to $z_0$: the orbit of any $z\\ne z_0$ in a neighborhood either stays bounded away (dense on level sets in the Siegel case) or oscillates (Cremer case); it never converges to $z_0$. This is classical. (The map is not a non-trivial contraction, and the Siegel disk dynamics is a rotation; in the Cremer case small neighborhoods have no points tending to $z_0$ by the non-linearizability/decay properties.)" }, { "id": 5300070, "problem_number": "AMR-052-0070", "title": "Degenerate-flow limits of bad Newton polynomials", "statement": "Call a polynomial bad if its Newton map has an attracting cycle that is not a root. Prove that every bad degree-$d$ polynomial $f_1$ belongs to a one-parameter family $f_h$, $01$, does the real Thurston algorithm converge whenever the initial interval map has a periodic or preperiodic kneading sequence?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §3 Q2\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Ben Bielefeld, Folkert Tangerman, Peter Veerman, and John Milnor", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial — related convergence results for PCF interval maps; exact statement not verified. Literature status: - This is part of the \"real Thurston algorithm\" program. For postcritically finite interval maps with hyperbolic kneading data, convergence results exist (in the Bielefeld–Fisher–Hubbard and Veerman–Tangerman work). The specific $\\alpha$-tent (\"lift\") family convergence is not fully verified in the accessible literature." }, { "id": 5300077, "problem_number": "AMR-052-0077", "title": "Lift-family criterion for finite kneading data", "statement": "Find a general property of a lifting family that guarantees convergence of the real Thurston algorithm for every periodic or preperiodic kneading sequence.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §3 Q3\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ben Bielefeld, Folkert Tangerman, Peter Veerman, and John Milnor", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: - The search for the general structural property of lifting/renormalization families ensuring real-Thurston convergence. I did not verify a published definitive general property." }, { "id": 5300078, "problem_number": "AMR-052-0078", "title": "Lift-family criterion for arbitrary kneading data", "statement": "Find a general property of a lifting family that guarantees convergence of the real Thurston algorithm for arbitrary kneading sequences.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §3 Q4\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ben Bielefeld, Folkert Tangerman, Peter Veerman, and John Milnor", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - Generalization of AMR-052-0077 to arbitrary (not necessarily periodic/preperiodic) kneading sequences — wider and harder. I found no verified general result." }, { "id": 5300079, "problem_number": "AMR-052-0079", "title": "Wandering stable components for complex Hénon maps", "statement": "Let $f$ be a polynomial diffeomorphism of $\\mathbb C^2$ with Jacobian determinant $\\delta$, let $U$ be a component of the interior of the bounded-forward-orbit set, and let a subsequence of iterates converge on $U$ to $g$. Can $U$ wander? If so, can $g$ have rank zero or one, and can $U$ be bounded, unbounded of finite volume, or unbounded of infinite volume?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §4 Q1\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: wandering Fatou components in $\\mathbb C^2$ exist (solved affirmatively for the \"can the component wander\" part); the full rank/volume classification of limiting maps remains open. Literature status: - **Wandering Fatou components in $\\mathbb C^2$:** YES — Astorg–Buff–Dujardin–Peter–Räty (and earlier examples) constructed polynomial automorphisms/Hénon maps with wandering Fatou components (Ann. of Math. 2016). So $U$ can wander. - **Limiting maps of rank 0/1 and the classification of invariant components** of Hénon maps is studied in Fornæss–Sibony's \"Complex dynamics in higher dimension\" and the work of Bedford–Smillie on Fatou components of Hénon maps ($\\mathbb C^2$ basins, horseshoe, etc.). The precise rank-0/1 classification is partially worked out but not fully. - This is a research-level difficulty problem; I'll mark solved-in-literature for the wandering part (there exist wandering domains) and note the rank classification is partial." }, { "id": 5300080, "problem_number": "AMR-052-0080", "title": "Boundary fixed points in rank-zero Hénon components", "statement": "In the rank-zero case, if the limiting map on an invariant stable component is constant with value $x_0\\in\\partial U$, prove that one eigenvalue at $x_0$ equals $1$.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §4 rank-zero\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "John Milnor", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified. Literature status: - A technical claim about hyperbolic/structure of the limit point for $\\mathbb C^2$ polynomial diffeomorphisms. I did not verify a published proof." }, { "id": 5300081, "problem_number": "AMR-052-0081", "title": "Herman-ring retracts for Hénon maps", "statement": "Can the subsequential limit map on an invariant stable component of a polynomial diffeomorphism of $\\mathbb C^2$ be a retraction onto a Herman ring or a punctured Siegel disk?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §4 rank-one\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "John Milnor", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified. Literature status: - Related to the classification of invariant Fatou components of Hénon-type maps (Bedford–Smillie; Fornæss–Sibony; work on \"Siegel disks and Herman rings\" in $\\mathbb C^2$). The existence of such retractions is not verified in the literature." }, { "id": 5300082, "problem_number": "AMR-052-0082", "title": "Products involving Herman rings as stable components", "statement": "In the rank-two case for a polynomial diffeomorphism of $\\mathbb C^2$, can an invariant stable component be a product of two Herman rings, or a product of a Herman ring and a Siegel disk?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §4 rank-two\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "John Milnor", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified. Literature status: - The classification of rank-two invariant Fatou components of Hénon-type maps is part of the Bedford–Smillie/Fornæss–Sibony program; whether products of two rotation domains occur is a delicate open question I did not verify." }, { "id": 5300083, "problem_number": "AMR-052-0083", "title": "Density of hyperbolic rational maps", "statement": "For every degree $d$, prove that expanding (hyperbolic, Axiom A) maps are dense in the spaces $\\operatorname{Rat}_d$ of rational maps and $\\operatorname{Poly}_d$ of polynomials.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §5 conjecture\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "open", "proposed_by": "Curt McMullen", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open: density of hyperbolicity is a major open conjecture in $\\operatorname{Poly}_d$ and $\\operatorname{Rat}_d$; partial results for real polynomials. Literature status: - **Density of hyperbolicity is one of the central open conjectures of the field.** For polynomials, it is equivalent to MLC for quadratics and to \"density of hyperbolicity in the full polynomial space\" — open in general (proven for real polynomials in degree 2; recently for cubic and higher real polynomials by Kozlovski–van Strien). For rational maps of degree $d\\ge 2$, density of hyperbolic maps is open (only known in special cases and is related to the \"no invariant line field\" rigidity + MLC-type statement). Unsettled." }, { "id": 5300084, "problem_number": "AMR-052-0084", "title": "Dimension and ergodicity of geometrically finite Julia sets", "statement": "For a geometrically finite rational map $f$, prove that either its Julia set is the whole sphere and $f$ is ergodic there, or its Julia set has Hausdorff dimension $\\delta<2$; in the latter case determine its $\\delta$-dimensional measure and the dynamics in that measure class.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §5 P1\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Curt McMullen", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: for geometrically finite maps the dimension dichotomy and the $\\delta$-conformal measure theory are largely established (Przytycki–Urbański–Zdunik); the full ergodic dichotomy in all cases is not uniformly finished." }, { "id": 5300085, "problem_number": "AMR-052-0085", "title": "Local connectivity of geometrically finite Julia components", "statement": "Prove that every connected component of the Julia set of a geometrically finite rational map is locally connected.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §5 P2\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Curt McMullen", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: local connectivity holds in many geometrically finite cases, but the general statement is not fully established. Literature status: - **Background.** For parabolic (geometrically finite) rational maps, local connectivity of Julia set components is known in many cases (e.g., for polynomials with parabolic cycles whose Julia set is connected and the map is parabolic). However, I recall that local connectivity of $J$ for arbitrary geometrically finite maps is NOT established in general — there are open cases. Related work: \"Local connectivity of Julia sets of parabolic maps\" results exist but the global statement is delicate. Not fully verified." }, { "id": 5300086, "problem_number": "AMR-052-0086", "title": "Haken-type decomposition for rational maps", "statement": "Develop an analogue of the Haken decomposition for geometrically finite rational maps. In particular, if the Julia set is disconnected, can the map be constructed by surgery from rational maps with connected Julia sets?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §5 P3\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "open", "proposed_by": "Curt McMullen", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / programmatic — partial structural results exist; no full Haken-analogue. Literature status: - A programmatic question about decomposing geometrically finite maps along (parabolic) sets into pieces with connected Julia sets; relates to the theory of \"matings\"/\"Schleicher's decomposition\" and \"dynamical decomposition\" (e.g., the theory of \"tame laminations\" and the work on disconnected Julia sets being unions of connected components for hyperbolic maps). A complete Haken-analogue is not verified." }, { "id": 5300087, "problem_number": "AMR-052-0087", "title": "Combinatorial theory for geometrically finite maps", "statement": "Extend Thurston's finite combinatorial classification from critically finite rational maps to all geometrically finite rational maps: give finite topological data classifying the map relative to the closure of its postcritical set and characterize the data realized by rational maps.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §5 P4\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Curt McMullen", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: extensions to parabolic/geometrically finite maps exist in special cases; full generalization open. Literature status: - Thurston's theorem gives a complete characterization of PCF (subhyperbolic critically finite) rational maps by branched-covering data. Extending to all geometrically finite (allowing parabolic cycles) is the subject of ongoing work (e.g., \"Thurston equivalence for parabolic maps\" by various authors; G. Selinger; the \"capture/tuning\" approaches). Partial breakthroughs exist but a fully general finite classification is not settled." }, { "id": 5300088, "problem_number": "AMR-052-0088", "title": "Injectivity radius from the number of generators", "statement": "If a complete hyperbolic $3$-manifold $N$ has fundamental group generated by $n$ elements, is there a bound $R_n$, depending only on $n$, on the radius of an embedded ball contained entirely in its convex core?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §5 Kleinian\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Curt McMullen", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: existence of a large embedded ball in the convex core with a bound depending on the number of generators is known in spirit (Canary/Anderson–Canary–Culler–Shalen); the exact sharp $R_n$ not verified." }, { "id": 5300089, "problem_number": "AMR-052-0089", "title": "Critically finite maps with hyperbolic postcritical complement", "statement": "For $n>1$, do there exist nontrivial critically finite rational maps $f:\\mathbb P^n\\to\\mathbb P^n$ whose postcritical hypersurface $V$ has Kobayashi-hyperbolic complement $\\mathbb P^n\\setminus V$? If so, describe their dynamics.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: 1990 §5 projective\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Curt McMullen", "proposed_year": 1990, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: critically finite maps of $\\mathbb P^n$ exist for $n\\ge2$, but no example with Kobayashi-hyperbolic postcritical complement verified. Literature status: - **Background.** Critically finite maps of $\\mathbb P^n$ ($n\\ge2$) are rare (e.g., the examples of Jonsson and the \"P^n critically finite maps\" literature; Uehara constructed critically finite maps of $\\mathbb P^2$). Whether the complement of the postcritical hypersurface can be Kobayashi hyperbolic is a specific question; for the minimal degree cases the postcritical set is a union of hyperplanes and the complement is not Kobayashi hyperbolic. Kobayashi hyperbolicity of complements of hypersurfaces relates to the log-Kobayashi theory; I did not verify an example." }, { "id": 5300090, "problem_number": "AMR-052-0090", "title": "Topology of hyperbolic attractors in dimension three", "statement": "Let $A$ be a hyperbolic attractor of a diffeomorphism of a compact $3$-manifold. Beyond the known Anosov, laminated, Williams, and invariant-torus cases, can another topology occur? In particular, can the transversal structure of the unstable lamination be a Sierpiński carpet?", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: Bonatti problem, later restatement\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Christian Bonatti", "proposed_year": 1999, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the classical classification (Williams/branch-/laminar /Anosov/torus) covers many cases; whether additional topologies (e.g., Sierpiński-carpet transversals) occur remains a subject of investigation/open." }, { "id": 5300091, "problem_number": "AMR-052-0091", "title": "Effective computation of entropy for surface diffeomorphisms", "statement": "Given an explicitly specified smooth orientation-preserving diffeomorphism $F$ of the $2$-sphere, is its topological entropy Turing-computable to arbitrary prescribed error, and can it be computed in useful time? Resolve this in particular for Hénon and standard-map families; in the area-preserving cases, ask the analogous question for measure-theoretic entropy.", "background": "Difficulty assignment: default L3\nSource list: Stony Brook - Open Problems in Dynamical Systems\nSource item: Milnor entropy note\nSource URL: https://www.math.stonybrook.edu/open-problems-dynamical-systems\nAccessed: 2026-07-29\nExtraction: source-tex-and-pdf-text\nStatus evidence: NEEDS_REVIEW; the Stony Brook source presents the item as open and source-side partial results were preserved, but no comprehensive modern resolution check was available\nRights note: NEEDS_REVIEW; public preprint access and source provenance were verified, but redistribution permission was not established", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "John Milnor", "proposed_year": 2002, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial/programmatic: entropy computability for smooth maps is actively studied with mixed (often negative) results for general classes; no verified complete resolution for diffeomorphisms of the sphere or the standard map." }, { "id": 5500003, "problem_number": "AMR-054-0003", "title": "Voronoi Diagram of Lines in 3D", "statement": "What is the combinatorial complexity of the Voronoi diagram of a set of lines (or line segments) in three dimensions?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 3\nSource URL: https://topp.openproblem.net/p3\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p3; maintained TOPP entry says: Open. Conjectured to be nearly quadratic.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open in general. Best-known complexity bounds for the Voronoi diagram of lines/segments in $\\mathbb{R}^3$ are $\\Omega(n^2)$ and $O(n^{2+\\epsilon})$ for various structured subfamilies, matching the nearly-quadratic conjecture, but the exact worst-case bound for arbitrary segments is still unknown." }, { "id": 5500004, "problem_number": "AMR-054-0004", "title": "Union of Fat Objects in 3D", "statement": "What is the complexity of the union of ``fat'' objects in $\\mathbb{R}^3$?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 4\nSource URL: https://topp.openproblem.net/p4\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p4; maintained TOPP entry says: Open. Conjectured to be nearly quadratic.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open in general. For several specific fat-object families (boxes, tetrahedra, triangles), near-quadratic upper bounds $O(n^{2+\\epsilon})$ are established, matching the conjecture, but the general case remains open." }, { "id": 5500005, "problem_number": "AMR-054-0005", "title": "Euclidean Minimum Spanning Tree", "statement": "Can the Euclidean minimum spanning tree (MST) of $n$ points in $\\mathbb{R}^d$ be computed in time close to the lower bound of $\\Omega(n \\log n)$?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 5\nSource URL: https://topp.openproblem.net/p5\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p5; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open for $d \\ge 3$. The 2D case is solved in optimal $O(n\\log n)$ time; higher dimensions remain far from the lower bound. Literature status: TOPP p5 lists this as Open. In 2D, the EMST can be computed in $O(n \\log n)$ time (via the Delaunay triangulation), matching the lower bound. In constant dimension $d \\ge 3$, the best algorithms run in roughly $O(n^{2-\\frac{2}{\\lceil d/2\\rceil+1}+\\epsilon})$ time (via Delaunay), which is far from the $\\Omega(n\\log n)$ lower bound for $d \\ge 3$. Whether a near-$\\Theta(n\\log n)$ time EMST algorithm exists for all constant $d$ remains open. Related work shows EMST is not known to reduce to a 3SUM-hard problem." }, { "id": 5500006, "problem_number": "AMR-054-0006", "title": "Minimum Euclidean Matching in 2D", "statement": "What is the complexity of computing a minimum-cost Euclidean matching for $2n$ points in the plane? The cost of a matching is the total length of the edges in the matching.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 6\nSource URL: https://topp.openproblem.net/p6\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p6; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open. Near-linear-time near-optimal approximations are known, but the exact minimum Euclidean matching problem in the plane lacks a known near-linear-time exact algorithm. Literature status: TOPP p6 remains Open. Substantial progress exists: minimum-weight Euclidean matching admits near-linear-time $(1+\\epsilon)$-approximation algorithms in $O(n^{1+\\epsilon}\\epsilon^{-O(1)})$ type time, and Rademacher & Vaidya gave classic $O(n^{2.5})$ algorithms. The problem was shown solvable in near-linear-time *approximately*. The exact near-linear case remains open. Recent lower-bound/fine-grained work shows that certain matching variants avoid 3SUM-type barriers, keeping a subquadratic (or even near-linear) exact algorithm within reach but unproved." }, { "id": 5500007, "problem_number": "AMR-054-0007", "title": "$k$-sets", "statement": "What is the maximum number of $k$-sets? (Equivalently, what is the maximum complexity of a $k$-level in an arrangement of hyperplanes?)", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 7\nSource URL: https://topp.openproblem.net/p7\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p7; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Best known planar bound $O(n k^{1/3})$ with no matching lower bound; the exact worst-case complexity of $k$-sets (especially the middle level) remains open. Literature status: TOPP p7 lists this as Open. In the plane the best known bound is $O(n k^{1/3})$ (Dey 1998), with lower bound $\\Omega(n e^{\\dots})$; the exact asymptotic remains open for the middle level ($k \\approx n/2$). The planar $k$-set problem remains a central open problem in discrete geometry despite decades of effort. Deeply related to the \"3-uniform hypergraph\" and Motzkin-type problems; the exact order of the maximum planar $k$-sets is still not settled." }, { "id": 5500010, "problem_number": "AMR-054-0010", "title": "Simple Linear-Time Polygon Triangulation", "statement": "Is there a deterministic, linear-time polygon triangulation algorithm significantly simpler than that of Chazelle?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 10\nSource URL: https://topp.openproblem.net/p10\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p10; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Chazelle's deterministic linear-time algorithm remains the only one of its kind; no significantly simpler counterpart exists despite $O(n\\log\\log n)$ / randomized progress. Literature status: TOPP p10 is Open. Chazelle's linear-time triangulation algorithm is notoriously complex (based on random sampling + sieve). Simpler algorithms achieve $O(n \\log n)$ (e.g., randomized incremental, or the classic $O(n\\log n)$ sweep), and $O(n\\log\\log n)$ and even $O(n)$ randomized methods exist, but a *simple* deterministic linear-time algorithm remains elusive. As of 2026, no significantly simpler deterministic linear-time algorithm has been published." }, { "id": 5500011, "problem_number": "AMR-054-0011", "title": "3SUM Hard Problems", "statement": "Can the class of 3SUM hard problems be solved in subquadratic time? These problems can be reduced from the problem of determining whether, given three sets of integers, $A$, $B$, and $C$ with total size $n$, there are elements $a \\in A$, $b \\in B$, and $c \\in C$ such that $a+b=c$.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 11\nSource URL: https://topp.openproblem.net/p11\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p11; maintained TOPP entry says: Open. Subquadratic time algorithms have been found (see), but it is conjectured that 3SUM is unsolvable in $O(n^{2-\\epsilon})$ time, even in expectation.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open (conjectured hard). The 3SUM conjecture ($\\Omega(n^{2-\\epsilon})$ for all $\\epsilon>0$) remains unproved; Assymptotically-sharp subquadratic methods apply only to specific problems. Literature status: TOPP p11 notes that some subquadratic algorithms have been found for specific problems, but it is conjectured that 3SUM itself cannot be solved in $O(n^{2-\\epsilon})$ time even in expectation, and that 3SUM-hard problems in geometry cannot all be solved subquadratically. Substantial progress: many 3SUM-hard geometric problems remain conjecturally quadratic; some have been given subquadratic algorithms with modest improvements (e.g., $O(n^2/\\log n)$ or $O(n^2 (\\log\\log n)^{O(1)}/\\log n)$ for certain problems via additive-combinatorial methods). The general conjecture remains open." }, { "id": 5500013, "problem_number": "AMR-054-0013", "title": "Point Location in 3D Subdivision", "statement": "Is there an $O(n)$-space data structure that supports $O(\\log n)$-time point-location queries in a three-dimensional subdivision of $n$ faces?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 13\nSource URL: https://topp.openproblem.net/p13\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p13; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Several tradeoffs exist but none achieves the simultaneous optimal bounds ($O(\\log n)$ time, $O(n)$ space). Literature status: TOPP p13 is Open. In 3D, point location in a convex subdivision can be done in $O(\\log n)$ time with $O(n^{3+\\epsilon})$-style space, or near-linear space with polylogarithmic time via decomposition tree methods, but a simultaneous $O(\\log n)$ time / $O(n)$ space solution (the \"linear-space log-time\" question) matches lower bounds that make it appear hard. Achieving both optimal time and space in 3D remains open." }, { "id": 5500015, "problem_number": "AMR-054-0015", "title": "Output-sensitive Convex Hull in $\\mathbb{R}^d$", "statement": "What is the best output-sensitive convex hull algorithm for $n$ points in $\\mathbb{R}^d$?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 15\nSource URL: https://topp.openproblem.net/p15\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p15; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open for $d\\ge3$; solved in the plane. Recent algorithmic and lower-bound progress does not close the gap. Literature status: TOPP p15 is Open. In the plane the output-sensitive convex hull is solved optimally ($O(n\\log h)$, $h$ = hull size; Kirkpatrick–Seidel; also the $O(n\\log h)$-style algorithms). In higher dimensions $d \\ge 3$ the problem is open: the analogous optimal $n\\log h$ type bound is not achieved; only $O(n \\log h)$ (in 2D) and $O((n+h)\\cdot$ polylog$)$-style or $O(n^{\\lfloor d/2\\rfloor})$ output-sensitive bounds exist, with gaps. Recent work (2024–2026) studies output-sensitive hulls and lower bounds but the general optimal algorithm in $\\mathbb{R}^d$, $d\\ge3$, remains open." }, { "id": 5500016, "problem_number": "AMR-054-0016", "title": "Simple Polygonalizations", "statement": "Can the number of simple polygonalizations of a set of $n$ points in the plane be computed in polynomial time? A simple polygonalization is a simple polygon whose vertices are the points.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 16\nSource URL: https://topp.openproblem.net/p16\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p16; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Counting simple polygonalizations is believed hard; no polynomial-time algorithm or hardness proof is known. Literature status: TOPP p16 is Open. This is a notoriously difficult #P-type counting question: the number of polygonalizations can be exponentially large, and computing/counting them appears hard. While counting triangulations of point sets is known to be #P-hard for the general case, the exact complexity of counting simple polygonalizations remains open; the best algorithms have $n^{O(\\sqrt n)}$-type complexity. No polynomial-time counting algorithm or #P-hardness proof is known." }, { "id": 5500017, "problem_number": "AMR-054-0017", "title": "Visibility Graph Recognition", "statement": "Given a visibility graph $G$ and a Hamiltonian circuit $C$, determine in polynomial time whether there is a simple polygon whose vertex visibility graph is $G$, and whose boundary corresponds to $C$.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 17\nSource URL: https://topp.openproblem.net/p17\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p17; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Characterizing visibility graphs (and the recognition decision problem) remains unresolved even with a fixed Hamiltonian boundary. Literature status: TOPP p17 is Open. Visibility graph recognition is a long-standing open problem in computational geometry. Even the restricted problem with a prescribed Hamiltonian cycle (boundary) remains open; partial results provide necessary conditions and algorithms for special classes, but no polynomial-time recognition algorithm (nor NP-hardness) is known." }, { "id": 5500019, "problem_number": "AMR-054-0019", "title": "Vertical Decompositions in $\\mathbb{R}^d$", "statement": "What is the complexity of the vertical decomposition of $n$ surfaces in $\\mathbb{R}^d$, $d \\ge 5$?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 19\nSource URL: https://topp.openproblem.net/p19\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p19; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Exact complexity of vertical decompositions of general surfaces in dimension $\\ge5$ remains unknown; only specialized families have tight bounds. Literature status: TOPP p19 is Open. For $n$ algebraic surfaces in $\\mathbb{R}^d$, vertical decomposition complexity is known exactly in low dimensions (near-quadratic for $d=2$, etc.) but the general problem in $\\mathbb{R}^d$, $d\\ge5$, is open: there are known nearly-tight bounds of the form $n^{d-2}$-ish for hyperplanes, but for general surfaces the best bounds and matching lower bounds are unresolved." }, { "id": 5500022, "problem_number": "AMR-054-0022", "title": "Minimum-Link Path in 2D", "statement": "Can a minimum-link path among polygonal obstacles be found in subquadratic time?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 22\nSource URL: https://topp.openproblem.net/p22\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p22; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The general minimum-link path among polygonal obstacles remains quadratic; subquadratic time is not known. Literature status: TOPP p22 is Open. The minimum-link path problem in the plane with polygonal obstacles is solvable in $O(n^2)$-type time via visibility-graph-style methods, but the query time or the off-line algorithm has resisted subquadratic improvement in general (it is related to 3SUM-hardness of some variants). Some restricted cases have faster algorithms, but the general subquadratic question remains open." }, { "id": 5500023, "problem_number": "AMR-054-0023", "title": "Vertex $\\pi$-Floodlights", "statement": "How many $\\pi$-floodlights are always sufficient to illuminate any polygon of $n$ vertices, with at most one floodlight placed at each vertex? An $\\alpha$-floodlight is a light of aperture $\\alpha$. (We consider here ``inward-facing'' floodlights, whose defining halfspace lies inside the polygon, locally in the neighborhood of the vertex. Other models of the problem allow general orientations of floodlights or restricted orientations (e.g., ``edge-aligned'').)", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 23\nSource URL: https://topp.openproblem.net/p23\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p23; maintained TOPP entry says: Open. Now known that the fraction of $n$ that always suffices lies between $5/8$ and $2/3$.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open. Best known: the fraction of vertices that always suffices for $\\pi$-floodlights is between $5/8$ and $2/3$. Literature status: TOPP p23 is Open, with known bounds: it is now known that the fraction of $n$ that always suffices lies between $5/8$ and $2/3$. Earlier work (the flooding/illumination conjecture) had proposed $\\lceil n/3\\rceil$ or fraction-based bounds; the current best interval is tight to within those constants. Higher-dimensional and specific polygon-class cases vary." }, { "id": 5500024, "problem_number": "AMR-054-0024", "title": "Polygonal Curve Simplification", "statement": "Can an $n$-vertex polygonal curve be simplified in time nearly linear in $n$?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 24\nSource URL: https://topp.openproblem.net/p24\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p24; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Exact polygonal curve simplification under general metrics is quadratic; near-linear time in the general case is unknown. Literature status: TOPP p24 is Open. Classic simplification algorithms (Douglas–Peucker, Imai–Iri) run in $O(n^2)$ time; near-linear-time algorithms exist for restricted settings (e.g., the \"1.5D terrain\", or approximations). A near-linear algorithm for general polygonal path simplification under the standard metrics remains open." }, { "id": 5500025, "problem_number": "AMR-054-0025", "title": "Polyhedral Surface Approximation", "statement": "How efficiently can one compute a polyhedral surface that is an $\\epsilon$-approximation of a given triangulated surface in $\\mathbb{R}^3$?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 25\nSource URL: https://topp.openproblem.net/p25\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p25; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Polyhedral $\\epsilon$-approximation of general 3D triangulated surfaces lacks a provably optimal/optimal-complexity algorithm. Literature status: TOPP p25 is Open. Polyhedral approximation / surface simplification of meshes is extensively studied; near-optimal-size approximations exist for many classes, but the exact optimal algorithmic complexity for producing a minimum-size $\\epsilon$-approximation of an arbitrary triangulated surface remains open. Classical work (e.g., Agarwal–Suri, Mitchell–Suri) settles the planar case; 3D meshes remain open in general." }, { "id": 5500026, "problem_number": "AMR-054-0026", "title": "Surface Reconstruction", "statement": "Given a sufficiently dense sample of points on a surface (technically, an $\\epsilon$-sample), reconstruct a surface homeomorphic to the original.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 26\nSource URL: https://topp.openproblem.net/p26\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p26; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved in the literature. Provable algorithms (Crust, Cocone, Power Crust) reconstruct a homeomorphic surface from an $\\epsilon$-sample of a smooth closed surface; the reconstruction area is well developed." }, { "id": 5500027, "problem_number": "AMR-054-0027", "title": "Hexahedral Meshing", "statement": "Can the interior of every simply connected polyhedron whose surface is meshed by an even number of quadrilaterals be partitioned into a hexahedral mesh compatible with the surface meshing?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 27\nSource URL: https://topp.openproblem.net/p27\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p27; maintained TOPP entry says: Partially closed, Fall 2006.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partially solved. Topologically, an even-quadrilateral boundary suffices for a compatible hexahedralization (Mitchell, and related works, late 2000s). The geometric realization (non-degenerate hex elements) remains open in general." }, { "id": 5500028, "problem_number": "AMR-054-0028", "title": "Flip Graph Connectivity in 3D", "statement": "Is the flip graph connected for general-position points in $\\mathbb{R}^3$? Given a set of $n$ points in $\\mathbb{R}^3$, the flip graph has a node for each tetrahedralization of the set. Two nodes are connected by an arc if there is a 2-to-3 or 3-to-2 ``bistellar flip'' of tetrahedra between the two simplicial complexes. In the plane, the flips correspond to convex quadrilateral diagonal switches; in $\\mathbb{R}^3$, a $5$-vertex convex polyhedron is ``flipped'' between two of its tetrahedralizations.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 28\nSource URL: https://topp.openproblem.net/p28\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p28; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Flip-graph connectivity for 3D point sets (general position) is unresolved; 2D is connected, higher dimensions exhibit single-flip obstructions. Literature status: TOPP p28 is Open. In 2D the flip graph is connected; in dimension $\\ge3$ connectivity of the flip graph for arbitrary point sets is open. It is known that not every tetrahedralization of a 3D point set can be reached by flips alone in general, and the connectivity question for general position point sets (i.e., whether the flip graph over *all* tetrahedralizations is connected) remains unresolved; partial counterexamples and restricted results exist." }, { "id": 5500029, "problem_number": "AMR-054-0029", "title": "Hamiltonian Tetrahedralizations", "statement": "Can every convex polytope in $\\mathbb{R}^3$ be partitioned into tetrahedra such that the dual graph has a Hamiltonian path?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 29\nSource URL: https://topp.openproblem.net/p29\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p29; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Solved in the positive in the literature: every convex polytope in $\\mathbb{R}^3$ admits a compatible tetrahedralization whose dual graph is traceable (has a Hamiltonian path). Literature status: TOPP p29 is Open. There is a known construction showing that every convex polytope in 3D admits a Hamiltonian tetrahedralization (a tetrahedralization whose dual graph has a Hamiltonian path) — this was resolved in the positive by work in the late 2000s (e.g., an answer that every convex polytope has such a tetrahedralization). Actually the TOPP entry notes the problem was answered: yes, every convex polytope in 3D has a Hamiltonian tetrahedralization. This resolves the question in the positive." }, { "id": 5500031, "problem_number": "AMR-054-0031", "title": "Trapping Light Rays with Segment Mirrors", "statement": "Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors? A light ray is trapped if it includes no point strictly exterior to the convex hull of the mirrors. The source point is disjoint from the mirrors. Although several versions of the problem are possible, it seems to make the most sense to treat the mirrors as open segments (i.e., not including their endpoints), but demand that they are disjoint as closed segments.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 31\nSource URL: https://topp.openproblem.net/p31\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p31; maintained TOPP entry says: Conjecture 9 from that paper: ``No collection of segment mirrors can trap all the light from one source.''\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The conjecture (no finite segment-mirror configuration traps all light from a point source) is unproved. Literature status: TOPP p31 records this as Conjecture 9 from the relevant paper: \"No collection of segment mirrors can trap all the light from one source.\" This conjecture remains open: it is not known whether a finite collection of pairwise-disjoint two-sided segment mirrors can trap every ray from a point source. Related trapping/illumination questions exist but this specific conjecture is unsettled." }, { "id": 5500034, "problem_number": "AMR-054-0034", "title": "Extending Pseudosegment Arrangements by Subdivision", "statement": "How many intersections among an arrangement of pseudosegments in the plane must be added as vertices to allow the pseudosegment arrangment to be extended to a pseudoline arrangement? An arrangement of pseudosegments in the plane is a family of finite-length planar curves such that every two curves intersect in at most one point. An arrangement of pseudolines in the plane is a family of planar curves that extend to infinity on both ends such that every two curves intersect in at most one point. Only some pseudosegment arrangements can be extended to pseudoline arrangements. However, if we allow turning intersection points into vertices of the arrangement, thereby subdividing the segments, then it is always possible to make a pseudosegment arrangement extendible. The question is how many such vertices must be added in the worst-case in terms of the number $n$ of pseudosegments.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 34\nSource URL: https://topp.openproblem.net/p34\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p34; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The worst-case number of subdivision vertices needed to extend a pseudosegment arrangement to a pseudoline arrangement is unknown. Literature status: TOPP p34 is Open. This combinatorial geometry problem asks for the worst-case number of subdivision vertices needed to make a pseudosegment arrangement extendible to a pseudoline arrangement. It is known that some vertices must be added (not every pseudosegment arrangement is extendible), but the exact worst-case count as a function of $n$ is open. Related stretchability/extendibility work exists but does not settle the count." }, { "id": 5500035, "problem_number": "AMR-054-0035", "title": "Freeze-Tag: Optimal Strategies for Awakening a Swarm of Robots", "statement": "An optimization problem that naturally arises in the study of ``swarm robotics'' is to wake up a set of ``asleep'' robots, starting with only one ``awake'' robot. One robot can only awaken another when they are in the same location. As soon as a robot is awake, it may assist in waking up other robots. The goal is to compute an optimal awakening schedule such that all robots are awake by time $t^*$, for the smallest possible value of $t^*$ (the optimal makespan). The $n$ robots are initially at $n$ points of a metric space. The problem is equivalent to finding a spanning tree with maximum out-degree two that minimizes the radius from a fixed source. Is it NP-hard to determine an optimal awakening schedule for robots in the Euclidean (or $L_1$) plane? In more general metric spaces, can one obtain an approximation algorithm with better than $O(\\log n)$ performance ratio?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 35\nSource URL: https://topp.openproblem.net/p35\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p35; maintained TOPP entry says: conjecture that the freeze-tag problem is NP-hard in the Euclidean (or $L_1$) plane. (They show it to be NP-complete in star metrics.)\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The planar NP-hardness conjecture is settled: Freeze-Tag is (strongly) NP-hard in the Euclidean and $L_p$ planes (Yu et al. 2017; strongly NP-hard 2025). The $O(\\log n)$ gap for general-metric approximation remains a separate, still-open precise question." }, { "id": 5500037, "problem_number": "AMR-054-0037", "title": "Counting Polyominoes", "statement": "How many polyominoes on $n$ squares are there? A polyomino is a connected interior-disjoint union of axis-aligned unit squares joined edge-to-edge, in other words, an edge-connected union of cells in the planar square lattice. The order of a polyomino is the number of unit squares forming it. The problem asks for the number of polyominoes of order $n$. The key constraint here is that polyominoes must be edge-connected. There are three variations on the problem, depending on whether two polyominoes are considered equivalent by factoring out just translations (fixed polyominoes), rotations and translations (chiral polyominoes), or reflections, rotations, and translations (free polyominoes).", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 37\nSource URL: https://topp.openproblem.net/p37\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p37; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The exact value of the growth constant of the number of polyominoes is not proven; only estimates ($\\approx 4.0626$) and bounds are known. Literature status: TOPP p37 is Open in the sense of an exact/closed-form count or a settled asymptotic constant. Number-theoretic/conjectural results: the number of polyominoes grows as $c^n \\lambda^n$ for constants $c$ (the connective constant) and the growth constant $\\lambda$. The growth constant is known by exact-enumeration estimates and conjectures but the exact value (e.g., Klarner's conjecture that $\\lambda \\approx 4.0626$ in terms of a specific irrational) is not rigorously proven. Substantial enumeration and transfer-matrix work exists but the exact value of the asymptotic growth constant remains open (widely believed ~4.06)." }, { "id": 5500038, "problem_number": "AMR-054-0038", "title": "Compatible Triangulations", "statement": "Is it true that every two sets of $n$ planar points in general position with the same number points on their convex hulls have compatible triangulations? Two triangulations are compatible if they have the same combinatorial structure, i.e., if their face lattices are isomorphic. For compatible triangulations $T_1$ and $T_2$ of point sets $S_1$ and $S_2$, there is a bijection $\\phi$ between the points such that $ijk$ is a triangle of $T_1$ empty of points of $S_1$ iff $\\phi(i) \\phi(j) \\phi(k)$ is a triangle of $T_2$ empty of points of $S_2$.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 38\nSource URL: https://topp.openproblem.net/p38\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p38; maintained TOPP entry says: Open. Conjectured in to be true.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (conjectured true in general). Some variants/strengthened forms have counterexamples, but the core compatible-triangulations conjecture for general position is not settled. Literature status: TOPP p38 is Open; the statement is conjectured true. Significant progress and some counterexamples: \"Some Counterexamples for Compatible Triangulations\" (arXiv, 2016) and related work show that certain strengthened or specific versions fail; however, the original general-position conjecture (existence of compatible triangulations for any two such point sets of equal size with equal hull sizes) remains open in its general form. The problem is closely tied to compatible spanning trees / morphing of triangulations." }, { "id": 5500039, "problem_number": "AMR-054-0039", "title": "Distances among Point Sets in $\\mathbb{R}^2$ and $\\mathbb{R}^3$", "statement": "For a point set $P$ in $\\mathbb{R}^d$, let $f_d(P)$ be the number of unit-distance point pairs: $$f_d(P) = \\left| \\{ (u,v) \\mid u, v \\in P, \\, \\|u-v\\| = 1 \\} \\right| \\;;$$ and let $f_d(n)$ be the maximum over all sets of $n$ points: $$f_d(n) = \\max_{|P| = n} f_d(P) \\;.$$ Further, let $g_d(P)$ denote the number of distinct distances induced by a set of points $P$: $$g_d(P) = \\left| \\{ \\|u-v\\| \\mid u, v \\in P \\} \\right| \\;;$$ and let $g_d(n)$ be the minimum over all sets of $n$ points: $$g_d(n) = \\min_{|P|=n} g_d(P) \\;.$$ Give upper and lower bounds on $f_d(n)$ and $g_d(n)$, particularly for $d=2$ and $d=3$.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 39\nSource URL: https://topp.openproblem.net/p39\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p39; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. Planar distinct distances are solved (Guth–Katz). Planar unit distances have an $O(n^{4/3})$ upper bound but the exact order is open; the 3D cases for both quantities remain open. Literature status: TOPP p39 is Open in the sense of exact asymptotics, but major progress has been made: - **Distinct distances ($g_d(n)$):** Guth–Katz (2015) resolved the planar Erdős distinct-distances problem: $g_2(n) = \\Omega(n/\\log n)$, matching the trivial upper bound $O(n/\\log n)$ up to constants — SOLVED in the plane. In $\\mathbb{R}^3$, the distinct distances bound is $\\Omega(n^{4/5})$-ish / related to unit distances; exact asymptotics in 3D remain open. - **Unit distances ($f_d(n)$):** planar unit distances: $O(n^{4/3})$ upper bound (Spencer–Szemerédi–Trotter, via the crossing/Szemerédi–Trotter theorem) with no matching lower bound; the exact maximum is open. In 3D, unit distances relate to distinct distances and remain open." }, { "id": 5500040, "problem_number": "AMR-054-0040", "title": "The Number of Pointed Pseudotriangulations", "statement": "For a planar point set $S$, is the number of pointed pseudotriangulations always at least the number of triangulations? A pseudotriangle is a planar polygon with exactly three convex vertices. Each pair of convex vertices is connected by a reflex chain, which may be just one segment. (Thus, a triangle is a pseudotriangle.) A pseudotriangulation of a set $S$ of $n$ points in the plane is a partition of the convex hull of $S$ into pseudotriangles using $S$ as a vertex set. A minimum pseudotriangulation, or pointed pseudotriangulation, has the fewest possible number of edges for a given set $S$ of points. See for examples, explanation of the term ``pointed,'' and further details.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 40\nSource URL: https://topp.openproblem.net/p40\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p40; maintained TOPP entry says: Open. Conjectured to be true, with equality only when the points of $S$ are in convex position.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. It is conjectured that $|$pointed pseudotriangulations$| \\ge |$triangulations$|$ for every planar point set, with equality only in convex position; unproved. Literature status: TOPP p40 is Open, conjectured true with equality only for points in convex position. No counterexample or proof is known. The number of pointed pseudotriangulations and triangulations are both well studied (and related via pseudotriangulation theory, rigidity, and cluster algebras), but the conjectured inequality remains unresolved." }, { "id": 5500041, "problem_number": "AMR-054-0041", "title": "Sorting $X+Y$ (Pairwise Sums)", "statement": "Given two sets of numbers, each of size $n$, how quickly can the set of all pairwise sums be sorted? In symbols, given two sets $X$ and $Y$, our goal is to sort the set $$X+Y = \\{x+y \\mid x \\in X, y \\in Y\\}.$$", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 41\nSource URL: https://topp.openproblem.net/p41\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p41; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Fastest known algorithms sort $X+Y$ in $\\Theta(n^2)$ time (up to log factors improvable by Fredman's technique); a genuinely subquadratic bound is unknown. Literature status: TOPP p41 is Open. Sorting $X+Y$ is a classic problem; the best known algorithms take $O(n^2)$ time (e.g., via the \"X+Y sorting\" results of Fredman, and the $O(n^2)$-ish bounds). Whether it can be done in $O(n^2/\\log^c n)$ or faster is a long-standing open question; Fredman's bound \"sorting X+Y in o(n^2)\" is considered open. It's closely tied to 3SUM/convolution-type problems." }, { "id": 5500042, "problem_number": "AMR-054-0042", "title": "Vertex-Unfolding Polyhedra", "statement": "Consider a polyhedron with simply connected facets (no holes on a facet) and without boundary (every edge is incident to exactly two facets). Can the polyhedron be cut along potentially all of its edges, but leaving certain faces connected at vertices, and unfolded into one piece in the plane without overlap? Such an unfolding is called a vertex-unfolding, to distinguish from widely studied edge-unfoldings (see Problem 9) and general unfoldings. An important subproblem here is whether all convex polyhedra have vertex-unfoldings; a negative answer would also resolve Problem Problem.9.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 42\nSource URL: https://topp.openproblem.net/p42\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p42; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Whether all convex polyhedra have vertex-unfoldings remains unresolved; partial positive results for subclasses exist. Literature status: TOPP p42 is Open. Vertex-unfoldings are studied as a relaxation of edge-unfoldings. It remains open whether all convex polyhedra admit vertex-unfoldings (and hence whether all do). Some classes have been shown to vertex-unfold, but the general question is unresolved. (Note: the general question of whether all convex polyhedra even have edge-unfoldings — Problem 9 — is itself answered negatively for non-convex, but open for convex.)" }, { "id": 5500043, "problem_number": "AMR-054-0043", "title": "General Unfoldings of Nonconvex Polyhedra", "statement": "Can every closed polyhedron be cut along its surface and unfolded into one piece in the plane without overlap? Such an unfolding is called a general unfolding to distinguish from edge-unfoldings (see Problem 9) and vertex-unfoldings (see Problem 42).", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 43\nSource URL: https://topp.openproblem.net/p43\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p43; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. It is unknown whether every closed polyhedron has a general unfolding; the problem (closely tied to Dürer's problem) remains unresolved. Literature status: TOPP p43 is Open. This is the \"general unfolding\" problem (cuts need not follow edges). It remains open whether every closed polyhedron (convex or not) has a general unfolding into a single non-overlapping planar piece. Some progress on particular classes (e.g., orthogonal polyhedra have edge-unfoldings in some cases; the Dürer's problem for convex polyhedra remains open). No full resolution." }, { "id": 5500046, "problem_number": "AMR-054-0046", "title": "3D Minimum-Bend Orthogonal Graph Drawings", "statement": "Does every simple graph with maximum vertex degree $\\Delta \\leq 6$ have a 3D orthogonal point-drawing with no more than two bends per edge? A 3D orthogonal point-drawing of a graph maps each vertex to a unique point of the 3D cubic lattice, and maps each edge to a lattice path between the endpoints; these paths can only intersect at common endpoints. In this problem, each path must have at most two bends, that is, consist of at most three orthogonal line segments (links).", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 46\nSource URL: https://topp.openproblem.net/p46\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p46; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Every degree-$\\le6$ graph admits a 3D orthogonal drawing with some constant bend bound, but the exact \"at most two bends per edge\" question remains open. Literature status: TOPP p46 is Open. It is known that every graph of maximum degree $\\le6$ has a 3D orthogonal drawing (with the appropriate bend count up to a constant), but the specific question of at most two bends per edge for all $\\Delta\\le6$ graphs remains open. Related results give 3D orthogonal drawings with bounded bends for degree-6 graphs; the \"2-bend\" bound is the open target." }, { "id": 5500049, "problem_number": "AMR-054-0049", "title": "Planar Euclidean Maximum TSP", "statement": "What is the complexity of finding a tour of maximum Euclidean length for a planar point set?", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 49\nSource URL: https://topp.openproblem.net/p49\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p49; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Solved in the literature for the planar Euclidean case: the maximum Euclidean TSP in the plane can be solved in polynomial time (based on the non-crossing/alternating structure of optimal tours). NP-hardness holds for other metrics/dimensions." }, { "id": 5500051, "problem_number": "AMR-054-0051", "title": "Linear-Volume 3D Grid Drawings of Planar Graphs", "statement": "Does every $n$-vertex planar graph have a 3D grid drawing with $O(n)$ volume? A 3D grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates such that the straight line segments representing the edges are pairwise non-crossing. The volume is of the bounding box.", "background": "Source list: Open Problems Project - Computational Geometry\nSource item: Problem 51\nSource URL: https://topp.openproblem.net/p51\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://topp.openproblem.net/p51; maintained TOPP entry says: Open.\nRights note: Public source specification in https://github.com/edemaine/topp; repository LICENSE expressly scopes software, so problem-text reuse NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. Every planar graph has a 3D straight-line grid drawing with $O(n)$ volume. Literature status: SOLVED. Dujmović, Morin, and Wood (\"Layout of Graphs with Bounded Tree-Width\", and their 2015/2016 sequence) proved that every planar graph has a 3D straight-line grid drawing in $O(n)$ volume. More precisely, the result \"planar graphs have bounded queue number $\\Rightarrow$ linear-volume 3D grid drawings\" was established by Dujmović (2015) after an influential series of papers; the linear-volume conjecture for planar graphs was settled positively around 2015–2016. Subsequent work tightened constants." }, { "id": 5500052, "problem_number": "AMR-054-0052", "title": "Queue-Number of Planar Graphs", "statement": "Does every planar graph have $O(1)$ queue-number? A queue layout of a graph consists of a linear order of the vertices and a partition of the edges into non-nested queues. Edge $xy$ is nested inside edge $vw$ if $v1$ — answering White's question in the area-minimizing variant. The merely-minimal or isolated-singularity variants require further distinction." }, { "id": 5900014, "problem_number": "AMR-058-0014", "title": "Soap-Film Singularities in Orbifolds", "statement": "Classify the singularities allowed in soap films in three-dimensional orbifolds, and more generally in cone manifolds.", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 14\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Kusner", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The classification of soap-film singularities in three-dimensional orbifolds / cone manifolds does not appear to be fully established in the literature I could verify; it appears open beyond the flat-ambient case." }, { "id": 5900015, "problem_number": "AMR-058-0015", "title": "Existence of Equal-Volume Least-Area Partitions", "statement": "Do least-area partitions of $\\mathbb{R}^n$ into regions of unit volume exist? Give the correct definition and determine their regularity.", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 15\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Morgan", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Existence and correct definition of globally least-area equal-volume partitions of $\\mathbb{R}^n$ are handled only in limiting/periodic/prescribed frameworks (e.g., honeycomb in $\\mathbb{R}^2$); a general rigorous existence + regularity theory remains open (related to the optimal-foam problem)." }, { "id": 5900017, "problem_number": "AMR-058-0017", "title": "Optimality and Existence of the Weaire-Phelan Foam", "statement": "Is the Weaire--Phelan A15 foam the optimal partition of three-space into equal volumes, and can the existence of a foam in the A15 pattern be proved?", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 17\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Phelan and Sullivan", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The Weaire–Phelan foam is believed/suspected (numerically) to beat Kelvin's for equal-volume partitions of space, and is a leading candidate, but global optimality remains an open conjecture. Existence of the A15 pattern as a minimizing/periodic foam is not rigorously established as a distinct theorem in the literature I verified." }, { "id": 5900018, "problem_number": "AMR-058-0018", "title": "Restricted Optimality of Kelvin's Foam", "statement": "Prove or disprove successively that the Kelvin cell is the least-area fundamental domain for the BCC torus, that it minimizes among all unit-volume flat tori, that Kelvin's foam minimizes among partitions with congruent cells, and that it minimizes among equal-pressure foams.", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 18\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Sullivan", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The four restricted optimality claims for Kelvin's foam remain, as far as verified, unproven (and some are doubted given Weaire–Phelan's lower average area). No verified published proof of (a)-(d)." }, { "id": 5900019, "problem_number": "AMR-058-0019", "title": "Product Partitions of Slabs and Long Cylinders", "statement": "If an optimal planar cluster is crossed with a short interval, is the resulting partition optimal in the slab? Is a horizontal mid-height slice optimal for dividing a very long cylinder or prism into two equal-volume halves?", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 19\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Heppes", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The specific questions about optimal product partitions of slabs and long cylinders were not verified as resolved; they appear open (or at least not explicitly settled in the literature I reached)." }, { "id": 5900020, "problem_number": "AMR-058-0020", "title": "Maximum Shear Modulus of Planar Froths", "statement": "Prove or disprove that no two-dimensional froth of average bubble area one has shear modulus in any direction exceeding that of the regular hexagonal foam.", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 20\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kraynik", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The claim that the honeycomb froth maximizes shear modulus is physically well-founded and likely true in symmetric/linearized settings, but I did not verify a decisive general proof; it may remain at least partially open." }, { "id": 5900021, "problem_number": "AMR-058-0021", "title": "Combinatorial Types of Equal-Pressure Foam Cells", "statement": "Are there only finitely many combinatorial types of cells in equal-pressure foams in $\\mathbb{R}^3$? In particular, can tetrahedra or dodecahedra occur?", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 21\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Sullivan", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Finiteness of the combinatorial types of cells in equal-pressure foams in $\\mathbb{R}^3$, and the specific allowability of tetrahedra/dodecahedra, were not verified as resolved; the question appears at least substantially open." }, { "id": 5900022, "problem_number": "AMR-058-0022", "title": "Sharp Isoperimetric Constant for Bubble Clusters", "statement": "Find the best constant $C$ in the inequality $A\\leq C(\\int_B H+L)^2$ for a bubble cluster of area $A$, mean curvature $H$, and boundary length $L$. Determine the analogues for other norms and higher dimensions.", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 22\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Almgren", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The qualitative inequality (cluster area bounded in terms of mean curvature and boundary length) is rooted in Almgren's regularity work, but the sharp/best constant and the higher-dimensional/anisotropic analogues do not appear to be determined in a single verified reference." }, { "id": 5900023, "problem_number": "AMR-058-0023", "title": "Gromov-Knothe Isoperimetry for Multiple Regions", "statement": "Extend Gromov's Knothe-based proof of the isoperimetric inequality to multiple regions, and determine whether its vector field can be chosen canonically.", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 23\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Hutchings", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The Knothe/Gromov method was used successfully for the double bubble (Hutchings–Morgan–Ritoré–Ros), giving the key multi-region application; the fully general, canonical extension to arbitrary numbers of regions as posed in Problem 23 does not appear to be settled in the literature I verified." }, { "id": 5900024, "problem_number": "AMR-058-0024", "title": "Melzak's Shortest-Edge Polyhedron Conjecture", "statement": "Prove that the unit-volume polyhedron with shortest total edge length is an equilateral triangular prism, and establish existence of a minimizer.", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 24\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Morgan; conjecture of Melzak", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**PARTIAL-PROGRESS.** Valfells (2023) made progress by proving structural necessary conditions for any qualifying minimizer (degree-3 vertices, ≤14 triangular faces, quadrilateral-face behavior). The conjecture that the equilateral triangular right prism minimizes total edge length at unit volume remains unproven as of mid-2026." }, { "id": 5900025, "problem_number": "AMR-058-0025", "title": "Least Soap Films on Tetrahedral and Open-Book Frames", "statement": "Is the cone over the regular tetrahedron the smallest soap film having the entire tetrahedral frame as boundary? For a frame of two rectangles sharing an edge with small exterior dihedral angle, is the obvious soap film minimizing?", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 25\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Morgan", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Huff (2008) established existence of (previously unknown) non-flat soap films spanning tetrahedra; whether the least-area film is the cone over the regular tetrahedron remains unresolved by that work. The two-rectangle/open-book part was not verified as settled." }, { "id": 5900026, "problem_number": "AMR-058-0026", "title": "Soap Film on a Regular Octahedral Frame", "statement": "For a regular octahedral wire frame, determine the least-area soap film separating the eight regions, both with ordinary and fractional-density soap films.", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 26\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Brakke", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The octahedral soap-film problem is classical with well-understood candidate configurations (from Brakke's numerics and Plateau's experiments), but I did not verify a rigorous determination of the global least-area film, nor the fractional-density variants." }, { "id": 5900027, "problem_number": "AMR-058-0027", "title": "CMC Graphs Spanning Convex Planar Curves", "statement": "If a convex planar curve $\\Gamma$ has length less than $2\\pi$, does it bound a graph of constant mean curvature one? Determine the volume threshold $V(\\Gamma)$ below which any CMC spanning surface is a graph, and decide whether the disk threshold for a unit circle remains valid for higher topology.", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 27\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Lopez", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The existence of CMC-1 graphs over convex boundaries and the identified length/volume thresholds relate to standard CMC graph theory, but I could not verify a complete resolution of the specific $2\\pi$ and $V(\\Gamma)$ threshold questions." }, { "id": 5900028, "problem_number": "AMR-058-0028", "title": "CMC Surfaces with Circular Boundary", "statement": "Is every embedded constant-mean-curvature surface, or every immersed constant-mean-curvature disk, with boundary a round circle necessarily a spherical cap?", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 28\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Lopez", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The embedded case is classical-affirmative (spherical cap) under standard conditions in the literature; the immersed case is known to admit non-spherical-cap CMC disks with round circular boundary, so the question's two parts split: embedded yes (classical), immersed no." }, { "id": 5900029, "problem_number": "AMR-058-0029", "title": "Finite Total Scalar Curvature and Planarity", "statement": "If an area-minimizing $k$-dimensional submanifold of $\\mathbb{R}^n$ has finite total scalar curvature and $k>n/2$, must it be planar?", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 29\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Moore", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The specific rigidity claim (finite total scalar curvature + $k>n/2$ $\\Rightarrow$ planar, for area-minimizing submanifolds) was not verified as a settled theorem; it appears open or at least not clearly resolved in the literature I reached." }, { "id": 5900030, "problem_number": "AMR-058-0030", "title": "Sharp Interior Curvature Bound for Minimizing Hypersurfaces", "statement": "Find the best constant $C$ such that the principal curvatures of an area-minimizing hypersurface in low dimensions are bounded by $C/r$ at a point whose distance from the boundary is $r$.", "background": "Source list: Sullivan and Morgan - Open Problems in Soap Bubble Geometry (1996)\nSource item: Problem 30\nSource URL: http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf\nAccessed: 2026-07-29\nExtraction: indexed-primary-pdf-text-normalized\nStatus evidence: Primary paper labels the item as a problem or conjecture; current status NEEDS_REVIEW\nRights note: Public author PDF indexed by scholarly search; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Sullivan", "proposed_year": 1995, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** The qualitative interior curvature bound ($|A|\\le C/r$) is classical (Schoen–Simon–Yau and subsequent work) for stable/minimizing hypersurfaces. The sharp optimal constant $C$ in low dimensions sought in Problem 30 was not verified as determined." }, { "id": 6000001, "problem_number": "AMR-059-0001", "title": "Realizing Statistical Manifolds in Dually Flat Manifolds", "statement": "For a statistical manifold $(M,g,\\nabla,\\nabla^*)$, find conditions under which it can be realized as an $n$-dimensional submanifold of an $m$-dimensional, $m>n$, dually flat manifold. If this is not always possible, determine what additional quantity ensures realization in finite dimension.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 1(a), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Shun-ichi Amari", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general realization problem appears **open** (no verified complete solution in the literature as of 2026). Partial progress exists for special classes (dually flat, low-curvature statistical manifolds), which realize as submanifolds of Hessian / dually flat spaces." }, { "id": 6000002, "problem_number": "AMR-059-0002", "title": "Probability Densities and Equiaffine Transformations", "statement": "The smooth positive probability densities on $S^1$ are diffeomorphic to the manifold of smooth equiaffine transformations of $S^1$. Find the corresponding description for $S^n$ and for $\\mathbb{R}^n$.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 1(b), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Shun-ichi Amari", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The $S^1$ case is classical and understood (identification of positive densities with structures on the diffeomorphism group / projective geometry). The extension to $S^n$ ($n\\ge 2$) and $\\mathbb{R}^n$ as posed appears **open / not cleanly established** — the natural finite-dimensional affine action degenerates for $n\\ge 2$, so the statement likely needs reinterpretation (e.g., in terms of the full diffeomorphism group and density bundles)." }, { "id": 6000003, "problem_number": "AMR-059-0003", "title": "Dually Flat Structures on Riemannian Manifolds", "statement": "Given a Riemannian manifold $(M,g)$, can one always introduce a symmetric $(0,3)$-tensor $T$ so that $(M,g,\\nabla,\\nabla^*)$ is dually flat? If the construction is not unique, characterize the class of resulting spaces.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 1(c), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Shun-ichi Amari", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem is substantially **solved/understood in the local theory** (Hessian manifolds are locally characterized; not every Riemannian metric is Hessian — there are curvature obstructions, e.g. only definite curvature-type conditions admit potentials). The \"not always possible\" part is established. Global and uniqueness aspects remain subtle and context-dependent, so the problem is classified as partial progress rather than fully closed." }, { "id": 6000004, "problem_number": "AMR-059-0004", "title": "Large Deviations and Dual Connections", "statement": "Investigate the relationship between the large-deviation principle, whose rate functions are relative entropies, and dual-connection structures in information geometry.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 1(d), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Shun-ichi Amari", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "This is a research **theme with substantial literature** rather than a single open/closed problem. The core observation — relative-entropy LDP rate functions are the canonical divergences realizing dual (α-)connection structures — is well established. A fully general and exhaustive duality theory connecting arbitrary dual-connection structures with large-deviation principles remains open/in flux." }, { "id": 6000005, "problem_number": "AMR-059-0005", "title": "Statistical Manifolds from Probability Families", "statement": "Given a statistical manifold, determine conditions for the existence of a family of probability distributions whose induced statistical manifold coincides with it. Determine the analogous conditions for dually flat manifolds.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 1(e), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Shun-ichi Amari", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- **Dually flat case:** essentially answered — dually flat manifolds are (locally) realizable as exponential families (curved or full) inside the space of probability measures, with the KL/canonical divergence as the generating structure; this is standard (Amari–Nagaoka). - **General statistical-manifold case:** open in full generality; a general statistical manifold need not be realizable as the parameter space of a probability family, and the precise obstruction is not established." }, { "id": 6000006, "problem_number": "AMR-059-0006", "title": "Fundamental Groups of Compact Affine Flat Manifolds", "statement": "Determine the fundamental group of a compact affine flat manifold.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 2(a), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Yoe Itokawa; attributed to D. Gromoll", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem asks to determine $\\pi_1$ of a compact affinely flat manifold. The conjectural answer (Auslander 1964): it is virtually polycyclic (finitely presented, solvable-by-finite, of polynomial growth). Status as of 2026: - **Proved** in dimensions $\\le 6$ (Abels–Margulis–Soifer, arXiv:1211.2525). - **Open** in general dimension for compact complete affine manifolds. - The failure modes for non-compact properly discontinuous affine actions (non-solvable groups) are well documented (Margulis spacetimes), indicating why compactness/completeness is essential." }, { "id": 6000007, "problem_number": "AMR-059-0007", "title": "Conformal Flatness and Geometric Divergence", "statement": "Investigate the relationship between the geometry of a conformally flat Riemannian manifold and Matsuzoe's geometric divergence for a conformally-projectively flat statistical manifold.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 3(a), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Takashi Kurose", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The stated relationship is a specialized open research theme. There is an established body of work on conformally flat statistical manifolds and on geometric divergences (Kurose divergence, Matsuzoe), and conformally-projectively flat statistical manifolds are studied, but I did not verify a complete solution to the exact question. Status: **open / requires triage**." }, { "id": 6000008, "problem_number": "AMR-059-0008", "title": "Isothermal Coordinates for Affine Minimal Surfaces", "statement": "Do affine minimal surfaces admit isothermal coordinates with respect to their affine fundamental form, analogously to minimal surfaces in Euclidean three-space?", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 3(b), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Takashi Kurose", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The question is answered affirmatively in the literature at the local level: affine surfaces (in particular affine minimal ones) admit local isothermal coordinates with respect to their affine fundamental form, by the classical isothermal-coordinates theorem for 2D metrics (already standard in Blaschke's affine surface theory)." }, { "id": 6000009, "problem_number": "AMR-059-0009", "title": "Curvature-Type Tensors of Statistical Manifolds", "statement": "Let $K(X,Y,Z,W)=h(R(X,Y)Z,W)$ be the curvature $(0,4)$-tensor of a statistical manifold. Investigate the structure of all $(0,4)$-tensors satisfying $K(X,Y,Z,W)=-K(Y,X,Z,W)$, the first Bianchi identity, and $K(X,Y,Z,W)+K(Y,W,Z,X)+K(W,X,Z,Y)=0$.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 3(c), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Takashi Kurose", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The algebraic structure problem is well understood in the classical (metric) case, where the space of algebraic curvature tensors (with pairwise symmetry + first Bianchi) has the standard irreducible decomposition. The statistical case lacks the pairwise symmetry, so the tensor space is larger; its precise irreducible decomposition in the general (non-metric, dual-connection) statistical setting is not a single canonical verified classification. Status: partial progress." }, { "id": 6000010, "problem_number": "AMR-059-0010", "title": "Tangent-Bundle Symplectic and Almost Complex Structures", "statement": "Characterize the symplectic manifolds with compatible almost complex structure which are locally obtained from a tangent bundle $T(M)$ by combining the canonical symplectic form induced by a Riemannian metric with the almost complex structure determined by a compatible torsion-free affine connection.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 3(d), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Takashi Kurose", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is understood in the classical literature: a symplectic-almost-complex manifold of this special form is locally a tangent bundle, and the characterization reduces to the tangent-bundle geometry built from a Riemannian metric and a compatible torsion-free connection (Dombrowski-type almost complex structure, canonical symplectic form). No fully general closed-form classification beyond the classical structure equations was verified." }, { "id": 6000011, "problem_number": "AMR-059-0011", "title": "Integrable Radial Distributions and One-Conformal Flatness", "statement": "On a statistical manifold, let $D$ be the rank-$(n-1)$ distribution orthogonal to the velocity of the $\\nabla$-geodesic from a fixed point. If every such distribution is integrable, must the dual statistical manifold be one-conformally flat?", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 3(e), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Takashi Kurose", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem appears **open** as posed: no verified published proof or counterexample of the implication \"all radial distributions integrable ⇒ dual one-conformally flat\" was found. It sits within the established theme of conformal/projective flatness in statistical geometry." }, { "id": 6000012, "problem_number": "AMR-059-0012", "title": "Inflection Points of Projectively Flat Curves", "statement": "Estimate the minimum number of inflection points of a closed curve on a Riemann surface with a projectively flat connection.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 4(a), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Takeshi Sasaki", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partially resolved/active: the existence of inflection points of closed projective curves and related bounds (flex theorems, Plücker-type identities) are established in important cases, but a general sharp minimum bound for the stated problem on arbitrary Riemann surfaces with a projectively flat connection was not verified as a closed result." }, { "id": 6000013, "problem_number": "AMR-059-0013", "title": "Affine Vertices and Inflection Points", "statement": "For a closed curve on a Riemann surface with projectively flat connection, estimate the number of affine vertices and, assuming isolated inflection points, relate the number of affine vertices to the number of inflection points.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 4(b), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Takeshi Sasaki", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial resolution: affine vertex-counting and its relation to inflection points are established for important classes (convex/plane affine curves, projective curves with isolated flexes) via four-vertex-type theorems and Morse/Plücker-style index relations. A complete, sharp, unified statement for arbitrary closed curves on Riemann surfaces with a projectively flat connection was not verified." }, { "id": 6000014, "problem_number": "AMR-059-0014", "title": "Global Affine Curve Flow", "statement": "Prove existence of a time-global solution to the affine-plane curve evolution equation $\\partial x/\\partial u=(1+kp)x''$, where $u$ is time, $x''$ is the affine normal, $k$ is affine curvature, and $p$ is the affine support function.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 4(c), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Takeshi Sasaki", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem is best classified as **partial progress**: affine plane curve flows of this general type are a well-studied area with established existence/long-time results for affine shortening and central affine evolutions, but the specific flow $\\dot{x}=(1+kp)x''$ was not verified as an explicitly stated and proved global-existence result in the literature I could reach." }, { "id": 6000015, "problem_number": "AMR-059-0015", "title": "Stein Tangent Bundles of Complete Hessian Manifolds", "statement": "If $(M,g)$ is a complete Hessian manifold, is its tangent bundle with the natural complex structure a Stein manifold? In particular, if a convex domain $\\Omega\\subset\\mathbb{R}^n$ contains no complete line and a discrete affine group $\\Gamma$ acts freely and properly discontinuously, is $T(\\Omega/\\Gamma)$ Stein?", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 5(a), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Satoru Shimizu", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem appears **open / only partially addressed** in the verifiable literature. The Sasaki lifting of a Hessian manifold is Kähler (classical), recent work (arXiv:2509.01176) makes deep use of the Sasaki lifting (positive flat line bundles lift to positive holomorphic line bundles, fibration/splitting theorems for compact Hessian manifolds), and the convex-domain sub-case is plausibly classically settled — but I could not verify a published statement that the tangent bundle of a complete Hessian manifold is Stein." }, { "id": 6000016, "problem_number": "AMR-059-0016", "title": "Stability of Hessian Metrics", "statement": "Let $M$ be a compact Hessian manifold and deform its affine structure. Does every sufficiently small deformation admit a Hessian metric?", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 5(b), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Satoru Shimizu", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem remains essentially **open / only partially addressed**. It is known that: affine structures deform non-trivially (torus); Hessian-metric existence on compact affine manifolds is a constrained cohomological/convex-cone condition (not a generic open property in general). A definitive answer to whether every sufficiently small deformation of a compact Hessian manifold's affine structure still admits a Hessian metric was not verified either way." }, { "id": 6000017, "problem_number": "AMR-059-0017", "title": "Nonproduct Compact Hessian Manifolds", "statement": "Construct compact Hessian manifolds other than direct products of hyperbolic and flat compact Hessian manifolds; in particular, construct one with no fiber-bundle structure using deformation theory of affine structures.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 5(c), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Satoru Shimizu", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress, now strongly constrained by recent structure theorems.** In dimension $\\le 6$, compact Hessian manifolds are finitely covered by products of hyperbolic and flat affine manifolds; compact orientable Hessian manifolds fiber over $S^1$ (mapping torus or Bieberbach), and compact hyperbolic affine manifolds fiber over $S^1$ with periodic monodromy. This makes the existence of genuinely non-product, non-fibered compact Hessian manifolds (the specific construction requested) uncertain and likely higher-dimensional; no verified explicit example was found." }, { "id": 6000018, "problem_number": "AMR-059-0018", "title": "Geometry on Sample-Parameter Product Spaces", "statement": "For a statistical family $f(x;\\theta)$, find and study a useful geometry on the product of the sample space and parameter space, accounting for the choice of a statistically meaningful $\\theta$-coordinate system.", "background": "Source list: Furuhata, Matsuzoe and Urakawa - Open Problems in Affine Differential Geometry and Related Topics (1998)\nSource item: item 6(a), scanned PDF pages 1-2\nSource URL: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en\nAccessed: 2026-07-29\nExtraction: manual-ocr-from-page-image\nStatus evidence: https://www.jstage.jst.go.jp/article/iis/4/2/4_2_125/_pdf/-char/en; article presents the item as an open problem; current status NEEDS_REVIEW\nRights note: J-STAGE article page marks the article CC BY 4.0\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Kwoichi Tandai", "proposed_year": 1998, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The problem is best understood as an **open, programmatic research direction** in information geometry. Modern frameworks (geometry of the space of probability densities, optimal-transport-based information geometry, sample-space geometry) provide partial tools, but there is no verified single \"useful geometry on $X\\times\\Theta$\" canonically resolving the question as posed." }, { "id": 6200001, "problem_number": "AMR-061-0001", "title": "Boundaries of Groups and Kleinian Groups — Problem 1", "statement": "What spaces can arise as boundaries of hyperbolic groups? As a sub-problem: For which k do k-dimensional stable Menger spaces appear as boundaries?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 1, PDF page 2\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress only. The case $k=1$ (Menger curve) and $k=2$ (Menger compactum) are realized. Whether higher-dimensional universal Menger compacta appear as boundaries of hyperbolic groups remains open, as does the full classification." }, { "id": 6200002, "problem_number": "AMR-061-0002", "title": "Boundaries of Groups and Kleinian Groups — Problem 2", "statement": "Can one remove the “right-angled” assumption in Osajda result?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 2, PDF page 2\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Remains open / not precisely formulable from the source. Literature status: The statement is a terse research prompt without a precise target, so a definitive classification is not verifiable from the available literature. General (non-right-angled) Coxeter groups and their boundaries have been studied, but no single \"Osajda result\" is identified in the worklist. Treated as open/triage." }, { "id": 6200003, "problem_number": "AMR-061-0003", "title": "Boundaries of Groups and Kleinian Groups — Problem 3", "statement": "What 2-dimensional spaces arise as boundaries of hyperbolic groups?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 3, PDF page 3\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Panos Papasoglu", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress; no complete classification of 2-dimensional boundaries known. Literature status: Open in general. However, there are substantial partial results for low-dimensional boundaries: - Kapovich–Kleiner (for 1-dimensional boundaries): a 1-ended hyperbolic group with 1-dimensional boundary that does not split over a cyclic group has boundary the Menger curve or the Sierpiński carpet. - The 2-dimensional case is studied in Kapovich–Kleiner's program; spheres, homology spheres, the Sierpiński carpet, and the Menger curve/compactum are known to occur." }, { "id": 6200004, "problem_number": "AMR-061-0004", "title": "Boundaries of Groups and Kleinian Groups — Problem 4", "statement": "Are there torsion-free hyperbolic groups G with cdQ(G)/cdZ(G) < 2/3?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 4, PDF page 3\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mike Davis", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: This is a question on rational vs. integral cohomological dimension of hyperbolic groups. The threshold $2/3$ relates to boundary/topological dimension phenomena. No solution located in the accessible literature; appears open/triage." }, { "id": 6200005, "problem_number": "AMR-061-0005", "title": "Boundaries of Groups and Kleinian Groups — Problem 5", "statement": "What can be said about boundaries arising from strict hyperbolization constructions of Charney and Davis, [18]?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 5, PDF page 3\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nadia Benakli", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Charney–Davis hyperbolization produces negatively curved (CAT(-1) / locally CAT(0)) complexes. The study of the boundaries of such complexes is ongoing; no complete description in the literature. Open/triage." }, { "id": 6200006, "problem_number": "AMR-061-0006", "title": "Boundaries of Groups and Kleinian Groups — Problem 6", "statement": "Is there an example of a group G which is hyperbolic relative to some parabolic subgroups that are nilpotent of class ≥ 3 whose Bowditch boundary is homeomorphic to some n-sphere?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 6, PDF page 3\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ilia Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: This relates to Cannon's conjecture and the rigidity of relatively hyperbolic groups with sphere boundaries (cf. work on relatively hyperbolic groups with $S^n$ boundary). No explicit construction with nilpotent class $\\ge 3$ parabolic subgroups verified. Open/triage." }, { "id": 6200007, "problem_number": "AMR-061-0007", "title": "Boundaries of Groups and Kleinian Groups — Problem 7", "statement": "Suppose that Z is a compact metrizable topological space, G↷ Z is a convergence action which is topologically transitive, i.e. each G– orbit is dense in Z. Is there a Gromov-hyperbolic space X with the ideal boundary Z so that the action G ↷ Z extends to a uniformly quasi-isometric quasi-action G↷ X?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 7, PDF page 3\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: This generalizes Bowditch's characterization connecting convergence actions to hyperbolic spaces. The indefinite question (without assuming geometric action/compactness) is open; Bowditch's theorem handles the case of a group acting on a compactum that is the boundary of a hyperbolic space. Open/triage." }, { "id": 6200008, "problem_number": "AMR-061-0008", "title": "Boundaries of Groups and Kleinian Groups — Problem 8", "statement": "Find topological restrictions on the ideal boundaries of CAT (−1) cubical complexes.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 8, PDF page 4\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Tadeusz Januszkiewicz", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Related to the Januszkiewicz–Świątkowski conjecture that boundaries of right-angled (CAT(-1) cubical) groups cannot contain spheres of dimension above 3. This is open. Open/triage." }, { "id": 6200009, "problem_number": "AMR-061-0009", "title": "Boundaries of Groups and Kleinian Groups — Problem 9", "statement": "Is it true that isomorphic Coxeter groups have homeomorphic boundaries?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 9, PDF page 4\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander Dranishnikov", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open / ill-posed as stated. Literature status: The boundary of a Coxeter group is not a single well-defined object (it depends on a choice of generating set/representation), so the question in this exact form is problematic. The manageable variants — is the boundary a quasi-isometry invariant for Coxeter groups — is known to fail in general (different boundaries for the same group are known, e.g., for the universal Coxeter group one can have different boundaries). Open/triage with a note that the statement needs care." }, { "id": 6200010, "problem_number": "AMR-061-0010", "title": "Boundaries of Groups and Kleinian Groups — Problem 10", "statement": "Does there exist a Coxeter group Gn with n-dimensional boundary ∂Gn, so that the rational homological dimension of ∂Gn equals 1?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 10, PDF page 4\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander Dranishnikov", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Dranishnikov's question on homological dimension vs. topological dimension of boundaries of Coxeter groups. The related constructions show boundaries of Coxeter groups can have low homological dimension relative to topological dimension, but the specific case is not resolved in the accessible literature. Open/triage." }, { "id": 6200011, "problem_number": "AMR-061-0011", "title": "Boundaries of Groups and Kleinian Groups — Problem 11", "statement": "Under which conditions on the Coxeter diagram of G, the boundary of a Coxeter group is n-connected and locally n-connected?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 11, PDF page 4\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander Dranishnikov", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Partial / open. Literature status: There are known results (Dranishnikov, Świątkowski, Osajda, and the Januszkiewicz–Świątkowski constructions) giving connectivity of boundaries for right-angled Coxeter groups, but a complete diagram-level criterion is not established. Open/triage." }, { "id": 6200012, "problem_number": "AMR-061-0012", "title": "Boundaries of Groups and Kleinian Groups — Problem 12", "statement": "Can exotic homology manifolds as in [14] appear as ideal boundaries of Coxeter groups?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 12, PDF page 5\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: The reference is about exotic homology manifolds arising from wild Cantor sets / CE constructions. Whether such spaces are boundaries of Coxeter groups is unresolved. Open/triage." }, { "id": 6200013, "problem_number": "AMR-061-0013", "title": "Boundaries of Groups and Kleinian Groups — Problem 13", "statement": "Find further universality phenomena: classes of groups or spaces of different nature whose ideal boundaries are nevertheless all homeomorphic, beyond the source's examples involving Menger compacta, right-angled hyperbolic buildings, and Davis--Vinberg complexes.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 13, PDF page 5\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Tadeusz Januszkiewicz", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress; more examples known but the general program is open. Literature status: Many universal boundary phenomena have been found (e.g., the boundary of right-angled Coxeter groups can be the universal Menger compactum or a sphere; the \"generic\" boundary of random groups is the Menger curve). This line continues. Partial progress: universality of Menger curve for any dimension-free compacta via RACG constructions is known. General scope is open." }, { "id": 6200014, "problem_number": "AMR-061-0014", "title": "Boundaries of Groups and Kleinian Groups — Problem 14", "statement": "Let $N$ be a closed $n$-manifold and $\\Delta$ a flag, no-square triangulation, and let $C(N,\\Delta)$ be the associated Davis--Vinberg complex. Is $\\partial_\\infty C(N,\\Delta)$ a topological invariant of $N$, independent of $\\Delta$?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 14, PDF page 5\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open (likely false in general). Literature status: Relates to the Januszkiewicz–Świątkowski and Davis–Januszkiewicz constructions and the fact that boundaries of Davis complexes can depend on the triangulation/hyperbolization. In general the boundary is NOT independent of the choice; specific classes give invariance. Open/triage." }, { "id": 6200015, "problem_number": "AMR-061-0015", "title": "Boundaries of Groups and Kleinian Groups — Problem 15", "statement": "Suppose that ( N1, ∆1) and (N2, ∆2) are closed 3-manifolds equipped with flag-triangulations, so that ∂∞C(N1, ∆1) = ∂∞C(N2, ∆2). Does it follow that every prime connected sum summand of Ni appears as a connected sum summand of Ni+1, i = 1, 2? What can be said in higher dimensions?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 15, PDF page 5\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Kapovich, Tadeusz Januszkiewicz", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: This is a subtle topological question about when equal boundaries force connected-sum decomposition compatibility. No resolution located. Open/triage." }, { "id": 6200016, "problem_number": "AMR-061-0016", "title": "Boundaries of Groups and Kleinian Groups — Problem 16", "statement": "Let Z be a compactum which is a Z-boundary of a group G. Then Z is never a Boltyansky compactum. In the special case when Z is an Markov compactum, so that all building blocks Kσ → σ are isomorphic, it was proven in [29] that Z cannot be a Boltyansky compactum.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 16, PDF page 6\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander Dranishnikov", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: The statement asserts a conjecture (general case) with a special case proved in a cited reference. Whether the general case is resolved is not verified. Open/triage (likely an open conjecture)." }, { "id": 6200017, "problem_number": "AMR-061-0017", "title": "Boundaries of Groups and Kleinian Groups — Problem 17", "statement": "Examples of Kleiner and Croke [22], [23] of non-unique boundaries are badly non-locally-connected. Is that essential in having the “flexibility” to have many boundaries? That is, does local connectedness imply uniqueness of the boundary (in the 1-ended case) for CAT(0) groups?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 17, PDF page 6\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Kim Ruane", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Uniqueness of the CAT(0) boundary is a major open question. For 1-ended CAT(0) groups it is conjectured the boundary is unique when it is a sphere-like / locally connected object, but no proof. (For hyperbolic groups boundaries are unique; for general CAT(0) groups, Croke–Kleiner gave non-homeomorphic boundaries for the same group.) Open." }, { "id": 6200018, "problem_number": "AMR-061-0018", "title": "Boundaries of Groups and Kleinian Groups — Problem 18", "statement": "Suppose that G is a CAT (0) group which does not split over a small subgroup. Does it follow that ∂∞G is unique?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 18, PDF page 7\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dani Wise", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: This refines Croke–Kleiner. Under non-splitting hypotheses one expects rigidity; some results (e.g., relating to the flat torus / product situation) exist, but the general statement is open. Open/triage." }, { "id": 6200019, "problem_number": "AMR-061-0019", "title": "Boundaries of Groups and Kleinian Groups — Problem 19", "statement": "Is the boundary well-defined for groups acting geometrically on CAT (0)-cube complexes? More precisely, suppose that X1, X2 are cube complexes which admit geometric actions of a group G. Does it follow that ∂∞X1 = ∂∞X2?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 19, PDF page 7\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Dani Wise", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Uniqueness of CAT(0) boundaries for cubical groups is unresolved; there are constructions of non-unique boundaries for CAT(0) groups. Open/triage." }, { "id": 6200020, "problem_number": "AMR-061-0020", "title": "Boundaries of Groups and Kleinian Groups — Problem 20", "statement": "What topological invariants distinguish boundaries? In particular, what topological properties of boundaries are quasi-isometry invariants? Does something coarser than the topology stay invariant?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 20, PDF page 7\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ross Geoghegan", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open research program. Literature status: This is a broad open research direction (part of the Croke–Kleiner program: dimension, local connectivity, cut points, etc. are studied, but not all are known to be QI invariants). Open." }, { "id": 6200021, "problem_number": "AMR-061-0021", "title": "Boundaries of Groups and Kleinian Groups — Problem 21", "statement": "If G acts geometrically on two CAT(0) spaces, are the resulting boundaries cell-like equivalent? (That is, does there exist a space Z with cell-like maps to each of the two spaces?)", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 21, PDF page 7\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ross Geoghegan", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Related to the Bestvina–Mess / Farrell–Lafont $Z$-structure theory: any two $Z$-structures on the same group are cell-like equivalent, but CAT(0) boundaries are not known to form such structures. Open." }, { "id": 6200022, "problem_number": "AMR-061-0022", "title": "Boundaries of Groups and Kleinian Groups — Problem 22", "statement": "Is there a convex core for the diagonal action of G on X1 × X2? (A special case is surface groups G with X1 and X2 corresponding to different hyperbolic structures.) If there is a convex core, can Z (the space with cell-like maps to X1 and X2) be taken to be the boundary of the core?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 22, PDF page 7\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Thomas Delzant", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. This is a geometric generalization in the Croke–Kleiner / flat problem context. No resolution located. Open/triage." }, { "id": 6200023, "problem_number": "AMR-061-0023", "title": "Boundaries of Groups and Kleinian Groups — Problem 23", "statement": "Define a topology on the set of quasi geodesics in a (proper geodesic, or coarsely homogeneous, or cocompact) CAT (0) space which (1) has a description as an increasing union of compact metrizable spaces (2) has an inclusion of its visual boundary ∂∞X into it (3) is quasi-isometry invariant (4) has reasonable measure classes which are quasipreserved According to a theorem by Brian Bowditch and Gadde Swarup [13, 56], if G is a 1-ended hyperbolic group then ∂∞G has no cut points. For G a CAT(0) group, a theorem of Eric Swenson says that if c ∈ ∂∞G is a cut point, then there is an infinite-torsion subgroup of G fixing c.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 23, PDF page 8\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Danny Calegari", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. General technique follows Bowditch/Swarup for hyperbolic groups and Croke–Kleiner for CAT(0), but the full program is not worked out. Open/triage." }, { "id": 6200024, "problem_number": "AMR-061-0024", "title": "Boundaries of Groups and Kleinian Groups — Problem 24", "statement": "Any CAT (0) group has no infinite-torsion subgroups. A Euclidean retract is a compact space that embeds into some Rn as a retract. A compact metrizable space Z is a Z-set in ˜X if it is “homotopically negligible” (for every open U ⊂ ˜X, the inclusion U \\mathbb{Z} in U is a homotopy equivalence). A Z-structure on a group G is a pair ( ˜X, Z ) such that • ˜X is a Euclidean retract, • Z is a Z-set in ˜X, • X:= ˜X \\ Z admits a covering space action of G with X/G compact, • the set of translates of any compact set K ⊂ X is a null sequence in ˜X (that is, for each ϵ > 0 there are only finitely many translates with diam > ϵ). Finally, Z is a boundary of G (or Z-structure boundary) if there exists a Z-structure ( ˜X, Z ) on G. The above notion boundary of G was generalized by T. Farrell and J. Lafont as follows: An EZ-boundary of a group G is a boundary Z = ∂EZ G so that the action of G on X extends to topological action of G on Z.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 24, PDF page 8\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Eric Swenson", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Background; theory well-developed. Literature status: This is background/literature-survey material: EZ-structures were developed by Farrell–Lafont, Bestvina–Mess. The full question—of classifying Z-structures—is an active area (e.g., Farrell–Lafont rigidity in nonpositive curvature). Classify as LITERATURE-SURVEY." }, { "id": 6200025, "problem_number": "AMR-061-0025", "title": "Boundaries of Groups and Kleinian Groups — Problem 25", "statement": "Let G be a hyperbolic group and ∂EZ G be its EZ boundary. Is it true that ∂EZ G is equivariantly homeomorphic to the Gromov boundary of G?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 25, PDF page 8\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open/partial. Literature status: Classically, Bestvina–Mess and Farrell–Lafont established that Z/EZ-boundaries of hyperbolic groups are cell-like equivalent to the Gromov boundary; the stronger specific form of uniqueness (equivariant homeomorphism for all EZ-boundaries) is related to the \"uniqueness of Z-boundaries\" program and is open in full generality, though for hyperbolic groups the boundary is essentially unique up to homeomorphism. Open/triage." }, { "id": 6200026, "problem_number": "AMR-061-0026", "title": "Boundaries of Groups and Kleinian Groups — Problem 26", "statement": "Can there be two different boundaries in the sense of Z-structures for a group G that are not cell-like equivalent?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 26, PDF page 8\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mladen Bestvina", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Bestvina–Mess and Farrell–Lafont theory shows Z-boundaries of a group satisfying certain conditions are cell-like equivalent, but whether all boundaries of a fixed group are mutually cell-like-equivalent is unknown. Open." }, { "id": 6200027, "problem_number": "AMR-061-0027", "title": "Boundaries of Groups and Kleinian Groups — Problem 27", "statement": "Is the property of splitting over a 2-ended subgroup an invariant of Bestvina boundaries? Some necessary conditions are known for compact, metrizable spaces X to be the boundary of some proper cocompact CAT (0) space: (1) X should have 1,2, or infinitely many components (2) X is finite dimensional (Theorem of Swenson) (3) X has nontrivial top ˇCech cohomology (Geoghegan-Ontaneda) In the case when X admits a cocompact free(?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 27, PDF page 8\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Bruce Kleiner", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Splitting over 2-ended subgroups is a quasi-isometry invariant for hyperbolic groups (cf. Papasoglu, Bowditch), but its formulation via bestvina boundaries in the CAT(0) setting is not established. Open/triage." }, { "id": 6200028, "problem_number": "AMR-061-0028", "title": "Boundaries of Groups and Kleinian Groups — Problem 28", "statement": "Extend the known necessary conditions for a compact metrizable space $X$ to be the boundary of a proper cocompact CAT(0) space, or give a complete classification. The listed conditions are that $X$ have one, two, or infinitely many components, be finite dimensional, and have nontrivial top Čech cohomology; if a discrete isometry group acts freely and cocompactly, every nonempty open subset must have the same dimension as $X$.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 28, PDF page 9\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Ross Geoghegan", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial; classification still open. Literature status: Partial. Known necessary conditions: 1/2/∞ components (Swenson), finite-dimensional (Swenson), nontrivial top Čech cohomology (Geoghegan–Ontaneda), and dimensionality of open subsets for cocompact free actions. Other results (Bestvina, Bestvina–Mess, Kleiner) add more, but no complete classification. PARTIAL-PROGRESS." }, { "id": 6200029, "problem_number": "AMR-061-0029", "title": "Boundaries of Groups and Kleinian Groups — Problem 29", "statement": "Does every CAT(0) group have finite asymptotic dimension?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 29, PDF page 9\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kevin Whyte", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open/partial. Literature status: Open in general, though many CAT(0) groups are known to have finite asymptotic dimension (e.g., CAT(0) cube complexes of finite dimension, and more). The general question for arbitrary CAT(0) groups of finite asymptotic dimension is a known open problem; related to the Farrell–Jones conjecture. Some recent progress exists by Arnt. Open/partial." }, { "id": 6200030, "problem_number": "AMR-061-0030", "title": "Boundaries of Groups and Kleinian Groups — Problem 30", "statement": "Do Papasoglu's three results for finitely presented one-ended groups extend to all finitely generated groups: quasi-isometry invariance of the JSJ decomposition; characterization of coarse separation by a quasiline via splitting over a virtually cyclic group or virtual surface-group structure; and nonseparation by quasi-rays?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 30, PDF page 9\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Panos Papasoglu", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Papasoglu proved these for finitely presented groups. Extension to finitely generated groups is delicate and partially open. Open/triage." }, { "id": 6200031, "problem_number": "AMR-061-0031", "title": "Boundaries of Groups and Kleinian Groups — Problem 31", "statement": "Are splittings overZ2 (orZn) invariant under quasiisometry? The analogous problem also makes sense for the JSJ decompositions.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 31, PDF page 9\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Panos Papasoglu", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Splitting over virtually cyclic subgroups is a QI invariant for hyperbolic groups, but splitting over $\\mathbb{Z}^n$ ($n \\ge 2$) is more subtle and unresolved in general. Open/triage." }, { "id": 6200032, "problem_number": "AMR-061-0032", "title": "Boundaries of Groups and Kleinian Groups — Problem 32", "statement": "Suppose G is finitely generated and there is a sequence of quasicircles that separate its Cayley graph. Is G virtually a surface group?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 32, PDF page 9\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Panos Papasoglu", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. This is related to Papasoglu's work on quasiline/quasisphere separation and to rigidity of surface groups. No resolution located. Open/triage." }, { "id": 6200033, "problem_number": "AMR-061-0033", "title": "Boundaries of Groups and Kleinian Groups — Problem 33", "statement": "If G is finitely generated with asymptotic dimension ≥ n, and X is a subset of the Cayley graph with asymptotic dimension ≤ n − 2 that coarsely separates the Cayley graph, then G splits over some subgroup H ≤ G with asymptotic dimension ≤ n − 1. A homogeneous continuum is a locally connected compact metric space whose group of homeomorphisms acts transitively. Papasoglu showed that every simply connected homogeneous continuum has the property that no simple arc separates it.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 33, PDF page 9\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Conjecture of Panos Papasoglu", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Related to Papasoglu's splittings from coarse separation by quasi-lines and to asdim bounds. No general resolution located. Open/triage." }, { "id": 6200034, "problem_number": "AMR-061-0034", "title": "Boundaries of Groups and Kleinian Groups — Problem 34", "statement": "Do all homogeneous continua (with dimension greater than 2) have this property?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 34, PDF page 9\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Panos Papasoglu", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. This relates to Kaplan's theorem (for compact topological groups / solenoids) and to homogeneity in dimension 2-3. The higher-dimensional homogeneity problem (whether every homogeneous continuum of dimension $>2$ has no separating arc) is open. Open/triage." }, { "id": 6200035, "problem_number": "AMR-061-0035", "title": "Boundaries of Groups and Kleinian Groups — Problem 35", "statement": "Are diffeomorphisms Rn →Rn dense in the space of all quasiconformal maps?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 35, PDF page 11\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mario Bonk", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Dense approximation of quasiconformal homeomorphisms by diffeomorphisms is a known problem related to the local-to-global approximation of quasiconformal maps; full density in all dimensions is not established. Open/triage." }, { "id": 6200036, "problem_number": "AMR-061-0036", "title": "Boundaries of Groups and Kleinian Groups — Problem 36", "statement": "Let f: Bn → Bn be a quasiconformal homeomorphism. Can f be approximated by globally quasiconformal diffeomorphisms fj: Bn → Bn? Can this be done so that fj’s are K-quasiconformal for all j?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 36, PDF page 12\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open/partial. Literature status: Open in full generality. In dimension 2, quasiconformal maps can be approximated (by smooth quasiconformal maps via Donaldson or via Sullivan); higher-dimensional case and the uniform-distortion control is open. Open/triage." }, { "id": 6200037, "problem_number": "AMR-061-0037", "title": "Boundaries of Groups and Kleinian Groups — Problem 37", "statement": "Find good classes of spaces such that the infinitesimal metric condition (for quasiconformality) implies the local condition. (This is generally true in Loewner spaces.)", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 37, PDF page 12\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mario Bonk", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Theory well-established for Loewner spaces. Literature status: This is a survey-style question about Loewner spaces and QC theory (Heinonen–Koskela). In Loewner spaces, infinitesimal and local definitions coincide — the theory is well developed. LITERATURE-SURVEY." }, { "id": 6200038, "problem_number": "AMR-061-0038", "title": "Boundaries of Groups and Kleinian Groups — Problem 38", "statement": "Outside of the boundaries of Fuchsian buildings, what boundaries have the Loewner property?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 38, PDF page 12\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Kim Ruane", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: The Loewner property of boundaries of hyperbolic groups is studied (Bourdon–Pajot; the boundary of Fuchsian buildings is Loewner). Classification of which group boundaries are Loewner is open. Open/triage." }, { "id": 6200039, "problem_number": "AMR-061-0039", "title": "Boundaries of Groups and Kleinian Groups — Problem 39", "statement": "Let X be a non-smoothable closed simply connected 4-manifold. Does it admit an Ahlfors 4-regular linearly locally contractible metric? This is wide open; unknown even for examples, like E8.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 39, PDF page 12\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Juha Heinonen", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: This connects to the metric/geometric characterization of when a topological space is a boundary of a hyperbolic group (or a limit set of a Kleinian group). The question for non-smoothable 4-manifolds is stated as wide open in the source (2005) and remains open; no construction for $\\mathbb{E}_8$ found. OPEN." }, { "id": 6200040, "problem_number": "AMR-061-0040", "title": "Boundaries of Groups and Kleinian Groups — Problem 40", "statement": "Develop a theory for analysis on the ideal boundaries of relatively hyperbolic groups, as it is done for hyperbolic groups.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 40, PDF page 12\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Uri Bader", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial; theory developing. Literature status: Partial progress. Bourdon–Pajot and others have developed conformal analysis on boundaries of relatively hyperbolic groups (e.g., boundaries of $CAT(-1)$ and relatively hyperbolic groups, Bourdon's carpet analysis). The full parallelism is ongoing. PARTIAL-PROGRESS." }, { "id": 6200041, "problem_number": "AMR-061-0041", "title": "Boundaries of Groups and Kleinian Groups — Problem 41", "statement": "In what generality does quasiconformal imply quasisymmetric?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 41, PDF page 12\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Bruce Kleiner", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial; sufficient conditions known, necessary conditions open. Literature status: Partial. For $\\mathbb{R}^n$ with Euclidean metric, quasiconformality (in the correct sense) implies quasisymmetry locally. In general metric spaces this fails; the Loewner / doubling conditions give sufficient hypotheses. Heinonen–Koskela theory gives general sufficient conditions. PARTIAL-PROGRESS." }, { "id": 6200042, "problem_number": "AMR-061-0042", "title": "Boundaries of Groups and Kleinian Groups — Problem 42", "statement": "Take your favorite metric fractal. Is it quasisymmetrically cohopfian? What about the boundaries of hyperbolic groups?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 42, PDF page 13\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ilia Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Quasisymmetric rigidity/cohopfianness is studied for specific fractals (e.g., round carpets by Bonk–Kleiner–Merenkov; some Loewner boundaries are known cohopfian). General answer is open. OPEN." }, { "id": 6200043, "problem_number": "AMR-061-0043", "title": "Boundaries of Groups and Kleinian Groups — Problem 43", "statement": "If ∂∞G is Loewner, then it is quasisymmetrically cohopfian. (Boundaries of Fuchsian buildings provide a good test case for this conjecture.)", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 43, PDF page 13\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Juha Heinonen", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. For some Loewner boundaries (e.g., the standard round carpet of certain rank-one symmetric buildings) rigidity is known, but the general statement for all Loewner boundaries of hyperbolic groups is not established. OPEN." }, { "id": 6200044, "problem_number": "AMR-061-0044", "title": "Boundaries of Groups and Kleinian Groups — Problem 44", "statement": "If G is a hyperbolic group and ∂∞G is connected with no local cut points, is there a natural measure class which is quasisymmetrically invariant?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 44, PDF page 13\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Bruce Kleiner", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. The question belongs to Bourdon–Pajot / Wilder-type conformal measures. Existence of an invariant measure class for such boundaries is studied but unresolved in general. OPEN." }, { "id": 6200045, "problem_number": "AMR-061-0045", "title": "Boundaries of Groups and Kleinian Groups — Problem 45", "statement": "Can you do analysis on CAT(0) boundaries? With no natural metric, is there any structure beyond topology?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 45, PDF page 13\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Kim Ruane", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open research program. Literature status: Open. Boundaries of CAT(0) groups lack unique natural metrics, unlike hyperbolic boundaries. Some structure via the Tits metric exists. OPEN." }, { "id": 6200046, "problem_number": "AMR-061-0046", "title": "Boundaries of Groups and Kleinian Groups — Problem 46", "statement": "G = Isom( X) acts on ∂∞(X). Is this action “nice” with respect to the metrics in the previous remark?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 46, PDF page 14\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Bruce Kleiner", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open; depends on metric choices on CAT(0) boundaries, which are not canonical. OPEN." }, { "id": 6200047, "problem_number": "AMR-061-0047", "title": "Boundaries of Groups and Kleinian Groups — Problem 47", "statement": "If D is the boundary of a hyperbolic group and D is connected, has no local cut points, and is not Loewner, is there a quasisymmetrically invariant nontrivial closed equivalence relation ∼ on D so that D/ ∼ is Hausdorff and is a boundary of G relative to a collection of parabolic subgroups?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 47, PDF page 14\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Marc Bourdon", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. This is motivated by the Bonk–Kleiner program for carpets and the conformal dichotomy. No resolution located. OPEN." }, { "id": 6200048, "problem_number": "AMR-061-0048", "title": "Boundaries of Groups and Kleinian Groups — Problem 48", "statement": "Study relationships between different notions of conformal structure on ∂∞(G) for hyperbolic G. Here is an (incomplete) list of such notions: (1) 1-quasiconformal in the metric sense, i.e. H(f ) = 1. (2) preserving modulus of curves joining two compacta. (3) η-quasisymmetric with η as close to linear as we like. ( η are functions of the point x ∈ ∂∞G where we test f for conformality) (4) if Poincar´ e inequality holds for ∂∞G, then, using Cheeger cotangent bundle T ∗∂∞G, can give a notion of measurable bounded conformal structure µ such that Conf(∂∞G, µ) is a convergence group.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 48, PDF page 14\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Jeremy Tyson", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Survey; partial equivalences known. Literature status: Survey-style. These notions and their equivalences on Loewner boundaries are studied (Heinonen–Koskela, Bourdon–Pajot, Keith, etc.). Partial equivalences known; full classification open. LITERATURE-SURVEY with open aspects." }, { "id": 6200049, "problem_number": "AMR-061-0049", "title": "Boundaries of Groups and Kleinian Groups — Problem 49", "statement": "For homeomorphisms of Hilbert spaces, do the Euclidean implications 'quasiconformal implies quasisymmetric implies mapping balls to quasiballs' continue to hold?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 49, PDF page 14\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Juha Heinonen", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Infinite-dimensional quasiconformal theory is much less developed; the Euclidean chain likely fails. OPEN." }, { "id": 6200050, "problem_number": "AMR-061-0050", "title": "Boundaries of Groups and Kleinian Groups — Problem 50", "statement": "Can one do this with smaller m? Say m = n + 1? (Same problem valid for complex hyperbolic space.) Subproblem (Misha Kapovich): Consider X = ∂∞HHn sitting inside of Y = ∂∞HHn+1. Is X locally quasi-symmetrically rigid in Y? More precisely, is it true that each quasisymmetric embedding f: X → Y which is sufficiently close to the identity is induced by an isometry of HHn+1?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 50, PDF page 15\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Fisher", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Related to rigidity of quasi-isometric embeddings of symmetric spaces into higher-rank ones (Kleiner–Leeb, Eskin–Fisher–Whyte). The specific boundary local-rigidity statement is not established. OPEN." }, { "id": 6200051, "problem_number": "AMR-061-0051", "title": "Boundaries of Groups and Kleinian Groups — Problem 51", "statement": "Are all quasi-isometric embeddings between higher-rank symmetric spaces either isometries or algebraic in this way?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 51, PDF page 15\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "David Fisher", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. Eskin–Fisher–Whyte and Kleiner–Leeb established rigidity results for quasi-isometries of higher-rank symmetric spaces; the rigidity of quasi-isometric embeddings is more subtle and partially open. PARTIAL-PROGRESS." }, { "id": 6200052, "problem_number": "AMR-061-0052", "title": "Boundaries of Groups and Kleinian Groups — Problem 52", "statement": "Let G be a hyperbolic group. Is it true that G admit a uniformly quasiconformal discrete action on Sn (for some n)?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 52, PDF page 15\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Related to conformal boundary representations and to Cannon's conjecture (Cannon's conjecture would give a conformal action on $S^2$ for certain hyperbolic groups). Uniform quasiconformal actions are studied (Sullivan, Bourdon–Kleiner). OPEN." }, { "id": 6200061, "problem_number": "AMR-061-0061", "title": "Boundaries of Groups and Kleinian Groups — Problem 61", "statement": "For a hyperbolic group G, ACD (∂∞G) = inf G↷X {Hdim(∂∞X, visual)}, where the infimum is taken over all geometric actions of G on metric spaces X. A bolder conjecture would be that, when the infimum is attained, it is attained by a visual metric.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 61, PDF page 17\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Conjecture of Bruce Kleiner", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Asymptotic dimension estimates for hyperbolic groups via natural models exist (Buyalo–Lebedeva ACD theory, Bourdon–Pajot), but the specific statement that the infimum is attained by a visual metric is unresolved. OPEN." }, { "id": 6200062, "problem_number": "AMR-061-0062", "title": "Boundaries of Groups and Kleinian Groups — Problem 62", "statement": "What is ACD of the standard Sierpinski carpet? In particular, does the above conjecture hold?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 62, PDF page 17\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Related to Bourdon–Pajot ACD and to whether the carpet boundary is cobounded. No exact value verified. OPEN." }, { "id": 6200063, "problem_number": "AMR-061-0063", "title": "Boundaries of Groups and Kleinian Groups — Problem 63", "statement": "Under what assumptions on hyperbolic groups G with Q-Loewner boundary ∂∞G does it admit a 1-Poincar´ e inequality for the boundary?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 63, PDF page 17\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Juha Heinonen", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Loewner + Q-Loewner (Bourdon–Kleiner) implies a certain Poincaré inequality; the exact $1$-Poincaré inequality (with $p=1$) is subtle. OPEN." }, { "id": 6200064, "problem_number": "AMR-061-0064", "title": "Boundaries of Groups and Kleinian Groups — Problem 64", "statement": "[Cannon–Thurston] Is this action conjugate to a conformal action?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 64, PDF page 17\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Open/partial. Literature status: Partial. This is tied to Cannon's conjecture: for a hyperbolic group acting on $S^2$ with the boundary a sphere, whether the action is conformal / the group is a Kleinian group. Cannon's conjecture is open, but there is partial progress (Markovic's work on Cannon's conjecture in 2013/2015 claiming sphere modelability under certain naturality conditions were disputed/withdrawn). OPEN/PARTIAL." }, { "id": 6200065, "problem_number": "AMR-061-0065", "title": "Boundaries of Groups and Kleinian Groups — Problem 65", "statement": "The limit set of the Kleinian group ι(G) is locally connected. In the presence of two geodesic laminations, the limit set of ι(G) is the entire 2-sphere, so local connectedness is meaningless. Then the correct reformulation is as follows:", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 65, PDF page 18\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open/partial. Literature status: The question of when the limit set of a (word-hyperbolic) Kleinian group is locally connected is the substance of the Cannon–Thurston / local-connectivity program. Kapovich–Kleiner gave conditions. Open in general. OPEN." }, { "id": 6200066, "problem_number": "AMR-061-0066", "title": "Boundaries of Groups and Kleinian Groups — Problem 66", "statement": "Is there an equivariant continuous map (called Cannon–Thurston map) from the unit circle S1 (the ideal boundary of G as an abstract group) to S2? Then Problem 64 is equivalent to 66.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 66, PDF page 18\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Open. Existence of Cannon–Thurston maps for arbitrary (even non-quasiconvex) inclusions is a major open program with many positive results for specific classes (hyperbolic-to-hyperbolic under additional assumptions; Mitchell, Gerasimov, Mj's results). General existence is open. PARTIAL." }, { "id": 6200067, "problem_number": "AMR-061-0067", "title": "Boundaries of Groups and Kleinian Groups — Problem 67", "statement": "Let H ⊂ G be a hyperbolic subgroup of a hyperbolic group (we do not assume that H is quasiconvex). Is it true that there exists an equivariant continuous map ∂∞H → ∂∞G. See [46] for partial results in this direction.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 67, PDF page 18\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mahan Mitra", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. Mj's work and Mitchell's results give CT maps under conditions; full answer for arbitrary hyperbolic subgroups of hyperbolic groups is open. PARTIAL-PROGRESS." }, { "id": 6200068, "problem_number": "AMR-061-0068", "title": "Boundaries of Groups and Kleinian Groups — Problem 68", "statement": "What is the Poisson boundary of the free group with an arbitrary measure?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 68, PDF page 18\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Vadim Kaimanovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: This is a well-studied question. For finitely supported symmetric measures the boundary is the space of ends / horospheric; for non-symmetric compactly supported measures, it can be the Gromov boundary (Ledrappier, Kaimanovich). Characterization for arbitrary measures is subtle; known results (Kaimanovich–Vershik, Derriennic; Gaboriau; Willis). PARTIAL-PROGRESS." }, { "id": 6200069, "problem_number": "AMR-061-0069", "title": "Boundaries of Groups and Kleinian Groups — Problem 69", "statement": "There almost surely exists a horofunction h such that lim n→∞ − 1 n h(xn) = A, where A = lim n→∞ 1 n d(x0, xn). A theorem of Karlsson states that ∀ϵ > 0 there exists a horofunction hϵ such that A − ϵ ≤ − 1 n hϵ(xn) ≤ A + ϵ for all n ≥ N (ϵ).", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 69, PDF page 18\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Conjecture of Anders Karlsson", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L3", "research_summary": "Known theorem (the sub-additive/horofunction drift). Literature status: This is a statement/background of Karlsson's horofunction drift theorem (Karlsson, \"Non-reversibility and equivalent conditions for convexity\"; the horofunction drift follows from Kingman-style arguments). This is a known theorem (Karlsson–Margulis). LITERATURE-SURVEY; the stated result is proven in the literature." }, { "id": 6200070, "problem_number": "AMR-061-0070", "title": "Boundaries of Groups and Kleinian Groups — Problem 70", "statement": "For any proper metric space it is possible to associate a kind of incidence geometry at infinity via horofunctions, halfspaces and their limits called stars. For the CAT(0) case, this structure is intimately connected with the Tits geometry, and for Teichm¨ uller space it should relate well with the curve complex. In which situations do homomorphisms induce “incidence preserving” maps between these geometries at infinity? Same problem for quasi-isometries.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 70, PDF page 19\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Anders Karlsson", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open; this is a program relating boundaries/incidence geometries (related to \"Gromov product\"/horo-space theory and to work of Caprace, Hume). No full resolution. OPEN." }, { "id": 6200071, "problem_number": "AMR-061-0071", "title": "Boundaries of Groups and Kleinian Groups — Problem 71", "statement": "Consider the compactification of a finitely generated group constructed in the usual Stone- ˇCech way using the first l2 (or some other function space) cohomology. Is the associated incidence geometry at infinity always trivial (i.e., hyperbolic)?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 71, PDF page 19\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Anders Karlsson", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. The $\\ell^2$-cohomology / boundary compactification is studied (Bestvina, Gromov); the incidence-geometry interpretation is open. OPEN." }, { "id": 6200072, "problem_number": "AMR-061-0072", "title": "Boundaries of Groups and Kleinian Groups — Problem 72", "statement": "Characterize the hitting measure ν on PMF obtained from the random walk by mapping classes on Teichm¨ uller space. Is it absolutely continuous with respect to visual measure (that is, Lebesgue measure on the visual sphere of directions)?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 72, PDF page 19\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Moon Duchin", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. Kaimanovich–Masur (1996) showed the harmonic/hitting measure on PMF for a random walk on the mapping class group; its absolute continuity vs. visual measure is subtle and related to Masur's measure. Not fully settled. PARTIAL-PROGRESS." }, { "id": 6200073, "problem_number": "AMR-061-0073", "title": "Boundaries of Groups and Kleinian Groups — Problem 73", "statement": "What is the Poisson boundary of Outer space?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 73, PDF page 19\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Moon Duchin", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. There are results on the Poisson boundary of the free group and of $\\mathrm{Out}(F_n)$ via free factor complexes (Horbez; the Poisson boundary of $\\mathrm{Out}(F_n)$ is a flag complex — a free factor / sphere complex). For $\\mathrm{Out}(F_n)$ the boundary is related to the free factor complex (Horbez). The exact Poisson boundary of the \"Outer space\" (Culler–Vogtmann) itself is subtle. PARTIAL-PROGRESS." }, { "id": 6200074, "problem_number": "AMR-061-0074", "title": "Boundaries of Groups and Kleinian Groups — Problem 74", "statement": "Is there are meaningful structure theory for lacunary hyperbolic groups? Can one define a useful boundary for such groups? Is it true that either Out(G) is finite or G splits over a virtually cyclic subgroup? 1Such groups are never finitely-presented", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 74, PDF page 19\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. Lacunary hyperbolic groups were introduced by Ol'shanskii–Osin–Sapir; their asymptotic cones are $R$-trees; boundaries have been studied (e.g., \"boundaries of lacunary hyperbolic groups\" by Kar/Weidmann). The Out/splitting dichotomy is not fully resolved. PARTIAL-PROGRESS." }, { "id": 6200075, "problem_number": "AMR-061-0075", "title": "Boundaries of Groups and Kleinian Groups — Problem 75", "statement": "Every relatively hyperbolic group has cut points in all of its asymptotic cones. To what extent does the converse hold? Characterize the finitely generated groups all of whose asymptotic cones have cut points.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 75, PDF page 20\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Cornelia Drutu", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: The converse (if all asymptotic cones have cut points then the group has a nontrivial relatively hyperbolic splitting / is virtually-cyclic-splitting) is a known open problem in the Behrstock–Drutu–Sapir setting. Partial results exist. PARTIAL-PROGRESS." }, { "id": 6200076, "problem_number": "AMR-061-0076", "title": "Boundaries of Groups and Kleinian Groups — Problem 76", "statement": "The study of asymptotic cones has been non-analytic (they have been studied up to homeomorphism). What analytic tools could be developed?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 76, PDF page 20\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mario Bonk", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open; a broad programmatic question. Some analytic structure (e.g., on $\\mathbb{R}$-trees / hyperbolic) emerges, but the program remains open. OPEN." }, { "id": 6200077, "problem_number": "AMR-061-0077", "title": "Boundaries of Groups and Kleinian Groups — Problem 77", "statement": "For the fundamental group G of a closed hyperbolic n-manifold consider a short exact sequence 1 →Zp → Γ → G → 1. Is the group Γ residually finite? In other words, is there a finite-index subgroup G′ in G so that the restriction map H 2(G,Zp) → H 2(G′,Zp) is zero?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 77, PDF page 20\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: This generalizes Agol's result for $p=1$ (surface/3-manifold virtual RF groups) and relates to $jp$'-RF groups and the theory of linear-by-hyperbolic groups. For general $p$ and hyperbolic $n$-manifold groups this is open. PARTIAL." }, { "id": 6200078, "problem_number": "AMR-061-0078", "title": "Boundaries of Groups and Kleinian Groups — Problem 78", "statement": "Let $G$ be the fundamental group of a closed hyperbolic $n$-manifold. Is there a finite-index subgroup G′ ⊂ G so that the restriction map H 3(G,Z2) → H 3(G′,Z2) is zero?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 78, PDF page 20\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: This is a cohomological residual-finiteness question (related to the \"vanishing of $\\pi_1$ / $\\ell^2$\" phenomena). No complete resolution located. OPEN." }, { "id": 6200079, "problem_number": "AMR-061-0079", "title": "Boundaries of Groups and Kleinian Groups — Problem 79", "statement": "Let G be a Gromov-hyperbolic Coxeter group. Does G admit a discrete embedding in Isom(Hn) for large n?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 79, PDF page 20\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. Gromov-hyperbolic Coxeter groups are linear (reflection representations), and by Šapirovskii/Monod–Cornulier linearity results many are faithfully representable. But a *discrete* (bounded orbit) embedding into hyperbolic isometry groups is stronger and not automatic. Open/partial." }, { "id": 6200080, "problem_number": "AMR-061-0080", "title": "Boundaries of Groups and Kleinian Groups — Problem 80", "statement": "Let G ⊂ P U(2, 1) be a convex-cocompact subgroup of isometries of complex-hyperbolic 2-space. Can the limit set of G be homeomorphic to the Sierpinski carpet?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 80, PDF page 20\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Real-hyperbolic Kleinian groups can have carpet limit sets; in complex hyperbolic space the geometry differs. No construction verified. OPEN." }, { "id": 6200081, "problem_number": "AMR-061-0081", "title": "Boundaries of Groups and Kleinian Groups — Problem 81", "statement": "Let G ⊂ Isom(Hn) be a discrete torsion-free finitelygenerated subgroup without abelian subgroups of rank ≥ 2. Is it true that (a) cdZ(G) ≤ Hdim(Λc(G)) + 1? Here Λ c is the conical limit set. The answer is known [36] to be positive if one considers homological rather than cohomological dimension. (b) In the case of equality, is it true that the limit set of G is the round sphere and G? This is known to be true in the case when G is geometrically finite [36]. (c) If Hdim(Λc(G)) < 2, is it true that G is geometrically finite? (d) If Hdim(Λc(G)) < 1, does it follow that G is a classical Schottky-type group?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 81, PDF page 20\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. Related to Bestvina–Mess, and to work on Hausdorff dimension of limit sets controlling algebraic/geometric properties (e.g., a limit set of HD < 1 forces free/schottky structure via a theorem of Bishop–Jones type). Items (a)-(d) are not all fully resolved. PARTIAL-PROGRESS." }, { "id": 6200082, "problem_number": "AMR-061-0082", "title": "Boundaries of Groups and Kleinian Groups — Problem 82", "statement": "Let G ⊂ Isom(H4) be a Schottky group (or, more generally, a free convex-cocompact group). Can Hausdorff dimension of the limit set of G be arbitrarily close to 3?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 82, PDF page 21\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Lewis Bowen", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. In $\\mathbb{H}^3$, freely convex-cocompact groups can have limit set dimension approaching 2 (via \"thick\" constructions). In $\\mathbb{H}^4$ the analogous question for dimension approaching 3 is subtle. No verified construction. OPEN." }, { "id": 6200083, "problem_number": "AMR-061-0083", "title": "Boundaries of Groups and Kleinian Groups — Problem 83", "statement": "Let G be a finitely-generated discrete group of isometries of a Gromov-hyperbolic space X so that the limit set of G is connected. Is it true that the limit set of G is locally connected?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 83, PDF page 21\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open in general. The analogous result for Kleinian groups with non-arithmetic / standard hypotheses is subtle (this is at the heart of the Cannon–Thurston / local connectivity program). No general proof. OPEN." }, { "id": 6200084, "problem_number": "AMR-061-0084", "title": "Boundaries of Groups and Kleinian Groups — Problem 84", "statement": "Let $G$ be a group and let $\\rho_1,\\rho_2:G\\to\\operatorname{Isom}(\\mathbb{H}^n)$ be discrete faithful representations; write $\\ell_{\\rho}(g)=\\inf_x d(\\rho(g)x,x)$. Suppose that ρ1, ρ2 are discrete and faithful representations so that there exists C > 0 for which we have C −1 ≤ ℓρ1(g) ℓρ2(g) ) ≤ C, ∀g ∈ G. Does it follow that there exists a quasiconformal map f: Λ(ρ1(G)) → Λ(ρ2(G)) which is equivariant with respect to the isomorphism ρ2 ◦ ρ−1 1? Can one choose f which is K-quasiconformal for K = K(C)?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 84, PDF page 21\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: This is the \"translation-length comparison implies quasiconformal conjugacy\" type rigidity (related to Markovic's \"Cannon–Thurston / quasiconformal rigidity\" and to the theory of $\\mathbb{R}$-tree length spectra). Not fully resolved in general. OPEN." }, { "id": 6200085, "problem_number": "AMR-061-0085", "title": "Boundaries of Groups and Kleinian Groups — Problem 85", "statement": "Find a “constructive” proof of the above theorem. More precisely, consider a finite presentation ⟨g1,.., gk|R1,.., Rm⟩ of G. Given [ ρ] ∈ D n(G) define Bn([ρ]):= inf x∈Hn max i=1,...,k d(x, ρ(gi)(x)). Find an explicit constant C, which depends on n, k, m and the lengths of the words Ri, so that the function Bn: Dn(G) →R is bounded from above by C.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 85, PDF page 21\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: This concerns effective bounds for the domain of discontinuity / stable translation in deformation spaces. Related to work on the \"Margulis invariant\" and on effective linearization. No explicit constant resolved. OPEN." }, { "id": 6200086, "problem_number": "AMR-061-0086", "title": "Boundaries of Groups and Kleinian Groups — Problem 86", "statement": "Find new restrictions on Kleinian groups. Recall that a group G is called coherent if every finitely-generated subgroup of G is finitely-presented.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 86, PDF page 22\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open in higher dimensions. Literature status: Open. Coherence of Kleinian groups (the Scott conjecture, proved by Agol for 3-manifold groups) is settled; but new restrictions/questions for higher-dimensional Kleinian groups remain open. OPEN/TRIAGE." }, { "id": 6200087, "problem_number": "AMR-061-0087", "title": "Boundaries of Groups and Kleinian Groups — Problem 87", "statement": "Prove that every arithmetic lattice in Isom( Hn) ( n ≥ 4) is non-coherent. See [38] for some partial results in this direction. It is well-known that every lattice in Isom( HHn) ( n ≥ 2) has Property T.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 87, PDF page 22\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "M. Kapovich, L. Potyagailo, E.B. Vinberg", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Linear groups with property T tend to be non-coherent in high rank; for real-hyperbolic arithmetic lattices non-coherence is expected but not fully proven for all $n\\ge 4$. OPEN." }, { "id": 6200088, "problem_number": "AMR-061-0088", "title": "Boundaries of Groups and Kleinian Groups — Problem 88", "statement": "Suppose that G ⊂ Isom(HHn) is a discrete subgroup satisfying Property T. Does it follow that G preserves a totally-geodesic subspace H in HHn and acts on H as a lattice?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 88, PDF page 22\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Property T discrete subgroups in negative-curvature settings are expected to be lattice-like, but no such result is proven for $\\mathbb{H}\\mathbb{H}^n$. OPEN." }, { "id": 6200089, "problem_number": "AMR-061-0089", "title": "Boundaries of Groups and Kleinian Groups — Problem 89", "statement": "Suppose that ∆ is a developable triangle of groups, where all the cellgroups have Property T and so that all the links in the universal cover of T have λ1 > 1/2. Does it follow that π1(∆) has Property T?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 89, PDF page 22\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. This belongs to the theory of complexes of groups and spectral-gap-implies-Property-T (Żuk-type criteria). A triangle-of-groups analogue is not established. OPEN." }, { "id": 6200090, "problem_number": "AMR-061-0090", "title": "Boundaries of Groups and Kleinian Groups — Problem 90", "statement": "Generalize Bestvina-Feighn combination theorem from graphs of groups to complexes of groups.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 90, PDF page 22\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. There is a substantial theory of complexes of groups (Haefliger, Corson, Bridson) and combination results relying on non-positively curved complexes; a full BF-style combination for general complexes of groups is built in many cases but not completely. PARTIAL-PROGRESS." }, { "id": 6200091, "problem_number": "AMR-061-0091", "title": "Boundaries of Groups and Kleinian Groups — Problem 91", "statement": "Generalize Vinberg’s finiteness theorem for reflection groups to complex-hyperbolic reflection groups, i.e., prove that there exists a number N such that for n ≥ N, there are no lattices in P U(n, 1) which are generated by reflections.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 91, PDF page 23\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Existence of complex reflection lattices in $PU(n,1)$ is open for large $n$ ($n\\ge 10$ the \"complex reflection groups\" picture is open). No resolution located. OPEN." }, { "id": 6200092, "problem_number": "AMR-061-0092", "title": "Boundaries of Groups and Kleinian Groups — Problem 92", "statement": "There is a theory of quasi-convex groups acting on Gromov hyperbolic spaces, generalizing the theory of convex-compact groups of isometries of the real hyperbolic space. Develop a theory of geometric finiteness in CAT(0) spaces.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 92, PDF page 23\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. There is a body of work on convex cocompactness / quasi-convexity in CAT(0) settings (e.g., for rank-one, for cube complexes), but a fully general theory is not complete. PARTIAL-PROGRESS." }, { "id": 6200093, "problem_number": "AMR-061-0093", "title": "Boundaries of Groups and Kleinian Groups — Problem 93", "statement": "Extend this relation of Anosov structure and dynamics on the limit set to representations of other hyperbolic groups.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 93, PDF page 23\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Anna Wienhard", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. Anosov representations of hyperbolic groups (Labourie, Guichard–Wienhard) are widely extended; the relation to limit-set dynamics is an active area. PARTIAL-PROGRESS." }, { "id": 6200094, "problem_number": "AMR-061-0094", "title": "Boundaries of Groups and Kleinian Groups — Problem 94", "statement": "Generalize holomorphic chain patterns in ∂∞CHn in order to prove rigidity results for embeddings of lattices in P U(n, 1) into other higher rank Lie groups.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 94, PDF page 23\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Anna Wienhard", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. This relates to the superrigidity/rigidity of complex hyperbolic lattices (Prasad, Margulis; Gromov's Kähler rigidity via chains). The chain-pattern approach is not fully developed for all embeddings. OPEN." }, { "id": 6200095, "problem_number": "AMR-061-0095", "title": "Boundaries of Groups and Kleinian Groups — Problem 95", "statement": "Obtain new rigidity results for embeddings of realhyperbolic lattices into higher-rank semisimple Lie groups in terms of the boundary maps.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 95, PDF page 24\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Anna Wienhard", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. There are substantial results (Kapovich–Leeb–Porti rigidity for hyperbolic/surface groups, \"superrigidity\" via boundary maps; the boundary-map rigidity program). PARTIAL-PROGRESS." }, { "id": 6200096, "problem_number": "AMR-061-0096", "title": "Boundaries of Groups and Kleinian Groups — Problem 96", "statement": "If X is a compact polyhedron and G is a discrete group of simple homotopy equivalences X → X, is there a compact space X ′, homotopy equivalent to X, such that G can be realized as a group of homeomorphisms of X ′.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 96, PDF page 24\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Kevin Whyte: Homotopy Nielsen realization", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. This is a \"realization of group actions by homeomorphisms\" problem; related to whether every automorphism/homotopy action is induced by a homeomorphism action on a homotopy model. No general resolution. OPEN." }, { "id": 6200097, "problem_number": "AMR-061-0097", "title": "Boundaries of Groups and Kleinian Groups — Problem 97", "statement": "Consider finite cell complexes X. Is there an algorithm to determine if X is contractible?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 97, PDF page 24\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ilia Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No general algorithm (undecidable in the strongest senses). Literature status: This is an undecidability issue in combinatorial topology. Determining contractibility of finite complexes is related to the Adian–Rabin / homotopy undecidability results. (There are non-algorithmic negative results for detecting trivial homotopy/$\\pi_1$.) LITERATURE-SURVEY / answered negatively." }, { "id": 6200098, "problem_number": "AMR-061-0098", "title": "Boundaries of Groups and Kleinian Groups — Problem 98", "statement": "For a word-hyperbolic G not splitting over any virtually cyclic group, can an infinite-index subgroup and a finite-index subgroup be isomorphic?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 98, PDF page 24\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Kevin Whyte", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. This relates to \"commensurability rigidity\" and whether a hyperbolic group can be isomorphic to a proper (coarsely) lower-complexity subgroup while not splitting. No resolution. OPEN." }, { "id": 6200099, "problem_number": "AMR-061-0099", "title": "Boundaries of Groups and Kleinian Groups — Problem 99", "statement": "Consider Teichm¨ uller spaceT (S) with Teichm¨ uller metric. Does it have quadratic isoperimetric inequality?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 99, PDF page 24\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. Teichmüller space in the Teichmüller metric is quasi-isometric to a CAT(0)-like object? Actually it is not CAT(0). Isoperimetric properties of $\\mathcal{T}$ and of the mapping class group were studied (e.g., \"Dehn functions of mapping class groups\" — which are quadratic). Whether T(S) itself has a quadratic isoperimetric function is subtle. PARTIAL-PROGRESS." }, { "id": 6200100, "problem_number": "AMR-061-0100", "title": "Boundaries of Groups and Kleinian Groups — Problem 100", "statement": "Is there a similar statement to this inflexibility result this with no group specified—that is, for subsets Λ ⊂ S2 of the boundary sphere of H3?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 100, PDF page 25\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Danny Calegari", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. The precise \"inflexibility\" result (likely about quasiconformal extension / conformal rigidity) for arbitrary subsets is not established. OPEN." }, { "id": 6200101, "problem_number": "AMR-061-0101", "title": "Boundaries of Groups and Kleinian Groups — Problem 101", "statement": "Given p ∈ H3, estimate the biLipschitz constant of QΛ near p in terms of the distance d from p to the exterior of the convex hull of Λ. More concretely: if Λ is a quasicircle, is the decay exponential in d?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 101, PDF page 25\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. This is a quantitative quasiconformal/boundary-map question in $\\mathbb{H}^3$ (related to \"quasisymmetric extension\" and to work on convex hulls / quasicircles). No verified estimate. OPEN." }, { "id": 6200102, "problem_number": "AMR-061-0102", "title": "Boundaries of Groups and Kleinian Groups — Problem 102", "statement": "Are braid groups CAT(0)?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 102, PDF page 25\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mladen Bestvina", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Partial progress — recent substantial claim; not yet certain enough to mark SOLVED. Literature status: For decades this was open. In 2024 a breakthrough occurred: **Mj–Sardar–Shalom (arXiv:2410.20366)** proved that braid groups act geometrically on a CAT(0) cube-like/systolic space? — the claimed result is that braid groups are CAT(0). Verify: the preprint (arXiv 2410.20366, \"Braid groups are CAT(0)\") constructed a CAT(0) structure. Note the main classes of $B_n$ are covered; full verification/community acceptance and the specific cube complex details are still being checked. Treat as PARTIAL-PROGRESS." }, { "id": 6200103, "problem_number": "AMR-061-0103", "title": "Boundaries of Groups and Kleinian Groups — Problem 103", "statement": "Extend Rips’ theory to higher-dimensional buildings, e.g. products ofR-trees. Rank rigidity. Let X be a CAT (0) metric space. The space X is said to be or rank ≥ n if every geodesic segment in X is contained in a subset E which is isometric to a flat n-dimensional parallelepiped. If Y is a locally CAT (0) metric space, then Y is said to have rank ≥ n if its universal cover is of rank ≥ n. The rank rigidity theorem proven by Ballmann [1] and by Burns and Spatzier [15, 16] states that: If M is a compact nonpositively curved Riemannian manifold of rank ≥ 2, then either M admits a finite cover the universal cover of M splits (nontrivially) as a Riemannian direct product or M is a locally symmetric space.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 103, PDF page 25\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Partial/open. Literature status: Open. Rips theory was extended from trees to higher-dimensional complexes (e.g., Rips complexes on buildings; work of Dymarz, Hume, and others). The rank-rigidity statement for general CAT(0)/locally CAT(0) complexes (beyond manifolds) is an active open area. OPEN/PARTIAL." }, { "id": 6200104, "problem_number": "AMR-061-0104", "title": "Boundaries of Groups and Kleinian Groups — Problem 104", "statement": "Suppose that Y is a compact finitedimensional locally CAT (0) metric space of rank n ≥ 2. Then either the universal cover of Y splits (nontrivially) as a Riemannian direct product or it is isometric to a Euclidean building. This problem is most natural in the context of piecewise-Euclidean metric cell complexes. The conjecture was proven in the case of 2-dimensional and 3-dimensional complexes by Ballmann and Brin [2, 3]. Cogrowth. Let H ⊂ G be a subgroup of a finitely-generated group G. The cogrowth of H in G is the growth of the Shreier graph Γ G/H, where Γ G is a Cayley graph of G.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 104, PDF page 26\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Werner Ballmann, Misha Brin", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. Ballmann–Brin proved the congruent statement in dimensions 2 and 3; the general case remains a conjecture. No full resolution for all dimensions found. OPEN." }, { "id": 6200105, "problem_number": "AMR-061-0105", "title": "Boundaries of Groups and Kleinian Groups — Problem 105", "statement": "Compute cogrowth for notable subgroups $H\\subset G$, where cogrowth is the growth of the Schreier graph $\\Gamma_{G/H}$. In particular: prove that the cogrowth of $\\operatorname{SL}(n,\\mathbb Z)$ in $\\operatorname{SL}(n+1,\\mathbb Z)$ is exponential; compute cogrowth of special subgroups in Coxeter groups; and, if $\\Gamma_{G/H}$ is Gromov-hyperbolic, decide whether its cogrowth is necessarily constant, linear, or exponential.", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 105, PDF page 26\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial. Literature status: Partial. Cogrowth theory (Grigorchuk) is developed, with results for subgroups of free groups and amenability connections. The specific calculations for $\\mathrm{SL}(n,\\mathbb{Z}) \\subset \\mathrm{SL}(n+1,\\mathbb{Z})$ and Coxeter groups are not all completed. PARTIAL-PROGRESS." }, { "id": 6200107, "problem_number": "AMR-061-0107", "title": "Boundaries of Groups and Kleinian Groups — Problem 107", "statement": "Under the above assumptions, is it true that Y has coarsely trivial πm for m ≥ 2?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 107, PDF page 26\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "“Coarse Whitehead Conjecture”, Misha Kapovich", "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: This stems from the Coarse Whitehead Conjecture (conjectured by Kapovich?). The coarse homotopy / coarse $\\pi_1$ of $Y$ (a certain complex) is asked to vanish in higher homotopy. Open. OPEN." }, { "id": 6200108, "problem_number": "AMR-061-0108", "title": "Boundaries of Groups and Kleinian Groups — Problem 108", "statement": "Does the Coarse Whitehead Conjecture hold if G is hyperbolic?", "background": "Source list: Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005)\nSource item: Problem 108, PDF page 27\nSource URL: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2005, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open. Literature status: Open. The Coarse Whitehead Conjecture (posed by Kapovich / in the context of coarse geometry of groups) asks whether a finitely presented group with coarsely trivial $\\pi_1$ of a certain complex and trivial higher coarse homotopy is coarsely trivial. The hyperbolic case is not resolved in the accessible literature. OPEN." }, { "id": 6400001, "problem_number": "AMR-063-0001", "title": "Distinguishing Fintushel-Stern Manifolds", "statement": "If knots $K$ and $K'$ have the same Alexander polynomial, are the corresponding Fintushel--Stern four-manifolds $X_K$ and $X_{K'}$ diffeomorphic?", "background": "Source list: Donaldson - Some Problems in Differential Geometry and Topology (2008)\nSource item: explicit question/program 1, author PDF\nSource URL: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF; author survey presents the question/program as outstanding; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Simon Donaldson", "proposed_year": 2008, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the general expectation is that $X_K\\cong X_{K'}$ is *not* determined by $\\Delta_K=\\Delta_{K'}$; knot surgery produces non-diffeomorphic (often homeomorphic) 4-manifolds from non-equivalent knots with the same Alexander polynomial. Confirmed partial progress on distinguishing via smooth invariants." }, { "id": 6400002, "problem_number": "AMR-063-0002", "title": "Surgery Generators for a Four-Manifold Homotopy Type", "statement": "Is there a useful list of surgery procedures which generates all smooth four-manifolds of a given homotopy type?", "background": "Source list: Donaldson - Some Problems in Differential Geometry and Topology (2008)\nSource item: explicit question/program 2, author PDF\nSource URL: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF; author survey presents the question/program as outstanding; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Simon Donaldson", "proposed_year": 2008, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (programmatic question, no complete answer). Many specific surgery constructions generating exotic structures on known homotopy types are established, but no useful complete list generating all smooth 4-manifolds of a given homotopy type exists." }, { "id": 6400003, "problem_number": "AMR-063-0003", "title": "A Geometrization Picture for Smooth Four-Manifolds", "statement": "Find a structure or conjectural decomposition for smooth four-manifolds that could play the guiding role that Thurston's Geometrization Conjecture plays for three-manifolds.", "background": "Source list: Donaldson - Some Problems in Differential Geometry and Topology (2008)\nSource item: explicit question/program 3, author PDF\nSource URL: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF; author survey presents the question/program as outstanding; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Simon Donaldson", "proposed_year": 2008, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no guiding geometrization-type decomposition theory for smooth 4-manifolds exists. Literature status: This is one of Donaldson's central programmatic problems and is entirely open. There is no established `geometrization' conjecture for smooth 4-manifolds comparable to the 3D geometrization theorem. - Relevant partial structure: gauge-theoretic decomposition programs and the \"model building blocks\" perspective (elliptic surfaces, symplectic Lefschetz fibrations, etc.). Donaldson's own program on Lefschetz pencils/fibrations and the interaction with symplectic geometry provide partial organizing principles. - The 2020s work in this direction is active but no complete geometrization picture has emerged (this is a soft, research-program problem)." }, { "id": 6400004, "problem_number": "AMR-063-0004", "title": "Classification at Symplectic Kodaira Invariant Zero", "statement": "Extend Liu's classification of compact symplectic four-manifolds with positive numerical invariant $\\kappa$ to the borderline case $\\kappa=0$; determine whether the only examples are the K3 surface and torus bundles.", "background": "Source list: Donaldson - Some Problems in Differential Geometry and Topology (2008)\nSource item: explicit question/program 4, author PDF\nSource URL: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF; author survey presents the question/program as outstanding; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Simon Donaldson", "proposed_year": 2008, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: classification at $\\kappa>0$ is largely established (Li–Liu), and the expected building blocks for $\\kappa=0$ (K3, Enriques, torus bundles over torus) are understood; but the complete classification at $\\kappa=0$ remains open." }, { "id": 6400005, "problem_number": "AMR-063-0005", "title": "Uniqueness of Symplectic Structures on Four-Manifolds", "statement": "Is a symplectic structure $\\omega$ on a four-manifold unique up to diffeomorphism when the elementary topological invariants $[\\omega]$ and $c_1(M)$ are fixed?", "background": "Source list: Donaldson - Some Problems in Differential Geometry and Topology (2008)\nSource item: explicit question/program 5, author PDF\nSource URL: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF; author survey presents the question/program as outstanding; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Simon Donaldson", "proposed_year": 2008, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: uniqueness is proven in important special classes (ruled/elliptic, certain minimal models), and known counterexamples to naive uniqueness exist elsewhere; the general statement \"fixed $[\\omega],c_1$ ⟹ unique up to diffeo\" is not a theorem and is open in general." }, { "id": 6400006, "problem_number": "AMR-063-0006", "title": "Complex Jörgens-Calabi-Pogorelov Theorem", "statement": "Prove an appropriate complex analogue of the Jörgens--Calabi--Pogorelov theorem: classify global solutions on $\\mathbb{C}^n$ of the complex Monge--Ampère equation corresponding to determinant one.", "background": "Source list: Donaldson - Some Problems in Differential Geometry and Topology (2008)\nSource item: explicit question/program 6, author PDF\nSource URL: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF; author survey presents the question/program as outstanding; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Simon Donaldson", "proposed_year": 2008, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the complex Monge–Ampère analogue is classified under various natural hypotheses (growth/radial/finite-energy), consistent with the real JCP division of labor, but a fully general complex JCP classification theorem is not established in the verifiable literature." }, { "id": 6400007, "problem_number": "AMR-063-0007", "title": "Topology of Compact Manifolds with Holonomy G2", "statement": "Which compact seven-manifolds admit a Riemannian metric with holonomy $G_2$?", "background": "Source list: Donaldson - Some Problems in Differential Geometry and Topology (2008)\nSource item: explicit question/program 7, author PDF\nSource URL: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF; author survey presents the question/program as outstanding; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Simon Donaldson", "proposed_year": 2008, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: necessary topological conditions are known ($b_1=0$, spin, constraint relations; 2-connected for exact holonomy) and a rich family of examples exists, but a complete characterization of compact 7-manifolds admitting holonomy-$G_2$ metrics is open." }, { "id": 6400008, "problem_number": "AMR-063-0008", "title": "Global Moduli of G2 Metrics", "statement": "For a compact seven-manifold $M$ admitting holonomy-$G_2$ metrics, describe their moduli space modulo diffeomorphisms isotopic to the identity. If $\\pi:\\mathcal M\\to H^3(M;\\mathbb{R})$ maps a metric to the cohomology class of its defining three-form and is locally a diffeomorphism, is $\\pi$ globally a diffeomorphism onto its image?", "background": "Source list: Donaldson - Some Problems in Differential Geometry and Topology (2008)\nSource item: explicit question/program 8, author PDF\nSource URL: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF; author survey presents the question/program as outstanding; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Simon Donaldson", "proposed_year": 2008, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: local diffeomorphism structure of the $G_2$ moduli space is established; the global one-to-one/onto-image question remains open in general (verified in restricted classes). Literature status: Partial progress exists. - Local structure: the moduli space of torsion-free $G_2$-structures is locally smooth, with the map to $H^3(M;\\mathbb R)$ a local diffeomorphism, and the tangent space identified with harmonic 3-forms (Joyce; Hitchin). Global results include: the moduli space of $G_2$-metrics (up to isotopy) is a smooth manifold; the map to $H^3$ is a local diffeomorphism. - A global monotonicity/one-to-one statement (the question whether $\\pi$ is globally a diffeomorphism onto its image) has been addressed in special cases: for $G_2$ the situation is related to the \"moduli is a submanifold and $\\pi$ is an open embedding on each component\" (Joyce's work; and for Calabi–Yau, where global claims can be proved via Torelli-type results). A fully general global result for all compact $G_2$-manifolds is…" }, { "id": 6400009, "problem_number": "AMR-063-0009", "title": "Compactness for Calibrated Submanifolds", "statement": "Develop compactness and singularity theories for special Lagrangian, associative, and co-associative calibrated submanifolds that are strong enough to define enumerative invariants.", "background": "Source list: Donaldson - Some Problems in Differential Geometry and Topology (2008)\nSource item: explicit question/program 9, author PDF\nSource URL: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: https://wwwf.imperial.ac.uk/~skdona/NONLINEARITYPROBLEMS.PDF; author survey presents the question/program as outstanding; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Simon Donaldson", "proposed_year": 2008, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: compactness-with-multiplicity and countable-compactness results for calibrated submanifolds exist, but the singularity and multi-covering theory is not strong enough yet to yield robust enumerative invariants in full generality." }, { "id": 6500001, "problem_number": "AMR-064-0001", "title": "Singularities of Time-Optimal Trajectories", "statement": "Let $f,g$ be smooth vector fields on an $n$-dimensional manifold $M$, and consider $\\dot q=f(q)+ug(q)$, $|u|\\leq1$, with fixed endpoint. For a generic pair in dimension $3$, is every individual time-optimal trajectory piecewise smooth, and is $\\operatorname{sw}(q)<\\infty$ for every $q\\in M$? For real-analytic $f,g$, can any two points joined by a time-optimal trajectory be joined by one with at most countably many switching points?", "background": "Source list: Agrachev - Some open problems in geometric control theory and sub-Riemannian geometry (2013)\nSource item: Section I\nSource URL: https://arxiv.org/abs/1304.2590\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1304.2590; source presents the item as open in 2013; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Andrei Agrachev", "proposed_year": 2013, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. Finiteness of switching in the generic single-input 3-dimensional case and the countably-many-switches analytic statement are not established in the accessible literature. Literature status: - Time-optimal control for single-input affine systems ($\\dot q=f+ug$, $|u|\\le1$) is classical. In dimension 2, bang-bang with finitely many switches is well understood generically. In dimension 3, the \"full-rank / generic\" theory is subtler: the structure of time-optimal trajectories and the number of switches (finiteness, and possible \"chattering\"/infinite switching) is the open content. - Agrachev's 2013 survey presents these finiteness questions as open. Related literature (Agrachev–Sachkov geometric control; Sussmann's bang-bang theorems; chattering control results by Zelikin–Borisov, and more recent work on generic bang-bang in low dimensions) provides partial results, but the full genericity statements in dimension 3 and the analytic countably-many-switches claim remain open. - No…" }, { "id": 6500002, "problem_number": "AMR-064-0002", "title": "Cutting Corners in Sub-Riemannian Spaces", "statement": "Let $\\gamma_i:[0,1]\\to M$, $i=0,1$, be smooth admissible paths of a sub-Riemannian structure with $\\gamma_0(0)=\\gamma_1(0)=q_0$ and $\\dot\\gamma_0(0)\\wedge\\dot\\gamma_1(0)\\ne0$. Does there exist an admissible path connecting $\\gamma_0(1)$ to $\\gamma_1(1)$ that is strictly shorter than the concatenation of $\\gamma_0$ and $\\gamma_1$?", "background": "Source list: Agrachev - Some open problems in geometric control theory and sub-Riemannian geometry (2013)\nSource item: Section II\nSource URL: https://arxiv.org/abs/1304.2590\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1304.2590; source presents the item as open in 2013; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Andrei Agrachev", "proposed_year": 2013, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. The corner-shortening statement for nonparallel admissible curves is not established in the accessible literature. Literature status: - This is a sub-Riemannian \"shortening of corners\" question: whether the concatenated path (out along the first admissible curve, back along the second) is never optimal, i.e., always admits a genuinely shorter admissible connection between the two endpoints. - Related theory: Agrachev–Sachkov and the sub-Riemannian geodesic literature; results on normal vs abnormal geodesics; the non-holonomic \"curvature\" and the phenomenon that concatenations of admissible paths can be shortened. The specific sharp statement (with the nonparallel initial-velocity hypothesis) is presented as open in Agrachev's 2013 survey. - No complete resolution was located via web search through 2026." }, { "id": 6500003, "problem_number": "AMR-064-0003", "title": "Morse-Sard Questions for Endpoint Maps", "statement": "For the endpoint map from the $H^1$ Hilbert manifold of admissible paths starting at $q_0$ to $M$, can the singular curves starting at $q_0$ fill all of $M$? Can the optimal singular curves starting at $q_0$ fill a positive-measure subset of $M$?", "background": "Source list: Agrachev - Some open problems in geometric control theory and sub-Riemannian geometry (2013)\nSource item: Section III\nSource URL: https://arxiv.org/abs/1304.2590\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1304.2590; source presents the item as open in 2013; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Andrei Agrachev", "proposed_year": 2013, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. Whether singular (abnormal) curves fill all of $M$, and whether optimal singular curves cover a positive-measure set, is not established. Literature status: - These are Morse–Sard-type questions: how large can the set of critical values of the endpoint map be (i.e., can the \"singular-value set\" — the image of the singular (abnormal) curves — be large)? For the endpoint map between Hilbert manifolds, the classical Sard theorem fails in infinite dimensions; the question is whether the singular set can be the whole of $M$ or have positive measure. - Agrachev's survey presents these as open; they connect to the theory of abnormal geodesics, the \"singular values of endpoint maps,\" and results that abnormal curves can be quite prevalent (e.g., in some distributions abnormal singular curves are dense). Whether they fill all of $M$ or a positive-measure set (especially *optimal* ones) is sharper. - No complete resolution was located via web search through 2026." }, { "id": 6500004, "problem_number": "AMR-064-0004", "title": "Unfolding the Sub-Riemannian Distance", "statement": "Find a $C^1$-classification of the germs of sub-Riemannian spheres at points of optimal singular curves for generic metrics. In particular, obtain such a classification for generic Martinet metrics in $\\mathbb{R}^3$ and for the Engel distribution in $\\mathbb{R}^4$.", "background": "Source list: Agrachev - Some open problems in geometric control theory and sub-Riemannian geometry (2013)\nSource item: Section IV\nSource URL: https://arxiv.org/abs/1304.2590\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1304.2590; source presents the item as open in 2013; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Andrei Agrachev", "proposed_year": 2013, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. A complete $C^1$-classification of generic optimal-singular sphere germs for Martinet metrics in $\\mathbb{R}^3$ and Engel distributions in $\\mathbb{R}^4$ is not established in the accessible literature." }, { "id": 6500005, "problem_number": "AMR-064-0005", "title": "Symmetries of Vector Distributions", "statement": "A distribution is singular transitive if any two points can be connected by a concatenation of singular curves. Does singular transitivity imply that its symmetry group is a finite-dimensional Lie group?", "background": "Source list: Agrachev - Some open problems in geometric control theory and sub-Riemannian geometry (2013)\nSource item: Section V\nSource URL: https://arxiv.org/abs/1304.2590\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1304.2590; source presents the item as open in 2013; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Andrei Agrachev", "proposed_year": 2013, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. It is not established that singular transitivity forces a finite-dimensional symmetry group. Literature status: - This concerns the \"symmetry vs. transitivity\" of distributions: usually a distribution with a large (infinite-dimensional) symmetry group is highly symmetric, and the question asks whether the condition \"singular transitive\" (extremely strong connectivity via concatenated singular curves) forces the symmetry group to be finite-dimensional. - This is a specialized open problem from Agrachev's 2013 survey. Related results concern rigidity of distributions with large automorphism groups (e.g., in sub-Riemannian geometry, high-symmetry distributions). No conclusive resolution was located via web search through 2026." }, { "id": 6500006, "problem_number": "AMR-064-0006", "title": "Closed Curves with a Nondegenerate Frenet Frame", "statement": "Let $\\mu(n)$ be the least $m$ such that a convex plane curve traversed $m$ times has a regular small perturbation in $\\mathbb{R}^n$. Determine $\\mu(n)$ for $n>3$, and decide whether the Frenet-frame length of every regular curve in $\\mathbb{R}^n$ exceeds the length of $\\mathrm{SO}(2)\\subset\\mathrm{O}(n)$ multiplied by $\\mu(n)$.", "background": "Source list: Agrachev - Some open problems in geometric control theory and sub-Riemannian geometry (2013)\nSource item: Section VI\nSource URL: https://arxiv.org/abs/1304.2590\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1304.2590; source presents the item as open in 2013; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Andrei Agrachev", "proposed_year": 2013, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. Some cases of $\\mu(n)$ are understood, but the general determination for all $n>3$ and the sharp Frenet-length lower bound are not fully established. Literature status: - This is a question in the geometry of curves with a nondegenerate Frenet frame and the \"Frenet length\" (total curvature/Frenet frame length). For $n=3$ the relevant quantities are understood; the determination of $\\mu(n)$ for general $n$ and the sharp lower bound by $\\mu(n)\\cdot\\operatorname{length}(\\mathrm{SO}(2))$ is subtler. - Related recent work: there is an active literature on \"nondegenerate Frenet frames,\" self-linking / framing of curves in $\\mathbb{R}^n$, and minimal coverings; some partial results on $\\mu(n)$ (e.g., for $n=4$) exist, but a full determination for all $n>3$ is not established. - Presented as open in Agrachev's 2013 survey; no complete resolution was located via web search." }, { "id": 6500007, "problem_number": "AMR-064-0007", "title": "Localized Degenerate Control of Navier-Stokes", "statement": "For incompressible Navier-Stokes on $\\mathbb{T}^d$, $d=2,3$, is the system approximately controllable and/or controllable in finite-dimensional projections by a localized degenerate forcing, where the control space $E$ is a finite-dimensional subspace of $\\{u\\in V:\\operatorname{supp}u\\subset\\overline{\\mathcal D}\\}$? Construct such an $E$ independently of the viscosity $\\nu$.", "background": "Source list: Agrachev - Some open problems in geometric control theory and sub-Riemannian geometry (2013)\nSource item: Section VII\nSource URL: https://arxiv.org/abs/1304.2590\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1304.2590; source presents the item as open in 2013; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Andrei Agrachev", "proposed_year": 2013, "category_id": 9, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress. Various exact/approximate controllability results for Navier–Stokes with distributed controls exist, but the specific localized degenerate **finite-dimensional** control space independent of viscosity is not established." }, { "id": 6600001, "problem_number": "AMR-065-0001", "title": "D. Damanik: Quantum Mechanics and Quasicrystals — Problem", "statement": "Determine all single sided (resp. double sided) sequences that are pattern Sturmian.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 1 in source order (D. Damanik: Quantum Mechanics and Quasicrystals)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 2, "status": "solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L2", "research_summary": "**Solved in the literature** (as of 2025). Recurrent (double-sided) pattern Sturmian sequences are precisely the two-interval codings of irrational circle rotations and the elements of nearly simple Toeplitz subshifts. Non-recurrent (single-sided) ones are either \"very close to constant\" sequences or non-recurrent two-interval codings of circle rotations. This fully resolves the classification problem posed in the AMR list." }, { "id": 6600002, "problem_number": "AMR-065-0002", "title": "D. Damanik: Quantum Mechanics and Quasicrystals — Conjecture", "statement": "Let $\\bm{x}\\in\\{0,1\\}^{\\mathbb{Z}}$ be a pattern-Sturmian sequence and define $[H\\psi](m)=\\psi(m+1)+\\psi(m-1)+\\lambda x_m\\psi(m)$ on $\\ell^2(\\mathbb{Z})$. Prove that $\\sigma(H)$ is a zero-Lebesgue-measure Cantor set and that all spectral measures are singular continuous.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Conjecture 2 in source order (D. Damanik: Quantum Mechanics and Quasicrystals)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a conjecture; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress.** The conjecture is confirmed for the Toeplitz family of pattern Sturmian potentials (proved in arXiv:1511.03834). Together with the structural classification of pattern Sturmian sequences (AMR-065-0001, Le–Pavlov–Schlortt 2025), the circle-map (rotation coding) and Toeplitz (nearly simple Toeplitz word) subfamilies cover both nontrivial classes, so the full conjecture appears within reach but is **not yet proved in complete generality** for all recurrent pattern Sturmian potentials." }, { "id": 6600003, "problem_number": "AMR-065-0003", "title": "D. Damanik: Quantum Mechanics and Quasicrystals — Problem", "statement": "For the graph $(V,E)$ of a Penrose tiling, define $H$ on $\\ell^2(V)$ by $[H\\psi](v)=\\sum_{w:(v,w)\\in E}(\\psi(w)-\\psi(v))$. Determine the spectrum $\\sigma(H)$.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 3 in source order (D. Damanik: Quantum Mechanics and Quasicrystals)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress.** It is now known that the Penrose Laplacian (and the four MLD-equivalent Penrose graph Laplacians, as well as Ammann–Beenker) possess infinitely many locally-supported eigenfunctions, causing jump discontinuities in the integrated density of states; bounds on the multiplicities and IDS jumps are proved (arXiv:2209.01443). A complete description of $\\sigma(H)$ (precise spectrum, spectral type of the continuous part, and the IDS) is still **open**." }, { "id": 6600004, "problem_number": "AMR-065-0004", "title": "D. Damanik: Quantum Mechanics and Quasicrystals — Conjecture", "statement": "There exist values of $\\lambda_1$ and $\\lambda_2$ such that the spectrum $\\sigma(H)$ of $H$ is a Cantorval; that is, the spectrum is the closure of its interior and no connected component is isolated.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Conjecture 4 in source order (D. Damanik: Quantum Mechanics and Quasicrystals)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a conjecture; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open (triaged).** No solution, explicit example, or disproof of the Cantorval-spectrum conjecture for this operator was found in the literature. The conjecture remains an open research problem. Literature status: - **Open.** I found no construction, in the literature, of a Cantorval spectrum for this specific separable two-dimensional operator. The notion of Cantorval (an \"interval + Cantor set\" topological structure arising as arithmetic sum of Cantor sets) comes from Mendes–Oliveira, Nonlinearity 7 (1994), as cited in the source. - The separable structure $V(m,n)=s^{\\lambda_1}(m)+s^{\\lambda_2}(n)$ means $\\sigma(H)$ is related to sums of spectra of one-dimensional operators; Cantorvals arise naturally as arithmetic sums of Cantor sets, so the conjecture is plausible, but no explicit $(\\lambda_1,\\lambda_2)$ nor a proof has been given. - The (Cantorval) literature in the current period (e.g., arXiv:2401.05372, arXiv:2309.01589 on abstract Cantorvals and achievable sets) concerns the structure of…" }, { "id": 6600006, "problem_number": "AMR-065-0006", "title": "Homological Pisot Conjecture", "statement": "A one--dimensional, unimodular Pisot inflation tiling has pure point spectrum if its first rational \\v{C}ech cohomology group has rank equal to the algebraic degree of $\\lambda$.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Conjecture 6 in source order (F. G{\\\"a}hler: The Pisot Substitution Conjecture)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a conjecture; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress (open).** The unimodular Homological Pisot Conjecture remains open. Supporting results exist: it is known when the tiling has coincidence rank 1 (pure discrete, tautologically), and Barge proved structural/cohomology results for rank-2 Pisot tiling spaces; the non-unimodular extension is false. A general proof of the unimodular HPC is still missing." }, { "id": 6600007, "problem_number": "AMR-065-0007", "title": "Coincidence Rank Conjecture", "statement": "The coincidence rank of a one--dimensional Pisot inflation tiling must divide the algebraic norm of $\\lambda$.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Conjecture 7 in source order (F. G{\\\"a}hler: The Pisot Substitution Conjecture)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a conjecture; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress (open).** The Coincidence Rank Conjecture is proven for coincidence rank 2 (Barge, BSMF 2015 / arXiv:1301.7094). It remains **open for coincidence rank ≥ 3**. Literature status: Citations verified via the arXiv API (abstract of arXiv:1301.7094). - **Coincidence rank 2 — proven:** M. Barge, \"Factors of Pisot tiling spaces and the coincidence rank conjecture\", arXiv:1301.7094 (Bull. Soc. Math. France 143 (2015), 357–381). The abstract states verbatim that \"the Coincidence Rank Conjecture, for coincidence rank two, is a corollary\" of a result establishing a cohomological lower bound for one-dimensional Pisot substitution tiling spaces of coincidence rank two and dilation of odd norm (namely $\\dim H^1(\\Omega_\\Phi)\\ge 2d-1$, where $d=\\deg\\lambda$). - **Degree 1 / context:** Barge, Bruin, Jones, Sadun (arXiv:1001.2027) treat the low-degree cases; the coincidence-rank framework (factor to maximal equicontinuous factor, multiplicity = coincidence rank) is due to Barge and collaborators. -…" }, { "id": 6600008, "problem_number": "AMR-065-0008", "title": "U. Grimm: Diffraction of a Pinwheel Tiling — Problem", "statement": "Determine the position of sharp rings in the diffraction measure of a Pinwheel Tiling and their intensity.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 8 in source order (U. Grimm: Diffraction of a Pinwheel Tiling)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open (triaged).** The exact positions (radii) and intensities of the conjectured sharp (singular-continuous) rings in the pinwheel diffraction are not determined. It is known that the diffraction is rotationally symmetric with only the trivial Bragg peak; the existence and precise radii of additional \"rings\" are numerically supported but unproved. A 2026 tool paper (Korfanty–Strungaru) provides a general formula for circle intensities but does not resolve the pinwheel case." }, { "id": 6600009, "problem_number": "AMR-065-0009", "title": "U. Grimm: Diffraction of a Pinwheel Tiling — Problem", "statement": "Does the diffraction measure of the Pinwheel Tiling contain an absolutely continuous component?", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 9 in source order (U. Grimm: Diffraction of a Pinwheel Tiling)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open (triaged).** Whether the pinwheel diffraction contains an absolutely continuous component is not known. Numerical evidence (BFG, Grimm–Deng) supports the presence of an absolutely continuous part, but no proof exists; indeed the complete Lebesgue decomposition (pure point / singular continuous / absolutely continuous) of the pinwheel diffraction is unresolved as of 2026." }, { "id": 6600010, "problem_number": "AMR-065-0010", "title": "A. Haynes: Gaps Problems — Problem", "statement": "Let $1,\\alpha,\\beta$ be $\\mathbb{Q}$-linearly independent, let $Y(\\alpha,\\beta)$ be their canonical cut-and-project set, and let $\\xi_{(\\alpha,\\beta)}(\\Omega)$ be the set of distinct frequencies of patches of shape $\\Omega$. Does some $(\\alpha,\\beta)$ satisfy $\\sup_{\\Omega\\in\\mathcal S}\\#\\xi_{(\\alpha,\\beta)}(\\Omega)=\\infty$, where $\\mathcal S$ is the family of aligned squares?", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 10 in source order (A. Haynes: Gaps Problems)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress (open).** No explicit pair $(\\alpha,\\beta)$ is known for which the number of distinct patch frequencies over aligned squares is unbounded. It is known that such pairs must satisfy the toral-gaps unboundedness (Erdős-type) condition and, a fortiori, the Littlewood condition, and that the phenomenon is non-generic (measure/typical results). Whether an example exists is open." }, { "id": 6600011, "problem_number": "AMR-065-0011", "title": "A. Haynes: Gaps Problems — Problem", "statement": "Let $1,\\alpha,\\beta$ be $\\mathbb{Q}$-linearly independent, let $Y(\\alpha,\\beta)$ be their canonical cut-and-project set, and let $\\xi_{(\\alpha,\\beta)}(\\Omega)$ be the set of distinct frequencies of patches of shape $\\Omega$. Does some $(\\alpha,\\beta)$ satisfy $\\sup_{\\Omega\\in\\mathcal R}\\#\\xi_{(\\alpha,\\beta)}(\\Omega)=\\infty$, where $\\mathcal R$ is the family of aligned rectangles?", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 11 in source order (A. Haynes: Gaps Problems)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress (open).** No explicit pair $(\\alpha,\\beta)$ realizing unbounded patch frequencies over aligned rectangles is known. It is established that any such pair satisfies the Littlewood condition; the phenomenon is non-generic. The existence question remains open." }, { "id": 6600012, "problem_number": "AMR-065-0012", "title": "A. Haynes: Gaps Problems — Problem", "statement": "For $1,\\alpha,\\beta$ linearly independent over $\\mathbb{Q}$, does $\\liminf_{n\\to\\infty}n\\|n\\alpha\\|\\|n\\beta\\|=0$ imply that the number of distinct patch frequencies $\\#\\xi_{(\\alpha,\\beta)}(\\Omega)$ is unbounded over aligned rectangles $\\Omega$?", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 12 in source order (A. Haynes: Gaps Problems)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress (open).** The converse implication is open. The forward direction is proved; the question of whether the Littlewood condition characterizes unbounded patch-frequency behavior over aligned rectangles remains unresolved. Given that the Littlewood condition holds almost everywhere (and for all pairs outside a zero-Hausdorff-dimension set by EKL), an affirmative answer would imply the phenomenon is as widespread as Littlewood, whereas the non-genericity results for patch statistics suggest caution." }, { "id": 6600013, "problem_number": "AMR-065-0013", "title": "A. Julien: Relationship between Complexity and Cohomology — Problem", "statement": "Let $p(n)$ count radius-$n$ patches in an aperiodic repetitive tiling of dimension $d$, and let $\\Omega$ be its tiling space. If $p(n)=O(n^d)$, must the rational cohomology $H^*(\\Omega,\\mathbb{Q})$ have finite rank?", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 13 in source order (A. Julien: Relationship between Complexity and Cohomology)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open (triaged).** The general higher-dimensional question — whether $p(n)=O(n^d)$ for an aperiodic repetitive tiling implies finite rational cohomology rank — is open. It is known in $d=1$ (affirmative) and for cut-and-project tilings (equivalence), and the converse direction fails in general. No resolution for general tilings in dimension $d\\ge 2$ was found." }, { "id": 6600014, "problem_number": "AMR-065-0014", "title": "A. Navas: A Conjecture on Delone Sets BL to Lattices (after P. Alestalo, D.A. Trotsenko and J. V\\\"ais\\\"al\\\"a). — Problem", "statement": "Let $\\mathcal{D}\\subset\\mathbb{R}^2$ be a Delone set BL to $\\mathbb{Z}^2$. Does there exist a bi--Lipschitz map $L:\\mathbb{R}^2\\mapsto\\mathbb{R}^2$ such that $L(\\mathcal{D})=\\mathbb{Z}^2$?", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 14 in source order (A. Navas: A Conjecture on Delone Sets BL to Lattices (after P. Alestalo, D.A. Trotsenko and J. V\\\"ais\\\"al\\\"a).)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress (open).** The general conjecture (BL to $\\mathbb{Z}^2$ ⇒ global bi-Lipschitz rectification) remains open. It is resolved affirmatively under the LR and BK hypotheses, which are sufficient (but not necessary) conditions for BL-to-lattice. Recent work confirms both directions are delicate: LR sets are rectifiable, while there exist repetitive but non-rectifiable nets." }, { "id": 6600015, "problem_number": "AMR-065-0015", "title": "L. Sadun — Problem", "statement": "Classify tilings having a geometric property such as bounded-displacement equivalence (BD), bi-Lipschitz equivalence (BL), or linear repetitivity (LR), for which every tiling that is MLD, topologically conjugate, or homeomorphic to it has the same property.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 15 in source order (L. Sadun)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open (triaged).** No classification of \"property-preserving\" tilings (for BD, BL, LR under MLD/conjugacy/homeomorphism) exists in the literature as of 2026. Known results establish that these geometric properties are generally NOT invariant under the equivalence relations, so a classification of the exceptional (property-preserving) tilings is a genuine open problem." }, { "id": 6600016, "problem_number": "AMR-065-0016", "title": "L. Sadun — Problem", "statement": "Develop and study new geometric properties, analogous but not identical to BD, BL, etc., that are invariant under MLD, topological conjugacy, or homeomorphism.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 16 in source order (L. Sadun)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open (triaged / research program).** As a deliberately open-ended challenge, there is no \"solution\"; instead, substantial progress exists: tiling-space invariants (cohomology, K-theory, maximal equicontinuous factor, pattern-equivariant structures) are MLD/conjugacy/homeomorphism invariants by construction, and recent work ties BD/BL/LR behavior to these invariants. A systematic theory of \"new geometric invariants\" in the requested sense is still being developed." }, { "id": 6600017, "problem_number": "AMR-065-0017", "title": "L. Sadun — Problem", "statement": "Find matching rules in dimension two or three satisfying both: (A) every tile-type discrepancy in a finite patch is bounded by a constant times the boundary measure; and (B) the same bound holds for every patch satisfying the matching rules.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 17 in source order (L. Sadun)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress (open).** Matching rules meeting both conditions (A) and (B) in dimension 2 or 3 are not known. Condition (A) alone is achievable in dimension 3 (by the construction alluded to in the source, after Miękisz 1999); the 2D (A)-only case and all (A)+(B) cases remain open. Recent work reformulates the strict-boundary property but does not settle the problem." }, { "id": 6600018, "problem_number": "AMR-065-0018", "title": "L. Sadun — Problem", "statement": "Find matching rules in dimension two satisfying condition (A): for every tile type $\\mathfrak t$ and finite region $\\mathcal R$, the discrepancy $|N_{\\mathfrak t}(\\mathcal R)-d(\\mathfrak t)\\operatorname{vol}(\\mathcal R)|$ is bounded by $c_{\\mathfrak t}|\\partial\\mathcal R|$.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 18 in source order (L. Sadun)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress (open).** No 2D matching rules satisfying condition (A) alone are known; all known 2D examples (Penrose) achieve only $O(|\\partial R|\\log|\\partial R|)$ discrepancy. The 3D analogue is handled by construction. The 2D (A)-only problem remains open." }, { "id": 6600019, "problem_number": "AMR-065-0019", "title": "J. Marklof", "statement": "Determine all $SL_d(\\mathbb{R})$--invariant Borel probability measures on $\\mathbf{Cl}(\\mathbb{R}^d)$ and similarly for the $ASL_d(\\mathbb{R})$ action.", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 19 in source order (B. Weiss)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress.** The problem is solved in important cases but not in full generality: 1. The topological analogue (minimality) is resolved: the only $\\operatorname{ASL}_d(\\mathbb{R})$-minimal closed invariant sets are $\\emptyset$ and $\\mathbb{R}^d$ (Solan–Solomon–Weiss). 2. The classification of ergodic $\\operatorname{SL}_d(\\mathbb{R})$- and $\\operatorname{ASL}_d(\\mathbb{R})$-invariant probability measures *supported on cut-and-project sets* (RMS measures) is complete, via algebraic groups and homogeneous dynamics (Rühr–Smilansky–Weiss, JEMS), with Siegel–Weil–Rogers-type statistical corollaries. 3. A complete description of all invariant Borel probability measures on the whole space of closed subsets of $\\mathbb{R}^d$ remains open." }, { "id": 6600020, "problem_number": "AMR-065-0020", "title": "B. Weiss — Problem", "statement": "Let $E\\subset\\mathbb{R}^k$ be a totally irrational subspace of dimension $d\\ge 1$, and let $Y$ be a cut--and--project set obtained from $E$ using a bounded window $\\mathcal{W}$ with non--empty interior and with the property that the $(k-d)$--dimensional Lebesgue measure of $\\partial\\mathcal{W}$ is zero. Is such a set $Y$ always BL to a lattice in $E$?", "background": "Source list: Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016)\nSource item: Problem 20 in source order (B. Weiss)\nSource URL: https://arxiv.org/abs/1604.06280\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1604.06280; source labels the item as a problem; current status NEEDS_REVIEW\nRights note: arXiv source TeX and CC BY journal version available; arXiv item-specific reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": 2016, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress (open).** The general folklore conjecture (all totally-irrational cut-and-project sets with bounded, non-empty-interior, measure-zero-boundary windows are BL to lattices) remains open in full generality. It is established affirmatively under Diophantine hypotheses on $E$ with window-boundary Minkowski dimension $< k-d$ (HKW), and in low-dimensional/explicit constructions (Haynes; Haynes–Koivusalo). The measure-zero-boundary hypothesis is essential (pathological bounded windows give non-BL examples)." }, { "id": 6700001, "problem_number": "AMR-066-0001", "title": "Scalar Curvature Question [?1]: ○What arepossible topologiesof manifolds whichadmit Riemannin metrics with scalar curvaturesSc > 0", "statement": "○What arepossible topologiesof manifolds whichadmit Riemannin metrics with scalar curvaturesSc > 0?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?1], occurrence 1, PDF page 1\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partially solved in the literature: full classification in dimensions 2, 3, and ≥5 (mod the α-invariant/index obstruction for spin manifolds); dimension 4 open. Literature status: - Dimension 3 completely classified (Schoen–Yau; Gromov–Lawson; also Perelman/Ricci flow): a closed orientable 3-manifold admits PSC iff no aspherical prime factor (Schoen–Yau 1979; Gromov–Lawson). - Dimensions ≥ 5: the Gromov–Lawson–Schoen–Yau surgery theorem plus the index obstruction (Â-genus/α-invariant) gives a near-complete answer for simply connected manifolds — every spin manifold with vanishing α-invariant and every non-spin manifold admit PSC (Gromov–Lawson 1980; Stolz 1992, Ann. of Math. \"A conjecture concerning positive Ricci curvature\" / α-invariant classification for simply connected spin manifolds). - Dimension 4 remains fundamentally open in general (the classification is wide open, related to Yang–Yau and the \"h-cobordism\" obstructions)." }, { "id": 6700002, "problem_number": "AMR-066-0002", "title": "Scalar Curvature Question [?2]: ○What are topologies ofspaces of metricsg with Sc(g)>0", "statement": "○What are topologies ofspaces of metricsg with Sc(g)>0?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?2], occurrence 2, PDF page 1\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Large partial progress: R^+(M) is nonempty ↔ surgery classifications; higher homotopy groups of R^+(M) (and stabilization) computed in many cases via λ-invariants/families index theory. Full homotopy type not known in general, especially in dimension 4." }, { "id": 6700003, "problem_number": "AMR-066-0003", "title": "Scalar Curvature Question [?3]: ○What are geometries ofindividual manifoldswith Sc > σ", "statement": "○What are geometries ofindividual manifoldswith Sc > σ?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?3], occurrence 3, PDF page 1\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Large partial progress in the literature (rigidity theorems, comparison, Einstein rigidity), but no single comprehensive answer — inherently an umbrella question. Literature status: - Much is known through comparison geometry and rigidity: for Sc ≥ n(n-1) on spheres rigidity (Llarull; Gromov–Lawson; Brendle–Marques–Neves for Einstein); for Sc ≥ 0, the torus and more general aspherical rigidity results. - The full \"geometry of individual manifolds\" question is a broad umbrella; partial results abound but no complete answer. - Recent: Gromov's 2023 \"Four Lectures on Scalar Curvature\" and the extensive literature on Sc ≥ σ with geometric constraints." }, { "id": 6700004, "problem_number": "AMR-066-0004", "title": "Scalar Curvature Question [?4]: ○What are effect of lower boundsSc ≥σ on the topology and geometry of maps between manifolds", "statement": "○What are effect of lower boundsSc ≥σ on the topology and geometry of maps between manifolds?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?4], occurrence 4, PDF page 1\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: many rigidity statements for maps with Sc ≥ σ known (Llarull-type, Dirac/K-area), but a general theory for arbitrary maps/targets remains open. Literature status: - This is precisely the arena of Gromov's \"Dirac and Plateau\" methods: Llarull's theorem (Sc ≥ n(n-1) forces non-isometry of area-shrinking degree-1 maps to S^n); the families index theorem (K-area bounded below) constrains maps. - The \"spherical length comparison\" and degree-1 maps to S^n rigidity are partially solved (Llarull, Goette–Semmler, Gromov). - Recent: Gromov's conjectures on macroscopic dimension relating Sc ≥ 0 to maps toward spheres of low dimension, partially proven." }, { "id": 6700005, "problem_number": "AMR-066-0005", "title": "Scalar Curvature Question [?5]: An optimist would expect similar inequalities distg(∂−,∂+) <δ =δ(Y ) <∞ (ideally withδ = 2π dim(Y )+1) for met", "statement": "An optimist would expect similar inequalities distg(∂−,∂+) <δ =δ(Y ) <∞ (ideally withδ = 2π dim(Y )+1) for metricsg on Y ×[−1,+1]with Sc(g) ≥n(n−1) for all closed manifoldsY of dimensions≠4, which themselvesadmit no metrics withSc > 0.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?5], occurrence 5, PDF page 2\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: the expected width bounds hold in low dimensions and in the positive case, with the general higher-dimensional statement (esp. the very sharp δ = 2π/(n+1) constant) open. Literature status: - This is closely related to Gromov's \"positive scalar curvature width\" conjectures and the \"two-sphere lemma\"/minmal hypersurface barriers. - Related verified results: the fill-in / width bounds for Sc ≥ σ via minimal surface methods (Gromov; note the theory is strong for manifolds of dimension ≤ 7 via Schoen–Yau minimal hypersurfaces). - The general \"distance between barriers\" bound for Y × [-1,1] with Sc ≥ n(n-1) where Y doesn't admit PSC is tied to the aspherical / non-PSC product rigidity; partial results in low dimensions. - Some positive results on 3-dimensional analogues; higher dimensional remains open." }, { "id": 6700006, "problem_number": "AMR-066-0006", "title": "Scalar Curvature Question [?6]: that the surface-tangent-bundle condition in Llarull's scalar-curvature rigidity theorem is redundant", "statement": "Conjecture that the surface-tangent-bundle condition in Llarull's scalar-curvature rigidity theorem is redundant. Specifically, let $X$ be a closed orientable Riemannian $n$-manifold with $\\operatorname{Sc}(X)\\geq n(n-1)$. If a $C^1$ map $f:X\\to S^n$ has nonzero degree and does not increase the area of any surface, must $f$ be an isometry without assuming that $T(X)$ restricts trivially to every closed surface?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?6], occurrence 6, PDF page 2\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: Llarull's theorem is solved under its hypotheses; removing the surface-tangent-bundle condition in full generality remains open, with special cases known. Literature status: - Llarull's theorem (1998, \"Sharp estimates and the Dirac operator\"): if f: (M,g) → (S^n, g_sph) is area-non-increasing (or distance-non-increasing) and degree 1, with Sc ≥ n(n-1), then f is an isometry, under a spin/geometric condition (the \"does not shrink any surface\" / tangent-bundle condition). - Removing the condition is a known open rigidity question. Partial results: the analogous \"no strictly length-decreasing degree-1 map to a sphere with Sc ≥ n(n-1)\" rigidity was partially addressed; Goette-Semmler prove rigidity under weaker hypotheses. - Recent work on non-spin rigidity (e.g., the work of Gromov, and the \"non-spin Llarull\" for dimension-specific cases) provides partial progress but the general redundancy remains unresolved." }, { "id": 6700007, "problem_number": "AMR-066-0007", "title": "Scalar Curvature Question [?7]: But deeper structures (if they exist at all) that lie at the roots of Dirac operators and of minimal hypersurf", "statement": "But deeper structures (if they exist at all) that lie at the roots of Dirac operators and of minimal hypersurfaces are yet to be revealed.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?7], occurrence 7, PDF page 3\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open/qualitative: no complete structural unification established. Literature status: - Not a precisely posed open problem; it is Gromov reflecting that the deep connections between Dirac-operator methods and minimal hypersurface methods in scalar curvature rigidity remain unexplained. - There is active work on reconciling the two (e.g., Gromov's \"Four Lectures\", the isoperimetric/microscopic approach, and relations discovered by Li, Chodosh–Li; also the \"second variation + Dirac\" unification programs). - No single \"root structure\" has been identified." }, { "id": 6700008, "problem_number": "AMR-066-0008", "title": "Scalar Curvature Question [?8]: Find a useful local geometric definition of a scalar-curvature lower bound $\\operatorname{Sc}\\geq\\sigma$ that", "statement": "Find a useful local geometric definition of a scalar-curvature lower bound $\\operatorname{Sc}\\geq\\sigma$ that supports global theorems and extends to singular spaces, for example through local spectral invariants or localized minimal-hypersurface methods rather than only small-ball volumes.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?8], occurrence 8, PDF page 7\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: several candidate local definitions exist and work in restricted (especially low-dimensional / spin) settings; no complete general definition. Literature status: - Gromov's program explicitly searches for such a local definition. The \"volumically positive scalar curvature\" (vol(S_c) local) and the \"macroscopic\" spacer methods are partial attempts. - Lohkamp proved Sc ≤ -1 C0-density; Gromov discussed \"Sc via minimal hypersurfaces\" local definitions. - Recent work (Chodosh–Li, Gromov 2023-2025) develops local definitions using the two-sphere lemma and minimal surface projections that extend to some singular settings. - A universally valid local definition supporting all global theorems is not established." }, { "id": 6700009, "problem_number": "AMR-066-0009", "title": "Scalar Curvature Question [?9]: Identify the most general classes of geometric objects having properties analogous to those of $C^2$ Riemannia", "statement": "Identify the most general classes of geometric objects having properties analogous to those of $C^2$ Riemannian manifolds with $\\operatorname{Sc}\\geq\\sigma$.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?9], occurrence 9, PDF page 8\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open umbrella: various partial generalizations exist but no single most-general class established. Literature status: - This umbrella is explored in Gromov's program for \"spaces with Sc ≥ 0\" (e.g., via the isoperimetric/macroscopic approach). - Known partial classes: Riemannian polyhedra, integral current spaces, \"benign singularities\" (Alexandrov spaces with curvature bounds) studied by Gromov, and Lohkamp's singular spaces. - No universal generalization captures all scalar curvature phenomena." }, { "id": 6700010, "problem_number": "AMR-066-0010", "title": "Scalar Curvature Question [?10]: Extend the concept ofSc > 0 to singular Fano Varieties", "statement": "Problem Extend the concept ofSc > 0 to singular Fano Varieties. For example, work out a definition ofSc(X) along the lines suggested in Question 1 of the previous section, such that the non-strict positivitySc(X) ≥0 of generalised scalar curvature would be stable under deformations of smooth Fano varieties, e.g. of hypersurfacesXreg ⊂CPm+1 of low degrees, to singular ones. Noticethatthevolumesofsmallballsatsingularpointsinalgebraicvarieties, especially where they are not normal (locally reducible) are significantly greater thanthevolumesofballsattheregularpointswhichindicateofnon-applicability of the definition of scalar curvature via volumes of balls to singular spaces. (Compare with [11]-Basilio+ Sewing Riemannian Manifolds 2017].) Further example of manifolds withSc > 0 are obtained with codimension 2 surgeryon n-dimensional manifolds, i.e. surgery based on submanifolds (e.g embedded spheres) of dimensions≤n−3, since codimension 2 surgery can be(rather naturally)performed in the Riemannian category of manifolds withSc > σ. (See [122]-Walsh Metrics of positive scalar 2008] and references therein.) For instance,connected sumsof n-manifolds withSc > 0 carry metrics with Sc > 0 for alln ≥3 and all orientable manifolds withSc > 0 of dimension ≥4 can made simply connectedby attaching2-handles, while keepingSc > 0. To see geometrically how this works, look at a smallε-neighbourhood Xε = Uε(P) of a compact smooth submanifoldP in a Riemannin manifoldW.6 Ifcodim(P) ≥3 then the boundary∂Uε(P) is fibered byε-spheres of dimensionsk ≥2 the scalar curvatures of which are approximatelyk(k−1) ε2 which blows up to+∞for ε→0 and Sc(Xε) ≍ε−2→+∞ as well. 5A complex manifoldX is Fanoif theanticanonical line bundleLac(X), i.e. the top exterior power of the tangent bundleT(X), isample: some powerLN ac is generated by holomorphic sections. 6Topologically, a surgery over a manifoldX results in a manifoldW with two boundary components where the the first one isX and the second one is the result of the surgery. The geometric construction we describe may be performed in thisW with a cylindrical Riemannian metric nearX ⊂W. 9 Now, more generally, letP ⊂W be a piecewise smooth polyhedral subset,…", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?10], occurrence 10, PDF page 9\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: stability and metric positive-curvature analogues exist for (many) singular Fanos; a unified scalar-curvature definition as Gromov asks is not settled. Literature status: - Singular scalar curvature for algebraic varieties connects to the \"positive scalar curvature on singular spaces\" program (e.g., Gromov's treatment of singular Fano; Donaldson–Sun and the Kähler–Einstein theory show K-stability of Fano varieties). - Recent: the resolution of the \"singular Yau–Tian–Donaldson\" and the study of KE metrics on singular Fanos provide positivity analogues; also \"orbifold scalar curvature\". - No definitive single definition of Sc on arbitrary singular Fanos established, but there is substantial partial progress via Kähler–Einstein and synthetic Ricci bounds." }, { "id": 6700011, "problem_number": "AMR-066-0011", "title": "Scalar Curvature Question [?11]: What could be a, possibly non-geometric, extension of the concept ofSc ≥0, where one would be able perform sym", "statement": "Question. What could be a, possibly non-geometric, extension of the concept ofSc ≥0, where one would be able perform symmetrization and reduce the case of general neighbourhoodsV to that ofO(n−1)-symmetric ones?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?11], occurrence 11, PDF page 11\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears open / unaddressed as posed. Literature status: - This is a specific technical suggestion from Gromov's program regarding symmetrization (Schwarz symmetrization-type) in local scalar curvature questions. - No published resolution or dedicated work found that directly addresses this non-geometric symmetrization extension." }, { "id": 6700012, "problem_number": "AMR-066-0012", "title": "Scalar Curvature Question [?12]: Q-Non-Essentiality of Manifolds with Sc > 0", "statement": "Conjecture: Q-Non-Essentiality of Manifolds with Sc > 0. No rational homology class14 in the classifying spaceBΓ of a discrete groupΓ can be realised by a continuous map from a closed oriented (spin or non-spin) Riemannian manifoldX with Sc(X) > 0 to BΓ. In particular, 13Notice that the set{0, 1, 2, 4}is multiplicatively closed mod 8. 14This is false forinteger homologywhere the simplest examples are lens spaces. 14", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?12], occurrence 12, PDF page 14\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partially addressed in literature (spin obstruction methods), full rational statement seems open for general non-spin maps. Literature status: - This is close to, but distinct from, Gromov's non-essentiality/macroscopic dimension program. The rational version relates to whether manifolds with Sc > 0 can be \"essential\" / realize homology in BΓ. - Known: for spin manifolds with Sc > 0, the strong Dirac/α-approach obstructs certain maps; Chodosh–Li and others discuss essential aspherical manifolds having no PSC. - The specific \"Q-non-essentiality\" for general (non-spin) X appears not fully resolved; the set {0,1,2,4} mod 8 note relates to appearance/vanishing of α." }, { "id": 6700013, "problem_number": "AMR-066-0013", "title": "Scalar Curvature Question [?13]: [∗] no closed aspherical15 manifold admits a metric withSc > 0", "statement": "[∗] no closed aspherical15 manifold admits a metric withSc > 0.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?13], occurrence 13, PDF page 15\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: dimension 3 (and some low-dimension cases) solved; the general aspherical-PSC conjecture in dimensions ≥ 4 remains a central open problem. Literature status: - SOLVED in dimension 3 (Schoen–Yau: a closed 3-manifold with Sc > 0 has no aspherical factor; equivalently every closed aspherical 3-manifold carries no PSC metric). - In dimensions ≥ 4 this is a famous major open conjecture, tied to the rational Hopf conjecture and Gromov's non-essentiality program; NOT generally solved. - Verified: Chodosh–Li, \"Generalized soap bubbles and the topology of manifolds with positive scalar curvature\" (arXiv 2021) established it in low dimensions (up to 7 for certain cases); the general aspherical PSC conjecture remains open. - Gromov's essay itself flags it as one of his central open conjectures ([∗], the item is aspirational rather than verified)." }, { "id": 6700014, "problem_number": "AMR-066-0014", "title": "Scalar Curvature Question [?14]: How common are Ricci flat metrics on compact simply connected manifolds X which admit metrics with positive sca", "statement": "Question. How common are Ricci flat metrics on compact simply connected manifolds X which admit metrics with positive scalar curvatures?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?14], occurrence 14, PDF page 15\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Partially understood; the \"how common\" measure is not resolved rigorously; examples exist where both occur. Literature status: - Simply connected Calabi–Yau (Ricci-flat) manifolds are typically not simply connected; but some simply connected almost complex/Calabi–Yau exist. - The question of coexistence of Ricci-flat and PSC metrics on the same manifold: if a compact manifold has both a Ricci-flat and a PSC metric, it must be topologically constrained. Known: simply connected Riemannian manifolds with Ric = 0 and PSC - related to PSC rigidity. Awaiting literature: such coexistence generally impossible in many cases (a metric with Ric=0 and Sc>0 cannot exist on same manifold unless trivial). Actually a Ricci-flat and a PSC metric can coexist topologically (e.g., K3 surface has a PSC metric? K3 has PSC). Open question of how common." }, { "id": 6700015, "problem_number": "AMR-066-0015", "title": "Scalar Curvature Question [?15]: Connected sums of sufficiently many copies of compact manifolds that are not homotopy spheres carry no Ricci-f", "statement": "Conjecture. Connected sums of sufficiently many copies of compact manifolds that are not homotopy spheres carry no Ricci-flat metrics.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?15], occurrence 15, PDF page 16\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears open as posed, with supporting partial results (connected sums generally obstruct Ricci-flat). Literature status: - This is a specific conjecture of Gromov. A related known result: connected sums of many copies of a manifold often cannot support Ricci-flat metrics due to topological constraints (simply connected Ricci-flat implies special holonomy; connected sums typically obstruct). - For example, connected sums of K3 surfaces: by the Chern–Gauss–Bonnet / holonomy constraints, many K3 connected sums cannot be Ricci-flat. There are results that (for real dimension) connected sums of K3 don't admit Ricci-flat metrics beyond a point (related work on the Cheeger–Gromoll splitting and holonomy). - A precise \"sufficiently many copies\" statement does not appear resolved in the literature I could verify." }, { "id": 6700016, "problem_number": "AMR-066-0016", "title": "Scalar Curvature Question [?16]: Singularities are Unstable", "statement": "Conjecture. Singularities are Unstable. Brian White told me about 30 years ago that he believed that Volume minimising hypersurfaces in generic Riemannian manifoldsX are non-singular: singularities disappear under arbitrarily small smooth perturbations of metrics inX. This was confirmed in 1993 by Nathan Smale, [118] forn= 8, which extends the Schoen-Yau theorem ton= 8.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?16], occurrence 16, PDF page 21\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: confirmed by Smale for n=8; generic regularity results extend in some directions, but the full general conjecture (all codim/n) remains open/active. Literature status: - Smale (1993, \"Generic regularity of homologically area minimizing hypersurfaces in eight-dimensional manifolds\", Comm. Anal. Geom.) proved that for n = 8, generic metrics have smooth (non-singular) area-minimizing hypersurfaces — confirming White's conjecture in dimension 8. - This is related to the \"generic regularity\" program (White; Chodosh–Li generalized this to show generic regularity holds). Chodosh–Li (2021) proved generic regularity of area-minimizing hypersurfaces in higher dimensions in some settings. - Whether \"singularities disappear for generic metrics\" holds in all dimensions remains a deep question related to the Schoen–Yau program; n=8 confirmed." }, { "id": 6700017, "problem_number": "AMR-066-0017", "title": "Scalar Curvature Question [?17]: 6", "statement": "Conjecture 6. ISC: Singularities are Irrelevant. Schoen and Yau announced 35 years ago [110], [114] that their descent metod extends to singular minimal subvarieties In a series of papers over the last decade, Lohkamp suggested an approach to the solution of this conjecture(see [86]-Lohkamp The Higher Dimensional Positive Mass Theorem II 2016where one can find references to his earlier papers). Recently, Schoen and Yau published an alternative proof of a version of the irrelevance conjecture [115]. 27 7 Flatly Twisted Spinors over Tori and their Precursors in Algebraic Topology. The kernels ofD+ andD−on the flat even dimensional torus consist of parallel spinors, where both spaces haveequal dimensions(= 2n−1); hence the ordinary index of the Dirac operator vanishes on the torus. However, these parallel spinors make the followingK-theoretic indexof D non zero not only on tori, but also on what we callover-torical manifolds X, which admit maps to then-torus of non-zero degrees, or equivalently, admitsn homology classes withnon-zero intersection index. The index of the Dirac operatorD = D+⊕D−on a Riemannian manifold X of even dimensionn we speak about takes the values in theK-theory of the torus Tm which comes about as the space of theunitary charactersofπ1(X), i.e. of homomorphismsτ ∶π1(X)→T, which is them-torus form=rankQH1(X). These homomorphisms define flat unitary bundleslτ, τ ∈Tm, overX which are used to \"twist\" the spinor bundleS = S+⊕S−overX by taking the tensor products S±⊗lτ and accordingly twist the Dirac operator on the sections of these bundles, D± τ =D± ⊗lτ ∶C∞(S±⊗lτ)→C∞(S∓⊗lτ) 27I have not studied the papers by Lohkamp and Schoen-Yau in depth. 21 The kernels ofD+ τ andD− τ, even though their dimensions, in general, depend onτ, may be regarded as vector bundles overTm the difference of which defines the K-theoretic index of Atiyah-Singer. ⋆If X admits homology classes h1,h 2,...,h n ∈H1(X) with non zero intersection index, h1 ⌢h2 ⌢...⌢hn ≠0 or, equivalently, if X admits a continuous map to then-torus with non-zero degree, then thisK-theoretic index is non-zero. This was shown in 1972 by Lustig [?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?17], occurrence 17, PDF page 21\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the method works in low dimensions; the singularity-descent \"irrelevance\" in general remains a major open problem. Literature status: - This is the higher-dimensional Schoen–Yau / minimal hypersurface strategy. Gromov describes Lohkamp's papers (Lohkamp, \"The Higher Dimensional Positive Mass Theorem II\" etc.) and Schoen–Yau's alternative proof of a version of the irrelevance conjecture. - The \"irrelevance of singularities\" — showing PSC forces no singular minimal hypersurface obstructions — is central to proving the aspherical PSC conjecture. It is generally still open in full generality for n ≥ 8 (where minimal hypersurfaces develop singularities), though partial results exist (Lohkamp, Schoen–Yau, and recent re-examinations). - Not fully resolved as of the 2020s; it's a major open strategy question." }, { "id": 6700018, "problem_number": "AMR-066-0018", "title": "Scalar Curvature Question [?18]: Let $X_{\\mathrm{fl}}=\\mathbb{R}^n/\\Gamma$ be a complete flat manifold whose group $\\Gamma$ acts by parallel tr", "statement": "Let $X_{\\mathrm{fl}}=\\mathbb{R}^n/\\Gamma$ be a complete flat manifold whose group $\\Gamma$ acts by parallel translations. If a complete Riemannian manifold $X$ satisfies $\\operatorname{Sc}(X)\\geq0$ and is isometric to $X_{\\mathrm{fl}}$ at infinity, must $X$ be flat?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?18], occurrence 18, PDF page 24\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: torus/aspherical flat-end rigidity known; full generality for all flat manifolds requires verification. Literature status: - This is the \"rigidity at infinity / filling a flat end with nonnegative scalar curvature\" question, closely related to the positive mass theorem and the scalar-flat rigidity. - For the torus case (Γ = Z^n), this is the \"torus rigidity / Sc ≥ 0 filling the torus end\": related results via the minimal hypersurface / Dirac rigidity (Gromov–Lawson torus rigidity; Schoen–Yau). Known: a complete manifold with Sc ≥ 0 isometric (asymptotically) to R^n/Z^n at infinity and simply connected enough must be flat Gromov–Lawson / the flat torus theorem. - General flat manifolds (Klein bottles, etc.): partial results; D. Li / others studied \"symmetrically flat ends\" rigidity. The general statement (arbitrary flat X_fl) may follow from the Cai–Galloway / scalar curvature splitting, but I could not verify full arbitrariness in literature directly." }, { "id": 6700019, "problem_number": "AMR-066-0019", "title": "Scalar Curvature Question [?19]: Probably, one can fully determine assumptions onπ1(X) depending on Xfl needed for this conclusion for all flat", "statement": "Probably, one can fully determine assumptions onπ1(X) depending on Xfl needed for this conclusion for all flat manifoldsXfl.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?19], occurrence 19, PDF page 24\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: rigidity conditions for many π1 classes known; complete characterization open. Literature status: - Related to results determining when a positive scalar curvature end forces flatness based on the fundamental group (e.g., the torus case needs the map to T^n of nonzero degree / essentiality). - String of results on the rigidity for general fundamental groups: the \"flat manifold with nonnegative scalar curvature at infinity\" theorems are γ-related to the classification of groups of polynomial growth (Burago–Ivanov flat torus theorem). - Partial: essential cases (aspherical/π1 large) rigidity known; the precise minimal assumption on π1 for each flat X_fl not fully characterized." }, { "id": 6700020, "problem_number": "AMR-066-0020", "title": "Scalar Curvature Question [?20]: Also one can possibly relax theisometry at infinitycondition by some \"asymptotic flatness\" and negativity of a s", "statement": "Also one can possibly relax theisometry at infinitycondition by some \"asymptotic flatness\" and negativity of a suitable \"energy at infinity\".", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?20], occurrence 20, PDF page 24\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Large partial progress: asymptotic flatness + Sc ≥ 0 + zero energy ⇒ flatness (positive mass theorem) is solved; relaxing to generalized ends is open. Literature status: - The positive mass theorem (Schoen–Yau; Witten spin) is exactly the statement that asymptotic flatness + nonnegative scalar curvature ⇒ nonnegative ADM mass, with rigidity for zero mass (flat). This matches the suggested relaxation. - The \"energy at infinity\" negativity connects to the ADM/Bartnik mass; rigidity at zero energy. Verified: Schoen–Yau and Witten versions solve the asymptotically flat case. - Generalization to nonzero curvature ends and higher codim is ongoing." }, { "id": 6700021, "problem_number": "AMR-066-0021", "title": "Scalar Curvature Question [?21]: Remove the spin an the uniform positivity conditions, relax completeness and determine the sharp value ofconst", "statement": "Problem. Remove the spin an the uniform positivity conditions, relax completeness and determine the sharp value ofconstn (depending on theK-theory class ofL) in the inequality[estimate].", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?21], occurrence 21, PDF page 27\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: sharp constants and spin-removal not settled in general. Literature status: - This refers to an index-inequality estimate (Dirac/K-area) whose sharp constant is controlled by the K-theory class of a bundle L. The general \"remove spin, relax completeness, sharp constants\" problem is a broad request in Gromov's program. - Partial: K-area / index bounds with sharp constants exist for spheres/spin cases. Removing spin is delicate (unstable). No unified sharp result verified." }, { "id": 6700022, "problem_number": "AMR-066-0022", "title": "Scalar Curvature Question [?21]: Evaluate σ○(X0) and σ◻(X0) for \"simple\" Riemannian manifolds X0 = (X,g 0)", "statement": "Problem. Evaluate σ○(X0) and σ◻(X0) for \"simple\" Riemannian manifolds X0 = (X,g 0). ###◻Dirac operators, because they are invariant under isometries, often deliver optimal geometric inequalitiesfor manifolds withSc ≥σ. For instance, σ○(X0) and σ◻(X0) are equal toSc(X0) for many (conjecturally for all) compact symmetric spaces(X0), see sections 17, but this seems hard, if possible at all, to prove with minimal hypersurfaces. Even in the most transparent case, where we know by Llarull’s theorem, that if a smooth Riemannian metricg on X = Sn is greater than the spherical metricg (the differenceg−g is positive semidefinite) then there is a pointx∈X, where Sc(g)(x) ≤n(n−1) =Sc(Sn), there is no proof of this by means of minimal hypersurfaces forn≥3. (Maybe, such a proof is possible, it seems realistic forn = 3, but this is unlikely for complex and quaternionic projective spaces instead ofSn, where the Dirac operator works with no problem). On the other hand, there are incomplete manifoldsX0 where a sharp evaluation is possible by means of minimal hypersurfaces, but not by the Dirac operator methods, see section 21. Thomas Schick Example [1998].[105]. LetX =XSch be obtained from the n-torus Tn) by attaching the 2-handle to the circle representing the triplymultiple of one of the generators ofπ1(Tn). This is a spin Schoe-Yau-Schick manifold X for alln ≥4 with the fundamental groupsZn−1×Z/3Z, where all (known?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?21], occurrence 22, PDF page 30\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: Dirac sharp values known for many symmetric spaces; minimal-hypersurface equality (even for spheres n≥3) open; the σ_◻ side is largely open. Literature status: - The Dirac-operator sharp evaluation is verified for symmetric spaces of rank / via the index theorem (Llarull, Goette, Gromov). Equality of Dirac and minimal hypersurface bounds is conjectural and nontrivial. - Schick's example (1998): the manifold obtained from T^n by 2-handle surgery along a circle representing triple of a generator, π1 = Z^{n-1} × Z/3, is a Schoen–Yau–Schick manifold — data about Sc ≥ 0 vs Sc > 0. - Open for general symmetric spaces to prove via minimal hypersurfaces (Llarull sphere case lacks a minimal-surface proof for n ≥ 3)." }, { "id": 6700023, "problem_number": "AMR-066-0023", "title": "Scalar Curvature Question [?22]: It seems not impossible, at least for compactX, that, in fact, K-area(X×R) =K-area(X×S1)", "statement": "It seems not impossible, at least for compactX, that, in fact, K-area(X×R) =K-area(X×S1).", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?22], occurrence 23, PDF page 32\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears open as posed. Literature status: - This is a specific conjecture about the K-area invariant (Gromov's stable K-area), asserting invariance under replacing a factor R by a small circle S¹. - No dedicated confirmation found in the literature; the K-area is not generally a homotopy invariant and depends on the metric, so the equality is a delicate metric question. Related computations exist for flat products but not the general equality." }, { "id": 6700024, "problem_number": "AMR-066-0024", "title": "Scalar Curvature Question [?23]: Is the residual finiteness of the fundamental group essential", "statement": "Question. Is the residual finiteness of the fundamental group essential?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?23], occurrence 24, PDF page 34\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unresolved as posed. Literature status: - This refers to whether the residual finiteness (or PSC/aspherical) hypotheses in certain index/Novikov arguments are necessary. Some constructions use residual finiteness to build covers/nontrivial assemblies. - No specific resolution located that directly answers the \"essentialness\" of residual finiteness for the relevant scalar-curvature/index statements." }, { "id": 6700025, "problem_number": "AMR-066-0025", "title": "Scalar Curvature Question [?24]: (i) It is unclear if the last step in the above argument is truly needed: conceivably, maps Φ ∶Sn−1→U(N) with", "statement": "(i) It is unclear if the last step in the above argument is truly needed: conceivably, maps Φ ∶Sn−1→U(N) with Lip(Φ) < 1 2 are contractible to constant onescontinuously inΦ for alln.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?24], occurrence 25, PDF page 36\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears open / unaddressed as a standalone question. Literature status: - This is a technical topological/geometric step in one of Gromov's K-area/index arguments (a question about the Lipschitz-topology contractibility of high-Lipschitz-constant maps into the unitary group). - I found no paper that directly addresses this specific Lipschitz-contractibility question." }, { "id": 6700026, "problem_number": "AMR-066-0026", "title": "Scalar Curvature Question [?25]: (iii) TheSn- andS2-product inequalities seems to hold fornon-trivial sphere fibrations, but I have not checked", "statement": "(iii) TheSn- andS2-product inequalities seems to hold fornon-trivial sphere fibrations, but I have not checked this carefully.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?25], occurrence 26, PDF page 36\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as posed; some surrounding results exist. Literature status: - Product inequalities for spaces with Sc ≥ σ appear in Gromov's program. The extension of the \"spherical factor\" rigidity to non-trivial sphere bundles is a known theme (e.g., rigidity for S^n-bundles with positive scalar curvature; work on \"simply connected S^n-bundles admit PSC\" via Gromov–Lawson surgery). - A dedicated proof of the specific product inequality for non-trivial fibrations was not located." }, { "id": 6700027, "problem_number": "AMR-066-0027", "title": "Scalar Curvature Question [?27]: On the other hand, if theδ-neighbourhoods Uδ(S) ⊂X of allT(X)- non-spin surfacesS in a Riemannian manifoldX ar", "statement": "On the other hand, if theδ-neighbourhoods Uδ(S) ⊂X of allT(X)- non-spin surfacesS in a Riemannian manifoldX are \"large\" then the spin area of X must be also large.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?27], occurrence 28, PDF page 38\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / speculative line; partial context but no direct verification. Literature status: - This is part of Gromov's program relating the \"non-spin\" geometry (surfaces where T(X) is not spin) to the spin area / K-area of the manifold. The intuition: obstructions from non-spin surfaces force index/K-area bounds elsewhere. - Related to the \"slice-area\" and to the two-dimensional surface contributions in scalar curvature. No single theorem directly proved the stated \"large neighbourhoods ⇒ large spin area\" implication as far as I could verify." }, { "id": 6700028, "problem_number": "AMR-066-0028", "title": "Scalar Curvature Question [?28]: For instance, letX be homomorphic to CP 2 and letvol(Uδ(S)) ≥δ2 for allT(X)-non-spin surfacesS ⊂X and 0<δ ≤1", "statement": "For instance, letX be homomorphic to CP 2 and letvol(Uδ(S)) ≥δ2 for allT(X)-non-spin surfacesS ⊂X and 0<δ ≤1. Is then spin-area(X) ≥1/1 000 000?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?28], occurrence 29, PDF page 38\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as a quantitative conjecture. Literature status: - This is a quantitative, essentially open estimate in Gromov's spin-area program connecting the volume growth of tube neighbourhoods of non-spin surfaces to the spin area (K-area) of the manifold. - No published verification of the specific CP² / 10⁻⁶ estimate found." }, { "id": 6700029, "problem_number": "AMR-066-0029", "title": "Scalar Curvature Question [?28]: Besides, similar results are expected for allsingular Alexandrov spaces with lower curvature boundsbut the Alm", "statement": "Besides, similar results are expected for allsingular Alexandrov spaces with lower curvature boundsbut the Almgren’s regularity theory has not been developed even for Alexandrov spaces withconical singularities, where the only apparent, yet instructive, case is that ofisolated singularities. ●≤κ0. If the injectivity radius ofX at a pointx∈Ym min ⊂X is ≥R and if the sectional curvatures ofX in theR-ball Bx(R) ⊂X are bounded from aboveby κ0 then them-volumevol(Ym min⋂Bx(R)) is bounded from below by the volume of theR-ballBm(R,κ 0) in the standardm-space with constant curvatureκ0 by the monotonicity formulafor minimal subvarieties. About the Balls.Neither●κ≥1 nor●≤κ0 directly apply to the ballsBn(R,κ 0) with constant curvatures; yet, by comparing their waists to those of spheres by means of suitableO(n)-equivariant mapsBn(R,κ 0)→Sn, one arrives at the expected values ( see section 3 in [?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?28], occurrence 30, PDF page 40\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: isolated singularity cases and some compactness handled; general conical regularity open. Literature status: - Regularity theory for minimal hypersurfaces in singular spaces / Alexandrov spaces with curvature bounds is an active topic. Almgren's regularity (for one-dimensional minimizers in metric spaces / geometric measure theory) extended to metric spaces by various authors; for higher dimensional minimal submanifolds in Alexandrov spaces the theory is less developed. - The \"hypersurface in Alexandrov spaces\" regularity and compactness are studied in works on \"Minimal hypersurfaces in Alexandrov spaces\" (e.g., by Stancu / others) and the recent significant progress on the isoperimetric problem in Alexandrov spaces. Still, the general Almgren regularity for conical singularities beyond isolated points remains incompletely developed." }, { "id": 6700030, "problem_number": "AMR-066-0030", "title": "Scalar Curvature Question [?30]: On the other hand, there probably exist compact simply connected n-dimensional manifolds for alln ≥4 with arbi", "statement": "On the other hand, there probably exist compact simply connected n-dimensional manifolds for alln ≥4 with arbitrarily prescribed (finite) values of theK-area and the slice-area.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?30], occurrence 31, PDF page 41\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as posed. Literature status: - This is a conjecture about realizing arbitrary prescribed finite values of independently-chosen K-area and slice-area invariants on simply connected manifolds in each dimension ≥ 4. - K-area (Gromov) and slice-area are metric-dependent invariants; whether they can be independently prescribed is not obviously resolved. No dedicated construction verified in the literature." }, { "id": 6700031, "problem_number": "AMR-066-0031", "title": "Scalar Curvature Question [?31]: the sharp values ofwidthn−m for these solids remains problematic for m≥2, (unless I missed some paper)", "statement": "the sharp values ofwidthn−m for these solids remains problematic for m≥2, (unless I missed some paper).", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?31], occurrence 32, PDF page 42\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: sharp values for m ≥ 2 generally unresolved. Literature status: - Width estimates (Gromov's \"width\" / Kumura-type) for metric balls with scalar curvature bounds are partially known; for spheres the widths of balls are known, and the \"two-sphere lemma\" gives the first nontrivial width. - Sharp values for higher co-widths (m ≥ 2) of general solids under Sc ≥ σ are largely open; some sharp results for low dimensions/balls." }, { "id": 6700032, "problem_number": "AMR-066-0032", "title": "Scalar Curvature Question [?32]: Waist-Width Inequality", "statement": "Conjecture: Waist-Width Inequality. All complete Riemannian n-manifolds X satisfy widthn−1(X) ≤constn⋅waistn−k+1(X). Contractibility Radius. This \"radius\", denoted contr(X,r), r > 0, of a metric spaceX is the infimum of the numbersR, such that everyr-ballBx(r) ⊂ X is contractible within the concentric ballBx(R) ⊃Bx(r) of radiusR. It is (almost) obvious that [41]: A. Complete Riemannian manifolds with cocompact isometry groups, e.g universal coverings of compact aspherical manifolds, havecontr(X,r) = ∞for all r > 0. B. If a completen-dimensional manifoldX satisfies contr(X,ri) ≤ri+1 for i= 1, 2,...,n and r1 ≤r2 ≤...≤rn+1 <∞, then fil.rad[X]≥r1/(n+ 1)!. C. It follows that the universal coverings ˜X of compact aspherical manifoldsX have fil.rad( ˜X) =∞. Consequently, these ˜X satisfy waistm( ˜X) =∞,m = 1, 2,...,n =dim(X), as well. 16 Standard Geometric and Topological Conjectures on Complete Manifolds with Positive Scalar Curvatures.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?32], occurrence 33, PDF page 43\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: waist inequalities for spheres/balls and macroscopic-dimensional cases known; general complete-manifold waist–width open. Literature status: - Gromov's \"waist\" theory: the sharp waist inequality for spheres/balls is solved (the paper by Gromov \"Isoperimetry of waists\"; and the \"width\" of spheres). The waist–width comparison for general manifolds is connected to the isoperimetric/width program. - Verified: Gromov's waist theorem (2003) resolves the sphere waist problem; the filling radius / waist relations. The general waist–width inequality across all complete manifolds is a broad conjecture with partial confirmations (Avvakumov–Karasev on waist for different distributions; more recent $\\mathbb{Z}/2$ waist results)." }, { "id": 6700033, "problem_number": "AMR-066-0033", "title": "Scalar Curvature Question [?34]: Bounds on Width and on the Macroscopic Dimension", "statement": "Conjecture. Bounds on Width and on the Macroscopic Dimension. Complete n-dimensional Riemannian manifoldsX with the scalar curvaturesSc(X) ≥σ > 0 satisfy macr.dim(X) ≤n−2. Moreover, m2, width n−2(X) ≤constnσ−1 2. where, in fact proving even the weaker inequality m1 widthn−1(X) ≤constnσ−1 2. would make one happy.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?34], occurrence 35, PDF page 44\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: some cases (dimension ≤ 4, spin assumptions) resolved; general statement open and central. Literature status: - The macroscopic dimension conjecture for Sc > 0 (Gromov) is a major open problem tied to essentiality. Known: for universal covers, macroscopic dimension ≤ n-2 is related to the non-spin / classical conjectures. Partial results by V. Kapovitch / and others for the macroscopic dimension of manifolds with positive scalar curvature; the bound follows in low dimensions and under extra assumptions. - There are counterexamples/limitations in special settings; the full statement is open. Verified literature discusses it as unresolved." }, { "id": 6700034, "problem_number": "AMR-066-0034", "title": "Scalar Curvature Question [?35]: Bound on the Filling Radius forSc ≥σ > 0", "statement": "Conjecture. Bound on the Filling Radius forSc ≥σ > 0., fil.rad[X]≤constn⋅( inf x∈X Sc(X)(x))−2. This, in view ofA,B,C from the previous section, yields the following. [?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?35], occurrence 36, PDF page 44\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: several ranges confirmed, general bound open. Literature status: - Gromov conjectured Sc ≥ σ ⇒ fil.rad bounded by const/√σ. Verified progress: for the universal covers and via macroscopic dimension; the \"filling radius\" (Gromov) of spheres is known. - Partial confirmations in low dimensions and under essentiality; general n-dim statement open. Recent work (e.g., by students of Gromov and other authors) gives progress linking Sc ≥ σ, filling radius, systolic bounds." }, { "id": 6700035, "problem_number": "AMR-066-0035", "title": "Scalar Curvature Question [?38]: Asphericity⇒K-Area =∞", "statement": "Conjecture. Asphericity⇒K-Area =∞. The universal coverings ˜X of compact aspherical manifoldsX satisfy K-area( ˜X) =∞. Notice that this inequality, even in a stabilised form, implies the strong Novikov conjecturefor π1(X), which is stronger than the non-asphericity for Sc ≥0 [101]. This makes it too good to be true; yet, no candidate for a counterexample is anywhere in sight. (e) Recall that a more comprehensive form of the asphericity conjecture [?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?38], occurrence 37, PDF page 45\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open (with important implications to Novikov); no counterexample known; partial spectral/index results exist. Literature status: - This is an extremely strong conjecture by Gromov. It is open in general and is known to imply/relate to the strong Novikov conjecture. Verified as open in the literature. - K-area of the universal cover being infinite encodes \"no K-theory vanishing\" and relates to the radius/waist of covers. Some partial results connect it to the Hilbert-space area and to the \"K-area\" lower bounds via the index theorem, but the full conjecture for all aspherical manifolds is unresolved." }, { "id": 6700036, "problem_number": "AMR-066-0036", "title": "Scalar Curvature Question [?39]: Let $B=B\\Gamma$ be the classifying space of a discrete countable group, and let $f:X\\to B$ be a continuous map", "statement": "Let $B=B\\Gamma$ be the classifying space of a discrete countable group, and let $f:X\\to B$ be a continuous map from a Riemannian manifold. Does there exist a compact subset $B_0\\subset B$ containing $f(X)$ such that the Fredholm coareas of certain nonzero multiples of pullback bundles become arbitrarily small? More precisely, given $\\varepsilon>0$, do there exist an integer $N\\ne0$ and a Fredholm bundle $(L,\\nabla)$ over $X$ with $\\|\\operatorname{curv}(\\nabla)\\|\\leq\\varepsilon$, such that $L$ is K-theoretically equivalent to the $N$th Whitney power of a pullback bundle $f^*(L_0)$?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?39], occurrence 38, PDF page 45\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open as posed. Literature status: - This is a technical K-theoretic \"coarea\" question in Gromov's program (Fredholm K-area machinery). It formalizes existence of almost-flat vector bundles in nontrivial K-theory classes. - No dedicated resolution located. Related: \"almost flat bundles\" and the Novikov conjecture context; the construction of almost flat bundles on aspherical covers is related to positive scalar curvature obstructions." }, { "id": 6700037, "problem_number": "AMR-066-0037", "title": "Scalar Curvature Question [?40]: Area Extremality and Rigidity of Symmetric and Einstein Spaces", "statement": "Conjecture Area Extremality and Rigidity of Symmetric and Einstein Spaces. All Riemannin manifolds with positive and parallel Ricci tensor, in particular all Symmetric and all Einstein SpacesX are area extremal and those of them which contain no local flatfactors are area rigid. For Einstein spaces, this agrees with local extremality lemma in [39], while an essential class of examples where the available proofs do not work are compact Lie groupsX with biinvarinat metricsg, where the tangent bundles are trivial and can not force non-vanishing of indices of natural Dirac operators over theseX.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?40], occurrence 39, PDF page 48\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: many rank-one symmetric and Einstein examples verified (sphere, projective, hyperbolic); general case (e.g., compact Lie groups) open. Literature status: - Area extremality/rigidity for symmetric spaces: verified famous cases (S^n rigidity: Brendle–Marques–Neves, and Gromov's area; CP^n/hyperbolic/Symmetric space rigidity via the Dirac operator or minimal hypersurfaces). - The general conjecture for all Einstein/symmetric spaces (especially Lie groups with bi-invariant metrics) is open; the tangent bundle being trivial defeats Dirac-index methods, so area rigidity there is not established. - Recent progress on Einstein spaces and the \"area rigidity\" for hyperbolic and spherical space forms." }, { "id": 6700038, "problem_number": "AMR-066-0038", "title": "Scalar Curvature Question [?41]: For instance, ifX =SO(n) withn≥5, then no known method can rule out metricsg ≥g on X with Sc(g) > Sc(g)", "statement": "For instance, ifX =SO(n) withn≥5, then no known method can rule out metricsg ≥g on X with Sc(g) > Sc(g).", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?41], occurrence 40, PDF page 48\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open as posed. Literature status: - This is Gromov's remark that for compact simple Lie groups SO(n) (n≥5), the bi-invariant metric's scalar curvature may possibly be increased by a metric g ≥ ḡ, and no known method (Dirac index, which needs nontrivial tangent bundle spinor bundle) rules it out. This is precisely the open case from item 0037. - Related: the family of metrics with scalar curvature ≥ that of the bi-invariant one on compact Lie groups; the \"curvature + variations\" questions. No resolution located." }, { "id": 6700039, "problem_number": "AMR-066-0039", "title": "Scalar Curvature Question [?41]: Are there compact manifoldsX which support metricsg withSc(g) > 0 but admit no area extremal or length extrema", "statement": "Are there compact manifoldsX which support metricsg withSc(g) > 0 but admit no area extremal or length extremal metricsg?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?41], occurrence 41, PDF page 48\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - For X with PSC, an extremal metric would be one achieving maximal area/length rigidity. Whether some PSC manifold fails to admit such an extremal metric is a specific open question of Gromov. - Note: manifolds admitting PSC have lots of PSC metrics; whether any is area/length-extremal is unclear. No resolution located." }, { "id": 6700040, "problem_number": "AMR-066-0040", "title": "Scalar Curvature Question [?42]: Can one \"effectively\" evaluate the minimal constantλ=λ(X,g)„ such that a given Riemannin manifoldX = (X,g), e", "statement": "Can one \"effectively\" evaluate the minimal constantλ=λ(X,g)„ such that a given Riemannin manifoldX = (X,g), e.g wheresect.curv(g) > 0, would support an area extremal (or a length extremal) metricg which would beλbiLipschitz equivalent tog, where such aλ were expressible in terms of the pinching constant in the case wheresect.curv(g) > 0?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?42], occurrence 42, PDF page 48\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This asks for quantitative control (bi-Lipschitz constant) relating a given metric to an extremal one near a prescribed geometric class. Related to the rigidity/quantitative stability of PSC metrics. - No specific resolution of the effective λ evaluation located." }, { "id": 6700041, "problem_number": "AMR-066-0041", "title": "Scalar Curvature Question [?43]: Would it be more prudent to replace the conditionSc(g) > 0 byRicci> 0", "statement": "Would it be more prudent to replace the conditionSc(g) > 0 byRicci> 0?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?43], occurrence 43, PDF page 48\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial/interpretive: Ric > 0 gives strong insights (Myers, Bonnet) but is much more restrictive; numerous manifolds separate the two notions. The \"prudence\" is a design choice, not resolved as a theorem." }, { "id": 6700042, "problem_number": "AMR-066-0042", "title": "Scalar Curvature Question [?45]: Spin Problem", "statement": "Spin Problem. All of the above only applies to spin maps f ∶X→X, for which the required twisted Dirac operator defined, and, as on similar occasions we met earlier, the necessity of the spin condition, say for equidimensional maps of degrees≠0, remains unsettled. Extremality via Lipschitz Maps.If a manifoldX admits noC2smooth length decreasing map X →X of degree≠0 which also strictly decreases the scalar curvature, then it admits no such Lipschitz map either my a simple approximation argument.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?45], occurrence 44, PDF page 52\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: spin rigidity well-understood; removing spin for general maps open. Literature status: - This is the long-standing question: do scalar-curvature rigidity / extremality statements for equidimensional nonzero-degree maps require spin? The torus/sphere index arguments need spin. For non-spin, results are sparse. - Some partial progress: non-spin versions of rigidity in low dimensions and specific settings; the general necessity of spin remains unsettled, as Gromov states. This is connected to the \"rational η-invariants\" and to showing the spin hypothesis is unavoidable." }, { "id": 6700043, "problem_number": "AMR-066-0043", "title": "Scalar Curvature Question [?46]: But it is unclear if this remain true with \"area\" in place of \"length\"", "statement": "But it is unclear if this remain true with \"area\" in place of \"length\".", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?46], occurrence 45, PDF page 52\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (case-dependent). Literature status: - This is a recurring theme in Gromov's program: area extremality is generally a stronger/different notion than length extremality; Llarull's theorem deals with area-non-increasing maps in the sphere case, while many index arguments give length versions. - Whether \"area\" versions of specific length/extremality results hold is case-by-case; no single resolution." }, { "id": 6700044, "problem_number": "AMR-066-0044", "title": "Scalar Curvature Question [?47]: Stabilisation of Extremality", "statement": "Conjecture: Stabilisation of Extremality.Let X0 be a compact area extremal Riemannin manifold. Then A. X0× Rm is area gap extremal for allm. 53 B. X0× R is area extremal. This can be confirmed for all known examples ofX0 the area extremality of which was established by a Dirac operator arguments sketched in the previous section. Let us do it in two cases. Start with A for X0 = Sn, let X be a complete orientable spin Riemannian manifold, letf ∶X→Sn× Rm be a smooth proper map. All we shall need ofRm for our purpose is thatSc(Rm) = 0 and K-Area(Rm) =∞. Assumeforsimplicity’ssakethat misandlet Lε0 bea Q-homologically essential unitary vector bundle withε0-flat connection which is flat at infinity. Assume for the same reason thatn is even and let S+ be the \"positive\" spinor bundle onSn. Let L! ε0 be the f-pullback of the tensor product S+ ⊗Lε0 and observe – this needs looking at the Llarull’s computation forRL!ε0 – that if the scalar curvature ofX at infinity is≥(n(n−1)+ε > 0 and if ε0 > 0 is much smaller thanε, then the twisted Dirac operatorD⊗Lε! 0 onX is strictly positive at infinity and ifdeg(f) ≠0 it has non-zero index relative the Dirac operator twisted with the pullback ofS+. Thus,X carries non-trivial twisted harmonic spinors. On the other hand, if the scalar curvature ofX at infinity is ≥(n(n−1)+ε> 0 on all ofX and εo <ε, then again, by looking at looking at LLarull’sRL!ε0 and sees thatX can’t carry such spinors and the proof follows. \"Subcomplete\" Extremality. LetXo be an open, not necessarily complete Riemannin manifold.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?47], occurrence 46, PDF page 53\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: confirmed for known Dirac-based examples; general statement open. Literature status: - Stabilization results: products with Euclidean factors preserve (or preserve-gap) extremality for manifolds established by Dirac-index methods (e.g., products with R^m of spheres and rigidity). This is largely verified for the known examples by the Dirac-operator argument Gromov sketches, since Sc(R^m) = 0 and K-area(R^m) = ∞. - The general stabilization conjecture for arbitrary area-extremal X₀ is open; the known examples (sphere; via Llarull-type) are confirmed." }, { "id": 6700045, "problem_number": "AMR-066-0045", "title": "Scalar Curvature Question [?48]: When does such anX0 is area extremal in the category of complete manifolds", "statement": "Question. When does such anX0 is area extremal in the category of complete manifolds?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?48], occurrence 47, PDF page 54\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open in general; several complete rigidity cases known. Literature status: - This concerns extending area extremality/rigidity to complete (noncompact) manifolds, where the index-theoretic arguments are subtler. Some rigidity results extend to complete manifolds (e.g., Brendle–Marques–Neves type for hyperbolic space; rigidity at infinity). - The precise characterization \"when X₀ is area extremal in the complete category\" is not resolved generally." }, { "id": 6700046, "problem_number": "AMR-066-0046", "title": "Scalar Curvature Question [?49]: The sphereSn minus Σo is area extremal in the \"subcomplete\" sense for all closed subsetsΣo ⊂Sn of topological", "statement": "Conjecture. The sphereSn minus Σo is area extremal in the \"subcomplete\" sense for all closed subsetsΣo ⊂Sn of topological dimensions k ≤1. 20 Lengths, Widths and Areas of Non-Complete Manifolds withSc > 0. To develop an adequate picture ofcomplete manifolds with scalar curvatures ≥σ one needs to understand geometric constraints imposed by the inequalitySc ≥σ on (bounded) domains in these manifolds. Thus, we look at possible sizes ofincomplete manifoldsX, e.g. (compact) manifolds with boundaries, where we do not, a priori, assume they are contained in complete manifolds with lower bounds on Sc. (We shall say something about theshapes rather than mere sizes of theseX in section 22) Asimpleexampleshowingwhatcanandwhatcannotbeexpected in this regard is the universal covering of the 2-sphere minus 2 opposite points times Rn−2, denoted ˜Σn π = the universal cover of S2∖{⋅⋅}× Rn−2 55 which satisfies: ●Sc( ˜Σn π) =Sc(S2) = 2, ●waist1[˜Σn π]=π and width1( ˜Σn π) ≤π. ●waistm[˜Σn π]=widthn−m( ˜Σn π) =∞for m= 2,...,n and K-area[˜Σn π]=∞as well. ●The r-interior ˜(Σn π)−r ⊂Σn π, that is the set of pointsx which are r-far from infinity, i.e. such that the closedr′-balls Bx(r′) ⊂Σn π for r′< r are compact, hassmall all waistm and K-area, actually, zero in the present example. r-Interior and Completeness. Metric completeness of anX in this terms is equivalent to non-emptiness of ther-interior ofX for r =+∞.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?49], occurrence 48, PDF page 55\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: example computations given; general estimates conjectural/partial. Literature status: - This is Gromov's \"incomplete manifolds / r-interior\" program. The explicit example Σ̃^n_π is from Gromov's essay. Linked to the \"waist of the r-interior\" shrinking; partial results via minimal hypersurfaces and the \"K-area of r-interior\". - The general principle that r-interiors of manifolds with Sc ≥ σ have small waist/K-area is a conjecture (item 0047 refers to it). Not fully resolved." }, { "id": 6700047, "problem_number": "AMR-066-0047", "title": "Scalar Curvature Question [?50]: All of the above is satisfied, modulo constants, for alln-manifolds, possibly incomplete and/or with boundaries", "statement": "Conjecture. All of the above is satisfied, modulo constants, for alln-manifolds, possibly incomplete and/or with boundaries, withSc(X) ≥σ > 0. Namely mbnd 1 widthn−1(X) ≤constnσ−1 2, mbnd 2 widthn−2(X−r) ≤constnσ−1 2 for r ≥constnσ−1 2, -bnd 2 waist2(X−r) ≤constnσ−1 for r ≥constnσ−1 2, [boundary estimate] K-area(X−r) ≤constnσ−1 for r ≥constnσ−1 2, where, of course, all theseconstn, especially the optimal ones, may be different. Among the first three inequalities, which generalise the corresponding conjecturesin section 16, a definite result is available only for mbnd 2 for 3-manifolds, which, similarly tom2, is proven with a use of minimal surfaces [54]. On the other hand, the inequality [boundary estimate], which, in the case whereX is complete spin, easily follows from the index theorem for the twisted Dirac operator, remains problematic fornon-complete, let them be spin,n-manifolds starting fromn = 3. (To make sense of this forn = 3 one should replaceK-area(X3 −r), which is zero by definition for oddn, byK-area(X3 −r× R).) It is tempting to try to reduce these conjectures, especially [boundary estimate], to the case of complete manifolds by extending (the metric on)X (onX× RN?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?50], occurrence 49, PDF page 56\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: spin-complete K-area bound and some width results proven; general non-complete/non-spin open. Literature status: - Verified: Gromov's width estimates under Sc ≥ σ are partially proven; the 3-manifold width result is due to Gromov (via minimal surfaces / the disks-to-spheres argument). The r-interior K-area bound for complete spin manifolds follows by the families index theorem — this is the Gromov–Lawson/relative index argument. - The general non-complete non-spin version is open, exactly as Gromov states." }, { "id": 6700048, "problem_number": "AMR-066-0048", "title": "Scalar Curvature Question [?51]: Extension Problem", "statement": "Extension Problem.LetX be a Riemanniann-manifold withSc(X) ≥σ > 0 and letσ−≤σ,r andr+ ≥r be positive numbers. Whendoesthereexistan n-dimensionalmanifold X+ withSc(X+) ≥ σ−, such that ther-interior X−r ⊂X isometrically embeds into the r+-interior (X+)−r+ ⊂X+?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?51], occurrence 50, PDF page 57\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open in general; some low-dimensional extension constructions known. Literature status: - This is Gromov's extension problem for incomplete manifolds with scalar curvature bounds. Related to \"filling\"/\"completion\" and the isometric embedding of manifolds with positive scalar curvature ends. - Partial: boundary-smoothing/extension results in low dimensions; the general existence question, in arbitrary dimension, is open." }, { "id": 6700049, "problem_number": "AMR-066-0049", "title": "Scalar Curvature Question [?52]: Completion by Extension", "statement": "Conjecture. Completion by Extension.If σ > σ−and r ≥constn(σ −σ−)−1 2 for some (large) constant constn, then the extension problem is solvable withr+ =∞: there exists a completeX+ withSc(X+) ≥σ−which isometrically containsX−r Let us introduce geometrically more transparent geometric invariants which may serve as lower bound to theK-area.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?52], occurrence 51, PDF page 57\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as posed. Literature status: - This is Gromov's completion conjecture: any sufficiently large r-interior of a manifold with Sc ≥ σ can be completed to a complete manifold with slightly smaller scalar curvature bound σ⁻. - Related to the \"Riemannian extension/fill-in\", Gromov's tiling constructions, and the \"positive scalar curvature complicates completion\" phenomenon. No full resolution located." }, { "id": 6700050, "problem_number": "AMR-066-0050", "title": "Scalar Curvature Question [?53]: Sharp Spherical Length Comparison Inequality", "statement": "Conjecture. Sharp Spherical Length Comparison Inequality. Spheres with finitely many punctures are length extremal. In fact – this is, probably equivalent– All Riemanninn-manifolds X, possibly non-complete and with boundaries, which haveSc(X) ≥Sc(Sn) =n(n−1) satisfy co-s.leng(X) ≤2π In plain words, ifSc(X) ≥n(n−1), then there is nono strictly distance decreasingproper maps fromX to Sn with non-zero degrees.54", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?53], occurrence 52, PDF page 57\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: complete spin spherical length comparison proven; boundary/non-complete case open. Literature status: - The \"no strictly distance-decreasing proper map to S^n of degree ≠ 0 under Sc ≥ n(n-1)\" rigidity is precisely Llarull-type and is related to the (partially proven) \"spherical length comparison\": for spin manifolds complete, an index argument (Gromov–Lawson/Llarull) shows distance-non-increasing degree-1 maps to S^n with Sc ≥ n(n-1) are isometries. - The statement allowing non-complete-with-boundary and \"proper\" maps is a broader Gromov conjecture; partial confirmation in the complete spin case; general open." }, { "id": 6700051, "problem_number": "AMR-066-0051", "title": "Scalar Curvature Question [?54]: ExtremalityofConcaveSphericalBalls", "statement": "Conjecture: ExtremalityofConcaveSphericalBalls. The balls B(R) ⊂Sn of radiiR≥π 2 are length extremal: no Riemannian metricg on such a ball which is greater than the spherical one (of constant curvature 1) can haveSc(g) > n(n−1) = Sc(Sn). [?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?54], occurrence 53, PDF page 57\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: sphere and some ball/hemisphere rigidity proven; general ball case open. Literature status: - The extremality of balls/hemispheres in S^n: hemispheres admit a proof of rigidity (Brendle–Marques–Nevens and the Llarull/Dirac). The \"no metric ≥ spherical on a hemisphere with Sc > n(n-1)\" is a form of positive-mass / Llarull result for hemispheres. - Partial: for the whole sphere, Llarull gives it; for hemispheres and balls, related rigidity via the boundary (first eigenvalue) and the \"Schoen–Yau/Huisken\" hemisphere rigidity exists in low dimensions. - General: not fully resolved for all radii ≥ π/2 and all dimensions." }, { "id": 6700052, "problem_number": "AMR-066-0052", "title": "Scalar Curvature Question [?55]: 18", "statement": "Conjecture 18. Interior Hemi-Spherical Area Inequality. The r-interiors of all compact Riemanninn-manifolds X with boundaries and withSc(X) ≥Sc(Sn) =n(n−1) satisfy for all r ≥π/2, co-s+ar(X−r) ≤2π∶ no strictly area decreasing proper mapX−r→Sn + of non-zero degree for r > π/2 exists. It is even unclear what happens in this regard to domains in closed spin manifoldsX. For instance:", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?55], occurrence 54, PDF page 58\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. Literature status: - This is a precise \"interior area inequality\" conjecture in Gromov's program, mixing the r-interior notion with the hemi-spherical area comparison (proper maps to the hemisphere). - No dedicated proof located; it is stated as a conjecture by Gromov and, to my knowledge, remains open." }, { "id": 6700053, "problem_number": "AMR-066-0053", "title": "Scalar Curvature Question [?56]: What are possible values of the co-s+areas of the complements ofr-balls in compact spin manifoldsX withSc(X) ≥", "statement": "What are possible values of the co-s+areas of the complements ofr-balls in compact spin manifoldsX withSc(X) ≥n(n−1)?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?56], occurrence 55, PDF page 58\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This is a quantitative \"co-area\" question in Gromov's program about the spherical/coarea invariants of the complement of balls in high-scalar-curvature spin manifolds. - No dedicated resolution located." }, { "id": 6700054, "problem_number": "AMR-066-0054", "title": "Scalar Curvature Question [?58]: there is no apparent non-trivial bound on the width ofX = Σn−1× [−1, 1]even we assume that thesectional curvat", "statement": "there is no apparent non-trivial bound on the width ofX = Σn−1× [−1, 1]even we assume that thesectional curvatureof X is = 1.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?58], occurrence 57, PDF page 59\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open/consistent: no nontrivial width bound is expected or known. Literature status: - Gromov notes that for the product Σ^{n-1} × [-1,1] with a fixed (sectional curvature 1? — this cannot literally be for arbitrary Σ) geometry, there is no nontrivial width bound. This likely reflects the fact that the width (distance between the two boundary components) of such a slab is not constrained by scalar curvature alone. - Consistent with the broader fact that scalar curvature controls \"waist/width\" only via the minimal-hypersurface mechanism and enters in codimension-1 ways, not giving a bound here." }, { "id": 6700055, "problem_number": "AMR-066-0055", "title": "Scalar Curvature Question [?59]: what is the (asymptotically forn→∞and/or fork→∞) sharp inequality for immersions of theseΣn−1 to spheres", "statement": "what is the (asymptotically forn→∞and/or fork→∞) sharp inequality for immersions of theseΣn−1 to spheres", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?59], occurrence 58, PDF page 59\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: sharp immersion bounds not determined. Literature status: - This connects to Gromov's work on curvature-bounded isometric immersions into spheres (e.g., the \"Gromov's curvature-bounded isometric immersions into Euclidean/sphere\" results and recent work by various authors). Sharp asymptotic constants for immersions of general Σ into S^N are not generally resolved." }, { "id": 6700056, "problem_number": "AMR-066-0056", "title": "Scalar Curvature Question [?60]: Also it is unclear if there are (non-trivial) inequalities of this kind for other exotic spheres", "statement": "Also it is unclear if there are (non-trivial) inequalities of this kind for other exotic spheres.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?60], occurrence 59, PDF page 59\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. Literature status: - Exotic spheres: many (in certain dimensions) admit PSC metrics (via Gromov–Lawson surgery, since they are null-cobordant / have vanishing α usually); whether various sharp rigidity/immersion inequalities hold is unclear. The distinguishing of exotic spheres by curvature is an active area (e.g., via minimal hypersurfaces there are sometimes differences). - No resolution of \"inequalities for exotic spheres\" located." }, { "id": 6700057, "problem_number": "AMR-066-0057", "title": "Scalar Curvature Question [?61]: Is then every immersion fromXj to the unit ball in RN satisfies supcurv(Xj↪BN(1) ⊂RN)) ≥ √ k for allN ≥n1+", "statement": "Is then every immersion fromXj to the unit ball in RN satisfies supcurv(Xj↪BN(1) ⊂RN)) ≥ √ k for allN ≥n1+....+nj+ 1?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?61], occurrence 60, PDF page 59\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This is in the theme of curvature-bounded isometric immersions into Euclidean balls (Gromov's \"curvature, curves, and isometric embeddings\" / the \"Gromov's rigidity on curvature-bounded immersions into unit balls\"; related to Conway's \"large solutions\" and the tangle/immersion theory). Partial results exist for rigidity of certain immersions into balls; the sharp supcurv ≥ √k statement not resolved generally." }, { "id": 6700058, "problem_number": "AMR-066-0058", "title": "Scalar Curvature Question [?62]: But it is also not impossible that all manifolds admit immersions into the unit ball in the Hilbert spaceR∞wit", "statement": "But it is also not impossible that all manifolds admit immersions into the unit ball in the Hilbert spaceR∞with principal curvatures bounded by a universal constant, say by 1 000 000.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?62], occurrence 61, PDF page 59\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. Literature status: - This is Gromov's conjecture about curvature-bounded immersions into Hilbert space / the unit ball of infinite-dimensional space. Related to Nash's C¹ embeddings and to Gromov's theory of curvature-bounded immersions; whether all manifolds embed into the finite-codimension unit ball with uniformly bounded curvature is a strong statement. - Partial: finite-dimensional analogues and curvature-bounded embedding results; the infinite-dimensional unit-ball version not resolved." }, { "id": 6700059, "problem_number": "AMR-066-0059", "title": "Scalar Curvature Question [?63]: Identify combinatorial typesPextr of convex polyhedra where all representativeP ∈P are extremal and also descr", "statement": "Problem. Identify combinatorial typesPextr of convex polyhedra where all representativeP ∈P are extremal and also describe extremal P of non-extremal typesPnonextr. Next, callP mean convexly extremalif there is noP′⊂Rn diffeomorphic toP and such that ●thefaces Q′ i ⊂P′correspondingtoall Qi ⊂P havemean.curv(Q′ i) ≥ 0, ●the dihedral angles ofP′, that are the angles between the tangent spacesTp′(Q′ i) andTp′(Q′ j) at the pointsp′on the(n−2)-faces Q′ ij =Q′ i∩Q′ j, satisfy ∠ij(P′) ≤∠ij(P), ●this angle inequality is strict at some point, i.e. there exits p′ 0 ∈Q′ ij in someQ′ ij, such that ∠(Tp′ 0(Q′ i),Tp′ 0(Q′ j)) <∠ij(P).", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?63], occurrence 62, PDF page 60\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. Literature status: - This is Gromov's program on extremal convex polyhedra (rigidity of polyhedra with respect to mean curvature and dihedral angles), related to the \"polyhedral comparison\" and positive scalar curvature / mean-convex hypersurfaces. - Specific classification of extremal combinatorial types is not resolved; related work on \"polyhedral scalar curvature\" and Alexandrov spaces is partial." }, { "id": 6700060, "problem_number": "AMR-066-0060", "title": "Scalar Curvature Question [?64]: Are all extremal convex polyhedraP are mean convexly extremal", "statement": "Question. Are all extremal convex polyhedraP are mean convexly extremal?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?64], occurrence 63, PDF page 60\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This is Gromov's specific question relating two notions of extremality for convex polyhedra. No resolution found in the literature." }, { "id": 6700061, "problem_number": "AMR-066-0061", "title": "Scalar Curvature Question [?65]: Is the regular Euclidean $3$-simplex mean-convexly extremal", "statement": "Is the regular Euclidean $3$-simplex mean-convexly extremal? Equivalently, can a simplex mapped facewise to it without decreasing distances have nonnegative face mean curvatures and no larger dihedral angles, with at least one angle strictly smaller?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?65], occurrence 64, PDF page 61\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This is a precise, concrete question about the mean-curvature/dihedral rigidity of the regular simplex. No resolution found in the literature." }, { "id": 6700062, "problem_number": "AMR-066-0062", "title": "Scalar Curvature Question [?66]: Probably, these equalities imply thatP is isometric to a Euclidean rectangular solidbut the approximation/smoo", "statement": "Probably, these equalities imply thatP is isometric to a Euclidean rectangular solidbut the approximation/smoothing is no good for proving this kind of rigidity.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?66], occurrence 65, PDF page 61\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This is a Gromov remark about rigidity of rectangular solids; the \"approximation/smoothing is no good\" indicates a need for a different rigidity argument. No resolution located." }, { "id": 6700063, "problem_number": "AMR-066-0063", "title": "Scalar Curvature Question [?67]: This suggests a possibility of definingSc(X) ≥0 for some singular spaces, X, e", "statement": "This suggests a possibility of definingSc(X) ≥0 for some singular spaces, X, e.g. for manifolds with continuous (bounded measurable?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?67], occurrence 66, PDF page 61\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: candidate definitions exist in restricted classes; no general one. Literature status: - Gromov's program explicitly aims at defining Sc ≥ 0 for metric/continuous metrics. Related partial frameworks: the \"volumically positive scalar curvature\" (for C⁰ metrics), and work on C⁰-metric scalar curvature (e.g., \"C⁰-metric positive scalar curvature\" and recent papers by Gromov and by others on C⁰ metrics and volume comparison). - A general definition for arbitrary continuous/bounded-measurable metrics supporting all global theorems is not settled." }, { "id": 6700064, "problem_number": "AMR-066-0064", "title": "Scalar Curvature Question [?68]: Let $\\widetilde X$ be the universal cover of a Riemannian $n$-manifold $X$ homeomorphic to the $n$-torus", "statement": "Let $\\widetilde X$ be the universal cover of a Riemannian $n$-manifold $X$ homeomorphic to the $n$-torus. Conjecture that $\\widetilde X$ has non-positive scalar curvature at infinity: it can be exhausted by overcubic domains $P_i\\subset \\widetilde X$ with corners whose codimension-one faces have positive mean curvature and whose dihedral angles are at most $\\pi/2$. Here overcubic means admitting a degree-one map to the $n$-cube that sends each $k$-face to a $k$-face.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?68], occurrence 67, PDF page 61\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: flat torus theorem solved via other means; the specific overcubic exhaustion conjecture remains a program-internal open statement. Literature status: - This is related to Gromov's conjecture that the universal cover of the torus (with Sc ≥ 0 by rigidity... actually the conjecture is part of proving flatness) — X̃ should have \"non-positive scalar curvature at infinity\" in the mean-convex/dihedral sense, giving a new proof that such X is flat (Schoen–Yau torus theorem). - The flat torus theorem (Schoen–Yau, Gromov–Lawson) is solved, but this particular exhaustion/mean-convex-dihedral formulation is a program-internal conjecture; partial constructions exist." }, { "id": 6700065, "problem_number": "AMR-066-0065", "title": "Scalar Curvature Question [?69]: Shrinking of Singularities", "statement": "Conjecture. Shrinking of Singularities. Let X be a compact orientable Riemanninn-manifold, f0 ∶X →Tn be a continuous map of non-zero degree, hi, i=0,1,2,..., be 1-dimensional homology classes in Tn which generate H1(Tn) and let εi →0 be positive numbers. Then the above (regularised for prudence) spaces Xi satisfy diam(X′ i)→0 for i→∞. 64 Moreover, theminimalhypersurfaces Yi ⊂X′ i becomenon-singular for sufficiently largei and the manifoldsX′ i as well asXi admit λibi-Lipschitz homeomorphisms to flatn-toriYi withλi→1 fori→∞. This conjecture implies that overtorical manifoldsX, (i.e. admitting maps toTn with non-zero degrees) withSc(X) ≥0 are, in fact, flat, since the scalar curvature is semicontinuous under \"Lipschitz limits\". (The proof of this semicontinuity forC0-convergence given in [49] automatically extends to the Lipschitz convergence.) (The arguments used in in [86] and those in [115], for the proof of this \"non-positivity\"Sc(X) >/0 probably yield the proof of the above Conjecture as well.) 24 Scalar Curvature and Mean Curvature. One may think of positive scalar curvature asRiemannian internalisation of the concept ofmean convexity, where a Riemannian manifoldY with boundary, e.g. a smooth domainY in a larger Riemannian manifold, is calledmean convexif the boundary∂Y has positive mean curvature. This \"internalisation\": is motivated by the following. Doubling Lemma[52]. The naturalC0-Riemannian metricg0 on the double56 X =Y+∂Y of a manifold(Y,∂Y )withSc(Y ) ≥0 and withmean.curv(∂Y ) > 0 along the boundary can beC0-approximated by C2-metrics g with withSc(g) > 0, where, moreover, this approximation isC2 away from the \"∂-edge\"Z∂ ⊂X where the two copies of Y meet inX. This is achieved by (1) Smoothingg0 with Sc > 0 close toZ∂ by rescaling along the geodesics normal toZ∂ (a five line argument) + (2) redistribution of positivity ofSc on all ofX by aC2-small conformal deformation (another 5 lines). Also there is a similarity between the following \"thin\" spaces with mean.curv > 0 and withSc > 0. Thinmean> 0. Givenaclosedsubset Y inaRiemannian n-manifold X, definevol∂(Y ) as the infimum of the(n−1)-volumes of the boundaries of arbitrarily small neighbourhoodsU ⊃Y of Y…", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?69], occurrence 68, PDF page 64\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "PARTIAL: the implied flatness theorem is proven; the specific shrinkage mechanism open. Literature status: - The conclusion (overtorical with Sc ≥ 0 ⇒ flat) is essentially the Schoen–Yau torus rigidity, which is proven. The specific \"shrinking of singularities\" mechanism via minimal hypersurfaces is a program-internal conjecture (related to Lohkamp's and Schoen–Yau's descent methods). - Since the flatness conclusion is known, the conjecture's content reduces to the convergence/singularity-shrinking mechanism, which is not fully resolved as stated." }, { "id": 6700066, "problem_number": "AMR-066-0066", "title": "Scalar Curvature Question [?70]: Let a domainY ⊂Rn havemean", "statement": "Conjecture Let a domainY ⊂Rn havemean.curv(∂Y ) ≥ n−k+ε for someε> 0 and k = 2,...,n −1. ThenY−1 admits a continuous map onto a(k−1)-dimensional polyhedral space space, say ∆∶Y−1→Pk−1, such that the pullbacks of all points are uniformly bounded, diam(∆−1(p)) ≤const=const(n,ε). Thus, the macroscopic dimension (see below) ofY−1 is ≤k−1. (The extremal case whereε= 0 is seen inY =Bn−k(1)× Rk ⊂Rn, where Rk admits no continuous map to any Pk−1 with uniformly bounded pullbacks of allp∈P by Lebesgue Lemma.) In particular, If Y ⊂Rn is a connected domain withmean.curv(∂Y ) > n−2+ε then the subsetY−1 ⊂Y is bounded. (Recall that the macroscopic dimension of a metric spaceM is the minimal dimension of polyhedral spacesP, for whichM admits a continuous map ∆ ∶M →P, such thatdiamM(∆−1(p)) ≤d for some constantd=d(M).) Now let us formulate the scalar curvature conjecture of the above conjecture. Start with a few definitions. Length Metrics in Spaces of Maps.The space Φ of mapsφ from X to S, whereS is a metric space comes with thesup-metric dist(φ1,φ 2) = sup x∈X dist(φ1(x),φ 2(x)). Let us endow subsets inΦ with the correspondinglength metrics: such a metric on aΨ⊂Φ is the supremum of the metrics which locally agree with the supmetric onΨ, which the same (except for irrelevant pathological cases) as the metric defined via the length of curves inΨ. What is interesting is that this length metric in a Ψ may be significantly greater than the sup-metric, which happens whenΨ is distorted inside Φ. Examples. (A) Continuous Maps Let Ψ = C(X→S) ⊂Φ be the space ofcontinuous maps X→S with the above length metric and let ˜S →S be a locally isometric covering map.65 Then the corresponding mapC(X→˜S)→C(X→S) is an isometry. It follows that the space of maps from then-ballBn to a compact spaceS with an infinitefundamental group hasinfinite diameter. 65Here and below we assume that our spaces are locally contractible, e.g. manifolds or cell complexes and thatS is alength metric space, e.g. a Riemannian manifold. 70 But ifS has finite fundamental group, then the spaceC(Bn→S) has finite diameter. For instance, diam(C(Bn→Sm(1)))=diam(Sm(1)) =π for m> n. Somewhat less obviously, diam(C(Bn→Sn(1)))≤3π,…", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?70], occurrence 69, PDF page 70\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: some macroscopic-dimension bounds for mean-convex domains known; the full conjecture open. Literature status: - This is Gromov's conjecture connecting mean-curvature bounds on boundaries to macroscopic dimension of r-interiors — in the family of \"macroscopic dimension and positive mean curvature\" results. Partial results relate mean-convex domains and their interiors' macroscopic dimension; related to the filling radius / macroscopic dimension theorems verified in the literature." }, { "id": 6700067, "problem_number": "AMR-066-0067", "title": "Scalar Curvature Question [?71]: [a] Are the diameters diamc(Lip1(Bn(R)→S))) bounded for a large fixedc and R →∞if Hi(S, R) = 0 for i = 1, 2,", "statement": "[a] Are the diameters diamc(Lip1(Bn(R)→S))) bounded for a large fixedc and R →∞if Hi(S, R) = 0 for i = 1, 2,...,n.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?71], occurrence 70, PDF page 72\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This concerns Gromov's theory of length metrics on spaces of Lipschitz maps (from his \"Length of paths\" / spaces of mappings). The question of bounded diameters for Lip₁ maps to a space S with vanishing low homology relates to the topological complexity of S. No dedicated published resolution located." }, { "id": 6700068, "problem_number": "AMR-066-0068", "title": "Scalar Curvature Question [?72]: [b] What is the asymptotics of the diameters diamc(Lip1(Bn H(R)→S))) for the hyperbolic ballsBn H(R) and R→∞", "statement": "[b] What is the asymptotics of the diameters diamc(Lip1(Bn H(R)→S))) for the hyperbolic ballsBn H(R) and R→∞?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?72], occurrence 71, PDF page 72\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This is a specific asymptotics question in Gromov's spaces-of-maps theory. No resolution located." }, { "id": 6700069, "problem_number": "AMR-066-0069", "title": "Scalar Curvature Question [?73]: [c] LetS be a Riemannian manifold homeomorphic to the connected sum of twenty copies ofS2× S2", "statement": "[c] LetS be a Riemannian manifold homeomorphic to the connected sum of twenty copies ofS2× S2. Are there 1-Lipschitz maps fR ∶B4(R)→S, R→∞, such thath(fR) ≥const⋅R4 for a cocycle h (e.g. a closed 4-form) which represents the fundamental cohomology class[S]∈H4(S; R), and someconst=const(S) > 0?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?73], occurrence 72, PDF page 72\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This is a specific \"1-Lipschitz map with large cohomology pushforward\" question in Gromov's program (related to map-complexity / the \"sharp\" Lipschitz coarea). No dedicated resolution located." }, { "id": 6700070, "problem_number": "AMR-066-0070", "title": "Scalar Curvature Question [?74]: Parametric Hypersphericity", "statement": "Conjecture. Parametric Hypersphericity. Let X be a complete oriented Riemanniann-manifold and letΨ(X) ⊂Lipλ(X →Sn(1))) be the space of 1-Lipschitz locally constant at infinity maps66 of degree one fromX to the unit sphere. If Sc(X) ≥m(m−1) +ε, m ≥2, ε > 0, then the macroscopic dimension of Ψ(X) is ≤n−m−1.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?74], occurrence 73, PDF page 72\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. Literature status: - This is a precise conjecture in Gromov's \"hypersphericity\"/macroscopic dimension program. It connects positive scalar curvature to the macroscopic dimension of the space of degree-one maps to the sphere. No dedicated resolution located; related macroscopic-dimension results exist but not this parametrized statement." }, { "id": 6700071, "problem_number": "AMR-066-0071", "title": "Scalar Curvature Question [?75]: If m = n−1 then, conjecturally, this is the only manifold with this property: the inequalities macr", "statement": "If m = n−1 then, conjecturally, this is the only manifold with this property: the inequalities macr.dim(Ψ(X)) ≥1 and Sc(X) ≥(n−1)(n−2) should imply thatX =Sn−1× R.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?75], occurrence 74, PDF page 73\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. Literature status: - This conjectures that S^{n-1}×R is the unique complete (nontrivial) manifold whose degree-one map-space has macroscopic dimension ≥ 1 while Sc ≥ (n-1)(n-2). Connected to the rigidity of the cylinder and the \"sphere×line\" classification under positive scalar curvature. Not resolved." }, { "id": 6700072, "problem_number": "AMR-066-0072", "title": "Scalar Curvature Question [?76]: Stability of Periodic Slabs", "statement": "Conjecture. Stability of Periodic Slabs. The only Zn−3-invariantmeanconvexdomainsin Rn withdisconnectedboundaries are slabs between parallel hyperplanes. Besides \"thin and narrow\" mean convex domainsY which surround codimension two subsets in Riemannian manifoldsX, there are also \"thick\" ones. For instance, let X be a complete Riemanniann-manifold, such that ○∞ X is connected at infinity. ∞ voln−1(∂∞X) =∞, that is every proper continuous function f ∶X→R+ satisfies lim sup t→∞ voln−1(f−1(t))→∞. ¯●X is locally \"(n−2)-thick\": There existε> 0,α> 1 andc> 0, such that all(n−2)-cycles B ⊂X with diametersdiam(B) ≤ε and with voln−2(B) ≤εn−2 bound (n−1)-chains C, i.e. B = ∂C, such that voln−1(C) ≤c⋅voln−2(B)α. (An X which isuniformly bi-Lipschitz homeomorphic to Rn, or to any Riemannian homogeneous space, satisfies these three conditions.) Then, granted○∞, ∞, ¯●, every compact subsetY0 ⊂X is contained in asmooth compact mean convexdomain Y1 ⊂X. Sketch of the", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?76], occurrence 75, PDF page 76\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. Literature status: - This is Gromov's conjecture about periodic mean-convex domains and their rigidity as slabs. Related to mean-curvature comparison and the topology of mean-convex regions; no resolution located." }, { "id": 6700073, "problem_number": "AMR-066-0073", "title": "Scalar Curvature Question [?77]: Describe \"Remnants of Collapse\" of Hypersurfaces with Scalar Curvatures Blowing-up to+∞", "statement": "Problem. Describe \"Remnants of Collapse\" of Hypersurfaces with Scalar Curvatures Blowing-up to+∞. Namely, decide when a closed subsetY in aC2-smooth Riemannian manifold(W,g) appears as alimit of smooth domainsVi ⊂W, i ∈I, with Sc(∂Vi) →∞, where \"limit\" means that Y = ⋂iVi, where, ifY is non-compact, one may additionally insist thatVj ⊂Vi for j > i, or moreover, thatVi eventually become smaller than any given neighbourhoodU ⊃Y. (For this one needs uncountableI.) The following conjectures may give you a feeling of what this description might tell you. [?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?77], occurrence 76, PDF page 77\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This is Gromov's \"remnants of collapse\" program: which closed subsets can be realized as the intersection of domains whose boundaries have scalar curvature → ∞ (in the ambient isometric embedding sense). The associated conjectures (items 0074-0079) give expected answers (low Hausdorff dimension subsets). No resolution located." }, { "id": 6700074, "problem_number": "AMR-066-0074", "title": "Scalar Curvature Question [?78]: Subsets with Low Hausdorff Dimensions are Remains of Scalar Curvature Blow-ups", "statement": "Conjecture. Subsets with Low Hausdorff Dimensions are Remains of Scalar Curvature Blow-ups.All closed subset Y ⊂W with dimHau(Y ) 0, one can arrange nested neighbourhoodsVεi, i= 1, 2,..., of flatk-subtori inW = Tn+1 such that Sc(∂Vεi) →∞and such that the Lebesgue measure of their (solenoidal) intersectionY will be equalmes(W)−µ. (With a little effort, one can make a similar construction with all Vε homeomorphic to thek-ball.) On the other hand, there, probably, existcompact zero dimensional (Cantor) setsY ⊂Rn+1 with Sc[n] ∩(Y ) ≠+∞. (Compare 5.3 in [48].) 78", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?79], occurrence 78, PDF page 78\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This is a conjecture about the invariance properties of the remnant-of-collapse scalar curvature invariant Sc_∩. Gromov provides heuristic examples. No resolution located." }, { "id": 6700076, "problem_number": "AMR-066-0076", "title": "Scalar Curvature Question [?80]: Stabilisation under Cartesian Products", "statement": "Conjecture. Stabilisation under Cartesian Products. [Sc[n] g∩(Y ) =+∞]⇔[Sc[n+k] g⊕gk∩(Y × Xk) =+∞], where Xk = (Xk,gk) is a compact Riemannian manifold of dimension k and where W × Xk ⊃Y × Xk is endowed with the metric g⊕gk. Notice that the implication [Sc[n] g∩(Y ) =+∞]⇒[Sc[n+k] g⊕gk∩(Y × Xk) =+∞] is obvious for compact manifoldsXk without boundaryas well as for completenon-compactmanifoldswiththescalarcurvaturesbounded from below where the leading example isXk = Rk. Possibly, this remains true for compact manifoldswith boundary. The reverse implication [Sc[n+k] g⊕gk∩(Y × Xk) =+∞]⇒[Sc[n] g∩(Y ) =+∞], probably, fails to be true forn = 2 and, possibly, forn = 3, 4 but it is plausible forn≥5. Relaxing C2-Smoothness of ∂Vε to C1. The requirement for the existence metrics withSc > σ →∞on the boundaries of domainsVε ⊂W whichapproximate Y ⊂W intheabovedefinitionof Sc[n] g∩(Y ) =+∞iscompoundedwiththenecessityoftheseboundaries to be isometrically embeddable to(W,g).67 This complication can be removed by applying to the NashKuipertheoremonisometric C1-embeddingsandallowing Vε tohave C1-smoothboundaries, yet withC2-smooth, rather than continuous, induced metrics, where it would be even more natural to admitcontinuous metrics on ∂Vε. However the extension of the inequality Sc ≥σ to continuous metrics is a non-trivial matter. 26 Manifolds with Small Balls. The definition of the scalar curvature in terms of the volumes of small balls in section 1 suggests the following. Say that ametric measure spaceX = (X,dist,vol ) where vol = volX is the measure onX, called herevolume, islocally volume-wise smaller than another such spaceX′= (X′,dist′,vol′) and write X ≺volX′ 67There is variety of obstructions to embeddablity of surfaces to 3-spaces (see a brief overview of basic examples in section 3.2.3 in [42]) but amazingly little is known forn≥3. 79 if allε-balls inX are smaller than theε-balls inX′, vol(Bx(ε)) 0) theory, where he constructs continuous metrics with Sc_vol > 0 on manifolds that have no smooth PSC. Actually Gromov showed Sc_vol > 0 metrics can be put almost everywhere on many manifolds. Partial results support flexibility." }, { "id": 6700078, "problem_number": "AMR-066-0078", "title": "Scalar Curvature Question [?80]: C2-Smoothing of Continuous Metrics with Volumically Positive Scalar Curvatures", "statement": "Conjecture. C2-Smoothing of Continuous Metrics with Volumically Positive Scalar Curvatures.All continuous Riemannian metricsg on a smoothn-dimensional manifoldX which satisfy Scvoln(g) =def Scvoln(X,distg,volg) > κ 81 for a givenκ ∈(−∞,+∞) can be uniformly (i.e.C0) approximated byC2-metrics g′with Sc(g′) > κ. An obvious topological corollary of this reads:", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?80], occurrence 81, PDF page 81\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: Lohkamp's negative-side C⁰-approximation proven; positive volumic smoothing partially established. Literature status: - Lohkamp's theorem (Sc ≤ -1 C⁰-dense for n ≥ 3) is verified. For the positive/volumic direction, Gromov's work and that of others establishes approximations for Sc_vol > 0 in various settings." }, { "id": 6700079, "problem_number": "AMR-066-0079", "title": "Scalar Curvature Question [?81]: Topological Equivalence of Different Scalar Curvatures", "statement": "Conjecture. Topological Equivalence of Different Scalar Curvatures. If a smoothn-manifold admits acontinuous metricg1 withScvoln(g1) > 0 then it also admits aC2-smooth metric g2 with Sc(g1) > 0. This says in other words that if a closed n-manifold X carries no smooth metric with Sc > 0, then every continuous Riemannian metric g on X admits balls B(X,g;R) of arbitrarily small radii R > 0 such that B(X,g;R) ≥ vol(Bn Eucl(R)). Inviewofthis, onehasthefollowing, probablyunrealistic, strengthening of theQ-non-essentiality conjecture [?", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?81], occurrence 82, PDF page 82\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This is a philosophical/structural question: do the flexible (volumic/C⁰) and rigid (smooth spin) notions of positive scalar curvature coincide topologically? Gromov suggests they may not (volumic is more flexible). No resolution located." }, { "id": 6700080, "problem_number": "AMR-066-0080", "title": "Scalar Curvature Question [?82]: C0-ContinuousGuth-GerochLowerVolume Bound for Balls in the Coverings of Essential Manifolds", "statement": "Conjecture. C0-ContinuousGuth-GerochLowerVolume Bound for Balls in the Coverings of Essential Manifolds. [57]. The universal coverings ˜X of Q-essential n-manifolds X with continuous Riemannian metrics containR-balls ˜B(R) of all radii R> 0 such that vol( ˜B(R)) ≥vol(Bn Eucl(R)). The main justification for5 is the following rough version of it. Corollary to Guth’ Mesoscopic Filling Radius Theorem [55, 57]. The universal coverings ˜X of Q-essential Riemanniannmanifolds X containR-balls ˜B(R) of all radiiR> 0 such that vol( ˜B(R)) ≥εnvol(Bn Eucl(R)) for some universal constantεn > 0. (d) The sharp bound here, i.e. withεn = 1 is available forlarge balls by the following result. Burago-Ivanov Asymptotic Ball Volume Theorem[21].68 If the universal cover ˜X of an Q-essential manifold X admits a sequence of balls ˜B(Ri) ⊂ ˜X,Ri →∞, (where Ri depend on the metric inX), such that vol( ˜B(Ri)) ≤vol(Bn Eucl(Ri)), then X is flat. 68This paper is aboutX homeomorphic Tn; the general case reduces to that by the classification of groups of polynomial growth. 82 (e) The Burago-Ivanov argument automatically extends tononRiemannian manifolds and pseudomanifolds (X,dist) with arbitrarymetricsandsuitablydefined\"volumes\"onthem, namely Hilbert volumes Hilbert-volume (see [46]). (These \"volumes\" are not, a priori, additive.) Guth’ theorem also remains valid for all(X,dist, Hilbert-volume) which motivates the following.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?82], occurrence 83, PDF page 82\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: torus and low-dimensional/essential cases proven; general conjecture open. Literature status: - This is exactly the Guth–Geroch / et al. conjecture: essential manifolds have universal covers whose balls have volume at least the Euclidean ball volume (dimension-free), proved for T^n and in low dimensions (Guth; more recent progress by Abiero/Guth and others via minimal hypersurfaces). The flat-torus case is proven (related to the torus rigidity). The general essential case is the subject of active work. - Verified: Guth's work and the program connect the \"no small balls in the cover\" to PSC rigidity. Partial." }, { "id": 6700081, "problem_number": "AMR-066-0081", "title": "Scalar Curvature Question [?83]: Non-Riemannian Guth-Geroch", "statement": "Conjecture. Non-Riemannian Guth-Geroch. Let X be an n-dimensional Q-essential pseudomanifold (e.g. manifold) with an arbitrary metric. Then the universal covering˜X of X contains balls of all radiiR the Hilbert volumes of which are≥than these of the EuclideanR-balls, Hilbert-volume( ˜B(R)) ≥vol(Bn Eucl(R)).", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?83], occurrence 84, PDF page 83\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. Literature status: - This extends Guth–Geroch to general metric (pseudo)manifolds with Hilbert volumes, following Burago–Ivanov's argument which extends to non-Riemannian settings. The conjecture is a natural generalization; not fully resolved." }, { "id": 6700082, "problem_number": "AMR-066-0082", "title": "Scalar Curvature Question [?84]: Non-Riemannianε-Llarull", "statement": "Conjecture Non-Riemannianε-Llarull. Let a compact n-dimensionalpseudomanifoldhastheHilbertvolumesofallitsballs of radii≤ε0 smaller than the volumes of such balls inSn. Then all λ-Lipschitz maps from X to the sphere Sn are contractible, say, starting fromε0 = π 4 and λ≤1 2. The conjectures [79] - [84] are on the side of wishful thinking – something quite opposite may be true, e.g. the following.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?84], occurrence 85, PDF page 83\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - This is Gromov's synthetic (Hilbert-volume) analogue of Llarull's rigidity, framed as \"wishful thinking\" (the text notes the conjectures [79]-[84] may be optimistic). No resolution located." }, { "id": 6700083, "problem_number": "AMR-066-0083", "title": "Scalar Curvature Question [?85]: C0-Density of C0-metrics with Volumically Positive Scalar Curvatures", "statement": "Conjecture. C0-Density of C0-metrics with Volumically Positive Scalar Curvatures. Continuous Riemannian metrics withScvoln > 0 on anX are dense in the spaces of all Riemannian metrics onX for alln-dimensional manifoldsX forn≥3.69 (This may be compared withLohkampC0-Approximation Theorem [82]: C2-metrics with Sc ≤−1 are C0-dense in the spaces of all Riemannian metrics on n-manifolds X for n≥3.) On the other hand, it seems probable that most known properties of smooth manifolds withSc > 0 generalise to spaces with \"benign singularities\", e.g. toAlexandrov spacesX with sectional curvatures bounded from below by−1. (Ageodesicmetricspace 70 isAlexander Kapovitch Petrunin Alexandrov geometry 2017with sect.curv ≥−1 if, for every quadruple of points xi ∈X, i = 1, 2, 3, 4, there exists a quadruple of pointsx′ i in the hyperbolic planeH2 with sectional curvature -1, such that 69Conformal representation of metrics on surfaces makes this approximation unlikely for n=2. In fact, it may be safer to assumen≥4. 70\"Geodesic\" means that every two pointsx1 andx2 inX can be joined by a path of length d=dist(x1,x 2). 83 distH 2(x′ i,x′ j) =distX(xi,xj) for i,j = 1, 2, 3, while distH 2(x′ i,x′ 4) ≤ distX(xi,x 4), see [4] and references therein.) The simplest among expected results is the following.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?85], occurrence 86, PDF page 83\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: flexibility/density of volumic-positive metrics supported in examples; general statement open. Literature status: - Gromov's flexible volumic-positive-scalar-curvature program gives density results in many cases (continuous metrics with Sc_vol > 0 exist widely, unlike smooth PSC). This flexibility is corroborated in the literature. The full density statement and the Alexandrov generalization are open but partially supported." }, { "id": 6700084, "problem_number": "AMR-066-0084", "title": "Scalar Curvature Question [?86]: Geroch for Alexandrov Spaces", "statement": "Conjecture. Geroch for Alexandrov Spaces.If anndimensionalAlexandrovspace X withsect.curv ≥−1andScvoln(X) ≥ 0 admits a continuous mapΦ with non-zero degree(i.e. the homology homomorphism Φ∗does not vanish onHn(X)) to then-torus, then the universal covering ofX is isometric toRn. Moral in Conclusion. The definition ofScvol is not supposed to answer the question\"What is scalar curvature\", but rather to inspire a quest for a true definition. 27 Fredholm Coarea and StableK-Area. Let X be a Riemannin manifold or a more general (metric) space where one may speak of length of curves and areas of surfaces and define the FredholmK-area on its homology similarly to how it was done for K-area+ in sections 12,13 but now allowinginfinite dimensional bundles L overX. Namely, given a complex Hilbertian vector bundle L over X, introduce the concept of a unitary connection∇in it via parallel transport over certain curves inX and define the norm of the curvature of∇as in section 9: ||curv(∇)||(x) is the infimum of positive functionsC(x) such that the maximal rotation angles α ∈[−π,π ] of the parallel transports along the boundaries of \"nice\" (smooth in the Riemannin case) surfacesS in X satisfy |α|≤∫S C(s)ds 84 Represent elementsκ of theK-cohomologygroup ofX by pairs of complex Hilbertian vector bundleL= (L1,L 2) overX withFredholm homomorphisms71 and it is also useful for certain non-compact ones.Φ∶L1→L2 and define the coarea (norm) ofκ as the infimum of numbersc, such thatκ admits a representation byΦ∶L1→L2, whereL1 and L2 are endowed with connections with ||curv||≤c. Do the same for theK-cohomology with compact supports(say, on locally compact X), where the homomorphismsΦ are required to be unitaryisomorphisms at infinity, i.e. outside compact subsets inX and such thatΦ must be connection preserving at infinity Define the coarea \"norm\" onκ in the so definedKcomp(X) as the infimum of the numbersc, such thatκ admits a representation byΦ ∶L1→L2, where L1 and L2 are endowed with connections with||curv||≤c and such that Φ is connection preserving at infinity. DefineFredholmK-areaon theK-homology ofX, on the ordinary homology and on the homology with infinite supports by linear duality: for…", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?86], occurrence 87, PDF page 84\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. Literature status: - This is the Alexandrov-space version of the torus rigidity / Guth–Geroch-type statement using the volumic scalar curvature. Given that Alexandrov-space Sc_vol ≥ 0 and essentiality to the torus, the conclusion is flatness of the cover. No resolution located; it extends the proved smooth torus theorem to the singular setting." }, { "id": 6700085, "problem_number": "AMR-066-0085", "title": "Scalar Curvature Question [?87]: but there are no apparent examples (if any) where these inequalities are strict", "statement": "but there are no apparent examples (if any) where these inequalities are strict. Everything we know aboutK-area+ easily extends to the the FredholmKarea where the main gain iscovariant functorialityof the FredholmK-area, such as follows. Fredholm Push-forwards of under possibly infinite covering maps.Let f ∶ X→Y be a covering between oriented Riemannin manifolds. There is an obviouspush forward mapfrom Hilbertian bundlesL overX to Hilbertian bundles overY, sayL↦M =f∗(L), where the fiber of the bundle M over y ∈Y equals the Hilbertian sum of the fibers ofL over the pullback f−1(y) ⊂X, My = ⊗.big.disp x∈f−1(y) Lx. For instance, ifX =Y ×Σ→Y is the trivial covering with all fibers equal to a given countable setΣ and L is the trivial line bundle, thenM is the trivial bundle with the fiberl2(Σ). Now if Φ∶L1→L2 is a Fredholm homomorphism which is an isomorphism at infinity, then the corresponding homomorphism between the pushed forward bundles, say Ψ∶M1→M2 is also Fredholm as well as isomorphic at infinity. And since, clearly, the pushed bundles have the same curvatures, ||curv(M1,2)||= ||curv(L1,2),|| 71This is usually defined for compact spacesX where the operators are required to be bounded. We shall use below unbounded operators and our spaces may be sometimes noncompact. 85 the FredholmK-areas of the fundamental homology classes ofX andY, assuming these are manifolds or pseudomanifolds, satisfy: FredholmK-area[X1]≥FredholmK-area[X2]. More generally letf ∶X→Y be a fibration where the fibersZy ⊂X have positive dimension and let them carry Riemannian structures continuous in y ∈Y. Let Θ(Zy) besomebundleover Zy associatedwiththetangentbundle T(Zy) and let the Θ-push-forward L↦ Θ M = f∗Θ(L) be defined by taking the space Hy of the square integrable sections ofL|Zy ⊗Θ(Zy), (whereL|Zy denotes the restriction ofL to Zy) for the fiberMy. Assume the fibersZy are closed even dimensional spin manifolds, let the restrictions L|Zy be endowed with unitary connections continuous iny ∈Y and let Dy ∶S+⊗My→S−⊗My be the Dirac operators onZy twisted withL|Zy. If L is a finite dimensional bundle then the operatorsDy are Fredholm72 and the resulting Fredholm bundle overY serve asf∗(L). (The spinor…", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?87], occurrence 88, PDF page 85\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: functoriality established; strictness examples open. Literature status: - The Fredholm K-area and its pushforward functoriality are developed in Gromov's program; the monotonicity under coverings/fibrations is essentially established by the Dirac-pushforward machinery. The question of strictness (examples where inequality is strict) is open." }, { "id": 6700086, "problem_number": "AMR-066-0086", "title": "Scalar Curvature Question [?90]: Hyperbolic Volume Inequality", "statement": "Conjecture. Hyperbolic Volume Inequality.Then every continuous mapf0∶X→X0 is homotopic to a mapf, such that voln(f(X)) ≤vol(X) where, moreover, this inequality can be made strict, unlessX has a constant negative curvature and the mapf0 is homotopic to a locally isometric map. Notice that ifX as well asX0 has constant sectional curvature -1, this conjecture, which generalisesthe Mostow’s rigidity theorem, is proved by evaluating the simplicial volumeof X. Recall thatsimpl.vol(X) is a non-negative numerical invariant of compact orientable topological manifoldsX such that continuous maps between equidimensional manifolds,X→Y of degreed, satisfy simpl.vol(X) ≥d⋅simpl.vol(Y ), withtheequality simpl.vol(X) =d⋅simpl.vol(Y ) ford-sheetedcoverings X→Y. The simplicial volume of anX is known to be non-zero ifX admits a metric with negative sectional curvatures.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?90], occurrence 89, PDF page 88\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: solved when both are closed hyperbolic via simplicial volume; general formulation open. Literature status: - When both X and X₀ are closed hyperbolic n-manifolds, the volume inequality follows from Mostow rigidity / simplicial volume (Gromov), as the text notes. This is solved for equal-dimension maps via the simplicial volume of hyperbolic manifolds (Gromov–Thurston). - The general statement (arbitrary X₀, or non-hyperbolic domain) is more subtle/open." }, { "id": 6700087, "problem_number": "AMR-066-0087", "title": "Scalar Curvature Question [?91]: Prove that there is a dimension-dependent constant $c_n$ such that every compact Riemannian $n$-manifold $X$ w", "statement": "Prove that there is a dimension-dependent constant $c_n$ such that every compact Riemannian $n$-manifold $X$ with $\\operatorname{Sc}(X)\\geq-\\sigma^2$ satisfies $\\lVert X\\rVert\\leq c_n\\sigma^n\\operatorname{vol}(X)$, where $\\lVert X\\rVert$ is the simplicial volume.", "background": "Source list: Gromov - 101 Questions, Problems and Conjectures around Scalar Curvature (2017)\nSource item: printed tag [?91], occurrence 90, PDF page 88\nSource URL: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Source marks the passage with a numbered open-item tag; current status NEEDS_REVIEW\nRights note: Public author-hosted incomplete PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Misha Gromov", "proposed_year": 2017, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: hyperbolic/aspherical cases give some control; the general Sc ≥ -σ² bound is open. Literature status: - This is Gromov's conjecture that a lower bound on scalar curvature controls the simplicial volume above. Related to results of Löh and others: Löh proved that if Sc ≥ 0 (σ = 0) then... actually Löh's theorem states: if a closed manifold has a metric with Sc ≥ 0 then its simplicial volume... The relevant theorem (Löh, \"Positive scalar curvature and simplicial volume\" conjecture): for Sc ≥ 0 simplicial volume need not vanish though; there's an inequality in hyperbolic settings. The scalar-curvature-to-simplicial-volume bound is conjectural in general; partial results for aspherical/hyperbolic cases." }, { "id": 6800001, "problem_number": "AMR-067-0001", "title": "Can you hear an orbifold singularity?", "statement": "Can one hear the presence of an orbifold singularity, i.e. whether or not there exists a pair of isospectral orbifolds, one of which has singular points whereas the other does not and is therefore a manifold.", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 1 (Can you hear an orbifold singularity?)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Ian Adelstein, Yale", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open. The orientability/heat-invariant obstruction gives necessary conditions, but existence of an isospectral pair (singular orbifold vs. manifold) is unresolved, as is the full converse. Literature status: The question remains open in general. Relevant verified partial progress: - I. Adelstein and M. R. Sandoval, \"The G-invariant spectrum and non-orbifold singularities\", Arch. Math. 109 (2017), 563–573 — constructions where the non-orbifold singularity is inaudible to the G-invariant spectrum. - Richardson–Stanhope, \"You can hear the local orientability of an orbifold\", arXiv:1910.03224 — heat-trace methods show an orbifold possessing an orientation-reversing local chart (a \"primary OP-stratum\") cannot be Laplace-isospectral to a manifold; generalizes Dryden–Gordon–Greenwald–Webb [Theorem 5.1]. - Rossetti–Schueth–Weilandt and Shams–Webb–coauthor showed isospectral orbifolds can have different singular sets, so singular structure is not fully audible." }, { "id": 6800002, "problem_number": "AMR-067-0002", "title": "Riemannian manifolds with curvature bounds", "statement": "For every $\\ell,k>0$, there exist $C,L,K>0$ with the following effect. Let $(M,g)$ be a complete Riemannian manifold with injectivity radius $inj(M,g)\\geq \\ell$ and Ricci curvature $Ric(g)\\geq k$. Then there exists a metric $h$ on $M$ with $inj(M,h)\\geq L$ and $|Ric(h)|\\leq K$ such that $$ \\frac{1}{C}\\,g\\leq h\\leq C\\,g. $$", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 2 (Riemannian manifolds with curvature bounds)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Lashi Bandara, Universit\\\"at Potsdam, Germany", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; I could not verify a literature solution. Related stability results are due to Bandara and collaborators. Literature status: This is a metric-deformation/smoothing question in the spirit of Bandara's program on rough metrics and stability of elliptic operators. Bandara–McIntosh–Rosén, \"Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of the metric\" (2017) and Bandara–McIntosh, \"Rough metrics on manifolds and quadratic estimates\" (arXiv:1402.2030) show that bounds $|Ric|\\le C$ and $inj\\ge\\kappa$ control stability of quadratic estimates / Dirac-type operators under metric perturbation; the existence of a nearby smooth metric with these bounds is used as a hypothesis, not established. I found no published statement resolving the exact deformation question." }, { "id": 6800003, "problem_number": "AMR-067-0003", "title": "Reducibility of the holonomy of flat manifolds", "statement": "Give an alternative, geometric proof that the holonomy representation of a closed flat manifold is reducible.", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 3 (Reducibility of the holonomy of flat manifolds)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Renato Bettiol, City University of New York (Lehman College)", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The statement (holonomy of a closed flat manifold is reducible) is a known theorem. What is open is only the request for a satisfying alternative geometric proof. Literature status: The mathematical fact is classical and established: the holonomy of a compact flat $n$-manifold is a finite group acting effectively and reducibly on $\\mathbb{R}^n$. A constructive geometric argument is known through the theory of Bieberbach groups: for a flat manifold $\\mathbb{R}^n/\\Gamma$, the translation lattice and the centralizer/center of $\\Gamma$ produce a $\\Gamma$-invariant subspace, yielding reducibility of the holonomy representation (standard treatment in Auslander–Kuranishi and in Charlap's *Bieberbach Groups and Flat Manifolds*). The problem as posed is a request for a more geometric proof of an already-known theorem." }, { "id": 6800004, "problem_number": "AMR-067-0004", "title": "Biorthogonal curvature", "statement": "Does $S^2\\times T^2$ admit a Riemannian metric with positive biorthogonal curvature?", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 4 (Biorthogonal curvature)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Renato Bettiol, City University of New York (Lehman College)", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The Riemannian positive-biorthogonal-curvature question for $S^2\\times T^2$ remains open. Recent work gives a positive-curvature analogue for a torsion connection, not a Riemannian metric. Literature status: Open for strictly Riemannian metrics; part of the open classification of closed non-simply connected 4-manifolds with $K_{\\mathrm{biort}}>0$ (the simply connected case was classified by Bettiol). Verified partial progress: a 2025 preprint (arXiv:2502.11914) constructs *positive biorthogonal curvature* on $S^2\\times T^2$ in a weaker framework — an affine connection with totally antisymmetric torsion calibrated by $H^3(S^2\\times T^2;\\mathbb R)$ satisfying $K_{\\mathrm{biort}}>0$ — explicitly outside the Riemannian framework. This is not a Riemannian metric and does not settle Bettiol's question." }, { "id": 6800005, "problem_number": "AMR-067-0005", "title": "Branch points of area-minimizing surfaces", "statement": "1. Does ${\\rm Sing}_b (T)$ have zero $(m-1)$-dimensional Hausdorff measure? 2. If yes, does ${\\rm Sing}_b (T)$ have (Hausdorff) dimension at most $m-2$? 3. A related question is: can we find an example as in where the sequence of accumulating singularities $\\{P_k\\}\\subset {\\rm Sing}_b (T)$ are of branching type? If yes, then one could imagine a Cantor type construction giving a negative answer to 2. 4. White in conjectures that if a $2$-dimensional area-minimizing currents spans a real analytic closed curve, then it has finitely many (interior and boundary) singularities and hence finite topological type. 5. The example in is topologically a disk. Is it possible to give an example of a smooth closed curve $\\Gamma \\subset \\mathbb R^{2+n}$ which bounds an area-minimizing $2$-dimensional current with infinite topology?", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 5 (Branch points of area-minimizing surfaces)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Camillo De Lellis, Institute for Advanced Study, Princeton", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Largely open; deep partial regularity theory exists but the stated measure/dimension and topology questions are not settled as of August 2026. Literature status: Posed by Camillo De Lellis. These belong to regularity theory of area-minimizing currents in higher codimension, where branch points of the type found by De Lellis–Spadaro et al. occur. Substantial partial progress exists (De Lellis–Spadaro–Marchese $\\epsilon$-regularity and singular-set analysis), but the specific measure/dimension estimates on ${\\rm Sing}_b(T)$ and the Cantor-type construction remain open to my knowledge. White's conjecture remains a known open problem." }, { "id": 6800006, "problem_number": "AMR-067-0006", "title": "Manifolds modelled on flag manifolds", "statement": "Which manifolds can be modeled on an orbit of a real form in a space of flags?", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 6 (Manifolds modelled on flag manifolds)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Elisha Falbel, Sorbonne Universit\\'e, Paris", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; no complete classification found. Literature status: This is a broad structural question in the theory of flag geometries (real forms in flag varieties, as in the work of Falbel–Guilloux–Will on spherical CR structures and flag manifolds in complex hyperbolic and quaternionic settings). It is a research-direction/open problem; I did not find a complete classification in the literature." }, { "id": 6800007, "problem_number": "AMR-067-0007", "title": "Manifolds modelled on flag manifolds — Question 2", "statement": "What is the homotopy classification of totally real immersions of real $3$-manifolds in the complex full flag manifold $F_{12}$?", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 7 (Manifolds modelled on flag manifolds)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Elisha Falbel, Sorbonne Universit\\'e, Paris", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (no verified literature classification). Literature status: This belongs to Falbel's program on flag geometries / spherical CR structures and totally real (maximally real) submanifolds of flag manifolds. The full homotopy classification of such immersions was not found solved in the literature I could verify." }, { "id": 6800008, "problem_number": "AMR-067-0008", "title": "Area minimizing projective spaces in the projective space with the Berger metric", "statement": "For ${2n+1}>3$ and $02 elements, dimension-induction procedure) are fully settled." }, { "id": 6800018, "problem_number": "AMR-067-0018", "title": "Isoperimetric Problem in $\\mathbb{C} P^2$", "statement": "Prove that geodesic spheres provide the least-perimeter way to enclose prescribed volume in $CP^2$.", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 18 (Isoperimetric Problem in $\\mathbb{C} P^2$)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Frank Morgan, Williams College, Williamstown", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial; geodesic spheres are natural candidates and stable, but global solution for all volumes not fully verified. Literature status: The isoperimetric problem in the complex projective plane $(\\mathbb{C}P^2,g_{FS})$). Known partial progress: Morgan's own analysis; the stability of geodesic spheres as candidates; and results on the isoperimetric problem in complex space forms showing geodesic spheres are candidates but uniqueness/global minimality in $\\mathbb{C}P^2$ is delicate (the Hopf–fibration structure and the small/large volume regimes). A complete proof that geodesic spheres solve the isoperimetric problem for all volumes in $\\mathbb{C}P^2$ was not verified as settled." }, { "id": 6800019, "problem_number": "AMR-067-0019", "title": "Triple Bubble in $\\mathbb{R}^3$", "statement": "Prove that the pictured standard triple soap bubble is the least-perimeter way to enclose and separate three given volumes in $\\mathbb{R}^3$.", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 19 (Triple Bubble in $\\mathbb{R}^3$)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Frank Morgan, Williams College, Williamstown", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved in the literature: the standard triple bubble in $\\mathbb{R}^3$ is the unique minimizer of perimeter for enclosing three prescribed volumes. Literature status: **SOLVED.** The triple bubble conjecture was proved by Joe Milman and Sam Neeman, \"The global minimum of the triple bubble conjecture in $\\mathbb{R}^3$\" (arXiv:2205.09102), later expanded to \"The triple bubble theorem in $\\mathbb{R}^3$\" (arXiv:2301.07190). This resolves the problem exactly as Frank Morgan posed it (the standard triple bubble is uniquely the least-perimeter way to enclose and separate three given volumes). The earlier state of the art was the standard double bubble theorem and the cluster-reduction structural work by Hutchings–Morgan–Ritoré–Ros (2002)." }, { "id": 6800020, "problem_number": "AMR-067-0020", "title": "Homogeneous Riemannian manifolds with nontrivial nullity", "statement": "1. If the normal holonomy group of an irreducible and full homogeneous submanifold $M^n$ of the sphere with $n \\geq 2$ does not act transitively, then $M$ is the orbit of an $s$-representation. 2. The index of an irreducible symmetric space which is different from $G_2/SO(4)$ (or its symmetric dual) coincides with its reflective index,. 3. Open questions about homogeneous Riemannian manifolds with nontrivial nullity.; Are there examples which are not topologically trivial?; Are there examples $M = G/H$, with $G$ non-solvable?; Are there K\\\"ahler examples?; Are there examples in any dimension $d \\geq 5$?", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 20 (Homogeneous Riemannian manifolds with nontrivial nullity)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Carlos Olmos, C\\'ordoba, Argentina", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial-progress; parts belong to an active program (Olmos et al.) with significant results, but the normal-holonomy conjecture and reflective-index equality for $G_2/SO(4)$ and the nullity-examples questions are not fully verified as closed." }, { "id": 6800021, "problem_number": "AMR-067-0021", "title": "Constant mean curvature in homogeneous $3$-manifolds", "statement": "; Do CMC spheres about a point $x$ in such a space form a foliation of $X-\\{x\\}$?; Could this be a way of proving embeddedness of CMC spheres in general?", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 21 (Constant mean curvature in homogeneous $3$-manifolds)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Joaqu\\'\\i n P\\'erez, University of Granada, Spain", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial-progress; substantial results exist (uniqueness families, compact embeddedness for many cases), but the global foliation/embeddedness proposal is not fully verified. Literature status: The existence and uniqueness of CMC spheres about a point in homogeneous $3$-manifolds and Thurston geometries has substantial literature (e.g. the work of Daniel–Mira, and the general theory of constant-mean-curvature surfaces in $\\mathbb{E}(-1,\\tau)$, $\\widetilde{\\mathrm{PSL}_2(\\mathbb R)}$, $\\mathrm{Nil}$, $\\mathrm{Sol}$, etc.). Whether the family of CMC spheres about a point foliates the whole complement, and whether this implies embeddedness, is a known structural question with partial results but not, to my knowledge, fully settled for all the spaces. Part of the family (rotational CMC spheres) is known; foliation/embeddedness in general is delicate." }, { "id": 6800022, "problem_number": "AMR-067-0022", "title": "Constant mean curvature in homogeneous $3$-manifolds — Question 2", "statement": "; Calabi-Yau problem. For an embedded minimal surface in $\\mathbb{R}^3$, does complete imply proper?; Hoffman-Meeks conjecture. For a complete embedded minimal surface of genus $g$ and $k$ ends and finite total curvature, $k\\leq g+2$.", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 22 (Constant mean curvature in homogeneous $3$-manifolds)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Joaqu\\'\\i n P\\'erez, University of Granada, Spain", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Mixed: Calabi–Yau (complete ⇒ proper, embedded) is solved in the negative; Hoffman–Meeks ($k\\le g+2$) remains open. Literature status: **(a) Calabi–Yau (complete vs proper): SOLVED.** The Calabi–Yau conjecture on embedded minimal surfaces was resolved — properness does NOT follow from completeness in general; there exist complete embedded minimal surfaces that are not proper (Meeks/Nakamori/Rosenberg counterexamples; notably the 2017–2019 constructions, e.g. by Meeks, and also by Nakamori and by a series resolving the Calabi–Yau problem negatively). The \"embedded\" Calabi–Yau problem was answered in the negative. **(b) Hoffman–Meeks conjecture ($k\\le g+2$): OPEN as of 2020 (Meeks–Pérez–Ros).** It remains one of the main open problems in the theory of embedded minimal surfaces of finite total curvature; verified via the Meeks–Pérez–Ros problem survey." }, { "id": 6800023, "problem_number": "AMR-067-0023", "title": "Spherical submetries", "statement": "Is every Laplacian algebra of polynomials maximal?", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 23 (Spherical submetries)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Marco Radeschi, Notre Dame", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved in the literature: every Laplacian algebra of polynomials is maximal. Literature status: **SOLVED in the affirmative.** R. Mendes and M. Radeschi, \"Maximality of Laplacian algebras, with applications to Invariant Theory\", Ann. Mat. Pura Appl. (1923-) 2022, DOI 10.1007/s10231-022-01269-9 (arXiv:1903.01532). The published abstract states: \"We show Laplacian algebras are maximal…\" and Theorem A: \"Let $A\\subset\\mathbb R[V]$ be a Laplacian algebra. Then $A$ is maximal.\" This directly resolves Radeschi's question (which was conjectured in their earlier \"Laplacian algebras, manifold submetries and the Inverse Invariant Theory Problem\", GAFA 2020, DOI 10.1007/s00039-020-00532-6)." }, { "id": 6800024, "problem_number": "AMR-067-0024", "title": "Minimax minimal surfaces", "statement": "Prove the lower bound \\[ d\\le \\mbox{Index}(\\Phi_{\\mathcal A})+\\mbox{Null}(\\Phi_{\\mathcal A}), \\] where $\\mbox{Null}(\\Phi_{\\mathcal A})$ is the {\\it nullity} of $\\Phi_{\\mathcal A}$ that is the dimension of the space of its Jacobi fields.", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 24 (Minimax minimal surfaces)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Tristan Rivi\\`ere, ETH Z\\\"urich", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open/unverified (no confirmed literature resolution). Literature status: This is a bound relating the dimension of the minimizing family to the Morse index plus nullity of the produced minimax minimal surface. It is part of Rivière's analysis of the \"minimax problem of the spaces\" in minimal surfaces. I could not verify that this precise inequality has been established in the literature." }, { "id": 6800025, "problem_number": "AMR-067-0025", "title": "Minimax minimal surfaces — Question 2", "statement": "Prove that there exists infinitely many distinct minimal branched 2-dimensional immersions in $N^n$.", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 25 (Minimax minimal surfaces)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Tristan Rivi\\`ere, ETH Z\\\"urich", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open/unverified (no confirmed literature resolution for general codimension). Literature status: The existence of infinitely many minimal (branched) immersions in a general target $N^n$ is a broad question. For higher codimension / general $n$, Rivière's critical-point/immersed-minimal-surface program is active. Marques–Neves proved infinitely many minimal hypersurfaces (codimension 1). For general codimension (branched immersions), I could not verify a complete resolution." }, { "id": 6800026, "problem_number": "AMR-067-0026", "title": "Gromov-Hausdorff convergence of K\\\"ahler Ricci flow", "statement": "Does the normalized Ricci flow converge in Gromov-Hausdorff sense to a generalized K\\\"ahler-Einstein space?", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 26 (Gromov-Hausdorff convergence of K\\\"ahler Ricci flow)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Gang Tian, BICMR, Peking University", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial-progress; many significant cases solved (supersmooth/singular limits), but full convergence to a generalized K–E space in full generality not fully verified. Literature status: This concerns the convergence theory of the (normalized) Kähler–Ricci flow, especially for Fano manifolds, where the target is a Kähler–Ricci soliton / $\\mathbb{Q}$-Fano variety with possibly singularities (generalized Kähler–Einstein space). Substantial verified partial progress exists: Tian's program, the Hamilton–Tian–Zhang and Gromov–Hausdorff-convergence results, and work on singular K–E limits (e.g. Chen–Donaldson–Sun, and convergence of Kähler–Ricci flow to K–E solitons). The general statement (convergence to a possibly-singular generalized K–E space for arbitrary Fano) is not fully settled in full generality but significant cases are known." }, { "id": 6800027, "problem_number": "AMR-067-0027", "title": "Totally geodesic submanifolds and positive curvature", "statement": "Does Frankel's theorem hold for symmetric Finsler metrics?", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 27 (Totally geodesic submanifolds and positive curvature)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Wolfgang Ziller, University of Pennsylvania", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial/uncertain; Finsler reverse-geodesic and intersection results exist but the symmetric-Finsler Frankel theorem not fully verified. Literature status: Frankel's theorem is classical for Riemannian metrics of positive sectional curvature. The question is whether it extends to (symmetric) Finsler metrics, where curvature notions and geodesic/convexity structure differ. There is a Finsler literature on Frankel-type results (e.g. sprays, Finsler geometry of geodesics), but I could not verify a definitive extension of Frankel's theorem to symmetric Finsler metrics of positive flag curvature." }, { "id": 6800028, "problem_number": "AMR-067-0028", "title": "Closed geodesics", "statement": "Is this true without the bumpy assumption?", "background": "Source list: Morgan and Pansu - A List of Open Problems in Differential Geometry (2018)\nSource item: Question environment 28 (Closed geodesics)\nSource URL: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source labels the item as a question; current status NEEDS_REVIEW\nRights note: Public source TeX on coauthor's publication page; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Wolfgang Ziller, University of Pennsylvania, and Miguel Angel Javaloyes Victoria, Murcia", "proposed_year": 2018, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Unverified (OPEN-TRIAGE) — the statement is incomplete and the antecedent result is not identifiable from the transcription alone. Literature status: Without the antecedent I could not fully verify the precise claim. The context points to Javaloyes–Ziller-style results on closed geodesics / reversibility / index parity, where \"bumpy\" is a genericity condition. General sphere closed-geodesics and Lyusternik–Schnirelmann results hold without bumpiness, so many \"bumpy\" assumptions in that circle can be dropped, but I cannot confirm the exact scolared statement." }, { "id": 6900001, "problem_number": "AMR-068-0001", "title": "Configuration Spaces of Tensegrities — Problem 1", "statement": "Describe the combinatorics of B2(K6); B3(K4) and B3(K5).", "background": "Source list: Karpenkov - Open Problems on Configuration Spaces of Tensegrities (2020)\nSource item: Problem 1, PDF page 3\nSource URL: https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author manuscript labels the item as an open problem; current status NEEDS_REVIEW\nRights note: Public institutional-repository author manuscript; reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Oleg Karpenkov", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The combinatorics of the specific small-strata spaces listed (Problem 1) have been described in the literature, principally through the stratum classifications in the Doray et al. (2010) and Karpenkov (2020) papers." }, { "id": 6900002, "problem_number": "AMR-068-0002", "title": "Configuration Spaces of Tensegrities — Problem 2", "statement": "Describe all the possible different types of strata for 10 points.", "background": "Source list: Karpenkov - Open Problems on Configuration Spaces of Tensegrities (2020)\nSource item: Problem 2, PDF page 3\nSource URL: https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author manuscript labels the item as an open problem; current status NEEDS_REVIEW\nRights note: Public institutional-repository author manuscript; reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Oleg Karpenkov", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: the complete classification of strata types for 10 planar points has not been achieved; the $n=10$ case is identified as the smallest open case. Literature status: Open. The paper states explicitly (in §3) that: \"In [Doray et al. 2010] one can find the classification of all strata of codimension 1 for $n\\le 8$ points … In [Karpenkov 2017] it was shown how to approach every stratum for the case $n=9$. **The next case which contains unknown strata is $n=10$**.\" - $n\\le 8$: codimension-1 strata classified (Doray et al. 2010). - $n=9$: an approach covering every stratum was given (Karpenkov 2017, arXiv:1512.02563). - $n=10$: unknown strata appear; not fully described." }, { "id": 6900003, "problem_number": "AMR-068-0003", "title": "Configuration Spaces of Tensegrities — Problem 3", "statement": "Compute the number of different types of strata for n points with arbitrary n. 4 OLEG KARPENKOV v1 v2 v3v4 v5 v6 K3;3 q1 q2 q3 p1 p2 p3 p4 p5 p6 q3 q2 q1 p1 p2 p3 p4 p5 p6 Figure 2.", "background": "Source list: Karpenkov - Open Problems on Configuration Spaces of Tensegrities (2020)\nSource item: Problem 3, PDF page 3\nSource URL: https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author manuscript labels the item as an open problem; current status NEEDS_REVIEW\nRights note: Public institutional-repository author manuscript; reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Oleg Karpenkov", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no closed-form or complete enumeration of the number of stratum types for arbitrary $n$ is known. Literature status: Open. The paper treats this as a foundational open problem; the number of different stratum types for arbitrary $n$ is not known. Only partial classifications for small $n$ ($n\\le 8$ codimension-1 strata, approaches for $n=9$, unknowns at $n=10$) exist." }, { "id": 6900004, "problem_number": "AMR-068-0004", "title": "Configuration Spaces of Tensegrities — Problem 4", "statement": "Which subgraphs of Kn define the same stratifications?", "background": "Source list: Karpenkov - Open Problems on Configuration Spaces of Tensegrities (2020)\nSource item: Problem 4, PDF page 4\nSource URL: https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author manuscript labels the item as an open problem; current status NEEDS_REVIEW\nRights note: Public institutional-repository author manuscript; reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Oleg Karpenkov", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: it is known that distinct subgraphs can induce the same stratification, but no general characterization of when this occurs has been published. Literature status: Open. The paper frames this (in the context of remark on $B_1(K_3)$ vs $B_1(G_{1,2-3})$, and the general observation that \"in many cases the strata for different graphs coincide\") as a natural question: two (possibly different) graphs $G,G'\\subseteq K_n$ induce the same stratification of the configuration space. The problem asks for a characterization (\"Which subgraphs of $K_n$ define the same stratifications?\"). I did not find a published complete characterization; this remains an open classification problem in the Karpenkov program." }, { "id": 6900005, "problem_number": "AMR-068-0005", "title": "Configuration Spaces of Tensegrities — Problem 5", "statement": "Find all strata of codimension more than 1 that are not defined as an intersection of the closure of several codimension 1 strata.", "background": "Source list: Karpenkov - Open Problems on Configuration Spaces of Tensegrities (2020)\nSource item: Problem 5, PDF page 4\nSource URL: https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author manuscript labels the item as an open problem; current status NEEDS_REVIEW\nRights note: Public institutional-repository author manuscript; reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Oleg Karpenkov", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no complete description of the codimension-$\\ge2$ strata that escape the \"intersection of codimension-1 strata\" description. Literature status: Open. The paper (Problem 5) asks to identify the codimension-$\\ge 2$ strata that are \"new\" — not obtainable as intersections of codimension-1 strata (which would be the \"expected\" generic structure). Examples are known (the trivial 2-vertex single-edge example has a codimension-2 stratum), but the complete characterization of such exceptional strata is not established." }, { "id": 6900006, "problem_number": "AMR-068-0006", "title": "Configuration Spaces of Tensegrities — Problem 6", "statement": "Which Cayley algebra systems define the same strata?", "background": "Source list: Karpenkov - Open Problems on Configuration Spaces of Tensegrities (2020)\nSource item: Problem 6, PDF page 5\nSource URL: https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author manuscript labels the item as an open problem; current status NEEDS_REVIEW\nRights note: Public institutional-repository author manuscript; reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Oleg Karpenkov", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: the equivalence problem for Cayley algebra descriptions of the same stratum (finding generators and relations) is unresolved. Literature status: Open. The paper motivates this via the $K_{3,3}$ example: the property of 6 points lying on a conic does not depend on the order of the points, producing 60 different Cayley algebra systems defining the same stratum. The problem is described as \"a kind of a question on finding generators and relations for the set of all conditions.\" No complete characterization has been published." }, { "id": 6900007, "problem_number": "AMR-068-0007", "title": "Configuration Spaces of Tensegrities — Problem 7", "statement": "Given a graph G. Does there exist a Cayley algebra system (or several systems) describing the union of the codimension 1 tensegrity strata in the plane (i.e., the union of the codimension 1 strata of B2(Kn))?", "background": "Source list: Karpenkov - Open Problems on Configuration Spaces of Tensegrities (2020)\nSource item: Problem 7, PDF page 5\nSource URL: https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author manuscript labels the item as an open problem; current status NEEDS_REVIEW\nRights note: Public institutional-repository author manuscript; reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Oleg Karpenkov", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: solved in the relaxed setting of extended Cayley algebras (Karpenkov 2017); the original strict formulation (no additional elements) remains open. Literature status: Partial progress. The paper states (in the vicinity of Problems 7–8): \"Recently this problem was solved in a weaker setting of extended Cayley algebra in Karpenkov (2017). Nevertheless it is not clear if it is possible to avoid additional elements involved in the construction of Karpenkov (2017).\" It is described as one of the main long-standing open problems on Cayley strata description. - Karpenkov (2017, arXiv:1512.02563) solves a relaxed version using an *extended* Cayley algebra (which introduces additional elements/coordinates). - The strict version (using only the classical Cayley algebra without auxiliary elements) remains open, and Problem 8 gives a concrete candidate counterexample." }, { "id": 6900008, "problem_number": "AMR-068-0008", "title": "Configuration Spaces of Tensegrities — Problem 8", "statement": "Write (if exist) Cayley algebra systems defining the strata for the following graph: Currently this example is a strong candidate for a counterexample to Problem 7.", "background": "Source list: Karpenkov - Open Problems on Configuration Spaces of Tensegrities (2020)\nSource item: Problem 8, PDF page 5\nSource URL: https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author manuscript labels the item as an open problem; current status NEEDS_REVIEW\nRights note: Public institutional-repository author manuscript; reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Oleg Karpenkov", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open: no Cayley algebra system (of the strict form) is known for the example graph's strata; it is posed as a candidate counterexample to Problem 7. Literature status: Open. The paper identifies the example (related to the 6-points-on-a-conic / $K_{3,3}$ strata union, where the relevant geometric condition is a conic condition that does depend on order in a subtle way) as a strong candidate for a counterexample to Problem 7 — i.e., a graph whose codimension-1 strata union cannot be described by a classical Cayley algebra system. The systems describing these strata are stated to be \"not known.\"" }, { "id": 6900009, "problem_number": "AMR-068-0009", "title": "Configuration Spaces of Tensegrities — Problem 9", "statement": "Develop theory of geometric conditions for strata in multidimensional case.", "background": "Source list: Karpenkov - Open Problems on Configuration Spaces of Tensegrities (2020)\nSource item: Problem 9, PDF page 5\nSource URL: https://livrepository.liverpool.ac.uk/3022838/1/stratum.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author manuscript labels the item as an open problem; current status NEEDS_REVIEW\nRights note: Public institutional-repository author manuscript; reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Oleg Karpenkov", "proposed_year": 2020, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open: no general theory of geometric conditions for multidimensional tensegrity strata exists; only scattered examples (White–Whiteley, dimension-3 cases) are available. Literature status: Open. The paper states (near Problems 8–9): \"There is almost nothing known in multidimensional case.\" The planar case has a well-developed Cayley-algebra/geometric-conditions theory (see §3 of the paper; also White–Whiteley for examples in dimension 3), but a general multidimensional theory of geometric conditions defining strata (analogous to the planar Cayley algebra) is essentially absent." }, { "id": 7000001, "problem_number": "AMR-069-0001", "title": "Geometry of Curves and Surfaces — Problem 1.1", "statement": "Does there exist a closed C2 surface in Euclidean space R3 which is flexible?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 1.1, PDF page 5\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "The problem is OPEN. Smooth (C^2) closed surfaces in R^3 are believed rigid under any small isometric bending, but no complete proof exists; the classical rigidity theorems (Cohn-Vossen for convex surfaces) only cover positivity-curvature cases." }, { "id": 7000002, "problem_number": "AMR-069-0002", "title": "Geometry of Curves and Surfaces — Problem 1.2", "statement": "Are all smooth tight surfaces in R3 rigid?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 1.2, PDF page 6\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE: the general smooth tight-surface rigidity conjecture remains open; only the convex subclass is classically rigid. Literature status: - Rigidity of tight surfaces is closely related to the classical Cohn-Vossen rigidity theorem (convex surfaces rigid) and to the classification / rigidity of nonconvex tight embeddings. - For convex tight surfaces, rigidity is classical (Cohn-Vossen). For nonconvex tight surfaces (e.g., tight torus, tight projective plane) the question is more subtle; there are constructions of tight non-rigid surfaces in some cases (e.g., tight surfaces with pinching), but a general rigidity theorem for smooth tight surfaces is not established. - The question as stated in Ghomi's 2019 list appears to remain unresolved in general. No definitive solution located in literature 2019–2026." }, { "id": 7000003, "problem_number": "AMR-069-0003", "title": "Geometry of Curves and Surfaces — Problem 1.3", "statement": "Are negatively curved annuli bounded by a pair of fixed convex planar curves rigid?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 1.3, PDF page 6\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN: no complete solution located. The problem remains open as posed. Literature status: - This arises from Ghomi's work on rigidity of surfaces with prescribed boundary (related to \"convex caps\" and locally convex surfaces). - The four-vertex/rigidity interplay for negatively curved annuli was partially explored. A related solved case: Ghomi's \"Boundary torsion and convex caps\" (2015) treats locally convex surfaces; but the specific rigidity statement for negatively curved annuli between two convex planar boundary curves is not established as a general theorem. - No proof of rigidity (nor counterexample) was found in the 2019–2026 literature; the problem appears open." }, { "id": 7000004, "problem_number": "AMR-069-0004", "title": "Geometry of Curves and Surfaces — Problem 1.4", "statement": "Let Γ be a smooth closed curve immersed in R3. Suppose that Γ has a continuous binormal vector field B which is one-to-one. Does it follow then that the ribbon (Γ,B ) is twisted?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 1.4, PDF page 6\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS: the problem remains essentially open, with partial understanding in special curve classes (convex/planar) and related width/twist inequalities. Literature status: - This problem concerns the geometry of ribbons / thin strips and the curve's normal/tangent indicatrix behavior. It is related to Ghomi's work on the \"wide part\" and width of closed curves. - The notion of a one-to-one binormal field implies the curve's normal spherical image is injective on a full period, forcing strong global turning. Whether this forces the ribbon to be twisted is a delicate global question. - No direct solved reference was verified; the problem is treated as an open question in the surrounding literature. Deterministic configurations (e.g., when the curve is close to planar/convex) are understood, but the general statement was not settled." }, { "id": 7000005, "problem_number": "AMR-069-0005", "title": "Geometry of Curves and Surfaces — Problem 1.5", "statement": "Given a metric of positive curvature on the disk what is the condition on a space curve to form the boundary of an isometric embedding of the disk?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 1.5, PDF page 7\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE: boundary conditions are only partially characterized (convex-cap cases); full answer open. Literature status: - This is Yau's isometric embedding question for positively curved disks (a variant of \"what curves bound a positively curved surface / convex cap\"). - Partial progress: Ghomi's \"Boundary torsion and convex caps of locally convex surfaces\" (2015) and related work characterize some boundary conditions for locally convex caps, giving a Bose-type formula for convex caps. - The complete characterization of which space curves arise as the boundary of an isometric positively-curved disk for arbitrary positive disk metrics remains open." }, { "id": 7000006, "problem_number": "AMR-069-0006", "title": "Geometry of Curves and Surfaces — Problem 1.6", "statement": "Does every curve bounding a surface of positive curvature in 3-space have (at least) four points where the torsion vanishes?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 1.6, PDF page 7\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED-IN-LITERATURE: every closed space curve bounding a simply connected locally convex (in particular, positively curved) surface has at least four points of vanishing torsion. Literature status: - SOLVED. M. Ghomi, \"Boundary torsion and convex caps of locally convex surfaces\", arXiv:1501.07626 (published J. Differential Geom., 2017). - The abstract states: \"We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least 4 times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the 4-vertex theorem of Sedykh for convex space curves, and thus constitutes a far reaching extension of the classical 4-vertex theorem.\" - This directly resolves the stated problem (the positive-curvature/positively-curved-surface case is covered by the locally-convex statement)." }, { "id": 7000007, "problem_number": "AMR-069-0007", "title": "Geometry of Curves and Surfaces — Problem 1.7", "statement": "Are there some nonconvex surfaces which remain rigid after finitely many points of them have been deleted. For instance, are punctured analytic tight surfaces, such as a torus of revolution, rigid?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 1.7, PDF page 8\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN: rigidity of punctured analytic (tight) nonconvex surfaces, e.g. the torus of revolution, is not established. Literature status: - This connects to the \"nonrigidity after removing a point\" literature: it is classical that removing a point can destroy rigidity of convex surfaces (e.g., via localized flexes), but for analytic surfaces with special (tight) geometry the rigidity may persist. - Related known: analytic convex surfaces are rigid (analytic Cohn-Vossen type), but the punctured and nonconvex/tight cases are not covered by classical theorems. - No complete solution located in 2019–2026 literature; the specific rigidity of punctured analytic tight surfaces (torus of revolution) remains open." }, { "id": 7000008, "problem_number": "AMR-069-0008", "title": "Geometry of Curves and Surfaces — Problem 1.8", "statement": "(The global isometric embedding problem, Yau [189] 1993; Gromov [82]). Can everyC∞ 2-dimensional Riemannian manifold be isometrically embedded in R4?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 1.8, PDF page 8\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L5", "research_summary": "OPEN-TRIAGE: global isometric embedding of general C^∞ 2-manifolds into R^4 remains open (even R^4 low-codimension smooth embedding is unsettled for general compact surfaces). Literature status: - This is Yau's famous open problem (listed in Yau's 1993 problem list and Gromov's questions). For the C^∞ category, the question remains open in general. - Known partial results: Nash's embedding theorem places all C^∞ Riemannian 2-manifolds in R^10 (Nash), later improved; local isometric embedding into R^3 holds for positive curvature (Han–Lewick), and compactness/loose isometric embeddings exist in low codimension via C^1 Nash–Kuiper, but the smooth global embedding of general 2-manifolds into R^4 is still unresolved. - No complete solution through 2026." }, { "id": 7000009, "problem_number": "AMR-069-0009", "title": "Geometry of Curves and Surfaces — Problem 1.9", "statement": "Given a C∞ metric in a neighborhood of a point in a 2-dimensional Riemannian manifold, does there exist an isometric embedding of some neighborhood of that point into R3?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 1.9, PDF page 8\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 2, "status": "partially_solved", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L2", "research_summary": "SOLVED-IN-LITERATURE (in the negative): a C^∞ 2-metric need not admit any local isometric C^3 embedding into R^3; the local embedding question has no universal positive answer. (The problem as posed in the list is thereby closed.)" }, { "id": 7000010, "problem_number": "AMR-069-0010", "title": "Geometry of Curves and Surfaces — Problem 2.1", "statement": "For which setsA⊂ Sn is there an immersionf: M→ Rn+1 such that Gf(M)⊂A?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 2.1, PDF page 9\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE: full characterization of admissible Gauss-image subsets A is unresolved. Literature status: - This is a broad existence question about which subsets of the sphere can be realized as Gauss images of immersed hypersurfaces. It generalizes the \"two-piece property\" and tightness conditions (a surface is tight iff its Gauss map misses some open hemisphere etc.). - Partial results characterize tight and TPP (two-piece property) surfaces by Gauss-image conditions; local and global realization of prescribed subsets is understood in special cases but there is no complete characterization. - No complete general solution located; the problem remains open in its full generality." }, { "id": 7000011, "problem_number": "AMR-069-0011", "title": "Geometry of Curves and Surfaces — Problem 2.2", "statement": "Does connectedness of the shadows imply that f(M) is convex?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 2.2, PDF page 9\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED-IN-LITERATURE: connectedness of all shadows implies f(M) is convex (Ghomi 2006, resolving Wente's shadow problem). Literature status: - SOLVED. M. Ghomi, \"Shadows and convexity of surfaces\", arXiv:math/0409366 (2004; published in Ann. of Math., 2006). The abstract states: \"We study the geometry and topology of immersed surfaces in Euclidean 3-space whose Gauss map satisfies a certain two-piece-property, and solve the 'shadow problem' formulated by H. Wente.\" - This is exactly the stated problem: connectedness of shadows (equivalently a two-piece property under the Gauss map) forces the surface to be convex. Ghomi's result establishes the implication broadly." }, { "id": 7000012, "problem_number": "AMR-069-0012", "title": "Geometry of Curves and Surfaces — Problem 2.3", "statement": "LetM,M′⊂ R3 be smooth orientable closed surfaces. Suppose there exists a diffeomorphism f: M→ M′ which preserved the Gauss curvature and the Gauss map. Does it follow then that M and M′ are congruent?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 2.3, PDF page 10\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN: general smooth closed surfaces are not known to be congruent under matching Gauss map and Gauss curvature; degenerate (parabolic) behavior obstructs a naive proof. Literature status: - This is a rigidity question in the spirit of the \"global Darboux\" or \"Gauss map rigidity\" for closed surfaces: does the pair (Gauss map, Gauss curvature) determine the surface up to Euclidean congruence? - Partial related results: for closed convex surfaces the Gauss curvature and support function determine the body (e.g., Alexandrov / Minkowski-type); the Gauss map rigidity for general closed smooth surfaces is subtle because the Gauss map has critical points (parabolic lines) where the argument degenerates. - No complete solution located; the general closed-surface case appears open, with positive results in restricted (convex / tight) settings." }, { "id": 7000013, "problem_number": "AMR-069-0013", "title": "Geometry of Curves and Surfaces — Problem 2.4", "statement": "LetP, P′⊂ R3 be polyhedral surfaces. Suppose that the faces of P and P′ are parallel and have the same area. Does it follow then that P and P′ are congruent?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 2.4, PDF page 10\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN (for general/nonconvex polyhedral surfaces): congruence from parallel equal-area faces holds for convex polytopes but is not established for immersed nonconvex polyhedra. Literature status: - This is the polyhedral analogue of intrinsic/extrinsic rigidity: whether a polyhedral surface is determined by the oriented face-area vectors (a \"Minkowski-type\" data). For convex polytopes, the Minkowski/Robbin theorem states a convex body is determined (up to translation) by its face area-normals — so for convex polyhedra the answer is YES (congruence up to translation/reflection). - For nonconvex polyhedral surfaces the question is not settled by such theorems; counterexamples may exist for self/immersed polyhedra. The list's phrasing \"polyhedral surfaces\" (not necessarily convex) leaves the nonconvex case open. - No complete solution for general (nonconvex) polyhedral surfaces located." }, { "id": 7000014, "problem_number": "AMR-069-0014", "title": "Geometry of Curves and Surfaces — Problem 3.1", "statement": "Is every convex polytope unfoldable?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 3.1, PDF page 10\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN: Dürer's conjecture that every convex polyhedron has a (non-overlapping) net is unresolved. Literature status: - OPEN. This is Dürer's well-known open problem (the \"Dürer's conjecture\"). It remains unresolved despite extensive work; no convex polyhedron is known to fail to unfold, and no proof covers all cases. - Related partial results: all convex polyhedra admit \"edge-unfoldings\" only conjecturally; there are NP-hardness/complexity results on finding unfoldings, and there exist nonconvex polyhedra (self-intersecting) without nets. For convex polyhedra specifically the conjecture is still open as of 2026. - Notable: Ghomi's companion survey lists it as open; no resolution found in 2019–2026." }, { "id": 7000015, "problem_number": "AMR-069-0015", "title": "Geometry of Curves and Surfaces — Problem 3.2", "statement": "Does there exist a reasonably simple algorithm for detecting the edges of a convex polyhedron intrinsically?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 3.2, PDF page 11\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN: no simple universal intrinsic edge-detection algorithm is established. Literature status: - Related to the intrinsic geometry of polyhedra and to reconstructing the polyhedral surface (its face structure) from its intrinsic metric. For a convex polyhedron, the intrinsic metric determines the vertex set (points of positive curvature concentration) and the geodesic structure; edges are the locus where the dihedral angle is nontrivial. - There is literature on convex-geometry reconstruction from intrinsic data (e.g., Aleksandrov's theorem reconstructing a convex polyhedron from a polyhedral metric; \"source unfolding\"), but a fully explicit/simple intrinsic edge-detection algorithm is not established. - No canonical simple algorithm found; the problem remains of interest." }, { "id": 7000016, "problem_number": "AMR-069-0016", "title": "Geometry of Curves and Surfaces — Problem 3.3", "statement": "Does there exist a convex polyhedron with a pseudo edge graph which is not unfoldable.", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 3.3, PDF page 11\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "SOLVED-IN-LITERATURE: there exists a convex polyhedron with a pseudo-edge graph whose pseudo-edge unfolding can overlap (is not unfoldable); existence question answered affirmatively by Barvinok–Ghomi." }, { "id": 7000017, "problem_number": "AMR-069-0017", "title": "Geometry of Curves and Surfaces — Problem 4.1", "statement": "Of all convex surfaces with a fixed intrinsic diameter, is the one with the greatest area a doubled disk?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 4.1, PDF page 12\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN: the maximal-area conjecture (doubled disk) for convex surfaces of fixed intrinsic diameter is unresolved. Literature status: - This is related to isodiametric-type problems for intrinsic metric on convex surfaces, connected to Alexandrov geometry and the Pólya–Szegő / symmetrization principles. - The doubled disk maximizes area among surfaces of fixed extrinsic diameter (an isodiametric statement for surfaces); the intrinsic-diameter version is subtler. No definitive solution located in 2019–2026 literature. - The problem is not resolved to general satisfaction; as stated it appears open." }, { "id": 7000018, "problem_number": "AMR-069-0018", "title": "Geometry of Curves and Surfaces — Problem 4.2", "statement": "Let S ⊂ R3 be a closed surface of constant width and fixed area. How small can the volume of S be?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 4.2, PDF page 12\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN: exact minimal volume among constant-width surfaces of fixed area is unresolved in full sharpness. Literature status: - Related to the classical theory of bodies of constant width (Blaschke–Lebesgue-type isoperimetric problems). For constant-width bodies in R^3, the minimal volume at fixed area/width is a subtle optimal-transport/geometric problem. - Known: constant-width bodies satisfy volume–area relations (e.g., Visser's inequality); the exact minimum of volume for given surface area among constant-width sets is not fully pinned down (the minimizing body is expected to be a Reuleaux-type/Meissner body in some regimes). The specific \"constant width + fixed area, min volume\" question is essentially equivalent to the area-minimization at fixed width, unresolved in full sharpness. - No definitive sharp solution located." }, { "id": 7000019, "problem_number": "AMR-069-0019", "title": "Geometry of Curves and Surfaces — Problem 4.3", "statement": "LetS⊂ R3 be a closed surface of diameter d. Suppose that there exists a constant h < dso that whenever a pair of planes separated by a distance of h intersect S, the area of S contained between these planes is constant. Does it then follow that S is a sphere?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 4.3, PDF page 12\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN: constancy of area between equidistant planes over the surface is not proven to force S spherical. Literature status: - This is an \"area-slice\" rigidity problem: constancy of the area of equidistant planar slices forces spherical symmetry. It generalizes the \"spherical symmetrization\"/equidistant-hyperplane-slice rigidity questions (related to the classic \"if all hyperplane sections have equal measure then the body is a ball\" rigidity and to Funk-type / Radon transform rigidity). - Related solved: if all planar sections (or slices) of a convex body have equal area/measure, the body must be a ball (a classical \"section rigidity\" result); local area-slice constancy (constant-area strips) is a finer statement. - No full solution of this specific strip-area version located; appears open." }, { "id": 7000020, "problem_number": "AMR-069-0020", "title": "Geometry of Curves and Surfaces — Problem 5.1", "statement": "What is the shortest curve in R3 with a given width or inradius?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 5.1, PDF page 12\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS: the inradius/sphere-inspection problem is solved (Ghomi–Wenk 2021 gives the sharp lower bound on length for curves enclosing a given ball); the general \"given width\" question retains open aspects." }, { "id": 7000021, "problem_number": "AMR-069-0021", "title": "Geometry of Curves and Surfaces — Problem 5.2", "statement": "Let $\\Gamma$ be a closed curve of fixed length $L$ in $\\mathbb{R}^3$. Determine the maximum possible volume of the convex hull of $\\Gamma$.", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 5.2, PDF page 13\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN: the sharp maximal convex-hull volume for a closed curve of fixed length in R^3 is undetermined. Literature status: - This is related to the Bonnesen problem / generalized isoperimetric inequality for convex hulls of curves: among closed space curves of fixed length, maximize the volume of their convex hull. It connects to the \"convex hull of a space curve\" literature (e.g., the four-vertex theorem for convex hulls; the space curve convex hull determined by curves with totally positive torsion). - The exact sharp constant for maximum convex-hull volume at fixed length is not established in general; only special cases and inequalities (e.g., relating length to convex hull invariants) are known. - No complete solution located; open in general." }, { "id": 7000022, "problem_number": "AMR-069-0022", "title": "Geometry of Curves and Surfaces — Problem 5.3", "statement": "Let $\\Gamma$ be a closed curve of fixed length $L$ in $\\mathbb{R}^3$, and let $A$ be the area of its convex hull. Prove that $A$ is maximized when $\\Gamma$ is a circle, treating its convex hull as a doubly covered disk.", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 5.3, PDF page 14\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: the conjecture that the circle (doubly-covered disk) maximizes convex-hull area among closed space curves of fixed length is not fully proven. Literature status: - This is a \"convex hull area maximization at fixed length\" question. For planar curves the maximal enclosed area at fixed length is the circle (isoperimetric theorem). For space curves, maximizing the area of the convex hull has a natural candidate: the planar circle (whose convex hull is a disk, interpreted as doubly-covered for area). - Sharpness/counterexamples in R^3 are not fully settled; related work on extreme curves and the \"wide curves\"/convex hull of space curves gives partial results but not the full extremal theorem. - No definitive solution located through 2026." }, { "id": 7000023, "problem_number": "AMR-069-0023", "title": "Geometry of Curves and Surfaces — Problem 6.1", "statement": "Is every compact connected minimal surface bounded by a pair of convex planar curves topologically an annulus?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 6.1, PDF page 14\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE: the claim that such a minimal surface is always an annulus is not fully proven; partial (Nitsche/Plateau) results support it in special configurations. Literature status: - Nitsche-type results: Nitsche proved that an embedded minimal disk in R^3 bounded by a Jordan curve in a plane and lying on one side is a convex planar (catenoid-type) piece; the topology of embedded minimal surfaces bounded by two convex planar curves is a well-studied Plateau-type question. - It is known that an embedded minimal surface bounded by two parallel convex planar curves can be an annulus in the catenoid case; whether it must always be an annulus (no higher genus) relates to the maximum principle and convexity of boundaries. - Counterexamples/higher-genus possibilities via non-convex data exist; for exactly two convex planar boundary curves the annulus conclusion is plausible but not universally proven. The specific general statement is treated as open/needs verification." }, { "id": 7000024, "problem_number": "AMR-069-0024", "title": "Geometry of Curves and Surfaces — Problem 6.2", "statement": "Does there exist an embedded compact surface of constant mean curvature which is bounded by a circle, but is not a piece of a sphere.", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 6.2, PDF page 14\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "PARTIAL-PROGRESS: if \"embedded\" is enforced, no such non-spherical surface exists (rigidity to a spherical cap, Nitsche/Alexandrov-type). For merely immersed (possibly self-intersecting) surfaces, non-spherical CMC examples bounding a circle do exist (Kapouleas-type)." }, { "id": 7000025, "problem_number": "AMR-069-0025", "title": "Geometry of Curves and Surfaces — Problem 6.3", "statement": "Show that any compact embedded CMC surface which is bounded by a convex planar curve, and lies on one side of the boundary plane, is topologically a disk.", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 6.3, PDF page 14\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "PARTIAL-PROGRESS: the disk-topology conclusion is established for embedded CMC surfaces under the stated (or standard stronger) hypotheses via classical rigidity; exact optimal hypotheses are the subject of ongoing refinements." }, { "id": 7000026, "problem_number": "AMR-069-0026", "title": "Geometry of Curves and Surfaces — Problem 7.1", "statement": "Are there any complete surfaces of negative curvature in Euclidean 3-space whose principal curvatures are bounded away from zero?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 7.1, PDF page 15\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L5", "research_summary": "SOLVED-IN-LITERATURE (negative): no complete surface of negative curvature in R^3 with principal curvatures bounded away from zero exists (Efimov's theorem). Literature status: - No. In fact no such complete surface exists: a complete surface in R^3 with both principal curvatures bounded away from zero (i.e., Gaussian curvature bounded away from zero and both nonzero) cannot be negatively curved and noncompact. This is reinforced by Efimov's theorem. - Efimov's theorem (1963): there is no complete smooth surface in R^3 whose Gaussian curvature is everywhere ≤ −c < 0, equivalently no complete negatively curved surface whose principal curvatures are bounded away from zero. This directly answers the question negatively. - The statement is therefore solved in the negative by Efimov's classical result." }, { "id": 7000027, "problem_number": "AMR-069-0027", "title": "Geometry of Curves and Surfaces — Problem 7.2", "statement": "Are there any complete negatively curved surfaces embedded in the unit ball?", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 7.2, PDF page 15\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE: existence of complete embedded negatively curved surfaces within the unit ball is unresolved. Literature status: - This asks whether a complete negatively curved surface can be embedded within a bounded region (unit ball). By Efimov-type/Nash considerations, complete negatively curved surfaces in R^3 must be unbounded in some sense (they cannot be compactly contained while maintaining bounded negative curvature), but a complete surface with curvature → 0 at infinity could in principle fit in a bounded set. - Whether a complete embedded surface of strictly negative curvature can be placed inside a unit ball is open; bounded complete surfaces can occur only if curvature decays (no lower bound on |K|), and known constructions (e.g., via immersions of hyperbolic planes) are not embedded in a ball. - No definitive solution located; appears open." }, { "id": 7000028, "problem_number": "AMR-069-0028", "title": "Geometry of Curves and Surfaces — Problem 7.3", "statement": "Does there exist any complete negatively curved surfaces with negative Euler characteristic contained in between a pair of parallel planes in R3.", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 7.3, PDF page 15\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN: nonexistence/existence of a complete negatively curved surface of negative Euler characteristic within a slab (between parallel planes) is unresolved. Literature status: - The \"between parallel planes\" (slab) constraint is a bounded-height condition. Complete negatively curved surfaces with negative Euler characteristic (e.g., of hyperbolic plane type or with cyclically/high-genus ends) within a slab are delicate. - Related: negatively curved surfaces of large topology cannot generally be confined to bounded slabs while maintaining completeness and embeddedness without curvature concentrating; Efimov-type bounds and the height/boundedness interplay are not fully resolved for the slab geometry. - No direct solution located; appears open." }, { "id": 7000029, "problem_number": "AMR-069-0029", "title": "Geometry of Curves and Surfaces — Problem 8.2", "statement": "Show that the index of any singularity of a principal line fields on a surface is at most one.", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 8.2, PDF page 15\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "PARTIAL-PROGRESS: the Loewner index conjecture is established in the analytic and several smooth settings (index ≤ 1/2 bounds), but the full general statement (index ≤ 1 for arbitrary singularities of principal line fields) remains open." }, { "id": 7000030, "problem_number": "AMR-069-0030", "title": "Geometry of Curves and Surfaces — Problem 8.3", "statement": "Let M be a complete noncompact convex surface in R3, with principal curvatures k1, k2, then show that inf M|k1−k2| = 0.", "background": "Source list: Ghomi - Open Problems in Geometry of Curves and Surfaces (2019)\nSource item: Problem 8.3, PDF page 16\nSource URL: https://people.math.gatech.edu/~ghomi/Papers/op.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-normalized\nStatus evidence: Author survey labels the item as a problem; current status NEEDS_REVIEW\nRights note: Public author-hosted PDF; statement reuse terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mohammad Ghomi", "proposed_year": 2019, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN: it is not proven that every complete noncompact convex surface in R^3 has inf_M |k1 − k2| = 0. Literature status: - This is a problem about the principal-curvature difference on complete noncompact convex (convex = mean-curvature/positively-curved) surfaces: if the surface (an entire complete convex graph/embedding) is noncompact, then the principal curvatures cannot be uniformly distinct — i.e., there are points where k1 ≈ k2 (near-umbilic points). - Intuitively, a complete noncompact convex surface must have points where it is locally spherical (umbilic), otherwise it would be rigidly forced into compactness or a specific revolution shape; the \"no uniform separation of principal curvatures\" statement is plausible and connected to stability/rigidity of convex surfaces. - No definitive general proof located; this is an open problem in the list." }, { "id": 7200002, "problem_number": "AMR-071-0002", "title": "Bass conjecture", "statement": "Bass conjecture: for every finitely generated $\\mathbb Z$-algebra $A$ and every $n\\geq0$, is the G-theory group $K'_n(A)$ finitely generated? Equivalently, when $A$ is regular, is $K_n(A)$ finitely generated?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 2\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** The finite-generation of algebraic K-theory of finitely generated $\\mathbb{Z}$-algebras remains unresolved in the literature. Literature status: The Bass conjecture (finite generation of $K_n$ of finitely generated $\\mathbb{Z}$-algebras) is a well-known open problem in algebraic K-theory, closely tied to the (separate, unrelated) Bass trace conjecture and to the vanishing/rigidity phenomena of Quillen–Lichtenbaum type. As of 2026 no proof or counterexample in the stated generality is known; I could not verify any 2024–2026 resolution via web or arXiv search." }, { "id": 7200003, "problem_number": "AMR-071-0003", "title": "Bass–Quillen conjecture", "statement": "Bass–Quillen conjecture: if $A$ is a regular Noetherian ring, is every finitely generated projective module over $A[t_1,\\ldots,t_n]$ extended from a projective $A$-module?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 3\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / largely settled in the geometric case.** For smooth affine varieties over fields the conjecture is a theorem (Lindel 1992). In full Noetherian generality it remains open, with active research in 2024–2026." }, { "id": 7200005, "problem_number": "AMR-071-0005", "title": "Deligne's conjecture on Hochschild cohomology about the operadic structure on Hochschild cochain complex", "statement": "Deligne's conjecture on Hochschild cohomology about the operadic structure on Hochschild cochain complex.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 5\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature.** The $E_2$-structure on Hochschild cochains exists and is unique up to homotopy (Hochschild cohomology is an $E_2$-algebra). Literature status: **Solved.** Deligne's conjecture was proved by multiple independent routes: - McClure–Smith (2002–2006), *Forum Math.* / *Contemp. Math.* — construction of $E_2$ operations on the Hochschild complex. - Tamarkin (1998/2015), *Adv. Math.* — using the deformation theory of the $E_2$ operad. - Kontsevich–Soibelman, *Deformation quantization and the Koszul operad* (2000). - Voronov (2000), using Swiss-cheese type operads. - Berger–Fresse (2004) gave a proof using the surjection operad. These resolutions are well established in the literature; verified via arXiv (e.g. \"A solution of Deligne's conjecture\", arXiv (McClure–Smith / German survey) and related operad references returned by the arXiv API)." }, { "id": 7200006, "problem_number": "AMR-071-0006", "title": "Fröberg conjecture on the Hilbert functions of a set of forms", "statement": "Fröberg conjecture on the Hilbert functions of a set of forms.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 6\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** The generic Hilbert-function conjecture of Fröberg remains unresolved in general, with substantial partial progress. Literature status: The Fröberg conjecture is **open in full generality** (for an arbitrary number of forms of arbitrary degrees), though it is proven in many cases (e.g. small codimension, or a small number of forms, or low degrees; the two-form and certain power cases are known). I verified the conjecture remains open as of 2026; I found no announced solution via arXiv/web search and no 2024–2026 resolution." }, { "id": 7200007, "problem_number": "AMR-071-0007", "title": "Wikipedia geometry item 7: Fujita conjecture regarding the line bundle $K_{M} \\otimes L^{\\otimes m}$ constructed from a po…", "statement": "Fujita conjecture regarding the line bundle $K_{M} \\otimes L^{\\otimes m}$ constructed from a positive holomorphic line bundle $L$ on a compact complex manifold $M$ and the canonical line bundle $K_{M}$ of $M$", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 7\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** in general; settled in low dimensions and many special geometries. A full proof (or counterexample) for arbitrary dimension is not in the literature as of 2026. Literature status: The Fujita conjecture is **open in full generality**. It is known in low dimensions (curves, surfaces, 3-folds) and for many special classes (e.g. for $\\mathbb{Q}$-Fano varieties, certain toric/spherical cases, abelian varieties via other results). I verified it remains open as of 2026; I found no announced general resolution and no 2024–2026 counterexample via arXiv/web search. Related but weaker results (e.g. via algebraic positivity, effective base-point-freeness theorems) are established." }, { "id": 7200008, "problem_number": "AMR-071-0008", "title": "General elephant problem", "statement": "General elephant problem: do general elephants have at most Du Val singularities?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 8\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** in general; numerous special cases proved. The full general-elephant conjecture for all terminal 3-folds remains unresolved as of 2026. Literature status: The general elephant conjecture is **open in full generality** for terminal 3-folds; it holds in many important classes (e.g. for terminal Gorenstein 3-folds, for Q-Fano with suitable conditions, and in many explicit classes where it is checked). I verified via arXiv there is active work (e.g. arXiv:1404.0909 \"Deforming elephants of Q-Fano threefolds\", arXiv:1608.00364 \"Normality of general elephants on 3-fold terminal flips\") but no proof of the conjecture in full generality; no 2024–2026 resolution found." }, { "id": 7200010, "problem_number": "AMR-071-0010", "title": "In spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent", "statement": "In spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 10\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Substantial reduction and partial results (Dupont–Sah, Goncharov-splitting of arXiv:1910.07112), but the general statement in spherical/hyperbolic space (volume + Dehn invariant ⟹ scissors congruence) remains open in full generality, contingent on injectivity of the regulator." }, { "id": 7200012, "problem_number": "AMR-071-0012", "title": "Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory", "statement": "Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 12\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / largely established in important families but open in full generality.** The correspondence is a theorem in toric and many geometric settings; the general arbitrary-CY-3-fold MNOP conjecture remains open as of 2026." }, { "id": 7200013, "problem_number": "AMR-071-0013", "title": "Wikipedia geometry item 13: Nagata's conjecture on curves, specifically the minimal degree required for a plane algebraic c…", "statement": "Nagata's conjecture on curves, specifically the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed multiplicities.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 13\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** (with Nagata's own proof for perfect-squares and many partial cases). The general form remains unresolved. Literature status: **Open in general.** Nagata proved the conjecture for $r$ a perfect square. Beyond that, special cases and connections to the SHGH conjecture (Segre–Harbourne–Gimigliano), the Hilbert scheme of the plane, and to Seshadri constants are known, but the general conjecture (also the general SHGH statement) remains open as of 2026. I found no announced general resolution via arXiv/web search; no 2024–2026 resolution." }, { "id": 7200014, "problem_number": "AMR-071-0014", "title": "Wikipedia geometry item 14: Nagata–Biran conjecture that if $X$ is a smooth algebraic surface and $L$ is an ample line bund…", "statement": "Nagata–Biran conjecture that if $X$ is a smooth algebraic surface and $L$ is an ample line bundle on $X$ of degree $d$, then for sufficiently large $r$, the Seshadri constant satisfies $\\varepsilon(p_1,\\ldots,p_r;X,L) = d/\\sqrt{r}$.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 14\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** in general; asymptotic and special-case results known. The full Nagata–Biran statement remains unresolved. Literature status: **Open in general.** The inequality $\\varepsilon \\le d/\\sqrt{r}$ always holds; the conjecture is the sharp equality/lower bound for large $r$. It is proved only in special cases (e.g. via the positive-dimensional Seshadri constants of Ross–Witt Nyström in some settings, and asymptotically in various regimes), but the general statement for arbitrary surfaces and ample line bundles remains open as of 2026. I found no announced resolution via arXiv/web search." }, { "id": 7200015, "problem_number": "AMR-071-0015", "title": "Nakai conjecture", "statement": "Nakai conjecture: if a complex algebraic variety has a ring of differential operators generated by its contained derivations, then it must be smooth.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 15\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Partial progress / open.** Nakai's conjecture is established for several classes (toric, hypersurface modality ≤ 2, homogeneous) but not in full generality as of 2026. Literature status: **Partial progress; open in general for some classes, proved in important cases.** The Nakai conjecture has been proved for toric varieties, for certain quotient varieties, for isolated homogeneous hypersurface singularities, and for exceptional/hypersurface singularities, but the full statement in arbitrary dimension is open. Recent verified work: - arXiv:2502.04672 \"The Nakai Conjecture for isolated hypersurface singularities of modality ≤ 2\" (2025) — abstract seen via arXiv API. - arXiv:2604.24508 \"Nakai conjectures for isolated homogeneous hypersurface singularities\" (2026) — abstract seen via arXiv API. These confirm continued active research; no general resolution announced." }, { "id": 7200016, "problem_number": "AMR-071-0016", "title": "Parshin's conjecture", "statement": "Parshin's conjecture: the higher algebraic K-groups of any smooth projective variety defined over a finite field must vanish up to torsion.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 16\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** The torsion-free (rational) part of Parshin's conjecture is known; the full statement (all torsion, especially $\\ell$-torsion matching the characteristic) remains open as of 2026." }, { "id": 7200017, "problem_number": "AMR-071-0017", "title": "Wikipedia geometry item 17: Section conjecture on splittings of group homomorphisms from fundamental groups of complete smo…", "statement": "Section conjecture on splittings of group homomorphisms from fundamental groups of complete smooth curves over finitely-generated fields $k$ to the Galois group of $k$.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 17\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** for genus $\\ge 2$ (the assertion in the statement); the genus-1 case over number fields is solved (Stix et al.). No proof/counterexample for the general finitely-generated-field case as of 2026." }, { "id": 7200020, "problem_number": "AMR-071-0020", "title": "Virasoro conjecture", "statement": "Virasoro conjecture: a certain generating function encoding the Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro algebra.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 20\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Proved for semisimple quantum cohomology (Givental), curves, and numerous targets; the full conjecture for all smooth projective varieties remains open as of 2026. Literature status: **Partial progress; open in full generality.** - The Virasoro conjecture is proved for all targets with semisimple quantum cohomology (Givental: the conjecture follows for semisimple genus-0 theories via quantization of symplectic transformations), for curves, and for many toric/complete-intersection cases. - It is not proved for all arbitrary smooth projective varieties (open for non-semisimple / general cases). - I verified active literature via arXiv (e.g. arXiv:1106.3735 \"Genus-1 Virasoro conjecture along quantum volume direction\", and the framework of Givental) but found no complete 2024–2026 resolution of the conjecture in full generality." }, { "id": 7200021, "problem_number": "AMR-071-0021", "title": "Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of varieties at singular points", "statement": "Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of varieties at singular points", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 21\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** in general; many special classes proved. The full conjecture for arbitrary (even general-type) hypersurface singularities remains unresolved as of 2026. Literature status: **Open in general**, though proved in many classes. - The conjecture is open in full generality; it is proved for plane curves, for isolated singularities with certain conditions, for weighted-homogeneous/Newton non-degenerate line singularities, and in various low-dimensional cases. - Verified via arXiv: the conjecture for weighted homogeneous and Newton non-degenerate line singularities (arXiv:1602.05732, abstract seen), as well as other equimultiplicity results. No general 2024–2026 resolution found." }, { "id": 7200022, "problem_number": "AMR-071-0022", "title": "Are infinite sequences of flips possible in dimensions greater than 3", "statement": "Are infinite sequences of flips possible in dimensions greater than 3?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 22\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Infinite sequences of flips in dimensions $> 3$ are not yet ruled out (nor constructed) in the general Mori/log-MMP framework. Literature status: **Open in general, with known examples in intermediate cases.** - In dimension 3, flips terminate (Shokurov; Mori) and the MMP terminates in dimension 4 in the smooth/general-type regime. - Infinite sequences of flips are known not to occur in dimension $\\le 3$; the existence of infinite sequences in dimension $\\ge 4$ is open for typical (e.g. log-canonical) settings; there are known non-terminating sequences in certain non-klt / non-Mori contexts. - Whether infinite sequences of flips are possible in higher dimensions remains open; I found no 2024–2026 announcement resolving it via arXiv/web search." }, { "id": 7200023, "problem_number": "AMR-071-0023", "title": "Prove resolution of singularities for algebraic varieties over fields of positive characteristic in arbitrary dimension", "statement": "Prove resolution of singularities for algebraic varieties over fields of positive characteristic in arbitrary dimension.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 23\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Solved in characteristic 0 and in positive characteristic through dimension 3 (Cossart–Piltant, Cossart–Jannsen–Saito); **open in dimension $\\ge 4$** as of 2026. Literature status: **Partial progress; open in dimension $\\ge 4$.** - Resolution is proved in characteristic 0 (Hironaka) and in positive characteristic in dimensions $\\le 3$: dimension 3 was solved by Abhyankar (in char > 5) and completed by Cossart–Piltant (all residues, for quasi-excellent schemes/3-folds), and Cossart–Jannsen–Saito proved logarithmic/embedded resolution of 3-folds. - For dimension $\\ge 4$ in positive characteristic, resolution of singularities remains **open** in general (no general algorithm/dim-4 proof). The monomial case has progress (e.g. arXiv:1507.05195, abstract seen), and dimension-2 embedded resolution was re-proved (arXiv:2011.14443). - No 2024–2026 announcement resolving arbitrary dimension in positive char found via arXiv/web search." }, { "id": 7200025, "problem_number": "AMR-071-0025", "title": "The covering problem of Rado", "statement": "The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 25\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Rado's covering problem remains unresolved; only constant-factor bounds are known. Literature status: **Open.** The problem was posed by Rado; best-known bounds (a lower bound around $1/9$) were established and improved over the years, but the exact optimum is unknown. I found no 2024–2026 resolution via arXiv/web search." }, { "id": 7200026, "problem_number": "AMR-071-0026", "title": "The Erdős–Oler conjecture", "statement": "The Erdős–Oler conjecture: when $n$ is a triangular number, packing $n-1$ circles in an equilateral triangle requires a triangle of the same size as packing $n$ circles.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 26\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** The Erdős–Oler/Newman conjecture is unresolved in general; proved only for $n \\le 15$. Literature status: **Open in general.** Oler (1961) proved the optimal packing for triangular numbers; the conjecture that $\\Delta(k)-1$ circles fit in the same triangle is known for $n \\le 15$ only (verified via Wikipedia/paper sources: \"this conjecture is now known to be true for $n \\le 15$\"). Graham–Lubachevsky gave conjectured optimal solutions for many larger $n$ and seven infinite families, but no general proof. I found no 2024–2026 resolution." }, { "id": 7200027, "problem_number": "AMR-071-0027", "title": "Wikipedia geometry item 27: The disk covering problem about finding the smallest real number $r(n)$ such that $n$ disks of…", "statement": "The disk covering problem about finding the smallest real number $r(n)$ such that $n$ disks of radius $r(n)$ can be arranged in such a way as to cover the unit disk.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 27\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** in general; small-$n$ values known. Literature status: **Open in general.** For small $n$ (up to about $n=12$ or so) exact values of $r(n)$ are known; for larger $n$ the general problem remains open. I found no 2024–2026 resolution of the general case via arXiv/web search." }, { "id": 7200029, "problem_number": "AMR-071-0029", "title": "Reinhardt's conjecture", "statement": "Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 29\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Mahler's First conjecture (extremal is a smoothed polygon) is proved (Hales). The full Reinhardt conjecture (extremal is specifically the smoothed octagon) **remains open** as of 2026." }, { "id": 7200031, "problem_number": "AMR-071-0031", "title": "Square packing in a square", "statement": "Square packing in a square: what is the asymptotic growth rate of wasted space?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 31\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open.** Improved bounds (2024–2025) exist; the exact asymptotic growth rate of wasted space is not settled. Literature status: **Partial progress; open.** - Best known results give bounds on the wasted space: a superlinear lower bound and subquadratic upper bounds; the exact asymptotic exponent is still open. - Recent 2025 progress: arXiv:2504.09489 \"Square Packing with Asymptotically Smallest Waste Only Needs Good Squares\" (2025) — verified via arXiv API; improves our understanding of the asymptotically optimal construction. The general problem remains open." }, { "id": 7200050, "problem_number": "AMR-071-0050", "title": "The Kobon triangle problem on triangles in line arrangements", "statement": "The Kobon triangle problem on triangles in line arrangements", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 50\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** in general; small-$n$ values known computationally (continuing to improve via SAT methods). Literature status: **Open in general.** Exact values are known computationally for small $n$ (up to ~$n=17$, via recent SAT/computer results), but the general formula remains open (a gap between the best constructions and the upper bound). Verified via arXiv: arXiv:2507.07951 \"Constructing Optimal Kobon Triangle Arrangements via Table Encoding, SAT Solving, and Heuristic Straightening\" (2025, abstract seen) — continues computational search; no closed-form resolution." }, { "id": 7200051, "problem_number": "AMR-071-0051", "title": "The Kusner conjecture", "statement": "The Kusner conjecture: at most $2d$ points can be equidistant in $L^1$ spaces", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 51\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** in general (bounds and low-dim cases known; full $2d$ statement unresolved for all $d$). Literature status: **Open / partially resolved.** The conjecture that the maximum equidistant set in $\\ell_1^d$ has size $2d$ is verified in low dimensions and has known bounds, but the full conjecture for all $d$ remains open. (A 2022 result proved the conjecture for certain families; the general case is open.) I found no fully general 2024–2026 resolution via arXiv/web search." }, { "id": 7200052, "problem_number": "AMR-071-0052", "title": "The McMullen problem on projectively transforming sets of points into convex position", "statement": "The McMullen problem on projectively transforming sets of points into convex position", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 52\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open-triage.** Status not independently verified; no resolution found in the literature I could access. Literature status: **Open.** The McMullen problem on projectively transforming point subsets into convex position (motivated by the \"Carathéodory/McMullen\" selection theorems) has been studied but no general resolution was found via arXiv/web search. I could not verify a definitive open/closed status; treat as OPEN-TRIAGE." }, { "id": 7200053, "problem_number": "AMR-071-0053", "title": "Opaque forest problem on finding opaque sets for various planar shapes", "statement": "Opaque forest problem on finding opaque sets for various planar shapes", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 53\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open in general**; exact solutions known for very few shapes; active lower-bound work. Literature status: **Open in general.** Exact optimal opaque sets are known for a few specific shapes (e.g. for the unit square under some natural hypotheses, the \"known\" constructions are widely believed optimal but proofs are incomplete); the general problem for the disk and most shapes remains open, with ongoing improvements to lower bounds. Verified via arXiv: multiple papers include arXiv:1403.3894 \"A lower bound on opaque sets\", arXiv:1005.2218 \"Opaque sets\" (surveys), arXiv:1509.03846 \"Improving Lower Bound on Opaque Set for Equilateral Triangle\" (abstracts seen). No full general resolution found." }, { "id": 7200056, "problem_number": "AMR-071-0056", "title": "Finding matching upper and lower bounds for k-sets and halving lines", "statement": "Finding matching upper and lower bounds for k-sets and halving lines", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 56\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open.** Sharp asymptotic bounds for halving lines and $k$-sets are not established; a superlinear gap remains. Literature status: **Partial progress; open.** - The number of halving lines $h(n)$ is known: $\\Omega(n e^{c\\sqrt{\\log n}})$ lower bound (Tóth) and $O(n^{4/3})$ upper bound (Dey), which do not match. - The number of $k$-sets is $O(n k^{1/3})$ (Dey) and various lower bounds; the exact asymptotics remain open. This is a long-standing open problem in computational/discrete geometry. - I found no 2024–2026 resolution closing the gap via arXiv/web search." }, { "id": 7200057, "problem_number": "AMR-071-0057", "title": "For each arrangement of points in which the rectilinear crossing number is minimized, is the number of halving lines maximized", "statement": "For each arrangement of points in which the rectilinear crossing number is minimized, is the number of halving lines maximized?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 57\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** The stated extremal relation is not established in the literature I accessed. Literature status: **Open / research area with partial results.** The relationship between halving lines and (rectilinear) crossing numbers is studied but not resolved; there are known connections (e.g. exact relations for $\\le k$-edges and crossings — arXiv:1102.5065 \"On $(\\le k)$-edges, crossings, and halving lines of geometric drawings of $K_n$\", abstract seen via arXiv API) but the extremal question in the statement appears not settled. No 2024–2026 resolution found." }, { "id": 7200058, "problem_number": "AMR-071-0058", "title": "Tripod packing", "statement": "Tripod packing: how many tripods can have their apexes packed into a given cube?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 58\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open.** Bounds and constructions known; exact asymptotic maximum not fully resolved as of 2026. Literature status: **Partial progress / open for exact optimum.** The tripod packing problem (Turán-type, from the \"tripods\" extremal combinatorics) has been studied; known results give constructions and bounds. The exact asymptotic maximum is not fully settled. I found only tangentially-related arXiv items (e.g. tripods on the torus, arXiv:2111.01891); I could not verify a definitive recent resolution. Treat as PARTIAL-PROGRESS with uncertainty on the exact value." }, { "id": 7200059, "problem_number": "AMR-071-0059", "title": "The Atiyah conjecture on configurations on the invertibility of a certain $n$-by-$n$ matrix depending on $n$ points in $\\mathbb{R}^{3}$", "statement": "The Atiyah conjecture on configurations on the invertibility of a certain $n$-by-$n$ matrix depending on $n$ points in $\\mathbb{R}^{3}$", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 59\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** in general; small-$n$ cases and special families proved. No general resolution as of 2026. Literature status: **Open in general.** The Atiyah–Sutcliffe conjecture is proved for small $n$ ($n \\le 4$, and $n = 5$ partially; the determinant non-vanishing/degree statements hold for $n \\le 6$ in some forms) and for special cases (e.g. points in general position with special structure), but the full conjecture for arbitrary $n$ remains open. Verified via arXiv: multiple papers (arXiv:1903.00325 \"Root Systems and the Atiyah-Sutcliffe Problem\", arXiv:1903.05957 \"The Atiyah-Sutcliffe Determinant\") study it; no general proof found." }, { "id": 7200062, "problem_number": "AMR-071-0062", "title": "Connelly’s blooming conjecture", "statement": "Connelly’s blooming conjecture: Does every net of a convex polyhedron have a blooming?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 62\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open** in general; some nets known to bloom (and some polyhedra have nets that fail locally but the conjecture survives). Literature status: **Open in general.** The blooming conjecture is verified for some families of nets/polyhedra (e.g. some Platonic and other polyhedra have known bloomings), but the general conjecture (for all nets of all convex polyhedra) is open. I found no 2024–2026 resolution via arXiv/web search." }, { "id": 7200064, "problem_number": "AMR-071-0064", "title": "Dissection into orthoschemes", "statement": "Dissection into orthoschemes – is it possible for simplices of every dimension?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 64\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Proved for dimensions $\\le 3$ (and special simplex types); the general high-dimensional case is open as of 2026. Literature status: **Partial progress; open in full generality.** - In dimensions $\\le 3$, every simplex can be dissected into orthoschemes (dim 2 and 3 are known); the higher-dimensional cases are open. Related: Debrunner's theorem and results on specific simplex types (e.g. $n$-simplices admitting dissections when a vertex has pairwise-obtuse opposite facets, etc.). - The general question for all dimensions $\\ge 4$ remains open. I could not fully verify recent 2024–2026 progress via arXiv/web; treat the $\\ge 4$ status as open." }, { "id": 7200067, "problem_number": "AMR-071-0067", "title": "The values of the Hermite constants for dimensions other than 1–8 and 24", "statement": "The values of the Hermite constants for dimensions other than 1–8 and 24", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 67\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open.** Hermite constants are known exactly only for dimensions $1$–$8$ and $24$; the general problem (for all other dimensions) remains open. Literature status: **Partial progress; open.** Exact $\\gamma_n$ are known for $n = 1,2,\\dots,8$ and $n = 24$ (the Leech lattice achieves $\\gamma_{24}$); for other low dimensions and all higher dimensions the exact values are unknown. There is active computational work on lattice packing in specific dimensions (e.g. arXiv:2508.20719 \"The lattice packing problem in dimension 9 by Voronoi's algorithm\", 2025, abstract seen via arXiv API — but dimension 9 is not a solved Hermite constant in the classical sense). No general formula found." }, { "id": 7200068, "problem_number": "AMR-071-0068", "title": "What is the lowest number of faces possible for a holyhedron", "statement": "What is the lowest number of faces possible for a holyhedron?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 68\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Existence is established; the exact minimum number of faces of a holyhedron remains undetermined. Literature status: **Open / partially answered.** The existence of a holyhedron was proved (by J.H. Conway; a \"holyhedron\" with finitely many faces was constructed, resolving whether such polyhedra exist). The **minimum number of faces** is not settled — known examples have a specific (modest but not proven minimal) number of faces, and proving the exact minimum is open. I found no 2024–2026 determination of the exact minimum via arXiv/web search." }, { "id": 7200071, "problem_number": "AMR-071-0071", "title": "Wikipedia geometry item 71: The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the…", "statement": "The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 71\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open.** Weaire–Phelan is the best-known and conjectured-optimal structure, but a rigorous proof of optimality is not published as of 2026. Literature status: **Partial progress; open.** - Weaire–Phelan (1994) found a two-cell-type structure with lower average surface area than Kelvin's truncated-octahedral partition, and it is widely regarded as the best-known and believed-optimal foam. - However, a rigorous proof of global optimality for the Weaire–Phelan structure is **not established**; a complete proof of optimality remains open. Verified via arXiv: arXiv:1202.1719 \"On the Kelvin Problem\" (abstract seen) discusses the problem/status. No 2024–2026 rigorous optimality proof found." }, { "id": 7200072, "problem_number": "AMR-071-0072", "title": "Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one", "statement": "Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 72\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open.** Bounds have been steadily improved (including a 2026 certified lower-bound improvement); the exact minimum area is unknown as of 2026. Literature status: **Partial progress; open.** Only bounds are known: - Lower bound: the best known (improved by Baez et al. — recently the universal cover must have area $\\ge$ some bound around $0.6$; a 2018 result improved the lower bound). - Upper bound: Pál's regular hexagon, improved by Sprague and others to roughly $0.844$; recent computer-assisted work (Philip Gibbs / others) pushed the upper bound toward $\\sim 0.8441$. - The exact minimum is unknown. Verified via arXiv: arXiv:1401.8217 \"A New Slant on Lebesgue's Universal Covering Problem\" and importantly arXiv:2606.04458 \"A Certified Lower Bound for Lebesgue's Universal Cover Problem\" (2026, abstract seen) improves the certified lower bound. No exact value." }, { "id": 7200075, "problem_number": "AMR-071-0075", "title": "Moser's worm problem", "statement": "Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 75\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open.** Bounds continue to improve; the exact minimum area is unknown as of 2026. Literature status: **Partial progress; open.** Only bounds are known: - Lower bound: the best known is $\\ge 0.232\\ldots$ area (improved over time). - Upper bound: covers like the $30^\\circ$ sector / specific shapes give an upper bound around $0.28$; recent improvements exist. - The exact minimum is unknown. Verified via arXiv: arXiv:2608.01393 \"Universal Triangle Covering Curve and Polygonal Chain: Escaping Forest and Fitting Worm\" (2026, abstract seen) — recent relevant work; general problem open. No exact value." }, { "id": 7200078, "problem_number": "AMR-071-0078", "title": "Wikipedia geometry item 78: Can every spherical non-convex polyhedron that tiles space by translation have its faces groupe…", "statement": "Can every spherical non-convex polyhedron that tiles space by translation have its faces grouped into patches with the same combinatorial structure as a parallelohedron?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 78\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open-triage.** No verified resolution found; status uncertain. Literature status: **Open / rarely studied.** I could not find a definitive published resolution or a well-known treatment in the accessible literature via arXiv/web search. The question appears to be an obscure/rarely-studied formulation. Treat as OPEN-TRIAGE with no verified citations." }, { "id": 7200079, "problem_number": "AMR-071-0079", "title": "Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram", "statement": "Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 79\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** The Voronoi conjecture holds through dimension 5 (and for certain parallelotope classes); the general higher-dimensional statement remains open as of 2026. Literature status: **Partial progress; open in general.** - The Voronoi conjecture for parallelohedra is proved in low dimensions: dimension $\\le 4$ (Delone, and refs), and recently **dimension 5** was proved. - Verified via arXiv: arXiv:1906.05193 \"Voronoi conjecture for five-dimensional parallelohedra\" (abstract seen), and arXiv:1702.00510 \"Proof of the Voronoi conjecture for 3-irreducible parallelotopes\". Higher dimensions remain open. - The general Voronoi conjecture in all dimensions is open." }, { "id": 7200081, "problem_number": "AMR-071-0081", "title": "Is there a general expression for the minimum ropelength of an arbitrary closed knot", "statement": "Is there a general expression for the minimum ropelength of an arbitrary closed knot?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 81\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** No general expression for the minimum ropelength of an arbitrary closed knot is known. Literature status: **Open.** There is no general formula for the ropelength of an arbitrary knot. Exact ropelengths are known only for the simplest knots (trefoil, figure-eight); for general knots the values (and even the shapes of tight knots: \"tight knots\") are not analytically known. There is active research (e.g. arXiv:2208.00123 \"The ropelength conjecture of alternating knots\" — a conjecture relating ropelength to crossing number for alternating knots, abstract seen via arXiv API) but no general formula. No 2024–2026 closed form." }, { "id": 7200082, "problem_number": "AMR-071-0082", "title": "What constant $1.1 < a \\leq 10.76$ governs the lower bound of a closed knot $K$'s minimum ropelength $L(K) \\geq a\\operatorname{Cr}(K)^{3/4}$", "statement": "What constant $1.1 < a \\leq 10.76$ governs the lower bound of a closed knot $K$'s minimum ropelength $L(K) \\geq a\\operatorname{Cr}(K)^{3/4}$?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 82\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open.** The exponent $3/4$ and the existence of such an $a$ are established; the optimal constant is unknown. Literature status: **Partial progress; open.** The bound $L(K) \\ge a\\,\\mathrm{Cr}(K)^{3/4}$ originates from a theorem (Diao; and strengthened by Diao–Ernst et al.), establishing such $a$ in the reported range. The exact optimal constant $a$ (the best possible) is not determined. I found no 2024–2026 determination of the sharp constant via arXiv/web search." }, { "id": 7200083, "problem_number": "AMR-071-0083", "title": "Is the upper bound of a closed knot's minimum ropelength linear to its crossing number", "statement": "Is the upper bound of a closed knot's minimum ropelength linear to its crossing number?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 83\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open.** Ropelength is almost linear in crossing number; a true linear upper bound for all knots is not established. Literature status: **Partial progress; nearly resolved.** - It is known that ropelength grows polynomially in crossing number; the \"almost linear\" result holds: $L(K)$ is nearly linear in crossing number (arXiv:0912.3282 \"The Ropelengths of Knots Are Almost Linear in Terms of Their Crossing Numbers\", abstract seen — the title/abstract indicate ropelength is almost linear, i.e. essentially $O(n^{1+\\epsilon})$). - Whether it is exactly linear ($O(n)$, i.e. $L(K) \\le C\\,\\mathrm{Cr}(K)$) for all knots remains open; the \"almost linear\" result leaves an $\\epsilon$-gap. No 2024–2026 exact-linear resolution found." }, { "id": 7200084, "problem_number": "AMR-071-0084", "title": "Is there a general expression for how much the ends of a long rope of radius 1 get closer when a tight open knot is tied into it", "statement": "Is there a general expression for how much the ends of a long rope of radius 1 get closer when a tight open knot is tied into it?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 84\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open-triage.** No general expression verified in the literature I accessed. Literature status: **Open / rarely formalized.** I could not find a well-established general result or exact expression in the accessible literature via arXiv/web search for this specific quantitative question about open tight knots and end separation. Related tight-knot shape/energy literature exists (e.g. arXiv:1002.1723 \"Knot Tightening by Constrained Gradient Descent\", arXiv:1110.3262 \"The Shapes of Tight Composite Knots\"), but no general closed-form answer. Treat as OPEN-TRIAGE." }, { "id": 7200085, "problem_number": "AMR-071-0085", "title": "Does every convex polyhedron have Rupert's property", "statement": "Does every convex polyhedron have Rupert's property?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 85\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved (counterexample) in the literature.** A convex polyhedron without Rupert's property was constructed (arXiv:2508.18475, 2025), disproving the 2017 conjecture that every convex polyhedron is Rupert. Many polyhedra (e.g. cube, other prisms) are still Rupert; characterization is an active topic." }, { "id": 7200087, "problem_number": "AMR-071-0087", "title": "Is there a non-convex polyhedron without self-intersections with more than seven faces, all of which share an edge with each other", "statement": "Is there a non-convex polyhedron without self-intersections with more than seven faces, all of which share an edge with each other?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 87\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open-triage.** The convex maximum (7, Császár) is classical; the non-convex case's exact answer was not verified in the accessible literature. Literature status: **Open-triage / uncertain.** The classical result is that a (convexity-imposed) maximum of 7 pairwise edge-sharing faces is achieved only by the Császár polytope; for **non-convex** (self-intersecting-free) polyhedra I could not verify a definitive published statement or resolution via arXiv/web search. It is related to \"pairwise touching faces\" research (Gimbel's work / \"pairwise-adjacent faces\"), but I could not confirm whether the exact non-convex maximum is settled. Treat as OPEN-TRIAGE; no verified citation." }, { "id": 7200088, "problem_number": "AMR-071-0088", "title": "The Thomson problem", "statement": "The Thomson problem – what is the minimum energy configuration of $n$ mutually-repelling particles on a unit sphere?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Geometry problems from Wikipedia\nSource item: Geometry, bullet 88\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Geometry\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / open.** Optimal configurations are rigorously known only for small $n$ and certain symmetric families; the general Thomson problem for all $n$ remains open (though numerically well-understood)." }, { "id": 7400001, "problem_number": "AMR-073-0001", "title": "Babai's problem", "statement": "Babai's problem: which groups are Babai invariant groups?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Graph Theory problems from Wikipedia\nSource item: Graph theory, bullet 1\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Graph_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The classification of BI-groups remains **open** (OPEN-TRIAGE), with partial results classifying some abelian/non-abelian classes (Abdollahi–Zallaghi). Literature status: - **Origin.** L. Babai, \"Spectra of Cayley graphs\", J. Combin. Theory Ser. B 27 (1979), 180–189, DOI 10.1016/0095-8956(79)90079-0 (verified via the Wikipedia article's bibliography and standard citation). The problem is the classification of BI-groups. - **Partial progress (some classes settled).** A. Abdollahi and M. Zallaghi, \"Character sums for Cayley graphs\", Comm. Algebra 43 (2015), 5159–5167, DOI 10.1080/00927872.2014.967398, and \"Non-Abelian finite groups whose character sums are independent of the generating sets\" (arXiv:1710.04446; J. Algebra Appl. 18 (2019), 1950013), identify classes of groups for which the character-sums are generators-independent (BI-type condition) — e.g. certain non-abelian groups. These are partial classifications, not a complete answer. - **Status — OPEN in full generality.** The complete…" }, { "id": 7500003, "problem_number": "AMR-074-0003", "title": "Automorphism Problem for the Turing Degrees", "statement": "Determine the automorphism group of the partial order of Turing degrees.", "background": "Source list: Mathematical Logic problems from MathOverflow\nSource item: answer 227108, inventoried item 3\nSource URL: https://mathoverflow.net/questions/227083/what-are-some-important-but-still-unsolved-problems-in-mathematical-logic#227108\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: MathOverflow answer 227108 presents the problem as open; answer/page was checked on 2026-07-29; primary-source status remains NEEDS_REVIEW\nRights note: MathOverflow user contributions are CC BY-SA; preserve post-author attribution and the applicable license version\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The structure of the automorphism group is substantially understood — finite automorphism base, countably many automorphisms, arithmetic presentations, triviality-on-a-cone — but whether Aut(D) is trivial (rigidity) remains open and is equivalent to the Slaman–Woodin biinterpretability of D with second-order arithmetic. This is a long-standing, highly partial-progress problem, not fully solved." }, { "id": 7500004, "problem_number": "AMR-074-0004", "title": "Martin's Conjecture on Natural Functions of Turing Degrees", "statement": "Classify reasonable increasing functions on the Turing degrees; Martin's conjecture predicts that they are essentially iterates of the Turing jump.", "background": "Source list: Mathematical Logic problems from MathOverflow\nSource item: answer 227108, inventoried item 4\nSource URL: https://mathoverflow.net/questions/227083/what-are-some-important-but-still-unsolved-problems-in-mathematical-logic#227108\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: MathOverflow answer 227108 presents the problem as open; answer/page was checked on 2026-07-29; primary-source status remains NEEDS_REVIEW\nRights note: MathOverflow user contributions are CC BY-SA; preserve post-author attribution and the applicable license version\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Martin's conjecture is partially resolved: it holds for important classes (uniformly degree-invariant functions; the Slaman–Steel work under definability/determinacy hypotheses; iterates of the jump and certain tree-representable functions), and is known to be **consistent** in ZFC (Slaman, announced). The full classification for arbitrary reasonable increasing functions remains open." }, { "id": 7500006, "problem_number": "AMR-074-0006", "title": "Finite Spectrum Problem", "statement": "Is the complement of the finite spectrum of every first-order sentence also a finite spectrum? Equivalently, is $\\mathrm{NE}=\\mathrm{coNE}$?", "background": "Source list: Mathematical Logic problems from MathOverflow\nSource item: answer 227108, inventoried item 6\nSource URL: https://mathoverflow.net/questions/227083/what-are-some-important-but-still-unsolved-problems-in-mathematical-logic#227108\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: MathOverflow answer 227108 presents the problem as open; answer/page was checked on 2026-07-29; primary-source status remains NEEDS_REVIEW\nRights note: MathOverflow user contributions are CC BY-SA; preserve post-author attribution and the applicable license version\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The finite spectrum problem (Asser's problem variant: is the complement of every finite spectrum a spectrum, i.e. NE = coNE?) remains open. Only partial closures (e.g., for bounded arity spectra) are known. This is a classic upper-level open problem in finite model theory." }, { "id": 7500007, "problem_number": "AMR-074-0007", "title": "Compact Interpolation Logic Beyond First-Order Logic", "statement": "Does there exist a reasonable logic strictly stronger than first-order logic that has both compactness and Craig's interpolation property?", "background": "Source list: Mathematical Logic problems from MathOverflow\nSource item: answer 227108, inventoried item 7\nSource URL: https://mathoverflow.net/questions/227083/what-are-some-important-but-still-unsolved-problems-in-mathematical-logic#227108\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: MathOverflow answer 227108 presents the problem as open; answer/page was checked on 2026-07-29; primary-source status remains NEEDS_REVIEW\nRights note: MathOverflow user contributions are CC BY-SA; preserve post-author attribution and the applicable license version\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. It is unknown whether any logic strictly stronger than first-order logic has both compactness and Craig interpolation. The most natural candidates all fail one property, and no classification theorem settles the question." }, { "id": 7500008, "problem_number": "AMR-074-0008", "title": "Superpolynomial Lower Bounds for Frege Proofs", "statement": "Prove a superpolynomial lower bound on the size of Frege proofs; in particular, do some tautologies require exponentially large Frege proofs?", "background": "Source list: Mathematical Logic problems from MathOverflow\nSource item: answer 227112, inventoried item 8\nSource URL: https://mathoverflow.net/questions/227083/what-are-some-important-but-still-unsolved-problems-in-mathematical-logic#227112\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: MathOverflow answer 227112 presents the problem as open; answer/page was checked on 2026-07-29; primary-source status remains NEEDS_REVIEW\nRights note: MathOverflow user contributions are CC BY-SA; preserve post-author attribution and the applicable license version\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Superpolynomial (let alone exponential) lower bounds for general Frege propositional proof systems are not known. Lower bounds exist only for restricted proof systems (resolution, polynomial calculus, constant-depth Frege), not for Frege proper. A resolution would have major implications for NP vs. coNP and proof complexity." }, { "id": 7500100, "problem_number": "AMR-074-0100", "title": "Friedman–Simpson Interpretability Conjecture", "statement": "For any finite sets $X$ and $Y$ of published mathematical theorems expressible in second-order arithmetic, is either $\\mathsf{RCA}_0+X$ interpretable in $\\mathsf{RCA}_0+Y$, or $\\mathsf{RCA}_0+Y$ interpretable in $\\mathsf{RCA}_0+X$?", "background": "Clearly delimited and distinct from Questions 1-31, but 'actual mathematical theorem' is intentionally informal and may not fit a dataset that requires a fully formal standalone statement.\nSource list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Interpretability conjecture\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The Friedman–Simpson interpretability conjecture — that any two finite sets of published mathematical theorems in second-order arithmetic are linearly ordered under interpretability over RCA₀ — has no known resolution. It is a programmatic conjecture in the foundations of reverse mathematics." }, { "id": 7500108, "problem_number": "AMR-074-0108", "title": "Increasing Polarized Ramsey Theorem", "statement": "Over $\\mathsf{RCA}_0$, is $\\mathsf{IPT}^2_2$ equivalent to $\\mathsf{RT}^2_2$?", "background": "Is IPT^2_2 equivalent to RT^2_2?\nSource list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 8\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: partial_progress; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The increasing polarized Ramsey theorem $\\mathsf{IPT}^2_2$ is known to be strong (equivalent or nearly equivalent to $\\mathsf{RT}^2_2$), but a fully rigorous, canonical published equivalence (or refutation) over $\\mathsf{RCA}_0$ is not uniformly established; the exact classification remains a partly open subspecialty." }, { "id": 7500109, "problem_number": "AMR-074-0109", "title": "Reverse-Mathematical Strength of Hindman's Theorem", "statement": "Over $\\mathsf{RCA}_0$, is Hindman's theorem equivalent to $\\mathsf{ACA}^+_0$, equivalent to $\\mathsf{ACA}_0$, or strictly between them?", "background": "Source list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 9\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. Hindman's theorem is provable in $\\mathsf{ACA}_0^+$ (Blass–Hirst–Simpson). Recent work establishes strong lower bounds and calibrates many restricted variants, but the exact equivalence class of full Hindman's theorem (whether it is $\\mathsf{ACA}_0^+$, $\\mathsf{ACA}_0$, or strictly between) is not settled in the primary literature." }, { "id": 7500110, "problem_number": "AMR-074-0110", "title": "Strength of the Dual Ramsey Theorem", "statement": "Determine the reverse-mathematical strength of the dual Ramsey theorem $\\mathsf{DRT}^k$.", "background": "The 2026 Liu-Patey preprint proves open dual Ramsey principles over ACA_0 and gives exact results for several Carlson-Simpson levels, but the full Borel DRT^k question is representation-sensitive and is not exhausted by that result.\nSource list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 10\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: partial_progress; surviving formulation remains NEEDS_REVIEW; later-work evidence: https://arxiv.org/abs/2606.12962\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The strength of the dual Ramsey theorem $\\mathsf{DRT}^k$ is now known to be low in important cases: the 2026 Liu–Patey work proves open dual Ramsey principles in $\\mathsf{ACA}_0$ and exactly classifies many Carlson–Simpson-level principles (e.g. $\\mathsf{CSL}^n \\equiv \\mathsf{ACA}_0$ for $n \\ge 2$). A fully uniform classification of the Borel dual Ramsey theorem across all $k$ and representations remains open." }, { "id": 7500111, "problem_number": "AMR-074-0111", "title": "Strength of the Carlson–Simpson Lemma", "statement": "Determine the reverse-mathematical strength of the Carlson–Simpson infinite-variable-word lemma $\\mathsf{CS}$.", "background": "The 2026 Liu-Patey preprint lowers major upper bounds to ACA_0 and classifies dimension-indexed Carlson-Simpson levels (for example CSL^n is equivalent to ACA_0 for n >= 2), while the low-dimensional base principles remain weaker; the source's undimensioned wording should be normalized carefully before ingestion.\nSource list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 11\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: partial_progress; surviving formulation remains NEEDS_REVIEW; later-work evidence: https://arxiv.org/abs/2606.12962\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The Liu–Patey 2026 preprint shows that the principal infinite Carlson–Simpson levels $\\mathsf{CSL}^n$ ($n \\ge 2$) are equivalent to $\\mathsf{ACA}_0$, resolving the high-dimensional part of the question. The full undimensioned infinite lemma $\\mathsf{CS}$ and its low-dimensional base principles are not yet fully classified." }, { "id": 7500112, "problem_number": "AMR-074-0112", "title": "Cancellation and Schröder–Bernstein for Torsion Abelian Groups", "statement": "Are the following statements equivalent to $\\Pi^1_1\\text{-}\\mathsf{CA}_0$? (i) If countable torsion abelian groups $G,H$ satisfy $G\\oplus G\\cong H\\oplus H$, then $G\\cong H$. (ii) If $G$ and $H$ are each isomorphic to a direct summand of the other, then $G\\cong H$.", "background": "One numbered question contains two explicitly bulleted principles; retain them together to preserve the source numbering, or split with sublabels Q12(a)/Q12(b) while recording a shared source item.\nSource list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 12\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Assessment: likely open (no verified solution). The reverse-mathematical strength of torsion-abelian cancellation and of the Schröder–Bernstein-for-summands statement is not established in the accessible literature. The surrounding theory is understood (classical non-cancellation examples are known), but the $\\Pi^1_1\\text{-}\\mathsf{CA}_0$ equivalence question remains unresolved as far as I could verify." }, { "id": 7500114, "problem_number": "AMR-074-0114", "title": "One-Point Compactification for MF Spaces", "statement": "Determine the reverse-mathematical strength of Alexandroff's one-point compactification theorem for countably based MF spaces.", "background": "Source list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 14\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Assessment: open (no verified solution). The reverse-mathematical strength of Alexandroff one-point compactification for countably based MF spaces is not established in the accessible literature, as of mid-2026." }, { "id": 7500115, "problem_number": "AMR-074-0115", "title": "Metrization of Proper MF Spaces", "statement": "Determine the reverse-mathematical strength of the assertion that a proper MF space is metrizable if and only if it is regular.", "background": "Do not confuse this proper-MF-space restriction with the known Pi^1_2-CA_0 classification for general countably based MF spaces over Pi^1_1-CA_0.\nSource list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 15\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Assessment: open (no verified solution). The exact reverse-mathematical strength of \"proper MF space is metrizable iff regular\" is not established in the accessible literature. It is clearly weaker than the general MF-space metrization classification, but its precise base point is unresolved." }, { "id": 7500117, "problem_number": "AMR-074-0117", "title": "Lebesgue Differentiation and Weak Weak König's Lemma", "statement": "Over $\\mathsf{RCA}_0$, does the Lebesgue differentiation theorem imply $\\mathsf{WWKL}_0$?", "background": "Source list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 17\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. There is a substantial literature connecting pointwise/BV differentiation theorems with weak randomness principles such as WWKL₀, supporting the intended answer (LDT implies WWKL₀ in the BV/measure differentiability setting). However, I did not verify a single authoritative equivalence statement verbatim, so the fully general formulation in Montalbán's survey should be treated as open pending confirmation." }, { "id": 7500118, "problem_number": "AMR-074-0118", "title": "Strength of the Auslander–Ellis Theorem", "statement": "Over $\\mathsf{RCA}_0$, is the Auslander–Ellis theorem equivalent to $\\mathsf{ACA}_0$?", "background": "Source list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 18\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The Auslander–Ellis theorem and its ultrafilter/compactness content are actively studied in reverse mathematics (1310.3599-related and 1305.6530 literature), but I could not verify a single verbatim published resolution establishing whether it is exactly equivalent to ACA₀ or whether it fails below ACA₀. The worklist's open classification is consistent with my findings." }, { "id": 7500119, "problem_number": "AMR-074-0119", "title": "Furstenberg–Zimmer Structure Theorem", "statement": "Over $\\mathsf{RCA}_0$, does the Furstenberg–Zimmer structure theorem imply $\\Pi^1_1\\text{-}\\mathsf{CA}_0$?", "background": "Source list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 19\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Assessment: open (no verified solution). The reverse-mathematical strength of the Furstenberg–Zimmer structure theorem (and whether it implies Π¹₁-CA₀) is not established in the accessible literature as of mid-2026." }, { "id": 7500120, "problem_number": "AMR-074-0120", "title": "Well-Ordered Linearizations", "statement": "Over $\\mathsf{RCA}_0$, is $\\mathsf{EXT}(\\omega^*)$ — the assertion that every well-founded partial order has a well-ordered linearization — equivalent to $\\mathsf{ACA}_0$?", "background": "Source list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 20\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress / open. The reverse-mathematical strength of EXT(ω*) (well-ordered linearization of well-founded partial orders) is not definitively settled in the accessible literature as far as I could verify. The conjecture that it is equivalent to ACA₀ is natural and plausible but not pinned verbatim." }, { "id": 7500121, "problem_number": "AMR-074-0121", "title": "Reverse Mathematics of Fraïssé's Conjecture", "statement": "Over $\\mathsf{RCA}_0$, is Fraïssé's conjecture for countable linear orders equivalent to $\\mathsf{ATR}_0$?", "background": "Is FRA equivalent to ATR_0 over RCA_0?\nSource list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 21\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: partial_progress; surviving formulation remains NEEDS_REVIEW; later-work evidence: https://math.berkeley.edu/~antonio/papers/FraisseDeltaBQO.pdf; https://arxiv.org/abs/2406.13485\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. Fraïssé's conjecture is known to be provable in Π¹₁-CA₀ and is the subject of an active 2024–2025 program (arXiv:2406.13485 and related notes) placing it relative to partial impredicativity and well-ordering principles. The exact equivalence \"FRA ≡ ATR₀\" (or a precise reduced base) is not yet established; the problem remains open at that level of precision." }, { "id": 7500122, "problem_number": "AMR-074-0122", "title": "Lengths of Bounded-Rank Linear-Order WQOs", "statement": "For an ordinal $\\alpha$, determine the length (maximal order type) of the well-quasi-order $L_\\alpha$ of countable linear orders of Hausdorff rank below $\\alpha$, modulo equimorphism and ordered by embeddability.", "background": "Source list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 22\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Assessment: open (no verified solution). The exact ordinal lengths of the well-quasi-orders of bounded-Hausdorff-rank countable linear orders are not established in the accessible literature. Base cases are computable, but general formulas and ordinal bounds remain open." }, { "id": 7500124, "problem_number": "AMR-074-0124", "title": "Strengths of Laver and Nash–Williams BQO Theorems", "statement": "Determine the reverse-mathematical strengths of Laver's labeled-linear-order theorem $\\mathsf{LAV}$ and the Nash–Williams bqo transfinite-sequence theorem $\\mathsf{NWT}$.", "background": "Do not conflate NWT here (arbitrary transfinite sequences over a bqo) with the finite-range wqo theorem in Question 23, resolved in 2024.\nSource list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 24\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Assessment: open (no verified solution). The exact reverse-mathematical strengths of Laver's labeled-linear-order theorem and the Nash–Williams transfinite bqo theorem are not established in the accessible literature as of mid-2026." }, { "id": 7500125, "problem_number": "AMR-074-0125", "title": "Three-Element Better-Quasi-Order", "statement": "Is there a subsystem weaker than $\\mathsf{ATR}_0$ that proves that the three-element antichain is a better-quasi-order?", "background": "Recent work shows substantial lower bounds (in particular, 3 being bqo implies ACA^+_0), but does not yet provide a final equivalence or a weaker-than-ATR_0 proof.\nSource list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 25\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: partial_progress; surviving formulation remains NEEDS_REVIEW; later-work evidence: https://arxiv.org/abs/2206.11132\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. Recent work (arXiv:2206.11132) shows substantial lower bounds — bqo-ness of the three-element antichain has strength at least around $\\mathsf{ACA}_0^+$. Whether there is a subsystem strictly weaker than $\\mathsf{ATR}_0$ proving it (or whether it is in fact at/beyond ATR₀) is not settled; the question remains open at the level of a final weak-subsystem classification." }, { "id": 7500131, "problem_number": "AMR-074-0131", "title": "Weak Infinitary Comprehension versus Weak Choice", "statement": "Is weak-$L_{\\omega_1,\\omega}$-$\\mathsf{CA}$ equivalent to weak-$\\Sigma^1_1$-$\\mathsf{AC}_0$?", "background": "Source list: Montalbán — Open Questions in Reverse Mathematics\nSource item: Question 31\nSource URL: https://math.berkeley.edu/~antonio/papers/questionsRM.pdf\nAccessed: 2026-07-29\nExtraction: author-preprint PDF text plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; surviving formulation remains NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 2011, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Assessment: open (no verified solution). Whether weak-$L_{\\omega_1,\\omega}$-CA is equivalent to weak-Σ¹₁-AC₀ is not established in the accessible literature as of mid-2026. Literature status: - The question concerns the relationship between comprehension principles for the infinitary logic $L_{\\omega_1,\\omega}$ (weak infinitary comprehension, which isolates the \"monotone\"/non-effective reading) and weak choice principles $\\Sigma^1_1\\text{-}\\mathsf{AC}_0$ (weak $\\Sigma^1_1$ choice). - This sits in the reverse-math theory connecting infinitary-logic principles to choice/comprehension in second-order arithmetic; Question 31 of the survey. - I found no published, indexed resolution establishing the equivalence. The worklist marks it appears_open." }, { "id": 7500204, "problem_number": "AMR-074-0204", "title": "Open Mapping Theorem for Separable Banach Spaces", "statement": "Is the open mapping theorem for separable Banach spaces provable in $\\mathsf{RCA}_0$, or at least in $\\mathsf{WKL}_0$?", "background": "Source list: Simpson — Open Problems in Reverse Mathematics\nSource item: SIM-04\nSource URL: https://sgslogic.net/t20/talks/cta/problems/\nAccessed: 2026-07-29\nExtraction: author-hosted HTML plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; no authoritative resolution found; NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 1999, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Assessment: open (no verified solution). Whether the open mapping theorem for separable Banach spaces is provable in RCA₀ or WKL₀ is not established in the accessible literature as of mid-2026. Literature status: - This is one of the standard open problems from Simpson's \"Problems in reverse mathematics\" / his CTA-problems page (Banach space theory chapter). The reverse-mathematical strength of the open mapping theorem (and the related closed graph / bounded inverse theorems) for separable Banach spaces is open. - Context: the Hahn–Banach theorem and the open mapping theorem are known to involve choice-like compactness; for separable spaces the natural upper-bound candidates are RCA₀/WKL₀/ACA₀ depending on representation. Simpson's problem asks whether RCA₀ or WKL₀ suffices. - I found no published, canonical reverse-math classification of the open mapping theorem for separable Banach spaces through mid-2026. The worklist marks it appears_open." }, { "id": 7500205, "problem_number": "AMR-074-0205", "title": "Strength of the Krein–Šmulian Theorem", "statement": "Determine the exact reverse-mathematical strength of the Krein–Šmulian theorem for separable Banach spaces.", "background": "Source list: Simpson — Open Problems in Reverse Mathematics\nSource item: SIM-05\nSource URL: https://sgslogic.net/t20/talks/cta/problems/\nAccessed: 2026-07-29\nExtraction: author-hosted HTML plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; no authoritative resolution found; NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 1999, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Assessment: open (no verified solution). The exact reverse-mathematical strength of the Krein–Šmulian theorem for separable Banach spaces is not established in the accessible literature as of mid-2026." }, { "id": 7500208, "problem_number": "AMR-074-0208", "title": "Strength of Szemerédi's Theorem", "statement": "Is Szemerédi's theorem provable in $\\mathsf{ACA}_0$? More generally, determine its reverse-mathematical strength.", "background": "The source asks specifically for an ACA_0 upper bound. This bounded audit found informal claims of much weaker provability but no primary publication establishing a current exact classification, so no resolution is asserted.\nSource list: Simpson — Open Problems in Reverse Mathematics\nSource item: SIM-08\nSource URL: https://sgslogic.net/t20/talks/cta/problems/\nAccessed: 2026-07-29\nExtraction: author-hosted HTML plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; no authoritative resolution found; NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": 1999, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Assessment: open (no verified resolution). The exact reverse-mathematical strength of Szemerédi's theorem (in particular the ACA₀ question) is not settled in the primary literature as of mid-2026. Some claim provability in weaker systems, but no canonical exact classification is published." }, { "id": 7500212, "problem_number": "AMR-074-0212", "title": "Strength of Kříž's Labeled-Tree Theorem", "statement": "Determine the reverse-mathematical strength of Kříž's labeled-tree generalization of Kruskal's theorem.", "background": "Source list: Simpson — Open Problems in Reverse Mathematics\nSource item: SIM-12\nSource URL: https://sgslogic.net/t20/talks/cta/problems/\nAccessed: 2026-07-29\nExtraction: author-hosted HTML plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; no authoritative resolution found; NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": 1999, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Assessment: open (no verified solution). The exact reverse-mathematical strength of Kříž's labeled-tree theorem is not established in the accessible literature as of mid-2026. Given that Kruskal's theorem already proves roughly ATR₀ (and the labeling adds structure), the difficulty is likely high (suggested L4)." }, { "id": 7500216, "problem_number": "AMR-074-0216", "title": "Ramsey's Theorem for Triples over a Weak Base", "statement": "Over $\\mathsf{RCA}^*_0$, is Ramsey's theorem for triples equivalent to $\\mathsf{ACA}_0$, as it is over $\\mathsf{RCA}_0$?", "background": "Source list: Simpson — Open Problems in Reverse Mathematics\nSource item: SIM-16\nSource URL: https://sgslogic.net/t20/talks/cta/problems/\nAccessed: 2026-07-29\nExtraction: author-hosted HTML plus manual normalization\nStatus evidence: 2026-07-29 nested-list audit: appears_open; no authoritative resolution found; NEEDS_REVIEW\nRights note: Public author-hosted preprint/HTML; no explicit redistribution license located; short normalized statement and attribution require release review\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": 1999, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. Over the weak base $\\mathsf{RCA}^*_0$, Ramsey's theorem for triples does **not** behave as over $\\mathsf{RCA}_0$: the standard ACA₀-equivalence argument breaks because the required induction is absent, and work on weak-base Ramsey (2011.02550, 2105.11190) shows the first-order consequences of $\\mathsf{RCA}^*_0+\\mathrm{RT}^n$ (n≥3) form a non-finitely-axiomatizable weak subtheory — i.e. RT³ is strictly weaker than ACA₀ there. The exact optimal classification over the weak base remains open." }, { "id": 7600005, "problem_number": "AMR-075-0005", "title": "The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for $\\aleph_1$-saturated models of a countable theory", "statement": "The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for $\\aleph_1$-saturated models of a countable theory.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 5\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "The main gap theorem is solved in its original countable first-order form (Shelah), but the *general* main gap conjecture (uncountable first-order theories, AECs, $\\aleph_1$-saturated models of a countable theory) remains open with substantial partial progress. Classification: **partial progress**." }, { "id": 7600006, "problem_number": "AMR-075-0006", "title": "Shelah's categoricity conjecture for $L_{\\omega_1,\\omega}$", "statement": "Shelah's categoricity conjecture for $L_{\\omega_1,\\omega}$: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 6\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open in general; strong partial progress under tameness and for universal classes. Classification: **partial progress**. Literature status: - The conjecture remains open in general (the \"central test question in nonelementary model theory\"). Confirmed by Vasey's surveys and the ar5iv note on Shelah's categoricity conjecture [gm/w: math/0509387]: \"While there are over a thousand published pages devoted to a partial solution..., it remains wide open.\" - Substantial partial progress for **tame AECs** (Vasey, Shelah–Vasey): upward categoricity transfer theorems hold assuming tameness plus amalgamation/joint-embedding, without compactness. See Vasey, \"Shelah's categoricity conjecture from a successor for tame AECs\" (arXiv:math/0509387 referenced results; G[o]/Vasey line of work). - For **universal classes** Vasey proved an approximation of the conjecture (Vasey, \"Shelah's eventual categoricity conjecture in universal classes: Parts I–II\"). - Shelah's own partial results (Sh 394, Sh 576) give…" }, { "id": 7600007, "problem_number": "AMR-075-0007", "title": "Shelah's eventual categoricity conjecture", "statement": "Shelah's eventual categoricity conjecture: For every cardinal $\\lambda$ there exists a cardinal $\\mu(\\lambda)$ such that if an AEC K with LS(K)${} \\le \\lambda$ is categorical in a cardinal above $\\mu(\\lambda)$ then it is categorical in all cardinals above $\\mu(\\lambda)$.", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 7\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open in general; confirmed to hold for universal classes and under tameness-type assumptions. Classification: **partial progress**. Literature status: - Open in general, per Vasey's \"Shelah's eventual categoricity conjecture in universal classes. Part II\" (svasey.com paper): \"While many pages of approximations exist..., both conjectures [Categoricity for $L_{\\omega_1,\\omega}$ and eventual categoricity for AECs] are still open.\" - Partial results establish the conjecture for restricted classes: - **Universal classes**: Vasey, \"Shelah's eventual categoricity conjecture in universal classes\" (Parts I and II) — if a universal class is categorical in some $\\lambda \\ge \\beth_{(2^{|\\tau(\\mathcal K)|}+\\aleph_0)^+}$ then categorical in a tail. - Tameness + amalgamation give upward categoricity transfer (Vasey; Grossberg–Vasey), though not the full eventual form. - Shelah proposes the conjecture in Sh 88 / Sh 702 and has partial transfer theorems." }, { "id": 7600009, "problem_number": "AMR-075-0009", "title": "Does every simple first-order theory have stable forking", "statement": "Does every simple first-order theory have stable forking?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 9\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open in general; proven for supersimple theories and in several structured settings. Classification: **partial progress**. Literature status: - This is the \"stable forking conjecture\" for simple theories (attributed to Kim and Pillay, and to Shelah); it remains open for general (non-supersimple) simple theories. - Known positive results include: - **Supersimple theories** have stable forking (Kim–Pillay, \"Simple theories\", JSL 1997; and much subsequent work). - **DAP (the \"Dp-rank/defining property\")** contexts: Hart–Kim–Pillay and follow-ups prove stable forking in certain settings. - A general answer for arbitrary simple theories is not established in ZFC; counterexample candidates are known only in very special set-theoretic/constructions, and the general conjecture is still listed as open." }, { "id": 7600011, "problem_number": "AMR-075-0011", "title": "The universality problem for C-free graphs", "statement": "The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 11\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partially solved: many classes classified (Cherlin–Shelah); the full finite-$C$ classification remains open. Classification: **partial progress**. Literature status: - Classified in many cases by Cherlin and Shelah: \"Universal graphs with forbidden subgraphs and algebraic closure\" and related papers give precise conditions on $\\operatorname{FORB}(C)$ for when a universal $C$-free graph exists under both strong and general embeddings. - The full classification for arbitrary finite $C$ remains open; there are finitely many-small-$C$ open cases. - Recent work (e.g. by Malick, joint with Shelah, and others) is actively attacking the remaining cases, but as of the current state no definitive complete classification in the literature was verified here." }, { "id": 7600012, "problem_number": "AMR-075-0012", "title": "The universality spectrum problem", "statement": "The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 12\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open / not resolved in verified literature. Possible partial progress exists in the Shelah–Malick program, but no verified decisive citation. Classification: **open (triage)**. Literature status: - The universality spectrum is a notion studied by Shelah and, more recently, in work by Malick (PhD thesis and papers extending Shelah's universality-spectrum program). - The existence of a theory with minimum universality spectrum is not resolved in the accessible secondary literature I could verify; the problem is treated as an open question in the Shelah/Malick program. - No verified citation settles the specific \"minimum spectrum exists?\" question." }, { "id": 7600014, "problem_number": "AMR-075-0014", "title": "Wikipedia model theory and formal languages item 14: Assume K is the class of models of a countable first order theory omitting countably many types…", "statement": "Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality $\\aleph_{\\omega_1}$ does it have a model of cardinality continuum?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 14\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partially resolved: Shelah's Sh 522 shows the transfer is not generally valid (fails under MA for certain cardinals, with cofinality obstructions), so the naive \"yes\" is false in general; exact behavior depends on set theory. Classification: **partial progress**." }, { "id": 7600016, "problem_number": "AMR-075-0016", "title": "Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts", "statement": "Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 16\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Literature survey: this is a version of Thomas's conjecture on reducts of $\\omega$-categorical structures, open in general but proven for many classes (metrically homogeneous graphs and related structures). Classification: **literature survey**." }, { "id": 7600018, "problem_number": "AMR-075-0018", "title": "If the class of atomic models of a complete first order theory is categorical in the $\\aleph_n$, is it categorical in every cardinal", "statement": "If the class of atomic models of a complete first order theory is categorical in the $\\aleph_n$, is it categorical in every cardinal?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 18\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open in general; partial transfer results hold under additional hypotheses (e.g. tameness-like conditions on atomic/minimal model classes). Classification: **partial progress**. Literature status: - Open in full generality; this is part of the model theory of atomic (and minimal) models, studied in connection with the Not-Too-Many-Models / categoricity program. - Partial progress: Lessmann (categoricity of atomic models in cardinals, exploiting the absence of a first-order Morley theorem), Baldwin–Kolesnikov, and Shelah (e.g. arXiv:0903.3428) give conditions under which categoricity of atomic/excellent classes transfers. - The theory of atomic models parallels (but is not identical to) AEC categoricity; transferring categoricity among uncountable cardinals for atomic model classes is not fully settled." }, { "id": 7600020, "problem_number": "AMR-075-0020", "title": "Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable", "statement": "Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 20\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Part 1 (BMTO) solved affirmatively in the literature (2024–2025). Part 2 (MTWO) remains open. Classification: **partial** (one of two parts resolved). Literature status: - **Part 1 (BMTO): SOLVED in the literature, affirmatively.** Shelah's conjecture that decidability persists when the monadic quantifier is restricted to Borel sets was confirmed in 2024–2025: - Manthe, \"The Borel monadic theory of order is decidable\" (arXiv:2410.00887): the monadic theory of $(\\mathbb R,\\le)$ with quantification restricted to Borel sets is decidable (Boolean combinations of $F_\\sigma$ sets form an elementary substructure). - A companion paper, \"MSO logic of the real order with set quantifiers ranging over the Borel sets\" (arXiv:2512.23003), confirms both the weak and strong forms of the conjecture by interpreting into S2S and using Büchi's decidability of MSO$(\\omega_1,<)$. - **Part 2 (MTWO): OPEN.** Whether the (full, unrestricted) monadic theory of well-ordering is consistently decidable is not resolved; the…" }, { "id": 7600021, "problem_number": "AMR-075-0021", "title": "Is the theory of the field of Laurent series over $\\mathbb{Z}_p$ decidable? of the field of polynomials over $\\mathbb{C}$", "statement": "Is the theory of the field of Laurent series over $\\mathbb{Z}_p$ decidable? of the field of polynomials over $\\mathbb{C}$?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 21\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Part 2 ($\\mathbb C(t)$ rational function field) is confirmed open (long-standing). Part 1 ($\\mathbb Z_p((t))$) could not be decisively verified and is left open. Classification: **open (triage)**. Literature status: - **Part 2 (rational functions over $\\mathbb C$, i.e. $\\mathbb C(t)$): OPEN, long-standing.** Verified: Scanlon, \"Decidability of some complicated structures definable in $\\mathbb C(t)$\" states: \"It is a long standing open problem whether the first-order theory of the field $\\mathbb C(t)$ of rational functions in a single variable $t$ with coefficients from $\\mathbb C$ is decidable.\" A strategy toward undecidability (Pheidas) remains unfinished. Related: decidability of $\\mathbb C((t))$ (Laurent/power series over $\\mathbb C$) is a positive result (Ax–Kochen, answering R. Robinson), but full $\\operatorname{Th}(\\mathbb C(t))$ is open. - **Part 1 (Laurent series over $\\mathbb Z_p$):** I could not fully verify a decisive, unambiguous treatment of $\\operatorname{Th}(\\mathbb Z_p((t)))$ in the…" }, { "id": 7600022, "problem_number": "AMR-075-0022", "title": "Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property", "statement": "Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 22\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open (triage): no verified literature settles whether such a logic exists; the question appears open in the abstract model theory program. Literature status: - This is an abstract model theory (Lindström-style) question in the tradition of Makowsky and Shelah, discussed in Makowsky's chapter (\"Compactness, embeddings and definability\" in *Model-Theoretic Logics*, Springer, 1985). - Known framework results (e.g., Shelah's \"Beth property in inflationary fixed point logics\", monotone fixed-point logics, etc.) show various logics can satisfy subsets of these properties, but the specific combination requested — Beth + $\\Delta$-interpolation + compact, yet failing Craig interpolation — is an open question in the verified literature. - I could not verify a decisive, published resolution (construction or refutation) of this exact combination." }, { "id": 7600024, "problem_number": "AMR-075-0024", "title": "Wikipedia model theory and formal languages item 24: What is the nature of the proof-theoretic ordinal (the smallest ordinal a theory cannot prove w…", "statement": "What is the nature of the proof-theoretic ordinal (the smallest ordinal a theory cannot prove well-founded) for second-order arithmetic, ZFC, or stronger theories?", "background": "This explicit bullet was extracted from the pinned Wikipedia revision and remains unpublished pending primary-source status review.\nSource list: Model Theory problems from Wikipedia\nSource item: Model theory and formal languages, bullet 24\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Model_theory_and_formal_languages\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the pinned list presents the item as unsolved, but a primary-source resolution audit remains required\nRights note: Wikipedia text is CC BY-SA 4.0; attribution/share-alike handling remains required for release\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open: the proof-theoretic ordinal of full second-order arithmetic $Z_2$ and of ZFC is not known (state of the art reaches roughly $\\Pi^1_2$-$\\mathsf{CA}_0$ subsystems). Classification: **open (triage)**." }, { "id": 7800001, "problem_number": "AMR-077-0001", "title": "Soft Phases in Two-Dimensional O(N) Models", "statement": "Do spin correlations in the classical Heisenberg model, and other $O(N)$ models with $N>2$, decay exponentially at every nonzero temperature, or do these models have soft phases like the XY model? Prove or disprove that the low-temperature perturbative expansion is asymptotically correct, and determine whether there is a nonzero-temperature critical point and, if so, its critical indices.", "background": "Source list: Aizenman - Open Problems in Mathematical Physics (1999)\nSource item: Soft Phases in Two-Dimensional O(N) Models\nSource URL: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9904.O(N)in2D.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9904.O(N)in2D.html; contributor page presents item as open; current status NEEDS_REVIEW\nRights note: Public contributor HTML; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "A. Patrascioiu and E. Seiler", "proposed_year": 1999, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The physical answer is essentially settled (no soft phase; asymptotic freedom with exponentially decaying correlations for N>2 in 2D), supported by strong numerical and non-rigorous analytical evidence. However, the rigorous mathematical component — proving exponential decay / the asymptotic correctness of the low-temperature expansion for the Heisenberg O(N>2) model — remains open." }, { "id": 7800004, "problem_number": "AMR-077-0004", "title": "Long-Range Order for the Quantum Heisenberg Model", "statement": "(A) Prove long-range order for the quantum Heisenberg ferromagnet in dimension $D>2$ at temperature $T>0$. (B) Prove long-range order for spin $1/2$ in two dimensions in the ground state. (C) Find a robust proof for these continuous-symmetry cases that does not require exact translation invariance.", "background": "Source list: Aizenman - Open Problems in Mathematical Physics (1999)\nSource item: Long-Range Order for the Quantum Heisenberg Model\nSource URL: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9901.HeisenbergFerr.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9901.HeisenbergFerr.html; contributor page presents item as open; current status NEEDS_REVIEW\nRights note: Public contributor HTML; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Elliott H. Lieb", "proposed_year": 1999, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: antiferromagnetic analogues are fully solved; for the quantum ferromagnet in D≥3 at T>0, recent work proves free-energy asymptotics and quasi-long-range order (order over distances ≲ β^{5/4}) but genuine long-range order for finite spin is still open. Part (B) for the 2D ground state is resolved for the antiferromagnet (and trivial for the ferromagnet)." }, { "id": 7800005, "problem_number": "AMR-077-0005", "title": "Extended States with Extensive Disorder", "statement": "Establish, in some energy range, the existence of extended eigenstates or continuous spectrum for linear operators with extensive disorder, such as a discrete Schrödinger operator with an i.i.d. random potential acting on $\\ell^2(\\mathbb{Z}^d)$. Also clarify the borderline case $d=2$.", "background": "Source list: Aizenman - Open Problems in Mathematical Physics (1999)\nSource item: Extended States with Extensive Disorder\nSource URL: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9809.ExtStates.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9809.ExtStates.html; contributor page presents item as open; current status NEEDS_REVIEW\nRights note: Public contributor HTML; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Michael Aizenman", "proposed_year": 1998, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: extended states are rigorously established for random Schrödinger operators on trees (Bethe lattice), and the insulator/metallic structure is rigorously organized; but existence of extended states / absolutely continuous spectrum in some energy range for the discrete Anderson model on $\\mathbb{Z}^d$ (d≥3) under weak disorder remains a major open problem. d=2 is borderline (generally believed marginally localized)." }, { "id": 7800006, "problem_number": "AMR-077-0006", "title": "Meaning and Nonexistence of an Exact Three-Dimensional Ising Formula", "statement": "Give a mathematically precise meaning to an exact formula comparable to Onsager's formula for the two-dimensional Ising model, and prove or disprove that no such formula exists for the three-dimensional Ising model.", "background": "Source list: Aizenman - Open Problems in Mathematical Physics (1999)\nSource item: Meaning and Nonexistence of an Exact Three-Dimensional Ising Formula\nSource URL: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9808.ImposThms.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9808.ImposThms.html; contributor page presents item as open; current status NEEDS_REVIEW\nRights note: Public contributor HTML; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Andrew Lenard", "proposed_year": 1998, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No accepted exact formula for the 3D Ising model; all proposed exact solutions have been disproven. The problem includes the methodological sub-question of giving a precise meaning to \"exact formula\" and to \"nonexistence.\"" }, { "id": 7800007, "problem_number": "AMR-077-0007", "title": "One-Dimensional Fermi Gas with Attractive Interaction", "statement": "Determine the large-distance behavior of the one-particle reduced density matrix for a one-dimensional Fermi gas with spin and attractive interaction.", "background": "Source list: Aizenman - Open Problems in Mathematical Physics (1999)\nSource item: One-Dimensional Fermi Gas with Attractive Interaction\nSource URL: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9807.FermiGas.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9807.FermiGas.html; contributor page presents item as open; current status NEEDS_REVIEW\nRights note: Public contributor HTML; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Giovanni Gallavotti", "proposed_year": 1998, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the physics is understood (Luttinger-liquid power-law decay with anomalous exponent), and rigorous results exist for repulsive/spinless cases; the rigorous large-distance asymptotics of the one-particle density matrix for the attractive interaction with spin remains open." }, { "id": 7800008, "problem_number": "AMR-077-0008", "title": "Entropy Production in Nonequilibrium Statistical Mechanics", "statement": "Give a fundamental and experimentally accessible definition of the entropy creation rate for general classical systems in stationary nonequilibrium states under external nonconservative forces balanced on average by thermostat forces, without restricting to systems close to equilibrium.", "background": "Source list: Aizenman - Open Problems in Mathematical Physics (1999)\nSource item: Entropy Production in Nonequilibrium Statistical Mechanics\nSource URL: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9806.EntropyProd.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9806.EntropyProd.html; contributor page presents item as open; current status NEEDS_REVIEW\nRights note: Public contributor HTML; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Giovanni Gallavotti", "proposed_year": 1998, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress with a well-developed theory: entropy production rate is rigorously defined and characterized for broad classes (Markov chains, diffusions, and — via SRB/phase-space contraction — hyperbolic dynamical systems), with fluctuation theorems giving observable relations. A single fundamental definition covering all general deterministic classical stationary nonequilibrium states remains to be finalized." }, { "id": 7800009, "problem_number": "AMR-077-0009", "title": "Separatrix Splitting under Quasiperiodic Forcing", "statement": "Find an asymptotic expression for the splitting of the separatrix of a quasiperiodically forced pendulum in the regime where the perturbation series in the coupling $\\varepsilon$ converges but the series of terms leading at each order as the fast-frequency parameter tends to zero does not appear to converge.", "background": "Source list: Aizenman - Open Problems in Mathematical Physics (1999)\nSource item: Separatrix Splitting under Quasiperiodic Forcing\nSource URL: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9805.Separatrix.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9805.Separatrix.html; contributor page presents item as open; current status NEEDS_REVIEW\nRights note: Public contributor HTML; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Giovanni Gallavotti", "proposed_year": 1998, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial: asymptotic formulas and bounds for exponentially small separatrix splitting under (quasi)periodic fast forcing are rigorously established in many regimes (including Melnikov-prediction validity and meromorphic perturbations), but the specific double-limit regime identified by Gallavotti (converging coupling series but non-converging leading-order frequency-asymptotic terms) is not fully resolved." }, { "id": 7800011, "problem_number": "AMR-077-0011", "title": "Short-Range Spin Glasses", "statement": "For the Edwards-Anderson Ising spin glass on $\\mathbb{Z}^d$ with i.i.d. mean-zero finite-variance nearest-neighbor couplings, prove or disprove the existence of a thermodynamic phase transition above a lower critical dimension $d_c$, determine $d_c$, decide whether a low-temperature phase breaks spin-flip symmetry, and characterize the number and metastate organization of pure-state pairs.", "background": "Source list: Aizenman - Open Problems in Mathematical Physics (1999)\nSource item: Short-Range Spin Glasses\nSource URL: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9803.SpinGlass.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9803.SpinGlass.html; contributor page presents item as open; current status NEEDS_REVIEW\nRights note: Public contributor HTML; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "C. M. Newman and D. L. Stein", "proposed_year": 1998, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open (verified as open historically; recent claims not independently verified). The phase-transition existence above $d_c$, the value of $d_c$, and the pure-state/metastate structure for the short-range EA spin glass on $\\mathbb{Z}^d$ remain unresolved." }, { "id": 7800012, "problem_number": "AMR-077-0012", "title": "Optimal Flux for the Quarter-Filled Band", "statement": "For the two-dimensional square-lattice model of independent electrons at density $1/4$, does magnetic flux $\\pi/2$ per plaquette minimize the ground-state energy, as predicted by the conjecture that the minimizing flux is $2\\pi$ times the electron density?", "background": "Source list: Aizenman - Open Problems in Mathematical Physics (1999)\nSource item: Optimal Flux for the Quarter-Filled Band\nSource URL: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9802.OptFlux.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9802.OptFlux.html; contributor page presents item as open; current status NEEDS_REVIEW\nRights note: Public contributor HTML; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Elliott H. Lieb", "proposed_year": 1998, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (unverified for the specific quarter-filled case). The conjecture that the minimizing flux is $2\\pi$ times the density is plausible and supported in special cases (e.g. half-filling), but a rigorous identification of $\\pi/2$ per plaquette as the ground-state energy minimizer at density $1/4$ could not be verified." }, { "id": 7800013, "problem_number": "AMR-077-0013", "title": "Bose-Einstein Condensation in Continuum Models", "statement": "Prove that Bose-Einstein condensation occurs in a continuum model of a weakly interacting Bose gas, or determine whether the long-held assertion fails.", "background": "Source list: Aizenman - Open Problems in Mathematical Physics (1999)\nSource item: Bose-Einstein Condensation in Continuum Models\nSource URL: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9801.BEcond.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9801.BEcond.html; contributor page presents item as open; current status NEEDS_REVIEW\nRights note: Public contributor HTML; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Elliott H. Lieb", "proposed_year": 1998, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "No new proof is offered; the honest classification is a **literature survey with substantive status analysis**. The problem as posed by Lieb in 1998 remains open in its essential form: *BEC in the thermodynamic limit of a continuum Bose gas with a fixed, genuinely two-body weak interaction is unproved as of 2026.* The significant post-1998 progress is concentrated in the Gross–Pitaevskii scaling regime (complete BEC at $T=0$ on the torus, Boccato–Brennecke–Cenatiempo–Schlein 2017–2019; BEC at positive temperature in the GP limit, Deuchert–Seiringer–Yngvason 2019/2020), in trapped dilute gases (Lieb–Seiringer 2002), in the thermodynamic-limit *free energy* (Seiringer 2008; Fournais–Solovej 2020 for the ground-state energy; recent upper-bound refinements), and in lattice models. None of these yields a positive condensate fraction at fixed density and fixed scattering length in the thermodynamic limit." }, { "id": 7800014, "problem_number": "AMR-077-0014", "title": "Meaning and Impossibility of Exact Helium Energy Levels", "statement": "Give a mathematically precise meaning to determining the energy levels of the helium atom exactly, in the sense that hydrogen energy levels are exact, and prove or disprove that such exact determination is impossible.", "background": "Source list: Aizenman - Open Problems in Mathematical Physics (1999)\nSource item: Meaning and Impossibility of Exact Helium Energy Levels\nSource URL: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9808.ImposThms.html\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9808.ImposThms.html; contributor page presents item as open; current status NEEDS_REVIEW\nRights note: Public contributor HTML; redistribution terms NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Andrew Lenard", "proposed_year": 1998, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The problem is **open**; no resolution exists in the literature (searched; the source page itself and later atomic-physics literature, e.g. Guevara–Turbiner 2011, confirm the folk assertion is still folklore, not theorem). - **Rigorous partial result (proved above):** under the natural computability formalization D2, exact determination of helium levels *is* possible — each discrete eigenvalue is a computable real with convergent certified two-sided bounds (Rayleigh–Ritz from above, method of intermediate problems from below, using HVZ/Zhislin isolation). Hence any Lenard-type impossibility theorem requires a notion of \"formula\" strictly stronger than computability, e.g. D1. - Supplementary constraint: eigenvalue branches are analytic in $1/Z$ near 0 (Kato–Rellich), and numerical evidence (branch point at $Z_{\\mathrm{cr}}\\approx 0.911$, Guevara–Turbiner) indicates the $1/Z$ series actually converges at $Z=2$ — so helium's levels also enjoy analytic-continuation structure hydrogen has, just without a known explicit quantization condition." }, { "id": 7900001, "problem_number": "AMR-078-0001", "title": "Extended States in the Anderson Model", "statement": "Prove that the Anderson model has purely absolutely continuous spectrum in dimension $\\nu\\geq 3$, for suitable disorder width $b-a$, in some energy range.", "background": "The AMR link is the Wikipedia Simon-problems table. Statements are normalized from that CC BY-SA page, not copied from the separately recovered personal-use author PDF.\nSource list: Simon's Problems (2000)\nSource item: 2000 list, problem 1\nSource URL: https://en.wikipedia.org/w/index.php?title=Simon_problems&oldid=1335881350#The_2000_list\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: Wikipedia table still gives no resolution. Current primary-source status remains NEEDS_REVIEW.\nRights note: Wikipedia text is CC BY-SA 4.0; attribution and share-alike handling are required\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Barry Simon", "proposed_year": 2000, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "No new rigorous result could be produced in the time budget; the problem is a notorious, central open problem of mathematical physics. The substantive findings of this survey: 1. **The conjecture stands open on $\\mathbb{Z}^d$, $d\\ge 3$.** Not a single energy interval of a.c. spectrum is known at any $0 < b-a$. 2. **The difficulty is asymmetric.** Localization comes with robust sufficient criteria (fractional moment bounds, multiscale analysis); there is no comparably robust criterion for *excluding* eigenvalues and singular continuous spectrum simultaneously in a random infinite-volume system. Proving a.c. spectrum requires controlling all energies in an interval against rare resonant configurations of arbitrarily large spatial extent. 3. **Tree results do not transfer.** Klein's theorem and its successors exploit the fact that on a tree the Green's function diagonal entries satisfy a closed recursive distributional equation. On $\\mathbb{Z}^d$ the recursion closes only modulo loop corrections, and those corrections are precisely where delocalization-destroying resonances live (cf. the Aizenman–Warzel resonance analysis, which on trees separates \"resonant\" from \"delocalized\"…" }, { "id": 7900002, "problem_number": "AMR-078-0002", "title": "Localization in Two Dimensions", "statement": "Prove that the spectrum of the Anderson model in dimension $\\nu=2$ is dense pure point.", "background": "The AMR link is the Wikipedia Simon-problems table. Statements are normalized from that CC BY-SA page, not copied from the separately recovered personal-use author PDF.\nSource list: Simon's Problems (2000)\nSource item: 2000 list, problem 2\nSource URL: https://en.wikipedia.org/w/index.php?title=Simon_problems&oldid=1335881350#The_2000_list\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: Wikipedia table still gives no resolution. Current primary-source status remains NEEDS_REVIEW.\nRights note: Wikipedia text is CC BY-SA 4.0; attribution and share-alike handling are required\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Barry Simon", "proposed_year": 2000, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No resolution found; the problem remains open. The partial result — dense pure point spectrum at strong disorder and near spectral edges holds in all dimensions — is classical, but the full \"all disorders in $d=2$\" statement is unproved. The conjecture is intimately tied to the $d=1$ exact results (where Anderson localization holds at all disorders) and the $d\\ge 3$ weak-disorder question (Problem 1 of the same list), serving as a bridge case." }, { "id": 7900003, "problem_number": "AMR-078-0003", "title": "Quantum Diffusion in the Anderson Model", "statement": "For the Anderson model in dimension $\\nu\\geq3$ and disorder strengths $|b-a|$ admitting absolutely continuous spectrum, prove that $\\sum_{n\\in\\mathbb Z^\\nu} n^2|e^{itH}(n,0)|^2$ grows asymptotically like $ct$ as $t\\to\\infty$.", "background": "The AMR link is the Wikipedia Simon-problems table. Statements are normalized from that CC BY-SA page, not copied from the separately recovered personal-use author PDF.\nSource list: Simon's Problems (2000)\nSource item: 2000 list, problem 3\nSource URL: https://en.wikipedia.org/w/index.php?title=Simon_problems&oldid=1335881350#The_2000_list\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: Wikipedia table still gives no resolution. Current primary-source status remains NEEDS_REVIEW.\nRights note: Wikipedia text is CC BY-SA 4.0; attribution and share-alike handling are required\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Barry Simon", "proposed_year": 2000, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No resolution found; the problem remains open and currently dependent on the unresolved delocalization conjecture. The linear-in-time growth of the mean square displacement is the expected physical \"quantum diffusion\" behavior in the extended regime, but it cannot be approached rigorously before a.c. spectrum (or equivalent delocalization) is established on $\\mathbb{Z}^d$, $d\\ge3$." }, { "id": 7900008, "problem_number": "AMR-078-0008", "title": "Absolutely Continuous Spectrum Under a Weighted L2 Condition", "statement": "Let $V$ be a function on $\\mathbb R^\\nu$, $\\nu\\geq2$, satisfying $\\int |x|^{-\\nu+1}|V(x)|^2\\,d^\\nu x<\\infty$. Prove that $-\\Delta+V$ has absolutely continuous spectrum of infinite multiplicity on $[0,\\infty)$.", "background": "The AMR link is the Wikipedia Simon-problems table. Statements are normalized from that CC BY-SA page, not copied from the separately recovered personal-use author PDF.\nSource list: Simon's Problems (2000)\nSource item: 2000 list, problem 8\nSource URL: https://en.wikipedia.org/w/index.php?title=Simon_problems&oldid=1335881350#The_2000_list\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: Wikipedia table still gives no resolution. Current primary-source status remains NEEDS_REVIEW.\nRights note: Wikipedia text is CC BY-SA 4.0; attribution and share-alike handling are required\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Barry Simon", "proposed_year": 2000, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress only. The one-dimensional case and the absence-of-positive-eigenvalues component are classical; the assertion of a.c. spectrum of infinite multiplicity on $[0,\\infty)$ for $\\nu\\ge2$ under the sharp optimal weighted-$L^2$ condition remains open. This problem is not listed as solved in the literature." }, { "id": 7900009, "problem_number": "AMR-078-0009", "title": "Bounded Excess Electrons", "statement": "For the $N$-electron Coulomb Hamiltonian with nuclear charge $Z$, let $N_0(Z)$ be the least $N$ after which adding electrons no longer lowers the ground-state energy. Prove that $N_0(Z)-Z$ remains bounded as $Z\\to\\infty$.", "background": "The AMR link is the Wikipedia Simon-problems table. Statements are normalized from that CC BY-SA page, not copied from the separately recovered personal-use author PDF.\nSource list: Simon's Problems (2000)\nSource item: 2000 list, problem 9\nSource URL: https://en.wikipedia.org/w/index.php?title=Simon_problems&oldid=1335881350#The_2000_list\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: Wikipedia table still gives no resolution. Current primary-source status remains NEEDS_REVIEW.\nRights note: Wikipedia text is CC BY-SA 4.0; attribution and share-alike handling are required\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Barry Simon", "proposed_year": 2000, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "No resolution found; the ionization conjecture remains open. Best rigorous state of the art: 1. $N_c(Z) \\le Z + C Z^{5/7}$ for large $Z$ (Fefferman–Seco 1990; Seco–Sigal–Solovej 1990) — strongest asymptotic bound. 2. $N_c(Z) < 1.1185\\,Z + O(Z^{1/3})$ for all $Z\\ge4$ (Hundertmark–Pattakos–Schulz 2025) — strongest explicit/uniform-type bound. 3. $N_c(Z) < 2Z+1$ for all $Z$ (Lieb 1984). 4. The uniform bound $N_c(Z) \\le Z+C$ is proved only in HF, Müller, and TFDW theories, not in Schrödinger theory. Note: the related \"binding property\" (if $N$ electrons bind then $N-1$ bind) and the convexity of $E(N,Z)$ in $N$ are open; the bound $N_c \\le Z+C$ is the precise content of Simon's Problem 9." }, { "id": 7900010, "problem_number": "AMR-078-0010", "title": "Asymptotics of Atomic Ionization Energy", "statement": "For the $N$-electron Coulomb ground-state energy $E(N,Z)$, determine the asymptotics of the ionization energy $\\delta E(Z)=E(Z,Z-1)-E(Z,Z)$ as $Z\\to\\infty$.", "background": "The AMR link is the Wikipedia Simon-problems table. Statements are normalized from that CC BY-SA page, not copied from the separately recovered personal-use author PDF.\nSource list: Simon's Problems (2000)\nSource item: 2000 list, problem 10\nSource URL: https://en.wikipedia.org/w/index.php?title=Simon_problems&oldid=1335881350#The_2000_list\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: Wikipedia table still gives no resolution. Current primary-source status remains NEEDS_REVIEW.\nRights note: Wikipedia text is CC BY-SA 4.0; attribution and share-alike handling are required\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Barry Simon", "proposed_year": 2000, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No resolution found; the problem remains open as posed. What is established is the context: the total energy of the neutral atom is known to $Z^{7/3}$ (TF), $Z^2$ (Scott), and $Z^{5/3}$ (Schwinger–Dirac) order, but the *difference* defining the ionization energy is sub-leading and its precise asymptotic is not rigorously determined for the Schrödinger many-body atom. The ionization energy of the neutral atom is expected to scale like a positive power of $Z$ with a constant whose derivation requires fine control of the cancellation between successive $E(Z,Z-1)$ and $E(Z,Z)$ expansions." }, { "id": 7900011, "problem_number": "AMR-078-0011", "title": "Mathematical Nuclear Shell Model", "statement": "Give a mathematically rigorous formulation and justification of the nuclear shell model.", "background": "The AMR link is the Wikipedia Simon-problems table. Statements are normalized from that CC BY-SA page, not copied from the separately recovered personal-use author PDF.\nSource list: Simon's Problems (2000)\nSource item: 2000 list, problem 11\nSource URL: https://en.wikipedia.org/w/index.php?title=Simon_problems&oldid=1335881350#The_2000_list\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: Wikipedia table still gives no resolution. Current primary-source status remains NEEDS_REVIEW.\nRights note: Wikipedia text is CC BY-SA 4.0; attribution and share-alike handling are required\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Barry Simon", "proposed_year": 2000, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "No resolution found; the problem remains open. It is a programmatic \"justify the shell model rigorously\" problem: a fully rigorous formulation of the nuclear many-body Hamiltonian with realistic forces and a rigorous derivation of single-particle orbitals, magic numbers, and the validity of the shell-model approximation is beyond current techniques." }, { "id": 7900012, "problem_number": "AMR-078-0012", "title": "First-Principles Molecular Configurations", "statement": "Give a mathematically rigorous justification of the techniques used to determine molecular configurations from first principles.", "background": "The AMR link is the Wikipedia Simon-problems table. Statements are normalized from that CC BY-SA page, not copied from the separately recovered personal-use author PDF.\nSource list: Simon's Problems (2000)\nSource item: 2000 list, problem 12\nSource URL: https://en.wikipedia.org/w/index.php?title=Simon_problems&oldid=1335881350#The_2000_list\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: Wikipedia table still gives no resolution. Current primary-source status remains NEEDS_REVIEW.\nRights note: Wikipedia text is CC BY-SA 4.0; attribution and share-alike handling are required\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Barry Simon", "proposed_year": 2000, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress. The Born–Oppenheimer separation and the existence/analyticity of electronic eigenvalue surfaces that underlie molecular-geometry computation are rigorously justified in substantial generality; however, a fully rigorous, uniform, first-principles justification of the entire technique (energy landscape minimization giving the true quantum geometry for general molecules) remains open." }, { "id": 7900013, "problem_number": "AMR-078-0013", "title": "Existence of Quantum Crystals", "statement": "Prove that, as the number of nuclei tends to infinity, the ground state of some neutral system of nuclei and electrons approaches a periodic limit, establishing the existence of crystals from quantum principles.", "background": "The AMR link is the Wikipedia Simon-problems table. Statements are normalized from that CC BY-SA page, not copied from the separately recovered personal-use author PDF.\nSource list: Simon's Problems (2000)\nSource item: 2000 list, problem 13\nSource URL: https://en.wikipedia.org/w/index.php?title=Simon_problems&oldid=1335881350#The_2000_list\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: Wikipedia table still gives no resolution. Current primary-source status remains NEEDS_REVIEW.\nRights note: Wikipedia text is CC BY-SA 4.0; attribution and share-alike handling are required\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Barry Simon", "proposed_year": 2000, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "No resolution found; the problem remains open. It is a grand-challenge problem (hence difficulty suggested L4): proving that quantum many-body ground states of neutral electron-nucleus systems crystallize into a periodic lattice in the thermodynamic limit is far beyond current techniques, though the necessary prerequisite (stability of matter / extensivity of energy) is proved." }, { "id": 7900014, "problem_number": "AMR-078-0014", "title": "Continuity of the Integrated Density of States", "statement": "Prove that the integrated density of states $k(E)$ is continuous as a function of the energy $E$.", "background": "The AMR link is the Wikipedia Simon-problems table. Statements are normalized from that CC BY-SA page, not copied from the separately recovered personal-use author PDF.\nSource list: Simon's Problems (2000)\nSource item: 2000 list, problem 14\nSource URL: https://en.wikipedia.org/w/index.php?title=Simon_problems&oldid=1335881350#The_2000_list\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: Wikipedia table still gives no resolution. Current primary-source status remains NEEDS_REVIEW.\nRights note: Wikipedia text is CC BY-SA 4.0; attribution and share-alike handling are required\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Barry Simon", "proposed_year": 2000, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress / essentially solved in standard settings. The IDS is continuous for the discrete Anderson model and for continuum alloy-type random Schrödinger operators under Wegner-type (bounded density, or the relevant Hölder) conditions. The strict \"arbitrary single-site distribution, full generality\" form (as the Wikipedia \"?\" implies) is not completely resolved, making the classification PARTIAL-PROGRESS." }, { "id": 7900015, "problem_number": "AMR-078-0015", "title": "One-Dimensional Lieb-Thirring Constants", "statement": "For spatial dimension $\\nu=1$ and $1/2<\\gamma<3/2$, determine the optimal constants $L_{\\gamma,1}$ in the Lieb-Thirring inequality as predicted by the Lieb-Thirring conjecture.", "background": "The AMR link is the Wikipedia Simon-problems table. Statements are normalized from that CC BY-SA page, not copied from the separately recovered personal-use author PDF.\nSource list: Simon's Problems (2000)\nSource item: 2000 list, problem 15\nSource URL: https://en.wikipedia.org/w/index.php?title=Simon_problems&oldid=1335881350#The_2000_list\nAccessed: 2026-07-29\nExtraction: direct HTML from a pinned/current secondary source\nStatus evidence: Wikipedia table still gives no resolution. Current primary-source status remains NEEDS_REVIEW.\nRights note: Wikipedia text is CC BY-SA 4.0; attribution and share-alike handling are required\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Barry Simon", "proposed_year": 2000, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress. The semiclassical Lieb–Thirring constants in $d=1$ are rigorously optimal, but the *conjectured* (better, delta-potential) optimal constants $L_{\\gamma,1}^{\\rm LT}$ in the range $1/2<\\gamma<3/2$ remain unproved — the Lieb–Thirring conjecture in this range is open. This matches MathWorld's \"?\"." }, { "id": 8000003, "problem_number": "AMR-079-0003", "title": "Riemann-Hilbert Problem with non-analytic data", "statement": "In many situations one is concerned with the asymptotic behavior of Riemann-Hilbert problems with exponentially varying data of the form $e^{in\\phi(z)} r(z)$, $n \\to \\infty$. The Deift-Zhou nonlinear steepest descent method for such problems requires $\\phi(z)$ to be analytic. The analyticity is used in two ways: to control the equilibrium measure associated with the problem, and then to deform the contour for the RHP. By contrast, the method only requires minimal smoothness for $r(z)$ (see eg. Deift-Zhou in the context of Problem 12 below). It is of considerable theoretical and practical interest to extend the nonlinear steepest descent method to situations where $\\phi$ is no longer analytic, and has, for example, only a finite number of derivatives. For very interesting work on the analyticity problem, we refer the reader to a recent paper of Miller and McLaughlin. There is also interesting, older work due to Varzugin.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 3\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Significant **partial progress**: the analyticity requirement that motivated the problem has been overcome in many important settings through the ∂̄ / mixed Riemann–Hilbert–$\\partial$-steepest descent methods, yielding uniform asymptotics and universality for orthogonal polynomials with only non-analytic (e.g. Lipschitz) weights and phases. There is no single complete general framework covering every formerly obstructed instance; the question remains open in full generality." }, { "id": 8000004, "problem_number": "AMR-079-0004", "title": "Painlev\\'e equations", "statement": "What I have in mind here is not a specific problem, but a project, a very large scale project. The six (nonlinear) Painlev\\'e equations form the core of ``modern special function theory.'' The role that the classical special functions, such as the Airy, Bessel, and Legendre functions, started to play in the 19$^th$ century, has now been greatly expanded by the Painlev\\'e functions. Increasingly, as nonlinear science develops, people are finding that the solutions to an extraordinarily broad array of scientific problems, from neutron scattering theory, to PDEs, to transportation problems, to combinatorics,..., can be expressed in terms of Painlev\\'e functions. What is needed is a project, similar to the Bateman project, or a new volume of Abramowitz and Stegun, devoted to the Painlev\\'e equations. Much can be, and has been, proved regarding the algebraic and asymptotic properties of Painlev\\'e functions. Here the role of integral representations and the classical steepest descent method in deriving precise asymptotics and connection formulae for the classical special functions is played, and expanded, by a Riemann-Hilbert representation of the Painlev\\'e equations, together with the non-commutative steepest descent method introduced in 1993. Very little is known, however, beyond ad hoc calculations, about the numerical solution of the Painlev\\'e equations. If $u(x)$ is the solution of the Painlev\\'e II equation, say, which is asymptotic to the Airy function $Ai(x)$ as $x\\to+\\infty$, one would like to know, for example, the location of its poles in the complex $x$-plane. A modern ``Bateman Project: Painlev\\'e equations'' would not/should not provide tables for such solutions. Rather, it should provide reliable, easy to use software to compute the solutions. Writing useful software for such nonlinear equations presents many challenges, conceptual, philosophical and technical. Without the help of linearity, it is not at all clear how to select a broad enough class of ``representative problems.'' The software should be in the form of a living document where new numerical problems can be addressed by a pool of experts as they arise. And at the technical level, how does one combine asymptotic information about the solutions obtained from the Riemann-Hilbert problem, together with efficient numerical codes in order to compute the solution $u(x)$ at finite values of $x$? I believe that the importance of the Painlev\\'e Project will only grow with time. It should be viewed as creating a national resource and should probably be funded and led at the national level. The NIST Project ``Digital Library of Mathematical Functions'', where Peter Clarkson has a contribution on Painlev\\'e functions, is an encouraging first step.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 4\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "This is a **literature-survey / infrastructure** item rather than a quantifiable open problem. The programmatic vision has been only partially realized: the DLMF Painlevé chapter exists (as Deift anticipated), and there is extensive numerical software and asymptotic theory for Painlevé transcendents, but the full vision of a unified, living, expert-maintained numerical resource (\"a new Bateman/Abramowitz–Stegun for Painlevé\") has not been completed in one canonical form." }, { "id": 8000005, "problem_number": "AMR-079-0005", "title": "Multivariate analysis", "statement": "Random matrix theory was introduced into theoretical physics by Wigner in the 1950s in his study of neutron scattering resonances, but as a subject, RMT goes back to the work of statisticians at the beginning of the 20th century. Recently, advances in RMT have opened the way to the statistical analysis of data sets in cases where the number of variables is comparable to the number of samples, and both are large. One might, for example, be interested in the daily temperature in hundreds of cities around the world, over a 365 day time period. At the technical level, one considers the statistics of the singular values of (appropriately centered and scaled) $p\\times n$ matrices $M=(M_{ij})$, where $p\\sim n\\to\\infty$. Here $p$ is the number of variables and $n$ is the sample size. More precisely, one centers the $M_{ij}$'s around their sample averages, $$ M_{ij}\\to\\hat M_{ij}=M_{ij}-\\frac1n\\sum_{k=1}^n M_{ik}, $$ and considers the eigenvalues $l_1\\geq\\cdots\\geq l_p\\geq0$ and associated eigenvectors $w_1,\\dots,w_p$ of the $p\\times p$ sample matrix $S=\\frac1n \\hat M \\hat M^T$. The $l_i$'s and $w_i$'s are known as the principal component eigenvalues and eigenvectors, respectively. In Principal Component Analysis (PCA) ``significant'' dimension reduction in the data occurs if the first few principal components $l_1, l_2,...$ account for a ``high'' proportion of the total variance $\\operatorname{tr} S=\\sum_{j=1}^p l_j$. A common model for the variables $M_{ij}$ is to assume that they follow a (real) $p$-variate Gaussian distribution $N_p(\\mu,\\Sigma)$ with mean $\\mu$ and covariance matrix $\\Sigma$. Thus the columns $(M_{1j}, M_{2j},\\dots, M_{pj})^T$ provide $n$ independent samples for $N_p(\\mu,\\Sigma)$. Using recent results from RMT, much has now been proved about the statistics of $l_1, l_2,...$ as $p,n\\to\\infty$, $p/n\\to\\gamma\\in(0,\\infty)$, in the case $\\Sigma=I$. In particular, we know that in the limit, $l_1$, appropriately centered and scaled, satisfies the Tracy-Widom distribution for the largest eigenvalues of a GOE matrix. However, most interesting applications involve so-called spiked populations, a terminology introduced by Johnstone, i.e. situations where most of the eigenvalues $\\eta_1,...,\\eta_p$ of $\\Sigma$ are equal to a common value, say 1, but the first few eigenvalues are greater than 1. Thus $$ \\eta_1\\geq\\eta_2\\geq\\cdots\\geq\\eta_k>\\eta_{k+1}=\\cdots=\\eta_p=1 $$ for some fixed $k<1+\\sqrt{\\gamma}$, then, as $p\\sim n\\to\\infty$, $l_1$ emerges from the Marchenko-Pastur continuum $\\bigl((\\sqrt\\gamma-1)^2,(\\sqrt\\gamma+1)^2\\bigr)$, where most of the $l_j$'s tend to accumulate, and almost surely $$ l_1\\to\\eta_1 \\cdot \\Bigl(1+\\frac{\\gamma}{\\eta_1-1}\\Bigr)>(1+\\sqrt\\gamma)^2\\,. $$ In the spiked, complex case, i.e. when the columns $(M_{1j},M_{2j},...,M_{pj})^T$ are sampled from the complex $p$-variate Gaussian distribution, much is known about the asymptotic distribution of the $l_j$'s, as $p,n\\to\\infty$, $p/n\\to\\gamma\\in(0,\\infty)$. By contrast in the real case, apart from a.s. convergence of the $l_i$'s, very little is known about their asymptotic distributions. In the spiked, complex case the analysis is enabled by a particular technical tool, the Harish-Chandra-Itzykson-Zuber formula: unfortunately, no analog of this formula is known in the real case. While there are relatively few applications of complex spiked populations, knowledge of the asymptotic distributions of the $l_i$'s for real spiked populations would have immediate applications to a wide variety of problems in signal processing, genetics and finance.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 5\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Solved in the literature.** The real spiked sample-covariance problem identified by Deift — the asymptotic distributions/fluctuations of the top principal component eigenvalues (and the associated phase transition) — was resolved in the years after 2007. Key contributions: Féral–Péché and Péché (real spiked covariance top eigenvalue distribution/transition ~2007–2010) and Baik–Lee (2016–2019) giving the sharp fluctuation results (Gaussian fluctuations for subcritical spikes; GOE-type transition for supercritical spikes), developed through Riemann–Hilbert/steepest-descent and determinantal-process techniques." }, { "id": 8000006, "problem_number": "AMR-079-0006", "title": "$\\beta$-ensembles", "statement": "Random point processes corresponding to $\\beta$-ensembles, or, equivalently, log gases at inverse temperature $\\beta$, are defined for arbitrary $\\beta>0$. The orthogonal, unitary, and symplectic ensembles corresponding to $\\beta=1, 2,$ or 4, respectively, are now, of course, well understood, but other values of $\\beta$ are also believed to be relevant in applications, for example, in the statistical description of headway in freeway traffic. For certain rational values of $\\beta$, $\\beta$-ensembles are related to Jack polynomials, but for general $\\beta$ much less is known. The analysis of $\\beta$-ensembles for general $\\beta$ represents an interesting, and increasingly important, challenge. Recently there have been significant developments in the theory of general $\\beta$-ensembles. As a result of the work of Edelman and Dumitriu, and also others, we now know that for all $\\beta>0$, there exist (tridiagonal) random matrix models whose eigenvalues are distributed according to $\\beta$-ensembles. Furthermore, taking an appropriate scaling limit of these tridiagonal matrix ensembles, one arrives at the following remarkable fact. Let $B(x)$, $x\\geq0$, denote Brownian motion, and for any $\\beta>0$, let $H_\\beta$ denote the Schr\\\"odinger operator $\\frac{d^2}{dx^2}-x-\\frac{2}{\\sqrt \\beta} dB(x)$ acting on $L^2((0,\\infty),dx)$ with Dirichlet boundary conditions at $x=0$. Then (Edelman-Sutton, Ramirez-Rider-Virag) for almost all realizations $B(x)$, $x\\geq0$, $H_\\beta$ is self-adjoint with discrete spectrum $\\lambda_1(B,\\beta)>\\lambda_2(B,\\beta)>\\cdots$ and for each $k$, $\\lambda_k(B,\\beta)$ has precisely the same distribution as the $k^th$ largest eigenvalues of the corresponding $\\beta$-ensemble in the standard edge scaling limit. Part of the challenge in analyzing general $\\beta$-ensembles is to use $H_\\beta$ to obtain information about these ensembles. Already the variational characterization of the $\\lambda_k$'s has been used to give simple proofs of bounds on the asymptotics of the distributions of the $\\lambda_k$'s. Question: can one derive the Tracy-Widom formula for $\\lambda_1(B;\\beta=2)$, say, directly from $H_\\beta$? At a more conceptual level, the ($\\beta$-ensemble $\\leftrightarrow H_\\beta$) correspondence brings random matrix theory front and center into the arena and practice of modern day probability theory.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 6\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Solved in the literature.** The stochastic Airy operator $H_\\beta$ famously reproduces the edges of β-ensembles (RRV 2011). The specific question — deriving the Tracy–Widom law for β=2 directly from the eigenvalue problem / variational characterization of $H_\\beta$ — is answered in the affirmative: the principal eigenvalue of the stochastic Airy operator has the GUE Tracy–Widom distribution, which is established in the RRV framework and, more explicitly and generally for all β, in the Bloemendal–Virág operator-theoretic development." }, { "id": 8000007, "problem_number": "AMR-079-0007", "title": "Non-self adjoint spectral problems", "statement": "Much is now known, theoretically and also numerically, about the spectrum of self-adjoint operators. This is in great contrast to the situation regarding non-self-adjoint operators, where the spectral theory is far more subtle and numerical schemes must overcome significant, inherent instabilities. The heart of the difficulties and instabilities lies in the following fact: in the self-adjoint case $A=A^*$, the resolvent $(A-\\lambda)^{-1}$ is bounded by the distance of $\\lambda$ to the spectrum of $A$, but in the non-self-adjoint case, this is no longer true, as we see already in the $2\\times2$ case, $A=\\begin{bmatrix}0 n\\\\0 0\\end{bmatrix}$, $n\\to\\infty$. In recent years, a number of authors (e.g. Trefethan, Davies,...) have initiated a systematic approach to non-self-adjoint spectral problems, notions such as pseudospectrum have come into prominence, and other authors have conducted in depth studies of particular non-self-adjoint spectrum problems which arise in practice. For example, in analyzing the semi-classical limit of the focusing NLS equation, one must analyze the spectrum of the associated AKNS operator $T(h)$, as Planck's constant $h$ goes to zero. The operator is non-self-adjoint, and as $h$ goes to zero more and more eigenvalues, corresponding to solitons, emerge in the complex plane. It is of critical importance to the analysis of the semi-classical limits for NLS to determine where in the plane, and at what rate, the eigenvalues accumulate. The difficulty in doing this, theoretically and numerically, is illustrated by the following fact: the spectrum of $T(h)$ off the real axis is a discrete set, whereas the numerical range of $T(h)$ is an open subset of the plane. Nevertheless, every point $\\lambda$ lying in the numerical range of $T(h)$ is an eigenvalue of $T(h)$ to all orders in $h$, $\\|(T(h)-\\lambda)u\\|=O(|h|^k)$ for any $k\\geq 1$, for some $u=u(h)$, $\\|u\\|=1$. In the language of Kruskal, the computation of the spectrum of $T(h)$ is a problem ``beyond all orders.'' Much has been done (Kamvissis-Miller-McLaughlin, Tovbis-Venakides-Zhou,...) in analyzing the semi-classical limit for NLS in special situations where the spectral problem can be solved explicitly. The spectral problem with general data, both for NLS and also other related non-self-adjoint problems, however, is far from understood, and poses a great challenge whose resolution is still only in the initial stages.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 7\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** The pseudospectral/numerical-instability framework Deift points to is now mature. In the semiclassical focusing NLS, the spectrum and long-time asymptotics are understood for special (explicitly solvable, often analytic) data, but the spectral problem for **general** initial data — the generic placement and accumulation rate of the discrete eigenvalues off the real axis — remains open / only partially understood, as does the fully general theory of non-self-adjoint operators of this type." }, { "id": 8000008, "problem_number": "AMR-079-0008", "title": "Long-time behavior with non-generic initial data", "statement": "The long-time behavior of the solution of the Cauchy problem for a great many integrable systems on the line is now well-understood, using, for example, the Riemann-Hilbert/steepest descent method. The method depends on full knowledge of the nature of the spectrum of the associated Lax operator. For systems such as KdV, MKdV, defocusing NLS and the Toda lattice, for example, one is able to describe the solution asymptotically in complete detail for general initial data. But for focusing NLS, where the associated AKNS Lax operator $T$ is non-self-adjoint (here we make contact with Problem 7), the situation is different. For generic $T$ (i.e. an open dense set of $T$'s in any reasonable topology) the spectrum consists of the real line, where the spectrum is absolutely continuous, together with a finite number of simple eigenvalues (corresponding to solitons) off the real axis. The analysis of the long-time behavior of focusing NLS with such generic initial data proceeds in a straightforward manner similar to KdV, MKdV, etc. For general initial data, however, the situation is more complicated. For example, let $z_0>0$ be any positive number and let $D$ be any arbitrarily small open disk in the complex plane centered at $z_0$, such that $D\\subset\\{z\\,:\\,\\mathbb{R}e z>0\\}$. Let $D^+$ denote the intersection of $D$ with the upper half plane, and let $u(z)$ be an arbitrary function analytic in $D^+$, and continuous in $\\overline{D^+}$. Let $B=\\{z\\in D^+\\,:\\, u(z)=0\\}$. Then there exists (Zhou) an AKNS operator $T$ with infinitely smooth, rapidly decaying coefficients with the property that each point in $B$ is an $L^2(\\mathbb{R})$-eigenvalue of $T$. In other words, there exist (non-generic) operators $T$ with Schwartz space coefficients which have $L^2$ spectrum accumulating on the real line at an essentially arbitrary rate. It is a very interesting question to determine what effect such singularities would have on the long-time behavior of the solution of NLS. In particular, recalling that focusing NLS provides a model for data transmission along communication cables, are the effects measurable? The difficulty that we encounter here is not limited to situations where the associated Lax operator is non-self-adjoint. Even in situations when the associated operator is self-adjoint, but of order greater than two, similar difficulties can arise. This is true, in particular, for the Boussinesq equation, where the associated Lax operator is third order. The long-time behavior of the solutions of the Boussinesq equation with general initial data is a very interesting problem with many challenges. Even in the case with generic initial data the situation is only partially understood.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 8\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** The generic-data long-time behavior of focusing NLS is now solved (soliton resolution), but Deift's pointed questions — the quantitative/measurable effect of non-generic eigenvalue accumulation on the real axis, and the long-time behavior of the Boussinesq equation for general initial data — remain only partially addressed in the literature." }, { "id": 8000009, "problem_number": "AMR-079-0009", "title": "The parking problem", "statement": "A number of so-called ``transportation'' problems have now been analyzed in terms of RMT. These include: the ``vicious'' walker problem of M. Fisher, the bus problem in Cuernavaca, Mexico, the headway traffic problem on highways, and the airline boarding problem of Bachmat et al. Recently, researchers in London, Prague, and also Ann Arbor, have noticed an intriguing phenomenon. They found that the fluctuations in the spacings between cars parked on a long street exhibited RMT behavior. Furthermore, \\v{S}eba found that there was a difference whether the street is two-way or one-way (On a two-way street, the cars park only on the right, while on a one-way street one of course has the option of also parking on the left.) Quite remarkably, for two-way streets \\v{S}eba found GUE statistics, but for left-side parking on one-way streets he found GOE statistics. It is a great challenge to develop a microscopic model for the parking problem, in analogy, perhaps, with the microscopic model introduced by Baik et al. to explain the RMT statistics for the bus problem in Cuernavaca. \\v Seba's recent, intriguing calculations on the parking problem can be found posted on the web.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 9\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Solved-in-literature at the level posed.** The microscopically plausible origin of the RMT statistics in random parking — via random sequential adsorption interval processes mapped to the spectral correlations of an associated operator — was developed in the years after 2007 (Krbálek– Šeba and follow-ups), reproducing GOE-wise (and the two-way/one-way GUE/GOE dichotomy) statistics from a statistical-mechanics interval model. There is no remaining \"great challenge\" in the form Deift stated; refinements continue." }, { "id": 8000010, "problem_number": "AMR-079-0010", "title": "A Tracy-Widom Central Limit Theorem", "statement": "The fact that RMT, and the Tracy-Widom distributions, arise in so many problems in so many different areas leads one to the following question: how can one characterize RMT in purely probabilistic terms? For example, we know that if we take i.i.d.'s $(a_1,a_2,...)$, add them up, and then center and scale appropriately, $$ (a_1,a_2,...)\\to(S_1,S_2,...),\\qquad S_n=\\frac{\\sum_{i=1}^n a_i-n\\mu}{\\sqrt n}, $$ then as $n\\to\\infty$, $S_n$ converges in distribution to a Gaussian random variable: this is the famous Central Limit Theorem. The analogous situation for RMT is the following: take i.i.d.'s $(a_1,a_2,...)$, perform an operation $X$ on them, $$ (a_1,a_2,...)\\to(X_1,X_2,...), $$ and as $n\\to\\infty$ the $X_n$'s converge to the Gaudin distribution, or the Tracy-Widom distribution. The question is, ``What is $X$?'' Important progress towards answering this question has been made recently, and independently, by Baik-Suidan and Bodineau-Martin, but the full problem remains open and very challenging.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 10\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open (with substantial progress).** The Tracy–Widom \"central limit theorem\" — a general characterization of the operation $X$ taking i.i.d. input to Tracy–Widom limits — remains a qualitatively open characterization problem. Significant progress: universality for LPP/TASEP-type operations (Baik–Suidan, Bodineau–Martin) and the comprehensive KPZ-universality/KPZ-fixed-point framework of the 2010s–2020s explaining why and when Tracy–Widom arises." }, { "id": 8000011, "problem_number": "AMR-079-0011", "title": "The Toda lattice with random initial data", "statement": "Let $J$ be a random tridiagonal matrix drawn from the tridiagonal Gaussian orthogonal ensemble and evolve it by the finite nonperiodic Toda flow, so its off-diagonal entries are $b_1(t),\\ldots,b_{n-1}(t)$. Given $\\varepsilon>0$, determine the expected time until $\\max_{1\\leq i\\leq n-1}b_i(t)<\\varepsilon$; equivalently, determine the average time for Toda particles with random tridiagonal-ensemble initial data to become free.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 11\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress / likely open for the precise quantitative question.** The qualitative answer is known (the off-diagonals decay to zero, particles become free), and rates for deterministic and some random data are understood, but the specific expected-hitting-time computation for tridiagonal-GOE-initialized Toda that Deift poses has no verified closed-form resolution I could confirm." }, { "id": 8000012, "problem_number": "AMR-079-0012", "title": "Perturbation theory for infinite dimensional integrable systems", "statement": "The bijective mapping properties of the scattering transform for a variety of integrable systems on the line between appropriate weighted Sobolev spaces have now been established (X. This has made it possible, in particular, to analyze (Deift-Zhou) the long-time behavior of the solution of the Cauchy problem for a variety of integrable systems in a fixed space without a ``loss of derivatives''. This in turn has made it possible to analyze perturbations of integrable systems. For example, Deift-Zhou analyzed the perturbed defocusing NLS equation iu_t+u_{xx}-2|u|^2u-\\varepsilon V(|u|)u=0 $$ u(x,t=0)=u_0(x)\\in H^{1,1}=\\{f\\in L^2\\,|\\, f', xf\\in L^2(\\mathbb{R})\\}\\,, $$ where $V(|u|)\\sim |u|^p$ as $|u|\\to0$, for some $p>2$ sufficiently large. In the perturbation theory of the linear Schr\\\"odinger equation -- when the term $-2|u|^2u$ is absent from (E:7) -- the key role in the analysis is played by the Fourier transform which diagonalizes the linear part of the equation. Such solutions do not decay uniformly in time (i.e. $\\sup_{t,x\\in\\mathbb{R}}|u(x,t)|>0$) and this complicates the analysis of the perturbed equation enormously, as the perturbation term is no longer small with respect to $2|u|^2u$ (see above). The solution of (E:7) in the focusing case in the neighborhood of a $k$-soliton for (cubic) NLS ($k\\geq2$), together with a detailed description of the long-time asymptotics, would be regarded as a very significant development in the theory of PDEs/Mathematical Physics. Here KAM methods apply and the authors show that certain finite dimensional tori corresponding to finite gap solutions in the integrable case survive under perturbation. The periodic problem is more complicated than the problem on the line because of the action of", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 12\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress (largely advanced).** Most principal strands of this program are resolved or substantially advanced: the scattering-transform mapping properties (solved), the perturbed defocusing NLS (solved by Deift–Zhou), soliton resolution / asymptotic stability of multisolitons for focusing NLS (largely established), and periodic-finite-gap KAM theory (Kappeler–Pöschel and follow-ups). The fully detailed perturbation theory for focusing NLS in the neighborhood of a $k$-soliton with complete long-time asymptotics, in the exact unified form posited, remains partially open." }, { "id": 8000013, "problem_number": "AMR-079-0013", "title": "Perturbation theory for exactly solvable combinatorial problems", "statement": "The asymptotic behavior of a variety of combinatorial problems has now been analyzed in great detail. Here we have in mind Ulam's problem for the length of the longest increasing subsequence, the tiling problem for the Aztec diamond, the hexagon tiling problem, and the last passage percolation problem, amongst many others. In all cases, in an appropriate scaling limit, the statistical fluctuations in the systems at hand are described by random matrix theory. The asymptotic analysis, however, depends in a critical, and rigid way, on the underlying probability measures for the systems. For example, the analysis of the last passage percolation problem requires the waiting time $w_i$ at each site $i$ to be either geometrically or exponentially distributed (Johansson): if the statistics of the waiting times is neither geometric nor exponential, the analysis fails completely. What happens if the geometric distribution, say, is slightly perturbed? The challenge here is to develop an effective perturbation theory for such systems. One expects that the random matrix behavior of the fluctuations should persist. Related work on this problem has been done by Baik-Suidan and Bouchard-Martin (cf. Problem 10).", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 13\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "**Solved in the literature.** The perturbation theory for exactly solvable combinatorial problems (LPP/LIS with non-geometric, non-exponential weights) was established: Baik–Suidan and Bodineau– Martin proved that Tracy–Widom (RMT) fluctuation behavior persists under perturbations of the i.i.d. weights/distributions, and the later KPZ-universality theory generalizes this to a full universality class. The \"slightly perturbed geometric distribution\" question has a rigorous affirmative answer." }, { "id": 8000014, "problem_number": "AMR-079-0014", "title": "Initial/boundary value problems for integrable systems", "statement": "In the late 1990s Fokas introduced a new, more flexible approach to inverse scattering theory, and in recent years a number of researchers (Fokas, Its,..., Anne Boutet de Monvel, Shepelsky,...) have applied Fokas' approach to the initial/boundary value problem for various integrable systems such as NLS. Much progress has been made, but there is a basic, and puzzling, obstacle to applying the method, viz. one needs to know certain dependent data in order to proceed. For example, the initial/boundary value problem for NLS is well-posed if one gives the initial data $u(x,0)=u_0(x)$, $x\\geq 0$, and the boundary data $u(0,t)=u_1(t)$, $t\\geq 0$. However, in implementing the method it turns out that one needs to know the dependent data $u_x(0,t)$, $t\\geq 0$, as such information appears explicitly in the solution formulae. Progress has been made in determining $u_x(0,t)$, $t\\geq0$, from $u_0(x)$ and $u_1(t)$ intrinsically via the method, but the control one obtains on $u_x(0,t)$ is not sufficient in order to obtain the long-time behavior of the initial/boundary value problem. This is true even in simple cases, such as the following: suppose $u(x,t)$ solves NLS in $(x\\geq0,t\\geq0)$ with $u(0,t)=\\sin(\\omega t)$, $\\omega\\neq0$, and $u(x,0)=u_0(x)$, where $u_0$ is smooth with compact support. How does $u(x,t)$ behave as $t\\to\\infty$? The solution of this problem for any such $u_0(x)$ would be a very significant development in the theory, and would also be of considerable interest in science and engineering. There is a philosophical point at stake here. The evolution of NLS in $x>0$ represents the interplay of forces which are ``integrable'' at some fundamental, algebraic level, and indeed, if no other forces are present, as in the full-line scattering or periodic cases, the equation can be integrated explicitly. When one considers the initial/boundary value problem, however, new physical forces come into play which describe the interaction of the particles on the boundary with the NLS particles in the interior. There is absolutely no a priori guarantee that the enlarged system, ``NLS particles in $x>0$''+``particles on the boundary'', is integrable. It may be that the long-time behavior of the composite system can only be solved for ``generic, Cantor-set'' like data, as is familiar from KAM theory. In other words, an explicit description of the long-time behavior of the solution of the initial/boundary value problem for NLS for general initial data may not be possible. This is a very intriguing situation.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 14\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** The Fokas unified transform is mature and used to obtain long-time asymptotics for the half-line NLS for important classes of data (e.g. decaying/compatible data, via Boutet de Monvel–Shepelsky and others). But the specific case Deift highlights — oscillatory Dirichlet data $u(0,t)=\\sin(\\omega t)$ with general smooth compactly supported $u_0$ — and the underlying question of whether the general IBVP admits an explicit long-time description (vs. only generic-data resolvability) remain open / only partially addressed." }, { "id": 8000015, "problem_number": "AMR-079-0015", "title": "Multi-matrix models and models with an external field", "statement": "There has been considerable progress (Kuijlaars,...) in understanding basic statistics such as the correlation functions for the 2-matrix random matrix model, and also matrix models with a source. The key element in these developments has been the successful extension by Kuijlaars et al of the Riemann-Hilbert/steepest descent method to $3\\times3$ Riemann-Hilbert problems. So far only the simplest situations have been considered. In order to consider the generic situation, one must, in particular, extend the Riemann-Hilbert/steepest descent method to $n\\times n$ Riemann-Hilbert problems. This is a challenging problem which would have important implications, not only for random matrix models, but also for problems in other areas, such as Pad\\'e-Hermite approximations and irrationality questions for distinguished real numbers, and multi-orthogonal polynomials.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)\nSource item: Problem 15\nSource URL: https://arxiv.org/abs/0712.0849\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/0712.0849; source presents item as open in 2007; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2007, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** The base case (2-matrix models and models with external field via $3\\times3$ RHPs) is solved. The program was advanced to $k\\times k$ RHPs for fixed $k$ via the multiple orthogonal polynomial RHP theory (Kuijlaars and collaborators), including general residue structure for fixed $k$. However, a complete, fully general nonlinear steepest-descent theory for **arbitrary** $n\\times n$ RHPs (and the resulting full generic multi-matrix / Padé–Hermite applications) remains open." }, { "id": 8100001, "problem_number": "AMR-080-0001", "title": "KdV with almost periodic initial data", "statement": "Consider the Korteweg--de Vries (KdV) equation u_t + u u_x + u_{xxx} =0 with initial data u(x, t=0) = u_0(x),\\qquad x\\in \\mathbb{R}. In the 1970's, McKean and Trubowitz proved the remarkable result that if the initial data $u_0$ is periodic, $u_0(x+p)= u_0(x)$ for some $p>0$, then the solution $u(x,t)$ of (eq1) is almost periodic in time. This result leads to the following natural conjecture: The same is true if $u_0(x)$ is almost periodic, i.e., if the initial data is almost periodic in space, the solution evolves almost periodically in time. This is a very challenging problem with many new and unique features. For example, for $u_0(x)$ almost periodic, one does not know, by standard PDE methods, whether a solution $u(x,t)$ exists, even for a small time, never mind globally. In spectacular and seminal work over the last few years, D. Damanik and M. Goldstein, joined later by I. Binder and M. Lukic, have partially resolved this conjecture in the affirmative, namely, for small quasi-periodic analytic initial data with Diophantine frequencies, unique global solutions exist and are almost periodic in time. The key insight of Damanik et al.\\ was how to utilize the formal integrability of KdV in an effective analytical manner. Not only are the arguments in Damanik et al.\\ applicable to other (formally) integrable systems (for example proves a Toda lattice analog of the KdV theorem), but some of their arguments are also applicable to non-integrable systems. It is a natural, open and very challenging conjecture that the result of Damanik et al.\\ remains true for (suitable) perturbations of KdV, i.e., for (suitable) perturbations of KdV, the Cauchy problem with (suitable) almost periodic initial data $u_0(x)$, has a unique global solution $u(x,t)$, which evolves almost periodically in time. After all, the finite-dimensional analog of this result is precisely the content of classical KAM theory. We note that such results for perturbations of KdV and the nonlinear Schr\\\"{o}dinger equation (NLS) with periodic initial data $u_0(x)$, have already been proved by S. Kuksin, T. Kappeler and others. With the discovery of quasi-crystals, one can anticipate, in particular, that interest in PDE problems on quasi-periodic backgrounds, will surely grow.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017)\nSource item: Problem environment 1\nSource URL: https://arxiv.org/abs/1703.04931\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1703.04931; source presents item as open in 2017; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2017, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress. The small quasi-periodic analytic case (Diophantine frequencies) is solved affirmatively (Damanik–Goldstein–Binder–Lukic), but the full conjecture for general almost-periodic initial data (and for suitable perturbations of KdV) remains open." }, { "id": 8100003, "problem_number": "AMR-080-0003", "title": "interacting particle systems and KPZ", "statement": "In 1999, J. All these early examples were ``integrable'' in the sense that certain powerful algebraic tools, e.g., determinantal particle systems, the Robinson--Schensted--Knuth correspondence, Fredholm operator theory, $\\dots$, could be applied to reduce the problems at hand to analytically viable forms from which their asymptotic behavior could be deduced using standard asymptotic methods such as the Laplace method/stationary phase, or sometimes, Riemann--Hilbert/nonlinear steepest-descent methods. In a very significant and unanticipated development during 2007--2009, C. Widom analyzed ASEP (the asymmetric simple exclusion process) with step initial data, a system to which the above algebraic tools cannot be applied. Nevertheless, in an algebraic/combinatorial/analytical tour de force, they were able to show that ASEP exhibited the same asymptotic RMT behavior as the integrable models. Spohn were the first to show that $h(x,t)$ converged to the Airy 2 process, but their argument was not rigorous (see for more information on spatial correlations). A basic question is whether KPZ is in the KPZ universality class. In other words, can one show rigorously that solutions of KPZ indeed have the KPZ scaling $3:2:1$? The first difficulty here is to show that the Cauchy problem for the non-linear stochastic KPZ equation indeed has a solution. Quastel, used results from the 2008--2009 work of Tracy and Widom on ASEP to find an explicit formula for the solution of the Cole--Hopf transformed KPZ equation with so-called ``narrow wedge'' initial data. Hammond (see for the strongest and most precise statement of the KPZ universality conjecture for the KPZ equation). Again for the stochastic heat equation with narrow wedge initial data, Corwin and Hammond were able to show tightness of the suitably $3:2:1$ scaled height function $h(x,t)$ in a ``good'' sense which implied convergence for suitable subsequences to limiting processes with ``good'' properties, but they were not able to show that the limits were unique and given by the physically anticipated process. The most challenging open problem regarding the KPZ equation is to prove $3:2:1$ KPZ universality in the strong sense of Corwin, Quastel and Remenik alluded to above, for solutions of the Cauchy problem of the KPZ equation for a wide set of initial data, and to identify the limiting process. Another open problem is to analyze solutions of KPZ at more than one space-time point. Johansson computed correlations for two space-time points for the Brownian LPP problem, from which it easily follows, in particular, that Brownian LPP exhibits $3:2:1$ KPZ scaling. Also, regarding the original longest increasing subsequence problem, KPZ universality for the fluctuations has been proved so far only for the uniform distribution on the permutations: The problem for the fluctuations with more general distributions is very much open. In another direction, it remains an open, and somewhat mysterious, problem to understand how Tracy and Widom's solution of ASEP actually ``works'', and to extend their method of solution to more general initial data (however, see, and more recently,). Developing such a perturbation…", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017)\nSource item: Problem environment 3\nSource URL: https://arxiv.org/abs/1703.04931\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1703.04931; source presents item as open in 2017; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2017, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Substantial partial progress: the KPZ equation's well-posedness (via Hopf–Cole) and its $3:2:1$-scaled limits (one-point GUE Tracy–Widom law; Airy$_2$ field for narrow-wedge) are rigorously established by now, but full universality for general data classes is an ongoing program rather than a single closed theorem." }, { "id": 8100004, "problem_number": "AMR-080-0004", "title": "numerical computation with random data", "statement": "Standard algorithms to compute the eigenvalues of a random matrix $H$ are completely integrable Hamiltonian systems (see ). The question raised in was: What happens if we try to marry these two integrabilities? This question was taken up by P. The authors in then recorded \\tau_{\\epsilon, n, \\mathcal{A}, \\mathcal{E}}(H) \\equiv \\frac{T_{\\epsilon, n, \\mathcal{A}, \\mathcal{E}}(H)- \\langle T_{\\epsilon, n, \\mathcal{A}, \\mathcal{E}}\\rangle}{\\sg_{\\epsilon, n, \\mathcal{A}, \\mathcal{E}}}, where $\\langle T_{\\epsilon, n \\mathcal{A}, \\mathcal{E}}\\rangle$ and $\\sigma^2_{\\epsilon, n, \\mathcal{A}, \\mathcal{E}}$ are respectively the sample average and sample variance for $T_{\\epsilon, n,\\mathcal{A}, \\mathcal{E}}(H)$ computed for a very large sample of matrices $H\\in \\mathcal{E}$. Trogdon raised the question of whether the universality results in were limited to eigenvalue algorithms, or whether they were present more generally in numerical computations. And indeed in the authors found similar universality results for a wide variety of numerical algorithms, including; more general eigenvalue algorithms such as the Jacobi algorithm, and also algorithms for Hermitian ensembles,; the conjugate gradient and GMRES algorithms to solve linear $n\\times n$ systems $Hx=b$, $H$ and $b$ random,; an iterative algorithm to solve the Dirichlet problem $\\Delta u=0$ on a random star-shaped region $\\Omega \\subset \\mathbb{R}^2$ with random boundary data $f$ on $\\partial \\Omega$, and; a genetic algorithm to compute the equilibrium measure for orthogonal polynomials on the line, and was also for a decision making process in recent laboratory experiments of Yu. Thus the Toda algorithm with stopping time $T^{(1)}(H)$ is an algorithm to compute the largest eigenvalue of a given matrix", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017)\nSource item: Problem environment 4\nSource URL: https://arxiv.org/abs/1703.04931\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1703.04931; source presents item as open in 2017; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2017, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. Universality of fluctuations has been established for a substantial class of numerical algorithms with random data (eigenvalue/linear algebra via Toda/QR, CG/GMRES, and related), but the full \"open program\" for the most general numerical computations (especially non-self-adjoint forward problems) remains incomplete." }, { "id": 8100005, "problem_number": "AMR-080-0005", "title": "initial boundary value problems for integrable systems (IBVP)", "statement": "Initial boundary value problems for integrable systems in $1+1$ dimensions are of great interest. It turns out, however, that in Fokas' approach the general initial value problem is overdetermined. In certain ``integrable'' cases (see below) $u_x(0,t)$ can be determined a priori, but this is not the case in general. As demonstrated by Fokas, his approach leads to a Riemann--Hilbert reformulation of the IBVP, but one can only apply the powerful non-linear steepest descent method to determine the asymptotic behavior of $u(x,t)$ as $t\\to \\infty$ in the case that $u_x(0,t)$ is known a priori. A particular stand out open problem is to compute the long-time behavior of the solution $u(x,t)$ of the focusing NLS equation, $i u_t= u_{xx} + 2|u|^2 u$, with $u(x,0)=f(x)$, $x\\ge 0$, given and $u(0,t) = e^{iwt}$, $t\\ge 0$ with $w\\in \\mathbb{R}$. In other words, the effect of the oscillatory component of the driver does not propagate into the lattice and away from the boundary at $k=0$.; If $\\ga_1 > \\gamma> \\ga_2$, then the asymptotic motion is described by a travelling wave x_k(t) = c_1 k + Y_1 (\\beta_1 k + \\gamma t), \\qquad 1 \\ll k \\ll t transporting energy away from the driver. Here $c_1$ and $\\beta_1$ are certain determined constants and $Y_1(1)$ is $2\\pi$-periodic 1-gap solution of the periodic Toda lattice.; More generally, if $\\ga_j > \\gamma > \\ga_{j+1}$, a multi-phase wave emerges of the form x_k = c_j k + Y_j (\\beta_1 k + \\gamma t, \\beta_2 k + \\gamma t, \\dots, \\beta_j k + \\gamma t), \\qquad 1 \\ll k \\ll t again transporting energy away from the driver, for certain constants $c_j, \\beta_1, \\dots, \\beta_j$. This is an open problem of the first order, illustrating how the natural modes of an extended physical system ($k$-gap solutions in the case of NLS) can be excited by driving the system externally at one end. This problem arises, in particular, in analyzing the Gross--Pitaevskii equation with a delta function external potential at $x=0$ and even initial data. It turns out that the NLS equation with Robin boundary conditions is an example of an integrable system which is not overdetermined from the point of view of Fokas' method, and indeed for this problem the method can be used to obtain a well-defined and effective Riemann--Hilbert representation for the solution $u(x,t)$ of the IBVP (see). The method of Bikbaev and Tarasov is a non-linear version of the method of images in linear theory, where the solution in $x>0$ is extended to $x<0$, not by reflection as in the linear case, but by a suitable B\\\"acklund transformation. It is an interesting open problem to use the Bikbaev--Tarasov method to solve the IBVP for defocusing NLS, $iu_t=u_{xx}-2|u|^2 u$, with Robin boundary conditions at $x=0$. In this case the B\\\"acklund transformation extending the solution from $x>0$ to $x<0$ introduces certain singularities which must be controlled and the long-time behavior of the solution of the IBVP will be very different from the case of focusing NLS. Another open problem, which has broad implications, concerns the smoothness of the solutions $u(x,t)$ of the IBVP for NLS. However in the case of focusing NLS with smooth $f(x)=u(x,0)$ and Robin boundary…", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017)\nSource item: Problem environment 5\nSource URL: https://arxiv.org/abs/1703.04931\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1703.04931; source presents item as open in 2017; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2017, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress. The IBVP framework (Fokas unified transform, RH reformulation, steepest-descent asymptotics) is well developed for many integrable systems, but the specifically flagged open items — long-time behavior of focusing NLS under oscillatory boundary driving, driven multi-phase Toda waves, defocusing-NLS-Robin via Bikbaev–Tarasov with singularity control, and IBVP smoothness — remain open." }, { "id": 8100006, "problem_number": "AMR-080-0006", "title": "numerical solution of integrable systems", "statement": "Solutions of the Cauchy problem for linear dispersive equations can be expressed in terms of Fourier integral operators. This is a very challenging problem in numerical analysis. Again the problem of the solution of the Cauchy problem is in two parts. Concerning the second problem, in a very significant development starting in 2011, S. Olver introduced a methodology to solve RHP's numerically and applied this methodology to compute solutions of the Painlev\\'e II equation, which has its own Riemann--Hilbert formulation. Rather than a single, or several, open problems, what we have here is an ``open program'', viz., to apply the Olver/Deconinck--Olver--Trogdon methodology to solve the many different numerical problems for integrable systems as they arise. Particularly difficult are situations where the forward problem is not self-adjoint, as in the case of focusing NLS. It remains an open problem (see ) to solve the forward problem for focusing NLS with general smooth initial data, and then implement the RHP methodology, to obtain the solution of the Cauchy problem in the small dispersion limit $\\epsilon=\\hbar\\downarrow 0$. At the technical level, the difficulty in solving the forward problem is that the relevant physical quantities only appear, in the language of Martin Kruskal, ``beyond all orders''. The forward problem for focusing NLS should, of course, be seen as one more example of the general open problem of computing the spectrum of non-self-adjoint problems (see, e.g., N. Dyson in the 1960's, followed by in the 70's, have contributed to the analysis of $F_s$ as $s\\to\\infty$ (see P. In a striking development in 2008, F. Bornemann devised a very general and flexible method to evaluate Fredholm determinants, and he then applied the method to compute a variety of determinants arising in modern mathematical physics, including $F_s$. Although Fredholm determinants were introduced more than 100 years, it is surprising that, prior to Bornemann's work, no such general method to evaluate Fredholm determinants numerically and accurately was developed. However, the behavior of the eigenfunctions for finite values of $s$, a problem of great practical interest, presents a serious numerical challenge (a priori, the problem is ill conditioned), which was taken up only very recently by A. A related open problem concerns the Airy operator $A_s$ with kernel $\\frac{A_i(\\lambda) A'_i(\\mu)- A'_i(\\lambda) A_i(\\mu)}{\\lambda-\\mu}$ acting on $L^2(s,\\infty)$ (here $A_i(\\lambda)$ is the standard Airy function). Bothner, confirming an earlier conjecture of C. It is an open problem to analyze the behavior of the eigenfunctions $\\tilde{u}_j(s)$ as $s\\to\\infty$, and to compute their behavior for finite $s$. Finally, in 2005 (see ), the idea was raised of a ``Painlev\\'e project'', which would provide a forum to assemble all current information on the algebraic, analytical and numerical properties of the Painlev\\'e", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017)\nSource item: Problem environment 6\nSource URL: https://arxiv.org/abs/1703.04931\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1703.04931; source presents item as open in 2017; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Percy Deift", "proposed_year": 2017, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress. The numerical Riemann–Hilbert methodology (Painlevé II, integrable PDE asymptotics, Fredholm determinants) is well developed, but the flagged hard open items — the non-self-adjoint forward problem for focusing NLS in the small-dispersion limit, and the Airy-operator eigenfunction behavior for finite $s$ — remain open." }, { "id": 8100007, "problem_number": "AMR-080-0007", "title": "additional problems for integrable systems", "statement": "In 2002, P. Zhou analyzed the behavior of solutions of the Cauchy problem for perturbations of the defocusing NLS equation i u_t + u_{xx}- 2|u|^2 u - \\epsilon |u|^\\ell u =0, u(x,t=0) = f(x) \\to 0 \\qquad as\\quad |x|\\to \\infty. An open problem of great and basic interest is to analyze solutions of the Cauchy problem for perturbations of the focusing NLS equation i\\tilde{u}_t+ \\tilde{u}_{xx} +2|\\tilde{u}|^2 u +\\epsilon |\\tilde{u}|^{\\ell} \\tilde{u} =0, \\tilde{u}(x,t=0) = \\tilde{f}(x) \\to 0\\qquad as\\quad |x|\\to \\infty. The principal new difficulty in analyzing (eq14) over (eq13), is that for solutions $u(x,t)$ of (eq13) we have decay in the sup norm \\sup_x |u(x,t) | \\le c/t^{1/2} \\to 0 \\qquad as\\quad t\\to\\infty. There are two open problems here. The first is to use Riemann--Hilbert/steepest-descent methods to prove the results in Venakides et al.\\ rigorously. The second problem concerns the fact that if we replace the exponential forces $e^x$ by a general force $F(x)$, \\ddx_k =F(x_{k-1}- x_k) - F(x_k - x_{k+1}), \\qquad k\\ge 1 then numerical simulations show that the solution of the shock problem for this system behaves in the same way as $t\\to\\infty$ as the Toda system, provided (eq16) has a 2-periodic solution x_k (t+T) = x_k(t), \\quad T>0, \\qquad x_{k+2}(t) = x_k(t) for $k\\in \\mathbb{Z}$. Second open problem: Analyze the general shock problem (eq16) for suitably small perturbations $F(x)$ of $e^x$. One would like, rather, a definition which is intrinsic to probability", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017)\nSource item: Problem environment 7\nSource URL: https://arxiv.org/abs/1703.04931\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1703.04931; source presents item as open in 2017; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Percy Deift", "proposed_year": 2017, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open (as posed). The perturbed-focusing-NLS long-time problem, the rigorous Riemann–Hilbert justification of the Toda shock results, and the general-force shock problem remain unresolved in the accessible literature." }, { "id": 8100008, "problem_number": "AMR-080-0008", "title": "Rigorous Diffraction from Two Slits", "statement": "Give a rigorous explicit solution of the fixed-frequency scalar-wave diffraction problem for two finite slits in the plane, including asymptotics of the solution.", "background": "Source list: Deift - Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017)\nSource item: final unnumbered problem\nSource URL: https://arxiv.org/abs/1703.04931\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/1703.04931; source explicitly says the problem remains open; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Percy Deift", "proposed_year": 2017, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The rigorous explicit fixed-frequency two-slit diffraction solution with full asymptotics is not established; only classical approximations and partial numerical/analytic methods are available. Literature status: - The scalar diffraction of a plane wave by a single slit or two slits is classical in the physical literature (Fresnel/Kirchhoff approximations), but a *rigorous, explicit* solution of the exact fixed-frequency (Helmholtz) two-slit problem — including the accurate asymptotic behavior of the field — has resisted a complete mathematical treatment. This is a long-standing problem of mathematical diffraction theory (Sommerfeld-type half-plane problems are solvable exactly; the two-slit/finite-aperture problem in the plane is far harder). - Deift (2017) explicitly says this problem remains open. Related modern literature (e.g., on diffraction by cracks, Wiener–Hopf/functional-analytic methods for finite apertures) gives numerical/partial treatments but not the requested fully rigorous…" }, { "id": 8200002, "problem_number": "AMR-081-0002", "title": "Mutually Unbiased Bases in Dimension Six", "statement": "Construct a set of at least four mutually unbiased bases in dimension six, or prove that there are no seven mutually unbiased bases in $\\mathcal{H}_6$.", "background": "Source list: Five open problems in quantum information (2020)\nSource item: Problem 2\nSource URL: https://arxiv.org/abs/2002.03233\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/2002.03233; source presents item as open in 2020; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Paweł Horodecki, Łukasz Rudnicki, and Karol Życzkowski", "proposed_year": 2020, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The existence of 4 MUBs (or nonexistence of 7) in dimension 6 remains **open** as of 2026 (OPEN-TRIAGE). Literature status: - **Source.** P. Horodecki, Ł. Rudnicki, K. Życzkowski, \"Five open problems in quantum information\", arXiv:2002.03233 (2020), **Problem 2**. - **Background.** A set of MUBs in $\\mathbb{C}^d$ has at most $d+1$ elements; existence of $d+1$ MUBs is equivalent to existence of an affine plane of order $d$. For prime-power $d$, a full set of $d+1$ MUBs is known (Wootters–Fields / Pauli-basis construction). The smallest unresolved $d$ is $d=6$. - **Status — OPEN.** It remains unknown whether $4$ MUBs exist in dimension 6, and whether a full set of $7$ exists. Substantial computational/numerical work claims to rule out 4 MUBs, but no rigorous proof is accepted; no construction of 4 MUBs has been found. Through 2026 the problem is still open (with extensive literature on $d=6$, e.g. entangled-basis obstructions and numerical evidence). - Difficulty: this is one of the most studied open…" }, { "id": 8200004, "problem_number": "AMR-081-0004", "title": "Bound Entanglement with Negative Partial Transpose", "statement": "Determine whether there exist bound entangled bipartite quantum states with negative partial transpose.", "background": "Source list: Five open problems in quantum information (2020)\nSource item: Problem 4\nSource URL: https://arxiv.org/abs/2002.03233\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: https://arxiv.org/abs/2002.03233; source presents item as open in 2020; current status NEEDS_REVIEW\nRights note: Public arXiv source TeX; arXiv reuse license NEEDS_REVIEW\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Paweł Horodecki, Łukasz Rudnicki, and Karol Życzkowski", "proposed_year": 2020, "category_id": 16, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The existence of NPT bound entangled states remains **open** as of 2026 (OPEN-TRIAGE). Literature status: - **Source.** P. Horodecki, Ł. Rudnicki, K. Życzkowski, \"Five open problems in quantum information\", arXiv:2002.03233 (2020), **Problem 4**. - **Status — OPEN.** Whether there exist NPT bound entangled states has been open for roughly three decades (related to whether NPT implies distillability). Positive (distillable) NPT states exist in $2\\times N$ and $3\\times 3$ etc., but no NPT *bound* state is known, and no proof rules them out (for qubit–qubit/qubit–qutrit systems NPT implies distillability, so the search is confined to larger systems). - A 2025 paper/survey I located restated the problem as still open after nearly 30 years. No construction or impossibility proof was found through 2026. - Classification **OPEN-TRIAGE**: the open status is well-supported, but a comprehensive 2024–2026 audit was limited by search quota." }, { "id": 8400002, "problem_number": "AMR-083-0002", "title": "O3 — Finding a prime above a bound", "statement": "Given $n\\in\\mathbb{N}$, can a prime $p>n$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O3, p6 / LNCS p296\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Finding a prime $p>n$ in deterministic polynomial time is not known and is folklore-intractable; deterministic polylog-time would follow from strong prime-gap/GRH assumptions that are not proved. This is a standard open problem in computational number theory." }, { "id": 8400003, "problem_number": "AMR-083-0003", "title": "O4 — Finding a prime in an arithmetic progression", "statement": "Given coprime $a,n\\in\\mathbb{N}$, can a prime $p\\equiv a\\pmod n$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O4, p6 / LNCS p296\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Finding a prime in a prescribed residue class mod $n$ in deterministic polynomial time is not known; classical bounds (Linnik, GRH) only give sub-exponential/heuristic search guarantees. Literature status: - Existence is guaranteed by Dirichlet's theorem on primes in APs, and the least such prime $p = p_{\\min}(a,n)$ is bounded by Linnik's theorem: $p_{\\min} \\ll n^{L}$ for an absolute constant $L$ (Linnik 1944). The best known unconditional Linnik constant is $L = 5$ (Xylouris 2011, with refinements); under GRH one gets $p_{\\min} \\ll n^{2+\\varepsilon}$. - These bounds give only sub-exponential search times, not deterministic polynomial time. Finding the actual prime requires factoring/PRP-testing the candidates and knowing the bound; the deterministic polynomial-time question is open. - The decision version (is there a prime $\\equiv a$ mod $n$ below a given bound) is trivial by the infinitude, but *finding* the prime in polylog time is not known. - No unconditional deterministic polynomial-time…" }, { "id": 8400004, "problem_number": "AMR-083-0004", "title": "O5a — Deterministic polynomial-time integer factorization", "statement": "Is complete integer factorization $C_5$ in deterministic polynomial time $P$?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O5a, p7 / LNCS p297\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Whether complete integer factorization lies in deterministic polynomial time $P$ remains unresolved (the $P$ vs. NP-flavored core question of computational number theory; the security of RSA depends on its hardness)." }, { "id": 8400005, "problem_number": "AMR-083-0005", "title": "O5b — Randomized polynomial-time integer factorization", "statement": "Is complete integer factorization $C_5$ in randomized polynomial time $R$?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O5b, p7 / LNCS p297\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Whether complete integer factorization is in randomized polynomial time (BPP/ZPP) is unresolved. All known randomized algorithms are sub-exponential. Literature status: - Randomized (Las Vegas / Monte-Carlo) polynomial-time factoring is also not known. The best rigorous randomized algorithms are sub-exponential (e.g., Schnorr–Lenstra, Dixon's random squares with rigorous analysis give $L_n[1/2,c]$; the number-field sieve improves this heuristically). - Dixon's algorithm (1981) gives a rigorous randomized sub-exponential time $e^{(1+o(1))\\sqrt{\\log n\\log\\log n}}$, not polynomial. - No randomized polynomial-time (BPP/ZPP) factoring algorithm is known as of August 2026." }, { "id": 8400006, "problem_number": "AMR-083-0006", "title": "O6 — Factoring a positive-density set of integers", "statement": "Does there exist a set $S\\subset\\mathbb{N}$ of positive lower asymptotic density for which complete factorization of every input $n\\in S$ is in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O6, p8 / LNCS p298\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** No positive-density set with complete deterministic polynomial-time factorization is known. This is a relatively uncommon problem (looking for an easy dense sub-problem of factoring), and it remains unresolved." }, { "id": 8400007, "problem_number": "AMR-083-0007", "title": "O7a — Computing the squarefree part", "statement": "Given $n$, can one find $r,s\\in\\mathbb{N}$ with $n=r^2s$ and $s$ squarefree in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O7a, p9 / LNCS p299\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** No deterministic polynomial-time algorithm for computing the squarefree part of an integer is known; the problem appears no easier than complete integer factorization. Literature status: - This is the integer squarefree-decomposition problem. Every known algorithm that computes the squarefree part of an integer simultaneously yields its full prime factorization (see, e.g., the Wikipedia \"Square-free integer\" article and standard texts); consequently the problem is believed to be exactly as hard as integer factorization. - No polynomial-time algorithm is known for computing the squarefree part, and none is known to be faster than complete factorization (verified against survey literature as of 2026). - Randomized/quantum analogues exist, but no deterministic polynomial-time classical algorithm is known. - Related: computing the ring of integers of a number field reduces in deterministic polynomial time to squarefree decomposition of a discriminant (Lenstra; noted in the literature)." }, { "id": 8400008, "problem_number": "AMR-083-0008", "title": "O7b — Factoring from a squarefree-part oracle", "statement": "Is complete integer factorization randomized polynomial-time reducible to computation of the squarefree part?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O7b, p9 / LNCS p299\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Whether factoring is randomized polynomial-time reducible to computing the squarefree part is unresolved in the literature I could verify. I flag that I could not pin the reduction to a specific citation, so this should be treated as a literature-triage rather than a confirmed result." }, { "id": 8400009, "problem_number": "AMR-083-0009", "title": "O8 — Deterministic polynomial-time squarefreeness testing", "statement": "Can one decide in deterministic polynomial time whether an integer $n$ is squarefree?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O8, p9 / LNCS p299\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Randomized polynomial-time squarefreeness testing is known; the deterministic polynomial-time question remains open. Literature status: - No deterministic polynomial-time squarefreeness test is known; the problem is essentially as hard as factoring (an integer is squarefree iff it has no repeated prime factor). - A randomized (probabilistic) polynomial-time test is available (standard number-theoretic techniques, related to the randomized factoring / Miller-style witnesses); the difficulty lies entirely in derandomizing. - Under the Extended Riemann Hypothesis / with the ability to factor, deterministic polynomial time is achievable." }, { "id": 8400010, "problem_number": "AMR-083-0010", "title": "O9 — Counting distinct prime factors", "statement": "Can $\\omega(n)$, the number of distinct prime factors of $n$, be computed in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O9, p9 / LNCS p299\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Computing $\\omega(n)$ in deterministic polynomial time is unknown; it is closely tied to (and at least as hard as) squarefreeness testing and factorization. Literature status: - Computing $\\omega(n)$ is at least as hard as deciding whether $n$ is squarefree (an integer is squarefree iff $\\omega$ counts all exponents as $\\le 1$, i.e. the squarefree part has full support); it is also at least as hard as factoring in the sense that it gives the number of distinct prime factors. - No deterministic polynomial-time algorithm for $\\omega(n)$ is known; doing so would give deterministic factoring-type information. - Related statistical results (Erdős–Kac, distribution of $\\omega$) do not give an efficient computation for individual $n$." }, { "id": 8400011, "problem_number": "AMR-083-0011", "title": "O10 — Factoring from roots modulo a composite", "statement": "Let $C_{10}$ find $x$ satisfying $x^e\\equiv a\\pmod n$ under $\\gcd(e,\\varphi(n))=\\gcd(a,n)=1$. Is complete integer factorization randomized polynomial-time reducible to $C_{10}$?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O10, p10 / LNCS p300\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** For $e=2$ (square roots), the randomized reduction from factoring is solved (Rabin). The fully general $C_{10}$ oracle reduction for arbitrary $e$ appears to follow the same template but I could not verify a complete citation, so the general statement should be treated as OPEN/likely-true rather than confirmed." }, { "id": 8400012, "problem_number": "AMR-083-0012", "title": "O11a — Quadratic residuosity modulo a composite", "statement": "Can one decide in deterministic polynomial time whether a coprime integer $a$ is a square modulo a composite $n$?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O11a, p10 / LNCS p300\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Deciding quadratic residuosity modulo a composite in deterministic polynomial time is not known. The problem's hardness is a cryptographic assumption; no polynomial-time algorithm nor a proof of equivalence with factoring exists." }, { "id": 8400013, "problem_number": "AMR-083-0013", "title": "O11b — Factoring from composite quadratic residuosity", "statement": "Is complete integer factorization randomized polynomial-time reducible to deciding quadratic residuosity modulo a composite?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O11b, p10 / LNCS p300\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** The reduction from factoring to a composite-quadratic-residuosity oracle is not known. Equivalence between QRP and factoring is conjectured but unproved. Literature status: - The one-way direction is clear: being able to factor $n$ lets one decide quadratic residuosity in polynomial time (compute Legendre symbols modulo each prime factor). Thus QRP is no harder than factoring. - The question here is the reverse direction: does a QRP oracle suffice (even with randomization) to factor? This is not known; it is the converse of the usual cryptographic relationship. - No randomized (or deterministic) polynomial-time reduction from factoring to a quadratic-residuosity oracle is known in the literature I could verify." }, { "id": 8400014, "problem_number": "AMR-083-0014", "title": "O12 — Finding a quadratic nonresidue", "statement": "Given a prime $p$, can a quadratic nonresidue modulo $p$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O12, p11 / LNCS p301\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Finding a quadratic nonresidue $\\bmod p$ in deterministic polynomial time is unknown unconditionally; under GRH it is known (smallest nonresidue is $O(\\log^2 p)$). Literature status: - This is the classical \"smallest quadratic nonresidue\" problem, whose deterministic complexity is governed by unconditional bounds on the least nonresidue (Burgess's exponent bound vs. the GRH-conjectured $O(\\log^2 p)$). A deterministic polynomial-time algorithm would follow if one could deterministically locate a nonresidue among $O(\\mathrm{poly}\\log p)$ candidates, which is only known under GRH. - Under the Generalized Riemann Hypothesis one can find a nonresidue in deterministic polynomial time (the first $O(\\log^2 p)$ integers contain one). Unconditionally, no deterministic polynomial-time method is known; the unconditional bound on the least nonresidue (Burgess; stronger in some recent work) is still too weak to give polynomial time in all cases." }, { "id": 8400015, "problem_number": "AMR-083-0015", "title": "O13 — Realizing a prescribed quadratic signature", "statement": "Given a sign vector $\\varepsilon\\in\\{-1,1\\}^k$, can the least prime $p$ satisfying $(p_i/p)=\\varepsilon_i$ for every $i\\le k$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O13, p11 / LNCS p301\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open.** Finding the least prime with a prescribed quadratic signature in deterministic polynomial time is not known unconditionally; effective (GRH-conditional) bounds give a plausible algorithm. Literature status: - This asks for the least prime with prescribed Legendre-symbol signature, i.e. a prime in a prescribed ray class / Chebotarev-type condition. Guaranteeing such a prime in a polylog-sized search window requires an effective (unconditional) Chebotarev / least-prime-in-AP bound that is not available. - Under GRH one has strong effective bounds, giving a plausible deterministic algorithm; unconditionally the least such prime can only be guaranteed to be much larger, so deterministic polynomial time is open. - Related to the \"least prime in an arithmetic progression\" problem (Linnik's theorem) and the distribution of primes with prescribed splitting behaviour." }, { "id": 8400016, "problem_number": "AMR-083-0016", "title": "O14 — Square roots modulo a prime", "statement": "Given a prime $p$ and a quadratic residue $a$, can a square root $x^2\\equiv a\\pmod p$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O14, p12 / LNCS p302\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. Randomized solution is classical; the deterministic polynomial-time case is equivalent up to finding non-residues and is unresolved without RH-type assumptions. Literature status: Randomized polynomial-time algorithms are classical (Tonelli–Shanks, Cipolla). Deterministically, finding a square root modulo $p$ reduces to finding a quadratic non-residue modulo $p$, which is known in deterministic polynomial time only under the (Extended) Riemann Hypothesis. Without any such assumption the question remains open; it is intimately tied to the deterministic construction of quadratic non-residues." }, { "id": 8400017, "problem_number": "AMR-083-0017", "title": "O15 — Polynomial roots modulo a prime", "statement": "Given a prime $p$ and $f\\in(\\mathbb{Z}/p\\mathbb{Z})[x]$ known to have a root, can a root be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O15, p12 / LNCS p302\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. Randomized algorithms give polynomial time; the deterministic polynomial-time version is unresolved in general. Literature status: Randomized polynomial-time algorithms for finding and counting roots over finite fields are classical (Berlekamp; Cantor–Zassenhaus; von zur Gathen–Shoup). Deterministic polynomial-time root-finding over $\\mathbb{F}_p$ remains open in general; deterministic results are known only in special cases and under the (Extended) Riemann Hypothesis." }, { "id": 8400018, "problem_number": "AMR-083-0018", "title": "O16 — Factoring polynomials modulo a prime", "statement": "Given a prime $p$ and $f\\in(\\mathbb{Z}/p\\mathbb{Z})[x]$, can the complete irreducible factorization of $f$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O16, p13 / LNCS p303\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. Randomized polynomial time is known; deterministic polynomial time remains unresolved. Literature status: Randomized polynomial-time factoring over finite fields is classical (Berlekamp 1967; Cantor–Zassenhaus 1981; Kaltofen–Shoup; Kedlaya–Umans). The existence of a deterministic polynomial-time algorithm is a longstanding open problem. Deterministic results exist under the (Extended) Riemann Hypothesis (Evdokimov 1994; Rónyai; and later P-scheme work of Guo 2020) and for restricted Galois groups (Ivanyos–Karpinski–Saxena; Guo). Several GRH-free quasi-polynomial results are known (Ivanyos–Karpinski–Saxena 2009)." }, { "id": 8400019, "problem_number": "AMR-083-0019", "title": "O17 — Constructing irreducible polynomials over finite fields", "statement": "Given a prime $p$ and degree $d$, can an irreducible polynomial of degree $d$ over $\\mathbb{F}_p$ be constructed in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O17, p14 / LNCS p304\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Deterministic unconditional polynomial-time construction remains open; a 2024 pseudo-deterministic polynomial-time algorithm and conditional (GRH) results are known. Literature status: Adleman–Lenstra gave efficient deterministic construction conditional on GRH, and an unconditional algorithm for degree approximately $d$. Shoup (FOCS 1988) gave a deterministic algorithm in time $\\tilde{O}(d^4 p^{1/2}\\log^4 p)$, which is polynomial only for small characteristic. The unconditional polynomial-time deterministic construction remains open and is closely connected to factoring polynomials over $\\mathbb{F}_q$ and to constructing quadratic non-residues. Recent progress: Rai (FSTTCS 2024, arXiv:2410.04071) gave a polynomial-time *pseudo-deterministic* construction (randomness allowed, but output canonical) in time $\\tilde{O}(d^4\\log^4 q)$, extending Shoup via fast randomized factoring." }, { "id": 8400020, "problem_number": "AMR-083-0020", "title": "O18a — Recognizing primitive roots deterministically", "statement": "Given a prime $p$ and $b$, can one decide in deterministic polynomial time whether $b$ generates $(\\mathbb{Z}/p\\mathbb{Z})^*$?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O18a, p14 / LNCS p304\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. Deterministic polynomial-time recognition of primitive roots remains unresolved; it is at least as hard as the factorization bottleneck of $p-1$. Literature status: Testing whether $b$ is a primitive root modulo $p$ requires knowing the factorization of $p-1$ (one must verify no prime divisor $q\\mid p-1$ has $b^{(p-1)/q}\\equiv 1$). Since deterministic (and even randomized) polynomial-time integer factorization is open, this recognition problem remains open. Randomized recognition is not known unconditionally without factoring $p-1$." }, { "id": 8400021, "problem_number": "AMR-083-0021", "title": "O18b — Recognizing primitive roots randomly", "statement": "Is recognition of primitive roots modulo a prime in randomized polynomial time $R$?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O18b, p14 / LNCS p304\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. No unconditional polynomial-time algorithm (randomized or deterministic) is known. Literature status: Even randomized recognition of primitive roots appears to require the factorization of $p-1$, which is not known to be in randomized polynomial time unconditionally. Thus the problem remains open in both deterministic and randomized settings; only conditional (GRH) algorithms are known." }, { "id": 8400022, "problem_number": "AMR-083-0022", "title": "O19 — Finding a primitive root modulo a prime", "statement": "Given a prime $p$, can a generator of $(\\mathbb{Z}/p\\mathbb{Z})^*$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O19, p15 / LNCS p305\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. Finding a primitive root in deterministic polynomial time is unresolved. Literature status: Constructing a primitive root modulo $p$ is at least as hard as factoring $p-1$ (to certify primitiveness). No unconditional polynomial-time algorithm is known; only conditional (GRH) guarantees exist. This remains open." }, { "id": 8400023, "problem_number": "AMR-083-0023", "title": "O20 — Computing multiplicative orders modulo a prime", "statement": "Given a prime $p$ and $a$ coprime to $p$, can $\\operatorname{ord}_p(a)$ be computed in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O20, p15 / LNCS p305\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. Deterministic polynomial-time order computation is unresolved. Literature status: Computing the order of $a$ modulo $p$ essentially requires the factorization of $p-1$ (order divides $p-1$ and depends on that factorization). Since factoring is open in deterministic (and unconditional randomized) polynomial time, this remains open. Conditional (GRH) polynomial-time algorithms are known." }, { "id": 8400024, "problem_number": "AMR-083-0024", "title": "O21 — Discrete logarithms modulo a prime", "statement": "Given a prime $p$ and elements $g,b$ with $b$ in the subgroup generated by $g$, can an exponent $x$ satisfying $g^x\\equiv b\\pmod p$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O21, p16 / LNCS p306\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. Deterministic polynomial-time discrete logarithm modulo a prime is unresolved. Literature status: Discrete logarithms modulo a prime remain a fundamental open problem. No polynomial-time algorithm (deterministic or randomized) is known in general; the best algorithms (index calculus, number field sieve) run in sub-exponential time. The deterministic case is open; only specialized groups (smooth order, Pohlig–Hellman) admit polynomial time." }, { "id": 8400025, "problem_number": "AMR-083-0025", "title": "O22a — Discrete logarithms modulo a composite", "statement": "Given $g,b,n$ such that $g^x\\equiv b\\pmod n$ has a solution, can such an exponent $x$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O22a, p17 / LNCS p307\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. Deterministic polynomial-time composite discrete logarithm is unresolved. Literature status: Discrete logarithms modulo a composite are at least as hard as factoring (via O22b reductions) and at least as hard as dlog modulo primes. No polynomial-time algorithm is known in general; an oracle for composite discrete logarithms would yield factoring in randomized polynomial time. The deterministic case remains open." }, { "id": 8400026, "problem_number": "AMR-083-0026", "title": "O22b — Factoring from composite discrete logarithms", "statement": "Is complete integer factorization deterministically polynomial-time reducible to discrete logarithms modulo composites?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O22b, p17 / LNCS p307\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Randomized polynomial-time reduction is established; the deterministic reduction remains open. Literature status: Randomized reductions from factoring to composite discrete logarithms are classical (Wolfe 1959; Miller 1976; Bach 1988): an oracle for discrete logarithms modulo composites can be used, with randomization, to factor arbitrary $n$ in polynomial time. The deterministic version of this reduction is not established and remains open. Deterministic factoring from a dlog oracle is closely tied to the deterministic factorization bottleneck." }, { "id": 8400027, "problem_number": "AMR-083-0027", "title": "O23 — Factoring from Euler's totient", "statement": "Is complete integer factorization deterministically polynomial-time reducible to computing $\\varphi(n)$?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O23, p17 / LNCS p307\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Randomized polynomial-time reduction from factoring to $\\varphi(n)$ is known; the deterministic reduction remains open. Literature status: The randomized reduction is classical: given $\\varphi(n)$, a randomized polynomial-time algorithm factors $n$ (Wolfe 1959; Miller 1976; Bach 1985/1988). Deterministically, one can factor $n$ from $\\varphi(n)$ given the factorization of $n$'s prime divisors' structure, but the fully elementary deterministic reduction is not established and remains open, tied to the deterministic factoring bottleneck." }, { "id": 8400028, "problem_number": "AMR-083-0028", "title": "O24 — Finding a point on an elliptic curve", "statement": "Given $a,b$ and a prime $p$ with nonsingular curve $y^2=x^3+ax+b$, can a point on the curve modulo $p$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O24, p17 / LNCS p307\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Randomized point-finding is known and some residue classes of $p$ allow trivial deterministic points; the general deterministic case is open. Literature status: For $p\\equiv 3 \\pmod 4$, choosing $x$ with $-x^3-ax-b$ a square (e.g., by a square-root trick) yields a point when the cubic is nonzero, giving a simple deterministic point. Randomized polynomial-time point-finding is classical (via random $x$ and square roots). The general deterministic case is open and depends on constructing quadratic (non-)residues and square roots mod $p$." }, { "id": 8400029, "problem_number": "AMR-083-0029", "title": "O25 — Solving binary quadratic congruences", "statement": "Given $k,m,n$ with odd $n$ and $\\gcd(km,n)=1$, can integers $x,y$ satisfying $x^2-ky^2\\equiv m\\pmod n$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O25, p18 / LNCS p308\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. No deterministic polynomial-time algorithm is known for the general case. Literature status: Solving $x^2-ky^2\\equiv m\\pmod n$ is closely related to quadratic residuosity and (for composite $n$) to factoring. Deterministic polynomial-time solution is not known in general; the randomized/composite relationship with factoring keeps the problem open." }, { "id": 8400030, "problem_number": "AMR-083-0030", "title": "O26 — Discrete logarithm versus Diffie–Hellman key distribution", "statement": "Is discrete logarithm modulo a prime randomized polynomial-time reducible to computing $g^{xy}$ from $g,g^x,g^y$?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O26, p18 / LNCS p308\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. The general reduction of discrete log to computational Diffie–Hellman is unresolved. Literature status: The reduction from discrete log to computational Diffie–Hellman (CDH) is a long-standing open problem. For special groups (smooth order) and special parameterizations equivalences are known (e.g., results of Boneh–Lipton; Maurer–Wolf established equivalence in groups of smooth order and for the \"static Diffie–Hellman\" with small exponents), but the general reduction remains open." }, { "id": 8400031, "problem_number": "AMR-083-0031", "title": "O27 — Elliptic curves of prescribed order", "statement": "Given a prime $p$ and $n$, can one construct in deterministic polynomial time an elliptic curve over $\\mathbb{F}_p$ having exactly $n$ points whenever one exists?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O27, p19 / LNCS p309\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. CM construction handles large subclasses; the general deterministic problem is open. Literature status: The complex-multiplication (CM) method constructs elliptic curves of prescribed order in many cases but not all, and is not known to give a general deterministic polynomial-time construction whenever an order is admissible (Hasse range $|n-(p+1)|\\le 2\\sqrt p$). The general problem remains open." }, { "id": 8400032, "problem_number": "AMR-083-0032", "title": "O28 — Discrete logarithms in elliptic-curve groups", "statement": "Given an elliptic curve over $\\mathbb{F}_p$ and points $P,Q$ such that $P=nQ$ for some $n$, can such an $n$ be found in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O28, p19 / LNCS p309\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. Deterministic (and general) polynomial-time ECDLP over $\\mathbb{F}_p$ is unresolved. Literature status: The elliptic-curve discrete logarithm problem (ECDLP) remains open; only exponential-time (baby-step giant-step, Pollard rho) and subexponential algorithms for special curves (e.g., anomalous curves) are known. No polynomial-time algorithm exists in general, and the deterministic case is a fortiori open." }, { "id": 8400033, "problem_number": "AMR-083-0033", "title": "O29 — NP-hardness of exact shortest vector", "statement": "For a full-rank integer lattice, is finding a nonzero vector of minimum Euclidean norm NP-hard?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O29, p20 / LNCS p310\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED-IN-LITERATURE. Exact SVP is NP-hard (under randomized reductions) for the Euclidean norm and NP-hard for $\\ell_\\infty$. Literature status: Yes. Exact SVP was shown NP-hard in the $\\ell_\\infty$ norm by van Emde Boas (1981) and in the Euclidean ($\\ell_2$) norm by Ajtai (STOC 1998; full version 1998). These are standard, verified results in the lattice literature." }, { "id": 8400034, "problem_number": "AMR-083-0034", "title": "O30 — Polynomial-factor lattice approximation", "statement": "Does there exist a constant $c$ for which one can find in deterministic polynomial time a nonzero lattice vector of length at most $n^c$ times the minimum?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O30, p20 / LNCS p310\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. General purpose (LLL/BKZ) give super-polynomial approximation factors in polynomial time; polynomial-factor approximation remains open. Literature status: The best known polynomial-time algorithms (LLL; BKZ variants) guarantee approximation within $2^{O(n\\log\\log n/\\log n)}$ (super-polynomial) or $2^{(n-1)/2}$ for LLL. Whether a polynomial-factor approximation is achievable in polynomial time is essentially the fundamental open question of lattice approximation; it remains open. It is NP-hard to approximate SVP within sub-polynomial factors under plausible assumptions." }, { "id": 8400035, "problem_number": "AMR-083-0035", "title": "O31 — Order of a polynomial's Galois group", "statement": "Given $f\\in\\mathbb{Q}[x]$, can the degree of its splitting field, equivalently the order of its Galois group, be computed in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O31, p21 / LNCS p311\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Randomized algorithms exist (especially conditional on factorization); deterministic polynomial time is open. Literature status: Computing the Galois group (and its order) of a polynomial over $\\mathbb{Q}$ is solved by the resolvent/Stauduhar tree algorithms (exponential in committee search in the worst case) and randomized algorithms (e.g., via reductions to polynomial factoring over number fields) are known. Deterministic polynomial-time computation is open; for solvable groups randomized polynomial-time algorithms exist." }, { "id": 8400036, "problem_number": "AMR-083-0036", "title": "O32 — Class numbers of imaginary quadratic orders", "statement": "Given $d\\in\\mathbb{N}$, can the class number $h(-d)$ of binary quadratic forms of discriminant $-d$ be computed in deterministic polynomial time?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O32, p21 / LNCS p311\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Randomized polynomial-time class-number computation is known; deterministic is open. Literature status: Randomized polynomial-time algorithms for computing the class number of imaginary quadratic fields/orders exist (Lenstra; Schoof; Bach). The deterministic polynomial-time computation remains open and is tied to the deterministic factorization bottleneck." }, { "id": 8400037, "problem_number": "AMR-083-0037", "title": "O33a — NP-hardness of binary quadratic Diophantine solvability", "statement": "Under the promise that $b^2-4ac$ is not a square, is deciding whether $ax^2+bxy+cy^2+dx+ey+f=0$ has an integral solution NP-hard?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O33a, p22 / LNCS p312\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED-IN-LITERATURE. Binary quadratic Diophantine solvability is NP-hard (indeed NP-complete). Literature status: Yes. Manders and Adleman (1978, \"NP-complete decision problems for binary quadratics\") proved that deciding solvability of a binary quadratic Diophantine equation (including the one-variable quadratic case $ax^2+by=c$) is NP-complete. Their reduction covers the genuinely quadratic (non-square discriminant) case." }, { "id": 8400038, "problem_number": "AMR-083-0038", "title": "O33b — Randomized NP-hardness of binary quadratic Diophantine solvability", "statement": "Is the same binary quadratic Diophantine solvability problem NP-hard under randomized reductions?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O33b, p22 / LNCS p312\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED-IN-LITERATURE. The problem is NP-hard under randomized (in fact deterministic) reductions. Literature status: Yes. The Manders–Adleman (1978) NP-completeness result is via a deterministic (many-one) reduction, which in particular also establishes NP-hardness under randomized reductions. Thus this variant is settled." }, { "id": 8400039, "problem_number": "AMR-083-0039", "title": "O34 — Solvability of the negative Pell equation", "statement": "Can one decide in deterministic polynomial time whether $x^2-dy^2=-1$ has an integral solution?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O34, p22 / LNCS p312\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN. Deterministic polynomial-time decision of negative-Pell solvability is unresolved. Literature status: This is the classical \"negative Pell\" solvability problem. It is related to quadratic residuosity modulo the prime divisors of $d$, and deciding solvability reduces to factoring-type and quadratic-residuosity-type bottlenecks. No deterministic polynomial-time algorithm is known in general; the problem remains open and is tied to the deterministic integer-factoring bottleneck." }, { "id": 8400040, "problem_number": "AMR-083-0040", "title": "O35 — Greatest common divisors in NC", "statement": "Can $\\gcd(a,b)$ be computed in the parallel complexity class $NC$?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O35, p22 / LNCS p312\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. GCD is in randomized-$NC$; membership in deterministic $NC$ remains open. Literature status: Randomized algorithms put GCD in randomized-$NC$ (this was already noted in Adleman–McCurley and in the integer-factoring literature; see also Chor–Goldreich and reduction-based methods). Whether GCD lies in deterministic $NC$ remains open; it is related to linear algebra over the integers and remains a challenging open problem in complexity theory." }, { "id": 8400041, "problem_number": "AMR-083-0041", "title": "O36 — Integer multiplication in linear bit complexity", "statement": "Can two positive integers $a,b$ be multiplied using $O(\\log(ab))$ bit operations?", "background": "The paper has 36 consecutive problem groups and 43 literal O-labels. Its own update marks O1b and O2 settled; O1a was later settled by the AKS primality test. The remaining historical candidates require current-status review.\n\nSource list: Adleman & McCurley - Open Problems in Number Theory Complexity II (1994)\nSource item: O36, p23 / LNCS p313\nSource URL: https://mccurley.org/papers/open.ps.gz\nAccessed: 2026-07-29\nExtraction: pdf recovered against coauthor PostScript\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leonard Adleman and Kevin McCurley", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS. Multiplication is now known in $O(n\\log n)$; whether it can be done in linear $O(n)$ bit operations remains open. Literature status: The best known algorithms are Harvey–van der Hoeven (2019/2021), who achieved integer multiplication in $O(n\\log n)$ time, where $n$ is the number of bits (published in Annals of Mathematics 2021). This is near-linear but strictly above linear; whether $O(n)$ (i.e., $O(\\log(ab))$) bit operations are achievable remains open." }, { "id": 8500001, "problem_number": "AMR-084-0001", "title": "Absolute bounds for rational Diophantine tuples", "statement": "Is there an absolute upper bound for the size of a rational Diophantine $m$-tuple, a set of nonzero rationals for which the product of every two distinct elements plus $1$ is a rational square?", "background": "All seven chapters of Dujella's maintained survey were inspected for current explicitly open questions; solved statements on the same pages were excluded.\n\nSource list: Dujella - Diophantine m-tuples (2002)\nSource item: Introduction\nSource URL: https://web.math.pmf.unizg.hr/~duje/dtuples.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Andrej Dujella", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "The problem remains **open**. There is no known absolute upper bound; unconditional and conditional (Lang-type) results give finiteness and a conjectural bound, but not an effective absolute constant." }, { "id": 8500002, "problem_number": "AMR-084-0002", "title": "Exceptional parameters without D(n)-quadruples", "statement": "For each $n\\in\\{-3,3,5,8,12,20\\}$, prove that no set of four distinct positive integers has property $D(n)$, meaning that every pairwise product plus $n$ is a square. (The page's original exceptional set also contained $-1$ and $-4$, which subsequent results on the same page settle.)", "background": "All seven chapters of Dujella's maintained survey were inspected for current explicitly open questions; solved statements on the same pages were excluded.\n\nSource list: Dujella - Diophantine m-tuples (2002)\nSource item: Section 3.1, conjecture following Theorem 3.2\nSource URL: https://web.math.pmf.unizg.hr/~duje/dtuples.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Andrej Dujella", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Not solved. Of the original exceptional set, $n=-1$ and $n=-4$ (plus all $n\\equiv2\\pmod4$) are settled; the six parameters $n\\in\\{-3,3,5,8,12,20\\}$ all remain open as of the current survey/open-problems list (2026)." }, { "id": 8500003, "problem_number": "AMR-084-0003", "title": "Finiteness of parameters admitting at most two D(n)-quadruples", "statement": "Let $U$ be the set of integers $n\\not\\equiv2\\pmod4$ for which there are at most two distinct $D(n)$-quadruples. Is $U$ finite?", "background": "All seven chapters of Dujella's maintained survey were inspected for current explicitly open questions; solved statements on the same pages were excluded.\n\nSource list: Dujella - Diophantine m-tuples (2002)\nSource item: Section 3.1, open question\nSource URL: https://web.math.pmf.unizg.hr/~duje/dtuples.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Andrej Dujella", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. The finiteness of $U$ (the set of nonsquare-$n$, $n\\not\\equiv2\\pmod4$, admitting at most two $D(n)$-quadruples) is not resolved in the literature. Literature status: - **Open.** Dujella's open-problems list (Problem 5.2) asks: \"Is the set of integers $n$ for which there exist at most two (three, four, ...) $D(n)$-quadruples finite?\" This is the exact question, listed as open in 2026. - Related supporting results: a conjecture of Dujella (Conjecture 3.1 — see AMR-084-0004) asserts that for nonsquare $n$ there are only finitely many $D(n)$-quadruples; Dujella proved that if $n\\equiv2\\pmod4$ there are at most two $D(n)$-quintuples etc. The finiteness of $U$ would be a (weaker) version of the finite-extensibility/small-quadruple-count finiteness theme. - Known structure: for $n\\equiv2\\pmod4$ there are at most two $D(n)$-quadruples; this is why $U$ is defined only for $n\\not\\equiv2\\pmod4$. The set $U$ includes the exceptional parameters $\\{-3,3,5,8,12,20\\}$ from AMR-084-0002 (where the count is 0).…" }, { "id": 8500004, "problem_number": "AMR-084-0004", "title": "Finiteness of D(n)-quadruples for nonsquare n", "statement": "For every nonzero integer $n$ that is not a square, are there only finitely many $D(n)$-quadruples?", "background": "All seven chapters of Dujella's maintained survey were inspected for current explicitly open questions; solved statements on the same pages were excluded.\n\nSource list: Dujella - Diophantine m-tuples (2002)\nSource item: Conjecture 3.1\nSource URL: https://web.math.pmf.unizg.hr/~duje/dtuples.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Andrej Dujella", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Not solved in general. The finiteness of the number of $D(n)$-quadruples for arbitrary nonsquare integer $n$ (Conjecture 3.1) remains **open**; only the special classes $n\\equiv2\\pmod4$, $n=-1$, $n=-4$ are proven." }, { "id": 8500005, "problem_number": "AMR-084-0005", "title": "Extremal parameters for D(n)-quintuples", "statement": "Determine the least positive integer $n_1$ and the greatest negative integer $n_2$ for which a $D(n_i)$-quintuple exists.", "background": "All seven chapters of Dujella's maintained survey were inspected for current explicitly open questions; solved statements on the same pages were excluded.\n\nSource list: Dujella - Diophantine m-tuples (2002)\nSource item: Section 3.2\nSource URL: https://web.math.pmf.unizg.hr/~duje/dtuples.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Andrej Dujella", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress. We know $n_1\\le256$ and $n_2\\ge-255$ (via explicit examples), but the least positive and greatest negative $n$ admitting a $D(n)$-quintuple are not determined. Literature status: - **Partial progress; exact values open.** Dujella's survey (Section 3.2) states: \"One may ask what is the least positive integer $n_1$, and what is the greatest negative integer $n_2$, for which there exist a Diophantine quintuple with the property $D(n_i)$. It is known that $n_1\\le256$ and $n_2\\ge-255$, since the sets $\\{1,33,105,320,18240\\}$ and $\\{5,21,64,285,6720\\}$ have the property $D(256)$, and the set $\\{8,32,77,203,528\\}$ has the property $D(-255)$.\" - So the current state is: $n_1 \\le 256$ and $n_2 \\ge -255$; the exact least $n_1$ and greatest $n_2$ are not determined. To settle the least positive case one would need to check all $n1$ satisfying\n$$\ny^m=f(x).\n$$\nThen either $m\\le C$, or else there is a proper subsum in\n$$\ny^m-b_1x^{n_1}-\\cdots-b_{L-1}x^{n_{L-1}}-b_L\n$$\nwhich vanishes.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 1.4\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Not independently verified. Appears to remain an open conjecture in the stated generality. Literature status: This is Shorey's Conjecture 1.4, in the circle of results on superelliptic equations $y^m=f(x)$ of polynomials with few terms (trinomials, general $L$-nomials). The governing results are of Shorey–Tijdeman type (bounds on integral points via Pell/Thue and linear forms in logarithms). I could not verify within the search budget whether this exact conjecture has been resolved; it belongs to a family where the \"subsum vanishing\" conclusions follow conditionally in special cases." }, { "id": 8700004, "problem_number": "AMR-086-0004", "title": "Conjecture 1.5", "statement": "Let $k\\ge 2$\nbe an integer and $\\alpha_1,\\ldots,\\alpha_n$ be non-zero elements in a field\n$K$ of zero characteristic, such that no quotient\n$\\alpha_i/\\alpha_j$ with $j\\not=i$ is a root of unity. Consider the\nfunction\n$$\nF(X_1,\\ldots,X_k)=\n\\det\\pmatrix{\n\\alpha_1^{X_1}&\\cdots&\\alpha_k^{X_1}\\cr\n\\vdots&\\ddots&\\vdots\\cr\n\\alpha_1^{X_k}&\\cdots&\\alpha_k^{X_k}\\cr\n}.\n$$\nThen the equation\n$$\nF(0,x_2,\\ldots,x_k)=0\n$$\nhas only finitely many solutions $(x_2,\\ldots,x_k)\\in\\mathbb{Z}^{k-1}$ such that\nin the corresponding determinant, all\n$(k-1)\\times k$ and all $k\\times (k-1)$ submatrices have rank $k-1$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 1.5\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Not independently resolved; treated as open. Literature status: This is Waldschmidt's Conjecture 1.5, a \"determinant\" exponential-Diophantine finiteness statement intimately tied to the theory of linear independence of logarithms and of unit/Subspace-type equations (the rank conditions exclude degenerate configurations so the problem is genuinely Diophantine). No primary-source resolution was located in this audit; the conjecture is not among the well-advertised solved results of the survey." }, { "id": 8700006, "problem_number": "AMR-086-0006", "title": "Conjecture 1.7", "statement": "If there is no prime in\nthe interval\n$[n+1,n+k]$, then the product $(n+1)\\cdots(n+k)$ has at least $k$ distinct prime\ndivisors.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 1.7\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / not directly resolved in the literature.** It is a consequence-type conjecture relating prime gaps to prime divisors of products of consecutive integers, in the same family as Grimm's conjecture, which remains open." }, { "id": 8700007, "problem_number": "AMR-086-0007", "title": "Conjecture 1.8 — Langevin", "statement": "Given an increasing sequence $n_10$, there\nexists a constant $C(\\varepsilon)>0$ such that, for any\nnonzero rational integers $a_1,\\ldots,a_m$, $b_1,\\ldots,b_m$\nwith\n$a_1^{b_1}\\cdots a_m^{b_m}\\not=1$,\n$$\n\\left|a_1^{b_1}\\cdots a_m^{b_m}-1\\right|\\ge\n{C(\\varepsilon)^m B\\over (|b_1|\\cdots |b_m|\\cdot |a_1|\\cdots\n|a_m|)^{1+\\varepsilon}},\n$$\nwhere\n$B=\\max_{1\\le i\\le m}|b_i|$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 2.5\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved**, with only weaker (effective but exponentially sloppy) bounds from linear-forms-in-logarithms theory. Literature status: This is a conjecture of Lang–Waldschmidt, a quantitative lower bound for the \"height\" of an $S$-unit minus 1, i.e., a lower bound for products $\\prod a_i^{b_i}-1$ with polynomial (rather than exponential) dependence on the sizes of the $a_i,b_i$. It is **open**; it is a strong form related to the $abc$ conjecture and to the \"generalized Catalan\"/Pillai family. Known results give lower bounds with weaker (exponentially large) constants via Baker's theory of linear forms in logarithms, not the polynomial form claimed. (Waldschmidt's own book on linear forms in logarithms discusses this.)" }, { "id": 8700014, "problem_number": "AMR-086-0014", "title": "Conjecture 2.6", "statement": "For any\n$\\varepsilon>0$, there is a constant $C(\\varepsilon)>0$ such that, for any positive\nintegers $x$,\n$y$,\n$p$, $q$ satisfying\n$x^p\\not= y^q$, the inequality\n$$\n|x^p-y^q| \\ge C(\\varepsilon)\\max\\{x^p, y^q\\}^{1-(1/p)-(1/q)-\\varepsilon}\n$$\nholds.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 2.6\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open / unresolved**; a quantitative Pillai-type lower bound for differences of perfect powers. Literature status: This is a quantitative strengthening of Pillai's conjecture relating to the size of differences of perfect powers. It generalizes Catalan/Pillai and is **open**. Verified context: the exponent $1-(1/p)-(1/q)$ reflects the \"expected\" counting; unconditional results (Tijdeman, linear forms in logarithms) give weaker exponents. The case $p=q=2$ recovers a Diophantine approximation-type statement for squares. No proof of this uniform bound was located." }, { "id": 8700015, "problem_number": "AMR-086-0015", "title": "Conjecture 2.7 — Hall", "statement": "If $x$\nand $y$ are positive integers with $y^2\\not=x^3$, then\n$$\n|y^2-x^3|\\ge C\\max\\{y^2,x^3\\}^{1/6}.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 2.7\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open.** Best unconditional exponent is much weaker than $1/6$; under ABC the conjecture holds. Literature status: This is **Hall's conjecture** (on the gap between a square and a nearby cube). It is **open**. Verified facts via web search: - Unconditionally one has $|y^2-x^3|\\gg \\max\\{y^2,x^3\\}^{1/2-\\varepsilon}$-type bounds (via linear forms in logarithms, Davenport et al.). - The conjectured exponent $1/6$ would follow from the $abc$ conjecture (Hall's conjecture is a consequence of ABC). - The \"Hall number\"/best known examples show the exponent cannot be pushed too far above; unconditionally $1/6$ is not known. Significantly, there is known numerical evidence and conditional (ABC) derivations; the exponent $1/6$ remains conjectural." }, { "id": 8700016, "problem_number": "AMR-086-0016", "title": "Conjecture 2.12", "statement": "Let $\\theta$ be real algebraic number of degree at least $3$.\nThen inequality (2.11) has infinitely\nmany solutions in integers $p$ and $q$ with $q>0$\nif and only if the integral\n$$\n\\int_1^\\infty \\psi(x) dx\n$$\ndiverges.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 2.12\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved** in the accessed literature; the precise approximation dichotomy for fixed algebraic numbers is delicate. Literature status: This is a **Khinchin-type / Duffin–Schaeffer style** question for algebraic numbers of degree $\\ge3$. Khinchin's theorem concerns measure (Lebesgue-almost-all) statements for all real $\\theta$; here the question restricts to a single algebraic $\\theta$, where the behavior is governed by Diophantine approximation properties tied to the degree. For algebraic $\\theta$ the classical results say $|\\theta-p/q|$ cannot be too small (Roth), and the precise divergence-convergence dichotomy for fixed algebraic $\\theta$ appears to be a delicate open question. No direct resolution located." }, { "id": 8700017, "problem_number": "AMR-086-0017", "title": "Conjecture 2.14 — Mahler", "statement": "There exists an absolute constant $c>0$ such that\n$$\n\\Vert \\log a\\Vert>a^{-c}\n$$\nfor all integers $a\\ge 2$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 2.14\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved.** Only far weaker lower bounds are known; the polynomial bound is not established (and possibly false). Literature status: This is a conjecture of Mahler (lower bound for the \"fractional part\" of $\\log a$). It is **open**. Verified context: unconditional results (via linear forms in logarithms / Baker) give the much weaker estimate $\\|\\log a\\|>\\exp(-C\\log a)$; the polynomial-type bound $a^{-c}$ is not known. It is related to the irrationality measure of $\\log a$ and to the $abc$ conjecture. (A negative answer is also plausible based on heuristic/known-theory considerations on the irrationality of $\\log a$, but no resolution was located.)" }, { "id": 8700018, "problem_number": "AMR-086-0018", "title": "Conjecture 2.15 — Mahler", "statement": "Let $(\\varepsilon_n)_{n\\ge 0}$ be a sequence of elements in $\\{0,1\\}$. Assume\nthat the real number\n$$\n\\sum_{n\\ge 0}\\varepsilon_n 3^{-n}\n$$\nis irrational, then it is transcendental.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 2.15\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Open**, with partial results for special (e.g., automatic) sequences; the general irrational $\\Rightarrow$ transcendental claim is unresolved. Literature status: This is a conjecture of Mahler (transcendence of \"Cantor-type\" numbers $\\sum \\varepsilon_n 3^{-n}$, the ternary analogue of the Davenport–Erdős / Mahler-type questions). The analogous statement for base 2 was studied by Mahler; the general conjecture is **open**. Verified partial results via web search: - The full statement (irrationality $\\Rightarrow$ transcendence for all $\\{0,1\\}$ sequences) is open. - Partial results are known: e.g., for a positive density of 0/1 sequences, or under growth/regularity assumptions on $\\varepsilon_n$, transcendence holds (results by Adamczewski–Bugeaud and others apply when the sequence is automatic/regular or has algebraic generating function; the general case is open). - Related to results on lacunary series and to the algebraic independence of such sums." }, { "id": 8700020, "problem_number": "AMR-086-0020", "title": "Conjecture 3.2 — Roy", "statement": "Let $k$ be a positive integer, $y_1,\\ldots,y_k$ complex numbers which are\nlinearly independent over $\\mathbb{Q}$, $\\alpha_1,\\ldots,\\alpha_k$ nonzero\ncomplex numbers and $s_0,s_1,t_0,t_1,u$ positive real numbers\nsatisfying\n$$\n\\max\\{1,t_0,2t_1\\} < \\min\\{s_0,2s_1\\}\n\\quad\\text{and}\\quad\n\\max\\{s_0,s_1+t_1\\} < u < {1\\over 2}(1+t_0+t_1).\n$$\nAssume that, for any sufficiently large positive integer $N$, there\nexists a nonzero polynomial $P_N\\in\\mathbb{Z}[X_0,X_1]$ with partial degree\n$\\le N^{t_0}$ in $X_0$, partial degree $\\le N^{t_1}$ in $X_1$ and\nheight $\\le e^N$ which satisfies\n$$\n\\Bigg| \\big(\\mathcal{D}^kP_N\\big)\n\\Big(\\sum_{j=1}^k m_jy_j,\n\\prod_{j=1}^k\\alpha_j^{m_j} \\Big) \\Bigg|\n\\le \\exp(-N^u)\n$$\nfor any nonnegative integers $k$, $m_1,\\ldots,m_k$ with $k\\le N^{s_0}$\nand $\\max\\{m_1,\\ldots,m_k\\}\\le N^{s_1}$. Then, we have\n$$\n\\operatorname{trdeg} \\mathbb{Q}(y_1,\\ldots,y_k, \\alpha_1,\\ldots,\\alpha_k) \\ge k.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.2\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved** in the accessed literature. Literature status: This is a conjecture of Roy quantifying transcendence/algebraic-independence from small values of a polynomial and its derivatives (a \"measure of algebraic independence\" type statement). It is a refinement in the circle of Roy's work on measures of algebraic independence and the strengthened forms of Schanuel-type statements. No proof of the exact statement was located; it appears **open**." }, { "id": 8700021, "problem_number": "AMR-086-0021", "title": "Conjecture 3.3 — Algebraic Independence of Logarithms of Algebraic Numbers", "statement": "Let $\\lambda_1,\\ldots, \\lambda_n$ be\n$\\mathbb{Q}$-linearly independent complex numbers. Assume that the\nnumbers\n$e^{\\lambda_1},\\ldots,e^{\\lambda_n}$ are algebraic.\nThen the\nnumbers $\\lambda_1,\\ldots, \\lambda_n$ are algebraically\nindependent over $\\mathbb{Q}$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.3\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open.** Linear independence (Baker) is known; algebraic independence is not. Literature status: This is a classical conjecture on the **algebraic independence of logarithms of algebraic numbers** (a special case of the conjecture coming from the Lindemann–Weierstrass-type / Schanuel circle). It is **open** for $n\\ge2$ (for $n=1$ it is Lindemann's theorem). Verified context: - Known: the $\\lambda_i$ are $\\mathbb{Q}$-linearly (and by Baker's theorem) linearly independent over the algebraic numbers; but algebraic independence is not known even for $n=2$. - Equivalently not known that $\\log\\alpha_1,\\log\\alpha_2$ are algebraically independent for multiplicatively independent algebraic $\\alpha_i$." }, { "id": 8700022, "problem_number": "AMR-086-0022", "title": "Conjecture 3.4 — Strong Four Exponentials Conjecture", "statement": "Let\n$x_1,x_2$ be two\n$\\overline{\\mathbb{Q}}$-linearly independent complex numbers and\n$y_1,y_2$ be also two\n$\\overline{\\mathbb{Q}}$-linearly independent complex numbers. Then at least\none of the four numbers\n$x_1y_1$, $x_1y_2$, $x_2y_1$, $x_2y_2$ does not belong to\n$\\widetilde{\\mathcal{L}}$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.4\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved**; the strong four exponentials conjecture over $\\widetilde{\\mathcal{L}}$ is not proved. Literature status: This is the **four exponentials conjecture** (in its \"strong\" form where $\\mathcal{L}$ is the ring generated by logarithms of algebraic numbers / exponential periods). The four exponentials conjecture is a classical **open** problem. The strong form with $\\widetilde{\\mathcal{L}}$ (involving abelian periods, per Waldschmidt) is a strengthening and is also **open**. The weak four exponentials theorem (asserting at least one of four certain numbers is transcendental) is known for the classical form, but the strong statement over $\\widetilde{\\mathcal{L}}$ is not." }, { "id": 8700023, "problem_number": "AMR-086-0023", "title": "Conjecture 3.5 — Strong Five Exponentials Conjecture", "statement": "Let $x_1, x_2$ be two $\\mathbb{Q}$-linearly independent complex\nnumbers and\n$y_1, y_2$ be also two $\\mathbb{Q}$-linearly independent complex\nnumbers. Further let $\\beta_{ij}$ \\ ($ i=1,2$, $j=1,2$),\n$\\gamma_1$ and\n$\\gamma_2$ be six algebraic numbers with $\\gamma_1\\not=0$.\nAssume that the five numbers\n$$\ne^{x_1y_1-\\beta_{11}},\\; e^{x_1y_2-\\beta_{12}},\\;\ne^{x_2y_1-\\beta_{21}},\\; e^{x_2y_2-\\beta_{22}},\\;\ne^{(\\gamma_1 x_1/x_2)-\\gamma_2}\n$$\nare algebraic.\nThen all five exponents vanish:\n$$\nx_iy_j=\\beta_{ij} \\quad (i=1,2, \\quad j=1,2)\n\\quad \\text{ and }\\quad \\gamma_1 x_1=\\gamma_2 x_2.\n$$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.5\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved.** Literature status: This is the **five exponentials conjecture** (strong form). It is a strengthening of the four exponentials conjecture and is **open**. The classical five exponentials theorem of Waldschmidt asserts a weaker statement (at least one nontrivial linear form is irrational); the strong \"vanish-or-dependence\" conclusion is not established. Related to the six exponentials and to the generalized algebraic-independence conjectures." }, { "id": 8700024, "problem_number": "AMR-086-0024", "title": "Conjecture 3.6 — Roy", "statement": "For any $4\\times 4$ skew-symmetric\nmatrix $\\mathrm{M}$ with\nentries in\n$\\mathcal{L}$ and rank $\\le 2$, either the rows of $\\mathrm{M}$ are\nlinearly dependent over $\\mathbb{Q}$, or the column space of\n$\\mathrm{M}$ contains a nonzero element of $\\mathbb{Q}^4$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.6\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved**; equivalent in spirit to the four exponentials conjecture. Literature status: This is a conjecture of Roy, a linear-algebra-restructured form of the four exponentials conjecture (it is exactly equivalent to a known form of the four exponentials / \"$4\\times4$ rank\" statement). It is **open**; it is a reformulation connected to the four exponentials conjecture and to measures of linear forms in logarithms. No resolution found." }, { "id": 8700026, "problem_number": "AMR-086-0026", "title": "Conjecture 3.8 — Gel’fond", "statement": "The two numbers\n$$\n\\log\\alpha\\quad\\text{and}\\quad \\alpha^\\beta\n$$\nare algebraically independent over $\\mathbb{Q}$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.8\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved**; special algebraic-independence cases are known, not the general log-vs-power statement. Literature status: This is a conjecture of Gel'fond, stated in Waldschmidt's survey. Gel'fond proved (1929) that $\\alpha^\\beta$ is transcendental for $\\beta$ irrational algebraic; he also proved that certain pairs are algebraically independent in special cases. The general claim that $\\log\\alpha$ and $\\alpha^\\beta$ (with $\\beta$ cubic) are algebraically independent is **open**. Known partial: Gel'fond proved algebraic independence of $\\alpha^{\\beta_1},\\alpha^{\\beta_2}$ for suitable $\\beta_i$; the specific log-vs-power independence is not known." }, { "id": 8700027, "problem_number": "AMR-086-0027", "title": "Conjecture 3.9 — Schneider", "statement": "The $d-1$\nnumbers\n$$\n\\alpha^\\beta,\\; \\alpha^{\\beta^2},\\ldots,\n\\alpha^{\\beta^{d-1}}\n$$\nare algebraically independent over $\\mathbb{Q}$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.9\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved**, with partial results for restricted $\\beta$. Literature status: This is a conjecture of Schneider (algebraic independence of the consecutive $\\beta$-powers of $\\alpha$). It is **open** beyond small cases. Known partial results (Gel'fond, later by others) prove algebraic independence of suitable pairs/families under conditions on $\\beta$, but the full statement for arbitrary algebraic $\\beta$ of degree $d$ is open, and even the number of independent quantities is limited." }, { "id": 8700028, "problem_number": "AMR-086-0028", "title": "Conjecture 3.10 — Gel’fond-Schneider", "statement": "The $d$ numbers\n$$\n\\log\\alpha,\\; \\alpha^\\beta,\\; \\alpha^{\\beta^2},\\ldots,\n\\alpha^{\\beta^{d-1}}\n$$\nare algebraically independent over $\\mathbb{Q}$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.10\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved.** Literature status: This is the Gel'fond–Schneider algebraic-independence conjecture, combining the log of $\\alpha$ with the powers of $\\alpha^\\beta$. It is **open**. Included among its cases, e.g., for $d=2$, the algebraic independence of $\\log\\alpha$ and $\\alpha^\\beta$ is itself open (see 0026). Only weak partial algebraic-independence results exist. This is a highlight open problem in the theory." }, { "id": 8700029, "problem_number": "AMR-086-0029", "title": "Conjecture 3.11 — $p$-adic analog of Lindemann-Weierstrass's Theorem", "statement": "Let $\\beta_1,\\ldots,\\beta_n$\nbe\n$p$-adic algebraic numbers in the domain of convergence of the $p$-adic\nexponential function $\\exp_p$. Then the\n$n$ numbers $\\exp_p\\beta_1,\\ldots, \\exp_p\\beta_n$ are algebraically\nindependent over $\\mathbb{Q}$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.11\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved**; only weak partial results known. Literature status: This is the **$p$-adic Lindemann–Weierstrass conjecture** for the $p$-adic exponential. Only weak partial results are known; even the $p$-adic Lindemann-type theorem (transcendence of $\\exp_p$ of a suitable nonzero algebraic) is not established in full generality (the $p$-adic Lindemann–Weierstrass is a long-open problem, since the $p$-adic exponential has a small convergence domain and known results compensate with height growth). The full algebraic-independence statement is **open**." }, { "id": 8700030, "problem_number": "AMR-086-0030", "title": "Conjecture 3.12 — $p$-adic analog of an algebraic independence result of Gel’fond", "statement": "Let $\\alpha$ be a non-zero\nalgebraic number in the domain of convergence of the $p$-adic logarithm\n$\\log_p$, and let\n$\\beta$ be a\n$p$-adic cubic algebraic number, such that $\\beta\\log_p\\alpha$ is in the\ndomain of convergence of the $p$-adic exponential function $\\exp_p$.\nThen the two numbers\n$$\n\\alpha^\\beta=\\exp_p(\\beta\\log_p\\alpha)\n\\quad\\text{and}\\quad\n\\alpha^{\\beta^2}=\\exp_p(\\beta^2\\log_p\\alpha)\n$$\nare algebraically independent\nover\n$\\mathbb{Q}$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.12\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved.** Literature status: This is the $p$-adic analog of a Gel'fond algebraic-independence result; the intended conclusion (algebraic independence of two $p$-adic exponentials) is **open** in general. The classical (complex) Gel'fond theorem for this setting is known, but the $p$-adic version with a cubic $\\beta$ is not established." }, { "id": 8700031, "problem_number": "AMR-086-0031", "title": "Conjecture 3.13 — Blum, Cucker, Shub and Smale", "statement": "Given an absolute constant $c$ and polynomials $P_1,\\ldots,P_m$ with a\ntotal of $N$ coefficients and no common complex zeros, there is no program\nto find, in at most $N^c$ step, the coefficients of polynomials\n$A_i$ satisfying B\\'ezout's relation\n$$\nA_1P_1+\\cdots+A_mP_m=1.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.13\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved** as a complexity lower bound; related to NP-hardness and effective Nullstellensatz questions in the BCSS model. Literature status: This is a **computational-complexity / algorithm** conjecture from the work of Blum–Cucker–Shub–Smale on the complexity of deciding existence / finding Bézout coefficients over the reals/complexes. It asserts an exponential lower bound for the Bézout identity problem (a version of an \"NP-hardness in the Blum-Shub-Smale model\" statement). It remains a research-level question tied to the computational complexity of finding Zariski-dense non-vanishing ideals and to effective Nullstellensatz complexity; no formal proof of the $N^c$-step lower bound for this exact problem was located." }, { "id": 8700032, "problem_number": "AMR-086-0032", "title": "Conjecture 3.14", "statement": "Let $\\Sigma$ be a finite subset of $\\mathbb{C}^n$ and\n$\\varepsilon$ a positive number. There exists a positive number\n$r_0(\\Sigma,\\varepsilon)$ such that, for any positive integer $t$ and any entire\nfunction $f$ in $\\mathbb{C}^n$ which vanishes on $\\Sigma$ with multiplicity $\\ge t$,\n$$\n\\Theta_f(r)\\ge \\omega_t(\\Sigma)-t\\varepsilon\n\\quad\\text{for}\\quad r\\ge r_0(\\Sigma,\\varepsilon).\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.14\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Survey item**, appears to remain unproved as stated in the reachable literature; the surrounding theory provides related (weaker) estimates. Literature status: This conjecture concerns the lower bounds for the counting/characteristic functions $\\Theta_f(r)$ of entire functions vanishing to high order on a finite set — a value-distribution / transcendence-measure refinement used in analytic proofs (e.g., of algebraic independence measures), related to work of Waldschmidt and others on $\\omega_t(\\Sigma)$ (transcendence-type exponents attached to a finite set). This precise asymptotic statement is a tool-oriented conjecture from the survey; I treated it as a survey item and did not find an isolated published resolution separate from the surrounding theory." }, { "id": 8700033, "problem_number": "AMR-086-0033", "title": "Conjecture 3.15 — Goncharov", "statement": "As a $\\mathbb{Q}$-algebra, $\\mathfrak{Z}$ is the direct sum of\n$\\mathfrak{Z}_p$ for $p\\ge 0$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.15\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Open**, with partial verification up to weight 16 and strong numerical support. Literature status: This is part of the **Goncharov/Zagier conjectures** on the dimension of the space of multiple zeta values of weight $p$: the conjecture $d_p=d_{p-2}+d_{p-3}$ (with $d_0=1,d_1=0,d_2=1$), implying $d_p$ grows like a Fibonacci-type sequence. Verified context: - Zagier proved this dimension formula up to weight **16** (dimension of the conjectured MZV space), and substantial numerical evidence supports it. - The full conjecture (all weights, and the direct-sum structure asserting Z is the direct sum of the graded pieces) is **open**. It is closely tied to (and would follow from) the algebraic-independence conjectures for zeta values." }, { "id": 8700034, "problem_number": "AMR-086-0034", "title": "Conjecture 3.16 — Zagier", "statement": "For $p\\ge 3$ we have\n$$\nd_p=d_{p-2}+d_{p-3}\n$$\nwith $d_0=1$, $d_1=0$, $d_2=1$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.16\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Open**, verified up to weight 16; equivalent in essence to algebraic-independence conjectures for odd zeta values. Literature status: This is **Zagier's conjecture** on the dimensions of the space of multiple zeta values. Verified context via web search: - Zagier verified $d_p$ for weights up to and including **16**. - The conjecture (with $d_p$ being the Fibonacci-like period-8 sequence) is **open** in general. It would follow from the conjectured algebraic independence of odd zeta values $\\zeta(3),\\zeta(5),\\ldots$ and $\\pi$." }, { "id": 8700035, "problem_number": "AMR-086-0035", "title": "Conjecture 3.17", "statement": "The numbers\n$\\pi$,\n$\\zeta(3),\\zeta(5),\\ldots,\\zeta(2n+1),\\ldots$ are algebraically\nindependent over $\\mathbb{Q}$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.17\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open.** Only irrationality/linear-independence-type partial results known. Literature status: This is a classical **open** conjecture on the algebraic independence of the odd zeta values and $\\pi$. Verified context: only very weak algebraic-independence/irrationality results are known ($\\zeta(3)$ irrational (Apéry), $\\zeta(5),\\zeta(7),\\zeta(9),\\zeta(11)$ irrational by work building on Zudilin; linear independence of $\\zeta(3),\\zeta(5),\\zeta(7)$ conjectural). Even the irrationality of infinitely many odd zeta values is open. The full algebraic-independence statement is far beyond current methods." }, { "id": 8700036, "problem_number": "AMR-086-0036", "title": "Conjecture 3.18", "statement": "At least three of the four numbers\n$$\n\\pi,\\; \\Gamma(1/5),\\; \\Gamma(2/5), \\; e^{\\pi\\sqrt 5}\n$$\nare algebraically independent over $\\mathbb{Q}$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.18\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved** as stated, though Nesterenko-type results cover related triples. Literature status: This is an algebraic-independence problem mixing $\\Gamma$-values, $\\pi$, and an exponential, related to Nesterenko's theorem (which gives algebraic independence of, e.g., $\\pi,e^{\\pi\\sqrt d}$ and related quantities). Nesterenko's theorem gives algebraic independence of $\\pi$, $e^{\\pi\\sqrt5}$ and certain $\\Gamma$-values in the spirit of the statement, but the exact \"at least 3 of these 4\" claim is a specific open refinement that I did not find resolved verbatim. Classified as open." }, { "id": 8700037, "problem_number": "AMR-086-0037", "title": "Conjecture 3.19 — Rohrlich", "statement": "$\\overline{G}$ is a universal odd distribution with values in groups where\nmultiplication by $2$ is invertible.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.19\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Open**, with partial progress in restricted settings (verification of distribution-relational structure for many Gamma-values). Literature status: This is the **Rohrlich–Lang conjectural structure** on relations among values of the Gamma function (Rodriquez Villegas / Deligne-type: \"Rohrlich's conjecture\" that all multiplicative $\\overline{\\mathbb{Q}}$-relations among Gamma values come from the distribution relations). Verified context: - The relevant expectational structure is that the odd zeta/period relations are governed by distribution relations. - Full resolution is **open**, though substantial partial results exist (e.g., work connecting Gamma relations to Belyi/regulator computations and the \"Rohrlich–Lang\" conjecture verified in restricted settings). The exact \"universal odd distribution\" form as stated is an open algebraic/tannakian conjecture." }, { "id": 8700038, "problem_number": "AMR-086-0038", "title": "Conjecture 3.20 — Nesterenko", "statement": "Let $\\tau\\in\\mathbb{C}$ have positive imaginary part.\nAssume that $\\tau$ is not quadratic.\nSet $q=e^{2i\\pi\\tau}$.\nThen at least $4$ of the $5$ numbers\n$$\n\\tau,\\; q,\\; P(q),\\; Q(q),\\; R(q)\n$$\nare algebraically independent.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.20\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Nesterenko's theorem yields algebraic independence degree $\\ge3$ (for algebraic $\\tau$); the conjectured degree $\\ge4$ is open. Literature status: This is Nesterenko's conjecture. **Nesterenko's theorem** (1996) proves that at least **3** of the 5 numbers are algebraically independent for algebraic $\\tau$, and in fact for algebraic $\\tau$ it yields that the transcendence degree of the field is $\\ge3$ (and at least 4 for the full statement is open). The conjecture generalizes this to all non-quadratic $\\tau$ and raises the count to 4; it is **open**. The known theorem (3, for algebraic $\\tau$) is the partial result." }, { "id": 8700039, "problem_number": "AMR-086-0039", "title": "Conjecture 3.21 — Bertolin", "statement": "Let $\\mathcal{E}_1,\\ldots,\\mathcal{E}_n$ be pairwise non isogeneous elliptic curves with modular invariants $j(\\mathcal{E}_h)$. For $h=1,\\ldots,n$, let $\\omega_{1h},\\omega_{2h}$ be a pair of fundamental periods of $\\wp_h$ with $\\eta_{1h},\\eta_{2h}$ the associated quasi-periods, $P_{ih}$ points on $\\mathcal{E}_h(\\mathbb{C})$ and $p_{ih}$ (resp. $d_{ih}$) elliptic integrals of the first (resp. second) kind associated to $P_{ih}$. Define $\\kappa_h=[k_h:\\mathbb{Q}]$ and let $d_h$ be the dimension of the $k_h$-subspace of $\\mathbb{C}/(k_h\\omega_{1h}+k_h\\omega_{2h})$ spanned by $p_{1h},\\ldots,p_{r_hh}$. Then the transcendence degree of the field\n$$\n\\mathbb{Q}\\Bigl(\\bigl\\{\nj(\\mathcal{E}_h), \\omega_{1h},\\omega_{2h}, \\eta_{1h},\\eta_{2h},\nP_{ih}, p_{ih}, d_{ih}\n\\bigr\\}_{1\\le i\\le r_h \\atop 1\\le h\\le n}\n\\Bigr)\n$$\nis at least\n$$\n2\\sum_{h=1}^n d_h\n+4\\sum_{h=1}^n \\kappa_h^{-1}-n+1.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.21\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved**, with weak partial results in the theory of periods. Literature status: This is a conjecture of Bertolin on the algebraic independence of the periods of several non-isogenous elliptic curves — a deep statement in the theory of periods, generalizing the Lindemann-type/algebraic-independence results for elliptic logarithms (and related to the \"Bertolin conjecture\" on periods). It is **open**; only weaker partial algebraic-structure results are known. This is a highlight open problem in the motivic theory of periods." }, { "id": 8700041, "problem_number": "AMR-086-0041", "title": "Conjecture 3.23", "statement": "Given an\nelliptic curve with Weierstrass equation\n$y^2=4x^3-g_2x-g_3$, a nonzero period $\\omega$, the associated quasi-period\n$\\eta$ of the zeta function and a complex number $u$ which is not a pole of\n$\\wp$, we have\n$$\n\\operatorname{trdeg}\\mathbb{Q}\n\\bigl(g_2,g_3,\\pi/\\omega,\\wp(u),\\zeta(u)-(\\eta/\\omega)u\\bigr)\\ge 2.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.23\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved** as stated, with related partial results (Chudnovsky-type) for sub-tuples. Literature status: This is a conjecture (attributed in the surrounding theory to Bertrand/Chudnovsky-type results) on the transcendence degree of fields generated by elliptic periods, the quasi-periods, and values of $\\wp$ and $\\zeta$. Chudnovsky's theorem gives related algebraic-independence results for $\\pi/\\omega$ and $g_2,g_3$; the specific degree $\\ge2$ for this full tuple is a refinement that I did not find resolved verbatim and appears **open** in general." }, { "id": 8700042, "problem_number": "AMR-086-0042", "title": "Conjecture 3.24 — Bertrand", "statement": "Let $q_1,\\ldots,q_n$ be nonzero algebraic numbers in the unit open disc such\nthat the $3n$ numbers\n$$\nJ(q_i), \\; DJ(q_i),\\; D^2J(q_i)\\qquad (i=1,\\ldots,n)\n$$\nare algebraically dependent over $\\mathbb{Q}$. Then there exist two indices $i\\not =\nj$ \\ ($1\\le i\\le n$, $1\\le j\\le n$) such that $q_i$ and $q_j$ are\nmultiplicatively dependent.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.24\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved.** Literature status: This is a conjecture of Bertrand on algebraic dependence of modular functions of several algebraic $q_i$ forcing multiplicative dependence among the $q_i$ — in the $q$-series/elliptic-modular-functions circle (related to Bourgain–Clozel–Kahane-type results on $q$-series and to Mahler's method for modular functions). It is **open** in the reachable literature; related work establishes linear/analytic dependence but not the multiplicative-dependence conclusion in full." }, { "id": 8700043, "problem_number": "AMR-086-0043", "title": "Conjecture 3.25 — Bertrand", "statement": "Let $q_1$ and $q_2$ be two nonzero algebraic numbers in the unit open disc.\nSuppose that there is an irreducible element $P\\in\\mathbb{Q}[X,Y]$ such that\n$$\nP\\bigl(J(q_1),J(q_2)\\bigr)=0.\n$$\nThen there exist a constant $c$ and a positive integer $s$ such that\n$P=c\\Phi_s$, where $\\Phi_s$ is the modular polynomial of level $s$. Moreover\n$q_1$ and $q_2$ are\nmultiplicatively dependent.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.25\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved** in the accessed literature; related work covers the modular-polynomial structural results only conditionally/partially. Literature status: This is a conjecture of Bertrand on the algebraic relations between values of a modular function $J$ at two algebraic arguments: it asserts that the only such relations are the classical modular-polynomial relations $\\Phi_s(J(q_1),J(q_2))=0$, which exactly correspond to multiplicative dependence of $q_1,q_2$. This is a known open problem in the transcendence theory of modular functions (a special case of the general \"Ramanujan/ modular-function independence\" questions); no full proof was located." }, { "id": 8700044, "problem_number": "AMR-086-0044", "title": "Conjecture 3.26", "statement": "Is there such a bound\ndepending polynomially on the degree and height of $P$?", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 3.26\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved** as a polynomial-bound refinement. Literature status: This is a refinement question in the transcendence theory of modular functions, asking whether the effective bounds can be polynomial in the degree and height. As a specific quantitative sub-question I did not find it resolved; the surrounding theory gives weaker (exponential) effective bounds. Classified as open/triage." }, { "id": 8700045, "problem_number": "AMR-086-0045", "title": "Question 3.27 — Mahler", "statement": "Are there entire transcendental functions $f(z)$ such\nthat if $x$ is a Liouville number then so is $f(x)$?", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Question 3.27\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved** (triage) — existence question for Liouville-preserving entire transcendental functions. Literature status: This is a question of Mahler about entire transcendental functions preserving the (Liouville) property of Liouville numbers. Verified context: related results exist showing that certain functions do or do not preserve irrationality measures/transcendence classes, but the specific existence question for Liouville-number preservation is a delicate problem in transcendental function theory. I did not find a definitive yes/no in the reachable literature; it appears **open/resolved-in-part** and requires verification. Marked OPEN-TRIAGE." }, { "id": 8700046, "problem_number": "AMR-086-0046", "title": "Conjecture 4.1 — Lehmer's Problem", "statement": "There exists\na positive absolute constant $c$ such that,\nfor any\nnonzero algebraic number $\\alpha$ which\nis not a root of unity,\n$$\n\\mathrm{M}(\\alpha)\\ge 1+c.\n$$\nEquivalently, there exists\na positive absolute constant $c$ such that,\nfor any\nnonzero algebraic number $\\alpha$ of degree at most $d$ which\nis not a root of unity,\n$$\n\\mathrm{h}(\\alpha)\\ge {c\\over d}.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 4.1\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open.** Best known unconditional lower bound is Dobrowolski's, well short of $c/d$. Literature status: This is **Lehmer's conjecture** (1933). It is **open**. Verified facts via web search: - Best unconditional lower bound: **Dobrowolski's theorem** $\\mathrm{h}(\\alpha)\\ge c\\,d^{-1}\\bigl(\\frac{\\log\\log d}{\\log d}\\bigr)^3$ (improved constants by Voutier and others). This is well short of $c/d$. - The conjecture is known for many restricted classes (e.g., $\\alpha$ with $\\mathrm{tr}_{\\mathbb{Q}(\\alpha)/\\mathbb{Q}}\\alpha$ \"large\", Salem/Schur-type, abelian, etc.), but not in general. - It is equivalent to several other famous open problems (e.g., the irrationality exponent/prime-counting forms)." }, { "id": 8700048, "problem_number": "AMR-086-0048", "title": "Conjecture 4.3 — Amoroso-David", "statement": "For each positive integer $n\\ge 1$ there\nexists a positive number $c(n)$ having the\nfollowing property. Let $\\alpha_1,\\ldots,\\alpha_n$ be\nmultiplicatively independent algebraic numbers. Define\n$D=[\\mathbb{Q}(\\alpha_1,\\ldots,\\alpha_n): \\mathbb{Q}]$. Then\n$$\n\\prod_{i=1}^n\\mathrm{h}(\\alpha_i)\\ge{c(n)\\over D}.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 4.3\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved**; only results with extra logarithmic factors are known. Literature status: This is a conjecture of **Amoroso–David** (a multidimensional generalization of Lehmer's conjecture for the product of heights of multiplicatively independent algebraic numbers, with exponent $-1$ on the degree $D$). It is **open**. Verified context: Amoroso–David proved results of the form $\\prod\\mathrm{h}(\\alpha_i)\\ge c(n)(\\log(3D)/D)^n$ or similar (with logarithmic factors), not the sharp $1/D$; the sharp exponent is open. It is strictly stronger than the $n=1$ Lehmer-type bound." }, { "id": 8700049, "problem_number": "AMR-086-0049", "title": "Conjecture 4.4 — Amoroso-David", "statement": "For each\npositive integer\n$n\\ge 1$ there exists a positive number $c(n)$ such that, if\n$\\underline{\\alpha}=(\\alpha_1,\\ldots,\\alpha_n)$ is a $n$-tuple of\nmultiplicatively independent algebraic numbers, then\n$$\n\\mathrm{h}(1\\colon \\alpha_1\\colon \\cdots\\colon\n\\alpha_n)\\ge{c(n)\\over\n\\omega(\\underline{\\alpha})}.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 4.4\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved**. Literature status: This is a further Amoroso–David conjecture, giving a lower bound with the *weight/exponent* $\\omega$ in the denominator rather than the degree. It is **open**; the known techniques (Amoroso–David, Bombieri–Masser–Zannier for the projective height) give bounds with extra logarithmic factors or weaker exponents. No proof of the sharp $1/\\omega$ form was located." }, { "id": 8700050, "problem_number": "AMR-086-0050", "title": "Conjecture 4.5 — Amoroso-David", "statement": "For each integer $n\\ge 1$ there exists a positive constant $c(n)$\nsuch that, for any algebraic subvariety $V$ of $\\mathbb{G}_m^n$ which is\ndefined over $\\mathbb{Q}$, which is $\\mathbb{Q}$-irreducible, and which is not a\nunion of translates of algebraic subgroups by torsion points, we have\n$$\n\\hat{h}(V)\\ge c(n)\\deg(V)^{(s-\\dim V-1)/(s-\\dim V)},\n$$\nwhere $s$ is the dimension of the smallest algebraic subgroup of\n$\\mathbb{G}_m^n$ containing $V$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 4.5\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Qualitative/effective lower bounds for $\\hat h(V)$ known (Bombieri–Masser–Zannier); the sharp exponent in the conjecture is open. Literature status: This is the **Amoroso–David (or Bombieri–Masser–Zannier-type) height conjecture** for subvarieties of the algebraic torus $\\mathbb{G}_m^n$: a lower bound for the (essential/minimal) canonical height of a subvariety not contained in a torsion translate. Verified context: **Bombieri–Masser–Zannier** (and Amoroso–David) proved effective lower bounds for such heights, but with exponents/definitions differing from the sharp conjectural form; the exact exponent conjectured here is **open** in full generality, though the qualitative statement (non-accumulation) is established. Partial progress exists." }, { "id": 8700051, "problem_number": "AMR-086-0051", "title": "Problem 4.6", "statement": "For $\\theta\\in(0,\\pi)$, define\n$$\nV_\\theta=\\{re^{it}\\; ;\\;\nr>0,\\; |t|>\\theta\\}.\n$$\nCompute $L(V_\\theta)$ in terms of $\\theta$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Problem 4.6\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved** (triage) — precise value of $L(V_\\theta)$ not verified in the accessible literature. Literature status: This is a concrete computation question within Waldschmidt's framework of approximation-invariant functions $L(\\cdot)$ of planar sets. I did not find the exact $L(V_\\theta)$ value recorded in the reachable literature; the general theory (Dubinin, and Waldschmidt's work on such invariants) treats related quantities but not necessarily this closed-form answer. Marked OPEN-TRIAGE pending literature verification." }, { "id": 8700052, "problem_number": "AMR-086-0052", "title": "Conjecture 4.7 — David-Hindry", "statement": "There exists a positive constant\n$c$, depending only on $A$ and $\\mathcal{L}$, such that for any $P\\in A(\\overline{\\mathbb{Q}})$\nwhich has infinite order modulo any abelian subvariety, we have\n$$\n\\hat{h}_{\\mathcal{L}}(P)\\ge c \\delta(P)^{-1}.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 4.7\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partial progress.** Weaker bounds ($\\delta(P)^{-\\text{const}}$) known; concessional sharp $\\delta(P)^{-1}$ is open. Literature status: This is the **David–Hindry conjecture** on the Lehmer-type lower bound for the Neron–Tate height of points on abelian varieties. It is **open** in general. Verified context: only weaker bounds of the form $\\hat h(P)\\gg \\delta(P)^{-c'}$ with $c'>1$ (and various logarithmic factors) are known (e.g., Masser's lower bounds for $A^g$; the conjecture's sharp exponent $-1$ is open). The conjecture is known for some special abelian varieties/fields." }, { "id": 8700053, "problem_number": "AMR-086-0053", "title": "Conjecture 4.11 — Wirsing and Schmidt", "statement": "For any positive integer\n$n$ and any real number $\\theta$ which is\neither transcendental or else is algebraic of degree $>n$, there exists a\npositive constant $c=c(n,\\theta)$ with the following property:\nthere exist infinitely many algebraic numbers $\\gamma$ of degree $\\le n$\nfor which\n$$\n0<|\\theta-\\gamma|\n\\exp\\{-C^n\nD^{n+2}(\\log A_1)\\cdots (\\log A_n)(\\log B+\\log D) (\\log D)\\}.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 4.15\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Partially addressed by the literature.** Effective lower bounds of this general shape are known (Baker's theorem and refinements); the sharp optimal form is open. Literature status: This is a conjectured **effective lower bound for linear forms in logarithms** asserting a specific (large-exponent but purely in $D,A_j,B$) bound. Verified context: **Baker's theorem** provides effective lower bounds of exactly this general shape (with constants/ exponents depending on $n$), and subsequent refinements (Matveev, etc.) prove such bounds. However, the *specific* constants and the exact form posed here (as an explicit conjecture) are not established verbatim; the field is basically solved in the \"some effective lower bound exists\" sense. The precise optimal/sharp value of the exponents is a quantitative refinement that remains a research question." }, { "id": 8700058, "problem_number": "AMR-086-0058", "title": "Conjecture 4.16 — Quantitative Refinement of Schanuel's Conjecture", "statement": "Let $x_1,\\ldots,x_n$ be $\\mathbb{Q}$-linearly independent\ncomplex numbers.\nAssume that for any $\\varepsilon>0$, there exists a positive number $H_0$ such\nthat, for any $H\\ge H_0$ and $n$-tuple $(h_1,\\ldots,h_n)$ of rational integers\nsatisfying $0<\\max\\{|h_1|,\\ldots,|h_n|\\}\\le H$, the inequality\n$$\n|h_1x_1+\\cdots+h_nx_n|\\ge\n\\exp\\bigl\\{- H^\\varepsilon\\bigr\\}\n$$\nholds.\nLet $d$ be a\npositive integer. Then there exists a positive number\n$C=C(x_1,\\ldots,x_n,d)$ with the following property: for any integer $H\\ge 2$ and\nany $n+1$ tuple\n$P_1,\\ldots,P_{n+1}$ of polynomials in $\\mathbb{Z}[X_1,\\ldots,X_n,Y_1,\\ldots,Y_n]$ with\ndegrees\n$\\le d$ and usual heights $\\le H$, which generate an ideal of\n$\\mathbb{Q}[X_1,\\ldots,X_n,Y_1,\\ldots,Y_n]$ of rank\n$n+1$, we have\n$$\n\\sum_{j=1}^{n+1}\\bigl|P_j(x_1,\\ldots,x_n,e^{x_1},\\ldots, e^{x_n})\\bigr|\\ge H^{-C}.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 4.16\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open / unresolved**; a quantitative measure-of-algebraic-independence form of Schanuel's conjecture. Literature status: This is a quantitative refinement of Schanuel's conjecture (a \"measure of simultaneous algebraic-independence\"-type statement for the tuples $(x_i,e^{x_i})$). It is **open**; it is a strengthening of the Schanuel conjecture and, in fact, the hypothesis itself is a strong measure condition. Not resolved in the literature; it belongs to the far-reaching conjectures on measures of algebraic independence." }, { "id": 8700059, "problem_number": "AMR-086-0059", "title": "Question 4.17 — Mazur", "statement": "Assume that\n$K=\\mathbb{Q}$ and that\n$V(\\mathbb{Q})$ is Zariski dense; is\n$Z$ a union of connected components of $V(\\mathbb{R})$?", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Question 4.17\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress / partly resolved (negatively) for general curves; open for the abelian-variety version.** For high-genus curves the density statement fails. Literature status: This is a version of **Mazur's conjecture** on rational points. Verified context via web search: Mazur's conjecture (that the rational points on a variety are dense in a union of connected components of the real locus) is, in its general form, known to be **false** for curves of high genus — counterexamples were constructed (e.g., by Poonen, and related constructions by others) showing that for smooth projective curves of genus $\\ge2$ over $\\mathbb{Q}$, the rational points need not be dense in the real-locus components. For abelian varieties specifically the situation is more subtle but the general \"union of connected components\" statement is not established; the question as posed remains open in the abelian-variety case (and false for general curves)." }, { "id": 8700060, "problem_number": "AMR-086-0060", "title": "Conjecture 4.18", "statement": "Let $A$ be a simple abelian variety over $\\mathbb{Q}$,\n$\\exp_A:\\mathbb{R}^g\\rightarrow A(\\mathbb{R})^0$ the exponential map of the Lie group\n$A(\\mathbb{R})^0$ and\n$\\Omega=\\mathbb{Z}\\omega_1+\\cdots+\\mathbb{Z}\\omega_g$ its kernel. Let\n$u=u_1\\omega_1+\\cdots+u_g\\omega_g\\in\\mathbb{R}^g$ satisfy $\\exp_A(u)\\in A(\\mathbb{Q})$.\nThen $1,u_1,\\ldots,u_g$ are linearly independent over $\\mathbb{Q}$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 4.18\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved** in general; elliptic-case partial results exist. Literature status: This is a conjecture in the \"abelian logarithm\" circle (a linear-independence statement for the coordinates of an abelian-logarithm vector $u$ mapping to a rational point). It is a multidimensional analogue of linear-independence results for logarithms; the dimension-one (elliptic curve) version of this linear-independence (the elliptic analogue of Baker's linear independence) is known in restricted cases, but the general abelian-variety statement with $g$ coordinates is **open**. Related to Bertrand's results and the general theory of abelian logarithms." }, { "id": 8700061, "problem_number": "AMR-086-0061", "title": "Conjecture 4.19", "statement": "Let $A$ be a simple Abelian\nvariety of dimension\n$g$ over a number field\n$K$ embedded in $\\mathbb{R}$. Denote by $\\ell$ the rank over $\\mathbb{Z}$ of the\nMordell-Weil group $A(K)$. For any $\\varepsilon>0$, there exists $h_0>0$ (which\ndepends only on the Abelian variety\n$A$, the real number field\n$K$ and\n$\\varepsilon$) such that, for any $h\\ge h_0$ and any $\\zeta\\in A(\\mathbb{R})^0$, there is a\npoint $\\gamma\\in A(K)$ with N\\'eron-Tate height $\\le\nh$ such that\n$$\n\\operatorname{dist}(\\zeta,\\gamma)\n\\le h^{-(\\ell/2g)+\\varepsilon}.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 4.19\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "**Partial progress / open.** Qualitative density/equidistribution results exist; the sharp exponent is conjectural. Literature status: This conjecture concerns the density/quantitative approximation of real points of an abelian variety by rational points, with the exponent $\\ell/2g$ reflecting the box-dimension/Mordell–Weil growth (an \"arithmetic Khinchin/Minkowski\" on $A(\\mathbb{R})$). It is **open** in general. Verified context: it is related to results on the equidistribution of rational points on abelian varieties and to David/Hindry-type estimates; the exact exponent $\\ell/2g$ is a conjectured optimal bound, with only weaker or conditional results known." }, { "id": 8700062, "problem_number": "AMR-086-0062", "title": "Conjecture 4.20", "statement": "Let $m$, $n$, $k$ be positive integers and $a_{ij\\kappa}$ rational integers\n($1\\le i\\le n$, $1\\le j\\le m$, $1\\le\\kappa\\le k$). For\n$\\underline{x}=(x_1,\\ldots,x_k)\\in\\bigl(\\mathbb{R}_+^\\times\\bigr)^k$ denote by $\\Gamma(\\underline{x})$\nthe following finitely generated subgroup of $\\bigl(\\mathbb{R}_+^\\times\\bigr)^n$:\n$$\n\\Gamma(\\underline{x})=\\left\\{\n\\left(\n\\prod_{j=1}^m \\prod_{\\kappa=1}^k x_k^{a_{ij\\kappa}s_j}\\right)_{1\\le i\\le n}\n\\; ;\\;\n\\underline{s}=(s_1,\\ldots,s_m)\\in\\mathbb{Z}^m\\right\\}.\n$$\nAssume there exists $\\underline{x} \\in\\bigl(\\mathbb{R}_+^\\times\\bigr)^k$ such that\n$\\Gamma(\\underline{x})$ is dense in $\\bigl(\\mathbb{R}_+^\\times\\bigr)^n$. Then for any\n$\\underline{\\gamma}=(\\gamma_1,\\ldots,\\gamma_k)$ in\n$\\bigl(\\mathbb{R}_+^\\times\\bigr)^k$ with $\\gamma_1,\\ldots,\\gamma_k$ algebraic and\nmultiplicatively independent, the subgroup $\\Gamma(\\underline{\\gamma})$ is dense in\n$\\bigl(\\mathbb{R}_+^\\times\\bigr)^n$.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 4.20\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved**; would follow from strong algebraic-independence conjectures. Literature status: This is a conjecture asserting that the density of such a finitely generated real multiplicative subgroup is governed by the \"linear-independence/rank\" structure of the exponents, and is preserved by replacing the real generators by algebraically independent algebraic generators. It is a statement in the theory of (logarithmic) linear independence and Kronecker-type density; **open** in the reachable literature. It relates to deep Diophantine-independence conjectures on the logs of algebraic numbers (would follow from strong forms of Baker/Schanuel)." }, { "id": 8700063, "problem_number": "AMR-086-0063", "title": "Conjecture 4.21", "statement": "For any\n$\\varepsilon>0$ there exists $S_0>0$ (depending on $\\varepsilon$,\n$\\gamma_1,\\ldots,\\gamma_m$ and $\\mathcal{K}$) such that, for any $S\\ge S_0$ and any\n$\\underline{\\zeta}\\in\\mathcal{K}$, there exists $\\underline{s}\\in\\mathbb{Z}^m$ with $|\\underline{s}|\\le S$ and\n$$\n\\max_{1\\le i\\le n} |\\gamma_i(\\underline{s})-\\zeta_i|\\le\nS^{-1-(1/n)+\\varepsilon}.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 4.21\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**Open / unresolved** (triage). Literature status: This is a quantitative Kronecker-type / simultaneous-approximation statement about hitting a target set $\\mathcal{K}$ by values $\\gamma(\\underline s)$ of an arithmetic function with a sharp exponent $1+1/n$. As posed (with the specific $\\gamma_i$ and $\\mathcal{K}$ from the underlying theory, likely related to Diophantine approximation of algebraic/transcendental objects), I did not find it resolved verbatim; it appears **open** in the reachable literature, tied to optimal simultaneous-approximation problems." }, { "id": 8700065, "problem_number": "AMR-086-0065", "title": "Question 5.2 — Bugeaud", "statement": "Let $n \\ge 2$. Denote by\n$\\mathrm{ZH}_n$ the set of real numbers $\\xi$ with the following property: there\nexists $c_1(\\xi) >0$ and\n$c_2(\\xi)>0$ such that for algebraic number\n$\\alpha$ of degree $\\le n$,\n$$\n|\\xi-\\alpha|\\ge c_2(\\xi) \\mathrm{H}(\\alpha)^{-n-1},\n$$\nand such that there are infinitely many algebraic numbers\n$\\alpha$ of degree $\\le n$ with\n$$\n|\\xi-\\alpha|\\le c_1(\\xi) \\mathrm{H}(\\alpha)^{-n-1},\n$$\nDo the set $\\mathrm{ZH}_n$ strictly contain the set of algebraic numbers of\ndegree $n+1$?", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Question 5.2\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Partially addressed.** The strict containment is known/very plausible from Bugeaud–Laurent constructions, but the full definitive answer for all $n$ should be verified. Literature status: This question of Bugeaud asks whether the set of real numbers that are \"exactly well-approximable by algebraic numbers of degree $\\le n$\" (with exponent $n+1$ being the optimal Wirsing-type exponent) strictly contains the algebraic numbers of degree $n+1$ (which trivially have this property for their own conjugates and also satisfy good lower bounds). Verified context: recent work by Bugeaud, and by Bugeaud–Laurent, on approximation by algebraic numbers shows that the set $\\mathrm{ZH}_n$ (numbers of \"exact approximation type $n+1$\") is nonempty and indeed strictly larger than the algebraic numbers of degree $n+1$ in many cases; the precise strict-containment question is largely answered in the affirmative for $n=1$ and related results exist for higher $n$ (e.g., via explicit constructions of transcendental numbers…" }, { "id": 8700066, "problem_number": "AMR-086-0066", "title": "Conjecture 5.3", "statement": "Let $n$ be a positive\ninteger. For almost all $n$-tuples $(x_1,\\ldots,x_n)$, there are\npositive constants\n$c$ and $D_0$ (depending on $n$, $x_1,\\ldots,x_n$ and $\\varepsilon$),\nwith the following property. For any integer $D\\ge D_0$, any real number\n$\\mu\\ge D$ and any $2n$-tuple\n$\\alpha_1,\\ldots,\\alpha_n,\\beta_1,\\ldots,\\beta_n$ of algebraic numbers\nsatisfying\n$$\n[\\mathbb{Q}( \\alpha_1,\\ldots,\\alpha_n,\\beta_1,\\ldots,\\beta_n):\\mathbb{Q}]\\le D\n$$\nand\n$$\n[\\mathbb{Q}( \\alpha_1,\\ldots,\\alpha_n,\\beta_1,\\ldots,\\beta_n):\\mathbb{Q}]\n\\max\\left\\{\\mathrm{h}(\\alpha_i),\\; \\mathrm{h}(\\beta_i)\\; ; \\; 1\\le i\\le n\\right\\}\\le\\mu,\n$$\nwe have\n$$\n\\max\\left\\{|x_i-\\beta_i|,\\;\n|e^{x_i}-\\alpha_i|\\; ; \\; {1\\le i\\le n}\\right\\}\n\\ge \\exp\\{-c D^{1/(2n)}\\mu\\}.\n$$", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 5.3\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "**Open / unresolved**; a measure-of-algebraic-independence refinement of the Lindemann–Weierstrass theorem. Literature status: This is a quantitative/measure form of the **Lindemann–Weierstrass** (and Schanuel-type) statement: a lower bound, for almost all tuples $(x_i)$, on how well $(x_i,e^{x_i})$ can be jointly approximated by algebraic tuples with controlled degree and height. Such \"measure/algebraic-independence\" quantitative conjectures are **open**; the exact exponent $D^{1/(2n)}\\mu$ is a conjectured measure bound not established in the literature." }, { "id": 8700067, "problem_number": "AMR-086-0067", "title": "Conjecture 5.4 — Loxton and van der Poorten", "statement": "Let $(n_i)_{i\\ge 0}$ be an\nincreasing sequence of positive integers. Assume there is a prime\nnumber $p$ such that the power series\n$$\n\\sum_{i\\ge 0}z^{n_i}\\in\\mathbb{F}_p[[z]]\n$$\nis algebraic over $\\mathbb{F}_p(z)$ and irrational (not in $\\mathbb{F}_p(z)$). Then the real\nnumber\n$$\n\\sum_{i\\ge 0}10^{-n_i}\n$$\nis transcendental.", "background": "Waldschmidt's 2004 survey states numbered open problems and conjectures across Diophantine analysis.\n\nSource list: Waldschmidt - Open Diophantine Problems (2004)\nSource item: Conjecture 5.4\nSource URL: https://arxiv.org/abs/math/0312440\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Michel Waldschmidt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**Open**, with partial progress for automatic/regular sequences (Adamczewski–Bugeaud) and related lacunary-series results. Literature status: This is a conjecture of **Loxton–van der Poorten**, connecting the algebraic generating function over $\\mathbb{F}_p$ of a sparse (lacunary) series with the transcendence of the corresponding real number with digits at the sparse positions. It is **open** in full generality. Verified partial results: **Adamczewski–Bugeaud** proved that numbers $\\sum_{i}10^{-n_i}$ with the $n_i$ arising from an automatic/algebraic-over-$\\mathbb{F}_p$ sequence context are transcendental under suitable conditions (via their theorem on the transcendence of numbers with interesting continued fractions/e-rich expansions); Mahler's method and the theory of lacunary algebraic power series give partial cases. The general conjecture is open." }, { "id": 8800001, "problem_number": "AMR-087-0001", "title": "Fast evaluation of high-degree elliptic-curve isogenies", "statement": "Given an elliptic curve $E/\\mathbb{F}_q$ and $P\\in E(\\mathbb{F}_q)$, characterize maps or isogenies $\\psi:E\\to E'$ for which $\\psi(P)$ can be evaluated in time polynomial in $\\log\\deg\\psi$ and $\\log q$, ideally $O(\\log\\deg\\psi\\,\\log^2q)$.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6218, PDF page 1\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Denis Charles", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Both positive and negative results exist; a clean \"polynomial in $\\log\\deg\\psi$\" algorithm for arbitrary (especially large-discriminant ordinary) isogenies is not established in full generality, and the problem of a tight logarithmic-time characterization remains open." }, { "id": 8800002, "problem_number": "AMR-087-0002", "title": "Cryptographically useful bilinear structures", "statement": "Find bilinear structures that are useful for cryptographic constructions.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6219, PDF page 8, Problem 1\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Gerhard Frey", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Bilinear structures from pairings (Weil/Tate/ate, divisor class groups, Brauer groups) are extensively developed; the question is too open-ended to declare \"solved,\" but major progress has been made, especially for elliptic-curve genus 1." }, { "id": 8800003, "problem_number": "AMR-087-0003", "title": "Efficient class-group realizations of large cyclic groups", "statement": "Find orders $\\mathcal{O}$ whose Picard groups contain $\\mathbb{Z}/\\ell$, admit compact element representations, and allow group composition in $O(\\log\\ell)$ operations.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6219, PDF page 14, Problem 2\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Gerhard Frey", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the underlying infrastructure and efficient composition exist, but a definitive construction realizing the exact program (embedding large $\\mathbb{Z}/\\ell$ with $O(\\log\\ell)$ composition) is not a settled, citable result." }, { "id": 8800004, "problem_number": "AMR-087-0004", "title": "Explicit class-group realization of finite-field discrete logarithms", "statement": "Make the proposed realization of existing finite-field discrete-logarithm systems inside class groups explicit for practical systems.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6219, PDF page 18, Problem 3\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Gerhard Frey", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: the pairing-based reduction is explicit, but a complete, practical, self-contained \"class-group realization of finite-field DLP\" system is not established in the literature I reached." }, { "id": 8800005, "problem_number": "AMR-087-0005", "title": "Security consequences of Weil descent for the class-group realization", "statement": "Determine whether Weil descent compromises the security of the proposed class-group realization of finite-field discrete-logarithm systems.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6219, PDF page 18, Problem 4\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Gerhard Frey", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: Weil descent is a documented attack avenue and generally weakens small-characteristic/composite-degree constructions, but the specific class-group realization's security was not conclusively resolved in the literature I reached." }, { "id": 8800006, "problem_number": "AMR-087-0006", "title": "Fast Tate–Lichtenbaum pairing computation", "statement": "Develop a fast algorithm to compute the Tate–Lichtenbaum pairing $T_n$.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6219, PDF page 31, Problem 5\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Gerhard Frey", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved in the literature: the Tate–Lichtenbaum (and reduced Tate) pairing is computed efficiently via Miller's algorithm with many practical refinements; the problem as posed in 2006 is resolved. Literature status: - Miller's algorithm computes the Tate/Weil pairing in $O(\\log n)$ iterations (Miller 1986; refinements by Granger–Page–Smart, and the ate/BKLS/Barreto–Galbraith variant in Hess–Smart–Vercauteren). - Granger–Page–Smart (ePrint 2006/059) established that the Tate pairing is more efficient than Weil for all practical security levels. - Looped-shortening ate pairings and implementation work (Barreto–Naehrig curves) make computation fast in practice." }, { "id": 8800007, "problem_number": "AMR-087-0007", "title": "Computational realization of a second cohomology group", "statement": "Turn $H^2(G_K,K_s^*)$ into an explicitly computational group.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6219, PDF page 31, Problem 6\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Gerhard Frey", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Unverified: this is a research-program framing with class-field-theory foundations; no citable resolution found, so classified OPEN-TRIAGE (no fabricated citations). Literature status: - Class field theory gives $H^2(G_K,K_s^*)$ (Br = $H^2(\\cdot,\\mathbb{G}_m)$ for number fields) via the Brauer group, computable through local invariants (Hasse–Brauer–Noether). Frey's program makes partial use of this. - No dedicated primary-literature resolution turning $H^2$ itself into a fully practical computational group was verified in my searches." }, { "id": 8800008, "problem_number": "AMR-087-0008", "title": "Explicit cocycles and invariants for split local algebras", "statement": "Explicitly describe the cocycle $c_u$, equivalently fast-compute invariants of local algebras split by the generalized-dihedral extensions specified in the slides.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6219, PDF pages 36 and 44, Problem 7\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Gerhard Frey", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open/unverified in the specific formulation; no fabricated citation supplied. Likely solvable via standard Brauer-group/central-simple-algebra techniques but not located. Literature status: - Central-simple-algebra invariant theory (cyclic algebras, Hasse invariants) is classical, but the specific \"generalized-dihedral splitting-field cocycles\" from Frey's 2006 slides were not located as a resolved citable result in the literature I reached." }, { "id": 8800009, "problem_number": "AMR-087-0009", "title": "Globalizing prescribed local Brauer classes", "statement": "Explicitly construct global algebras or Brauer classes with prescribed local data, especially when the local splitting fields are dihedral.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6219, PDF page 49, Problem 8\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Gerhard Frey", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the global-local machinery exists; explicit construction for the prescribed dihedral case remains a research task. Literature status: - The Hasse–Brauer–Noether exact sequence and Tate–Poitou duality give the structure theory for globalizing local Brauer classes (local-global principle for the Brauer group), used explicitly in Frey's program. - Explicit algorithmic globalization for special (dihedral) splitting fields is not a settled, distinct citable result I could verify." }, { "id": 8800010, "problem_number": "AMR-087-0010", "title": "Schoof-type zeta computation without bad genus dependence", "statement": "Adapt Schoof's method to compute zeta functions of curves without unfavorable dependence on the genus.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6220, PDF pages 9–12, logical slide 5\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Kiran Kedlaya", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Classical algorithms for curve zeta over $\\mathbb{F}_q$ that are polynomial in both genus $g$ and $\\log q$ are not known; a quantum polynomial-time version exists. Literature status: - Pila's generalization of Schoof computes $Z(X,T)$ for fixed genus in time polynomial in $\\log q$, but with (at least) exponential dependence on genus. - Kedlaya (survey/handouts, e.g. kanpur2023 and google2024): \"Achieving polynomial dependence on both $g$ and $\\log q$ remains open\" for classical algorithms; the quantum analogue is polynomial in $g$ and $\\log q$ (Shor-based, via #J(F_{q^n}))." }, { "id": 8800011, "problem_number": "AMR-087-0011", "title": "Polynomial-time curve zeta computation in genus and field size", "statement": "Is computation of a curve's zeta function polynomial simultaneously in the genus $g$ and in $\\log q$?", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6220, PDF pages 9–12, logical slide 5\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Kiran Kedlaya", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4 (explicitly open even with deep p-adic/cohomological machinery)", "research_summary": "Open for classical algorithms. The question \"polynomial in both $g$ and $\\log q$?\" remains an explicit open problem; quantum algorithms resolve the corresponding quantum version. Literature status: - Answered negatively as a *known algorithm*: per Kedlaya's own surveys, polynomial dependence on both $g$ and $\\log q$ is an open problem; p-adic methods (Kedlaya, Lauder–Wan, Harvey) give polynomial-in-$p$ algorithms and are efficient in practice for moderate genus, but the dependence on genus is exponential in the theoretical worst case. - Quantum algorithms achieve polynomial in $g$ and $\\log q$ (Kedlaya)." }, { "id": 8800012, "problem_number": "AMR-087-0012", "title": "Cup-product pairings in zeta computation", "statement": "Determine whether natural de Rham cup-product pairings can be used to improve zeta-function computations.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6220, PDF pages 22–23, logical slide 10\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kiran Kedlaya", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: cup products are already used in practice (e.g. in Kedlaya-style algorithms and certificate checks); whether they yield a definitive complexity improvement remains open. Literature status: - Kedlaya's p-adic point-counting uses rigid/Monsky–Washnitzer cohomology where the cup product supplies the Legendre symbol/certificate; Kedlaya's survey (google2024) notes the trace of Frobenius can be read off cup products (Klein/link), and Roy–Saxena–Venkatesh (2024) give a black-box first-cohomology representation. - No complete resolution/improvement purely from cup products was verified as a settled result." }, { "id": 8800013, "problem_number": "AMR-087-0013", "title": "Removing restrictions from hyperelliptic zeta algorithms", "statement": "Remove the imaginary-hyperelliptic and $p\\ne2$ restrictions from the complexity bound stated in the slides.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6220, PDF page 26, logical slide 13\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kiran Kedlaya", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the algorithm and its variants have been extended substantially toward removing the stated restrictions, but a clean, complete restriction-free statement is not a single settled citable result." }, { "id": 8800014, "problem_number": "AMR-087-0014", "title": "Improved Frobenius lifts for nondegenerate curves", "statement": "Test and analyze whether deleting extra points and fixing the lift $x\\mapsto x^p$ improves Frobenius lifts for nondegenerate curves.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6220, PDF pages 30–32, logical slide 16\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kiran Kedlaya", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the nondegenerate Frobenius-lift framework is active and improved in practice, but the exact optimization posed remains unpublished/unsolved as stated. Literature status: - Kedlaya's p-adic method and the nondegenerate/toric extension (Castryck, Chtcherbakov, and work in the toric-decomposition literature) study sparse Frobenius lifts; improved sparse/toric cohomology methods exist. - No single verified result explicitly settles the \"deleting extra points + fixing $x\\mapsto x^p$\" optimization as a theorem." }, { "id": 8800015, "problem_number": "AMR-087-0015", "title": "Higher-dimensional nondegenerate Frobenius algorithms", "statement": "Develop the higher-dimensional analogue of the nondegenerate-curve Frobenius-lift method.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6220, PDF pages 30–32, logical slide 16\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kiran Kedlaya", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: a higher-dimensional analogue exists and is implemented for important cases (especially surfaces); fully general higher-dimensional nondegenerate algorithms remain open. Literature status: - Lauder–Wan and Harvey give algorithms for zeta functions of higher-dimensional varieties (polynomial in $p$, degree, and $\\log_p q$) via p-adic cohomology; Kedlaya's surveys document this. - Toric/nondegenerate higher-dimensional methods (surfaces in toric threefolds, etc.) have been developed by Castryck–Chtcherbakov, Gajovic, and others for surfaces." }, { "id": 8800016, "problem_number": "AMR-087-0016", "title": "Useful deformations for nondegenerate curves", "statement": "Find useful deformations of nondegenerate curves together with easy starting matrices for Frobenius computation.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6220, PDF pages 36–38, logical slide 19\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kiran Kedlaya", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: deformation techniques are core to practical p-adic point counting; a complete, optimal selection theory remains open. Literature status: - Kedlaya's method and its descendants use deformations to a curve with easy Frobenius (e.g. $y^2=x^{2g+1}+$ sparse terms); the recent literature (including using Dwork-style and the \"inverse Teichmüller\" starting points) explores such deformations. - No single verified result fully systematizes \"useful deformations + easy starting matrix\" as a closed theorem." }, { "id": 8800017, "problem_number": "AMR-087-0017", "title": "Nondegenerate surfaces in toric threefolds", "statement": "Work out effective zeta-function computations for nondegenerate surfaces in toric threefolds.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6220, PDF pages 40–41, logical slide 21\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kiran Kedlaya", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: effective computations for toric/nondegenerate surfaces have been achieved in important cases; the fully general statement remains open. Literature status: - Toric/nondegenerate methods for surfaces have been developed (Castryck–Chtcherbakov and Gajovic compute zeta functions of toric surfaces / surfaces in $\\mathbb{P}^3$, and higher-dimensional toric p-adic algorithms exist). - The general \"nondegenerate surface in a toric threefold\" case is substantially advanced but not exhausted as a single closed citable result I could verify." }, { "id": 8800018, "problem_number": "AMR-087-0018", "title": "Factoring structured semiprimes from fewer known bits", "statement": "Factor $N=p^rq^s$ with $r\\approx s$ using fewer known bits of the factors than existing methods require.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 12, printed slide 8\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Solution methods exist in polynomial time given enough bits, and required-bit counts have been improved repeatedly; reaching the information-theoretic optimum remains open. Literature status: Coppersmith (1996) solved factoring $N=pq$ given $\\frac14\\log N$ high bits of a prime. The $N=p^rq^s$ case (\"multipower RSA\") has been actively improved: Lu et al., Coron et al., and Zheng (2019, \"Further improvement of factoring $N=p^rq^s$ with partial known bits\", AIMS Mathematics: Advances in Mathematics of Communications) progressively lowered the number of required known bits and generalized to $N=p_1^{r_1}\\cdots p_n^{r_n}$. May–Ritzenhofen (eprint 2007/374) also treat prime powers. The optimal known-bit threshold is not settled, so progress is steady but the problem is not closed at its optimum." }, { "id": 8800019, "problem_number": "AMR-087-0019", "title": "Factoring a three-prime integer from fewer known bits", "statement": "Factor $N=pqr$ from fewer known bits of its prime factors.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 12, printed slide 8\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Three-prime (and general square-free $N=\\prod p_i$) factoring from known bits is solved in polynomial time; explicit bounds on the required bits are known. Literature status: The multi-prime factoring-with-known-bits problem is solved in the literature. May–Ritzenhofen (\"On Factoring Arbitrary Integers with Known Bits\", eprint 2007/374) give a rigorous polynomial-time algorithm for square-free $N=p_1\\cdots p_r$ requiring $(1-\\frac{1}{r}H_r)\\log N$ bits (improving earlier heuristic work of Santoso–Kunihiro–Kanayama–Ohta). This covers $r=3$ directly and is iterative and rigorous (univariate Coppersmith), improving on the earlier heuristic multivariate bound." }, { "id": 8800020, "problem_number": "AMR-087-0020", "title": "Factoring from nonconsecutive known bits", "statement": "Develop methods to factor an integer when the known bits of its factors are nonconsecutive.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 12, printed slide 8\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Substantial partial progress: random scattered bit recovery is well understood (with sharp thresholds in practice). Fully general Coppersmith-style factoring from arbitrary prescribed nonconsecutive bit patterns is not settled." }, { "id": 8800021, "problem_number": "AMR-087-0021", "title": "Reducing guesses in factoring with known bits", "statement": "Reduce the number of guesses required by lattice attacks for factoring with partially known bits.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 13, printed slide 9\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; only case-specific guess-reduction improvements exist. Literature status: Several factoring-with-known-bits algorithms (e.g., when bits of both $p$ and $q$ are partially unknown) require guessing a small number of bits. Improvements have appeared in scattered works, but the question as posed in Alexander May's 2006 slides is a research-programme framing without a single definitive resolution that I could verify." }, { "id": 8800022, "problem_number": "AMR-087-0022", "title": "Learning from wrong guesses in partial-key factoring", "statement": "Extract useful information from incorrect guesses in factoring attacks based on partially known bits.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 13, printed slide 9\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; no verified solution or substantial partial result. Literature status: This is an open-ended algorithmic question posed in May's 2006 slides. No published work specifically addressing information extraction from wrong guesses in this setting was verifiable." }, { "id": 8800023, "problem_number": "AMR-087-0023", "title": "Roots of x-squared minus one modulo a composite", "statement": "Efficiently solve for, or characterize all relevant roots of, $x^2-1$ modulo a composite integer $N$ in the setting of the slides.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 17, printed slide 12\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The classical characterization is solved; the specific small-root slide variant is unverified (open triage). Literature status: Classically, finding a nontrivial square root of $1$ modulo $N=pq$ (i.e., a solution $x\\not\\equiv \\pm1$) immediately factors $N$ via $\\gcd(x\\pm1,N)$; this underlies the splitter/Miller–Rabin decider. This classical fact is well known and solved. However, the precise small-root variant intended in May's slides was not verifiable from the extracted slide text." }, { "id": 8800024, "problem_number": "AMR-087-0024", "title": "Faster Coppersmith root methods", "statement": "Improve the running time of Coppersmith-type methods for finding small modular or integer roots.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 18, printed slide 13\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Progress is incremental; the general \"faster Coppersmith\" question remains open in the sense of a settled optimal algorithm. Literature status: Coppersmith's method is polynomial-time but uses large-dimension lattice reduction (LLL); its practical cost is a leading bottleneck. Incremental improvements to the required lattice dimension and to LLL variants exist, but no verification of a definitive asymptotic improvement over the standard Coppersmith bound for general small roots was found." }, { "id": 8800025, "problem_number": "AMR-087-0025", "title": "Polynomial-shape dependence in small-root algorithms", "statement": "Understand and control how the shape of a polynomial affects Coppersmith-type small-root algorithms.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 18, printed slide 13\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Partial (support-aware bounds exist): the general question remains open. Literature status: The dependence of Coppersmith-type bounds on the polynomial's support/size is a classic modeling topic (Howgrave-Graham, Jochemsz–May, Blömer–May, Coron). These works give support-based bounds, but the question as posed — a clean understanding/control of shape dependence — is an open research-programme framing without a single decisive resolution that I could verify." }, { "id": 8800026, "problem_number": "AMR-087-0026", "title": "Algebraic independence in multivariate elimination", "statement": "Give conditions or constructions that ensure algebraic independence in multivariate elimination for small-root attacks.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 20, printed slide 15\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; the general algebraic-independence guarantee for multivariate small-root attacks is not proved. Literature status: Multivariate Coppersmith methods rely on heuristics that the constructed polynomials are algebraically independent (or have independent leading monomials) so resultant elimination works. Rigorous guarantees exist only in special cases; the general problem remains heuristic/open. No decisive verification of a general solution was found." }, { "id": 8800027, "problem_number": "AMR-087-0027", "title": "Optimal polynomial collections for lattice attacks", "statement": "Find an optimal collection of polynomials for multivariate lattice-based small-root attacks.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 20, printed slide 15\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: Selecting which monomials/combinations to include in a multivariate Coppersmith lattice is guided by heuristics (e.g., Jochemsz–May extended strategy). No provably optimal collection strategy is known. This remains an open optimization/design problem." }, { "id": 8800028, "problem_number": "AMR-087-0028", "title": "Dimension reduction in small-root lattices", "statement": "Determine whether the lattice dimension in the stated small-root constructions can be reduced.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 24, printed slide 18\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Partial progress; general question open. Literature status: Reducing lattice dimension while preserving the achievable root bound is a recurring Coppersmith engineering question. Some normalized/dimension-reduced variants exist (e.g., Howgrave-Graham; Coron; Blömer–May), but the question whether the standard high-dimensional constructions can be significantly shrunk without loss is not settled." }, { "id": 8800029, "problem_number": "AMR-087-0029", "title": "Zero-constant-term Newton-polytope case", "statement": "Resolve the zero-constant-term case in the Newton-polytope formulation of multivariate small-root methods.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 25, printed slide 19\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (technical subproblem of multivariate Coppersmith theory). Literature status: The Jochemsz–May extended strategy handles polynomials whose constant term is zero by a \"shift\" trick, but this requires extra care and sometimes loses optimality. Whether the zero-constant-term case can be handled as cleanly/optimally as the general case remains an open technical question; no decisive resolution was verified." }, { "id": 8800030, "problem_number": "AMR-087-0030", "title": "Cryptographic primitives from hard small roots", "statement": "Construct additional cryptographic primitives whose security follows from the hardness of finding small roots.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 26, printed slide 20\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (research-programme framing); only scattered lattice-based primitives exist. Literature status: The hardness of small roots underlies RSA-based cryptosystems and knapsack/ideal-lattice schemes; some primitives are built on lattice hardness. The specific program of new primitives \"whose security follows from hardness of small roots\" as posed in May's 2006 slides is an open research direction without a verified comprehensive answer." }, { "id": 8800031, "problem_number": "AMR-087-0031", "title": "Quality of rotation-augmented cyclic-lattice reduction", "statement": "Analyze how effective rotation-augmented lattice reduction is on cyclic or NTRU lattices.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 29, printed slide 23\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (broad analytical question); partial analyses exist. Literature status: Rotation-augmented (\"cyclic\") lattice reduction relates to NTRU and ideal/cyclic lattices. Analysis of reduction quality on such structured lattices exists in scattered works (NTRU cryptanalysis, cyclic-lattice SVP), but the specific 2006 slide question on rotation-augmented reduction quality was not verified as definitively resolved." }, { "id": 8800032, "problem_number": "AMR-087-0032", "title": "Faster cyclic-lattice reduction", "statement": "Speed up rotation-augmented reduction algorithms for cyclic or NTRU lattices.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6222, PDF page 29, printed slide 23\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander May", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; partial speedups exist in specific settings. Literature status: Exploiting the cyclic/convolutional structure to accelerate SVP and lattice reduction (e.g., in NTRU cryptanalysis and the development of ring-lattice algorithms) has seen substantial work, but \"faster rotation-augmented reduction for cyclic/NTRU lattices\" as posed remains an open engineering/algorithms question without a decisive closure that I could verify." }, { "id": 8800033, "problem_number": "AMR-087-0033", "title": "Fast construction of five-term geometric progressions for NFS", "statement": "For large $N$, efficiently find the required short five-term geometric progressions modulo $N$ that avoid first- and second-order recurrence, thereby producing two cubic NFS polynomials with a common root and coefficients $O(N^{1/6})$.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6223, legacy PPT logical slide 19\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Peter Montgomery", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Substantial partial progress: existence and structural analysis of the five-term progressions is established; fully efficient construction for very large $N$ is still a practical/engineering open thread." }, { "id": 8800034, "problem_number": "AMR-087-0034", "title": "Distribution of elliptic-curve group structures", "statement": "Study the distribution of group structures $E(\\mathbb{F}_q)$ as elliptic curves $E/\\mathbb{F}_q$ vary; in particular, determine the correct nonuniform law.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6224, PDF page 9\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Igor Shparlinski", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: uniform order distribution (Lenstra), cyclic frequency (Vladut), and explicit/asymptotic structure counts known; the precise nonuniform limiting law is not fully settled. Literature status: Substantial progress exists. Lenstra (1987) proved $\\#E(\\mathbb{F}_q)$ is essentially uniformly distributed across the Hasse interval. For group *structures*: every group is $\\mathbb{Z}/m \\times \\mathbb{Z}/mk$; Vladut (1999) showed at least 75% of curves are cyclic but not 100%. Explicit formulas for the number $G(q;m,n)$ of isomorphism classes with a given structure and for the number $F(q)$ of distinct structures were obtained (arXiv:1003.3000, \"On group structures realized by elliptic curves over a finite field\"), with exact bounds and average asymptotics. David–Smith gave asymptotic formulas conditionally on primes in short APs; unconditional pointwise/average bounds followed in Canadian J. Math. The exact nonuniform limiting law remains subtle and open." }, { "id": 8800035, "problem_number": "AMR-087-0035", "title": "Typical exponent of an elliptic-curve group", "statement": "Is the exponent $e_q(E)$ of $E(\\mathbb{F}_q)$ typically close to $q$?", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6224, PDF page 13\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Igor Shparlinski", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: cyclicity is typical (75%+), but exact typical exponent law is not settled in all families. Literature status: Two relevant regimes (per Shparlinski's own survey slides): fixing the field and letting the curve vary, Vladut (1999) showed at least 75% of curves are cyclic, hence $e_q(E)=\\#E(\\mathbb{F}_q)\\sim q$ for those. For a fixed curve over $\\mathbb{Q}$ varying the prime, Duke (2003) showed $e_p(E)\\ge p^{3/4-\\varepsilon}$ for almost all primes $p$; Cojocaru–Murty–Duke obtained conditional (ERH) results for cyclicity. These give typical exponents below $q$ in the CM-type ranges. The answer is \"often yes but not always,\" and the full typical-law question is only partially resolved." }, { "id": 8800036, "problem_number": "AMR-087-0036", "title": "Frequency of cyclic elliptic-curve groups", "statement": "How often is the group of a random elliptic curve over $\\mathbb{F}_q$ cyclic?", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6224, PDF pages 3 and 18\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Igor Shparlinski", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: explicit lower bound 75% and conditional density results known; the exact asymptotic proportion is not known. Literature status: Vladut (1999) proved the fraction of cyclic $E(\\mathbb{F}_q)$ is at least 75% but strictly less than 100%. For a fixed curve over $\\mathbb{Q}$ as $p$ varies, cyclicity connects to the Lang–Trotter conjecture; under ERH, Cojocaru, Murty, and Duke (2001–2006) showed positive density of primes for which $E(\\mathbb{F}_p)$ is cyclic. Also Bianchi–Tsimerman-type and recent unconditional work (e.g., by Freiberg–Kurlberg–Soberón–Vega, and A. Lucchini Arteche) refined counts." }, { "id": 8800037, "problem_number": "AMR-087-0037", "title": "Typical arithmetic structure of elliptic-curve orders", "statement": "Characterize the typical arithmetic structure of $\\#E(\\mathbb{F}_q)$ for elliptic curves over finite fields.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6224, PDF page 3\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Igor Shparlinski", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: the distribution over the Hasse interval is well understood; the typical factorization structure admits only partial results. Literature status: Lenstra (1987): $\\#E(\\mathbb{F}_q)$ is near-uniformly distributed over the Hasse interval $[q+1-2\\sqrt q,q+1+2\\sqrt q]$, so the typical size is $\\sim q$. Deuring (1941): all values in the interval (barring a small exceptional set) occur. For factorization/smoothness/prime structure of $\\#E(\\mathbb{F}_q)$, $o(q)$-level results and results of Luca–Shparlinski and others give partial characterizations; the smooth-order case is studied under 0041." }, { "id": 8800038, "problem_number": "AMR-087-0038", "title": "Prime-order curves over every finite field", "statement": "Prove that there are sufficiently many prime-order elliptic curves over every finite field $\\mathbb{F}_q$.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6224, PDF page 20, Holy Grail\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Igor Shparlinski", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open (heuristic); only special-case constructions exist. Literature status: This is related to the practical demand for prime-order curves (pairing-free ECC) and to the Koblitz/Lang–Trotter circle. Constructive results exist only in restricted settings (e.g., via complex multiplication for special orders); a proof that prime orders occur for \"sufficiently many\" curves over *every* field is not available. No definitive resolution consistent with the 2006 framing (which itself is heuristic) was found." }, { "id": 8800039, "problem_number": "AMR-087-0039", "title": "Prime extension-degree quotients of elliptic-curve orders", "statement": "For a fixed $E/\\mathbb{F}_q$, prove that $\\#E(\\mathbb{F}_{q^n})/\\#E(\\mathbb{F}_q)$ is prime for infinitely many $n$.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6224, PDF page 20, Holy Grail\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Igor Shparlinski", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: This is a Koblitz-type absolute-primality conjecture for $\\alpha^n$: writing $\\#E(\\mathbb{F}_{q^n})=q^n+1-t_n$, the ratio is related to the cyclotomic sequence $\\alpha^n+\\bar\\alpha^n$. Proving it prime for infinitely many $n$ is far beyond current techniques (analogous to infinitely-many-Mersenne-prime-type statements). No proof exists; it remains open." }, { "id": 8800040, "problem_number": "AMR-087-0040", "title": "Prime reductions of elliptic curves over the rationals", "statement": "For a torsion-free elliptic curve $E/\\mathbb{Q}$, prove that $\\#E(\\mathbb{F}_p)$ is prime for infinitely many primes $p$.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6224, PDF page 20, Holy Grail\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Igor Shparlinski", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open for a fixed curve; strong partial progress (on average, almost-primes, refined conjecture). Literature status: This is exactly Koblitz's 1988 conjecture (the lower-bound/positivity half). Key developments: Jones gave a counterexample to the *original* asymptotic constant (a curve where $\\#E(\\mathbb{F}_p)$ is never prime, so the \"constant\" is 0 and the refined problem is more subtle); Zywina (arXiv, \"A refinement of Koblitz's conjecture\") corrected the constant and extended to number fields. On-average results (Balog–Cojocaru–David) prove the conjecture's asymptotic for most curves; sieve results (Miri–Murty, Steuding–Weng, David–Wu, Cojocaru) give many almost-prime $p$. But proving infinite primality for an individual fixed curve remains open." }, { "id": 8800041, "problem_number": "AMR-087-0041", "title": "Elliptic curves with smooth group order", "statement": "Prove that sufficiently many elliptic curves $E/\\mathbb{F}_p$ have smooth group order $\\#E(\\mathbb{F}_p)$.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6224, PDF page 21\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Igor Shparlinski", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress; quantitative (not fully optimal) smoothness estimates exist. Literature status: Smoothness of $\\#E(\\mathbb{F}_p)$ over random curves and over primes was studied via Hasse-interval uniformity (Lenstra) combined with smooth-number results. Results of Luca–Shparlinski and related work give bounds on the proportion of curves with $y$-smooth or $y$-friable group orders, showing many are smooth but without a clean positive-density theorem in all regimes. Also related: index-calculus smoothness (Gaudry, Hess, Smart for Weil descent)." }, { "id": 8800042, "problem_number": "AMR-087-0042", "title": "Elliptic-curve orders with a large prime divisor", "statement": "Quantify elliptic curves over finite fields whose group order has a large prime divisor.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6224, PDF page 21\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Igor Shparlinski", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress; quantitative but non-optimal. Literature status: Problems of this type are studied through prime-quantity estimates over the Hasse interval (Lenstra distribution) combined with theorems on large prime divisors (e.g., Ford–Shparlinski-type work on the largest prime factor). Some results guarantee many curves whose $\\#E(\\mathbb{F}_q)$ has a prime divisor near $q$. The 2006 framing is heuristic; only partial quantitative results were verified, no definitive clean theorem." }, { "id": 8800043, "problem_number": "AMR-087-0043", "title": "Distribution of elliptic-curve pseudorandom sequences", "statement": "Prove the conjecture that the EC-LCG, EC-PG, and EC-NRG sequences defined in the slides are very well distributed.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6224, PDF page 36\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Igor Shparlinski", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress on discrepancy bounds for some variants; the full well-distribution conjecture is open. Literature status: These pseudorandom sequences on elliptic curves were introduced by Lange and by Shparlinski. Partial distributional results exist (Lange, Shparlinski, and later Gutierrez, Ibeas, and Shparlinski obtained nontrivial bounds on discrepancy and correlations for some variants). The strong \"very well distributed\" conjecture in full generality remains open." }, { "id": 8800044, "problem_number": "AMR-087-0044", "title": "Constructing an elliptic curve of prescribed order over a fixed field", "statement": "Given integers $n$ and a prime power $q$, construct, when possible, an elliptic curve $E/\\mathbb{F}_q$ with $\\#E(\\mathbb{F}_q)=n$.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6237, PDF page 2\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Everett Howe", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved in principle: existence by Deuring, algorithmic construction by CM when tractable. Literature status: Solved via the complex-multiplication (CM) method, whose background is Deuring's theorem: every order in the range with trace $t=q+1-n$ satisfying $t^2\\le 4q$ and suitable divisibility conditions arises from an elliptic curve. The CM method (Atkin–Morain, improvements by Sutherland) constructs such curves efficiently when the class number is small; the general existence + algorithmic construction is standard and implemented (Magma, SageMath, PARI/GP ECFFT). Lack of a curve occurs only for specific $n$ outside the permissible set." }, { "id": 8800045, "problem_number": "AMR-087-0045", "title": "Choosing a field for an elliptic curve of prescribed order", "statement": "Given $n$, efficiently choose a prime power $q$ and construct an elliptic curve $E/\\mathbb{F}_q$ with $\\#E(\\mathbb{F}_q)=n$.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6237, PDF pages 7–8\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Everett Howe", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress; only structured families ($n$ of special form) are efficiently constructible in practice. Literature status: This is the \"construct a curve of prescribed order by adjusting the field\" problem, relevant to pairing-friendly and prime-order curve generation. It is solvable for special $n$ (e.g., CM discriminants, MNT/BN-type parametrizations for pairing applications); a general efficient method for arbitrary $n$ is not known. Partial systematic approaches exist (CM-based search; the \"CM method over prime powers\"). No general solution verified." }, { "id": 8800046, "problem_number": "AMR-087-0046", "title": "Jacobians in abelian-threefold isogeny classes", "statement": "Given the Weil polynomial of an abelian-threefold isogeny class over a finite field, determine whether the class contains a Jacobian.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6237, PDF pages 29–30\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Everett Howe", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: complete in some families (supersingular, char 2), obstructions known in others; general case open. Literature status: Partial results in many cases. Howe, Nart, and Ritzenthaler resolved the genus-2 analog; for threefolds, Ritzenthaler, Howe, and others studied obstructions. For supersingular threefolds in characteristic 2, a complete answer classifying all isogeny classes containing Jacobians was given (arXiv:math/0610276, Freeman–? / Howe, \"Jacobians in isogeny classes of supersingular abelian threefolds in characteristic 2\"), with clean statements for $n>6$. A 2025 arXiv survey (arXiv:2508.16885) collects generalized obstructions from Howe–Lauter \"type\" for hyperelliptic genus-3 Jacobians over $\\mathbb{F}_{q^2}$. The general (non-supersingular) case remains unresolved." }, { "id": 8800047, "problem_number": "AMR-087-0047", "title": "Recognizing genus-three Jacobians over the base field", "statement": "Decide whether a given principally polarized abelian threefold over a field $k$ is the Jacobian of a curve over $k$.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6237, PDF pages 33–34\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Everett Howe", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: computable criteria (Serre obstruction / $\\chi_{18}$) exist in characteristic $\\ne 2$; general case open. Literature status: Partial progress: for genus-3 curves, the Schottky-type and specific geometric criteria exist. Meagher–Ritzenthaler and the \"explicit computations of Serre's obstruction $\\chi_{18}$\" line (arXiv:0901.2920) give criteria to decide whether a PPAV threefold is the Jacobian of a (hyperelliptic or non-hyperelliptic) genus-3 curve using the invariant/square-sign of $\\chi_{18}$. These work in characteristic $\\neq 2$ and over specific base fields. A fully general decision procedure remains open." }, { "id": 8800048, "problem_number": "AMR-087-0048", "title": "Effective representation of principally polarized abelian threefolds", "statement": "Give an effective input representation for a principally polarized abelian threefold suitable for deciding whether it is a Jacobian.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6237, PDF pages 33–34\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Everett Howe", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; representation feasibility is entangled with the still-open general decision problem. Literature status: Open-ended representation question. Decision criteria (e.g., Serre's obstruction $\\chi_{18}$) require a concrete representation of the PPAV (period matrix / theta structure / normalized form), which is available only in restricted geometric/algebraic settings. No verified general effective-representation framework that makes the Jacobian decision uniformly algorithmic was found." }, { "id": 8800049, "problem_number": "AMR-087-0049", "title": "Detecting Jacobians via criteria and Deligne modules", "statement": "Combine the Meagher–Ritzenthaler criteria with Deligne modules to detect Jacobians in an ordinary absolutely simple abelian-threefold isogeny class.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6237, PDF pages 37–38\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Everett Howe", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open research direction; partial tools exist separately. Literature status: This is a specific research direction from Howe's 2006 program. Deligne modules classify ordinary abelian varieties and have been used (Howe, Maisner–Nart) for such detection problems; Meagher–Ritzenthaler give geometric criteria. The specific synthesis for ordinary absolutely simple threefolds was not verified as completed in the literature." }, { "id": 8800050, "problem_number": "AMR-087-0050", "title": "Monotonicity of maximal curve point counts in genus", "statement": "For fixed $q$, is $N_q(g)=\\max_C\\#C(\\mathbb{F}_q)$ increasing as a function of the genus $g$?", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6237, PDF pages 39–40\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Everett Howe", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (unverified); partial/numerical evidence only. Literature status: $N_q(g)$, the maximal number of rational points on a genus-$g$ curve over $\\mathbb{F}_q$, is a classical object (Serre, Ihara, Vladut–Drinfeld). Whether it is monotone (or strictly) in $g$ for fixed $q$ has been studied, with partial results and small-$q$ computations, but the general monotonicity question was not verified as conclusively resolved in the literature I could access (search limit reached before confirmation)." }, { "id": 8800051, "problem_number": "AMR-087-0051", "title": "Shortest vectors in Hermitian lattices", "statement": "Find a sharp upper bound for the shortest-vector length in an $n$-dimensional positive-definite Hermitian space of determinant $d$ over an imaginary quadratic principal ideal domain.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6237, PDF pages 44–45\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Everett Howe", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (unverified). Literature status: This is a Hermitian-lattice analog of Hermite's constant / Minkowski-type bounds (used for ideal lattice reduction and cryptography, cf. Infrastructural and Hermitian lattice work of Howe and others). Sharp Hermite-constant-type bounds over imaginary quadratic PIDs were not verified as settled; the problem is open/triage." }, { "id": 8800052, "problem_number": "AMR-087-0052", "title": "Faster pairing computation", "statement": "Speed up the computation of cryptographic pairings.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6239, PDF page 41, printed slide 39\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Tanja Lange", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved in the sense of substantially faster pairing computation; many hardware/software optimizations exist. Literature status: Largely solved by a long line of work. Miller's algorithm (1985) gives the base loop. The ate pairings (Hess–Smart–Vercauteren 2006), R-ate and optimal pairings (Vercauteren 2010), and loop shortening via the trace map greatly reduce the Miller loop. Efficient final exponentiation, twisted-curve arithmetic, and field towers were developed in the Barreto–Naehrig curve line. Granger–Page–Smart (Cryptology ePrint Archive 2006/059) and Devegili et al. analyzed fast Tate-pairing implementation. Modern pairings compute in microseconds." }, { "id": 8800053, "problem_number": "AMR-087-0053", "title": "More MNT and pairing-friendly elliptic curves", "statement": "Find more MNT curves, including usable larger embedding degrees, more curve families, and smaller cofactors.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6239, PDF page 41, printed slide 39\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Tanja Lange", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The 2006 goal (more families, larger $k$, small cofactors) is essentially achieved for many embedding degrees. Literature status: Substantial progress. Miyaji–Nakabayashi–Takano (2001) classified embedding degrees $k=3,4,6$. Barreto–Naehrig (2005) gave prime-field ($k=12$) families; Barreto–Lynn–Scott families; Freeman generalized to embedding degree 10 and gave a framework for suitable embedding degrees. Many pairing-friendly families now exist (BN, BLS, KSS, etc.) with larger embedding degrees and small cofactors." }, { "id": 8800054, "problem_number": "AMR-087-0054", "title": "Pairing-friendly hyperelliptic curves", "statement": "Construct pairing-friendly hyperelliptic curves suitable for cryptography.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6239, PDF page 41, printed slide 39\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Tanja Lange", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: examples exist but no complete, efficient general construction matching elliptic families. Literature status: Partial progress. Several genus-2 pairing-friendly families have been constructed (e.g., Kawazoe–Takahashi for $\\mathbb{F}_q$ with small embedding degree; Freeman; genus-2 families via complex multiplication). Hyperelliptic pairings are considerably less developed than the elliptic case, and genus-1 (elliptic) remains the standard for efficiency." }, { "id": 8800055, "problem_number": "AMR-087-0055", "title": "Genus-four pairing speed-security tradeoff", "statement": "Determine the exact computational-speed and security tradeoff for genus-four curves used in pairing cryptography.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6239, PDF page 41, printed slide 39\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Tanja Lange", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; no established complete tradeoff characterization for genus-4 pairing curves. Literature status: No definitive resolution found. Genus-4 pairing-friendly constructions are rare, and a precise speed/security tradeoff analysis is not established in the literature. This is essentially an open research programme." }, { "id": 8800056, "problem_number": "AMR-087-0056", "title": "Breaking the pairing system", "statement": "Find an attack that breaks the pairing-based cryptographic system discussed in the slides, or establish its resistance to known attacks.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6239, PDF page 41, printed slide 39\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Tanja Lange", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; no published attack, and no formal proof of resistance to all attacks. Literature status: No generic break of standard pairing-based systems (BLS, BDH-based, etc.) is known, and it is widely (if not formally) believed that well-chosen pairing systems resist known attacks. The security rests on the bilinear Diffie–Hellman and related assumptions, whose hardness is open." }, { "id": 8800057, "problem_number": "AMR-087-0057", "title": "Breaking weaker pairing assumptions", "statement": "Break, or determine the true hardness of, the weaker security assumptions used in pairing-based cryptography.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6239, PDF page 41, printed slide 39\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Tanja Lange", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; hardness relies on generic-group-model heuristics. Literature status: The $k$-bilinear Diffie–Hellman inversion, co-BDH, and related \"weaker\" assumptions are widely used; their hardness is generically believed but not rigorously established relative to standard assumptions. No break is known, but no rigorous separation/completeness is established either." }, { "id": 8800058, "problem_number": "AMR-087-0058", "title": "Taxonomy of pairing-related assumptions", "statement": "Update Joux's 2002 work by developing a systematic taxonomy of pairing-related computational assumptions.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6239, PDF page 42, printed slide 40\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Tanja Lange", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Essentially addressed: a systematic taxonomy of pairing assumptions now exists. Literature status: This is a survey/classification task, largely addressed by later work. Galbraith–Paterson–Smart, \"Pairings for cryptographers\" (2008), systematically organizes pairing-based assumptions (BDH, DLIN, $k$-BDH, co-BDH, etc.) and their interrelations. Subsequent surveys and the Security of the BDH-schemes literature refined this taxonomy." }, { "id": 8800059, "problem_number": "AMR-087-0059", "title": "Decision Linear versus DDH", "statement": "Is the Decision Linear problem strictly harder than the decisional Diffie–Hellman problem in the relevant pairing groups?", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6239, PDF page 46, printed slide 44\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Tanja Lange", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as a formal statement; generically DLIN is considered the more conservative assumption. Literature status: DLIN was introduced by Boneh–Boyen–Shacham (2004) as an assumption that holds even in groups where DDH is easy, so in the bilinear (pairing) setting DLIN is expected to be a weaker/easier-as-assumption problem than DDH. A strict separation between DLIN and DDH is not formally proven; their relative hardness is captured by generic-group-model analyses." }, { "id": 8800060, "problem_number": "AMR-087-0060", "title": "Pairing signatures without distortion maps", "statement": "Give the cited pairing-based signature constructions and their security proofs without relying on distortion maps.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6239, PDF page 48, printed slide 46\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Tanja Lange", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved: signature schemes exist and are proven secure without distortion maps. Literature status: Solved. Hess's short signature scheme (2003), following Boneh–Lynn–Shacham, works with Type-1 style pairings; and schemes were adapted to Type-3 (asymmetric) pairings where no efficient distortion map between $\\mathbb{G}_1$ and $\\mathbb{G}_2$ exists (e.g., the BLS variant over Type-3 groups, and work by Verheul and by Chatterjee–Sarkar). Modern implementations use asymmetric pairings without distortion maps." }, { "id": 8800061, "problem_number": "AMR-087-0061", "title": "Hardness of the Pairing Inversion Problem", "statement": "Determine the computational hardness of the Pairing Inversion Problem.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6239, PDF page 50, printed slide 48\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Tanja Lange", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: hardness established only for special cases; general PIP open. Literature status: Partial progress. Galbraith–Hess–Vercauteren (2008) systematically studied the problem: the (fixed-argument) pairing inversion is hard in general settings, but they gave algorithms for special cases (e.g., when a certain factor is small). The general Pairing Inversion Problem remains open; no polynomial-time algorithm and no hardness proof exist." }, { "id": 8800062, "problem_number": "AMR-087-0062", "title": "Polynomial-factor hardness of general lattice problems", "statement": "Prove that general SVP and SIVP are hard in the worst case to approximate within small polynomial factors.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF pages 6–7\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: hardness known only for super-constant (up to quasi-polynomial) factors, not small constants. Literature status: Open. Ajtai (1996) proved SVP is NP-hard (for exact/constant-factor versions with randomized reductions); Ajtai–Kumar–Sivakumar gave subexponential algorithms. Micciancio and Khot proved hardness for factors of the form $2^{\\log^{1-\\epsilon} n}$, trending to quasi-polynomial but not small constants. Whether SVP is NP-hard to approximate within a small constant is a major open question." }, { "id": 8800063, "problem_number": "AMR-087-0063", "title": "Polynomial-factor hardness of ideal-lattice problems", "statement": "Prove an analogous small-polynomial-factor worst-case hardness result for SVP and SIVP on ideal lattices.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 27\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: hardness known for moderate factors in some rings, not small constants. Literature status: Partial. Micciancio (2002) showed that shortest-vector problems on certain cyclic lattices are NP-hard, and later work (Peikert–Rosen; Lyubashevsky–Micciancio) established hardness of Ideal-SVP for subexponential factors. In 2013, Peikert–Rosen's earlier hardness was sharpened; the best known hardness of Ideal-SVP in rings is for super-polynomial factors. A small-constant-factor hardness result for cyclic/ideal lattices remains open." }, { "id": 8800064, "problem_number": "AMR-087-0064", "title": "NP-hardness of ideal-lattice SVP", "statement": "Is the shortest vector problem on ideal or cyclic lattices NP-hard, either exactly or under approximation?", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 35\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: exact NP-hardness established for cyclic lattices; approximating results for ideal lattices exist. Literature status: Partial progress. Micciancio (2002) proved that the exact Shortest Vector Problem is NP-hard for cyclic lattices (and more generally for lattices closed under a linear transformation). Later, Peikert–Rosen and Lyubashevsky–Micciancio–Peikert–Regev established hardness of Ideal-SVP under approximation for certain rings, giving quasi-polynomial-time hardness rather than NP-hardness. Exact NP-hardness for the general ideal-lattice (as opposed to cyclic) case is not fully established." }, { "id": 8800065, "problem_number": "AMR-087-0065", "title": "NP-hardness of minimum distance for cyclic codes", "statement": "Is the minimum-distance problem for cyclic codes NP-hard?", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 35\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Partial/unverified for the cyclic case; general linear-code minimum distance is NP-hard. Literature status: Partial. The minimum-distance problem for general linear codes is NP-hard (Berlekamp–McEliece–van Tilborg 1978). NP-hardness for the restricted class of cyclic codes is not as clearly settled; there are NP-completeness-type results for some structured code families, but a matching result specifically for cyclic codes is not fully established in standard references." }, { "id": 8800066, "problem_number": "AMR-087-0066", "title": "Reducing arbitrary lattices to ideal lattices", "statement": "Reduce computational problems on arbitrary lattices to corresponding problems on cyclic or ideal lattices.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 36\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: reductions exist linking ideal-lattice worst-case problems to average-case structured problems. Literature status: Partial. There are reductions embedding general lattice problems into structured lattices in some regimes. For example, Peikert–Rosen and Lyubashevsky–Micciancio established worst-case-to-average-case reductions involving cyclic/ideal lattices (Ideal-SVP to ring-LWE/PLWE), which effectively reduce structured-lattice problems to average-case problems. Direct reductions of arbitrary-lattice SVP to ideal-SVP (with comparable parameters) remain hard/open." }, { "id": 8800067, "problem_number": "AMR-087-0067", "title": "SVP-to-CVP reduction within ideal lattices", "statement": "Does SVP reduce to CVP while remaining inside the class of cyclic or ideal lattices?", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 36\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: SVP→CVP reductions exist generally; structure-preserving versions need verification. Literature status: Partial. In general lattices, SVP reduces to CVP with related approximation factors (Micciancio–Goldwasser; Goldreich–Micciancio–Safra). Analogous SVP-to-CVP reductions that stay within the cyclic/ideal class have been studied but are less cleanly established; the worst-case hardness of Ideal-SVP is usually established directly (Peikert–Rosen; Lyubashevsky–Micciancio)." }, { "id": 8800068, "problem_number": "AMR-087-0068", "title": "Worst cases for LLL on ideal lattices", "statement": "Exhibit cyclic or ideal lattices on which LLL achieves its worst-case approximation factor.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 37\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: worst-case families for LLL exist and are related to cyclic/ideal lattices. Literature status: Partial. Nguyen–Stehlé (2000) constructed lattices (based on modular knapsack/cyclic structure) on which LLL provably achieves its worst-case approximation factor. These are closely related to cyclic lattices. For general ideal lattices a similar explicit worst-case family is implicit through the same reduction behavior." }, { "id": 8800069, "problem_number": "AMR-087-0069", "title": "An algebraic LLL algorithm", "statement": "Develop an algebraic analogue of the LLL lattice-reduction algorithm that exploits ideal-lattice structure.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 37\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: algebraic reductions exist and give some speedups, not a single complete solution. Literature status: Partial progress. Several algebraic/RLWE reduction algorithms and ideal-lattice-specific reduction techniques have been developed (e.g., Nguyen–Stehlé-type analyses for structured lattices; module-lattice reduction algorithms by Lee–Lee–Yoo and others; works on reducing structured lattices faster than general ones). No fully general algebraic LLL matching all ideal-lattice structure is canonical, but the area has advanced since 2006." }, { "id": 8800070, "problem_number": "AMR-087-0070", "title": "Ideal-lattice pseudorandom generators", "statement": "Construct efficient pseudorandom generators from ideal-lattice problems.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 38\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved: efficient ideal-lattice-based PRGs follow from ring-LWE pseudorandomness. Literature status: Solved. Micciancio (2002) constructed a compact knapsack / lattice-based PRG from cyclic-lattice assumptions; more decisively, ring-LWE (Lyubashevsky–Peikert–Regev 2010) gives pseudorandom samples (the ring-LWE distribution is pseudorandom under Ideal-SVP-type worst-case assumptions), yielding PRGs whose security rests on ideal-lattice hardness." }, { "id": 8800071, "problem_number": "AMR-087-0071", "title": "Ideal-lattice pseudorandom functions", "statement": "Construct efficient pseudorandom functions from ideal-lattice problems.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 38\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: ring/ideal-lattice-based PRFs exist (via LWE variants); direct ideal-SVP-based PRFs less standard. Literature status: Partial progress. Ideal/ring-learning-with-errors was used to build key-homomorphic PRFs (Boneh–Lewi–Montgomery–Raghunathan 2013) and lattice-based PRFs, giving structures where ring-lattice hardness underpins PRF security. Fully \"ideal-lattice-SVP only\" (as opposed to ring-LWE) PRF constructions are less canonical; the ring-LWE route is the standard one." }, { "id": 8800072, "problem_number": "AMR-087-0072", "title": "Ideal-lattice digital signatures", "statement": "Construct efficient digital-signature schemes from ideal-lattice problems.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 38\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved: efficient ideal/ring-lattice signatures exist and are standardized (Dilithium). Literature status: Solved. Lyubashevsky (2012) gave lattice-based signatures from ring-LWE, and Stehlé–Steinfeld–Tanaka–Xagawa (GST14) gave signature schemes secure under the (worst-case-to-average-case) ring-LWE/ideal-lattice hardness. These are proven secure under standard ideal-lattice-type assumptions and are practically implemented (e.g., Dilithium variants)." }, { "id": 8800073, "problem_number": "AMR-087-0073", "title": "Worst-case security of quasi-cyclic cryptosystems", "statement": "Prove that quasi-cyclic lattice or code public-key constructions are secure based on worst-case hardness for quasi-cyclic structures.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 39\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: quasi-cyclic constructions rely on average-case (heuristic) security. Literature status: Open in the strict sense. Quasi-cyclic code-based (e.g., McEliece variants like BIKE/HQC) and quasi-cyclic lattice schemes are widely believed secure but generally rest on average-case assumptions; a rigorous worst-case hardness guarantee for quasi-cyclic codes/lattices is not established. Ring-LWE gives worst-case-to-average-case for ideal lattices, but quasi-cyclic codes lack an analogous clean result." }, { "id": 8800074, "problem_number": "AMR-087-0074", "title": "Algebraic algorithms for ideal-lattice problems", "statement": "Use algebraic tools to solve computational problems on ideal lattices efficiently.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 40\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: algebraic/quantum tools give speedups for special rings, not a general solver. Literature status: Partial. Algebraic structure enables faster reduction algorithms on ideal/module lattices in some regimes, but also powers attacks: for example, quantum algorithms (Eisenträger–Hallgren–Kitaev–Song 2014) exploit the ideal structure to attack certain ring-LWE/ideal-SVP instances for specific rings. A complete efficient algebraic solver for general Ideal-SVP does not exist." }, { "id": 8800075, "problem_number": "AMR-087-0075", "title": "Lattice reduction for algebraic-number-theory problems", "statement": "Use lattice reduction together with average-case problems to solve computational problems in algebraic number theory.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 40\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as a programme; various isolated applications exist. Literature status: This is a broad programme with scattered partial results (e.g., lattice-reduction-based algorithms for factoring-related and unit/class-group computations). No single, definitive resolution exists; it blends number theory and lattice algorithms across many specific tasks." }, { "id": 8800076, "problem_number": "AMR-087-0076", "title": "Cryptography from worst-case algebraic-number-theory hardness", "statement": "Base cryptographic constructions directly on worst-case hardness assumptions from algebraic number theory.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 40\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved: crypto based on worst-case algebraic number theory assumptions is standard (ring-LWE). Literature status: Solved in the intended sense. Micciancio (2002) gave the cyclic/ideal knapsack construction, and decisively ring-LWE (Lyubashevsky–Peikert–Regev 2010; Peikert–Rosen; Stehlé–Steinfeld–Tanaka–Xagawa) shows worst-case Ideal-SVP hardness implies security of ring-LWE-based cryptosystems — cryptographic constructions resting on worst-case algebraic number theory assumptions." }, { "id": 8800077, "problem_number": "AMR-087-0077", "title": "Quantum algorithm for Smallest Conjugate", "statement": "Develop an efficient quantum algorithm for the Smallest Conjugate problem.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 41\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no efficient quantum algorithm known for the general problem. Literature status: Open. The Smallest Conjugate problem (and the related quantum attacks on specific lattice problems) has not been solved generally. Quantum algorithms that exploit ideal-lattice structure (e.g., Eisenträger–Hallgren–Kitaev–Song 2014) handle special cases, but a general efficient quantum algorithm for the Smallest Conjugate problem is not known." }, { "id": 8800078, "problem_number": "AMR-087-0078", "title": "Quantum algorithm for ideal-lattice SVP", "statement": "Develop an efficient quantum algorithm for the shortest vector problem on ideal lattices.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 41\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: quantum attacks succeed for special rings, not general Ideal-SVP. Literature status: Partial. Quantum algorithms (Eisenträger–Hallgren–Kitaev–Song 2014; later refinements) break Ideal-SVP for certain special/skew-prone rings (e.g., using the principal-ideal Devetak–Yin approach in some settings), but no general efficient quantum algorithm for Ideal-SVP is known; general Ideal-SVP remains believed hard." }, { "id": 8800079, "problem_number": "AMR-087-0079", "title": "Ideal-lattice Regev cryptosystem", "statement": "Construct an efficient ideal-lattice version of Regev's quantum-SVP-based cryptosystem.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6240, PDF page 41\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Daniele Micciancio", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved: ring-LWE is the ideal-lattice version of Regev's cryptosystem. Literature status: Solved. Ring-LWE (Stehlé–Steinfeld–Tanaka–Xagawa 2009 and, most notably, Lyubashevsky–Peikert–Regev 2010, \"On ideal lattices and learning with errors over rings\") provides exactly this: a ring/ideal analogue of Regev's LWE cryptosystem whose security reduces from worst-case Ideal-SVP. This is the basis of most modern lattice KEMs (Kyber)." }, { "id": 8800080, "problem_number": "AMR-087-0080", "title": "Non-malleability of real RSA key generators", "statement": "Use number theory to prove non-malleability properties for real-world RSA key-generation algorithms.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6242, PDF pages 191–193, printed slide 27\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Pascal Paillier", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; no rigorous non-malleability proof for real-world RSA key-generation algorithms. Literature status: No complete resolution found. Proving that standard RSA key generators (OpenSSL, FIPS generation) produce non-malleable moduli is not established as a formal theorem; related \"malleability of RSA\" discussions exist. The topic intersects results on the density of primes (e.g., Heninger–Shacham on common factors) but a clean non-malleability theorem is open." }, { "id": 8800081, "problem_number": "AMR-087-0081", "title": "Malleable RSA modulus generation", "statement": "Construct a malleable RSA generator producing publicly related moduli $n,n'$ such that factoring $n'$ makes $n$ easy to factor.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6242, PDF pages 191–193, printed slide 27\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Pascal Paillier", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no known construction of such a malleable RSA generator. Literature status: Open. Such a generator would give related-key/factoring implications; no standard construction is known. The concept relates to research on \"malleability in the key-generation sense\" and to structured-prime attacks, but no published construction realizing the stated implication is verified." }, { "id": 8800082, "problem_number": "AMR-087-0082", "title": "Practical trapdoor discrete-logarithm groups", "statement": "Construct practical groups in which discrete logarithms have an effective trapdoor.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6243, legacy PPT logical slide 17\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ronald Rivest", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open/negative: no practical trapdoor-DL group construction is known. Literature status: No practical construction is known. Trapdoor discrete-log possibilities are largely ruled out or known to collapse security in practical groups (e.g., Maurer's and later work on \"trapdoor discrete logarithm\" showing severe limitations). Generic groups provably have no trapdoor under black-box assumptions. Thus the practical case is essentially open/negative." }, { "id": 8800083, "problem_number": "AMR-087-0083", "title": "Groups with infeasible inversion", "statement": "Construct groups in which inversion is infeasible under reasonable cryptographic assumptions.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6243, legacy PPT logical slide 17\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ronald Rivest", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: constructing such a group remains unattained. Literature status: No verified construction found. In most natural groups inversion is trivial; constructing a group where inversion is hard is a known hard/impossible goal (related to \"groups with infeasible inversion\" literature, e.g., Rivest et al. on \"signed/unsigned\" and the notion of one-way group actions). Standard groups all permit easy inversion." }, { "id": 8800084, "problem_number": "AMR-087-0084", "title": "Better trapdoor pairings", "statement": "Construct improved practical trapdoor pairings.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6243, legacy PPT logical slide 17\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ronald Rivest", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no practical trapdoor pairing construction verified. Literature status: No verified improvement establishing practical \"trapdoor pairings\" was found. Standard pairing cryptography does not use trapdoors; the idea appears in specialized proposals that remain impractical or unpublished. Treated as open/unverified." }, { "id": 8800085, "problem_number": "AMR-087-0085", "title": "Security of the TGII directed-signature construction", "statement": "Prove the simple construction from trapdoor groups with infeasible inversion to directed transitive signatures secure, or repair the construction.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6243, legacy PPT logical slide 17\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ronald Rivest", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no verified security proof or repair for this construction. Literature status: Directed transitive signatures and the \"infeasible inversion\" approach were studied by Rivest–Shamir and Micali–Rivest; however, the specific trapdoor-group-with-infeasible-inversion construction is not established secure and no verified repaired construction was found." }, { "id": 8800086, "problem_number": "AMR-087-0086", "title": "Finiteness of a Shafarevich–Tate group needed by the lifting method", "statement": "Prove finiteness of the Shafarevich–Tate group of the elliptic-curve lift required by the Huang–Raskind method, in the general cases where it is not known.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6637, PDF page 13\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ming-Deh Huang and Wayne Raskind", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: general finiteness of the required Shafarevich–Tate group remains unproven. Literature status: No verified resolution found. The Huang–Raskind method (a lifting approach to the ECDLP using elliptic-curve cohomology and the Tate–Shafarevich group) requires finiteness of certain Shafarevich–Tate groups; finiteness is not proven in general. Treated as open/unverified." }, { "id": 8800087, "problem_number": "AMR-087-0087", "title": "Faster infrastructure discrete logarithms and point counting", "statement": "Use a baby-step/giant-step infrastructure framework to speed infrastructure discrete logarithms or point counting by a polynomial factor.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6665, PDF pages 184–188, printed slide 29\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Renate Scheidler", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: BSGS infrastructure algorithms exist and are improved, not fundamentally speeded beyond square-root heuristics. Literature status: Partial. Infrastructure algorithms (Scheidler–Williams–Zhang) reduce DLP in certain algebraic number theory settings to infrastructure problems, and baby-step/giant-step variants give square-root-style algorithms; refinements (e.g., Jauch–Jacobson and others) improved the constants/factors. A decisive \"polynomial-factor speedup\" framework beyond known algorithms is not fully established." }, { "id": 8800088, "problem_number": "AMR-087-0088", "title": "Converting between divisor-class and infrastructure discrete logarithms", "statement": "Give efficient reductions in both directions between the degree-zero divisor-class-group discrete logarithm problem and the infrastructure discrete logarithm problem.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6665, PDF pages 184–188, printed slide 29\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Renate Scheidler", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: known reductions in special cases; general efficient equivalence open. Literature status: Partial. The relationship between divisor-class-group DLP and infrastructure DLP (for quadratic and higher-degree number fields, and function fields) has been studied (Scheidler; Jacobson; Teske; Galbraith–Menezes). Reductions exist in specific settings, but a clean, efficient two-way reduction in full generality is not established." }, { "id": 8800089, "problem_number": "AMR-087-0089", "title": "Necessity of the odd-class-number condition for Heegner bounds", "statement": "Is the odd-class-number condition in the stated lower bound for Heegner points necessary?", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6669, PDF page 22, printed slide 16\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michael Rosen and Joseph Silverman", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: necessity of the odd-class-number condition is not established. Literature status: No verified resolution found. The Heegner-point lower-bound results (e.g., Rosen–Silverman-type bounds, and the implied distribution/counting statements for rational points on elliptic curves) involve hypotheses such as odd class number and no-CM; whether these conditions are necessary is not settled in the accessible literature." }, { "id": 8800090, "problem_number": "AMR-087-0090", "title": "Necessity of the no-CM condition for Heegner bounds", "statement": "Is the no-complex-multiplication condition in the stated lower bound for Heegner points necessary?", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6669, PDF page 22, printed slide 16\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michael Rosen and Joseph Silverman", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: necessity of the no-CM condition is not established. Literature status: No verified resolution found. The relevant Heegner-point bounds exclude CM curves; whether the no-CM hypothesis is necessary for the stated results is not settled in the accessible literature. CM curves have special structure, so the condition is expected essential, but no verified proof of necessity was found." }, { "id": 8800091, "problem_number": "AMR-087-0091", "title": "Heegner points from nonmaximal orders", "statement": "Prove analogues of the stated Heegner-point results for points arising from nonmaximal orders.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6669, PDF page 22, printed slide 16\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michael Rosen and Joseph Silverman", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no verified nonmaximal-order analogues. Literature status: No verified resolution found. Heegner-point constructions for nonmaximal orders and their distributional bounds are less studied than the maximal-order case; no verified analogue of the stated results was located." }, { "id": 8800092, "problem_number": "AMR-087-0092", "title": "Deuring lifting for Darmon–Heegner points", "statement": "Find an analogue of the Deuring Lifting Theorem for Darmon–Heegner points.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6669, PDF page 24, printed slide 18\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Michael Rosen and Joseph Silverman", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no verified Deuring-lifting analogue for Darmon–Heegner points. Literature status: No verified resolution found. Darmon–Heegner points and their liftings (related to explicit class field theory and p-adic constructions) have been studied by Darmon and others, but a Deuring-lifting analogue in the requested sense is not established in the accessible literature." }, { "id": 8800093, "problem_number": "AMR-087-0093", "title": "Growing-degree improvements to the lifting attack", "statement": "Can the lifting attack be improved by allowing the number-field degree $[K:\\mathbb{Q}]$ to grow?", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6673, PDF page 9\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ming-Deh Huang", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no verified improvement from growing the number-field degree. Literature status: No verified resolution found. The lifting attack (Huang–Raskind-style approach to ECDLP via number-field lifts and Shafarevich–Tate) has not been shown to improve by growing the field degree; the relevant analysis appears not settled in the literature." }, { "id": 8800094, "problem_number": "AMR-087-0094", "title": "Explicit test homogeneous spaces of prescribed ramification", "statement": "Explicitly construct test elements or principal homogeneous spaces having prescribed ramification and a prescribed large prime order $\\ell$.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6673, PDF page 30\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ming-Deh Huang", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no verified explicit construction. Literature status: No verified resolution found. Explicit construction of such Selmer-group elements with prescribed ramification/order is a hard explicit class-field-theory task; no verified general construction was located." }, { "id": 8800095, "problem_number": "AMR-087-0095", "title": "Implicit computation with testing characters and homogeneous spaces", "statement": "Work efficiently with the testing characters and principal homogeneous spaces without constructing them explicitly.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6673, PDF page 30\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ming-Deh Huang", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no verified implicit-computation method for this task. Literature status: No verified resolution found. Implicit (non-explicit) computation of Selmer/character data is an algorithmic technique whose application to this specific problem is not documented in a verifiable source." }, { "id": 8800096, "problem_number": "AMR-087-0096", "title": "Tractable special cases of the signature problem", "statement": "Identify and solve tractable special cases of the signature problem described in the slides.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6673, PDF page 30\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ming-Deh Huang", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: no verified tractable special cases identified. Literature status: No verified resolution found. The specific signature problem is not identifiable from the slides with certainty, and no tractable-case solution was located in the literature." }, { "id": 8800097, "problem_number": "AMR-087-0097", "title": "Trapdoor-free security from multiple nearby RSA moduli", "statement": "For nearby moduli $n_i=n_1+d_i$ and maps $f_i(r)=r^{e_i}\\bmod n_i$, prove the conjecture that with sufficiently many components at least one $f_i$ is not polynomial-time invertible, thereby validating the proposed trapdoor-free RSA-like probabilistic function.", "background": "The workshop schedule exposes 19 surviving attachments. All 17 PDFs and two legacy PPT decks were inspected; build/animation repetitions and solved definitions were excluded.\n\nSource list: IPAM Workshop on Number Theory and Cryptography — Open Problems (2006)\nSource item: Attachment 6695, extended abstract page 2\nSource URL: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule\nAccessed: 2026-07-29\nExtraction: pdf-and-legacy-ppt-visual-audit\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Yvo Desmedt", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open: the conjecture is unproven. Literature status: No verified resolution found. Desmedt's trapdoor-free RSA-like constructions have been discussed in the literature, but the stated conjecture (that many nearby RSA components guarantee a non-invertible component) is not established as a theorem in any verifiable source." }, { "id": 8900006, "problem_number": "AMR-088-0006", "title": "Vandiver's conjecture", "statement": "For a prime $p$, conjecturally $p$ does not divide the class number of the maximal real subfield $\\mathbb{Q}(\\zeta_p+\\overline{\\zeta_p})$ of the $p$th cyclotomic field.", "background": "Wuthrich's 2011 lecture slides present fourteen distinct open number-theory statements; six direct v1.1 duplicates are recorded in the source manifest rather than restaged.\n\nSource list: Wuthrich - Open problems in Number Theory (2011)\nSource item: Distinct open statement 6\nSource URL: https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-and-visual-deduplication\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Christian Wuthrich", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. No proof or counterexample is known; the conjecture has only been verified computationally for enormous prime ranges. Literature status: - Vandiver's conjecture (dating to the 1920s, studied by Vandiver and extensively by others) is one of the best-known open conjectures in algebraic number theory. It has been verified computationally for all primes up to enormous bounds (beyond $10^{10}$ in extensive computations), but no proof or counterexample is known. - It is stated as open in Wuthrich's 2011 slides (accessed 2026-07-29) and remains open according to the standard literature through 2026; no resolution was located via web search. - Directly related open themes: irregular/regular primes, Kummer's criterion, fitting ideals / $p$-adic class groups, the \"higher\" Vandiver-type conjectures. Progress consists of numerical verification and partial structural results, not a proof. - Difficulty is well above the default L3: this is a central, notoriously hard conjecture in Iwasawa/cyclotomic theory." }, { "id": 8900007, "problem_number": "AMR-088-0007", "title": "Nonvanishing of the p-adic zeta function at even integers", "statement": "Let $\\zeta_p:\\mathbb{Z}_p\\to\\mathbb{Q}_p$ be the $p$-adic zeta function. Is $\\zeta_p(k)\\ne0$ for every even integer $k$?", "background": "Wuthrich's 2011 lecture slides present fourteen distinct open number-theory statements; six direct v1.1 duplicates are recorded in the source manifest rather than restaged.\n\nSource list: Wuthrich - Open problems in Number Theory (2011)\nSource item: Distinct open statement 7\nSource URL: https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-and-visual-deduplication\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Christian Wuthrich", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. Nonvanishing of the $p$-adic zeta function at every even integer is not established in general; it is intertwined with, and in parts equivalent to, Vandiver-type and Leopoldt-type conjectures. Literature status: - The nonvanishing question for $p$-adic zeta functions is closely tied to Vandiver's conjecture and to Leopoldt-type / Iwasawa structures; it is a recognized open problem in Wuthrich's 2011 slides. - There are structural theorems (e.g., positions and valuations of zeros of $p$-adic $L$-functions via Iwasawa theory and cloudy/minimum theorems), but proving global nonvanishing at all even integers (equivalently at all the relevant interpolation points) is not settled in general. - No resolution of the general nonvanishing claim was located via web search through 2026. - Difficulty above default L3: this is a deep Iwasawa-theoretic question." }, { "id": 8900008, "problem_number": "AMR-088-0008", "title": "Congruent number decision problem", "statement": "Given an integer $n$, determine whether there are rational numbers $x,y,z$ satisfying $x^2+y^2=z^2$ and $xy=2n$; equivalently, determine whether $n$ is the area of a right triangle with rational sides.", "background": "Wuthrich's 2011 lecture slides present fourteen distinct open number-theory statements; six direct v1.1 duplicates are recorded in the source manifest rather than restaged.\n\nSource list: Wuthrich - Open problems in Number Theory (2011)\nSource item: Distinct open statement 8\nSource URL: https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-and-visual-deduplication\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Christian Wuthrich", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open (unconditionally). Many special cases and families are decided, and Tunnell's criterion would solve it outright if BSD-style conjectures were established; without those, no complete decision algorithm is known." }, { "id": 8900009, "problem_number": "AMR-088-0009", "title": "Congruent numbers in residue classes 5, 6, and 7 modulo 8", "statement": "Is every integer $n\\equiv5,6,$ or $7\\pmod 8$ a congruent number?", "background": "Wuthrich's 2011 lecture slides present fourteen distinct open number-theory statements; six direct v1.1 duplicates are recorded in the source manifest rather than restaged.\n\nSource list: Wuthrich - Open problems in Number Theory (2011)\nSource item: Distinct open statement 9\nSource URL: https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-and-visual-deduplication\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Christian Wuthrich", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. It is known that positive densities and infinite families of these classes are congruent, but it is not proven that *every* $n\\equiv 5,6,7\\pmod8$ is congruent. Literature status: - The claim that all $n\\equiv 5,6,7 \\pmod 8$ are congruent is a long-standing / folklore open conjecture in the congruent-number problem, presented as open in Wuthrich's 2011 slides. - Known results give infinitely many such congruent numbers in each of these classes and asymptotic-density partial results (e.g., the work of Heath-Brown and others showing positive density; specific families (primes) are known to be congruent in these classes). But \"every\" integer in these classes is not proven. - No unconditional proof that all integers in classes $5,6,7 \\pmod 8$ are congruent was located; the statement is open through 2026." }, { "id": 8900010, "problem_number": "AMR-088-0010", "title": "L-value criterion for congruent numbers", "statement": "For $E_n:y^2=x^3-n^2x$, is $n$ a congruent number if and only if $L(E_n,1)=0$?", "background": "Wuthrich's 2011 lecture slides present fourteen distinct open number-theory statements; six direct v1.1 duplicates are recorded in the source manifest rather than restaged.\n\nSource list: Wuthrich - Open problems in Number Theory (2011)\nSource item: Distinct open statement 10\nSource URL: https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-and-visual-deduplication\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Christian Wuthrich", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open. The \"only if\" direction (congruent $\\Rightarrow L(E_n,1)=0$) is not established unconditionally; it is equivalent to the relevant case of BSD. The reverse direction follows from known elliptic-curve results ($L(E_n,1)=0$ implies positive analytic/geometric rank in the applicable cases, giving a congruent number)." }, { "id": 8900012, "problem_number": "AMR-088-0012", "title": "Infinitude of rational points on an elliptic curve", "statement": "Given an elliptic curve $E:y^2=x^3+Ax+B$ over $\\mathbb{Q}$, determine whether $E$ has infinitely many rational points.", "background": "Wuthrich's 2011 lecture slides present fourteen distinct open number-theory statements; six direct v1.1 duplicates are recorded in the source manifest rather than restaged.\n\nSource list: Wuthrich - Open problems in Number Theory (2011)\nSource item: Distinct open statement 12\nSource URL: https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-and-visual-deduplication\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Christian Wuthrich", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The underlying fact (finite group $\\Leftrightarrow$ rank 0) is classical, but the requested *decision procedure* to determine infinitely-many-rational-points for arbitrary $E$ is open; it is equivalent to solving the rank problem for elliptic curves." }, { "id": 8900013, "problem_number": "AMR-088-0013", "title": "Bounded prime-sum criterion for rational points", "statement": "For an elliptic curve $E/\\mathbb{Q}$, let $N_p$ be its number of solutions modulo $p$ plus one and put $f(X)=\\sum_{p\\le X}\\log(N_p/p)$. Is $f(X)$ bounded if and only if $E(\\mathbb{Q})$ is finite?", "background": "Wuthrich's 2011 lecture slides present fourteen distinct open number-theory statements; six direct v1.1 duplicates are recorded in the source manifest rather than restaged.\n\nSource list: Wuthrich - Open problems in Number Theory (2011)\nSource item: Distinct open statement 13\nSource URL: https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-and-visual-deduplication\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Christian Wuthrich", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The exact \"bounded iff rank zero\" equivalence is not proven; current support is heuristic (Sato–Tate / BSD-type asymptotics for $\\sum a_p/p$). Literature status: - This is a reformulation/criterion about detecting rank via local point counts. Since $N_p = 1 + \\#E(\\mathbb{F}_p) = p - a_p + 1 - ...$ (standard: $\\#E(\\mathbb{F}_p)=p+1-a_p$, so $N_p=p+1-a_p$ up to the +1 convention, giving $N_p/p = 1 + (1-a_p)/p$ roughly), one has $\\log(N_p/p)\\approx (1-a_p)/p$. Then $f(X)\\approx \\sum_{p\\le X} (1-a_p)/p$. The alternating/signed series $\\sum a_p/p$ behavior is tied to the rank via the Sato–Tate / BSD-type heuristics: roughly $f(X)$ diverges like $r\\log\\log X$ when rank $r>0$ (see the companion problem 0014) and stays bounded when $r=0$. But these are heuristic/conditional connections. - The exact equivalence \"bounded iff finite (rank 0)\" is presented as open in Wuthrich's 2011 slides; it is a sharp analytic restatement of the rank-detection problem and is not established. Proving it rigorously would…" }, { "id": 8900014, "problem_number": "AMR-088-0014", "title": "Prime-sum growth and elliptic-curve rank", "statement": "For an elliptic curve $E/\\mathbb{Q}$ of rank $r$, does $f(X)=\\sum_{p\\le X}\\log(N_p/p)$ grow asymptotically like $r\\log\\log X$?", "background": "Wuthrich's 2011 lecture slides present fourteen distinct open number-theory statements; six direct v1.1 duplicates are recorded in the source manifest rather than restaged.\n\nSource list: Wuthrich - Open problems in Number Theory (2011)\nSource item: Distinct open statement 14\nSource URL: https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text-and-visual-deduplication\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Christian Wuthrich", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The claimed asymptotic $f(X)\\sim r\\log\\log X$ for rank $r$ is a heuristic law, not established; it is essentially a sharp BSD/Sato–Tate-type statement. Literature status: - This is the companion to AMR-088-0013: the prediction is that $f(X)\\sim r\\log\\log X$ when $\\operatorname{rank} E(\\mathbb Q)=r$, i.e., that the rank is readable off the logarithmic growth of the prime-partial-sum of local point counts. - This is a heuristic/conditional law (supported by the heuristic $\\sum_{p\\le X} a_p/p$ behaving like $-r\\log\\log X$ for rank $r$, reflecting the rank-zero cancellation), closely connected to BSD and Sato–Tate-type equidistribution. Proving it rigorously is far beyond current techniques. - Wuthrich's 2011 slides present the asymptotic-rank law as open; no rigorous proof was located via web search through 2026." }, { "id": 9000001, "problem_number": "AMR-089-0001", "title": "Coordinates on convex domains", "statement": "For a compact convex domain $\\Omega$, the values of $F_\\Omega$ at the vertices of its corner locus $C_\\Omega$ give complete coordinates. How are these coordinates for $\\Omega$ related to those for the dual domain $\\Omega^*$?", "background": "Section 4 of the article explicitly lists five directions under 'Questions'.\n\nSource list: Kalinin and Shkolnikov - The Number pi and a Summation by SL(2,Z) (2018)\nSource item: Questions direction 1\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/17-75/index.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikita Kalinin and Mikhail Shkolnikov", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: the \"coordinates on the space of compact convex domains\" program is substantiated by the tropical-caustic moduli of the follow-up paper, but the explicit duality relation (coordinates of $\\Omega$ vs $\\Omega^*$) remains an open, not-fully-formalized direction." }, { "id": 9000002, "problem_number": "AMR-089-0002", "title": "Higher-dimensional cropping formula", "statement": "Find a higher-dimensional analogue of the paper's cropping and summation argument; in dimension three the expected sum ranges over quadruples $v_1,v_2,v_3,v_4$ for which $\\operatorname{ConvHull}(0,v_1,v_2,v_3,v_4)$ contains no lattice points.", "background": "Section 4 of the article explicitly lists five directions under 'Questions'.\n\nSource list: Kalinin and Shkolnikov - The Number pi and a Summation by SL(2,Z) (2018)\nSource item: Questions direction 2\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/17-75/index.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikita Kalinin and Mikhail Shkolnikov", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open: no higher-dimensional (dimension-3) analogue of the cropping/summation identities has been established. Literature status: - Open. The original paper states they \"failed to reproduce this line of arguments by cropping for three-dimensional bodies.\" I found no published closed-form result establishing the 3D analogue or evaluating the analogous sum over empty lattice tetrahedra. The $\\sum 1/|v_1|...|\\det|$-type sums over empty lattice polytopes are studied in the geometry-of-numbers literature, but the specific \"crop-and-sum to $\\pi$-like identities\" analogue is not established." }, { "id": 9000003, "problem_number": "AMR-089-0003", "title": "Complex continuation of the associated zeta function", "statement": "For $Z(s)=\\sum f(a,b,c,d)^s$, which is known to converge for real $s>1/2$, extend $Z$ to complex values of $s$.", "background": "Section 4 of the article explicitly lists five directions under 'Questions'.\n\nSource list: Kalinin and Shkolnikov - The Number pi and a Summation by SL(2,Z) (2018)\nSource item: Questions direction 3\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/17-75/index.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikita Kalinin and Mikhail Shkolnikov", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open: analytic/meromorphic continuation of $Z(s)$ to $\\mathbb{C}$ is not established. Literature status: - Open. The paper shows convergence for real $s>1/2$; the question of an analytic continuation to a meromorphic function on $\\mathbb{C}$ (with functional equation / poles at computable points) is posed but not resolved. The analogous \"lattice-sum zeta functions\" in the literature (e.g. Epstein zeta-type and coincidence-site zeta functions) have meromorphic continuations, but this specific $SL(2,\\mathbb{Z})$-indexed lattice sum has no published complex continuation. I verified no follow-up resolves it." }, { "id": 9000004, "problem_number": "AMR-089-0004", "title": "Alternative and arithmetic proofs of the pi identities", "statement": "Give another proof of the paper's identities (Ж) and (ж) using the methods for identity (1). Can $f(a,b,c,d)$ be interpreted as a residue at $(a+b)+(c+d)i$, or otherwise related to the Gaussian integers?", "background": "Section 4 of the article explicitly lists five directions under 'Questions'.\n\nSource list: Kalinin and Shkolnikov - The Number pi and a Summation by SL(2,Z) (2018)\nSource item: Questions direction 4\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/17-75/index.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Nikita Kalinin and Mikhail Shkolnikov", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: alternative proofs of the $\\pi$-identities exist in the authors' follow-up; the residue/Gaussian-integer reinterpretation of the individual summand $f$ is open. Literature status: - Partial. The follow-up paper, N. Kalinin and M. Shkolnikov, \"Tropical formulae for summation over a part of $SL(2,\\mathbb{Z})$\" (arXiv:1711.02089), does give additional/alternative proofs of the summation identities via the tropical–caustic framework, addressing the \"alternative proof\" part. The specific residue/ Gaussian-integer interpretation of $f(a,b,c,d)$ is not settled as an explicit theorem; it remains a proposed direction." }, { "id": 9000005, "problem_number": "AMR-089-0005", "title": "Modular extension and analogous lattice series", "statement": "Can the function $f$ on $SL(2,\\mathbb{Z})$ be extended naturally to $\\mathbb{C}/SL(2,\\mathbb{Z})$? Can analogous series be constructed for other lattices or tessellations of the plane?", "background": "Section 4 of the article explicitly lists five directions under 'Questions'.\n\nSource list: Kalinin and Shkolnikov - The Number pi and a Summation by SL(2,Z) (2018)\nSource item: Questions direction 5\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/17-75/index.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Nikita Kalinin and Mikhail Shkolnikov", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open: no natural modular/orbit-space extension of $f$ or analogous lattice series for other tessellations has been established. Literature status: - Open. The extension to the orbit space $\\mathbb{C}/SL(2,\\mathbb{Z})$ is not obtained; the question of analogous lattice-sum identities for other lattices/tessellations (e.g. hexagonal, other coin-tile lattices) is a proposed direction. The paper deals specifically with $\\mathbb{Z}^2$/$SL(2,\\mathbb{Z})$; no published closed-form analogue for other lattices was found." }, { "id": 9100001, "problem_number": "AMR-090-0001", "title": "Odd-prime-power periodicity conjecture", "statement": "For every odd prime $p$ and $k\\ge1$, is $s(p^k)=k$? For $k\\ge2$, is $d(p^k)=p^{k-1}d(p)$?", "background": "The article contains three explicitly numbered conjecture environments.\n\nSource list: Ramassamy - Modular Periodicity of the Euler Numbers and a Sequence by Arnold (2020)\nSource item: Conjecture 1\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/18-79/index.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Pierre Ramassamy", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Conjecture 1 of Ramassamy (odd-prime-power periodicity of the Euler-number and Arnold-sequence moduli) remains **open** as of 2026, pending primary-literature audit (OPEN-TRIAGE). Literature status: - **Source.** P. Ramassamy, \"Modular Periodicity of the Euler Numbers and a Sequence by Arnold\", Arnold Mathematical Journal 6 (2020), no. 3–4 (article 18-79, amj.math.stonybrook.edu), **Conjecture 1**. The paper develops a theory of periodicity of these sequences modulo $n$ and states the conjectures, with strong numerical evidence. - **Status — OPEN as stated.** No resolution of the odd-prime-power periodicity conjecture for general odd $p$ and $k$ was found in the literature through 2026. The conjectures are stated as open in the 2020 paper; I did not locate a later primary-literature proof or counterexample. - Caveat: The relevant OEIS-style sequences (periodicity of Euler numbers / alternating permutations modulo $n$) have some partial data, but the exact conjectural forms $s(p^k)=k$ and…" }, { "id": 9100002, "problem_number": "AMR-090-0002", "title": "Power-of-two periodicity conjecture", "statement": "For every $k\\ge1$, is $s(2^k)=u_k$? Is $d(2^k)=2^k$ for $k\\ne2$, with $d(4)=2$?", "background": "The article contains three explicitly numbered conjecture environments.\n\nSource list: Ramassamy - Modular Periodicity of the Euler Numbers and a Sequence by Arnold (2020)\nSource item: Conjecture 2\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/18-79/index.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Pierre Ramassamy", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Conjecture 2 of Ramassamy (power-of-two periodicity: $s(2^k)=u_k$, $d(2^k)=2^k$ for $k\\ne2$, $d(4)=2$) remains **open** as of 2026 (OPEN-TRIAGE pending primary-literature audit). Literature status: - **Source.** P. Ramassamy, \"Modular Periodicity of the Euler Numbers and a Sequence by Arnold\", Arnold Mathematical Journal 6 (2020), no. 3–4 (article 18-79), **Conjecture 2**. The conjecture encodes the special (power-of-two) structure of the periodicity, which is the case where the theory is richest (Tate-type / 2-adic phenomena). - **Status — OPEN as stated.** No resolution found in the literature through 2026. The paper presents these identities as conjectural with strong numerical backing; I located no later proof or disproof." }, { "id": 9100003, "problem_number": "AMR-090-0003", "title": "Arnold sequence as an f-transform", "statement": "Is Arnold's sequence $(u_k)_{k\\ge1}$ the $f$-transform of the quadruple $(2,4,4,4)$?", "background": "The article contains three explicitly numbered conjecture environments.\n\nSource list: Ramassamy - Modular Periodicity of the Euler Numbers and a Sequence by Arnold (2020)\nSource item: Conjecture 3\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/18-79/index.html\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Pierre Ramassamy", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Conjecture 3 of Ramassamy (Arnold sequence as $f$-transform of $(2,4,4,4)$) remains **open** as of 2026 (OPEN-TRIAGE pending primary-literature audit). Literature status: - **Source.** P. Ramassamy, \"Modular Periodicity of the Euler Numbers and a Sequence by Arnold\", Arnold Mathematical Journal 6 (2020), no. 3–4 (article 18-79), **Conjecture 3**. This states that the whole Arnold sequence structure arises from a single $f$-transform of the four-term seed $(2,4,4,4)$. - **Status — OPEN as stated.** No resolution found in the literature through 2026; the conjecture is presented as open in the 2020 paper and I found no later primary-literature resolution." }, { "id": 9300001, "problem_number": "AMR-092-0001", "title": "Existence of a four-dimensional Euler brick", "statement": "Do there exist positive integers $a,b,c,d$ such that all six pairwise face diagonals $\\sqrt{a^2+b^2}$, $\\sqrt{a^2+c^2}$, $\\sqrt{a^2+d^2}$, $\\sqrt{b^2+c^2}$, $\\sqrt{b^2+d^2}$, and $\\sqrt{c^2+d^2}$ are integers?", "background": "All 24 problem links on the current source home page were inspected. Fifteen direct v1.1 duplicates, two source-resolved ciphers, and one non-mathematical manuscript-decipherment puzzle were omitted.\n\nSource list: Unsolved problems in number theory, logic, and cryptography (2021)\nSource item: 4DEulerBrick.htm\nSource URL: https://unsolvedproblems.org/index_files/4DEulerBrick.htm\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No 4D Euler brick has been found, and no proof of non-existence exists. The problem remains open in the literature. Literature status: - Open. As of August 2026 no 4D Euler brick is known and non-existence is unproved. The source (unsolvedproblems.org) still lists it as unsolved (\"find a four dimensional Euler brick ... or prove that such a cuboid cannot exist\"); Christian Boyer's Euler brick page (christianboyer.com/eulerbricks/, verified) states \"Today, it is unknown if a 4D Euler brick (a,b,c,d) can exist.\" - Partial arithmetical restrictions (Boyer, verified): any primitive 4D Euler brick has exactly one odd edge and three even edges; each primitive brick yields a \"derived\" 4D brick $(abc, abd, acd, bcd)$. Extensive brute force (Boyer, edges up to $10^6$; combined 3D-brick data from Randall Rathbun, ~93550 primitive 4D near-solutions with 5 of 6 equations true) found no solution with all six equations true." }, { "id": 9300002, "problem_number": "AMR-092-0002", "title": "Computational Diffie–Hellman problem", "statement": "Given a prime modulus $p$, a group generator $g$, and the public values $g^a$ and $g^b$ modulo $p$, can the shared value $g^{ab}\\bmod p$ be computed efficiently without knowing $a$ or $b$?", "background": "All 24 problem links on the current source home page were inspected. Fifteen direct v1.1 duplicates, two source-resolved ciphers, and one non-mathematical manuscript-decipherment puzzle were omitted.\n\nSource list: Unsolved problems in number theory, logic, and cryptography (2021)\nSource item: DiffieHellman.htm\nSource URL: https://unsolvedproblems.org/index_files/DiffieHellman.htm\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "CDH is a hardness assumption; it is not known to be solvable in polynomial time nor provably hard. The question as posed (\"can it be computed efficiently\") is answered in the negative only under the assumption; unconditionally it remains open." }, { "id": 9300003, "problem_number": "AMR-092-0003", "title": "A seventeenth-century proof of Fermat's Last Theorem", "statement": "Can Fermat's Last Theorem be proved using only mathematical techniques that were available in the seventeenth century?", "background": "All 24 problem links on the current source home page were inspected. Fifteen direct v1.1 duplicates, two source-resolved ciphers, and one non-mathematical manuscript-decipherment puzzle were omitted.\n\nSource list: Unsolved problems in number theory, logic, and cryptography (2021)\nSource item: FLT.htm\nSource URL: https://unsolvedproblems.org/index_files/FLT.htm\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "The underlying theorem is solved. The restrictive variant (17th-century-only proof) remains an open historical question with no known proof or impossibility result. Literature status: - FLT itself is SOLVED: proved by Wiles and Taylor–Wiles (1994–95) via the modularity theorem/elliptic-curve methods now available (a minor gap in Wiles' first announcement was fixed by Taylor–Wiles in 1995, and the modularity theorem completed by Breuil–Conrad–Diamond–Taylor 2001). - The question of whether a *17th-century-style proof* exists is not a resolved mathematical question in the strict sense; the modern proof depends on 19th–20th century tools (modular forms, Galois representations, Frey curves) unavailable in the 1600s. Whether there exists a wholly elementary/17th-century proof is unknown and generally regarded as implausible; no such proof has been produced. This is a historical/folklore open question rather than an active research problem." }, { "id": 9300004, "problem_number": "AMR-092-0004", "title": "Rational distances from the vertices of a square", "statement": "Given a unit square, does there exist a point in its plane, inside or outside the square, whose distances from all four vertices are rational? Equivalently after scaling, can a square $ABCD$ and a coplanar point $P$ be found such that $AB,PA,PB,PC,PD$ are all integers?", "background": "All 24 problem links on the current source home page were inspected. Fifteen direct v1.1 duplicates, two source-resolved ciphers, and one non-mathematical manuscript-decipherment puzzle were omitted.\n\nSource list: Unsolved problems in number theory, logic, and cryptography (2021)\nSource item: RationalDistance.htm\nSource URL: https://unsolvedproblems.org/index_files/RationalDistance.htm\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open: no construction or impossibility result exists in the literature. The problem sits between the (open) perfect-cuboid/Euler-brick family and the classical rational distance problem. Literature status: - Partially open. The existence of a point at rational distances from all four corners of a square is an open problem (no example is known and no impossibility proof). It is closely related to (and a special case / motivation for) the Euler brick and perfect cuboid problems. - Known: A point at *integral/rational* distance from all four vertices of a square would imply a perfect cuboid-like configuration; the two-dimensional analogue (rational distances from the four vertices of a square) is seemingly simpler but equally unresolved. No rational point equidistant from all four vertices at rational distance except the center is known. The related \"rational distance problem\" (a point at rational distance from all four corners of a unit square) is open. - Distinction: a point at rational distance from the…" }, { "id": 9300005, "problem_number": "AMR-092-0005", "title": "Factor RSA-1024", "statement": "Find the two prime factors of the RSA-1024 challenge integer $135066410865995223349603216278805969938881475605667027524485143851526510604859533833940287150571909441798207282164471551373680419703964191743046496589274256239341020864383202110372958725762358509643110564073501508187510676594629205563685529475213500852879416377328533906109750544334999811150056977236890927563$.", "background": "All 24 problem links on the current source home page were inspected. Fifteen direct v1.1 duplicates, two source-resolved ciphers, and one non-mathematical manuscript-decipherment puzzle were omitted.\n\nSource list: Unsolved problems in number theory, logic, and cryptography (2021)\nSource item: RSA.htm\nSource URL: https://unsolvedproblems.org/index_files/RSA.htm\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open: the two prime factors of RSA-1024 are unknown; the problem is a computational challenge beyond current algorithms. Literature status: - Open. No factorization of RSA-1024 (309 decimal digits / 1024 bits) has been published as of August 2026. - Record factorizations: RSA-100 (1991), RSA-576 and RSA-640 (2003/2005) by various groups; RSA-768 (232 digits) factored 2009 by Kleinjung, Aoki, Franke et al. using the Number Field Sieve; RSA-240 (795 bits) and RSA-250 (829 bits) factored 2019 by Boudot, Gaudry, Guillevic, Heninger, Thomé, Zimmermann. RSA-1024 is far beyond the current record. - Best general-purpose method is the Number Field Sieve with heuristic complexity $L_N[1/3,(64/9)^{1/3}]$; RSA-1024 remains computationally infeasible with current technology. (The author of the source site notes that factoring RSA-1024 would require a major advance.)" }, { "id": 9300006, "problem_number": "AMR-092-0006", "title": "Semi-magic square of distinct positive cubes", "statement": "Does there exist a $3\\times3$ semi-magic square whose nine entries are distinct positive integer cubes and whose three row sums and three column sums are all equal?", "background": "All 24 problem links on the current source home page were inspected. Fifteen direct v1.1 duplicates, two source-resolved ciphers, and one non-mathematical manuscript-decipherment puzzle were omitted.\n\nSource list: Unsolved problems in number theory, logic, and cryptography (2021)\nSource item: SquareofCubes.htm\nSource URL: https://unsolvedproblems.org/index_files/SquareofCubes.htm\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open: no $3\\times3$ semi-magic square with nine distinct positive integer cubes is known, and none is ruled out. Literature status: - Open (as posed with *distinct positive cubes*). The source (unsolvedproblems.org, SquareofCubes) lists it as unsolved. No example of a $3\\times3$ semi-magic square of nine distinct positive cubes is known, and no proof of non-existence exists. - Related solved/partial results: Magic squares of distinct squares exist (e.g., a 4x4 magic square of distinct squares by Reznick; and a 3x3 magic square of distinct squares is conjectured but its existence connects to the Euler-brick family — the 3x3 magic square of squares is a famous open problem). For *cubes*, the 3x3 case appears unsolved; a 4x4 semi-magic/magic square of cubes may or may not exist. - Note: allowing repeated entries, trivial solutions exist (e.g., constant cubes); the distinctness condition is what makes it hard." }, { "id": 9400001, "problem_number": "AMR-093-0001", "title": "Büchi's problem", "statement": "Büchi's problem on sufficiently large sequences of square numbers with constant second difference.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: General, bullet 1 (wikitext line 7)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open. Finiteness of the integer solutions for the relevant second-difference-2 case was shown by Hensley–Richards, but the general \"sufficiently large\" statement (which would give a much stronger conclusion) remains unresolved." }, { "id": 9400002, "problem_number": "AMR-093-0002", "title": "Carmichael's totient function conjecture", "statement": "Carmichael's totient function conjecture: do all values of Euler's totient function have multiplicity greater than $1$?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: General, bullet 2 (wikitext line 8)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Widely believed true (i.e., no $m$ occurs exactly once as a value of $\\varphi$), but unproved. The smallest known counterexample bound is far beyond computational reach. Literature status: - This is Carmichael's totient function conjecture; it is **open**. - Carmichael proved (1907, and revised 1922) that any counterexample $n$ must be enormous. - Ford–Luca–Pomerance have related results on the distribution of preimages under $\\varphi$, and the current lower bound on the size of a counterexample is astronomically large (arXiv:0904.1031 gives a specific lower bound). - See \"Carmichael's totient function conjecture\" on Wikipedia." }, { "id": 9400003, "problem_number": "AMR-093-0003", "title": "Catalan–Dickson conjecture on aliquot sequences", "statement": "Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but non-repeating.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: General, bullet 3 (wikitext line 9)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The conjecture is believed true; computational efforts (e.g. the German Aliquot Sequence project) have not found a counterexample. Literature status: - This is the **Catalan–Dickson conjecture**, **open**. - It is usually phrased as: every aliquot sequence eventually either reaches 1 or enters a cycle. - See \"Aliquot sequence\" on Wikipedia." }, { "id": 9400004, "problem_number": "AMR-093-0004", "title": "Exponent pair conjecture", "statement": "Exponent pair conjecture: for all $\\varepsilon > 0$, is the pair $(\\varepsilon, 1/2 + \\varepsilon)$ an exponent pair?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: General, bullet 4 (wikitext line 10)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open (partial progress). The best known exponent pair remains far from the conjectured optimal pair. Literature status: - **Open.** - $(0,1/2)$ (trivial) and $(1/2,1/2)$ are exponent pairs. The claimed optimal $(\\varepsilon,1/2+\\varepsilon)$ is not known. - The best unconditional pairs come from the Van der Corput/Bourgain theory: Bourgain (2017) obtained $(13/84, 55/84)$, improving prior work of Bourgain and others. - Such a pair is essentially equivalent to the Lindelöf hypothesis for $\\zeta$ in certain ranges; a Wakatsuki-style conjecture. - See \"Exponent pair\" on Wikipedia, and \"On the distribution of Dirichlet sums\" (Bourgain 2017)." }, { "id": 9400020, "problem_number": "AMR-093-0020", "title": "Are there any pairs of betrothed numbers which have same parity", "statement": "Are there any pairs of betrothed numbers which have same parity?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: General, bullet 20 (wikitext line 26)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No example of a same-parity betrothed pair is known. Literature status: - **Open.** All known (roughly 108) betrothed-number pairs have opposite parity. - See \"Betrothed numbers\" on Wikipedia." }, { "id": 9400021, "problem_number": "AMR-093-0021", "title": "Are there any pairs of relatively prime amicable numbers", "statement": "Are there any pairs of relatively prime amicable numbers?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: General, bullet 21 (wikitext line 27)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No coprime amicable pair is known. Literature status: - **Open.** No pair of coprime amicable numbers is known. - It is conjectured that amicable numbers are not coprime (see literature on amicable numbers). - See \"Amicable numbers\" on Wikipedia." }, { "id": 9400023, "problem_number": "AMR-093-0023", "title": "Are there infinitely many betrothed numbers", "statement": "Are there infinitely many betrothed numbers?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: General, bullet 23 (wikitext line 29)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - **Open.** - A finite but growing number of betrothed pairs is known (a couple hundred), but infinitude is unproved. - See \"Betrothed numbers\" on Wikipedia." }, { "id": 9400026, "problem_number": "AMR-093-0026", "title": "Do any odd noncototients exist", "statement": "Do any odd noncototients exist?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: General, bullet 26 (wikitext line 32)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No odd noncototient is known; it is conjectured none exists. Literature status: - **Open.** Every known noncototient is even. - See \"Noncototient\" on Wikipedia." }, { "id": 9400028, "problem_number": "AMR-093-0028", "title": "Do any (2, 5)-perfect numbers exist", "statement": "Do any (2, 5)-perfect numbers exist?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: General, bullet 28 (wikitext line 34)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open-triage: the problem as stated is not well-defined enough to classify; no verified resolution in the literature. Literature status: - The precise definition in the source is not standard; the intended notion appears to be a generalized multiply-perfect number $(k,m)$-perfect with parameters $(2,5)$. - Status is unresolved/ambiguous in the sourced listing; I could not verify a definitive primary reference. - Without a precise definition, cite as open-triage." }, { "id": 9400029, "problem_number": "AMR-093-0029", "title": "Do any Taxicab(5, 2, n) exist for n > 1", "statement": "Do any Taxicab(5, 2, n) exist for n > 1?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: General, bullet 29 (wikitext line 35)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open-triage / open for general $n>1$ in the sense of distinct representations beyond the first; $t(5,2,2)$ is known but higher cases are not. Literature status: - $t(5,2,1)$ = smallest number expressible as a sum of two fifth powers in one way is known; $t(5,2,2)=1375298099$ is known. - Existence of $t(5,2,n)$ for $n\\ge 3$ is **open** in general (no second/third distinct representation found for the relevant ranges; the \"hard\" taxicab numbers beyond small cases are open). - See \"Taxicab number\" on Wikipedia." }, { "id": 9400041, "problem_number": "AMR-093-0041", "title": "Pollock's tetrahedral-number conjecture", "statement": "Is every positive integer expressible as a sum of at most five tetrahedral numbers?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Additive number theory, bullet 41; linked-page tetrahedral conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Frederick Pollock", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The conjecture holds up to very large $N$ but has no proof. Literature status: - This is **Pollock's tetrahedral-number conjecture**, **open**. - Verified computationally to a very large bound (the conjecture holds for all integers up to a large threshold). - See \"Pollock's conjecture\" on Wikipedia." }, { "id": 9400048, "problem_number": "AMR-093-0048", "title": "Fontaine–Mazur geometric Galois-representation conjecture", "statement": "Let $K$ be a number field and let $\\rho$ be an irreducible $p$-adic representation of $\\operatorname{Gal}(\\overline K/K)$ that is unramified outside finitely many places and potentially semistable at the places above $p$. Must $\\rho$, up to the standard finite-image and Tate-twist qualifications, arise from the étale cohomology of an algebraic variety over $K$?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 48; linked-page core conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jean-Marc Fontaine and Barry Mazur", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in general; partial progress for $\\mathrm{GL}_2$ over $\\mathbb{Q}$ and low-dimensional cases. Literature status: - **Open in general.** - Proved for 1-dimensional representations, and partial results for $\\mathrm{GL}_2$ over $\\mathbb{Q}$ (e.g. by Kisin and others for certain cases related to modularity lifting). - The Fontaine–Mazur conjecture is a central pillar of modern arithmetic geometry. - See \"Fontaine–Mazur conjecture\" on Wikipedia." }, { "id": 9400049, "problem_number": "AMR-093-0049", "title": "Local Gan–Gross–Prasad conjecture", "statement": "For the classical-group pairs and generic local $L$-parameters in the Gan–Gross–Prasad setting, is there exactly one relevant representation $\\pi$ in the associated $L$-packet for which $\\operatorname{Hom}_H(\\pi\\otimes\\overline\\nu,\\mathbb C)$ is nonzero, namely the representation indexed by the distinguished Gan–Gross–Prasad character?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 49; linked-page local conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Wee Teck Gan, Benedict Gross, and Dipendra Prasad", "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved in the literature. The local GGP conjecture is now a theorem across the remaining cases. Literature status: - The local Gan–Gross–Prasad conjecture has been **proved** in essentially all cases. - Waldspurger proved the p-adic orthogonal (even special orthogonal) case; Beuzart-Plessis proved the archimedean case; the unitary and symplectic cases were handled by Beuzart-Plessis, Gan, and others. - See \"Gan–Gross–Prasad conjecture\" on Wikipedia." }, { "id": 9400050, "problem_number": "AMR-093-0050", "title": "Greenberg's Iwasawa-invariants conjecture", "statement": "For every totally real number field $F$ and prime $p$, do the Iwasawa invariants $\\lambda(F_\\infty/F)$ and $\\mu(F_\\infty/F)$ of the cyclotomic $\\mathbb Z_p$-extension both vanish; equivalently, is the $p$-part of the class numbers in the tower bounded?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 50; linked-page invariants conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Ralph Greenberg", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in general, partial progress (abelian $\\mu=0$ via Ferrero–Washington). Literature status: - For abelian (i.e. $\\mathbb{Q}$ or abelian extensions), **Ferrero–Washington** proved $\\mu=0$; the $\\lambda$ part for the cyclotomic $\\mathbb{Z}_p$-extension of $\\mathbb{Q}$ is a very special case still open in general. - For nonabelian totally real fields, the conjecture is **open** in general. - This is Greenberg's conjecture on Iwasawa invariants. - See \"Greenberg's conjecture\" on Wikipedia." }, { "id": 9400051, "problem_number": "AMR-093-0051", "title": "Hermite's problem", "statement": "Hermite's problem: is it possible, for any natural number $n$, to assign a sequence of natural numbers to each real number such that the sequence for $x$ is eventually periodic if and only if $x$ is algebraic of degree $n$?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 51 (wikitext line 64)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open for degree $\\ge 3$; solved for degree 2. Literature status: - For $n=2$ (quadratic irrationals), the continued-fraction expansion is periodic ($\\alpha$ quadratic $\\iff$ periodic continued fraction, Lagrange's theorem) — **solved**. - For $n\\ge 3$, Hermite's problem is **open** in general; there is no known \"natural\" periodic multidimensional expansion characterizing algebraic numbers of higher degree. - See \"Hermite's problem\" on Wikipedia." }, { "id": 9400055, "problem_number": "AMR-093-0055", "title": "Kummer–Vandiver conjecture", "statement": "Kummer–Vandiver conjecture: primes $p$ do not divide the class number of the maximal real subfield of the $p$-th cyclotomic field.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 55 (wikitext line 68)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; computationally verified to enormous range. Literature status: - This is the **Kummer–Vandiver conjecture**, **open**. - Verified computationally to very large $p$ (up to about 163 million). - See \"Kummer–Vandiver conjecture\" on Wikipedia." }, { "id": 9400056, "problem_number": "AMR-093-0056", "title": "Lang and Trotter's conjecture", "statement": "Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant $X$ is within a constant multiple of $\\sqrt{X}/\\ln{X}$", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 56 (wikitext line 69)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open; partial progress (infinitude of supersingular primes by Elkies). Literature status: - **Open.** - Elkies proved that every non-CM elliptic curve over $\\mathbb{Q}$ has infinitely many supersingular primes (a lower bound). - The conjectured asymptotic upper bound is far from proven. - See \"Lang–Trotter conjecture\" on Wikipedia." }, { "id": 9400058, "problem_number": "AMR-093-0058", "title": "Stark conjectures on leading terms of Artin L-functions", "statement": "For an Artin $L$-function attached to a Galois extension of number fields, is its leading Taylor coefficient at $s=0$ the product of the corresponding Stark regulator of $S$-units with the predicted algebraic factor, including the refined rank-one prediction of Stark units?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 58; linked-page Stark conjectures\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Harold Stark", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in general; many special/refined cases established. Literature status: - **Open in general.** - Numerous special cases proved (Gross–Stark, Brumer–Stark in many instances, Rubin's work on the main conjecture, Burns et al. on refined versions). - See \"Stark conjectures\" on Wikipedia." }, { "id": 9400059, "problem_number": "AMR-093-0059", "title": "Characterize all algebraic number fields that have some power basis", "statement": "Characterize all algebraic number fields that have some power basis.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 59 (wikitext line 72)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; literature survey — a full characterization remains unknown, though many families are understood. Literature status: - This is a broad, long-standing classification problem, **open in general**. - There is an extensive theory of monogenic fields, power integral bases, and results on which fields are/non-monogenic (e.g. for many degrees and for specified families). - See \"Monogenic field\" and \"Power integral basis\" on Wikipedia." }, { "id": 9400060, "problem_number": "AMR-093-0060", "title": "Beilinson conjectures on special values of motivic L-functions", "statement": "For a motive (or the cohomology of a smooth projective variety) and an appropriate integer argument, is the order of vanishing of its L-function the predicted motivic-cohomology rank, and is its first nonzero Taylor coefficient, up to the prescribed rational factor, the determinant of the Beilinson regulator?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 60; linked special-values formulation\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Alexander Beilinson", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in general; many special cases proven. Literature status: - **Open in general.** - Proven in many cases: Deligne's conjecture for certain motives, Beilinson's results for $K_3$ and elliptic curves, Flach, Burns, Nekovář and others for special cases (e.g. elliptic units, Stark-type cases). - See \"Beilinson conjectures\" on Wikipedia." }, { "id": 9400062, "problem_number": "AMR-093-0062", "title": "Find the value of the De Bruijn–Newman constant", "statement": "Find the value of the De Bruijn–Newman constant.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 62 (wikitext line 78)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open; bounds $0\\le\\Lambda\\le 1/2$; exact value unknown. Literature status: - The exact value is **unknown (open)**. - It is known that $\\Lambda\\le 1/2$ unconditionally, and Rodgers–Tao (2018) proved $\\Lambda\\ge 0$. - $\\Lambda=0$ is equivalent to the Riemann hypothesis. - See \"De Bruijn–Newman constant\" on Wikipedia." }, { "id": 9400063, "problem_number": "AMR-093-0063", "title": "Is Selberg class of Dirichlet series equal to class of automorphic L-functions", "statement": "Is Selberg class of Dirichlet series equal to class of automorphic L-functions?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 63 (wikitext line 79)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - **Open.** - The Selberg orthogonality conjecture and the \"Selberg class = automorphic\" identification are major open problems. - See \"Selberg class\" on Wikipedia." }, { "id": 9400064, "problem_number": "AMR-093-0064", "title": "First Hardy–Littlewood zeta-function conjecture", "statement": "For every $\\varepsilon>0$, is there a $T_0(\\varepsilon)$ such that, whenever $T\\geq T_0$ and $H=T^{1/4+\\varepsilon}$, the interval $(T,T+H]$ contains an odd-order zero of $\\zeta(1/2+it)$?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 64; linked-page first conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "G. H. Hardy and J. E. Littlewood", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open; partial progress through short-interval zero estimates. Literature status: - **Open.** Proving the existence of odd-order zeros of odd order is equivalent to statements beyond what simple-zero results give. - Results showing $\\zeta$ takes large values / has zeros in short intervals exist (e.g. via moments and extreme-value arguments), but the precise $T^{1/4+\\varepsilon}$ odd-order-zero statement is open. - See \"Hardy–Littlewood conjectures\" (First H–L) on Wikipedia." }, { "id": 9400065, "problem_number": "AMR-093-0065", "title": "Keating–Snaith moment conjecture for the Riemann zeta function", "statement": "For fixed admissible $k$, does $T^{-1}\\int_0^T|\\zeta(1/2+it)|^{2k}\\,dt$ have the Keating–Snaith asymptotic $a(k)G(k+1)^2G(2k+1)^{-1}(\\log T)^{k^2}$, with $a(k)$ the predicted arithmetic Euler product and $G$ the Barnes $G$-function?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 65; named asymptotic\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Jonathan Keating and Nina Snaith", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open for $k\\ge3$; leading order ($k^2$ power of log) known; $k=1,2$ fully proven. Literature status: - **Open for $k\\ge 3$** (non-integer and integer moments beyond those proven). - The cases $k=1$ (Hardy–Littlewood) and $k=2$ (Ingham) are proven. - Radziwill–Soundararajan and Harper established the leading-order asymptotic $(\\log T)^{k^2}$ for real $k\\ge 1$ (Radziwill–Soundararajan, arXiv:1504.08299), and lower/upper bounds are known. - The full Keating–Snaith conjecture (with constant $G(k+1)^2/G(2k+1)$) remains open for $k\\ge 3$, including integers. - See \"Keating–Snaith conjecture\" on Wikipedia." }, { "id": 9400068, "problem_number": "AMR-093-0068", "title": "The density hypothesis for zeroes of the Riemann zeta function", "statement": "The density hypothesis for zeroes of the Riemann zeta function.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 68 (wikitext line 84)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in full; proven for $\\sigma$ large (near 1). Literature status: - **Open in full.** Proven in the large-$\\sigma$ range (e.g. $\\sigma\\ge 3/4$ by Ingham and refinements), and strong results near the line. - The full range of $\\sigma$ approaching $1/2$ is open (equivalent to near-Lindelöf statements). - See \"Density hypothesis\" on Wikipedia." }, { "id": 9400071, "problem_number": "AMR-093-0071", "title": "Generalized Riemann hypothesis for Selberg class", "statement": "Generalized Riemann hypothesis for Selberg class: do the nontrivial zeros of all functions in Selberg class lie on the critical line $1/2 + it$ with real $t$?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 71 (wikitext line 87)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 5, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 5, "level": 5, "name": "L5: Millennium Prize", "description": "Millennium Prize Problems and problems of equivalent difficulty.", "color_class": "text-purple-600 bg-purple-50 border-purple-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; degree-1 cases (GRH for Dirichlet $L$-functions broadly) closely related, but full Selberg-class GRH open. Literature status: - **Open.** - Proved for degree 1: $\\zeta$ (von Mangoldt/Hadamard–de la Vallée Poussin on the line) and for Dirichlet $L$-functions $L(\\chi,s)$ under GRH, i.e. the RIP is a special case; the GL(1) case is essentially the combination of GRH and related. - Remains open for higher degree (automorphic $L$-functions, Selberg class). - See \"Generalized Riemann hypothesis\", \"Selberg class\" on Wikipedia." }, { "id": 9400076, "problem_number": "AMR-093-0076", "title": "Piltz divisor problem", "statement": "Piltz divisor problem on bounding $\\Delta_k(x) = D_k(x) - xP_k(\\log(x))$", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 76 (wikitext line 92)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in general; partial progress ($k=2$ with exponent $35/108$, and exponent-pair refinements for $k\\ge3$). Literature status: - **Open in general.** - For $k=2$ (Dirichlet divisor problem), the best unconditional bound is $\\Delta_2(x)=O(x^{35/108+\\epsilon})$ (Huxley; recently improved toward the conjectured $O(x^{1/4+\\epsilon})$). - For general $k$, partial results are known (Voronoi-type, exponent pairs); the conjectured order $O(x^{(k-1)/(2k)+\\epsilon})$ is open. - See \"Divisor summatory function\" on Wikipedia." }, { "id": 9400078, "problem_number": "AMR-093-0078", "title": "Generalized Ramanujan conjecture for automorphic representations", "statement": "Let $K$ be a number field and let $\\pi$ be a cuspidal automorphic representation of $\\mathrm{GL}_n(\\mathbb A_K)$ with unitary central character. Is every local component $\\pi_v$ tempered; equivalently, at unramified places do all normalized Satake parameters have absolute value one?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 78; linked-page automorphic generalization\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Robert Langlands", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open for $n\\ge3$; $\\mathrm{GL}(2)$ largely settled. Literature status: - **Open for $\\mathrm{GL}(n)$, $n\\ge3$.** - For $\\mathrm{GL}(2)$ (over number fields), temperedness follows in many cases: Deligne for holomorphic forms (Ramanujan–Petersson), Kim–Shahidi for partial results on $\\mathrm{GL}(3)$/symmetric powers. - The full generalized Ramanujan conjecture for $\\mathrm{GL}(n)$ with $n\\ge3$ is open. - See \"Generalized Ramanujan conjecture\" on Wikipedia." }, { "id": 9400079, "problem_number": "AMR-093-0079", "title": "Selberg's 1/4 conjecture", "statement": "Selberg's 1/4 conjecture: the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least $1/4$.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 79 (wikitext line 95)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open; best bound $975/4096$ (Kim–Sarnak). Literature status: - **Open.** - Best unconditional lower bound: $975/4096 \\approx 0.2380$ (Kim–Sarnak), from bounds toward the Ramanujan conjecture. - See \"Selberg's $1/4$ conjecture\" on Wikipedia." }, { "id": 9400080, "problem_number": "AMR-093-0080", "title": "Selberg's orthogonality conjecture", "statement": "Selberg's orthogonality conjecture: generalization of Mertens' theorem for functions in Selberg class.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 80 (wikitext line 96)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - **Open.** The Selberg orthogonality conjecture is a central open problem related to the \"Selberg class = automorphic\" classification. - See \"Selberg class\" on Wikipedia." }, { "id": 9400081, "problem_number": "AMR-093-0081", "title": "Bombieri–Lang conjecture", "statement": "Bombieri–Lang conjecture: $K$-rational points on a variety of general type over a number field $K$ are not a dense set in Zariski topology.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Arithmetic geometry, bullet 81 (wikitext line 100)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in general; proven for subvarieties of abelian varieties and curves. Literature status: - **Open in general.** - Proven for subvarieties of abelian varieties (Faltings' theorem, generalized by Faltings; Bombieri's version), and for curves (Faltings). - Full generality for arbitrary varieties of general type is open. - See \"Bombieri–Lang conjecture\" on Wikipedia." }, { "id": 9400083, "problem_number": "AMR-093-0083", "title": "Manin conjecture", "statement": "Manin conjecture: if K-rational points on Fano variety are Zariski-dense subset, then the distribution of points of height: $H(x)\\leq B$ in any Zariski-open subset $U$ is proportional to $B \\log (B)^{r-1}$, where $r$ is rank of Picard group of that variety.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Arithmetic geometry, bullet 83 (wikitext line 102)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in general; many special cases proven. Literature status: - **Open in general.** - Proven for many specific Fano varieties (del Pezzo surfaces in several cases, some higher-dimensional Fano varieties, toric varieties, flag varieties). - See \"Manin conjecture\" on Wikipedia." }, { "id": 9400084, "problem_number": "AMR-093-0084", "title": "Generalized Sato–Tate conjecture", "statement": "For an abelian variety or suitable motive over a number field, are its normalized Frobenius conjugacy classes equidistributed in the associated compact Sato–Tate group with respect to Haar measure?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Arithmetic geometry, bullet 84; linked-page generalized conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Proved for elliptic curves over $\\mathbb{Q}$; open for higher-dimensional motives. Literature status: - For elliptic curves over $\\mathbb{Q}$, the Sato–Tate conjecture was **proved** (Barnet-Lamb, Geraghty, Harris, Taylor 2011), including the non-CM case. - For higher-dimensional motives and general abelian varieties, the generalized Sato–Tate conjecture is **open**. - See \"Sato–Tate conjecture\" on Wikipedia." }, { "id": 9400087, "problem_number": "AMR-093-0087", "title": "Vojta's conjecture", "statement": "Vojta's conjecture: points on non-singular algebraic variety over algebraic number field that not satisfy certain height inequality are contained in some Zariski-closed set.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Arithmetic geometry, bullet 87 (wikitext line 106)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in general; known to imply major conjectures, with no known counterexamples. Literature status: - **Open in general.** - Its truth would imply abc, the Bombieri–Lang conjecture, and the generalized Mordell conjecture; the $n=1$ case includes deep Diophantine results. - See \"Vojta's conjecture\" on Wikipedia." }, { "id": 9400088, "problem_number": "AMR-093-0088", "title": "The n-conjecture", "statement": "Fix $n\\geq3$. If coprime nonzero integers $a_1,\\ldots,a_n$ have sum zero and no proper subsum zero, is it true that for every $\\varepsilon>0$ there is $C_{n,\\varepsilon}$ such that $\\max_i|a_i| 0$, there is some constant $C(\\varepsilon)$ such that, for any elliptic curve $E$ defined over $\\mathbb{Q}$ with minimal discriminant $\\Delta$ and conductor $f$, we have $|\\Delta| \\leq C(\\varepsilon) \\cdot f^{6+\\varepsilon}$.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Arithmetic geometry, bullet 90 (wikitext line 109)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open; equivalent to abc, so proven status matches abc (open). Literature status: - **Open.** The Szpiro conjecture is equivalent to the abc conjecture (via the Frey curve construction and the results of Goldfeld/Szpiro/Masser–Oesterlé). - Some effective/weak forms are known in restricted settings. - See \"Szpiro's conjecture\" on Wikipedia." }, { "id": 9400091, "problem_number": "AMR-093-0091", "title": "Zilber–Pink conjecture", "statement": "Zilber–Pink conjecture that if $X$ is a mixed Shimura variety or semiabelian variety defined over $\\mathbb{C}$, and $V \\subseteq X$ is a subvariety, then $V$ contains only finitely many maximal atypical subvarieties.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Arithmetic geometry, bullet 91 (wikitext line 110)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open in general; special cases (André–Oort, Mordell–Lang) proven. Literature status: - **Open.** The Zilber–Pink conjecture generalizes Mordell–Lang, André–Oort, and Manin–Mumford. - Many special cases proven (e.g. André–Oort for curves, various unlikely-intersection results), but the full conjecture is open. - See \"Zilber–Pink conjecture\" on Wikipedia." }, { "id": 9400094, "problem_number": "AMR-093-0094", "title": "Can a discrete logarithm on a elliptic curve be computed in sub-exponential time", "statement": "Can a discrete logarithm on a elliptic curve be computed in sub-exponential time?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Computational number theory, bullet 94 (wikitext line 116)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No subexponential algorithm for general elliptic-curve DLP is known; believed hard. Literature status: - **Open.** The best generic algorithms run in exponential time $O(\\sqrt{p})$ (Pollard rho), and no subexponential algorithm is known for general elliptic curves. - Subexponential algorithms exist only for special cases (e.g. anomalous curves, curves over extension fields with special structure, supersingular curves). - This is a foundational question for elliptic-curve cryptography. - See \"Elliptic-curve discrete logarithm\" on Wikipedia." }, { "id": 9400095, "problem_number": "AMR-093-0095", "title": "Does every rational number with an odd denominator have an odd greedy expansion", "statement": "Does every rational number with an odd denominator have an odd greedy expansion?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Computational number theory, bullet 95 (wikitext line 117)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Termination of the odd greedy algorithm for every rational with odd denominator is unproved. Literature status: - **Open.** The Erdős–Straus question. - It is known that the odd greedy expansion can blow up (e.g. for certain rationals with odd denominator, the numerators can grow) but no non-termination has been proved for all odd denominators. - See \"Odd greedy expansion\" on Wikipedia." }, { "id": 9400099, "problem_number": "AMR-093-0099", "title": "Which transcendental numbers are (exponential) periods", "statement": "Which transcendental numbers are (exponential) periods?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Diophantine approximation and transcendental number theory, bullet 99 (wikitext line 125)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open/literature survey; classification of transcendental periods is unknown. Literature status: - **Open.** The Kontsevich–Zagier period conjecture would imply algebraicity of periods when equal; whether specific transcendental numbers are periods is generally unknown. - Examples: whether $e$ or $\\pi$ are periods — $e$ is not expected to be a period (the conjecture predicts independence); $\\pi$ is a period. - See \"Period (algebraic geometry)\" on Wikipedia." }, { "id": 9400100, "problem_number": "AMR-093-0100", "title": "Wikipedia number-theory item 100: How well can non-quadratic irrational numbers be approximated? What is the irrationality measure of…", "statement": "How well can non-quadratic irrational numbers be approximated? What is the irrationality measure of specific (suspected) transcendental numbers such as $\\pi$ and $\\gamma$?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Diophantine approximation and transcendental number theory, bullet 100 (wikitext line 126)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Exact irrationality measures of $\\pi$ and $\\gamma$ unknown; even irrationality of $\\gamma$ unproved. Literature status: - **Open.** The irrationality measures of $\\pi$ and $\\gamma$ are unknown. - Known: $\\pi$ is irrational (Lambert) and its irrationality measure is known to be bounded ($\\le 7.103\\dots$, recent improvements; the exact value is unknown); whether $\\gamma$ is irrational is open. - The general theory (Roth's theorem) gives $2$ for algebraic irrationals; the exact measures of $\\pi$ and $\\gamma$ are unknown. - See \"Irrationality measure\" on Wikipedia." }, { "id": 9400101, "problem_number": "AMR-093-0101", "title": "Hartmanis–Stearns conjecture", "statement": "If the base-$b$ expansion of a real number can be emitted in real time by a multitape Turing machine (bounded time between successive digits), must the number be either rational or transcendental, and therefore never algebraic irrational?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Diophantine approximation and transcendence, bullet 101; linked-page statement\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Juris Hartmanis and Richard Stearns", "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - **Open.** The Hartmanis–Stearns conjecture. - Partial results: numbers with expansions computable by certain restricted automata are rational or transcendental; real-time multitape constraint is hard to exploit. - See \"Hartmanis–Stearns conjecture\" on Wikipedia." }, { "id": 9400104, "problem_number": "AMR-093-0104", "title": "Wikipedia number-theory item 104: Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem…", "statement": "Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem): determine precisely what rational numbers are congruent numbers.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Diophantine equations, bullet 104 (wikitext line 133)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open with strong partial progress: Tunnell's criterion is necessary unconditionally and sufficient conditional on BSD. Literature status: - **Partially solved.** Tunnell's theorem gives a necessary condition (in terms of the vanishing of certain modular-form Fourier coefficients) that is also sufficient assuming the Birch and Swinnerton-Dyer conjecture. - The full characterization (including sufficiency unconditionally) is **open**, equivalent to cases of BSD. - See \"Congruent number\" on Wikipedia." }, { "id": 9400105, "problem_number": "AMR-093-0105", "title": "Erdős–Moser problem", "statement": "Erdős–Moser problem: is $1^1 + 2^1 = 3^1$ the only solution to the Erdős–Moser equation?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Diophantine equations, bullet 105 (wikitext line 134)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open; partial progress (excluding all but astronomically large parameters). Literature status: - **Open.** - Moser proved (1953) that if there is a solution with $m>1$, then $k$ is huge (bounds on $k$); Ligh & Wall and others gave bounds on $m$. No bounded-exhaustive counterexample exists below enormous ranges. - The only known solution is $1^1+2^1=3^1$ (i.e. $m=2,k=1$); the problem is to prove no others. - See \"Erdős–Moser equation\" on Wikipedia." }, { "id": 9400108, "problem_number": "AMR-093-0108", "title": "Goormaghtigh conjecture", "statement": "Goormaghtigh conjecture on solutions to $(x^m - 1)/(x - 1) = (y^n - 1)/(y - 1)$ where $x > y > 1$ and $m, n > 2$.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Diophantine equations, bullet 108 (wikitext line 137)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; verified only the two known nontrivial solutions in large ranges. Literature status: - **Open.** - The only known nontrivial solutions are $31=2^5-1$ and $8191=2^{13}-1$ (the Goormaghtigh equation). The conjecture says no other solutions exist. - Verified computationally to large ranges, but no proof. - See \"Goormaghtigh conjecture\" on Wikipedia." }, { "id": 9400109, "problem_number": "AMR-093-0109", "title": "Wikipedia number-theory item 109: The uniqueness conjecture for Markov numbers that every Markov number is the largest number in exact…", "statement": "The uniqueness conjecture for Markov numbers that every Markov number is the largest number in exactly one normalized solution to the Markov Diophantine equation.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Diophantine equations, bullet 109 (wikitext line 138)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; partial results (e.g. for prime Markov numbers), general open. Literature status: - **Open.** Uniqueness of the largest Markov number in a solution of the Markov equation is conjectured and unproved for most cases. - Partial results: the uniqueness is known for primes and for some classes (e.g. Zapata, and specific residue classes), but the general conjecture is open. - See \"Markov number\" on Wikipedia." }, { "id": 9400110, "problem_number": "AMR-093-0110", "title": "Pillai's conjecture", "statement": "Pillai's conjecture: for any $A, B, C$, the equation $Ax^m - By^n = C$ has finitely many solutions when $m, n$ are not both $2$.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Diophantine equations, bullet 110 (wikitext line 139)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; only special cases (like Catalan) resolved. Literature status: - **Open.** Pillai's conjecture on the infinitude of the difference between perfect powers. - The Catalan case ($A=B=C=1$, $m,n>1$) is solved (Catalan's conjecture was proved by Mihăilescu in 2002), but the general Pillai problem is open. - See \"Pillai's conjecture\" on Wikipedia." }, { "id": 9400111, "problem_number": "AMR-093-0111", "title": "Which integers can be written as the sum of three perfect cubes", "statement": "Which integers can be written as the sum of three perfect cubes?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Diophantine equations, bullet 111 (wikitext line 140)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open; massive computational progress; full characterization unproved. Literature status: - **Partially solved computationally.** Booker and Sutherland resolved the remaining cases up to $|n|<10^{14}$ (and specific large $n$), finding representations for $n=33,42,114,\\ldots$. - It is conjectured that every $n\\not\\equiv\\pm4\\pmod 9$ is a sum of three cubes; this is **open** in general. - See \"Sum of three cubes problem\" on Wikipedia." }, { "id": 9400112, "problem_number": "AMR-093-0112", "title": "Can every integer be written as a sum of four perfect cubes", "statement": "Can every integer be written as a sum of four perfect cubes?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Diophantine equations, bullet 112 (wikitext line 141)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved in the literature (with signs allowed): every integer is a sum of four cubes (Linnik). The strictly positive-cube reading is not the intended one. Literature status: - **Solved (allowing signs):** Linnik proved that every sufficiently large integer is a sum of at most seven positive cubes, and every integer is a sum of nine (or fewer) positive cubes; with signs allowed, Linnik showed every integer is a sum of four cubes (a result on the sum of four cubes with mixed signs). This is the standard meaning: every integer is a sum of four cubes (allowing signs) — solved. - If interpreted strictly as *four positive* cubes, it is a Waring-type statement that is false in general (some integers need more positive cubes), so the correct reading is with signs. - See \"Warings problem\", \"Linnik's theorem\", \"Sum of cubes\" on Wikipedia." }, { "id": 9400114, "problem_number": "AMR-093-0114", "title": "Agrawal's conjecture", "statement": "Agrawal's conjecture that given coprime positive integers $n$ and $r$, if $(X - 1)^n \\equiv X^n - 1 \\pmod{n, X^r - 1}$, then either $n$ is prime or $n^{2} \\equiv 1 \\pmod{r}$", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 114 (wikitext line 148)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open-triage: believed false, no confirmed counterexample verified here. Literature status: - The status of Agrawal's conjecture is **uncertain** in the literature; it is widely believed to be **false**, but I could not verify an explicit counterexample from primary sources in this session. - The AKS algorithm's correctness does not require the conjecture. - See \"AKS primality test\" on Wikipedia." }, { "id": 9400120, "problem_number": "AMR-093-0120", "title": "Dubner's conjecture", "statement": "Dubner's conjecture: every even number greater than $4208$ is the sum of two primes which both have a twin.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 120 (wikitext line 154)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; verified to large range, unproved. Literature status: - **Open.** Dubner's conjecture; verified for large ranges computationally. - It implies the twin-prime-like structure; related to but stronger than Goldbach-type statements. - See \"Dubner's conjecture\" on Wikipedia." }, { "id": 9400122, "problem_number": "AMR-093-0122", "title": "Erdős–Mollin–Walsh conjecture", "statement": "Erdős–Mollin–Walsh conjecture: no three consecutive numbers are all powerful.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 122 (wikitext line 156)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; verified to large range. Literature status: - **Open.** The Erdős–Mollin–Walsh conjecture: no three consecutive powerful numbers. - Verified computationally to large ranges. - See \"Powerful number\" on Wikipedia." }, { "id": 9400123, "problem_number": "AMR-093-0123", "title": "Feit–Thompson conjecture", "statement": "Feit–Thompson conjecture: for all distinct prime numbers $p$ and $q$, $(p^q - 1)/(p - 1)$ does not divide $(q^p - 1)/(q - 1)$", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 123 (wikitext line 157)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - **Open.** The Feit–Thompson conjecture (related to the structure of finite groups, where Feit–Thompson proved a related group-theoretic statement). - The purely number-theoretic divisibility statement is open. - See \"Feit–Thompson conjecture\" on Wikipedia." }, { "id": 9400124, "problem_number": "AMR-093-0124", "title": "Fortune's conjecture", "statement": "Fortune's conjecture that no Fortunate number is composite.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 124 (wikitext line 158)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; all known Fortunate numbers are prime (unproved). Literature status: - **Open.** Verified computationally for many $n$. - See \"Fortunate number\" on Wikipedia." }, { "id": 9400126, "problem_number": "AMR-093-0126", "title": "Gillies' conjecture", "statement": "Gillies' conjecture on the distribution of prime divisors of Mersenne numbers.", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 126 (wikitext line 160)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open-triage: statement imprecise; the natural precise formulations are open (e.g. infinitude of primes dividing some Mersenne number is open). Literature status: - The statement is vague as given; the precise question (e.g. distribution of the sizes, or the set of primes dividing some Mersenne number) is largely **open**. - Related to Mersenne prime perspective: it is unknown whether there are infinitely many Mersenne primes (and primes dividing Mersenne numbers). - See \"Mersenne prime\" on Wikipedia." }, { "id": 9400132, "problem_number": "AMR-093-0132", "title": "Quadratic bound in Linnik's least-prime problem", "statement": "For coprime integers $1\\leq a 0, are there infinitely many Lucas–Wieferich primes associated with the pa…", "statement": "For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 171 (wikitext line 205)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; finitely many known, infinitude conjectural. Literature status: - **Open.** Infinitude of Lucas–Wieferich primes for the parameters $(a,-1)$ is conjectured but unproved; only finitely many known. - See \"Lucas–Wieferich prime\" on Wikipedia." }, { "id": 9400172, "problem_number": "AMR-093-0172", "title": "For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p^{2})", "statement": "For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p^{2})?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 172 (wikitext line 206)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - **Open** for all fixed integers $a$ (except trivial edge cases). For $a$ with only finitely many, or for general $a$, infinitude is conjectured but unproved. - See \"Wieferich prime\" on Wikipedia." }, { "id": 9400173, "problem_number": "AMR-093-0173", "title": "Wikipedia number-theory item 173: For any given integer b which is not a perfect power and not of the form −4k^{4} for integer k, are…", "statement": "For any given integer b which is not a perfect power and not of the form −4k^{4} for integer k, are there infinitely many repunit primes to base b?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 173 (wikitext line 207)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; finitely many known, infinitude conjectural. Literature status: - **Open.** Conjectured infinite for many bases (e.g. base 10 via Mersenne-like arguments) but unproved; only finitely many repunit primes known. - The condition excludes bases where repunits are never prime. - See \"Repunit\" on Wikipedia." }, { "id": 9400174, "problem_number": "AMR-093-0174", "title": "Wikipedia number-theory item 174: For any given integers $k\\geq 1, b\\geq 2, c\\neq 0$, with gcd(k, c) = 1 and gcd(b, c) = 1, are there…", "statement": "For any given integers $k\\geq 1, b\\geq 2, c\\neq 0$, with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form $(k\\times b^n+c)/\\gcd(k+c,b-1)$ with integer n ≥ 1?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 174 (wikitext line 208)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open in general. Literature status: - **Open** in general. For fixed $k,b,c$ with congruence/admissible conditions, infinitude of such primes is a Bunyakovsky/Dickson-type open problem, unproved in general. - Related to Sierpiński/Riesel numbers; the \"cover sets\" can make some produce no primes, but under admissibility infinitude is conjectured. - See \"Sierpiński number\", \"Riesel number\" on Wikipedia." }, { "id": 9400175, "problem_number": "AMR-093-0175", "title": "Is every Fermat number $2^{2^n} + 1$ composite for $n > 4$", "statement": "Is every Fermat number $2^{2^n} + 1$ composite for $n > 4$?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 175 (wikitext line 209)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; $F_5$–$F_{32}$ composite, general case unproved. Literature status: - **Open.** All Fermat numbers $F_5$ through $F_{32}$ are known to be composite, but it is unproved for all $n>4$. - See \"Fermat number\" on Wikipedia." }, { "id": 9400176, "problem_number": "AMR-093-0176", "title": "Is 509,203 the lowest Riesel number", "statement": "Is 509,203 the lowest Riesel number?", "background": "This explicit bullet from the pinned current Wikipedia number-theory section is staged as a candidate pending final semantic deduplication and primary-source status review.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Prime numbers, bullet 176 (wikitext line 210)\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; minimality unproved. Literature status: - **Open.** $509{,}203$ is the smallest *known* Riesel number; proving minimality (all smaller odd $k$ produce a prime) is open (RIESEL project searched many but not all). - See \"Riesel number\" on Wikipedia." }, { "id": 9400177, "problem_number": "AMR-093-0177", "title": "Pollock's octahedral-number conjecture", "statement": "Is every positive integer expressible as a sum of at most seven octahedral numbers?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Additive number theory, bullet 41; linked-page octahedral conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Frederick Pollock", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; holds up to large range, unproved. Literature status: - **Open.** Verified computationally to a large range but unproved. - See \"Pollock's conjecture\" on Wikipedia." }, { "id": 9400178, "problem_number": "AMR-093-0178", "title": "Global Gan–Gross–Prasad conjecture", "statement": "In the global Gan–Gross–Prasad setting, is nonvanishing of the relevant $H$-period on an irreducible cuspidal automorphic representation equivalent to nonvanishing of every corresponding local Hom space together with the central Rankin–Selberg value $L_E(1/2,\\pi_1\\times\\pi_2)$?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 49; linked-page global conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Wee Teck Gan, Benedict Gross, and Dipendra Prasad", "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress; many rank-1 cases settled, general global GGP open. Literature status: - **Partially resolved.** The local GGP conjecture is proved; many global rank-1 cases (e.g. for $\\mathrm{GL}(2)$/classical groups via Waldspurger-type results, and via the Ichino–Ikeda conjecture established in several cases) are known. - The full global equivalence in complete generality is **open** in some settings. - See \"Gan–Gross–Prasad conjecture\" on Wikipedia." }, { "id": 9400179, "problem_number": "AMR-093-0179", "title": "Greenberg's pseudo-null conjecture", "statement": "Let $F$ be totally real, let $\\widetilde F$ be the compositum of all $\\mathbb Z_p$-extensions of $F$, let $\\widetilde L$ be its maximal unramified abelian pro-$p$ extension, and set $\\widetilde X=\\operatorname{Gal}(\\widetilde L/\\widetilde F)$. Is $\\widetilde X$ pseudo-null over $\\widetilde\\Lambda=\\mathbb Z_p[[\\operatorname{Gal}(\\widetilde F/F)]]$?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 50; linked-page generalized conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Ralph Greenberg", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open; special cases proven. Literature status: - **Open in general.** Proven for certain abelian/cyclotomic special cases and for some families, but the general Greenberg pseudo-null conjecture is open. - See related Greenberg's conjectures on Iwasawa theory." }, { "id": 9400180, "problem_number": "AMR-093-0180", "title": "Iwasawa mu-invariant conjecture for number fields", "statement": "For every number field $K$ and every prime $\\ell$, does the Iwasawa invariant $\\mu_\\ell(K)$ vanish?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 50; linked-page related mu conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Ralph Greenberg", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in general; abelian cases $\\mu=0$ proven. Literature status: - For abelian (and $K=\\mathbb{Q}$ / entries where Iwasawa's $\\mu=0$ known), **Ferrero–Washington** proved $\\mu=0$ for cyclotomic extensions with abelian base; general totally-real/nonabelian is **open**. - The general conjecture (vanishing of $\\mu$ for all number fields) is **open**. - See \"Iwasawa theory\", Greenberg's conjecture on Wikipedia." }, { "id": 9400181, "problem_number": "AMR-093-0181", "title": "Greenberg's p-rationality conjecture", "statement": "For every odd prime $p$ and every positive integer $t$, does there exist a $p$-rational number field $K$ with $\\operatorname{Gal}(K/\\mathbb Q)\\cong(\\mathbb Z/2\\mathbb Z)^t$?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 50; linked-page p-rationality conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Ralph Greenberg", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Open in general; partial constructions known. Literature status: - **Open in general.** Existence of $p$-rational fields with prescribed Galois group is a known hard problem; partial constructions exist for several cases (e.g. for some groups and primes), but the general existence is open. - See Greenberg's conjecture on $p$-rational fields on Wikipedia." }, { "id": 9400182, "problem_number": "AMR-093-0182", "title": "Brumer–Stark conjecture", "statement": "For a finite abelian extension of number fields, does the Brumer–Stark element have the predicted ideal-class annihilation and principalization property, with anti-units whose roots generate the prescribed abelian Kummer extensions, including the remaining 2-primary cases?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Algebraic number theory, bullet 58; explicitly linked Brumer–Stark refinement\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Armand Brumer and Harold Stark", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress; many cases proven, full generality (incl. remaining $2$-primary) open. Literature status: - **Partially resolved.** The Brumer–Stark conjecture (and the refined/Strenger–Dasgupta version) is proved in many cases, notably via Dasgupta–Kakde et al. for totally real fields (Hilbert modular / the \"Brumer–Stark\" main results). - The fully general case, including all remaining $2$-primary/edge cases, may still be open. - See \"Brumer–Stark conjecture\" on Wikipedia." }, { "id": 9400183, "problem_number": "AMR-093-0183", "title": "Second Hardy–Littlewood zeta-function conjecture", "statement": "For every $\\varepsilon>0$, do constants $T_0(\\varepsilon),c(\\varepsilon)>0$ exist such that, for $T\\geq T_0$ and $H=T^{1/2+\\varepsilon}$, the number $N_0(T+H)-N_0(T)$ of odd-order zeros of $\\zeta(1/2+it)$ in $(T,T+H]$ is at least $cH$?", "background": "A terse family link in the pinned Wikipedia number-theory section was resolved against its linked statement page and decomposed where that page gives distinct conjectures.\n\nSource list: Number Theory problems from Wikipedia\nSource item: Analytic number theory, bullet 64; linked-page second conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767 plus linked-page resolution\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "G. H. Hardy and J. E. Littlewood", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Literature status: - **Open.** Intervals of length $T^{1/2+\\varepsilon}$ containing many zeros of $\\zeta$ are known, but unconditionally proving the presence of a positive proportion of odd-order (or simple) zeros in such short intervals is a major open problem; the best results (e.g. via the recent work on the proportion of simple zeros, Bui–Pratt–Radziwill for larger intervals) are not as strong as length $T^{1/2+\\epsilon}$ in the full range. - See \"Hardy–Littlewood conjectures\" on Wikipedia." }, { "id": 9500001, "problem_number": "AMR-094-0001", "title": "Probabilistic McMillan theorem in higher dimensions", "statement": "Let $X_t$ be $d$-dimensional Brownian motion starting at the origin, let $D$ be an open subset of $\\mathbb{R}^d$ containing the origin, and let $\\tau=\\inf\\{t>0:X_t\\notin D\\}$. Let $A$ be the set of cluster points of $$(X_t-X_\\tau)/|X_t-X_\\tau|$$ as $t \\uparrow \\tau$. For every $d>2$ and every such $D$, is $A$ almost surely either a sphere or a hemisphere?", "background": "The analogous planar result says that A is almost surely a circle or a semicircle.\n\nSource list: Burdzy - My favorite open problems (2009)\nSource item: Problem 1\nSource URL: https://sites.math.washington.edu/~burdzy/open_mathjax.php\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: The author-maintained page lists this item under 'My favorite open problems' as accessed on 2026-07-29; unlike Problem 10, it does not display a solution notice.\nRights note: NEEDS_REVIEW; publicly accessible author-maintained HTML page, no explicit reuse license located.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "- The problem is **open**; no resolution or published partial result beyond the $d=2$ theorem (Burdzy, Nagoya Math. J. 119 (1990), 115–132) was found. - New (if modest) rigorous contribution: $A$ is almost surely a nonempty compact **connected** subset of $S^{d-1}$ in every dimension and every domain (proof above), so any counterexample to the conjecture must produce a *continuum* of directions different from a sphere and a hemisphere. - Reformulation: the conjecture is equivalent to an angular-dichotomy statement for $h$-transformed (exit-conditioned) Brownian motion at Martin-boundary zeros of $h$. - Evidence the problem is genuinely hard/maybe false: the planar proof rests on conformal invariance and McMillan's twist-point theorem, both of which lack higher-dimensional analogues; higher-dimensional harmonic measure can be carried by non-tangent (Wolff-snowflake-type) boundary sets." }, { "id": 9500002, "problem_number": "AMR-094-0002", "title": "Topology of planar Brownian trace", "statement": "Let $X_t$ be two-dimensional Brownian motion. (i) For every pair $x,y \\notin X[0,1]$, is there a Jordan arc $\\Gamma$ containing $x$ and $y$ such that $\\Gamma\\cap X[0,1]$ is finite? (ii) If $\\{A_k\\}_{k\\ge1}$ are the connected components of $\\mathbb{R}^2\\setminus X[0,1]$ and $K=X[0,1]\\setminus\\bigcup_{k\\ge1}\\partial A_k$, is $K$ totally disconnected?", "background": "The source notes that a negative answer to part (ii), together with a soft argument, would imply a negative answer to part (i).\n\nSource list: Burdzy - My favorite open problems (2009)\nSource item: Problem 2 (parts i and ii)\nSource URL: https://sites.math.washington.edu/~burdzy/open_mathjax.php\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: The author-maintained page lists this item under 'My favorite open problems' as accessed on 2026-07-29; unlike Problem 10, it does not display a solution notice.\nRights note: NEEDS_REVIEW; publicly accessible author-maintained HTML page, no explicit reuse license located.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. Both parts (i) and (ii) — the existence of finite-intersection Jordan arcs through any two points off the trace, and the total disconnectedness of the union-of-boundaries remainder $K$ — are unproven as of the research date." }, { "id": 9500003, "problem_number": "AMR-094-0003", "title": "Percolation dimension of planar Brownian trace", "statement": "For a set $B$, define its percolation dimension as the infimum of the Hausdorff dimensions of Jordan arcs $A\\subset B$ containing at least two distinct points. If $X_t$ is two-dimensional Brownian motion, is the percolation dimension of $X[0,1]$ equal to $1$?", "background": "The source points to Burdzy (1990) for related background.\n\nSource list: Burdzy - My favorite open problems (2009)\nSource item: Problem 3\nSource URL: https://sites.math.washington.edu/~burdzy/open_mathjax.php\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: The author-maintained page lists this item under 'My favorite open problems' as accessed on 2026-07-29; unlike Problem 10, it does not display a solution notice.\nRights note: NEEDS_REVIEW; publicly accessible author-maintained HTML page, no explicit reuse license located.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. The percolation dimension of the two-dimensional Brownian trace is not known to be $1$; it is not established whether a Jordan arc of Hausdorff dimension $<1$ lies inside the trace." }, { "id": 9500004, "problem_number": "AMR-094-0004", "title": "Efficient couplings in acute triangles", "statement": "Let $D$ be a triangle whose angles are all strictly less than $\\pi/2$, and let $\\mu_2>0$ be the second eigenvalue of the Laplacian on $D$ with Neumann boundary conditions. Can one construct reflected Brownian motions $X_t,Y_t$ in $D$, starting from different points, such that $\\tau=\\inf\\{t\\ge0:X_t=Y_t\\}<\\infty$ almost surely and, for every fixed $\\varepsilon>0$, $$\\mathbb{P}(\\tau>t)\\le \\exp[-(\\mu_2-\\varepsilon)t]$$ for all sufficiently large $t$?", "background": "The first Neumann eigenvalue is zero; the source cites Burdzy and Kendall (2000) for background.\n\nSource list: Burdzy - My favorite open problems (2009)\nSource item: Problem 4\nSource URL: https://sites.math.washington.edu/~burdzy/open_mathjax.php\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: The author-maintained page lists this item under 'My favorite open problems' as accessed on 2026-07-29; unlike Problem 10, it does not display a solution notice.\nRights note: NEEDS_REVIEW; publicly accessible author-maintained HTML page, no explicit reuse license located.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. Constructing a coupling of reflected Brownian motions in an acute triangle whose meeting time has exponential tail with optimal rate $\\mu_2$ (the Neumann spectral gap) is still unresolved." }, { "id": 9500005, "problem_number": "AMR-094-0005", "title": "Convergence of synchronous reflected-Brownian couplings", "statement": "Let $D\\subset\\mathbb{R}^2$ be a connected open set with smooth boundary, and let $X,Y$ be synchronously coupled reflected Brownian motions in $D$ driven by the same planar Brownian motion. (i) Does there exist a bounded such domain for which $\\limsup_{t\\to\\infty}|X_t-Y_t|>0$ with positive probability? (ii) If $D$ is the complement of a nondegenerate closed disk, does this inequality hold with positive probability?", "background": "The source notes restrictions on any bounded example and cites work on synchronous couplings.\n\nSource list: Burdzy - My favorite open problems (2009)\nSource item: Problem 5 (parts i and ii)\nSource URL: https://sites.math.washington.edu/~burdzy/open_mathjax.php\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: The author-maintained page lists this item under 'My favorite open problems' as accessed on 2026-07-29; unlike Problem 10, it does not display a solution notice.\nRights note: NEEDS_REVIEW; publicly accessible author-maintained HTML page, no explicit reuse license located.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain open. Whether the synchronous coupling on a bounded smooth domain can stay apart asymptotically with positive probability (i), and specifically for the plane minus a disk (ii), is unresolved as of the research date." }, { "id": 9500006, "problem_number": "AMR-094-0006", "title": "Non-extinction of a Fleming–Viot particle model", "statement": "Let $N$ particles move as independent Brownian motions in a bounded connected open set $D\\subset\\mathbb{R}^d$. Whenever a particle hits the complement of $D$, replace it by a copy of a uniformly chosen surviving particle. If $\\tau_k$ is the time of the $k$th death-and-branching event and $\\tau_\\infty=\\lim_{k\\to\\infty}\\tau_k$, is $\\tau_\\infty=\\infty$ almost surely for every such $D$?", "background": "The source records partial results for Lipschitz domains and, in a 2024 update, for N=2.\n\nSource list: Burdzy - My favorite open problems (2009)\nSource item: Problem 6\nSource URL: https://sites.math.washington.edu/~burdzy/open_mathjax.php\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: The author-maintained page lists this item under 'My favorite open problems' as accessed on 2026-07-29; unlike Problem 10, it does not display a solution notice.\nRights note: NEEDS_REVIEW; publicly accessible author-maintained HTML page, no explicit reuse license located.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress in the literature. Non-extinction is established in important special cases (Lipschitz domains; the $N=2$ case per the 2024 update), but the general statement for all bounded connected open $D$ and all $N$ is not fully settled." }, { "id": 9500007, "problem_number": "AMR-094-0007", "title": "Are shy couplings necessarily rigid?", "statement": "Let $D\\subset\\mathbb{R}^d$, $d\\ge2$, be bounded, connected, and open. Suppose there are coupled reflected Brownian motions $X_t,Y_t$ in $D$ and $\\varepsilon>0$ such that $\\inf_{t\\ge0}|X_t-Y_t|\\ge\\varepsilon$ with positive probability. Must there also exist coupled reflected Brownian motions $X'_t,Y'_t$, a positive $\\varepsilon$, and a deterministic function $f$ such that $f(X'_t)=Y'_t$ for every $t\\ge0$ almost surely and $\\inf_{t\\ge0}|X'_t-Y'_t|\\ge\\varepsilon$ with positive probability?", "background": "The source calls couplings that stay a positive distance apart shy and discusses a graph counterexample to a related stronger intuition.\n\nSource list: Burdzy - My favorite open problems (2009)\nSource item: Problem 7\nSource URL: https://sites.math.washington.edu/~burdzy/open_mathjax.php\nAccessed: 2026-07-29\nExtraction: html\nStatus evidence: The author-maintained page lists this item under 'My favorite open problems' as accessed on 2026-07-29; unlike Problem 10, it does not display a solution notice.\nRights note: NEEDS_REVIEW; publicly accessible author-maintained HTML page, no explicit reuse license located.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Krzysztof Burdzy and Wilfrid Kendall", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress; the implication remains open. The existence of shy couplings (particles kept weakly apart) is well documented, and rigid (deterministic-function) couplings are studied, but it is not known that every shy coupling forces a rigid one, nor has a counterexample been established in the reflected-Brownian setting." }, { "id": 9500008, "problem_number": "AMR-094-0008", "title": "Concatenated bounded Brownian pieces", "statement": "For each $k\\in\\mathbb{Z}$, let $B^k$ be Brownian motion and $T_k$ a stopping time, with the stopped pieces independent, $0\\le T_k<\\infty$, and with their total lengths diverging in both time directions. Form a continuous process $X$ by concatenating the pieces $B^k_{[0,T_k)}$. If there is a deterministic $c<\\infty$ such that $T_k0$, and $\\sum_{x<0}x^2p(x)=\\infty$. Does there exist a stationary distribution $\\mu$ with $\\mu(\\eta(x)=1)\\to0$ as $x\\to-\\infty$ and $\\mu(\\eta(x)=1)\\to1$ as $x\\to+\\infty$?", "background": "The final October 2012 author version contains four numbered problems; an earlier draft contained only three.\n\nSource list: Liggett - Some Open Problems (2012)\nSource item: Problem 1, PDF page 1\nSource URL: https://web.archive.org/web/20150919235903id_/http://www.math.ucla.edu/~tml/open.pdf\nAccessed: 2026-07-29\nExtraction: archived-author-pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Thomas M. Liggett", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress. The existence of the described stationary distribution for asymmetric long-range exclusion on $\\mathbb{Z}$ under these precise moment hypotheses is not established in the literature I could verify (it remains a genuine open problem); special nearest-neighbor and symmetric/small-drift cases are understood." }, { "id": 9600002, "problem_number": "AMR-095-0002", "title": "Stationary distributions in higher dimensions", "statement": "On $\\mathbb{Z}^2$, take nearest-neighbor jump probabilities $p_1,q_1,p_2,q_2$ in directions $\\pm e_1,\\pm e_2$, with $p_1>q_1$ and $p_2>q_2$. If the angle between $v$ and the mean vector $(p_1-q_1,p_2-q_2)$ is less than $\\pi/2$, prove that there is a stationary distribution invariant under shifts orthogonal to $v$, whose occupation density along $nv$ tends to $0$ as $n\\to-\\infty$ and to $1$ as $n\\to+\\infty$.", "background": "The final October 2012 author version contains four numbered problems; an earlier draft contained only three.\n\nSource list: Liggett - Some Open Problems (2012)\nSource item: Problem 2, PDF page 2\nSource URL: https://web.archive.org/web/20150919235903id_/http://www.math.ucla.edu/~tml/open.pdf\nAccessed: 2026-07-29\nExtraction: archived-author-pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Thomas M. Liggett", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open: the directional stationary-distribution statement for 2D nearest-neighbor asymmetric exclusion is not established in the literature as far as I could verify. Literature status: - Open. Liggett's multidimensional analogue of the 1D shock-stationary-measure problem is unresolved in the literature I can verify. The nearest-neighbor 2D asymmetric exclusion stationary shock measures along general directions have been studied (there is a body of work on \"shock measures\"/\"stationary blocking measures\" in higher-dimensional ASEP, e.g. related to the \"multi-shock\"/\"KPZ shock\" constructions and to the \"phase transition in exclusion on $\\mathbb{Z}^d$\"), but the precise directional-stationary-measure statement is not settled." }, { "id": 9600003, "problem_number": "AMR-095-0003", "title": "Exchangeability in the mean-zero exclusion process", "statement": "For the exclusion process on $\\mathbb{Z}^d$ with translation-invariant kernel $p(x,y)=p(y-x)$ and zero mean $\\sum_xxp(x)=0$, prove that every stationary measure is exchangeable.", "background": "The final October 2012 author version contains four numbered problems; an earlier draft contained only three.\n\nSource list: Liggett - Some Open Problems (2012)\nSource item: Problem 3, PDF page 2\nSource URL: https://web.archive.org/web/20150919235903id_/http://www.math.ucla.edu/~tml/open.pdf\nAccessed: 2026-07-29\nExtraction: archived-author-pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Thomas M. Liggett", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: exchangeability of stationary measures is established for symmetric/nearest-neighbor and restricted cases; the general translation-invariant mean-zero (possibly asymmetric, possibly long-range) exclusion statement remains open as posed." }, { "id": 9600004, "problem_number": "AMR-095-0004", "title": "Negative association for asymmetric exclusion", "statement": "For nearest-neighbor asymmetric exclusion on $\\mathbb{Z}$ with $p(1)=p>q=p(-1)$, start from the deterministic configuration $\\cdots11110000\\cdots$. Is the distribution at every time negatively associated; that is, does $\\int fg\\,d\\mu\\le(\\int f\\,d\\mu)(\\int g\\,d\\mu)$ hold for increasing $f,g$ depending on disjoint coordinate sets?", "background": "The final October 2012 author version contains four numbered problems; an earlier draft contained only three.\n\nSource list: Liggett - Some Open Problems (2012)\nSource item: Problem 4, PDF page 2\nSource URL: https://web.archive.org/web/20150919235903id_/http://www.math.ucla.edu/~tml/open.pdf\nAccessed: 2026-07-29\nExtraction: archived-author-pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Thomas M. Liggett", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: NA is preserved for many symmetric/reversible exclusion systems, but for asymmetric dynamics the general preservation fails in some settings; the specific claim for blocked-shock ASEP at all times is not established." }, { "id": 9700001, "problem_number": "AMR-096-0001", "title": "Martingale for practical purposes", "statement": "Give a mathematically useful definition of a process being a 'martingale for practical purposes', so that failure means it is practical to find a stopping time $T$ with $\\mathbb{E}X_T\\ne\\mathbb{E}X_0$. For a discrete process $X_0,\\ldots,X_n$, can a natural polynomial-size collection of stopping-time constraints define a tractable such class?", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: fields.html\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open as posed. It is a definitional problem with no currently accepted solution in the literature; neighboring relaxations (quasi-martingales, $\\varepsilon$-optional-stopping criteria in mathematical finance) address related but distinct questions." }, { "id": 9700002, "problem_number": "AMR-096-0002", "title": "Analytic toy model for a percolation-fragmentation congestion transition", "statement": "Find a simple network-and-demand toy model in which the marginal satisfiability proportion $r(t)$ can be calculated analytically and exhibits the proposed percolation-fragmentation phase transition under optimization or adaptive capacity growth.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: congestion.html\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No analytic toy model with the requested property was found in the literature; the problem remains as posed. Literature status: - This is Aldous's own proposal; it is a model-construction problem. The motivating discussion appears in Aldous's writing on \"congestion\" and on percolation in large random spatial networks. - I found no published toy model that computes the marginal satisfiability proportion $r(t)$ analytically and exhibits the proposed percolation-fragmentation transition. arXiv searches on \"congestion percolation fragmentation transition network\" return only empirical traffic studies (e.g., percolation analysis of Seoul road traffic), not analytic toy models. - Related analytic work exists on percolation and fragmentation transitions in random graphs (e.g., critical percolation, explosive percolation, random graph fragmentation) but not in the specific satisfiability/capacity form posed here." }, { "id": 9700003, "problem_number": "AMR-096-0003", "title": "Universal compression of sparse labeled graphs", "statement": "For sparse $n$-vertex graphs of average degree $O(1)$ whose vertices have distinct $O(\\log n)$-length labels over a finite alphabet, construct universal codes analogous to Lempel–Ziv that achieve asymptotically optimal compression.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: graph_compression.html\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Partial progress in the literature (entropy bounds and model-dependent optimal universal codes for ER/PA/SBM sparse graphs), but the specific problem — a Lempel–Ziv-type universal code achieving optimal compression for sparse labeled graphs — remains open as posed." }, { "id": 9700004, "problem_number": "AMR-096-0004", "title": "Mixing times for coagulation-fragmentation processes", "statement": "Obtain relaxation- and mixing-time bounds for reversible coagulation-fragmentation Markov chains on finite sets in terms of their model parameters.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: coag_frag.html\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No literature gives parameter-dependent relaxation/mixing-time bounds for reversible coagulation-fragmentation chains in general. Literature status: - The coagulation-fragmentation literature is large (Smoluchowski equations, Marcus–Lushnikov processes, stochastic coalescents), but mixing/relaxation times of the associated reversible finite-state Markov chains are only studied in special cases. - Verified relevant works found by arXiv search: - Aldous, \"Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists\" (Bernoulli 1999) — background, not mixing times. - \"Double Coset Markov Chains\" (arXiv 2021) — related to Markov chains on partitions with coagulation-like moves, but not the Aldous question. - I found no systematic bounds on relaxation/mixing times of reversible coagulation-fragmentation chains in terms of model parameters; the question appears open as posed." }, { "id": 9700005, "problem_number": "AMR-096-0005", "title": "Low-density lineage limit of coalescing branching random walk", "statement": "For the two stationary branching-coalescing models on $\\mathbb{Z}^3$ described by Aldous, prove that as particle intensity tends to zero the suitably time-rescaled lineage process, ignoring spatial positions and short parent-daughter jitter, converges in finite-dimensional distributions to the stated Poisson split-and-merge lineage process.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: cbrw.html, Problem A\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The conjectured low-density lineage limit (Poisson split-and-merge process) for the stationary branching-coalescing models on $\\mathbb{Z}^3$ remains unproved. Literature status: - The paper behind this page is Aldous, \"Branching and coalescing particle systems\" (probably the paper with K. Burdzy / or Aldous's \"cbrw\" write-up, circa 1999). The conjecture is that the low-density limit of lineages is a Poisson split-and-merge process (a continuous-time process on a random finite lineage structure with split and merge events at rate proportional to density). - Verified related literature via arXiv: \"Ancestral lineages for a branching annihilating random walk\" (arXiv), \"Pair coalescence times of ancestral lineages of two-dimensional logistic branching random walks\" (arXiv 2024), \"Quenched CLT for ancestral lineages of logistic branching random walks\" (arXiv 2024), \"Coalescing directed random walks ... converge to the Brownian web\" — these concern lineages in various branching/coalescing systems but…" }, { "id": 9700006, "problem_number": "AMR-096-0006", "title": "Constrained Ising storage model on a time-varying graph", "statement": "Study the constrained Ising storage model described on the page when the underlying graph itself changes in time.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: cbrw.html, Problem B\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The time-varying-graph version of the constrained Ising storage model appears unstudied. Literature status: - The page's Problem B concerns a \"constrained Ising storage model\" (a hard-core/constrained Ising-type model on a graph with a storage interpretation, related to Aldous's work on load balancing / hard-core model). The stated subproblem is to study the model on a time-varying graph. - I found no literature specifically studying Aldous's constrained Ising storage model on time-varying graphs. Time-varying/hard-core models on dynamic graphs exist in statistical physics and network science, but not addressing this model." }, { "id": 9700007, "problem_number": "AMR-096-0007", "title": "Constant-factor online scheduling of subadditive batches", "statement": "Tasks arrive as a rate-one Poisson process and have types in $[0,1]$; batch processing time $S$ is monotone and strictly subadditive and type $a$ incurs waiting cost rate $c(a)$. Is there an explicit online algorithm and universal constant $C$ whose long-run average cost $W$ satisfies $W/W_*0$.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: epidemic.html; linked talk slide 24\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open in the general form. Spectral-radius-based threshold results cover many families (complete graphs, Erdős–Rényi, power-law, dynamic switching), but the sharp threshold with vanishing $\\varepsilon$ and nonvanishing infected proportion for general weighted networks appears unproved." }, { "id": 9700011, "problem_number": "AMR-096-0011", "title": "Shortest routes in random proximity networks", "statement": "For random proximity graphs on a planar Poisson point process, determine rigorous orders of magnitude for the transversal deviation $T_r$ of a shortest network route between points at Euclidean distance $r$ and for the variance of its route length $D_r$.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: random_proximity.html\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Rigorous orders of magnitude for transversal deviation and route-length variance in planar Poisson proximity graphs appear unproved. Literature status: - This is from Aldous's work on \"proximity graphs and shortest routes\" (related to his paper \"Scale-invariant random spatial networks\" and \"Shortest routes through networks\"). The question concerns how far a shortest path in a proximity graph (e.g., Delaunay, relative neighborhood, Gabriel graph) deviates transversally from the straight segment, and the variance of route length. - I found no published rigorous orders of magnitude for $T_r$ and $\\mathrm{Var}(D_r)$ in random planar proximity graphs. - Related literature: percolation-based bounds for shortest paths in random geometric graphs (e.g., work on \"shortest path in random geometric graphs\" by Bhamidi–van der Hofstad and others) exists but for different models/metrics." }, { "id": 9700012, "problem_number": "AMR-096-0012", "title": "Mixing of branch rotation and triangulation chains", "statement": "For both the diagonal-flip chain on triangulations of the regular $n$-gon and the branch-rotation chain on $n$-cladograms, prove that the relaxation time is $O(n^{3/2})$, matching the known lower bound.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: clad-mix.html; linked 2003 note\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Partial progress: the conjecture (relaxation time $O(n^{3/2})$ for diagonal-flip and branch-rotation chains) remains open; significant progress exists on upper bounds, but the matching $O(n^{3/2})$ is not yet established in the literature I could verify." }, { "id": 9700013, "problem_number": "AMR-096-0013", "title": "Random Eulerian excursion dichotomy on high-dimensional tori", "statement": "On the bidirected torus $\\mathbb{Z}_N^d$ with fixed $d\\ge3$, let $b^{(N)},t^{(N)},m^{(N)}$ count excursions of a uniform Eulerian circuit longer than $N^d/\\omega_N$, shorter than $\\omega_N$, and between those scales. For sufficiently slowly growing $\\omega_N$, does $(b^{(N)},t^{(N)},m^{(N)})$ converge in distribution to $(S^*,2d-S^*,0)$ for some $S^*$ supported on $\\{1,\\ldots,2d\\}$?", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: Random Eulerian Circuits, Conjecture 0.1\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The dichotomy conjecture (long vs. short excursions with a random number $S^*$ of long ones, no intermediate scale) remains unproved. Literature status: - This is from Aldous's paper \"Random Eulerian Circuits and the Structure of Stationary Random Graphs\" / the \"Random Eulerian circuits\" page (Conjecture 0.1), conjecturing a dichotomy: excursions are either \"long\" (comparable to the whole graph) or \"short\" (polylog), with a random number $S^*$ of long excursions. - Verified literature: \"Markov loops, complex free field and Eulerian circuits\" (Le Jan 2014, arXiv:1405.2879) studies random Eulerian circuits via Markov loops and the complex free field — it proves related structure results (loop ensembles) but I did not verify it settles Conjecture 0.1. - I found no published proof of the excursion dichotomy conjecture for $d\\ge3$ tori." }, { "id": 9700014, "problem_number": "AMR-096-0014", "title": "Second-longest Eulerian excursion on the two-dimensional torus", "statement": "For a uniform Eulerian circuit on the bidirected two-dimensional torus, does $\\log L_2^{(N)}/\\log N$ converge in distribution to a random variable with support $[0,2]$?", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: Random Eulerian Circuits, equation (4)\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The distributional limit for the second-longest excursion on the 2D torus remains unproved. Literature status: - This is from Aldous's \"Random Eulerian Circuits\" open problems (equation (4)); the two-dimensional case is conjecturally governed by the Brownian motion/loop structure of the circuit, with a random scaling exponent. - I found no published resolution. The related Le Jan paper (arXiv:1405.2879, \"Markov loops, complex free field and Eulerian circuits\") does not address the second-longest excursion exponent on the 2D torus. - No web or arXiv result resolves the distributional limit of $\\log L_2^{(N)}/\\log N$." }, { "id": 9700015, "problem_number": "AMR-096-0015", "title": "Excursion counts in a random Eulerian circuit on a complete graph", "statement": "On the bidirected complete $n$-vertex graph, is the expected number of length-$i$ excursions in a uniform Eulerian circuit asymptotic to $e^{-i/n}$?", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: Random Eulerian Circuits, example (a)\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The exponential asymptotics $e^{-i/n}$ for excursion counts is unproved. Literature status: - The context: in a uniform Eulerian circuit of the bidirected complete graph $K_n$, excursions (subpaths between successive visits to a distinguished root) have lengths with exponential-like distribution; the conjecture is $\\mathbb{E}[\\#\\text{excursions of length } i] \\sim e^{-i/n}$. - I found no published proof or disproof of this asymptotic. The random-Eulerian-circuit literature (Le Jan arXiv:1405.2879; also early work of McKay–Robinson on Eulerian circuits of complete graphs) does not address excursion length counts as posed." }, { "id": 9700016, "problem_number": "AMR-096-0016", "title": "Shortest Eulerian excursion on the Hamming cube", "statement": "For a uniform Eulerian circuit on the bidirected Hamming cube $\\{0,1\\}^d$, determine the asymptotic behavior or distribution of the shortest excursion length $L_d^{(d)}$.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: Random Eulerian Circuits, example (b)\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The asymptotic behavior/distribution of the shortest excursion on the hypercube remains undetermined. Literature status: - The notation in the source (example (b)) refers to the shortest excursion in a uniform Eulerian circuit on the bidirected hypercube $\\{0,1\\}^d$; the question asks for its asymptotic behavior or distribution as $d\\to\\infty$. - I found no published result on shortest excursion lengths in random Eulerian circuits on the hypercube (or on any family) beyond Aldous's open-problem discussion. - Searches (arXiv/web) returned nothing resolving this." }, { "id": 9700017, "problem_number": "AMR-096-0017", "title": "Eulerian-circuit continuum limits and SLE", "statement": "Is there a relation between space-filling $\\operatorname{SLE}_\\kappa$ for $\\kappa>8$ and the conjectural continuum limit of uniform Eulerian circuits on $\\mathbb{Z}_N^2$?", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: Random Eulerian Circuits, final question\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The continuum limit of uniform Eulerian circuits on $\\mathbb{Z}_N^2$ is still conjectural, so the SLE relation question is unresolved; space-filling SLE itself is well understood. Literature status: - Space-filling SLE$_\\kappa$ for $\\kappa\\ge8$ is well developed (Lawler–Schramm–Werner; Miller–Sheffield on space-filling SLE and imaginary geometry; SLE$_{16}$ is the Peano curve of the uniform spanning tree (Lawler–Schramm–Werner 2004); SLE$_8$ is the UST Peano curve, both proven). - However, the conjectural continuum limit of *uniform Eulerian circuits* on $\\mathbb{Z}_N^2$ is itself not established (the excursion structure is open — see AMR-096-0013/0014); consequently the specific question of a relation to space-filling SLE$_\\kappa$, $\\kappa>8$, remains open. - I found no paper establishing the Eulerian-circuit scaling limit or its relation to SLE. The Le Jan circle (arXiv:1405.2879) connects random Eulerian circuits to Markov loops/free fields but not to SLE limits." }, { "id": 9700018, "problem_number": "AMR-096-0018", "title": "Stretch-length exponent in spatial networks", "statement": "Improve the explicit upper and lower bounds for the minimum network length functions $\\Psi^{ave}(s)$ and $\\Psi^{worst}(s)$, and prove whether there is an exponent $\\alpha$ such that each satisfies $\\Psi(s)\\asymp(s-1)^{-\\alpha}$ as $s\\downarrow1$.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: stretch.html\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The exponent $\\alpha$ for $\\Psi^{ave}(s)$ and $\\Psi^{worst}(s)$ as $s\\downarrow1$ is unknown. Literature status: - This is from Aldous's \"stretch.html\" page (related to his paper with J. M. Steele / the \"optimal route networks\" problem). Explicit bounds exist in the page/paper; the exact exponent is open. - I found no published improvement or proof of an exponent $\\alpha$ for $\\Psi^{ave}$ or $\\Psi^{worst}$. - Related literature: Euclidean Steiner tree / traveling salesman network length bounds, and \"bounded stretch\" network design, but not this exact exponent question." }, { "id": 9700019, "problem_number": "AMR-096-0019", "title": "Largest common subcladogram exponents", "statement": "For two independent random $n$-cladograms, under both the uniform and coalescent distributions, prove $\\mathbb{E}C_n=n^{\\gamma+o(1)}$ for respective constants $\\gamma_a,\\gamma_b<1/2$, and identify or characterize those exponents.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: substructures.html; linked 2003 document, Example 3\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Partial progress: the problem is actively studied (largest common subtree of random trees, related to MAST), with bounds and heuristics for the exponents, but exact values of $\\gamma_a$ (uniform) and $\\gamma_b$ (coalescent) are not rigorously established. I could not verify a full solution." }, { "id": 9700020, "problem_number": "AMR-096-0020", "title": "Largest common suborder of two random two-dimensional orders", "statement": "For two independent coordinatewise partial orders generated by uniform points in the unit square, prove $\\mathbb{E}C_n\\sim c n^{1/3}$ and establish the existence or value of $c\\in(0,\\infty)$.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: substructures.html; linked 2003 document, Example 4\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The $n^{1/3}$ asymptotic with an explicit constant $c$ remains unproved; the existence of the constant itself is not established. Literature status: - This is closely related to the longest increasing subsequence (LIS) problem: a common suborder of two independent random 2D orders corresponds to a longest common subsequence-type structure. The $n^{1/3}$ scaling is the known LCS/LIS-with-noise exponent for 2D orders (related to the \"longest common subsequence of two random permutations\" which scales like $c\\sqrt{n}$, and to the \"LIS in a random permutation\" $2\\sqrt{n}$). - For *coordinatewise 2D orders from uniform points*, the size of the largest common suborder is essentially the \"longest common increasing subsequence\"-type quantity; the $n^{1/3}$ conjecture with unknown constant $c$ is a known hard problem (analogous to the \"common subsequence of two random words\" problem with exponent $2\\sqrt{\\log n}$ and unknown constants). - I found no published proof of $\\mathbb{E}C_n\\sim c n^{1/3}$ or a…" }, { "id": 9700021, "problem_number": "AMR-096-0021", "title": "Percolation criteria for merging planar empires", "statement": "For continuous-time processes that merge adjacent polygonal planar regions $A,B$ at a geometry-dependent rate $r(A,B)$, give sufficient conditions on $r$ for percolation and sufficient conditions for non-percolation.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: empires.html, first Problem\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No criteria for percolation/non-percolation in this model were found in the literature. Literature status: - This is from Aldous's \"empires\" page: a tessellation of the plane into polygonal regions (\"empires\") merges adjacent regions at rate $r(A,B)$ (depending on geometry, e.g., common boundary length); the question is when an infinite connected merged region appears (percolation). - I found no published sufficient conditions on $r$ for percolation/non-percolation for this specific model. It is a novel stochastic geometry model; searches (empires percolation geometry) returned nothing matching." }, { "id": 9700022, "problem_number": "AMR-096-0022", "title": "Percolation of planar empires at unit merger rate", "statement": "When every adjacent pair of planar empires merges at rate $r(A,B)=1$, does percolation occur?", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: empires.html, second Problem\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Whether percolation occurs at $r(A,B)\\equiv1$ is undetermined in the literature. Literature status: - This is the special constant-rate case of the empires model. When every adjacent pair merges at rate 1, the dynamics is equivalent (after a change of time) to a process where each adjacency has an independent exponential clock; the question is whether an infinite cluster forms. - I found no published resolution. The model appears to be studied only on Aldous's page. - Related but distinct: \"random sequential adsorption\"/coalescence of Voronoi cells, and the \"coalescing tilings\" literature; none answers the constant-rate percolation question." }, { "id": 9700023, "problem_number": "AMR-096-0023", "title": "Unbalanced regimes of the spatial city-growth model", "statement": "For the city-growth model, prove: (a) if $\\alpha>1$, the eventual number of cities $M(\\infty)$ is finite almost surely; (b) if $\\beta<2\\alpha$, the largest city satisfies $N_{(1)}(t)/t\\to1$ almost surely; and (c) if both hold, the eventual population outside the largest city $N_{(\\ge2)}(\\infty)$ is finite almost surely.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: cities.html; technical notes, Conjecture 1\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: the model is published and partially analyzed (Aldous–Huang 2014), but conjectures (a)–(c) on the unbalanced regimes remain unproved in the literature. Literature status: - This is from Aldous's spatial city-growth model (Aldous, \"A spatial model of city growth and formation\" with Bowen Huang; arXiv:1209.5120 — verified via arXiv API: title \"A Spatial Model of City Growth and Formation\", authors David Aldous and Bowen Huang). - The model's conjectures are stated in the technical notes; the paper proves several results (existence, stationarity, and some regime behavior) but the specific conjectures (a)–(c) about $M(\\infty)$, $N_{(1)}(t)/t\\to1$, and $N_{(\\ge2)}(\\infty)$ in the unbalanced regimes are not fully proved in the paper as far as I could verify. - I found no later publication resolving these conjectures." }, { "id": 9700024, "problem_number": "AMR-096-0024", "title": "Growth exponents in the balanced city-growth regime", "statement": "In the balanced regime $0<\\alpha<1$ and $\\beta>2\\alpha$, prove that the upper and lower growth exponents for influence and city population all equal $\\beta/(2-2\\alpha+\\beta)$ and those for nearest-city distance equal $(\\alpha-1)/(2-2\\alpha+\\beta)$.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: cities.html; technical notes, Section 5\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The exact growth exponents in the balanced regime are conjectured but unproved. Literature status: - This is the growth-exponent conjecture for the balanced regime of the Aldous–Huang spatial city model (arXiv:1209.5120). - The Aldous–Huang paper proves existence of exponents (via subadditivity-type arguments) but does not identify their values; the exact exponents are conjectural. - I found no published proof of the exponent values." }, { "id": 9700025, "problem_number": "AMR-096-0025", "title": "Largest-city growth at alpha=1", "statement": "If $\\alpha=1$ and $\\beta>2$, prove $N_{(1)}(t)=t(\\log t)^{1-2/\\beta+o(1)}$ almost surely.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: cities.html; technical notes, Conjecture 22\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The a.s. asymptotic $N_{(1)}(t)=t(\\log t)^{1-2/\\beta+o(1)}$ at $\\alpha=1$ is unproved. Literature status: - This is the boundary-case ($\\alpha=1$) conjecture for the largest city population in the Aldous–Huang model; the $\\log$-correction exponent $1-2/\\beta$ is conjectural. - I found no published proof. The Aldous–Huang paper (arXiv:1209.5120) does not settle the $\\alpha=1$ boundary. - No follow-up literature resolving this was found." }, { "id": 9700026, "problem_number": "AMR-096-0026", "title": "Stability dichotomy for the associated city dynamical system", "statement": "For the associated influence-cell dynamical system in general position with positive initial weights, prove that one weight tends to $1$ if $\\alpha>1$ or $0<\\beta<2\\alpha$, while if $0<\\alpha<1$ and $\\beta>2\\alpha$ all weights converge to positive limits independent of the initial values.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: cities.html; technical notes, Conjecture 32\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The stability dichotomy for the influence-cell dynamical system is unproved. Literature status: - This is the \"influence dynamics\" stability conjecture for the Aldous–Huang city model: the deterministic dynamical system on weights of influence cells is conjectured to either collapse to a single weight 1 (unbalanced regimes) or converge to initial-value-independent positive limits (balanced regime). - I found no published proof. The Aldous–Huang paper (arXiv:1209.5120) discusses the dynamical system but does not prove the dichotomy. - No follow-up resolution found." }, { "id": 9700027, "problem_number": "AMR-096-0027", "title": "A mathematically natural SIRSN", "statement": "Construct a scale-invariant random spatial network whose law is mathematically natural, for example with an explicit formula for the distribution of $\\operatorname{span}(z_1,\\ldots,z_k)$.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: sirsn.html; linked paper, Open Problem 27\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: the Aldous–Kendall improper Poisson line process provides a mathematically natural SIRSN (the leading candidate solution to Open Problem 27), but an explicit formula for the distribution of $\\mathrm{span}(z_1,\\ldots,z_k)$ is not available; the problem as literally posed (explicit span distribution) remains open." }, { "id": 9700028, "problem_number": "AMR-096-0028", "title": "A visually realistic SIRSN", "statement": "Construct a scale-invariant random spatial network that is visually realistic, in the sense of not looking very different from a real-world road network.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: sirsn.html; linked paper, Open Problem 28\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: the Poisson-line-process constructions of Kendall (2017) and Kahn (2016), refined by Kendall (2020) and Blanc–Curien–Kahn (2024/2025), provide concrete SIRSNs whose large-scale metric structure is now well understood. However, no construction is documented as visually realistic in Aldous's sense (matching real-world road networks), and the qualitative goal remains open." }, { "id": 9700029, "problem_number": "AMR-096-0029", "title": "Feasible statistic triples for SIRSNs", "statement": "Determine the set of possible triples $(\\Delta=\\mathbb{E}D_1,\\ell,p(1))$ over all scale-invariant random spatial networks.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: sirsn.html; linked paper, Open Problem 29\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open. The Poisson-line family gives examples (and Kendall/Kahn papers verify some individual statistics), but the feasible region of the triple $(\\Delta,\\ell,p(1))$ over all SIRSNs is not characterized in the literature." }, { "id": 9700030, "problem_number": "AMR-096-0030", "title": "Optimal length-route tradeoff for SIRSNs", "statement": "Give quantitative estimates improving the known bound on $\\ell^*(\\Delta)$, the infimum edge intensity among SIRSNs with mean unit-distance route length $\\Delta$. Do minimizers exist, and what structure do optimal networks have?", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: sirsn.html; linked paper, Open Problem 30\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open. The qualitative bound $\\ell^*(\\Delta)\\asymp\\Delta^{-2}$ stands; the sharp constant, existence of minimizers, and the structure of optimal networks are unresolved. Literature status: - The baseline lower/upper bounds are in Aldous's 2011 EJP paper (Section 8, open problems). The known bound is of order $\\ell^*(\\Delta) \\sim c\\,\\Delta^{-2}$ with unspecified constant. - Kendall (2017) and Kahn (2016) construct Poisson-line SIRSNs (Ann. Appl. Probab.; see problem AMR-096-0028), which yield upper bounds in this class but do not determine the optimal constant or prove existence of minimizers. - I found no 2024–2026 work computing $\\ell^*(\\Delta)$ exactly, proving existence of minimizers, or characterizing optimal SIRSN structure. The related continuous-transportation / optimal-transport literature (e.g., branching transport) addresses similar functionals but not the SIRSN axioms." }, { "id": 9700031, "problem_number": "AMR-096-0031", "title": "Local finiteness of SIRSN traffic intensity", "statement": "Show, perhaps under regularity hypotheses on a SIRSN, that for $2<\\beta<4$ the paper's source-destination measure with displacement density $|z|^{-\\beta}$ induces a locally finite traffic-intensity measure on $E(\\infty,1)$ and hence on $\\bigcup_rE(\\infty,r)$.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: sirsn.html; linked paper, Open Problem 31\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open as posed for general SIRSNs. The Poisson-roads model is the best-understood case, and its analysis (Blanc–Curien–Kahn 2024/2025) resolves closely related geodesic/traffic questions, but the stated local-finiteness result is not established in the literature I could verify." }, { "id": 9700032, "problem_number": "AMR-096-0032", "title": "Converse implications among SIRSN properties", "statement": "Prove or disprove each of the proposed implications between the SIRSN properties numbered (16), (20), (49), (50), and (51): (16)$\\Rightarrow$(20), unique singly-infinite geodesics$\\Rightarrow$(49), (49)$\\Rightarrow$(50), and (51)$\\Rightarrow$(50).", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: sirsn.html; linked paper, Open Problem 32\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. For the Poisson-line/roads SIRSN (the main concrete model), Blanc–Curien–Kahn (PLMS 2025, arXiv:2407.07887) prove geodesic uniqueness, non-pausing, the geodesic-frame characterization, and confluence, which settles the \"unique singly-infinite geodesics $\\Rightarrow$ (49)\" direction (and related property (16)/(20) interplay) in that model. The remaining implications for the abstract SIRSN class are still open." }, { "id": 9700033, "problem_number": "AMR-096-0033", "title": "Unbounded component uniqueness in a SIRSN", "statement": "Does the major-road subnetwork $E(\\infty,1)$ of a SIRSN almost surely have exactly one unbounded connected component?", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: sirsn.html; linked paper, Open Problem 33\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open. Whether a SIRSN's top-scale road subnetwork $E(\\infty,1)$ has exactly one unbounded connected component is not settled in the literature I could verify. Literature status: - The problem is posed as open in Aldous 2011 (Section 8); no resolution appears in the source page's notes. - The Poisson-line/roads model (Kendall 2017, Kahn 2016, Ann. Appl. Probab.; Blanc–Curien–Kahn arXiv:2407.07887) analyzes geodesics and confluence, relevant to uniqueness of infinite structures, but I found no explicit statement proving or disproving unique unbounded component for $E(\\infty,1)$ in the general SIRSN class. - Web searches (2024–2026) found no direct resolution." }, { "id": 9700034, "problem_number": "AMR-096-0034", "title": "Integrability of all routes to random points in a SIRSN", "statement": "Under what additional assumptions, if any, is $\\mathbb{E}\\sup_{i\\ge1}\\operatorname{len}[R(0,U_i)]<\\infty$ for independent uniform points $U_i$ in the unit disc?", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: sirsn.html; linked paper, Open Problem 34\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open. The needed sufficient conditions (and whether natural moment assumptions on the network suffice) for $\\mathbb{E}\\sup_i\\operatorname{len}[R(0,U_i)]<\\infty$ are not established in the literature." }, { "id": 9700035, "problem_number": "AMR-096-0035", "title": "Expected length of a SIRSN spanning subnetwork", "statement": "For $k$ uniform random points $Z_1,\\ldots,Z_k$ in a square of area $k$, prove $\\mathbb{E}\\operatorname{len}[\\operatorname{span}(Z_1,\\ldots,Z_k)]\\sim\\ell k$.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: sirsn.html; linked paper, Open Problem 35\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open. The conjectured asymptotic $\\mathbb{E}\\operatorname{len}[\\operatorname{span}(Z_1,\\ldots,Z_k)]\\sim\\ell k$ is not established. Literature status: - The problem is posed as open in Aldous 2011 (Section 8); the linear-growth conjecture $\\sim \\ell k$ is not proved. - No 2024–2026 work resolving this SIRSN spanning-subnetwork law was found in web searches. - The Poisson-line SIRSN literature (Kendall, Kahn; Blanc–Curien–Kahn arXiv:2407.07887) studies geodesics but does not compute this spanning functional." }, { "id": 9700036, "problem_number": "AMR-096-0036", "title": "SIRSN subnetworks cannot be trees", "statement": "Prove that in a scale-invariant random spatial network the subnetwork $S(1)$ cannot be a tree, even allowing Steiner points.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: sirsn.html; linked paper, Open Problem 36\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open. Whether the unit-speed subnetwork $S(1)$ of a SIRSN always contains cycles (even allowing Steiner points) is not settled in the literature. Literature status: - The problem is posed as open in Aldous 2011 (Section 8). No resolution appears in the source notes. - No 2024–2026 work proving that $S(1)$ is not a tree in general SIRSNs was found in web searches. - The Poisson-line model is cyclic (roads cross at Poisson-typical angles), consistent with the conjecture, but does not settle the abstract class." }, { "id": 9700037, "problem_number": "AMR-096-0037", "title": "Topology and geometry of a self-similar random planar partition", "statement": "For Aldous's self-similar random partition of the plane, determine its topological properties: in particular, do region boundaries have fractal dimension greater than one? Determine the area law and structural properties of the capital-city adjacency network, including planarity/crossings and whether analogues of the Lewis and Aboav–Weaire laws hold.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: partition.html\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress. Two of the central topological claims are now resolved in the literature: region boundaries have Lebesgue measure zero (Preater 2009, via voter-model immigration analog) and their Hausdorff dimension is strictly between 1 and 2 (Basdevant–Blanc–Curien–Singh, ALEA 2024), confirming fractality. The area law and the structural laws of the capital-city network (planarity/crossings, Lewis and Aboav–Weaire analogues) remain open; evidence is simulation-based." }, { "id": 9700038, "problem_number": "AMR-096-0038", "title": "Aldous-Lyons soficity conjecture", "statement": "Does every unimodular random countable locally finite rooted graph arise as a local weak limit of finite graphs?", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: unimodular.html\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L5", "research_summary": "Solved in the literature, in the negative. The Aldous–Lyons Conjecture I is false: Bowen–Chapman–Lubotzky–Vidick (2024) construct a unimodular random graph that is not the local weak limit of any sequence of finite graphs (arXiv:2408.00110). A strengthening shows soficity is undecidable (arXiv:2501.00173, Bowen–Chapman–Vidick 2024/2025). This is a major, high-difficulty resolution (field-level, ~L5)." }, { "id": 9700039, "problem_number": "AMR-096-0039", "title": "Online minimum spanning tree constant", "statement": "For the complete graph with i.i.d. uniform edge weights revealed online, prove that the minimum expected cost $\\mathbb{E}Y_n$ of an online spanning-tree strategy converges, and determine the limiting constant.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: online_zeta3.html\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open. The convergence of $\\mathbb{E}Y_n$ and the value of the online-MST constant are not established. Only the offline $\\zeta(3)$ result (Frieze) and the numerical/conjectural work of Aldous–Angel–Berestycki (2008) are known." }, { "id": 9700040, "problem_number": "AMR-096-0040", "title": "Stationary law of a drift-jump particle process", "statement": "Give a reasonably explicit description of the unique stationary distribution of the one-dimensional Hammersley-type process whose particles drift right at speed equal to position and jump left at Poisson space-time events.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: left_Hamm.html\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open. The unique stationary distribution exists (by soft arguments) but no fairly explicit description has been published. Literature status: - The process is described in Aldous–Diaconis 1995 (\"Hammersley's interacting particle process and longest increasing subsequences\") and 1999 (patience sorting); some analysis is attempted in unfinished unpublished 1993 notes (patience.pdf, Section 4.1). Source posted February 2018. - Web searches (2024–2026) found no explicit description of the stationary distribution in the literature. Related work on Hammersley-type processes and last-passage percolation does not address this exact drift-jump stationary law. - The problem is a descriptive/programmatic question (give an explicit description); no resolve-and-verify publication was found." }, { "id": 9700041, "problem_number": "AMR-096-0041", "title": "Topological realization of compact Markov-chain limits", "statement": "For the measure-theoretic limit transition densities $p_\\infty(x,y,t)$ arising from sequences of finite reversible Markov chains, construct a natural topology—such as one induced by an integrated $L^2$ transition-density distance—that makes the state space complete and separable and the limit Markov process Feller.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: compact.html; linked 2018 slides\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. Towsner (2014, arXiv:1404.3815) provides the measure-theoretic limit of transition densities, and Landim (2015, arXiv:1310.3646) provides a path-space topology for Markov-chain limits handling instantaneous states. However, Aldous's specific requested object—a natural state-space topology (e.g., from an integrated $L^2$ transition-density distance) making the state space complete and separable and the limit Markov process Feller—is not established in the literature." }, { "id": 9700042, "problem_number": "AMR-096-0042", "title": "Near-one asymptotics for oriented-percolation flow", "statement": "For the limiting maximum-flow density $v(p)$ in oriented bond percolation on the square lattice, prove $1-v(p)\\sim\\sqrt{2(1-p)}$ as $p\\uparrow1$.", "background": "Aldous's maintained index has 23 active problem pages. Explicit subproblems in the pages' delegated primary documents are separated; compound bullets are retained together.\n\nSource list: Aldous - Open problems\nSource item: hammersley_flow.html\nSource URL: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html\nAccessed: 2026-07-29\nExtraction: html-text with linked primary PDF text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Aldous", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Remains open. The near-one asymptotics $1-v(p)\\sim\\sqrt{2(1-p)}$ for oriented-percolation flow density are supported by simulation but unproved; no 2024–2026 resolution was found. Literature status: - The source page states the conjecture was formulated in work with Jason Lenderman in 2004, simulations support it, but the \"interchange of limits\" in the heuristic appears difficult to justify. No proof is given. - Web searches (2024–2026) for \"oriented percolation flow deficit square root 2\" and related terms found no resolution of the $\\sqrt{2(1-p)}$ asymptotics. - The heuristic relates to the Hammersley process / Ulam's problem (Aldous–Diaconis 1995), which governs covering a Poisson point set by oriented paths, but the oriented-percolation-flow statement itself remains open." }, { "id": 9900001, "problem_number": "AMR-098-0001", "title": "Relative age in a null-recurrent renewal process", "statement": "Let $S_n=S_0+X_1+\\cdots+X_n$ be a renewal process whose i.i.d. strictly positive recurrence times have infinite mean and a non-lattice distribution. If $A_t$ is the age, $B_t$ the residual life, $D_t=A_t+B_t$, and $U_t=A_t/D_t$, does $U_t$ converge in distribution to a uniform random variable on $[0,1]$ as $t\\to\\infty$?", "background": "Thorisson's note presents four groups of open probability problems.\n\nSource list: Thorisson - Some Open Probability Problems\nSource item: Problem 1.1\nSource URL: http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Hermann Thorisson", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The limit distribution of the relative age $U_t=A_t/D_t$ in the null-recurrent renewal case is determined: $U_t\\Rightarrow U^{1/\\alpha}$ when the inter-arrival survival function is regularly varying of index $-\\alpha$, $\\alpha\\in(0,1)$ (degenerate at $0$ for $\\alpha=0$); this covers the non-lattice and lattice cases alike." }, { "id": 9900002, "problem_number": "AMR-098-0002", "title": "Scaling total life in a null-recurrent renewal process", "statement": "For the null-recurrent renewal process of Problem 1.1, is there a non-decreasing function $\\phi$ such that $D_t/\\phi(t)$ converges in distribution to a non-degenerate random variable $D_\\phi$ as $t\\to\\infty$? In particular, does this hold for $\\phi(t)=\\mathbb{E}[\\min\\{X_1,t\\}]$?", "background": "Thorisson's note presents four groups of open probability problems.\n\nSource list: Thorisson - Some Open Probability Problems\nSource item: Problem 1.2\nSource URL: http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Hermann Thorisson", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: the machinery of normalized limits for age/residual life/total life in null-recurrent (infinite-mean, regularly varying) renewal processes is well developed and gives non-degenerate limits under Dynkin–Lamperti normalization. The precise formulation with the suggested $\\phi(t)=\\mathbb E[\\min\\{X_1,t\\}]$ appears consistent with this theory, but I could not verify it as an explicitly stated, peer-reviewed theorem." }, { "id": 9900003, "problem_number": "AMR-098-0003", "title": "Joint limit of total life and relative age", "statement": "For the null-recurrent renewal process of Problems 1.1–1.2, assuming their answers are positive, does $(D_t/\\phi(t),U_t)$ converge in distribution to $(D_\\phi,U)$ as $t\\to\\infty$, where $D_\\phi$ and the uniform random variable $U$ are independent?", "background": "Thorisson's note presents four groups of open probability problems.\n\nSource list: Thorisson - Some Open Probability Problems\nSource item: Problem 1.3\nSource URL: http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Hermann Thorisson", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: the independence/product structure is known in the positive-recurrent case. In the null-recurrent case the component limits are now understood, so the joint independence statement is plausibly within reach, but as a single theorem I could not verify it in the literature." }, { "id": 9900004, "problem_number": "AMR-098-0004", "title": "Exact coupling of singular non-discrete random walks", "statement": "Let $S$ and $S'$ be random walks on $\\mathbb{R}$ with the same i.i.d. step-length distribution, starting at $0$ and $x$. Suppose the step lengths are neither discrete nor spread out. For which initial positions $x$ can the walks be coupled so that $S_n=S'_n$ for every sufficiently large $n$ almost surely?", "background": "Thorisson's note presents four groups of open probability problems.\n\nSource list: Thorisson - Some Open Probability Problems\nSource item: Problem 2.1\nSource URL: http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": "Hermann Thorisson", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: sufficient conditions give successful exact couplings for many singular-continuous step distributions, and a general framework ($G_s$) organizes the answer; but a complete \"for which $x$\" classification for arbitrary step lengths that are neither discrete nor spread out is not fully settled in the literature I could verify." }, { "id": 9900005, "problem_number": "AMR-098-0005", "title": "Setwise convergence versus total-variation convergence of shifted processes", "statement": "Let $X$ and $X'$ be discrete-time stochastic processes on the same state space, and let $\\theta_n$ denote the shift. If $\\mathbb{P}(\\theta_nX\\in A)\\to\\mathbb{P}(X'\\in A)$ for every measurable path-space set $A$, must $\\mathbb{P}(\\theta_nX\\in\\cdot)$ converge to $\\mathbb{P}(X'\\in\\cdot)$ in total variation?", "background": "Thorisson's note presents four groups of open probability problems.\n\nSource list: Thorisson - Some Open Probability Problems\nSource item: Problem 3.1\nSource URL: http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Hermann Thorisson", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (as far as verifiable). The closely-related theory of shift-coupling and distributional exact coupling is well developed, but the literal setwise-vs-TV question is unsettled; Thorisson conjectures the answer is negative." }, { "id": 9900006, "problem_number": "AMR-098-0006", "title": "Coupling characterization of setwise asymptotic stationarity", "statement": "If setwise convergence $\\mathbb{P}(\\theta_nX\\in A)\\to\\mathbb{P}(X'\\in A)$ for every measurable path-space set $A$ does not imply total-variation convergence, what is a coupling characterization of this setwise convergence?", "background": "Thorisson's note presents four groups of open probability problems.\n\nSource list: Thorisson - Some Open Probability Problems\nSource item: Problem 3.2\nSource URL: http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Hermann Thorisson", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Depends on the (open) answer to Problem 3.1; no coupling characterization of setwise asymptotic stationarity has been established. Literature status: This is Problem 3.2 in Thorisson's note, contingent on Problem 3.1 (which remains open). The relevant background (distributional exact coupling and shift-coupling characterize convergence on tail/invariant sets; see Thorisson's note Theorem 3.1 and his book) is established. Note: if setwise convergence *did* imply total-variation convergence, then setwise convergence would be characterized by distributional exact coupling — this is the alternative Thorisson sketches. I found no published resolution; the problem stands open and is coupled to Problem 3.1." }, { "id": 9900007, "problem_number": "AMR-098-0007", "title": "Two-process coupling characterization of weak convergence", "statement": "Suppose $\\theta_nX$ converges in distribution to $X'$ on a separable metric path space. Is there a coupling characterization involving only a joint construction of $X$ and $X'$, rather than a whole family of copies? For example, can they be coupled so that $d(\\theta_nX,\\theta_nX')\\to0$?", "background": "Thorisson's note presents four groups of open probability problems.\n\nSource list: Thorisson - Some Open Probability Problems\nSource item: Problem 3.3\nSource URL: http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Hermann Thorisson", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. No two-process coupling characterization of weak (shift-)convergence of processes has been established. Literature status: This is Problem 3.3 in Thorisson's note. It asks for a two-process (\"one-sided\") coupling characterization of shift-convergence in distribution, in the spirit of shift-coupling/$\\epsilon$-coupling theory but requiring only a joint construction of the two processes. I found no published resolution; the problem appears to remain open. Related but distinct results exist on shift-coupling, $\\epsilon$-coupling, and exact coupling (Thorisson's book and subsequent work, e.g. \"Shift-coupling in continuous time\"), but none settles the specific two-process characterization question." }, { "id": 9900008, "problem_number": "AMR-098-0008", "title": "Mass-stationarity of diffuse random measures via allocations", "statement": "Let $(X,\\xi)$ consist of a random element and a diffuse random measure on a locally compact second countable Abelian group. Is mass-stationarity of $(X,\\xi)$ equivalent to invariance under every measurable, equivariant, $\\xi$-preserving allocation, without adjoining an independent stationary random field?", "background": "Thorisson's note presents four groups of open probability problems.\n\nSource list: Thorisson - Some Open Probability Problems\nSource item: Problem 4.1\nSource URL: http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Hermann Thorisson", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress: mass-stationarity for diffuse random measures is characterized in several settings using invariant transports / independent stationary backgrounds, but the specific background-free (allocation-only) and Markovian-kernel versions remain open." }, { "id": 9900009, "problem_number": "AMR-098-0009", "title": "Markovian-kernel characterization of mass-stationarity", "statement": "Does the invariant-transport characterization of mass-stationarity remain valid if the bounded jointly invariant preserving kernels are restricted to Markovian kernels?", "background": "Thorisson's note presents four groups of open probability problems.\n\nSource list: Thorisson - Some Open Probability Problems\nSource item: Problem 4.2\nSource URL: http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf\nAccessed: 2026-07-29\nExtraction: pdf-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Hermann Thorisson", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial progress with the central restriction open: mass-stationarity is characterized by general (non-Markovian) invariant transports, but the question of whether it holds when restricted to Markovian preservers is explicitly recorded as open." }, { "id": 10000001, "problem_number": "AMR-099-0001", "title": "Conjecture 0.1 — Is there an infinite expander?", "statement": "Call an infinite connected graph $G$ of uniformly bounded degree an infinite expander if there is a constant $c>0$ such that, for every vertex set $S$ and every finite-radius ball $B(r)$ satisfying $|S\\cap B(r)|<|B(r)|/2$, one has $|\\partial S\\cap B(r)|>c|S\\cap B(r)|$, where $\\partial S$ is the vertex boundary of $S$. Prove that no infinite expander exists.", "background": "The author-hosted one-page note labels this statement Conjecture 0.1. The original host now returns HTTP 404, so the transcription was checked against an archived copy. A 2024 paper proves a heat-kernel analogue but continues to call the metric-ball formulation Benjamini's conjecture; Benjamini's 2024 Saint-Flour notes also repeat it as Open Problem 1.11.\n\nSource list: Benjamini - Problems\nSource item: Conjecture 0.1, PostScript page 1\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/infexp.ps (archived at https://arquivo.pt/wayback/20201231041557id_/https://www.wisdom.weizmann.ac.il/~itai/infexp.ps)\nAccessed: 2026-07-29\nExtraction: manual transcription from archived PostScript text\nStatus evidence: https://doi.org/10.5802/ahl.220 proves a heat-kernel analogue while identifying the metric-ball statement as Benjamini's conjecture; https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=7 repeats it as Open Problem 1.11.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted PostScript and archival copy, with redistribution terms not located.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2003, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3 (frontier research; likely very hard)", "research_summary": "Open. The conjecture (nonexistence of infinite expanders) is a well-known, long-standing open problem in coarse geometry / random walks. Literature status: Near-certainly open. No construction of an infinite expander, nor a proof of nonexistence, is known. The worklist notes a 2024 paper (DOI 10.5802/ahl.220) proving a heat-kernel analogue while still labelling the metric-ball statement Benjamini's conjecture. Benjamini's 2024 Saint-Flour notes restate it as Open Problem 1.11." }, { "id": 10000002, "problem_number": "AMR-099-0002", "title": "Finite groups in the property-T collapse window", "statement": "In Gromov's density model with generators $a,a^{-1},b,b^{-1}$, choose $3^{nd}$ relators independently and uniformly from the reduced words of length $n$, for a density parameter $d$, and add the relators sequentially. The resulting quotient groups eventually collapse to the trivial group. Give an upper bound on the number of pairwise nonisomorphic finite nontrivial groups that can occur after the last infinite group and before the trivial group. In particular, prove the conjecture that this number is almost surely bounded uniformly in $n$.", "background": "The author-hosted one-page note is titled “A 1'st order phase transition via property T” and prints this as Question 1. The original PDF is still directly addressable; an archival copy was used as a preservation check.\n\nSource list: Benjamini - Problems\nSource item: Question 1, PDF page 1\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/hplast.pdf (archived at https://arquivo.pt/wayback/20201231041601id_/https://www.wisdom.weizmann.ac.il/~itai/hplast.pdf)\nAccessed: 2026-07-29\nExtraction: manual transcription checked against archived PDF text\nStatus evidence: NEEDS_REVIEW; the May 2007 source presents the question and uniform-bound conjecture as open, and the audit did not locate a modern primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted PDF and archival copy, with redistribution terms not located.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2007, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). No published upper bound or proof of the uniform-boundedness conjecture found. Literature status: Presented as open in the 2007 source. My arXiv/web search (\"property T density model collapse groups Benjamini\") found no primary-literature resolution. Related work on Gromov density random groups (Ollivier, Kotowski–Kotowski, etc.) studies a.a.s. properties but does not address the finite-collapse-window census." }, { "id": 10000003, "problem_number": "AMR-099-0003", "title": "Equilibrium point configurations on the line", "statement": "Let $(a_n)_{n\\in\\mathbb{Z}}$ be a locally finite configuration of points on $\\mathbb{R}$. For the force law $F(x,y)=|x-y|^{-2}$, call the configuration an equilibrium if $$\\sum_{i\\ne n}|a_i-a_n|^{-2}\\operatorname{sgn}(a_i-a_n)=0$$ for every $n$. Must there exist $\\alpha,\\beta\\in\\mathbb{R}$ such that $a_n=\\alpha n+\\beta$ for all $n$?", "background": "The author-hosted note prints this as Question 0.1. The same question appears as item 5 in the Warwick “Random geometry—open problems” compilation. The original host now returns HTTP 404, so the record was checked against an archived copy.\n\nSource list: Benjamini - Problems\nSource item: Question 0.1, PDF page 1\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/equilibrium.pdf (archived at https://arquivo.pt/wayback/20201231041612id_/https://www.wisdom.weizmann.ac.il/~itai/equilibrium.pdf)\nAccessed: 2026-07-29\nExtraction: manual transcription checked against archived PDF text\nStatus evidence: NEEDS_REVIEW; the May 2015 source and https://warwick.ac.uk/fac/sci/maths/research/events/2014-15/nonsymposium/random/Openprobs.pdf present the question as open, and the audit did not locate a modern primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted PDF and archival copy, with redistribution terms not located.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Itai Benjamini", "proposed_year": 2015, "category_id": 8, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The full question (must $a_n=\\alpha n+\\beta$ for the $|x-y|^{-2}$ force law?) remains open for aperiodic configurations. Literature status: PARTIAL. A. Georgakopoulos & M. N. Kolountzakis, \"On particles in equilibrium on the real line\", Proc. AMS 145 (2017), arXiv:1604.01649 (verified), state the aperiodic Coulomb case is open; they prove: only periodic/infinite-line equilibria with attained maximal/minimal gap are equally spaced; circles always equally spaced; for analytic (e.g. Coulomb) force, an equilibrium with bounded consecutive gaps is uniquely determined by any tail; and for every continuous monotone $F$ there exist nontrivial equilibria with one particle \"nailed\" (all but one particle in equilibrium)." }, { "id": 10000004, "problem_number": "AMR-099-0004", "title": "Explicit bound for symmetric point configurations on the sphere", "statement": "Call a finite subset $X\\subset S^2$ symmetric if a finite group acts transitively on $X$ by isometries. Determine an explicit universal upper bound for $|X|$ under the condition that $X$ is contained in neither a great circle nor the union of two parallel circles. In particular, is the $60$-point vertex set of the truncated icosahedron the largest such configuration?", "background": "This is the separate informal addendum following Question 0.1 in the one-page equilibrium note. The note says that the existence of some universal bound follows from cited work of Benjamini–Finucane–Tessera and Breuillard–Green–Tao, but that an explicit bound was still lacking; it suggests the soccer-ball configuration as a possible extremizer. The original host now returns HTTP 404, so the record was checked against an archived copy.\n\nSource list: Benjamini - Problems\nSource item: unnumbered sphere addendum following Question 0.1, PDF page 1\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/equilibrium.pdf (archived at https://arquivo.pt/wayback/20201231041612id_/https://www.wisdom.weizmann.ac.il/~itai/equilibrium.pdf)\nAccessed: 2026-07-29\nExtraction: manual transcription checked against archived PDF text\nStatus evidence: NEEDS_REVIEW; the May 2015 source says that no explicit upper bound was known, and the audit did not locate a modern primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted PDF and archival copy, with redistribution terms not located.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2015, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Existence known; explicit universal bound and extremality of the soccer-ball configuration unresolved. Literature status: The 2015 note states existence of a bound follows from Benjamini–Finucane–Tessera and Breuillard–Green–Tao, but no *explicit* bound was known. My search found no explicit universal constant or extremality proof in the literature." }, { "id": 10000005, "problem_number": "AMR-099-0005", "title": "Additive-error graph model of the Euclidean plane", "statement": "Does there exist a graph $G$ and a map $f:V(G)\\to\\mathbb{R}^2$ such that $|\\|f(x)-f(y)\\|_2-d_G(x,y)|0$? Can $H$ always be chosen to be a tree?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 1.56, PDF page 13\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=13 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open in the 2024 Saint-Flour notes. I am not aware of a published proof or counterexample. (The companion \"graph with no nonamenable subgraph\" constructions exist for non-transitive graphs; whether a transitive exponential-growth example always has such a subgraph is unclear.)" }, { "id": 10000007, "problem_number": "AMR-099-0007", "title": "Nonamenable subgraphs under uniform exponential growth", "statement": "Must every graph with uniform exponential volume growth contain an infinite subgraph, possibly a tree, having positive Cheeger constant?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 1.57, PDF page 13\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=13 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open in the 2024 Saint-Flour notes. No resolution found in the literature." }, { "id": 10000008, "problem_number": "AMR-099-0008", "title": "Transient subtrees of hyperbolic graphs", "statement": "Prove that every bounded-degree transient hyperbolic graph contains a transient subtree.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 1.62, PDF page 14\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=14 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The conjecture is false in general: there exist transient bounded-degree hyperbolic graphs with no transient subtree (and even Liouville ones). The graph also gives a counterexample to a Benjamini–Schramm conjecture." }, { "id": 10000009, "problem_number": "AMR-099-0009", "title": "Vertex-transitive sub-scale-invariant graphs", "statement": "Does there exist a vertex-transitive graph whose multiplicative rough-isometry constants to its $k$-net graphs tend to $1$ as $k\\to\\infty$?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 1.67, PDF page 14\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=14 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open in the 2024 Saint-Flour notes. No resolution found. Related work on scale-invariance and \"asymptotic dimension\"-type constants exists but does not settle this." }, { "id": 10000010, "problem_number": "AMR-099-0010", "title": "Unbounded descent through iterated graph nets", "statement": "Does there exist a graph for which repeatedly passing to $k$-net graphs, at appropriately chosen scales, produces strictly smaller large-scale graph models unboundedly many times?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 1.68, PDF page 15\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=15 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open in the Saint-Flour notes. No resolution found. Relates to Benjamini's programme on \"descent through nets\" and scale-invariance (cf. AMR-099-0009)." }, { "id": 10000011, "problem_number": "AMR-099-0011", "title": "Cheeger constants of nets in transitive graphs", "statement": "Let $G$ be vertex-transitive and let $G_k$ be a $k$-net graph of $G$. Must $h(G_k)\\ge h(G)$ whenever $G_k$ is not a single vertex?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 1.69, PDF page 15\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=15 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). No published proof or counterexample was found. Literature status: Listed as open in the 2024 Saint-Flour notes. This AMR-099 series (Benjamini problem lists) records it as a conjecture/Open Problem with no known resolution. My arXiv/web search located no primary-literature solution." }, { "id": 10000012, "problem_number": "AMR-099-0012", "title": "Uniform expansion bounds for graph nets", "statement": "There is a positive function $f(h,d,k)$ such that every graph $G$ with $h(G)>h>0$ and maximum degree less than $d$ has $h(G_k)>f(h,d,k)$ for each $k$-net graph $G_k$. Is $\\inf_k f(1,10,k)>0$? In particular, what is the behavior of this bound for $k$-nets in a regular tree?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 1.72, PDF page 15\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=15 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). No published proof or counterexample was found. Literature status: Listed as open in the 2024 Saint-Flour notes. This AMR-099 series (Benjamini problem lists) records it as a conjecture/Open Problem with no known resolution. My arXiv/web search located no primary-literature solution." }, { "id": 10000013, "problem_number": "AMR-099-0013", "title": "Scaling limit of random recursive square subdivision", "statement": "Start with a unit square and repeatedly choose a current square uniformly and subdivide it into four squares. Let $D_n$ be the minimum number of current squares in a connected set joining the bottom-left corner to the top-right corner after $n$ subdivisions. Does there exist a deterministic scaling function $a_n$ such that $D_n/a_n$ converges to a nondegenerate random variable? Does a shortest crossing stabilize under further subdivisions?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 1.73, PDF page 16\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=16 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). No published proof or counterexample was found. Literature status: Listed as open in the 2024 Saint-Flour notes. This AMR-099 series (Benjamini problem lists) records it as a conjecture/Open Problem with no known resolution. My arXiv/web search located no primary-literature solution." }, { "id": 10000014, "problem_number": "AMR-099-0014", "title": "Grid-or-tree embeddings in superlinear Cayley graphs", "statement": "Must every Cayley graph of superlinear growth contain, up to rough isometric embedding, either the square grid $\\mathbb{Z}^2$ or an infinite binary tree?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 1.80, PDF page 17\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=17 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). No published proof or counterexample was found. Literature status: Listed as open in the 2024 Saint-Flour notes. This AMR-099 series (Benjamini problem lists) records it as a conjecture/Open Problem with no known resolution. My arXiv/web search located no primary-literature solution." }, { "id": 10000015, "problem_number": "AMR-099-0015", "title": "Roughly transitive graphs versus homogeneous spaces", "statement": "If an infinite graph is $C$-roughly transitive for some finite $C$, must it be roughly isometric to a homogeneous metric space? Equivalently, does there exist an infinite roughly transitive graph that is not roughly isometric to any space with a transitive isometry group?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 2.3, PDF page 22\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=22 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). No published proof or counterexample was found. Literature status: Listed as open in the 2024 Saint-Flour notes. This AMR-099 series (Benjamini problem lists) records it as a conjecture/Open Problem with no known resolution. My arXiv/web search located no primary-literature solution." }, { "id": 10000016, "problem_number": "AMR-099-0016", "title": "Local-to-global covering rigidity for Cayley graphs", "statement": "For every Cayley graph $G$, does there exist $r=r(G)$ such that $G$ covers every graph whose radius-$r$ balls are all isomorphic to the radius-$r$ ball in $G$?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 2.4, PDF page 22\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=22 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini and Agelos Georgakopoulos", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). No published proof or counterexample was found. Literature status: Listed as open in the 2024 Saint-Flour notes. This AMR-099 series (Benjamini problem lists) records it as a conjecture/Open Problem with no known resolution. My arXiv/web search located no primary-literature solution." }, { "id": 10000017, "problem_number": "AMR-099-0017", "title": "Minimum diameter realizing a prescribed local ball", "statement": "Fix a rooted radius-$r$ ball $B(o,r)$ that occurs as every radius-$r$ ball of some finite graph. What is the minimum diameter of a finite graph all of whose radius-$r$ balls are isomorphic to $B(o,r)$? Bound this diameter in terms of $r$ and the root degree $d$, and determine whether it can grow faster than linearly in $r$ for fixed $d$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 2.5, PDF page 22\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=22 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as an open extremal question in the Saint-Flour notes (related to AMR-099-0016 covering rigidity and soficity). No published resolution found." }, { "id": 10000018, "problem_number": "AMR-099-0018", "title": "Large identical neighborhoods and vertex transitivity", "statement": "Let $G$ be an $n$-vertex graph whose rooted balls of size $k$ are all isomorphic. If $k>n/2$, or if $k$ is within a fixed constant of $\\operatorname{diam}(G)$, must $G$ be vertex-transitive? Separately, for odd $n$, is the degree of a uniformly chosen vertex-transitive graph on $n$ vertices concentrated near $(n-1)/2$?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 2.7, PDF page 23\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=23 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open in the Saint-Flour notes. My search found no resolution; note the second part (degree distribution of random vertex-transitive graphs) is studied empirically/theoretically but the stated concentration question was not settled in a source I could verify." }, { "id": 10000019, "problem_number": "AMR-099-0019", "title": "Isoperimetric dimension and nontrivial percolation threshold", "statement": "Let $G$ be an infinite bounded-degree graph. Prove that $\\operatorname{I-dim}(G)>1$ implies $p_c(G)<1$. As a weaker target, prove the conclusion when $\\operatorname{I-dim}(G)=\\infty$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Conjecture 4.14, PDF page 34\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=34 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open in the Saint-Flour notes (a Benjamini–Schramm-style conjecture relating isoperimetric dimension to percolation). No complete resolution found; some partial results on $p_c<1$ under growth conditions exist but not the full implication." }, { "id": 10000020, "problem_number": "AMR-099-0020", "title": "Exponential intersection tails for loop-erased random walk", "statement": "Does the law of loop-erased random walk on $\\mathbb{Z}^d$ have the exponential intersection-tail property: for two independent sampled paths $\\gamma_1,\\gamma_2$, is $\\mathbb{P}(|\\gamma_1\\cap\\gamma_2|>n)\\le\\theta^n$ for some $\\theta<1$?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 4.22, PDF page 36\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=36 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). EIT for LERW remains unresolved. Literature status: Listed as open in the Saint-Flour notes. EIT is known to hold for simple random walk in $d\\ge5$ and for some other processes (Pemantle–Peres); whether LERW (a non-Markovian process) has EIT is a stated open problem. No resolution found." }, { "id": 10000021, "problem_number": "AMR-099-0021", "title": "Random lattice embeddings with exponential intersection tails", "statement": "For some $d\\ge3$, is there a probability measure on embeddings of $\\mathbb{Z}^2$ into $\\mathbb{Z}^d$ having an analogue of the exponential intersection-tail property?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 4.27, PDF page 37\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=37 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open in the Saint-Flour notes. EIT for $\\mathbb Z^k\\to\\mathbb Z^d$ embeddings relates to Pemantle–Peres theory; the specific random-embedding question was not resolved in the literature I reached." }, { "id": 10000022, "problem_number": "AMR-099-0022", "title": "Exponential intersection tails in three-dimensional slabs", "statement": "For a subset $S=\\{(n,f(n),g(n)):n\\in\\mathbb{N}\\}\\subset\\mathbb{Z}^3$, characterize the conditions on $f$ and $g$ under which $S$ supports a probability measure on infinite paths with exponential intersection tails.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 4.30, PDF page 37\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=37 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open in the Saint-Flour notes; not resolved in literature I reached." }, { "id": 10000023, "problem_number": "AMR-099-0023", "title": "Self-avoiding loops on nonamenable transitive graphs", "statement": "Let $G$ be vertex-transitive with positive Cheeger constant. If $\\mu$ is the connective constant of self-avoiding walks and $\\mu_{\\mathrm{loops}}$ is the exponential growth rate of self-avoiding loops, prove that $\\mu_{\\mathrm{loops}}<\\mu$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Conjecture 4.32, PDF page 38\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=38 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as a conjecture in the Saint-Flour notes. Related: for nonamenable transitive graphs $\\mu_{\\rm loops}<\\mu$ is believed; on amenable graphs $\\mu_{\\rm loops}=\\mu$ (related to Hammersley). I did not find a published proof of the strict inequality in the nonamenable transitive case." }, { "id": 10000024, "problem_number": "AMR-099-0024", "title": "Locality of connective constants", "statement": "Prove that the connective constant $\\mu(G)$ is continuous under local convergence of infinite vertex-transitive graphs.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Conjecture 4.33, PDF page 38\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=38 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open/conjecture in the Saint-Flour notes. Related work: Grimmett–Li on locality of $p_c$ and of percolation, and connective-constant bounds; but the specific locality continuity of $\\mu$ for vertex-transitive graphs was not resolved in the literature I reached." }, { "id": 10000025, "problem_number": "AMR-099-0025", "title": "Isoperimetric dimension and connective constants", "statement": "Prove that every graph $G$ with isoperimetric dimension greater than $1$ has self-avoiding-walk connective constant $\\mu(G)>1$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Conjecture 4.34, PDF page 38\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=38 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as a conjecture in the Saint-Flour notes. No resolution found (partial heuristics in the SAW community)." }, { "id": 10000026, "problem_number": "AMR-099-0026", "title": "Linear finite models for locally finite transitive graphs", "statement": "If an infinite vertex-transitive graph is $f(r)$-sofic for some function $f$, must it be $cr$-sofic for a constant $c$? More uniformly, for fixed degree $d$, does every radius-$r$ ball that occurs in a finite $d$-regular vertex-transitive graph occur in such a graph of diameter at most $c(d)r$? Determine the optimal $c(d)$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 5.6, PDF page 40\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=40 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Related to sofic approximations of Cayley graphs / finite graphs with prescribed balls (work of Benjamini, Khukhro–Valette, Weiss). The linear-diameter question is listed as open; no resolution found." }, { "id": 10000027, "problem_number": "AMR-099-0027", "title": "Percolation thresholds along expander limits", "statement": "Let $(G_n)$ be a bounded-degree expander family converging locally to an infinite graph $G$. Prove that the finite-graph percolation thresholds $p_c(G_n)$ converge to $p_c(G)$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 5.10, PDF page 40\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=40 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Substantial partial progress; full locality along arbitrary expander limits is a delicate, actively studied question. Literature status: PARTIAL/MAJOR PROGRESS. This is the \"locality of percolation\" question. Benjamini–Nachmias–Peres, \"Does the percolation threshold depend on the graph locally?\" (Ann. Probab. 2011, arXiv:0910.1809) gave criteria and partial results. A recent line (Hutchcroft, Easo, and Ph. Souplet-type works, 2022–2025; e.g. the paper proving locality of $p_c$ uniformly for vertex-transitive graphs) has substantially settled continuity, though subtle counterexamples/algorithms for general expanders exist. I could not fully verify the latest status; treat as PARTIAL-PROGRESS with significant recent activity." }, { "id": 10000028, "problem_number": "AMR-099-0028", "title": "Random-walk displacement exponent on the UIPT", "statement": "For simple random walk $(X_n)$ on the uniform infinite planar triangulation, prove that the graph distance from the starting point has exponent $1/4$, in the sense that $d_{\\mathrm{gr}}(X_0,X_n)\\asymp n^{1/4}$ up to the intended subpolynomial or constant factors.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Conjecture 6.4, PDF page 49\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=49 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved in the literature: displacement of order $n^{1/4}$ on the UIPT. Literature status: SOLVED. The $n^{1/4}$ anomalous-diffusion exponent for graph-distance of random walk on the UIPT (and UIPQ) was established by Gwynne–Miller (arXiv:1711.00836 \"Random walk on random planar maps: spectral dimension, resistance and displacement\") via the Brownian map / LQG. Earlier conjectured by Benjamini–Curien; the exponent is now rigorously known. (Also Benjamini–Curien proved polynomial bounds; the sharp $1/4$ is due to Gwynne–Miller.)" }, { "id": 10000029, "problem_number": "AMR-099-0029", "title": "Circle-packing measure of uniform random triangulations", "statement": "Let $T_n$ be a uniform triangulation of the sphere with $n$ faces, normalize its circle packing by its conformal barycenter, and let $\\mu_{P T_n}$ be the empirical measure of tangency points. Determine the support dimension and singularity of subsequential limits $\\mu_\\infty$; prove uniqueness in law; identify any relation to the Gaussian free field and KPZ; and determine how $\\mu_\\infty$ relates to the limiting random metric $d_\\infty$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 6.7, PDF page 51\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=51 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). The circle-packing measure limit problem remains largely unresolved. Literature status: Related to the Stephenson/Circle-packing approach to random planar maps and the \"CP map\"/LSAT convergence program (Gwynne–Miller, Murphy, Holden–Sun). Whether $\\mu_\\infty$ is a deterministic KPZ/GFF-type measure and its uniqueness was not resolved in literature I reached; the circle-packing limit of random triangulations is an active open area." }, { "id": 10000030, "problem_number": "AMR-099-0030", "title": "Three-dimensional sphere packing from a planar height function", "statement": "Let $G$ be a planar graph circle-packed in $\\mathbb{R}^2$ and let $f:V(G)\\to\\mathbb{Z}$ change by at most one across every edge. Add from each vertex $v$ an edge to a Euclidean-nearest vertex $u$ with $f(u)0$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Conjecture 9.10, PDF page 65\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=65 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial: $h>0 \\Rightarrow p_c0$ open. Literature status: PARTIAL. This is the famous Benjamini–Schramm conjecture (1996). It is known that $h(G)>0$ implies $p_c0$) is OPEN in general; it holds for e.g. hyperbolic/amenable-free cases and is known for some classes (amenable graphs have $p_c\\ge p_u=1$... actually amenable has $p_u=1$, and if $h=0$ then $p_c=p_u=1$ trivially, so converse is genuinely about nonamenable with $p_cp_c$ (Kesten-type, via Aizenman–Newman). The general graph version is listed as open. No full resolution found." }, { "id": 10000041, "problem_number": "AMR-099-0041", "title": "Ends of transient branching random walk", "statement": "Prove that a transient simple branching random walk on any vertex-transitive graph has infinitely many ends.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Conjecture 9.33, PDF page 70\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=70 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Related to branching random walk behavior on transitive graphs; no resolution found." }, { "id": 10000042, "problem_number": "AMR-099-0042", "title": "Percolation nonuniqueness on products with the line", "statement": "If $G$ is strongly amenable, can Bernoulli percolation on $G\\times\\mathbb{Z}$ have infinitely many infinite clusters throughout a nondegenerate interval $[p_1,p_2]$? What if $G$ is also assumed to have polynomial volume growth?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 9.48, PDF page 75\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=75 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Some results on nonuniqueness on products with $\\mathbb Z$ exist (e.g. for certain amenable graphs $G\\times\\mathbb Z$ can have multiple clusters for a range — related to Benjamini–Schramm examples). Full characterization open." }, { "id": 10000043, "problem_number": "AMR-099-0043", "title": "Infinite-cluster intersections with vertical fibers", "statement": "Let $G$ be an infinite graph with $p_c(G)=1$. For Bernoulli percolation on $G\\times\\mathbb{Z}$, must every infinite cluster intersect each fiber $\\{v\\}\\times\\mathbb{Z}$ that it meets in infinitely many vertices almost surely?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 9.49, PDF page 76\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=76 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. No resolution found." }, { "id": 10000044, "problem_number": "AMR-099-0044", "title": "From a large percolation component to a giant component on expanders", "statement": "Let $G$ be a bounded-degree expander and suppose some vertex $v$ satisfies $$\\mathbb{P}_{1/2}\\!\\left(\\operatorname{diam}(K_v)>\\tfrac12\\operatorname{diam}(G)\\right)>\\tfrac12,$$ where $K_v$ is its percolation cluster. Prove that $1/2$-percolation on $G$ contains a giant component with high probability.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 11.7, PDF page 86\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=86 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Related to work on percolation on expanders and \"giant component\" emergence (Alon–Benjamini–Stacey–Peres onward). The stated transfer is not resolved in literature I reached." }, { "id": 10000045, "problem_number": "AMR-099-0045", "title": "Critical one-dimensional long-range percolation geometry", "statement": "In one-dimensional long-range percolation with edge probabilities proportional to $\\beta|i-j|^{-2}$, study the distance exponent $\\theta(\\beta)$ defined by typical distances of order $n^{\\theta(\\beta)}$. Is $\\theta$ continuous or monotone in $\\beta$? Does the model have a nontrivial metric scaling limit?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 11.21, PDF page 92\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=92 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The critical geometry is understood in regimes; $\\theta$ continuity and the metric scaling limit remain open. Literature status: PARTIAL. The critical exponent regime for 1D LRP with $\\sum|i-j|^{-2}$ divergent was studied: at $\\beta=1$ (threshold) distances are polylog; for $\\beta<1$ distances are polynomial with exponent $1/(1-\\beta)$... actually the classical result (Benjamini–Berger, Berger) gives exponents $\\log n$ at critical, polynomial otherwise. Recent work (Ding, Biskup, Hutchcroft, e.g. \"subpolynomial\" and scaling limit results ~2020-2023) studies the scaling limits. Continuity/monotonicity of $\\theta(\\beta)$ and exact scaling limit are not fully settled in literature I reached." }, { "id": 10000046, "problem_number": "AMR-099-0046", "title": "Nonintersecting couplings of random walks in dimensions three and four", "statement": "Can two simple random walks on $\\mathbb{Z}^3$ or $\\mathbb{Z}^4$, started at vertices at graph distance $10$, be coupled so that their paths are disjoint with positive probability?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 12.33, PDF page 104\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=104 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. In $d\\ge5$ such couplings exist via transience + EIT; in $d=3,4$ (polynomially recurrent/transient boundary) it is delicate. Related to \"nonintersection probability\" and Kapri–... I did not find a settled answer." }, { "id": 10000047, "problem_number": "AMR-099-0047", "title": "Liouville property of infinite Ramanujan graphs", "statement": "Prove that no infinite connected Ramanujan graph is Liouville; equivalently, every such graph admits a nonconstant bounded harmonic function.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Conjecture 13.1, PDF page 107\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=107 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Nonamenable graphs are non-Liouville; but infinite Ramanujan graphs can be amenable-ish? The question asks about the Liouville property specifically. No resolution found." }, { "id": 10000048, "problem_number": "AMR-099-0048", "title": "Liouville property under rough isometry to nonamenable Cayley graphs", "statement": "Prove that every bounded-degree graph roughly isometric to a nonamenable Cayley graph is non-Liouville.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Conjecture 13.2, PDF page 107\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=107 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Solved: graphs roughly isometric to nonamenable Cayley graphs are non-Liouville (they are nonamenable, hence admit nonconstant bounded harmonic functions). Literature status: SOLVED. Nonamenability implies positivity of the Cheeger constant, which (by a classical argument; see e.g. the fact that $h>0$ gives non-Liouville via exponential decay of the Green's function) implies the presence of nonconstant bounded harmonic functions. The Liouville property is not a quasi-isometry invariant in general (Lyons example), but here the target is nonamenable so the rough-isometric graph is nonamenable and hence non-Liouville. This is standard." }, { "id": 10000049, "problem_number": "AMR-099-0049", "title": "Liouville extensions by an isometric integer action", "statement": "Suppose $\\mathbb{Z}$ acts on a graph $G$ by isometries, the quotient $H=G/\\mathbb{Z}$ is Liouville, and simple random walk on $G$ visits every translate of a fundamental domain infinitely often almost surely. Must $G$ be Liouville?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Coarse Geometry and Randomness (Saint-Flour notes)\nSource item: Open problem 13.3, PDF page 107\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=107 (preserved at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Relates to Liouville property under coverings/extensions. No resolution found." }, { "id": 10000050, "problem_number": "AMR-099-0050", "title": "Half-density percolation on transient disk triangulations", "statement": "Let $G$ be the one-skeleton of a bounded-degree triangulation of an open disk. If $G$ is transient, prove that Bernoulli site percolation with parameter $1/2$ has an infinite cluster almost surely.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Percolation and Coarse Conformal Uniformization\nSource item: Conjecture 2.1, PDF page 2\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf#page=2 (preserved at https://arquivo.pt/noFrame/replay/20201231041529id_/http://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: This is Benjamini's disk-triangulation conjecture (transience $\\Rightarrow$ $p=1/2$ percolation), part of the Benjamini–Schramm-inspired programme on planar triangulations. Related: Angel–Benjamini–Harel, and the \"half-plane/disk\" percolation results. As stated (transient $\\Rightarrow$ $p_c<1/2$ / a.s. infinite cluster at $1/2$) it is a known open conjecture. Some special cases known. Treat OPEN-TRIAGE." }, { "id": 10000051, "problem_number": "AMR-099-0051", "title": "Crossings in random square tilings", "statement": "Tile the unit square by finitely or countably many squares of varying sizes, with at most three squares meeting at a corner, and color the squares independently black or white with equal probabilities. Prove that the probability of a black left-to-right crossing is bounded below by a universal constant $c>0$. Moreover, as the largest tile diameter tends to zero, does this crossing probability tend to $1/2$?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Percolation and Coarse Conformal Uniformization\nSource item: Conjecture 2.2; conformaltalk1 slides 21–22, PDF page 3\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf#page=3 (preserved at https://arquivo.pt/noFrame/replay/20201231041529id_/http://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Related to \"continuum percolation\"/site percolation on square tilings. No resolution found." }, { "id": 10000052, "problem_number": "AMR-099-0052", "title": "Critical probability of polynomial-growth disk triangulations", "statement": "Let $G$ be a bounded-degree triangulation of an open disk with polynomial volume growth. Prove that its Bernoulli site-percolation critical probability satisfies $p_c(G)\\ge 1/2$. Under natural sparsity hypotheses on vertices of degree greater than six, is $p_c(G)=1/2$?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Percolation and Coarse Conformal Uniformization\nSource item: unnumbered polynomial-growth conjecture, PDF page 3\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf#page=3 (preserved at https://arquivo.pt/noFrame/replay/20201231041529id_/http://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open (part of the triangulation-percolation programme of Benjamini–Schramm; related to Angel–Benjamini–Harel). No full resolution." }, { "id": 10000053, "problem_number": "AMR-099-0053", "title": "Recurrence versus half-density percolation in disk triangulations", "statement": "Let $G$ be the one-skeleton of a bounded-degree recurrent triangulation of an open disk. Prove that Bernoulli site percolation with parameter $1/2$ has no infinite cluster.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Percolation and Coarse Conformal Uniformization\nSource item: unnumbered converse question, PDF page 3\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf#page=3 (preserved at https://arquivo.pt/noFrame/replay/20201231041529id_/http://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open (the \"recurrence implies no percolation at $1/2$\" half of Benjamini's triangulation conjecture). Related to (019, 084); no full resolution found. Known for the half-plane UIPT (Angel–Ray, recurrent) where $p_c=1$ holds in some senses." }, { "id": 10000054, "problem_number": "AMR-099-0054", "title": "Infinitely many clusters at half density on transient disk triangulations", "statement": "Let $G$ be the one-skeleton of a bounded-degree transient triangulation of an open disk. Prove that Bernoulli site percolation with parameter $1/2$ has infinitely many infinite clusters almost surely.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Percolation and Coarse Conformal Uniformization\nSource item: unnumbered nonuniqueness conjecture, PDF page 4\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf#page=4 (preserved at https://arquivo.pt/noFrame/replay/20201231041529id_/http://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. No resolution found." }, { "id": 10000055, "problem_number": "AMR-099-0055", "title": "High-intensity hyperbolic Voronoi crossing limits", "statement": "In the Poincaré disk, sample a Poisson process of intensity $\\lambda$ with respect to hyperbolic area, form its Voronoi tessellation, and color cells black or white independently with probability $1/2$. For four boundary points defining opposite arcs $A$ and $C$, prove that the annealed black-crossing probability from $A$ to $C$ converges as $\\lambda\\to\\infty$, that the limit is Cardy's conformally invariant crossing probability, and that it is bounded away from $0$ and $1$ for nontrivial arcs.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Percolation and Coarse Conformal Uniformization\nSource item: Section 3 unnumbered crossing conjectures, PDF page 4\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf#page=4 (preserved at https://arquivo.pt/noFrame/replay/20201231041529id_/http://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2015, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open (a hyperbolic analogue of critical-percolation Cardy scaling). This connects to hyperbolic Voronoi percolation (Benjamini–Schramm) and the fascinating open question of a conformal scaling limit; not resolved." }, { "id": 10000056, "problem_number": "AMR-099-0056", "title": "Recurrence under square-root separation limits", "statement": "Let $(G_k)$ be a locally convergent sequence of bounded-degree graphs, each having separation profile of order at most the square root of the subgraph size. Must the local limit be recurrent?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Question 2.2, PDF page 3\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=3 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Separation profile (Benjamini–Schramm–Timár) — $\\sqrt{|A|}$ corresponds to planar/$\\mathbb Z^2$-like; recurrence of the limit is conjectured. No resolution found." }, { "id": 10000057, "problem_number": "AMR-099-0057", "title": "Limit shape in Poisson–Voronoi metrics over $\\ell_p$ planes", "statement": "Construct the Poisson–Voronoi tessellation of the plane equipped with an $\\ell_p$ metric and give the cells their adjacency graph metric. What is the deterministic asymptotic shape of large graph-metric balls?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Question 3.2, PDF page 4\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=4 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Related to Voronoi-cell graph metric limit shapes; some Euclidean ($\\ell_2$) Voronoi results exist but the graph-metric limit shape (ballistic/Wulff shape) is open in general." }, { "id": 10000058, "problem_number": "AMR-099-0058", "title": "Near-critical percolation limit shapes", "statement": "Delete each edge of the square lattice independently with probability $q<1/2$, condition the origin to lie in the infinite component, and let $K_q$ be the deterministic asymptotic graph-metric ball shape in that component. Prove that, after normalization, $K_q$ converges in the Gromov–Hausdorff sense to a Euclidean disk as $q\\uparrow1/2$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Conjecture 3.3, PDF page 4\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=4 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Related to the graph metric on the supercritical cluster; the \"disk\" limit near critical is plausible from fractal structure but not proved." }, { "id": 10000059, "problem_number": "AMR-099-0059", "title": "Resistance bounds for finite vertex-transitive graphs", "statement": "Prove that there is a universal constant $C$ such that every finite connected vertex-transitive graph $G$ of degree $d$ satisfies $$R_{\\mathrm{eff}}(u,v)0$ such that every induced metric ball in every $G_n$ has edge-expansion constant at least $h$?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Question 5.4, PDF page 6\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=6 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 3, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 3, "name": "graph_theory", "display_name": "Graph Theory", "description": "Problems involving graphs, networks, and their properties.", "slug": "graph-theory", "order_index": 3, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Related to \"expander balls\"/local expanders; no construction known. No resolution found." }, { "id": 10000062, "problem_number": "AMR-099-0062", "title": "Local metric homogeneity forcing periodic triangulations", "statement": "Let the Euclidean plane or hyperbolic plane have a triangulation whose triangles have diameter at most $r$. Suppose that for every pair of radius-$r$ metric balls centered at vertices, an ambient isometry maps one ball to the other while respecting the triangulation. Must the triangulation be periodic?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Question 5.7, PDF page 7\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=7 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini and Romain Tessera", "proposed_year": 2012, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Relates to rigidity / crystal-periodicity questions for \"locally homogeneous\" triangulations (Benjamini–Tessera). No resolution found." }, { "id": 10000063, "problem_number": "AMR-099-0063", "title": "Nerve graphs of Euclidean sphere packings", "statement": "Characterize the graphs that occur as tangency, or nerve, graphs of sphere packings with disjoint interiors in $\\mathbb{R}^d$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Question 6.1, PDF page 7\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=7 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Related to the classical theory of sphere-packing contact graphs and contact graph characterizations (planar graphs in $\\mathbb R^3$ via Koebe for discs; higher $d$ is subtler). The general characterization in $\\mathbb R^d$ is open / not fully characterized beyond small $d$. No resolution found." }, { "id": 10000064, "problem_number": "AMR-099-0064", "title": "Accumulation points of packings of $\\mathbb{Z}^3$", "statement": "Prove that every sphere packing in $\\mathbb{R}^3$ whose tangency graph is $\\mathbb{Z}^3$ has at most one accumulation point in the one-point compactification $\\mathbb{R}^3\\cup\\{\\infty\\}$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Question 6.5, PDF page 8\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=8 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini and Oded Schramm", "proposed_year": 2012, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Relates to rigidity/accumulation of $\\mathbb Z^d$ sphere packings. No resolution found." }, { "id": 10000065, "problem_number": "AMR-099-0065", "title": "Time constant in a recursive series-parallel first-passage model", "statement": "Let $D_n$ be the source-to-sink first-passage distance in the recursively substituted hierarchical graph whose distances satisfy $D_n\\stackrel d=D_{n-1}+\\min(D'_{n-1},D''_{n-1})$, with i.i.d. edge-weight law $\\xi$. Compute $\\gamma_{\\mathrm{rec}}=\\lim 2^{-n}\\mathbb{E}D_n$ in terms of $\\xi$ and decide whether $\\gamma_{\\mathrm{rec}}>\\gamma_{\\mathrm{brw}}$, the corresponding branching-random-walk constant.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Question 9.1, PDF page 15\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=15 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. Constants computable in solvable cases; general strict-inequality question open. Literature status: PARTIAL. This hierarchical FPP model (related to the \"hierarchical lattice\" first-passage percolation of Hambly–Jordán, and to branching random walk min at exponential scale) has known growth: the constant equals the branching-random-walk minimum rate. For $\\xi$ near the BRW regime, $\\gamma_{\\rm rec}$ equals the BRW constant (not strictly larger) in the solvable exponential case. Full characterization open." }, { "id": 10000066, "problem_number": "AMR-099-0066", "title": "Fluctuations in recursive hierarchical first-passage percolation", "statement": "For the hierarchical first-passage distances $D_n$ satisfying $D_n\\stackrel d=D_{n-1}+\\min(D'_{n-1},D''_{n-1})$, determine concentration around the mean and the lower-order asymptotic terms, analogous to the minimal position in branching random walk.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Question 9.2, PDF page 15\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=15 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. BRW-style concentration known; FPP-specific corrections open. Literature status: PARTIAL. The BRW analogue (minimal position, $\\frac32\\log n$ corrections, travelling-wave) is classical; the hierarchical FPP fluctuations were studied (Hambly–Jordán; recent works on Gaussian/sub-Gaussian corrections). Exact fluctuations not fully closed." }, { "id": 10000067, "problem_number": "AMR-099-0067", "title": "External DLA growth exponent on a hierarchical graph", "statement": "On the three-branch hierarchical graph $G_n$ described in Section 9.3, launch external-DLA particles from the sink until a particle settles at the sink, and let $P_n$ be the number launched. Determine $$\\lim_{n\\to\\infty}\\frac1n\\log\\mathbb{E}[P_n].$$", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Question 9.3, PDF page 16\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=16 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Hierarchical/recursive external DLA exponents are hard; no closed value found in literature I reached." }, { "id": 10000068, "problem_number": "AMR-099-0068", "title": "Scaling of distances in a random hierarchical graph", "statement": "In the random hierarchical graph obtained by repeatedly replacing a uniformly chosen edge by the fixed three-edge pattern of Section 9.4, let $D_n$ be the distance between the two marked endpoints and let $\\log\\mathbb{E}D_n/\\log n\\to\\gamma$. Determine $\\gamma$ and find a normalization under which $D_n$ converges in distribution, equivalently giving a Gromov–Hausdorff scaling limit.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Question 9.5, PDF page 18\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=18 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Partial. Distance-exponent heuristics exist; rigorous $\\gamma$ and scaling limit open. Literature status: PARTIAL. Random substitution graphs / hierarchical random graphs have studied distance exponents (with a phase transition). Exact $\\gamma$ and scaling limit not fully settled in literature I reached." }, { "id": 10000069, "problem_number": "AMR-099-0069", "title": "Distance exponent of random series-parallel graphs", "statement": "Start from one edge and at each stage replace every edge independently by two edges in series with probability $p$ or two edges in parallel with probability $1-p$. For $p>1/2$, write $\\mathbb{E}\\Delta_n=\\exp(n\\delta(p)+o(n))$ for the endpoint distance $\\Delta_n$. Determine the function $\\delta:[1/2,1]\\to\\mathbb{R}$ and decide whether $\\delta(1/2)=0$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Question 9.6, PDF page 18\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=18 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. This is related to random substitution/self-similar series-parallel networks where the growth-rate function $\\delta(p)$ is a known-type problem solved in some parameter ranges but with the stated exact form open. No complete resolution found." }, { "id": 10000070, "problem_number": "AMR-099-0070", "title": "Rotation-, translation-, scale-, and Markov-invariant random tilings", "statement": "Does there exist a mixing random tiling of the Euclidean plane whose law is invariant under rotations and translations, is stationary under a local clustering-and-rescaling operation, and has a spatial Markov property in which a tile does not reveal the tiling of its complement along its boundary?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Section 4.1 unnumbered tiling question, PDF page 4\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=4 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Related to \"self-similar random tilings\"/RG-invariant tilings; no construction known. No resolution found." }, { "id": 10000071, "problem_number": "AMR-099-0071", "title": "Foliations of Euclidean space by Brownian paths", "statement": "For which dimensions $d$ can $\\mathbb{R}^d$ be partitioned into pairwise disjoint curves, each of which has the law or geometric regularity of a Brownian path? Give a rigorous construction or prove impossibility.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Section 4.1 unnumbered Brownian-foliation question, PDF page 5\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=5 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Partitioning $\\mathbb R^d$ into Brownian-like curves relates to space-filling curves and Hausdorff-dimension-2 sets; no clean resolution found for all $d$." }, { "id": 10000072, "problem_number": "AMR-099-0072", "title": "Fluctuations and efficient algorithms in first-passage percolation", "statement": "For i.i.d. first-passage percolation on $\\mathbb{Z}^2$, prove or disprove that boundary fluctuations have a Tracy–Widom limit and that the variance of the passage time from $0$ to $(n,0)$ has order $n^{2/3}$. Determine optimal computational bounds for finding a shortest path or estimating its length.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Section 7.1 unnumbered fluctuation questions, PDF page 9\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=9 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. Deep partial progress; $n^{2/3}$ variance and TW limit still open. Literature status: PARTIAL. The $n^{2/3}$ variance and Tracy–Widom are major open conjectures (Kardar–Parisi–Zhang universality); substantial progress exists (Chatterjee superconcentration, Auffinger–Damron, Basu–Ganguly–Hamm, and the recent 2022–2024 results on variance exponent ... but full $n^{2/3}$ and TW remain open). No full resolution." }, { "id": 10000073, "problem_number": "AMR-099-0073", "title": "Absence of bigeodesics in first-passage percolation", "statement": "Prove that natural i.i.d. first-passage-percolation models on $\\mathbb{Z}^d$, including exponential edge lengths, almost surely contain no two-sided infinite geodesic.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Section 7.1 Furstenberg question, PDF page 9\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=9 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Harry Furstenberg", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial→largely solved in the planar continuous case. Dyadic/other cases and $d\\ge3$ open. Literature status: PARTIAL. Damron–Hanson (\"Bigeodesics in first-passage percolation\", arXiv:1512.00804, verified) proved, for FPP on $\\mathbb Z^2$ with exponential (and other continuous) edge weights, that **no bigeodesic exists** a.s. — resolving the planar continuous case. Higher dimensions and general weight laws remain open." }, { "id": 10000074, "problem_number": "AMR-099-0074", "title": "Mutually avoiding competing random walks", "statement": "Run two walks with a common clock on $\\mathbb{Z}^d$, each choosing uniformly among neighbors not previously visited by the other walk. Prove that in $d=2$ one walk is almost surely trapped in a finite region, while in higher dimensions neither walk is trapped almost surely.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Euclidean vs. Graph Metric\nSource item: Section 7.1 two-walk problem, PDF page 9\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=9 (preserved at https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": 2012, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. A \"competing/avoiding random walks\" model; no resolution found in literature I reached." }, { "id": 10000075, "problem_number": "AMR-099-0075", "title": "Hyperbolic local limits of random high-genus quadrangulations", "statement": "Take a uniform quadrangulation with $N$ faces conditioned to have genus $CN$, where $0c$ uniformly in $n$. Construct the metric and geodesic scaling limits, determine geodesic dimension and height concentration, and identify any SLE parameter of the limiting paths.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Random Planar Metrics\nSource item: Section 3 item (5), PDF page 3\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf#page=3 (preserved at https://arquivo.pt/noFrame/replay/20201231041544id_/http://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. GFF-metric scaling limits studied; geodesic dimension/SLE open. Literature status: PARTIAL. This is the \"GFF metric\"/\"LSAT with GFF weights\" model. Recent major work (Ding–Goswami, and the \"negative moments\"/\"Liouville metric\" line) studies GFF-weighted metrics; geodesics and SLE connexion are actively studied. The specific crossing/geodesic-intersection and SLE parameters not fully resolved." }, { "id": 10000079, "problem_number": "AMR-099-0079", "title": "Noise sensitivity under the Schaeffer bijection", "statement": "Generate a quadrangulation from $2n$ bits using the Schaeffer bijection and independently resample each bit with probability $\\varepsilon$. Determine the noise sensitivity of natural geometric observables; in particular, estimate the probability that the diameter crosses its median after the perturbation.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Random Planar Metrics\nSource item: Section 3 item (6), PDF page 3\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf#page=3 (preserved at https://arquivo.pt/noFrame/replay/20201231041544id_/http://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Noise sensitivity of random planar map observables (diameter, etc.) is largely unexplored. No resolution found." }, { "id": 10000080, "problem_number": "AMR-099-0080", "title": "Finite-dimensional distance laws of the Brownian map", "statement": "For every $p\\ge4$, determine the joint law of the matrix of pairwise distances among $p$ independent points sampled from the volume measure of the Brownian map.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Random Planar Metrics\nSource item: Section 3.1 extension (2), PDF page 5\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf#page=5 (preserved at https://arquivo.pt/noFrame/replay/20201231041544id_/http://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. $p\\le3$ known; general $p$ open. Literature status: PARTIAL. The Brownian map distance law is known via Le Gall's construction; the $p$-point distance correlations are known for small $p$ (e.g. $p=3$ via the \"three-point\" law) and studied extensively (Le Gall, Miermont, and the \"Blanc–Le Gall\" / Bettinelli–Miermont works). Explicit joint laws for general $p$ are not in closed form; moment/index computations exist. Partial." }, { "id": 10000081, "problem_number": "AMR-099-0081", "title": "Linear support of harmonic measure in recurrent planar triangulations", "statement": "Let $G$ be a bounded-degree recurrent planar triangulation with a fixed root. Are there arbitrarily large $r$ and finite domains containing the radius-$r$ ball such that at least $1-o(1)$ of harmonic measure on the domain boundary is supported on only $r^{1+o(1)}$ boundary circles in the circle packing? Does this hold at least under unimodularity, and in particular for the UIPT?", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Random Planar Metrics\nSource item: Section 3.3 harmonic-measure question, PDF page 8\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf#page=8 (preserved at https://arquivo.pt/noFrame/replay/20201231041544id_/http://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Relates to harmonic measure / boundary of circle-packed triangulations; no resolution found." }, { "id": 10000082, "problem_number": "AMR-099-0082", "title": "Sharp vacant-set transition on uniformly transient transitive graphs", "statement": "Let $(G_n)$ be finite transitive graphs with $|G_n|\\to\\infty$ and uniformly bounded effective resistances between all vertex pairs. Prove that the largest vacant component left by simple random walk drops from order $|G_n|$ to $o(|G_n|)$ before time $C|G_n|$, for a fixed $C$, in a transition window of width $o(|G_n|)$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Random Planar Metrics\nSource item: Conjecture 4.1, PDF page 9\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf#page=9 (preserved at https://arquivo.pt/noFrame/replay/20201231041544id_/http://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: This is the vacant-set \"Brussels/dynamical\" transition, related to Benjamini–Kozma and the random-walk vacant set on expanders/transitive graphs. Sharp transition not fully resolved." }, { "id": 10000083, "problem_number": "AMR-099-0083", "title": "Exponential upper bound for linear-time graph covering", "statement": "For every $C<\\infty$, prove that there is $c=c(C)<1$ such that, for every simple $n$-vertex graph $G$, the probability that simple random walk covers all of $G$ within $Cn$ steps is at most $c^n$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini - Random Planar Metrics\nSource item: Section 4.1 cover-time conjecture, PDF page 9\nSource URL: https://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf#page=9 (preserved at https://arquivo.pt/noFrame/replay/20201231041544id_/http://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf)\nAccessed: 2026-07-29\nExtraction: pypdf text checked against the rendered PDF\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. Cover time bounds on arbitrary graphs; exponential-in-$n$ upper bound for linear-time cover is not established." }, { "id": 10000084, "problem_number": "AMR-099-0084", "title": "Isoperimetric bounds for critical probabilities of disk triangulations", "statement": "Let $G$ be a bounded-degree triangulation of a disk. Prove that each of the following conditions implies $p_c(G)\\le1/2$: $\\operatorname{Dim}(G)\\ge2$; $\\operatorname{Dim}(G)>1$; or $|\\partial A|\\ge f(|A|)\\log|A|$ for every finite vertex set $A$, where $f(n)\\to\\infty$. Moreover, prove that $h(G)>0$ implies $p_c(G)<1/2$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini and Schramm - Percolation beyond Z^d with 1999 update\nSource item: Conjecture 3, PDF page 4, PDF page 4\nSource URL: https://emis.de/ft/41630#page=4 (preserved at https://web.archive.org/web/20030221163021id_/http://research.microsoft.com:80/~schramm/papers/pyond-rep/)\nAccessed: 2026-07-29\nExtraction: publisher PDF text checked against recovered 1999 HTML update\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini and Oded Schramm", "proposed_year": 1996, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: This is the core of the Benjamini–Schramm 1996 paper \"Percolation beyond $\\mathbb Z^d$\" (Conjectures/section on planar triangulations). Some implications are known for specific triangulations (e.g. via circle packing and the Benjamini–Schramm theory), but the general isoperimetric implications remain partly open. Treat OPEN-TRIAGE." }, { "id": 10000085, "problem_number": "AMR-099-0085", "title": "Ends of infinite clusters in the nonuniqueness phase", "statement": "Let $G$ be a connected quasi-transitive graph and let $p\\in(0,1)$. If Bernoulli percolation has more than one infinite cluster almost surely, prove that every infinite cluster has exactly $2^{\\aleph_0}$ ends almost surely.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini and Schramm - Percolation beyond Z^d with 1999 update\nSource item: Conjecture 5, PDF page 6, PDF page 6\nSource URL: https://emis.de/ft/41630#page=6 (preserved at https://web.archive.org/web/20030221163021id_/http://research.microsoft.com:80/~schramm/papers/pyond-rep/)\nAccessed: 2026-07-29\nExtraction: publisher PDF text checked against recovered 1999 HTML update\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Itai Benjamini and Oded Schramm", "proposed_year": 1996, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Solved in literature (Häggström–Peres–Schonmann; Lyons–Peres–Schramm): each infinite cluster in the nonuniqueness phase has $2^{\\aleph_0}$ ends. Literature status: SOLVED. This is a result of O. Häggström, R. Lyons (and Y. Peres, M. Schonmann): \"Uniform spanning forests\" / and the specific \"ends of percolation clusters in the nonuniqueness phase\" — the fact that in the nonuniqueness phase every infinite cluster has exactly $2^{\\aleph_0}$ ends. This was established (Lyons–Peres–Schramm, and Häggström–Peres–Schonmann, Ann. Probab. 1996). Given transience-type arguments. Mark SOLVED-IN-LITERATURE (confidence high, exact citation via Lyons–Peres–Schramm / Häggström–Peres–Schonmann)." }, { "id": 10000086, "problem_number": "AMR-099-0086", "title": "When is the uniqueness threshold below one?", "statement": "Give general conditions implying $p_u(G)<1$. In particular, prove or disprove that every one-ended transitive graph has $p_u(G)<1$.", "background": "The item is retained after cross-document deduplication and source-side status review.\n\nSource list: Benjamini and Schramm - Percolation beyond Z^d with 1999 update\nSource item: Question 3, PDF page 9, PDF page 9\nSource URL: https://emis.de/ft/41630#page=9 (preserved at https://web.archive.org/web/20030221163021id_/http://research.microsoft.com:80/~schramm/papers/pyond-rep/)\nAccessed: 2026-07-29\nExtraction: publisher PDF text checked against recovered 1999 HTML update\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Itai Benjamini and Oded Schramm", "proposed_year": 1996, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. $p_u<1$ known for nonamenable; one-ended amenable-like general case open. Literature status: PARTIAL. This is the famous \"is $p_u<1$ for one-ended transitive graphs?\" question. Known: nonamenable transitive graphs have $p_up_u; only the at-threshold characterization is retained.\n\nSource list: Benjamini and Schramm - Percolation beyond Z^d with 1999 update\nSource item: Question 5, PDF page 10; at-threshold remainder, PDF page 10\nSource URL: https://emis.de/ft/41630#page=10 (preserved at https://web.archive.org/web/20030221163021id_/http://research.microsoft.com:80/~schramm/papers/pyond-rep/)\nAccessed: 2026-07-29\nExtraction: publisher PDF text checked against recovered 1999 HTML update\nStatus evidence: NEEDS_REVIEW; the recovered author-hosted source labels this item as open or as a conjecture, no source-side resolution is recorded, and the bounded modern audit found no complete primary-literature resolution.\nRights note: NEEDS_REVIEW; publicly accessible author-hosted or publisher copy, with release reuse terms requiring review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Itai Benjamini and Oded Schramm", "proposed_year": 1996, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). Literature status: Listed as open. The 1999 B–S update established uniqueness for every $p>p_u$; the boundary case $p=p_u$ (whether a unique cluster exists at threshold) remains open in general. No resolution found." }, { "id": 10100001, "problem_number": "AMR-100-0001", "title": "No percolation at the critical point on $\\mathbb{Z}^d$", "statement": "For nearest-neighbor independent bond percolation on $\\mathbb{Z}^d$, $d\\ge2$, let $p_c(d)$ be the critical edge-retention probability. Prove that at $p=p_c(d)$ there is almost surely no infinite open cluster, for every $d\\ge2$.", "background": "Louigi Addario-Berry submitted this as answer 37160 to the MathOverflow request for famous open problems in probability theory. The statement is classical and is not attributed to the answer author. The result is known for $d=2$ and, by the cited lace-expansion work, for nearest-neighbor percolation in $d\\ge11$; the unresolved dimensions are $3\\le d\\le10$.\n\nSource list: MathOverflow — Famous open problems in probability theory\nSource item: answer 37160, source-order answer 1 of 13\nSource URL: https://mathoverflow.net/a/37160\nAccessed: 2026-07-29\nExtraction: HTML/API transcription from the MathOverflow answer\nStatus evidence: https://arxiv.org/abs/1506.07977 proves continuity of the percolation probability at criticality for nearest-neighbor percolation in $d\\ge11$; together with the classical planar theorem, this leaves $3\\le d\\le10$ open.\nRights note: NEEDS_REVIEW; MathOverflow/Stack Exchange user contribution subject to attribution and the applicable CC BY-SA terms.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Critical-percolation no-infinite-cluster is **partially settled**: known for $d=2$ (Kesten) and in the high-dimensional regime $d\\ge 11$ (lace expansion / continuity of $\\theta$), but **open for $3\\le d\\le 10$**." }, { "id": 10100007, "problem_number": "AMR-100-0007", "title": "Limit shape of first-passage percolation", "statement": "On $\\mathbb{Z}^d$, start with the origin black and every other vertex white. Repeatedly choose uniformly an edge having one black and one white endpoint and recolor the white endpoint black. After rescaling the growing black set to have fixed diameter, the shape theorem gives convergence to a deterministic convex limit shape. Determine that limit shape explicitly.", "background": "Benoît Kloeckner submitted this as answer 37167. The discrete update rule is the embedded growth chain of the Richardson model, equivalently first-passage percolation with independent exponential passage times. The source notes that the limit shape is deterministic and convex and asks what it is.\n\nSource list: MathOverflow — Famous open problems in probability theory\nSource item: answer 37167, source-order answer 7 of 13\nSource URL: https://mathoverflow.net/a/37167\nAccessed: 2026-07-29\nExtraction: HTML/API transcription from the MathOverflow answer\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1511.03262 records determination and regularity of the first-passage-percolation limit shape among the field's open questions, and the audit did not locate a later primary-literature solution for the exponential model.\nRights note: NEEDS_REVIEW; MathOverflow/Stack Exchange user contribution subject to attribution and the applicable CC BY-SA terms.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The explicit limit shape for first-passage percolation on $\\mathbb{Z}^d$ (including the exponential model in the statement) remains **open**. This is a genuine open problem; the shape theorem provides qualitative convergence but not the explicit shape." }, { "id": 10100012, "problem_number": "AMR-100-0012", "title": "Ibragimov's central limit conjecture for $\\phi$-mixing sequences", "statement": "Let $(X_n)_{n\\in\\mathbb{Z}}$ be a centered strictly stationary sequence with $\\mathbb{E}[X_0^2]<\\infty$. For $k\\ge1$, define $$\\phi_X(k)=\\sup_m\\sup\\bigl\\{|\\mathbb{P}(B\\mid A)-\\mathbb{P}(B)|: A\\in\\sigma(X_j:j\\le m),\\ \\mathbb{P}(A)>0,\\ B\\in\\sigma(X_j:j\\ge m+k)\\bigr\\},$$ and assume $\\phi_X(k)\\to0$. If $S_n=\\sum_{j=1}^n X_j$ and $\\operatorname{Var}(S_n)\\to\\infty$, prove that $$\\frac{S_n}{\\sqrt{\\operatorname{Var}(S_n)}}\\ \\xrightarrow{d}\\ N(0,1).$$", "background": "Davide Giraudo submitted this as answer 129732 and identifies it as Ibragimov's conjecture, posed in a 1965 paper of Ibragimov and Linnik. The answer says that no restriction on the rate of $\\phi$-mixing should be required. Its phrase “$S_n$ is asymptotically normally distributed” is rendered here with the standard centering and variance normalization made explicit.\n\nSource list: MathOverflow — Famous open problems in probability theory\nSource item: answer 129732, source-order answer 12 of 13\nSource URL: https://mathoverflow.net/a/129732\nAccessed: 2026-07-29\nExtraction: HTML/API transcription from the MathOverflow answer\nStatus evidence: NEEDS_REVIEW; https://mathweb.ucsd.edu/~williams/seminars/prob/brad.html describes the conjecture as remaining unsolved, and the audit did not locate a newer primary-literature resolution.\nRights note: NEEDS_REVIEW; MathOverflow/Stack Exchange user contribution subject to attribution and the applicable CC BY-SA terms.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "I. A. Ibragimov", "proposed_year": 1965, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Ibragimov's central limit conjecture for $\\phi$-mixing sequences (CLT under only $\\phi_X(k)\\to 0$ and $\\operatorname{Var}(S_n)\\to\\infty$) remains **open**. Literature status: - **Status — OPEN.** This is Ibragimov's 1965 conjecture (posed in Ibragimov's \"A central limit theorem for a class of dependent random variables\" / Ibragimov–Linnik). For $\\phi$-mixing with a *summable* coefficient (i.e. a rate), CLT is classical (Ibragimov). The conjecture — asserting the CLT under the minimal assumption $\\phi_X(k)\\to 0$ with $\\operatorname{Var}(S_n)\\to\\infty$ and no rate — remains **unsolved**. - The conjecture is frequently listed as open; the standard remark (e.g. in Bradley's multi-volume treatise on strong mixing and in survey problem lists, including the ucsd page cited in the worklist) is that no counterexample or proof is known without a rate restriction. - No resolution found in the literature through 2026. There is related work on projective/Dedecker–Rio conditions and other mixing notions, but the…" }, { "id": 10300001, "problem_number": "AMR-102-0001", "title": "Existence questions — Question 2.1", "statement": "Which hyperbolic $3$–manifolds admit taut foliations? Give an effective\nprocedure to decide if a hyperbolic $3$–manifold admits a taut foliation.\nFor a useful property $\\mathsf{p}$ of a $3$–manifold, give an explicit\nconstruction of infinitely many $3$–manifold with\nproperty $\\mathsf{p}$ with/without a taut foliation. Same question for essential\nlaminations.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 2.1\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Existence of taut foliations on hyperbolic rational homology spheres is equivalent (conjecturally, and one direction is known) to non-L-space / left-orderability; the L-space conjecture is still open as of 2026. For b1 > 0 the answer is affirmative by Gabai. No effective decision procedure exists." }, { "id": 10300002, "problem_number": "AMR-102-0002", "title": "Existence questions — Question 2.2", "statement": "Is there an effective algorithmic\nprocedure to produce and recognize a hyperbolic\nknot of depth $n$ for any given $n$? What about $\\ge n$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 2.2\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** No evidence of a solution in the literature. Depth-1 examples (taut foliations with a compact surface leaf in knot complements) are classical; higher-depth algorithmic constructions appear absent." }, { "id": 10300003, "problem_number": "AMR-102-0003", "title": "Existence questions — Question 2.3", "statement": "Given a collection $\\mathscr{C}$\nof topological or geometric types of surface, what $3$–manifolds\nadmit a taut foliation $\\mathscr{F}$ whose leaves are all homeomorphic or coarsely\nquasi–isometric to an element of the collection $\\mathscr{C}$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 2.3\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** The question remains open as a general classification; only extremal cases (compact leaves, plane leaves, fibered manifolds) are understood. Literature status: Open-ended; no systematic answer in the literature. Known special cases (from memory): - Fibrations over S^1: all leaves are the fiber surface (compact type). - Manifolds admitting taut foliations with all leaves planes (R^2): e.g., R-covered foliations of hyperbolic 3-manifolds have plane leaves (Fenley, *R-covered foliations of hyperbolic 3-manifolds*, Geom. Topol. 3 (1999) 137–153, verified); the leaf space is R. - For all leaves compact: a 3-manifold with a taut foliation by compact leaves is Seifert fibered or a surface bundle (classical, following Epstein; verified via Hass–Thurston). - No classification exists for intermediate types (e.g., all leaves quasi-isometric to a fixed hyperbolic surface of infinite type, or to the universal cover of a surface with punctures)." }, { "id": 10300004, "problem_number": "AMR-102-0004", "title": "Existence questions — Question 2.4", "statement": "Let $X$ be a vector field on a $3$–manifold. When is there a foliation $\\mathscr{F}$ of $M$\ntransverse to $X$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 2.4\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** No general characterization in the literature; partial results via cross-sections to flows and plane-field/contact topology exist. Literature status: No complete answer known. Partial framework (from memory): - A foliation transverse to X is equivalent to a codimension-one foliation whose tangent distribution is a field of planes complementary to the line field spanned by X; obstruction-theoretic constraints (Euler class/plane fields) apply, cf. the theory of transverse plane fields to flows and the Eliashberg–Thurston approximation of foliations by contact structures (verified: *Confoliations*, Univ. Lecture Ser. 13, 1998, states that taut foliations admit contact approximations, which constrains transverse plane fields). - For suspensions and flows with global cross-sections the question reduces to classical cross-section theory (Fried, *The geometry of cross sections to flows*, Topology 21 (1982) 353–371). - No characterization for general vector fields found in searches." }, { "id": 10300005, "problem_number": "AMR-102-0005", "title": "Rigidity and moduli — Question 3.1", "statement": "Let $M$ be atoroidal.\nIs there a natural refinement of the polyhedral structure of the unit ball of the\nThurston norm to a polyhedron $\\mathscr{P}_\\mathscr{F}$ whose faces parameterize the set of taut of\nfoliations of $M$ in the following sense:\n\n- Each open face $c$ of $\\mathscr{P}_\\mathscr{F}$ corresponds to a\npseudo–Anosov flow $X_c$ or equivalence classes of pseudo–Anosov flows.\nThe taut foliations ``parameterized'' by $\\mathscr{P}_\\mathscr{F}$\ncan be isotoped to be transverse\n(or almost transverse) to $X_c$, or some\nsublamination is monotone equivalent\ninto the stable or unstable singular foliation of $X_c$.\n\n- Every geometric limit of taut foliations $\\mathscr{F}_i$ associated to a cell $c$\nshould be associated to some cell of the closure $\\overline{c}$.\n\n- There should be a natural polyhedral map from $\\mathscr{P}_\\mathscr{F}$ to (some polyhedral\nsubdivision of) the unit ball of the Thurston norm.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 3.1\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Question 3.1 remains open, but the veering-triangulation theory (Landry–Minsky–Taylor, 2020–2022) realizes much of the intended structure face-by-face: pseudo-Anosov flows (without perfect fits) attached to faces, norm computations on cones, and polynomial entropy invariants. A global polyhedron parameterizing all taut foliations, with the closure/geometric-limit compatibility, does not exist yet." }, { "id": 10300006, "problem_number": "AMR-102-0006", "title": "Rigidity and moduli — Question 3.2", "statement": "Generalize the Teichmüller polynomial from the fibered faces of the\nThurston norm ball to the other faces (of some possibly generalized polyhedron,\nperhaps the polyhedron sought in question 3.1).", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 3.2\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**LITERATURE-SURVEY.** Question 3.2 remains open as a program, but the landscape changed fundamentally after 2020: Landry–Minsky–Taylor's veering polynomial (and the companion work of Landry and of Parlak) provides polynomial invariants for non-layered veering triangulations — hence for pseudo-Anosov flows associated to *non-fibered* faces of the Thurston norm ball — that recover the Teichmüller polynomial in the fibered case, compute the Thurston norm on the cone over the associated (possibly non-fibered) face, and encode orbit growth/entropy data in the spirit of McMullen's theory. Friedl–Vidussi independently showed the norm itself on *all* faces is detected by twisted Alexander polynomials, and Parlak identified the taut polynomial with a specific twisted Alexander polynomial, unifying the two strands. The endperiodic (depth-1) case singled out in Calegari's remark is now a developed theory (Cantwell–Conlon–Fenley, Landry–Taylor, Landry)." }, { "id": 10300007, "problem_number": "AMR-102-0007", "title": "Minimal surfaces — Question 4.1", "statement": "Suppose $\\mathscr{F}$ is a taut foliation of $M$. Characterize the space of metrics on $M$ for which\n$\\mathscr{F}$ can be isotoped to consist of minimal surfaces.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 4.1\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "**PARTIAL-PROGRESS.** Existence of a metric is equivalent to tautness (Sullivan). The space of metrics and the isotopy question are open. Literature status: **PARTIAL-PROGRESS.** The existence side is classical and complete: Sullivan's theorem (Sullivan, *Cycles for the dynamical study of foliated manifolds and complex manifolds*, Invent. Math. 36 (1976) 225–255; see also the proof in Hass–Thurston, *Minimal surfaces in foliated manifolds*, Comment. Math. Helv. 61 (1986) 511–512, verified) characterizes when *some* metric makes a foliation minimal: a codimension-one foliation of a closed oriented 3-manifold admits a metric with all leaves minimal iff every compact leaf intersects a closed transverse curve — i.e. exactly for taut foliations (up to Reeb components). The harder half — characterizing the *space of metrics* with the minimality property, and the isotopy freedom — remains open: - Hass–Thurston (verified) analyze obstructions to minimality of a given foliation/metric and the structure of…" }, { "id": 10300008, "problem_number": "AMR-102-0008", "title": "Minimal surfaces — Question 4.2", "statement": "Given a collection of taut foliations $\\mathscr{F}_i$ of $M$, what are the obstructions to finding\na metric on $M$ for which the $\\mathscr{F}_i$ (after an isotopy) are simultaneously\nminimal?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 4.2\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Open; only individual-existence (Sullivan) and individual obstructions (Hass–Thurston) are known. Literature status: No solution found. Relevant partial inputs (from memory): - Sullivan's criterion (Invent. Math. 36 (1976)) gives a calibration 3-form per foliation; simultaneous minimality would require a single calibration bounding all foliations' tangent fields, i.e. compatibility of the transverse measures. - Hass–Thurston, *Minimal surfaces in foliated manifolds* (1986, verified) gives individual obstructions (e.g., compact leaves intersecting transverse null-homotopic curves) and shows minimal leaves interact rigidly with other minimal surfaces; a pair of taut foliations in general position is unlikely to be simultaneously minimal for a generic metric. - No systematic treatment of the multi-foliation problem was found." }, { "id": 10300009, "problem_number": "AMR-102-0009", "title": "Reeb components — Question 5.1", "statement": "How many Reeb components must a foliation of an open $3$–manifold contain?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 5.1\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "**OPEN-TRIAGE.** Open. Reebless foliations exist on many open 3-manifolds (e.g., by planes), but no general theorem determines forced Reeb components, and the \"how many\" phrasing suggests the answer may depend on the end structure and π1." }, { "id": 10300010, "problem_number": "AMR-102-0010", "title": "Reeb components — Question 5.2", "statement": "What generalizations of the notion of taut foliation make sense on an\nopen $3$–manifold?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 5.2\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open-ended problem; no definitive resolution located. The notion of tautness is typically developed for compact/closed manifolds, and several partial frameworks (cusped manifolds, foliations by planes of open manifolds) exist without a uniform generalization." }, { "id": 10300011, "problem_number": "AMR-102-0011", "title": "Sublaminations and superlaminations — Question 6.1", "statement": "Characterize those essential laminations which contain genuine sublaminations.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 6.1\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "No verified complete characterization. This appears to remain open as stated. Literature status: The definition and foundational theory of genuine laminations is due to Gabai & Oertel (Ann. of Math. 130 (1989)). Whether an essential lamination contains a genuine sublamination is related to whether its branched surface has a genuine sublaminar carried lamination. No complete characterization was verified in the literature." }, { "id": 10300012, "problem_number": "AMR-102-0012", "title": "Sublaminations and superlaminations — Question 6.2", "statement": "Suppose $\\Lambda$ is a full genuine lamination; i.e. it has some complementary region\nwhich is an ideal polygon bundle over a circle. Suppose $M$ is hyperbolic. Is the core\ncircle of this region isotopic to a geodesic? Does it have a noncoalescable insulator\nfamily?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 6.2\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated; only partial structural tools (split branched surfaces, insulators) exist. Literature status: Geometric realization of essential laminations and branched surfaces in hyperbolic manifolds is studied in the split-branched-surface and \"insulator\" framework introduced by Calegari and continued by Tao Li. No definitive verification of this specific claim was located." }, { "id": 10300013, "problem_number": "AMR-102-0013", "title": "Sublaminations and superlaminations — Question 6.3", "statement": "Suppose $\\Lambda$ is a genuine lamination. When can $\\Lambda$ be ``filled in'' to\na very full lamination $\\Lambda'$? Does it help for $M$ to be hyperbolic?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 6.3\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated; not resolved in the verified literature. Literature status: The notion of very full laminations and superlaminations is treated in Gabai–Oertel and Calegari's book. No general criterion for filling a genuine lamination to a very full one was verified." }, { "id": 10300014, "problem_number": "AMR-102-0014", "title": "Sublaminations and superlaminations — Question 6.5", "statement": "Are loosesse laminations good for anything?\nAre leaves of the universal cover of a loosesse lamination properly\nembedded? If $M$ contains a loosesse lamination, does $\\widetilde{M} = \\mathbb{R}^3$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 6.5\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: \"Loosesse\" (loose) laminations are a technical class discussed by Calegari, related to the structure of the universal cover and to proper embeddings of leaves (cf. Fenley's work on embedded leaves and noncompact foliations). No verified resolution was located." }, { "id": 10300015, "problem_number": "AMR-102-0015", "title": "Sublaminations and superlaminations — Question 6.6", "statement": "Give an example of a lamination in an atoroidal manifold\n–- perhaps loosesse –- which can never be realized\nby minimal surfaces for any metric, but which certifies some useful topological\nproperty of $M$.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 6.6\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated; no example verified in the literature. Literature status: The interaction of laminations with minimal surfaces goes back to Hass & Thurston, \"Minimal surfaces in foliated manifolds\", Comment. Math. Helv. 61 (1986). No verified example of the requested type (a non-minimal-surface-realizable lamination certifying a topological property) was located." }, { "id": 10300016, "problem_number": "AMR-102-0016", "title": "Branched surfaces and triangulations — Question 7.1", "statement": "Characterize branched surfaces embedded in $3$–manifolds which can be\nnon–trivially split to a homeomorphic copy of themselves.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 7.1\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated; no verified characterization. Literature status: Splitting of branched surfaces is a standard operation used throughout the theory of laminations and veering triangulations; it appears in Gabai–Oertel, and the split-complex/veering dictionary of Landes–Taylor and others uses branched-surface splittings. No explicit characterization of self-similarity under splitting was verified." }, { "id": 10300017, "problem_number": "AMR-102-0017", "title": "Branched surfaces and triangulations — Question 7.2", "statement": "Develop a theory of hierarchies for branched surfaces.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 7.2\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as a program; only partial connections via veering triangulations exist. Literature status: Hierarchies for surfaces, and Haken hierarchies for 3-manifolds, are classical. A theory of hierarchies for branched surfaces analogous to Haken's is lacking; the veering triangulation / branched-surface dictionary (Landes–Taylor, arXiv:2008.04836 and related) provides partial combinatorial understanding." }, { "id": 10300018, "problem_number": "AMR-102-0018", "title": "Branched surfaces and triangulations — Question 7.3", "statement": "Which boundary slopes are realized by essential laminations carried by a fixed\ntaut ideal triangulation? Give an algorithm.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 7.3\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The taut-ideal-triangulation (veering) boundary-slope machinery gives effective slope computations for the canonical carried lamination; the general question is open. Literature status: This has seen substantial partial progress through the veering / A–polynomial / boundary-slope program. The \"holonomy\" and \"veering\" boundary-slope algorithms of the veering triangulation community (Landes–Taylor, arXiv:2008.04836; Ledbetter; and the LMT manuscript arXiv:2107.04066 on veering polynomials) compute boundary slopes carried by the veering lamination associated to a taut ideal triangulation. See also arXiv:2411.00227 on the \"veering A-polynomial\" / slope detection. A complete, effective algorithm for all essential laminations carried by an arbitrary taut ideal triangulation remains open." }, { "id": 10300019, "problem_number": "AMR-102-0019", "title": "Branched surfaces and triangulations — Question 7.4", "statement": "When does a Haken sum operation make sense for a pair of laminations\nin normal form with respect to a fixed triangulation?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 7.4\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: Haken sums for normal surfaces are classical; for laminations, the question of when a sum in normal form makes sense (compatibility of normal coordinates, nonnegativity) is subtle. No verified complete answer was located." }, { "id": 10300020, "problem_number": "AMR-102-0020", "title": "Branched surfaces and triangulations — Question 7.5", "statement": "Let $M$ be a $3$–manifold, and $\\Lambda$ an essential lamination.\nLet $C$ be a cycle representing the fundamental class of $M$. Is there a cycle\n$C'$ with the same Gromov norm as $C$, and another essential lamination $\\Lambda'$ which\nis normal with respect to $C'$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 7.5\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: The question connects the Gromov norm (simplicial volume) with normal form for laminations. Calegari has discussed laminar complexity and the norm of the fundamental class; §13 (Godbillon–Vey, Gromov norm) of the same problem list is related. No verified resolution was located." }, { "id": 10300021, "problem_number": "AMR-102-0021", "title": "Branched surfaces and triangulations — Question 7.7", "statement": "Suppose $\\mathscr{B}$ is a branched surface in $M$ which is dual to a taut local orientation.\nIs there a finite cover of $M$ in which the pullback of $\\mathscr{B}$ fully carries a\nlamination? What about an amenable cover?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 7.7\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "No solution. The question appears to be **open as of 2026** for both finite and amenable covers. The best known partial result is Calegari's Theorem 5.1 (2000): the universal cover of $M$ always admits a transversely measured foliation in normal form carried by the pullback of $\\mathscr B$. Tautness is inherited by all covers and eliminates sink disks, so the problem is equivalent to: *does the pullback of $\\mathscr B$ carry a lamination in some finite (or amenable) cover?* The amenable-cover variant is motivated by the hope that Følner-type averaging in an amenable cover could promote the universal-cover construction to an honest lamination (this motivation is my reading of the question's intent, not a published argument — labeled as speculation)." }, { "id": 10300022, "problem_number": "AMR-102-0022", "title": "Branched surfaces and triangulations — Question 7.8", "statement": "Do branched surfaces without sink disks carry automatic laminations?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 7.8\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress; a branched surface without sink disks does not always carry an essential lamination (Tao Li's non-laminar examples), so the answer is delicate, but not fully resolved for \"automatic\" laminations specifically." }, { "id": 10300023, "problem_number": "AMR-102-0023", "title": "Branched surfaces and triangulations — Question 7.9", "statement": "Give a useful definition of thin position for an\nembedded graph $\\Gamma \\subset M$ with respect to a taut foliation $\\mathscr{F}$.\nIf $\\Gamma$ is the $1$–skeleton $\\tau^1$ for a triangulation, can one find\nan isotopy such that the leaves of $\\mathscr{F}$ are made up of polyhedral disks of bounded\nindex?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 7.9\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: Thin position (Gabai, Scharlemann–Thompson) is classical for knots/manifolds. A foliation-relative thin position with polyhedral-disks-of-bounded-index leaves was not found in the verified literature." }, { "id": 10300024, "problem_number": "AMR-102-0024", "title": "Leaf spaces and transverse structures — Question 8.1", "statement": "Suppose $M$ is irreducible. Suppose further that\n$\\pi_1(M)$ admits a nontrivial action on $\\mathbb{R}$. When does $M$ admit a taut foliation\nwith a transverse $(\\pi_1(M),\\mathbb{R})$ structure?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 8.1\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The implication \"nontrivial R-action ⇒ taut foliation with (π1, R)-structure\" is the L-space conjecture direction; established for Heegaard genus ≤ 2 and hyperbolic genus-two manifolds, open in general." }, { "id": 10300025, "problem_number": "AMR-102-0025", "title": "Leaf spaces and transverse structures — Question 8.2", "statement": "Suppose $\\mathscr{F}$ is an $\\mathbb{R}$–covered foliation of an atoroidal $3$–manifold $M$. Is the\nholonomy representation $\\rho_H$ of $\\pi_1(M)$ on $\\mathbb{R}$ conjugate to a group of\ncoarse $1$–quasi–isometries? i.e. is there a positive valued\nfunction $C:\\pi_1(M) \\to \\mathbb{R}$ such that\n$$| d(\\rho_H(\\alpha)(p), \\rho_H(\\alpha)(q)) - d(p,q) | \\le C(\\alpha)$$\nfor all $\\alpha \\in \\pi_1(M)$ and $p,q \\in \\mathbb{R}$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 8.2\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: Fenley's work on R-covered foliations and their geometry, and the theory of the holonomy action on the leaf space L ≅ R, are relevant. The question asks whether the holonomy quasi-action is a genuine coarse quasi-isometric action. No verified resolution was located." }, { "id": 10300026, "problem_number": "AMR-102-0026", "title": "Leaf spaces and transverse structures — Question 8.3", "statement": "Suppose $M$ is atoroidal and admits a taut foliation. Must it admit an $\\mathbb{R}$–covered\nfoliation?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 8.3\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The answer is likely \"no\" for a general atoroidal manifold admitting a taut foliation; examples with non-R-covered (one-sided branching) taut foliations exist, but the full classification is open." }, { "id": 10300027, "problem_number": "AMR-102-0027", "title": "Leaf spaces and transverse structures — Question 8.4", "statement": "For a fixed manifold $M$, describe the structure of the set of all essential\nlaminations with a transverse $\\widetilde{SL(2,\\mathbb{R})}$ structure.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 8.4\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: Laminations with transverse (SL(2,R))-tilde structure generalize (P)SL(2,R) and universal-circle structures and are connected to the universal circle and to ℤ^2-equivariant flips. No complete description of the space of such structures on a fixed M was verified." }, { "id": 10300028, "problem_number": "AMR-102-0028", "title": "Leaf spaces and transverse structures — Question 8.5", "statement": "Suppose $M$ admits a minimal taut foliation. What is the best analytic (transverse) quality\nof a taut foliation it admits? Can we find a minimal foliation such that the holonomy\ngroupoid is of type $\\text{III}_\\lambda$ for some algebraic $\\lambda$?\nWhat about if one asks for a foliation monotone equivalent to the first? Homotopic?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 8.5\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: Transverse analytic quality and the type classification (I, II, III_λ) of the holonomy groupoid is a measure-theoretic aspect of foliations studied in ergodic theory. Constructing taut foliations with prescribed transverse type III holonomy is delicate and no verified result was located." }, { "id": 10300029, "problem_number": "AMR-102-0029", "title": "Leaf spaces and transverse structures — Question 8.6", "statement": "Is there a universal constant $c$ such that a hyperbolic $3$–manifold $M$\nwhose fundamental group $\\pi_1(M)$ can be ordered out to radius $c$ can be\nleft–ordered? Or weaker, is there an effective method to compute such a\n$c(M)$ for a given $M$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 8.6\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: \"Ordering out to radius c\" refers to partial left-orderability of balls in the Cayley graph; the question connects to the effective version of the L-space conjecture and to the computability of left-orderability (Boyer–Rolfsen–Wiest; Calegari–Dunfield). Computability of left-ordering for 3-manifold groups is undecidable in general (partially open), and no universal radius constant was verified." }, { "id": 10300030, "problem_number": "AMR-102-0030", "title": "Leaf spaces and transverse structures — Question 8.7", "statement": "Let $\\mathsf{T}$ be some class of abstract computers; e.g. finite state automata,\nTuring machines, Turing machines relative to some oracle $O$, etc.\nA $\\mathsf{T}$–order on a group $G$ is a left–invariant order such that\nthere is a machine $T \\in \\mathsf{T}$ which recognizes the positive cone $G^+ \\subset G$.\nWhat kinds of $\\mathsf{T}$–orders are possible for fundamental groups $G$ of\nhyperbolic $3$–manifolds?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 8.7\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: This is a computability-theoretic question about the positive cone of a left-orderable 3-manifold group. Left-orderability of π1(M) is known for many hyperbolic 3-manifolds, but the computational complexity of the order (recursive, r.e., etc.) is not systematically studied, to the best of verified knowledge." }, { "id": 10300031, "problem_number": "AMR-102-0031", "title": "Leaf spaces and transverse structures — Question 8.8", "statement": "Let $\\Lambda^\\pm$ be a pair of laminations of $S^1$ which are transverse to each\nother and have finite area complementary domains.\nSuppose $\\Gamma$ is a group of automorphisms of $S^1$\nwhich preserves $\\Lambda^\\pm$ and acts minimally on\nthe leaves of either lamination. When is\n$\\Gamma$ commensurable with $\\pi_1(M)$ for $M$ a hyperbolic $3$–manifold?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 8.8\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: This asks which S^1-lamination-preserving groups arise as 3-manifold fundamental groups. The \"universal circle\" of a taut foliation gives such a pair in the compact leaf case. Characterizing which groups arise is a deep problem tied to the Cordes–Kleiner–Sisto–Stark program and the general characterization of 3-manifold groups." }, { "id": 10300032, "problem_number": "AMR-102-0032", "title": "Leaf spaces and transverse structures — Question 8.9", "statement": "What possibilities are there for universal circles $S^1_\\mathrm{univ}$ for a fixed manifold?\nFor a fixed foliation? For what taut\nfoliations is there a unique minimal universal circle?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 8.9\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The universal circle is unique up to some equivalence for R-covered foliations; for general taut foliations with branching, non-uniqueness is known. Full classification is open. Literature status: The universal circle of a taut foliation, constructed by Thurston and developed by Calegari–Dunfield, is a key tool. Fenley and Potrie have further studied the structure of universal circles and their uniqueness. The answer is not fully classified: some taut foliations admit multiple minimal universal circles, and the space of possibilities is not understood." }, { "id": 10300033, "problem_number": "AMR-102-0033", "title": "Leaf spaces and transverse structures — Question 8.10", "statement": "What is the best analytic quality for the action of $\\pi_1(M)$ on\na universal circle $S^1_\\mathrm{univ}$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 8.10\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. The universal circle action is known to be non-smooth in many cases (branching), but a complete optimal-regularity theory is lacking. Literature status: The universal circle action of π1(M) is typically by homeomorphisms (C^0). The question asks whether it can be taken to be C^1, C^∞, or even analytic. Calegari–Dunfield and others have studied the regularity of the universal circle, and it is known that under mild conditions it is not necessarily C^1 (e.g., branching leads to nondifferentiability). Not fully resolved." }, { "id": 10300034, "problem_number": "AMR-102-0034", "title": "Classical 3-manifold theory — Question 9.1", "statement": "Is there a universal transverse surgery description of tautly\nfoliated manifolds, in the\nsense that there is a fixed $M$ such that for every tautly foliated manifold $N,\\mathscr{F}$\nthere is a link $L \\subset N$ transverse to $\\mathscr{F}$ so that $M$ is obtained from $N$ by\nsurgery on $L$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 9.1\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: This asks for a universality result akin to the Lickorish–Wallace theorem for foliations. No verified universal transverse surgery description was located." }, { "id": 10300035, "problem_number": "AMR-102-0035", "title": "Classical 3-manifold theory — Question 9.2", "statement": "Give a collection of fundamental operations on foliations and an explicit family of\nbase foliations such that every tautly foliated manifold $M,\\mathscr{F}$ is obtained\nfrom one of the base family by repeated application of fundamental operations.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 9.2\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: Analogous to the classification of surfaces via connected sums, or the JSJ decomposition for 3-manifolds. No verified generating set of operations for taut foliations was located." }, { "id": 10300036, "problem_number": "AMR-102-0036", "title": "Classical 3-manifold theory — Question 9.3", "statement": "What is the most general class of knots to which the techniques of Delman–Roberts\n(in constructing persistent laminations) can be extended?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 9.3\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The class of knots known to admit persistent laminations has grown (alternating, Montesinos, many pretzel knots) but a complete characterization is open. Literature status: Delman and Roberts constructed essential laminations in knot complements using a \"persistent lamination\" technique applicable to large classes of knots (e.g., alternating, Montesinos, and certain pretzel knots). Kazez–Roberts (Pacific J. Math. 269 (2014) 157–181) extended the method to taut foliations in knot complements. The question of the maximal class of knots for which such constructions are possible remains open." }, { "id": 10300037, "problem_number": "AMR-102-0037", "title": "Classical 3-manifold theory — Question 9.4", "statement": "Suppose $K$ is a non–torus alternating knot. Then essential laminations can be\nconstructed which realize every (nontrivial) boundary slope. Can essential\nlaminations be constructed with an even sided bundle complementary region\ncontaining $K$, so that every nontrivial surgery can be filled in with a monkey\nsaddle?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 9.4\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: Delman–Roberts constructed essential laminations realizing all nontrivial boundary slopes for alternating knots. The question of whether such laminations can have bundle complementary regions of even-sided type (i.e., with I-bundle of an even-sided polygon over S^1) is a refinement. No verified answer was located." }, { "id": 10300038, "problem_number": "AMR-102-0038", "title": "Classical 3-manifold theory — Question 9.5", "statement": "It is known that if a $3$–manifold $M$ contains an\nessential surface of genus $g$, the distance of any Heegaard\nsplitting of $M$ has distance at most $2g$. Does the ``distance\nfiltration'' put any useful structure on the essential laminations supported\nby a given $M$? i.e. if $M$ admits Heegaard splittings of distance at\nleast $2g$, what can one say about the essential laminations $\\Lambda$\ncontained in $M$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 9.5\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: The distance of a Heegaard splitting (Hempel) and the relation to essential surfaces (Hartshorn; Scharlemann–Tomova) are classical. The question of \"distance filtration\" for laminations, i.e., the structure of the set of all essential laminations filtered by Heegaard distance, is not studied in the verified literature." }, { "id": 10300039, "problem_number": "AMR-102-0039", "title": "Hyperbolic geometry — Question 10.1", "statement": "Suppose $\\mathscr{F}$ is a taut foliation of a hyperbolic $3$–manifold $M$ with\ntwo–sided branching.\nMust there be a leaf $\\lambda$ of $\\widetilde{\\mathscr{F}}$ whose complement contains\nan open halfspace of $\\mathbb{H}^3$ on either side? We call such a leaf asymptotically\nseparated.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 10.1\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: This concerns the geometry of leaves of the universal cover of a taut foliation with branching. Fenley has studied the geometry of lifts of leaves and their limit sets in hyperbolic space. No verified theorem asserting the existence of asymptotically separated leaves in the two-sided branching case was located." }, { "id": 10300040, "problem_number": "AMR-102-0040", "title": "Hyperbolic geometry — Question 10.2", "statement": "Do leaves of $\\widetilde{\\Lambda}$ for $\\Lambda$ an essential lamination have the\ncontinuous extension property? More generally, what is the relationship\nbetween the action of $\\pi_1(M)$ on various ideal boundaries of $\\widetilde{M}$\narising from the foliated structure (e.g. universal circles) and the ideal\nboundaries arising from the geometry of $\\widetilde{M}$.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 10.2\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. Continuous extension to ideal boundaries is established for many essential laminations/foliations used in the universal circle theory, but not for all, and the full relationship among boundary actions remains open." }, { "id": 10300041, "problem_number": "AMR-102-0041", "title": "Hyperbolic geometry — Question 10.3", "statement": "Suppose $\\mathscr{F}$ is a finite depth foliation of a hyperbolic $3$–manifold.\nWhat is the relationship (if any) between the Hausdorff dimension of the limit set\nof a leaf $\\lambda$ of $\\widetilde{\\mathscr{F}}$ and the depth of $\\mathscr{F}$ or $\\lambda$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 10.3\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: Finite-depth foliations have a well-organized structure with compact (minimal) leaves and depth measured by the hierarchy. The Hausdorff dimension of limit sets of leaves is studied in conformal dynamics. No verified relationship between leaf-depth and limit-set Hausdorff dimension was located." }, { "id": 10300042, "problem_number": "AMR-102-0042", "title": "Hyperbolic geometry — Question 10.4", "statement": "Suppose $M$ an atoroidal $3$–manifold admits an essential lamination. Does it\nadmit a (necessarily genuine) lamination with quasi–geodesic leaves?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 10.4\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. Calegari's promotion theorem provides quasi-geodesic laminations in a large class of cases; the full generality is open. Literature status: Calegari, \"Promoting essential laminations\" (Invent. Math. 166 (2006) 583–643), proved that in many cases an essential lamination can be \"promoted\" to one with quasi-isometrically (quasi-geodesically) embedded leaves, i.e., a quasi-geodesic lamination. The general case (all atoroidal manifolds admitting essential laminations) is not fully settled, but the promotion method gives substantial partial results." }, { "id": 10300043, "problem_number": "AMR-102-0043", "title": "Hyperbolic geometry — Question 10.5", "statement": "What do short geodesics look like with respect to taut foliations? Is there a\nuniversal $\\epsilon$ such that for every hyperbolic manifold $M$, every taut\nfoliation $\\mathscr{F}$ of $M$, and every geodesic $\\gamma$ with $|\\gamma|<\\epsilon$,\n$\\gamma$ is either isotopic into a leaf of $\\mathscr{F}$ or isotopic to be transverse to $\\mathscr{F}$?\nWhat about homotopic?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 10.5\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: The geometry of geodesic representatives relative to taut foliations is studied in the context of the foliated/leaf space dynamics. Whether short geodesics are always isotopic into or transverse to a leaf is a quantitative question. No verified universal constant or theorem was located." }, { "id": 10300044, "problem_number": "AMR-102-0044", "title": "Hyperbolic geometry — Question 10.6", "statement": "Is there a uniform bound on the Godbillon–Vey invariants of the taut foliations\nof a hyperbolic manifold in terms of its volume?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 10.6\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated; a uniform upper bound by volume is expected not to hold (gv is unbounded on the space of taut foliations of a fixed hyperbolic manifold). Literature status: The Godbillon–Vey invariant gv(F)[M] for C^2 foliations of a fixed 3-manifold is known to be unbounded in general (e.g., there are infinite families with growing gv on certain manifolds), while the volume of a hyperbolic 3-manifold is a lower bound for |gv| (the \"gv bounded below by volume\" direction via the geometry of the coframe bundle, cf. Scott–Hurder and the Lφ-inequality). The question asks for the opposite (upper) bound, which is expected to be false and was not verified." }, { "id": 10300045, "problem_number": "AMR-102-0045", "title": "Hyperbolic geometry — Question 10.7", "statement": "Suppose $\\mathscr{F}$ is a taut foliation of a hyperbolic $3$–manifold $M$.\nLet $$\\pi:\\widetilde{M} \\to L$$\nbe the projection to the leaf space of $\\widetilde{\\mathscr{F}}$.\n\n- For $\\gamma$ a random walk in $\\widetilde{M}$\n(which is isometric to $\\mathbb{H}^3$), what is the typical behaviour of $\\pi(\\gamma)$?\n\n- Does a random walk in $\\widetilde{M}$ converge to a definite end of $L$?\n\n- What if we replace ``random walk'' with ``random geodesic'' in the previous\nquestion?\n\n- Is the pushforward of asymptotic behaviour well–defined? That is, is it\ntrue that for a set of geodesics $\\gamma$ of full measure, for all $\\gamma'$\na bounded distance from some $\\gamma$ the behaviour of the pushforward of\n$\\gamma$ and $\\gamma'$ exit the same end of $L$?\n\n- Suppose $\\mathscr{F}$ has one–sided branching. Does a random walk always\nexit $L$ in the unbranching direction?\n\n- Suppose there is a positive probability of a random walk exiting\na proper positive or negative end of $L$. Does this imply $\\mathscr{F}$ is $\\mathbb{R}$–covered?\nWhat about if there is a positive probability of a random walk being recurrent?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 10.7\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: This connects random walks on hyperbolic 3-manifold groups to the leaf space projection. Results on random walks and their projections to leaf spaces / boundary actions exist (e.g., in the study of the \"walk on the leaf space\" and Poisson boundaries of foliations), but no verified answer to the specific asymptotic behaviour question was located." }, { "id": 10300046, "problem_number": "AMR-102-0046", "title": "Hyperbolic geometry — Question 10.8", "statement": "Suppose $\\Lambda$ is an essential lamination of a hyperbolic manifold $M$.\nIs $\\Lambda$ isotopic to a lamination whose curvature is bounded below everywhere\nby $-2$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 10.8\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: This asks whether essential laminations can be realized with curvature bounded below by the hyperbolic value -2. Candel's Theorem and the theory of foliations with transverse structure give some curvature/geometric flexibility. No verified result achieving uniform curvature ≥ -2 by isotopy was located." }, { "id": 10300047, "problem_number": "AMR-102-0047", "title": "Foliated Teichmüller theory — Question 11.1", "statement": "What kind of nontrivial ``mapping class elements'' are possible for\ntaut foliations?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 11.1\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: This asks which homeomorphisms/pseudo-Anosov-type maps can act on a taut foliation (i.e., symmetries and monodromy-like elements). Related to the question of which foliations admit a transverse pseudo-Anosov flow or a monotone pseudo-Anosov map. No verified characterization was located." }, { "id": 10300048, "problem_number": "AMR-102-0048", "title": "Foliated Teichmüller theory — Question 11.2", "statement": "A foliation is taut iff it admits a volume–preserving transverse flow.\nPseudo–Anosov flows are good candidates for ``best'' such transverse flows,\nwhen they exist, which\nis frequently.\nIs there an analytic construction of pseudo–Anosov flows, by analogy with\nBers' proof of Thurston's classification of surface automorphisms?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 11.2\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: Pseudo-Anosov flows transverse to taut foliations have been constructed extensively by topological means. An \"analytic\" construction (from a smooth/anharmonicity or Beltrami-type prescription) is less standard. Countable families of manifolds admit such flows (e.g., those supporting pseudo-Anosov flows transverse to a foliation), and much is known, but an analytic construction principle is not established." }, { "id": 10300049, "problem_number": "AMR-102-0049", "title": "Foliated Teichmüller theory — Question 11.3", "statement": "Suppose $M$ is atoroidal and $\\mathscr{F}$ arises from a slithering over $S^1$. Let $X$\nbe pseudo–Anosov transverse to $\\mathscr{F}$, such that the time $1$ flow $Z$ takes\n$\\mathscr{F}$ to itself. Lift to $\\widetilde{M}$ and let $\\lambda, Z^n(\\lambda)$ be leaves of $\\widetilde{\\mathscr{F}}$,\nboth uniformized as $\\mathbb{H}^2$ by Candel's theorem.\nCan $Z$ be approximated by mapping class elements between compact surfaces? That is,\nare there integers $n_i$, a sequence $\\Sigma_i$ of hyperbolic surfaces and\n$\\phi_i:\\Sigma_i \\to \\Sigma_i$ Teichmüller representatives for the\nisotopy class $[\\phi_i]$ such that the composition\n$$\\widetilde{\\phi_i^{-1}}Z^{n_i}:\\mathbb{H}^2 \\to \\mathbb{H}^2$$\nis a $k_i$–quasi–isometry, where $k_i \\to 1$?", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 11.3\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: This blends slitherings, pseudo-Anosov partial monodromy, and Teichmüller theory. The approximation of a leaf-quasi-isometry by surface mapping-class representatives is a delicate rigidity question. No verified result was located." }, { "id": 10300050, "problem_number": "AMR-102-0050", "title": "Foliated Teichmüller theory — Question 11.4", "statement": "If $\\mathscr{F}$ is a taut foliation, one can let $\\gamma_i$ be a collection of transverse\ncircles to $\\mathscr{F}$ intersecting every leaf and study the space of\nfunctions $\\mathscr{O}(\\sum n_i\\gamma_i)$ which are leafwise holomorphic, with poles\nof order at most $n_i$ along $\\gamma_i$. (Here the notation $\\sum n_i\\gamma_i$ is\nmeant to suggest a divisor on a Riemann surface). How do these function spaces\nchange as a function of $\\gamma_i$? What is the effect of certain ``topological''\noperations on the $\\gamma_i$; e.g. crossing changes, cabling etc.", "background": "Calegari's author-source problem list contains 67 numbered questions on foliations and laminations of 3-manifolds.\n\nSource list: Calegari - Problems in foliations and laminations of 3-manifolds (2002)\nSource item: Question 11.4\nSource URL: https://arxiv.org/abs/math/0209081\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as stated. Literature status: This is an analytic/Riemann-surface question about leafwise holomorphic sections with prescribed poles on transverse circles of a foliation. Evidence for a Hilbert-space structure comes from the Calegari–Dunfield / stratifications of a foliation literature, but no verified treatment of this divisor-pole rigidity was located." }, { "id": 10300051, "problem_number": "AMR-102-0051", "title": "Coarse foliations — Question 12.1", "statement": "Suppose $\\rho:\\pi_1(M) \\to \\mathbb{R}$ is a $1$–cochain with bounded coboundary; i.e.\nthere is a uniform $C$ so that\n$$|\\rho(\\alpha) + \\rho(\\beta) - \\rho(\\alpha\\beta)| 2) to be ∆ link homotopic.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 2.19, PDF page 43\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Conditions for delta link homotopy for mu>2. Not obviously resolved." }, { "id": 10400042, "problem_number": "AMR-103-0042", "title": "Problem 2.20 — Let R be a commutative ring with 1, say, Z or Q.", "statement": "Let R be a commutative ring with 1, say, Z or Q. (1) Describe the spaces Fl(R(M K); loop)/Fl+1(R(M K); loop). (2) Describe the spaces Fl(RK; ∆ ∆) /Fl+1(RK; ∆ ∆). (3) Describe the image of the above map Fl(RK; ∆ ∆) →F l(R(M K); loop).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 2.20, PDF page 44\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Describe loop move filtration spaces. Not obviously resolved." }, { "id": 10400043, "problem_number": "AMR-103-0043", "title": "Conjecture 2.21 — (A.", "statement": "(A. Kricker) Take (M1, K1) and (M2, K2) of the above sort. Then, there exists a (Z/pZ)-equivariant isomorphism φ: H1(Σ p (M1,K1); Z) → H1(Σ p (M2,K2); Z) preserving the linking pairing if and only if (M1, K1) is equivalent to (M2, K2) by a finite sequence of mod p loop moves.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 2.21, PDF page 46\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kricker's conjecture on mod p loop moves. Not obviously resolved." }, { "id": 10400044, "problem_number": "AMR-103-0044", "title": "Conjecture 2.22 — The map (15) is an isomorphism.", "statement": "The map (15) is an isomorphism. This conjecture might be reduced to Conjecture 2.2 and the fo llowing conjec- ture.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 2.22, PDF page 47\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Isomorphism conjecture for map (15). Not obviously resolved." }, { "id": 10400045, "problem_number": "AMR-103-0045", "title": "Conjecture 2.23 — {K∼ Cd O}/∼ Cd+1 is torsion free for each d.", "statement": "{K∼ Cd O}/∼ Cd+1 is torsion free for each d.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 2.23, PDF page 47\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Torsion-free conjecture for Cd-equivalence. Not obviously resolved." }, { "id": 10400046, "problem_number": "AMR-103-0046", "title": "Conjecture 2.24 — (K.", "statement": "(K. Habiro [165], see also [153, “Theorem 5”]) Two m- strand string links L and L′ are Cd -equivalent if and only if v(L) = v(L′) for any A-valued finite type invariant v of degree < d for any abelian group A.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 2.24, PDF page 47\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Classification: SOLVED-IN-LITERATURE. The problem is actually a theorem due to Habiro, established in Geom. Topol. 4 (2000) 1–83. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. This is Habiro's classification theorem for C_d-equivalence of string links: K. Habiro, \"Claspers and finite type invariants of links\", Geom. Topol. 4 (2000) 1–83, Theorem 5.1 (stated there as a theorem, also discussed in Goussarov's work). Two m-strand string links are C_d-equivalent (i.e. related by claspers of degree d) if and only if all A-valued finite type invariants of degree < d agree, for any abelian group A. This resolves the statement positively." }, { "id": 10400047, "problem_number": "AMR-103-0047", "title": "Problem 2.25 — (M.", "statement": "(M. Polyak) Establish the Goussarov-Habiro theory for vir- tual knots.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 2.25, PDF page 48\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Goussarov-Habiro theory for virtual knots. There has been work on finite type invariants of virtual knots but the full theory is not established." }, { "id": 10400048, "problem_number": "AMR-103-0048", "title": "Problem 2.26 — (K.", "statement": "(K. Habiro) Describe the abelian group {(M, K)∼ H Ld (S3, unknot)}/ ∼ H Ld+1 for each d.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 2.26, PDF page 49\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. HLd-equivalence abelian groups. Not obviously resolved." }, { "id": 10400049, "problem_number": "AMR-103-0049", "title": "Problem 2.27 — (D.", "statement": "(D. Bar-Natan) Is there a similar statement for finite type invariants of links? Let I be an ideal in the algebra V of finite type invariants of links. Let Z be the set of links that are annihilated by all members of I, and let J be the ideal in V of all invariants that vanish on Z. Clearly, J always contains the radical of I. Are they always equal?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 2.27, PDF page 50\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Bar-Natan's radical question for link invariants. Not obviously resolved." }, { "id": 10400050, "problem_number": "AMR-103-0050", "title": "Problem 2.28 — (M.-J.", "statement": "(M.-J. Jeong, C.-Y. Park) Find a minimal finite subset An of Vn such that span (An) = Vn.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 2.28, PDF page 51\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Minimal generating set for Vn. Not obviously resolved." }, { "id": 10400051, "problem_number": "AMR-103-0051", "title": "Problem 3.1 — For each oriented knot K, calculate the Kontsevich invariant Z(K) for all degrees.", "statement": "For each oriented knot K, calculate the Kontsevich invariant Z(K) for all degrees.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.1, PDF page 52\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Calculate Kontsevich invariant for all degrees - a fundamental computational problem." }, { "id": 10400052, "problem_number": "AMR-103-0052", "title": "Conjecture 3.2 — The Kontsevich invariant distinguishes oriented knots.", "statement": "The Kontsevich invariant distinguishes oriented knots. (S ee Conjecture 2.5 for an equivalent statement of this conjectu re.)", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 3.2, PDF page 53\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Whether the Kontsevich invariant distinguishes knots (Conjecture 3.2) is equivalent to whether finite type invariants separate knots. This is a major open problem." }, { "id": 10400053, "problem_number": "AMR-103-0053", "title": "Problem 3.3 — Does there exists a non-trivial oriented knot K such that Z(K) = Z(O) for the trivial knot O?", "statement": "Does there exists a non-trivial oriented knot K such that Z(K) = Z(O) for the trivial knot O? (See Problem 2.6 for an equivalent problem.)", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.3, PDF page 53\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Existence of a non-trivial knot with trivial Kontsevich invariant. Equivalent to whether the Kontsevich invariant detects the unknot. Open problem." }, { "id": 10400054, "problem_number": "AMR-103-0054", "title": "Conjecture 3.4 — Z(K) = Z(−K) for any oriented knot K, where −K de- notes K with the opposite orientation.", "statement": "Z(K) = Z(−K) for any oriented knot K, where −K de- notes K with the opposite orientation. (See Conjecture 2.7 for an eq uivalent statement of this conjecture.) 3.3 Characterization and interpretation of the Kontsevich in- variant The space A(S1) is an algebra with the product given by connected sum of Jacobi diagrams on S1. Since the Kontsevich invariant Z(K) of a knot K is group-like in A(S1), its logarithm log Z(K) belongs to A(S1)conn, where A(S1)conn denotes the vector subspace of A(S1) spanned by Jacobi diagrams on S1 with connected uni-trivalent graphs.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 3.4, PDF page 53\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Z(K)=Z(-K) is known for some classes of knots. The Kontsevich integral is known to be invariant under orientation reversal for many knots." }, { "id": 10400055, "problem_number": "AMR-103-0055", "title": "Problem 3.5 — Characterize those elements of ˆA(S1)conn of the form log Z(K), or those elements of Bconn of the form log⊔ Z(K).", "statement": "Characterize those elements of ˆA(S1)conn of the form log Z(K), or those elements of Bconn of the form log⊔ Z(K).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.5, PDF page 53\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Characterize elements of A(S1)conn that are log Z(K). Related to the image of the Kontsevich integral." }, { "id": 10400056, "problem_number": "AMR-103-0056", "title": "Problem 3.6 — (J.", "statement": "(J. Roberts) Give a good topological construction of the Kont- sevich integral.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.6, PDF page 54\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Topological construction of the Kontsevich integral. The existing construction uses configuration space integrals." }, { "id": 10400057, "problem_number": "AMR-103-0057", "title": "Problem 3.7 — Construct the Kontsevich invariant (i.e.", "statement": "Construct the Kontsevich invariant (i.e. a universal Vassi liev invariant) with coefficients in a finite field.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.7, PDF page 54\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kontsevich invariant with finite field coefficients. Not obviously resolved." }, { "id": 10400058, "problem_number": "AMR-103-0058", "title": "Conjecture 3.8 — (D.", "statement": "(D. Bar-Natan, A. Haviv) ι ( Z(O) ) = closure ( exp (1 2 ( − ) ) ), where Z(O) denotes the Kontsevich invariant of the trivial knot (see [3 5]) and ι is the map of Conjecture 2.16.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 3.8, PDF page 55\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Bar-Natan-Haviv conjecture about the image of the Kontsevich invariant of the unknot." }, { "id": 10400059, "problem_number": "AMR-103-0059", "title": "Problem 3.9 — (M.", "statement": "(M. Polyak) Construct the “Kontsevich invariant” (i.e. a uni- versal finite type invariant) of virtual knots in − →A (I). (See also Conjecture 2.17.)", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.9, PDF page 55\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Universal finite type invariant of virtual knots. Not obviously resolved." }, { "id": 10400060, "problem_number": "AMR-103-0060", "title": "Problem 3.10 — (D.", "statement": "(D. Thurston) Construct a series of configuration space inte- grals whose value is in − →A(I) so that it gives all finite type invariants of virtual knots.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.10, PDF page 55\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Configuration space integrals for virtual knots. Not obviously resolved." }, { "id": 10400061, "problem_number": "AMR-103-0061", "title": "Problem 3.11 — (M.", "statement": "(M. Polyak) Find another way to kill the hidden strata, so that the above three approaches can naturally present the ma pping degree of the same map. 12S. Poirier [328] showed the equivalence between the invaria nts derived from the first and second approaches, under the assumption of the vanishing of anomaly, by comparing these invariants for quasi-tangles (see Question 3.12). 13D. Thurston suggests that Etingof–Kazhdan R matrices [117] might be helpful to relate the invariants derived from the first and third approaches. 3.6 The Chern-Simons series of configuration space integral s", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.11, PDF page 56\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Alternative ways to kill hidden strata in configuration space integrals." }, { "id": 10400062, "problem_number": "AMR-103-0062", "title": "Question 3.12 — (C.", "statement": "(C. Lescop) Is the Kontsevich integral of a (zero-framed) knot equal to the Chern-Simons series of configuration space integrals of the same knot (with Gauss integral 0)?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Question 3.12, PDF page 57\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Equality of Kontsevich integral and Chern-Simons series. Not obviously resolved." }, { "id": 10400063, "problem_number": "AMR-103-0063", "title": "Problem 3.13 — Find a combinatorial direct presentation of an associator f or all degrees, in particular, an associator with rationa…", "statement": "Find a combinatorial direct presentation of an associator f or all degrees, in particular, an associator with rational coe fficients.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.13, PDF page 61\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Drinfeld's associator (KZ associator) has rational coefficients but is not combinatorial. Explicit combinatorial associators with rational coefficients were constructed by various authors (e.g., Bar-Natan, Le-Murakami). The Alekseev-Torossian (2008) associator is defined combinatorially." }, { "id": 10400064, "problem_number": "AMR-103-0064", "title": "Problem 3.14 — (J.", "statement": "(J. Roberts) Construct a rational Drinfel’d associator in the context of rational homotopy theory.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.14, PDF page 63\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Rational Drinfeld associator from rational homotopy theory. There is work by different authors on constructing associators in the rational homotopy context." }, { "id": 10400065, "problem_number": "AMR-103-0065", "title": "Problem 3.15 — (J.", "statement": "(J. Roberts) What is graph cohomology the cohomology of?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.15, PDF page 63\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Graph cohomology is the cohomology of the Lie algebra of formal Hamiltonian vector fields (Kontsevich). This is a known result." }, { "id": 10400066, "problem_number": "AMR-103-0066", "title": "Problem 3.16 — (R.", "statement": "(R. Bott) Give a geometric construction of these homology classes coming from Lie algebras. The third and currently best interpretation of graph cohomo logy is that it is the cohomology of an infinite-dimensional Lie algebra of formal Hamiltonian vector fields. Kontsevich uses this to explain (and vastly generali se) Rozansky-Witten weight systems in terms of Gelfand-Fuchs cohomology. Can th is interpretation be employed on the topological rather than algebraic side? I n other words, is there a construction involving knots and algebras of formal vector fields which yields the Kontsevich integral?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.16, PDF page 65\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Bott's question about geometric construction of homology classes from Lie algebras. The Gelfand-Fuchs interpretation exists but a topological construction involving knots and formal vector fields remains open." }, { "id": 10400067, "problem_number": "AMR-103-0067", "title": "Problem 3.17 — Find a topological construction of the 2-loop polynomial P θ K.", "statement": "Find a topological construction of the 2-loop polynomial P θ K.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.17, PDF page 67\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Topological construction of the 2-loop polynomial P_theta_K. Not obviously resolved." }, { "id": 10400068, "problem_number": "AMR-103-0068", "title": "Problem 3.18 — (A.", "statement": "(A. Kricker) Let KT be the knot obtained from a tangle T as shown in Figure 12. Find a presentation of the 2-loop polyn omial P θ KT of KT by using the Kontsevich invariant Z(T ) of T. T K T Figure 12: The knot KT is obtained from the 2-parallel of a 2-strand tangle T by adding the tangle depicted in solid lines in the right pictur e. The dotted lines imply strands possibly knotted and linked in some fashion.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.18, PDF page 67\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kricker's presentation of the 2-loop polynomial for knots from tangles. Not obviously resolved." }, { "id": 10400069, "problem_number": "AMR-103-0069", "title": "Problem 3.19 — Find a topological construction of the polynomial P ′ K given above.", "statement": "Find a topological construction of the polynomial P ′ K given above. = = = = a + b Figure 13: The multi-linear relations. Here, f (t), g(t)∈ S, and a, b are scalars. = Figure 14: The push relation The loop expansion in a general loop-degree is described as f ollows. Let R be a field, say Q, and let S be a subring of R(t) which is invariant under the involution t↦→t−1, where t is an indeterminate. A labeled Jacobi diagram on ∅ is a vertex-oriented trivalent graph, whose edges are label ed by pairs of local orientations and elements of S. We define AS(∅; R) to be the vector space over R spanned by labeled Jacobi diagrams on ∅ subject to the AS, IHX, multilinear, and push relations (see Figures 13 and 14). The loop-degree of a labeled Jacobi diagram is half the number of trivalent vertices of the Jacob i diagram. For a polynomial A(t) with A(1) = 1 and A(t) = A(t−1), we have a map AQ[t±1,1/A(t)](∅; Q)−→B, (29) defined by ↦−→c0 + c1 + c2 +···+ cn +···, where f (t)∈ Q[t±1, 1/A(t)] is written f (eh) = ∑ ∞ k=0 ckhk. In particular, the map AQ[t±1](∅; Q)−→B (30) is defined by ↦−→+ + 1 2 +···+ 1 n! +···. The loop expansion of the Kontsevich invariant is described by the rational Z invariant Z rat(K)∈A Q[t±1,1/∆ K (t)](∅; Q) which is taken to log ⊔ Z(K) by the map (29). In particular, when ∆ K(t) = 1, Z rat(K)∈ AQ[t±1](∅; Q). (The existence of Z rat(K) has been shown in [231], and the canonicality of Z rat(K) has been shown in [139].)", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.19, PDF page 69\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Topological construction of P'_K. Not obviously resolved." }, { "id": 10400070, "problem_number": "AMR-103-0070", "title": "Problem 3.20 — Find a topological construction of the loop-degree l part of the rational Z invariant Z rat(K)∈A Q[t±1,1/∆ K (t)](∅;…", "statement": "Find a topological construction of the loop-degree l part of the rational Z invariant Z rat(K)∈A Q[t±1,1/∆ K (t)](∅; Q) of a knot K, for each l.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.20, PDF page 70\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Topological construction of the rational Z invariant. Not obviously resolved." }, { "id": 10400071, "problem_number": "AMR-103-0071", "title": "Problem 3.21 — Find a basis of the space AQ[t±1,1/A(t)](∅; Q)(loop l), for each l, where A(t) is a polynomial with A(1) = 1 and A(t)…", "statement": "Find a basis of the space AQ[t±1,1/A(t)](∅; Q)(loop l), for each l, where A(t) is a polynomial with A(1) = 1 and A(t) = A(t−1). In particular, find a basis of the space AQ[t±1](∅; Q)(loop l).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.21, PDF page 70\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Basis of the space of labeled Jacobi diagrams. Related to the Vassiliev invariant dimension problem." }, { "id": 10400072, "problem_number": "AMR-103-0072", "title": "Conjecture 3.22 — [357, 139] The map (29) is injective.", "statement": "[357, 139] The map (29) is injective. In particular, the map (30) is injective.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 3.22, PDF page 70\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Injectivity of the loop expansion map. Not obviously resolved." }, { "id": 10400073, "problem_number": "AMR-103-0073", "title": "Problem 3.23 — (T.", "statement": "(T. Kohno) Construct explicitly a universal invariant of finite type for links in Σ × [0, 1] with values in AΣ. In the case of genus 0 the above problem is solved by Kontsevic h integral. In higher genus case a suggestion for a construction of a univer sal invariant was given by Deligne at Oberwolfach meeting 1995. In the case of a punctured surface the problem was solved by Andersen, Mattes and Reshe tikhin. Let G be a simple Lie group and MG(Σ) the moduli space of G flat connections on Σ. The space of smooth functions on MG(Σ) denoted by C(MG(Σ)) has a structure of a Poisson algebra coming from a symplectic stru cture on MG(Σ). The algebraAΣ has also a Poisson algebra structure (see [8]). If each compo nent ofAΣ is colored by a representation of G, then there is a natural Poisson algebra homomorphism τ:AΣ → C(MG(Σ)). Problem 3.23 is related to the following problem.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.23, PDF page 71\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Universal invariant for surfaces of higher genus. Deligne's suggestion. Andersen-Mattes-Reshetikhin solved the punctured surface case. The higher genus case is partially addressed." }, { "id": 10400074, "problem_number": "AMR-103-0074", "title": "Problem 3.24 — (T.", "statement": "(T. Kohno) Give a deformation quantization of the Poisson algebraAΣ which descends to a deformation quantization of C(MG(Σ)). The above problem will give a new insight on quantization of MG(Σ). It would also be interesting to investigate a relation to the ge ometric quantization ofMG(Σ).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.24, PDF page 71\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Deformation quantization of Poisson algebra A_Sigma and C(M_G(Sigma)). Related to quantization of moduli spaces. There is extensive literature on quantization of character varieties." }, { "id": 10400075, "problem_number": "AMR-103-0075", "title": "Problem 3.25 — (T.", "statement": "(T. Kohno) Clarify the relation between a deformation quan- tization of C(MG(Σ)) at a special parameter and the space of conformal blocks in WZW models. Section 3.10 was written by T. Kohno.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.25, PDF page 71\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Relation between deformation quantization of C(M_G(Sigma)) and conformal blocks in WZW models. This is a well-studied connection in the Chern-Simons/WZW correspondence." }, { "id": 10400076, "problem_number": "AMR-103-0076", "title": "Problem 3.26 — (T.", "statement": "(T. Kohno) Determine the image and the kernel of the above map τ. The space of conformal blocks in WZW model is defined as the spa ce of coin- variant tensors in the following way. Let p1,···, pn be marked points on Σ and H1,···, Hn be representations of the affine Lie algebra ˆg. The space of conformal blocks is defined to be the set of linear forms φ: H1⊗···⊗Hn−→ C invariant under the action of meromorphic functions with va lues in g with poles at most at p1,···, pn, where the action is defined by the Laurent expansion at these points. There is a twisted version of the above constru ction, where the above meromorphic functions are replaced by meromorphic se ctions of a g local system.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.26, PDF page 72\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Image and kernel of the map tau from A_Sigma to C(M_G(Sigma)). Partially understood." }, { "id": 10400077, "problem_number": "AMR-103-0077", "title": "Problem 3.27 — (T.", "statement": "(T. Kohno) Compute the holonomy of the space of conformal blocks of the twisted WZW model. In particular, determine th e action of the braid group of Σ on the space of conformal blocks for each G flat connection on Σ. There is also a notion of the algebra of chord diagrams on n strings with horizontal chord on Σ, which we shall denote by An(Σ).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.27, PDF page 72\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Holonomy of conformal blocks and braid group actions. The braid group action on conformal blocks is a well-studied subject." }, { "id": 10400078, "problem_number": "AMR-103-0078", "title": "Problem 3.28 — (T.", "statement": "(T. Kohno) Let Pn(Σ) denote the pure braid group of Σ with n strings. Does there exist an injective multiplicative homo morphism θ: Pn(Σ) →A n(Σ) defined over Q?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 3.28, PDF page 72\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Whether there is an injective multiplicative homomorphism from the pure braid group of Sigma to the chord diagram algebra." }, { "id": 10400079, "problem_number": "AMR-103-0079", "title": "Problem 4.1 — Calculate S2,∞(M ) for each oriented 3-manifold M.", "statement": "Calculate S2,∞(M ) for each oriented 3-manifold M. Find a convenient methodology to calculate it.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.1, PDF page 74\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Compute S2,infinity(M) for all 3-manifolds. Partial results exist for various manifolds." }, { "id": 10400080, "problem_number": "AMR-103-0080", "title": "Problem 4.2 — (J.", "statement": "(J. Przytycki) Incompressible tori and 2-spheres in M yield torsion in S2,∞(M ) [339]. It is a question of fundamental importance whether other surfaces can yield torsion as well.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.2, PDF page 74\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Przytycki's question about torsion from non-torus surfaces. Not obviously resolved." }, { "id": 10400081, "problem_number": "AMR-103-0081", "title": "Conjecture 4.3 — If every closed incompressible surface in M is parallel to ∂M, then S2,∞(M ) is torsion free.", "statement": "If every closed incompressible surface in M is parallel to ∂M, then S2,∞(M ) is torsion free.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 4.3, PDF page 74\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Torsion-free conjecture for S2,infinity when no closed incompressible surfaces. Not obviously resolved." }, { "id": 10400082, "problem_number": "AMR-103-0082", "title": "Problem 4.4 — (J.", "statement": "(J. Przytycki) Compute S2,∞(F0,3× S1).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.4, PDF page 75\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Compute S2,infinity(F_{0,3}xS^1). Not obviously resolved." }, { "id": 10400083, "problem_number": "AMR-103-0083", "title": "Problem 4.5 — Let F be a surface and I an interval.", "statement": "Let F be a surface and I an interval. Describe the algebra S2,∞(F× I).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.5, PDF page 75\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Describe S2,infinity(FxI) algebra structure. Not obviously resolved." }, { "id": 10400084, "problem_number": "AMR-103-0084", "title": "Problem 4.6 — Calculate the skein homology based on the Kauffman bracket skein relation.", "statement": "Calculate the skein homology based on the Kauffman bracket skein relation.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.6, PDF page 75\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Calculate skein homology based on Kauffman bracket. Related to Khovanov homology." }, { "id": 10400085, "problem_number": "AMR-103-0085", "title": "Problem 4.7 — We define the sl3 skein module Ssl3(M ) of an oriented 3- manifold M by the defining relations of the sl3 linear skei…", "statement": "We define the sl3 skein module Ssl3(M ) of an oriented 3- manifold M by the defining relations of the sl3 linear skein [233, 323]. Calculate Ssl3(M ) of each 3-manifold M.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.7, PDF page 75\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Calculate sl3 skein module. Partial results exist." }, { "id": 10400086, "problem_number": "AMR-103-0086", "title": "Problem 4.8 — Calculate S3(M ) for each oriented 3-manifold M.", "statement": "Calculate S3(M ) for each oriented 3-manifold M. Find a con- venient methodology to calculate it.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.8, PDF page 76\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Calculate S3(M) for all 3-manifolds. Not obviously resolved." }, { "id": 10400087, "problem_number": "AMR-103-0087", "title": "Problem 4.9 — Let F be a surface and I an interval.", "statement": "Let F be a surface and I an interval. Describe the algebra S3(F× I).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.9, PDF page 76\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Describe S3(FxI) algebra. Not obviously resolved." }, { "id": 10400088, "problem_number": "AMR-103-0088", "title": "Problem 4.10 — Calculate S3,∞(M ) for each oriented 3-manifold M.", "statement": "Calculate S3,∞(M ) for each oriented 3-manifold M. Find a convenient methodology to calculate it.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.10, PDF page 76\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Calculate S3,infinity(M). Not obviously resolved." }, { "id": 10400089, "problem_number": "AMR-103-0089", "title": "Problem 4.11 — Calculate the higher skein modules based on the Kauffman skein relation W 3,∞ i (M ) and ˆW 3,∞(M ) (see below for the…", "statement": "Calculate the higher skein modules based on the Kauffman skein relation W 3,∞ i (M ) and ˆW 3,∞(M ) (see below for their definitions).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.11, PDF page 77\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Calculate higher Kauffman skein modules. Not obviously resolved." }, { "id": 10400090, "problem_number": "AMR-103-0090", "title": "Problem 4.12 — Construct invariants of 3-manifolds via a linear skein theo ry based on the Kauffman skein module.", "statement": "Construct invariants of 3-manifolds via a linear skein theo ry based on the Kauffman skein module.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.12, PDF page 77\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Construct 3-manifold invariants from Kauffman skein module. Not obviously resolved." }, { "id": 10400091, "problem_number": "AMR-103-0091", "title": "Problem 4.13 — Calculate HS q(M ) for each 3-manifold M.", "statement": "Calculate HS q(M ) for each 3-manifold M.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.13, PDF page 78\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Calculate HS_q(M) for all 3-manifolds. Not obviously resolved." }, { "id": 10400092, "problem_number": "AMR-103-0092", "title": "Problem 4.14 — (J.", "statement": "(J. Przytycki) (i) Find generators of S4,∞(S3, R). (ii) For which parameters of the (4,∞) skein and framing relations, trivial links are linearly independent in S4,∞(S3; R)? (iii) For which parameters of the (4,∞) skein and framing relations, the trivial knot is not representing a torsion element of S4,∞(S3, R)? A generalization of the Montesinos-Nakanishi conjecture [ 345] said that S4,∞(S3, R) is generated by trivial links and that the (4,∞) skein module (suit- ably defined) for n-tangles is generated by ∏ n−1 i=1 (3i +1) certain basic n-tangles. This would give a generating set for the (4,∞) skein module of S3 or D3 with 2 n boundary points (for n-tangles). However, the Montesinos-Nakanishi 3-move conjecture has been disproved by M.Dabkowski and J.H.Przytycki in February 2002 [99] and [342]. Therefore ∏ n−1 i=1 (3i + 1) is only the lower bound for the number of generators. In [345] we extensively analyze the possibilities that triv ial links are linearly independent; if b∞ = 0, then this may happen only if b0b1 = b2b3. These leads to the following conjecture (cases (1)–(2)):", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.14, PDF page 78\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Przytycki's questions about S4,infinity generators. The Montesinos-Nakanishi 3-move conjecture was disproved (Dabkowski-Przytycki 2002)." }, { "id": 10400093, "problem_number": "AMR-103-0093", "title": "Conjecture 4.15 — (J.", "statement": "(J. Przytycki, see [286]) (1) There is a polynomial invariant of unoriented links, P1(L)∈ Z[x, t] which satisfies: (i) Initial conditions: P1(Tn) = tn, where Tn is a trivial link of n components. (ii) Skein relation P1(L0) + xP1(L1)− xP1(L2)− P1(L3) = 0 where L0, L1, L2, L3 is a standard, unoriented skein quadruple ( Li+1 is obtained from Li by a right-handed half twist on two arcs involved in Li; compare Figure 15.) (2) There is a polynomial invariant of unoriented framed links, P2(L) ∈ Z[A±1, t] which satisfies: (i) Initial conditions: P2(Tn) = tn, (ii) Framing relation: P2(L(1)) =−A3P2(L) where L(1) is obtained from a framed link L by a positive half twist on its framing. (iii) Skein relation: P2(L0) + A(A2 + A−2)P2(L1) + (A2 + A−2)P2(L2) + AP2(L3) = 0. (3) There is a rational function invariant of unoriented framed links, P3(L)∈ Z[a±1, x, y, (x + y + xy + y2)−1] which satisfies: (i) Initial conditions: P3(Tn) = ( −a3(x+y+xy+x2)+a7(x+y+1)2−a−1 x+y+xy+y2 )n−1, (ii) Framing relation: P3(L(1)) = aP3(L), (iii) Skein relation: P3(L0)+axP3(L1)+a2yP3(L2)−a3(x+y+1)P3(L3) = 0. (4) The invariant predicted in (1) (respectively (2) and (3)) is not uniquely defined (if it exists). Note that a solution to (3) becomes a solution to (1) under the substitution a = 1, x =−y and that a solution to (3) becomes a solution to (2) under the substitution a =−A3, x =−1− A−4, y = A−4 + A−8. As for the uniqueness of (4), note that all such invariants agree on trivial links and therefore they agree on the space spanned by trivial links in the related cubic ske in module. The above conjectures assume that b∞ = 0 in our skein relation. Let consider the possibility that b∞ is invertible in R. Using the “denominator” of our skein relation (the first line of Figure 16) we get the relation whic h allows to compute the effect of adding a trivial component to a link L (we write tn for the trivial link Tn ): (a−3b3 + a−2b2 + a−1b1 + b0 + b∞t)L = 0. (33) When considering the “numerator” of the relation and its mir ror image (Figure 16) we obtain formulas for Hopf link summands, and because un oriented Hopf link is amphicheiral we can eliminate it from our equations t o get the formula (34): b3(L#H) + (ab2 + b1t + a−1b0 + ab∞)L = 0. b0(L#H) + (a−1b1 + b2t + ab3 + a2b∞)L = 0. ((b0b1− b2b3)t + (a−1b2 0− ab2 3) + (ab0b2− a−1b1b3) + b∞(ab0− a2b3))L = 0. (34) It is possible that (33) and (34) are the only relations in the module. Precisely, we ask whether S4,∞(S3; R) is the quotient ring R[t]/(I) where ti represents the trivial link of i components andI is the ideal generated by (33) and (34) for L = t. The substitution which realizes the relations is: b0 = b3 = a = 1, b1 = b2 = x, b∞ = y. This may lead to the polynomial invariant of unoriented lin ks in S3 with values in Z[x, y] and the skein relation L3+xL2+xL1+L0+yL∞ = 0.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 4.15, PDF page 79\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Existence of polynomial invariants for cubic skein relation. Not obviously resolved." }, { "id": 10400094, "problem_number": "AMR-103-0094", "title": "Problem 4.16 — (J.", "statement": "(J. Przytycki) For which coefficients of the (4,∞) skein rela- tion is the number of Fox 7-colorings measured by the (4,∞) skein module? Figure 16", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.16, PDF page 80\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Fox 7-colorings from (4,infinity) skein module. Not obviously resolved." }, { "id": 10400095, "problem_number": "AMR-103-0095", "title": "Problem 4.17 — Calculate Wk(M ) for each 3-manifold M.", "statement": "Calculate Wk(M ) for each 3-manifold M.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.17, PDF page 82\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Calculate Wk(M) for all 3-manifolds. Not obviously resolved." }, { "id": 10400096, "problem_number": "AMR-103-0096", "title": "Problem 4.18 — Define a skein module of 3-manifolds, and calculate it.", "statement": "Define a skein module of 3-manifolds, and calculate it.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 4.18, PDF page 82\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Define and calculate a skein module of 3-manifolds. Not obviously resolved." }, { "id": 10400097, "problem_number": "AMR-103-0097", "title": "Problem 5.1 — Classify the isomorphism classes of connected quandles of o rder n for each positive integer n.", "statement": "Classify the isomorphism classes of connected quandles of o rder n for each positive integer n. See Table 4 for a list of connected quandles of order n for some n. n # Connected quandles of order n Self-dual Not self-dual 1 1 A trivial quandle 2 0 3 1 R3 4 1 Λ 2/(t2 + t + 1) 5 3 R5 Λ 5/(t− 2), its dual 6 2 2 subquandles of Conj( S4) 7 5 R7 Λ 7/(t− 2), Λ 7/(t− 3), their duals 8 ≥ 3 An abelian extension Λ 2/(t3 + t + 1), its dualof Λ 2/(t2 + t + 1) 9 8 R9, Λ 3/(t2− t + 1), Λ 9/(t− 2), Λ 3/(t2 + t− 1), R3× R3, Λ 3/(t2 + 1) their duals 10 ≥ 1 A subquandle of Conj( S5) 11 9 R11 Λ 11/(t− a) ( a = 2, 3,···, 9) 12 ≥ 2 R3× ( Λ 2/(t2 + t + 1) ), An icosahedral quandle 13 11 R13 Λ 13/(t− a) ( a = 2, 3,···, 11) 14 ≥ 0 15 ≥ 4 R3× R5, R3× ( Λ 5/(t− 2) ), its dualA subquandle of Conj( S5)... Prime p p− 2 Rp Λ p/(t− a) ( a = 2, 3,···, p− 2) Table 4: A table of some connected quandles. The second colum n shows the numbers of isomorphism classes of connected quandles of order n. We denote Z[t±1]/(n) by Λ n. Conj(Sm) denotes the conjugation quandle of the mth symmetric group Sm. An icosahedral quandle is a quandle whose elements are the vertices of an icosahedro n such that Sx of each element x is given by a rotation of the icosahedron centered at x.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 5.1, PDF page 84\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Classification of connected quandles of finite order. Partial results for small orders exist." }, { "id": 10400098, "problem_number": "AMR-103-0098", "title": "Problem 5.2 — Describe (the number of) representations of a knot quandle t o a fixed connected quandle of finite order, say, by usi…", "statement": "Describe (the number of) representations of a knot quandle t o a fixed connected quandle of finite order, say, by using knot in variants known so far, or by reducing the problem to the case of smaller targe t quandles.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 5.2, PDF page 86\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Describe representations of knot quandle to a fixed connected quandle. Not obviously resolved." }, { "id": 10400099, "problem_number": "AMR-103-0099", "title": "Conjecture 5.3 — Let hX be as above.", "statement": "Let hX be as above. Then, log hX is not a Vassiliev invariant, unless it is constant. 5.3 (Co)homology of quandles Second cohomology classes of a quandle are used in order to de fine quandle cocycle invariants of knots. They are introduced as follows. Let A be an abelian group, written additively, and let C n(X; A) be the abelian group consisting of maps X n→ A, where X n denotes the direct product of n copies of X. We put C 1 Q(X; A) = C 1(X; A), C 2 Q(X; A) ={f∈ C 2(X; A)| f (x, x) = 0 for any x∈ X}, C 3 Q(X; A) ={g∈ C 3(X; A)| g(x, x, y) = 0 and g(x, y, y) = 0 for any x, y∈ X}. The coboundary operators di: C i Q(X; A)→ C i+1 Q (X; A) are given by d1f (x, y) = f (x)− f (x∗ y), d2g(x, y, z) = g(x, z)− g(x, y)− g(x∗ y, z) + g(x∗ z, y∗ z), for f∈ C 1 Q(X; A) and g∈ C 2 Q(X; A). We define the second quandle cohomol- ogy group by H 2 Q(X; A) = (kernel d2)/(image d1). It is known that H 2 Q(X; A) is isomorphic to Hom ( H Q 2 (X); A ) by the universal coefficient theorem, noting that H Q 1 (X) is free abelian (see [82]). Here, H Q 2 (X) denotes the second homol- ogy group of the dual complex of {C ⋆ Q(X; Z), d⋆}. See [82] for the definition of the nth quandle (co)homology group. Therefore, to obtain H 2 Q(X; A) for any A, it is sufficient to compute H Q 2 (X). Connected quandle X Order H Q 2 (X) H Q 3 (X) R3 3 0 Z/3Z Z[t±1]/(2, t2 + t + 1) 4 Z/2Z Z/2Z⊕ Z/4Z R5 5 0 Z/5Z Z[t±1]/(5, t− 2) 0 0 R7 0 Z/7Z Z[t±1]/(7, t− 2) 7 0 0 Z[t±1]/(7, t− 3) 0 0 Z[t±1]/(2, t3 + t + 1) 8 0 Z/2Z R9 0 Z/9Z Z[t±1]/(9, t− 2) 0 Z/3Z Z[t±1]/(3, t2 + 1) 9 Z/3Z (Z/3Z)3 Z[t±1]/(3, t2− t + 1) Z/3Z Z/3Z⊕ Z/9Z Z[t±1]/(3, t2 + t− 1) 0 0 Z[t±1]/(p, t− a) p 0for any prime p and any a̸= 0, 1∈ Z/pZ Table 5: The cohomologies of the quandles, except for the las t one, in the table are due to [264]. From a table in [264] we omit one of two dual quandles and quandles that are not connected (see remarks on Problem 5.6). The 2nd homology of Z[t±1]/(p, t− a) is due to [284]. See [264, 284] for computations of cohomolog y groups of some more quandles.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 5.3, PDF page 87\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Conjecture about hX and homology. Not obviously resolved." }, { "id": 10400100, "problem_number": "AMR-103-0100", "title": "Problem 5.4 — Compute H Q 2 (X) for each connected quandle X.", "statement": "Compute H Q 2 (X) for each connected quandle X. More gener- ally, find a convenient methodology to compute quandle (co)h omology groups. See Table 5 for some quandle homology groups given in [264]; s ee also [284] for computations of quandle cohomology groups of many Alexa nder quandles. There are maple programs [185] for computing quandle cohomo logy groups.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 5.4, PDF page 88\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Compute HQ^2(X) for connected quandles. Partial results exist for specific quandles." }, { "id": 10400101, "problem_number": "AMR-103-0101", "title": "Problem 5.5 — (J.S.", "statement": "(J.S. Carter) Compute H Q i (Sm n ) of Sm n which denotes the quandle of the nth symmetric group with the binary operation given by x∗ y = y−mxym. 5.4 Quandle cocycle invariant The quandle cocycle invariant, introduced in [79, 80], is de fined as follows. For α∈ H 2(X; A) we choose a 2-cocycle φ representing α. Any representation of a knot quandle Q(K) to X is presented by a coloring of a knot diagram of K, where a coloring of an oriented knot diagram is a map of the set of over-arcs of it to X satisfying the condition depicted in the pictures of (35) at each crossing of the knot diagram. We define the weight of a crossing of a colo red diagram by W ( x x∗y y ) = φ(x, y)∈ A, W ( y x x∗y) = φ(x, y)−1∈ A, (35) where we write A multiplicatively here. The quandle cocycle invariant of a knot K is defined by Φ α(K) = ∑ C ∏ τ W (τ,C)∈ Z[A], where the sum runs over all coloring C of a diagram of K, and the product runs over all crossing τ of the diagram, and Z[A] denotes the group ring of A. Φ α(K) only depends on K and α.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 5.5, PDF page 88\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Compute HQ^i of symmetric group quandle. Not obviously resolved." }, { "id": 10400102, "problem_number": "AMR-103-0102", "title": "Problem 5.6 — Compute the quandle cocycle invariant Φ α(K) of each knot K for a second cohomology class α of a connected quandle.", "statement": "Compute the quandle cocycle invariant Φ α(K) of each knot K for a second cohomology class α of a connected quandle.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 5.6, PDF page 89\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Compute quandle cocycle invariants. Many computations exist for specific knots and quandles." }, { "id": 10400103, "problem_number": "AMR-103-0103", "title": "Problem 5.7 — Find relations between quandle cocycle invariants and knot in- variants known so far, such as quantum invariants.", "statement": "Find relations between quandle cocycle invariants and knot in- variants known so far, such as quantum invariants.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 5.7, PDF page 91\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Relations between quandle cocycle invariants and quantum invariants. Some relations known." }, { "id": 10400104, "problem_number": "AMR-103-0104", "title": "Problem 5.8 — Compute H 2 Q(X; A) for each X -module A.", "statement": "Compute H 2 Q(X; A) for each X -module A.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 5.8, PDF page 92\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Compute quandle cohomology with module coefficients. Not obviously resolved." }, { "id": 10400105, "problem_number": "AMR-103-0105", "title": "Problem 5.9 — Let the notation be as above.", "statement": "Let the notation be as above. Then, extending the definition o f the quandle cocycle invariant, define a knot invariant assoc iated with α, which is, roughly speaking, an invariant obtained by counting rep resentations of a knot quandle Q(K) to X with information whether each representation can lift to a representation Q(K)→ Y. 5.5 Quantum quandles", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 5.9, PDF page 92\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Lifted quandle cocycle invariant. Not obviously resolved." }, { "id": 10400106, "problem_number": "AMR-103-0106", "title": "Problem 5.10 — (M.", "statement": "(M. Polyak) Define a quantum quandle.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 5.10, PDF page 92\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Define quantum quandle. Not obviously resolved." }, { "id": 10400107, "problem_number": "AMR-103-0107", "title": "Problem 5.11 — (C.", "statement": "(C. Rourke, B. Sanderson) Is there a natural quandle space whose cohomology groups are the quandle cohomology groups?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 5.11, PDF page 93\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Natural quandle space whose cohomology is quandle cohomology. Not obviously resolved." }, { "id": 10400108, "problem_number": "AMR-103-0108", "title": "Conjecture 5.12 — (R.", "statement": "(R. Fenn, C. Rourke, B. Sanderson) H3(Rp)∼ = Z⊕ Z/pZ for p prime.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 5.12, PDF page 93\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Classification: SOLVED-IN-LITERATURE. The conjectured homology computation was proved by Niebrzydowski and Przytycki (2006/2009). Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Conjecture 5.12 (Fenn–Rourke–Sanderson): H^R_3(R_p) ≅ Z ⊕ Z/pZ for p prime (rack homology of the dihedral/rotation rack). Solved by M. Niebrzydowski and J. H. Przytycki, \"Homology of dihedral quandles\", J. Pure Appl. Algebra 213 (2009) 742–755 (arXiv:math/0611803, 2006): they prove H^R_3(R_p) = Z ⊕ Z_p for p odd prime, using Mochizuki's computation H^Q_3(R_p; Z_p) = Z_p and the decomposition H^R_3(X) ≅ H^Q_3(X) ⊕ H^Q_2(X) ⊕ Z·O_X^2. They also show H^R_n(R_p) contains Z_p for n ≥ 3 and prove related results for R_3." }, { "id": 10400109, "problem_number": "AMR-103-0109", "title": "Problem 6.1 — ([188, Problem 3]) Is the representation of the braid group inside the Temperley-Lieb algebra faithful?", "statement": "([188, Problem 3]) Is the representation of the braid group inside the Temperley-Lieb algebra faithful?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 6.1, PDF page 95\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Faithful for n ≤ 4; open for n ≥ 5. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kirby's Problem 3: is the representation of the braid group inside the Temperley-Lieb algebra faithful? Partial progress: Bigelow ('Does the Jones polynomial detect the unknot?', J. Knot Theory Ramifications 11 (2002) 493–505, arXiv:math/0012086) proved the TL, Jones, and Burau representations of B4 are simultaneously faithful or unfaithful. The Burau representation of B4 was proved faithful by V. Bharathram, J. Birman, T. Brendle (arXiv:2607.05283, July 2026), hence the Temperley-Lieb representation of B4 is faithful. For n ≤ 3 faithfulness is classical. For n ≥ 5 the question remains open; the case n = 5 is linked (via Bigelow's Conjecture) to the existence of non-trivial knots with Jones/HOMFLY polynomial equal to 1." }, { "id": 10400110, "problem_number": "AMR-103-0110", "title": "Problem 6.2 — Is the Burau representation of B4 faithful?", "statement": "Is the Burau representation of B4 faithful?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 6.2, PDF page 96\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Classification: SOLVED-IN-LITERATURE. The Burau representation of B4 is faithful, proved by Bharathram–Birman–Brendle (2026). Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. The Burau representation of B4 was long an outstanding open question (faithful for n ≤ 3 by Magnus–Peluso; unfaithful for n ≥ 5 by Moody, Long–Paton, Bigelow). It was SOLVED in July 2026: V. Bharathram, J. S. Birman, T. E. Brendle, \"The Burau representation of the braid group is faithful for n = 4\", arXiv:2607.05283 (July 2026). The main theorem proves ρ_4 is faithful, using point-pushing subgroups of the disk's mapping class group; an immediate corollary is that the Jones representation of B4 is also faithful." }, { "id": 10400111, "problem_number": "AMR-103-0111", "title": "Problem 6.3 — (S.J.", "statement": "(S.J. Bigelow) Is the action of B6 on V 6 2 faithful?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 6.3, PDF page 96\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Faithfulness of the action of B6 on V^6_2. Not obviously resolved." }, { "id": 10400112, "problem_number": "AMR-103-0112", "title": "Problem 6.4 — (S.J.", "statement": "(S.J. Bigelow) Generalise Lawrence’s construction to obtain the irreducible representations of the BMW algebra.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 6.4, PDF page 97\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Generalize Lawrence's construction for BMW algebra irreps. Not obviously resolved." }, { "id": 10400113, "problem_number": "AMR-103-0113", "title": "Problem 6.5 — (S.J.", "statement": "(S.J. Bigelow) Find a larger family of irreducible representa- tions of Bn which includes those coming from the BMW algebra.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 6.5, PDF page 97\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Find larger family of irreducible representations of Bn. Not obviously resolved." }, { "id": 10400114, "problem_number": "AMR-103-0114", "title": "Problem 6.6 — Classify all irreducible representations of Bn.", "statement": "Classify all irreducible representations of Bn.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 6.6, PDF page 97\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Complete classification of irreducible representations of Bn is a major open problem in representation theory." }, { "id": 10400115, "problem_number": "AMR-103-0115", "title": "Problem 6.7 — (S.J.", "statement": "(S.J. Bigelow) Is there a faithful representation of Bn into a group of matrices over ¯Q?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 6.7, PDF page 98\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Existence of a faithful representation of Bn into matrices over Q-bar. Not obviously resolved." }, { "id": 10400116, "problem_number": "AMR-103-0116", "title": "Problem 7.1 — (see [220, Problem 3.108]) Does there exist a closed 3-manifold M, other than S3, such that τ SO(3) r (M ) = τ SO(3)…", "statement": "(see [220, Problem 3.108]) Does there exist a closed 3-manifold M, other than S3, such that τ SO(3) r (M ) = τ SO(3) r (S3) for all odd r≥ 3?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.1, PDF page 100\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Existence of a closed 3-manifold other than S3 with same SO(3) quantum invariants. Not obviously resolved." }, { "id": 10400117, "problem_number": "AMR-103-0117", "title": "Problem 7.2 — (S.K.", "statement": "(S.K. Hansen, T. Takata) Find pairs of non-homeomorphic rational homology 3-spheres that can be distinguished by th eir quantum G invariants τ G r or their quantum P G invariants τ P G r for some level r and some simply connected compact simple Lie group G but not by their LMO invariants.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.2, PDF page 100\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Finding pairs of 3-manifolds distinguished by quantum invariants but not by LMO. Not obviously resolved." }, { "id": 10400118, "problem_number": "AMR-103-0118", "title": "Problem 7.3 — (S.K.", "statement": "(S.K. Hansen, T. Takata) Do the family of quantum G invari- ants τ G r or the family of quantum P G invariants τ P G r, G running through all simply connected compact simple Lie groups and r running through all allowed levels, separate rational homology 3-spheres? How well do t hese families of invariants separate closed oriented 3-manifolds?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.3, PDF page 101\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Whether quantum invariants separate rational homology spheres. Not obviously resolved." }, { "id": 10400119, "problem_number": "AMR-103-0119", "title": "Problem 7.4 — Find a 3-dimensional topological interpretation of quantu m in- variants of 3-manifolds.", "statement": "Find a 3-dimensional topological interpretation of quantu m in- variants of 3-manifolds.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.4, PDF page 101\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. 3D topological interpretation of quantum invariants. Witten's Chern-Simons theory provides a 3D interpretation, but a purely topological one is sought." }, { "id": 10400120, "problem_number": "AMR-103-0120", "title": "Conjecture 7.5 — For non-vanishing $\\tau_r^G(M)$, the absolute value $|\\tau_r^G(M)|$ depends only on the fundamental group $\\pi_1(M)$.", "statement": "For non-vanishing $\\tau_r^G(M)$, the absolute value $|\\tau_r^G(M)|$ depends only on the fundamental group $\\pi_1(M)$.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 7.5, PDF page 102\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Conjecture that |tau_r^G(M)| depends only on pi_1(M). Not obviously resolved." }, { "id": 10400121, "problem_number": "AMR-103-0121", "title": "Conjecture 7.6 — (The perturbative expansion conjecture) The asymptotic expansion of Z G k (M ) of a closed oriented 3-manifold M is g…", "statement": "(The perturbative expansion conjecture) The asymptotic expansion of Z G k (M ) of a closed oriented 3-manifold M is given by Z G k (M ) ∼ k→∞ e−π√ −1(dim G)(1+b1(M ))/4 × ∫ [A]∈M e2π√−1rCS(A)r(h1 A−h0 A)/2e−2π√ −1(IA/4+(h0 A+h1 A)/8)τM (A)1/2 × exp   ∞∑ l=1 clk−l (2l)!(3l)! ∑ e(Γ)= −l ZΓ (M, A) |Aut(Γ)|  , putting r = k + h ∨, where the right hand side can be given in the mathemat- ical viewpoint in certain cases, as mentioned above, but whi ch needs further interpretation in general.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 7.6, PDF page 105\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Perturbative expansion conjecture. The asymptotic expansion of quantum invariants is understood in the framework of perturbative Chern-Simons theory, at least formally." }, { "id": 10400122, "problem_number": "AMR-103-0122", "title": "Conjecture 7.7 — (The asymptotic expansion conjecture, J.E.", "statement": "(The asymptotic expansion conjecture, J.E. Andersen [6]) Let{c0 = 0, c1,···, cm} be the set of values of the Chern-Simons functional of flat G connections on a closed oriented 3-manifold M. There exist dj∈ Q, ˜Ij∈ Q/Z, vj∈ R+, and ae j∈ C for j = 0, 1,···, m and e = 1, 2, 3,···such that ( r = k + h ∨ ) Z G k (M ) ∼ r→∞ m∑ j=0 e2π√ −1rcj rdj eπ√−1 ˜Ij/4vj ( 1 + ∞∑ e=1 ae jr−e), that is, for all E = 0, 1, 2,..., there exists a constant cE such that ⏐ ⏐ ⏐Z G k (M )− m∑ j=0 e2π√ −1rcj rdj eπ√−1 ˜Ij/4vj ( 1 + E∑ e=1 ae jr−e) ⏐ ⏐ ⏐≤ cErd−E−1 for all r = 2, 3, 4,···. Here, d = max{d0,···, dm}.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 7.7, PDF page 106\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Asymptotic expansion conjecture (Andersen). Proven for mapping tori and Seifert fibered spaces." }, { "id": 10400123, "problem_number": "AMR-103-0123", "title": "Problem 7.8 — (J.E.", "statement": "(J.E. Andersen) If such an expansion exists, understand how it is related to the expansion of Ohtsuki and the expansion of Habiro. It will of course be important to establish, that an expansio n of this type ex- ists, however, of far greater importance will be to give inde pendent topological meaning to the many resulting new invariants, e.g. to prove t hat the phases are the Chern-Simons values cj. From the discussion above on the semi-classical approximation we derive the following conjecture:", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.8, PDF page 106\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Andersen's problem about the measure tau_M(A)^{1/2}." }, { "id": 10400124, "problem_number": "AMR-103-0124", "title": "Conjecture 7.9 — (Topological interpretations of the dj ’s) Let Mj be the union of components of the moduli space of flat connections…", "statement": "(Topological interpretations of the dj ’s) Let Mj be the union of components of the moduli space of flat connections M which has Chern-Simons value cj. Then dj = 1 2 max A∈Mj (h1 A− h0 A), where max here means the maximum value that (h1 A−h0 A) assumes on a Zariski open subset of Mj. Note that this conjecture might be rather optimistic, and ma y only hold in the non-degenerate cases. However, we do not know of any cases wh ere it fails (see [136]).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 7.9, PDF page 106\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Topological interpretation of the exponents dj." }, { "id": 10400125, "problem_number": "AMR-103-0125", "title": "Conjecture 7.10 — (The growth rate conjecture) Let d = max{d0,..., dn}.", "statement": "(The growth rate conjecture) Let d = max{d0,..., dn}. Then|Z G r (M )| = O(rd). It is well known that the quantum invariants only grows like r to some power. The power is bounded from above by some simple function (depe nding on G) of the Heegaard genus of the manifold.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 7.10, PDF page 107\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Growth rate conjecture. The growth rate of quantum invariants is bounded by a function of Heegaard genus." }, { "id": 10400126, "problem_number": "AMR-103-0126", "title": "Conjecture 7.11 — There is a construct of the right measure, say τM (A)1/2 for A∈M i, from the square root of the Reidemeister torsion…", "statement": "There is a construct of the right measure, say τM (A)1/2 for A∈M i, from the square root of the Reidemeister torsion generaliz ing the non-degenerate case explained above and such that eπ√ −1 ˜Ij/4vj = ∫ A∈Mi eπ√−1(−2IA+h0 A+h1 A)/4τM (A)1/2. Conjectures 7.7 and 7.9 together with Conjecture 7.11 were fi rst proved for mapping tori of all finite order diffeomorphisms of all surfac es of genus at least two in [6]. Recently, Conjecture 7.7 was proved for all Seife rt fibered spaces in [168] by supplementing the calculations in [353] and [354] w ith the need analytic estimates.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 7.11, PDF page 107\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Construction of the measure tau_M(A)^{1/2} from Reidemeister torsion. Related to the work of Witten, Freed, etc." }, { "id": 10400127, "problem_number": "AMR-103-0127", "title": "Conjecture 7.12 — (H.", "statement": "(H. Murakami [294]) For any closed 3-manifold M, 2π √ −1·o-lim N →∞ log τ SU (2) N (M ) N = CS(M ) + √ −1vol(M ), where vol(M ) and CS(M ) denote the hyperbolic volume 23 and the Chern- Simons invariant24 of M respectively, and o- lim denotes the “optimistic limit” introduced in [294]. 23When M is not hyperbolic, we define vol( M ) to be v3||M ||, where ||M || is the simplicial volume and v3 is the hyperbolic volume of the regular ideal tetrahedron. 24It is also conjectured (see Problem 7.16) that there exists a n appropriate definition of CS(M ) of any closed 3-manifold M, though CS( M ) is defined only for hyperbolic 3-manifolds M at present.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 7.12, PDF page 109\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Complex Volume Conjecture for SU(2). The optimistic limit of log tau_{SU(2)}_N / N is expected to give CS + i Vol." }, { "id": 10400128, "problem_number": "AMR-103-0128", "title": "Problem 7.13 — (H.", "statement": "(H. Murakami) Calculate o- lim log τ SU (2) N (M ) N for Seifert fibered 3-manifolds M.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.13, PDF page 110\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Calculate optimistic limit for Seifert fibered spaces. Some computations exist." }, { "id": 10400129, "problem_number": "AMR-103-0129", "title": "Problem 7.14 — (D.", "statement": "(D. Thurston) Find a series of invariants of a 3-manifold (de- pending on roots of unity) that grows as its hyperbolic volum e (or its simplicial volume).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.14, PDF page 111\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Series of invariants growing as hyperbolic volume. Not obviously resolved." }, { "id": 10400130, "problem_number": "AMR-103-0130", "title": "Problem 7.15 — (D.", "statement": "(D. Thurston) Find a correct generalization of the volume conjecture to other non-compact Lie groups.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.15, PDF page 111\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Volume conjecture for other non-compact Lie groups. Not obviously resolved." }, { "id": 10400131, "problem_number": "AMR-103-0131", "title": "Problem 7.16 — (S.", "statement": "(S. Morita [228]) Define the Chern-Simons invariant CS(M ) as a topological invariant of any closed oriented 3-manifol d M, and of any knot (link) complement M in a closed 3-manifold. This problem includes two problems: to define CS( M ) (topologically or com- binatorially) as a topological invariant, and to define it fo r non-hyperbolic 3- manifolds.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.16, PDF page 111\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Topological definition of Chern-Simons invariant for all 3-manifolds. Various approaches exist but a fully satisfactory definition remains open." }, { "id": 10400132, "problem_number": "AMR-103-0132", "title": "Problem 7.17 — (T.", "statement": "(T. Ohtsuki) Give a “complex structure” to the set of 3- manifolds. More precisely, find an embedding (or, an immersi on) of the set of 3-manifolds to some complex variety such that its restric tion to the set {NK;(p,q)| p2 + q2 > >0} can be extended to a holomorphic map of the above mentioned complex parameter for any (hyperbolic) knot K in any 3-manifold N. We would expect some structures of the set of 3-manifolds suc h as mentioned in Problems 7.17 and Problem 10.16. Such structures would yiel d new viewpoints in the study of (the set of, and invariants of) 3-manifolds.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.17, PDF page 112\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. 'Complex structure' on the set of 3-manifolds. Not obviously resolved." }, { "id": 10400133, "problem_number": "AMR-103-0133", "title": "Problem 7.18 — (S.", "statement": "(S. Baseilhac, R. Benedetti) Generalize the construction of the QHI for flat principal G-bundles, for Lie groups G different from B. Section 7.4 was written by S. Baseilhac and R. Benedetti.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.18, PDF page 113\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Generalize QHI for other Lie groups. Not obviously resolved." }, { "id": 10400134, "problem_number": "AMR-103-0134", "title": "Problem 7.19 — (S.", "statement": "(S. Baseilhac, R. Benedetti) Fix (W, L) and vary ρ. Study KN as a function of the bundle, that is as a function defined on the character variety of W with respect to B: regularity, fibers, and so on.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.19, PDF page 114\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Study KN as function on character variety. Not obviously resolved." }, { "id": 10400135, "problem_number": "AMR-103-0135", "title": "Problem 7.20 — (S.", "statement": "(S. Baseilhac, R. Benedetti) Specialize Problem 7.19 to bun- dles coming from the ordinary cohomology as above. For real a dditive ones, analyze the behaviour of the QHI with respect to Thurston’s n orm. Are they constant on the faces of the corresponding unit sphere?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.20, PDF page 114\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Specialize QHI to cohomology bundles. Not obviously resolved." }, { "id": 10400136, "problem_number": "AMR-103-0136", "title": "Problem 7.21 — (S.", "statement": "(S. Baseilhac, R. Benedetti) Understand the ‘phase factor’ (i.e. the ambiguity due to N -th roots of unity) of the state sum HN (T ). Possi- bly derive from it an invariant for (W, L, ρ) endowed with some extra-structure, thus refining KN (W, L, ρ).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.21, PDF page 114\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Phase factor of state sum HN(T). Not obviously resolved." }, { "id": 10400137, "problem_number": "AMR-103-0137", "title": "Problem 7.22 — (S.", "statement": "(S. Baseilhac, R. Benedetti) Determine a suitable (2 + 1) ‘decorated’ cobordism theory supporting a (non purely topo logical) QFT con- taining the already defined QHI. Study in particular the beha viour of the QHI with respect to connected sums.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.22, PDF page 115\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Decorated cobordism theory for QHI. Not obviously resolved." }, { "id": 10400138, "problem_number": "AMR-103-0138", "title": "Problem 7.23 — (S.", "statement": "(S. Baseilhac, R. Benedetti) Develop a 4-dimensional theory of QHI based on Turaev’s shadow theory.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.23, PDF page 115\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. 4-dimensional QHI based on shadow theory. Not obviously resolved." }, { "id": 10400139, "problem_number": "AMR-103-0139", "title": "Problem 7.24 — (S.", "statement": "(S. Baseilhac, R. Benedetti) Determine the actual relation- ship between KN (S3,·) and the coloured Jones polynomial JN (·) (evaluated at ω = exp(2iπ/N) and normalized by JN (unknot) = 1 ), as functions of links.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.24, PDF page 115\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Relation between QHI and colored Jones polynomial. Not obviously resolved." }, { "id": 10400140, "problem_number": "AMR-103-0140", "title": "Conjecture 7.25 — (S.", "statement": "(S. Baseilhac, R. Benedetti) (Real Volume Conjecture for QHI) For any triple (W, L, ρ) one has: lim N →∞ (2π/N 2) log(|KN (W, L, ρ)|) = Im R ( cI (W, L, ρ) ).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 7.25, PDF page 116\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Real Volume Conjecture for QHI. Not obviously resolved." }, { "id": 10400141, "problem_number": "AMR-103-0141", "title": "Problem 7.26 — For each rational homology 3-sphere M, calculate τ SO(3)(M ) and τ P SU (N )(M ) for all degrees.", "statement": "For each rational homology 3-sphere M, calculate τ SO(3)(M ) and τ P SU (N )(M ) for all degrees.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.26, PDF page 117\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Calculate tau^{SO(3)} and tau^{PSU(N)} for all degrees for rational homology spheres." }, { "id": 10400142, "problem_number": "AMR-103-0142", "title": "Problem 7.27 — (J.", "statement": "(J. Roberts) Explain the appearance of modular forms in the Witten invariants.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.27, PDF page 118\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Appearance of modular forms in Witten invariants. This is related to the modularity of quantum invariants at roots of unity (work of Zagier, Lawrence, etc.)." }, { "id": 10400143, "problem_number": "AMR-103-0143", "title": "Problem 7.28 — Characterize those elements of Z[[q−1]] of the form τ SO(3)(M ) of integral homology 3-spheres M.", "statement": "Characterize those elements of Z[[q−1]] of the form τ SO(3)(M ) of integral homology 3-spheres M.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.28, PDF page 118\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Characterize tau^{SO(3)} of integral homology spheres. Not obviously resolved." }, { "id": 10400144, "problem_number": "AMR-103-0144", "title": "Conjecture 7.29 — (K.", "statement": "(K. Habiro, T. Le) For each g as above, there is a (unique) invariant I g(M )∈ R1 of an integral homology 3-sphere M such that for each root of unity ζ of order r divisible by d we have I g(M ) ⏐ ⏐ q=ζ = τ g ζ (M ).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 7.29, PDF page 119\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Habiro-Le's unified invariant conjecture. This is related to the Habiro ring and the theory of integral quantum invariants. Habiro's work on the cyclotomic expansion." }, { "id": 10400145, "problem_number": "AMR-103-0145", "title": "Conjecture 7.30 — (K.", "statement": "(K. Habiro) Suppose that Conjecture 7.29 would hold. For a new indeterminate t, set R′ 1 = lim←−nR1[t]/((t− q)(t− q2)···(t− qn)) Then there exists an invariant I sl(M )∈ R′ 1 of an integral homology 3-sphere M such that I sl(M )|t=qn = I sln(M ) for any n≥ 1, where we set I sl1(M ) = 1.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 7.30, PDF page 120\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Habiro's unified sl_n invariant. Not obviously resolved." }, { "id": 10400146, "problem_number": "AMR-103-0146", "title": "Problem 7.31 — Characterize those elements of Habiro’s expansion (45) of τ SO(3)(M ) of integral homology 3-spheres M.", "statement": "Characterize those elements of Habiro’s expansion (45) of τ SO(3)(M ) of integral homology 3-spheres M.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 7.31, PDF page 120\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Characterize Habiro's expansion of tau^{SO(3)}. Not obviously resolved." }, { "id": 10400147, "problem_number": "AMR-103-0147", "title": "Problem 8.1 — Find (and classify) all TQFT’s.", "statement": "Find (and classify) all TQFT’s.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.1, PDF page 122\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Classification of all TQFTs in 3 dimensions is a major open problem." }, { "id": 10400148, "problem_number": "AMR-103-0148", "title": "Problem 8.2 — Find (and classify) all modular categories.", "statement": "Find (and classify) all modular categories. For a TQFT ( V, Z), put P(V,Z )(t) =∑ ∞ g=0 ( dimV (Σ g) ) tg, where Σ g denotes a closed surface of genus g. The following problem is a refinement of Problem 8.1.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.2, PDF page 122\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Classification of all modular categories is a major open problem. Related to the classification of TQFTs." }, { "id": 10400149, "problem_number": "AMR-103-0149", "title": "Problem 8.3 — (1) Characterize the power series of the form P(V,Z )(t).", "statement": "(1) Characterize the power series of the form P(V,Z )(t). (2) For each power series P (t) (satisfying the characterization of (1)), classify all TQFT’s (V, Z) such that P(V,Z )(t) = P (t).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.3, PDF page 123\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Characterize power series of TQFT dimensions and classify TQFTs with given series." }, { "id": 10400150, "problem_number": "AMR-103-0150", "title": "Problem 8.4 — Find other spin TQFT’s.", "statement": "Find other spin TQFT’s.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.4, PDF page 123\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Find other spin TQFTs beyond the known examples." }, { "id": 10400151, "problem_number": "AMR-103-0151", "title": "Problem 8.5 — Formulate and find spin c TQFT’s.", "statement": "Formulate and find spin c TQFT’s.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.5, PDF page 124\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Formulate and find spin^c TQFTs." }, { "id": 10400152, "problem_number": "AMR-103-0152", "title": "Problem 8.6 — (V.", "statement": "(V. Turaev) (1) Extend HQFT’s to spin and spin c settings. (2) Find algebra structures behind spin and spin c HQFT’s in dimension 1+1.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.6, PDF page 124\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Extend HQFTs to spin and spin^c settings." }, { "id": 10400153, "problem_number": "AMR-103-0153", "title": "Problem 8.7 — (V.", "statement": "(V. Turaev) Study (spin and spin c ) HQFT’s with the target space K(H, 2) in dimensions 1 + 1, 2 + 1, and 3 + 1 for H = ZN.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.7, PDF page 124\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Study HQFTs with target K(H,2) in various dimensions." }, { "id": 10400154, "problem_number": "AMR-103-0154", "title": "Problem 8.8 — Find a geometric construction of a TQFT using H 0(MΣ,L⊗k).", "statement": "Find a geometric construction of a TQFT using H 0(MΣ,L⊗k). Namely, find a geometric way to associate a vector in H 0(MΣ,L⊗k) to a 3- manifold M with ∂M = Σ.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.8, PDF page 124\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Geometric construction of TQFT using H^0(M_Sigma, L^k)." }, { "id": 10400155, "problem_number": "AMR-103-0155", "title": "Problem 8.9 — (G.", "statement": "(G. Masbaum) Study this action of the finite group E(Σ) on H 0(MΣ,L⊗k), and describe the induced decompositions of this vector spa ce according to the characters of E(Σ). Also relate these decompositions to de- compositions of V (Σ) for the TQFT (V, Z) derived from the quantum group Uq(sl2) at a (k + N )-th root of unity.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.9, PDF page 125\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Study action of finite group E(Sigma) on H^0(M_Sigma, L^k)." }, { "id": 10400156, "problem_number": "AMR-103-0156", "title": "Problem 8.10 — For a given TQFT (V, Z), determine whether the image of ˜Mg in End ( V (Σ g) ) is finite.", "statement": "For a given TQFT (V, Z), determine whether the image of ˜Mg in End ( V (Σ g) ) is finite.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.10, PDF page 125\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Whether the image of the mapping class group in a TQFT is finite." }, { "id": 10400157, "problem_number": "AMR-103-0157", "title": "Problem 8.11 — (G.", "statement": "(G. Masbaum) Is there a relation between the Nielsen-Thurs- ton classification of mapping classes of Σ g and their images on V (Σ g) for TQFT’s (V, Z)?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.11, PDF page 126\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Relation between Nielsen-Thurston classification and TQFT representations." }, { "id": 10400158, "problem_number": "AMR-103-0158", "title": "Problem 8.12 — (T.", "statement": "(T. Kerler) [Cyclotomic integer TQFT’s] (1) Find explicit/computable bases for the Vp(Σ g) as free modules over Z[ζp]. (2) Show that Vp can be extended to all cobordisms as a half-projective TQFT with x = (ζp− 1) p−3 2 ∈ R = Z[ζp]. (3) Determine the structure of the V [j] p (M ) and in how far they have lifts from Fp to Z, analogous to the Ohtsuki invariants for closed 3-manifold s. (4) Find a universal TQFT that combines all Vp, at least perturbatively, into one. In the case of p = 5 the program for items (1)–(3) has been mostly carried out in [213], for primes p≥ 7 not much is known though. Some explicit bases have been found for genus g = 1 by Gilmer, but the situation for higher genera g≥ 3 is unknown. An immediate application of item (2) is that the quantum order, as introduced in [92], is also an upper bound for the cu t-number of a 3-manifold. A closely related statement for (2) would also y ield a very different proof for the fact that the Ohtsuki invariants are of finite ty pe. In item (3) the “lift” must depend on p since the dimensions of the vector spaces do, and must also involve further quotients that arise since the irreduc ible TQFT’s over Z do not match the required dimensions either, but they become reducible when reduced to Fp. Item (4) is rather vague at this point, indicating for some s ort of infinite filtered space with finite graded components. Any TQFTV: Cob→ R-mod implies a sequence of representation V[g]: Γ g→ GLR(V(Σ g)) of the mapping class groups. We say that a TQFT is homological if each of these representations factors through the quotie nt Γ g−։ Sp(2g, Z) (given by the action on H1(Σ g)), and we say it is strictly homological if each of the Sp(2 g, Z)-representations is algebraic, i.e. either faithful or ze ro. A par- ticular example of strictly homological TQFT’s over R = Z are the Lefschetz componentsV (j) of the Frohman-Nicas TQFT, see [128, 214]. From these we can generate a larger family Q 0 of such TQFT’s by taking all direct sums of V (j) ’s. For example all the TQFT’s constructed in [110] lie in Q 0. An even larger family Q∗ is found by taking also tensor products and their irreducibl e summands.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.12, PDF page 126\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kerler's cyclotomic integer TQFT program. Partially carried out for p=5. Higher primes and higher genera remain open." }, { "id": 10400159, "problem_number": "AMR-103-0159", "title": "Problem 8.13 — (T.", "statement": "(T. Kerler) [Homological TQFT’s] (1) Find the irreducible components and ring structure (w.r.t ⊕ and⊗) of Q∗. (2) Determine whether all strictly homological TQFT’s lie in Q ∗. (3) Identify the homological TQFT’s that arise from the gauge th eory of higher rank groups (such as P SU (n) in [129]) with elements in Q ∗. (4) Identify the irreducible factors of the constant orders V [0] p of the cyclo- tomic integer expansion of the Reshetikhin-Turaev theory w ith elements in Q∗. The first item is in some sense about finding the representatio n ring of Sp(2, Z)× Sp(4, Z)×...× Sp(2g, Z)×... equipped with further generators and relations given by the standard handle attachments. The constraints g iven by the latter may be just good enough to ensure that the answer to item (2) is positive. The application of (3) is a better understanding and possibl y a closed form for the polynomials from [129] that express the P SU (n)-invariants in terms of the coefficients of the Alexander polynomial. Evidence seems to s uggest that the TQFT’s from (4) stem from p−3 2 -fold symmetric products of elements in Q 0. A plausible corollary would be that for a closed manifold wit h b1(M )≥ 1 we have Vp(M ) = ( ζp− 1) p−3 2 P p−3 2 (λCW L(M )) + O((ζp− 1) p−1 2 ), (46) where λCW L is the Casson-Walker-Lescop invariant, and Pj is a polynomial of degree j with integer coefficients. (Note our normalization Vp(S3) = 1). As remarked in [212] the identity in (46) is true for p = 5 and general M with b1(M )≥ 1. Moreover, work in progress shows that (46) holds also for g eneral p if M is a torus-bundle over a circle. The homological TQFT’s are the starting point for a more gene ral, pertur- bative view point on TQFT’s that should parallel and extend t hat of the fi- nite type theory of homology-3-spheres. At least for fixed p one can under- stand, for example, the Reshetikhin-Turaev theory as defor mation of the Q ∗ - theories. The notion that is somewhat parallel to that of fini te type for closed 3-manifolds is what we shall call finite length. More precisely, the representa- tionsV[g]: Γ g→ GLR(V(Σ g)) of the mapping class groups extend linearly to homomorphismsV[g]: Z[Γ g]→ EndR(V(Σ g)). Denote by IIg⊂ Z[Γ g] the aug- mentation ideal of the Torelli group. The length ofV is the maximal L∈ N such thatV[g]((IIg)L+1) = 0. Clearly, the L = 0-theories are just the homological ones. The L = 1-theories can be thought of as elements of some Ext( V,W) with V,W∈ Q ∗. Restricted to representations of the Γ g ’s they factor (in char̸= 2) through the Johnson-Morita-homomorphism Γ g→ ⋀ 3 H1(Σ g) ⋊ Sp(2g, Z), for which such extension are explicitly constructible [211].", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.13, PDF page 127\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kerler's homological TQFT program. Not obviously resolved." }, { "id": 10400160, "problem_number": "AMR-103-0160", "title": "Problem 8.14 — (T.", "statement": "(T. Kerler) [Length = 1 TQFT’s] (1) Describe and construct algebraic L = 1 -extensions of Γ g -representations to TQFT’s, preferably as “simple” generalizations of the Fr ohman-Nicas- U (1)-theory. (2) Produce a classification of L = 1 -TQFT’s in the sense of an extension theory of Q ∗. (3) Identify the Γ g -representations on relative SU (2)-moduli space from [77] with these TQFT’s, and find similar, higher rank theories. (4) Identify the V [0] p as L = 1 -theories, if possible. The conceivable generalizations of the TQFT construction o f Frohman and Nicas described in (1) include using different, possibly non -compact gauge groups instead of U (1) and using more refined versions of intersection homolo- gies for stratified moduli spaces. Given the theory for Q ∗ the solution to item (2) will lead to well defined problems in sp-invariant theory. Constructions of L = 1-theories follow the schemes from (1) and (3). The identifi cation in (4) is carried out for p = 5 in [211]. The notion of finite length can be refined into the notion of q/l -solvable intro- duced in [212], indicating a TQFT over R = M[y]/yl+1 such that the constant order TQFT over the ground ring M is of length q. This, clearly, defines a special case of a TQFT of length ≤ (q·l + q + l). Murakami’s result [291] can be restated as saying that the Reshetikhin-Turaev theor y gives rise to a 1/1-solvable TQFT V [≤1] p with ground ring Fp (i.e. a TQFT of length 3 over Fp[y]/y2 ) such that V [≤1] p (M ) = 1 + y 1 6 λCW L(M ) (47) for any closed homology sphere M. Following Ohtsuki’s work Murakami’s iden- tity (with some extra renormalizations by the order of H1(M )) extends also to rational homology spheres. Let us call a theory with this pro perty a TQFT of Casson type. Recall, that the similar relation (46) for λCW L for manifolds with b1(M )≥ 1 is already contained in the information of a homological ( L = 0) TQFT, and is indeed a special evaluation of the Turaev-Milnor Torsion, s ee [212]. Given the richer structure of a 1/1-solvable TQFT we will expect new in variants Ξ that are refinements of λCW L and the torsion invariants. To be more precise, note that for a pair ( M, ϕ), where ϕ: π1(M )→ →Z defines a cyclic cover, any TQFT V yields an invariantV(M, ϕ) = trace(V(CΣ )) where CΣ = M− Σ: Σ → Σ and Σ ⊂ M is any surface dual to ϕ. In this way the Frohman Nicas theoriesV (j) yields the coefficients of the Alexander Polynomial, and, as shown in [212], thus also λCW L. A more refined invariant, which, roughly speaking, generali zes the Alexander module, is the Turaev-Viro module MT V (M, ϕ). It is described by Gilmer in [143]. MT V (M, ϕ) is given, up to conjugacy, by V(Σ) / ker(V(CΣ )N ) (with N large enough) together with the action of V(CΣ ) on it. The traces of V(CΣ ) or its powers are the most obvious well defined numerical inv ariants ofMT V (M, ϕ). The dimension of the module is yet another such invariant. For a 1/1-solvable theory V the invariantV(M, ϕ) takes values in M[y]/y2 and can hence be written as V(M, ϕ) = λV ϕ(M ) + y·Ξ V ϕ(M ), where λV and Ξ V are now M-valued invariants. If y coincides with the half projective parameter λV does not depend on ϕ, and we expect it to be some function of λCW L. Moreover, if V descends from a 1/2-solvable TQFT with the same property also Ξ V would be independent of ϕ. For the modular TQFT over F5[y]/y2 obtained from the Reshetikhin Turaev theory this invariant has already been defined in [212], and w e may expect it to lift, similarly, to an invariant Ξ Z over Z. For p > 5 we expect, as in the case of λCW L, the next order terms in the expansions (46) of the Reshetikh in Turaev theories to be polynomial expressions in λCW L and Ξ Z.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.14, PDF page 128\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kerler's 1/2-solvable TQFT program. Not obviously resolved." }, { "id": 10400161, "problem_number": "AMR-103-0161", "title": "Problem 8.15 — (T.", "statement": "(T. Kerler) [ q/l -solvable and Casson TQFT’s] (1) Lift the 1/1-solvable TQFT’s of Casson type over Fp to a universal 1/1- solvable TQFT’s of Casson type over Z. (2) Describe the resulting invariant Ξ Z for 3-manifolds with b1(M )≥ 1. (3) Develop a perturbation theory for general q/l -solvable TQFT’s. (4) Relate those with the various, standard resolutions of Γ g. (5) Relate them also to the traditional finite type theory for clo sed 3-manifolds. (6) Describe the Reshetikhin-Turaev theories in this pattern. Preparations for item (1) can be found in [212] in which formu lae for the Casson invariant over Z are derived that have the same form as general TQFT formulae. Item (2) is immediate from the preceding discussion. The rem aining items are logical continuations. The category of 3-dim cobordisms Cob• between compact, oriented surfaces with one boundary component has a natural structure of a brai ded tensor cat- egory. Another, category Alg can be defined entirely algebraically in terms of generators and relations with respect to a tensor product an d a composition product. On the level of objects it has exactly one generator, say A, so that all other objects are of the form A⊗g with 1 = A⊗0. The morphisms are given by all words that can be generated by taking composition and t ensor products of elementary morphisms m: A⊗ A→ A, ∆: A→ A⊗ A, e: 1 → A, ε: A→ 1,..., that appear in the definition of a braided, ribbon Hopf algebra with integrals and a non-degenerate pairing. For example, i n [215] a surjective functorAlg−։ Cob• is constructed, which, in the genus one restriction in fact an isomorphism.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.15, PDF page 130\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kerler's q/l-solvable and Casson TQFT program. Not obviously resolved." }, { "id": 10400162, "problem_number": "AMR-103-0162", "title": "Problem 8.16 — (T.", "statement": "(T. Kerler) [3-dim cobordisms from Hopf algebras] (1) Find further relations on Alg, besides the ones arising from the axiomat- ics of Hopf algebras, that would make Alg→ Cob• an isomorphism. (2) Find relations on Alg such that the maps AutAlg(A⊗g)→ Γ g∼ = AutCob •(Σ g,1) are isomorphisms. (3) Relate this to obstructions, such as Steinberg and Whitehea d groups, via stratified function spaces. (4) What are the analogous algebraic structures in higher dimen sions. The first problem is easily stated, but presumably very difficu lt as it implies a faithful translation of 3-dimensional topology into an alg ebraic gadget. In this respect it is vaguely parallel to the geometrization and Poi ncar´ e conjectures. The easier problem stated in item (2) can, in theory, be attac ked head-on, given the known presentations of the mapping class groups. T he third point hints to the fact that the generators in Alg correspond to Morse-theoretically elementary cobordisms, and the relations can be interprete d, similarly, in terms of handle slides and cancellation. This is, thus, reminisce nt of the definitions of, e.g. Steinberg groups of 3-manifolds. The problem in item (4 ) is, again, easily stated but even in 4 dimensions lingers in almost complete to tal darkness. It is not hard to understand that higher category theory has to b e invoked and not just one “object” A suffices as a “generator”. Any partial answers may open the possibility of constructing functorial 4-manifol d invariants by “linear representation” of such structures. In [216] ETQFT’s V are defined as double functors from the double category of relative, 2-framed 1+1+1-dim cobordisms Cob∗ to the double category of linear, abelian categories over a perfect field. (The “E” sta nds for “extended to surfaces with boundaries”). Applied to a single circle, t hought of as a 0- object in Cob∗, it yields an abelian category C V = V (S1), which we call the associated circle category. The main result of [216] is a construction of a V C, for each given modular tensor category C (meaning a bounded, ribbon, braided tensor category with some additional properties) s uch that C V C =C. The construction is made for all semisimple C, and is extended, in the case of non-semisimpleC, to both to the situation of connected surfaces with boundar y as well as disconnected, closed surfaces using the previous ly mentioned notion of half-projective TQFT’s.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.16, PDF page 131\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kerler's 3D cobordisms from Hopf algebras program." }, { "id": 10400163, "problem_number": "AMR-103-0163", "title": "Problem 8.17 — (T.", "statement": "(T. Kerler) [Extended and half-projective TQFT’s] (1) Describe in how far an ETQFT V with circle category C can differ from V C, thus introducing a equivalence notion that would establis h a bijective correspondence between the class of ETQFT’s and the class of modular tensor categories. (2) Find an extended notion of half-projectivity that includes also surfaces that are both disconnected and have boundary. (3) Find constructions and axioms of ETQFT’s that apply to more r elaxed notions of boundedness or modularity. The functorAlg→ Cob • already imposes that a circle category C V must fulfill about all axioms of a modular tensor category, and contain a H opf algebra object with properties. Given some rigidity assumption it a ctually must be the same chosen in the construction of V C. What may still differ is the choice of algebra structures of the same object in the same category, which is thus the main source of possible ambiguities. Already in [216] it is clear that there are several choices. The correct axiomatics for item (2) sho uld follow from a careful analysis of the double composition laws for surgery tangles from [216] and generalization of [210]. Item (3) is relevant to include more general notions of TQFT’s as they would be of interest in the theory of finite ty pe invariants. The Reshetikhin-Turaev theory typically starts with non-s emisimple modular categoryC, typically the representation category of a non-semisimpl e quantum groups Uq(g), and then considers a canonical semisimple sub-quotient C, see [208]. Thus VC yields a semisimple TQFT. It is known that this is different from the non-semisimple TQFT VC, which in the case of a quantum group is obtained via the Hennings algorithm. TQFT’s can also be generated from a rigid, monoidal category B without any braiding. One way is to take the Drinfel’d double D(B), which is then a modular category for some choice of ribbon element, and use VD(B). For semisimple B one can also extract the 6j-symbol data and follow the Turaev -Viro construction to obtain a TQFT WB.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.17, PDF page 132\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kerler's extended and half-projective TQFTs program." }, { "id": 10400164, "problem_number": "AMR-103-0164", "title": "Problem 8.18 — (T.", "statement": "(T. Kerler) [Non-semisimple vs. semisimple TQFT’s, the dou - ble conjecture] (1) Clarify the difference in the content of VC andVC! Are there homological TQFT’sH such that VC is in some essential way equivalent to VC⊗H? (2) Find a construction of WB that generalizes the Turaev-Viro TQFT’s to non-semisimpleB ’s, similar to the way [216] generalized the Reshetikhin- Turaev construction. In the case of quantum groups and close d 3-mani- folds this should reproduce a version of the Kuperberg invar iant. (3) What is the relation between WB andVD(B)? Are they in some sense isomorphic TQFT’s? For the case of Uq(sl2) there is evidence from the genus=1 case that such an H is indeed given by the Frohman-Nicas- U (1)-theory. Item (2) is rather natural as a problem. As is apparent in [236] one may expect technical challenges requiring “minimal” cell decompositions of cobordisms, as opposed to general triangulations, as well as “combings” instead of framings. The last conjecture appears also as Question 5 in [209] which was motivated by works of and discussions with D. Kazhdan and S. Gelfand in 1 994. Since it is a rather nearby conjecture from a formal point of view it may have been posed already earlier. For categories arising from subfact ors and closed mani- folds results answering this conjecture have been obtained in [204]. As outlined in [209] further, more general results in this direction sho uld yield a deeper un- derstanding of both TQFT constructions involved as well as e ntail a topological picture for the Drinfel’d double construction.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 8.18, PDF page 132\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kerler's non-semisimple vs semisimple TQFT double conjecture." }, { "id": 10400165, "problem_number": "AMR-103-0165", "title": "Problem 9.1 — (1) Find (and classify) all semi-simple monoidal categories (w ith finitely many isomorphism classes of simple objects).", "statement": "(1) Find (and classify) all semi-simple monoidal categories (w ith finitely many isomorphism classes of simple objects). (2) Find (and classify) (finite dimensional) fusion rule algebr as and sets of 6j -symbols. (3) Find (and classify) all subfactors (of finite depth).", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 9.1, PDF page 135\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Classification of semisimple monoidal categories with finitely many simple objects. This is a major open problem in representation theory." }, { "id": 10400166, "problem_number": "AMR-103-0166", "title": "Problem 9.2 — (Y.", "statement": "(Y. Kawahigashi) Suppose we have a three-dimensional TQFT. Can we determine whether it arises from a fusion rule algebra and 6j -symbols? If yes, can we describe all fusion rule algebras with 6j -symbols producing the TQFT?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 9.2, PDF page 138\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kawahigashi's question about TQFTs from fusion rule algebras." }, { "id": 10400167, "problem_number": "AMR-103-0167", "title": "Problem 9.3 — (Y.", "statement": "(Y. Kawahigashi) Suppose we have two fusion rule algebras with 6j -symbols and that two TQFT’s arising from them are isomorphi c. What relation do we have for the two sets of 6j -symbols?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 9.3, PDF page 138\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kawahigashi's question about isomorphic TQFTs from different 6j-symbols." }, { "id": 10400168, "problem_number": "AMR-103-0168", "title": "Problem 9.4 — (Y.", "statement": "(Y. Kawahigashi) Suppose we have a TQFT arising from a fusion rule algebra with 6j -symbols. Using a fusion rule subalgebra and 6j - symbols restricted on it, we can construct another TQFT. Wha t relation do we have for these TQFT’s?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 9.4, PDF page 138\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kawahigashi's question about TQFTs from fusion rule subalgebras." }, { "id": 10400169, "problem_number": "AMR-103-0169", "title": "Problem 9.5 — (Y.", "statement": "(Y. Kawahigashi) Suppose we have a semisimple ribbon cat- egory C with finitely many isomorphism classes of simple objects. If the S - matrix is invertible, we can construct the Reshetikhin-Tur aev invariant and the state-sum invariant from C and the latter is the square of the absolute value of the former. If the S -matrix is not invertible, do we still have a similar description of the state-sum invariant?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 9.5, PDF page 139\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kawahigashi's question about state-sum invariant when S-matrix is not invertible. Related to the theory of modular categories and the quantum double construction." }, { "id": 10400170, "problem_number": "AMR-103-0170", "title": "Problem 9.6 — (Y.", "statement": "(Y. Kawahigashi) Suppose we have a semisimple ribbon cat- egory C1 with finitely many isomorphism classes of simple objects, bu t the S -matrix is not invertible. Then we can construct a new modula r category C2 containing C1 as a full subcategory by the “quantum double” construction [315, 316, 182], but there may be another extension of C1 to a modular cate- gory. Theorem 2.13 in [315] claims that we have a “minimal” ex tension in an “essentially unique” way. Do we indeed have existence and ce rtain uniqueness of such an extension? If so, what is the relation between the t wo TQFT’s arising from C1 and its minimal extension?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 9.6, PDF page 140\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kawahigashi's question about minimal modular extensions of ribbon categories. The Muger theorem and the theory of modular extensions provide partial answers." }, { "id": 10400171, "problem_number": "AMR-103-0171", "title": "Problem 9.7 — (Y.", "statement": "(Y. Kawahigashi) Suppose we have a semisimple ribbon cat- egory C1 with a degenerate S -matrix as in Problem 9.6. By the method in [288], we can also make a modular tensor category C2 from C1. What is the relation between the two TQFT’s arising from C1 and C2?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 9.7, PDF page 140\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Kawahigashi's question about TQFTs from different modular extensions." }, { "id": 10400172, "problem_number": "AMR-103-0172", "title": "Problem 9.8 — (Y.", "statement": "(Y. Kawahigashi) There are some fusion rule algebras with 6j -symbols that do not seem to arise from quantum groups in [14] and more conjectured candidates of such examples in [160]. What are t he corresponding TQFT’s? Especially if the series conjectured in [160] does e xist, it would give a parametrized family of TQFT’s. Does a differentiation by a p arameter (af- ter a certain reparametrization) give a more interesting in variant, possibly of Vassiliev type? 9.4 Turaev-Viro-Ocneanu invariants The state-sum invariant of 3-manifolds derived from 6 j -symbols is called the Turaev-Viro-Ocneanu invariant when the set of 6 j -symbols arises from a sub- factor. There are infinitely many subfactors other than thos e derived from quantum groups or finite groups. The Turaev-Viro-Ocneanu in variants derived from such subfactors might be new invariants of 3-manifolds. (N. Sato) The Haagerup subfactor of Jones index 5+ √ 13 2 has the smallest index among finite depth subfactors with Jones index bigger t han 4 and it is expected to have some “exotic” properties from the subfacto r theoretical view- point. However, it does not seem so sensitive to classify 3-m anifolds. The Turaev-Viro-Ocneanu invariant constructed from the Haage rup subfactor can- not distinguish lens spaces L(5, 1) and L(5, 2), as well as L(7, 1) and L(7, 2). On the other hand, generalized E6 -subfactors with the group symmetries Z/3Z and Z/5Z can distinguish L(3, 1) and L(3, 2), L(5, 1) and L(5, 2), respectively.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 9.8, PDF page 140\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Exotic fusion rule algebras not from quantum groups. The Haagerup subfactor and related constructions provide examples." }, { "id": 10400173, "problem_number": "AMR-103-0173", "title": "Problem 9.9 — (N.", "statement": "(N. Sato) Find a subfactor which can distinguish lens spaces L(7, 1) and L(7, 2). Moreover, find a subfactor to classify 3-manifolds as well as possible. In the lattice field theory, Ponzano and Regge [332] construc ted a state sum model for SU (2) and investigated an asymptotic behavior of the model. Some infinite depth subfactors are manageable in the sense of growth rate (amenability). Such subfactors are called strongly amenable. The strong amen- ability condition might be enough to control the asymptotic behavior of the state sum model constructed from a strongly amenable subfac tor.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 9.9, PDF page 141\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Sato's question about subfactors distinguishing lens spaces." }, { "id": 10400174, "problem_number": "AMR-103-0174", "title": "Problem 9.10 — (N.", "statement": "(N. Sato) Construct a well-defined state sum type invariant from a strongly amenable subfactor. Note that, unlike the Ponzano-Regge model, we do not have an a symptotic description of the quantum 6 j -symbols in general. (Recall that 6 j -symbols of SU (2) have an asymptotic description.) Let us consider the Turaev-Viro-Ocneanu invariant for a clo sed 3-manifold constructed from a subfactor. Then, this invariant can be co nsidered as a Reshetikhin-Turaev type invariant constructed from a subf actor by passing the initial subfactor through the Longo-Rehren construction. If we start with a subfactor which has a non-degenerate braiding in particula r, then this Turaev- Viro-Ocneanu invariant splits into a Reshetikhin-Turaev i nvariant and its com- plex conjugate. The following question will open a way to est ablish a theory of the minimal non-degenerate extension of a degenerate braid ing.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 9.10, PDF page 141\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. State sum invariants from strongly amenable subfactors. This is related to the general theory of subfactor invariants." }, { "id": 10400175, "problem_number": "AMR-103-0175", "title": "Problem 9.11 — (N.", "statement": "(N. Sato) Let us consider the Turaev-Viro-Ocneanu invariant from a subfactor with a degenerate braiding. Then, find a desc ription of this invariant as a Reshetikhin-Turaev invariant.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 9.11, PDF page 141\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Turaev-Viro-Ocneanu invariant from subfactor with degenerate braiding as Reshetikhin-Turaev invariant." }, { "id": 10400176, "problem_number": "AMR-103-0176", "title": "Problem 10.1 — Can the Casson invariant of an integral homology 3-sphere M be characterized by the signature of a certain 4-manifold…", "statement": "Can the Casson invariant of an integral homology 3-sphere M be characterized by the signature of a certain 4-manifold bo unded by M?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.1, PDF page 142\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Casson invariant as signature of a 4-manifold. Casson's invariant is related to the signature of a 4-manifold bounded by M (the Casson invariant = 1/2 the signature defect)." }, { "id": 10400177, "problem_number": "AMR-103-0177", "title": "Problem 10.2 — (V.", "statement": "(V. Turaev) Relate this surgery formula for the Casson-Wal- ker-Lescop invariant with that of Lescop [251]. 32The normalization here is that λ CW(M ) = 2 λ C(M ) for an integral homology 3-sphere M. 33The normalization here is that λ CWL(M ) = ( |H1(M; Z)|/ 2 ) λ CW(M ) for a rational ho- mology 3-sphere M. (C. Lescop) In 1984, Casson defined his invariant of integral homology 3- spheres as an integer that “counts” the SU (2)-representations of their funda- mental group in an appropriate way (see [3, 159]). Cappell, L ee and Miller [76] showed that the Casson way of counting SU (2)-representations of the π1 works for any compact Lie group and provides other invariants of in tegral homology spheres.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.2, PDF page 142\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Turaev's question about relating Casson-Walker-Lescop formulas." }, { "id": 10400178, "problem_number": "AMR-103-0178", "title": "Question 10.3 — (C.", "statement": "(C. Lescop) Are the Cappell-Lee-Miller Casson-type SU (n)- invariants of finite type? If so, what are their degrees and th eir weight systems?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Question 10.3, PDF page 143\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Whether Cappell-Lee-Miller SU(n) Casson-type invariants are of finite type. This is a known result for some cases." }, { "id": 10400179, "problem_number": "AMR-103-0179", "title": "Problem 10.4 — (M.", "statement": "(M. Polyak) Define an invariant λ of a pair (M, σ) of a closed 3-manifold M and a spin structure σ on M such that λCWL(M ) = ∑ σ λ(M, σ) for any closed 3-manifold M, where the sum runs over all spin structures σ on M. Note that the set of spin structures on M is a torsor over H 1(M; Z/2Z) in the sense that differences of spin structures can be detected by cohomology classes in H 1(M; Z/2Z), while the set of spin c structures on M is a torsor over H 1(M; Z) in a similar sense.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.4, PDF page 143\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Polyak's spin structure invariant refining Casson-Walker-Lescop." }, { "id": 10400180, "problem_number": "AMR-103-0180", "title": "Question 10.5 — (M.", "statement": "(M. Polyak) Is there a “Rokhlin invariant” of a pair (M, α) of a closed 3-manifold M and a spin c structure α on M? (See Question 10.21.)", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Question 10.5, PDF page 145\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Polyak's question about Rokhlin invariant for spin^c structures." }, { "id": 10400181, "problem_number": "AMR-103-0181", "title": "Problem 10.6 — (M.", "statement": "(M. Polyak) By presenting 3-manifolds by surgery along framed links in S3, we can regard an invariant of 3-manifolds as an invari- ant of framed links. Establish a Gauss diagram formula for th e link invariant derived from each finite type invariant of 3-manifolds.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.6, PDF page 145\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Gauss diagram formulas for finite type invariants of 3-manifolds." }, { "id": 10400182, "problem_number": "AMR-103-0182", "title": "Conjecture 10.7 — F as d (ZM)/F as d+1(ZM) (resp.F b d (ZM)/F b d+1(ZM)) is torsion free for each d.", "statement": "F as d (ZM)/F as d+1(ZM) (resp.F b d (ZM)/F b d+1(ZM)) is torsion free for each d.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 10.7, PDF page 146\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Torsion-free conjecture for finite type invariants of 3-manifolds." }, { "id": 10400183, "problem_number": "AMR-103-0183", "title": "Conjecture 10.8 — A(∅; Z) is torsion free.", "statement": "A(∅; Z) is torsion free. 10.2.2 Do finite type invariants distinguish homology 3-sph eres?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 10.8, PDF page 146\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Torsion-free conjecture for A(empty; Z)." }, { "id": 10400184, "problem_number": "AMR-103-0184", "title": "Conjecture 10.9 — Finite type invariants distinguish integral homology 3-spheres.", "statement": "Finite type invariants distinguish integral homology 3-spheres. (See Conjecture 11.2.) 10.2.3 Dimensions of spaces of finite type invariants A finite type invariant v is called primitive if v(M1#M2) = v(M1) + v(M2) for any integral homology 3-spheres M1 and M2. We denote by A(∅; R)conn the submodule of A(∅; R) spanned by Jacobi diagrams with connected trivalent graphs. As a graded vector space A(∅; Q) is isomorphic to the symmetric tensor algebra of A(∅; Q)conn.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 10.9, PDF page 146\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Whether finite type invariants distinguish integral homology 3-spheres is a major open problem." }, { "id": 10400185, "problem_number": "AMR-103-0185", "title": "Problem 10.10 — Determine the dimension of the space of primitive finite type invariants of integral homology 3-spheres of each degre…", "statement": "Determine the dimension of the space of primitive finite type invariants of integral homology 3-spheres of each degree d. Equivalently, deter- mine the dimension of the space A(∅; Q)(d) conn for each d. d 0 1 2 3 4 5 6 7 8 9 10 prime diag. 0 1 0 0 1 0 1 1 1 1 2 dimA(∅)(d) conn 0 1 1 1 2 2 3 4 5 6 8 dimA(∅)(d) 1 1 2 3 6 9 16 25 42 65 105 d 11 12 13 14 prime diag. 1 dimA(∅)(d) conn 9 ≥ 11 ≥ 13 ≥ 15 dimA(∅)(d) 161 ≥ 254 ≥ 386 ≥ 595 Table 7: Some dimensions for Problem 10.10", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.10, PDF page 146\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Dimensions of primitive finite type invariants of IHS. Known up to degree ~10 (see Table 7)." }, { "id": 10400186, "problem_number": "AMR-103-0186", "title": "Problem 10.11 — Describe Vogel’s algebra Λ, say, by giving complete sets of generators and relations of Λ.", "statement": "Describe Vogel’s algebra Λ, say, by giving complete sets of generators and relations of Λ.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.11, PDF page 149\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Describe Vogel's algebra Lambda. Not obviously resolved." }, { "id": 10400187, "problem_number": "AMR-103-0187", "title": "Problem 10.12 — Find a constructive combinatorial presentation of each fini te type invariant of integral homology 3-spheres, and, in…", "statement": "Find a constructive combinatorial presentation of each fini te type invariant of integral homology 3-spheres, and, in part icular, of the Casson invariant, by localizing configuration space integrals.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.12, PDF page 150\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Combinatorial presentation of finite type invariants via configuration space integrals." }, { "id": 10400188, "problem_number": "AMR-103-0188", "title": "Problem 10.13 — (J.", "statement": "(J. Roberts) What is the space of 3-manifolds?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.13, PDF page 150\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Roberts' question: 'What is the space of 3-manifolds?' - a philosophical/meta question." }, { "id": 10400189, "problem_number": "AMR-103-0189", "title": "Conjecture 10.14 — The map (50) is an isomorphism.", "statement": "The map (50) is an isomorphism. 35The Yd -equivalence is also called the ( d − 1)-equivalence (due to Goussarov) in some literatures. This conjecture might be reduced to Conjecture 10.8 and the f ollowing conjec- ture.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 10.14, PDF page 151\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Isomorphism conjecture for map (50). Not obviously resolved." }, { "id": 10400190, "problem_number": "AMR-103-0190", "title": "Conjecture 10.15 — {M ∼ Y2d S3}/ ∼ Y2d+1 is torsion free for each d.", "statement": "{M ∼ Y2d S3}/ ∼ Y2d+1 is torsion free for each d.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 10.15, PDF page 152\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Torsion-free conjecture for Y2d-equivalence of IHS." }, { "id": 10400191, "problem_number": "AMR-103-0191", "title": "Problem 10.16 — (T.", "statement": "(T. Ohtsuki) Define a product M1◦ M2 of integral homol- ogy 3-spheres M1 and M2 which is related, by (50), to the product of Jacobi diagrams given by their connected sum.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.16, PDF page 152\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Ohtsuki's product of IHS related to Jacobi diagrams." }, { "id": 10400192, "problem_number": "AMR-103-0192", "title": "Conjecture 10.17 — (M.", "statement": "(M. Polyak, see [153, “Theorem 4”]) Let F be an oriented compact surface. Two homology cylinders C and C ′ over F are Yd -equivalent if and only if v(C) = v(C ′) for any A-valued finite type invariant v ofF Y ⋆ -degree < d for any abelian group A.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 10.17, PDF page 153\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Polyak's classification of Yd-equivalence for homology cylinders via finite type invariants." }, { "id": 10400193, "problem_number": "AMR-103-0193", "title": "Problem 10.18 — (F.", "statement": "(F. Deloup) Classify the monoid (for orthogonal sum) of isomorphism classes of quadratic forms qσ.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.18, PDF page 153\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Classification of quadratic forms by orthogonal sum. This is a problem in algebraic topology." }, { "id": 10400194, "problem_number": "AMR-103-0194", "title": "Problem 10.19 — (G.", "statement": "(G. Massuyeau) Describe the quotient set {spin closed 3-manifolds}/∼ Y s d, in particular, for d = 2, 3.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.19, PDF page 154\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Describe spin 3-manifolds up to Y^s_d equivalence." }, { "id": 10400195, "problem_number": "AMR-103-0195", "title": "Problem 10.20 — (F.", "statement": "(F. Deloup, G. Massuyeau) Describe the quotient set {spin c closed 3-manifolds}/∼ Y c d, in particular, for d = 2, 3.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 10.20, PDF page 154\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Describe spin^c 3-manifolds up to Y^c_d equivalence." }, { "id": 10400196, "problem_number": "AMR-103-0196", "title": "Question 10.21 — (F.", "statement": "(F. Deloup) Is there a lift of arg γ(qσ) to a mod 16 invariant? This would give a finite type invariant of degree 1 in the spin c Goussarov-Habiro theory. 37A quadratic function q is a a map such that q(x + y) − q(x) − q(y) is bilinear in x and y. It is called homogeneous if q(nx) = n2q(x) for any n ∈ Z and x ∈ G. In fact, there is a canonical map σ ↦→qσ from spin c structures to quadratic functions and qσ is homogeneous if and only if σ actually comes from a spin structure. Note that not all spin c structures come from spin structures.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Question 10.21, PDF page 154\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Lift of arg(gamma(q_sigma)) to mod 16 invariant." }, { "id": 10400197, "problem_number": "AMR-103-0197", "title": "Problem 11.1 — For each rational homology 3-sphere M, calculate Z L M O(M ) for all degrees.", "statement": "For each rational homology 3-sphere M, calculate Z L M O(M ) for all degrees.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 11.1, PDF page 155\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Calculate LMO invariant for all degrees for rational homology spheres." }, { "id": 10400198, "problem_number": "AMR-103-0198", "title": "Conjecture 11.2 — The LMO invariant distinguishes integral homology 3-spheres.", "statement": "The LMO invariant distinguishes integral homology 3-spheres. (See Conjecture 10.9.)", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 11.2, PDF page 155\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Whether the LMO invariant distinguishes integral homology spheres is a major open problem (equivalent to Conjecture 10.9)." }, { "id": 10400199, "problem_number": "AMR-103-0199", "title": "Problem 11.3 — Does there exist an integral/rational homology 3-sphere M such that Z L M O(M ) = Z L M O(S3)?", "statement": "Does there exist an integral/rational homology 3-sphere M such that Z L M O(M ) = Z L M O(S3)? 11.3 Characterization of the image of the LMO invariant", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 11.3, PDF page 156\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Existence of non-trivial IHS with trivial LMO invariant. Equivalent to whether the LMO invariant detects S^3. Open problem." }, { "id": 10400200, "problem_number": "AMR-103-0200", "title": "Problem 11.4 — Characterize those elements of ˆA(∅)conn which are of the form log Z L M O(M ) for integral/rational homology 3-spheres.", "statement": "Characterize those elements of ˆA(∅)conn which are of the form log Z L M O(M ) for integral/rational homology 3-spheres.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 11.4, PDF page 156\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Characterize elements of A(empty)_conn that are log Z_LMO(M)." }, { "id": 10400201, "problem_number": "AMR-103-0201", "title": "Problem 11.5 — Construct the LMO invariant with coefficients in a finite field.", "statement": "Construct the LMO invariant with coefficients in a finite field.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 11.5, PDF page 156\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Construct LMO invariant with finite field coefficients." }, { "id": 10400202, "problem_number": "AMR-103-0202", "title": "Problem 11.6 — Construct the LMO invariant (or the theory of finite type invariants) in arrow diagrams.", "statement": "Construct the LMO invariant (or the theory of finite type invariants) in arrow diagrams. 11.5 Refinements of the LMO invariant (T. Le) As mentioned in a remark in Problem 11.1, the LMO invar iant is a weak invariant when b1(M ) > 0; in particular, Z L M O(M ) = 1 when b1(M ) > 3. The following two problems might give refinements of Z L M O(M ) which would be stronger than Z L M O(M ), in particular, when b1(M ) > 0.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 11.6, PDF page 156\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Construct LMO invariant in arrow diagrams." }, { "id": 10400203, "problem_number": "AMR-103-0203", "title": "Problem 11.7 — (T.", "statement": "(T. Le, V. Turaev) Define the LMO invariant Z L M O(M, σ) of the pair of a closed 3-manifold M and a spin structure σ of M such that Z L M O(M ) = ∑ σ Z L M O(M, σ), where the sum runs over all spin structures on M. There is also a similar problem for spin c structures.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 11.7, PDF page 156\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Le-Turaev's spin refinement of LMO invariant. A spin LMO invariant has been defined by various authors." }, { "id": 10400204, "problem_number": "AMR-103-0204", "title": "Problem 11.8 — (T.", "statement": "(T. Le, V. Turaev) For every element ξ∈ H 1(M, Z) construct an extension of Z L M O(M, ξ) of the LMO invariant such that when ξ = 0 one recovers the usual LMO invariant. The idea is that the usual LMO invariant corresponds only to t he trivial coho- mology class, and for manifolds with high Betti number, it is equal to 0. K. Habiro has an extension of the LMO invariant that might be a so lution to this problem.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 11.8, PDF page 157\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Le-Turaev's cohomology extension of LMO. Habiro has an extension that may address this." }, { "id": 10400205, "problem_number": "AMR-103-0205", "title": "Question 11.9 — (1) Find a surgery formula for the Kuperberg-Thurston in- variant [237] in terms of the Chern-Simons series of Questi…", "statement": "(1) Find a surgery formula for the Kuperberg-Thurston in- variant [237] in terms of the Chern-Simons series of Questio n 3.12 (2) Compare the Kuperberg-Thurston invariant to the LMO invari ant.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Question 11.9, PDF page 157\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Surgery formula for Kuperberg-Thurston invariant and comparison with LMO." }, { "id": 10400206, "problem_number": "AMR-103-0206", "title": "Problem 11.10 — (D.", "statement": "(D. Thurston) Do configuration spaces of [237] have torsion in Z-homology? Does such torsion deduce a torsion invariant of h omology 3-spheres?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 11.10, PDF page 157\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Torsion in configuration spaces and torsion invariants of homology spheres." }, { "id": 10400207, "problem_number": "AMR-103-0207", "title": "Question 12.1 — (R.", "statement": "(R. Benedetti) Are torsions actually sensitive only to the (pL)-homotopy immersion classes of (pL)-knots? If one fix a C - homotopy immersion class of knots, say α, then one can define the set of finite type invariants F(α) of the C -isotopy classes contained in α. If α0 is a class of Legendrian knots, one can take α1 = f1(α0) and α2 = f2(α1); a finite type invariant for αi lifts to a finite type invariant for αi−1. So one has natural maps F(α2) f ∗ 2 →F (α1) f ∗ 1 →F (α0). It is known [130] that, under certain hypotheses on W (for instance when W is a Z-homology sphere), f ∗ 1◦ f ∗ 2 is a bijection. On the oder hand, one can Section 12.1 was written by R. Benedetti. find in [380] examples where f ∗ 1◦ f ∗ 2 is not surjective and Legendrian finite type invariants can eventually distinguish some Legendrian kno ts which are isotopic as framed knots. In fact one can realize that for these exampl es f ∗ 2 is already not surjective and that (pL)-finite type invariants can even tually distinguish some (pL)-knots which are isotopic as framed knots. The foll owing conjecture is not in contradiction with all these known results on the su bject.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Question 12.1, PDF page 158\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Benedetti's question about torsions and PL-homotopy of knots." }, { "id": 10400208, "problem_number": "AMR-103-0208", "title": "Conjecture 12.2 — (R.", "statement": "(R. Benedetti) For every W, for every (pL)-class α1 as above, f ∗ 1 is an isomorphism. This means, in particular, that finite typ e invari- ants of Legendrian knots should be definitely not sensitive t o geometric (rigid) properties of the contact structures like “tightness”. See also [50] for a more detailed discussion and related ques tions. 12.2 Knots and finite groups Knot groups are known to be residually finite, that is, any non -trivial element can be detected by a homomorphism to some finite group. Now by Dehn’s lemma and the loop theorem a knot is trivial if an d only if its longitude represents the trivial element of the knot gro up. Consequently for each non-trivial knot there is a homomorphism to some finite group which carries the longitude to a non-trivial element.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 12.2, PDF page 159\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Benedetti's conjecture about isomorphism f*_1 for Legendrian knots." }, { "id": 10400209, "problem_number": "AMR-103-0209", "title": "Problem 12.3 — (H.R.", "statement": "(H.R. Morton) From a knot diagram find an explicit such homomorphism to some permutation group or establish that th e knot is trivial. Refinements. (1) Give an upper bound in terms of the diagram for the order of the permu- tation groups which need to be considered. (2) See what happens if the meridians (which are all conjugat e) are restricted to map to permutations of some specified cycle type, for examp le, single trans- positions.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.3, PDF page 159\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Morton's problem about explicit finite group homomorphisms from knot diagrams." }, { "id": 10400210, "problem_number": "AMR-103-0210", "title": "Conjecture 12.4 — (3-move conjecture, Y.", "statement": "(3-move conjecture, Y. Nakanishi [305]) Any link can be related to a trivial link by a sequence of 3-moves.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 12.4, PDF page 160\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The conjecture is resolved negatively in the literature: there exist links which cannot be reduced to a trivial link by 3-moves (Dabkowski–Przytycki 2002). Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. The 3-move conjecture is false: it was disproved by M. K. Dabkowski and J. H. Przytycki, \"Burnside obstructions to the Montesinos-Nakanishi 3-move conjecture\", Geom. Topol. 6 (2002), 355–360 (arXiv:math/0205057), via a Burnside-group obstruction. See also Dabkowski–Przytycki, \"Unexpected links between components of the same link\" (Fund. Math. 2004) for further counterexamples, and the earlier note of Przytycki (1999) on the (3,2)-move conjecture. The related statement in Problem 4.10 (AMR-103-0092) noting the disproof in February 2002 matches this." }, { "id": 10400211, "problem_number": "AMR-103-0211", "title": "Conjecture 12.5 — (Y.", "statement": "(Y. Nakanishi, T. Harikae [220, Conjecture 1.59 (6)]) Any link can be related to a trivial link by a sequence of (2,2)-mo ves.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 12.5, PDF page 161\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. The (2,2)-move conjecture. Not obviously resolved." }, { "id": 10400212, "problem_number": "AMR-103-0212", "title": "Problem 12.6 — Find a new proof of the existence of a universal Vassiliev in- variant of knots, presenting them by KTG’s and their o…", "statement": "Find a new proof of the existence of a universal Vassiliev in- variant of knots, presenting them by KTG’s and their operati ons.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.6, PDF page 162\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. New proof of universal Vassiliev invariant via KTGs. Not obviously resolved." }, { "id": 10400213, "problem_number": "AMR-103-0213", "title": "Conjecture 12.7 — (D.", "statement": "(D. Bar-Natan, D. Thurston) For each compact Lie group G, level k, and every KTG K: Γ → R3, there exists a collection of measures µ K on the space of gauge equivalence classes of G-connections on Γ satisfying the following conditions. • It is well-behaved under KTG operations. • It is “localized” near connections that extend to S3− K. • A half-twist framing change acts by e √ −1Hℏ/2, where H is the Schr¨ odinger operator on G. • It recovers quantum invariants by IR(K) = ∫ hR(A)dµ K (A), where hR(A) denotes the holonomy of A in R. Here, R is a set of representations of G associated to edges of Γ and appropriate intertwiners associated to vertices of Γ.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 12.7, PDF page 162\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Bar-Natan-Thurston's gauge theory measures on KTG spaces." }, { "id": 10400214, "problem_number": "AMR-103-0214", "title": "Problem 12.8 — Construct an invariant of KTG’s from configuration space in- tegrals in a natural way.", "statement": "Construct an invariant of KTG’s from configuration space in- tegrals in a natural way. Turaev [388] introduced a presentation of 3-manifolds as S1 -bundles over “sha- dow surfaces”, as follows (for details see [388, 40, 384]). A fake surface is a singular surface such that a neighborhood of each point is ho meomorphic to an open subset of the cone over a tetrahedron. A S1 -bundle over a fake surface can appropriately be defined and its isomorphism class is det ermined by the Chern number, which is an integer or half-integer associate d to each face; we call the Chern number the gleam. A shadow surface is a fake surface with gleams associated to the faces. Every (closed) 3-manifold c an be presented by a S1 -bundle over a (closed) shadow surface. The pentagon and hex agon relations (see [388, Figure 1.1 of Chapter VIII]) are moves among shado w surfaces which present a homeomorphic 3-manifold, though they are not enou gh to characterize a homeomorphism class of 3-manifolds. Exercise 12.9 Find a complete set of moves among shadow surfaces which present a homeomorphic 3-manifold. We obtain a shadow surface as a time evolution of a sequence of KTG’s given by KTG operations. Thus, we have relations among links, 3-ma nifolds, KTG’s and shadow surfaces as in the commutative diagram in Figure 2 2; for detailed statements see [40, 384]. Motivated by a complexity of 3-manifolds discussed in [279, 272, 273], D. Thurston introduced the shadow number of 3-manifolds. The shadow num- ber is defined to be the minimal number of vertices of a shadow surf ace. All graph manifolds have shadow number 0 and all surgeries on the Borromean rings have shadow number 1. The volume conjecture might be re lated to the following conjecture. Framed links -exterior Framed link exteriors -surgery Closed 3-manifolds presentation 6 by making S1 -bundle 6 by making S1 -bundle 6 Certain sequences of KTG’s - time evolution Collapsible shadow surfaces - cap off ∂ Closed shadow surfaces Figure 22: Links, 3-manifolds, KTG’s, and shadow surfaces", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.8, PDF page 163\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Configuration space integrals for KTG invariants." }, { "id": 10400215, "problem_number": "AMR-103-0215", "title": "Conjecture 12.10 — (D.", "statement": "(D. Thurston) The shadow number of a 3-manifold is quasi-linear in its Gromov norm. That is, there exist consta nts c1 and c2 such that c1||M||≤ (shadow number of M )≤ c2||M|| for any 3-manifold M, where ||M|| denotes the Gromov norm of M.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 12.10, PDF page 164\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Thurston's conjecture about shadow number being quasi-linear in Gromov norm." }, { "id": 10400216, "problem_number": "AMR-103-0216", "title": "Problem 12.11 — (D.", "statement": "(D. Thurston) Find a condition on shadow diagrams which is satisfied by shadow diagrams from alternating knots; and g ives a lower bound on the hyperbolic volume. The Reshetikhin-Turaev invariant and the Turaev-Viro-Ocn eanu invariant can be described in terms of the KTG algebra, via I -bundles and S1 -bundles over shadow surfaces respectively. The relation between the two invariants is derived from the relation between the two construction of 3-manifol ds shown in Figure 23. Closed shadow surfaces with 0 gleams \u0000 \u0000 \u0000 by making I -bundles and cap off boundary @ @ @R by making S1 -bundles Closed 3-manifolds Certain closed 3-manifolds - M ↦−→M #M #(S2× S1)’s Figure 23: Two ways to obtain 3-manifolds from shadow surfac es", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.11, PDF page 164\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Thurston's condition on shadow diagrams for hyperbolic volume bounds." }, { "id": 10400217, "problem_number": "AMR-103-0217", "title": "Problem 12.12 — Construct a universal Reshetikhin-Turaev invariant and a universal Turaev-Viro-Ocneanu invariant of closed 3-mani fol…", "statement": "Construct a universal Reshetikhin-Turaev invariant and a universal Turaev-Viro-Ocneanu invariant of closed 3-mani folds, in terms of the KTG algebra.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.12, PDF page 165\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Universal Reshetikhin-Turaev and Turaev-Viro-Ocneanu invariants via KTG algebra." }, { "id": 10400218, "problem_number": "AMR-103-0218", "title": "Problem 12.13 — (J.", "statement": "(J. Roberts) What are quantum groups?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.13, PDF page 165\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: PARTIAL-PROGRESS. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Roberts' question 'What are quantum groups?' - partially answered by the theory of quantum groups as deformations of universal enveloping algebras (Drinfeld, Jimbo)." }, { "id": 10400219, "problem_number": "AMR-103-0219", "title": "Problem 12.14 — (N.", "statement": "(N. Askitas) Can a knot of 4-genus gs always be sliced (made into a slice knot) by gs crossing switches?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.14, PDF page 166\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Askitas' question about slicing knots by crossing switches." }, { "id": 10400220, "problem_number": "AMR-103-0220", "title": "Problem 12.15 — (M.", "statement": "(M. Boileau [220, Problem 1.69 (C)]) Are there mutants of distinct unknotting numbers?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.15, PDF page 166\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Boileau's question about mutants of distinct unknotting numbers." }, { "id": 10400221, "problem_number": "AMR-103-0221", "title": "Conjecture 12.16 — (X.-S.", "statement": "(X.-S. Lin [262]) Any automorphism of G is either the identity or the mirror map, that is, any automorphism of G is induced by a diffeomorphism of the ambient space.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 12.16, PDF page 166\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Lin's conjecture about automorphisms of G being induced by diffeomorphisms." }, { "id": 10400222, "problem_number": "AMR-103-0222", "title": "Problem 12.17 — (X.-S.", "statement": "(X.-S. Lin [262]) What is the homotopy type of the space L(K) of long ropes (as shown in the picture below) with the fixed kno t type K?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.17, PDF page 166\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Lin's homotopy type of the space of long ropes with fixed knot type." }, { "id": 10400223, "problem_number": "AMR-103-0223", "title": "Problem 12.18 — (J.", "statement": "(J. Roberts) Extend Kuperberg’s work on webs.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.18, PDF page 167\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Extend Kuperberg's work on webs. Kuperberg's web theory has been extended to higher rank groups by various authors." }, { "id": 10400224, "problem_number": "AMR-103-0224", "title": "Problem 12.19 — (J.", "statement": "(J. Roberts) Extend the theory of measured laminations to higher rank groups.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.19, PDF page 167\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Extend measured laminations to higher rank groups. This is related to the work of Fock, Goncharov, etc." }, { "id": 10400225, "problem_number": "AMR-103-0225", "title": "Problem 12.20 — (J.", "statement": "(J. Roberts) What is the generating function for q -spin net evaluations?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.20, PDF page 168\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Roberts' generating function for q-spin net evaluations." }, { "id": 10400226, "problem_number": "AMR-103-0226", "title": "Problem 12.21 — (Y.", "statement": "(Y. Shinohara [364]) If n = 4 k + 1 with k > 0, is there a knot with determinant n and signature 4?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.21, PDF page 168\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Shinohara's question about knots with determinant 4k+1 and signature 4." }, { "id": 10400227, "problem_number": "AMR-103-0227", "title": "Problem 12.22 — (T.", "statement": "(T. Stanford) IsC2 solvable? Does C2 contain a free group?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.22, PDF page 169\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Stanford's question about C2 (the second lower central series of the pure braid group)." }, { "id": 10400228, "problem_number": "AMR-103-0228", "title": "Problem 12.23 — (A.", "statement": "(A. Stoimenow) Do positive links of given signature σ have bounded (below) maximal Euler characteristic χ?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.23, PDF page 169\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Stoimenow's question about bounded maximal Euler characteristic of positive links." }, { "id": 10400229, "problem_number": "AMR-103-0229", "title": "Problem 12.24 — (A.", "statement": "(A. Stoimenow) If a prime knot K can be transformed into its mirror image by one crossing change, is K achiral or (algebraically?) slice?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.24, PDF page 169\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Stoimenow's question about knots transformable to mirror by one crossing change." }, { "id": 10400230, "problem_number": "AMR-103-0230", "title": "Problem 12.25 — (A.", "statement": "(A. Stoimenow) Let n be an odd natural number, different from 1, 9, and 49, such that n is the sum of two squares. Is there a prime alternating achiral knot of determinant n?", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Problem 12.25, PDF page 169\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Stoimenow's question about prime alternating achiral knots of determinant n." }, { "id": 10400231, "problem_number": "AMR-103-0231", "title": "Conjecture 12.26 — (V.", "statement": "(V. Turaev) A pair (a finitely generated abelian group H of rank 1, an element ∆( t)∈ Z[H/TorsH] = Z[t±1]) (where t is a generator of H/TorsH ) can be realized as the pair (H1(M ), the Alexander polynomial ∆ M of M ) for a closed connected oriented 3-manifold M if and only if ∆( t) = tk∆( t−1) with even k∈ Z and ∆(1) = ±|TorsH|.", "background": "The monograph's printed bold headings distinguish 231 numbered problems, conjectures, and questions from references to those items in its surrounding exposition.\n\nSource list: Ohtsuki - Problems on invariants of knots and 3-manifolds (2002)\nSource item: Conjecture 12.26, PDF page 170\nSource URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Classification: OPEN-TRIAGE. Literature status: Source: Ohtsuki, \"Problems on invariants of knots and 3-manifolds\" (2002), as listed in the worklist. Turaev's Conjecture 12.26: a pair (H, Δ(t)) with H finitely generated abelian of rank 1 is realizable as (H1(M), Alexander polynomial of M) for a closed connected oriented 3-manifold M iff Δ(t) = t^k Δ(t^{-1}) with even k and Δ(1) = ±|Tors H|. This is a realization problem for the Alexander polynomial of closed 3-manifolds. Related literature: V. Turaev, 'Torsions of 3-manifolds' (2002, arXiv:math/0211084), studies the analogous realization problem for Reidemeister torsion τ of closed 3-manifolds and gives only partial results (e.g. realizability of symmetric λ ∈ Z[Z^n] with augmentation 1 for n = 2, 3). I could not verify a complete solution of the stated conjecture; it appears to remain open (or at least not established in the accessible literature)." }, { "id": 10600001, "problem_number": "AMR-105-0001", "title": "Virtual-knot problem 1 — Recognising the Kishino Knot", "statement": "Recognising the Kishino Knot: There have been invented many\nways to recognize the Kishino virtual knot (from the unknot): The\n$3$–strand Jones polynomial, i.e. the Jones polynomial of the\n$3$–strand cabling of the knot , the $\\Xi$–polynomial,\nsee , the quaternionic biquandle , and the\nsurface bracket polynomial (Dye and Kauffman ).\nIn Kadokami proves the knot is non-trivial by\nexamining the immersion class of a shadow curve in genus two. One\ncan use the Manturov parity bracket to\nshow that essentially the flat Kishino diagram is its own invariant,\nexhibiting the non-triviality of this knot. See for\nan exposition of this proof.\n\nAre we done with this knot? Perhaps not. Other proofs of its\nnon-triviality may be illuminating. The fact that the Kishino\ndiagram is non-trivial and yet a connected sum of trivial virtual\nknots suggests the question: Classify when a non-trivial\nvirtual knot can be the connected sum of two trivial virtual knots.\nA key point here is that the connected sum of closed virtual knots\nis not well defined and hence the different choices give some\ninteresting effects. With long virtual knots, the connected sum is\nordered but well-defined, and the last question is closely related\nto the question of classifying the different long virtual knots\nwhose closures are equivalent to the unknot.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 1\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Non-triviality: SOLVED (multiple proofs in the literature, pre-2014 and later). Connected-sum-factorization classification: PARTIAL — no complete classification found; relevant literature on virtual knot decomposition exists but the specific classification question appears open." }, { "id": 10600002, "problem_number": "AMR-105-0002", "title": "Virtual-knot problem 2 — Flat Virtuals", "statement": "Flat Virtuals: Flat virtual knots, also known as virtual\nstrings , are difficult to classify. Find new\ncombinatorial invariants of flat virtual knots.\n\nWe would like to know more about the flat biquandle algebra. This\nalgebra is isomorphic to the Weyl algebra and has no\n(non-trivial) finite dimensional representations. An example that\ngoes beyond the usual restrictions is the (very simple) affine\nbiquandle used in to construct the Affine Index\npolynomial invariant of virtual knots. One can make small examples\nof the flat biquandle algebra, that detect some flat linking beyond\nmod $2$ linking numbers, but the absence of other finite dimensional\nrepresentations presents a problem.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 2\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: new invariants exist; classification still open. The specific algebra question about the flat biquandle algebra appears open. Literature status: Substantial progress, no complete classification. - Flat virtual knots remain not fully classified. New invariants: semiquandle coloring invariants and u-polynomials (arXiv:2411.xxxx, 2024); polynomial invariants for flat virtual links (arXiv:0512.xxxx, 2005); flat-virtual invariants (arXiv:2403.xxxx, 2024); \"Connected sum and crossing numbers of flat virtual knots\" (arXiv:2312.xxxx, 2023); \"Equivalence of flat-virtual diagrams\" (arXiv:2410.xxxx, 2024). - Flat virtual braid groups and their representations studied (arXiv:2503.xxxx 2025; arXiv:2010.xxxx 2020; arXiv:2306.xxxx 2023). - The flat biquandle algebra question (finite-dimensional representations) appears unaddressed in the recent literature I could find." }, { "id": 10600003, "problem_number": "AMR-105-0003", "title": "Virtual-knot problem 3 — The Flat Hierarchy", "statement": "The Flat Hierarchy: The flat hierarchy is constructed for any\nordinal $\\alpha$. We label flat crossings with members of this\nordinal. In a flat third Reidemeister move, a line with two $a$\nlabels can slide across a crossing labeled $b$ only if $a$ is\ngreater than $b.$ This generalizes the usual to theory of flat\nvirtual diagrams to a system with arbitrarily many different types\nof flat crossings. Classify the diagrams in this hierarchy. This\nconcept is due to Kauffman (unpublished). A first step in working\nwith the flat hierarchy can be found in .", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 3\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open and unstudied; no literature located. Literature status: No work found on this specific ordinal-labelled flat hierarchy in the literature I could search. The concept remains attributed to Kauffman (unpublished)." }, { "id": 10600004, "problem_number": "AMR-105-0004", "title": "Virtual-knot problem 4 — Virtuals and the Theory of Doodles", "statement": "Virtuals and the Theory of Doodles: Compare flat theories of\nvirtual knots with theories of doodles. A doodle is represented by a\nflat diagram in the plane. Reidemeister moves of type I and II are\nallowed but not type III. So a triple point must not be allowed.\nKhovanov has associated a group to doodles . Commutator\nidentities can also be associated to a doodle. Cobordism of doodles\nis defined by an immersed surface without triple points. Cobordism\nclasses represent elements of a free abelian group. There is\nundoubtably a rich seam of results which could be found by\ninvestigating doodles. For example there is no reason not to have\nvirtual crossings as well as flat crossings. This would give doodles\non a surface of higher genus, see .", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 4\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: doodle theory (including virtual doodles and complete invariants in special settings) is an active area; a full comparison of flat virtual theories with doodle theories, as framed here, is not documented as complete." }, { "id": 10600005, "problem_number": "AMR-105-0005", "title": "Virtual-knot problem 5 — Virtual Three Manifolds", "statement": "Virtual Three Manifolds: There is a theory of virtual\n$3$–manifolds constructed as formal equivalence classes of virtual\ndiagrams modulo generalized Kirby moves, see . From this\npoint of view, there are two equivalences for ordinary\n$3$–manifolds: homeomorphisms and virtual equivalence. Do these\nequivalences coincide? That is, given two ordinary three manifolds,\npresented by surgery on framed links $K$ and $L$, suppose that $K$\nand $L$ are equivalent through the virtual Kirby calculus. Does this\nimply that they are equivalent through the classical Kirby calculus?\n\nWhat is a virtual $3$-manifold?: That is, give an\ninterpretation of these equivalence classes in the domain of\ngeometric topology.\n\nConstruct another theory of virtual $3$–manifolds by\nperforming surgery on links in thickened surfaces $S_{g}\\times\n\\mathbb{R}$ considered up to stabilization. Will this theory\ncoincide to that proposed by Kauffman and Dye ?", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 5\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: the three questions appear open; one complexity paper exists but its content was not verified. Literature status: Very little literature found. - \"Complexity of virtual 3-manifolds\" (Martelli? arXiv:1609.xxxx, 2016) — I could not verify authorship; the arXiv API returned the title only. This paper exists and discusses virtual 3-manifolds, but I did not verify its content. - No paper found resolving the coincidence question (1)." }, { "id": 10600006, "problem_number": "AMR-105-0006", "title": "Virtual-knot problem 6 — Welded Knots", "statement": "Welded Knots: We would like to understand welded\nknots . It is well known\nthat if we admit forbidden moves to the virtual link diagrams, each\nvirtual knot can be transformed to the unknot. If we allow only one\nforbidden move (e.g. the upper one), then there are lots of\ndifferent equivalence classes of knots. In fact the fundamental\ngroup and the quandle of the virtual diagram are invariant under the\nupper forbidden move. The resulting equivalence classes are called\nwelded knots. Similarly, welded braids were studied\nin , and every welded knot is the closure of a welded\nbraid. The question is to construct good invariants of welded knots\nand, if possible, to classify them. In a mapping is\nconstructed from welded knots to ambient isotopy classes of\nembeddings of tori (ribbon tori to be exact) in four dimensional\nspace, and it is proved that this mapping is an isomorphism from the\ncombinatorial fundamental group (in fact the quandle) of the welded\nknot to the fundamental group of the complement of the corresponding\ntorus embedding in four-space. Is this Satoh mapping faithful from\nequivalence classes of welded knots (links) to ambient isotopy\nclasses of ribbon torus embeddings in four-space?\n\nAnother interesting question is the following: If a welded knot has\ntrivial fundamental group, does this imply that the knot is trivial\nas a welded knot?", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 6\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL: good invariants exist and Tube map faithfulness is partially resolved (injective on certain classes); the general faithfulness question and the trivial-group characterization remain open. Literature status: Partial progress on all parts. - Tube map: Audoux, \"On the welded Tube map\" (arXiv:1408.xxxx, 2014) — proves the Tube map is injective on a certain class and studies its kernel; \"Some generalizations of Satoh's Tube map\" (arXiv:2103.xxxx, 2021). Faithfulness in general remains open. - \"Non-triviality of welded knots and ribbon torus-knots\" (arXiv:2403.xxxx, 2024); \"An unknotting invariant for welded knots\" (arXiv:2008.xxxx, 2020). - Homotopy classification of ribbon tubes and welded string links (arXiv:1407.xxxx, 2014); \"Bridge numbers and meridional ranks of knotted surfaces and welded knots\" (arXiv:2111.xxxx, 2021); \"Welded graphs, Wirtinger groups and knotted punctured spheres\" (arXiv:2311.xxxx, 2023). - The trivial-group question for welded knots: no definitive resolution found; related…" }, { "id": 10600007, "problem_number": "AMR-105-0007", "title": "Virtual-knot problem 7 — Long Knots and Long Flat Knots", "statement": "Long Knots and Long Flat Knots: Enlarge the long knot\ninvariant structure proposed in . Can one get new\nclassical knot invariants from the approach in this paper? Bring\ntogether the ideas from with the biquandle\nconstruction from to obtain more powerful invariants of\nlong knots. Long flat virtual knots can be studied via a powerful\nremark due to Turaev (in conversation) to the effect that one can\nassociate to a given long flat virtual knot diagram $F$ a descending diagram $D(F)$ (by always going over before going under\nin resolving the flat (non-virtual) crossings in the diagram). The\nlong virtual knot type of $D(F)$ is an invariant of the long flat\nknot $F.$ This means that one can apply any other invariant $I$ of\nvirtual knots that one likes to $D(F)$ and $I[D(F)]$ will be an\ninvariant of the long flat $F.$ It is quite interesting to do sample\ncalculations of such invariants and this situation underlines the\ndeeper problem of finding a full classification of long flat knots.\n\nBartholomew, Fenn, S. Kamada and N. Kamada have applied quaternion\ninvariants to long virtual knots, see .\n\nSee also Sec. the cited item for more questions about long\nvirtual knots and long flat virtual knots.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 7\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: invariants of long virtual knots are rich and growing; full classification of long flat knots remains open. Literature status: Active area with progress; classification open. - \"Prime Decomposition and Non-Commutativity in the Monoid of Long Virtual Knots\" (arXiv:1311.xxxx, 2013). - \"The intersection polynomials of a long virtual knot I/II\" (arXiv:2512.xxxx, 2025) — new invariants. - \"Biquandle longitude invariant of long virtual knots\" (arXiv:0709.xxxx, 2007); \"Finite-Type Invariants of order one for long virtual knots\" (arXiv:1602.xxxx, 2016). - \"Band-Passes and Long Virtual Knot Concordance\" (arXiv:1603.xxxx, 2016)." }, { "id": 10600008, "problem_number": "AMR-105-0008", "title": "Virtual-knot problem 8 — Virtual Biquandle", "statement": "Virtual Biquandle: Construct presentations of the virtual\nbiquandle with the a linear (non-commutative) representation at\nclassical crossings and some interesting structure at virtual\ncrossings. A start has been made by Bartholomew and Fenn .\nIn this paper various biquandle decorations are made at classical\nand virtual crossings which were found by a computer search.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 8\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: no published resolution found. Literature status: No literature found specifically resolving this construction problem. Biquandle/virtual-biquandle theory exists (Bartholomew–Fenn computer searches; Nelson's biquandle census), but the specific \"linear representation at classical crossings + structure at virtual crossings\" construction was not located." }, { "id": 10600009, "problem_number": "AMR-105-0009", "title": "Virtual-knot problem 9 — Virtual braids", "statement": "Virtual braids: Is there a birack such that its action on\nvirtual braids is faithful?\n\nIs the invariant of virtual braids in , see\nalso , faithful?\n\nThe action defined by linear biquandles is not faithful. This\nalmost certainly means that the corresponding linear invariants of\nvirtual knots and links are not faithful (see\nalso ).", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 9\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: no faithful birack action found; representation theory of virtual braid groups is actively studied with many non-faithful examples; the faithful-action question appears open. Literature status: Partial progress; faithfulness of the virtual braid group representations is an active topic. - \"Representations of virtual braids by automorphisms and virtual knot groups\" (Bardakov et al., arXiv:1603.xxxx, 2016) — a virtual braid group representation by automorphisms of a free group with a large kernel. - \"Virtual and universal braid groups, their quotients and representations\" (arXiv:2107.xxxx, 2021). - \"Representations of flat virtual braids by automorphisms of free group\" (arXiv:2306.xxxx, 2023). - \"Maps from braids to virtual braids and braid representations\" (arXiv:2210.xxxx, 2022); \"The nontrivial kernel of Manturov-Nikonov map from classical braids to virtual braids\" (arXiv:2603.xxxx, 2026). - Linear biquandle actions are known to be non-faithful (as noted in the problem)." }, { "id": 10600010, "problem_number": "AMR-105-0010", "title": "Virtual-knot problem 10 — The Fundamental Biquandle", "statement": "The Fundamental Biquandle: Does the fundamental biquandle,\nsee classify virtual links up to mirror images? (We know\nthat the biquandle has the same value on the orientation reversed\nmirror image where the mirror stands perpendicular to the plane\n(see ).\n\nAre there good examples of weak biquandles which are not strong?\nThis problem is solved in .\n\nWe would like to know more about the algebra with 2 generators\n$A,\\,B$ and one relation $[B,(A-1)(A,B)]=0$ (see ). It is\nassociated to the linear case.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 10\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: classification by fundamental biquandle appears open; the algebra question also appears open. Literature status: The fundamental biquandle question is open. The weak-vs-strong biquandle sub-problem is noted in the survey as already solved (citation elided). No resolution of the classification question was found." }, { "id": 10600011, "problem_number": "AMR-105-0011", "title": "Virtual-knot problem 11 — Virtualization and Unit Jones Polynomial", "statement": "Virtualization and Unit Jones Polynomial: Suppose the knot $K$\nis classical and not trivial. Suppose that ${\\tilde K}$ (obtained\nfrom $K$ by virtualizing a subset of its crossings) is not trivial\nand has a unit Jones polynomial, $V(\\tilde K)=1$. Is it possible\nthat ${\\tilde K}$ is classical (i.e. isotopic through virtual\nequivalence to a classical knot)?\n\nSuppose $K$ is a virtual knot diagram with unit Jones polynomial. Is\n$K$ equivalent to a classical diagram via virtual equivalence plus\ncrossing virtualization? (Recall that by crossing virtualization, we\nmean flanking a classical crossing by two virtual crossings. This\noperation does not affect the value of the Jones polynomial.)\n\nGiven two classical knots $K$ and $K',$ if $K$ can be obtained from\n$K'$ by a combination of crossing virtualization and virtual\nReidemeister moves, then is $K$ classically equivalent to $K'?$\n\nIf the above two questions have affirmative answers, then the only\nclassical knot with unit Jones polynomial is the unknot.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 11\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No resolution found. This cluster appears open in the literature I could find. Related: Jones polynomials of long virtual knots (arXiv:2012.xxxx, 2020); checkerboard colorable virtual knots with Jones polynomial studies (arXiv:0008.xxxx, 2000)." }, { "id": 10600012, "problem_number": "AMR-105-0012", "title": "Virtual-knot problem 12 — Virtual Quandle Homology", "statement": "Virtual Quandle Homology: Study virtual quandle homology in\nanalogy to quandle homology .", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 12\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: no dedicated literature found; appears open. Literature status: I could not locate literature specifically on \"virtual quandle homology\". Quandle homology theory (Carter–Kamada–Saito) is well developed; virtual analogues exist in scattered form (e.g., via virtual quandles and rack spaces), but the arXiv search returned no dedicated paper." }, { "id": 10600013, "problem_number": "AMR-105-0013", "title": "Virtual-knot problem 13 — Khovanov Homology", "statement": "Khovanov Homology: Construct a generalization of the Khovanov\ncomplex for the case of virtual knots that will work for arbitrary\nvirtual diagrams. Investigate the Khovanov homology constructed\nin . The main construction in this approach uses\nan orientable atom condition to give a Khovanov homology over the\nintegers for large classes of virtual links. The import of our\nquestion, is to investigate this structure and to possibly find a\nway to do Khovanov homology for all virtual knots over the ring of\nintegers. Similar questions can be raised for the presently evolving\nnew classes of Khovanov homology theories related to other quantum\ninvariants (cf. ).\n\nBy a K-full virtual knot we mean a knot for which there exists\na diagram such that the leading (the lowest, or both) term comes\nfrom the $B$-state. Analogously, one defines the Kho-full knot\nrelative to the Khovanov invariant. Call such diagrams optimal\ndiagrams. (It is easy to find knots which are neither K-full nor\nKho-full.)\n\nClassify all K-full (Kho-full) knots.\n\nAre optimal diagrams always minimal with respect to the number of\nclassical crossings?\n\nClassify all diagram moves that preserve optimality.\n\nIs it true that if a classical knot $K$ has minimal classical\ndiagram with $n$ crossings then any virtual diagram of $K$ has at\nleast $n$ classical crossings?\n\nCan any virtual knot have torsion in the $B$-state of the Khovanov\nhomology (the genuine leading term of some diagram)? Here we use the\nformulation of Khovanov homology given in .\n\nThe behaviour of the lowest and the leading term of the Kauffman\nbracket for virtual knots was studied in\nand and .", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 13\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: integral Khovanov homology for virtuals exists via several constructions (Manturov; Khovanov–Rozansky over integers; KLS for thickened surfaces); the classification questions (K-full knots, optimal diagram minimality, B-state torsion) appear open." }, { "id": 10600014, "problem_number": "AMR-105-0014", "title": "Virtual-knot problem 14 — Brauer algebra", "statement": "Brauer algebra: The appropriate domain for the virtual\nrecoupling theory is to place the Jones–Wenzl projectors in the\nBrauer algebra. That is, when we add virtual crossings to the\nTemperley–Lieb Algebra to obtain “Virtual Temperley–Lieb\nAlgebra” the result is the Brauer algebra of all connections from\n$n$ points to $n$ points (see ). What is the structure\nof the projectors in this context? Can a useful algebraic\ngeneralization of the classical recoupling theory be formulated?", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 14\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: projectors in the virtual TL algebra are studied (2021); a full recoupling theory in the Brauer algebra context does not appear to be documented. Literature status: Partial progress. - \"Virtual Extension of Temperley–Lieb Algebra\" (arXiv:0610.xxxx, 2006) — establishes the virtual TL = Brauer algebra connection. - \"Projectors in the Virtual Temperley-Lieb Algebra\" (arXiv:2103.xxxx, 2021) — directly studies projectors in the virtual TL algebra. - \"Anyonic Topological Quantum Computation and the Virtual Braid Group\" (arXiv:0909.xxxx, 2009) — recoupling/anyonic context." }, { "id": 10600015, "problem_number": "AMR-105-0015", "title": "Virtual-knot problem 15 — Virtual Alternating Knots", "statement": "Virtual Alternating Knots: Define and classify alternating\nvirtual knots.\n\nFind an analogue of the Tait flyping conjecture and prove it.\nCompare .\n\nClassify all alternating weaves on surfaces (without stabilization).", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 15\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL: flyping analogue SOLVED (Kindred 2022); many classical results extended to alternating virtual links; the weave classification on surfaces is addressed in 2020 work; full classification of alternating virtual knots remains a research program." }, { "id": 10600016, "problem_number": "AMR-105-0016", "title": "Virtual-knot problem 16 — Crossing Number", "statement": "Crossing Number: One of the most important problems in knot\ntheory is the problem of finding the minimal crossing number for a\ngiven knot. It can be estimated by means of various invariants, the\nsimplest one, perhaps, being the Kauffman bracket estimate:\n$$\n\\mathrm{span}\\langle K\\rangle \\leqslant 4n-4g,\n$$\nHere $\\mathrm{span}$ stands for the difference between the leading\ndegree and the lowest degree non-zero terms in the Laurent\npolynomial defined by the Kauffman bracket, $n$ is the crossing\nnumber, $g$ is the atom genus for the knot $K$ (also called the\nTuraev genus).\n\nIn the case $g=0$, we get the celebrated\nMurasugi–Kauffman–Thistlethwaite's theorem on minimality of\nalternating knot diagrams.\n\nIs there is a way to prove that the atom genus $g$ is minimal for a\nconcrete knot diagram? Then this together with a sharp estimate for\nthe span of the Kauffman bracket guarantees the minimality of the\ndiagram.\n\nThe estimate $4n-4g$ is exact for adequate diagrams (after\nThistlethwaite). It is exact for many diagrams which are not\nadequate however: there are lots of minimal diagrams where the span\nof the Kauffman bracket “drops down”, for example, for torus\n$(p,q)$ knots for $p>q>2$.\n\nWhy is the span of the Kauffman bracket smaller than expected for\nsuch classes of knots?\n\nOne of possible explanations may be that the projection surface is\nwrongly chosen. For a torus knot, it is more convenient to consider\nit as being projected to the torus with no crossing points rather\nthan to the plane with many crossing points. Thus the following\nproblem arises: How do we use the Kauffman bracket, Thistlethwaite\nspanning tree, atoms and other invariants to get estimates for the\nnumber of crossings when projecting the knot diagram not to the\nplane but rather to some surfaces of higher genera?\n\nCertainly, when applying smoothing on the plane we get the final\nexpansion of the Kauffman bracket as a linear combination of the\nbracket for the unknot; we shall get expression in terms of the\nbracket of some “basic” torus knots, which makes the problem\nharder.\n\nCan we get estimates from Thistlethwaite's spanning tree and/or from\nthe Khovanov homology?\n\nWhich minimality theorems can be proved in this direction?\n\nIs it possible to get an estimate for the atom genus when the atom\nis considered not in the neighborhood of the plane but rather in a\nneighborhood of the surface the knot is projected to and the\nthickness of the Khovanov homology does not exceed $2+g$?", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 16\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL: the core phenomenon is understood for alternating/adequate classes; the general questions (atom genus minimality proofs, span drop, higher-genus projections) remain open. Literature status: Partial. The Murasugi–Kauffman–Thistlethwaite minimality theorem for alternating diagrams and its adequate-diagram generalizations are classical; virtual analogues exist via \"Classical results for alternating virtual links\" (arXiv:2204.xxxx, 2022) and Khovanov homology of alternating virtual links (arXiv:1904.xxxx, 2019). The span-drop phenomenon for torus knots and the higher-genus projection problem remain open research directions." }, { "id": 10600017, "problem_number": "AMR-105-0017", "title": "Virtual-knot problem 17 — Crossing number problems", "statement": "Crossing number problems: For each virtual link $L$, there are\nthree crossing numbers: the minimal number $C$ of classical\ncrossings, the minimal number $V$ of virtual crossings, and the\nminimal total number $T$ of crossings for representatives of $L$.\nThere are also a number of unknotting numbers: The classical\nunknotting number is the number of crossing switches needed to\nunknot the knot (using any diagram for the knot). The virtual\nunknotting number is the number of crossings one needs to convert\nfrom classical to virtual (by direct flattening) in order to unknot\nthe knot (using any virtual diagram for the knot). Very little is\nknown. Find out more about the virtual unknotting number.\n\nWhat is the relationship between the least number of virtual\ncrossings and the least genus in a surface representation of the\nvirtual knot.\n\nIs it true that $T=V+L$?\n\nIs there any algorithm for finding $V$ for some class of virtual\nknots. For $T$, this is partially done for two classes of links:\nquasialternating and some other, see . For classical\nlinks and alternating diagrams see .\n\nAre there some (non–trivial) upper and lower bounds for $T,\\,V,\\,L$\ncoming from virtual knot polynomials (see )?", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 17\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: polynomial bounds on V exist (arrow polynomial, writhe polynomial); virtual unknotting numbers computed for special classes; T = V + L remains open; algorithms for V remain open. Literature status: Partial progress on many parts. - Lower bounds on V and surface genus: \"Lower bounds on virtual crossing number and minimal surface genus\" (arXiv:0904.xxxx, 2009); \"Virtual Crossing Number and the Arrow Polynomial\" (arXiv:0810.xxxx, 2008); \"On Virtual Crossing Number Estimates For Virtual Links\" (arXiv:0811.xxxx, 2008); \"On Virtual Crossing Numbers for Virtual Knots\" (arXiv:1107.xxxx, 2011); \"A note on the writhe polynomial and the virtual crossing number\" (arXiv:1805.xxxx, 2018). - Virtual unknotting numbers: \"Virtual unknotting numbers of certain virtual torus knots\" (arXiv:1701.xxxx, 2017). - T = V + L: no resolution found." }, { "id": 10600018, "problem_number": "AMR-105-0018", "title": "Virtual-knot problem 18 — Wild Virtuals", "statement": "Wild Virtuals: Create the category of “wild virtual knots”\nand establish its axiomatics. In particular, one needs a theorem\nthat states when a wild equivalence of tame virtual links implies a\ntame equivalence of these links.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 18\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE: appears completely open. Literature status: No literature found. This is a foundational open problem with essentially no published work (searches returned nothing relevant)." }, { "id": 10600019, "problem_number": "AMR-105-0019", "title": "Virtual-knot problem 19 — Vassiliev Invariants", "statement": "Vassiliev Invariants: Understand the connection between\nvirtual knot polynomials and the Vassiliev knot invariants of\nvirtual knots (in Kauffman's sense). Some of that was done\nin .\n\nThe key question about this collection of invariants is this: Does every Vassiliev invariant of finite type, for classical knots\nextend to an invariant of finite type for long virtual knots? Here\nwe mean the problem in the sense of the formulation given\nin . In it was pointed out that there is a\nnatural notion of Vassiliev invariants for virtual knots that has a\ndifferent notion of finite type from that given in . This\nalternate formulation needs further investigation.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 19\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: the theory of finite-type invariants of virtual knots is well developed; the specific extension question (every classical finite-type invariant extends to long virtual knots) is addressed in the literature in special cases but I did not find a definitive full resolution." }, { "id": 10600020, "problem_number": "AMR-105-0020", "title": "Virtual-knot problem 20 — Embeddings of Surfaces", "statement": "Embeddings of Surfaces: Given a non-trivial virtual knot $K$.\nProve that there exists a minimal realization of $K$ in\n$N=S_{g}\\times I$ and an unknotted embedding of\n$N\\subset\\mathbb{R}^{3}$ such that the obtained classical knot in\n$\\mathbb{R}^{3}$ is not trivial. (This problem is partially solved\nby Dye in .", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 20\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No literature found resolving this. Dye's partial solution predates the survey; searches returned nothing newer." }, { "id": 10600021, "problem_number": "AMR-105-0021", "title": "Virtual-knot problem 21 — Non-Commutativity and Long Knots", "statement": "Non-Commutativity and Long Knots: It is known that any\nclassical long knot commutes with any long knot. This is definitely\nnot the case in the virtual category, see paper by Kamadas, Fenn et\nal. However the commutativity of classical long knots may be\nincorporated into the commutativity of virtual knots if the\nfollowing question is true. Is it true that if $K$ and $K'$ are long\nknots and $K{\\#} K'$ is isotopic to $K'{\\#} K$ then there exists a\nvirtual long knot $L$, classical long knots $Q,\\,Q'$, and\nnon-negative integer numbers $m,n$ such that\n$$\nK=L^{m} {\\#} Q, \\quad K'=L^{n} {\\#} Q',\n$$\nwhere by $L^{m}$ we mean the connected sum of $m$ copies of the same\nknot?", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 21\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: the specific statement appears open; related monoid structure results exist. Literature status: Partial context: \"Prime Decomposition and Non-Commutativity in the Monoid of Long Virtual Knots\" (arXiv:1311.xxxx, 2013) studies the monoid structure. No resolution of the specific factorization statement was found." }, { "id": 10600022, "problem_number": "AMR-105-0022", "title": "Virtual-knot problem 22 — The Rack Space", "statement": "The Rack Space: The rack space was invented by Fenn, Rourke\nand Sanderson . The homology of the rack\nspace has been considered by the above authors and Carter, Kamada,\nSaito . For low dimensions, the homology has the following\ninteresting interpretations. Two dimensional cycles are represented\nby virtual link diagrams consistently colored by the rack, and three\ndimensional cycles by the same but with the regions also colored.\nSee the thesis of Greene . So virtual links can give,\nin this way, information about classical knots! For the second\nhomology of the dihedral rack, the results are given in Greene's\nthesis. We now know that for a prime $p$ the third homology has a\nfactor $\\mathbb{Z}_p$, see .\n\nAnother line of enquiry is to look at properties of the birack\nspace and associated homology.\n\nBeyond problem number the cited item we list new\nproblems that are added to this revised problem list.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 22\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: the rack-space homology program exists with known results; the birack space question appears open. Literature status: Partial progress documented in the problem itself (Greene's thesis; Z_p factor in third homology of dihedral rack). No newer resolution found for the birack space homology question." }, { "id": 10600023, "problem_number": "AMR-105-0023", "title": "Virtual-knot problem 23 — Find new geometric/topological interpretations for the Jones polynomial and for Khovanov homology.", "statement": "Find new geometric/topological interpretations for the Jones\npolynomial and for Khovanov homology.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 23\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: open-ended; no resolution expected or found. Literature status: This is a broad open-ended research direction; no \"solution\" possible. Relevant recent work: Khovanov homology and homotopy types (Lipshitz–Sarkar program), Rasmussen invariants for virtual knots (arXiv:1603.xxxx, 2016), Khovanov homology of alternating virtual links (arXiv:1904.xxxx, 2019)." }, { "id": 10600024, "problem_number": "AMR-105-0024", "title": "Virtual-knot problem 24 — Does it follow that $K$ and $K'$ are equivalent as classical knots?", "statement": "Let $VKT/Z$ denote virtual knot theory modulo $Z$-equivalence, as\ndefined in the section above on virtual knot theory. Recall that two\nvirtual diagrams that are $Z$-equivalent have the same Jones\npolynomial. It is known that classical knot theory embeds in virtual\nknot theory. Does classical knot theory embed in $VKT/Z?$ That is,\nsuppose that $K$ and $K'$ are classical knot diagrams, and suppose\nthat $K$ and $K'$ are equivalent using virtual moves and $Z$-moves.\nDoes it follow that $K$ and $K'$ are equivalent as classical\nknots?", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 24\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No literature found on VKT/Z embedding. The Z-equivalence search returned nothing relevant." }, { "id": 10600025, "problem_number": "AMR-105-0025", "title": "Virtual-knot problem 25 — Rotational virtual knot theory introduced in is virtual knot theory without the first virtual move (thus one does not…", "statement": "Rotational virtual knot theory introduced in is virtual\nknot theory without the first virtual move (thus one does not allow\nthe addition or deletion of a virtual curl). This version of virtual\nknot theory is significant because all quantum link invariants\noriginally defined for classical links extend to rotational virtual\nknot theory. This theory has begun to be\nexplored and deserves further exploration. We\nhave formulated a version of the bracket polynomial for rotational\nvirtuals that assigns variables according to the absolute value of\nthe Whitney degree of state curves. See Figure the cited item for\nexamples of non-trivial rotational virtual links.\n\nThe rotational bracket can be generalised in the fashion\nof the Arrow Polynomial. Beyond this there are all the quantum link\ninvariants and how they behave on rotational virtuals. We are in the\nprocess of writing a new paper on rotational virtual knot theory. A\nnatural class of invariants for rotational virtuals arises as\nquantum invariants associated with finite dimensional\nquasi-triangular Hopf algebras. This harks back to an early\nproject that needs further articulation.\nIn that work with Radford we articulate invariants that immediately\ngeneralize to invariants of rotational virtual knots. These\ninvariants are defined via integrals on finite dimensional Hopf\nalgebras and should be studied for their own sake. We would also\nlike to know about the possibility of categorification of finite\ndimensional Hopf algebras and, in particular, the meaning of\ncategorifying a right integral.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 25\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: the theory is established (2015 paper); specific sub-problems (categorification of integrals) appear open. Literature status: - \"Rotational Virtual Knots and Quantum Link Invariants\" (Kauffman–Manturov, arXiv:1509.xxxx, 2015) — foundational paper on the theory. - \"Centrality and the KRH Invariant\" (arXiv:2107.xxxx, 2021) — studies the KRH invariant relevant here. - Categorification of Hopf-algebra integrals: no dedicated paper found." }, { "id": 10600026, "problem_number": "AMR-105-0026", "title": "Virtual-knot problem 26 — surface arrow invariant", "statement": "The arrow polynomial generalizes to a more powerful invariant of\nvirtual knots by defining a version of it for knots in specific\nthickened surfaces. The arrow polynomial for a knot in a thickened\nsurface has state variables that take into account the isotopy class\nof the state curve and its arrow number as well. This gives a very\nstrong invariant of knots in thickened surfaces and it can be\napplied to virtual knots by using a minimal surface representative\nfor the virtual knot. Investigating this surface arrow\ninvariant generalizes previous work on the surface bracket\npolynomial .", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 26\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: categorifications and surface variants exist; the specific surface arrow invariant program appears partially developed. Literature status: - \"On two categorifications of the arrow polynomial for virtual knots\" (arXiv:0906.xxxx, 2009). - \"The homological arrow polynomial for virtual links\" (arXiv:2207.xxxx, 2022). - \"On arrow polynomials of checkerboard colorable virtual links\" (arXiv:2002.xxxx, 2020). - \"Surface pole bracket polynomials of virtual knots and twisted knots\" (arXiv:1401.xxxx, 2014) — surface generalizations. - The specific thickened-surface arrow invariant appears explored within these programs; no dedicated paper found specifically on \"arrow polynomial for knots in thickened surfaces\" beyond the above." }, { "id": 10600027, "problem_number": "AMR-105-0027", "title": "Virtual-knot problem 27 — Study concordance and cobordism invariants of virtual knots.", "statement": "Study concordance and cobordism invariants of virtual knots. In\nparticular, solve the question of virtual knots up to\npass-equivalence (taking a direct generalization of\nclassical pass equivalence that gives the Arf invariant of knots).\nUnderstanding this specific problem will promote our understanding\nof concordance of virtual knots.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 27\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: pass-equivalence and Arf-type invariants for (long) virtual knots are studied (2016, 2024); the concordance group is actively investigated; full solution of the pass-equivalence classification is not documented." }, { "id": 10600028, "problem_number": "AMR-105-0028", "title": "Virtual-knot problem 28 — Biquandles", "statement": "Biquandles: The first two problems are old chestnuts.\n\nGive a descriptive representation of the free biquandle.\n\nGive a topological explanation of the fundamental biquandle of a\nknot.\n\nIs there such a thing as a free partial biquandle: probably not.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 28\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No dedicated literature found on these specific questions. General constructions of biquandles exist (arXiv:1908.xxxx, 2019); the free biquandle representation question was not located." }, { "id": 10600029, "problem_number": "AMR-105-0029", "title": "Virtual-knot problem 29 — Khovanov homology for virtual knots and links works directly with mod-$2$ coefficients.", "statement": "Khovanov homology for virtual knots and links works directly with\nmod-$2$ coefficients. With mod-2 coefficients there are no technical\ndifficulties associated with the fact that a virtual state curve can\nresmooth to a single curve. Over the integers, this is a problem\nthat has to be dealt with. An integral Khovanov homology for\nvirtual knots has been constructed by Manturov . It is\nalso the case that the work of Khovanov–Rozansky\ncategorifiying infinitely many specializations of the Homflypt\npolynomial is an integral invariant for virtual knots. In principle,\nthe Khovanov–Rozansky work solves the problem of integral Khovanov\nhomology for virtuals, but to see this in terms of Khovanov's\noriginal definition is a very good technical problem. Solve this\nproblem to clarify the Manturov construction.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 29\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "SOLVED-IN-LITERATURE: integral Khovanov homology for virtual knots exists via Manturov and via Khovanov–Rozansky/KLS-type constructions; the \"very good technical problem\" of relating them is documented as substantially accomplished, though a fully explicit dictionary to Khovanov's original definition is a technical exercise rather than an open problem." }, { "id": 10600030, "problem_number": "AMR-105-0030", "title": "Virtual-knot problem 30 — In we have studied a mod-2 categorification of the arrow polynomial and discovered many baffling examples of pairs of…", "statement": "In we have studied a mod-2 categorification of\nthe arrow polynomial and discovered many baffling examples of pairs\nof virtual knots that are discriminated by this link homology and\nnot distinguished by Khovanov homology mod-2 or the arrow\npolynomial. We want to understand this new link homology and to that\nend will do more computations and will try to understand the\nstructure of this homology theory for virtual links.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 30\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: the categorifications exist (2009, 2022); understanding the structure and full computations continues. Literature status: - \"On two categorifications of the arrow polynomial for virtual knots\" (arXiv:0906.xxxx, 2009) — the two categorifications. - \"The homological arrow polynomial for virtual links\" (arXiv:2207.xxxx, 2022) — newer homological arrow polynomial." }, { "id": 10600031, "problem_number": "AMR-105-0031", "title": "Virtual-knot problem 31 — Make a systematic study of Vassiliev invariants for virtual knots and links.", "statement": "Make a systematic study of Vassiliev invariants for virtual knots\nand links. There is ongoing work on this problem.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 31\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: systematic study is ongoing; no completion documented. Literature status: Ongoing work exists. - \"On Vassiliev Invariants of Virtual Knots\" (arXiv:2208.xxxx, 2022). - \"A Lattice of Finite-Type Invariants of Virtual Knots\" (arXiv:1303.xxxx, 2013); \"Some Dimensions of Spaces of Finite Type Invariants of Virtual Knots\" (arXiv:0909.xxxx, 2009); \"Vassiliev Invariants from Parity Mappings\" (arXiv:1203.xxxx, 2012); \"Parity and Exotic Combinatorial Formulae for Finite-Type Invariants of Virtual Knots\" (arXiv:1002.xxxx, 2010)." }, { "id": 10600032, "problem_number": "AMR-105-0032", "title": "Virtual-knot problem 32 — Generalize virtual knot theory to virtual 2-spheres in 4-space.", "statement": "Generalize virtual knot theory to virtual 2-spheres in 4-space.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 32\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: virtual surface theory exists in scattered form (spinning constructions, Gauss diagram invariants); a full generalization of virtual knot theory to virtual 2-spheres is not documented as complete." }, { "id": 10600033, "problem_number": "AMR-105-0033", "title": "Virtual-knot problem 33 — Find a way to effectively compute the Kauffman–Radford–Hennings (KRH) invariants for three-manifolds via right integr…", "statement": "Find a way to effectively compute the Kauffman–Radford–Hennings\n(KRH) invariants for three-manifolds via right\nintegrals on finite dimensional Hopf algebras . Find a way\nto categorify the Kauffman–Radford–Hennings invariants for all\nfinite dimensional quasitriangular Hopf algebras. The KRH invariants\nare actually invariants of rotational virtual links that are also,\nafter normalization, invariant under the moves of the Kirby\ncalculus. In this way, they give invariants of three manifolds and\nvirtual three manifolds. The formulation of the KRH invariants in\nterms of integrals for finite dimensional Hopf algebras is very\nelegant, but computation is very difficult. Thus these invariants\nform a challenge for both virtual and classical knot theory. It is\nworth pointing out again, that the proper domain for all\nquantum link invariants is rotational virtual knot theory. Thus,\nthis problem about the KRH invariants is a test case for the\nstructure of quantum link invariants as a whole.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 33\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: computations of Hennings-type invariants exist for restricted quantum groups; KRH centrality studied (2021); categorification appears open. Literature status: - \"Centrality and the KRH Invariant\" (arXiv:2107.xxxx, 2021) — studies the KRH invariant. - \"Quantum Invariants of Links and 3-Manifolds with Boundary defined via Virtual Links\" (arXiv:2108.xxxx, 2021; examples arXiv:2203.xxxx, 2022) — computations of related invariants via virtual links. - \"Logarithmic Hennings invariants for restricted quantum sl(2)\" (arXiv:1705.xxxx, 2017); \"Integrality and Gauge Dependence of Hennings TQFTs\" (arXiv:1305.xxxx, 2013). - Categorification of KRH: no resolution found." }, { "id": 10600034, "problem_number": "AMR-105-0034", "title": "Virtual-knot problem 34 — Create a combinatorial homotopy theory for Khovanov homology so that the Khovanov homology of a knot or link is equiv…", "statement": "Create a combinatorial homotopy theory for Khovanov homology so that\nthe Khovanov homology of a knot or link is equivalent to the\nhomotopy type of an abstract complex (or category) associated with\nthe knot of link. This problem is meant in the sprit of Bar-Natan's\nreformulation of Khovanov homology as an abstract chain homotopy\nclass of a categorical chain complex associated with the knot or\nlink. Recent work of Lifshitz and Sarkar goes very far\nin this direction. Of course we would like a deeper connection, and\nwe would like to have the theory working for virtual knots and\nlinks.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 34\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: the Lipshitz–Sarkar program provides homotopy types (now extended to thickened surfaces); a full combinatorial homotopy theory for virtual knots in the Bar-Natan spirit is not documented as complete." }, { "id": 10600035, "problem_number": "AMR-105-0035", "title": "Virtual-knot problem 35 — In we give a formula for the Kauffman polynomial, due to Jaeger, that expresses this invariant as a state sum over or…", "statement": "In we give a formula for the Kauffman polynomial, due\nto Jaeger, that expresses this invariant as a state sum over\noriented, partially smoothed links associated with the given link.\nthese states are evaluated by using the Homfly polynomial.\nGeneralize the Jaeger point of view to obtain a categorification of\nthe Kauffman polynomial by using chain complexes for these states\nthat are derived from the Khovanov–Rozansky categorification of the\nHomflypt polynomial.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 35\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: categorifications of SO(2N) Kauffman-type polynomials exist; the full Jaeger-state categorification appears addressed in progress reports but not completed. Literature status: - \"Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial\" (arXiv:0701.xxxx, 2007) — a categorification of the SO(2N) Kauffman polynomial, directly relevant. - \"On Jaeger's HOMFLY-PT expansions, branching rules and link homology: a progress report\" (arXiv:1309.xxxx, 2013) — progress on the Jaeger approach." }, { "id": 10600036, "problem_number": "AMR-105-0036", "title": "Virtual-knot problem 36 — electrical", "statement": "In we show how, by translating between knots and\nplanar graphs (the checkerboard and medial constructions) one can\nassociate a signed graph to a classical knot and that, with a choice\nof nodes and an interpretation of the signs as generalized\nelectrical conductance, the conductance between two nodes of the\ngraph is an invariant of isotopy of the knot restricted to move in\nthe complement of these nodes. The moves on the graphs involve\nreplacements of pedant loops and edges, series and parallel\nreplacements and a star–triangle exchange move. These moves can be\napplied to abstract (possibly non-planar) graphs. As a result we\nobtain a generalization of knot theory (similar in spirit but quite\ndifferent from virtual knot theory) by taking the electrical\nequivalence classes of signed graphs. Call this theory $EG_{\\pm}.$\nThe conductance remains an invariant of these Electric–Knots\nand shows that the theory is highly non-trivial. This formulation of\nElectric Knots gives rise to many problems. Find new invariants of\nGraph-Knots. Note that if we take a planar immersion of a signed\ngraph, then the induced cyclic orders of edges at the nodes of the\nplanar embedding give a natural way (analogous to the medial graph)\nto associate a virtual knot diagram to the immersion. Explore this\nrelationship of Electric Knots and virtual knots.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 36\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: no published follow-up found. Literature status: No literature found on \"electric knots\" as such. The electrical network paradigm is classical (Kirchhoff, conductance, star-triangle), and its graph-knot connections were explored in the cited source; no dedicated follow-up was located." }, { "id": 10600037, "problem_number": "AMR-105-0037", "title": "Virtual-knot problem 37 — It has been pointed out that the chain complex for Knot Floer Homology (categorifying the Alexander–Conway polynomial…", "statement": "It has been pointed out that the chain complex for Knot Floer\nHomology (categorifying the Alexander–Conway polynomial)\nis generated by the states described in Formal Knot\nTheory . These states involve an assignment of pointers to\neach region of the diagram (with two adjacent regions omitted) such\nthat each pointer marks a unique crossing in the diagram. The key\nproblem about this complex is that while the chain groups are\ndescribed via knot diagram combinatorics, the differential in the\ncomplex seem to require high dimensional contact geometry. Even\nthough there is now a combinatorial definition of Knot Floer\nhomology via arc diagrams, this is an unsatisfactory state of\naffairs. The problem is to find a combinatorial definition of the\ndifferential on the complex generated by the Formal Knot Theory\nstates.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 37\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: combinatorial models for knot Floer homology exist (spanning tree model, arc diagrams); the specific \"Formal Knot Theory states\" differential question is addressed in related work but the fully satisfying combinatorial definition in the stated sense is still a research topic." }, { "id": 10600038, "problem_number": "AMR-105-0038", "title": "Virtual-knot problem 38 — The Kauffman bracket polynomial of a virtual diagram can have the leading term, i.e., the term having the highest pos…", "statement": "The Kauffman bracket polynomial of a virtual diagram can have the\nleading term, i.e., the term having the highest possible degree,\nequal to zero.\n\nAssume that we have a diagram for which the Kauffman bracket\npolynomial has the nonzero leading term. Classify moves\n(compositions of the generalized Reidemeister moves) establishing\nthe equivalence in the class of diagrams with nonzero leading terms.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 38\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No literature found on this specific classification. Related: \"Local transformations and functorial maps\" (arXiv:2301.xxxx, 2023) and parity-based bracket studies exist, but nothing resolving the classification." }, { "id": 10600039, "problem_number": "AMR-105-0039", "title": "Virtual-knot problem 39 — We still do not know whether the free knot whose Gauss diagram is a heptagon (i.e.", "statement": "We still do not know whether the free knot whose Gauss diagram is a\nheptagon (i.e. consists of 7 chords each of which is linked with\nprecisely two adjacent ones) is trivial.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 39\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No resolution found. Searches for \"free knot heptagon\" and related returned nothing." }, { "id": 10600040, "problem_number": "AMR-105-0040", "title": "Virtual-knot problem 40 — Given a chord diagram $D$ we can construct the following formal chain complex.", "statement": "Given a chord diagram $D$ we can construct the following formal\nchain complex. Formally speaking, this complex will look like a\nsimplicial complex in all dimensions except zero, but one can get\nrid of this by using duality arguments.\n\nIn dimension $-1$ there is just one chain denoted by $\\emptyset$. In\ndimension $0$, the chains are in one-to-one correspondence with\nunoriented chords of the diagram $D$; in dimension $1$ the chains\nare in one-to-one correspondence with pairs of unlinked chords, for\ndimension $2$ we take triples of pairwise unlinked chords, etc.\n\nThe differential acts on the $k$-tuple, $\\{a_{1},\\dots, a_{k}\\}$ of\npairwise unlinked chords to the $(k+1)$-tuple $\\{a_{1},\\dots,\na_{k},a_{l}\\}$ obtained by the addition of a chord unlinked with\nall the previous ones. The sign is taken in a standard way to make\nthe complex well-defined with integral coefficients. Such a complex\nis naturally defined by the intersection graph of the chord diagram.\n\nExample.\nLet the chord diagram consist of $2n$ chords, where the the\nfollowing pairs of chords are linked: $(1,2)$, $(3,4)$, $(5,6),\n\\dots,(2n-1,2n)$. Then the corresponding simplicial complex is the\n$(n-1)$-sphere (the $n$-th join of $S^{0}$.\n\nCan we get any other simplicial complexes this way other than just\nbouquets of spheres? An affirmative answer to this question may lead\nto some homological calculations which are very easy for computer\nimplementation. This problem was motivated several years ago by an\nattempt to construct the spectrification of the Khovanov homology.\nIt turns out that if some (virtual) knot in the $A$-state has\nexactly one circle then its Khovanov homology in the lowest quantum\ngrading looks exactly as described above. Thus, this approach may be\nuseful for understanding the Khovanov homology of chord diagrams as\nwell as some other approaches to the Khovanov homology\nspectrification obtained by recent work with Lipshitz.\n\nOn the other hand, this problem is motivated the construction of\ngraph-link theory. Indeed, every triangulated manifold admits a\nshallow triangulation such that together with any 1-frame of a\n$k$-simplex, it contains the whole simplex.\n\nConsidering the one-dimensional frame of such a triangulation as the\nnon-intersection graph of some chord diagrams, we get exactly the\nchord diagram, whose lower term Khovanov homology looks as described\nabove. The problem is that not every graph is an intersection graph\n(neither any graph is a non-intersection graph) of any chord\ndiagram. This leads to the graph-link theory for which all possible\nmanifolds appear in the lower degree homology theory. This leads to\nsome lower homotopy of graph links. The next problem can be\nformulated as follows.\n\nGiven two diagrams of (virtual) knots or links with exactly one\ncircle in the $A$-state. Which combinations of Reidemeister moves on\nthe diagrams have to be be considered in such a way that whenever\ntwo diagrams generate equivalent knots, there exists a chain of\nmoves from one diagram to the other with all intermediate diagrams\nhaving single-circle $A$-state and such that chord diagram homology\n(and homotopy) corresponding to those intermediate diagrams do not\nchange under such moves?\n\nIt is worth comparing these thoughts with the recent work of\nLipschitz on the spectrification of Khovanov homology. Possibly, the\nBloom–Nikonov approach to Khovanov\nhomology can be used here.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 40\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No resolution found. This is a research direction motivated by Khovanov homology spectrification (cf. Lipshitz–Sarkar) and graph-link theory; no dedicated paper was located." }, { "id": 10600041, "problem_number": "AMR-105-0041", "title": "Virtual-knot problem 41 — Which quantum invariants extend to virtual knots themselves without restrictions?", "statement": "Which quantum invariants extend to virtual knots themselves without\nrestrictions? Certainly, there are such ones, e.g., the Kauffman\nbracket and the Jones polynomial.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 41\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: the answer is well understood — quantum invariants extend elegantly to rotational virtuals (Kauffman–Manturov 2015), which is the \"natural\" restriction-free domain; the Kauffman bracket and Jones polynomial extend to all virtuals." }, { "id": 10600042, "problem_number": "AMR-105-0042", "title": "Virtual-knot problem 42 — One can consider braids with even numbers of strands.", "statement": "One can consider braids with even numbers of strands. Markov's moves\nchange the parity of the number of strands. Can one reformulate\nMarkov's theorem in such a way that only braids with even numbers of\nstrands take part in this formulation?", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 42\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: the specific reformulation appears open. Literature status: No literature found specifically on this reformulation. Markov theorems for virtual braids exist (\"Virtual Braids\", arXiv:0407.xxxx, 2004; \"Virtual Braids and the L-Move\", arXiv:0507.xxxx, 2005; \"Markov Theorem For Free Links\", arXiv:1112.xxxx, 2011; \"A Markov's theorem for extended welded braids and links\", arXiv:1705.xxxx, 2017), but the even-strand restriction was not located." }, { "id": 10600043, "problem_number": "AMR-105-0043", "title": "Virtual-knot problem 43 — In his wonderful paper , which firstly had the title “Knot Floer Homotopy”, Sarkar constructs a cell complex.", "statement": "In his wonderful paper , which firstly had the title\n“Knot Floer Homotopy”, Sarkar constructs a cell complex. The\nhomology of this complex coincides with the Floer homology. In this\nwork the following main ingredients are used:\nManolescu–Ozsváth–Sarkar–Szab\\'o–Thurston approach\nallowing one to define Heegaard–Floer homology by using grid\ndiagrams and shellability. Roughly speaking, if complexes look like\norder complexes, then they can be replaced with cell (simplicial)\ncomplexes homotopically equivalent to them.\n\nTry to do the same things for Khovanov homology. In the initial\nKhovanov definition this problem seems to be very difficult, since\nthe set of chains and states can hard be ordered. However, taking\ninto account Bloom's construction this problem looks more\noptimistic.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 43\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: Sarkar's analogues for Khovanov homology are substantially achieved by Lipshitz–Sarkar (and extended to thickened surfaces); the fully combinatorial/simplicial presentation in Bloom's style for all virtuals is a continuing effort." }, { "id": 10600044, "problem_number": "AMR-105-0044", "title": "Virtual-knot problem 44 — A complete invariant is used for proving that one theory is a part of another theory.", "statement": "A complete invariant is used for proving that one theory is a part\nof another theory. For example, the fact that the set of classical\nknots is a part of the set of virtual knots, is true for the same\nreason as a quandle with peripheral structure can be extended from\nclassical knots into virtual knots. Analogous statements can be done\nfor classical and virtual braids (the extension of Hurwitz's\naction).\n\nSo, we get an actual problem: how can one construct a complete\ninvariant for virtual knots?", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 44\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No complete invariant for virtual knots has been constructed. Related: \"Quotient Quandles and the Fundamental Latin Alexander Quandle\" (arXiv:1404.xxxx, 2014); virtual knot groups are not complete. The classification of virtual knots (up to Gauss diagrams) is finite in low crossing numbers but the theoretical complete invariant question remains open." }, { "id": 10600045, "problem_number": "AMR-105-0045", "title": "Virtual-knot problem 45 — To prove that the invariant $\\mathcal{F}$ constructed by V.", "statement": "To prove that the invariant $\\mathcal{F}$ constructed by\nV. O. Manturov for virtual braids, is complete for virtual braids\nwith more than two strands.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 45\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No resolution found. The completeness of Manturov's braid invariant appears open. Related representation and invariant work: \"Representations of virtual braids by automorphisms and virtual knot groups\" (arXiv:1603.xxxx, 2016)." }, { "id": 10600046, "problem_number": "AMR-105-0046", "title": "Virtual-knot problem 46 — In it was showed how one could determine non-invertibility of long virtual knots and non-commutativity of long virtua…", "statement": "In it was showed how one could determine\nnon-invertibility of long virtual knots and non-commutativity of\nlong virtual knots. This method used two type crossings: an early\nundercrossing and an early overcrossing. Further, for two types we\nhave different operations, these operations are similar but not\ncoincide (these operations are related to quandles). Regretfully,\nthis method gives nearly nothing for classical knots. But it is\nuniversal: instead of the construction of quandle with two\noperations one can consider other ways of constructing knot\ninvariants. In these ways we partition the set of crossings into two\nsubsets, and for each subset we use an operation. Moreover, two\noperations are different enough to determine non-invertibility, and\nin the same time are similar to give an invariant structure. In the\nlevel of the Kauffman bracket polynomial this construction does not\nwork. We get two problems.\n\n\nTry to do the same for other objects than quandles, for example:\nnonlinear quandles, quandles with rings having non-unique\ndecomposition of multiples.\n\n\nTry to apply it on a “categorified level”: use the usual chain\nspace for Khovanov homology and in it two different differentials\nbut commute with each other (these differentials correspond to\ncircle and star operations). These two different differentials could\narise from usual and odd Khovanov homologies in the case when we\nhave an identification (but maybe not canonical) between chain\nspaces of these complexes. It is expected that these invariants\ncould recognize non-invertibility of knots.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 46\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: parity-based and two-operation methods exist for non-invertibility/non-commutativity; the specific categorified two-differential construction appears open. Literature status: - \"On the non-invertibility of virtual knots\" / parity-based two-operation methods: \"Vassiliev Invariants from Parity Mappings\" (arXiv:1203.xxxx, 2012); \"Free Knots, Groups, and Finite-Type Invariants\" (arXiv:1004.xxxx, 2010). - Non-invertibility of virtual knots: \"Virtual Covers of Links\" (arXiv:1405.xxxx, 2014); virtual knot chirality studies. - The two-differential categorified construction: no direct paper found." }, { "id": 10600047, "problem_number": "AMR-105-0047", "title": "Virtual-knot problem 47 — It is well known that flat virtual knots are easily algorithmically recognizable, see .", "statement": "It is well known that flat virtual knots are easily algorithmically\nrecognizable, see . The absence of a geometric approach to\nthe definition of free knots leads to the lack of the methods for\nsolving the following three problems.\n\n\nIs it true that (any) connected sum of free knots is trivial?\n\n\nAre free knots algorithmically recognizable?\n\n\nProve that free knots do not commute in the general case.\n\nIn the first case we suggest that the answer is affirmative, and the\nproof should be a certain modification of the analogous proof for\nvirtual knots (see, e.g., )\n\nIn the second case we suggest that the answer is negative. It seems\nto us that free knots are a very complicated object, where one can\nconstruct models for many logical constructions.\n\nThe third problem seems to be solvable by using (and, possibly,\nextending) standard methods in the free knot theory.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 47\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: invariant theory for free knots is well developed; the three specific questions (connected-sum triviality, algorithmic recognition, non-commutativity) appear open, with non-commutativity likely accessible." }, { "id": 10600048, "problem_number": "AMR-105-0048", "title": "Virtual-knot problem 48 — The functorial mapping was constructed, by means of it we constructed the map from the set of virtual knots to the se…", "statement": "The functorial mapping was constructed, by means of it we\nconstructed the map from the set of virtual knots to the set of\nvirtual knots with orientable atoms.\n\nCan one construct an analogous map which projects the set of virtual\nknots to the set of classical knots? For example, one can try to\nfind an index by using characteristic classes. Here only one\ncondition which arises under the third Reidemeister move should\nhold.\n\nNote that in the case of homologically trivial knots we can consider\na two-branched covering like a projection.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 48\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: functorial maps from virtual knots to classical knots exist (2010); full structural understanding continues. Literature status: - \"A Functorial Map from Virtual Knots to Classical Knots and Generalisations of Parity\" (arXiv:1011.xxxx, 2010) — exactly constructs such a functorial map to classical knots. - \"Local transformations and functorial maps\" (arXiv:2301.xxxx, 2023)." }, { "id": 10600049, "problem_number": "AMR-105-0049", "title": "Virtual-knot problem 49 — Turaev constructed the map from the set of long flat knots to the set of long virtual knots.", "statement": "Turaev constructed the map from the set of long flat knots to the\nset of long virtual knots. Can one construct any map from the set of\nlong free knots to the set of long virtual knots? This question can\nbe extended and one can try to construct any map which “enlarge”\nthe structure.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 49\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No literature found constructing such a map." }, { "id": 10600050, "problem_number": "AMR-105-0050", "title": "Virtual-knot problem 50 — The problem about cobordisms in sections: Let us have a free knot and its cobordism.", "statement": "The problem about cobordisms in sections: Let us have a free knot\nand its cobordism. Construct a parity on the given free knot, which\nis defined by using only this cobordism and respect only moves\ninside this cobordism.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 50\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE / borderline: the parity-cobordism program exists; the specific construction is not documented as completed. Literature status: - \"Parity and Cobordisms of Free Knots\" (arXiv:1001.xxxx, 2010); \"Cobordisms of Free Knots and Gauss Words\" (arXiv:0904.xxxx, 2009); \"A Sliceness Criterion for Odd Free Knots\" (arXiv:1707.xxxx, 2017). - The specific cobordism-defined parity construction appears addressed in the parity-cobordism program but not as a closed problem." }, { "id": 10600051, "problem_number": "AMR-105-0051", "title": "Virtual-knot problem 51 — Prove or disprove the conjecture about the non-uniqueness of minimal representative of a free link, i.e.", "statement": "Prove or disprove the conjecture about the non-uniqueness of minimal\nrepresentative of a free link, i.e. there exists a free link having\nseveral minimal representatives.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 51\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: - \"Minimal Diagrams of Free Knots\" (arXiv:1008.xxxx, 2010) — discusses minimal diagrams but does not resolve the uniqueness question. - No resolution found." }, { "id": 10600052, "problem_number": "AMR-105-0052", "title": "Virtual-knot problem 52 — If $X$ is the free rack, is $\\Gamma X$ a cat(0) space?", "statement": "If $X$ is the free rack, is $\\Gamma X$ a cat(0) space? A positive\nanswer would imply that all its higher homotopy groups are trivial.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 52\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No literature found on this question." }, { "id": 10600053, "problem_number": "AMR-105-0053", "title": "Virtual-knot problem 53 — Is there any algorithm for recognition whether two graph-links are equivalent or not?", "statement": "Is there any algorithm for recognition whether two graph-links are\nequivalent or not? Our conjecture is “no”. The idea behind that is\nthat graph-links are complicated enough, and possibly, they contain\nsufficiently many degrees of freedom to include something like\nTuring machines. All mathematics is roughly split into the\nrecognizable one (low-dimensional topology, hyperbolic groups,\ndecidability) and the non-recognizable one (topology of dimension\n$4$ and higher, arbitrary finitely presentable groups, Turing\nmachines etc). The usual argument for undecidability for finitely\npresented group allows one to construct some universal (semi)groups\nwhich includes the apparatus of the Turing machines. We think that\ngraph-links are enough complicated to include similar things: they\ncontain arbitrary graphs, and Reidemeister moves, in principle,\ncould play the role of “rules for formal languages” or “group\nrelations”. It seems unlikely that Reidemeister moves collapse\ngraph-links to anything simple enough because of the parity\nconsiderations: there are “irreducibly odd” graph-links which are\nstable in the sense that every equivalent graph contains the initial\ngraph as a subgraph.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 53\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "PARTIAL: graph-link theory is developed; the recognition/undecidability question appears open (as conjectured). Literature status: - Graph-link theory: \"Graph-Links\" (arXiv:1001.xxxx, 2010); \"Introduction to Graph-Link Theory\" (arXiv:0810.xxxx, 2008); \"Khovanov homology of graph-links\" (arXiv:1005.xxxx, 2010); \"Checkerboard graph links and simply laced Dynkin diagrams\" (arXiv:1907.xxxx, 2019). - No progress on the undecidability question found." }, { "id": 10600054, "problem_number": "AMR-105-0054", "title": "Virtual-knot problem 54 — Can one construct a projection from the set of graph-links to the set of realizable graph-links?", "statement": "Can one construct a projection from the set of graph-links to the\nset of realizable graph-links?", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 54\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No literature found on this specific projection." }, { "id": 10600055, "problem_number": "AMR-105-0055", "title": "Virtual-knot problem 55 — Construct a parity on graph-links by using Bouchet's criterion about the realizability of a graph.", "statement": "Construct a parity on graph-links by using Bouchet's criterion about\nthe realizability of a graph. Try to find a parity which is\nresponsible for the cyclically 6-edge connection, see .", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 55\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No literature found on this specific construction. Bouchet's interlacement/realizability framework exists (interlace polynomials; arXiv:0209.xxxx 2002; arXiv:0606.xxxx 2006), but not applied to graph-link parity as posed." }, { "id": 10600056, "problem_number": "AMR-105-0056", "title": "Virtual-knot problem 56 — Construct generalizations of the Frobenius extension and the Rasmussen for “rigid” graph-links with orientable atoms.", "statement": "Construct generalizations of the Frobenius\nextension and the Rasmussen for “rigid”\ngraph-links with orientable atoms.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 56\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: the pieces exist (Frobenius Khovanov for graph-links; virtual Rasmussen); the specific generalization to rigid graph-links is not documented as done. Literature status: - \"Khovanov homology of graph-links\" (arXiv:1005.xxxx, 2010) — Khovanov homology for graph-links using Frobenius extensions. - \"On the virtual Rasmussen invariant\" (arXiv:1603.xxxx, 2016) — Rasmussen invariant for virtual knots. - The specific combination for rigid graph-links is not documented as complete." }, { "id": 10600057, "problem_number": "AMR-105-0057", "title": "Virtual-knot problem 57 — Is it true that two equivalent realizable graph-links are equivalent in the class of realizable graph-links?", "statement": "Is it true that two equivalent realizable graph-links are equivalent\nin the class of realizable graph-links? If it is not true, then\nconstruct an example.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 57\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No literature found resolving this." }, { "id": 10600058, "problem_number": "AMR-105-0058", "title": "Virtual-knot problem 58 — Construct a group for graph-links which is analogous to the group from .", "statement": "Construct a group for graph-links which is analogous to the group\nfrom .", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 58\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open. Literature status: No literature found on a graph-link group (in the \"Reidemeister Moves and Groups\" sense; cf. arXiv:1412.xxxx, 2014)." }, { "id": 10600059, "problem_number": "AMR-105-0059", "title": "Virtual-knot problem 59 — Construct “graph-braids”.", "statement": "Construct “graph-braids”.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 59\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "OPEN-TRIAGE: appears open (and distinct from the classical graph braid group literature). Literature status: No literature on \"graph-braids\" in this sense. (Note: \"graph braid groups\" is an unrelated topic — motion groups of points on graphs; e.g., arXiv:0907.xxxx 2009, arXiv:2210.xxxx 2022 — not the intended object.)" }, { "id": 10600060, "problem_number": "AMR-105-0060", "title": "Virtual-knot problem 60 — Problems on Free Knot Cobordism", "statement": "Problems on Free Knot Cobordism: The methods used for proving\nthe fact that the invariant $L$ gives an obstruction to\nthe sliceness are not immediately generalized for obtaining lower\nestimates on the slice genus of free knots. Two reasons are as\nfollows. First, when we define a parity and justified\nparity for double lines on $\\mathcal{D}$, we chose an arbitrary\ncurve connecting the two preimages of the point on the double curve.\nWe assert that any two curves connecting these two preimages (and\nbehaving correctly in neighborhoods of the ends) are homotopic. It\nis true in the case of cobordism of genus zero, but in the case of a\nsurface of an arbitrary genus $h$ it is, indeed, not true.\n\nThus, in order to define a parity for double lines we have to impose\nsome restrictions on the spanning surface: we have to require that\nthe cohomology class dual to the graph $\\Psi$ was\n$\\mathbb{Z}_{2}$–homologically trivial. The significance of the\nproperty for even-valent graphs to be\n$\\mathbb{Z}_{2}$–homologically trivial is closely connected with\natoms (for details see Sec. the cited item\nand ).\n\nAnother problem is that the Reeb graph of an arbitrary Morse\nfunction (not necessarily corresponding to the disc) is not\nnecessarily a disc. Consider Fig. the cited item.\n\nThus, starting from a free knot $K$ for which, we say, $L(K)=8$, we\n(in principle) can turn it by a Morse bifurcation into free\ntwo-component link consisting of two free knots $K_{1}$ and\n$K_{2}$, for which $L(K_{i})=4$, and then by another Morse\nbifurcation we can reconstruct this trivial link into the unknot.\nThe invariant $L$ is not an obstruction to this, since the sum of\n$4$ and $-4$ is zero.\n\nIn some cases we can overcome these two difficulties for cobordisms\n(of arbitrary genus).\n\nLet $\\mathcal{D}_{g}$ be a surface with boundary $S^1$. Obviously,\nthe collection of double lines of $\\mathcal{D}_{g}$ defines a\nrelative $\\mathbb{Z}_{2}$–homology class $\\kappa\\in\nH_1(\\mathcal{D}_g,S^1;\\mathbb{Z}_2)$. This homology class is an\nobstruction for the surface to be checkerboard-colorable; also, this\nis an obstruction for well-definedness of even/odd double lines.\n\nNamely, if we look at the definition of an even/odd double line: we\nsee that there is an ambiguity in the choice of path connecting two\npreimages of a generic point on the double line. For the case of a\ndisc cobordism, the parity of double lines is well defined, because\nall such curves are homotopic. For $\\mathcal{D}_{g}$ the unique\nobstruction to this well-definedness is the class $\\kappa$.\n\nWe call a cobordism of genus $g$ checkerboard (or atomic\\/) if the corresponding class $\\kappa$ vanishes.\n\nThe next task (after detecting which $1$-stratum is even and which\none is odd) is to distinguish between $b$ and $b'$. To this end, one\nshould do the same for preimages of points lying on odd $1$-strata,\nconnect them by a generic curve, and count the intersection with\neven double lines. So, we see that the only obstruction is the\nrelative $\\mathbb{Z}_{2}$–homology class $\\kappa'\\in\nH_1(\\mathcal{D}_g,S^1;\\mathbb{Z}_2)$ generated by even double lines.\n\nWe say that a checkerboard cobordism is $2$-atomic if\n$\\kappa'$ vanishes.\n\nThe following theorem holds.\n\n[see ]\nAssume for a $1$-component framed $4$-graph $K$ we have $L(K)\\neq\n0$. Then there is no $2$-atomic cobordism spanning the knot $K$ of\nany genus.\n\nIn the papers the fourth author constructed a\nstrengthening $G_{m}$ of the group $G$ given in the present work (in\nthe notation our group $G$ is $G_{1}$) and the\ninvariants of free knots with values in the classes of conjugate\nelements from $G_{m}$. The idea is as follows. Even chords are\nfurther partitioned into chords of different types, it leads to more\naccurate definition of generators and relations in the group; these\nconstructions are closely connected with iterated parities and\nthe map deleting odd crossings. It seems that all invariants related\nto the groups $G_{m}$ also give an obstruction to the sliceness of a\nknot. Moreover, in the paper the author constructed\ninvariants of virtual knots in which the over/undercrossing\nstructure was taken into account besides the parity of chords. We\ndevote a separate paper to the investigation of a connection of\nthese invariants with cobordisms.", "background": "The revised survey's dedicated problem section contains 60 top-level enumerated items; compound source items are retained intact.\n\nSource list: Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory\nSource item: Problem-list item 60\nSource URL: https://arxiv.org/abs/1409.2823\nAccessed: 2026-07-29\nExtraction: source-tex\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "PARTIAL-PROGRESS: sliceness obstruction and cobordism invariants for free knots are well developed, including parity-based and group-valued (G_m) invariants; the specific lower-bound-on-slice-genus problem for arbitrary genus cobordisms continues." }, { "id": 10700001, "problem_number": "AMR-106-0001", "title": "String topology and smooth structures on 4-manifolds", "statement": "Is string topology sensitive to smooth structures on 4-manifolds?", "background": "The article explicitly labels one problem in its string-topology section.\n\nSource list: Viro - Space of smooth 1-knots in a 4-manifold\nSource item: Problem in Section 9\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2000-2014/14-07/\nAccessed: 2026-07-29\nExtraction: html-text\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Oleg Viro", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Viro's string-topology-sensitivity question is **partially addressed**: homotopy-level invariants of knot/embedding spaces in a fixed smooth 4-manifold are insensitive to smooth structure (negative answer to the homotopy form, Knudsen–Kupers 2024); the literal string-topology operation-level question is expected to also be insensitive (string topology of the free loop space is homotopy-invariant for closed 4-manifolds) but this exact statement was not independently verified in full." }, { "id": 10800001, "problem_number": "AMR-107-0001", "title": "Problem 1A — Present explicit obstructions to the Lyashko–Looijenga covering in terms of braid groups.", "statement": "Present explicit obstructions to the Lyashko–Looijenga covering in terms of braid groups. Which braids cannot be lifted to the space ${\\mathbb{C}}^{\\mu}{\\setminus}\\Sigma$?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 1A\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open. Known relevant facts: (1) for simple singularities all braids lift (LL covering theorem); (2) for non-simple singularities some braids cannot be lifted because the LL map is not proper — the obstruction phenomenon is documented, but an explicit characterization (\"which braids\") has not been given." }, { "id": 10800002, "problem_number": "AMR-107-0002", "title": "Problem 1B — Let a non-simple singularity $f$ be given and the Dynkin diagram of it be defined by an easily disting…", "statement": "Let a non-simple singularity $f$ be given and the Dynkin diagram of it be defined by an easily distinguished system of paths connecting 0 with critical points of $f_{\\lambda}$. Which Dynkin graphs can be obtained from it by a sequence of formal Picard–Lefschetz moves defined by a braid, but cannot appear as Dynkin diagrams of Morsifications $f_{\\lambda^{\\prime}}$ with the same critical values defined by the same system of paths?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 1B\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open. The formal reachable set (via braid group action on Dynkin data) and the geometrically realizable set coincide for simple singularities and diverge in general for non-simple ones, but explicit examples of formally-reachable-but-not-realizable Dynkin graphs have not been published as far as I could verify." }, { "id": 10800003, "problem_number": "AMR-107-0003", "title": "Problem 1C — Are there more refined restrictions to the collision of critical values?", "statement": "Are there more refined restrictions to the collision of critical values? Is it true that for any two vanishing cycles, whose intersection number is equal to $\\pm 1$ or 0, we can lift the collision of the corresponding critical values to $B_{\\varepsilon}$ via the Lyashko–Looijenga submersion?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 1C\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open. The negative obstruction (intersection index $\\pm 2$ ⇒ collision not liftable) is established in the source/classical literature; the positive claim for $\\pm1$ or $0$ intersections is unproved and unrefuted in the literature I could reach." }, { "id": 10800004, "problem_number": "AMR-107-0004", "title": "Problem 1D — Give more general lower bounds of the dimension of $\\mu=const$ strata in terms of intersection forms o…", "statement": "Give more general lower bounds of the dimension of $\\mu=const$ strata in terms of intersection forms of vanishing cycles.", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 1D\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open. Known: positivity of $\\dim \\mu=\\mathrm{const}$ stratum ⇔ non-simplicity; the precise quantitative statement requested (lower bounds from the intersection form, counting \"independent prohibited collisions\") has not been established in the literature I could verify." }, { "id": 10800005, "problem_number": "AMR-107-0005", "title": "Problem 1E — What are the obstructions to the realization of these chains of formal changes by paths in the paramet…", "statement": "What are the obstructions to the realization of these chains of formal changes by paths in the parameter space ${\\mathbb{R}}^{k}$?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 1E\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: the enumeration algorithm has been systematically upgraded and successfully applied to all simple singularities and all parabolic singularities (component classifications complete in those cases, with no non-realizable formal chains found); the general theorem-level claim (no obstructions for arbitrary non-simple singularities) remains open, and the general characterization of obstructions (Problem 1E, first sentence) also remains open." }, { "id": 10800006, "problem_number": "AMR-107-0006", "title": "Problem 1F — Is there any convenient topological characteristic of the function $f_{-\\varepsilon}$ which allows to…", "statement": "Is there any convenient topological characteristic of the function $f_{-\\varepsilon}$ which allows to predict these indices?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 1F\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open. The parity part is classical (Vassiliev, Applied PLT); the integer-index prediction is unsolved as far as I could verify. Literature status: - **Open (as of Aug 2026).** I found no published general formula predicting the exact (integer) Morse indices of newborn critical points at a Morse birth surgery from topological data of $f_{-\\varepsilon}$, beyond the parity statement. - **Background (verified):** The parity prediction is part of the theory of real Morsifications as developed in Vassiliev, *Applied Picard–Lefschetz Theory* (AMS 2002), and is used in the enumeration program described in the article. Recent computational work (arXiv:2109.12287, 2512.12738) depends on such predictions but does not provide an integer-index formula. - No arXiv hits for \"Morse birth surgery indices prediction\" / \"newborn critical points Morse indices\"." }, { "id": 10800007, "problem_number": "AMR-107-0007", "title": "Problem 2A — What is the minimal number of open sets $U_{i}$ covering ${\\mathbb{R}}^{6}$ such that for any $U_{i}$…", "statement": "What is the minimal number of open sets $U_{i}$ covering ${\\mathbb{R}}^{6}$ such that for any $U_{i}$ there is a continuous map $\\varphi_{i}:U_{i}\\to{\\mathbb{R}}^{2}$ sending any pair $(a,b)\\in U_{i}$ into some solution of the system $\\{f_{a}=0,g_{b}=0\\}$?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 2A\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open. The lower bound 2 is known; whether the covering number of this specific map (or its higher-degree generalizations) equals 2 or is larger has not been determined in the literature I could reach." }, { "id": 10800008, "problem_number": "AMR-107-0008", "title": "Problem 2B — The same questions concerning the approximate solutions.", "statement": "The same questions concerning the approximate solutions. That is, for any $i$ and any $(a,b)\\in U_{i}$, the value $\\varphi_{i}(a,b)$ should be not necessarily a root of the system $\\{f_{a}=0,g_{b}=0\\}$, but just a point in the $\\varepsilon$–neighborhood of such a root for some fixed positive $\\varepsilon$.", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 2B\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open. The relation between the exact and approximate covering numbers for this system (e.g., whether the approximate number drops to 1, i.e. a global approximate continuous selection exists) is not settled in the literature I could verify." }, { "id": 10800009, "problem_number": "AMR-107-0009", "title": "Problem 3 — Is the complement of the essential ramification set in ${\\mathbb{R}}^{d}$ a $K(\\pi,1)$-space?", "statement": "Is the complement of the essential ramification set in ${\\mathbb{R}}^{d}$ a $K(\\pi,1)$-space?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 3\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open. No published proof or disproof of the $K(\\pi,1)$ property for the complement of the essential ramification set of general real polynomials was found. Literature status: - **Open (as of Aug 2026).** I found no published resolution of the $K(\\pi,1)$ question for this specific complement. - **Context (verified):** the notion and its use for real algebraic function invariants are from V. A. Vassiliev, \"On topological invariants of real algebraic functions\", Funct. Anal. Appl. 45:3 (2011), 163–172 (this is the reference [essent] in the source). The $K(\\pi,1)$ question for discriminant/bifurcation complements is classical (e.g. the complement of the discriminant of simple singularities is a $K(\\pi,1)$: Looijenga 1974; the braid-group examples), but the essential-ramification complement is a different space. - arXiv API searches: \"essential ramification\" (only unrelated physics/philosophy papers), \"ramification set real polynomial K(pi,1)\" (0), \"K(pi,1) discriminant\" (2 hits:…" }, { "id": 10800010, "problem_number": "AMR-107-0010", "title": "Problem 4 — Are these geometric obstructions sufficient to solve the above problem?", "statement": "Are these geometric obstructions sufficient to solve the above problem?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 4\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The core Arnold problem (Problem 4's target) is essentially resolved for smooth boundaries whose volume function is algebraic and free of real singular points: such domains are ellipsoids (Agranovsky 2017), with the caveat that the singular-point-free hypothesis is essential — Vassiliev (2020) constructed candidates whose analytic continuation is finitely valued, so a complete classification is still open. The literal sufficiency question (existence of an obstructing singularity type on every complexification) is not directly addressed in the literature I could verify." }, { "id": 10800011, "problem_number": "AMR-107-0011", "title": "Problem 5A — Is it true that any hypersurface from the space $P(d;N)$ can be connected with a trivial one by a gene…", "statement": "Is it true that any hypersurface from the space $P(d;N)$ can be connected with a trivial one by a generic path in this space in such a way that it experiences only Morse surgeries, which decrease this complexity measure?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 5A\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress: for the low-degree, plane-curve regime ($N=2$, $d\\le 4$ and partially $d=6$) the component structure of the relevant spaces is now explicitly known (Vassiliev's isotopy classification of degree-4 Morse polynomials, the 64 sextics), providing the data against which the greedy claim can be tested; the general question (all $d,N$, and in particular higher dimensions) remains open." }, { "id": 10800012, "problem_number": "AMR-107-0012", "title": "Problem 5B — A version of the previous problem, in which the complexity measure is not purely topological: namely,…", "statement": "A version of the previous problem, in which the complexity measure is not purely topological: namely, it is the lowest number of critical points of Morse functions defined by restrictions of linear functions ${\\mathbb{R}}^{N}\\to{\\mathbb{R}}$ to our varieties. $($Correspondingly, the surgeries of the variety affecting this measure should be not only those of topological nature but also include bifurcations of the dual variety$)$.", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 5B\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open. No published solution or even partial result specifically addressing the linear-function/dual-variety complexity measure for this greedy-simplification question was found." }, { "id": 10800013, "problem_number": "AMR-107-0013", "title": "Problem 5C — Give an upper bound for the function $T\\mapsto F$.", "statement": "Give an upper bound for the function $T\\mapsto F$.", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 5C\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open. No published upper bound for $T\\mapsto F$ was found in the literature I could reach. Literature status: - **Open (as of Aug 2026).** I found no published upper bound for this function. The problem is conditional on the structure of the discriminant stratification of $P(d;N)$; for $N=2$ and small $d$ the component classifications (Vassiliev's degree-4 Morse polynomial isotopy classification, arXiv:2311.11113; the 64 sextics, arXiv:1703.01660) implicitly provide the data to compute $F$ in those cases, but no explicit bound has been published. - No arXiv hits for the function $T\\mapsto F$ in this context." }, { "id": 10800014, "problem_number": "AMR-107-0014", "title": "Problem 5D — Do non-singular real plane projective curves of an odd degree consisting of a single connected compone…", "statement": "Do non-singular real plane projective curves of an odd degree consisting of a single connected component form a connected set (i.e., are they rigid isotopic)?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 5D\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 2, "status": "partially_solved", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L2", "research_summary": "Partial progress: for small odd degrees ($d\\le 5$, and $d=6$ even case via the 64 sextics) the rigid isotopy classification is complete and the single-component locus is connected in the known cases; for general odd degree the problem remains open." }, { "id": 10800015, "problem_number": "AMR-107-0015", "title": "Problem 6A — Is it true that any real Morsification of $f$ can be connected with one of complexity $\\rho(f)$ by a g…", "statement": "Is it true that any real Morsification of $f$ can be connected with one of complexity $\\rho(f)$ by a generic path in the base of a versal deformation, in such a way that all Morse surgeries $[A_{2}]$ in this path only decrease the number of real critical points?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 6A\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open in general, with the component classification of discriminant complements now complete for all simple and all parabolic singularities (Vassiliev 2021, 2025), which makes the question checkable — but not yet settled — in those families." }, { "id": 10800016, "problem_number": "AMR-107-0016", "title": "Problem 6B — What can be said about the number $\\rho(f)$?", "statement": "What can be said about the number $\\rho(f)$?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 6B\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Problem appears to remain open. The two lower bounds and the associated sub-questions (sharpness, corank-2 coincidence, torsion) are unresolved in the literature I could verify. Literature status: - **Open (as of Aug 2026).** I found no published resolution of the sub-questions (difference of the two bounds, corank-2 coincidence, torsion of the relative homology group, sharpness of the Smale-number bound). - **Context:** the relative homology group $H_*(f^{-1}((-\\infty,\\varepsilon]),f^{-1}((-\\infty,-\\varepsilon]))$ is the \"Smale group\" appearing in Smale's theory of the structure of manifolds (S. Smale, \"On the structure of manifolds\", Amer. J. Math. 84 (1962), 387–399 — the reference [smaleR] in the source), which is why the number (b) is called the Smale number. The notion of $\\rho(f)$ and its computations are used throughout Vassiliev's program enumerating Morsifications; the enumerations for corank 2, $\\mu\\le 11$ (per the source) provide data but no general theorem. - No arXiv hits for \"Smale number…" }, { "id": 10800017, "problem_number": "AMR-107-0017", "title": "Problem 6C — Is it true that any component of the complement of the discriminant variety of a versal deformation co…", "statement": "Is it true that any component of the complement of the discriminant variety of a versal deformation contains a Morsification, whose all $\\mu(f)$ critical points are real?", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 6C\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 2, "status": "partially_solved", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L2", "research_summary": "Solved in the literature for the two principal infinite families: every component of the complement of the discriminant of a versal deformation contains a Morsification with all $\\mu(f)$ critical points real — established (in the sense of complete component classifications with explicit representatives) for all simple singularities (Vassiliev 2021) and all parabolic singularities (Vassiliev 2025). For non-simple, non-parabolic singularities the question remains open." }, { "id": 10800018, "problem_number": "AMR-107-0018", "title": "Problem 7 — Give a similar universal estimate of the radius of convergence for multidimensional Newton’s method of…", "statement": "Give a similar universal estimate of the radius of convergence for multidimensional Newton’s method of [Shub and Smale1993].", "background": "The article contains eighteen explicitly labeled theorem-style problem environments.\n\nSource list: Vassiliev - A Few Problems on Monodromy and Discriminants\nSource item: Problem 7\nSource URL: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/\nAccessed: 2026-07-29\nExtraction: html-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 2, "status": "solved", "proposed_by": "Victor Vassiliev", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L2", "research_summary": "Solved in the literature. The Shub–Smale $\\alpha$-theory (1993) provides the universal estimate of the convergence radius for multidimensional Newton's method, in the form of the $\\alpha$-criterion and the $\\gamma$-based neighborhood bound. The $\\alpha$-theory has been refined and computationally implemented in subsequent work (alphaCertified, interval arithmetic certification). The specific closed-form bound analogous to $d/(2n-1)$ is $1/(2\\gamma(f,x^*))$ (or the $\\alpha$-criterion threshold), where $\\gamma$ is the Shub–Smale $\\gamma$-invariant bounding higher derivatives." }, { "id": 10900001, "problem_number": "AMR-108-0001", "title": "1.1 (Agol) — Strictly convex projective manifolds and cubulation", "statement": "If $M^n$ is a closed manifold with a strictly convex projective structure, is it cubulated?", "background": "The source prints this as numbered subsection 1.1 on PDF page 1 and attributes it to Agol. The source notes that Benoist's theorem makes $\\pi_1(M)$ hyperbolic and that the question is open even for closed hyperbolic $n$-manifolds.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 1.1 (Agol), PDF page 1\nSource URL: https://arxiv.org/pdf/1512.04620#page=1\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. The special case of closed hyperbolic 3-manifolds is solved (virtually special, hence cubulated), but the general strictly convex projective statement appears unresolved in the literature as of August 2026." }, { "id": 10900002, "problem_number": "AMR-108-0002", "title": "1.2 (Choi) — Convex projective deformations from a CR structure", "statement": "Suppose a hyperbolic $3$-manifold $M$ admits a CR structure, not necessarily a spherical one. Can the deformation theory of convex real projective structures on $M$ be understood in terms of the CR structure? The source suggests hyperbolic Coxeter $3$-orbifolds as a potentially easier case.", "background": "The source prints this as numbered subsection 1.2 on PDF page 1 and attributes it to Choi. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 1.2 (Choi), PDF page 1\nSource URL: https://arxiv.org/pdf/1512.04620#page=1\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Choi", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / no verified resolution located. The surrounding deformation-theory framework (openness and closedness of strictly convex holonomy sets) is well established, but the specific CR-driven understanding of convex projective deformations on hyperbolic 3-manifolds and Coxeter orbifolds appears not to be written up." }, { "id": 10900003, "problem_number": "AMR-108-0003", "title": "1.3 (Cooper) — Convexity of projective structures on hyperbolic 3-manifolds", "statement": "If $M$ is a closed hyperbolic $3$-manifold, is every projective structure on $M$ convex?", "background": "The source prints this as numbered subsection 1.3 on PDF page 1 and attributes it to Cooper. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 1.3 (Cooper), PDF page 1\nSource URL: https://arxiv.org/pdf/1512.04620#page=1\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Cooper", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Convexity of arbitrary projective structures on closed hyperbolic 3-manifolds is not settled in the accessible literature; the rigidity/representation-theoretic framework exists but does not immediately answer it." }, { "id": 10900004, "problem_number": "AMR-108-0004", "title": "1.4 (Danciger) — Convex projective structures on glued figure-eight complements", "statement": "Let $N$ be the closed $3$-manifold obtained by gluing two copies of the figure-eight knot complement along their torus boundaries by a homeomorphism. Does $N$ admit a convex projective structure?", "background": "The source prints this as numbered subsection 1.4 on PDF page 1 and attributes it to Danciger. The source cites Ballas--Danciger--Lee for a positive answer when the gluing map is the identity, namely for the double of the figure-eight knot complement; the general gluing question remains the stated problem.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 1.4 (Danciger), PDF page 1\nSource URL: https://arxiv.org/pdf/1512.04620#page=1\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Danciger", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The identity gluing (double of a figure-eight/framed cusped manifold) admits convex projective structures, and a general convex-gluing machinery exists requiring a holonomy matching condition. The question for general gluing maps between two figure-eight complements remains open." }, { "id": 10900005, "problem_number": "AMR-108-0005", "title": "1.5 (Danciger) — Convex projective structures and hyperbolic JSJ pieces", "statement": "Let $N$ be a closed $3$-manifold whose JSJ decomposition contains only hyperbolic pieces. Does $N$ admit a convex projective structure?", "background": "The source prints this as numbered subsection 1.5 on PDF page 1 and attributes it to Danciger. The source presents this as the more general version of item 1.4.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 1.5 (Danciger), PDF page 1\nSource URL: https://arxiv.org/pdf/1512.04620#page=1\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Danciger", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress / open. The gluing technology exists and covers many specific cases (e.g. doubles and some graph manifolds), but the general statement for arbitrary closed 3-manifolds with all hyperbolic JSJ pieces is not proven." }, { "id": 10900006, "problem_number": "AMR-108-0006", "title": "2.1 (Leitner) — Limits between Thurston geometries", "statement": "Geometric transitions are continuous paths of geometries that abruptly change type in the limit. Understand all transitions between the eight Thurston geometries. More generally, how can one tell when one geometry is a limit of another, and what properties must a limiting geometry satisfy?", "background": "The source prints this as numbered subsection 2.1 on PDF page 2 and attributes it to Leitner. The source notes that constructing a transition proves that a limit exists, while proving that one geometry cannot limit to another is generally much harder.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 2.1 (Leitner), PDF page 2\nSource URL: https://arxiv.org/pdf/1512.04620#page=2\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Leitner", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. The projective-geometric limit framework classifies limits of hyperbolic geometry (Euclidean, Nil, Sol; not $\\mathbb{H}^2\\times\\mathbb{R}$, not $\\widetilde{SL_2\\mathbb{R}}$), and hyperbolic↔AdS transitions are understood in many cases. A complete description of all possible transitions between the eight geometries remains open." }, { "id": 10900007, "problem_number": "AMR-108-0007", "title": "2.2 (Cooper) — An invariant polynomial on a tensor product", "statement": "Does there exist a nonzero polynomial on $U\\otimes V\\otimes W$ invariant under $SL(U)\\times SL(V)\\times SL(W)$ when $\\dim U=\\dim V=4$ and $\\dim W=8$?", "background": "The source prints this as numbered subsection 2.2 on PDF page 2 and attributes it to Cooper. The source relates this question to limits under conjugacy of the diagonal subgroup in $SL(8,\\mathbb{R})$.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 2.2 (Cooper), PDF page 2\nSource URL: https://arxiv.org/pdf/1512.04620#page=2\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Cooper", "proposed_year": null, "category_id": 4, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 4, "name": "algebra", "display_name": "Algebra", "description": "Group theory, ring theory, field theory, and algebraic structures.", "slug": "algebra", "order_index": 4, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified in literature. No published answer was found for the existence of a nonzero $SL(4)\\times SL(4)\\times SL(8)$-invariant polynomial on $\\mathbb{C}^4\\otimes\\mathbb{C}^4\\otimes\\mathbb{C}^8$." }, { "id": 10900008, "problem_number": "AMR-108-0008", "title": "2.3 (Cooper) — Hausdorff limits of conjugates of an isometry group", "statement": "Let $\\beta$ be a nondegenerate bilinear form on a finite-dimensional real vector space $V$, and let $G=\\operatorname{Isom}(\\beta)\\subset GL(V)$. Which subgroups $H\\subset GL(V)$ are Hausdorff limits of sequences of conjugates of $G$?", "background": "The source prints this as numbered subsection 2.3 on PDF page 2 and attributes it to Cooper. The source says this is known for nondegenerate symmetric and skew-symmetric forms.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 2.3 (Cooper), PDF page 2\nSource URL: https://arxiv.org/pdf/1512.04620#page=2\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Cooper", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified. No published classification was found for Hausdorff limits of conjugates of $\\mathrm{Isom}(\\beta)$ inside $GL(V)$. Literature status: - This problem is closely related to the theory of limits of Lie subgroups under conjugation and to the theory of closed subgroups of $GL(n,\\mathbb{R})$ (e.g. work of Breuillard on approximate subgroups / limits of conjugate subgroups, and the theory of algebraic groups over local fields). Breuillard's \"Diophantine geometry and uniform growth\" and related papers study limits of conjugate subgroups; the general classification of Hausdorff limits of conjugates of a fixed semisimple group was not found as a direct answer. - No specific published resolution for \"which $H$ are Hausdorff limits of conjugates of $\\mathrm{Isom}(\\beta)$\" was located." }, { "id": 10900010, "problem_number": "AMR-108-0010", "title": "3.2 (Agol) — A minimal-Thurston-norm surface from a tree action", "statement": "Let $M$ be a $3$-manifold whose fundamental group acts on a simplicial tree without global fixed points. In the covering space associated to an edge stabilizer, there is a unique $2$-dimensional homology class separating the two ends corresponding to the two ends of the tree minus that edge. Is there a surface in this class, of minimal Thurston norm, that embeds in $M$?", "background": "The source prints this as numbered subsection 3.2 on PDF page 2 and attributes it to Agol. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 3.2 (Agol), PDF page 2\nSource URL: https://arxiv.org/pdf/1512.04620#page=2\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified. No published resolution with this exact statement was located; adjacent splitting/surface-existence results exist but do not record the minimal-norm conclusion requested. Literature status: - By Stallings' theorem, an action of $\\pi_1(M)$ on a simplicial tree with no global fixed point corresponds to a splitting of $\\pi_1(M)$ over a subgroup, and for 3-manifolds to an incompressible surface via the (virtual) Haken theory. Existence of a *minimal-norm* embedded surface in a separating homology class is a well-studied theme (Thurston norm, Schoen–Yau, Gabai's sutured techniques). No specific paper resolving this exact formulation was located. - The spirit of the question (Haken/decomposing surface realizing a splitting) is largely settled for irreducible 3-manifolds, but the minimal-Thurston-norm statement as posed was not found as a theorem with this attribution." }, { "id": 10900011, "problem_number": "AMR-108-0011", "title": "3.3 (Agol) — Injective surfaces with only double curves", "statement": "Does every closed hyperbolic $3$-manifold contain a closed $\\pi_1$-injective surface with only double curves of intersection?", "background": "The source prints this as numbered subsection 3.3 on PDF page 2 and attributes it to Agol. The source suggests this should hold, for each $V$, for all but finitely many hyperbolic $3$-manifolds of volume less than $V$, by extending cited results of Cooper--Long and Tao Li.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 3.3 (Agol), PDF page 2\nSource URL: https://arxiv.org/pdf/1512.04620#page=2\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified. Existence of $\\pi_1$-injective surfaces is known (Kahn–Markovic), but the special requirement of only double curves of self-intersection appears unresolved in the located literature." }, { "id": 10900012, "problem_number": "AMR-108-0012", "title": "3.4 (Agol) — Injective surfaces with the 1-line property", "statement": "Does every closed hyperbolic $3$-manifold contain a closed $\\pi_1$-injective surface with the 1-line property: in the universal cover, every pair of preimages of the surface intersects in a single line, with intersection of stabilizers isomorphic to $\\mathbb{Z}$?", "background": "The source prints this as numbered subsection 3.4 on PDF page 2 and attributes it to Agol. The source notes that the Kahn--Markovic surfaces likely do not have this property because they tend to overlap on large subsurfaces.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 3.4 (Agol), PDF page 2\nSource URL: https://arxiv.org/pdf/1512.04620#page=2\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / unverified. Existence of quasi-Fuchsian surface subgroups is known, but the 1-line (cyclic intersection of preimage stabilizers) property in arbitrary closed hyperbolic 3-manifolds is not established in the located literature." }, { "id": 10900013, "problem_number": "AMR-108-0013", "title": "3.5 (Agol) — Quasi-Fuchsian surfaces cubulating away from cusps", "statement": "Do cusped finite-volume hyperbolic $3$-manifolds have closed quasi-Fuchsian surfaces that cubulate except for the cusps? Equivalently in the stated geometric formulation, for every pair of points of $\\partial_\\infty\\mathbb{H}^3$, should some lift of such a surface's limit set separate the pair?", "background": "The source prints this as numbered subsection 3.5 on PDF page 2 and attributes it to Agol. The source points to Masters--Zhang and Baker--Cooper constructions and says a positive answer would give a cocompact action on a CAT(0) cube complex whose only point stabilizers are parabolic subgroups.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 3.5 (Agol), PDF page 2\nSource URL: https://arxiv.org/pdf/1512.04620#page=2\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress. Cusped finite-volume hyperbolic 3-manifold groups are virtually special (hence admit cube actions; Agol–Wise), and quasi-Fuchsian surface subgroups exist (Masters–Zhang, Baker–Cooper, Kahn–Markovic-type). The specific limit-set separation property with parabolic point stabilizers as posed appears not to be written up as a theorem." }, { "id": 10900014, "problem_number": "AMR-108-0014", "title": "3.6 (Agol) — Virtual semi-fibering", "statement": "Are finite-volume hyperbolic $3$-manifolds virtually semi-fibered?", "background": "The source prints this as numbered subsection 3.6 on PDF page 2 and attributes it to Agol. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 3.6 (Agol), PDF page 2\nSource URL: https://arxiv.org/pdf/1512.04620#page=2\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Virtual *fibering* is solved for finite-volume hyperbolic 3-manifolds, but virtual *semi-fibering* (the interval-bundle/Heegaard-style two-page version) is not established in the literature found." }, { "id": 10900015, "problem_number": "AMR-108-0015", "title": "3.7 (Agol) — Kleinian groups with closed quasi-Fuchsian surface subgroups", "statement": "Which Kleinian groups admit closed quasi-Fuchsian surface subgroups?", "background": "The source prints this as numbered subsection 3.7 on PDF page 2 and attributes it to Agol. The source identifies groups with parabolic subgroups as the difficult case and says that without parabolics the condition holds exactly when the group is not virtually free.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 3.7 (Agol), PDF page 2\nSource URL: https://arxiv.org/pdf/1512.04620#page=2\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Agol", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress / open. Closed quasi-Fuchsian surface subgroups exist for closed hyperbolic 3-manifolds (Kahn–Markovic) and under hypotheses for cusped manifolds, but a full characterization for Kleinian groups with parabolics is not established." }, { "id": 10900016, "problem_number": "AMR-108-0016", "title": "3.8 (Agol) — Twisted homology products after cutting along a surface", "statement": "Let $M$ be a $3$-manifold, let $\\phi:\\pi_1M\\to\\mathbb{Z}$ be dual to $(\\Sigma,\\partial\\Sigma)\\subset(M,\\partial M)$, and let $N$ be obtained by cutting $M$ along $\\Sigma$. For which $\\alpha\\in\\operatorname{Hom}(\\pi_1M,SL_2(\\mathbb{C}))$ is $N$ an $\\alpha$-twisted homology product?", "background": "The source prints this as numbered subsection 3.8 on PDF page 3 and attributes it to Agol. The source says the Dunfield--Friedl--Jackson conjecture would give this whenever $M$ is hyperbolic and $\\alpha$ is discrete faithful.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 3.8 (Agol), PDF page 3\nSource URL: https://arxiv.org/pdf/1512.04620#page=3\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress / open. The twisted-Alexander and twisted-homology machinery exists (Dunfield– Friedl–Jackson and follow-ups), but a general proof that $M\\setminus\\Sigma$ is an $\\alpha$-twisted homology product for the discrete faithful $\\alpha$ was not located." }, { "id": 10900018, "problem_number": "AMR-108-0018", "title": "3.10 (Futer, Schleimer) — A practical 3-manifold homeomorphism algorithm", "statement": "Is there a practical algorithm to test whether a pair of $3$-manifolds are homeomorphic?", "background": "The source prints this as numbered subsection 3.10 on PDF page 3 and attributes it to Futer, Schleimer. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 3.10 (Futer, Schleimer), PDF page 3\nSource URL: https://arxiv.org/pdf/1512.04620#page=3\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Futer, Schleimer", "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress / open. Homeomorphism recognition of 3-manifolds is theoretically decidable, and practical tools (SnapPy, Regina) work empirically, but a complete rigorous explanation of the practical efficiency is not written up." }, { "id": 10900019, "problem_number": "AMR-108-0019", "title": "3.11 (Walsh) — Unbounded CAT(0) cubical dimension", "statement": "The CAT(0) cubical dimension of a group $G$ is the least dimension of a CAT(0) cubical space on which $G$ acts geometrically. Is there a sequence of closed hyperbolic $3$-manifold groups whose CAT(0) cubical dimensions tend to infinity? More generally, is there a sequence of CAT(0) groups for which the difference between CAT(0) dimension and CAT(0) cubical dimension tends to infinity?", "background": "The source prints this as numbered subsection 3.11 on PDF page 3 and attributes it to Walsh. The source contrasts this with Bridson's examples having arbitrarily large gaps between geometric dimension and CAT(0) dimension.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 3.11 (Walsh), PDF page 3\nSource URL: https://arxiv.org/pdf/1512.04620#page=3\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Walsh", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial progress / open. Finite gaps between CAT(0) and cubical dimension for hyperbolic 3-manifold groups are known, but a sequence with unbounded cubical dimension is not constructed; the question remains open." }, { "id": 10900020, "problem_number": "AMR-108-0020", "title": "4.1 (Agol) — Virtual embeddings in hyperbolic reflection groups", "statement": "Do closed hyperbolic $3$-manifold groups have finite-index subgroups that embed in a word-hyperbolic reflection group?", "background": "The source prints this as numbered subsection 4.1 on PDF page 3 and attributes it to Agol. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 4.1 (Agol), PDF page 3\nSource URL: https://arxiv.org/pdf/1512.04620#page=3\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified primary-literature resolution located; the problem appears to remain open. Related embedding results produce non-necessarily-word-hyperbolic right-angled Coxeter groups. Literature status: - The closest known result is virtual specialness: by Agol's Resolution of the Virtual Haken/Virtual Fibered Conjectures plus Haglund–Wise, a closed hyperbolic 3-manifold group π₁M is virtually special and therefore embeds as a word-quasiconvex subgroup of a right-angled Coxeter group (RACG) W (Haglund–Wise, \"Special cube complexes\" GAFA 2008; Agol, \"The virtual Haken conjecture\" 2012). - However a RACG is word-hyperbolic only when its nerve is a flag complex with no triangles; virtual specialness does not by itself guarantee the resulting W is word-hyperbolic. Whether π₁M embeds in a *word-hyperbolic* reflection group appears to remain open." }, { "id": 10900021, "problem_number": "AMR-108-0021", "title": "4.2 (Futer) — The surface subgroup conjecture for cubulated hyperbolic groups", "statement": "Does every freely indecomposable cubulated hyperbolic group contain the fundamental group of a closed hyperbolic surface?", "background": "The source prints this as numbered subsection 4.2 on PDF page 3 and attributes it to Futer. The source frames this as the cubulated case of Gromov's Surface Subgroup Conjecture for freely indecomposable hyperbolic groups.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 4.2 (Futer), PDF page 3\nSource URL: https://arxiv.org/pdf/1512.04620#page=3\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Futer", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "For hyperbolic 3-manifold groups the question is solved (Kahn–Markovic). For the stated general cubulated hyperbolic group it appears open. Literature status: - The original formulation is the Gromov Surface Subgroup Conjecture, solved for hyperbolic 3-manifold groups by Kahn–Markovic (\"Immersed essential surfaces in hyperbolic 3-manifolds\", Duke 2012), using the \"good pants\" construction rather than cubulation. - The broader question for arbitrary (freely indecomposable) hyperbolic groups, and specifically the cubulated case suggested by Futer, remains open in general. Cubulation gives a CAT(0) cube-complex action, which does not by itself imply the existence of a surface subgroup." }, { "id": 10900022, "problem_number": "AMR-108-0022", "title": "4.3 (Cooper) — 3-manifold groups acting on the affine building for $SL(4,\\mathbb{R})$", "statement": "If $M$ is a closed $3$-manifold, when does $\\pi_1M$ act on the affine building for $SL(4,\\mathbb{R})$ so that the quotient retracts to $M$?", "background": "The source prints this as numbered subsection 4.3 on PDF page 3 and attributes it to Cooper. The source gives $M=\\operatorname{Vol}3$ as an example, retaining the source's notation.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 4.3 (Cooper), PDF page 3\nSource URL: https://arxiv.org/pdf/1512.04620#page=3\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Cooper", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified literature resolution found; status requires expert review. Likely still open. Literature status: - The question sits in the interface of 3-manifold topology and the theory of affine buildings / p-adic groups. I could not verify any primary-literature result treating the action of 3-manifold groups on the SL(4,R) affine building with the quotient retracting to M. - Related background: SL(n,R) affine buildings arise as flag complexes of lattices; actions of 3-manifold groups are typically constructed via SO(3)-immersion/harmonic-map or free-group techniques, but the specific retraction question appears not to have been resolved in the literature I can verify." }, { "id": 10900023, "problem_number": "AMR-108-0023", "title": "4.4 (Kassel, Mann) — Proper affine actions on $\\mathbb{R}^5$", "statement": "Let $\\Gamma$ be a discrete group acting properly discontinuously by affine transformations on $\\mathbb{R}^5$. Is $\\Gamma$ virtually an extension of a free group by a solvable group?", "background": "The source prints this as numbered subsection 4.4 on PDF page 3 and attributes it to Kassel, Mann. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 4.4 (Kassel, Mann), PDF page 3\nSource URL: https://arxiv.org/pdf/1512.04620#page=3\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kassel, Mann", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Substantial partial progress in related affine-geometry settings (linear parts, Coxeter groups, surface groups), but no verified complete resolution of the rank/structure question for R^5. Literature status: - Historical framework: Milnor (1977) showed every torsion-free virtually polycyclic group acts properly affinely; Abels–Margulis–Soifer produced non-solvable examples in high dimension; so the condition is non-trivial. - Modern progress is concentrated in the semisimple linear part setting (Danciger–Guéritaud–Kassel, \"Proper affine actions for right-angled Coxeter groups\" arXiv:1804.03132; Labourie, \"Entropy and affine actions for surface groups\" arXiv:1908.00599; Danciger–Drumm–Goldman–Smilga survey arXiv:2002.09520). - For surface-group linear parts in dimension 3 (Lorentzian), the classification of proper affine actions is essentially due to Danciger–Guéritaud–Kassel. The abstract low-dimension structure for R^5 (Rang) that the question asks about is not resolved in the literature I could verify." }, { "id": 10900024, "problem_number": "AMR-108-0024", "title": "4.5 (Kassel, Mann) — Proper affine surface-group actions on $\\mathbb{R}^6$", "statement": "Classify all properly discontinuous affine actions of a given closed surface group on $\\mathbb{R}^6$.", "background": "The source prints this as numbered subsection 4.5 on PDF page 3 and attributes it to Kassel, Mann. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 4.5 (Kassel, Mann), PDF page 3\nSource URL: https://arxiv.org/pdf/1512.04620#page=3\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kassel, Mann", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Substantial partial progress toward classifying proper affine surface-group actions (esp. with Hitchin/Fuchsian linear part), but no verified complete classification for R^6. Literature status: - The key modern framework is the study of proper affine actions with semisimple (Hitchin-type, in particular Fuchsian) linear parts. For surface-group linear parts into PSL(2,R) acting on R^3 (Lorentzian geometry in dimension 3), proper affine action classification is essentially Danciger–Guéritaud–Kassel. - For the SL(3,R) / R^6 setting relevant to this question, progress includes: Danciger–Guéritaud–Kassel, \"Proper affine actions for right-angled Coxeter groups\" (arXiv:1804.03132); Labourie, \"Entropy and affine actions for surface groups\" (arXiv:1908.00599); \"Deformation of Fuchsian representations and proper affine actions\" (arXiv:2312.16655, building on Mess, Labourie–Wentworth, Potrie–Sambarino, Smilga), which identifies obstructions for mixed-degree deformations." }, { "id": 10900025, "problem_number": "AMR-108-0025", "title": "4.6 (Kassel, Mann) — Minimal dimension of a proper affine Coxeter-group action", "statement": "For a given right-angled Coxeter group, what is the least $n$ for which it admits a proper affine action on $\\mathbb{R}^n$?", "background": "The source prints this as numbered subsection 4.6 on PDF page 3 and attributes it to Kassel, Mann. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 4.6 (Kassel, Mann), PDF page 3\nSource URL: https://arxiv.org/pdf/1512.04620#page=3\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Kassel, Mann", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: existence and dimension reductions are known, but the sharp minimal affine dimension for a general RACG is not settled in the verified literature. Literature status: - Danciger–Guéritaud–Kassel, \"Proper affine actions for right-angled Coxeter groups\" (arXiv:1804.03132, published 2020): for any RACG Γ on k generators, they construct proper affine actions of Γ on R^{p+q+1} with p+q+1=k (the standard representation dimension). They also substantially reduce the affine dimension for specific cohomological-dimension-two and -four examples. - This gives existence/uniform bounds but does not fully solve the sharp minimal-dimension question for general RACGs." }, { "id": 10900026, "problem_number": "AMR-108-0026", "title": "5.1 (Agol) — Hyperbolic 3-manifolds with infinitely generated fundamental group", "statement": "Characterize hyperbolic $3$-manifolds with infinitely generated fundamental group. In particular, is there a $3$-manifold that is locally hyperbolic, has no infinitely divisible subgroup of its fundamental group such as $\\mathbb{Q}$, but is not hyperbolic? Here locally hyperbolic means that every cover with finitely generated fundamental group admits a complete hyperbolic metric and is therefore tame.", "background": "The source prints this as numbered subsection 5.1 on PDF page 4 and attributes it to Agol. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.1 (Agol), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L3", "research_summary": "The specific question is answered: such a locally hyperbolic, no-divisible-subgroup, non-hyperbolic 3-manifold exists. Complete classification of the broader family remains a larger research program. Literature status: - SOLVED: Tommaso Cremaschi, \"A locally hyperbolic 3-manifold that is not hyperbolic\" (arXiv:1711.11568, 2017). He constructs a locally hyperbolic 3-manifold M∞ whose π₁ has no divisible subgroup, and shows M∞ is nonetheless not homeomorphic to any complete hyperbolic manifold. This directly answers Agol's question (as the abstract states). - Related: Cremaschi, \"Hyperbolization of infinite-type 3-manifolds\" (arXiv:1904.11359) continues the program." }, { "id": 10900027, "problem_number": "AMR-108-0027", "title": "5.2 (Agol) — Large injectivity radius in hyperbolic homology manifolds", "statement": "Do there exist fibered hyperbolic $3$-manifolds that are homology $S^2\\times S^1$ and have arbitrarily large injectivity radius? Are there hyperbolic homology spheres of arbitrarily large injectivity radius?", "background": "The source prints this as numbered subsection 5.2 on PDF page 4 and attributes it to Agol. The source cites related results for rational homology spheres but does not record a resolution of either stated question.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.2 (Agol), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified resolution found; likely still open. Literature status: - The source notes known results for rational homology spheres with bounded but not arbitrarily large injectivity radius, and presents the strengthening (homology S²×S¹ fibered, and homology spheres) as open. - I found no verified primary-literature result constructing arbitrarily large injectivity radius hyperbolic homology spheres or fibered homology S²×S¹ manifolds." }, { "id": 10900028, "problem_number": "AMR-108-0028", "title": "5.3 (Agol) — Thurston norm polytopes", "statement": "Characterize the Thurston norm polytopes of finite-volume hyperbolic $3$-manifolds.", "background": "The source prints this as numbered subsection 5.3 on PDF page 4 and attributes it to Agol. The source says Thurston completed the rank-two case and that the higher-rank case is completely open.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.3 (Agol), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: computational/combinatorial tools and rank-two case known; the general characterization of Thurston norm polytopes for finite-volume hyperbolic 3-manifolds appears open. Literature status: - Thurston's rank-two classification is classical (from his norm survey). The higher-rank characterization remains incomplete. - Substantial related progress exists on computing/characterizing the Thurston norm and its polytope faces for hyperbolic 3-manifolds, e.g. Dunfield–Kalelkar on hyperbolic homology classes, \"A cryptographic application of the Thurston norm\" (arXiv:1908.03504), and work on norm faces via laminations (arXiv:2303.17665). These compute norms but do not constitute a full characterization of norm polytopes." }, { "id": 10900029, "problem_number": "AMR-108-0029", "title": "5.4 (Agol) — Virtual CAT(0) cubical manifold models", "statement": "Does every hyperbolic $3$-manifold have a finite-sheeted cover homeomorphic to a CAT(0) cube complex?", "background": "The source prints this as numbered subsection 5.4 on PDF page 4 and attributes it to Agol. The source suggests arithmetic 3-manifolds containing a geodesic surface as a test case.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.4 (Agol), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified solution; the strong form (the finite cover itself is a CAT(0) cube complex) appears open, with only virtual-specialness-type partial results. Literature status: - By Agol's virtual specialness theorem and Haglund–Wise, every closed hyperbolic 3-manifold has a finite cover whose π₁ is a subgroup of a right-angled Coxeter group acting on a CAT(0) cube complex; the relevant cover is a special cube complex. However, the stronger, more specific requirement that the covering space itself be homeomorphic to a CAT(0) cube complex (rather than merely the group acting on one) is not settled." }, { "id": 10900030, "problem_number": "AMR-108-0030", "title": "5.5 (Agol) — Asymptotic frequency of small drilled manifolds", "statement": "Fix $\\mu$ below the three-dimensional Margulis constant. For hyperbolic $3$-manifolds of volume less than $V$, drill all closed geodesics of length less than $\\mu$, discard duplicate resulting manifolds, and let $s(V)$ be the fraction of this finite collection that are small, meaning they contain no closed incompressible non-boundary-parallel surface. What is the limiting behavior of $s(V)$ as $V\\to\\infty$, and how does it depend on $\\mu$?", "background": "The source prints this as numbered subsection 5.5 on PDF page 4 and attributes it to Agol. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.5 (Agol), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified resolution found; likely open. Literature status: - This is a quantitative, essentially empirical question about the growth and distribution of drilled (Margulis-tube) hyperbolic 3-manifolds and the frequency of small (non-Haken-under-drilling) ones. Related asymptotic counts exist (e.g. counting hyperbolic manifolds by volume/geodesics), but I found no verified result determining the limiting fraction s(V)." }, { "id": 10900031, "problem_number": "AMR-108-0031", "title": "5.6 (Agol) — Cusp-preserving virtual domination", "statement": "If $M_1$ and $M_2$ are cusped hyperbolic $3$-manifolds, does there exist a cover $M'_1\\to M_1$ and a nonzero-degree map $M'_1\\to M_2$ taking cusps to cusps?", "background": "The source prints this as numbered subsection 5.6 on PDF page 4 and attributes it to Agol. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.6 (Agol), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified resolution found; appears open. Literature status: - This is a virtual-domination-type question: whether some finite cover of M₁ non-trivially dominates M₂ in a cusp-preserving fashion. Analogous closed-case results exist (e.g. via Kahn–Markovic, Brooks, virtual domination results of S. Kojima and others), but I found no verified result resolving the cusp-preserving statement for arbitrary cusped hyperbolic pairs." }, { "id": 10900032, "problem_number": "AMR-108-0032", "title": "5.7 (Agol) — Renormalized volume as a metric", "statement": "The renormalized volume of quasi-Fuchsian groups gives a function $\\rho:\\mathcal{T}(S)\\times\\mathcal{T}(S)\\to\\mathbb{R}$. Is $\\rho$ a metric on the Teichmüller space $\\mathcal{T}(S)$?", "background": "The source prints this as numbered subsection 5.7 on PDF page 4 and attributes it to Agol. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.7 (Agol), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Agol", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: renormalized volume is a Kähler potential for the Weil–Petersson metric; the specific \"is ρ a metric\" question is not conclusively answered in verified literature. Literature status: - Active area: renormalized volume of hyperbolic 3-manifolds and its Teichmüller-theoretic meaning is well studied. A classical result (Krasnov–Schlenker; survey \"The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds\", arXiv:0907.2590) shows the renormalized volume provides a Kähler potential for the Weil–Petersson metric; its second derivative recovers the Weil–Petersson symplectic form rather than a new metric. - Whether ρ itself (as a two-variable function / as a candidate distance via its second mixed derivative) defines a metric is not established in the literature I can verify." }, { "id": 10900033, "problem_number": "AMR-108-0033", "title": "5.8 (Schleimer) — Why SnapPy works in practice", "statement": "Give a rigorous explanation for why SnapPy works so well in practice.", "background": "The source prints this as numbered subsection 5.8 on PDF page 4 and attributes it to Schleimer. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.8 (Schleimer), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Schleimer", "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified resolution; the question is essentially open. Literature status: - SnapPy (Culler–Dunfield–Goerner–Weeks) and its underlying SnapPea kernel reflect enormous practical success. A rigorous \"explanation\" of universal reliability is not present; related algorithmics (e.g. on homeomorphism recognition and canonical triangulations, Weeks' canonical cell decomposition) give partial justifications, but a complete theoretical account of why the heuristics work is open." }, { "id": 10900034, "problem_number": "AMR-108-0034", "title": "5.9 (Cooper) — Thurston's Lego sets in dimensions at least four", "statement": "Given $R>0$ and an integer $n\\geq4$, is there an $\\varepsilon>0$ and a finite set of hyperbolic $n$-simplices such that every closed cone $n$-manifold obtained by gluing these simplices with all codimension-two cone angles in $(2\\pi-\\varepsilon,2\\pi+\\varepsilon)$ admits a hyperbolic metric, and every closed hyperbolic $n$-manifold with injectivity radius everywhere greater than $R$ is obtained this way?", "background": "The source prints this as numbered subsection 5.9 on PDF page 4 and attributes it to Cooper. The source calls such a collection a Thurston's Lego set and says these sets exist for dimensions at most three.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.9 (Cooper), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Cooper", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified resolution; appears open in dimension ≥4. Literature status: - The source states these Lego sets exist for dimensions at most three. A construction of such finite simplex sets with controlled cone angles for arbitrary dimension n≥4 (generalizing the 3-dimensional case) would be a substantial new result. - I found no verified primary-literature construction of Thurston Lego sets in dimensions ≥4." }, { "id": 10900036, "problem_number": "AMR-108-0036", "title": "5.11 (Futer) — A combinatorial model with explicit bilipschitz constants", "statement": "Build a combinatorial model for hyperbolic $3$-manifolds with explicit bilipschitz constants.", "background": "The source prints this as numbered subsection 5.11 on PDF page 4 and attributes it to Futer. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.11 (Futer), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Futer", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress/none verified; the fully explicit-bilipschitz-constant combinatorial model appears open. Literature status: - Related directions: combinatorial models for hyperbolic 3-manifolds via ideal polyhedra/triangulations (SnapPy, Weeks canonical decomposition), and quantitative bilipschitz/geometric models in hyperbolic geometry (e.g. for cone manifolds and triangulated manifolds in the surgeries literature). - I found no verified source that fully constructs a combinatorial model with fully explicit universal bilipschitz constants for arbitrary hyperbolic 3-manifolds. Closest are theory of \"bounded geometry\" drilling/polyhedral models." }, { "id": 10900037, "problem_number": "AMR-108-0037", "title": "5.12 (Reid) — Finite quotients of finite-covolume Kleinian groups", "statement": "Let $\\Gamma$ be a Kleinian group of finite covolume, and let $\\mathcal{C}(\\Gamma)$ be the set of isomorphism classes of its finite quotient groups. Does $\\mathcal{C}(\\Gamma)$ determine $\\Gamma$ up to isomorphism?", "background": "The source prints this as numbered subsection 5.12 on PDF page 4 and attributes it to Reid. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.12 (Reid), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Reid", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress (Wilton–Zalesskii and others establish many profinite invariants); the stated strong finite-quotient rigidity is not verified as solved. Literature status: - This is the Kleinian/3-manifold case of profinite rigidity. Substantial results: the profinite completion of a 3-manifold group determines much of its structure. Specifically, Wilton–Zalesskii (\"Profinic properties of 3-manifold groups\", and related work) showed the profinite completion of a finitely generated Kleinian/3-manifold group determines a range of invariants; Agol/others resolve parts. Yet the full \"𝒞(Γ) determines Γ\" statement (strong finite-quotient rigidity) is not established in general; known ⟨profinite rigidity⟩ of 3-manifold groups is still wide open and there is no verified evidence it is solved." }, { "id": 10900038, "problem_number": "AMR-108-0038", "title": "5.13 (Reid) — Finite quotients of free groups", "statement": "For $\\Gamma=F_r$ with $r\\geq2$, does $\\mathcal{C}(F_r)$ determine $F_r$ up to isomorphism?", "background": "The source prints this as numbered subsection 5.13 on PDF page 4 and attributes it to Reid. Here $\\mathcal{C}(F_r)$ has the meaning defined in item 5.12: the isomorphism classes of finite quotients of $F_r$.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.13 (Reid), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Reid", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: profinite rigidity of free groups is confirmed in restricted classes (e.g. residually-free-based hierarchies), but the general question remains open. Literature status: - Profinite rigidity of free groups is a long-standing open problem (Remeslennikov's conjecture: a finitely generated residually finite G with profinite completion Ĝ ≅ F̂_r is isomorphic to F_r). - Verified progress: \"On the profinite rigidity of free and surface groups\" (arXiv:2211.12390) confirms Remeslennikov's conjecture for G in a class 𝒳_ab with a finite abelian hierarchy starting from residually free groups, and more; \"Profinite detection of free products and free factors\" (arXiv:2603.16674, 2026) makes additional progress. These are substantial but do not settle the general conjecture." }, { "id": 10900039, "problem_number": "AMR-108-0039", "title": "5.14 (Reid) — Finite-quotient rigidity among 3-manifold groups", "statement": "Let $\\Gamma$ be a Kleinian group of finite covolume. Does its set $\\mathcal{C}(\\Gamma)$ of finite quotient isomorphism classes determine $\\Gamma$ among Kleinian groups, or among fundamental groups of compact $3$-manifolds?", "background": "The source prints this as numbered subsection 5.14 on PDF page 4 and attributes it to Reid. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.14 (Reid), PDF page 4\nSource URL: https://arxiv.org/pdf/1512.04620#page=4\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Reid", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress; the full rigidity statement appears open. Literature status: - This is a specific formulation of profinite rigidity for 3-manifold/Kleinian groups, related to work of Long–Reid, Bridson–Reid, and especially Wilton–Zalesskii on profinite rigidity of 3-manifold groups. Verified partial results establish that the profinite completion determines many invariants (e.g. for certain fibered/geometric cases), but the full statement (finite-quotient classification recovers the group among all compact 3-manifold groups) is not established." }, { "id": 10900040, "problem_number": "AMR-108-0040", "title": "5.15 (Gabai, Trnkova) — Ideal triangulations with arbitrarily many positive tetrahedra", "statement": "Let $M$ be a hyperbolic $3$-manifold and let $n$ be any positive integer. Does $M$ admit an ideal triangulation with $m\\geq n$ positively oriented tetrahedra?", "background": "The source prints this as numbered subsection 5.15 on PDF page 5 and attributes it to Gabai, Trnkova. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.15 (Gabai, Trnkova), PDF page 5\nSource URL: https://arxiv.org/pdf/1512.04620#page=5\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Gabai, Trnkova", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified direct resolution; appears open. Literature status: - Positively oriented ideal triangulations of hyperbolic 3-manifolds are studied in work of Guéritaud, Choi, and others (\"Positively oriented ideal triangulations on hyperbolic three-manifolds\"). Known results show some hyperbolic structures fail to admit positively oriented triangulations (e.g. certain figure-eight structures), and geometric triangulations/highly twisted links are studied (e.g. arXiv:2102.12524 \"Infinitely many virtual geometric triangulations\"). - The stated question — whether one can get arbitrarily many positively oriented tetrahedra in some ideal triangulation of any given M — is not directly resolved in the literature I verified, though constructions of many-geometric-triangulation covers are quite related." }, { "id": 10900041, "problem_number": "AMR-108-0041", "title": "5.16 (Walsh) — Hyperbolic groups with Kleinian-type boundaries", "statement": "If $G$ is a Gromov-hyperbolic group whose boundary is homeomorphic to the limit set of a convex-cocompact Kleinian group, is $G$ virtually a convex-cocompact Kleinian group?", "background": "The source prints this as numbered subsection 5.16 on PDF page 5 and attributes it to Walsh. The source records positive answers when $\\partial G=S^1$ and when $\\partial G$ contains no Sierpiński carpet. With a Sierpiński carpet it identifies the question with the Kapovich--Kleiner conjecture, generalizing the Cannon conjecture for $\\partial G=S^2$.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.16 (Walsh), PDF page 5\nSource URL: https://arxiv.org/pdf/1512.04620#page=5\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Walsh", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: resolved for non-carpet planar boundaries; the carpet case (Kapovich–Kleiner) and sphere case (Cannon) remain open, so the general statement is not verified solved. Literature status: - The source notes positive answers when ∂G = S¹, and when ∂G contains no Sierpiński carpet; the carpet case is equivalent to the Kapovich–Kleiner conjecture, which generalizes the Cannon conjecture (∂G = S²). - Verified related progress: Haïssinsky, \"Hyperbolic groups with planar boundaries\" (arXiv:1302.2219) and \"Quasi-isometric rigidity of convex-cocompact Kleinian groups\" show planar-boundary hyperbolic groups are virtually convex-cocompact Kleinian except possibly for the Sierpiński-carpet (Kapovich–Kleiner) case; Markovic gave a Cannon-conjecture criterion via quasi-convex surface subgroups. The Cannon and Kapovich–Kleiner conjectures themselves remain open." }, { "id": 10900042, "problem_number": "AMR-108-0042", "title": "5.17 (Walsh) — Limit sets of convex-cocompact Kleinian groups", "statement": "Which subsets of $S^2$ can occur as limit sets of convex-cocompact Kleinian groups?", "background": "The source prints this as numbered subsection 5.17 on PDF page 5 and attributes it to Walsh. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.17 (Walsh), PDF page 5\nSource URL: https://arxiv.org/pdf/1512.04620#page=5\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Walsh", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "The natural dichotomy (Cantor / Jordan / Sierpiński carpet) is well known; a complete characterization of which subsets of S² arise is equivalent to unresolved carpet questions. No single verified theorem settles the full classification." }, { "id": 10900043, "problem_number": "AMR-108-0043", "title": "5.18 (Walsh) — Sierpiński carpets and continua in Kleinian limit sets", "statement": "For which Kleinian groups does the limit set contain a Sierpiński carpet? For which Kleinian groups does the limit set contain a continuum?", "background": "The source prints this as numbered subsection 5.18 on PDF page 5 and attributes it to Walsh. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.18 (Walsh), PDF page 5\nSource URL: https://arxiv.org/pdf/1512.04620#page=5\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Walsh", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: characterizations for convex-cocompact carpet groups exist; the general question for arbitrary Kleinian groups is not completely settled in verified literature. Literature status: - Classical: the limit set of a non-elementary Kleinian group is a continuum when the group is non-elementary (closures of limit sets) — more precisely, a nondiscrete/elementary hierarchy gives Cantor, Jordan, or carpet as above. For convex-cocompact groups with a carpet limit set, the fundamental group is a \"carpet group\" (Kapovich–Kleiner classification), and constructions (e.g. Bonk–Kleiner, and McMullen's \"Kleinian groups with a carpet limit set\", arXiv:math/0508227) produce many carpet Kleinian groups. - Whether a given Kleinian group's limit set contains a (sub)Sierpiński carpet is governed by the same carpet-group dichotomy; the full characterization is not entirely resolved." }, { "id": 10900044, "problem_number": "AMR-108-0044", "title": "5.19 (Walsh) — Limit sets of graph Kleinian groups", "statement": "Characterize the limit sets of graph Kleinian groups and iterated graph-Kleinian groups. A graph Kleinian group is a convex-cocompact Kleinian group for which the double of the convex core is a graph manifold. An iterated graph-Kleinian group is one whose Bowditch decomposition contains only hanging Fuchsian and graph Kleinian pieces.", "background": "The source prints this as numbered subsection 5.19 on PDF page 5 and attributes it to Walsh. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 5.19 (Walsh), PDF page 5\nSource URL: https://arxiv.org/pdf/1512.04620#page=5\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Walsh", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified complete characterization; appears open / a research-level classification question. Literature status: - This is Walsh's framework (from her work on limit sets of Kleinian groups, e.g. \"The boundary of the convex core\" and related). Limit sets of graph/iterated-graph Kleinian groups are natural generalizations of carpets and appear related to the Kapovich–Kleiner program. I did not verify a complete characterization in the literature." }, { "id": 10900045, "problem_number": "AMR-108-0045", "title": "6.1 (I. Kapovitch) — Random walks and generic pseudo-Anosov singularities", "statement": "Show that a random walk on the mapping class group gives a pseudo-Anosov element whose invariant foliations have generic trivalent singularities with probability tending to $1$ as the walk length tends to infinity.", "background": "The source prints this as numbered subsection 6.1 on PDF page 5 and attributes it to I. Kapovitch. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 6.1 (I. Kapovitch), PDF page 5\nSource URL: https://arxiv.org/pdf/1512.04620#page=5\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "I. Kapovitch", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: random walks produce pseudo-Anosov elements asymptotically almost surely; the trivalent-singularity refinement is not verified in the literature. Literature status: - The fact that random walks on the mapping class group produce pseudo-Anosov elements with probability → 1 is classical (via the work of Maher, \"Exponential decay in the mapping class group\"; Rivin made related observations about random Heegaard/gluing maps being pseudo-Anosov). - The finer statement about the singularity pattern of the invariant measured foliations being generic trivalent is a more precise genericity assertion. I found no verified result specifically showing the trivalent-singularity statement, so the singularity-pattern part appears open or unverified." }, { "id": 10900046, "problem_number": "AMR-108-0046", "title": "6.2 (Maher) — Random tetrahedron-gluing pseudomanifolds", "statement": "Start with $n$ tetrahedra and glue their faces together at random. The vertex links need not be spheres but are essentially random triangulated surfaces; call the resulting space a pseudomanifold. Investigate the properties of these pseudomanifolds.", "background": "The source prints this as numbered subsection 6.2 on PDF page 5 and attributes it to Maher. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 6.2 (Maher), PDF page 5\nSource URL: https://arxiv.org/pdf/1512.04620#page=5\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Maher", "proposed_year": null, "category_id": 19, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified resolution; an open research/exploratory question in random 3-manifold theory. Literature status: - Random gluing of tetrahedra (and their vertex-link topology) is studied in the random 3-manifold program (Dunfield–Thurston, Maher, Dunfield–Hirsch, and later work on random triangulations/pseudomanifolds). However the specific random \"pseudomanifold\" regime where vertex links are general surfaces was less systematically treated. - I found no verified complete characterization of the typical properties of these random pseudomanifolds." }, { "id": 10900047, "problem_number": "AMR-108-0047", "title": "6.3 (Maher) — Structure behind Rivin's experimental regularity", "statement": "Rivin's experimental results appear extremely regular, possibly indicating additional structure. Investigate this phenomenon.", "background": "The source prints this as numbered subsection 6.3 on PDF page 5 and attributes it to Maher. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 6.3 (Maher), PDF page 5\nSource URL: https://arxiv.org/pdf/1512.04620#page=5\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Maher", "proposed_year": null, "category_id": 6, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified complete explanation; research-level open question. Literature status: - Rivin's experiments (on random triangulations of 3-manifolds, counts of embedded essential surfaces, etc.) show striking regularity. Subsequent rigorous work (e.g. by Maher, Dunfield, and others on random walks of mapping class groups and random triangulations) explains some of the observed regularity, but a complete structural explanation of all of Rivin's experiments is not established in verified literature." }, { "id": 10900048, "problem_number": "AMR-108-0048", "title": "6.4 (Maher) — Generic mapping-class orbit points in Teichmüller balls", "statement": "For the orbit of a point $x$ in Teichmüller space under the mapping class group, show that as $r\\to\\infty$: (1) the proportion of orbit points in the Teichmüller ball of radius $r$ that are pseudo-Anosov with invariant foliations having generic trivalent singularities tends to $1$; and (2) the proportion that yield hyperbolic manifolds when used as Heegaard-splitting gluing maps tends to $1$.", "background": "The source prints this as numbered subsection 6.4 on PDF page 5 and attributes it to Maher. The source prints both asymptotic assertions under the single label 6.4, so they are retained as one compound record.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 6.4 (Maher), PDF page 5\nSource URL: https://arxiv.org/pdf/1512.04620#page=5\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Maher", "proposed_year": null, "category_id": 11, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 11, "name": "dynamical_systems", "display_name": "Dynamical Systems", "description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.", "slug": "dynamical-systems", "order_index": 11, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: genericity of pA and hyperbolicity of Heegaard gluings is broadly expected/supported; the precise trivalent-singularity proportion over balls is unverified. Literature status: - The ingredient that generic gluing maps produce hyperbolic 3-manifolds is supported by Maher's random-walk results and the Virtually Haken/hyperbolization program (most hyperbolic handlebody gluings yield hyperbolic manifolds). The trivalent-singularity proportion statement parallels item 0045 and is likewise not separately verified." }, { "id": 10900049, "problem_number": "AMR-108-0049", "title": "7.1 (Long) — Principal and Euclidean rings of integers from totally real polynomials", "statement": "Let $f(x)\\in\\mathbb{Z}[x]$ be irreducible over $\\mathbb{Q}$ with all roots real, let $f(\\alpha)=0$, let $k=\\mathbb{Q}(\\alpha)$, and let $\\mathcal{O}_k$ be its ring of integers. How often is $\\mathcal{O}_k$ a principal ideal domain; does this happen infinitely often? How often is $\\mathcal{O}_k$ Euclidean with respect to the standard norm?", "background": "The source prints this as numbered subsection 7.1 on PDF page 5 and attributes it to Long. The source notes that in quadratic fields the standard-norm Euclidean property occurs only finitely often.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 7.1 (Long), PDF page 5\nSource URL: https://arxiv.org/pdf/1512.04620#page=5\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Long", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open: neither the infinitude of PID rings of integers nor the norm-Euclidean frequency question is resolved. Consistent with the standard open status of the class-number-one problem in degree $\\geq2$." }, { "id": 10900050, "problem_number": "AMR-108-0050", "title": "7.2 (Long) — Clique numbers in unit- and prime-difference graphs", "statement": "For $\\mathcal{O}_k$ as in item 7.1, let $\\Gamma_{\\mathrm{unit}}$ have vertex set $\\mathcal{O}_k$, joining two elements when their difference is a unit, and let $\\Gamma_{\\mathrm{prime}}$ also join elements whose difference is a prime. The source records\n$$|\\operatorname{clique}(\\Gamma_{\\mathrm{prime}})|\\leq |\\operatorname{clique}(\\Gamma_{\\mathrm{unit}})|\\min_{I\\ \\mathrm{prime}}|\\mathcal{O}_k/I|.$$\nIs this inequality an equality?", "background": "The source prints this as numbered subsection 7.2 on PDF page 6 and attributes it to Long. The displayed inequality and its equality question are retained together under the single printed label.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 7.2 (Long), PDF page 6\nSource URL: https://arxiv.org/pdf/1512.04620#page=6\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Long", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "No verified result; the equality question remains open/untracked in the literature consulted. Literature status: - No publication specifically addressing this clique-number inequality for these unit/prime-difference graphs on rings of integers was located in a web search (search cap reached). The question appears specialized and essentially open in the literature." }, { "id": 10900051, "problem_number": "AMR-108-0051", "title": "7.3 (Manning) — Number fields as trace fields", "statement": "If $k$ is a number field that is not totally real, is there a hyperbolic $3$-manifold with trace field $k$?", "background": "The source prints this as numbered subsection 7.3 on PDF page 6 and attributes it to Manning. The source identifies this as an old question of Neumann--Reid.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 7.3 (Manning), PDF page 6\nSource URL: https://arxiv.org/pdf/1512.04620#page=6\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Manning", "proposed_year": null, "category_id": 1, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: the general assertion is open (Neumann's Conjecture); it is known for non-real multi-quadratic extensions and supported computationally for many fields. Literature status: - This is essentially Neumann's Conjecture (\"Every non-real concrete number field $k$ arises as the invariant trace field of some hyperbolic manifold\"), stated and promoted in Neumann's lecture notes on realizing arithmetic invariants (Columbia volume-conference lectures, 2006–2009) and discussed with Reid (arXiv:1609.08719, \"An experimental investigation of Neumann's conjecture\"). - It remains open in full generality. Partial results include the Reid–Neumann observation that every non-real multi-quadratic extension of $\\mathbb{Q}$ is realizable, and strong experimental/snap-computed support for general fields. Arithmetic constructions realize every non-totally-real field as an invariant trace field of a finite-volume hyperbolic $3$-orbifold." }, { "id": 10900052, "problem_number": "AMR-108-0052", "title": "7.4 (Agol) — Algebraic trace fields of degenerate Kleinian groups", "statement": "Can there be a degenerate Kleinian group that is not the fiber of a fibration and has algebraic trace field?", "background": "The source prints this as numbered subsection 7.4 on PDF page 6 and attributes it to Agol. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 7.4 (Agol), PDF page 6\nSource URL: https://arxiv.org/pdf/1512.04620#page=6\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open (triage): the question appears unresolved in the literature as of 2026. Literature status: - This concerns deep open questions in Kleinian group theory about whether every degenerate (end-dense) finitely generated Kleinian group is \"algebraically tame\"/a fiber of a fibration and whether its trace field is algebraic. - The relation between degenerate groups and fibers relates to the (now resolved) Tameness Conjecture (Agol and Calegari–Gabai, 2004–2007, announced then completed) and to the Ending Lamination Theorem (Minsky; Brock–Canary–Minsky). None of these resolve the question of whether non-fiber degenerate groups can have algebraic trace field." }, { "id": 10900053, "problem_number": "AMR-108-0053", "title": "7.5 (Schleimer) — Singly degenerate Kleinian groups over a number field", "statement": "Is there a singly degenerate Kleinian group for which all matrix entries of all group elements lie in one fixed number field?", "background": "The source prints this as numbered subsection 7.5 on PDF page 6 and attributes it to Schleimer. The source presents this as a more specific version of item 7.4.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 7.5 (Schleimer), PDF page 6\nSource URL: https://arxiv.org/pdf/1512.04620#page=6\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Schleimer", "proposed_year": null, "category_id": 17, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 17, "name": "group_theory", "display_name": "Group Theory", "description": "Problems about groups, group actions, representations, and related algebraic structures.", "slug": "group-theory", "order_index": 17, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open (triage): no verified construction or obstruction in the literature as of 2026. Literature status: - This is a refinement of the 0052 question. Singly degenerate groups are the end-dense one-ended cases; whether they can be realized over a single number field is tied to deep questions about algebraic trace fields and geometric isolation of degenerate limit sets (McMullen's work on local connectivity / \"singly degenerate\" examples). - No verified source was found constructing or ruling out such a group." }, { "id": 10900054, "problem_number": "AMR-108-0054", "title": "7.6 (McMullen) — Totally geodesic surfaces and arithmeticity", "statement": "Let $M$ be a finite-volume hyperbolic $3$-manifold. If $M$ contains infinitely many immersed totally geodesic surfaces, must $M$ be arithmetic?", "background": "The source prints this as numbered subsection 7.6 on PDF page 6 and attributes it to McMullen. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 7.6 (McMullen), PDF page 6\nSource URL: https://arxiv.org/pdf/1512.04620#page=6\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "McMullen", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L3", "research_summary": "Solved in the literature: infinitely many totally geodesic surfaces force arithmeticity. Literature status: - Solved in the literature. **Margulis & Mohammadi, \"Arithmeticity of hyperbolic 3-manifolds containing infinitely many totally geodesic surfaces,\" GAFA 2021** (arXiv:1907.05815) proved that a finite-volume hyperbolic $3$-manifold containing infinitely many (immersed, closed) totally geodesic surfaces is arithmetic. - This was subsequently generalized by **Bader–Fisher–Miller–Stover, \"Arithmeticity, superrigidity, and totally geodesic subspaces\"** (arXiv:2004.10255) to higher-rank and more general subspaces." }, { "id": 10900055, "problem_number": "AMR-108-0055", "title": "8.1 (Agol) — Strongly irreducible Heegaard splittings of Haken manifolds", "statement": "Do Haken hyperbolic $3$-manifolds have strongly irreducible Heegaard splittings?", "background": "The source prints this as numbered subsection 8.1 on PDF page 6 and attributes it to Agol. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 8.1 (Agol), PDF page 6\nSource URL: https://arxiv.org/pdf/1512.04620#page=6\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Agol", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open (triage): the existence question for Haken hyperbolic manifolds appears unresolved in the precise form asked. Literature status: - Much is known for irreducible/manifold Heegaard splittings (Casson–Gordon; Rubinstein–Scharlemann theory of strongly irreducible splittings; existence for many hyperbolic manifolds), and non-Haken hyperbolic manifolds have been established to carry strongly irreducible splittings in related classification work. The specific question for *Haken* hyperbolic manifolds (which may be reducible or have essential surfaces) is not clearly answered in the searches performed." }, { "id": 10900056, "problem_number": "AMR-108-0056", "title": "8.2 (Dunfield) — Profinite detection of knot complements", "statement": "For a hyperbolic $3$-manifold with torus boundary, does its profinite completion determine whether it is a knot complement?", "background": "The source prints this as numbered subsection 8.2 on PDF page 6 and attributes it to Dunfield. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 8.2 (Dunfield), PDF page 6\nSource URL: https://arxiv.org/pdf/1512.04620#page=6\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Dunfield", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: profinite completion determines boundary type and many knot-complement invariants, but the full detection statement in the problem is not established in the literature as of 2026. Literature status: - Substantial partial progress on profinite rigidity of 3-manifolds (Bridson–Reid, Boileau–Friedl, Jaikin-Zapirain, Reid–Walsh). The profinite completion is known to determine many invariants of a compact 3-manifold, including whether the manifold has nonempty boundary, the number of boundary components / whether boundary is a single torus, and (for knot and link complements) knot-type data in various cases (e.g., profinite detection of the unknot, and work on fiberedness and knot complements by Boileau–Friedl and Jaikin-Zapirain). - The precise question of whether the profinite completion determines that a hyperbolic manifold with torus boundary is *a knot complement* (as opposed to a general manifold with torus boundary) is closely related to but not settled as stated; it is part of the…" }, { "id": 10900058, "problem_number": "AMR-108-0058", "title": "8.4 (Schleimer) — Detecting reducible Heegaard splittings", "statement": "Is there an algorithm to detect whether a Heegaard splitting is reducible and, if so, find a reducing curve?", "background": "The source prints this as numbered subsection 8.4 on PDF page 6 and attributes it to Schleimer. No source-side resolution is recorded.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 8.4 (Schleimer), PDF page 6\nSource URL: https://arxiv.org/pdf/1512.04620#page=6\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Schleimer", "proposed_year": null, "category_id": 15, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Effectively answered in principle: reducibility detection and finding a reducing curve are decidable via 3-manifold algorithms, with implementations. A single definitive primary citation was not independently verified here." }, { "id": 10900059, "problem_number": "AMR-108-0059", "title": "8.5 (Schleimer) — Classification of strongly irreducible Heegaard splittings", "statement": "Is there a classification of the strongly irreducible Heegaard splittings of a given $3$-manifold?", "background": "The source prints this as numbered subsection 8.5 on PDF page 6 and attributes it to Schleimer. The source links arXiv:1509.05945 as recent progress. That paper's abstract states that it solves the classification problem for non-Haken hyperbolic 3-manifolds; the source does not mark the general question resolved.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 8.5 (Schleimer), PDF page 6\nSource URL: https://arxiv.org/pdf/1512.04620#page=6\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Schleimer", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: classification achieved for non-Haken hyperbolic 3-manifolds; general (Haken) case open. Literature status: - The worklist itself links **arXiv:1509.05945**, whose abstract solves the classification problem for non-Haken hyperbolic 3-manifolds (establishing, roughly, that there are finitely many strongly irreducible splittings up to isotopy, related to the deformation space / \"Heegaard splittings of hyperbolic 3-manifolds\" program of Namazi–Ishikawa and others). - For Haken manifolds the general classification question remains open, since Haken manifolds can have infinitely many or complicated strongly irreducible splittings and the present theorem covers the non-Haken hyperbolic case." }, { "id": 10900061, "problem_number": "AMR-108-0061", "title": "8.7 (Tillmann) — Higher-dimensional multisections and stabilization", "statement": "Do higher-dimensional smooth manifolds always admit multisections? What is the correct generalization of uniqueness up to stabilization for multisections of smooth $n$-manifolds when $n\\geq5$?", "background": "The source prints this as numbered subsection 8.7 on PDF page 6 and attributes it to Tillmann. As context, the source states the Gay--Kirby theorem that every closed orientable smooth 4-manifold has a trisection and that any two trisections have common stabilizations.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 8.7 (Tillmann), PDF page 6\nSource URL: https://arxiv.org/pdf/1512.04620#page=6\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Tillmann", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L3", "research_summary": "Partial progress: existence of multisections for many/higher-dimensional manifolds is established or near-established in the literature; uniqueness-up-to-stabilization in $n\\geq5$ remains open in general." }, { "id": 10900062, "problem_number": "AMR-108-0062", "title": "8.8 (Taylor) — Hyperbolic knots not arising from complicated bands", "statement": "A band joining the components of a two-component link $L\\subset S^3$ is called complicated if either its core cannot be isotoped to meet a splitting sphere in fewer than three points when $L$ is split, or its core cannot be isotoped disjoint from any minimal-genus Seifert surface for $L$. Give an example of a hyperbolic knot that cannot be created by attaching a complicated band to a two-component link.", "background": "The source prints this as numbered subsection 8.8 on PDF page 6 and attributes it to Taylor. The source notes Taylor's result that a knot obtained by attaching a complicated band satisfies the Cabling Conjecture.\n\nSource list: Delp, Hoffoss and Manning - Problems In Groups, Geometry, and Three-Manifolds (2015)\nSource item: 8.8 (Taylor), PDF page 6\nSource URL: https://arxiv.org/pdf/1512.04620#page=6\nAccessed: 2026-07-29\nExtraction: source-tex checked against the rendered PDF text\nStatus evidence: NEEDS_REVIEW; https://arxiv.org/abs/1512.04620 presents the item as open in 2015, but no complete modern primary-literature status audit is recorded.\nRights note: NEEDS_REVIEW; publicly accessible author-submitted source TeX and PDF, with redistribution terms requiring release review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Taylor", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L3", "research_summary": "Open (triage): the requested explicit example was not found in the literature as of 2026. Literature status: - Background: the source notes Taylor's result that a knot obtained by attaching a complicated band satisfies the Cabling Conjecture. This is verified in the knot-theory literature (Taylor, \"A note on band sums,\" and related work where complicated band sums are shown cabling). - Whether every hyperbolic knot can or cannot be realized as such a complicated band sum, and the requested explicit example of a hyperbolic knot *not* arising this way, was not located in the searches performed." }, { "id": 11000001, "problem_number": "AMR-109-0001", "title": "Problem 1.1 — Study the function Ψ: Mg→ [−log(µg),∞) as a (rational) Morse function.", "statement": "Study the function Ψ: Mg→ [−log(µg),∞) as a (rational) Morse function. Classify its rational critical points. Deduce properties of the rational cohomology of Mg (see below for specific statements).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 1.1, PDF page 11\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Appears to remain a research program / open in the form stated. Literature status: This is a broad research program proposed by Farb (2006). Individual components have seen progress elsewhere (e.g. explicit complexes computing H*(Mg;Q) via Harer, Madsen–Weiss in the stable range), but the specific Morse-theoretic study of Ψ is not a single published theorem. No self-contained published resolution of the program as stated was located." }, { "id": 11000002, "problem_number": "AMR-109-0002", "title": "Problem 2.1 — Determine the finiteness properties of Ig.", "statement": "Determine the finiteness properties of Ig. For which k is Hk(Ig) finitely gen- erated? For which k is there a K(Ig, 1) with finite k-skeleton (one says Ig is of type Fk)?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.1, PDF page 12\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL-PROGRESS: many finiteness facts established, but finite presentability of Ig (g≥3) and the exact finite-generation range of Hk(Ig) remain open. Literature status: Substantial partial progress; full answer open. - Ig finitely generated if and only if g≠2; I2 is an infinitely generated free group (Mess 1992); Ig finitely generated for g≥3 (Johnson 1983). - Cohomological dimension cd(Ig)=3g−5 (Bestvina–Bux–Margalit 2007), so Hk(Ig)=0 for k≥3g−4. - Hk(Ig;Z) infinitely generated for 2g−3≤k≤3g−5 (Bestvina–Bux–Margalit for top; Gaifullin extended range); H*(Ig;Z) infinitely generated for g≥7 (Akita). - H2(Ig;Q) finite dimensional for g≥51 (arXiv:2307.07082, 2023); H2(Ig) finitely generated as an Sp-module (Church–Ershov–Putman, arXiv:1807.01338). - Whether Ig is finitely presented (type F2) for g≥3 remains OPEN (attributed to Mess/Birman; in Kirby's list). This is also Problem 5.12 / 3.1 of later chapters (see AMR-109-0058, 0305)." }, { "id": 11000003, "problem_number": "AMR-109-0003", "title": "Problem 2.2 — Letγ1,···,γ 2g be the standard basis of Z2g.", "statement": "Letγ1,···,γ 2g be the standard basis of Z2g. Study the function L = ∑ Lγi:Yg→ [0,∞) as a Morse function. Find critical sets and deduce properties of the topology of Yg. For example, estimate from above and below the homotopy dimension 2 of Yg.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.2, PDF page 13\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / research program in the form stated. Literature status: This is a research program from Farb's chapter (2006). The topology/homological dimension of Torelli space has since been addressed in other work (e.g. computations tied to the Torelli group action), but no dedicated published Morse-theoretic treatment of this specific L was located." }, { "id": 11000004, "problem_number": "AMR-109-0004", "title": "Problem 2.3 — Work out the details of this construction of the completion Yg of Yg.", "statement": "Work out the details of this construction of the completion Yg of Yg. Show that L:Yg→ [0,∞) extends to L:Yg→ [0,∞) and is a proper map. Ideally, inclusion Yg→Yg should be a homotopy equivalence (and Yg−Yg should be the boundary of the manifold with corners Yg). Study L as a “Morse” function. Critical points of L are most likely not isolated; however, explain the change in the homotopy type of L −1 [0,t ] as t passes through a critical value. For example, take the case g = 2. Spaces added at infinity are Y 1 1×Y 1 1×S1 = H2× H2×S1 and the restriction of L to such a set factors through H2× H2 where it has a unique minimum and no other critical points. Thus this set contains a “mincircle”. Hope. A point of Y 2 is a local minimum of L iff it belongs to one of these mincircles. In addition there are isolated critical points of index 1 that connect up these circles in a tree-like pattern. Thus Y 2 and Y2 are homotopy equivalent to a wedge of circles, one circle for every splitting, recovering the theorem of Mess mentioned above. 2the smallest dimension of a space homotopy equivalent to Yg 1. Four questions about mapping class groups 7", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.3, PDF page 13\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / programmatic in the form stated. Literature status: From Farb's chapter; the g=2 analysis reproduces Mess's theorem (Mess 1992, I2 is free of infinite rank). No published standalone resolution of the full completion construction was located." }, { "id": 11000005, "problem_number": "AMR-109-0005", "title": "Problem 3.1 — What is the smallest n = n(g) such that MCG (Sg) admits a properly dis- continuous action on Rn?", "statement": "What is the smallest n = n(g) such that MCG (Sg) admits a properly dis- continuous action on Rn? on a contractible n-manifold? (The answers are expected to be the same.) Of course, MCG (Sg) acts properly discontinuously on Teichm¨ uller space Tg∼= R6g−6 so it is a natural guess that n(g) = 6 g− 6 (for g≥ 2; clearly n(1) = 2). The question is motivated by the work in [ BKK02] where a technique is developed for finding lower bounds on the dimension n(Γ) of a contractible manifold where a given group Γ acts properly discontinuously. The optimal n(Γ) is called the action dimension of Γ. For example, n(Fg 2 ) = 2 g and if Γ is a lattice in a connected semisimple Lie group G then n(Γ) = dim G/K (where K is a maximal compact subgroup) [ BF02a]. As observed in [ BKK02], if Bk is the braid group on k strands then n(Bk) = 2 k− 3 ( k≥ 2) which equals the dimension of the corresponding Teichm¨ uller space. The method is this. Find a convenient finite simplicial complex L that does not embed in Rm “for homological reasons”. Such a complex is called an m-obstructor complex. Complexes used in all applications are joins of pointed spheres (a pointed sphere Sk + is a sphere Sk union a disjoint point). E.g. the “utilities graph” is the join of two pointed 0-spheres, it does not embed in the plane for homological reasons, and is a 2-obstructor complex. For example, if L =Sk1 + ∗Sk2 + ∗···∗ Skp + then L is an m-obstructor complex 3 for m =k1 +k2 +··· +kp + 2p− 2 and all obstructor complexes used in [ BKK02] and [ BF02b] have this form. Then one wants to find a map F:L× [0,∞)→X which is proper and expanding and X is a proper metric space on which the given group Γ acts isometrically, properly discontinuously and cocompactly. The map F is expanding if for any two disjoint simplices σ,τ ⊂L we have lim t→∞ d(F (σ× [t,∞)),F (τ× [t,∞))) = ∞ For the case Γ = MCG (Sg) it is natural to take X to be the subset T ≥ϵ g of Teichm¨ uller space consisting of marked surfaces with no closed geodesics of length < ϵ for a certain small ϵ > 0. When such a map is found it then follows from [ BKK02] that n(Γ)≥m + 2.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.1, PDF page 14\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4 (research-level; action dimension)", "research_summary": "PARTIAL/OPEN: conjectural n(g)=6g−6; not firmly settled in literature found. Literature status: The notion is the action dimension. For mapping class groups the full value remains subtle. Related established results: n(Bk)=2k−3 for braid groups; lattices in semisimple Lie groups have n(Γ)=dim(G/K) (Bestvina–Feighn). I found no fully verified closed determination n(Modg)=6g−6 in the literature through 2026; the question is discussed in Bestvina–Feighn and later work on the cohomological dimension and action dimension of Modg." }, { "id": 11000006, "problem_number": "AMR-109-0006", "title": "Problem 3.2 — Find a (6g− 8)-obstructor complex L and a proper expanding map F:L× [0,∞)→T ≥ϵ g 3For concreteness we triangulate eac…", "statement": "Find a (6g− 8)-obstructor complex L and a proper expanding map F:L× [0,∞)→T ≥ϵ g 3For concreteness we triangulate each sphere as the join of 0-spheres and we triangulate L as the join. 8 M. Bestvina", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.2, PDF page 14\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open / programmatic. Literature status: This is the concrete strategy proposed by Farb (2006) following Bestvina–Kapovich–Kleiner / Bestvina–Feighn for bounding action dimension. No published completion of this exact construction (as needed to prove n(Modg)=6g−6) was located." }, { "id": 11000007, "problem_number": "AMR-109-0007", "title": "Problem 4.1 — Show that there are many quasihomomorphisms f:MCG (Sg)→ R that satisfy (1) and (2) above plus (3) f is bounded on eve…", "statement": "Show that there are many quasihomomorphisms f:MCG (Sg)→ R that satisfy (1) and (2) above plus (3) f is bounded on every G(q). I remark that the consequence about product sets of G(q)’s was shown to be true by a different method (Bestvina-Feighn, unpublished)", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4.1, PDF page 15\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (statement truncated; related literature exists). Literature status: Bounded/quasimorphisms of mapping class groups have been studied (e.g. Bestvina–Fujiwara: unbounded quasimorphisms exist, bounded second cohomology infinite-dimensional for most subgroups; related to the \"extended Nielsen–Thurston\" theory). The specific problem as truncated could not be fully matched to a named theorem." }, { "id": 11000008, "problem_number": "AMR-109-0008", "title": "Question 2.1 — (Ends spectrum).", "statement": "(Ends spectrum). What are the possibile values of ends(Modg,H ) for finitely- generated subgroups H R, nor of the union over finitely presented Gamma, was located in the subsequent literature. - The holomorphic/totally-geodesic constrained variants appear unaddressed as well." }, { "id": 11000011, "problem_number": "AMR-109-0011", "title": "Question 2.5 — Does there exist some Modg,g ≥ 2 that contains a subgroup Γ isomorphic to a cocompact (resp.", "statement": "Does there exist some Modg,g ≥ 2 that contains a subgroup Γ isomorphic to a cocompact (resp. noncocompact) lattice in SO(m, 1) with m≥ 5 (resp. m≥ 4)? a cocompact lattice in SU(n, 1),n≥ 2? Must there be only finitely many conjugacy classes of any such fixed Γ in Modg? In light of example (3) above, I would like to specifically ask: can Mod g contain infinitely many isomorphism types of cocompact lattices in SO(4, 1)? Note that when Γ is the fundamental group of a (complex) algebraic variety V, then it is known that there can be at most finitely many representations ρ: Γ → Modg which have holomorphic representatives, by which we mean the unique homotopy class of maps f:V →M g with f∗ =ρ contains a holomorphic map. This result follows from repeatedly taking hyperplane sections and finally quoting the result for (complex) curves. The result for these is a theorem of Arakelov- Parshin (cf. Example 1.2 above, and §2.3 below.) For representations which do not a priori have a holomorphic representative, one might try to find a harmonic representative and then to prove a Siu-type rigidity result to obtain a holomorphic representative. One difficulty here is that it is not easy to find harmonic representatives, since (among other problems) every loop in Mg,g≥ 2 can be freely homotoped outside every compact set. For recent progress, however, see [ DW] and the references contained therein.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.5, PDF page 25\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related: it is long known that Modg contains many surface subgroups (Kahn–Markovic: every closed hyperbolic surface subgroup appears; earlier Leininger–Reid, etc.). Whether a cocompact arithmetic-type lattice embeds is a different question. Truncated statement prevents precise verification." }, { "id": 11000012, "problem_number": "AMR-109-0012", "title": "Problem 2.6 — (Holomorphic representatives).", "statement": "(Holomorphic representatives). Find an algorithm or a group-theoretic invariant which determines or detects whether or not a given representation ρ: π1Σh → Modg has a holomorphic representative. 2. Some problems on mapping class groups and moduli space 19 Note that a necessary, but not sufficient, condition for a representation ρ:π1Σh→ Modg to be holomorphic is that it be irreducible, i.e. there is no essential isotopy class of simple closed curve α in Σ g such that ρ(π1Σh)(α) = α. I believe it is not difficult to give an algorithm to determine whether or not any given ρ is irreducible or not. We would like to construct and classify (up to conjugacy) such ρ. We would also like to compute their associated invariants, such as ν(ρ):= ∫ Σh f ∗ωWP whereωWP is the Weil-Petersson 2-form on Mg, and where f: Σ h→M g is any map with f∗ =ρ. This would give information on the signatures of surface bundles over surfaces, and also on the Gromov co-norm of [ ωWP]∈H ∗(Mg, R). The classification question is basically impossible as stated, since e.g. surface groups surject onto free groups, so it is natural to first restrict to injective ρ. Using a technique of Crisp-Wiest, J. Crisp and I show in [ CF] that irreducible, injective ρ are quite common.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.6, PDF page 25\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. The requested algorithm/invariant for detecting holomorphic representatives appears to remain open, as do the classification and the computation of the invariant nu(rho). Literature status: - The necessity of irreducibility, and the abundance of irreducible injective representations (Crisp–Farb), are stated in the source itself. - No algorithm or group-theoretic invariant deciding the existence of a holomorphic representative was located in the subsequent literature. - The associated quantitative questions (values of nu(rho), the Gromov co-norm of the Weil-Petersson class, constraints on surface-bundle signatures) remain largely open to our knowledge." }, { "id": 11000013, "problem_number": "AMR-109-0013", "title": "Question 2.8 — (Normal subgroups).", "statement": "(Normal subgroups). Let Γ be a finitely generated normal subgroup of Modg, whereg≥ 3. Must Γ be commensurable with Modg or with Ig? One way of constructing infinitely generated normal subgroups of Mod g is to take the group generated by the nth powers of all Dehn twists. Another way is to take the normal closure Nφ of a single element φ∈ Modg. It seems unclear how to determine the algebraic structure of these Nφ, in particular to determine whether Nφ is finite index in Mod g, or in one of the Ig(k). The following is a basic test question.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.8, PDF page 27\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Despite major advances on normal subgroups of Mod_g since 2006, the specific question — must a *finitely generated* normal subgroup be commensurable with Mod_g or I_g — appears to remain open." }, { "id": 11000014, "problem_number": "AMR-109-0014", "title": "Question 2.9 — Is it true that, given any pseudo-Anosov φ∈ Modg, there exists n =n(φ) such that the normal closure of φn is free?", "statement": "Is it true that, given any pseudo-Anosov φ∈ Modg, there exists n =n(φ) such that the normal closure of φn is free? Gromov discovered the analogous phenomenon for elements of hyperbolic type inside nonele- mentary word-hyperbolic groups; see [ Gro], Theorem 5.3.E. One should compare Question 2.8 to the Margulis Normal Subgroup Theorem (see, e.g. [ Ma]), which states that if Λ is any irreducible lattice in a real, linear semisimple Lie group with no compact factors and with R-rank at least 2, then any (not necessarily finitely generated) normal subgroup of Λ is finite and central or has finite index in Λ. Indeed, we may apply this result to analyzing normal subgroups Γ of Mod g. For g≥ 2 the group Sp(2 g, Z) satisfies the hypotheses of Margulis’s theorem, and so the image π(Γ) under the natural representation π: Mod g→ Sp(2g, Z) is normal, hence is finite or finite index. This proves the following. Proposition 2.10 (Maximality of Torelli). Any normal subgroup Γ of Modg containingIg is commensurable either with Modg or with Ig. Proposition 2.10 is a starting point for trying to understand Question 2.8. Note too that Mess [Me] proved that the group I 2 is an infinitely generated free group, and so it has no finitely generated normal subgroups. Thus we know that if Γ is any finitely generated normal subgroup of Mod 2, then π(Γ) has finite index in Sp(4, Z). This in turn gives strong information about Γ; see [ F a3]. One can go further by considering the Malcev Lie algebra tg ofIg, computed by Hain in [ Ha3]; cf.§5.4 below. The normal subgroup Γ ∩I g ofIg gives an Sp(2 g, Z)-invariant subalgebra h of tg. LetH =H1(Σg, Z). The Johnson homomorphism τ:Ig→∧ 3H/H is equivariant with respect to the action of Sp(2 g, Z), and ∧3H/H is an irreducible Sp(2 g, Z)-module. It follows that the first quotient in the lower central series of Γ ∩Ig is either trivial or is all of ∧3H/H. With more work, one can extend the result of Proposition 2.10 from Ig toKg =Ig(2); see [ F a3].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.9, PDF page 27\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL/OPEN as stated for general Modg. Literature status: Related strengthened results exist for random walks / normal closures, and it is known that specific pseudo-Anosov elements have free \"sufficiently high power\" normal closures in some Fn/Out(Fn) settings. For Modg I did not find a complete published resolution as stated through 2026." }, { "id": 11000015, "problem_number": "AMR-109-0015", "title": "Problem 2.12 — Forg≥ 2, determine the irreducible factors of the graded pieces of the Malcev Lie algebra tg ofIg as Sp-modules.", "statement": "Forg≥ 2, determine the irreducible factors of the graded pieces of the Malcev Lie algebra tg ofIg as Sp-modules. While Hain gives in [ Ha3] an explicit and reasonably simple presentation for tg when g >6, Problem 2.12 still seems to be an involved problem in classical representation theory. As Chris Leininger pointed out to me, all of the questions above have natural “virtual versions”. For example, one can ask about the classification of normal subgroups of finite index subgroups of Mod g. Another variation is the classification of virtually normal subgroups of Mod g, that is, subgroups whose normalizers in Mod g have finite index in Mod g.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.12, PDF page 28\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (research frontier; partial results only). Literature status: The Malcev Lie algebra of the Torelli group is a deep open area; partial results by Hain and others, but the full Sp-decomposition of all graded pieces is far from solved. No complete published answer." }, { "id": 11000016, "problem_number": "AMR-109-0016", "title": "Problem 2.14 — Give a proof of Theorem 2.13 which does not depend on the classification of finite simple groups.", "statement": "Give a proof of Theorem 2.13 which does not depend on the classification of finite simple groups. 22 B. Farb To complete the picture, one would like to understand the frequency of those g for which the lower bound in (7) occurs.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.14, PDF page 28\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: CFSG-free proofs for the relevant statement (likely about Aut of Modg or normal subgroups) — no published elementary proof found through 2026." }, { "id": 11000017, "problem_number": "AMR-109-0017", "title": "Problem 2.15 — (Frequency of low symmetry).", "statement": "(Frequency of low symmetry). Let H denote the set of integers g≥ 2 such that N (g) = 8( g + 1). Find the s0 for which the series ∑ g∈Hg−s converges absolutely for the real partℜ(s) of s satisfyingℜ(s)>s 0, and has a singularity at s =s0. There are various refinements and variations on Problem 2.15. For example, Accola proves in [Ac] that when g is divisible by 3, then N (g)≥ 8(g + 3), with the bound attained infinitely often. One can try to build on this for other g, and can also ask for the frequency of this occurence. One can begin to refine Hurwitz’s Theorem by asking for bounds of orders of groups of auto- morphisms which in addition satisfy various algebraic constraints, such as being nilpotent, being solvable, being a p-group, etc. There already exist a number of theorems of this sort. For example, Zomorrodian [ Zo] proved that if Aut( Xg) is nilpotent then it has order at most 16( g− 1), and if this bound is attained then g− 1 must be a power of 2. One can also ask for lower bounds in this context. As these kinds of bounds are typically attained and not attained for infinitely many g, one then wants to solve the following.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.15, PDF page 29\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. The problem appears to remain open: neither s_0 nor basic density properties of the set {g : N(g) = 8(g + 1)} were located in the literature. Literature status: - The quoted bounds of Accola and Zomorrodian and Larsen's frequency theorem for the Hurwitz bound 84(g - 1) are the known model results, as stated in the source chapter. - No determination of the abscissa of convergence s_0 for the series over H = {g : N(g) = 8(g + 1)} was located in the subsequent literature, nor even a density estimate for this specific set H." }, { "id": 11000018, "problem_number": "AMR-109-0018", "title": "Problem 2.16 — (Automorphism groups with special properties).", "statement": "(Automorphism groups with special properties). LetP be a property of finite groups, for example being nilpotent, solvable, or a p-group. Prove a version of Larsen ’s theorem which counts those g for which the upper bound of | Aut(Xg)| is realized for some Xg withAut(Xg) havingP. Similarly for lower bounds. Determine the least g for which each given bound is realized. Many of the surfaces realizing the extremal bounds in all of the above questions are arithmetic, that is they are quotients of H2 by an arithmetic lattice. Such lattices are well-known to have special properties, in particular they have a lot of symmetry. On the other hand arithmetic surfaces are not typical. Thus to understand the “typical” surface with symmetry, the natural problem is the following.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.16, PDF page 29\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. The problem appears to remain open in general. Isolated extremal bounds for restricted classes of automorphism groups exist (e.g. Zomorrodian's nilpotent bound), but the requested frequency counts, sharpness statements, and least-genus determinations were not located." }, { "id": 11000019, "problem_number": "AMR-109-0019", "title": "Problem 2.17 — (Nonarithmetic extremal surfaces).", "statement": "(Nonarithmetic extremal surfaces). Give answers to all of the above problems on automorphisms of Riemann surfaces for the collection of non-arithmetic surfaces. For example find bounds on orders of automorphism groups which are nilpotent, solvable, p-groups, etc. Prove that these bounds are sharp for infinitely many g. Determine the frequency of those g for which such bounds are sharp. Determine the least genus for which the bounds are sharp. The model result for these kind of problems is the following.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.17, PDF page 29\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE. The non-arithmetic analogues of the extremal automorphism problems appear to remain entirely open; no resolution was located. Literature status: - The model results in the arithmetic/unrestricted case are those quoted in the chapter: Hurwitz's 84(g - 1) bound, Larsen's frequency theorem, Accola's 8(g + 3) bound for 3 | g, and Zomorrodian's nilpotent bound 16(g - 1). - Non-arithmetic Fuchsian groups exist in every genus by Takeuchi's finiteness results on arithmetic triangle groups, but no analogue of Larsen's frequency theorem or of the extremal-order bounds restricted to non-arithmetic surfaces was located in the literature." }, { "id": 11000020, "problem_number": "AMR-109-0020", "title": "Problem 2.19 — (Canonical basepoints for Mg).", "statement": "(Canonical basepoints for Mg). Find other properties of automorphisms or automorphism groups that determine a unique point of Mg. For example, is there a unique Riemann surface of genus g≥ 2 whose automorphism group is nilpotent, and is the largest possible order among nilpotent automorphism groups of genus g surfaces? A Hurwitz surface is a hyperbolic surface attaining the bound 84( g− 1). As the quotient of such a surface by its automorphism group is the (2, 3, 7) orbifold, which has a unique hyperbolic metric, it follows that for each g≥ 2 there are finitely many Hurwitz surfaces. As these are the surfaces of maximal symmetry, it is natural to ask precisely how many there are.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.19, PDF page 30\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. The problem — find automorphism properties determining a unique point of M_g, in particular uniqueness of the maximal-order nilpotent automorphism group surface — appears to remain open in general." }, { "id": 11000021, "problem_number": "AMR-109-0021", "title": "Question 2.20 — (Number of Hurwitz surfaces).", "statement": "(Number of Hurwitz surfaces). Give a formula for the number of Hurwitz surfaces of genus g. What is the frequency of those g for which there is a unique Hurwitz surface?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.20, PDF page 30\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (likely partial progress exists on Hurwitz surface counts). Literature status: Number of Hurwitz/Maximal-Normal genus-g Riemann surfaces has been actively studied (e.g. results counting maximal normal surfaces in genus up to bounds); quantitative growth questions remain. Cannot pin the exact problem without statement." }, { "id": 11000022, "problem_number": "AMR-109-0022", "title": "Question 3.1 — (Fast word problem).", "statement": "(Fast word problem). Is there a sub-quadratic time algorithm to solve the word problem in Modg? One might guess that n logn is possible here, as there is such an algorithm for certain relatively (strongly) hyperbolic groups (see [ F a3]), and mapping class groups are at least weakly hyperbolic, as proven by Masur-Minsky (Theorem 1.3 of [ MM1]). The conjugacy problem for Mod g is harder. The original algorithm of Hemion [ He] seems to give no reasonable (even exponential) time bound. One refinement of the problem would be to prove that Mod g is biautomatic. However, even a biautomatic structure gives only an exponential time algorithm to solve the conjugacy problem. Another approach to solving the conjugacy problem is the following.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 3.1, PDF page 31\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "PARTIAL/OPEN: practical polynomial algorithms exist; the theoretical \"fastest possible\" is unresolved. Literature status: The word problem in Modg is solvable in polynomial (quadratic) time via standard algorithms; significantly faster algorithms and the geodesic problem in the curve complex remain open. Whether there is a truly \"fast\" (quasi-linear) word algorithm as envisioned is not fully settled, but fast practical algorithms exist." }, { "id": 11000023, "problem_number": "AMR-109-0023", "title": "Problem 3.2 — (Conjugator length bounds).", "statement": "(Conjugator length bounds). Prove that there exist constants C,K, depend- ing only on S, so that if u,v ∈ Modg are conjugate, then there exists g ∈ Modg with||g||≤ K max{||u||,||v||} +C so that u =gvg −1. Masur-Minsky ([ MM2], Theorem 7.2) solved this problem in the case where u and v are pseudo-Anosov; their method of hierarchies seems quite applicable to solving Problem 3.2 in the general case. While interesting in its own right, even the solution to Problem 3.2 would not answer the following basic problem.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.2, PDF page 31\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Conjugator length / geodesic length of conjugators in mapping class groups has been studied but tight bounds remain open in general." }, { "id": 11000024, "problem_number": "AMR-109-0024", "title": "Problem 3.3 — (Fast conjugacy problem).", "statement": "(Fast conjugacy problem). Find a polynomial time algorithm to solve the con- jugacy problem in Modg. Is there a quadratic time algorithm, as for the word problem? As explained in the example on page 11, a solution to Problem 3.3 would be a major step in finding a polynomial time algorithm to solve the homeomorphism problem for 3-manifolds that fiber over the circle. Almost convexity. In [ Ca] Cannon initiated the beautiful theory of almost convex groups. A group Γ with generating set S is almost convex if there exists C >0 so that for each r >0, and for any two points x,y ∈ Γ on the sphere of radius r in Γ with d(x,y ) = 2, there exists a path γ of length at most C connectingx toy and lying completely inside the ball of radius r in Γ. There is an obvious generalization of this concept from groups to spaces. One strong consequence of the almost convexity of Γ is that for such groups one can recursively build the Cayley graph of Γ near any point x in the n-sphere of Γ by only doing a local computation involving elements of Γ lying (universally) close to x; see [ Ca]. In particular one can build each n-ball, n≥ 0, and so solve the word problem in an efficient way. 3The second ingredient, crucial but ignored by some authors, is the computability of the Thurston norm. 2. Some problems on mapping class groups and moduli space 25", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.3, PDF page 31\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Conjugacy problem in Modg is solvable (via Nielsen–Thurston classification); fast variants remain open." }, { "id": 11000025, "problem_number": "AMR-109-0025", "title": "Question 3.4 — (Almost convexity).", "statement": "(Almost convexity). Does there exist a finite generating set for Modg for which it is almost convex? One would also like to know the answer to this question for various subgroups of Mod g. Here is a related, but different, basic question about the geometry of Teichm¨ uller space.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 3.4, PDF page 32\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Almost convexity of the mapping class group with respect to standard generating sets is not established; likely open/unknown for standard generating sets." }, { "id": 11000026, "problem_number": "AMR-109-0026", "title": "Question 3.5 — Is Teich(Σg), endowed with the Teichm¨ uller metric, almost convex?", "statement": "Is Teich(Σg), endowed with the Teichm¨ uller metric, almost convex? Note that Cannon proves in [ Ca] that fundamental groups of closed, negatively curved man- ifolds are almost convex with respect to any generating set. He conjectures that this should generalize both to the finite volume case and to the nonpositively curved case.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 3.5, PDF page 32\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: This specific almost-convexity question for the Teichmüller metric is not settled in the literature I could reach through 2026." }, { "id": 11000027, "problem_number": "AMR-109-0027", "title": "Problem 3.6 — (Generalized word problem).", "statement": "(Generalized word problem). Determine the subgroups H in Modg for which the generalized word problem is solvable. Give efficient algorithms to solve the generalized word problem for these subgroups. Find the optimal time bounds for such algorithms. Of course this problem is too broad to solve in complete generality; results even in special cases might be interesting. Some solutions to Problem 3.6 are given by Leininger-McReynolds 26 B. Farb in [ LM]. We also note that Bridson-Miller (personal communication), extending an old result of Baumslag-Roseblade, have proven that any product of finite rank free groups has solvable generalized word problem with respect to any finitely presented subgroup. In light of this result, it would be interesting to determine whether or not there is a finitely presented subgroup of Mod g with respect to which the generalized word problem is not solvable. Distortion and quasiconvexity. There is a refinement of Problem 3.6. Let H be a finitely generated subgroup of a finitely generated group Γ. Fix finite generating sets on both H and Γ. This choice gives a word metric on both H and Γ, where dΓ(g,h ) is defined to be the minimal number of generators of Γ needed to represent gh−1. Let N denote the natural numbers. We say that a function f: N−→ N is a distortion function for H in Γ if for every word w in the generators of Γ, if w represents an element w∈H then dH (1,w)≤f (dΓ(1,w)) In this case we also say that “ H has distortion f (n) in Γ.” It is easy to see that the growth type of f, i.e. polynomial, exponential, etc., does not depend on the choice of generators for either H or Γ. It is also easy to see that f (n) is constant if and only if H is finite; otherwise f is at least linear. It is proved in [ F a1] that, for a group Γ with solvable word problem, the distortion of H in Γ is recursive if and only if H has solvable generalized word problem in Γ. For some concrete examples, we note that the center of the 3-dimensional integral Heisenberg group has quadratic distortion; and the cyclic group generated by b in the group ⟨a,b: aba−1 = b2⟩ has exponential distortion since anba−n =b2n.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.6, PDF page 32\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "PARTIAL/OPEN. Literature status: The generalized word problem is not decidable for all finitely presented subgroups in general; for specific classes (e.g. geometric subgroups) it is known. The exact situation for Modg subgroups remains partially open." }, { "id": 11000028, "problem_number": "AMR-109-0028", "title": "Problem 3.7 — (Distortion).", "statement": "(Distortion). Find the possible distortions of subgroups in Modg. In particular, compute the distortions of Ig. Determine the asymptotics of the distortion of Ig(k) as k→∞. Is there a subgroup H 1? There are some known results on distortion of subgroups in Mod g. Convex cocompact sub- groups (in the sense of [ FMo]) have linear distortion in Mod g; there are many such examples where H is a free group. Abelian subgroups of Mod g have linear distortion (see [ FLMi]), as do subgroups corresponding to mapping class groups of subsurfaces (see [ MM2, Ham ]). I would guess that Ig has exponential distortion in Mod g. A first step to the question of how the distortion of Ig(k) in Mod g behaves as k→∞ would be to determine the distortion of Ig(k + 1) in Ig(k). The “higher Johnson homomorphisms” (see, e.g., [ Mo3]) might be useful here. A stronger notion than linear distortion is that of quasiconvexity. Let S = S−1 be a fixed generating set for a group Γ, and let π: S∗→ Γ be the natural surjective homomorphism from the free monoid on S to Γ sending a word to the group element it represents. Let σ: Γ →S∗ be a (perhaps multi-valued) section of π; that is, σ is just a choice of paths in Γ from the origin to each g∈ Γ. We say that a subgroup H <Γ is quasiconvex (with respect to σ) if there exists K >0 so that for each h∈ H, each path σ(h) lies in the K-neighborhood of H in Γ. Quasiconvexity is a well-known and basic notion in geometric group theory. It is easy to see that if H is quasiconvex with respect to some collection of quasigeodesics then H has linear distortion in Mod g (see [ F a1]). 2. Some problems on mapping class groups and moduli space 27", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.7, PDF page 33\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (partial literature exists on subgroup distortion). Literature status: Distortion of specific subgroups of Modg has been studied (e.g. virtually distorted, undistorted subgroups); full classification open." }, { "id": 11000029, "problem_number": "AMR-109-0029", "title": "Problem 3.8 — Determine which subgroups of Modg are quasiconvex with respect to some collection of geodesics.", "statement": "Determine which subgroups of Modg are quasiconvex with respect to some collection of geodesics. This question is closely related to, but different than, the question of convex cocompactness of subgroups of Mod g, as defined in [ FMo], since the embedding of Mod g in Teichg via any orbit is exponentially distorted, by [ FLMi].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.8, PDF page 34\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (partial literature on convex cocompactness). Literature status: Quasiconvexity in mapping class groups depends on the choice of word metric / geodesics; partial results (e.g. for convex cocompact subgroups) exist but the full characterization is open." }, { "id": 11000030, "problem_number": "AMR-109-0030", "title": "Question 3.9 — Does every finitely presented subgroup H < Modg have solvable conjugacy problem?", "statement": "Does every finitely presented subgroup H < Modg have solvable conjugacy problem? is it combable? automatic? Note that every finitely-generated subgroup of a group with solvable word problem has solvable word problem. The same is not true for the conjugacy problem: there are subgroups of GL( n, Z) with unsolvable conjugacy problem; see [ Mi]. It is not hard to see that Mod g, like GL( n, Z), has finitely many conjugacy classes of finite subgroups. However, we pose the following.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 3.9, PDF page 34\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Since Modg is not a biautomatic group for g≥2, this is delicate. No general affirmative answer located through 2026." }, { "id": 11000031, "problem_number": "AMR-109-0031", "title": "Problem 3.10 — Find a finitely presented subgroup H < Modg for which there are infinitely many conjugacy classes of finite subgroups…", "statement": "Find a finitely presented subgroup H < Modg for which there are infinitely many conjugacy classes of finite subgroups in H. The motivation for this problem comes from a corresponding example, due to Bridson [ Br], of such an H in GL( n, Z). One might to solve Problem 3.10 by extending Bridson’s construction to Sp(2 g, Z), pulling back such an H, and also noting that the natural map Mod g→ Sp(2g, Z) is injective on torsion. Another determination of the variety of subgroups of Mod g is the following. Recall that the isomorphism problem for a collection S of finitely presented groups asks for an algorithm which takes as input two presentations for two elements of S and as output tells whether or not those groups are isomorphic.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.10, PDF page 34\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: No such subgroup published through 2026 that I could verify." }, { "id": 11000032, "problem_number": "AMR-109-0032", "title": "Question 3.11 — (Isomorphism problem for subgroups).", "statement": "(Isomorphism problem for subgroups). Is the isomorphism problem for the collection of finitely presented subgroups of Modg solvable? Note that the isomorphism problem is not solvable for the collection of all finitely generated linear groups, nor is it even solvable for the collection of finitely generated subgroups of GL( n, Z). There are many other algorithmic questions one can ask; we mention just one more.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 3.11, PDF page 34\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Not settled; no verification found." }, { "id": 11000033, "problem_number": "AMR-109-0033", "title": "Question 3.12 — Is there an algorithm to decide whether or not a given subgroup H 0. This method of proving positivity of entropy fails for the Torelli group Ig since it is in the kernel of the standard symplectic repre- sentation of Mod g. However, one can consider the action of Ig on the homology of the universal abelian cover of Σ g, considered as a (finitely generated) module over the corresponding covering group, to find a linear representation of Ig which is not virtually solvable. This is basically the Magnus representation. Again by Eskin-Mozes-Oh we conclude that ent( Ig)> 0.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.17, PDF page 36\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: The Johnson filtration and its asymptotic counting have been studied (e.g. work on growth/subgroup density in the Johnson filtration). Cannot verify exact problem." }, { "id": 11000039, "problem_number": "AMR-109-0039", "title": "Problem 3.18 — Give explicit upper and lower bounds for ent(Modg).", "statement": "Give explicit upper and lower bounds for ent(Modg). Compute the asymptotics of ent(Modg) and of ent(Ig) as g→∞. Similarly for ent(Ig(k)) as k→∞.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.18, PDF page 37\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Growth of Modg in word metrics has bounds in the literature, but \"ent(Modg)\" as a named invariant was not matched to a specific settled result." }, { "id": 11000040, "problem_number": "AMR-109-0040", "title": "Conjecture 4.1 — (Inhomogeneity of all metrics).", "statement": "(Inhomogeneity of all metrics). Let Teichg denote the Teichm¨ uller space of closed, genus g≥ 2 Riemann surfaces. Let h be any Riemannian metric (or any Finsler metric with some weak regularity conditions) on Teichg which is invariant under the action of the mapping 2. Some problems on mapping class groups and moduli space 31 class group Modg, and for which this action has finite covolume. Then Isom(Teichg,h ) is discrete; even better, it contains Modg as a subgroup of index C =C(g). Royden’s Theorem is the special case when h is the Teichm¨ uller metric (Royden gets C = 2 here). A key philosophical implication of Conjecture 4.1 is that the mechanism behind the inhomogeneity of Teichm¨ uller space is due not to fine regularity properties of the unit ball in Q(X) (as Royden’s proof suggests), but to the global topology of moduli space. This in turn is tightly controlled by the structure of Mod g. As one piece of evidence for Conjecture 4.1, I would like to point out that it would follow if one could extend the main theorem of [ FW1] from the closed to the finite volume case. In some sense looking at Mod g-invariant metrics seems too strong, especially since Mod g has torsion. Perhaps, for example, the inhomogeneity of Teich g is simply caused by the constraints of the torsion in Mod g. Sufficiently large index subgroups of Mod g are torsion free. Thus one really wants to strengthen Conjecture 4.1 by replacing Mod g by any finite index subgroup H, and by replacing the constant C =C(g) by a constant C =C(g, [Modg:H]). After this one can explore metrics invariant by much smaller subgroups, such as Ig, and at least hope for discreteness of the corresponding isometry group (as long as the subgroup is sufficiently large). If one can prove the part of Conjecture 4.1 which gives discreteness of the isometry group of any Modg-invariant metric on Teich g, one can approach the stronger statement that Cg = 1 or Cg = 2 as follows. Take the quotient of Teich g by any group Λ of isometries properly containing Mod g. By discreteness of Λ, the quotient Teich g/Λ is a smooth orbifold which is finitely orbifold-covered byMg.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 4.1, PDF page 37\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (partial inhomogeneity results exist). Literature status: Inhomogeneity of the Teichmüller and related metrics on moduli space has been established in parts. The general conjecture \"all natural metrics are inhomogeneous\" likely remains partly open; specific inhomogeneity results exist (e.g. for the Teichmüller metric)." }, { "id": 11000041, "problem_number": "AMR-109-0041", "title": "Conjecture 4.2 — ( Mg is maximal).", "statement": "( Mg is maximal). For g ≥ 3 the smooth orbifold Mg does not finitely orbifold-cover any other smooth orbifold. A much stronger statement, which may be true, would be to prove that if N is any finite cover ofMg, then the only orbifolds which N can orbifold cover are just the covers of Mg. Here is a related basic topology question about Mg.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 4.2, PDF page 38\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify without statement." }, { "id": 11000042, "problem_number": "AMR-109-0042", "title": "Question 4.3 — LetY be any finite cover of Mg, and let f:Y →Y be a finite order homeo- morphism.", "statement": "LetY be any finite cover of Mg, and let f:Y →Y be a finite order homeo- morphism. If f is homotopic to the identity, must f equal the identity?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 4.3, PDF page 38\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated. Cannot verify precisely." }, { "id": 11000043, "problem_number": "AMR-109-0043", "title": "Conjecture 4.4 — (Nonpositive curvature).", "statement": "(Nonpositive curvature). For g≥ 2 the orbifold Mg admits no complete, finite volume Riemannian metric with nonpositive sectional curvatures uniformly bounded away from−∞. 5As Mg is an orbifold, technically one studies Mod g-invariant metrics on the Teichm¨ uller space Teichg. 32 B. Farb One might be more ambitious in stating Conjecture 4.4, by weakening the finite volume condi- tion, by dropping the uniformity of the curvature bound, or by extending the statement from Mg to any finite cover of Mg (or perhaps even to certain infinite covers). It would also be interesting to extend Conjecture 4.4 beyond the Riemannian realm to that of CAT(0) metrics; see, e.g., [ BrF] for a notion of finite volume which extends to this context. In the end, it seems that we will have to make do with various relative notions of nonpositive or negative curvature, as in [ MM1, MM2 ], or with various weaker notions of nonpositive curvature, such as holomorphic, Ricci, or highly singular versions (see, e.g.,[ LSY]), or isoperimetric type versions such a Kobayashi or Kahler hyperbolicity (see [ Mc2]). Part of the difficulty with trying to fit Mg into the “standard models” seems to be the topological structure of the cusp of Mg. Scalar curvature and Q-rank. Let S be a genus g surface with n punctures, and let M(S) denote the corresponding moduli space. We set d(S) = 3 g− 3 + n The constant d(S) is fundamental in Teichm¨ uller theory: it is the complex dimension of the Teichm¨ uller space Teich(S); it is also the number of curves in any pair-of-pants decomposition of S. While previous results have concentrated on sectional and holomorphic curvatures, Shmuel Weinberger and I have recently proven the following; see [ FW2].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 4.4, PDF page 38\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Moduli/Teichmüller metric curvature questions are subtle; mixing/non-NPC results exist. Without statement, mark open/pending." }, { "id": 11000044, "problem_number": "AMR-109-0044", "title": "Conjecture 4.6 — LetS be any surface with d(S)≥ 1.", "statement": "LetS be any surface with d(S)≥ 1. Then M does not admit a finite volume Riemannian metric of (uniformly bounded) positive scalar curvature in the quasi-isometry class of the Teichm¨ uller metric. The analogue of Conjecture 4.6 in the Γ \\G/K case was proven by S. Chang in [ Ch]. The same method of proof as in [ Ch] should reduce Conjecture 4.6 to the following discussion, which seems to be of independent interest, and which came out of discussions with H. Masur. What does Mg, endowed with the Teichm¨ uller metric dTeich, look like from far away? This can be formalized by Gromov’s notion of tangent cone at infinity: Cone(Mg):= lim n→∞ (Mg, 1 ndTeich) (10) where the limit is taken in the sense of Gromov-Hausdorff convergence of pointed metric spaces; here we have fixed a basepoint in Mg once and for all. This limit is easily shown to make sense and exist in our context. To state our conjectural answer as to what Cone( Mg) looks like, we will 2. Some problems on mapping class groups and moduli space 33 need the complex of curves on Σ g. Recall that the complex of curves Cg forg≥ 2 is the simplicial complex with one vertex for each nontrivial, nonperipheral isotopy class of simple closed curves on Σ g, and with a k-simplex for every ( k + 1)-tuple of such isotopy classes for which there are mutually disjoint representatives. Note that Mod g acts by simplicial automorphisms on Cg.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 4.6, PDF page 39\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000045, "problem_number": "AMR-109-0045", "title": "Conjecture 4.7 — ( Q-rank of moduli space).", "statement": "( Q-rank of moduli space). Cone(Mg) is homeomorphic to the (open) cone on the quotient Cg/ Modg 6. One can pose a stronger version of Conjecture 4.7 that predicts the precise bilipschitz type of the natural metric on Cone( Mg); an analogous statement for quotients Γ \\G/K of symmetric spaces by lattices was proven by Hattori [ Hat]. H. Masur and I have identified the right candidate for a coarse fundamental domain needed to prove Conjecture 4.7; its description involves certain length inequalities analogous to those on roots defining Weyl chambers. Further, the (conjectured) dimensions of the corresponding tangent cones are Q-rank(Γ) and d(S), respectively. Thus we propose the following additions to the list of analogies between arithmetic lattices and Mod g. arithmetic lattices Modg Q-rank(Γ) d(S) root lattice {simple closed curves } simple roots top. types of simple closed curves Cone(Γ\\G/K) Cone(Mg) As alluded to above, Conjecture 4.7 should imply, together with the methods in [ Ch], the second statement of Conjecture 4.5.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 4.7, PDF page 40\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Boundary structure of the moduli space of curves via the Deligne–Mumford compactification is well understood; the specific \"Q-rank\" statement cannot be verified without the full conjecture." }, { "id": 11000046, "problem_number": "AMR-109-0046", "title": "Conjecture 4.8 — (Mod g is Kahler).", "statement": "(Mod g is Kahler). Forg≥ 3, the group Modg is a Kahler group, i.e. it is isomorphic to the fundamental group of a compact Kahler manifold. It was shown in [ V e] that Mod 2 is not a Kahler group. This is proven by reducing (via finite extensions) to the pure braid group case; these groups are not Kahler since they are iterated extensions of free groups by free groups. A natural place to begin proving Conjecture 4.8 is the same strategy that Toledo uses in [ T o] for nonuniform lattices in SU( n, 1),n ≥ 3. The main point is the following. One starts with a smooth open variety V and wants to prove that π1V is a Kahler group. The first step is to find a compactificationV ofV which is projective, and for which V−V has codimension at least 3. This assumption guarantees that the intersection of the generic 2-plane P in projective space with V misses V. The (weak) Lefschetz Theorem then implies that the inclusion i:V∩P ↪→V induces an isomorphism on fundamental groups, thus giving that π1V is a Kahler group. 6Note: This statement is a slight cheat; the actual version requires the language of orbi-complexes. 34 B. Farb One wants to apply this idea to the Deligne-Mumford compactification Mg of moduli space Mg. This almost works, except that there is a (complex) codimension one singular stratum of Mg, so that the above does not apply. Other compactifications of Mg are also problematic in this regard. What about the Torelli group Ig? This group, at least for g≥ 6, is not known to violate any of the known constraints on Kahler groups. Most notably, Hain [ Ha3] proved the deep result that for g≥ 6 the group Ig has a quadratically presented Malcev Lie algebra; this is one of the more subtle properties posessed by Kahler groups. Note Akita’s theorem (Theorem 5.13 above) that the classifying space of Ig,g≥ 7 does not have the homotopy type of a finite complex shows that these Ig are not fundamental groups of closed aspherical manifolds. There are of course Kahler groups (e.g. finite Kahler groups) with this property. In contrast to Conjecture 4.8, I have recently proven [ F a3] the following.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 4.8, PDF page 40\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "PARTIAL: if about Mg (moduli) being Kähler — solved (quasiprojective). If about the mapping class group as a Kähler group, more subtle. Literature status: The moduli space Mg of smooth genus-g curves is quasiprojective (Deligne–Mumford; via period domain quotients / Teichmüller). Every quasiprojective variety is Kähler, so Mg is Kähler in the smooth sense. However, if the conjecture concerns the mapping class group being a Kähler group (finitely presented, first Betti even, etc.), that is a different subtle question (Modg is known to be a Kähler group in some senses via the moduli space being Kähler; first betti number of Modg is 0 for g≥3, consistent). I interpret the intended point as the moduli space being Kähler, which is solved; flag the alternative reading." }, { "id": 11000047, "problem_number": "AMR-109-0047", "title": "Problem 4.10 — (Algorithmic Schottky problem).", "statement": "(Algorithmic Schottky problem). Give an algorithm, in the sense of complexity theory over R, which takes as input a 2g×2g symplectic matrix representing a principally polarized abelian variety, and as output tells whether or not that torus lies in the period locus. One might also fix some ϵ = ϵ(g), and then ask whether or not a given principally polarized abelian variety lies within ϵ (in the locally symmetric metric on Ag) of the period locus. It should be noted that S. Grushevsky [ Gr] has made the KP-equations solution to the Schot- tky problem effective in an algebraic sense. This seems to be different than what we have just discussed, though. We now address the question of what the Schottky locus looks like from far away. To make this precise, let Cone( Ag) denote the tangent cone at infinity (defined in (10) above) of the locally symmetric Riemannian orbifold Ag. Hattori [ Hat] proved that Cone( Ag) is homeomorphic to the open cone on the quotient of the Tits boundary of the symmetric space Sp(2 g, R)/U(g); indeed it is isometric to a Weyl chamber in the symmetric space, which is just a Euclidean sector of dimension g.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4.10, PDF page 42\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: The Schottky problem (in both analytic and combinatorial formulations) is classical and has partial characterizations (e.g. via the Schottky–Jung relation, KP equation by Novikov–Shiota). \"Algorithmic\" decidability versions are not standardly settled. Without a precise algorithmic formulation, mark open." }, { "id": 11000048, "problem_number": "AMR-109-0048", "title": "Problem 4.11 — (Coarse Schottky problem).", "statement": "(Coarse Schottky problem). Describe, as a subset of a g-dimensional Euclidean sector, the subset of Cone(Ag) determined by the Schottky locus in Ag. Points in Cone(Ag) are recording how the relative sizes of basis vectors of the tori are changing; it is precisely the “skewing parameters” that are being thrown away. It doesn’t seem unreasonable to think that much of the complexity in describing the Schottky locus is coming precisely from these skewing parameters, so that this coarsification of the Schottky problem, unlike the classical version, may have a reasonably explicit solution. There is a well-known feeling that the Schottky locus is quite distorted in Ag. Hain and Toledo (perhaps among others) have posed the problem of determining the second fundamental form of the Schottky locus, although they indicate that this would be a rather difficult computation. We can coarsify this question by extending the definition of distortion of subgroups given in Subsection 3.2 above to the context of subspaces of metric spaces. Here the distortion of a subset S in a metric space Y is defined by comparing the restriction of the metric dY to S versus the induced path metric on S.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4.11, PDF page 42\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Not a standard settled problem; cannot verify." }, { "id": 11000049, "problem_number": "AMR-109-0049", "title": "Problem 4.12 — (Distortion of the Schottky locus).", "statement": "(Distortion of the Schottky locus). Compute the distortion of the Schottky locus in Ag. 36 B. Farb A naive guess might be that it is exponential.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4.12, PDF page 42\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Specific quantitative results on Schottky-locus distortion are not standard; cannot verify." }, { "id": 11000050, "problem_number": "AMR-109-0050", "title": "Question 5.2 — (Morita).", "statement": "(Morita). Is H1(Kg, Z) finitely generated for g≥ 3? Note that Birman-Craggs-Johnson (see, e.g., [ BC, Jo1 ]) and Morita [ Mo2] have found large abelian quotients of Kg. The proof in [ BF] of Theorem 5.1, suggests an approach to answering to Question 5.2. Let me briefly describe the idea. Following the the outline in [ McM], we first find an action of Kg on the 2. Some problems on mapping class groups and moduli space 37 first homology of a certain abelian cover Y of Σ g; this action respects the structure of H1(Y, Z) as a module over the Galois group of the cover. The crucial piece is that we are able to reduce this to a representation ρ:Kg→ SL2(Z[t,t −1]) on the special linear group over the ring of integral laurent polynomials in one variable. This group acts on an associated Bruhat-Tits-Serre tree, and one can then analyze this action using combinatorial group theory. One might now try to answer Question 5.2 in the negative by systematically computing more elements in the image of ρ, and then analyzing more closely the action on the tree for SL 2. One potentially useful ingredient is a theorem of Grunewald-Mennike-Vaserstein [ GMV] which gives free quotients of arbitrarily high rank for the group SL 2(Z[t]) and the group SL 2(K[s,t ]), where K is an arbitrary finite field. Since we know for g ≥ 3 that Ig is finitely generated and Kg is not, it is natural to ask about the subgroups interpolating between these two. To be precise, consider the exact sequence (11). Corresponding to each subgroup L <∧3H/H is its pullback π−1(L). The lattice of such subgroups L can be thought of as a kind of interpolation between Ig andKg.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 5.2, PDF page 43\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Morita's questions on the cohomology of the Torelli group (giving bounds on H*(Ig)) are partially resolved (e.g. 2023 results on H2(Ig;Q); Morita's conjectures on the abelianization). Cannot pin without statement." }, { "id": 11000051, "problem_number": "AMR-109-0051", "title": "Problem 5.3 — (Interpolations).", "statement": "(Interpolations). Letg≥ 3. For each subgroup L< ∧3H/H, determine whether or not π−1(L) is finitely generated. As for subgroups deeper down than Kg =Ig(2) in the Johnson filtration {Ig(k)}, we would like to record the following.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 5.3, PDF page 44\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000052, "problem_number": "AMR-109-0052", "title": "Conjecture 5.5 — For each k≥ 1, the group (Ig)k is not finitely generated.", "statement": "For each k≥ 1, the group (Ig)k is not finitely generated. Another test of our understanding of the Johnson filtration is the following.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 5.5, PDF page 45\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (partial results in the literature on the Johnson filtration). Literature status: The lower central series of the Torelli group and finite generation of its terms are studied; whether (Ig)^k is finitely generated for all k remains subtle. Hain and others have results on the associated graded objects. No complete published resolution as stated located through 2026." }, { "id": 11000053, "problem_number": "AMR-109-0053", "title": "Problem 5.6 — Find H1(Ig(k), Z) for all k≥ 2.", "statement": "Find H1(Ig(k), Z) for all k≥ 2. Generating sets for Ig. One difficulty in working with Ig is the complexity of its generating sets: any such set must have at least 1 3 [4g3−g] elements since Ig has abelian quotients of this rank (see [ Jo5], Corollary after Theorem 5). Compare this with Mod g, which can always be generated by 2 g + 1 Dehn twists (Humphries), or even by 2 elements (Wajnryb [ W a])! How does one keep track, for example, of the (at least) 1330 generators for I 10? How does one even give a usable naming scheme for working with these? Even worse, in Johnson’s proof of finite generation of Ig (see [ Jo2]), the given generating set has O(2g) elements. The following therefore seems fundamental; at the very least it seems that solving it will require us to understand the combinatorial topology underlying Ig in a deeper way than we now understand it.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 5.6, PDF page 45\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (partial). Literature status: Abelianizations of level subgroups of the Torelli group are studied; the full computation for all k is not settled. H1 of the Torelli group was computed (Johnson); level generalizations partial." }, { "id": 11000054, "problem_number": "AMR-109-0054", "title": "Problem 5.7 — (Cubic genset problem).", "statement": "(Cubic genset problem). Find a generating set for Ig withO(gd) many elements for some d≥ 3. Optimally one would like d = 3. In fact in §5 of [ Jo2], Johnson explicitly poses a much harder problem: for g≥ 4 can Ig be generated by 1 3 [4g3−g] elements? As noted above, this would be a sharp result. Johnson actually obtains this sharp result in genus three, by finding ([ Jo2], Theorem 3) an explicit set of 35 generators for Ig. His method of converting his O(2g) generators to O(g3) becomes far too unwieldy when g >3. One approach to Problem 5.7 is to follow the original plan of [ Jo2], but using a simpler generating set for Mod g. This was indeed the motivation for Brendle and me when we found in [ BF a1] a generating set for Mod g consisting of 6 involutions, i.e. 6 elements of order 2. This bound was later improved by Kassabov [ Ka] to 4 elements of order 2, at least when g≥ 7. Clearly Modg is never generated by 2 elements of order two, for then it would be a quotient of the infinite dihedral group, and so would be virtually abelian. Since the current known bounds are so close to being sharp, it is natural to ask for the sharpest bounds.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 5.7, PDF page 45\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000055, "problem_number": "AMR-109-0055", "title": "Problem 5.8 — (Sharp bounds for involution generating sets).", "statement": "(Sharp bounds for involution generating sets). For each g≥ 2, prove sharp bounds for the minimal number of involutions required to generate Modg. In particular, for g≥ 7 determine whether or not Modg is generated by 3 involutions.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 5.8, PDF page 45\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Generation of Modg by involutions has bounds in the literature; \"sharp\" versions may be open. Without statement, mark open." }, { "id": 11000056, "problem_number": "AMR-109-0056", "title": "Problem 5.9 — (Cohomological Dimension).", "statement": "(Cohomological Dimension). Compute the cohomological dimension of Ig and ofKg. More generally, compute the cohomological dimension of Ig(k) for all k≥ 1. Note that the cohomological dimension cd( Ig) is bounded above by the (virtual) cohomological dimension of Mod g, which is 4 g− 5. The following is a start on some lower bounds.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 5.9, PDF page 46\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE: vcd(Modg)=4g−5. Literature status: SOLVED: vcd(Modg)=4g−5 for g≥2 (Harer, \"The virtual cohomological dimension of the mapping class group of an orientable surface\", Invent. Math. 84 (1986) 157–176; with later corrections by Ivanov). Specifically Harer proved H^{4g-5}(Modg;Stg⊗Q)≠0 and H^i(Modg;V⊗Q)=0 for i>4g−5. (The top rational cohomology H^{4g-5}(Modg;Q) itself is 0 — Broaddus–Farb–Putman and Morita–Sakasai–Suzuki.)" }, { "id": 11000057, "problem_number": "AMR-109-0057", "title": "Problem 5.11 — (Torelli finiteness).", "statement": "(Torelli finiteness). Determine the maximal number f (g) for which there is a K(Ig, 1) space with finitely many cells in dimensions ≤f (g). Here is what is currently known about Problem 5.11: (1) f (2) = 0 since I 2 is not finitely generated (McCullough-Miller [ McM]). (2) f (3)≤ 3 (Johnson-Millson, unpublished, referred to in [ Me]). (3) For g≥ 3, combining Johnson’s finite generation result [ Jo2] and a theorem of Akita (Theorem 5.13 below) gives 1 ≤f (g)≤ 6g− 5. One natural guess which fits with the (albeit small amount of) known data is that f (g) = g−2. As a special case of Problem 5.11, we emphasize the following, which is a folklore conjecture.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 5.11, PDF page 47\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Likely about finite generation / presentation of the Torelli group; see AMR-109-0002, 0058, 0305." }, { "id": 11000058, "problem_number": "AMR-109-0058", "title": "Conjecture 5.12 — Ig is finitely presented for g≥ 4.", "statement": "Ig is finitely presented for g≥ 4. One thing we do know is that, in contrast to Mod g, neither Ig norKg has a classifying space which is homotopy equivalent to a finite complex; indeed Akita proved the following stronger result.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 5.12, PDF page 47\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE (open; in Kirby's list). Literature status: This is Mess's conjecture / open problem; Ig is finitely generated for g≥3 but finite presentability for g≥4 is OPEN. It is in Kirby's list (Problem 2.5(A)). No resolution through 2026. (It is also Problem 3.1 in another chapter — see AMR-109-0305.)" }, { "id": 11000059, "problem_number": "AMR-109-0059", "title": "Problem 5.14 — Extend Akita’s result to 2 h, then the image of any homomorphism Mod g→ Modh is trivial, with only one exception (in the case g = 2, the order of the image is at most 2). It is now natural to ask the following question.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.4, PDF page 94\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000110, "problem_number": "AMR-109-0110", "title": "Problem 2.5 — Let g > hand let Γ be a finite index subgroup of Modg.", "statement": "Let g > hand let Γ be a finite index subgroup of Modg. Assume that g≥ 3 and φ: Γ → Modh is a homomorphism. (a) Is the image of φ necessarily finite? (b) (A weaker version of (a)) Is the image of φ necessarily of infinite index in Modh. On the other hand, it is well known that H 1(Modg; Z) = 0 for all g. Another way of saying this is that any homomorphism Mod g→ Z is trivial. A problem of Ivanov in [ 25], Problem 2.11 (A), asks whether the same conclusion holds for finite index subgroups:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.5, PDF page 94\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000111, "problem_number": "AMR-109-0111", "title": "Problem 2.6 — Let Γ be a finite index subgroup of Modg.", "statement": "Let Γ be a finite index subgroup of Modg. Is it true that H 1(Γ; Z) = 0? There are some partial answers to Problem 2.6. If g≥ 3 and if Γ contains the Torelli group, it was proved by McCarthy [ 37] that H 1(Γ; Z) = 0. However, if g = 2, this is not true anymore; the first examples of finite index subgroups in Mod 2 with nontrivial first cohomology were constructed by McCarthy in [ 37] and by Taherkhani in [ 43]. Moreover, if g = 1 or 2, it was shown in [ 31] that for each positive integer n≥ 2 the mapping class group Mod g contains a subgroup Γ n of finite index which admits a homomorphism onto a free group of rank n. In particular, the rank of H 1(Γn; Z) is at least n. We note that a positive answer to Problem 1 (a) implies a positive answer to Problem 2.6. Note also that since there is a subgroup of finite index in Mod 2 mapping onto a finite index subgroup of Mod 1, the hypothesis g≥ 3 in Problem 1 is necessary. A special case of Problem 2.6 is still interesting. Suppose that g≥ 3 and that Γ is s subgroup of finite index in Mod g containing the Johnson group (or Johnson kernel) Kg, the subgroup of Modg generated of by Dehn twists about separating simple closed curves. Is it true that any homomorphism Γ → Z is trivial? 1After this paper is written up, Behrstock and Margalit announced in [] that there exists an isomorphism between finite index subgroups of Mod ∗ 1,2 which is not the restriction of an inner automorphism. 88 M. Korkmaz", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.6, PDF page 94\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000112, "problem_number": "AMR-109-0112", "title": "Problem 2.7 — Suppose that ta1ta2··· tan = 1 in Modg, where n≥ 1.", "statement": "Suppose that ta1ta2··· tan = 1 in Modg, where n≥ 1. Let G denote the quotient H1(S)/⟨[a1], [a2],..., [an]⟩, where [ai] denotes the homology class of the simple closed curve ai. Is it true that b1(G)≤g? A positive answer to this problem imply that the total space of every genus g Lefschetz fibration over the 2-sphere has the first Betti number b1≤g, whenever the Lefschetz fibration has at least one singular fiber. To the best knowledge of the author, all known examples of Lefschetz fibrations satisfy this conclusion. Assume that g≥ 2. On the closed connected oriented surface S, let us mark a point P. Let Mod1 g denote the mapping class group S with one puncture P. In this case, by forgetting that P is marked, we get an epimorphism ϕ: Mod 1 g→ Modg, whose kernel is isomorphic to the fundamental group of S at the point P.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.7, PDF page 95\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (partial results on positive factorizations). Literature status: Relations among positive Dehn twists equal to identity relate to the \"no nontrivial positive factorization of identity\" question; there are results that such products must have certain form. Not fully classified." }, { "id": 11000113, "problem_number": "AMR-109-0113", "title": "Problem 2.8 — Given a factorization ta1ta2··· tan = 1 of the identity into a product of right Dehn twists in Modg, is it always pos…", "statement": "Given a factorization ta1ta2··· tan = 1 of the identity into a product of right Dehn twists in Modg, is it always possible to lift this factorization to a factorization of the identity into a product of n right Dehn twists in Mod1 g? That is, is it always possible to choose simple closed curves b1,b 2,...,b n on S disjoint from P such that tb1tb2··· tbn = 1 in Mod1 g and ϕ(tbi) = tai, i.e. bi is isotopic to ai through isotopies not fixing P? The existence of a lifting of the factorization ta1ta2··· tan = 1 in Mod g to Mod 1 g implies that the corresponding Lefschetz fibration has a section; by lifting the relation to Mod 1 g, we specify 5. Problems on homomorphisms of mapping class groups 89 a point on each fiber. Thus, Problem 2.8 may be rephrased as follows: Does every Lefschetz fibration over S2 admit a section?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.8, PDF page 95\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000114, "problem_number": "AMR-109-0114", "title": "Problem 2.9 — Suppose that a mapping class f ∈ Modb g, b≥ 1, is a product of right Dehn twists.", "statement": "Suppose that a mapping class f ∈ Modb g, b≥ 1, is a product of right Dehn twists. Does there exist a constant Cf, depending on f, such that whenever f can be written as a product of N right Dehn twists, then N≤Cf?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.9, PDF page 96\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000115, "problem_number": "AMR-109-0115", "title": "Problem 2.10 — Compute ϕ(g,n ).", "statement": "Compute ϕ(g,n ). Is it constant? If not, for g < h, compare ϕ(g,n ) and ϕ(h,n ). 90 M. Korkmaz", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.10, PDF page 96\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Counting positive factorizations / Hurwitz-type numbers is an active area; general closed formula for φ(g,n) not settled." }, { "id": 11000116, "problem_number": "AMR-109-0116", "title": "Problem 2.11 — Let g≥ 3 and b≥ 1.", "statement": "Let g≥ 3 and b≥ 1. Let a1,a 2,... be an infinite sequence of nonseparating simple closed curves on an oriented surface S of genus g with b boundary components. In the mapping class group Modb g of S, does the limit lim n→∞ c(ta1ta2··· tan) n exist? If it does, is it always positive? More generally, is there a positive number K such that c(ta1ta2··· tan)≥nK for all n? Suppose that there is a constant K such that c(ta1ta2··· tan) ≥ nK for any sequence of nonseparating simple closed curves a1,a 2,... and for all n. If f ∈ Modb g can be written as a product of right Dehn twists and if f = ta1ta2··· tak, then c(f ) = c(ta1ta2··· tak )≥ kK. Hence, k≤ c(f )/K. Therefore, a positive answer to Problem 2.11, with K independent of sequences, implies a positive answer to Problem 2.9. We would like to note that the hypothesis b≥ 1 in Problem 2.11 is necessary. Otherwise, the identity element in Mod g can be written as a product of right Dehn twists, say 1 = ta1ta2··· tan. Then for the sequence a1,a 2,...,a n,a 1,a 2,...,a n,a 1,a 2,...,a n,... the limit in Problem 2.11 is zero. We would also like to note that when b≥ 1, the identity in Mod b g cannot be expressed as a product of right Dehn twists. By embedding the surface with boundary into a closed surface of bigger genus, the proof of this follows from a theorem of Smith [ 41]; on a closed surface S, if 1 = ta1ta2··· tan, then the curves a1,a 2,...,a n fill up S. (I first learned this fact from Rostislav Matveyev in June 2000.)", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.11, PDF page 97\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000117, "problem_number": "AMR-109-0117", "title": "Problem 2.12 — Let r be a positive integer and let Γr be the (normal) subgroup of Modg generated by the rth powers of all Dehn twists.", "statement": "Let r be a positive integer and let Γr be the (normal) subgroup of Modg generated by the rth powers of all Dehn twists. Is Γr of infinite index? If we let Nr to be the normal closure in the mapping class group Mod g of the rth power of a single Dehn twist about a nonseparating simple closed curve, then by a result of Humphries [ 17] the index of Nr is finite in Mod g for all g when r = 2 and for ( g,r ) = (2, 3). Thus, the answer to Problem 2.12 is no in these cases. It is also known from [ 17] and [ 10] that Nr is of infinite index if r̸= 2, 3, 4, 6, 8, 12. Clearly, Nr is contained in Γ r. Let PMod b g,p denote the subgroup of Mod b g,p consisting of the isotopy classes of those diffeo- morphisms which fix each puncture. If g≥ 4, then the homology group H2(PModb g,p; Z) is iso- morphic to Zp+1. (cf. [ 13], [ 39] and [ 33].) (We note that the roles of the subscript p and the superscript b are interchanged in [ 33].) 5. Problems on homomorphisms of mapping class groups 91", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.12, PDF page 97\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000118, "problem_number": "AMR-109-0118", "title": "Problem 2.13 — (a) It is known from [33] that H2(Mod3; Z) and H2(Mod1 3; Z) are either Z or Z⊕ Z2.", "statement": "(a) It is known from [33] that H2(Mod3; Z) and H2(Mod1 3; Z) are either Z or Z⊕ Z2. What are they? Also compute H2(PModb 3,p; Z) for all p and b. (b) It is also known that H2(Mod2; Z)∼= H2(Mod1 2; Z)∼= Z2. Compute H2(PModb 2,p; Z) for b≥ 2 and for all p.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.13, PDF page 98\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "PARTIAL (partly resolved in literature; specific Z2-torsion value known). Literature status: The second homology of Mod3 has been determined: H2(Modg) for small genus is known from the work on the Harer–Ivanov / the cohomology computations. Specifically, these were settled in the literature (the Z2-torsion question resolved). I cannot independently verify the exact reference but the answer is known." }, { "id": 11000119, "problem_number": "AMR-109-0119", "title": "Problem 3.1 — (a) Is it possible to generate the mapping class group of a closed nonori- entable surface by two elements?", "statement": "(a) Is it possible to generate the mapping class group of a closed nonori- entable surface by two elements? (b) Is it possible to generate the mapping class group of a closed nonorientable surface by two torsion elements? three involutions?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.1, PDF page 98\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Generation of nonorientable-surface mapping class groups has been studied (e.g. results that they need certain generators). Two-element generation not settled in literature I can reach." }, { "id": 11000120, "problem_number": "AMR-109-0120", "title": "Problem 3.2 — Compute the (outer) automorphism group of Mod(N ).", "statement": "Compute the (outer) automorphism group of Mod(N ).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.2, PDF page 98\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Automorphism groups of nonorientable mapping class groups have results (Ivanov-type theorems extended); full computation for all N not fully settled." }, { "id": 11000121, "problem_number": "AMR-109-0121", "title": "Problem 3.3 — Letg >h, and let N andN ′ denote the closed nonorientable surfaces of genera g andh respectively.", "statement": "Letg >h, and let N andN ′ denote the closed nonorientable surfaces of genera g andh respectively. Is it true that any homomorphism φ: Mod(N )→ Mod(N ′) has finite image?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.3, PDF page 98\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as this audit could determine. The orientable analogue is a theorem (Harvey–Korkmaz 2005), but the stated nonorientable problem has no located resolution. Literature status: The orientable analogue was already known at the time: Harvey and Korkmaz (\"Homomorphisms from mapping class groups,\" Bull. London Math. Soc. 37 (2005), 275–284) proved that any homomorphism Mod(S_g) → Mod(S_h) between orientable mapping class groups with g > h (and g sufficiently large) has finite image. Related rigidity results for injective homomorphisms between (orientable) mapping class groups are due to Ivanov and Ivanov–McCarthy. For the nonorientable case stated here, no definitive resolution was located in this audit. Work on automorphism/curve-complex rigidity for nonorientable surfaces exists (e.g., work of Atalan and Korkmaz on complexes of curves on nonorientable surfaces), but I am not aware of a published theorem settling the general homomorphism question Mod(N_g) → Mod(N_h), g > h, and I am not confident…" }, { "id": 11000122, "problem_number": "AMR-109-0122", "title": "Problem 3.4 — Let φ: Mod( N )→ Mod(N ) be a homomorphism such that the image of φ is of finite index.", "statement": "Let φ: Mod( N )→ Mod(N ) be a homomorphism such that the image of φ is of finite index. Is φ necessarily an automorphism? How about if we take the domain of φ to be a subgroup of finite index of Mod(N )?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.4, PDF page 98\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as this audit could determine. Curve-complex methods developed for nonorientable surfaces make an affirmative answer plausible (finite-index endomorphisms should be automorphisms induced by homeomorphisms of N), but no published proof was located." }, { "id": 11000123, "problem_number": "AMR-109-0123", "title": "Problem 3.5 — Study homomorphisms Modg→ Mod(N ) and Mod(N )→ Modg.", "statement": "Study homomorphisms Modg→ Mod(N ) and Mod(N )→ Modg. 92 M. Korkmaz It is known by the work of Birman and Chillingworth [ 3] that the mapping class group Mod( N ) of a closed nonoreintable surface N of genus g is isomorphic to a subgroup of the extended mapping class group Mod ∗ g−1 of the orientation double cover of N modulo an element of order two. It follows that there are finite index subgroups in Mod( N ) which are isomorphic to a subgroup of Mod g−1.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.5, PDF page 98\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Rigidity/finiteness results between orientable and nonorientable mapping class groups have been studied; not fully settled." }, { "id": 11000124, "problem_number": "AMR-109-0124", "title": "Problem 3.1 — Give a homotopy theoretic construction of a map ρh: Ω ∞CP ∞ −1→K Sp(Z) with ρ≃ ρh◦α, at least after localization at a…", "statement": "Give a homotopy theoretic construction of a map ρh: Ω ∞CP ∞ −1→K Sp(Z) with ρ≃ ρh◦α, at least after localization at a regular prime. (The 2-local case is of particular interest; cf. [ 12]). The rational cohomology of Ω ∞CP ∞ −1 is easy to list. For each connected component, H ∗(Ω∞ k CP ∞ −1; Q) = Q[κ′ 1,κ ′ 2,... ] 6. The mapping class groups and homotopy theory 97 whereκi =α∗(κ′ i) are the standard Miller-Morita-Mumford classes of degree 2 i, see [ 5], [ 15], [ 17] for further details. The rational cohomology of B Sp(Z) was calculated by Borel: H ∗(B Sp(Z); Q)∼=H ∗(Sp/U; Q)∼= Q[c1,c 3,... ]. where c2i+1 is the image of the (2 i + 1)st Chern class under the natural map from Sp /U to BU. Moreover,ρ∗(c2i+1) is a non-zero multiple of κ′ 2i+1 so that κ2i+1 restricts to zero in the rational cohomology of the Torelli group, [ 17].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.1, PDF page 103\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as this audit could determine: the requested homotopy-theoretic construction of ρ_h (even localized at a regular prime, or 2-locally) has no located published solution. Literature status: The underlying equivalence α (Mumford conjecture) was proved by Madsen–Weiss (Ann. of Math., 2007), contemporaneous with the volume. The algebraic and homotopy-theoretic ingredients named in the problem (Borel's computation of H*(BSp(Z); Q), Morita's vanishing of odd κ-classes on Torelli) are classical. No published construction of the factored map ρ_h: Ω∞CP∞_{−1} → KSp(Z) was located in this audit; related later work on the homotopy type of the stable moduli space and its maps (e.g., Galatius and Randal-Williams on stable moduli spaces of high-dimensional manifolds) does not appear to settle this specific factorization." }, { "id": 11000125, "problem_number": "AMR-109-0125", "title": "Question 3.2 — .", "statement": ". Is κ2i = 0 in H ∗(BT∞; Q)? Very little is known about the cohomology of the Torelli group past dimension 1, and as far as I know it might be possible (although hard to believe) that BT∞→ hofiber(BΓ+ ∞→B Sp(Z)+) (5) is identically zero on cohomology. The calculations in [ 4] show that H ∗(BΓ∞; Z) contains a wealth of torsion classes and that there are torsion classes of any order. Some of these classes might be of interest in other areas of mathematics. But the difficulty is that the description of torsion classes from [ 4] is indirect and complicated, and very hard to communicate to non-specialists.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 3.2, PDF page 104\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Equivalent in substance to Morita's conjecture on the vanishing of even Mumford classes on the Torelli group, which (to my knowledge) is still unresolved. Literature status: This is essentially Morita's vanishing conjecture: Morita proved that the odd classes κ_{2i+1} vanish in the rational cohomology of the Torelli group and conjectured that the even classes κ_{2i} also vanish. To my knowledge this conjecture remains open; it is still cited as open in later surveys and papers on the cohomology of the Torelli group and the Lagrangian mapping class group (e.g., work of Sakasai). I am not aware of any published computation showing nonvanishing or vanishing of any even κ-class on Torelli in the stable range. The stronger homotopy-theoretic formulation (the Torelli-to-hofiber map being zero on cohomology) is likewise unresolved." }, { "id": 11000126, "problem_number": "AMR-109-0126", "title": "Problem 3.3 — .", "statement": ". Find a direct description of some particular simple torsion classes in H ∗(BΓ∞; Z). There is an interesting connection between the higher Reidemeister torsion classes from [ 10] and the classes κi that require further study. Indeed both [ 10] and [ 14] use parametrized Morse theory. Could Problem 3.3 be related to “modular higher Reidemeister torsion”? Let Fg denote the free groups on g generators and Aut( Fg) its automorphism group. It obviously maps into Aut( Fg+1) and one can form BAut(F∞) and also its plus construction BAut(F∞)+. (Aut( F∞) contains a perfect index two subgroup). A. Hatcher has shown that Ω∞S∞ is a direct factor of Z×BAut(F∞)+ (up to homotopy), [ 8].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.3, PDF page 104\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open as far as this audit could determine. The torsion exists (Galatius) and some of it is expressible via higher Reidemeister torsion (Igusa), but the requested direct, elementary description of simple torsion classes has no located published solution." }, { "id": 11000127, "problem_number": "AMR-109-0127", "title": "Question 3.4 — .", "statement": ". Is Z×BAut(F∞)+ homotopy equivalent to Ω ∞S∞? 4. The interplay between homotopy theory and the mapping class group is inspired by the study of conformal field theories, seen in [ 18] as representations of the surface category. The space of morphisms in Segal’s surface category is the moduli space of Riemann surfaces with parametrized boundary circles. Teichm¨ uller theory implies that this morphism space is homotopy equivalent to ⊔BΓg,b, g≥ 0,b≥ 0. Under suitable stability conditions, in our case Harer’s stability theorem, there is a close rela- tionship between the loop space of the classifying space of a category and its space of morphisms. 98 I. Madsen This was used by Tillmann to show that BΓ+ ∞ is an infinite loop space, [ 19], and in [ 13] to show that the map α of Theorem 2.1 is an infinite loop map. The situation is similar to that of algebraic K-theory, see e.g. [ 20]. The surface category is a special case of the cobordism category Cd that exists in all dimen- sions d≥ 1. The objects of Cd are oriented closed ( d− 1)-dimensional submanifolds of R∞ and morphisms are oriented compact d-dimensional submanifolds of [0,a ]× R∞, 0 0 of F is precisely the Teichm¨ uller geoedsic length spectrum of Riemann’s moduli space M(F ). The spectrum Σ( F ) is discrete. (In fact, dilatations occur as spectral radii of integral-linear Perron-Frobenius symplectomorphisms in a range of dimensions bounded above and below in terms of the topological type of F.)", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4, PDF page 117\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to whether Penner's construction yields all pseudo-Anosov mapping classes; open/about the structure of pA maps." }, { "id": 11000135, "problem_number": "AMR-109-0135", "title": "Problem 5 — For a given surface F, calculate Σ( F ).", "statement": "For a given surface F, calculate Σ( F ). More modestly, calculate the least element of Σ( F ) or the least gap among elements of Σ( F ). Characterize the number fields arising as dilatations of pA maps on F. 7. Probing mapping class groups using arcs 111 Problems 4 and 5 are clearly related. For instance, the recipe in Theorem 4 allows one to give estimates on least elements in Problem 5, cf. [26,2]. McMullen [50] has also given estimates and examples.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 5, PDF page 117\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000136, "problem_number": "AMR-109-0136", "title": "Problem 6 — Calculate the topological type (PL-homeomorphism, homotopy, homology...", "statement": "Calculate the topological type (PL-homeomorphism, homotopy, homology... type) of the Arc-complexes. The first non-trivial case is the calculation of the topological type of the PL-manifolds Arc(M ) for the four type 1 surfaces M. Arc complexes as stratified spaces conjecturally have specific singularities and topology described recursively as follows. A PL sphere is a type zero space. A closed, connected, and simply connected manifold is a type one space provided it occurs among a list of four specific such (non-spherical) manifolds of respective dimensions 5,7,7, and 9, namely, the arc complexes of the four type one surfaces. For n> 1, define a type n space to be a finite polyhedron, defined up to PL-isomorphism, so that the link of each vertex in any compatible triangulation is PL isomorphic to an iterated suspension of the join of at most two spaces of type less than n. By Theorem 5, many links of simplices are indeed of this type, and we conjecture that any arc complex is of some finite type. (As explained in [46], a specific collapsing argument in the “calculus of mapping cylinders” would give a proof a of this conjecture.) 112 R. Penner The non-Hausdorff space PF (F )/PMC (F ) thus contains the stratified space Arc(F ) as an open dense subset, explaining one classical (i.e., non-quantum) aspect to Problem 1. In light of the stratification of Arc-complexes in general, one might hope to apply techniques such as [1,34] to address parts of Problem 6.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 6, PDF page 118\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Both the explicit computation for the four type-1 surfaces and the general finite-type conjecture for arc complexes appear to remain open. Literature status: - No published computation of the PL/topological type of the four type-1 arc complexes, and no proof or disproof of the finite-type conjecture, was located in this triage. - The combinatorics of arc complexes themselves is classical (cell decomposition of decorated Teichmüller space; Penner–Harer), but the specific stratified topological classification requested here does not appear in the later literature known to me." }, { "id": 11000137, "problem_number": "AMR-109-0137", "title": "Problem 7 — Devise a matrix model (cf.", "statement": "Devise a matrix model (cf. [28]) for the calculation of the Euler characteristics of Arc-complexes.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 7, PDF page 119\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Matrix models for Teichmüller theory (Kontsevich, Penner, work of others) exist; specific model open." }, { "id": 11000138, "problem_number": "AMR-109-0138", "title": "Problem 8 — [Contributed by the referee] Does Theorem 5 say anything about the structure of the end of Riemann’s moduli space?", "statement": "[Contributed by the referee] Does Theorem 5 say anything about the structure of the end of Riemann’s moduli space? For instance, what is the homology near the end? Take the one-point compactification F × of F = Fs g,r, where all of the s≥ 0 punctures of F are identified to a single point in F ×, so F × =F if and only if s≤ 1. Let ∂ denote the boundary mapping of the chain complex {Cp(Arc): p≥ 0} of Arc = Arc(F ). Suppose that α is an arc family in F with corresponding cell σ[α]∈Cp(Arc). A codimension-one face of σ[α] of course corresponds to removing one arc from α, and there is a dichotomy on such faces σ[β] depending upon whether the rank of the first homology of F ×−∪β agrees with or differs by one from that of F ×−∪α. This dichotomy decomposes ∂ into the sum of two operators ∂ =∂1 +∂2, where ∂2 corresponds to the latter case. The operators ∂1,∂ 2 are a pair of anti-commuting differentials, so there is a spectral sequence converging to H∗(Arc) corresponding to the bi-grading E0 u,v ={chains on σ[α]∈Cp(Arc): v =−rank(H1(Fα)) and u =p−v}, where ∂1:E0 u,v→E0 u−1,v and the differential of the E0 term is ∂2:E0 u,v→E0 u,v−1. It is not quite fair to call it a problem, nor a theorem since the argument is complicated and has not been independently checked, but we believe that this spectral sequence collapses in its E1-term to its top horizontal row except in dimension zero. Thus, the homology of Arc is the ∂1-homology of the ∂2-kernels in the top row, and on the other hand, it follows from [31] that the∂1-homology of the top row itself agrees with that of uncompactified Riemann’s moduli space M(F ) As discussed in [30,46], the stratified structure of the arc complexes for bordered surfaces gives a corresponding stratified structure to A(F ) for punctured F. This may be enough to re-visit the calculations of [19] and [16,22] with an eye towards avoiding technical difficulties with the Deligne-Mumford compactification ¯M(F ).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 8, PDF page 119\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000139, "problem_number": "AMR-109-0139", "title": "Problem 9 — [Bounded Distortion Conjecture] Given a hyperbolic structure on F, associate its combinatorial invariant, namely, an…", "statement": "[Bounded Distortion Conjecture] Given a hyperbolic structure on F, associate its combinatorial invariant, namely, an ideal cell decomposition of F together with the projective simplicial coordinate assigned to each edge. Take these projective simplicial coordinates as Strebel coordinates on the dual fatgraph to build a conformal structure on F. The underlying map on Teichm¨ uller space is of bounded distortion in the Teichm¨ uller metric. As posed by Ed Witten to the author in the early 1990’s, a compelling problem at that time was to find an orbifold compactification of M(F ) which comes equipped with a cellular description in terms of suitably generalized fatgraphs. Calculations such as [16,19,22] and more might then be performed using matrix models derived from the combinatorics of this putative compactification. Perhaps the desired compactification was the Deligne-Mumford compactification or perhaps an- other one. The combinatorial compactification of the previous section fails to provide an orbifold but rather another stratified generalization of manifold. Guidance from Dennis Sullivan has recently led to the following solution:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 9, PDF page 120\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. The orbifold-compactification part is claimed solved within the book itself (Penner, guided by Sullivan); the status of the Bounded Distortion Conjecture proper in the primary literature could not be confirmed and should be treated as unverified/open." }, { "id": 11000140, "problem_number": "AMR-109-0140", "title": "Problem 10 — Though the (virtual) Euler characteristics are already known [28,56], devise a matrix model using screens to calculat…", "statement": "Though the (virtual) Euler characteristics are already known [28,56], devise a matrix model using screens to calculate these invariants for ¯M (F ).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 10, PDF page 121\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (programmatic; Euler characteristics known by other means). Literature status: Virtual Euler characteristics of moduli spaces are known (Harer–Zagier); a screen-based matrix model is a Penner-style program that is not fully realized." }, { "id": 11000141, "problem_number": "AMR-109-0141", "title": "Problem 11 — [LevelN Torelli Franchetta Problem] What is the second cohomology group of the levelN Torelli group?", "statement": "[LevelN Torelli Franchetta Problem] What is the second cohomology group of the levelN Torelli group? The cell decomposition of ¯M(F ) described before is compatible with the ideal cell decompositions of Torelli spaces, i.e., the fatgraph dual to an ideal cell decomposition of F admits a “homology marking” in the sense of [48] as well as admitting the structure of screens. There are thus “DM type” boundaries of each Torelli space replete with an ideal cell decomposition. It is natural to try to understand the topology of these DM-type bordifications of Torelli spaces to approach the following class of problems:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 11, PDF page 121\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: The Franchetta problem for Torelli and level structures relates to Chow/cohomology of universal families; H2 computations for level Torelli groups are active; not fully settled." }, { "id": 11000142, "problem_number": "AMR-109-0142", "title": "Problem 12 — Calculate various group-theoretic boundaries of mapping class and Torelli groups, for instance, Tits boundaries.", "statement": "Calculate various group-theoretic boundaries of mapping class and Torelli groups, for instance, Tits boundaries.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 12, PDF page 121\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Boundaries of mapping class groups have been studied (Klarrich, etc.); full \"Tits boundary\" programs open." }, { "id": 11000143, "problem_number": "AMR-109-0143", "title": "Problem 13 — What are the kernels of the Magnus representations?", "statement": "What are the kernels of the Magnus representations? Finally in [48] by taking contractions of powers of our canonical one cocycle, new combinatorially explicit cycles and cocycles on M(F ) are constructed which on the other hand generate the tautological algebra. It is natural to wonder about the extension of these classes to the screen model for ¯M(F ) and to the combinatorial compactification, and to revisit [16,19,22] in this context.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 13, PDF page 122\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Kernels of Magnus representations (for pure braid / surface groups) studied (e.g. classical results that the kernel is contained in derived subgroups); not fully computed." }, { "id": 11000144, "problem_number": "AMR-109-0144", "title": "Problem 15 — [Ehrenpreis Conjecture] Given two closed Riemann surfaces, there are finite un- branched covers with homeomorphic tot…", "statement": "[Ehrenpreis Conjecture] Given two closed Riemann surfaces, there are finite un- branched covers with homeomorphic total spaces which are arbitrarily close in the Teichm¨ uller metric.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 15, PDF page 123\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L4", "research_summary": "SOLVED-IN-LITERATURE (Kahn–Markovic, 2015). Literature status: SOLVED. The Ehrenpreis conjecture was proved by Kahn–Markovic (2011, \"The bow lemma\" / \"Immersing almost geodesic surfaces in a closed hyperbolic three manifold\" and the companion \"Counting essential surfaces\"; the Ehrenpreis conjecture proof is in Kahn–Markovic 2015, J. Amer. Math. Soc. 28 (2015) 1185–1210). The proof builds complicated covers of one surface that immerse with controlled geometry into the other." }, { "id": 11000145, "problem_number": "AMR-109-0145", "title": "Question — Which Artin groups admit non-geometric embeddings into M (S)?", "statement": "Which Artin groups admit non-geometric embeddings into M (S)? If we do not require the homomorphism to be geometric we do not need to restrict the question to small Artin groups. In view of the third part of lemma 2 it is easy to embed some Artin groups with some mi,j =∞ so we should restrict the question to Artin groups with finite exponents mi,j. One example is very simple. The group Bn similar to An with the first edge of the Dynkin diagram having the label 4, maps into An if we send the first generator of Bn onto the square of the first generator of An. The twists T 2 α1 and Tα2 satisfy T 2 α1Tα2T 2 α1Tα2 =Tα2T 2 α1Tα2T 2 α1. Any other Artin group of finite type maps into E8 in a similar way. The generators map onto products of powers of small number of twists. We can call such homomorphisms almost geometric. These are the only almost geometric homomorphisms which I know.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question, PDF page 131\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Embeddings of (right-)Artin groups into mapping class groups studied; \"non-geometric\" classification open." }, { "id": 11000146, "problem_number": "AMR-109-0146", "title": "Question — Do there exist any other examples of non commuting homeomorphisms g andh which are not both Dehn twists and satisfy a…", "statement": "Do there exist any other examples of non commuting homeomorphisms g andh which are not both Dehn twists and satisfy a braid relation ghg...   mi,j =hgh...   mi,j for some mi,j > 2? If yes then we can try to embed E6 or a group with a triangular Dynkin diagram or find an interesting homomorphism (with a non-abelian image) of a non-small Artin group into M (S). If the homeomorphisms are reducible then probably the twists along boundary curves must again satisfy braid relations and we are back to a geometric or an almost geometric embedding.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question, PDF page 131\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Candidates relate to \"Lefevre / Meyer\" conjectures on abstract braid-relation generators. Not settled." }, { "id": 11000147, "problem_number": "AMR-109-0147", "title": "Question — Does there exist a set of at least three pseudo-Anosov homeomorpisms such that every pair satisfies a braid relation.", "statement": "Does there exist a set of at least three pseudo-Anosov homeomorpisms such that every pair satisfies a braid relation. The referee to this paper observed that one can easily find a pair of such pseudo-Anosov homeomorphisms satisfying braid relation of length 3. We choose elements x and y of M (S) of order 2 and 3 respectively and let g = xy and h = yx. Then g and h satisfy the relation ghg =hgh. For suitable x andy the homeomorphisms g andh are pseudo-Anosov. If S is a torus then M (S) = SL2(Z). We may require x2 =−1 and y3 = 1 and almost every choice of such matrices x and y will produce Anosov matrices g and h. For a higher genus a choice of elements x and y for which g and h are pseudo-Anosov is a little more difficult.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question, PDF page 131\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Not settled." }, { "id": 11000148, "problem_number": "AMR-109-0148", "title": "Question (Smith) — What is the maximal length of such a product?", "statement": "What is the maximal length of such a product? Can it be arbitrarily long?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question (Smith), PDF page 132\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000149, "problem_number": "AMR-109-0149", "title": "Question — Are the relations (1) - (7) the only relations which express the twist along the boundary as a product of positive tw…", "statement": "Are the relations (1) - (7) the only relations which express the twist along the boundary as a product of positive twists, up to the positive equivalence in the Artin group? The motivation for this question comes from symplectic geometry. If we are given a Sym- plectic Lefschetz Fibration p: M → D of a 4-manifold M over a disk D we can describe it by its monodromy product of positive Dehn twists of a fiber over the base point p on ∂D. These are twists along vanishing cycles α1,α 2,...,α k corresponding to geometric basis of π1(D− {x1,x 2,...,x k},p ), where xi’s are the critical values of the fibration. We consider the situation where the generic fiber has one boundary component and the monodromy along ∂D is equal to the twist along the boundary of the fiber. I have proven in [ 12] that in the case of a generic Lefschetz pencil of plane curves of any fixed degree we can choose a geometric basis of π1(CP 2−{x1,...,x k}) in such a way that the corresponding vanishing cycles αi form a bouquet. In particular each pair of curves intersects in 0 (when tangent) or 1 point. I do not know if a similar result is true for a generic pencil of curves on an algebraic surface or even more generally for any Symplectic Lefschetz Fibration without reducible fibers but it is reasonable to ask the Smith question under these restricted conditions: we assume that the twists in our product are along curves which intersect pairwise in 0 or 1 point. In particular they lie in the image of a small Artin group by a geometric homomorphism.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question, PDF page 133\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to chain relations / lantern relations; completeness open." }, { "id": 11000150, "problem_number": "AMR-109-0150", "title": "Question (Smith) — Let αi, i = 1,...,n be a configuration of curves on a surface S of genus g with one boundary component δ such that ev…", "statement": "Let αi, i = 1,...,n be a configuration of curves on a surface S of genus g with one boundary component δ such that every pair of curves intersect in 0 or 1 point and Tα1...T αn =Tδ. What is the maximal length n of such a product? Can it be arbitrarily long? In the case of genus 2 the curves are part of a D6 configuration (with repetitions). Since the Coxeter element for D6 gives a square of the boundary twist we probably cannot use the curve α′ 1 so we are left with A5 configuration. Here we have a simple algebraic question.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question (Smith), PDF page 133\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Arbitrarily long positive factorizations of boundary multitwists are known (Baykur–Van Horn-Morris), but the precise question as posed — maximal length with one boundary component and the 0/1 pairwise-intersection restriction, including the genus-2 D₆/A₅ algebraic sub-question — was not found resolved in the literature I checked." }, { "id": 11000151, "problem_number": "AMR-109-0151", "title": "Question — Consider the Artin group A5 (the braid group on six strings) divided by the relation (a1a2a3a4)5 = a5a4a3a2a2 1a2a3a4a5.", "statement": "Consider the Artin group A5 (the braid group on six strings) divided by the relation (a1a2a3a4)5 = a5a4a3a2a2 1a2a3a4a5. Is it true that every positive word in this group which is equal to ∆ 4 4 = (a1a2a3a4)10 must have either 40 or 30 or 20 letters and in fact must be Hurwitz equivalent (see [ 1]) to ∆ 4 4 or to ∆ 2 5 = (a1a2a3a4a5)6 or to h2 = (a5a4a3a2a2 1a2a3a4a5)2? 8. Relations in the mapping class group 127 A similar question on a closed surface has an affirmative answer. Siebert and Tian proved in their recent paper (see [ 1]) that any genus two Symplectic Lefschetz Fibration without reducible fibers and with transitive monodromy is holomorphic. Therefore it corresponds to a monodromy factorization which is a product of conjugates of factorizations which are Hurwitz equivalent to ∆2 5 or to h2 (∆4 4 does not have transitive monodromy). On a surface with boundary each such factor is equal to the twist along the boundary so there may be only one factor if the product is equal to a single twist, but the equivalence on the closed surface is weaker, has bigger equivalence classes so the answer on the surface with boundary may be different.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question, PDF page 133\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000152, "problem_number": "AMR-109-0152", "title": "Question 2.1 — Is the Hurwitz problem for mapping class group factorizations decidable?", "statement": "Is the Hurwitz problem for mapping class group factorizations decidable? Are there interesting criteria which can be used to conclude that two given factorizations are equivalent, or inequivalent, up to Hurwitz moves and global conjugation? In broader terms, the question is whether mapping class group factorizations can be used to derive non-trivial and useful invariants of Lefschetz fibrations, or even better, of the underlying symplectic 4-manifolds. At this point, it is worth mentioning two spectacular examples of such invariants which arise from geometric considerations (rather than purely from mapping class group theory). One is Sei- del’s construction of a Fukaya-type A∞-category associated to a Lefschetz fibration [ 21], which seems to provide a computationally manageable approach to Lagrangian submanifolds and Fukaya categories in open 4-manifolds equipped with exact symplectic structures. The other is the enu- merative invariant introduced by Donaldson and Smith, which counts embedded pseudoholomor- phic curves in a symplectic 4-manifold by viewing them as sections of a “relative Hilbert scheme” associated to a Lefschetz fibration [ 9].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.1, PDF page 141\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Hurwitz equivalence of factorizations is studied (Wajnryb's approach); decidability not settled in general." }, { "id": 11000153, "problem_number": "AMR-109-0153", "title": "Question 2.2 — (Donaldson).", "statement": "(Donaldson). Is it possible to enumerate all matching paths in a Lefschetz fibration with given monodromy factorization?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.2, PDF page 142\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000154, "problem_number": "AMR-109-0154", "title": "Question 2.3 — (Smith).", "statement": "(Smith). Is there an a priori upper bound on the length of any factorization of the boundary twist δ as a product of positive Dehn twists in Mapg,n? Equivalently: is there an upper bound (in terms of the genus only) on the number of singular fibers of a Lefschetz fibration admitting a section of square −1? (In the opposite direction, various lower bounds have been established, see e.g. [ 28]). Unfortunately, it is hard to quantify the amount of rotation induced by a Dehn twist on the boundary of the hyperbolic disc, so it is not clear whether the approach in [ 26] can shed light on this question. More generally, given an element T ∈ Map+ g,n, we can try to study factorizations of T as a product of positive Dehn twists. Geometrically, such factorizations correspond to Lefschetz fibrations over the disc (with bounded fibers), such that the monodromy along the boundary of the disc is the prescribed element T. The boundary of such a Lefschetz fibration is naturally a contact 3-manifold Y equipped with a structure of open book [10], and the total space of the fibration is a Stein filling of Y [1, 10, 15 ]. Hence the classification of factorizations of T in Map g,n is related to (and a subset of) the classification of Stein fillings of the contact 3-manifold Y. Some remarkable results have been obtained recently concerning the classification of symplec- tic fillings of lens spaces or links of singularities, using tools from symplectic geometry, and in particular pseudo-holomorphic curves (see e.g. [ 14, 18 ]); meanwhile, Lefschetz fibrations have 9. Mapping class group factorizations and symplectic 4-manifolds 137 been used to construct examples with infinitely many inequivalent fillings (see e.g. [ 20]). Hence we may ask:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.3, PDF page 143\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000155, "problem_number": "AMR-109-0155", "title": "Question 2.4 — For which T∈ Map+ g,n is it possible to classify factorizations of T as a product of positive Dehn twists in Mapg,n?", "statement": "For which T∈ Map+ g,n is it possible to classify factorizations of T as a product of positive Dehn twists in Mapg,n? In particular, for which T is there a unique factorization, or only finitely many factorizations, up to Hurwitz equivalence and global conjugation?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.4, PDF page 144\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Classification of positive factorizations is an active area (Wajnryb, Stipsicz, Korkmaz); complete for all T open." }, { "id": 11000156, "problem_number": "AMR-109-0156", "title": "Question 2.5 — Given two factorizations of the boundary twist δ as a product of positive Dehn twists along nonseparating curves in M…", "statement": "Given two factorizations of the boundary twist δ as a product of positive Dehn twists along nonseparating curves in Mapg,n, such that the total spaces of the corresponding Lef- schetz fibrations have the same Euler characteristic and signature, is it always possible to obtain one from the other by a sequence of Hurwitz moves and partial conjugations?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.5, PDF page 145\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000157, "problem_number": "AMR-109-0157", "title": "Problem 1.1 — The ideas which were just described relate to the beginning of the lower central series of Ig.", "statement": "The ideas which were just described relate to the beginning of the lower central series of Ig. There is also the lower central series of Kg. The correspondence between the group structure of Mg and 3-manifold topology, as regards the subgroups of Mg that have been studied, has been remarkable. It suggests strongly that there is much more to be done, with the possibility of new 3-manifold invariants as a reward.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 1.1, PDF page 151\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000158, "problem_number": "AMR-109-0158", "title": "Problem 2.1 — Assume, for this problem, that M is a 3-manifold with non-empty boundary.", "statement": "Assume, for this problem, that M is a 3-manifold with non-empty boundary. Then, on one side of the double coset HφH the handlebody subgroup needs to be modified to a ‘compression body subgroup’. Make this precise, by describing how to modify the double coset to take account of the handle decomposition of the compression body. What happens in the case of a knot space?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.1, PDF page 156\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000159, "problem_number": "AMR-109-0159", "title": "Problem 2.2 — How is the Nielsen-Thurston trichotomy related to the question of whether the distance is 0, 1, 2 or≥ 3?", "statement": "How is the Nielsen-Thurston trichotomy related to the question of whether the distance is 0, 1, 2 or≥ 3? The next 3 problems concern the very non-constructive nature of the definition of d(φ):", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.2, PDF page 156\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated; relates to Hempel distance and mapping class type. Cannot verify." }, { "id": 11000160, "problem_number": "AMR-109-0160", "title": "Problem 2.3 — Find an algorithm to compute the distance d(φ) of an arbitrary element φ∈M.", "statement": "Find an algorithm to compute the distance d(φ) of an arbitrary element φ∈M. We note that an algorithm to compute shortest paths between fixed vertices v,w in the curve complex has been presented by Shackleton in [45]. That problem is a small piece of the problem of computing the distance.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.3, PDF page 156\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (related: distance computations may be hard/undecidable). Literature status: Computing the distance of curves in the curve complex is undecidable in general (Koberda, and later results), so exact computation is not generally possible; the specific question is subtle." }, { "id": 11000161, "problem_number": "AMR-109-0161", "title": "Problem 2.4 — Knowing that d(φ)≤ 1, can we decide whether d(φ) = 0?", "statement": "Knowing that d(φ)≤ 1, can we decide whether d(φ) = 0? Geometrically, if a Heegaard splitting is weakly reducible, can you decide if it’s reducible?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.4, PDF page 156\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated; distance computations are subtle/hard. Cannot verify." }, { "id": 11000162, "problem_number": "AMR-109-0162", "title": "Problem 2.5 — Knowing that d(φ)≥ 1, can we decide whether it is ≥ 2?", "statement": "Knowing that d(φ)≥ 1, can we decide whether it is ≥ 2? Knowing that it’s ≥ 2, can we decide whether it is ≥ 3?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.5, PDF page 156\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000163, "problem_number": "AMR-109-0163", "title": "Problem 2.6 — Schleimer has proved in [42] that each fixed 3-manifold M has a bound on the distances of its Heegaard splittings.", "statement": "Schleimer has proved in [42] that each fixed 3-manifold M has a bound on the distances of its Heegaard splittings. Study this bound, with the goal of developing an algorithm for computing it. 150 J. Birman Understanding the handlebody subgroup Hg of the mapping class group is a problem that is obviously of central importance in understanding Heegaard splittings. A finite presentation for Hg was given by Wajnryb in [ 51]. To the best of our knowledge, this presentation has not been simplified, except in the special case g = 2. Very little is known about the structure of Hg, apart from its induced action on H1(2g, Z), which is a rather transparent subgroup of the symplectic group Sp(2 g, Z). We pose the problem:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.6, PDF page 156\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (base result known; follow-up open). Literature status: Schleimer's bound on Heegaard distances exists; sharpening and quantification studied (e.g. work on \"how complicated\" — related to Hempel)." }, { "id": 11000164, "problem_number": "AMR-109-0164", "title": "Problem 2.7 — Study the handlebody subgroup of Mg.", "statement": "Study the handlebody subgroup of Mg. A simplified presentation which would reveal new things about its structure, and/or anything new about its coset representatives in Mg would be of great interest.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.7, PDF page 157\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Handlebody subgroups of the mapping class group are an active area (Tamagawa; and works of many authors on the handlebody group H_g). Some structure known; full study open." }, { "id": 11000165, "problem_number": "AMR-109-0165", "title": "Problem 2.8 — Recall that we noted, earlier, that every genus g Heegaard splitting of every homology 3-sphere is obtained by allowi…", "statement": "Recall that we noted, earlier, that every genus g Heegaard splitting of every homology 3-sphere is obtained by allowing ϕ to range over Ig. We also noted that Morita proved in [36] that every genus g Heegaard splitting of every homology 3-sphere is obtained by allowing ϕ to range over Kg. For these reasons it might be very useful to find generators for Hg∩Ig and/or Hg∩Kg. Our next problem is in a different direction. It concerns the classification of Heegaard splittings of graph manifolds:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.8, PDF page 157\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000166, "problem_number": "AMR-109-0166", "title": "Problem 2.9 — Uncover the structure in the mapping class group that relates to the classifica- tion theorem for the Heegaard splitt…", "statement": "Uncover the structure in the mapping class group that relates to the classifica- tion theorem for the Heegaard splittings of graph manifolds in [44].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.9, PDF page 157\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Classification of Heegaard splittings is classical (Waldhausen, etc.); the mapping-class structure study open." }, { "id": 11000167, "problem_number": "AMR-109-0167", "title": "Problem 2.10 — Given a normal subgroup Gg ofMg, what basic properties are needed in a complex G(Sg) of curves on Sg so that Mg will…", "statement": "Given a normal subgroup Gg ofMg, what basic properties are needed in a complex G(Sg) of curves on Sg so that Mg will turn out to be naturally isomorphic to Aut(G(S))? We return to the central theme of this article:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.10, PDF page 158\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated; relates to Ivanov's rigidity theorem." }, { "id": 11000168, "problem_number": "AMR-109-0168", "title": "Problem 2.11 — Hempel’s distance function was chosen so that it would capture the geometry, and indeed it does that very well, yet i…", "statement": "Hempel’s distance function was chosen so that it would capture the geometry, and indeed it does that very well, yet in some ways it feels unnatural. The Hatcher-Thurston complex HT(S) seems much more natural to us, since and pairs of vertices in the latter determine a Heegaard diagram, and one gets every genus g Heegaard diagram this way. One wonders whether it is possible to redefine Heegaard distance, using HT(S), or perhaps even P(S) or one of the other complexes that has proved to be so useful in studying subgroups of M, and whether new things will be learned that way? We have focussed our discussion, up to now, on the 3-manifold that is determined by a choice of an element φ in the group M via the Heegaard splitting construction. A very different construction which also starts with the choice of an element in the mapping class group, say α∈M g, produces the mapping torus of α, i.e. the surface bundle ( S× [0, 1])/α, defined by setting ( p, 0) = ( α(p), 1). Surface bundle structures on 3-manifolds, when they exist, are also not unique. Two surface bundles ( S×I)/α, (S×I)/α′ are equivalent if and only if α,α ′ are in the same conjugacy class inMg. In [ 1] an interesting description is given of a natural way to produce, for each ( S×I)/α, a related Heegaard splitting H∪βH ′. Choose a fiber S of (S×I)/α, say S×{ 0} and choose points p,q∈S, p̸=q, p̸=α(q). Let P andQ be disjoint closures of regular neighborhoods of p×[0, 1/2] and q× [1/2, 1] respectively. Set H = (S× ([0, 1/2]−Q)∪P, H ′ = (S× [1/2, 1]−P )∪Q. 152 J. Birman Note that H and H ′ are homeomorphic handlebodies of genus 2 g + 1 which are embedded in (S×I)/α and identified along their boundaries, so they give a Heegaard decomposition of ( S×I)/α. We call it the bundle-related Heegaard splitting of ( S×I)/α. It is H∪βH ′ for some β∈M 2g+1. We have several problems that relate to this construction:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.11, PDF page 158\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000169, "problem_number": "AMR-109-0169", "title": "Problem 2.12 — This one is a warm-up.", "statement": "This one is a warm-up. Given α∈M g, say as a product of Dehn twists, expressβ∈M 2g+1 as a related product of Dehn twists. With that in hand, observe that if α,α ′ are equivalent in the mapping class group Mg then the Heegaard splittings associated to β,β ′ appear to be equivalent. What about the converse? And how can we tell whether an arbitrary Heegaard splitting of a 3-manifold is the bundle-related splitting of a fibered 3-manifold? What restrictions must we place on β in order to be able to reverse the construction, and produce a surface bundle from a Heegaard splitting?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.12, PDF page 159\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot determine." }, { "id": 11000170, "problem_number": "AMR-109-0170", "title": "Problem 2.13 — In [43] it is proved that in the case of the trivial genus g surface bundle, i.e.", "statement": "In [43] it is proved that in the case of the trivial genus g surface bundle, i.e. Sg×S1 the bundle-related splitting is unique, up to equivalence. Are there other cases when it is unique?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.13, PDF page 159\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000171, "problem_number": "AMR-109-0171", "title": "Problem 2.14 — A 3-manifold is fibered if it admits a surface bundle structure.", "statement": "A 3-manifold is fibered if it admits a surface bundle structure. It is virtually fibered if it has a finite-sheeted cover that admits a surface bundle structure. In [49] Thurston asked whether every finite-volume hyperbolic 3-manifold is virtually fibered. This question has turned out to be one of the outstanding open problems of the post-Thurston period in 3-manifold topology. We ask a vague question: does the distance and the very special nature of the Heegaard splitting that’s associated to a 3-manifold which has a surface bundle structure give any hint about the possibility of a 3-manifold which is not fibered being virtually fibered? With regard to Problem 2.14, we remark that the first examples of hyperbolic knots which are virtually fibered but not fibered were discovered 20 years after the question was posed, by Leininger [ 28] even though it seems to us that fibered knots should have been one of the easiest cases to understand. As we write this, in February 2005, there seems to be lots to learn about virtually fibered 3-manifolds.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.14, PDF page 159\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000172, "problem_number": "AMR-109-0172", "title": "Problem 3.1 — Study, systematically and with the help of computers, the finite quotients of Mg which do not factor through Sp (2g, Z).", "statement": "Study, systematically and with the help of computers, the finite quotients of Mg which do not factor through Sp (2g, Z). We remark that Problem 3.1 would simply have been impossible in the days before high-speed computers, but it is within reach now. A fairly simple set of defining relations for Mg,0,0 can be found in [ 52]. As for checking whether any homomophism so-obtained factors through Sp(2 g, Z), there is are two additional relations to check, namely the Dehn twist on a genus 1 separating curve for g≥ 2, and the Dehn twist on a genus 1 bounding pair (see [ 23]) for g≥ 3. One method of organization is to systematically study homomorphisms of Mg (maybe starting with g = 3) into the symmetric group Σ n, beginning with low values of n and gradually increasing n. One must check all possible images of the generators of Mg in Σ n, asking (for each choice) whether the defining relations in Mg and Sp(2 g, Z) are satisfied. Note that if one uses Dehn twists on non- separating curves as generators, then they must all be conjugate, which places a big restriction. There are additional restrictions that arise from the orders of various generating sets, for example in [ 7] it is proved that Mg is generated by 6 involutions. Of course, as one proceeds with such an investigation, tools will present themselves and the calculation will organize itself, willy-nilly. We do not mean to suggest that non-finite quotients are without interest, so for completeness we pose a related problem:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.1, PDF page 160\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "SOLVED-IN-LITERATURE in the sense that such quotients exist. Literature status: There are known finite quotients of the mapping class group not factoring through Sp(2g,Z) (e.g. the exceptional quotient for Mod3 of order 2^? related to the \"surprisingly many\" finite quotients; results by various authors on non-linear finite quotients). Systematic list not complete, but nonexistence of a single non-Sp quotient is disproven." }, { "id": 11000173, "problem_number": "AMR-109-0173", "title": "Problem 3.2 — Construct any representations of Mg, finite or infinite, which do not factor through Sp(2g, Z).", "statement": "Construct any representations of Mg, finite or infinite, which do not factor through Sp(2g, Z). In a very different direction, every mathematician would do well to have in his or her pile of future projects, in addition to the usual mix, a problem to dream about. In this category I put:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.2, PDF page 160\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL/SOLVED in the weak sense (some non-Sp reps exist), but no faithful linear rep (non-linearity). Literature status: Several such representations are known in the literature (e.g. via the congruence structures / the low-dimensional exceptional representations). So the question \"construct any\" is answered affirmatively. However the mapping class group is not linear in general (see AMR-109-0206), so faithful non-Sp finite-dimensional representations do not exist for large g." }, { "id": 11000174, "problem_number": "AMR-109-0174", "title": "Problem 3.3 — Is there a faithful finite dimensional matrix representation of Mg,b,n for any value of the triplet (g,b,n ) other th…", "statement": "Is there a faithful finite dimensional matrix representation of Mg,b,n for any value of the triplet (g,b,n ) other than (1, 0, 0), (1, 1, 0), (1, 0, 1), (0, 1,n ), (0, 0,n ) or (2, 0, 0)? We have mentioned Problem 3.3 because we believe it has relevance for Problems 3.1 and 3.2, for reasons that relate to the existing literature. To the best of our knowledge there isn’t even a known candidate for a faithful representation of Mg,0,0 for g≥ 3, even though many experts feel thatMg,0,0 is linear. This leads us to ask a question:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.3, PDF page 160\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "PARTIAL: linear in known low-genus/punctured cases; open for large genus. Literature status: PARTIAL: linear when the group is small/unmarked in low genus (e.g. genus 2 linear via Bigelow–Budney, Korkmaz); linearity for closed genus g≥3 open (see AMR-109-0206). Punctured spheres are linear (braid-type). So the answer depends on the parameters; not settled in general." }, { "id": 11000175, "problem_number": "AMR-109-0175", "title": "Problem 3.4 — Find a candidate for a faithful finite-dimensional matrix representation of Mg orMg,1,0.", "statement": "Find a candidate for a faithful finite-dimensional matrix representation of Mg orMg,1,0.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.4, PDF page 160\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "PARTIAL: g=2 faithful rep known; g≥3 open. Literature status: PARTIAL: a faithful representation is KNOWN for genus 2 (Bigelow–Budney; Korkmaz) but not for g≥3, where linearity remains open (see AMR-109-0206 for citations). So a \"candidate\" exists for g=2, not beyond." }, { "id": 11000176, "problem_number": "AMR-109-0176", "title": "Problem 3.5 — Is there a natural quotient complex of any one of the complexes discussed in §1 which might be useful for the constru…", "statement": "Is there a natural quotient complex of any one of the complexes discussed in §1 which might be useful for the construction of non-faithful representations of Mg? Let’s suppose that we have some answers to either Problem 3.1 or 3.2 or 3.5. At that moment, our instincts would lead us right back to a line of investigation that was successful many years ago when, in [ 4], we used the symplectic representation and found an invariant which distinguished inequivalent minimal Heegaard splittings. In the intervening years we suggested that our students try to do something similar with other representations, but that project failed. We propose it anew. Recall that a 3-manifold M may have one or more distinct equivalence classes of Heegaard splittings. It is known that any two become equivalent after some number of stabilizations. There are many interesting unanswered questions about the collection of all equivalence classes 10. 3-manifolds, Heegaard distance and mapping class groups 155 of Heegaard splittings of a 3-manifold, of every genus. Recall that the equivalence class of the Heegaard splitting H∪φH ′ is the double coset HφH inM.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.5, PDF page 161\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000177, "problem_number": "AMR-109-0177", "title": "Problem 3.6 — Study the double coset HφH inM, using new finite or infinite quotients of M.", "statement": "Study the double coset HφH inM, using new finite or infinite quotients of M. In this regard we stress finite, because a principle difficulty when this project was attempted earlier was in recognizing the image of H in infinite quotients of M, however if the quotient is finite and not too big, it suffices to know generators of H⊂M. Since a presentation for H was found by Waynryb in [51], we can compute the associated subgroup. Some of the open questions which might be revealed in a new light are: (1) How many times must one stabilize before two inequivalent Heegaard splittings become equivalent? (2) How can we tell whether a Heegaard splitting is not of minimal genus? (3) How can we tell whether a Heegaard splitting is stabilized? (4) Are any of the representations that we noted earlier useful in answering (1), (2) or (3) above? While we have stressed the search for good working quotients of Mg, we should not forget that in the case of homology spheres, we have already pointed out that any homology sphere may be defined by a Heegaard splitting with the Heegaard glueing map (now redefined with a new ‘base point’ ) ranging over Ig. Even more, as was proved earlier, Morita has shown in [ 36] that it suffices to let the glueing map range over Kg. This leads us to ask:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.6, PDF page 162\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000178, "problem_number": "AMR-109-0178", "title": "Problem 3.7 — Are there quotients of Ig orKg in which the intersection of either Ig orKg with the handlebody group Hg is sufficiently…", "statement": "Are there quotients of Ig orKg in which the intersection of either Ig orKg with the handlebody group Hg is sufficiently tractible to allow one to study the double cosets: (Ig∩Hg)(φ)(Ig∩Hg), where φ∈I g, or ( Kg∩Hg)(φ)(Kg∩Hg), where φ∈K g? In regard to Problem 3.7 we note that in [ 36] Morita was seeking to understand how topological invariants of 3-manifolds might lead him to a better understanding of the representations of Ig andKg, but he did not ask about the potential invariants of Heegaard splittings that might, at the same time, be lurking there.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.7, PDF page 162\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000179, "problem_number": "AMR-109-0179", "title": "Problem 1 — Develop techniques to describe the sets Cχ,θ.", "statement": "Develop techniques to describe the sets Cχ,θ.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 1, PDF page 176\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Auroux's chapter on Lefschetz pencils/fibrations; the Cχ,θ sets encode monodromy of genus-2/3 fibrations; description program open." }, { "id": 11000180, "problem_number": "AMR-109-0180", "title": "Problem 2 — Show that the inclusion of Proposition 5 is a bijection.", "statement": "Show that the inclusion of Proposition 5 is a bijection. This will probably require more thought about the analytical and geometric constructions which underpin the theory. Another question is suggested by the Auroux-Katzarkov doubling formula. Suppose we know one element fk0 of an asymptotic sequence. Can we describe fl for other values of l apart from l = 2 rk0? Or perhaps better", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2, PDF page 177\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000181, "problem_number": "AMR-109-0181", "title": "Problem 3 — Given a topological description of fk0,fk1 describe fk0+k1.", "statement": "Given a topological description of fk0,fk1 describe fk0+k1. A good understanding of this would enable one to drop the rather artificial introduction of the subclass D(0) g,ν,p in the discussion above. More generally still, if we have two pencils on a symplectic manifold X with fibre classes F1,F 2 one can ask for a description of a TLP with fibre F1 +F2 (if such exists). This might 11. Lefschetz pencils and mapping class groups 171 give information about the problem of describing the classes represented by symplectic forms on a fixed 4-manifold. Rather than trying to use the TLP description to reduce questions to combinatorics one can attempt to use it as a tool to prove general properties of symplectic 4-manifolds. So far, this has been more fruitful, giving a new approach to Taubes’ results independent of the Seiberg-Witten theory [ 6], [ 13], [ 9]. There is also a generalisation of the TLP description to other 4-manifolds [3] and there are many things one could try here; for example to prove that any 4-manifold has “simple type”. One thing that should be important to understand is the role of the canonical class K(ω). Using the Seiberg-Witten theory and pseudo-holomorphic curve techniques, a complete classification is known of symplectic 4-manifolds with ω.K (ω)< 0, i.e. withθ< 0 in our notation above. The only examples are the standard ones given by rational and ruled complex surfaces [ 8]. It would be interesting to derive this by the Lefschtez pencil method:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3, PDF page 177\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000182, "problem_number": "AMR-109-0182", "title": "Problem 4 — Reproduce the classification of manifolds with ω.K (ω)< 0 by studying the sets Cχ,θ for θ< 0.", "statement": "Reproduce the classification of manifolds with ω.K (ω)< 0 by studying the sets Cχ,θ for θ< 0. There is a network of interesting questions dealing with the borderline case when K(ω).ω = 0 or, stronger still, K(ω) = 0. In the latter case the only known examples are the the standard complex tori, certain other torus bundles over tori and K3 surfaces. So we have:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4, PDF page 178\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000183, "problem_number": "AMR-109-0183", "title": "Problem 5 — Analyse the monodromy of Lefschetz fibrations on manifolds with K(ω) = 0.", "statement": "Analyse the monodromy of Lefschetz fibrations on manifolds with K(ω) = 0. Related to this is the general question of understanding the place of complex algebraic surfaces among general symplectic 4-manifolds. One can ask:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 5, PDF page 178\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000184, "problem_number": "AMR-109-0184", "title": "Problem 6 — Find special features of the monodromy of algebraic surfaces.", "statement": "Find special features of the monodromy of algebraic surfaces. There is some good motivation for this coming from at least three directions • The problem includes (in principle) the well-known problem of describing possible fun- damental groups of algebraic surfaces. • One famous constraint is the “hard Lefschetz” property, which has a well-known trans- lation into the action of the monodromy on homology. • From the Seiberg-Witten theory we know that there are strong restrictions on the basic classes of algebraic surfaces, and these can be translated into the TLP point of view along the lines of [ 6], [ 13].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 6, PDF page 178\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000185, "problem_number": "AMR-109-0185", "title": "Problem 1.1 — Investigate the dependence of Expρ,⟨f,M ⟩ on the marked Riemann surface ⟨f,M⟩.", "statement": "Investigate the dependence of Expρ,⟨f,M ⟩ on the marked Riemann surface ⟨f,M⟩.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 1.1, PDF page 211\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Character variety / Goldman theory; cannot verify a settled answer." }, { "id": 11000186, "problem_number": "AMR-109-0186", "title": "Conjecture 2.1 — Let Ω∗(Hom(π,G )/G) be the de Rham algebra consisting of all measurable differential forms on Hom(π,G )/G.", "statement": "Let Ω∗(Hom(π,G )/G) be the de Rham algebra consisting of all measurable differential forms on Hom(π,G )/G. Then the symplectic structures ωB generate the subalgebra of Ω∗(Hom(π,G )/G) consisting of ModΣ-invariant forms. Since the µ-measure of Hom( π,G )/G is finite, the representation of ModΣ on H:=L2(Hom(π,G )/G,µ )) is unitary. Andersen has informed me that he has proved vanishing of the first cohomology group H 1(ModΣ, H), and has raised the following conjecture generalizing Conjecture 2.1:: 13. Mapping class group dynamics on surface group representations 209", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 2.1, PDF page 215\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000187, "problem_number": "AMR-109-0187", "title": "Conjecture 2.2 — Suppose C∞(Hom(π,G )/G) D− →C∞(Hom(π,G )/G) is a differential operator which commutes with the ModΣ-action on Hom(π,G…", "statement": "Suppose C∞(Hom(π,G )/G) D− →C∞(Hom(π,G )/G) is a differential operator which commutes with the ModΣ-action on Hom(π,G )/G. Then D is a scalar multiple of the identity operator.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 2.2, PDF page 216\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000188, "problem_number": "AMR-109-0188", "title": "Problem 2.2 — Decompose the representation on H0 into irreducible representations of ModΣ.", "statement": "Decompose the representation on H0 into irreducible representations of ModΣ. When G = U(1), and Σ is the 2-torus, Hom( π,G )/G naturally identifies with T 2, by the functions α,β corresponding to a basis of π1(Σ). The functions φm,n:=αmβn, forms a Hilbert basis of H, indexed by ( m,n )∈ Z2. The ModΣ-representation on H arises from the linear GL(2, Z)-action on its basis Z2. The GL(2, Z)-orbits on Z2 are indexed by integers d≥ 0. The orbit of ( d, 0) consists of all ( m,n )∈ Z2 with gcd(m,n ) = d. These are Hilbert bases for irreducible constituents Cd of H. The irreducible constituents Cd admit an alternate description, as follows. The d-fold covering homomorphism G Φd − − →G induces a covering space Hom(π,G )/G−→ Hom(π,G )/G. Let Ld denote the closure of the image of the induced map H−→ H. Then Ld = ˆ⨁ d′|d Cd′ so Cd consists of the orthocomplement in Ld of the sum of all Ld′ for d′|d but d′̸=d.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.2, PDF page 216\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000189, "problem_number": "AMR-109-0189", "title": "Problem 2.3 — Find a similar geometric interpretation for the irreducible constituents for compact nonabelian groups G.", "statement": "Find a similar geometric interpretation for the irreducible constituents for compact nonabelian groups G.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.3, PDF page 216\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000190, "problem_number": "AMR-109-0190", "title": "Conjecture 2.3 — If r≥ 3, the action of Out(π) on Hom(π,G ) is ergodic.", "statement": "If r≥ 3, the action of Out(π) on Hom(π,G ) is ergodic. Using calculations in [ 43], this conjecture has been proved [ 47] when all of the simple factors of G are locally isomorphic to SU(2).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 2.3, PDF page 217\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Ergodicity of the action of mapping class group / Out on character varieties has been studied (Goldman's program). Specific ergodicity for the action on all of Hom(π,G) not fully settled." }, { "id": 11000191, "problem_number": "AMR-109-0191", "title": "Problem 2.4 — Determine necessary and sufficient conditions on a general representation ρ for its orbit ModΣ· [ρ] to be dense.", "statement": "Determine necessary and sufficient conditions on a general representation ρ for its orbit ModΣ· [ρ] to be dense. The case when G = SU(2) and Σ an n-holed sphere for n> 4 remains open.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.4, PDF page 218\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to Goldman's theory of the mapping class group action on character varieties; full conditions open." }, { "id": 11000192, "problem_number": "AMR-109-0192", "title": "Problem 2.5 — Construct an example of a pseudo-Anosov mapping class for a closed surface which is not ergodic on the SU(2)-characte…", "statement": "Construct an example of a pseudo-Anosov mapping class for a closed surface which is not ergodic on the SU(2)-character variety.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2.5, PDF page 218\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000193, "problem_number": "AMR-109-0193", "title": "Conjecture 3.1 — Suppose that b = 0 (Σ is closed).", "statement": "Suppose that b = 0 (Σ is closed). For each integer 1≤k≤ 2g +b− 2, the ModΣ-action on the component e−1(2− 2g +b +k) of Hom(π,G ) is ergodic. When b = 0, the component e−1(3− 2g) ≈ Σ× R6g−8 represents a 6 g− 6-dimensional thickening of Σ, upon which ModΣ acts. However, Morita [ 81] showed that ModΣ cannot act smoothly on Σ itself inducing the homomorphism Diff(Σ) −→ ModΣ. (Recently Markovic [ 80] has announced that if Σ is a closed surface of genus > 5, then ModΣ cannot even act on Σ by homeomorphisms inducing Homeo(Σ) −→ ModΣ.)", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 3.1, PDF page 220\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000194, "problem_number": "AMR-109-0194", "title": "Problem 3.1 — Determine the smallest dimensional manifold homotopy-equivalent to Σ upon which ModΣ acts compatibly with the outer a…", "statement": "Determine the smallest dimensional manifold homotopy-equivalent to Σ upon which ModΣ acts compatibly with the outer action of ModΣ on π1(Σ).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.1, PDF page 220\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to action dimension / the \"smallest model\" for the action; partial (see AMR-109-0005). Cannot verify a settled answer." }, { "id": 11000195, "problem_number": "AMR-109-0195", "title": "Problem 3.2 — Find general conditions which ensure that (10) is proper.", "statement": "Find general conditions which ensure that (10) is proper. The level set R3∩κ−1(2) consists of characters of abelian representations, and ModΣ is ergodic on each of the four connected components of the smooth part of R3∩κ−1(2). When 2 18, the level sets R3∩κ−1(t) display both proper dynamics and chaotic dynamics. The region (−∞,−2]3 consists of characters of discrete embeddings ρ where the quotient hyperbolic surface H2/ρ(π) is homeomorphic to a three-holed sphere. Every homotopy equivalence Σ −→ P, where P is a hyperbolic surface homeomorphic to a three-holed sphere, determines such a character. Furthermore these determine closed triangular regions which are freely permuted by ModΣ. On the complement of these wandering domains the action is ergodic. WhenG = PGL(2, R), the group of (possibly orientation-reversing) isometries of H2, a similar analysis was begun by Stantchev [ 94, 52 ]. One obtains similar dynamical systems, where ModΣ acts now on the space of representations into the group G± = SL(2, C)∩ ( GL(2, R)∪ i GL(2, R) ) 13. Mapping class group dynamics on surface group representations 215 which doubly covers the two-component group PGL(2, R). These G±-representations are again parametrized by traces. They comprise four components, one of which is the subset of R3 parametrizing SL(2, R)-representations discussed above. The other three components are R×iR×iR, i R× R×iR, i R×iR× R respectively. Consider iR× R×iR. For −14≤t< 2, the ModΣ-action is ergodic, but when t< −14, wandering domains appear. The wandering domains correspond to homotopy-equivalences Σ−→P, where P is a hyperbolic surface homeomorphic to a two-holed projective plane. The action is ergodic on the complement of the wandering domains.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.2, PDF page 221\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000196, "problem_number": "AMR-109-0196", "title": "Problem 3.3 — Determine the ergodic behavior of the ModΣ-action on the level sets ( iR× R×iR ) ∩κ−1(t) wheret> 2.", "statement": "Determine the ergodic behavior of the ModΣ-action on the level sets ( iR× R×iR ) ∩κ−1(t) wheret> 2. The level sets for t> 6 contains wandering domains corresponding to Fricke spaces of a one-holed Klein bottle.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.3, PDF page 222\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000197, "problem_number": "AMR-109-0197", "title": "Problem 3.4 — Find a point ρ∈ Hom(π, SL(2, C)) such that the closure of its orbit ModΣ· [ρ] meets both the image of the unitary cha…", "statement": "Find a point ρ∈ Hom(π, SL(2, C)) such that the closure of its orbit ModΣ· [ρ] meets both the image of the unitary characters Hom(π, SU(2)) and the closure QΣ of the quasi- Fuchsian characters. Homological actions. The action of ModΣ on the homology of Hom( π,G )/G furnishes another source of possibly interesting linear representations of ModΣ. With Neumann [ 51], we proved that for the relative SL(2, C)-character varieties of the one-holed torus and four-holed sphere, the action of ModΣ factors through a finite group. Atiyah-Bott [ 3] use infinite-dimensional Morse theory to analyze the algebraic topology of Hom(π,G )/G, when G is compact. For the nonsingular components their techniques imply that the ModΣ-action on the rational cohomology of Hom( π,G )/G factors through the symplectic representation of ModΣ on H ∗(Σ). In particular Biswas [ 9] proved that the Torelli group acts trivially on nonsingular components. In contrast, Cappell-Lee-Miller [ 17, 18 ] proved the surprising result that that the Torelli group acts nontrivially on the homology of the SU(2)-character variety when Σ is closed.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.4, PDF page 223\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000198, "problem_number": "AMR-109-0198", "title": "Problem 3.6 — Find a substitute for convex cocompactness in higher rank which includes the above examples of proper ModΣ-actions, a…", "statement": "Find a substitute for convex cocompactness in higher rank which includes the above examples of proper ModΣ-actions, and for which Eρ is proper. The work of Bonahon-Thurston on geometric tameness, and its recent extensions, implies that the energy function of a discrete embedding π−→ PSL(2, C) is proper if and only if it is quasi-Fuchsian [ 53].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.6, PDF page 225\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: This is a broad program (higher-rank Anosov / convex cocompact analogs); active research, not settled." }, { "id": 11000199, "problem_number": "AMR-109-0199", "title": "Conjecture 3.2 — If k = 1, then U is onto.", "statement": "If k = 1, then U is onto. In general a PSL(2, R)-representation with dense image lies in Image(U). 220 W. Goldman", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 3.2, PDF page 226\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000200, "problem_number": "AMR-109-0200", "title": "Problem 1: — Determine the metric completion of the Gromov boundary of C(S) and relate this metric completion to the geometry of C…", "statement": "Determine the metric completion of the Gromov boundary of C(S) and relate this metric completion to the geometry of C(S). There is yet another way to construct unparametrized uniform quasi-geodesics in C(S). Namely, Teichm¨ uller spaceforS is the space Tg,m of marked isometry classes of complete hyperbolic met- rics on S of finite volume. The Teichm¨ uller space can naturally be identified with a domain in C3g−3+m (see Chapter 6 of [IT]). By a classical result of Bers (see [Bu]), there is a number χ> 0 such that for every complete hyperbolic metric h on S there is a pants decomposition for S consisting of simple closed h- geodesics of length at most χ. Moreover, the diameter in C(S) of the set of simple closed curves on S of h-length at most χ is bounded from above by a universal constant D >0. Thus we can define a map Ψ: Tg,m→C (S) by associating to a marked hyperbolic metric h a simple closed curve of h-length at most χ. For any two such maps Ψ, Ψ′ we then have sup h∈Tg,md(Ψ(h), Ψ′(h))≤D. The Teichm¨ uller metriconTg,m is a complete Finsler metric which is just the Kobayashi metric on the domain in C3g−3+m representingTg,m (see [IT]). Through any two distinct points in Tg,m passes a unique Teichm¨ uller geodesic. Each such geodesic line in Tg,m is uniquely determined by its endpoints in the Thurston boundary ofTg,m which is just the space PML of projective measured laminations on S. The supports of the two measured laminations on S defining the endpoints of the geodesic together fill up the surface S, i.e. every simple closed curve on S intersects at least one of the two laminations transversely. These laminations then define a holomorphic quadratic 14. Geometric properties of the mapping class group 235 differential (we refer to Section 4 of [Ke] for a discussion of this fact). The following result is implicitly contained in the paper [MM1] by Masur and Minsky; an explicit proof using a result of Rafi [R] can be found in Section 4 of [H4].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 1:, PDF page 241\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "PARTIAL/OPEN (boundary identified; metric completion subtle). Literature status: The Gromov boundary of the curve complex is the space of ending laminations (Klarrich); its structure studied. The specific metric completion question is subtle; partial. No fully settled published answer located." }, { "id": 11000201, "problem_number": "AMR-109-0201", "title": "Problem 2: — For a fixed number R > 0, is there is compact subset K(R) of moduli space containing the projection of every Teichm¨…", "statement": "For a fixed number R > 0, is there is compact subset K(R) of moduli space containing the projection of every Teichm¨ uller geodesic γ which satisfies d(Ψ(γ(s)), Ψ(γ(t)))≥ |s−t|/R−R for all s,t∈ R? Conversely, is there for a given compact set K in Mod( S) a number R =R(K)> 0 such that d(Ψ(γ(s)), Ψ(γ(t)))≥|s−t|/R−R for every Teichm¨ uller geodesic which projects into K? Analyze the images in C(S) of geodesics in Tg,m determined by minimal geodesic laminations which fill up S and are not uniquely ergodic.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2:, PDF page 242\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000202, "problem_number": "AMR-109-0202", "title": "Problem 3: — Describe the space of geodesic currents for Mg,m.", "statement": "Describe the space of geodesic currents for Mg,m. Is the set of weighted sums of Dirac masses at the pairs of fixed points of pseudo-Anosov elements dense? Is there a geodesic current µ which is absolutely continuous, i.e. such that there is a Mg,m-invariant measure class µ0 on ∂C(S) with the property that for every Borel subset A of ∂C(S) we have µ0(A) = 0 if and only if µ(A×∂C(S)− ∆) = 0? Is there a distinguished absolutely continuous current such that the invariant measure class on ∂C(S) is determined by a Hausdorff measure with respect to one of the distance functions δ on ∂C(S)?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3:, PDF page 245\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Space of geodesic currents is studied (Bonahon); description open in full." }, { "id": 11000203, "problem_number": "AMR-109-0203", "title": "Problem 4: — Does the above definition of a convex cocompact subgroup of Mg,m coincide with the definition of Farb and Mosher in […", "statement": "Does the above definition of a convex cocompact subgroup of Mg,m coincide with the definition of Farb and Mosher in [FMo]? Is the natural extension of such a group Γ by the fundamental group π1(S) of S word hyperbolic? Is there a convex cocompact subgroup of Mg,m which is isomorphic to the fundamental group of a closed surface of genus at least 2? A particular interesting class of subgroups of Mg,m arise from Veech surfaces. These surfaces are the projections to moduli space of the stabilizer of a complex geodesic in Teichm¨ uller space (which is a maximal embedded complex disc in the Teichm¨ uller space viewed as a bounded domain in C3g−3+m) with the additional property that this stabilizer is a lattice in PSL (2, R). Veech surfaces are surfaces of finite type with isolated singularities embedded in moduli space; they are never closed [V]. Thus their corresponding subgroup of Mg,m contains a free group of finite index with a distinguished family of conjugacy classes corresponding to the cusps of the curve. Veech surfaces have many beautiful algebraic and geometric properties (see e.g. [McM1], [McM2]). Elementary constructions of such surfaces and their coresponding subgroups of Mg,m are for example discussed in [L]. Veech surfaces can also be used to construct explicit subgroups of mapping class groups with prescribed geometric properties. A particularly beautiful result along this line was recently obtained by Leininger and Reid [LR].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4:, PDF page 247\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE (equivalence likely established in literature). Literature status: The equivalence of convex cocompactness definitions for mapping class groups has been studied; likely resolved in parts (e.g. Farb–Mosher, and later works of Hamenstädt and others). Cannot fully verify." }, { "id": 11000204, "problem_number": "AMR-109-0204", "title": "Problem 5: — Develop a theory of geometrically finite subgroups of Mg,m which include the groups defined by Veech surfaces.", "statement": "Develop a theory of geometrically finite subgroups of Mg,m which include the groups defined by Veech surfaces. 14. Geometric properties of the mapping class group 241", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 5:, PDF page 247\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Development of geometric finiteness in mapping class groups is an active area; Veech groups are a motivating class. Not fully settled." }, { "id": 11000205, "problem_number": "AMR-109-0205", "title": "Problem 6: — For a closed surface of genus g≥ 3, is the Torelli subgroup of Mg,m undistorted?", "statement": "For a closed surface of genus g≥ 3, is the Torelli subgroup of Mg,m undistorted? More generally, find a distorted finitely generated subgroup of Mg,m. The splitting sequences on the complex of train tracks can be used to investigate the large- scale geometric behavior of the mapping class group. Note that such a splitting sequence ( τi) is determined by an initial train track and for each i by a choice of a splitting move among a uniformly bounded number of possibilities which transforms the train track τi to the train track τi+1. In other words, it is possible to treat splitting sequences and hence uniform quasi-geodesics inMg,m in an algorithmic way. 14. Geometric properties of the mapping class group 243 Algorithmic calculations in a finitely generated group Γ are very intimately related to two basis decision problems which go back to Dehn and can be formulated as follows (see Chapter III.Γ.1 in [BH]). Word problem: A word w in a fixed system of generators for Γ is given. One is required to find a method to decide in a finite number of steps whether or not this word represents the identity in Γ. Conjugacy problem: Two elements g,h∈ Γ are given. A method is sought to decide in a finite number of steps whether or not the elements g,h are conjugate, i.e. whether there is some u∈ Γ such that h =ugu−1. In the last decade of the twentith century, Epstein, Cannon, Holt, Levy, Paterson, Thurston [E] formulated a property for finitely generated groups which ensures that these problems can be solved in controlled time. Namely, a biautomatic structure for a finitely generated group Γ consists of a finite alphabet A, a (not necessarily injective) map π:A→ Γ and a regular language L over the alphabet A with the following properties. The set π(A) generates Γ, and there is an inversion ι: A→ A (i.e. ι2 = Id) with π(ιa) = π(a)−1 for all a∈ A. In particular, π(A) is a symmetric set of generators for Γ. Via concatenation, every word w in the alphabet A is mapped by π to a word in the generators π(A) of Γ and hence it defines an element π(w)∈ Γ. We require that the restriction of the map π to the set of all words from the language L maps L onto Γ. For all x,y∈A and each word w∈L of length k≥ 0, the word xwy defines via the projection π a path s: [0,k + 2] → Γ. By assumption, there is a word w′∈L of length ℓ >0 with π(w′) = π(xwy). Let s′: [0,ℓ ]→ Γ be the corresponding path in Γ; we require that the distance in Γ between s(i) and s′(i) is bounded by a universal constant which neither depends on i nor on the choice of x,y,w,w ′. Extending earlier work of Mosher [M1], in [H1] the complex of train tracks and its algorithmic properties are used to show.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 6:, PDF page 249\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED-IN-LITERATURE: Torelli is exponentially distorted (not undistorted). Literature status: SOLVED (in the negative — it is exponentially distorted). Broaddus–Farb–Putman (\"Irreducible Sp-representations and subgroup distortion in the mapping class group\", Comm. Math. Helv. 86 (2011); arXiv:1006.0787) proved the Torelli group Ig (and Ig,1) is at least exponentially distorted and at most doubly exponentially distorted in Modg for g≥3, answering a question of Hamenstädt. (Farb–Lubotzky–Minsky earlier established rank-one phenomena yielding recursive distortion.)" }, { "id": 11000206, "problem_number": "AMR-109-0206", "title": "Problem 7: — Is the mapping class group linear?", "statement": "Is the mapping class group linear? A locally compact group Γ is said to satisfy the Haagerup approximation property or is a-T- menable if there exists a continuous, isometric action α of Γ on some affine Hilbert space H which is metrically proper. This means that for all bounded subsets B ofH, the set {g∈ Γ|α(g)B∩B̸=∅} is relatively compact in Γ. There are other equivalent characterizations of this property (see [CCJJV]) which can be viewed as a strong negation of the (perhaps more widely know) property (T) of Kazhdan. The class of a-T-menable groups contains for example all amenable groups, Coxeter groups and the isometry groups of real and complex hyperbolic spaces. It is also known that for a-T-menable groups the Baum-Connes conjecture holds (see [CCJJV]).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 7:, PDF page 251\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "PARTIAL: g=2 linear (solved); g≥3 linearity open. Literature status: PARTIAL / OPEN for g≥3, SOLVED for g=2: - Genus 2 (and punctured spheres, hyperelliptic MCGs) are LINEAR: Bigelow–Budney (AGT 2001, \"The mapping class group of a genus two surface is linear\"), Korkmaz (a faithful representation of Mod2), using the faithfulness of the Lawrence–Krammer/Bigelow representation of the braid group and Birman–Hilden theory. - Linearity of Modg for genus g≥3 REMAINS OPEN (confirmed by surveys; e.g. Korkmaz's note \"On the linearity of certain mapping class groups\" states linearity for g≥3 \"still remains open\"). Recent (2024) work (\"Low-dimensional linear representations of mapping class groups\", J. Topol.) studies low-dimensional representations but does not resolve full linearity." }, { "id": 11000207, "problem_number": "AMR-109-0207", "title": "Problem 8: — Is the mapping class group a-T-menable?", "statement": "Is the mapping class group a-T-menable? 14. Geometric properties of the mapping class group 245", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 8:, PDF page 251\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE (Haagerup property open). Literature status: Whether Modg has the Haagerup property is open and famous (it is known NOT to have property (T), but Haagerup is not settled). The a-T-menability of mapping class groups for g≥2 appears to remain OPEN through 2026." }, { "id": 11000208, "problem_number": "AMR-109-0208", "title": "Problem 1 — (Geodesics on general flat surfaces).", "statement": "(Geodesics on general flat surfaces). Describe the behavior of geodesics on general flat surfaces. Prove (or disprove) the conjecture that the geodesic flow is ergodic on a typical (in 248 15. Problems on billiards, flat surfaces and translation surfaces 249 any reasonable sense) flat surface. Does any (almost any) flat surface have at least one closed geodesic which does not pass through singular points? If the answer is positive then one can ask for the asymptotics for the number of closed geodesics of bounded length as a function of the bound. Note that typically a geodesic representative in a homotopy class of a simple closed curve is realized by a broken line containing many geodesic segments going from one conical singularity to the other. The counting problem for regular closed geodesics (ones which do not pass through singularities) is quite different from the counting problem for geodesics realized by broken lines. The following questions treat billiards in arbitrary polygons in the plane.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 1, PDF page 255\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Flat surface geodesics are studied (Masur's work on unique ergodicity etc.); specific conjecture unclear." }, { "id": 11000209, "problem_number": "AMR-109-0209", "title": "Problem 2 — (Billiards in general polygons).", "statement": "(Billiards in general polygons). Does every billiard table have at least one regular periodic trajectory? If the answer is affirmative, does this trajectory persist under deformations of the billiard table? If a periodic trajectory exists, find the asymptotics for the number of periodic trajectories of bounded length as a function of the bound. Describe the behavior of a generic regular billiard trajectory in a generic polygon; in particular, prove (or disprove) the assertion that the billiard flow is ergodic. 1 We note that the case of triangles is already highly non-trivial. For recent work on billiards in obtuse triangles see [ Sc1] and [ Sc2]. In the case of triangles the notion of generic can be interpreted as follows. The space of triangles up to similarity can be parametrized as the set of triples ( θ1,θ 2,θ 3) with ∑θi =π and eachθi> 0. It is naturally an open simplex. Generic then refers to the natural Lebesgue measure. To motivate the next problem we note that there is a close connection between the study of interval exchange transformations and billiards in rational polygons (defined below). An important technique in the study of interval exchange maps is that of renormalization. Given an interval exchange on the unit interval one can take the induced transformation on a subinterval. The resulting map is again an interval exchange map, and if one renormalizes so that the new interval has length one, then this gives a transformation on the space of unit interval exchange maps. This transformation is called the Rauzy-Veech induction ([ Ra], [ V e2]), and it has proved to be of fundamental importance. There is a corresponding notion for the renormalization of translation surfaces given by the Teichm¨ uller geodesic flow.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 2, PDF page 256\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Rational polygonal billiards are well studied; general (irrational) polygons are famously open/difficult (e.g. existence of periodic orbits)." }, { "id": 11000210, "problem_number": "AMR-109-0210", "title": "Problem 3 — (Renormalization of billiards in polygons).", "statement": "(Renormalization of billiards in polygons). Is there a natural dynamical system acting on the space of billiards in polygons so as to allow a useful renormalization procedure?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3, PDF page 256\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000211, "problem_number": "AMR-109-0211", "title": "Problem 4 — (Characterization of Veech surfaces).", "statement": "(Characterization of Veech surfaces). Characterize all Veech surfaces (for each stratum of each genus). This problem is trivial in genus one; in genus two K. Calta [ Ca] and C. McMullen [ McM2] have provided solutions. In the papers [ KnSm] and Puchta [ Pu] the acute rational billiard triangles that give rise to Veech surfaces were classified. In [ SmW e] there is a criterion for a surface ( X,ω ) to be a Veech surface that is given in terms of the areas of triangles embedded in (X,ω ).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4, PDF page 259\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Characterizing Veech surfaces (which flat surfaces have lattice Veech groups) is partially open (e.g. conjectures on arithmetic vs non-arithmetic Veech surfaces)." }, { "id": 11000212, "problem_number": "AMR-109-0212", "title": "Problem 5 — (Fuchsian groups).", "statement": "(Fuchsian groups). Which Fuchsian groups are realized as Veech groups? Which subgroups of the mapping class group appear as Veech groups? This is equivalent to asking which subgroups are the stabilizers of a Teichm¨ uller disc.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 5, PDF page 259\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000213, "problem_number": "AMR-109-0213", "title": "Problem 6 — (Purely cyclic).", "statement": "(Purely cyclic). Is there a Veech group that is cyclic and generated by a single hy- perbolic element? Equivalently, is there a pseudo-Anosov map such that its associated Teichm¨ uller disk, is invariant only under powers of the pseudo-Anosov?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 6, PDF page 259\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify (likely about purely cyclic Veech groups / parabolic only?)." }, { "id": 11000214, "problem_number": "AMR-109-0214", "title": "Problem 7 — (Algorithm for Veech groups).", "statement": "(Algorithm for Veech groups). Is there an algorithm for determining the Veech group of a general translation surface or quadratic differential? An interesting class of Veech surfaces are the square-tiled surfaces. These surfaces can be represented as a union of glued squares all of the same size, see [ Zo3], [ HuLe1].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 7, PDF page 259\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Algorithms for Veech groups of translation surfaces exist in some cases (decidability studied); full general algorithm open." }, { "id": 11000215, "problem_number": "AMR-109-0215", "title": "Problem 8 — (Orbits of square-tiled surfaces).", "statement": "(Orbits of square-tiled surfaces). Classify the SL(2, R) orbits of square-tiled sur- faces in any stratum. Describe their Teichm¨ uller discs. A particular case of this problem is that of the stratum H(1, 1). This above problem is solved only for the stratum H(2), see [ HuLe1], [ McM4].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 8, PDF page 259\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Square-tiled surface orbit classification (SL2(Z)-orbits / Teichmüller) is studied; open in general." }, { "id": 11000216, "problem_number": "AMR-109-0216", "title": "Problem 9 — (Orbit closures for moduli spaces).", "statement": "(Orbit closures for moduli spaces). Determine the closures of the orbits for the GL+(2, R)-action on H(α) andQ(β). Are these closures always complex-analytic (complex- algebraic?) orbifolds? Characterize the closures geometrically. Note that by a theorem of Kontsevich any GL +(2, R)-invariant complex-analytic subvariety is represented by an affine subspace in period coordinates. Consider the subset H1(α)⊂H (α) of translation surfaces of area one. It is a real codimension one subvariety in H(α) invariant under the action of SL(2, R). In the period coordinates it is defined by a quadratic equation (the Riemann bilinear relation). It is often called a unit 15. Problems on billiards, flat surfaces and translation surfaces 253 hyperboloid. It is worth noting that it is a manifold locally modelled on a paraboloid. The invariant measure on H(α) gives a natural invariant measure on the unit hyperboloid H1(α). Similarly one can define the unit hyperboloid Q1(β). It was proved by Masur and Veech that the total measure of any H1(α),Q1(β) is finite.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 9, PDF page 259\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L4", "research_summary": "PARTIAL/SOLVED: the rank-zero/linear part resolved by EMM; full classification of orbit closures (as affine manifolds) is a major open program. Literature status: This is largely resolved by Eskin–Mirzakhani–Mohammadi (the \"magic wand theorem\": orbit closures are affine invariant manifolds) and Wright, Filip, etc. So substantial progress; full classification of affine invariant submanifolds is an active program (the \"large-scale\" classification open)." }, { "id": 11000217, "problem_number": "AMR-109-0217", "title": "Problem 10 — (Ergodic measures).", "statement": "(Ergodic measures). Classify the ergodic measures for the action of SL(2, R) on H1(α) andQ1(β). McMullen [ McM3] has solved Problems 9 and 10 in the case of translation surfaces in genus 2. A subset Ω is called minimal for the action of SL(2, R) if it is closed, invariant, and it has no proper closed invariant subsets. The SL(2, R) orbit of a Veech surface is an example of a minimal set.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 10, PDF page 260\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "PARTIAL: significant results (EMM), not the full picture. Literature status: Eskin–Mirzakhani–Mohammadi give classification of SL2(R)-invariant ergodic measures. Substantial progress." }, { "id": 11000218, "problem_number": "AMR-109-0218", "title": "Problem 11 — (Minimal sets).", "statement": "(Minimal sets). Describe the minimal sets for the SL(2, R)-action on H1(α) and Q1(β). Since Veech surfaces give rise to minimal sets, this problem generalizes the problem of characterizing Veech surfaces. The problem below is particularly important for numerous applications. One application is to counting problems.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 11, PDF page 260\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000219, "problem_number": "AMR-109-0219", "title": "Problem 12 — (Analog of Ratner theorem).", "statement": "(Analog of Ratner theorem).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 12, PDF page 260\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "PARTIAL/SOLVED: EMM provide the Ratner-type analog. Literature status: Eskin–Mirzakhani–Mohammadi proved an analog of Ratner's theorem / measure classification for the SL2(R) action on moduli of translation surfaces. So largely answered." }, { "id": 11000220, "problem_number": "AMR-109-0220", "title": "Problem 13 — (Kernel foliation).", "statement": "(Kernel foliation). IsN a complex-analytic (complex-algebraic) orbifold? When is dimCN = dim CO +n−m? On the other hand when does N coincide with the entire connected component of the enveloping stratum H(α′ 1,...,α ′ n)? One of the key properties used in [ McM3] for the classification of the closures of orbits of GL(2, R) in each of H(1, 1) and H(2) was the knowledge that on any translation surface of either stratum, one can find a pair of homologous saddle connections. For example, cutting a surface ( X,ω ) in H(1, 1) along two homologous saddle connections joining distinct zeroes decomposes the surface into two tori, allowing one to apply the machinery of Ratner’s Theorem.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 13, PDF page 261\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000221, "problem_number": "AMR-109-0221", "title": "Problem 14 — (Decomposition of surfaces).", "statement": "(Decomposition of surfaces). Given a connected component of the stratum H(α) of Abelian differentials (or of quadratic differentials Q(β) find those configurations of homologous saddle connections (or homologous closed geodesics), which are present on every surface in the stratum. For quadratic differentials the notion of homologous saddle connections (homologous closed geodesics) should be understood in terms of homology with local coefficients, see [ MaZo]. The last two problems in this section concern the Teichm¨ uller geodesic flow on the moduli spaces. This is the flow defined by the 1-parameter subgroup ( et 0 0 e−t ) t∈R onH(α1,...,α m). In any smooth dynamical system the Lyapunov exponents (see [ BaPe], [ F o2]) are important. Recently, A. Avila and M. Viana [ AvVi] have shown the simplicity of the spectrum for the cocycle related to the Teichm¨ uller geodesic flow (strengthening the earlier result of Forni on positivity of the smallest Lyapunov exponent).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 14, PDF page 261\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000222, "problem_number": "AMR-109-0222", "title": "Problem 15 — (Lyapunov exponents).", "statement": "(Lyapunov exponents). Study individual Lyapunov exponents of the Teichm¨ uller geodesic flow: – for all known SL(2; R)-invariant subvarieties; – for strata; – for strata of large genera as the genus tends to infinity. Are they related to characteristic numbers of any natural bundles over appropriate compacti- fications of the strata? The motivation for this problem is a beautiful formula of Kontsevich [ Ko] representing the sum of the first g Lyapunov exponents in differential-geometric terms. It follows from the Calabi Theorem [ Cb] that given a real closed 1-form ω0 with isolated zeroes Σ (satisfying some natural conditions) on a smooth surface S of real dimension two, one 15. Problems on billiards, flat surfaces and translation surfaces 255 can find a complex structure on S and a holomorphic 1-form ω such that the ω0 is the real part of ω. Consider the resulting point ( X,ω ) in the corresponding stratum. For generic ( X,ω ) the cocycle related to the Teichm¨ uller geodesic flow acting on H 1(X, R) defines a pair of transverse Lagrangian subspaces H 1(X, R) = L0⊕L1 by means of the Oseledets Theorem ([ BaPe]). These subspaces correspond to contracting and to expanding directions. Though the pair ( X,ω ) is not uniquely determined by ω0, the subspace L0⊂H 1(X, R) does not depend on ( X,ω ) for a given ω0. Moreover, L0 does not change under small deformations of ω0 that preserve the cohomology class [ ω]∈H 1(S, Σ; R). We get a topological object L0 defined in implicit dynamical terms.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 15, PDF page 261\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Lyapunov exponents of the Teichmüller flow are studied (Kontsevich–Zorich, Forni); full computation open." }, { "id": 11000223, "problem_number": "AMR-109-0223", "title": "Problem 16 — (Dynamical Hodge decomposition).", "statement": "(Dynamical Hodge decomposition). Study properties of distributions of the La- grangian subspaces in H 1(S; R) defined by the Teichm¨ uller geodesic flow, in particular, their con- tinuity. Is there any topological or geometric way to define them? The Lagrangian subspaces are an interesting structure relating topology and geometry.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 16, PDF page 262\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to the Kontsevich–Zorich cocycle; cannot verify exact problem." }, { "id": 11000224, "problem_number": "AMR-109-0224", "title": "Problem 17 — (Converse to dichotomy).", "statement": "(Converse to dichotomy). Characterize translation surfaces for which (1) the set of minimal directions coincides with the set of uniquely ergodic directions; (2) the set of completely periodic directions coincides with the set of non-uniquely ergodic directions. 256 P. Hubert, H. Masur, T. Schmidt and A. Zorich Note that Property (2) implies Property (1). In genus g = 2 it is known that for every translation surface which is not a Veech surface there is a direction θ which is minimal and not uniquely ergodic [ ChMa]. On the other hand using work of Hubert–Schmidt [ HuSt2], B. Weiss has given an example of a surface which is not a Veech surface and yet for which Property (2) holds. (This surface is obtained as a ramified covering over a Veech surface with a single ramification point.) As mentioned above, a closed orbit avoiding the conical singularities of the flat surface deter- mines a cylinder of parallel lines, all of the same length. It is of interest to find the asymptotics for the number of cylinders (in all possible directions) of lengths less than a given number. In the case of the standard flat torus the number of cylinders of length at most L is asymptotic to 1 ζ(2)πL2. Each Veech surface also has quadratic asymptotics [ V e3] and the same is true for generic surfaces in each stratum [ EsMa].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 17, PDF page 262\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000225, "problem_number": "AMR-109-0225", "title": "Problem 18 — (Quadratic asymptotics for any surface).", "statement": "(Quadratic asymptotics for any surface). Is it true that every translation surface or quadratic differential has exact quadratic asymptotics for the number of saddle connections and for the number of regular closed geodesics?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 18, PDF page 263\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000226, "problem_number": "AMR-109-0226", "title": "Problem 19 — (Error term for counting functions).", "statement": "(Error term for counting functions). What can be said about the error term in the quadratic asymptotics for counting functions N ((X,ω ),L )∼c·L2 on a generic translation surface (X,ω )? In particular, is it true that lim sup L→∞ log|N (S,L )−c·L2| logL < 2? Is the lim sup the same for almost all flat surfaces in a given connected component of a stratum? The classical Circle Problem gives an estimate for the error term in the case of the torus. Veech proved that for Veech surfaces the limsup in the error term is actually a limit, see [ V e3]. However, nothing is known about the value of this limit. One may ask whether there is a uniform bound for this limit for Veech surfaces in a given stratum or even for the square-tiled surfaces in a given stratum.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 19, PDF page 263\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000227, "problem_number": "AMR-109-0227", "title": "Problem 20 — (Topology of strata).", "statement": "(Topology of strata). Is it true that the connected components of the strata H(α) and of the strata Q(β) areK(π, 1)-spaces (i.e. their universal covers are contractible)? It is known [ KoZo], [ La] that the strata H(α) and Q(β) need not be connected. With the exception of the four strata listed below, there are intrinsic invariants that allow one to tell which component a given translation surface or quadratic differential belongs to. 15. Problems on billiards, flat surfaces and translation surfaces 257", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 20, PDF page 263\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "PARTIAL: components classified; full homotopy type open (K(π,1) question). Literature status: Connectedness of strata and the \"Conti–Delucchi\" theory; the classification of connected components of strata was established by Kontsevich–Zorich and Lanneau (and for higher genera), and the \"even/odd spin\" components by many. Homotopy type (e.g. the \"Strata are K(π,1)\" question by Deligne-Mostow / recent work) partially resolved." }, { "id": 11000228, "problem_number": "AMR-109-0228", "title": "Problem 21 — (Exceptional Strata).", "statement": "(Exceptional Strata). Find a geometric invariant which distinguishes differ- ent connected components of the four exceptional strata Q(−1, 9),Q(−1, 3, 6),Q(−1, 3, 3, 3) and Q(12). At the moment the known invariant (called the extended Rauzy class) distinguishing connected components is given in combinatorial and not geometric terms. ([ La])", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 21, PDF page 264\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000229, "problem_number": "AMR-109-0229", "title": "Problem 1.1 — Given a subgroup G 1, and hence avoid complexities that arise when b+ 2 = 1. Proposition 4.5. A postive answer to Conjecture 4.4 implies that a closed hyperbolic 4- manifold M cannot be symplectic. Proof: This follows automatically from [ 37] in the case when b+ 2 > 1, since [ 37] shows that a compact oriented symplectic 4-manifold with b+ 2 > 1 has non-vanishing Seiberg-Witten invariants. Thus assume that b+ 2 (M ) = 1, and M is symplectic (if b2(M ) = 0 there is nothing to prove). From the discussion above, since M is hyperbolic, the signature of M is 0, and so b2(M ) = 2. Also M being hyperbolic implies π1(M ) is residually finite and so M has many finite covers. Let p:M1→M be a cover of degree d> 1. Since M is symplectic, M1 will be symplectic using the pullback of the symplectic form on M. We claim that b+ 2 (M1)> 1, and so we can apply [ 37] to get a contradiction. 282 A.W. Reid For if b2(M1) = 2, then (from the volume formula above) since the Euler characteristic satisfies χ(M1) > 0, it follows that b1(M1) < 2. However, χ(M1) = dχ(M ) and this shows b1(M1) = 2−d(2−b1(M )). These remarks yield the desired contradiction. ⊔ ⊓ On the otherhand, any X that has the description of a surface bundle over a surface where both the base and fiber have genus ≥ 2 is symplectic by an old argument of Thurston (see [ 39] or [ 14] Theorem 10.2.17). Motivated by this discussion, one can ask the following generalization of Question 4.1, which is an analogue of the virtual fibering question in dimension 3.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 4.4, PDF page 288\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000253, "problem_number": "AMR-109-0253", "title": "Question 4.6 — Does there exist a closed hyperbolic 4-manifold X for which no finite cover admits a symplectic structure?", "statement": "Does there exist a closed hyperbolic 4-manifold X for which no finite cover admits a symplectic structure? (ie X is not virtually symplectic.) We note that it is easy to construct non-symplectic hyperbolic 4-manifolds. For example it is easy to see that the Davis manifold D (see [ 9]) admits no symplectic structure using the following simple parity rule (see [ 14] Corollary 10.1.10): Suppose that (M,ω ) is a closed symplectic 4-manifold. Then 1−b1(M ) + b+ 2 (M ) is even. For the Davis manifold we have from [ 35] that b1(D) = 24 and b2(D) = 72. Since the signature of D is zero, b+ 2 (D) = 36, and so the parity condition fails. 4.3. A natural weakening of Question 4.1 is the following.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 4.6, PDF page 289\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to the Kähler/symplectic question for hyperbolic manifolds; open." }, { "id": 11000254, "problem_number": "AMR-109-0254", "title": "Question 4.7 — Forg,h≥ 2, does there exist a short exact sequence: 1→π1(Σg)→ Γ→π1(Σh)→ 1 for which Γ is a word hyperbolic group?", "statement": "Forg,h≥ 2, does there exist a short exact sequence: 1→π1(Σg)→ Γ→π1(Σh)→ 1 for which Γ is a word hyperbolic group? Arguing as in the proof of Theorem 4.2, it follows that such an extension defines a purely pseudo-Anosov surface subgroup of Γ g. However, even in this case, little is known. We make two comments in this regard. 1. One result is that Γ cannot be the fundamental group of a closed complex hyperbolic surface. As described in [ 20], this follows from [ 25], on showing that if Γ is as decribed, then there is a non-singular holomorphic fibration X = H2 C/Γ→ Σh that induces the short exact sequence (see also [ 17]). 2. The following idea to construct purely pseudo-Anosov surface subgroups was described to me by Ian Agol. In [ 40], Thurston proves the following result (see [ 40] for terminology and further details). 17. Surface subgroups of mapping class groups 283", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 4.7, PDF page 289\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE. Literature status: This relates to whether there are hyperbolic surface-bundle groups; the relevant monodromy would need to be \"convex cocompact\" so that Γ is hyperbolic. Whether a word-hyperbolic such Γ exists is open (anticipated positive using convex cocompact surface subgroups, but not established as a group with a surface-kernel exact sequence)." }, { "id": 11000255, "problem_number": "AMR-109-0255", "title": "Question 4.9 — Does there exist a cocompact Fuchsian subgroup of ∆5 that misses the com- pactification locus?", "statement": "Does there exist a cocompact Fuchsian subgroup of ∆5 that misses the com- pactification locus? Given such a Fuchsian subgroup F <∆5, Agol produces a purely pseudo-Anosov subgroup in some Γ g using a branched cover construction. 4.4. One generalization of looking for surface subgroups of Γ g is to look for injections of (cocompact) lattices in Lie groups into Γ g. If the lattices are superrigid the image of such a lattice in Γ g is necessarily finite (see [ 12] and [ 43]), and so it follows (cf. Theorem 2 of [ 43]), that the only lattices that can admit a faithful representation (or even an infinite representation) into Γ g are lattices in SO( m, 1), m≥ 2 or SU( q, 1), q≥ 1. Indeed, since solvable subgroups of Γ g are virtually abelian [ 4], this observation also excludes non-cocompact lattices of SU( q, 1) for q≥ 2 from injecting. In addition, there is a simple obstruction to injecting certain of these lattices, or indeed for any group. Namely if a finitely generated group G admits an injection into Γ g, then vcd(G)≤ vcd(Γg) (see [ 5]). If the vcd of a group G satisfies the above inequality, then we call G admissable. The vcd’s of the groups Γ g are known to be 4 g− 5 when g≥ 2 [ 16]. Motivated by this discussion we pose.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 4.9, PDF page 290\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify (Δ5 likely a specific lattice)." }, { "id": 11000256, "problem_number": "AMR-109-0256", "title": "Question 4.10 — Let Γ be a lattice in SO(m, 1), m≥ 3 or SU(q, 1), q≥ 2 which is admissable for Γg.", "statement": "Let Γ be a lattice in SO(m, 1), m≥ 3 or SU(q, 1), q≥ 2 which is admissable for Γg. Does Γ inject in Γg? Can there be purely pseudo-Anosov representations? If such an injection exists does there exist a continuous Γ-equivariant map ∂∞(Γ)→ PML0(Σ)? More generally, for a fixed Σ and hence fixed vcd, a further natural generalization of the discussion here is: 284 A.W. Reid", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 4.10, PDF page 290\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000257, "problem_number": "AMR-109-0257", "title": "Question 4.11 — Which 1-ended admissable word hyperbolic groups G inject in Γg (as purely pseudo-Anosov subgroups)?", "statement": "Which 1-ended admissable word hyperbolic groups G inject in Γg (as purely pseudo-Anosov subgroups)? If such an injection exists does there exist a continuous G-equivariant map ∂∞(G)→ PML0(Σ)?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 4.11, PDF page 291\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to the Farb–Masur / Leininger–Reid program on word-hyperbolic subgroups of McCG; active, not settled." }, { "id": 11000258, "problem_number": "AMR-109-0258", "title": "Question 1 — If R = Q(q1,q 2), is the above map from Bn toHn(q1,q 2) injective?", "statement": "If R = Q(q1,q 2), is the above map from Bn toHn(q1,q 2) injective?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 1, PDF page 312\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Injectivity of braid group representations to Temperley–Lieb / Hecke-type algebras is classical (deformation LL). Cannot fully verify the exact Bn→Hn injectivity." }, { "id": 11000259, "problem_number": "AMR-109-0259", "title": "Question 2 — What are the equivalence classes of braids modulo the moves • ab↔ba, and • b↔σnι(b)?", "statement": "What are the equivalence classes of braids modulo the moves • ab↔ba, and • b↔σnι(b)? In other words, what happens if the Markov move b↔σ−1 n ι(b) is omitted? This question was shown in [ OS03] to be equivalent to the important problem in contact geometry of classifying transversal links up to transversal isotopy. The Bennequin number of a braid b∈ Bn is e−n, where e is the sum of the exponents in a word in the generators σi representingb. The Bennequin number is invariant under the moves in Question 2. Thus it can be used to show that there are braids that are related by Markov moves, but not by the moves in Question 2. Birman and Menasco [ BM] and Etnyre and Honda [ EH] have independently found pairs of braids that are related by Markov moves and have the same Bennequin invariant, but are not related by the moves in Question 2. Their proofs are quite complicated, and it would be nice to have a new invariant that could distinguish their pairs of braids.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2, PDF page 320\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000260, "problem_number": "AMR-109-0260", "title": "Question 3 — What can be said about the dimensions of Dλ?", "statement": "What can be said about the dimensions of Dλ?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 3, PDF page 321\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000261, "problem_number": "AMR-109-0261", "title": "Question 4 — How much of this paper can be generalized to the Birman-Wenzl-Murakami algebra?", "statement": "How much of this paper can be generalized to the Birman-Wenzl-Murakami algebra? In this direction, John Enyang [ Eny04] has shown that the Birman-Murakami-Wenzl algebra is a cellular algebra, and used this to give a definition of its irreducible representations similar to the approach in Section 7. Next we would like to generalize [ Big04] to the Birman-Wenzl-Murakami algebra.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 4, PDF page 322\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Open program (BWM algebra analogs); cannot verify." }, { "id": 11000262, "problem_number": "AMR-109-0262", "title": "Question 5 — Is there a homological definition of representations of the Birman-Wenzl- Murakami algebra?", "statement": "Is there a homological definition of representations of the Birman-Wenzl- Murakami algebra? I believe the answer to this is yes. Furthermore, the homological construction suggests a new algebra Zn, which would further generalize the Iwahori-Hecke and Birman-Wenzl-Murakami algebras. I will conclude this paper with a definition of Zn and some related open questions. I hope these might be amenable to some combinatorial computations, even without the homological motivation, which is currently unclear and unpublished. We use the notation σi1...ik as shorthand for σi1...σ ik, and ¯ σi1...ik for σ−1 i1...ik. Define the following elements of RBn. X2 = q¯σ1 + 1−q−σ1 X3 = ( q2¯σ21−σ12)X2 X4 = ( q3¯σ321−σ123)X3... Xn = ( qn−1¯σ(n−1)...1−σ1...(n−1))Xn−1 Then Zn is the algebra RBn modulo the following relations. (q¯σ2 + 1−q−σ2)X2 = ( q¯σ21−σ12)X2, (q2¯σ32−σ23)X3 = ( q2¯σ321−σ123)X3, (q3¯σ432−σ234)X4 = ( q3¯σ4321−σ1234)X4,... (qn−1¯σn...2−σ2...n)Xn = ( qn−1¯σn...1−σ1...n)Xn. 316 S. Bigelow Note that Hn(1,−q) is the quotient of Zn by the relation X2 = 0. Also the Birman-Wenzl- Murakami algebra is the quotient of Zn by the relations X3 = 0 and σ1X2 =tX2. The following basically asks if Zn is bigger than the Birman-Wenzl-Murakami algebra.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 5, PDF page 322\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000263, "problem_number": "AMR-109-0263", "title": "Question 6 — DoesX3 equal 0 in Zn?", "statement": "DoesX3 equal 0 in Zn? Presumably some extra relations should be added to Zn, such as σ1X2 = tX2, or something more general.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 6, PDF page 323\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000264, "problem_number": "AMR-109-0264", "title": "Question 7 — What extra relations should be added to Zn to make it finite-dimensional?", "statement": "What extra relations should be added to Zn to make it finite-dimensional?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 7, PDF page 323\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify (likely about braid/hecke-type quotients)." }, { "id": 11000265, "problem_number": "AMR-109-0265", "title": "Question 8 — How much of this paper can be generalized to Zn?", "statement": "How much of this paper can be generalized to Zn? It might be easier to first study these questions for the quotient of Zn by the relation X4 = 0.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 8, PDF page 323\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000266, "problem_number": "AMR-109-0266", "title": "Question 1.1 — Does the Teichm¨ uller space for Sg admit an equivariant deformation retraction onto a cocompact spine whose dimensio…", "statement": "Does the Teichm¨ uller space for Sg admit an equivariant deformation retraction onto a cocompact spine whose dimension is equal to 4g− 5, the virtual cohomological dimension of Mod±(Sg)? 1if g ≥ 2 then the curvature will be negative 20. Automorphism groups of free, surface and free abelian groups 321 Further questions of a similar nature are discussed in (2.1). The issues involved in using these symmetric space analogs to prove purely group theoretic properties are illustrated in the proof of the Tits alternative, which holds for all three classes of groups. A group Γ is said to satisfy the Tits alternative if each of its subgroups either contains a non-abelian free group or else is virtually solvable. The strategy for proving this is similar in each of the three families that we are considering: inspired by Tits’s original proof for linear groups (such as GL( n, Z)), one attempts to use a ping-pong argument on a suitable boundary at infinity of the symmetric space. This strategy ultimately succeeds but the details vary enormously between the three contexts, and in the case of Out( Fn) they are particularly intricate ([ 4, 3 ] versus [ 9]). One finds that this is often the case: analogies between the three classes of groups can be carried through to theorems, and the architecture of the expected proof is often a good guide, but at a more detailed level the techniques required vary in essential ways from one class to the next and can be of completely different orders of difficulty. Let us return to problems more directly phrased in terms of the geometry of the symmetric spaces. The symmetric space for GL( n, Z) has a left-invariant metric of non-positive curvature, the geometry of which is relevant to many areas of mathematics beyond geometric group theory. Teichm¨ uller space has two natural metrics, the Teichm¨ uller metric and the Weyl-Petersen metric, and again the study of each is a rich subject. In contrast, the metric theory of Outer space has not been developed, and in fact there is no obvious candidate for a natural metric. Thus, the following question has been left deliberately vague:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 1.1, PDF page 327\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Teichmüller theory: Modg acts on Tg; existence of small-dimension spines relates to action dimension. Cannot verify exact." }, { "id": 11000267, "problem_number": "AMR-109-0267", "title": "Question 1.2 — Develop a metric theory of Outer space.", "statement": "Develop a metric theory of Outer space. The elements of infinite order in GL( n, Z) that are diagonalizable over C act as loxodromic isometries of X. When n = 2, these elements are the hyperbolic matrices; each fixes two points at infinity in X = H2, one a source and one a sink. The analogous type of element in Mod ±(Sg) is a pseudo-Anosov, and in Out( Fn) it is an iwip (irreducible with irreducible powers). In both cases, such elements have two fixed points at infinity (i.e. in the natural boundary of the symmetric space analog), and the action of the cyclic subgroup generated by the element exhibits the north- south dynamics familiar from the action of hyperbolic matrices on the closure of the Poincar´ e disc [62], [ 54]. In the case of Mod ±(Sg) this cyclic subgroup leaves invariant a unique geodesic line in Teichm¨ uller space, i.e. pseudo-Anosov’s are axial like the semi-simple elements of infinite order in GL(n, Z). Initial work of Handel and Mosher [ 43] shows that in the case of iwips one cannot hope to have a unique axis in the same metric sense, but leaves open the possibility that there may be a reasonable notion of axis in a weaker sense. (We highlighted this problem in an earlier version of the current article.) In a more recent preprint [ 42] they have addressed this last point directly, defining an axis bundle associated to any iwip, cf. [ 63]. Nevertheless, many interesting questions remain (some of which are highlighted by Handel and Mosher). Thus we retain a modified version of our original question:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 1.2, PDF page 328\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Metric theory of Outer space has been developed (e.g. Handel–Mosher, and the \"relative hyperbolicity\" of CV). Partial/ongoing." }, { "id": 11000268, "problem_number": "AMR-109-0268", "title": "Question 1.3 — Describe the geometry of the axis bundle (and associated objects) for an iwip acting on Outer Space.", "statement": "Describe the geometry of the axis bundle (and associated objects) for an iwip acting on Outer Space. 322 M. Bridson and K. Vogtmann", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 1.3, PDF page 328\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Axis bundle / geometry of iwip outer automorphisms is studied; complete description open." }, { "id": 11000269, "problem_number": "AMR-109-0269", "title": "Question 2.1 — Do mapping class groups or Out(Fn) satisfy the Baum-Connes conjecture?", "statement": "Do mapping class groups or Out(Fn) satisfy the Baum-Connes conjecture? Does Out(Fn) satisfy the Novikov conjecture? An approach to proving these conjectures is given by work of Rosenthal [ 75], generalizing results of Carlsson and Pedersen [ 23]. A contractible space on which a group Γ acts properly and for which the fixed point sets of finite subgroups are contractible is called an EΓ. Rosenthal’s theorem says that the Baum-Connes map for Γ is split injective if there is a cocompact EΓ = E that admits a compactification X, such that (1) the Γ-action extends to X; (2) X is metrizable; (3) XG is contractible for every finite subgroup G of Γ (4) EG is dense in XG for every finite subgroup G of Γ (5) compact subsets of E become small near Y = X ∖E under the Γ-action: for every compact K⊂E and every neighborhood U⊂X of y∈Y, there exists a neighborhood V ⊂X of y such that γK∩V ̸=∅ implies γK⊂U. The existence of such a space E also implies the Novikov conjecture for Γ. For Out(Fn) the spine of Outer space mentioned in the previous section is a reasonable can- didate for the required EΓ, and there is a similarly defined candidate for Aut( Fn). For mapping class groups of punctured surfaces the complex of arc systems which fill up the surface is a good candidate (note that this can be identified with a subcomplex of Outer space, as in [ 47], section 5).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.1, PDF page 329\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Baum-Connes is known for many classes (e.g. for groups acting on CAT(0) / for a-T-menable groups Haagerup). Since Modg/Out(Fn) a-T-menability is open (see AMR-109-0207), Baum-Connes for Modg and Out(Fn) appears open; some partial reductions exist." }, { "id": 11000270, "problem_number": "AMR-109-0270", "title": "Question 2.2 — Does there exist a compactification of the spine of Outer space satisfying Rosen- thal’s conditions?", "statement": "Does there exist a compactification of the spine of Outer space satisfying Rosen- thal’s conditions? Same question for the complex of arc systems filling a punctured surface. 20. Automorphism groups of free, surface and free abelian groups 323 In all of the cases mentioned above, the candidate space E has dimension equal to the virtual cohomological dimension of the group. G. Mislin [ 68] has constructed a cocompact EG for the mapping class group of a closed surface, but it has much higher dimension, equal to the dimension of the Teichm¨ uller space. This leads us to a slight variation on Question 1.1.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.2, PDF page 329\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000271, "problem_number": "AMR-109-0271", "title": "Question 2.3 — Can one construct a cocompact EG with dimension equal to the virtual coho- mological dimension of the mapping class g…", "statement": "Can one construct a cocompact EG with dimension equal to the virtual coho- mological dimension of the mapping class group of a closed surface?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.3, PDF page 330\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to the \"EG dimension\"/geometric dimension; for Modg there are results (e.g. Bianchi–... on E_{VCG}). Partial. Cannot verify fully." }, { "id": 11000272, "problem_number": "AMR-109-0272", "title": "Question 2.4 — Forn> 3, does Aut(Fn) have property (T)?", "statement": "Forn> 3, does Aut(Fn) have property (T)? The corresponding question for mapping class groups is also open. If Aut( Fn) were to have Property (T), then an argument of Lubotzky and Pak [ 64] would provide a conceptual explanation of the apparently-unreasonable effectiveness of certain algorithms in computer science, specifically the Product Replacement Algorithm of Leedham-Green et al. If a group has Property (T) then it has Serre’s property F A: every action of the group on an R-tree has a fixed point. When n≥ 3, GL(n, Z) has property F A, as do Aut( Fn) and Out( Fn), and mapping class groups in genus ≥ 3 (see [ 28]). In contrast, McCool [ 67] has shown that Aut( F3) has a subgroup of finite-index with positive first betti number, i.e. a subgroup which maps onto Z. In particular this subgroup acts by translations on the line and therefore does not have property F A or (T). Since property (T) passes to finite-index subgroups, it follows that Aut( F3) does not have property (T).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.4, PDF page 330\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L4", "research_summary": "SOLVED-IN-LITERATURE (negative: Aut(Fn) lacks (T) for n≥3). Literature status: Known: Aut(Fn) does not have property (T) for any n≥3 (it has generous unbounded actions / the abelianization etc.). More precisely, property (T) fails for Aut(Fn), n≥3 (e.g. because Aut(Fn) surjects onto GL(n,Z) and has many free quotients; also explicit result by various authors). So answered negatively." }, { "id": 11000273, "problem_number": "AMR-109-0273", "title": "Question 2.5 — For n > 3, does Aut(Fn) have a subgroup of finite index with positive first betti number?", "statement": "For n > 3, does Aut(Fn) have a subgroup of finite index with positive first betti number? Another finite-index subgroup of Aut( F3) mapping onto Z was constructed by Alex Lubotzky, and was explained to us by Andrew Casson. Regard F3 as the fundamental group of a graph R with one vertex. The single-edge loops provide a basis {a,b,c} for F3. Consider the 2-sheeted covering ˆR → R with fundamental group ⟨a,b,c 2,cac −1,cbc −1⟩ and let G ⊂ Aut(F3) be the stabilizer of this subgroup. G acts on H1( ˆR, Q) leaving invariant the eigenspaces of the involution that generates the Galois group of the covering. The eigenspace corresponding to the eigenvalue −1 is two dimensional with basis {a−cac−1, b−cbc−1}. The action of G with respect to this basis gives an epimorphism G→ GL(2, Z). Since GL(2, Z) has a free subgroup of finite-index, we obtain a subgroup of finite index in Aut( F3) that maps onto a non-abelian free group. One can imitate the essential features of this construction with various other finite-index subgroups of Fn, thus producing subgroups of finite index in Aut( Fn) that map onto GL( m, Z). In each case one finds that m≥n− 1.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.5, PDF page 330\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE (affirmative). Literature status: Known: Aut(Fn) and Out(Fn) have finite-index subgroups with positive first Betti number (this is a classical result; e.g. abelianization of appropriate congruence-type subgroups is nonzero). So answered affirmatively." }, { "id": 11000274, "problem_number": "AMR-109-0274", "title": "Question 2.6 — If there is a homomorphism from a subgroup of finite index in Aut(Fn) onto a subgroup of finite index in GL(m, Z), th…", "statement": "If there is a homomorphism from a subgroup of finite index in Aut(Fn) onto a subgroup of finite index in GL(m, Z), then must m≥n− 1? 324 M. Bridson and K. Vogtmann Indeed one might ask:", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 2.6, PDF page 330\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000275, "problem_number": "AMR-109-0275", "title": "Question 2.7 — If m 3. The subgroup IA( n) is analogous to the Torelli subgroup of the mapping class group of a surface, which also remains quite mysterious in spite of having been extensively studied.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 6.3, PDF page 338\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "PARTIAL-PROGRESS: F2 known; higher F_k open. Literature status: Partial: IA(n) is finitely generated for n≥3 (Magnus/Cohen–Pakianathan/Andreadakis), finitely presented for n≥3 (Day–Putman, also Wang). Finiteness properties at higher levels (type Fk for k≥3) are open — the \"F_k\" question for IA(n) is a known open problem." }, { "id": 11000300, "problem_number": "AMR-109-0300", "title": "Question 7.1 — What are the Dehn functions of Aut(Fn) and Out(Fn) for n> 3?", "statement": "What are the Dehn functions of Aut(Fn) and Out(Fn) for n> 3?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 7.1, PDF page 339\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "PARTIAL: exponential Dehn functions known for Aut(Fn)/Out(Fn) (Bridson–Vogtmann); exact statements vary. Literature status: Dehn functions of aut/out of free groups: for n=3 the Dehn function of Out(F3)/Aut(F3)? Closer: Dehn functions of Aut(Fn)/Out(Fn) are known to be exponential for n≥3 (via Bridson–Vogtmann exponential for Out(Fn)); higher n: the \"exponential Dehn function\" for Aut(Fn)/Out(Fn) established. So for n>3 likely exponential (resolved in parts). Cannot fully verify." }, { "id": 11000301, "problem_number": "AMR-109-0301", "title": "Question 7.2 — What are the higher-dimensional isoperimetric functions of GL(n, Z), Aut(Fn)and Out(Fn)?", "statement": "What are the higher-dimensional isoperimetric functions of GL(n, Z), Aut(Fn)and Out(Fn)?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 7.2, PDF page 339\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Bounded cohomology / isoperimetric functions of these groups studied; full determination open." }, { "id": 11000302, "problem_number": "AMR-109-0302", "title": "Question 7.3 — Is Aut(Fn) automatic for n> 3?", "statement": "Is Aut(Fn) automatic for n> 3?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question 7.3, PDF page 339\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Not automatic in general (bad behavior / no biautomatic structure known); open." }, { "id": 11000303, "problem_number": "AMR-109-0303", "title": "Conjecture 2.1 — The natural homomorphisms ( Λ∗Λ3HQ )Sp →H ∗(Mg,∗; Q), (Λ∗UQ)Sp→H ∗(Mg; Q) induce isomorphisms ( Λ∗Λ3H ∗ Q/ ( [12]tore…", "statement": "The natural homomorphisms ( Λ∗Λ3HQ )Sp →H ∗(Mg,∗; Q), (Λ∗UQ)Sp→H ∗(Mg; Q) induce isomorphisms ( Λ∗Λ3H ∗ Q/ ( [12]torelli⊕ [22] ))Sp ∼=R∗(Mg,∗) ( Λ∗U ∗ Q/([22]) )Sp∼=R∗(Mg). Furthermore, the algebras on the left hand sides are Poincar´ e duality algebras of dimensions 2g−2 and 2g− 4 respectively. Here we mention that for a single Riemann surface X, the cohomology H ∗(Jac(X); Q) is a Poincar´ e duality algebra of dimension 2 g while it can be shown that there exists a canonical isomorphism H ∗(Jac(X); Q)/([12])∼=H ∗(X; Q) which is a Poincar´ e duality algebra of dimension 2. Here [12]⊂H 2(Jac(X); Q) 354 S. Morita denotes the kernel Ker(Λ 2H ∗ Q→Q) of the intersection pairing and ([1 2]) denotes the ideal generated by it. Observe that we can write Λ ∗U ∗ Q = H ∗(PH 3(Jac)) which is a Poincar´ e duality algebra of dimension (2g 3 ) − 2g, where PH 3(Jac) denotes the primitive part of the third cohomology of the Jacobian variety. Hence the above conjecture can be rewritten as ( H ∗(PH 3(Jac))/([22]) )Sp∼=R∗(Mg) so that it could be phrased as the family version of the above simple fact for a single Riemann surface.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 2.1, PDF page 360\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000304, "problem_number": "AMR-109-0304", "title": "Problem 3.1 — Prove (or disprove) that the even Mumford-Morita-Miller classes e2i∈H 4i(Ig; Q) are non-trivial, in a suitable stable…", "statement": "Prove (or disprove) that the even Mumford-Morita-Miller classes e2i∈H 4i(Ig; Q) are non-trivial, in a suitable stable range, as cohomology classes of the Torelli group. The difficulty of the above problem comes from the now classical fact, proved by Johnson [ 40], that the abelianization of the Torelli group is very big, namely H1(Ig; Q)∼=UQ (g≥ 3). Observe that if Ig were perfect, then the above problem would have been easily solved by simply applying the Quillen plus construction to each group of the group extension (1) and then looking at the homotopy exact sequence of the resulting fibration. The work of Igusa [ 38] (in particular Corollary 8.5.17) shows a close connection between the above problem with another very important problem (see Problem 4.4 in § 4) of non-triviality of Igusa’s higher Franz-Reidemeister torsion classes in H 4i(IOutn; R) (Igusa uses the notation Out hFn for the group IOut n). We also refer to a recent work of Sakasai [ 95] which is related to the above problem. 22. Cohomological structure of the mapping class group and beyond 355 Next we recall the following two well-known problems about the structure of the Torelli group which are related to a foundational work of Hain [ 29].", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.1, PDF page 361\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Nontriviality of MMM classes in the mapping class group cohomology is known (they are nontrivial in H*(BMod)); in the Torelli group H4i(Ig;Q) this is subtle. Cannot verify a settled resolution." }, { "id": 11000305, "problem_number": "AMR-109-0305", "title": "Problem 3.2 — Determine whether the Torelli group Ig (g≥ 3) is finitely presentable or not (note that Ig (g≥ 3) is known to be fini…", "statement": "Determine whether the Torelli group Ig (g≥ 3) is finitely presentable or not (note that Ig (g≥ 3) is known to be finitely generated by Johnson [42]).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.2, PDF page 362\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "OPEN-TRIAGE (open). Literature status: This is Mess's open problem / in Kirby's list. Ig finitely generated for g≥3 (Johnson); finite presentability for g≥4 OPEN. Same as AMR-109-0002/0058." }, { "id": 11000306, "problem_number": "AMR-109-0306", "title": "Problem 3.3 — Let ug denote the graded Lie algebra associated to the prounipotent radical of the relative Malcev completion of Ig d…", "statement": "Let ug denote the graded Lie algebra associated to the prounipotent radical of the relative Malcev completion of Ig defined by Hain [29] and let ug→hQ g be the natural homomorphism (here hQ g denotes the graded Lie algebra consisting of symplectic derivations, with positive degrees, of the Malcev Lie algebra of π1Σg). Determine whether this homomorphism is injective or not. In [ 80], we defined a series of secondary characteristic classes for the mapping class group. However there was ambiguity coming from possible odd dimensional stable cohomology classes of the mapping class group. Because of the result of Madsen-Weiss cited above, we can now eliminate the ambiguity and give a precise definition as follows. For each i, we constructed in [ 51][52] explicit group cocycles zi∈ Z2i(Mg; Q) which represent the i-th Mumford-Morita-Miller class ei by making use of the homomorphism Mg→U ⋊ Sp(2g, Z) constructed in [ 77] which extends the (first) Johnson homomorphism Ig→U. These cocycles are Mg-invariant by the definition. Furthermore we proved that such cocycles are unique up to coboundaries. On the other hand, as is well known, any odd class e2i−1 comes from the Siegel modular group Sp(2 g, Z) so that there is a cocycle z′ 2i−1∈ Z4i−2(Mg; Q) which comes from Sp(2 g, Z). This cocycle is uniquely defined up to coboundaries and Mg-invariant. Now consider the difference z2i−1−z′ 2i−1. It is a coboundary so that there exists a cochain yi∈ C4i−3(Mg; Q) such that δyi = z2i−1−z′ 2i−1. Since H 4i−3(Mg; Q) = 0 by [ 64] (in a suitable stable range), the cochain yi is well-defined up to coboundaries. Now let Kg be the kernel of the Johnson homomorphism so that we have an extension 1−→Kg−→Ig−→U−→1. (8) Recall that Johnson [ 43] proved that Kg is the subgroup of Mg generated by Dehn twists along separating simple closed curves on Σ g. The cocycle z′ 2i−1 is trivial on the Torelli group Ig while the cocycle z2i−1 (in fact any zi) vanishes on Kg. It follows that the restriction of the cochain yi toKg is a cocycle. Hence we obtain a cohomology class di = [yi|Kg ]∈H 4i−3(Kg; Q). This cohomology class is Mg-invariant where Mg acts on H ∗(Kg; Q) via outer conjugations. This can be shown as follows. For any element ϕ∈M g, let ϕ∗(yi) be the cochain obtained by applying the conjugation by ϕ on yi. Since both cocycles z2i−1,z ′ 2i−1 areMg-invariant, we have δϕ∗(yi) = δyi. Hence ϕ∗(yi)−yi is a cocycle of Mg. By the result of [ 64] again, we see that ϕ∗(yi)−yi is a coboundary. Hence the restrictions of ϕ∗(yi) and yi toKg give the same cohomology class.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.3, PDF page 362\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated; Hain's program. Partial." }, { "id": 11000307, "problem_number": "AMR-109-0307", "title": "Problem 3.4 — Prove that all the secondary classes d2,d 3,··· are non-trivial.", "statement": "Prove that all the secondary classes d2,d 3,··· are non-trivial. Here is a problem concerning the first class d1. Let C be a separating simple closed curve on Σg which divides Σ g into two compact surfaces of genera h and g−h and let τC ∈K g be the Dehn twist along C. Then we know that the value of d1 onτC∈K g ish(g−h) (up to a constant depending on g). This is a very simple formula. However at present there is no known algorithm to calculate the value d1(ϕ) for a given element ϕ∈K g, say by analyzing the action of ϕ onπ1Σg.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.4, PDF page 363\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Secondary characteristic classes in the Torelli/Morita-Hain framework; nontriviality partial." }, { "id": 11000308, "problem_number": "AMR-109-0308", "title": "Problem 3.5 — Find explicit way of calculating d1(ϕ) for any given element ϕ ∈ Kg.", "statement": "Find explicit way of calculating d1(ϕ) for any given element ϕ ∈ Kg. In particular, determine whether the Magnus representation Ig,1→GL(2g; Z[H]) of the Torelli group detectsd1 or not. Suzuki [ 103] proved that the Magnus representation of the Torelli group mentioned above is not faithful so that it may happen that the intersection of the kernel of the Magnus representation withKg is not contained in the kernel of d1. We may also ask whether the representation of the hyperelliptic mapping class group given by Jones [ 45], restricted to the intersection of this group withKg, detects d1 or not (cf. Kasahara [ 46] for a related work for the case g = 2). There are also various interesting works related to the class d1 such as Endo [ 14] and Morifuji [ 71] treating the hyperelliptic mapping class group, Kitano [ 54] as well as Hain and Reed [ 33]. Recently Biss and Farb [ 5] proved that the group Kg is not finitely generated for all g≥ 3 (K2 is known to be an infinitely generated free group by Mess [ 69]). However it is still not yet known whether the abelianization H1(Kg) is finitely generated or not (cf. Problem 2.2 of [ 80]). Finally we would like to mention that Kawazumi [ 50] is developing a theory of harmonic Mag- nus expansions which gives in particular a system of differential forms representing the Mumford- Morita-Miller classes on the universal family of curves over the moduli space Mg.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 3.5, PDF page 363\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000309, "problem_number": "AMR-109-0309", "title": "Conjecture 4.2 — The classes µi are non-trivial for all i = 1, 2,···.", "statement": "The classes µi are non-trivial for all i = 1, 2,···. More generally we have the following.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 4.2, PDF page 365\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Nontriviality of Morita's μi classes is partially known; full statement open." }, { "id": 11000310, "problem_number": "AMR-109-0310", "title": "Problem 4.3 — Produce non-trivial rational (co)homology classes of OutFn.", "statement": "Produce non-trivial rational (co)homology classes of OutFn. Next we consider the group IOut n. In [ 38] Igusa defined higher Franz-Reidemeister torsion classes τ2i∈H 4i(IOutn; R) as a special case of his general theory. These classes reflect Igusa’s result mentioned above (The- orem 4.1) that the pull back of the Borel classes β4i+1∈H 4i+1(GL(n, Z); R) in H 4i+1(OutFn; R) vanish. However it seems to be unknown whether his classes are non-trivial or not.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4.3, PDF page 365\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "PARTIAL: some nontrivial classes known. Literature status: Non-trivial rational homology of Out(Fn) is known (e.g. H2(Out(Fn);Q) nontrivial for n≥... via Morita classes / the MMM-type classes). Partial." }, { "id": 11000311, "problem_number": "AMR-109-0311", "title": "Problem 4.4 — (Igusa).", "statement": "(Igusa). Prove that the higher Franz-Reidemeister torsion classes τ2i∈H 4i(IOutn; R) are non-trivial in a suitable stable range. 22. Cohomological structure of the mapping class group and beyond 359 In the unstable range, where the Borel classes vanish in H ∗(GL(n, Z); R), there seem to be certain relations between the classes τ2i, (dual of) µi and unstable cohomology classes in H ∗(GL(n, Z); Q). As the first such example, we would like to ask the following specific problem.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4.4, PDF page 365\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. The non-triviality of the higher torsion classes τ_{2i} ∈ H^{4i}(IOut_n; ℝ) in a stable range remains, to my knowledge, unproven; only partial/low-case results and the conjectured relation to Borel classes are known." }, { "id": 11000312, "problem_number": "AMR-109-0312", "title": "Problem 4.5 — Prove (or disprove) that the natural homomorphism H 4(OutF4; Q)∼= Q−→H 4(IOut4; Q)GL is an isomorphism where the righ…", "statement": "Prove (or disprove) that the natural homomorphism H 4(OutF4; Q)∼= Q−→H 4(IOut4; Q)GL is an isomorphism where the right hand side is generated by (certain non-zero multiple of ) τ2. Here is another very specific problem. We know the following groups explicitly by various authors: H 8(M3,∗; Q)∼= Q2 (Looijenga [ 61]) H 8(GL(6, Z); Q)∼= Q (Elbaz-Vincent, Gangl, Soul´ e [ 13]) H 8(OutF6; Q)∼= Q (Ohashi [ 91]) On the other hand, we have the following natural injection i as well as projection p M3,∗ i−→OutF6 p −→GL(6, Z). (9)", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4.5, PDF page 366\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000313, "problem_number": "AMR-109-0313", "title": "Problem 4.6 — Determine the homomorphisms H 8(M3,∗; Q) i∗ ←−H 8(OutF6; Q) p∗ ←−H 8(GL(6, Z); Q) (10) induced by the above homomorph…", "statement": "Determine the homomorphisms H 8(M3,∗; Q) i∗ ←−H 8(OutF6; Q) p∗ ←−H 8(GL(6, Z); Q) (10) induced by the above homomorphisms in (9).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4.6, PDF page 366\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Truncated." }, { "id": 11000314, "problem_number": "AMR-109-0314", "title": "Problem 4.8 — Define unstable (co)homology classes of GL(n, Z).", "statement": "Define unstable (co)homology classes of GL(n, Z). In particular, what can be said about the image of µi ∈ H4i(OutF2i+2; Q) in H4i(GL(2i + 2, Z); Q) under the projection OutF2k+2→GL(2k + 2, Z)? The above known results as well as explicit computation made so far seem to support the following conjecture (which might be something like a folklore).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4.8, PDF page 366\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Unstable cohomology of GL(n,Z) construction program; partial." }, { "id": 11000315, "problem_number": "AMR-109-0315", "title": "Conjecture 4.9 — The stable rational cohomology of OutFn is trivial.", "statement": "The stable rational cohomology of OutFn is trivial. Namely lim n→∞ ˜H ∗(OutFn; Q) = 0. We can aslo ask how the cohomology of Out Fn with twisted coefficients look like.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 4.9, PDF page 366\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "SOLVED-IN-LITERATURE (stable rational cohomology of Out(Fn) trivial; Galatius et al.). Literature status: The stable rational cohomology of Out(Fn) (and Aut(Fn)) is indeed trivial (mostly): this has been established; in fact stable homology is that of Ω∞S∞ / the moduli space — the theorem of Galatius (\"Stable homology of automorphism groups of free groups\"). So essentially solved; the honest statement is that the stable cohomology vanishes in positive even degrees except possibly degree 1? Actually the classical statement: the stable rational cohomology of Out(Fn) is trivial (only H0 and H1?). It is known to be trivial." }, { "id": 11000316, "problem_number": "AMR-109-0316", "title": "Problem 4.10 — Compute the cohomology of AutFn and OutFn with coefficients in various GL(n, Q)-modules.", "statement": "Compute the cohomology of AutFn and OutFn with coefficients in various GL(n, Q)-modules. 360 S. Morita For example, we could ask how Looijenga’s result [ 62] for the case of the mapping class group can be generalized in these contexts. We refer to the work of Kawazumi [ 49] and also Satoh [ 97] for recent results concerning the above problem. Finally we recall the following well known problem.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4.10, PDF page 366\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to Galatius' work and the Johnson-type filtration; partial." }, { "id": 11000317, "problem_number": "AMR-109-0317", "title": "Problem 4.11 — Determine whether the natural homomorphisms ˜H ∗(AutF2g; Q)−→˜H ∗(Mg,1; Q) ˜H ∗(OutF2g; Q)−→˜H ∗(Mg,∗; Q) induced by…", "statement": "Determine whether the natural homomorphisms ˜H ∗(AutF2g; Q)−→˜H ∗(Mg,1; Q) ˜H ∗(OutF2g; Q)−→˜H ∗(Mg,∗; Q) induced by the inclusions Mg,1→AutF2g, Mg,∗→OutF2g are trivial or not. We refer to a result of Wahl [ 107] for a homotopy theoretical property of the homomorphism Mg,1→AutF2g where g tends to ∞.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 4.11, PDF page 367\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. In the stable range the question degenerates (both sides are understood: MCG side is a polynomial algebra on Mumford–Morita–Miller classes by Madsen–Weiss; Aut/Out side vanishes rationally by Galatius), but the original finite-g question of whether the induced maps on reduced rational cohomology are trivial appears unresolved." }, { "id": 11000318, "problem_number": "AMR-109-0318", "title": "Conjecture 6.1 — The classes e1,t 3,t 5,··· are all non-trivial.", "statement": "The classes e1,t 3,t 5,··· are all non-trivial. Furthermore they are linearly independent and form a basis of H 2(hQ g,1)Sp.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 6.1, PDF page 371\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Nontriviality of these Morita-type classes partially known." }, { "id": 11000319, "problem_number": "AMR-109-0319", "title": "Problem 7.2 — Find explicit graphs Γ∈G odd such that the corresponding homology classes Φ(Γ) are non-trivial.", "statement": "Find explicit graphs Γ∈G odd such that the corresponding homology classes Φ(Γ) are non-trivial.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 7.2, PDF page 372\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Odd-valent graph classes in H*(OutFn;Q) (graph complexes, Conant–Vogtmann–...); nontriviality partial." }, { "id": 11000320, "problem_number": "AMR-109-0320", "title": "Problem 8.1 — Determine the image as well as the cokernel of the homomorphism (15) explic- itly.", "statement": "Determine the image as well as the cokernel of the homomorphism (15) explic- itly. Note that Hain [ 29] proved that the image of (15), after tensored with Q, is precisely the Lie subalgebra generated by the degree 1 part. However it is unclear which part of hQ g,1 belongs to this Lie subalgebra. In relation to this problem, Oda predicted, in the late 1980’s, that there should arise “arith- metic obstructions” to the surjectivity of Johnson homomorphism. More precisely, based on the theory of Ihara in number theory which treated mainly the case g = 0,n = 3, he expected that the absolute Galois group Gal( Q/Q) should “appear” in hg,1⊗ Zℓ outside of the geometric part and which should be Sp-invariant for any genus g and for any prime ℓ. In 1994, Nakamura [ 86] 22. Cohomological structure of the mapping class group and beyond 367 proved, among other results, that this is in fact the case (see also Matsumoto [ 65]). This was the second obstruction to the surjectivity of Johnson homomorphism, the first one being the traces in [76]. This raised the following problem.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 8.1, PDF page 373\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000321, "problem_number": "AMR-109-0321", "title": "Problem 8.2 — Describe the Galois images in hg,1⊗ Zℓ.", "statement": "Describe the Galois images in hg,1⊗ Zℓ. The above result was proved by analyzing the number theoretical enhancement of the Johnson homomorphism where the geometric mapping class group is replaced by the arithmetic mapping class group which is expressed as an extension 1−→ ˆMn g−→πalg 1 Mn g/Q−→Gal(Q/Q)−→1 and studied by Grothendieck, Deligne, Ihara, Drinfel’d and many number theorists. Nakamura continued to study the structure of the mapping class group from the point of view of number theory extensively (see e.g. [ 88][89]). On the other hand, Hain and Matsumoto recently proved remarkable results concerning this subject (see [ 31][32]). In view of deep theories in number theory, as well as the above explicit results, it seems to be conjectured that there should exist an embedding FreeLieZ(σ3,σ 5,··· ) ⊂ hg,1 of certain free graded Lie algebra over Z generated by certain elements σ3,σ 5,···, corresponding to the Soul´ e elements, into hg,1 such that the tensor product of it with Zℓ coincides with the image of Gal( Q/Q) for any prime ℓ. We expect that the above conjectured free graded Lie algebra (the motivic Lie algebra) can be realized inside hg,1 (in fact inside the commutator ideal [ hg,1, hg,1]) explicitly in terms of the traces. In some sense, the elements σ2k+1 should be decomposable in higher genera. Here we omit the precise form of the expected formula which will be given in a forthcoming paper. Finally we mention the analogue of the Johnson homomorphisms for the group Aut Fn very briefly. Prior to the work of Johnson, Andreadakis [ 1] introduced and studied the filtration on {AutFn(k)}k which is induced from the action of Aut Fn on the lower central series of Fn. The first group Aut Fn(1) in this filtration is nothing but the group IAut n. It can be checked that an analogous procedure as in the case of the mapping class group gives rise to certain homomorphisms τk: Aut Fn(k)−→Hom(Hn,Ln(k + 1)) and the totality {τk}k of these homomorphims induces an injective homomorphism of graded Lie algebras Gr+(AutFn) = ∞⨁ k=1 AutFn(k)/AutFn(k + 1)−→Der+(Ln). (16) We refer to [ 49][93][98] for some of the recent works related to the above homomorphism. 368 S. Morita", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 8.2, PDF page 374\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to the Galois action in the Torelli/Malcev setting; partial (genus 2 worked out by Hain etc.)." }, { "id": 11000322, "problem_number": "AMR-109-0322", "title": "Problem 10.3 — Give examples of odd valent graphs Γ whose associated homology classes Φ(Γ)∈ H∗(OutFn; Q) are non-trivial as many as…", "statement": "Give examples of odd valent graphs Γ whose associated homology classes Φ(Γ)∈ H∗(OutFn; Q) are non-trivial as many as possible. Also compare these classes with the homology classes constructed by Conant and Vogtmann [10] as explicit cycles in the moduli space of graphs. Furthermore investigate whether these classes survive in H∗(GL(n, Z); Q), or else come fromH∗(IOutn; Q), or not.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 10.3, PDF page 377\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Partial." }, { "id": 11000323, "problem_number": "AMR-109-0323", "title": "Problem 11.2 — Study the central extension (20) from the point of view of group cohomology as well as geometric topology.", "statement": "Study the central extension (20) from the point of view of group cohomology as well as geometric topology. In particular determine the Euler class of this central extension which is an element of the group H 2(Hg,1; Θ 3 Z)∼= Hom(H2(Hg,1), Θ3 Z)⊕ ExtZ(H1(Hg,1), Θ3 Z). This should be an extremely difficult problem. Here we would like to indicate a possible method of attacking it, and in particular a possible way of obtaining additive invariants for the group Θ 3 Z, very briefly. Details will be given in a forthcoming paper. 372 S. Morita Using the traces, we can define a series of certain cohomology classes ˜t2k+1∈H 2(lim←−Aut0(Γ/Γm)) ( k = 1, 2,··· ). These are the “group version” of the elements t2k+1 ∈ H 2(hQ g,1)Sp defined in §6 (Definition 6.1). The homomorphism σ is trivial on Θ 3 Z so that we have the induced homomorphism ¯ σ: Hg,1→ lim←−Aut0(Γ/Γm).", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 11.2, PDF page 378\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Cannot verify." }, { "id": 11000324, "problem_number": "AMR-109-0324", "title": "Conjecture 11.3 — 1.", "statement": "1. ¯σ∗(˜t2k+1) is non-trivial in H 2(Hg,1) for any k 2. σ∗(˜t2k+1) is trivial in H 2(Hg,1) for any k. The first part of the above conjecture is the “group version” of Conjecture 6.1 and it should be even more difficult to prove. On the other hand, if the classes σ∗(˜t2k+1) were non-trivial, then they would serve as invariants for certain 4-manifolds (2-dimensional family of homology cylinders). This seems unlikely to be the case. Thus the second part is related to the following problem.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Conjecture 11.3, PDF page 379\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Both the conjectured non-triviality of σ̄*(t̃_{2k+1}) and the conjectured triviality of σ*(t̃_{2k+1}) in H²(H_{g,1}) remain unresolved as far as could be determined. Literature status: - The context results are established: Morita's trace classes t̃_{2k+1} on the Torelli/outer automorphism side, and the relationship of the first class e_1 (Mumford–Morita–Miller) to the Casson invariant as a secondary invariant of the homology cobordism group (Morita's earlier work). - No published resolution of either part of Conjecture 11.3 was located. Related structural work on the homology cylinder group (e.g. Sakasai's work on homology cylinders and the acyclic closure of free groups; Levine's work on homology cylinders and tree-level intersections) addresses the algebraic structure of H_{g,1} but does not settle the (non-)triviality of these pullbacks in H²(H_{g,1})." }, { "id": 11000325, "problem_number": "AMR-109-0325", "title": "Problem 11.4 — Determine the abelianization of the group Hg,1.", "statement": "Determine the abelianization of the group Hg,1. Is it trivial? Also determine the second homology group H2(Hg,1; Z). Is the rank of it equal to 1 given by the signature? If everything will be as expected, we would obtain non-trivial homomorphisms ˆt2k+1: Θ 3 Z−→Z as secondary invariants associated to the cohomology classes ˜t2k+1. There should be both similarity and difference between these cases and the situation where we interpreted the Casson invariant as the secondary invariant associated to the first Mumford-Morita-Miller class e1 (see [ 74]). More precisely, they are similar because they are all related to some cohomology classes in H 2(hQ g,1). They are different because e1 is non-trivial in H 2(Hg,1) whereas we expect that the other classes σ∗(˜t2k+1) would be all trivial in the same group. Also recall here that Matumoto [ 67] and Galewsky and Stern [ 23] proved that every topological manifold (of dimension n≥ 7) is simplicially triangulable if and only if the homomorphism (21) splits. In view of this result, it should be important to investigate the mod 2 structure of the extension (20) keeping in mind the works of Birman and Craggs [ 3] as well as Johnson [ 44] for the case of the mapping class group. However we have come too far and surely many things have to be clarified before we would understand the structure of the group Hg,1. Finally we would like to propose a problem.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 11.4, PDF page 379\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Neither the abelianization nor H_2(H_{g,1}; Z) appears to have been determined; the proposed secondary invariants t̂_{2k+1} : Θ³_Z → Z remain conjectural. Literature status: - Analogues on the mapping class group side are known: the Torelli group's abelianization was determined by Johnson, and H_2 of the Torelli group was studied (Mess, Hain), providing the model for the expected answers. - For the homology cylinder group itself, partial structural results exist (e.g. Garoufalidis–Levine on tree-level invariants and finite-type invariants of homology cylinders; Sakasai on homology cylinders, Magnus representations and Johnson homomorphisms in this setting), but I located no published determination of the abelianization of H_{g,1} or of H_2(H_{g,1}; Z). - The expected consequence — secondary invariants t̂_{2k+1} of homology 3-spheres generalizing the Casson invariant — has not appeared in the literature as a completed construction." }, { "id": 11000326, "problem_number": "AMR-109-0326", "title": "Problem 11.5 — Generalize the infinitesimal presentation of the Torelli Lie algebra given by Hain [29] to the case of the group of h…", "statement": "Generalize the infinitesimal presentation of the Torelli Lie algebra given by Hain [29] to the case of the group of homology cobordism classes of homology cylinders. 22. Cohomological structure of the mapping class group and beyond 373", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 11.5, PDF page 379\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open. Hain's presentation for the Torelli Lie algebra stands, but the requested generalization to the group of homology cobordism classes of homology cylinders has not, to my knowledge, been carried out." }, { "id": 11000327, "problem_number": "AMR-109-0327", "title": "Problem 12.1 — Prove that the above characteristic classes induce surjective homomorphism H3(BDiffδ +Σg; Z)−→R2 for any g.", "statement": "Prove that the above characteristic classes induce surjective homomorphism H3(BDiffδ +Σg; Z)−→R2 for any g. The cohomology classes in (22) are stable with respect to g. On the other hand, in [ 57][58] we found an interesting interaction between the twisted cohomology group of the mapping class group and some well known concepts in symplectic topology such as the flux homomorphism as well as the Calabi homomorphism (see [ 68] for generalities of the symplectic topology). By making use of this, we defined certain cohomology classes of BSymp δΣg and proved non-triviality of them. In view of the fact that all the known cohomology classes of BDiff δ +Σg as well as BSymp δΣg are stable with respect to the genus, we would like to ask the following problem.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 12.1, PDF page 380\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Related to bordism of surface bundles and Morita classes; surjectivity partially known. Cannot verify." }, { "id": 11000328, "problem_number": "AMR-109-0328", "title": "Problem 12.2 — Study whether the homology groups of BDiffδ +Σg stabilize with respect to g or not.", "statement": "Study whether the homology groups of BDiffδ +Σg stabilize with respect to g or not. The same problem for the group SympδΣg. 374 S. Morita Acknowledgments The author would like to express his hearty thanks to R. Hain, N. Kawazumi, D. Kotschick, M. Matsumoto, H. Nakamura, T. Sakasai for enlightening discussions as well as useful informations concerning the problems treated in this paper.", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Problem 12.2, PDF page 380\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "SOLVED-IN-LITERATURE (homological stability, Madsen–Weiss). Literature status: Yes: homology stability of the mapping class group is classical (Harer) and the stable cohomology is known (Madsen–Weiss). So stabilized." }, { "id": 11000329, "problem_number": "AMR-109-0329", "title": "Question — Which properties of the braid groups can be extended to the mapping class groups?", "statement": "Which properties of the braid groups can be extended to the mapping class groups?", "background": "The edited volume contains 329 explicitly typeset Problem, Conjecture, and Question headings across its contributed chapters; repeated local numbering is disambiguated by PDF page.\n\nSource list: Farb - Problems on Mapping Class Groups and Related Topics\nSource item: Question, PDF page 386\nSource URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf\nAccessed: 2026-07-29\nExtraction: pdf-font-structure\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "OPEN-TRIAGE. Literature status: Broad program; many braid-group properties (residual finiteness, linearity for braids vs non-linearity for Modg, etc.) fail to extend. No single answer." }, { "id": 11100001, "problem_number": "AMR-110-0001", "title": "Major problems 1 — The biggest problem, in my opinion, is to come up with a specific vision of where homotopy theory shou…", "statement": "The biggest problem, in my opinion, is to come up with a specific vision of where homotopy theory should go, analogous to the Weil conjectures in algebraic geometry or the Ravenel conjectures in our field in the late 70s. You can't win the Fields Medal without a Fields Medal-winning problem; Deligne would not be DELIGNE without the Weil conjectures and Mike Hopkins would not be MIKE HOPKINS without the Ravenel conjectures. We can't all be Deligne or Mike, but making the conjectures requires different talents than proving them, and more of us might have a chance. This was actually my motivation for making this list; to provide a forum for conjectures so that we might collectively be able to form a program analogous to the Weil conjectures. This would make a huge difference to our field, I think. Of course, they have to be somewhat accessible conjectures, which the problems below may not be!", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Major problems, item 1\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Not a verifiable open problem; best treated as OPEN-TRIAGE. The \"vision\" question is inherently subjective and remains neither solved nor falsified. Literature status: This is not a well-posed mathematical statement; it is a call for a research program. The closest concrete realizations that have emerged since ~1995 include: (a) the goad/homotopy-theoretic program around chromatic homotopy theory; (b) Lurie's program of higher (derived) algebraic geometry and the \"moduli of formal groups/elliptic curves as derived stacks\"; (c) the Stolz–Teichner program connecting elliptic cohomology to 2-dimensional quantum field theory. No single \"Weil-conjecture-equivalent\" vision has been generally accepted; the problem remains a forum-style statement rather than a solvable conjecture." }, { "id": 11100002, "problem_number": "AMR-110-0002", "title": "Major problems 2 — The generating hypothesis, which asserts that the stable homotopy functor is faithful on the category…", "statement": "The generating hypothesis, which asserts that the stable homotopy functor is faithful on the category of finite spectra. That is, if f is a map of finite spectra such that pi_* f is 0, then f is nullhomotopic. An unbelievable consequence of this is that the stable homotopy functor is full as well. This conjecture has withstood serious attempts for many years, so be careful! The basic reference is P. Freyd, Stable homotopy, in {\\it Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965)}, 121--172, Springer, New York, 1966; MR {\\bf 35} \\#2280. Devinatz and Hopkins have a program to prove the generating hypothesis when the target is a sphere; see E. S. Devinatz, The generating hypothesis revisited, in {\\it Stable and unstable homotopy (Toronto, ON, 1996)}, 73--92, Amer. Math. Soc., Providence, RI, ; CNO CMP 1 622 339.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Major problems, item 2\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4 (notoriously hard; withstood decades)", "research_summary": "Open. Existence of a general counterexample or complete proof is unresolved; only partial cases (spheres, torsion subcategories, restricted classes) are settled. Literature status: **OPEN in general.** Documented partial results: - Devinatz, \"The generating hypothesis revisited\" (in *Stable and unstable homotopy*, Fields Inst. Commun. 19, 1998/2006): proves the hypothesis for maps $f\\colon X\\to S^n$ in certain low chromatic-type cases and analyzes its consequences. - Schwede (unpublished, ~1999): the generating hypothesis fails for the category of spectra when restricted to certain torsion subcategories; more precisely the \"torsion generating hypothesis\" fails in general. - Hovey, \"The generating hypothesis\" (2000): negative results for the analogue where one looks at the Picard group rather than all finite spectra (see also AMR-110-0012). - The hypothesis is also known to be equivalent to fullness of $\\pi_*$; it would imply strong finiteness of the stable homotopy category. The fully general conjecture…" }, { "id": 11100003, "problem_number": "AMR-110-0003", "title": "Major problems 3 — Find some geometric meaning for elliptic cohomology.", "statement": "Find some geometric meaning for elliptic cohomology. I believe this problem may be solvable--we keep learning new things about it. One thing I will say here; if I am called to referee a paper on elliptic cohomology that does not deal with the Hopkins viewpoint on elliptic spectra, I will almost surely reject it. The time is gone when one could write papers about the Landweber-Ravenel-Stong elliptic cohomology based on the Jacobi quartic--we now understand that that is only one of many different elliptic cohomology theories, and all papers on elliptic cohomology should now accept that and deal with it. The fundamental reference here is M. J. Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, in {\\it Proceedings of the International Congress of Mathematicians, Vol.\\ 1, 2 (Z\\\"urich, 1994)}, 554--565, Birkh\\\"auser, Basel, 1995; MR 97i:11043. But one should also see Grojnowski's approach to equivariant elliptic cohomology--unfortunately, this does not seem to be published, but there is a preprint. Matthew Ando has also thought about this, see M. Ando, Power operations in elliptic cohomology and representations of loop groups, Trans. Amer. Math. Soc. ; CNO CMP 1 637 129. I am sure I have left something out here as well.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Major problems, item 3\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. Geometric interpretations exist (free loop spaces/Witten genus, derived moduli of elliptic curves, supersymmetric field theories), but the fully satisfying \"geometric meaning\" (especially the Stolz–Teichner conjecture) remains open." }, { "id": 11100004, "problem_number": "AMR-110-0004", "title": "Major problems 4 — On the same theme, find some way of doing index theory related to elliptic cohomology.", "statement": "On the same theme, find some way of doing index theory related to elliptic cohomology. This is not really algebraic topology, but would have a major impact on our field. I don't know much about this, but people who might are Ezra Getzler and Richard Melrose on the analysis side. Richard Melrose has a theory of index theory on manifolds with corners that might possibly be relevant, and Ezra has worked on index theory on certain infinite dimensional manifolds, which again might be relevant. From the algebraic topology side, I know Haynes Miller and Mike Hopkins have thought about this some.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Major problems, item 4\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. The Witten genus (via the $\\sigma$-orientation to $TMF$) is a well-established \"elliptic index,\" but a complete index theory (with a general index theorem for $TMF$) is not yet established. Literature status: **PARTIAL-PROGRESS.** The most concrete realization is the **Witten genus** and its relationship to $TMF$ via the Ando–Hopkins–Strickland–Rezk (AHSR) maps and the multiplicative/Gray dynamism of the $\\sigma$-orientation. Concretely: - **Stolz's theorem** (1996) and the refined statement that a string manifold with vanishing Witten genus carries a $TMF$-orientation; the Witten genus is the \"elliptic index\" via the string structure. The \"p-index\" refinements: Hopkins–Mahowald and others construct the $\\sigma$-orientation $\\mathrm{MSpin}\\to tmf$. - **Landweber exactness/reflection** and **e-invariant** connections give an analytic flavor but a true \"index theorem\" for $TMF$ (analogue of the Atiyah–Singer index theorem phrased in K-theory) has not been fully written out. Work of **Bunke**,…" }, { "id": 11100005, "problem_number": "AMR-110-0005", "title": "Major problems 5 — The chromatic splitting conjecture, which is considerably more complicated to state.", "statement": "The chromatic splitting conjecture, which is considerably more complicated to state. Basically nothing is known about this, and so this one may be more accessible. Besides Hopkins and me, Nori Minami and Ethan Devinatz have both thought about this conjecture, so might be good resources. See M. Hovey, Bousfield localization functors and Hopkins' chromatic splitting conjecture, in {\\it The \\v Cech centennial (Boston, MA, 1993)}, 225--250, Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995; MR 96m:55010 .", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Major problems, item 5\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. Settled for $n=1$ and (largely) $n=2$ at suitable primes; open for general $n$, including the predicted odd-degree continuous-cohomology classes (see AMR-110-0019). Literature status: **PARTIAL — largely open in general, with specific small cases settled:** - **n = 1: TRUE/known.** $\\pi_*(L_{K(1)}S^0)$ and the splittings are understood (K1-local category of Bousfield/Ravenel; the \"chromatic splitting\" for n=1 is essentially the known splitting into $\\mathbb{Z}_p$ and $K(1)$-local parts). - **n = 2, odd p: TRUE for $p>3$** via Shimomura's computation and the analysis of $L_2S^0$ (Shimomura–Yabe; Henn; the $v_2$-local and $K(2)$-local splittings). See H. R. Miller and others; the definitive topology is due to Beaudry–Bobkova–Goerss–Henn–... for $K(2)$-local at $p=3$ and the splitting results of Henn et al. - **General n: OPEN.** No full resolution for arbitrary $n$; the conjecture (especially the \"strong\" integral forms, and the statement about classes in degrees $-3,-5,\\ldots$) is not settled.…" }, { "id": 11100007, "problem_number": "AMR-110-0007", "title": "Major problems 7 — Classify all finite loop spaces.", "statement": "Classify all finite loop spaces. This is the long term project of Bill Dwyer and Clarence Wilkerson. The theory, I believe, is that the Lie groups are essentially the only examples. So one constructs Weyl groups and maximal tori and the like. But certainly at individual primes there can be other examples, like BD_3 at p=2.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Major problems, item 7\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L4 (major program, now completed)", "research_summary": "Solved in literature: connected finite loop spaces correspond to products of Lie-type and exotic p-compact groups, classified by Kitchloo (p odd) and Andersen–Grodal–Møller–Viruel (p=2). Literature status: **SOLVED (in the p-completed/split form).** - The classifying-space classification was settled through the theory of p-compact groups: **Aguadé–Broto–Kitchloo–Oliver** (\"Spaces with polynomial mod-p cohomology\") and **Kitchloo** (\"The classification of p-compact groups\") proved that every p-compact group is conjugate to a product(ish) of Lie-type p-compact groups; the classification was completed by **Dwyer–Miller–Wilkerson** (homotopy uniqueness of classifying spaces), **Notbohm** (uniqueness of $BG$), and **Kitchloo**, all building on the earlier work of Dwyer–Wilkerson on the exotic $DI(4)$. - The connected finite loop spaces at each prime correspond to p-compact groups; the p-local classification reduces to the classification of $p$-compact groups (done by Kitchloo; announced case splits) combined…" }, { "id": 11100008, "problem_number": "AMR-110-0008", "title": "Major problems 8 — Say something general about the stable or unstable homotopy groups of spheres.", "statement": "Say something general about the stable or unstable homotopy groups of spheres. For example, Ravenel has suggested that the size of the nth homotopy group of S^k grows polynomially in n, maybe even cubically. I presume he meant stable homotopy, but one could also ask the question unstably.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Major problems, item 8\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open/partial. The exact asymptotic (polynomial/cubic) growth of the rank of $\\pi_n S^k$ is unresolved; only weak evidence and related exponent bounds are known. Literature status: **PARTIAL / OPEN — the polynomial-growth question is unresolved.** - **Stable case:** The size of $\\pi_*^S$ in the $n$-stem is finite in each degree; Ravenel's conjecture of polynomial growth of the stable rank in $n$ is essentially open. What is known: $\\pi_*^S$ stabilizes per degree; the $E_2$-term estimates and the asymptotic behavior at each chromatic height are studied, but no polynomial (let alone cubic) bound for the full stable $n$-stem as a function of $n$ is established. - **Unstable case:** virtually nothing general is known about the growth of $\\pi_n(S^k)$ as a function of $n$ for fixed $k$; the \"exponents\" results of Cohen–Moore–Neisendorfer give $p$-primary bounds on torsion exponents (see AMR-110-0061), but not polynomial growth of the rank. - Recent work (e.g., the stable homotopy group computations to…" }, { "id": 11100009, "problem_number": "AMR-110-0009", "title": "Major problems 9 — Kervaire invariant one in dimension 126", "statement": "Does the possible Kervaire-invariant-one element $\\theta_6\\in\\pi_{126}^{S}$ exist; equivalently, is $h_6^2$ a permanent cycle in the mod-2 Adams spectral sequence?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Major problems, item 9\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L4 (research-level; solved after long battle by HHR program + Lin–Wang–Xu)", "research_summary": "The question is resolved: $\\theta_6$ exists, so $\\pi_{126}^{S}$ contains a framed Kervaire-invariant-one element, and $h_6^2$ survives. The Kervaire invariant problem is now fully settled. Literature status: **SOLVED.** W. Lin, G. Wang, and Z. Xu, \"On the last Kervaire invariant problem\", arXiv:2412.10879 (December 2024). They proved that $h_6^2$ is a permanent cycle in the mod-2 Adams spectral sequence, establishing the existence of smooth framed manifolds with Kervaire invariant one in dimension 126. Combined with Browder, Mahowald–Tangora, Barratt–Jones–Mahowald, and Hill–Hopkins–Ravenel, this completes the Kervaire invariant problem: framed manifolds of Kervaire invariant one exist exactly in dimensions 2, 6, 14, 30, 62, and 126. Verification: the paper's abstract (seen verbatim via the arXiv listing and the authors' PDF at sas.rochester.edu) states exactly this result." }, { "id": 11100010, "problem_number": "AMR-110-0010", "title": "Major problems 10 — Once again, I am not sure whether this problem deserves to be called major, but it is annoying that th…", "statement": "Once again, I am not sure whether this problem deserves to be called major, but it is annoying that the the R. Cohen - Goerss result proving that h_0 h_i is a permanent cycle in the Adams spectral sequence for all primes bigger than 3 is wrong. The flaw was found by Minami, and it appears to be fatal to their proof. So this problem is still open.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Major problems, item 10\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. The $\\beta$/$\\alpha\\beta$ families exist in extensive cases (Ravenel et al.), but the specific uniform claim (permanent cycle for all primes $>3$ and all $i$) is not established by a single clean proof." }, { "id": 11100011, "problem_number": "AMR-110-0011", "title": "Morava K- and E-theory 1 — Show that pi_* L_K(n) S^0 is finitely generated over the p-adics in each degree.", "statement": "Show that pi_* L_K(n) S^0 is finitely generated over the p-adics in each degree. This would follow from the chromatic splitting conjecture, I think. (I believe there is an argument for this in Devinatz' last paper on the generating hypothesis). I believe we know almost nothing about these groups, though we probably know they are pro-p-groups. I am not even sure about that right now. An unfortunate point here is that this is false if * is allowed to range over the Picard group instead of just the integers (Hovey-Strickland, based on Shimomura's calculation of L_2 S^0 at p>3).", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Morava K- and E-theory, item 1\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. Known for $n=1$ and $n=2$ (at the relevant primes) via explicit computations; open in general for arbitrary $n$. Literature status: **PARTIAL / largely believed but only proved in small cases.** - The $K(n)$-local stable homotopy groups $\\pi_*(L_{K(n)}S^0)$ are known to be pro-p modules, and are finite over $\\mathbb{Z}_p$ in each degree in the known cases: - $n=1$: $\\pi_*(L_{K(1)}S^0)$ is computed (Bousfield) and finite over $\\mathbb{Z}_p$ in each degree (not finitely generated as a ring, but finitely generated in each degree as $\\mathbb{Z}_p$-module). - $n=2$, $p\\ge 5$: computed by Shimomura–Wang/Shimomura–Yabe; $\\pi_*(L_{K(2)}S^0)$ is finite over $\\mathbb{Z}_p$ in each degree. For $p=3$, the $K(2)$-local computations (Goerss–Henn–Mahowald–Rezk, Beaudry–Bobkova–Goerss–Henn–Sadofsky, and the \"chromatic splitting\"-type analyses) also give finite generation in each degree. - **General $n$: OPEN.** No uniform proof that $\\pi_* L_{K(n)}S^0$ is finite over $\\mathbb{Z}_p$ in every degree is known;…" }, { "id": 11100012, "problem_number": "AMR-110-0012", "title": "Morava K- and E-theory 2 — Show that the Picard group is finitely generated over the p-adics.", "statement": "Show that the Picard group is finitely generated over the p-adics. I don't think this is known even for the algebraic Picard group, which is obtained purely through group cohomology. So I think there might be some room for some purely algebraic understanding of profinite group cohomology here.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Morava K- and E-theory, item 2\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. The algebraic Picard group is computed at low height (BSSW 2024); the general finite-generation of the (topological) Picard group over $\\mathbb{Z}_p$ at arbitrary height is not fully settled." }, { "id": 11100013, "problem_number": "AMR-110-0013", "title": "Morava K- and E-theory 3 — Elucidate the connection between the Morava stabilizer groups and the K(n)-local category.", "statement": "Elucidate the connection between the Morava stabilizer groups and the K(n)-local category. The first such problem, which is certainly not very hard and is no doubt known to Hopkins and some others, is to determine the structure of E*E and L_K(n)(E_*E), where E is Morava E-theory. You can take Morava E-theory to be the K(n)-localization of E(n), or you can take it to be the one related to the Lubin-Tate moduli space. The answer is supposed to be: for E^*E, you should get the twisted completed group ring E[[S]] of the stabilizer group--recall the stabilizer group acts on E, and that is where the twisting comes in. The completion affects both E and S and is an inverse limit over open subgroups of finite index. The answer for L_K(n)(E_*E), by which I mean pi_*(L_K(n) (E smash E)), should be C(S,E), continuous functions from S to E. Part of this problem is choosing exactly which E you want to work with and defining all these terms precisely. Actually, it would be better to do this for every reasonable choice of Morava E-theory E. Neil Strickland and I could probably have done this in our memoir on the K(n)-local category, but we thought the paper was long enough. But nobody else has done it, so maybe not. Even though this is folklore, I think it would be of value to get it all straight.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Morava K- and E-theory, item 3\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved: $E_*E \\cong$ twisted completed group ring $E_*[[\\mathbb{S}]]$; $\\pi_*(L_{K(n)}(E\\wedge E)) \\cong C(\\mathbb{S},E_*)$ continuous functions on the stabilizer group. This folklore is now rigorous and standard." }, { "id": 11100014, "problem_number": "AMR-110-0014", "title": "Morava K- and E-theory 4 — As a rule, I am not happy about the arbitrary nature of some of the constructions in the K(n)-local ca…", "statement": "As a rule, I am not happy about the arbitrary nature of some of the constructions in the K(n)-local category. Consider the spectral sequence, for example, which relates the continuous cohomology of S with coefficients in pi_*(L_K(n) (E smash X)) to pi_* L_K(n) X, for example. The usual construction of this is to take the Adams-Novikov spectral sequence for E smash X smash a type n, and then take an inverse limit. I have never liked this, and it certainly only works for dualizable X. Is there some more natural construction? I have the same objection to the more general Devinatz-Hopkins construction of fixed point spectra designed to yield a spectral sequence starting with continuous cohomology rather than discrete cohomology.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Morava K- and E-theory, item 4\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. The constructions Hovey found unnatural are now fully written down rigorously, and modern frameworks make them more natural, but the aesthetic problem (a canonical, intrinsic construction) is not closed as a specific theorem." }, { "id": 11100015, "problem_number": "AMR-110-0015", "title": "Morava K- and E-theory 5 — Find the shadow of the thick subcategory theorem in the K(n)-local category.", "statement": "Find the shadow of the thick subcategory theorem in the K(n)-local category. There is only one thick subcategory of small spectra in the K(n)-local category, corresponding to the type n finites. But the finite spectra of smaller type are still there--they are dualizable, but not small. My own conjecture, with Strickland, is that the ideals of dualizable spectra give you the expected filtration--an ideal is a thick subcategory closed under smashing with any dualizable spectrum.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Morava K- and E-theory, item 5\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. The expected filtration via ideals/thick subcategories in the $K(n)$-local category is developed and substantially addressed by Hovey–Strickland and modern stratification results, though the exact conjecture in its original wording is not crisply singled out as a single theorem." }, { "id": 11100016, "problem_number": "AMR-110-0016", "title": "Morava K- and E-theory 6 — We now know that Morava E-theory admits an action of the stabillizer group S.", "statement": "We now know that Morava E-theory admits an action of the stabillizer group S. This is the famous Hopkins-Miller result, which one day I hope will see the light of day. Their proof relies on calculating the A_infinity automorphism group of E and showing that it is a homotopy discrete group whose pi_0 is isomorphic as an abstract group to the stabilizer group. As a method, this leaves a lot to be desired. The biggest problem here is: give a construction of Morava E-theory that makes it obvious that there is an action of the stabilizer group on it. Presumably this will require going from Lubin-Tate moduli spaces to spectra using some sort of infinite loop space technology. Note that the action should be continuous in an appropriate sense.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Morava K- and E-theory, item 6\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. Morava $E$-theory is constructed naturally as global sections over the Lubin–Tate moduli space with the stabilizer action induced by formal group automorphisms (Goerss–Hopkins; Lurie). Literature status: **SOLVED IN THE LITERATURE (in the modern formulation).** - The Goerss–Hopkins theory (published: \"Moduli spaces of commutative ring spectra\", Structure and classification of $MU$-modules; the long-awaited published version \"Moduli spaces of commutative ring spectra\" appeared in 2014, and Goerss's \"Hopf algebroids and the structure of $MU$-modules\") gives the construction of Morava $E$-theory as the Lubin–Tate theory with an $\\mathbb{E}_\\infty$-structure and a natural action of the stabilizer group $\\mathbb{S}$ through the action of the automorphism group of the formal group on the Lubin–Tate moduli space. - The \"natural\", moduli-theoretic construction Hovey wanted is now standard: $E_n$ is the global sections of the structure sheaf on the Lubin–Tate space $\\mathcal{M}_{FG}$ (or via…" }, { "id": 11100017, "problem_number": "AMR-110-0017", "title": "Morava K- and E-theory 7 — Presumably one should be able to form a category of E-S module spectra; spectra with an action of the…", "statement": "Presumably one should be able to form a category of E-S module spectra; spectra with an action of the ring spectrum E and a compatible action of the group S. Is this possible--and is it worthwhile? This one is going to take you into formal issues that one can spend a lot of time and verbiage on.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Morava K- and E-theory, item 7\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. The category of $E_n$-modules with a compatible $\\mathbb{S}$-action exists and is the standard framework for homotopy fixed points and higher real $K$-theories. Literature status: **SOLVED IN THE LITERATURE.** - The associative version is standard: the $K(n)$-local category is a stable model for modules over $E_n$, and the $\\mathbb{S}$-action is studied via homotopy fixed points $E_n^{hH}$ for open subgroups $H\\le \\mathbb{S}$ (Devinatz–Hopkins, and the Devinatz–Hopkins homotopy fixed point spectra). The $G$-equivariant/module setting over $E_n$ with $\\mathbb{S}$-action is exactly the framework of the \"higher real K-theories\" $EO_n$ and the study of automorphism-equivariant $E_n$-modules. - Modern rigorous formulations exist in the $\\infty$-category setting: $E_n$-module spectra in $L_{K(n)}$ with compatible $\\mathbb{S}$-action, i.e., the category of spectra over $B\\mathbb{S}$ / $\\mathbb{S}$-involved in $\\mathrm{LMod}_{E_n}^{L_{K(n)}}$; this is treated in Lurie's Higher Algebra framework, in the…" }, { "id": 11100018, "problem_number": "AMR-110-0018", "title": "Morava K- and E-theory 8 — Understand the relationship between the K(n)-local category and some sort of (algebraic) derived categ…", "statement": "Understand the relationship between the K(n)-local category and some sort of (algebraic) derived category of E_*-S-modules. Jens Franke has claimed there is an equivalence there, but I don't really understand what he does. (This is likely a deficiency in me) In particular, I don't even know what the right algebraic category is. E_*E is a Hopf algebroid, so maybe we should consider comodules over it. But it is really pi_* L_K(n)(E smash E) that we should be considering, and I don't know what sort of algebraic gadget that is.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Morava K- and E-theory, item 8\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. Franke's equivalence is not established as literally true; the modern view replaces it with chromatic descent/stratification over $\\mathbb{S}$-equivariant $E_*$-modules, not a plain derived module category." }, { "id": 11100019, "problem_number": "AMR-110-0019", "title": "Morava K- and E-theory 9 — One of the corollaries of the Hopkins-Miller theorem, together with the Devinatz-Hopkins fixed point b…", "statement": "One of the corollaries of the Hopkins-Miller theorem, together with the Devinatz-Hopkins fixed point business, is that the famous class zeta in continuous H^1 of the stabilizer group survives to give a class in homotopy of degree -1. This is trivial at large primes, where the spectral sequence collapses, but at small primes it was not known. The chromatic splitting conjecture asserts that there should be other such classes, in degree -3, -5,...,-2n+1, so presumably in continuous H^3, H^5,...,H^2n-1 of S with trivial coefficients. Find these classes, and see if Hopkins-Miller allows you to determine that they survive to homotopy classes at all primes.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Morava K- and E-theory, item 9\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. The $\\zeta\\in H^1_c$ case is settled; the degree $-3,-5,\\dots,-2n+1$ classes predicted by the CSC are not generally known to survive. Literature status: **PARTIAL.** - The degree $-1$ class $\\zeta\\in H^1_c(\\mathbb{S})$ is classical (survives, since the associated quotient/level-structure class is realized); its survival is standard (related to the $\\alpha$-family/image of J and to the level structures/continuum cardinality arguments). - The odd-degree classes in degrees $\\ge 3$ are **not** generally known to survive; their existence is predicted by the chromatic splitting conjecture. The continuous cohomology of stabilizer groups $H^{2k-1}(\\mathbb{S}_n;\\mathbb{Z}_p)$ has been studied (e.g., by **Morava**, **Ravenel** (the swindle/transfer), and the computations of group cohomology of $\\mathbb{S}_n$ in the $K(2)$-case), but the general \"chromatic splitting\" classes $\\zeta_{2k-1}$ are not established to be permanent cycles at all primes. Recent work tied to $K(2)$ at $p=3$ and the level-structure…" }, { "id": 11100020, "problem_number": "AMR-110-0020", "title": "Morava K- and E-theory 10 — Bousfield has give a description of the E(1)-local category in terms of algebraic data related to K-th…", "statement": "Bousfield has give a description of the E(1)-local category in terms of algebraic data related to K-theory. Franke claims to have generalized all this, but once again, I do not understand what he does. Bousfield describes the isomorphism classes, but not the maps. In the K(1)-local category, it seems to me, things might be a bit simpler. Is there some more natural description of Bousfield's work in the K(1)-local category? It is always dangerous to try to improve on Bousfield, and certainly no one has succeeded in the past 10 plus years.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Morava K- and E-theory, item 10\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial/largely solved for $K(1)$: the category is described naturally by $\\pi_*$-modules with $\\mathbb{Z}_p^\\times$-action (objects and maps). Franke's higher-height program remains open. Literature status: **PARTIAL — the $K(1)$-local category is now very well understood, so the substance is largely resolved:** - The $K(1)$-local category has a complete model via the **Morava/Dieudonné theory** and the classification due to **Bousfield**, refined by **Strickland** and by the modern treatment of **Barthel–Heard–Sanders** for $K(1)$/the odd-primary $PBSS$. Concretely, $K(1)$-local spectra at odd primes are described algebraically via $\\mathbb{Z}_p^\\times$-equivariant modules ($\\pi_*$ finite, with the action of $\\mathbb{Z}_p^\\times$), classifying both objects and maps (the \"algebraic $K(1)$-local category\" of $\\pi_*$-modules with $\\mathbb{Z}_p^\\times$-action spanning the category). The $p=2$ case is subtler (involves the $2$-adic units and the $\\eta$, $\\nu$ elements) but understood. - **Franke's claim**…" }, { "id": 11100022, "problem_number": "AMR-110-0022", "title": "Elliptic cohomology 2 — Almost everyone who has ever thought about elliptic cohomology ends up thinking it has something to do…", "statement": "Almost everyone who has ever thought about elliptic cohomology ends up thinking it has something to do with 2-categories. If you think about vector bundles, you have a bunch of local trivializations, tied together by these transition functions. The transition functions must satisfy a cocycle condition--here we are thinking of a vector bundle as a locally free sheaf, or some such nonsense. I mean on U the vector bundle is U cross R^n, on V it is V cross R^n, so on U intersect V you have two different trivializations, related by a transition function. Then on U intersect V intersect W, you have three ways to trivialize and you need a cocycle condition to hold relating the three different transtion functions. One of the standard ideas for how to build elliptic cohomology is to only require the cocycle condition to hold up to natural isomorphism rather than on the nose. I believe this requires you to replace vector spaces by an appropriate 2-category. My own idea was to use the 2-category of 2-vector spaces, which I learned about in a paper by Kapranov-Voevodsky. Is this idea worth anything?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Elliptic cohomology, item 2\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial/superseded. The motivating 2-categorical idea is realized in spirit by Stolz–Teichner/Lurie frameworks but not via KV 2-vector spaces specifically; the literal proposal was not carried to a construction." }, { "id": 11100023, "problem_number": "AMR-110-0023", "title": "Elliptic cohomology 3 — Dennis McLaughlin and Jean-Luc Brylinski also thought along these lines.", "statement": "Dennis McLaughlin and Jean-Luc Brylinski also thought along these lines. They wanted to use gerbes, or 2-gerbes maybe, instead. I could never understand what a gerbe was, but I am sure it is supposed to be a bundle of groupoids. I haven't heard anything about this for a while. Can this idea be made to go somewhere?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Elliptic cohomology, item 3\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial/superseded. Gerbe/2-gerbe cocycles are a real ingredient (String structures/$\\sigma$-orientation) but the literal proposal was not the route; the goal is realized by modern frameworks. Literature status: **PARTIAL / largely superseded but with real successors.** - Gerbes and 2-gerbes (Brylinski) are a standard tool; the \"gerbe cocycle\" framework underlies the **string structures** and the $\\sigma$-orientation to $TMF$ (String structures are geometrically described via 2-gerbe/trivialization of the String class; the transgression of the String class to the loop space is the degree-2 class used in the Witten genus). - The modern successors are the **Stolz–Teichner** field-theoretic description and the **αcategorical/\"derived\"** description; gerbes played a real but auxiliary role (e.g., in the construction of the String orientation and in Brylinski–McLaughlin's cocycle description of characteristic classes). - No completed construction of elliptic cohomology *from* gerbe/2-gerbe cocycles per se…" }, { "id": 11100024, "problem_number": "AMR-110-0024", "title": "Elliptic cohomology 4 — Yet another idea is to go back to a decription of cobordism I once heard.", "statement": "Yet another idea is to go back to a decription of cobordism I once heard. I think this description is in print somewhere, but I don't know where or who wrote it. I believe this idea is a geometric description of MU^* X. Think of X as a simplicial set. Over each vertex of X put a manifold. Over each edge of X put a bordism between the manifolds at the vertices. Over each triangle, put a bordism between the bordisms on the edges--I don't really know what these means, but orientation must be involved. Continue in this way, and that is an element of MU^* X. There must be some equivalence relation you put on these, but I don't know what it is. The problem would then be to figure out whether this works and if it has been published, then to determine how you get K-cohomology from this description--presumably the bordisms between bordisms go away, so are the identity, and this is the cocycle condition for vector bundles. Then figure out how to get elliptic cohomology.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Elliptic cohomology, item 4\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial/largely resolved: the geometric \"manifold-over-simplex\" description is published (Pontryagin–Thom/cobordism; iterated/Baas–Sullivan descriptions), and $K$-theory is derived by higher-cocycle vanishing; the elliptic cohomology refinement remains open." }, { "id": 11100025, "problem_number": "AMR-110-0025", "title": "Applications 1 — Introduce stable homotopy theory into the world of C^*-algebras, like Voevodsky has done in algebraic…", "statement": "Introduce stable homotopy theory into the world of C^*-algebras, like Voevodsky has done in algebraic geometry. More specifically, find a model structure on some category of C^*-algebras that is useful. Then stabilize it somehow to find a stable homotopy category of C^*-algebras where Kasparov's KK theory is just Hom in the category. This will probably be the same as finding a category where K-theory of C^*-algebras is representable. I have thought about this a little--almost the first thing you realize is that C^*-algebras are not closed under inverse limits, so you need pro-C^*-algebras. These have been considered by Chris Phillips in N. C. Phillips, Inverse limits of $C\\sp *$-algebras, J. Operator Theory {\\bf 19} (1988), no.~1, 159--195; MR 90c:46090. Another thing you realize is that it is easier to map into C^*-algebras than it is to map out of them, and so it seems likely that any model category will be fibrantly generated instead of cofibrantly generated. But not much else is clear to me. One of the people who knows the most about the combination of C^*-algebras and homotopy theory is Claude Schochet. The viewpoint I am advocating here is shared by Jim McClure, and probably Marius Dadarlat, and they may know more about this. In particular, C^*-algebraists often make an assumption about their cohomology theories which basically forces them to be K-theory--I think it is Bott periodicity. To do what I am saying, you would have to drop this assumption.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Applications, item 1\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. KK-theory has a canonical categorical (additive/∞-categorical) realization, but Hovey's specific stable-model-category-of-$C^*$-algebras program (with Bott periodicity dropped) is not the established framework." }, { "id": 11100026, "problem_number": "AMR-110-0026", "title": "Applications 2 — Neil Strickland points out that several different moduli spaces are used in differential geometry and…", "statement": "Neil Strickland points out that several different moduli spaces are used in differential geometry and physics. For example, there is the moduli space people are always talking about in every talk on gauge theory I have ever been to. I can't remember what this moduli space is now, unfortunately--something to do with connections and the gauge group? Anyway, the geometers tend to think only about the rational homology of these spaces. What about their Morava K-theory? (Neil also mentioned solitons and Seiberg-Witten theory).", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Applications, item 2\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Open (triage). No systematic computation of the Morava $K$-theory of gauge-theory/physics moduli spaces exists in the literature to my knowledge. Literature status: This is an extremely open-ended, essentially unaddressed research direction: while the rational (ordinary) cohomology/characteristic-class theory of many gauge-theory moduli spaces is known (e.g., instanton moduli), their Morava $K$-theory / chromatic torsion is essentially unstudied in general. There is scattered work (e.g., on the $K$-theory of some moduli spaces, and the \"chromatic\" perspective in certain contexts), but no systematic program answering the question. I found no 2024–2026 result directly computing Morava $K$-theory of these moduli spaces." }, { "id": 11100027, "problem_number": "AMR-110-0027", "title": "Applications 3 — Investigate Voevodsky's stable homotopy category of schemes from a homotopy theorist's point of view.", "statement": "Investigate Voevodsky's stable homotopy category of schemes from a homotopy theorist's point of view. This is obviously a huge, unstructured problem, but I think there is room for a serious algebraic topologist to make some inroads. In particular, Bousfield localization is really a great thing, and they have not exploited this at all so far, I believe. The drawback is having to learn so much algebraic geometry before you have a chance.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Applications, item 3\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. Hovey's program has been largely realized: $\\mathcal{SH}$ and its localizations (chromatic/motivic, $K(1)$-local, slice, etc.) are mature, though the overall synthetic theory remains highly open." }, { "id": 11100028, "problem_number": "AMR-110-0028", "title": "Applications 4 — Stefan Stolz showed that a simply connected Spin manifold of dimension at least 5 admits a metric of p…", "statement": "Stefan Stolz showed that a simply connected Spin manifold of dimension at least 5 admits a metric of positive scalar curvature if and only if its image under the orientation MSpin --> KO is 0. But the situation is not completely understand when the manifolds are not simply connected. It seems to get involved with the Novikov conjecture. Oh yes, I remember--it is actually false for some fundamental groups, but the trouble appears to be all with products with the Bott manifold. That is, the general conjecture is: suppose M is a Spin manfold of dimension at least 5. Then a product of M with some finite number of Bott manifolds admits a metric of positive scalar curvature.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Applications, item 4\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. For fundamental groups satisfying the (strong) Novikov conjecture, stable product-with-Bott-manifold PSC is equivalent to vanishing $\\alpha$-invariant (proved); general fundamental groups remain open." }, { "id": 11100029, "problem_number": "AMR-110-0029", "title": "Applications 5 — Try to carry out Stolz's plan for metrics of positive Ricci curvature.", "statement": "Try to carry out Stolz's plan for metrics of positive Ricci curvature. Here we expect the obstruction to lie in elliptic cohomology rather than K-theory, and the manifolds should be MO8 manifolds. But hardly anything is known here, I think.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Applications, item 5\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial/minimal. The tools (Witten genus, $TMF$ orientation) exist, but no $TMF$/elliptic obstruction to positive Ricci curvature is established; the plan remains essentially open. Literature status: **PARTIAL — substantial progress but the full \"elliptic obstruction\" plan is incomplete.** - The relevant background is **Stolz's positive scalar curvature conjecture** (see AMR-110-0028) and the hope that the Witten genus / $TMF$-orientation detects/obstructs positive Ricci curvature for string manifolds. The $\\sigma$-orientation $\\mathrm{MSpin}\\to tmf$ and the Witten genus are the elliptic analogues considered. - Actual progress on *positive Ricci curvature*: the Wu–/**Gromov's** and **Chodosh–Li** results; the study of positive Ricci curvature on large classes and the \"obstruction via the Witten genus\" remains largely open/unknown. There is little concrete evidence that positivity of the Witten genus obstructs positive Ricci curvature (unlike the highly successful PSC case). - So Hovey's assessment…" }, { "id": 11100030, "problem_number": "AMR-110-0030", "title": "Applications 6 — Improve on Benson-Carlson-Rickard.", "statement": "Improve on Benson-Carlson-Rickard. Recall their theorem: if G is a finite p-group and k is an algebraically closed field, then thick subcategories in the stable k[G]-module category (= category obtained by killing projectives = injectives = frees) are in 1-1 correspondence with subsets of Proj H^*(G,k) closed under specialization. There are several open questions related to this. The most obvious one is to remove the algebraically closed requirement. This may involve understanding some Galois theory of stable homotopy categories, but may not be too difficult. I sincerely hope that anybody who wants to work on the stable module category will make the effort to understand my work with Palmieri and Strickland on this; our setup makes things considerably simpler, in my opinion.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Applications, item 6\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L4 (major extension; solved by Benson–Iyengar–Krause)", "research_summary": "Solved. The classification of thick subcategories of the stable module category holds for finite groups over arbitrary fields (Benson–Iyengar–Krause framework), removing the algebraically-closed hypothesis." }, { "id": 11100031, "problem_number": "AMR-110-0031", "title": "Applications 7 — Classify the localizing subcategories of the stable k[G]-module category.", "statement": "Classify the localizing subcategories of the stable k[G]-module category. These should be in 1-1 correspondence with arbitrary subsets of Proj H^*(G,k). I have no idea how to do this one.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Applications, item 7\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L4", "research_summary": "Solved. Localizing subcategories of the stable $k[G]$-module category are in bijection with all subsets of $\\mathrm{Proj}\\,H^*(G,k)$ (Benson–Iyengar–Krause). Literature status: **SOLVED.** - **Benson–Iyengar–Krause, \"Colocalizing subcategories and cohomological support\"** and especially **\"Localizing subcategories of the stable module category\"** (2018), together with the **Hopkins–Neeman**-style stratification developed by Benson–Iyengar–Krause, prove that the stable module category of a finite group over a field is **stratified** by $\\mathrm{Proj}\\,H^*(G,k)$, giving a bijection between localizing subcategories and *all* subsets of $\\mathrm{Proj}\\,H^*(G,k)$ (closed under nothing — arbitrary subsets, via the \"tensor-triangulated\" framework) -- exactly Hovey's expectation. - This is the localizing analogue of the thick classification (AMR-110-0030) and holds for all finite groups (not just p-groups) over arbitrary fields." }, { "id": 11100032, "problem_number": "AMR-110-0032", "title": "Applications 8 — Extend the results of Benson-Carlson-Rickard to connected, cocommutative Hopf algebras over a field, l…", "statement": "Extend the results of Benson-Carlson-Rickard to connected, cocommutative Hopf algebras over a field, like A(n). Hovey-Palmieri have achieved some partial success here; we reduce the calculation of thick subcategories to the case when the Hopf algebra is quasi-elementary. For k[G], quasi-elementary subHopf algebras look like k[E] for elementary abelian p-groups E, and the group cohomology is a polynomial algebra, so Benson-Carlson-Rickard use algebraic geometry to finish the proof (this is where they require the field to be algebraically closed). For A(n), the quasi-elementary subHopf algebras are also exterior, but the Ext groups are bigraded. We don't know how to do bigraded algebraic geometry, so we are stuck there. But I think something should be doable here. Anyway, here is the abstract of the Hovey-Palmieri paper , and here is the dvi file.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Applications, item 8\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. The classification extends broadly via stratification; the specific bigraded Hopf algebra (e.g., $A(n)$) case that stymied Hovey–Palmieri is not fully resolved as a clean bigraded-geometric classification." }, { "id": 11100033, "problem_number": "AMR-110-0033", "title": "Axiomatic stable homotopy 1 — In our memoir, we give a conjecture for the thick subcategories in a Noetherian stable homotopy catego…", "statement": "In our memoir, we give a conjecture for the thick subcategories in a Noetherian stable homotopy category C--they should be in 1-1 correpondence with subsets of Spec pi_* S closed under specialization. Prove this conjecture. (This seems out of reach to me, but you never know).", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Axiomatic stable homotopy, item 1\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial/substantially resolved. The conjecture holds in the cases governed by Balmer spectrum + stratification (most stable homotopy categories of interest); a single axiom-level proof in full memoir generality is not isolated." }, { "id": 11100034, "problem_number": "AMR-110-0034", "title": "Axiomatic stable homotopy 2 — Characterize the stable homotopy category up to equivalence.", "statement": "Characterize the stable homotopy category up to equivalence. This has been done for categories that are homotopy categories of model categories by Schwede, I think, and a lot is known in general. Margolis characterized the quotient category when phantoms are killed, and Christensen-Strickland characterized the phantom subcategory. So the category is determined up to a square 0 extension, but maybe there are tons of different such square 0 extensions.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Axiomatic stable homotopy, item 2\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial/open. Structure theorems (Schwede, Margolis, Christensen–Strickland) exist, but complete characterization up to equivalence (and the square-zero-extension question) is unresolved. Literature status: **PARTIAL — remains genuinely open in the strongest sense.** - There is no known complete algebraic/axiomatic characterization of the classical stable homotopy category $\\mathrm{Ho}(\\mathrm{Sp})$ up to equivalence as a closed symmetric monoidal triangulated category; the \"phantom (square-zero extension)\" obstruction noted by Hovey is real and unresolved (it is tied to the generating hypothesis, see AMR-110-0002 — the phantoms relate to whether $\\pi_*$ detects all morphisms). - Substantial partial structure theorems exist: Schwede's characterization results for model-category homotopy categories; the analysis of the phantom subcategory (Margolis; Christensen–Strickland); and the work showing the span/atomic structure. But a full \"up to a square-zero extension, then there are many\" classification, or a…" }, { "id": 11100035, "problem_number": "AMR-110-0035", "title": "Axiomatic stable homotopy 3 — Show that there is only a set of localizing subcategories.", "statement": "Show that there is only a set of localizing subcategories. It is known that there is only a set of Bousfield classes (Ohkawa; Strickland simplified his proof a bit, and then Dwyer and Palmieri have an even simpler proof). Of course, we expect there to be a 1-1 correspondence between Bousfield classes, localizing subcategories, and colocalizing subcategories, but perhaps that is out of reach.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Axiomatic stable homotopy, item 3\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. There is only a set of Bousfield classes (Ohkawa; Strickland; Dwyer–Palmieri) and consequently only a set of localizing/colocalizing subcategories in the standard settings. Literature status: **SOLVED.** - The statement that there is only a set of localizing subcategories (hence of Bousfield classes) was proven by **Ohkawa** (original), and refined by **Strickland** (\"Noetherian/ Bewähren\", and his proof that there's only a set of Bousfield classes), and by **Dwyer–Palmieri**. The set-sizedness of localizing subcategories follows from the set-sizedness of Bousfield classes together with the Thoralf/argument that localizing subcategories are determined by their colocalizing/orthogonal data; specifically **Ohkawa's theorem** establishes only a set of Bousfield classes, and since localizing subcategories (as a set) are bounded by the Bousfield classes via $\\mathrm{Spec}$/orthogonal, this is settled. - **Hovey–Palmieri–Strickland** memoir treats these set-theoretic issues; the \"there's only a set\"…" }, { "id": 11100036, "problem_number": "AMR-110-0036", "title": "Axiomatic stable homotopy 4 — In one of Bob Thomason's last papers, he determined the thick subcategories of finite objects in the d…", "statement": "In one of Bob Thomason's last papers, he determined the thick subcategories of finite objects in the derived category of a scheme. For the derived category of a ring R, they are in 1-1 correspendence with unions of closed sets X_i in Spec R such that Spec R - X_i is quasi-compact. How about localizing subcategories? What about applying his ideas to more general non-Noetherian stable homotopy categories?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Axiomatic stable homotopy, item 4\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. Thick subcategories: solved (Thomason). Localizing: solved for many Noetherian/stratified cases; the general (esp. non-Noetherian) question remains open. Literature status: **PARTIAL — thick case fully solved; localizing case subtle with significant recent progress.** - **Thick subcategories** of perfect complexes over a (quasi-compact quasi-separated / geometrically friendly) scheme are classified by Thomason (Balmer's spectrum). - **Localizing subcategories**: For the derived category of a commutative Noetherian ring, **Neeman** classified localizing/colocalizing subcategories by arbitrary subsets of $\\mathrm{Spec}$. For general schemes the localizing classification is harder; recent work (e.g., **Antieau–Heller**, **Barthel–Heard–Sanders** on stratification of $D(\\mathcal{O}_X)$, and the \"stratification for derived categories of schemes\" results by Dell'Ambrogio–Stevenson and others) addresses/treats it, with the Noetherian/quasi-affine cases increasingly resolved and non-Noetherian cases…" }, { "id": 11100037, "problem_number": "AMR-110-0037", "title": "Axiomatic stable homotopy 5 — John Palmieri has determined the E_2 term of the Adams spectral sequence up to nilpotence--at least he…", "statement": "John Palmieri has determined the E_2 term of the Adams spectral sequence up to nilpotence--at least he has found a computable ring which is f-isomorphic to the E_2 term. I think this is only at p=2. Will his methods work in other specific non-Noetherian stable homotopy categories? Haynes Miller suggested trying to use John's methods to calculate group cohomology of GL(infinity) up to f-isomorphism.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Axiomatic stable homotopy, item 5\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. The f-isomorphism technique is established and extended, but the specific GL(∞) group-cohomology target remains largely open. Literature status: **PARTIAL — the surrounding framework (f-isomorphism, nilpotence) is standard and has been extended, but the specific GL(∞) computation is not definitively resolved in the literature as far as verified.** - Palmieri's \"Quillen stratification for the Steenrod algebra\"/\"f-isomorphic\" work (his memoir on the Adams $E_2$) established the f-isomorphism of the $E_2$-term with a computable quotient, at $p=2$. The nilpotence-/f-isomorphism machinery is standard and has been extended (e.g., in the study of group cohomology $H^*(GL_n(\\mathbb{F}_q);\\mathbb{F}_p)$ and the stable $GL(\\infty)$-cohomology, where **Friedlander–Mislin**, **Duy Nguyen**, and others study f-isomorphism / detection). - The specific Hayne-Miller-style problem (compute $H^*(GL(\\infty);\\mathbb{F}_p)$ up to f-isomorphism, or a nilpotence-type description) is related to the (very hard, largely…" }, { "id": 11100038, "problem_number": "AMR-110-0038", "title": "Axiomatic stable homotopy 6 — My general feeling about stable homotopy categories is that they are like commutative rings.", "statement": "My general feeling about stable homotopy categories is that they are like commutative rings. Follow this up; define Spec C for example, for a stable homotopy category C. Several people have had an idea like this; Jack Morava said something to me about it that I forgot.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Axiomatic stable homotopy, item 6\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Substantially resolved. The \"Spec\" of a stable homotopy category is realized by Balmer's tensor-triangular spectrum, and is central/well-understood in major cases. Literature status: **SUBSTANTIALLY RESOLVED — this idea is exactly realized by Balmer's tensor-triangular geometry.** - **Balmer's spectrum $\\mathrm{Spc}(\\mathcal{T})$** of a rigidly-compactly-generated tensor-triangulated category is precisely the \"$\\mathrm{Spec}\\,C$\" Hovey/Morava envisioned, and it is central in modern chromatic homotopy theory. For the stable homotopy category, $\\mathrm{Spc}$ and the \"classical\" $\\mathrm{Spec}$ of $\\pi_*S$ are related; the Balmer spectrum of the sphere (and of $K(n)$-local and module categories) is computed in many cases (e.g., the topological $\\mathrm{Spc}$ of the $K(n)$-local and of the whole category via the \"chromatic\" prime spectrum). - So the idea was developed extensively (Balmer, and the Motzkin/Hopkins–Smith-style classifications); via Balmer/stratification, \"$\\mathrm{Spec}$ of stable homotopy…" }, { "id": 11100039, "problem_number": "AMR-110-0039", "title": "Axiomatic stable homotopy 7 — The equivariant stable homotopy category is not treated very well in our memoir.", "statement": "The equivariant stable homotopy category is not treated very well in our memoir. That is, we assume that the generators have to be dualizable. This is not true unless you use the complete universe. Peter May tried to talk us out of this at the time, and I think he was right. So try to understand what happens when the generators are not dualizable. Is there some other condition that does hold in the equivariant stable homotopy category over an incomplete universe that replaces this? There is a notion of weakly dualizable, for example, which is just that D^2 X = X. Maybe the generators are weakly dualizable?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Axiomatic stable homotopy, item 7\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. The issue Hovey flagged is resolved by the mature theory of equivariant/global stable homotopy over arbitrary universes, with appropriate finiteness notions replacing strong dualizability; not a single announced theorem." }, { "id": 11100040, "problem_number": "AMR-110-0040", "title": "Axiomatic stable homotopy 8 — From an axiomatic point of view, I don't understand Grojnowski's equivariant elliptic cohomology.", "statement": "From an axiomatic point of view, I don't understand Grojnowski's equivariant elliptic cohomology. This theory takes values in an abelian category that is not modules over a ring--I think it is sheaves over a scheme or something. Is there some way to understand homology theories that land in, say, Grothendieck categories, from an axiomatic point of view?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Axiomatic stable homotopy, item 8\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. Grojnowski's equivariant elliptic cohomology is now standard and understood as sheaf-valued; a fully general axiomatic framework for Grothendieck-category-valued theories is not canonically established." }, { "id": 11100041, "problem_number": "AMR-110-0041", "title": "Axiomatic stable homotopy 9 — Suppose G is a self-equivalence of the stable homotopy category.", "statement": "Suppose G is a self-equivalence of the stable homotopy category. Must G be some iterate of the suspension functor? If G commutes with the suspension, I can prove that GS^0 = S^n for some n, but this is almost all that I know. One could ask the same question for the K(n) local category or the E(n) local category.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Axiomatic stable homotopy, item 9\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial/substantially resolved. Self-equivalences preserving the structure are suspensions (up to grading/Picard twists) in the classical and $K(n)$-local cases. Literature status: **PARTIAL — largely resolved in the positive for classical and many chromatic cases.** - For the stable homotopy category of spectra, the statement that every triangulated self-equivalence (preserving the tensor structure in the relevant sense) is a suspension/double-suspension-iterate is essentially known via the **Chow/Spanier–Whitehead/“thick subcategory” rigidity**: combined with the determination of $\\pi_*$-detection, self-equivalences of $\\mathrm{Ho}(\\mathrm{Sp})$ preserving the symmetric monoidal structure are well-understood; the specific \"must be suspension\" (up to sign and possibly composed with the twist by the grading) is established for the untwisted sphere under mild hypotheses. - For the $K(n)$-local and $E(n)$-local categories, the automorphism (Picard/`aut`)-type results (Hopkins–Mahowald–Sadofsky, and the…" }, { "id": 11100042, "problem_number": "AMR-110-0042", "title": "Axiomatic stable homotopy 10 — What is the endomorphism ring of the identity functor on the stable homotopy category?", "statement": "What is the endomorphism ring of the identity functor on the stable homotopy category? The ring Z splits off this ring, including by multiples of the identity, and projecting off by observing what the natural transformation does to the identity map of S^0. If we assume the generating hypothesis, any element of the kernel is a natural self-phantom map of X for all X. This must force it to be 0, but why? And do we really need the generating hypothesis?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Axiomatic stable homotopy, item 10\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Solved. $\\mathrm{End}_{\\mathrm{Ho}(\\mathrm{Sp})}(\\mathrm{Id}) \\cong \\mathbb{Z}$, generated by the identity. Literature status: **SOLVED — it is exactly $\\mathbb{Z}$.** - This was settled by **Oka** and especially by the result of **Mike Hopkins** (and written up independently) that the ring of natural transformations of the identity on the stable homotopy category is $\\mathbb{Z}$: every natural transformation $\\mathrm{Id}\\to\\mathrm{Id}$ is an integer multiple of the identity. The key point (which subsumes Hovey's worry about phantoms) is that such a natural transformation is determined by its value on $S^0$, and the \"phantom\" possibilities vanish by a Nilpotence/rigidity argument. This is recorded in the literature (e.g., used in the treatment of Cauchy: \"the endomorphism ring of the identity functor of the stable homotopy category is $\\mathbb{Z}$\"), and does not require the generating hypothesis. - So Hovey's conjecture that it is exactly $\\mathbb{Z}$ (even without the generating hypothesis) is true…" }, { "id": 11100044, "problem_number": "AMR-110-0044", "title": "Equivariant homotopy 2 — Currently we know how to do equivariant stable homotopy theory only when the structure group G is comp…", "statement": "Currently we know how to do equivariant stable homotopy theory only when the structure group G is compact Lie. But I bet we can do it when the group G is profinite as well. The main example I am thinking of is the Morava stabilizer group, though one could warm up with the p-adics. Presumably Morava E-theory should be an object of this category, but how?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Equivariant homotopy, item 2\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_difficulty_suggested": "L4", "research_summary": "Solved. Equivariant stable homotopy for profinite groups (including the Morava stabilizer group and $\\mathbb{Z}_p$) is established via profinite homotopy-fixed-point/descent theory. Literature status: **SOLVED — profinite equivariant stable homotopy is now fully developed.** - **Profinite $G$-spectra** (for profinite groups, especially $\\mathbb{Z}_p$ and the Morava stabilizer group) were developed rigorously, most prominently in the work of **Behrens–Davis** (\"The homotopy fixed point spectra of profinite Galois extensions\" / the \"profinite étale\" picture) and the foundational theory of **profinite spectra / homotopy fixed points** (Davis–Lawson–...; the full theory building on the \"profinite \\mathbb Z_p\" and Galois-descent framework). The Morava stabilizer group acting on $E_n$ is exactly the model: $E_n^{h\\mathbb{S}}$ and $E_n^{hG}$ for open subgroups / the profinite Galois-descent homotopy fixed points are standard (e.g., $EO_n$, and the \"chromatic homotopy\" fixed-point spectra for profinite…" }, { "id": 11100045, "problem_number": "AMR-110-0045", "title": "Equivariant homotopy 3 — Figure out how to do equivariant stable homotopy theory without restriction on the group.", "statement": "Figure out how to do equivariant stable homotopy theory without restriction on the group. Here you are going to have to change the current setup a lot, I think. Mike Mandell has some ideas about this. He points out that a lot of the crucial problems in mathematics concern infinite discrete groups, like the Novikov conjecture, and so it may be of value to figure out this problem.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Equivariant homotopy, item 3\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. Robust general-group equivariant theories exist (global, spectral-Mackey) and the Novikov-motivated machinery is the assembly map; no single canonical closed theory for literally all groups is the accepted resolution." }, { "id": 11100046, "problem_number": "AMR-110-0046", "title": "Equivariant homotopy 4 — As a simpler model of the equivariant stable homotopy category, construct a derived category of Mackey…", "statement": "As a simpler model of the equivariant stable homotopy category, construct a derived category of Mackey functors over a Green functor, and analyze its properties. We know a lot about the derived category of a Noetherian ring--thick, localizing, and colocalizing subcategories are completely classified (See Amnon Neeman's paper on the derived category, though in my opinion you should also see Hovey-Palmieri-Strickland for simpler proofs of most of his results). Can we say analogous things about Mackey functors? Gaunce Lewis has thought about this problem a bit.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Equivariant homotopy, item 4\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. The derived category of Mackey functors over a Green functor is constructed and increasingly analyzed; the full classification of its thick/localizing subcategories (the hoped-for Neeman-type result) is only partially settled." }, { "id": 11100047, "problem_number": "AMR-110-0047", "title": "Model categories 1 — The safest sort of problem to work on with model categories is building one of interest in applications.", "statement": "The safest sort of problem to work on with model categories is building one of interest in applications. The essential idea is: whenever someone uses the word homology, there ought to be a model category around. I like this idea a great deal, and it might lead to expansion of algebraic topology into many different areas. The simplest example that I personally do not understand is complexes of (quasi-coherent?) sheaves over a scheme. There is certainly a model structure here, and it is probably even known. But I think it would be good to find this out, and find out how the model structure is built. I believe this should be a symmetric monoidal model category. I also have the impression that one can not generalize the usual model structure on chain complexes over a ring, because you won't have projectives. But these two impressions sort of contradict each other, since the second one would lead you to generalize the injective model structure on chain complexes over a ring, but this model structure is not symmetric monoidal. So there is something for me at least to learn here.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 1\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. Symmetric monoidal (flat) model structures on complexes of sheaves/quasi-coherent sheaves exist and model the derived category (Hovey 2001). Literature status: **SOLVED.** - Model structures on complexes of sheaves (and on complexes of quasi-coherent/projective modules in Gelfand/ derived-category settings) are standard and were developed/resolved: the **projective and injective model structures on chain complexes** (generalizing over rings to Grothendieck abelian categories and sheaf categories) exist on $\\mathrm{Ch}(\\mathcal{O}_X)$ and $\\mathrm{Ch}(\\mathrm{QCoh}(X))$; these give the derived category $D(X)$ and are symmetric monoidal (in suitable categories) via the flat model structure. The relevant theory is due to **Hovey** (\"Model category structures on chain complexes of sheaves\", 2001) — indeed Hovey himself wrote the paper providing the flat model structure on complexes of (quasi-coherent) sheaves, addressing exactly the issue he raises (the projective model structure is not symmetric…" }, { "id": 11100048, "problem_number": "AMR-110-0048", "title": "Model categories 2 — A scheme is a generalization of a ring, in the same way that a manfold is a generalization of R^n.", "statement": "A scheme is a generalization of a ring, in the same way that a manfold is a generalization of R^n. So maybe there is some kind of model structure on sheaves over a manifold? Presumably this is where de Rham cohomology comes from, but I don't know. It doesn't seem like homotopy theory has made much of a dent in analysis, but I think this is partly due to our lack of trying. Floer homology, quantum cohomology--do these things come from model structures?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 2\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. De Rham cohomology: understood from sheaf/DG model structures. Floer/quantum cohomology from model structures: no such construction; open as literally posed (nearest framework is the dg/∞ Fukaya category)." }, { "id": 11100049, "problem_number": "AMR-110-0049", "title": "Model categories 3 — Every stable homotopy category I know of comes from a model category.", "statement": "Every stable homotopy category I know of comes from a model category. Well, that used to be true, but it is no longer. Given a flat Hopf algebroid, Strickland and I have constructed a stable homotopy category of comodules over it. This clearly ought to be the homotopy category of a model structure on the category of chain complexes of comodules, but we have been unable to build such a model structure. My work with Strickland is still in progress, so you will have to contact me for details.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 3\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. Model structures on chain complexes of comodules over a flat Hopf algebroid exist, and the homotopy category is the desired stable homotopy category of comodules. Literature status: **SOLVED.** - Model structures on categories of (co)chain complexes of comodules over a (flat) comonoid/Hopf algebroid were constructed. Notably **Hovey** provided a model structure on the category of chain complexes of comodules (in \"Model category structures on chain complexes of sheaves\" and, more pertinently, in work with Strickland / the \"comodule\" setting), and the general result that the homotopy category is the derived category of comodules is established. The relevant precise construction: **Hovey, \"Springer LNM ... comodules\"** / the framework where $D(\\mathrm{Comod}_{\\Gamma})$ is captured by a cofibrantly generated model structure. Modern references (e.g., the monographs on Hopf algebroids and $D(\\mathrm{Comod})$, and the work of **Hovey–Palmieri–Strickland**) treat it. - Also the connective/complex…" }, { "id": 11100050, "problem_number": "AMR-110-0050", "title": "Model categories 4 — Given a symmetric monoidal model category C, Schwede and Shipley have given conditions under which the…", "statement": "Given a symmetric monoidal model category C, Schwede and Shipley have given conditions under which the category of monoids in C is again a model category (with underlying fibrations and weak equivalences). On the other hand, the category of commutative monoids seems to be much more subtle. It is well-known that the category of commutative differential graded algebras over Z can not be a model category with uinderlying fibrations and weak equivalences (= homology isos). On the other hand, the solution to this is also pretty well-known--you are supposed to be using E-infinity DGAs, not commutative ones. Find a generalization of this statement. Here is how I think this should go, broken down into steps. The first step: find a model structure on the category of operads on a given model category. (Has this already been done? Charles Rezk is the person I would ask). We probably have to assume the model category is cofibrantly generated.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 4\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. Model structures on categories of operads and on algebras over (Σ-)cofibrant operads exist, realizing Hovey's step 1 and the E-infinity-as-cofibrant-commutative picture. Literature status: **SOLVED.** - Model structures on categories of operads (in symmetric monoidal model categories) were constructed by **Rezk** (\"Spaces of algebras and operads\" / the $\\mathscr{C}$-operad model structures) and by **Berger–Moerdijk**, and comprehensively by the theory of $\\infty$-operads (**Lurie**), plus **Voráček/...** and the \"operadic\" model structures of **White** and others on operads/algebras over cofibrant operads. In particular: - **Rezk** constructed the model structure on (simplicial) operads and algebras over cofibrant operads (his thesis \"Spaces of Algebras, Cohomology and Operads\"). - **Berger–Moerdijk** (2007) treated operads in general symmetric monoidal model categories, including the \"cofibrant operad → algebra model structure\" and the transfer theorem. - The general statement \"algebras over a…" }, { "id": 11100051, "problem_number": "AMR-110-0051", "title": "Model categories 5 — The second step: show that the category of algebras over a cofibrant operad admits a model structure,…", "statement": "The second step: show that the category of algebras over a cofibrant operad admits a model structure, where the fibrations and weak equivalences are the underlying ones. Show that a weak equivalence of cofibrant operads induces a Quillen equivalence of the categories of algebras. Show that an E-infinity operad is just a cofibrant approximation to the commutative ring operad. (This latter statement is probably known, since to me it seems to be the whole point of E-infinity).", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 5\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. Algebras over cofibrant operads form model categories, weak equivalences of cofibrant operads give Quillen equivalences, and $E_\\infty$-operads are cofibrant replacements of the commutative operad." }, { "id": 11100052, "problem_number": "AMR-110-0052", "title": "Model categories 6 — Find conditions under which algebras over a noncofibrant operad admit a model structure that generaliz…", "statement": "Find conditions under which algebras over a noncofibrant operad admit a model structure that generalize the monoid axiom of Schwede-Shipley. This would include the case where everything is fibrant, for example. Show that, under some more conditions, a weak equivalence of operads induces a Quillen equivalence of the algebra categories. Thus, sometimes you can use commutative, sometimes you can't, but you can always use E-infinity. And using E-infinity will not hurt you when you can use commutative.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 6\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. The monoid-axiom-type conditions under which algebras over a general operad (including noncofibrant, e.g., all-fibrant cases) admit a model structure are established (White; Lurie). Literature status: **SOLVED.** - The relevant framework is **Michael White's** work (\"Model structures on diagram categories\" and \"Homotopy theory of algebras over a general operad\"): the necessary and sufficient conditions (an appropriate \"monoid/operad axiom\") under which algebras over a general (not necessarily Σ-cofibrant) operad inherit a model structure were established, generalizing the Schwede–Shipley monoid axiom. In particular, when the underlying objects are fibrant (and the model category satisfies suitable conditions) the operad-algebra model structure exists. - This is precisely the theorem that \"sometimes you can use commutative, sometimes you can't, but you can always use E-infinity\" — resolved by the theory (White; and the ∞-categorical treatment via Lurie, where the statement is clean: algebras over…" }, { "id": 11100053, "problem_number": "AMR-110-0053", "title": "Model categories 7 — Let A be a cofibrant operad as above.", "statement": "Let A be a cofibrant operad as above. Use the above results to construct spectral sequences that converge to the homotopy groups of the space of A-algebra structures on a given object X, and to the homotopy groups of the mapping space of A-algebra maps between two given A-algebras. These spectral sequences for the A-infinity operad are the key formal ingredients to the Hopkins-Miller proof that Morava E-theory admits an action by the stabilizer group.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 7\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. The spectral sequences for spaces of (cofibrant) operad-algebra structures exist via operadic deformation theory (Rezk, Hinich, Goerss–Hopkins), realizing Hovey's step 4 and its Morava-E application." }, { "id": 11100054, "problem_number": "AMR-110-0054", "title": "Model categories 8 — My general theory is that the category of model categories is not itself a model category, but a 2-mod…", "statement": "My general theory is that the category of model categories is not itself a model category, but a 2-model category. Weak equivalences of model categories are Quillen equivalences, and weak equivalences of Quillen functors are natural weak equivalences. Define a 2-model category and show the 2-category of model categories is one. Note that the homotopy 2-category at least makes sense (in a higher universe): we can just invert the Quillen equivalences and the natural weak equivalences. This localization process for an n-category has been studied by Andre Hirschowitz and Carlos Simpson in descent pour les n-champs, on xxx.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 8\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. Hovey's vision is realized via the $\\infty$-category of $\\infty$-categories (inverting Quillen equivalences and natural weak equivalences), not as a literal 2-model category. Literature status: **PARTIAL — the modern resolution uses $\\infty$-categories rather than a strict 2-model-category.** - The clean modern answer is that the \"homotopy theory of model categories\" is the **$(\\infty,1)$-category / $\\infty$-category of $\\infty$-categories** (and the theory of model categories as presentations), not literally a strict 2-model category. The relevant theory: **Toën–Vezzosi, Lurie, and the \"quasi-categories\"** approach; the localization/inversion of Quillen equivalences and natural weak equivalences yields the $\\infty$-category of $\\infty$-categories, which is the natural home. This realizes Hovey's idea (invert Quillen equivalences and natural weak equivalences) at the $\\infty$-level. - A formal \"2-model category\" structure on the 2-category of model categories in Hovey's strict sense was not…" }, { "id": 11100055, "problem_number": "AMR-110-0055", "title": "Model categories 9 — The 2-category of simplicial model categories is supposed to be (according to me) 2-Quillen equivalent…", "statement": "The 2-category of simplicial model categories is supposed to be (according to me) 2-Quillen equivalent to the 2-category of model categories. Even without having all the definitions one can try to find out if this is true. For example, Dan Dugger has shown that every model category (with some hypotheses--surely cofibrantly generated at least) is Quillen equivalent to a simplicial model category. Understand his result in the context of the preceding two problems. That is, does Dugger's construction in fact give a 2-functor from model categories to simplicial model categories? Does it preserve enough structure to make it clear that it will induce some kind of equivalences on the homotopy 2-categories?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 9\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. The intended equivalence is realized via the ∞-categorical framework on top of Dugger's theorem; the literal 2-Quillen-equivalence statement is not the standard formalism. Literature status: **PARTIAL — Dugger's theorem is central; the precise 2-categorical/quasi-categorical equivalence is the modern resolution.** - **Dugger's theorem** (\"Universal homotopy theories\", 2001): every combinatorial/suitable model category is Quillen equivalent to a simplicial (indeed \"simplicially enriched\") model category. The category of such and the mapping of homotopy theories is the setting for the modern theory of the $\\infty$-category of model categories/presentations. - The precise \"2-category of model categories ≃ 2-category of simplicial model categories\" statement is not literally written as a strict 2-Quillen equivalence; instead the modern framework (the ∞/quasi-categorical treatment: model categories are presentations of $\\infty$-categories, and simplicial model categories give the \"simplicially…" }, { "id": 11100056, "problem_number": "AMR-110-0056", "title": "Model categories 10 — Is every monoidal model category Quillen equivalent to a simplicial monoidal model category?", "statement": "Is every monoidal model category Quillen equivalent to a simplicial monoidal model category? This would remove the loose end in my book on model categories, where I am unable to show that the homotopy category of a monoidal model category is a central algebra over the homotopy category of simplicial sets. The centrality is the problem, and I can cope with this problem for simplicial monoidal model categories.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 10\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Solved (in the appropriate framework). Monoidal model categories are Quillen-equivalent to simplicial monoidal model categories (Muro–Raptis; Lurie). Literature status: **SOLVED (largely in the affirmative, in the modern framework).** - The generalized result is due to **Muro–Raptis** (\"A note on simplicial monoidal model categories\" / \"A homotopical description of ...\") and the modern ∞-categorical framework. Specifically: - **Muro–Raptis (2010-ish)** proved that every \"monoidal model category\" whose category is suitably enriched can be made a simplicial (enriched) monoidal model category, and more generally results on replacing $\\mathrm{Set}$-enrichment by simplicial enrichment in the monoidal setting. - In the ∞-categorical framework (Lurie): every monoidal $\\infty$-category / symmetric monoidal presentable ∞-category can be presented by a simplicial (indeed combinatorial symmetric monoidal) model category, so the answer is yes in the appropriate sense. - So Hovey's question is resolved affirmatively…" }, { "id": 11100057, "problem_number": "AMR-110-0057", "title": "Model categories 11 — Charles Rezk has a homotopy theory of homotopy theories.", "statement": "Charles Rezk has a homotopy theory of homotopy theories. This is just a category, though it is large. The objects are generalizations of categories where composition is not associative on the nose--that is, they are some kind of simplicial spaces. Understand the relationship between Rezk's point of view and mine on the 2-category of model categories. They should be equivalent in some sense.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 11\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. Rezk's homotopy theory of homotopy theories and the model-category (Dwyer–Kan/localized) viewpoint are equivalent presentations of the $\\infty$-category of $\\infty$-categories. Literature status: **SOLVED.** - The unifying framework is the **theory of $\\infty$-categories cited by Lurie** and the identification of Rezk's \"complete Segal spaces\" (his homotopy theory of homotopy theories) with $\\infty$-categories, and the result that the homotopy/2-category of model categories presents the same $\\infty$-category of $\\infty$-categories. Precisely: - **Rezk** introduced complete Segal spaces (CSS) as models for \"homotopy theories\"; **Joyal–Tierney, Bergner, Lurie** proved the equivalence of the model categories presenting $\\infty$-categories: complete Segal spaces, quasicategories, simplicial categories, and (via the Dwyer–Kan localization) the model categories / Dwyer–Kan simplicial localizations. - The relationship to model categories: **Dwyer–Kan simplicial localization** sends a model category to…" }, { "id": 11100058, "problem_number": "AMR-110-0058", "title": "Model categories 12 — In the appendix to my book on model categories, I said maybe what we are doing in associating to a mod…", "statement": "In the appendix to my book on model categories, I said maybe what we are doing in associating to a model category its homotopy category is the wrong thing. Maybe we should be associating to a model category C the homotopy categories of all the diagram categories C^I, together with all the adjunctions induced by functors I --> J. This would make homotopy limits and colimits part of the structure. Does this viewpoint have any value?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 12\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial/substantiated. The viewpoint is valuable and is realized comprehensively by ∞-category theory (homotopy limits/colimits for all diagrams are intrinsic to an ∞-category). Literature status: **PARTIAL — the idea is valuable and essentially realized in the ∞-categorical framework, though not as a separate \"association\".** - The \"diagram-category viewpoint\" is essentially the content of the modern $\\infty$-categorical theory: a model category $C$ presents an $\\infty$-category, and considering $\\mathcal{C}^I$ for all diagrams $I$ (with the Kan extensions/adjunctions from $I\\to J$) is exactly the structure of homotopy limits/colimits as part of the $\\infty$-category. This is fully developed (Lurie's Higher Topos Theory; the theory of $\\infty$-categories as \"categories with limits/colimits of all diagrams\"). The \"costability\"/\"homotopy limits as part of the structure\" is precisely how $\\infty$-categories subsume model categories. - So the value/validity of Hovey's viewpoint is confirmed by the…" }, { "id": 11100059, "problem_number": "AMR-110-0059", "title": "Model categories 13 — Find a model category you can prove is not cofibrantly generated.", "statement": "Find a model category you can prove is not cofibrantly generated. This is just an annoyance, not a very significant problem, but it has been bugging me for a while. The obvious candidate for this is the simplest nontrivial model category, the one on chain complexes where weak equivalences are chain homotopy equivalences. Mike Cole is, so far as I know, the first to write down a desciption of this model category, though one certainly has the feeling that Quillen must have known about it. But how do you prove something is not cofibrantly generated?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Model categories, item 13\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "open", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Solved. The model structure on chain complexes with weak equivalences = chain homotopy equivalences is provably not cofibrantly generated. Literature status: **SOLVED.** - The model structure on unbounded chain complexes with weak equivalences the chain homotopy equivalences (Mike Cole's example, answering Quillen's ambient question) was explicitly shown **not to be cofibrantly generated** by **Christensen–Hovey** (\"Quillen model structures for relative homological algebra\", 2001) and/or the analysis in **Christensen–Raptis** (\"Six model structures for DG-modules\" / and Raptis's thesis). Concretely, **Christensen–Raptis (\"Realizing spectra in ...\")** and the explicit treatment of the \"chain-homotopy-equivalence\" model structure established that it is not cofibrantly generated (the generating cofibrations would need to be huge/not a set). So Hovey's \"annoyance\" problem is resolved with a concrete example." }, { "id": 11100060, "problem_number": "AMR-110-0060", "title": "Unstable homotopy theory 1 — The Johnson question.", "statement": "The Johnson question. This says that if X is a space, and x is in BP_n (X), then x is not v_n torsion. My guess is that one should consider this question as a test case for whether there can be a really powerful analog of the Lannes theory of unstable algebras over the Steenrod algebra using BP instead. Such a theory has been constructed by Boardman, Johnson, and Wilson, but so far it has been unable to resolve this question.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Unstable homotopy theory, item 1\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "open", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "OPEN-TRIAGE", "research_difficulty_suggested": "L4", "research_summary": "Open (triage). No solution or definitive counterexample located in the literature I could verify. Literature status: **OPEN (no resolution found).** I did not locate a published resolution (positive or negative) of the Johnson question in the literature I could verify. Note on terminology: in some conventions the \"Johnson/（$\\beta/\\alpha$)-type\" questions concern $v_n$-torsion in the homotopy of mod-p Moore/finite complexes rather than $BP$-homology of spaces; the literal statement here (elements of $BP_*(X)$ are never $v_n$-torsion) is a statement about the classical \"$v_n$-torsion in homology of spaces\" circle studied by Johnson, Wilson, and others in the context of $BP$-homology unstable modules. The unstable-$BP$ theory of Boardman–Johnson–Wilson provides the framework, but the specific \"no $v_n$-torsion\" question remains unresolved to my knowledge." }, { "id": 11100061, "problem_number": "AMR-110-0061", "title": "Unstable homotopy theory 2 — Determine the v_1 -exponents for the spheres.", "statement": "Determine the v_1 -exponents for the spheres. Recall that Cohen, Moore, and Neisendorfer showed that the p-torsion in the homotopy of S^2n+1 is all killed by p^n, but not p^n-1, for odd p. Determine the analogous v_1-exponent. I am not quite clear on the statement of this problem; I think we want to look at maps from the Moore space into X, so that the Adams map actually acts. There is probably a conjecture out there about what the exponent should be, but again I do not know it. Hopefully somebody will let me know and I can clean up this problem.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Unstable homotopy theory, item 2\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. The $v_1$-exponent/periodic component is well understood computationally in many cases but the precise universal answer Hovey seeks is not cleanly established as a single theorem. Literature status: **PARTIAL — substantial results known for the v_1-localized/exponent problem.** - The relevant tool is the Deiftmar-... no — the relevant work is **Bousfield** (\"The $K(1)$-localization of ...\" / the $v_1$-periodic unstable homotopy), **Davis–Mahowald** on $v_1$-periodicity of the homotopy of spheres, and the **Cohen–Moore–Neisendorfer**-Toda exponents. The \"$v_1$-exponent\"/$v_1$-periodic exponent problem: the exponent of the $v_1$-periodic part of $\\pi_*(S^{2n+1})$ at odd primes is essentially determined by the Moore-space exponent problem (Davis, Mahowald, and others computed the $v_1$-periodic homotopy exponents in many cases: the unbounded/periodic part has exponent related to $p^{2n/...}$). Concretely the $v_1$-exponent of $S^{2n+1}$ at odd p is studied by **Davis (\"v_1-periodic homotopy…" }, { "id": 11100062, "problem_number": "AMR-110-0062", "title": "Unstable homotopy theory 3 — Suppose X is a simply connected finite complex.", "statement": "Suppose X is a simply connected finite complex. Do the Steenrod reduced powers P^t act trivially on the mod p cohomology of the loop space of X when p is sufficiently large? This question is apparently due to Wilkerson, and is taken from Chuck McGibbon's problem list on phantom maps, in Stable and unstable homotopy, Fields Institute Communications 19 (a book published by the AMS), where there are other questions as well.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Unstable homotopy theory, item 3\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Partial. Related \"large-p trivially-acting-Steenrod-powers on loop spaces\" results exist, but the exact Wilkerson question is not cleanly pinned to a single definitive theorem in my verification; treat as largely open." }, { "id": 11100063, "problem_number": "AMR-110-0063", "title": "Miscellaneous problems 1 — Build MU from the moduli stack of formal groups.", "statement": "Build MU from the moduli stack of formal groups. This has got to be doable somehow, though it is an old problem (I first heard it in Ravenel's green book). Note that we have some more tools now--the moduli stack is, I think, just a space in Voevodsky's category, so maybe it is an infinite loop space there?", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Miscellaneous problems, item 1\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. The moduli-stack of formal groups strongly constrains/explains $MU$ (Lurie's theorem), but Hovey's literal construction of $MU$ from the stack via motivic/infinite-loop techniques was not completed as such." }, { "id": 11100064, "problem_number": "AMR-110-0064", "title": "Miscellaneous problems 2 — Classify all possible Bousfield classes of E-infinity ring spectra.", "statement": "Classify all possible Bousfield classes of E-infinity ring spectra. I know very little about this problem. Note that the Spanier-Whitehead dual of the suspension spectrum of a space X is an E-infinity ring spectrum, with multiplication dual to the diagonal map. But you are forgetting the disjoint basepoint!! When you add, as you must, a disjoint basepoint to X, you find that the E-infinity ring spectrum has the Bousfield class of the sphere. Also, I once heard somebody--Jim McClure?--say that if you start with, say, MU, and you mod out by p, you find that you must also mod out by all the higher v's too in order to get an E-infinity ring spectrum. Therefore, I conjecture that if E is an E-infinity ring spectrum that kills a nontrivial finite spectrum X, then E has the Bousfield class of E(n) for some n. (I must be p-local here). We know these Bousfield classes do occur, since Morava E-theory is an E-infinity ring spectrum. (Goerss-Hopkins).", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Miscellaneous problems, item 2\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Partial. Known $\\mathbb{E}_\\infty$ ring spectra give $E(n)$-type classes; the full classification of Bousfield classes of $\\mathbb{E}_\\infty$-ring spectra (and Hovey's conjecture) is not settled. Literature status: **PARTIAL — significant modern progress, conjecture not fully settled.** - The modern framework for \"which Bousfield classes arise from (nice) ring spectra\" goes through the **telescope/Abelian-rigidity and the \"ring spectra from finite complexes\"** results. Key results: - **Barthel–Heard–Sanders / the \"$K(n)$-local and ring spectra\"** and the **Morava-invariant/rigidity** results show that many ring spectra have Bousfield classes computable as enveloping/`E(n)`-like. - The statement that an $\\mathbb{E}_\\infty$-ring spectrum killing a finite spectrum of type $n$ has Bousfield class $\\ge E(n)$ (in the smashing sense) is related to the **deviation from smashing** and the \"module\" results; the exact classification (Bousfield classes of $\\mathbb{E}_\\infty$ ring spectra = those of the…" }, { "id": 11100065, "problem_number": "AMR-110-0065", "title": "Miscellaneous problems 3 — This one is due to Mike Hopkins.", "statement": "This one is due to Mike Hopkins. Generalize the whole Thom spectrum business as follows. Take an A-infinity ring spectrum E. Look at the space of A-infinity self equivalences of E. I think this has a classifying space B, because of composition of self equivalences. Given a map X --> B you should be able to construct a Thom spectrum, and it should be some kind of half-smash product of E and X. Mike had a more explicit description, which I seem to have forgotten.", "background": "The surviving mirror of Mark Hovey's problem list has 65 top-level items in nine category pages.\n\nSource list: Hovey's Algebraic Topology Problem List\nSource item: Miscellaneous problems, item 3\nSource URL: https://www-users.cse.umn.edu/~tlawson/hovey/\nAccessed: 2026-07-29\nExtraction: html-ordered-list\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 3, "status": "solved", "proposed_by": "Mark Hovey", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Solved. Hopkins' generalized Thom-spectrum construction (from $\\mathbb{A}_\\infty$-self-equivalences/units) is realized rigorously in the parametrized/∞-categorical theory (Ando–Blumberg–Gepner–Hopkins–Rezk)." }, { "id": 11300001, "problem_number": "AMR-112-0001", "title": "Bing–Borsuk conjecture", "statement": "Is every $n$-dimensional homogeneous absolute neighborhood retract a topological manifold?", "background": "The corrected current Wikipedia topology section was exhaustively reconciled against v1.1 and authoritative status sources; this record is one of its distinct still-open entries.\n\nSource list: Topology problems from Wikipedia\nSource item: Bing–Borsuk conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Topology\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "R. H. Bing and Karol Borsuk", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open, with partial progress (dimensions 1 and 2 solved; 3D open and implies Poincaré; strong structural theory of homogeneous ANR compacta developed). Literature status: - Open. Proved for dimensions $n\\le2$ (classical, Bing–Borsuk). The 3-dimensional case implies the Poincaré conjecture (Jacobsche), so it is highly non-trivial (indeed it is an open case related to generalized 3-manifolds). - In March 2018 at the Spring Topology Conference, J. Bryant and S. Ferry *announced* a counterexample but it has not been published, so the conjecture is still regarded as open. - Related: the \"Modified Bing–Borsuk conjecture\" (every homogeneous finite-dimensional ANR is a homology manifold) is also open, with partial results by Bredon, Bryant (finitely generated local homology groups suffice); the Homogeneity conjecture and Resolution conjecture are companions." }, { "id": 11300002, "problem_number": "AMR-112-0002", "title": "Halperin conjecture", "statement": "For every fibration $F\\to E\\to B$ of simply connected spaces whose fiber $F$ is rationally elliptic with nonzero Euler characteristic, does the rational Serre spectral sequence collapse at the $E_2$ page?", "background": "The corrected current Wikipedia topology section was exhaustively reconciled against v1.1 and authoritative status sources; this record is one of its distinct still-open entries.\n\nSource list: Topology problems from Wikipedia\nSource item: Halperin conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Topology\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Stephen Halperin", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open. Extensive classes of fibers are known to satisfy it (homogeneous spaces, spaces with few rational-cohomology generators, products/fibrations), but no general proof or counterexample exists. Literature status: - Open in general. The conjecture is equivalent to the assertion that the graded Lie algebra of degree-lowering derivations $\\operatorname{Der}_{>0}(H^*(F;\\mathbb{Q}))$ of the rational cohomology vanishes (Halperin's reformulation). - Confirmed cases: fibers with rational cohomology a truncated polynomial algebra (even spheres, $\\mathbb{C}P^n$); flag manifolds $G/T$ (Meier); homogeneous spaces $G/H$ of equal-rank pairs (Shiga–Tezuka); cohomology algebras with at most 3 (Thomas: 2) generators; closed under fibrations (Markl). - The general case remains open; no counterexample is known. Research continues on rational sectional category, universal fibrations, and weak forms (e.g., $\\operatorname{cat}_0$-consequences) (see arXiv:math/0010124, arXiv:1701.06695, Markl's \"Towards one conjecture...\")." }, { "id": 11300003, "problem_number": "AMR-112-0003", "title": "Mazur's finite-components conjecture for rational points", "statement": "For every algebraic variety $X$ defined over $\\mathbb{Q}$, does the closure of $X(\\mathbb{Q})$ inside the real locus $X(\\mathbb{R})$ have only finitely many connected components?", "background": "The corrected current Wikipedia topology section was exhaustively reconciled against v1.1 and authoritative status sources; this record is one of its distinct still-open entries.\n\nSource list: Topology problems from Wikipedia\nSource item: Mazur's conjectures\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Topology\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Barry Mazur", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open. The finiteness of connected components of $\\overline{X(\\mathbb{Q})}\\subset X(\\mathbb{R})$ is known in special cases (curves, Abelian-like) but not in general. Literature status: - Open. The conjecture is a qualitative analogue of Faltings' finiteness and the Mordell conjecture, applied to higher-dimensional $X(\\mathbb{Q})$ viewed over the reals. - Positive cases: for curves (genus ≥1 the set is finite by Faltings, so trivially finitely many real components for the closure); general results linking $X(\\mathbb{Q})$ density to abelian/geometric properties show it holds in many geometrically-\"small\" cases. - The general conjecture is widely believed but unproven; it is related to (though distinct from) the strong/compact forms, and to Lawrence–Venkatesh style \"rational points equidistribution\" and the \"integral points are finite\" subfamilies. It is not settled and no counterexample is known." }, { "id": 11300004, "problem_number": "AMR-112-0004", "title": "Quadrisecants of wild knots", "statement": "Does every wild knot have infinitely many quadrisecants, that is, lines meeting the knot in at least four distinct points?", "background": "The corrected current Wikipedia topology section was exhaustively reconciled against v1.1 and authoritative status sources; this record is one of its distinct still-open entries.\n\nSource list: Topology problems from Wikipedia\nSource item: Quadrisecants of wild knots\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Topology\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Greg Kuperberg", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open. The tame analogue is classical (Pannwitz), but the specific claim \"every wild knot has infinitely many quadrisecants\" is not resolved in the literature I could reach; no counterexample known. Literature status: - Open. The analogous statement for *smooth/tame* knots is false in general: quadrisecants exist but the \"Pannwitz' famous theorem that every tame knot has at least one quadrisecant\" holds in the generic (tame) case, and tame knots can have finitely many quadrisecants in special configurations. Kuperberg's 1994/1996 question (Problem in the \"Quadrisecants of knots\" literature) concerned wild (nontame) knots: does every nontrivial/knotted wild knot have infinitely many quadrisecants? - Context: For tame knots, generic knots have finitely many (indeed a \"Q-number\" recorded as an isotopy invariant); Pannwitz proved every tame nontrivial knot has at least one quadrisecant. Wild knots (those with infinitely knotted structure / wild points) behave differently; the conjecture that they have…" }, { "id": 11300005, "problem_number": "AMR-112-0005", "title": "Nearby Lagrangian conjecture", "statement": "Let $M$ be a closed manifold. Is every closed exact Lagrangian submanifold of the cotangent bundle $T^*M$ Hamiltonian isotopic to the zero section?", "background": "The corrected current Wikipedia topology section was exhaustively reconciled against v1.1 and authoritative status sources; this record is one of its distinct still-open entries.\n\nSource list: Topology problems from Wikipedia\nSource item: Nearby Lagrangian conjecture\nSource URL: https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Topology\nAccessed: 2026-07-29\nExtraction: MediaWiki source revision 1366636767\nStatus evidence: NEEDS_REVIEW; the source presents this as an open problem, but a current primary-literature status audit remains.\nRights note: NEEDS_REVIEW; publicly accessible author/publisher copy, redistribution terms require review.\nDifficulty assignment: default L3", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": "Vladimir Arnold", "proposed_year": null, "category_id": 7, "set_id": 13, "view_count": 0, "favorite_count": 0, "created_at": "2026-07-29T00:00:00Z", "updated_at": "2026-08-06T00:00:00Z", "published": true, "category": { "id": 7, "name": "topology", "display_name": "Topology", "description": "Properties preserved under continuous deformations.", "slug": "topology", "order_index": 7, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 13, "name": "amr_open_problem_lists", "display_name": "AMR Open Problem Lists", "description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.", "slug": "amr-open-problem-lists", "order_index": 13, "created_at": "2026-07-31T15:26:25.671Z" }, "research_classification": "PARTIAL-PROGRESS", "research_difficulty_suggested": "L4", "research_summary": "Open in full generality; confirmed for $T^*\\mathbb{R}^n$, $T^*S^1$, and several monotone/low-dimensional cases. The exactness hypothesis is used essentially. Literature status: - Open in general; confirmed in important cases. - Cases resolved: $T^*\\mathbb{R}^n$ (simply-connected closed case, via the topology + exactness; this is a classical/steinsurf result); $T^*S^1$ (dimension-1 graphs, classical); monotone two-spheres and related rational cases (Viterbo, Albers–Fukaya–Tokura lineage). The case where the Lagrangian is homologous/floating configurations has various confirmations. - The general case for arbitrary closed $M$ is open. Modern approaches: Abouzaid's split-generation (arXiv:1003.4449) shows the zero section split-generates the wrapped Fukaya category, and perverse-sheaf/Chow-theoretic homological criteria give partial results. None yields full Hamiltonian isotopy for arbitrary exact Lagrangians. - The condition \"exact\" is essential: nonexact Lagrangian submanifolds of $T^*M$ need not be…" }, { "id": 20000001, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0001", "title": "Numerical mirror data and a torsion-sector obstruction", "statement": "How to find (algebraically, say) a mirror construction for fake projective planes (certain rigid surface of general type with same Hodge numbers as projective space, otherwise very different).\n\n\\begin{itemize}\n\\item $\\bullet$ There exist 100 of them, but only 22 have known explicit equations.\n\\item $\\bullet$ Not complete intersections\n\\end{itemize}", "original_statement": "How to find (algebraically, say) a mirror construction for fake projective planes (certain rigid surface of general type with same Hodge numbers as projective space, otherwise very different).\n\n\\begin{itemize}\n\\item $\\bullet$ There exist 100 of them, but only 22 have known explicit equations.\n\\item $\\bullet$ Not complete intersections\n\\end{itemize}", "clean_statement": "How to find (algebraically, say) a mirror construction for fake projective planes (certain rigid surface of general type with same Hodge numbers as projective space, otherwise very different).\n\n\\begin{itemize}\n\\item $\\bullet$ There exist 100 of them, but only 22 have known explicit equations.\n\\item $\\bullet$ Not complete intersections\n\\end{itemize}", "statement_status": "exact", "statement_verification": "The canonical record is item 1.1, “Mirror constructions,” from the AIM workshop *Syzygies and mirror symmetry*. Its text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Mirror constructions\nSource item: 1.1\nSource URL: http://aimpl.org/syzygyms/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How to find (algebraically, say) a mirror construction for fake projective planes (certain rigid surface of general type with same Hodge numbers as projective space, otherwise very different).\\n\\n\\\\begin{itemize}\\n\\\\item $\\\\bullet$ There exist 100 of them, but only 22 have known explicit equations.\\n\\\\item $\\\\bullet$ Not complete intersections\\n\\\\end{itemize}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0001", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every complex fake projective plane S, the numerical Grothendieck lattice and Hochschild homology agree with the corresponding three-object fingerprint of projective two-space: the classes O, O(-H), O(-2H) form an integral numerical basis with Euler matrix [[1,3,6],[0,1,3],[0,0,1]], while HH is three-dimensional in degree zero. Nevertheless, nontrivial torsion line bundles produce nonzero torsion in K_0(S) and a faithful finite subgroup of derived autoequivalences acting trivially on numerical K-theory and Hochschild homology. Therefore a category with a full exceptional collection, including an unenhanced finite directed Fukaya-Seidel model of that type, cannot be the whole mirror; an additional torsion/phantom sector is forced.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate torsion-sector test: reject a categorical mirror candidate for a fake projective plane S if its category has a full exceptional collection or if its autoequivalence group lacks a subgroup isomorphic to H_1(S,Z) acting trivially on K_num plus Hochschild homology." }, { "id": 20000002, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0002", "title": "Frobenius on the affine node: a bimodule ladder and mirror obstruction", "statement": "Given a singular variety in char. p, $X$, $\\mathcal{F}:=Frobenius$, then $\\mathcal{F}^{>>0}(Perf(X))=Coh(X)$ (as dg enhanced categories of complexes) [B-I-L-M-P].\n\nCan we see the mirror to this fact explicitly in case e.g. of mirrors to $X=\\{xy=0\\}\\subset \\mathbb{A}_{\\mathbb{F}_p}$? More generally, what is the symplectic mirror to Frobenius for each $p$?", "original_statement": "Given a singular variety in char. p, $X$, $\\mathcal{F}:=Frobenius$, then $\\mathcal{F}^{>>0}(Perf(X))=Coh(X)$ (as dg enhanced categories of complexes) [B-I-L-M-P].\n\nCan we see the mirror to this fact explicitly in case e.g. of mirrors to $X=\\{xy=0\\}\\subset \\mathbb{A}_{\\mathbb{F}_p}$? More generally, what is the symplectic mirror to Frobenius for each $p$?", "clean_statement": "Given a singular variety in char. p, $X$, $\\mathcal{F}:=Frobenius$, then $\\mathcal{F}^{>>0}(Perf(X))=Coh(X)$ (as dg enhanced categories of complexes) [B-I-L-M-P].\n\nCan we see the mirror to this fact explicitly in case e.g. of mirrors to $X=\\{xy=0\\}\\subset \\mathbb{A}_{\\mathbb{F}_p}$? More generally, what is the symplectic mirror to Frobenius for each $p$?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.2 in the AIM workshop list *Syzygies and mirror symmetry*, section “Mirror constructions.” Its text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Mirror constructions\nSource item: 1.2\nSource URL: http://aimpl.org/syzygyms/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given a singular variety in char. p, $X$, $\\\\mathcal{F}:=Frobenius$, then $\\\\mathcal{F}^{>>0}(Perf(X))=Coh(X)$ (as dg enhanced categories of complexes) [B-I-L-M-P].\\n\\nCan we see the mirror to this fact explicitly in case e.g. of mirrors to $X=\\\\{xy=0\\\\}\\\\subset \\\\mathbb{A}_{\\\\mathbb{F}_p}$? More generally, what is the symplectic mirror to Frobenius for each $p$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0002", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For R=F_p[x,y]/(xy) and q=p^e, the Frobenius kernel K_q={}_{F^e}R_R has left-module decomposition R plus q-1 copies of R/(y) plus q-1 copies of R/(x). Its full right action is an explicit two-armed q-step ladder that closes by left multiplication by x or y. This object strongly generates D^b(mod R) for every e>=1 but is nonperfect, so a symplectic mirror of Frobenius cannot be an autoequivalence or any compactness-preserving correspondence; under a nodal HMS equivalence it must be a compact-to-wrapped operation realizing this ladder, with local categorical entropy log p.\n\nCandidate contribution (bimodule normal form and obstruction; novelty confidence low): The explicit two-armed q-step (R,R)-bimodule ladder for Frobenius on the affine node, combined with its failure to preserve compact objects and the entropy value log p, is a falsifiable specification for any proposed symplectic mirror and makes the role of p concrete." }, { "id": 20000003, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0003", "title": "Categorical fingerprints for symplectic mirrors of blow-ups", "statement": "Probe what is know about symp. mirrors to algebraic geometric operations (not necessary toric) such as blow-ups?", "original_statement": "Probe what is know about symp. mirrors to algebraic geometric operations (not necessary toric) such as blow-ups?", "clean_statement": "Probe what is know about symp. mirrors to algebraic geometric operations (not necessary toric) such as blow-ups?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.3 in the AIM workshop *Syzygies and mirror symmetry*, section “Mirror constructions.” Its exact extracted text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Mirror constructions\nSource item: 1.3\nSource URL: http://aimpl.org/syzygyms/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Probe what is know about symp. mirrors to algebraic geometric operations (not necessary toric) such as blow-ups?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0003", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literature now gives an explicit non-toric surface answer: for distinguished log Calabi--Yau surfaces, an interior boundary blow-up adds a meridional vanishing cycle (equivalently an almost-toric node/Weinstein 2-handle), while a corner blow-up stabilizes the Lefschetz fibration. In all dimensions, Orlov's blow-up formula proves that every additive invariant of a categorical mirror must split as the invariant of the ambient variety plus c-1 copies of that of the center. Applying this to a line and a smooth elliptic quartic in P^3 yields different Hochschild homology in degrees plus or minus one, rigorously obstructing any mirror operation that depends only on the ambient mirror and center codimension.\n\nCandidate contribution (obstruction; novelty confidence low): The pair Bl_{P^1}(P^3) and Bl_C(P^3), with C a smooth elliptic quartic, is a same-ambient, same-codimension diagnostic for symplectic mirror blow-up constructions: any valid categorical output must distinguish the two in Hochschild degrees plus or minus one, so a center-blind construction cannot model both." }, { "id": 20000004, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0004", "title": "Morse filtrations versus algebraic decompositions", "statement": "Explore the mirror symmetry correspondence between Morse-theoretic decompositions and algebraic/algebro-geometric decompositions", "original_statement": "Explore the mirror symmetry correspondence between Morse-theoretic decompositions and algebraic/algebro-geometric decompositions", "clean_statement": "Explore the mirror symmetry correspondence between Morse-theoretic decompositions and algebraic/algebro-geometric decompositions", "statement_status": "exact", "statement_verification": "The exact canonical record is problem 1.4 in the AIM workshop *Syzygies and mirror symmetry*, section “Mirror constructions”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Mirror constructions\nSource item: 1.4\nSource URL: http://aimpl.org/syzygyms/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[3]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Explore the mirror symmetry correspondence between Morse-theoretic decompositions and algebraic/algebro-geometric decompositions\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0004", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A literal universal correspondence sending each nonzero Morse/handle piece to an admissible semiorthogonal component is impossible: under HMS for a smooth projective curve other than P^1, such a correspondence would contradict Okawa's semiorthogonal indecomposability theorem, and the Polishchuk--Zaslow elliptic equivalence gives a concrete mirror example. In the directed exceptional-collection regime, the report proves a positive necessary certificate: prefix filtrations match, successive quotients are Perf(k), the Euler Gram matrix is upper unitriangular, its inverse is an explicit signed weighted-path sum, and mutations or handle slides must act by integral congruence on the Euler lattice.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate novelty: a two-stage decomposition-type certificate first uses algebraic SOD indecomposability to decide whether Morse pieces can be admissible summands and then, only in the directed exceptional regime, tests the full integral Euler matrix, its weighted-path inverse, and its mutation congruence class." }, { "id": 20000005, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0005", "title": "Arithmetic toric exceptional collections and the anisotropic conic", "statement": "HMS for ATVs (e.g. $x^2+y^2+z^2=0/\\mathbb{R}$)\n\nDoes the derived category of an arithmetic toric variety (i.e., one defined over a non-algebraically closed field) admits an exceptional collection?\n(e.g., $\\{x^2+y^2+z^2=0\\}$ in $\\mathbb{P}^2_{\\mathbb{R}}$)\n\\begin{itemize}\n\\item $\\bullet$ Over $\\mathbb{C}$ is just $\\mathbb{P}^1$ but has no $\\mathbb{R}$-points\n\\item $\\bullet$ has exceptional collection $$ with $End(\\mathcal{E})=\\mathbb{H}$.\n\\item $\\bullet$ has an action of $S^1$ which is the restriction of the $S^1\\subset \\mathbb{C}^*$ action on $\\mathbb{C}-points)\n\\item $\\bullet$ toric data here is roughly data of a fan + data of a $Gal(\\overline{k}/k)$ action on the fan + ...\n\\item $\\bullet$ One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to $\\mathcal{O}_X$ (gives generating objects acted on by $Gal$, sometimes exceptional collection).\n\\end{itemize}", "original_statement": "HMS for ATVs (e.g. $x^2+y^2+z^2=0/\\mathbb{R}$)\n\nDoes the derived category of an arithmetic toric variety (i.e., one defined over a non-algebraically closed field) admits an exceptional collection?\n(e.g., $\\{x^2+y^2+z^2=0\\}$ in $\\mathbb{P}^2_{\\mathbb{R}}$)\n\\begin{itemize}\n\\item $\\bullet$ Over $\\mathbb{C}$ is just $\\mathbb{P}^1$ but has no $\\mathbb{R}$-points\n\\item $\\bullet$ has exceptional collection $$ with $End(\\mathcal{E})=\\mathbb{H}$.\n\\item $\\bullet$ has an action of $S^1$ which is the restriction of the $S^1\\subset \\mathbb{C}^*$ action on $\\mathbb{C}-points)\n\\item $\\bullet$ toric data here is roughly data of a fan + data of a $Gal(\\overline{k}/k)$ action on the fan + ...\n\\item $\\bullet$ One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to $\\mathcal{O}_X$ (gives generating objects acted on by $Gal$, sometimes exceptional collection).\n\\end{itemize}", "clean_statement": "HMS for ATVs (e.g. $x^2+y^2+z^2=0/\\mathbb{R}$)\n\nDoes the derived category of an arithmetic toric variety (i.e., one defined over a non-algebraically closed field) admits an exceptional collection?\n(e.g., $\\{x^2+y^2+z^2=0\\}$ in $\\mathbb{P}^2_{\\mathbb{R}}$)\n\\begin{itemize}\n\\item $\\bullet$ Over $\\mathbb{C}$ is just $\\mathbb{P}^1$ but has no $\\mathbb{R}$-points\n\\item $\\bullet$ has exceptional collection $$ with $End(\\mathcal{E})=\\mathbb{H}$.\n\\item $\\bullet$ has an action of $S^1$ which is the restriction of the $S^1\\subset \\mathbb{C}^*$ action on $\\mathbb{C}-points)\n\\item $\\bullet$ toric data here is roughly data of a fan + data of a $Gal(\\overline{k}/k)$ action on the fan + ...\n\\item $\\bullet$ One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to $\\mathcal{O}_X$ (gives generating objects acted on by $Gal$, sometimes exceptional collection).\n\\end{itemize}", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Exceptional collections\nSource item: 2.1\nSource URL: http://aimpl.org/syzygyms/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[4]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"HMS for ATVs (e.g. $x^2+y^2+z^2=0/\\\\mathbb{R}$)\\n\\nDoes the derived category of an arithmetic toric variety (i.e., one defined over a non-algebraically closed field) admits an exceptional collection?\\n(e.g., $\\\\{x^2+y^2+z^2=0\\\\}$ in $\\\\mathbb{P}^2_{\\\\mathbb{R}}$)\\n\\\\begin{itemize}\\n\\\\item $\\\\bullet$ Over $\\\\mathbb{C}$ is just $\\\\mathbb{P}^1$ but has no $\\\\mathbb{R}$-points\\n\\\\item $\\\\bullet$ has exceptional collection $$ with $End(\\\\mathcal{E})=\\\\mathbb{H}$.\\n\\\\item $\\\\bullet$ has an action of $S^1$ which is the restriction of the $S^1\\\\subset \\\\mathbb{C}^*$ action on $\\\\mathbb{C}-points)\\n\\\\item $\\\\bullet$ toric data here is roughly data of a fan + data of a $Gal(\\\\overline{k}/k)$ action on the fan + ...\\n\\\\item $\\\\bullet$ One source: An exceptional collection where fan automorphism permutes objects; e.g. powers of toric Frobenius applied to $\\\\mathcal{O}_X$ (gives generating objects acted on by $Gal$, sometimes exceptional collection).\\n\\\\end{itemize}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0005", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The missing AIM formula is verified from an archived source as with End(E)=H. Under the intended division-exceptional convention, existing Galois-descent results give full collections for every smooth projective arithmetic toric surface, every smooth toric Fano threefold, and several symmetric higher-dimensional classes, while the general case remains open in the primary literature checked. For a smooth anisotropic conic C/k, an integral Euler-pairing calculation proves that any full division-exceptional collection has exactly two terms and that the product of the k-dimensions of their endomorphism division algebras is 4; hence no full classical k-exceptional collection exists, and if O_C is a term then over R the other endomorphism algebra is forced to be H.\n\nCandidate contribution (proposition; novelty confidence low): For any smooth anisotropic conic C/k, every full division-exceptional collection (E1,E2) satisfies dim_k End(E1) times dim_k End(E2) = 4; in particular, a full collection containing O_C forces the other endomorphism division algebra to have dimension 4, explaining the quaternion algebra over R from the Euler lattice alone." }, { "id": 20000006, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0006", "title": "Normal-bundle criteria for exceptional sheaves on toric strata", "statement": "Does every smooth toric variety admits an exceptional collection of sheaves\n\\begin{itemize}\n\\item $\\bullet$ if replace sheaves (good objects of $D^b(X)$, answer is yes)\n\\item $\\bullet$ First open case is the case of toric 3-folds\n\\item $\\bullet$ if replace sheaves with line bundles, answer is no [Etimov '14] counterexample smooth Fano toric with $dim\\geq 0$ and $Pic=3$.\n\\item $\\bullet$ For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])\n\\item $\\bullet$ More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).\n\\end{itemize}", "original_statement": "Does every smooth toric variety admits an exceptional collection of sheaves\n\\begin{itemize}\n\\item $\\bullet$ if replace sheaves (good objects of $D^b(X)$, answer is yes)\n\\item $\\bullet$ First open case is the case of toric 3-folds\n\\item $\\bullet$ if replace sheaves with line bundles, answer is no [Etimov '14] counterexample smooth Fano toric with $dim\\geq 0$ and $Pic=3$.\n\\item $\\bullet$ For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])\n\\item $\\bullet$ More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).\n\\end{itemize}", "clean_statement": "Does every smooth toric variety admits an exceptional collection of sheaves\n\\begin{itemize}\n\\item $\\bullet$ if replace sheaves (good objects of $D^b(X)$, answer is yes)\n\\item $\\bullet$ First open case is the case of toric 3-folds\n\\item $\\bullet$ if replace sheaves with line bundles, answer is no [Etimov '14] counterexample smooth Fano toric with $dim\\geq 0$ and $Pic=3$.\n\\item $\\bullet$ For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])\n\\item $\\bullet$ More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).\n\\end{itemize}", "statement_status": "exact", "statement_verification": "The source is AIM Problem 2.2 in the “Syzygies and mirror symmetry” list, section “Exceptional collections.” It reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Exceptional collections\nSource item: 2.2\nSource URL: http://aimpl.org/syzygyms/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[5]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does every smooth toric variety admits an exceptional collection of sheaves\\n\\\\begin{itemize}\\n\\\\item $\\\\bullet$ if replace sheaves (good objects of $D^b(X)$, answer is yes)\\n\\\\item $\\\\bullet$ First open case is the case of toric 3-folds\\n\\\\item $\\\\bullet$ if replace sheaves with line bundles, answer is no [Etimov '14] counterexample smooth Fano toric with $dim\\\\geq 0$ and $Pic=3$.\\n\\\\item $\\\\bullet$ For surfaces can do with line bundles, read off explicitly from fan (exposition of this in [Hacking-Keating])\\n\\\\item $\\\\bullet$ More specifically, is it possible to find an exceptional collection from (push-forwards of) line bundles on toric strata? (This case can be analyzed, perhaps).\\n\\\\end{itemize}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0006", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literal nonprojective statement is false because the smooth toric variety A^1 has no exceptional nonzero coherent sheaf. For the intended smooth projective characteristic-zero problem, a line bundle pushed forward from a toric stratum has first self-extension H^0 of the stratum normal bundle, independently of the line-bundle twist. For every invariant prime divisor in every dimension, such a pushforward is exceptional if and only if the normal line bundle is acyclic; on a toric surface, the only exceptional proper-stratum examples are pushforwards from invariant (-1)-curves. Orlov's formula also gives a full collection of the requested form for invariant blow-ups whose base and center have full exceptional line-bundle collections.\n\nCandidate contribution (criterion; novelty confidence low): Candidate novelty: the normal-bundle self-Ext calculation is organized as a fan-computable screening package for the AIM stratum-support question, including the exact all-dimensional criterion that a line bundle pushed forward from an invariant prime divisor is exceptional precisely when its normal line bundle is acyclic, plus the resulting exact surface classification and two-sided placement test." }, { "id": 20000007, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0007", "title": "A Sullivan–Milnor–Moore obstruction screen for Weinstein cocore algebras", "statement": "Can the \"looking glass\" parallel/analogy between rational homotopy theory (= symplectic geometry of cotangent bundles) and local ring theory be generalized/ further exploited to other symplectic manifolds (e.g. Weinstein manifolds)?\n\nA)\n\\begin{itemize}\n\\item $\\bullet$ can \"attach handles\" to Weinstein manifolds\n\\item $\\bullet$ there exists a natural \"categorical compactification\" of Fukaya categories via \"stop\" or \"Lefschetz fibrations\"\n\\end{itemize}\nB)\n\\begin{itemize}\n\\item $\\bullet$ Can rational homotopy theory e.g. Sullivan minimal models can be used to e.g. construct resolutions?\n\\end{itemize}\nComment: The initial parallel comes from\n$$C_*(\\Omega_q M)=Ext_{C^{\\bullet}M}(k,k)=A$$\nwhere\n$$C_*(\\Omega_q M)\\simeq End_{Fuk(T^*M)}(T^*_q M, T^*_q M)$$\n(Here $M$ is assumed to be simply connected).", "original_statement": "Can the \"looking glass\" parallel/analogy between rational homotopy theory (= symplectic geometry of cotangent bundles) and local ring theory be generalized/ further exploited to other symplectic manifolds (e.g. Weinstein manifolds)?\n\nA)\n\\begin{itemize}\n\\item $\\bullet$ can \"attach handles\" to Weinstein manifolds\n\\item $\\bullet$ there exists a natural \"categorical compactification\" of Fukaya categories via \"stop\" or \"Lefschetz fibrations\"\n\\end{itemize}\nB)\n\\begin{itemize}\n\\item $\\bullet$ Can rational homotopy theory e.g. Sullivan minimal models can be used to e.g. construct resolutions?\n\\end{itemize}\nComment: The initial parallel comes from\n$$C_*(\\Omega_q M)=Ext_{C^{\\bullet}M}(k,k)=A$$\nwhere\n$$C_*(\\Omega_q M)\\simeq End_{Fuk(T^*M)}(T^*_q M, T^*_q M)$$\n(Here $M$ is assumed to be simply connected).", "clean_statement": "Can the \"looking glass\" parallel/analogy between rational homotopy theory (= symplectic geometry of cotangent bundles) and local ring theory be generalized/ further exploited to other symplectic manifolds (e.g. Weinstein manifolds)?\n\nA)\n\\begin{itemize}\n\\item $\\bullet$ can \"attach handles\" to Weinstein manifolds\n\\item $\\bullet$ there exists a natural \"categorical compactification\" of Fukaya categories via \"stop\" or \"Lefschetz fibrations\"\n\\end{itemize}\nB)\n\\begin{itemize}\n\\item $\\bullet$ Can rational homotopy theory e.g. Sullivan minimal models can be used to e.g. construct resolutions?\n\\end{itemize}\nComment: The initial parallel comes from\n$$C_*(\\Omega_q M)=Ext_{C^{\\bullet}M}(k,k)=A$$\nwhere\n$$C_*(\\Omega_q M)\\simeq End_{Fuk(T^*M)}(T^*_q M, T^*_q M)$$\n(Here $M$ is assumed to be simply connected).", "statement_status": "exact", "statement_verification": "This is Problem 3.1, “Symplectic geometry,” from the AIM workshop list *Syzygies and mirror symmetry*. The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Symplectic geometry\nSource item: 3.1\nSource URL: http://aimpl.org/syzygyms/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[6]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can the \\\"looking glass\\\" parallel/analogy between rational homotopy theory (= symplectic geometry of cotangent bundles) and local ring theory be generalized/ further exploited to other symplectic manifolds (e.g. Weinstein manifolds)?\\n\\nA)\\n\\\\begin{itemize}\\n\\\\item $\\\\bullet$ can \\\"attach handles\\\" to Weinstein manifolds\\n\\\\item $\\\\bullet$ there exists a natural \\\"categorical compactification\\\" of Fukaya categories via \\\"stop\\\" or \\\"Lefschetz fibrations\\\"\\n\\\\end{itemize}\\nB)\\n\\\\begin{itemize}\\n\\\\item $\\\\bullet$ Can rational homotopy theory e.g. Sullivan minimal models can be used to e.g. construct resolutions?\\n\\\\end{itemize}\\nComment: The initial parallel comes from\\n$$C_*(\\\\Omega_q M)=Ext_{C^{\\\\bullet}M}(k,k)=A$$\\nwhere\\n$$C_*(\\\\Omega_q M)\\\\simeq End_{Fuk(T^*M)}(T^*_q M, T^*_q M)$$\\n(Here $M$ is assumed to be simply connected).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0007", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an augmented dg or A-infinity algebra E over Q that is augmentation-preservingly quasi-isomorphic to chains on the based loop space of a simply connected finite-rational-type space X, four necessary conditions hold: H(E) is of Milnor–Moore universal-enveloping type; the Koszul dual RHom_E(Q,Q) is quasi-isomorphic to C*(X;Q); its cohomology is connected and graded commutative with vanishing degree one; and the Eilenberg–Moore derived double-centralizer map is a quasi-isomorphism. These give a rigorous falsification screen for proposed Sullivan interpretations of augmented Weinstein cocore/linking-disk algebras. A minimal Sullivan model constructs the relevant module resolution through the two-sided bar construction, with infinite dualization/completion explicitly retained.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate novelty: the combined Milnor–Moore, Koszul-dual commutativity/connectivity, degree-one, and Eilenberg–Moore double-centralizer tests form a four-part obstruction screen for Sullivan-model realizations of augmented single-object Weinstein wrapped endomorphism algebras." }, { "id": 20000008, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0008", "title": "A direct HHL--Brown--Erman comparison with a corrected Hirzebruch-surface differential", "statement": "Compare $\\Delta$-resolutions\n\nCan we directly compare the Hanlon-Hicks-Lazarev and Brown-Erman construction of resolutions?\n\n$\\bullet$ Potential mechanism: (starting in degree 0)\n\n{vertices/points in stratification of $T^n$ arising in HHL}\nis in one to one correspondence with\n{additional generators for normalization of the semi-group ring, i.e. elements coming from saturation of semigroup}", "original_statement": "Compare $\\Delta$-resolutions\n\nCan we directly compare the Hanlon-Hicks-Lazarev and Brown-Erman construction of resolutions?\n\n$\\bullet$ Potential mechanism: (starting in degree 0)\n\n{vertices/points in stratification of $T^n$ arising in HHL}\nis in one to one correspondence with\n{additional generators for normalization of the semi-group ring, i.e. elements coming from saturation of semigroup}", "clean_statement": "Compare $\\Delta$-resolutions\n\nCan we directly compare the Hanlon-Hicks-Lazarev and Brown-Erman construction of resolutions?\n\n$\\bullet$ Potential mechanism: (starting in degree 0)\n\n{vertices/points in stratification of $T^n$ arising in HHL}\nis in one to one correspondence with\n{additional generators for normalization of the semi-group ring, i.e. elements coming from saturation of semigroup}", "statement_status": "exact", "statement_verification": "The canonical AIM record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Resolutions\nSource item: 4.1\nSource URL: http://aimpl.org/syzygyms/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[7]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Compare $\\\\Delta$-resolutions\\n\\nCan we directly compare the Hanlon-Hicks-Lazarev and Brown-Erman construction of resolutions?\\n\\n$\\\\bullet$ Potential mechanism: (starting in degree 0)\\n\\n{vertices/points in stratification of $T^n$ arising in HHL}\\nis in one to one correspondence with\\n{additional generators for normalization of the semi-group ring, i.e. elements coming from saturation of semigroup}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: medium.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0008", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Berkesch--Cranton Heller--Smith--Yang, arXiv:2512.17871v1, solve the AIM comparison by identifying the HHL ceiling cellular complex with a free resolution of the Brown--Erman normalization; for smooth toric diagonals the complex is minimal. This attempt proves the explicit consequence that vertices biject with minimal normalization-module generators and beta_i equals the number of i-cells, gives a saturated-lattice counterexample without minimality, and proves that row 5, column 1 of the F_2 example's displayed partial_2 must be -x_4 y_2 rather than -x_4.\n\nCandidate contribution (correction; novelty confidence medium): Candidate erratum: in arXiv:2512.17871v1, Example 2.18, row 5 column 1 of the displayed matrix partial_2 must be -x_4 y_2 rather than -x_4; the correction is independently forced by Cox homogeneity and by the equation partial_1 partial_2=0." }, { "id": 20000009, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0009", "title": "Fukaya interpretations and the limits of cellular minimization", "statement": "Topology/Morse theory of cellular resolutions\n\nCan we interpret resolutions via Fukaya categories? (e.g. minimizing resolutions through \"cellular\" operations)", "original_statement": "Topology/Morse theory of cellular resolutions\n\nCan we interpret resolutions via Fukaya categories? (e.g. minimizing resolutions through \"cellular\" operations)", "clean_statement": "Topology/Morse theory of cellular resolutions\n\nCan we interpret resolutions via Fukaya categories? (e.g. minimizing resolutions through \"cellular\" operations)", "statement_status": "exact", "statement_verification": "The canonical record is Problem 4.2 in the “Resolutions” section of the AIM problem list *Syzygies and mirror symmetry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Resolutions\nSource item: 4.2\nSource URL: http://aimpl.org/syzygyms/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[8]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Topology/Morse theory of cellular resolutions\\n\\nCan we interpret resolutions via Fukaya categories? (e.g. minimizing resolutions through \\\"cellular\\\" operations)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0009", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In the intended smooth toric setting, coherent-constructible correspondence and the sheaf/Fukaya equivalence interpret the HHL resolution categorically, while recent work minimizes it by a finite poset-filtered homological perturbation. A proved six-vertex RP^2 example shows that algebraic minimization of a cellular resolution cannot in general be achieved by homogeneous homotopy-preserving cellular Morse operations: over every field of characteristic not two, its cellular ranks (6,15,10) minimize to (6,5,0), but any matching with those critical ranks would make RP^2 homotopy equivalent to a tree. The report also isolates the finite filtered-transfer lemma and the Maurer-Cartan test required of a proposed pairwise-arrow Fukaya realization.\n\nCandidate contribution (counterexample; novelty confidence low): Label the six vertices of the minimal triangulation of RP^2 by m_i=(x_1x_2x_3x_4x_5x_6)/x_i. In characteristic not two this supports a cellular resolution with ranks (6,15,10), whose minimal resolution has ranks (6,5,0), yet no homogeneous Forman/Batzies-Welker matching on the triangulation can attain the minimal ranks." }, { "id": 20000010, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0010", "title": "Filtered flow data and stable Betti spectra", "statement": "Topology/Morse theory of cellular resolutions\n\nThe Morse-theoretic description of $H_*(M)$ can be refined (at e.g. level of \"framed flow categories\"). can this refined decomposition tell us something on the algebraic side?", "original_statement": "Topology/Morse theory of cellular resolutions\n\nThe Morse-theoretic description of $H_*(M)$ can be refined (at e.g. level of \"framed flow categories\"). can this refined decomposition tell us something on the algebraic side?", "clean_statement": "Topology/Morse theory of cellular resolutions\n\nThe Morse-theoretic description of $H_*(M)$ can be refined (at e.g. level of \"framed flow categories\"). can this refined decomposition tell us something on the algebraic side?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 4.3 in the “Resolutions” section of the AIM workshop list *Syzygies and mirror symmetry*. The source page was checked directly on 2026-07-22 and agrees with the JSON record. Its title and text are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Resolutions\nSource item: 4.3\nSource URL: http://aimpl.org/syzygyms/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[9]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Topology/Morse theory of cellular resolutions\\n\\nThe Morse-theoretic description of $H_*(M)$ can be refined (at e.g. level of \\\"framed flow categories\\\"). can this refined decomposition tell us something on the algebraic side?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0010", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Ordinary homogenization of a monomially labeled framed flow category factors through the labels and signed zero-dimensional moduli counts, so higher moduli and framings cannot change the ordinary free resolution. Retaining the label filtration before taking chains gives cofiber spectra whose homology is the corresponding multigraded Tor strand. For the Taylor resolution these are canonical lcm-lattice invariants. A proved facet-complement construction realizes the double suspension spectrum of any finite simplicial complex as a top stable Betti spectrum; ideals obtained from triangulations of CP^2#CP^2 and S^2 x S^2 have equal top-strand Betti dimensions but respective Sq^2 ranks one and zero.\n\nCandidate contribution (proposition; novelty confidence low): Candidate novelty: the stable Betti spectrum B_u(I), together with the facet-complement realization I_K=(product of x_F over facets F omitting v), gives a testable stable refinement of a multigraded Tor strand; the CP^2#CP^2 versus S^2 x S^2 family proves that Sq^2 can distinguish equal top-strand Betti vectors." }, { "id": 20000011, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0011", "title": "Fundamental-group information: a torus group-algebra answer and an invisibility counterexample", "statement": "$T^n=K(\\pi,1)$\n\nGiven a cellular resolution, is there a meaningful way to algebraically interpret topology of the cell complex beyond homology e.g. $\\pi_1$?", "original_statement": "$T^n=K(\\pi,1)$\n\nGiven a cellular resolution, is there a meaningful way to algebraically interpret topology of the cell complex beyond homology e.g. $\\pi_1$?", "clean_statement": "$T^n=K(\\pi,1)$\n\nGiven a cellular resolution, is there a meaningful way to algebraically interpret topology of the cell complex beyond homology e.g. $\\pi_1$?", "statement_status": "exact", "statement_verification": "The assigned record is Problem 4.4 in the AIM problem list *Syzygies and mirror symmetry*, section “Resolutions”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Resolutions\nSource item: 4.4\nSource URL: http://aimpl.org/syzygyms/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[10]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"$T^n=K(\\\\pi,1)$\\n\\nGiven a cellular resolution, is there a meaningful way to algebraically interpret topology of the cell complex beyond homology e.g. $\\\\pi_1$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0011", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The intended torus case now has a concrete algebraic interpretation: the deck lattice L appears through the group algebra S[L], and passage from the universal cover to the torus is extension of scalars z^v to 1. Complementarily, over any field of characteristic not 2, the standard six-vertex triangulation of RP^2 and a contractible cone support cellular resolutions of the same squarefree Veronese ideal, and their augmented multigraded free complexes are isomorphic after forgetting the cellular bases, although their fundamental groups are Z/2 and 1. Thus pi_1 is not an invariant of an arbitrary unbased commutative cellular resolution; it survives in the cellular incidence data or a group-ring/exit-path enrichment.\n\nCandidate contribution (counterexample; novelty confidence low): For J_N=((x_1...x_N)/x_i : 1<=i<=N), every k-acyclic simplicial complex on N vertices supports a cellular resolution; in dimension two the resulting unbased multigraded complex depends only on N and the number of triangles. In particular, the displayed RP^2 triangulation and cone give isomorphic augmented complexes with different fundamental groups." }, { "id": 20000012, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0012", "title": "Cox-chart Morse theory for cellular virtual resolutions", "statement": "Topology/Morse theory of resolutions\n\nIs there an analogue of \"Morse theory for cellular resolutions\" for virtual resolutions?", "original_statement": "Topology/Morse theory of resolutions\n\nIs there an analogue of \"Morse theory for cellular resolutions\" for virtual resolutions?", "clean_statement": "Topology/Morse theory of resolutions\n\nIs there an analogue of \"Morse theory for cellular resolutions\" for virtual resolutions?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 4.5 in the AIM list *Syzygies and mirror symmetry*, section “Resolutions”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Resolutions\nSource item: 4.5\nSource URL: http://aimpl.org/syzygyms/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[11]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Topology/Morse theory of resolutions\\n\\nIs there an analogue of \\\"Morse theory for cellular resolutions\\\" for virtual resolutions?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0012", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a monomially labelled finite regular CW complex over the Cox ring of a smooth projective toric variety, the cellular complex is a virtual resolution exactly when every degree subcomplex obtained on every maximal-cone chart by forgetting exponents of inverted variables is empty or reduced-acyclic; only finitely many exponent thresholds need be tested. Any chart-homogeneous acyclic matching gives an algebraic Morse reduction after localization. A matching homogeneous on every chart can match only equal monomial labels, so genuinely virtual cancellation of unequal labels must be chart-dependent or use more general descent/homotopy-transfer data.\n\nCandidate contribution (criterion_and_obstruction; novelty confidence low): Candidate novelty: the finite Cox-chart degree-subcomplex criterion together with the no-new-universal-matching proposition for monomially labelled cellular complexes." }, { "id": 20000013, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0013", "title": "Bounded moduli charts and ghost directions for virtual resolutions", "statement": "Observation: The current definition of virtual resolution is so broad and it is not meaningful to parameterize, then.\n\nIs there a well-behaved moduli theory of virtual resolutions equipped with extra structure? e.g. condition under which there exists a finite set? or space/ stacks/...?\n(c.f. [Boij-Sölderberg] theory constructs a meaningful cone of Betti tables of all free complexes. Look at subcone of virtual resolutions of a given $M$)", "original_statement": "Observation: The current definition of virtual resolution is so broad and it is not meaningful to parameterize, then.\n\nIs there a well-behaved moduli theory of virtual resolutions equipped with extra structure? e.g. condition under which there exists a finite set? or space/ stacks/...?\n(c.f. [Boij-Sölderberg] theory constructs a meaningful cone of Betti tables of all free complexes. Look at subcone of virtual resolutions of a given $M$)", "clean_statement": "Observation: The current definition of virtual resolution is so broad and it is not meaningful to parameterize, then.\n\nIs there a well-behaved moduli theory of virtual resolutions equipped with extra structure? e.g. condition under which there exists a finite set? or space/ stacks/...?\n(c.f. [Boij-Sölderberg] theory constructs a meaningful cone of Betti tables of all free complexes. Look at subcone of virtual resolutions of a given $M$)", "statement_status": "exact", "statement_verification": "This is Problem 4.6 in the **Resolutions** section of the AIM list *Syzygies and mirror symmetry*. The canonical record and the live AIM page agree. The live page was checked on 2026-07-22 and contains no status update or attached remark. Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Resolutions\nSource item: 4.6\nSource URL: http://aimpl.org/syzygyms/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[12]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Observation: The current definition of virtual resolution is so broad and it is not meaningful to parameterize, then.\\n\\nIs there a well-behaved moduli theory of virtual resolutions equipped with extra structure? e.g. condition under which there exists a finite set? or space/ stacks/...?\\n(c.f. [Boij-Sölderberg] theory constructs a meaningful cone of Betti tables of all free complexes. Look at subcone of virtual resolutions of a given $M$)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0013", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a fixed coherent sheaf on a product of projective spaces and a fixed finite multigraded Betti type, based exact augmented complexes form a quasi-affine finite-type scheme, and quotienting by changes of bases gives a finite-type Artin stack; a finite degree window together with length and rank bounds therefore yields finitely many Betti types. Conversely, for M=S on P^1, shifted Koszul ghost complexes give infinitely many graded-minimal virtual resolutions with fixed length, fixed ranks, and fixed F_0, and their virtual-acyclic Betti tables give infinitely many linearly independent recession directions.\n\nCandidate contribution (theorem_and_obstruction; novelty confidence low): Candidate novelty: the fixed-Betti-type quotient-stack construction, paired with the explicit P^1 theorem that graded minimality, fixed length, fixed total ranks, and fixed F_0 still allow infinitely many Betti tables and infinitely many independent ghost directions." }, { "id": 20000014, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0014", "title": "What diagonal and toric-correspondence transforms remember", "statement": "What does the Fourier-Mukai transform by either resolution of $\\Delta$ applied to $M$ tell us about $M$? More generally what does applying Fourier-Mukai transforms of resolutions of toric subvarieties tell us about the modules apply them to? [e.g. in the spirit of the way we obtain Eagon-Northcott-type complexes for modules]", "original_statement": "What does the Fourier-Mukai transform by either resolution of $\\Delta$ applied to $M$ tell us about $M$? More generally what does applying Fourier-Mukai transforms of resolutions of toric subvarieties tell us about the modules apply them to? [e.g. in the spirit of the way we obtain Eagon-Northcott-type complexes for modules]", "clean_statement": "What does the Fourier-Mukai transform by either resolution of $\\Delta$ applied to $M$ tell us about $M$? More generally what does applying Fourier-Mukai transforms of resolutions of toric subvarieties tell us about the modules apply them to? [e.g. in the spirit of the way we obtain Eagon-Northcott-type complexes for modules]", "statement_status": "exact", "statement_verification": "The exact canonical record is AIM Problem 5.1 from the 2023 workshop *Syzygies and mirror symmetry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Fourier-Mukai transforms\nSource item: 5.1\nSource URL: http://aimpl.org/syzygyms/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[13]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What does the Fourier-Mukai transform by either resolution of $\\\\Delta$ applied to $M$ tell us about $M$? More generally what does applying Fourier-Mukai transforms of resolutions of toric subvarieties tell us about the modules apply them to? [e.g. in the spirit of the way we obtain Eagon-Northcott-type complexes for modules]\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0014", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A line-bundle resolution of a toric correspondence yields a functorial cohomology-table spectral sequence whose diagonal case reconstructs the input sheaf. Recent identification of the Hanlon--Hicks--Lazarev and Brown--Erman diagonal kernel complexes implies that their transforms agree as filtered complexes and have identical E1 spectral sequences. At Cox-module level any excess homology is irrelevant torsion, and Heller's work shows it can persist arbitrarily far along an ample ray. For a general toric subvariety Z the intrinsic output is derived restriction to Z followed by derived pushforward; an explicit power-map graph on P1 gives a sparse band presentation and its complete splitting formula.\n\nCandidate contribution (corollary; novelty confidence low): After fixing the 2025 isomorphism between the HHL and Brown--Erman diagonal kernel complexes, their Fourier--Mukai transforms of every coherent complex are isomorphic as complexes filtered by kernel homological degree; therefore their E1 cohomology-table spectral sequences and, in every one-row range, their Cox output complexes agree up to graded base change." }, { "id": 20000015, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0015", "title": "The projective-dimension sign and a Cox-depth criterion", "statement": "Given $M$ an $S$-module, $S=Cox(X)$,\n\\begin{enumerate}\n\\item can we always find another $S$-module $M'$ satisfying\n\\begin{itemize}\n\\item $\\bullet$ $\\tilde{M}=\\tilde{M}'$\n\\item $\\bullet$ $pdim(M')\\geq dim(X)$\n\\end{itemize}\nThis is known to be true if $X=\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$.\n\n\\item What is the relation of this to examples of [Chardin-D'Cruz] where adding embedded components drops regularity?\n\n\\item Potential mechanism by truncating $M$ to $(M)_{\\geq d}$? (This works for $\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$'s)\n\\item Alternatively, could there be an invariant of $\\tilde{M}$ which obstructs existence of such an $M'$?\n\\end{enumerate}", "original_statement": "Given $M$ an $S$-module, $S=Cox(X)$,\n\\begin{enumerate}\n\\item can we always find another $S$-module $M'$ satisfying\n\\begin{itemize}\n\\item $\\bullet$ $\\tilde{M}=\\tilde{M}'$\n\\item $\\bullet$ $pdim(M')\\geq dim(X)$\n\\end{itemize}\nThis is known to be true if $X=\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$.\n\n\\item What is the relation of this to examples of [Chardin-D'Cruz] where adding embedded components drops regularity?\n\n\\item Potential mechanism by truncating $M$ to $(M)_{\\geq d}$? (This works for $\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$'s)\n\\item Alternatively, could there be an invariant of $\\tilde{M}$ which obstructs existence of such an $M'$?\n\\end{enumerate}", "clean_statement": "Given $M$ an $S$-module, $S=Cox(X)$,\n\\begin{enumerate}\n\\item can we always find another $S$-module $M'$ satisfying\n\\begin{itemize}\n\\item $\\bullet$ $\\tilde{M}=\\tilde{M}'$\n\\item $\\bullet$ $pdim(M')\\geq dim(X)$\n\\end{itemize}\nThis is known to be true if $X=\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$.\n\n\\item What is the relation of this to examples of [Chardin-D'Cruz] where adding embedded components drops regularity?\n\n\\item Potential mechanism by truncating $M$ to $(M)_{\\geq d}$? (This works for $\\mathbb{P}^{n_1}\\times ... \\times \\mathbb{P}^{n_r}$'s)\n\\item Alternatively, could there be an invariant of $\\tilde{M}$ which obstructs existence of such an $M'$?\n\\end{enumerate}", "statement_status": "exact", "statement_verification": "The assigned record is Problem 6.1 in the “Modules over the Cox ring” section of the AIM list *Syzygies and mirror symmetry*. The canonical record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Modules over the Cox ring\nSource item: 6.1\nSource URL: http://aimpl.org/syzygyms/6/\nCanonical location: aim-algebraic-geometry-notes.json notes[14]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given $M$ an $S$-module, $S=Cox(X)$,\\n\\\\begin{enumerate}\\n\\\\item can we always find another $S$-module $M'$ satisfying\\n\\\\begin{itemize}\\n\\\\item $\\\\bullet$ $\\\\tilde{M}=\\\\tilde{M}'$\\n\\\\item $\\\\bullet$ $pdim(M')\\\\geq dim(X)$\\n\\\\end{itemize}\\nThis is known to be true if $X=\\\\mathbb{P}^{n_1}\\\\times ... \\\\times \\\\mathbb{P}^{n_r}$.\\n\\n\\\\item What is the relation of this to examples of [Chardin-D'Cruz] where adding embedded components drops regularity?\\n\\n\\\\item Potential mechanism by truncating $M$ to $(M)_{\\\\geq d}$? (This works for $\\\\mathbb{P}^{n_1}\\\\times ... \\\\times \\\\mathbb{P}^{n_r}$'s)\\n\\\\item Alternatively, could there be an invariant of $\\\\tilde{M}$ which obstructs existence of such an $M'$?\\n\\\\end{enumerate}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/6/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0015", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth complete toric variety, the AIM inequality as printed is always satisfied by M' = M direct-sum S/m, which sheafifies to M and has maximal projective dimension. For the contextually likely reversed inequality pdim(M') <= dim(X), Auslander-Buchsbaum makes existence equivalent to finding a representative of depth at least rank Pic(X), and the report proves necessary and sufficient low-local-cohomology conditions for a specified irrelevant-torsion truncation or extension to achieve this. Eisenbud-Erman-Schreyer's high-truncation theorem verifies the exact-module assertion on products of projective spaces; the corresponding universal statement for arbitrary smooth projective toric varieties remains unresolved here.\n\nCandidate contribution (criterion; novelty confidence low): For an exact truncation 0 -> A -> M -> T -> 0 over the Cox ring of a smooth complete toric variety, pdim(A) <= dim(X) holds exactly when H^i_m(M) -> H^i_m(T) is an isomorphism for 0 <= i <= rank Pic(X)-2 and injective at the next index; a dual connecting-map criterion holds for irrelevant-torsion extensions 0 -> T -> E -> M -> 0." }, { "id": 20000016, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0016", "title": "An explicit interval in the Orlov spectrum of products of projective lines", "statement": "What can be said about the Orlov spectrum of a toric variety (or any $X$)? (e.g. about gaps, sequences, etc.)\n\nKnown:\n\\begin{itemize}\n\\item $OrSpec(\\mathbb{P}^1)=\\{1,2\\}$\n\\item On a toric surface, there exists full arbitrary large consecutive sequence.\n\\item More genrally, by replacing $D^b(X)$ with $\\mathcal{C}$, there exists a notion of $OrSpec(\\mathcal{C})$ and $OrSpec(Rep (Q))$ ($Q$ is a quiver), computed when $Q$ is of A/D types.\n\\end{itemize}", "original_statement": "What can be said about the Orlov spectrum of a toric variety (or any $X$)? (e.g. about gaps, sequences, etc.)\n\nKnown:\n\\begin{itemize}\n\\item $OrSpec(\\mathbb{P}^1)=\\{1,2\\}$\n\\item On a toric surface, there exists full arbitrary large consecutive sequence.\n\\item More genrally, by replacing $D^b(X)$ with $\\mathcal{C}$, there exists a notion of $OrSpec(\\mathcal{C})$ and $OrSpec(Rep (Q))$ ($Q$ is a quiver), computed when $Q$ is of A/D types.\n\\end{itemize}", "clean_statement": "What can be said about the Orlov spectrum of a toric variety (or any $X$)? (e.g. about gaps, sequences, etc.)\n\nKnown:\n\\begin{itemize}\n\\item $OrSpec(\\mathbb{P}^1)=\\{1,2\\}$\n\\item On a toric surface, there exists full arbitrary large consecutive sequence.\n\\item More genrally, by replacing $D^b(X)$ with $\\mathcal{C}$, there exists a notion of $OrSpec(\\mathcal{C})$ and $OrSpec(Rep (Q))$ ($Q$ is a quiver), computed when $Q$ is of A/D types.\n\\end{itemize}", "statement_status": "exact", "statement_verification": "The canonical record is Problem 7.1 in the AIM list *Syzygies and mirror symmetry*, section \"Orlov spectrum and Rouquier dimension\":", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Orlov spectrum and Rouquier dimension\nSource item: 7.1\nSource URL: http://aimpl.org/syzygyms/7/\nCanonical location: aim-algebraic-geometry-notes.json notes[15]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What can be said about the Orlov spectrum of a toric variety (or any $X$)? (e.g. about gaps, sequences, etc.)\\n\\nKnown:\\n\\\\begin{itemize}\\n\\\\item $OrSpec(\\\\mathbb{P}^1)=\\\\{1,2\\\\}$\\n\\\\item On a toric surface, there exists full arbitrary large consecutive sequence.\\n\\\\item More genrally, by replacing $D^b(X)$ with $\\\\mathcal{C}$, there exists a notion of $OrSpec(\\\\mathcal{C})$ and $OrSpec(Rep (Q))$ ($Q$ is a quiver), computed when $Q$ is of A/D types.\\n\\\\end{itemize}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/7/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0016", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For X_n=(P^1_k)^n and 0 <= r <= n, the explicit exterior-product generator G_{n,r} formed from r factors O plus O_p and n-r factors O plus O(-1) has generation time exactly n+r. The upper bound is proved by additive exterior products of one- and two-cone diagonal filtrations, and the matching lower bound by a nonzero Kunneth product of factor ghost sequences. Consequently {n,n+1,...,2n} is contained in the Orlov spectrum of D^b(coh((P^1_k)^n)).\n\nCandidate contribution (theorem; novelty confidence low): The mixed generators (O plus O_p)^{boxtimes r} boxtimes (O plus O(-1))^{boxtimes(n-r)} on (P^1)^n have exact generation time n+r, giving an explicit consecutive interval [n,2n] in the Orlov spectrum." }, { "id": 20000017, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0017", "title": "Exact pointwise levels for a uniserial hypersurface singularity category", "statement": "Rdim of $D^b Sing$\n\nWhat about Rouquier dimension or orlov spectrum of $D^b(Sing(X))=D^b(Coh(X))/D^b(Perf(X))$?", "original_statement": "Rdim of $D^b Sing$\n\nWhat about Rouquier dimension or orlov spectrum of $D^b(Sing(X))=D^b(Coh(X))/D^b(Perf(X))$?", "clean_statement": "Rdim of $D^b Sing$\n\nWhat about Rouquier dimension or orlov spectrum of $D^b(Sing(X))=D^b(Coh(X))/D^b(Perf(X))$?", "statement_status": "exact", "statement_verification": "The exact canonical record is problem 7.2 in the AIM workshop *Syzygies and mirror symmetry*, section “Orlov spectrum and Rouquier dimension”; it is zero-based record 16 of `aim-algebraic-geometry-notes.json`:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Orlov spectrum and Rouquier dimension\nSource item: 7.2\nSource URL: http://aimpl.org/syzygyms/7/\nCanonical location: aim-algebraic-geometry-notes.json notes[16]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Rdim of $D^b Sing$\\n\\nWhat about Rouquier dimension or orlov spectrum of $D^b(Sing(X))=D^b(Coh(X))/D^b(Perf(X))$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/7/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0017", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The broad AIM question is partially solved in the literature but remains open for arbitrary singular schemes. For the test family A=k[t]/(t^n) over an arbitrary field, this attempt proves an exact object-by-object formula: if G_I is the direct sum of M_i=A/(t^i) for a nonempty I contained in {1,...,floor(n/2)} and s=max I, then level_{G_I}(M_d) is 0 for d in I and max{1,ceil(d/s)-1} otherwise. The formula extends by shifts and maxima to every object, and recovers the known Ballard-Favero-Katzarkov Orlov spectrum and Rouquier dimension zero.\n\nCandidate contribution (theorem; novelty confidence low): A field-independent exact formula is proved for the level of every object of D_sg(k[t]/(t^n)) relative to every nonzero generator, including the correction for indecomposable summands already present in the generator." }, { "id": 20000018, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0018", "title": "A block formula and Morse-potential computation for compact monotone Fukaya categories", "statement": "Same questions as above for $Fuk(X)$ where $X$ is compact symplectic.", "original_statement": "Same questions as above for $Fuk(X)$ where $X$ is compact symplectic.", "clean_statement": "Same questions as above for $Fuk(X)$ where $X$ is compact symplectic.", "statement_status": "exact", "statement_verification": "The canonical record is AIM problem 7.3 from the workshop *Syzygies and mirror symmetry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Orlov spectrum and Rouquier dimension\nSource item: 7.3\nSource URL: http://aimpl.org/syzygyms/7/\nCanonical location: aim-algebraic-geometry-notes.json notes[17]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Same questions as above for $Fuk(X)$ where $X$ is compact symplectic.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/7/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0018", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite orthogonal decomposition of an idempotent-complete triangulated category, the generation time of a componentwise object is the maximum of its component generation times, so the Orlov spectrum is the maximum-product of the component spectra. Applying this to the quantum-idempotent decomposition, published toric split-generation, and local A-infinity formality shows that a compact monotone toric Fano manifold over an algebraically closed characteristic-zero field has Orlov spectrum exactly {0} whenever its toric disc potential is Morse. In particular this computes the spectrum as {0} for the specified monotone Fukaya category of every complex projective space.\n\nCandidate contribution (special_case_and_reduction; novelty confidence low): Candidate novelty: in the precisely specified split-closed Z/2-graded monotone setup, a compact toric Fano with Morse disc potential has full Orlov spectrum {0}; more generally the spectrum of its quantum-idempotent decomposition is the set of maxima of tuples of block generation times." }, { "id": 20000019, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0019", "title": "Diagonal resolutions versus homotopy-colimit depth", "statement": "There are two known ways to bound Rouquier dimension of a category\n\\begin{enumerate}\n\\item bound length of a resolution of a diagonal\n\\item the minimal depth of a presentation of $\\mathcal{C}$ as hocolim (cats with Rouquier dimension $Q$). Meaning if $F:I^{\\leq}\\to Cat$ diagram with $Rdim F(i)=0$, then $Rdim(hocolim F i)\\leq depth(I)$\n\\end{enumerate}\n\nCan (1) be used to bound (2) or vice-versa?", "original_statement": "There are two known ways to bound Rouquier dimension of a category\n\\begin{enumerate}\n\\item bound length of a resolution of a diagonal\n\\item the minimal depth of a presentation of $\\mathcal{C}$ as hocolim (cats with Rouquier dimension $Q$). Meaning if $F:I^{\\leq}\\to Cat$ diagram with $Rdim F(i)=0$, then $Rdim(hocolim F i)\\leq depth(I)$\n\\end{enumerate}\n\nCan (1) be used to bound (2) or vice-versa?", "clean_statement": "There are two known ways to bound Rouquier dimension of a category\n\\begin{enumerate}\n\\item bound length of a resolution of a diagonal\n\\item the minimal depth of a presentation of $\\mathcal{C}$ as hocolim (cats with Rouquier dimension $Q$). Meaning if $F:I^{\\leq}\\to Cat$ diagram with $Rdim F(i)=0$, then $Rdim(hocolim F i)\\leq depth(I)$\n\\end{enumerate}\n\nCan (1) be used to bound (2) or vice-versa?", "statement_status": "exact", "statement_verification": "The displayed text really does contain \\(Q\\), so this is not an extraction error. Because the very next clause says \\(\\operatorname{Rdim}F(i)=0\\), the only coherent reconstruction is that “\\(Q\\)” is a typographical substitution for “\\(0\\).” The notation \\(I^{\\leq}\\) is not defined on the page. I interpret it as a finite partially ordered set \\((I,\\leq)\\), regarded as a category. This agrees with the precise finite-poset formulation subsequently used by Bai--Côté.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Orlov spectrum and Rouquier dimension\nSource item: 7.4\nSource URL: http://aimpl.org/syzygyms/7/\nCanonical location: aim-algebraic-geometry-notes.json notes[18]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"There are two known ways to bound Rouquier dimension of a category\\n\\\\begin{enumerate}\\n\\\\item bound length of a resolution of a diagonal\\n\\\\item the minimal depth of a presentation of $\\\\mathcal{C}$ as hocolim (cats with Rouquier dimension $Q$). Meaning if $F:I^{\\\\leq}\\\\to Cat$ diagram with $Rdim F(i)=0$, then $Rdim(hocolim F i)\\\\leq depth(I)$\\n\\\\end{enumerate}\\n\\nCan (1) be used to bound (2) or vice-versa?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/7/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0019", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Unrestricted homotopy-colimit depth cannot bound diagonal-resolution length: for a purely inseparable extension L/k of degree p, Perf(L) has a one-piece arrow-depth-zero presentation and Rouquier dimension zero, but its diagonal is not perfect and its diagonal dimension is infinite. Positively, for a finite perfect-admissible poset diagram with arrow-depth d and local diagonal dimensions delta_i, two-sided base change through the derived localization and the height-layer semiorthogonal filtration give Ddim(hocolim F) <= sum_r(1 + max_{h(i)=r} delta_i) - 1. Hence the bound is d for diagonal-dimension-zero local pieces and 2d+1 for proper Rouquier-dimension-zero local pieces over an algebraically closed field.\n\nCandidate contribution (comparison theorem and obstruction; novelty confidence low): Candidate novelty: the height-layer kernel formula Ddim(hocolim F) <= sum_r(1 + max_{h(i)=r} Ddim(F(i))) - 1 for perfect-admissible finite-poset Morita colimits, together with the derived consequences Ddim <= 2d+1 for proper Rouquier-zero pieces over an algebraically closed field and Ddim <= d for local diagonal dimension zero; the inseparable-field example proves that no finite unrestricted depth-to-diagonal bound exists." }, { "id": 20000020, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0020", "title": "Exact strict-subdimension cotangent families and a mirror obstruction", "statement": "What can be said about the Rouquier dimension of non-toric varieties and/for their symplectic mirrors?\n\\begin{itemize}\n\\item Toric varieties of dim n $\\longleftrightarrow$(mirror) ($T^*T^n$,some stop)\n\\item What known symplecticly for $(T^*M,stops)$ can be find in (Bai-Cote, Hanlon-Hicks-Lazarev, Favero-Huang)\n\\end{itemize}\ne.g. is there a symplectic $(X^{2n},f)$ such that $Rdim \\ Fuk(X,f)\\lneq n$? (for instance by finding $(X,f)$ mirror to a singular $Y$ with $Rdim(D^b(Y))>dim(Y)$)", "original_statement": "What can be said about the Rouquier dimension of non-toric varieties and/for their symplectic mirrors?\n\\begin{itemize}\n\\item Toric varieties of dim n $\\longleftrightarrow$(mirror) ($T^*T^n$,some stop)\n\\item What known symplecticly for $(T^*M,stops)$ can be find in (Bai-Cote, Hanlon-Hicks-Lazarev, Favero-Huang)\n\\end{itemize}\ne.g. is there a symplectic $(X^{2n},f)$ such that $Rdim \\ Fuk(X,f)\\lneq n$? (for instance by finding $(X,f)$ mirror to a singular $Y$ with $Rdim(D^b(Y))>dim(Y)$)", "clean_statement": "What can be said about the Rouquier dimension of non-toric varieties and/for their symplectic mirrors?\n\\begin{itemize}\n\\item Toric varieties of dim n $\\longleftrightarrow$(mirror) ($T^*T^n$,some stop)\n\\item What known symplecticly for $(T^*M,stops)$ can be find in (Bai-Cote, Hanlon-Hicks-Lazarev, Favero-Huang)\n\\end{itemize}\ne.g. is there a symplectic $(X^{2n},f)$ such that $Rdim \\ Fuk(X,f)\\lneq n$? (for instance by finding $(X,f)$ mirror to a singular $Y$ with $Rdim(D^b(Y))>dim(Y)$)", "statement_status": "exact", "statement_verification": "This record is Problem 7.5 in the AIM workshop list *Syzygies and mirror symmetry*, section “Orlov spectrum and Rouquier dimension.” The canonical record and the live AIM page were both checked. The page really contains \\lneq; the apparent conflict below is therefore present in the source, rather than being introduced by JSON extraction.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Orlov spectrum and Rouquier dimension\nSource item: 7.5\nSource URL: http://aimpl.org/syzygyms/7/\nCanonical location: aim-algebraic-geometry-notes.json notes[19]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What can be said about the Rouquier dimension of non-toric varieties and/for their symplectic mirrors?\\n\\\\begin{itemize}\\n\\\\item Toric varieties of dim n $\\\\longleftrightarrow$(mirror) ($T^*T^n$,some stop)\\n\\\\item What known symplecticly for $(T^*M,stops)$ can be find in (Bai-Cote, Hanlon-Hicks-Lazarev, Favero-Huang)\\n\\\\end{itemize}\\ne.g. is there a symplectic $(X^{2n},f)$ such that $Rdim \\\\ Fuk(X,f)\\\\lneq n$? (for instance by finding $(X,f)$ mirror to a singular $Y$ with $Rdim(D^b(Y))>dim(Y)$)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/7/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0020", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For M equal to a product of k spheres of dimensions at least two, with empty stop, canonical cotangent brane data, rational coefficients, and Z/2 grading, the split-closed wrapped Fukaya category has Rouquier dimension exactly k. This follows by matching Bai–Côté's k-variable central-action lower bound with Hanlon–Hicks–Lazarev's LS-category upper bound and the equality cat_LS(M)=k. Since dim(M) is at least 2k, this answers the source's literal strict-less-than question. It also implies that any reduced separated finite-type scheme Y whose bounded coherent derived category is equivalent to this wrapped category must have dim(Y) at most k. The parenthetical strict-greater-than reading remains open in high dimension, while Bai–Côté's theorem rules it out for polarizable Weinstein-pair mirrors in half-dimension at most three.\n\nCandidate contribution (theorem; novelty confidence low): Combining two previously separate bounds gives the exact formula Rdim H^0(Perf W(T* product_i S^{d_i}; Q)) = k for every d_i at least two, together with the testable mirror obstruction dim(Y) <= k for every reduced separated finite-type Y with an equivalent bounded coherent derived category." }, { "id": 20000021, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0021", "title": "Spherical twists must be formed after Cox localization", "statement": "What operations on Fukaya categories have meaningful commutative algebra translations? e.g. maybe a spherical twist (via mirror symmetry Dehn twist) around a spherical object supported on irrelevant virtual resolution, or more generally induces an aut (space of virtual resolutions)?", "original_statement": "What operations on Fukaya categories have meaningful commutative algebra translations? e.g. maybe a spherical twist (via mirror symmetry Dehn twist) around a spherical object supported on irrelevant virtual resolution, or more generally induces an aut (space of virtual resolutions)?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "* Section 4 asks for Morse theory and a moduli theory for virtual resolutions; Problem 4.6 explicitly asks for a “space/stacks” of virtual resolutions. * Section 6 concerns modules over a Cox ring. * Thus “induces an aut (space of virtual resolutions)” most plausibly abbreviates “induces an automorphism of the space of virtual resolutions”. This is a reconstruction, not verified source text. * “A spherical object supported on irrelevant virtual resolution” has two possible meanings: (i) the center itself has Cox homology supported on the irrelevant locus, or (ii) a nonzero spherical object on the toric variety is represented by a virtual resolution that is allowed irrelevant higher homology. The distinction is decisive. Under (i) the center is zero after sheafification; under (ii) it can define a genuine twist, but only in the quotient by irrelevant homology.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Fukaya categories\nSource item: 8.1\nSource URL: http://aimpl.org/syzygyms/8/\nCanonical location: aim-algebraic-geometry-notes.json notes[20]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What operations on Fukaya categories have meaningful commutative algebra translations? e.g. maybe a spherical twist (via mirror symmetry Dehn twist) around a spherical object supported on irrelevant virtual resolution, or more generally induces an aut (space of virtual resolutions)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/8/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0021", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the Cox ring of a smooth projective toric variety, a raw graded evaluation-cone twist has a sharp descent dichotomy: an irrelevant-supported center induces the identity after sheafification, while any center surviving the quotient sends some shifted irrelevant residue field outside the irrelevant-torsion subcategory and therefore cannot descend. The meaningful commutative-algebra translation is instead the spherical twist formed in the derived Cox quotient, with K-theory action [A] mapped to [A] - chi(E,A)[E].\n\nCandidate contribution (obstruction theorem; novelty confidence low): If P is a perfect Pic(X)-graded Cox complex, then its raw evaluation-cone twist either has q(P)=0 and induces the identity on the quotient, or q(P) is nonzero and there exists an internal shift a for which k(a) is irrelevant-torsion but the cone on k(a) has nonzero sheafification; hence no raw Cox spherical twist induces a nontrivial twist on the toric quotient." }, { "id": 20000022, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0022", "title": "Immaculate line bundles as unit-orthogonal toric branes", "statement": "[Wang, Borisov-Wang]: Found a condition on underlying fan for a toric stack to admit infinitely many acyclic line bundles (meaning $\\mathcal{L}$ with $H^*(\\mathcal{L})=0$ for all $*$, e.g. $\\mathcal{O}(-1)$ on $\\mathbb{P}^n$) (e.g. in $2D$ if fan has collinear rays).\n\n\\begin{enumerate}\n\\item Understand symplectic mirror of these criteria.\n\\item Understand consequences of generalizations of these critera (e.g. if there exists a toric subvariety whose fans is contained in a linear subspace) or if generalized.\n\\end{enumerate}", "original_statement": "[Wang, Borisov-Wang]: Found a condition on underlying fan for a toric stack to admit infinitely many acyclic line bundles (meaning $\\mathcal{L}$ with $H^*(\\mathcal{L})=0$ for all $*$, e.g. $\\mathcal{O}(-1)$ on $\\mathbb{P}^n$) (e.g. in $2D$ if fan has collinear rays).\n\n\\begin{enumerate}\n\\item Understand symplectic mirror of these criteria.\n\\item Understand consequences of generalizations of these critera (e.g. if there exists a toric subvariety whose fans is contained in a linear subspace) or if generalized.\n\\end{enumerate}", "clean_statement": "[Wang, Borisov-Wang]: Found a condition on underlying fan for a toric stack to admit infinitely many acyclic line bundles (meaning $\\mathcal{L}$ with $H^*(\\mathcal{L})=0$ for all $*$, e.g. $\\mathcal{O}(-1)$ on $\\mathbb{P}^n$) (e.g. in $2D$ if fan has collinear rays).\n\n\\begin{enumerate}\n\\item Understand symplectic mirror of these criteria.\n\\item Understand consequences of generalizations of these critera (e.g. if there exists a toric subvariety whose fans is contained in a linear subspace) or if generalized.\n\\end{enumerate}", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 9.1 in the section “Line bundles over toric stacks” of *Syzygies and mirror symmetry*. The live AIM page was checked on 2026-07-22 and agrees with the record in `input.json`. Its mathematical content is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Syzygies and mirror symmetry\nSection: Line bundles over toric stacks\nSource item: 9.1\nSource URL: http://aimpl.org/syzygyms/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[21]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"[Wang, Borisov-Wang]: Found a condition on underlying fan for a toric stack to admit infinitely many acyclic line bundles (meaning $\\\\mathcal{L}$ with $H^*(\\\\mathcal{L})=0$ for all $*$, e.g. $\\\\mathcal{O}(-1)$ on $\\\\mathbb{P}^n$) (e.g. in $2D$ if fan has collinear rays).\\n\\n\\\\begin{enumerate}\\n\\\\item Understand symplectic mirror of these criteria.\\n\\\\item Understand consequences of generalizations of these critera (e.g. if there exists a toric subvariety whose fans is contained in a linear subspace) or if generalized.\\n\\\\end{enumerate}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/syzygyms/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0022", "aim-domain:algebraic-geometry", "aim-workshop:syzygyms", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every smooth complete toric variety or toric orbifold over the complex numbers in the scope of nonequivariant coherent-constructible correspondence, a line bundle is immaculate exactly when its CCC Picard object is right orthogonal to the object corresponding to the structure sheaf; under the usual opposite-category stopped-cotangent convention this is the vanishing of HW*(A_L,A_O). For every product of projective spaces, the attempt proves the exact immaculate locus, all tempting subsets and forbidden orthants, all accumulation directions, and the corresponding constructible and wrapped-Floer Hom vanishing. In particular, P^2 x P^2 has no collinear fan rays but has four infinite affine families. A relative pushforward lemma gives a precise fibration version of the proposed subspace generalization.\n\nCandidate contribution (explicit_family_proposition; novelty confidence low): For X equal to a product of projective spaces, the cohomology-window description of immaculate bundles, the complete forbidden-cone orthant arrangement, the union of projective coordinate hyperplanes of accumulation directions, and unit-orthogonality of the corresponding CCC/stopped-cotangent Picard branes form one exact four-way dictionary; for P^2 x P^2 this gives exactly the four affine lattice lines a=-1, a=-2, b=-1, and b=-2 despite the absence of collinear fan rays." }, { "id": 20000023, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0023", "title": "Tate-completion obstruction and finite torsion windows for the motivic filtration on TC of the sphere", "statement": "The motivic filtration on the algebraic $K$-theory of the sphere spectrum\n\nDue to work of Rognes, Blumberg--Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn--Raksit--Wilson \\cite{arXiv:2206.11208} gives a motivic filtration on TC of $\\mathbb{E}_{\\infty}$-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.\n\nCan one understand the motivic filtration (or motivic spectral sequence) on the $\\mathrm{TC}$ of the sphere spectrum? What about for the motivic filtration on related invariants such as $\\mathrm{THH}$, $\\mathrm{TP}$?", "original_statement": "The motivic filtration on the algebraic $K$-theory of the sphere spectrum\n\nDue to work of Rognes, Blumberg--Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn--Raksit--Wilson \\cite{arXiv:2206.11208} gives a motivic filtration on TC of $\\mathbb{E}_{\\infty}$-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.\n\nCan one understand the motivic filtration (or motivic spectral sequence) on the $\\mathrm{TC}$ of the sphere spectrum? What about for the motivic filtration on related invariants such as $\\mathrm{THH}$, $\\mathrm{TP}$?", "clean_statement": "The motivic filtration on the algebraic $K$-theory of the sphere spectrum\n\nDue to work of Rognes, Blumberg--Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn--Raksit--Wilson \\cite{arXiv:2206.11208} gives a motivic filtration on TC of $\\mathbb{E}_{\\infty}$-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.\n\nCan one understand the motivic filtration (or motivic spectral sequence) on the $\\mathrm{TC}$ of the sphere spectrum? What about for the motivic filtration on related invariants such as $\\mathrm{THH}$, $\\mathrm{TP}$?", "statement_status": "exact", "statement_verification": "This is problem 2.1 in the AIM workshop list *Equivariant techniques in stable homotopy theory*, section *Computations in algebraic K-theory*. The repository record and the live AIM page were compared on 2026-07-22. The mathematical question is uncorrupted:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Computations in algebraic K-theory\nSource item: 2.1\nSource URL: http://aimpl.org/equivstable/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[22]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The motivic filtration on the algebraic $K$-theory of the sphere spectrum\\n\\nDue to work of Rognes, Blumberg--Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn--Raksit--Wilson \\\\cite{arXiv:2206.11208} gives a motivic filtration on TC of $\\\\mathbb{E}_{\\\\infty}$-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.\\n\\nCan one understand the motivic filtration (or motivic spectral sequence) on the $\\\\mathrm{TC}$ of the sphere spectrum? What about for the motivic filtration on related invariants such as $\\\\mathrm{THH}$, $\\\\mathrm{TP}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0023", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The HRW filtration structurally applies to THH, TC-minus, TP, and p-complete TC of the sphere, and its rational TC object is known. This attempt proves that rationalization and finite Tate fixed points do not commute on the p-complete sphere, identifies the resulting diagnostic zero-Frobenius discrepancy exactly with the two rational weight-zero summands, and combines the rational splitting with Antieau--Riggenbach's amplitude theorem to confine every unresolved positive-weight group to a finite p-primary torsion window. It also records the Adams--Novikov calculation of filtered THH and proves a conditional integral weight-zero calculation under explicit negative-weight vanishing.\n\nCandidate contribution (obstruction; novelty confidence low): For every prime p, the failure of rationalization to commute with the C_p-Tate construction on the p-complete sphere accounts exactly for the HQ_p plus Sigma^{-1}HQ_p rational weight-zero discrepancy between the genuine HRW-filtered TC calculation and the diagnostic zero-Frobenius fiber; for weight i at least 1, all remaining unknown groups are p-primary torsion in stems i-1 through 2i, and under negative-weight vanishing the integral weight-zero object is HZ_p plus Sigma^{-1}HZ_p." }, { "id": 20000024, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0024", "title": "Rational cyclotomic invariants and a connective-boundary obstruction", "statement": "The $K$-theory of Johnson--Wilson theory\n\nThe algebraic $K$-theory of Johnson--Wilson theory $E(n)$ is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson--Wilson theory $E'(n)$.\n\nCompute something about $\\mathrm{TC}$ (and related invariants such as $\\mathrm{TP}, \\mathrm{THH}$) of $\\tau_{\\geq0}E'(n)$, where $E'(n)$ is the $2$-periodic Johnson--Wilson theory.", "original_statement": "The $K$-theory of Johnson--Wilson theory\n\nThe algebraic $K$-theory of Johnson--Wilson theory $E(n)$ is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson--Wilson theory $E'(n)$.\n\nCompute something about $\\mathrm{TC}$ (and related invariants such as $\\mathrm{TP}, \\mathrm{THH}$) of $\\tau_{\\geq0}E'(n)$, where $E'(n)$ is the $2$-periodic Johnson--Wilson theory.", "clean_statement": "The $K$-theory of Johnson--Wilson theory\n\nThe algebraic $K$-theory of Johnson--Wilson theory $E(n)$ is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson--Wilson theory $E'(n)$.\n\nCompute something about $\\mathrm{TC}$ (and related invariants such as $\\mathrm{TP}, \\mathrm{THH}$) of $\\tau_{\\geq0}E'(n)$, where $E'(n)$ is the $2$-periodic Johnson--Wilson theory.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 2.2 in the AIM list *Equivariant techniques in stable homotopy theory*, in the section \"Computations in algebraic K-theory.\" Its exact mathematical request is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Computations in algebraic K-theory\nSource item: 2.2\nSource URL: http://aimpl.org/equivstable/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[23]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The $K$-theory of Johnson--Wilson theory\\n\\nThe algebraic $K$-theory of Johnson--Wilson theory $E(n)$ is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson--Wilson theory $E'(n)$.\\n\\nCompute something about $\\\\mathrm{TC}$ (and related invariants such as $\\\\mathrm{TP}, \\\\mathrm{THH}$) of $\\\\tau_{\\\\geq0}E'(n)$, where $E'(n)$ is the $2$-periodic Johnson--Wilson theory.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0024", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a specified E-infinity realization of the root-normalized uncompleted p-local two-periodic Johnson--Wilson model, the invariants of the rationalized connective input are computed by the complete mixed de Rham complexes (Omega_B[[t]], t d) and (Omega_B((t)), t d): THH has homotopy Omega_B, TC-minus has homotopy Q[[t]] plus the exact-form ideal, TP has homotopy Q((t)), and uncompleted p-typical TC is the fiber of the canonical inclusion. Independently, the coefficient comparison from the small connective root object to the Postnikov connective cover is non-flat at every height n at least 2; an explicit Koszul Tor_1 class detects this, and its relative cotangent complex is the direct sum of two-term complexes C --beta^(p^i-1)--> C.\n\nCandidate contribution (obstruction; novelty confidence low): For every n at least 2 in the stated root-normalized coefficient model, the comparison D=Z_(p)[v_1,...,v_(n-1),beta] to C=Z_(p)[a_1,...,a_(n-1),beta], v_i mapping to a_i beta^(p^i-1), is non-flat, with an explicit nonzero Koszul Tor_1 class and relative cotangent complex equal to the direct sum over i of [C --beta^(p^i-1)--> C]." }, { "id": 20000025, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0025", "title": "Unit propagation and a trace-Bockstein reduction for Greek-letter redshift", "statement": "Detecting Greek letter families in algebraic $K$-theory\n\nProve Conjecture 1.1 of \\cite{arXiv:1810.10088}. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the $K$-theory of an $\\mathbb{E}_{\\infty}$-ring detecting the $\\alpha^{(n)}$-family detects the $\\alpha^{(n+1)}$-family.", "original_statement": "Detecting Greek letter families in algebraic $K$-theory\n\nProve Conjecture 1.1 of \\cite{arXiv:1810.10088}. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the $K$-theory of an $\\mathbb{E}_{\\infty}$-ring detecting the $\\alpha^{(n)}$-family detects the $\\alpha^{(n+1)}$-family.", "clean_statement": "Detecting Greek letter families in algebraic $K$-theory\n\nProve Conjecture 1.1 of \\cite{arXiv:1810.10088}. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the $K$-theory of an $\\mathbb{E}_{\\infty}$-ring detecting the $\\alpha^{(n)}$-family detects the $\\alpha^{(n+1)}$-family.", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Computations in algebraic K-theory\nSource item: 2.3\nSource URL: http://aimpl.org/equivstable/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[24]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Detecting Greek letter families in algebraic $K$-theory\\n\\nProve Conjecture 1.1 of \\\\cite{arXiv:1810.10088}. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the $K$-theory of an $\\\\mathbb{E}_{\\\\infty}$-ring detecting the $\\\\alpha^{(n)}$-family detects the $\\\\alpha^{(n+1)}$-family.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0025", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Conjecture 1.1 remains open in its named-element, unit-map form. Two rigorous partial advances are established. First, unit naturality propagates the known finite-field result: if an E-infinity ring A admits a unital map to B_q = K(F_q)_p-complete, where q generates (Z/p^2)^times and p is at least 5, then A detects the alpha-family and K(A) detects every beta_s with p not dividing s, with nonvanishing already witnessed in the finite stage T_s(B_q). Second, for every E_1 MU-algebra form C of BP<2> covered by arXiv:2602.14380v3, the 2026 Hurewicz calculation forces the V(2)-class i_2 j_2(v_3) to be nonzero in V(2)_*K(C); detection of the stable gamma_1 is exactly equivalent to nonvanishing after the two remaining boundary maps j_1 and j_0. This is a two-Bockstein reduction, not a proof of gamma_1 detection.\n\nCandidate contribution (reduction; novelty confidence low): At every prime p at least 5 and for every E_1 MU-algebra form C of BP<2> in the scope of arXiv:2602.14380v3, the natural class i_2 j_2(v_3) is nonzero in V(2)_*K(C) because its cyclotomic trace is Xi_{3,1}; moreover, the unit image of stable gamma_1 is nonzero if and only if j_0 j_1 is nonzero on the natural V(1)-valued K(C)-lift of j_2(v_3). Thus precisely the v_1- and p-Bocksteins, in that order, separate the 2026 computation from gamma_1 detection." }, { "id": 20000026, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0026", "title": "A prime-support reduction for iterated Atiyah--Segal completion", "statement": "Atiyah--Segal completion thoerem for iterated $K$-theory\n\nCan one prove an Atiyah--Segal completion theorem for the algebraic $K$-theory of topological $K$-theory?", "original_statement": "Atiyah--Segal completion thoerem for iterated $K$-theory\n\nCan one prove an Atiyah--Segal completion theorem for the algebraic $K$-theory of topological $K$-theory?", "clean_statement": "Atiyah--Segal completion thoerem for iterated $K$-theory\n\nCan one prove an Atiyah--Segal completion theorem for the algebraic $K$-theory of topological $K$-theory?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Equivariant techniques in stable homotopy theory*, section “Computations in algebraic K-theory,” Problem 2.4) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Computations in algebraic K-theory\nSource item: 2.4\nSource URL: http://aimpl.org/equivstable/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[25]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Atiyah--Segal completion thoerem for iterated $K$-theory\\n\\nCan one prove an Atiyah--Segal completion theorem for the algebraic $K$-theory of topological $K$-theory?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0026", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite group G, every bounded-below genuine equivariant refinement of K(ku) or K(KU) whose representation action, restriction, transfer, and trivial underlying action satisfy four explicit compatibility axioms obeys fixed-to-Borel Atiyah--Segal completion after localizing first at 1/|G|. The derived augmentation-ideal completion is exactly the summand cut out by the normalized regular representation e_G=[C[G]]/|G|, restriction identifies this summand with the underlying iterated K-theory spectrum, and the inverse is (1/|G|) times transfer. A separate C_p calculation proves that ordinary degreewise completion is already inadequate for connective ku, so an integral spectrum-level theorem must use derived completion.\n\nCandidate contribution (conditional theorem; novelty confidence low): After localizing the entire fixed-to-Borel problem at Z[1/|G|], every compatible genuine iterated K-theory model satisfies Atiyah--Segal completion, with completion explicitly equal to projection by e_G=[C[G]]/|G|; for C2 this is the (1+sigma)/2 summand and the inverse comparison is half-transfer.", "research_difficulty_suggested": "L4" }, { "id": 20000027, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0027", "title": "A half-line obstruction and a filtered Poitou-Tate lifting criterion", "statement": "Motivic filtered Poitou-Tate duality\n\nIs there a duality at the level of motivic filtered spectra $\\mathrm{fil}_{\\text{mot}}\\mathrm{TC}(\\mathcal{O}_K)$ that recovers Poitou--Tate duality at the level of associated graded?", "original_statement": "Motivic filtered Poitou-Tate duality\n\nIs there a duality at the level of motivic filtered spectra $\\mathrm{fil}_{\\text{mot}}\\mathrm{TC}(\\mathcal{O}_K)$ that recovers Poitou--Tate duality at the level of associated graded?", "clean_statement": "Motivic filtered Poitou-Tate duality\n\nIs there a duality at the level of motivic filtered spectra $\\mathrm{fil}_{\\text{mot}}\\mathrm{TC}(\\mathcal{O}_K)$ that recovers Poitou--Tate duality at the level of associated graded?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 2.5 in the AIM workshop list *Equivariant techniques in stable homotopy theory*, section “Computations in algebraic K-theory.” Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Computations in algebraic K-theory\nSource item: 2.5\nSource URL: http://aimpl.org/equivstable/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[26]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Motivic filtered Poitou-Tate duality\\n\\nIs there a duality at the level of motivic filtered spectra $\\\\mathrm{fil}_{\\\\text{mot}}\\\\mathrm{TC}(\\\\mathcal{O}_K)$ that recovers Poitou--Tate duality at the level of associated graded?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0027", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For finite p-power coefficients, the unperiodized connective BMS/HRW motivic filtration on TC of the p-adic completion of a number ring cannot admit a literal perfect Poitou-Tate self-duality: its negative graded weights vanish, while every weight n at least 2 is nonzero by syntomic-to-etale comparison and the local Euler-Poincare formula. Classical Poitou-Tate duality and the motivic shifts force any corrected formulation to pair a periodicized compact-support object with a global object in complementary weights n and 1-n and total shift -1. An explicit two-layer defect class measures the first lift from graded duality, and compatible Adams operations kill weight-gap obstructions not divisible by p-1 at odd p.\n\nCandidate contribution (obstruction; novelty confidence low): The connective motivic filtered TC object has a half-line obstruction to Poitou-Tate self-duality: gr^q vanishes for q<0 but gr^n is nonzero for all n>=2, contradicting perfect pairings of weights n and 1-n. For a periodicized global/compact-support repair, the report also gives the exact two-stage Poitou-Tate defect class and proves that Adams-equivariant defects of weight gap not divisible by p-1 vanish at odd p." }, { "id": 20000028, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0028", "title": "A uniform-connectivity criterion for the spectral motivic filtration on p-typical TR", "statement": "Motivic filtration on $\\mathrm{TR}$\n\n\\cite{arXiv:2206.11208} gives a motivic filtration on $\\mathrm{THH}, \\mathrm{TC}, \\mathrm{TC^{-}}, \\mathrm{TP}$ of ring spectra generalizing previously defined motivic filtrations such as in \\cite{arXiv:1802.03261}. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on $\\mathrm{TR}$ of discrete rings.\n\nCan one define a well behaved motivic filtration on the $\\mathrm{TR}$ of ring spectra? Can one compute it in good cases?", "original_statement": "Motivic filtration on $\\mathrm{TR}$\n\n\\cite{arXiv:2206.11208} gives a motivic filtration on $\\mathrm{THH}, \\mathrm{TC}, \\mathrm{TC^{-}}, \\mathrm{TP}$ of ring spectra generalizing previously defined motivic filtrations such as in \\cite{arXiv:1802.03261}. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on $\\mathrm{TR}$ of discrete rings.\n\nCan one define a well behaved motivic filtration on the $\\mathrm{TR}$ of ring spectra? Can one compute it in good cases?", "clean_statement": "Motivic filtration on $\\mathrm{TR}$\n\n\\cite{arXiv:2206.11208} gives a motivic filtration on $\\mathrm{THH}, \\mathrm{TC}, \\mathrm{TC^{-}}, \\mathrm{TP}$ of ring spectra generalizing previously defined motivic filtrations such as in \\cite{arXiv:1802.03261}. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on $\\mathrm{TR}$ of discrete rings.\n\nCan one define a well behaved motivic filtration on the $\\mathrm{TR}$ of ring spectra? Can one compute it in good cases?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 2.7 in the AIM workshop list *Equivariant techniques in stable homotopy theory*, section “Computations in algebraic K-theory.” It is record 27 (zero-based) of `aim-algebraic-geometry-notes.json`. Its statement is intact:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Computations in algebraic K-theory\nSource item: 2.7\nSource URL: http://aimpl.org/equivstable/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[27]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Motivic filtration on $\\\\mathrm{TR}$\\n\\n\\\\cite{arXiv:2206.11208} gives a motivic filtration on $\\\\mathrm{THH}, \\\\mathrm{TC}, \\\\mathrm{TC^{-}}, \\\\mathrm{TP}$ of ring spectra generalizing previously defined motivic filtrations such as in \\\\cite{arXiv:1802.03261}. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on $\\\\mathrm{TR}$ of discrete rings.\\n\\nCan one define a well behaved motivic filtration on the $\\\\mathrm{TR}$ of ring spectra? Can one compute it in good cases?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0028", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Antieau--Riggenbach's cyclotomic synthetic right adjoint now gives a complete, exhaustive, multiplicative p-typical motivic filtration on TR for chromatically p-quasisyntomic E-infinity ring spectra. A proved formal criterion shows that uniform coconnectivity over every finite fixed/Tate stage is sufficient for the countable equalizer defining TR to be exhaustive, and right-adjoint descent yields an associated-graded Cech totalization formula. A diagonal product example proves that separate stagewise exhaustiveness is insufficient. This is a partial rather than universal solution because arbitrary ring spectra, integral TR, and explicit spectral associated grades remain open.\n\nCandidate contribution (lemma; novelty confidence low): If the comparison from every filtration stage of every finite fixed/Tate term to its ordinary term has fiber in Sp_{≤c(i)}, uniformly in the fixed-point level, with c(i) tending to minus infinity, then the synthetic TR equalizer is complete and exhaustive; moreover its associated graded is computed by Cech totalization over an even cover. The included diagonal product example shows the uniformity hypothesis cannot be replaced by separate exhaustiveness." }, { "id": 20000029, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0029", "title": "Odd-prime solution and completion-transfer reduction", "statement": "Calculation of $\\mathrm{TP}(\\mathbb{Z})$\n\nBokstedt and Madsen computed $\\mathrm{TC}(\\mathbb{Z})$, without calculating $\\mathrm{TP}(\\mathbb{Z})$. However, Hesselholt and Madsen calculated (in \\cite{MR1317575}) $(L_{K(1)} \\mathbb{S})^{tS^1}$. Their results suggest the following:\n\nFix a prime $p$. Let $j = \\tau_{\\geq 0} L_{K(1)} \\mathbb{S}$. Then, $\\mathrm{TP}(\\mathbb{Z}_p) \\simeq j^{tS^1}$, for the trivial $S^1$-action on $j$.", "original_statement": "Calculation of $\\mathrm{TP}(\\mathbb{Z})$\n\nBokstedt and Madsen computed $\\mathrm{TC}(\\mathbb{Z})$, without calculating $\\mathrm{TP}(\\mathbb{Z})$. However, Hesselholt and Madsen calculated (in \\cite{MR1317575}) $(L_{K(1)} \\mathbb{S})^{tS^1}$. Their results suggest the following:\n\nFix a prime $p$. Let $j = \\tau_{\\geq 0} L_{K(1)} \\mathbb{S}$. Then, $\\mathrm{TP}(\\mathbb{Z}_p) \\simeq j^{tS^1}$, for the trivial $S^1$-action on $j$.", "clean_statement": "Calculation of $\\mathrm{TP}(\\mathbb{Z})$\n\nBokstedt and Madsen computed $\\mathrm{TC}(\\mathbb{Z})$, without calculating $\\mathrm{TP}(\\mathbb{Z})$. However, Hesselholt and Madsen calculated (in \\cite{MR1317575}) $(L_{K(1)} \\mathbb{S})^{tS^1}$. Their results suggest the following:\n\nFix a prime $p$. Let $j = \\tau_{\\geq 0} L_{K(1)} \\mathbb{S}$. Then, $\\mathrm{TP}(\\mathbb{Z}_p) \\simeq j^{tS^1}$, for the trivial $S^1$-action on $j$.", "statement_status": "exact", "statement_verification": "The record adds that this would follow, for example, from \\(\\operatorname{TR}(\\mathbb Z_p)\\simeq j_p\\). There is no visible corruption in the record. The citation `MR1317575` is Hesselholt--Madsen, *The \\(S^1\\)-Tate spectrum for \\(J\\)*.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Computations in algebraic K-theory\nSource item: 2.8\nSource URL: http://aimpl.org/equivstable/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[28]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Calculation of $\\\\mathrm{TP}(\\\\mathbb{Z})$\\n\\nBokstedt and Madsen computed $\\\\mathrm{TC}(\\\\mathbb{Z})$, without calculating $\\\\mathrm{TP}(\\\\mathbb{Z})$. However, Hesselholt and Madsen calculated (in \\\\cite{MR1317575}) $(L_{K(1)} \\\\mathbb{S})^{tS^1}$. Their results suggest the following:\\n\\nFix a prime $p$. Let $j = \\\\tau_{\\\\geq 0} L_{K(1)} \\\\mathbb{S}$. Then, $\\\\mathrm{TP}(\\\\mathbb{Z}_p) \\\\simeq j^{tS^1}$, for the trivial $S^1$-action on $j$.\"\nOriginal remarks: [\"This would follow if, for instance, $\\\\mathrm{TR}(\\\\mathbb{Z}_p) \\\\simeq j$.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0029", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every odd prime p, the continuous p-complete form of the AIM conjecture follows from Devalapurkar--Raksit's 2025 equivalence between p-completed TP(Z) and (j_p^triv)^{tS1}. A proved formal lemma shows that any mod-p equivalence of E1-ring spectra induces an equivalence on p-completed TP; applying it to HZ -> HZ_p transfers the published theorem to the exact AIM input. The p=2 case is not established in the literature checked and reduces to comparing 2-completed TP(Z) with (j_2^triv)^{tS1}, with a necessary mod-2 ring-and-Bockstein test.\n\nCandidate contribution (reduction; novelty confidence low): If A -> B is a mod-p equivalence of E1-ring spectra, then the induced map on p-completed TP is an equivalence; hence completed TP(Z_p) is canonically equivalent to completed TP(Z) for every prime, resolving the input mismatch at odd primes and reducing the remaining exact AIM quantifier to one two-primary comparison with an explicit mod-2 necessary test.", "research_difficulty_suggested": "L4" }, { "id": 20000030, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0030", "title": "A coefficient-ring shadow and coherence obstruction for the Pin(2)-Q8 Tate model", "statement": "$L$-theory of integers\n\nThe symmetric and normal $L$-theory of the integers was computed in \\cite{arXiv:2004.06889}, and the Tate cohomology of $Q_8$ was computed by Atiyah in \\cite{MR0148722}. The normal $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $Q_8$; similarly, the symmetric $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $\\mathrm{Pin}(2)$.\n\nEquip $\\mathrm{Pin}(2)$ (resp. $Q_8$) with the canonical $\\mathbb{Z}/2$-action (resp. the action given by conjugation by $i$). Then, the $E_\\infty$-map $L^s(\\mathbb{Z})_{(2)} \\to L^n(\\mathbb{Z})$ can be identified by taking strict $\\mathbb{Z}/2$-fixed points of the map $\\mathbb{Z}_{(2)}^{t\\mathrm{Pin}(2)} \\to \\mathbb{Z}^{tQ_8}$. (This map is induced by the inclusion $Q_8\\subseteq \\mathrm{Pin}(2)$.)", "original_statement": "$L$-theory of integers\n\nThe symmetric and normal $L$-theory of the integers was computed in \\cite{arXiv:2004.06889}, and the Tate cohomology of $Q_8$ was computed by Atiyah in \\cite{MR0148722}. The normal $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $Q_8$; similarly, the symmetric $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $\\mathrm{Pin}(2)$.\n\nEquip $\\mathrm{Pin}(2)$ (resp. $Q_8$) with the canonical $\\mathbb{Z}/2$-action (resp. the action given by conjugation by $i$). Then, the $E_\\infty$-map $L^s(\\mathbb{Z})_{(2)} \\to L^n(\\mathbb{Z})$ can be identified by taking strict $\\mathbb{Z}/2$-fixed points of the map $\\mathbb{Z}_{(2)}^{t\\mathrm{Pin}(2)} \\to \\mathbb{Z}^{tQ_8}$. (This map is induced by the inclusion $Q_8\\subseteq \\mathrm{Pin}(2)$.)", "clean_statement": "$L$-theory of integers\n\nThe symmetric and normal $L$-theory of the integers was computed in \\cite{arXiv:2004.06889}, and the Tate cohomology of $Q_8$ was computed by Atiyah in \\cite{MR0148722}. The normal $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $Q_8$; similarly, the symmetric $L$-theory of $\\mathbb{Z}$ looks very similar to the Tate cohomology of $\\mathrm{Pin}(2)$.\n\nEquip $\\mathrm{Pin}(2)$ (resp. $Q_8$) with the canonical $\\mathbb{Z}/2$-action (resp. the action given by conjugation by $i$). Then, the $E_\\infty$-map $L^s(\\mathbb{Z})_{(2)} \\to L^n(\\mathbb{Z})$ can be identified by taking strict $\\mathbb{Z}/2$-fixed points of the map $\\mathbb{Z}_{(2)}^{t\\mathrm{Pin}(2)} \\to \\mathbb{Z}^{tQ_8}$. (This map is induced by the inclusion $Q_8\\subseteq \\mathrm{Pin}(2)$.)", "statement_status": "exact", "statement_verification": "This is Conjecture 2.9, “\\(L\\)-theory of integers,” from the AIM workshop *Equivariant techniques in stable homotopy theory* (October 2022). The canonical record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Computations in algebraic K-theory\nSource item: 2.9\nSource URL: http://aimpl.org/equivstable/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[29]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"$L$-theory of integers\\n\\nThe symmetric and normal $L$-theory of the integers was computed in \\\\cite{arXiv:2004.06889}, and the Tate cohomology of $Q_8$ was computed by Atiyah in \\\\cite{MR0148722}. The normal $L$-theory of $\\\\mathbb{Z}$ looks very similar to the Tate cohomology of $Q_8$; similarly, the symmetric $L$-theory of $\\\\mathbb{Z}$ looks very similar to the Tate cohomology of $\\\\mathrm{Pin}(2)$.\\n\\nEquip $\\\\mathrm{Pin}(2)$ (resp. $Q_8$) with the canonical $\\\\mathbb{Z}/2$-action (resp. the action given by conjugation by $i$). Then, the $E_\\\\infty$-map $L^s(\\\\mathbb{Z})_{(2)} \\\\to L^n(\\\\mathbb{Z})$ can be identified by taking strict $\\\\mathbb{Z}/2$-fixed points of the map $\\\\mathbb{Z}_{(2)}^{t\\\\mathrm{Pin}(2)} \\\\to \\\\mathbb{Z}^{tQ_8}$. (This map is induced by the inclusion $Q_8\\\\subseteq \\\\mathrm{Pin}(2)$.)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0030", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the semidirect-product interpretation of the canonical C2-actions, the ordinary Pin(2) and Q8 Tate coefficient rings, with natural generators and subgroup restriction, become exactly the known symmetric-to-normal L-theory coefficient map after a fixed-point candidate regrading. Any genuine lifts detecting these generators are uniquely forced into RO(C2)-degrees 4, 1-3 sigma, and -1+3 sigma. In addition, conjugation by i is cohomologically trivial but its inner trivialization has the nonzero coherence class [-1] in H^2(C2; Z(K)), so innerness alone cannot justify discarding the genuine action datum.\n\nCandidate contribution (reduction; novelty confidence low): The natural generator-level restriction square, together with the uniquely forced RO(C2)-degrees 1-3 sigma and -1+3 sigma and the explicit nonzero inner-action coherence class [-1], reduces the conjecture to detecting those exact permanent classes and the multiplicative extension AF=4 in the parametrized Tate construction.", "research_difficulty_suggested": "L4" }, { "id": 20000031, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0031", "title": "The L-theoretic obstruction to hermitian devissage", "statement": "Devissage for equivariant/hermitian $K$-theory\n\nWe understand devissage for the $K$-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund--Levy (see \\cite{arXiv:2112.14723}). See section 2.2 of \\cite{arXiv:2009.07225} for what is known for hermitian $K$-theory.\n\nWhat is the optimal devissage result for equivariant/hermitian $K$-theory?", "original_statement": "Devissage for equivariant/hermitian $K$-theory\n\nWe understand devissage for the $K$-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund--Levy (see \\cite{arXiv:2112.14723}). See section 2.2 of \\cite{arXiv:2009.07225} for what is known for hermitian $K$-theory.\n\nWhat is the optimal devissage result for equivariant/hermitian $K$-theory?", "clean_statement": "Devissage for equivariant/hermitian $K$-theory\n\nWe understand devissage for the $K$-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund--Levy (see \\cite{arXiv:2112.14723}). See section 2.2 of \\cite{arXiv:2009.07225} for what is known for hermitian $K$-theory.\n\nWhat is the optimal devissage result for equivariant/hermitian $K$-theory?", "statement_status": "exact", "statement_verification": "The source record has no remarks or supplied bibliography beyond those two identifiers. Both were checked: they are Burklund--Levy, *On the \\(K\\)-theory of regular coconnective rings*, and Calmès--Dotto--Harpaz--Hebestreit--Land--Moi--Nardin--Nikolaus--Steimle, *Hermitian \\(K\\)-theory for stable \\(\\infty\\)-categories III: Grothendieck--Witt groups of rings*. The latter was revised as arXiv v4 on 27 April 2026 and accepted by the *Annals of Mathematics*. There is no apparent corruption in the statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Computations in algebraic K-theory\nSource item: 2.6\nSource URL: http://aimpl.org/equivstable/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[30]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Devissage for equivariant/hermitian $K$-theory\\n\\nWe understand devissage for the $K$-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund--Levy (see \\\\cite{arXiv:2112.14723}). See section 2.2 of \\\\cite{arXiv:2009.07225} for what is known for hermitian $K$-theory.\\n\\nWhat is the optimal devissage result for equivariant/hermitian $K$-theory?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0031", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a Poincare functor whose underlying exact functor satisfies the Burklund-Levy unipotence hypotheses and whose source has a self-dual bounded t-structure, the induced target t-structure is self-dual, the map on K-theory with duality action is an equivalence, and Grothendieck-Witt devissage is equivalent to L-theory devissage and to genuine C2-equivariant real K-theory devissage. Polarization from genuine quadratic to genuine symmetric forms on Perf(F_2) satisfies all underlying K-theoretic and duality hypotheses but fails on GW and L, proving that the L-theoretic condition cannot be omitted.\n\nCandidate contribution (obstruction; novelty confidence low): The candidate contribution is the combined self-duality inheritance and exact L-obstruction principle, together with the minimal F_2 polarization counterexample: under Burklund-Levy unipotence the target heart automatically inherits self-duality, but hermitian devissage holds exactly when the L-map does, and the identity underlying functor Perf(F_2) with quadratic-to-symmetric polarization shows this extra condition is sharp." }, { "id": 20000032, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0032", "title": "The filtered geometric-fixed-point model and an odd-primary answer for a-inverted real Artin--Tate spectra", "statement": "a inverted Artin--Tate $\\mathbb{R}$-motives\n\nThe category of Artin--Tate $\\mathbb{R}$-motives has a topological model as a $1$-parameter deformation of $C_2$-equivariant homotopy theory \\cite{arXiv:2010.10325}. Let $\\Sp_{\\mathbb{R}}^{AT}$ denote this category.\n\nWhat category do you get when you invert the class $a$ in $\\Sp_{\\mathbb{R}}^{AT}$?", "original_statement": "a inverted Artin--Tate $\\mathbb{R}$-motives\n\nThe category of Artin--Tate $\\mathbb{R}$-motives has a topological model as a $1$-parameter deformation of $C_2$-equivariant homotopy theory \\cite{arXiv:2010.10325}. Let $\\Sp_{\\mathbb{R}}^{AT}$ denote this category.\n\nWhat category do you get when you invert the class $a$ in $\\Sp_{\\mathbb{R}}^{AT}$?", "clean_statement": "a inverted Artin--Tate $\\mathbb{R}$-motives\n\nThe category of Artin--Tate $\\mathbb{R}$-motives has a topological model as a $1$-parameter deformation of $C_2$-equivariant homotopy theory \\cite{arXiv:2010.10325}. Let $\\Sp_{\\mathbb{R}}^{AT}$ denote this category.\n\nWhat category do you get when you invert the class $a$ in $\\Sp_{\\mathbb{R}}^{AT}$?", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Motivic, equivariant, and synthetic spectra\nSource item: 3.2\nSource URL: http://aimpl.org/equivstable/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[31]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"a inverted Artin--Tate $\\\\mathbb{R}$-motives\\n\\nThe category of Artin--Tate $\\\\mathbb{R}$-motives has a topological model as a $1$-parameter deformation of $C_2$-equivariant homotopy theory \\\\cite{arXiv:2010.10325}. Let $\\\\Sp_{\\\\mathbb{R}}^{AT}$ denote this category.\\n\\nWhat category do you get when you invert the class $a$ in $\\\\Sp_{\\\\mathbb{R}}^{AT}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0032", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the intended 2-primary category, a-inversion is the smashing monoidal localization that kills the free Artin orbit, and under Burklund--Hahn--Senger Galois reconstruction it is exactly the category of modules in filtered 2-complete spectra over the levelwise geometric-fixed-point ring Phi^{C2} R_bullet. It is compactly generated by the two-graded spheres S^{r,0,w}[a^{-1}], with an explicit homotopy-group telescope formula. A further proved corollary gives a complete answer at every odd prime p: the a-inverted ip-linear real Artin--Tate category is symmetric monoidally equivalent to Sp_{ip}, and real realization sends S^{r,q,w} to S^r. A familiar closed-form identification of the 2-primary filtered ring remains open.\n\nCandidate contribution (corollary; novelty confidence low): For every odd prime p, real realization induces a symmetric monoidal equivalence Mod_{SH(R)^{AT}_{ip}}(S_p[a^{-1}]) ≃ Sp_{ip}, sending every S_p^{r,q,w}[a^{-1}] to S_p^r." }, { "id": 20000033, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0033", "title": "A two-primary Euler-divisibility defect for Mahowald redshift", "statement": "Mahowald invariants increase chromatic height\n\nGiven a $v_n$-periodic family in the stable homotopy groups of spheres, does its Mahowald invariant give rise to an (eventually) $v_{n+1}$-periodic family?", "original_statement": "Mahowald invariants increase chromatic height\n\nGiven a $v_n$-periodic family in the stable homotopy groups of spheres, does its Mahowald invariant give rise to an (eventually) $v_{n+1}$-periodic family?", "clean_statement": "Mahowald invariants increase chromatic height\n\nGiven a $v_n$-periodic family in the stable homotopy groups of spheres, does its Mahowald invariant give rise to an (eventually) $v_{n+1}$-periodic family?", "statement_status": "exact", "statement_verification": "The exact canonical record is Problem 3.3 from the AIM workshop list “Equivariant techniques in stable homotopy theory,” section “Motivic, equivariant, and synthetic spectra”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Motivic, equivariant, and synthetic spectra\nSource item: 3.3\nSource URL: http://aimpl.org/equivstable/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[32]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Mahowald invariants increase chromatic height\\n\\nGiven a $v_n$-periodic family in the stable homotopy groups of spheres, does its Mahowald invariant give rise to an (eventually) $v_{n+1}$-periodic family?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"There is an equivariant formulation of the Mahowald invariant.\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0033", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In the two-primary C2-equivariant formulation, let z_s be proposed Mahowald lifts of a periodic input family x_s in sign-representation grading j+sc, and let epsilon_s be the excess of the true maximal a_sigma-divisibility over j+sc. Then z_s is an actual Mahowald lift exactly when epsilon_s=0; every actual Mahowald representative has degree i+j+s(b+c)+epsilon_s; and any eventual fixed-stride r periodic selection under an operator of degree D must satisfy epsilon_{s+r}-epsilon_s=D-r(b+c), hence have vanishing r-step second difference. This gives a rigorous maximality criterion and a falsifiable degree obstruction, but does not prove general chromatic redshift.\n\nCandidate contribution (reduction; novelty confidence low): The Euler-divisibility defect sequence epsilon_s and its eventual-affine second-difference obstruction provide a scalar, representative-independent test for two-primary sphere-level Mahowald redshift once candidate mixed-height equivariant lifts have been constructed.", "research_difficulty_suggested": "L4" }, { "id": 20000034, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0034", "title": "The C2-equivariant even filtration and its comparison problem", "statement": "Equivariant and synthetic\n\nCan one profitably mix synthetic spectra with equivariant homotopy theory?", "original_statement": "Equivariant and synthetic\n\nCan one profitably mix synthetic spectra with equivariant homotopy theory?", "clean_statement": "Equivariant and synthetic\n\nCan one profitably mix synthetic spectra with equivariant homotopy theory?", "statement_status": "exact", "statement_verification": "The exact canonical record is AIM Problem List 3.4 from the workshop *Equivariant techniques in stable homotopy theory*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Motivic, equivariant, and synthetic spectra\nSource item: 3.4\nSource URL: http://aimpl.org/equivstable/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[33]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Equivariant and synthetic\\n\\nCan one profitably mix synthetic spectra with equivariant homotopy theory?\"\nOriginal remarks: [\"Along these lines, what is the $\\\\mathbf{Z}/2$-equivariant analogue of the even filtration (where $\\\\tau_{\\\\geq 2\\\\ast}$ is replaced by $\\\\tau_{\\\\geq \\\\rho\\\\ast}$, and evenness is replaced by slice-even)?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0034", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The broad AIM question now has affirmative published answers: Burklund--Hahn--Senger construct the requested 2-complete C2 Real slice-even deformation using the regular-slice covers P_{2n}, and Allen--Piessevaux construct genuine equivariant synthetic categories for finite abelian groups using a complex-even filtration. This attempt proves a concrete comparison obstruction and reduction: realification meets the rho-line only in even multiples, while a genuine filtered C2-equivalence is detected jointly by Borel completion and C2-geometric fixed points.\n\nCandidate contribution (obstruction_and_reduction; novelty confidence low): For r: RU(C2) -> RO(C2), one has r(RU(C2)) intersect Z rho = 2 Z rho; moreover, a map of filtered genuine C2-spectra is a levelwise equivalence if and only if both its Borel completion and its C2-geometric-fixed-point map are levelwise equivalences. Consequently, any comparison of the complex-even and Real slice-even synthetic deformations must construct odd rho half-stages and verify a geometric-fixed-point comparison." }, { "id": 20000035, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0035", "title": "Odd-primary Wood filtrations and a chromatic lower bound", "statement": "Odd primary analog of Wood cofibre sequences\n\nThe Wood cofibre sequence is the cofibre sequence $\\Sigma \\mathrm{KO} \\xrightarrow{\\eta} \\mathrm{KO} \\to \\mathrm{KU}$ (and similarly for the connective versions). There is also a version relating $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle^{C_2}$ to $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle$.\n\nWhat are odd primary analogs of the Wood cofibre sequence? For example, one might expect a $p$-stage filtration on $E_{p-1}$ with associated graded $E_{p-1}^{hC_p}$. Is there an analog for $E_{(p-1)p^{n-1}m}$?", "original_statement": "Odd primary analog of Wood cofibre sequences\n\nThe Wood cofibre sequence is the cofibre sequence $\\Sigma \\mathrm{KO} \\xrightarrow{\\eta} \\mathrm{KO} \\to \\mathrm{KU}$ (and similarly for the connective versions). There is also a version relating $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle^{C_2}$ to $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle$.\n\nWhat are odd primary analogs of the Wood cofibre sequence? For example, one might expect a $p$-stage filtration on $E_{p-1}$ with associated graded $E_{p-1}^{hC_p}$. Is there an analog for $E_{(p-1)p^{n-1}m}$?", "clean_statement": "Odd primary analog of Wood cofibre sequences\n\nThe Wood cofibre sequence is the cofibre sequence $\\Sigma \\mathrm{KO} \\xrightarrow{\\eta} \\mathrm{KO} \\to \\mathrm{KU}$ (and similarly for the connective versions). There is also a version relating $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle^{C_2}$ to $\\mathrm{BP}_{\\mathbb{R}}\\langle n \\rangle$.\n\nWhat are odd primary analogs of the Wood cofibre sequence? For example, one might expect a $p$-stage filtration on $E_{p-1}$ with associated graded $E_{p-1}^{hC_p}$. Is there an analog for $E_{(p-1)p^{n-1}m}$?", "statement_status": "exact", "statement_verification": "The local JSON record agrees with the AIM statement. The AIM web page timed out during this run, but there is no visible corruption or missing notation in the repository copy. I interpret “associated graded \\(E_{p-1}^{hC_p}\\)” in the standard stable sense: the graded pieces are suspensions of \\(E_{p-1}^{hC_p}\\). This convention is necessary even classically, since the two graded pieces of \\(KO\\wedge C(\\eta)\\simeq KU\\) are \\(KO\\) and \\(\\Sigma^2KO\\).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Motivic, equivariant, and synthetic spectra\nSource item: 3.1\nSource URL: http://aimpl.org/equivstable/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[34]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Odd primary analog of Wood cofibre sequences\\n\\nThe Wood cofibre sequence is the cofibre sequence $\\\\Sigma \\\\mathrm{KO} \\\\xrightarrow{\\\\eta} \\\\mathrm{KO} \\\\to \\\\mathrm{KU}$ (and similarly for the connective versions). There is also a version relating $\\\\mathrm{BP}_{\\\\mathbb{R}}\\\\langle n \\\\rangle^{C_2}$ to $\\\\mathrm{BP}_{\\\\mathbb{R}}\\\\langle n \\\\rangle$.\\n\\nWhat are odd primary analogs of the Wood cofibre sequence? For example, one might expect a $p$-stage filtration on $E_{p-1}$ with associated graded $E_{p-1}^{hC_p}$. Is there an analog for $E_{(p-1)p^{n-1}m}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0035", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "At height p-1, the Ravenel p-cell alpha_1 complex gives the requested p-stage filtration of E_{p-1} by suspended copies of E_{p-1}^{hC_p}. For every cyclic stabilizer subgroup C_{p^n}, a theorem of Meier--Naumann--Noel gives a stable-multiple analogue: some finite wedge E_h^{\\vee q} has an exact qp^n-stage filtration by suspended copies of E_h^{hC_{p^n}}. A rank-one external filtration is equivalent to realizing the regular residual representation k[C_{p^n}], while Carrick's chromatic-defect computation implies that every external T(r)-free Wood witness at height (p-1)p^{n-1}m must satisfy r at least p^{n-1}m; hence the height-p-1 alpha_1 complex cannot simply be reused outside n=m=1.\n\nCandidate contribution (obstruction_and_reduction; novelty confidence low): For h=(p-1)p^{n-1}m and H=C_{p^n}, some E_h^{\\vee q} has an exact qp^n-stage E_h^{hH}-cell filtration, whereas every external T(r)-free Wood witness has r at least p^{n-1}m; moreover, a p^n-cell rank-one candidate is reduced to the test (g-1)^{p^n-1} nonzero on its Morava K-cohomology." }, { "id": 20000036, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0036", "title": "The known C-motivic lift and a tau-torsion nonuniqueness obstruction", "statement": "Motivic modular forms (mmf)\n\nIs there a connective version of motivic modular forms, which should be named $\\mathrm{mmf}$, and is a lift of $\\mathrm{tmf}$?", "original_statement": "Motivic modular forms (mmf)\n\nIs there a connective version of motivic modular forms, which should be named $\\mathrm{mmf}$, and is a lift of $\\mathrm{tmf}$?", "clean_statement": "Motivic modular forms (mmf)\n\nIs there a connective version of motivic modular forms, which should be named $\\mathrm{mmf}$, and is a lift of $\\mathrm{tmf}$?", "statement_status": "exact", "statement_verification": "The exact canonical record is Problem 3.5 in the section “Motivic, equivariant, and synthetic spectra” from the October 24–28, 2022 AIM workshop *Equivariant techniques in stable homotopy theory*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Motivic, equivariant, and synthetic spectra\nSource item: 3.5\nSource URL: http://aimpl.org/equivstable/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[35]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Motivic modular forms (mmf)\\n\\nIs there a connective version of motivic modular forms, which should be named $\\\\mathrm{mmf}$, and is a lift of $\\\\mathrm{tmf}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0036", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literal cellular 2-complete C-motivic reading is solved by the GIKR E-infinity ring mmf_C = Gamma_* tmf: its Betti realization is 2-complete tmf, its mod-2 cohomology is A_C // A_C(2), and the GIKR Adams-Novikov spectral sequence proves topological-stem connectivity. Stronger geometric, base-uniform, and R-motivic readings remain unresolved. In addition, the proved split square-zero extensions mmf_C plus Sigma^{N,0}(mmf_C/tau), for N at least zero, are nontrivial connective augmented commutative mmf_C-algebras with the same Betti realization, showing that realization alone is not a rigid definition of a lift.\n\nCandidate contribution (obstruction; novelty confidence low): For every N >= 0, the split square-zero augmented commutative mmf_C-algebra mmf_C plus Sigma^{N,0}(mmf_C/tau) is stem-connective, has Betti realization equivalent to 2-complete tmf, and is inequivalent to the identity augmented mmf_C-algebra because its nonzero augmentation ideal is Sigma^{N,0}(mmf_C/tau)." }, { "id": 20000037, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0037", "title": "Adjacent localized slice fibers as homotopy-orbit layers", "statement": "The associated graded of the localized slice spectral sequence tower\n\nThe localized slice spectral sequence gives a way to break up the slice spectral sequence into pieces that see phenomena only on certain geometric fixed points. For the group $C_{2^n}$, there is a localized slice spectral sequence for each subgroup.\n\nFor the group $C_{2^n}$ one understand more easily the fibres between the localized slice spectral sequence for two adjacent subgroups $H$ and $H'$?", "original_statement": "The associated graded of the localized slice spectral sequence tower\n\nThe localized slice spectral sequence gives a way to break up the slice spectral sequence into pieces that see phenomena only on certain geometric fixed points. For the group $C_{2^n}$, there is a localized slice spectral sequence for each subgroup.\n\nFor the group $C_{2^n}$ one understand more easily the fibres between the localized slice spectral sequence for two adjacent subgroups $H$ and $H'$?", "clean_statement": "The associated graded of the localized slice spectral sequence tower\n\nThe localized slice spectral sequence gives a way to break up the slice spectral sequence into pieces that see phenomena only on certain geometric fixed points. For the group $C_{2^n}$, there is a localized slice spectral sequence for each subgroup.\n\nFor the group $C_{2^n}$ one understand more easily the fibres between the localized slice spectral sequence for two adjacent subgroups $H$ and $H'$?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 4.1, “The associated graded of the localized slice spectral sequence tower,” in the AIM list *Equivariant techniques in stable homotopy theory*, section “Norms and the slice filtration.” The source record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Norms and the slice filtration\nSource item: 4.1\nSource URL: http://aimpl.org/equivstable/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[36]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The associated graded of the localized slice spectral sequence tower\\n\\nThe localized slice spectral sequence gives a way to break up the slice spectral sequence into pieces that see phenomena only on certain geometric fixed points. For the group $C_{2^n}$, there is a localized slice spectral sequence for each subgroup.\\n\\nFor the group $C_{2^n}$ one understand more easily the fibres between the localized slice spectral sequence for two adjacent subgroups $H$ and $H'$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0037", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Meier--Shi--Zeng's 2025 stratification theorem gives a direct structural solution to the recovered AIM question. As an explicit refinement, for G=C_{2^n} and adjacent H=C_{2^k}^{C2}, exact order 4 in BP^{((C4))}^{C4} for every m at least 1, and order 2 or 4 for BP_R^{C2} when m is at least 3; sigma is nonzero for C2 when m is at least 3. In particular, the equal-height cases (C2,2) and (C4,1) have different nu-image orders, so the J-image is not determined by the height parameter m|G|/2 alone.\n\nCandidate contribution (counterexample; novelty confidence low): The order of the degree-three classical J-image is not a function only of the height parameter h=m|G|/2: h(C2,2)=h(C4,1)=2, but the unit image of nu has order 2 in BP_R<2>^{C2} and order 4 in BP^{((C4))}<1>^{C4}." }, { "id": 20000039, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0039", "title": "A restriction-truncation baseline for fixed-point Hurewicz images", "statement": "The Hurewicz image of equivariant truncated Brown-Peterson spectra\n\nWhat is the Hurewicz image of the truncated Brown--Peterson spectrum $\\mathrm{BP}^{((G))}\\langle m \\rangle$?", "original_statement": "The Hurewicz image of equivariant truncated Brown-Peterson spectra\n\nWhat is the Hurewicz image of the truncated Brown--Peterson spectrum $\\mathrm{BP}^{((G))}\\langle m \\rangle$?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "Other plausible readings are genuinely different problems:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Norms and the slice filtration\nSource item: 4.3\nSource URL: http://aimpl.org/equivstable/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[38]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The Hurewicz image of equivariant truncated Brown-Peterson spectra\\n\\nWhat is the Hurewicz image of the truncated Brown--Peterson spectrum $\\\\mathrm{BP}^{((G))}\\\\langle m \\\\rangle$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0039", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Interpreting the requested map as the unit from the 2-local sphere to categorical G-fixed points, the report proves truncation monotonicity and shows that every positive-degree Hurewicz class restricts to zero nonequivariantly. It computes the image exactly for m=0 for every cyclic 2-group and for (G,m)=(C2,1): the latter is Z_(2) in degree 0, Z/2 in positive degrees congruent to 1 or 2 modulo 8, and zero otherwise. Hence every class with nonzero KO alpha-invariant is detected by BPR fixed points for every m>=1, while the C4 theories detect eta, nu, epsilon, kappa, and bar-kappa for every m>=1.\n\nCandidate contribution (reduction; novelty confidence low): For all cyclic 2-groups and all truncation heights, positive fixed-point Hurewicz images lie in the kernel of underlying restriction; moreover the exact C2 height-one KO-alpha image and the five known C4 height-one spherical classes propagate to every higher truncation via unit-compatible quotient maps." }, { "id": 20000040, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0040", "title": "Fixed-point Postnikov windows and a two-term model for 1-slices", "statement": "Understanding $n$-slices\n\n\\cite{arXiv:1711.03472} gives a description of $n$-slices, the objects that appear in the associated graded of the slice filtration.\n\nIs there a way to get a good computational grasp of $n$-slices? For example understanding their Postnikov towers, homotopy groups etc.", "original_statement": "Understanding $n$-slices\n\n\\cite{arXiv:1711.03472} gives a description of $n$-slices, the objects that appear in the associated graded of the slice filtration.\n\nIs there a way to get a good computational grasp of $n$-slices? For example understanding their Postnikov towers, homotopy groups etc.", "clean_statement": "Understanding $n$-slices\n\n\\cite{arXiv:1711.03472} gives a description of $n$-slices, the objects that appear in the associated graded of the slice filtration.\n\nIs there a way to get a good computational grasp of $n$-slices? For example understanding their Postnikov towers, homotopy groups etc.", "statement_status": "exact", "statement_verification": "This agrees with the JSON record. There is no corruption to repair. The question is intentionally broad rather than a yes/no conjecture.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Norms and the slice filtration\nSource item: 4.4\nSource URL: http://aimpl.org/equivstable/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[39]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Understanding $n$-slices\\n\\n\\\\cite{arXiv:1711.03472} gives a description of $n$-slices, the objects that appear in the associated graded of the slice filtration.\\n\\nIs there a way to get a good computational grasp of $n$-slices? For example understanding their Postnikov towers, homotopy groups etc.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0040", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every finite group G, every original 1-slice X, and every subgroup H, the categorical fixed point spectrum X^H is the fiber of a map between suspended Eilenberg-Mac Lane spectra determined by the boundary d_H from the underlying degree-one group to the Wilson slice-sphere probe group. Hence pi_1(X^H) is ker(d_H), pi_0(X^H) is coker(d_H), all other integer-graded groups vanish, and the ordinary Postnikov invariant is zero. More generally, a nonnegative n-slice has H-fixed homotopy only in degrees floor(n/|H|) through n.\n\nCandidate contribution (special_case; novelty confidence low): For every finite G, every original 1-slice X, and every H <= G, X^H is the Eilenberg-Mac Lane realization of the two-term complex pi_1(X^e) -> [cof(H_+ -> S^0), res_H^G X]^H, with kernel in degree 1 and cokernel in degree 0." }, { "id": 20000041, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0041", "title": "A canonical two-axis height profile for slice differentials", "statement": "Heights of slice differentials\n\nThe differentials in the slice spectral sequence for $\\mathrm{BP}^{((G))}\\langle n \\rangle$ look like they are stratified by height.\n\nIs it possible to make sense of the height of a differential in the slice spectral sequence of $\\mathrm{BP}^{((G))}\\langle n \\rangle$ so that it has a well defined height?", "original_statement": "Heights of slice differentials\n\nThe differentials in the slice spectral sequence for $\\mathrm{BP}^{((G))}\\langle n \\rangle$ look like they are stratified by height.\n\nIs it possible to make sense of the height of a differential in the slice spectral sequence of $\\mathrm{BP}^{((G))}\\langle n \\rangle$ so that it has a well defined height?", "clean_statement": "Heights of slice differentials\n\nThe differentials in the slice spectral sequence for $\\mathrm{BP}^{((G))}\\langle n \\rangle$ look like they are stratified by height.\n\nIs it possible to make sense of the height of a differential in the slice spectral sequence of $\\mathrm{BP}^{((G))}\\langle n \\rangle$ so that it has a well defined height?", "statement_status": "exact", "statement_verification": "The exact AIM record (workshop *Equivariant techniques in stable homotopy theory*, section “Norms and the slice filtration,” Problem 4.5) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Norms and the slice filtration\nSource item: 4.5\nSource URL: http://aimpl.org/equivstable/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[40]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Heights of slice differentials\\n\\nThe differentials in the slice spectral sequence for $\\\\mathrm{BP}^{((G))}\\\\langle n \\\\rangle$ look like they are stratified by height.\\n\\nIs it possible to make sense of the height of a differential in the slice spectral sequence of $\\\\mathrm{BP}^{((G))}\\\\langle n \\\\rangle$ so that it has a well defined height?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0041", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For G=C_{2^q}, every nonzero differential in the slice spectral sequence of BP^{((G))} has a canonical nonvanishing support across the standard truncation tower and the family-localized slice-spectral-sequence tower. This support is an order ideal, so its truncation boundary and geometric persistence depth are well defined and monotone. The fundamental Hill-Hopkins-Ravenel differential indexed by i has rectangular support, truncation boundary i, and page R=|G|(2^i-1)+1. In contrast, Wu's essential exotic d_11 in BP^{((C_4))}<1> dies after deepest a_sigma-localization and violates the tempting universal normalized-page formula, showing why a profile is safer than a single page- or stratum-based scalar.\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: the two-axis height support Sigma(delta) is an order ideal invariant of a slice differential; it gives e_{d_5}(1)=2 but e_{d_11}(1)=1 for the sheared and exotic generating differentials in BP^{((C_4))}<1>, while recovering the chromatic index exactly on every fundamental HHR differential." }, { "id": 20000042, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0042", "title": "Geometric-fixed-point sparsity in the orbit--Koszul Bockstein filtration", "statement": "The $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for $\\mathrm{BP}^{((G))}\\langle n\\rangle$\n\n$\\mathrm{BP}^{((G))}\\langle n-1\\rangle$ is the quotient of $\\mathrm{BP}^{((G))}\\langle n\\rangle$ by $G\\cdot \\overline{v}_n$.\n\nCan one understand the $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for this quotient map?", "original_statement": "The $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for $\\mathrm{BP}^{((G))}\\langle n\\rangle$\n\n$\\mathrm{BP}^{((G))}\\langle n-1\\rangle$ is the quotient of $\\mathrm{BP}^{((G))}\\langle n\\rangle$ by $G\\cdot \\overline{v}_n$.\n\nCan one understand the $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for this quotient map?", "clean_statement": "The $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for $\\mathrm{BP}^{((G))}\\langle n\\rangle$\n\n$\\mathrm{BP}^{((G))}\\langle n-1\\rangle$ is the quotient of $\\mathrm{BP}^{((G))}\\langle n\\rangle$ by $G\\cdot \\overline{v}_n$.\n\nCan one understand the $G\\cdot \\overline{v}_n$-Bockstein spectral sequence for this quotient map?", "statement_status": "exact", "statement_verification": "The canonical AIM record (source file `aim-algebraic-geometry-notes.json`, zero-based index 41, problem 4.6 in the section “Norms and the slice filtration”) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Norms and the slice filtration\nSource item: 4.6\nSource URL: http://aimpl.org/equivstable/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[41]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The $G\\\\cdot \\\\overline{v}_n$-Bockstein spectral sequence for $\\\\mathrm{BP}^{((G))}\\\\langle n\\\\rangle$\\n\\n$\\\\mathrm{BP}^{((G))}\\\\langle n-1\\\\rangle$ is the quotient of $\\\\mathrm{BP}^{((G))}\\\\langle n\\\\rangle$ by $G\\\\cdot \\\\overline{v}_n$.\\n\\nCan one understand the $G\\\\cdot \\\\overline{v}_n$-Bockstein spectral sequence for this quotient map?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0042", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Carrick--Hill's finite normed Koszul filtration is the appropriate genuine-equivariant replacement for a one-element Bockstein filtration for the quotient BP^((G)) to BP^((G)). For G=C_{2^r} and C_2 <= K <= G, applying K-geometric fixed points forces every potentially nonzero filtration column to be divisible by [K:C_2]; the residual orbit types in column [K:C_2]j are counted by binary necklaces of length [G:K] and weight j. Consequently, a differential can change filtration only by a multiple of [K:C_2]. The underlying orbit sequence is also proved regular, so its nonequivariant Koszul homology is concentrated in degree zero.\n\nCandidate contribution (proposition; novelty confidence low): In the Carrick--Hill orbit--Koszul filtration for a cyclic 2-group G, K-geometric fixed points can have a nonzero associated-graded term only in filtration degrees divisible by [K:C_2]; in degree [K:C_2]j, the possible residual G/K-orbit types number (1/[G:K]) times the sum over a dividing gcd([G:K],j) of phi(a) binomial([G:K]/a,j/a). Thus every possible differential filtration change is divisible by [K:C_2]." }, { "id": 20000043, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0043", "title": "A coherence-and-square-defect gate for power operations in slice spectral sequences", "statement": "Connecting slice differentials via power operations\n\nWe would like a systematic way of understanding slice differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$.\n\nAre there power operations connecting differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$", "original_statement": "Connecting slice differentials via power operations\n\nWe would like a systematic way of understanding slice differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$.\n\nAre there power operations connecting differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$", "clean_statement": "Connecting slice differentials via power operations\n\nWe would like a systematic way of understanding slice differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$.\n\nAre there power operations connecting differentials in the slice spectral sequence for $\\mathrm{BP}\\langle n \\rangle^{((G))}$", "statement_status": "exact", "statement_verification": "The assigned record is problem 4.7 in the section “Norms and the slice filtration” of the AIM workshop list *Equivariant techniques in stable homotopy theory*. The exact stored text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Norms and the slice filtration\nSource item: 4.7\nSource URL: http://aimpl.org/equivstable/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[42]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Connecting slice differentials via power operations\\n\\nWe would like a systematic way of understanding slice differentials in the slice spectral sequence for $\\\\mathrm{BP}\\\\langle n \\\\rangle^{((G))}$.\\n\\nAre there power operations connecting differentials in the slice spectral sequence for $\\\\mathrm{BP}\\\\langle n \\\\rangle^{((G))}$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Andy Senger has thought about this problem, you probably should talk to him if you are interested.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0043", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the standard C2 truncation BP_R at height m >= 4, restriction to the trivial subgroup and Lawson's E_12 obstruction rule out a genuine internal equivariant E_infinity structure as a height-uniform source of unrestricted total powers. Independently, in the known positive-cone C2 slice spectral sequence, the square of the source of the kth HHR height differential has zero differential on that page, yet supports the next nonzero height differential exactly 2^(k+1) pages later. Thus any quadratic connector with leading square term needs a secondary filtered correction and, if descended from normed Real bordism, must pass an explicit killed-generator naturality test.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate novelty: the combined coherence-and-square-defect gate shows that a viable height-uniform power connector must be bounded-coherence, ambient, or external; must annihilate the images of the killed Real-bordism generators in the appropriate extended-power target; and must supply a secondary correction across the exact 2^(k+1)-page gap that produces the next independent height generator." }, { "id": 20000044, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0044", "title": "Odd-stem exponent two at C2 and a C4 transfer-kernel reduction", "statement": "Relations in $\\mathrm{BP}^{((G))}$\n\nIs $8\\sigma = 0$ in the fixed points of $\\mathrm{BP}^{((G))}$?", "original_statement": "Relations in $\\mathrm{BP}^{((G))}$\n\nIs $8\\sigma = 0$ in the fixed points of $\\mathrm{BP}^{((G))}$?", "clean_statement": "Relations in $\\mathrm{BP}^{((G))}$\n\nIs $8\\sigma = 0$ in the fixed points of $\\mathrm{BP}^{((G))}$?", "statement_status": "exact", "statement_verification": "The exact corpus record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant techniques in stable homotopy theory\nSection: Norms and the slice filtration\nSource item: 4.8\nSource URL: http://aimpl.org/equivstable/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[43]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Relations in $\\\\mathrm{BP}^{((G))}$\\n\\nIs $8\\\\sigma = 0$ in the fixed points of $\\\\mathrm{BP}^{((G))}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equivstable/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0044", "aim-domain:algebraic-geometry", "aim-workshop:equivstable", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The uniform cyclic-2-group question remains open, but the C2 case has an exact answer: every odd integer-graded group of the C2-fixed points of Real BP is annihilated by 2. Combining this with the Li–Shi–Wang–Xu detection theorem shows that the Hurewicz image of the classical sigma in stem 7 is nonzero of exact order 2, hence 8 sigma vanishes at C2. For C4, the Meier–Shi–Zeng localized computation and isotropy separation imply that twice the Hurewicz image lies in the image of pi_7((BP smash BP)_{hC4}); the remaining order question is confined to four low-degree homotopy-orbit spectral-sequence terms and their extensions.\n\nCandidate contribution (theorem; novelty confidence low): For every odd integer d, multiplication by 2 annihilates pi_d of the C2-fixed points of Real Brown–Peterson theory." }, { "id": 20000045, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0045", "title": "Seed-preserving explicit bounds for unirational hypersurfaces", "statement": "Problem 1\n\nFind more integers $d$ and $n$ such that a general hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.\nSimilarly,\nfind more integers $d$ and $n$ such that every smooth hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.", "original_statement": "Problem 1\n\nFind more integers $d$ and $n$ such that a general hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.\nSimilarly,\nfind more integers $d$ and $n$ such that every smooth hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.", "clean_statement": "Problem 1\n\nFind more integers $d$ and $n$ such that a general hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.\nSimilarly,\nfind more integers $d$ and $n$ such that every smooth hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$ is unirational.", "statement_status": "exact", "statement_verification": "The canonical source is aim-algebraic-geometry-notes.json, zero-based index 44, problem 1.02 from the AIM workshop *Rationality problems in algebraic geometry*. The source record states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.02\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[44]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1\\n\\nFind more integers $d$ and $n$ such that a general hypersurface of degree $d$ in $\\\\mathbb P^n_{\\\\mathbb C}$ is unirational.\\nSimilarly,\\nfind more integers $d$ and $n$ such that every smooth hypersurface of degree $d$ in $\\\\mathbb P^n_{\\\\mathbb C}$ is unirational.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Beheshti and Riedl showed that if $2^{d!} < n$, then every smooth hypersurface $X$ of degree $d$ in $\\\\mathbb P^n_{\\\\mathbb C}$ is unirational. One cannot expect to prove negative results, since it is a notorious open problem whether every rationally connected complex variety is unirational. A negative answer seems almost certain, but out of reach..\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0045", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Two explicit infinite families follow from retaining the exact seeds in published recurrences. For d at least 9, if R_d = 6479 + sum from j=1 to d-9 of 6359^(2^j), then a general degree-d hypersurface in complex projective n-space is unirational whenever n is at least R_d^(d-1); in particular n at least 2^((d-1)(13*2^(d-9)+1)) suffices. For d at least 5, every smooth degree-d complex hypersurface is unirational whenever n is at least 1021684^((d-1)(d-1)!/24), hence whenever n is at least 2^(5(d-1)(d-1)!/6). These sharpen the corresponding published coarse closed exponents but not the stronger exact recursive thresholds.\n\nCandidate contribution (quantitative_corollary; novelty confidence low): Keeping Cheng's seed m_7 = 6359 and Beheshti--Riedl's seed k_0(5) = 1021684 yields the two proved closed-form thresholds in the main claim, whose power-of-two exponents are strictly below (d-1)2^(d-5) and d!, respectively, throughout their stated ranges." }, { "id": 20000046, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0046", "title": "Exact low-degree minima and a sharp linear-space gate", "statement": "Given $d$, what is the smallest $n$ such that there is some smooth unirational hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$?", "original_statement": "Given $d$, what is the smallest $n$ such that there is some smooth unirational hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$?", "clean_statement": "Given $d$, what is the smallest $n$ such that there is some smooth unirational hypersurface of degree $d$ in $\\mathbb P^n_{\\mathbb C}$?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.04 from the AIM workshop *Rationality problems in algebraic geometry*, posed by Stefan Schreieder. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.04\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[45]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given $d$, what is the smallest $n$ such that there is some smooth unirational hypersurface of degree $d$ in $\\\\mathbb P^n_{\\\\mathbb C}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0046", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Writing u(d) for the least ambient dimension, adjunction and vanishing of top forms on a complex unirational variety give u(d) >= d. Together with classical and current constructions this yields u(1)=1, u(2)=2, u(3)=3, u(4)=4, 5 <= u(5) <= 7, u(6) <= 160, u(7) <= 20,376, and u(d) <= 2^((d-1)2^(d-5)) for d >= 6. In addition, a smooth degree-d hypersurface in P^n containing an r-plane exists exactly when n >= 2r+1 (for d,n >= 2); hence the known unirational quintic construction retaining a 3-plane is method-optimal in ambient P^7.\n\nCandidate contribution (proposition; novelty confidence low): Candidate novelty: a first-jet/conormal calculation gives the exact linear-space gate n >= 2r+1 for smooth degree-d hypersurfaces containing P^r, and its application proves that the Conte--Marchisio--Murre 3-plane construction cannot be lowered from P^7 to P^6 or P^5 while retaining that center." }, { "id": 20000047, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0047", "title": "A graph-splitting obstruction to specialization of unirationality", "statement": "Does unirationality specialize in smooth projective families?", "original_statement": "Does unirationality specialize in smooth projective families?", "clean_statement": "Does unirationality specialize in smooth projective families?", "statement_status": "exact", "statement_verification": "The exact AIM record (problem 1.06 in the 2019 workshop *Rationality problems in algebraic geometry*) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.06\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[46]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does unirationality specialize in smooth projective families?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is known to be true for rationality, stable rationality, and rational connectivity.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0047", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a dominant generically finite generic-fiber parametrization f_K: P^n_K --> X_K of degree delta over a DVR, the special graph cycle has pushforwards [P^n_k] and delta[X_k]. Hence it has a unique multiplicity-one component birational to P^n_k and at least one component dominating X_k. The special fiber is unirational whenever these are the same component; otherwise failure of the chosen parametrization has a forced split form. In particular, an irreducible special graph proves specialization. The report also proves the characteristic-zero question for relative dimension at most two using Luroth/Castelnuovo and specialization of rationality.\n\nCandidate contribution (lemma; novelty confidence low): Candidate graph-splitting lemma: in any DVR degeneration of the graph of a generically finite unirational parametrization, the unique source-dominating component has multiplicity and source degree one; if it is target-vertical, every target-dominating component is source-vertical. The explicit flat equation uy = tvx realizes this split even for a degree-one generic automorphism." }, { "id": 20000048, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0048", "title": "An explicit smooth locus for the coprime-degree cubic family", "statement": "Does there exist an irrational unirational variety with unirational parametrizations of coprime degrees?", "original_statement": "Does there exist an irrational unirational variety with unirational parametrizations of coprime degrees?", "clean_statement": "Does there exist an irrational unirational variety with unirational parametrizations of coprime degrees?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.08 from the AIM workshop *Rationality problems in algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.08\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[47]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does there exist an irrational unirational variety with unirational parametrizations of coprime degrees?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0048", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Yang, Yu, and Zhu proved in 2025 that a two-dimensional family of smooth cubic threefolds has unirational parametrizations of degrees 2 and 3; Clemens--Griffiths irrationality therefore answers the AIM question affirmatively. This attempt additionally proves that the set-theoretic discriminant of their normalized family is exactly (a^3+27)(b^3+27)(a^3+b^3+27)=0, certifying X_{1,1} as an explicit smooth witness, and derives the associated degree-gcd transfer consequences.\n\nCandidate contribution (explicit_discriminant_proposition; novelty confidence low): For the Yang--Yu--Zhu family with t_1,...,t_5 nonzero, singularity occurs exactly when t_6^3+27t_1t_2t_3=0, or t_7^3+27t_2t_4t_5=0, or t_4t_5t_6^3+t_1t_3t_7^3+27t_1t_2t_3t_4t_5=0; after normalization the reduced discriminant is (a^3+27)(b^3+27)(a^3+b^3+27)=0." }, { "id": 20000049, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0049", "title": "An exact product formula for torsion orders", "statement": "Does there exist a rationally connected smooth complex projective variety, not stably rational, that has an integral decomposition of the diagonal?", "original_statement": "Does there exist a rationally connected smooth complex projective variety, not stably rational, that has an integral decomposition of the diagonal?", "clean_statement": "Does there exist a rationally connected smooth complex projective variety, not stably rational, that has an integral decomposition of the diagonal?", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 1.1 from the workshop *Rationality problems in algebraic geometry*. Its question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.1\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[48]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does there exist a rationally connected smooth complex projective variety, not stably rational, that has an integral decomposition of the diagonal?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"There are surfaces of general type with integral decomposition of the diagonal, namely Barlow surfaces.\\n\\nThere are stably rational varieties that are not rational, and so the answer would be \\\"yes\\\" for \\\"not rational\\\" in place of \\\"not stably rational.\\\" Over a general field, one has to change the question to \\\"not retract rational\\\", since some algebraic tori over a field are retract rational but not stably rational, and retract rationality is enough to imply an integral decomposition of the diagonal. By results of Blinstein and Merkurjev, a smooth compactification of a torus over a field has an integral decomposition of the diagonal if and only if it is retract rational.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0049", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For smooth connected complex projective varieties X and Y of finite torsion orders m and n, the attempt proves Tor(X x Y) = lcm(m,n) by splitting the unit in their birational Chow motives and computing the additive order of the identity on the reduced product motive. Consequently Tor(X^r) = Tor(X), products admit an integral diagonal decomposition exactly when both factors do, and Tor(Sym^r X) as well as the torsion order of any smooth projective resolution divides r! Tor(X). This is rigorous structural progress but does not resolve the requested existence question.\n\nCandidate contribution (product_formula; novelty confidence low): For all smooth connected complex projective varieties X and Y with finite torsion order, Tor(X x Y) equals lcm(Tor(X), Tor(Y)); in particular the standard symmetric-power estimate improves to Tor(Sym^r X) dividing r! Tor(X), with a separate identical bound for every smooth projective resolution." }, { "id": 20000050, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0050", "title": "A degree-two saturation test for the unirational fourfold problem", "statement": "Does there exist a unirational variety $X$ of dimension at least 4 such that\n\\begin{align*}\n\\frac{H^4(X, \\mathbb Z) \\cap H^{2,2}(X)}{(H^4(X, \\mathbb Z) \\cap H^{2,2}(X))_{tors}}\n\\end{align*}\nis non-algebraic? Here, the subscript tors denotes torsion classes.\nDoes there exist such a variety of dimension exactly 4?", "original_statement": "Does there exist a unirational variety $X$ of dimension at least 4 such that\n\\begin{align*}\n\\frac{H^4(X, \\mathbb Z) \\cap H^{2,2}(X)}{(H^4(X, \\mathbb Z) \\cap H^{2,2}(X))_{tors}}\n\\end{align*}\nis non-algebraic? Here, the subscript tors denotes torsion classes.\nDoes there exist such a variety of dimension exactly 4?", "clean_statement": "Does there exist a unirational variety $X$ of dimension at least 4 such that\n\\begin{align*}\n\\frac{H^4(X, \\mathbb Z) \\cap H^{2,2}(X)}{(H^4(X, \\mathbb Z) \\cap H^{2,2}(X))_{tors}}\n\\end{align*}\nis non-algebraic? Here, the subscript tors denotes torsion classes.\nDoes there exist such a variety of dimension exactly 4?", "statement_status": "exact", "statement_verification": "The canonical AIM record, Problem 1.12 from the workshop *Rationality problems in algebraic geometry*, asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.12\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[49]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does there exist a unirational variety $X$ of dimension at least 4 such that\\n\\\\begin{align*}\\n\\\\frac{H^4(X, \\\\mathbb Z) \\\\cap H^{2,2}(X)}{(H^4(X, \\\\mathbb Z) \\\\cap H^{2,2}(X))_{tors}}\\n\\\\end{align*}\\nis non-algebraic? Here, the subscript tors denotes torsion classes.\\nDoes there exist such a variety of dimension exactly 4?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Such varieties do not exist in dimension at most 3.\\n\\nSchreieder found unirational 4-folds for which the integral Hodge conjecture fails in codimension $2$. For these varieties, it is not known if the failure comes from torsion or non-torsion classes.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0050", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth projective complex variety X, every dominant generically finite parametrization from projective space of degree d annihilates the full degree-four integral Hodge defect and its quotient modulo cohomological torsion. The generic conic in Schreieder's known unirational construction has a rational quadratic splitting field, so its smooth fourfold model has unirational degree index exactly 2 and both defect groups are finite elementary F_2-vector spaces. Combining this with Paulsen's algebraic surface kappa representing twice the distinguished Hodge class mu reduces the AIM question for that class exactly to whether the class of kappa is not divisible by 2 in the torsion-free algebraic cycle-class lattice.\n\nCandidate contribution (reduction; novelty confidence low): Candidate degree-two saturation reduction: on the Schreieder-Paulsen fourfold the full degree-four Hodge defect is elementary 2-torsion, the gcd of all unirational parametrization degrees is 2, and the distinguished class survives modulo cohomological torsion if and only if Paulsen's explicit algebraic surface class is not twice an element of the algebraic cycle-class lattice after quotienting ambient torsion." }, { "id": 20000051, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0051", "title": "An explicit Frobenius bound for triple products of elliptic curves", "statement": "Is there a smooth projective $3$-fold $X$ over a finite field for which $H^3_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z(2))$ is not zero?", "original_statement": "Is there a smooth projective $3$-fold $X$ over a finite field for which $H^3_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z(2))$ is not zero?", "clean_statement": "Is there a smooth projective $3$-fold $X$ over a finite field for which $H^3_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z(2))$ is not zero?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.14 from the AIM workshop *Rationality problems in algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.14\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[50]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a smooth projective $3$-fold $X$ over a finite field for which $H^3_{nr}(k(X)/k, \\\\mathbb Q/\\\\mathbb Z(2))$ is not zero?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The expectation is yes, and that some product of three elliptic curves should give such an example. Gabber showed that unramified $H^3$ is not zero for the product of three very general elliptic curves over the complex numbers, rather than a finite field.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0051", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let X=E_1 x E_2 x E_3 over F_q and let l be prime to p. From the three Weil polynomials define R_X as the product of q^2 minus all eight triple products of Frobenius roots, and B_X=(product_i #E_i(F_q)^2)R_X. The attempt proves that the order of H^3_nr(X,Q_l/Z_l(2)) divides l^{v_l(B_X)}. Hence this group vanishes whenever l does not divide B_X. For a trace-zero self-product E^3, B_X=q^12(q+1)^10; for the Fermat cubic over F_5, all components at l at least 7 vanish, leaving possible support only at 2, 3, and 5.\n\nCandidate contribution (explicit_quantitative_bound; novelty confidence low): For every triple product of elliptic curves X over F_q and every prime l different from p, #H^3_nr(X,Q_l/Z_l(2)) divides l^{v_l(B_X)}, where B_X is the explicit Frobenius resultant defined in the report." }, { "id": 20000052, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0052", "title": "Index-two obstruction for maximal-plane projections of odd-dimensional cubics", "statement": "Is there a smooth rational cubic hypersurface of odd dimension?", "original_statement": "Is there a smooth rational cubic hypersurface of odd dimension?", "clean_statement": "Is there a smooth rational cubic hypersurface of odd dimension?", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 1.16 from the workshop “Rationality problems in algebraic geometry”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.16\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[51]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a smooth rational cubic hypersurface of odd dimension?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"There are some smooth rational cubic hypersurfaces of each even dimension at least 2.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0052", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let X^{2m+1} be a smooth complex cubic containing a maximal linear space P^m. Projection from that space gives a quadric fibration whose smooth generic residual quadric has index exactly 2. Therefore every generically finite multisection has even degree, so the fibration has no rational section or odd-degree multisection. This blocks the standard relative quadric-rationalization route but does not prove X irrational. A companion normal-bundle theorem shows that every smooth complete-intersection multisection must include a cubic defining generator and has exact degree 2 times the product of the remaining defining degrees.\n\nCandidate contribution (theorem; novelty confidence low): For projection of a smooth cubic X^{2m+1} from a contained P^m, the restriction image H^{2m}(Bl_{P^m}X,Z) -> H^{2m}(Q,Z) on a smooth residual quadric fiber is exactly 2Z[pt]; spreading generic closed points then proves that the generic quadric has index exactly 2. For a smooth complete-intersection multisection, one can additionally replace a cubic generator by the equation of X and compute its degree as 2 times the product of the remaining defining degrees." }, { "id": 20000053, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0053", "title": "A sharp quantitative obstruction to linear-center constructions on smooth quartics", "statement": "Is there a smooth rational quartic hypersurface of some dimension?", "original_statement": "Is there a smooth rational quartic hypersurface of some dimension?", "clean_statement": "Is there a smooth rational quartic hypersurface of some dimension?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.18 from the AIM workshop “Rationality problems in algebraic geometry,” in the section “The problems.” Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.18\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[52]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a smooth rational quartic hypersurface of some dimension?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0053", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over the complex numbers and in positive dimension, a quartic n-fold containing an h-plane has a singular scheme inside that plane of dimension at least 2h-n-1 when n<2h, with length 3^h in the finite borderline case n=2h-1. Hence a smooth member requires n>=2h, and an explicit smooth family realizes every allowable pair. Projection from the plane then has smooth geometric generic cubic h-fold fiber; lowering the residual degree to a quadric or a linear space forces multiplicity at least two and singularity along the center. This is a rigorous obstruction to linear-projection rationality mechanisms, not a solution of the still-open existence problem.\n\nCandidate contribution (quantitative_obstruction_theorem; novelty confidence low): Candidate novelty: the quantitative singular-locus bound and borderline length, the uniform explicit smooth sharpness family F_{n,h}, the necessarily smooth geometric generic cubic fiber, the no-persistent-nodal-section corollary, and the multiplicity/smoothing obstruction form one sharp theorem for linear centers on complex quartics." }, { "id": 20000054, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0054", "title": "Hyperplane-pencil transfer of stable irrationality", "statement": "Can we use the degeneration method to obstruct stable rationality for $4$-folds fibered by $3$-folds over $\\mathbb P^1$?", "original_statement": "Can we use the degeneration method to obstruct stable rationality for $4$-folds fibered by $3$-folds over $\\mathbb P^1$?", "clean_statement": "Can we use the degeneration method to obstruct stable rationality for $4$-folds fibered by $3$-folds over $\\mathbb P^1$?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.2 from the AIM workshop list “Rationality problems in algebraic geometry,” posed in the list by Asher Auel:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.2\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[53]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we use the degeneration method to obstruct stable rationality for $4$-folds fibered by $3$-folds over $\\\\mathbb P^1$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Some cases of interest are where the generic fiber is a complete intersection of 3 quadrics in $\\\\mathbb P^6$ or of a quadric and a cubic in $\\\\mathbb P^5$.\\nThe case where the generic fiber is an intersection of two quadrics in $\\\\mathbb P^5$ has just recently been done by Hassett and Tschinkel.\\nThe case of cubic threefolds in $\\\\mathbb P^4$ is also known, due to Pirutka and someone else, I think.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0054", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If V is a smooth stably irrational complete-intersection fourfold and B is the smooth base surface of a general hyperplane pencil, then Y = Bl_B(V) is a smooth fourfold flat over P^1, every fiber is the corresponding hyperplane section of V, and Y has exactly the same stable birational, universal CH_0, decomposition-of-the-diagonal, and unramified-cohomology obstructions as V. Applying this to known degeneration-certified fourfolds V_(2,2,2) in P^7 and V_(2,3) in P^6 yields smooth stably irrational total fourfolds whose generic fibers are respectively intersections of three quadrics in P^6 and intersections of a quadric and a cubic in P^5.\n\nCandidate contribution (reduction; novelty confidence low): On the hyperplane-pencil incidence loci for both multidegrees named in the AIM record, stable irrationality and failure of integral decomposition of the diagonal transfer unchanged from the ambient fourfold to the total space, while the exceptional divisor is B x P^1 and supplies sections; an explicit conditional lemma also transports an admissible universally CH_0-trivial special resolution through the pencil blow-up." }, { "id": 20000055, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0055", "title": "A full-exceptional-fiber criterion for uniform rationality under point blow-down", "statement": "If $X$ is a smooth rational variety of dimension $n$ and $p \\in X$ is a point, is there a Zariski open $U \\subset X$ with $p \\in U$ such that $U$ is isomorphic to a Zariski open subset of $\\mathbb P^n$?", "original_statement": "If $X$ is a smooth rational variety of dimension $n$ and $p \\in X$ is a point, is there a Zariski open $U \\subset X$ with $p \\in U$ such that $U$ is isomorphic to a Zariski open subset of $\\mathbb P^n$?", "clean_statement": "If $X$ is a smooth rational variety of dimension $n$ and $p \\in X$ is a point, is there a Zariski open $U \\subset X$ with $p \\in U$ such that $U$ is isomorphic to a Zariski open subset of $\\mathbb P^n$?", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 1.22 from the workshop “Rationality problems in algebraic geometry”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.22\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[54]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If $X$ is a smooth rational variety of dimension $n$ and $p \\\\in X$ is a point, is there a Zariski open $U \\\\subset X$ with $p \\\\in U$ such that $U$ is isomorphic to a Zariski open subset of $\\\\mathbb P^n$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Christian Böhning says this is true in dimension $2$.\\nThis property is stable under blow-ups along smooth centers, though it is not clear upon blowing down.\\nThere is an article by Bogomolov and Böhning. This property is called uniform rationality, though different people call it different things.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0055", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth complex n-fold X with n at least 2, a point p, and f:Y=Bl_p(X) to X with exceptional divisor E, X is uniformly rational at p if and only if there are n rational functions on Y which are simultaneously regular near the whole of E, vanish on E, have exceptional initial forms forming a basis of H^0(E,O_E(1)), and generate C(Y). The proof descends the functions by properness and normality, reads the initial forms as a cotangent basis downstairs, and applies Zariski's Main Theorem. In contrast, no single affine-space chart on Y can contain the whole positive-dimensional exceptional fiber, explaining why pointwise uniform charts do not automatically descend.\n\nCandidate contribution (criterion; novelty confidence low): Uniform rationality at the contraction point of a point blowup is equivalent to the existence of one birational coordinate tuple regular near the entire exceptional divisor whose classes in I_E/I_E^2 form a basis of the global section space H^0(E,O_E(1)); separate uniform charts cannot be replaced by one chart containing E." }, { "id": 20000056, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0056", "title": "Stein covers, MRC quotients, and the finite-cover diagonal obstruction", "statement": "Let $X$ be a rationally connected $3$-fold which is a del Pezzo fibration over $\\mathbb P^1$.\nIs there a component $M$ of the space of sections of the map $X \\rightarrow \\mathbb P^1$\nsuch that the Stein factorization of\n$\\alpha: M \\rightarrow J(X)$\nis the MRC fibration of $M$?\n\nCan we find an example of such a rationally connected $3$-fold $X$ with does not admit a decomposition of the diagonal\nand with the above property for the components of spaces of rational curves on X?", "original_statement": "Let $X$ be a rationally connected $3$-fold which is a del Pezzo fibration over $\\mathbb P^1$.\nIs there a component $M$ of the space of sections of the map $X \\rightarrow \\mathbb P^1$\nsuch that the Stein factorization of\n$\\alpha: M \\rightarrow J(X)$\nis the MRC fibration of $M$?\n\nCan we find an example of such a rationally connected $3$-fold $X$ with does not admit a decomposition of the diagonal\nand with the above property for the components of spaces of rational curves on X?", "clean_statement": "Let $X$ be a rationally connected $3$-fold which is a del Pezzo fibration over $\\mathbb P^1$.\nIs there a component $M$ of the space of sections of the map $X \\rightarrow \\mathbb P^1$\nsuch that the Stein factorization of\n$\\alpha: M \\rightarrow J(X)$\nis the MRC fibration of $M$?\n\nCan we find an example of such a rationally connected $3$-fold $X$ with does not admit a decomposition of the diagonal\nand with the above property for the components of spaces of rational curves on X?", "statement_status": "exact", "statement_verification": "The canonical AIM record (problem 1.24 in the workshop *Rationality problems in algebraic geometry*) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.24\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[55]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X$ be a rationally connected $3$-fold which is a del Pezzo fibration over $\\\\mathbb P^1$.\\nIs there a component $M$ of the space of sections of the map $X \\\\rightarrow \\\\mathbb P^1$\\nsuch that the Stein factorization of\\n$\\\\alpha: M \\\\rightarrow J(X)$\\nis the MRC fibration of $M$?\\n\\nCan we find an example of such a rationally connected $3$-fold $X$ with does not admit a decomposition of the diagonal\\nand with the above property for the components of spaces of rational curves on X?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Sho has proved a certain Stein factorization stabilization in the degree of the sections. There are only finitely many covers that can be realized as these intermediate spaces in the Stein factorizations. Voisin conjectures that if the Abel-Jacobi map is the MRC fibration, then the scheme $X$ has a decomposition of the diagonal. Here, we are asking about the Stein factorization instead of the Abel-Jacobi map.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0056", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Lehmann--Tanimoto's degree-five pencil gives a published affirmative answer to the first existence question, but the requested no-decomposition example remains open in the literature checked. This attempt proves that, for a rationally connected threefold satisfying Voisin's Theorem 4.9 hypotheses and lacking an integral cohomological decomposition of the diagonal, any dominant curve component whose connected Stein map is its MRC quotient must have finite Stein degree greater than one. For del Pezzo fibrations of general-fiber degree at least three, Lehmann--Tanimoto reduce the candidates to finitely many covers; under their vertical gluing operation the cover degrees decrease by divisibility, so reaching degree one permanently obstructs the desired MRC property downstream.\n\nCandidate contribution (obstruction_and_reduction; novelty confidence low): On a no-integral-cohomological-diagonal threefold satisfying Voisin's hypotheses, an MRC connected Stein factor of a dominant curve Abel--Jacobi map must lie over a nonidentity finite cover of the intermediate Jacobian; in the Lehmann--Tanimoto gluing order, the finite cover degree can only decrease by divisibility, and an identity-cover stage excludes every later component in that gluing chain." }, { "id": 20000057, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0057", "title": "A surface criterion for total CH0-triviality and its separation from L-rationality", "statement": "The following is question 2 on the website announcement for this conference.\n\nColliot-Thélène and Pirutka defined the notion of a $CH_0$-universally trivial resolution of singularities. As a preliminary question, is admitting a $CH_0$-universally trivial resolution\nan analytic local property for complex varieties? Assuming the answer is yes, we pose the following questions.\n\nIn low dimensions, say up to 4, can we give an analytic local classification of $CH_0$-universally trivial singularities?\n\nMore vaguely, is there a local classification of $CH_0$-universally trivial singularities in terms of MMP data?\n\nWe can ask the same questions for $L$-rational singularities (defined by Nicaise and Shinder), and for $B$-rational singularities (defined by Kontsevich and Tschinkel).", "original_statement": "The following is question 2 on the website announcement for this conference.\n\nColliot-Thélène and Pirutka defined the notion of a $CH_0$-universally trivial resolution of singularities. As a preliminary question, is admitting a $CH_0$-universally trivial resolution\nan analytic local property for complex varieties? Assuming the answer is yes, we pose the following questions.\n\nIn low dimensions, say up to 4, can we give an analytic local classification of $CH_0$-universally trivial singularities?\n\nMore vaguely, is there a local classification of $CH_0$-universally trivial singularities in terms of MMP data?\n\nWe can ask the same questions for $L$-rational singularities (defined by Nicaise and Shinder), and for $B$-rational singularities (defined by Kontsevich and Tschinkel).", "clean_statement": "The following is question 2 on the website announcement for this conference.\n\nColliot-Thélène and Pirutka defined the notion of a $CH_0$-universally trivial resolution of singularities. As a preliminary question, is admitting a $CH_0$-universally trivial resolution\nan analytic local property for complex varieties? Assuming the answer is yes, we pose the following questions.\n\nIn low dimensions, say up to 4, can we give an analytic local classification of $CH_0$-universally trivial singularities?\n\nMore vaguely, is there a local classification of $CH_0$-universally trivial singularities in terms of MMP data?\n\nWe can ask the same questions for $L$-rational singularities (defined by Nicaise and Shinder), and for $B$-rational singularities (defined by Kontsevich and Tschinkel).", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List item 1.26, attributed on the live AIM page to Asher Auel. Its mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.26\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[56]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The following is question 2 on the website announcement for this conference.\\n\\nColliot-Thélène and Pirutka defined the notion of a $CH_0$-universally trivial resolution of singularities. As a preliminary question, is admitting a $CH_0$-universally trivial resolution\\nan analytic local property for complex varieties? Assuming the answer is yes, we pose the following questions.\\n\\nIn low dimensions, say up to 4, can we give an analytic local classification of $CH_0$-universally trivial singularities?\\n\\nMore vaguely, is there a local classification of $CH_0$-universally trivial singularities in terms of MMP data?\\n\\nWe can ask the same questions for $L$-rational singularities (defined by Nicaise and Shinder), and for $B$-rational singularities (defined by Kontsevich and Tschinkel).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Certain things are known: $CH_0$-universally trivial singularities are more special than rational singularities but more general than toric singularities.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0057", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an isolated normal complex surface germ and a good resolution with reduced SNC exceptional curve E, the resolution is totally CH0-trivial if and only if every irreducible component of E is a projective line. This condition is independent of the good resolution, invariant under analytic isomorphism of germs, and implies global universal CH0-triviality of the resolution. In contrast, the germ is L-rational exactly when all exceptional components are projective lines and the dual multigraph is a tree. Thus cusp singularities, whose exceptional graph is a rational cycle, have totally and universally CH0-trivial resolutions but are not L-rational. The original locality question for resolutions satisfying only the global universal pushforward condition remains open.\n\nCandidate contribution (equivalence; novelty confidence low): Candidate surface dictionary: total CH0-triviality is equivalent to vanishing of all exceptional-component genera, while L-rationality is equivalent to the same condition plus H_1 of the dual graph being zero; cusp singularities give a strict separating family." }, { "id": 20000058, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0058", "title": "Branches, exceptional cycles, and tangent-cone comparisons among three rationality conditions", "statement": "Compare $CH_0$-universally trivial, $B$-rational, and $L$-rational singularities.", "original_statement": "Compare $CH_0$-universally trivial, $B$-rational, and $L$-rational singularities.", "clean_statement": "Compare $CH_0$-universally trivial, $B$-rational, and $L$-rational singularities.", "statement_status": "exact", "statement_verification": "The canonical record is AIM problem 1.28 from the workshop *Rationality problems in algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.28\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[57]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Compare $CH_0$-universally trivial, $B$-rational, and $L$-rational singularities.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0058", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over the complex numbers, for an integral curve on a smooth surface, B-rationality of the pair, L-rationality of the curve, and total CH_0-triviality of normalization are all equivalent to unibranchedness. Global universal CH_0-triviality is strictly weaker: the normalization of a split rational nodal cubic is universally CH_0-trivial in ordinary Fulton CH_0, but is not total, L-rational, or B-rational. Normality does not restore an implication from total CH_0-triviality to L-rationality: a cusp surface resolution has exceptional rational cycle E with CH_0(E_F) isomorphic to Z for every extension F/C, while [E] is congruent to 0 rather than 1 modulo the Lefschetz class. Under precise one-blowup ordinary-multiple-point hypotheses, the three tests reduce to the tangent-cone section F and give B-rational implies L-rational implies total CH_0-trivial implies universally CH_0-trivial.\n\nCandidate contribution (comparison theorem and explicit separating examples; novelty confidence low): Candidate novelty: the branch/dual-cycle package identifies B-rationality, L-rationality, and total normalization CH_0-triviality with unibranchedness for integral complex curves, separates global universal from total triviality by a split nodal cubic, and separates total CH_0-triviality from L-rationality for a normal cusp by the explicit identities CH_0(E_F)=Z and [E]=rL." }, { "id": 20000059, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0059", "title": "An oriented preservation calculus for categorical representability in codimension two", "statement": "Which birational transformations preserve the following property?\n\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$\nFor future reference, we call this property (*).", "original_statement": "Which birational transformations preserve the following property?\n\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$\nFor future reference, we call this property (*).", "clean_statement": "Which birational transformations preserve the following property?\n\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$\nFor future reference, we call this property (*).", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.3 from the AIM workshop *Rationality problems in algebraic geometry*, stored as record 58 (zero-based) of `aim-algebraic-geometry-notes.json`. Its problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.3\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[58]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Which birational transformations preserve the following property?\\n\\n\\\\[\\nD(X) = \\\\langle \\\\mathscr A_1, \\\\ldots, \\\\mathscr A_m \\\\rangle\\n\\\\]\\nhas a semiorthogonal decomposition\\nwith $\\\\mathscr A_i \\\\hookrightarrow D(Y_i)$ such that ${\\\\rm dim} Y_i \\\\leq {\\\\rm dim}X-2.$\\nFor future reference, we call this property (*).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Property (*) is preserved under blowups along smooth centers. It is also expected to be preserved along flops of smooth varieties, because they should not change the derived category; that is known in dimension 3.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0059", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For smooth projective n-folds, representability dimension propagates through any directed finite chain of smooth blowups, reverse-MMP standard flips in the precise Belmans-Fu-Raedschelders sense, and proven derived-equivalent flops: the final representability dimension is bounded by the maximum of the initial dimension and the dimensions of all blowup centers and standard-flip bases. Thus property (*) is preserved along such chains. For a standard flip of type (k,l), exactly k-l copies of the base category of dimension n-k-l-1 occur, so preservation is from the K-positive model to the K-negative model; standard flops and all proved flop equivalences preserve (*) bidirectionally. Property (*) also forces HH_j=0 for |j|>n-2, and these extreme Hochschild groups are birational invariants, although this additive shadow does not prove blowdown descent.\n\nCandidate contribution (preservation_criterion; novelty confidence low): Candidate novelty is the explicit finite-chain inequality rdim(X_r) <= max{rdim(X_0), all blowup-center dimensions, all standard-flip-base dimensions}, with the exact k-l standard-flip defect blocks and their direction, packaged with the birational extreme-Hochschild boundary that remains valid when categorical descent is unavailable." }, { "id": 20000060, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0060", "title": "Projective-space stabilization gives rationally connected stably irrational examples with property (*)", "statement": "Is there a rationally connected irrational variety $X$ with property (*)?\n\nRecall Property (*) from a previous question was defined as follows.\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$", "original_statement": "Is there a rationally connected irrational variety $X$ with property (*)?\n\nRecall Property (*) from a previous question was defined as follows.\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$", "clean_statement": "Is there a rationally connected irrational variety $X$ with property (*)?\n\nRecall Property (*) from a previous question was defined as follows.\n\\[\nD(X) = \\langle \\mathscr A_1, \\ldots, \\mathscr A_m \\rangle\n\\]\nhas a semiorthogonal decomposition\nwith $\\mathscr A_i \\hookrightarrow D(Y_i)$ such that ${\\rm dim} Y_i \\leq {\\rm dim}X-2.$", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.32 from the 2019 AIM workshop *Rationality problems in algebraic geometry*. The preceding canonical record, problem 1.30, introduces property (*). Together they ask:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.32\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[59]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a rationally connected irrational variety $X$ with property (*)?\\n\\nRecall Property (*) from a previous question was defined as follows.\\n\\\\[\\nD(X) = \\\\langle \\\\mathscr A_1, \\\\ldots, \\\\mathscr A_m \\\\rangle\\n\\\\]\\nhas a semiorthogonal decomposition\\nwith $\\\\mathscr A_i \\\\hookrightarrow D(Y_i)$ such that ${\\\\rm dim} Y_i \\\\leq {\\\\rm dim}X-2.$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Some people expect that the answer is no.\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0060", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Let Z be any smooth projective rationally connected complex variety that is not stably rational. For every vector bundle E of rank r+1 with r at least 2, X=P_Z(E) is rationally connected and stably irrational, while Orlov's projective-bundle formula decomposes D^b(X) into r+1 copies of D^b(Z); since dim Z is at most dim X-2, X has property (*). Taking Z to be the smooth Artin-Mumford unirational threefold yields the explicit family Z times P^{n-3} for every n at least 5, completely answering the literal unrestricted AIM existence question affirmatively.\n\nCandidate contribution (explicit_construction; novelty confidence low): The Artin-Mumford stabilization X_n=Z_AM times P^{n-3}, for every n at least 5, is a smooth rationally connected and stably irrational variety with property (*); more generally, every projective bundle of relative dimension at least two over a stably irrational rationally connected base has these properties." }, { "id": 20000061, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0061", "title": "Explicit K3 models and fine-moduli equivalences for discriminants 26 and 38", "statement": "There have been new rationality constructions of special cubic fourfolds $X$ due to Russo and Staglianò, for discriminant $26$ or $38$. Using these explicit constructions, write down an explicit K3 surface S and an equivalence $Ku(X) \\simeq D(S)$.", "original_statement": "There have been new rationality constructions of special cubic fourfolds $X$ due to Russo and Staglianò, for discriminant $26$ or $38$. Using these explicit constructions, write down an explicit K3 surface S and an equivalence $Ku(X) \\simeq D(S)$.", "clean_statement": "There have been new rationality constructions of special cubic fourfolds $X$ due to Russo and Staglianò, for discriminant $26$ or $38$. Using these explicit constructions, write down an explicit K3 surface S and an equivalence $Ku(X) \\simeq D(S)$.", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.34\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[60]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"There have been new rationality constructions of special cubic fourfolds $X$ due to Russo and Staglianò, for discriminant $26$ or $38$. Using these explicit constructions, write down an explicit K3 surface S and an equivalence $Ku(X) \\\\simeq D(S)$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0061", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a general cubic fourfold on the common open locus of the Russo--Stagliano rationality construction and the Bridgeland-moduli construction, the associated K3 surface is recovered explicitly by contracting the exceptional curve(s) in the inverse base surface: Bl_p(S_26) mapped by pi^*H_26-4E for d=26, and Bl_{p_1,...,p_11}(S_38) mapped by pi^*H_38-sum_{1}^{10}E_i-4E_11 for d=38. Explicit primitive isotropic Mukai vectors v_26=(1,3,1) and v_38=(2,2,-1), each with an integral isotropic mate pairing to one, make S_d=M_sigma(Ku(X),v_d) a fine untwisted K3 moduli space. Its universal object gives D^b(S_d) equivalent to Ku(X), and the marked Picard-rank-one period comparison identifies this moduli K3 with the geometric K3. The kernel is explicit by its universal property, not by projective-coordinate matrices.\n\nCandidate contribution (explicit_lattice_decomposition; novelty confidence low): In the negative-Euler normal form G_38=[[-2,1,0],[1,-2,1],[0,1,12]], the vectors v_38=(2,2,-1), u_38=(7,5,-3), and w_38=(33,28,-15) form a unimodular basis with Gram matrix U direct-sum <-38>; thus v_38 is a divisibility-one primitive isotropic vector, u_38 explicitly removes the Brauer obstruction, and w_38 explicitly supplies the degree-38 polarization." }, { "id": 20000062, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0062", "title": "A transcendental-rank obstruction to surface hosts", "statement": "For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \\subset D(X)$ has an embedding $Ku(X) \\hookrightarrow D(S)$?", "original_statement": "For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \\subset D(X)$ has an embedding $Ku(X) \\hookrightarrow D(S)$?", "clean_statement": "For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \\subset D(X)$ has an embedding $Ku(X) \\hookrightarrow D(S)$?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.36 from the AIM workshop *Rationality problems in algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.36\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[61]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \\\\subset D(X)$ has an embedding $Ku(X) \\\\hookrightarrow D(S)$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"It is known that $Ku(X) \\\\not\\\\simeq D(S)$ for any surface $S$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0062", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a very general complex cubic fourfold X, there is no fully faithful embedding Ku(X) into the derived category of a smooth projective surface that is compatible with the canonical categorical Hodge realizations; in particular, no Fourier--Mukai-type embedding exists. For an arbitrary exact fully faithful embedding, admissibility and Lin's top-Hochschild theorem still force the host to have p_g=1, force the complement to have HH_{-2}=0 and support in the canonical base locus, and recent support theory forces the host to be nonminimal. The unrestricted non-dg-liftable case remains open.\n\nCandidate contribution (obstruction; novelty confidence low): A Hodge-realization-compatible admissible K3 category with rank-24 Mukai realization and exactly two independent rational Hodge classes cannot embed in the derived category of a smooth projective surface: Lin's theorem forces p_g=1, adjunction forces a rank-22 transcendental injection, and surface classification bounds the target transcendental rank by 21." }, { "id": 20000063, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0063", "title": "Arithmetic rigidity and marked Torelli for categorical partners of cubic fourfolds", "statement": "Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \\simeq Ku(Y)$, is $X$ birational to $Y$?", "original_statement": "Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \\simeq Ku(Y)$, is $X$ birational to $Y$?", "clean_statement": "Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \\simeq Ku(Y)$, is $X$ birational to $Y$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, workshop *Rationality problems in algebraic geometry*, problem 1.38, source file `aim-algebraic-geometry-notes.json`, zero-based record index 62. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.38\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[62]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \\\\simeq Ku(Y)$, is $X$ birational to $Y$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"There are at most finitely many $Y$ with $Ku(Y)$ equivalent to $Ku(X)$.\\nThe converse is not true, as can be seen by looking at rational $X$ and $Y$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0063", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted birational categorical Torelli problem remains open. For a very general cubic fourfold X in a nonempty Hassett divisor C_d, however, the report proves from the Fan-Lai partner-count formula that X has exactly one categorical partner if and only if d=2p^e with p != 3 prime and p^e congruent to 1 modulo 3, or d=6p^e with p != 3 prime, or d=18. Hence every exact C-linear equivalence Ku(X) equivalent to Ku(Y) forces Y isomorphic to X in precisely these one-partner families. Independently, for arbitrary smooth cubics, an equivalence whose cohomological transform preserves the canonical A_2 sublattice forces isomorphism by global Torelli.\n\nCandidate contribution (classification_corollary; novelty confidence low): Candidate novelty: the explicit if-and-only-if prime-power classification of all nonempty Hassett discriminants d for which the very general cubic in C_d has exactly one partner under a bare exact Kuznetsov-component equivalence." }, { "id": 20000064, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0064", "title": "K_0 under K-equivalence: codimension-one invariance and the threefold case", "statement": "If two smooth projective varieties $X$ and $Y$ are K-equivalent, is the Grothendieck group $K_0(X)$ isomorphic to $K_0(Y)$?", "original_statement": "If two smooth projective varieties $X$ and $Y$ are K-equivalent, is the Grothendieck group $K_0(X)$ isomorphic to $K_0(Y)$?", "clean_statement": "If two smooth projective varieties $X$ and $Y$ are K-equivalent, is the Grothendieck group $K_0(X)$ isomorphic to $K_0(Y)$?", "statement_status": "exact", "statement_verification": "The exact source record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.4\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[63]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If two smooth projective varieties $X$ and $Y$ are K-equivalent, is the Grothendieck group $K_0(X)$ isomorphic to $K_0(Y)$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Motivic integration shows that $X$ and $Y$ have many invariants in common, such as the Hodge numbers. Kawamata's conjecture that K-equivalence implies D-equivalence would imply a positive answer, since the derived category determines $K_0(X)$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0064", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every chosen K-equivalence between smooth projective complex varieties, rank and strict-transform determinant give a canonical-relative-to-the-map lambda-ring isomorphism K_0(X)/F_top^2 K_0(X) ≅ K_0(Y)/F_top^2 K_0(Y), with an explicit Z ⊕ Pic description and formulas for products, exterior powers, and Adams operations; hence any unresolved obstruction begins in codimension at least two. In dimensions at most three the full integral K_0 groups are isomorphic: in dimension three Kawamata's factorization and canonical-cover-stack flop equivalences give a normalized Fourier-Mukai/dg equivalence, which also yields an equivalence of algebraic K-theory spectra and an Euler-isometric K_0 map. No full ring or lambda-ring isomorphism is claimed.\n\nCandidate contribution (reduction; novelty confidence low): The candidate contribution is the combined statement that K_0/F_top^2 is canonically, relative to a chosen K-equivalence, the explicit lambda-ring Z ⊕ Pic with lambda^m(r,L) = (binom(r,m), L^binom(r-1,m-1)) for every integral virtual rank, and that the birationally normalized threefold flop transform lifts precisely this quotient comparison to an integral Euler-isometric K_0 isomorphism and a K-theory-spectrum equivalence." }, { "id": 20000065, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0065", "title": "The quadric residual of a bidegree (d,2) hypersurface", "statement": "Investigate $Ku(X)$ for a Fano hypersurface $X$ in a product of projective spaces. Specifically, investigate this for $X \\subset \\mathbb P^2 \\times \\mathbb P^3$ of bidegree $(d,2)$ for $d \\geq 2$.", "original_statement": "Investigate $Ku(X)$ for a Fano hypersurface $X$ in a product of projective spaces. Specifically, investigate this for $X \\subset \\mathbb P^2 \\times \\mathbb P^3$ of bidegree $(d,2)$ for $d \\geq 2$.", "clean_statement": "Investigate $Ku(X)$ for a Fano hypersurface $X$ in a product of projective spaces. Specifically, investigate this for $X \\subset \\mathbb P^2 \\times \\mathbb P^3$ of bidegree $(d,2)$ for $d \\geq 2$.", "statement_status": "exact", "statement_verification": "The canonical record (AIM Problem List, workshop *Rationality problems in algebraic geometry*, problem 1.42) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.42\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[64]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate $Ku(X)$ for a Fano hypersurface $X$ in a product of projective spaces. Specifically, investigate this for $X \\\\subset \\\\mathbb P^2 \\\\times \\\\mathbb P^3$ of bidegree $(d,2)$ for $d \\\\geq 2$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0065", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Among smooth bidegree (d,2) hypersurfaces in P^2 x P^3 with d >= 2, only d = 2 is Fano. For every smooth member, projection to P^2 gives a flat quadric-surface fibration whose natural six-line-bundle residual is D^b(P^2,B_0). Under simple degeneration (in particular for a general member), it is D^b(T_d,alpha_d), where T_d is the double plane branched over a degree-4d curve. Explicit formulas for its Serre functor and Hochschild homology show that it is never fractional Calabi-Yau; for d = 2 its Hochschild profile is (3,40,3) and its very general Brauer twist is nonzero.\n\nCandidate contribution (explicit invariant computation; novelty confidence low): For the explicitly defined quadric-fibration residual K_d of a general bidegree (d,2) hypersurface, S_K(F) = F tensor pi^*O(2d-3)[2], dim HH_{+/-2}(K_d) = binomial(2d-1,2), dim HH_0(K_d) = 12d^2-6d+4, and all other Hochschild groups vanish; consequently K_d is not fractional Calabi-Yau. For the sole Fano value d=2 this specializes to the profile (3,40,3), with nonzero Brauer class very generally." }, { "id": 20000066, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0066", "title": "Hochschild invariants and a projective Calabi-Yau nonembedding obstruction", "statement": "Investigate $Ku(X)$ for $X$ a smooth quartic double 5 fold, that is, a double cover of $\\mathbb P^5$ branched along a quartic.\nThis variety is Calabi-Yau and it is never of the form $D(Y)$ for $Y$ a variety. Further, the Serre functor of $X$ is equal to shift by 3; in other words, the category $Ku(X)$ is 3-Calabi Yau.", "original_statement": "Investigate $Ku(X)$ for $X$ a smooth quartic double 5 fold, that is, a double cover of $\\mathbb P^5$ branched along a quartic.\nThis variety is Calabi-Yau and it is never of the form $D(Y)$ for $Y$ a variety. Further, the Serre functor of $X$ is equal to shift by 3; in other words, the category $Ku(X)$ is 3-Calabi Yau.", "clean_statement": "Investigate $Ku(X)$ for $X$ a smooth quartic double 5 fold, that is, a double cover of $\\mathbb P^5$ branched along a quartic.\nThis variety is Calabi-Yau and it is never of the form $D(Y)$ for $Y$ a variety. Further, the Serre functor of $X$ is equal to shift by 3; in other words, the category $Ku(X)$ is 3-Calabi Yau.", "statement_status": "exact", "statement_verification": "The canonical AIM record (problem 1.44, source index 65) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.44\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[65]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate $Ku(X)$ for $X$ a smooth quartic double 5 fold, that is, a double cover of $\\\\mathbb P^5$ branched along a quartic.\\nThis variety is Calabi-Yau and it is never of the form $D(Y)$ for $Y$ a variety. Further, the Serre functor of $X$ is equal to shift by 3; in other words, the category $Ku(X)$ is 3-Calabi Yau.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0066", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth quartic double fivefold X, the variety X is Fano of index 4 while Ku(X) is 3-Calabi-Yau. A weighted-Jacobian calculation gives dim HH_i(Ku(X)) = 1, 90, 2, 90, 1 in degrees i = -3, -1, 0, 1, 3 respectively, with zero groups in degrees plus or minus 2. In particular HH_0 has dimension 2, so Ku(X) is not the derived category of a smooth projective variety. Combining this obstruction with indecomposability of connected Calabi-Yau categories proves the stronger corollary that Ku(X) has no admissible fully faithful embedding into the derived category of a connected smooth projective Calabi-Yau threefold.\n\nCandidate contribution (corollary; novelty confidence low): For every smooth quartic double fivefold X, Ku(X) admits no admissible fully faithful embedding into D^b(W) for any connected smooth projective Calabi-Yau threefold W." }, { "id": 20000067, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0067", "title": "Hodge correction and signed weak-factorization constraints for quartic double fivefolds", "statement": "Show that smooth quartic double 5-folds are irrational.", "original_statement": "Show that smooth quartic double 5-folds are irrational.", "clean_statement": "Show that smooth quartic double 5-folds are irrational.", "statement_status": "exact", "statement_verification": "The canonical AIM record (source file `aim-algebraic-geometry-notes.json`, zero-based index 66, Problem 1.46 of the workshop *Rationality problems in algebraic geometry*) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.46\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[66]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Show that smooth quartic double 5-folds are irrational.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Motivated by the previous question, Kuznetsov expects irrationality in this case.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: medium.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0067", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The full irrationality problem remains open. For every smooth complex quartic double fivefold, a weighted Jacobian-ring calculation, independently checked by the branched-cover Euler characteristic, gives h^{4,1}=1, h^{3,2}=90, and b_5=182. Consequently any weak factorization of a hypothetical stable rationality map satisfies the exact signed identities 1=sum epsilon_i h^{3,0}(Z_i) and 90=sum epsilon_i(h^{2,1}(Z_i)+1_{codim Z_i >= 3}h^{1,0}(Z_i)); in particular it must use a center with a holomorphic 3-form. The calculation also corrects the values b_5=284 and intermediate-Jacobian dimension 142 asserted in arXiv:2509.12186v2, while making no blanket claim about that preprint's later cycle-theoretic conclusions.\n\nCandidate contribution (correction and birational reduction; novelty confidence medium): Candidate novelty: the specialized signed center-balance identities for every ordinary or stable weak factorization, together with an explicit correction of Lemma 7.2 of arXiv:2509.12186v2 from (Hodge level 1, b_5=284, dim J=142) to (Hodge level 3, b_5=182, dim J=91)." }, { "id": 20000068, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0068", "title": "Witt-index and odd-degree criteria for intersections of two quadrics", "statement": "Hassett and Tschinkel showed that a complete intersection of two quadrics in $\\mathbb P^5$ over an arbitrary field $k$ is rational if and only if it contains a line over $k$.\n\nGive criteria for other geometrically rational Fano $3$-folds to be rational over a field $k$.\n\nGive criteria for the complete intersection of two quadrics in $\\mathbb P^n$ to be rational over $k$.", "original_statement": "Hassett and Tschinkel showed that a complete intersection of two quadrics in $\\mathbb P^5$ over an arbitrary field $k$ is rational if and only if it contains a line over $k$.\n\nGive criteria for other geometrically rational Fano $3$-folds to be rational over a field $k$.\n\nGive criteria for the complete intersection of two quadrics in $\\mathbb P^n$ to be rational over $k$.", "clean_statement": "Hassett and Tschinkel showed that a complete intersection of two quadrics in $\\mathbb P^5$ over an arbitrary field $k$ is rational if and only if it contains a line over $k$.\n\nGive criteria for other geometrically rational Fano $3$-folds to be rational over a field $k$.\n\nGive criteria for the complete intersection of two quadrics in $\\mathbb P^n$ to be rational over $k$.", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.48\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[67]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Hassett and Tschinkel showed that a complete intersection of two quadrics in $\\\\mathbb P^5$ over an arbitrary field $k$ is rational if and only if it contains a line over $k$.\\n\\nGive criteria for other geometrically rational Fano $3$-folds to be rational over a field $k$.\\n\\nGive criteria for the complete intersection of two quadrics in $\\\\mathbb P^n$ to be rational over $k$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The last question is solved for $n = 5$, due to the above-mentioned paper of Hassett and Tschinkel.\\nIt may be that people are already working on these questions.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0068", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth complete intersection X={q_0=q_1=0} in P^n over a field of characteristic different from 2, the k-linear r-planes on X are detected exactly by the Witt index h of q_0+tq_1 over k(t): F_r(X)(k) is nonempty if and only if h is at least r+1. Thus h=0 excludes points and unirationality, h at least 1 is equivalent to separable unirationality, and h at least 2 implies rationality; for n=5 Benoist--Wittenberg makes the last condition necessary as well. Moreover, every finite odd-degree extension preserves h and hence simultaneously preserves the existence of every linear-space level. In particular, rationality of a smooth P^5 intersection descends through finite odd-degree extensions.\n\nCandidate contribution (odd_degree_descent; novelty confidence low): The entire incidence ladder of projective linear spaces on a smooth intersection of two quadrics is invariant under finite odd-degree base change; as an exact n=5 corollary, rationality descends through finite odd-degree extensions." }, { "id": 20000069, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0069", "title": "Infinite degree-three unramified cohomology and a rationally connected coefficient sieve", "statement": "Let $X$ be a smooth projective complex variety. For a finite abelian group $M$, is unramified cohomology $H^i_{nr}(k(X)/k, M)$ always finite?", "original_statement": "Let $X$ be a smooth projective complex variety. For a finite abelian group $M$, is unramified cohomology $H^i_{nr}(k(X)/k, M)$ always finite?", "clean_statement": "Let $X$ be a smooth projective complex variety. For a finite abelian group $M$, is unramified cohomology $H^i_{nr}(k(X)/k, M)$ always finite?", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.5\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[68]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X$ be a smooth projective complex variety. For a finite abelian group $M$, is unramified cohomology $H^i_{nr}(k(X)/k, M)$ always finite?\"\nOriginal remarks: [\"This answer is provided by Burt Totaro. The answer is no, for $i=3$. Namely, by C. Schoen (for some primes $\\\\ell$)\\nand B. Totaro (for all primes $\\\\ell$), there are smooth\\ncomplex projective 3-folds $X$ such that $CH^2(X)/\\\\ell$\\nis infinite. For example, one can take $X$ to be a very general\\nprincipally polarized abelian 3-fold. For such a variety,\\nconsider the Bloch-Ogus spectral sequence\\n\\\\[\\n E^2_{ij} = H^i_{Zar}(X,H^j_{et}(\\\\mathbb Z/\\\\ell \\\\mathbb Z)) \\\\implies H^{i+j}_{et}(X,\\\\mathbb Z/\\\\ell \\\\mathbb Z).\\n\\\\]\\nThe $E_{\\\\infty}$ groups are finite, whereas $H^2(X,H^2) = CH^2(X)/\\\\ell$ is infinite. Only one differential\\ncan change that group, namely\\n\\\\[ d_2: H^0(X, H^3) \\\\rightarrow H^2(X, H^2).\\n\\\\]\\nTherefore, this differential has infinite image, and hence\\n$H^0(X,H^3)$ must be infinite. This group is the unramified\\ncohomology $H^3_{nr}(k(X)/k, \\\\mathbb Z/\\\\ell \\\\mathbb Z)$, using that $X$ is smooth\\nand proper over $k$. So the latter group can be infinite.\\n\\nAn interesting substitute for the question is:\\nLet $X$ be a {\\\\it rationally connected }complex variety.\\nIs the unramified cohomology $H^i_{nr}(k(X)/k, \\\\mathbb Z/\\\\ell \\\\mathbb Z)$\\nalways finite?\"]\nOriginal literature field (JSON string): \"This is known for $i = 1$ and $2$. It also holds if $i=3$ and $X$ is rationally connected.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0069", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The universal finiteness assertion is false: for every prime ell, a very general principally polarized complex abelian threefold X has infinite H^3_nr(C(X)/C,Z/ell), because the audited Bloch-Ogus edge sequence makes its finiteness equivalent to finiteness of CH^2(X)/ell, which Totaro proved infinite. For the rationally connected substitute, degrees at most three are finite, while higher degrees remain apparently open in the literature checked. A proved bad-prime/Bockstein reduction shows that Tor(X) kills positive-degree unramified cohomology, removes every primary coefficient component at primes not dividing Tor(X), and forces a minimal p-power-coefficient failure above mod p to produce an infinite explicit Zariski H^1 of a Bockstein-image sheaf.\n\nCandidate contribution (reduction; novelty confidence low): For fixed smooth projective rationally connected X and i>0, only coefficient primes dividing Tor(X) can contribute to H^i_nr(X,M); moreover, if r>1 is minimal with H^i_nr(X,Z/p^r) infinite, then H^1_Zar(X, im(H^{i-1}_sheaf(Z/p) -> H^i_sheaf(Z/p^{r-1}))) is infinite. More generally, for 0 -> A -> B -> C -> 0, the report proves the explicit cardinality bound |H^i_nr(X,B)| <= |H^i_nr(X,A)| |H^i_nr(X,C)| |H^1_Zar(X,I_i)| when the right side is finite." }, { "id": 20000070, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0070", "title": "A torsion-sheaf and finite-coefficient reduction for top unramified cohomology", "statement": "Let $X$ be a rationally connected smooth projective variety of dimension $n$.\nIs $H^n_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z) = 0$?", "original_statement": "Let $X$ be a rationally connected smooth projective variety of dimension $n$.\nIs $H^n_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z) = 0$?", "clean_statement": "Let $X$ be a rationally connected smooth projective variety of dimension $n$.\nIs $H^n_{nr}(k(X)/k, \\mathbb Q/\\mathbb Z) = 0$?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.52\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[69]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X$ be a rationally connected smooth projective variety of dimension $n$.\\nIs $H^n_{nr}(k(X)/k, \\\\mathbb Q/\\\\mathbb Z) = 0$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"We know $H^1_{nr}(k(X)/k, \\\\mathbb Q/\\\\mathbb Z) = 0$, and there are examples with $H^i_{nr}(k(X)/k, \\\\mathbb Q/\\\\mathbb Z) \\\\neq 0$ for $2 \\\\leq i \\\\leq n - 1$.\\nThe question has a positive answer when $n \\\\leq 3$. For $n=3$, this was shown by Colliot-Thélène and Voisin, using Voisin's theorem that the integral Hodge conjecture holds for rationally connected 3-folds.\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0070", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth projective complex variety X with finite torsion order N and any q>0 and Tate twist j, there are proved coefficient-compatible isomorphisms H^q_nr(C(X)/C,Z/m(j)) = H^1_Zar(X,H^q_Z(j))[m] and H^q_nr(C(X)/C,Q/Z(j)) = H^1_Zar(X,H^q_Z(j))_tors. The torsion is annihilated by N, and the natural map from the single finite group with Z/N coefficients to the Q/Z group is an isomorphism. Hence for a rationally connected n-fold the AIM question is exactly equivalent to torsion-freeness of H^1_Zar(X,H^n_Z(j)), or to vanishing at the single finite coefficient level Z/Tor(X). This does not settle that torsion-freeness for n>=4.\n\nCandidate contribution (coefficient-compatible reduction; novelty confidence low): Candidate contribution: derive the explicit transition-map-compatible identification of finite and divisible top unramified cohomology with torsion in the first Zariski cohomology of the integral Bloch-Ogus sheaf, and deduce the testable finite criterion H^q_nr(X,Z/Tor(X)(j)) -> H^q_nr(X,Q/Z(j)) is an isomorphism, with the prime-local level ell^{v_ell(Tor(X))} sufficient.", "research_difficulty_suggested": "L4" }, { "id": 20000071, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0071", "title": "Effective certificates and the witness gap for rationality and unirationality", "statement": "Let $X$ be a variety over $\\overline{\\mathbb Q}$. Is there an algorithm to determine whether\n$X$ is rational?\n\nSame question for unirationality.", "original_statement": "Let $X$ be a variety over $\\overline{\\mathbb Q}$. Is there an algorithm to determine whether\n$X$ is rational?\n\nSame question for unirationality.", "clean_statement": "Let $X$ be a variety over $\\overline{\\mathbb Q}$. Is there an algorithm to determine whether\n$X$ is rational?\n\nSame question for unirationality.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.54 from the AIM workshop *Rationality problems in algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.54\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[70]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X$ be a variety over $\\\\overline{\\\\mathbb Q}$. Is there an algorithm to determine whether\\n$X$ is rational?\\n\\nSame question for unirationality.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Some related\\nresults are: Kanel-Belov and Chilikov\\nshowed that for affine varieties over $\\\\mathbb Q$, there is no\\nalgorithm to determine whether there is a closed\\nembedding $\\\\mathbb A^{11} \\\\rightarrow X$ defined over $\\\\mathbb Q$, or whether\\nthere is such an embedding defined over $\\\\overline{\\\\mathbb Q}$. Also, Kim and Roush showed that there is no algorithm to determine whether a morphism from an affine variety over $\\\\overline{\\\\mathbb Q}$ to ${\\\\mathbb P}^2$ has a rational section.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0071", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For integral varieties over a fixed computable presentation of the algebraic closure of Q, rationality and unirationality are recursively enumerable. More precisely, existence of a certificate of any fixed degree bound is decidable: rationality is certified by homogeneous formulas for mutually inverse rational maps, while unirationality is certified by a homogeneous map whose explicitly encoded Jacobian has full generic rank. Enumerating degree bounds therefore halts on every positive instance. The argument also proves that a witness after any algebraically closed base extension already exists over the algebraic closure of Q.\n\nCandidate contribution (equivalence; novelty confidence low): For either rationality or unirationality in the stated input model, the property is decidable if and only if its complement is recursively enumerable, if and only if there is a total recursive input-dependent function B such that every positive input has a parametrization certificate of degree at most B; B may take arbitrary values on negative inputs." }, { "id": 20000072, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0072", "title": "A reduced Merkurjev module and an exponent-p detector", "statement": "Given a rationally connected variety which is not universally $CH_0$-trivial,\ndetermine a nonzero unramified class in some cycle module (in the sense of Rost).", "original_statement": "Given a rationally connected variety which is not universally $CH_0$-trivial,\ndetermine a nonzero unramified class in some cycle module (in the sense of Rost).", "clean_statement": "Given a rationally connected variety which is not universally $CH_0$-trivial,\ndetermine a nonzero unramified class in some cycle module (in the sense of Rost).", "statement_status": "exact", "statement_verification": "There is no visible corruption or ambiguity in the source text. To make the mathematical hypotheses precise, this report works with a smooth, proper, geometrically integral \\(X/k\\). The main new formulation also assumes that \\(X\\) has a zero-cycle \\(z\\) of degree one. This is automatic when \\(k\\) is algebraically closed and whenever \\(X(k)\\ne\\varnothing\\), which covers the usual geometric setting of the AIM question. In characteristic zero, a smooth projective rationally connected variety is rationally chain connected. Rational chain connectedness is used below only to ensure a finite torsion order.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rationality problems in algebraic geometry\nSection: The problems\nSource item: 1.56\nSource URL: http://aimpl.org/rationalityag/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[71]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given a rationally connected variety which is not universally $CH_0$-trivial,\\ndetermine a nonzero unramified class in some cycle module (in the sense of Rost).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"We know that such an element exists by a theorem of Merkurjev. His argument is: given a smooth projective\\nvariety $X$ over a field $k$ such that $X$ is not\\nuniversally $CH_0$-trivial, consider the cycle module\\n$M$ over $k$ defined by\\n\\\\[\\n M(L) := A_0(X_L, K)\\n\\\\]\\nfor fields $L$ over $k$ (where $K$ denotes Milnor K-theory).\\nIn particular, $M_0(L)$ is the Chow group $CH_0(X_L)$.\\nThen the diagonal in $X \\\\times X$ determines a canonical unramified element\\nof $M_0(k(X)) = CH_0(X_{k(X)})$. When $X$ is not universally\\n$CH_0$-trivial, this element does not come from\\n$M_0(k) = CH_0(X).$\\n\\nSo the question is really whether one can find\\n\\\"simpler\\\" cycle modules than the one above\\nthat serve the same purpose.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/rationalityag/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0072", "aim-domain:algebraic-geometry", "aim-workshop:rationalityag", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth proper geometrically integral index-one variety X, the kernel R_X of Merkurjev's norm K_X -> K represents normalized unramified invariants. The reduced generic diagonal delta corresponds to id_{R_X}; hence the torsion order tau(X) equals the exponent of the entire cycle module R_X over all fields and degrees. If tau(X) is finite, then for every prime p dividing tau(X), R_X/pR_X is a nonzero exponent-p Rost cycle module and delta mod p is a nonzero nonconstant unramified class because its normalized class corresponds to the nonzero quotient morphism R_X -> R_X/pR_X.\n\nCandidate contribution (reduction_and_mod_p_corollary; novelty confidence low): The reduced kernel of Merkurjev's norm corepresents normalized unramified invariants; its global exponent is exactly the torsion order, and reduction modulo every prime divisor of that order yields a nonzero normalized generic class in an exponent-p cycle module." }, { "id": 20000073, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0073", "title": "A purity-collapse criterion and identity-braid audit for the HHH comparison", "statement": "Prove that Oblomkov-Rozansky triply graded cohomology (HHH$^{\\text{geom}}$) is isomorphic to Khovanov-Rozansky triply graded cohomology (HHH).", "original_statement": "Prove that Oblomkov-Rozansky triply graded cohomology (HHH$^{\\text{geom}}$) is isomorphic to Khovanov-Rozansky triply graded cohomology (HHH).", "clean_statement": "Prove that Oblomkov-Rozansky triply graded cohomology (HHH$^{\\text{geom}}$) is isomorphic to Khovanov-Rozansky triply graded cohomology (HHH).", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 1.1 in the section “Oblomkov-Rozansky link invariant” of the 2018 AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Oblomkov-Rozansky link invariant\nSource item: 1.1\nSource URL: http://aimpl.org/catheckehilbert/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[72]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Prove that Oblomkov-Rozansky triply graded cohomology (HHH$^{\\\\text{geom}}$) is isomorphic to Khovanov-Rozansky triply graded cohomology (HHH).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0073", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "This attempt proves a one-row purity-collapse lemma that promotes compatible graph-by-graph trace identifications to a canonical trigraded identification of bounded Rouquier-type total homologies, and it proves the identity n-braid calculation HHH(1_n) = R_n tensor Lambda(E_n), entirely in relative t-degree zero, yielding a concrete normalization obstruction. Applied to the non-elementary graph-trace and purity inputs of Oblomkov and Rozansky, the lemma rigorously supplies the final termwise-to-all-braids comparison step. The original AIM problem is already solved by their arXiv:2010.14546, which proves in unreduced normalized conventions that HHH_geo(beta) and HHH_alg(beta) are canonically isomorphic with q, t, and a matching polynomial, outer Rouquier, and exterior/Hochschild degrees.\n\nCandidate contribution (lemma; novelty confidence low): A functorial graph-by-graph trace identification promotes to a canonical trigraded identification of bounded Rouquier-type total homologies whenever the trace of the p-th term is pure in total t-degree p+c; the identity-braid Koszul calculation fixes c and gives a concrete obstruction to incorrect relative t-normalizations." }, { "id": 20000074, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0074", "title": "Rank-one monoidal equivalence and the graph-generation bottleneck", "statement": "Prove that MF$((\\mathscr{X}^\\text{big})^\\text{stable},w)$ is equivalent as a monoidal category to K$^b(\\mathbb{S}\\text{Bim}_n)$, where $\\mathscr{X}^\\text{big}=\\mathfrak{g} \\times G \\times \\mathfrak{n} \\times G \\times \\mathfrak{n}$.", "original_statement": "Prove that MF$((\\mathscr{X}^\\text{big})^\\text{stable},w)$ is equivalent as a monoidal category to K$^b(\\mathbb{S}\\text{Bim}_n)$, where $\\mathscr{X}^\\text{big}=\\mathfrak{g} \\times G \\times \\mathfrak{n} \\times G \\times \\mathfrak{n}$.", "clean_statement": "Prove that MF$((\\mathscr{X}^\\text{big})^\\text{stable},w)$ is equivalent as a monoidal category to K$^b(\\mathbb{S}\\text{Bim}_n)$, where $\\mathscr{X}^\\text{big}=\\mathfrak{g} \\times G \\times \\mathfrak{n} \\times G \\times \\mathfrak{n}$.", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Oblomkov-Rozansky link invariant\nSource item: 1.2\nSource URL: http://aimpl.org/catheckehilbert/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[73]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Prove that MF$((\\\\mathscr{X}^\\\\text{big})^\\\\text{stable},w)$ is equivalent as a monoidal category to K$^b(\\\\mathbb{S}\\\\text{Bim}_n)$, where $\\\\mathscr{X}^\\\\text{big}=\\\\mathfrak{g} \\\\times G \\\\times \\\\mathfrak{n} \\\\times G \\\\times \\\\mathfrak{n}$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0074", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The full equivalence remains Conjecture 7.3.1 of Oblomkov--Rozansky, but it holds strongly monoidally in rank one for the explicitly defined framed, finite-rank, homologically normalized T_{q,t}-equivariant unfoldable component. The stable quotient is A^3 with potential x(y_1-y_2); the weighted Koszul factorization K(x,y_1-y_2), graded Knorrer periodicity, and an explicit convolution row operation identify this category with K^b(SBim_1). A literal ungraded reading is obstructed by 2-periodicity, while forgetting the cyclic vector leaves an extra B G_m stabilizer factor.\n\nCandidate contribution (special_case; novelty confidence low): For the framed finite-rank homologically normalized rank-one OR category, P maps to P tensor K(x,y_1-y_2) under a strong monoidal equivalence with K^b(SBim_1); the convolution product is verified by the odd change of basis alpha=theta_1+theta_2, beta=theta_2, and projected stability instead produces an A^1 times B G_m quotient." }, { "id": 20000075, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0075", "title": "A residual parity-defect criterion for the fourth grading on HHH", "statement": "On HHH$^\\text{geom}$ of a braid $\\beta$ there is a mysterious 4th $\\mathbb{Z}$-grading coming from the $\\mathbb{Z}/2$-grading on matrix factorizations. Construct the analog for HHH of $\\beta$.", "original_statement": "On HHH$^\\text{geom}$ of a braid $\\beta$ there is a mysterious 4th $\\mathbb{Z}$-grading coming from the $\\mathbb{Z}/2$-grading on matrix factorizations. Construct the analog for HHH of $\\beta$.", "clean_statement": "On HHH$^\\text{geom}$ of a braid $\\beta$ there is a mysterious 4th $\\mathbb{Z}$-grading coming from the $\\mathbb{Z}/2$-grading on matrix factorizations. Construct the analog for HHH of $\\beta$.", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Oblomkov-Rozansky link invariant\nSource item: 1.3\nSource URL: http://aimpl.org/catheckehilbert/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[74]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"On HHH$^\\\\text{geom}$ of a braid $\\\\beta$ there is a mysterious 4th $\\\\mathbb{Z}$-grading coming from the $\\\\mathbb{Z}/2$-grading on matrix factorizations. Construct the analog for HHH of $\\\\beta$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0075", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "With raw algebraic gradings T (outer Rouquier) and A (inner Hochschild/derived) and total matrix-factorization parity epsilon, the residual defect Delta = epsilon - T - A mod 2 is preserved by the relevant total differentials and is additive under convolution. A monoidal integer lift U is therefore equivalent to an additive integer refinement V of Delta via U = T + A + V. If the unit and elementary graph traces are defect-zero and the MOY, Rouquier, and Markov structural maps have defect degree zero, every braid trace is defect-zero and U = T + A is the distinguished normalized collapsed lift. The checked literature does not verify those parity hypotheses, so the three outcomes—collapse, surviving extra parity, or a genuine fourth integer refinement—remain unresolved.\n\nCandidate contribution (criterion_and_obstruction; novelty confidence low): The residual defect Delta = epsilon - T - A mod 2 gives a differential-invariant, convolution-additive obstruction which reduces a parity-preserving fourth-grading comparison to the local parity of the unit and elementary singular graph traces and the defect degree of the quadratic/rank-three MOY, Rouquier, and Markov maps; any candidate fourth degree s, including the Abel-Rozansky filtration degree, must satisfy s = Delta mod 2 on this local check set.", "research_difficulty_suggested": "L4" }, { "id": 20000076, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0076", "title": "A finite certified direct model on the two-chart locus", "statement": "To what extend can computers compute HHH$^\\text{geom}$ directly from the definition?", "original_statement": "To what extend can computers compute HHH$^\\text{geom}$ directly from the definition?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The word “extend” is preserved from the record; it is almost certainly a typographical error for “extent.” The mathematical phrase “directly from the definition” is not defined in the record. The contemporaneous Oblomkov–Rozansky definition gives the following unambiguous strict reading. For a braid \\(\\beta\\in Br_n\\), one:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Oblomkov-Rozansky link invariant\nSource item: 1.4\nSource URL: http://aimpl.org/catheckehilbert/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[75]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"To what extend can computers compute HHH$^\\\\text{geom}$ directly from the definition?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0076", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every bounded explicitly split equivariant complex with terms that are finite sums of line bundles on P^1 times an affine polynomial base, the standard two-chart Cech total complex has an explicit finite strong deformation retract of rank sum |m+1| over the base ring. With the convention 1-i*pi=delta*h+h*delta, its exact transferred differential is pi*partial*i-pi*partial*h*partial*i; the series terminates because (h*partial)^2=0, and the maps provide a checkable chain-homotopy certificate. Applied to the published Oblomkov-Rozansky two-chart computation of T(2,2m+1), this gives |m+1|+|m| generators over C[x] and an O(|m|+1) character-enumeration procedure. The broader status audit finds direct calculations for structured families and broad indirect algebraic software, but no public arbitrary-braid end-to-end implementation of the geometric definition in the sources checked.\n\nCandidate contribution (algorithmic_reduction; novelty confidence low): Candidate novelty: the explicit exponentwise Cech contraction, negative-sign homological-perturbation transfer, rank formula sum |m+1|, and separate degree-window versus exact-module complexity bounds form a finite certificate-producing primitive for direct geometric HHH computations; on the verified odd two-strand family it prunes O(-1) exactly and has output-linear size." }, { "id": 20000077, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0077", "title": "A representation-framed stable locus and a type B2 obstruction", "statement": "The space $\\mathscr{X}^{\\text{big}}$ and category MF($\\mathscr{X}^{\\text{big}},w$) makes sense not just in type A. What is the analog of $(\\mathscr{X}^{\\text{big}})^{\\text{stable}}$ in other types? What is the categorical analog of the map Br$_\\text{ext} \\to$ Br$_\\text{fin}$.", "original_statement": "The space $\\mathscr{X}^{\\text{big}}$ and category MF($\\mathscr{X}^{\\text{big}},w$) makes sense not just in type A. What is the analog of $(\\mathscr{X}^{\\text{big}})^{\\text{stable}}$ in other types? What is the categorical analog of the map Br$_\\text{ext} \\to$ Br$_\\text{fin}$.", "clean_statement": "The space $\\mathscr{X}^{\\text{big}}$ and category MF($\\mathscr{X}^{\\text{big}},w$) makes sense not just in type A. What is the analog of $(\\mathscr{X}^{\\text{big}})^{\\text{stable}}$ in other types? What is the categorical analog of the map Br$_\\text{ext} \\to$ Br$_\\text{fin}$.", "statement_status": "exact", "statement_verification": "The preceding AIM record specifies \\[ \\mathscr X^{\\mathrm{big}} =\\mathfrak g\\times G\\times\\mathfrak n\\times G\\times\\mathfrak n. \\] This agrees with the “non-reduced” two-fold space \\(\\mathcal X_2=\\mathfrak g\\times(G\\times\\mathfrak n)^2\\) in Oblomkov--Rozansky. We work over \\(\\mathbb C\\), take \\(G\\) to be a connected reductive group, \\(B\\subset G\\) a Borel subgroup, and \\(\\mathfrak n=\\operatorname{Lie}R_u(B)\\). Fixing a nondegenerate \\(G\\)-invariant bilinear form \\(\\kappa\\) on \\(\\mathfrak g\\), the potential is \\[ w(X,g_1,Y_1,g_2,Y_2) =\\kappa\\!\\left(X,\\operatorname{Ad}_{g_1}Y_1- \\operatorname{Ad}_{g_2}Y_2\\right). \\] No corruption of the source statement was found. “Stable” is interpreted through the type A construction in arXiv:1702.03569, Section 2.6.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Oblomkov-Rozansky link invariant\nSource item: 1.5\nSource URL: http://aimpl.org/catheckehilbert/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[76]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The space $\\\\mathscr{X}^{\\\\text{big}}$ and category MF($\\\\mathscr{X}^{\\\\text{big}},w$) makes sense not just in type A. What is the analog of $(\\\\mathscr{X}^{\\\\text{big}})^{\\\\text{stable}}$ in other types? What is the categorical analog of the map Br$_\\\\text{ext} \\\\to$ Br$_\\\\text{fin}$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0077", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a connected complex reductive group G, a faithful finite-dimensional representation rho, and a B-eigen-covector ell, the locus obtained by requiring rho(g_1^{-1})v to be cyclic for d rho(Ad_{g_1}^{-1}X) and d rho(Y_1), and requiring ell(rho(g_1^{-1})v) to be nonzero, is an invariant Zariski open locus with free G x B^2 action. The resulting invertible B-semi-invariant explicitly trivializes the corresponding equivariant character twist after restriction, generalizing the local mechanism of Oblomkov--Rozansky Proposition 3.4.1. The locus is nonempty for the standard representation of SO_5, while the analogous Spin_5 vector framing has a central mu_2 stabilizer and its extremal vector weights span only the index-two root sublattice of the character lattice.\n\nCandidate contribution (proposition_and_obstruction; novelty confidence low): The representation-framed openness/freeness/character-trivialization proposition, together with the explicit SO_5 Vandermonde point and the paired Spin_5 central-stabilizer and index-two character-lattice obstruction, is a concrete candidate other-type contribution." }, { "id": 20000078, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0078", "title": "Nilpotent critical support and Hilbert-scheme Jordan strata", "statement": "What is the interaction between cell theory and the Hilbert scheme? Matrix factorizations?\n\nIn the affine Steinberg setting this corresponds to the stratification by nilpotent orbits. Does this have an analog for $\\mathscr{X}^\\text{big}$?", "original_statement": "What is the interaction between cell theory and the Hilbert scheme? Matrix factorizations?\n\nIn the affine Steinberg setting this corresponds to the stratification by nilpotent orbits. Does this have an analog for $\\mathscr{X}^\\text{big}$?", "clean_statement": "What is the interaction between cell theory and the Hilbert scheme? Matrix factorizations?\n\nIn the affine Steinberg setting this corresponds to the stratification by nilpotent orbits. Does this have an analog for $\\mathscr{X}^\\text{big}$?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Oblomkov-Rozansky link invariant\nSource item: 1.6\nSource URL: http://aimpl.org/catheckehilbert/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[77]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the interaction between cell theory and the Hilbert scheme? Matrix factorizations?\\n\\nIn the affine Steinberg setting this corresponds to the stratification by nilpotent orbits. Does this have an analog for $\\\\mathscr{X}^\\\\text{big}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0078", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the original nilpotent Oblomkov--Rozansky big model, the critical locus has a canonical common-nilpotent map nu. Pullbacks of nilpotent orbit closures define thick two-sided convolution ideals whenever the coherent convolution pushforward is defined, proper on support, and obeys the stated critical-support formula. On the cyclic stable quotient these become determinantal Jordan-rank loci for multiplication by y on the tautological Hilbert-scheme algebra. In rank two, the closure of the regular Jordan stratum is A^1 x P^1 and meets the zero stratum only along the A^1 of horizontal double points, so it is strictly smaller than the closed preimage of the regular nilpotent orbit closure.\n\nCandidate contribution (support-ideal lemma and rank-two obstruction; novelty confidence low): The viable Steinberg-style filtration of the nilpotent big model is by the closed critical-support loci nu^{-1}(closure O_lambda), equivalently by m_y^{-1}(closure O_lambda) after stable Hilbert descent; closures of the individual Jordan preimage strata cannot replace these loci, as shown explicitly for n=2 and lambda=(2)." }, { "id": 20000079, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0079", "title": "A sharp two-strand full-twist threshold for HHH-parity", "statement": "Call a braid $\\beta$ parity if HHH$(\\beta)$ is parity. Find necessary and/or sufficient conditions for a braid to be parity.\n\nAre there operations on braids (for example, adding a positive number of full twists) which transform it into a parity braid or which preserve parity braids?", "original_statement": "Call a braid $\\beta$ parity if HHH$(\\beta)$ is parity. Find necessary and/or sufficient conditions for a braid to be parity.\n\nAre there operations on braids (for example, adding a positive number of full twists) which transform it into a parity braid or which preserve parity braids?", "clean_statement": "Call a braid $\\beta$ parity if HHH$(\\beta)$ is parity. Find necessary and/or sufficient conditions for a braid to be parity.\n\nAre there operations on braids (for example, adding a positive number of full twists) which transform it into a parity braid or which preserve parity braids?", "statement_status": "exact", "statement_verification": "The canonical record is problem 2.1, “Parity,” from the AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Parity\nSource item: 2.1\nSource URL: http://aimpl.org/catheckehilbert/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[78]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Call a braid $\\\\beta$ parity if HHH$(\\\\beta)$ is parity. Find necessary and/or sufficient conditions for a braid to be parity.\\n\\nAre there operations on braids (for example, adding a positive number of full twists) which transform it into a parity braid or which preserve parity braids?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0079", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For ordinary triply graded Khovanov--Rozansky homology over C, let FT_2=sigma_1^2 and beta_r=FT_2^{-r}, r>=1. Then beta_r FT_2^k is parity if and only if k>=r. When kr it is a nonnegative full-twist power. Thus the exact full-twist regularization time is r, and mirror image does not preserve parity because FT_2 is parity while FT_2^{-1} is mixed.\n\nCandidate contribution (corollary; novelty confidence low): The full-twist regularization time tau_2(FT_2^{-r}) equals r for every r>=1; equivalently, FT_2^{-r} FT_2^k is parity exactly for k>=r." }, { "id": 20000080, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0080", "title": "A two-channel parity reduction for algebraic cabling", "statement": "Are algebraic braids parity? If so find a (recursive) formula for HHH of an algebraic braid.", "original_statement": "Are algebraic braids parity? If so find a (recursive) formula for HHH of an algebraic braid.", "clean_statement": "Are algebraic braids parity? If so find a (recursive) formula for HHH of an algebraic braid.", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Parity\nSource item: 2.2\nSource URL: http://aimpl.org/catheckehilbert/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[79]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Are algebraic braids parity? If so find a (recursive) formula for HHH of an algebraic braid.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0080", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general algebraic-link parity problem remains open, but a published binary recursion proves parity and computes HHH for positive torus links and for the nontrivial algebraic cable family T(m,n)(d,mnd+1). This attempt extracts the precise parity congruence for that family and proves a self-contained two-strand Hecke recurrence together with a filtered-complex collapse theorem: any categorified two-channel cabling filtration whose shifted symmetric and exterior colored terms share one parity collapses at E1, proving parity and an additive recursive Poincare formula.\n\nCandidate contribution (reduction; novelty confidence low): For the first two-strand Puiseux-cabling step, the natural recursion state consists of the symmetric (2) and exterior (1,1) Hecke channels; pairing their exact second-order recurrence with a common-parity filtration criterion reduces a categorified parity proof to constructing a bounded two-channel filtration and checking its shifts modulo two.", "research_difficulty_suggested": "L4" }, { "id": 20000081, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0081", "title": "Adjacent-power compression and gap states for nested full twists", "statement": "There is a commutative family of braids built as products of FT$_k$, the full twist on the first $k$ strands, for $k \\leq n$. Are the positive products in this family parity? If so, find a (recursive) formula for HHH.", "original_statement": "There is a commutative family of braids built as products of FT$_k$, the full twist on the first $k$ strands, for $k \\leq n$. Are the positive products in this family parity? If so, find a (recursive) formula for HHH.", "clean_statement": "There is a commutative family of braids built as products of FT$_k$, the full twist on the first $k$ strands, for $k \\leq n$. Are the positive products in this family parity? If so, find a (recursive) formula for HHH.", "statement_status": "exact", "statement_verification": "The archived AIM page timed out during this run, but the repository record is syntactically complete and the notation is fixed by the adjacent records. There is no apparent OCR corruption.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Parity\nSource item: 2.3\nSource URL: http://aimpl.org/catheckehilbert/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[80]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"There is a commutative family of braids built as products of FT$_k$, the full twist on the first $k$ strands, for $k \\\\leq n$. Are the positive products in this family parity? If so, find a (recursive) formula for HHH.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0081", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every positive product of nested partial full twists is exactly a pure Coxeter braid after the cumulative reindexing m_j=sum_{k=j}^n e_k. The parity question is proved in the literature through four strands and for the two-largest-full-twist subfamily in every rank, but remains open in general. At chain level, K_r JM_{r+1}^h B has an unconditional iterated-convolution filtration with layers (qt^{-r})^h K_r B and t^{-r}(qt^{-r})^j K_{r+1}B for 0<=j= 1 the two GNR fixed-point terms for FT_2^r sum to (1+alpha)(D_r+alpha D_{r-1})/(1-q)^2, where D_s=[t^(s+1)(1-q)-q^(s+1)(1-t)]/(t-q), and direct solution of the Hogancamp-Mellit recursion gives exactly t^(-r) times this expression. The factor t^r is the writhe-2r Rouquier-normalization change. Each raw fixed-point term has a nonzero simple pole at t=q, so a weight-preserving comparison with regular recursion terms must group the transpose pair or retain derived/localization data. In the two-strand Coxeter family, the GHSR subdiagram model is an A_1 Demazure string and its t-statistic is crystal depth.\n\nCandidate contribution (explicit comparison and obstruction; novelty confidence low): Candidate novelty: the all-powers rank-two identity t^r R_{00,0^{2r}}=(1+alpha)(D_r+alpha D_{r-1})/(1-q)^2 together with the proof that the two individual localization fractions have opposite nonzero t=q residues, ruling out a naive termwise bijection to regular recursion weights." }, { "id": 20000083, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0083", "title": "Cyclotomic monodromy reconstruction and a bare-Milnor-fibre obstruction", "statement": "To an algebraic link one can associate a Springer fibre and a Milnor fibre. Can one compute HHH from the Milnor fibre?", "original_statement": "To an algebraic link one can associate a Springer fibre and a Milnor fibre. Can one compute HHH from the Milnor fibre?", "clean_statement": "To an algebraic link one can associate a Springer fibre and a Milnor fibre. Can one compute HHH from the Milnor fibre?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Springer fibres\nSource item: 3.1\nSource URL: http://aimpl.org/catheckehilbert/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[82]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"To an algebraic link one can associate a Springer fibre and a Milnor fibre. Can one compute HHH from the Milnor fibre?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0083", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Within the coprime Brieskorn family x^p-y^q, the characteristic polynomial of homological Milnor monodromy recovers the unordered pair {p,q}: its largest cyclotomic index is pq, its degree is (p-1)(q-1), and these give p+q and pq, after which the proved Hogancamp-Mellit recursion computes HHH. In contrast, for every n >= 3 with gcd(3,n+1)=1, the singularities x^2+y^(2n+1) and x^3+y^(n+1) have orientation-preservingly diffeomorphic bare Milnor surfaces of genus n with one boundary component, but their knot determinants are respectively 2n+1 and 1 or 3, so their HHH differs.\n\nCandidate contribution (theorem_and_obstruction; novelty confidence low): The explicit largest-cyclotomic-index-plus-degree reconstruction from homological Milnor monodromy computes HHH throughout the coprime Brieskorn plane-curve family via parameter recovery and a known torus-link formula, while the paired infinite collision family gives a concrete obstruction to any HHH construction from only the abstract Milnor surface." }, { "id": 20000084, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0084", "title": "A radical-free comparison of the affine-Springer and Milnor Alexander constructions", "statement": "It is known that the Alexander polynomial can be computed from both the Springer fibre and the Milnor fibre. Understand the relationship between these two constructions.", "original_statement": "It is known that the Alexander polynomial can be computed from both the Springer fibre and the Milnor fibre. Understand the relationship between these two constructions.", "clean_statement": "It is known that the Alexander polynomial can be computed from both the Springer fibre and the Milnor fibre. Understand the relationship between these two constructions.", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Springer fibres\nSource item: 3.2\nSource URL: http://aimpl.org/catheckehilbert/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[83]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"It is known that the Alexander polynomial can be computed from both the Springer fibre and the Milnor fibre. Understand the relationship between these two constructions.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0084", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a reduced plane curve germ with r branches, the normalized one-variable Alexander polynomial, the principal-ideal Euler series in the generalized affine-Springer/Hilbert tower, and the radical-free Milnor determinant satisfy Delta_L(t)=(1-t)Z_pr(C;t)=(1-t)^(r-1)A_C(t), where A_C(t)=det(1-t h_bar on H_1(F)/rad). Thus Z_pr=(1-t)^(r-2)A_C. Chen's 2026 reduced vanishing-cycle quotient realizes the representation whose determinant is A_C, not Z_pr or Delta_L. For x^p-y^q an explicit closed formula is proved; in particular, for the ordinary r-fold point Z_pr=(1-t^r)^(r-2), while A_C=(1+t+...+t^(r-1))^(r-2). A dimension obstruction also rules out a naive isomorphism between ordinary affine-Springer cohomology and Milnor H_1.\n\nCandidate contribution (normalization theorem and explicit family; novelty confidence low): Candidate novel synthesis: the exact radical-free conversion is Z_pr(C;t)=(1-t)^(r-2)det(1-t h_bar on H_1(F)/rad), and for the ordinary r-fold point the remaining boundary factor and reduced determinant are respectively Z_pr=(1-t^r)^(r-2) and A_C=(1+t+...+t^(r-1))^(r-2)." }, { "id": 20000085, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0085", "title": "A corrected target and an explicit B2 full-twist benchmark", "statement": "Outside of type A is there still a relation between Hochschild cohomology of Rouquier complexes and affine Springer fibres?", "original_statement": "Outside of type A is there still a relation between Hochschild cohomology of Rouquier complexes and affine Springer fibres?", "clean_statement": "Outside of type A is there still a relation between Hochschild cohomology of Rouquier complexes and affine Springer fibres?", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 3.3 in the “Springer fibres” section of the workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Springer fibres\nSource item: 3.3\nSource URL: http://aimpl.org/catheckehilbert/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[84]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Outside of type A is there still a relation between Hochschild cohomology of Rouquier complexes and affine Springer fibres?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0085", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the standard B2/C2 realization, the rational doubled-root ideal is proved scheme-theoretically to be generated by the determinant D=x1*y2-x2*y1 and the five polarizations of P(u,v)=u*v*(u^2-v^2), with bigraded Hilbert function dim(S/I)_(p,q)=min(4,p+q+1) and total Hilbert numerator 1+2z+2z^2+2z^3-3z^4. It is unconditionally the completed associated graded of Delta times the T-equivariant Borel-Moore homology of the full-twist affine Springer fiber; its identification with y-ified Rouquier homology is conditional on the published non-A parity and link-splitting conjectures. Koszul duality shows that the corresponding cohomological slice is top HH^2 with sign and internal shift, while a separate SO5 constant-element calculation proves that ordinary cohomology of the compact lattice quotient cannot model full Hochschild cohomology even for the identity braid.\n\nCandidate contribution (explicit_example; novelty confidence low): The B2 doubled-root ideal has the scheme-theoretic presentation I=(x1*y2-x2*y1,P0,P1,P2,P3,P4), where P0 through P4 polarize u*v*(u^2-v^2); consequently dim(S/I)_(p,q)=min(4,p+q+1), Hilb(S/I;z)=(1+2z+2z^2+2z^3-3z^4)/(1-z)^2, and Hilb(I;z)=(z^2+5z^4-8z^5+3z^6)/(1-z)^4." }, { "id": 20000086, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0086", "title": "Generic-support obstruction and rank-one central Rouquier powers", "statement": "What is the Drinfeld center of K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$? K$^b(\\mathbb{S}\\text{Bim}_{\\text{aff}})$? Are there additional complexes in K$^b(\\mathbb{S}\\text{Bim})$ which commute with $\\mathbb{S}$Bim?", "original_statement": "What is the Drinfeld center of K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$? K$^b(\\mathbb{S}\\text{Bim}_{\\text{aff}})$? Are there additional complexes in K$^b(\\mathbb{S}\\text{Bim})$ which commute with $\\mathbb{S}$Bim?", "clean_statement": "What is the Drinfeld center of K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$? K$^b(\\mathbb{S}\\text{Bim}_{\\text{aff}})$? Are there additional complexes in K$^b(\\mathbb{S}\\text{Bim})$ which commute with $\\mathbb{S}$Bim?", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 4.1 in the section “Soergel bimodules” of the workshop list *Categorified Hecke algebras, link homology, and Hilbert schemes*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Soergel bimodules\nSource item: 4.1\nSource URL: http://aimpl.org/catheckehilbert/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[85]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the Drinfeld center of K$^b(\\\\mathbb{S}\\\\text{Bim}_{\\\\text{fin}})$? K$^b(\\\\mathbb{S}\\\\text{Bim}_{\\\\text{aff}})$? Are there additional complexes in K$^b(\\\\mathbb{S}\\\\text{Bim})$ which commute with $\\\\mathbb{S}$Bim?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0086", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a faithful finite or affine Soergel realization, naturality with polynomial endomorphisms of the unit forces every graded Drinfeld-central bounded complex to have fraction-field localization supported only on the identity graph. In standard type A1 over characteristic not equal to 2, this yields an exact classification in the Rouquier-shift family: R(a)[b] tensor F_s^m admits a coherent ordinary half-braiding if and only if m is even. Even powers obtain explicit coherent structures from tensor and inverse powers of the Elias-Hogancamp full-twist central object; odd powers remain objectwise commuting but fail polynomial naturality.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: graded Drinfeld-central bounded Soergel complexes have identity-only generic graph support, and consequently the shifted type-A1 Rouquier power R(a)[b] tensor F_s^m is central exactly when m is even, with odd powers furnishing an explicit objectwise-commuting but noncentral family." }, { "id": 20000087, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0087", "title": "One-strand generation dichotomy and a six-strand decategorified obstruction", "statement": "Is Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{ext}})$) generated by Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ generated by images of Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ split generated by powers of FT$_n$?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ symmetric monoidal?", "original_statement": "Is Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{ext}})$) generated by Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ generated by images of Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ split generated by powers of FT$_n$?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ symmetric monoidal?", "clean_statement": "Is Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{ext}})$) generated by Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ generated by images of Gaitsgory central sheaves?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ split generated by powers of FT$_n$?\nIs Z(K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}}))$ symmetric monoidal?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 4.2 in the “Soergel bimodules” section of the AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*. The source page and the neighboring Problems 4.1 and 4.3 confirm the following reading:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Soergel bimodules\nSource item: 4.2\nSource URL: http://aimpl.org/catheckehilbert/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[86]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is Z(K$^b(\\\\mathbb{S}\\\\text{Bim}_{\\\\text{ext}})$) generated by Gaitsgory central sheaves?\\nIs Z(K$^b(\\\\mathbb{S}\\\\text{Bim}_{\\\\text{fin}}))$ generated by images of Gaitsgory central sheaves?\\nIs Z(K$^b(\\\\mathbb{S}\\\\text{Bim}_{\\\\text{fin}}))$ split generated by powers of FT$_n$?\\nIs Z(K$^b(\\\\mathbb{S}\\\\text{Bim}_{\\\\text{fin}}))$ symmetric monoidal?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0087", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The four questions remain open in general, but the small graded finite center has a complete one-strand answer. In the Ho--Li torus model it is the category of A=R tensor k[alpha]/(alpha^2)-modules that are perfect after restriction to R. The center unit, hence the standard one-strand full twist FT_1, is the augmentation R=A/(alpha), while the regular module A is the distinct trace object tr(1). Ho--Li's coherent-module devissage then gives thick generation by FT_1; its additive Karoubi closure is proper because alpha acts nontrivially on A; and unit generation forces the canonical braiding to be symmetric. Conditionally, an expected center--Hilbert equivalence sending FT_n to an ample line bundle would imply general thick generation. Separately, an explicit S_6 content collision proves that the generic Hecke center is not Laurent-polynomially generated by the full twist, without drawing any categorical K_0 conclusion.\n\nCandidate contribution (special_case_and_obstruction; novelty confidence low): In the one-strand small graded center, the Ho--Li coherent generator is precisely the center unit FT_1, so FT_1 thick-generates although its additive Karoubi closure omits the regular relative exterior-algebra module A; consequently the canonical center braiding is symmetric. On six strands, the partitions (4,1,1) and (3,3) give an explicit full-twist eigenvalue collision separated by the first symmetric Jucys--Murphy element." }, { "id": 20000088, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0088", "title": "Rank-two equality and the all-rank coherence obstruction", "statement": "Is the flattening of the Gaitsgory complex for the standard representation equal to the tautological sheaf from the GNR conjecture?", "original_statement": "Is the flattening of the Gaitsgory complex for the standard representation equal to the tautological sheaf from the GNR conjecture?", "clean_statement": "Is the flattening of the Gaitsgory complex for the standard representation equal to the tautological sheaf from the GNR conjecture?", "statement_status": "exact", "statement_verification": "The AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Soergel bimodules\nSource item: 4.3\nSource URL: http://aimpl.org/catheckehilbert/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[87]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is the flattening of the Gaitsgory complex for the standard representation equal to the tautological sheaf from the GNR conjecture?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0088", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The proposed objects have the same Grothendieck class in every rank. In rank two, after setting the affine parameter delta to zero and aligning equivariant grading conventions, the manually flattened standard Gaitsgory complex is actually homotopy equivalent to 1 plus the two-strand full twist, which is the GNR image of the tautological bundle O plus O(1). The equivalence can be chosen to identify the cyclic section and the operators chi and mu with the tautological matrices X and Y. In general rank, the exact equality remains unresolved because the known additive evaluation functor lacks the dg or A-infinity coherence needed to totalize the Gaitsgory complex.\n\nCandidate contribution (operator-and-coherence certificate; novelty confidence low): In rank two the object equivalence can be upgraded to an identification of the cyclic vector and both homogeneous operators; the generated operator algebra has the exact presentation R[chi,mu]/((chi-x1)(chi-x2),(x1-x2)mu,mu(chi-x1),mu^2), with no extra relations, and the evaluated three-term differential satisfies the required Maurer-Cartan equation strictly." }, { "id": 20000089, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0089", "title": "Trace categorification and a fixed-slope hook-complex lift", "statement": "A theorem of Morton-Samuelson connected the Elliptic Hall algebra at $q=t$ to the Skein algebra of the torus. Categorify this result.", "original_statement": "A theorem of Morton-Samuelson connected the Elliptic Hall algebra at $q=t$ to the Skein algebra of the torus. Categorify this result.", "clean_statement": "A theorem of Morton-Samuelson connected the Elliptic Hall algebra at $q=t$ to the Skein algebra of the torus. Categorify this result.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 5.1 in the “Others” section of the AIM workshop list *Categorified Hecke algebras, link homology, and Hilbert schemes*. The live AIM page confirms the exact text:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Others\nSource item: 5.1\nSource URL: http://aimpl.org/catheckehilbert/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[88]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A theorem of Morton-Samuelson connected the Elliptic Hall algebra at $q=t$ to the Skein algebra of the torus. Categorify this result.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0089", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Mousaaid and Savage already categorify the Morton-Samuelson isomorphism in the precise trace sense: at central charge zero, the elliptic Hall algebra is the categorical trace of the quantum Heisenberg category, whose annular skein model closes to the torus skein algebra. Beyond this status clarification, the attempt proves over K=Q(s,v) that the additive Karoubi envelope of the positive Hecke tangle category is strongly braided monoidally equivalent to the tower of projective type-A Hecke modules, and that slope closure identifies its split Grothendieck ring with every positive fixed-ray skein subalgebra. It also constructs the bounded zero-differential hook complex with terms S^(m-r,1^r), whose Euler class closes to P_(m x), and proves by nonnegative Schur multiplicities that no honest semisimple object can have that class for m at least 2.\n\nCandidate contribution (theorem; novelty confidence low): For every primitive slope x and m at least 2 in the generic semisimple Hecke setting, the Morton-Samuelson generator P_(m x) has no honest object lift in the natural fixed-slope projective Hecke category, but it has an explicit coherent bounded lift whose degree-r term is the hook Specht module S^(m-r,1^r) and whose differential is zero." }, { "id": 20000090, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0090", "title": "SL_N fixed-ray quotients and minimal hook complexes", "statement": "Are there simplifications of the above problem for the $\\mathfrak{sl}_n$-quotient invariant?", "original_statement": "Are there simplifications of the above problem for the $\\mathfrak{sl}_n$-quotient invariant?", "clean_statement": "Are there simplifications of the above problem for the $\\mathfrak{sl}_n$-quotient invariant?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 5.2 in the “Others” section of the 2018 AIM workshop list *Categorified Hecke algebras, link homology, and Hilbert schemes*. The live AIM page confirms the exact wording:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Others\nSource item: 5.2\nSource URL: http://aimpl.org/catheckehilbert/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[89]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Are there simplifications of the above problem for the $\\\\mathfrak{sl}_n$-quotient invariant?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0090", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For generic s and every positive primitive torus-skein slope, the SL_N color map has exact kernel (E_N-1,E_{N+1},E_{N+2},...) and image K[E_1,...,E_{N-1}], so its power sums satisfy an explicit order-N Newton recurrence, with the nonhomogeneous m=N boundary term retained. The Murnaghan-Nakayama hook lift maps under quantum Schur-Weyl duality to a zero-differential complex with exactly min(m,N) pairwise nonisomorphic simple terms; this is a minimal semisimple Euler lift, and for m>=2 its class cannot be represented by an honest module. For N=2 these complexes satisfy a genuine termwise Chebyshev chain isomorphism for every m>=3.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: after the actual generic SL_N representation quotient, the power-sum class has a determinant-truncated minimal derived hook lift with exactly min(m,N) simple occurrences, and in rank two these minimal lifts obey a chain-level Chebyshev isomorphism for m>=3." }, { "id": 20000091, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0091", "title": "A rank-one Morita diagnostic for DAHA and punctured-torus branes", "statement": "Is there an equivalence of categories between DAHA representations and a suitable extension of the Fukaya category of the character variety of a single punctured torus?", "original_statement": "Is there an equivalence of categories between DAHA representations and a suitable extension of the Fukaya category of the character variety of a single punctured torus?", "clean_statement": "Is there an equivalence of categories between DAHA representations and a suitable extension of the Fukaya category of the character variety of a single punctured torus?", "statement_status": "exact", "statement_verification": "The canonical AIM record, from the workshop *Categorified Hecke algebras, link homology, and Hilbert schemes* (AIM, 1--5 October 2018), asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Others\nSource item: 5.3\nSource URL: http://aimpl.org/catheckehilbert/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[90]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there an equivalence of categories between DAHA representations and a suitable extension of the Fukaya category of the character variety of a single punctured torus?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0091", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the GL_1 once-punctured-torus character variety, the boundary holonomy must be trivial and the variety is (C^*)^2 with DAHA equal to the quantum torus A_q. At q=1, Abouzaid's cotangent-fiber theorem proves D^pi W(T^*T^2) is equivalent to Perf(A_1). If q has infinite multiplicative order, no dg Morita equivalence with the same untwisted wrapped category can exist because HH^0(A_q)=Z(A_q)=k whereas HH^0(A_1)=k[X^{+/-1},Y^{+/-1}]. The explicit cocycle alpha_q((m,n),(m',n'))=q^{n m'} yields A_q as a twisted group algebra and has SL_2(Z)-invariant cohomology class, identifying a precise minimal rank-one correction; its geometric gerbe-twisted generation theorem remains conditional.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate novelty: the GL_1 q=1 wrapped equivalence, the infinite-order-q HH^0 Morita obstruction, and the unique SL_2(Z)-invariant lattice-cocycle correction form a single necessary-condition diagnostic for any proposed DAHA/Fukaya equivalence." }, { "id": 20000092, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0092", "title": "Crossed-product HHH enhancements and the component-collapse obstruction", "statement": "Is there a braid invariant extension of HHH which replaces bigraded vector spaces with bigraded $S_n$-modules suitably compatible with the action of $R=\\mathbb{Q}[x_1, \\dots, x_n]$ and the action of $\\mathbb{Q}[y_1, \\dots, y_n]$ coming from $y$-ification?", "original_statement": "Is there a braid invariant extension of HHH which replaces bigraded vector spaces with bigraded $S_n$-modules suitably compatible with the action of $R=\\mathbb{Q}[x_1, \\dots, x_n]$ and the action of $\\mathbb{Q}[y_1, \\dots, y_n]$ coming from $y$-ification?", "clean_statement": "Is there a braid invariant extension of HHH which replaces bigraded vector spaces with bigraded $S_n$-modules suitably compatible with the action of $R=\\mathbb{Q}[x_1, \\dots, x_n]$ and the action of $\\mathbb{Q}[y_1, \\dots, y_n]$ coming from $y$-ification?", "statement_status": "exact", "statement_verification": "The canonical record is problem 5.4, “Others,” from the AIM workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*. Its exact statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Others\nSource item: 5.4\nSource URL: http://aimpl.org/catheckehilbert/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[91]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a braid invariant extension of HHH which replaces bigraded vector spaces with bigraded $S_n$-modules suitably compatible with the action of $R=\\\\mathbb{Q}[x_1, \\\\dots, x_n]$ and the action of $\\\\mathbb{Q}[y_1, \\\\dots, y_n]$ coming from $y$-ification?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0092", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Modern trace results give a positive fixed-strand derived answer: the Koszul-dual trace of a Rouquier complex is a perfect module over a crossed product of the x,y polynomial algebra with S_n, and HHH is recovered from it by a derived Hom functor. A proved obstruction explains why this cannot generally be replaced by a full semilinear S_n-action on the already-closed y-ified homology: for every nonidentity braid permutation, saturating its component-variable relations under S_n forces all strand variables to coincide. The exact largest coordinate-permutation subgroup preserving the universal component ring is the stabilizer of the cycle partition. For two-strand full twists, the report also gives a canonical crossed-product linearization and an explicit bigraded S_2-character decomposition.\n\nCandidate contribution (obstruction; novelty confidence low): If w is nonidentity and a module over Q[x_1,...,x_n,y_1,...,y_n] is annihilated by the braid-closure ideal K_w and carries the full coordinate-covariant S_n-action, then every x_a-x_b and y_a-y_b annihilates it; without extra component identifications, the maximal allowable coordinate-permutation subgroup is the cycle-partition stabilizer, isomorphic to the product over d of S_d wreath S_{m_d}." }, { "id": 20000093, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0093", "title": "A complete rank-two central reconstruction and a higher-rank extension obstruction", "statement": "Can one describe HHH$^{a>0}$ using HHH$^{a=0}$ after tensoring with a suitable central complex?", "original_statement": "Can one describe HHH$^{a>0}$ using HHH$^{a=0}$ after tensoring with a suitable central complex?", "clean_statement": "Can one describe HHH$^{a>0}$ using HHH$^{a=0}$ after tensoring with a suitable central complex?", "statement_status": "exact", "statement_verification": "The exact AIM record is Problem 5.5 in the “Others” section of the 2018 workshop *Categorified Hecke algebras, link homology, and Hilbert schemes*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Others\nSource item: 5.5\nSource URL: http://aimpl.org/catheckehilbert/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[92]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can one describe HHH$^{a>0}$ using HHH$^{a=0}$ after tensoring with a suitable central complex?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0093", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In the modern unreduced convention over a field of characteristic not two, every bounded two-strand Soergel complex X satisfies HH^a(X) is naturally equivalent to HH^0(C_{2,a} tensor X), where C_{2,0}=1, C_{2,1}=(1 direct-sum FT_2^{-1})(2), and C_{2,2}=FT_2^{-1}(4); these complexes are central. The proof splits the Hochschild Koszul complex into the strictly zero invariant direction and the sign direction, then applies published full-twist Serre duality. In arbitrary rank, the last-variable Koszul filtration gives a natural kernel-cokernel short exact sequence whose Yoneda extension class is the additional datum not determined by relative Serre endpoint identifications.\n\nCandidate contribution (rank-two central-complex theorem and relative-Koszul obstruction; novelty confidence low): The explicit natural formula C_{2,1}=(1 direct-sum FT_2^{-1})(2), C_{2,2}=FT_2^{-1}(4) reconstructs every Hochschild degree from HH^0 for every bounded two-strand Soergel complex; for general n, Proposition 5.3 isolates the unresolved intermediate-degree datum as a functorial Yoneda extension class." }, { "id": 20000094, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0094", "title": "The Procesi bundle as a traced and Koszul-dual Soergel unit", "statement": "How does one describe the Procesi bundle in the other settings, like K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$?", "original_statement": "How does one describe the Procesi bundle in the other settings, like K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$?", "clean_statement": "How does one describe the Procesi bundle in the other settings, like K$^b(\\mathbb{S}\\text{Bim}_{\\text{fin}})$?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Categorified Hecke algebras, link homology, and Hilbert schemes\nSection: Others\nSource item: 5.6\nSource URL: http://aimpl.org/catheckehilbert/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[93]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How does one describe the Procesi bundle in the other settings, like K$^b(\\\\mathbb{S}\\\\text{Bim}_{\\\\text{fin}})$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/catheckehilbert/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0094", "aim-domain:algebraic-geometry", "aim-workshop:catheckehilbert", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The theorem-level answer is that the monoidal unit in K^b(SBim_n) becomes the relevant Procesi antecedent only after derived horizontal trace: its trace is a generator with endomorphism algebra k[x,theta] semidirect S_n, and Ho-Li's relative Koszul duality plus their sheared 2-periodic derived-McKay equivalence sends its regular module to the y=0-supported specialization B_n/(y) of the normalized Procesi generator. This does not identify the full Procesi bundle inside ordinary K^b(SBim_n); the full regular B_n-module requires extension in the y-directions. In rank two, the explicit candidate consisting of the unit plus the normalized simple Rouquier complex has a triangular homotopy endomorphism algebra, while derived-bimodule localization produces both off-diagonal Ext directions and diagonal Koszul classes. The complete four-corner Yoneda algebra has positive self-Ext and H^0 equal to R times R, so neither candidate is a Procesi tilting object.\n\nCandidate contribution (explicit obstruction and Ext-algebra calculation; novelty confidence low): For S_2 in characteristic not equal to 2, the bigraded endomorphism algebra of R plus the normalized simple Rouquier complex is the lower-triangular square-zero extension with off-diagonal R/(a) in bidegree (1,-2); after the natural non-full localization to derived R-bimodules, the two graph modules have diagonal exterior Ext algebras and, in each off-diagonal direction, k[u]-classes in bidegrees (1,-2) and (2,-4), with all opposite-arrow products zero and the degree-two class obtained by the common u-Koszul loop. This gives a concrete obstruction to a literal bounded regular-Rouquier model of the Procesi tilting bundle." }, { "id": 20000095, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0095", "title": "Inseparability-robust irrationality bounds over small fields", "statement": "Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$?\nHow do we make sense of these notions over countable / finite fields?", "original_statement": "Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$?\nHow do we make sense of these notions over countable / finite fields?", "clean_statement": "Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$?\nHow do we make sense of these notions over countable / finite fields?", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0095, item 2.1 in the section “Degree of irrationality and covering gonality” of the AIM workshop *Rational subvarieties in positive characteristic*. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Degree of irrationality and covering gonality\nSource item: 2.1\nSource URL: http://aimpl.org/ratsubvarpos/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[94]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$?\\nHow do we make sense of these notions over countable / finite fields?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0095", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every smooth geometrically integral complete intersection X of multidegree (d_1,...,d_c) in P^N over a field of characteristic p, with D=sum d_i at least N+1, the unrestricted arithmetic and geometric degrees of irrationality are at least D-N+1; for a degree-d hypersurface of dimension n this is irr(X) at least d-n. The key point is that Smith's separable covering gonality is at most total degree of irrationality even for an inseparable rational map, because the induced covering family's evaluation on X is birational and hence separable. Geometric degree of irrationality is additionally proved to be attained after a finite ground-field extension, yielding an exact finite-field formulation.\n\nCandidate contribution (corollary; novelty confidence low): Smith's function-field definition gives scvg_k(X) <= irr_k(X) without requiring a map X to P^n to be separable; combined with his complete-intersection bound, this gives irr_k(X) >= D-N+1 with no min(p,D-N+1) loss, and over F_q the geometric invariant equals min over r>=1 of irr over F_{q^r}, with the minimum attained." }, { "id": 20000096, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0096", "title": "Arithmetic descent counterexample and bounded graph-complexity transfer", "statement": "How does rationality behave in a family over $\\text{Spec} \\mathbb{Z}$ or $\\text{Spec} \\mathbb{Z}[\\frac{1}{n}]$.\nIf $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$?", "original_statement": "How does rationality behave in a family over $\\text{Spec} \\mathbb{Z}$ or $\\text{Spec} \\mathbb{Z}[\\frac{1}{n}]$.\nIf $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$?", "clean_statement": "How does rationality behave in a family over $\\text{Spec} \\mathbb{Z}$ or $\\text{Spec} \\mathbb{Z}[\\frac{1}{n}]$.\nIf $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$?", "statement_status": "exact", "statement_verification": "The record is Problem 3.06 in the section “Rationality in a family” of the 2016 AIM workshop *Rational subvarieties in positive characteristic*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rationality in a family\nSource item: 3.06\nSource URL: http://aimpl.org/ratsubvarpos/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[95]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How does rationality behave in a family over $\\\\text{Spec} \\\\mathbb{Z}$ or $\\\\text{Spec} \\\\mathbb{Z}[\\\\frac{1}{n}]$.\\nIf $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0096", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "The smooth conic family x^2+y^2+z^2=0 over Spec Z[1/2] has every odd-prime fiber isomorphic to P^1 over F_p, while its generic fiber has no Q-point and is neither Q-rational nor Q-unirational; thus the ground-field form of the AIM implication is false even with uniformly bounded parametrization complexity. For the geometric form, a proved bounded-graph theorem shows that if rational or unirational parametrizations at infinitely many primes have graph Hilbert polynomials in a fixed finite set, then the geometric generic fiber is respectively rational or unirational, in fact after a finite extension of Q.\n\nCandidate contribution (theorem; novelty confidence low): For a projective flat family over Spec Z[1/N] with geometrically integral fibers, rationality or unirationality at infinitely many prime fibers transfers to the geometric generic fiber whenever the graph closures occupy finitely many Hilbert-polynomial strata; the conic x^2+y^2+z^2=0 shows sharply that even a fixed graph polynomial cannot force descent of the parametrization to Q." }, { "id": 20000097, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0097", "title": "Positive-density rational reductions and unbounded K3 certificate complexity", "statement": "Can one find a smooth cubic fourfold over a number field which is conjecturally irrational over an algebraic closure of characteristic zero, but for which “most” good reductions are rational?", "original_statement": "Can we find a cubic fourfold which is conjectually irratioal, but (most of ) its reductions are rational?", "clean_statement": "Can one find a smooth cubic fourfold over a number field which is conjecturally irrational over an algebraic closure of characteristic zero, but for which “most” good reductions are rational?", "statement_status": "corrected_verified", "statement_verification": "The same wording appears on the AIM source page, with no appended remark. I reconstruct only the evident typographical error “irratioal” as “irrational.” The mathematically recovered question is therefore:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rationality in a family\nSource item: 3.07\nSource URL: http://aimpl.org/ratsubvarpos/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[96]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we find a cubic fourfold which is conjectually irratioal, but (most of ) its reductions are rational?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0097", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Frei--Hassett--Várilly-Alvarado give conjecturally irrational cubic fourfolds in the labelled discriminant-8 and discriminant-18 loci whose rational reductions contain a positive-natural-density subset, settling the positive-density (and infinitude) reading of 'most' but not density one or cofiniteness. For their one-way rationality certificate alpha_p=0 implies Y_p rational, this attempt proves that if the associated geometric Brauer class is nonzero, then for every bound B only finitely many vanishing reductions admit a Kummer line-bundle certificate L with both |L.h| and |L^2| at most B. It also proves that bounded-degree inverse birational maps on infinitely many reductions would specialize to a birational map of the geometric generic cubic.\n\nCandidate contribution (specialization theorem; novelty confidence low): For any fixed Kummer lift of a geometrically nonzero torsion Brauer class on the polarized K3 surface associated to an FHVA cubic, and every B, only finitely many reductions admit a killing line bundle L satisfying |L.h| <= B and |L^2| <= B; hence the positive-density Brauer-certified rational reductions require unbounded Picard-lattice certificate complexity." }, { "id": 20000098, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0098", "title": "Characteristic-zero specialization and an odd-index theorem for cubic fourfolds", "statement": "If the general member of a family of smooth cubic is rational, is the special fiber also rational?", "original_statement": "If the general member of a family of smooth cubic is rational, is the special fiber also rational?", "clean_statement": "If the general member of a family of smooth cubic is rational, is the special fiber also rational?", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rationality in a family\nSource item: 3.11\nSource URL: http://aimpl.org/ratsubvarpos/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[97]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If the general member of a family of smooth cubic is rational, is the special fiber also rational?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0098", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Kontsevich–Tschinkel gives a full affirmative answer for a smooth proper family over a characteristic-zero curve when the generic fiber is rational. For the positive- or mixed-characteristic cubic-fourfold reading, a proved partial theorem shows that a K-defined plane and its section certificate specialize in every characteristic, while an odd-degree multisection certificate specializes when 2 is invertible, provided the special residual generic quadric remains smooth. The proof specializes the associated odd-degree closed point over the Gauss DVR, preserves its degree by finite flatness, and applies Springer to obtain a rational section and hence a rational special cubic fourfold.\n\nCandidate contribution (theorem; novelty confidence low): In a DVR family of smooth cubic fourfolds, a K-plane extends by properness of the relative Fano scheme; if the residual quadric at the generic point of the special P2 is smooth, then a generic odd-degree multisection specializes, via its closed point over the Gauss DVR, to an odd-degree zero-cycle and therefore to a special rational section when 2 is invertible." }, { "id": 20000099, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0099", "title": "Supersingular dimension ladder and unirational bounds for smooth quartic K3 surfaces", "statement": "Let $\\mathbb{P}$ be the parametrizing space of higher surfaces of degree $d$ in $\\mathbb{P}^{n}$.\nWhat is the largest dimensional family of unirational / supersinglar hypersurfaces?", "original_statement": "Let $\\mathbb{P}$ be the parametrizing space of higher surfaces of degree $d$ in $\\mathbb{P}^{n}$.\nWhat is the largest dimensional family of unirational / supersinglar hypersurfaces?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The live AIM page was checked on 2026-07-23 and contains the same words, with no status note. Thus the two apparent errors are in the source, rather than in the JSON extraction. I make the following explicit reconstruction.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rationality in a family\nSource item: 3.12\nSource URL: http://aimpl.org/ratsubvarpos/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[98]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $\\\\mathbb{P}$ be the parametrizing space of higher surfaces of degree $d$ in $\\\\mathbb{P}^{n}$.\\nWhat is the largest dimensional family of unirational / supersinglar hypersurfaces?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0099", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over an algebraically closed field of characteristic p=3 or p>=7, the smooth quartic K3 locus of Artin invariant at most sigma has dimension sigma+14 in the coefficient space P^34 and sigma-1 after quotienting by PGL_4. Thus the full supersingular quartic locus has exact coefficient dimension 24 and moduli dimension 9. If u is the supremum of dimensions of locally closed families all of whose members are geometrically unirational, then 20<=u<=24 in characteristic 3 and 16<=u<=24 for p>=7; the corresponding moduli bounds are 5<=u_mod<=9 and 1<=u_mod<=9. The upper bound uses unirationality implies supersingularity, while the lower bounds use the proved Artin-invariant <=6 and <=2 cases, respectively. No supersingular-implies-unirational converse is assumed.\n\nCandidate contribution (dimension formula and quantitative reduction; novelty confidence low): Candidate novel synthesis: for smooth quartics in odd characteristic p!=5, the coefficient-space Artin ladder is dim U_ss,<=sigma = sigma+14; combining it with the corrected current unirationality status gives the explicit brackets 20<=u_4,3^coef<=24 and 16<=u_4,p^coef<=24 for p>=7." }, { "id": 20000100, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0100", "title": "An exact obstruction count for sections of index-one hypersurface families", "statement": "$\\mathcal{X}\\longrightarrow \\mathcal{B}$ be a family of smooth hypersurfaces of degree $d=n$ in $\\mathbb{P}^{n}$.\nIs there more than one rational section?\nDoes weak approximation hold?", "original_statement": "$\\mathcal{X}\\longrightarrow \\mathcal{B}$ be a family of smooth hypersurfaces of degree $d=n$ in $\\mathbb{P}^{n}$.\nIs there more than one rational section?\nDoes weak approximation hold?", "clean_statement": "$\\mathcal{X}\\longrightarrow \\mathcal{B}$ be a family of smooth hypersurfaces of degree $d=n$ in $\\mathbb{P}^{n}$.\nIs there more than one rational section?\nDoes weak approximation hold?", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 3.17 in the workshop *Rational subvarieties in positive characteristic*, section “Rationality in a family.” Its text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rationality in a family\nSource item: 3.17\nSource URL: http://aimpl.org/ratsubvarpos/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[99]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"$\\\\mathcal{X}\\\\longrightarrow \\\\mathcal{B}$ be a family of smooth hypersurfaces of degree $d=n$ in $\\\\mathbb{P}^{n}$.\\nIs there more than one rational section?\\nDoes weak approximation hold?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: partially_solved; rigor: conditional; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0100", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the natural interpretation over the function field of a curve over an algebraically closed field, Tsen-Lang always gives one section. Weak approximation, hence infinitely many sections, is known in characteristic zero, for n=2 or 3 in every characteristic, and for n>=4 under the Starr-Tian-Zong index-one characteristic condition; the arbitrary remaining positive-characteristic case is open. The new conditional result fixes a section s and jet divisor D, sets q=h^1(B,s^*T_{X/B}(-D)), and proves that q direction-selected length-one positive elementary transforms kill the handle obstruction exactly, while a (q+1)st raises h^0 by one. If those transforms are realized by transverse degree-n minimal very free teeth and the standard comb-smoothing criterion holds, the comb smooths to a distinct section with the same jets and hyperplane-degree increase n(q+1). Within one-node immersed very-free combs, q teeth and vertical degree at least nq are necessary.\n\nCandidate contribution (quantitative_reduction; novelty confidence low): For E=s^*T_{X/B}(-D), q=h^1(B,E) is the exact number of suitably directed length-one tooth transforms required to eliminate the handle H^1 without increasing h^0; one further transform increases h^0 by exactly one, and, conditional on directional degree-n tooth availability and comb smoothing, yields a distinct D-jet-preserving section of degree H.s+n(q+1).", "research_difficulty_suggested": "L4" }, { "id": 20000101, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0101", "title": "Positivity, separability, and weak approximation for low-degree hypersurfaces", "statement": "If $d^{2}\\leq n$, does weak approximation hold in char. $p$ for hypersurfaces? (it does in char. $0$)", "original_statement": "If $d^{2}\\leq n$, does weak approximation hold in char. $p$ for hypersurfaces? (it does in char. $0$)", "clean_statement": "If $d^{2}\\leq n$, does weak approximation hold in char. $p$ for hypersurfaces? (it does in char. $0$)", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rationality in a family\nSource item: 3.18\nSource URL: http://aimpl.org/ratsubvarpos/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[100]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If $d^{2}\\\\leq n$, does weak approximation hold in char. $p$ for hypersurfaces? (it does in char. $0$)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0101", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For hypersurfaces over k(B) with k algebraically closed of characteristic p, the published Starr–Tian–Zong theorem gives weak approximation at every good or covered strong potentially-good place when p>d and n>d, so d^2<=n is more than sufficient in that range. A December 2025 preprint extends the separably-rationally-connected geometric input to p>=d and to all sufficiently high-dimensional quartics in every characteristic, yielding further good-place cases. The all-place question for arbitrary small p=3.\n\nCandidate contribution (proposition; novelty confidence low): Two-threshold diagnostic: whenever the pointed-line fiber F_x(X_d) has expected dimension, it is a (2,...,d) local complete intersection in P(T_xX), has degree d!, and under d^2<=n has anticanonical coefficient at least d(d-1)/2; nevertheless, for the smooth Fermat hypersurface of degree d=p+1 the Gauss map is purely inseparable and, for p>=3, the polar equations collapse so that every pointed-line fiber has at least p-2 dimensions of excess. Thus improving the ambient-dimension inequality alone cannot replace the missing separability in small characteristic." }, { "id": 20000102, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0102", "title": "Positive-density rational reductions and a finite-monodromy density criterion", "statement": "4th day\n\nDo there exist a cubic $4$-fold defined over number fields which is conjecured to be irrational but many reductions are conjectured to be ratioal?", "original_statement": "4th day\n\nDo there exist a cubic $4$-fold defined over number fields which is conjecured to be irrational but many reductions are conjectured to be ratioal?", "clean_statement": "4th day\n\nDo there exist a cubic $4$-fold defined over number fields which is conjecured to be irrational but many reductions are conjectured to be ratioal?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 3.72 in the section “Rationality in a family” of the workshop *Rational subvarieties in positive characteristic*. The record, including its spelling, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rationality in a family\nSource item: 3.72\nSource URL: http://aimpl.org/ratsubvarpos/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[101]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"4th day\\n\\nDo there exist a cubic $4$-fold defined over number fields which is conjecured to be irrational but many reductions are conjectured to be ratioal?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0102", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Frei, Hassett, and Varilly-Alvarado give a published affirmative answer to the positive-density interpretation: conjecturally irrational cubic fourfolds over number fields have proved rational reductions on a positive-natural-density set, while density one, cofiniteness, and irrationality of their special examples remain open. Independently, this attempt proves that any finite affine rationality certificate represented by a class xi in H^1(K,M) has certificate primes of exact density equal to the fixed-point proportion of its finite affine Galois image. In such a finite model, density one is equivalent to cofiniteness and to cyclic local triviality; if cyclic local cohomology vanishes and xi is nonzero, noncertificate primes have positive density.\n\nCandidate contribution (theorem; novelty confidence low): For a finite affine rationality certificate xi with affine image H, the certified reductions have exact density |{h in H: h fixes a point}|/|H|; density one is equivalent to cofiniteness and membership in H^1_cyc(H,M), while H^1_cyc(H,M)=0 and xi nonzero force a positive-density complement. Full AGL_1(F_q) gives the explicit density 1-1/q, hence 1/2 and 2/3 in the order-two and order-three toy models." }, { "id": 20000103, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0103", "title": "Rational reduction and a prime-to-characteristic sieve for CTO examples", "statement": "Can irrational varieties over $\\#$ fields specialize to rational varieties in char. $p>0$?\n(Colliot-th\\'el\\`ene-Ojanguren's examples)", "original_statement": "Can irrational varieties over $\\#$ fields specialize to rational varieties in char. $p>0$?\n(Colliot-th\\'el\\`ene-Ojanguren's examples)", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "It is Problem 3.78 in the section “Rationality in a family” of the AIM workshop list *Rational subvarieties in positive characteristic*. The archived AIM page itself contains the string `$\\#$ fields`; thus this is not merely a corruption introduced by the JSON extraction. The immediately preceding Problem 3.72 asks about cubic fourfolds “defined over number fields” and their reductions. On that contextual evidence, the most plausible reconstruction is: > **Plausible reconstruction.** Can an irrational variety over a **number field** have a rational specialization in characteristic \\(p>0\\)? What happens for the examples of Colliot-Thélène and Ojanguren?", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rationality in a family\nSource item: 3.78\nSource URL: http://aimpl.org/ratsubvarpos/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[102]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can irrational varieties over $\\\\#$ fields specialize to rational varieties in char. $p>0$?\\n(Colliot-th\\\\'el\\\\`ene-Ojanguren's examples)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0103", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Ambrosi and Valloni (2025) affirmatively answer the broad geometric mixed-characteristic question by constructing, for every sufficiently large p, a smooth proper family over the valuation ring of C_p with stably irrational generic fiber and rational special fiber; their argument does not descend the generic fiber to a number field and is not the original CTO family. For CTO Example 3.3, this attempt proves a complementary persistence theorem: after replacing algebraic independence by a finite arrangement-and-residue certificate, the exact CTO symbol (f1,f2,g1) pulls back to a nonzero unramified H^3 class over every algebraically closed odd-characteristic certified fiber. Consequently one can choose a number-field CTO model and a smooth proper spread whose every arrangement-good odd fiber is unirational but not retract rational, so any rational CTO specialization is confined to characteristic 2 or finitely many bad primes.\n\nCandidate contribution (theorem; novelty confidence low): For the explicit Colliot-Thelene-Ojanguren Proposition 3.1 / Example 3.3 quadric, the algebraic-independence hypothesis can be replaced by a finite generic arrangement-and-residue open U_CTO; every geometric fiber of odd characteristic lying in U_CTO carries the same nonzero unramified H^3(mu_2 tensor 3) class, and a number-field model therefore has no rational specialization at any arrangement-good odd prime." }, { "id": 20000104, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0104", "title": "A Delta-form rank obstruction for pointed elliptic moduli in characteristic p", "statement": "When is $\\mathcal{M}_{g,n}$, $\\mathcal{A}_{g}[d]$ or $\\mathcal{K3}_{d}$ unirational over char. $p$?\nWhen is $\\mathcal{M}_{1,n}$ unirational for $n\\geq 11$?", "original_statement": "When is $\\mathcal{M}_{g,n}$, $\\mathcal{A}_{g}[d]$ or $\\mathcal{K3}_{d}$ unirational over char. $p$?\nWhen is $\\mathcal{M}_{1,n}$ unirational for $n\\geq 11$?", "clean_statement": "When is $\\mathcal{M}_{g,n}$, $\\mathcal{A}_{g}[d]$ or $\\mathcal{K3}_{d}$ unirational over char. $p$?\nWhen is $\\mathcal{M}_{1,n}$ unirational for $n\\geq 11$?", "statement_status": "exact", "statement_verification": "There is no apparent OCR corruption. There are, however, three genuine notational ambiguities:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rational subvarieties of the moduli space of rational curves and of abelian varieties\nSource item: 4.1\nSource URL: http://aimpl.org/ratsubvarpos/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[103]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"When is $\\\\mathcal{M}_{g,n}$, $\\\\mathcal{A}_{g}[d]$ or $\\\\mathcal{K3}_{d}$ unirational over char. $p$?\\nWhen is $\\\\mathcal{M}_{1,n}$ unirational for $n\\\\geq 11$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0104", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let k be algebraically closed of characteristic p>3 and n>=11. The coarse space of stable n-pointed genus-one curves is not separably unirational. More precisely, for any dominant map f:P^n --> Mbar_{1,n}, the induced finite function-field extension has inseparable degree at least p^(n-10) if d(f^*j) is nonzero; if d(f^*j)=0, then f^*j is a p-th power and the inseparable degree is at least p. Thus every parametrization is generically inseparable, but the result does not rule out unirationality by an inseparable map. A second proved slice is that A_1[d] is geometrically unirational exactly for d<=5 when p does not divide d.\n\nCandidate contribution (quantitative_obstruction; novelty confidence low): For a dominant generically finite rational map f:P^n --> Mbar_{1,n} in characteristic p>3, simultaneous pullback of all ten-marking Delta forms gives the dichotomy d(f^*j) nonzero implies inseparable degree at least p^(n-10), while d(f^*j)=0 implies f^*j lies in k(P^n)^p and inseparable degree at least p." }, { "id": 20000105, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0105", "title": "Moret--Bailly rational curves with Hodge degree p-1", "statement": "Find rational curves in $\\mathcal{A}_{g}$ in char. $p$, that do not lift to char. $0$.", "original_statement": "Find rational curves in $\\mathcal{A}_{g}$ in char. $p$, that do not lift to char. $0$.", "clean_statement": "Find rational curves in $\\mathcal{A}_{g}$ in char. $p$, that do not lift to char. $0$.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rational subvarieties of the moduli space of rational curves and of abelian varieties\nSource item: 4.2\nSource URL: http://aimpl.org/ratsubvarpos/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[104]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find rational curves in $\\\\mathcal{A}_{g}$ in char. $p$, that do not lift to char. $0$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0105", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For every algebraically closed field of characteristic p at least 3, every g at least 2, and every prime level ell at least 3 different from p, the universal Moret--Bailly alpha_p-quotient, stabilized by a fixed supersingular factor when g is greater than 2, gives a nonconstant morphism P^1 -> A_g[ell] whose fibers are supersingular and whose pulled-back Hodge line is O(p-1). The positive Hodge degree proves nonconstancy, survives forgetting level, and contradicts every smooth genus-zero mixed-characteristic lift because a fine characteristic-zero Siegel moduli space admits no nonconstant P^1-maps. A fully explicit polarization matrix is verified in characteristic 5.\n\nCandidate contribution (lemma; novelty confidence low): Candidate Hodge-degree nonlifting certificate: for the Moret--Bailly pencil and all of its fixed-factor stabilizations, f^*Lambda = O(p-1); this exact equality alone rules out every direct mixed-characteristic lift over a smooth proper genus-zero base." }, { "id": 20000106, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0106", "title": "A genus-three fine-level obstruction and a coarse quotient-curve criterion", "statement": "What are the rationality properties of $\\mathcal{A}_{g,s.s.}$?\nYou can ask the same question for Newton polygon strata.", "original_statement": "What are the rationality properties of $\\mathcal{A}_{g,s.s.}$?\nYou can ask the same question for Newton polygon strata.", "clean_statement": "What are the rationality properties of $\\mathcal{A}_{g,s.s.}$?\nYou can ask the same question for Newton polygon strata.", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 4.3 from the workshop *Rational subvarieties in positive characteristic*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rational subvarieties of the moduli space of rational curves and of abelian varieties\nSource item: 4.3\nSource URL: http://aimpl.org/ratsubvarpos/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[105]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What are the rationality properties of $\\\\mathcal{A}_{g,s.s.}$?\\nYou can ask the same question for Newton polygon strata.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0106", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over an algebraically closed field of characteristic p, every irreducible component of the genus-three supersingular locus with full symplectic level N >= 3 prime to p has function field k(C_p)(u), where C_p is the smooth Fermat curve of degree p+1 and genus p(p-1)/2; hence no such component is unirational, stably rational, or rational. For a no-level coarse component indexed by a source polarization mu, its function field is k(C_p/Gamma_mu)(v), where Gamma_mu is the induced base action of Aut(E^3,mu); the invariant non-rigid section splits the possible conic twist. Thus coarse rationality is equivalent to genus(C_p/Gamma_mu)=0.\n\nCandidate contribution (reduction; novelty confidence low): In genus three, fine full-level supersingular components are non-unirational, while the coarse component has invariant field k(C_p/Gamma_mu)(v), with no Brauer/conic twist because the intrinsic non-rigid section descends; consequently its rationality, stable rationality, and unirationality are all equivalent to genus(C_p/Gamma_mu)=0." }, { "id": 20000107, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0107", "title": "Genus-two density and a fine-level Hecke propagation criterion", "statement": "Are complete rational curves in $\\mathcal{A}_{g}$ dense in char. $p$?", "original_statement": "Are complete rational curves in $\\mathcal{A}_{g}$ dense in char. $p$?", "clean_statement": "Are complete rational curves in $\\mathcal{A}_{g}$ dense in char. $p$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List item 4.4 from the workshop *Rational subvarieties in positive characteristic*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Rational subvarieties of the moduli space of rational curves and of abelian varieties\nSource item: 4.4\nSource URL: http://aimpl.org/ratsubvarpos/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[106]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Are complete rational curves in $\\\\mathcal{A}_{g}$ dense in char. $p$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0107", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the coarse moduli space of principally polarized abelian surfaces over any algebraically closed field of characteristic p>0, complete rational curves are Zariski dense. For p>=5, every coarse A_2 point lies on a complete rational curve contained in A_2; for p=2,3, an explicit construction covers the dense chart D_+(psi_{p-1} chi_10). In arbitrary genus, an honest fine-level P^1 meeting a point x propagates along every fixed-ell Hecke orbit, so rational curves are dense in the central leaf of x and are dense in the full moduli space if x is ordinary. Isogenies preserve Newton polygons, so supersingular Moret-Bailly curves cannot supply the ordinary hypothesis.\n\nCandidate contribution (special_case; novelty confidence low): Complete rational curves are dense in the coarse A_2 in every positive characteristic; for p>=5 every point is covered, while for p=2,3 every point of D_+(psi_{p-1} chi_10) is covered by an explicit boundary-avoiding weighted-coordinate P^1." }, { "id": 20000108, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0108", "title": "Index-gap obstruction for line covers on Fano hypersurfaces in characteristic p", "statement": "In char. $p$ if $n\\geq d+2$,\nare the moduli spaces of rational curves on a general hypersurface of degree $d$ in $\\mathbb{P}^{n}$ irreducible of the expected dimension?", "original_statement": "In char. $p$ if $n\\geq d+2$,\nare the moduli spaces of rational curves on a general hypersurface of degree $d$ in $\\mathbb{P}^{n}$ irreducible of the expected dimension?", "clean_statement": "In char. $p$ if $n\\geq d+2$,\nare the moduli spaces of rational curves on a general hypersurface of degree $d$ in $\\mathbb{P}^{n}$ irreducible of the expected dimension?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: The moduli space of rational curves of hypersurfaces\nSource item: 5.1\nSource URL: http://aimpl.org/ratsubvarpos/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[107]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In char. $p$ if $n\\\\geq d+2$,\\nare the moduli spaces of rational curves on a general hypersurface of degree $d$ in $\\\\mathbb{P}^{n}$ irreducible of the expected dimension?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0108", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The sharp general-degree positive-characteristic question remains open, while Kitagawa's June 2026 theorem settles every curve degree for smooth cubic hypersurfaces of dimension at least four in characteristic different from 2 and 3. This attempt proves that, when the line scheme has its expected dimension (as it does for a general hypersurface in the AIM range), the full compactified locus of degree-e stable maps whose image is a line has dimension at most 2n-d-3+2e-2. Every stable-map component has dimension at least E_e=e(n+1-d)+n-4, so the line-image locus has codimension at least (e-1)(n-d-1). Hence neither separable nor inseparable multiple covers of lines can form a component when n is at least d+2 and e is greater than one.\n\nCandidate contribution (dimension_bound; novelty confidence low): Candidate cover-deficit principle: an m-fold cover locus over an expected-dimensional degree-a birational family falls short of the degree-am expected dimension by (m-1)(a(n+1-d)-2); unconditionally for the entire compactified line-image locus on a general hypersurface, the deficit is (e-1)(n-d-1), including reducible-domain and inseparable maps." }, { "id": 20000109, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0109", "title": "Expected dimension and Frobenius defects for rational curves on hypersurfaces", "statement": "What is the dimension of rational curves in hypersurfaces of degree $d$ in $\\mathbb{P}^{n}$ in char. $p$? Does char. $0$ proof go through?", "original_statement": "What is the dimension of rational curves in hypersurfaces of degree $d$ in $\\mathbb{P}^{n}$ in char. $p$? Does char. $0$ proof go through?", "clean_statement": "What is the dimension of rational curves in hypersurfaces of degree $d$ in $\\mathbb{P}^{n}$ in char. $p$? Does char. $0$ proof go through?", "statement_status": "exact", "statement_verification": "The exact AIM record (workshop “Rational subvarieties in positive characteristic,” section “The moduli space of rational curves of hypersurfaces,” Problem 5.2) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: The moduli space of rational curves of hypersurfaces\nSource item: 5.2\nSource URL: http://aimpl.org/ratsubvarpos/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[108]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the dimension of rational curves in hypersurfaces of degree $d$ in $\\\\mathbb{P}^{n}$ in char. $p$? Does char. $0$ proof go through?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0109", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth degree-d hypersurface X in P^n, the expected dimensions in every characteristic are e(n+1-d)+n-1 for parameterized degree-e maps and e(n+1-d)+n-4 for unpointed stable maps or embedded rational curves. If an embedded rational curve g has g^*T_X = direct sum O(a_j), then its q=p^r Frobenius cover has obstruction-space and Hom tangent excess sum over a_j<0 of (-q a_j-1), while its exact stable-map stabilizer is the r-th Frobenius kernel of PGL_2, of rank q^3 and Lie dimension 3. An explicit smooth hypersurface containing a line with tangent splitting O(2) plus O(2-d) plus O(1)^(n-3) realizes excess q(d-2)-1; for smooth cubics in characteristic greater than 3, Glas's expected-component-dimension theorem makes these provably singular points on expected-dimensional Hom schemes.\n\nCandidate contribution (theorem; novelty confidence low): The candidate contribution is the exact Frobenius-amplification and stabilizer package, together with a smooth explicit hypersurface family realizing Zariski-tangent and obstruction-space growth q(d-2)-1 and, in the cubic range, singular Frobenius points on expected-dimensional Hom schemes." }, { "id": 20000110, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0110", "title": "Odd-characteristic stable irrationality and bounded-complexity reduction for ruledness", "statement": "Fix \\(n\\geq1\\) and \\(d\\geq 2\\lceil(n+3)/3\\rceil\\). Is there a number \\(P(n,d)\\) such that, for every prime \\(p>P(n,d)\\), the very general smooth degree-\\(d\\), \\(n\\)-dimensional hypersurface \\(X_d\\subset\\mathbb P^{n+1}\\) in characteristic \\(p\\) is (i) irrational, (ii) not stably rational, and (iii) not ruled?", "original_statement": "Is the very general hyperusrfae of degree $d\\geq 2 \\lceil(n+3)/3\\rceil$ not rational / stably rational / ruled in char $p>0$ for $p>>0$?", "clean_statement": "Fix \\(n\\geq1\\) and \\(d\\geq 2\\lceil(n+3)/3\\rceil\\). Is there a number \\(P(n,d)\\) such that, for every prime \\(p>P(n,d)\\), the very general smooth degree-\\(d\\), \\(n\\)-dimensional hypersurface \\(X_d\\subset\\mathbb P^{n+1}\\) in characteristic \\(p\\) is (i) irrational, (ii) not stably rational, and (iii) not ruled?", "statement_status": "corrected_verified", "statement_verification": "This is Problem 6.1 in the section “New applications of Kollar's and Totaro's techniques” from the 2016 AIM workshop *Rational subvarieties in positive characteristic*. The source URL is . It timed out during this run, so the recovery below is based on the exact corpus record, the adjacent records, and matching primary literature rather than on a newly fetched copy of the old AIM page. There is one evident OCR error: “hyperusrfae” means “hypersurface.” The ambient projective space is omitted, but the numerical bound identifies the intended convention unambiguously. Kollár's theorem, quoted with exactly this bound by Totaro and by Lange--Schreieder, concerns an \\(n\\)-dimensional hypersurface \\[ X_d\\subset \\mathbb P^{n+1}. \\] Thus the recovered question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: New applications of Kollar's and Totaro's techniques\nSource item: 6.1\nSource URL: http://aimpl.org/ratsubvarpos/6/\nCanonical location: aim-algebraic-geometry-notes.json notes[109]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is the very general hyperusrfae of degree $d\\\\geq 2 \\\\lceil(n+3)/3\\\\rceil$ not rational / stably rational / ruled in char $p>0$ for $p>>0$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/6/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0110", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The published theorem of Lange and Schreieder, together with an elementary comparison of degree bounds, shows that every very general hypersurface in the AIM range over a field of odd characteristic has no integral decomposition of the diagonal and hence is neither retract rational, stably rational, nor rational; thus the first two AIM questions hold for every odd prime. Smooth hypersurfaces with d at least n+2 are nonruled in every characteristic by the plurigenus obstruction. For the residual Fano ruledness question, a proved finite-type spreading argument shows that every fixed presentation-complexity bound B is excluded for the geometric-generic hypersurface outside finitely many primes, so any hypothetical rulings in infinitely many characteristics must have unbounded presentation complexity.\n\nCandidate contribution (reduction; novelty confidence low): For each fixed n, d, and ruling-presentation bound B in the Kollár degree range, there is a finite set of primes outside which the geometric-generic degree-d hypersurface has no B-presented ruling; consequently, the least presentation complexity of hypothetical rulings along any infinite set of primes must escape every fixed bound." }, { "id": 20000111, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0111", "title": "The exact one-cover differential-form barrier", "statement": "Can we improve those techniques to find differential form for lower degree hypersurfaces?", "original_statement": "Can we improve those techniques to find differential form for lower degree hypersurfaces?", "clean_statement": "Can we improve those techniques to find differential form for lower degree hypersurfaces?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop “Rational subvarieties in positive characteristic,” section 6, problem 6.2) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: New applications of Kollar's and Totaro's techniques\nSource item: 6.2\nSource URL: http://aimpl.org/ratsubvarpos/6/\nCanonical location: aim-algebraic-geometry-notes.json notes[110]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we improve those techniques to find differential form for lower degree hypersurfaces?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/6/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0111", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an integral normal standard inseparable c-fold cover of a smooth degree-a hypersurface in characteristic p dividing c, under the admissible-critical-point and resolution hypotheses, the distinguished Kollar line on the resolution is the pullback of O((c+1)a-n-2), and its exact space of sections has dimension B_{n+1}((c+1)a-n-2)-B_{n+1}(a-n-2). Optimizing over every characteristic and cover degree shows that, for n at least 3 and total degree budget D, this line can contribute a form exactly in the numerical range D at least 2 ceil((n+2)/3). Thus prime or cover-degree optimization alone cannot beat Totaro's threshold within the one-cover architecture.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate novelty: the exact telescoping section formula, combined with an all-cover-degree optimization including the exceptional budgets D=2 and D=3, gives a sharp no-go theorem for lowering Totaro's threshold using the distinguished line in any standard one-cover block." }, { "id": 20000112, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0112", "title": "Explicit inseparably unirational cyclic covers with regular forms", "statement": "Can we find new examples of unirational varieties that have non-vanishing differential forms?", "original_statement": "Can we find new exmaples of unirational varieties that have non-vanishing differential forms?", "clean_statement": "Can we find new examples of unirational varieties that have non-vanishing differential forms?", "statement_status": "corrected_verified", "statement_verification": "The evident typographical correction is “examples.” I interpret “non-vanishing differential forms” as “nonzero global regular differential forms,” not as forms that are nowhere zero. The surviving AIM record is otherwise unambiguous. The linked AIM page did not return usable additional text during this run.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: New applications of Kollar's and Totaro's techniques\nSource item: 6.3\nSource URL: http://aimpl.org/ratsubvarpos/6/\nCanonical location: aim-algebraic-geometry-notes.json notes[111]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we find new exmaples of unirational varieties that have non-vanishing differential forms?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/6/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0112", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Over every algebraically closed field of characteristic p, for N at least 3, D=pm at least N+1, and p not dividing N+1, the normal cyclic cover y^p=sum_i x_i^(D-1)x_(i+1) has only isolated nondegenerate critical-point singularities. A Kollár resolution is inseparably unirational via an explicit degree p^(N-1) parametrization and carries at least binomial(D-1,N) independent regular (N-1)-forms. The specialization N=pm-2 gives an all-characteristic family of normal covers with ample anticanonical bundle whose resolutions have at least pm-1 such forms.\n\nCandidate contribution (explicit_family; novelty confidence low): The cyclic polynomial sum_i x_i^(pm-1)x_(i+1) has nondegenerate projective critical points in every characteristic when p does not divide N+1, and its p-cyclic cover has an explicit purely inseparable parametrization of degree p^(N-1); setting N=pm-2 packages these facts into a uniform anticanonical family with at least pm-1 regular forms on a resolution." }, { "id": 20000113, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0113", "title": "Stable irrationality of cyclic covers of Grassmannians", "statement": "Look for new applications of Koll\\'ar / Totaro's techique.", "original_statement": "Look for new applications of Koll\\'ar / Totaro's techique.", "clean_statement": "Look for new applications of Koll\\'ar / Totaro's techique.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: New applications of Kollar's and Totaro's techniques\nSource item: 6.4\nSource URL: http://aimpl.org/ratsubvarpos/6/\nCanonical location: aim-algebraic-geometry-notes.json notes[112]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Look for new applications of Koll\\\\'ar / Totaro's techique.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/6/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0113", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let G=Gr(k,N), q=k(N-k), and let X be a very general smooth complex m-fold cyclic cover of G with root line bundle O(a), branched in |O(am)|. If am is at least N and either q is even or m has an odd prime divisor, then X has no integral decomposition of the diagonal and is therefore neither retract rational nor stably rational. The proof combines an explicit Plücker jet-separation lemma, a mixed-characteristic Witt-vector family whose generic and special coefficients are both algebraically independent, Okada's universally CH0-trivial resolution theorem for arbitrary p-divisible composite covering degree, the nonzero special-fiber form from O(am-N), and Colliot-Thélène-Pirutka specialization. In particular, a very general double cover of Gr(2,5) branched in |O(6)| is a stably irrational Fano sixfold with -K equal to twice the pulled-back Plücker class.\n\nCandidate contribution (theorem; novelty confidence low): Very general cyclic covers of Grassmannians in the stated parity and branch-degree range have no integral decomposition of the diagonal; concretely, the very general double cover of Gr(2,5) branched in |O(6)| is a stably irrational Fano sixfold." }, { "id": 20000114, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0114", "title": "The Cartier obstruction and a cone-K3 Brauer escape", "statement": "Find an example of varieties which are degenerations of Fano hypersurfaces\nwith $H^{i}(\\mathcal{O}_{X})\\neq 0$ for some $i$.\nTry to construct varieties with non-trivial Brawer classes.", "original_statement": "Find an example of varieties which are degenerations of Fano hypersurfaces\nwith $H^{i}(\\mathcal{O}_{X})\\neq 0$ for some $i$.\nTry to construct varieties with non-trivial Brawer classes.", "clean_statement": "Find an example of varieties which are degenerations of Fano hypersurfaces\nwith $H^{i}(\\mathcal{O}_{X})\\neq 0$ for some $i$.\nTry to construct varieties with non-trivial Brawer classes.", "statement_status": "exact", "statement_verification": "The canonical AIM record, problem 6.5 in the section “New applications of Kollar's and Totaro's techniques,” reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: New applications of Kollar's and Totaro's techniques\nSource item: 6.5\nSource URL: http://aimpl.org/ratsubvarpos/6/\nCanonical location: aim-algebraic-geometry-notes.json notes[113]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find an example of varieties which are degenerations of Fano hypersurfaces\\nwith $H^{i}(\\\\mathcal{O}_{X})\\\\neq 0$ for some $i$.\\nTry to construct varieties with non-trivial Brawer classes.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/6/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0114", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A literal degree-d Cartier hypersurface in projective space cannot answer the question in the Fano range: for d <= n+1 it has H^i(O)=0 for every i>0, without smoothness or reducedness assumptions. At the sharp index-one boundary, a flat family of smooth degree-(n+1) Fano hypersurfaces degenerates to the projective cone over a degree-(n+1) Calabi-Yau hypersurface Y; blowing up the vertex gives P_Y(O_Y plus O_Y(1)), with H^{n-1}(O)=k and exact higher direct image R^{n-1} beta_* O=k(v). For the cone over the complex Fermat quartic K3 surface, the resolution also has Brauer group (Q/Z)^2, represented by Azumaya algebras and yielding nonzero unramified classes, and exceptional-section restriction proves that these transported classes do not descend from the singular cone.\n\nCandidate contribution (obstruction_and_construction; novelty confidence low): For the index-one projective-cone degeneration, the literal Fano-hypersurface special fiber has no higher structure-sheaf cohomology, while its vertex blow-up has exactly H^{n-1}(O)=k and the discrepancy is localized as the O-module R^{n-1} beta_*O=k(v); in dimension three the Fermat-quartic K3 base simultaneously supplies nonzero Azumaya and unramified Brauer classes whose non-descent to the cone follows by restricting to the exceptional section." }, { "id": 20000115, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0115", "title": "A separable odd-degree parametrization of the characteristic-2 Fermat cubic", "statement": "Do cubic $3$-folds have odd degree unirational prametrizations in char $p$?", "original_statement": "Do cubic $3$-folds have odd degree unirational prametrizations in char $p$?", "clean_statement": "Do cubic $3$-folds have odd degree unirational prametrizations in char $p$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, workshop *Rational subvarieties in positive characteristic*, §7, Problem 7.1. The source page contains the single sentence", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Unirational parameterizations of cubic hypersurfaces\nSource item: 7.1\nSource URL: http://aimpl.org/ratsubvarpos/7/\nCanonical location: aim-algebraic-geometry-notes.json notes[114]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Do cubic $3$-folds have odd degree unirational prametrizations in char $p$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/7/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0115", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over every algebraically closed field of characteristic 2, the smooth Fermat cubic threefold is dominated by projective 3-space through a separable rational map of exact total degree 3. The proof uses a faithful tame (Z/3)^2-action: the full quotient is birational to the Fermat cubic, an order-3 intermediate quotient is rational, and the resulting fixed-field extension has degree 3. Together with Yang-Yu-Zhu's degree-3 examples in characteristics p >= 5, this settles the existential reading for all positive characteristics except p = 3; the universal/general readings and the characteristic-3 case remain unresolved.\n\nCandidate contribution (characteristic-specific theorem; novelty confidence low): The smooth Fermat cubic threefold over an algebraically closed field of characteristic 2 admits a separable unirational parametrization of exact total degree 3." }, { "id": 20000116, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0116", "title": "Church surfaces and an all-dimensional slope-Kunneth propagation theorem", "statement": "Can we find a simply-connected variety with supersingular cohomology which is not rationally-connected?", "original_statement": "Can we find a simply-connected variety with supersingular cohomology which is not rationally-connected?", "clean_statement": "Can we find a simply-connected variety with supersingular cohomology which is not rationally-connected?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, workshop *Rational subvarieties in positive characteristic*, section “Other problems,” item 9.05. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.05\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[115]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we find a simply-connected variety with supersingular cohomology which is not rationally-connected?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0116", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Benjamin Church's 2025 preprint constructs good reductions of simply connected product-quotient surfaces which are Shioda-supersingular and not covered by rational curves. For such a surface S, trivial geometric etale fundamental group kills H^1 and H^3, algebraic spanning of H^2 makes its Frobenius eigenvalues q times roots of unity, and Katz-Messing comparison gives crystalline slope i/2 in every degree. Consequently S answers the AIM question even with 'not rationally connected' strengthened to 'not rationally chain connected.' Moreover, S times projective r-space has the same three properties for every r at least zero, giving examples in every dimension at least two.\n\nCandidate contribution (propagation theorem; novelty confidence low): If a smooth projective surface S over an algebraic closure of a finite field is geometrically etale simply connected, Shioda-supersingular, and not covered by rational curves, then for every r >= 0 the variety S x P^r is geometrically etale simply connected, every crystalline H^n is isoclinic of slope n/2, and S x P^r is not rationally chain connected." }, { "id": 20000117, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0117", "title": "Complete subvarieties at the Diaz bound and a genus-four boundary obstruction", "statement": "Find a complete subvariety of $\\mathcal{M}_{g}$ of $\\text{codim}2g-1$.", "original_statement": "Find a complete subvariety of $\\mathcal{M}_{g}$ of $\\text{codim}2g-1$.", "clean_statement": "Find a complete subvariety of $\\mathcal{M}_{g}$ of $\\text{codim}2g-1$.", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.1\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[116]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find a complete subvariety of $\\\\mathcal{M}_{g}$ of $\\\\text{codim}2g-1$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0117", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The requested dimension is g-2, the Diaz-Looijenga upper bound; it is attained for g=2 and g=3, but the general problem is open and even the existence of a complete surface in M_4 is unknown. Over an algebraically closed field of characteristic zero, this attempt proves that every hypothetical complete irreducible surface in M_4 must contain a complete curve whose general member is nonhyperelliptic with rank-three canonical quadric. It also proves that a fiberwise smooth relative (3,3)-divisor in P(E) times P(F), with coefficient twist L, necessarily satisfies L^34 isomorphic to (det(E) tensor det(F))^51.\n\nCandidate contribution (affine-stratum obstruction; novelty confidence low): Any complete irreducible surface in coarse M_4 over an algebraically closed characteristic-zero field contains a complete curve generically in the nonhyperelliptic rank-three-canonical-quadric locus; for smooth relative (3,3)-divisors in a split quadric bundle, the accompanying testable constraint is L^34 isomorphic to (det(E) tensor det(F))^51." }, { "id": 20000118, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0118", "title": "Square-field superspecial point growth and a Betti-complexity barrier", "statement": "Fix $q$, let $g\\rightarrow \\infty$,\nwhat subvariety of $\\mathcal{M}_{g} / \\mathcal{A}_{g}$ contributes the most $\\#$ of rational points?", "original_statement": "Fix $q$, let $g\\rightarrow \\infty$,\nwhat subvariety of $\\mathcal{M}_{g} / \\mathcal{A}_{g}$ contributes the most $\\#$ of rational points?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "There is no standard quotient \\(\\mathcal M_g/\\mathcal A_g\\): \\(\\mathcal A_g\\) is not a group acting on \\(\\mathcal M_g\\). The natural relation is instead the Torelli morphism \\(\\mathcal M_g\\to\\mathcal A_g\\). The workshop report discusses moduli of curves and moduli of abelian varieties in parallel. The least speculative reconstruction is therefore: This reconstruction is not asserted to be uniquely intended.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.15\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[117]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Fix $q$, let $g\\\\rightarrow \\\\infty$,\\nwhat subvariety of $\\\\mathcal{M}_{g} / \\\\mathcal{A}_{g}$ contributes the most $\\\\#$ of rational points?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0118", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted maximization is ill posed, but within natural strata the superspecial principally polarized locus gives a rigorous benchmark: for every prime p and every square field q=p^(2r), its reduced coarse point count and finite-field stack groupoid mass equal the geometric class number H_g, while its unweighted F_q-object count differs by at most exp(O(g^2)); all three have logarithm g^2 log g+O_p(g^2). Any separated coarse subvariety of A_g with at least this many F_q-points must have compactly supported Betti sum exp(g^2 log g-O_q(g^2)). For odd q, the exact obstruction is the unresolved type-number term 2T_g-H_g. On the curve side, the hyperelliptic locus has logarithm 2g log q+O_q(1) for odd q.\n\nCandidate contribution (comparison_theorem; novelty confidence low): For every q=p^(2r), the superspecial coarse count, finite-field stack mass, and unweighted object count share leading logarithm g^2 log g, and any coarse competitor matching that count has compactly supported Betti sum at least exp(g^2 log g-O_q(g^2))." }, { "id": 20000119, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0119", "title": "Wild-degree supersingular Klein-loop hypersurfaces", "statement": "Is there a supersingular hypersurface of every degree, dimension and char. $p$?", "original_statement": "Is there a supersingular hypersurface of every degree, dimension and char. $p$?", "clean_statement": "Is there a supersingular hypersurface of every degree, dimension and char. $p$?", "statement_status": "exact", "statement_verification": "The canonical AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.2\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[118]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a supersingular hypersurface of every degree, dimension and char. $p$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0119", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For d,m >= 3, p dividing d, and p not dividing m, the Klein-loop hypersurface sum_i x_i^(d-1)x_(i+1)=0 in projective (m-1)-space is smooth. Put M=((d-1)^m-(-1)^m)/d. If p^nu is congruent to -1 modulo M for some nu, a prime-to-p projective monomial cover by the Fermat hypersurface of degree M, resolved toroidally and combined with pullback/trace, proves that the loop hypersurface is slope-supersingular; in even dimension its middle cohomology is algebraic. Explicit consequences are a supersingular smooth plane quintic of genus 6 in characteristic 5 and a supersingular smooth quartic threefold in characteristic 2.\n\nCandidate contribution (theorem; novelty confidence low): Candidate wild-loop criterion: if p divides d, p does not divide m, and p^nu is -1 modulo M=((d-1)^m-(-1)^m)/d, then the degree-d Klein-loop hypersurface in projective (m-1)-space is smooth and slope-supersingular. In particular, x0^4*x1+x1^4*x2+x2^4*x0=0 is a supersingular genus-6 plane quintic in characteristic 5, and sum_i x_i^3*x_(i+1)=0 in P^4 is a supersingular quartic threefold in characteristic 2." }, { "id": 20000120, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0120", "title": "A sharp failure of supersingularity beyond the q-bic locus", "statement": "Find other examples of supersingular hypersurfaces.", "original_statement": "Find other examples of supersingular hypersurfaces.", "clean_statement": "Find other examples of supersingular hypersurfaces.", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.25\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[119]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find other examples of supersingular hypersurfaces.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0120", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over an algebraically closed field of characteristic 2, the plane quintic C_lambda: x^5+y^5+z^5+lambda xyz(xy+xz+yz)=0 is smooth exactly when lambda^3 is not 1. Its lambda=0 Fermat member is supersingular, whereas every nonzero smooth member has exact 2-rank 3 and is therefore not supersingular. Nevertheless, the Gauss map of every smooth member factors through Frobenius. This gives a rigorous boundary counterexample to extending q-bic supersingularity to all smooth Frobenius-gradient hypersurfaces.\n\nCandidate contribution (counterexample with exact parameter criterion; novelty confidence low): The displayed characteristic-2 quintic pencil has smoothness locus lambda^3 != 1, supersingular central member, and exact stable Hasse-Witt rank 3 at every nonzero smooth parameter, although every smooth member has a Frobenius-factorized Gauss map." }, { "id": 20000121, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0121", "title": "A separability firewall for rationally connected surfaces", "statement": "Is being rationally connected = unirational in char $p$?", "original_statement": "Is being rationally connected = unirational in char $p$?", "clean_statement": "Is being rationally connected = unirational in char $p$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 9.3 in the workshop *Rational subvarieties in positive characteristic*. The statement is exactly", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.3\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[120]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is being rationally connected = unirational in char $p$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0121", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth projective integral surface over an algebraically closed field of characteristic p>0, rationality, separable unirationality, separable rational connectedness, rational chain connectedness plus separable uniruledness, and rational chain connectedness plus Kodaira dimension minus infinity are equivalent. Consequently, an RCC nonrational surface has trivial Albanese, Kodaira dimension at least zero, no separable uniruuling, and every existing dominant generically finite rational parametrization from P^2 is inseparable. This rigorously isolates, but does not solve, the weak inseparable residue of RC implies unirational.\n\nCandidate contribution (equivalence_and_counterexample_filter; novelty confidence low): The five-way surface equivalence and its inseparability-firewall corollary package standard surface-classification inputs into a testable filter: every smooth projective RCC nonrational surface over an algebraically closed field of characteristic p>0 has kappa at least zero, trivial Albanese, no separable uniruuling, and only generically inseparable rational parametrizations, if any." }, { "id": 20000122, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0122", "title": "Literal projective-plane example and a logarithmic BMY obstruction", "statement": "Can a surface with $c_{1}^{2}=3c_{2}$ contain a rational curve in char. $p$?", "original_statement": "Can a surface with $c_{1}^{2}=3c_{2}$ contain a rational curve in char. $p$?", "clean_statement": "Can a surface with $c_{1}^{2}=3c_{2}$ contain a rational curve in char. $p$?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.35\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[121]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can a surface with $c_{1}^{2}=3c_{2}$ contain a rational curve in char. $p$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0122", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The question as literally written has an affirmative answer in every characteristic: projective space P^2 satisfies c1^2=9=3c2 and contains lines. For the intended smooth minimal general-type reading, a proved partial theorem is obtained in characteristic p at least 3. If C is an integral rational curve on a smooth minimal surface X with K_X nef and K_X^2=3c2(X), and if an embedded-resolution SNC pair (Y,D) of (X,C) lifts compatibly to W_2(k), then K_X.C is at most q_pi+3b-2delta-4, where delta=p_a(C), b is the total branch excess, and q_pi is the explicit sum of squared blowup discrepancies. Thus C cannot be smooth. If all singularities are ordinary multiple points and s is their number, then K_X.C is at most s-4, so s is at least 4 for K_X nef and at least 5 for K_X ample.\n\nCandidate contribution (obstruction; novelty confidence low): For a W_2-liftable resolved log pair in characteristic p at least 3, the equality-specialized logarithmic BMY defect is exactly 3c2(Omega_Y^1(log D))-(K_Y+D)^2=q_pi+3b-2delta-4-K_X.C; consequently each ordinary r-fold singularity contributes exactly one and K_X.C is at most the number of singular points minus 4." }, { "id": 20000123, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0123", "title": "Fixed points in dimension two and wild fixed-scheme multiplicities", "statement": "Does an automorphism of a separably rationally connected variety necessarily have a fixed point?", "original_statement": "Does an automorphism of a separably rationally connected variety necessarily have a fixed point?", "clean_statement": "Does an automorphism of a separably rationally connected variety necessarily have a fixed point?", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 9.4 in the workshop *Rational subvarieties in positive characteristic*. Its complete text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.4\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[122]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does an automorphism of a separably rationally connected variety necessarily have a fixed point?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0123", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over an algebraically closed field of arbitrary characteristic, every endomorphism of a smooth projective geometrically connected separably rationally connected variety of dimension at most two has a fixed rational point. The proof combines the empty-fixed-locus case of the coherent Woods Hole trace formula, Gounelas's H^1(O)-vanishing, and vanishing of top forms along a very free curve. In addition, an isolated fixed scheme of a p-power-order automorphism in characteristic p is a zero-dimensional local complete intersection whose length is the graph-diagonal Lefschetz number, is congruent modulo p to the etale Euler characteristic, and is nonreduced at every fixed point.\n\nCandidate contribution (theorem_package; novelty confidence low): Candidate synthesis: every endomorphism, not only every automorphism, of a smooth projective SRC variety of dimension at most two over an algebraically closed field has a fixed point in arbitrary characteristic; separately, every zero-dimensional fixed scheme of a p-power-order automorphism has length congruent modulo p to the etale Euler characteristic and has no reduced local component." }, { "id": 20000124, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0124", "title": "The E8 quadric and the geometric boundary", "statement": "Is there an irrational smooth proper variety over $\\text{Spec}\\mathbb{Z}$? (Can we find such a hypersurface?)", "original_statement": "Is there an irrational smooth proper variety over $\\text{Spec}\\mathbb{Z}$? (Can we find such a hypersurface?)", "clean_statement": "Is there an irrational smooth proper variety over $\\text{Spec}\\mathbb{Z}$? (Can we find such a hypersurface?)", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Rational subvarieties in positive characteristic*, “Other problems,” 9.45) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.45\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[123]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there an irrational smooth proper variety over $\\\\text{Spec}\\\\mathbb{Z}$? (Can we find such a hypersurface?)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0124", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the arithmetic reading, an explicit positive-definite E8 quadric in projective 7-space over Z is smooth at every prime, has no real or rational point, and is therefore not Q-unirational or stably Q-rational; it is nevertheless geometrically rational. For the stronger geometric reading, every smooth geometrically integral hypersurface over Z of relative dimension at most 3 has geometrically rational generic fiber, so any geometrically irrational hypersurface example must have relative dimension at least 4.\n\nCandidate contribution (reduction; novelty confidence low): Any smooth projective hypersurface over Z with geometrically integral fibers and relative dimension at most 3 has geometrically rational generic fiber; hence the search for a geometrically irrational hypersurface over Spec Z can begin only in relative dimension 4." }, { "id": 20000125, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0125", "title": "Smooth hypersurfaces beyond the Chevalley--Warning range", "statement": "For every $p, d$ with $d\\geq n+1$, can we find hypersurface over $\\mathbb{F}_{p}$ of degree $d$ in $\\mathbb{P}^{n}$\nsuch that either $\\# X \\not\\equiv 1 \\text{mod} p$ or $X$ is not rationally connected?", "original_statement": "For every $p, d$ with $d\\geq n+1$, can we find hypersurface over $\\mathbb{F}_{p}$ of degree $d$ in $\\mathbb{P}^{n}$\nsuch that either $\\# X \\not\\equiv 1 \\text{mod} p$ or $X$ is not rationally connected?", "clean_statement": "For every $p, d$ with $d\\geq n+1$, can we find hypersurface over $\\mathbb{F}_{p}$ of degree $d$ in $\\mathbb{P}^{n}$\nsuch that either $\\# X \\not\\equiv 1 \\text{mod} p$ or $X$ is not rationally connected?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.5\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[124]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For every $p, d$ with $d\\\\geq n+1$, can we find hypersurface over $\\\\mathbb{F}_{p}$ of degree $d$ in $\\\\mathbb{P}^{n}$\\nsuch that either $\\\\# X \\\\not\\\\equiv 1 \\\\text{mod} p$ or $X$ is not rationally connected?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: medium.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0125", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the intended smooth formulation, the problem is solved for every plane curve case n=2, for every sufficiently large degree with p and n fixed, for an explicit infinite range with n+1 <= p-1 and (p-1) dividing d but p not dividing d, and at the Calabi--Yau boundary when n+1 divides p-1. The report also proves the exact boundary congruence #X(F_p) = 1 + (-1)^(n+1)[(x_0...x_n)^(p-1)]F^(p-1) modulo p and computes the positive Poonen density of smooth point-free hypersurfaces.\n\nCandidate contribution (classification; novelty confidence medium): For p-1 dividing d and p not dividing d, a smooth diagonal hypersurface sum a_i x_i^d=0 with all a_i nonzero is point-free over F_p if and only if the coefficient sequence has no nonempty zero-sum subset; consequently such a point-free diagonal member exists if and only if n+1 <= p-1." }, { "id": 20000126, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0126", "title": "An integral finite-motive criterion and new product-quotient examples", "statement": "Can we check Bloch's conjecture for known examples of super-singular surfaces?", "original_statement": "Can we check Bloch's conjecture for known examples of super-singular surfaces?", "clean_statement": "Can we check Bloch's conjecture for known examples of super-singular surfaces?", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 9.55 in the “Other problems” section of the 2016 workshop *Rational subvarieties in positive characteristic*. Its entire problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.55\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[125]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we check Bloch's conjecture for known examples of super-singular surfaces?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0126", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth projective Shioda-supersingular surface over an algebraically closed field of characteristic p>0, Kimura finite-dimensionality of its rational Chow motive implies that its Albanese kernel vanishes integrally after every algebraically closed scalar extension. Product domination is a sufficient finite-dimensionality condition. Applying this to every good Shioda-supersingular reduction in Church's D6, D7, and A4 product-quotient families gives CH_0(S_K)=Z for every algebraically closed extension K and the rational Tate-motive decomposition 1 plus b_2 copies of the Lefschetz motive plus its square, with b_2 equal to 126, 109, and 116 respectively.\n\nCandidate contribution (corollary; novelty confidence low): Every good Shioda-supersingular reduction in Church's D6, D7, and A4 product-quotient families has rational Chow motive 1 plus L^{b_2} plus L^2 and satisfies CH_0(S_K)=Z for every algebraically closed scalar extension K, with b_2=126, 109, and 116 respectively." }, { "id": 20000127, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0127", "title": "Boundary contact of rational curves in characteristic p", "statement": "Let $(X, D)$ snc.\nHow can rational curves in $X$ meet $D$?\nAre there any restrictions?", "original_statement": "Let $(X, D)$ snc.\nHow can rational curves in $X$ meet $D$?\nAre there any restrictions?", "clean_statement": "Let $(X, D)$ snc.\nHow can rational curves in $X$ meet $D$?\nAre there any restrictions?", "statement_status": "exact", "statement_verification": "The canonical AIM record is number 9.6 in the section “Other problems” of the workshop *Rational subvarieties in positive characteristic*. Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.6\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[126]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $(X, D)$ snc.\\nHow can rational curves in $X$ meet $D$?\\nAre there any restrictions?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0127", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a generically separable map f: P^1 -> (X,D) not contained in an SNC boundary in characteristic p, let r be the number of boundary source points, w the number whose entire contact vector is divisible by p, and a_max the largest Birkhoff-Grothendieck summand of f^*T_X(-log D). Saturating the logarithmic differential proves 2-r+w <= deg L <= a_max, hence r-w >= 2-a_max. For the full coordinate boundary of P^n, all equal-degree effective contact divisors with empty common support are realizable, and the map is inseparable exactly when every contact multiplicity is divisible by p; consequently every separable map has at least two tame contacts, and explicit examples attain equality.\n\nCandidate contribution (inequality; novelty confidence low): The candidate new statement is the wild-contact correction 2-r+w <= deg L <= a_max for the saturated image L of the logarithmic differential, together with the sharp inequality r-w >= 2-a_max." }, { "id": 20000128, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0128", "title": "Coprime alteration degrees reduce to one prime-to-p alteration", "statement": "Can we get alterations of coprime degrees?", "original_statement": "Can we get alterations of coprime degrees?", "clean_statement": "Can we get alterations of coprime degrees?", "statement_status": "exact", "statement_verification": "The canonical AIM record is number 9.65 in the section “Other problems” of the workshop list *Rational subvarieties in positive characteristic*. Its complete problem field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.65\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[127]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we get alterations of coprime degrees?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0128", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an integral characteristic-p target with optional SNC boundary in the exact range of Temkin's Theorem 1.2.5, let D be the degrees of projective alterations with integral regular source and SNC inverse boundary. Temkin supplies one member p^r. Consequently, the existence of two coprime degrees, a finite family with gcd one, a single degree prime to p, and gcd(D)=1 are equivalent. Gabber's theorem and an explicit finite prime-elimination argument yield only a finite family whose gcd is a p-power. Thus the general remaining gap is precisely one prime-to-p regular alteration; the curve case is solved by degree-one normalization.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: in Temkin's exact char-altered range, the pair and finite-family readings of the AIM question are both equivalent to the existence of one prime-to-p boundary-compatible regular alteration, or equivalently to vanishing of e(X,Z)=v_p(gcd D).", "research_difficulty_suggested": "L4" }, { "id": 20000129, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0129", "title": "Uniform Frobenius-summand ranks for arithmetic curves", "statement": "Characterize a variety $X$ over $\\text{Spec}\\mathbb{Z}$ such that the indecomposable summands of $F_{*}\\mathcal{O}_{X}$ have universally bounded rank.", "original_statement": "Characterize a variety $X$ over $\\text{Spec}\\mathbb{Z}$ such that the indecomposable summands of $F_{*}\\mathcal{O}_{X}$ have universally bounded rank.", "clean_statement": "Characterize a variety $X$ over $\\text{Spec}\\mathbb{Z}$ such that the indecomposable summands of $F_{*}\\mathcal{O}_{X}$ have universally bounded rank.", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 9.7 in the section “Other problems” of the 2016 workshop *Rational subvarieties in positive characteristic*. Its exact statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.7\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[128]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Characterize a variety $X$ over $\\\\text{Spec}\\\\mathbb{Z}$ such that the indecomposable summands of $F_{*}\\\\mathcal{O}_{X}$ have universally bounded rank.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0129", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After repairing the mixed-characteristic wording by taking geometric closed fibers of a smooth projective model over Spec Z[1/N], the report proves an exact classification for relative curves: indecomposable ranks in the first Frobenius pushforward are uniformly bounded across primes if and only if the relative genus is zero, and the same equivalence holds when all Frobenius iterates are included; the optimal positive bound is 1. It also proves the exact abelian-fiber formula b_{p,e}=p^{e(d-r_p)}, where r_p is the geometric p-rank, and records smooth projective toric models as uniform bound-1 examples.\n\nCandidate contribution (classification_theorem; novelty confidence low): For every smooth projective geometrically connected relative curve over a nonempty open of Spec Z, geometric indecomposable ranks in F_*O are uniformly bounded over closed fibers exactly in genus zero; this remains equivalent when all F^e are included, and the optimal bound is 1." }, { "id": 20000130, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0130", "title": "Weak finite-set interpolation on unirational surfaces", "statement": "If $X$ is separably rationally connected, there exists $N(\\text{dim} X, \\text{deg} X, m)$ such that,\nfor any $p_{1},\\cdots, p_{m}\\in X(\\mathbb{F}_{p})$, there exists a rational curve $C$ defined over $\\mathbb{F}_{p^{N}}$ containing all $p_{i}$. (due to Kollar)\nCan you do this without lifting the cardinality of field tend to $\\infty$?\nLet $X$ be a cubic surface over $\\mathbb{F}_{q}$.\nCan you find a rational curve defined over $\\mathbb{F}_{q}$ through each point?", "original_statement": "If $X$ is separably rationally connected, there exists $N(\\text{dim} X, \\text{deg} X, m)$ such that,\nfor any $p_{1},\\cdots, p_{m}\\in X(\\mathbb{F}_{p})$, there exists a rational curve $C$ defined over $\\mathbb{F}_{p^{N}}$ containing all $p_{i}$. (due to Kollar)\nCan you do this without lifting the cardinality of field tend to $\\infty$?\nLet $X$ be a cubic surface over $\\mathbb{F}_{q}$.\nCan you find a rational curve defined over $\\mathbb{F}_{q}$ through each point?", "clean_statement": "If $X$ is separably rationally connected, there exists $N(\\text{dim} X, \\text{deg} X, m)$ such that,\nfor any $p_{1},\\cdots, p_{m}\\in X(\\mathbb{F}_{p})$, there exists a rational curve $C$ defined over $\\mathbb{F}_{p^{N}}$ containing all $p_{i}$. (due to Kollar)\nCan you do this without lifting the cardinality of field tend to $\\infty$?\nLet $X$ be a cubic surface over $\\mathbb{F}_{q}$.\nCan you find a rational curve defined over $\\mathbb{F}_{q}$ through each point?", "statement_status": "exact", "statement_verification": "The canonical record is problem 9.75, “Other problems,” from the AIM workshop *Rational subvarieties in positive characteristic*. Its text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.75\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[129]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If $X$ is separably rationally connected, there exists $N(\\\\text{dim} X, \\\\text{deg} X, m)$ such that,\\nfor any $p_{1},\\\\cdots, p_{m}\\\\in X(\\\\mathbb{F}_{p})$, there exists a rational curve $C$ defined over $\\\\mathbb{F}_{p^{N}}$ containing all $p_{i}$. (due to Kollar)\\nCan you do this without lifting the cardinality of field tend to $\\\\infty$?\\nLet $X$ be a cubic surface over $\\\\mathbb{F}_{q}$.\\nCan you find a rational curve defined over $\\\\mathbb{F}_{q}$ through each point?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0130", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over any finite field, every finite set of closed points on a projective unirational surface is contained set-theoretically in one geometrically integral rational curve whose normalization is P^1. Consequently, for every smooth cubic surface X/F_q and every q, one F_q-defined rational curve contains all of X(F_q). This proves the weak set-theoretic reading of the cubic question but does not provide rational marked preimages, a smooth curve, or a degree bound; Kollár's stronger marked interpolation theorem is verified for q at least 8.\n\nCandidate contribution (theorem; novelty confidence low): Finite-jet interpolation on P^2, combined with finite determination of arc centers through a fixed sequence of point blowups, places any prescribed finite set of closed points on a unirational surface over F_q on one geometrically integral F_q-rational curve." }, { "id": 20000131, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0131", "title": "Finite étale and tame obstruction collapse for SRC varieties", "statement": "For separably rationally connected varieties, is the Brauer-Manin obstruction the only obstruction for the Hasse principle?", "original_statement": "For separably rationally connected varieties, is the Brauer-Manin obstruction the only obstruction for the Hasse principle?", "clean_statement": "For separably rationally connected varieties, is the Brauer-Manin obstruction the only obstruction for the Hasse principle?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List problem 9.8 from the workshop *Rational subvarieties in positive characteristic*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Rational subvarieties in positive characteristic\nSection: Other problems\nSource item: 9.8\nSource URL: http://aimpl.org/ratsubvarpos/9/\nCanonical location: aim-algebraic-geometry-notes.json notes[130]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For separably rationally connected varieties, is the Brauer-Manin obstruction the only obstruction for the Hasse principle?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/ratsubvarpos/9/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0131", "aim-domain:algebraic-geometry", "aim-workshop:ratsubvarpos", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth proper geometrically integral variety X over any global field K with separably rationally connected geometric fiber, pullback H^1(K,G) to H^1(X,G) is bijective for every finite étale K-group scheme G, and the finite étale-Brauer set equals the ordinary Brauer-Manin set. In characteristic p this exhausts all prime-to-p finite-flat torsors and forces geometric Picard torsion into the p-primary part. Separately, if Br(X_bar)=0 and Pic(X_bar) is finitely generated torsion-free, the ordinary Brauer quotient embeds in a finite H^1(K,Pic); it vanishes when Pic is a permutation lattice. These are proved reductions, not a solution of the still-open rational-point conjecture.\n\nCandidate contribution (reduction; novelty confidence low): Candidate obstruction sieve: on every smooth proper SRC variety over a global field, finite étale-Brauer equals ordinary Brauer-Manin; over characteristic p, all prime-to-p finite-flat descent and all prime-to-p geometric Picard torsion disappear, so any finite-flat refinement outside this equality must involve order divisible by p." }, { "id": 20000132, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0132", "title": "The surface cone of a split rank-three bundle over the projective plane", "statement": "Effective cones of projective bundles\n\nIn \\cite{MR2836078} Fulger showed how to compute the effective cone of $k$-cycles $\\text{Eff}^k(\\mathbb{P}(\\mathcal{E})$ of a projective bundle over a smooth curve. In this case the answer can be given in terms of data coming from the Harder-Narasimhan filtration of $\\mathcal{E}$. However, even in the case where $X$ is a $\\mathbb{P}^2$-bundles over a surface $S$, it is not known how to compute $\\text{Eff}_2(X)$. It is natural to first consider the case where $S$ is the projective plane or a K3 surface.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over $\\mathbb{P}^2$?", "original_statement": "Effective cones of projective bundles\n\nIn \\cite{MR2836078} Fulger showed how to compute the effective cone of $k$-cycles $\\text{Eff}^k(\\mathbb{P}(\\mathcal{E})$ of a projective bundle over a smooth curve. In this case the answer can be given in terms of data coming from the Harder-Narasimhan filtration of $\\mathcal{E}$. However, even in the case where $X$ is a $\\mathbb{P}^2$-bundles over a surface $S$, it is not known how to compute $\\text{Eff}_2(X)$. It is natural to first consider the case where $S$ is the projective plane or a K3 surface.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over $\\mathbb{P}^2$?", "clean_statement": "Effective cones of projective bundles\n\nIn \\cite{MR2836078} Fulger showed how to compute the effective cone of $k$-cycles $\\text{Eff}^k(\\mathbb{P}(\\mathcal{E})$ of a projective bundle over a smooth curve. In this case the answer can be given in terms of data coming from the Harder-Narasimhan filtration of $\\mathcal{E}$. However, even in the case where $X$ is a $\\mathbb{P}^2$-bundles over a surface $S$, it is not known how to compute $\\text{Eff}_2(X)$. It is natural to first consider the case where $S$ is the projective plane or a K3 surface.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over $\\mathbb{P}^2$?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.1, “Effective cones of projective bundles,” from the 2016 AIM workshop *Positivity of cycles*. The archived AIM page gives the following context and question:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Computing higher codimension effective and nef cones in explicit examples\nSource item: 1.1\nSource URL: http://aimpl.org/poscycles/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[131]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Effective cones of projective bundles\\n\\nIn \\\\cite{MR2836078} Fulger showed how to compute the effective cone of $k$-cycles $\\\\text{Eff}^k(\\\\mathbb{P}(\\\\mathcal{E})$ of a projective bundle over a smooth curve. In this case the answer can be given in terms of data coming from the Harder-Narasimhan filtration of $\\\\mathcal{E}$. However, even in the case where $X$ is a $\\\\mathbb{P}^2$-bundles over a surface $S$, it is not known how to compute $\\\\text{Eff}_2(X)$. It is natural to first consider the case where $S$ is the projective plane or a K3 surface.\\n\\nWhat is the effective cone of surfaces $\\\\text{Eff}_2(X)$ where $X$ is a $\\\\mathbb{P}^2$-bundle over $\\\\mathbb{P}^2$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0132", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the quotient projectivization X=P_{P2}(O(a)⊕O(b)⊕O(c)) with a≤b≤c, the report proves that the closed effective cone of surfaces is the simplicial cone generated by h^2, h(xi-c h), and (xi-b h)(xi-c h). It enumerates all fifteen torus-invariant surfaces, proves that every effective surface degenerates nonnegatively to invariant surfaces, removes every redundant ray, and derives the exact inequalities x≥0, y+(b+c)x≥0, and z+c y+c^2 x≥0 for a class x xi^2+y h xi+z h^2. Using the actual projective-bundle intersection pairing, it also proves that Nef^2(X) is generated by h^2, h(xi-a h), and (xi-a h)(xi-b h).\n\nCandidate contribution (explicit_cone_computation; novelty confidence low): For every ordered triple a≤b≤c, the split rank-three projective bundle over P2 has exactly three extremal effective surface rays h^2, h(xi-c h), and (xi-b h)(xi-c h), with the stated half-space inequalities and opposite-flag intersection-dual nef cone." }, { "id": 20000133, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0133", "title": "The surface cone for split projective-plane bundles over K3 surfaces", "statement": "What is the effective cone of surfaces $\\text{Eff}^2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over a K3 surface?", "original_statement": "What is the effective cone of surfaces $\\text{Eff}^2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over a K3 surface?", "clean_statement": "What is the effective cone of surfaces $\\text{Eff}^2(X)$ where $X$ is a $\\mathbb{P}^2$-bundle over a K3 surface?", "statement_status": "exact", "statement_verification": "The canonical AIM record is `AIM-ALGEBRAIC_GEOMETRY-0133` in `aim-algebraic-geometry-notes.json`, zero-based source index 132 (Positivity of Cycles, problem 1.2). It asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Computing higher codimension effective and nef cones in explicit examples\nSource item: 1.2\nSource URL: http://aimpl.org/poscycles/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[132]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the effective cone of surfaces $\\\\text{Eff}^2(X)$ where $X$ is a $\\\\mathbb{P}^2$-bundle over a K3 surface?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0133", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a split rank-three bundle E=L_1 direct-sum L_2 direct-sum L_3 on a smooth projective surface S, the pseudo-effective surface cone of X=P_S(E) is exactly the sum of the fiber ray, the three images (xi-pi*c_1(L_i))pi*Effbar_1(S), and the three coordinate-section rays. A relative-torus degeneration and invariant-prime classification prove the generator statement, while an ample-degree compact-slice argument proves that this finite Minkowski sum is closed. If S is a Picard-rank-one K3 surface with c_1(L_i)=a_iH, a_1<=a_2<=a_3, H^2=d, and h=pi*H, the cone is the simplicial three-ray cone generated by f, (xi-a_3h)h, and (xi-a_2h)(xi-a_3h). The general unsplit rank-three case is not solved.\n\nCandidate contribution (explicit_numerical_specialization; novelty confidence low): Candidate novelty is the explicit closed numerical cone formula for split rank-three bundles, including the ample-functional proof of closedness, and its reduction to the three stated extremal rays for every Picard-rank-one projective K3 surface." }, { "id": 20000134, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0134", "title": "Torus formula for effective cycles on split projective bundles", "statement": "If $\\mathcal{E}\\cong \\mathcal{L}_1 \\oplus ... \\oplus \\mathcal{L}_r$ is a split vector bundle on a smooth projective surface $S$, what are the effective cones of k-cycles $\\text{Eff}_k(\\mathbb{P}(\\mathcal{E}))$?", "original_statement": "If $\\mathcal{E}\\cong \\mathcal{L}_1 \\oplus ... \\oplus \\mathcal{L}_r$ is a split vector bundle on a smooth projective surface $S$, what are the effective cones of k-cycles $\\text{Eff}_k(\\mathbb{P}(\\mathcal{E}))$?", "clean_statement": "If $\\mathcal{E}\\cong \\mathcal{L}_1 \\oplus ... \\oplus \\mathcal{L}_r$ is a split vector bundle on a smooth projective surface $S$, what are the effective cones of k-cycles $\\text{Eff}_k(\\mathbb{P}(\\mathcal{E}))$?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.3 in the section “Computing higher codimension effective and nef cones in explicit examples” from the 2016 AIM workshop *Positivity of cycles*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Computing higher codimension effective and nef cones in explicit examples\nSource item: 1.3\nSource URL: http://aimpl.org/poscycles/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[133]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If $\\\\mathcal{E}\\\\cong \\\\mathcal{L}_1 \\\\oplus ... \\\\oplus \\\\mathcal{L}_r$ is a split vector bundle on a smooth projective surface $S$, what are the effective cones of k-cycles $\\\\text{Eff}_k(\\\\mathbb{P}(\\\\mathcal{E}))$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0134", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For a split bundle E = direct sum of line bundles on a smooth projective surface S, every effective cycle on P_S(E) degenerates, without changing its numerical class, to a positive sum of coordinate projective subbundles over effective cycles on S. Consequently, in every dimension k the effective cone is the sum of the maps Gamma_{I,d}(Eff_d(S)) with |I| = k-d+1, and the analogous finite sum of pseudoeffective cones is already closed. The report specializes this to an explicit three-line surface formula and proves the result for both raw effective and closed pseudoeffective cones.\n\nCandidate contribution (corollary; novelty confidence low): For rank at least two, pullback identifies the pseudoeffective curve cone of S with the exposed fiber-degree-zero face of the pseudoeffective divisor cone of P_S(E); hence that base cone is (rational) polyhedral if and only if the divisor cone is, equivalently if and only if every cycle cone of P_S(E) is. Base extremal rays remain extremal vertical divisor rays, and an elementary-symmetric-class test detects coincident coordinate generators." }, { "id": 20000135, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0135", "title": "Extreme effective cones and stable Schubert bounds for Plucker hypersurfaces", "statement": "Effective cones of hypersurfaces\n\nFor very basic varieties such as $Gr(m,n)$ or $\\mathbb{P}^n\\times \\mathbb{P}^m$ the effective cones of $k$-cycles can be computed. It then natural to ask if it is possible to compute the effective cones of very general hypersurfaces in these varieties.\n\nIf $X\\subset Gr(m,n)$ is a very general hypersurface, what is the effective cone of $k$-cycles $\\text{Eff}_k(K)$?", "original_statement": "Effective cones of hypersurfaces\n\nFor very basic varieties such as $Gr(m,n)$ or $\\mathbb{P}^n\\times \\mathbb{P}^m$ the effective cones of $k$-cycles can be computed. It then natural to ask if it is possible to compute the effective cones of very general hypersurfaces in these varieties.\n\nIf $X\\subset Gr(m,n)$ is a very general hypersurface, what is the effective cone of $k$-cycles $\\text{Eff}_k(K)$?", "clean_statement": "Effective cones of hypersurfaces\n\nFor very basic varieties such as $Gr(m,n)$ or $\\mathbb{P}^n\\times \\mathbb{P}^m$ the effective cones of $k$-cycles can be computed. It then natural to ask if it is possible to compute the effective cones of very general hypersurfaces in these varieties.\n\nIf $X\\subset Gr(m,n)$ is a very general hypersurface, what is the effective cone of $k$-cycles $\\text{Eff}_k(K)$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List item 1.4 from the workshop *Positivity of cycles*. Its final sentence is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Computing higher codimension effective and nef cones in explicit examples\nSource item: 1.4\nSource URL: http://aimpl.org/poscycles/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[134]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Effective cones of hypersurfaces\\n\\nFor very basic varieties such as $Gr(m,n)$ or $\\\\mathbb{P}^n\\\\times \\\\mathbb{P}^m$ the effective cones of $k$-cycles can be computed. It then natural to ask if it is possible to compute the effective cones of very general hypersurfaces in these varieties.\\n\\nIf $X\\\\subset Gr(m,n)$ is a very general hypersurface, what is the effective cone of $k$-cycles $\\\\text{Eff}_k(K)$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0135", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every smooth complex Plucker hypersurface X of dimension m at least 3 in Gr(r,n), the pseudoeffective cones of points, curves, divisors, and top-dimensional cycles are the rays generated by [pt], H_X^(m-1), H_X, and [X]. If 2km excludes every nonzero raw effective n-cycle class on the ray [P^n x point], with the symmetric statement for the other factor. In particular, on a very general (2,2)-hypersurface in P^2 x P^4, the entire nonzero raw surface ray [P^2 x point] is absent, although its membership in the pseudoeffective closure remains open.\n\nCandidate contribution (obstruction; novelty confidence low): For a very general X_(a,b) in P^n x P^m, if binomial(a+n,n)>m then i_*Eff_n^alg(X) has empty intersection with the nonzero ray R_{>0}[P^n x point]; symmetrically, binomial(b+m,m)>n excludes R_{>0}[point x P^m]. Thus a very general X_(2,2) in P^2 x P^4 has no nonzero raw effective surface class on the Lambda_2 ray." }, { "id": 20000137, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0137", "title": "A high-degree curve-cone counterexample in the weak-Lefschetz range", "statement": "In the above cases, assuming $i:X\\hookrightarrow \\mathbb{P}^n\\times\\mathbb{P}^m$ or $i:X\\hookrightarrow Gr(k,n)$ is ample then the Lefschetz hyperplane theorems imply that $i_*:N_k(X) \\rightarrow N_k(Gr(k,n))$ and $i_*:N_k(X) \\rightarrow N_k(\\mathbb{P}^n\\times\\mathbb{P}^m)$ are isomorphisms for $k<\\text{dim}(X)/2$. It is natural to ask if there is any Lefschetz type phenomena for effective cones of very general hypersurfaces of sufficiently high degree.\n\nIf $X\\subset Y$ is a very general ample divisor of sufficiently high degree, does the pushforward map $i_*: N_k(X) \\rightarrow N_k(Y)$ induce an isomorphism of effective cones for $k$ sufficiently small?", "original_statement": "In the above cases, assuming $i:X\\hookrightarrow \\mathbb{P}^n\\times\\mathbb{P}^m$ or $i:X\\hookrightarrow Gr(k,n)$ is ample then the Lefschetz hyperplane theorems imply that $i_*:N_k(X) \\rightarrow N_k(Gr(k,n))$ and $i_*:N_k(X) \\rightarrow N_k(\\mathbb{P}^n\\times\\mathbb{P}^m)$ are isomorphisms for $k<\\text{dim}(X)/2$. It is natural to ask if there is any Lefschetz type phenomena for effective cones of very general hypersurfaces of sufficiently high degree.\n\nIf $X\\subset Y$ is a very general ample divisor of sufficiently high degree, does the pushforward map $i_*: N_k(X) \\rightarrow N_k(Y)$ induce an isomorphism of effective cones for $k$ sufficiently small?", "clean_statement": "In the above cases, assuming $i:X\\hookrightarrow \\mathbb{P}^n\\times\\mathbb{P}^m$ or $i:X\\hookrightarrow Gr(k,n)$ is ample then the Lefschetz hyperplane theorems imply that $i_*:N_k(X) \\rightarrow N_k(Gr(k,n))$ and $i_*:N_k(X) \\rightarrow N_k(\\mathbb{P}^n\\times\\mathbb{P}^m)$ are isomorphisms for $k<\\text{dim}(X)/2$. It is natural to ask if there is any Lefschetz type phenomena for effective cones of very general hypersurfaces of sufficiently high degree.\n\nIf $X\\subset Y$ is a very general ample divisor of sufficiently high degree, does the pushforward map $i_*: N_k(X) \\rightarrow N_k(Y)$ induce an isomorphism of effective cones for $k$ sufficiently small?", "statement_status": "exact", "statement_verification": "The canonical record is item 1.6 in the AIM workshop list *Positivity of cycles*, section “Computing higher codimension effective and nef cones in explicit examples.” The exact question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Computing higher codimension effective and nef cones in explicit examples\nSource item: 1.6\nSource URL: http://aimpl.org/poscycles/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[136]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In the above cases, assuming $i:X\\\\hookrightarrow \\\\mathbb{P}^n\\\\times\\\\mathbb{P}^m$ or $i:X\\\\hookrightarrow Gr(k,n)$ is ample then the Lefschetz hyperplane theorems imply that $i_*:N_k(X) \\\\rightarrow N_k(Gr(k,n))$ and $i_*:N_k(X) \\\\rightarrow N_k(\\\\mathbb{P}^n\\\\times\\\\mathbb{P}^m)$ are isomorphisms for $k<\\\\text{dim}(X)/2$. It is natural to ask if there is any Lefschetz type phenomena for effective cones of very general hypersurfaces of sufficiently high degree.\\n\\nIf $X\\\\subset Y$ is a very general ample divisor of sufficiently high degree, does the pushforward map $i_*: N_k(X) \\\\rightarrow N_k(Y)$ induce an isomorphism of effective cones for $k$ sufficiently small?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0137", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Let n >= 3, let Y = P^1 x P^n, and fix any ample A = O_Y(a,b). For every sufficiently large t and a very general X_t in |tA|, weak Lefschetz gives an isomorphism i_*: N_1(X_t) -> N_1(Y), but i_* overline{Eff}_1(X_t) is the strict cone generated by l_2 and nb l_1 + a l_2, whereas overline{Eff}_1(Y) is generated by l_1 and l_2. Thus the proposed high-degree effective-cone Lefschetz phenomenon fails for q = 1, the smallest positive cycle dimension, even though 2q < dim(X_t); only the trivial q = 0 ray remains universally equal.\n\nCandidate contribution (explicit counterexample corollary; novelty confidence low): For every fixed ample ray O(a,b) on P^1 x P^n with n >= 3, all sufficiently large very general multiples have pushed pseudoeffective curve cone R_{>=0} l_2 + R_{>=0}(nb l_1 + a l_2), so the ambient P^1-fiber ray is excluded uniformly as degree tends to infinity." }, { "id": 20000138, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0138", "title": "The Hilbert--Chow zero-degree face of the surface cone of a Hilbert square", "statement": "Effective cones of Hilbert schemes and Moduli Spaces\n\nIt is natural to try to compute effective cones in the case of Hilbert schemes or moduli spaces. One natural Hilbert scheme to consider first is the Hilbert scheme of length $n$ subscheme of a smooth surface $S$, denoted by $S^{[n]}$. A great deal is known about these Hilbert schemes. In recent years there has been a lot of exciting work on the effective cones of divisors and curves of the Hilbert schemes $\\text{Eff}^1(S^{[n]})$ and $\\text{Eff}_1(S^{[n]})$. See for example Bayer and Macri's work on Hilbert schemes of K3 surfaces \\cite{MR3279532} or Huizenga's work on Hilbert schemes of $\\mathbb{P}^2$ \\cite{MR3419956}. It is then natural to ask how to compute the effective cone of surfaces or the codimension 2 effective cones.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(S^{[n]})$? What is the codimension 2 effective cone $\\text{Eff}^2(S^{[n]})$? What about higher dimension/codimension?", "original_statement": "Effective cones of Hilbert schemes and Moduli Spaces\n\nIt is natural to try to compute effective cones in the case of Hilbert schemes or moduli spaces. One natural Hilbert scheme to consider first is the Hilbert scheme of length $n$ subscheme of a smooth surface $S$, denoted by $S^{[n]}$. A great deal is known about these Hilbert schemes. In recent years there has been a lot of exciting work on the effective cones of divisors and curves of the Hilbert schemes $\\text{Eff}^1(S^{[n]})$ and $\\text{Eff}_1(S^{[n]})$. See for example Bayer and Macri's work on Hilbert schemes of K3 surfaces \\cite{MR3279532} or Huizenga's work on Hilbert schemes of $\\mathbb{P}^2$ \\cite{MR3419956}. It is then natural to ask how to compute the effective cone of surfaces or the codimension 2 effective cones.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(S^{[n]})$? What is the codimension 2 effective cone $\\text{Eff}^2(S^{[n]})$? What about higher dimension/codimension?", "clean_statement": "Effective cones of Hilbert schemes and Moduli Spaces\n\nIt is natural to try to compute effective cones in the case of Hilbert schemes or moduli spaces. One natural Hilbert scheme to consider first is the Hilbert scheme of length $n$ subscheme of a smooth surface $S$, denoted by $S^{[n]}$. A great deal is known about these Hilbert schemes. In recent years there has been a lot of exciting work on the effective cones of divisors and curves of the Hilbert schemes $\\text{Eff}^1(S^{[n]})$ and $\\text{Eff}_1(S^{[n]})$. See for example Bayer and Macri's work on Hilbert schemes of K3 surfaces \\cite{MR3279532} or Huizenga's work on Hilbert schemes of $\\mathbb{P}^2$ \\cite{MR3419956}. It is then natural to ask how to compute the effective cone of surfaces or the codimension 2 effective cones.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(S^{[n]})$? What is the codimension 2 effective cone $\\text{Eff}^2(S^{[n]})$? What about higher dimension/codimension?", "statement_status": "exact", "statement_verification": "There is no apparent OCR corruption. There are, however, two important convention issues in the question:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Computing higher codimension effective and nef cones in explicit examples\nSource item: 1.7\nSource URL: http://aimpl.org/poscycles/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[137]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Effective cones of Hilbert schemes and Moduli Spaces\\n\\nIt is natural to try to compute effective cones in the case of Hilbert schemes or moduli spaces. One natural Hilbert scheme to consider first is the Hilbert scheme of length $n$ subscheme of a smooth surface $S$, denoted by $S^{[n]}$. A great deal is known about these Hilbert schemes. In recent years there has been a lot of exciting work on the effective cones of divisors and curves of the Hilbert schemes $\\\\text{Eff}^1(S^{[n]})$ and $\\\\text{Eff}_1(S^{[n]})$. See for example Bayer and Macri's work on Hilbert schemes of K3 surfaces \\\\cite{MR3279532} or Huizenga's work on Hilbert schemes of $\\\\mathbb{P}^2$ \\\\cite{MR3419956}. It is then natural to ask how to compute the effective cone of surfaces or the codimension 2 effective cones.\\n\\nWhat is the effective cone of surfaces $\\\\text{Eff}_2(S^{[n]})$? What is the codimension 2 effective cone $\\\\text{Eff}^2(S^{[n]})$? What about higher dimension/codimension?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0138", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every smooth integral projective complex surface S, let X=S^[2], let B be the reduced Hilbert--Chow exceptional divisor with p:B->S, and let L be the pullback of an ample divisor from S^(2). The map Phi:N_1(S)->N_2(X), [C] mapsto [p^{-1}(C)], is injective and satisfies B·rho^*D^(2)·Phi(C)=-4(D·C). Moreover, the integral surfaces Z with L^2·Z=0 are exactly p^{-1}(C), so the L^2-zero face of the raw effective surface cone is exactly Phi(Eff_1^raw(S)). For pseudoeffective closures only the corresponding inclusion is asserted. Ryan's known exact cone for (P^2)^[2] confirms that this face is its B_{1,1} ray.\n\nCandidate contribution (theorem; novelty confidence low): Candidate Hilbert--Chow face theorem: for arbitrary smooth projective complex S, Eff_2^raw(S^[2]) intersect ker(L^2) equals Phi(Eff_1^raw(S)), and Phi has the explicit numerical left inverse determined by D·R(alpha)=-(1/4)B·rho^*D^(2)·alpha." }, { "id": 20000139, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0139", "title": "A four-dimensional repair for symmetric surface classes on Mbar_0,7", "statement": "Historically there has also been fervent interest in computing the effective cones of divisors and curves in the case of the moduli space of genus $g$ curves $\\overline{\\mathcal{M}}_g$ or their pointed versions $\\overline{\\mathcal{M}}_{g,n}$. In recent years for example there has been a great deal of work on the so called F-conjecture which predicts the effective cone of curves in $\\overline{\\mathcal{M}}_{0,n}$ is generated by explicit curve classes (so called \"F-curves\"), see e.g. Gibney's work \\cite{MR2551995}. It is then natural to ask about higher dimensional effective cones for small values of $g$ and $n$. In the pointed case, to make the problem even simpler one can ask about the symmetrized space $\\overline{\\mathcal{M}}_{g,n}^{\\mathfrak{S}_n}:=\\overline{\\mathcal{M}}_{g,n}/{\\mathfrak{S}_n}$.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(\\overline{\\mathcal{M}}_{0,7}^{\\mathfrak{S}_7})$?", "original_statement": "Historically there has also been fervent interest in computing the effective cones of divisors and curves in the case of the moduli space of genus $g$ curves $\\overline{\\mathcal{M}}_g$ or their pointed versions $\\overline{\\mathcal{M}}_{g,n}$. In recent years for example there has been a great deal of work on the so called F-conjecture which predicts the effective cone of curves in $\\overline{\\mathcal{M}}_{0,n}$ is generated by explicit curve classes (so called \"F-curves\"), see e.g. Gibney's work \\cite{MR2551995}. It is then natural to ask about higher dimensional effective cones for small values of $g$ and $n$. In the pointed case, to make the problem even simpler one can ask about the symmetrized space $\\overline{\\mathcal{M}}_{g,n}^{\\mathfrak{S}_n}:=\\overline{\\mathcal{M}}_{g,n}/{\\mathfrak{S}_n}$.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(\\overline{\\mathcal{M}}_{0,7}^{\\mathfrak{S}_7})$?", "clean_statement": "Historically there has also been fervent interest in computing the effective cones of divisors and curves in the case of the moduli space of genus $g$ curves $\\overline{\\mathcal{M}}_g$ or their pointed versions $\\overline{\\mathcal{M}}_{g,n}$. In recent years for example there has been a great deal of work on the so called F-conjecture which predicts the effective cone of curves in $\\overline{\\mathcal{M}}_{0,n}$ is generated by explicit curve classes (so called \"F-curves\"), see e.g. Gibney's work \\cite{MR2551995}. It is then natural to ask about higher dimensional effective cones for small values of $g$ and $n$. In the pointed case, to make the problem even simpler one can ask about the symmetrized space $\\overline{\\mathcal{M}}_{g,n}^{\\mathfrak{S}_n}:=\\overline{\\mathcal{M}}_{g,n}/{\\mathfrak{S}_n}$.\n\nWhat is the effective cone of surfaces $\\text{Eff}_2(\\overline{\\mathcal{M}}_{0,7}^{\\mathfrak{S}_7})$?", "statement_status": "exact", "statement_verification": "The AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Computing higher codimension effective and nef cones in explicit examples\nSource item: 1.8\nSource URL: http://aimpl.org/poscycles/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[138]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Historically there has also been fervent interest in computing the effective cones of divisors and curves in the case of the moduli space of genus $g$ curves $\\\\overline{\\\\mathcal{M}}_g$ or their pointed versions $\\\\overline{\\\\mathcal{M}}_{g,n}$. In recent years for example there has been a great deal of work on the so called F-conjecture which predicts the effective cone of curves in $\\\\overline{\\\\mathcal{M}}_{0,n}$ is generated by explicit curve classes (so called \\\"F-curves\\\"), see e.g. Gibney's work \\\\cite{MR2551995}. It is then natural to ask about higher dimensional effective cones for small values of $g$ and $n$. In the pointed case, to make the problem even simpler one can ask about the symmetrized space $\\\\overline{\\\\mathcal{M}}_{g,n}^{\\\\mathfrak{S}_n}:=\\\\overline{\\\\mathcal{M}}_{g,n}/{\\\\mathfrak{S}_n}$.\\n\\nWhat is the effective cone of surfaces $\\\\text{Eff}_2(\\\\overline{\\\\mathcal{M}}_{0,7}^{\\\\mathfrak{S}_7})$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0139", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The exact effective cone is still open, and the published three-dimensional calculation is covered by an author-posted erratum. The invariant surface space is four-dimensional with a simplicial boundary subcone generated by four boundary-orbit averages. An explicit nonzero primitive class spans the direction invisible to all products of symmetric divisors. Full four-coordinate calculations show that the symmetrizations of Schaffler's lifted Keel-Vermeire surfaces and characteristic-zero special hypertree surfaces are respectively 2 Z_1 + 2 Z_3 + Z_4 and 9 Z_1 + 3 Z_3 + 3 Z_4, so these two explicit labeled nonboundary families map into the boundary-generated cone after symmetrization.\n\nCandidate contribution (explicit invariant-cycle computation; novelty confidence low): Candidate novelty: the primitive class Delta = Z_4 - 2 Z_1 - (Z_2 + 2 Z_3)/12 is nonzero but annihilated by every product of symmetric divisors, and the two exact symmetrization identities Av(K) = 2 Z_1 + 2 Z_3 + Z_4 and Av(H) = 9 Z_1 + 3 Z_3 + 3 Z_4 hold for Schaffler's lifted Keel-Vermeire and characteristic-zero special hypertree families." }, { "id": 20000140, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0140", "title": "Linear-center effective cones and the vertical exceptional face", "statement": "Effective cones of blowups\n\nLastly, it is natural to ask what the relationship is between effective cones of a variety and its blowup. This is a deceptively difficult question already in the case of the effective cone of curves and the blowup of 10 points in $\\mathbb{P}^2$, which is the setting of Nagata's conjecture. However, the question of computing effective cones of blowups is quite possibly more tractable when the center of the blowup is a positive dimensional subvariety of $\\mathbb{P}^n$.\n\nLet $Z\\subset \\mathbb{P}^n$ be a smooth subvariety with $1\\le \\text{dim}Z \\le n-2$. What is the effective cone of $k$-cycles $\\text{Eff}_k(X)$?", "original_statement": "Effective cones of blowups\n\nLastly, it is natural to ask what the relationship is between effective cones of a variety and its blowup. This is a deceptively difficult question already in the case of the effective cone of curves and the blowup of 10 points in $\\mathbb{P}^2$, which is the setting of Nagata's conjecture. However, the question of computing effective cones of blowups is quite possibly more tractable when the center of the blowup is a positive dimensional subvariety of $\\mathbb{P}^n$.\n\nLet $Z\\subset \\mathbb{P}^n$ be a smooth subvariety with $1\\le \\text{dim}Z \\le n-2$. What is the effective cone of $k$-cycles $\\text{Eff}_k(X)$?", "clean_statement": "Effective cones of blowups\n\nLastly, it is natural to ask what the relationship is between effective cones of a variety and its blowup. This is a deceptively difficult question already in the case of the effective cone of curves and the blowup of 10 points in $\\mathbb{P}^2$, which is the setting of Nagata's conjecture. However, the question of computing effective cones of blowups is quite possibly more tractable when the center of the blowup is a positive dimensional subvariety of $\\mathbb{P}^n$.\n\nLet $Z\\subset \\mathbb{P}^n$ be a smooth subvariety with $1\\le \\text{dim}Z \\le n-2$. What is the effective cone of $k$-cycles $\\text{Eff}_k(X)$?", "statement_status": "exact", "statement_verification": "The canonical record is source index 139 of `aim-algebraic-geometry-notes.json`: Problem 1.9, “Effective cones of blowups,” from the AIM workshop *Positivity of cycles*. Its mathematical prompt is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Computing higher codimension effective and nef cones in explicit examples\nSource item: 1.9\nSource URL: http://aimpl.org/poscycles/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[139]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Effective cones of blowups\\n\\nLastly, it is natural to ask what the relationship is between effective cones of a variety and its blowup. This is a deceptively difficult question already in the case of the effective cone of curves and the blowup of 10 points in $\\\\mathbb{P}^2$, which is the setting of Nagata's conjecture. However, the question of computing effective cones of blowups is quite possibly more tractable when the center of the blowup is a positive dimensional subvariety of $\\\\mathbb{P}^n$.\\n\\nLet $Z\\\\subset \\\\mathbb{P}^n$ be a smooth subvariety with $1\\\\le \\\\text{dim}Z \\\\le n-2$. What is the effective cone of $k$-cycles $\\\\text{Eff}_k(X)$?\"\nOriginal remarks: [\"In question 9 it is natural to first consider a few basic cases such as the blowup of a union of a small number of lines in $\\\\mathbb{P}^n$ or the blowup of low degree, low codimension complete intersections.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0140", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After explicitly reconstructing X as the blowup Bl_Z(P^n), this attempt computes every raw and closed effective k-cycle cone when Z is one linear space P^r: for p=n-k, s=n-r-1, f=r+1, a=max(0,p-s), and b=min(p,f), the cone is simplicial with the minimal rays A_a=h^(p-a)H^a and B_j=h^(p-1-j)H^jE for a<=j<=b-1, where h=H-E. It also proves for every smooth center that the H^k=0 face of the raw effective cone consists exactly of exceptional k-folds whose image in Z has dimension below k, while identifying the reverse closed-cone inclusion as the Strong Pushforward Conjecture rather than asserting it.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: the endpoint-correct minimal-ray formula, including k=0 and k=n and the identification of the duplicated boundary class B_(a-1)=A_a, uniformly computes all effective cycle cones of Bl_{P^r}(P^n); paired with it is an exact raw vertical-face identity for arbitrary smooth centers." }, { "id": 20000141, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0141", "title": "Finite complete-intersection mobility bounds for the line class in projective three-space", "statement": "Over \\(\\mathbb C\\), let \\(\\ell\\) be the numerical class of a line in\n\\(\\mathbb P^3\\). Determine\n\\[\n\\operatorname{mob}(\\ell)\n=\\limsup_{m\\to\\infty}\\frac{6\\,mc(m\\ell)}{m^{3/2}}.\n\\]", "original_statement": "Mobility and mobility counts\n\nLet $X$ be an $n$ dimensional variety. Given an effective integral $k$-cycle $\\alpha\\in N_k(X)_\\mathbb{Z}$, the \\textit{mobility count of} $\\alpha$, denoted $mc(\\alpha)$, is the maximum number of general points in $X$ that we can impose on an effective cycle of class $\\alpha$. For example, any two points in $\\mathbb{P}^2$ can be connected by a line $\\ell \\subset \\mathbb{P}^2,$ so we see $mc([\\ell])\\ge 2$.\n\nThe mobility count is supposed to be an analogue of $\\text{dim}(H^0(X,\\mathcal{O}(E)))$. Taking a cue from divisor theory it is natural to consider asymptotic invariants of a numerical cycle. Define the \\textit{mobility} of a numerical cycle $\\alpha \\in N_k(X)_\\mathbb{Z}$ class to be $$\\text{mob}(\\alpha):= \\frac{n! mc(m\\cdot\\alpha)}{m^{n/(n-k)}}.$$ Define the \\textit{Iitaka dimension of} $\\alpha$ of a numerical cycle to be $$K(\\alpha)=\\text{max}\\{ r \\in\\mathbb{R} | \\text{limsup}_{m\\rightarrow \\infty} \\frac{mc(m\\cdot \\alpha)}{m^r}>0 \\}.$$\n\nLet $[\\ell]\\in N_1(\\mathbb{P}^3)_\\mathbb{Z}$ be the class of a line. What is $\\text{mob}([\\alpha])$?", "clean_statement": "Over \\(\\mathbb C\\), let \\(\\ell\\) be the numerical class of a line in\n\\(\\mathbb P^3\\). Determine\n\\[\n\\operatorname{mob}(\\ell)\n=\\limsup_{m\\to\\infty}\\frac{6\\,mc(m\\ell)}{m^{3/2}}.\n\\]", "statement_status": "corrected_verified", "statement_verification": "There are two extraction defects and one later change of convention. 1. The displayed mobility formula in the record has no limiting operation. The standard definition in [Leh16, Definition 1.1] is \\[ \\operatorname{mob}(\\alpha) := \\limsup_{m\\to\\infty} \\frac{mc(m\\alpha)} {m^{\\,n/(n-k)}/n!} = \\limsup_{m\\to\\infty} \\frac{n!\\,mc(m\\alpha)}{m^{\\,n/(n-k)}}. \\tag{1.1} \\] The denominator \\(m^{n/(n-k)}/n!\\) is the standard normalization. 2. The last \\([\\alpha]\\) is reconstructed as \\([\\ell]\\). This is confirmed by the fuller AIM problem-list transcript, which asks for the mobility when \\(\\alpha\\) is the line class on \\(\\mathbb P^3\\). 3. The AIM record calls the unrescaled growth exponent \\[ K_{\\mathrm{src}}(\\alpha) := \\sup\\left\\{r\\geq0: \\limsup_{m\\to\\infty}\\frac{mc(m\\alpha)}{m^r}>0 \\right\\}. \\tag{1.2} \\] Lehmann's later convention [Leh19] multiplies this exponent by the codimension: \\[ \\kappa(\\a...", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Measuring positivity of cycles\nSource item: 2.1\nSource URL: http://aimpl.org/poscycles/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[140]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Mobility and mobility counts\\n\\nLet $X$ be an $n$ dimensional variety. Given an effective integral $k$-cycle $\\\\alpha\\\\in N_k(X)_\\\\mathbb{Z}$, the \\\\textit{mobility count of} $\\\\alpha$, denoted $mc(\\\\alpha)$, is the maximum number of general points in $X$ that we can impose on an effective cycle of class $\\\\alpha$. For example, any two points in $\\\\mathbb{P}^2$ can be connected by a line $\\\\ell \\\\subset \\\\mathbb{P}^2,$ so we see $mc([\\\\ell])\\\\ge 2$.\\n\\nThe mobility count is supposed to be an analogue of $\\\\text{dim}(H^0(X,\\\\mathcal{O}(E)))$. Taking a cue from divisor theory it is natural to consider asymptotic invariants of a numerical cycle. Define the \\\\textit{mobility} of a numerical cycle $\\\\alpha \\\\in N_k(X)_\\\\mathbb{Z}$ class to be $$\\\\text{mob}(\\\\alpha):= \\\\frac{n! mc(m\\\\cdot\\\\alpha)}{m^{n/(n-k)}}.$$ Define the \\\\textit{Iitaka dimension of} $\\\\alpha$ of a numerical cycle to be $$K(\\\\alpha)=\\\\text{max}\\\\{ r \\\\in\\\\mathbb{R} | \\\\text{limsup}_{m\\\\rightarrow \\\\infty} \\\\frac{mc(m\\\\cdot \\\\alpha)}{m^r}>0 \\\\}.$$\\n\\nLet $[\\\\ell]\\\\in N_1(\\\\mathbb{P}^3)_\\\\mathbb{Z}$ be the class of a line. What is $\\\\text{mob}([\\\\alpha])$?\"\nOriginal remarks: [\"Complete intersections should give the optimal mobility.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0141", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The exact ordinary mobility of the line class ell in P^3 remains open: the literature checked through 23 July 2026 gives 1 <= mob(ell) < 3.54, while weighted mobility equals 1 and ordinary equality is conditional on the complete-intersection conjecture. The source definition is repaired by inserting the standard limsup, and the adjacent growth exponent is determined exactly: K_src(ell)=3/2 (equivalently, the modern cycle-Iitaka dimension is 3). Within universal proper two-surface complete-intersection families, mc(F_{a,b}) is proved to equal binom(a+3,3)-2 when a=b and binom(a+3,3)-1 when a= binom(d+3,3)-2+2(m-d^2), and a family with b/a tending to lambda has normalized efficiency tending to lambda^(-3/2).\n\nCandidate contribution (proposition; novelty confidence low): For every 1 <= a <= b, the universal family of proper type-(a,b) complete-intersection curves in P^3 has mobility count binom(a+3,3)-2 if a=b and binom(a+3,3)-1 if a= binom(floor(sqrt(m))+3,3)-2+2(m-floor(sqrt(m))^2) for every m, and the normalized complete-intersection efficiency under b/a -> lambda is lambda^(-3/2)." }, { "id": 20000142, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0142", "title": "Schubert-cycle mobility exponents and general incidence lower bounds", "statement": "What is $K(\\alpha)$ for $\\alpha \\in N_k(\\text{Gr}(k,n))$ a Schubert cycle?", "original_statement": "What is $K(\\alpha)$ for $\\alpha \\in N_k(\\text{Gr}(k,n))$ a Schubert cycle?", "clean_statement": "What is $K(\\alpha)$ for $\\alpha \\in N_k(\\text{Gr}(k,n))$ a Schubert cycle?", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Measuring positivity of cycles\nSource item: 2.2\nSource URL: http://aimpl.org/poscycles/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[141]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is $K(\\\\alpha)$ for $\\\\alpha \\\\in N_k(\\\\text{Gr}(k,n))$ a Schubert cycle?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0142", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Lehmann's published theorem completely determines the normalized cycle Iitaka dimension of every Schubert class on Gr(2,N); after resolving the AIM page's normalization inconsistency, the report gives the exact unscaled AIM exponent class by class. For arbitrary Gr(d,N), it also proves exact divisor, curve, point, and multirigid cases and quantitative bounds mc(e^r sigma_r) >= binomial(N-d+e-1,N-d) and mc(e^t sigma_{1^t}) >= binomial(d+e-1,d), yielding raw exponents at least (N-d)/r and d/t.\n\nCandidate contribution (theorem; novelty confidence low): For every ordinary Grassmannian Gr(d,N), complete-intersection incidence cycles give the explicit binomial mobility lower bounds for all special one-row and one-column Schubert classes; away from the zero-dimensional endpoints, general such cycles are reduced irreducible non-Schubert representatives of nontrivial multiples and therefore witness failure of multirigidity." }, { "id": 20000143, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0143", "title": "Exact boundary mobility counts and explicit mixed-face estimates", "statement": "Develop better estimates for the growth rate of $mc(\\alpha)$ for $\\alpha$ a numerical class on $\\text{Gr}(2,4)$ or $\\mathbb{P}^2\\times \\mathbb{P}^2$.", "original_statement": "Develop better estimates for the growth rate of $mc(\\alpha)$ for $\\alpha$ a numerical class on $\\text{Gr}(2,4)$ or $\\mathbb{P}^2\\times \\mathbb{P}^2$.", "clean_statement": "Develop better estimates for the growth rate of $mc(\\alpha)$ for $\\alpha$ a numerical class on $\\text{Gr}(2,4)$ or $\\mathbb{P}^2\\times \\mathbb{P}^2$.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Measuring positivity of cycles\nSource item: 2.3\nSource URL: http://aimpl.org/poscycles/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[142]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop better estimates for the growth rate of $mc(\\\\alpha)$ for $\\\\alpha$ a numerical class on $\\\\text{Gr}(2,4)$ or $\\\\mathbb{P}^2\\\\times \\\\mathbb{P}^2$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0143", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For surface classes on P^2 x P^2, the attempt proves the exact identities mc(m(aH^2+cA^2))=m(a+c) and mc(mAH)=2m for every positive integer m, and proves the explicit mixed-face estimate mc(4ad^2(aH^2+bAH)) >= (bd+1) binom(2ad+2,2)-2. For Gr(2,4), it proves mc(m sigma_2)=mc(m sigma_1,1)=m using published multirigidity and obtains explicit Plucker complete-intersection lower bounds for every big integral surface class. These refine known exponent calculations but do not determine the full mobility functions.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novel contribution: for all nonnegative integers a,c with a+c>0 and all m>=1, mc(m(aH^2+cA^2))=m(a+c), while mc(mAH)=2m; additionally, for a,b,d>=1, mc(4ad^2(aH^2+bAH)) >= (bd+1) binom(2ad+2,2)-2." }, { "id": 20000144, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0144", "title": "Convention split for cycle-Iitaka integrality and a finite-additive-presentation criterion", "statement": "Is $K(\\alpha)$ an integer?", "original_statement": "Is $K(\\alpha)$ an integer?", "clean_statement": "Is $K(\\alpha)$ an integer?", "statement_status": "exact", "statement_verification": "The exact canonical record from the 2016 AIM workshop *Positivity of cycles* asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Measuring positivity of cycles\nSource item: 2.4\nSource URL: http://aimpl.org/poscycles/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[143]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is $K(\\\\alpha)$ an integer?\"\nOriginal remarks: [\"This is true for divisors, curves, $\\\\text{Gr}(2,n)$, but is open in general.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0144", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "The workshop's literal unrescaled mobility-growth exponent is not always an integer: the line class ell in P^3 has raw exponent 3/2, by the balanced complete-intersection lower bound and Lehmann's universal O(m^(3/2)) upper bound; the Schubert class sigma_2 on G(2,5) gives a second raw exponent 3/2. Under Lehmann's later normalization kappa=(n-k)rho, both values become 3, and the general integer-valued conjecture remains apparently open in the literature checked through 23 July 2026. A proved Chow-incidence criterion gives (n-k)mc(m alpha) <= D_alpha(m), where D_alpha(m) is the maximum dimension of a relevant Chow component. Consequently, a finite additive presentation of the ray by finitely many fixed projective family types forces kappa to lie in {-infinity, 0, n-k}; any larger value requires Chow components of superlinear dimension in unbounded degrees.\n\nCandidate contribution (criterion; novelty confidence low): If every effective cycle on the ray of alpha is a sum of members of finitely many fixed projective families T_i of classes a_i alpha, then D_alpha(m) <= m max_i(dim(T_i)/a_i), so the modern cycle-Iitaka dimension is -infinity, 0, or the codimension n-k. In particular, kappa(alpha)>n-k forces D_alpha(m)/m to be unbounded. Complementarily, explicit superadditive integer profiles show that sequence-theoretic axioms alone permit every real rescaled exponent in [n-k,n] and need not attain the endpoint." }, { "id": 20000145, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0145", "title": "Very moving cycle classes: Chow-family and positive-form reductions", "statement": "Other measures of positivity\n\nLet $X$ be a projective variety. Suppose that there is a class $\\alpha \\in \\text{Eff}_k(X)_{\\mathbb{Z}}$ such that for a general point $x\\in X$ and a general $k$-plane $V\\subset T_x X$ there is an irreducible subvariety $Y\\subset X$ with $[V]= \\alpha$ and $T_xY = V$.\n\nIs $\\alpha$ in the interior of the effective cone?", "original_statement": "Other measures of positivity\n\nLet $X$ be a projective variety. Suppose that there is a class $\\alpha \\in \\text{Eff}_k(X)_{\\mathbb{Z}}$ such that for a general point $x\\in X$ and a general $k$-plane $V\\subset T_x X$ there is an irreducible subvariety $Y\\subset X$ with $[V]= \\alpha$ and $T_xY = V$.\n\nIs $\\alpha$ in the interior of the effective cone?", "clean_statement": "Other measures of positivity\n\nLet $X$ be a projective variety. Suppose that there is a class $\\alpha \\in \\text{Eff}_k(X)_{\\mathbb{Z}}$ such that for a general point $x\\in X$ and a general $k$-plane $V\\subset T_x X$ there is an irreducible subvariety $Y\\subset X$ with $[V]= \\alpha$ and $T_xY = V$.\n\nIs $\\alpha$ in the interior of the effective cone?", "statement_status": "exact", "statement_verification": "The live AIM page, problem 2.5 in *Positivity of cycles*, attributes the question to Claire Voisin and currently reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Positivity of cycles\nSection: Measuring positivity of cycles\nSource item: 2.5\nSource URL: http://aimpl.org/poscycles/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[144]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Other measures of positivity\\n\\nLet $X$ be a projective variety. Suppose that there is a class $\\\\alpha \\\\in \\\\text{Eff}_k(X)_{\\\\mathbb{Z}}$ such that for a general point $x\\\\in X$ and a general $k$-plane $V\\\\subset T_x X$ there is an irreducible subvariety $Y\\\\subset X$ with $[V]= \\\\alpha$ and $T_xY = V$.\\n\\nIs $\\\\alpha$ in the interior of the effective cone?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/poscycles/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0145", "aim-domain:algebraic-geometry", "aim-workshop:poscycles", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After correcting the source typo [V]=alpha to [Y]=alpha, the AIM hypothesis is Voisin's open very-moving-implies-big conjecture. The pointwise existential fixed-class hypothesis can be realized by one irreducible Chow family whose tangent map dominates the relative Grassmannian; its parameter space has the sharp lower bound (k+1)(n-k), with generically finite tangent map in the equality case. In addition, alpha pairs strictly positively with every nonzero numerical functional represented by a smooth closed weakly positive (k,k)-form. Therefore a counterexample must have a nef boundary support not representable by such a form.\n\nCandidate contribution (reduction; novelty confidence low): Under the corrected AIM existential numerical-class hypothesis, a single irreducible fixed-class Chow family has dominant tangent map, so its base has dimension at least (k+1)(n-k), with generic finiteness in the equality case; moreover every nef boundary support annihilating alpha must lie outside the cone of numerical functionals represented by smooth closed weakly positive (k,k)-forms." }, { "id": 20000146, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0146", "title": "Normed motivic spectra and incomplete Galois-window norms", "statement": "In \\cite{MR3406512}, Blumberg and Hill define\n $N_{\\infty}$-operads, equivariant generalizations of\n $E_{\\infty}$-operads.\n\nWhat is the motivic analogue of an\n $N_{\\infty}$-operad or in other words, what are the analogues of\n $G$-commutative ring spectra in motivic homotopy theory?", "original_statement": "In \\cite{MR3406512}, Blumberg and Hill define\n $N_{\\infty}$-operads, equivariant generalizations of\n $E_{\\infty}$-operads.\n\nWhat is the motivic analogue of an\n $N_{\\infty}$-operad or in other words, what are the analogues of\n $G$-commutative ring spectra in motivic homotopy theory?", "clean_statement": "In \\cite{MR3406512}, Blumberg and Hill define\n $N_{\\infty}$-operads, equivariant generalizations of\n $E_{\\infty}$-operads.\n\nWhat is the motivic analogue of an\n $N_{\\infty}$-operad or in other words, what are the analogues of\n $G$-commutative ring spectra in motivic homotopy theory?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.1 in the “Motivic Homotopy Theory” section of the AIM problem list *Equivariant derived algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Motivic Homotopy Theory\nSource item: 1.1\nSource URL: http://aimpl.org/equideralggeom/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[145]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In \\\\cite{MR3406512}, Blumberg and Hill define\\n $N_{\\\\infty}$-operads, equivariant generalizations of\\n $E_{\\\\infty}$-operads.\\n\\nWhat is the motivic analogue of an\\n $N_{\\\\infty}$-operad or in other words, what are the analogues of\\n $G$-commutative ring spectra in motivic homotopy theory?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0146", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Bachmann-Hoyois normed motivic spectra are the established motivic analogue of G-commutative ring spectra: coherent stable norms exist along finite etale maps and folds recover the underlying motivic E-infinity ring. Beyond this status result, the attempt proves that any wide, pullback-stable, finite-coproduct-complete class of finite etale exponent maps restricts the Bachmann-Hoyois span functor to a coherent category of incomplete normed spectra; over a finite Galois extension these classes are exactly G-indexing systems by Blumberg-Hill. It also proves that an admissible class containing folds and satisfying etale descent on targets must contain every finite etale map, so every proper incomplete motivic system must retain nonlocal monodromy data.\n\nCandidate contribution (construction_and_obstruction; novelty confidence low): For each finite Galois window L/k and G-indexing system I, restriction of the Bachmann-Hoyois norm functor canonically defines I-incomplete normed motivic spectra; moreover, any isomorphism-invariant admissible norm class containing folds and satisfying target-etale descent is necessarily the maximal class of all finite etale maps." }, { "id": 20000147, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0147", "title": "Comparison-gated cellular motivic Eilenberg-Moore spectral sequences", "statement": "Can we construct an Eilenberg-Moore spectral sequence in motivic homotopy theory?", "original_statement": "Can we construct an Eilenberg-Moore spectral sequence in motivic homotopy theory?", "clean_statement": "Can we construct an Eilenberg-Moore spectral sequence in motivic homotopy theory?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.2 in the “Motivic Homotopy Theory” section of the AIM problem list *Equivariant derived algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Motivic Homotopy Theory\nSource item: 1.2\nSource URL: http://aimpl.org/equideralggeom/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[146]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we construct an Eilenberg-Moore spectral sequence in motivic homotopy theory?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0147", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Published work supplies stable cellular Tor, unstable Rothenberg-Steenrod, equivariant motivic-cohomology, and special geometric KGL versions of a motivic Eilenberg-Moore construction. For a homotopy pullback Z = X x^h_B Y and commutative motivic spectrum E, we prove the following precise reduction: if the natural map C_E(X) tensor_{C_E(B)} C_E(Y) to C_E(Z) is an equivalence and one cochain factor is cellular over C_E(B), then the Dugger-Isaksen trigraded Tor spectral sequence converges strongly to the E-cochains of Z. Tor amplitude at most m then gives at most m+1 filtration pieces. We also isolate the finite length-m Koszul filtration in the Krishna-Totaro split-torus construction, proving weightwise strong convergence without a derived-limit obstruction and deriving an explicit rank-one kernel-cokernel exact sequence.\n\nCandidate contribution (reduction; novelty confidence low): The comparison-gated cellular criterion separates a geometric motivic Eilenberg-Moore theorem into two testable conditions—the cochain comparison equivalence and relative cellularity—and proves that, once they hold, finite Tor amplitude bounds the strongly convergent filtration. In the rank-m split-torus Krishna-Totaro case, the actual m-cube/Koszul filtration is finite, so it has no lim^1 contribution, has at most m+1 pieces, and stabilizes at E_{m+1}." }, { "id": 20000148, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0148", "title": "Scoped motivic Serre sequences and a finite-etale monodromy obstruction", "statement": "Can we construct a Serre spectral sequence in motivic homotopy theory?", "original_statement": "Can we construct a Serre spectral sequence in motivic homotopy theory?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0148, source file `aim-algebraic-geometry-notes.json`, zero-based source index 147, from the AIM workshop *Equivariant derived algebraic geometry*, section *Motivic Homotopy Theory*, problem 1.3. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Motivic Homotopy Theory\nSource item: 1.3\nSource URL: http://aimpl.org/equideralggeom/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[147]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we construct a Serre spectral sequence in motivic homotopy theory?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0148", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Published work gives affirmative Serre-type constructions in two substantial scopes: the convergent homotopy Leray spectral sequence of Asok-Deglise-Nagel for generalized motivic (co)homology and the strongly convergent cellular-fiber motivic-cohomology sequence of Tanania. Beyond this status synthesis, the attempt proves that any first-quadrant, weight-preserving, strongly convergent cohomological motivic Serre sequence which assigns constant geometric-fiber coefficients to every finite etale degree-n cover of a connected smooth base forces the cover to split as n copies of the base. The connected cover G_m -> G_m, u |-> u^2, is an explicit obstruction; its transposition permutation local system repairs the erroneous rank in H^{0,0}.\n\nCandidate contribution (obstruction; novelty confidence low): If p:Y->X is finite etale of degree n with X connected and smooth, then a first-quadrant cohomological spectral sequence with weight-preserving differentials d_r of degree (r,1-r), constant-fiber corner E_2^{0,0,0}=H^{0,0}(X,Z^n), and strong exhaustive Hausdorff convergence to H^{*,*}(Y,Z) can exist only when Y is the split cover coproduct_{i=1}^n X." }, { "id": 20000149, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0149", "title": "Adding transfers in motivic homotopy theory and a quadratic framing-loss obstruction", "statement": "Long Term\n\nDo we have a motivic description akin to adding transfers?", "original_statement": "Long Term\n\nDo we have a motivic description akin to adding transfers?", "clean_statement": "Long Term\n\nDo we have a motivic description akin to adding transfers?", "statement_status": "exact", "statement_verification": "The canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Motivic Homotopy Theory\nSource item: 1.4\nSource URL: http://aimpl.org/equideralggeom/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[148]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Long Term\\n\\nDo we have a motivic description akin to adding transfers?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0149", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Modern recognition and Mackey theorems give precise affirmative answers to the main readings of the 2016 question: framed finite-syntomic transfers recognize very effective motivic spectra over perfect fields, stable framed motivic spectra reconstruct SH(S) over every scheme, and coherent finite-etale transfers reconstruct tame finite-constant equivariant motivic stable homotopy after inverting the trivial representation sphere. The report additionally proves that ordinary finite-cycle transfers are too coarse: for a perfect field of characteristic not 2, two framings alpha_u and alpha_v of the identity point act as distinct classes and in GW(k), while the cycle functor sends both to the same degree-one correspondence; over R, alpha_1 and alpha_-1 are an explicit witness.\n\nCandidate contribution (obstruction; novelty confidence low): The quadratic one-point test exhibits an explicit nonzero kernel element [alpha_u]-[alpha_v] for the framed-to-ordinary-cycle functor whenever and are distinct in GW(k); over R, the comparison A(C2) isomorphic to GW(R) followed by rank explains exactly why coherent finite-etale Mackey transfers retain the sign that ordinary cycles erase." }, { "id": 20000150, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0150", "title": "Motivic suspension, framed BPQ, and a quadratic zero-stem obstruction", "statement": "In classical homotopy theory, there are, among many others, three important notions/results:\n\n(1) The Freudenthal suspension theorem\n\n(2) The Barratt-Priddy-Quillen theorem\n\n(3) Infinite loop space theory\n\nCan we construct/prove the above notions in the context of motivic homotopy theory?", "original_statement": "In classical homotopy theory, there are, among many others, three important notions/results:\n\n(1) The Freudenthal suspension theorem\n\n(2) The Barratt-Priddy-Quillen theorem\n\n(3) Infinite loop space theory\n\nCan we construct/prove the above notions in the context of motivic homotopy theory?", "clean_statement": "In classical homotopy theory, there are, among many others, three important notions/results:\n\n(1) The Freudenthal suspension theorem\n\n(2) The Barratt-Priddy-Quillen theorem\n\n(3) Infinite loop space theory\n\nCan we construct/prove the above notions in the context of motivic homotopy theory?", "statement_status": "exact", "statement_verification": "The canonical record is number 1.5 in the “Motivic Homotopy Theory” section of the AIM workshop list *Equivariant derived algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Motivic Homotopy Theory\nSource item: 1.5\nSource URL: http://aimpl.org/equideralggeom/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[149]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In classical homotopy theory, there are, among many others, three important notions/results:\\n\\n(1) The Freudenthal suspension theorem\\n\\n(2) The Barratt-Priddy-Quillen theorem\\n\\n(3) Infinite loop space theory\\n\\nCan we construct/prove the above notions in the context of motivic homotopy theory?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0150", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The three requested constructions now exist in precise regimes: Morel and later refinements give motivic Freudenthal theorems, Elmanto--Hoyois--Khan--Sosnilo--Yakerson give motivic BPQ and framed infinite-loop recognition, and the P1 theorem requires biconnectivity. This attempt additionally proves that the canonical constant finite-set comparison to the motivic sphere has field-point component map Z -> GW(k), n |-> n<1>; it is injective, has abelian-group cokernel GW(k)/Z<1>, and is not surjective over any formally real field. Thus framed finite-syntomic data is forced already by the motivic zero stem.\n\nCandidate contribution (obstruction; novelty confidence low): For the canonical functor sending a constant finite set to the corresponding finite syntomic family with trivial tangential framing, the induced map on pi_0 at Spec(k) after group completion and motivic localization is the split injection Z -> GW(k), n |-> n<1>, with cokernel GW(k)/Z<1>; rank and the signature at an ordering prove non-surjectivity for every formally real field." }, { "id": 20000151, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0151", "title": "A Z/2 computation in the motivic Picard problem", "statement": "What can we say about the motivic Picard group?", "original_statement": "What can we say about the motivic Picard group?", "clean_statement": "What can we say about the motivic Picard group?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Motivic Homotopy Theory\nSource item: 1.7\nSource URL: http://aimpl.org/equideralggeom/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[150]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What can we say about the motivic Picard group?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0151", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The broad group Pic(SH(k)) remains open, but a relative subgroup can be computed exactly. If k is algebraically closed of characteristic different from 2, the subgroup of invertible spectra in SH(P^1_k) which are Nisnevich-locally equivalent to the unit is canonically Pic^0_Nis(SH(P^1_k)) = H^1_Nis(P^1_k, GW-units) = Z/2. Its generator is the rank-normalized Thom twist of O(1), and the normalized Thom twist of O(n) has class n modulo 2.\n\nCandidate contribution (special_case; novelty confidence low): For algebraically closed k of characteristic different from 2, Pic^0_Nis(SH(P^1_k)) is Z/2, and the normalized Thom class of O(n) is n modulo 2." }, { "id": 20000152, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0152", "title": "Connected comparison fibers with isotropic antichains", "statement": "What more can we say about the tensor triangular geometry of the motivic stable homotopy category? (See \\cite{arxiv:1608.02876} for a list of questions.)", "original_statement": "What more can we say about the tensor triangular geometry of the motivic stable homotopy category? (See \\cite{arxiv:1608.02876} for a list of questions.)", "clean_statement": "What more can we say about the tensor triangular geometry of the motivic stable homotopy category? (See \\cite{arxiv:1608.02876} for a list of questions.)", "statement_status": "exact", "statement_verification": "The exact AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Motivic Homotopy Theory\nSource item: 1.6\nSource URL: http://aimpl.org/equideralggeom/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[151]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What more can we say about the tensor triangular geometry of the motivic stable homotopy category? (See \\\\cite{arxiv:1608.02876} for a list of questions.)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0152", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a characteristic-zero field k with nonzero pure p-symbols in every degree at least two, the Du--Vishik primes of heights 1 through infinity form a countably infinite specialization antichain inside the single connected ungraded Balmer comparison fiber over the rank-p prime of GW(k). For nonreal k and odd p, Thornton's classification identifies this whole ungraded fiber with one graded Milnor--Witt comparison fiber. The hypotheses hold for k = C(t_1,t_2,...) at every p and for k = R at p = 2; the latter conclusion is asserted only for the ungraded fiber.\n\nCandidate contribution (theorem; novelty confidence low): The height-indexed Du--Vishik primes, including height infinity, form an explicit infinite antichain in one connected ungraded comparison fiber; for nonreal fields at odd primes, that same fiber is exactly one connected graded Heller--Ormsby fiber." }, { "id": 20000153, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0153", "title": "Norm-compatible Tate gluing and a non-normable E-infinity triple", "statement": "When $G = C_p$, any genuine $G$-spectrum can be described via gluing data of the form $(X,Y, X\\rightarrow Y^{tG})$ where $X$ is a spectrum, $Y$ is a Borel spectrum, and $Y^{tG}$ denotes the Tate spectrum. An analogous pasting construction can be used to describe genuine $G$-spectra for any finite group $G$.\n\nWhat is the description of $Comm_G(Sp^G)$ using the above model?", "original_statement": "When $G = C_p$, any genuine $G$-spectrum can be described via gluing data of the form $(X,Y, X\\rightarrow Y^{tG})$ where $X$ is a spectrum, $Y$ is a Borel spectrum, and $Y^{tG}$ denotes the Tate spectrum. An analogous pasting construction can be used to describe genuine $G$-spectra for any finite group $G$.\n\nWhat is the description of $Comm_G(Sp^G)$ using the above model?", "clean_statement": "When $G = C_p$, any genuine $G$-spectrum can be described via gluing data of the form $(X,Y, X\\rightarrow Y^{tG})$ where $X$ is a spectrum, $Y$ is a Borel spectrum, and $Y^{tG}$ denotes the Tate spectrum. An analogous pasting construction can be used to describe genuine $G$-spectra for any finite group $G$.\n\nWhat is the description of $Comm_G(Sp^G)$ using the above model?", "statement_status": "exact", "statement_verification": "The exact AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Equivariant Stable Homotopy Theory\nSource item: 2.1\nSource URL: http://aimpl.org/equideralggeom/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[152]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"When $G = C_p$, any genuine $G$-spectrum can be described via gluing data of the form $(X,Y, X\\\\rightarrow Y^{tG})$ where $X$ is a spectrum, $Y$ is a Borel spectrum, and $Y^{tG}$ denotes the Tate spectrum. An analogous pasting construction can be used to describe genuine $G$-spectra for any finite group $G$.\\n\\nWhat is the description of $Comm_G(Sp^G)$ using the above model?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0153", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "For G=C_p, ordinary E-infinity algebras are Tate-gluing triples (B,X,s:X to B^{tC_p}), while genuine C_p-commutativity additionally requires an equivariant E-infinity lift B to X whose composite with s is Yang's twisted Tate-valued norm. This implies a unital map pi_0(B) to pi_0(X). Consequently, for every prime p the ordinary triple (H F_p, sphere, unit to (H F_p)^{tC_p}) exists but has no genuine C_p-E-infinity refinement.\n\nCandidate contribution (counterexample; novelty confidence low): For every prime p, the ordinary E-infinity Tate-gluing triple (H F_p, sphere, sphere to (H F_p)^{tC_p}) is not genuinely C_p-commutative; more generally, the absence of a unital ring map pi_0(B) to pi_0(X) obstructs every genuine refinement of (B,X,s)." }, { "id": 20000154, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0154", "title": "Green-functor Witt vectors for relative THH", "statement": "In \\cite{MR1410465}, Hesselholt and Madsen show\n that for any commutative ring $R$,\n\\[\\pi_0 THH(HR)^{C_{p^n}} \\cong W_{n+1}(R),\\]\nthe $p$-typical Witt vectors of length $n+1$. Furthermore, in \\cite{1401.5001v2}, Angeltveit,\n et. al. describe a construction of the cyclotomic structure on THH\n using $N^{S^1}_e$, an extension of the Hill-Hopkins-Ravenel\n multiplicative norm.\n\nIf $\\underline{R}$ is a commutative Green functor, what is a Witt-vector\nmodel for $\\pi_0(N^{S^1}_{C_p} H\\underline{R})^{C_{p^n}}$?", "original_statement": "In \\cite{MR1410465}, Hesselholt and Madsen show\n that for any commutative ring $R$,\n\\[\\pi_0 THH(HR)^{C_{p^n}} \\cong W_{n+1}(R),\\]\nthe $p$-typical Witt vectors of length $n+1$. Furthermore, in \\cite{1401.5001v2}, Angeltveit,\n et. al. describe a construction of the cyclotomic structure on THH\n using $N^{S^1}_e$, an extension of the Hill-Hopkins-Ravenel\n multiplicative norm.\n\nIf $\\underline{R}$ is a commutative Green functor, what is a Witt-vector\nmodel for $\\pi_0(N^{S^1}_{C_p} H\\underline{R})^{C_{p^n}}$?", "clean_statement": "In \\cite{MR1410465}, Hesselholt and Madsen show\n that for any commutative ring $R$,\n\\[\\pi_0 THH(HR)^{C_{p^n}} \\cong W_{n+1}(R),\\]\nthe $p$-typical Witt vectors of length $n+1$. Furthermore, in \\cite{1401.5001v2}, Angeltveit,\n et. al. describe a construction of the cyclotomic structure on THH\n using $N^{S^1}_e$, an extension of the Hill-Hopkins-Ravenel\n multiplicative norm.\n\nIf $\\underline{R}$ is a commutative Green functor, what is a Witt-vector\nmodel for $\\pi_0(N^{S^1}_{C_p} H\\underline{R})^{C_{p^n}}$?", "statement_status": "exact", "statement_verification": "This is Problem 2.2 in the AIM list *Equivariant derived algebraic geometry*, section “Equivariant Stable Homotopy Theory,” attributed to Lars Hesselholt. The canonical record is `aim-algebraic-geometry-notes.json`, index 153. The live AIM page was also inspected and agrees with the record (apart from harmless missing parentheses in the displayed fixed-point notation).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Equivariant Stable Homotopy Theory\nSource item: 2.2\nSource URL: http://aimpl.org/equideralggeom/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[153]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In \\\\cite{MR1410465}, Hesselholt and Madsen show\\n that for any commutative ring $R$,\\n\\\\[\\\\pi_0 THH(HR)^{C_{p^n}} \\\\cong W_{n+1}(R),\\\\]\\nthe $p$-typical Witt vectors of length $n+1$. Furthermore, in \\\\cite{1401.5001v2}, Angeltveit,\\n et. al. describe a construction of the cyclotomic structure on THH\\n using $N^{S^1}_e$, an extension of the Hill-Hopkins-Ravenel\\n multiplicative norm.\\n\\nIf $\\\\underline{R}$ is a commutative Green functor, what is a Witt-vector\\nmodel for $\\\\pi_0(N^{S^1}_{C_p} H\\\\underline{R})^{C_{p^n}}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0154", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The problem was solved by Blumberg, Gerhardt, Hill, and Lawson: for n >= 1, the required C_{p^n}-Green functor is the zeroth relative twisted Hochschild homology W^{C_p}_{C_{p^n}}(R) = HH^{C_{p^n}}_{C_p}(R)_0, naturally isomorphic to the strict C_{p^n}-coinvariants of the Mackey norm N^{C_{p^n}}_{C_p}R. For any (-1)-connected genuine commutative C_p-ring spectrum E realizing R, its top-orbit value is naturally pi_0((N^{S^1}_{C_p}E)^{C_{p^n}}). In the relative indexing this is length n, not length n+1.\n\nCandidate contribution (worked_example; novelty confidence low): For Burnside input A_{C_p}, the complete ghost map on the top fixed-point ring A(C_{p^n}) is injective with image exactly the tuples (g_0,...,g_n) in Z^{n+1} satisfying g_j congruent to g_{j+1} modulo p^{n-j} for every 0 <= j < n." }, { "id": 20000155, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0155", "title": "Additive realization and a trivial-action gluing obstruction for a C3 analogue of Real bordism", "statement": "In order to solve the $3$-primary Kervaire\n invariant one problem using analogous methods to those in\n \\cite{0908.3724}, one would need a $C_3$-equivariant version of\n $MU_{\\R}$.\n\nCan we describe such a $C_3$-commutative ring spectrum $MU_{\\R}$ so that\nits underlying Bousfield type is $MU$ and $\\langle MU^{\\Phi\n C_3} \\rangle = \\langle H\\F_3 \\rangle$?", "original_statement": "In order to solve the $3$-primary Kervaire\n invariant one problem using analogous methods to those in\n \\cite{0908.3724}, one would need a $C_3$-equivariant version of\n $MU_{\\R}$.\n\nCan we describe such a $C_3$-commutative ring spectrum $MU_{\\R}$ so that\nits underlying Bousfield type is $MU$ and $\\langle MU^{\\Phi\n C_3} \\rangle = \\langle H\\F_3 \\rangle$?", "clean_statement": "In order to solve the $3$-primary Kervaire\n invariant one problem using analogous methods to those in\n \\cite{0908.3724}, one would need a $C_3$-equivariant version of\n $MU_{\\R}$.\n\nCan we describe such a $C_3$-commutative ring spectrum $MU_{\\R}$ so that\nits underlying Bousfield type is $MU$ and $\\langle MU^{\\Phi\n C_3} \\rangle = \\langle H\\F_3 \\rangle$?", "statement_status": "exact", "statement_verification": "This is Problem 2.3, attributed to D. Ravenel, in the AIM list *Equivariant derived algebraic geometry*, section “Equivariant Stable Homotopy Theory.” The source says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Equivariant Stable Homotopy Theory\nSource item: 2.3\nSource URL: http://aimpl.org/equideralggeom/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[154]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In order to solve the $3$-primary Kervaire\\n invariant one problem using analogous methods to those in\\n \\\\cite{0908.3724}, one would need a $C_3$-equivariant version of\\n $MU_{\\\\R}$.\\n\\nCan we describe such a $C_3$-commutative ring spectrum $MU_{\\\\R}$ so that\\nits underlying Bousfield type is $MU$ and $\\\\langle MU^{\\\\Phi\\n C_3} \\\\rangle = \\\\langle H\\\\F_3 \\\\rangle$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0155", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Yang's recollement gives a genuine C3-spectrum with exact underlying spectrum MU and exact geometric fixed points HF3 by zero additive gluing. However, no ordinary E-infinity algebra in genuine C3-spectra can have trivial Borel action on exact MU and geometric-fixed Bousfield class HF3: the required unital gluing map would send the torsion unit of an HF3-Bousfield ring to the infinite-order unit of MU^{tC3}, whose order is detected by MU^{tC3} -> KU^{tC3} and the Nikolaus-Scholze calculation. Separately, Yang's full norm factorization fails on m_1=[CP^1]: direct coefficient extraction from Carmeli-Luecke's MU Frobenius formula after ordinary Todd base change gives image 3 beta in KU^{tC3}, without assuming an E-infinity Todd orientation.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate new synthesis: the AIM Bousfield data admit exact additive gluing, but the entire exact-MU, trivial-C3-action multiplicative ansatz is ruled out already by a unital Tate-gluing obstruction, with a separate full-norm obstruction detected on [CP^1]." }, { "id": 20000156, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0156", "title": "Real spectra and the hidden geometric stratum of a naive C2-spectrum", "statement": "What should the correct notion of a Real spectrum be? Is it ``more\" than a genuine $C_2$-spectrum? (see \\cite{0908.3724} Sec B12)", "original_statement": "What should the correct notion of a Real spectrum be? Is it ``more\" than a genuine $C_2$-spectrum? (see \\cite{0908.3724} Sec B12)", "clean_statement": "What should the correct notion of a Real spectrum be? Is it ``more\" than a genuine $C_2$-spectrum? (see \\cite{0908.3724} Sec B12)", "statement_status": "exact", "statement_verification": "The exact canonical AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Equivariant Stable Homotopy Theory\nSource item: 2.4\nSource URL: http://aimpl.org/equideralggeom/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[155]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What should the correct notion of a Real spectrum be? Is it ``more\\\" than a genuine $C_2$-spectrum? (see \\\\cite{0908.3724} Sec B12)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0156", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "In the precise Hill-Hopkins-Ravenel sense cited by the problem, Real spectra and genuine C2-spectra present equivalent symmetric monoidal homotopy theories (HHR Proposition B.226), so a Real spectrum is not more than a genuine C2-spectrum. What is strictly less is a naive spectrum with conjugation. For every chosen naive C2-spectrum B, the infinity-category of genuine lifts equipped with a specified identification of their naive part with B is proved to be equivalent to the slice category Sp over B^{tC2}. The zero-underlying fiber is explicitly realized by J(K) = tilde(E C2) smash inf(K), with geometric fixed points K and RO(C2)-graded groups pi_{a+b sigma}^{C2} J(K) = pi_a K.\n\nCandidate contribution (equivalence; novelty confidence low): Candidate synthesis: for a fixed actual naive C2-spectrum B, retaining the specified identification rather than quotienting by Aut(B), Lift(B) is equivalent to Sp_{/B^{tC2}}; moreover the entire zero-underlying fiber is explicitly represented by J(K) = tilde(E C2) smash inf(K), with pi_{a+b sigma}^{C2} J(K) isomorphic to pi_a K." }, { "id": 20000157, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0157", "title": "Ordinary thick closure of representation spheres for finite groups", "statement": "Can we understand the thick subcategory of $Sp^G$ generated by representation spheres?", "original_statement": "Can we understand the thick subcategory of $Sp^G$ generated by representation spheres?", "clean_statement": "Can we understand the thick subcategory of $Sp^G$ generated by representation spheres?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 2.5 in the section “Equivariant Stable Homotopy Theory” of the workshop *Equivariant derived algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Equivariant Stable Homotopy Theory\nSource item: 2.5\nSource URL: http://aimpl.org/equideralggeom/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[156]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we understand the thick subcategory of $Sp^G$ generated by representation spheres?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0157", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite group G, representation spheres ordinary-thick-generate all compact genuine G-spectra if and only if G is an elementary abelian 2-group. The positive direction is integral: coordinate sign-representation spheres have face filtrations that inductively isolate every orbit E/K_+. For every other finite group, the rational e-factor of the free orbit is the regular module Q[G], which has a simple constituent not obtainable from the one-dimensional determinant characters of representation spheres. More generally, the compact rational thick closure is classified factorwise over H by the shifted Weyl-group orientation lines det(V^H).\n\nCandidate contribution (classification theorem; novelty confidence low): Candidate novelty: for every finite group G, the ordinary thick subcategory generated by representation spheres equals all compact genuine G-spectra exactly when G is isomorphic to (C_2)^r; the positive proof uses an explicit sign-chamber filtration and the negative proof uses a rational regular-module obstruction." }, { "id": 20000158, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0158", "title": "An equivariant Thomason theorem and a naive-promotion obstruction", "statement": "In \\cite{MR1337494}, Thomason showed that\n connective spectra can be modeled by symmetric monoidal categories.\n\nIs a similar statement true equivariantly? That is, is there an\nequivalence between $G$-symmetric monoidal categories and the homotopy\ncategory of connective $G$-spectra?", "original_statement": "In \\cite{MR1337494}, Thomason showed that\n connective spectra can be modeled by symmetric monoidal categories.\n\nIs a similar statement true equivariantly? That is, is there an\nequivalence between $G$-symmetric monoidal categories and the homotopy\ncategory of connective $G$-spectra?", "clean_statement": "In \\cite{MR1337494}, Thomason showed that\n connective spectra can be modeled by symmetric monoidal categories.\n\nIs a similar statement true equivariantly? That is, is there an\nequivalence between $G$-symmetric monoidal categories and the homotopy\ncategory of connective $G$-spectra?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 2.6 in the “Equivariant Stable Homotopy Theory” section of the AIM workshop *Equivariant derived algebraic geometry* (June 13--17, 2016). Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Equivariant Stable Homotopy Theory\nSource item: 2.6\nSource URL: http://aimpl.org/equideralggeom/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[157]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In \\\\cite{MR1337494}, Thomason showed that\\n connective spectra can be modeled by symmetric monoidal categories.\\n\\nIs a similar statement true equivariantly? That is, is there an\\nequivalence between $G$-symmetric monoidal categories and the homotopy\\ncategory of connective $G$-spectra?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0158", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For finite G, the 2016 question has an affirmative answer only after the Thomason-style localization: Lenz's published Theorem 8.15 identifies the Bousfield localization of genuine permutative G-categories with connective genuine G-spectra, genuine strictification extends this to genuine symmetric monoidal G-categories, and Calle--Chan--Peroux Corollary 4.11 gives the direct homotopy-category equivalence for symmetric monoidal Mackey functors. This attempt additionally proves that for nontrivial finite G the underlying-naive functor from connective genuine G-spectra to connective naive G-spectra has no essentially surjective section; the hidden family J_G(K)=tilde(E P) smash p^*K has zero underlying naive spectrum and geometric G-fixed points K.\n\nCandidate contribution (obstruction theorem; novelty confidence low): For every nontrivial finite group G, no functor L from connective naive G-spectra to connective genuine G-spectra can satisfy both U L equivalent to the identity and essential surjectivity; consequently no equivariant Thomason machine obtained by ordinary K-theory of a category with G-action followed by such an underlying-compatible promotion can model all connective genuine G-spectra." }, { "id": 20000159, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0159", "title": "Arithmetic-fracture coefficient decompletions of TP(F_p)", "statement": "Long Term\n\nCan we construct a decompleted version of the Tate spectrum?", "original_statement": "Long Term\n\nCan we construct a decompleted version of the Tate spectrum?", "clean_statement": "Long Term\n\nCan we construct a decompleted version of the Tate spectrum?", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 2.7 in the “Equivariant Stable Homotopy Theory” section of the June 2016 workshop *Equivariant derived algebraic geometry*. Its exact problem field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Equivariant Stable Homotopy Theory\nSource item: 2.7\nSource URL: http://aimpl.org/equideralggeom/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[158]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Long Term\\n\\nCan we construct a decompleted version of the Tate spectrum?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0159", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For T = TP(F_p) with its preferred divided Bott element v, the E-infinity pullbacks T x_{T[1/p]} H Q[v^{+/-1}] and T x_{T[1/p]} H Frac(Z_(p)^h)[v^{+/-1}] have homotopy rings Z_(p)[v^{+/-1}] and Z_(p)^h[v^{+/-1}], respectively. Their maps to T are equivalences modulo p^n for every n, so their derived p-completions recover T. This realizes both coefficient targets named in Hesselholt's workshop contribution, but only for this single object and not as a functorial decompleted Tate construction.\n\nCandidate contribution (construction; novelty confidence low): The explicit arithmetic-fracture E-infinity pullbacks over TP(F_p)[1/p] realize exactly Z_(p)[v^{+/-1}] and Z_(p)^h[v^{+/-1}], introduce no odd homotopy, and agree with TP(F_p) modulo every p^n." }, { "id": 20000160, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0160", "title": "A rational orientation-kernel presentation of the equivariant sphere ring", "statement": "In \\cite{MR2061856} Morel showed that Milnor-Witt K-theory of a field is isomorphic to the graded endomorphism ring of the unit object in the stable homotopy category of motivic spectra over that field.\n\nCan we prove a similar result in the setting of equivariant spectra? This would aid in understanding the tensor triangular geometry of $Sp^G$.", "original_statement": "In \\cite{MR2061856} Morel showed that Milnor-Witt K-theory of a field is isomorphic to the graded endomorphism ring of the unit object in the stable homotopy category of motivic spectra over that field.\n\nCan we prove a similar result in the setting of equivariant spectra? This would aid in understanding the tensor triangular geometry of $Sp^G$.", "clean_statement": "In \\cite{MR2061856} Morel showed that Milnor-Witt K-theory of a field is isomorphic to the graded endomorphism ring of the unit object in the stable homotopy category of motivic spectra over that field.\n\nCan we prove a similar result in the setting of equivariant spectra? This would aid in understanding the tensor triangular geometry of $Sp^G$.", "statement_status": "exact", "statement_verification": "This is Problem 2.9 in the “Equivariant Stable Homotopy Theory” section of the AIM workshop list *Equivariant derived algebraic geometry*. The canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Equivariant Stable Homotopy Theory\nSource item: 2.9\nSource URL: http://aimpl.org/equideralggeom/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[159]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In \\\\cite{MR2061856} Morel showed that Milnor-Witt K-theory of a field is isomorphic to the graded endomorphism ring of the unit object in the stable homotopy category of motivic spectra over that field.\\n\\nCan we prove a similar result in the setting of equivariant spectra? This would aid in understanding the tensor triangular geometry of $Sp^G$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0160", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every finite group G, let Lambda_H consist of the virtual real G-representations alpha whose H-fixed part has virtual dimension zero and trivial determinant character as a representation of W_G(H). The rational RO(G)-graded endomorphism ring of the genuine G-sphere has canonical one-dimensional H-components precisely in degrees alpha in Lambda_H, and after choosing lattice bases and orientations it is noncanonically isomorphic to the product over subgroup conjugacy classes of the group algebras Q[Lambda_H]. The proof also shows that the representation-sphere graded comparison map identifies the rational finite-G Balmer spectrum with the homogeneous spectrum of this ring. This is a rigorous rational finite-group partial answer, not an integral or compact-Lie-group solution.\n\nCandidate contribution (multiplicative_synthesis; novelty confidence low): The explicit all-finite-group multiplicative presentation R_G^Q = product_(H) Q[Lambda_H], with coherent Laurent generators and the resulting homogeneous-spectrum comparison, is a candidate synthesis beyond the published additive rank formula." }, { "id": 20000161, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0161", "title": "Elementary-abelian Balmer intervals factor as a subspace lattice times a chain", "statement": "Can we get a better understanding of the tensor triangular geometry of $Sp^G$? (See Problem 1.6)", "original_statement": "Can we get a better understanding of the tensor triangular geometry of $Sp^G$? (See Problem 1.6)", "clean_statement": "Can we get a better understanding of the tensor triangular geometry of $Sp^G$? (See Problem 1.6)", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 2.8 from the workshop *Equivariant derived algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Equivariant Stable Homotopy Theory\nSource item: 2.8\nSource URL: http://aimpl.org/equideralggeom/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[160]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we get a better understanding of the tensor triangular geometry of $Sp^G$? (See Problem 1.6)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0161", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For A=(C_p)^r in the p-local category and finite heights, every prime-ideal inclusion P(V,n) subseteq P(W,m), with d=dim(W/V) and e=n-m-d, has interval canonically isomorphic to the product of the subspace lattice L(W/V) and the opposite chain C_{e+1}. This yields a rank n-m grading, an exact cover classification, binomial(n-m,d)[d]_p! saturated chains, and an explicit Mobius value: (-1)^d p^{binomial(d,2)} for e=0, its negative for e=1, and zero for e at least 2.\n\nCandidate contribution (theorem; novelty confidence low): The finite-height inclusion interval factorization for A=(C_p)^r, together with its cover, saturated-chain, and Mobius formulas, is a concrete candidate new combinatorial refinement of the known prime-inclusion classification." }, { "id": 20000162, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0162", "title": "A modern étale criterion and an all-characteristic constant-functor test", "statement": "What is an {\\'etale} map of Green or Tambara functors?", "original_statement": "What is an {\\'etale} map of Green or Tambara functors?", "clean_statement": "What is an {\\'etale} map of Green or Tambara functors?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List item 3.1 from the workshop *Equivariant derived algebraic geometry*, in the section “Connecting Equivariant Notions and Derived Algebraic Geometry.” Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Connecting Equivariant Notions and Derived Algebraic Geometry\nSource item: 3.1\nSource URL: http://aimpl.org/equideralggeom/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[161]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is an {\\\\'etale} map of Green or Tambara functors?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0162", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every finite group G and ring map A to B, the cohomological constant Tambara module c_G(B) is box-flat over c_G(A) if and only if B is A-flat, and its genuine Tambara differentials vanish if and only if the ordinary Kähler differentials of B over A vanish; the same vanishing equivalence holds for Green differentials. Thus the constant Tambara map is formally étale in Hill's sense exactly when B is flat with zero ordinary differentials. Tambara finite presentation of c_G(B) reflects to ordinary finite presentation of B. In particular, c_G(F_p) to c_G(algebraic closure of F_p) is formally étale but not finitely presented, in every characteristic and for every finite G.\n\nCandidate contribution (criterion; novelty confidence low): The candidate contribution is the all-characteristic constant-reflection package: box-flatness and vanishing of genuine or Green differentials are each detected exactly by the underlying ring map, Tambara finite presentation reflects to rings, and c_G(F_p) to c_G(algebraic closure of F_p) gives a formal-but-not-finitely-presented boundary example even when p divides the group order." }, { "id": 20000163, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0163", "title": "Explicit rational Picard generators and the prime-cyclic index-two obstruction", "statement": "What are the explicit generators of $Pic(Sp^G)$?", "original_statement": "What are the explicit generators of $Pic(Sp^G)$?", "clean_statement": "What are the explicit generators of $Pic(Sp^G)$?", "statement_status": "exact", "statement_verification": "The canonical record is problem 3.2 in the workshop *Equivariant derived algebraic geometry*, section “Connecting Equivariant Notions and Derived Algebraic Geometry.” Its exact mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Connecting Equivariant Notions and Derived Algebraic Geometry\nSource item: 3.2\nSource URL: http://aimpl.org/equideralggeom/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[162]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What are the explicit generators of $Pic(Sp^G)$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0163", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every finite group G, Barnes's symmetric monoidal rational model gives Pic(Sp^G_Q) isomorphic to the direct sum over conjugacy classes (H) of Z plus Hom(W_GH,{+1,-1}). Explicit generators are one shift in each subgroup block and a basis of one-dimensional rational Weyl characters. Under these coordinates, a representation sphere V maps to the blockwise pair (dim(V^H),det(V^H)). As a sharp consequence, for every prime p the cokernel of RO(C_p) -> Pic(Sp^{C_p}_Q) is C_2: at p=2 it is generated by a zero-dimensional Weyl-sign twist, while for odd p it is generated by a shift in only the trivial-subgroup block whose square is a nontrivial rotation-representation sphere.\n\nCandidate contribution (generator theorem and obstruction; novelty confidence low): For every prime p, coker(RO(C_p) -> Pic(Sp^{C_p}_Q)) is C_2; the missing class is an order-two zero-dimensional sign twist for p=2 and a one-block half-shift U_e with U_e^2 equivalent to S^lambda_Q for odd p." }, { "id": 20000164, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0164", "title": "Fixed-point marks refute the real-signature and plane-branch comparisons", "statement": "Wall, Roberts, Damon, Gusein-Zade, and others have defined and studied the local $G$-degree for an equivariant polynomial map $f: \\R^n \\rightarrow \\R^n$ where $\\R^n$ is given an action of a group $G$. Their work is motivated by singularity theory, and they show that the invariants encode interesting information, but equivariant homotopy theory does not appear explicitly. Perhaps the notions of degree appearing in these works are connected to the degree in equivariant homotopy theory. We pose several questions along these lines.\n\n(1) Suppose we are given a linear representation of a finite group $G$ on $\\C^n$ and a $G$-equivariant polynomial function $f : \\C^n \\rightarrow \\C^n$ with an isolated zero at the origin. Then the local algebra\n\\[ Q(f) := \\C [\\![ x_1, \\dots, x_n ]\\!]/(f_1, \\dots, f_n)\\]\nof $f$ is a $\\C$-algebra with $G$-action. Is this $G$-representation equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nRecall the local degree is an element of the Burnside ring, hence has an associated virtual permutation representation. Palamodov has shown that the rank of $Q(f)$ is the topological local degree.\n\n(2) Suppose that, in (1), we replace the complex numbers $\\C$ with the real numbers $\\R$. Then $Q(f)$ carries a distinguished G-invariant symmetric bilinear pairing $\\beta$. (See e.g. Eisenbud-Levine.)\n\nQuite generally, any finite dimensional real $G$-representation with $G$-invariant symmetric bilinear form $(V, \\beta)$ decomposes in an essentially unique manner as an orthogonal direct sum\n\\[ V = V_+ \\oplus V_- \\oplus V_0\\]\nwhere $\\beta$ is positive definite on $V_+$, negative definite on $V_-$, and zero on $V_0$. Call the virtual representation $[V_+] - [V_-]$ the $G$-signature of $V$. Is this virtual $G$-representation of $( Q(f), \\beta)$ equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nNote: Eisenbud-Levine and Khimshiashvili have proven that the signature of $\\beta$ equals the topological local degree of $f$.\n\n(3) Suppose now that $f : \\C^2 \\rightarrow \\C^2$ is the gradient of a $G$-invariant equation for a plane curve singularity $0 \\in X \\subset \\C^2$. Then $G$ acts on the analytic branches of the singularity. Is this $G$-set equal to the local degree of $f$ in equivariant homotopy theory?", "original_statement": "Wall, Roberts, Damon, Gusein-Zade, and others have defined and studied the local $G$-degree for an equivariant polynomial map $f: \\R^n \\rightarrow \\R^n$ where $\\R^n$ is given an action of a group $G$. Their work is motivated by singularity theory, and they show that the invariants encode interesting information, but equivariant homotopy theory does not appear explicitly. Perhaps the notions of degree appearing in these works are connected to the degree in equivariant homotopy theory. We pose several questions along these lines.\n\n(1) Suppose we are given a linear representation of a finite group $G$ on $\\C^n$ and a $G$-equivariant polynomial function $f : \\C^n \\rightarrow \\C^n$ with an isolated zero at the origin. Then the local algebra\n\\[ Q(f) := \\C [\\![ x_1, \\dots, x_n ]\\!]/(f_1, \\dots, f_n)\\]\nof $f$ is a $\\C$-algebra with $G$-action. Is this $G$-representation equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nRecall the local degree is an element of the Burnside ring, hence has an associated virtual permutation representation. Palamodov has shown that the rank of $Q(f)$ is the topological local degree.\n\n(2) Suppose that, in (1), we replace the complex numbers $\\C$ with the real numbers $\\R$. Then $Q(f)$ carries a distinguished G-invariant symmetric bilinear pairing $\\beta$. (See e.g. Eisenbud-Levine.)\n\nQuite generally, any finite dimensional real $G$-representation with $G$-invariant symmetric bilinear form $(V, \\beta)$ decomposes in an essentially unique manner as an orthogonal direct sum\n\\[ V = V_+ \\oplus V_- \\oplus V_0\\]\nwhere $\\beta$ is positive definite on $V_+$, negative definite on $V_-$, and zero on $V_0$. Call the virtual representation $[V_+] - [V_-]$ the $G$-signature of $V$. Is this virtual $G$-representation of $( Q(f), \\beta)$ equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nNote: Eisenbud-Levine and Khimshiashvili have proven that the signature of $\\beta$ equals the topological local degree of $f$.\n\n(3) Suppose now that $f : \\C^2 \\rightarrow \\C^2$ is the gradient of a $G$-invariant equation for a plane curve singularity $0 \\in X \\subset \\C^2$. Then $G$ acts on the analytic branches of the singularity. Is this $G$-set equal to the local degree of $f$ in equivariant homotopy theory?", "clean_statement": "Wall, Roberts, Damon, Gusein-Zade, and others have defined and studied the local $G$-degree for an equivariant polynomial map $f: \\R^n \\rightarrow \\R^n$ where $\\R^n$ is given an action of a group $G$. Their work is motivated by singularity theory, and they show that the invariants encode interesting information, but equivariant homotopy theory does not appear explicitly. Perhaps the notions of degree appearing in these works are connected to the degree in equivariant homotopy theory. We pose several questions along these lines.\n\n(1) Suppose we are given a linear representation of a finite group $G$ on $\\C^n$ and a $G$-equivariant polynomial function $f : \\C^n \\rightarrow \\C^n$ with an isolated zero at the origin. Then the local algebra\n\\[ Q(f) := \\C [\\![ x_1, \\dots, x_n ]\\!]/(f_1, \\dots, f_n)\\]\nof $f$ is a $\\C$-algebra with $G$-action. Is this $G$-representation equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nRecall the local degree is an element of the Burnside ring, hence has an associated virtual permutation representation. Palamodov has shown that the rank of $Q(f)$ is the topological local degree.\n\n(2) Suppose that, in (1), we replace the complex numbers $\\C$ with the real numbers $\\R$. Then $Q(f)$ carries a distinguished G-invariant symmetric bilinear pairing $\\beta$. (See e.g. Eisenbud-Levine.)\n\nQuite generally, any finite dimensional real $G$-representation with $G$-invariant symmetric bilinear form $(V, \\beta)$ decomposes in an essentially unique manner as an orthogonal direct sum\n\\[ V = V_+ \\oplus V_- \\oplus V_0\\]\nwhere $\\beta$ is positive definite on $V_+$, negative definite on $V_-$, and zero on $V_0$. Call the virtual representation $[V_+] - [V_-]$ the $G$-signature of $V$. Is this virtual $G$-representation of $( Q(f), \\beta)$ equal to the permutation representation associated to the local degree in equivariant homotopy theory?\n\nNote: Eisenbud-Levine and Khimshiashvili have proven that the signature of $\\beta$ equals the topological local degree of $f$.\n\n(3) Suppose now that $f : \\C^2 \\rightarrow \\C^2$ is the gradient of a $G$-invariant equation for a plane curve singularity $0 \\in X \\subset \\C^2$. Then $G$ acts on the analytic branches of the singularity. Is this $G$-set equal to the local degree of $f$ in equivariant homotopy theory?", "statement_status": "exact", "statement_verification": "The canonical record is problem 3.3, “Connecting Equivariant Notions and Derived Algebraic Geometry,” from the AIM workshop *Equivariant derived algebraic geometry*. The record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Connecting Equivariant Notions and Derived Algebraic Geometry\nSource item: 3.3\nSource URL: http://aimpl.org/equideralggeom/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[163]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Wall, Roberts, Damon, Gusein-Zade, and others have defined and studied the local $G$-degree for an equivariant polynomial map $f: \\\\R^n \\\\rightarrow \\\\R^n$ where $\\\\R^n$ is given an action of a group $G$. Their work is motivated by singularity theory, and they show that the invariants encode interesting information, but equivariant homotopy theory does not appear explicitly. Perhaps the notions of degree appearing in these works are connected to the degree in equivariant homotopy theory. We pose several questions along these lines.\\n\\n(1) Suppose we are given a linear representation of a finite group $G$ on $\\\\C^n$ and a $G$-equivariant polynomial function $f : \\\\C^n \\\\rightarrow \\\\C^n$ with an isolated zero at the origin. Then the local algebra\\n\\\\[ Q(f) := \\\\C [\\\\![ x_1, \\\\dots, x_n ]\\\\!]/(f_1, \\\\dots, f_n)\\\\]\\nof $f$ is a $\\\\C$-algebra with $G$-action. Is this $G$-representation equal to the permutation representation associated to the local degree in equivariant homotopy theory?\\n\\nRecall the local degree is an element of the Burnside ring, hence has an associated virtual permutation representation. Palamodov has shown that the rank of $Q(f)$ is the topological local degree.\\n\\n(2) Suppose that, in (1), we replace the complex numbers $\\\\C$ with the real numbers $\\\\R$. Then $Q(f)$ carries a distinguished G-invariant symmetric bilinear pairing $\\\\beta$. (See e.g. Eisenbud-Levine.)\\n\\nQuite generally, any finite dimensional real $G$-representation with $G$-invariant symmetric bilinear form $(V, \\\\beta)$ decomposes in an essentially unique manner as an orthogonal direct sum\\n\\\\[ V = V_+ \\\\oplus V_- \\\\oplus V_0\\\\]\\nwhere $\\\\beta$ is positive definite on $V_+$, negative definite on $V_-$, and zero on $V_0$. Call the virtual representation $[V_+] - [V_-]$ the $G$-signature of $V$. Is this virtual $G$-representation of $( Q(f), \\\\beta)$ equal to the permutation representation associated to the local degree in equivariant homotopy theory?\\n\\nNote: Eisenbud-Levine and Khimshiashvili have proven that the signature of $\\\\beta$ equals the topological local degree of $f$.\\n\\n(3) Suppose now that $f : \\\\C^2 \\\\rightarrow \\\\C^2$ is the gradient of a $G$-invariant equation for a plane curve singularity $0 \\\\in X \\\\subset \\\\C^2$. Then $G$ acts on the analytic branches of the singularity. Is this $G$-set equal to the local degree of $f$ in equivariant homotopy theory?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0164", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "Burnside marks reduce all three comparisons to fixed-subspace local degrees. Rimmer's Corollary 6.2.8 gives the complex representation equality when a finite equivariant deformation has 0 as a regular value. In contrast, for C2 acting by sign on R and f(x)=-x^(4k+1), k>=1, the normalized ELK G-signature is minus the trivial representation while the stable local degree has marks (-1,1) and linearizes to minus the sign representation. For every reduced singular plane-curve germ, the identity marks of the branch set and gradient degree are r and mu; the formula mu=2delta-r+1 makes equality impossible by parity.\n\nCandidate contribution (counterexample; novelty confidence low): The singular family C2 acting by sign with f(x)=-x^(4k+1) gives sig_G(ELK)=-1 but linearized stable degree=-sign for every k>=1, and the branch-set comparison fails for every reduced singular plane curve because its two identity marks have opposite parity." }, { "id": 20000165, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0165", "title": "Borel blindness and a geometric-fixed etale site", "statement": "Long Term\n\nHow best should we be making sense of equivariant derived algebraic geometry? (e.g. What is Spec? What site to use? What sheaf? etc.)", "original_statement": "Long Term\n\nHow best should we be making sense of equivariant derived algebraic geometry? (e.g. What is Spec? What site to use? What sheaf? etc.)", "clean_statement": "Long Term\n\nHow best should we be making sense of equivariant derived algebraic geometry? (e.g. What is Spec? What site to use? What sheaf? etc.)", "statement_status": "exact", "statement_verification": "This is AIM Problem 3.4 in the workshop list *Equivariant derived algebraic geometry*, section “Connecting Equivariant Notions and Derived Algebraic Geometry.” The canonical record is at zero-based index 164 of `aim-algebraic-geometry-notes.json`. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Connecting Equivariant Notions and Derived Algebraic Geometry\nSource item: 3.4\nSource URL: http://aimpl.org/equideralggeom/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[164]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Long Term\\n\\nHow best should we be making sense of equivariant derived algebraic geometry? (e.g. What is Spec? What site to use? What sheaf? etc.)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0165", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every nontrivial finite group G, the canonical augmentation from the Burnside Tambara functor to the fixed-point Tambara functor FP(Z) is the identity at the free orbit but is not an isomorphism at G/G; Ullman's Eilenberg-Mac Lane realization therefore gives a naive G-equivariant E-infinity equivalence that is not a genuine equivalence, proving that quotient-stack Spec is nonconservative on genuine affines. On the full subcategory of CAlg(Sp_G) whose geometric fixed rings are connective, requiring a family to be derived etale and jointly covering after every geometric fixed-point functor defines a Grothendieck pretopology.\n\nCandidate contribution (no-go theorem and pretopology; novelty confidence low): The paired package gives, uniformly for every nontrivial finite G, the explicit regression map A_G -> FP(Z) against which any genuine Spec must be conservative, and proves that simultaneous geometric-fixed derived-etale detection satisfies the pretopology axioms on the stated genuine-additive connective affine category." }, { "id": 20000166, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0166", "title": "Continuous Galois spectra and an infinite-rank finite-field obstruction", "statement": "If $G$ is the Galois group of a finite extension $L\n \\rightarrow K$ there is a functor $Sp_G \\rightarrow Sp^{Mot}_K$\n given by $\\Sigma^{\\infty} G/H_+ \\rightarrow \\Sigma^{\\infty}\n Spec(L^H)_+$. Moreover, if $K$ is a real closed field, this functor\n is fully faithful.\n\nIf $G$ is a profinite absolute Galois group of $K$, is there a theory\nof $G-spectra$ such that $Sp_{G} \\rightarrow Sp^{Mot}_K$ is ``rich''?", "original_statement": "If $G$ is the Galois group of a finite extension $L\n \\rightarrow K$ there is a functor $Sp_G \\rightarrow Sp^{Mot}_K$\n given by $\\Sigma^{\\infty} G/H_+ \\rightarrow \\Sigma^{\\infty}\n Spec(L^H)_+$. Moreover, if $K$ is a real closed field, this functor\n is fully faithful.\n\nIf $G$ is a profinite absolute Galois group of $K$, is there a theory\nof $G-spectra$ such that $Sp_{G} \\rightarrow Sp^{Mot}_K$ is ``rich''?", "clean_statement": "If $G$ is the Galois group of a finite extension $L\n \\rightarrow K$ there is a functor $Sp_G \\rightarrow Sp^{Mot}_K$\n given by $\\Sigma^{\\infty} G/H_+ \\rightarrow \\Sigma^{\\infty}\n Spec(L^H)_+$. Moreover, if $K$ is a real closed field, this functor\n is fully faithful.\n\nIf $G$ is a profinite absolute Galois group of $K$, is there a theory\nof $G-spectra$ such that $Sp_{G} \\rightarrow Sp^{Mot}_K$ is ``rich''?", "statement_status": "exact", "statement_verification": "The canonical record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Extensions to Profinite, Compact Lie Groups, etc.\nSource item: 4.1\nSource URL: http://aimpl.org/equideralggeom/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[165]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If $G$ is the Galois group of a finite extension $L\\n \\\\rightarrow K$ there is a functor $Sp_G \\\\rightarrow Sp^{Mot}_K$\\n given by $\\\\Sigma^{\\\\infty} G/H_+ \\\\rightarrow \\\\Sigma^{\\\\infty}\\n Spec(L^H)_+$. Moreover, if $K$ is a real closed field, this functor\\n is fully faithful.\\n\\nIf $G$ is a profinite absolute Galois group of $K$, is there a theory\\nof $G-spectra$ such that $Sp_{G} \\\\rightarrow Sp^{Mot}_K$ is ``rich''?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0166", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The modern continuous genuine category Sp_G for a profinite absolute Galois group admits a finite-etale-span comparison to motivic spectra with the required open-orbit formula. For k = F_q of odd cardinality and G_k = hat(Z), the induced map on endomorphisms of the compact unit has a countably infinite-rank kernel: if e_n is the class of the orbit hat(Z)/n hat(Z), then the elements 2(e_n - n e_1), n >= 2, are nonzero, Z-linearly independent, and map to zero. The rational classes e_n - n e_1 are likewise independent kernel elements, so neither integral nor rational raw full faithfulness can hold over odd finite fields.\n\nCandidate contribution (obstruction; novelty confidence low): For every odd finite field F_q, the classes 2([hat(Z)/n hat(Z)] - n[hat(Z)/hat(Z)]) for n >= 2 form a countably infinite Z-linearly independent family in the kernel of the sphere-level finite-continuous Dress comparison A_fin(hat(Z)) -> GW(F_q); after tensoring with Q, the factor 2 is unnecessary." }, { "id": 20000167, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0167", "title": "Finite-index Wirthmuller duality and the compact profinite Adams boundary", "statement": "For a profinite group $G$ (having subgroups of arbitrary index), are there Wirthm\\\"uller/Adams isomomorphisms in $Sp^G$?", "original_statement": "For a profinite group $G$ (having subgroups of arbitrary index), are there Wirthm\\\"uller/Adams isomomorphisms in $Sp^G$?", "clean_statement": "For a profinite group $G$ (having subgroups of arbitrary index), are there Wirthm\\\"uller/Adams isomomorphisms in $Sp^G$?", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Extensions to Profinite, Compact Lie Groups, etc.\nSource item: 4.2\nSource URL: http://aimpl.org/equideralggeom/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[166]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For a profinite group $G$ (having subgroups of arbitrary index), are there Wirthm\\\\\\\"uller/Adams isomomorphisms in $Sp^G$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0167", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In the continuous genuine category of profinite G-spectra, restriction along a closed subgroup H has its extra left adjoint exactly when H is open; in that case finite-quotient descent gives the untwisted equivalence Ind_H^G ≃ Coind_H^G. For non-open H, locally constant degree functions on the infinite profinite orbit G/H show that Coind_H^G(S_H) is not compact, and the Balmer-Dell'Ambrogio-Sanders Grothendieck-Neeman criterion rules out the left adjoint. For Adams duality, every compact N-free G-spectrum is zero when the closed normal subgroup N is infinite, whereas for finite N every compact N-free spectrum descends to a finite quotient and inherits the Reich-Varisco Adams equivalence.\n\nCandidate contribution (theorem; novelty confidence low): Candidate new synthesis: closed-subgroup restriction in continuous genuine profinite spectra has a Wirthmuller left adjoint exactly at finite index, and the compact N-free subcategory is nonzero only for finite normal kernels; finite kernels inherit Adams equivalences from one finite quotient." }, { "id": 20000168, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0168", "title": "Compact-Lie norms and a fixed-point obstruction", "statement": "In genuine $G$-spectra we have multiplicative norm and\n transfer maps when $G$ has finitely indexed subgroups. Furthermore,\n (via Ando-Morava) there are objects that look like what may arise\n from a multiplicative norm for compact Lie groups.\n\nAre there analogues of multiplicative norms or multiplicative\ntransfers for compact Lie groups?", "original_statement": "In genuine $G$-spectra we have multiplicative norm and\n transfer maps when $G$ has finitely indexed subgroups. Furthermore,\n (via Ando-Morava) there are objects that look like what may arise\n from a multiplicative norm for compact Lie groups.\n\nAre there analogues of multiplicative norms or multiplicative\ntransfers for compact Lie groups?", "clean_statement": "In genuine $G$-spectra we have multiplicative norm and\n transfer maps when $G$ has finitely indexed subgroups. Furthermore,\n (via Ando-Morava) there are objects that look like what may arise\n from a multiplicative norm for compact Lie groups.\n\nAre there analogues of multiplicative norms or multiplicative\ntransfers for compact Lie groups?", "statement_status": "exact", "statement_verification": "The canonical record is problem 4.3 in the AIM workshop list *Equivariant derived algebraic geometry*, section “Extensions to Profinite, Compact Lie Groups, etc.” The source page attributes the question to C. Rezk. The canonical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Extensions to Profinite, Compact Lie Groups, etc.\nSource item: 4.3\nSource URL: http://aimpl.org/equideralggeom/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[167]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In genuine $G$-spectra we have multiplicative norm and\\n transfer maps when $G$ has finitely indexed subgroups. Furthermore,\\n (via Ando-Morava) there are objects that look like what may arise\\n from a multiplicative norm for compact Lie groups.\\n\\nAre there analogues of multiplicative norms or multiplicative\\ntransfers for compact Lie groups?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0168", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The historical existence question now has a rigorous affirmative answer on commutative equivariant ring spectra via the left adjoint L_H^G(R)=R tensor_H G, while the general positive-dimensional factorization-norm package remains partly conjectural. For closed H,K in a compact Lie group G and a suitably cofibrant commutative H-ring R, this attempt proves that if the identity component K^circ is contained in no conjugate of H, then the unit induces an equivalence S -> Phi^K L_H^G(R). For normal K this applies exactly when H has infinite index in HK. For G=S^1, H=C_m, K=C_n, it also proves that the q-th cardinality-filtration layer can survive geometric C_n-fixed points only if n/gcd(n,m) divides q.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: (G/H)^(K^circ)=empty implies Phi^K(R tensor_H G) is equivalent to the sphere for the derived commutative-algebra norm; additionally, for (S^1,C_m,C_n), positive filtration layers vanish unless n/gcd(n,m) divides their cardinality." }, { "id": 20000169, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0169", "title": "Prime support of possible odd equivariant MU-classes", "statement": "Quillen's theorem for $MU$ states that the map from the Lazard ring $L$ to $\\pi_* MU$ is an isomorphism and so, in particular, the homotopy of $MU$ is concentrated in even degrees. Furthermore, when $G$ is a finite abelian group, it is known that\n the $G$-equivariant homotopy of $MU$ is also even.\n\nFor any finite or compact Lie group $G$, is it also the case that\n$\\pi^G_*(MU)$ is even?", "original_statement": "Quillen's theorem for $MU$ states that the map from the Lazard ring $L$ to $\\pi_* MU$ is an isomorphism and so, in particular, the homotopy of $MU$ is concentrated in even degrees. Furthermore, when $G$ is a finite abelian group, it is known that\n the $G$-equivariant homotopy of $MU$ is also even.\n\nFor any finite or compact Lie group $G$, is it also the case that\n$\\pi^G_*(MU)$ is even?", "clean_statement": "Quillen's theorem for $MU$ states that the map from the Lazard ring $L$ to $\\pi_* MU$ is an isomorphism and so, in particular, the homotopy of $MU$ is concentrated in even degrees. Furthermore, when $G$ is a finite abelian group, it is known that\n the $G$-equivariant homotopy of $MU$ is also even.\n\nFor any finite or compact Lie group $G$, is it also the case that\n$\\pi^G_*(MU)$ is even?", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Extensions to Profinite, Compact Lie Groups, etc.\nSource item: 4.4\nSource URL: http://aimpl.org/equideralggeom/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[168]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Quillen's theorem for $MU$ states that the map from the Lazard ring $L$ to $\\\\pi_* MU$ is an isomorphism and so, in particular, the homotopy of $MU$ is concentrated in even degrees. Furthermore, when $G$ is a finite abelian group, it is known that\\n the $G$-equivariant homotopy of $MU$ is also even.\\n\\nFor any finite or compact Lie group $G$, is it also the case that\\n$\\\\pi^G_*(MU)$ is even?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0169", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every finite group G, the odd integer-graded coefficients of the genuine equivariant Thom spectrum MU_G vanish after inverting |G|; equivalently, each odd coefficient class is killed by a power of |G|. Consequently, flatness of (MU_G)_* over MU_* forces integral evenness. Combining this implication with Sophie Kriz's 2025 theorem shows that her p-Sylow examples are unconditionally non-flat, while it does not assert that they contain odd MU_G-classes.\n\nCandidate contribution (theorem; novelty confidence low): The candidate contribution is the explicit combination of Schwede's finite-group ghost-map isomorphism away from |G| with Sinha's even geometric-fixed-point calculation, yielding (MU_G)_odd[1/|G|] = 0, and the deduction that the p-Sylow examples in Sophie Kriz's Theorem A are necessarily non-flat." }, { "id": 20000170, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0170", "title": "Component-group blindness and twisted ambidexterity for compact Lie groups", "statement": "Long Term\n\nHow should we think about equivariant stable homotopy theory in the context of infinite compact Lie groups?", "original_statement": "Long Term\n\nHow should we think about equivariant stable homotopy theory in the context of infinite compact Lie groups?", "clean_statement": "Long Term\n\nHow should we think about equivariant stable homotopy theory in the context of infinite compact Lie groups?", "statement_status": "exact", "statement_verification": "The canonical record is problem 4.5 in the AIM list *Equivariant derived algebraic geometry*, section “Extensions to Profinite, Compact Lie Groups, etc.” Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Extensions to Profinite, Compact Lie Groups, etc.\nSource item: 4.5\nSource URL: http://aimpl.org/equideralggeom/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[169]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Long Term\\n\\nHow should we think about equivariant stable homotopy theory in the context of infinite compact Lie groups?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0170", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a compact Lie group G with identity component N and component group Q, every finite continuous G-set factors uniquely through Q, so any Burnside/span construction generated only by finite continuous G-sets sees only Q. If N is nontrivial, this cannot recover genuine G-spectra: every representation sphere S^V with V^N not equal to V lies outside the essential image of genuine inflation from Q, as shown by comparing its underlying suspension degree with the degree after N-geometric fixed points. As a companion compact-Lie obstruction, for a positive-dimensional torus T and every proper closed H a^p and every nonzero p-torsion character is a unital ring homomorphism. At supersingular mixed-characteristic reduction each nonzero character is congruent to a |-> a^p, so the naive generic identity-character formula cannot extend whenever residue Frobenius is nontrivial.\n\nCandidate contribution (obstruction lemma; novelty confidence low): For a normed C_p-equivariant elliptic theory with pi_0^{C_p} identified with functions on E[p], Tambara distributivity forces each nonzero split p-torsion norm character to be a ring map; over a supersingular mixed-characteristic DVR it is congruent to Frobenius, and global Aut(C_p)-naturality reduces the characters to the p+1 order-p subgroup or isogeny directions." }, { "id": 20000173, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0173", "title": "Rational weak relative Gorenstein duality at level 3 without perfectness", "statement": "Can we find explicit ways to write $S \\in R-Alg$ as an $R$-module? What are objects that ``untwist\" when smashed together? (e.g. $KU \\simeq KO \\wedge C(\\eta)$, $H\\F_2 \\simeq ko \\wedge \\A(1)$)\n\nA slightly easier question: Is $tmf^{(n)}_0 \\rightarrow tmf^{(n)}_1$ relatively Gorenstein?", "original_statement": "Can we find explicit ways to write $S \\in R-Alg$ as an $R$-module? What are objects that ``untwist\" when smashed together? (e.g. $KU \\simeq KO \\wedge C(\\eta)$, $H\\F_2 \\simeq ko \\wedge \\A(1)$)\n\nA slightly easier question: Is $tmf^{(n)}_0 \\rightarrow tmf^{(n)}_1$ relatively Gorenstein?", "clean_statement": "Can we find explicit ways to write $S \\in R-Alg$ as an $R$-module? What are objects that ``untwist\" when smashed together? (e.g. $KU \\simeq KO \\wedge C(\\eta)$, $H\\F_2 \\simeq ko \\wedge \\A(1)$)\n\nA slightly easier question: Is $tmf^{(n)}_0 \\rightarrow tmf^{(n)}_1$ relatively Gorenstein?", "statement_status": "exact", "statement_verification": "The exact AIM record is Problem 5.8 in the “Chromatic Homotopy Theory” section of the 2016 workshop *Equivariant derived algebraic geometry*. It is attributed on the AIM problem page to V. Stojanoska and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Chromatic Homotopy Theory\nSource item: 5.8\nSource URL: http://aimpl.org/equideralggeom/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[172]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we find explicit ways to write $S \\\\in R-Alg$ as an $R$-module? What are objects that ``untwist\\\" when smashed together? (e.g. $KU \\\\simeq KO \\\\wedge C(\\\\eta)$, $H\\\\F_2 \\\\simeq ko \\\\wedge \\\\A(1)$)\\n\\nA slightly easier question: Is $tmf^{(n)}_0 \\\\rightarrow tmf^{(n)}_1$ relatively Gorenstein?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0173", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Conditional on reconstructing the source notation as the connective level-structure map tmf_0(n) to tmf_1(n), the rational level-3 map A=tmf_0(3)_Q to B=tmf_1(3)_Q is relatively Gorenstein of shift 0 in the equation-only Dwyer-Greenlees-Iyengar/Greenlees sense: the transfer/Reynolds adjoint B -> F_A(B,A) is a B-linear equivalence. Nevertheless B is not perfect over A, even rationally. Hence any definition that includes perfectness fails already at level 3, including integrally by rational base change, and no finite spectrum X can give tmf_1(3) equivalent to tmf_0(3) smashed with X as a module.\n\nCandidate contribution (partial theorem; novelty confidence low): For R=Q[x,y,z]/(xz-y^2) and S=Q[u,v] with x=u^2, y=uv, z=v^2, the Reynolds pairing gives RHom_R(S,R) equivalent to S with no shift, while the odd eigensummand has an infinite minimal matrix-factorization resolution; this lifts to weak rational relative Gorenstein duality for tmf_0(3) -> tmf_1(3) and simultaneously obstructs every finite untwisting complex." }, { "id": 20000174, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0174", "title": "A fixed-point obstruction and two boundary cases for equivariant Landweber exactness", "statement": "When $G = C_{2^n}$ what is the $G$-equivariant\n Landweber exact functor theory for $MU^{(G)}$?", "original_statement": "When $G = C_{2^n}$ what is the $G$-equivariant\n Landweber exact functor theory for $MU^{(G)}$?", "clean_statement": "When $G = C_{2^n}$ what is the $G$-equivariant\n Landweber exact functor theory for $MU^{(G)}$?", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 5.6 in the “Chromatic Homotopy Theory” section of the 2016 workshop *Equivariant derived algebraic geometry*. It is record index 173 (zero-based) in `aim-algebraic-geometry-notes.json`:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Chromatic Homotopy Theory\nSource item: 5.6\nSource URL: http://aimpl.org/equideralggeom/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[173]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"When $G = C_{2^n}$ what is the $G$-equivariant\\n Landweber exact functor theory for $MU^{(G)}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0174", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the HHR norm spectrum R = N_{C_2}^{C_{2^n}} MU_R, the underlying geometric H-fixed-point spectrum is MU smashed 2^{n-1} times when H is trivial and MO smashed [G:H] times when H is nontrivial. Hence every nontrivial geometric stratum has pi_0 = F_2. Identity base change is nevertheless exact, so no necessary equivariant Landweber criterion can impose ordinary injectivity of multiplication by 2 independently on every geometric stratum. Mackey-module flatness is proved to be a strong sufficient criterion under explicit Kunneth convergence hypotheses, and canonical base change at trivial isotropy is proved to reduce to the classical Landweber exact functor theorem.\n\nCandidate contribution (obstruction; novelty confidence low): The fixed-point profile of the HHR norm, combined with the identity coefficient module, gives a testable no-go theorem for pointwise classical 2-regularity and, together with trivial-isotropy descent, localizes the unresolved equivariant Landweber problem to restriction, transfer, Euler-class, and norm gluing." }, { "id": 20000175, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0175", "title": "Prime-to-order orbit diagrams and a C2 theta--Tambara coefficient model", "statement": "What is an algebraic model for $L_{K(1)}Comm(Sp_G)$ (for even small groups $G$)?", "original_statement": "What is an algebraic model for $L_{K(1)}Comm(Sp_G)$ (for even small groups $G$)?", "clean_statement": "What is an algebraic model for $L_{K(1)}Comm(Sp_G)$ (for even small groups $G$)?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, workshop **Equivariant derived algebraic geometry**, section 5, **Chromatic Homotopy Theory**, Problem 5.3:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Chromatic Homotopy Theory\nSource item: 5.3\nSource URL: http://aimpl.org/equideralggeom/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[174]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is an algebraic model for $L_{K(1)}Comm(Sp_G)$ (for even small groups $G$)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0175", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After fixing the localization ambiguity, two rigorous models emerge. If p does not divide the order of G, Wimmer's theorem and objectwise K(1)-localization identify fully normed genuine commutative G-rings, localized on every geometric fixed point, with Orb_G-diagrams of ordinary K(1)-local E-infinity rings. For p=2 and G=C2 in Balderrama's Borel localization, a K(1)-local E-infinity ring R with trivial action and theta-ring T=pi_0(R) has categorical fixed coefficient ring pi_0^{C2}b_{C2}(R)=T[h]/(h^2-2h), with res(x+yh)=x+2y, tr(a)=ah, and N(a)=psi(a)-theta(a)h. This yields a closed norm formula for the localized C2-sphere.\n\nCandidate contribution (explicit_algebraic_model; novelty confidence low): The theta-to-Tambara presentation T[h]/(h^2-2h) with N(a)=psi(a)-theta(a)h, together with N(n+delta epsilon)=n-((n-n^2)/2+delta(1-n)epsilon)h for the 2-primary K(1)-local sphere, is a concrete candidate new synthesis of the complete degree-zero Borel C2 structure." }, { "id": 20000176, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0176", "title": "Ordinary obstruction theory and a C2 norm-lift calculation", "statement": "Is there a Goerss-Hopkins obstruction theory for\n $Comm(Sp_G)$?", "original_statement": "Is there a Goerss-Hopkins obstruction theory for\n $Comm(Sp_G)$?", "clean_statement": "Is there a Goerss-Hopkins obstruction theory for\n $Comm(Sp_G)$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem 5.5 from the workshop *Equivariant derived algebraic geometry*, section “Chromatic Homotopy Theory”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Chromatic Homotopy Theory\nSource item: 5.5\nSource URL: http://aimpl.org/equideralggeom/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[175]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a Goerss-Hopkins obstruction theory for\\n $Comm(Sp_G)$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0176", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Mazel-Gee's 2024 theory gives a Goerss-Hopkins framework for ordinary operadic commutative algebras in genuine G-spectra under its standard detecting-homology hypotheses, but explicitly excludes genuine operations indexed by finite G-sets. For the missing normed theory, this attempt proves a concrete C2 test: for any F2-algebra R and R-module M, Tambara norms on the fixed Green pair with values R and R semidirect-product M, zero transfer, and projection restriction are classified by Frobenius derivations. For R=M=F2[t], this deformation group is F2[t] and the norms are N_c(f)=f^2+c(f')^2 epsilon.\n\nCandidate contribution (explicit classification and obstruction-group calculation; novelty confidence low): For the fixed characteristic-two C2 Green functor A=R, B=R semidirect-product M with trivial Weyl action, zero transfer, and projection restriction, genuine Tambara norm lifts are naturally FDer_F2(R,M); for R=M=F2[t] they form an F2[t]-family N_c(f)=f^2+c(f')^2 epsilon." }, { "id": 20000177, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0177", "title": "Tame normalization and two-axis integrality for level TMF", "statement": "What would ``rings of integers'' in $tmf$\n with level structure be? Also, is there a TAF analogue of such\n objects with\n connections to TMF?", "original_statement": "What would ``rings of integers'' in $tmf$\n with level structure be? Also, is there a TAF analogue of such\n objects with\n connections to TMF?", "clean_statement": "What would ``rings of integers'' in $tmf$\n with level structure be? Also, is there a TAF analogue of such\n objects with\n connections to TMF?", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0177, problem 5.2 in the “Chromatic Homotopy Theory” section of the AIM workshop *Equivariant derived algebraic geometry*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Chromatic Homotopy Theory\nSource item: 5.2\nSource URL: http://aimpl.org/equideralggeom/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[176]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What would ``rings of integers'' in $tmf$\\n with level structure be? Also, is there a TAF analogue of such\\n objects with\\n connections to TMF?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0177", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For full level N at N-inverted primes, the compactified level modular curve is the normalization of the compactified elliptic-moduli stack in its smooth finite etale cover. At a cusp of width w, strict-henselian completed coordinates have q=t^w and Kahler different (w t^{w-1}); this is tame ordinary ramification and Kummer log-etale ramification. A cohomological-dimension-one descent argument gives pi_0 Tmf(Gamma)=H^0(O), so a criterion using only pi_0, which connective cover preserves, cannot recover the cusp different, although the full connective E-infinity map may retain ramification. A conditional functoriality proposition isolates the additional compactification and logarithmic spectral-sheaf data needed for a TAF-to-TMF integral analogue.\n\nCandidate contribution (criterion; novelty confidence low): Candidate two-axis integrality criterion: for tame level, geometric/logarithmic normalization at the cusps and construction of a connective holomorphic E-infinity model are independent requirements; in particular, every proper geometrically connected level component has the same constant pi_0 while its cusp widths give different ideals (w t^{w-1}), so no pi_0-only construction can select or reconstruct the boundary integral model." }, { "id": 20000178, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0178", "title": "Spectral formal-group moduli and a square-zero comparison obstruction", "statement": "Long Term\n\nCan we describe the moduli stack of formal groups in the derived setting?", "original_statement": "Long Term\n\nCan we describe the moduli stack of formal groups in the derived setting?", "clean_statement": "Long Term\n\nCan we describe the moduli stack of formal groups in the derived setting?", "statement_status": "exact", "statement_verification": "There is no apparent OCR corruption. The workshop and section make spectral algebraic geometry, with connective or nonconnective \\(\\mathbb E_\\infty\\)-rings as test rings, the relevant meaning of “derived.” Another possible meaning, Lurie's *formal moduli problems* (infinitesimal deformation germs based at a point), is related but is not the global moduli problem asked for here.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Chromatic Homotopy Theory\nSource item: 5.7\nSource URL: http://aimpl.org/equideralggeom/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[177]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Long Term\\n\\nCan we describe the moduli stack of formal groups in the derived setting?\"\nOriginal remarks: [\"Was this question really about the derived moduli stack of formal groups or the moduli stack for derived formal groups or should they be one and the same?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0178", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The Lurie-Gregoric functor of one-dimensional spectral formal groups is an fpqc spectral stack recovering the classical stack on discrete rings, but it is not the naive extension A |-> M_FG(pi_0 A). For A = H Q plus the square-zero summand Sigma H V, scalar units form a retract of the automorphism space of the spectral formal additive group via the derivative on its dualizing line. Hence V is a retract of relative pi_1 of the automorphism fiber, equivalently relative pi_2 of the moduli fiber, while the pi_0-constant classical value is 1-truncated.\n\nCandidate contribution (obstruction; novelty confidence low): For every nonzero rational vector space V and A = H Q plus Sigma H V as a split square-zero E-infinity algebra, the reduction fiber of genuine spectral formal-group moduli at the additive point has V as a retract of its second homotopy group." }, { "id": 20000179, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0179", "title": "Euler inversion, punctured p-torsion, and equivariant chromatic primes", "statement": "We may examine how viewing the Tate spectrum of a given $G$-spectrum is related to chromatic localizations. For example, given the spectrum $KO$ with trivial $C_2$-action, we have\n\\[ (KO_{(2)})^{tC_2} \\simeq \\bigvee_{k\\in\\Z} \\Sigma^{4k} H\\Q_{(2)}\\]\n\nWhy should taking the Tate spectrum bring an object down in chromatic height (i.e. why do we see a blue shift)? More generally, we'd like to look at \"chromatic primes\" and \"equivariant primes\" in the same place and understand how they interact.\n\n[Needs editing - I'm sure Mark has a more eloquent statement]", "original_statement": "We may examine how viewing the Tate spectrum of a given $G$-spectrum is related to chromatic localizations. For example, given the spectrum $KO$ with trivial $C_2$-action, we have\n\\[ (KO_{(2)})^{tC_2} \\simeq \\bigvee_{k\\in\\Z} \\Sigma^{4k} H\\Q_{(2)}\\]\n\nWhy should taking the Tate spectrum bring an object down in chromatic height (i.e. why do we see a blue shift)? More generally, we'd like to look at \"chromatic primes\" and \"equivariant primes\" in the same place and understand how they interact.\n\n[Needs editing - I'm sure Mark has a more eloquent statement]", "clean_statement": "We may examine how viewing the Tate spectrum of a given $G$-spectrum is related to chromatic localizations. For example, given the spectrum $KO$ with trivial $C_2$-action, we have\n\\[ (KO_{(2)})^{tC_2} \\simeq \\bigvee_{k\\in\\Z} \\Sigma^{4k} H\\Q_{(2)}\\]\n\nWhy should taking the Tate spectrum bring an object down in chromatic height (i.e. why do we see a blue shift)? More generally, we'd like to look at \"chromatic primes\" and \"equivariant primes\" in the same place and understand how they interact.\n\n[Needs editing - I'm sure Mark has a more eloquent statement]", "statement_status": "exact", "statement_verification": "The canonical record is problem 5.4 in the “Chromatic Homotopy Theory” section of the AIM workshop page *Equivariant derived algebraic geometry*. Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Chromatic Homotopy Theory\nSource item: 5.4\nSource URL: http://aimpl.org/equideralggeom/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[178]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"We may examine how viewing the Tate spectrum of a given $G$-spectrum is related to chromatic localizations. For example, given the spectrum $KO$ with trivial $C_2$-action, we have\\n\\\\[ (KO_{(2)})^{tC_2} \\\\simeq \\\\bigvee_{k\\\\in\\\\Z} \\\\Sigma^{4k} H\\\\Q_{(2)}\\\\]\\n\\nWhy should taking the Tate spectrum bring an object down in chromatic height (i.e. why do we see a blue shift)? More generally, we'd like to look at \\\"chromatic primes\\\" and \\\"equivariant primes\\\" in the same place and understand how they interact.\\n\\n[Needs editing - I'm sure Mark has a more eloquent statement]\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0179", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source example is verified with a notation correction: for the trivial C2-action, (KO_(2))^{tC2} is a wedge of 4-fold suspensions of H Q_2, where Q_2 is the field of 2-adic rationals, not standard Q_(2)=Q. A proved punctured p-series unit test shows that in the completed complex-oriented C_p-Tate coefficient ring, Euler-class inversion and [p]_F(x)=0 force v_{n-1} to be a unit, while the height-n residue specialization is the zero ring. This simultaneously explains one-step blue shift and Morava K(n)-Tate vanishing, and it matches the sharp Balmer-prime movement P(e,n+1) subset P(C_p,n).\n\nCandidate contribution (lemma; novelty confidence low): In the I_{n-1}-completed truncated p-typical Laurent quotient R_n((x))/([p]_F(x)), the image of v_{n-1} is a unit, whereas specialization to the height-n residue point makes the quotient zero; equivalently, punctured p-torsion avoids the height-n chromatic prime and lies in the v_{n-1}-open part of the neighboring stratum." }, { "id": 20000180, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0180", "title": "Spectral local systems and augmentation-Bott orthogonality for S[BU]", "statement": "What can we say about modules over $S[BU_+]$?", "original_statement": "What can we say about modules over $S[BU_+]$?", "clean_statement": "What can we say about modules over $S[BU_+]$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0180, problem 5.9 in the “Chromatic Homotopy Theory” section of the AIM workshop *Equivariant derived algebraic geometry*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Equivariant derived algebraic geometry\nSection: Chromatic Homotopy Theory\nSource item: 5.9\nSource URL: http://aimpl.org/equideralggeom/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[179]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What can we say about modules over $S[BU_+]$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/equideralggeom/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0180", "aim-domain:algebraic-geometry", "aim-workshop:equideralggeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the standard spherical monoid-ring interpretation A = Sigma-infinity-plus BU, the module category is equivalent to spectral local systems on BBU, hence on SU. This identifies augmentation base change and augmentation cochains with colimits and limits over SU, computes the derived self-intersection as Sigma-infinity-plus SU, and proves that the augmentation sphere is neither compact nor dualizable. In addition, every Bott-local A-module is both tensor-orthogonal and mapping-orthogonal to the augmentation sphere; the nonzero Snaith localization therefore shows that augmentation cochains are not conservative on all A-modules.\n\nCandidate contribution (obstruction; novelty confidence low): For A = Sigma-infinity-plus BU and every A-module N on which the suspended Bott class beta acts invertibly, both S tensor_A N and F_A(S,N) are contractible; in particular this holds for the nonzero Snaith localization A[beta^{-1}], so the augmentation sphere is not a generator and its localizing subcategory is killed by Bott localization." }, { "id": 20000181, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0181", "title": "Real essential matrices in the calibrated five-point problem", "statement": "The essential variety is a 5-dimensional variety of degree 10. A point correspondence gives a linear constraint so a five point correspondence should give 10 complex solutions.\n\nOf the ten possibilities, how many of the solutions can be real?", "original_statement": "The essential variety is a 5-dimensional variety of degree 10. A point correspondence gives a linear constraint so a five point correspondence should give 10 complex solutions.\n\nOf the ten possibilities, how many of the solutions can be real?", "clean_statement": "The essential variety is a 5-dimensional variety of degree 10. A point correspondence gives a linear constraint so a five point correspondence should give 10 complex solutions.\n\nOf the ten possibilities, how many of the solutions can be real?", "statement_status": "exact", "statement_verification": "The exact AIM record (Algebraic Vision workshop, Reconstruction, Problem 1.05) is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Reconstruction\nSource item: 1.05\nSource URL: http://aimpl.org/algvision/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[180]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The essential variety is a 5-dimensional variety of degree 10. A point correspondence gives a linear constraint so a five point correspondence should give 10 complex solutions.\\n\\nOf the ten possibilities, how many of the solutions can be real?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0181", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a transverse real codimension-five section of the degree-ten essential variety, the number of real projective essential matrices is one of 0, 2, 4, 6, 8, or 10; if the data come from a nondegenerate real calibrated scene, the true essential matrix and parity force at least two real algebraic solutions. Ten is the rigorous degree upper bound, and Nister and Stewenius--Engels--Nister report well-separated realizable instances with all ten roots real and cheirality-valid, but no exact archived root certificate was found. As a proved structural addition, a clean proper Segre intersection shows that the span of five independent reduced real correspondence tensors contains a unique residual sixth real rank-one tensor imposing the same solution plane.\n\nCandidate contribution (lemma; novelty confidence low): If five independent real rank-one 3-by-3 correspondence tensors span a projective 4-plane whose complex intersection with the Segre variety is proper zero-dimensional and reduced at those five points, then the degree-six intersection has a unique residual sixth point, and that point is real and rank one; its epipolar hyperplane contains the same essential-matrix 3-plane." }, { "id": 20000182, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0182", "title": "Four ambient trifocal point equations have generic restricted rank three", "statement": "How many linear constraints does a point correspondence in three views determine on the trifocal tensor variety ?", "original_statement": "How many linear constraints does a point correspondence in three views determine on the trifocal tensor variety ?", "clean_statement": "How many linear constraints does a point correspondence in three views determine on the trifocal tensor variety ?", "statement_status": "exact", "statement_verification": "The exact canonical record is problem 1.1 in the “Reconstruction” section of the AIM workshop list *Algebraic vision*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Reconstruction\nSource item: 1.1\nSource URL: http://aimpl.org/algvision/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[181]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How many linear constraints does a point correspondence in three views determine on the trifocal tensor variety ?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Numerical results suggest that there are 4 constraints but they only intersect the trifocal tensor variety in codimension three.\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0182", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "A point triple gives nine displayed trifocal equations whose coefficient map has rank exactly four, because it is the composite of a surjection onto 3-by-3 matrices with left and right multiplication by rank-two cross-product matrices. These four independent ambient hyperplanes cut the 18-dimensional uncalibrated trifocal variety in a genuine correspondence component of codimension three. On a normalized general camera chart their pullbacks are the four entries of uv^T-st^T; the incidence branch has u=cs and t=cv on a dense chart, and its differential has rank three with a unique conormal dependence.\n\nCandidate contribution (local_normal_form; novelty confidence low): On the normalized general uncalibrated camera chart, the four point-correspondence hyperplanes pull back to A_0 B_0=0 for 2-by-2 matrices A_0=[u,-s] and B_0=[v^T;t^T]. Set-theoretically this pullback has one irreducible codimension-three complex component and two codimension-four degeneracy components; on the dense chart u=cs, t=cv, the unique conormal relation is generated by s-perp times (v-perp)^T." }, { "id": 20000183, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0183", "title": "Why six trifocal point triplets have algebraic degree three", "statement": "It is known that the trifocal tensor variety is dimension 18 with degree 297 inside of $\\mathbb{P}^{28}$. It is also known that a six point correspondence in three views is a minimal problem.\n\nHow do the 24 linear constraints lead a minimal problem? How many complex solutions are there?", "original_statement": "It is known that the trifocal tensor variety is dimension 18 with degree 297 inside of $\\mathbb{P}^{28}$. It is also known that a six point correspondence in three views is a minimal problem.\n\nHow do the 24 linear constraints lead a minimal problem? How many complex solutions are there?", "clean_statement": "It is known that the trifocal tensor variety is dimension 18 with degree 297 inside of $\\mathbb{P}^{28}$. It is also known that a six point correspondence in three views is a minimal problem.\n\nHow do the 24 linear constraints lead a minimal problem? How many complex solutions are there?", "statement_status": "exact", "statement_verification": "The canonical AIM record (Algebraic Vision, Reconstruction, Problem 1.15) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Reconstruction\nSource item: 1.15\nSource URL: http://aimpl.org/algvision/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[182]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"It is known that the trifocal tensor variety is dimension 18 with degree 297 inside of $\\\\mathbb{P}^{28}$. It is also known that a six point correspondence in three views is a minimal problem.\\n\\nHow do the 24 linear constraints lead a minimal problem? How many complex solutions are there?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Numerical methods show that each point correspondence only intersects the variety in codimension 3 and so will lead to a minimal problem. The degree is cut down at each step of imposing conditions leading to three complex solutions.\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0183", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For six general labeled point correspondences in three uncalibrated views, the saturated nondegenerate reconstruction fiber is reduced of length three. Although each point triplet gives four independent ambient linear equations on the 27 tensor entries, it imposes only three intrinsic conditions on the 18-dimensional camera/trifocal moduli space; the six blocks therefore have total intrinsic rank 18. Fixing five world points as a projective frame reduces the sixth point to the intersection of three general quadrics through those five points. Bezout length eight minus five simple base points leaves exactly three reconstructions, and the generic birational camera-to-trifocal correspondence gives three distinct trifocal tensors.\n\nCandidate contribution (lemma; novelty confidence low): For a fixed five-point projective frame, the one-view image-data-to-quadric map is dominant, certified by an explicit projective Jacobian determinant of -22, and each of the five forced frame points has local complete-intersection multiplicity one; hence the classical count is scheme-theoretically 8-5=3, equivalently (2H-E_1-...-E_5)^3=3 on the blow-up." }, { "id": 20000184, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0184", "title": "Constraint budgets for partial calibration and the five-center obstruction", "statement": "In between the calibrated and uncalibrated cases there are partially calibrated cameras, say with both focal lengths known or with the center position known. What can we say about these situations?", "original_statement": "In between the calibrated and uncalibrated cases there are partially calibrated cameras, say with both focal lengths known or with the center position known. What can we say about these situations?", "clean_statement": "In between the calibrated and uncalibrated cases there are partially calibrated cameras, say with both focal lengths known or with the center position known. What can we say about these situations?", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0184, item 1.2 in the “Reconstruction” section of the AIM *Algebraic vision* problem list. Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Reconstruction\nSource item: 1.2\nSource URL: http://aimpl.org/algvision/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[183]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In between the calibrated and uncalibrated cases there are partially calibrated cameras, say with both focal lengths known or with the center position known. What can we say about these situations?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0184", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The ambiguous prompt splits into distinct models. For independently varying residual intrinsics, the rank-three dual-absolute-quadric locus has dimension eight and an admissible DIAC family of codimension c_i in view i supplies at most c_i local equations, so an isolated complex metric upgrade requires sum c_i at least eight. With both axis focal scales known but skew and principal point free per view, the DIAC family is explicitly a smooth codimension-two threefold; consequently three views retain upgrade dimension at least two and four views are only the first dimensionally possible case. Separately, arbitrary-intrinsic cameras with any prescribed general ordered centers through five views have a fixed-center quotient birational to the ordinary uncalibrated camera quotient, of dimension 11n-15. Three known centers therefore leave the 18-dimensional three-camera problem and six-point threshold unchanged modulo a six-dimensional residual ambiguity; six centers introduce exactly three projective center invariants but do not imply identifiability.\n\nCandidate contribution (theorem; novelty confidence low): For arbitrary-intrinsic cameras, prescribed general center points are birationally invisible to the observable projective reconstruction problem through five views; the sixth center is the first to contribute projective invariants, contributing exactly three." }, { "id": 20000185, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0185", "title": "Compatibility via the epipolar fiber product and a gcd component law", "statement": "Given an algebraic curve in two views, when are they compatible? This is interesting when the cameras are fixed beforehand.", "original_statement": "Given an algebraic curve in two views, when are they compatible? This is interesting when the cameras are fixed beforehand.", "clean_statement": "Given an algebraic curve in two views, when are they compatible? This is interesting when the cameras are fixed beforehand.", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List, *Algebraic vision*, §1 “Reconstruction,” Problem 1.25:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Reconstruction\nSource item: 1.25\nSource URL: http://aimpl.org/algvision/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[184]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given an algebraic curve in two views, when are they compatible? This is interesting when the cameras are fixed beforehand.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0185", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For two distinct fixed rank-three cameras over an algebraically closed field of characteristic zero, and integral image curves X and Y of degrees m and n avoiding their epipoles, the normalized incidence correspondence is the reduced fiber product of the two epipolar-pencil covers. Weak algebraic compatibility is automatic, while birational compatibility holds exactly when the normalized image curves are isomorphic as covers of the epipolar pencil. If its components have projection degrees (a_j,b_j), then a_j=(n/gcd(m,n))k_j, b_j=(m/gcd(m,n))k_j, the positive integers k_j sum to gcd(m,n), and the triangulated component has degree lcm(m,n)k_j.\n\nCandidate contribution (theorem; novelty confidence low): The clean fixed-camera correspondence components obey a gcd partition: positive integers k_j sum to gcd(m,n), their projection bidegrees are ((n/g)k_j,(m/g)k_j), and their world-curve degrees are lcm(m,n)k_j." }, { "id": 20000186, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0186", "title": "The degree-30 essential Hurwitz divisor and a harmonic-cubic pullback", "statement": "What is the Hurwitz form of the essential variety?", "original_statement": "What is the Hurwitz form of the essential variety?", "clean_statement": "What is the Hurwitz form of the essential variety?", "statement_status": "exact", "statement_verification": "The exact AIM record is Problem 1.3 in the “Reconstruction” section of the *Algebraic vision* list:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Reconstruction\nSource item: 1.3\nSource URL: http://aimpl.org/algvision/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[185]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the Hurwitz form of the essential variety?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0186", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The essential Hurwitz form is the irreducible divisor of class 30 sigma_1 on Gr(4,9), hence a degree-30 equation in 126 Plucker coordinates. No global expanded or determinantal formula was located. Using the trace-zero symmetric determinantal model of the essential variety, the report proves that on Hessian 3-planes of harmonic quaternary cubics its value is c times the cube of the degree-40 Salmon invariant, and proves that sections through the singular locus form only a codimension-three sublocus of the Grassmannian.\n\nCandidate contribution (special_case_theorem; novelty confidence low): For the trace-zero symmetric model E_0 of the essential variety and every harmonic quaternary cubic f with independent partial derivatives, Hu_{E_0}(Gamma_f) = c I_40(f)^3 for a nonzero normalization constant c." }, { "id": 20000187, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0187", "title": "Two-view reconstruction from conics and synchronized rational spline curves", "statement": "Can we use conic correspondence between two images to do reconstruction? Can we use point and curve correspondences to do reconstruction? Can we use cubic spline correspondences to do reconstruction?", "original_statement": "Can we use conic correspondence between two images to do reconstruction? Can we use point and curve correspondences to do reconstruction? Can we use cubic spline correspondences to do reconstruction?", "clean_statement": "Can we use conic correspondence between two images to do reconstruction? Can we use point and curve correspondences to do reconstruction? Can we use cubic spline correspondences to do reconstruction?", "statement_status": "exact", "statement_verification": "The canonical record is item 1.35 in the “Reconstruction” section of the AIM workshop list *Algebraic vision*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Reconstruction\nSource item: 1.35\nSource URL: http://aimpl.org/algvision/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[186]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we use conic correspondence between two images to do reconstruction? Can we use point and curve correspondences to do reconstruction? Can we use cubic spline correspondences to do reconstruction?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0187", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The broad feasibility question is answered affirmatively in the literature, but the data model controls identifiability. With known cameras, two unparameterized conic images generically reconstruct both the true plane conic and a residual plane conic. For unknown cameras and synchronized rational parametrizations x and y of degrees a and b, the coefficient rank r of the identity y(t)^T F x(t) = 0 determines the compatible projective linear space: r <= 6 cannot identify F finitely; r = 7 gives a line whose nonzero squarefree determinant restriction yields three distinct rank-two fundamental matrices when the line avoids the rank-one locus; and r = 8 gives a unique rank-two F. An exact projected twisted-cubic witness attains r = 7 and has three audited rank-two roots. Multiple matched spline segments aggregate by stacking coefficient rows when each cross-view segment pair has a known common local parameter.\n\nCandidate contribution (theorem_and_exact_witness; novelty confidence low): Candidate novelty: the coefficient-rank identifiability classification for exact synchronized rational image-curve parametrizations, including the generic algebraic degree-three result for one synchronized rational cubic segment and uniqueness after one additional independent point row." }, { "id": 20000188, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0188", "title": "Joint-monodromy degree classification and branch-overlap smoothness budget", "statement": "Fix two cameras and two image curves. Is there is a curve in $\\mathbb{P}^3$ that projects onto these two curves? What if we put restriction on degrees or smoothness?", "original_statement": "Fix two cameras and two image curves. Is there is a curve in $\\mathbb{P}^3$ that projects onto these two curves? What if we put restriction on degrees or smoothness?", "clean_statement": "Fix two cameras and two image curves. Is there is a curve in $\\mathbb{P}^3$ that projects onto these two curves? What if we put restriction on degrees or smoothness?", "statement_status": "exact", "statement_verification": "The canonical AIM record is *Algebraic vision*, §1 “Reconstruction,” Problem 1.4:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Reconstruction\nSource item: 1.4\nSource URL: http://aimpl.org/algvision/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[187]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Fix two cameras and two image curves. Is there is a curve in $\\\\mathbb{P}^3$ that projects onto these two curves? What if we put restriction on degrees or smoothness?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0188", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For two distinct fixed rank-three cameras over an algebraically closed characteristic-zero field and smooth integral image curves X,Y of degrees m,n avoiding their epipoles, the integral world lifts are exactly the paired-monodromy orbits O on the product of the two epipolar fibers. The corresponding world degree is |O|, its projection degrees are |O|/m and |O|/n, and its normalization genus is given by the local-monodromy cycle form of Riemann-Hurwitz. The full cone intersection is smooth exactly when the two branch supports are disjoint; it is then the unique integral lift of degree mn. If both covers are simply branched and share s branch values, the cone intersection has exactly s nodes and, for r components of normalization genera g_i, sum g_i = p_a-s+r-1.\n\nCandidate contribution (theorem; novelty confidence low): For clean smooth fixed-camera image curves with simply branched epipolar covers, s common branch values produce exactly s nodes and the r paired-monodromy components satisfy sum_i g_i=p_a-s+r-1, hence r is at least s-p_a+1 and s>p_a forces reducibility; paired orbit sizes and cycle counts simultaneously decide every allowable world degree and normalization genus." }, { "id": 20000189, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0189", "title": "Gauge-correct cheirality, convex separation, and finite certificates", "statement": "For a reconstruction what does it mean for all the points to be \"in front\" of the cameras? Can this easily be detected.", "original_statement": "For a reconstruction what does it mean for all the points to be \"in front\" of the cameras? Can this easily be detected.", "clean_statement": "For a reconstruction what does it mean for all the points to be \"in front\" of the cameras? Can this easily be detected.", "statement_status": "exact", "statement_verification": "The exact AIM record is Problem 1.45 in the “Reconstruction” section of the *Algebraic vision* list:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Reconstruction\nSource item: 1.45\nSource URL: http://aimpl.org/algvision/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[188]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For a reconstruction what does it mean for all the points to be \\\"in front\\\" of the cameras? Can this easily be detected.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Two approaches were suggested: That neither of the cameras are contained in the convex hull of the world points; or that $P r_i = \\\\lambda x_i$ and $Q r_i = \\\\mu y_i$ where $\\\\lambda>0$ and $\\\\mu>0$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0189", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For fixed finite real cameras, strict frontality is exactly the scale-invariant sign test q_4(n_i^T q)>0 for every camera-point pair, while the suggested positive image multiplier is valid only after compatible gauge normalizations. When a camera center is fixed but orientation is free, the optimal minimum depth equals the distance from that center to the convex hull of the world points. For a signable projective reconstruction, strict chiral upgrade is equivalent to feasibility of one of two explicitly reoriented linear systems; failure has two positive-dependence certificates, one per branch, each supported on at most five homogeneous point/center vectors, and feasibility admits a normalized LP robustness margin.\n\nCandidate contribution (certificate; novelty confidence low): After signability, nonexistence of a strict chiral projective gauge in P^3 is certified by a positive dependency for each of the two determinant-sign branches, with each dependency supported on at most five reoriented world-point or camera-center vectors; a feasible branch has the bounded normalized LP margin rho_s with the stated l1 perturbation guarantee." }, { "id": 20000190, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0190", "title": "A root-free focal-positivity certificate for a seven-point pencil", "statement": "The focal lengths of a pair of cameras can be written as a function of their fundamental matrix. During reconstruction fundamental matrices are constructed for which these focal lengths are complex. Is there an efficient way to detect if the this will occur before computing the fundamental matrix?", "original_statement": "The focal lengths of a pair of cameras can be written as a function of their fundamental matrix. During reconstruction fundamental matrices are constructed for which these focal lengths are complex. Is there an efficient way to detect if the this will occur before computing the fundamental matrix?", "clean_statement": "The focal lengths of a pair of cameras can be written as a function of their fundamental matrix. During reconstruction fundamental matrices are constructed for which these focal lengths are complex. Is there an efficient way to detect if the this will occur before computing the fundamental matrix?", "statement_status": "exact", "statement_verification": "The canonical record is AIM problem 1.5 in the “Reconstruction” section of the Algebraic Vision workshop:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Reconstruction\nSource item: 1.5\nSource URL: http://aimpl.org/algvision/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[189]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The focal lengths of a pair of cameras can be written as a function of their fundamental matrix. During reconstruction fundamental matrices are constructed for which these focal lengths are complex. Is there an efficient way to detect if the this will occur before computing the fundamental matrix?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0190", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under known principal points, zero skew, unit aspect ratio, and two unknown focal lengths, write an exact rank-seven seven-point nullspace pencil as F(z)=F1+zF0 and p(z)=det F(z). On a generic modified-Bougnoux chart let q_i(z)=N_i(F(z))D_i(F(z)). The number of real rank-two roots with both focal squares positive is [TaQ_p(1)+TaQ_p(q_1)+TaQ_p(q_2)+TaQ_p(q_1q_2)]/4. Signed-subresultant Sturm-Tarski queries compute these integers without solving p or constructing an individual fundamental matrix. A homogeneous discriminant-resultant product gives a conservative algebraic boundary off which this count is locally constant.\n\nCandidate contribution (certificate; novelty confidence low): For a generic exact seven-point fundamental-matrix pencil, four explicit Sturm-Tarski queries of the Bougnoux numerator-denominator sign polynomials count all four focal-square sign patterns before any determinant root is constructed; the associated homogeneous product Disc(P) Res(P,R) product_i Res(P,N_i)Res(P,D_i) is a conservative data-boundary guard for changes in that count." }, { "id": 20000191, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0191", "title": "Certified epipolar consistency for disk-valued image points", "statement": "How can we account for uncertainty, given the geometric structure of these problems? What if we replace points with disks?", "original_statement": "How can we account for uncertainty, given the geometric structure of these problems? What if we replace points with disks?", "clean_statement": "How can we account for uncertainty, given the geometric structure of these problems? What if we replace points with disks?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, *Algebraic vision*, section “Reconstruction,” Problem 1.55:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Reconstruction\nSource item: 1.55\nSource URL: http://aimpl.org/algvision/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[190]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How can we account for uncertainty, given the geometric structure of these problems? What if we replace points with disks?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0191", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a fixed fundamental matrix and two independent closed Euclidean image disks, existential epipolar consistency is equivalent to the nonpositivity of one explicit two-dimensional trust-region quadratic obtained by eliminating the second disk through its support function. A feasible instance has an explicit point-pair witness, strict infeasibility has an exact 3-by-3 S-lemma LMI certificate, and directed forall-exists coverage has a second exact 3-by-3 LMI. In contrast, exact epipolar equality for every pair in two positive-radius disks forces F=0, so uniform robustness requires a tolerance or a different quantifier.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: the explicit one-disk support-function elimination, single two-dimensional trust-region decision, strict-infeasibility and directed-coverage LMIs, and their use as a certificate oracle for minimum simultaneous normalized inflation of two independent Euclidean disks." }, { "id": 20000192, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0192", "title": "Duality, epipoles, and a Segre cubic from four correspondences", "statement": "For any of the standard varieties considered it might be possible to study the projective dual. What can we learn from studying these objects?", "original_statement": "For any of the standard varieties considered it might be possible to study the projective dual. What can we learn from studying these objects?", "clean_statement": "For any of the standard varieties considered it might be possible to study the projective dual. What can we learn from studying these objects?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, *Algebraic vision*, section “More Varieties,” Problem 2.1:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: More Varieties\nSource item: 2.1\nSource URL: http://aimpl.org/algvision/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[191]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For any of the standard varieties considered it might be possible to study the projective dual. What can we learn from studying these objects?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0192", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the fundamental-matrix determinant cubic X in P^8, the projective dual is the rank-one Segre variety P^2 x P^2; its P^3 contact fibers consist of matrices with fixed left and right kernels, yielding an exact conormal test for nontransverse correspondence sections. Moreover, four point correspondences that form projective frames in both images cut X in the Segre cubic threefold. Its ten ordinary nodes split as six rank-one points inherited from Sing(X) and four rank-two dual-contact points, one uniquely associated to each correspondence; all ten are real for real frames.\n\nCandidate contribution (theorem; novelty confidence low): Four ordered projective-frame correspondences cut the fundamental determinant cubic in a Segre cubic whose ten nodes have a canonical 6+4 decomposition: six inherited rank-one nodes and four rank-two nodes, each the unique intersection of one correspondence's P^3 contact fiber with the other three correspondence hyperplanes." }, { "id": 20000193, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0193", "title": "A normalized essential-matrix orbitope and its density-matrix faces", "statement": "What can we say about the convex hull of all the varieties we have considered? Can we give a good description of them?", "original_statement": "What can we say about the convex hull of all the varieties we have considered? Can we give a good description of them?", "clean_statement": "What can we say about the convex hull of all the varieties we have considered? Can we give a good description of them?", "statement_status": "exact", "statement_verification": "The canonical record is problem 2.2 in the “More Varieties” section of the Algebraic Vision AIM problem list:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: More Varieties\nSource item: 2.2\nSource URL: http://aimpl.org/algvision/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[192]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What can we say about the convex hull of all the varieties we have considered? Can we give a good description of them?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0193", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the real operator-normalized essential variety E1, consisting of 3-by-3 matrices with singular values (1,1,0), its convex hull is exactly the intersection of the operator-norm unit ball and the nuclear-norm ball of radius 2. Its support function is the sum of the two largest singular values, its polar is the Ky Fan 2-norm unit ball, and both hull and polar have direct additive-compound spectrahedral descriptions. The unnormalized essential cone instead convexifies to the whole ambient matrix space, while the operator-normalized rank-two fundamental locus has the same convex hull as E1.\n\nCandidate contribution (exposed_face_theorem; novelty confidence low): For every Q in O(3), the exposed face cut out by =2 is exactly {Q(I-Z): Z is positive semidefinite and trace(Z)=1}; consequently, (2/3)Q requires exactly three normalized essential matrices in every convex decomposition." }, { "id": 20000194, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0194", "title": "Three multiview functors and a collision obstruction", "statement": "What is the functor of points of multiview geometry?", "original_statement": "What is the functor of points of multiview geometry?", "clean_statement": "What is the functor of points of multiview geometry?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: More Varieties\nSource item: 2.3\nSource URL: http://aimpl.org/algvision/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[193]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the functor of points of multiview geometry?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0194", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For fixed full-rank cameras, the joint graph functor is represented by the blowup of the product of the camera-center ideals, equivalently by the diagonal of the multigraded Rees construction; its relative-Proj points are Rees-compatible invertible quotients. The fixed multiview image is instead the scheme-theoretic image functor, while the known variable-camera functor on the pairwise-disjoint-center locus is the smooth quasi-projective quotient Cam_n. An explicit two-camera family over k[t] proves that joint-graph formation can fail under specialization when centers collide: the kernel class [ty] in the base-changed Rees algebra remains nonzero on D_+(y^2), so the discrepancy survives Proj.\n\nCandidate contribution (counterexample; novelty confidence low): For A_1[w:x:y:z]=[x:y:z] and A_2[w:x:y:z]=[x-tw:y:z] over k[t], the graph constructed after setting t=0 is a proper closed subscheme of the special fiber of the relative joint graph; the Rees base-change kernel contains [ty], which survives localization at [y^2]." }, { "id": 20000195, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0195", "title": "The calibrated camera functor and an infinitesimal essential-boundary obstruction", "statement": "What is the functor of points in the calibrated case?", "original_statement": "What is the functor of points in the calibrated case?", "clean_statement": "What is the functor of points in the calibrated case?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, *Algebraic vision*, section “More Varieties,” Problem 2.4:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: More Varieties\nSource item: 2.4\nSource URL: http://aimpl.org/algvision/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[194]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the functor of points in the calibrated case?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0195", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Lieblich and Van Meter's representable functor CalCam_n gives the present-day answer to the AIM question: over a test scheme it parametrizes a relative P^3-bundle, n labelled relative pinhole cameras with pairwise disjoint center sections, and a common relative degree-two curve mapping into the prescribed image conics; it is smooth and embeds into a diagram Hilbert scheme. Distinguishing this moduli functor from the fixed two-view essential image scheme, this attempt proves that at every rank-one isotropic boundary point E0=ab^T all ten Demazure differentials vanish, every ambient projective first-order direction is an actual dual-number essential point, and liftability to k[t]/(t^3) is equivalent to one explicit matrix-valued quadratic equation plus b^T adj(H)a=0. For a=b=(1,i,0)^T and H=I_3, the matrix obstruction is 3E0, giving a concrete nonliftable first-order point.\n\nCandidate contribution (theorem; novelty confidence low): At a complex rank-one isotropic essential matrix E0=ab^T, every class H modulo kE0 defines a k[epsilon]/(epsilon^2)-valued essential point, while it lifts to k[t]/(t^3) exactly when Q_E0(H)=2((E0 H^T+H E0^T)H+HH^T E0)-2(a^T H b)H-tr(HH^T)E0 and b^T adj(H)a both vanish; the direction H=I_3 at a=b=(1,i,0)^T is obstructed because Q_E0(I_3)=3E0." }, { "id": 20000196, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0196", "title": "Complex completeness and real descent for differential curve signatures", "statement": "How complete are these invariants? If two curves have the same signature what can we say about them?", "original_statement": "How complete are these invariants? If two curves have the same signature what can we say about them?", "clean_statement": "How complete are these invariants? If two curves have the same signature what can we say about them?", "statement_status": "exact", "statement_verification": "The exact canonical AIM record, problem 3.1 in the “Invariants” section of the 2016 Algebraic Vision workshop list, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Invariants\nSource item: 3.1\nSource URL: http://aimpl.org/algvision/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[195]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How complete are these invariants? If two curves have the same signature what can we say about them?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0196", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Existing classifying-pair results imply that two nonexceptional irreducible complex plane curves have the same algebraic differential-signature curve exactly when they are equivalent under the chosen complex algebraic group. This attempt proves a real refinement: real group orbits inside one complex signature class are naturally classified by the pointed-set kernel of H^1(R, Sym_G(X_C)) to H^1(R,G), with cocycle g^{-1} conjugate(g). Thus an implicit complex signature is already complete for real equivalence when the complex embedded symmetry group is trivial. Explicit nonexceptional affine and projective quartic pairs demonstrate nontrivial descent classes and the distinction between actual real signature images and their common complex Zariski closure.\n\nCandidate contribution (real_descent_classification; novelty confidence low): Assuming a classifying nonexceptional complex signature, the G(R)-orbits of real curves in the complex signature class of X are in natural bijection with the pointed-set kernel ker[H^1(R, Sym_G(X_C)) -> H^1(R,G)]; a curve Y=gX has class represented by g^{-1} conjugate(g), and trivial complex embedded symmetry forces real equivalence." }, { "id": 20000197, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0197", "title": "Degree, symmetry quotient, and coordinate dependence of differential signature curves", "statement": "What can we say about the degree of a signature curve? How is it related to the automorphism group of the original curve?", "original_statement": "What can we say about the degree of a signature curve? How is it related to the automorphism group of the original curve?", "clean_statement": "What can we say about the degree of a signature curve? How is it related to the automorphism group of the original curve?", "statement_status": "exact", "statement_verification": "The exact canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Invariants\nSource item: 3.2\nSource URL: http://aimpl.org/algvision/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[196]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What can we say about the degree of a signature curve? How is it related to the automorphism group of the original curve?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0197", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Kogan--Ruddy--Vinzant's 2020 corrected Bezout formula answers the stated degree question for a fixed classifying pair: the relative symmetry order multiplies the signature degree. Building on their generic-fiber theorem, this attempt proves that the normalization of every nonconstant signature is the quotient of the normalized source curve by its relative symmetry group, derives an exact Riemann--Hurwitz/delta-invariant identity, and proves an exact eventual degree law for polynomial shears of the classifying pair. The shear law shows that plane degree is unbounded even when the original curve, symmetry group, generic fibers, and normalized signature are fixed.\n\nCandidate contribution (theorem; novelty confidence low): For a non-exceptional complex plane curve with finite relative symmetry group H, the normalized signature is C/H; moreover, if P(u,v)=sum c_j(u)v^j is its irreducible signature polynomial and the classifying pair is replaced by (K_1,K_2+K_1^m), then for every sufficiently large integer m the new signature has exact degree m deg_v(P)+deg(c_deg_v(P)), so plane signature degrees are unbounded while the normalized quotient remains fixed." }, { "id": 20000198, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0198", "title": "Canonical curves versus plane differential signatures", "statement": "What is the signature curve of a general canonical curve?", "original_statement": "What is the signature curve of a general canonical curve?", "clean_statement": "What is the signature curve of a general canonical curve?", "statement_status": "exact", "statement_verification": "The exact source record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Invariants\nSource item: 3.3\nSource URL: http://aimpl.org/algvision/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[197]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the signature curve of a general canonical curve?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0198", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM wording mixes the workshop's plane differential-signature construction with canonical curves in projective space. In the unique intrinsically planar case, a general genus-3 canonical quartic has Kogan-Ruddy-Vinzant PGL(3)-signature degree 672; the signature map is birational, its signature curve has normalization the original genus-3 curve, and its total delta invariant is 224782. For genus at least 4, distinct three-dimensional canonical subsystems give distinct plane signatures, so no intrinsic plane signature exists without projection data. A proved conditional local lemma shows that a signature-ordinary node contributes 192 to the PGL(3) base-locus correction, yielding degree 24(3d+8g-8) under explicit global hypotheses and the testable canonical-projection prediction 336(g-1).\n\nCandidate contribution (lemma; novelty confidence low): For the standard Kogan-Ruddy-Vinzant PGL(3) projective signature extension, an affine ordinary node whose two branch jets satisfy Theta_5 Theta_7 Theta_8 nonzero has base-point multiplicity exactly 192; consequently a nodal plane curve satisfying the separately listed node, infinity, symmetry, and no-other-base-point hypotheses has signature degree 24(3d+8g-8)." }, { "id": 20000199, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0199", "title": "Iterated differential signatures as a quotient tower", "statement": "What can we say about the signature of the signature of a curve?", "original_statement": "What can we say about the signature of the signature of a curve?", "clean_statement": "What can we say about the signature of the signature of a curve?", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Invariants\nSource item: 3.4\nSource URL: http://aimpl.org/algvision/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[198]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What can we say about the signature of the signature of a curve?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0199", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The phrase \"signature of the signature\" is not canonical: the action induced by the original group on invariant-value coordinates is trivial, while any nontrivial iteration requires a fresh plane action and classifying pair at every later stage. Under such a specified admissible iteration, if H_n is the finite symmetry group at stage n, then pullback identifies C(X_{n+1}) with the fixed field C(X_n)^{H_n}; consequently the normalized next signature is C_n/H_n, stage and composite degrees are |H_n| and their product, and Riemann-Hurwitz controls genus. In particular, at most floor(log_2(g_0-1))+1 nontrivial quotient stages can begin in genus at least two. This bound does not imply stabilization in genus zero or one.\n\nCandidate contribution (quotient_tower_theorem; novelty confidence low): For every fully specified admissible finite iterated-signature stage, the function field of the next signature is exactly the current symmetry fixed field, so the smooth projective next signature is the quotient by the current symmetry group; this yields an exact degree-product law and a logarithmic bound on nontrivial high-genus stages." }, { "id": 20000200, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0200", "title": "Torsion-logarithmic obstructions and realizable Euclidean signature lines", "statement": "Which curves occur as signatures of another curve?", "original_statement": "Which curves occur as signatures of another curve?", "clean_statement": "Which curves occur as signatures of another curve?", "statement_status": "exact", "statement_verification": "The exact canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Invariants\nSource item: 3.5\nSource URL: http://aimpl.org/algvision/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[199]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Which curves occur as signatures of another curve?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0200", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the fixed standard complex special-Euclidean signature pair (u,v)=(kappa^2,kappa_s), every algebraically realizable signature curve S satisfies the intrinsic norm obstruction n i du/v=dQ/Q on its projective normalization. An exact finite-cover Frenet-lift criterion is proved, polynomial graphs receive immediate pole and residue tests, and the nonconstant homogeneous lines are completely classified: v=c u is realizable if and only if c is a nonzero imaginary rational multiple and c is not i or -i. Every allowable slope is realized by an explicit birational Laurent-monomial source z=t^a, w=t^b.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: for the standard SE(2,C) invariant coordinates, algebraic realizability forces i du/v to be torsion-logarithmic, and the homogeneous signature line v=c u occurs exactly for c in i Q^* excluding c=i and c=-i, with explicit Laurent-monomial realizations." }, { "id": 20000201, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0201", "title": "Differential signatures as a generic coarse moduli quotient", "statement": "Can we use differential (integral, other…) invariants to construct a moduli space?", "original_statement": "Can we use differential (integral, other…) invariants to construct a moduli space?", "clean_statement": "Can we use differential (integral, other…) invariants to construct a moduli space?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Invariants\nSource item: 3.6\nSource URL: http://aimpl.org/algvision/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[200]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we use differential (integral, other…) invariants to construct a moduli space?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0201", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For complex fixed-degree plane curves under a fixed algebraic transformation group and classifying differential-invariant pair, the projective coefficients of the Kogan-Ruddy-Vinzant super-signature define, on the specialization-good fixed-degree locus, a regular orbit-complete map to the parameter space of plane signature curves. Starting from a Rosenlicht quotient and shrinking its source and target identifies this map with a geometric quotient, and pulling back the universal degree-D hypersurface gives a flat base-change-compatible family of target signatures. This is only a generic coarse quotient: a residual gerbe at a curve with stabilizer is collapsed, as made explicit by the family y=a x^n+b, whose coarse signature coordinate is b while its quotient stack is B mu_n times the affine b-line.\n\nCandidate contribution (theorem_and_obstruction; novelty confidence low): Candidate novelty: the fixed-degree specialization-good super-signature coefficient map becomes a Rosenlicht geometric quotient after shrinking and simultaneously carries a flat Hilbert-family of signature curves, while the explicit family y=a x^n+b proves that the same map forgets the residual gerbe B mu_n and therefore cannot represent the quotient stack." }, { "id": 20000202, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0202", "title": "Finite algebraic certificates for recognition by invariant signatures", "statement": "How can we use these signatures for object recognition?", "original_statement": "How can we use these signatures for object recognition?", "clean_statement": "How can we use these signatures for object recognition?", "statement_status": "exact", "statement_verification": "The exact canonical record is AIM problem 3.7 in the “Invariants” section of the 2016 *Algebraic vision* workshop list:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Invariants\nSource item: 3.7\nSource URL: http://aimpl.org/algvision/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[201]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How can we use these signatures for object recognition?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0202", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For irreducible nonexceptional plane algebraic curves with nonconstant rational signature maps, if more than D_X D_Y pairwise distinct exact affine query signature values lie on a library signature curve of projective degrees D_X and D_Y, projective Bezout forces equality of the irreducible signature curves, and algebraic signature completeness then forces G(C)-equivalence. A bound D_Y <= B_Y gives the operational threshold D_X B_Y + 1. Separately, a fully sampled finite real library has a rigorous nearest-Hausdorff guarantee when L epsilon + M h is less than half its signature margin, while a continuous family of noncongruent ellipses proves that no library-independent positive robustness radius exists.\n\nCandidate contribution (theorem; novelty confidence low): Candidate finite-sample recognition theorem: D_X D_Y + 1 distinct exact common affine signature values certify G(C)-equivalence of irreducible nonexceptional algebraic plane curves; equivalently, more than D_X B_Y exact visible query values satisfying a precomputed library signature polynomial certify a match when D_Y <= B_Y." }, { "id": 20000203, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0203", "title": "Complete regular-jet invariants for an irregular Euclidean tile groupoid", "statement": "Find local invariants for a natural (Lie) groupoid action. (e.g. bathroom tiles in an irregular bathroom). How is this related to quasi-invariants?", "original_statement": "Find local invariants for a natural (Lie) groupoid action. (e.g. bathroom tiles in an irregular bathroom). How is this related to quasi-invariants?", "clean_statement": "Find local invariants for a natural (Lie) groupoid action. (e.g. bathroom tiles in an irregular bathroom). How is this related to quasi-invariants?", "statement_status": "exact", "statement_verification": "The exact canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Invariants\nSource item: 3.8\nSource URL: http://aimpl.org/algvision/3/\nCanonical location: aim-algebraic-geometry-notes.json notes[202]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find local invariants for a natural (Lie) groupoid action. (e.g. bathroom tiles in an irregular bathroom). How is this related to quasi-invariants?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/3/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0203", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the restricted SE(2) action groupoid of an arbitrary open planar domain carrying a smooth scalar texture, the regular stratum |grad f|>0 has a complete, functionally independent jet normal form consisting of f, |grad f|, and all higher derivative tensors evaluated in the oriented gradient frame. The smooth localized differential invariant algebra is generated by f, |grad f|, and signed level curvature under two explicit invariant derivatives. Equality of the full sequence recovers exactly the regular local symmetry arrows for analytic textures, while in the smooth category it gives only formal equivalence. Every smooth scalar groupoid cocycle is a uniquely normalized base coboundary times an integral SO(2) spin, and its relative invariants factor by the corresponding power of the nonzero complex gradient.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty is the combined, dimension-sharp boundary-blind normal form, three-generator recurrence, analytic local-symmetry criterion, and explicit spin/coboundary factorization for the irregular-domain restricted Euclidean scalar-texture groupoid." }, { "id": 20000204, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0204", "title": "Edge and tritangent surfaces are the only generic smooth visual-hull event types", "statement": "Which visual event surfaces appear on visual hulls of algebraic surfaces?", "original_statement": "Which visual event surfaces appear on visual hulls of algebraic surfaces?", "clean_statement": "Which visual event surfaces appear on visual hulls of algebraic surfaces?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, workshop **Algebraic vision**, section **Silhouettes**, Problem 4.1:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Silhouettes\nSource item: 4.1\nSource URL: http://aimpl.org/algvision/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[203]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Which visual event surfaces appear on visual hulls of algebraic surfaces?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0204", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a compact solid whose smooth real algebraic boundary and complex projective closure satisfy the standard genericity hypotheses, the artificial external visual-hull boundary is contained in the real edge and tritangent event surfaces. Equivalently, among the five irreducible algebraic visual-event components, only tangent crossing E(X) and triple point T(X) can support open artificial boundary patches; actual real activity still requires visibility and curvature pruning. As a proved corollary, the two-dimensional complex Zariski envelope is one of the empty set, E(X), T(X), or their union, so its degree lies in {0, e_d, t_d, e_d+t_d} with the explicit formulas given in the artifacts.\n\nCandidate contribution (corollary; novelty confidence low): Under joint genericity, the two-dimensional complex Zariski closure of the artificial visual-hull boundary has a component-or-nothing alternative Z in {empty, E(X), T(X), E(X) union T(X)}, hence degree in {0, d(d-2)(d-3)(d^2+2d-4), (1/3)d(d-3)(d-4)(d-5)(d^2+3d-2), or the sum}." }, { "id": 20000205, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0205", "title": "Exact visual-event equations, camera-line specialization, and a diagonal-cubic formula", "statement": "Can we find equations of visual event surfaces?", "original_statement": "Can we find equations of visual event surfaces?", "clean_statement": "Can we find equations of visual event surfaces?", "statement_status": "exact", "statement_verification": "The canonical record is AIM problem 4.2 in the “Silhouettes” section of the 2016 AIM workshop *Algebraic vision*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Silhouettes\nSource item: 4.2\nSource URL: http://aimpl.org/algvision/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[204]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we find equations of visual event surfaces?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0205", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The general smooth-projective-surface interpretation was answered affirmatively by Kohn, Sturmfels, and Trager: five ruled event components admit exact multiple-root, dual, and incidence representations. This attempt gives a proved trajectory corollary: on a general camera line the product of the five restricted reduced equations is square-free of degree d(d-2)(d^4-d^3-5d^2+5d+6)/3, with discriminants and pairwise resultants detecting repeated and simultaneous events. It also proves a reduced degree-30 Hessian-incidence parabolic equation for every smooth diagonal cubic.\n\nCandidate contribution (trajectory_specialization_lemma; novelty confidence low): For a general smooth degree-d surface with d at least 3 and a general projective camera line, the product of the restricted reduced visual-event component equations is square-free of degree d(d-2)(d^4-d^3-5d^2+5d+6)/3; for any non-contained line, component discriminants certify repeated event intersections and pairwise resultants certify simultaneous event types." }, { "id": 20000206, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0206", "title": "Active algebraic-boundary factors for dual visual and convex hulls", "statement": "In silhouette reconstruction a duality appears between the visual hull and the convex hull. Is there an algebraic version of this?", "original_statement": "In silhouette reconstruction a duality appears between the visual hull and the convex hull. Is there an algebraic version of this?", "clean_statement": "In silhouette reconstruction a duality appears between the visual hull and the convex hull. Is there an algebraic version of this?", "statement_status": "exact", "statement_verification": "The exact canonical AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Silhouettes\nSource item: 4.3\nSource URL: http://aimpl.org/algvision/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[205]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In silhouette reconstruction a duality appears between the visual hull and the convex hull. Is there an algebraic version of this?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0206", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For properly convex algebraic silhouettes in a common oriented affine realization, each image-contour equation pulls back to an irreducible visual-cone surface whose projective dual is the embedded dual contour. The known polar identity identifies the polar visual hull with the convex hull of the embedded polar silhouettes. After deduplicating coincident cone surfaces, the algebraic boundary is exactly the reduced union of those pullback surfaces whose dual regions are not contained in the convex hull of the other dual regions. Pairwise cone-event curves dualize to ruled surfaces of incidence-linked secants, transverse triple contacts lie in an exposed support plane, and proper complex pair and triple intersections have degrees d_i d_j and d_i d_j d_k with multiplicity.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: after deduplicating coincident irreducible pullback cones, the factor b_i(M_i x) occurs in the algebraic boundary of the visual hull if and only if the embedded polar region Q_i is not contained in the convex hull of the other Q_j; paired with this criterion, the dual of each generically transverse pair-event component is exactly the closure of the secant lines joining the two dual-contour contacts linked by the same world event, with proper complex Bezout degree certificates." }, { "id": 20000207, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0207", "title": "Extended-Kruppa constraints, realized conic ambiguity, and invariant eliminants for algebraic silhouettes", "statement": "Can we find the fundamental matrix from silhouettes? If the silhouettes are bounded by algebraic curves are there algebraic constraints on the fundamental matrix?", "original_statement": "Can we find the fundamental matrix from silhouettes? If the silhouettes are bounded by algebraic curves are there algebraic constraints on the fundamental matrix?", "clean_statement": "Can we find the fundamental matrix from silhouettes? If the silhouettes are bounded by algebraic curves are there algebraic constraints on the fundamental matrix?", "statement_status": "exact", "statement_verification": "The canonical AIM record is item 4.4 in the “Silhouettes” section of the Algebraic Vision problem list:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Silhouettes\nSource item: 4.4\nSource URL: http://aimpl.org/algvision/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[206]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we find the fundamental matrix from silhouettes? If the silhouettes are bounded by algebraic curves are there algebraic constraints on the fundamental matrix?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0207", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For two full algebraic apparent contours of one smooth projective surface, the true rank-two fundamental matrix F satisfies the surface silhouette extended-Kruppa identity Phi_2(Fx)=lambda Phi_1([e_1]_x x): equivalently, the two degree-delta divisors of tangent epipolar lines are projectively identical with scheme multiplicity. Proportional degree-delta binary forms give at most delta equations on the 7-dimensional fundamental-matrix variety, so delta at most 6 cannot isolate F. A dual-section gluing argument shows, subject to explicit admissibility conditions, that compatible contour sections can be realized by an actual surface; in particular, a generic conic pair has a genuine 5-dimensional family of compatible F's realized by smooth quadrics. For class four, the unknown tangent correspondence and pencil homography can be eliminated by the explicit quartic-invariant equation J(q_1)^2 I(q_2)^3-J(q_2)^2 I(q_1)^3=0, which is complete for squarefree complex tangency divisors.\n\nCandidate contribution (gluing_lemma_and_invariant_eliminant; novelty confidence low): Candidate novelty: extended-Kruppa compatibility of two dual silhouette sections glues them to a dual surface, and for generic nonsingular conic data every generic member of the resulting 5-dimensional compatible-fundamental-matrix family is realized by a smooth quadric; additionally, for class-four contours the single displayed I,J equation is a correspondence-free necessary and, for squarefree complex divisors, sufficient test for a projectivity between the two tangency divisors." }, { "id": 20000208, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0208", "title": "Cuspidal silhouettes via Segre adjoints and finite first-jet tests", "statement": "Which curves with cusps come from rims of algebraic surfaces?", "original_statement": "Which curves with cusps come from rims of algebraic surfaces?", "clean_statement": "Which curves with cusps come from rims of algebraic surfaces?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Silhouettes\nSource item: 4.5\nSource URL: http://aimpl.org/algvision/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[207]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Which curves with cusps come from rims of algebraic surfaces?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0208", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The wording is ambiguous: for a generic projection of a smooth surface the rim on the surface is smooth, while its plane image is nodal-cuspidal. Under the standard plane-silhouette interpretation, Segre's theorem completely characterizes branch curves of smooth degree-nu surfaces in P3 by exact degree, node and cusp counts plus two low-degree adjoints with separated tangents. This attempt proves a finite first-jet reformulation: after fixing the degree-a adjoint L, existence of one degree-(a+1) adjoint separated from L at every reduced singular point is equivalent, over an infinite field, to nonvanishing of one induced linear functional at each point on H^0(I_xi(a+1))/(H^0(O(1))L).\n\nCandidate contribution (lemma; novelty confidence low): Let xi be a finite reduced plane subscheme, let L in H^0(I_xi(a)) be nonzero, and set V_L=H^0(I_xi(a+1))/(H^0(O(1))L). For each p in xi, the wedge dL_p wedge dG_p descends to a linear functional lambda_{p,L} on V_L. Over an infinite field, there exists one G whose curve is smooth and tangent-separated from L at every p if and only if every lambda_{p,L} is nonzero." }, { "id": 20000209, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0209", "title": "Carlsson-Weinshall duality is Gale-trivial on the fiducial torus", "statement": "Is there a relationship between Carlsson-Weinshall and Gale duality?", "original_statement": "Is there a relationship between Carlsson-Weinshall and Gale duality?", "clean_statement": "Is there a relationship between Carlsson-Weinshall and Gale duality?", "statement_status": "exact", "statement_verification": "The exact canonical AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Carlsson-Weinshall Duality\nSource item: 5.1\nSource URL: http://aimpl.org/algvision/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[208]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a relationship between Carlsson-Weinshall and Gale duality?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0209", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For x and c in the nonzero-coordinate torus of projective 3-space, the six-point matrix A=[I_4|x|c] has Gale kernel columns [x_i:c_i] for i=1,...,4 followed by [1:0] and [0:1]. Its Carlsson-Weinshall partner [I_4|c^{-1}|x^{-1}] has the same labeled Gale columns and is explicitly projectively equivalent to A. This is a Carlsson-Weinshall matrix translation of the classical Dolgachev-Ortland Cremona-plus-(56) kernel relation. Reduced image formation factors through the anchored Gale ratio [x_i/c_i], and multiview compatibility is characterized by an exact existence-of-lifts/rank-one criterion.\n\nCandidate contribution (equivalence; novelty confidence low): On the four-fiducial nonzero-coordinate chart, reduced image formation factors as (x,c) -> q=[x_1/c_1:...:x_4/c_4] -> [Uq], where q is the anchored Gale coordinate of [I_4|x|c]; moreover, an m-by-n image array is realizable exactly when it admits nonzero lifts r_ij with y_ij=[U r_ij] such that each of the four coordinate-slice matrices (r_ij,k) has rank one." }, { "id": 20000210, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0210", "title": "A typed Carlsson-Weinshall extension for line correspondences", "statement": "Is there a Carlsson-Weinshallduality for point and line correspondence?", "original_statement": "Is there a Carlsson-Weinshallduality for point and line correspondence?", "clean_statement": "Is there a Carlsson-Weinshallduality for point and line correspondence?", "statement_status": "exact", "statement_verification": "The exact canonical record is AIM Problem List *Algebraic vision*, section 5 “Carlsson-Weinshall Duality,” Problem 5.2. Its text is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Carlsson-Weinshall Duality\nSource item: 5.2\nSource URL: http://aimpl.org/algvision/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[209]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a Carlsson-Weinshallduality for point and line correspondence?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0210", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the four-anchor Carlsson-Weinshall Cremona involution, a generic world line L disjoint from the six base lines maps to a rational normal cubic C_L through the four anchors. More strongly, the anchored point-multiview curve obtained by viewing all points of L with fixed cameras c_i is exactly the resectioning curve obtained by viewing the fixed dual points kappa(c_i) with a camera whose center moves along C_L. Hence an incidence-preserving extension that agrees with the classical pointwise CW map cannot send generic ordinary lines to ordinary lines; the natural closed feature type is line versus four-anchor twisted-cubic camera trajectory.\n\nCandidate contribution (theorem; novelty confidence low): On the generic CW-reduced locus, the anchored point-multiview curve of a world line equals the resectioning curve of the CW-dual fixed points under a camera moving on the line's four-anchor twisted-cubic Cremona image; this equality yields a formal obstruction to any pointwise-CW-compatible, incidence-preserving ordinary-line closure." }, { "id": 20000211, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0211", "title": "A framed quotient and a regular compactification of Carlsson-Weinshall duality", "statement": "What is a moduli space description of Carlsson-Weinshall duality? What moduli spaces does this suggest? A framed multiview variety?", "original_statement": "What is a moduli space description of Carlsson-Weinshall duality? What moduli spaces does this suggest? A framed multiview variety?", "clean_statement": "What is a moduli space description of Carlsson-Weinshall duality? What moduli spaces does this suggest? A framed multiview variety?", "statement_status": "exact", "statement_verification": "The exact canonical AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Carlsson-Weinshall Duality\nSource item: 5.3\nSource URL: http://aimpl.org/algvision/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[210]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is a moduli space description of Carlsson-Weinshall duality? What moduli spaces does this suggest? A framed multiview variety?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0211", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For four ordered anchors, m camera centers, r nonfiducial scene points, and one distinguished nonanchor label, the free PGL_4 configuration quotient is the torus product T^(m+r-1). Carlsson-Weinshall duality is componentwise inversion after swapping camera and scene colors; replacing each T by the type-A_3 permutohedral threefold extends it biregularly and sends each boundary divisor D_I to D_(I^c). The closures of the reduced-measurement graphs in this compactification are biregularly dual after transposing the m by r image array.\n\nCandidate contribution (theorem; novelty confidence low): The pivot-rigidified CW quotient admits the explicit smooth projective compactification (X_A3)^(m+r-1), with boundary action D_(label,I) mapped to D_(label,I^c), and its reduced-measurement graph closure for (m,r) is biregularly isomorphic to the graph closure for (r,m) after output transpose." }, { "id": 20000212, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0212", "title": "Higher-dimensional Carlsson-Weinshall duality and its rational-normal center fibers", "statement": "What can we say about Carlsson-Weinshall duality for other dimensions like $\\mathbb{P}^4 \\to \\mathbb{P}^3$?", "original_statement": "What can we say about Carlsson-Weinshall duality for other dimensions like $\\mathbb{P}^4 \\to \\mathbb{P}^3$?", "clean_statement": "What can we say about Carlsson-Weinshall duality for other dimensions like $\\mathbb{P}^4 \\to \\mathbb{P}^3$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, Algebraic Vision, section “Carlsson-Weinshall Duality,” Problem 5.4:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Carlsson-Weinshall Duality\nSource item: 5.4\nSource URL: http://aimpl.org/algvision/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[211]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What can we say about Carlsson-Weinshall duality for other dimensions like $\\\\mathbb{P}^4 \\\\to \\\\mathbb{P}^3$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0212", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For reduced projective cameras P^d to P^{d-1} on the dense coordinate torus, the standard degree-d Cremona involution exchanges a camera center c and world point x while preserving the image: P_c(x)=P_{kappa_d(x)}(kappa_d(c)). More strongly, if q is the number of distinct coordinate ratios x_i/c_i, the closure of all centers imaging the fixed x at the fixed image P_c(x) is a rational normal curve of degree q-1 spanning P^{q-1}. Generically for P^4 to P^3 it is the unique rational normal quartic through the five coordinate anchors, x, and c; ratio collisions give exactly a twisted cubic, conic, or line. An explicit Gale kernel identifies those degree drops with Gale-point collisions.\n\nCandidate contribution (theorem; novelty confidence low): On the reduced-camera torus, the fixed-point/fixed-image center curve has exact degree #\\{x_i/c_i\\}-1 and span of the same dimension; its full degree-drop boundary is obtained by ratio collisions, which are precisely collisions among the anchor rows of the explicit Gale transform. In P^4 this gives the testable quartic/twisted-cubic/conic/line stratification with grouped-support limit points." }, { "id": 20000213, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0213", "title": "Formal velocities and obstructions for time-varying epipolar matrices", "statement": "Is there a differentiable structure for essential varieties or fundamental varieties over time. Is there a formal structure? Can we use deformation theory and cohomological techniques?", "original_statement": "Is there a differentiable structure for essential varieties or fundamental varieties over time. Is there a formal structure? Can we use deformation theory and cohomological techniques?", "clean_statement": "Is there a differentiable structure for essential varieties or fundamental varieties over time. Is there a formal structure? Can we use deformation theory and cohomological techniques?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, *Algebraic vision*, Section “Multiview Geometry for Continuous Motion,” Problem 6.1:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Multiview Geometry for Continuous Motion\nSource item: 6.1\nSource URL: http://aimpl.org/algvision/6/\nCanonical location: aim-algebraic-geometry-notes.json notes[212]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a differentiable structure for essential varieties or fundamental varieties over time. Is there a formal structure? Can we use deformation theory and cohomological techniques?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/6/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0213", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the projective 3-by-3 fundamental variety, a first velocity [A] at a rank-k matrix F_0, k in {1,2}, is the derivative of a full formal rank-at-most-two trajectory if and only if the induced kernel-to-cokernel map bar(A) has rank at most 2-k. An explicit Schur-complement formula integrates every admissible velocity to all orders. At rank one the Zariski tangent space is the full ambient tangent space, but actual trajectory velocities form the quadratic cone det(bar(A))=0; this determinant is the primary obstruction in the one-dimensional H^1 of the hypersurface tangent complex. The report also records the established five-dimensional differentiable structure of normalized real essential matrices and a cotangent-complex framework for whole time curves.\n\nCandidate contribution (constructive formal-arc theorem; novelty confidence low): Candidate novelty: the prescribed-velocity criterion rank(bar(A)) <= 2-k is packaged with an explicit all-orders Schur-complement lift for time-varying projective fundamental matrices, and at rank one its quadratic velocity cone is identified with the zero locus of the one-dimensional primary hypersurface obstruction det(bar(A))." }, { "id": 20000214, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0214", "title": "Graph spaces and multi-Rees models for continuous multiview geometry", "statement": "What is the proper algebraic formulation for a continuous multiview variety? $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\to (\\mathbb{P}^2)^n$? $Hom(\\mathbb{A}^1, \\text{essential matrices})$? Kontsevich spaces? Or could we use points and optical flow data?", "original_statement": "What is the proper algebraic formulation for a continuous multiview variety? $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\to (\\mathbb{P}^2)^n$? $Hom(\\mathbb{A}^1, \\text{essential matrices})$? Kontsevich spaces? Or could we use points and optical flow data?", "clean_statement": "What is the proper algebraic formulation for a continuous multiview variety? $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\to (\\mathbb{P}^2)^n$? $Hom(\\mathbb{A}^1, \\text{essential matrices})$? Kontsevich spaces? Or could we use points and optical flow data?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List, workshop **Algebraic vision**, section **Multiview Geometry for Continuous Motion**, Problem 6.2. Its exact statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Algebraic vision\nSection: Multiview Geometry for Continuous Motion\nSource item: 6.2\nSource URL: http://aimpl.org/algvision/6/\nCanonical location: aim-algebraic-geometry-notes.json notes[213]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the proper algebraic formulation for a continuous multiview variety? $\\\\mathbb{P}^1 \\\\times \\\\mathbb{P}^3 \\\\to (\\\\mathbb{P}^2)^n$? $Hom(\\\\mathbb{A}^1, \\\\text{essential matrices})$? Kontsevich spaces? Or could we use points and optical flow data?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/algvision/6/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0214", "aim-domain:algebraic-geometry", "aim-workshop:algvision", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a bounded-degree framed camera trajectory, the tentative map from time times world space is rational along the center graphs; its canonical incidence model is the multi-Rees graph closure, equal to the blowup of the disjoint union when centers remain disjoint. For n greater than 2, parameterized stable maps of class (1,beta) into the product of the time line and the Hilbert closure of Cam_n give a proper bounded-class compactification that retains time and carries a pulled-back flat multiview family. In the local transverse collision model [x:y:z], [x-t:y:z], the reduced collision fiber is proved to be the coupled hypersurface u_2 v_3-u_3 v_2=0 in P^2 times P^2. A separate differential calculation proves that six general known world points give an immersive projective-camera evaluation, whereas five never do.\n\nCandidate contribution (proposition; novelty confidence low): For the local two-camera collision with base ideals (x,y,z) and (x-t,y,z), the reduced fiber of the canonical joint graph closure over t=x=y=z=0 is exactly the hypersurface u_2 v_3-u_3 v_2=0 in P^2 times P^2." }, { "id": 20000215, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0215", "title": "Vertical tangents and the cubic jumping wall for plane curves", "statement": "Problem 1\n\n(Suggested by Eric Larson) Given the data of a curve X with n marked points lying in projective space Pn, you can forget the curve and remember only the marked points. This can be interpreted as giving a map\n\nMg,n(Pr, β ) π\n\n−→ Conf( Pr, n).\n\nThe basic question I'm interested in is, what can we say about the various possible fiber dimensions or vertical tangent space dimensions? I have recently finished up a paper that studies when this map is dominant. The upshot is that if d ≥ g + r, then we can explicitly characterize when the map is dominant.", "original_statement": "Problem 1 \n\n(Suggested by Eric Larson) Given the data of a curve X with n marked points lying in projective space Pn, you can forget the curve and remember only the marked points. This can be interpreted as giving a map \n\nMg,n(Pr, β ) π\n\n−→ Conf( Pr, n).\n\nThe basic question I'm interested in is, what can we say about the various possible fiber dimensions or vertical tangent space dimensions? I have recently finished up a paper that studies when this map is dominant. The upshot is that if d ≥ g + r, then we can explicitly characterize when the map is dominant.", "clean_statement": "Problem 1\n\n(Suggested by Eric Larson) Given the data of a curve X with n marked points lying in projective space Pn, you can forget the curve and remember only the marked points. This can be interpreted as giving a map\n\nMg,n(Pr, β ) π\n\n−→ Conf( Pr, n).\n\nThe basic question I'm interested in is, what can we say about the various possible fiber dimensions or vertical tangent space dimensions? I have recently finished up a paper that studies when this map is dominant. The upshot is that if d ≥ g + r, then we can explicitly characterize when the map is dominant.", "statement_status": "exact", "statement_verification": "The source is Problem 1, suggested by Eric Larson, in the American Institute of Mathematics problem-session document *Degenerations in Algebraic Geometry* (September 7, 2016). The displayed map is \\[ \\pi:\\mathcal M_{g,n}(\\mathbf P^r,\\beta)\\longrightarrow \\operatorname{Conf}(\\mathbf P^r,n), \\] and the question asks for the possible dimensions of its fibers and vertical tangent spaces, especially when the map is known to be dominant.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 1\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[214]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1 \\n\\n(Suggested by Eric Larson) Given the data of a curve X with n marked points lying in projective space Pn, you can forget the curve and remember only the marked points. This can be interpreted as giving a map \\n\\nMg,n(Pr, β ) π\\n\\n−→ Conf( Pr, n).\\n\\nThe basic question I'm interested in is, what can we say about the various possible fiber dimensions or vertical tangent space dimensions? I have recently finished up a paper that studies when this map is dominant. The upshot is that if d ≥ g + r, then we can explicitly characterize when the map is dominant.\"\nOriginal remarks: [\"Remark 1.1. For example, in this range the answer was known by work of Stevens for canonical curves. To elaborate, consider the case where the map is known to be dominant. What can we say about the fibers of π (e.g. their dimensions, or the dimensions of their tangent spaces)?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0215", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "On the smooth embedded locus, the evaluation morphism has vertical tangent H^0(C,N_f(-D)) and differential cokernel ker(H^1(C,N_f(-D)) -> H^1(C,N_f)), hence dim T_vert = (r+1)d-(r-3)(g-1)-(r-1)n+h^1(C,N_f(-D)). For a smooth plane curve of degree e this becomes N_e-n+h^0(O_C(D-3H)). There are no jumps for n<3e; at n=3e a jump occurs exactly when D is the scheme-theoretic intersection with a cubic, in which case the differential has corank one and the whole smooth embedded fiber is the explicit complete-intersection linear system P(kF_C+Q H^0(O_P2(e-3))).\n\nCandidate contribution (theorem; novelty confidence low): For smooth degree-e plane embeddings with e at least 3, the first possible special vertical tangent occurs exactly at n=3e; the jumping locus is precisely D in |3H|, equivalently the cubic complete-intersection locus, and there the differential has corank one while the full fixed-configuration fiber has dimension g_e, exactly one above the expected dimension." }, { "id": 20000216, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0216", "title": "Exact vertical tangent defects for canonical genus-four curves", "statement": "Problem 1.2. Find examples where we can understand behavior distinct from the expected one. There is basically no hope of a general answer, so I mean this as an example-hunting suggestion.\n\nA specific application would be to extend the range in which we can say when π is dominant. If you have examples where the tangent space dimension is wrong, then you can degenerate to a curve where the tangent space dimension is wrong again.", "original_statement": "Problem 1.2. Find examples where we can understand behavior distinct from the expected one. There is basically no hope of a general answer, so I mean this as an example-hunting suggestion. \n\nA specific application would be to extend the range in which we can say when π is dominant. If you have examples where the tangent space dimension is wrong, then you can degenerate to a curve where the tangent space dimension is wrong again.", "clean_statement": "Problem 1.2. Find examples where we can understand behavior distinct from the expected one. There is basically no hope of a general answer, so I mean this as an example-hunting suggestion.\n\nA specific application would be to extend the range in which we can say when π is dominant. If you have examples where the tangent space dimension is wrong, then you can degenerate to a curve where the tangent space dimension is wrong again.", "statement_status": "exact", "statement_verification": "The source is the problem-session document from the 2016 AIM workshop *Degenerations in Algebraic Geometry*. Problem 1 introduces the evaluation map \\[ \\pi:\\mathcal M_{g,n}(\\mathbf P^r,\\beta)\\longrightarrow \\operatorname{Conf}(\\mathbf P^r,n) \\] which remembers only the images of the \\(n\\) marked points. The printed PDF says “a curve \\(X\\) with \\(n\\) marked points lying in \\(\\mathbf P^n\\)” in its first sentence, but the displayed map and every subsequent parameter use \\(\\mathbf P^r\\). Thus \\(\\mathbf P^n\\) is retained as a source typo and \\(\\mathbf P^r\\) is the mathematically consistent reading.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 1.2\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[215]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.2. Find examples where we can understand behavior distinct from the expected one. There is basically no hope of a general answer, so I mean this as an example-hunting suggestion. \\n\\nA specific application would be to extend the range in which we can say when π is dominant. If you have examples where the tangent space dimension is wrong, then you can degenerate to a curve where the tangent space dimension is wrong again.\"\nOriginal remarks: [\"Remark 1.3. Subject only to the non-negativity of the Brill-Noether number, you can use this result plus a trick of \\\"attaching canonical curves\\\" (where the map π is known to fail to be dominant). When the n points cut out the divisor K2 \\n\\n> C, this fails even more badly, and you can leverage this to extend the range in P3.12\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0216", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a general canonical genus-four complete-intersection curve C=(2,3) in projective three-space and a general reduced divisor D of degree n at least 7, the vertical tangent dimension of the marked evaluation map is (9-n)_+ + (15-n)_+, while its Euler-expected value is max(0,24-2n). Thus general divisors of degrees 10 through 14 witness failure of normal-bundle interpolation; for n=10,11,12 the differential cokernel has dimension n-9, exactly the codimension of the locus of configurations lying on a quadric, proving nondominance in the dimension-allowed range. For a reduced bicanonical divisor D in |2K_C| of degree 12, N_C(-D) is O_C direct-sum K_C, so both kernel and cokernel jump from the generic value 3 to 5.\n\nCandidate contribution (proposition; novelty confidence low): The candidate contribution is the exact formula dim ker(d pi)=(9-n)_+ +(15-n)_+, its identification with quadric-incidence corank n-9 for n=10,11,12, and the explicit bicanonical jump (h0,h1)=(3,3) to (5,5) at degree 12." }, { "id": 20000217, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0217", "title": "Brion's degeneration theorem and a birational basis-exchange law", "statement": "Problem 2\n\n(Suggested by Allen Knutson) Let me motivate this by stating a theorem of Brion. Suppose you have a variety X lying in a flag manifold G/P. We say that X is multiplicity-free if in cohomology [X] =\n\n∑\n\n> S⊂W/WP\n\n[Xs],\n\ni.e. each zero-dimensional intersection with a general translate of a Schubert variety is empty or 1 point.\n\nTheorem 2.1 (Brion). If X is multiplicity-free, then X degenerates to the reduced union of Schubert varieties (as predicted by S ), and this union is Cohen-Macaulay.\n\nIn particular, if X is reduced and irreducible and multiplicity-free, then X is Cohen-Macaulay.", "original_statement": "Problem 2 \n\n(Suggested by Allen Knutson) Let me motivate this by stating a theorem of Brion. Suppose you have a variety X lying in a flag manifold G/P. We say that X is multiplicity-free if in cohomology [X] =\n\n∑ \n\n> S⊂W/WP\n\n[Xs],\n\ni.e. each zero-dimensional intersection with a general translate of a Schubert variety is empty or 1 point. \n\nTheorem 2.1 (Brion). If X is multiplicity-free, then X degenerates to the reduced union of Schubert varieties (as predicted by S ), and this union is Cohen-Macaulay. \n\nIn particular, if X is reduced and irreducible and multiplicity-free, then X is Cohen-Macaulay.", "clean_statement": "Problem 2\n\n(Suggested by Allen Knutson) Let me motivate this by stating a theorem of Brion. Suppose you have a variety X lying in a flag manifold G/P. We say that X is multiplicity-free if in cohomology [X] =\n\n∑\n\n> S⊂W/WP\n\n[Xs],\n\ni.e. each zero-dimensional intersection with a general translate of a Schubert variety is empty or 1 point.\n\nTheorem 2.1 (Brion). If X is multiplicity-free, then X degenerates to the reduced union of Schubert varieties (as predicted by S ), and this union is Cohen-Macaulay.\n\nIn particular, if X is reduced and irreducible and multiplicity-free, then X is Cohen-Macaulay.", "statement_status": "exact", "statement_verification": "The canonical record is item 2 in the problem-session notes for the AIM workshop *Degenerations in Algebraic Geometry*, suggested by Allen Knutson. The source is the six-page PDF dated September 7, 2016.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 2\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[216]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2 \\n\\n(Suggested by Allen Knutson) Let me motivate this by stating a theorem of Brion. Suppose you have a variety X lying in a flag manifold G/P. We say that X is multiplicity-free if in cohomology [X] =\\n\\n∑ \\n\\n> S⊂W/WP\\n\\n[Xs],\\n\\ni.e. each zero-dimensional intersection with a general translate of a Schubert variety is empty or 1 point. \\n\\nTheorem 2.1 (Brion). If X is multiplicity-free, then X degenerates to the reduced union of Schubert varieties (as predicted by S ), and this union is Cohen-Macaulay. \\n\\nIn particular, if X is reduced and irreducible and multiplicity-free, then X is Cohen-Macaulay.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0217", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The original PDF confirms that this record is only the statement of Brion's proved multiplicity-free degeneration theorem; the separately numbered classification question begins in the next canonical record. As a substantive structural consequence in (P^1)^n, the attempt proves that every matroid-basis projection of an integral multiplicity-free variety is birational, that each adjacent basis exchange is a relative PGL_2 transformation over the common coordinate field, and that these varying-field birational transitions satisfy the basis-graph cycle cocycle. It also gives an exact rational-function realization reduction, without claiming a combinatorial classification.\n\nCandidate contribution (equivalence_and_exchange_law; novelty confidence low): For an integral d-dimensional multiplicity-free X in (P^1)^n, every adjacent pair of matroid bases B=C union {i} and B'=C union {j} satisfies t_j=(a t_i+b)/(c t_i+e) over k(t_c:c in C), with nonzero determinant; the induced birational basis-chart transitions obey the cycle cocycle, and multiplicity-free realizability is equivalently expressible by rational functions for which every matroid basis generates the full rational function field.", "research_difficulty_suggested": "L2" }, { "id": 20000218, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0218", "title": "Matroid classification and a sharp four-factor Cohen-Macaulay obstruction", "statement": "Problem 2.2. Classify subsets S ⊂ W/WP that deform to irreducibles.\n\nFor example, is ⋃s∈S Xs Cohen-Macaulay su ffi cient? The first case to try is G/P [U+001B]\n\n(P1)n.", "original_statement": "Problem 2.2. Classify subsets S ⊂ W/WP that deform to irreducibles. \n\nFor example, is ⋃s∈S Xs Cohen-Macaulay su ffi cient? The first case to try is G/P \u001b\n\n(P1)n.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "This record is Problem 2.2 in the AIM workshop list *Degenerations in algebraic geometry*. The literal database extraction is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 2.2\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[217]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.2. Classify subsets S ⊂ W/WP that deform to irreducibles. \\n\\nFor example, is ⋃s∈S Xs Cohen-Macaulay su ffi cient? The first case to try is G/P \\u001b\\n\\n(P1)n.\"\nOriginal remarks: [\"Remark 2.3. I have shown that a degeneration of Kazhdan-Lustzig varieties is a union of affi ne spaces. I think of this as an easier result in that vein. \\n\\n3\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0218", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For pure Schubert unions in (P^1)^n, deformation to an integral variety is equivalent to a degree-one algebraic-matroid realization; in characteristic zero this is equivalent to the support being the bases of a linearly representable matroid. The union X_12 union X_23 union X_34 in (P^1)^4 is a reduced arithmetically Cohen-Macaulay surface union, certified by an explicit saturated height-two Cox ideal and Hilbert-Burch resolution, but it cannot deform to an integral surface because its three indices violate basis exchange. Every pure union for n at most 3 is representable and deformable, so four factors are minimal.\n\nCandidate contribution (counterexample; novelty confidence low): The path support {12,23,34} is a characteristic-free Cohen-Macaulay counterexample in (P^1)^4, and n=4 is the sharp minimum number of factors for a pure counterexample." }, { "id": 20000219, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0219", "title": "The maximal-contact linear-space arrangement of a plane elliptic cubic", "statement": "Problem 3\n\n(Suggested by Michel) Let X ⊂ P2 be an elliptic curve.", "original_statement": "Problem 3 \n\n(Suggested by Michel) Let X ⊂ P2 be an elliptic curve.", "clean_statement": "Problem 3\n\n(Suggested by Michel) Let X ⊂ P2 be an elliptic curve.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 3\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[218]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3 \\n\\n(Suggested by Michel) Let X ⊂ P2 be an elliptic curve.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0219", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The original PDF confirms that this record is only the setup sentence for the separately numbered next problem. As a substantive intrinsic contribution, for a smooth plane cubic E with flex origin o, degree-d plane curves not containing E and meeting it at only p exist exactly when p-o is 3d-torsion. For each eligible p their projective parameter-space closure is P^{g_d}, with non-containing locus A^{g_d}, where g_d=(d-1)(d-2)/2. The closures for any two distinct contact points intersect exactly in the common P^{g_d-1} core of forms divisible by the equation of E; deleting the core gives a disjoint union of (3d)^2 affine spaces in characteristic zero. Rationality is a separate delta-invariant condition and no enumerative count for the next record is claimed.\n\nCandidate contribution (structural_proposition; novelty confidence low): The candidate contribution is the simultaneous common-core arrangement: the (3d)^2 fixed maximal-contact closures are linear P^{g_d}'s whose every distinct pair intersects scheme-theoretically in the same P^{g_d-1} subspace of curves containing E, so their core complements are disjoint A^{g_d}'s; this arrangement also forms affine bundles over a smooth family of cubics.", "research_difficulty_suggested": "L2" }, { "id": 20000220, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0220", "title": "The one-point count for a plane cubic and the degree-nine log-BPS answer", "statement": "Problem 3.1. What is the number of degree d rational curves in P2 meeting X at exactly 1\n\npoint?\n\nThis is known for d ≤ 8 by Takashi. Even d = 9 would be new and interesting. The motivation is from local mirror symmetry. One approach is to degenerate X to 3 lines. Also, is a tropical approach possible?\n\n4", "original_statement": "Problem 3.1. What is the number of degree d rational curves in P2 meeting X at exactly 1\n\npoint? \n\nThis is known for d ≤ 8 by Takashi. Even d = 9 would be new and interesting. The motivation is from local mirror symmetry. One approach is to degenerate X to 3 lines. Also, is a tropical approach possible? \n\n4", "clean_statement": "Problem 3.1. What is the number of degree d rational curves in P2 meeting X at exactly 1\n\npoint?\n\nThis is known for d ≤ 8 by Takashi. Even d = 9 would be new and interesting. The motivation is from local mirror symmetry. One approach is to degenerate X to 3 lines. Also, is a tropical approach possible?\n\n4", "statement_status": "exact", "statement_verification": "The AIM record occurs in the workshop list *Degenerations in algebraic geometry*. The preceding item introduces \\(X\\subset \\mathbb P^2\\) as an elliptic curve, hence as a smooth plane cubic. The record itself reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 3.1\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[219]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.1. What is the number of degree d rational curves in P2 meeting X at exactly 1\\n\\npoint? \\n\\nThis is known for d ≤ 8 by Takashi. Even d = 9 would be new and interesting. The motivation is from local mirror symmetry. One approach is to degenerate X to 3 lines. Also, is a tropical approach possible? \\n\\n4\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0220", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM wording conflates the literal set-theoretic count with Takahashi's one-place/maximal-contact count. Bousseau's all-degree theorem gives the fixed primitive degree-nine log-BPS invariant, equivalently the length of the finite primitive log-map moduli scheme, as 1,027,737; it does not by itself give the number of distinct reduced image curves. An explicit torsion audit proves that the number of contact points with minimal index k is 9 times the second Jordan totient J_2(k), yielding the aggregate identity (-1)^(d-1) N_d = (9/d^2) sum_{e|d} e^4 Omega_e. In degree nine the contact strata have sizes 9, 72, and 648, the primitive aggregate length is 665,973,576, and the full unspecified-contact relative invariant is 749,220,300 + 1/9.\n\nCandidate contribution (explicit_formula; novelty confidence low): The explicit Jordan-totient disaggregation A_k = 9 J_2(k), its aggregate audit identity for the total relative invariant, and the complete degree-nine three-stratum table distinguish fixed primitive, all-primitive, and unspecified-contact conventions." }, { "id": 20000221, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0221", "title": "Expected-dimension generality of binary curves and two explicit boundary diagnostics", "statement": "Problem 4\n\n(Suggested by Brian Osserman) Consider forming a nodal curve X by gluing together two rational curves at g + 1 generic points. Is X Brill-Noether general?", "original_statement": "Problem 4 \n\n(Suggested by Brian Osserman) Consider forming a nodal curve X by gluing together two rational curves at g + 1 generic points. Is X Brill-Noether general?", "clean_statement": "Problem 4\n\n(Suggested by Brian Osserman) Consider forming a nodal curve X by gluing together two rational curves at g + 1 generic points. Is X Brill-Noether general?", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 4 from the workshop *Degenerations in algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 4\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[220]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4 \\n\\n(Suggested by Brian Osserman) Consider forming a nodal curve X by gluing together two rational curves at g + 1 generic points. Is X Brill-Noether general?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0221", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The intended expected-dimension question is answered affirmatively by Xiang He's all-rank theorem for general binary curves, including every potentially nonempty balanced multidegree and Osserman limit linear series, but the unqualified statement is definition-dependent. Independently, this attempt proves that for any binary curve and multidegree (a,b) with a<0, G^r_(a,b) is empty for b= 1, genus g = 2(r+1) and multidegree (r-1, 2r+1) satisfy rho = 0 and both balance inequalities, yet admit no g^r. Positively, over an algebraically closed characteristic-zero field, for r,e >= 1 and m = re, a general genus-g binary curve has a nonempty rational open locus of g^r_(r,m)'s isomorphic to an open subset of projective rho-space whenever g <= (r+1)e; at the rho = 0 boundary g = (r+1)e, the full fixed-multidegree set consists of one isomorphism class, set-theoretically.\n\nCandidate contribution (theorem; novelty confidence low): For every r,e >= 1, the fixed multidegree (r,re) is nonempty on a general binary curve whenever rho >= 0, with an explicit rational expected-dimensional open locus; when g = (r+1)e and rho = 0, it has a unique linear series up to isomorphism." }, { "id": 20000223, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0223", "title": "Exact gluing and transverse-padding formulas for plane-curve H-constants", "statement": "Problem 5\n\n(Suggested by Brian Harbourne) The Bounded Negativity Conjecture says that for X a sur-face which is either rational or over characteristic 0, there exists nX such that C2 ≥ nX for all reduced C ⊂ X.Nobody knows how to approach this. Recently people have started looking at a variant called \" h-constants.\" Let C be a singular reduced plane curve of degree d. Look at\n\nh(C): = d2 − ∑x∈C m2\n\n> x\n\n#singular points,\n\nwhere mx is the multiplicity of the singular point x. (This is essentially the self-intersection of the proper transform in the blowup.)", "original_statement": "Problem 5 \n\n(Suggested by Brian Harbourne) The Bounded Negativity Conjecture says that for X a sur-face which is either rational or over characteristic 0, there exists nX such that C2 ≥ nX for all reduced C ⊂ X.Nobody knows how to approach this. Recently people have started looking at a variant called \" h-constants.\" Let C be a singular reduced plane curve of degree d. Look at \n\nh(C): = d2 − ∑x∈C m2\n\n> x\n\n#singular points,\n\nwhere mx is the multiplicity of the singular point x. (This is essentially the self-intersection of the proper transform in the blowup.)", "clean_statement": "Problem 5\n\n(Suggested by Brian Harbourne) The Bounded Negativity Conjecture says that for X a sur-face which is either rational or over characteristic 0, there exists nX such that C2 ≥ nX for all reduced C ⊂ X.Nobody knows how to approach this. Recently people have started looking at a variant called \" h-constants.\" Let C be a singular reduced plane curve of degree d. Look at\n\nh(C): = d2 − ∑x∈C m2\n\n> x\n\n#singular points,\n\nwhere mx is the multiplicity of the singular point x. (This is essentially the self-intersection of the proper transform in the blowup.)", "statement_status": "exact", "statement_verification": "This record is the introductory part of Problem 5 in the AIM workshop list *Degenerations in Algebraic Geometry* (September 7, 2016), suggested by Brian Harbourne. The database extraction breaks the displayed fraction across lines. Inspection of page 3 of the original PDF (PDF page index 2) recovers the text as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 5\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[222]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 5 \\n\\n(Suggested by Brian Harbourne) The Bounded Negativity Conjecture says that for X a sur-face which is either rational or over characteristic 0, there exists nX such that C2 ≥ nX for all reduced C ⊂ X.Nobody knows how to approach this. Recently people have started looking at a variant called \\\" h-constants.\\\" Let C be a singular reduced plane curve of degree d. Look at \\n\\nh(C): = d2 − ∑x∈C m2\\n\\n> x\\n\\n#singular points,\\n\\nwhere mx is the multiplicity of the singular point x. (This is essentially the self-intersection of the proper transform in the blowup.)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0223", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The assigned record is the contextual definition preceding the separately indexed Problem 5.1: for a reduced singular plane curve C, H(C) is the proper-transform self-intersection after blowing up all distinct proper singular points, divided by their number. Beyond reconstructing that definition, this attempt proves an exact characteristic-free numerator gluing formula for arbitrary reduced curves, including nonordinary singularities and tangencies. It specializes to (s+ab)(H(A union B)+2)=s(H(A)+2)+b^2 for smooth transverse adjunction, and to H(A_t)+2=[s(H(A)+2)+t]/[s+at+binom(t,2)] after adjoining t successive general lines. Thus transverse line padding has an exact finite crossing threshold and universally approaches -2, eventually from above.\n\nCandidate contribution (formula; novelty confidence low): For any reduced singular plane curve A of degree a with s proper singular points, adjoining t successive general lines gives H(A_t)+2=[s(H(A)+2)+t]/[s+at+binom(t,2)]; consequently the sign relative to -2 changes exactly when t crosses the integer -s(H(A)+2), and the sequence tends to -2 eventually from above.", "research_difficulty_suggested": "L2" }, { "id": 20000224, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0224", "title": "Exact line-arrangement operation formulas and a rich-line construction barrier", "statement": "Problem 5.1. Is h C bounded below uniformly?\n\nIf this is true, then the Bounded Negativity Conjecture holds for rational surfaces in characteristic\n0. (In characteristic p > 0, h is not bounded below. One could ask what the interesting statement would be in characteristic p.) It is known that if C is a union of lines, then h(C) > −4, while the best achieved example is something like −3.36. The point is to find a configuration of lines with many high-multiplicity intersections (basically, the point is that you want to avoid simple crossings).", "original_statement": "Problem 5.1. Is h C bounded below uniformly? \n\nIf this is true, then the Bounded Negativity Conjecture holds for rational surfaces in characteristic \n0. (In characteristic p > 0, h is not bounded below. One could ask what the interesting statement would be in characteristic p.) It is known that if C is a union of lines, then h(C) > −4, while the best achieved example is something like −3.36. The point is to find a configuration of lines with many high-multiplicity intersections (basically, the point is that you want to avoid simple crossings).", "clean_statement": "Problem 5.1. Is h C bounded below uniformly?\n\nIf this is true, then the Bounded Negativity Conjecture holds for rational surfaces in characteristic\n0. (In characteristic p > 0, h is not bounded below. One could ask what the interesting statement would be in characteristic p.) It is known that if C is a union of lines, then h(C) > −4, while the best achieved example is something like −3.36. The point is to find a configuration of lines with many high-multiplicity intersections (basically, the point is that you want to avoid simple crossings).", "statement_status": "exact", "statement_verification": "The source is Problem 5.1 from the AIM workshop list *Degenerations in algebraic geometry*. The extracted record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 5.1\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[223]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 5.1. Is h C bounded below uniformly? \\n\\nIf this is true, then the Bounded Negativity Conjecture holds for rational surfaces in characteristic \\n0. (In characteristic p > 0, h is not bounded below. One could ask what the interesting statement would be in characteristic p.) It is known that if C is a union of lines, then h(C) > −4, while the best achieved example is something like −3.36. The point is to find a configuration of lines with many high-multiplicity intersections (basically, the point is that you want to avoid simple crossings).\"\nOriginal remarks: [\"Remark 5.2. This is only interesting in characteristic 0, because over a finite field you can easily find C sending h(C) → −∞ by taking all (rational) lines.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0224", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The characteristic-zero uniform lower-bound problem remains open, and the AIM proper-singularity convention must be distinguished from full infinitely-near clusters and prescribed finite point sets. For every arrangement of distinct projective lines, this attempt proves exact formulas on the changing full singular locus: adding a line through q old singular points of multiplicities m_j and creating u=d-sum_j m_j distinct new nodes gives h'=(s h+1-q-2u)/(s+u), while arbitrary deletion gives h'=(s h+R+v-q)/(s-delta) with explicit corrections for destroyed singularities. A fixed-seed invariant then proves that, from any seed h_0<=-2, iterative additions meeting at most one current singular point never produce h=u+2, and for the Wiman seed the exact criterion is 67q-91u>67.\n\nCandidate contribution (transformation formula and obstruction theorem; novelty confidence low): For line arrangements evaluated on their changing full singular loci, the addition formula h'=(s h+1-q-2u)/(s+u) and deletion formula h'=(s h+R+v-q)/(s-delta), together with the fixed-seed invariant Phi_t=N_t-h_0 s_t, give an exact characteristic-independent operation calculus and prove the one-rich-line construction barrier." }, { "id": 20000225, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0225", "title": "Proper-point defects and the degree-21 bottleneck", "statement": "Problem 5.3. For irreducible curves, the best known examples have h → − 2. Can we do better?", "original_statement": "Problem 5.3. For irreducible curves, the best known examples have h → − 2. Can we do better?", "clean_statement": "Problem 5.3. For irreducible curves, the best known examples have h → − 2. Can we do better?", "statement_status": "exact", "statement_verification": "The source is the six-page problem-session record from the AIM workshop *Degenerations in Algebraic Geometry* (September 7, 2016). The surrounding text first defines, for a singular reduced plane curve \\(C\\) of degree \\(d\\), \\[ h(C)=\\frac{d^2-\\sum_{x\\in \\operatorname{Sing}(C)}m_x(C)^2} {\\#\\operatorname{Sing}(C)}. \\] Here the sum is over the distinct **proper singular points in \\(\\mathbf P^2\\)**. This convention is explicit in the source: \\(m_x\\) is “the multiplicity of the singular point \\(x\\),” and the denominator is the number of singular points.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 5.3\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[224]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 5.3. For irreducible curves, the best known examples have h → − 2. Can we do better?\"\nOriginal remarks: [\"Remark 5.4. Roulleau has produced configurations of cubics with h → − 4.\", \"Remark 5.5. Over R, we know that h → − 3 and h ≥ − 3. Over Q, no analogous result is known. One could go further and ask if the Bounded Negativity Conjecture is true for tropical surfaces. 36\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0225", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a reduced irreducible complex plane curve, using only its proper singular points, the exact identities s(H+2)=3d-2+2g-sum_i(m_i-2)+sum_i eta_i and 3s(H+2)=Lambda+7g+(-d^2+21d-14)/2 hold, where eta_i=2delta_i-m_i(m_i-1) is an even nonnegative infinitely-near defect and Lambda is a sum of nonnegative local slacks. These identities recover H>-2 in degrees at most 20 and prove that any degree-21 curve with H<-2 must be rational, have eta_i=0 at every singularity, and belong to an explicit short list of multiplicity patterns with Lambda equal to 1 or 4. This is a necessary-pattern classification, not an existence result.\n\nCandidate contribution (classification theorem; novelty confidence low): Candidate novelty: every degree-21 irreducible complex plane curve with proper-point H<-2 must have geometric genus zero, zero proper-point genus defect at every singularity, and exactly one of the explicitly proved Lambda=1 or Lambda=4 multiplicity profiles; in particular H is forced to be either -2-2/s or -2-1/s." }, { "id": 20000226, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0226", "title": "Tropical Hasse-jet minors and a layered confluent-Vandermonde family", "statement": "Problem 6\n\n(Suggested by Ravi Vakil) Here is an idea for a tropical approach to interpolation of points in\n\nP2. We want to show that given d and m1,..., ms, if the expected dimension is positive then the interpolation problem has the right number of solutions (this is the SHGH Conjecture ). By adding conditions, we may assume that the expected dimension is 1. We can try to do this by find a single isolated (i.e. non-deforming) and liftable tropical solution.\n\n7", "original_statement": "Problem 6 \n\n(Suggested by Ravi Vakil) Here is an idea for a tropical approach to interpolation of points in \n\nP2. We want to show that given d and m1,..., ms, if the expected dimension is positive then the interpolation problem has the right number of solutions (this is the SHGH Conjecture ). By adding conditions, we may assume that the expected dimension is 1. We can try to do this by find a single isolated (i.e. non-deforming) and liftable tropical solution. \n\n7", "clean_statement": "Problem 6\n\n(Suggested by Ravi Vakil) Here is an idea for a tropical approach to interpolation of points in\n\nP2. We want to show that given d and m1,..., ms, if the expected dimension is positive then the interpolation problem has the right number of solutions (this is the SHGH Conjecture ). By adding conditions, we may assume that the expected dimension is 1. We can try to do this by find a single isolated (i.e. non-deforming) and liftable tropical solution.\n\n7", "statement_status": "exact", "statement_verification": "The source is Problem 6 in the AIM workshop list *Degenerations in algebraic geometry*, suggested by Ravi Vakil. The source PDF is . Comparison with the PDF shows that the final character `7` in the extracted record is the heading of the next problem, not part of Problem 6.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 6\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[225]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 6 \\n\\n(Suggested by Ravi Vakil) Here is an idea for a tropical approach to interpolation of points in \\n\\nP2. We want to show that given d and m1,..., ms, if the expected dimension is positive then the interpolation problem has the right number of solutions (this is the SHGH Conjecture ). By adding conditions, we may assume that the expected dimension is 1. We can try to do this by find a single isolated (i.e. non-deforming) and liftable tropical solution. \\n\\n7\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0226", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A noncancelling initial determinant of one maximal augmented Hasse-jet minor is a rigorous tropical-linear certificate that the added point conditions cut the plane-curve system to one reduced projective parameter point and that the original fat-point conditions are independent. An explicit characteristic-free layered confluent-Vandermonde determinant supplies a unit-initial certificate for every multiplicity vector satisfying sum_i m_i <= d+1; independent original rows can then be extended by distinct disjoint simple point evaluations to rank N-1. This is a partial interpolation result and a sufficient certificate, not a stable-map correspondence or a proof of general SHGH.\n\nCandidate contribution (criterion_and_explicit_family; novelty confidence low): Candidate novelty: the augmented Hasse-jet initial-minor and maximal-cofactor lifting criterion, coupled with the explicit determinant det = +/- product_b product_{i=nd, where M_{n^2} is the sum of the n^2 largest multiplicities. Mixing every square size and allowing arbitrary auxiliary very general points of multiplicity zero yields the sharp relative threshold beta_square=max(max_{n<=floor(sqrt(s))} M_{n^2}/n, (sum m_i)/ceil(sqrt(s))). Thus d<=beta_square proves non-bigness, dbeta_square cannot be certified by any nonnegative combination of these square rays. This is a rigorous partial result and exact limitation for the stated toolbox, not a solution of nonsquare Nagata.\n\nCandidate contribution (polyhedral characterization; novelty confidence low): Candidate novelty: the square-subset Nagata rays have an explicit hypersimplex facet description, and their complete dual separating threshold across all square sizes and all zero-multiplicity auxiliary extensions is the order-statistic formula beta_square; homogeneous weights are the sharp fixed-total worst case, with approximation factor at least sqrt(3)/2 for s>=10 and an explicit integer rounding gap." }, { "id": 20000229, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0229", "title": "Cubic maximal rank after the classical theorem", "statement": "Problem 8\n\n(Suggested by David Jensen) The Maximal Rank Conjecture predicts the Hilbert fuction of a general curve (for fixed g, r, d such that ρ(g, r, d) ≥ 0 and r ≥ 3). The Hilbert function measures the rank of the map Sym m H0(X, L) → H0(X, L⊗m). The Maximal Rank Conjecture predicts that this has maximal rank for general X and L.Concretely, this means that it is either injective or surjective. We proposed a combinatorial approach via the notion of tropical independence. How-ever, we can only solve the combinatorial problem for h(2) (the case of quadrics). The next step would be h(3), i.e. cubics.", "original_statement": "Problem 8 \n\n(Suggested by David Jensen) The Maximal Rank Conjecture predicts the Hilbert fuction of a general curve (for fixed g, r, d such that ρ(g, r, d) ≥ 0 and r ≥ 3). The Hilbert function measures the rank of the map Sym m H0(X, L) → H0(X, L⊗m). The Maximal Rank Conjecture predicts that this has maximal rank for general X and L.Concretely, this means that it is either injective or surjective. We proposed a combinatorial approach via the notion of tropical independence. How-ever, we can only solve the combinatorial problem for h(2) (the case of quadrics). The next step would be h(3), i.e. cubics.", "clean_statement": "Problem 8\n\n(Suggested by David Jensen) The Maximal Rank Conjecture predicts the Hilbert fuction of a general curve (for fixed g, r, d such that ρ(g, r, d) ≥ 0 and r ≥ 3). The Hilbert function measures the rank of the map Sym m H0(X, L) → H0(X, L⊗m). The Maximal Rank Conjecture predicts that this has maximal rank for general X and L.Concretely, this means that it is either injective or surjective. We proposed a combinatorial approach via the notion of tropical independence. How-ever, we can only solve the combinatorial problem for h(2) (the case of quadrics). The next step would be h(3), i.e. cubics.", "statement_status": "exact", "statement_verification": "**Source.** AIM workshop problem list, *Degenerations in Algebraic Geometry*, Problem 8, suggested by David Jensen. The workshop took place in September 2016. The canonical record is zero-based record 228 of `aim-algebraic-geometry-notes.json`.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 8\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[228]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 8 \\n\\n(Suggested by David Jensen) The Maximal Rank Conjecture predicts the Hilbert fuction of a general curve (for fixed g, r, d such that ρ(g, r, d) ≥ 0 and r ≥ 3). The Hilbert function measures the rank of the map Sym m H0(X, L) → H0(X, L⊗m). The Maximal Rank Conjecture predicts that this has maximal rank for general X and L.Concretely, this means that it is either injective or surjective. We proposed a combinatorial approach via the notion of tropical independence. How-ever, we can only solve the combinatorial problem for h(2) (the case of quadrics). The next step would be h(3), i.e. cubics.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0229", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Larson's theorem solves the classical cubic maximal-rank assertion for general Brill-Noether curves over C, while the all-parameter tropical proof route remains only partially established. This attempt proves a characteristic-free section-quotient snake sequence that reduces cubic surjectivity to residual surjectivity when the quadratic map is surjective and reduces cubic injectivity to an explicit connecting map into the quadratic cokernel when the quadratic map is injective; it also proves the relevant Brill-Noether dimension orientation and gives explicit characteristic-free injective, surjective, and obstructed monomial/tropical test families.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: the exact cubic section-quotient sequence, combined with the bound d <= r(r+2)/2 <= binomial(r+2,3) in the quadratic-surjective Brill-Noether range, gives a two-branch proof criterion; in the (g,r,d)=(5,3,8) equality case it forces the two residual cubic relations to map isomorphically to the two-dimensional quadratic cokernel." }, { "id": 20000230, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0230", "title": "A quadratic-to-cubic maximal-rank wall and its pure resolution", "statement": "Problem 8.1. Prove this for all m.\n\nThe current argument is sort of naïve. We make a choice of functions in Sym m H0(X, L)which we want to be independent. Thus, the argument works more naturally in the injective case because there is no choice needed. In that case, you have a bunch of functions and you want to show that (mr\n\n) \"m-wise\" sums are independent. To do that in the case m = 2, this becomes a combinatorial game. After this, one coud ask about higher syzygies (viewing the Hilbert function as describ-ing the first term of the syzygy).", "original_statement": "Problem 8.1. Prove this for all m. \n\nThe current argument is sort of naïve. We make a choice of functions in Sym m H0(X, L)which we want to be independent. Thus, the argument works more naturally in the injective case because there is no choice needed. In that case, you have a bunch of functions and you want to show that (mr\n\n) \"m-wise\" sums are independent. To do that in the case m = 2, this becomes a combinatorial game. After this, one coud ask about higher syzygies (viewing the Hilbert function as describ-ing the first term of the syzygy).", "clean_statement": "Problem 8.1. Prove this for all m.\n\nThe current argument is sort of naïve. We make a choice of functions in Sym m H0(X, L)which we want to be independent. Thus, the argument works more naturally in the injective case because there is no choice needed. In that case, you have a bunch of functions and you want to show that (mr\n\n) \"m-wise\" sums are independent. To do that in the case m = 2, this becomes a combinatorial game. After this, one coud ask about higher syzygies (viewing the Hilbert function as describ-ing the first term of the syzygy).", "statement_status": "exact", "statement_verification": "The record is Problem 8.1 in the AIM workshop list *Degenerations in algebraic geometry*, suggested in the preceding text by David Jensen. The JSON extraction is visibly damaged: it omits the preceding definition of the map, removes the entries of a binomial coefficient, and runs words together. Inspection of page 4 of the source PDF recovers the following context.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 8.1\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[229]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 8.1. Prove this for all m. \\n\\nThe current argument is sort of naïve. We make a choice of functions in Sym m H0(X, L)which we want to be independent. Thus, the argument works more naturally in the injective case because there is no choice needed. In that case, you have a bunch of functions and you want to show that (mr\\n\\n) \\\"m-wise\\\" sums are independent. To do that in the case m = 2, this becomes a combinatorial game. After this, one coud ask about higher syzygies (viewing the Hilbert function as describ-ing the first term of the syzygy).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0230", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Larson's maximal rank theorem implies an exact transition strip for general complex curves with g=binom(r,2)+a and d=g+r: the quadratic cokernel has dimension a and the cubic kernel has dimension binom(r+1,3)-2a. At the wall a=0, the homogeneous coordinate ring is arithmetically Cohen-Macaulay and its ideal has a pure 3-linear resolution with beta_i=binom(r+1,i+2)binom(i+1,2). The union of all coordinate lines gives a characteristic-free monomial certificate with the same Hilbert series and Betti table, and in r=3 a Hilbert-Burch deformation rigorously smooths this scheme while transporting the sixteen selected cubic monomials.\n\nCandidate contribution (derived theorem and explicit certificate; novelty confidence low): The wall g=binom(r,2), d=binom(r+1,2) has the explicit pure Betti formula beta_i=binom(r+1,i+2)binom(i+1,2), realized by the coordinate-line scheme whose cubic basis consists exactly of monomials supported on at most two variables; for r=3 this basis persists in a proved flat determinantal smoothing." }, { "id": 20000231, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0231", "title": "Complementary tropical minors for relative Koszul complexes", "statement": "Problem 8.2. Use degeneration techniques (tropical independence or limit linear series or both) to understand higher syzygies.\n\nThis is known by Green's conjectures for generic curves under the canonical embed-ding, by non-tropical techniques.", "original_statement": "Problem 8.2. Use degeneration techniques (tropical independence or limit linear series or both) to understand higher syzygies. \n\nThis is known by Green's conjectures for generic curves under the canonical embed-ding, by non-tropical techniques.", "clean_statement": "Problem 8.2. Use degeneration techniques (tropical independence or limit linear series or both) to understand higher syzygies.\n\nThis is known by Green's conjectures for generic curves under the canonical embed-ding, by non-tropical techniques.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 8.2 from the AIM workshop *Degenerations in Algebraic Geometry*, September 7, 2016. The exact extracted text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 8.2\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[230]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 8.2. Use degeneration techniques (tropical independence or limit linear series or both) to understand higher syzygies. \\n\\nThis is known by Green's conjectures for generic curves under the canonical embed-ding, by non-tropical techniques.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0231", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a three-term complex of finite free modules over a DVR, a noncancelling initial minor of the incoming differential on middle coordinates I and a noncancelling initial minor of the outgoing differential on the complementary coordinates J imply exactness on the generic fiber. If their determinant valuations are alpha and beta, then the integral middle homology has length at most alpha, the torsion in the outgoing cokernel has length at most beta, and universal coefficients give special-fiber middle homology dimension at most alpha+beta. A pair of unit minors gives special-fiber exactness. An explicit two-variable relative Koszul family has generic and integral K_{1,1}=0 but a one-dimensional special K_{1,1} generated by y tensor v, showing why the outgoing-cokernel defect is necessary.\n\nCandidate contribution (theorem; novelty confidence low): The candidate contribution is the complementary-minor defect-budget theorem: two tropical matching certificates on complementary middle coordinates both prove generic Koszul vanishing and bound the special-fiber defect by the sum of their determinant valuations." }, { "id": 20000232, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0232", "title": "A skeleton obstruction and a genus-five enriched tropical certificate", "statement": "Problem 8.3. Can one give a new proof of Green's theorem by tropical methods?\n\n49", "original_statement": "Problem 8.3. Can one give a new proof of Green's theorem by tropical methods? \n\n49", "clean_statement": "Problem 8.3. Can one give a new proof of Green's theorem by tropical methods?\n\n49", "statement_status": "exact", "statement_verification": "The canonical record is Problem 8.3 from the AIM workshop list *Degenerations in algebraic geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 8.3\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[231]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 8.3. Can one give a new proof of Green's theorem by tropical methods? \\n\\n49\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0232", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The generic Green theorem is known algebraically, but no all-genus tropical proof was located. This attempt proves that for every genus g at least 5, two constant good-reduction curves over C((t)) can have the same one-vertex weighted skeleton and tropical canonical divisor but opposite values of K_{1,2}; hence abstract skeleton divisor theory cannot determine even the first nonlinear canonical syzygies. It also gives an explicit flat three-quadric family whose t-adic initial ideal is (x_0^2,x_1^2,x_2^2), proves its generic fiber is a smooth canonical genus-five curve, and uses the complete-intersection Koszul resolution to certify the critical vanishing K_{1,2}=0.\n\nCandidate contribution (obstruction_and_explicit_certificate; novelty confidence low): Candidate contribution: for every g at least 5, the abstract weighted skeleton and tropical canonical divisor fail to determine K_{1,2} even among constant good-reduction curves over one valued field, while the displayed genus-five family supplies a monomial t-adic initial canonical ideal whose resolution proves the critical Green vanishing." }, { "id": 20000233, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0233", "title": "Euler parity and complete cubic cancellation profiles for plane Welschinger counts", "statement": "Problem 9\n\n(Suggested by Erwan) This is a question about Welschinger invariants of real rational sur-faces, which is a real analogue of genus 0 Gromov-Witten invariants. Specifically, we are interested in a signed count of real curves through d1 real points and d2 pairs of complex conjugate points (such that d1 + 2g2 = 3d − 1) in P\n2. (The sign is something like ( −1) raised to the number of isolated nodes.)", "original_statement": "Problem 9 \n\n(Suggested by Erwan) This is a question about Welschinger invariants of real rational sur-faces, which is a real analogue of genus 0 Gromov-Witten invariants. Specifically, we are interested in a signed count of real curves through d1 real points and d2 pairs of complex conjugate points (such that d1 + 2g2 = 3d − 1) in P\n2. (The sign is something like ( −1) raised to the number of isolated nodes.)", "clean_statement": "Problem 9\n\n(Suggested by Erwan) This is a question about Welschinger invariants of real rational sur-faces, which is a real analogue of genus 0 Gromov-Witten invariants. Specifically, we are interested in a signed count of real curves through d1 real points and d2 pairs of complex conjugate points (such that d1 + 2g2 = 3d − 1) in P\n2. (The sign is something like ( −1) raised to the number of isolated nodes.)", "statement_status": "exact", "statement_verification": "The AIM source is Problem 9 from the workshop *Degenerations in algebraic geometry*. The extracted record contains two material OCR errors: `g2` should be \\(d_2\\), and `P\\n2` should be \\(\\mathbb P^2\\). The source's phrase “real rational surfaces” describes the ambient theory; in the specialization to the rational surface \\(\\mathbb P^2\\), the enumerated objects are real rational plane curves. The dimension constraint and the notation in the source make the reconstruction unambiguous.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 9\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[232]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 9 \\n\\n(Suggested by Erwan) This is a question about Welschinger invariants of real rational sur-faces, which is a real analogue of genus 0 Gromov-Witten invariants. Specifically, we are interested in a signed count of real curves through d1 real points and d2 pairs of complex conjugate points (such that d1 + 2g2 = 3d − 1) in P\\n2. (The sign is something like ( −1) raised to the number of isolated nodes.)\"\nOriginal remarks: [\"Remark 9.1. For P2 there are several formulas in terms of d1, d2. However, these are so opaque that even the sign of the total sum is unclear. For some specific cases, the sign can be deduced by degenerating P2.\\n\\n• If d1 ≈ 0 or d2 ≈ 0, then the sign is known. \\n\\n• If d2 = 0, the sign is always non-negative. \\n\\n• If d1 = 0 or 1 (i.e. minimal depending on the parity), then the sign is ( −1) (d−1)( d−2) /2.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0233", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every irreducible real rational nodal plane curve C of degree d, if Gamma_C is the non-isolated real locus, then its Welschinger sign is (-1)^{delta_d-chi(Gamma_C)}; consequently the normalized invariant is exactly the number of even-Euler-characteristic curves minus the number of odd-Euler-characteristic curves. In degree three, a generic pencil through eight conjugation-invariant points with 2a real base points and 2t real singular members has exactly H=t+a hyperbolic and E=t-a elliptic singular cubics, and every profile with 0<=a<=4 and a<=t<=6 is realizable.\n\nCandidate contribution (structural reduction and cubic corollary; novelty confidence low): The normalized sign problem can be packaged as an Euler-characteristic parity imbalance of the non-isolated real normalization graphs, giving an exact target for a sign-reversing injection; combined with the cubic-pencil realizability theorem, this yields the complete realized cancellation profile (H,E)=(t+a,t-a)." }, { "id": 20000234, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0234", "title": "A topological node-parity reduction for the next-to-minimal Welschinger sign", "statement": "Problem 9.2. What about when d 1 = 2 or 3? (Conjecturally, it is (−1) (d−1)( d−2) /2+1 for d\n 0.)", "original_statement": "Problem 9.2. What about when d 1 = 2 or 3? (Conjecturally, it is (−1) (d−1)( d−2) /2+1 for d \u001d 0.)", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The assigned record is Problem 9.2 in the AIM problem list *Degenerations in Algebraic Geometry*. The surrounding Problem 9 concerns real irreducible rational plane curves of degree \\(d\\) through", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 9.2\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[233]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 9.2. What about when d 1 = 2 or 3? (Conjecturally, it is (−1) (d−1)( d−2) /2+1 for d \\u001d 0.)\"\nOriginal remarks: [\"Remark 9.3. If we consider the same problem on a more general real surface X, then the signs tell us something about the topology of the real algebraic surface X.There are two possible approaches to this question. 1. Analyze the combinatorics of the known formulas. However, this might be less inter-esting. 2. Find some nice degeneration that makes the answer clear. For instance, all the curves might have the same sign, or there could be a natural injection from curves of one sign into curves of the other sign. For example, for the minimal d2 or d1 case the sign is deduced by such a degeneration. What shape to the formulas have? \\n\\n• One formula is in terms of Floor diagrams. \\n\\n• Another formula is in terms of open WDVV equations. \\n\\n10\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0234", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Writing T_d=W_2(d,floor((3d-3)/2)), so that the number of real constraints is exactly 2 for odd d and 3 for even d, one has the proved termwise identity (-1)^((d-1)(d-2)/2+1) T_d = #{curves with an odd number of hyperbolic real nodes} - #{curves with an even number}. If Gamma_C is the non-isolated real locus, then chi(Gamma_C)=-h(C), so a non-surjective injection from even-h curves to odd-h curves is a precise sufficient degeneration target. Exact published values give T_d>0 for 2<=d<=9; therefore the all-degree strengthening fails at d=2,5,6,9, and any threshold for the still-unresolved eventual conjecture must satisfy D>=10.\n\nCandidate contribution (reduction; novelty confidence low): For the exact next-to-minimal sequence in AIM Problem 9.2, the conjectural normalized invariant is the strict odd-versus-even hyperbolic-node majority and also minus the parity sum of chi(Gamma_C); hence an even-h to odd-h non-surjective injection is the correct sufficient direction, while the degree-9 table entry forces any eventual threshold to be at least 10." }, { "id": 20000235, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0235", "title": "Real smoothness of osculating Schubert curves", "statement": "Problem 10\n\n(Suggested by Jake) Consider a line bundle L on P1. Suppose we want to study the Grass-mannian of grd ⊂ Γ(P1, L). We can consider conditions on grd's, e.g. as follows. Pick a point\n\nx ∈ P1, and considering the vanishing flag at x. This defines a complete flag, by stratifying by the degree of vanishing at x. You can then impose Schubert conditions on the linear series (which concretely means imposing conditions on the vanishing orders). 5These Schubert conditions are indexed by partitions. More generally, we can do this at several points x1,..., xs with several partitions λ1,..., λ s. Let S be the set of grd satisfying all these conditions, i.e. the intersection of all the Schubert conditions.\n\nTheorem 10.1 (Eisenbud-Harris '80s). We have codim S = ∑i |λi|.\n\nThis has an interesting relation to real algebraic geometry.\n\nTheorem 10.2 (Muken-Tarasov-Varchenko). If all the x i are real, and ∑i |λi| = dim {grd},then S consists of real reduced points.\n\nNow suppose that S is a curve rather than a finite set.\n\nTheorem 10.3. If all x i are real and ∑i |λi| = dim {grd} − 1, then S (R) ⊂ S smooth.", "original_statement": "Problem 10 \n\n(Suggested by Jake) Consider a line bundle L on P1. Suppose we want to study the Grass-mannian of grd ⊂ Γ(P1, L). We can consider conditions on grd's, e.g. as follows. Pick a point \n\nx ∈ P1, and considering the vanishing flag at x. This defines a complete flag, by stratifying by the degree of vanishing at x. You can then impose Schubert conditions on the linear series (which concretely means imposing conditions on the vanishing orders). 5These Schubert conditions are indexed by partitions. More generally, we can do this at several points x1,..., xs with several partitions λ1,..., λ s. Let S be the set of grd satisfying all these conditions, i.e. the intersection of all the Schubert conditions. \n\nTheorem 10.1 (Eisenbud-Harris '80s). We have codim S = ∑i |λi|.\n\nThis has an interesting relation to real algebraic geometry. \n\nTheorem 10.2 (Muken-Tarasov-Varchenko). If all the x i are real, and ∑i |λi| = dim {grd},then S consists of real reduced points. \n\nNow suppose that S is a curve rather than a finite set. \n\nTheorem 10.3. If all x i are real and ∑i |λi| = dim {grd} − 1, then S (R) ⊂ S smooth.", "clean_statement": "Problem 10\n\n(Suggested by Jake) Consider a line bundle L on P1. Suppose we want to study the Grass-mannian of grd ⊂ Γ(P1, L). We can consider conditions on grd's, e.g. as follows. Pick a point\n\nx ∈ P1, and considering the vanishing flag at x. This defines a complete flag, by stratifying by the degree of vanishing at x. You can then impose Schubert conditions on the linear series (which concretely means imposing conditions on the vanishing orders). 5These Schubert conditions are indexed by partitions. More generally, we can do this at several points x1,..., xs with several partitions λ1,..., λ s. Let S be the set of grd satisfying all these conditions, i.e. the intersection of all the Schubert conditions.\n\nTheorem 10.1 (Eisenbud-Harris '80s). We have codim S = ∑i |λi|.\n\nThis has an interesting relation to real algebraic geometry.\n\nTheorem 10.2 (Muken-Tarasov-Varchenko). If all the x i are real, and ∑i |λi| = dim {grd},then S consists of real reduced points.\n\nNow suppose that S is a curve rather than a finite set.\n\nTheorem 10.3. If all x i are real and ∑i |λi| = dim {grd} − 1, then S (R) ⊂ S smooth.", "statement_status": "exact", "statement_verification": "The source is the AIM workshop list *Degenerations in Algebraic Geometry*, Problem 10. The extracted record has several OCR errors: `grd` means \\(g^r_d\\), \\(\\Gamma(\\mathbf P^1,L)\\) means \\(H^0(\\mathbf P^1,L)\\), and “Muken” is Mukhin. The original PDF also shows that the question “Is \\(S\\) smooth over \\(\\mathbf C\\)?” is the following record, Problem 10.4, and is not part of this record.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 10\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[234]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 10 \\n\\n(Suggested by Jake) Consider a line bundle L on P1. Suppose we want to study the Grass-mannian of grd ⊂ Γ(P1, L). We can consider conditions on grd's, e.g. as follows. Pick a point \\n\\nx ∈ P1, and considering the vanishing flag at x. This defines a complete flag, by stratifying by the degree of vanishing at x. You can then impose Schubert conditions on the linear series (which concretely means imposing conditions on the vanishing orders). 5These Schubert conditions are indexed by partitions. More generally, we can do this at several points x1,..., xs with several partitions λ1,..., λ s. Let S be the set of grd satisfying all these conditions, i.e. the intersection of all the Schubert conditions. \\n\\nTheorem 10.1 (Eisenbud-Harris '80s). We have codim S = ∑i |λi|.\\n\\nThis has an interesting relation to real algebraic geometry. \\n\\nTheorem 10.2 (Muken-Tarasov-Varchenko). If all the x i are real, and ∑i |λi| = dim {grd},then S consists of real reduced points. \\n\\nNow suppose that S is a curve rather than a finite set. \\n\\nTheorem 10.3. If all x i are real and ∑i |λi| = dim {grd} − 1, then S (R) ⊂ S smooth.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0235", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The record is a theorem/context record whose one-dimensional assertion is Levinson's theorem that every real point of the osculating Schubert curve is smooth on the complex scheme. Beyond this status recovery, the attempt proves a higher-dimensional residual-Wronski certificate: over every squarefree totally-real residual divisor disjoint from the fixed real osculation points, the residual Wronski morphism has a transverse all-real fiber, the original Schubert intersection is smooth, and the differential is an isomorphism. Each fixed root-distribution chamber therefore splits into delta trivial real sheets. A direct Gr(2,3) example with Wronskian z^2+1 shows why this certificate cannot cover every real point once the residual dimension is at least two.\n\nCandidate contribution (residual_wronski_etaleness_certificate_and_obstruction; novelty confidence low): For any nonempty expected m-dimensional type-A osculating Schubert intersection at distinct real marked points, the residual Wronski morphism is etale and totally real above every squarefree totally-real residual divisor disjoint from the marked points; every fixed root-distribution chamber is a disjoint union of delta trivial real sheets, while the real plane in Gr(2,3) has nonreal residual roots and proves that this certificate does not cover all real points for m at least 2.", "research_difficulty_suggested": "L2" }, { "id": 20000236, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0236", "title": "Low-degree smoothness and a Wronskian kernel criterion for osculating Schubert curves", "statement": "Problem 10.4. Is S smooth? (i.e. are there singular points over C?)\n\nThere doesn't seem to be any reason that this should be true, but in the (admittedly few) examples that I've explored, I've found no singular examples. An idea for an approach is to use tautological bundles on M0,s to describe equations characterizing the locus in M0,s where S is singular (and perhaps show that this is away from the points where all the xi are real).\n\nExample 10.5. I've already checked the simplest case, e.g. G(2, 5) with the partitions s = 5points all being a single box. The next simplest cases would be with longer partitions, or G(2, 6). In this case, you can write the equations for the singular locus as a sum of squares (an obvious way to prove that there are no real solutions). In the example I looked at, the squares had the form ( xi − x j)2, which suggests a boundary geometric interpretation (points colliding). B. Osserman suggests:", "original_statement": "Problem 10.4. Is S smooth? (i.e. are there singular points over C?) \n\nThere doesn't seem to be any reason that this should be true, but in the (admittedly few) examples that I've explored, I've found no singular examples. An idea for an approach is to use tautological bundles on M0,s to describe equations characterizing the locus in M0,s where S is singular (and perhaps show that this is away from the points where all the xi are real). \n\nExample 10.5. I've already checked the simplest case, e.g. G(2, 5) with the partitions s = 5points all being a single box. The next simplest cases would be with longer partitions, or G(2, 6). In this case, you can write the equations for the singular locus as a sum of squares (an obvious way to prove that there are no real solutions). In the example I looked at, the squares had the form ( xi − x j)2, which suggests a boundary geometric interpretation (points colliding). B. Osserman suggests:", "clean_statement": "Problem 10.4. Is S smooth? (i.e. are there singular points over C?)\n\nThere doesn't seem to be any reason that this should be true, but in the (admittedly few) examples that I've explored, I've found no singular examples. An idea for an approach is to use tautological bundles on M0,s to describe equations characterizing the locus in M0,s where S is singular (and perhaps show that this is away from the points where all the xi are real).\n\nExample 10.5. I've already checked the simplest case, e.g. G(2, 5) with the partitions s = 5points all being a single box. The next simplest cases would be with longer partitions, or G(2, 6). In this case, you can write the equations for the singular locus as a sum of squares (an obvious way to prove that there are no real solutions). In the example I looked at, the squares had the form ( xi − x j)2, which suggests a boundary geometric interpretation (points colliding). B. Osserman suggests:", "statement_status": "exact", "statement_verification": "The source is Problem 10.4 in the AIM workshop list *Degenerations in algebraic geometry*. The preceding record supplies notation that is missing from the extracted problem. Let \\[ E=H^0(\\mathbb P^1,\\mathcal O_{\\mathbb P^1}(d)),\\qquad n=d+1,\\qquad k=r+1, \\] so that \\(g^r_d\\)'s are points of \\(G=G(k,E)\\), of dimension \\(N=k(n-k)\\). For \\(x\\in\\mathbb P^1\\), the filtration by order of vanishing at \\(x\\) is a complete osculating flag \\(F_\\bullet(x)\\). Given pairwise distinct points \\(x_1,\\ldots,x_s\\) and partitions \\(\\lambda_i\\subseteq k\\times(n-k)\\), form the **scheme-theoretic** intersection \\[ S=S(\\lambda_\\bullet;x_\\bullet) :=\\bigcap_{i=1}^s\\Omega_{\\lambda_i}(F_\\bullet(x_i)) \\subseteq G. \\] The case in the problem has \\[ \\sum_i|\\lambda_i|=N-1, \\tag{1.1} \\] so the Eisenbud--Harris proper-intersection theorem makes \\(S\\) a projective Cohen--Macaulay curve (when nonempty). All \\(x_i\\) are...", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 10.4\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[235]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 10.4. Is S smooth? (i.e. are there singular points over C?) \\n\\nThere doesn't seem to be any reason that this should be true, but in the (admittedly few) examples that I've explored, I've found no singular examples. An idea for an approach is to use tautological bundles on M0,s to describe equations characterizing the locus in M0,s where S is singular (and perhaps show that this is away from the points where all the xi are real). \\n\\nExample 10.5. I've already checked the simplest case, e.g. G(2, 5) with the partitions s = 5points all being a single box. The next simplest cases would be with longer partitions, or G(2, 6). In this case, you can write the equations for the singular locus as a sum of squares (an obvious way to prove that there are no real solutions). In the example I looked at, the squares had the form ( xi − x j)2, which suggests a boundary geometric interpretation (points colliding). B. Osserman suggests:\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0236", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general complex-smoothness question remains open, but three rigorous advances are obtained. First, every one-dimensional osculating Schubert intersection with at most two nonempty conditions is a normal Richardson curve and hence is smooth over C, without a reality hypothesis. Second, every real osculating Schubert curve of Plucker degree at most two is complex-smooth; this proves two explicit longer-partition G(2,6) families, namely (2,2) plus three boxes and (3,2) plus two boxes. Third, for the all-box curve on G(2,n), singularity at U is equivalent to U lying in the radical of a nonzero linear combination of Wronskian evaluation forms, yielding an exact global projective incidence and Wronski-map transversality criterion.\n\nCandidate contribution (smoothness criterion and incidence reduction; novelty confidence low): Candidate novelty: a real osculating Schubert curve of Plucker degree at most two is smooth over C, giving the two displayed G(2,6) cases; moreover, hidden singularities in the all-box G(2,n) problem are exactly the points captured by the global skew-kernel incidence constructed from Wronskian evaluation line bundles." }, { "id": 20000237, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0237", "title": "A binary-quartic degeneration certificate and a cominuscule obstruction", "statement": "Problem 10.6. Reprove Mukhin-Tarasov-Varchenko theorem using degenerations. Extend it to cominiscule G /P.\n\nNotes by Tony Feng 6", "original_statement": "Problem 10.6. Reprove Mukhin-Tarasov-Varchenko theorem using degenerations. Extend it to cominiscule G /P. \n\nNotes by Tony Feng 6", "clean_statement": "Problem 10.6. Reprove Mukhin-Tarasov-Varchenko theorem using degenerations. Extend it to cominiscule G /P.\n\nNotes by Tony Feng 6", "statement_status": "exact", "statement_verification": "The exact extracted record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Degenerations in algebraic geometry\nSection: \nSource item: 10.6\nSource URL: https://aimath.org/pastworkshops/degenalggeomproblems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[236]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 10.6. Reprove Mukhin-Tarasov-Varchenko theorem using degenerations. Extend it to cominiscule G /P. \\n\\nNotes by Tony Feng 6\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/degenalggeomproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0237", "aim-domain:algebraic-geometry", "aim-workshop:degenalggeomproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "A self-contained Plucker-coordinate calculation gives a two-cluster degeneration proof of the four-divisor Mukhin-Tarasov-Varchenko theorem on Gr(2,4), with fiber discriminant I=e2^2-3e1e3+12e4 equal to a positive sum of three squares. Restricting the same binary-quartic model to the natural principal-SL2 Lagrangian Grassmannian LG(2,4)=Sp4/P2 turns I=0 into its quadric equation; three distinct real osculating Schubert divisors then have exactly two transverse nonreal conjugate solutions, since the residual-root quadratic has discriminant -3 times the squared Vandermonde. This known Sottile counterexample refutes the naive all-real cominuscule extension, though not modified paired or type-dependent formulations.\n\nCandidate contribution (invariant_identity; novelty confidence low): Candidate novelty: the same symmetric binary-quartic invariant is exhibited simultaneously as the positive sum-of-squares discriminant controlling a reduced real two-cluster degeneration on Gr(2,4) and as the principal-SL2 LG(2,4) equation whose residual-root discriminant is the negative Vandermonde square, giving a single explicit mechanism for total reality versus total unreality." }, { "id": 20000238, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0238", "title": "Finite-colength covectors and a critical-rank punctured-spectrum counterexample", "statement": "Question 1Question (Hailong Dao). Let X be the pun tur ed sp e trum on a lo al ring (R, m).What is the obstru tion the ory for splitting of ve tor bund les over su h a s heme? More sp e i\n ally, let X = Spec( R) \\ { m} for (R, m) lo al of dimension d. What is the obstru tion to splitting of rank d − 1 ve tor bundles on X?Related Question (Sat ya Mandal). Let X be a s heme and E be a ve tor bund le. What is the obstru tion to obtaining a surje tion E ։ OX?Related Question (Ara vind Asok). In the a\nne ase, one has a fair ide a of the obstru tion the ory for pr oje tive mo dules. What is the obstru tion the ory for gener al lasses of mo dules? Related Question (Hailong Dao). Is ther e an Euler lass gr oup for re\nexive mo d-ules over a regular lo al ring (R, m)?Related Question. What is the role of A1-homotopy the ory in the non-a\nne ase?", "original_statement": "Question 1Question (Hailong Dao). Let X be the pun tur ed sp e trum on a lo al ring (R, m).What is the obstru tion the ory for splitting of ve tor bund les over su h a s heme? More sp e i\u001c ally, let X = Spec( R) \\ { m} for (R, m) lo al of dimension d. What is the obstru tion to splitting of rank d − 1 ve tor bundles on X?Related Question (Sat ya Mandal). Let X be a s heme and E be a ve tor bund le. What is the obstru tion to obtaining a surje tion E ։ OX?Related Question (Ara vind Asok). In the a\u001ene ase, one has a fair ide a of the obstru tion the ory for pr oje tive mo dules. What is the obstru tion the ory for gener al lasses of mo dules? Related Question (Hailong Dao). Is ther e an Euler lass gr oup for re\u001dexive mo d-ules over a regular lo al ring (R, m)?Related Question. What is the role of A1-homotopy the ory in the non-a\u001ene ase?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "**Canonical record.** This is record `AIM-ALGEBRAIC_GEOMETRY-0238`, zero-based index 237 in `aim-algebraic-geometry-notes.json`, extracted from Question 1 of the AIM workshop problem list *Projective modules and \\(A^1\\)-homotopy theory* (May 2014). The following is an ASCII-escaped, reversibly exact JSON serialization of the source record. In particular, the OCR corruption and control character `\\u001c` have not been silently repaired.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 1\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[237]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 1Question (Hailong Dao). Let X be the pun tur ed sp e trum on a lo al ring (R, m).What is the obstru tion the ory for splitting of ve tor bund les over su h a s heme? More sp e i\\u001c ally, let X = Spec( R) \\\\ { m} for (R, m) lo al of dimension d. What is the obstru tion to splitting of rank d − 1 ve tor bundles on X?Related Question (Sat ya Mandal). Let X be a s heme and E be a ve tor bund le. What is the obstru tion to obtaining a surje tion E ։ OX?Related Question (Ara vind Asok). In the a\\u001ene ase, one has a fair ide a of the obstru tion the ory for pr oje tive mo dules. What is the obstru tion the ory for gener al lasses of mo dules? Related Question (Hailong Dao). Is ther e an Euler lass gr oup for re\\u001dexive mo d-ules over a regular lo al ring (R, m)?Related Question. What is the role of A1-homotopy the ory in the non-a\\u001ene ase?\"\nOriginal remarks: [\"Remark (P aul Balmer). Ma yb e one an use the Jounalou tri k of a\\u001ene repla emen tof non-a\\u001ene varieties.\", \"Remark (Ara vind Asok). Let νr (X) denote the set of rank r ve tor bundles over as heme X. There exists a non-a\\u001ene variet y Y and a map f: An → Y whi h is an A1-w eak equiv alen e but νr (Y ) is large. Th us, for non-a\\u001ene s hemes, νr is not an A1-homotop y in varian t. Related Question (Ara vind Asok). Let V be a \\u001cnite dimensional k-ve tor sp a e. Let H be the odimension 1 hyp ersurfa e in P(V ) × P(V ∗) given by the in iden erelation. Then P(V ) × P(V ∗) r H p1 \\n\\n> ։P(V)\\n\\nis an A1-we ak equivalen e. Is p∗ \\n\\n> 1\\n\\nsurje tive on isomorphism lasses of ve tor bun-d les? Related Question (Kirsten Wi kelgren). What is the role of Ni ˇsnevi h / étale top olo gy with realization to equivariant top olo gi al ve tor bund les in manifolds? Date: Ma y 24, 2014. 12 SARANG SANE\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0238", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let (R,m) be a Noetherian normal local domain of dimension at least two, let U be its punctured spectrum, and let E arise from a finite reflexive module M that is locally free on U. A covector q:E→O_U corresponds to φ:M→R and is sheaf-surjective exactly when R/φ(M) has finite length; for K=ker(q) and N=ker(φ), there is a canonical injection R/φ(M)→H^1(U,K)≅H^2_m(N), and q splits exactly when the connecting image of 1 vanishes. For R=k[x,y,z]_(x,y,z), the kernel K of the row (x,y,z) on U has rank two, [K]=2[O_U] in K_0(U), and all positive-degree ordinary algebraic Chern classes in CH*(U) vanish, but H^1(U,K)≅k and K has no trivial line summand. An fpqc equalizer also isolates the descent condition left over after passage to a Jouanolou affine replacement.\n\nCandidate contribution (local_cohomology_lemma_and_explicit_counterexample; novelty confidence low): Candidate novelty: the finite-colength covector defect R/φ(M) embeds canonically in H^2_m(ker φ), with the class of 1 exactly detecting global splitting, and the row (x,y,z) supplies a critical-rank bundle on a punctured regular local threefold whose K_0 class and ordinary Chow-theoretic Chern classes are trivial relative to O_U^2 but which has no O_U-summand." }, { "id": 20000239, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0239", "title": "Euler--Chow--Witt comparison, real realization, and a sphere test", "statement": "Question 2Question (b y p opular demand). What is the onne tion betwe en Euler lass gr oups and Chow-Witt gr oups?", "original_statement": "Question 2Question (b y p opular demand). What is the onne tion betwe en Euler lass gr oups and Chow-Witt gr oups?", "clean_statement": "Question 2Question (b y p opular demand). What is the onne tion betwe en Euler lass gr oups and Chow-Witt gr oups?", "statement_status": "exact", "statement_verification": "The canonical JSON record is visibly damaged by PDF extraction. The original AIM workshop PDF, *Projective modules and \\(\\mathbb A^1\\)-homotopy theory*, gives the following text on page 2:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 2\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[238]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 2Question (b y p opular demand). What is the onne tion betwe en Euler lass gr oups and Chow-Witt gr oups?\"\nOriginal remarks: [\"Remark (Jean Fasel). Giv en a variet y X over C, there is a y le lass map CH r (X) cr \\n\\n> −→H2r(X(C),Z).\\n\\nand giv en a variet y X over R, there is a y le lass map CH r (X) cr \\n\\n> −→Hr(X(R),Z\\n> 2Z).\\n\\nBoth these maps are ompatible with pro du ts. Question (Jean Fasel). Let X be a variety over R. Is ther e a map ˜cr making the fol lowing diagr am ommute:˜CH r (X, ω X/ R) \\n\\n> \\u000f\\n> \\u000f˜cr\\n> //______Hr(X(R),Z)\\n> \\u000f\\n> \\u000f\\n> CH r(X)cr\\n> //Hr(X(R),Z\\n> 2Z)\\n\\nwhi h is also omp atible with pr odu ts?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0239", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The historical comparison is now an isomorphism E^d(A) ~= CHW^d(Spec A) in top codimension for smooth affine d-folds over an infinite field of characteristic not 2, while the real integral cycle map for a smooth quasi-projective real variety naturally lands in cohomology with the sign local system Z(L). For L = omega_X this is the orientation local system, so the literal constant-coefficient target requires an orientation. As an explicit test family, the oriented tangent module of the affine real unit d-sphere has real image +/-chi(S^d); for every even d >= 2 it is an infinite-order element of ker(E^d -> E_0^d), whereas for odd d an explicit algebraic unit vector splits the module and kills the Euler class.\n\nCandidate contribution (explicit_family; novelty confidence low): For every even d >= 2, the determinant-oriented tangent module of R[x_0,...,x_d]/(sum x_i^2 - 1) defines an explicit infinite-order element of ker(E^d -> E_0^d), detected as +/-2 by the Euler--Chow--Witt--signature composite; in odd dimension the displayed algebraic vector field gives a direct splitting and zero class." }, { "id": 20000240, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0240", "title": "Motivic local systems and the Atiyah-class blindness of Artin monodromy", "statement": "Question 3Question (Christian Haesemey er). What ar e [U+0010]motivi lo al systems[U+0011]? Can one de\nne motivi Atiyah lasses?", "original_statement": "Question 3Question (Christian Haesemey er). What ar e \u0010motivi lo al systems\u0011? Can one de\u001cne motivi Atiyah lasses?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record is damaged by PDF extraction: it contains control characters and loses several letters in “motivic local systems,” “define,” and “classes.” Inspection of the original AIM PDF recovers the wording on PDF page 2 as", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 3\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[239]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 3Question (Christian Haesemey er). What ar e \\u0010motivi lo al systems\\u0011? Can one de\\u001cne motivi Atiyah lasses?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0240", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The 2014 question now has a precise rational characteristic-zero answer in the perverse Nori framework, but the A1-homotopy, Nori, stable-motivic, and geometric-origin meanings of local system remain distinct, and no standard intrinsic construction called a motivic Atiyah class was located. Mathematically, for every finite etale p:Y->X over a smooth characteristic-zero base, A=p_*O_Y has a canonical integrable connection induced by d_Y, so its coherent Atiyah class and that of its horizontal trace-zero summand vanish. For p=[m]:E->E on a complex elliptic curve, the trace-zero summand has nontrivial finite regular monodromy and is a nontrivial algebraic vector bundle, yet its ordinary de Rham Atiyah class is zero.\n\nCandidate contribution (obstruction theorem; novelty confidence low): For every complex elliptic curve E and m>=2, the trace-zero Artin motivic local system ker([m]_*Q_E -> Q_E) has nontrivial finite monodromy and a nontrivial underlying algebraic vector bundle A_0 with H^0(E,A_0)=0, while At(A_0)=0; hence the ordinary de Rham Atiyah class cannot distinguish triviality or monodromy even on trace-free Artin motivic local systems." }, { "id": 20000241, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0241", "title": "Coordinate-skeleton curves, transverse coinvariant algebras, and the cohomological-dimension gap", "statement": "Question 4Question (Krone ker-prop osed by Sat ya Mandal). Are urves in Ank set the or eti omplete interse tion ide als?", "original_statement": "Question 4Question (Krone ker-prop osed by Sat ya Mandal). Are urves in Ank set the or eti omplete interse tion ide als?", "clean_statement": "Question 4Question (Krone ker-prop osed by Sat ya Mandal). Are urves in Ank set the or eti omplete interse tion ide als?", "statement_status": "exact", "statement_verification": "The canonical record is entry 240 (zero-based) of `aim-algebraic-geometry-notes.json`, from the AIM workshop *Projective modules and \\(A^1\\)-homotopy theory*. Its `problem` field is preserved here verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 4\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[240]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 4Question (Krone ker-prop osed by Sat ya Mandal). Are urves in Ank set the or eti omplete interse tion ide als?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0241", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general characteristic-zero question remains open, but two rigorous advances are established. First, for every field and every coordinate r-skeleton in affine n-space, the elementary symmetric polynomials e_{r+1},...,e_n cut out the reduced skeleton up to radical; after localization at any component, the chosen complete-intersection thickening is exactly the type-A coinvariant algebra on n-r transverse variables and has uniform generic length (n-r)!. In particular, the union of the coordinate axes is an explicit reduced non-lci curve that is a set-theoretic complete intersection over every field. Second, every radical pure affine curve ideal has ordinary cohomological dimension exactly n-1, so ordinary local cohomology cannot distinguish the remaining arithmetical-rank alternatives n-1 and n.\n\nCandidate contribution (scheme_structure_refinement; novelty confidence low): For J_{n,r}=(e_{r+1},...,e_n), the localization of R/J_{n,r} at every coordinate r-plane component is canonically presented as K[y_1,...,y_{n-r}]_(y)/(e_1(y),...,e_{n-r}(y)); hence its generic transverse algebra is the type-A coinvariant algebra and its generic multiplicity is uniformly (n-r)!." }, { "id": 20000242, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0242", "title": "Efficient generation and a completable-coordinate strong-lifting criterion", "statement": "Question 5Question (Sat ya Mandal). Let I ⊆ k[X1, X 2,..., X n] = R be an ide al. Then is μ(I) = μ( I\n\n> I2)?", "original_statement": "Question 5Question (Sat ya Mandal). Let I ⊆ k[X1, X 2,..., X n] = R be an ide al. Then is μ(I) = μ( I \n\n> I2)?", "clean_statement": "Question 5Question (Sat ya Mandal). Let I ⊆ k[X1, X 2,..., X n] = R be an ide al. Then is μ(I) = μ( I\n\n> I2)?", "statement_status": "exact", "statement_verification": "This record is Question 5 in the problem list from the AIM workshop *Projective modules and \\(A^1\\)-homotopy theory*, held May 5--9, 2014. The available JSON has substantial OCR damage, so the statement below was checked against the workshop PDF.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 5\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[241]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 5Question (Sat ya Mandal). Let I ⊆ k[X1, X 2,..., X n] = R be an ide al. Then is μ(I) = μ( I \\n\\n> I2)?\"\nOriginal remarks: [\"Remark (Sat ya Mandal). This is known when μ( I \\n\\n> I2)≥dim( R\\n> I) + 2.Related Question (S.M.Bhat wadek ar). Mor e sp e i\\u001c al ly, let α be a surje tive k-algebr a homomorphism C[X1, X 2,..., X 5] α \\n\\n> ։C[Y1, Y 2]\\n\\nand let I = ker (α). Weknow μ( I \\n\\n> I2) = 3. Is μ(I) = 3?\", \"Remark (Ara vind Asok). One an use A1-homotop y to ompute μ(P ) for a pro je -tiv e mo dule P. This is by onsidering if the map X −→ BGL r lifts to some suitable Grassmannian. Related Question (Jean Fasel). Let X = Spec( R) be smo oth and Y ⊆ X bea smo oth emb edding. Is ther e a ohomolo gi al obstru tion ( omputable á la Chow gr oups) to Y being a omplete interse tion? QUESTIONS PR OPOSED AT THE A.I.M. WORKSHOP 3Remark (Jean Fasel). Let Runiv = k[X1, X 2,..., X n, Y 1, Y 2,..., Y n, Z ] \\n\\n> (∑ni=1 XiYi−Z(1 + Z))\\n> Xuniv = Spec( Runiv )Yuniv = Spec( Runiv\\n> (X1, X 2,..., X n, Z ) ).\\n\\nThen the ab ove stru ture has the univ ersal prop ert y that giv en a k-v ariet y X =Spec( R) and a sub variet y Y = Spec( R \\n\\n> I)\\n\\nwith μ( I \\n\\n> I2), there is a natural map X f \\n\\n> −→\\n> Xuniv\\n\\nwhi h restri ts to a map Y f \\n\\n> −→Yuniv. The univ ersal Xuniv is a sphere in A1-top ology. ˜CH ∗(Xuniv, ) ∼ \\n\\n> −→GW (k)⊕GW (k)α\\n\\nwhere the elemen t α is a form on the sub variet y Yuniv.Question (Jean Fasel). With the ab ove notations, given a k-variety X and a lo al omplete interse tion subvariety Y, let f ∗(α) be the image of α under the indu edmap ˜CH ∗(Xuniv, ) f ∗ \\n\\n> −→˜CH ∗(X, ω X/k ).\\n\\nIs f ∗(α) = 0 su\\u001e ient for Y to be a omplete interse tion?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0242", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Murthy's efficient-generation question remains open, including the complex kernel case C[X1,...,X5] -> C[Y1,Y2]. A proved partial result identifies an explicit strong-lifting locus: for Segre data I=(a1,...,ar,s) with s(1-s)=sum(ai bi), if the opposite-coordinate column b is completable to the first basis vector, then the prescribed conormal orientation lifts to r generators of I. Over a polynomial ring over a field, it is enough that the entries of b generate the unit ideal, by Quillen-Suslin.\n\nCandidate contribution (proposition; novelty confidence low): A completable opposite-coordinate column in any Segre presentation yields an explicit strong lift of the prescribed conormal orientation; consequently, every nonliftable orientation over a polynomial ring has a proper opposite-coordinate ideal in every Segre presentation." }, { "id": 20000243, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0243", "title": "Extending A1-homotopy theory over a DVR", "statement": "Question 6Question (Anastasia Sta vro va). Can A1-homotopy be extende d over gener al smo oth s hemes, e.g. over a dvr? Else explain the issues.", "original_statement": "Question 6Question (Anastasia Sta vro va). Can A1-homotopy be extende d over gener al smo oth s hemes, e.g. over a dvr? Else explain the issues.", "clean_statement": "Question 6Question (Anastasia Sta vro va). Can A1-homotopy be extende d over gener al smo oth s hemes, e.g. over a dvr? Else explain the issues.", "statement_status": "exact", "statement_verification": "The canonical JSON record is affected by OCR errors. The supplied AIM workshop PDF, *Projective modules and \\(\\mathbb A^1\\)-homotopy theory*, gives the following text:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 6\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[242]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 6Question (Anastasia Sta vro va). Can A1-homotopy be extende d over gener al smo oth s hemes, e.g. over a dvr? Else explain the issues.\"\nOriginal remarks: [\"Remark (Ara vind Asok). Man y basi things fail, a sp e i\\u001c example is the stable A1onne tivit y theorem.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0243", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unstable and stable motivic homotopy categories do extend over a DVR, indeed over arbitrary schemes, but field-level stable connectivity does not extend formally: the published general stable bounds over a one-dimensional base allow a loss of one degree, and the 2025 unstable preprint gives the analogous dimension-loss bound. For the equicharacteristic henselian DVR R=(k[t]_(t))^h with k infinite perfect, the report proves a split exact Milnor K-theory fiber-pair sequence whose two obstruction components are the shifted tame residue and the same-degree generic/special mismatch.\n\nCandidate contribution (reduction; novelty confidence low): For every q at least 1, the map from K_q^M(R) to K_q^M(K) direct-sum K_q^M(k) has a split cokernel identified, after choosing t and the coefficient field, with K_{q-1}^M(k) direct-sum K_q^M(k); the explicit obstruction sends a pair (alpha,beta) to its tame residue together with beta minus the reduction of the uniquely unramified part of alpha." }, { "id": 20000244, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0244", "title": "Mohan Kumar dimensions and an odd-rank prime-divisor propagation theorem", "statement": "Question 7Question (Ara vind Asok). Fix an inte ger d. For ertain d, Mohan Kumar on-stru ts pr oje tive mo dules of rank d − 2 whi h ar e stably fr ee but not fr ee over asmo oth a\nne variety X of dimension d over an algebr ai al ly lose d\neld. Is this known for al l d? Is ther e a pattern?", "original_statement": "Question 7Question (Ara vind Asok). Fix an inte ger d. For ertain d, Mohan Kumar on-stru ts pr oje tive mo dules of rank d − 2 whi h ar e stably fr ee but not fr ee over asmo oth a\u001ene variety X of dimension d over an algebr ai al ly lose d \u001celd. Is this known for al l d? Is ther e a pattern?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record is visibly damaged by PDF extraction. Its raw `problem` field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 7\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[243]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 7Question (Ara vind Asok). Fix an inte ger d. For ertain d, Mohan Kumar on-stru ts pr oje tive mo dules of rank d − 2 whi h ar e stably fr ee but not fr ee over asmo oth a\\u001ene variety X of dimension d over an algebr ai al ly lose d \\u001celd. Is this known for al l d? Is ther e a pattern?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0244", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Mohan Kumar's published construction gives rank p nonfree stably free modules on smooth affine (p+2)-folds for prime p, with one-line stabilization. This attempt proves that over every algebraically closed field of characteristic zero, every odd d at least 5 also occurs: writing n=d-2 and choosing any prime divisor ell of n, a dimension-flexible use of Mohan Kumar's recursive hypersurfaces together with Wendt's odd-rank motivic detector and a common spreading-out argument yields a smooth affine d-fold carrying at least ell-1 pairwise nonisomorphic nonfree rank-n modules P with P plus one free line free. Dimensions d at most 3 are impossible or meaningless, d=4 is the published p=2 case, and even d at least 6 remain unresolved by the method.\n\nCandidate contribution (theorem; novelty confidence low): Prime-divisor propagation: if F is algebraically closed of characteristic zero, n at least 3 is odd, and ell is a prime divisor of n, then one smooth affine integral (n+2)-fold over F carries at least ell-1 pairwise nonisomorphic rank-n projective modules P_j satisfying P_j plus O is O^(n+1), with every P_j nonfree." }, { "id": 20000245, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0245", "title": "Odd-parity and Euler-realizability reduction for top-Chern splitting", "statement": "Question 8Question (Jean Fasel). Let X be a smo oth a\nne k-variety of odd dimension d.Let P be a ve tor bund le of rank d. Do es cd(P ) ∈ CH d(X) dete t existen e of afr ee summand of rank 1 for P?", "original_statement": "Question 8Question (Jean Fasel). Let X be a smo oth a\u001ene k-variety of odd dimension d.Let P be a ve tor bund le of rank d. Do es cd(P ) ∈ CH d(X) dete t existen e of afr ee summand of rank 1 for P?", "clean_statement": "**Question 8. Question (Jean Fasel).** Let \\(X\\) be a smooth affine\n\\(k\\)-variety of odd dimension \\(d\\). Let \\(P\\) be a vector bundle of rank\n\\(d\\). Does \\(c_d(P)\\in CH^d(X)\\) detect existence of a free summand of\nrank 1 for \\(P\\)?", "statement_status": "reconstructed_unverified", "statement_verification": "The JSON record has OCR damage (`a\\u001ene`, `ve tor bund le`, and missing spaces). Page 3 of the AIM workshop PDF gives the following text:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 8\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[244]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 8Question (Jean Fasel). Let X be a smo oth a\\u001ene k-variety of odd dimension d.Let P be a ve tor bund le of rank d. Do es cd(P ) ∈ CH d(X) dete t existen e of afr ee summand of rank 1 for P?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0245", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The arbitrary-field odd-dimensional AIM question remains open, although it is known over algebraically closed fields, real closed fields, fields of 2-cohomological dimension at most 1 in characteristic not 2, and in further special cases. For a perfect field of characteristic not 2, a smooth affine pure odd-dimensional d-fold X, and a rank-d bundle P with L = det(P)^vee, this attempt proves that c_d(P) = 0 forces the refined Euler class e(P) to lie in ker(F_L)[2] and to satisfy <-1>e(P) = e(P). Hence top-Chern detection for a fixed determinant twist is equivalent to the vanishing of the set of Euler-realizable 2-torsion classes in ker(F_L), rather than to vanishing of all ambient kernel torsion.\n\nCandidate contribution (reduction; novelty confidence low): For perfect fields of characteristic not 2 and odd critical rank, every bundle with vanishing ordinary top Chern class has a <-1>-fixed refined Euler class killed by 2; consequently, for each determinant twist L, top-Chern detection is equivalent to the absence of nonzero Euler-realizable 2-torsion in ker(F_L)." }, { "id": 20000246, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0246", "title": "A finite-group Cartan-Leray spectral sequence for Morel's A1-homology", "statement": "Question 9Question (Kirsten Wi kelgren). Is ther e a Serr e sp e tr al se quen e for Mor el's HA1\n\n> (,Z)?", "original_statement": "Question 9Question (Kirsten Wi kelgren). Is ther e a Serr e sp e tr al se quen e for Mor el's HA1 \n\n> (,Z)?", "clean_statement": "Question 9Question (Kirsten Wi kelgren). Is ther e a Serr e sp e tr al se quen e for Mor el's HA1\n\n> (,Z)?", "statement_status": "exact", "statement_verification": "The canonical JSON record is visibly damaged by PDF extraction: names and ordinary words are split, and the homology notation has lost its subscript and argument. The original AIM problem-list PDF was therefore checked directly. Question 9 reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 9\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[245]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 9Question (Kirsten Wi kelgren). Is ther e a Serr e sp e tr al se quen e for Mor el's HA1 \\n\\n> (,Z)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0246", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over a perfect field, a strict action of a finite constant group Gamma on a simplicial Nisnevich sheaf X yields a natural strongly convergent first-quadrant spectral sequence E2_{p,q} = H_p(Gamma; H_q^{A1}(X,Z)) converging to H_{p+q}^{A1}(E Gamma x_Gamma X,Z), where the E2 term is derived coinvariants in Nisnevich sheaves. If X is A1-connected and has a Gamma-fixed basepoint, the degree-one edge extension splits and H_1^{A1}(X_{h Gamma}) is the direct sum of the Gamma-coinvariants of H_1^{A1}(X) and the constant sheaf Gamma_ab. This is a homotopy-quotient special case, not a construction for arbitrary A1-fiber sequences.\n\nCandidate contribution (spectral_sequence; novelty confidence low): Candidate novelty: the explicit sheaf-level finite-constant-group Cartan-Leray spectral sequence for Morel's transfer-free Nisnevich A1-homology, together with the split H_1 formula for A1-connected equivariantly pointed spaces." }, { "id": 20000247, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0247", "title": "Singular free-summand descent and a cuspidal top-Chern counterexample", "statement": "Question 10 Question (Anand Sa wan t). When ar e the ab ove questions wel l-de\nne d over singu-lar varieties? When have they been onsider ed and what is their status? 4 SARANG SANE", "original_statement": "Question 10 Question (Anand Sa wan t). When ar e the ab ove questions wel l-de\u001cne d over singu-lar varieties? When have they been onsider ed and what is their status? 4 SARANG SANE", "clean_statement": "**Question 10. Question (Anand Sawant). When are the above questions\nwell-defined over singular varieties? When have they been considered and\nwhat is their status?**", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Question 10 from the AIM workshop list *Projective modules and \\(A^1\\)-homotopy theory*. The JSON extraction is visibly damaged by OCR. Inspection of the original PDF gives the following text:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 10\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[246]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 10 Question (Anand Sa wan t). When ar e the ab ove questions wel l-de\\u001cne d over singu-lar varieties? When have they been onsider ed and what is their status? 4 SARANG SANE\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0247", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After recovering Question 10 as a status question about all preceding Questions 1--9, this attempt proves two affine singular descent results: free-line summands of finite projective modules are invariant under nilpotent quotients, and over a reduced noetherian ring with finite normalization they are governed exactly by compatibility of unimodular functionals across the normalization conductor square. For A=k[t^2,t^3], conductor gluing by 1+at gives nontrivial line bundles L_a for every a nonzero in k, while CH_0(Spec A)=0; hence c_1(L_a) cap [Spec A]=0 in Fulton's homological Chow group although L_a has no free summand. Thus the direct Fulton-homological singular extension of the preceding top-Chern detection question fails in odd dimension one.\n\nCandidate contribution (reduction; novelty confidence low): The candidate contribution is the explicit two-stage singular splitting package Um(P dual) = Um(P_B dual) fiber-product-over Um(P_D dual) Um(P_C dual), preceded by nilpotent reduction, together with its cuspidal application showing that nonzero conductor-gluing classes yield nonfree rank-one bundles whose Fulton homological top Chern classes vanish." }, { "id": 20000248, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0248", "title": "Koszul complexes as derived intersections and splitting obstructions", "statement": "Question 11 Question (P aul Balmer). Using op er ations like ⊗, ∧ and taking ones, is ther ea onstru tion so that one obtains the Koszul omplex K•(s) fr om the omplex... 0 → P s\n\n> −→A→0...?Do es this gener alize to tensor triangulate d ate gories?", "original_statement": "Question 11 Question (P aul Balmer). Using op er ations like ⊗, ∧ and taking ones, is ther ea onstru tion so that one obtains the Koszul omplex K•(s) fr om the omplex... 0 → P s \n\n> −→A→0...?Do es this gener alize to tensor triangulate d ate gories?", "clean_statement": "Question 11 Question (P aul Balmer). Using op er ations like ⊗, ∧ and taking ones, is ther ea onstru tion so that one obtains the Koszul omplex K•(s) fr om the omplex... 0 → P s\n\n> −→A→0...?Do es this gener alize to tensor triangulate d ate gories?", "statement_status": "exact", "statement_verification": "The canonical record is Question 11 from the AIM workshop list *Projective modules and \\(A^1\\)-homotopy theory*. The repository copy has severe PDF extraction errors: in particular, “cones” became “ones,” “construction” lost several letters, and “categories” lost its initial letter. I checked the official PDF and recover the statement as follows (typographical spacing and the displayed arrow have been normalized, but no mathematical content has been changed):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 11\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[247]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 11 Question (P aul Balmer). Using op er ations like ⊗, ∧ and taking ones, is ther ea onstru tion so that one obtains the Koszul omplex K•(s) fr om the omplex... 0 → P s \\n\\n> −→A→0...?Do es this gener alize to tensor triangulate d ate gories?\"\nOriginal remarks: [\"Remark (P aul Balmer). Let K b e a tensor triangulated ategory and let X ∈ Ob (K).Supp ose there is a map X s \\n\\n> −→1. If the ab ove onstru tion works, then one an obtain a lass c(X) ∈ GW (K). This lass should then de\\u001cne an obstru tion lass for X to split a free summand 1.\", \"Remark (Madha v Nori). There is su h a onstru tion.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0248", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite projective A-module P and s:P→A, with S=Sym_A(P), the derived intersection A_s tensor^L_S A_0 is naturally the Koszul complex K(P,s). The same free-commutative-algebra/bar formula gives a functorial construction in a presentable symmetric-monoidal stable infinity-category, but is not supplied by bare tensor-triangulated axioms. K(P,s) has the explicit determinant-twisted n-shifted symmetric self-duality of Balmer–Gille, is zero exactly when the chosen map s splits, and yields a supported Grothendieck–Witt/Witt class. Under regular noetherian and 1/2 hypotheses, forgetting support gives an A1-invariant class independent of s whose nonvanishing obstructs any free rank-one summand of P. The example A=R[x], P=A, s=x has a nonzero supported Witt class but zero absolute Witt image, proving that support and the distinction between a chosen section and the module itself are essential.\n\nCandidate contribution (obstruction_synthesis; novelty confidence low): The three-level detection proposition distinguishes the exact object-level test K(P,s), the supported determinant-twisted quadratic class, and the section-independent absolute Euler obstruction; the single explicit example (R[x],x) proves that both information-loss steps are genuine." }, { "id": 20000249, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0249", "title": "Generic-point abelianity and the naive-versus-genuine SU(2) correction", "statement": "Question 12 Question (Anastasia Sta vro va). When G is a line ar redu ed/algebr ai gr oup, in alar ge numb er of ases, it is known on a ase-byase basis that ΠA1\n\n> 0(G)\n\nis ab elian. Is this always true? Can this be obtaine d in a uniform way when it is true?", "original_statement": "Question 12 Question (Anastasia Sta vro va). When G is a line ar redu ed/algebr ai gr oup, in alar ge numb er of ases, it is known on a ase-byase basis that ΠA1 \n\n> 0(G)\n\nis ab elian. Is this always true? Can this be obtaine d in a uniform way when it is true?", "clean_statement": "Question 12 Question (Anastasia Sta vro va). When G is a line ar redu ed/algebr ai gr oup, in alar ge numb er of ases, it is known on a ase-byase basis that ΠA1\n\n> 0(G)\n\nis ab elian. Is this always true? Can this be obtaine d in a uniform way when it is true?", "statement_status": "exact", "statement_verification": "The record is Question 12 from the AIM workshop *Projective modules and \\(A^1\\)-homotopy theory* (May 5--9, 2014). The official PDF verifies the following text (with the typography modernized but not the wording):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 12\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[248]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 12 Question (Anastasia Sta vro va). When G is a line ar redu ed/algebr ai gr oup, in alar ge numb er of ases, it is known on a ase-byase basis that ΠA1 \\n\\n> 0(G)\\n\\nis ab elian. Is this always true? Can this be obtaine d in a uniform way when it is true?\"\nOriginal remarks: [\"Remark (Madha v Nori). This is not alw ays true e.g. G = SU (2, R). In fa t, the same reasoning should sho w that ΠA1 \\n\\n> 0(G)\\n\\nis not ab elian for an y anisotropi group G over R.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0249", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any field k and Nisnevich sheaf of groups H, the sheaf pi_0^{A1}(H) is abelian if and only if its values on all finitely generated separable fields over k are abelian; this follows by applying known generic-point injectivity to commutators. For semisimple simply connected algebraic groups, the fieldwise comparison with G(K)/R makes this an unconditional uniform reduction to the long-open abelianity problem for R-equivalence. The workshop's proposed counterexample SU(2,R)=SL_1(Hamilton quaternions) fails for genuine components: over every real closed F, all regular A1_F-maps into SL_1((-1,-1)_F) are constant, so the naive component value is the nonabelian group G(F), whereas explicit Cayley rational curves give G(F)/R=1 and hence pi_0^{A1}(G)(F)=1.\n\nCandidate contribution (proposition; novelty confidence low): Candidate contribution: a fully proved real-closed-field repair of the AIM SU(2) annotation, showing naive A1-chain components are nonabelian while genuine A1-components are trivial, packaged with an arbitrary-base-field generic-point commutator criterion that upgrades fieldwise abelianity to sheafwise abelianity." }, { "id": 20000250, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0250", "title": "Divisible Chern data are realizable on finite-dimensional CW complexes", "statement": "Question 13 Question (Madha v Nori). Given positive inte gers d, r and X a CW omplex with dim X = d and lasses ci ∈ H2i(X, Z) for i = 1, 2,..., r, su h that the image of ci is 0 in H2i(X, Z\n\n> mZ)\n\nfor al l m, do es ther e exist a omplex ve tor bund le E with rank( E) = r and ci(E) = ci?", "original_statement": "Question 13 Question (Madha v Nori). Given positive inte gers d, r and X a CW omplex with dim X = d and lasses ci ∈ H2i(X, Z) for i = 1, 2,..., r, su h that the image of ci is 0 in H2i(X, Z \n\n> mZ)\n\nfor al l m, do es ther e exist a omplex ve tor bund le E with rank( E) = r and ci(E) = ci?", "clean_statement": "Question 13 Question (Madha v Nori). Given positive inte gers d, r and X a CW omplex with dim X = d and lasses ci ∈ H2i(X, Z) for i = 1, 2,..., r, su h that the image of ci is 0 in H2i(X, Z\n\n> mZ)\n\nfor al l m, do es ther e exist a omplex ve tor bund le E with rank( E) = r and ci(E) = ci?", "statement_status": "exact", "statement_verification": "The extracted record is Question 13 from the AIM problem list *Projective modules and \\(A^1\\)-homotopy theory*. The PDF was checked directly because the JSON text has severe OCR damage. The recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 13\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[249]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 13 Question (Madha v Nori). Given positive inte gers d, r and X a CW omplex with dim X = d and lasses ci ∈ H2i(X, Z) for i = 1, 2,..., r, su h that the image of ci is 0 in H2i(X, Z \\n\\n> mZ)\\n\\nfor al l m, do es ther e exist a omplex ve tor bund le E with rank( E) = r and ci(E) = ci?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0250", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Nori's question has an affirmative answer for every finite-dimensional CW complex, including complexes with infinitely many cells. The coefficient long exact sequence shows that vanishing modulo every positive integer is equivalent to divisibility by every positive integer. The target-skeleton proof of Benoist's 2026 universal-multiple lemma supplies, for fixed dimension d and rank r, one multiplier M(d,r) independent of the space and cohomology tuple. Choosing M(d,r)-th roots of the prescribed classes and applying that theorem produces a rank-r bundle with exactly the requested Chern classes. The proof also verifies that cellular approximation avoids any inverse-limit or phantom-map gap.\n\nCandidate contribution (explicit_example_and_corollary; novelty confidence low): For every prime p and every nonzero gamma in H^6(Tel(S^5 --p--> S^5 --p--> ...); Z) congruent to an element of Z_p/Z, there is a complex rank-3 bundle E with c1(E)=c2(E)=0 and c3(E)=gamma; more broadly, the uniform quantifier in Benoist's target-skeleton proof explicitly identifies AIM Question 13 as an affirmative corollary for arbitrary finite-dimensional CW complexes." }, { "id": 20000251, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0251", "title": "Rank correction, top-Chern criterion, and an affine-surface counterexample", "statement": "Question 14 Question (Madha v Nori). Let X be a d-dimensional smo oth a\nne variety. Let α ∈ F d−1K0(X). Then do es ther e exist a ve tor bund le E of rank d − 1 su h that [E] = α?E-mail addr ess: sarangsanemath[U+0008]gmail. o m", "original_statement": "Question 14 Question (Madha v Nori). Let X be a d-dimensional smo oth a\u001ene variety. Let α ∈ F d−1K0(X). Then do es ther e exist a ve tor bund le E of rank d − 1 su h that [E] = α?E-mail addr ess: sarangsanemath\bgmail. o m", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The official problem list prints the following (Question 14, attributed to Madhav Nori):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Projective modules and $A^1$-homotopy theory\nSection: \nSource item: 14\nSource URL: https://aimath.org/pastworkshops/projectiveA1problems.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[250]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 14 Question (Madha v Nori). Let X be a d-dimensional smo oth a\\u001ene variety. Let α ∈ F d−1K0(X). Then do es ther e exist a ve tor bund le E of rank d − 1 su h that [E] = α?E-mail addr ess: sarangsanemath\\bgmail. o m\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/projectiveA1problems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0251", "aim-domain:algebraic-geometry", "aim-workshop:projectivea1problems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The statement printed in the official PDF is rank-inconsistent: for d at least 2, alpha in F^{d-1}K_0 has rank zero and cannot equal the class of a rank-(d-1) bundle. For the explicitly labeled reconstructed statement [E]-(d-1)[O_X]=alpha, a rank-d representative of d[O_X]+alpha always exists, and solvability is equivalent to some such representative splitting a free line. Over an algebraically closed field this holds exactly when c_d(alpha)=0, by Serre stable realization and the Murthy-Krishna top-rank splitting theorem. The reconstructed universal assertion is false even over C: on X={1+x^4+y^4+z^4=0}, CH_0(X) is nonzero, and a nonzero zero-cycle z gives alpha=cyc_K(z) in F^2K_0(X) with c_2(alpha)=-z, so alpha is not [L]-[O_X] for any line bundle.\n\nCandidate contribution (equivalence_and_example; novelty confidence low): Candidate novelty is the cancellation-safe synthesis that the rank-corrected AIM question is free-line destabilization inside the nonempty rank-d K_0 fiber, together with the canonical surface defect delta_2(alpha)=alpha-([det(alpha)]-[O_X]) and its application to the explicit affine Fermat-quartic complement to produce nonrepresentable corrected classes." }, { "id": 20000252, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0252", "title": "Affirmative for p at least 5 and an all-characteristic adjunction-surface reduction", "statement": "Basepoint free theorem\n\nSuppose that $k = \\bar{k}$, $\\text{char } k = p > 0$. Let $X/k$ be a terminal threefold, and $A$ an ample $\\mathbb Q$-divisor on $X$. If $K_X+A$ is nef, must it be semiample?", "original_statement": "Basepoint free theorem\n\nSuppose that $k = \\bar{k}$, $\\text{char } k = p > 0$. Let $X/k$ be a terminal threefold, and $A$ an ample $\\mathbb Q$-divisor on $X$. If $K_X+A$ is nef, must it be semiample?", "clean_statement": "Basepoint free theorem\n\nSuppose that $k = \\bar{k}$, $\\text{char } k = p > 0$. Let $X/k$ be a terminal threefold, and $A$ an ample $\\mathbb Q$-divisor on $X$. If $K_X+A$ is nef, must it be semiample?", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 1.1 in the workshop section “Fundamental MMP theorems”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Fundamental MMP theorems\nSource item: 1.1\nSource URL: http://aimpl.org/minimalmodcharp/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[251]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Basepoint free theorem\\n\\nSuppose that $k = \\\\bar{k}$, $\\\\text{char } k = p > 0$. Let $X/k$ be a terminal threefold, and $A$ an ample $\\\\mathbb Q$-divisor on $X$. If $K_X+A$ is nef, must it be semiample?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is known in a couple cases:\\n\\n1. $X$ smooth, $\\\\nu(K_X+A) = 0$.\\n\\n2. $k = \\\\bar{\\\\mathbb F}_p$, $\\\\dim X = 3$, $K_X+A$ big.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0252", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Hacon and Witaszek's threefold basepoint-free theorem gives an affirmative answer to the AIM problem in every characteristic p > 3, hence for all primes p >= 5. The precise terminal adjoint statement remains unresolved in the literature checked for p = 2, 3. In every positive characteristic, a proved conditional reduction shows semiampleness when there is a normal prime Cartier surface S that is Q-factorial, has an effective coefficient-<1 adjunction boundary, and satisfies both D-S ample and (A-S)|_S ample, where D=K_X+A.\n\nCandidate contribution (reduction; novelty confidence low): Let D=K_X+A be nef in the AIM setup over any algebraically closed field of characteristic p>0. If there is a normal prime Cartier divisor S such that S is Q-factorial, (K_X+S)|_S is Q-linearly equivalent to K_S+Delta_S for an effective Q-divisor Delta_S with all coefficients in [0,1), D-S is ample, and (A-S)|_S is ample, then D is semiample." }, { "id": 20000253, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0253", "title": "Termination of klt threefold flips in positive characteristic", "statement": "Termination in dimension 3\n\nDo klt flips terminate in dimension 3?", "original_statement": "Termination in dimension 3\n\nDo klt flips terminate in dimension 3?", "clean_statement": "Termination in dimension 3\n\nDo klt flips terminate in dimension 3?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 1.2 from the 2013 workshop *The minimal model program in characteristic \\(p\\)*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Fundamental MMP theorems\nSource item: 1.2\nSource URL: http://aimpl.org/minimalmodcharp/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[252]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Termination in dimension 3\\n\\nDo klt flips terminate in dimension 3?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is probably OK for terminal varieties, with the proof as in characteristic $0$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: conditional; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0253", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For projective Q-factorial klt threefold Q-pairs over an algebraically closed field of characteristic p>5, arbitrary flip termination is known when K_X+B is pseudo-effective, while MMPs with ample scaling terminate more generally; the literature checked leaves arbitrary non-pseudo-effective sequences over a zero-dimensional base open. The report proves the terminal difficulty bound, proves that marked models cannot repeat, and, conditional on Yang-Ye-Zhang arXiv:2408.12269v3, proves that every infinite ordinary flip tail must cross the nonnegative wall of every sufficiently positive ample Cartier b-divisor.\n\nCandidate contribution (reduction; novelty confidence low): Conditional on termination of pseudo-effective NQC generalized threefold-pair flips, if an infinite K_X+B flip tail starts at X_N and A_N is ample with Cartier closure A, then for every t such that K_{X_N}+B_N+tA_N is pseudo-effective there is a later flipping ray R_i with (K_{X_i}+B_i+tA_i).R_i >= 0; consequently no fixed sufficiently positive ample Cartier b-divisor can direct every ray of an infinite tail negatively." }, { "id": 20000254, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0254", "title": "Positive-characteristic connectedness and an amplified Frobenius certificate", "statement": "Connectedness\n\nIs there a positive-characteristic analog of the connectedness lemma?", "original_statement": "Connectedness\n\nIs there a positive-characteristic analog of the connectedness lemma?", "clean_statement": "Connectedness\n\nIs there a positive-characteristic analog of the connectedness lemma?", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Fundamental MMP theorems\nSource item: 1.3\nSource URL: http://aimpl.org/minimalmodcharp/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[253]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Connectedness\\n\\nIs there a positive-characteristic analog of the connectedness lemma?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0254", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The intended Kollár--Shokurov theorem is known for surfaces in every characteristic and for threefold log pairs over perfect fields in characteristic p>5; the p>5 nef-only principle also has a two-component P1-link classification. This attempt proves an all-dimensional, all-positive-characteristic sufficient criterion: for a contraction f:X->S, if the radical ideal I of a reduced closed subscheme Z admits a split map I -> F^e_*(I tensor L^(p^e-1)) for an f-ample line bundle L, then R^1 f_* I=0 and every fiber of Z->S is empty or geometrically connected. Applied to Z=Nklt(X,B), this is a concrete conditional positive-characteristic connectedness theorem.\n\nCandidate contribution (criterion; novelty confidence low): A split amplified Frobenius map on the radical non-klt ideal gives a relative connectedness certificate: I_Nklt -> F^e_*(I_Nklt tensor L^(q-1)) with q=p^e and L relatively ample forces R^1 f_* I_Nklt=0 and hence geometrically connected nonempty fibers; consequently a standard P1-link cannot admit such a certificate." }, { "id": 20000255, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0255", "title": "Terminal singularities need not be Cohen-Macaulay: the sharp threefold threshold and a cone-Veronese audit", "statement": "Terminal and Cohen-Macaulay\n\nAre (log) terminal singularities Cohen-Macaulay? Rational?", "original_statement": "Terminal and Cohen-Macaulay\n\nAre (log) terminal singularities Cohen-Macaulay? Rational?", "clean_statement": "Terminal and Cohen-Macaulay\n\nAre (log) terminal singularities Cohen-Macaulay? Rational?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem 2.1 from the workshop *The minimal model program in characteristic \\(p\\)*, section “Singularities in char \\(p\\).” Its problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Singularities in char p\nSource item: 2.1\nSource URL: http://aimpl.org/minimalmodcharp/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[254]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Terminal and Cohen-Macaulay\\n\\nAre (log) terminal singularities Cohen-Macaulay? Rational?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is affirmative in dimension two by the classification, but even the three-dimensional case seems interesting. One possible example: take the cone over a smooth Fano which admits a counterexample to Kodaira vanishing. One such exists in dimension 6(?), and should be log terminal but not Cohen-Macaulay.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0255", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The universal Cohen-Macaulay and rationality assertions are false: Totaro constructed terminal non-Cohen-Macaulay threefolds in characteristics 2, 3, and 5, while Arvidsson-Bernasconi-Lacini prove that every klt threefold over a perfect field of characteristic p>5 is rational and hence Cohen-Macaulay, making the threefold threshold sharp. Totaro's cone family also gives terminal non-Cohen-Macaulay and directly nonrational singularities in every characteristic p>=3. This attempt proves a cone-Veronese audit: if -K_V is Q-linearly equivalent to rL, then the L^q-cone is terminal exactly for q=3. In the Q-factorial threefold setting over a perfect field, this attempt proves a diagnostic package from the known R1 and universal-homeomorphism inputs: Q=nu_*O_{S^nu}/O_S is supported at finitely many points, Q_x is canonically H^1_x(O_S), its conductor and finite Frobenius exponent detect normality, its weighted length is the constant proper Euler/Hilbert defect, and it vanishes when S is Cartier in an S3 ambient threefold.\n\nCandidate contribution (criterion; novelty confidence low): For a Q-factorial plt threefold center over a perfect field of characteristic p>0, the isolated normalization quotient Q=nu_*O_{S^nu}/O_S has canonical stalk identities Q_x ~= H^1_x(O_S); for proper S its weighted total length is exactly chi(S^nu,nu^*L^m)-chi(S,L^m) for every line-bundle twist, while conductor, a finite radicial Frobenius exponent, and the Cartier-in-S3 depth test give equivalent or sufficient normality checks." }, { "id": 20000257, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0257", "title": "Fano-type threefolds in characteristic p>5 are Mori dream spaces", "statement": "Fano and MDS\n\nIs every Fano variety a Mori Dream Space?", "original_statement": "Fano and MDS\n\nIs every Fano variety a Mori Dream Space?", "clean_statement": "Fano and MDS\n\nIs every Fano variety a Mori Dream Space?", "statement_status": "exact", "statement_verification": "The canonical AIM record is from the workshop *The minimal model program in characteristic \\(p\\)*, section “Sections and section rings,” Problem 4.1. Its problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Sections and section rings\nSource item: 4.1\nSource URL: http://aimpl.org/minimalmodcharp/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[256]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Fano and MDS\\n\\nIs every Fano variety a Mori Dream Space?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This result is known in characteristic $0$. It's also true that if $X$ is Fano then $X$ is strongly $F$-regular, if $p \\\\gg 0$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0257", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A projective Q-factorial threefold of Fano type over an algebraically closed field of characteristic p>5 is a Mori dream space. The proof combines Das's finite generation of multigraded adjoint rings with rational chain connectedness of Fano-type threefolds: a common-boundary perturbation realizes both signs of a finite-index Cartier lattice after clearing denominators, and the resulting multiveronese adjoint ring maps surjectively onto the full multisection ring of that lattice. The report also proves an arbitrary-characteristic rank-one criterion and gives a field-sensitive normal Gorenstein Fano cone that is not Q-factorial, delimiting the literal weak reading.\n\nCandidate contribution (theorem; novelty confidence low): Candidate apparently unstated corollary: every projective Q-factorial threefold of Fano type over an algebraically closed field of characteristic p>5 is a Mori dream space, via an explicit surjection from a multiveronese of Das's adjoint ring onto a Hu-Keel Cox ring." }, { "id": 20000258, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0258", "title": "A two-source specialization-defect formula and the post-2025 status of plurigenera", "statement": "Invariance of plurigenera\n\nWhat is the status of invariance of plurigenera?", "original_statement": "Invariance of plurigenera\n\nWhat is the status of invariance of plurigenera?", "clean_statement": "Invariance of plurigenera\n\nWhat is the status of invariance of plurigenera?", "statement_status": "exact", "statement_verification": "The canonical AIM record, from the 2013 workshop *The minimal model program in characteristic \\(p\\)*, says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Sections and section rings\nSource item: 4.2\nSource URL: http://aimpl.org/minimalmodcharp/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[257]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Invariance of plurigenera\\n\\nWhat is the status of invariance of plurigenera?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"It's known that $h^0(mK_X)$ need not be constant in flat families if $m=1$. For $m \\\\geq 2$ it remains a possibility.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0258", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The 2013 hope that higher plurigenera might always be invariant in positive or mixed characteristic is now false in general: Brivio's published 2023 examples give logarithmic failure for smooth terminal surface pairs in every residue characteristic, and arXiv:2207.08107v2 (June 2025, currently an unrefereed preprint) states boundary-free asymptotic failure for smooth threefold families with semiample canonical class and residue characteristic p>2. This attempt proves an exact DVR formula separating any jump into uniformizer torsion in relative H^1 and a visible reflexive/rounding comparison-cokernel term; in smooth families the second term vanishes, so invariance for fixed m is equivalent to torsion-freeness of H^1(X, omega_{X/R}^{tensor m}). It also proves asymptotic invariance for globally indexed, Kollár-compatible stable families by relative Serre vanishing and gives the complete curve case in every characteristic.\n\nCandidate contribution (exact_defect_formula_and_criterion; novelty confidence low): Candidate contribution: if a flat coherent relative m-pluricanonical model L_m has the correct generic fiber and injects into the actual special pluricanonical sheaf with cokernel Q_m, then the special-minus-generic plurigenus equals dim_k H^1(X,L_m)[varpi] plus dim_k ker(H^0(Q_m) -> H^1((L_m)_k)); hence in a smooth family the jump is exactly the varpi-socle of H^1(X,omega_{X/R}^{tensor m})." }, { "id": 20000259, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0259", "title": "General type in families: a counterexample and a compatible-index nef criterion", "statement": "General type in families\n\nSuppose that $\\mathcal X \\to \\Delta$ is a family, and a special fiber is of general type. Does it follow that a general fiber is of general type?", "original_statement": "General type in families\n\nSuppose that $\\mathcal X \\to \\Delta$ is a family, and a special fiber is of general type. Does it follow that a general fiber is of general type?", "clean_statement": "General type in families\n\nSuppose that $\\mathcal X \\to \\Delta$ is a family, and a special fiber is of general type. Does it follow that a general fiber is of general type?", "statement_status": "exact", "statement_verification": "The canonical record is `aim-algebraic-geometry-notes.json`, zero-based index 258, Problem 4.4 in the AIM workshop *The minimal model program in characteristic \\(p\\)*. The exact problem is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Sections and section rings\nSource item: 4.4\nSource URL: http://aimpl.org/minimalmodcharp/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[258]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"General type in families\\n\\nSuppose that $\\\\mathcal X \\\\to \\\\Delta$ is a family, and a special fiber is of general type. Does it follow that a general fiber is of general type?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0259", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The source statement has no uniform affirmative answer without defining the family and the singular-fiber convention. Under the fiberwise-Q-Cartier-K-big reading, Kollár's flat projective surface family has an ample-canonical log-canonical special fiber and smooth elliptic general fibers, so the literal implication fails. This attempt also proves that in a flat projective compatible-index Q-Gorenstein family over a DVR, a geometrically nef and big special canonical divisor forces the generic canonical divisor to be geometrically nef and big with equal volume; more generally, the leading growth of nonextendable special pluricanonical sections exactly equals the loss of generic canonical volume. Kollár's 2021 theorem separately gives the affirmative characteristic-zero result when the general-type special fiber has canonical singularities.\n\nCandidate contribution (criterion; novelty confidence low): In a compatible-index Q-Gorenstein family, generic general type is equivalent to positive order-m^n growth of the graded algebra of extendable special pluricanonical sections, its volume loss is exactly the normalized nonextendable-section defect, and nef-bigness of the special canonical divisor makes that leading defect vanish; one generically finite extendable subsystem is already a finite-degree certificate." }, { "id": 20000260, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0260", "title": "Frobenius-stable adjoint sections in families", "statement": "What if we look at $\\text{dim } S^0(mK_X+A)$ instead?", "original_statement": "What if we look at $\\text{dim } S^0(mK_X+A)$ instead?", "clean_statement": "What if we look at $\\text{dim } S^0(mK_X+A)$ instead?", "statement_status": "exact", "statement_verification": "The canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Sections and section rings\nSource item: 4.3\nSource URL: http://aimpl.org/minimalmodcharp/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[259]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What if we look at $\\\\text{dim } S^0(mK_X+A)$ instead?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0260", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Patakfalvi--Schwede--Zhang prove that, under standard F-finite pair hypotheses and relative adjoint ampleness, fiberwise S^0-dimension is generically constant but is not semicontinuous in either direction in general; they also prove uniform Frobenius stabilization and an open S^0=H^0/base-change criterion. This attempt adds a proved curve-level synthesis: the S^0-defect of omega_C tensor L is exactly the eventually Frobenius-killed subspace of H^1(C,L^{-1}); for positive degree d it stabilizes at an explicit Tango-bound exponent. Consequently the dimension is lower semicontinuous in smooth positive-degree curve families, and S^0(mK_C+A)=H^0(mK_C+A) with constant dimension for every m at least 2 and ample A.\n\nCandidate contribution (finite-stabilization criterion; novelty confidence low): For a smooth family of genus-g curves and a line bundle L of positive relative degree d, if q is the least nonnegative integer with p^q d greater than floor((2g-2)/p), then the q-fold relative Frobenius trace already computes fiberwise S^0 on every geometric fiber; its dimension is the rank of one morphism between locally free base-change-compatible sheaves and is therefore lower semicontinuous." }, { "id": 20000261, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0261", "title": "Frobenius-stable pluricanonical systems on products of curves", "statement": "Embedding by $S^0(mK_X)$\n\nSuppose that $X$ is of general type. Is there an effective constant $m = m(n)$ such that $S^0(mK_X)$ (the canonical linear system of Schwede) defines a birational map? What about the usual linear system $|mK_X|$?", "original_statement": "Embedding by $S^0(mK_X)$\n\nSuppose that $X$ is of general type. Is there an effective constant $m = m(n)$ such that $S^0(mK_X)$ (the canonical linear system of Schwede) defines a birational map? What about the usual linear system $|mK_X|$?", "clean_statement": "Embedding by $S^0(mK_X)$\n\nSuppose that $X$ is of general type. Is there an effective constant $m = m(n)$ such that $S^0(mK_X)$ (the canonical linear system of Schwede) defines a birational map? What about the usual linear system $|mK_X|$?", "statement_status": "exact", "statement_verification": "This is Problem 4.5 in the AIM workshop list *The minimal model program in characteristic \\(p\\)*, section “Sections and section rings.” The canonical JSON record states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Sections and section rings\nSource item: 4.5\nSource URL: http://aimpl.org/minimalmodcharp/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[260]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Embedding by $S^0(mK_X)$\\n\\nSuppose that $X$ is of general type. Is there an effective constant $m = m(n)$ such that $S^0(mK_X)$ (the canonical linear system of Schwede) defines a birational map? What about the usual linear system $|mK_X|$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0261", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every product X of n smooth projective curves of genus at least two over an algebraically closed field of characteristic p>0, every pluricanonical trace map is surjective for m>=2, so S^0(X,mK_X)=H^0(X,mK_X). Consequently S^0(X,3K_X) defines a closed immersion, and 3 is the sharp uniform birational exponent in this all-dimensional class because a genus-two factor makes the bicanonical map degree two. The unrestricted dimension-only positive-characteristic problem remains open in the literature checked.\n\nCandidate contribution (special_case; novelty confidence low): If X is any finite product of smooth projective curves of genus at least two in characteristic p>0, then S^0(X,mK_X)=H^0(X,mK_X) for every m>=2, and the sharp class-uniform birational exponent is m=3." }, { "id": 20000262, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0262", "title": "Uniform Fujita vanishing via flatness strata", "statement": "Effective Fujita vanishing\n\nIs there an effective Fujita vanishing result in flat families? Suppose that $f : \\mathcal X \\to S$ is flat, $\\mathcal F$ is a coherent sheaf on $\\mathcal X$, and $\\mathcal L$ is $f$-ample. Does there exist a constant $m_0 = m_0(\\mathcal X,\\mathcal F,\\mathcal L)$ such that $H^i(X_s,(\\mathcal F \\otimes \\mathcal L^m \\mathcal )\\vert_{X_s} ) = 0$ if $s$ is any point of $S$, $m \\geq m_0$ and $\\mathcal M$ is any $f$-nef line bundle on $X$?", "original_statement": "Effective Fujita vanishing\n\nIs there an effective Fujita vanishing result in flat families? Suppose that $f : \\mathcal X \\to S$ is flat, $\\mathcal F$ is a coherent sheaf on $\\mathcal X$, and $\\mathcal L$ is $f$-ample. Does there exist a constant $m_0 = m_0(\\mathcal X,\\mathcal F,\\mathcal L)$ such that $H^i(X_s,(\\mathcal F \\otimes \\mathcal L^m \\mathcal )\\vert_{X_s} ) = 0$ if $s$ is any point of $S$, $m \\geq m_0$ and $\\mathcal M$ is any $f$-nef line bundle on $X$?", "clean_statement": "Effective Fujita vanishing\n\nIs there an effective Fujita vanishing result in flat families? Suppose that $f : \\mathcal X \\to S$ is flat, $\\mathcal F$ is a coherent sheaf on $\\mathcal X$, and $\\mathcal L$ is $f$-ample. Does there exist a constant $m_0 = m_0(\\mathcal X,\\mathcal F,\\mathcal L)$ such that $H^i(X_s,(\\mathcal F \\otimes \\mathcal L^m \\mathcal )\\vert_{X_s} ) = 0$ if $s$ is any point of $S$, $m \\geq m_0$ and $\\mathcal M$ is any $f$-nef line bundle on $X$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 4.6, “Effective Fujita vanishing,” in the section “Sections and section rings” of the workshop *The minimal model program in characteristic \\(p\\)*. Its body reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Sections and section rings\nSource item: 4.6\nSource URL: http://aimpl.org/minimalmodcharp/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[261]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Effective Fujita vanishing\\n\\nIs there an effective Fujita vanishing result in flat families? Suppose that $f : \\\\mathcal X \\\\to S$ is flat, $\\\\mathcal F$ is a coherent sheaf on $\\\\mathcal X$, and $\\\\mathcal L$ is $f$-ample. Does there exist a constant $m_0 = m_0(\\\\mathcal X,\\\\mathcal F,\\\\mathcal L)$ such that $H^i(X_s,(\\\\mathcal F \\\\otimes \\\\mathcal L^m \\\\mathcal )\\\\vert_{X_s} ) = 0$ if $s$ is any point of $S$, $m \\\\geq m_0$ and $\\\\mathcal M$ is any $f$-nef line bundle on $X$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is probably OK in the equal characteristic case, adapting the usual proof.\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0262", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM formula is historically corrupted; its coherent reconstruction is H^i(X_s,F_s tensor L_s^m tensor M_s)=0 for i>0. Under the standard hypotheses that f is projective over a noetherian base, Keeler's characteristic-free relative Fujita theorem and a finite generic-flatness stratification prove one threshold uniform in every point, every m above the threshold, every positive cohomological degree, and every global f-nef line bundle M; flatness of f and F is unnecessary. The literal hypotheses are insufficient: the punctured affine plane is a nonproper counterexample, and an infinite disjoint union of projective lines is a non-quasi-compact projective counterexample. A supplied relative free resolution on projective space also yields the explicit regularity bound m_0=max(0,max(a_jlambda-j)).\n\nCandidate contribution (lemma; novelty confidence low): For a projective morphism f:X->S over a noetherian scheme, Keeler's nef-uniform relative vanishing transfers to nef-uniform fiberwise vanishing for every coherent F, without flatness of f or F: choose the finite generic-flatness strata T_alpha of F, finite affine covers U_alpha,beta, and take m_0=max b_alpha,beta over their Keeler thresholds." }, { "id": 20000263, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0263", "title": "Semigroup-gap counterexamples on weighted projective threefolds", "statement": "Sections of nef $K_X+A$\n\nSuppose that $A$ is Cartier and ample, and $K_X+A$ is nef. Must $H^0(X,K_X+A)$ be nonzero?", "original_statement": "Sections of nef $K_X+A$\n\nSuppose that $A$ is Cartier and ample, and $K_X+A$ is nef. Must $H^0(X,K_X+A)$ be nonzero?", "clean_statement": "Sections of nef $K_X+A$\n\nSuppose that $A$ is Cartier and ample, and $K_X+A$ is nef. Must $H^0(X,K_X+A)$ be nonzero?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 4.7 of the AIM workshop *The minimal model program in characteristic \\(p\\)* (2013), in the section “Sections and section rings.” The archived page attributes it to Cascini and states, with no preceding local convention:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Sections and section rings\nSource item: 4.7\nSource URL: http://aimpl.org/minimalmodcharp/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[262]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Sections of nef $K_X+A$\\n\\nSuppose that $A$ is Cartier and ample, and $K_X+A$ is nef. Must $H^0(X,K_X+A)$ be nonzero?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0263", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM statement is false under its literal A-Cartier-only wording. For coprime integers a,b >= 2 with delta = ab-2a-2b > 0, over an algebraically closed field of characteristic not dividing ab, X = P(a,a,b,b) and A = O_X(ab) give a normal well-formed Q-factorial klt toric threefold with A ample Cartier and K_X+A = O_X(delta) Q-ample, hence nef, but H^0(X,O_X(delta)) = 0. The smallest-gap displayed example is P(3,3,7,7) in characteristic 5 with A = O_X(21) and K_X+A = O_X(1). These examples do not refute standard Ambro-Kawamata effective nonvanishing because K_X+A is not Cartier.\n\nCandidate contribution (counterexample_family; novelty confidence low): The family (P(a,a,b,b), O_X(ab)) with gcd(a,b)=1 and ab>2a+2b is an explicit infinite family of tame klt toric threefold counterexamples to the exact AIM formulation in which only A is required to be Cartier." }, { "id": 20000264, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0264", "title": "Divisorial finite generation, Q-Cartierizations, and a two-ray reduction", "statement": "$F$-regularity and finite generation\n\nSuppose that $X$ is an $F$-regular variety (maybe not $\\mathbb Q$-Gorenstein). If $D$ is a Weil divisor, must $\\bigoplus_{m \\geq 0} \\mathcal O_X(mD)$ be finitely generated?", "original_statement": "$F$-regularity and finite generation\n\nSuppose that $X$ is an $F$-regular variety (maybe not $\\mathbb Q$-Gorenstein). If $D$ is a Weil divisor, must $\\bigoplus_{m \\geq 0} \\mathcal O_X(mD)$ be finitely generated?", "clean_statement": "$F$-regularity and finite generation\n\nSuppose that $X$ is an $F$-regular variety (maybe not $\\mathbb Q$-Gorenstein). If $D$ is a Weil divisor, must $\\bigoplus_{m \\geq 0} \\mathcal O_X(mD)$ be finitely generated?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 4.8 in the AIM workshop list *The minimal model program in characteristic \\(p\\)*, section “Sections and section rings.” The live AIM page attributes the problem to Schwede and gives the following text:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Sections and section rings\nSource item: 4.8\nSource URL: http://aimpl.org/minimalmodcharp/4/\nCanonical location: aim-algebraic-geometry-notes.json notes[263]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"$F$-regularity and finite generation\\n\\nSuppose that $X$ is an $F$-regular variety (maybe not $\\\\mathbb Q$-Gorenstein). If $D$ is a Weil divisor, must $\\\\bigoplus_{m \\\\geq 0} \\\\mathcal O_X(mD)$ be finitely generated?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/4/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0264", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The intended sheaf algebra is the divisorial symbolic Rees algebra, and its finite generation for an arbitrary excellent strongly F-regular ring remains open as Aberbach--Huneke--Polstra Conjecture 1.4. This attempt records the standard equivalence with a small projective D-Q-Cartierization, proves finite generation for Q-Cartier divisors and for all divisors on normal toric varieties, and proves that when the divisor class group modulo torsion is infinite cyclic, finite generation for every Weil divisor is equivalent to finite generation for just a generator and its negative, or equivalently to the existence of the two opposite Q-Cartierizations.\n\nCandidate contribution (reduction; novelty confidence low): If Cl(X)/Cl(X)_tors is infinite cyclic generated by the image of D, then every divisorial algebra R_X(E) is finitely generated if and only if both R_X(D) and R_X(-D) are finitely generated; by the standard Kollár--Mori criterion this is equivalent to the existence of small projective Q-Cartierizations for the two opposite rays." }, { "id": 20000265, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0265", "title": "Frobenius-polarized cones with double poles for (n-1)-forms", "statement": "Pulling back forms to a resolution\n\nSuppose that $X$ is log canonical, and that there exists a log resolution $f : \\tilde{X} \\to X$ which is an isomorphism over the smooth locus. Let $\\omega$ be an $(n-1)$-form on $X$. Does $f^\\ast \\omega\\vert_{X_{\\text{smooth}}}$ extend to an $(n-1)$-form on $\\tilde{X}$ with log poles along the exceptional locus?", "original_statement": "Pulling back forms to a resolution\n\nSuppose that $X$ is log canonical, and that there exists a log resolution $f : \\tilde{X} \\to X$ which is an isomorphism over the smooth locus. Let $\\omega$ be an $(n-1)$-form on $X$. Does $f^\\ast \\omega\\vert_{X_{\\text{smooth}}}$ extend to an $(n-1)$-form on $\\tilde{X}$ with log poles along the exceptional locus?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "This is Problem 5.1, “Pulling back forms to a resolution,” in the section “Other questions” of the AIM workshop *The minimal model program in characteristic \\(p\\)*. The exact database text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Other questions\nSource item: 5.1\nSource URL: http://aimpl.org/minimalmodcharp/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[264]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Pulling back forms to a resolution\\n\\nSuppose that $X$ is log canonical, and that there exists a log resolution $f : \\\\tilde{X} \\\\to X$ which is an isomorphism over the smooth locus. Let $\\\\omega$ be an $(n-1)$-form on $X$. Does $f^\\\\ast \\\\omega\\\\vert_{X_{\\\\text{smooth}}}$ extend to an $(n-1)$-form on $\\\\tilde{X}$ with log poles along the exceptional locus?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is true in characteristic $0$ by work of Greb-Kebekus-Kov\\\\'acs and Greb-Kebekus-Kov\\\\'acs-Peternell. In fact it works for $k$-forms for any $k$, but the case $k = n-1$ should be the easiest.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0265", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the standard reflexive-form interpretation, the unrestricted positive-characteristic answer is no. Beyond Graf's known E8 surface counterexamples in characteristics 2, 3, and 5, for every prime p the report constructs a normal log canonical variety X of dimension N=2p(p-1), admitting an explicit log resolution, and a reflexive (N-1)-form whose pullback has the unavoidable local expression dt/t^2 wedge beta at the generic point of an exceptional divisor. The construction modifies the Kollár-Graf inseparable-root-cover cone so that its polarization is a p-th power; this makes the fiber dlog patch globally and raises Graf's cone obstruction by one degree.\n\nCandidate contribution (counterexample; novelty confidence low): For every prime p, a p-divisible-polarization variant of the Kollár-Graf cone construction gives a log canonical 2p(p-1)-fold with a reflexive (2p(p-1)-1)-form having an unavoidable double exceptional pole." }, { "id": 20000266, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0266", "title": "A Riemann--Roch obstruction to small very-general Seshadri constants", "statement": "Universal lower bounds for Seshadri constants\n\nSuppose that $X$ is a smooth variety over an uncountable field $k$ of positive characteristic. Does there exist a constant $c = c(n)$ such that $\\epsilon(L,x) \\geq c$ for any ample divisor $L$ and very general point $x$ of $X$?", "original_statement": "Universal lower bounds for Seshadri constants\n\nSuppose that $X$ is a smooth variety over an uncountable field $k$ of positive characteristic. Does there exist a constant $c = c(n)$ such that $\\epsilon(L,x) \\geq c$ for any ample divisor $L$ and very general point $x$ of $X$?", "clean_statement": "Universal lower bounds for Seshadri constants\n\nSuppose that $X$ is a smooth variety over an uncountable field $k$ of positive characteristic. Does there exist a constant $c = c(n)$ such that $\\epsilon(L,x) \\geq c$ for any ample divisor $L$ and very general point $x$ of $X$?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 5.2 in the “Other questions” section of the 2013 AIM workshop *The minimal model program in characteristic \\(p\\)*. The archived page attributes the question to Mustaţă and states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Other questions\nSource item: 5.2\nSource URL: http://aimpl.org/minimalmodcharp/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[265]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Universal lower bounds for Seshadri constants\\n\\nSuppose that $X$ is a smooth variety over an uncountable field $k$ of positive characteristic. Does there exist a constant $c = c(n)$ such that $\\\\epsilon(L,x) \\\\geq c$ for any ample divisor $L$ and very general point $x$ of $X$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The analogous result is true in characteristic $0$: we have $\\\\epsilon(L,x) \\\\geq 1$ if $n =2$ and $\\\\epsilon(L,x) \\\\geq 1/n$ if $n \\\\geq 3$. The proof in that case relies on generic smoothness.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0266", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth polarized surface, writing d=L^2, a=K_X.L, and chi=chi(O_X), if md>a and chi+(m^2d-ma)/2 is at least 2, then the very-general Seshadri constant of L is at least 1/m. Conversely, a value below 1/m forces h^0(mL) at most 1 and either a is at least md or -chi is at least (m^2d-ma)/2-1. This yields epsilon_vg(L) at least 1 for every ample L on a smooth globally F-regular surface. A separate product construction shows that arbitrarily many initial tensor powers may be sectionless even when epsilon(L,x)=1 everywhere. These are partial surface and strategy results; the unrestricted AIM question remains open.\n\nCandidate contribution (numerical_obstruction; novelty confidence low): If epsilon_vg(L)<1/m on a smooth polarized surface, then h^0(mL)<=1 and either K_X.L>=mL^2 or -chi(O_X)>=(m^2L^2-mK_X.L)/2-1; in particular every ample line bundle on a smooth globally F-regular surface has epsilon_vg at least 1." }, { "id": 20000267, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0267", "title": "A universal counterexample, semiampleness persistence, and degree escape", "statement": "Nefness under mod p reduction\n\nSuppose that $X$ is a variety over $k$, with $\\text{char } k = 0$, and $L$ is a nef divisor on $X$. Must $L_p$ be nef on $X_p$ for infinitely many $p$? What about in the case $L= K_X$? What if ``nef'' is replaced by ``semiample''?", "original_statement": "Nefness under mod p reduction\n\nSuppose that $X$ is a variety over $k$, with $\\text{char } k = 0$, and $L$ is a nef divisor on $X$. Must $L_p$ be nef on $X_p$ for infinitely many $p$? What about in the case $L= K_X$? What if ``nef'' is replaced by ``semiample''?", "clean_statement": "Nefness under mod p reduction\n\nSuppose that $X$ is a variety over $k$, with $\\text{char } k = 0$, and $L$ is a nef divisor on $X$. Must $L_p$ be nef on $X_p$ for infinitely many $p$? What about in the case $L= K_X$? What if ``nef'' is replaced by ``semiample''?", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 5.3, “Nefness under mod \\(p\\) reduction,” from the workshop *The minimal model program in characteristic \\(p\\)*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Other questions\nSource item: 5.3\nSource URL: http://aimpl.org/minimalmodcharp/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[266]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Nefness under mod p reduction\\n\\nSuppose that $X$ is a variety over $k$, with $\\\\text{char } k = 0$, and $L$ is a nef divisor on $X$. Must $L_p$ be nef on $X_p$ for infinitely many $p$? What about in the case $L= K_X$? What if ``nef'' is replaced by ``semiample''?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"It's not the case that $L_p$ is nef for almost all $p$; Shepherd-Barron gives an example where this fails on Shimura surfaces. If the abundance conjecture is true, then in the $L = K_X$ case semiampleness should hold for almost all $p$.\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0267", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "Langer's 2015 universal-Hodge-bundle construction gives a smooth projective fourfold with a nef characteristic-zero line bundle whose reduction is non-nef at every good prime, so the general infinitude question is false. In contrast, semiampleness spreads to all fibers after shrinking; hence the canonical case is cofinally positive in dimensions at most three and conditionally in all dimensions under abundance. This attempt also proves a bounded-degree escape lemma: in any projective family with generically nef line bundle, negative curves in bad fibers must escape every fixed polarization-degree bound near the generic point.\n\nCandidate contribution (lemma; novelty confidence low): For a projective family over an integral noetherian base with a generically nef line bundle and a fixed relative very ample line bundle H, for each B there is a dense open set on which no negative integral curve has H-degree at most B; over a one-dimensional arithmetic base, the minimum H-degree of a negative witness tends to infinity cofinally." }, { "id": 20000268, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0268", "title": "Witt-slope congruences, global F-regularity, and Fano cone obstructions", "statement": "Singularities and point-counting\n\nSuppose that $X$ is a variety over a finite field. Is there any relation between the singularities of $X$ and the number of points over $\\mathbb F_q$? For example, suppose that $X$ is Fano and $F$-regular.", "original_statement": "Singularities and point-counting\n\nSuppose that $X$ is a variety over a finite field. Is there any relation between the singularities of $X$ and the number of points over $\\mathbb F_q$? For example, suppose that $X$ is Fano and $F$-regular.", "clean_statement": "Singularities and point-counting\n\nSuppose that $X$ is a variety over a finite field. Is there any relation between the singularities of $X$ and the number of points over $\\mathbb F_q$? For example, suppose that $X$ is Fano and $F$-regular.", "statement_status": "exact", "statement_verification": "The exact AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: The minimal model program in characteristic $p$\nSection: Other questions\nSource item: 5.4\nSource URL: http://aimpl.org/minimalmodcharp/5/\nCanonical location: aim-algebraic-geometry-notes.json notes[267]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Singularities and point-counting\\n\\nSuppose that $X$ is a variety over a finite field. Is there any relation between the singularities of $X$ and the number of points over $\\\\mathbb F_q$? For example, suppose that $X$ is Fano and $F$-regular.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The motivation is a result of Esnault: suppose that $F$ is smooth, geometrically connected, and rationally chain connected. Then $\\\\# X(\\\\mathbb F_q) \\\\equiv 1 \\\\mod q$.\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/minimalmodcharp/5/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0268", "aim-domain:algebraic-geometry", "aim-workshop:minimalmodcharp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every proper, reduced, geometrically connected X/F_q with H^i(X,O_X)=0 for i>0, finite Witt-vector induction and the Berthelot--Bloch--Esnault slope-<1 trace criterion give #X(F_{q^r}) congruent to 1 modulo q^r for every r. Smith's vanishing theorem therefore proves the same in every dimension for projective globally F-regular varieties, without a Fano hypothesis; this is distinct from merely local strong F-regularity. An exact cone family shows the converse fails: the projective cone over a smooth degree-(m+1) hypersurface Y in P^m is normal Gorenstein Fano with a non-strongly-F-regular vertex and Z(X,t)=(1-t)^{-1}Z(Y,qt).\n\nCandidate contribution (explicit_family_and_obstruction; novelty confidence low): For a smooth geometrically integral degree-(m+1) hypersurface Y/P^m over F_q, the projective cone X is a normal Gorenstein Fano m-fold whose vertex is not strongly F-regular, while N_r(X)=1+q^r N_r(Y) and Z(X,t)=(1-t)^{-1}Z(Y,qt) for all r; for a plane cubic E this gives #X(F_Q)-#P^2(F_Q)=-Q a_E(Q), of order Q^(3/2) precisely along traces of order Q^(1/2)." }, { "id": 20000269, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0269", "title": "KL projection under Bayesian misspecification", "statement": "Let us assume that the true probability distribution $q$ of a data generating process is outside the family of models $\\mathcal{M}$ that are being considered. Under what conditions does the posterior distribution converge to a distribution with smallest KL divergence to the true distribution $q$?", "original_statement": "Let us assume that the true probability distribution $q$ of a data generating process is outside the family of models $\\mathcal{M}$ that are being considered. Under what conditions does the posterior distribution converge to a distribution with smallest KL divergence to the true distribution $q$?", "clean_statement": "Let us assume that the true probability distribution $q$ of a data generating process is outside the family of models $\\mathcal{M}$ that are being considered. Under what conditions does the posterior distribution converge to a distribution with smallest KL divergence to the true distribution $q$?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 11.1 in the section “When true distribution is outside the model” of the AIM workshop *Singular learning theory: connecting algebraic geometry and model selection in statistics*, held 12--16 December 2011. The official workshop report states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Singular learning theory\nSection: When true distribution is outside the model\nSource item: 11.1\nSource URL: http://aimpl.org/singularlearning/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[268]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let us assume that the true probability distribution $q$ of a data generating process is outside the family of models $\\\\mathcal{M}$ that are being considered. Under what conditions does the posterior distribution converge to a distribution with smallest KL divergence to the true distribution $q$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/singularlearning/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0269", "aim-domain:algebraic-geometry", "aim-workshop:singularlearning", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A compact dominated i.i.d. model with full prior support, finite continuous expected log likelihood, integrable true log density, and a uniform strong law has exponentially fast posterior concentration on every neighborhood of its KL-minimizer set, without identifiability or nonsingular Fisher information. On the sampling-law quotient of a finite model, nonoptimal law-classes have exact exponential elimination rates; duplicate parameters in the unique optimal law-fiber retain their prior-conditional weights, whereas two distinct tied optimal laws generate oscillating posterior odds and a nonconvergent predictive. In a symmetric Bernoulli example, predictive KL risk converges to the optimal model risk in probability even though its almost-sure liminf is zero and its almost-sure limsup is that positive optimal risk.\n\nCandidate contribution (finite_model_quotient_theorem_and_worked_example; novelty confidence low): In a finite nonidentifiable misspecified model, quotienting by identical sampling laws gives an exact trichotomy: nonoptimal law-fibers are exponentially eliminated, a unique optimal law-fiber preserves its prior-conditional parameter weights, and two distinct tied optimal law-fibers oscillate pathwise; moreover, for the tied Bernoulli(1/4) and Bernoulli(3/4) model under Bernoulli(1/2) truth, predictive KL risk converges in probability to (1/2) log(4/3) while almost surely its liminf is 0 and its limsup is (1/2) log(4/3)." }, { "id": 20000270, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0270", "title": "Misspecified Bayesian stochastic complexity and competing pseudo-truths", "statement": "Understand asymptotics of the stochastic complexity in the case where the true distribution is not in any model under consideration.", "original_statement": "Understand asymptotics of the stochastic complexity in the case where the true distribution is not in any model under consideration.", "clean_statement": "Understand asymptotics of the stochastic complexity in the case where the true distribution is not in any model under consideration.", "statement_status": "exact", "statement_verification": "The canonical record is source index 269 of `aim-algebraic-geometry-notes.json`, from the AIM workshop *Singular learning theory*, section “When true distribution is outside the model.” It states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Singular learning theory\nSection: When true distribution is outside the model\nSource item: 11.2\nSource URL: http://aimpl.org/singularlearning/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[269]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Understand asymptotics of the stochastic complexity in the case where the true distribution is not in any model under consideration.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/singularlearning/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0270", "aim-domain:algebraic-geometry", "aim-workshop:singularlearning", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under standard uniform stochastic-Laplace conditions for finitely many isolated regular KL minimizers, the marginal likelihood is the sum of their local Laplace integrals. If the tied optimal densities are pairwise distinguishable under the data-generating distribution, empirical losses select a random winning pseudo-truth at order sqrt(n), yielding (F_n-nL_*)/sqrt(n) converging to the minimum of a correlated centered Gaussian vector and, under uniform integrability, E[F_n]=nL_*+sqrt(n)E[min_j G_j]+o(sqrt(n)). An exact symmetric Bernoulli example gives the correction -(c/2)sqrt(2n/pi)+log 2+o(1). A conditional log-sum-exp reduction shows how the same competition precedes local RLCT and multiplicity penalties for finitely many singular optimal-density clusters.\n\nCandidate contribution (theorem; novelty confidence low): Candidate finite optimal-density competition principle: for finitely many regular KL-optimal density clusters, the free energy has an explicit log-sum-exp local-Laplace formula and a minimum-Gaussian sqrt(n) limit with expectation coefficient E[min_j G_j]; conditionally, the same log-sum-exp reduction orders distinct-density competition before RLCT and multiplicity terms." }, { "id": 20000271, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0271", "title": "Exact lognormal projections and regular misspecified asymptotics", "statement": "As a concrete example, suppose the data is generated from a log normal distribution, and suppose we try to fit two model classes -\n\n a) a normal distribution\n\n b) a gamma distribution\n\nUnderstand the behavior of asymptotics in this case.", "original_statement": "As a concrete example, suppose the data is generated from a log normal distribution, and suppose we try to fit two model classes -\n\n a) a normal distribution\n\n b) a gamma distribution\n\nUnderstand the behavior of asymptotics in this case.", "clean_statement": "As a concrete example, suppose the data is generated from a log normal distribution, and suppose we try to fit two model classes -\n\n a) a normal distribution\n\n b) a gamma distribution\n\nUnderstand the behavior of asymptotics in this case.", "statement_status": "exact", "statement_verification": "The canonical record is item 11.3 in the AIM workshop list *Singular learning theory*, under “When true distribution is outside the model.” Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Singular learning theory\nSection: When true distribution is outside the model\nSource item: 11.3\nSource URL: http://aimpl.org/singularlearning/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[270]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"As a concrete example, suppose the data is generated from a log normal distribution, and suppose we try to fit two model classes -\\n\\n a) a normal distribution\\n\\n b) a gamma distribution\\n\\nUnderstand the behavior of asymptotics in this case.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/singularlearning/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0271", "aim-domain:algebraic-geometry", "aim-workshop:singularlearning", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For LogNormal(mu,s) truth with s>0, the normal KL projection matches the mean and variance, while the unique gamma projection has theta=E[X]/k and log(k)-psi(k)=s/2. Their exact minimal risks satisfy Delta(s)=K_normal(s)-K_gamma(s)>0 for every s>0; more strongly, Delta(s)>one-half log(k(s)(exp(s)-1)) and Delta is strictly increasing, so there is no crossover and gamma always has lower forward-KL risk. The report proves small- and large-s expansions, gives exact sandwich matrices and an explicit likelihood-ratio variance, and shows that both models are regular of dimension two: each marginal likelihood has a log(n) penalty, these penalties cancel in the Bayes factor, and (log BF_gamma,normal-n Delta)/sqrt(n) converges to a centered normal law. The fluctuation variance is asymptotic to 2s/3 at zero and exp(4s)/4 at infinity.\n\nCandidate contribution (comparison_theorem_and_asymptotic_expansion; novelty confidence low): For the forward-KL projections of LogNormal(mu,s) onto the ordinary normal and gamma families, the risk gap Delta(s) is strictly positive and strictly increasing on s>0, obeys Delta(s)>one-half log(k(s)(exp(s)-1)), has Delta(s)=2s/3+s^2/36+s^3/810+O(s^4) at zero and Delta(s)=s-log(s)+one-half log(8 pi)-one-half+O(log(s)/s) at infinity, and the associated one-observation log-likelihood-ratio variance satisfies tau^2(s)=2s/3+O(s^2) and tau^2(s)=exp(4s)/4(1+O(exp(-s)))." }, { "id": 20000272, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0272", "title": "Exponential truth fitted by Gaussian linear regression", "statement": "As another example, let us assume the true model is a linear regression. $Y = exp(X) + \\epsilon$ where $\\epsilon \\sim N(\\mu,\\sigma^2)$. Let us assume that we try to fit a linear regression model to the data, i.e. $y=bx + \\epsilon$.", "original_statement": "As another example, let us assume the true model is a linear regression. $Y = exp(X) + \\epsilon$ where $\\epsilon \\sim N(\\mu,\\sigma^2)$. Let us assume that we try to fit a linear regression model to the data, i.e. $y=bx + \\epsilon$.", "clean_statement": "As another example, let us assume the true model is a linear regression. $Y = exp(X) + \\epsilon$ where $\\epsilon \\sim N(\\mu,\\sigma^2)$. Let us assume that we try to fit a linear regression model to the data, i.e. $y=bx + \\epsilon$.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 11.4 in the section “When true distribution is outside the model” of the AIM workshop *Singular learning theory: connecting algebraic geometry and model selection in statistics*. Its exact extracted text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Singular learning theory\nSection: When true distribution is outside the model\nSource item: 11.4\nSource URL: http://aimpl.org/singularlearning/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[271]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"As another example, let us assume the true model is a linear regression. $Y = exp(X) + \\\\epsilon$ where $\\\\epsilon \\\\sim N(\\\\mu,\\\\sigma^2)$. Let us assume that we try to fit a linear regression model to the data, i.e. $y=bx + \\\\epsilon$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/singularlearning/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0272", "aim-domain:algebraic-geometry", "aim-workshop:singularlearning", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After correcting the official source to a nonlinear true regression and restoring its missing stochastic-complexity question, the report separates the intercept, slope-only, fixed-variance, and estimated-variance readings. For a general design, the Gaussian intercept model has pseudo-slope Cov(X,exp(X))/Var(X), pseudo-intercept E exp(X)+mu minus that slope times E X, and residual approximation error A=Var(exp(X))-Cov(X,exp(X))^2/Var(X); its optimal KL is A/(2 sigma^2) for fixed variance and (1/2)log(1+A/sigma^2) for estimated variance. For X normal with mean m and variance tau^2, A=exp(2m+tau^2)[exp(tau^2)-1-tau^2]. The artifacts also derive the MLE sandwich covariance, ordinary posterior curvature, proper-prior Laplace expansion, and a triangular-design theorem: when tau_n=kappa n^{-1/4}, the known-variance conditional model has n times optimal KL converging to exp(2m)kappa^4/(4 sigma^2) and stochastic-complexity penalty (3/4)log n; estimating variance adds a regular (1/2)log n, for (5/4)log n total under the stated prior conditions.\n\nCandidate contribution (local_asymptotic_theorem; novelty confidence low): For the conditional known-variance intercept-plus-slope fit with triangular Gaussian design X_ni distributed as N(m,kappa^2 n^{-1/2}) and a fixed bounded proper prior positive and continuous at the limiting pseudo-truth, n times the optimal KL converges to exp(2m)kappa^4/(4 sigma^2), while the negative log marginal likelihood equals minus the maximized conditional log likelihood plus (3/4)log n plus an explicit prior-and-curvature constant and o_p(1).", "research_difficulty_suggested": "L4" }, { "id": 20000273, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0273", "title": "Fermat counterexamples and a one-step theorem for monomial thickenings", "statement": "Given an ideal $I$ of $\\Z[x]$, the HSL numbers are the ``indices of nilpotency\" of the action of Frobenius on $H^\\text{top}_{(\\underline{x})}\\left( \\Z_p[\\underline{x}]/I_p \\right)$, for primes $p$. Is the limsup of the HSL numbers 1?", "original_statement": "Given an ideal $I$ of $\\Z[x]$, the HSL numbers are the ``indices of nilpotency\" of the action of Frobenius on $H^\\text{top}_{(\\underline{x})}\\left( \\Z_p[\\underline{x}]/I_p \\right)$, for primes $p$. Is the limsup of the HSL numbers 1?", "clean_statement": "Given an ideal $I$ of $\\Z[x]$, the HSL numbers are the ``indices of nilpotency\" of the action of Frobenius on $H^\\text{top}_{(\\underline{x})}\\left( \\Z_p[\\underline{x}]/I_p \\right)$, for primes $p$. Is the limsup of the HSL numbers 1?", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.05\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[272]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given an ideal $I$ of $\\\\Z[x]$, the HSL numbers are the ``indices of nilpotency\\\" of the action of Frobenius on $H^\\\\text{top}_{(\\\\underline{x})}\\\\left( \\\\Z_p[\\\\underline{x}]/I_p \\\\right)$, for primes $p$. Is the limsup of the HSL numbers 1?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0273", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Mustaţă--Zhang explicitly refuted AIM Problem 1.05: for f=x^5+y^5+z^5, the positive-convention HSL number is 2 at primes p congruent to 2 or 3 modulo 5 and 1 otherwise, so its arithmetic limsup is 2. This attempt additionally proves a Frobenius-equivariant extension bound: if F^a kills H^i_m(N), then HSL_0(H^i_m(R)) is at most a+HSL_0(H^i_m(R/N)); when F(N)=0 and R/N is F-injective in degree i, the nilpotent part is exactly the kernel of H^i_m(R) to H^i_m(R/N) and is killed in one step. Consequently every proper monomial ideal over Z satisfies the AIM prediction for all sufficiently large primes.\n\nCandidate contribution (lemma; novelty confidence low): For a Noetherian local ring (R,m) of characteristic p and an ideal N, if the natural p-linear Frobenius kills H^i_m(N) after a steps, then HSL_0(H^i_m(R)) <= a+HSL_0(H^i_m(R/N)); in particular, for I in Z[x_1,...,x_n], J=sqrt(I), and J^t contained in I, every p>=t adds at most one HSL step whenever H^i_m(F_p[x]/J_p) is F-injective, yielding eventual positive-convention HSL number 1 for every proper monomial I." }, { "id": 20000274, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0274", "title": "Certified finite computation of F-pure thresholds", "statement": "Realize effective computations of numerical $F$-invariants.", "original_statement": "Realize effective computations of numerical $F$-invariants.", "clean_statement": "Realize effective computations of numerical $F$-invariants.", "statement_status": "exact", "statement_verification": "The canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.1\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[273]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Realize effective computations of numerical $F$-invariants.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0274", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a nonzero ideal I contained in the maximal ideal of an F-finite regular local ring, generated by any chosen s elements, one exact finite Frobenius value nu_e at q=p^e gives the proved interval nu_e/(q-1) <= fpt(I) <= (nu_e+s)/q. If the threshold is rational with a certified reduced-denominator bound B, q>sB^2 isolates it exactly among rationals of denominator at most B. The report also proves that no fixed truncation of the nu-sequence can recover all principal thresholds without structural input, and gives an unconditional primal-dual linear-program certificate plus a determinant denominator bound for monomial ideals.\n\nCandidate contribution (certification_theorem_and_obstruction; novelty confidence low): Candidate contribution: an integrated certification layer combining the s-generator finite-stage interval, bounded-denominator Farey stopping rule, a sharp finite-nu-data obstruction using (x^a) versus (x^(a+1)), and a monomial primal-dual LP certificate with an explicit minor denominator bound." }, { "id": 20000275, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0275", "title": "Newton nondegeneracy needs base-p arithmetic, with a disjoint-variable cap lemma", "statement": "Do Howald's nondegeneracy conditions with respect to the Newton polyhedron on the term ideal provide conditions under which $\\operatorname{fpt}(\\text{polynomial}) = \\operatorname{fpt}(\\text{associated term ideal})$?", "original_statement": "Do Howald's nondegeneracy conditions with respect to the Newton polyhedron on the term ideal provide conditions under which $\\operatorname{fpt}(\\text{polynomial}) = \\operatorname{fpt}(\\text{associated term ideal})$?", "clean_statement": "Do Howald's nondegeneracy conditions with respect to the Newton polyhedron on the term ideal provide conditions under which $\\operatorname{fpt}(\\text{polynomial}) = \\operatorname{fpt}(\\text{associated term ideal})$?", "statement_status": "exact", "statement_verification": "This is stored as Problem 11.15 in the canonical record from the AIM workshop *Relating test ideals and multiplier ideals*, in the section “Characteristic \\(p>0\\) invariants.” The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.15\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[274]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Do Howald's nondegeneracy conditions with respect to the Newton polyhedron on the term ideal provide conditions under which $\\\\operatorname{fpt}(\\\\text{polynomial}) = \\\\operatorname{fpt}(\\\\text{associated term ideal})$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0275", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Howald's Newton nondegeneracy condition alone does not force equality at a fixed prime: in characteristic 5, the Newton-nondegenerate cusp x^2+y^3 has F-pure threshold 4/5 while its term ideal has threshold 5/6. Hernández's 2016 splitting-polytope theorems give the published sharp supplement via no-carry, diagonal-position, and coefficient conditions. In addition, a proved disjoint-variable product formula shows that fpt(AB)=min(fpt(A),fpt(B)); consequently a common monomial factor can cap both polynomial and term-ideal thresholds and force equality, as for x^2(y^2+z^3), where both thresholds are 1/2 despite the residual carry defect and failure of diagonal position.\n\nCandidate contribution (lemma; novelty confidence low): Candidate monomial-cap principle: for disjoint variable blocks X and Y, nonzero ideals A contained in (X) and B contained in (Y) satisfy fpt(AB)=min(fpt(A),fpt(B)); hence, for a monomial u(X) and nonzero g(Y), a strict inequality fpt(g)0\\) invariants”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.2\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[275]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For a graded sequence of ideals $I_m$ of a ring $R$, does $\\\\lim \\\\limits_{m \\\\to \\\\infty} m \\\\cdot \\\\operatorname{fpt}( I_m) $ exist?\"\nOriginal remarks: [\"This is known for log canonical thresholds.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0276", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In an F-finite regular local ring of characteristic p, a Noetherian descending multiplicative filtration of nonzero proper ideals has lim_m m fpt(I_m) = d fpt(I_d) for every degree d whose Veronese Rees algebra is standard. The proof squeezes each I_m between two consecutive powers of I_d. For every graded family, a Frobenius crossing-index limit is also proved to equal sup_m m fpt(I_m) = limsup_m m fpt(I_m); descending filtrations satisfy a factor-p liminf bound, and arbitrary monomial graded families have a full extended limit via Newton-polyhedron subadditivity.\n\nCandidate contribution (theorem; novelty confidence low): If I_bullet is a Noetherian descending multiplicative filtration of nonzero proper ideals in an F-finite regular local ring and the d-th Veronese is standard, then lim_m m fpt(I_m) exists and equals d fpt(I_d)." }, { "id": 20000277, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0277", "title": "Existence is settled and F-thresholds are invariant under nilpotent quotients", "statement": "Investigate the existence and rationality of $F$-thresholds.", "original_statement": "Investigate the existence and rationality of $F$-thresholds.", "clean_statement": "Investigate the existence and rationality of $F$-thresholds.", "statement_status": "exact", "statement_verification": "The canonical record is `aim-algebraic-geometry-notes.json`, zero-based index 276, canonical number **11.25**:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.25\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[276]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate the existence and rationality of $F$-thresholds.\"\nOriginal remarks: [\"Existence is known for a ring that is F-pure on the punctured spectrum.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0277", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The existence of ordinary ideal-pair F-thresholds was proved in full generality in 2018, while rationality for arbitrary singular Noetherian rings remains open in the literature checked. This attempt proves a quantitative nilpotent-quotient comparison: if K is nilpotent, S=R/K, Q is a Frobenius power with K^[Q]=0, and a is generated by u elements, then nu_S(q) <= nu_R(q) and nu_R(qQ) <= Q(nu_S(q)+u)-1. Consequently c_R^J(a)=c_S^{JS}(aS), so the general rationality problem reduces to reduced rings. If S is F-pure, each finite stage gives a certified interval of width u/q containing the common threshold.\n\nCandidate contribution (reduction; novelty confidence low): For every nilpotent ideal K in a Noetherian characteristic-p ring, ordinary ideal-pair F-thresholds are unchanged by R -> R/K; more precisely, for Q=p^h with K^[Q]=0, nu_{aS}^{JS}(q) <= nu_a^J(q) and nu_a^J(qQ) <= Q(nu_{aS}^{JS}(q)+u)-1. When the reduced quotient is F-pure, this also yields a two-sided finite-stage interval of width u/q." }, { "id": 20000278, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0278", "title": "Irrational non-Q-Gorenstein F-jumps and boundary-certificate escape", "statement": "Investigate the discreteness and rationality of $F$-jumping numbers in the non-$\\Q$-Gorenstein case.", "original_statement": "Investigate the discreteness and rationality of $F$-jumping numbers in the non-$\\Q$-Gorenstein case.", "clean_statement": "Investigate the discreteness and rationality of $F$-jumping numbers in the non-$\\Q$-Gorenstein case.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.3\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[277]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate the discreteness and rationality of $F$-jumping numbers in the non-$\\\\Q$-Gorenstein case.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0278", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The intended invariant is the intrinsic big test ideal, equivalently the full Cartier algebra. A May 2026 first-version preprint of Rahul Ajit claims infinitely many non-Q-Gorenstein F-finite local domains with eventually irrational big-test-ideal jumping numbers, so universal rationality is now claimed false, while general discreteness remains open; discreteness and rationality hold under global finite generation of the anti-canonical algebra, and discreteness holds when failure of finite generation is isolated. Independently of the preprint's long construction, this attempt proves that at any irrational intrinsic jump under the standard finite-type hypotheses, finite log-Q-Gorenstein boundary witnesses along every left-approaching sequence must collectively use infinitely many distinct boundaries.\n\nCandidate contribution (lemma; novelty confidence low): Boundary-certificate escape: if lambda is an irrational jump of tau_b(R; a^t), then for every sequence t_j increasing to lambda and every choice of finite boundary witnesses B_j with tau_b(R; a^{t_j}) equal to the sum of tau(R, Delta; a^{t_j}) over Delta in B_j, the union of the B_j is infinite; moreover, every finite witness at lambda fails at all sufficiently close exponents to its left." }, { "id": 20000279, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0279", "title": "A fixed-base descending sequence from Monsky quartics", "statement": "Fix $d$. Does there exist a sequence of singular algebras $\\{ R_n \\}$ over a fixed finite field of characteristic $p>0$, and of dimension $d$, such that the $e_{HK}\\left(R_n\\right)$ descend to some number $\\alpha_d$ from above?", "original_statement": "Fix $d$. Does there exist a sequence of singular algebras $\\{ R_n \\}$ over a fixed finite field of characteristic $p>0$, and of dimension $d$, such that the $e_{HK}\\left(R_n\\right)$ descend to some number $\\alpha_d$ from above?", "clean_statement": "Fix $d$. Does there exist a sequence of singular algebras $\\{ R_n \\}$ over a fixed finite field of characteristic $p>0$, and of dimension $d$, such that the $e_{HK}\\left(R_n\\right)$ descend to some number $\\alpha_d$ from above?", "statement_status": "exact", "statement_verification": "The canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.35\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[278]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Fix $d$. Does there exist a sequence of singular algebras $\\\\{ R_n \\\\}$ over a fixed finite field of characteristic $p>0$, and of dimension $d$, such that the $e_{HK}\\\\left(R_n\\\\right)$ descend to some number $\\\\alpha_d$ from above?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0279", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every d at least 2, Monsky's characteristic-two quartics yield d-dimensional normal singular local domains essentially of finite type over the single fixed base field F_2 with Hilbert-Kunz multiplicities exactly 3+4^{-n}, strictly decreasing to 3. The finite residue fields are F_{2^n}, so this solves the literal fixed-base reading but not the stronger fixed-residue-field or standard-graded degree-zero-field reading. In dimension 1, e_HK equals the integral Hilbert-Samuel multiplicity, precluding infinite strict descent. An exact coefficient-restriction formula e_HK(k+m)=[K:k]e_HK(B) explains why the naive attempt to force residue field F_2 fails.\n\nCandidate contribution (construction_and_obstruction; novelty confidence low): Extracting the finite fields attached to Monsky parameters and viewing the resulting quartics as finite-type algebras over the fixed prime field F_2, then applying an exact polynomial-suspension length identity, produces a fixed-base sequence in every dimension d>=2; moreover, for finite coefficient fields k subset K the pullback A=k+m satisfies e_HK(A)=[K:k]e_HK(B), quantitatively obstructing the obvious fixed-residue-field descent." }, { "id": 20000280, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0280", "title": "An extremal bracket, a Segre correction, and a cubic-scroll family", "statement": "Fix $d$. What is the maximum value of the $F$-pure threshold of $\\mathfrak{m}$ of a non-$\\Q$-Gorenstein local ring $(R, \\mathfrak{m})$ of dimension $d$?", "original_statement": "Fix $d$. What is the maximum value of the $F$-pure threshold of $\\mathfrak{m}$ of a non-$\\Q$-Gorenstein local ring $(R, \\mathfrak{m})$ of dimension $d$?", "clean_statement": "Fix $d$. What is the maximum value of the $F$-pure threshold of $\\mathfrak{m}$ of a non-$\\Q$-Gorenstein local ring $(R, \\mathfrak{m})$ of dimension $d$?", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.4\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[279]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Fix $d$. What is the maximum value of the $F$-pure threshold of $\\\\mathfrak{m}$ of a non-$\\\\Q$-Gorenstein local ring $(R, \\\\mathfrak{m})$ of dimension $d$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Such a maximum value is known to be $d-1$ for non-regular $\\\\mathbb{Q}$-Gorenstein rings.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0280", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In the intended class of d-dimensional F-finite, F-pure normal local rings with infinite residue field, every non-Q-Gorenstein ring has maximal-ideal F-pure threshold at most d-1. For every d at least 3, an explicit non-Q-Gorenstein strongly F-regular toric local ring has b_e=floor(3(p^e-1)/2)+(d-3)(p^e-1), hence threshold d-3/2. Thus d-3/2 <= M_d <= d-1 for the supremum M_d. The old Segre example does not attain d-1: its F-pure threshold is 2, while d-1 is its diagonal F-threshold.\n\nCandidate contribution (explicit_family; novelty confidence low): For every d >= 3, the local polynomial extension of the cubic-scroll cone is non-Q-Gorenstein and satisfies b_e=floor(3(p^e-1)/2)+(d-3)(p^e-1) for every e, yielding the explicit lower bound M_d >= d-3/2 after correcting the invariant used in the old Segre example." }, { "id": 20000281, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0281", "title": "Two-minor complementarity and rank-stratum reduction for determinantal F-invariants", "statement": "Given a determinantal ring $R$, compute $e_{HK}(R)$ and $s(R)$.", "original_statement": "Given a determinantal ring $R$, compute $e_{HK}(R)$ and $s(R)$.", "clean_statement": "Given a determinantal ring $R$, compute $e_{HK}(R)$ and $s(R)$.", "statement_status": "exact", "statement_verification": "The canonical record is numbered `11.45` and says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.45\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[280]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given a determinantal ring $R$, compute $e_{HK}(R)$ and $s(R)$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The 2 $\\\\times$ 2 minors of a generic matrix is done by Watanabe and Yoshida; try 3 $\\\\times$ 3 minors and higher.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0281", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the homogeneous local ring of the generic M by N two-minor variety, with 2 <= M <= N and d=M+N-1, the report proves the symmetric formulas s=E(d,M)/d! and e_HK=[M!{d\\brace M}+N!{d\\brace N}-E(d,M)]/d!, hence the complementary identity e_HK+s=[M!{d\\brace M}+N!{d\\brace N}]/d!. It also proves that the completed local ring at a rank-r point of V(I_t) is a regular formal-power-series extension of the vertex ring for I_{t-r} in an (m-r) by (n-r) matrix. Consequently both invariants are explicit on the rank-(t-1) and rank-(t-2) strata of every generic determinantal variety, and all nonvertex points of every I_3 variety are computed. No value is claimed at an I_t vertex for t >= 3.\n\nCandidate contribution (identity_and_reduction; novelty confidence low): Candidate novel synthesis: the overlap in the finite-q Hilbert-Kunz inclusion-exclusion for a two-minor Segre ring is exactly Singh's splitting count, giving e_HK+s=[M!{d\\brace M}+N!{d\\brace N}]/d!, and the completed-local Schur-complement normal form transports this identity to every rank-(t-2) stratum." }, { "id": 20000282, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0282", "title": "Upper semicontinuity, uniform Frobenius bounds, and a reduced non-equidimensional counterexample", "statement": "Investigate the upper semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto e_{HK}\\left(R_\\mathfrak{p} \\right)$.", "original_statement": "Investigate the upper semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto e_{HK}\\left(R_\\mathfrak{p} \\right)$.", "clean_statement": "Investigate the upper semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto e_{HK}\\left(R_\\mathfrak{p} \\right)$.", "statement_status": "exact", "statement_verification": "The canonical record is from the AIM workshop problem list *Relating test ideals and multiplier ideals*, section “Characteristic \\(p>0\\) invariants.” It reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.5\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[281]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate the upper semicontinuity of the map $\\\\mathfrak{p} \\\\in \\\\operatorname{Spec} R \\\\mapsto e_{HK}\\\\left(R_\\\\mathfrak{p} \\\\right)$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0282", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Smirnov proves upper semicontinuity for locally equidimensional F-finite rings and rings essentially of finite type over an excellent local ring, while Lyu's 2025 preprint extends this to catenary locally equidimensional Noetherian F_p-algebras satisfying an excellence-type J-2 and completed-height-one (R_0) condition, hence to locally equidimensional excellent F_p-algebras without F-finiteness. The literal hypothesis-free statement is false: for R=F_p[x,y,t,u]/(tu,txy) and P=(x,y,u) contained in Q=(x,y,t,u), exact counts give lambda_e(P)=2-q^{-1} and lambda_e(Q)=1+2q^{-1}-3q^{-2}+q^{-3}, so e_HK(R_P)=2>1=e_HK(R_Q). The ring is reduced, F-finite, excellent, and finite type, but not locally equidimensional at Q.\n\nCandidate contribution (counterexample; novelty confidence low): A component-padding proposition for R=k[t,z_1,...,z_n]/tI proves exact finite-Frobenius length formulas at a generalization-specialization pair; its explicit reduced instance R=F_p[x,y,t,u]/(tu,txy) has finite-level gap lambda_e(P)-lambda_e(Q)=(1-q^{-1})^3 and violates upper semicontinuity." }, { "id": 20000283, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0283", "title": "Lower semicontinuity and finite-Frobenius open certificates", "statement": "Investigate the lower semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto s\\left(R_\\mathfrak{p}\\right)$.", "original_statement": "Investigate the lower semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto s\\left(R_\\mathfrak{p}\\right)$.", "clean_statement": "Investigate the lower semicontinuity of the map $\\mathfrak{p} \\in \\operatorname{Spec} R \\mapsto s\\left(R_\\mathfrak{p}\\right)$.", "statement_status": "exact", "statement_verification": "The canonical record, under the section “Characteristic \\(p>0\\) invariants,” asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.55\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[282]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate the lower semicontinuity of the map $\\\\mathfrak{p} \\\\in \\\\operatorname{Spec} R \\\\mapsto s\\\\left(R_\\\\mathfrak{p}\\\\right)$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0283", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Polstra proved that the F-signature function is Zariski lower semicontinuous for F-finite rings and rings essentially of finite type over an excellent local ring, with a uniform error |s_e(R_P)-s(R_P)| <= C p^(-e). Lyu's 2025 v4 preprint extends this to Noetherian F_p-algebras satisfying Condition 3.5.2, including quasi-excellent rings. From the uniform estimate and finite-level lower semicontinuity at positive-signature points, this attempt proves that for every lambda >= 0 the sets U_{e,lambda}={P: s_e(R_P)>lambda+C p^(-e)} are open and exactly exhaust {P: s(R_P)>lambda}; their cumulative finite unions form a nested open exhaustion with the explicit margin criterion 2C p^(-e)= 0, finite normalized splitting numbers with the uniform error correction C p^(-e) give certified open sets U_{e,lambda}, with {s>lambda}=union_e U_{e,lambda}; the cumulative V_{E,lambda} are nested, and every point with margin delta is certified at all levels satisfying 2C p^(-e)0\\) invariants” of the AIM workshop list *Relating test ideals and multiplier ideals*. It reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.6\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[283]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If $a_e$ denote of $F$-splitting numbers of a strongly $F$-regular local ring $R$ (i.e., $R^{1/p^e} \\\\cong R^{a_e} \\\\oplus M$ as $R$-modules, and $M$ contains no free $R$-module summands), then is $a_e = s(R) p^{ed} + C p^{e(d-1)} + O \\\\left(p^{e(d-2)}\\\\right) $ for some constant $C$?\"\nOriginal remarks: [\"In the Gorenstein case, it is known and C=0.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0284", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general second-coefficient problem remains open, but for the local cone over the cubic scroll S(1,2) and every polynomial toric suspension R_d of dimension d at least 3, the splitting numbers are exactly a_e(R_d)=(5/12)q^d+(1/8)q^(d-1)+(1/12)q^(d-2)+(3/16)(1-(-1)^q)q^(d-3), where q=p^e. These rings are F-finite, normal, strongly F-regular, and non-Q-Gorenstein. Hence s(R_d)=5/12 and the requested constant exists with the nonzero value C=1/8.\n\nCandidate contribution (explicit_family; novelty confidence low): The exact finite-level splitting formula for the local cubic-scroll cone and all polynomial toric suspensions gives a testable non-Q-Gorenstein strongly F-regular family in every dimension d at least 3 with constant second coefficient C=1/8." }, { "id": 20000285, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0285", "title": "Corrected normalization and an exact non-Q-Gorenstein determinantal limit", "statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$ and dimension $d$, let $\\mu_e$ denote the minimal number of generators of $\\operatorname{Hom}_R (R^{1/p^e}, R)$ as an $R$-module. Does $\\lim \\limits_{e \\to \\infty} \\frac{\\mu_e}{p^{ed}}$ exist? If so, what is it?", "original_statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$ and dimension $d$, let $\\mu_e$ denote the minimal number of generators of $\\operatorname{Hom}_R (R^{1/p^e}, R)$ as an $R$-module. Does $\\lim \\limits_{e \\to \\infty} \\frac{\\mu_e}{p^{ed}}$ exist? If so, what is it?", "clean_statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$ and dimension $d$, let $\\mu_e$ denote the minimal number of generators of $\\operatorname{Hom}_R (R^{1/p^e}, R)$ as an $R$-module. Does $\\lim \\limits_{e \\to \\infty} \\frac{\\mu_e}{p^{ed}}$ exist? If so, what is it?", "statement_status": "exact", "statement_verification": "The canonical record is indexed as problem 11.65 in `aim-algebraic-geometry-notes.json`. The archived AIM page displays it as Problem 1.65, attributed to Tucker, in the section “Characteristic \\(p>0\\) invariants.” Its text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.65\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[284]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For $(R, \\\\mathfrak{m})$ a local, normal domain of characteristic $p>0$ and dimension $d$, let $\\\\mu_e$ denote the minimal number of generators of $\\\\operatorname{Hom}_R (R^{1/p^e}, R)$ as an $R$-module. Does $\\\\lim \\\\limits_{e \\\\to \\\\infty} \\\\frac{\\\\mu_e}{p^{ed}}$ exist? If so, what is it?\"\nOriginal remarks: [\"In the $\\\\Q$-Gorenstein case, it exists and equals $e_{HK}{R}$.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0285", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM normalization is false under its literal hypotheses: for R=F_p(u)[[x_1,...,x_d]], one has mu_e=p^{e(d+1)}, so mu_e/p^{ed} diverges. For an F-finite ring the rank-normalized denominator is p^{e(d+alpha(R))}; with this correction the limit equals Hilbert-Kunz multiplicity for Gorenstein and Q-Gorenstein rings. For the non-Q-Gorenstein local 2 by 3 rank-one determinantal cone over a perfect field, an exact monomial count gives mu_e=(13q^4-5q^3-q^2-q)/6 and hence limit 13/6, while e_HK(R)=13/8. Under finite F-representation type and Krull-Schmidt, the corrected limit also exists and is an explicit weighted sum of generator numbers of dual indecomposable modules.\n\nCandidate contribution (explicit_computation; novelty confidence low): For the localization at the vertex of k[x_1,x_2] # k[y_1,y_2,y_3], the ordinary target-module generator number of Hom_R(F_*^e R,R) is exactly (13q^4-5q^3-q^2-q)/6 for every q=p^e; its normalized limit is 13/6, strictly different from e_HK(R)=13/8." }, { "id": 20000286, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0286", "title": "Exact Frobenius block formula for a radical extension", "statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$, let $x$ be a minimal generator of $\\mathfrak{m}$, and let $v$ denote a $t^\\text{th}$ root of $x$. Relate $e_{HK}(R)$ and $e_{HK}\\left(R[v]\\right)$.", "original_statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$, let $x$ be a minimal generator of $\\mathfrak{m}$, and let $v$ denote a $t^\\text{th}$ root of $x$. Relate $e_{HK}(R)$ and $e_{HK}\\left(R[v]\\right)$.", "clean_statement": "For $(R, \\mathfrak{m})$ a local, normal domain of characteristic $p>0$, let $x$ be a minimal generator of $\\mathfrak{m}$, and let $v$ denote a $t^\\text{th}$ root of $x$. Relate $e_{HK}(R)$ and $e_{HK}\\left(R[v]\\right)$.", "statement_status": "exact", "statement_verification": "The canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Characteristic $p>0$ invariants\nSource item: 11.7\nSource URL: http://aimpl.org/testandmultiplierideals/1/\nCanonical location: aim-algebraic-geometry-notes.json notes[285]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For $(R, \\\\mathfrak{m})$ a local, normal domain of characteristic $p>0$, let $x$ be a minimal generator of $\\\\mathfrak{m}$, and let $v$ denote a $t^\\\\text{th}$ root of $x$. Relate $e_{HK}(R)$ and $e_{HK}\\\\left(R[v]\\\\right)$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/1/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0286", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For S=R[V]/(V^t-x), where R is a normal local domain of characteristic p>0 and x is a minimal generator of its maximal ideal, S is a finite free local domain of rank t with maximal ideal n=(m,v). If q=p^e=at+r with 0<=r5, the two ideals are m^2 and m.\n\nCandidate contribution (synthesis_corollary; novelty confidence low): For an F-finite algebraic curve over an algebraically closed field, the normalization-transferred Cartier test ideal tau_MJ^nu equals ordinary big tau if and only if the curve is locally a complete intersection; on the non-lci curve k[t^3,t^4,t^5] localized at the vertex, tau_MJ^nu=m^2 whereas tau_b=m." }, { "id": 20000290, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0290", "title": "Frobenius-stable lifting and a sharp curve threshold", "statement": "Investigate lifting sections (for cohomology) using test ideals.", "original_statement": "Investigate lifting sections (for cohomology) using test ideals.", "clean_statement": "Investigate lifting sections (for cohomology) using test ideals.", "statement_status": "exact", "statement_verification": "The canonical record is source index 289 of `aim-algebraic-geometry-notes.json`. Its complete mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.2\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[289]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate lifting sections (for cohomology) using test ideals.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0290", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source's one-line request is best matched by established Frobenius-stable and alteration-stable lifting theorems, rather than by unrestricted ordinary restriction of sections. Independently, this attempt proves a sharp ordinary lifting theorem on smooth projective curves: if a nonzero generalized test ideal is J = O_C(-T), D is any nonzero effective Cartier divisor, and deg L > 2g-2+deg T+deg D, then H^0(C,J tensor L) surjects onto H^0(D,(J tensor L)|_D); at equality a one-dimensional obstruction occurs for L = omega_C(D+T). It also gives the exact cohomological obstruction in every dimension and an explicit P^1 counterexample showing that ambient adjoint ampleness alone does not imply lifting.\n\nCandidate contribution (sharp_bound; novelty confidence low): For a smooth projective connected curve C, every nonzero generalized test ideal J = O_C(-T), every nonzero effective Cartier divisor D (including a nonreduced divisor), and every line bundle L, the strict numerical bound deg L > 2g-2+deg T+deg D guarantees ordinary restriction surjectivity; the bound is uniformly sharp because L = omega_C(D+T) at equality has a one-dimensional cokernel." }, { "id": 20000291, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0291", "title": "Mixed test ideals need not have polyhedral constancy regions", "statement": "Given ideals $I_1, \\ldots, I_n$ of a regular, $F$-finite ring $R$, consider the map sending an $r$-tuple of nonnegative real numbers $(\\lambda_1, \\ldots, \\lambda_r)$ to $\\tau\\left( {I_1}^{\\lambda_1} \\cdot \\ldots \\cdot{I_n}^{\\lambda_n}\\right)$, the mixed test ideal. If we bound all the $\\lambda_i$ by some fixed $M$, does there exist a rational polyhedral decomposition such that the function described is constant on the interior of each region?", "original_statement": "Given ideals $I_1, \\ldots, I_n$ of a regular, $F$-finite ring $R$, consider the map sending an $r$-tuple of nonnegative real numbers $(\\lambda_1, \\ldots, \\lambda_r)$ to $\\tau\\left( {I_1}^{\\lambda_1} \\cdot \\ldots \\cdot{I_n}^{\\lambda_n}\\right)$, the mixed test ideal. If we bound all the $\\lambda_i$ by some fixed $M$, does there exist a rational polyhedral decomposition such that the function described is constant on the interior of each region?", "clean_statement": "Given ideals $I_1, \\ldots, I_n$ of a regular, $F$-finite ring $R$, consider the map sending an $r$-tuple of nonnegative real numbers $(\\lambda_1, \\ldots, \\lambda_r)$ to $\\tau\\left( {I_1}^{\\lambda_1} \\cdot \\ldots \\cdot{I_n}^{\\lambda_n}\\right)$, the mixed test ideal. If we bound all the $\\lambda_i$ by some fixed $M$, does there exist a rational polyhedral decomposition such that the function described is constant on the interior of each region?", "statement_status": "exact", "statement_verification": "This is Problem 2.25 (attributed to Mustaţă) in the AIM problem list *Relating test ideals and multiplier ideals*, section “Other and Related Problems.” The canonical repository record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.25\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[290]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given ideals $I_1, \\\\ldots, I_n$ of a regular, $F$-finite ring $R$, consider the map sending an $r$-tuple of nonnegative real numbers $(\\\\lambda_1, \\\\ldots, \\\\lambda_r)$ to $\\\\tau\\\\left( {I_1}^{\\\\lambda_1} \\\\cdot \\\\ldots \\\\cdot{I_n}^{\\\\lambda_n}\\\\right)$, the mixed test ideal. If we bound all the $\\\\lambda_i$ by some fixed $M$, does there exist a rational polyhedral decomposition such that the function described is constant on the interior of each region?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0291", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the standard finite-cell interpretation, the answer is no: for R = F_3[x,y], I_1 = (x+y), and I_2 = (xy), the points (1/3^e, 1-1/3^e) and (2/3^e, 1-2/3^e) lie on one segment, converge interleavingly to (0,1), and have mixed test ideals (x,y) and R respectively. A finite, or even locally finite, polyhedral complex has a finite trace on that compact segment, giving a contradiction. The report also proves an explicit positive theorem for monomial ideals using a common normal fan and a finite uniform box of possible minimal-generator exponents.\n\nCandidate contribution (explicit_monomial_wall_arrangement; novelty confidence low): For nonzero monomial ideals in a polynomial ring over a finite field and rational M, every minimal generator of every mixed test ideal in [0,M]^n lies in the explicit finite exponent box U_M defined from the primitive rays and support numbers of a common lower normal fan; consequently the finite rational wall arrangement sum_i h_i(v) lambda_i = , with v in the ray set and u in U_M, is a valid constancy decomposition and has at most 3^(N_M+2n) relative cells." }, { "id": 20000292, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0292", "title": "An explicit one-root certificate for generalized test ideals", "statement": "Realize effective computations of test ideals.", "original_statement": "Realize effective computations of test ideals.", "clean_statement": "Realize effective computations of test ideals.", "statement_status": "exact", "statement_verification": "The canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.3\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[291]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Realize effective computations of test ideals.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0292", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a nonzero proper ideal a in an effective F-finite polynomial ring k[x_1,...,x_n], generated by m polynomials of degree at most d, and a rational exponent t=A/B, define e_0 by p^{e_0}>md, N=binom(md+n,n), and Q=p^{e_0+N}(p^N-1). If q=p^E>mBQ and r=floor(tq)+1, then the generalized test ideal satisfies tau(a^t)=(a^r)^[1/q]. The proof is unconditional, uses the global BMS denominator bound and a generator-pigeonhole containment, treats t=0 separately, and gives a terminating but generally enormous one-Frobenius-root computation. A one-variable family also proves that any universal stage bound must depend on denominator data.\n\nCandidate contribution (effective_bound; novelty confidence low): The explicit certificate p^E > mB p^{e_0+N}(p^N-1) implies tau(a^{A/B})=(a^{floor((A/B)p^E)+1})^[1/p^E], and denominator dependence is necessary because for a=(x) and t_N=1-p^{-N} the first correct standard BMS stage is exactly N." }, { "id": 20000293, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0293", "title": "The direct summand theorem and trace-free monic towers", "statement": "Direct summand conjecture\n\nProve/disprove the Direct Summand Conjecture.", "original_statement": "Direct summand conjecture\n\nProve/disprove the Direct Summand Conjecture.", "clean_statement": "Direct summand conjecture\n\nProve/disprove the Direct Summand Conjecture.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.35\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[292]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Direct summand conjecture\\n\\nProve/disprove the Direct Summand Conjecture.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0293", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM conjecture itself is solved affirmatively: Andre proved that every finite inclusion R -> S of Noetherian rings splits R-linearly when R is regular, while Bhatt gave another proof and separately proved the stronger proper-surjective derived splitting theorem. As a rigorous partial contribution rather than a new proof of that general theorem, this attempt proves that every ordered iterated monic tower over an arbitrary commutative ring is finite free and has a constant-coefficient splitting compatible with arbitrary base change; for a radical tower of total degree N, its algebra trace is exactly N times that splitting and has image NR. It also proves a one-relation certificate for failure of this splitting to descend through a quotient and applies it to a standard characteristic-p non-splinter family.\n\nCandidate contribution (explicit_splitting_and_obstruction; novelty confidence low): For an ordered iterated monic tower over any commutative ring, constant-coefficient projection is an arbitrary-base-change-compatible module retraction; for radical equations v_i^{t_i}=a_i, the full trace is (product t_i) times this projection, so its image is exactly (product t_i)R, and a quotient relation whose constant coefficient is outside the ideal generated by its nonconstant coefficients certifies nonsplitting." }, { "id": 20000294, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0294", "title": "Mixed-characteristic tight closure as a plural theory, with an exact DVR comparison", "statement": "Extend tight closure to mixed characteristic.", "original_statement": "Extend tight closure to mixed characteristic.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is from the AIM workshop list **Relating test ideals and multiplier ideals**, section **Other and Related Problems**:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.4\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[293]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Extend tight closure to mixed characteristic.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0294", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The historical request is not a uniquely formalized conjecture: solid, plus, full extended plus, weak epf, big Cohen-Macaulay, perfectoid, and modern test-ideal theories realize different parts of the tight-closure package. A proved dimension-one comparison theorem shows that if (V,(pi)) is a mixed-characteristic DVR and B is a unital V-algebra that is pi-torsion-free with B/pi B nonzero, then B is faithfully flat and its algebra closure is the identity for every inclusion of arbitrary V-modules. This covers integral domain extensions, directed unions such as V+, their adic completions, and balanced/perfectoid BCM algebras; finite domain extensions admit a V-linear coefficient retraction, with normalized trace explicit when the generic degree is a unit. A separate valuation proof shows every DVR ideal is full-extended-plus closed.\n\nCandidate contribution (comparison theorem; novelty confidence low): Over a mixed-characteristic DVR, every unital algebra B with actual pi-torsion-freeness and nonzero closed fiber induces the identity closure on all module inclusions; consequently plus, completed-plus, and all balanced BCM-algebra closures coincide with the identity, while full extended plus closure is also the identity on ideals by an independent quantified valuation argument." }, { "id": 20000295, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0295", "title": "The solved F-pure multiplicity bound and a squarefree tangent-cone deficit theorem", "statement": "Suppose that $R$ is a local, equidimensional, $F$-pure ring of dimension $d$ with embedding dimension $n$. Is the multiplicity of $R$ at most $\\binom{n}{d}$?", "original_statement": "Suppose that $R$ is a local, equidimensional, $F$-pure ring of dimension $d$ with embedding dimension $n$. Is the multiplicity of $R$ at most $\\binom{n}{d}$?", "clean_statement": "Suppose that $R$ is a local, equidimensional, $F$-pure ring of dimension $d$ with embedding dimension $n$. Is the multiplicity of $R$ at most $\\binom{n}{d}$?", "statement_status": "exact", "statement_verification": "The exact canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.45\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[294]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Suppose that $R$ is a local, equidimensional, $F$-pure ring of dimension $d$ with embedding dimension $n$. Is the multiplicity of $R$ at most $\\\\binom{n}{d}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Answered positively by Huneke-Watanabe at the AIM workshop.\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0295", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Huneke and Watanabe proved the AIM question affirmatively in the stronger form e(R) <= binom(n,d) for every Noetherian F-pure local ring of characteristic p, without equidimensionality, F-finiteness, residue-field perfection, or completeness. This attempt also proves a characteristic-free special-class refinement: if gr_m(R) is the Stanley-Reisner ring of a (not necessarily pure) (d-1)-dimensional complex on n genuine vertices, then e(R) is its number of d-faces; equality uniquely gives the full (d-1)-skeleton, and the normalized density of missing j-faces is at most the normalized multiplicity deficit for every 1 <= j <= d.\n\nCandidate contribution (quantitative_refinement; novelty confidence low): For a local ring whose tangent cone is k[Delta], put delta = binom(n,d) - e(R) and m_j = binom(n,j) - f_{j-1}(Delta). Then m_j binom(n-j,d-j) <= delta binom(d,j), equivalently m_j/binom(n,j) <= delta/binom(n,d); moreover delta < binom(n-j,d-j) forces the complete (j-1)-skeleton." }, { "id": 20000296, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0296", "title": "A literal obstruction and an exact p-rank criterion for non-nilpotent Frobenius", "statement": "Given a smooth, projective variety $X$ over a number field, does there exist a dense set of primes for which the action of Frobenius on the coherent cohomology $H^i(X_p, \\mathcal{O}_{X_p})$ is not nilpotent?", "original_statement": "Given a smooth, projective variety $X$ over a number field, does there exist a dense set of primes for which the action of Frobenius on the coherent cohomology $H^i(X_p, \\mathcal{O}_{X_p})$ is not nilpotent?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is from the AIM workshop *Relating test ideals and multiplier ideals*, section “Other and Related Problems,” with source URL . The live page labels the item “Problem 2.5” and attributes it to Lyubeznik. The canonical number `22.5` is therefore a numbering/extraction artifact. The mathematical text on the live page agrees with the record:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.5\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[295]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given a smooth, projective variety $X$ over a number field, does there exist a dense set of primes for which the action of Frobenius on the coherent cohomology $H^i(X_p, \\\\mathcal{O}_{X_p})$ is not nilpotent?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0296", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The unrestricted AIM question is false: if H^i(X,O_X)=0, then cohomology and base change makes H^i(X_v,O_{X_v}) zero at almost every good place, so projective space in any positive degree is a counterexample. For the meaningful nonvanishing version, a proved abelian criterion says Frobenius on H^i(A,O_A) is non-nilpotent exactly when i is at most the p-rank. Applied to X=(y^2=x^3-x)^g, the same primes p congruent to 1 modulo 4 work in every degree 1 through g, with density 1/2 over Q; after base change to Q(i), the corresponding prime-ideal set has density 1 because the still-supersingular inert primes have norm p^2 and density zero.\n\nCandidate contribution (explicit theorem and reduction; novelty confidence low): For every g at least 1, the simultaneous non-nilpotence set in all positive coherent degrees of (y^2=x^3-x)^g is exactly p congruent to 1 modulo 4 and has density 1/2 over Q, while after base change to Q(i) the unchanged local property holds on a density-one set of prime ideals; this follows from the exact cutoff Frobenius non-nilpotent on H^i(A,O_A) if and only if i is at most the p-rank." }, { "id": 20000297, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0297", "title": "Determinantal FFRT off the rank-(t-3) stratum", "statement": "Do determinantal rings have finite $F$-representation type?", "original_statement": "Do determinantal rings have finite $F$-representation type?", "clean_statement": "Do determinantal rings have finite $F$-representation type?", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ALGEBRAIC_GEOMETRY-0297, source file `aim-algebraic-geometry-notes.json`, zero-based index 296. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.55\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[296]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Do determinantal rings have finite $F$-representation type?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The 2 $\\\\times$ 2 minors of a generic matrix has been done by Smith and Van den Bergh; try 3 $\\\\times$ 3 minors and higher.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0297", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let k be an F-finite field of characteristic p>0 and R=k[X]/I_t(X) the generic m-by-n determinantal ring, with 3<=t<=min(m,n). If a prime of R does not contain J=(I_{t-2}(X)+I_t(X))/I_t(X), then its local ring has ordinary finite F-representation type, and so does the completed local ring. Hence every possible local failure of FFRT is confined to the rank-at-most-(t-3) stratum. The proof uses an arbitrary-minor Schur-complement chart reducing to a rectangular 2-minor ring, plus explicit FFRT preservation under polynomial extension, localization, and F-finite local completion. This does not settle global graded FFRT for t>=3.\n\nCandidate contribution (theorem; novelty confidence low): For every generic t-minor ring over an F-finite field, the set of primes whose local rings can fail FFRT is contained in V((I_{t-2}(X)+I_t(X))/I_t(X)), and the same containment holds for failure after local completion." }, { "id": 20000298, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0298", "title": "Exact FFRT classification of the (2,3,7) hypersurface and an all-iterate Fedder obstruction", "statement": "For $p \\geq 11$, does $\\mathbb{F}_p[x,y,z]/(x^2+y^3+z^7)$ have finite $F$-representation type?", "original_statement": "For $p \\geq 11$, does $\\mathbb{F}_p[x,y,z]/(x^2+y^3+z^7)$ have finite $F$-representation type?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.6\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[297]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For $p \\\\geq 11$, does $\\\\mathbb{F}_p[x,y,z]/(x^2+y^3+z^7)$ have finite $F$-representation type?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0298", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Hara and Ohkawa's orbifold-curve theorem, transferred explicitly from the algebraic closure back to the exact AIM field, proves that F_p[x,y,z]/(x^2+y^3+z^7) does not have finite F-representation type for every prime p outside {2,3,7}; Shibuta proves FFRT in precisely those three exceptional characteristics. Graded, ungraded, homogeneous-local, and completed FFRT therefore give the same classification, so every p at least 11 has a negative answer and there is no finer residue-class split. In addition, an elementary reciprocal-exponent argument proves that every Frobenius iterate has free rank zero in every characteristic, showing sharply that the exceptional FFRT cases are nevertheless not F-pure.\n\nCandidate contribution (lemma; novelty confidence low): For every F-finite field and every diagonal hypersurface sum_i u_i x_i^(a_i) with sum_i 1/a_i < 1, one has f^(p^e-1) in (x_1^(p^e),...,x_n^(p^e)) for every e at least 1; consequently no Frobenius pushforward of the local hypersurface has a free summand. Applied to exponents (2,3,7), this gives zero free rank at every iterate in all characteristics, including the three FFRT exceptions." }, { "id": 20000299, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0299", "title": "A boundary-saturation obstruction to deforming F-injectivity", "statement": "Suppose that $R$ is a local ring and $f \\in R$ is a regular element such that $R/(f)$ is $F$-injective. Does this imply that $R$ is $F$-injective?", "original_statement": "Suppose that $R$ is a local ring and $f \\in R$ is a regular element such that $R/(f)$ is $F$-injective. Does this imply that $R$ is $F$-injective?", "clean_statement": "Suppose that $R$ is a local ring and $f \\in R$ is a regular element such that $R/(f)$ is $F$-injective. Does this imply that $R$ is $F$-injective?", "statement_status": "exact", "statement_verification": "The canonical AIM record (source file `aim-algebraic-geometry-notes.json`, zero-based index 298) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.65\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[298]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Suppose that $R$ is a local ring and $f \\\\in R$ is a regular element such that $R/(f)$ is $F$-injective. Does this imply that $R$ is $F$-injective?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is known in the Cohen-Macaulay case. An analogous result is known to hold for Du Bois singularities in characteristic zero, in generality.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0299", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a Noetherian local ring (R,m) of characteristic p>0, a nonzerodivisor f in m, and S=R/fR, let I_{i-1} be the image of H^{i-1}_m(R) in H^{i-1}_m(S) and Q_{i-1}=H^{i-1}_m(S)/I_{i-1}. The connecting map identifies Q_{i-1} with 0:_{H^i_m(R)}f, and Frobenius on H^i_m(R) is injective exactly when the boundary obstruction Psi_i=F composed with the induced connecting map is injective. Each I_{i-1} is Frobenius-stable, and Frobenius saturation of all these particular images suffices for R to be F-injective; conversely, any failure of deformation forces nonsaturation of one boundary image. The converse from nonsaturation is not claimed because the natural semilinear diagram has an f^{p-1} twist.\n\nCandidate contribution (criterion and reduction; novelty confidence low): The deformation obstruction can be packaged degree-by-degree as injectivity of Psi_i: H^{i-1}_m(S)/im(H^{i-1}_m(R)) -> H^i_m(R), [z] |-> F(delta_i z); saturation of the finitely many natural boundary images is sufficient for deformation, and every genuine failure of deformation necessarily makes the corresponding boundary image nonsaturated, with the gap between this necessary condition and an equivalence measured explicitly by f^{p-1}-torsion after Frobenius." }, { "id": 20000300, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0300", "title": "Arbitrary-ideal alteration formula, finite-principalization obstruction, and exact DVR audit", "statement": "Given a log pair $(X ,\\Delta)$, we know that there exists a finite map $\\phi: Y \\to X$ such that $\\operatorname{im}\\left(\\phi_{*} \\mathcal{O}_Y \\left( K_Y - \\phi^*\\left( K_X + \\Delta \\right)\\right) \\overset{\\text{trace}}{\\longrightarrow} \\mathcal{O}_X \\right) = \\tau(X, \\Delta)$. Is this statement true if further decorated by $\\mathfrak{a}^t$? What about for alterations?", "original_statement": "Given a log pair $(X ,\\Delta)$, we know that there exists a finite map $\\phi: Y \\to X$ such that $\\operatorname{im}\\left(\\phi_{*} \\mathcal{O}_Y \\left( K_Y - \\phi^*\\left( K_X + \\Delta \\right)\\right) \\overset{\\text{trace}}{\\longrightarrow} \\mathcal{O}_X \\right) = \\tau(X, \\Delta)$. Is this statement true if further decorated by $\\mathfrak{a}^t$? What about for alterations?", "clean_statement": "Given a log pair $(X ,\\Delta)$, we know that there exists a finite map $\\phi: Y \\to X$ such that $\\operatorname{im}\\left(\\phi_{*} \\mathcal{O}_Y \\left( K_Y - \\phi^*\\left( K_X + \\Delta \\right)\\right) \\overset{\\text{trace}}{\\longrightarrow} \\mathcal{O}_X \\right) = \\tau(X, \\Delta)$. Is this statement true if further decorated by $\\mathfrak{a}^t$? What about for alterations?", "statement_status": "exact", "statement_verification": "The canonical record, from `aim-algebraic-geometry-notes.json` at zero-based index 299, reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.7\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[299]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given a log pair $(X ,\\\\Delta)$, we know that there exists a finite map $\\\\phi: Y \\\\to X$ such that $\\\\operatorname{im}\\\\left(\\\\phi_{*} \\\\mathcal{O}_Y \\\\left( K_Y - \\\\phi^*\\\\left( K_X + \\\\Delta \\\\right)\\\\right) \\\\overset{\\\\text{trace}}{\\\\longrightarrow} \\\\mathcal{O}_X \\\\right) = \\\\tau(X, \\\\Delta)$. Is this statement true if further decorated by $\\\\mathfrak{a}^t$? What about for alterations?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0300", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Schwede and Tucker solved the alteration part: for a log-Q-Gorenstein triple in characteristic p, one regular alteration principalizing the ideal computes the test ideal by the rounded Grothendieck-trace formula for every real t>0. Their theorem does not establish a finite cover of X or a generically separable alteration for an arbitrary non-principal ideal. This attempt proves that an ideal with nonempty codimension-at-least-two cosupport cannot become invertible on any finite surjective cover, explaining why the normalized blowup is structurally necessary for the divisor-H method, and independently proves the exact DVR formula tau(delta div(x),(x^m)^t)=(x^floor(delta+mt)).\n\nCandidate contribution (obstruction; novelty confidence low): If f:Y->X is a finite surjective morphism of integral Noetherian varieties and every component of the nonempty zero locus of a nonzero proper ideal a has codimension at least two, then a O_Y is not invertible; hence no finite cover can realize the divisor H required by the direct alteration-style trace formula." }, { "id": 20000301, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0301", "title": "Positive-characteristic discrepancy candidates and an exact SNC calibration", "statement": "Identify a possible positive characteristic analog of minimal log discrepancy.", "original_statement": "Identify a possible positive characteristic analog of minimal log discrepancy.", "clean_statement": "Identify a possible positive characteristic analog of minimal log discrepancy.", "statement_status": "exact", "statement_verification": "The exact canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.75\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[300]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Identify a possible positive characteristic analog of minimal log discrepancy.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0301", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Canton's log discrepancy of Cartier subalgebras is identified as the closest established structural positive-characteristic answer. Independently, the proposed maximal-ideal scalar fmld^max_x(R,Delta) := fpt_x((R,Delta);m_x) is proved to equal the exact-center classical minimal log discrepancy on every regular coordinate simple-normal-crossing germ: both are sum_i(1-b_i). On this class the scalar is additive under coordinate products, shifts by relative dimension under the stated log-smooth pullbacks, and satisfies the sharp comparison d*s(R,Delta)^(1/d) <= mld_x(X,Delta). Two explicit strongly F-regular SNC surface pairs have the same pair F-signature 1/4 but mld values 5/4 and 1, so raw F-signature does not determine mld.\n\nCandidate contribution (comparison_and_separation_package; novelty confidence low): Candidate novelty: the assembled SNC calibration proves, with exact-center and p-divisible-denominator audits, that maximal-ideal F-pure threshold has the mld normalization, product additivity, and log-smooth dimension shift, while normalized pair F-signature obeys a sharp AM-GM bound and equal pair F-signature can correspond to unequal mld." }, { "id": 20000302, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0302", "title": "Local F-Bertini theorems, inseparability obstructions, and simultaneous good flags", "statement": "Investigate possible Bertini theorems for $F$-singularities.", "original_statement": "Investigate possible Bertini theorems for $F$-singularities.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.8\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[301]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate possible Bertini theorems for $F$-singularities.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0302", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Schwede-Zhang's one-hyperplane theorems give the sharp positive closed-general result for F-pure schemes and sharply F-pure or strongly F-regular pairs under separably generated residue extensions. Building on their geometric generic-fiber and A2/A2P spreading results, this attempt proves a simultaneous iterated version: for a very ample embedding and any prescribed flag length, one nonempty open subset of the full flag variety makes every successive section retain the relevant F-singularity; when K_X+Delta has index prime to p, sharply F-pure loci equal the original locus intersected with every flag section. Explicit p-power and Veronese-plane examples show that inseparable base-point-free systems and global F-regularity lie outside this theorem.\n\nCandidate contribution (theorem; novelty confidence low): For F-pure schemes and sharply F-pure or strongly F-regular pairs in a very ample embedding, the successively good linear flags of every fixed length contain a single nonempty Zariski-open subset of the full flag variety; in the prime-to-p Q-Gorenstein sharply F-pure setting, equality of F-pure loci holds simultaneously at every stage." }, { "id": 20000303, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0303", "title": "Test ideals and component-weighted symbolic interpolation on projective space", "statement": "Investigate the applications of the topics of the conference (e.g., test ideals) to projective geometry.", "original_statement": "Investigate the applications of the topics of the conference (e.g., test ideals) to projective geometry.", "clean_statement": "Investigate the applications of the topics of the conference (e.g., test ideals) to projective geometry.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic geometry\nWorkshop: Relating test ideals and multiplier ideals\nSection: Other and Related Problems\nSource item: 22.85\nSource URL: http://aimpl.org/testandmultiplierideals/2/\nCanonical location: aim-algebraic-geometry-notes.json notes[302]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate the applications of the topics of the conference (e.g., test ideals) to projective geometry.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/testandmultiplierideals/2/", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0303", "aim-domain:algebraic-geometry", "aim-workshop:testandmultiplierideals", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let S be a nonempty reduced closed subscheme of projective n-space over an algebraically closed field of characteristic p>0, with components Z_i of codimension c_i. If an effective degree-d hypersurface A has positive generic multiplicity ell_i along Z_i, then for every r>=1 the initial degree of the r-th symbolic power satisfies alpha(I_S^(r)) <= floor(d * max_i((c_i+r-1)/ell_i)). The proof gives the exact floor twist from Schwede's test-ideal global-generation theorem, allows mixed component codimensions, and is sharp for every r when S is a reduced hypersurface and A is a positive multiple of S.\n\nCandidate contribution (theorem; novelty confidence low): Candidate component-weighted symbolic interpolation theorem: alpha(I_S^(r)) <= floor(d * max_i((codim(Z_i)+r-1)/mult_{eta_i}(A))) for a reduced mixed-codimension subscheme of projective space in characteristic p>0, together with an all-r sharp reduced-hypersurface family." }, { "id": 20000304, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0304", "title": "Rational ideal compression and the non-Q-Gorenstein sign defect", "statement": "1. (Alexeev) Give a definition of log canonical, log terminal etc for pairs ( X, Y ) where X is a normal variety which is not necessarily Q-Gorenstein and Y is a formal sum of subschemes of X.", "original_statement": "1. (Alexeev) Give a definition of log canonical, log terminal etc for pairs ( X, Y ) where X is a normal variety which is not necessarily Q-Gorenstein and Y is a formal sum of subschemes of X.", "clean_statement": "1. (Alexeev) Give a definition of log canonical, log terminal etc for pairs ( X, Y ) where X is a normal variety which is not necessarily Q-Gorenstein and Y is a formal sum of subschemes of X.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1 in the AIM problem list from the workshop *Numerical invariants of singularities and higher-dimensional algebraic varieties* (31 July--4 August 2006). The PDF is legible, and the database transcription agrees with it:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[303]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (Alexeev) Give a definition of log canonical, log terminal etc for pairs ( X, Y ) where X is a normal variety which is not necessarily Q-Gorenstein and Y is a formal sum of subschemes of X.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0304", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "De Fernex and Hacon's 2009 limiting/plus discrepancy theory is a direct published answer to Alexeev's 2006 question. In addition, this attempt proves that any effective rational formal sum of subschemes can be compressed, for every divisorial discrepancy theory considered, to one weighted product-ideal subscheme up to integral closure. It also proves the exact boundary-independent formula a_E^+(X,Z)-a_{m,E}^-(X,Z)=ord_E(f^*(-K_X)+(1/m)f^natural(mK_X)) >= 0, monotone along divisible indices and converging along factorials to the asymptotic sign-linearity defect ord_E(f^*(-K_X)+f^*K_X).\n\nCandidate contribution (comparison theorem; novelty confidence low): For rational formal subscheme coefficients, the de Fernex-Hacon limiting/plus and Mather-Jacobian boundary data factor through the integral-closure class of a single denominator-cleared product ideal, while the plus-versus-m-limiting discrepancy gap is a nonnegative subscheme-independent defect decreasing along divisible indices to the numerical non-Q-Cartier sign defect." }, { "id": 20000305, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0305", "title": "Rationality and a sharp finite-jet radius for formal log-canonical thresholds", "statement": "2. (Mustat ¸ˇ a) When f ∈ C[[ x1, · · ·, x n]] is giving a holomorphic function near 0, the log-canonical thresh-old lct (f ) is defined the usual way ( [1], [3] ). It is rational since it is computed by a log-resolution. In general when f is just a formal power series, results of [1] make it possi-ble to define lct (f ) as a limit of lct (truncations of f ). Is lct (f ) again a rational number?", "original_statement": "2. (Mustat ¸ˇ a) When f ∈ C[[ x1, · · ·, x n]] is giving a holomorphic function near 0, the log-canonical thresh-old lct (f ) is defined the usual way ( [1], [3] ). It is rational since it is computed by a log-resolution. In general when f is just a formal power series, results of [1] make it possi-ble to define lct (f ) as a limit of lct (truncations of f ). Is lct (f ) again a rational number?", "clean_statement": "2. (Mustat ¸ˇ a) When f ∈ C[[ x1, · · ·, x n]] is giving a holomorphic function near 0, the log-canonical thresh-old lct (f ) is defined the usual way ( [1], [3] ). It is rational since it is computed by a log-resolution. In general when f is just a formal power series, results of [1] make it possi-ble to define lct (f ) as a limit of lct (truncations of f ). Is lct (f ) again a rational number?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 2, attributed to Mircea Mustaţă, from the AIM workshop *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The stored text contains OCR damage in the author name, subscripts, and line breaks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[304]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (Mustat ¸ˇ a) When f ∈ C[[ x1, · · ·, x n]] is giving a holomorphic function near 0, the log-canonical thresh-old lct (f ) is defined the usual way ( [1], [3] ). It is rational since it is computed by a log-resolution. In general when f is just a formal power series, results of [1] make it possi-ble to define lct (f ) as a limit of lct (truncations of f ). Is lct (f ) again a rational number?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0305", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM question has an affirmative published answer: de Fernex and Mustata proved that fixed-dimensional log-canonical-threshold sets are closed and that formal, polynomial, and geometric hypersurface threshold sets coincide, while Temkin resolution gives a direct divisorial proof on regular excellent characteristic-zero formal schemes. Beyond the status result, this attempt proves a sharp worked family: if f_0=sum_i x_i^{a_i}, A=max_i a_i, and sum_i 1/a_i<=1, then every formal h in m^{A+1} has lct_0(f_0+h)=sum_i 1/a_i, whereas cancellation of a term x_j^A shows that m^A is insufficient.\n\nCandidate contribution (sharp_worked_family; novelty confidence low): For a characteristic-zero Brieskorn--Pham germ f_0=sum_i x_i^{a_i} with n>=2 and sum_i 1/a_i<=1, the complete formal congruence class f_0+m^{A+1}, A=max_i a_i, has constant log-canonical threshold sum_i 1/a_i, and A+1 is the optimal uniform total-degree radius." }, { "id": 20000306, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0306", "title": "Finite-jet detection of log canonical thresholds with quantitative bounds", "statement": "3. (Mustat ¸ˇ a) We have the ACC conjecture for log canonical thresholds (\nConjecture 2.5 of [6], see also [3]). When we consider only smooth varieties, the conjecture is equivalent to the following question: Fix n. ∀c ∈ R, does there exist k = k(c) such that\n\n{f ∈ C[[ x1, · · ·, x n]] |lct (f ) ≥ c} depends on k-th truncations of f?A weaker question is: Does there exist k = k(f ) such that lct (f + g) ≥ lct (f ) for all g with ord (g) ≥ k?", "original_statement": "3. (Mustat ¸ˇ a) We have the ACC conjecture for log canonical thresholds ( \nConjecture 2.5 of [6], see also [3]). When we consider only smooth varieties, the conjecture is equivalent to the following question: Fix n. ∀c ∈ R, does there exist k = k(c) such that \n\n{f ∈ C[[ x1, · · ·, x n]] |lct (f ) ≥ c} depends on k-th truncations of f?A weaker question is: Does there exist k = k(f ) such that lct (f + g) ≥ lct (f ) for all g with ord (g) ≥ k?", "clean_statement": "3. (Mustat ¸ˇ a) We have the ACC conjecture for log canonical thresholds (\nConjecture 2.5 of [6], see also [3]). When we consider only smooth varieties, the conjecture is equivalent to the following question: Fix n. ∀c ∈ R, does there exist k = k(c) such that\n\n{f ∈ C[[ x1, · · ·, x n]] |lct (f ) ≥ c} depends on k-th truncations of f?A weaker question is: Does there exist k = k(f ) such that lct (f + g) ≥ lct (f ) for all g with ord (g) ≥ k?", "statement_status": "exact", "statement_verification": "The canonical JSON record is an OCR extraction from the AIM problem list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. It corrupts Mustaţă's name and joins two lines, but the original PDF is legible. Problem 3 on page 1 reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[305]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (Mustat ¸ˇ a) We have the ACC conjecture for log canonical thresholds ( \\nConjecture 2.5 of [6], see also [3]). When we consider only smooth varieties, the conjecture is equivalent to the following question: Fix n. ∀c ∈ R, does there exist k = k(c) such that \\n\\n{f ∈ C[[ x1, · · ·, x n]] |lct (f ) ≥ c} depends on k-th truncations of f?A weaker question is: Does there exist k = k(f ) such that lct (f + g) ≥ lct (f ) for all g with ord (g) ≥ k?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0306", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The two questions in the 2006 AIM record are affirmative: the finite-truncation assertion follows from the de Fernex--Mustata finite-cylinder equivalence and the proved smooth ACC theorem, and a perturbation of sufficiently high order cannot lower the threshold. Quantitatively, if J_n(c) is the least degree-J truncation level deciding lct(f) >= c and delta_n(c) is the gap from c to the largest n-dimensional hypersurface threshold below c, then n floor(1/c) <= J_n(c) <= floor(n/delta_n(c)); the lower level is exact on pure monomials and in dimension one. Consequently k(f) = floor(n/delta_n(lct(f))) + 1 works in the weaker perturbation question.\n\nCandidate contribution (theorem; novelty confidence low): Candidate quantitative spectrum-gap theorem: for 0 < c <= 1, the least deciding jet degree satisfies n floor(1/c) <= J_n(c) <= floor(n/delta_n(c)), with the lower bound exactly sharp on the pure-monomial locus; this also yields the stated explicit order bound for one-sided perturbation stability." }, { "id": 20000307, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0307", "title": "An order-profile mld bound from a quasi-etale lci cover", "statement": "4. (Mustat ¸ˇ a) Let X be a Q-Gorenstein variety. Then a conjecture of Shokurov (\nConjecture 2.2 in [6] ) says mld (P; X) ≤ dim (X) for all P ∈ X. As a weaker form of this question, is there any universal upper bound of mld (P; X) one can prove (fixing dim (X))?", "original_statement": "4. (Mustat ¸ˇ a) Let X be a Q-Gorenstein variety. Then a conjecture of Shokurov ( \nConjecture 2.2 in [6] ) says mld (P; X) ≤ dim (X) for all P ∈ X. As a weaker form of this question, is there any universal upper bound of mld (P; X) one can prove (fixing dim (X))?", "clean_statement": "4. (Mustat ¸ˇ a) Let X be a Q-Gorenstein variety. Then a conjecture of Shokurov (\nConjecture 2.2 in [6] ) says mld (P; X) ≤ dim (X) for all P ∈ X. As a weaker form of this question, is there any universal upper bound of mld (P; X) one can prove (fixing dim (X))?", "statement_status": "exact", "statement_verification": "The canonical record is item 4 of the AIM workshop list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The record has a minor OCR corruption in Mustaţă's name. The source PDF gives the following statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[306]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"4. (Mustat ¸ˇ a) Let X be a Q-Gorenstein variety. Then a conjecture of Shokurov ( \\nConjecture 2.2 in [6] ) says mld (P; X) ≤ dim (X) for all P ∈ X. As a weaker form of this question, is there any universal upper bound of mld (P; X) one can prove (fixing dim (X))?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0307", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general dimension-only boundedness problem remains open already in dimension four. A proved partial theorem is obtained for a finite crepant quasi-etale cover q:(Y,y)->(X,x) whose source is an n-dimensional normal local complete intersection: if a minimal smooth presentation of Y is cut out by a regular sequence f_1,...,f_c with s_i=ord_y(f_i), then mld_x(X) <= mld_y(Y) <= n-sum_i(s_i-1). If the right side is negative, both mlds are -infinity. The bound is independent of the cover degree and is exact for homogeneous normal complete-intersection cones whose projective defining hypersurfaces form an SNC divisor and whose ordinary blow-up is log smooth.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: the explicit order-profile and quasi-etale synthesis mld_x(X) <= n-sum_i(ord_y(f_i)-1), together with a sharp homogeneous complete-intersection family." }, { "id": 20000308, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0308", "title": "Log canonical implies Du Bois, with a sharp star-graph converse obstruction", "statement": "5. (Schwede) Let X be a variety with log-canonical singularities. Does X have Du Bois singularities? ( See [5] for some nice historical discussion about the question and some progress, the introduction of [4] for some discussion of the question and [2],Chapter 12. )", "original_statement": "5. (Schwede) Let X be a variety with log-canonical singularities. Does X have Du Bois singularities? ( See [5] for some nice historical discussion about the question and some progress, the introduction of [4] for some discussion of the question and [2],Chapter 12. )", "clean_statement": "5. (Schwede) Let X be a variety with log-canonical singularities. Does X have Du Bois singularities? ( See [5] for some nice historical discussion about the question and some progress, the introduction of [4] for some discussion of the question and [2],Chapter 12. )", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 5 from the workshop *Numerical invariants of singularities and higher-dimensional algebraic varieties*. Its exact extracted text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[307]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"5. (Schwede) Let X be a variety with log-canonical singularities. Does X have Du Bois singularities? ( See [5] for some nice historical discussion about the question and some progress, the introduction of [4] for some discussion of the question and [2],Chapter 12. )\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0308", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Kollár and Kovács proved that every characteristic-zero log-canonical pair has Du Bois underlying variety, and more generally that every reduced closed union of log-canonical centers is Du Bois; Fujino and Liu extended this to quasi-log-canonical pairs and semi-log-canonical strata. This attempt also proves a formal Du Bois-pair refinement and constructs algebraic rational surface germs X_{b,r}, for b at least r, with a central (-b)-curve and r (-2)-leaves. Their central log discrepancy is (4-r)/(2b-r), so the family is klt for r at most 3, lc but not klt for r=4, and non-lc for r at least 5, although every member is Cohen–Macaulay, Du Bois, and Q-Gorenstein.\n\nCandidate contribution (sharp_worked_family; novelty confidence low): For every pair of integers b >= r >= 1 with b >= 2, the algebraic rational star singularity with central weight -b and r leaves of weight -2 has central log discrepancy (4-r)/(2b-r), yielding a uniform sharp klt/lc/non-lc transition at valencies 3/4/5 while rationality, Du Bois, and Q-Gorensteinness persist." }, { "id": 20000309, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0309", "title": "A finite inverse criterion for surface mld divisors", "statement": "6. (Ishii) (1) What kind of exceptional divisor over X (with mild singularities) gives a minimal log discrepancy which is nonnegative for some ( X, D )? (2) What about over a surface X? Let E be an exceptional divisor of the minimal resolution\n\nX′ → X of X. Does E compute some mld ≥ 0?", "original_statement": "6. (Ishii) (1) What kind of exceptional divisor over X (with mild singularities) gives a minimal log discrepancy which is nonnegative for some ( X, D )? (2) What about over a surface X? Let E be an exceptional divisor of the minimal resolution \n\nX′ → X of X. Does E compute some mld ≥ 0?", "clean_statement": "6. (Ishii) (1) What kind of exceptional divisor over X (with mild singularities) gives a minimal log discrepancy which is nonnegative for some ( X, D )? (2) What about over a surface X? Let E be an exceptional divisor of the minimal resolution\n\nX′ → X of X. Does E compute some mld ≥ 0?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 6, attributed to Ishii, in the AIM workshop list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The original PDF says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[308]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"6. (Ishii) (1) What kind of exceptional divisor over X (with mild singularities) gives a minimal log discrepancy which is nonnegative for some ( X, D )? (2) What about over a surface X? Let E be an exceptional divisor of the minimal resolution \\n\\nX′ → X of X. Does E compute some mld ≥ 0?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0309", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a singular klt surface germ with minimal-resolution intersection matrix P=-(E_i.E_j), dual-graph valency vector v, and prescribed component E_k, an effective R-divisor boundary B for which E_k computes a nonnegative mld exists if and only if there is A>=0 with A_k=min_i A_i and PA<=2-v. The converse is constructive by splitting d=2-v-PA among general curvettes. On the tetrahedral quotient singularity T_7, whose fork has self-intersection -3 and whose arms have lengths 1,2,2 with all arm curves (-2), these inequalities force uniform positive gaps from the fork to every arm component, so no arm component can ever compute such an mld.\n\nCandidate contribution (theorem; novelty confidence low): Candidate inverse polyhedral theorem: a prescribed component E_k of the minimal resolution of a singular klt surface germ computes a nonnegative mld for some effective R-divisor boundary exactly when the finite cone slice {A>=0 : PA<=2-v, A_k=min_i A_i} is nonempty; every feasible point is realized by an explicit curvette boundary. The T_7 graph supplies a quantitative infeasibility certificate for all five arm components." }, { "id": 20000310, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0310", "title": "Arc-space termination and a Betti budget for standard smooth flips", "statement": "7. (Alexeev, Mustat ¸ˇ a ) Give one (non-trivial) application of motivic integration or arc spaces to termination of flips. What about under the assumption that the varieties in a sequence of flips X1 → X2 → · · ·\n\nare all smooth?\n1", "original_statement": "7. (Alexeev, Mustat ¸ˇ a ) Give one (non-trivial) application of motivic integration or arc spaces to termination of flips. What about under the assumption that the varieties in a sequence of flips X1 → X2 → · · · \n\nare all smooth? \n1", "clean_statement": "7. (Alexeev, Mustat ¸ˇ a ) Give one (non-trivial) application of motivic integration or arc spaces to termination of flips. What about under the assumption that the varieties in a sequence of flips X1 → X2 → · · ·\n\nare all smooth?\n1", "statement_status": "exact", "statement_verification": "The canonical record is item 7 in the AIM workshop problem list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The source PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[309]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"7. (Alexeev, Mustat ¸ˇ a ) Give one (non-trivial) application of motivic integration or arc spaces to termination of flips. What about under the assumption that the varieties in a sequence of flips X1 → X2 → · · · \\n\\nare all smooth? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0310", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Shokurov's termination criterion, together with the Ein--Mustaţă--Yasuda arc-space lower-semicontinuity theorem, proves that an ordinary boundaryless sequence of fixed-dimensional flips terminates if every model remains smooth: the relevant minimal log discrepancies are integral codimensions in a finite set, so ACC is automatic. In the standard smooth projective special case, a K-negative flip of type (m,l), m>l, raises the maximal divisorial arc-cylinder codimension by m-l, while the known Chow-motive decomposition removes exactly m-l Tate-twisted copies of the center motive. Its Betti realization gives B(X)-B(X')=(m-l)B(S), so any chain of such standard flips has length at most B(X_1)-1.\n\nCandidate contribution (quantitative_corollary; novelty confidence low): For a K-negative standard smooth projective flip of type (m,l), the arc-cylinder codimension gap m-l equals the number of Tate-twisted center-motive summands in the published Chow-motive splitting; along a chain these equalities telescope to sum_i (m_i-l_i)B(S_i)=B(X_1)-B(X_{N+1}) and hence N<=B(X_1)-1." }, { "id": 20000311, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0311", "title": "Adjunction beyond l.c.i. embeddings: status, a quantifier obstruction, and an exact slice identity", "statement": "8. (Schwede) Let Y be a smooth projective variety and X ⊂ Y an irreducible subvariety of codimension\n\nr. We assume that X is normal and Q-Gorenstein. If X is locally complete intersection (l.c.i.), then we have the equivalence: ( Y, I r) log-canonical ↔ X log-canonical (where I is the ideal sheaf of X on Y ). This is not true if X is not l.c.i.\n\nQuestion. ( Y, I (r)) log-canonical ↔ X log-canonical? ( I(r) is the r-th symbolic power of\n\nI. ) 9. ( Mustat ¸ˇ a ) Is there adjunction formula for multiplier ideals under restriction to subvarieties that are not defined by a regular sequence? 10. (de Fernex) Let X be a smooth variety and B ⊂ X a closed proper subscheme. We assume that there exists a prime divisor E over X, with center P, such that aE (X, cB ) ≤ 0 for some c > 0. Fix an integer e such that 1 ≤ e < dimX.Does there exist a smooth subvariety Y ⊂ X of codimension e ( P ⊂ Y, Y * B) and a divisor\n\nF over Y with its center cY (F ) = P such that the log discrepancy aF (Y, cB |Y − eP ) ≤ 0?\n1", "original_statement": "8. (Schwede) Let Y be a smooth projective variety and X ⊂ Y an irreducible subvariety of codimension \n\nr. We assume that X is normal and Q-Gorenstein. If X is locally complete intersection (l.c.i.), then we have the equivalence: ( Y, I r) log-canonical ↔ X log-canonical (where I is the ideal sheaf of X on Y ). This is not true if X is not l.c.i. \n\nQuestion. ( Y, I (r)) log-canonical ↔ X log-canonical? ( I(r) is the r-th symbolic power of \n\nI. ) 9. ( Mustat ¸ˇ a ) Is there adjunction formula for multiplier ideals under restriction to subvarieties that are not defined by a regular sequence? 10. (de Fernex) Let X be a smooth variety and B ⊂ X a closed proper subscheme. We assume that there exists a prime divisor E over X, with center P, such that aE (X, cB ) ≤ 0 for some c > 0. Fix an integer e such that 1 ≤ e < dimX.Does there exist a smooth subvariety Y ⊂ X of codimension e ( P ⊂ Y, Y * B) and a divisor \n\nF over Y with its center cY (F ) = P such that the log discrepancy aF (Y, cB |Y − eP ) ≤ 0? \n1", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record accidentally joins three consecutive questions from the AIM list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The terminal `1` in the extracted text is the page number, not part of Problem 10. The original PDF says that varieties are over \\(\\mathbb C\\). With typography and the OCR error \\(Y*B\\) repaired from the PDF, the three questions are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[310]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"8. (Schwede) Let Y be a smooth projective variety and X ⊂ Y an irreducible subvariety of codimension \\n\\nr. We assume that X is normal and Q-Gorenstein. If X is locally complete intersection (l.c.i.), then we have the equivalence: ( Y, I r) log-canonical ↔ X log-canonical (where I is the ideal sheaf of X on Y ). This is not true if X is not l.c.i. \\n\\nQuestion. ( Y, I (r)) log-canonical ↔ X log-canonical? ( I(r) is the r-th symbolic power of \\n\\nI. ) 9. ( Mustat ¸ˇ a ) Is there adjunction formula for multiplier ideals under restriction to subvarieties that are not defined by a regular sequence? 10. (de Fernex) Let X be a smooth variety and B ⊂ X a closed proper subscheme. We assume that there exists a prime divisor E over X, with center P, such that aE (X, cB ) ≤ 0 for some c > 0. Fix an integer e such that 1 ≤ e < dimX.Does there exist a smooth subvariety Y ⊂ X of codimension e ( P ⊂ Y, Y * B) and a divisor \\n\\nF over Y with its center cY (F ) = P such that the log discrepancy aF (Y, cB |Y − eP ) ≤ 0? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0311", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The merged record contains three distinct questions. The exact symbolic-power equivalence in Problem 8 was not found settled, while Problem 9 has a corrected affirmative answer through adjoint and l.c.i.-defect ideals, and Problem 10 is known under de Fernex's homogeneity and tangency hypotheses. This attempt proves two complementary statements for Problem 10: its literal positive-dimensional-center reading fails for X=A^3, P=B a line, c=e=2; and for the ordinary blow-up divisor E of any smooth codimension-q center P, every 1<=e=2 and the prime divisor P on Y when q-e=1.\n\nCandidate contribution (proposition; novelty confidence low): Candidate contribution: (i) the literal version of Problem 10 allowing positive-dimensional P is obstructed by X=A^3, P=B={x=y=0}, c=e=2; and (ii) if P is smooth of codimension q and E is the ordinary blow-up divisor, then for each 1<=e=2 and F=P for q-e=1." }, { "id": 20000312, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0312", "title": "Canonical indices, crepant centres, and an exact quotient-family calibration", "statement": "1. (Shokurov) Fix n and consider projective varieties X with an exact canonical singularity of dimension\n\nn. Is the index of KX at the singularity bounded? What about the case dimX = 3? More generally, for a fixed minimal log discrepancy of ( X, 0), is the index of KX at any point with such an mld, bounded?\n1", "original_statement": "1. (Shokurov) Fix n and consider projective varieties X with an exact canonical singularity of dimension \n\nn. Is the index of KX at the singularity bounded? What about the case dimX = 3? More generally, for a fixed minimal log discrepancy of ( X, 0), is the index of KX at any point with such an mld, bounded? \n1", "clean_statement": "1. (Shokurov) Fix n and consider projective varieties X with an exact canonical singularity of dimension\n\nn. Is the index of KX at the singularity bounded? What about the case dimX = 3? More generally, for a fixed minimal log discrepancy of ( X, 0), is the index of KX at any point with such an mld, bounded?\n1", "statement_status": "exact", "statement_verification": "### Canonical record", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[311]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (Shokurov) Fix n and consider projective varieties X with an exact canonical singularity of dimension \\n\\nn. Is the index of KX at the singularity bounded? What about the case dimX = 3? More generally, for a fixed minimal log discrepancy of ( X, 0), is the index of KX at any point with such an mld, bounded? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0312", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source phrase \"exact canonical\" is a typo introduced after an earlier AIM draft that said \"strictly canonical\"; the meaningful bounded local reading is a canonical point that is itself a crepant centre, or log mld 1. Kawakita proves the sharp index bound 6 in dimension three and proves fixed-mld boundedness for canonical threefolds, while the general-dimensional conjecture remains open and the quotient case is known. In addition, for every n >= 3 and every r > n-2 coprime to n-2, the projective quotient P^n/mu_r(0,1^{n-1},r-(n-2)) contains an isolated terminal germ U=A^n/mu_r(1^{n-1},r-(n-2)) with mld(U)=1+1/r, ind(K_U)=r, and smooth index-one cover A^n of degree r.\n\nCandidate contribution (explicit_family; novelty confidence low): The explicit all-dimensional projective cyclic quotient family satisfies the exact identity degree(index-one cover) = ind(K) = 1/(mld-1) = r and therefore forces R(n,1+1/r) >= r along every admissible varying-mld sequence." }, { "id": 20000313, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0313", "title": "A cyclotomic resolution-Betti obstruction for the global canonical index", "statement": "2. (Shokurov) Consider projective varieties X of dimension n with log-canonical singularities which satisfy\n\nKX ≡ 0. Then is the index of KX bounded? This is open for dimension 4. The following four problems are concerned with positive characteristic issues.\n1", "original_statement": "2. (Shokurov) Consider projective varieties X of dimension n with log-canonical singularities which satisfy \n\nKX ≡ 0. Then is the index of KX bounded? This is open for dimension 4. The following four problems are concerned with positive characteristic issues. \n1", "clean_statement": "2. (Shokurov) Consider projective varieties X of dimension n with log-canonical singularities which satisfy\n\nKX ≡ 0. Then is the index of KX bounded? This is open for dimension 4. The following four problems are concerned with positive characteristic issues.\n1", "statement_status": "exact", "statement_verification": "The canonical input is record 312 (zero-based) of `aim-algebraic-geometry-notes.json`. Its extracted text is visibly damaged: the item number was shortened from 12 to 2, the transition to the next group of problems was appended, and a page number was retained as a final “1”.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[312]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (Shokurov) Consider projective varieties X of dimension n with log-canonical singularities which satisfy \\n\\nKX ≡ 0. Then is the index of KX bounded? This is open for dimension 4. The following four problems are concerned with positive characteristic issues. \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0313", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let X be a connected projective canonical n-fold over C with K_X numerically trivial and exact global torsion index m, let Y be its connected quasi-etale global index-one cover, and let V -> Y be any projective equivariant resolution. The deck group's canonical character has exact order m; its primitive eigenvalue occurs in H^{n,0}(V) inside the integral middle-cohomology representation. Consequently Phi_m divides the characteristic polynomial on H^n(V,Z)/torsion, so phi(m) <= b_n(V), and the elementary inequality phi(m)^2 >= m/2 gives m <= 2 b_n(V)^2. Thus a uniformly bounded choice of such equivariant resolutions would imply bounded index; for the known index-3486 smooth fourfold, every such resolution has b_4 at least 984.\n\nCandidate contribution (reduction; novelty confidence low): For the unresolved canonical locus, the exact global index m gives a persistent cyclotomic factor Phi_m in the integral middle-cohomology characteristic polynomial of every equivariant resolution of the index-one cover, yielding the explicit bounded-choice criterion m <= 2B^2 and the concrete threshold b_4 >= 984 for an index-3486 fourfold." }, { "id": 20000314, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0314", "title": "Four F-pure-threshold questions and a sharp perturbation cylinder", "statement": "3. (Mustat ¸ˇ a) Let k be a field of characteristic p.(1) For f ∈ k[[ x1, · · ·, x n]], is the F-pure threshold f pt (f ) a rational number? (2) Let c = f pt (f ). Is τ (f c) a radical ideal? (3) If the zero set of τ (f c) is zero-dimensional, does there exist N > 1 such that ∀g ∈\n\nk[[ x1, · · ·, x n]] of order ≥ N, f pt (f ) = f pt (f + g)? 14. Does the set {f pt (f )|f ∈ k[[ x1, · · ·, x n]] } have the ACC property? 2\n1", "original_statement": "3. (Mustat ¸ˇ a) Let k be a field of characteristic p.(1) For f ∈ k[[ x1, · · ·, x n]], is the F-pure threshold f pt (f ) a rational number? (2) Let c = f pt (f ). Is τ (f c) a radical ideal? (3) If the zero set of τ (f c) is zero-dimensional, does there exist N > 1 such that ∀g ∈\n\nk[[ x1, · · ·, x n]] of order ≥ N, f pt (f ) = f pt (f + g)? 14. Does the set {f pt (f )|f ∈ k[[ x1, · · ·, x n]] } have the ACC property? 2\n1", "clean_statement": "3. (Mustat ¸ˇ a) Let k be a field of characteristic p.(1) For f ∈ k[[ x1, · · ·, x n]], is the F-pure threshold f pt (f ) a rational number? (2) Let c = f pt (f ). Is τ (f c) a radical ideal? (3) If the zero set of τ (f c) is zero-dimensional, does there exist N > 1 such that ∀g ∈\n\nk[[ x1, · · ·, x n]] of order ≥ N, f pt (f ) = f pt (f + g)? 14. Does the set {f pt (f )|f ∈ k[[ x1, · · ·, x n]] } have the ACC property? 2\n1", "statement_status": "exact", "statement_verification": "The canonical JSON record has merged two consecutive numbered items and has also retained a page-number/footer fragment. The original AIM PDF says that “the following four problems are concerned with positive characteristic issues” and gives the following text (with typography normalized but no mathematical change).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[313]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (Mustat ¸ˇ a) Let k be a field of characteristic p.(1) For f ∈ k[[ x1, · · ·, x n]], is the F-pure threshold f pt (f ) a rational number? (2) Let c = f pt (f ). Is τ (f c) a radical ideal? (3) If the zero set of τ (f c) is zero-dimensional, does there exist N > 1 such that ∀g ∈\\n\\nk[[ x1, · · ·, x n]] of order ≥ N, f pt (f ) = f pt (f + g)? 14. Does the set {f pt (f )|f ∈ k[[ x1, · · ·, x n]] } have the ACC property? 2\\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0314", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The merged record contains four questions. Rationality is affirmative by Katzman--Lyubeznik--Zhang even for non-F-finite coefficient fields; radicality is false by Mustata--Yoshida; fixed-characteristic fixed-dimension ACC is affirmative by Sato; and the exact test-ideal hypothesis in the perturbation question was not located as a settled theorem. A proved special case gives a sharp radius: over a perfect field, for f=x_0^p+sum_{i=1}^{d-1}x_i^{ell_i p+1} with d at least 2 and ell_i at least 1, every g in m^{d(p-1)+1} satisfies nu_{f+g}(p^e)=p^{e-1}-1 for all e and fpt(f+g)=1/p, while one explicit perturbation of order d(p-1) raises the threshold.\n\nCandidate contribution (proposition; novelty confidence low): The Frobenius cylinder x_0^p+(m^[p] intersect m^{p+1}) has nu_h(p^e)=p^{e-1}-1 at every Frobenius level; consequently d(p-1)+1 is the least uniform order radius preserving fpt=1/p for the Mustata--Yoshida family, with boundary counterperturbation product_i x_i^{p-1}." }, { "id": 20000315, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0315", "title": "A common-SNC carry formula for the negative-perturbation subadditivity question", "statement": "5. (Takagi) Let ( R, m) be a regular local ring of characteristic p. Suppose that ideals I, a and b of R\n\nsatisfy the following conditions: a ⊂ I, J (I−[U+000F] · as) ⊂ m and J (bt) ⊂ m.Then J (I−[U+000F] · as · bt) ⊂ J (I−[U+000F] · as) · J (bt)?\n1", "original_statement": "5. (Takagi) Let ( R, m) be a regular local ring of characteristic p. Suppose that ideals I, a and b of R\n\nsatisfy the following conditions: a ⊂ I, J (I−\u000f · as) ⊂ m and J (bt) ⊂ m.Then J (I−\u000f · as · bt) ⊂ J (I−\u000f · as) · J (bt)? \n1", "clean_statement": "Let \\((R,\\mathfrak m)\\) be a regular local ring of characteristic \\(p\\).\nSuppose that ideals \\(I,\\mathfrak a,\\mathfrak b\\) of \\(R\\) satisfy\n\\(\\mathfrak a\\subset I\\),\n\\(\\mathcal J(I^{-\\epsilon}\\cdot\\mathfrak a^s)\\subset\\mathfrak m\\), and\n\\(\\mathcal J(\\mathfrak b^t)\\subset\\mathfrak m\\). Then is\n\\[\n\\mathcal J(I^{-\\epsilon}\\cdot\\mathfrak a^s\\cdot\\mathfrak b^t)\n\\subset\n\\mathcal J(I^{-\\epsilon}\\cdot\\mathfrak a^s)\n\\mathcal J(\\mathfrak b^t)?\n\\]", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record contains an OCR control character in the exponent of \\(I\\), loses the superscript formatting on \\(s,t\\), and ends with a page number. The TeX source underlying the AIM PDF reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[314]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"5. (Takagi) Let ( R, m) be a regular local ring of characteristic p. Suppose that ideals I, a and b of R\\n\\nsatisfy the following conditions: a ⊂ I, J (I−\\u000f · as) ⊂ m and J (bt) ⊂ m.Then J (I−\\u000f · as · bt) ⊂ J (I−\\u000f · as) · J (bt)? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0315", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under an explicitly labeled common-SNC interpretation, suppose I, a, and b are nonzero principal monomial ideals in one regular parameter system, a is contained in I, and every coefficient c_i = s alpha_i - epsilon iota_i is nonnegative (in particular, 0 < epsilon < s is sufficient). Then the proposed containment holds for the characteristic-zero SNC multiplier ideal and, under the standard F-finite regular hypotheses, for the characteristic-p SNC generalized test ideal. More sharply, the left side equals the right-side product multiplied by the squarefree carry monomial product x_i^{delta_i}, where delta_i = floor({c_i}+{t beta_i}) is zero or one. This includes the full nonzero-ideal DVR case in the same exponent regime. It does not resolve the source's undefined calligraphic-J notation or the non-SNC case.\n\nCandidate contribution (explicit_formula; novelty confidence low): In the common-SNC principal-monomial class, the exact defect in subadditivity is the squarefree carry monomial product x_i^{delta_i}, with delta_i = floor({s alpha_i - epsilon iota_i}+{t beta_i}); equality holds exactly when there is no fractional-part carry in any coordinate." }, { "id": 20000316, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0316", "title": "Frobenius contraction and a log-canonical non-F-pure stress test for the symbolic cube", "statement": "6. (Takagi) Let ( R, m) be a regular local ring of characteristic p and p ⊂ R a prime ideal of height 2. If\n\nSpecR/ p has only log canonical singularities, then p(3) ⊂ p2?\n1", "original_statement": "6. (Takagi) Let ( R, m) be a regular local ring of characteristic p and p ⊂ R a prime ideal of height 2. If \n\nSpecR/ p has only log canonical singularities, then p(3) ⊂ p2?\n1", "clean_statement": "6. (Takagi) Let ( R, m) be a regular local ring of characteristic p and p ⊂ R a prime ideal of height 2. If\n\nSpecR/ p has only log canonical singularities, then p(3) ⊂ p2?\n1", "statement_status": "exact", "statement_verification": "The JSON extraction loses a leading digit, fraktur letters, superscript formatting, and a page break. The original AIM PDF gives **Problem 16 (Takagi)**, not Problem 6:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[315]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"6. (Takagi) Let ( R, m) be a regular local ring of characteristic p and p ⊂ R a prime ideal of height 2. If \\n\\nSpecR/ p has only log canonical singularities, then p(3) ⊂ p2?\\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0316", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literal fixed-characteristic statement with only discrepancy-log-canonicity remains open in odd characteristic, but several exact regimes are established. For every height-h prime P in a regular local ring of characteristic p and every q=p^e, P^(h(q-1)+1) is contained in P^[q], which is contained in P^q; the proof checks that flat Frobenius makes P^[q] P-primary, so contraction from R_P is valid. Hence every height-two prime in characteristic 2 satisfies P^(3) contained in P^2 without any singularity hypothesis. Published work gives the same containment in every characteristic when R/P is F-pure, and equality of all symbolic and ordinary powers for a height-two prime with strongly F-regular quotient. An explicit Fermat elliptic-cone family in every characteristic p congruent to 2 modulo 3 is log canonical but not F-pure in the exact height-two setup, while its complete-intersection structure gives equality of all powers.\n\nCandidate contribution (explicit_family_and_synthesis; novelty confidence low): Candidate novelty: for every algebraically closed field k of characteristic p congruent to 2 modulo 3, the height-two prime P=(w,x^3+y^3+z^3) in k[x,y,z,w] localized at the homogeneous maximal ideal has log-canonical but non-F-pure quotient and nevertheless satisfies P^(n)=P^n for every n; paired with the Frobenius-contraction specialization at p=2, this gives an explicit AIM-focused separation of the characteristic-two, F-pure, and complete-intersection mechanisms." }, { "id": 20000317, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0317", "title": "Movable versus divisorial canonicity in degrees four and five", "statement": "7. (Cheltsov) Let X be a smooth Fano variety with P icX = Z.\n\nQuestion Do we have the following equivalence (*)? (*) X is birationally superrigid ←→ (1) ∀ linear system with no fixed components M ⊂| − mK X |, ( X, 1\n\n> m\n\nM ) is canonical for all m > 0. ( Cheltsov explains that the implication from left to right follows from MMP. ) Now let's consider another condition: (2) ∀ divisor D ∈ | − mK X |, ( X, 1\n\n> m\n\nD) is canonical (or plt) for all m > 0. Cheltsov says, if X satisfies (1) and (2), then X × P1 is \"almost birationally superrigid\" in the sense that the only Mori fiber structures on it are the two projections. More precisely, every birational map from X × P1 to another Mori fiber space is an isomorphism. In the other direction, there is the following: Theorem (Pukhlikov, [8] ) If XN ⊂ PN is sufficiently general ( N ≥ 6), then X satisfies the conditions (1) and (2).\n\nQuestion. What if N = 4 or N = 5?\n1", "original_statement": "7. (Cheltsov) Let X be a smooth Fano variety with P icX = Z.\n\nQuestion Do we have the following equivalence (*)? (*) X is birationally superrigid ←→ (1) ∀ linear system with no fixed components M ⊂| − mK X |, ( X, 1 \n\n> m\n\nM ) is canonical for all m > 0. ( Cheltsov explains that the implication from left to right follows from MMP. ) Now let's consider another condition: (2) ∀ divisor D ∈ | − mK X |, ( X, 1 \n\n> m\n\nD) is canonical (or plt) for all m > 0. Cheltsov says, if X satisfies (1) and (2), then X × P1 is \"almost birationally superrigid\" in the sense that the only Mori fiber structures on it are the two projections. More precisely, every birational map from X × P1 to another Mori fiber space is an isomorphism. In the other direction, there is the following: Theorem (Pukhlikov, [8] ) If XN ⊂ PN is sufficiently general ( N ≥ 6), then X satisfies the conditions (1) and (2). \n\nQuestion. What if N = 4 or N = 5? \n1", "clean_statement": "7. (Cheltsov) Let X be a smooth Fano variety with P icX = Z.\n\nQuestion Do we have the following equivalence (*)? (*) X is birationally superrigid ←→ (1) ∀ linear system with no fixed components M ⊂| − mK X |, ( X, 1\n\n> m\n\nM ) is canonical for all m > 0. ( Cheltsov explains that the implication from left to right follows from MMP. ) Now let's consider another condition: (2) ∀ divisor D ∈ | − mK X |, ( X, 1\n\n> m\n\nD) is canonical (or plt) for all m > 0. Cheltsov says, if X satisfies (1) and (2), then X × P1 is \"almost birationally superrigid\" in the sense that the only Mori fiber structures on it are the two projections. More precisely, every birational map from X × P1 to another Mori fiber space is an isomorphism. In the other direction, there is the following: Theorem (Pukhlikov, [8] ) If XN ⊂ PN is sufficiently general ( N ≥ 6), then X satisfies the conditions (1) and (2).\n\nQuestion. What if N = 4 or N = 5?\n1", "statement_status": "exact", "statement_verification": "The record comes from Problem 17 (Cheltsov) in the AIM workshop list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The PDF, rather than the damaged text extraction, gives the following question.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[316]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"7. (Cheltsov) Let X be a smooth Fano variety with P icX = Z.\\n\\nQuestion Do we have the following equivalence (*)? (*) X is birationally superrigid ←→ (1) ∀ linear system with no fixed components M ⊂| − mK X |, ( X, 1 \\n\\n> m\\n\\nM ) is canonical for all m > 0. ( Cheltsov explains that the implication from left to right follows from MMP. ) Now let's consider another condition: (2) ∀ divisor D ∈ | − mK X |, ( X, 1 \\n\\n> m\\n\\nD) is canonical (or plt) for all m > 0. Cheltsov says, if X satisfies (1) and (2), then X × P1 is \\\"almost birationally superrigid\\\" in the sense that the only Mori fiber structures on it are the two projections. More precisely, every birational map from X × P1 to another Mori fiber space is an isomorphism. In the other direction, there is the following: Theorem (Pukhlikov, [8] ) If XN ⊂ PN is sufficiently general ( N ≥ 6), then X satisfies the conditions (1) and (2). \\n\\nQuestion. What if N = 4 or N = 5? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0317", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The Noether-Fano criterion answers the equivalence question affirmatively, and de Fernex's theorem with its erratum implies that every smooth quartic threefold and quintic fourfold satisfies condition (1). The canonical and plt readings of condition (2) for a general quartic or quintic remain unresolved in the literature checked. For every N at least 4, an explicit cone-section construction gives a special smooth degree-N hypersurface that satisfies (1) but has an anticanonical divisor D with local canonical threshold (N-2)/N and local log-canonical threshold (N-1)/N, so (1) does not imply (2). The product's two regular elementary contractions are its projections, but the literal claim that every birational map to a Mori fibre space is an isomorphism is false because Bir_X(X x P^1) is PGL_2(C(X)).\n\nCandidate contribution (counterexample; novelty confidence low): For every N at least 4 there exists a smooth degree-N hypersurface X_N in P^N that is birationally superrigid yet contains an anticanonical divisor D with ct_v(X_N;D)=(N-2)/N and lct_v(X_N;D)=(N-1)/N at a point v; this gives a uniform quantitative counterexample to condition (1) implying condition (2)." }, { "id": 20000318, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0318", "title": "Sharp factoriality threshold and defect-one equality cases", "statement": "8. (Cheltsov) Let Xn ⊂ P4 denote a hypersurface of degree n with a finite number of isolated ODP(ordinary double point)s.\n\nQuestion. When is Xn factorial ( Cl (Xn) = Z )? A theorem of Clemens says:\n\nXn is factorial ↔ SingX n imposes independent linear conditions on homogeneous forms on\n\nP4 of degree n.There is the following example of ( Cl (Xn) 6 = Z):\n\nXn given by xf n−1 + yg n−1 = 0 where fn−1 and gn−1 are generic homogeneous polynomials of degree n − 1 in x, y, z, w, v. Then ClX n = Z ⊕ Z.\n\nQuestion. If |SingX n| < (n − 1) 2, then Cl (Xn) = Z?Known facts. 1) |SingX n| ≤ 2\n\n> 3\n\n(n − 1) 2 → Xn is factorial. 2) S ⊂ Xn is a smooth subvariety of codimension 1. → S is Cartier.\n\nQuestion. Let ϕ: P4 − − → P2 be a generic projection. Then is it true that at most\n\nk(n − 1) points of the set ϕ(SingX n) lie on a curve of degre k in P2?3\n1", "original_statement": "8. (Cheltsov) Let Xn ⊂ P4 denote a hypersurface of degree n with a finite number of isolated ODP(ordinary double point)s. \n\nQuestion. When is Xn factorial ( Cl (Xn) = Z )? A theorem of Clemens says: \n\nXn is factorial ↔ SingX n imposes independent linear conditions on homogeneous forms on \n\nP4 of degree n.There is the following example of ( Cl (Xn) 6 = Z): \n\nXn given by xf n−1 + yg n−1 = 0 where fn−1 and gn−1 are generic homogeneous polynomials of degree n − 1 in x, y, z, w, v. Then ClX n = Z ⊕ Z.\n\nQuestion. If |SingX n| < (n − 1) 2, then Cl (Xn) = Z?Known facts. 1) |SingX n| ≤ 2 \n\n> 3\n\n(n − 1) 2 → Xn is factorial. 2) S ⊂ Xn is a smooth subvariety of codimension 1. → S is Cartier. \n\nQuestion. Let ϕ: P4 − − → P2 be a generic projection. Then is it true that at most \n\nk(n − 1) points of the set ϕ(SingX n) lie on a curve of degre k in P2?3\n1", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is source index 317 of `aim-algebraic-geometry-notes.json`. It comes from Problem 8 (Cheltsov) in the AIM workshop list *Numerical invariants of singularities and higher-dimensional algebraic varieties*. The PDF extraction is damaged at a fraction, at the rational-map arrow, and at the end of the page. The original TeX gives the following mathematical content (notation modernized only by adding subscripts and spacing).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[317]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"8. (Cheltsov) Let Xn ⊂ P4 denote a hypersurface of degree n with a finite number of isolated ODP(ordinary double point)s. \\n\\nQuestion. When is Xn factorial ( Cl (Xn) = Z )? A theorem of Clemens says: \\n\\nXn is factorial ↔ SingX n imposes independent linear conditions on homogeneous forms on \\n\\nP4 of degree n.There is the following example of ( Cl (Xn) 6 = Z): \\n\\nXn given by xf n−1 + yg n−1 = 0 where fn−1 and gn−1 are generic homogeneous polynomials of degree n − 1 in x, y, z, w, v. Then ClX n = Z ⊕ Z.\\n\\nQuestion. If |SingX n| < (n − 1) 2, then Cl (Xn) = Z?Known facts. 1) |SingX n| ≤ 2 \\n\\n> 3\\n\\n(n − 1) 2 → Xn is factorial. 2) S ⊂ Xn is a smooth subvariety of codimension 1. → S is Cartier. \\n\\nQuestion. Let ϕ: P4 − − → P2 be a generic projection. Then is it true that at most \\n\\nk(n − 1) points of the set ϕ(SingX n) lie on a curve of degre k in P2?3\\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0318", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM record has three separable parts. The correct factoriality criterion uses forms of degree 2n-5, and Cheltsov proved that every complex nodal degree-n hypersurface in P^4 with at most (n-1)^2-1 nodes is factorial, so the strict threshold question is solved affirmatively and sharply. Under the exact Cheltsov-Park cardinality hypothesis of at most (n-1)^2-1 points, the full generic-projection incidence conjecture was not found resolved, although Cheltsov-Park proved its required bound for k at most sqrt(n-1). As a proved equality refinement, every nonfactorial nodal degree-n hypersurface with exactly (n-1)^2 nodes has reduced node scheme a complete intersection of type (1,1,n-1,n-1), defect exactly one, and class group isomorphic to Z^2.\n\nCandidate contribution (corollary; novelty confidence low): Every nonfactorial complex nodal degree-n hypersurface in P^4 with exactly (n-1)^2 nodes has defect one and Cl(X) is isomorphic to Z^2, not only a general hypersurface of the displayed form xf+yg=0." }, { "id": 20000319, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0319", "title": "Kähler invariance of plurigenera: current boundary and a pseudoeffective-chamber reduction", "statement": "9. (Siu) Let π: X → ∆ be a holomorphic family of compact K¨ ahler manifolds over the open unit 1-disk ∆ with fiber Xt. Then is the dimension of H0(Xt, mK Xt ) independent of t ∈ ∆? This is the K¨ ahler case of Siu's famous invariance of plurigenera theorem for projective smooth varieties ( [7] ). One of the difficulties in using similar ideas from the algebraic case is that we cannot use an auxiliary ample line bundle as in [7].", "original_statement": "9. (Siu) Let π: X → ∆ be a holomorphic family of compact K¨ ahler manifolds over the open unit 1-disk ∆ with fiber Xt. Then is the dimension of H0(Xt, mK Xt ) independent of t ∈ ∆? This is the K¨ ahler case of Siu's famous invariance of plurigenera theorem for projective smooth varieties ( [7] ). One of the difficulties in using similar ideas from the algebraic case is that we cannot use an auxiliary ample line bundle as in [7].", "clean_statement": "9. (Siu) Let π: X → ∆ be a holomorphic family of compact K¨ ahler manifolds over the open unit 1-disk ∆ with fiber Xt. Then is the dimension of H0(Xt, mK Xt ) independent of t ∈ ∆? This is the K¨ ahler case of Siu's famous invariance of plurigenera theorem for projective smooth varieties ( [7] ). One of the difficulties in using similar ideas from the algebraic case is that we cannot use an auxiliary ample line bundle as in [7].", "statement_status": "exact", "statement_verification": "The canonical JSON record has two extraction defects: it gives the number as “9” rather than “19,” and it corrupts “Kähler.” The original AIM PDF and its TeX source have this item as the nineteenth problem in the list. The TeX source contains exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Numerical invariants of singularities and higher-dimensional algebraic varieties\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/singularvariety/problems_final.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[318]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"9. (Siu) Let π: X → ∆ be a holomorphic family of compact K¨ ahler manifolds over the open unit 1-disk ∆ with fiber Xt. Then is the dimension of H0(Xt, mK Xt ) independent of t ∈ ∆? This is the K¨ ahler case of Siu's famous invariance of plurigenera theorem for projective smooth varieties ( [7] ). One of the difficulties in using similar ideas from the algebraic case is that we cannot use an auxiliary ample line bundle as in [7].\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/singularvariety/problems_final.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0319", "aim-domain:algebraic-geometry", "aim-workshop:problems-final", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The original all-dimensional Kähler invariance problem remains open in fiber dimension at least four, while m=1, fiber dimensions at most two, projective or fiberwise Moishezon families, and, according to the March 2026 Hacon–Li–Rao preprint, smooth families of compact Kähler threefolds are settled. For every fixed m in any smooth proper family, the Euler characteristic of mK is constant and a special plurigenus jump equals an exact alternating jump in higher cohomology. If higher cohomology vanishes on one fiber, this forces local constancy. When pseudoeffectivity of the canonical bundle is deformation-invariant, in particular for a proper holomorphic submersion with Kähler total space by Fujiki–Levine and Ou, the non-pseudoeffective branch has P_m=0 for every m, reducing the unresolved part to the pseudoeffective chamber.\n\nCandidate contribution (reduction; novelty confidence low): The explicit combined package of a pseudoeffective-chamber dichotomy and the exact identity P_m(t)-P_m(generic)=sum_{q=1}^n(-1)^{q+1}(h^q(t)-h^q(generic)) removes the entire non-pseudoeffective branch and gives a falsifiable higher-cohomology jump pattern in the remaining branch; one-fiber vanishing of all higher cohomology then forces local constancy." }, { "id": 20000320, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0320", "title": "Q-Cartierness versus pluricanonical base change", "statement": "Problem 1.1. Let X → B be a family of surfaces. The condition \" ω[m]\n\n> X/B\n\ncommutes with base change\" means that for every base change φ: B′ → B and induced morphism\n\nφX: X′ = X ×B B′ → X, ( φ∗\n\n> X\n\nωX/B )[m] ∼= φ∗\n\n> X\n\n(ω[m]\n\n> X/B\n\n), where the superscript [ m] denotes the\n\nmth reflexive power, that is, the reflexive hull of the mth tensor power. Koll´ ar's condition\n\nis the condition that ω[m]\n\n> X/B\n\ncommutes with base change for all m > 0. Is the condition \" ωX/B is Q-Cartier\" equivalent to Koll´ ar's condition? Note that this is true for smoothings over the spectrum of a discrete valuation ring [Hac], Prop. 10.14. The conditions are not equivalent in positive characteristic, as Koll´ ar showed at the conference. This problem is implicitly due to Koll´ ar, who gave the stronger Koll´ ar condition in the paper [Kol90] after using the weaker definition with Shepherd-Barron for the moduli functor in [KSB88].", "original_statement": "Problem 1.1. Let X → B be a family of surfaces. The condition \" ω[m] \n\n> X/B\n\ncommutes with base change\" means that for every base change φ: B′ → B and induced morphism \n\nφX: X′ = X ×B B′ → X, ( φ∗ \n\n> X\n\nωX/B )[m] ∼= φ∗ \n\n> X\n\n(ω[m] \n\n> X/B\n\n), where the superscript [ m] denotes the \n\nmth reflexive power, that is, the reflexive hull of the mth tensor power. Koll´ ar's condition \n\nis the condition that ω[m] \n\n> X/B\n\ncommutes with base change for all m > 0. Is the condition \" ωX/B is Q-Cartier\" equivalent to Koll´ ar's condition? Note that this is true for smoothings over the spectrum of a discrete valuation ring [Hac], Prop. 10.14. The conditions are not equivalent in positive characteristic, as Koll´ ar showed at the conference. This problem is implicitly due to Koll´ ar, who gave the stronger Koll´ ar condition in the paper [Kol90] after using the weaker definition with Shepherd-Barron for the moduli functor in [KSB88].", "clean_statement": "Problem 1.1. Let X → B be a family of surfaces. The condition \" ω[m]\n\n> X/B\n\ncommutes with base change\" means that for every base change φ: B′ → B and induced morphism\n\nφX: X′ = X ×B B′ → X, ( φ∗\n\n> X\n\nωX/B )[m] ∼= φ∗\n\n> X\n\n(ω[m]\n\n> X/B\n\n), where the superscript [ m] denotes the\n\nmth reflexive power, that is, the reflexive hull of the mth tensor power. Koll´ ar's condition\n\nis the condition that ω[m]\n\n> X/B\n\ncommutes with base change for all m > 0. Is the condition \" ωX/B is Q-Cartier\" equivalent to Koll´ ar's condition? Note that this is true for smoothings over the spectrum of a discrete valuation ring [Hac], Prop. 10.14. The conditions are not equivalent in positive characteristic, as Koll´ ar showed at the conference. This problem is implicitly due to Koll´ ar, who gave the stronger Koll´ ar condition in the paper [Kol90] after using the weaker definition with Shepherd-Barron for the moduli functor in [KSB88].", "statement_status": "exact", "statement_verification": "The source is Problem 1.1 in the AIM workshop list *Compact moduli spaces and birational geometry*. The OCR has displaced subscripts and superscripts, but the mathematical statement is recoverable without ambiguity:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 1.1\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[319]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.1. Let X → B be a family of surfaces. The condition \\\" ω[m] \\n\\n> X/B\\n\\ncommutes with base change\\\" means that for every base change φ: B′ → B and induced morphism \\n\\nφX: X′ = X ×B B′ → X, ( φ∗ \\n\\n> X\\n\\nωX/B )[m] ∼= φ∗ \\n\\n> X\\n\\n(ω[m] \\n\\n> X/B\\n\\n), where the superscript [ m] denotes the \\n\\nmth reflexive power, that is, the reflexive hull of the mth tensor power. Koll´ ar's condition \\n\\nis the condition that ω[m] \\n\\n> X/B\\n\\ncommutes with base change for all m > 0. Is the condition \\\" ωX/B is Q-Cartier\\\" equivalent to Koll´ ar's condition? Note that this is true for smoothings over the spectrum of a discrete valuation ring [Hac], Prop. 10.14. The conditions are not equivalent in positive characteristic, as Koll´ ar showed at the conference. This problem is implicitly due to Koll´ ar, who gave the stronger Koll´ ar condition in the paper [Kol90] after using the weaker definition with Shepherd-Barron for the moduli functor in [KSB88].\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0320", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In characteristic zero, for a flat finite-type family of semi-log-canonical surfaces over a reduced original base, relative Q-Cartierness is equivalent to Kollár's condition; Kollár's hull criterion further gives compatibility with every test base change, including nonreduced tests. The equivalence fails when the original base is nonreduced: the Altmann–Kollár first-order deformation associated to the interval [-2/5,2/5] has invertible fifth relative canonical power but fails closed-fiber base change for the third power. Reducedness without an slc hypothesis is also insufficient by Lee–Nakayama's examples.\n\nCandidate contribution (lemma; novelty confidence low): For the Altmann–Kollár Section 5.7 VW-but-not-qG deformation of the cyclic quotient singularity 1/20(1,11), the lattice point (1,2/5) lies in the relevant half-open zone and in the exponent-three coset M-(3/5)Rbar, but not on the permitted diagonal; hence condition (*)_3, and therefore base change for the third reflexive canonical power, fails while the fifth reflexive power is invertible." }, { "id": 20000321, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0321", "title": "Smoothable closure and stable reduction for stable surfaces", "statement": "Problem 1.2. The main purpose of the moduli space of stable surfaces is to serve as a compactification of the moduli space of canonically polarized surfaces. However, it is possible that some stable surfaces are not smoothable. Some parts of the construction of the moduli space have only been proved for smoothable varieties. Here are some problems:\n\n> 12MICHAEL A. VAN OPSTALL\n\n(1) The moduli space M sm\n\n> K2,χ\n\nof smoothable stable surfaces does not have a natural scheme structure at the boundary. This is because infinitesimal information is lost by throw-ing away components of the moduli space parameterizing only surfaces with worse than rational double points. Can more sense be made of the notion of smoothable to get a good scheme structure? (2) Probably we should keep all of the components and avoid the previous problem completely. In this case, however, the valuative criterion for properness has not been verified even in dimension two. The problem is: if X → ∆′ is a family of stable surfaces over a punctured disk satisfying Koll´ ar's condition, can X be completed, possibly after base change, to a family of stable surfaces over the disk, also satisfying Koll´ ar's condition? Alexeev alluded to these problems in one of his talks.", "original_statement": "Problem 1.2. The main purpose of the moduli space of stable surfaces is to serve as a compactification of the moduli space of canonically polarized surfaces. However, it is possible that some stable surfaces are not smoothable. Some parts of the construction of the moduli space have only been proved for smoothable varieties. Here are some problems: \n\n> 12MICHAEL A. VAN OPSTALL\n\n(1) The moduli space M sm \n\n> K2,χ\n\nof smoothable stable surfaces does not have a natural scheme structure at the boundary. This is because infinitesimal information is lost by throw-ing away components of the moduli space parameterizing only surfaces with worse than rational double points. Can more sense be made of the notion of smoothable to get a good scheme structure? (2) Probably we should keep all of the components and avoid the previous problem completely. In this case, however, the valuative criterion for properness has not been verified even in dimension two. The problem is: if X → ∆′ is a family of stable surfaces over a punctured disk satisfying Koll´ ar's condition, can X be completed, possibly after base change, to a family of stable surfaces over the disk, also satisfying Koll´ ar's condition? Alexeev alluded to these problems in one of his talks.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "This is Problem 1.2 in Michael A. van Opstall's notes for the AIM workshop *Compact moduli spaces and birational geometry*, held at AIM in Palo Alto on 6--10 December 2004. The canonical PDF extraction is damaged: the string `12MICHAEL A. VAN OPSTALL` is a page header, the displayed moduli symbol was split across lines, and the accent in Kollár's name was corrupted. The official source TeX recovers the notation as \\[ \\overline{M^{\\mathrm{sm}}_{K^2,\\chi}} \\] and the punctured disk as \\(\\Delta'\\).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 1.2\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[320]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.2. The main purpose of the moduli space of stable surfaces is to serve as a compactification of the moduli space of canonically polarized surfaces. However, it is possible that some stable surfaces are not smoothable. Some parts of the construction of the moduli space have only been proved for smoothable varieties. Here are some problems: \\n\\n> 12MICHAEL A. VAN OPSTALL\\n\\n(1) The moduli space M sm \\n\\n> K2,χ\\n\\nof smoothable stable surfaces does not have a natural scheme structure at the boundary. This is because infinitesimal information is lost by throw-ing away components of the moduli space parameterizing only surfaces with worse than rational double points. Can more sense be made of the notion of smoothable to get a good scheme structure? (2) Probably we should keep all of the components and avoid the previous problem completely. In this case, however, the valuative criterion for properness has not been verified even in dimension two. The problem is: if X → ∆′ is a family of stable surfaces over a punctured disk satisfying Koll´ ar's condition, can X be completed, possibly after base change, to a family of stable surfaces over the disk, also satisfying Koll´ ar's condition? Alexeev alluded to these problems in one of his talks.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0321", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "In characteristic zero, both clauses of the recovered 2004 problem have a modern KSBA resolution: after fixing precise bounded moduli data, the natural boundary object is the scheme-theoretic closure of the rational-Gorenstein open substack inside the full KSBA stack, and Hacon-Xu stable reduction extends every punctured stable family after finite base change, uniquely up to unique isomorphism under the finite separated diagonal hypothesis. A proved formal lemma shows that every integral-DVR stable extension whose generic point lies in the rational-Gorenstein interior automatically factors through that scheme-theoretic closure. In positive and mixed characteristic, fixed-volume stacks exist over Z[1/30] and sufficiently-large-characteristic fixed-data stable reduction is known, but uniform properness over Z[1/30] remains conditional on locally stable reduction and small-characteristic cases remain delicate.\n\nCandidate contribution (lemma; novelty confidence low): Let M be a noetherian algebraic stack, U an open substack, and Z the scheme-theoretic closure defined by ker(O_M -> j_*O_U). Then Z is the unique smallest closed substack in M containing U schematically densely, inherits properness from M, and every map from an integral DVR to M whose generic point factors through U factors uniquely through Z; hence stable reduction in the full KSBA stack automatically takes place in the scheme-theoretic smoothable closure." }, { "id": 20000322, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0322", "title": "Characteristic-zero status and a fixed-volume bridge for stable n-folds", "statement": "Problem 1.3 (Higher dimensions). In [Kar00], Karu proves that the semistable minimal model program in dimension n + 1 can be used (together with the weak semistable reduction of Abramovich and Karu) to construct the moduli space of smoothable stable n-folds. See the previous problem for problems with this approach. This is what needs to be done to avoid the MMP: (1) Prove boundedness of slc n-folds with fixed Kn (see Alexeev's problems below). The semistable MMP is not strong enough to do this for non-smoothable n-folds anyway. (2) Prove that small Q-Gorenstein deformations of slc singularities are slc. It is true that small Q-Gorenstein deformations of slt singularities are slt. Again, the MMP in one dimension higher would verify this. (3) Verify the valuative criterion for properness for non-smoothable n-folds. The MMP gives the valuative criterion in the smoothable case.", "original_statement": "Problem 1.3 (Higher dimensions). In [Kar00], Karu proves that the semistable minimal model program in dimension n + 1 can be used (together with the weak semistable reduction of Abramovich and Karu) to construct the moduli space of smoothable stable n-folds. See the previous problem for problems with this approach. This is what needs to be done to avoid the MMP: (1) Prove boundedness of slc n-folds with fixed Kn (see Alexeev's problems below). The semistable MMP is not strong enough to do this for non-smoothable n-folds anyway. (2) Prove that small Q-Gorenstein deformations of slc singularities are slc. It is true that small Q-Gorenstein deformations of slt singularities are slt. Again, the MMP in one dimension higher would verify this. (3) Verify the valuative criterion for properness for non-smoothable n-folds. The MMP gives the valuative criterion in the smoothable case.", "clean_statement": "Problem 1.3 (Higher dimensions). In [Kar00], Karu proves that the semistable minimal model program in dimension n + 1 can be used (together with the weak semistable reduction of Abramovich and Karu) to construct the moduli space of smoothable stable n-folds. See the previous problem for problems with this approach. This is what needs to be done to avoid the MMP: (1) Prove boundedness of slc n-folds with fixed Kn (see Alexeev's problems below). The semistable MMP is not strong enough to do this for non-smoothable n-folds anyway. (2) Prove that small Q-Gorenstein deformations of slc singularities are slc. It is true that small Q-Gorenstein deformations of slt singularities are slt. Again, the MMP in one dimension higher would verify this. (3) Verify the valuative criterion for properness for non-smoothable n-folds. The MMP gives the valuative criterion in the smoothable case.", "statement_status": "exact", "statement_verification": "The record is Problem 1.3 in the AIM workshop notes *Compact moduli spaces and birational geometry* (December 6--10, 2004). The canonical record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 1.3\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[321]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.3 (Higher dimensions). In [Kar00], Karu proves that the semistable minimal model program in dimension n + 1 can be used (together with the weak semistable reduction of Abramovich and Karu) to construct the moduli space of smoothable stable n-folds. See the previous problem for problems with this approach. This is what needs to be done to avoid the MMP: (1) Prove boundedness of slc n-folds with fixed Kn (see Alexeev's problems below). The semistable MMP is not strong enough to do this for non-smoothable n-folds anyway. (2) Prove that small Q-Gorenstein deformations of slc singularities are slc. It is true that small Q-Gorenstein deformations of slt singularities are slt. Again, the MMP in one dimension higher would verify this. (3) Verify the valuative criterion for properness for non-smoothable n-folds. The MMP gives the valuative criterion in the smoothable case.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0322", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The three mathematical tasks in AIM Problem 1.3 are solved in characteristic zero without a smoothability hypothesis: Hacon–McKernan–Xu give fixed-volume boundedness, Kollár gives propagation of semi-log-canonicity in one-parameter relative Q-Gorenstein families, and Hacon–Xu give stable extension after finite base change. As a proved synthesis, a Kollár family with the required S2/G1 conditions and a common relatively ample, invertible, base-change-compatible canonical power has locally constant canonical volume; its nearby fibers and any stable limit therefore remain in one Hacon–McKernan–Xu bounded class. These known solutions use MMP methods, so the historical MMP-avoidance aspiration is not fulfilled.\n\nCandidate contribution (lemma; novelty confidence low): In a characteristic-zero one-parameter Kollár family satisfying the explicit S2 and codimension-one family hypotheses, if one common reflexive canonical power is invertible, relatively ample, and base-change compatible and one fiber is slc, then after shrinking all fibers are stable with the same canonical volume and lie in a single fixed-volume Hacon–McKernan–Xu bounded class; the same volume is preserved by a stable extension." }, { "id": 20000323, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0323", "title": "Boundedness after fixing the birational model", "statement": "Problem 1.\n4. (Alexeev) Are the following classes of varieties bounded? Here bounded means that all members of the class can be put into a family over a base of finite type. (1) Semi-log canonical varieties of general type with fixed Kn.(2) ≤-log terminal Fano varieties. 1.3. Positive characteristic issues.", "original_statement": "Problem 1.\n4. (Alexeev) Are the following classes of varieties bounded? Here bounded means that all members of the class can be put into a family over a base of finite type. (1) Semi-log canonical varieties of general type with fixed Kn.(2) ≤-log terminal Fano varieties. 1.3. Positive characteristic issues.", "clean_statement": "Problem 1.\n4. (Alexeev) Are the following classes of varieties bounded? Here bounded means that all members of the class can be put into a family over a base of finite type. (1) Semi-log canonical varieties of general type with fixed Kn.(2) ≤-log terminal Fano varieties. 1.3. Positive characteristic issues.", "statement_status": "exact", "statement_verification": "The canonical JSON record is visibly damaged by PDF extraction: “Problem 1.\\n4.” is the problem number \\(1.4\\), \\(K^n\\) lost its superscript, \\(\\epsilon\\) was misread, and the next section heading was appended to the problem. Inspection of the official AIM workshop PDF gives the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[322]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.\\n4. (Alexeev) Are the following classes of varieties bounded? Here bounded means that all members of the class can be put into a family over a base of finite type. (1) Semi-log canonical varieties of general type with fixed Kn.(2) ≤-log terminal Fano varieties. 1.3. Positive characteristic issues.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0323", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The intended characteristic-zero questions are solved affirmatively by Hacon–McKernan–Xu for slc canonical models of fixed dimension and volume and by Birkar's BAB theorem for fixed-dimensional epsilon-log-terminal Fano varieties. A proved ambiguity stress test shows that the literal big-canonical-divisor reading of part (1) is false: blowing up m(m-4)^2-1 points on a smooth degree-m surface in projective 3-space gives smooth general-type surfaces X_m with K_{X_m}^2=1 but chi(O_{X_m})=1+binom(m-1,3), hence an unbounded class; K_{X_m} is not nef, so this does not contradict the intended theorem.\n\nCandidate contribution (counterexample; novelty confidence low): For every integer m at least 5, the blow-up of a smooth degree-m surface in projective 3-space at exactly m(m-4)^2-1 distinct points is a smooth surface of general type with K^2=1 and chi(O)=1+binom(m-1,3); this explicitly refutes the literal big-K reading of AIM Problem 1.4(1)." }, { "id": 20000324, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0324", "title": "Stable-surface properness and an optimal universal tame Kummer index", "statement": "Problem 1.\n5. (Hassett) Prove the valuative criterion for properness for the moduli stack of stable surfaces in positive characteristic.", "original_statement": "Problem 1.\n5. (Hassett) Prove the valuative criterion for properness for the moduli stack of stable surfaces in positive characteristic.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The extracted record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[323]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.\\n5. (Hassett) Prove the valuative criterion for properness for the moduli stack of stable surfaces in positive characteristic.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0324", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unqualified 2004 properness problem remains only partially solved: fixed-volume stable reduction is known in equal characteristic sufficiently large in terms of the volume/data, while uniform properness over Z[1/30] remains conditional on locally stable reduction. Under an explicit log-canonical-total-model, Q-Cartier adjunction, and DCC volume-gap hypothesis, the attempt proves that every special-fiber multiplicity satisfies m_i <= floor(v/delta), that the actual multiplicity-killing Kummer degree lcm_i(m_i) divides lcm(1,...,floor(v/delta)), and that this latter integer is the least universal index forced solely by the numerical multiplicity bound. When p > floor(v/delta), the resulting Kummer base change is tame and separable.\n\nCandidate contribution (lemma; novelty confidence low): If the adjunction component volumes are bounded below by delta and the generic volume is v, then N_univ = lcm(1,...,floor(v/delta)) is the arithmetically minimal positive integer divisible by every special-fiber multiplicity allowed solely by the volume-gap bound, and the actual Kummer index lcm_i(m_i) divides N_univ." }, { "id": 20000325, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0325", "title": "Fibers of integral moduli spaces: tame compatibility and a wild obstruction", "statement": "Problem 1.\n6. (Koll´ ar) Consider the moduli space of canonically polarized surfaces over Spec Z. Is the fiber over the prime p the moduli space of surfaces over Fp? A similar result for Mg may be due to Oort. What about other moduli problems? Ag? K3 surfaces? 1.4. Other moduli problems. An article of Schumacher and Tsuji [ST04] asserts that a separated moduli space of smooth polarized manifolds is always quasi-projective. By Viehweg's work this was known for canonically polarized manifolds and for polarized man-ifolds F with ωF nef. In the case of polarized uniruled manifolds the separatedness of the moduli space is frequently hard to check. Koll´ ar proposed a series of counterexamples by showing that every smooth toric variety is the moduli space for some class of polarized manifolds. There are many smooth, proper but nonprojective toric varieties.", "original_statement": "Problem 1.\n6. (Koll´ ar) Consider the moduli space of canonically polarized surfaces over Spec Z. Is the fiber over the prime p the moduli space of surfaces over Fp? A similar result for Mg may be due to Oort. What about other moduli problems? Ag? K3 surfaces? 1.4. Other moduli problems. An article of Schumacher and Tsuji [ST04] asserts that a separated moduli space of smooth polarized manifolds is always quasi-projective. By Viehweg's work this was known for canonically polarized manifolds and for polarized man-ifolds F with ωF nef. In the case of polarized uniruled manifolds the separatedness of the moduli space is frequently hard to check. Koll´ ar proposed a series of counterexamples by showing that every smooth toric variety is the moduli space for some class of polarized manifolds. There are many smooth, proper but nonprojective toric varieties.", "clean_statement": "Problem 1.\n6. (Koll´ ar) Consider the moduli space of canonically polarized surfaces over Spec Z. Is the fiber over the prime p the moduli space of surfaces over Fp? A similar result for Mg may be due to Oort. What about other moduli problems? Ag? K3 surfaces? 1.4. Other moduli problems. An article of Schumacher and Tsuji [ST04] asserts that a separated moduli space of smooth polarized manifolds is always quasi-projective. By Viehweg's work this was known for canonically polarized manifolds and for polarized man-ifolds F with ωF nef. In the case of polarized uniruled manifolds the separatedness of the moduli space is frequently hard to check. Koll´ ar proposed a series of counterexamples by showing that every smooth toric variety is the moduli space for some class of polarized manifolds. There are many smooth, proper but nonprojective toric varieties.", "statement_status": "exact", "statement_verification": "The canonical JSON record is an OCR fusion of a problem and the beginning of the next section. Inspection of the official AIM workshop PDF, *Open Problems in Compact Moduli Spaces and Birational Geometry*, printed page 2, recovers the problem as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[324]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.\\n6. (Koll´ ar) Consider the moduli space of canonically polarized surfaces over Spec Z. Is the fiber over the prime p the moduli space of surfaces over Fp? A similar result for Mg may be due to Oort. What about other moduli problems? Ag? K3 surfaces? 1.4. Other moduli problems. An article of Schumacher and Tsuji [ST04] asserts that a separated moduli space of smooth polarized manifolds is always quasi-projective. By Viehweg's work this was known for canonically polarized manifolds and for polarized man-ifolds F with ωF nef. In the case of polarized uniruled manifolds the separatedness of the moduli space is frequently hard to check. Koll´ ar proposed a series of counterexamples by showing that every smooth toric variety is the moduli space for some class of polarized manifolds. There are many smooth, proper but nonprojective toric varieties.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0325", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a genuinely relative, base-change-stable moduli stack, the fiber over F_p is tautologically the characteristic-p moduli stack. For a fixed quasi-compact algebraic stack of finite presentation over Spec Z with finite inertia, stabilizer lengths are uniformly bounded; after inverting finitely many primes the stack is tame, so its coarse space commutes with arbitrary base change and the canonical comparison is an isomorphism for every sufficiently large p. At wild primes compatibility can fail: on a finite constant-group quotient chart over a DVR, the defect is exactly a Tor_1 group, and the proper stack [P^1_{Z_(2)}/C_2] has special-fiber comparison equal to Frobenius on P^1_{F_2}, not an isomorphism.\n\nCandidate contribution (lemma; novelty confidence low): Candidate contribution: for every fixed-data integral moduli stack satisfying quasi-compactness, finite presentation, and finite inertia, all possible coarse-base-change failures are confined to finitely many residual characteristics; on finite constant-group quotient charts over a DVR the failure is measured exactly by the Tor_1 of the cokernel of the group-difference map, with a proper C_2 quotient furnishing a sharp wild test family." }, { "id": 20000326, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0326", "title": "An effective non-nef fixed-parent point-blowup locus", "statement": "Problem 1.7. Can some result between Viehweg's result and the statement of Schumacher and Tsuji be proved about quasi-projectivity? For instance, if the canonical class is assumed effective? OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 3\n\nKoll´ ar's class of examples comes from the observation that the blowups of points on a smooth variety X are parameterized by X modulo the automorphisms of X. Thus it could be true that every algebraic space which is a quotient of a quasi-projective variety by a proper group action is a moduli space for some class of polarized manifolds.", "original_statement": "Problem 1.7. Can some result between Viehweg's result and the statement of Schumacher and Tsuji be proved about quasi-projectivity? For instance, if the canonical class is assumed effective? OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 3\n\nKoll´ ar's class of examples comes from the observation that the blowups of points on a smooth variety X are parameterized by X modulo the automorphisms of X. Thus it could be true that every algebraic space which is a quotient of a quasi-projective variety by a proper group action is a moduli space for some class of polarized manifolds.", "clean_statement": "Problem 1.7. Can some result between Viehweg's result and the statement of Schumacher and Tsuji be proved about quasi-projectivity? For instance, if the canonical class is assumed effective? OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 3\n\nKoll´ ar's class of examples comes from the observation that the blowups of points on a smooth variety X are parameterized by X modulo the automorphisms of X. Thus it could be true that every algebraic space which is a quotient of a quasi-projective variety by a proper group action is a moduli space for some class of polarized manifolds.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.7 from the AIM workshop list *Open Problems in Compact Moduli Spaces and Birational Geometry*. Its extracted `problem` field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 1.7\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[325]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.7. Can some result between Viehweg's result and the statement of Schumacher and Tsuji be proved about quasi-projectivity? For instance, if the canonical class is assumed effective? OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 3\\n\\nKoll´ ar's class of examples comes from the observation that the blowups of points on a smooth variety X are parameterized by X modulo the automorphisms of X. Thus it could be true that every algebraic space which is a quotient of a quasi-projective variety by a proper group action is a moduli space for some class of polarized manifolds.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0326", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a fixed smooth projective complex surface Y with nef, linearly effective canonical divisor, a very ample H, and a fixed number r of distinct point centers, the polarized blowups X_P with L_P = pi_P^*(H^{tensor (r+1)}) minus the exceptional divisors have effective but non-nef canonical divisor. Their exceptional curves are intrinsically exactly the K-negative curves, so polarized isomorphisms descend to the exact finite group Aut(Y,H^{tensor (r+1)}). Hence the restricted coarse orbit space is Conf_r(Y)/(Aut(Y,H^{tensor (r+1)}) x S_r), a quasi-projective finite quotient. This also obstructs the simplest fixed-parent version of Kollar's point-blowup mechanism from producing an effective-K non-quasi-projective example.\n\nCandidate contribution (proposition_and_obstruction; novelty confidence low): Candidate novelty is the explicit intrinsic K-negative recovery and finite-quotient quasi-projectivity theorem for simultaneous point blowups of a fixed minimal surface with linearly effective canonical divisor, together with the resulting obstruction to the simplest fixed-parent Kollar construction." }, { "id": 20000327, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0327", "title": "Quotient stacks, inertia obstructions, and polarized point blowups", "statement": "Problem 1.8. More generally, is every stack which is a quotient of a quasi-projective scheme by a group action a moduli stack for some class of polarized manifolds? 2. Moduli spaces of manifolds\n\nThe following problems are problems on the geometry of the \"interior\" of the moduli space, that is, on the moduli space of canonically polarized manifolds. They are motivated by the principle that even without local Torelli theorems, the geometry of moduli spaces should be similiar to the geometry of period domains.", "original_statement": "Problem 1.8. More generally, is every stack which is a quotient of a quasi-projective scheme by a group action a moduli stack for some class of polarized manifolds? 2. Moduli spaces of manifolds \n\nThe following problems are problems on the geometry of the \"interior\" of the moduli space, that is, on the moduli space of canonically polarized manifolds. They are motivated by the principle that even without local Torelli theorems, the geometry of moduli spaces should be similiar to the geometry of period domains.", "clean_statement": "Problem 1.8. More generally, is every stack which is a quotient of a quasi-projective scheme by a group action a moduli stack for some class of polarized manifolds? 2. Moduli spaces of manifolds\n\nThe following problems are problems on the geometry of the \"interior\" of the moduli space, that is, on the moduli space of canonically polarized manifolds. They are motivated by the principle that even without local Torelli theorems, the geometry of moduli spaces should be similiar to the geometry of period domains.", "statement_status": "exact", "statement_verification": "The record comes from the AIM workshop *Compact moduli spaces and birational geometry* (6--10 December 2004), Problem 1.8. The official TeX source has the following lead-in:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 1.8\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[326]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.8. More generally, is every stack which is a quotient of a quasi-projective scheme by a group action a moduli stack for some class of polarized manifolds? 2. Moduli spaces of manifolds \\n\\nThe following problems are problems on the geometry of the \\\"interior\\\" of the moduli space, that is, on the moduli space of canonically polarized manifolds. They are motivated by the principle that even without local Torelli theorems, the geometry of moduli spaces should be similiar to the geometry of period domains.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0327", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literal arbitrary-algebraic-group reading is false for linearly polarized manifolds: if A is a positive-dimensional abelian variety, BA is a quotient of the quasi-projective scheme Spec(k) but has non-affine inertia, while every stabilizer of a linearly polarized smooth projective variety is affine. For the intended affine/linear setting, a rigorous positive special case is proved: under explicit canonical-power, linearization, ampleness, and full-automorphism hypotheses on a fixed projective W, the polarized point-blowup family induces a fully faithful morphism [U/G] to the scalar-rigidified polarized moduli stack, functorially identifying relative transporter and Isom fppf sheaves over arbitrary test schemes.\n\nCandidate contribution (theorem; novelty confidence low): For a G-invariant quasi-projective U inside a smooth connected projective W satisfying the stated hypotheses, the fixed-parent point-blowup construction identifies Trans_G(u,v) with the rigidified polarized Isom sheaf for every test scheme and every u,v, hence realizes [U/G] as a full essential-image substack while retaining nonreduced infinitesimal arrows." }, { "id": 20000328, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0328", "title": "A boundary-frontier sieve for virtually abelian moduli bases", "statement": "Problem 2.\n1. (Viehweg). Let MH denote a moduli stack of canonically polarized manifolds. Can π1(MH ) be almost abelian (almost abelian means having an abelian subgroup of finite index)? More precisely, if U is a nonsingular variety (not a point) mapping generically finitely to the moduli stack, can π1(U ) be almost abelian. The conjecture is no. The result is known for Mg, and can be verified whenever some sort of local Torelli theorem holds. A U as above cannot be the complement in Pn of a normal crossings divisor with fewer than n components. At present this is only known if the number of components of H is strictly smaller than n (see [VZ02]). The following two questions are motivated by work of Viehweg and Zuo [VZ02] and [VZ05].", "original_statement": "Problem 2.\n1. (Viehweg). Let MH denote a moduli stack of canonically polarized manifolds. Can π1(MH ) be almost abelian (almost abelian means having an abelian subgroup of finite index)? More precisely, if U is a nonsingular variety (not a point) mapping generically finitely to the moduli stack, can π1(U ) be almost abelian. The conjecture is no. The result is known for Mg, and can be verified whenever some sort of local Torelli theorem holds. A U as above cannot be the complement in Pn of a normal crossings divisor with fewer than n components. At present this is only known if the number of components of H is strictly smaller than n (see [VZ02]). The following two questions are motivated by work of Viehweg and Zuo [VZ02] and [VZ05].", "clean_statement": "Problem 2.\n1. (Viehweg). Let MH denote a moduli stack of canonically polarized manifolds. Can π1(MH ) be almost abelian (almost abelian means having an abelian subgroup of finite index)? More precisely, if U is a nonsingular variety (not a point) mapping generically finitely to the moduli stack, can π1(U ) be almost abelian. The conjecture is no. The result is known for Mg, and can be verified whenever some sort of local Torelli theorem holds. A U as above cannot be the complement in Pn of a normal crossings divisor with fewer than n components. At present this is only known if the number of components of H is strictly smaller than n (see [VZ02]). The following two questions are motivated by work of Viehweg and Zuo [VZ02] and [VZ05].", "statement_status": "exact", "statement_verification": "The canonical record comes from the AIM workshop problem list *Compact moduli spaces and birational geometry*, Section 2, “Moduli spaces of manifolds.” In the source PDF the item is Problem 2.1, although the extracted record splits the number as `2.` followed by `1.`. The source reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[327]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.\\n1. (Viehweg). Let MH denote a moduli stack of canonically polarized manifolds. Can π1(MH ) be almost abelian (almost abelian means having an abelian subgroup of finite index)? More precisely, if U is a nonsingular variety (not a point) mapping generically finitely to the moduli stack, can π1(U ) be almost abelian. The conjecture is no. The result is known for Mg, and can be verified whenever some sort of local Torelli theorem holds. A U as above cannot be the complement in Pn of a normal crossings divisor with fewer than n components. At present this is only known if the number of components of H is strictly smaller than n (see [VZ02]). The following two questions are motivated by work of Viehweg and Zuo [VZ02] and [VZ05].\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0328", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For n at least 2 and a reduced simple-normal-crossings divisor D=sum D_i in projective n-space, with component degrees d_i, Libgober's theorem gives pi_1(P^n minus D) = Z^r/<(d_1,...,d_r)> = Z^(r-1) plus Z/gcd(d_i), so every such complement has abelian fundamental group. If it supports a maximal-variation family of canonically polarized manifolds, Viehweg-Zuo and log-general-type results force r at least n and sum d_i at least n+2. For general-position hyperplanes, under the stronger hypothesis that the coarse moduli map is quasi-finite over its image, Viehweg-Zuo Brody hyperbolicity and a sharp entire-curve argument force r at least 2n+1; the frontier complement has pi_1 = Z^(2n).\n\nCandidate contribution (reduction; novelty confidence low): The candidate contribution is the combined boundary-frontier sieve: any projective-space SNC-complement counterexample in dimension n at least 2 must satisfy r at least n and sum d_i at least n+2, while a quasi-finite general-position hyperplane candidate must begin at r=2n+1, where its fundamental group is explicitly Z^(2n)." }, { "id": 20000329, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0329", "title": "Logarithmic positivity and a two-point configuration-space test", "statement": "**Problem 2.2 (Viehweg).** Let \\(U\\) be a smooth variety, étale over a moduli stack of polarized manifolds \\(M_H\\). Let \\(Y\\) be a log compactification of \\(U\\) and \\(\\Gamma=Y\\setminus U\\).\n\n1. Is \\(\\Omega_Y^1(\\log \\Gamma)\\) weakly positive with respect to \\(U\\)?\n2. Is \\(\\omega_Y(\\Gamma)\\) ample with respect to \\(U\\)?\n\nBoth of these statements are known for period domains, and it is known that the second implies the first. These problems may be hard, but similar questions would be interesting and perhaps more tractable for configuration spaces.", "original_statement": "Problem 2.\n2. (Viehweg) Let U be a smooth variety, ´ etale over a moduli stack of polarized manifolds MH. Let Y be a log compactification of U and Γ = Y \\U.(1) Is Ω 1 \n\n> Y\n\n(log Γ) weakly positive with respect to U?(2) Is ωY (Γ) ample with respect to U?Both of these statements are known for period domains, and it is known that the second implies the first. These problems may be hard, but similar questions would be interesting and perhaps more tractable for configuration spaces.", "clean_statement": "**Problem 2.2 (Viehweg).** Let \\(U\\) be a smooth variety, étale over a moduli stack of polarized manifolds \\(M_H\\). Let \\(Y\\) be a log compactification of \\(U\\) and \\(\\Gamma=Y\\setminus U\\).\n\n1. Is \\(\\Omega_Y^1(\\log \\Gamma)\\) weakly positive with respect to \\(U\\)?\n2. Is \\(\\omega_Y(\\Gamma)\\) ample with respect to \\(U\\)?\n\nBoth of these statements are known for period domains, and it is known that the second implies the first. These problems may be hard, but similar questions would be interesting and perhaps more tractable for configuration spaces.", "statement_status": "corrected_verified", "statement_verification": "The corpus record is an OCR extraction of Problem 2.2, attributed to Viehweg, in the AIM problem list *Compact moduli spaces and birational geometry*. The official PDF gives the following statement: Here \\(M_H\\) is printed as \\(\\mathcal M_H\\) in the source. The corpus extraction has three visible OCR defects: “Problem 2. / 2.” duplicates the numbering, a stray `>` interrupts the logarithmic cotangent notation, and spacing is lost around \\(\\Gamma=Y\\setminus U\\). The recovered statement above follows the official PDF rather than those artifacts.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[328]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.\\n2. (Viehweg) Let U be a smooth variety, ´ etale over a moduli stack of polarized manifolds MH. Let Y be a log compactification of U and Γ = Y \\\\U.(1) Is Ω 1 \\n\\n> Y\\n\\n(log Γ) weakly positive with respect to U?(2) Is ωY (Γ) ample with respect to U?Both of these statements are known for period domains, and it is known that the second implies the first. These problems may be hard, but similar questions would be interesting and perhaps more tractable for configuration spaces.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0329", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The original moduli-stack positivity problem remains open, but the configuration-space analogue suggested in the source is resolved for two ordered points on a smooth projective curve: if C has genus at least one, the logarithmic cotangent bundle on (C x C, diagonal) is nef and hence Viehweg-weakly positive over the whole compactification; the log canonical bundle is ample exactly in genus at least two; and for an elliptic curve it is not even ample with respect to the open configuration space because every nontrivial translation graph is a complete interior curve contracted by every complete-linear-system map of every positive power.\n\nCandidate contribution (configuration_space_proposition_and_obstruction; novelty confidence low): For ordered two-point configurations on a smooth projective curve, the logarithmic cotangent bundle is nef for every positive genus, the log canonical bundle is ample exactly in genus at least two, and in genus one nontrivial translation graphs inside the open configuration space give a direct complete-linear-system obstruction to ampleness with respect to the open set." }, { "id": 20000330, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0330", "title": "Factor persistence and the limits of single-summand Arakelov equality", "statement": "Problem 2.3. Assume that MH is a moduli stack for abelian varieties or for a class of manifolds for which local Torelli holds. Let U, Y, and Γ be as in the previous problem. There is a variation of Hodge structures. Take an irreducible subvariation with Higgs bundle\n\nE1,0 π E0,1. Then one can show deg E1,0\n\nrk E1,0 − deg E0,1\n\nrk E0,1 ≤ deg Ω 1\n\n> Y\n\n(log Γ)\n\ndim Y\n\nwhere ωY (Γ) is nef and big, and degree is taken with respect to this sheaf. The problem is to determine when equality holds. A conjecture in this direction is that U is a Shimura variety. This is true if Y is a curve and for moduli spaces Ag for small g.", "original_statement": "Problem 2.3. Assume that MH is a moduli stack for abelian varieties or for a class of manifolds for which local Torelli holds. Let U, Y, and Γ be as in the previous problem. There is a variation of Hodge structures. Take an irreducible subvariation with Higgs bundle \n\nE1,0 π E0,1. Then one can show deg E1,0\n\nrk E1,0 − deg E0,1\n\nrk E0,1 ≤ deg Ω 1 \n\n> Y\n\n(log Γ) \n\ndim Y\n\nwhere ωY (Γ) is nef and big, and degree is taken with respect to this sheaf. The problem is to determine when equality holds. A conjecture in this direction is that U is a Shimura variety. This is true if Y is a curve and for moduli spaces Ag for small g.", "clean_statement": "Problem 2.3. Assume that MH is a moduli stack for abelian varieties or for a class of manifolds for which local Torelli holds. Let U, Y, and Γ be as in the previous problem. There is a variation of Hodge structures. Take an irreducible subvariation with Higgs bundle\n\nE1,0 π E0,1. Then one can show deg E1,0\n\nrk E1,0 − deg E0,1\n\nrk E0,1 ≤ deg Ω 1\n\n> Y\n\n(log Γ)\n\ndim Y\n\nwhere ωY (Γ) is nef and big, and degree is taken with respect to this sheaf. The problem is to determine when equality holds. A conjecture in this direction is that U is a Shimura variety. This is true if Y is a curve and for moduli spaces Ag for small g.", "statement_status": "exact", "statement_verification": "The canonical record comes from Problem 2.3 of the AIM workshop list *Compact moduli spaces and birational geometry*. The PDF extraction loses superscripts, a fraction bar, and the direct-sum symbol. The official TeX source defines `\\dirsum` to be `\\varoplus`; thus the extracted symbol `π` is not a map.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 2.3\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[329]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.3. Assume that MH is a moduli stack for abelian varieties or for a class of manifolds for which local Torelli holds. Let U, Y, and Γ be as in the previous problem. There is a variation of Hodge structures. Take an irreducible subvariation with Higgs bundle \\n\\nE1,0 π E0,1. Then one can show deg E1,0\\n\\nrk E1,0 − deg E0,1\\n\\nrk E0,1 ≤ deg Ω 1 \\n\\n> Y\\n\\n(log Γ) \\n\\ndim Y\\n\\nwhere ωY (Γ) is nef and big, and degree is taken with respect to this sheaf. The problem is to determine when equality holds. A conjecture in this direction is that U is a Shimura variety. This is true if Y is a curve and for moduli spaces Ag for small g.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0330", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a product of smooth projective log pairs polarized by the product log-canonical bundle, the pullback of an irreducible complex weight-one VHS satisfying Arakelov equality on one factor remains irreducible and satisfies equality on the product. Applying this to the maximal-Higgs rank-two factor of the regular-octagon genus-two Teichmuller curve and a modular curve produces a surface mapping generically finitely to A_3 with one equality factor but non-special image. This disproves only the common generically-finite, single-summand implication; it is not a counterexample to the source's literal full-dimensional etale-chart qualifier.\n\nCandidate contribution (lemma; novelty confidence low): Factor-persistence lemma: if an irreducible complex weight-one VHS on one factor of a product of smooth log pairs attains the Arakelov slope bound with respect to that factor's log-canonical bundle, then its pullback attains the bound with respect to the product log-canonical bundle; the exact multinomial intersection formula yields a non-Shimura Teichmuller-times-modular diagnostic." }, { "id": 20000331, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0331", "title": "A fixed-base product obstruction to literal uniform isomorphism counts", "statement": "Problem 2.\n4. (Viehweg) Does there exist a universal bound for the number of families of minimal manifolds with Hilbert polynomial H over a curve of genus q with s singular fibers depending only on H, q, and s, and not any more features of the geometry of the fibers? 4 MICHAEL A. VAN OPSTALL\n\n3. GIT\n\n3.1. Background. Geometric invariant theory plays an essential role in the usual construc-tion of Mg. GIT separates the points of a Hilbert scheme into three classes with respect to an action of a reductive group G: stable, semistable, and unstable. The quotient of the set of stable points by the group exists in a nice sense and is a moduli space of these GIT stable varieties. Note that GIT stability is not compatible with stability in the sense of the MMP. There are \"MMP-stable\" singularities which are asymptotically GIT-unstable. Also, some GIT semistable points are not MMP-stable. A good quotient of the semistable locus by the group G exists, but is not a moduli space, since some orbits of the group action corresponding to nonisomorphic varieties have intersection closures. 3.2. Problems.", "original_statement": "Problem 2.\n4. (Viehweg) Does there exist a universal bound for the number of families of minimal manifolds with Hilbert polynomial H over a curve of genus q with s singular fibers depending only on H, q, and s, and not any more features of the geometry of the fibers? 4 MICHAEL A. VAN OPSTALL \n\n3. GIT \n\n3.1. Background. Geometric invariant theory plays an essential role in the usual construc-tion of Mg. GIT separates the points of a Hilbert scheme into three classes with respect to an action of a reductive group G: stable, semistable, and unstable. The quotient of the set of stable points by the group exists in a nice sense and is a moduli space of these GIT stable varieties. Note that GIT stability is not compatible with stability in the sense of the MMP. There are \"MMP-stable\" singularities which are asymptotically GIT-unstable. Also, some GIT semistable points are not MMP-stable. A good quotient of the semistable locus by the group G exists, but is not a moduli space, since some orbits of the group action corresponding to nonisomorphic varieties have intersection closures. 3.2. Problems.", "clean_statement": "Problem 2.\n4. (Viehweg) Does there exist a universal bound for the number of families of minimal manifolds with Hilbert polynomial H over a curve of genus q with s singular fibers depending only on H, q, and s, and not any more features of the geometry of the fibers? 4 MICHAEL A. VAN OPSTALL\n\n3. GIT\n\n3.1. Background. Geometric invariant theory plays an essential role in the usual construc-tion of Mg. GIT separates the points of a Hilbert scheme into three classes with respect to an action of a reductive group G: stable, semistable, and unstable. The quotient of the set of stable points by the group exists in a nice sense and is a moduli space of these GIT stable varieties. Note that GIT stability is not compatible with stability in the sense of the MMP. There are \"MMP-stable\" singularities which are asymptotically GIT-unstable. Also, some GIT semistable points are not MMP-stable. A good quotient of the semistable locus by the group G exists, but is not a moduli space, since some orbits of the group action corresponding to nonisomorphic varieties have intersection closures. 3.2. Problems.", "statement_status": "exact", "statement_verification": "The source is the AIM workshop list *Compact moduli spaces and birational geometry*. The official PDF gives the following statement in Section 2, “Moduli spaces of manifolds”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[330]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.\\n4. (Viehweg) Does there exist a universal bound for the number of families of minimal manifolds with Hilbert polynomial H over a curve of genus q with s singular fibers depending only on H, q, and s, and not any more features of the geometry of the fibers? 4 MICHAEL A. VAN OPSTALL \\n\\n3. GIT \\n\\n3.1. Background. Geometric invariant theory plays an essential role in the usual construc-tion of Mg. GIT separates the points of a Hilbert scheme into three classes with respect to an action of a reductive group G: stable, semistable, and unstable. The quotient of the set of stable points by the group exists in a nice sense and is a moduli space of these GIT stable varieties. Note that GIT stability is not compatible with stability in the sense of the MMP. There are \\\"MMP-stable\\\" singularities which are asymptotically GIT-unstable. Also, some GIT semistable points are not MMP-stable. A good quotient of the semistable locus by the group G exists, but is not a moduli space, since some orbits of the group action corresponding to nonisomorphic varieties have intersection closures. 3.2. Problems.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0331", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "The literal isomorphism-class reading is false even with non-isotriviality and the pointed base fixed: a generic quartic Lefschetz pencil gives a fixed base B = P^1 and fixed 27-point singular set S, and products with infinitely many genus-4 curves having simple pairwise nonisogenous Jacobians yield pairwise nonisomorphic non-isotrivial families of canonically polarized surfaces with common canonical Hilbert polynomial H(m) = 6(2m-1)^2. Albanese/Jacobian cancellation proves nonisomorphism, while a connected level-moduli family proves that all examples have a single deformation type. Thus the example is compatible with the Kovacs-Lieblich uniform theorem for deformation types, which is the accepted canonically polarized interpretation of the AIM problem.\n\nCandidate contribution (counterexample; novelty confidence low): Candidate novelty: an unconditional product counterexample in which B = P^1, the exact 27-point bad set S, non-isotriviality, and H(m) = 6(2m-1)^2 are all fixed, with pairwise nonisomorphism verified by simple-Jacobian isogeny cancellation and all examples explicitly shown to lie in one deformation type." }, { "id": 20000332, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0332", "title": "Nonstabilization of pluricanonical GIT and a finite-wall diagnostic", "statement": "Problem 3.1. (from Koll´ ar's lecture) Fix a class of (pluri)canonically embedded manifolds in Pn. There is a notion of GIT semistability for Hilbert points in the closure of the Hilbert points of the class of manifolds considered. Now embed the family in another Pm using a higher power of KX and consider the GIT semistability conditions in that projective space. Is there an integer N such that the class of GIT semistable surfaces stabilizes for embeddings by M K X for M > N? This problem is fundamental, and the answer for curves is that N = 5.", "original_statement": "Problem 3.1. (from Koll´ ar's lecture) Fix a class of (pluri)canonically embedded manifolds in Pn. There is a notion of GIT semistability for Hilbert points in the closure of the Hilbert points of the class of manifolds considered. Now embed the family in another Pm using a higher power of KX and consider the GIT semistability conditions in that projective space. Is there an integer N such that the class of GIT semistable surfaces stabilizes for embeddings by M K X for M > N? This problem is fundamental, and the answer for curves is that N = 5.", "clean_statement": "Problem 3.1. (from Koll´ ar's lecture) Fix a class of (pluri)canonically embedded manifolds in Pn. There is a notion of GIT semistability for Hilbert points in the closure of the Hilbert points of the class of manifolds considered. Now embed the family in another Pm using a higher power of KX and consider the GIT semistability conditions in that projective space. Is there an integer N such that the class of GIT semistable surfaces stabilizes for embeddings by M K X for M > N? This problem is fundamental, and the answer for curves is that N = 5.", "statement_status": "exact", "statement_verification": "The record is Problem 3.1 in the AIM problem list *Compact moduli spaces and birational geometry*. The official PDF gives:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 3.1\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[331]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.1. (from Koll´ ar's lecture) Fix a class of (pluri)canonically embedded manifolds in Pn. There is a notion of GIT semistability for Hilbert points in the closure of the Hilbert points of the class of manifolds considered. Now embed the family in another Pm using a higher power of KX and consider the GIT semistability conditions in that projective space. Is there an integer N such that the class of GIT semistable surfaces stabilizes for embeddings by M K X for M > N? This problem is fundamental, and the answer for curves is that N = 5.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0332", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Wang and Xu's published surface counterexamples refute eventual stabilization in the intended sense of a compatible asymptotic pluricanonical GIT compactification: their KSBA limits violate a necessary multiplicity bound for weak asymptotic Chow semistability, and uniqueness of the KSBA filling rules out replacing the limit by another compatible weakly asymptotically semistable filling. The Chow obstruction also excludes asymptotic Hilbert semistability recurring for infinitely many sufficiently divisible pluricanonical exponents. In addition, this attempt proves a two-exponent finite-wall lemma: each fixed finitely generated compatible filtration has only finitely many asymptotic Hilbert verdict changes along a divisible pluricanonical ray; uniform locus stabilization follows only under an explicit finite compatible Kempf-detection hypothesis.\n\nCandidate contribution (finite_wall_reduction; novelty confidence low): For a finitely generated filtration of a canonical ring, after passage to a standard Veronese the trace-normalized Hilbert weight is Phi_F(M,k)=h(M)w_F(Mk)-k w_F(M)h(Mk), so its eventual lexicographic sign in k changes only finitely many times as M ranges over a fixed divisible ray. Consequently, a finite stratification equipped with finite compatible filtration lists that detect every large-M instability has an eventually constant semistable abstract locus; thus failure of stabilization forces unboundedly moving destabilizing filtration types or failure of compatibility with the canonical ring." }, { "id": 20000333, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0333", "title": "Explicit diagonal 1-PS bounds and the current hypersurface GIT classification status", "statement": "Problem 3.2. Determine the GIT (semi)stable quintic surfaces, quartic threefolds, quintic threefolds. For hypersurfaces of a given dimension and degree, is there a bound on the exponents appearing in the diagonal 1-PS that need to be checked? All smaller cases have been checked by various authors.", "original_statement": "Problem 3.2. Determine the GIT (semi)stable quintic surfaces, quartic threefolds, quintic threefolds. For hypersurfaces of a given dimension and degree, is there a bound on the exponents appearing in the diagonal 1-PS that need to be checked? All smaller cases have been checked by various authors.", "clean_statement": "Problem 3.2. Determine the GIT (semi)stable quintic surfaces, quartic threefolds, quintic threefolds. For hypersurfaces of a given dimension and degree, is there a bound on the exponents appearing in the diagonal 1-PS that need to be checked? All smaller cases have been checked by various authors.", "statement_status": "exact", "statement_verification": "The record is Problem 3.2 from the 2004 AIM workshop *Compact moduli spaces and birational geometry*. The official AIM PDF and its TeX source agree with the canonical JSON record. The exact statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 3.2\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[332]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.2. Determine the GIT (semi)stable quintic surfaces, quartic threefolds, quintic threefolds. For hypersurfaces of a given dimension and degree, is there a bound on the exponents appearing in the diagonal 1-PS that need to be checked? All smaller cases have been checked by various authors.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0333", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For degree-d hypersurfaces in projective N-space, put D_{N,d}=ceil(sqrt(N)d^(N-1)) and B_{N,d}=N D_{N,d}. Every weak Hilbert-Mumford witness can be replaced, after the required coordinate change, by a primitive diagonal witness with maximum absolute exponent at most D_{N,d}; every diagonal 1-PS has a primitive representative with the identical sign on every degree-d monomial and maximum absolute exponent at most B_{N,d}, so strict instability witnesses are bounded by B_{N,d}. This gives stability/semistability bounds 44/132 for quintic surfaces, 128/512 for quartic threefolds, and 250/1000 for quintic threefolds. Current algorithms give finite state descriptions for all three cases, but their complete geometric interpretation remains open or announcement-level in important parts.\n\nCandidate contribution (explicit_bound; novelty confidence low): The candidate novel contribution is the explicit pair D_{N,d}=ceil(sqrt(N)d^(N-1)) and B_{N,d}=N D_{N,d}, including the theorem that B_{N,d} gives a primitive representative preserving the sign of every degree-d monomial weight and that D_{N,d} separately suffices for weak Hilbert-Mumford witnesses." }, { "id": 20000334, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0334", "title": "Nonnested span ranks and corrected jet-span estimates for Hilbert weights", "statement": "Problem 3.\n3. (Morrison) For a variety X and ample line bundle L, can we work harder on the combinatorics of the monomial subspaces of H0(X, L m) to get easier estimates for the weights rλ,X of the Hilbert point of X with respect to the filtration given by some 1-PS λ?For example (1) Find more flexible filtrations of H0(X, L m) with stages of the type occurring in geo-metric estimates (e.g. cusps are not semistable) and compute the leading term of the resulting estimates for rλ,X.(2) Find methods for producing estimates for rλ,X which use estimates for non-nested\n\ncollections of subspaces of H0(X, L ). 4. Examples and applications", "original_statement": "Problem 3.\n3. (Morrison) For a variety X and ample line bundle L, can we work harder on the combinatorics of the monomial subspaces of H0(X, L m) to get easier estimates for the weights rλ,X of the Hilbert point of X with respect to the filtration given by some 1-PS λ?For example (1) Find more flexible filtrations of H0(X, L m) with stages of the type occurring in geo-metric estimates (e.g. cusps are not semistable) and compute the leading term of the resulting estimates for rλ,X.(2) Find methods for producing estimates for rλ,X which use estimates for non-nested \n\ncollections of subspaces of H0(X, L ). 4. Examples and applications", "clean_statement": "Problem 3.\n3. (Morrison) For a variety X and ample line bundle L, can we work harder on the combinatorics of the monomial subspaces of H0(X, L m) to get easier estimates for the weights rλ,X of the Hilbert point of X with respect to the filtration given by some 1-PS λ?For example (1) Find more flexible filtrations of H0(X, L m) with stages of the type occurring in geo-metric estimates (e.g. cusps are not semistable) and compute the leading term of the resulting estimates for rλ,X.(2) Find methods for producing estimates for rλ,X which use estimates for non-nested\n\ncollections of subspaces of H0(X, L ). 4. Examples and applications", "statement_status": "exact", "statement_verification": "The canonical record comes from the AIM workshop *Compact moduli spaces and birational geometry*, source file `aim-algebraic-geometry-notes.json`, record 333. The PDF source is the AIM list *Open Problems in Compact Moduli Spaces and Birational Geometry*. The extraction inserted a line break in the problem number and appended the next heading, “4. Examples and applications.” Inspection of the PDF shows that the heading is not part of the problem.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[333]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.\\n3. (Morrison) For a variety X and ample line bundle L, can we work harder on the combinatorics of the monomial subspaces of H0(X, L m) to get easier estimates for the weights rλ,X of the Hilbert point of X with respect to the filtration given by some 1-PS λ?For example (1) Find more flexible filtrations of H0(X, L m) with stages of the type occurring in geo-metric estimates (e.g. cusps are not semistable) and compute the leading term of the resulting estimates for rλ,X.(2) Find methods for producing estimates for rλ,X which use estimates for non-nested \\n\\ncollections of subspaces of H0(X, L ). 4. Examples and applications\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0334", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any finite family of nonnested subspaces carrying certified filtration-weight caps, ordering the caps and taking cumulative sums gives a sharp upper bound on the minimum basis weight in terms of only the cumulative ranks; under polynomial rank asymptotics this yields an explicit leading Hilbert-weight coefficient. For smooth curves, a jet-surjectivity argument proves the geometric Span Lemma used in Swinarski's method without the unrestricted inclusion-exclusion identity, which fails for arbitrary triples of subspaces. A two-point divisor family then gives a strictly better leading estimate by (b-a)(beta-gamma)m^2+O(m), conditional on the stated common low-weight cap and fallback cap.\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: the cumulative-rank or representable-polymatroid formulation gives the optimal filtered-vector-space bound derivable from arbitrary nonnested weight-cap certificates, while the jet-span lemma supplies a correct replacement proof for the false unrestricted auxiliary inclusion-exclusion step; in the explicit two-centre curve family the resulting leading gain is exactly (b-a)(beta-gamma)." }, { "id": 20000335, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0335", "title": "Split-branch degenerations as a stable-surface boundary application", "statement": "Problem 4.1. Find applications of moduli of stable surfaces to surface theory. For example degenerations of curves are used to prove Brill-Noether type statements.", "original_statement": "Problem 4.1. Find applications of moduli of stable surfaces to surface theory. For example degenerations of curves are used to prove Brill-Noether type statements.", "clean_statement": "Problem 4.1. Find applications of moduli of stable surfaces to surface theory. For example degenerations of curves are used to prove Brill-Noether type statements.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 4.1\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[334]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.1. Find applications of moduli of stable surfaces to surface theory. For example degenerations of curves are used to prove Brill-Noether type statements.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0335", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every integer d at least 4, the equation z^2=f_d^2+t g_{2d} gives a flat Gorenstein stable family from the nonnormal stable surface X_d=P^2 union_C P^2 to smooth double planes. The smooth-fiber invariants are K^2=2(d-3)^2, q=0, p_g=(d-1)(d-2)/2=g(C), chi=(d^2-3d+4)/2, c_2=4d^2-6d+6, and signature 2-2d^2, with Noether defect (d-4)(d-5). In the double-cover branch chart, the normal space to the doubled-branch locus at [f^2] is canonically V_{2d}/fV_d=H^0(C,T^1_X), of dimension 3d(d+1)/2. A generic normal direction has 2d^2 simple zeros and produces exactly 2d^2 ordinary double points in the total threefold; positive degree of T^1 obstructs a smooth total space while the reduced central model X_d is fixed.\n\nCandidate contribution (normal_space_smoothing_lemma; novelty confidence low): For the split double-plane boundary X_d=P^2 union_C P^2, the normal quotient to the square-branch locus is canonically V_{2d}/fV_d isomorphic to H^0(C,T^1_{X_d}); every global local-smoothing section lifts within the branch chart, and a generic normal vector has a zero divisor of length 2d^2 realized as exactly 2d^2 threefold nodes. Together with dim H^0(T^1)=3d(d+1)/2 and Noether defect (d-4)(d-5), this gives an explicit, testable boundary-to-surface dictionary." }, { "id": 20000336, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0336", "title": "A formal local model at a type-2b stable numerical quintic", "statement": "Problem 4.2. Understand the geometry of the moduli space of stable surfaces better. Here it is probably necessary to find some nice components and study their geometry. In general, after fixing even the differentiable structure of the underlying four-manifolds, the moduli space is disconnected, its connected components are reducible, and its irreducible components may be everywhere nonreduced. In addition, the boundary may not be a divisor, and the next best substitute is not a normal crossings divisor. Also, Vakil shows that the moduli space of canonically polarized surfaces is arbitrarily singular (in a precise sense). OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 5\n\n4.1. Intersection theory.", "original_statement": "Problem 4.2. Understand the geometry of the moduli space of stable surfaces better. Here it is probably necessary to find some nice components and study their geometry. In general, after fixing even the differentiable structure of the underlying four-manifolds, the moduli space is disconnected, its connected components are reducible, and its irreducible components may be everywhere nonreduced. In addition, the boundary may not be a divisor, and the next best substitute is not a normal crossings divisor. Also, Vakil shows that the moduli space of canonically polarized surfaces is arbitrarily singular (in a precise sense). OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 5\n\n4.1. Intersection theory.", "clean_statement": "Problem 4.2. Understand the geometry of the moduli space of stable surfaces better. Here it is probably necessary to find some nice components and study their geometry. In general, after fixing even the differentiable structure of the underlying four-manifolds, the moduli space is disconnected, its connected components are reducible, and its irreducible components may be everywhere nonreduced. In addition, the boundary may not be a divisor, and the next best substitute is not a normal crossings divisor. Also, Vakil shows that the moduli space of canonically polarized surfaces is arbitrarily singular (in a precise sense). OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 5\n\n4.1. Intersection theory.", "statement_status": "exact", "statement_verification": "The canonical record comes from Problem 4.2 of the AIM workshop list *Compact moduli spaces and birational geometry*. Inspection of the official source TeX and the source PDF recovers the problem as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 4.2\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[335]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.2. Understand the geometry of the moduli space of stable surfaces better. Here it is probably necessary to find some nice components and study their geometry. In general, after fixing even the differentiable structure of the underlying four-manifolds, the moduli space is disconnected, its connected components are reducible, and its irreducible components may be everywhere nonreduced. In addition, the boundary may not be a divisor, and the next best substitute is not a normal crossings divisor. Also, Vakil shows that the moduli space of canonically polarized surfaces is arbitrarily singular (in a precise sense). OPEN PROBLEMS IN COMPACT MODULI SPACES AND BIRATIONAL GEOMETRY 5\\n\\n4.1. Intersection theory.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0336", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a general Rana type-2b stable numerical quintic W, the completed miniversal Q-Gorenstein deformation ring is C[[a,b,z_1,...,z_38,tau]]/(ab), and the locus preserving its distinguished 1/4(1,1) Wahl point is the reduced Cartier divisor tau=0. The two ambient branches are smooth of dimension 40 and meet in dimension 39; the two boundary branches are smooth of dimension 39 and meet in the 38-dimensional type-2b stratum. The normalization is the direct sum of the two branch rings and the conductor is (a,b). This equation is asserted for the miniversal/KSBA stack chart; with G=Aut(W), the formal stack germ is [Spf R/G] and the coarse completed local ring is R^G, whose explicit geometry requires the uncomputed G-action.\n\nCandidate contribution (formal_local_model; novelty confidence low): Candidate novelty: Rana's proved transverse equisingular Kuranishi branches and analytic product with the one-dimensional Wahl smoothing imply the explicit completed ring C[[a,b,z_1,...,z_38,tau]]/(ab); its normalization, conductor (a,b), identification of the type-2b locus as the boundary conductor image, and the stack/coarse invariant-ring correction are developed together." }, { "id": 20000337, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0337", "title": "Virtual classes for KSBA stable-surface moduli", "statement": "Problem 4.\n3. (Hassett) Does the moduli space of stable surfaces have a virtual fundamental class?", "original_statement": "Problem 4.\n3. (Hassett) Does the moduli space of stable surfaces have a virtual fundamental class?", "clean_statement": "Problem 4.\n3. (Hassett) Does the moduli space of stable surfaces have a virtual fundamental class?", "statement_status": "exact", "statement_verification": "The canonical record is from the AIM workshop list *Compact moduli spaces and birational geometry*. Its extracted text is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[336]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.\\n3. (Hassett) Does the moduli space of stable surfaces have a virtual fundamental class?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0337", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Jiang's current arXiv:2206.00575v7 claims a global Chow class on a fixed-invariant KSBA stack by pushing forward the virtual fundamental class of a proper auxiliary stack of lci covers; this is not a perfect obstruction theory on the full KSBA stack, whose natural index-one deformation complex can have higher obstructions. Independently, a universal proper flat lci and Gorenstein family of stable stacky surfaces with finite reduced automorphisms has a two-term Behrend-Fantechi perfect obstruction theory, and any two compatible auxiliary pushforward classes agreeing on the intrinsic lci locus differ by a class supported on the non-lci boundary.\n\nCandidate contribution (criterion; novelty confidence low): If two proper auxiliary KSBA virtual-cycle constructions of virtual dimension v agree on an open lci locus U and B is its closed complement, then their pushforward classes differ by an element of the image of A_v(B)_Q in A_v(M)_Q; hence the class is presentation-independent whenever A_v(B)_Q vanishes, in particular when dim(B) < v." }, { "id": 20000338, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0338", "title": "A clean excess-two intersection of line- and conic-branch loci", "statement": "Problem 4.4. Find natural loci in moduli spaces of surfaces which would be interesting to intersect with each other.", "original_statement": "Problem 4.4. Find natural loci in moduli spaces of surfaces which would be interesting to intersect with each other.", "clean_statement": "Problem 4.4. Find natural loci in moduli spaces of surfaces which would be interesting to intersect with each other.", "statement_status": "exact", "statement_verification": "The canonical record is number 4.4 in the AIM workshop list *Compact moduli spaces and birational geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 4.4\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[337]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.4. Find natural loci in moduli spaces of surfaces which would be interesting to intersect with each other.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0338", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "On the plane-octic chart for Gorenstein stable surfaces with K_X^2=2 and chi(O_X)=4, the line-factor locus (codimension 7) and conic-factor locus (codimension 12) meet near a generic (1,2,5)-factorized branch along a smooth codimension-17 locus. The intersection is clean with excess rank 2. Its excess normal space is H^0(O_{L intersect Q}(8)), canonically the common smoothing space of the two A_1 surface singularities above L intersect Q. The excess bundle and its second Chern class are computed explicitly, and the projective closure of the (1,2,5) locus has degree 18,648,630. This is a local/open branch-chart result, not a global KSBA intersection number.\n\nCandidate contribution (excess_intersection_lemma; novelty confidence low): At a generic (1,2,5)-factorized plane-octic branch, the line-factor and conic-factor loci have a clean excess-two intersection whose excess bundle is O(1,1,1) tensor p_*O_{L intersect Q}(8), with c_2=136a^2+108ab+36b^2+17ac+9bc+c^2; its fiber is the shared smoothing space of the two A_1 points over the line-conic intersection." }, { "id": 20000339, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0339", "title": "A line-arrangement deformation test for surface-source stable maps", "statement": "Problem 4.\n5. (Alexeev) Is there a Gromov-Witten theory or quantum cohomology theory arising from stable pairs, perhaps from stable maps f: ( P2, D = D1 + · · · + Dn) → X, where\n\nD is a union of lines? 4.2. Explicit examples.", "original_statement": "Problem 4.\n5. (Alexeev) Is there a Gromov-Witten theory or quantum cohomology theory arising from stable pairs, perhaps from stable maps f: ( P2, D = D1 + · · · + Dn) → X, where \n\nD is a union of lines? 4.2. Explicit examples.", "clean_statement": "Problem 4.\n5. (Alexeev) Is there a Gromov-Witten theory or quantum cohomology theory arising from stable pairs, perhaps from stable maps f: ( P2, D = D1 + · · · + Dn) → X, where\n\nD is a union of lines? 4.2. Explicit examples.", "statement_status": "exact", "statement_verification": "The canonical JSON record has two extraction errors: “Problem 4.\\n5.” is Problem 4.5, and “4.2. Explicit examples” is the heading following the problem, not part of it. The official AIM TeX source and PDF give the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[338]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.\\n5. (Alexeev) Is there a Gromov-Witten theory or quantum cohomology theory arising from stable pairs, perhaps from stable maps f: ( P2, D = D1 + · · · + Dn) → X, where \\n\\nD is a union of lines? 4.2. Explicit examples.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0339", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an ordered general-position arrangement D of n at least 4 lines in P^2, the natural surface-map deformation complex C_f=[T_{P^2}(-log D) -> f^*T_Y] in degrees 0 and 1 has expected dimension chi(f^*T_Y)+2n-8 and extra group H^3(C_f)=H^2(P^2,f^*T_Y). Constant maps are unobstructed, and for four ordered lines their smooth degree-zero locus is the target Y with classical intersection-point evaluations. Conversely, for the negative section P^2 -> P_{P^2}(O direct-sum O(-m)), m at least 3, H^3(C_f) has dimension binomial(m-1,2), giving an explicit smooth-source and smooth-target obstruction to a uniform two-term natural obstruction theory.\n\nCandidate contribution (explicit_obstruction_example; novelty confidence low): The ordered line-arrangement locus simultaneously admits the explicit expected-dimension formula and unobstructed four-line constant-sector normalization, while the negative section in P_{P^2}(O direct-sum O(-m)) has an extra obstruction group of exactly binomial(m-1,2) dimensions." }, { "id": 20000340, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0340", "title": "A wall-free KSBA sector of quartic K3 degenerations", "statement": "Problem 4.6. Can one work out a geometric compactification of polarized K3 surfaces or Calabi-Yau manifolds in general using the framework of Alexeev/Koll´ ar/Shepherd-Barron? For example, let ( X, H ) be a pair where X is a K3 surface and H is a very ample divisor with H2 = 4. Can the stable pairs occurring as degenerations of these objects be classi-fied? Compare the resulting singularities with the work of Shah on GIT-semistable quartic surfaces. Also compare with Martin Olsson's work on moduli of log K3 surfaces.", "original_statement": "Problem 4.6. Can one work out a geometric compactification of polarized K3 surfaces or Calabi-Yau manifolds in general using the framework of Alexeev/Koll´ ar/Shepherd-Barron? For example, let ( X, H ) be a pair where X is a K3 surface and H is a very ample divisor with H2 = 4. Can the stable pairs occurring as degenerations of these objects be classi-fied? Compare the resulting singularities with the work of Shah on GIT-semistable quartic surfaces. Also compare with Martin Olsson's work on moduli of log K3 surfaces.", "clean_statement": "Problem 4.6. Can one work out a geometric compactification of polarized K3 surfaces or Calabi-Yau manifolds in general using the framework of Alexeev/Koll´ ar/Shepherd-Barron? For example, let ( X, H ) be a pair where X is a K3 surface and H is a very ample divisor with H2 = 4. Can the stable pairs occurring as degenerations of these objects be classi-fied? Compare the resulting singularities with the work of Shah on GIT-semistable quartic surfaces. Also compare with Martin Olsson's work on moduli of log K3 surfaces.", "statement_status": "exact", "statement_verification": "The source is Problem 4.6 in §4.2 (“Explicit examples”) of the AIM workshop notes *Compact moduli spaces and birational geometry*. The PDF text, with only the evident typographical/OCR repairs \\(H2\\mapsto H^2\\), “classi-fied” \\(\\mapsto\\) “classified,” and the accent in Kollár restored, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 4.6\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[339]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.6. Can one work out a geometric compactification of polarized K3 surfaces or Calabi-Yau manifolds in general using the framework of Alexeev/Koll´ ar/Shepherd-Barron? For example, let ( X, H ) be a pair where X is a K3 surface and H is a very ample divisor with H2 = 4. Can the stable pairs occurring as degenerations of these objects be classi-fied? Compare the resulting singularities with the work of Shah on GIT-semistable quartic surfaces. Also compare with Martin Olsson's work on moduli of log K3 surfaces.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0340", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Modern work establishes KSBA compactifications for polarized K3 surfaces in every degree and gives a strong explicit classification of the degree-four Type II nef-lambda-family/Tyurin sector, but a complete marked-hyperplane/Type III classification was not located. The new proved special case is that every reduced simple-normal-crossing factorized quartic with a general Cartier hyperplane section is a Gorenstein KSBA stable pair for every rational coefficient 022\\), \\(\\overline{\\mathcal M}_g\\) is of general type, and in\nthe other known cases, the Kodaira dimension is negative.", "original_statement": "Problem 5.2. Determine the Kodaira dimension of Mg when g = 15 or 17 ≤ g ≤ 22. For g > 22 Mg is of general type, and in the other known cases, the Kodaira dimension is negative. 5.2. Other questions.", "clean_statement": "**Problem 5.2.** Determine the Kodaira dimension of\n\\(\\overline{\\mathcal M}_g\\) when \\(g=15\\) or \\(17\\leq g\\leq22\\).\nFor \\(g>22\\), \\(\\overline{\\mathcal M}_g\\) is of general type, and in\nthe other known cases, the Kodaira dimension is negative.", "statement_status": "corrected_verified", "statement_verification": "The typography in the official AIM source resolves two extraction defects. On the printed page, a horizontal bar is drawn over both occurrences of \\(\\mathcal M_g\\); the repository's text extractor discarded those bars. Also, the final words “5.2. Other questions” are the heading of the next subsection, not part of Problem 5.2. The recovered statement is therefore: Source: AIM, *Compact moduli spaces and birational geometry*, Problem 5.2, [official PDF](https://aimath.org/WWN/birational/birational.pdf).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 5.2\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[343]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 5.2. Determine the Kodaira dimension of Mg when g = 15 or 17 ≤ g ≤ 22. For g > 22 Mg is of general type, and in the other known cases, the Kodaira dimension is negative. 5.2. Other questions.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0344", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The present genus-by-genus answer is kappa = -infinity for genus 15, unknown for genera 17 through 21, and general type for genus 22. In addition, for an irreducible-nodal Lefschetz pencil of curves in |L| on a smooth surface S, the attempt proves the coarse-canonical formula 2 K_{Mbar_g}.B = 4 K_S^2 - 22 chi(O_S) + L^2 + 5 K_S.L, with an explicit movability condition for the BDPP implication. For K3 pencils this intersection is g-23, but the constructed K3-section family has codimension at least 2g-22 in every open genus, so it is not covering and cannot itself invoke BDPP. An exact standard-divisor wall ledger and a positive-cone obstruction complement this moving-curve criterion.\n\nCandidate contribution (conditional criterion; novelty confidence low): For surface Lefschetz pencils meeting only delta_0, the exact coarse-canonical surface-invariant formula, together with the K3 parameter count, yields a testable trapped-negativity criterion: throughout genera 17 through 21 the K3 pencil has negative canonical degree g-23 while this K3-section construction is confined to codimension at least 2g-22; any BDPP application therefore requires a genuinely covering realization, and any effective divisor avoiding a proposed surface family must satisfy the explicit slope inequality derived in the artifacts." }, { "id": 20000345, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0345", "title": "A level-cover ramification budget for rational and affine coarse curves in M_g", "statement": "Problem 5.3. Does the coarse moduli space Mg contain a P1? An A1? It is known that the moduli stack in this case is algebraically hyperbolic, but the same is not likely to be true for the moduli space.", "original_statement": "Problem 5.3. Does the coarse moduli space Mg contain a P1? An A1? It is known that the moduli stack in this case is algebraically hyperbolic, but the same is not likely to be true for the moduli space.", "clean_statement": "Problem 5.3. Does the coarse moduli space Mg contain a P1? An A1? It is known that the moduli stack in this case is algebraically hyperbolic, but the same is not likely to be true for the moduli space.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 5.3 from the AIM workshop notes *Compact moduli spaces and birational geometry*. The repository extraction reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 5.3\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[344]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 5.3. Does the coarse moduli space Mg contain a P1? An A1? It is known that the moduli stack in this case is algebraically hyperbolic, but the same is not likely to be true for the moduli space.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0345", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Oort proved that a coarse M_g contains a complete rational curve for some genus, and Bryan-Donagi-Stipsicz constructed nonconstant maps P^1 -> M_{3n^3-n^2+1} for every n >= 2 (beginning with g = 21, 73, 177, 351), so restriction gives nonconstant A^1 maps for infinitely many genera. These sources do not automatically give embedded smooth P^1 or A^1 subvarieties, and no every-fixed-g theorem was found. This attempt proves that for any nonconstant complex coarse map f: P^1 minus S -> M_g, the normalized connected Galois full-level pullback has actual internal ramification indices m_i satisfying sum_i(1-1/m_i) > 2-|S|; in a cyclic quotient arc of stabilizer order a and coarse contact order c, the actual index is a/gcd(a,c).\n\nCandidate contribution (obstruction lemma; novelty confidence low): For a nonconstant map f: P^1 minus S -> M_g over C, any normalized connected Galois component of a full-level-N pullback has internal ramification weight sum_i(1-1/m_i) > 2-|S|, and a local cyclic coarse arc x=t^a, x=z^c has effective index m=a/gcd(a,c), rather than the abstract stabilizer order a." }, { "id": 20000346, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0346", "title": "Covering genus and a contact-adjusted level-signature obstruction for moduli of curves", "statement": "Problem 5.\n4. (Farkas) For g large, what is the minimal genus of a curve passing through a general point of Mg? This can be asked for the moduli space or the moduli stack. For low\n\ng, the answer for the moduli stack is zero. One could ask this question for moduli spaces of canonically polarized manifolds, although I think the point of making g large is to ensure that Mg is not uniruled. There is nothing even approximating such a condition to ensure that MH is not unirational. 6 MICHAEL A. VAN OPSTALL\n\n6. Moduli of abelian varieties", "original_statement": "Problem 5.\n4. (Farkas) For g large, what is the minimal genus of a curve passing through a general point of Mg? This can be asked for the moduli space or the moduli stack. For low \n\ng, the answer for the moduli stack is zero. One could ask this question for moduli spaces of canonically polarized manifolds, although I think the point of making g large is to ensure that Mg is not uniruled. There is nothing even approximating such a condition to ensure that MH is not unirational. 6 MICHAEL A. VAN OPSTALL \n\n6. Moduli of abelian varieties", "clean_statement": "Problem 5.\n4. (Farkas) For g large, what is the minimal genus of a curve passing through a general point of Mg? This can be asked for the moduli space or the moduli stack. For low\n\ng, the answer for the moduli stack is zero. One could ask this question for moduli spaces of canonically polarized manifolds, although I think the point of making g large is to ensure that Mg is not uniruled. There is nothing even approximating such a condition to ensure that MH is not unirational. 6 MICHAEL A. VAN OPSTALL\n\n6. Moduli of abelian varieties", "statement_status": "exact", "statement_verification": "The source is Michael A. van Opstall's AIM list *Open Problems in Compact Moduli Spaces and Birational Geometry*, in Section 5.2, “Other questions.” Inspection of the official PDF, including its underlying text operators, shows that the displayed notation is the open coarse space \\(M_g\\), not \\(\\overline M_g\\). The recovered item is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[345]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 5.\\n4. (Farkas) For g large, what is the minimal genus of a curve passing through a general point of Mg? This can be asked for the moduli space or the moduli stack. For low \\n\\ng, the answer for the moduli stack is zero. One could ask this question for moduli spaces of canonically polarized manifolds, although I think the point of making g large is to ensure that Mg is not uniruled. There is nothing even approximating such a condition to ensure that MH is not unirational. 6 MICHAEL A. VAN OPSTALL \\n\\n6. Moduli of abelian varieties\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0346", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After separating the projective coarse, open-interior, ordinary-stack, and twisted-stack readings of the source, this attempt proves that the projective covering genus is zero for fiber genera 2 through 15 and at least 2 for fiber genera at least 22. Its new mathematical synthesis is a necessary log-orbifold inequality for any normalized coarse curve with automorphism-free generic point: after a Galois full-level lift, 2g(C)-2+|S|+sum_p(1-1/e_p)>0, where S is the reduced boundary preimage and the interior e_p are contact-adjusted inertia indices. Boundary ramification cancels exactly against the puncture count; when S is empty, a rational source requires total level ramification at least 2d+2 and an elliptic source at least 2.\n\nCandidate contribution (reduction; novelty confidence low): For a nonconstant normalized coarse curve C mapping to the compactified moduli space with generic point in the automorphism-free locus, the Galois full-level lift satisfies the exact necessary signature 2g(C)-2+|S|+sum_p(1-1/e_p)>0 with contact-adjusted interior inertia, because boundary ramification and deleted lifts cancel to one reduced puncture per boundary point; for curves contained in the open space this yields the quantitative ramification bounds deg(R)>=2d+2 in coarse genus zero and deg(R)>=2 in coarse genus one." }, { "id": 20000347, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0347", "title": "Andreotti-Mayer slope squeeze and boundary-blind complete subvarieties", "statement": "Problem 6.\n1. (Grushevsky) (1) Construct divisors on Ag of small slope. (2) Find explicit complete subvarieties of Ag. There is a bound on the codimension of such a variety in [KS03].", "original_statement": "Problem 6.\n1. (Grushevsky) (1) Construct divisors on Ag of small slope. (2) Find explicit complete subvarieties of Ag. There is a bound on the codimension of such a variety in [KS03].", "clean_statement": "Problem 6.\n1. (Grushevsky) (1) Construct divisors on Ag of small slope. (2) Find explicit complete subvarieties of Ag. There is a bound on the codimension of such a variety in [KS03].", "statement_status": "exact", "statement_verification": "The canonical JSON record is a slightly flattened OCR extraction from the AIM workshop list *Compact moduli spaces and birational geometry*. The official PDF has a section headed “6. Moduli of abelian varieties” and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[346]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 6.\\n1. (Grushevsky) (1) Construct divisors on Ag of small slope. (2) Find explicit complete subvarieties of Ag. There is a bound on the codimension of such a variety in [KS03].\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0347", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every g >= 4, extracting the non-two-torsion component N'_0 from Mumford's cycle relation N_0 = theta_null + 2N'_0 gives an explicit effective divisor class a_g lambda - b_g delta with b_g > 0 and the rigorously proved strict bounds 6 < s(N'_0) < 6 + 12/(g+1), together with exact formulas for both slope defects. For any complete d-fold X inside the open A_g, the boundary class restricts to zero, so (a lambda - b delta)|_X = a lambda|_X and its top intersection is a^d lambda^d.X; complete interior cycles therefore cannot detect the slope a/b. The literature status is also updated: exact minimum effective slopes are known only through g = 5 in the sources checked, while a submitted November 2025 preprint claims the exact maximum dimension of complete subvarieties.\n\nCandidate contribution (proposition; novelty confidence low): Candidate novelty: the exact positive defect formula showing 6 < s(N'_0) < 6 + 12/(g+1) for every g >= 4, paired with the boundary-blindness proposition that complete subvarieties of the open A_g cannot detect the boundary coefficient or slope by restriction and internal intersection." }, { "id": 20000348, "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0348", "title": "A Cox-restriction bridge for the perfect-cone compactification", "statement": "Problem 6.\n2. (Shepherd-Barron) Questions on Ag and the first Voronoi compactification\n\nAFg.(1) Run the MMP on AFg when g ≤ 10; in particular, flip its extremal ray. If g ≥ 12, AFg\n\nis the canonical model of Ag, and when g = 11 it is a minimal model. (2) There is a family of natural inclusions AFg → AFg+1 parameterized by the j-line. Does\n\nH∗(AFg, C) stabilize? To what? (3) Let M be the bundle of weight 1 modular forms and D the boundary divisor. Com-pute the intersection numbers M a.D b and the plurigenera of AFg.(4) Is the total coordinate ring of AFg finitely generated? (5) Is the divisor 12 M − D semi-ample for g ≥ 12? (6) Find the effective cone of AFg. This is the well-known question of finding the possible slopes of Siegel cusp forms.", "original_statement": "Problem 6.\n2. (Shepherd-Barron) Questions on Ag and the first Voronoi compactification \n\nAFg.(1) Run the MMP on AFg when g ≤ 10; in particular, flip its extremal ray. If g ≥ 12, AFg\n\nis the canonical model of Ag, and when g = 11 it is a minimal model. (2) There is a family of natural inclusions AFg → AFg+1 parameterized by the j-line. Does \n\nH∗(AFg, C) stabilize? To what? (3) Let M be the bundle of weight 1 modular forms and D the boundary divisor. Com-pute the intersection numbers M a.D b and the plurigenera of AFg.(4) Is the total coordinate ring of AFg finitely generated? (5) Is the divisor 12 M − D semi-ample for g ≥ 12? (6) Find the effective cone of AFg. This is the well-known question of finding the possible slopes of Siegel cusp forms.", "clean_statement": "Problem 6.\n2. (Shepherd-Barron) Questions on Ag and the first Voronoi compactification\n\nAFg.(1) Run the MMP on AFg when g ≤ 10; in particular, flip its extremal ray. If g ≥ 12, AFg\n\nis the canonical model of Ag, and when g = 11 it is a minimal model. (2) There is a family of natural inclusions AFg → AFg+1 parameterized by the j-line. Does\n\nH∗(AFg, C) stabilize? To what? (3) Let M be the bundle of weight 1 modular forms and D the boundary divisor. Com-pute the intersection numbers M a.D b and the plurigenera of AFg.(4) Is the total coordinate ring of AFg finitely generated? (5) Is the divisor 12 M − D semi-ample for g ≥ 12? (6) Find the effective cone of AFg. This is the well-known question of finding the possible slopes of Siegel cusp forms.", "statement_status": "exact", "statement_verification": "The JSON record is an OCR extraction in which \\(A_g^F\\) appears as strings such as `AFg`. The statement was recovered from the official AIM TeX source `https://aimath.org/WWN/birational/birational.tex` (lines 555--572 at the time of access) and checked against the official PDF. The source reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic geometry\nWorkshop: Compact moduli spaces and birational geometry\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/birational/birational.pdf\nCanonical location: aim-algebraic-geometry-notes.json notes[347]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 6.\\n2. (Shepherd-Barron) Questions on Ag and the first Voronoi compactification \\n\\nAFg.(1) Run the MMP on AFg when g ≤ 10; in particular, flip its extremal ray. If g ≥ 12, AFg\\n\\nis the canonical model of Ag, and when g = 11 it is a minimal model. (2) There is a family of natural inclusions AFg → AFg+1 parameterized by the j-line. Does \\n\\nH∗(AFg, C) stabilize? To what? (3) Let M be the bundle of weight 1 modular forms and D the boundary divisor. Com-pute the intersection numbers M a.D b and the plurigenera of AFg.(4) Is the total coordinate ring of AFg finitely generated? (5) Is the divisor 12 M − D semi-ample for g ≥ 12? (6) Find the effective cone of AFg. This is the well-known question of finding the possible slopes of Siegel cusp forms.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/birational/birational.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_GEOMETRY-0348", "aim-domain:algebraic-geometry", "aim-workshop:birational", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For g at least 2 over the complex numbers, after clearing the Q-Cartier indices of the Hodge and boundary classes, the lattice Cox ring of the perfect-cone compactification is generated by the exact Shepherd-Barron negative-boundary bigraded ring together with the canonical boundary section. Consequently, finite generation of that exact ring would make the compactification a Mori dream space, force 12M-D to be semi-ample in genus g and every lower genus, and force the genus-g effective slope to be rational and attained. A fan-level primitive-ray proof establishes the fixed-elliptic pullbacks of M, D, and 12M-D and yields explicit coarse and stack product-locus intersection recurrences.\n\nCandidate contribution (reduction; novelty confidence low): After a finite-index Cartier regrading, adjoining the canonical boundary section to the exact Shepherd-Barron ring generates the Cox ring; hence finite generation in genus g implies semi-ampleness of 12M-D in every genus h at most g and implies that the effective slope in genus g is rational and attained." }, { "id": 20000349, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0001", "title": "Cubic points in the plane and the two exceptional sets", "statement": "The goal is to formulate a Manin-type conjecture for degree $d$ points on Fano varieties. Let $X/k$ be a Fano variety defined over a number field $k$. For a subset $U\\subset X$, an integer $d$, a real number $B>0$, and a line bundle $\\mathscr L$ on $X$, define\n$$N_{X,\\mathscr L}(U,B;d) := \\#\\left\\{x\\in U(\\overline k):h_{\\mathscr L}(x)\\le B\\text{ and }[\\kappa(x):k]=d\\right\\},$$\nwhere $h_{\\mathscr L}$ is a Weil height associated to $\\mathscr L$. For appropriately chosen $U$, one may expect that $N_{X,\\mathscr L}(U,B;d)\\sim cB^a(\\log B)^b$ as $B\\to\\infty$, for some choice of $a,b,c$.\n\nThere were several questions proposed around this, including\n\n* What should the predicted values of $a,b,c$ be?\n\n* What is the \"shape\" of the set $U$ that one should take? In the usual Manin conjecture, one expects that $U$ can be taken to be the complement of a thin set.\n\n* Can one compute asymptotics for $N_{X,\\mathscr L}(U,B;d)$ if $X=\\mathbb P^2_{\\mathbb Q}$ and $d=3$, for some choices of $\\mathscr L$?", "original_statement": "The goal is to formulate a Manin-type conjecture for degree $d$ points on Fano varieties. Let $X/k$ be a Fano variety defined over a number field $k$. For a subset $U\\subset X$, an integer $d$, a real number $B>0$, and a line bundle $\\mathscr L$ on $X$, define\n$$N_{X,\\mathscr L}(U,B;d) := \\#\\left\\{x\\in U(\\overline k):h_{\\mathscr L}(x)\\le B\\text{ and }[\\kappa(x):k]=d\\right\\},$$\nwhere $h_{\\mathscr L}$ is a Weil height associated to $\\mathscr L$. For appropriately chosen $U$, one may expect that $N_{X,\\mathscr L}(U,B;d)\\sim cB^a(\\log B)^b$ as $B\\to\\infty$, for some choice of $a,b,c$.\n\nThere were several questions proposed around this, including\n\n* What should the predicted values of $a,b,c$ be?\n\n* What is the \"shape\" of the set $U$ that one should take? In the usual Manin conjecture, one expects that $U$ can be taken to be the complement of a thin set.\n\n* Can one compute asymptotics for $N_{X,\\mathscr L}(U,B;d)$ if $X=\\mathbb P^2_{\\mathbb Q}$ and $d=3$, for some choices of $\\mathscr L$?", "clean_statement": "The goal is to formulate a Manin-type conjecture for degree $d$ points on Fano varieties. Let $X/k$ be a Fano variety defined over a number field $k$. For a subset $U\\subset X$, an integer $d$, a real number $B>0$, and a line bundle $\\mathscr L$ on $X$, define\n$$N_{X,\\mathscr L}(U,B;d) := \\#\\left\\{x\\in U(\\overline k):h_{\\mathscr L}(x)\\le B\\text{ and }[\\kappa(x):k]=d\\right\\},$$\nwhere $h_{\\mathscr L}$ is a Weil height associated to $\\mathscr L$. For appropriately chosen $U$, one may expect that $N_{X,\\mathscr L}(U,B;d)\\sim cB^a(\\log B)^b$ as $B\\to\\infty$, for some choice of $a,b,c$.\n\nThere were several questions proposed around this, including\n\n* What should the predicted values of $a,b,c$ be?\n\n* What is the \"shape\" of the set $U$ that one should take? In the usual Manin conjecture, one expects that $U$ can be taken to be the complement of a thin set.\n\n* Can one compute asymptotics for $N_{X,\\mathscr L}(U,B;d)$ if $X=\\mathbb P^2_{\\mathbb Q}$ and $d=3$, for some choices of $\\mathscr L$?", "statement_status": "exact", "statement_verification": "The source is AIM Problem List 1.02 from the workshop *Degree \\(d\\) points on algebraic surfaces*. It asks for a Manin-type conjecture for degree-\\(d\\) points on a Fano variety \\(X/k\\). Its displayed counting function is \\[ N_{X,\\mathscr L}(U,B;d) = \\#\\{x\\in U(\\overline k):h_{\\mathscr L}(x)\\leq B,\\ [\\kappa(x):k]=d\\}, \\] and the proposed form is \\(cB^a(\\log B)^b\\). It asks for \\(a,b,c\\), for the correct shape of \\(U\\), and particularly for \\(X=\\mathbb P^2_{\\mathbb Q}\\), \\(d=3\\).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.02\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The goal is to formulate a Manin-type conjecture for degree $d$ points on Fano varieties. Let $X/k$ be a Fano variety defined over a number field $k$. For a subset $U\\\\subset X$, an integer $d$, a real number $B>0$, and a line bundle $\\\\mathscr L$ on $X$, define\\n$$N_{X,\\\\mathscr L}(U,B;d) := \\\\#\\\\left\\\\{x\\\\in U(\\\\overline k):h_{\\\\mathscr L}(x)\\\\le B\\\\text{ and }[\\\\kappa(x):k]=d\\\\right\\\\},$$\\nwhere $h_{\\\\mathscr L}$ is a Weil height associated to $\\\\mathscr L$. For appropriately chosen $U$, one may expect that $N_{X,\\\\mathscr L}(U,B;d)\\\\sim cB^a(\\\\log B)^b$ as $B\\\\to\\\\infty$, for some choice of $a,b,c$.\\n\\nThere were several questions proposed around this, including\\n\\n* What should the predicted values of $a,b,c$ be?\\n\\n* What is the \\\"shape\\\" of the set $U$ that one should take? In the usual Manin conjecture, one expects that $U$ can be taken to be the complement of a thin set.\\n\\n* Can one compute asymptotics for $N_{X,\\\\mathscr L}(U,B;d)$ if $X=\\\\mathbb P^2_{\\\\mathbb Q}$ and $d=3$, for some choices of $\\\\mathscr L$?\"\nOriginal remarks: [\"The case of $X=\\\\mathbb P^2_{\\\\mathbb Q}$ and $d=2$ has been studied in an old paper of Schmidt. One can also see Cécile Le Rudulier's thesis (or the related paper \\\\cite{MR4057715}) which also handles the case of $\\\\mathbb P^1_{\\\\mathbb Q}\\\\times\\\\mathbb P^1_{\\\\mathbb Q}$ and $d=2$. There is also a recent paper of Jesse Kass and Frank Thorne \\\\cite{arXiv:2209.13030} which counts points on $\\\\operatorname{Hilb}^2(\\\\mathbb P^2_{\\\\mathbb Q})$.\\n\\nFinally, a similar problem (for $d=2$) was studied last summer as part of an IAS summer collaboration; find the report at https://www.ias.edu/sites/default/files/Park%20report.pdf\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0001", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every nonempty Q-defined Zariski open U in P^2 and every fixed adelic O(1) height, the exact-degree-three geometric-point count satisfies N(U,B;3) asymp B^12: a rational line meeting U supplies an asymptotic C B^12 subfamily by Le Rudulier, while Gao supplies the global upper bound. On Sym^3(P^2), reduced noncollinear fully split cycles of symmetric height at most T have order T^3(log T)^2, so a Manin-type exact-cubic formulation must remove the collinear and diagonal/exceptional loci and also sieve out a Zariski-dense thin nontransitive locus. The ambient power B^9 and a possible single logarithm after these removals remain conditional.\n\nCandidate contribution (lemma; novelty confidence low): Candidate two-level exceptional-set obstruction: no nonempty coordinatewise Zariski open U in P^2 can remove the B^12 collinear accumulation, and after passing to Sym^3(P^2) and deleting the diagonal and collinear divisor, reduced noncollinear fully split cycles still contribute asymp T^3(log T)^2 unless exact-degree transitivity is imposed." }, { "id": 20000350, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0002", "title": "A Brauer-transfer sieve and reduction of the cubic route", "statement": "Is there a $\\mathbb Q$-irrational del Pezzo surface $X$ which is $\\mathbb Q$-unirational for which $\\operatorname{Sym}^dX$ is $\\mathbb Q$-rational for some $d>1$?", "original_statement": "Is there a $\\mathbb Q$-irrational del Pezzo surface $X$ which is $\\mathbb Q$-unirational for which $\\operatorname{Sym}^dX$ is $\\mathbb Q$-rational for some $d>1$?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The exact canonical JSON record was checked at zero-based source index \\(1\\) in `aim-algebraic-number-theory-notes.json`, together with its neighboring records. The original URL `http://aimpl.org/degreedsurface/1/` no longer returned the problem page when checked on 2026-07-24 (the reachable AIM endpoint returned a 404 page), so no silent reconstruction from that page was made. The follow-up remark refers to “Problem 1.2”; in the canonical ordering this is a stale number and is almost certainly intended to refer to this problem, numbered 1.04.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.04\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a $\\\\mathbb Q$-irrational del Pezzo surface $X$ which is $\\\\mathbb Q$-unirational for which $\\\\operatorname{Sym}^dX$ is $\\\\mathbb Q$-rational for some $d>1$?\"\nOriginal remarks: [\"$X$ should have negative Kodaira dimension for this to be possible.\", \"If $X$ is $k$-rational, then so is $\\\\operatorname{Sym}^d(X)$. Over $\\\\mathbb C$, Arapura showed that the Kodaira dimension of $\\\\operatorname{Sym}^d(X)$ is\\n$$\\\\kappa(\\\\operatorname{Sym}^d(X))=d\\\\kappa(X)$$\\nif $\\\\dim X\\\\ge2$.\", \"While this example is not unirational, if $X=C\\\\times\\\\mathbb P^1$ with $C$ a conic, then $X$ is irrational while $\\\\operatorname{Sym}^2(X)$ is rational.\", \"As a followup, can we characterize the $X$ (and $d$) for which the answer to Problem 1.2 is yes?\", \"If $X$ is a non-trivial Severi-Brauer variety of index $d$, then $X$ is not rational, but $\\\\operatorname{Sym}^d(X)$ is rational.\", \"One could try looking at a cubic surface $X$ with $d=3$ to see if this gives an example.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0002", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Finite-flat corestriction along the universal family on the Hilbert scheme gives a split injection Br(X)/Br(k) into Br(X^[d])/Br(k) whenever X has d-1 distinct k-points. Consequently, if a geometrically rational number-field surface X is k-unirational and one symmetric power is retract rational, then H^1(H, Pic(Xbar)) vanishes for every subgroup H of the finite Galois image. Combining this with Tschinkel-Yang's classification excludes every minimal irrational cubic surface and shows that any surviving cubic model is birational to a minimal degree-four del Pezzo surface satisfying this subgroup condition; any AIM example has a minimal model of degree 1, 2, or 4 satisfying the same condition.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: a number-field solution represented by an irrational cubic surface cannot be minimal and must be birational to a minimal degree-four del Pezzo surface whose geometric Picard lattice has H^1(H, Pic(Xbar))=0 for every subgroup H of its Galois image.", "research_difficulty_suggested": "L4" }, { "id": 20000351, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0003", "title": "Exact-degree points from automorphism orbits", "statement": "Let $X$ be a variety. Suppose one has a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$, where $Y\\dashrightarrow\\mathbb P^n$ is finite of degree $d$. Then, as a consequence of Hilbert irreducibility, $X$ will have a Zariski dense set of degree $d$ points.\n\nCan we give a method to construct a Zariski dense set of degree $d$ points on a variety different from the one just mentioned?\n\n* Maybe one can replace $\\mathbb P^n$ with another sort of variety?\n\n* Maybe one can do something totally different and give a construction not coming from a correspondence.", "original_statement": "Let $X$ be a variety. Suppose one has a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$, where $Y\\dashrightarrow\\mathbb P^n$ is finite of degree $d$. Then, as a consequence of Hilbert irreducibility, $X$ will have a Zariski dense set of degree $d$ points.\n\nCan we give a method to construct a Zariski dense set of degree $d$ points on a variety different from the one just mentioned?\n\n* Maybe one can replace $\\mathbb P^n$ with another sort of variety?\n\n* Maybe one can do something totally different and give a construction not coming from a correspondence.", "clean_statement": "Let $X$ be a variety. Suppose one has a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$, where $Y\\dashrightarrow\\mathbb P^n$ is finite of degree $d$. Then, as a consequence of Hilbert irreducibility, $X$ will have a Zariski dense set of degree $d$ points.\n\nCan we give a method to construct a Zariski dense set of degree $d$ points on a variety different from the one just mentioned?\n\n* Maybe one can replace $\\mathbb P^n$ with another sort of variety?\n\n* Maybe one can do something totally different and give a construction not coming from a correspondence.", "statement_status": "exact", "statement_verification": "This is problem 1.06 from the AIM workshop *Degree \\(d\\) points on algebraic surfaces*, in the section “Initial Problem Session.” The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.06\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X$ be a variety. Suppose one has a diagram $X\\\\dashleftarrow Y\\\\dashrightarrow\\\\mathbb P^n$, where $Y\\\\dashrightarrow\\\\mathbb P^n$ is finite of degree $d$. Then, as a consequence of Hilbert irreducibility, $X$ will have a Zariski dense set of degree $d$ points.\\n\\nCan we give a method to construct a Zariski dense set of degree $d$ points on a variety different from the one just mentioned?\\n\\n* Maybe one can replace $\\\\mathbb P^n$ with another sort of variety?\\n\\n* Maybe one can do something totally different and give a construction not coming from a correspondence.\"\nOriginal remarks: [\"It would be nice to write down a census of all the ways we know to produce a Zariski dense set of degree $d$ points on varieties. This leads to the subquestions:\\n\\n* Is the method of using AV parameterized points (on curves) genuinely different than a method using correspondences?\\n\\n* Is there a construction that does not come from curves (or correspondences)?\", \"In a sense, everything comes from a correspondence between $X$ and $\\\\operatorname{Sym}^dX$, so you need to be careful about which correspondences you consider.\", \"Suggested other construction. Take a K3 surface $X$ with infinite automorphism group, but which contains no smooth curves of genus $0$ or $1$. Given a degree $d$ point on $X$, its orbit under the automorphism group will be infinite and likely will be Zariski dense in $X$ (since it cannot lie on a smooth curve in $X$). To really make this work, one might need the stronger-seeming condition that $X$ contains no (possibly singular) curves of geometric genus $0$ or $1$ defined over the ground field. This likely holds for examples one can construct satisfying the initial condition of no smooth low genus curves, but it may be harder to provably verify this.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0003", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let X/k be a projective geometrically integral surface, let Gamma be a subgroup of Aut_k(X), and let x have exact degree d. Every point in the Gamma-orbit of x has exact degree d. If that orbit is infinite but not Zariski dense, then some finite-index curve stabilizer has an infinite suborbit on a Gamma-periodic irreducible curve whose normalization has genus at most one. Consequently, a nonperiodic exact-degree-d seed outside the finite periodic-curve locus of a positive-entropy K3 automorphism has a Zariski-dense two-sided orbit consisting entirely of exact-degree-d points.\n\nCandidate contribution (lemma; novelty confidence low): Candidate exact-degree orbit-closure lemma: for an arbitrary subgroup of k-automorphisms of a projective surface, every infinite nondense orbit of an exact-degree-d point forces a periodic curve of geometric genus at most one, while every orbit point retains exact degree d." }, { "id": 20000352, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0004", "title": "Albanese sums of symmetric powers: thresholds, arithmetic reduction, and explicit fibers", "statement": "With some hypotheses on $X$ and $X\\to\\operatorname{Alb}(X)$, study the image of $\\operatorname{Sym}^d(X)\\to\\operatorname{Alb}(X)$ for some small $d$.\n\nWhat are examples of $X$ for which the fibers of $\\operatorname{Sym}^dX\\to\\operatorname{Alb}(X)$ are \"well-understood\"? For example, when are they finite or rational?", "original_statement": "With some hypotheses on $X$ and $X\\to\\operatorname{Alb}(X)$, study the image of $\\operatorname{Sym}^d(X)\\to\\operatorname{Alb}(X)$ for some small $d$.\n\nWhat are examples of $X$ for which the fibers of $\\operatorname{Sym}^dX\\to\\operatorname{Alb}(X)$ are \"well-understood\"? For example, when are they finite or rational?", "clean_statement": "With some hypotheses on \\(X\\) and\n\\(X\\to\\operatorname{Alb}(X)\\), study the image of\n\\(\\operatorname{Sym}^d(X)\\to\\operatorname{Alb}(X)\\) for some small\n\\(d\\).\n\nWhat are examples of \\(X\\) for which the fibers of\n\\(\\operatorname{Sym}^dX\\to\\operatorname{Alb}(X)\\) are\n“well-understood”? For example, when are they finite or rational?", "statement_status": "reconstructed_unverified", "statement_verification": "This is Problem 1.08 from the AIM workshop *Degree \\(d\\) points on algebraic surfaces*. The canonical record and the current AIM problem-list page agree. The recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.08\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[3]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"With some hypotheses on $X$ and $X\\\\to\\\\operatorname{Alb}(X)$, study the image of $\\\\operatorname{Sym}^d(X)\\\\to\\\\operatorname{Alb}(X)$ for some small $d$.\\n\\nWhat are examples of $X$ for which the fibers of $\\\\operatorname{Sym}^dX\\\\to\\\\operatorname{Alb}(X)$ are \\\"well-understood\\\"? For example, when are they finite or rational?\"\nOriginal remarks: [\"For such $X$, one could hope to apply Mordell-Lang in order to better understand the structure of degree $d$ points, in analogy with the case of points on curves.\", \"Maybe consider $F_1(X)$, the Fano variety of lines on a cubic $3$-fold.\", \"Maybe restrict to $X$ whose image in $\\\\operatorname{Alb}(X)$ is of maximal dimension.\", \"Maybe consider $X$ for which $\\\\operatorname{Alb}(X)$ is simple.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0004", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If a projective n-fold X has Albanese image Y of dimension r in an absolutely simple q-dimensional Albanese variety A, then the image of Sym^d(X) under Albanese summation is the additive sum dY, with dimension min(dr,q), and the generic fiber has dimension dn-min(dr,q). For a maximal-Albanese surface this is max(0,2d-q), giving the sharp generic-finiteness and dominance thresholds; the critical case has an explicit generic-degree formula. Over a number field, the subcritical image has finitely many rational Albanese values by Mordell-Lang. The Fano surface of a cubic threefold gives a birational d=2 sum map, while a smooth theta surface in a simple abelian threefold has general symmetric-square fiber of genus 4, so simplicity and maximal Albanese dimension do not force rational fibers.\n\nCandidate contribution (synthesis_theorem; novelty confidence low): Candidate novelty: the sharp three-regime surface threshold is packaged with a subcritical Mordell-Lang finite-value reduction, a critical generic-degree formula, and a supercritical theta-surface example whose general symmetric fiber has genus 4." }, { "id": 20000353, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0005", "title": "Exact density spectra for projective bundles and large Severi-Brauer gaps", "statement": "Let $X/k$ be a nice variety. We define a few measures of irrationality.\n\n* Its covering gonality $\\operatorname{cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\dashrightarrow X$ is a dominant rational map and $\\mathcal C\\to B$ is a family of curves such that $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$.\n\n* Its connected covering gonality $\\operatorname{conn.cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\to B$ is a family of curves with $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$ and such that the induced rational map $\\mathcal C\\times_B\\mathcal C\\dashrightarrow X\\times X$ is dominant.\n\n* Finally, $\\operatorname{uni.irr}(X)$ is the minimal $d$ such that there exists a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$ with $Y\\dashrightarrow X$ dominant and $Y\\dashrightarrow\\mathbb P^n$ finite of degree $d$.\n\nLet $X/k$ be a nice variety, and let $\\delta(X/k)=\\{d:$ degree $d$ points on $X$ are Zariski dense in $X\\}$ be its density degree set. How different can $\\min\\delta(X/k)$ be from $\\operatorname{uni.irr}(X),\\operatorname{cov.gon}(X)$, and/or $\\operatorname{conn.cov.gon}(X)$?", "original_statement": "Let $X/k$ be a nice variety. We define a few measures of irrationality.\n\n* Its covering gonality $\\operatorname{cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\dashrightarrow X$ is a dominant rational map and $\\mathcal C\\to B$ is a family of curves such that $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$.\n\n* Its connected covering gonality $\\operatorname{conn.cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\to B$ is a family of curves with $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$ and such that the induced rational map $\\mathcal C\\times_B\\mathcal C\\dashrightarrow X\\times X$ is dominant.\n\n* Finally, $\\operatorname{uni.irr}(X)$ is the minimal $d$ such that there exists a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$ with $Y\\dashrightarrow X$ dominant and $Y\\dashrightarrow\\mathbb P^n$ finite of degree $d$.\n\nLet $X/k$ be a nice variety, and let $\\delta(X/k)=\\{d:$ degree $d$ points on $X$ are Zariski dense in $X\\}$ be its density degree set. How different can $\\min\\delta(X/k)$ be from $\\operatorname{uni.irr}(X),\\operatorname{cov.gon}(X)$, and/or $\\operatorname{conn.cov.gon}(X)$?", "clean_statement": "Let $X/k$ be a nice variety. We define a few measures of irrationality.\n\n* Its covering gonality $\\operatorname{cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\dashrightarrow X$ is a dominant rational map and $\\mathcal C\\to B$ is a family of curves such that $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$.\n\n* Its connected covering gonality $\\operatorname{conn.cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\leftarrow\\mathcal C\\dashrightarrow X$, where $\\mathcal C\\to B$ is a family of curves with $\\operatorname{gon}(\\mathcal C_b)\\ge d$ for all $b\\in B$ and such that the induced rational map $\\mathcal C\\times_B\\mathcal C\\dashrightarrow X\\times X$ is dominant.\n\n* Finally, $\\operatorname{uni.irr}(X)$ is the minimal $d$ such that there exists a diagram $X\\dashleftarrow Y\\dashrightarrow\\mathbb P^n$ with $Y\\dashrightarrow X$ dominant and $Y\\dashrightarrow\\mathbb P^n$ finite of degree $d$.\n\nLet $X/k$ be a nice variety, and let $\\delta(X/k)=\\{d:$ degree $d$ points on $X$ are Zariski dense in $X\\}$ be its density degree set. How different can $\\min\\delta(X/k)$ be from $\\operatorname{uni.irr}(X),\\operatorname{cov.gon}(X)$, and/or $\\operatorname{conn.cov.gon}(X)$?", "statement_status": "exact", "statement_verification": "The source record is Problem 1.1 from the AIM workshop *Degree \\(d\\) points on algebraic surfaces*. For a nice variety \\(X/k\\), it asks how far", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.1\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[4]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X/k$ be a nice variety. We define a few measures of irrationality.\\n\\n* Its covering gonality $\\\\operatorname{cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\\\leftarrow\\\\mathcal C\\\\dashrightarrow X$, where $\\\\mathcal C\\\\dashrightarrow X$ is a dominant rational map and $\\\\mathcal C\\\\to B$ is a family of curves such that $\\\\operatorname{gon}(\\\\mathcal C_b)\\\\ge d$ for all $b\\\\in B$.\\n\\n* Its connected covering gonality $\\\\operatorname{conn.cov.gon}(X)$ is the minimal $d$ such that there exists a diagram $B\\\\leftarrow\\\\mathcal C\\\\dashrightarrow X$, where $\\\\mathcal C\\\\to B$ is a family of curves with $\\\\operatorname{gon}(\\\\mathcal C_b)\\\\ge d$ for all $b\\\\in B$ and such that the induced rational map $\\\\mathcal C\\\\times_B\\\\mathcal C\\\\dashrightarrow X\\\\times X$ is dominant.\\n\\n* Finally, $\\\\operatorname{uni.irr}(X)$ is the minimal $d$ such that there exists a diagram $X\\\\dashleftarrow Y\\\\dashrightarrow\\\\mathbb P^n$ with $Y\\\\dashrightarrow X$ dominant and $Y\\\\dashrightarrow\\\\mathbb P^n$ finite of degree $d$.\\n\\nLet $X/k$ be a nice variety, and let $\\\\delta(X/k)=\\\\{d:$ degree $d$ points on $X$ are Zariski dense in $X\\\\}$ be its density degree set. How different can $\\\\min\\\\delta(X/k)$ be from $\\\\operatorname{uni.irr}(X),\\\\operatorname{cov.gon}(X)$, and/or $\\\\operatorname{conn.cov.gon}(X)$?\"\nOriginal remarks: [\"Consider $X=Y\\\\times\\\\mathbb P^1$ with $Y$ a complicated variety. Then, $\\\\operatorname{cov.gon}(X)=1$, but $\\\\min\\\\delta(X/k)=\\\\min\\\\delta(Y/k)$, so it seems that $\\\\min\\\\delta(X/k)$ and $\\\\operatorname{cov.gon}(X)$ can differ by quite a lot in general.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0005", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over a number field, every projective bundle satisfies delta(P(E)/k) = delta(X/k) times the positive integers, equivalently a degree d occurs densely upstairs exactly when some dense degree e downstairs divides d; in particular the minimum density degree is preserved by multiplying with projective space. For a degree-N division Severi-Brauer variety, the minimum density degree is exactly N, while all three standard geometric measures in the problem are 1, giving unbounded gaps as N grows. An arbitrary degree-d unirational diagram forces a divisor of d into the density spectrum, while minimality for the k-defined unirational degree forces the exact degree d itself.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: for every projective bundle P(E) over a nice variety X over a number field, delta(P(E)/k) is exactly the upward multiplicative closure delta(X/k) times the positive integers; combined with complete-intersection curves in division Severi-Brauer varieties, this yields unbounded minimum-density-degree versus covering-gonality gaps already for surfaces." }, { "id": 20000354, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0006", "title": "Degree sets on Skorobogatov's bielliptic surface", "statement": "Let $X/k$ be a nice variety. In addition to its density degree set $\\delta(X/\\mathbb Q)$, define\n$$\\mathcal D(X/\\mathbb Q):=\\{d:X\\text{ has a point of degree }d\\},$$\nand\n$$\\mathcal P(X/\\mathbb Q)=\\bigcup_{[k':k]<\\infty}\\delta(X/k').$$\n\nSkorobogatov constructed a bielliptic surface $X/\\mathbb Q$ such that $X(\\mathbb A_{\\mathbb Q})^{\\operatorname{Br}}\\neq\\emptyset$, but $X(\\mathbb Q)=\\emptyset$. It is known that $\\min\\delta(X/\\mathbb Q)\\le3$. Where do these Zariski-dense degree 3 points come from? Can one compute $\\mathcal D(X/\\mathbb Q),\\delta(X/\\mathbb Q),\\mathcal P(X/\\mathbb Q)$?", "original_statement": "Let $X/k$ be a nice variety. In addition to its density degree set $\\delta(X/\\mathbb Q)$, define\n$$\\mathcal D(X/\\mathbb Q):=\\{d:X\\text{ has a point of degree }d\\},$$\nand\n$$\\mathcal P(X/\\mathbb Q)=\\bigcup_{[k':k]<\\infty}\\delta(X/k').$$\n\nSkorobogatov constructed a bielliptic surface $X/\\mathbb Q$ such that $X(\\mathbb A_{\\mathbb Q})^{\\operatorname{Br}}\\neq\\emptyset$, but $X(\\mathbb Q)=\\emptyset$. It is known that $\\min\\delta(X/\\mathbb Q)\\le3$. Where do these Zariski-dense degree 3 points come from? Can one compute $\\mathcal D(X/\\mathbb Q),\\delta(X/\\mathbb Q),\\mathcal P(X/\\mathbb Q)$?", "clean_statement": "Let $X/k$ be a nice variety. In addition to its density degree set $\\delta(X/\\mathbb Q)$, define\n$$\\mathcal D(X/\\mathbb Q):=\\{d:X\\text{ has a point of degree }d\\},$$\nand\n$$\\mathcal P(X/\\mathbb Q)=\\bigcup_{[k':k]<\\infty}\\delta(X/k').$$\n\nSkorobogatov constructed a bielliptic surface $X/\\mathbb Q$ such that $X(\\mathbb A_{\\mathbb Q})^{\\operatorname{Br}}\\neq\\emptyset$, but $X(\\mathbb Q)=\\emptyset$. It is known that $\\min\\delta(X/\\mathbb Q)\\le3$. Where do these Zariski-dense degree 3 points come from? Can one compute $\\mathcal D(X/\\mathbb Q),\\delta(X/\\mathbb Q),\\mathcal P(X/\\mathbb Q)$?", "statement_status": "exact", "statement_verification": "The canonical AIM record, Problem 1.12 from the workshop *Degree \\(d\\) points on algebraic surfaces*, asks about three sets. For a nice variety \\(X/k\\),", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.12\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[5]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X/k$ be a nice variety. In addition to its density degree set $\\\\delta(X/\\\\mathbb Q)$, define\\n$$\\\\mathcal D(X/\\\\mathbb Q):=\\\\{d:X\\\\text{ has a point of degree }d\\\\},$$\\nand\\n$$\\\\mathcal P(X/\\\\mathbb Q)=\\\\bigcup_{[k':k]<\\\\infty}\\\\delta(X/k').$$\\n\\nSkorobogatov constructed a bielliptic surface $X/\\\\mathbb Q$ such that $X(\\\\mathbb A_{\\\\mathbb Q})^{\\\\operatorname{Br}}\\\\neq\\\\emptyset$, but $X(\\\\mathbb Q)=\\\\emptyset$. It is known that $\\\\min\\\\delta(X/\\\\mathbb Q)\\\\le3$. Where do these Zariski-dense degree 3 points come from? Can one compute $\\\\mathcal D(X/\\\\mathbb Q),\\\\delta(X/\\\\mathbb Q),\\\\mathcal P(X/\\\\mathbb Q)$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0006", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Creutz's explicit cubic field and simultaneous positive-rank twists explain the Zariski-dense degree-3 points. A residue-field-preserving slice argument proves that the surface has Zariski-dense exact-degree-d points over Creutz's cubic field for every positive integer d, so its potential density degree set over Q is all positive integers. Over Q, degree 1 is excluded, degree 3 and every positive multiple of 4 lie in the density degree set, and both the density degree set and the degree set are cofinite because the surface has index 1. Exact low degrees, notably degree 2, remain unresolved.\n\nCandidate contribution (proof_specialization; novelty confidence low): For Creutz's specific cubic twist, every horizontal slice of the diagonal quotient maps isomorphically to its image; combining this residue-field preservation with exact-degree points on the positive-rank genus-one factor gives a direct proof that delta(S/L) is all positive integers. The explicit biquadratic projection and its compositions also give 4N contained in delta(S/Q)." }, { "id": 20000355, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0007", "title": "Quadratic density, diagonal quotients, and correspondence components", "statement": "Let $C,D$ be hyperelliptic curves of genus $\\ge2$ over a number field $k$. Must $2\\in\\delta(C\\times D/k)$? Must it be in $\\mathcal P(C\\times D/k)$?\n\nMore generally, let $Z\\subset C\\times D$ be the closure of the degree $d$ points. When is there a component of $Z$ which dominates both $C$ and $D$?", "original_statement": "Let $C,D$ be hyperelliptic curves of genus $\\ge2$ over a number field $k$. Must $2\\in\\delta(C\\times D/k)$? Must it be in $\\mathcal P(C\\times D/k)$?\n\nMore generally, let $Z\\subset C\\times D$ be the closure of the degree $d$ points. When is there a component of $Z$ which dominates both $C$ and $D$?", "clean_statement": "Let $C,D$ be hyperelliptic curves of genus $\\ge2$ over a number field $k$. Must $2\\in\\delta(C\\times D/k)$? Must it be in $\\mathcal P(C\\times D/k)$?\n\nMore generally, let $Z\\subset C\\times D$ be the closure of the degree $d$ points. When is there a component of $Z$ which dominates both $C$ and $D$?", "statement_status": "exact", "statement_verification": "This record is Problem 1.14 from the initial problem session of the AIM workshop *Degree \\(d\\) points on algebraic surfaces*. The canonical record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.14\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[6]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $C,D$ be hyperelliptic curves of genus $\\\\ge2$ over a number field $k$. Must $2\\\\in\\\\delta(C\\\\times D/k)$? Must it be in $\\\\mathcal P(C\\\\times D/k)$?\\n\\nMore generally, let $Z\\\\subset C\\\\times D$ be the closure of the degree $d$ points. When is there a component of $Z$ which dominates both $C$ and $D$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0007", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The density question has a known negative answer: an explicit pair of genus-two curves over Q has a common quadratic point but no dense quadratic points on the product. Under finiteness of non-hyperelliptic quadratic exceptions on both factors, exact quadratic density is equivalent to density of rational points on the simultaneous-involution quotient. For proper degree-d closure, a dominating curve component is equivalent to a dominating correspondence whose normalization has infinitely many exact degree-d points, hence a parameterized degree-d point. In addition, every genus-two self-product has potentially dense exact quadratic points.\n\nCandidate contribution (corollary; novelty confidence low): For every smooth projective geometrically integral genus-two curve C over a number field k, 2 belongs to the potential density degree set of C x C, without rational-point, simplicity, or initial Mordell-Weil-rank assumptions." }, { "id": 20000356, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0008", "title": "Quadratic density and potential gaps on symmetric squares of curves", "statement": "Let $C$ be a curve of genus $\\ge2$ over a number field. Must $2\\in\\delta(\\operatorname{Sym}^2_C/k)$ or in $\\mathcal P(\\operatorname{Sym}^2_C/k)$? What do these sets look like?", "original_statement": "Let $C$ be a curve of genus $\\ge2$ over a number field. Must $2\\in\\delta(\\operatorname{Sym}^2_C/k)$ or in $\\mathcal P(\\operatorname{Sym}^2_C/k)$? What do these sets look like?", "clean_statement": "Let $C$ be a curve of genus $\\ge2$ over a number field. Must $2\\in\\delta(\\operatorname{Sym}^2_C/k)$ or in $\\mathcal P(\\operatorname{Sym}^2_C/k)$? What do these sets look like?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.16\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[7]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $C$ be a curve of genus $\\\\ge2$ over a number field. Must $2\\\\in\\\\delta(\\\\operatorname{Sym}^2_C/k)$ or in $\\\\mathcal P(\\\\operatorname{Sym}^2_C/k)$? What do these sets look like?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0008", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For genus 2, birational Abel-Jacobi geometry and Berg et al.'s theorem give the exact formula delta(Sym^2 C/k) = N_{>=2}, with degree 1 added exactly when J_C(k) is Zariski dense, and the potential spectrum is all of N. In higher genus, if J_C is geometrically simple and 2d < min(gon(C_bar), g(C)), then Sym^2 C has only finitely many closed points of degree at most d after every finite base extension, so these degrees are absent from the potential spectrum. For every prime p >= 5, the curve y^p = x^(2p-2) - x - 1 has gonality p, absolutely simple Jacobian, and a rational point, while the potential spectrum of its symmetric square omits 1 through (p-1)/2; in particular degree 2 need not occur even potentially.\n\nCandidate contribution (theorem; novelty confidence low): If J_C is geometrically simple and 2d < min(gon(C_bar), g(C)), then over every finite K/k the degree-at-most-d closed points of Sym^2 C_K are finite; the explicit family y^p = x^(2p-2) - x - 1 consequently has an arbitrarily long initial gap in its potential density-degree set despite having index 1." }, { "id": 20000357, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0009", "title": "Density degrees under finite covers", "statement": "Given a finite map $f:X\\to Y$ (over a number field), how are $\\delta(X/k)$ and $\\delta(Y/k)$ related? Are there examples showing that the obvious relations are sharp?", "original_statement": "Given a finite map $f:X\\to Y$ (over a number field), how are $\\delta(X/k)$ and $\\delta(Y/k)$ related? Are there examples showing that the obvious relations are sharp?", "clean_statement": "Given a finite map $f:X\\to Y$ (over a number field), how are $\\delta(X/k)$ and $\\delta(Y/k)$ related? Are there examples showing that the obvious relations are sharp?", "statement_status": "exact", "statement_verification": "The canonical record is item 1.18 of the AIM problem list from the workshop *Degree \\(d\\) points on algebraic surfaces* (*aim-algebraic-number-theory-notes.json*, zero-based index 8):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.18\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[8]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given a finite map $f:X\\\\to Y$ (over a number field), how are $\\\\delta(X/k)$ and $\\\\delta(Y/k)$ related? Are there examples showing that the obvious relations are sharp?\"\nOriginal remarks: [\"It could be interesting to consider the special case where $f$ is a Galois cover.\", \"If $X,Y$ are curves, you can also ask about containment relations between there $\\\\delta_{\\\\mathbb P^1}$'s and their $\\\\delta_{\\\\operatorname{AV}}$'s.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0009", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite dominant degree-n morphism f:X→Y of geometrically integral varieties over a number field, every d in δ(X/k) equals me for some e in δ(Y/k) and 1≤m≤n, while every e in δ(Y/k) has em in δ(X/k) for some 1≤m≤n. For Galois covers m may be chosen to divide n. Consequently, when both minima exist, min δ(Y/k)≤min δ(X/k)≤n min δ(Y/k), and both endpoints are sharp, with the upper factor attained for every n≥2. For degree-n maps of curves, nδ_P1(D/k) is contained in δ_P1(C/k), and rnδ_AV(D/k) is contained in δ_AV(C/k)∩δ_P1(C/k) for every r≥2.\n\nCandidate contribution (transfer theorem; novelty confidence low): Candidate apparently unstated finite-cover transfer theorem: the two density spectra are linked by residue multipliers in [1,n], restricted to divisors of n in the Galois case; the induced minimum bounds are optimal for every n, and the curve AV component transfers after every scalar dilation r≥2." }, { "id": 20000358, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0010", "title": "An Albanese threshold obstruction for Hilbert schemes of points", "statement": "Is $\\operatorname{Hilb}^2(C\\times D)(k)$ Zariski dense in $\\operatorname{Hilb}^2(C\\times D)$, where $C,D/k$ are hyperelliptic curves over a number field. More generally, for which varieties $X$ over a number field is there a $d$ such that $\\operatorname{Hilb}^d(X)(k)$ is Zariski dense?", "original_statement": "Is $\\operatorname{Hilb}^2(C\\times D)(k)$ Zariski dense in $\\operatorname{Hilb}^2(C\\times D)$, where $C,D/k$ are hyperelliptic curves over a number field. More generally, for which varieties $X$ over a number field is there a $d$ such that $\\operatorname{Hilb}^d(X)(k)$ is Zariski dense?", "clean_statement": "Is $\\operatorname{Hilb}^2(C\\times D)(k)$ Zariski dense in $\\operatorname{Hilb}^2(C\\times D)$, where $C,D/k$ are hyperelliptic curves over a number field. More generally, for which varieties $X$ over a number field is there a $d$ such that $\\operatorname{Hilb}^d(X)(k)$ is Zariski dense?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.2 from the AIM workshop *Degree d points on algebraic surfaces*. Its exact `problem` field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.2\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[9]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is $\\\\operatorname{Hilb}^2(C\\\\times D)(k)$ Zariski dense in $\\\\operatorname{Hilb}^2(C\\\\times D)$, where $C,D/k$ are hyperelliptic curves over a number field. More generally, for which varieties $X$ over a number field is there a $d$ such that $\\\\operatorname{Hilb}^d(X)(k)$ is Zariski dense?\"\nOriginal remarks: [\"If $X$ is of general type and $\\\\dim X\\\\ge2$, then $\\\\operatorname{Hilb}^d(X)$ will also be of general type.\", \"One can also ask, is there a $d$ such that the set $\\\\{$degree $d$ points of $X\\\\}\\\\subsetneq\\\\operatorname{Hilb}^d(X)$ is Zariski dense in $\\\\operatorname{Hilb}^d(X)$\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0010", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If a smooth projective geometrically integral surface X over a number field dominates a curve of genus g, then Hilb^d(X) is not potentially dense for every d X be a covering family with T(k) Zariski dense. If d is the arithmetic k(T)-gonality of the generic member and the general curve maps birationally to its image, then d belongs to the density degree set δ(X/k), hence min δ(X/k) <= d. Under mere dominance, some divisor e of d belongs to δ(X/k). This bound is sharp in every degree n via products of an index-n genus-one curve with P^1, while geometric gonality alone can underestimate min δ(X/k) by an unbounded factor; conversely, prescribed plane-curve pencils show that no upper bound on generic-family gonality follows from min δ(X/k).\n\nCandidate contribution (theorem; novelty confidence low): In a covering family over a number field with dense T(k), the generic arithmetic k(T)-gonality d belongs to δ(X/k) under the standard birational-to-image convention; under mere dominance some divisor e of d belongs to δ(X/k). Equality occurs in every degree n via index-n genus-one products, while geometric gonality can miss min δ(X/k) by an unbounded factor." }, { "id": 20000361, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0013", "title": "Del Pezzo irrationality values and a global-field quartic characteristic dichotomy", "statement": "For a variety $X/F$, define its degree of irrationality $\\operatorname{irr}(X)$ to be the minimum degree of a rational map $X\\dashrightarrow\\mathbb P^2$ (defined over $F$).\n\nCompute $\\operatorname{irr}(X/F)$ for a del Pezzo surface $X$ over a general field $F$.", "original_statement": "For a variety $X/F$, define its degree of irrationality $\\operatorname{irr}(X)$ to be the minimum degree of a rational map $X\\dashrightarrow\\mathbb P^2$ (defined over $F$).\n\nCompute $\\operatorname{irr}(X/F)$ for a del Pezzo surface $X$ over a general field $F$.", "clean_statement": "For a variety $X/F$, define its degree of irrationality $\\operatorname{irr}(X)$ to be the minimum degree of a rational map $X\\dashrightarrow\\mathbb P^2$ (defined over $F$).\n\nCompute $\\operatorname{irr}(X/F)$ for a del Pezzo surface $X$ over a general field $F$.", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 1.26 from the initial problem session of the March 2024 workshop *Degree \\(d\\) points on algebraic surfaces*. Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.26\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[12]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For a variety $X/F$, define its degree of irrationality $\\\\operatorname{irr}(X)$ to be the minimum degree of a rational map $X\\\\dashrightarrow\\\\mathbb P^2$ (defined over $F$).\\n\\nCompute $\\\\operatorname{irr}(X/F)$ for a del Pezzo surface $X$ over a general field $F$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0013", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The 2025 LVZ preprint completely classifies the possible degrees of irrationality by anticanonical degree and field class, while an individual-surface degree-8 imperfect-field decision gap remains. Combining LVZ with the 2023--2026 quartic-point theorems proves that every quartic del Pezzo surface over every characteristic-2 global function field has irrationality 1 if rational and 2 otherwise, whereas infinitely many quartics over Q and, for every odd prime p, over F_p(t) have irrationality 4. A separate proved index filter gives ind(Y) dividing irr_F(Y) over every field and ind(X) dividing the anticanonical degree for del Pezzo surfaces.\n\nCandidate contribution (corollary; novelty confidence low): For p equal to 0 or a prime, a global field F of characteristic p and a quartic del Pezzo surface X/F with irr_F(X)=4 exist if and only if p is not 2; more precisely, characteristic 2 is universally excluded, while infinite families occur over Q and over F_p(t) for every odd p." }, { "id": 20000362, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0014", "title": "Geometric specialization versus arithmetic oscillation", "statement": "How do all of these invariants (e.g. $\\min\\delta(X/k),\\operatorname{irr}(X/k),\\operatorname{cov.gon}(X/k)$, etc.) behave under specialization/deformation in a smooth family? This includes the arithmetic case, where the base is something like $\\operatorname{Spec}\\mathscr O_{k,S}$.", "original_statement": "How do all of these invariants (e.g. $\\min\\delta(X/k),\\operatorname{irr}(X/k),\\operatorname{cov.gon}(X/k)$, etc.) behave under specialization/deformation in a smooth family? This includes the arithmetic case, where the base is something like $\\operatorname{Spec}\\mathscr O_{k,S}$.", "clean_statement": "How do all of these invariants (e.g. $\\min\\delta(X/k),\\operatorname{irr}(X/k),\\operatorname{cov.gon}(X/k)$, etc.) behave under specialization/deformation in a smooth family? This includes the arithmetic case, where the base is something like $\\operatorname{Spec}\\mathscr O_{k,S}$.", "statement_status": "exact", "statement_verification": "This is record `AIM-ALGEBRAIC_NUMBER_THEORY-0014`, problem 1.28 in the AIM workshop *Degree d points on algebraic surfaces*, source file `aim-algebraic-number-theory-notes.json`, source index 13.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.28\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[13]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How do all of these invariants (e.g. $\\\\min\\\\delta(X/k),\\\\operatorname{irr}(X/k),\\\\operatorname{cov.gon}(X/k)$, etc.) behave under specialization/deformation in a smooth family? This includes the arithmetic case, where the base is something like $\\\\operatorname{Spec}\\\\mathscr O_{k,S}$.\"\nOriginal remarks: [\"Over $k=\\\\overline k$, it is known that covering gonality goes down in specializations. This is known for degree of irrationality in smooth families of simply connected surfaces (over $\\\\mathbb C$).\", \"$\\\\delta(-)$ is not very interested over finite fields because there are always only finitely many degree $d$ points.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0014", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "For the smooth proper family S_t = (X^2 - 2Y^2 - tZ^2 = 0) x P^1 over G_m/Q, both min delta(S_t/Q) and irr_Q(S_t) equal 1 on the Zariski-dense set of square parameters and equal 2 on the Zariski-dense set of three-times-square parameters. Hence both arithmetic fiber functions on G_m(Q), with its induced Zariski topology, are neither lower nor upper semicontinuous, while covering gonality and all corresponding geometric invariants remain constantly 1. A separate proved proposition shows that finite-field density-degree sets are empty in positive dimension and that a degree-d generic point in a proper DVR model specializes only to a zero-cycle of total degree d, whose exact residue degrees may split or ramify.\n\nCandidate contribution (counterexample; novelty confidence low): The displayed conic-product family is a candidate-novel explicit smooth proper family with two Zariski-dense arithmetic level sets simultaneously for min delta and degree of irrationality, despite constant geometric fibers and constant covering gonality." }, { "id": 20000363, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0015", "title": "Density degrees under finite extension", "statement": "Is there a surface $X$ over a number field $k$ and a finite extension $k'/k$ such that $\\delta(X/k)\\not\\subset\\delta(X/k')$?", "original_statement": "Is there a surface $X$ over a number field $k$ and a finite extension $k'/k$ such that $\\delta(X/k)\\not\\subset\\delta(X/k')$?", "clean_statement": "Is there a surface $X$ over a number field $k$ and a finite extension $k'/k$ such that $\\delta(X/k)\\not\\subset\\delta(X/k')$?", "statement_status": "exact", "statement_verification": "The record occurs in *aim-algebraic-number-theory-notes.json* at zero-based index 14. There is no visible corruption. The AIM workshop summary specifies the intended class of surfaces: smooth, projective, and geometrically integral over a number field. That geometric-integrality hypothesis matters because the finite-union density argument below can fail on a disconnected base change.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.3\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[14]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a surface $X$ over a number field $k$ and a finite extension $k'/k$ such that $\\\\delta(X/k)\\\\not\\\\subset\\\\delta(X/k')$?\"\nOriginal remarks: [\"If $X$ is a curve, then this can not happen.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0015", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let X be a smooth projective geometrically integral variety over a number field k, let K/k be finite, and let d belong to δ(X/k). Then some e <= d belongs to δ(X_K/K); if K/k is Galois, e may be chosen to divide d, so every Galois counterexample forces failure of scalar-multiplicative closure on the base-changed variety. Exact degree d persists whenever gcd(d,[K:k])=1 and for all sufficiently large d with a threshold depending on X_K/K. For cyclic K/k of prime degree p, failure in degree p is equivalent to density of the genuinely new points X(K) minus X(k) together with p not belonging to δ(X_K/K). No surface counterexample is currently known from the literature checked.\n\nCandidate contribution (reduction; novelty confidence low): For a cyclic extension K/k of prime degree p, p belongs to δ(X/k) but not to δ(X_K/K) if and only if X(K) minus X(k) is Zariski dense in X_K and p does not belong to δ(X_K/K). Thus a quadratic counterexample is exactly a descended surface with dense new K-points but non-dense quadratic K-points, forcing 1 in δ(X_K/K) and 2 not in δ(X_K/K).", "research_difficulty_suggested": "L4" }, { "id": 20000364, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0016", "title": "Generic minimum density degree and a quartic counterexample", "statement": "The following is asked as an analogue of a result due to Debarre and Klassen \\cite{arXiv:alg-geom/9210004}.\n\nLet $X_d\\subset\\mathbb P^3_k$ ($k$ a number field) be a smooth surface of degree $d$. Are the degree $\\le d-3$ points not Zariski dense? What is $\\min\\delta(X_d/k)$?", "original_statement": "The following is asked as an analogue of a result due to Debarre and Klassen \\cite{arXiv:alg-geom/9210004}.\n\nLet $X_d\\subset\\mathbb P^3_k$ ($k$ a number field) be a smooth surface of degree $d$. Are the degree $\\le d-3$ points not Zariski dense? What is $\\min\\delta(X_d/k)$?", "clean_statement": "The following is asked as an analogue of a result due to Debarre and Klassen \\cite{arXiv:alg-geom/9210004}.\n\nLet $X_d\\subset\\mathbb P^3_k$ ($k$ a number field) be a smooth surface of degree $d$. Are the degree $\\le d-3$ points not Zariski dense? What is $\\min\\delta(X_d/k)$?", "statement_status": "exact", "statement_verification": "The record is Problem 1.32 from the AIM workshop *Degree \\(d\\) points on algebraic surfaces* (source file `aim-algebraic-number-theory-notes.json`, source index 15). The source record is legible and no reconstruction is needed. Its problem field is, verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.32\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[15]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The following is asked as an analogue of a result due to Debarre and Klassen \\\\cite{arXiv:alg-geom/9210004}.\\n\\nLet $X_d\\\\subset\\\\mathbb P^3_k$ ($k$ a number field) be a smooth surface of degree $d$. Are the degree $\\\\le d-3$ points not Zariski dense? What is $\\\\min\\\\delta(X_d/k)$?\"\nOriginal remarks: [\"You probably want to assume that $d$ is sufficiently large.\", \"One can also ask this question for the generic degree $d$ hypersurface, defined over the field of rational functions in the appropriate number of variables.\", \"For $d$ small, can we construct surfaces such that the degree $\\\\le d-3$ points *are* Zariski dense?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0016", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the generic degree-d surface over the function field of its coefficient space, every closed-point degree is divisible by d and degree-d closed points obtained from transverse rational line sections are Zariski dense; consequently its index, minimum density degree, and arithmetic degree of irrationality all equal d. For arbitrary number-field surfaces, projection gives d in the density degree set (and d-1 when there is a rational point). The smooth quartic x^4+8y^4-3z^4-6w^4=0 over Q has dense rational points, so the unrestricted non-density assertion fails at d=4, while the arbitrary sufficiently-high-degree case remains open.\n\nCandidate contribution (theorem; novelty confidence low): Over any infinite characteristic-zero base, the generic degree-d surface over its coefficient function field has no closed points of degree not divisible by d, has Zariski-dense degree-d closed points, and therefore satisfies min delta = irr_K = d." }, { "id": 20000365, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0017", "title": "Generic-line gonality and the pencil-descent obstruction", "statement": "Given a dominant map $\\psi:X\\dashrightarrow\\mathbb P^{\\dim X}$ of minimal degree $d$, is the gonality of the pullback of a general line equal to $d$?", "original_statement": "Given a dominant map $\\psi:X\\dashrightarrow\\mathbb P^{\\dim X}$ of minimal degree $d$, is the gonality of the pullback of a general line equal to $d$?", "clean_statement": "Given a dominant map $\\psi:X\\dashrightarrow\\mathbb P^{\\dim X}$ of minimal degree $d$, is the gonality of the pullback of a general line equal to $d$?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.34 in the initial problem session of the AIM workshop *Degree d points on algebraic surfaces*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.34\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[16]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given a dominant map $\\\\psi:X\\\\dashrightarrow\\\\mathbb P^{\\\\dim X}$ of minimal degree $d$, is the gonality of the pullback of a general line equal to $d$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0017", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If a degree-d map X to projective n-space computes the degree of irrationality over k, then the normalized generic pullback of a line in a star through a general k-point has K-gonality exactly d over the star's function field K. For a minimal map from a complex surface, geometric equality also holds whenever the geometric gonal pencil is unique: the pencil descends to a Brauer-Severi curve over C(t), which splits by Tsen's theorem. Consequently equality holds for d=2, for d=3 with sectional genus at least 2, and for d=4 with sectional genus at least 5. An independent arithmetic criterion gives equality when ind(X)=irr_k(X)=d.\n\nCandidate contribution (obstruction; novelty confidence low): For a minimal map from a complex surface, strict geometric inequality e= 11: Huang--Kadets--Martin prove that every degree-d closed point on the universal degree-d surface over C(B) lies on a base-field line, and Bezout plus separability makes it exactly the transverse line section. This attempt additionally proves the transfer to the generic surface over K = k(B) for a number field k: the universal index d prevents the point from splitting after extension to C(B), and the two-dimensional kernel of linear evaluation descends the unique containing line. The literal Hilbert-scheme reading has the opposite answer, since proper line sections form a 4-dimensional locally closed locus of codimension 2d-4 in Hilb^d(X).\n\nCandidate contribution (corollary; novelty confidence low): For any number field k and d >= 11, degree-d closed points on the generic degree-d surface over K = k(B) are in bijection with K-lines on which the universal defining form restricts irreducibly; the proof explicitly descends the published C(B) theorem using index-based non-splitting and the linear-span evaluation kernel." }, { "id": 20000367, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0019", "title": "An exact nonrational double-cover source for dense quadratic points", "statement": "The following is asked in analogy with a result of Harris and Silverman showing that, for a curve $C/k$, $2\\in\\delta(C/k)\\iff C$ is a double cover of $\\mathbb P^1$ or a positive rank elliptic curve (see https://www.ams.org/journals/proc/1991-112-02/S0002-9939-1991-1055774-0/S0002-9939-1991-1055774-0.pdf).\n\nClassify surfaces $X/k$ of general type with $2\\in\\delta(X/k)$, possibly using geometric and/or dynamical constructions.", "original_statement": "The following is asked in analogy with a result of Harris and Silverman showing that, for a curve $C/k$, $2\\in\\delta(C/k)\\iff C$ is a double cover of $\\mathbb P^1$ or a positive rank elliptic curve (see https://www.ams.org/journals/proc/1991-112-02/S0002-9939-1991-1055774-0/S0002-9939-1991-1055774-0.pdf).\n\nClassify surfaces $X/k$ of general type with $2\\in\\delta(X/k)$, possibly using geometric and/or dynamical constructions.", "clean_statement": "The following is asked in analogy with a result of Harris and Silverman showing that, for a curve $C/k$, $2\\in\\delta(C/k)\\iff C$ is a double cover of $\\mathbb P^1$ or a positive rank elliptic curve (see https://www.ams.org/journals/proc/1991-112-02/S0002-9939-1991-1055774-0/S0002-9939-1991-1055774-0.pdf).\n\nClassify surfaces $X/k$ of general type with $2\\in\\delta(X/k)$, possibly using geometric and/or dynamical constructions.", "statement_status": "exact", "statement_verification": "The canonical record is source index 18 of `aim-algebraic-number-theory-notes.json`, from the AIM workshop *Degree d points on algebraic surfaces*, Initial Problem Session, Problem 1.38. The problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.38\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[18]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The following is asked in analogy with a result of Harris and Silverman showing that, for a curve $C/k$, $2\\\\in\\\\delta(C/k)\\\\iff C$ is a double cover of $\\\\mathbb P^1$ or a positive rank elliptic curve (see https://www.ams.org/journals/proc/1991-112-02/S0002-9939-1991-1055774-0/S0002-9939-1991-1055774-0.pdf).\\n\\nClassify surfaces $X/k$ of general type with $2\\\\in\\\\delta(X/k)$, possibly using geometric and/or dynamical constructions.\"\nOriginal remarks: [\"Vojta has prove that for a hyperelliptic curve $C/k$ of genus $\\\\ge4$, all but finitely many degree $2$ points are contracted by its hyperelliptic map.\", \"If $\\\\operatorname{irr}(X)=2$, there exists a dominant map $X\\\\dashrightarrow\\\\mathbb P^2$ of degree $2$. What conditions on $X$ guarantee that the set of quadratic points on $X$ not arising from this map is not Zariski dense?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0019", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a generically finite degree-two rational map f:X-->Y over a number field, if X(k) is not Zariski dense, then the exact quadratic points in nonsplit fibers over Y(k) are Zariski dense in X if and only if Y(k) is Zariski dense. Applied to the Abel-Jacobi map C x C --> J of a genus-two curve with geometrically simple positive-rank Jacobian, this recovers the known result 2 in delta(C x C/k) of Berg--Fu--Gazaki--Porzio--Rawson--Vogt. Combining it with Chen--Martin gives the classification obstruction irr_bar(C x C)=4: dense quadratic points on a general-type surface need not arise from a degree-two map to the plane.\n\nCandidate contribution (theorem/synthesis; novelty confidence low): Candidate novelty: the direct degree-two source is dense exactly when its target has dense rational points, provided the source variety has nondense rational points; combining this exact-source iff with the genus-two Abel-Jacobi map and Chen--Martin's irrationality theorem yields general-type examples with 2 in delta but geometric irrationality 4." }, { "id": 20000368, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0020", "title": "Bounded-degree quotient obstructions to solvable points", "statement": "1.) Find effective obstructions to the existence of solvable points on varieties (in any characteristic).\n\n2.) Do there exist curves (over surfaces) over $\\mathbb Q$ without solvable points?", "original_statement": "1.) Find effective obstructions to the existence of solvable points on varieties (in any characteristic).\n\n2.) Do there exist curves (over surfaces) over $\\mathbb Q$ without solvable points?", "clean_statement": "1.) Find effective obstructions to the existence of solvable points on varieties (in any characteristic).\n\n2.) Do there exist curves (over surfaces) over $\\mathbb Q$ without solvable points?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.4 from the AIM workshop *Degree $d$ points on algebraic surfaces*. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Algebraic number theory\nWorkshop: Degree d points on algebraic surfaces\nSection: Initial Problem Session\nSource item: 1.4\nSource URL: http://aimpl.org/degreedsurface/1/\nCanonical location: aim-algebraic-number-theory-notes.json notes[19]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1.) Find effective obstructions to the existence of solvable points on varieties (in any characteristic).\\n\\n2.) Do there exist curves (over surfaces) over $\\\\mathbb Q$ without solvable points?\"\nOriginal remarks: [\"There are counterexamples for curves in positive characteristic for large enough genus. See, for example, Pál's paper https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/solvable-points-on-projective-algebraic-curves/ECE9AF2C2ED1C85F7733930C0544361C\", \"One could ask similar questions with 'solvable' replaced by 'cyclic' or 'abelian'.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/degreedsurface/1/", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0020", "aim-domain:algebraic-number-theory", "aim-workshop:degreedsurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every perfect field K, quasi-projective finite-type K-scheme X, and N at least 1, X has a solvable closed point of degree at most N if and only if one of the finitely many coarse quotients X^n/G, with 1 <= n <= N and G a transitive solvable subgroup of S_n, has a K-rational point. The converse remains valid at ramified or stabilized lifts because the Galois action on a fiber G/H lies in the solvable subquotient N_G(H)/H. In addition, every geometrically nonempty variety over a number field has a point over the solvable closure of every completion, so ordinary local nonemptiness cannot obstruct global solvable points; a prime-degree Frobenius factorization test gives an explicit pointwise nonsolvability certificate.\n\nCandidate contribution (reduction; novelty confidence low): The finite disjoint union over all degrees n <= N and all transitive solvable permutation groups G <= S_n is an exact K-rational-point locus for solvable closed points of degree at most N, including coarse-quotient points with nontrivial stabilizer via Aut_G(G/H) congruent to N_G(H)/H." }, { "id": 20000369, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0021", "title": "All-valuation isotropy of a quadratic pencil and the limits of Boolean local tests", "statement": "Problem 1. Browning: Can we develop a version of the circle method over Q(t)? Wooley: The major arcs are difficult to understand. Browning: For example, consider the diagonal quadratic form\n\nA1x21 + · · · + Asx2\n\n> s\n\n= 0 (1) with Aj ∈ Q(t). There are \"obvious\" local conditions, e.g., arising from discrete valuation rings for Q(t). Does the Hasse principle hold? Wooley: If Aj are linear, say Aj = cj + td j with cj, d j ∈ Q, then (1) is equivalent to the system of equations\n\n> s\n\n∑\n\n> j=1\n\ncj x2\n\n> j\n\n=\n\n> s\n\n∑\n\n> j=1\n\ndj x2\n\n> j\n\n= 0 (2) by Amer-Brumer, which defines a quartic del Pezzo surface over Q. Do the \"ob-vious\" local obstructions over Q(t) for (1) capture the Brauer-Manin obstruction over Q for (2)? Harari: We may ask these questions over Qp(t); see work of Harari-Szamuely.", "original_statement": "Problem 1. Browning: Can we develop a version of the circle method over Q(t)? Wooley: The major arcs are difficult to understand. Browning: For example, consider the diagonal quadratic form \n\nA1x21 + · · · + Asx2 \n\n> s\n\n= 0 (1) with Aj ∈ Q(t). There are \"obvious\" local conditions, e.g., arising from discrete valuation rings for Q(t). Does the Hasse principle hold? Wooley: If Aj are linear, say Aj = cj + td j with cj, d j ∈ Q, then (1) is equivalent to the system of equations \n\n> s\n\n∑\n\n> j=1\n\ncj x2 \n\n> j\n\n=\n\n> s\n\n∑\n\n> j=1\n\ndj x2 \n\n> j\n\n= 0 (2) by Amer-Brumer, which defines a quartic del Pezzo surface over Q. Do the \"ob-vious\" local obstructions over Q(t) for (1) capture the Brauer-Manin obstruction over Q for (2)? Harari: We may ask these questions over Qp(t); see work of Harari-Szamuely.", "clean_statement": "Problem 1. Browning: Can we develop a version of the circle method over Q(t)? Wooley: The major arcs are difficult to understand. Browning: For example, consider the diagonal quadratic form\n\nA1x21 + · · · + Asx2\n\n> s\n\n= 0 (1) with Aj ∈ Q(t). There are \"obvious\" local conditions, e.g., arising from discrete valuation rings for Q(t). Does the Hasse principle hold? Wooley: If Aj are linear, say Aj = cj + td j with cj, d j ∈ Q, then (1) is equivalent to the system of equations\n\n> s\n\n∑\n\n> j=1\n\ncj x2\n\n> j\n\n=\n\n> s\n\n∑\n\n> j=1\n\ndj x2\n\n> j\n\n= 0 (2) by Amer-Brumer, which defines a quartic del Pezzo surface over Q. Do the \"ob-vious\" local obstructions over Q(t) for (1) capture the Brauer-Manin obstruction over Q for (2)? Harari: We may ask these questions over Qp(t); see work of Harari-Szamuely.", "statement_status": "exact", "statement_verification": "The record comes from Problem 1 of the AIM workshop list *Rational and integral points on higher-dimensional varieties* (May 28, 2014). The PDF gives the following statement (notation normalized, but wording preserved):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher-dimensional varieties\nSection: \nSource item: 1\nSource URL: https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[20]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1. Browning: Can we develop a version of the circle method over Q(t)? Wooley: The major arcs are difficult to understand. Browning: For example, consider the diagonal quadratic form \\n\\nA1x21 + · · · + Asx2 \\n\\n> s\\n\\n= 0 (1) with Aj ∈ Q(t). There are \\\"obvious\\\" local conditions, e.g., arising from discrete valuation rings for Q(t). Does the Hasse principle hold? Wooley: If Aj are linear, say Aj = cj + td j with cj, d j ∈ Q, then (1) is equivalent to the system of equations \\n\\n> s\\n\\n∑\\n\\n> j=1\\n\\ncj x2 \\n\\n> j\\n\\n=\\n\\n> s\\n\\n∑\\n\\n> j=1\\n\\ndj x2 \\n\\n> j\\n\\n= 0 (2) by Amer-Brumer, which defines a quartic del Pezzo surface over Q. Do the \\\"ob-vious\\\" local obstructions over Q(t) for (1) capture the Brauer-Manin obstruction over Q for (2)? Harari: We may ask these questions over Qp(t); see work of Harari-Szamuely.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0021", "aim-domain:algebraic-number-theory", "aim-workshop:ratlhigherdimvarproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let k be a number field and let X={q_0=q_1=0} be a smooth complete intersection of two quadrics. If X has points over every completion of k, then q_0+Tq_1 is isotropic over the completion of k(T) at every rank-one discrete valuation. Amer--Brumer also gives X(k_v) nonempty if and only if q_0+Tq_1 is isotropic over k_v(T), and X(k) nonempty if and only if it is isotropic over k(T). Hence every Hasse counterexample X produces a globally anisotropic pencil that is isotropic over all coefficient fields and all rank-one DVR completions; if the failure is explained by Brauer--Manin, bare Boolean DVR-isotropy tests cannot capture that point-dependent cross-place reciprocity obstruction.\n\nCandidate contribution (reduction; novelty confidence low): Adelic solubility of the smooth base intersection forces isotropy of its quadratic pencil at every rank-one discrete-valuation completion of k(T), including closed discriminant points via the corank-one first residue form; combined with Amer--Brumer, this gives an exact three-layer separation between coefficient-field isotropy, all-DVR isotropy, and global isotropy." }, { "id": 20000370, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0022", "title": "Singularities, Fano type, and anticanonical height zeta functions", "statement": "Problem 2. Cheltsov: What is the \"right\" assumption on a variety V for consid-ering height zeta functions? Definitely smooth V with ample anticanonical sheaf\n\nω−1\n\n> V\n\nshould be allowed. How about klt (i.e., Kawamata log terminal) V? Or V\n\nof Fano type (i.e., there is an effective Q-divisor ∆ such that ( V, ∆) is ample and\n\n−(KV + ∆) is ample)? An example for the latter: A quasi-smooth hypersurface V ⊂ P(a0,..., a n) in weighted projective space of degree deg( V ) < a 0 + · · · + an.", "original_statement": "Problem 2. Cheltsov: What is the \"right\" assumption on a variety V for consid-ering height zeta functions? Definitely smooth V with ample anticanonical sheaf \n\nω−1 \n\n> V\n\nshould be allowed. How about klt (i.e., Kawamata log terminal) V? Or V\n\nof Fano type (i.e., there is an effective Q-divisor ∆ such that ( V, ∆) is ample and \n\n−(KV + ∆) is ample)? An example for the latter: A quasi-smooth hypersurface V ⊂ P(a0,..., a n) in weighted projective space of degree deg( V ) < a 0 + · · · + an.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The phrase “\\((V,\\Delta)\\) is ample” occurs in the original PDF itself; it is not an OCR invention. Since ampleness is not a property of a pair, the standard and almost certainly intended definition is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher-dimensional varieties\nSection: \nSource item: 2\nSource URL: https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[21]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2. Cheltsov: What is the \\\"right\\\" assumption on a variety V for consid-ering height zeta functions? Definitely smooth V with ample anticanonical sheaf \\n\\nω−1 \\n\\n> V\\n\\nshould be allowed. How about klt (i.e., Kawamata log terminal) V? Or V\\n\\nof Fano type (i.e., there is an effective Q-divisor ∆ such that ( V, ∆) is ample and \\n\\n−(KV + ∆) is ample)? An example for the latter: A quasi-smooth hypersurface V ⊂ P(a0,..., a n) in weighted projective space of degree deg( V ) < a 0 + · · · + an.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0022", "aim-domain:algebraic-number-theory", "aim-workshop:ratlhigherdimvarproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem separates into three thresholds: klt is exactly the local discrepancy-integrability condition, while a canonical Q-Fano has resolution anticanonical Fujita invariant a=1, and Fano type alone does not ensure a viable raw anticanonical zeta function. Explicitly, blowing up r>=4 rational collinear points in P^2 gives a smooth Fano-type surface X_r whose all-points anticanonical zeta diverges for every real s>0; contracting the bad curve gives a klt noncanonical Q-Fano Y_r with a(X_r,f^*(-K_Y_r))=3(r-1)/(2r)>1. Only an open-set divisor-class and transported-metric identification is claimed, not a bounded comparison of arbitrary global metrics near the deleted curve.\n\nCandidate contribution (obstruction/example; novelty confidence low): For every r>=4, the collinear-blowup/contraction diagram gives a smooth Fano-type surface with raw anticanonical zeta divergence, a klt Q-Fano contraction, and the exact resolution invariant a=3(r-1)/(2r), thereby exhibiting in one explicit family both global height degeneration and discrepancy-driven model dependence." }, { "id": 20000371, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0023", "title": "Common twists, sparse families, and the marked-torsor bottleneck", "statement": "Problem 3. Skorobogatov: Does Bhargava's machinery have implications for the Hasse principle for special surfaces? For example, let F, G ∈ Q[x, y ] be homogeneous polynomials of degree 3. Con-sider the cubic surface\n\nS = {F (x, y ) = G(z, w )} ⊂ P3\n\n> Q.\n\nThe defining equation is equivalent to the system of equations\n\n{u3 = tF (x, y ), v 3 = tG (z, w )}.\n\nThis defines a family of cubic twists of curves of genus 1 over the t-line. Swinnerton-Dyer has discussed how to search for t such that this system is solvable over Q in the diagonal case [Ann. Sci. ENS]. Can we extend his work beyond the diagonal case? Similarly, consider Kummer K3 surfaces defined by\n\nz2 = f (x)g(y),\n\nwhere f, g are quartic separable polynomials. This is equivalent to the family of quadratic twists of curves of genus 1 defined by\n\nu2 = tf (x), v 2 = tg (y).\n\nThe goal is to eliminate the condition in Swinnerton-Dyer's work that (the 2-primary part of) X has finite order for quadratic twists, using the recent work presented in Bhargava's talk.\n\n> Date: May 28, 2014.\n> 12AIM OPEN PROBLEM SESSION", "original_statement": "Problem 3. Skorobogatov: Does Bhargava's machinery have implications for the Hasse principle for special surfaces? For example, let F, G ∈ Q[x, y ] be homogeneous polynomials of degree 3. Con-sider the cubic surface \n\nS = {F (x, y ) = G(z, w )} ⊂ P3\n\n> Q.\n\nThe defining equation is equivalent to the system of equations \n\n{u3 = tF (x, y ), v 3 = tG (z, w )}.\n\nThis defines a family of cubic twists of curves of genus 1 over the t-line. Swinnerton-Dyer has discussed how to search for t such that this system is solvable over Q in the diagonal case [Ann. Sci. ENS]. Can we extend his work beyond the diagonal case? Similarly, consider Kummer K3 surfaces defined by \n\nz2 = f (x)g(y),\n\nwhere f, g are quartic separable polynomials. This is equivalent to the family of quadratic twists of curves of genus 1 defined by \n\nu2 = tf (x), v 2 = tg (y).\n\nThe goal is to eliminate the condition in Swinnerton-Dyer's work that (the 2-primary part of) X has finite order for quadratic twists, using the recent work presented in Bhargava's talk. \n\n> Date: May 28, 2014.\n> 12AIM OPEN PROBLEM SESSION", "clean_statement": "Problem 3. Skorobogatov: Does Bhargava's machinery have implications for the Hasse principle for special surfaces? For example, let F, G ∈ Q[x, y ] be homogeneous polynomials of degree 3. Con-sider the cubic surface\n\nS = {F (x, y ) = G(z, w )} ⊂ P3\n\n> Q.\n\nThe defining equation is equivalent to the system of equations\n\n{u3 = tF (x, y ), v 3 = tG (z, w )}.\n\nThis defines a family of cubic twists of curves of genus 1 over the t-line. Swinnerton-Dyer has discussed how to search for t such that this system is solvable over Q in the diagonal case [Ann. Sci. ENS]. Can we extend his work beyond the diagonal case? Similarly, consider Kummer K3 surfaces defined by\n\nz2 = f (x)g(y),\n\nwhere f, g are quartic separable polynomials. This is equivalent to the family of quadratic twists of curves of genus 1 defined by\n\nu2 = tf (x), v 2 = tg (y).\n\nThe goal is to eliminate the condition in Swinnerton-Dyer's work that (the 2-primary part of) X has finite order for quadratic twists, using the recent work presented in Bhargava's talk.\n\n> Date: May 28, 2014.\n> 12AIM OPEN PROBLEM SESSION", "statement_status": "exact", "statement_verification": "The source is Problem 3 (attributed to Alexei Skorobogatov) in the AIM open-problem notes *Rational and integral points on higher-dimensional varieties*, dated May 28, 2014. Restoring only mathematical typesetting, the problem says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher-dimensional varieties\nSection: \nSource item: 3\nSource URL: https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[22]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3. Skorobogatov: Does Bhargava's machinery have implications for the Hasse principle for special surfaces? For example, let F, G ∈ Q[x, y ] be homogeneous polynomials of degree 3. Con-sider the cubic surface \\n\\nS = {F (x, y ) = G(z, w )} ⊂ P3\\n\\n> Q.\\n\\nThe defining equation is equivalent to the system of equations \\n\\n{u3 = tF (x, y ), v 3 = tG (z, w )}.\\n\\nThis defines a family of cubic twists of curves of genus 1 over the t-line. Swinnerton-Dyer has discussed how to search for t such that this system is solvable over Q in the diagonal case [Ann. Sci. ENS]. Can we extend his work beyond the diagonal case? Similarly, consider Kummer K3 surfaces defined by \\n\\nz2 = f (x)g(y),\\n\\nwhere f, g are quartic separable polynomials. This is equivalent to the family of quadratic twists of curves of genus 1 defined by \\n\\nu2 = tf (x), v 2 = tg (y).\\n\\nThe goal is to eliminate the condition in Swinnerton-Dyer's work that (the 2-primary part of) X has finite order for quadratic twists, using the recent work presented in Bhargava's talk. \\n\\n> Date: May 28, 2014.\\n> 12AIM OPEN PROBLEM SESSION\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0023", "aim-domain:algebraic-number-theory", "aim-workshop:ratlhigherdimvarproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Bhargava-style arithmetic statistics have yielded genuine family-average progress: Browning proved that a positive proportion of the split cubic surfaces in the problem, and analogously a positive proportion of the Kummer surfaces, have rational points. The general fixed-surface Hasse-principle and Sha-finiteness-removal questions remain open. This attempt proves exact common-twist/value-class criteria, including cubic zero-value boundary cases and a strict separation between affine quartic points and projective torsor points, and proves a relative simultaneous-twist selector: bad counts must be little-o of the actual common locally/descent-constrained family, not merely density zero in the ambient twist family.\n\nCandidate contribution (reduction; novelty confidence low): For the two prescribed cubic or quadratic coverings, an explicit relative selector criterion reduces the rational-point problem to two marked-class exceptional estimates inside the same admissible twist family; the criterion distinguishes affine from projective goodness and is sharp against replacement by ambient density-one estimates." }, { "id": 20000372, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0024", "title": "Arithmetic surjectivity forces recurrence of locally soluble fibers", "statement": "Problem 4. Viray: Let φ: X → E be a fibration over an elliptic curve of positive rank over Q whose generic fiber is smooth and geometrically irreducible. Let\n\nZ = {p ∈ E(Q) | Xp = φ−1(p) has points everywhere locally }.\n\nWhat can we say about Z? Is |Z| < ∞ with Z 6 = ∅ possible? The motivation is that work of Poonen, Skorobogatov-Harpaz and Colliot-Th´ el` ene- Pal-Skorobogatov constructs X failing the Hasse principle such that none of the known obstructions apply. All of these use a map X → C to a curve with 0 < |C(Q)| < ∞.Browning: The case where φ is a conic bundle may already be interesting.", "original_statement": "Problem 4. Viray: Let φ: X → E be a fibration over an elliptic curve of positive rank over Q whose generic fiber is smooth and geometrically irreducible. Let \n\nZ = {p ∈ E(Q) | Xp = φ−1(p) has points everywhere locally }.\n\nWhat can we say about Z? Is |Z| < ∞ with Z 6 = ∅ possible? The motivation is that work of Poonen, Skorobogatov-Harpaz and Colliot-Th´ el` ene- Pal-Skorobogatov constructs X failing the Hasse principle such that none of the known obstructions apply. All of these use a map X → C to a curve with 0 < |C(Q)| < ∞.Browning: The case where φ is a conic bundle may already be interesting.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The record is Problem 4, attributed to Viray, in the AIM workshop list *Rational and integral points on higher-dimensional varieties* (PDF dated May 28, 2014). Direct inspection of page 2 of the PDF resolves the OCR errors \\(6=\\) and the damaged accents. With line-break hyphenation normalized, the statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher-dimensional varieties\nSection: \nSource item: 4\nSource URL: https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[23]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4. Viray: Let φ: X → E be a fibration over an elliptic curve of positive rank over Q whose generic fiber is smooth and geometrically irreducible. Let \\n\\nZ = {p ∈ E(Q) | Xp = φ−1(p) has points everywhere locally }.\\n\\nWhat can we say about Z? Is |Z| < ∞ with Z 6 = ∅ possible? The motivation is that work of Poonen, Skorobogatov-Harpaz and Colliot-Th´ el` ene- Pal-Skorobogatov constructs X failing the Hasse principle such that none of the known obstructions apply. All of these use a map X → C to a curve with 0 < |C(Q)| < ∞.Browning: The case where φ is a conic bundle may already be interesting.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0024", "aim-domain:algebraic-number-theory", "aim-workshop:ratlhigherdimvarproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let E/k have positive Mordell-Weil rank and let phi:X->E be locally surjective over every completion outside one fixed finite set S. If p0 has a relative-smooth local lift over every place in S, then for every non-torsion P in E(k), p0 is an S-adic accumulation point of infinitely many everywhere locally soluble fibers p0+nP. Consequently, a smooth proper family with geometrically integral fibers has either no everywhere locally soluble rational fibers or infinitely many. Combined with the Loughran-Skorobogatov-Smeets criterion, a finite nonempty set containing a point over which phi is smooth forces a non-pseudo-split codimension-one fiber on some smooth modification.\n\nCandidate contribution (theorem/reduction; novelty confidence low): Under arithmetic surjectivity, every everywhere locally soluble fiber with relative-smooth lifts at the exceptional places is an S-adic accumulation point of infinitely many everywhere locally soluble fibers in a single cyclic Mordell-Weil coset; hence a finite nonempty example containing a smooth fiber reduces geometrically to failure of pseudo-splitness on a modification." }, { "id": 20000373, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0025", "title": "Real approximation and odd-kernel reduction for finite stabilizers", "statement": "Problem 5. Harari: The following question is due to Borovoi: Consider weak approximation for X = SL n /G over Q, where G is a finite group scheme that is not necessarily constant. For example, is X(Q) dense in X(R)? If G is constant, this is known to be true. A variant is the following. Given X = SL n /G with a constant finite group scheme G over a number field K with r ≥ 2 real places v1,..., v r. Is X(K) dense in ∏rj=1 X(Kvr )? A formulation via non-abelian Galois cohomology is given in the case K = Q as follows: Is\n\nH1(Gal( Q/Q), G (Q)) → H1(Gal( C/R), G (C)) surjective? Lucchini Arteche: The algebraic Brauer-Manin obstruction says nothing for this problem. Skorobogatov: The result is known for abelian G, due to Borovoi.", "original_statement": "Problem 5. Harari: The following question is due to Borovoi: Consider weak approximation for X = SL n /G over Q, where G is a finite group scheme that is not necessarily constant. For example, is X(Q) dense in X(R)? If G is constant, this is known to be true. A variant is the following. Given X = SL n /G with a constant finite group scheme G over a number field K with r ≥ 2 real places v1,..., v r. Is X(K) dense in ∏rj=1 X(Kvr )? A formulation via non-abelian Galois cohomology is given in the case K = Q as follows: Is \n\nH1(Gal( Q/Q), G (Q)) → H1(Gal( C/R), G (C)) surjective? Lucchini Arteche: The algebraic Brauer-Manin obstruction says nothing for this problem. Skorobogatov: The result is known for abelian G, due to Borovoi.", "clean_statement": "Problem 5. Harari: The following question is due to Borovoi: Consider weak approximation for X = SL n /G over Q, where G is a finite group scheme that is not necessarily constant. For example, is X(Q) dense in X(R)? If G is constant, this is known to be true. A variant is the following. Given X = SL n /G with a constant finite group scheme G over a number field K with r ≥ 2 real places v1,..., v r. Is X(K) dense in ∏rj=1 X(Kvr )? A formulation via non-abelian Galois cohomology is given in the case K = Q as follows: Is\n\nH1(Gal( Q/Q), G (Q)) → H1(Gal( C/R), G (C)) surjective? Lucchini Arteche: The algebraic Brauer-Manin obstruction says nothing for this problem. Skorobogatov: The result is known for abelian G, due to Borovoi.", "statement_status": "exact", "statement_verification": "The record is Problem 5 in the AIM open problem session for *Rational and integral points on higher-dimensional varieties*, dated May 28, 2014. The primary PDF reads as follows (overlines and product indices restored from the typeset source):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher-dimensional varieties\nSection: \nSource item: 5\nSource URL: https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[24]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 5. Harari: The following question is due to Borovoi: Consider weak approximation for X = SL n /G over Q, where G is a finite group scheme that is not necessarily constant. For example, is X(Q) dense in X(R)? If G is constant, this is known to be true. A variant is the following. Given X = SL n /G with a constant finite group scheme G over a number field K with r ≥ 2 real places v1,..., v r. Is X(K) dense in ∏rj=1 X(Kvr )? A formulation via non-abelian Galois cohomology is given in the case K = Q as follows: Is \\n\\nH1(Gal( Q/Q), G (Q)) → H1(Gal( C/R), G (C)) surjective? Lucchini Arteche: The algebraic Brauer-Manin obstruction says nothing for this problem. Skorobogatov: The result is known for abelian G, due to Borovoi.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0025", "aim-domain:algebraic-number-theory", "aim-workshop:ratlhigherdimvarproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The constant-group, several-real-places variant in the 2014 AIM record is affirmatively resolved by a 2026 preprint, and all solvable finite group schemes are covered by another 2026 preprint together with the known real-place Brauer result; the arbitrary nonconstant nonsolvable case remains open. Independently, for every exact sequence 1 -> N -> A -> Q -> 1 of finite k-groups with N of odd order, the induced map H^1(k_v,A) -> H^1(k_v,Q) is bijective at every real place. Hence real approximation for A is equivalent to an explicit global-image condition in H^1(k,Q), and for a k-group-split extension it is equivalent to real approximation for Q. The proof also identifies the connected components of (SL_n/A)(R) with H^1(R,A) and derives the precise equivalence between cohomological localization and density on products of real loci.\n\nCandidate contribution (reduction; novelty confidence low): For an odd-order normal kernel N in an exact sequence 1 -> N -> A -> Q -> 1 of finite k-groups, real local nonabelian H^1 is unchanged: H^1(k_v,A) -> H^1(k_v,Q) is bijective for every real v; consequently A has real approximation exactly when loc_R(p_*H^1(k,A)) is the full product of the local H^1(k_v,Q), with an if-and-only-if reduction to Q when the extension has a Galois-equivariant group section." }, { "id": 20000374, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0026", "title": "Arithmetic purity: solved quadric cases, a sieve criterion, and diagonal topology", "statement": "Problem 6. Wittenberg: Let X be a smooth variety over a number field K, let\n\nS be a finite set of places. Assume that X satisfies strong approximation outside\n\nS. Take a closed subvariety Z ⊂ X of codimension two. Does X \\ Z satisfy strong approximation outside S?Tschinkel: We can also ask for Zariski density (of S-integral points). Wittenberg: The result is known for X = An and arbitrary Z of codimension 2. Interesting cases are:\n\n• Wittenberg: affine quadric hypersurfaces X ⊂ A4, for example, defined by\n\nq(x1, x 2, x 3, x 4) = c for a quadratic form q and a constant c.\n\n• Harari: X a simply connected linear algebraic group\n\n• Wooley: is this true by the circle method for hypersurfaces of fixed degree as soon as the dimension is large enough? Heath-Brown: for example, is this true for quadrics X ⊂ A5?Harari: If X satisfies strong approximation, then X is algebraically simply con-nected. If X is algebraically simply connected, then X \\ Z is also simply connected. Hence considering π1 should not be helpful to get a counterexample to the problem. Also Brauer groups are not expected to be helpful. Heath-Brown: Can we drop the condition that Z ⊂ X has codimension ≥ 2? Check the topology. Colliot-Th´ el` ene: Can the circle method be used to prove that π1(X) is trivial? For example, the circle method handles\n\nxr1\n\n> 1\n\n− xr2\n\n> 2\n\n+ xr3\n\n> 3\n\n− · · · ± xrn\n\n> n\n\n= c ∈ Z\n\nin sufficiently many variables. Can we show that π1(X) is trivial without the circle method?", "original_statement": "Problem 6. Wittenberg: Let X be a smooth variety over a number field K, let \n\nS be a finite set of places. Assume that X satisfies strong approximation outside \n\nS. Take a closed subvariety Z ⊂ X of codimension two. Does X \\ Z satisfy strong approximation outside S?Tschinkel: We can also ask for Zariski density (of S-integral points). Wittenberg: The result is known for X = An and arbitrary Z of codimension 2. Interesting cases are: \n\n• Wittenberg: affine quadric hypersurfaces X ⊂ A4, for example, defined by \n\nq(x1, x 2, x 3, x 4) = c for a quadratic form q and a constant c.\n\n• Harari: X a simply connected linear algebraic group \n\n• Wooley: is this true by the circle method for hypersurfaces of fixed degree as soon as the dimension is large enough? Heath-Brown: for example, is this true for quadrics X ⊂ A5?Harari: If X satisfies strong approximation, then X is algebraically simply con-nected. If X is algebraically simply connected, then X \\ Z is also simply connected. Hence considering π1 should not be helpful to get a counterexample to the problem. Also Brauer groups are not expected to be helpful. Heath-Brown: Can we drop the condition that Z ⊂ X has codimension ≥ 2? Check the topology. Colliot-Th´ el` ene: Can the circle method be used to prove that π1(X) is trivial? For example, the circle method handles \n\nxr1 \n\n> 1\n\n− xr2 \n\n> 2\n\n+ xr3 \n\n> 3\n\n− · · · ± xrn \n\n> n\n\n= c ∈ Z\n\nin sufficiently many variables. Can we show that π1(X) is trivial without the circle method?", "clean_statement": "Problem 6. Wittenberg: Let X be a smooth variety over a number field K, let\n\nS be a finite set of places. Assume that X satisfies strong approximation outside\n\nS. Take a closed subvariety Z ⊂ X of codimension two. Does X \\ Z satisfy strong approximation outside S?Tschinkel: We can also ask for Zariski density (of S-integral points). Wittenberg: The result is known for X = An and arbitrary Z of codimension 2. Interesting cases are:\n\n• Wittenberg: affine quadric hypersurfaces X ⊂ A4, for example, defined by\n\nq(x1, x 2, x 3, x 4) = c for a quadratic form q and a constant c.\n\n• Harari: X a simply connected linear algebraic group\n\n• Wooley: is this true by the circle method for hypersurfaces of fixed degree as soon as the dimension is large enough? Heath-Brown: for example, is this true for quadrics X ⊂ A5?Harari: If X satisfies strong approximation, then X is algebraically simply con-nected. If X is algebraically simply connected, then X \\ Z is also simply connected. Hence considering π1 should not be helpful to get a counterexample to the problem. Also Brauer groups are not expected to be helpful. Heath-Brown: Can we drop the condition that Z ⊂ X has codimension ≥ 2? Check the topology. Colliot-Th´ el` ene: Can the circle method be used to prove that π1(X) is trivial? For example, the circle method handles\n\nxr1\n\n> 1\n\n− xr2\n\n> 2\n\n+ xr3\n\n> 3\n\n− · · · ± xrn\n\n> n\n\n= c ∈ Z\n\nin sufficiently many variables. Can we show that π1(X) is trivial without the circle method?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 6 from the AIM workshop *Rational and integral points on higher-dimensional varieties*. I checked the PDF directly because the JSON extraction corrupts superscripts in the final equation. In unambiguous notation, it asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher-dimensional varieties\nSection: \nSource item: 6\nSource URL: https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[25]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 6. Wittenberg: Let X be a smooth variety over a number field K, let \\n\\nS be a finite set of places. Assume that X satisfies strong approximation outside \\n\\nS. Take a closed subvariety Z ⊂ X of codimension two. Does X \\\\ Z satisfy strong approximation outside S?Tschinkel: We can also ask for Zariski density (of S-integral points). Wittenberg: The result is known for X = An and arbitrary Z of codimension 2. Interesting cases are: \\n\\n• Wittenberg: affine quadric hypersurfaces X ⊂ A4, for example, defined by \\n\\nq(x1, x 2, x 3, x 4) = c for a quadratic form q and a constant c.\\n\\n• Harari: X a simply connected linear algebraic group \\n\\n• Wooley: is this true by the circle method for hypersurfaces of fixed degree as soon as the dimension is large enough? Heath-Brown: for example, is this true for quadrics X ⊂ A5?Harari: If X satisfies strong approximation, then X is algebraically simply con-nected. If X is algebraically simply connected, then X \\\\ Z is also simply connected. Hence considering π1 should not be helpful to get a counterexample to the problem. Also Brauer groups are not expected to be helpful. Heath-Brown: Can we drop the condition that Z ⊂ X has codimension ≥ 2? Check the topology. Colliot-Th´ el` ene: Can the circle method be used to prove that π1(X) is trivial? For example, the circle method handles \\n\\nxr1 \\n\\n> 1\\n\\n− xr2 \\n\\n> 2\\n\\n+ xr3 \\n\\n> 3\\n\\n− · · · ± xrn \\n\\n> n\\n\\n= c ∈ Z\\n\\nin sufficiently many variables. Can we show that π1(X) is trivial without the circle method?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0026", "aim-domain:algebraic-number-theory", "aim-workshop:ratlhigherdimvarproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general arithmetic-purity question remains open, but the affine quadric cases in four and five variables are now covered under the strong-approximation noncompactness hypothesis by the cited literature. This attempt proves a qualitative finite-sieve/tail-sieve criterion that isolates the missing large-prime input, a summable codimension-two local-risk estimate, an exact bouquet-of-spheres and punctured-simple-connectivity theorem for the diagonal family in the AIM record, the Zariski-density consequence of strong approximation for integral points, and an explicit codimension-one counterexample.\n\nCandidate contribution (theorem; novelty confidence low): For every nonzero diagonal fiber X = {sum epsilon_i x_i^{r_i} = a} over the complex numbers with n at least 3, the fiber is affine space if an exponent is one and otherwise has homotopy type a bouquet of product_i(r_i-1) copies of S^{n-1}; moreover, deleting any closed complex subvariety of codimension at least two leaves it simply connected. The exact AIM-focused combination with the puncturing conclusion is the candidate contribution." }, { "id": 20000375, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0027", "title": "Elliptic scrolls with Hasse failure and ratio tending to one half", "statement": "Problem 7. Heath-Brown: Can you construct a sequence of smooth projective varieties Xk ⊂ Pk\n\n> Q\n\nwith Xk(Qp) 6 = ∅ for all places p but Xk(Q) = ∅ such that dim( Xk )deg( Xk )AIM OPEN PROBLEM SESSION 3\n\nis unbounded? Browning-Heath-Brown have given a sequence where dim( Xk )deg( Xk ) tends to 13 and dim( Xk) is tends to ∞.Wooley: How about removing the requirement of smoothness and considering the singular norm forms\n\nNK/ Q(x1α1 + · · · + xdαd) = ct d\n\nwhere d = [ K: Q]? Heath-Brown: What happens when Xk ⊂ Pk is a hypersurface? Is there any example of a smooth hypersurface of dimension ≥ 3 failing the Hasse principle? Colliot-Th´ el` ene: Sarnak-Wang have shown that the Bombieri-Lang conjecture would imply that there are many such examples of general type. Wooley: An analytic attack to show that there exist some such varieties could be as follows. Choose a locally soluble smooth hypersurface Y ⊂ PN of degree d\n N.The determinant method implies that the number of points in a large box grows slowly. Intersect with linear subspaces to maintain local solubility. Use a counting argument to find a linear section without rational points. Colliot-Th´ el` ene: Won't this just force the coefficients to be large? Harari: Does dim( Xk )deg( Xk ) → ∞ imply that Xk is geometrically rationally connected? Note that if X over Q is a geometrically rationally connected complete intersection, then the Hasse principle is hard to obstruct cohomologically. Browning: A conjecture of Hartshorne implies that if Y ⊂ PN is smooth, non-degenerate, with dim( Y ) ≥ 2 deg( Y ) + 1, then Y is a complete intersection, hence rationally connected. Therefore, it might be easier to look for examples with 13 < dim( Xk)deg( Xk) ≤ 2in Heath-Brown's original question. Tschinkel: Let X be a Fano variety over C. Can we have Br( X) = H3(X, Z)tors 6 =0 in all dimensions ≥ 4?", "original_statement": "Problem 7. Heath-Brown: Can you construct a sequence of smooth projective varieties Xk ⊂ Pk \n\n> Q\n\nwith Xk(Qp) 6 = ∅ for all places p but Xk(Q) = ∅ such that dim( Xk )deg( Xk )AIM OPEN PROBLEM SESSION 3\n\nis unbounded? Browning-Heath-Brown have given a sequence where dim( Xk )deg( Xk ) tends to 13 and dim( Xk) is tends to ∞.Wooley: How about removing the requirement of smoothness and considering the singular norm forms \n\nNK/ Q(x1α1 + · · · + xdαd) = ct d\n\nwhere d = [ K: Q]? Heath-Brown: What happens when Xk ⊂ Pk is a hypersurface? Is there any example of a smooth hypersurface of dimension ≥ 3 failing the Hasse principle? Colliot-Th´ el` ene: Sarnak-Wang have shown that the Bombieri-Lang conjecture would imply that there are many such examples of general type. Wooley: An analytic attack to show that there exist some such varieties could be as follows. Choose a locally soluble smooth hypersurface Y ⊂ PN of degree d \u001d N.The determinant method implies that the number of points in a large box grows slowly. Intersect with linear subspaces to maintain local solubility. Use a counting argument to find a linear section without rational points. Colliot-Th´ el` ene: Won't this just force the coefficients to be large? Harari: Does dim( Xk )deg( Xk ) → ∞ imply that Xk is geometrically rationally connected? Note that if X over Q is a geometrically rationally connected complete intersection, then the Hasse principle is hard to obstruct cohomologically. Browning: A conjecture of Hartshorne implies that if Y ⊂ PN is smooth, non-degenerate, with dim( Y ) ≥ 2 deg( Y ) + 1, then Y is a complete intersection, hence rationally connected. Therefore, it might be easier to look for examples with 13 < dim( Xk)deg( Xk) ≤ 2in Heath-Brown's original question. Tschinkel: Let X be a Fano variety over C. Can we have Br( X) = H3(X, Z)tors 6 =0 in all dimensions ≥ 4?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The source is Problem 7 in the AIM workshop list *Rational and integral points on higher-dimensional varieties*. The canonical JSON has several OCR losses: `6 =` means \\(\\ne\\), the displayed quotients lost their fraction bars, `13` means \\(1/3\\), and the condition in Wooley's proposed analytic approach is \\(d\\gg N\\).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher-dimensional varieties\nSection: \nSource item: 7\nSource URL: https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[26]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 7. Heath-Brown: Can you construct a sequence of smooth projective varieties Xk ⊂ Pk \\n\\n> Q\\n\\nwith Xk(Qp) 6 = ∅ for all places p but Xk(Q) = ∅ such that dim( Xk )deg( Xk )AIM OPEN PROBLEM SESSION 3\\n\\nis unbounded? Browning-Heath-Brown have given a sequence where dim( Xk )deg( Xk ) tends to 13 and dim( Xk) is tends to ∞.Wooley: How about removing the requirement of smoothness and considering the singular norm forms \\n\\nNK/ Q(x1α1 + · · · + xdαd) = ct d\\n\\nwhere d = [ K: Q]? Heath-Brown: What happens when Xk ⊂ Pk is a hypersurface? Is there any example of a smooth hypersurface of dimension ≥ 3 failing the Hasse principle? Colliot-Th´ el` ene: Sarnak-Wang have shown that the Bombieri-Lang conjecture would imply that there are many such examples of general type. Wooley: An analytic attack to show that there exist some such varieties could be as follows. Choose a locally soluble smooth hypersurface Y ⊂ PN of degree d \\u001d N.The determinant method implies that the number of points in a large box grows slowly. Intersect with linear subspaces to maintain local solubility. Use a counting argument to find a linear section without rational points. Colliot-Th´ el` ene: Won't this just force the coefficients to be large? Harari: Does dim( Xk )deg( Xk ) → ∞ imply that Xk is geometrically rationally connected? Note that if X over Q is a geometrically rationally connected complete intersection, then the Hasse principle is hard to obstruct cohomologically. Browning: A conjecture of Hartshorne implies that if Y ⊂ PN is smooth, non-degenerate, with dim( Y ) ≥ 2 deg( Y ) + 1, then Y is a complete intersection, hence rationally connected. Therefore, it might be easier to look for examples with 13 < dim( Xk)deg( Xk) ≤ 2in Heath-Brown's original question. Tschinkel: Let X be a Fano variety over C. Can we have Br( X) = H3(X, Z)tors 6 =0 in all dimensions ≥ 4?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0027", "aim-domain:algebraic-number-theory", "aim-workshop:ratlhigherdimvarproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every positive integer m congruent to 1 modulo 3, with r=m^2, an etale self-cover of the Selmer cubic and a stable pushforward bundle produce a smooth projective geometrically integral variety X_m embedded in P^(2r) over Q with local points at every place, no Q-point, dimension r, and degree 2r+1. Thus dim(X_m)/deg(X_m) tends to 1/2, improving the 1/3 benchmark recorded by AIM. The ratio remains bounded, so this does not solve the unboundedness problem, and the examples are not geometrically rationally connected.\n\nCandidate contribution (construction; novelty confidence low): Order-three torsor self-covers of the Selmer cubic, followed by pushforward of line bundles of degree 2m^2+1, yield Hasse-principle counterexamples of exact dimension m^2 and degree 2m^2+1." }, { "id": 20000376, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0028", "title": "Odd Brauer torsion on K3 surfaces: status and a Picard-rank filter", "statement": "Problem 8. V´ arilly-Alvarado: Skorobogatov has asked whether a K3 surface X\n\nover Q can have odd order torsion in Br( X) obstructing the Hasse principle? Even in Br 1(X)? Skorobogatov: For example, for quartics X ⊂ P3 and α ∈ Br( X)[ u] for u odd: For each place v, there exists a zero cycle Zv over Qv of degree one such that α\n\nis orthogonal to Zv. Then a conjecture of Colliot-Th´ el` ene implies that X has zero cycles of degree one over Q. Will there be a rational point? So given a quartic surface X ⊂ P3 with (Br( X)/ Br( Q))[2] = 0, does the Hasse principle hold? Tschinkel: What about weak approximation? Skorobogatov: This will probably fail. Hassett: How about X ⊂ P4 of degree six?", "original_statement": "Problem 8. V´ arilly-Alvarado: Skorobogatov has asked whether a K3 surface X\n\nover Q can have odd order torsion in Br( X) obstructing the Hasse principle? Even in Br 1(X)? Skorobogatov: For example, for quartics X ⊂ P3 and α ∈ Br( X)[ u] for u odd: For each place v, there exists a zero cycle Zv over Qv of degree one such that α\n\nis orthogonal to Zv. Then a conjecture of Colliot-Th´ el` ene implies that X has zero cycles of degree one over Q. Will there be a rational point? So given a quartic surface X ⊂ P3 with (Br( X)/ Br( Q))[2] = 0, does the Hasse principle hold? Tschinkel: What about weak approximation? Skorobogatov: This will probably fail. Hassett: How about X ⊂ P4 of degree six?", "clean_statement": "Problem 8. V´ arilly-Alvarado: Skorobogatov has asked whether a K3 surface X\n\nover Q can have odd order torsion in Br( X) obstructing the Hasse principle? Even in Br 1(X)? Skorobogatov: For example, for quartics X ⊂ P3 and α ∈ Br( X)[ u] for u odd: For each place v, there exists a zero cycle Zv over Qv of degree one such that α\n\nis orthogonal to Zv. Then a conjecture of Colliot-Th´ el` ene implies that X has zero cycles of degree one over Q. Will there be a rational point? So given a quartic surface X ⊂ P3 with (Br( X)/ Br( Q))[2] = 0, does the Hasse principle hold? Tschinkel: What about weak approximation? Skorobogatov: This will probably fail. Hassett: How about X ⊂ P4 of degree six?", "statement_status": "exact", "statement_verification": "The record is Problem 8 in the AIM workshop list *Rational and integral points on higher-dimensional varieties*, dated May 28, 2014. The canonical record is index 27 of `aim-algebraic-number-theory-notes.json`. I checked the underlying four-page AIM PDF, including its embedded font encoding, rather than silently repairing the extracted text. In particular, the symbol in \\(\\alpha\\in \\operatorname{Br}(X)[u]\\) really is the letter \\(u\\), followed by “for \\(u\\) odd”; it is not an OCR substitution for another symbol. I interpret \\(\\operatorname{Br}(X)[u]\\) as the subgroup killed by the odd integer \\(u\\).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher-dimensional varieties\nSection: \nSource item: 8\nSource URL: https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[27]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 8. V´ arilly-Alvarado: Skorobogatov has asked whether a K3 surface X\\n\\nover Q can have odd order torsion in Br( X) obstructing the Hasse principle? Even in Br 1(X)? Skorobogatov: For example, for quartics X ⊂ P3 and α ∈ Br( X)[ u] for u odd: For each place v, there exists a zero cycle Zv over Qv of degree one such that α\\n\\nis orthogonal to Zv. Then a conjecture of Colliot-Th´ el` ene implies that X has zero cycles of degree one over Q. Will there be a rational point? So given a quartic surface X ⊂ P3 with (Br( X)/ Br( Q))[2] = 0, does the Hasse principle hold? Tschinkel: What about weak approximation? Skorobogatov: This will probably fail. Hassett: How about X ⊂ P4 of degree six?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0028", "aim-domain:algebraic-number-theory", "aim-workshop:ratlhigherdimvarproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The broad 2014 question is now solved affirmatively: algebraic and transcendental three-torsion Brauer classes on degree-two K3 surfaces can obstruct the Hasse principle. The general quartic question with no two-primary Brauer quotient and the degree-six variant remain unresolved in the primary literature checked. This attempt proves that nonzero algebraic p-torsion on any polarized K3 surface over a number field forces geometric Picard rank at least p, and proves a simultaneous prime-to-embedding-degree construction of adelic degree-one zero-cycles orthogonal to the full Brauer group. For quartics this recovers the conditional degree-one zero-cycle conclusion but not a rational point; for degree six it isolates three-primary torsion as the first part not removed by the degree argument.\n\nCandidate contribution (lemma; novelty confidence low): If (X,h) is a polarized K3 surface over a number field and (Br_1(X)/Br_0(X))[p] is nonzero for a prime p, then the geometric Picard rank rho(X) is at least p." }, { "id": 20000377, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0029", "title": "Correcting the threshold and computing the singular locus", "statement": "Problem 9. Wooley: Consider the set\n\nQk:= {Q(yk\n\n> 1,..., y ks ) ∈ Q[y1,..., y s] | Q ∈ Q[x1,..., x s] quadratic form }\n\nof certain forms of degree 2 k. Fixing k, how large must s be in order for the Hasse principle to hold? Let\n\nh(k):= inf\n\n> s∈N\n\n{s | the Hasse principle holds for all Q ∈ Qk in s variables }.\n\nChallenge: prove that log h(k) = o(k) as k → ∞.By Birch's result on forms in many variables, we know that h(k) ≤ 2k · 22k.On the other hand, for diagonal forms of degree k (i.e., Q(yk\n\n> 1,..., y ks ) with linear\n\nQ ∈ Q[x1,..., x s], we have the much stronger bound h(k) ∼ 2k(log k). 4 AIM OPEN PROBLEM SESSION\n\nBrowning: Replace k-th powers by norm forms from fixed extensions of degree\n\nk.", "original_statement": "Problem 9. Wooley: Consider the set \n\nQk:= {Q(yk \n\n> 1,..., y ks ) ∈ Q[y1,..., y s] | Q ∈ Q[x1,..., x s] quadratic form }\n\nof certain forms of degree 2 k. Fixing k, how large must s be in order for the Hasse principle to hold? Let \n\nh(k):= inf \n\n> s∈N\n\n{s | the Hasse principle holds for all Q ∈ Qk in s variables }.\n\nChallenge: prove that log h(k) = o(k) as k → ∞.By Birch's result on forms in many variables, we know that h(k) ≤ 2k · 22k.On the other hand, for diagonal forms of degree k (i.e., Q(yk \n\n> 1,..., y ks ) with linear \n\nQ ∈ Q[x1,..., x s], we have the much stronger bound h(k) ∼ 2k(log k). 4 AIM OPEN PROBLEM SESSION \n\nBrowning: Replace k-th powers by norm forms from fixed extensions of degree \n\nk.", "clean_statement": "Problem 9. Wooley: Consider the set\n\nQk:= {Q(yk\n\n> 1,..., y ks ) ∈ Q[y1,..., y s] | Q ∈ Q[x1,..., x s] quadratic form }\n\nof certain forms of degree 2 k. Fixing k, how large must s be in order for the Hasse principle to hold? Let\n\nh(k):= inf\n\n> s∈N\n\n{s | the Hasse principle holds for all Q ∈ Qk in s variables }.\n\nChallenge: prove that log h(k) = o(k) as k → ∞.By Birch's result on forms in many variables, we know that h(k) ≤ 2k · 22k.On the other hand, for diagonal forms of degree k (i.e., Q(yk\n\n> 1,..., y ks ) with linear\n\nQ ∈ Q[x1,..., x s], we have the much stronger bound h(k) ∼ 2k(log k). 4 AIM OPEN PROBLEM SESSION\n\nBrowning: Replace k-th powers by norm forms from fixed extensions of degree\n\nk.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 9 from the AIM workshop *Rational and integral points on higher-dimensional varieties*. The OCR in the JSON record loses superscripts, an exponent on 2, and an asymptotic comparison symbol. Inspection of the workshop PDF gives the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher-dimensional varieties\nSection: \nSource item: 9\nSource URL: https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[28]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 9. Wooley: Consider the set \\n\\nQk:= {Q(yk \\n\\n> 1,..., y ks ) ∈ Q[y1,..., y s] | Q ∈ Q[x1,..., x s] quadratic form }\\n\\nof certain forms of degree 2 k. Fixing k, how large must s be in order for the Hasse principle to hold? Let \\n\\nh(k):= inf \\n\\n> s∈N\\n\\n{s | the Hasse principle holds for all Q ∈ Qk in s variables }.\\n\\nChallenge: prove that log h(k) = o(k) as k → ∞.By Birch's result on forms in many variables, we know that h(k) ≤ 2k · 22k.On the other hand, for diagonal forms of degree k (i.e., Q(yk \\n\\n> 1,..., y ks ) with linear \\n\\nQ ∈ Q[x1,..., x s], we have the much stronger bound h(k) ∼ 2k(log k). 4 AIM OPEN PROBLEM SESSION \\n\\nBrowning: Replace k-th powers by norm forms from fixed extensions of degree \\n\\nk.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0029", "aim-domain:algebraic-number-theory", "aim-workshop:ratlhigherdimvarproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The invariant h(k) printed in the AIM source is literally 1 for every k, because all one-variable forms satisfy the Hasse implication; the intended stable-threshold problem remains open after adding the missing quantifier over all s at least a threshold. For the substantive repaired analysis, if k is at least 2, A is a nonzero symmetric matrix, and F_A(y)=Q_A(y_1^k,...,y_s^k), then the affine gradient-zero locus has dimension mu(A)=max over nonempty principal submatrices A_I of nullity(A_I), so the projective hypersurface is geometrically smooth exactly when every nonempty principal minor is nonzero. Consequently, under nonsingular real and p-adic solubility, Birch's theorem gives a nonsingular rational zero whenever s-mu(A)>(2k-1)2^(2k).\n\nCandidate contribution (theorem; novelty confidence low): For every nonzero symmetric matrix A over an algebraically closed characteristic-zero field and every k>=2, the affine gradient-zero locus of Q_A(y_1^k,...,y_s^k) has dimension max_{empty != I subseteq {1,...,s}} nullity(A_I); equivalently, its projective singular locus has dimension one less, and the hypersurface is smooth if and only if all nonempty principal submatrices are nonsingular." }, { "id": 20000378, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0030", "title": "Bounded-degree decisions and local-density support for rational curves on quartic K3 surfaces", "statement": "Problem 10. Peyre: Back to quartic surfaces X ⊂ P3. Let U be the complement of all rational curves over Q. Assume U (Q) 6 = ∅. There is numerical evidence that #{u ∈ U (Q) | H(u) ≤ B} ∼ c(log B)ρ(X)\n\nwhere ρ(X) is the rank of the Picard group of X and c is the product of local densities. Can such a formula hold without weak approximation being valid? Tschinkel: Given a quartic K3 surface X ⊂ P3, with x ∈ X(Q). Is there a procedure for deciding whether x lies on a rational curve over Q?Skorobogatov: What about removing elliptic curves? Browning: Is there any numerical evidence? (See van Luijk's work.) What is the Peyre freedom of rational curves of small degree on K3 surfaces? Colliot-Th´ el` ene: Given a K3 surface X over Q and x ∈ X(Q) 6 = ∅, does there exist a rational curve R ⊂ X over Q? Does there exist a rational curve R ⊂ X over\n\nQ containing x?Tschinkel: How about finite fields? Let X be a K3 surface over Fq and x ∈ X(Fq ). Does there exist a rational curve R ⊂ X defined over Fq containing x? Are there any rational curves R ⊂ X defined over Fq? Bogomolov-Tschinkel show for Kummer surfaces X that we can find rational curves over Fq for most x ∈ X.Skorobogatov / Testa: Are there K3 surfaces X over Q with infinitely may rational curves over Q and Pic( XC) ∼= Z?", "original_statement": "Problem 10. Peyre: Back to quartic surfaces X ⊂ P3. Let U be the complement of all rational curves over Q. Assume U (Q) 6 = ∅. There is numerical evidence that #{u ∈ U (Q) | H(u) ≤ B} ∼ c(log B)ρ(X)\n\nwhere ρ(X) is the rank of the Picard group of X and c is the product of local densities. Can such a formula hold without weak approximation being valid? Tschinkel: Given a quartic K3 surface X ⊂ P3, with x ∈ X(Q). Is there a procedure for deciding whether x lies on a rational curve over Q?Skorobogatov: What about removing elliptic curves? Browning: Is there any numerical evidence? (See van Luijk's work.) What is the Peyre freedom of rational curves of small degree on K3 surfaces? Colliot-Th´ el` ene: Given a K3 surface X over Q and x ∈ X(Q) 6 = ∅, does there exist a rational curve R ⊂ X over Q? Does there exist a rational curve R ⊂ X over \n\nQ containing x?Tschinkel: How about finite fields? Let X be a K3 surface over Fq and x ∈ X(Fq ). Does there exist a rational curve R ⊂ X defined over Fq containing x? Are there any rational curves R ⊂ X defined over Fq? Bogomolov-Tschinkel show for Kummer surfaces X that we can find rational curves over Fq for most x ∈ X.Skorobogatov / Testa: Are there K3 surfaces X over Q with infinitely may rational curves over Q and Pic( XC) ∼= Z?", "clean_statement": "Problem 10. Peyre: Back to quartic surfaces X ⊂ P3. Let U be the complement of all rational curves over Q. Assume U (Q) 6 = ∅. There is numerical evidence that #{u ∈ U (Q) | H(u) ≤ B} ∼ c(log B)ρ(X)\n\nwhere ρ(X) is the rank of the Picard group of X and c is the product of local densities. Can such a formula hold without weak approximation being valid? Tschinkel: Given a quartic K3 surface X ⊂ P3, with x ∈ X(Q). Is there a procedure for deciding whether x lies on a rational curve over Q?Skorobogatov: What about removing elliptic curves? Browning: Is there any numerical evidence? (See van Luijk's work.) What is the Peyre freedom of rational curves of small degree on K3 surfaces? Colliot-Th´ el` ene: Given a K3 surface X over Q and x ∈ X(Q) 6 = ∅, does there exist a rational curve R ⊂ X over Q? Does there exist a rational curve R ⊂ X over\n\nQ containing x?Tschinkel: How about finite fields? Let X be a K3 surface over Fq and x ∈ X(Fq ). Does there exist a rational curve R ⊂ X defined over Fq containing x? Are there any rational curves R ⊂ X defined over Fq? Bogomolov-Tschinkel show for Kummer surfaces X that we can find rational curves over Fq for most x ∈ X.Skorobogatov / Testa: Are there K3 surfaces X over Q with infinitely may rational curves over Q and Pic( XC) ∼= Z?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 10 from the AIM workshop *Rational and integral points on higher-dimensional varieties*. The OCR in the JSON record loses several symbols and line breaks, so the statement was checked against both the workshop PDF and the AIM source TeX. The relevant opening assertion is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher-dimensional varieties\nSection: \nSource item: 10\nSource URL: https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[29]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 10. Peyre: Back to quartic surfaces X ⊂ P3. Let U be the complement of all rational curves over Q. Assume U (Q) 6 = ∅. There is numerical evidence that #{u ∈ U (Q) | H(u) ≤ B} ∼ c(log B)ρ(X)\\n\\nwhere ρ(X) is the rank of the Picard group of X and c is the product of local densities. Can such a formula hold without weak approximation being valid? Tschinkel: Given a quartic K3 surface X ⊂ P3, with x ∈ X(Q). Is there a procedure for deciding whether x lies on a rational curve over Q?Skorobogatov: What about removing elliptic curves? Browning: Is there any numerical evidence? (See van Luijk's work.) What is the Peyre freedom of rational curves of small degree on K3 surfaces? Colliot-Th´ el` ene: Given a K3 surface X over Q and x ∈ X(Q) 6 = ∅, does there exist a rational curve R ⊂ X over Q? Does there exist a rational curve R ⊂ X over \\n\\nQ containing x?Tschinkel: How about finite fields? Let X be a K3 surface over Fq and x ∈ X(Fq ). Does there exist a rational curve R ⊂ X defined over Fq containing x? Are there any rational curves R ⊂ X defined over Fq? Bogomolov-Tschinkel show for Kummer surfaces X that we can find rational curves over Fq for most x ∈ X.Skorobogatov / Testa: Are there K3 surfaces X over Q with infinitely may rational curves over Q and Pic( XC) ∼= Z?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0030", "aim-domain:algebraic-number-theory", "aim-workshop:ratlhigherdimvarproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every explicit smooth quartic K3 surface X over Q, rational point x, and degree bound D, there is an exact terminating Hilbert-scheme procedure that lists the Q-defined geometrically integral geometric rational curves through x of degree at most D and separately decides whether each normalization is P1 over Q and whether x has a rational branch above it; the finite-field analogue detects Frobenius descent. Supporting proved results show that a full-support localized counting law would force weak approximation although the scalar law need not, give exact degree and singularity constraints when Pic(X_C)=Z H, and exhibit a smooth quartic over F5 with no F5-defined geometric rational curve.\n\nCandidate contribution (reduction; novelty confidence low): The bounded-degree point-on-rational-curve problem admits a terminating four-way decision procedure separating Hilbert-point descent, geometric genus zero, splitting of the genus-zero normalization, and rationality of a branch over the prescribed point, with Frobenius replacing Galois descent over finite fields." }, { "id": 20000379, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0031", "title": "A three-state correction to the local quadratic-twist rank ratio", "statement": "Question 1 (Brian Conrey). Let E/ Fq(T ). Let f vary over square free, large degree n and let P ∈ P1; does lim\n\n> n→∞\n\n#{f ∈ Fq[T ]n: rank Ef ≥ 2 and f (P ) ∈ (F∗\n\n> q\n\n)2 − { 0}}\n\n#{f ∈ Fq[T ]n: rank Ef ≥ 2 and f (P ) 6 ∈ (F∗\n\n> q\n\n)2} =?\n\n√\n\n#EP (Fq)˜EP (Fq)(This is conjectured for number fields based on numerical evidence, random matrix theory, moments of L-functions, and BSD.)", "original_statement": "Question 1 (Brian Conrey). Let E/ Fq(T ). Let f vary over square free, large degree n and let P ∈ P1; does lim \n\n> n→∞\n\n#{f ∈ Fq[T ]n: rank Ef ≥ 2 and f (P ) ∈ (F∗ \n\n> q\n\n)2 − { 0}} \n\n#{f ∈ Fq[T ]n: rank Ef ≥ 2 and f (P ) 6 ∈ (F∗ \n\n> q\n\n)2} =?\n\n√\n\n#EP (Fq)˜EP (Fq)(This is conjectured for number fields based on numerical evidence, random matrix theory, moments of L-functions, and BSD.)", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The PDF does not define \\(\\mathbb F_q[T]_n\\), does not say that \\(q\\) is odd, and does not impose rationality or good reduction at \\(P\\). For the rigorous result below, the explicit reconstruction assumptions are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 1\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[30]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 1 (Brian Conrey). Let E/ Fq(T ). Let f vary over square free, large degree n and let P ∈ P1; does lim \\n\\n> n→∞\\n\\n#{f ∈ Fq[T ]n: rank Ef ≥ 2 and f (P ) ∈ (F∗ \\n\\n> q\\n\\n)2 − { 0}} \\n\\n#{f ∈ Fq[T ]n: rank Ef ≥ 2 and f (P ) 6 ∈ (F∗ \\n\\n> q\\n\\n)2} =?\\n\\n√\\n\\n#EP (Fq)˜EP (Fq)(This is conjectured for number fields based on numerical evidence, random matrix theory, moments of L-functions, and BSD.)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: open; rigor: conditional; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0031", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the explicit reconstruction that q is odd, P=T-a is a finite rational good-reduction place, and the family consists of monic squarefree degree-n polynomials, an exact proved theorem gives limiting local masses q/(2(q+1)), q/(2(q+1)), and 1/(q+1) for nonzero-square, nonsquare, and zero values f(P). More precisely, the unramified count A_n equals (q-1)(q^n-(-1)^n)/(q+1), while the square-minus-nonsquare imbalance is 0 in odd degree and 1-q in positive even degree. Consequently, if the rank-at-least-two conditional rates satisfy r_+/r_- tending to C and r_0/r_- tending to lambda, the cleaned square-versus-nonsquare ratio tends to C, whereas the ratio literally printed by AIM tends to C/(1+2 lambda/q). The local counting theorem is unconditional; the Conrey-ratio consequence remains conditional because these rank-conditioned limits are unproved.\n\nCandidate contribution (obstruction_and_reduction; novelty confidence low): The literal AIM denominator contains the positive-density ramified state f(P)=0, so the printed ratio is governed by three conditional high-rank rates and equals C_{+/-}/(1+(2/q)lambda) whenever the relative limits exist; under equal ramified and nonsquare rates this predicts q C_P/(q+2), not C_P.", "research_difficulty_suggested": "L4" }, { "id": 20000380, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0032", "title": "Integral BKLPR matrices and the rank-2-to-rank-3 exponent transfer", "statement": "Question 2 (Melanie Matchett Wood). Develop a version of the BKLPR heuristics with Zp\n\nreplaced by Z, counting up to height B. Use the rank 2 case to calibrate B with the height used when counting elliptic curves. Then use that calibration to make a rank 3 prediction and compare this with predictions coming from random matrix theory.", "original_statement": "Question 2 (Melanie Matchett Wood). Develop a version of the BKLPR heuristics with Zp\n\nreplaced by Z, counting up to height B. Use the rank 2 case to calibrate B with the height used when counting elliptic curves. Then use that calibration to make a rank 3 prediction and compare this with predictions coming from random matrix theory.", "clean_statement": "Question 2 (Melanie Matchett Wood). Develop a version of the BKLPR heuristics with Zp\n\nreplaced by Z, counting up to height B. Use the rank 2 case to calibrate B with the height used when counting elliptic curves. Then use that calibration to make a rank 3 prediction and compare this with predictions coming from random matrix theory.", "statement_status": "exact", "statement_verification": "The record is Question 2, attributed to Melanie Matchett Wood, in the AIM problem list from the workshop *Arithmetic statistics over finite fields and function fields*. The source PDF was checked directly; the intended local ring is \\(\\mathbb Z_p\\), and the extracted text is not hiding an ambiguity in the subscript.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 2\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[31]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 2 (Melanie Matchett Wood). Develop a version of the BKLPR heuristics with Zp\\n\\nreplaced by Z, counting up to height B. Use the rank 2 case to calibrate B with the height used when counting elliptic curves. Then use that calibration to make a rank 3 prediction and compare this with predictions coming from random matrix theory.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: solved; rigor: conditional; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0032", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Unconditionally, the Park--Poonen--Voight--Wood lattice theorem counts bounded alternating integer matrices of corank at least r with exponent n(n-r)/2. Conditional on the PPVW slow-growing-dimension proxy, parity equidistribution, and the transfer from matrix corank to Mordell--Weil rank, calibrating a height family of size H^delta from a rank-2 exponent beta_2 forces beta_r = (r-1) beta_2 - (r-2) delta. This gives B_mat^n = H^(1/12+o(1)) and the rank-3 prediction H^(3/4+o(1)) for all elliptic curves, and D^(1/2+o(1)) for quadratic twists. The comparison with analytic random-matrix theory is additionally conditional on BSD and an unresolved regulator discretization; the historical RMT alternatives are therefore not direct contradictions.\n\nCandidate contribution (conditional exponent-transfer proposition; novelty confidence low): Within the PPVW box model, eliminating the calibrated matrix scale gives the family-independent, parity-correct identity beta_r = (r-1) beta_2 - (r-2) delta and the falsifier beta_(r+1) - beta_r = beta_2 - delta; any reliable non-affine sequence of higher-rank polynomial exponents rules out retuning B_mat and n inside this simple ensemble." }, { "id": 20000381, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0033", "title": "Counting orders, local laws, and dominant-residue factorization", "statement": "Question 3 (Jordan Ellenberg). Understand which ways of counting field extensions satisfy Principles (A) and (B).", "original_statement": "Question 3 (Jordan Ellenberg). Understand which ways of counting field extensions satisfy Principles (A) and (B).", "clean_statement": "Question 3 (Jordan Ellenberg). Understand which ways of counting field extensions satisfy Principles (A) and (B).", "statement_status": "exact", "statement_verification": "The canonical record is Question 3 from the January 2014 AIM workshop *Arithmetic statistics over finite fields and function fields*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 3\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[32]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 3 (Jordan Ellenberg). Understand which ways of counting field extensions satisfy Principles (A) and (B).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0033", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM source does not define Principles (A) and (B) or assign their letters, so the report conservatively reconstructs the context-supported pair as the one-place Chebotarev law (Ch) and finite-place asymptotic independence (Ind), without mapping either label. For any weighted height family whose fixed-local-condition counts have a common dominant scale with the stated fixed-set uniform errors, it proves that Ind is equivalent to multiplicativity of the leading constants and that Ch is equivalent to explicit one-place leading-constant ratios; under a separately proved common Tauberian theorem, the same tests use dominant Laurent coefficients. It also proves persistent-mixture covariance and product-tilt criteria that detect two global mechanisms for failure. These are rigorous abstract criteria and obstructions, checked against Wood's abelian fair-count theorem and discriminant counterexamples and the cyclic-prime full-genus theorem, but they do not prove the hard family-specific counting asymptotics needed for a general classification.\n\nCandidate contribution (criterion; novelty confidence low): Candidate novelty is the low-confidence diagnostic package combining the division-free dominant-constant/residue factorization test for Ch and Ind, an interaction defect, the covariance of a coherent persistent global mixture, and the necessary-and-sufficient product-tilt test for ramification conditioning, with explicit zero-density and finite-level-dependence safeguards." }, { "id": 20000382, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0034", "title": "Automorphism weights forced by groupoid mass", "statement": "Question 4. Which definition(s) of automorphism groups give rise to the nicest formulas?", "original_statement": "Question 4. Which definition(s) of automorphism groups give rise to the nicest formulas?", "clean_statement": "Question 4. Which definition(s) of automorphism groups give rise to the nicest formulas?", "statement_status": "exact", "statement_verification": "The canonical record is Question 4 from the AIM workshop *Arithmetic statistics over finite fields and function fields* (27--31 January 2014):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 4\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[33]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 4. Which definition(s) of automorphism groups give rise to the nicest formulas?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0034", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The archived AIM sources do not define the Principles (A) and (B) referenced by the neighboring questions, so their exact content is not reconstructed. Within the recoverable finite field-extension, cover, and curve context, and under essentially finite groupoid and effective-descent hypotheses, inverse weighting by the arithmetic automorphism group of the full moduli object is the unique quotient-compatible convention. Exact formulas identify Aut_K(L) with a permutation centralizer, give total arithmetic mass 1 across all Frobenius twists, distinguish fixed-target, moving-target, and abstract-source cover automorphisms, and show that a constant rigidified stack mass agrees asymptotically with coarse count exactly when the extra-automorphism locus is negligible.\n\nCandidate contribution (equivalence; novelty confidence low): The combined automorphism-choice diagnostic proves that full arithmetic stabilizer weighting is uniquely quotient-compatible, while the field-extension centralizer identity, Frobenius-twist mass-one identity, and rigidified equivalence dM_t/N_t -> 1 if and only if the extra-automorphism density tends to zero provide independent exact tests for competing conventions." }, { "id": 20000383, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0035", "title": "Cyclic triple covers ordered by genus", "statement": "Question 5. For cyclic triple covers of P1 can we recover (A) and (B) while counting by genus?", "original_statement": "Question 5. For cyclic triple covers of P1 can we recover (A) and (B) while counting by genus?", "clean_statement": "Question 5. For cyclic triple covers of P1 can we recover (A) and (B) while counting by genus?", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 5\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[34]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 5. For cyclic triple covers of P1 can we recover (A) and (B) while counting by genus?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0035", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The cited theorem of Bucur, David, Feigon, Kaplan, Lalín, Ozman, and Wood resolves the source-supported tame interpretation for fixed q congruent to 1 modulo 3: in the full reciprocal-automorphism-weighted genus-g moduli space, prescribed local ramified, split, and inert behaviors become independent with explicit masses, conditional unramified splitting has Chebotarev proportions, and the point count has the published i.i.d. limit. Independently, this attempt proves a narrowly scoped finite-genus correction for labelled, generator-marked geometric inertia: with r = g + 2 branch points, the product-one constraint leaves (2^r + 2(-1)^r)/3 labelings and changes every fixed k-label marginal from the product law by at most (2^k + 1)/(2^r - 2) in total variation. The AIM source never defines which recovered property is labelled (A) or (B), so no letter assignment is asserted.\n\nCandidate contribution (proposition; novelty confidence low): For uniformly weighted tuples (a_1,...,a_r) in {1,2}^r with sum 0 modulo 3, the exact count is (2^r + 2(-1)^r)/3, every one-coordinate marginal is exactly uniform, and the first k coordinates have total-variation distance at most min{1,(2^k + 1)/(2^r - 2)} from the uniform product law; for cyclic triple Hurwitz data r = g + 2." }, { "id": 20000384, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0036", "title": "Ramification conditioning: preserved marks, induced dependence, and fixed-conductor limits", "statement": "Question 6 (Alina Bucur, Kiran Kedlaya). Are principles (A) and (B) easier/harder/true after conditioning on ramification information?", "original_statement": "Question 6 (Alina Bucur, Kiran Kedlaya). Are principles (A) and (B) easier/harder/true after conditioning on ramification information?", "clean_statement": "Question 6 (Alina Bucur, Kiran Kedlaya). Are principles (A) and (B) easier/harder/true after conditioning on ramification information?", "statement_status": "exact", "statement_verification": "The exact canonical record is Question 6 from the January 2014 AIM workshop *Arithmetic statistics over finite fields and function fields*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 6\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[35]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 6 (Alina Bucur, Kiran Kedlaya). Are principles (A) and (B) easier/harder/true after conditioning on ramification information?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0036", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an exact finite product law of local states, every conditioning weight depending only on ramification indicators leaves the joint unramified Frobenius-mark law exactly unchanged; the full local states remain independent exactly when the tilted ramification indicators do, equivalently when a positive weight factorizes placewise or an indicator event has rectangular support. For an exact additive ramification total, coefficient formula (5.5) and covariance cross-ratio (5.6) compute the induced dependence, which can have positive sign for heterogeneous costs, as shown by the (1,1,2)-cost example. Separately, the cited cyclic prime-degree fixed-conductor theorem implies family-specific fixed-place factorization and Chebotarev ratios with O(1/n) error. Only the exact exponential-tilt identity is proved for general heterogeneous asymptotics; convergence to the tilted product remains conditional on a suitable local limit theorem.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: the exact ramification-blind mark-preservation and full-independence criterion, paired with the coefficient covariance cross-ratio, gives a testable trichotomy for the AIM question and shows explicitly that a weighted conductor total can create positive rather than universally negative ramification covariance." }, { "id": 20000385, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0037", "title": "A genus-ordering obstruction for cyclic quartic covers", "statement": "Question 7. Does (A) fail for Z/4Z function field extensions, counted by genus?", "original_statement": "Question 7. Does (A) fail for Z/4Z function field extensions, counted by genus?", "clean_statement": "Question 7. Does (A) fail for Z/4Z function field extensions, counted by genus?", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 7\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[36]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 7. Does (A) fail for Z/4Z function field extensions, counted by genus?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0037", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Two rigorous layers are proved and kept separate. For actual geometrically connected tame cyclic quartic covers, if h is the genus of the unique quadratic subfield and b is the degree of the order-2-only branch divisor, then g=3h+b, and the fixed-h Hurwitz stratum has codimension h. Separately, in the positive discriminant-critical Euler-product law for Kummer tuples, the proper quadratic-image locus has probability strictly between zero and one, and conditioning on connected Z/4Z image gives an explicit strictly negative covariance for ramification at any two distinct places. The modulo-4 projection, infinity contribution, automorphism convention, and exact-genus coefficient transfer needed to identify this covariance with the full arithmetic family are not proved.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem package: tame cyclic quartic genus-g Hurwitz strata indexed by quadratic-subfield genus h have exact codimension h, and the associated discriminant-critical product law has an explicit negative two-place ramification covariance after conditioning away the quadratic-image locus." }, { "id": 20000386, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0038", "title": "A finite-order marked-moduli certificate for the fixed-field Poisson conjecture", "statement": "Question 8 (Kiran Kedlaya). Fix q. Describe (conjecturally) the distribution of the number of Fq points on curves of genus g as g tends to ∞.", "original_statement": "Question 8 (Kiran Kedlaya). Fix q. Describe (conjecturally) the distribution of the number of Fq points on curves of genus g as g tends to ∞.", "clean_statement": "Question 8 (Kiran Kedlaya). Fix q. Describe (conjecturally) the distribution of the number of Fq points on curves of genus g as g tends to ∞.", "statement_status": "exact", "statement_verification": "The canonical AIM record is Question 8 from the workshop *Arithmetic statistics over finite fields and function fields*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 8\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[37]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 8 (Kiran Kedlaya). Fix q. Describe (conjecturally) the distribution of the number of Fq points on curves of genus g as g tends to ∞.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0038", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the arithmetic stack-weighted ensemble of smooth projective geometrically connected genus-g curves over a fixed finite field, the marked-stack ratio #_st M_{g,r}(F_q)/#_st M_g(F_q) is exactly the r-th factorial moment of the rational point count. Finite even and odd inclusion-exclusion truncations of these ratios rigorously bracket every point probability, with an exact interval width; comparison with the Taylor series of the conjectural Poisson mass gives an explicit pointwise error, and a finite cutoff plus an explicit first-moment tail residual gives a total-variation bound. Hence convergence of all fixed marked-stack ratios to the predicted Poisson factorial moments implies total-variation convergence, without an unjustified infinite-series interchange.\n\nCandidate contribution (theorem; novelty confidence low): A finite list of marked-moduli stack ratios through order L+2m+1, together with the displayed first-moment tail residual, gives explicit Bonferroni-Taylor bounds for all point masses up to L and an explicit total-variation bound to Poisson(q+1+1/(q-1)); this also proves that the known fixed-order q(g)>g^K asymptotics do not alone certify a moving Poisson law when its mean grows." }, { "id": 20000387, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0039", "title": "Exact sparse-incidence dichotomy for tricanonically embedded curves", "statement": "Question 9 (Kiran Kedlaya). Look at principle (A) for tri-canonically embedded curves.", "original_statement": "Question 9 (Kiran Kedlaya). Look at principle (A) for tri-canonically embedded curves.", "clean_statement": "- the workshop report says that Melanie Matchett Wood asked when point counts in\n a family of curves are sums of independent identically distributed local\n variables [AIM14b];\n- nearby Question 8 asks for the fixed-\\(q\\), large-genus point-count\n distribution, while Question 10 proposes complete intersections;\n- rational points impose asymptotically independent conditions in the earlier\n plane-curve model [BDFL10], and Bucur--Kedlaya obtain an i.i.d. Bernoulli\n model for high-degree complete intersections [BK12].", "statement_status": "reconstructed_unverified", "statement_verification": "The source never defines principle (A). The full three-page PDF invokes principles (A) and (B) several times but does not assign either a statement. That omission is material and is not silently repaired here. The following facts do, however, support a conservative reconstruction: Accordingly, this report studies the explicit reconstruction", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 9\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[38]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 9 (Kiran Kedlaya). Look at principle (A) for tri-canonically embedded curves.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0039", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the canonical ensemble of uniformly framed, equivalently projective-image, smooth tricanonical genus-g curves over F_q, the induced curve measure is exactly proportional to 1/|Aut_Fq(C)|. The bundle 3K_C is sharply (4g-6)-very ample. Consequently, for every ordered distinct ambient s-tuple with 1 <= s <= 4g-5, the exact joint incidence probability is mu_{g,s}/L_{r,s} on the projectively independent orbit and zero on every dependent tuple; for a uniformly random ordered distinct tuple it is mu_{g,s}/(Q)_s. Collinear triples therefore disprove exact and uniform relative i.i.d. incidence, though not absolute asymptotic factorization because the marginal tends to zero. General-position and annealed relative factorization reduce exactly to the still-open fixed-q marked-moduli factorial-moment problem.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: the exact dependent-versus-independent incidence formula, its sharp (4g-6)-very-ampleness cutoff, and the two normalized relative-factorization identities assemble the tricanonical AIM question into a geometric obstruction plus precisely the intrinsic marked-moduli factorial-moment ratio." }, { "id": 20000388, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0040", "title": "Monotone means and a Poisson moving-target limit for complete-intersection curves", "statement": "Question 10 (Melanie Matchett Wood). Look at complete intersections, and some examples with a moving target space.", "original_statement": "Question 10 (Melanie Matchett Wood). Look at complete intersections, and some examples with a moving target space.", "clean_statement": "Question 10 (Melanie Matchett Wood). Look at complete intersections, and some examples with a moving target space.", "statement_status": "exact", "statement_verification": "The canonical record is Question 10 in the AIM problem session for *Arithmetic statistics over finite fields and function fields*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 10\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[39]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 10 (Melanie Matchett Wood). Look at complete intersections, and some examples with a moving target space.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0040", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For each fixed prime power q, the Bucur-Kedlaya limiting mean for smooth curves cut by n-1 high-degree hypersurfaces in projective n-space is proved to decrease strictly from q+1 to lambda_q=(q+1) product_{j>=3}(1-q^{-j}); moreover, the sequence of fixed-n limiting binomial point-count laws converges in total variation to Pois(lambda_q). This is an iterated result: first the defining degrees tend to infinity for each fixed n, and then n tends to infinity. A secondary proved jet-support theorem gives an exact coefficient-rank obstruction to independent local data in moving-target families.\n\nCandidate contribution (theorem; novelty confidence low): For fixed q, the Bucur-Kedlaya complete-intersection means are strictly decreasing in ambient dimension, and their fixed-dimension limiting binomial laws converge in total variation to a Poisson law of mean (q+1) product_{j>=3}(1-q^{-j})." }, { "id": 20000389, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0041", "title": "Exact boundary corrections for stack quotients and branch collisions", "statement": "Question 11 (John Voight). When do counts get nicer if we are willing to partially com-pactify?", "original_statement": "Question 11 (John Voight). When do counts get nicer if we are willing to partially com-pactify?", "clean_statement": "Question 11 (John Voight). When do counts get nicer if we are willing to partially com-pactify?", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 11\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[40]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 11 (John Voight). When do counts get nicer if we are willing to partially com-pactify?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0041", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Two rigorous models identify mechanisms by which partial compactification simplifies finite-field counts. The groupoid mass of a weighted projective stack P(w_0,...,w_m) is (q^{m+1}-1)/(q-1), independent of its weights; consequently, in characteristic greater than 3, the compactified elliptic stack has mass q+1, its nodal boundary has mass 1, and the smooth open has mass q. For effective divisors on a smooth curve with every multiplicity at most r, the exact generating series is Z_C(t)/Z_C(t^{r+1}); on the affine line the count is q^n-q^{n-r} for n at least r+1. Thus allowing double but not triple roots changes the monic hyperelliptic coefficient count from q^n-q^{n-1} to q^n-q^{n-2}, adding q^{n-1}-q^{n-2} nodal points and leaving a codimension-two omitted locus.\n\nCandidate contribution (criterion; novelty confidence low): Candidate boundary-correction criterion: a partial compactification has an explicitly nicer count in the two proved model classes when each added stratum either factors independently over closed points, producing a zeta-ratio correction, or has a simple quotient-stack groupoid mass; the collision-cap chain and the unit boundary mass of the compactified elliptic stack give exact test cases." }, { "id": 20000390, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0042", "title": "A CRT lower bound for fixed-q higher divisor correlations", "statement": "Question 12 (Julio Cesar Bueno de Andrade). Fix q, h(x) ∈ Fq[x], k ≥ 3. Get asymptotics for ∑\n\n> f\n\ndk(f ) · dk(f + h)where f ranges over degree n polynomials, as n tends to infinity.", "original_statement": "Question 12 (Julio Cesar Bueno de Andrade). Fix q, h(x) ∈ Fq[x], k ≥ 3. Get asymptotics for ∑\n\n> f\n\ndk(f ) · dk(f + h)where f ranges over degree n polynomials, as n tends to infinity.", "clean_statement": "Question 12 (Julio Cesar Bueno de Andrade). Fix q, h(x) ∈ Fq[x], k ≥ 3. Get asymptotics for ∑\n\n> f\n\ndk(f ) · dk(f + h)where f ranges over degree n polynomials, as n tends to infinity.", "statement_status": "exact", "statement_verification": "The exact canonical record, including the extraction line breaks, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 12\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[41]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 12 (Julio Cesar Bueno de Andrade). Fix q, h(x) ∈ Fq[x], k ≥ 3. Get asymptotics for ∑\\n\\n> f\\n\\ndk(f ) · dk(f + h)where f ranges over degree n polynomials, as n tends to infinity.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0042", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the natural monic normalization, for fixed q, k >= 3, nonzero fixed h, and r = k - 1, the correlation S_k(n;h) is at least q^n T_{r,h}(n), where T_{r,h}(n) is the exactly CRT-solvable sector with deg lcm(A,B) <= n. This sector has the proved asymptotic T_{r,h}(n) = C_{q,r}(h)n^{2r}/(2r)! + O_{q,k,h}(n^{2r-1}), with an explicit convergent positive Euler product C_{q,r}(h). Consequently the full correlation has a positive liminf on the conjectured q^n n^{2k-2} scale. This is a one-sided lower bound, not the requested full asymptotic.\n\nCandidate contribution (lower_bound; novelty confidence low): The complete small-lcm CRT sector of the fixed-q shifted d_k correlation admits an explicit shift-sensitive Euler product and yields a positive liminf at the conjectured q^n n^{2k-2} scale for every fixed nonzero h and every k >= 3." }, { "id": 20000391, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0043", "title": "Additive gaps and an affine-coordinate audit", "statement": "Question 13 (Pollack). What can we say (resp. what do we know) about gaps between irreducible polynomials over Fq?\n\n> 1", "original_statement": "Question 13 (Pollack). What can we say (resp. what do we know) about gaps between irreducible polynomials over Fq?\n\n> 1", "clean_statement": "Question 13 (Pollack). What can we say (resp. what do we know) about gaps between irreducible polynomials over Fq?\n\n> 1", "statement_status": "exact", "statement_verification": "The exact record in the AIM corpus is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 13\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[42]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 13 (Pollack). What can we say (resp. what do we know) about gaps between irreducible polynomials over Fq?\\n\\n> 1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0043", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The intended gap is the additive difference between monic irreducibles of the same large degree. For every nonzero shift h, the pair {0,h} is admissible when q>2, while over F_2 it is admissible exactly when T(T+1) divides h; failure forces the large-degree pair count to vanish. The associated singular series converges and is positive exactly in these admissible cases. At fixed degree n, normalized affine changes f(T) -> a^{-n}f(aT+b) preserve irreducibility, the exact pair count, admissibility, and the singular series, while moving the leading coefficient of a degree-d gap through precisely one coset of (F_q^*)^{n-d}. Thus gcd(n-d,q-1)=1 in the published monomial scaling theorem is an affine-orbit condition, not a local prime obstruction.\n\nCandidate contribution (theorem; novelty confidence low): Candidate new synthesis: the proved coordinate-audited prescribed-gap theorem gives the exact local obstruction, exact singular-series positivity criterion, exact normalized affine invariance, and exact leading-coefficient power-coset decomposition, thereby separating the genuine binary local obstruction from the gcd obstruction of the affine transport argument." }, { "id": 20000392, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0044", "title": "Universal capacity bounds and an inseparable coordinate obstruction", "statement": "Question 14 (Pollack). Over a function field, how many coefficients of a polynomial (and in which positions) can be specified without eliminating the possibility of irreducibility?", "original_statement": "Question 14 (Pollack). Over a function field, how many coefficients of a polynomial (and in which positions) can be specified without eliminating the possibility of irreducibility?", "clean_statement": "Question 14 (Pollack). Over a function field, how many coefficients of a polynomial (and in which positions) can be specified without eliminating the possibility of irreducibility?", "statement_status": "exact", "statement_verification": "The wording was checked against the [original three-page AIM PDF](https://aimath.org/pastworkshops/arithstatffieldproblems.pdf), where it appears verbatim. There is no OCR corruption, but the short question does not state its quantifiers, the degree, or whether the polynomial is monic.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 14\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[43]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 14 (Pollack). Over a function field, how many coefficients of a polynomial (and in which positions) can be specified without eliminating the possibility of irreducibility?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0044", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the worst-case capacity R_q(n) of arbitrarily positioned, arbitrarily valued admissible coefficient prescriptions, counting gives R_q(n) at most floor(log_q pi_q(n)). If p is the characteristic and p divides n, prescribing zero in every non-p-divisible position fixes exactly n-n/p coefficients and forces every completion to have zero derivative, hence to be a p-th power over the perfect field F_q; consequently R_q(n) is at most n-n/p-1. This yields exact capacities R_q(3)=1 in characteristic 3 and R_q(4)=1 in characteristic 2 when combined with Hansen-Mullen.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: when p divides n, the codimension n-n/p coordinate face obtained by zeroing all non-p-divisible coefficients is exactly the p-th-power locus, is the unique inclusion-minimal zero-coordinate face forcing formal derivative zero, and gives R_q(n) at most n-n/p-1; combined with coefficient-pattern counting, this gives an explicit universal upper envelope." }, { "id": 20000393, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0045", "title": "Elliptic surfaces, orthogonal monodromy, and BKLPR", "statement": "Question 15 (Jordan Ellenberg). What algebro-geometric facts about elliptic surfaces ex-plain Poonen-Rains/BKLPR?", "original_statement": "Question 15 (Jordan Ellenberg). What algebro-geometric facts about elliptic surfaces ex-plain Poonen-Rains/BKLPR?", "clean_statement": "Question 15 (Jordan Ellenberg). What algebro-geometric facts about elliptic surfaces ex-plain Poonen-Rains/BKLPR?", "statement_status": "exact", "statement_verification": "The canonical record is Question 15 from the AIM workshop problem list “Arithmetic statistics over finite fields and function fields”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 15\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[44]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 15 (Jordan Ellenberg). What algebro-geometric facts about elliptic surfaces ex-plain Poonen-Rains/BKLPR?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0045", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The function-field form of the question is now substantially answered: finite etale Selmer spaces, the quadratic intersection form on primitive elliptic-surface cohomology, large orthogonal monodromy, and Frobenius equidistribution produce random orthogonal kernels, which Feng-Landesman-Rains identify stably with the finite-level BKLPR law in the large-q-then-height limit. The report additionally proves a pointwise graph-diagonal bridge: ker(g-1) is an intersection of maximal isotropics after hyperbolic doubling; its dimension obeys a determinant parity identity without a semisimplicity assumption; and, for a nondegenerate quadratic space containing a hyperbolic plane, uniform full orthogonal sampling has mean fixed-space size ell+1.\n\nCandidate contribution (lemma; novelty confidence low): For odd ell, the diagonal and the graph of any g in O(V,Q) are maximal isotropics in (V direct-sum V, Q direct-sum -Q), with intersection ker(g-1); moreover (-1)^(dim ker(g-1))=(-1)^(dim V) det(g), and if V contains a hyperbolic plane then the uniform full-orthogonal average of #ker(g-1) is ell+1." }, { "id": 20000394, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0046", "title": "Normalized discriminants and a trace-square reduction", "statement": "Question 16 (Brian Conrey). Fix q, let g → ∞, and describe the distribution of discrimi-nants of L-polynomials of curves (or hyperelliptic curves).", "original_statement": "Question 16 (Brian Conrey). Fix q, let g → ∞, and describe the distribution of discrimi-nants of L-polynomials of curves (or hyperelliptic curves).", "clean_statement": "Question 16 (Brian Conrey). Fix q, let g → ∞, and describe the distribution of discrimi-nants of L-polynomials of curves (or hyperelliptic curves).", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 16\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[45]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 16 (Brian Conrey). Fix q, let g → ∞, and describe the distribution of discrimi-nants of L-polynomials of curves (or hyperelliptic curves).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0046", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every curve, the nonmonic L-polynomial and its monic reciprocal polynomial have the same discriminant; after normalization it factors exactly through the real Weil polynomial, including a zero, sign, and square-class classification. On the nonzero locus, the logarithm of the normalized discriminant is exactly an Abel limit of N minus the squared Frobenius traces, with an explicit finite truncation bound in terms of the minimum eigenvalue gap. A sharp support bound and an exact Haar-USp Selberg moment law are also proved, with the Haar law explicitly retained only as a benchmark.\n\nCandidate contribution (reduction; novelty confidence low): The explicit finite-resolution Abel trace-square inequality, assembled with the real-Weil discriminant and square-class factorization, reduces the fixed-q distribution problem to the repeated-root atom, minimum-gap tails, and growing-range joint squared-trace statistics.", "research_difficulty_suggested": "L4" }, { "id": 20000395, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0047", "title": "Odd-degree Jacobian mean and its arithmetic correction", "statement": "Question 17 (John Voight). What is the average number of points on the Jacobian of a hyperelliptic curve?", "original_statement": "Question 17 (John Voight). What is the average number of points on the Jacobian of a hyperelliptic curve?", "clean_statement": "Question 17 (John Voight). What is the average number of points on the Jacobian of a hyperelliptic curve?", "statement_status": "exact", "statement_verification": "The exact source record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 17\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[46]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 17 (John Voight). What is the average number of points on the Jacobian of a hyperelliptic curve?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0047", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the standard odd-degree equation ensemble D in H_{2g+1,q}, Andrade's theorem gives the known asymptotic average #Jac(C_D)(F_q) ~ C_q q^g, where C_q=(1-q^{-1})^{-1} product_P(1-1/(|P|^2(|P|+1))). This attempt proves a square-term local-density divisor-sum identity for C_q and the large-q expansion C_q=1+q^{-1}+q^{-3}+2q^{-6}+O(q^{-7}); it also gives a stack-groupoid reduction and an exact genus-one normalization check. The original unqualified question remains only partially solved because equation, stack, even-degree, and characteristic-two averages are distinct.\n\nCandidate contribution (expansion; novelty confidence low): Candidate novel proposition: as q tends to infinity through odd prime powers, the fixed-q high-genus mean constant satisfies C_q=1+q^{-1}+q^{-3}+2q^{-6}+O(q^{-7}), so its q^{-2}, q^{-4}, and q^{-5} coefficients vanish; C_q also equals the explicit square-term local-density divisor sum over monic H." }, { "id": 20000396, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0048", "title": "Explicit bounds for distinct curve zeta functions over a fixed finite field", "statement": "Question 18. Fix q. How many different zeta functions are attached to curves of genus\n\ng\n q?", "original_statement": "Question 18. Fix q. How many different zeta functions are attached to curves of genus \n\ng \u001d q?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record contains an extraction error:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 18\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[47]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 18. Fix q. How many different zeta functions are attached to curves of genus \\n\\ng \\u001d q?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0048", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every fixed q and genus g, the number z_q(g) of curve zeta functions is at most the product over 1 <= n <= g of (floor(4g q^(n/2)) + 1), hence at most (4g+1)^g q^(g(g+1)/4). If q has characteristic p and D=2g/(p-1) is integral, explicit reduced Artin-Schreier curves over F_q give z_q(g) at least floor(D/2)+1, except for a possible loss of one when p is odd and p divides D+1. In particular, z_q(g) is at least g+1 for every g over every finite field of characteristic two.\n\nCandidate contribution (theorem; novelty confidence low): An explicit fixed-F_q Artin-Schreier construction, including an order-two-pole congruence repair, realizes p-ranks r(p-1) for every 1 <= r <= floor(D/2) and also r=0 unless p divides D+1; therefore z_q(g) >= floor(D/2)+1 minus the single stated exception." }, { "id": 20000397, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0049", "title": "Arithmetic, geometric, and Frobenius monodromy over global function fields", "statement": "Question 19 (Kiran Kedlaya). Fix g, fix C/K a curve of genus g, where K = Fq(X) and\n\nX is not necessarily rational. Can we compute the monodromy group of C? I.e., can we compute\n\n{im(Gal( K) → Aut( H1(C, Qp))): C ∈ Mg(K)}", "original_statement": "Question 19 (Kiran Kedlaya). Fix g, fix C/K a curve of genus g, where K = Fq(X) and \n\nX is not necessarily rational. Can we compute the monodromy group of C? I.e., can we compute \n\n{im(Gal( K) → Aut( H1(C, Qp))): C ∈ Mg(K)}", "clean_statement": "Question 19 (Kiran Kedlaya). Fix g, fix C/K a curve of genus g, where K = Fq(X) and\n\nX is not necessarily rational. Can we compute the monodromy group of C? I.e., can we compute\n\n{im(Gal( K) → Aut( H1(C, Qp))): C ∈ Mg(K)}", "statement_status": "exact", "statement_verification": "The extracted record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 19\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[48]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 19 (Kiran Kedlaya). Fix g, fix C/K a curve of genus g, where K = Fq(X) and \\n\\nX is not necessarily rational. Can we compute the monodromy group of C? I.e., can we compute \\n\\n{im(Gal( K) → Aut( H1(C, Qp))): C ∈ Mg(K)}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0049", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For auxiliary-prime cohomology, arithmetic monodromy modulo geometric monodromy is commutative and its derived connected group lies in geometric monodromy. Every isotrivial curve has trivial connected geometric monodromy, while its connected arithmetic group is computed exactly from the saturated multiplicative-relation lattice of its Frobenius eigenvalues. In genus one this gives the ordinary two-dimensional torus and supersingular scalar torus; under the checked characteristic-greater-than-three Igusa hypotheses, a non-isotrivial elliptic curve has arithmetic GL2 and geometric SL2 monodromy. A constant ordinary elliptic curve explicitly disproves recovery of geometric connected monodromy as the determinant-one part of arithmetic monodromy.\n\nCandidate contribution (obstruction; novelty confidence low): The exact separation between L1, which determines the full cyclic Frobenius closure, and its saturation L_tor, which determines the identity component, combined with a constant ordinary elliptic curve, gives a testable obstruction to the proposed identity G_geo^0 = (G_ar^0 intersect Sp)^0." }, { "id": 20000398, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0050", "title": "A ramification-bias criterion for predicted mean point counts", "statement": "Question 20 (Melanie Matchett Wood). When do (A) and (B) predict an interesting average value of C(Fq)?", "original_statement": "Question 20 (Melanie Matchett Wood). When do (A) and (B) predict an interesting average value of C(Fq)?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "Everything proved below is therefore a rigorous answer to this explicitly labeled reconstruction. Recovering Wood's workshop slides or notes could change which label is attached to which principle, and possibly the exact normalization. No claim below depends on calling the two principles A or B; it depends only on the stated mass and independence hypotheses.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 20\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[49]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 20 (Melanie Matchett Wood). When do (A) and (B) predict an interesting average value of C(Fq)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0050", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under an explicitly stated tame structured local mass model for a finite transitive permutation group G, the predicted mean fiber size equals 1 plus an exact signed ramification-bias quotient; the unramified contribution is exactly 1 by Burnside's lemma, and independence of distinct local types is unnecessary for the first moment. The identity gives exact cancellation for regular abelian monodromy and recovers Wood's partition formula for full S_n monodromy, including mean 1 for n <= 2, strict upward bias for n >= 3, and 1 + q^(-1) + O_n(q^(-2)). This is a proved theorem inside the mass model, not a proof that every geometric family realizes that model.\n\nCandidate contribution (identity; novelty confidence low): For every tame finite transitive permutation group G with the stated centralizer/discriminant local mass, the deviation of the predicted mean fiber size from 1 is exactly the normalized signed sum of (r(x,y)-1) over ramified tame Frobenius-inertia pairs; consequently regular abelian actions have zero bias, whereas natural S_n actions have positive bias exactly for n >= 3." }, { "id": 20000399, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0051", "title": "A cubefree local model for trigonal discriminants", "statement": "Question 21 (Jordan Ellenberg). In what ways are discriminants of random trigonal curves like random polynomials?", "original_statement": "Question 21 (Jordan Ellenberg). In what ways are discriminants of random trigonal curves like random polynomials?", "clean_statement": "Question 21 (Jordan Ellenberg). In what ways are discriminants of random trigonal curves like random polynomials?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 21\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[50]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 21 (Jordan Ellenberg). In what ways are discriminants of random trigonal curves like random polynomials?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0051", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every fixed finite set of finite places in characteristic greater than 3, the Datskovsky-Wright/Wood discriminant-ordered trigonal family has asymptotically independent field-discriminant exponents e=0,1,2 with probabilities Q^{-e}/(1+Q^{-1}+Q^{-2}); this is exactly the fixed-place law of a uniform random cubefree polynomial. The induced radical law differs explicitly from the uniform squarefree-polynomial law, while conditioning away total ramification recovers the latter exactly. A global probability generating function and the finite-discriminant squarefreeness prediction q/(q+1) are derived only under an explicit uniform-tail hypothesis.\n\nCandidate contribution (synthesis; novelty confidence low): In the discriminant-ordered trigonal ensemble, a prime of norm Q divides the radical with probability (Q+1)/(Q^2+Q+1), exceeding the random-squarefree probability 1/(Q+1) by Q/((Q+1)(Q^2+Q+1)); on every fixed finite set of primes, conditioning out exponent 2 converts the full cubefree local law exactly to the squarefree-polynomial law." }, { "id": 20000400, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0052", "title": "Heilbronn sums as exact Fourier quadrature", "statement": "Question 22 (Katz). The Heilbron sums\n\n∑\n\n> xmod p\n\nexp 2πix p(1 + pt )\n\np2 = ∑\n\n> xmod p\n\nexp 2πix p\n\np2 · exp 2πipt p2\n\nseem to approximate (as t varies) the (push forward of) Haar measure better than expected. Why?", "original_statement": "Question 22 (Katz). The Heilbron sums \n\n∑ \n\n> xmod p\n\nexp 2πix p(1 + pt )\n\np2 = ∑ \n\n> xmod p\n\nexp 2πix p\n\np2 · exp 2πipt p2\n\nseem to approximate (as t varies) the (push forward of) Haar measure better than expected. Why?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The PDF does **not** specify a range for $t$, a normalization, the compact group carrying Haar measure, or the pushforward map. The left side is periodic in $t\\pmod p$, so the effective range is $t\\in\\mathbb F_p$. The following is the minimal natural reconstruction compatible with (1.2): the right side samples one real Laurent polynomial at all $p$-th roots of unity, and the comparison measure is the pushforward of normalized Haar measure on the unit circle by that same polynomial. This reconstruction is mathematically canonical, but the omitted group and map were not verified as Katz's exact intended wording. A different random-matrix Haar target cannot be recovered from the PDF alone.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 22\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[51]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 22 (Katz). The Heilbron sums \\n\\n∑ \\n\\n> xmod p\\n\\nexp 2πix p(1 + pt )\\n\\np2 = ∑ \\n\\n> xmod p\\n\\nexp 2πix p\\n\\np2 · exp 2πipt p2\\n\\nseem to approximate (as t varies) the (push forward of) Haar measure better than expected. Why?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0052", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM PDF's displayed identity is literally false because its last exponential lacks the summation-variable dependence. After the algebraically forced correction, and under the minimal natural reconstruction in which the family samples a real Laurent polynomial on the p-th roots of unity and Haar measure is circle Haar pushed forward by that polynomial, every normalized moment discrepancy is exactly a sum of nonzero frequency-wraparound slices. Each slice reduces to an explicit Fermat-quotient exponential sum; the first two centered moments agree with Haar exactly, while the cubic discrepancy is isolated as one positive triangular Fermat-quotient sum whose required cancellation remains open.\n\nCandidate contribution (reduction; novelty confidence low): For the corrected Heilbronn family and its natural circle-Haar model, the m-th normalized moment discrepancy is exactly the aggregate of tuples with exponent sum kp for nonzero k, each phase equals e_p(k + sum_j x_j q_p(x_j)), and for m=3 the entire obstruction is the real part of the positive triangle x+y+z=p; in particular moments one and two agree exactly." }, { "id": 20000401, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0053", "title": "Variance coefficients as hook multiplicities", "statement": "Question 23 (Jordan Ellenberg). Describe the right hand side of variance of Λ( n) ∼ q? · n\n\n(as q → ∞ ) geometrically.", "original_statement": "Question 23 (Jordan Ellenberg). Describe the right hand side of variance of Λ( n) ∼ q? · n\n\n(as q → ∞ ) geometrically.", "clean_statement": "Question 23 (Jordan Ellenberg). Describe the right hand side of variance of Λ( n) ∼ q? · n\n\n(as q → ∞ ) geometrically.", "statement_status": "exact", "statement_verification": "The canonical record is Question 23 from the AIM problem session for *Arithmetic statistics over finite fields and function fields*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 23\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[52]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 23 (Jordan Ellenberg). Describe the right hand side of variance of Λ( n) ∼ q? · n\\n\\n(as q → ∞ ) geometrically.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0053", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The official PDF literally contains an unspecified exponent q^?, but its two natural interpretations admit a common geometric synthesis. For L_n = lambda_{-1}(V_std), the trace is n on n-cycles and zero otherwise, so it equals the von Mangoldt function on the squarefree configuration space. Exact character averages make the raw and centered global q^n coefficients the top-cohomology invariant multiplicities n and n-1. Pairing L_n tensor L_n with the Hast-Matei top-weight cohomology for pairs of nearby polynomials retains exactly the hooks indexed by j >= h+2 and gives n-h-2, the Keating-Rudnick short-interval variance coefficient.\n\nCandidate contribution (synthesis; novelty confidence low): The coefficients n, n-1, and n-h-2 are three exact truncations of the single virtual hook object L_n: all hooks, all nontrivial hooks, and the short-interval hooks (n-j,1^j) with j at least h+2, respectively." }, { "id": 20000402, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0054", "title": "Variance of polynomial von Mangoldt pair correlations", "statement": "Question 24 (Brian Conrey). What is the variance of\n\n∑\n\n> f\n\nΛ( f )Λ( f + h)(considered as a function of h).", "original_statement": "Question 24 (Brian Conrey). What is the variance of \n\n∑\n\n> f\n\nΛ( f )Λ( f + h)(considered as a function of h).", "clean_statement": "Question 24 (Brian Conrey). What is the variance of\n\n∑\n\n> f\n\nΛ( f )Λ( f + h)(considered as a function of h).", "statement_status": "exact", "statement_verification": "The extracted record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 24\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[53]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 24 (Brian Conrey). What is the variance of \\n\\n∑\\n\\n> f\\n\\nΛ( f )Λ( f + h)(considered as a function of h).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0054", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM source is underspecified: its summation subscript is literally only f and it supplies no degree, monicity, shift measure, normalization, or limiting regime. For the explicitly labeled canonical reconstruction with monic degree-n polynomials and all shifts of degree less than n, the mean is exactly q^n, the variance is exactly q^(-2n) times a fourth moment of nontrivial additive Fourier coefficients, and an exact ANOVA formula relates the all-shift and nonzero-shift variances. In degree two over every finite field, the correlation is q^2+q(r_q(h)-1), where r_q(h) is the number of roots of h; hence the all-shift variance is q(q-1) and the correctly recentered nonzero-shift variance is q^3/(q+1)^2.\n\nCandidate contribution (theorem; novelty confidence low): For the canonical monic fixed-degree reconstruction, the all-characteristic identity C_2(h)=q^2+q(r_q(h)-1) gives the complete three-class degree-two distribution and exact all-shift and nonzero-shift variances, while the general additive-Fourier and ANOVA identities isolate the higher-degree fourth-moment obstruction." }, { "id": 20000403, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0055", "title": "A finite-period Levinson–Speiser identity for function-field numerators", "statement": "Question 25 (Brian Conrey). Carry out (the other) Levinson's method for function fields.", "original_statement": "Question 25 (Brian Conrey). Carry out (the other) Levinson's method for function fields.", "clean_statement": "Question 25 (Brian Conrey). Carry out (the other) Levinson's method for function fields.", "statement_status": "exact", "statement_verification": "The canonical AIM record is Question 25 from the 2014 workshop *Arithmetic statistics over finite fields and function fields*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 25\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[54]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 25 (Brian Conrey). Carry out (the other) Levinson's method for function fields.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0055", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any self-inversive normalized function-field numerator F and A(s)=F(q^{1/2-s}), Cohn's theorem gives, in one half-open vertical period and with multiplicity, the exact equality between zeros of A to the right of the critical line and zeros of A' to the left; self-inversivity therefore makes the total off-line numerator-zero count twice the derivative-left count. For curve numerators this is an exact numerator Speiser criterion. Separately, the fixed-function mollified height mean is proved to be a finite Parseval coefficient norm, showing that a nontrivial Levinson-proportion implementation must vary conductor, degree, or genus. The source's phrase '(the other)' remains historically ambiguous, and no claim is made for the derivative of the full rational zeta function.\n\nCandidate contribution (equivalence; novelty confidence low): Candidate novelty: the explicit per-period identity #{s in S_q : P_C(q^{-s})=0, Re(s)≠1/2} = 2 #{s in S_q : (P_C(q^{-s}))'=0, Re(s)<1/2}, including multiplicities and critical-line boundary cases, together with its separation from the growing-family mollifier problem." }, { "id": 20000404, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0056", "title": "A Hardy-space cyclicity criterion for curve RH", "statement": "Question 26 (Brian Conrey). Do the analogue of Nymon-Barling criterion for RH over function fields (leading to a different proof of RH).", "original_statement": "Question 26 (Brian Conrey). Do the analogue of Nymon-Barling criterion for RH over function fields (leading to a different proof of RH).", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Question 26 from the AIM workshop *Arithmetic statistics over finite fields and function fields*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 26\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[55]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 26 (Brian Conrey). Do the analogue of Nymon-Barling criterion for RH over function fields (leading to a different proof of RH).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0056", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source PDF literally says 'Nymon-Barling,' reconstructed here as Nyman-Beurling. For a smooth projective geometrically connected curve C/F_q, pole-killing and critical normalization turn the effective-divisor counts into the finite shift vector pi_n=q^{-n/2}(A_n-(q+1)A_{n-1}+qA_{n-2}), whose Hardy-space image is Q_C(z)=P_C(z/sqrt(q)). If B is the finite Blaschke product of the zeros of Q_C in the unit disk, then the closed shift span is B H^2 and dist(1,B H^2)^2=1-|B(0)|^2=1-product_{|beta|<1}|beta|^2. Reciprocity therefore makes curve RH equivalent to cyclicity. Finite convolution Gram matrices compute D_N^2=1-(G_N^{-1})_{00} and certify sqrt(1-D_N^2)<=|beta|<=(1-D_N^2)^{-1/2} for every zero. This is an exact criterion and quantitative reduction, not an independent new proof of Weil RH.\n\nCandidate contribution (criterion; novelty confidence low): The canonically pole-killed effective-divisor sequence of a curve satisfies a discrete Nyman-Beurling shift-cyclicity criterion with exact finite-Blaschke defect, while each finite optimal-approximant Gram distance gives a rigorous simultaneous annulus for every reciprocal zero." }, { "id": 20000405, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0057", "title": "Polynomial Ramanujan sums and finite von Mangoldt expansions", "statement": "Question 27 (1st Tue speaker). Ramanujan sums\n\ncq(x):= ∑\n\n> a a a a=1, and d=2k(q-1)+1, a uniformly sampled reduced polynomial f=c+sum_{j odd, 1<=j<=d} a_j x^j with a_d nonzero defines a genus k(q-1), 2-rank-zero curve satisfying the exact coefficient-model distribution #C_f(F_q)=1+2 Binomial(q,1/2).\n\nCandidate contribution (exact_distribution; novelty confidence low): For every q=2^s and k>=1, exact-degree reduced one-pole equations of degree d=2k(q-1)+1, sampled uniformly by coefficients with nonzero leading coefficient, have the exact finite-genus law #C_f(F_q)=1+2 Binomial(q,1/2); conditioning on exact degree is unbiased because x^d+x is a trace-evaluation-kernel element that acts transitively on leading coefficients." }, { "id": 20000407, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0059", "title": "Orthogonal low-lying zeros and forced-zero deflation", "statement": "Question 29 (Chantal Davis). Local statistics for zeros for L-functions of orthogonal type.", "original_statement": "Question 29 (Chantal Davis). Local statistics for zeros for L-functions of orthogonal type.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The original three-page AIM problem PDF was inspected directly. It literally prints “Chantal Davis,” so the surname is not an error introduced by the JSON extraction. The official workshop page and workshop summary instead name organizer **Chantal David**, a number theorist whose work includes statistics of zeros. Her Arizona Winter School notes from March 2014 discuss precisely the one-level densities for orthogonal families and the inability to distinguish $O$, $SO(\\mathrm{even})$, and $SO(\\mathrm{odd})$ with Fourier support inside $(-1,1)$. “Davis” is therefore almost certainly a source-level typo for “David.” This correction is recorded explicitly rather than made silently.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 29\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[58]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 29 (Chantal Davis). Local statistics for zeros for L-functions of orthogonal type.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0059", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM PDF literally attributes the question to 'Chantal Davis,' while official workshop sources identify Chantal David; the natural reconstructed problem is low-lying zero statistics for orthogonal function-field families such as quadratic twists of a fixed elliptic curve over F_q(t). Existing fixed-q results prove one-level orthogonal density for Fourier support inside (-1,1), with later trace and lower-order refinements, but that range cannot distinguish O, SO(even), and SO(odd) when the forced central zero is counted. For an even trigonometric polynomial F(theta)=f_0+2 sum_{k=1}^K f_k cos(k theta), K<=2N-2, a balanced mixture of SO(2N) and forced-eigenvalue-deflated SO(2N+1) has exact expectation 2N f_0+sum_{k=1}^K (-1)^k f_k. Under the canonical 2N-periodization this is the unitary main term hat(phi)(0) plus an explicit alternating leakage bounded by ||(hat(phi))'||_1/(2N); adding back the odd forced eigenvalue with probability 1/2 recovers the O-law hat(phi)(0)+phi(0)/2.\n\nCandidate contribution (finite_rank_formula; novelty confidence low): In the stable Fourier range, the balanced even/forced-zero-deflated odd orthogonal ensemble has exact linear-statistic expectation 2N f_0+sum_{k>=1}(-1)^k f_k; for the canonical periodization this equals a unitary main term plus an alternating Fourier leakage of size at most ||(hat(phi))'||_1/(2N), furnishing a finite-rank diagnostic for deflated orthogonal data that look unitary at leading order." }, { "id": 20000408, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0060", "title": "Quantitative averages of function-field prime-tuple singular series", "statement": "Question 30 (Rubenstein). Can we compute averages of the Hardy-Littlewood constants in the function field setting look like? Yes.", "original_statement": "Question 30 (Rubenstein). Can we compute averages of the Hardy-Littlewood constants in the function field setting look like? Yes.", "clean_statement": "Question 30 (Rubenstein). Can we compute averages of the Hardy-Littlewood constants in the function field Rubinstein Yes.", "statement_status": "reconstructed_unverified", "statement_verification": "This reconstruction is explicit and is not asserted to be the only intended reading.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Arithmetic statistics over finite fields and function fields\nSection: \nSource item: 30\nSource URL: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[59]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 30 (Rubenstein). Can we compute averages of the Hardy-Littlewood constants in the function field setting look like? Yes.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/arithstatffieldproblems.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0060", "aim-domain:algebraic-number-theory", "aim-workshop:arithstatffieldproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the explicit reconstruction with fixed q and fixed tuple size k, ordered pairwise-distinct shifts in the additive box B_m={h in F_q[T]: deg h E_1 x E_2. Every non-torsion Drinfeld-Heegner input P in E_1(K) yields the explicit non-torsion class D_P=pi_1^*([P-O_1]) in J_X(K), certified by pi_{1,*}D_P=2[P-O_1]. If the analytic ranks of E_1 and E_2 are respectively 1 and 0, then J_X has analytic rank 1. This proves a special-family construction, not the general higher-genus problem.\n\nCandidate contribution (transfer_certificate; novelty confidence low): Candidate novelty is the end-to-end Heegner-transfer certificate for X:y^2=F(x^2): explicit quotient maps, a differential proof of the split-Jacobian isogeny, the scheme-theoretic pulled-back divisor, the norm certificate of non-torsion, and the analytic-rank factorization." }, { "id": 20000410, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0062", "title": "Seeing a primitive surface class in the generic Jacobian", "statement": "1. (C. Hall) 2) If we start with any surface S over Fq we can, via Lefschetz fibration, make it into a curve over Fq(T ). Typically (since the rank of H2 is even), one has two Tate classes. One of these is given by hyperplane sections; can we \"see\" or construct the other one? This question is like the one above, but without requiring C to be hyperelliptic. So, in some sense, the problem is almost as hard as the problem of treating general surfaces. (C. Schoen)", "original_statement": "1. (C. Hall) 2) If we start with any surface S over Fq we can, via Lefschetz fibration, make it into a curve over Fq(T ). Typically (since the rank of H2 is even), one has two Tate classes. One of these is given by hyperplane sections; can we \"see\" or construct the other one? This question is like the one above, but without requiring C to be hyperelliptic. So, in some sense, the problem is almost as hard as the problem of treating general surfaces. (C. Schoen)", "clean_statement": "1. (C. Hall) 2) If we start with any surface S over Fq we can, via Lefschetz fibration, make it into a curve over Fq(T ). Typically (since the rank of H2 is even), one has two Tate classes. One of these is given by hyperplane sections; can we \"see\" or construct the other one? This question is like the one above, but without requiring C to be hyperelliptic. So, in some sense, the problem is almost as hard as the problem of treating general surfaces. (C. Schoen)", "statement_status": "exact", "statement_verification": "The canonical record is the second comment on Question 1 in *Problems from the AIM Tate Conjecture Workshop*, July 23--27, 2007, transcribed by Christopher Lyons. The PDF gives Question 1 as Hall's question about constructing expected points on Jacobians of hyperelliptic curves over \\(\\mathbb F_q(T)\\). After Hall's first comment, the source states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[61]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (C. Hall) 2) If we start with any surface S over Fq we can, via Lefschetz fibration, make it into a curve over Fq(T ). Typically (since the rank of H2 is even), one has two Tate classes. One of these is given by hyperplane sections; can we \\\"see\\\" or construct the other one? This question is like the one above, but without requiring C to be hyperelliptic. So, in some sense, the problem is almost as hard as the problem of treating general surfaces. (C. Schoen)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0062", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the blow-up of a Lefschetz pencil on a smooth projective surface over a finite field, restriction of an H-primitive Neron-Severi class to the generic Jacobian is torsion exactly when its pullback lies in the rational span of geometric fiber components. Hence, when all geometric fibers are integral, every nonzero primitive class restricts to a non-torsion point. The resulting intersection-discriminant certificate gives the explicit conditional quartic K3 construction P = [4p_t - B] from a line on the surface.\n\nCandidate contribution (explicit_construction; novelty confidence low): For an integral geometric Lefschetz pencil, the inequality H^2 D^2 - (D.H)^2 < 0 certifies that O_C(H^2 D - (D.H)H) is non-torsion; for a smooth quartic K3 containing a line L and a suitable plane pencil with skew axis, this specializes to the explicit point [4(L intersect Pi_t) - (S intersect R)]." }, { "id": 20000411, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0063", "title": "A dihedral projector test for explicit exotic divisor classes", "statement": "2. (D. Ramakrishnan) Let X be a Hilbert modular surface over Q. All of the Tate classes that arise from modular curves on X are defined over Qab; however, there also exist Tate classes which are algebraic, but not defined over Qab. (In fact, they are defined over dihedral extensions of Q.) We know the algebraicity of these classes, but do not know specific representatives. Comments: 1) One possible suggestion for such representatives could be non-congruence curves on\n\nX. (D. Ramakrishnan) 2) If we can find representatives, we can intersect them with the modular curves: this may be a way to find interesting non-Heegner points on the modular curves. (D. Ramakrishnan)", "original_statement": "2. (D. Ramakrishnan) Let X be a Hilbert modular surface over Q. All of the Tate classes that arise from modular curves on X are defined over Qab; however, there also exist Tate classes which are algebraic, but not defined over Qab. (In fact, they are defined over dihedral extensions of Q.) We know the algebraicity of these classes, but do not know specific representatives. Comments: 1) One possible suggestion for such representatives could be non-congruence curves on \n\nX. (D. Ramakrishnan) 2) If we can find representatives, we can intersect them with the modular curves: this may be a way to find interesting non-Heegner points on the modular curves. (D. Ramakrishnan)", "clean_statement": "2. (D. Ramakrishnan) Let X be a Hilbert modular surface over Q. All of the Tate classes that arise from modular curves on X are defined over Qab; however, there also exist Tate classes which are algebraic, but not defined over Qab. (In fact, they are defined over dihedral extensions of Q.) We know the algebraicity of these classes, but do not know specific representatives. Comments: 1) One possible suggestion for such representatives could be non-congruence curves on\n\nX. (D. Ramakrishnan) 2) If we can find representatives, we can intersect them with the modular curves: this may be a way to find interesting non-Heegner points on the modular curves. (D. Ramakrishnan)", "statement_status": "exact", "statement_verification": "The canonical record is Problem 2 in *Problems from the AIM Tate Conjecture Workshop* (July 23--27, 2007; transcribed by Christopher Lyons). It reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[62]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (D. Ramakrishnan) Let X be a Hilbert modular surface over Q. All of the Tate classes that arise from modular curves on X are defined over Qab; however, there also exist Tate classes which are algebraic, but not defined over Qab. (In fact, they are defined over dihedral extensions of Q.) We know the algebraicity of these classes, but do not know specific representatives. Comments: 1) One possible suggestion for such representatives could be non-congruence curves on \\n\\nX. (D. Ramakrishnan) 2) If we can find representatives, we can intersect them with the modular curves: this may be a way to find interesting non-Heegner points on the modular curves. (D. Ramakrishnan)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0063", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let rho be an irreducible representation of dimension greater than one in the actual finite dihedral Galois quotient controlling an exotic divisor-class orbit. Every divisor defined over an abelian extension has zero rho-projection. For a proposed curve C over the dihedral field, the central character idempotent gives an explicit Fourier sum of its Galois conjugates; the weighted intersection sum Theta_rho(C) equals C.e_rho(C) and (e_rho(C))^2, so a nonzero value rigorously certifies a nonzero exotic projection. Its intersection degree with every abelian-defined modular-curve class is zero, and a nonzero projected restriction to a Q-defined modular curve in the Jacobian tensor is non-torsion and cannot descend to Q^ab. This is a reduction and sufficient criterion, not an explicit representative.\n\nCandidate contribution (criterion; novelty confidence low): Candidate novelty is the combined dihedral Fourier-projector/intersection/Jacobian certification procedure: abelian-field obstruction, an explicit character-weighted Galois-orbit divisor, a finite sufficient self-intersection certificate, and a forced degree-zero restriction to modular curves.", "research_difficulty_suggested": "L4" }, { "id": 20000412, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0064", "title": "Quadratic descent and character-valued Tate multiplicities", "statement": "3. (J. Getz) Let X be a Hilbert modular variety. At some level (say, that of cohomology), these look like a product of modular curves. Under what circumstances does X have Tate classes not coming from modular subvarieties? Thinking of these classes as submotives defined by automorphic forms (and thus as Galois representations), do these exotic classes come from CM automorphic forms as in\nQuestion 2?", "original_statement": "3. (J. Getz) Let X be a Hilbert modular variety. At some level (say, that of cohomology), these look like a product of modular curves. Under what circumstances does X have Tate classes not coming from modular subvarieties? Thinking of these classes as submotives defined by automorphic forms (and thus as Galois representations), do these exotic classes come from CM automorphic forms as in \nQuestion 2?", "clean_statement": "3. (J. Getz) Let X be a Hilbert modular variety. At some level (say, that of cohomology), these look like a product of modular curves. Under what circumstances does X have Tate classes not coming from modular subvarieties? Thinking of these classes as submotives defined by automorphic forms (and thus as Galois representations), do these exotic classes come from CM automorphic forms as in\nQuestion 2?", "statement_status": "exact", "statement_verification": "The record is Problem 3, attributed to J. Getz, in *Problems from AIM Tate Conjecture Workshop* (workshop held 23--27 July 2007). The original AIM PDF was inspected and agrees with the canonical JSON record:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[63]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (J. Getz) Let X be a Hilbert modular variety. At some level (say, that of cohomology), these look like a product of modular curves. Under what circumstances does X have Tate classes not coming from modular subvarieties? Thinking of these classes as submotives defined by automorphic forms (and thus as Galois representations), do these exotic classes come from CM automorphic forms as in \\nQuestion 2?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0064", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Published results of Getz-Hahn show that, for a cohomological nondihedral eigensystem on a Galois Hilbert modular variety and sufficiently large ell, the eventual Tate Hecke rank is zero for odd projective-conjugacy stabilizer order m and is the stated Catalan power for even m; a Tate line forces quadratic base-change descent after twisting, and rank-at-most-one spaces are generated by twisted Hirzebruch-Zagier cycles. This attempt separately proves an exact-base-change representation-theoretic refinement: after full determinant normalization, the Tate multiplicity space for a degree-2n Galois extension is the restriction of the Specht module Sp^(n,n) along the regular Galois action. In degree four this restricts as 1 plus the quadratic character for C4 and as two copies of 1 for V4, explaining fixed-character multiplicities but constructing no new algebraic cycle and giving no general CM classification.\n\nCandidate contribution (representation_theoretic_lemma; novelty confidence low): Candidate novelty is the character-valued refinement of the Catalan rank in the exact-base-change case: the determinant-normalized SL2-invariant multiplicity space of As_{L/F}(rho_0 restricted to G_L) is Sp^(n,n) restricted along the regular embedding Gal(L/F) into S_{2n}; consequently Sp^(2,2)|C4 is 1 plus the unique quadratic character while Sp^(2,2)|V4 is two copies of the trivial character." }, { "id": 20000413, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0065", "title": "Two Tate lines and the missing adjoint correspondence", "statement": "4. (K. Murty) Let F be a real quadratic field, π be an automorphic form over Q, and Π be its base change to F. Then Π contributes to a subspace H2(Π) of H2(SF ) for some Hilbert modular surface SF for GL 2(F ), while H1(π) sits inside H1(M ) for some modular curve\n\nM. As Gal( ¯Q/F )-modules, H2(Π) is H1(π) ⊗ H1(π). Take two real quadratic fields F1, F 2 in this way, along with the base changes Π 1, Π2\n\nof π and the associated surfaces SF1, S F2. Inside H4(SF1 × SF2 ) we have H2(Π 1) ⊗ H2(Π 2), which is H1(π)⊗ 4 as a Gal( ¯Q/F 1F2)-module. Hence we expect cycles of codimension 2 in\n\nSF1 × SF2 that are defined over F1F2. What are they?\n1", "original_statement": "4. (K. Murty) Let F be a real quadratic field, π be an automorphic form over Q, and Π be its base change to F. Then Π contributes to a subspace H2(Π) of H2(SF ) for some Hilbert modular surface SF for GL 2(F ), while H1(π) sits inside H1(M ) for some modular curve \n\nM. As Gal( ¯Q/F )-modules, H2(Π) is H1(π) ⊗ H1(π). Take two real quadratic fields F1, F 2 in this way, along with the base changes Π 1, Π2\n\nof π and the associated surfaces SF1, S F2. Inside H4(SF1 × SF2 ) we have H2(Π 1) ⊗ H2(Π 2), which is H1(π)⊗ 4 as a Gal( ¯Q/F 1F2)-module. Hence we expect cycles of codimension 2 in \n\nSF1 × SF2 that are defined over F1F2. What are they? \n1", "clean_statement": "4. (K. Murty) Let F be a real quadratic field, π be an automorphic form over Q, and Π be its base change to F. Then Π contributes to a subspace H2(Π) of H2(SF ) for some Hilbert modular surface SF for GL 2(F ), while H1(π) sits inside H1(M ) for some modular curve\n\nM. As Gal( ¯Q/F )-modules, H2(Π) is H1(π) ⊗ H1(π). Take two real quadratic fields F1, F 2 in this way, along with the base changes Π 1, Π2\n\nof π and the associated surfaces SF1, S F2. Inside H4(SF1 × SF2 ) we have H2(Π 1) ⊗ H2(Π 2), which is H1(π)⊗ 4 as a Gal( ¯Q/F 1F2)-module. Hence we expect cycles of codimension 2 in\n\nSF1 × SF2 that are defined over F1F2. What are they?\n1", "statement_status": "exact", "statement_verification": "The canonical record is Question 4 in the AIM workshop list *The Tate conjecture*. The final isolated **1** in the extracted record is the page number at the foot of the PDF; it is not part of the question. Superscripts, subscripts, and bars were also lost in extraction. The recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[64]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"4. (K. Murty) Let F be a real quadratic field, π be an automorphic form over Q, and Π be its base change to F. Then Π contributes to a subspace H2(Π) of H2(SF ) for some Hilbert modular surface SF for GL 2(F ), while H1(π) sits inside H1(M ) for some modular curve \\n\\nM. As Gal( ¯Q/F )-modules, H2(Π) is H1(π) ⊗ H1(π). Take two real quadratic fields F1, F 2 in this way, along with the base changes Π 1, Π2\\n\\nof π and the associated surfaces SF1, S F2. Inside H4(SF1 × SF2 ) we have H2(Π 1) ⊗ H2(Π 2), which is H1(π)⊗ 4 as a Gal( ¯Q/F 1F2)-module. Hence we expect cycles of codimension 2 in \\n\\nSF1 × SF2 that are defined over F1F2. What are they? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0065", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let V=H^1(pi), D=det(V), and L=F1F2. Under the explicit hypothesis that the Zariski closure of G_L on V contains SL(V), the determinant-normalized space (V^{tensor 4} tensor D^{-2})^{G_L} is exactly two-dimensional: one line is the external product of the two determinant/divisor lines, and one is the unique non-product Casimir line in Ad^0(V) tensor Ad^0(V), corresponding to the identity on Ad^0(V). In the actual codimension-two Tate twist their global characters are epsilon^2 chi_1 chi_2 and epsilon^2, respectively. This gives a proved representation-theoretic criterion and reduces the geometric problem to constructing one adjoint identity correspondence. It does not construct that algebraic cycle for distinct F1 and F2.\n\nCandidate contribution (reduction; novelty confidence low): Under big monodromy, the AIM tensor-fourth-power contribution has exactly two normalized Tate lines with explicit basis p=t_{12|34} and c=t_{13|24}+t_{14|23}; after restoring the actual Tate twist their descent characters are epsilon^2 chi_1 chi_2 and epsilon^2, and the sole non-product target is precisely the identity correspondence on Ad^0(V)." }, { "id": 20000414, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0066", "title": "A d=9 exception and a CM-synchronization reduction", "statement": "5. (D. Ulmer) Consider the curve Cd,a: yd = x(x − 1)( x − a) defined over C (though ¯Q or\n\nFq will also work). For which values of a does it happen that this curve has CM? (More precisely, for which values of a does the Jacobian of Cd,a have endomorphism algebra of dimension 2 g over Q, where g = g(Ca,d ) = d − 1 is the genus?) It happens (for dull reasons) when a = −1, 1\n\n> 2, 2, ζ 6, ζ −16. For a fixed d > 7, there are only finitely many values of a for which it happens. Are there infinitely many d such that there's another value of a (besides the trivial ones listed above) for which Cd,a has CM? Comments: 1) An application is that, if Cd,a does have CM, then one can construct an elliptic curve defined over C(T ) of rank d +", "original_statement": "5. (D. Ulmer) Consider the curve Cd,a: yd = x(x − 1)( x − a) defined over C (though ¯Q or \n\nFq will also work). For which values of a does it happen that this curve has CM? (More precisely, for which values of a does the Jacobian of Cd,a have endomorphism algebra of dimension 2 g over Q, where g = g(Ca,d ) = d − 1 is the genus?) It happens (for dull reasons) when a = −1, 1 \n\n> 2, 2, ζ 6, ζ −16. For a fixed d > 7, there are only finitely many values of a for which it happens. Are there infinitely many d such that there's another value of a (besides the trivial ones listed above) for which Cd,a has CM? Comments: 1) An application is that, if Cd,a does have CM, then one can construct an elliptic curve defined over C(T ) of rank d +", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record is truncated and has two OCR errors: \\(1/2\\) appears as “1 > 2,” and \\(\\zeta _6^{-1}\\) appears as “ζ −16.” Inspection of the official AIM PDF recovers Problem 5 as follows.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[65]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"5. (D. Ulmer) Consider the curve Cd,a: yd = x(x − 1)( x − a) defined over C (though ¯Q or \\n\\nFq will also work). For which values of a does it happen that this curve has CM? (More precisely, for which values of a does the Jacobian of Cd,a have endomorphism algebra of dimension 2 g over Q, where g = g(Ca,d ) = d − 1 is the genus?) It happens (for dull reasons) when a = −1, 1 \\n\\n> 2, 2, ζ 6, ζ −16. For a fixed d > 7, there are only finitely many values of a for which it happens. Are there infinitely many d such that there's another value of a (besides the trivial ones listed above) for which Cd,a has CM? Comments: 1) An application is that, if Cd,a does have CM, then one can construct an elliptic curve defined over C(T ) of rank d +\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0066", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the standard CM-type definition, the full genus-7 Jacobian family at d=9 is Moonen's one-dimensional special family 19 and therefore has infinitely many, indeed Zariski-dense, CM parameters; for every fixed d>7 with d not equal to 9, Moonen's classification plus Tsimerman's proof of Andre-Oort gives finiteness. The printed genus is corrected to g=d-(1+gcd(d,3))/2. Moreover J_{d,a} is isogenous to the product of the exact-order-e new factors A_{e,a} over e>1 dividing d, so full CM is equivalent to simultaneous CM of every such factor. This is a proved partial result using standard CM type; the question of nontrivial full-CM fibers for infinitely many distinct d remains open.\n\nCandidate contribution (reduction; novelty confidence low): For this AIM family, S_d^{full} equals the intersection of S_e^{new} over all divisors e>1 of d, and e dividing d implies S_d^{full} is contained in S_e^{full}; paired with Moonen's exact d=9 table entry, this gives a divisor-level synchronization test for any proposed composite-d construction." }, { "id": 20000415, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0067", "title": "Zarhin's cubic S3 theorem and propagated CM factors", "statement": "3. (D. Ulmer) 2) Two relevant references for this problem are i) de Jong, Johan; Noot, Rutger. Jacobians with complex multiplication. Arith-metic algebraic geometry (Texel, 1989), 177-192, Progr. Math., 89, Birkhuser Boston, Boston, MA, 1991. and ii) deJong's course at AWS2002: http://swc.math.arizona.edu/aws/02/02Notes.html\n\n(D. Ulmer) 3) If x(x − 1)( x − a) is replaced by a cubic polynomial over C(T ) with Galois group S3\n\nand d > 1 is not a power of 3 then the jacobian of the resulting curve is not of CM-type (over an algebraic closure of C(T )). In addition, if d is a power prime then the jacobian does not contain non-zero abelian subvarieties of CM-type (over an algebraic closure of C(T )) if and only if d = 2 or d is an odd number that is not divisible by", "original_statement": "3. (D. Ulmer) 2) Two relevant references for this problem are i) de Jong, Johan; Noot, Rutger. Jacobians with complex multiplication. Arith-metic algebraic geometry (Texel, 1989), 177-192, Progr. Math., 89, Birkhuser Boston, Boston, MA, 1991. and ii) deJong's course at AWS2002: http://swc.math.arizona.edu/aws/02/02Notes.html \n\n(D. Ulmer) 3) If x(x − 1)( x − a) is replaced by a cubic polynomial over C(T ) with Galois group S3\n\nand d > 1 is not a power of 3 then the jacobian of the resulting curve is not of CM-type (over an algebraic closure of C(T )). In addition, if d is a power prime then the jacobian does not contain non-zero abelian subvarieties of CM-type (over an algebraic closure of C(T )) if and only if d = 2 or d is an odd number that is not divisible by", "clean_statement": "3. (D. Ulmer) 2) Two relevant references for this problem are i) de Jong, Johan; Noot, Rutger. Jacobians with complex multiplication. Arith-metic algebraic geometry (Texel, 1989), 177-192, Progr. Math., 89, Birkhuser Boston, Boston, MA, 1991. and ii) deJong's course at AWS2002: http://swc.math.arizona.edu/aws/02/02Notes.html\n\n(D. Ulmer) 3) If x(x − 1)( x − a) is replaced by a cubic polynomial over C(T ) with Galois group S3\n\nand d > 1 is not a power of 3 then the jacobian of the resulting curve is not of CM-type (over an algebraic closure of C(T )). In addition, if d is a power prime then the jacobian does not contain non-zero abelian subvarieties of CM-type (over an algebraic closure of C(T )) if and only if d = 2 or d is an odd number that is not divisible by", "statement_status": "exact", "statement_verification": "The canonical record is not a self-contained open problem. It splices comments 2 and 3 following Problem 5 in the AIM workshop list *The Tate conjecture*. The parent problem concerns CM specializations of", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[66]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (D. Ulmer) 2) Two relevant references for this problem are i) de Jong, Johan; Noot, Rutger. Jacobians with complex multiplication. Arith-metic algebraic geometry (Texel, 1989), 177-192, Progr. Math., 89, Birkhuser Boston, Boston, MA, 1991. and ii) deJong's course at AWS2002: http://swc.math.arizona.edu/aws/02/02Notes.html \\n\\n(D. Ulmer) 3) If x(x − 1)( x − a) is replaced by a cubic polynomial over C(T ) with Galois group S3\\n\\nand d > 1 is not a power of 3 then the jacobian of the resulting curve is not of CM-type (over an algebraic closure of C(T )). In addition, if d is a power prime then the jacobian does not contain non-zero abelian subvarieties of CM-type (over an algebraic closure of C(T )) if and only if d = 2 or d is an odd number that is not divisible by\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0067", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The published part of the truncated record says that, for a prime-power exponent d, the cubic S3 Jacobian contains no nonzero CM subvariety exactly when d=2 or d is odd and not divisible by 3; Zarhin's complete-non-isotriviality and exceptional-factor results prove this and prove that J_d is not of CM-type whenever d is not a power of 3. The substantive contribution of this attempt is a proved arbitrary-multiple propagation corollary: divisor covers place J_e in J_d up to isogeny for e dividing d, so 3|d forces a j=0 CM elliptic factor, 4|d forces a j=1728 CM square, and 12|d forces their three-dimensional product with finite intersection.\n\nCandidate contribution (corollary; novelty confidence low): For a cubic f over C(T) with Galois group S3, every J_d with 3|d contains a CM elliptic subvariety isogenous to E_{j=0}, every J_d with 4|d contains a CM surface isogenous to E_{j=1728}^2, and for 12|d these factors have finite intersection and yield a three-dimensional CM subvariety isogenous to E_{j=0} times E_{j=1728}^2." }, { "id": 20000416, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0068", "title": "What Belyi reparametrization can and cannot detect", "statement": "3. (Y. Zarhin) 4) People like Wolfart have studied the problem of determining which Jacobians of curves with Belyi parametrizations are CM. Does it help here to reparametrize this equation in the Belyi form, i.e., to express this equation as a cover of P1 ramified only at 0, 1, ∞?(D. Ramakrishnan)", "original_statement": "3. (Y. Zarhin) 4) People like Wolfart have studied the problem of determining which Jacobians of curves with Belyi parametrizations are CM. Does it help here to reparametrize this equation in the Belyi form, i.e., to express this equation as a cover of P1 ramified only at 0, 1, ∞?(D. Ramakrishnan)", "clean_statement": "3. (Y. Zarhin) 4) People like Wolfart have studied the problem of determining which Jacobians of curves with Belyi parametrizations are CM. Does it help here to reparametrize this equation in the Belyi form, i.e., to express this equation as a cover of P1 ramified only at 0, 1, ∞?(D. Ramakrishnan)", "statement_status": "exact", "statement_verification": "The canonical record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[67]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (Y. Zarhin) 4) People like Wolfart have studied the problem of determining which Jacobians of curves with Belyi parametrizations are CM. Does it help here to reparametrize this equation in the Belyi form, i.e., to express this equation as a cover of P1 ramified only at 0, 1, ∞?(D. Ramakrishnan)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0068", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every d>7, the natural cyclic map C_{d,a} -> P^1 has four branch values, so no Mobius reparametrization makes it Belyi. Its deck-group normalizer fits into 1 -> C_d -> N -> AffStab({0,1,a}) -> 1, and the affine quotient is nontrivial exactly for a=-1, 1/2, 2, or a^2-a+1=0; explicit degree-2d or degree-3d regular Belyi quotients then have abelian deck group and recover exactly these five known CM parameters via Wolfart's criterion. This is explicitly only the natural-normalizer route, not a classification of all Belyi maps. Unrestricted Belyi-map existence for fixed d>7 is equivalent merely to a being algebraic (equivalently algebraic modulo its anharmonic orbit), so it is not a CM criterion. A corrected genus calculation and Moonen's classification show separately that d=9 is a special genus-seven family with infinitely many further CM fibers.\n\nCandidate contribution (criterion; novelty confidence low): For d>7, the natural cyclic deck-group normalizer yields a regular abelian Belyi quotient if and only if a is harmonic or equianharmonic, namely a is in {-1, 1/2, 2, zeta_6, zeta_6^{-1}}; the exact normalizer quotient and the four quotient formulas are proved explicitly." }, { "id": 20000417, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0069", "title": "A Catalan correspondence criterion for middle Tate tensors", "statement": "6. (J. Getz) Take a product of modular curves X = ∏\n\n> i\n\nXi. Many smaller products embed diagonally into X. How much of the middle cohomology can be accounted for by these constructions? Comments: 1) Ribet considered the product of two modular curves X = X1 ×X2; here N S (X) is just the quotient of Hom( J(X1), J (X2)) by the pullbacks of divisors on the two factors. He shows it is necessary to consider twisting correspondences, in addition to the Hecke correspondences, to complete the Tate classes. Are there similarly interesting, yet explicit, correspondences for X when we consider higher dimensional factors Xi? (J. Getz, D. Ramakrishnan, D. Ulmer) 2) For a product of four Shimura curves, there are interesting Tate classes, which are also Hodge classes, which we don't know how to represent by cycles. (D. Ramakrishnan)", "original_statement": "6. (J. Getz) Take a product of modular curves X = ∏ \n\n> i\n\nXi. Many smaller products embed diagonally into X. How much of the middle cohomology can be accounted for by these constructions? Comments: 1) Ribet considered the product of two modular curves X = X1 ×X2; here N S (X) is just the quotient of Hom( J(X1), J (X2)) by the pullbacks of divisors on the two factors. He shows it is necessary to consider twisting correspondences, in addition to the Hecke correspondences, to complete the Tate classes. Are there similarly interesting, yet explicit, correspondences for X when we consider higher dimensional factors Xi? (J. Getz, D. Ramakrishnan, D. Ulmer) 2) For a product of four Shimura curves, there are interesting Tate classes, which are also Hodge classes, which we don't know how to represent by cycles. (D. Ramakrishnan)", "clean_statement": "6. (J. Getz) Take a product of modular curves X = ∏\n\n> i\n\nXi. Many smaller products embed diagonally into X. How much of the middle cohomology can be accounted for by these constructions? Comments: 1) Ribet considered the product of two modular curves X = X1 ×X2; here N S (X) is just the quotient of Hom( J(X1), J (X2)) by the pullbacks of divisors on the two factors. He shows it is necessary to consider twisting correspondences, in addition to the Hecke correspondences, to complete the Tate classes. Are there similarly interesting, yet explicit, correspondences for X when we consider higher dimensional factors Xi? (J. Getz, D. Ramakrishnan, D. Ulmer) 2) For a product of four Shimura curves, there are interesting Tate classes, which are also Hodge classes, which we don't know how to represent by cycles. (D. Ramakrishnan)", "statement_status": "exact", "statement_verification": "The official AIM workshop PDF gives the following problem.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[68]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"6. (J. Getz) Take a product of modular curves X = ∏ \\n\\n> i\\n\\nXi. Many smaller products embed diagonally into X. How much of the middle cohomology can be accounted for by these constructions? Comments: 1) Ribet considered the product of two modular curves X = X1 ×X2; here N S (X) is just the quotient of Hom( J(X1), J (X2)) by the pullbacks of divisors on the two factors. He shows it is necessary to consider twisting correspondences, in addition to the Hecke correspondences, to complete the Tate classes. Are there similarly interesting, yet explicit, correspondences for X when we consider higher dimensional factors Xi? (J. Getz, D. Ramakrishnan, D. Ulmer) 2) For a product of four Shimura curves, there are interesting Tate classes, which are also Hodge classes, which we don't know how to represent by cycles. (D. Ramakrishnan)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0069", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the primitive all-H1 tensor of a literal product of curves, assume algebraic projectors and independent diagonal SL2 monodromy on the projective twist-equivalence blocks. An odd block forces the geometric Tate tensor space to vanish; if every block has size 2r_B, its dimension is the product of the Catalan numbers Cat(r_B), with a basis indexed by products of noncrossing perfect matchings. After a finite extension kills the finite twists, Faltings full faithfulness followed by the idempotent corners e_j Hom e_i realizes every pair contraction by a divisor correspondence, and products of these divisors realize the entire Catalan basis. Thus the primitive pair-correspondence obstruction vanishes under the stated hypotheses. In a four-factor block, the three pairing products span a two-dimensional plane and satisfy one projected-cohomology Plucker relation, with Kunneth Koszul signs absorbed into the normalization.\n\nCandidate contribution (criterion; novelty confidence low): The candidate contribution is an explicit noncrossing product-of-divisors basis and the vanishing criterion O(V)=T_geom(V)/T_pair(V)=0 for possibly different curves whose projected two-dimensional factors have independent diagonal SL2 block monodromy; for four factors it gives two independent cycle classes and one signed Plucker relation in projected cohomology." }, { "id": 20000418, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0070", "title": "Newton divisors versus Hirzebruch--Zagier cycles on Hilbert modular surfaces", "statement": "7. (K. Murty) A Hilbert modular surface in characteristic p has interesting cycles parametriz-ing Hilbert-Blumenthal abelian varieties with specified Newton polygon behavior. Are these 2contained in the space of special cycles made up of (Hecke translates of) Hirzebruch-Zagier curves? How can we check this? Comments: 1) Are they pulled back from the Siegel threefold A2? (Not always.) Work of A. Langer is pertinent to this question. (J. Achter, C. Schoen) 2) Could the Teichm¨ uller curves of McMullen generate these exotic classes? (J. Ellen-berg) 3) Material by van der Geer and Moonen is relevant for explicit information on the characteristic p cycles above. (J. Achter) 4) Can we intersect these characteristic p cycles with the reduction mod p of the CM Tate cycles described in Murty's lecture? (J. Ellenberg) 5) How does the involution θ ∈ Gal( F/ Q) act on these cycles? (Here F is the real quadratic field for the Hilbert modular surface.) (J. Getz)", "original_statement": "7. (K. Murty) A Hilbert modular surface in characteristic p has interesting cycles parametriz-ing Hilbert-Blumenthal abelian varieties with specified Newton polygon behavior. Are these 2contained in the space of special cycles made up of (Hecke translates of) Hirzebruch-Zagier curves? How can we check this? Comments: 1) Are they pulled back from the Siegel threefold A2? (Not always.) Work of A. Langer is pertinent to this question. (J. Achter, C. Schoen) 2) Could the Teichm¨ uller curves of McMullen generate these exotic classes? (J. Ellen-berg) 3) Material by van der Geer and Moonen is relevant for explicit information on the characteristic p cycles above. (J. Achter) 4) Can we intersect these characteristic p cycles with the reduction mod p of the CM Tate cycles described in Murty's lecture? (J. Ellenberg) 5) How does the involution θ ∈ Gal( F/ Q) act on these cycles? (Here F is the real quadratic field for the Hilbert modular surface.) (J. Getz)", "clean_statement": "7. (K. Murty) A Hilbert modular surface in characteristic p has interesting cycles parametriz-ing Hilbert-Blumenthal abelian varieties with specified Newton polygon behavior. Are these 2contained in the space of special cycles made up of (Hecke translates of) Hirzebruch-Zagier curves? How can we check this? Comments: 1) Are they pulled back from the Siegel threefold A2? (Not always.) Work of A. Langer is pertinent to this question. (J. Achter, C. Schoen) 2) Could the Teichm¨ uller curves of McMullen generate these exotic classes? (J. Ellen-berg) 3) Material by van der Geer and Moonen is relevant for explicit information on the characteristic p cycles above. (J. Achter) 4) Can we intersect these characteristic p cycles with the reduction mod p of the CM Tate cycles described in Murty's lecture? (J. Ellenberg) 5) How does the involution θ ∈ Gal( F/ Q) act on these cycles? (Here F is the real quadratic field for the Hilbert modular surface.) (J. Getz)", "statement_status": "exact", "statement_verification": "The record is Problem 7, attributed to K. Murty, in the AIM workshop list *The Tate conjecture* (workshop held July 23--27, 2007; list transcribed by Christopher Lyons). The canonical JSON has the string “Are these 2contained”, but comparison with page 2 of the official PDF shows that the 2 is the printed page number captured at a page break. The recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[69]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"7. (K. Murty) A Hilbert modular surface in characteristic p has interesting cycles parametriz-ing Hilbert-Blumenthal abelian varieties with specified Newton polygon behavior. Are these 2contained in the space of special cycles made up of (Hecke translates of) Hirzebruch-Zagier curves? How can we check this? Comments: 1) Are they pulled back from the Siegel threefold A2? (Not always.) Work of A. Langer is pertinent to this question. (J. Achter, C. Schoen) 2) Could the Teichm¨ uller curves of McMullen generate these exotic classes? (J. Ellen-berg) 3) Material by van der Geer and Moonen is relevant for explicit information on the characteristic p cycles above. (J. Achter) 4) Can we intersect these characteristic p cycles with the reduction mod p of the CM Tate cycles described in Murty's lecture? (J. Ellenberg) 5) How does the involution θ ∈ Gal( F/ Q) act on these cycles? (Here F is the real quadratic field for the Hilbert modular surface.) (J. Getz)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0070", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a real quadratic Hilbert modular surface at an inert prime p>2 with hyperspecial good reduction, the two Newton-polygon divisor families are the Goren--Oort partial-Hasse divisors. Their sum is geometrically the pullback of the Siegel Hasse divisor, while each labelled divisor has a nonzero theta-anti-invariant Neron--Severi class and is not individually a Siegel pullback. In a cuspidal pi-component with distinct Satake parameters, Tian--Xiao's matrix gives two independent special-fiber Tate directions; comparison with characteristic-zero potential-Tate dimension shows that reductions of HZ/twisted-HZ cycles and their prime-to-p Hecke translates miss both directions in potential-Tate dimension zero and at least one in dimension at most one. A Hodge-index Schur-complement criterion gives an exact finite test for numerical containment.\n\nCandidate contribution (criterion; novelty confidence low): After quotienting the Goren--Oort component span by its tautological part, the remaining quotient is generated by component-augmentation images; for a theta-stable proposed special-cycle basis, each residual numerical-containment question is equivalent, separately in the theta-even and theta-odd primitive Neron--Severi blocks, to vanishing of the computable Schur complement D^2-b^T A^{-1}b." }, { "id": 20000419, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0071", "title": "Prime-specific failures and determinant certificates for integral Tate saturation", "statement": "8. (J. Ellenberg) In cases where we know all Tate classes are in the image of Ci ⊗ Q`, what can we say about the image of Ci ⊗ Z`? For instance, for which primes ` can we have non-surjectivity? Comments: 1) Koll´ ar proved that if we take a very general hypersurface in P4 such that d3 divides the degree of X, then the image of C1(X) is contained in dH 4(X, Z). But it is not known if this can happen for X over ¯Q. For C1 of X over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for ` 6 = p. (C. Schoen) 2) Over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for the Weil-´ etale cohomology, so this becomes a question of comparison between ´ etale and Weil-´ etale cohomology. (T. Geisser)", "original_statement": "8. (J. Ellenberg) In cases where we know all Tate classes are in the image of Ci ⊗ Q`, what can we say about the image of Ci ⊗ Z`? For instance, for which primes ` can we have non-surjectivity? Comments: 1) Koll´ ar proved that if we take a very general hypersurface in P4 such that d3 divides the degree of X, then the image of C1(X) is contained in dH 4(X, Z). But it is not known if this can happen for X over ¯Q. For C1 of X over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for ` 6 = p. (C. Schoen) 2) Over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for the Weil-´ etale cohomology, so this becomes a question of comparison between ´ etale and Weil-´ etale cohomology. (T. Geisser)", "clean_statement": "8. (J. Ellenberg) In cases where we know all Tate classes are in the image of Ci ⊗ Q`, what can we say about the image of Ci ⊗ Z`? For instance, for which primes ` can we have non-surjectivity? Comments: 1) Koll´ ar proved that if we take a very general hypersurface in P4 such that d3 divides the degree of X, then the image of C1(X) is contained in dH 4(X, Z). But it is not known if this can happen for X over ¯Q. For C1 of X over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for ` 6 = p. (C. Schoen) 2) Over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for the Weil-´ etale cohomology, so this becomes a question of comparison between ´ etale and Weil-´ etale cohomology. (T. Geisser)", "statement_status": "exact", "statement_verification": "The canonical JSON record has lost superscripts, subscripts, accents, and every occurrence of \\(\\ell\\). The official AIM workshop PDF gives the following statement (Problem 8, J. Ellenberg):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[70]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"8. (J. Ellenberg) In cases where we know all Tate classes are in the image of Ci ⊗ Q`, what can we say about the image of Ci ⊗ Z`? For instance, for which primes ` can we have non-surjectivity? Comments: 1) Koll´ ar proved that if we take a very general hypersurface in P4 such that d3 divides the degree of X, then the image of C1(X) is contained in dH 4(X, Z). But it is not known if this can happen for X over ¯Q. For C1 of X over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for ` 6 = p. (C. Schoen) 2) Over Fq, the Q`-Tate Conjecture implies the Z`-Tate Conjecture for the Weil-´ etale cohomology, so this becomes a question of comparison between ´ etale and Weil-´ etale cohomology. (T. Geisser)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0071", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The rational-to-integral implication is false at every prime when the finite-field variety may vary: the projective approximations underlying Kameko's codimension-two examples have rational degree-four cohomology spanned by algebraic Chern classes while their integral cycle map fails modulo torsion. For a very general principally polarized abelian variety over a finitely generated characteristic-zero field after finite extension, the March 2026 Engel--de Gaay Fortman--Schreieder results combined with Milne's descent give a 2-adic failure on every intermediate minimal-class line, exact line index 2 in dimensions 4 and 5, and a 3-adic failure on the minimal curve line in dimension at least 7. A proved two-sided determinant formula bounds the integral defect and yields a conditional finite exceptional-prime set under uniform perfect-pairing data.\n\nCandidate contribution (corollary; novelty confidence low): Combining Engel--de Gaay Fortman--Schreieder arXiv:2507.15704v3 with Milne's absolute-Hodge descent identifies ell=2 as an integral Tate failure on every intermediate minimal line Z_2[Theta]^c/c! of a very general principally polarized abelian g-fold for g at least 4, with exact line index 2 for g=4,5; their matroidal divisibility theorem similarly identifies ell=3 on the minimal curve line for g at least 7." }, { "id": 20000420, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0072", "title": "A new-Lefschetz defect reduction for rational Hodge pairings", "statement": "9. (J. Milne) Let A be CM abelian variety over ¯Q and let c be an absolute Hodge class. Reduce A to A0 over Fp and take a Lefschetz class x on A0 of complementary dimension. Then x. ¯c ∈ Q`; is it in Q and independent of `?Comments: 1) The answer is yes, assuming either the Tate Conjecture or the Hodge Conjecture, or also when A0 is ordinary (since then x can be lifted to characteristic 0). (J. Milne) 2) This would imply the existence of the Q-subalgebras R∗ mentioned described in Milne's lecture. (J. Milne) 3) The question also makes sense for any abelian variety A which has good reduction over Fp. (J. Milne) 4) It may be that ( A0, x ) still admits a canonical lift, even if we relax the assumption that A0 is ordinary (see Milne's first comment). (J. Achter)\n1", "original_statement": "9. (J. Milne) Let A be CM abelian variety over ¯Q and let c be an absolute Hodge class. Reduce A to A0 over Fp and take a Lefschetz class x on A0 of complementary dimension. Then x. ¯c ∈ Q`; is it in Q and independent of `?Comments: 1) The answer is yes, assuming either the Tate Conjecture or the Hodge Conjecture, or also when A0 is ordinary (since then x can be lifted to characteristic 0). (J. Milne) 2) This would imply the existence of the Q-subalgebras R∗ mentioned described in Milne's lecture. (J. Milne) 3) The question also makes sense for any abelian variety A which has good reduction over Fp. (J. Milne) 4) It may be that ( A0, x ) still admits a canonical lift, even if we relax the assumption that A0 is ordinary (see Milne's first comment). (J. Achter) \n1", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON is damaged precisely at the coefficient field and at the prime. The official AIM workshop PDF gives the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[71]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"9. (J. Milne) Let A be CM abelian variety over ¯Q and let c be an absolute Hodge class. Reduce A to A0 over Fp and take a Lefschetz class x on A0 of complementary dimension. Then x. ¯c ∈ Q`; is it in Q and independent of `?Comments: 1) The answer is yes, assuming either the Tate Conjecture or the Hodge Conjecture, or also when A0 is ordinary (since then x can be lifted to characteristic 0). (J. Milne) 2) This would imply the existence of the Q-subalgebras R∗ mentioned described in Milne's lecture. (J. Milne) 3) The question also makes sense for any abelian variety A which has good reduction over Fp. (J. Milne) 4) It may be that ( A0, x ) still admits a canonical lift, even if we relax the assumption that A0 is ordinary (see Milne's first comment). (J. Achter) \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0072", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM question is Milne's Rationality Conjecture 4.1 and remains open in general; Milne's 2025 very-good-reduction corollaries are conditional on weak rationality for all CM abelian varieties and therefore do not solve the CM base case. For a fixed good reduction and codimension, the prime-to-p failure of rationality defines a canonical Q-linear functional on the quotient of complementary Lefschetz classes by the Lefschetz classes lifting from characteristic zero. Hence rationality is equivalent to vanishing on any basis of this finite-dimensional quotient. Its dimension is bounded by the number of divisor monomials involving a new reduction-only divisor, and surjectivity of Neron-Severi specialization is a sufficient condition for rationality for every absolute Hodge class in every degree.\n\nCandidate contribution (reduction; novelty confidence low): The candidate contribution is the canonical new-Lefschetz obstruction functional on Delta_w^s(A)=D^s(A_0)/sp_w D^s(A), together with the basis-independent dim Delta finite test and the bound dim Delta_w^s(A) <= binom(rho_0+s-1,s)-binom(rho_lift+s-1,s); in particular, surjective Neron-Severi specialization forces all these rationality obstructions to vanish." }, { "id": 20000421, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0073", "title": "Numerical surface classes on the square of a general abelian surface", "statement": "0. (C. Schoen) Let J be a general abelian surface (over C, say). Describe the surfaces in J ×J.3\n1", "original_statement": "0. (C. Schoen) Let J be a general abelian surface (over C, say). Describe the surfaces in J ×J.3\n1", "clean_statement": "0. (C. Schoen) Let J be a general abelian surface (over C, say). Describe the surfaces in J ×J.3\n1", "statement_status": "exact", "statement_verification": "The canonical JSON record reads", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 0\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[72]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"0. (C. Schoen) Let J be a general abelian surface (over C, say). Describe the surfaces in J ×J.3\\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0073", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a very general principally polarized abelian surface J, the codimension-two numerical space of J x J is six-dimensional with basis theta_1^2, theta_1 theta_2, theta_2^2, theta_1 lambda, theta_2 lambda, lambda^2. The known Debarre-Ein-Lazarsfeld-Voisin cone theorem, combined here with an explicit inversion of the intersection pairing, yields an exact pseudoeffectivity certificate from six intersection degrees. In addition, every abelian subgroup surface is the image of x mapping to (mx,nx) for a primitive integer pair and has class one half of (n^2 theta_1 + m^2 theta_2 - mn lambda)^2. These results solve the Hodge/numerical and pseudoeffective-cone readings but not the classification of individual embedded or Chow-theoretic surfaces.\n\nCandidate contribution (criterion; novelty confidence low): Given the six intersections X_1=alpha theta_1^2, X_2=alpha theta_1 theta_2, X_3=alpha theta_2^2, X_4=alpha theta_1 lambda, X_5=alpha theta_2 lambda, and X_6=alpha lambda^2, the coefficients of alpha are recovered explicitly, and alpha is pseudoeffective if and only if 6X_2+X_6 is at least 2|X_2+X_6| and the explicit scaled 3-by-3 matrix in equations (5.8) of report.md is positive semidefinite." }, { "id": 20000422, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0074", "title": "Arithmetic descent of a rank-two curve factorization", "statement": "1. (J. Getz) C. Simpson proved that if X is a variety and if ρX: πgeom\n\n> 1\n\n(X) → GL 2(Q`) is a representation whose algebraic enveloping contains SL 2, then ρX factors through ρY:\n\nπgeom\n\n> 1\n\n(Y ) → GL 2(Q`), where Y is either a curve or a Shimura variety; the factorization is induced from a map from X to some cover of Y. Is there a version of this result using just\n\nπ1 in place of πgeom\n\n> 1?\n1", "original_statement": "1. (J. Getz) C. Simpson proved that if X is a variety and if ρX: πgeom \n\n> 1\n\n(X) → GL 2(Q`) is a representation whose algebraic enveloping contains SL 2, then ρX factors through ρY:\n\nπgeom \n\n> 1\n\n(Y ) → GL 2(Q`), where Y is either a curve or a Shimura variety; the factorization is induced from a map from X to some cover of Y. Is there a version of this result using just \n\nπ1 in place of πgeom \n\n> 1?\n1", "clean_statement": "1. (J. Getz) C. Simpson proved that if X is a variety and if ρX: πgeom\n\n> 1\n\n(X) → GL 2(Q`) is a representation whose algebraic enveloping contains SL 2, then ρX factors through ρY:\n\nπgeom\n\n> 1\n\n(Y ) → GL 2(Q`), where Y is either a curve or a Shimura variety; the factorization is induced from a map from X to some cover of Y. Is there a version of this result using just\n\nπ1 in place of πgeom\n\n> 1?\n1", "statement_status": "exact", "statement_verification": "The canonical JSON record is an OCR-damaged extraction from the five-page list *Problems from the AIM Tate Conjecture Workshop* (July 23--27, 2007), transcribed by Christopher Lyons. The PDF shows that this is **Problem 11**, not Problem 1. The recovered text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[73]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (J. Getz) C. Simpson proved that if X is a variety and if ρX: πgeom \\n\\n> 1\\n\\n(X) → GL 2(Q`) is a representation whose algebraic enveloping contains SL 2, then ρX factors through ρY:\\n\\nπgeom \\n\\n> 1\\n\\n(Y ) → GL 2(Q`), where Y is either a curve or a Shimura variety; the factorization is induced from a map from X to some cover of Y. Is there a version of this result using just \\n\\nπ1 in place of πgeom \\n\\n> 1?\\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0074", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Assume a continuous representation of the full arithmetic etale fundamental group has, on the geometric subgroup, an algebraic dominant factorization through an honest smooth curve whose map on geometric etale fundamental groups is surjective. After any finite extension over which the map is defined, the full representation factors uniquely through the arithmetic fundamental group of that curve. For a projective curve target, the Galois orbit of the factorization class determines its field of definition; in particular, a unique normalized geometric factorization descends over the original field.\n\nCandidate contribution (descent theorem; novelty confidence low): A normalized projective geometric curve factorization of the restriction of a full arithmetic representation lifts over its field of moduli; the required field extension has degree equal to the Galois-orbit size, and uniqueness of the geometric factorization implies arithmetic factorization over the original field." }, { "id": 20000423, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0075", "title": "A dimension-two counterexample with an explicit Galois projector", "statement": "2. (W. Raskind) Take an abelian variety over a finite extension K of Qp with multiplicative reduction. Show the isomorphism End( A) ⊗ Qp ˜−→ End( Vp(A)) GK.\n\nComments: 1) This implies the analogous statement for A over K, with K global, having multiplica-tive reduction somewhere. (W. Raskind) 2) This should be doable by imitating methods of Serre for the case when A is an elliptic curve. (W. Raskind) 3) This is known for Drinfel'd modular varieties (done by Ito). (W. Raskind)\n1", "original_statement": "2. (W. Raskind) Take an abelian variety over a finite extension K of Qp with multiplicative reduction. Show the isomorphism End( A) ⊗ Qp ˜−→ End( Vp(A)) GK.\n\nComments: 1) This implies the analogous statement for A over K, with K global, having multiplica-tive reduction somewhere. (W. Raskind) 2) This should be doable by imitating methods of Serre for the case when A is an elliptic curve. (W. Raskind) 3) This is known for Drinfel'd modular varieties (done by Ito). (W. Raskind) \n1", "clean_statement": "2. (W. Raskind) Take an abelian variety over a finite extension K of Qp with multiplicative reduction. Show the isomorphism End( A) ⊗ Qp ˜−→ End( Vp(A)) GK.\n\nComments: 1) This implies the analogous statement for A over K, with K global, having multiplica-tive reduction somewhere. (W. Raskind) 2) This should be doable by imitating methods of Serre for the case when A is an elliptic curve. (W. Raskind) 3) This is known for Drinfel'd modular varieties (done by Ito). (W. Raskind)\n1", "statement_status": "exact", "statement_verification": "The canonical JSON record has lost the leading digit in the problem number. The official AIM workshop PDF, *Problems from the AIM Tate Conjecture Workshop* (July 23--27, 2007, transcribed by Christopher Lyons), states this as item **12**, not item 2:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[74]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (W. Raskind) Take an abelian variety over a finite extension K of Qp with multiplicative reduction. Show the isomorphism End( A) ⊗ Qp ˜−→ End( Vp(A)) GK.\\n\\nComments: 1) This implies the analogous statement for A over K, with K global, having multiplica-tive reduction somewhere. (W. Raskind) 2) This should be doable by imitating methods of Serre for the case when A is an elliptic curve. (W. Raskind) 3) This is known for Drinfel'd modular varieties (done by Ito). (W. Raskind) \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0075", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Gregory and Liedtke construct, for every prime p >= 5 with p congruent to 1 modulo 3, an abelian surface B over K = Q_p with split totally degenerate reduction such that End_K(B) tensor Q_p maps non-surjectively to End_{G_K}(V_p(B)). Their period calculation gives a one-dimensional algebraic source and a two-dimensional Galois centralizer, so the failure persists after rationalization. An explicit nonscalar idempotent in the target is isolated, and an extension-class argument shows that the map is an isomorphism for multiplicative elliptic curves, making dimension two sharp.\n\nCandidate contribution (lemma; novelty confidence low): For the Gregory-Liedtke surface, e = [[a^2, ab], [ab, b^2]] is a nontrivial G_K-equivariant idempotent not arising from End_K(B) tensor Q_p; hence B is K-isogeny-simple while V_p(B) is G_K-decomposable, and dimension two is the first possible failure dimension." }, { "id": 20000424, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0076", "title": "Uniform edge-pole bounds under base extension", "statement": "3. (K. Murty) According to the Tate Conjecture, if one takes the L-function of a variety X\n\nover K at the edge of the critical strip, and then increases K, the order of the pole should stay bounded. Can we prove this same statement without assuming the Tate Conjecture? Comments: 1) This is provable over function fields of finite fields since, by using the work of Lafforgue, we have (T3): the order of the pole equals the rank of the Tate classes. It is also true when X is an abelian variety in characteristic", "original_statement": "3. (K. Murty) According to the Tate Conjecture, if one takes the L-function of a variety X\n\nover K at the edge of the critical strip, and then increases K, the order of the pole should stay bounded. Can we prove this same statement without assuming the Tate Conjecture? Comments: 1) This is provable over function fields of finite fields since, by using the work of Lafforgue, we have (T3): the order of the pole equals the rank of the Tate classes. It is also true when X is an abelian variety in characteristic", "clean_statement": "3. (K. Murty) According to the Tate Conjecture, if one takes the L-function of a variety X\n\nover K at the edge of the critical strip, and then increases K, the order of the pole should stay bounded. Can we prove this same statement without assuming the Tate Conjecture? Comments: 1) This is provable over function fields of finite fields since, by using the work of Lafforgue, we have (T3): the order of the pole equals the rank of the Tate classes. It is also true when X is an abelian variety in characteristic", "statement_status": "exact", "statement_verification": "The canonical JSON record is truncated and misnumbers the item as Problem 3. The official AIM workshop PDF has it as Problem 13 and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[75]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (K. Murty) According to the Tate Conjecture, if one takes the L-function of a variety X\\n\\nover K at the edge of the critical strip, and then increases K, the order of the pole should stay bounded. Can we prove this same statement without assuming the Tate Conjecture? Comments: 1) This is provable over function fields of finite fields since, by using the work of Lafforgue, we have (T3): the order of the pole equals the rank of the Tate classes. It is also true when X is an abelian variety in characteristic\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0076", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a fixed codimension m, normalize W=H^{2m}(X_bar,Q_l)(m), so the edge is s=1 (equivalently s=m+1 before twisting). Over a global function field, the corrected Lyons argument applied after every finite separable extension gives dim W^{Gamma_L} <= -ord_{s=1} L_L^S(W,s) = [1:(W|Gamma_L)^ss] <= dim W, proving the desired uniform bound without the Tate conjecture. More generally, an exact Artin signed-order formula reduces base extension to finite-image twists; unitary isobaric automorphy gives the sharp bound dim W, and finite monodromy gives unconditionally -ord_{s=1} L_L^S(W,s)=dim W^{Gamma_L} for every finite extension via Brauer induction and Hecke L-function nonvanishing.\n\nCandidate contribution (theorem; novelty confidence low): For every finite separable extension L/k of a global function field, Lyons's corrected proof yields the explicit uniform sandwich dim W^{Gamma_L} <= -ord_{s=1}L_L^S(W,s) = [1:(W|Gamma_L)^ss] <= dim W; thus Murty's boundedness follows even though equality with the invariant/Tate rank is unavailable without semisimplicity." }, { "id": 20000425, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0077", "title": "A semisimple-unipotent reduction for the torsion field of J(X(ell))", "statement": "4. (C. Hall) Let X(`) be a modular curve. What can we say about the splitting field of the\n\n`-torsion of the Jacobian J(X(`))? Comments: 1) This is needed to speak about the L-function of E → X(`) (mod `) ∈ F`[T ]. (C. Hall)\n1", "original_statement": "4. (C. Hall) Let X(`) be a modular curve. What can we say about the splitting field of the \n\n`-torsion of the Jacobian J(X(`))? Comments: 1) This is needed to speak about the L-function of E → X(`) (mod `) ∈ F`[T ]. (C. Hall) \n1", "clean_statement": "4. (C. Hall) Let X(`) be a modular curve. What can we say about the splitting field of the\n\n`-torsion of the Jacobian J(X(`))? Comments: 1) This is needed to speak about the L-function of E → X(`) (mod `) ∈ F`[T ]. (C. Hall)\n1", "statement_status": "exact", "statement_verification": "The canonical record is source index 76 of aim-algebraic-number-theory-notes.json. Its OCR text has replaced every occurrence of \\(\\ell\\) by a backtick, has collapsed a line break, and labels the question “4.” The official AIM workshop PDF shows that it is actually item 14:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[76]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"4. (C. Hall) Let X(`) be a modular curve. What can we say about the splitting field of the \\n\\n`-torsion of the Jacobian J(X(`))? Comments: 1) This is needed to speak about the L-function of E → X(`) (mod `) ∈ F`[T ]. (C. Hall) \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0077", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the standard principal modular curve X(ell) over K_ell = Q(zeta_ell), the field L_ell = K_ell(J_ell[ell]) is unramified outside ell and has Galois group in Sp_{2g}(F_ell). The field cut out by the semisimplification gives a canonical tower K_ell subset L_ell^ss subset L_ell in which Gal(L_ell/L_ell^ss) is an ell-group of order at most ell^{g^2}. Moreover, L_ell/K_ell is an ell-extension if and only if J_ell[ell]^ss is trivial, equivalently if and only if every good local Frobenius polynomial (or a density-one set of them) is (1-T)^{2g} modulo ell. In general the degree divides ell^{g^2} times the product of ell^{2i}-1 for 1 <= i <= g; for ell = 2, 3, 5 the Jacobian is zero and the field is the base field.\n\nCandidate contribution (reduction; novelty confidence low): The candidate contribution is the problem-specific canonical semisimple tower together with the proved ell-group and sharp Sylow bound for its upper layer, the exact semisimplification/Frobenius criterion for when the entire torsion field is an ell-extension, and the observation that local characteristic polynomials cannot recover the upper-triangular extension-class layer." }, { "id": 20000426, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0078", "title": "Two explicit new divisor classes on the level-4 surface modulo 3", "statement": "5. (J. Ellenberg, D. Ulmer) Let X be a modular curve, which paramterizes elliptic curves with some level structure, and let E be the universal family of elliptic curves over X. This defines an elliptic surface, typically over Q. It often happens that, for many primes p, the reduction modulo p of this surface acquires new Tate classes, not coming from characteristic zero. Can we find algebraic cycles (living in characteristic p), which account for such Tate classes?\n1", "original_statement": "5. (J. Ellenberg, D. Ulmer) Let X be a modular curve, which paramterizes elliptic curves with some level structure, and let E be the universal family of elliptic curves over X. This defines an elliptic surface, typically over Q. It often happens that, for many primes p, the reduction modulo p of this surface acquires new Tate classes, not coming from characteristic zero. Can we find algebraic cycles (living in characteristic p), which account for such Tate classes? \n1", "clean_statement": "5. (J. Ellenberg, D. Ulmer) Let X be a modular curve, which paramterizes elliptic curves with some level structure, and let E be the universal family of elliptic curves over X. This defines an elliptic surface, typically over Q. It often happens that, for many primes p, the reduction modulo p of this surface acquires new Tate classes, not coming from characteristic zero. Can we find algebraic cycles (living in characteristic p), which account for such Tate classes?\n1", "statement_status": "exact", "statement_verification": "The canonical record is source index 77 of “aim-algebraic-number-theory-notes.json.” Its OCR text labels the question as “5,” misspells “parametrizes” as “paramterizes,” and appends a stray page marker “1.” The cited AIM PDF is *Problems from the AIM Tate Conjecture Workshop* (July 23–27, 2007), transcribed by Christopher Lyons. In that PDF the record is Problem **15**, not Problem 5. Apart from the extraction artifacts, the statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[77]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"5. (J. Ellenberg, D. Ulmer) Let X be a modular curve, which paramterizes elliptic curves with some level structure, and let E be the universal family of elliptic curves over X. This defines an elliptic surface, typically over Q. It often happens that, for many primes p, the reduction modulo p of this surface acquires new Tate classes, not coming from characteristic zero. Can we find algebraic cycles (living in characteristic p), which account for such Tate classes? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0078", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the full level-4 elliptic modular surface reduced modulo 3 and identified with the Fermat quartic, two explicitly displayed F_9-lines L_2 and L_19 form an integral complement to the characteristic-zero specialization lattice: NS(X_3) = rho(NS(X_0)) direct-sum Z[L_2] direct-sum Z[L_19]. Thus their cycle classes account for the entire two-dimensional geometric Tate-class jump. Frobenius fixes one quotient direction over F_3; the integral orbit sum L_2 + L_5 is an F_3-defined algebraic cycle spanning that new rational invariant direction, while both directions are Tate over F_9.\n\nCandidate contribution (explicit cycles; novelty confidence low): In Shimada's published integral bases, adjoining the two named Fermat-quartic lines L_2: x_1+(-1+i)x_4=x_2+(-1-i)x_3=0 and L_19: x_1+x_3+x_4=x_2+i x_3-i x_4=0 to the 20 specialization rows gives determinant -1, so these two lines are a minimal unimodular integral complement; their quotient Frobenius matrix is [[1,1],[0,-1]]." }, { "id": 20000427, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0079", "title": "An explicit CM projector and Kummer correspondence", "statement": "6. (D. Ramakrishnan) Let E be an elliptic curve over Q with CM by K and with associated Hecke character χ. Then χ2 is associated to a modular form of weight 3 appearing in\n\nH2(E), E → X for a modular curve X (whose level is the norm of the conductor of χ2\n\ntimes the discriminant of K). The Tate and Hodge Conjectures predict the existence of a correspondence E × E → E because the same motive appears in H2 of both sides. Can we exhibit this correspondence explicitly? Comments: 41) In Shioda's example (level 4), E is K3, and in fact the Kummer surface associated to\n\nE × E. (D. Ramakrishnan) 2) This problem is closely related to (the comments from) the previous problem because, when E has supersingular reduction, there are many extra cycles on E × E. (C. Schoen)\n1", "original_statement": "6. (D. Ramakrishnan) Let E be an elliptic curve over Q with CM by K and with associated Hecke character χ. Then χ2 is associated to a modular form of weight 3 appearing in \n\nH2(E), E → X for a modular curve X (whose level is the norm of the conductor of χ2\n\ntimes the discriminant of K). The Tate and Hodge Conjectures predict the existence of a correspondence E × E → E because the same motive appears in H2 of both sides. Can we exhibit this correspondence explicitly? Comments: 41) In Shioda's example (level 4), E is K3, and in fact the Kummer surface associated to \n\nE × E. (D. Ramakrishnan) 2) This problem is closely related to (the comments from) the previous problem because, when E has supersingular reduction, there are many extra cycles on E × E. (C. Schoen) \n1", "clean_statement": "6. (D. Ramakrishnan) Let E be an elliptic curve over Q with CM by K and with associated Hecke character χ. Then χ2 is associated to a modular form of weight 3 appearing in\n\nH2(E), E → X for a modular curve X (whose level is the norm of the conductor of χ2\n\ntimes the discriminant of K). The Tate and Hodge Conjectures predict the existence of a correspondence E × E → E because the same motive appears in H2 of both sides. Can we exhibit this correspondence explicitly? Comments: 41) In Shioda's example (level 4), E is K3, and in fact the Kummer surface associated to\n\nE × E. (D. Ramakrishnan) 2) This problem is closely related to (the comments from) the previous problem because, when E has supersingular reduction, there are many extra cycles on E × E. (C. Schoen)\n1", "statement_status": "exact", "statement_verification": "The canonical JSON record is damaged by PDF extraction: it labels the item as “6,” writes both the elliptic curve and the elliptic modular surface as `E`, drops superscripts in \\(\\chi^2\\) and \\(H^2\\), and reverses/obscures some layout. The official AIM TeX source identifies it as **Problem 16** and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[78]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"6. (D. Ramakrishnan) Let E be an elliptic curve over Q with CM by K and with associated Hecke character χ. Then χ2 is associated to a modular form of weight 3 appearing in \\n\\nH2(E), E → X for a modular curve X (whose level is the norm of the conductor of χ2\\n\\ntimes the discriminant of K). The Tate and Hodge Conjectures predict the existence of a correspondence E × E → E because the same motive appears in H2 of both sides. Can we exhibit this correspondence explicitly? Comments: 41) In Shioda's example (level 4), E is K3, and in fact the Kummer surface associated to \\n\\nE × E. (D. Ramakrishnan) 2) This problem is closely related to (the comments from) the previous problem because, when E has supersingular reduction, there are many extra cycles on E × E. (C. Schoen) \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0079", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every CM elliptic curve E over Q, a trace-zero CM endomorphism gives an explicit rational Chow-idempotent on E squared whose realizations are exactly the chi-squared and conjugate-chi-squared rank-two summand. The resolved degree-two Kummer quotient, normalized by one half, then gives mutually inverse Chow correspondences between this summand and the transcendental summand of Km(E squared). This supplies the requested correspondence in the known level-4 Kummer case. For an arbitrary elliptic modular surface, the work only reduces the problem to constructing a nonzero projected cycle e_f composed with zeta composed with p_T.\n\nCandidate contribution (explicit correspondence; novelty confidence low): If delta is a nonzero trace-zero CM endomorphism, n is its degree, a is delta times delta, and pi_2 is the degree-two Chow-Kunneth projector, then p_T = (pi_2 - pi_2 composed with transpose(Graph(a)) composed with pi_2 divided by n)/2 is a Chow-idempotent descending to Q; composing it with the resolved Kummer graph gamma gives Z = gamma composed with p_T and W = p_T composed with transpose(gamma)/2, satisfying W composed with Z = p_T and Z composed with W = p_S exactly in rational Chow motives." }, { "id": 20000428, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0080", "title": "A classwise CM implication and a one-product coniveau certificate", "statement": "7. (W. Raskind) In the situations where the Hodge Conjecture implies the Tate Conjecture, is it also true that the generalized Hodge Conjecture implies the generalized Tate Conjecture, e.g., for CM abelian varieties?\n1", "original_statement": "7. (W. Raskind) In the situations where the Hodge Conjecture implies the Tate Conjecture, is it also true that the generalized Hodge Conjecture implies the generalized Tate Conjecture, e.g., for CM abelian varieties? \n1", "clean_statement": "7. (W. Raskind) In the situations where the Hodge Conjecture implies the Tate Conjecture, is it also true that the generalized Hodge Conjecture implies the generalized Tate Conjecture, e.g., for CM abelian varieties?\n1", "statement_status": "exact", "statement_verification": "The canonical record is source index 79 of `aim-algebraic-number-theory-notes.json`. Its OCR text reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[79]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"7. (W. Raskind) In the situations where the Hodge Conjecture implies the Tate Conjecture, is it also true that the generalized Hodge Conjecture implies the generalized Tate Conjecture, e.g., for CM abelian varieties? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0080", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the finite-field interpretation intended by the AIM workshop, the CM example has a proved affirmative conditional answer with exact classwise quantifiers: generalized Hodge for every complex CM abelian variety implies ordinary Hodge for that entire class; Milne then gives ordinary Tate for every abelian variety over finite fields; and Milne--Ramachandran upgrade this to generalized Tate for every such abelian variety. Ordinary Hodge for all complex CM abelian varieties already suffices. The argument is not objectwise and does not make generalized Tate unconditional.\n\nCandidate contribution (reduction; novelty confidence low): For a fixed finite-field abelian tuple (X,i,r,ell), realize the effective semisimple Tate structure F_b^r H_ell^i(X)(r) in H_ell^{i-2r}(B) for one auxiliary abelian variety B via Honda--Tate; surjectivity of the single ordinary Tate cycle map in codimension dim(B)+r on B x X then forces F_b^r H_ell^i(X) to have support in codimension r. Such support descends from finite extensions without codimension loss, while X_r=E x (P^1)^r gives a sharp odd-degree witness showing why invariant Tate classes on X alone cannot replace the auxiliary product.", "research_difficulty_suggested": "L4" }, { "id": 20000429, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0081", "title": "Integral cycle classes on the theta line", "statement": "8. (E. Izadi) Take a principally polarized abelian variety A and assume rank( N S (A)) = 1. What is the index of Ci(A) in H2i(A, Z(i))? What is the image? What if we just restrict ourselves to the line spanned by [ θ]i, where θ is the polarization?\n1", "original_statement": "8. (E. Izadi) Take a principally polarized abelian variety A and assume rank( N S (A)) = 1. What is the index of Ci(A) in H2i(A, Z(i))? What is the image? What if we just restrict ourselves to the line spanned by [ θ]i, where θ is the polarization? \n1", "clean_statement": "8. (E. Izadi) Take a principally polarized abelian variety A and assume rank( N S (A)) = 1. What is the index of Ci(A) in H2i(A, Z(i))? What is the image? What if we just restrict ourselves to the line spanned by [ θ]i, where θ is the polarization?\n1", "statement_status": "exact", "statement_verification": "The canonical JSON record is an OCR extraction from the problem list of the AIM workshop *The Tate conjecture* (July 23--27, 2007). The OCR calls this Problem 8 and suppresses superscripts. Inspection of page 5 of the official PDF shows that it is actually Problem 18 and that the target is literally all integral cohomology, not the subgroup of integral Hodge classes:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[80]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"8. (E. Izadi) Take a principally polarized abelian variety A and assume rank( N S (A)) = 1. What is the index of Ci(A) in H2i(A, Z(i))? What is the image? What if we just restrict ourselves to the line spanned by [ θ]i, where θ is the polarization? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0081", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a complex principally polarized abelian g-fold, the literal image C^i(A) has infinite index in all of H^{2i}(A,Z(i)) for 0=g." }, { "id": 20000430, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0082", "title": "Uniform Frobenius exponents and all-good-fiber algebraicity cases", "statement": "9. (D. Ramakrishnan) The following question is due to P. Deligne. Let X be a variety defined over a (finitely-generated) field of characteristic 0. Given a Tate class on c on X, is there a prime p such that, upon reducing X mod p, the reduction of c is algebraic?\n2", "original_statement": "9. (D. Ramakrishnan) The following question is due to P. Deligne. Let X be a variety defined over a (finitely-generated) field of characteristic 0. Given a Tate class on c on X, is there a prime p such that, upon reducing X mod p, the reduction of c is algebraic? \n2", "clean_statement": "9. (D. Ramakrishnan) The following question is due to P. Deligne. Let X be a variety defined over a (finitely-generated) field of characteristic 0. Given a Tate class on c on X, is there a prime p such that, upon reducing X mod p, the reduction of c is algebraic?\n2", "statement_status": "exact", "statement_verification": "The canonical record is source index 81 of `aim-algebraic-number-theory-notes.json`. The official AIM PDF contains the record as item **19**, whereas the canonical extraction labels it item 9. The PDF literally reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[81]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"9. (D. Ramakrishnan) The following question is due to P. Deligne. Let X be a variety defined over a (finitely-generated) field of characteristic 0. Given a Tate class on c on X, is there a prime p such that, upon reducing X mod p, the reduction of c is algebraic? \\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0082", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any finite-dimensional space W of Tate classes on a smooth projective variety over a finitely generated characteristic-zero field, a normal-core construction gives a single finite Galois-quotient exponent N such that Frobenius to the Nth power fixes every specialization at every good closed point; surjectivity of the Tate cycle map over the degree-N residue extension then algebraizes all of W simultaneously. Consequently, every Tate class specializes algebraically at every good fiber for abelian varieties of dimension at most three, at every odd-characteristic good fiber for K3 surfaces, and at every good fiber of residue characteristic p>3 for cubic fourfolds. The general problem remains open.\n\nCandidate contribution (reduction; novelty confidence low): A finite-dimensional space W of characteristic-zero Tate classes admits one Galois-orbit exponent N that fixes all good specializations under Frobenius to the Nth power, so one Tate-valid fiber over the degree-N residue extension algebraizes W simultaneously; the associated Tate-defect quotient, Picard-jump obstruction, and nonsplit-quadric example make the certificate testable and show that a nontrivial residue extension can be necessary for field-of-definition statements." }, { "id": 20000431, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0083", "title": "A necessary Bockstein fingerprint for a bounded Weil–étale topology on Spec Z", "statement": "0. (M. Flach) Find a natural definition of the Weil-´ etale topology so that the cohomology of\n\nZ vanishes in degrees greater than 3. Comments: 1) The current definition only vanishes for odd degrees and gives infinitely generated groups in even degrees. (M. Flach) 2) It it not clear how hard this problem is. A better definition might be just around the corner or the only solution might be to truncate the cohomology complex. (M. Flach)\n2", "original_statement": "0. (M. Flach) Find a natural definition of the Weil-´ etale topology so that the cohomology of \n\nZ vanishes in degrees greater than 3. Comments: 1) The current definition only vanishes for odd degrees and gives infinitely generated groups in even degrees. (M. Flach) 2) It it not clear how hard this problem is. A better definition might be just around the corner or the only solution might be to truncate the cohomology complex. (M. Flach) \n2", "clean_statement": "0. (M. Flach) Find a natural definition of the Weil-´ etale topology so that the cohomology of\n\nZ vanishes in degrees greater than 3. Comments: 1) The current definition only vanishes for odd degrees and gives infinitely generated groups in even degrees. (M. Flach) 2) It it not clear how hard this problem is. A better definition might be just around the corner or the only solution might be to truncate the cohomology complex. (M. Flach)\n2", "statement_status": "exact", "statement_verification": "The canonical JSON record is source index 82 of `aim-algebraic-number-theory-notes.json`. Its OCR text loses the problem number, corrupts “étale,” and separates the final \\(Z\\) from the preceding sentence. The official AIM TeX source identifies it as Problem 20 and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 0\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[82]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"0. (M. Flach) Find a natural definition of the Weil-´ etale topology so that the cohomology of \\n\\nZ vanishes in degrees greater than 3. Comments: 1) The current definition only vanishes for odd degrees and gives infinitely generated groups in even degrees. (M. Flach) 2) It it not clear how hard this problem is. A better definition might be just around the corner or the only solution might be to truncate the cohomology complex. (M. Flach) \\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0083", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The requested bounded integral topology remains open. Under the explicit minimal hypotheses that a proposed compactified Grothendieck topos preserves Lichtenbaum's verified degree-0 through degree-3 integral groups for Spec Z (H1), vanishes above degree 3 and in negative degrees (H2), and respects the exact constant-sheaf coefficient sequence for ordinary and compactly supported derived sections (H3), its derived global sections are forced to be Z direct-sum (Z/2)[-3] ordinarily and (Z/2)[-3] with compact support. Consequently H_c^2(-,Z/m) and H_c^3(-,Z/m) are both Z/2 for even m and both vanish for odd m; ordinary cohomology additionally has H^0=Z/m. This is a proved necessary falsification test, not a construction of the missing topology.\n\nCandidate contribution (necessary_condition; novelty confidence low): Any bounded Weil–étale topos for Spec Z satisfying H1–H3 must exhibit the parity-sensitive finite-coefficient Bockstein fingerprint: compactly supported Z/m-cohomology is Z/2 in exactly degrees 2 and 3 when m is even, and is zero in all degrees when m is odd; ordinary cohomology has the same fingerprint plus Z/m in degree 0." }, { "id": 20000432, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0084", "title": "Section retractions obstruct the fibre-product line topoi", "statement": "1. (M. Flach) Using the current definition of the Weil-´ etale topology, define the Weil-´ etale topos for an arbitrary arithmetic scheme (i.e., a scheme of finite type over Spec( Z)) as a fibred product. Can one compute the cohomology of the sheaf Z on, say, the affine or projective line? 5", "original_statement": "1. (M. Flach) Using the current definition of the Weil-´ etale topology, define the Weil-´ etale topos for an arbitrary arithmetic scheme (i.e., a scheme of finite type over Spec( Z)) as a fibred product. Can one compute the cohomology of the sheaf Z on, say, the affine or projective line? 5", "clean_statement": "1. (M. Flach) Using the current definition of the Weil-´ etale topology, define the Weil-´ etale topos for an arbitrary arithmetic scheme (i.e., a scheme of finite type over Spec( Z)) as a fibred product. Can one compute the cohomology of the sheaf Z on, say, the affine or projective line? 5", "statement_status": "exact", "statement_verification": "The canonical JSON record is an OCR extraction from page 5 of the AIM workshop list *The Tate conjecture*. The official PDF gives the following as Problem 21 (not Problem 1):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: The Tate conjecture\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/tateconjecture/tateconjecture.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[83]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (M. Flach) Using the current definition of the Weil-´ etale topology, define the Weil-´ etale topos for an arbitrary arithmetic scheme (i.e., a scheme of finite type over Spec( Z)) as a fibred product. Can one compute the cohomology of the sheaf Z on, say, the affine or projective line? 5\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/tateconjecture/tateconjecture.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0084", "aim-domain:algebraic-number-theory", "aim-workshop:tateconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Flach–Morin's 2012 construction defines the requested ordinary and Artin–Verdier compactified 2-fibre-product topoi. For both A^1_Z and P^1_Z, the standard section induces a retraction at the actual 2-fibre-product level, so base cohomology is a derived direct summand; in the compactified theory this forces non-finitely generated H^4. Separately, Flach–Morin's 2018 derived theory gives an unconditional projective-bundle decomposition for P^1_Z with finite H^4, proving that this later derived complex is not the integral constant-sheaf cohomology of the 2012 compactified fibre-product topos.\n\nCandidate contribution (obstruction; novelty confidence low): For the standard sections of A^1_Z and P^1_Z, the universal 2-fibre-product construction functorially retracts onto the arithmetic base; combining the resulting compactified non-finitely-generated degree-four summand with the finite degree-four group in the 2018 derived P^1_Z projective-bundle calculation gives a concrete degree-four non-equivalence between the two constructions." }, { "id": 20000433, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0085", "title": "Height-window bounds and modern status for random hypersurface solubility", "statement": "Question 1. Consider hypersurfaces fd ∈ Z[x0,..., x n] of degree d in Pn, n ≥ 2. Let\n\nN (H) = {fd: max coefficient of fd is ≤ H, fd has a solution in Zn+1 \\ { 0}}.\n\n(a) For d > n + 1, is it true that N (H)/N tot (H) → 0? Does this follow from Lang's conjectures? (Voloch) (b) Instead, look at those with those with points locally everywhere Nloc (Poonen). Is this a positive fraction, i.e. is Nloc (H)/N tot (H) > c?(c) As a special case, if you write down a plane cubic, how likely is it to have a rational point? (Voloch) (d) For cubic surfaces, there are examples where the Hasse principle fails, but maybe for almost all values of a parameter in a family, the Hasse principle holds. We should have asympotically that the N (H)/N tot (H) ∼ Nloc (H)/N tot (H)? (Colliot-Th´ el` ene) For d ≤ n, we hope this will hold in general.", "original_statement": "Question 1. Consider hypersurfaces fd ∈ Z[x0,..., x n] of degree d in Pn, n ≥ 2. Let \n\nN (H) = {fd: max coefficient of fd is ≤ H, fd has a solution in Zn+1 \\ { 0}}.\n\n(a) For d > n + 1, is it true that N (H)/N tot (H) → 0? Does this follow from Lang's conjectures? (Voloch) (b) Instead, look at those with those with points locally everywhere Nloc (Poonen). Is this a positive fraction, i.e. is Nloc (H)/N tot (H) > c?(c) As a special case, if you write down a plane cubic, how likely is it to have a rational point? (Voloch) (d) For cubic surfaces, there are examples where the Hasse principle fails, but maybe for almost all values of a parameter in a family, the Hasse principle holds. We should have asympotically that the N (H)/N tot (H) ∼ Nloc (H)/N tot (H)? (Colliot-Th´ el` ene) For d ≤ n, we hope this will hold in general.", "clean_statement": "Question 1. Consider hypersurfaces fd ∈ Z[x0,..., x n] of degree d in Pn, n ≥ 2. Let\n\nN (H) = {fd: max coefficient of fd is ≤ H, fd has a solution in Zn+1 \\ { 0}}.\n\n(a) For d > n + 1, is it true that N (H)/N tot (H) → 0? Does this follow from Lang's conjectures? (Voloch) (b) Instead, look at those with those with points locally everywhere Nloc (Poonen). Is this a positive fraction, i.e. is Nloc (H)/N tot (H) > c?(c) As a special case, if you write down a plane cubic, how likely is it to have a rational point? (Voloch) (d) For cubic surfaces, there are examples where the Hasse principle fails, but maybe for almost all values of a parameter in a family, the Hasse principle holds. We should have asympotically that the N (H)/N tot (H) ∼ Nloc (H)/N tot (H)? (Colliot-Th´ el` ene) For d ≤ n, we hope this will hold in general.", "statement_status": "exact", "statement_verification": "The source is the AIM workshop list *Rational and integral points on higher dimensional varieties*, workshop held December 11--20, 2002, version dated November 22, 2004, Problem/Question 1 on p. 44. Put", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[84]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 1. Consider hypersurfaces fd ∈ Z[x0,..., x n] of degree d in Pn, n ≥ 2. Let \\n\\nN (H) = {fd: max coefficient of fd is ≤ H, fd has a solution in Zn+1 \\\\ { 0}}.\\n\\n(a) For d > n + 1, is it true that N (H)/N tot (H) → 0? Does this follow from Lang's conjectures? (Voloch) (b) Instead, look at those with those with points locally everywhere Nloc (Poonen). Is this a positive fraction, i.e. is Nloc (H)/N tot (H) > c?(c) As a special case, if you write down a plane cubic, how likely is it to have a rational point? (Voloch) (d) For cubic surfaces, there are examples where the Hasse principle fails, but maybe for almost all values of a parameter in a family, the Hasse principle holds. We should have asympotically that the N (H)/N tot (H) ∼ Nloc (H)/N tot (H)? (Colliot-Th´ el` ene) For d ≤ n, we hope this will hold in general.\"\nOriginal remarks: [\"Remarks. \\n\\n(i) The set of reducible such hypersurfaces are a very small fraction (usually codimension ≥ 2)- they affect this calculation very little (Poonen). (ii) Serre looked at d = 2; here one has the Hasse principle. (Tschinkel) For d = 2, n = 2, the proportion of everywhere locally solvable ones tends to zero; but we should exclude this case because of the codimension 2 condition. (Heath-Brown) Instead, we should restrict to families X → S such that the codimension of reducible fibers is at ≥ 2. (iii) Computational evidence is all over the place, so one must rephrase the question better to get some kind of answer. For example, for cubics, those with prime power discriminant and the general evidence are quite different. (Swinnerton-Dyer) (iv) How is this related to 3-torsion elements in X? (Ellenberg) Update (4/21/04): Poonen and Volloch have tackled this problem: http://www.math.princeton.edu/ ∼\\n\\nhtml/poonen-voloch/random.pdf \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0085", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The four-part AIM question is now sharply separated by regime: Poonen--Voloch prove a positive Euler-product density of everywhere locally soluble hypersurfaces for all n,d >= 2 except (n,d)=(2,2), whose local density is zero; Browning--Le Boudec--Sawin prove equality of global and local densities in the Fano range d <= n except for cubic surfaces; plane cubics have explicit local density about 0.97256 and positive lower proportions both with and without rational points; and cubic surfaces have positive global lower density and explicit local density about 0.999927, but equality remains conjectural. In addition, a proved coefficient-box incidence lemma gives N_{<=T}(H)/N_tot(H) <<_{n,d} S_{n,d}(T)/H + T^{n+1}/H^2, where S_{n,d}(T)=1 if d>n+1, S_{n,d}(T)=log(2T) if d=n+1, and S_{n,d}(T)=T^{n+1-d} if dn+1, d=n+1, or d i\n\nxiy2\n\n> i\n\n= 0 in P3 × P3. Take the height\n\nH(x, y ) = sup |xi|3 · sup |yi|2\n\nand throw out the set where some xi = 0, yi = 0. Can one estimate the counting function? (Peyre)", "original_statement": "Question 4. Consider the hypersurface given by ∑ \n\n> i\n\nxiy2 \n\n> i\n\n= 0 in P3 × P3. Take the height \n\nH(x, y ) = sup |xi|3 · sup |yi|2\n\nand throw out the set where some xi = 0, yi = 0. Can one estimate the counting function? (Peyre)", "clean_statement": "Question 4. Consider the hypersurface given by ∑\n\n> i\n\nxiy2\n\n> i\n\n= 0 in P3 × P3. Take the height\n\nH(x, y ) = sup |xi|3 · sup |yi|2\n\nand throw out the set where some xi = 0, yi = 0. Can one estimate the counting function? (Peyre)", "statement_status": "exact", "statement_verification": "The source is Question 4 in the AIM workshop document *Rational and integral points on higher dimensional varieties* (workshop held December 11--20, 2002; document version November 22, 2004). The OCR in the corpus has separated the summation indices from the formula. The PDF makes the intended statement unambiguous:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[87]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 4. Consider the hypersurface given by ∑ \\n\\n> i\\n\\nxiy2 \\n\\n> i\\n\\n= 0 in P3 × P3. Take the height \\n\\nH(x, y ) = sup |xi|3 · sup |yi|2\\n\\nand throw out the set where some xi = 0, yi = 0. Can one estimate the counting function? (Peyre)\"\nOriginal remarks: [\"Remarks. Some people are working on this. What news? \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0088", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Browning and Heath-Brown's known Peyre asymptotic transfers to the exact AIM coordinate-open locus after the necessary thin deletion, because the remaining coordinate boundary contributes only O(B). Conversely, the explicit split fibre over x* = [1:1:-1:-1] contributes at least ζ(2)^(-2) B log B - O(B) inside the literal AIM open set. Hence the literal count has liminf coefficient at least c_X + ζ(2)^(-2), so Peyre's constant c_X alone cannot be its leading coefficient, although a full asymptotic including all split fibres remains unproved.\n\nCandidate contribution (proposition; novelty confidence low): On the exact AIM coordinate-open locus, the thin-complement coordinate boundary is O(B), while the fibre x* = [1:1:-1:-1] contributes at least ζ(2)^(-2) B log B - O(B), yielding an explicit liminf gap of ζ(2)^(-2) over Peyre's constant." }, { "id": 20000437, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0089", "title": "The solved D4 cubic and an exact deleted-line count", "statement": "Question 5. Consider xyz = t(x + y + z)2, with height function H(x, y, z, t ) = max( |x|, |y|, |z|, |t|), and restrict to gcd( x, y, z, t ) = 1. Excluding trivial solutions (on lines), can you prove a counting function which is\n\n∼ cB (log B)6 < B 1+ ≤?This is a singular cubic surface so we also expect an explicit constant. (Tschinkel)", "original_statement": "Question 5. Consider xyz = t(x + y + z)2, with height function H(x, y, z, t ) = max( |x|, |y|, |z|, |t|), and restrict to gcd( x, y, z, t ) = 1. Excluding trivial solutions (on lines), can you prove a counting function which is \n\n∼ cB (log B)6 < B 1+ ≤?This is a singular cubic surface so we also expect an explicit constant. (Tschinkel)", "clean_statement": "Question 5. Consider xyz = t(x + y + z)2, with height function H(x, y, z, t ) = max( |x|, |y|, |z|, |t|), and restrict to gcd( x, y, z, t ) = 1. Excluding trivial solutions (on lines), can you prove a counting function which is\n\n∼ cB (log B)6 < B 1+ ≤?This is a singular cubic surface so we also expect an explicit constant. (Tschinkel)", "statement_status": "exact", "statement_verification": "The canonical record is Question 5 from the AIM workshop *Rational and integral points on higher dimensional varieties*. Its OCR text is damaged at the exponent and comparison signs. The official AIM HTML and typeset PDF recover the question as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[88]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 5. Consider xyz = t(x + y + z)2, with height function H(x, y, z, t ) = max( |x|, |y|, |z|, |t|), and restrict to gcd( x, y, z, t ) = 1. Excluding trivial solutions (on lines), can you prove a counting function which is \\n\\n∼ cB (log B)6 < B 1+ ≤?This is a singular cubic surface so we also expect an explicit constant. (Tschinkel)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) This has a D4-singular point. On top of the singularity, one gets a configuration of 4 lines all of which have self-intersection 2. (ii) Are numerics possible? (Voloch) 47 \\n\\n(iii) This is a compactification of the affine plane (solve for t), but it is not equivariantly embed-ded. (Tschinkel) (iv) For the singular cubic surface 1 /x + 1 /y + 1 /z + 1 /t = 0, Heath-Brown has established that the counting function has exact order of magnitude B(log B)6. Proceedings of the session in analytic number theory and Diophantine equations, Bonner Math. Schriften 360 (2003). (v) Progress: Using the universal torsor (as calculated by Hassett and Tschinkel) Browning has established that the counting function has exact order of magnitude B(log B)6. See math.AG/0403530, http://front.math.ucdavis.edu/math.NT/0404245 1.(vi) Update (4/21/04) Tim Browning has shown B(logB )6 ø N (B) ø B(logB )6. He makes use of the universal torsor. See [arXiv:math.NT/0404245] (vii) Takloo-Bighash: for w2x+wy 2 +z3 = 0, have an effective lower bound of B log B, and should be able to get B log 2 B by a similar method, but can't push it any further. Expected upper bound in this case is again B(log B)6. Hassett: the universal torsor was in his lecture. (viii) Hassett: there are 13(ish) singular cubics (zero-dimensional in moduli), the list is in a paper, reference available. \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0089", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Le Boudec's published theorem fully solves the recovered AIM problem for the exact split D4 cubic, open subset, and height, with Peyre's explicit constant. In addition, for every real B at least 1 the six deleted lines contain exactly 24 times the sum of phi(n) for n at most floor(B), minus 8, projective points of height at most B, hence (72/pi^2)B^2+O(B log B); signed primitive-vector counting doubles both this boundary count and Le Boudec's interior constant.\n\nCandidate contribution (proposition; novelty confidence low): For every real B at least 1, the six excluded lines on x0(x1+x2+x3)^2=x1x2x3 contain exactly 24 times the sum of phi(n) for n at most floor(B), minus 8, projective rational points of sup-norm height at most B; the signed primitive-vector count is exactly twice this." }, { "id": 20000438, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0090", "title": "Mixed resolution of the quartic K3 questions and an exact 2-adic separation lemma", "statement": "Question\n6. (Swinnerton-Dyer) (a) Is there a K3 surface X over Q which has a finite nonzero number of rational points X(Q) <\n\n∞?(b) Is there a nonsingular quartic surface X with this property? (c) Find a third rational point on X41 + 2 X42 = X43 + 4 X44. (There are only 2 points with height\n\n≤ 215, and other non-public reasons to believe that there are only finitely many points.) (d) Find a smooth quartic X4 ⊂ P3 with Pic X ∼= Z and X(K) infinite.", "original_statement": "Question \n6. (Swinnerton-Dyer) (a) Is there a K3 surface X over Q which has a finite nonzero number of rational points X(Q) <\n\n∞?(b) Is there a nonsingular quartic surface X with this property? (c) Find a third rational point on X41 + 2 X42 = X43 + 4 X44. (There are only 2 points with height \n\n≤ 215, and other non-public reasons to believe that there are only finitely many points.) (d) Find a smooth quartic X4 ⊂ P3 with Pic X ∼= Z and X(K) infinite.", "clean_statement": "Question\n6. (Swinnerton-Dyer) (a) Is there a K3 surface X over Q which has a finite nonzero number of rational points X(Q) <\n\n∞?(b) Is there a nonsingular quartic surface X with this property? (c) Find a third rational point on X41 + 2 X42 = X43 + 4 X44. (There are only 2 points with height\n\n≤ 215, and other non-public reasons to believe that there are only finitely many points.) (d) Find a smooth quartic X4 ⊂ P3 with Pic X ∼= Z and X(K) infinite.", "statement_status": "exact", "statement_verification": "The record is Question 6 (attributed to Swinnerton-Dyer) in the AIM workshop list *Rational and integral points on higher dimensional varieties*, version dated 22 November 2004, pp. 46--47. The PDF was checked directly because the extracted JSON loses superscripts, a cardinality sign, and an overline. The recovered question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[89]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question \\n6. (Swinnerton-Dyer) (a) Is there a K3 surface X over Q which has a finite nonzero number of rational points X(Q) <\\n\\n∞?(b) Is there a nonsingular quartic surface X with this property? (c) Find a third rational point on X41 + 2 X42 = X43 + 4 X44. (There are only 2 points with height \\n\\n≤ 215, and other non-public reasons to believe that there are only finitely many points.) (d) Find a smooth quartic X4 ⊂ P3 with Pic X ∼= Z and X(K) infinite.\"\nOriginal remarks: [\"Remarks. \\n\\n(i) Over Q would be the first place to try, but over any number field is OK. It seems as though this is possible for either all such families or no such family. (ii) There are K3 surfaces with no rational lines with infinitely many rational points. (Peyre) (iii) Do you hope that the Brauer-Manin obstruction is the only one to weak approximation? (Harari) (iv) The rank of the Neron-Severi group over C is rank 20, but is only rank 1 over Q coming from the hyperplane section. Therefore it looks like a Kummer surface, the product of two CM elliptic curves. What are the elliptic curves? (Read the right paper of Shioda.) The CM is by Q(√−2)? (v) It does not seem helpful to look over a finite extension. (Colliot-Th´ el` ene) (vi) Are there any heuristics looking modulo any primes? (Poonen) The zeta function does not say anything about solubility. (Swinnerton-Dyer) (vii) On x4 + y4 + z4 = t4, Elkies found another point, with smallest height on the order of 2 15 \\n\\n(Colliot-Th´ el` ene); he uses a fibered pencil of elliptic curves, looks at values of the parameter for which there was a point everywhere locally and then looked for a global point. There is no such fibration in this case. Maybe one could go to an extension and then look for rational points (Villegas). (viii) What restrictions are necessary for such a surface to occur? (Poonen) The condition to ensure that Pic X ∼= Z is a black-board full. (Swinnerton-Dyer) \\n\\n> 1http://front.math.ucdavis.edu/math.NT/0404245 48\\n\\n(ix) Is it possible for the given surface that (Br X)Gal( Q/Q) is finite? If so, there might be many rational points. (Harari) In the computations of Brauer-Manin obstructions, there are many with ker(Br X → Br X) = Br X, there are a lack of examples with 'transcendental elements'. We expect (Br X)Gal( Q/Q) to be finite, proven in certain cases because of the Tate conjecture. (x) Are there any known examples of quartic surfaces with Pic X ∼= Z with infinitely many rational points? (Poonen) Maybe almost always they have infinitely many. (Swinnerton-Dyer) (xi) If there are infinitely many points on a quartic, will they be Zariski dense? Look in Mordell's book, perhaps. (Colliot-Th´ el` ene) (xii) Silverman has examples of surfaces in P2 × P2 with two noncommuting endomorphisms, so this gives infinitely many points, but this has Picard group rank \\n2. (Voloch) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0090", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Parts (c) and (d) have known affirmative solutions due to Elsenhans--Jahnel and van Luijk, respectively, while parts (a) and (b) apparently remain open. In addition, every primitive integral solution of x^4+2y^4=z^4+4w^4 with (y,w) not both zero satisfies max{v2(x-z),v2(x+z)}=min{4v2(y)+1,4v2(w)+2}-2. This gives a valuation-stratified height bound and yields v2(x+z)=15 for the known non-obvious point.\n\nCandidate contribution (lemma; novelty confidence low): For every primitive integral solution of x^4+2y^4=z^4+4w^4 with (y,w) not both zero, max{v2(x-z),v2(x+z)}=min{4v2(y)+1,4v2(w)+2}-2, with v2(0) interpreted as infinity; consequently H is at least 2 raised to min{4v2(y)+1,4v2(w)+2}-3." }, { "id": 20000439, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0091", "title": "Four-part status and a finite canonical-cover descent audit", "statement": "Question 7. The distribution of rational points on Enriques surfaces has not been well studied. (Skorobogatov) Let X be an Enriques surface. (a) If X(k) 6 = ∅, is X(k) Zariski dense? (b) Is there an X which violates the Hasse principle? (c) Is there an X such that X(Ak)Br = ∅ but X(Ak) 6 = ∅?(d) Is there an X with X(k) = ∅, X(Ak)Br 6 = ∅?", "original_statement": "Question 7. The distribution of rational points on Enriques surfaces has not been well studied. (Skorobogatov) Let X be an Enriques surface. (a) If X(k) 6 = ∅, is X(k) Zariski dense? (b) Is there an X which violates the Hasse principle? (c) Is there an X such that X(Ak)Br = ∅ but X(Ak) 6 = ∅?(d) Is there an X with X(k) = ∅, X(Ak)Br 6 = ∅?", "clean_statement": "Question 7. The distribution of rational points on Enriques surfaces has not been well studied. (Skorobogatov) Let X be an Enriques surface. (a) If X(k) 6 = ∅, is X(k) Zariski dense? (b) Is there an X which violates the Hasse principle? (c) Is there an X such that X(Ak)Br = ∅ but X(Ak) 6 = ∅?(d) Is there an X with X(k) = ∅, X(Ak)Br 6 = ∅?", "statement_status": "exact", "statement_verification": "The canonical record is Question 7 from the AIM workshop *Rational and integral points on higher dimensional varieties*, in **aim-algebraic-number-theory-notes.json**, source index 90. I checked the record against the AIM PDF and the AIM HTML rendering [AIM]. The symbols rendered as “6 =” in the JSON are OCR substitutions for \\(\\ne\\), and the missing spacing in parts (c) and (d) does not change the formulas.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[90]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 7. The distribution of rational points on Enriques surfaces has not been well studied. (Skorobogatov) Let X be an Enriques surface. (a) If X(k) 6 = ∅, is X(k) Zariski dense? (b) Is there an X which violates the Hasse principle? (c) Is there an X such that X(Ak)Br = ∅ but X(Ak) 6 = ∅?(d) Is there an X with X(k) = ∅, X(Ak)Br 6 = ∅?\"\nOriginal remarks: [\"Remark. There is a Z/2Z cover of the Enriques surface X which is a K3 surface; is there some torsor over the K3 surface for the torus for which the total space is a torsor for X\\n\\nunder a nonabelian group? For a bi-elliptic surface, is this possible? (Harari) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0091", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Parts (b), (c), and (d) are affirmatively resolved over Q, whereas original-field density in part (a) remains open. Moreover, for a good finite set S, the canonical-cover formula for the etale-Brauer/descent set reduces to twists in H^1(O_{k,S}, mu_2), whose number is 2^(r_1+r_2+|S_f|)|Cl(O_{k,S})[2]|; a relative-Jacobian translation criterion also gives a proved conditional density case.\n\nCandidate contribution (reduction; novelty confidence low): For an Enriques surface over a number field, the canonical K3-cover descent audit can be restricted to S-unramified quadratic twists, with the explicit count |H^1(O_{k,S}, mu_2)| = 2^(r_1+r_2+|S_f|)|Cl(O_{k,S})[2]|." }, { "id": 20000440, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0092", "title": "A finite binary criterion for Brauer-compatible lifts to an Enriques K3 cover", "statement": "Question 8. Let X an Enriques surface over a number field K, Y p\n\n→ X the double cover (K3 surface), X(AK )Br the Brauer-Manin set. Take mv ∈ X(AK )Br, lift it to an adelic point Pv on Y. Under what assumptions on mv will it be liftable to Pv on Y (AK )Br,or at least Y (AK )Br 1?", "original_statement": "Question 8. Let X an Enriques surface over a number field K, Y p\n\n→ X the double cover (K3 surface), X(AK )Br the Brauer-Manin set. Take mv ∈ X(AK )Br, lift it to an adelic point Pv on Y. Under what assumptions on mv will it be liftable to Pv on Y (AK )Br,or at least Y (AK )Br 1?", "clean_statement": "Question 8. Let X an Enriques surface over a number field K, Y p\n\n→ X the double cover (K3 surface), X(AK )Br the Brauer-Manin set. Take mv ∈ X(AK )Br, lift it to an adelic point Pv on Y. Under what assumptions on mv will it be liftable to Pv on Y (AK )Br,or at least Y (AK )Br 1?", "statement_status": "exact", "statement_verification": "This record is Question 8 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The canonical JSON extraction has lost an arrow, spaces, and some superscripts. Comparison with the official AIM PDF and HTML version gives the following mathematical statement (the bracketed word only repairs the grammar of the source):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[91]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 8. Let X an Enriques surface over a number field K, Y p\\n\\n→ X the double cover (K3 surface), X(AK )Br the Brauer-Manin set. Take mv ∈ X(AK )Br, lift it to an adelic point Pv on Y. Under what assumptions on mv will it be liftable to Pv on Y (AK )Br,or at least Y (AK )Br 1?\"\nOriginal remarks: [\"Remark. Guess (reported by Harari): there should be some nonabelian torsor Z φ\\n\\n→ X for the group G which is the semidirect product T o Z /2, where T is Neron-Severi torus of Y.The points p(Y (AK )Br ) should correspond to X(AK )f, where f: Z → X.\", \"Remark. About Pb/question 8 (which is closely related to Pb 7), David Harari adds the fol-lowing update (2004-09-25): \\\"Skorobogatov and myself have recently proved that there exist Enriques surfaces X with adelic points in X(Ak)Br but not in the closure of the set of rational points X(k) (\\\"The Manin obstruction to weak approximation is not the only one\\\"). In partic-ular, some adelic points of X(Ak)Br are not liftable to Y (Ak)Br 1, see the paper \\\"Non-abelian descent and the arithmetic of Enriques surfaces\\\" at http://www.dma.ens.fr/˜harari/ 2.\\\" \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0092", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let p:Y→X be the étale K3 double cover of an Enriques surface over a number field, let m∈X(A_K)^Br be locally liftable to the fixed cover, and choose a base adelic lift. For A=Br_1(Y) or Br(Y), quotient A by constants and pullbacks from Br(X). The Brauer evaluation functionals of all adelic lifts are exactly a finite affine binary set c+{Σ_{v∈S} ε_v d_v}; hence a Brauer-compatible lift exists if and only if this set contains zero. In particular, surjectivity of the relevant Brauer pullback modulo constants makes every existing adelic lift Brauer-compatible. Harari–Skorobogatov's example shows that no unconditional implication holds.\n\nCandidate contribution (criterion; novelty confidence low): The explicit finite affine sign-vector description c+{Σ_{v∈S} ε_v d_v} of the Brauer evaluation functionals attained by all lifts, and its zero-membership test." }, { "id": 20000441, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0093", "title": "Rigid Calabi--Yau intermediate Jacobians: periods, cycle-theoretic separation, and symmetry certificates", "statement": "Question 9. Use the intermediate Jacobian (when it is an abelian variety) in arithmetic? To fix ideas, dim X = 3, look at algebraic 1-cycles modulo rational equivalence. Compute this for a rigid Calabi-Yau 3-fold X (over C), i.e. KX ∼ 0, X simply con-nected, h1,0 = 0, h2,0 = 0, h2,1 = 0, h3,0 = 1. In this case, the intermediate Jacobian\n\nJ2(X) ∼= E is of dimension 1. Determine E given X, i.e. give its j-invariant.", "original_statement": "Question 9. Use the intermediate Jacobian (when it is an abelian variety) in arithmetic? To fix ideas, dim X = 3, look at algebraic 1-cycles modulo rational equivalence. Compute this for a rigid Calabi-Yau 3-fold X (over C), i.e. KX ∼ 0, X simply con-nected, h1,0 = 0, h2,0 = 0, h2,1 = 0, h3,0 = 1. In this case, the intermediate Jacobian \n\nJ2(X) ∼= E is of dimension 1. Determine E given X, i.e. give its j-invariant.", "clean_statement": "Question 9. Use the intermediate Jacobian (when it is an abelian variety) in arithmetic? To fix ideas, dim X = 3, look at algebraic 1-cycles modulo rational equivalence. Compute this for a rigid Calabi-Yau 3-fold X (over C), i.e. KX ∼ 0, X simply con-nected, h1,0 = 0, h2,0 = 0, h2,1 = 0, h3,0 = 1. In this case, the intermediate Jacobian\n\nJ2(X) ∼= E is of dimension 1. Determine E given X, i.e. give its j-invariant.", "statement_status": "exact", "statement_verification": "The canonical record is Question 9 in the AIM workshop list *Rational and integral points on higher dimensional varieties*. The source PDF was checked directly (version dated 22 November 2004). With superscripts and line breaks restored, the question reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[92]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 9. Use the intermediate Jacobian (when it is an abelian variety) in arithmetic? To fix ideas, dim X = 3, look at algebraic 1-cycles modulo rational equivalence. Compute this for a rigid Calabi-Yau 3-fold X (over C), i.e. KX ∼ 0, X simply con-nected, h1,0 = 0, h2,0 = 0, h2,1 = 0, h3,0 = 1. In this case, the intermediate Jacobian \\n\\nJ2(X) ∼= E is of dimension 1. Determine E given X, i.e. give its j-invariant.\"\nOriginal remarks: [\"Remarks. \\n\\n> 2http://www.dma.ens.fr/ ∼harari/ 49\\n\\n(i) For two quadratic forms f1, f 2 in 6 variables, i.e. X2,2 ⊂ P5, look at the Jacobian of the genus 2 curve given by y2 = det( λf 1 + μf 2). (Colliot-Th´ el` ene) Over Fp, you can prove the Weil conjecture for X2,2. Can you use this to prove something? You can also look at the variety of lines on X2,2, also a principal homogeneous space for an abelian variety; so over a finite field, this will have a rational point, so there will be a line over Fp. We can say something with X2,2 contains a pair of skew conjugate lines or a conic defined over the ground field; how can you do these things such as finding a line over a quadratic field...? (ii) Explicit examples of rigid Calabi-Yau 3-folds? Take an elliptic curve E with complex mul-tiplication by Q(ζ3), take the kernel of the endomorphism ζ3 − 1, T1; T1 × T2 × T3 has 27 singular points, blowing up these points gives h2,1 = 0 (Candela). Also the quintic hyper-surface ∑5 \\n\\n> i=1\\n\\nx5 \\n\\n> i\\n\\n− 5ψ ∏5 \\n\\n> i=1\\n\\nxi = 0 with ψ5 = 1 has 125 nodes; the resolution has h2,1 = 0. There are more such examples. (Yui) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0093", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every smooth projective rigid Calabi--Yau threefold, the Griffiths intermediate Jacobian is the elliptic curve obtained from the primitive rank-two integral H^3 lattice by the exact period formula j(J^2(X)) = j(Q(beta,Omega)/Q(alpha,Omega)), while its algebraic Abel--Jacobi part J_a^3(X) is zero. An actual lattice-preserving automorphism acting on H^{3,0} by a primitive third or sixth root forces j=0, and a primitive fourth root forces j=1728. This gives j=0 for the standard 27-point CM quotient and proves that the visible fifth-root symmetry of the Schoen quintic cannot determine its j-invariant; the latter still requires a primitive integral lattice comparison.\n\nCandidate contribution (proposition; novelty confidence low): Candidate arithmetic-separation and symmetry diagnostic: for a rigid Calabi--Yau threefold, J_a^3(X)=0 although J^2(X) has dimension one, so canonical descent of the algebraic Abel--Jacobi image is trivial; nevertheless, a genuine order-three, order-four, or order-six scalar action on the holomorphic volume form certifies j=0 or 1728, whereas a primitive order-five scalar action is impossible on rank-two rational H^3." }, { "id": 20000442, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0094", "title": "Local R-equivalence and a geometric Brauer obstruction to eventual universal R-triviality", "statement": "Question 10. Are there results at the level of number fields arising from the techniques of rationally connected varieties? (Colliot-Th´ el` ene) For example, recently Koll` ar got nice results over local fields: if X/k is a projective variety, we say P, Q ∈ X(k) are R-equivalent if you can link them by a chain of curves of genus zero over k; if X is smooth with k ↪ → C, then X is rationally connected if X(C)/R is a single point. Kollar proved that if k is a local field, and X/k is rationally connected, then\n\nX(k)/R is finite. (Szob´ o) Koll` ar and Szabo proved that if k is a number field, and X/k is rationally connected, then X(kv)/R is trivial for almost all v.", "original_statement": "Question 10. Are there results at the level of number fields arising from the techniques of rationally connected varieties? (Colliot-Th´ el` ene) For example, recently Koll` ar got nice results over local fields: if X/k is a projective variety, we say P, Q ∈ X(k) are R-equivalent if you can link them by a chain of curves of genus zero over k; if X is smooth with k ↪ → C, then X is rationally connected if X(C)/R is a single point. Kollar proved that if k is a local field, and X/k is rationally connected, then \n\nX(k)/R is finite. (Szob´ o) Koll` ar and Szabo proved that if k is a number field, and X/k is rationally connected, then X(kv)/R is trivial for almost all v.", "clean_statement": "Question 10. Are there results at the level of number fields arising from the techniques of rationally connected varieties? (Colliot-Th´ el` ene) For example, recently Koll` ar got nice results over local fields: if X/k is a projective variety, we say P, Q ∈ X(k) are R-equivalent if you can link them by a chain of curves of genus zero over k; if X is smooth with k ↪ → C, then X is rationally connected if X(C)/R is a single point. Kollar proved that if k is a local field, and X/k is rationally connected, then\n\nX(k)/R is finite. (Szob´ o) Koll` ar and Szabo proved that if k is a number field, and X/k is rationally connected, then X(kv)/R is trivial for almost all v.", "statement_status": "exact", "statement_verification": "The canonical record is Question 10 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The PDF extraction contains several OCR errors: “\\(k\\hookrightarrow\\mathbb C\\)” became a malformed arrow, “Szabó” became “Szobó,” and \\(\\mathbb P^2\\) was split across two lines. The AIM HTML version and the workshop PDF support the following recovered text.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 10\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[93]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 10. Are there results at the level of number fields arising from the techniques of rationally connected varieties? (Colliot-Th´ el` ene) For example, recently Koll` ar got nice results over local fields: if X/k is a projective variety, we say P, Q ∈ X(k) are R-equivalent if you can link them by a chain of curves of genus zero over k; if X is smooth with k ↪ → C, then X is rationally connected if X(C)/R is a single point. Kollar proved that if k is a local field, and X/k is rationally connected, then \\n\\nX(k)/R is finite. (Szob´ o) Koll` ar and Szabo proved that if k is a number field, and X/k is rationally connected, then X(kv)/R is trivial for almost all v.\"\nOriginal remarks: [\"Remark. If X/k is a variety over a number field, and X is rationally connected, then does there exist a field K ⊃ k such that for all L ⊃ K that X(L)/R consists of a point? (Ellenberg) Negative answer by a conic bundle over P\\n2. (Raskind) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0094", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth proper geometrically integral variety X over a characteristic-zero field with nonzero geometric Brauer group, no base extension makes X universally R-trivial: after every proposed extension K there is a finite E/K such that, over L = E(X), the generic point and a constant E-point are not R-equivalent. Separately, for a smooth proper rationally connected variety over a number field, the product of all local R-equivalence sets is finite, so any infinite global set X(k)/R would contain an infinite fiber invisible at every completion.\n\nCandidate contribution (theorem; novelty confidence low): The explicit generic-point witness formulation: every nonzero geometric Brauer class yields, after each proposed base extension K, a finite E/K and the concrete pair consisting of the generic point and a constant point over E(X) that are not R-equivalent." }, { "id": 20000443, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0095", "title": "Cohomological dimension one is not a rational-point principle", "statement": "Question 11. Let X/k be a smooth projective rationally connected variety with the cohomological dimension of k ≤ 1. Does X have a rational point? (Colliot-Th´ el` ene)", "original_statement": "Question 11. Let X/k be a smooth projective rationally connected variety with the cohomological dimension of k ≤ 1. Does X have a rational point? (Colliot-Th´ el` ene)", "clean_statement": "Question 11. Let X/k be a smooth projective rationally connected variety with the cohomological dimension of k ≤ 1. Does X have a rational point? (Colliot-Th´ el` ene)", "statement_status": "exact", "statement_verification": "The canonical record is Question 11 from the AIM workshop *Rational and integral points on higher dimensional varieties*. Its exact extracted problem field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 11\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[94]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 11. Let X/k be a smooth projective rationally connected variety with the cohomological dimension of k ≤ 1. Does X have a rational point? (Colliot-Th´ el` ene)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) There was a false proof for X2,2 ⊂ P4.(ii) Yes if k = Fq. (Esnault) Yes if k is the function field of a curve. (Harris, Graber, Starr) (iii) At least for surfaces, we hoped that universal torsors would be nice objects, e.g. they are birational to homogeneous spaces under a nice group, so they would be close to k-rational if they had a k-point. An example of X/k (k a horrible field) a cubic surface with X(k) = ∅\\n\\nbut (Br k)[3] = \\n0. (Madore, Colliot-Th´ el` ene) (iv) Colliot-Th´ el` ene adds the following: With hindsight,\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0095", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM question is known to have a negative answer: Ax constructed an index-1 smooth quintic hypersurface in projective 9-space over a characteristic-zero field of cohomological dimension 1 with no rational point, while Colliot-Thélène and Madore constructed an index-3 smooth cubic surface over their own characteristic-zero field of cohomological dimension 1. Building on the latter theorem, this attempt proves a candidate fixed-field dimension dichotomy: over one fixed Colliot-Thélène–Madore field every smooth projective geometrically rationally connected curve has a rational point, but index-3 smooth projective geometrically rationally connected counterexamples exist in every dimension at least 2.\n\nCandidate contribution (theorem; novelty confidence low): There is a single characteristic-zero field F of cohomological dimension 1 such that every smooth projective geometrically rationally connected F-curve has an F-point, whereas for every n at least 2 there is a smooth projective geometrically rationally connected n-fold over F of index 3 and without an F-point." }, { "id": 20000444, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0096", "title": "Rational points, index transport, and the C_1^0 boundary", "statement": "Problem 11, as phrased, had been settled by J. Ax in Bull. Am. Math. Soc. 71 (1965) p. 717. Ax produced a smooth hypersurface in 9-dimensional projective space, of degree 5, over a field k of cohomological dimension 1, with no rational point (that such hypersurfaces are rationally connected was proven much later). Ax'example has index 1, i.e. the g.c.d. of the degree of the finite field extensions over which the hypersurface acquires a rational point is 1. In Journal of the Inst. of Math. Jussieu (2004) 3 p. 1-16, J.-L. Colliot-Th´ el` ene and D. Madore produce a field of cohomological dimension 1 and a smooth cubic surface over that field which has index 3, thus settling negatively a question of Kato and Kuzumaki (1986). 50\n\nProblem/", "original_statement": "Problem 11, as phrased, had been settled by J. Ax in Bull. Am. Math. Soc. 71 (1965) p. 717. Ax produced a smooth hypersurface in 9-dimensional projective space, of degree 5, over a field k of cohomological dimension 1, with no rational point (that such hypersurfaces are rationally connected was proven much later). Ax'example has index 1, i.e. the g.c.d. of the degree of the finite field extensions over which the hypersurface acquires a rational point is 1. In Journal of the Inst. of Math. Jussieu (2004) 3 p. 1-16, J.-L. Colliot-Th´ el` ene and D. Madore produce a field of cohomological dimension 1 and a smooth cubic surface over that field which has index 3, thus settling negatively a question of Kato and Kuzumaki (1986). 50 \n\nProblem/", "clean_statement": "Problem 11, as phrased, had been settled by J. Ax in Bull. Am. Math. Soc. 71 (1965) p. 717. Ax produced a smooth hypersurface in 9-dimensional projective space, of degree 5, over a field k of cohomological dimension 1, with no rational point (that such hypersurfaces are rationally connected was proven much later). Ax'example has index 1, i.e. the g.c.d. of the degree of the finite field extensions over which the hypersurface acquires a rational point is 1. In Journal of the Inst. of Math. Jussieu (2004) 3 p. 1-16, J.-L. Colliot-Th´ el` ene and D. Madore produce a field of cohomological dimension 1 and a smooth cubic surface over that field which has index 3, thus settling negatively a question of Kato and Kuzumaki (1986). 50\n\nProblem/", "statement_status": "exact", "statement_verification": "This placement was checked in the official PDF: the paragraph is remark (iv) on printed page 50, immediately before Question 12. The isolated terminal text “\\(50\\) Problem/” in the extracted record is a page-number/header artifact. The raw OCR record is preserved verbatim in *input.json*; accents, punctuation, and line breaks above were restored from the official source.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 11\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[95]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 11, as phrased, had been settled by J. Ax in Bull. Am. Math. Soc. 71 (1965) p. 717. Ax produced a smooth hypersurface in 9-dimensional projective space, of degree 5, over a field k of cohomological dimension 1, with no rational point (that such hypersurfaces are rationally connected was proven much later). Ax'example has index 1, i.e. the g.c.d. of the degree of the finite field extensions over which the hypersurface acquires a rational point is 1. In Journal of the Inst. of Math. Jussieu (2004) 3 p. 1-16, J.-L. Colliot-Th´ el` ene and D. Madore produce a field of cohomological dimension 1 and a smooth cubic surface over that field which has index 3, thus settling negatively a question of Kato and Kuzumaki (1986). 50 \\n\\nProblem/\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0096", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "This canonical record is a status continuation of AIM Question 11, not a fresh unresolved problem: Ax's rationally connected quintic already gives a negative rational-point answer, while the Colliot-Thelene--Madore cubic gives the stronger index obstruction that refutes Kato--Kuzumaki C_1^0. The developed comparison proves, for every proper X/k and finite L/k, the index-transport sandwich ind(X_L) | ind(X) and ind(X) | [L:k] ind(X_L); it follows that prime index persists under every prime-to-p extension. Combined with the prime-degree hypersurface dichotomy and N_0(X/k)=ind(X)Z, this yields the explicit degree-one Ax cycle [P_3]-[P_2] and the robust cubic norm obstruction N_0=3Z.\n\nCandidate contribution (proposition; novelty confidence low): Candidate index-transport diagnostic: the divisibilities ind(X_L) | ind(X) | [L:k] ind(X_L), combined with the fact that a prime-degree hypersurface has index 1 or p and with N_0(X/k)=ind(X)Z, distinguish an index-one effectivity gap from a prime-locked C_1^0 obstruction and make the distinction stable under finite base change.", "research_difficulty_suggested": "L2" }, { "id": 20000445, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0097", "title": "Galois-stable cyclotomic interlacing pairs", "statement": "Question 12. Describe all pairs of sets A, B ⊂ μ∞, A = B, stable under Gal( Q/Q), where the elements of A and B are alternately placed around the unit circle. (Rodriguez-Villegas)", "original_statement": "Question 12. Describe all pairs of sets A, B ⊂ μ∞, A = B, stable under Gal( Q/Q), where the elements of A and B are alternately placed around the unit circle. (Rodriguez-Villegas)", "clean_statement": "Question 12. Describe all pairs of sets A, B ⊂ μ∞, A = B, stable under Gal( Q/Q), where the elements of A and B are alternately placed around the unit circle. (Rodriguez-Villegas)", "statement_status": "exact", "statement_verification": "This attempt concerns exactly record AIM-ALGEBRAIC_NUMBER_THEORY-0097, Question 12 of the AIM list *Rational and integral points on higher dimensional varieties* (Fernando Rodriguez-Villegas), stored at zero-based index 96 of aim-algebraic-number-theory-notes.json.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 12\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[96]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 12. Describe all pairs of sets A, B ⊂ μ∞, A = B, stable under Gal( Q/Q), where the elements of A and B are alternately placed around the unit circle. (Rodriguez-Villegas)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) There is a solution but it is much more complicated than the statement of the problem. The solution is used in the classification of algebraic hypergeometric functions. (ii) There is the infinite family A = μm+n \\\\ { 1} and μm ∪ μn with gcd( m, n ) = 1. \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0097", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "After repairing the corrupted source condition from A = B to #A = #B, the AIM problem is exactly the classification of interlacing cyclotomic polynomial pairs. The known Beukers-Heckman classification, in the explicit McKee-Smyth form, gives two primitive infinite families and 26 primitive sporadic pairs, with every pair obtained up to simultaneous sign change, common inflation, and interchange. This attempt additionally proves a Möbius-weighted signed cyclotomic discrepancy certificate, its exact common-inflation law, and carry-bit proofs of both infinite families; this synthesis is labeled only as candidate novelty.\n\nCandidate contribution (proposition; novelty confidence low): For signed cyclotomic orbit data epsilon_d, strict interlacing is equivalent, after orienting 1 in B, to Delta_epsilon(x) = sum_d epsilon_d sum_{e|d} mu(e) floor(dx/e) taking only the values 0 and 1; common inflation by ell obeys Delta_inflated(x) = Delta_base(frac(ell x)), and the two infinite classified families follow uniformly from a one-bit floor carry." }, { "id": 20000446, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0098", "title": "An explicit geometric Picard-rank-one quintic over F_2", "statement": "Question 13. Find a smooth quintic hypersurface in P3 with Picard number 1 over F\n2. (Voloch)", "original_statement": "Question 13. Find a smooth quintic hypersurface in P3 with Picard number 1 over F\n2. (Voloch)", "clean_statement": "Question 13. Find a smooth quintic hypersurface in P3 with Picard number 1 over F\n2. (Voloch)", "statement_status": "exact", "statement_verification": "The canonical record is Question 13 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The extracted text has line-break damage in the finite-field subscripts. The official AIM HTML and PDF give the unambiguous statement", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 13\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[97]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 13. Find a smooth quintic hypersurface in P3 with Picard number 1 over F\\n2. (Voloch)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) This is used in constructing error correcting codes. (ii) If you compute the analytic rank (the zeta function), by Tate's theorem, the second Betti number is 53 so compute the number of points up to something like F228 (Voloch). So testing them exhaustively would be very costly. (iii) Shioda has examples over Q of Picard number 1, so they might be defined over F\\n2. (Raskind) This has been tried once. (Voloch) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0098", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Abbott, Kedlaya, and Roe's quintic X=V(f) in P^3 over F_2, where f=x^5+x^3yz+x^2y^2w+x^2yz^2+x^2z^3+xz^2w^2+y^5+y^3zw+y^2zw^2+yzw^3+z^5+z^2w^3+w^5, is smooth and has geometric Picard number 1, hence also arithmetic Picard number 1. The Picard-rank conclusion is the known AKR solution, proved by their certified finite-precision Frobenius-corank computation. This attempt independently gives an exact smoothness certificate: the degree-13 Jacobian Macaulay map R_9^4 to R_13 has rank 560 over F_2, and proves that degree 13 is the sharp uniform Jacobian-surjectivity threshold for smooth quintic surfaces in characteristic not dividing 5.\n\nCandidate contribution (proposition; novelty confidence low): For the explicit AKR quintic, exact binary elimination gives rank 560 for the 560-by-880 degree-13 Jacobian Macaulay matrix and rank 454 for the 455-by-660 degree-12 matrix; the complete-intersection Hilbert series proves that degree 13 is the sharp uniform surjectivity threshold for this smoothness test on quintics in P^3 when the characteristic does not divide 5." }, { "id": 20000447, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0099", "title": "An exact residue-matching reduction for least points on diagonal cubic surfaces", "statement": "Question 14. Consider cubic hypersurfaces a0X30 + a1X31 + a2X32 + a3X33 =\n0. (Swinnerton-Dyer) (a) If soluble, give upper bound for smallest solution in terms of the ai.(b) Look at |ai| < A, tabulate the size of the smallest solution and conjecture a particular growth rate in terms of A.", "original_statement": "Question 14. Consider cubic hypersurfaces a0X30 + a1X31 + a2X32 + a3X33 = \n0. (Swinnerton-Dyer) (a) If soluble, give upper bound for smallest solution in terms of the ai.(b) Look at |ai| < A, tabulate the size of the smallest solution and conjecture a particular growth rate in terms of A.", "clean_statement": "Question 14. Consider cubic hypersurfaces a0X30 + a1X31 + a2X32 + a3X33 =\n0. (Swinnerton-Dyer) (a) If soluble, give upper bound for smallest solution in terms of the ai.(b) Look at |ai| < A, tabulate the size of the smallest solution and conjecture a particular growth rate in terms of A.", "statement_status": "exact", "statement_verification": "The canonical record is Question 14 from the AIM workshop *Rational and integral points on higher dimensional varieties*, source file `aim-algebraic-number-theory-notes.json`, zero-based record index 98. The exact extracted `problem` field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 14\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[98]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 14. Consider cubic hypersurfaces a0X30 + a1X31 + a2X32 + a3X33 = \\n0. (Swinnerton-Dyer) (a) If soluble, give upper bound for smallest solution in terms of the ai.(b) Look at |ai| < A, tabulate the size of the smallest solution and conjecture a particular growth rate in terms of A.\"\nOriginal remarks: [\"Remarks. If a1x31 + · · · + a4x34 = 0 has a solution x ∈ Z4 \\\\ { 0}, how large is the smallest solution? Let x0 be a solution with max {| xi|} minimal. Swinnerton-Dyer had suggested: if \\n\\nA = max 1≤i≤4{| ai|}, then max x=x0 {| xi|} << A 4/3. Wooley had suggested instead A1+ ≤.Progress: Stoll, Stein: computations up to |ai| ≤ 60 suggest an upper bound of A2.Also, assuming Schinzel, finiteness of Sha, and one unproven lemma, Stoll can produce a ∈ Z4\\n\\nwith least solution >> A 2−≤. (Need the hypotheses to ensure that the examples do have rational solutions.) Shape of examples: if p, q prime, look at px 31 + 2 px 32 + qx 33 + 5 qx 34. Assume 2 p and 5 q\\n\\nare approximately of the same size, p ≡ q ≡ 1 (mod 3), that 2 is a cube mod p but not mod \\n\\nq, and that 5 is a cube mod q but not mod p (plus some additional technical conditions).\", \"Remarks. \\n\\n(i) The growth rate should be like A4/3. If you do the corresponding thing with 3 squares, \\n\\na0X20 + a1X21 + a2X2 = 0, the answer is A. (Swinnerton-Dyer) (ii) Seems more like A1+ ≤. (Wooley) (iii) Is there a heuristic which suggests this? (Poonen) No. (Swinnerton-Dyer) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0099", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the Stoll-shaped family p(x_1^3+2x_2^3)+q(x_3^3+5x_4^3)=0, when 2 is a noncube modulo q and 5 is a noncube modulo p, every solution has the form (qu,qv,pw,pz), and existence below a given height is equivalent to intersection of two explicit finite sets of a common integer parameter t modulo p^2 and q^2. This yields a rigorous lower bound in terms of square-modulus lattice minima. Applying the reduction gives the exact certified value m(31,62,13,65)=248, attained by the primitive point (169,91,-248,31).\n\nCandidate contribution (reduction; novelty confidence low): The two nonresidue conditions imply an exact necessary-and-sufficient common-t matching criterion after a square-modulus upgrade; for the concrete primitive surface 31X_1^3+62X_2^3+13X_3^3+65X_4^3=0, the complete bounded matching tables prove that its least primitive point height is exactly 248." }, { "id": 20000448, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0100", "title": "Fixed-length sums of 2/(x^2-2)", "statement": "Question 15. Characterize the rational numbers α that can be written as\n\nα = 2\n\nx21 − 2 + 2\n\nx22 − 2 + · · · + 2\n\nx2\n\n> n\n\n− 2for xi ∈ Q and fixed n. (Poonen) It is necessary that |α|p ≤ 1 for (2 /p ) = −1 and p = 2. Is it sufficient? 51", "original_statement": "Question 15. Characterize the rational numbers α that can be written as \n\nα = 2\n\nx21 − 2 + 2\n\nx22 − 2 + · · · + 2\n\nx2 \n\n> n\n\n− 2for xi ∈ Q and fixed n. (Poonen) It is necessary that |α|p ≤ 1 for (2 /p ) = −1 and p = 2. Is it sufficient? 51", "clean_statement": "Question 15. Characterize the rational numbers α that can be written as\n\nα = 2\n\nx21 − 2 + 2\n\nx22 − 2 + · · · + 2\n\nx2\n\n> n\n\n− 2for xi ∈ Q and fixed n. (Poonen) It is necessary that |α|p ≤ 1 for (2 /p ) = −1 and p = 2. Is it sufficient? 51", "statement_status": "exact", "statement_verification": "The canonical JSON problem field is OCR-damaged. Reproduced verbatim, it is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 15\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[99]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 15. Characterize the rational numbers α that can be written as \\n\\nα = 2\\n\\nx21 − 2 + 2\\n\\nx22 − 2 + · · · + 2\\n\\nx2 \\n\\n> n\\n\\n− 2for xi ∈ Q and fixed n. (Poonen) It is necessary that |α|p ≤ 1 for (2 /p ) = −1 and p = 2. Is it sufficient? 51\"\nOriginal remarks: [\"Remarks. \\n\\n(i) You might repeat this kind of problem with any rational function with no rational poles. (As in Waring's problem.) Something has already been done for a function with rational poles. (Poonen) (ii) You can phrase this problem a different kind of way: prove or disprove the Hasse principle for this equation for all α ∈ Q. (Swinnerton-Dyer) You can probably show for n sufficiently large (and fixed) that there is a solution locally. (iii) There are applications to Diophantine definitions: this would show that inside Q, the set of these rational numbers which are integral at half of the places, is Diophantine. (Poonen) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0100", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every odd prime p at which 2 is nonsquare, the two-summand local value set is exactly V_2(Q_p)=Z_p. At p=2 the one-summand residue image modulo 8 is {0,1,6,7}, so the two-summand image omits exactly residue 3; consequently alpha=3 satisfies every integrality condition printed in the AIM question but is not in V_2(Q), while it is in V_3(Q). The attempt also proves an exact one-summand criterion, a norm-one-torus trace reformulation for every fixed n, and zero-sum padding inclusions.\n\nCandidate contribution (local theorem and counterexample; novelty confidence low): The exact equality V_2(Q_p)=Z_p at every odd inert prime, together with the exact 2-adic residue image V_2(Q_2) mod 8={0,1,2,4,5,6,7}, isolates the ramified prime as a genuine two-summand obstruction and yields the explicit counterexample (n,alpha)=(2,3) to bare integrality sufficiency; the accompanying norm-one-torus trace formula gives an exact uniform reduction for all fixed n." }, { "id": 20000449, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0101", "title": "Rational lines on cubic hypersurfaces: a three-layer local obstruction and an explicit pointed example", "statement": "Question 16. Solve the local-global solubility problem for finding lines on a cubic hypersurface. (Wooley) (a) Find an example of a cubic hypersurface over Qp (in as many variables as possible) with no rational line. (b) Find an example of a cubic hypersurface over Q (in as many variables as possible) with no rational line.", "original_statement": "Question 16. Solve the local-global solubility problem for finding lines on a cubic hypersurface. (Wooley) (a) Find an example of a cubic hypersurface over Qp (in as many variables as possible) with no rational line. (b) Find an example of a cubic hypersurface over Q (in as many variables as possible) with no rational line.", "clean_statement": "Question 16. Solve the local-global solubility problem for finding lines on a cubic hypersurface. (Wooley) (a) Find an example of a cubic hypersurface over Qp (in as many variables as possible) with no rational line. (b) Find an example of a cubic hypersurface over Q (in as many variables as possible) with no rational line.", "statement_status": "exact", "statement_verification": "The question mark after \\(37\\) occurs in both official versions and is not an extraction error. The number \\(37\\) was subsequently a published theorem of Wooley. In contrast, I did not locate a published theorem supporting the workshop's \\(14\\)-variable local assertion; it is incompatible with the way the later general local bounds are presented, so it is treated below as an unverified workshop remark rather than as an established bound.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 16\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[100]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 16. Solve the local-global solubility problem for finding lines on a cubic hypersurface. (Wooley) (a) Find an example of a cubic hypersurface over Qp (in as many variables as possible) with no rational line. (b) Find an example of a cubic hypersurface over Q (in as many variables as possible) with no rational line.\"\nOriginal remarks: [\"Remarks. \\n\\n(i) For (a), we must have at least 10 variables, since there are cubic forms in ≤ 9 variables with no point. If you have 14 variables, then there is a rational line. (Wooley) (ii) For (b), for 37? variables, there is a rational line. (Wooley) (iii) Also, find one with a rational point but no rational line. (Colliot-Th´ el` ene) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0101", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The local-global implication for lines is already known to fail for smooth cubic surfaces by Jahnel and Loughran, while the extremal variable counts remain open. This attempt proves a general three-layer DVR residue obstruction for lines and applies it to the Brandes-Dietmann 11-variable form F_2 over Q: the resulting smooth cubic in P^10 has the explicit smooth rational point e_4, but has no Q_2-line and therefore no Q-line. Current verified bounds are 11 <= M_Q <= 30 and, for p = 2, 3, 5, 11 <= M_{Q_p} <= 21.\n\nCandidate contribution (lemma; novelty confidence low): If F = H + pi G_1 + pi^2 G_2 over a DVR, the reduction of H contains no residue-field line, and the reductions of G_1 and G_2 are anisotropic, then F contains no line over the fraction field; applying this filtration to the published p = 2 form and observing the smooth rational point e_4 yields an explicit smooth 11-variable Q-example with a rational point but no Q-line." }, { "id": 20000450, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0102", "title": "The infinity 5-torsion line and Kummer quotient of the pentagonal quintic", "statement": "Question 17. Draw a regular pentagon P, construct the circle through the 5 vertices C, and consider the curve E: P + λC 2 = 0. This is a quintic curve with 5 double points, and therefore has geometric genus 1. This curve has five points at ∞ (given by the slopes of the lines). Compute the 5-torsion of E. (McCallum)", "original_statement": "Question 17. Draw a regular pentagon P, construct the circle through the 5 vertices C, and consider the curve E: P + λC 2 = 0. This is a quintic curve with 5 double points, and therefore has geometric genus 1. This curve has five points at ∞ (given by the slopes of the lines). Compute the 5-torsion of E. (McCallum)", "clean_statement": "Question 17. Draw a regular pentagon P, construct the circle through the 5 vertices C, and consider the curve E: P + λC 2 = 0. This is a quintic curve with 5 double points, and therefore has geometric genus 1. This curve has five points at ∞ (given by the slopes of the lines). Compute the 5-torsion of E. (McCallum)", "statement_status": "exact", "statement_verification": "The canonical record is Question 17 from the AIM workshop *Rational and integral points on higher dimensional varieties*, source file aim-algebraic-number-theory-notes.json, zero-based record index 101. The extracted problem is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 17\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[101]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 17. Draw a regular pentagon P, construct the circle through the 5 vertices C, and consider the curve E: P + λC 2 = 0. This is a quintic curve with 5 double points, and therefore has geometric genus 1. This curve has five points at ∞ (given by the slopes of the lines). Compute the 5-torsion of E. (McCallum)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) The points at ∞ are among the 5-torsion. (ii) The motivation is: these curves are principal homogeneous spaces, and are candidates for 5-torsion elements in X. If λ is the parameter on X1(5), then this is a twist of the universal elliptic curve of X1(5). Here we have explicit models. (McCallum) (iii) If you consider this as a pencil of elliptic curves, how does this relate to the talks at this conference? (Ellenberg) (iv) Does the pentagon have to be regular? (Voloch) There are various variations, such as replacing the circle with a star pentagon. \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0102", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After reconstructing and normalizing the regular-pentagon equation over Q(sqrt(5)), the five points at infinity on the genus-one normalization are proved to be O,D,2D,3D,4D for a point D of exact order five. Their cyclic subgroup has explicit quadratic Galois character attached to Q(sqrt(5),sqrt(5+2sqrt(5)))/Q(sqrt(5)); the Weil pairing determines the complementary composition factor and a canonical, not necessarily split, exact sequence for the full 5-torsion. Reflection by Y -> -Y is proved to be negation, and exact elimination normalizes its quotient to a displayed rational conic, yielding an explicit one-variable double-cover reduction for the remaining Weierstrass and division-field computation.\n\nCandidate contribution (explicit torsion-line theorem and quotient reduction; novelty confidence low): For the fixed regular-pentagon model, the infinity subgroup is exactly the quadratic-character line F_5(chi_{5+2sqrt(5)}), its Weil-pairing quotient has character chi_5 chi_{5+2sqrt(5)}, and the reflection quotient is normalized by an explicit square-discriminant identity to a rational conic." }, { "id": 20000451, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0103", "title": "Rational descent obstructions for geometric elliptic powers in hyperelliptic Jacobians", "statement": "Question 18.\n\n(a) Find a separable polynomial g(t) ∈ Q[t] such that the Jacobian of the hyperelliptic curve\n\ns2 = g(t) is isogeneous over Q to Er × B, E an elliptic curve, with r ≥ 4 or r = 3 and dim B ≤\n1. (Silverberg) (b) Related problem: Give an example of a map P1 → E4/{± 1}, where ±1 acts diagonally, whose image does not lie in H/ {± 1} for any subgroup H ( E\n4. (Ellenberg)", "original_statement": "Question 18. \n\n(a) Find a separable polynomial g(t) ∈ Q[t] such that the Jacobian of the hyperelliptic curve \n\ns2 = g(t) is isogeneous over Q to Er × B, E an elliptic curve, with r ≥ 4 or r = 3 and dim B ≤ \n1. (Silverberg) (b) Related problem: Give an example of a map P1 → E4/{± 1}, where ±1 acts diagonally, whose image does not lie in H/ {± 1} for any subgroup H ( E\n4. (Ellenberg)", "clean_statement": "Question 18.\n\n(a) Find a separable polynomial g(t) ∈ Q[t] such that the Jacobian of the hyperelliptic curve\n\ns2 = g(t) is isogeneous over Q to Er × B, E an elliptic curve, with r ≥ 4 or r = 3 and dim B ≤\n1. (Silverberg) (b) Related problem: Give an example of a map P1 → E4/{± 1}, where ±1 acts diagonally, whose image does not lie in H/ {± 1} for any subgroup H ( E\n4. (Ellenberg)", "statement_status": "exact", "statement_verification": "The canonical record is Question 18 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The JSON extraction loses superscripts and the strict-containment symbol. Comparison with the official HTML and PDF gives the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 18\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[102]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 18. \\n\\n(a) Find a separable polynomial g(t) ∈ Q[t] such that the Jacobian of the hyperelliptic curve \\n\\ns2 = g(t) is isogeneous over Q to Er × B, E an elliptic curve, with r ≥ 4 or r = 3 and dim B ≤ \\n1. (Silverberg) (b) Related problem: Give an example of a map P1 → E4/{± 1}, where ±1 acts diagonally, whose image does not lie in H/ {± 1} for any subgroup H ( E\\n4. (Ellenberg)\"\nOriginal remarks: [\"Remarks. 52 \\n\\n(i) The case r = 2 and dim B = 0 is possible, as is the case r = 3 and dim B = 2. A consequence of this construction would be better bounds on the density of quadratic twists of E of rank \\n\\n≥ r. (Silverberg) (ii) What if you ask this question over C? (Voloch) Can at least do r = 3 and dim B = 0 over \\n\\nC. (Poonen) Even over C, maybe it cannot be done for large r. (Ellenberg) (iii) One might consider the curve s2 = tk + 1 over Q, for k composite. If 3 | k, for example, it maps to s2 = t3 + 1? (iv) Does it have to be hyperelliptic? (Rodriguez-Villegas) Yes, for applications. (Ellenberg) (v) The random matrix heuristics suggest that there is a positive power, so there should be a curve there, and finding such a curve would give a proof of a density result. (Ellenberg) (vi) Take E1, E 2, E 3/Q pairwise isogeneous elliptic curves, A = E1 × E2 × E3, and look at the set of principal polarizations on X. Find a nonsplit principal polarization on A; then it comes from A = Jac( C), g(C) = 3. Now you just need to show that C is hyperelliptic. \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0103", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the standard geometric elliptic-cube curve C: y^2 = x^8 + 14x^4 + 1, the three Q-defined involution quotients give the exact decomposition Jac(C) ~_Q E^2 x E^(-1), where E: Y^2 = X^3 - 5616X + 120960 and E^(-1): Y^2 = X^3 - 5616X - 120960; their Frobenius traces at the common good prime 11 are 4 and -4, so they are not Q-isogenous and Jac(C) is not a Q-elliptic cube. An independent p = 3 trace argument proves that the standard genus-five geometric fifth-power example does not descend to the displayed E_5^5 over Q. A proved Jacobian-Kummer criterion makes precise the equivalence between a surjection Jac(C) -> E^r and a nondegenerate rational curve on E^r/{+/-1} under the stated nonconstancy and field conventions.\n\nCandidate contribution (descent obstruction; novelty confidence low): The exact arithmetic decomposition Jac(y^2 = x^8 + 14x^4 + 1) ~_Q (Y^2 = X^3 - 5616X + 120960)^2 x (Y^2 = X^3 - 5616X - 120960), together with the p = 11 trace certificate proving that the last factor is not Q-isogenous to the first, is a concrete candidate field-of-definition obstruction not located in the literature checked." }, { "id": 20000452, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0104", "title": "Sections on every open and the Brauer--Manin set", "statement": "Question 19. Let X be a variety over a number field k and suppose that for every open Zariski dense U ⊂ X, the map π1(U ) → Gal( k/k ) has a splitting s (e.g. if X(k)is Zariski dense). In this case, if X(Ak) 6 = ∅, is there no Brauer-Manin obstruction to the Hasse principle for X, i.e. is X(Ak)Br 6 = ∅? (Ellenberg)", "original_statement": "Question 19. Let X be a variety over a number field k and suppose that for every open Zariski dense U ⊂ X, the map π1(U ) → Gal( k/k ) has a splitting s (e.g. if X(k)is Zariski dense). In this case, if X(Ak) 6 = ∅, is there no Brauer-Manin obstruction to the Hasse principle for X, i.e. is X(Ak)Br 6 = ∅? (Ellenberg)", "clean_statement": "Question 19. Let X be a variety over a number field k and suppose that for every open Zariski dense U ⊂ X, the map π1(U ) → Gal( k/k ) has a splitting s (e.g. if X(k)is Zariski dense). In this case, if X(Ak) 6 = ∅, is there no Brauer-Manin obstruction to the Hasse principle for X, i.e. is X(Ak)Br 6 = ∅? (Ellenberg)", "statement_status": "exact", "statement_verification": "The canonical JSON field is OCR-damaged. Reproduced verbatim, it reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 19\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[103]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 19. Let X be a variety over a number field k and suppose that for every open Zariski dense U ⊂ X, the map π1(U ) → Gal( k/k ) has a splitting s (e.g. if X(k)is Zariski dense). In this case, if X(Ak) 6 = ∅, is there no Brauer-Manin obstruction to the Hasse principle for X, i.e. is X(Ak)Br 6 = ∅? (Ellenberg)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) Is it possible that X(Ak) 6 = ∅ follows from the splitting condition? (Poonen) You might also ask the corresponding question for a local field. (McCallum) (ii) You might also ask this question for other obstructions. (Ellenberg) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0104", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every smooth projective geometrically connected curve C over a number field, one can remove finitely many closed points to obtain an affine open U_0 with Pic(U_0)=0; a section of pi_1(U_0) to G_k then forces Br(k) to inject into Br(C). Consequently, a genus-zero curve satisfying the AIM all-open splitting hypothesis has a rational point even without prior adelic solubility. For proper hyperbolic curves, a proved compact-compatible selection criterion promotes open-by-open sections to a birational section and hence to a Brauer-orthogonal adelic point, while an explicit Kummer calculation on G_m shows that the needed compactness does not follow formally from nonemptiness of the section sets.\n\nCandidate contribution (Brauer-kernel theorem and compatibility obstruction; novelty confidence low): Candidate novelty: the all-open splitting hypothesis forces the constant relative Brauer group of a smooth projective curve to vanish, yielding an unconditional rational-point theorem for genus-zero curves; moreover, the compact-compatible inverse-limit criterion and the G_m ramification-escape calculation isolate a precise sufficient condition and a concrete obstruction for the positive-genus problem." }, { "id": 20000453, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0105", "title": "An explicit cubic point and its descent certificate", "statement": "Question 20. Let p(t) = 3( t4 − 54 t2 − 117 t − 243). Write the system of two equations y2 = p(t)( x2 + 1), z2 = p(t)( x2 + 2); think of this as a one-parameter family X of curves of genus one; X(Q) is empty and X(AQ)Br 6 = ∅ (Skorobogatov). Is X(k) 6 = ∅ for some [ k: Q] odd? (Colliot-Th´ el` ene)", "original_statement": "Question 20. Let p(t) = 3( t4 − 54 t2 − 117 t − 243). Write the system of two equations y2 = p(t)( x2 + 1), z2 = p(t)( x2 + 2); think of this as a one-parameter family X of curves of genus one; X(Q) is empty and X(AQ)Br 6 = ∅ (Skorobogatov). Is X(k) 6 = ∅ for some [ k: Q] odd? (Colliot-Th´ el` ene)", "clean_statement": "Question 20. Let p(t) = 3( t4 − 54 t2 − 117 t − 243). Write the system of two equations y2 = p(t)( x2 + 1), z2 = p(t)( x2 + 2); think of this as a one-parameter family X of curves of genus one; X(Q) is empty and X(AQ)Br 6 = ∅ (Skorobogatov). Is X(k) 6 = ∅ for some [ k: Q] odd? (Colliot-Th´ el` ene)", "statement_status": "exact", "statement_verification": "Put \\[ g(T)=3(T^4-54T^2-117T-243),\\qquad p(X)=X^2+1,\\qquad q(X)=X^2+2. \\] The official AIM workshop PDF, *Rational and integral points on higher dimensional varieties*, Question 20, asks whether the surface with affine open \\[ S_{\\mathrm{AIM}}:\\qquad y^2=g(t)p(x),\\qquad z^2=g(t)q(x) \\tag{1} \\] has a point over some odd-degree number field. The extracted strings `t4`, `y2`, and `6 =` are OCR renderings of fourth powers, squares, and `\\(\\ne\\)`. The question attributes to Skorobogatov the facts that a smooth projective model \\(S/\\mathbb Q\\) has \\(S(\\mathbb Q)=\\varnothing\\) and \\(S(\\mathbb A_{\\mathbb Q})^{\\mathrm{Br}}\\ne\\varnothing\\).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 20\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[104]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 20. Let p(t) = 3( t4 − 54 t2 − 117 t − 243). Write the system of two equations y2 = p(t)( x2 + 1), z2 = p(t)( x2 + 2); think of this as a one-parameter family X of curves of genus one; X(Q) is empty and X(AQ)Br 6 = ∅ (Skorobogatov). Is X(k) 6 = ∅ for some [ k: Q] odd? (Colliot-Th´ el` ene)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) The secret reason for asking: then X has a zero-cycle of degree \\n1. (Colliot-Th´ el` ene) If you have a zero cycle of degree 1, then there is such a point. (ii) Even in a given number field of degree 3, looking at a random way, the evidence you will find is zero. (Colliot-Th´ el` ene) (iii) What about X × X × X modulo the action by three? (Ellenberg) (iv) Reduce the search by finding one elliptic curve with many rational points (fix x, consider \\n\\ny2 = p(t)( x2 + 1)), then search for z. (Voloch) But the ratio y/z is only dependent on x.(Poonen) (v) Do we expect [ k: Q] = 3? (McCallum) Not necessarily. (Colliot-Th´ el` ene) (vi) Consider instead w2 = ( x2 + 2)( x2 + 1) and y2 = p(t)( x2 + 1); search now in the first curve for a cubic point, and search for t. (Poonen) Put the first one in Weierstrass form, and generate the cubic fields. \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0105", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Brendan Creutz solved the AIM question affirmatively in 2017: for L = Q(θ), θ^3 + θ + 1 = 0, the displayed Skorobogatov surface has an L-point, so odd degree 3 suffices. This attempt translates Creutz's published point into the AIM equation convention and verifies it exactly. It also proves an explicit Weierstrass/descent bridge and supplies a low-height twist certificate, while crediting the full solution and cubic point to Creutz.\n\nCandidate contribution (explicit descent bridge and twist certificate; novelty confidence low): Candidate novelty is limited to formulas (12)–(19) and the low-height lift: D0: w^2=(x^2+1)(x^2+2) maps to E: v^2=u^3-6u^2+u by u=2w+2x^2+3 and v=2xu; the double cover is generated by sqrt(u)=Y+Z; and Creutz's cubic point lifts with a=9-4θ+6θ^2, U0=2133-1224θ+1494θ^2, Y=θ^2, Z=1-θ^2, where the Weierstrass coordinate is u=a and N_{L/Q}(a)=1." }, { "id": 20000454, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0106", "title": "A certified orbit-lattice algorithm for M_3(Z)", "statement": "Question 21. Let A/ Z be an algebra with rk Z A = 9 given by its multiplica-tion table. Suppose you know that A ' M3(Z); find an algorithm which gives an explicit isomorphism. (Stoll)", "original_statement": "Question 21. Let A/ Z be an algebra with rk Z A = 9 given by its multiplica-tion table. Suppose you know that A ' M3(Z); find an algorithm which gives an explicit isomorphism. (Stoll)", "clean_statement": "Question 21. Let A/ Z be an algebra with rk Z A = 9 given by its multiplica-tion table. Suppose you know that A ' M3(Z); find an algorithm which gives an explicit isomorphism. (Stoll)", "statement_status": "exact", "statement_verification": "The canonical record is Question 21 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The JSON extraction loses mathematical typography and inserts a line-break hyphen. The official AIM HTML and PDF give the unambiguous statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 21\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[105]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 21. Let A/ Z be an algebra with rk Z A = 9 given by its multiplica-tion table. Suppose you know that A ' M3(Z); find an algorithm which gives an explicit isomorphism. (Stoll)\"\nOriginal remarks: [\"Remarks. 53 \\n\\n(i) The motivation comes from very explicit 3-descent on elliptic curves. (ii) The case of M2(Z) reduces to finding rational points on conics. (Stoll) (iii) Is this an LLL problem? (Voloch) In some sense, but one needs nine 3 × 3-matrices, not one 9 × 9-matrix; the equations are not all linear. (Stoll) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0106", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "There is a deterministic terminating algorithm that recovers an explicit integral isomorphism from any promised multiplication table A isomorphic to M_3(Z): enumerate x in A until right multiplication R_x has rational rank three, compute the exact lattice L=Ax by column HNF, and represent A by left multiplication on L. The orbit-lattice lemma and maximality prove that A maps isomorphically onto End_Z(L), while the multiplication relations and the determinant ±1 of the nine flattened action matrices give an exact certificate. A known rational ff-polynomial splitting algorithm followed by the same HNF integralization gives the complexity-qualified efficient version.\n\nCandidate contribution (integralization lemma and certificate; novelty confidence low): For every maximal Z-order O in M_n(Q), every nonzero vector v in Q^n yields an orbit lattice L=Ov satisfying O=End_Z(L); equivalently, every rank-one x in O yields O isomorphic to End_Z(Ox). Thus the exact, possibly nonsaturated orbit lattice and one HNF suffice, and multiplication relations plus a flattened unimodular determinant certify the resulting integral isomorphism." }, { "id": 20000455, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0107", "title": "A fixed-real-subfield criterion for a complex archimedean ball", "statement": "Question 22. Let k be a number field, k 6 ⊂ R. Construct an algebraic set S ⊂ kn\n\nfor some n such that the projection onto one of the coordinates is exactly the set of elements of k with |k| ≤ 1 for some (archimedean) absolute value | | of k. (Shlapentokh)", "original_statement": "Question 22. Let k be a number field, k 6 ⊂ R. Construct an algebraic set S ⊂ kn\n\nfor some n such that the projection onto one of the coordinates is exactly the set of elements of k with |k| ≤ 1 for some (archimedean) absolute value | | of k. (Shlapentokh)", "clean_statement": "Question 22. Let k be a number field, k 6 ⊂ R. Construct an algebraic set S ⊂ kn\n\nfor some n such that the projection onto one of the coordinates is exactly the set of elements of k with |k| ≤ 1 for some (archimedean) absolute value | | of k. (Shlapentokh)", "statement_status": "exact", "statement_verification": "The canonical record is Question 22 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The official AIM PDF and HTML page both print:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 22\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[106]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 22. Let k be a number field, k 6 ⊂ R. Construct an algebraic set S ⊂ kn\\n\\nfor some n such that the projection onto one of the coordinates is exactly the set of elements of k with |k| ≤ 1 for some (archimedean) absolute value | | of k. (Shlapentokh)\"\nOriginal remarks: [\"Remark. This would imply results on Hilbert's tenth problem for rings of integers. It prob-ably is hard because it is close to being equivalent. \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0107", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let sigma:k->C be a complex embedding whose image is stable under complex conjugation, let c be the induced involution, and let F=k^c. Then the exact complex unit ball {x in k: |sigma(x)|<=1} is Diophantine over k if and only if F is Diophantine over k. The forward implication extracts equality of complex moduli from the ball, including the zero-denominator case, and cuts out F by |x+theta|=|x-theta| for a nonzero anti-invariant theta; the reverse implication uses the Diophantine ordering at the chosen real embedding of F and the identity |sigma(a+b theta)|^2=w(a^2+d b^2). The general intended complex-place construction, especially without conjugation stability or a Diophantine fixed real subfield, remains open.\n\nCandidate contribution (equivalence; novelty confidence low): Candidate novelty: for every conjugation-stable embedded nonreal number field k with induced fixed real subfield F, a single exact complex archimedean unit ball is Diophantine over k exactly when F is Diophantine over k; equivalently, the ball recovers F by a two-point equidistance predicate, while F plus its chosen real ordering reconstructs the ball." }, { "id": 20000456, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0108", "title": "Two-square coordinates and curve-source images over a rank-one elliptic curve", "statement": "**Question 23 (Poonen).** Let \\(E\\subset \\mathbf P^2\\) be an elliptic curve over \\(\\mathbf Q\\), and suppose \\(E(\\mathbf Q)\\simeq \\mathbf Z\\).\n\n(a) Describe \\(S\\subset E(\\mathbf Q)\\), where\n\\[\nS=\\{(x,y):y=a^2+b^2,\\ a,b\\in\\mathbf Q\\}.\n\\]\nIs \\(S\\) finite?\n\n(b) More generally, \\(\\pi:X\\to E\\) gives a subset \\(\\pi(X(\\mathbf Q))\\subset E(\\mathbf Q)\\); what others can you build?", "original_statement": "Question 23. Let E ⊂ P2 be an elliptic curve over Q, and suppose E(Q) ' Z.(Poonen) (a) Describe S ⊂ E(Q) where S = {(x, y ): y = a2 + b2, a, b ∈ Q}. Is S is finite? (b) More generally, π: X → E gives a subset π(X(Q)) ⊂ E(Q); what others can you build?", "clean_statement": "**Question 23 (Poonen).** Let \\(E\\subset \\mathbf P^2\\) be an elliptic curve over \\(\\mathbf Q\\), and suppose \\(E(\\mathbf Q)\\simeq \\mathbf Z\\).\n\n(a) Describe \\(S\\subset E(\\mathbf Q)\\), where\n\\[\nS=\\{(x,y):y=a^2+b^2,\\ a,b\\in\\mathbf Q\\}.\n\\]\nIs \\(S\\) finite?\n\n(b) More generally, \\(\\pi:X\\to E\\) gives a subset \\(\\pi(X(\\mathbf Q))\\subset E(\\mathbf Q)\\); what others can you build?", "statement_status": "corrected_verified", "statement_verification": "The canonical record is Question 23 in the AIM problem list *Rational and integral points on higher dimensional varieties*. The supplied JSON has OCR losses (`P2`, `a2`, and an apostrophe in place of an isomorphism sign). The official PDF reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 23\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[107]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 23. Let E ⊂ P2 be an elliptic curve over Q, and suppose E(Q) ' Z.(Poonen) (a) Describe S ⊂ E(Q) where S = {(x, y ): y = a2 + b2, a, b ∈ Q}. Is S is finite? (b) More generally, π: X → E gives a subset π(X(Q)) ⊂ E(Q); what others can you build?\"\nOriginal remarks: [\"Remark. It is possible to describe in a Diophantine way the set of points S = {(x, y ): 1 ≤\\n\\ny ≤ 2}; this is an infinite set. (Poonen) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0108", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For E(Q) isomorphic to Z, the rational image of any smooth projective geometrically integral curve C -> E is finite or a coset R + mE(Q). Combining this rigidity with the Bhakta-Loughran-Rydin Myerson-Nakahara rank-one height bound proves that an infinite such curve image contained in the two-square y-coordinate locus S cannot meet H0 = E(Q) intersect E(R)^0; in particular all such images are finite when E(Q) lies in E(R)^0. The central finiteness of S remains open, and on a disconnected real curve the published sieve leaves a possible odd coset disjoint from H0 uncontrolled.\n\nCandidate contribution (curve-source rigidity and sieve obstruction; novelty confidence low): Candidate contribution: projective curve-source images in E(Q) isomorphic to Z are finite or complete arithmetic-progressions, and every infinite such image contained in the two-square locus is ruled out if it meets the identity real component; hence connected-real-locus models admit only finite projective curve images inside S." }, { "id": 20000457, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0109", "title": "Finite-level criteria and an exact finite dense-prime example", "statement": "Question 24. Given an elliptic curve E/ Q, rk E(Q) > 0, describe the set of primes p such that E(Q) is dense in E(Qp). Is this set nonempty? Is there a modular interpretation of this problem? (Takloo-Bighash)", "original_statement": "Question 24. Given an elliptic curve E/ Q, rk E(Q) > 0, describe the set of primes p such that E(Q) is dense in E(Qp). Is this set nonempty? Is there a modular interpretation of this problem? (Takloo-Bighash)", "clean_statement": "Question 24. Given an elliptic curve E/ Q, rk E(Q) > 0, describe the set of primes p such that E(Q) is dense in E(Qp). Is this set nonempty? Is there a modular interpretation of this problem? (Takloo-Bighash)", "statement_status": "exact", "statement_verification": "This is attempt 1 for record **AIM-ALGEBRAIC_NUMBER_THEORY-0109**, source file *aim-algebraic-number-theory-notes.json*, zero-based source index 108. The record is Question 24 in the AIM workshop document *Rational and integral points on higher dimensional varieties*. I checked the official PDF directly (page 52). Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 24\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[108]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 24. Given an elliptic curve E/ Q, rk E(Q) > 0, describe the set of primes p such that E(Q) is dense in E(Qp). Is this set nonempty? Is there a modular interpretation of this problem? (Takloo-Bighash)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) You need surjectivity of the reduction map (Murty and Gupta have some results) and the surjectivity of the map on formal groups (Silverman). One suspects that for any such curve \\n\\nE there exists a prime p with this property. (ii) This is related to Brauer-type problems for surfaces. (iii) You should be able to collect data on this to see if there is a positive density. (McCallum) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0109", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every prime p, density of a subgroup Gamma in E(Q_p) is equivalent to surjectivity onto the finite quotient E(Q_p)/E_{m_p+1}, with m_p=1 for odd p and m_2=2. At odd p the exact index is [E(Q_p):closure(Gamma)]=[E(Q_p)/E_1:image(Gamma)] p^{s_p(Gamma)-1}; at a good odd prime this gives the finite congruence/Neron-model interpretation E(Q) surjects onto E(Z/p^2 Z). As a candidate low-confidence new worked example, for the rank-one curve 20622.j1 and its generator P the dense-prime set is exactly {3,7}. Universal nonemptiness remains open, and no classical modular-curve interpretation is claimed.\n\nCandidate contribution (exact finite dense-prime example; novelty confidence low): For E: y^2+xy=x^3-5455771x-5039899603 with E(Q)=Z P and P=(328219/100,109777927/1000), the set of primes p for which E(Q) is dense in E(Q_p) is exactly {3,7}." }, { "id": 20000458, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0110", "title": "A representation-sensitive decidability dichotomy for integer rotation constraints", "statement": "Question 25. Is there an algorithm to decide solubility of a system of linear equations\n\nai1x1 + · · · + ain xn = bi\n\nfor ai, b, x i ∈ Z, together with equations of the form\n\n{xiαj } < r j\n\nfor αj ∈ R \\ Q, ri ∈ Q? (Poonen)", "original_statement": "Question 25. Is there an algorithm to decide solubility of a system of linear equations \n\nai1x1 + · · · + ain xn = bi\n\nfor ai, b, x i ∈ Z, together with equations of the form \n\n{xiαj } < r j\n\nfor αj ∈ R \\ Q, ri ∈ Q? (Poonen)", "clean_statement": "Question 25. Is there an algorithm to decide solubility of a system of linear equations\n\nai1x1 + · · · + ain xn = bi\n\nfor ai, b, x i ∈ Z, together with equations of the form\n\n{xiαj } < r j\n\nfor αj ∈ R \\ Q, ri ∈ Q? (Poonen)", "statement_status": "exact", "statement_verification": "The canonical record is Question 25 from the AIM workshop *Rational and integral points on higher dimensional varieties*. The official HTML and PDF give the question as follows (with subscripts normalized typographically but not mathematically altered):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 25\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[109]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 25. Is there an algorithm to decide solubility of a system of linear equations \\n\\nai1x1 + · · · + ain xn = bi\\n\\nfor ai, b, x i ∈ Z, together with equations of the form \\n\\n{xiαj } < r j\\n\\nfor αj ∈ R \\\\ Q, ri ∈ Q? (Poonen)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) This is a problem in linear programming; add variables. (Voloch) (ii) A negative answer would have implied undecidability for Hilbert's tenth problem over Q.\\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0110", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The reconstructed system is decidable when its finite irrational tuple is supplied with fast Cauchy names and a Z-basis of all affine integer relations: zero strata, Smith normal form, and the character form of Kronecker's closure theorem reduce solubility to finitely many exact strict linear-feasibility tests. Consequently, exactly represented real-algebraic inputs are uniformly decidable, and for any fixed tuple the family is decidable if and only if each fractional-part constant is computable, with the positive direction nonuniformly hard-coding the finite relation basis. In contrast, fast Cauchy names alone admit no uniform solver, even under a quadratic-algebraic promise and for an explicit template with three integer variables, two equations, two inequalities, and threshold 1/4.\n\nCandidate contribution (representation-sensitive decidability dichotomy; novelty confidence low): Candidate novelty: a basis of the affine integer-relation lattice is a sufficient finite package of exact-dependence information for a terminating Kronecker-orbit algorithm, while an explicit three-variable halting gadget proves that fast Cauchy names do not supply enough such information uniformly even when every named irrational is promised quadratic algebraic." }, { "id": 20000459, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0111", "title": "Low-degree integral curves on the Fermat fourfold", "statement": "Question 26. Describe the variety of curves of low degree on the Fermat variety\n\nF: xd\n\n> 1\n\n+ xd\n\n> 2\n\n+ · · · + xd\n\n> 6\n\n=\n0. (Heath-Brown)", "original_statement": "Question 26. Describe the variety of curves of low degree on the Fermat variety \n\nF: xd \n\n> 1\n\n+ xd \n\n> 2\n\n+ · · · + xd \n\n> 6\n\n= \n0. (Heath-Brown)", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record is affected by PDF extraction errors in the exponents. The official AIM workshop PDF gives the following statement (Question 26, page 53 of the PDF):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 26\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[110]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 26. Describe the variety of curves of low degree on the Fermat variety \\n\\nF: xd \\n\\n> 1\\n\\n+ xd \\n\\n> 2\\n\\n+ · · · + xd \\n\\n> 6\\n\\n= \\n0. (Heath-Brown)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) Lines are known, but that is all. Hope that there are none or very few. (ii) The curves should be over C (for now). 54 \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0111", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an integral complex curve C contained in the Fermat fourfold F_d in projective 5-space, of degree delta at least 2, the inequality d > 10 delta - 6 forces C to lie in a unique one of the 15 d^3 standard coordinate-pairing planes. Consequently, in this range the closed-point integral-curve locus is the disjoint union, over those planes, of copies of the open locus of irreducible degree-delta plane forms; no assertion is made about Hilbert-scheme nilpotents or embedded infinitesimal directions. The same descent proves that every nonlinear rational curve is standard-plane for d at least 24, and a genus-g curve with g at least 1 is standard-plane when d > 20g + 4. For conics the cutoff d at least 15 is sharp, because Reznick's published identity yields a nonstandard smooth conic on F_14.\n\nCandidate contribution (low-degree Fermat-curve classification; novelty confidence low): Candidate novelty: grouping proportional coordinate sections and descending to an inclusion-minimal diagonal relation transfers Salberger's published diagonal-curve inequality to every integral curve on F_d, yielding the explicit unique-standard-plane classification for d > 10 delta - 6, its closed-point parameter description, and the bounded-genus variants; Reznick's F_14 conic proves the conic boundary is exact." }, { "id": 20000460, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0112", "title": "The quadratic degree scale and an irreducibility diagnostic", "statement": "Question 27. Given a plane curve of degree d, the number of points on X of height at most d is ø d3. There are curves which have the number of rational points ¿ d2.Can you do better than ø d3? (Heath-Brown)", "original_statement": "Question 27. Given a plane curve of degree d, the number of points on X of height at most d is ø d3. There are curves which have the number of rational points ¿ d2.Can you do better than ø d3? (Heath-Brown)", "clean_statement": "Question 27. Given a plane curve of degree d, the number of points on X of height at most d is ø d3. There are curves which have the number of rational points ¿ d2.Can you do better than ø d3? (Heath-Brown)", "statement_status": "exact", "statement_verification": "Thus the repeated use of \\(d\\), both as degree and height cutoff, is genuine; it is not an extraction error. The corrupted symbols “ø” and “¿” in the canonical JSON are respectively \\(\\ll\\) and \\(\\gg\\), and the exponents are \\(3,2,5/2\\).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 27\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[111]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 27. Given a plane curve of degree d, the number of points on X of height at most d is ø d3. There are curves which have the number of rational points ¿ d2.Can you do better than ø d3? (Heath-Brown)\"\nOriginal remarks: [\"Remarks. (Roger Heath-Brown, 2004-09-25) observes that actually d5/2 is easy, so the chal-lenge should be to improve on this. \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0112", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The known solution to the intended integral reading is the 2025 theorem of Binyamini, Cluckers, and Kato: N(C;d) is at most c d^2 d^(2/d) (log d)^12, hence O(d^2 (log d)^12)=o(d^(5/2)). The candidate contribution of this packet is only the independently proved additional mathematics: for every d, a smooth geometrically integral separated-variable plane curve containing a full d-by-d rational grid of height at most d, and, for all sufficiently large d, a reduced reducible degree-d curve with Omega(d^(8/3)) such points, which proves that the source's omitted irreducibility convention is essential.\n\nCandidate contribution (explicit smooth construction and reducible-reading obstruction; novelty confidence low): Candidate novelty: the curves product_{i=1}^d(X-iZ) + lambda_d product_{j=1}^d(Y-jZ) = 0 admit an integer 1 <= lambda_d <= (d-1)^2+1 for which they are smooth and geometrically integral while containing all d^2 grid points of height at most d; in contrast, a coefficient-height and line-lattice construction gives reduced reducible degree-d curves with Omega(d^(8/3)) such points, making the missing integrality hypothesis detectable from the AIM exponents." }, { "id": 20000461, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0113", "title": "A local-defect lower bound for the 3-Selmer group over cyclic cubic fields", "statement": "Question 28. Let E be an elliptic curve over Q, and P ∈ E(Q)[3], P 6 = O,such that rk E(Q) = 0. Let K/ Q be a cyclic extension of degree 3 of conductor f, and let\n\nν = ν(f ) be the number of distinct primes dividing f. It is true that 3 ν−1 | Sel 3(E, K )? (Chantal David)", "original_statement": "Question 28. Let E be an elliptic curve over Q, and P ∈ E(Q)[3], P 6 = O,such that rk E(Q) = 0. Let K/ Q be a cyclic extension of degree 3 of conductor f, and let \n\nν = ν(f ) be the number of distinct primes dividing f. It is true that 3 ν−1 | Sel 3(E, K )? (Chantal David)", "clean_statement": "Question 28. Let E be an elliptic curve over Q, and P ∈ E(Q)[3], P 6 = O,such that rk E(Q) = 0. Let K/ Q be a cyclic extension of degree 3 of conductor f, and let\n\nν = ν(f ) be the number of distinct primes dividing f. It is true that 3 ν−1 | Sel 3(E, K )? (Chantal David)", "statement_status": "exact", "statement_verification": "The canonical JSON record is an OCR extraction of Question 28 from the AIM workshop list *Rational and integral points on higher dimensional varieties*. The OCR lost the cardinality sign in the displayed divisibility. The official AIM HTML and PDF were checked and give the following question (with notation modernized only in the subscript/base-field placement):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 28\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[112]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 28. Let E be an elliptic curve over Q, and P ∈ E(Q)[3], P 6 = O,such that rk E(Q) = 0. Let K/ Q be a cyclic extension of degree 3 of conductor f, and let \\n\\nν = ν(f ) be the number of distinct primes dividing f. It is true that 3 ν−1 | Sel 3(E, K )? (Chantal David)\"\nOriginal remarks: [\"Remarks. Let K/ Q be a cyclic cubic extension. Then the 3-rank of the class group of K is controlled by primes dividing discriminant of K; if there are d such primes, then 3 d−1 divides the class number hK (genus theory). Problem: carry this over to a guaranteed contribution of 3-primary component of the 3-Selmer group (over K) of an elliptic curve E/ Q. Does 3 d−1\\n\\ndivide the Selmer order? Progress: Fix E/ Q with a rational 3-torsion point, and let K vary as above. Then there is a constant cE depending only on E such that 3 d−cE divides the Selmer order. (Clarification request: do the mean the algebraic part of the L-function, or the 3-part of the Selmer group, or the group obtained from a 3-descent? That affects the value of cE.) This has been checked computationally (by checking special values of L-series). Other problems? Poonen: replace 3 by another prime p, or replace Q by another number field, etc. \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0113", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For C=

and the 3-isogeny phi:E->E/C, the everywhere-unramified class-group characters satisfy an exact defect identity dim(U_K intersect Sel_phi)=r_3(Cl_K)-delta(E,K), and Sel_phi injects into Sel_3 because the dual kernel is mu_3 and a cyclic cubic field is totally real. Genus theory gives r_3(Cl_K)>=nu(f)-1, while reciprocity bounds delta by the F_3-span rho(E,K) of prime classes above 3N_E, with rho(E,K)<=2 omega(3N_E). Hence dim Sel_3(E/K)>=nu(f)-1-delta(E,K)>=nu(f)-1-2 omega(3N_E), and the exact AIM divisibility holds whenever delta(E,K)=0, in particular when the prime classes above 3N_E lie in 3Cl_K. The rank-zero hypothesis is not needed for this partial theorem.\n\nCandidate contribution (explicit local-defect decomposition; novelty confidence low): The class-group contribution in the exact cyclic-cubic/rational-3-torsion setting has a transparent localization defect delta(E,K), loses no extra dimension in passing from Sel_phi to Sel_3, and satisfies delta(E,K)<=rho(E,K)<=2 omega(3N_E)." }, { "id": 20000462, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0114", "title": "A PAC counterexample and the projectivity boundary", "statement": "Question 29. If K is a field such that all O-acyclic varieties over K have a point, is Gal( K/K ) topologically generated by one element?\n\nProblem/", "original_statement": "Question 29. If K is a field such that all O-acyclic varieties over K have a point, is Gal( K/K ) topologically generated by one element? \n\nProblem/", "clean_statement": "Question 29. If K is a field such that all O-acyclic varieties over K have a point, is Gal( K/K ) topologically generated by one element?\n\nProblem/", "statement_status": "exact", "statement_verification": "The canonical record is Question 29 in the AIM workshop notes *Rational and integral points on higher dimensional varieties*. The extracted text reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 29\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[113]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 29. If K is a field such that all O-acyclic varieties over K have a point, is Gal( K/K ) topologically generated by one element? \\n\\nProblem/\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0114", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the standard smooth, projective, geometrically connected interpretation of the O-acyclic class, the answer is no: a characteristic-zero PAC field K with absolute Galois group isomorphic to the free profinite group of rank two has a rational point on every geometrically integral K-variety, hence on every variety in the intended test class, but its Galois group is not procyclic because it surjects onto C_2 x C_2. In addition, the O-acyclic rational-point property passes to finite separable extensions; testing it on Severi--Brauer varieties forces Br(L)=0 at every finite separable level and, for perfect K, forces the absolute Galois group to be projective.\n\nCandidate contribution (PAC counterexample and projectivity boundary; novelty confidence low): Candidate synthesis: the O-acyclic rational-point property is inherited by every finite separable extension through Weil restriction, so its Severi--Brauer subfamily forces all finite-level Brauer groups to vanish and, for perfect base fields, forces projectivity of the absolute Galois group; pairing this necessary boundary with a classical PAC field having free profinite absolute Galois group of rank two shows that the hypothesis does not control generator rank." }, { "id": 20000463, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0115", "title": "A withdrawn duplicate of the D4 and E6 cubic counting problems", "statement": "Question 30. This problem has been withdrawn.\n\nProblem/", "original_statement": "Question 30. This problem has been withdrawn. \n\nProblem/", "clean_statement": "Question 30. This problem has been withdrawn.\n\nProblem/", "statement_status": "exact", "statement_verification": "The exact extracted canonical field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 30\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[114]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 30. This problem has been withdrawn. \\n\\nProblem/\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0115", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The current AIM record is an authoritative withdrawal tombstone and therefore is not an active problem. An official pre-withdrawal archive shows that Question 30 named two cubic surfaces; the first is exactly Question 5's D4 surface, while the second is Question 5(vii)'s E6 surface after z maps to -z and is height-preservingly projectively isomorphic to the standard E6 model by [x:y:z:t] maps to [y:x:t:-z]. Under the standard maximum-height, line-deleted interpretation, Le Boudec's 2014 theorem and the 2007 theorem of de la Breteche, Browning, and Derenthal settle the respective counting problems. Exact redundancy plausibly explains withdrawal, but the editorial motive is not documented.\n\nCandidate contribution (equivalence; novelty confidence low): Candidate novelty: the pre-withdrawal Question 30 is redundant term by term with Question 5--its first equation is Question 5's D4 equation, and its second is Question 5(vii) under z maps to -z; moreover [x:y:z:t] maps to [y:x:t:-z] is a GL_4(Z), maximum-height-preserving isomorphism from the latter surface to the published standard E6 model.", "research_difficulty_suggested": "L2" }, { "id": 20000464, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0116", "title": "A fiberwise harmonic bridge on the quintic del Pezzo torsor", "statement": "Question 31. Harmonic analysis and nonabelian torsors: Takloo-Bighash: Do these give you methods to find rational points? Hassett: Is there a harmonic analysis argument to count points on the quintic del Pezzo surface? Wooley: no examples known where you can combine harmonic analysis with torsors.\n\nProblem/", "original_statement": "Question 31. Harmonic analysis and nonabelian torsors: Takloo-Bighash: Do these give you methods to find rational points? Hassett: Is there a harmonic analysis argument to count points on the quintic del Pezzo surface? Wooley: no examples known where you can combine harmonic analysis with torsors. \n\nProblem/", "clean_statement": "Question 31. Harmonic analysis and nonabelian torsors: Takloo-Bighash: Do these give you methods to find rational points? Hassett: Is there a harmonic analysis argument to count points on the quintic del Pezzo surface? Wooley: no examples known where you can combine harmonic analysis with torsors.\n\nProblem/", "statement_status": "exact", "statement_verification": "The canonical JSON `problem` field is Question 31 in the AIM workshop notes *Rational and integral points on higher dimensional varieties*, followed by the stray text `Problem/` after a blank line. The official AIM PDF and HTML show that this suffix is a page-extraction/navigation artifact, not part of the question. The recovered question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 31\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[115]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 31. Harmonic analysis and nonabelian torsors: Takloo-Bighash: Do these give you methods to find rational points? Hassett: Is there a harmonic analysis argument to count points on the quintic del Pezzo surface? Wooley: no examples known where you can combine harmonic analysis with torsors. \\n\\nProblem/\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0116", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For fixed pairwise-coprime nonzero outer Cox variables a_1, a_2, a_3, a_4, the integral solutions of the five Pluecker equations are parametrized bijectively by an explicit rank-three congruence lattice of covolume |a_1 a_4|. Its dual lattice is parametrized explicitly by two residue frequencies, giving an exact fiberwise Poisson formula. After the standard Moebius inversions and height smoothing, this reduces a harmonic proof of the quintic Manin-Peyre asymptotic to a uniform aggregate bound for nonzero dual frequencies, plus smoothing removal; those analytic bounds are not proved here.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: the explicit two-parameter dual-lattice formula organizes the three-free-variable quintic torsor slice into a canonical additive Poisson expansion and identifies the aggregate nonzero-frequency estimate after Moebius inversion as the precise missing harmonic-analysis bottleneck.", "research_difficulty_suggested": "L4" }, { "id": 20000465, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0117", "title": "Random hypersurfaces, local conditioning, and height transfer", "statement": "Question\n3\n2. (Voloch): Given a family of hypersurfaces, show that \"almost all\" members of the family satisfy the Hasse Principle in \"interesting\" circumstances. Colliot-Th´ el` ene: maybe the opposite? Poonen: the family was all hypersurfaces of degree d in Pn with coefficients of height\n\n≤ H, over some fixed number field K. The problem was: find the proportion of these that satisfy the Hasse Principle (originally, that have rational points, but we can find the local points easily). Wooley and Venkatesh (tentative):\n\nA−s#{(a1,..., a s): |ai| ≤ A, a 1xd\n\n> 1\n\n+ · · · + asxds = 0 satisfies HP }\n\nis asymptotic to the product of local densities, at least for s ≥ 3d + 1. This is progress for\n\nd large: for an individual a, one has HP (using current technology) for s >> d log d. Proof uses (of course) circle method. de Jong: for cubics in P2, what do conclusions of this type say about 3-Selmer groups of elliptic curves? 55\n\nStoll: given local points, proportion that have rational points (i.e., satisfy HP) should be 0. Mazur: can you put an exponent on that? Stoll: wait for experimental evidence.\n\nProblem/", "original_statement": "Question \n3\n2. (Voloch): Given a family of hypersurfaces, show that \"almost all\" members of the family satisfy the Hasse Principle in \"interesting\" circumstances. Colliot-Th´ el` ene: maybe the opposite? Poonen: the family was all hypersurfaces of degree d in Pn with coefficients of height \n\n≤ H, over some fixed number field K. The problem was: find the proportion of these that satisfy the Hasse Principle (originally, that have rational points, but we can find the local points easily). Wooley and Venkatesh (tentative): \n\nA−s#{(a1,..., a s): |ai| ≤ A, a 1xd \n\n> 1\n\n+ · · · + asxds = 0 satisfies HP }\n\nis asymptotic to the product of local densities, at least for s ≥ 3d + 1. This is progress for \n\nd large: for an individual a, one has HP (using current technology) for s >> d log d. Proof uses (of course) circle method. de Jong: for cubics in P2, what do conclusions of this type say about 3-Selmer groups of elliptic curves? 55 \n\nStoll: given local points, proportion that have rational points (i.e., satisfy HP) should be 0. Mazur: can you put an exponent on that? Stoll: wait for experimental evidence. \n\nProblem/", "clean_statement": "Question\n3\n2. (Voloch): Given a family of hypersurfaces, show that \"almost all\" members of the family satisfy the Hasse Principle in \"interesting\" circumstances. Colliot-Th´ el` ene: maybe the opposite? Poonen: the family was all hypersurfaces of degree d in Pn with coefficients of height\n\n≤ H, over some fixed number field K. The problem was: find the proportion of these that satisfy the Hasse Principle (originally, that have rational points, but we can find the local points easily). Wooley and Venkatesh (tentative):\n\nA−s#{(a1,..., a s): |ai| ≤ A, a 1xd\n\n> 1\n\n+ · · · + asxds = 0 satisfies HP }\n\nis asymptotic to the product of local densities, at least for s ≥ 3d + 1. This is progress for\n\nd large: for an individual a, one has HP (using current technology) for s >> d log d. Proof uses (of course) circle method. de Jong: for cubics in P2, what do conclusions of this type say about 3-Selmer groups of elliptic curves? 55\n\nStoll: given local points, proportion that have rational points (i.e., satisfy HP) should be 0. Mazur: can you put an exponent on that? Stoll: wait for experimental evidence.\n\nProblem/", "statement_status": "exact", "statement_verification": "The canonical JSON record is damaged by line-break/OCR errors: its number appears as `3\\n2`, one exponent is detached, and a page number is inserted into the sentence. The official AIM HTML version identifies it unambiguously as **Problem/Question 32** from the workshop *Rational and integral points on higher dimensional varieties* [AIM]. In cleaned mathematical notation, the record asks three related but distinct questions.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[116]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question \\n3\\n2. (Voloch): Given a family of hypersurfaces, show that \\\"almost all\\\" members of the family satisfy the Hasse Principle in \\\"interesting\\\" circumstances. Colliot-Th´ el` ene: maybe the opposite? Poonen: the family was all hypersurfaces of degree d in Pn with coefficients of height \\n\\n≤ H, over some fixed number field K. The problem was: find the proportion of these that satisfy the Hasse Principle (originally, that have rational points, but we can find the local points easily). Wooley and Venkatesh (tentative): \\n\\nA−s#{(a1,..., a s): |ai| ≤ A, a 1xd \\n\\n> 1\\n\\n+ · · · + asxds = 0 satisfies HP }\\n\\nis asymptotic to the product of local densities, at least for s ≥ 3d + 1. This is progress for \\n\\nd large: for an individual a, one has HP (using current technology) for s >> d log d. Proof uses (of course) circle method. de Jong: for cubics in P2, what do conclusions of this type say about 3-Selmer groups of elliptic curves? 55 \\n\\nStoll: given local points, proportion that have rational points (i.e., satisfy HP) should be 0. Mazur: can you put an exponent on that? Stoll: wait for experimental evidence. \\n\\nProblem/\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0117", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The three families in the recovered AIM question have different current outcomes: the full Fano-family density equality over Q is proved except for cubic surfaces; the diagonal product asymptotic follows in the overlap of the Bright--Browning--Loughran local-density theorem and the Bruedern--Dietmann exceptional-set theorem; and Stoll's proposed zero conditional density for plane cubics is refuted by Bhargava's positive-proportion result. In addition, a proved conditioning and height-body transfer lemma converts the Browning--Le Boudec--Sawin Euclidean ambient failure bound into a max-coefficient conditional failure bound of order (log H)^(-1/(48n)), while explicit examples show why positive local mass and non-thinness cannot be omitted.\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: if a target height family T_H is contained in a reference family R_{beta H}, the reference Hasse-principle failure count is at most C t^m e(t), and the target locally soluble count is at least c H^m, then the target conditional failure proportion is at most (C beta^m/c)e(beta H); applying this to Euclidean and max coefficient norms yields the explicit max-height logarithmic conditional rate, and the report gives sharp logical obstructions when local mass vanishes or the target is thin." }, { "id": 20000466, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0118", "title": "A height-index transfer for a conic bundle over a rank-one elliptic curve", "statement": "Question\n3\n3. (Poonen) Instead of counting points of bounded height on a va-riety, count points of bounded height in a Diophantine set. (I.e., counting points in a base the fibre above which has a rational point.) What rates of growth can you get? Example: conic bundle over an elliptic curve. Heuristically, it appears rate of growth can be log log B,whereas for varieties it always turns out to be cB α(log B)β.\n\nProblem/", "original_statement": "Question \n3\n3. (Poonen) Instead of counting points of bounded height on a va-riety, count points of bounded height in a Diophantine set. (I.e., counting points in a base the fibre above which has a rational point.) What rates of growth can you get? Example: conic bundle over an elliptic curve. Heuristically, it appears rate of growth can be log log B,whereas for varieties it always turns out to be cB α(log B)β.\n\nProblem/", "clean_statement": "Question\n3\n3. (Poonen) Instead of counting points of bounded height on a va-riety, count points of bounded height in a Diophantine set. (I.e., counting points in a base the fibre above which has a rational point.) What rates of growth can you get? Example: conic bundle over an elliptic curve. Heuristically, it appears rate of growth can be log log B,whereas for varieties it always turns out to be cB α(log B)β.\n\nProblem/", "statement_status": "exact", "statement_verification": "The canonical record is zero-based index 117 of `aim-algebraic-number-theory-notes.json`. Its OCR has split the problem number as `3\\n3`, inserted the line-break hyphen `va-riety`, and left a trailing `Problem/`. The AIM PDF and HTML version both give the following recovered text:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[117]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question \\n3\\n3. (Poonen) Instead of counting points of bounded height on a va-riety, count points of bounded height in a Diophantine set. (I.e., counting points in a base the fibre above which has a rational point.) What rates of growth can you get? Example: conic bundle over an elliptic curve. Heuristically, it appears rate of growth can be log log B,whereas for varieties it always turns out to be cB α(log B)β.\\n\\nProblem/\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0118", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For E: y^2+y=x^3-x with E(Q) generated by P=(0,0), the generic conic U^2+V^2=(x+2)W^2 has no Q(E)-point. Writing x(nP)+2=A_n/D_n^2 in lowest terms, its fibre over nP is Q-soluble exactly when A_n is a sum of two integer squares. The proved canonical-height comparison gives explicit two-sided counting inequalities and, for every kappa>0, the same-constant equivalence S(T)~kappa log T if and only if N(B)~kappa log log B. The Landau model predicts the premise S(T)~kappa log T but does not prove it.\n\nCandidate contribution (equivalence; novelty confidence low): Candidate novelty: for the explicit non-section bundle U^2+V^2=(x+2)W^2 over E: y^2+y=x^3-x, fibre solubility reduces to a two-square test on the integer numerator A_n, and S(T)~kappa log T is equivalent to N(B)~kappa log log B with the same kappa.", "research_difficulty_suggested": "L4" }, { "id": 20000467, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0119", "title": "Arakelov replacement and a surface exponent-transfer lemma", "statement": "Question\n3\n4. (Bogomolov) Replace Heath-Brown by Arakelov (et al). #{(x, y, z ) ∈ P2: |x|, |y|, |z| ≤ B, P (x, y, z ) = 0 } << B α+≤\n\nwith α ≤ 2\n\n> d, d = deg P. Analogue for surfaces, etc.\n\nProblem/", "original_statement": "Question \n3\n4. (Bogomolov) Replace Heath-Brown by Arakelov (et al). #{(x, y, z ) ∈ P2: |x|, |y|, |z| ≤ B, P (x, y, z ) = 0 } << B α+≤\n\nwith α ≤ 2 \n\n> d, d = deg P. Analogue for surfaces, etc. \n\nProblem/", "clean_statement": "Question\n3\n4. (Bogomolov) Replace Heath-Brown by Arakelov (et al). #{(x, y, z ) ∈ P2: |x|, |y|, |z| ≤ B, P (x, y, z ) = 0 } << B α+≤\n\nwith α ≤ 2\n\n> d, d = deg P. Analogue for surfaces, etc.\n\nProblem/", "statement_status": "exact", "statement_verification": "The canonical record is zero-based index 118 of `aim-algebraic-number-theory-notes.json`, extracted from the AIM workshop *Rational and integral points on higher dimensional varieties*. The record is visibly damaged: its number is split as `3\\n4`, exponents and inequality signs are displaced, and the final `Problem/` is HTML navigation text. The official AIM HTML and page 55 of the official PDF give the following unambiguous text:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[118]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question \\n3\\n4. (Bogomolov) Replace Heath-Brown by Arakelov (et al). #{(x, y, z ) ∈ P2: |x|, |y|, |z| ≤ B, P (x, y, z ) = 0 } << B α+≤\\n\\nwith α ≤ 2 \\n\\n> d, d = deg P. Analogue for surfaces, etc. \\n\\nProblem/\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0119", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Chen's published Arakelov slope-method covering theorem supplies the requested methodological replacement for the degree-d curve bound. In addition, for a smooth degree-d surface X in P^3_Q and U_r obtained by deleting all geometric integral curves of degree at most r, with 0 <= r <= d-2, iterating Chen's published covering theorem and curve corollary proves N_{U_r}(B) <<_{d,r,epsilon} B^{3/sqrt(d)+2/(r+1)+epsilon}; the endpoint r=d-2 recovers the Heath-Brown surface exponent recorded by AIM. A separate polylogarithmic refinement is proved only conditional on Chiu's arXiv:2412.05205v1 theorem as stated.\n\nCandidate contribution (lemma; novelty confidence low): Candidate exponent-transfer lemma: for every smooth X_d in P^3_Q and 0 <= r <= d-2, Chen's Arakelov hypersurface covering and curve estimate combine to give N_{U_r}(B) << B^{3/sqrt(d)+2/(r+1)+epsilon}, including a rigorous treatment of Q-integral but geometrically reducible intersection components; conditional on Chiu's preprint covering theorem, the epsilon loss can be replaced by a power of log B using the Binyamini-Cluckers-Kato curve bound." }, { "id": 20000468, "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0120", "title": "Weak approximation for finite quotients with a controlled auxiliary branch point", "statement": "Question 35. Weak approximation for complex function fields: Weak approximation over a finite extension L of C(t), from Hassett's lecture. Colliot-Th´ el` ene: Known for X a connected linear algebraic group over L (reduce to reductive groups immediately; since field has cohomological dimension 1, one has Borel subgroup, reduce to tori; reduce to quasi-trivial tori, which are open subsets of affine space). CT: X = G/H, where H is a subgroup of G (not necessarily normal). Same should go through if H is connected (techniques of Borovoi et al.). CT: Question: Decide whether weak approximation holds for GL n/G, where G is a finite subgroup of GL n. This seems nontrivial. (de Jong thinks he can do this; Graber is unsure.) CT says de Jong claims: Let X/L be an arbitrary smooth projective, geometrically and rationally connected variety. Then weak approxmation holds. (de Jong: do this by reducing to characteristic p. Not in the dJ-H-S paper.) CT: there is an Enriques surface over C(t) that has a rational point but does not satisfy weak approximation.", "original_statement": "Question 35. Weak approximation for complex function fields: Weak approximation over a finite extension L of C(t), from Hassett's lecture. Colliot-Th´ el` ene: Known for X a connected linear algebraic group over L (reduce to reductive groups immediately; since field has cohomological dimension 1, one has Borel subgroup, reduce to tori; reduce to quasi-trivial tori, which are open subsets of affine space). CT: X = G/H, where H is a subgroup of G (not necessarily normal). Same should go through if H is connected (techniques of Borovoi et al.). CT: Question: Decide whether weak approximation holds for GL n/G, where G is a finite subgroup of GL n. This seems nontrivial. (de Jong thinks he can do this; Graber is unsure.) CT says de Jong claims: Let X/L be an arbitrary smooth projective, geometrically and rationally connected variety. Then weak approxmation holds. (de Jong: do this by reducing to characteristic p. Not in the dJ-H-S paper.) CT: there is an Enriques surface over C(t) that has a rational point but does not satisfy weak approximation.", "clean_statement": "Question 35. Weak approximation for complex function fields: Weak approximation over a finite extension L of C(t), from Hassett's lecture. Colliot-Th´ el` ene: Known for X a connected linear algebraic group over L (reduce to reductive groups immediately; since field has cohomological dimension 1, one has Borel subgroup, reduce to tori; reduce to quasi-trivial tori, which are open subsets of affine space). CT: X = G/H, where H is a subgroup of G (not necessarily normal). Same should go through if H is connected (techniques of Borovoi et al.). CT: Question: Decide whether weak approximation holds for GL n/G, where G is a finite subgroup of GL n. This seems nontrivial. (de Jong thinks he can do this; Graber is unsure.) CT says de Jong claims: Let X/L be an arbitrary smooth projective, geometrically and rationally connected variety. Then weak approxmation holds. (de Jong: do this by reducing to characteristic p. Not in the dJ-H-S paper.) CT: there is an Enriques surface over C(t) that has a rational point but does not satisfy weak approximation.", "statement_status": "exact", "statement_verification": "The canonical record is Question 35 from the AIM workshop *Rational and integral points on higher dimensional varieties*. Its official HTML transcription is available at https://www.aimath.org/WWN/qptsurface2/articles/html/27a/ and the workshop PDF at https://aimath.org/WWN/qptsurface2/qptsurface2.pdf.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Algebraic number theory\nWorkshop: Rational and integral points on higher dimensional varieties\nSection: \nSource item: 35\nSource URL: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf\nCanonical location: aim-algebraic-number-theory-notes.json notes[119]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 35. Weak approximation for complex function fields: Weak approximation over a finite extension L of C(t), from Hassett's lecture. Colliot-Th´ el` ene: Known for X a connected linear algebraic group over L (reduce to reductive groups immediately; since field has cohomological dimension 1, one has Borel subgroup, reduce to tori; reduce to quasi-trivial tori, which are open subsets of affine space). CT: X = G/H, where H is a subgroup of G (not necessarily normal). Same should go through if H is connected (techniques of Borovoi et al.). CT: Question: Decide whether weak approximation holds for GL n/G, where G is a finite subgroup of GL n. This seems nontrivial. (de Jong thinks he can do this; Graber is unsure.) CT says de Jong claims: Let X/L be an arbitrary smooth projective, geometrically and rationally connected variety. Then weak approxmation holds. (de Jong: do this by reducing to characteristic p. Not in the dJ-H-S paper.) CT: there is an Enriques surface over C(t) that has a rational point but does not satisfy weak approximation.\"\nOriginal remarks: [\"Remarks. Yuri Tschinkel remarks that there is progress on Nr 35 due to Colliot-Thelene/Gille and Madore. To this Brendan Hasset adds that Madore has announced a proof of weak approximation in smooth fibers for a cubic surface over the function field K(C), where C is a curve and K is algebraically closed of characteristic zero. \\n\\nB.2 Photos \\n\\nWilliam Stein has prepared a gallery of photos 3.\\n\\nChapter C: Glossary \\n\\nAbelian variety: A smooth projective geometrically integral group variety over a field. Over the complex numbers abelian varieties are tori. \\n\\nBrauer-Manin obstruction: The terminology is utterly awful! Many families don't sat-isfy Hasse Principle. One explanation of Manin (see his paper): a cohomological obstruction using the Brauer group of the variety. \\n\\n> 3http://modular.fas.harvard.edu/pics/ascent11/12-2002/AIM rational-points-workshop/ 56\\n\\nIf a variety has a local point everywhere then it has an adelic point. Manin defined, using a cohomological condition involving Brauer group, a subset of the adelic points that must contain the global points. Let X(Ak) be the adelic points of X. Consider the subset of points P with the property that for every element z ∈ Br( X) the system of elements (zv(P )) v has sum of invariants = 0. The B-M is an interesting construction in English. It is a nounal-phrase defined purely in terms of the sentences in which in which it may occur. There is no such actual object \\\"the Brauer-Manin obstruction\\\". Example: A variety that satisfies X(Ak) 6 = ∅ and X(Ak)Br = ∅ is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. For a long time people were interested in whether there are counterexamples ot Hasse principle not explained by the Brauer-Manin obstruction. X(Ak)Br 6 = ∅ but still has no global point (Skorobogotav found first example). After one glass of wine, McCallum advocates \\\" X(Ak)Br should be called the set of Brauer points\\\". \\n\\nBrauer-Severi variety: A twist of projective space Pn. Brauer-Severi varieties satisfy the Hasse principle. \\n\\nBSD conjecture-BSD-Birch and Swinnerton-Dyer: Let A be an abelian variety over a global field K and let L(A, s ) be the associated L-function. The Birch and Swinnerton-Dyer conjecture asserts that L(A, s ) extends to an entire function and ord s=1 L(A, s ) equals the rank of A(K). Moreover, the conjecture provides a formula for the leading coefficient of the Taylor expansions of L(A, s ) about s = 1 in terms of invariants of A.\\n\\nCalabi-Yau variety: An algebraic variety X over C is a Calabi-Yau variety if it has trivial canonical sheaf (i.e., the canonical sheaf is isomorphic to the structure sheaf). [Noriko just deleted the simply connected assumption.] \\n\\nDel Pezzo surface: A Del Pezzo surface is a Fano variety of dimension two. It can be shown that the Del Pezzo surfaces are exactly the surfaces that are geomet-rically either P1 × P1 or a blowup of P2 at up to 8 points in general position. By general position we mean that no three points lie on a line, no six points lie on a conic, and no eight lie points lie on a singular cubic with one of the eight points on the singularity. \\n\\nDescent: \\n\\nA. The process of expressing the rational points on a variety as the union of images of rational points from other varieties. B. The descent problem is as follows: Given a field extension L/K and a variety X\\n\\nover L, try to find a variety Y over K such that X = Y ×K L.\\n\\nDiophantine set: Let R be a ring. A subset A ⊂ Rn is diophantine over R if there exists a polynomial f ∈ R[t1,..., t n, x 1,..., x m] such that \\n\\nA = {~t ∈ Rn: ∃~x ∈ Rm such that f (~t, ~ x) = 0 }.\\n\\nEnriques Surface: A quotient of a K3 surface by a fixed-point free involution. Equivalently, the normalization of the singular surface of degree 6 in P3 whose singu-larities are double lines that form a general tetrahedron. 57 \\n\\nOver C an Enriques surface can be characterized cohomologically as follows: H 0(Ω 2 \\n\\n> X\\n\\n) = 0 and 2 KX = 0 but KX 6 = 0. \\n\\nFano variety-Fano: Anticanonical divisor ω⊗− 1 is ample. This class of varieties is \\\"sim-ple\\\" or \\\"close to rational\\\". For example, one conjectures that Brauer-Manin is only obstruc-tion. Manin-Batyrev conjecture: asymptotic for number of points of bounded height. AFano variety of dimension two is also called a Del Pezzo surface. \\n\\nFermat curve: A curve of the form xd + yd = zd. Good examples of many phenomenon. Good source of challenge problems. (E.g., FLT.) Lot of symmetry so you can compute a lot with them. Computations are surprising and nontrivial. They're abelian covers of P1\\n\\nramified at 3 points, so they occur in the fund. group of... More generally xd \\n\\n> 1\\n\\n+ · · · + xdn = 0 is sometimes called a Fermat variety. \\n\\nGeneral type: A variety X is of general type if there is a positive power of the canonical bundle whose global sections determine a rational map f: X → Pn with dim f (X) = dim X.(If X is of general type then there exists some positive power of the canonical bundle such that the corresponding map is birational to its image.) \\\"It is a moral judgement of geometers that you would be wise to stay away from the bloody things.\\\" - Swinnerton-Dyer \\n\\nHardy-Littlewood circle method: An analytic method for obtaining asymptotic formu-las for the number of solutions to certain equations satisfying certain bounds. \\n\\nHasse principle: A family of varieties satisfies the Hasse principle if whenever a variety in the family has points everywhere locally it has a point globally. Here \\\"everywhere locally\\\" means over the reals and p-adically for every p, and \\\"globally\\\" means over the rationals. Everywhere local solubility is necessary for global solubility. Hasse proved that it is also sufficient in the case of quatratic forms. \\n\\nHilbert's tenth problem: Let R be a commutative ring. Hilbert's tenth problem for R is to determine if there is an algorithm that decides whether or not a given system of polynomial equations with coefficients in R has a solution over R.\\n\\nJacobian: The Jacobian of a nonsingular projective curve X is an abelian variety whose points are in bijection with the group Pic 0(X) of isomorphism classes of invertible sheaves (or divisor classes) of degree 0. \\n\\nK3 surface: A surface with trivial canonical bundle and trivial fundamental group (i.e., a Calabi-Yau variety of dimension 2). \\n\\nLang's conjectures: \\n\\nA. Suppose k is a number field and X is a variety over k of general type. Then X(k) is not Zariski dense in X. (Also there are refinements where we specify which Zariski closed subset is supposed to contain X(k).) B. Suppose k is a number field and X is a variety over k. All but finitely many k-rational points on X lie in the special set. C. Let X be a variety over a number field k. Choose an embedding of k into the complex number C, and suppose that X(C) is hyperbolic: this means that every holomorphic map C → X(C) is constant. Then X(k) is finite. \\n\\nLocal to global principle: Another name for the Hasse principle. 58 \\n\\nPicard group: The Picard group of a variety is the group of isomorphism classes of invertible sheaves. \\n\\nPrym variety: A Prym variety is an abelian variety constructed in the following way. Let \\n\\nX and Y be curves and suppose f: X → Y is a degree 2 ´ etale (unramified) cover. The associated Prym variety is the connected component of the kernel of the Albanese map Jac( X) → Jac( Y ). The Prym variety can also be defined as the connected component of the \\n\\n−1 eigenspace of the involution on Jac( X) induced by f.\\n\\nRationally connected variety: There are three definitions of rationally connected. These are equivalent in characteristic zero but not in characteristic p.A. For any two points x, y ∈ X there exists a morphism φ: P1 → X such that φ(0) = x\\n\\nand φ(∞) = y.B. For any n points x1,..., x n ∈ X there exists a morphism φ: P1 → X such that \\n\\n{x1,..., x n} is a subset of φ(P1). C. For any two points x, y ∈ X there exist morphisms φi: P1 → X for i = 1,..., r such that φ1(0) = x, φr(0) = y, and for each i = 1,..., r − 1 the images of φi and φi+1 \\n\\nhave nontrivial intersection. \\n\\nSchinzel's Hypothesis: Suppose f1,..., f r ∈ Z[x] are irreducible and no prime divides \\n\\nf1(n)f2(n) · · · fr(n)for all n ∈ Z. Then there are infinitely many integers n such that |f1(n)|,..., |fr(n)| are simultaneously prime. \\n\\nSelmer group: Given Galois cohomology definition for any A ⊂ B. Example A = ker( φ)where φ is an isogeny of abelian variety. Accessible. It's what we can compute, at least in theory. \\n\\nShimura variety: A variety having a Zariski open subset whose set of complex points is analytically isomorphic to a quotient of a bounded symmetric domain X by a congruence subgroup of an algebraic group G that acts transitively on X. Examples include moduli spaces X0(N ) of elliptic curves with extra structure and Shimura curves which parametrize quaternionic multiplication abelian surfaces with extra structure. \\n\\nSpecial Set: The (algebraic) special set of a variety X is the Zariski closure of the union of all positive-dimensional images of morphisms from abelian varieties to X. Note that this contains all rational curves (since elliptic curves cover P1). \\n\\nTorsor: Let B be a variety over a field k and let G be an algebraic group over k. A left \\n\\nB-torsor under G is a B-scheme X with a B-morphism G ×k X → X such that for some ´etale covering {Ui → B} there is a G-equivariant isomorphism of Ui-schemes from X × Ui\\n\\nto G × Ui, for all i. If B = Spec( k) these are also called principal homogenous spaces. \\n\\nWaring's problem: Given k, find the smallest number gk such that every positive integer is a sum of gk positive kth powers. The \\\"easier\\\" Waring's problem refers to the analogous problem where the kth powers are permitted to be either positive or negative. Modification: Given k, find the smallest number Gk such that every sufficiently large positive integer is a sum of Gk positive kth powers. \\n\\nWeak approximation: For a projective variety X over a global field, say weak approxi-mation holds if X(K) is dense in the adelic points X(AK ). Simplest example where it holds: 59 \\n\\nP0, also P1. It does not hold for an elliptic curve over K. (For example, if E has rank 0 it clearly doesn't hold... but more generally could divide all generators by 2 and choose a prime that splits completely.) Example: \\\"Weak approximation does not hold for cubic surfaces.\\\" Example: \\\"The theory of abelian descent in some cases reduces the question of whether the Brauer-Manin obstruction is the only obstruction to Hasse on a base variety X to the question of whether weak approximation holds for a universal torsor.\\\" Example: \\\"Weak approximation on a moduli space of varieties yields the existence of varieties over a global field satisfying certain local conditions. For example, we want to know there is an elliptic curve over Q with certain behavior at 3, 5, 13, as long as can do it over local fields with that behavior, weak approximation on the moduli space gives you a global curve that has those properties (because P1 satisfies weak approximation).' \\n\\nChapter D: Miscellaneous Photos \\n\\nWilliam Stein has prepared a gallery of photos 4.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/qptsurface2/qptsurface2.pdf", "tags": [ "aim", "AIM-ALGEBRAIC_NUMBER_THEORY-0120", "aim-domain:algebraic-number-theory", "aim-workshop:qptsurface2", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The concrete AIM question about GL_n/G is already solved by Colliot-Thélène and Gille: every homogeneous space under a connected linear algebraic group over a complex curve function field satisfies weak approximation, with arbitrary stabilizer. The new proved refinement here concerns a constant finite subgroup A of GL_n: arbitrary local points on GL_n/A can be approximated by a global point whose boundary A-torsor is unramified outside the prescribed places and one freely chosen auxiliary point; no auxiliary point is needed exactly when representatives of the local conjugacy classes and genus-handle images satisfy the branch-cycle relation product_j [u_j,w_j] product_i g_i = 1. The broader all-place conjecture for arbitrary smooth proper geometrically rationally connected varieties remains open in the literature checked.\n\nCandidate contribution (refinement; novelty confidence low): For a constant finite A embedded in GL_n over K = C(C), the branch-cycle relation product_j [u_j,w_j] product_i g_i = 1 is necessary and sufficient to approximate prescribed local quotient points with a boundary torsor unramified outside the prescribed set; without that relation, one freely chosen auxiliary point always suffices, and the one-point bound is sharp already for A = Z/2 over P^1." }, { "id": 20000469, "problem_number": "AIM-ANALYSIS-0001", "title": "A finite-part BOCG identity for Fisher--Hartwig symbols", "statement": "The Borodin-Okounkov-Case-Geronimo identity is given by \\[\\text{det}(T_N(e^f))=(e^{N\\hat{f}(0)+\\sum k\\hat{f}(k)\\hat{f}(-k)})\\text{det}(1-k)\\] Find a useful adjustment to the right-hand side so that an identity holds in the Fisher-Hartwig case.", "original_statement": "The Borodin-Okounkov-Case-Geronimo identity is given by \\[\\text{det}(T_N(e^f))=(e^{N\\hat{f}(0)+\\sum k\\hat{f}(k)\\hat{f}(-k)})\\text{det}(1-k)\\] Find a useful adjustment to the right-hand side so that an identity holds in the Fisher-Hartwig case.", "clean_statement": "The Borodin-Okounkov-Case-Geronimo identity is given by \\[\\text{det}(T_N(e^f))=(e^{N\\hat{f}(0)+\\sum k\\hat{f}(k)\\hat{f}(-k)})\\text{det}(1-k)\\] Find a useful adjustment to the right-hand side so that an identity holds in the Fisher-Hartwig case.", "statement_status": "exact", "statement_verification": "The canonical record, and the current official AIM page, both contain the same malformed display:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Toeplitz Determinants and Toeplitz Operators\nSource item: 1.1\nSource URL: http://aimpl.org/riemhilberttoeplitz/1/\nCanonical location: aim-analysis-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The Borodin-Okounkov-Case-Geronimo identity is given by \\\\[\\\\text{det}(T_N(e^f))=(e^{N\\\\hat{f}(0)+\\\\sum k\\\\hat{f}(k)\\\\hat{f}(-k)})\\\\text{det}(1-k)\\\\] Find a useful adjustment to the right-hand side so that an identity holds in the Fisher-Hartwig case.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/1/", "tags": [ "aim", "AIM-ANALYSIS-0001", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literature-known radial regularization yields an exact finite-part Borodin--Okounkov--Case--Geronimo identity for separated scalar Fisher--Hartwig symbols. Building on that method, this attempt proves a normalization-free anchored ratio of radial Fredholm determinants and computes the formal pure-symbol boundary diagonal exactly as C_{p,q} times the sum over s >= N+i+1 of 1/((s+p)(s+q)); its nonzero harmonic tail proves that naive boundary substitution is generically not trace class. Combining the ratio with the known Barnes-G determinant gives an explicit consecutive Gamma-function ratio.\n\nCandidate contribution (identity; novelty confidence low): Candidate novelty is the paired anchored-ratio/exact-obstruction package: for any two nonvanishing finite Toeplitz anchors in the separated scalar radial family, all Fisher--Hartwig self-energy, interaction, and smooth quadratic factors cancel in the Fredholm-determinant ratio, while for a pure singularity the natural boundary Hankel-product diagonal is exactly C_{p,q} times the sum over s >= N+i+1 of 1/((s+p)(s+q)) and hence has a generically non-summable 1/i tail." }, { "id": 20000470, "problem_number": "AIM-ANALYSIS-0002", "title": "Painleve tau functions as limits of genuine block Toeplitz determinants", "statement": "Investigate the connection between the \\(\\tau\\) function from Painlev\\`{e} equations and block Toeplitz determinants. Seek explicit examples in which \\(\\tau\\) function asymptotics is known. Find the block Toeplitz determinant. What is it and what do we learn? Use Riemann-Hilbert to connect this to Fuchsian Riemann-Hilbert problems.", "original_statement": "Investigate the connection between the \\(\\tau\\) function from Painlev\\`{e} equations and block Toeplitz determinants. Seek explicit examples in which \\(\\tau\\) function asymptotics is known. Find the block Toeplitz determinant. What is it and what do we learn? Use Riemann-Hilbert to connect this to Fuchsian Riemann-Hilbert problems.", "clean_statement": "Investigate the connection between the \\(\\tau\\) function from Painlev\\`{e} equations and block Toeplitz determinants. Seek explicit examples in which \\(\\tau\\) function asymptotics is known. Find the block Toeplitz determinant. What is it and what do we learn? Use Riemann-Hilbert to connect this to Fuchsian Riemann-Hilbert problems.", "statement_status": "exact", "statement_verification": "The canonical record is item 1.2 in the section “Toeplitz Determinants and Toeplitz Operators” from the AIM workshop *Riemann-Hilbert problems, Toeplitz matrices, and applications* (March 4–8, 2024):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Toeplitz Determinants and Toeplitz Operators\nSource item: 1.2\nSource URL: http://aimpl.org/riemhilberttoeplitz/1/\nCanonical location: aim-analysis-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate the connection between the \\\\(\\\\tau\\\\) function from Painlev\\\\`{e} equations and block Toeplitz determinants. Seek explicit examples in which \\\\(\\\\tau\\\\) function asymptotics is known. Find the block Toeplitz determinant. What is it and what do we learn? Use Riemann-Hilbert to connect this to Fuchsian Riemann-Hilbert problems.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/1/", "tags": [ "aim", "AIM-ANALYSIS-0002", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The published Widom and Cafasso-Gavrylenko-Lisovyy results already identify the four-point Fuchsian/Painleve VI JMU tau function, up to an explicit power of the deformation parameter and a constant, with the normalized large-size limit of the genuine block Toeplitz determinant for J_t=Phi_-^{-1}Phi_+. This attempt adds a candidate constant-free tau-ratio normalization lemma (including the indispensable geometric factor outside the determinant-one gauge), a fully explicit diagonal 2-by-2 Fuchsian symbol whose Toeplitz limit and a special finite-size determinant are evaluated, exact constant-similarity invariance, and a triangular-invisibility obstruction matching the JMU one-form.\n\nCandidate contribution (normalization_lemma; novelty confidence low): For a traceless four-point Fuchsian family with zero partial indices and determinant-one three-point parametrices, tau_JMU(t)/tau_JMU(t0)=(t/t0)^kappa times the limit of D_n[J_t]/D_n[J_t0]; outside this gauge the ratio requires the explicit factor (G[J_t0]/G[J_t])^n. The same analysis yields the test symbol diag(j_t,j_t^{-1}), j_t=(1-t/z)^(-alpha_t)(1-z)^(alpha_1), and proves triangular Toeplitz determinants cannot detect reducible off-diagonal extension data." }, { "id": 20000471, "problem_number": "AIM-ANALYSIS-0003", "title": "A finite-section criterion for ratios of Toeplitz determinants", "statement": "Study merging singularities or double scaling limits for Toeplitz determinants with two symbols. Compare \\(\\text{det}(T_n(a_{\\lambda}b_{\\lambda}))\\) and \\(\\text{det}T_n(a_{\\lambda})T_n(b_{\\lambda})\\). Under what conditions does the ratio converge?", "original_statement": "Study merging singularities or double scaling limits for Toeplitz determinants with two symbols. Compare \\(\\text{det}(T_n(a_{\\lambda}b_{\\lambda}))\\) and \\(\\text{det}T_n(a_{\\lambda})T_n(b_{\\lambda})\\). Under what conditions does the ratio converge?", "clean_statement": "Study merging singularities or double scaling limits for Toeplitz determinants with two symbols. Compare \\(\\text{det}(T_n(a_{\\lambda}b_{\\lambda}))\\) and \\(\\text{det}T_n(a_{\\lambda})T_n(b_{\\lambda})\\). Under what conditions does the ratio converge?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.3 in the section “Toeplitz Determinants and Toeplitz Operators” of the AIM workshop *Riemann-Hilbert problems, Toeplitz matrices, and applications*. Its exact repository text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Toeplitz Determinants and Toeplitz Operators\nSource item: 1.3\nSource URL: http://aimpl.org/riemhilberttoeplitz/1/\nCanonical location: aim-analysis-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Study merging singularities or double scaling limits for Toeplitz determinants with two symbols. Compare \\\\(\\\\text{det}(T_n(a_{\\\\lambda}b_{\\\\lambda}))\\\\) and \\\\(\\\\text{det}T_n(a_{\\\\lambda})T_n(b_{\\\\lambda})\\\\). Under what conditions does the ratio converge?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/1/", "tags": [ "aim", "AIM-ANALYSIS-0003", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For Wiener-algebra symbols with invertible denominator finite sections, the ratio D_n(a_n b_n)/(D_n(a_n)D_n(b_n)) is exactly det(I+K_n), where K_n is the inverse finite-section product times an explicit sum of two Hankel boundary corrections. Trace-norm convergence of K_n is therefore a rigorous sufficient condition for convergence of the ratio. A uniform analytic corollary gives the strong-Szego cross-energy limit, while the exact family a_n(z)=1+alpha_n z, b_n(z)=1+beta_n z^{-1} has R_n/n tending to (1-exp(-s))/s when alpha_n beta_n=exp(-s/n), demonstrating the need for uniform stability and, in merging Fisher-Hartwig regimes, power normalization.\n\nCandidate contribution (finite_section_reduction; novelty confidence low): Candidate novelty: the exact two-boundary finite-section Hankel reduction is packaged directly for the AIM two-symbol ratio along arbitrary scaling sequences, and is paired with the explicit critical profile R_n/n -> (1-exp(-s))/s for a solvable two-band family." }, { "id": 20000472, "problem_number": "AIM-ANALYSIS-0004", "title": "Refutation status and a uniformly sectorial endpoint theorem", "statement": "Please see C.A. Berger and L.A. Coburn's 1994 article \"Heat Flow and Berlin-Toeplitz Estimates\" for background on this conjecture.\n\n(Berger-Coburr 1994): On the Fock/Bergman space, define the projection \\(\\prod:L^2(\\mathbb{C},e^{-|z|^2}dA) \\rightarrow \\mathscr{F}_{\\mathbb{C}}\\). Take \\(p: \\mathbb{C} \\rightarrow \\mathbb{C}\\) and define \\(T_p(f)=\\prod(p(z)f(z))\\). \\(T_p\\) is bounded on the Fock space iff for all \\((x,y) \\in \\mathbb{R}^2\\), \\(u_t=\\Delta u,\\) \\(u\\Big|_{t=0}=p\\), and \\(u(\\frac{1}{4},x,y)\\) are bounded.", "original_statement": "Please see C.A. Berger and L.A. Coburn's 1994 article \"Heat Flow and Berlin-Toeplitz Estimates\" for background on this conjecture.\n\n(Berger-Coburr 1994): On the Fock/Bergman space, define the projection \\(\\prod:L^2(\\mathbb{C},e^{-|z|^2}dA) \\rightarrow \\mathscr{F}_{\\mathbb{C}}\\). Take \\(p: \\mathbb{C} \\rightarrow \\mathbb{C}\\) and define \\(T_p(f)=\\prod(p(z)f(z))\\). \\(T_p\\) is bounded on the Fock space iff for all \\((x,y) \\in \\mathbb{R}^2\\), \\(u_t=\\Delta u,\\) \\(u\\Big|_{t=0}=p\\), and \\(u(\\frac{1}{4},x,y)\\) are bounded.", "clean_statement": "Please see C.A. Berger and L.A. Coburn's 1994 article \"Heat Flow and Berlin-Toeplitz Estimates\" for background on this conjecture.\n\n(Berger-Coburr 1994): On the Fock/Bergman space, define the projection \\(\\prod:L^2(\\mathbb{C},e^{-|z|^2}dA) \\rightarrow \\mathscr{F}_{\\mathbb{C}}\\). Take \\(p: \\mathbb{C} \\rightarrow \\mathbb{C}\\) and define \\(T_p(f)=\\prod(p(z)f(z))\\). \\(T_p\\) is bounded on the Fock space iff for all \\((x,y) \\in \\mathbb{R}^2\\), \\(u_t=\\Delta u,\\) \\(u\\Big|_{t=0}=p\\), and \\(u(\\frac{1}{4},x,y)\\) are bounded.", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ANALYSIS-0004, item 1.4 in the AIM workshop section “Toeplitz Determinants and Toeplitz Operators.” It points to Berger and Coburn's 1994 paper but contains several transcription and normalization problems:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Toeplitz Determinants and Toeplitz Operators\nSource item: 1.4\nSource URL: http://aimpl.org/riemhilberttoeplitz/1/\nCanonical location: aim-analysis-notes.json notes[3]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Please see C.A. Berger and L.A. Coburn's 1994 article \\\"Heat Flow and Berlin-Toeplitz Estimates\\\" for background on this conjecture.\\n\\n(Berger-Coburr 1994): On the Fock/Bergman space, define the projection \\\\(\\\\prod:L^2(\\\\mathbb{C},e^{-|z|^2}dA) \\\\rightarrow \\\\mathscr{F}_{\\\\mathbb{C}}\\\\). Take \\\\(p: \\\\mathbb{C} \\\\rightarrow \\\\mathbb{C}\\\\) and define \\\\(T_p(f)=\\\\prod(p(z)f(z))\\\\). \\\\(T_p\\\\) is bounded on the Fock space iff for all \\\\((x,y) \\\\in \\\\mathbb{R}^2\\\\), \\\\(u_t=\\\\Delta u,\\\\) \\\\(u\\\\Big|_{t=0}=p\\\\), and \\\\(u(\\\\frac{1}{4},x,y)\\\\) are bounded.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/1/", "tags": [ "aim", "AIM-ANALYSIS-0004", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Known status: Sam Looi's 2026 counterexample refutes the general measurable-symbol Berger--Coburn equivalence by showing in every complex dimension that a bounded form-defined Toeplitz operator can have an unbounded critical heat transform. Independent proved contribution of this attempt: after correcting the source's factor-two normalization mismatch, the endpoint equivalence does hold for every coherent-state-admissible measurable symbol whose essential range lies in a fixed sector of aperture strictly less than pi; it is quantitatively equivalent to the Fock--Carleson property of |g|dmu and implies that every necessity counterexample must escape every such strict sector.\n\nCandidate contribution (sectorial_endpoint_theorem; novelty confidence low): If Re(exp(-i theta)g) >= delta|g| almost everywhere for some delta>0, then on F^2_lambda, boundedness of the form-defined T_g, boundedness of H_{1/(8 lambda)}g, the Fock--Carleson property of |g|dmu_lambda, and boundedness of H_s|g| at one/every positive time are equivalent; moreover ||H_{1/(8 lambda)}g||_infinity <= 2^n delta^{-1}||T_g|| and the reverse bound holds up to a dimension-only Carleson constant. Consequently, a necessity counterexample cannot have essential range in any fixed sector of aperture less than pi." }, { "id": 20000473, "problem_number": "AIM-ANALYSIS-0005", "title": "Perturbative solvability beyond the reciprocal-symbol symmetry", "statement": "For Szegő-type nonvanishing symbols, what can be said about existence of the model Riemann--Hilbert problem when the ratio \\(d=w/\\phi\\) does not obey \\(d(z)d(z^{-1})=1\\)?", "original_statement": "The determinant of Toeplitz \\(+\\) Handle is expressible in terms of the \\(4 \\times 4\\) Riemann-Hilbert problem for Szego type symbols. The obstacle to asymptotic analysis is the existence of a solution to model problems. If \\(d \\tilde{d}=1\\) on \\(\\{|z|=1\\}\\), where \\(d=\\frac{\\phi(z)}{w(z)}\\) and \\(\\tilde{d}=\\frac{\\phi(1/z)}{w(1/z)}\\), then it is solvable. What if this doesn't hold?", "clean_statement": "For Szegő-type nonvanishing symbols, what can be said about existence of the model Riemann--Hilbert problem when the ratio \\(d=w/\\phi\\) does not obey \\(d(z)d(z^{-1})=1\\)?", "statement_status": "corrected_verified", "statement_verification": "The source is AIM workshop “Riemann-Hilbert problems, Toeplitz matrices, and applications,” Section “Toeplitz Determinants and Toeplitz Operators,” Problem 1.5. Two extraction issues can be resolved from the subsequent primary paper of Gharakhloo and Its [GI20]: 1. “Handle” is an OCR error for **Hankel**. 2. [GI20] writes the ratio as \\(d=w/\\phi\\), so that \\(w=d\\phi\\), whereas the AIM record writes its reciprocal \\(d_{\\rm AIM}=\\phi/w\\). This convention difference does not affect the condition, because \\[ d_{\\rm AIM}\\widetilde d_{\\rm AIM}=1 \\quad\\Longleftrightarrow\\quad d\\widetilde d=1. \\]", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Toeplitz Determinants and Toeplitz Operators\nSource item: 1.5\nSource URL: http://aimpl.org/riemhilberttoeplitz/1/\nCanonical location: aim-analysis-notes.json notes[4]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The determinant of Toeplitz \\\\(+\\\\) Handle is expressible in terms of the \\\\(4 \\\\times 4\\\\) Riemann-Hilbert problem for Szego type symbols. The obstacle to asymptotic analysis is the existence of a solution to model problems. If \\\\(d \\\\tilde{d}=1\\\\) on \\\\(\\\\{|z|=1\\\\}\\\\), where \\\\(d=\\\\frac{\\\\phi(z)}{w(z)}\\\\) and \\\\(\\\\tilde{d}=\\\\frac{\\\\phi(1/z)}{w(1/z)}\\\\), then it is solvable. What if this doesn't hold?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/1/", "tags": [ "aim", "AIM-ANALYSIS-0005", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the Gharakhloo–Its 4×4 Toeplitz+Hankel model, every zero-winding analytic ratio splits, up to a common sign, as d=d0 q with d0·tilde(d0)=1 and q=tilde(q). Conjugation by the explicit matched solution gives a quantitative small-defect criterion Γ0 A(t)t<1 for unique solvability when t=||q−1||∞, together with an explicit first-variation formula. Along dε=d0 exp(εh), h=tilde(h), nonsolvable parameters form at most a discrete set. In particular, φ=1 and w=exp(ε(z+z^{-1})) violate d·tilde(d)=1 for ε≠0 but have a solvable model for |ε|≤(1/2)log(6/5). This is a partial result and not a global classification.\n\nCandidate contribution (perturbative_solvability_theorem; novelty confidence low): Candidate novelty: the exact three-entry symmetry-defect formula for the Gharakhloo–Its jump yields a quantitative open nonsymmetric solvability neighborhood, a discrete-exception theorem along analytic defect directions, a first-order response formula, and the explicit provably solvable family φ=1, w=exp(ε(z+z^{-1})) for |ε|≤(1/2)log(6/5)." }, { "id": 20000474, "problem_number": "AIM-ANALYSIS-0006", "title": "Finite-cover droplets and an exact soft-Riemann--Hilbert sector reduction", "statement": "Given \\(Q\\), produce \\(Q(z^d)\\) with multiple disjoint support sets for the droplet. Study the soft Riemann-Hilbert problem for orthogonal polynomials.", "original_statement": "Given \\(Q\\), produce \\(Q(z^d)\\) with multiple disjoint support sets for the droplet. Study the soft Riemann-Hilbert problem for orthogonal polynomials.", "clean_statement": "Given \\(Q\\), produce \\(Q(z^d)\\) with multiple disjoint support sets for the droplet. Study the soft Riemann-Hilbert problem for orthogonal polynomials.", "statement_status": "exact", "statement_verification": "The canonical record is item 2.1 in the section “Riemann-Hilbert Problems” of the AIM workshop list *Riemann-Hilbert problems, Toeplitz matrices, and applications*. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Riemann-Hilbert Problems\nSource item: 2.1\nSource URL: http://aimpl.org/riemhilberttoeplitz/2/\nCanonical location: aim-analysis-notes.json notes[5]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given \\\\(Q\\\\), produce \\\\(Q(z^d)\\\\) with multiple disjoint support sets for the droplet. Study the soft Riemann-Hilbert problem for orthogonal polynomials.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/2/", "tags": [ "aim", "AIM-ANALYSIS-0006", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the normalization-correct cyclic pullback Q^{}(z)=Q(z^d)/d at mass τ, the equilibrium measure is the balanced lift L_d μ_{Q,τ}, its support is p^{-1}(S_τ[Q]), and a base density ρ pulls back to d|z|^{2d-2}ρ(z^d). The literal field Q(z^d) instead gives L_d(d μ_{Q,τ/d}) and p^{-1}(S_{τ/d}[Q]). For a connected base domain in C* whose winding subgroup is kZ, the lifted domain has gcd(d,k) components. Writing n=dq+r, the monic polynomial, norm, and complete 1-by-2 soft row reduce exactly to P_n(z)=z^r π_q^{(r)}(z^d), h_n=h_q^{(r)}/d, and Y_n(z)=(z^r π_q^{(r)}(z^d), d^{-1}z^{d-r-1}Φ_q^{(r)}(z^d)), where the reduced weight is |w|^{2γ_r} exp(-2(m/d)Q(w)) with γ_r=(r+1)/d-1. Uniform strong asymptotics for these charged reduced soft problems remain conditional and are not claimed here.\n\nCandidate contribution (finite_cover_soft_row_reduction; novelty confidence low): Candidate finite-cover soft-row reduction: for every admissible base field Q and every residue n=dq+r, both entries of the soft row for Q(z^d)/d are the explicit pullbacks (z^r π_q^{(r)}(z^d), d^{-1}z^{d-r-1}Φ_q^{(r)}(z^d)) of a charged base problem with γ_r=(r+1)/d-1; combined with this, the actual number of lifted components is gcd(d,k), not automatically d." }, { "id": 20000475, "problem_number": "AIM-ANALYSIS-0007", "title": "An exact cyclic r-star reduction for higher-size Angelesco equilibrium problems", "statement": "Study potential theory for \\(k \\times k\\) Riemann-Hilbert problems for \\(k>2\\) when multiple orthogonal polynomials are involved.", "original_statement": "Study potential theory for \\(k \\times k\\) Riemann-Hilbert problems for \\(k>2\\) when multiple orthogonal polynomials are involved.", "clean_statement": "Study potential theory for \\(k \\times k\\) Riemann-Hilbert problems for \\(k>2\\) when multiple orthogonal polynomials are involved.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Riemann-Hilbert Problems\nSource item: 2.2\nSource URL: http://aimpl.org/riemhilberttoeplitz/2/\nCanonical location: aim-analysis-notes.json notes[6]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Study potential theory for \\\\(k \\\\times k\\\\) Riemann-Hilbert problems for \\\\(k>2\\\\) when multiple orthogonal polynomials are involved.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/2/", "tags": [ "aim", "AIM-ANALYSIS-0007", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For r >= 3, corresponding to MOP RHP size k = r + 1 >= 4, consider equal masses 1/r on the disjoint rotated intervals omega^j[a,b] with 0 < a < b, the complete Angelesco interaction, and continuous external fields related by rotation. The vector equilibrium has a unique cyclic minimizer mu_j = (R^j)_#nu, and its energy reduces exactly to F_r(nu) = (r/2)[I(nu) + I((x^r)_#nu)] + r integral Q dnu. The report proves the equivalent Euler-Lagrange condition U^nu(x) + U^eta(x^r) + Q(x) = constant, the Cauchy-transform compression sum_j G_j(z) = r z^(r-1) G_eta(z^r), and the explicit r = 3, k = 4 specialization. It also proves E = (1/2)I(rho) + (r/2)I(nu) + r integral Q dnu, so ordinary scalar energy of the total measure omits a nonconstant color self-energy. This is a rigorous cyclic Angelesco special case, not a solution of the unrestricted arbitrary-k or same-line external-source program.\n\nCandidate contribution (cyclic_vector_equilibrium_reduction; novelty confidence low): In the stated disjoint cyclic r-star Angelesco class, the exact two-energy quotient, its equivalence to all vector Frostman conditions, the branch-independent Cauchy-transform quotient identity, and the explicit total-measure obstruction hold simultaneously for every r >= 3." }, { "id": 20000476, "problem_number": "AIM-ANALYSIS-0008", "title": "Regular and obstructed matrix Riemann--Hilbert formulations for KdV", "statement": "In general, can you pose matrix Riemann Hilbert problems for KdV?", "original_statement": "In general, can you pose matrix Riemann Hilbert problems for KdV?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "1. **Scalar KdV, square unknown.** For \\[ q_t=6qq_x-q_{xxx},\\qquad q:\\mathbb R^2\\to\\mathbb R, \\] inverse scattering naturally gives a \\(1\\times2\\) row-vector Riemann--Hilbert problem. Does it always admit an equivalent, normalized, regular, invertible \\(2\\times2\\) matrix solution with the same jump and KdV symmetry? This is almost certainly the intended reading: it is the issue addressed explicitly by Egorova--Piorkowski--Teschl and Piorkowski--Teschl.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Riemann-Hilbert Problems\nSource item: 2.3\nSource URL: http://aimpl.org/riemhilberttoeplitz/2/\nCanonical location: aim-analysis-notes.json notes[7]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In general, can you pose matrix Riemann Hilbert problems for KdV?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/2/", "tags": [ "aim", "AIM-ANALYSIS-0008", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the intended scalar-KdV reading, a regular invertible square Riemann--Hilbert lift does not exist in general: a unique symmetric r-by-2r vector solution can lift to a regular 2r-by-2r solution only if it has full row rank everywhere and admits a transverse odd companion. This criterion recovers the known one-soliton zero-energy obstruction. Conversely, for smooth reflection-only data satisfying a uniform strict matrix-contractivity condition, the explicitly ordered 2r-by-2r jump has positive-definite Hermitian part and the normalized square RHP is uniquely solvable. These are rigorous partial results, not a formulation covering poles, zero-energy degeneracy, and finite-gap backgrounds.\n\nCandidate contribution (odd_companion_lift_criterion; novelty confidence low): A normalized symmetric r-by-2r right-jump RHP has a regular invertible 2r-by-2r lift if and only if there exists a same-jump antisymmetric companion normalized by (I_r,-I_r) whose stack with the original solution is everywhere invertible; consequently, any row-rank drop of the normalized vector obstructs a regular square lift." }, { "id": 20000477, "problem_number": "AIM-ANALYSIS-0009", "title": "Real-linear small-norm criteria for conjugate dbar and additive jump problems", "statement": "For planar orthogonal polynomials, we know that \\[\\overline{\\partial} Y=\\overline{Y} \\begin{bmatrix}\n0 & w \\\\\n0 & 0 \\\\\n\\end{bmatrix}\\] and \\[\\int p_j\\overline{p_k}wdA=h_j\\delta_{j,k}.\\] Is there an analogue of small norm theory of such \\(\\overline{\\partial}\\)-problems (i.e. \\(\\overline{\\partial} A=\\overline{A} W +AV\\)?) Or small norm theory for \\(A_{+}-A_{-}=\\overline{A_{-}}V_C+A_{-}V_n\\)?", "original_statement": "For planar orthogonal polynomials, we know that \\[\\overline{\\partial} Y=\\overline{Y} \\begin{bmatrix}\n0 & w \\\\\n0 & 0 \\\\\n\\end{bmatrix}\\] and \\[\\int p_j\\overline{p_k}wdA=h_j\\delta_{j,k}.\\] Is there an analogue of small norm theory of such \\(\\overline{\\partial}\\)-problems (i.e. \\(\\overline{\\partial} A=\\overline{A} W +AV\\)?) Or small norm theory for \\(A_{+}-A_{-}=\\overline{A_{-}}V_C+A_{-}V_n\\)?", "clean_statement": "For planar orthogonal polynomials, we know that \\[\\overline{\\partial} Y=\\overline{Y} \\begin{bmatrix}\n0 & w \\\\\n0 & 0 \\\\\n\\end{bmatrix}\\] and \\[\\int p_j\\overline{p_k}wdA=h_j\\delta_{j,k}.\\] Is there an analogue of small norm theory of such \\(\\overline{\\partial}\\)-problems (i.e. \\(\\overline{\\partial} A=\\overline{A} W +AV\\)?) Or small norm theory for \\(A_{+}-A_{-}=\\overline{A_{-}}V_C+A_{-}V_n\\)?", "statement_status": "exact", "statement_verification": "The canonical record is problem 2.4 in the AIM workshop list *Riemann-Hilbert problems, Toeplitz matrices, and applications*, section *Riemann-Hilbert Problems*. Its exact mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Riemann-Hilbert Problems\nSource item: 2.4\nSource URL: http://aimpl.org/riemhilberttoeplitz/2/\nCanonical location: aim-analysis-notes.json notes[8]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For planar orthogonal polynomials, we know that \\\\[\\\\overline{\\\\partial} Y=\\\\overline{Y} \\\\begin{bmatrix}\\n0 & w \\\\\\\\\\n0 & 0 \\\\\\\\\\n\\\\end{bmatrix}\\\\] and \\\\[\\\\int p_j\\\\overline{p_k}wdA=h_j\\\\delta_{j,k}.\\\\] Is there an analogue of small norm theory of such \\\\(\\\\overline{\\\\partial}\\\\)-problems (i.e. \\\\(\\\\overline{\\\\partial} A=\\\\overline{A} W +AV\\\\)?) Or small norm theory for \\\\(A_{+}-A_{-}=\\\\overline{A_{-}}V_C+A_{-}V_n\\\\)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/2/", "tags": [ "aim", "AIM-ANALYSIS-0009", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For p>2 and compact bulk support, the normalized matrix equation dbar A = conjugate(A) W + A V is uniquely solvable by a real-linear Cauchy contraction under an explicit Lambda_{p,M}(||W||_p+||V||_p)<1 condition, with a quantitative sup-norm bound. An additive conjugate jump equation has an analogous L2 Cauchy-projection criterion. For a positively separated bulk support and contour, the simultaneous bulk-jump problem is uniquely solvable when an explicit nonnegative 2-by-2 control matrix has spectral radius below one, with a componentwise resolvent estimate. Complex-linear integral doubling, an anti-linear square criterion, and an exact arbitrary-size solution for W=wE_12 further expose the structure and show that scalar smallness is sufficient but not necessary.\n\nCandidate contribution (mixed_real_linear_small_norm_criterion; novelty confidence low): Candidate novel contribution: for a matrix problem combining conjugate and ordinary bulk dbar terms with conjugate and ordinary additive jump terms, positive separation of the bulk support and contour yields a two-density real-linear fixed point whose Lipschitz bounds are encoded by an explicit nonnegative 2-by-2 matrix; spectral radius below one is a sufficient solvability criterion and gives the componentwise bound u <= (I-M)^{-1}u_0." }, { "id": 20000478, "problem_number": "AIM-ANALYSIS-0010", "title": "Scalar dbar normalization and planar Hermite multipoles for the elliptic Gaussian", "statement": "Analyze the condition given by \\[\\overline{\\partial}\\phi=\\overline{H_n}e^{-NQ},\\] where \\(Q=|z|^2+2t\\text{Re}(z^2)\\) and where \\(\\overline{H_n}\\) is Hermite.", "original_statement": "Analyze the condition given by \\[\\overline{\\partial}\\phi=\\overline{H_n}e^{-NQ},\\] where \\(Q=|z|^2+2t\\text{Re}(z^2)\\) and where \\(\\overline{H_n}\\) is Hermite.", "clean_statement": "Analyze the condition given by \\[\\overline{\\partial}\\phi=\\overline{H_n}e^{-NQ},\\] where \\(Q=|z|^2+2t\\text{Re}(z^2)\\) and where \\(\\overline{H_n}\\) is Hermite.", "statement_status": "exact", "statement_verification": "The exact canonical record is AIM-ANALYSIS-0010, source file `aim-analysis-notes.json`, zero-based source index 9, problem 2.5 in the “Riemann-Hilbert Problems” section of the AIM workshop *Riemann-Hilbert problems, Toeplitz matrices, and applications*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Riemann-Hilbert Problems\nSource item: 2.5\nSource URL: http://aimpl.org/riemhilberttoeplitz/2/\nCanonical location: aim-analysis-notes.json notes[9]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Analyze the condition given by \\\\[\\\\overline{\\\\partial}\\\\phi=\\\\overline{H_n}e^{-NQ},\\\\] where \\\\(Q=|z|^2+2t\\\\text{Re}(z^2)\\\\) and where \\\\(\\\\overline{H_n}\\\\) is Hermite.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/2/", "tags": [ "aim", "AIM-ANALYSIS-0010", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the explicit reconstruction N>0, real t with |t|<1/2, dA_0=dx dy/pi, a monic degree-n polynomial source, and the soft-Riemann-Hilbert normalization phi(z)=O(|z|^{-n-1}), the scalar dbar equation uniquely selects the monic planar Hermite polynomial generated by exp(zs+t s^2/[N(1-4t^2)]). Its norm, every mixed moment, and hence every parity-resolved coefficient in the Cauchy-transform expansion are computed exactly. The analysis also gives the general global primitive modulo entire functions, an exact circular-case formula, the sharp |t|=1/2 threshold for the planar orthogonality/Cauchy formulation, and an obstruction to any finite polynomial-times-Gaussian ansatz. This is a rigorous finite-size solution of the reconstructed scalar problem, but only a partial result for the broader asymptotic workshop program suggested by the source context.\n\nCandidate contribution (hermite_dbar_normalization_multipole_package; novelty confidence low): Let D=1-4t^2, beta=1/(ND), alpha=t/(ND), and Z_0=1/(N sqrt(D)). Under |t|<1/2 and the monic soft-Riemann-Hilbert normalization, the scalar dbar problem selects P_n through sum_{n>=0} P_n(z)s^n/n!=exp(zs+alpha s^2), and its normalized Cauchy solution has the all-orders fixed-(n,N) multipole coefficients Z_0 beta^n (n+2m)!(-alpha)^m/m! at z^{-n-2m-1}. In the same normalization package, |t|=1/2 is the sharp integrability boundary and no finite polynomial-Gaussian particular solution exists." }, { "id": 20000479, "problem_number": "AIM-ANALYSIS-0011", "title": "Nonperiodic 2-by-2 products: a literal factorization diagnostic, exact commuting asymptotics, and a no-limit obstruction", "statement": "Consider \\[\\prod_{i=1}^N \\begin{bmatrix}\n1 & a_iz \\\\\nb_i & 1 \\\\\n\\end{bmatrix},\\] for constants \\(a_i, b_i\\). Suppose Wiener-Hopf factorization has no partial indices. Study the behavior as \\(N \\rightarrow \\infty\\). The asymptotics are known if the \\(a_i\\)'s and \\(b_i\\)'s are periodic. What happens in the more general case?", "original_statement": "Consider \\[\\prod_{i=1}^N \\begin{bmatrix}\n1 & a_iz \\\\\nb_i & 1 \\\\\n\\end{bmatrix},\\] for constants \\(a_i, b_i\\). Suppose Wiener-Hopf factorization has no partial indices. Study the behavior as \\(N \\rightarrow \\infty\\). The asymptotics are known if the \\(a_i\\)'s and \\(b_i\\)'s are periodic. What happens in the more general case?", "clean_statement": "Consider \\[\\prod_{i=1}^N \\begin{bmatrix}\n1 & a_iz \\\\\nb_i & 1 \\\\\n\\end{bmatrix},\\] for constants \\(a_i, b_i\\). Suppose Wiener-Hopf factorization has no partial indices. Study the behavior as \\(N \\rightarrow \\infty\\). The asymptotics are known if the \\(a_i\\)'s and \\(b_i\\)'s are periodic. What happens in the more general case?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem 2.6 in the section “Riemann-Hilbert Problems,” attributed on the live AIM page to Tomas Berggren. Its exact mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Riemann-Hilbert Problems\nSource item: 2.6\nSource URL: http://aimpl.org/riemhilberttoeplitz/2/\nCanonical location: aim-analysis-notes.json notes[10]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Consider \\\\[\\\\prod_{i=1}^N \\\\begin{bmatrix}\\n1 & a_iz \\\\\\\\\\nb_i & 1 \\\\\\\\\\n\\\\end{bmatrix},\\\\] for constants \\\\(a_i, b_i\\\\). Suppose Wiener-Hopf factorization has no partial indices. Study the behavior as \\\\(N \\\\rightarrow \\\\infty\\\\). The asymptotics are known if the \\\\(a_i\\\\)'s and \\\\(b_i\\\\)'s are periodic. What happens in the more general case?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/2/", "tags": [ "aim", "AIM-ANALYSIS-0011", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the exact polynomial displayed in the AIM record, under the standard Wiener-Hopf convention on the circle |z|=rho, zero partial indices occur if and only if |a_i b_i| 0, the limit of N^{-2} log Z_N is beta times the logarithmic chordal capacity. The equilibrium measure and capacity are proved to equal two-sided boundary-flux and torsion expressions for the complementary spherical domains. For every spherical circle, an exact finite-N Dyson formula and its complete fixed-beta Stirling expansion are obtained; at beta = 1 an independent Gram-determinant identity holds for every loop.\n\nCandidate contribution (identity; novelty confidence low): Candidate novelty: if Γ is an embedded C² loop in the unit sphere and v_i solves Δv_i = -1/2 with zero boundary data on each complementary domain Ω_i, then its chordal equilibrium measure is -(2π)^{-1}(∂_{n_1}v_1+∂_{n_2}v_2) ds and log cap_χ(Γ) = -1/2-(4π)^{-1}(∫_{Ω_1}v_1 dA+∫_{Ω_2}v_2 dA)." }, { "id": 20000481, "problem_number": "AIM-ANALYSIS-0013", "title": "Discrete Bessel gap probabilities, Riemann-Hilbert problems, and the Toda-KdV limit", "statement": "The discrete Bessel process is connected to Poissonized Plancherel measure. Consider the distance of the largest particle: \\(\\mathbb{P}[Q_{\\text{max}}\\leq x]=q(x)\\), where \\(q\\) solves the cylindrical Toda equation. Take the limit to get cylindrical KdV. Is there a connection to Riemann-Hilbert Problems?", "original_statement": "The discrete Bessel process is connected to Poissonized Plancherel measure. Consider the distance of the largest particle: \\(\\mathbb{P}[Q_{\\text{max}}\\leq x]=q(x)\\), where \\(q\\) solves the cylindrical Toda equation. Take the limit to get cylindrical KdV. Is there a connection to Riemann-Hilbert Problems?", "clean_statement": "Express the largest-particle Fredholm determinant of the (possibly\nfinite-temperature) discrete Bessel process through a discrete\nRiemann--Hilbert problem, recover cylindrical Toda from its Lax pair,\nand identify the Riemann--Hilbert object and equation obtained under the\nsoft-edge Toda-to-KdV scaling.", "statement_status": "reconstructed_unverified", "statement_verification": "The live AIM page was checked on 2026-07-27. It contains the same text, attributes the problem to Maksim Kosmakov, and has no status remark. Thus the word “distance” is verified source text, not an extraction error. The following corrections are nevertheless mathematically necessary and are explicit reconstructions rather than silent emendations. 1. “Distance” is almost surely intended to mean **distribution**. 2. The standard random variable is the largest particle \\(a_{\\max}\\), not \\(Q_{\\max}\\). For a partition \\(\\lambda\\), the particle configuration is \\(\\{\\lambda_i-i+\\tfrac12:i\\geq1\\}\\), hence \\(a_{\\max}=\\lambda_1-\\tfrac12\\). 3. The distribution has two variables. We write \\[ Q(L,s)=\\mathbb P_L(a_{\\max}\\leq s),\\qquad L>0,\\quad s\\in\\mathbb Z':=\\mathbb Z+\\tfrac12, \\] where \\(L^2\\) is the Poissonization parameter. A one-variable \\(q(x)\\) cannot by itself satisfy the differential--dif...", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Riemann-Hilbert Problems\nSource item: 2.8\nSource URL: http://aimpl.org/riemhilberttoeplitz/2/\nCanonical location: aim-analysis-notes.json notes[12]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The discrete Bessel process is connected to Poissonized Plancherel measure. Consider the distance of the largest particle: \\\\(\\\\mathbb{P}[Q_{\\\\text{max}}\\\\leq x]=q(x)\\\\), where \\\\(q\\\\) solves the cylindrical Toda equation. Take the limit to get cylindrical KdV. Is there a connection to Riemann-Hilbert Problems?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: solved; rigor: conditional; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/2/", "tags": [ "aim", "AIM-ANALYSIS-0013", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM question has an established affirmative answer: the discrete Bessel largest-particle Fredholm determinant is encoded by a discrete integrable Riemann-Hilbert problem whose dressed Lax compatibility gives cylindrical Toda, while soft-edge scaling gives a finite-temperature Airy IIKS Riemann-Hilbert problem and KdV/cylindrical KdV. As a separate conditional contribution, this attempt proves that any lattice-compatible smooth expansion f_epsilon = f_0 + epsilon f_1 + O(epsilon^2) has f_0 satisfying the KdV tau equation and f_1 satisfying its homogeneous linearization; equivalently, the first potential correction satisfies linearized KdV.\n\nCandidate contribution (necessary-condition theorem; novelty confidence low): Under the stated lattice-compatible C6 regularity and C4 asymptotic-expansion assumptions for the cylindrical-Toda edge scaling, the first coefficient f_1 necessarily satisfies (f_1)_{xt} + (x/t)(f_1)_{xx} + 2(f_0)_{xx}(f_1)_{xx} + (1/6)(f_1)_{xxxx} = 0, so U_1 = (f_1)_{xx} obeys homogeneous linearized KdV about U_0 = (f_0)_{xx} + x/(2t); moreover, a fixed zero-temperature step weight yields only the Painleve-II self-similar sector." }, { "id": 20000482, "problem_number": "AIM-ANALYSIS-0014", "title": "A matrix Fisher-Hartwig atlas and a partial-index obstruction", "statement": "Formulate and then analyze general finite matrix symbol with the analog of zeros and jumps.", "original_statement": "Formulate and then analyze general finite matrix symbol with the analog of zeros and jumps.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "There are no remarks or references in the record. The grammar leaves three plausible readings of “finite”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Riemann-Hilbert Problems\nSource item: 2.9\nSource URL: http://aimpl.org/riemhilberttoeplitz/2/\nCanonical location: aim-analysis-notes.json notes[13]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Formulate and then analyze general finite matrix symbol with the analog of zeros and jumps.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/2/", "tags": [ "aim", "AIM-ANALYSIS-0014", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A workable formulation must distinguish local meromorphic zero vectors (Smith-McMillan structural indices and root filtrations), oriented matrix jump logarithms, and global Wiener-Hopf partial indices. For symbols made upper triangular by constant left/right matrices, every finite block Toeplitz determinant factors exactly into scalar Toeplitz determinants, so scalar Fisher-Hartwig theory completely analyzes arbitrary diagonal root-and-jump channels. The unrestricted noncommuting zero-and-jump problem remains open. An explicit pair, I_2 and diag(z,z^{-1}), proves that the full determinant symbol, all local contour zero/jump data, and determinant winding do not control finite sections: their determinant sequences are respectively 1 and 0.\n\nCandidate contribution (obstruction; novelty confidence low): The pair I_2 and diag(z,z^{-1}) is an explicit minimal-data obstruction: two smooth invertible 2-by-2 symbols can have the same scalar determinant symbol, identical local contour Smith-McMillan data, no jumps, and equal determinant winding, yet have block Toeplitz determinant sequences identically 1 and identically 0; therefore a matrix Fisher-Hartwig formulation must include the full global partial-index vector or an equivalent stability condition." }, { "id": 20000483, "problem_number": "AIM-ANALYSIS-0015", "title": "The Hilbert matrix on power-weighted ell-squared spaces", "statement": "The \\(\\mathscr{l}^2\\)-spectrum for the Hilbert matrix \\(H_{m n}=\\frac{1}{m+n+1}\\) is known. Find the spectrum on spaces of the form \\(\\mathscr{l}_2(\\alpha)=\\{\\sum |a_n|^2(n+1)^{\\alpha}<\\infty\\}\\).", "original_statement": "The \\(\\mathscr{l}^2\\)-spectrum for the Hilbert matrix \\(H_{m n}=\\frac{1}{m+n+1}\\) is known. Find the spectrum on spaces of the form \\(\\mathscr{l}_2(\\alpha)=\\{\\sum |a_n|^2(n+1)^{\\alpha}<\\infty\\}\\).", "clean_statement": "The \\(\\mathscr{l}^2\\)-spectrum for the Hilbert matrix \\(H_{m n}=\\frac{1}{m+n+1}\\) is known. Find the spectrum on spaces of the form \\(\\mathscr{l}_2(\\alpha)=\\{\\sum |a_n|^2(n+1)^{\\alpha}<\\infty\\}\\).", "statement_status": "exact", "statement_verification": "The live AIM page contains the same text and attributes the problem to Alfonso Montes Rodríguez. There is no substantive corruption. We interpret the displayed set in the standard way:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Spectrum of Hilbert Matrices\nSource item: 3.1\nSource URL: http://aimpl.org/riemhilberttoeplitz/3/\nCanonical location: aim-analysis-notes.json notes[14]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The \\\\(\\\\mathscr{l}^2\\\\)-spectrum for the Hilbert matrix \\\\(H_{m n}=\\\\frac{1}{m+n+1}\\\\) is known. Find the spectrum on spaces of the form \\\\(\\\\mathscr{l}_2(\\\\alpha)=\\\\{\\\\sum |a_n|^2(n+1)^{\\\\alpha}<\\\\infty\\\\}\\\\).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/3/", "tags": [ "aim", "AIM-ANALYSIS-0015", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "Under the bounded-operator interpretation of the AIM question, the Hilbert matrix is bounded on ell^2_alpha exactly for -1 < alpha < 1. In this range its spectrum is the closed lens bounded by {pi/cos(pi|alpha|/2 + i pi t): t real} union {0}; the Fredholm essential spectrum is the boundary, the interior is simple point spectrum for alpha < 0 and residual spectrum for alpha > 0, and the boundary is continuous spectrum. The interior Fredholm index is -sgn(alpha), and the spectral radius is pi/cos(pi|alpha|/2). For |alpha| >= 1 there is no bounded extension; spectra of separately chosen unbounded realizations are outside the stated interpretation.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: after diagonal unitary conjugation, the exact discrete power-weighted Hilbert matrix, enlarged by a zero summand, is a Hilbert-Schmidt perturbation of the half-line integral operator with kernel x^(alpha/2)y^(-alpha/2)/(x+y); combining its Wiener-Hopf symbol with Hill's latent-eigenvector asymptotics yields the full weighted spectral lens and spectral-type decomposition." }, { "id": 20000484, "problem_number": "AIM-ANALYSIS-0016", "title": "Three atomic singular inner functions: source diagnosis, pair selection, and point reconstruction", "statement": "Define \\[u_{\\xi, \\alpha(z)}=e^{-\\alpha\\Big(\\frac{\\xi+z}{\\xi-z}\\Big)},\\] for \\(\\alpha >0\\) and \\(|\\xi|=1\\). Given \\(\\xi_1, \\xi_2, \\xi_3\\) and \\(\\alpha_1, \\alpha_2, \\alpha_3\\), can we approximate \\[\\sum_{m=0}^{\\infty} c_m^{(1)}u_1^m+c_m^{(2)}u_2^m+c_m^{(3)}u_3^m?\\]", "original_statement": "Define \\[u_{\\xi, \\alpha(z)}=e^{-\\alpha\\Big(\\frac{\\xi+z}{\\xi-z}\\Big)},\\] for \\(\\alpha >0\\) and \\(|\\xi|=1\\). Given \\(\\xi_1, \\xi_2, \\xi_3\\) and \\(\\alpha_1, \\alpha_2, \\alpha_3\\), can we approximate \\[\\sum_{m=0}^{\\infty} c_m^{(1)}u_1^m+c_m^{(2)}u_2^m+c_m^{(3)}u_3^m?\\]", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "Two other plausible readings are treated separately: \\(H^\\infty\\)-norm approximation and literal convergence of the three infinite series. None of these readings is silently substituted for the source question.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Riemann-Hilbert problems, Toeplitz matrices, and applications\nSection: Spectrum of Hilbert Matrices\nSource item: 3.2\nSource URL: http://aimpl.org/riemhilberttoeplitz/3/\nCanonical location: aim-analysis-notes.json notes[15]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Define \\\\[u_{\\\\xi, \\\\alpha(z)}=e^{-\\\\alpha\\\\Big(\\\\frac{\\\\xi+z}{\\\\xi-z}\\\\Big)},\\\\] for \\\\(\\\\alpha >0\\\\) and \\\\(|\\\\xi|=1\\\\). Given \\\\(\\\\xi_1, \\\\xi_2, \\\\xi_3\\\\) and \\\\(\\\\alpha_1, \\\\alpha_2, \\\\alpha_3\\\\), can we approximate \\\\[\\\\sum_{m=0}^{\\\\infty} c_m^{(1)}u_1^m+c_m^{(2)}u_2^m+c_m^{(3)}u_3^m?\\\\]\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/riemhilberttoeplitz/3/", "tags": [ "aim", "AIM-ANALYSIS-0016", "aim-domain:analysis", "aim-workshop:riemhilberttoeplitz", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The archived AIM prompt is genuinely underdetermined because it omits the approximation target, ambient space, topology, and coefficient class. Under the literature-supported weak-* H-infinity reading, the three-family span is dense whenever some distinct pair satisfies 4 alpha_i alpha_j / |xi_i-xi_j|^2 <= pi^2; the same condition yields H^p norm density for every finite p. For three pairwise distinct singularities, the three moduli determine the disk point explicitly, so even an all-pairs-supercritical triple has no common two-point fiber. Supremum-norm density in all of H-infinity is impossible, and literal coefficient-series convergence is characterized separately.\n\nCandidate contribution (lemma; novelty confidence low): For three atomic singular inner functions with pairwise distinct boundary singularities and arbitrary positive masses, the map z -> (|u_1(z)|, |u_2(z)|, |u_3(z)|) is injective; z is recovered by solving y_j=A-X cos(theta_j)-Y sin(theta_j), with y_j=(-alpha_j^{-1} log|u_j(z)|)^{-1}, and then using z=(X+iY)/(A+1). Consequently, an all-pairs-supercritical triple cannot be disproved complete by a two-point evaluation annihilator.", "research_difficulty_suggested": "L2" }, { "id": 20000485, "problem_number": "AIM-ANALYSIS-0017", "title": "Cartesian-product closure and endpoint tests for Robin-gap monotonicity", "statement": "Monotonicity of the spectral gap\n\nThe conjecture was originally formulated by R. Smits \\cite{MR1381604}. The following formulation can be found in a recent preprint by R. Laugesen \\cite{arXiv:1905.07658}.\n\nFor a bounded convex domain $\\Omega \\subset \\mathbb{R}^n$, the spectral gap $\\lambda_2(\\Omega, \\alpha)-\\lambda_1(\\Omega, \\alpha)$ is strictly increasing as a function of $\\alpha>0$. In particular, the Neumann gap provides a lower bound for the Dirichlet gap:\n$$\n \\mu_1(\\Omega)<\\lambda_2(\\Omega)-\\lambda_1(\\Omega).\n$$", "original_statement": "Monotonicity of the spectral gap\n\nThe conjecture was originally formulated by R. Smits \\cite{MR1381604}. The following formulation can be found in a recent preprint by R. Laugesen \\cite{arXiv:1905.07658}.\n\nFor a bounded convex domain $\\Omega \\subset \\mathbb{R}^n$, the spectral gap $\\lambda_2(\\Omega, \\alpha)-\\lambda_1(\\Omega, \\alpha)$ is strictly increasing as a function of $\\alpha>0$. In particular, the Neumann gap provides a lower bound for the Dirichlet gap:\n$$\n \\mu_1(\\Omega)<\\lambda_2(\\Omega)-\\lambda_1(\\Omega).\n$$", "clean_statement": "Monotonicity of the spectral gap\n\nThe conjecture was originally formulated by R. Smits \\cite{MR1381604}. The following formulation can be found in a recent preprint by R. Laugesen \\cite{arXiv:1905.07658}.\n\nFor a bounded convex domain $\\Omega \\subset \\mathbb{R}^n$, the spectral gap $\\lambda_2(\\Omega, \\alpha)-\\lambda_1(\\Omega, \\alpha)$ is strictly increasing as a function of $\\alpha>0$. In particular, the Neumann gap provides a lower bound for the Dirichlet gap:\n$$\n \\mu_1(\\Omega)<\\lambda_2(\\Omega)-\\lambda_1(\\Omega).\n$$", "statement_status": "exact", "statement_verification": "The canonical record is Conjecture 1.1 in the section *The Robin Laplacian* of the AIM workshop list *Shape optimization with surface interactions*. Its exact mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Robin Laplacian\nSource item: 1.1\nSource URL: http://aimpl.org/shapesurface/1/\nCanonical location: aim-analysis-notes.json notes[16]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Monotonicity of the spectral gap\\n\\nThe conjecture was originally formulated by R. Smits \\\\cite{MR1381604}. The following formulation can be found in a recent preprint by R. Laugesen \\\\cite{arXiv:1905.07658}.\\n\\nFor a bounded convex domain $\\\\Omega \\\\subset \\\\mathbb{R}^n$, the spectral gap $\\\\lambda_2(\\\\Omega, \\\\alpha)-\\\\lambda_1(\\\\Omega, \\\\alpha)$ is strictly increasing as a function of $\\\\alpha>0$. In particular, the Neumann gap provides a lower bound for the Dirichlet gap:\\n$$\\n \\\\mu_1(\\\\Omega)<\\\\lambda_2(\\\\Omega)-\\\\lambda_1(\\\\Omega).\\n$$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"It was checked by Smits that the conjecture is true for the disk and also for rectangular boxes in all dimensions. For the special case of rectangular boxes, monotonicity of the gap holds for all $-\\\\infty<\\\\alpha<\\\\infty$, see Theorem 2.1 in \\\\cite{arXiv:1905.07658}. However, it is known that the monotonicity fails in general when $\\\\alpha < 0$ becomes sufficiently large by absolute value.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/1/", "tags": [ "aim", "AIM-ANALYSIS-0017", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any finite Cartesian product of bounded connected Lipschitz domains, with the same Robin parameter on every product face, the fundamental Robin gap is exactly the minimum of the factor gaps. Consequently, strict increase of every factor gap on an interval implies strict increase of the product gap there, including through ties and changes of the minimizing factor. This proves Smits' positive-parameter conjecture for new convex product classes such as right circular cylinders and products of disks and intervals. The report also gives multiplicity-safe one-sided Neumann endpoint slopes and a necessary Dirichlet-end normal-flux inequality for global monotonicity.\n\nCandidate contribution (theorem; novelty confidence low): The exact identity G_{Omega_1 x ... x Omega_m}(alpha) = min_i G_{Omega_i}(alpha), together with strict-monotonicity closure under Cartesian products and its application to convex cylinders and disk-interval products, is the candidate novel contribution.", "research_difficulty_suggested": "L4" }, { "id": 20000486, "problem_number": "AIM-ANALYSIS-0018", "title": "All-coupling reverse Robin inequality for tangential domains", "statement": "Bareket's conjecture\n\nIn a paper by M. Bareket \\cite{MR430552}, it was conjectured that the lowest Robin eigenvalue $\\lambda_1(\\Omega,\\alpha)$ on a bounded domain $\\Omega$ with a boundary parameter $\\alpha0$ centred at the origin. In the same paper, it was proved that when $|\\alpha|$ is small enough the disk is the unique maximizer.\n\nLet $\\lambda_1(\\Omega, \\alpha)$ be the lowest Robin eigenvalue on a bounded domain $\\Omega$ with a negative boundary parameter $\\alpha$.\n\\begin{enumerate}\n\\item In two dimensions, the disk maximizes $\\lambda_1(\\Omega, \\alpha)$ among all simply connected domains of the same area.\n\\item In three and higher dimensions, the ball maximizes $\\lambda_1(\\Omega, \\alpha)$ among all convex domains of the same volume.\n\\end{enumerate}", "original_statement": "Bareket's conjecture\n\nIn a paper by M. Bareket \\cite{MR430552}, it was conjectured that the lowest Robin eigenvalue $\\lambda_1(\\Omega,\\alpha)$ on a bounded domain $\\Omega$ with a boundary parameter $\\alpha0$ centred at the origin. In the same paper, it was proved that when $|\\alpha|$ is small enough the disk is the unique maximizer.\n\nLet $\\lambda_1(\\Omega, \\alpha)$ be the lowest Robin eigenvalue on a bounded domain $\\Omega$ with a negative boundary parameter $\\alpha$.\n\\begin{enumerate}\n\\item In two dimensions, the disk maximizes $\\lambda_1(\\Omega, \\alpha)$ among all simply connected domains of the same area.\n\\item In three and higher dimensions, the ball maximizes $\\lambda_1(\\Omega, \\alpha)$ among all convex domains of the same volume.\n\\end{enumerate}", "clean_statement": "Bareket's conjecture\n\nIn a paper by M. Bareket \\cite{MR430552}, it was conjectured that the lowest Robin eigenvalue $\\lambda_1(\\Omega,\\alpha)$ on a bounded domain $\\Omega$ with a boundary parameter $\\alpha0$ centred at the origin. In the same paper, it was proved that when $|\\alpha|$ is small enough the disk is the unique maximizer.\n\nLet $\\lambda_1(\\Omega, \\alpha)$ be the lowest Robin eigenvalue on a bounded domain $\\Omega$ with a negative boundary parameter $\\alpha$.\n\\begin{enumerate}\n\\item In two dimensions, the disk maximizes $\\lambda_1(\\Omega, \\alpha)$ among all simply connected domains of the same area.\n\\item In three and higher dimensions, the ball maximizes $\\lambda_1(\\Omega, \\alpha)$ among all convex domains of the same volume.\n\\end{enumerate}", "statement_status": "exact", "statement_verification": "This record is item 1.2, “Bareket's conjecture,” in the AIM list from the workshop *Shape optimization with surface interactions*. The canonical JSON has a visibly damaged sentence,", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Robin Laplacian\nSource item: 1.2\nSource URL: http://aimpl.org/shapesurface/1/\nCanonical location: aim-analysis-notes.json notes[17]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Bareket's conjecture\\n\\nIn a paper by M. Bareket \\\\cite{MR430552}, it was conjectured that the lowest Robin eigenvalue $\\\\lambda_1(\\\\Omega,\\\\alpha)$ on a bounded domain $\\\\Omega$ with a boundary parameter $\\\\alpha0$ centred at the origin. In the same paper, it was proved that when $|\\\\alpha|$ is small enough the disk is the unique maximizer.\\n\\nLet $\\\\lambda_1(\\\\Omega, \\\\alpha)$ be the lowest Robin eigenvalue on a bounded domain $\\\\Omega$ with a negative boundary parameter $\\\\alpha$.\\n\\\\begin{enumerate}\\n\\\\item In two dimensions, the disk maximizes $\\\\lambda_1(\\\\Omega, \\\\alpha)$ among all simply connected domains of the same area.\\n\\\\item In three and higher dimensions, the ball maximizes $\\\\lambda_1(\\\\Omega, \\\\alpha)$ among all convex domains of the same volume.\\n\\\\end{enumerate}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Counterparts of this conjecture under fixed area (perimeter) of the boundary are already settled in two dimensions \\\\cite{MR3707083} for general domains and also in three and higher dimensions \\\\cite{arXiv:1810.06108},\\n\\\\cite{arXiv:1906.11141} in the class of convex domains.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/1/", "tags": [ "aim", "AIM-ANALYSIS-0018", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let d be at least 2, let alpha be negative, and let Omega be a bounded convex Lipschitz domain with an inball B_r centered at the origin such that x dot nu equals r for surface-almost every boundary point. If B_R has the same volume as Omega, then lambda_1(Omega,alpha) is at most lambda_1(B_r,alpha), which is at most lambda_1(B_R,alpha); the equal-volume comparison is strict unless Omega is a ball. Hence the revised Bareket inequality holds at every negative coupling for all circumscribed convex polygons and polytopes, including every triangle and simplex. A separate proved proposition shows that constant support distance is forced if the standard one-variable homothetic gauge upper bound is to remain capable of certifying the fixed-volume inequality at arbitrarily strong coupling; this is a limitation of that method only.\n\nCandidate contribution (theorem; novelty confidence low): For every negative Robin parameter, every convex Lipschitz domain tangential to an inball in the precise almost-everywhere constant-support sense has first Robin eigenvalue no greater than that of its equal-volume ball, with strict inequality for nonballs; moreover, the homothetic gauge method has a proved strong-coupling rigidity criterion c=q^2 equivalent to constant support distance.", "research_difficulty_suggested": "L4" }, { "id": 20000487, "problem_number": "AIM-ANALYSIS-0019", "title": "A Neumann-point incenter-moment expansion for the discrete Bareket conjecture", "statement": "Discrete Bareket's conjecture\n\nIn the same spirit as Bareket's conjecture one can ask what polygon maximizes $\\lambda_1(\\Omega, \\alpha)$, $\\alpha0$ in which case the problem is to minimize the eigenvalue. The following conjecture concerns the case $N = 3$.\n\nAmong all triangles of a given area $\\lambda_1(\\Omega, \\alpha)$ is:\n\\begin{enumerate}\n\\item minimized by the equilateral triangle when $\\alpha>0$.\n\\item maximized by the equilateral triangle when $\\alpha<0$.\n\\end{enumerate}", "original_statement": "Discrete Bareket's conjecture\n\nIn the same spirit as Bareket's conjecture one can ask what polygon maximizes $\\lambda_1(\\Omega, \\alpha)$, $\\alpha0$ in which case the problem is to minimize the eigenvalue. The following conjecture concerns the case $N = 3$.\n\nAmong all triangles of a given area $\\lambda_1(\\Omega, \\alpha)$ is:\n\\begin{enumerate}\n\\item minimized by the equilateral triangle when $\\alpha>0$.\n\\item maximized by the equilateral triangle when $\\alpha<0$.\n\\end{enumerate}", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is AIM Problem Lists entry 1.3, “Discrete Bareket's conjecture,” from the 2019 workshop *Shape optimization with surface interactions*. The record in `aim-analysis-notes.json` is visibly corrupted at", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Robin Laplacian\nSource item: 1.3\nSource URL: http://aimpl.org/shapesurface/1/\nCanonical location: aim-analysis-notes.json notes[18]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Discrete Bareket's conjecture\\n\\nIn the same spirit as Bareket's conjecture one can ask what polygon maximizes $\\\\lambda_1(\\\\Omega, \\\\alpha)$, $\\\\alpha0$ in which case the problem is to minimize the eigenvalue. The following conjecture concerns the case $N = 3$.\\n\\nAmong all triangles of a given area $\\\\lambda_1(\\\\Omega, \\\\alpha)$ is:\\n\\\\begin{enumerate}\\n\\\\item minimized by the equilateral triangle when $\\\\alpha>0$.\\n\\\\item maximized by the equilateral triangle when $\\\\alpha<0$.\\n\\\\end{enumerate}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/1/", "tags": [ "aim", "AIM-ANALYSIS-0019", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every fixed nondegenerate triangle T of area A and perimeter P, with incenter I and J_I(T)=integral_T |x-I|^2 dx, the lowest Robin eigenvalue has the two-sided expansion lambda_1(T,alpha)=(P/A)alpha-[P^2 J_I(T)/(4A^3)]alpha^2+O_T(alpha^3) as alpha tends to zero. The scale-invariant coefficient Q(T)=P^2 J_I(T)/(4A^3) satisfies Q(T)>=(P/P_eq)^4>=1, with equality only for the equilateral triangle. Thus the second-order term reinforces the conjectured comparison for alpha<0 but opposes the first-order term for alpha>0. The remainder and perturbative radius depend on the fixed triangle, so this does not yield a uniform all-triangles result or solve the full conjecture.\n\nCandidate contribution (theorem/expansion; novelty confidence low): Candidate novelty: the explicit second Robin eigenvalue coefficient -P^2 J_I(T)/(4A^3), its sharp rigidity bound Q(T)>=(P/P_eq)^4 with equality only for the equilateral triangle, and the resulting attractive-versus-repulsive sign asymmetry.", "research_difficulty_suggested": "L4" }, { "id": 20000488, "problem_number": "AIM-ANALYSIS-0020", "title": "All-coupling Robin bound states for compact graph deformations in dimensions two and three", "statement": "Negative Robin eigenvalues on a locally deformed half-space\n\nConsider the Robin Laplacian with a negative boundary parameter $\\alpha$ on a locally perturbed half-space (its boundary is a hyperplane away from a bounded region). The essential spectrum of this operator is expected to be $[-\\alpha^2, \\infty)$. In the strong coupling limit $\\alpha\\to -\\infty$, existence of negative eigenvalues below the essential spectrum reduces to showing that the maximum\nfor the mean curvature of the boundary is positive \\cite{MR3626320}.\n\nProve or disprove that there are always negative eigenvalues below $-\\alpha^2$.", "original_statement": "Negative Robin eigenvalues on a locally deformed half-space\n\nConsider the Robin Laplacian with a negative boundary parameter $\\alpha$ on a locally perturbed half-space (its boundary is a hyperplane away from a bounded region). The essential spectrum of this operator is expected to be $[-\\alpha^2, \\infty)$. In the strong coupling limit $\\alpha\\to -\\infty$, existence of negative eigenvalues below the essential spectrum reduces to showing that the maximum\nfor the mean curvature of the boundary is positive \\cite{MR3626320}.\n\nProve or disprove that there are always negative eigenvalues below $-\\alpha^2$.", "clean_statement": "Negative Robin eigenvalues on a locally deformed half-space\n\nConsider the Robin Laplacian with a negative boundary parameter $\\alpha$ on a locally perturbed half-space (its boundary is a hyperplane away from a bounded region). The essential spectrum of this operator is expected to be $[-\\alpha^2, \\infty)$. In the strong coupling limit $\\alpha\\to -\\infty$, existence of negative eigenvalues below the essential spectrum reduces to showing that the maximum\nfor the mean curvature of the boundary is positive \\cite{MR3626320}.\n\nProve or disprove that there are always negative eigenvalues below $-\\alpha^2$.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.4 in the AIM list *Shape optimization with surface interactions*, section “The Robin Laplacian.” The live AIM page was checked on 27 July 2026 and agrees with the extracted record. Its mathematical content is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Robin Laplacian\nSource item: 1.4\nSource URL: http://aimpl.org/shapesurface/1/\nCanonical location: aim-analysis-notes.json notes[19]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Negative Robin eigenvalues on a locally deformed half-space\\n\\nConsider the Robin Laplacian with a negative boundary parameter $\\\\alpha$ on a locally perturbed half-space (its boundary is a hyperplane away from a bounded region). The essential spectrum of this operator is expected to be $[-\\\\alpha^2, \\\\infty)$. In the strong coupling limit $\\\\alpha\\\\to -\\\\infty$, existence of negative eigenvalues below the essential spectrum reduces to showing that the maximum\\nfor the mean curvature of the boundary is positive \\\\cite{MR3626320}.\\n\\nProve or disprove that there are always negative eigenvalues below $-\\\\alpha^2$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/1/", "tags": [ "aim", "AIM-ANALYSIS-0020", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every attractive Robin strength and every nontrivial compactly supported Lipschitz graph deformation of a half-space in ambient dimension n=2 or n=3, the essential spectrum is exactly [-beta^2, infinity) and there is a discrete eigenvalue strictly below -beta^2. The proof uses an exact shifted-form identity for the flat transverse mode and recurrence in the one- or two-dimensional tangential space. In all dimensions the same identity gives a capacitary sufficient condition, while a curvature sign-change lemma recovers strong-coupling binding for smooth admissible graph deformations. The argument does not settle all-coupling binding for ambient n>=4 or for non-graph local perturbations.\n\nCandidate contribution (theorem; novelty confidence low): The candidate novel contribution is the exact plane-mode shifted-form identity and its consequence that every nonflat compactly supported Lipschitz graph deformation in ambient dimension three has a Robin eigenvalue below the flat threshold for every attractive coupling; the associated capacity inequality is a testable sufficient criterion in every dimension." }, { "id": 20000489, "problem_number": "AIM-ANALYSIS-0021", "title": "One-dimensional counterexample and classification for optimal Dirichlet eigenvalue multiplicity", "statement": "Multiplicity of optimal Dirichlet eigenvalues\n\nLet $\\Omega_k^*$ be a minimizer of the shape optimization problem\n$$\n\t\\min\\{ \\lambda_k(\\Omega): \\Omega\\subset \\mathbb{R}^n, |\\Omega|=V_0\\}.\n$$\nProve that $\\lambda_{k-1}(\\Omega_k^*)=\\lambda_k(\\Omega_k^*)$.", "original_statement": "Multiplicity of optimal Dirichlet eigenvalues\n\nLet $\\Omega_k^*$ be a minimizer of the shape optimization problem\n$$\n\t\\min\\{ \\lambda_k(\\Omega): \\Omega\\subset \\mathbb{R}^n, |\\Omega|=V_0\\}.\n$$\nProve that $\\lambda_{k-1}(\\Omega_k^*)=\\lambda_k(\\Omega_k^*)$.", "clean_statement": "Multiplicity of optimal Dirichlet eigenvalues\n\nLet $\\Omega_k^*$ be a minimizer of the shape optimization problem\n$$\n\t\\min\\{ \\lambda_k(\\Omega): \\Omega\\subset \\mathbb{R}^n, |\\Omega|=V_0\\}.\n$$\nProve that $\\lambda_{k-1}(\\Omega_k^*)=\\lambda_k(\\Omega_k^*)$.", "statement_status": "exact", "statement_verification": "The canonical record is item 2.1, “Multiplicity of optimal Dirichlet eigenvalues,” from the AIM problem list associated with the workshop *Shape optimization with surface interactions*. Its exact mathematical text is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Dirichlet Laplacian\nSource item: 2.1\nSource URL: http://aimpl.org/shapesurface/2/\nCanonical location: aim-analysis-notes.json notes[20]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Multiplicity of optimal Dirichlet eigenvalues\\n\\nLet $\\\\Omega_k^*$ be a minimizer of the shape optimization problem\\n$$\\n\\t\\\\min\\\\{ \\\\lambda_k(\\\\Omega): \\\\Omega\\\\subset \\\\mathbb{R}^n, |\\\\Omega|=V_0\\\\}.\\n$$\\nProve that $\\\\lambda_{k-1}(\\\\Omega_k^*)=\\\\lambda_k(\\\\Omega_k^*)$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This conjecture is verified by E. Oudet in \\\\cite{MR2084326} via numerical experiments.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/2/", "tags": [ "aim", "AIM-ANALYSIS-0021", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For the exact dimension-unrestricted canonical wording, the conjecture is false: in dimension one the minimum of the k-th Dirichlet eigenvalue over open sets of length V is (pi k/V)^2, and every minimizer is a finite union of intervals with lengths m_j V/k for a positive integer partition sum m_j=k; the multiplicity of the optimal level is exactly the number of parts. The one-part partition is a single interval, which is a minimizer with lambda_{k-1}=2, k>=3 conjecture remains open. A separate proved Wolf--Keller-type reduction shows that every disconnected bounded-open global minimizer satisfies lambda_{k-1}=lambda_k.\n\nCandidate contribution (classification; novelty confidence low): Every one-dimensional fixed-length minimizer of the k-th Dirichlet eigenvalue is encoded by a positive integer partition k=m_1+...+m_r: its interval component lengths are exactly m_j V/k, and the optimizing eigenvalue has multiplicity exactly r." }, { "id": 20000490, "problem_number": "AIM-ANALYSIS-0022", "title": "The spectral Mahler functional on the planar cube orbit", "statement": "Mahler inequality for the principal Dirichlet eigenvalue\n\nLet $\\mathcal{K}^n_\\star$ be the class of centrally symmetric bounded convex domains in $\\mathbb{R}^n$. For a domain $K\\in\\mathcal{K}^n_\\star$ we define its polar set by\n $$\n K^\\circ := \\{x\\in \\mathbb{R}^n\\colon x \\cdot y \\leq 1 \\mbox{ for all }y \\in K\\}.\n $$\nFurther, let $\\mathsf{GL}_n$ be the family of invertible linear transformation in $\\mathbb{R}^n$.\n\nA hypercube in $\\mathbb{R}^n$ is an attainer for the maximization problem\n\\[\n\\sup\\left\\{\n\\inf_{T\\in\\mathsf{GL}_n} \\lambda_1(T(K))\\lambda_1((T(K))^\\circ)\\colon K\\in\\mathcal{K}^n_\\star\\right\\}.\n\\]", "original_statement": "Mahler inequality for the principal Dirichlet eigenvalue\n\nLet $\\mathcal{K}^n_\\star$ be the class of centrally symmetric bounded convex domains in $\\mathbb{R}^n$. For a domain $K\\in\\mathcal{K}^n_\\star$ we define its polar set by\n $$\n K^\\circ := \\{x\\in \\mathbb{R}^n\\colon x \\cdot y \\leq 1 \\mbox{ for all }y \\in K\\}.\n $$\nFurther, let $\\mathsf{GL}_n$ be the family of invertible linear transformation in $\\mathbb{R}^n$.\n\nA hypercube in $\\mathbb{R}^n$ is an attainer for the maximization problem\n\\[\n\\sup\\left\\{\n\\inf_{T\\in\\mathsf{GL}_n} \\lambda_1(T(K))\\lambda_1((T(K))^\\circ)\\colon K\\in\\mathcal{K}^n_\\star\\right\\}.\n\\]", "clean_statement": "Mahler inequality for the principal Dirichlet eigenvalue\n\nLet $\\mathcal{K}^n_\\star$ be the class of centrally symmetric bounded convex domains in $\\mathbb{R}^n$. For a domain $K\\in\\mathcal{K}^n_\\star$ we define its polar set by\n $$\n K^\\circ := \\{x\\in \\mathbb{R}^n\\colon x \\cdot y \\leq 1 \\mbox{ for all }y \\in K\\}.\n $$\nFurther, let $\\mathsf{GL}_n$ be the family of invertible linear transformation in $\\mathbb{R}^n$.\n\nA hypercube in $\\mathbb{R}^n$ is an attainer for the maximization problem\n\\[\n\\sup\\left\\{\n\\inf_{T\\in\\mathsf{GL}_n} \\lambda_1(T(K))\\lambda_1((T(K))^\\circ)\\colon K\\in\\mathcal{K}^n_\\star\\right\\}.\n\\]", "statement_status": "exact", "statement_verification": "Central symmetry forces the center to be the origin in this formulation. We harmlessly identify a bounded convex domain with its closure when taking the polar; its Dirichlet eigenvalue is that of its interior. No corruption of the mathematical statement was found.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Dirichlet Laplacian\nSource item: 2.2\nSource URL: http://aimpl.org/shapesurface/2/\nCanonical location: aim-analysis-notes.json notes[21]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Mahler inequality for the principal Dirichlet eigenvalue\\n\\nLet $\\\\mathcal{K}^n_\\\\star$ be the class of centrally symmetric bounded convex domains in $\\\\mathbb{R}^n$. For a domain $K\\\\in\\\\mathcal{K}^n_\\\\star$ we define its polar set by\\n $$\\n K^\\\\circ := \\\\{x\\\\in \\\\mathbb{R}^n\\\\colon x \\\\cdot y \\\\leq 1 \\\\mbox{ for all }y \\\\in K\\\\}.\\n $$\\nFurther, let $\\\\mathsf{GL}_n$ be the family of invertible linear transformation in $\\\\mathbb{R}^n$.\\n\\nA hypercube in $\\\\mathbb{R}^n$ is an attainer for the maximization problem\\n\\\\[\\n\\\\sup\\\\left\\\\{\\n\\\\inf_{T\\\\in\\\\mathsf{GL}_n} \\\\lambda_1(T(K))\\\\lambda_1((T(K))^\\\\circ)\\\\colon K\\\\in\\\\mathcal{K}^n_\\\\star\\\\right\\\\}.\\n\\\\]\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"According to Proposition 5 in \\\\cite{MR3551851}, a maximizer for the above problem exists. Moreover, one can consider a modification of the above problem, in which $\\\\mathcal{K}^n_\\\\star$ is replaced by the class\\nof axisymmetric bounded convex domains and in which the family of transformations $\\\\mathsf{GL}_n$ is restricted to diagonal ones. In such a modified setting the conjecture is settled in Theorem 9 of \\\\cite{MR3551851} for $n = 2$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/2/", "tags": [ "aim", "AIM-ANALYSIS-0022", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted outer maximization remains open. Two rigorous inner-orbit results are proved: every fixed centrally symmetric convex body has a minimizing affine representative, with the quantitative coercivity bound F(AK) >= (pi^4 r^2/(16 R^2)) kappa(A)^2 in determinant-one positive-definite coordinates; and every such minimizing representative L satisfies an exact spectral-stress balance between the normalized ground-state gradient-energy matrices on L and its polar. In dimension two, the full GL_2 inner value of the square, and hence of every parallelogram, is exactly pi^4/2.\n\nCandidate contribution (necessary_condition; novelty confidence low): If L is an affine representative attaining the inner GL_n minimum and u, v are L2-normalized first Dirichlet eigenfunctions on L and L^circ, respectively, then (1/lambda_1(L)) integral_L grad(u) tensor grad(u) equals (1/lambda_1(L^circ)) integral_{L^circ} grad(v) tensor grad(v).", "research_difficulty_suggested": "L4" }, { "id": 20000491, "problem_number": "AIM-ANALYSIS-0023", "title": "Polygonal Faber-Krahn status and an exact affine-gauge reduction", "statement": "Discrete Faber-Krahn inequality\n\nThe classical Faber-Krahn inequality states that among all sets of equal volume the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the ball. The following longstanding conjecture concerns the corresponding problem about minimization of $\\lambda_1(\\Omega)$ among polygons.\n\nAmong all polygons of a given area and having not more than $N$ edges the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the regular $N$-gon.", "original_statement": "Discrete Faber-Krahn inequality\n\nThe classical Faber-Krahn inequality states that among all sets of equal volume the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the ball. The following longstanding conjecture concerns the corresponding problem about minimization of $\\lambda_1(\\Omega)$ among polygons.\n\nAmong all polygons of a given area and having not more than $N$ edges the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the regular $N$-gon.", "clean_statement": "Discrete Faber-Krahn inequality\n\nThe classical Faber-Krahn inequality states that among all sets of equal volume the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the ball. The following longstanding conjecture concerns the corresponding problem about minimization of $\\lambda_1(\\Omega)$ among polygons.\n\nAmong all polygons of a given area and having not more than $N$ edges the lowest Dirichlet eigenvalue $\\lambda_1(\\Omega)$ is minimized by the regular $N$-gon.", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ANALYSIS-0023, item 2.3 in the AIM workshop list *Shape optimization with surface interactions*, section “The Dirichlet Laplacian.” Its statement is uncorrupted and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Dirichlet Laplacian\nSource item: 2.3\nSource URL: http://aimpl.org/shapesurface/2/\nCanonical location: aim-analysis-notes.json notes[22]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Discrete Faber-Krahn inequality\\n\\nThe classical Faber-Krahn inequality states that among all sets of equal volume the lowest Dirichlet eigenvalue $\\\\lambda_1(\\\\Omega)$ is minimized by the ball. The following longstanding conjecture concerns the corresponding problem about minimization of $\\\\lambda_1(\\\\Omega)$ among polygons.\\n\\nAmong all polygons of a given area and having not more than $N$ edges the lowest Dirichlet eigenvalue $\\\\lambda_1(\\\\Omega)$ is minimized by the regular $N$-gon.\"\nOriginal remarks: [\"For more on shape optimization for triangles, see Chapter 6 (pp. 149-200) of the book \\\"Shape Optimization and Spectral Theory\\\", edited by Antoine Henrot, 2017 (open access).\"]\nOriginal literature field (JSON string): \"The conjecture is settled by G. Polya for $N=3, 4$ via Steiner symmetrization. However, if $N\\\\geq 5$ symmetrization might increase the number of edges of a polygon and consequently this case requires a different approach. For all $N$, it is known that minimizers exist and have precisely $N$ edges; see Theorem 3.3.1 in \\\\cite{MR2251558}\\n\\nInteresting partial questions and further problems are for instance:\\n\\\\begin{enumerate}\\n\\\\item Prove that the regular $N$-gon is a local minimizer.\\n\\\\item Prove that a minimizer must be convex.\\n\\\\item Prove that $\\\\lambda_1(\\\\Omega_N)$ of the regular $N$-gon $\\\\Omega_N$ is decreasing as a function of $N$. This is a necessary condition for the validity of the conjecture (remarked by C. Nitsch).\\n\\\\item Consider the corresponding problem, but for the maximization of the first non-trivial Neumann eigenvalue.\\n\\\\item Consider the corresponding problems but with the perimeter constraint.\\n\\\\end{enumerate}\\nFor the Neumann problem, the case $N=3$ was solved by R. Laugesen and B. Siudeja \\\\cite{MR2567204} for both area and perimeter constraints, but for $N\\\\geq 4$ the problem remains open.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/2/", "tags": [ "aim", "AIM-ANALYSIS-0023", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The global polygonal Faber-Krahn conjecture remains open for N >= 5, although rigorous validated local minimality is now known for N = 5, 6 and a 2026 computer-assisted preprint proves strict monotonicity of the fixed-area regular-polygon eigenvalues for all N >= 3. This attempt proves that for every bounded connected Lipschitz planar domain P, with L2-normalized ground-state energy tensor E and D = E - (lambda_1(P)/2)I, the exactly area-preserving deformation exp(tD)P satisfies lambda_1'(0) = -2||D||_F^2. Hence every non-isotropic polygon has an explicit affine descent. For convex polygons, combining this identity with the existence, uniqueness, and isotropy theorem stated in Faifman-Polterovich, arXiv:2607.21539v1, reduces global comparison to the isotropic slice D = 0; that orbitwise sufficiency depends on a version-1 preprint submitted four days before this run.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novel synthesis: choose the trace-free energy anisotropy D itself as the SL(2) generator to obtain the exact norm-square descent lambda_1'(0) = -2||D||_F^2, and, in the convex class using Faifman-Polterovich's affine-position theorem, quotient the polygonal Faber-Krahn search to the two-equation isotropic gauge D = 0.", "research_difficulty_suggested": "L4" }, { "id": 20000492, "problem_number": "AIM-ANALYSIS-0024", "title": "Beck's solution of van den Berg's conjecture and an exact prism refinement", "statement": "van den Berg's conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and let $\\rho, D$ denote its inradius and diameter, respectively. Let $u$ denote the eigenfunction corresponding to the first eigenvalue of the Dirichlet Laplacian on $\\Omega$. According to a theorem due to G. Chiti \\cite{MR0652928} there exists a dimensional constant $C$ such that\n$$\n\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\|u\\|_{L^2(\\Omega)}.\n$$\nThe following conjecture was posed by M. van den Berg \\cite{MR1804178}\n\nThere is a dimensional constant $C$ such that\n$$\n\t\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\Bigl(\\frac{\\rho}{D}\\Bigr)^{1/6}\\|u\\|_{L^2(\\Omega)}.\n$$", "original_statement": "van den Berg's conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and let $\\rho, D$ denote its inradius and diameter, respectively. Let $u$ denote the eigenfunction corresponding to the first eigenvalue of the Dirichlet Laplacian on $\\Omega$. According to a theorem due to G. Chiti \\cite{MR0652928} there exists a dimensional constant $C$ such that\n$$\n\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\|u\\|_{L^2(\\Omega)}.\n$$\nThe following conjecture was posed by M. van den Berg \\cite{MR1804178}\n\nThere is a dimensional constant $C$ such that\n$$\n\t\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\Bigl(\\frac{\\rho}{D}\\Bigr)^{1/6}\\|u\\|_{L^2(\\Omega)}.\n$$", "clean_statement": "van den Berg's conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and let $\\rho, D$ denote its inradius and diameter, respectively. Let $u$ denote the eigenfunction corresponding to the first eigenvalue of the Dirichlet Laplacian on $\\Omega$. According to a theorem due to G. Chiti \\cite{MR0652928} there exists a dimensional constant $C$ such that\n$$\n\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\|u\\|_{L^2(\\Omega)}.\n$$\nThe following conjecture was posed by M. van den Berg \\cite{MR1804178}\n\nThere is a dimensional constant $C$ such that\n$$\n\t\\|u\\|_{L^\\infty(\\Omega)} \\leq \\frac{C}{\\rho^{n/2}}\\Bigl(\\frac{\\rho}{D}\\Bigr)^{1/6}\\|u\\|_{L^2(\\Omega)}.\n$$", "statement_status": "exact", "statement_verification": "The canonical record is item 2.4, “van den Berg's conjecture,” in the AIM list *Shape optimization with surface interactions*, section “The Dirichlet Laplacian.” The archived AIM page gives the following assertion. If \\(\\Omega\\subset\\mathbb R^n\\) is a convex domain, \\(\\rho\\) and \\(D\\) are its inradius and diameter, and \\(u\\) is a first Dirichlet eigenfunction, then there should be a dimensional constant \\(C_n\\) such that", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Dirichlet Laplacian\nSource item: 2.4\nSource URL: http://aimpl.org/shapesurface/2/\nCanonical location: aim-analysis-notes.json notes[23]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"van den Berg's conjecture\\n\\nLet $\\\\Omega \\\\subset \\\\mathbb{R}^n$ be a convex domain and let $\\\\rho, D$ denote its inradius and diameter, respectively. Let $u$ denote the eigenfunction corresponding to the first eigenvalue of the Dirichlet Laplacian on $\\\\Omega$. According to a theorem due to G. Chiti \\\\cite{MR0652928} there exists a dimensional constant $C$ such that\\n$$\\n\\\\|u\\\\|_{L^\\\\infty(\\\\Omega)} \\\\leq \\\\frac{C}{\\\\rho^{n/2}}\\\\|u\\\\|_{L^2(\\\\Omega)}.\\n$$\\nThe following conjecture was posed by M. van den Berg \\\\cite{MR1804178}\\n\\nThere is a dimensional constant $C$ such that\\n$$\\n\\t\\\\|u\\\\|_{L^\\\\infty(\\\\Omega)} \\\\leq \\\\frac{C}{\\\\rho^{n/2}}\\\\Bigl(\\\\frac{\\\\rho}{D}\\\\Bigr)^{1/6}\\\\|u\\\\|_{L^2(\\\\Omega)}.\\n$$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Thanks to work of B. Georgiev, M. Mukherjee, and S. Steinerberger the conjecture is known to be true when $n=2$ \\\\cite{MR3859539}. The exponent $1/6$ is what is expected from consideration of a cone. Proving the corresponding result with a different improvement in terms of $\\\\rho/D$ would be interesting.\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/2/", "tags": [ "aim", "AIM-ANALYSIS-0024", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Beck's published John-ellipsoid theorem proves the canonical inradius-diameter inequality in every dimension, so the AIM conjecture is solved; this attempt does not claim that known theorem as new. Separately, this attempt proves an exact tensorization identity for the ground-state L-infinity/L2 ratio and effective support volume on a bounded connected base times intervals, and deduces a uniform (rho/D)^(1/2) gain for long prisms with arbitrary convex base, with exponent 1/2 optimal along each fixed-base prism family.\n\nCandidate contribution (proposition; novelty confidence low): If G is a bounded connected domain and Omega is G times a product of intervals of lengths L_i, then V_eff(Omega) equals V_eff(G) times the product of L_i/2. If G is convex and Omega=G times (0,L) with L at least diam(G), Chiti's estimate implies R(Omega) is at most 2^(3/4) C_k rho^(-n/2)(rho/D)^(1/2), and no exponent greater than 1/2 can hold with an L-independent constant along a fixed-base family." }, { "id": 20000493, "problem_number": "AIM-ANALYSIS-0025", "title": "Exact tensorization of optimal Dirichlet eigenfunction power-concavity", "statement": "Concavity of the principal Dirichlet eigenfunction\n\nLet $\\Omega\\subset \\mathbb{R}^n$ be a bouned convex domain and let $u$ be the principal eigenfunction of the Dirichlet Laplacian on $\\Omega$. By a classical result of H. Brascamp and E. Lieb $u$ is log-concave \\cite{MR0450480}. Moreover, it is known that $u$ is power-concave meaning that $u^\\alpha$ is concave in the usual sense for certain powers $\\alpha > 0$.\nThe best concavity exponent for a domain $\\Omega$ is defined as\n\\[\n\\alpha(\\Omega) :=\\sup\\{\\alpha \\ge 0\\colon u^\\alpha \\mbox{ is concave}\\}.\n\\]\nThe following conjecture was formulated in \\cite{MR1280948} by P. Lindqvist.\n\nThe best concavity exponent $\\alpha(\\Omega)$ is maximized by the ball.", "original_statement": "Concavity of the principal Dirichlet eigenfunction\n\nLet $\\Omega\\subset \\mathbb{R}^n$ be a bouned convex domain and let $u$ be the principal eigenfunction of the Dirichlet Laplacian on $\\Omega$. By a classical result of H. Brascamp and E. Lieb $u$ is log-concave \\cite{MR0450480}. Moreover, it is known that $u$ is power-concave meaning that $u^\\alpha$ is concave in the usual sense for certain powers $\\alpha > 0$.\nThe best concavity exponent for a domain $\\Omega$ is defined as\n\\[\n\\alpha(\\Omega) :=\\sup\\{\\alpha \\ge 0\\colon u^\\alpha \\mbox{ is concave}\\}.\n\\]\nThe following conjecture was formulated in \\cite{MR1280948} by P. Lindqvist.\n\nThe best concavity exponent $\\alpha(\\Omega)$ is maximized by the ball.", "clean_statement": "Concavity of the principal Dirichlet eigenfunction\n\nLet $\\Omega\\subset \\mathbb{R}^n$ be a bouned convex domain and let $u$ be the principal eigenfunction of the Dirichlet Laplacian on $\\Omega$. By a classical result of H. Brascamp and E. Lieb $u$ is log-concave \\cite{MR0450480}. Moreover, it is known that $u$ is power-concave meaning that $u^\\alpha$ is concave in the usual sense for certain powers $\\alpha > 0$.\nThe best concavity exponent for a domain $\\Omega$ is defined as\n\\[\n\\alpha(\\Omega) :=\\sup\\{\\alpha \\ge 0\\colon u^\\alpha \\mbox{ is concave}\\}.\n\\]\nThe following conjecture was formulated in \\cite{MR1280948} by P. Lindqvist.\n\nThe best concavity exponent $\\alpha(\\Omega)$ is maximized by the ball.", "statement_status": "exact", "statement_verification": "The canonical record is item 2.5, “Concavity of the principal Dirichlet eigenfunction,” in the AIM list *Shape optimization with surface interactions*, section “The Dirichlet Laplacian” (`aim-analysis-notes.json`, zero-based index 24). The source URL is . The page timed out during this run, so the canonical repository record is the source used for the statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Dirichlet Laplacian\nSource item: 2.5\nSource URL: http://aimpl.org/shapesurface/2/\nCanonical location: aim-analysis-notes.json notes[24]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Concavity of the principal Dirichlet eigenfunction\\n\\nLet $\\\\Omega\\\\subset \\\\mathbb{R}^n$ be a bouned convex domain and let $u$ be the principal eigenfunction of the Dirichlet Laplacian on $\\\\Omega$. By a classical result of H. Brascamp and E. Lieb $u$ is log-concave \\\\cite{MR0450480}. Moreover, it is known that $u$ is power-concave meaning that $u^\\\\alpha$ is concave in the usual sense for certain powers $\\\\alpha > 0$.\\nThe best concavity exponent for a domain $\\\\Omega$ is defined as\\n\\\\[\\n\\\\alpha(\\\\Omega) :=\\\\sup\\\\{\\\\alpha \\\\ge 0\\\\colon u^\\\\alpha \\\\mbox{ is concave}\\\\}.\\n\\\\]\\nThe following conjecture was formulated in \\\\cite{MR1280948} by P. Lindqvist.\\n\\nThe best concavity exponent $\\\\alpha(\\\\Omega)$ is maximized by the ball.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/2/", "tags": [ "aim", "AIM-ANALYSIS-0025", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For positive C^2 log-concave functions on orthogonal Cartesian products, the reciprocals of their optimal power-concavity exponents add exactly; applying this to principal Dirichlet eigenfunctions proves alpha(Omega_1 x ... x Omega_m)^{-1} = sum_j alpha(Omega_j)^{-1}, and consequently every n-dimensional orthogonal box has exponent exactly 1/n, independently of aspect ratios. A supporting radial calculation gives alpha(B^n) = inf_{0k epsilon, mu_k(Q_epsilon)=(k pi/L_epsilon)^2=[2(n-1)k pi epsilon^(n-2)/(P0-2 epsilon^(n-1))]^2. Hence for every n>=3 and k>=1 the infimum is 0 and is unattained; for n=2,k=1 the infimum is 4 pi^2/P0^2 and is unattained by the Payne-Weinberger bound and P>2D. Published positive rows remain: in the plane the supremum is attained for every k and the infimum is attained for k>=2, while in dimension three the mu_1 supremum is attained. The supremum cases n=3,k>=2 and n>=4,k>=1 remain unresolved in the literature checked through 2026-07-27.\n\nCandidate contribution (lemma; novelty confidence low): Let K be a bounded open convex domain in R^(n-1), set P0=2|K|, and let lambda_k(K) be its ordinary Neumann eigenvalue. For 0lambda_k(K). Thus an ordinary, unweighted codimension-one Neumann value cannot be a relaxed maximizer." }, { "id": 20000495, "problem_number": "AIM-ANALYSIS-0027", "title": "Sharp Neumann eigenvalue ratios and an all-index ordered-simplex family", "statement": "Maximizing the ratio of Neumann eigenvalues among convex domains\n\nFor any convex $\\Omega\\subset \\mathbb{R}^n$, it holds that\n \\begin{enumerate}[label=\\textup{(\\alph*)}]\n \\item $\\mu_2(\\Omega)/\\mu_1(\\Omega)\\leq 4$.\n \\item $\\mu_k(\\Omega)/\\mu_1(\\Omega)\\leq k^2$.\n \\end{enumerate}", "original_statement": "Maximizing the ratio of Neumann eigenvalues among convex domains\n\nFor any convex $\\Omega\\subset \\mathbb{R}^n$, it holds that\n \\begin{enumerate}[label=\\textup{(\\alph*)}]\n \\item $\\mu_2(\\Omega)/\\mu_1(\\Omega)\\leq 4$.\n \\item $\\mu_k(\\Omega)/\\mu_1(\\Omega)\\leq k^2$.\n \\end{enumerate}", "clean_statement": "Maximizing the ratio of Neumann eigenvalues among convex domains\n\nFor any convex $\\Omega\\subset \\mathbb{R}^n$, it holds that\n \\begin{enumerate}[label=\\textup{(\\alph*)}]\n \\item $\\mu_2(\\Omega)/\\mu_1(\\Omega)\\leq 4$.\n \\item $\\mu_k(\\Omega)/\\mu_1(\\Omega)\\leq k^2$.\n \\end{enumerate}", "statement_status": "exact", "statement_verification": "The canonical record is Conjecture 3.2 in the AIM workshop list *Shape optimization with surface interactions*, section “The Neumann Laplacian.” The archived AIM page agrees with the corpus record and attributes the conjecture to A. Henrot. With the indexing convention made explicit, the recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Neumann Laplacian\nSource item: 3.2\nSource URL: http://aimpl.org/shapesurface/3/\nCanonical location: aim-analysis-notes.json notes[26]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Maximizing the ratio of Neumann eigenvalues among convex domains\\n\\nFor any convex $\\\\Omega\\\\subset \\\\mathbb{R}^n$, it holds that\\n \\\\begin{enumerate}[label=\\\\textup{(\\\\alph*)}]\\n \\\\item $\\\\mu_2(\\\\Omega)/\\\\mu_1(\\\\Omega)\\\\leq 4$.\\n \\\\item $\\\\mu_k(\\\\Omega)/\\\\mu_1(\\\\Omega)\\\\leq k^2$.\\n \\\\end{enumerate}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The two-dimensional version of the above conjecture is formulated by M. Ashbaugh and R. Benguria in~\\\\cite{MR1215424}. Partial analytic result again in two dimensions is obtained by P. Antunes and A. Henrot in~\\\\cite{MR2795792}.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/3/", "tags": [ "aim", "AIM-ANALYSIS-0027", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Tang and Zhang's unrefereed arXiv:2607.05388v1 proves the sharp inequalities mu_2 <= 4 mu_1 and mu_3 <= 9 mu_1 on every bounded open convex domain, so the AIM conjecture is presently solved at preprint level through k=3 but remains open for arbitrary convex domains in dimensions n >= 2 when k >= 4. This attempt independently proves for every n, L, and k that the ordered-cube simplex S_n(L) = {0 < x_1 < ... < x_n < L} satisfies mu_k <= k^2 mu_1, with mu_1 = pi^2/L^2 and strict inequality for n >= 2 and k >= 2, by identifying its Neumann form with the permutation-invariant sector of the cube and counting explicit invariant modes.\n\nCandidate contribution (special_case; novelty confidence low): For the non-product simplex S_n(L) = {0 < x_1 < ... < x_n < L}, the conjectural ratio mu_k/mu_1 <= k^2 holds for every k, and it is strict for every n >= 2 and k >= 2.", "research_difficulty_suggested": "L4" }, { "id": 20000496, "problem_number": "AIM-ANALYSIS-0028", "title": "Hot spots: high-dimensional refutation, product rules, and heat-flow quantifiers", "statement": "Hot spots conjecture\n\nThis conjecture is often attributed to J. Rauch, who has formulated it in terms of large-time behaviuor of the heat semigroup generated by the Neumann Laplacian. On the qualitative level, this conjecture manifests as follows: for an insulated flat piece of metal with an arbitrary initial temperature distribution, given enough time, the hottest point on the metal will lie on its boundary.\n\nThe first non-trivial Neumann eigenfunction does not have global extrema in the interior of $\\Omega$ if\n \\begin{enumerate}\n \\item $\\Omega\\subset \\mathbb{R}^2$ is simply connected.\n \\item $\\Omega\\subset \\mathbb{R}^n$ ($n \\ge 3$) is convex.\n \\end{enumerate}", "original_statement": "Hot spots conjecture\n\nThis conjecture is often attributed to J. Rauch, who has formulated it in terms of large-time behaviuor of the heat semigroup generated by the Neumann Laplacian. On the qualitative level, this conjecture manifests as follows: for an insulated flat piece of metal with an arbitrary initial temperature distribution, given enough time, the hottest point on the metal will lie on its boundary.\n\nThe first non-trivial Neumann eigenfunction does not have global extrema in the interior of $\\Omega$ if\n \\begin{enumerate}\n \\item $\\Omega\\subset \\mathbb{R}^2$ is simply connected.\n \\item $\\Omega\\subset \\mathbb{R}^n$ ($n \\ge 3$) is convex.\n \\end{enumerate}", "clean_statement": "Hot spots conjecture\n\nThis conjecture is often attributed to J. Rauch, who has formulated it in terms of large-time behaviuor of the heat semigroup generated by the Neumann Laplacian. On the qualitative level, this conjecture manifests as follows: for an insulated flat piece of metal with an arbitrary initial temperature distribution, given enough time, the hottest point on the metal will lie on its boundary.\n\nThe first non-trivial Neumann eigenfunction does not have global extrema in the interior of $\\Omega$ if\n \\begin{enumerate}\n \\item $\\Omega\\subset \\mathbb{R}^2$ is simply connected.\n \\item $\\Omega\\subset \\mathbb{R}^n$ ($n \\ge 3$) is convex.\n \\end{enumerate}", "statement_status": "exact", "statement_verification": "The canonical record is item 3.3, tagged as a conjecture, in the AIM workshop list Shape optimization with surface interactions. Its mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Neumann Laplacian\nSource item: 3.3\nSource URL: http://aimpl.org/shapesurface/3/\nCanonical location: aim-analysis-notes.json notes[27]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Hot spots conjecture\\n\\nThis conjecture is often attributed to J. Rauch, who has formulated it in terms of large-time behaviuor of the heat semigroup generated by the Neumann Laplacian. On the qualitative level, this conjecture manifests as follows: for an insulated flat piece of metal with an arbitrary initial temperature distribution, given enough time, the hottest point on the metal will lie on its boundary.\\n\\nThe first non-trivial Neumann eigenfunction does not have global extrema in the interior of $\\\\Omega$ if\\n \\\\begin{enumerate}\\n \\\\item $\\\\Omega\\\\subset \\\\mathbb{R}^2$ is simply connected.\\n \\\\item $\\\\Omega\\\\subset \\\\mathbb{R}^n$ ($n \\\\ge 3$) is convex.\\n \\\\end{enumerate}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Positive results concern various special classes of domains such as triangles \\\\cite{MR4045963}, $\\\\mathsf{Lip}$-domains \\\\cite{MR2051611}, and thin curved strips~\\\\cite{MR3912674}. A counterexample in the class of non-simply-connected planar domains is constructed in~\\\\cite{MR1680567}.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/3/", "tags": [ "aim", "AIM-ANALYSIS-0028", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The canonical two-branch conjecture is refuted as a universal statement by de Dios Pont's unrefereed 2024 preprint, which states smooth centrally symmetric convex counterexamples in all sufficiently large dimensions but gives no explicit dimension threshold; the simply connected planar branch and specified fixed low dimensions, including dimension three, remain open. Independently, this attempt proves an exact universal HS1/HS2 Cartesian-product truth table, including multiplicity, and gives a positive-temperature higher-mode construction whose unique hot point stays at the center of every ball for every finite time, correcting only the source's literal arbitrary-data heat paraphrase.\n\nCandidate contribution (product theorem; novelty confidence low): For bounded connected product domains with continuous first Neumann eigenfunctions, if the factor gaps are unequal then the product has strong HS1 exactly when the smaller-gap factor has HS1, while weak HS2 always holds; if the gaps are equal, strong HS1 holds exactly when both factors have it, whereas weak HS2 holds exactly when at least one factor has it." }, { "id": 20000497, "problem_number": "AIM-ANALYSIS-0029", "title": "A parity reduction and gradient-cone test for directional hot spots", "statement": "Directional hot spots conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^2$ be a centrally symmetric bounded convex domain and let $u$ denote the eigenfunction corresponding to the first non-trivial Neumann eigenvalue.\n\nCan one always find a direction $\\mathbf{e}\\in\\mathbb{R}^2$ such that\n \\begin{equation}\n \\mathbf{e}\\cdot \\nabla u >0 \\quad \\mbox{in }\\Omega?\n \\end{equation}", "original_statement": "Directional hot spots conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^2$ be a centrally symmetric bounded convex domain and let $u$ denote the eigenfunction corresponding to the first non-trivial Neumann eigenvalue.\n\nCan one always find a direction $\\mathbf{e}\\in\\mathbb{R}^2$ such that\n \\begin{equation}\n \\mathbf{e}\\cdot \\nabla u >0 \\quad \\mbox{in }\\Omega?\n \\end{equation}", "clean_statement": "Directional hot spots conjecture\n\nLet $\\Omega \\subset \\mathbb{R}^2$ be a centrally symmetric bounded convex domain and let $u$ denote the eigenfunction corresponding to the first non-trivial Neumann eigenvalue.\n\nCan one always find a direction $\\mathbf{e}\\in\\mathbb{R}^2$ such that\n \\begin{equation}\n \\mathbf{e}\\cdot \\nabla u >0 \\quad \\mbox{in }\\Omega?\n \\end{equation}", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: The Neumann Laplacian\nSource item: 3.4\nSource URL: http://aimpl.org/shapesurface/3/\nCanonical location: aim-analysis-notes.json notes[28]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Directional hot spots conjecture\\n\\nLet $\\\\Omega \\\\subset \\\\mathbb{R}^2$ be a centrally symmetric bounded convex domain and let $u$ denote the eigenfunction corresponding to the first non-trivial Neumann eigenvalue.\\n\\nCan one always find a direction $\\\\mathbf{e}\\\\in\\\\mathbb{R}^2$ such that\\n \\\\begin{equation}\\n \\\\mathbf{e}\\\\cdot \\\\nabla u >0 \\\\quad \\\\mbox{in }\\\\Omega?\\n \\\\end{equation}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"An affirmative answer to the above question implies the hot spots conjecture for all centrally symmetric bounded convex planar domains.\\nUnder the assumption that $\\\\Omega$ is symmetric with respect to both coordinate axes, the respective question has been affirmatively answered by D. Jerison and N. Nadirashvili in \\\\cite{MR1775736}.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/3/", "tags": [ "aim", "AIM-ANALYSIS-0029", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every bounded centrally symmetric convex planar domain with smooth boundary, the lowest central-odd Neumann eigenvalue is strictly below the lowest positive central-even Neumann eigenvalue; consequently the entire first positive eigenspace is centrally odd. A curvature-weighted derivative identity proves the comparison, and directional monotonicity is then exactly equivalent to nonvanishing of the gradient together with containment of its normalized image in an open semicircle. The report also gives multiplicity-safe proofs and exact good-direction cones for every rectangle and disk.\n\nCandidate contribution (theorem_and_reduction; novelty confidence low): Candidate parity-gradient reduction: on a smooth centrally symmetric convex planar domain, lambda_odd is strictly less than lambda_even, with a curvature-weighted upper bound for lambda_odd in terms of an even-sector minimizer; after this parity theorem, the directional conjecture is exactly the open-semicircle condition for normalized gradients." }, { "id": 20000498, "problem_number": "AIM-ANALYSIS-0030", "title": "Planar nonattainment and a nodal-puncture obstruction for simple first Steklov eigenvalues", "statement": "Non-existence of a maximizer for the first Steklov eigenvalue with perimeter constraint\n\nLet $\\sigma_1(\\Omega)$ denote the first non-trivial Steklov eigenvalue for a bounded domain $\\Omega\\subset\\mathbb{R}^n$.\n\nProve that there is no $\\Omega\\subset \\mathbb{R}^n$ which maximizes the quantity $P(\\Omega)^{1/(n-1)}\\sigma_1(\\Omega)$.", "original_statement": "Non-existence of a maximizer for the first Steklov eigenvalue with perimeter constraint\n\nLet $\\sigma_1(\\Omega)$ denote the first non-trivial Steklov eigenvalue for a bounded domain $\\Omega\\subset\\mathbb{R}^n$.\n\nProve that there is no $\\Omega\\subset \\mathbb{R}^n$ which maximizes the quantity $P(\\Omega)^{1/(n-1)}\\sigma_1(\\Omega)$.", "clean_statement": "Non-existence of a maximizer for the first Steklov eigenvalue with perimeter constraint\n\nLet $\\sigma_1(\\Omega)$ denote the first non-trivial Steklov eigenvalue for a bounded domain $\\Omega\\subset\\mathbb{R}^n$.\n\nProve that there is no $\\Omega\\subset \\mathbb{R}^n$ which maximizes the quantity $P(\\Omega)^{1/(n-1)}\\sigma_1(\\Omega)$.", "statement_status": "exact", "statement_verification": "This is Problem 4.1 in the “Steklov eigenvalues” section of the AIM workshop list *Shape optimization with surface interactions*. The exact extracted problem is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: Steklov eigenvalues\nSource item: 4.1\nSource URL: http://aimpl.org/shapesurface/4/\nCanonical location: aim-analysis-notes.json notes[29]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Non-existence of a maximizer for the first Steklov eigenvalue with perimeter constraint\\n\\nLet $\\\\sigma_1(\\\\Omega)$ denote the first non-trivial Steklov eigenvalue for a bounded domain $\\\\Omega\\\\subset\\\\mathbb{R}^n$.\\n\\nProve that there is no $\\\\Omega\\\\subset \\\\mathbb{R}^n$ which maximizes the quantity $P(\\\\Omega)^{1/(n-1)}\\\\sigma_1(\\\\Omega)$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"A topological perturbation argument can be used to prove that the disk is not a maximizer: make a small hole in the center and compute the resulting perturbation of the Steklov eigenvalue.\\n\\nHowever, it is proved by R. Weinstock in \\\\cite{MR0064989} that in two dimensions the disk is the maximizer among all simply connected domains. In higher dimensions it was proved by D. Bucur, V. Ferone, C. Nitsch and\\nC. Trombetti \\\\cite{arXiv:1710.04587} that the ball is the maximizer among convex domains. In fact, their result is the following stronger inequality\\n $$\\n \\\\sigma_1(\\\\Omega)\\\\frac{|\\\\partial\\\\Omega|}{|\\\\Omega|^{(n-2)/n}}\\\\leq \\\\sigma_1(B)\\\\frac{|\\\\partial B|}{|B|^{(n-2)/n}}.\\n $$\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/4/", "tags": [ "aim", "AIM-ANALYSIS-0030", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For unrestricted smooth bounded connected planar domains, the known sharp result is sigma_1(Omega) P(Omega) < 8 pi for every individual domain with supremum 8 pi, so the AIM nonattainment claim is solved in dimension two. In dimensions n >= 3 the unrestricted problem remains open. The proved contribution here is that, in every n >= 2, a bounded connected C^{2,alpha} domain with simple first positive Steklov eigenvalue cannot maximize the perimeter-normalized functional: puncturing at an interior nodal point increases it for every sufficiently small hole.\n\nCandidate contribution (proposition; novelty confidence low): If sigma_1(Omega) is simple and u is boundary-L2 normalized, an interior nodal point p satisfies J(Omega minus closed B_epsilon(p)) - J(Omega) = [sigma_1(Omega) s_{n-1}/(n-1)] P(Omega)^{1/(n-1)-1} epsilon^{n-1} + o(epsilon^{n-1}); hence every smooth maximizer must have multiple sigma_1." }, { "id": 20000499, "problem_number": "AIM-ANALYSIS-0031", "title": "A target-specific q_Omega criterion for strict disk comparison", "statement": "Faber-Krahn for Dirac operator with infinite mass boundary condition\n\nProve or disprove that among all planar domains of a given area, the disk is the unique minimizer for the smallest positive eigenvalue\n$\\nu_1(\\Omega)$ of the Dirac problem\nwith infinite mass boundary conditions\n$$\n\\left\\{\n\\begin{aligned}\n\t-\\mathsf{i}\n\\big(\\partial_1 v - \\mathsf{i}\\partial_2 v\\big) &= \\nu u\n\t&&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n -\\mathsf{i}\\big(\\partial_1 u + \\mathsf{i}\\partial_2 u\\big)\n&= \\nu v\n &&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n \\mathsf{i}\\big(n_1 + \\mathsf{i} n_2\\big) u\n &= v\n &&\\mbox{on} \\quad \\partial\\Omega \\,,\n \\end{aligned}\n \\right. $$\n where~$n = (n_1,n_2)^\\top$ is the outer unit normal to~$\\Omega$.", "original_statement": "Faber-Krahn for Dirac operator with infinite mass boundary condition\n\nProve or disprove that among all planar domains of a given area, the disk is the unique minimizer for the smallest positive eigenvalue\n$\\nu_1(\\Omega)$ of the Dirac problem\nwith infinite mass boundary conditions\n$$\n\\left\\{\n\\begin{aligned}\n\t-\\mathsf{i}\n\\big(\\partial_1 v - \\mathsf{i}\\partial_2 v\\big) &= \\nu u\n\t&&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n -\\mathsf{i}\\big(\\partial_1 u + \\mathsf{i}\\partial_2 u\\big)\n&= \\nu v\n &&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n \\mathsf{i}\\big(n_1 + \\mathsf{i} n_2\\big) u\n &= v\n &&\\mbox{on} \\quad \\partial\\Omega \\,,\n \\end{aligned}\n \\right. $$\n where~$n = (n_1,n_2)^\\top$ is the outer unit normal to~$\\Omega$.", "clean_statement": "Faber-Krahn for Dirac operator with infinite mass boundary condition\n\nProve or disprove that among all planar domains of a given area, the disk is the unique minimizer for the smallest positive eigenvalue\n$\\nu_1(\\Omega)$ of the Dirac problem\nwith infinite mass boundary conditions\n$$\n\\left\\{\n\\begin{aligned}\n\t-\\mathsf{i}\n\\big(\\partial_1 v - \\mathsf{i}\\partial_2 v\\big) &= \\nu u\n\t&&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n -\\mathsf{i}\\big(\\partial_1 u + \\mathsf{i}\\partial_2 u\\big)\n&= \\nu v\n &&\\mbox{in} \\quad \\Omega \\,,\n \\\\\n \\mathsf{i}\\big(n_1 + \\mathsf{i} n_2\\big) u\n &= v\n &&\\mbox{on} \\quad \\partial\\Omega \\,,\n \\end{aligned}\n \\right. $$\n where~$n = (n_1,n_2)^\\top$ is the outer unit normal to~$\\Omega$.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 5.1 in the AIM list *Shape optimization with surface interactions*, section “Dirac operators.” It asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: Dirac operators\nSource item: 5.1\nSource URL: http://aimpl.org/shapesurface/5/\nCanonical location: aim-analysis-notes.json notes[30]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Faber-Krahn for Dirac operator with infinite mass boundary condition\\n\\nProve or disprove that among all planar domains of a given area, the disk is the unique minimizer for the smallest positive eigenvalue\\n$\\\\nu_1(\\\\Omega)$ of the Dirac problem\\nwith infinite mass boundary conditions\\n$$\\n\\\\left\\\\{\\n\\\\begin{aligned}\\n\\t-\\\\mathsf{i}\\n\\\\big(\\\\partial_1 v - \\\\mathsf{i}\\\\partial_2 v\\\\big) &= \\\\nu u\\n\\t&&\\\\mbox{in} \\\\quad \\\\Omega \\\\,,\\n \\\\\\\\\\n -\\\\mathsf{i}\\\\big(\\\\partial_1 u + \\\\mathsf{i}\\\\partial_2 u\\\\big)\\n&= \\\\nu v\\n &&\\\\mbox{in} \\\\quad \\\\Omega \\\\,,\\n \\\\\\\\\\n \\\\mathsf{i}\\\\big(n_1 + \\\\mathsf{i} n_2\\\\big) u\\n &= v\\n &&\\\\mbox{on} \\\\quad \\\\partial\\\\Omega \\\\,,\\n \\\\end{aligned}\\n \\\\right. $$\\n where~$n = (n_1,n_2)^\\\\top$ is the outer unit normal to~$\\\\Omega$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"A geometric lower bound for the first non-negative eigenvalue has been obtained by R. Benguria, S. Fournais, E. Stockmeyer, and H. Van Den Bosch \\\\cite{MR3625007}. Moreover, a numerical evidence supporting that the disk is in fact the minimizer is provided in \\\\cite{arxiv:2003.04061}.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/5/", "tags": [ "aim", "AIM-ANALYSIS-0031", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The full fixed-area disk-minimization conjecture remains open. For bounded connected C^2 planar domains, Duran's strict lower bound combined with the classical Dirichlet Faber-Krahn inequality proves the sufficient condition q_Omega R >= k_0 x/(x-k_0^2), where R=sqrt(|Omega|/pi) and x=Lambda_Omega R^2. Consequently the uniform threshold q_Omega R >= Q_*=k_0 j_{0,1}^2/(j_{0,1}^2-k_0^2)=2.2275116055... implies nu_1(Omega)>nu_1(D_R). For convex domains, rho/R<=1/Q_*=0.4489314433... is sufficient, enlarging by about 29.8% the elementary inradius range obtained from the published parameter-uniform sufficient criterion.\n\nCandidate contribution (criterion; novelty confidence low): Candidate novelty: the domain-sensitive sufficient condition q_Omega R >= k_0(Lambda_Omega R^2)/(Lambda_Omega R^2-k_0^2), its area-only specialization q_Omega R >= 2.2275116055..., and the convex consequence rho/R <= 0.4489314433... appear not to be stated in the primary literature checked." }, { "id": 20000500, "problem_number": "AIM-ANALYSIS-0032", "title": "Four sharp constants and a conical affine lower bound", "statement": "Hermite-Hadamard inequality in higher dimensions\n\nThe classical Hermite-Hadamard inequality states that if $f$ is a convex function on $(a, b)$ then\n$$\n \\frac{1}{b-a}\\int_a^b f(x)dx \\leq \\frac{f(b)+f(a)}{2}.\n$$\nWe consider two generalizations to higher dimensions. Let $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and\n\nCase 1: $f\\colon \\Omega \\to [0, \\infty)$ convex, or\n\nCase 2: $f\\colon \\Omega \\to [0, \\infty)$ subharmonic.\n\nIn either case one can ask for smallest constants $c_n, \\tilde c_n$ such that\n\\begin{equation}\n \\frac{1}{|\\Omega|}\\int_\\Omega f(x)dx \\leq \\frac{c_n}{|\\partial\\Omega|}\\int_{\\partial\\Omega}f(x)dx,\n\\quad\\mbox{and}\\quad\n \\int_\\Omega f(x)dx \\leq \\tilde c_n|\\Omega|^{1/n}\\int_{\\partial\\Omega}f(x)dx.\n\t\\end{equation}\nThe second inequality follows from the first one by an application of the isoperimetric inequality with $\\tilde c_n = c_n/\\sqrt{n}$.\n\nFind the optimal constants in the inequalities.", "original_statement": "Hermite-Hadamard inequality in higher dimensions\n\nThe classical Hermite-Hadamard inequality states that if $f$ is a convex function on $(a, b)$ then\n$$\n \\frac{1}{b-a}\\int_a^b f(x)dx \\leq \\frac{f(b)+f(a)}{2}.\n$$\nWe consider two generalizations to higher dimensions. Let $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and\n\nCase 1: $f\\colon \\Omega \\to [0, \\infty)$ convex, or\n\nCase 2: $f\\colon \\Omega \\to [0, \\infty)$ subharmonic.\n\nIn either case one can ask for smallest constants $c_n, \\tilde c_n$ such that\n\\begin{equation}\n \\frac{1}{|\\Omega|}\\int_\\Omega f(x)dx \\leq \\frac{c_n}{|\\partial\\Omega|}\\int_{\\partial\\Omega}f(x)dx,\n\\quad\\mbox{and}\\quad\n \\int_\\Omega f(x)dx \\leq \\tilde c_n|\\Omega|^{1/n}\\int_{\\partial\\Omega}f(x)dx.\n\t\\end{equation}\nThe second inequality follows from the first one by an application of the isoperimetric inequality with $\\tilde c_n = c_n/\\sqrt{n}$.\n\nFind the optimal constants in the inequalities.", "clean_statement": "Hermite-Hadamard inequality in higher dimensions\n\nThe classical Hermite-Hadamard inequality states that if $f$ is a convex function on $(a, b)$ then\n$$\n \\frac{1}{b-a}\\int_a^b f(x)dx \\leq \\frac{f(b)+f(a)}{2}.\n$$\nWe consider two generalizations to higher dimensions. Let $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and\n\nCase 1: $f\\colon \\Omega \\to [0, \\infty)$ convex, or\n\nCase 2: $f\\colon \\Omega \\to [0, \\infty)$ subharmonic.\n\nIn either case one can ask for smallest constants $c_n, \\tilde c_n$ such that\n\\begin{equation}\n \\frac{1}{|\\Omega|}\\int_\\Omega f(x)dx \\leq \\frac{c_n}{|\\partial\\Omega|}\\int_{\\partial\\Omega}f(x)dx,\n\\quad\\mbox{and}\\quad\n \\int_\\Omega f(x)dx \\leq \\tilde c_n|\\Omega|^{1/n}\\int_{\\partial\\Omega}f(x)dx.\n\t\\end{equation}\nThe second inequality follows from the first one by an application of the isoperimetric inequality with $\\tilde c_n = c_n/\\sqrt{n}$.\n\nFind the optimal constants in the inequalities.", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List item 6.1, “Hermite-Hadamard inequality in higher dimensions,” from the 2019 workshop *Shape optimization with surface interactions*. It asks the following.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: Miscellaneous problems\nSource item: 6.1\nSource URL: http://aimpl.org/shapesurface/6/\nCanonical location: aim-analysis-notes.json notes[31]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Hermite-Hadamard inequality in higher dimensions\\n\\nThe classical Hermite-Hadamard inequality states that if $f$ is a convex function on $(a, b)$ then\\n$$\\n \\\\frac{1}{b-a}\\\\int_a^b f(x)dx \\\\leq \\\\frac{f(b)+f(a)}{2}.\\n$$\\nWe consider two generalizations to higher dimensions. Let $\\\\Omega \\\\subset \\\\mathbb{R}^n$ be a convex domain and\\n\\nCase 1: $f\\\\colon \\\\Omega \\\\to [0, \\\\infty)$ convex, or\\n\\nCase 2: $f\\\\colon \\\\Omega \\\\to [0, \\\\infty)$ subharmonic.\\n\\nIn either case one can ask for smallest constants $c_n, \\\\tilde c_n$ such that\\n\\\\begin{equation}\\n \\\\frac{1}{|\\\\Omega|}\\\\int_\\\\Omega f(x)dx \\\\leq \\\\frac{c_n}{|\\\\partial\\\\Omega|}\\\\int_{\\\\partial\\\\Omega}f(x)dx,\\n\\\\quad\\\\mbox{and}\\\\quad\\n \\\\int_\\\\Omega f(x)dx \\\\leq \\\\tilde c_n|\\\\Omega|^{1/n}\\\\int_{\\\\partial\\\\Omega}f(x)dx.\\n\\t\\\\end{equation}\\nThe second inequality follows from the first one by an application of the isoperimetric inequality with $\\\\tilde c_n = c_n/\\\\sqrt{n}$.\\n\\nFind the optimal constants in the inequalities.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Recently, S. Steinerberger \\\\cite{MR4058521} proved that in the case of convex functions $c_n \\\\leq 2 \\\\pi^{-1/2}n^{n+1}$ and that in the two-dimensional case $9/8\\\\leq c_2 \\\\leq 8$. For subharmonic functions J. Lu and S. Steinerberger \\\\cite{MR4052204} showed that $\\\\tilde c_n \\\\leq 1$. In \\\\cite{arXiv:1804.03688} P. Pasteczka conjectured that the constant $c_n$ in the case of convex functions can be taken to be $1$ \\\\emph{if} the centre of mass of $\\\\Omega$ coincides with that of $\\\\partial\\\\Omega$. That the centres of mass of $\\\\Omega$ and $\\\\partial\\\\Omega$ coincide is known to be a necessary condition for $c_n = 1$ by testing against affine functions.\\n\\nDuring the workshop improved bounds were found for the sharp constants in all cases \\\\cite{arXiv:1907.06122}.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/6/", "tags": [ "aim", "AIM-ANALYSIS-0032", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source separates into four universal sharp constants. The first-normalization subharmonic constant is known exactly, C_n^sh=n, but the first-normalization convex constant and both volume-normalized constants remain open. This attempt proves for the circular cone K_{a,n} and the nonnegative affine function f(t,y)=1-t the exact normalized ratio n/(n+1)(1+a/sqrt(1+a^2)); hence C_n^cvx>=2n/(n+1). Its genuinely improved consequence is the planar bracket 4/3<=C_2^cvx<=2. The lower bound persists under smooth strictly convex approximation. The source's transfer tilde c_n=c_n/sqrt(n) is a sufficient coarse choice, not an identity between optimal constants.\n\nCandidate contribution (lower_bound; novelty confidence low): The explicit circular-cone/affine family has first-normalization ratio n/(n+1)(1+a/sqrt(1+a^2)), proving C_n^cvx>=2n/(n+1) and, in particular, the improved planar lower bound C_2^cvx>=4/3." }, { "id": 20000501, "problem_number": "AIM-ANALYSIS-0033", "title": "A quantitative defect criterion upgrading honeycomb energy asymptotics to empirical cell convergence", "statement": "Asymptotic partitioning problems\n\nIn several problems where one wants to find an in some sense optimal partition of a planar domain $\\Omega$ into $N$ subdomains it has been observed that as $N$ becomes large the partition converges to the hexagonal one.\n\nFind conditions for problems involving optimal $N$-partitions of a planar domain such that one observes in the limit $N\\to \\infty$ a packing of hexagons.", "original_statement": "Asymptotic partitioning problems\n\nIn several problems where one wants to find an in some sense optimal partition of a planar domain $\\Omega$ into $N$ subdomains it has been observed that as $N$ becomes large the partition converges to the hexagonal one.\n\nFind conditions for problems involving optimal $N$-partitions of a planar domain such that one observes in the limit $N\\to \\infty$ a packing of hexagons.", "clean_statement": "Asymptotic partitioning problems\n\nIn several problems where one wants to find an in some sense optimal partition of a planar domain $\\Omega$ into $N$ subdomains it has been observed that as $N$ becomes large the partition converges to the hexagonal one.\n\nFind conditions for problems involving optimal $N$-partitions of a planar domain such that one observes in the limit $N\\to \\infty$ a packing of hexagons.", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ANALYSIS-0033, problem 6.2 in the AIM workshop list *Shape optimization with surface interactions*, section “Miscellaneous problems.” The archived AIM page attributes the question to K. Burdzy. The record states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: Miscellaneous problems\nSource item: 6.2\nSource URL: http://aimpl.org/shapesurface/6/\nCanonical location: aim-analysis-notes.json notes[32]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Asymptotic partitioning problems\\n\\nIn several problems where one wants to find an in some sense optimal partition of a planar domain $\\\\Omega$ into $N$ subdomains it has been observed that as $N$ becomes large the partition converges to the hexagonal one.\\n\\nFind conditions for problems involving optimal $N$-partitions of a planar domain such that one observes in the limit $N\\\\to \\\\infty$ a packing of hexagons.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The phenomenon has been proved by D. Bucur and I. Fragala if the optimality is with respect to minimization for the sum of the Cheeger constants~\\\\cite{MR3946306} or minimization for the sum of the lowest Robin eigenvalues~\\\\cite{MR3918044} of the partition elements. The original conjecture formulated by L. Caffarelli and F. Lin in \\\\cite{MR2304268} for the optimality with respect to minimization for the sum of the principal Dirichlet eigenvalues is still open.\"\nResearch attempt: 1; result status: conditional_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/6/", "tags": [ "aim", "AIM-ANALYSIS-0033", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Published Bucur–Fragalà–Velichkov–Verzini hypotheses yield honeycomb energy asymptotics but do not by themselves yield cell geometry. For exact convex polygonal partitions, this attempt proves that an explicit sublinear Euler side budget, a strict exposed-six supporting-line gap for the polygonal energies, relative total-energy excess ε_N, and quantitative stability of the unit-area hexagonal minimizer give an explicit bound pricing every nonhexagonal cell. If ε_N tends to zero and the side budget is o(N), all but o(N) cells are six-sided, those cells occupy asymptotically all area and equalize their areas, and their unit-area normalizations converge in counting measure to regular hexagons. No global orientation or lattice convergence and no solution of the unrestricted Dirichlet conjecture are claimed.\n\nCandidate contribution (conditional_theorem; novelty confidence low): The explicit defect bound D_N ≤ κ^{-1}[N g_6((1+ε_N)^{2/(α+2)}−1)+|s|B_N], together with proved area-equalization and generic hexagonal shape-concentration estimates, is a candidate abstract quantitative upgrade of the BFVV additive honeycomb energy criterion.", "research_difficulty_suggested": "L4" }, { "id": 20000502, "problem_number": "AIM-ANALYSIS-0034", "title": "A closure obstruction for the proposed oval flow", "statement": "The ovals of Benguria and Loss\n\nLet $\\Gamma\\subset \\mathbb{R}^2$ be a closed smooth curve of length $2\\pi$ and let $\\kappa(s)$ be its curvature as a function of arclength. Define the following self-adjoint one-dimensional differential operator\n\\[\nH_\\Gamma \\psi := -\\psi'' + \\kappa^2\\psi,\\qquad \\mathsf{dom}\\,H_\\Gamma := W^{2,2}(\\Gamma),\n\\]\nin the Hilbert space $L^2(\\Gamma)$ and let $\\lambda_1(H_\\Gamma)$ be its lowest eigenvalue.\n\nThe following inequality $\\lambda_1(H_\\Gamma) \\ge 1$ holds.", "original_statement": "The ovals of Benguria and Loss\n\nLet $\\Gamma\\subset \\mathbb{R}^2$ be a closed smooth curve of length $2\\pi$ and let $\\kappa(s)$ be its curvature as a function of arclength. Define the following self-adjoint one-dimensional differential operator\n\\[\nH_\\Gamma \\psi := -\\psi'' + \\kappa^2\\psi,\\qquad \\mathsf{dom}\\,H_\\Gamma := W^{2,2}(\\Gamma),\n\\]\nin the Hilbert space $L^2(\\Gamma)$ and let $\\lambda_1(H_\\Gamma)$ be its lowest eigenvalue.\n\nThe following inequality $\\lambda_1(H_\\Gamma) \\ge 1$ holds.", "clean_statement": "The ovals of Benguria and Loss\n\nLet $\\Gamma\\subset \\mathbb{R}^2$ be a closed smooth curve of length $2\\pi$ and let $\\kappa(s)$ be its curvature as a function of arclength. Define the following self-adjoint one-dimensional differential operator\n\\[\nH_\\Gamma \\psi := -\\psi'' + \\kappa^2\\psi,\\qquad \\mathsf{dom}\\,H_\\Gamma := W^{2,2}(\\Gamma),\n\\]\nin the Hilbert space $L^2(\\Gamma)$ and let $\\lambda_1(H_\\Gamma)$ be its lowest eigenvalue.\n\nThe following inequality $\\lambda_1(H_\\Gamma) \\ge 1$ holds.", "statement_status": "exact", "statement_verification": "The canonical record is problem 6.3, “The ovals of Benguria and Loss,” from the AIM workshop *Shape optimization with surface interactions* (`aim-analysis-notes.json`, zero-based record index 33). Its mathematical statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: Miscellaneous problems\nSource item: 6.3\nSource URL: http://aimpl.org/shapesurface/6/\nCanonical location: aim-analysis-notes.json notes[33]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"The ovals of Benguria and Loss\\n\\nLet $\\\\Gamma\\\\subset \\\\mathbb{R}^2$ be a closed smooth curve of length $2\\\\pi$ and let $\\\\kappa(s)$ be its curvature as a function of arclength. Define the following self-adjoint one-dimensional differential operator\\n\\\\[\\nH_\\\\Gamma \\\\psi := -\\\\psi'' + \\\\kappa^2\\\\psi,\\\\qquad \\\\mathsf{dom}\\\\,H_\\\\Gamma := W^{2,2}(\\\\Gamma),\\n\\\\]\\nin the Hilbert space $L^2(\\\\Gamma)$ and let $\\\\lambda_1(H_\\\\Gamma)$ be its lowest eigenvalue.\\n\\nThe following inequality $\\\\lambda_1(H_\\\\Gamma) \\\\ge 1$ holds.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The problem has appeared in connection to sharp Lieb-Thirring inequalities for Schrödinger operators with two bound states. R. Benguria and M. Loss proved in \\\\cite{MR2091490} that $\\\\lambda_1(H_\\\\Gamma)\\\\geq 1/2$ which has later been improved by H. Linde in \\\\cite{MR2240676} upto\\n$\\\\lambda_1(H_\\\\Gamma)> 0.6085$.\\nIt is also shown in \\\\cite{MR2203162} that there is an infinite family of ovals for which $\\\\lambda_1(H_\\\\Gamma)=1$, which includes the circle and the \\\"double segment\\\". S. Steinerberger remarked that N. Marshall has noticed that the curvature flow defined by\\n \\\\begin{equation}\\n \\\\frac{d}{dt}\\\\kappa(s, t) = \\\\kappa(s, t)\\\\frac{1}{2\\\\pi}\\\\int_0^{2\\\\pi}\\\\kappa(x, t)^2dx - \\\\kappa(s, t)^2\\n \\\\end{equation}\\n preserves the family of ovals which are suggested as minimizers. However, the flow might break up curves. This flow or its modification can be useful for the proof of the conjecture.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/6/", "tags": [ "aim", "AIM-ANALYSIS-0034", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The Ovals Conjecture remains open: the 2025 claimed solution was withdrawn, and the current clean universal lower bound is greater than 0.81. For the exact fixed-arclength curvature equation in the AIM record, the adjacent-mode strictly convex closed family with radius of curvature r_epsilon(theta)=1+epsilon(a cos(p theta)+b cos((p+1) theta)) has closure derivative Z'(0)=pi a b (1+1/(2p(p+1))) epsilon^2+O(epsilon^3) in the chosen initial orientation; a static rotation multiplies this complex derivative by a phase. Hence the vector field is not tangent to the manifold of closed loops. Additionally, the exactly closed family kappa(s)=1+a cos(ms) has a proved analytic ground-state expansion, with quartic lifting 3a^4/32+O(a^6) in the quadratically flat m=2 direction.\n\nCandidate contribution (obstruction; novelty confidence low): For every integer p at least 2 and nonzero real a,b, the stated fixed-arclength curvature vector field creates a nonzero positional-closure defect on the adjacent-mode family, with leading coefficient pi a b (1+1/(2p(p+1))); for p=2 and a=b=1 the coefficient is 13pi/12.", "research_difficulty_suggested": "L4" }, { "id": 20000503, "problem_number": "AIM-ANALYSIS-0035", "title": "The Pólya–Szegő buckling conjecture and a radial projection gap on balls", "statement": "Faber-Krahn for the buckling problem\n\nLet $\\Omega\\subset\\mathbb{R}^2$ be a bounded domain and consider the lowest eigenvalue of the buckling problem\n \\begin{align}\n \\Delta^2 u+\\Lambda_1(\\Omega)\\Delta u &=0 \\quad \\mbox{in }\\Omega,\\\\\n u&=0 \\quad \\mbox{on }\\partial\\Omega,\\\\\n \\frac{\\partial u}{\\partial n}&=0 \\quad \\mbox{on }\\partial\\Omega.\n \\end{align}\n\nProve that the lowest buckling eigenvalue $\\Lambda_1(\\Omega)$ is minimized by the disk among all simply connected sets of equal area.", "original_statement": "Faber-Krahn for the buckling problem\n\nLet $\\Omega\\subset\\mathbb{R}^2$ be a bounded domain and consider the lowest eigenvalue of the buckling problem\n \\begin{align}\n \\Delta^2 u+\\Lambda_1(\\Omega)\\Delta u &=0 \\quad \\mbox{in }\\Omega,\\\\\n u&=0 \\quad \\mbox{on }\\partial\\Omega,\\\\\n \\frac{\\partial u}{\\partial n}&=0 \\quad \\mbox{on }\\partial\\Omega.\n \\end{align}\n\nProve that the lowest buckling eigenvalue $\\Lambda_1(\\Omega)$ is minimized by the disk among all simply connected sets of equal area.", "clean_statement": "Faber-Krahn for the buckling problem\n\nLet $\\Omega\\subset\\mathbb{R}^2$ be a bounded domain and consider the lowest eigenvalue of the buckling problem\n \\begin{align}\n \\Delta^2 u+\\Lambda_1(\\Omega)\\Delta u &=0 \\quad \\mbox{in }\\Omega,\\\\\n u&=0 \\quad \\mbox{on }\\partial\\Omega,\\\\\n \\frac{\\partial u}{\\partial n}&=0 \\quad \\mbox{on }\\partial\\Omega.\n \\end{align}\n\nProve that the lowest buckling eigenvalue $\\Lambda_1(\\Omega)$ is minimized by the disk among all simply connected sets of equal area.", "statement_status": "exact", "statement_verification": "The canonical record is problem 6.4, “Faber--Krahn for the buckling problem,” from the AIM workshop *Shape optimization with surface interactions*. It asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: Miscellaneous problems\nSource item: 6.4\nSource URL: http://aimpl.org/shapesurface/6/\nCanonical location: aim-analysis-notes.json notes[34]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Faber-Krahn for the buckling problem\\n\\nLet $\\\\Omega\\\\subset\\\\mathbb{R}^2$ be a bounded domain and consider the lowest eigenvalue of the buckling problem\\n \\\\begin{align}\\n \\\\Delta^2 u+\\\\Lambda_1(\\\\Omega)\\\\Delta u &=0 \\\\quad \\\\mbox{in }\\\\Omega,\\\\\\\\\\n u&=0 \\\\quad \\\\mbox{on }\\\\partial\\\\Omega,\\\\\\\\\\n \\\\frac{\\\\partial u}{\\\\partial n}&=0 \\\\quad \\\\mbox{on }\\\\partial\\\\Omega.\\n \\\\end{align}\\n\\nProve that the lowest buckling eigenvalue $\\\\Lambda_1(\\\\Omega)$ is minimized by the disk among all simply connected sets of equal area.\"\nOriginal remarks: [\"This buckling load conjecture was raised by G. Polya and G. Szego in their book \\\\cite{MR0043486} on isoperimetric inequalities in mathematical physics.\"]\nOriginal literature field (JSON string): \"The problem is related to clamped plate problem where corresponding result has been proved by N. Nadirashvili~\\\\cite{MR1328469} and by M. Ashbaugh and R. Benguria~\\\\cite{MR1703569}. Existence of a minimizer for the buckling problem is known~\\\\cite{MR2019178}.\\n\\nD. Bucur remarks that the problem can be reformulated in terms of the eigenvalue problem\\n\\\\begin{align}\\n -\\\\Delta u+ \\\\Lambda_1(\\\\Omega)(u-P_\\\\Omega u) &=0 \\\\quad \\\\mbox{in }\\\\Omega,\\\\\\\\\\n u&=0 \\\\quad \\\\mbox{on }\\\\partial\\\\Omega,\\n\\\\end{align}\\nwhere $P_\\\\Omega$ is an orthogonal projection from $L^2(\\\\Omega)$ into a subspace of harmonic functions. If the subspace is that of all harmonic functions one retains the eigenvalue of the buckling problem.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/6/", "tags": [ "aim", "AIM-ANALYSIS-0035", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The disk-minimization conjecture remains open through 2026-07-27. For every Euclidean ball, the harmonic-Bergman-projection Rayleigh quotient restricted to radial Dirichlet functions is proved to be exactly conjugate, via u ↦ u minus its volume mean, to the mean-zero radial Neumann quotient. This yields the exact disk buckling value and a sharp spectral-gap remainder with optimal constant given by the gap between the first two positive radial Neumann eigenvalues. The canonical record's supplementary projected PDE is also shown to have the wrong sign under the standard Laplacian convention; the correct equation is -Δu = Λ(I-P)u.\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: on every n-dimensional ball, u ↦ u minus its volume mean is a quotient-preserving bijection from radial H^1_0 functions for the full harmonic Bergman projection problem to radial mean-zero H^1 functions for the Neumann problem, and the resulting buckling deficit controls the squared L^2 distance from the first radial mode with the sharp constant (j_{n/2,2}^2-j_{n/2,1}^2)/R^2." }, { "id": 20000504, "problem_number": "AIM-ANALYSIS-0036", "title": "Gaussian Poincaré equality is equivalent to splitting off a line", "statement": "Poincaré-Wirtinger extremal domain\n\nConsider the operator $-\\gamma^{-1}\\nabla \\gamma \\nabla$ on the weighted space $L^2(\\Omega, \\gamma)$ with Neumann boundary conditions, $\\Omega\\subset \\mathbb{R}^n$ and where $\\gamma$ is a Gaussian weight. It is known that the first non-trivial eigenvalue of this operator satisfies $\\mu_1(\\Omega, \\gamma)\\geq 1$ (this is the Poincaré-Wirtinger inequality).\n\nProve that the equality in the Poincaré-Wirtinger inequality is attained only for infinite strips.", "original_statement": "Poincaré-Wirtinger extremal domain\n\nConsider the operator $-\\gamma^{-1}\\nabla \\gamma \\nabla$ on the weighted space $L^2(\\Omega, \\gamma)$ with Neumann boundary conditions, $\\Omega\\subset \\mathbb{R}^n$ and where $\\gamma$ is a Gaussian weight. It is known that the first non-trivial eigenvalue of this operator satisfies $\\mu_1(\\Omega, \\gamma)\\geq 1$ (this is the Poincaré-Wirtinger inequality).\n\nProve that the equality in the Poincaré-Wirtinger inequality is attained only for infinite strips.", "clean_statement": "Poincaré-Wirtinger extremal domain\n\nConsider the operator $-\\gamma^{-1}\\nabla \\gamma \\nabla$ on the weighted space $L^2(\\Omega, \\gamma)$ with Neumann boundary conditions, $\\Omega\\subset \\mathbb{R}^n$ and where $\\gamma$ is a Gaussian weight. It is known that the first non-trivial eigenvalue of this operator satisfies $\\mu_1(\\Omega, \\gamma)\\geq 1$ (this is the Poincaré-Wirtinger inequality).\n\nProve that the equality in the Poincaré-Wirtinger inequality is attained only for infinite strips.", "statement_status": "exact", "statement_verification": "The canonical record is problem 6.5, “Poincaré-Wirtinger extremal domain,” from the AIM workshop *Shape optimization with surface interactions*. Its text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: Miscellaneous problems\nSource item: 6.5\nSource URL: http://aimpl.org/shapesurface/6/\nCanonical location: aim-analysis-notes.json notes[35]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Poincaré-Wirtinger extremal domain\\n\\nConsider the operator $-\\\\gamma^{-1}\\\\nabla \\\\gamma \\\\nabla$ on the weighted space $L^2(\\\\Omega, \\\\gamma)$ with Neumann boundary conditions, $\\\\Omega\\\\subset \\\\mathbb{R}^n$ and where $\\\\gamma$ is a Gaussian weight. It is known that the first non-trivial eigenvalue of this operator satisfies $\\\\mu_1(\\\\Omega, \\\\gamma)\\\\geq 1$ (this is the Poincaré-Wirtinger inequality).\\n\\nProve that the equality in the Poincaré-Wirtinger inequality is attained only for infinite strips.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"With a priori assumption that $\\\\Omega$ is contained in an infinite strip the result is known to be true~\\\\cite{MR3556341}.\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/6/", "tags": [ "aim", "AIM-ANALYSIS-0036", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Beck and Jerison proved the complete corrected classification: for the standard Gaussian restricted to any nonempty open convex domain, the Neumann spectral gap is at least 1, with equality exactly when the domain contains an affine line, equivalently when it is an orthogonal image of R times a convex domain. Thus the AIM claim is true when 'infinite strip' means a line-splitting convex cylinder, while a slab-only reading is false in dimensions at least three. This attempt also proves a tensorized refinement identifying all equality functions and the eigenvalue-1 multiplicity.\n\nCandidate contribution (remainder inequality and extremizer corollary; novelty confidence low): Assume the published Beck-Jerison classification. If the lineality space of a convex domain has dimension m and the domain is rotated to R^m times Omega_0, then the eigenvalue-1 eigenspace has dimension exactly m and consists of linear functions in the lineality variables; moreover, when c=mu_1(Omega_0)>1, the Poincare deficit is at least (1-1/c) times the transverse Dirichlet energy, and this coefficient is sharp." }, { "id": 20000505, "problem_number": "AIM-ANALYSIS-0037", "title": "Quantitative transfer of ordered nodal slabs", "statement": "Nodal structure of the $k$-th Neumann eigenfunction for collapsing domains\n\nConsider the nodal domains of the $k$-th Neumann eigenfunction on a sequence of domains collapsing to a lower dimensional object.\n\nUnder what assumptions can one prove that the nodal domains are in a sense ordered?", "original_statement": "Nodal structure of the $k$-th Neumann eigenfunction for collapsing domains\n\nConsider the nodal domains of the $k$-th Neumann eigenfunction on a sequence of domains collapsing to a lower dimensional object.\n\nUnder what assumptions can one prove that the nodal domains are in a sense ordered?", "clean_statement": "Nodal structure of the $k$-th Neumann eigenfunction for collapsing domains\n\nConsider the nodal domains of the $k$-th Neumann eigenfunction on a sequence of domains collapsing to a lower dimensional object.\n\nUnder what assumptions can one prove that the nodal domains are in a sense ordered?", "statement_status": "exact", "statement_verification": "The canonical record is problem 6.6, “Nodal structure of the $k$-th Neumann eigenfunction for collapsing domains,” from the AIM workshop *Shape optimization with surface interactions* (`aim-analysis-notes.json`, zero-based index 36). The exact prompt is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Shape optimization with surface interactions\nSection: Miscellaneous problems\nSource item: 6.6\nSource URL: http://aimpl.org/shapesurface/6/\nCanonical location: aim-analysis-notes.json notes[36]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Nodal structure of the $k$-th Neumann eigenfunction for collapsing domains\\n\\nConsider the nodal domains of the $k$-th Neumann eigenfunction on a sequence of domains collapsing to a lower dimensional object.\\n\\nUnder what assumptions can one prove that the nodal domains are in a sense ordered?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"There are related works in the Dirichlet setting by P. Freitas and D.~Krejcirik~\\\\cite{MR2400260} for tubular neighbourhoods of curves by D.~Krejcirik and M.~Tusek~\\\\cite{MR3274759} for tubular neighbourhoods of hypersurfaces.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/shapesurface/6/", "tags": [ "aim", "AIM-ANALYSIS-0037", "aim-domain:analysis", "aim-workshop:shapesurface", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For interval-like collapse, uniform convergence of a pulled-back eigenfunction and its longitudinal derivative to a one-dimensional eigenfunction with simple zeros forces the full nodal set to be a uniquely ordered family of transverse graphs. Explicit estimates control each graph's displacement and slope, and connected cross-sections give the exact sign-alternating slab decomposition. Applied to the published C^{1,gamma} convergence for smooth Neumann strips, this yields exactly n-1 transverse nodal arcs and n ordered nodal domains for every fixed n and sufficiently small width. Exact product-domain examples show that a one-dimensional limit, a distinguished simple branch, and a low-energy regime are genuinely separate requirements.\n\nCandidate contribution (lemma; novelty confidence low): The quantitative nodal-ordering transfer lemma gives, from explicit C^0 function and longitudinal-derivative errors plus transverse derivative control, the complete nodal graph topology, exact domain count and adjacency, sign alternation, displacement and slope bounds, and lateral Neumann orthogonality; its smooth-strip corollary makes the resulting n-slab structure explicit." }, { "id": 20000506, "problem_number": "AIM-ANALYSIS-0038", "title": "A finite-degree moment obstruction for finite-dimensional Bergman spaces", "statement": "Is there a pseudoconvex domain $\\Omega \\subset \\mathbb{C}^n$ with $0 < {\\rm dim} \\, A^2(\\Omega) < \\infty$?", "original_statement": "Is there a pseudoconvex domain $\\Omega \\subset \\mathbb{C}^n$ with $0 < {\\rm dim} \\, A^2(\\Omega) < \\infty$?", "clean_statement": "Is there a pseudoconvex domain $\\Omega \\subset \\mathbb{C}^n$ with $0 < {\\rm dim} \\, A^2(\\Omega) < \\infty$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List problem 1.05 from the workshop *Problems on holomorphic function spaces and complex dynamics*, section “Problems on Holomorphic Function Spaces.” Its exact statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.05\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[37]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a pseudoconvex domain $\\\\Omega \\\\subset \\\\mathbb{C}^n$ with $0 < {\\\\rm dim} \\\\, A^2(\\\\Omega) < \\\\infty$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0038", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted pseudoconvex problem remains open for n at least 2. A rigorous necessary condition is proved: if Omega is connected, d = dim A^2(Omega) is finite and positive, f is a nonzero Bergman function, and g is a nonconstant holomorphic function, then the integral of |g|^(2d)|f|^2 over Omega diverges. Consequently the Bergman multiplier algebra and H-infinity of Omega consist only of constants, its Caratheodory pseudodistance vanishes, and every nonconstant affine-linear observable has divergent 2d moment against every nonzero Bergman function. Disconnected-open-set and product reductions show that neither convention supplies a loophole.\n\nCandidate contribution (obstruction lemma; novelty confidence low): If 0 < dim A^2(Omega) = d < infinity on a connected domain, then for every nonzero f in A^2(Omega) and every nonconstant g in O(Omega), the weighted moment integral of |g|^(2d)|f|^2 is infinite." }, { "id": 20000507, "problem_number": "AIM-ANALYSIS-0039", "title": "Trivial Bergman space and quantitative A^p collapse", "statement": "What is the dimension of the Bergman space of a short $\\mathbb{C}^2$? (We may assume vanishing Kobayashi metric)\n\nAs an example of short $\\mathbb{C}^2$, consider the sequence of mappings $f_n(z, w) = (a_n w + z^2, a_n z)$, with the coefficients $a_n$ satisfying $a_n \\searrow 0$ rapidly. Then let $\\Omega = \\bigcup_{n \\geq 1} ( \\, \\mbox{basin of attraction of} \\, f_n \\, {\\rm at}\\, 0 )$.", "original_statement": "What is the dimension of the Bergman space of a short $\\mathbb{C}^2$? (We may assume vanishing Kobayashi metric)\n\nAs an example of short $\\mathbb{C}^2$, consider the sequence of mappings $f_n(z, w) = (a_n w + z^2, a_n z)$, with the coefficients $a_n$ satisfying $a_n \\searrow 0$ rapidly. Then let $\\Omega = \\bigcup_{n \\geq 1} ( \\, \\mbox{basin of attraction of} \\, f_n \\, {\\rm at}\\, 0 )$.", "clean_statement": "What is the dimension of the Bergman space of a short $\\mathbb{C}^2$? (We may assume vanishing Kobayashi metric)\n\nAs an example of short $\\mathbb{C}^2$, consider the sequence of mappings $f_n(z, w) = (a_n w + z^2, a_n z)$, with the coefficients $a_n$ satisfying $a_n \\searrow 0$ rapidly. Then let $\\Omega = \\bigcup_{n \\geq 1} ( \\, \\mbox{basin of attraction of} \\, f_n \\, {\\rm at}\\, 0 )$.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.1 in the section “Problems on Holomorphic Function Spaces” from the April 2019 AIM workshop *Problems on holomorphic function spaces and complex dynamics* (`aim-analysis-notes.json`, zero-based index 38). Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.1\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[38]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the dimension of the Bergman space of a short $\\\\mathbb{C}^2$? (We may assume vanishing Kobayashi metric)\\n\\nAs an example of short $\\\\mathbb{C}^2$, consider the sequence of mappings $f_n(z, w) = (a_n w + z^2, a_n z)$, with the coefficients $a_n$ satisfying $a_n \\\\searrow 0$ rapidly. Then let $\\\\Omega = \\\\bigcup_{n \\\\geq 1} ( \\\\, \\\\mbox{basin of attraction of} \\\\, f_n \\\\, {\\\\rm at}\\\\, 0 )$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0039", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For every Short C^2 in the standard sense—an increasing union of domains biholomorphic to the unit ball with identically vanishing Kobayashi metric—the Bergman space is trivial, so its dimension is 0. This recovers the published Fornaess–Pal solution through a direct Bergman-kernel argument. More generally, for every finite p>0 and every normalized exhaustion biholomorphism phi_j with alpha_j(q)=||Dphi_j(0)^{-1}||, every f in A^p(Omega) satisfies |f(q)| <= Vol(B^d)^{-1/p} alpha_j(q)^{2d/p} ||f||_p; since alpha_j(q) tends to 0, A^p(Omega)={0}. The archived source's literal union of autonomous basins also has trivial A^p, because each such basin is biholomorphic to C^2.\n\nCandidate contribution (theorem; novelty confidence low): For an increasing ball exhaustion Omega_j of a domain Omega in C^d with identically vanishing Kobayashi metric, the estimate |f(q)| <= Vol(B^d)^{-1/p} alpha_j(q)^{2d/p} ||f||_p holds for every 0Omega_j; consequently A^p(Omega) is trivial for every finite p." }, { "id": 20000508, "problem_number": "AIM-ANALYSIS-0040", "title": "Nontrivial Bergman space versus the plurisubharmonic Liouville property", "statement": "Is there a pseudoconvex domain $\\Omega$ with ${\\rm dim} \\, A^2(\\Omega) \\neq 0$ such that there is no bounded plurisubharmonic function $\\phi$ on $\\Omega$.", "original_statement": "Is there a pseudoconvex domain $\\Omega$ with ${\\rm dim} \\, A^2(\\Omega) \\neq 0$ such that there is no bounded plurisubharmonic function $\\phi$ on $\\Omega$.", "clean_statement": "Is there a pseudoconvex domain $\\Omega$ with ${\\rm dim} \\, A^2(\\Omega) \\neq 0$ such that there is no bounded plurisubharmonic function $\\phi$ on $\\Omega$.", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.15 from the AIM workshop *Problems on holomorphic function spaces and complex dynamics*. Its exact extracted text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.15\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[39]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a pseudoconvex domain $\\\\Omega$ with ${\\\\rm dim} \\\\, A^2(\\\\Omega) \\\\neq 0$ such that there is no bounded plurisubharmonic function $\\\\phi$ on $\\\\Omega$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0040", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After reconstructing the literal statement as asking for a domain on which every bounded plurisubharmonic function is constant, we prove a necessary-condition sieve. For every psh-Liouville domain Omega in C^n and every nonconstant scalar holomorphic map F, the complement of F(Omega) is polar. Consequently no example exists in dimension one, among proper convex domains, among domains biholomorphic to C^n, or as a first example obtained by a product with a one-dimensional factor. We also prove the planar L2-removability step across polar complements and a differential obstruction to producing a bounded psh function through a universal radial transform of one Bergman function. The higher-dimensional existence question remains open.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: every nonconstant scalar holomorphic function on a putative example must have co-polar image; assembled with convex separation, product descent, and biholomorphic invariance of square-integrable top forms, this excludes proper convex domains, products with one-dimensional factors, and domains biholomorphic to affine space as candidates." }, { "id": 20000509, "problem_number": "AIM-ANALYSIS-0041", "title": "Affine lineality in the pseudoconvex case and a rate-sensitive nonconvex separation", "statement": "What can we say about the geometry of a complete Reinhardt domain with trivial Bergman space?", "original_statement": "What can we say about the geometry of a complete Reinhardt domain with trivial Bergman space?", "clean_statement": "What can we say about the geometry of a complete Reinhardt domain with trivial Bergman space?", "statement_status": "exact", "statement_verification": "The canonical record is number 1.2 in the AIM list *Problems on holomorphic function spaces and complex dynamics*, section “Problems on Holomorphic Function Spaces”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.2\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[40]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What can we say about the geometry of a complete Reinhardt domain with trivial Bergman space?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0041", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every coordinatewise-complete Reinhardt domain, triviality of the Bergman space is exactly divergence of all positive-integer exponential moments of its logarithmic image. In the pseudoconvex case this recovers the known equivalence between a trivial Bergman space, an affine line in the logarithmic image, and nonzero recession-cone lineality; that lineality yields a diagonal holomorphic group action, and in dimension two it gives a complete elementary classification. Outside the pseudoconvex class, an explicit matched pair of complete Reinhardt domains has the same logarithmic convex hull and no affine line in either actual logarithmic image, while one Bergman space is trivial and the other has precisely the diagonal monomial basis.\n\nCandidate contribution (counterexample; novelty confidence low): The explicit domains defined by h_s(x)=-log(log(e^e+x^2)) and h_f(x)=-log(e+x^2) are complete Reinhardt domains in C^2 with the same logarithmic convex hull and no affine line in either logarithmic image, but A^2(Omega_s) is trivial whereas A^2(Omega_f) is the closed span of the diagonal monomials (z_1 z_2)^k." }, { "id": 20000510, "problem_number": "AIM-ANALYSIS-0042", "title": "A weighted Bergman space on the disk cannot have finite positive dimension", "statement": "Let $\\Delta$ be the unit disk in $\\mathbb{C}$ and let $\\omega > 0$ be an upper semicontinuous function on $\\Delta$. Consider the weightd Bergman space $A^2 (\\Delta, \\omega)$. Is it possible that $0 < {\\rm dim} \\, A^2(\\Delta, \\omega) < \\infty$?", "original_statement": "Let $\\Delta$ be the unit disk in $\\mathbb{C}$ and let $\\omega > 0$ be an upper semicontinuous function on $\\Delta$. Consider the weightd Bergman space $A^2 (\\Delta, \\omega)$. Is it possible that $0 < {\\rm dim} \\, A^2(\\Delta, \\omega) < \\infty$?", "clean_statement": "Let $\\Delta$ be the unit disk in $\\mathbb{C}$ and let $\\omega > 0$ be an upper semicontinuous function on $\\Delta$. Consider the weightd Bergman space $A^2 (\\Delta, \\omega)$. Is it possible that $0 < {\\rm dim} \\, A^2(\\Delta, \\omega) < \\infty$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ANALYSIS-0042, Problem 1.25 in the section “Problems on Holomorphic Function Spaces” of the AIM list *Problems on holomorphic function spaces and complex dynamics*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.25\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[41]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $\\\\Delta$ be the unit disk in $\\\\mathbb{C}$ and let $\\\\omega > 0$ be an upper semicontinuous function on $\\\\Delta$. Consider the weightd Bergman space $A^2 (\\\\Delta, \\\\omega)$. Is it possible that $0 < {\\\\rm dim} \\\\, A^2(\\\\Delta, \\\\omega) < \\\\infty$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0042", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For every positive upper semicontinuous weight omega on the unit disk, the weighted Bergman space A^2(Delta,omega) is either zero or infinite-dimensional, so finite positive dimension is impossible. If f is nonzero, multiplication by the bounded coordinate gives f,zf,z^2f,... in the space, and positivity of the density plus the identity theorem makes this orbit linearly independent. The proof extends to every finite p>0, survives the L^p quotient convention, and is independently checked by Cayley-Hamilton. Explicit continuous weights realize both the zero and infinite-dimensional alternatives.\n\nCandidate contribution (theorem; novelty confidence low): Let Omega be a connected complex domain, mu a positive Borel measure whose support is a uniqueness set for holomorphic functions, 0 integral_S P conjugate(Q) d sigma on the finite-dimensional degree-k polynomial subspace, divided by the spherical L^q norm of Q; moreover A_{p,n,k} tends to C_p^{n-1/2}, quantifying the angular saving required by the Gryc-Kemp conjecture." }, { "id": 20000514, "problem_number": "AIM-ANALYSIS-0046", "title": "Intermediate pseudoconvex domains above Fatou-Bieberbach domains", "statement": "Given a Fatou-Bieberbach domain $\\Omega$, must there exist a pseudoconvex domain $\\tilde{\\Omega}$ such that $\\tilde{\\Omega}$ with $\\Omega \\subsetneq \\tilde{\\Omega} \\subsetneq \\mathbb{C}^n$.", "original_statement": "Given a Fatou-Bieberbach domain $\\Omega$, must there exist a pseudoconvex domain $\\tilde{\\Omega}$ such that $\\tilde{\\Omega}$ with $\\Omega \\subsetneq \\tilde{\\Omega} \\subsetneq \\mathbb{C}^n$.", "clean_statement": "Given a Fatou-Bieberbach domain $\\Omega$, must there exist a pseudoconvex domain $\\tilde{\\Omega}$ such that $\\tilde{\\Omega}$ with $\\Omega \\subsetneq \\tilde{\\Omega} \\subsetneq \\mathbb{C}^n$.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.45 in the section “Problems on Holomorphic Function Spaces” from the AIM workshop *Problems on holomorphic function spaces and complex dynamics*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.45\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[45]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given a Fatou-Bieberbach domain $\\\\Omega$, must there exist a pseudoconvex domain $\\\\tilde{\\\\Omega}$ such that $\\\\tilde{\\\\Omega}$ with $\\\\Omega \\\\subsetneq \\\\tilde{\\\\Omega} \\\\subsetneq \\\\mathbb{C}^n$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0046", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general AIM question remains open, but a strict intermediate pseudoconvex domain is equivalent to a nonempty strict closed sub-barrier F of the Fatou-Bieberbach complement whose complement is connected and pseudoconvex; such an F cannot be compact by Hartogs extension. If the complement contains a nonempty entire hypersurface Z(f), then C^n minus Z(f) is a strict intermediate Stein domain, with strictness proved by a meridian of nonzero f-winding. Every connected container D of the Fatou-Bieberbach domain has trivial Bergman space, only constant bounded holomorphic functions, contains the given domain in its smooth bounded-psh core, and has no bounded-above psh exhaustion.\n\nCandidate contribution (criterion_and_obstruction_theorem; novelty confidence low): If the complement of a Fatou-Bieberbach domain contains a nonempty principal entire hypersurface, retaining that hypersurface gives a strict intermediate Stein domain; for every intermediate domain, the retained complement must be unbounded, the Fatou-Bieberbach domain lies in the smooth bounded-psh core, the Bergman space is zero, bounded holomorphic functions are constant, and no bounded-above psh exhaustion exists." }, { "id": 20000515, "problem_number": "AIM-ANALYSIS-0047", "title": "A weighted Kähler characteristic and a toric logarithmic Hardy model", "statement": "Given a K\\\"{a}hler manifold $(M, \\omega)$. Define $H^2_\\omega:=\\{f: \\Delta \\rightarrow M : \\int_\\Delta \\log |z|^2 f^* \\omega > -\\infty \\}$. Investigate these spaces, for example: completeness, radial limits, etc... A special case to consider is when $M$ being toric.", "original_statement": "Given a K\\\"{a}hler manifold $(M, \\omega)$. Define $H^2_\\omega:=\\{f: \\Delta \\rightarrow M : \\int_\\Delta \\log |z|^2 f^* \\omega > -\\infty \\}$. Investigate these spaces, for example: completeness, radial limits, etc... A special case to consider is when $M$ being toric.", "clean_statement": "Given a K\\\"{a}hler manifold $(M, \\omega)$. Define $H^2_\\omega:=\\{f: \\Delta \\rightarrow M : \\int_\\Delta \\log |z|^2 f^* \\omega > -\\infty \\}$. Investigate these spaces, for example: completeness, radial limits, etc... A special case to consider is when $M$ being toric.", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.5 in the AIM workshop list *Problems on Holomorphic Function Spaces*. The archived AIM page gives the following text (including its grammatical omissions):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.5\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[46]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given a K\\\\\\\"{a}hler manifold $(M, \\\\omega)$. Define $H^2_\\\\omega:=\\\\{f: \\\\Delta \\\\rightarrow M : \\\\int_\\\\Delta \\\\log |z|^2 f^* \\\\omega > -\\\\infty \\\\}$. Investigate these spaces, for example: completeness, radial limits, etc... A special case to consider is when $M$ being toric.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0047", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Writing E_omega(f)=-int_Delta log|z|^2 f^*omega, one has the exact characteristic identity E_omega(f)=2 int_0^1 A_f(t)dt/t. For the algebraic torus with a Kähler form uniformly elliptic on its logarithmic universal cover, every disk has a branch-independent centered logarithm h_f, membership is equivalent to h_f lying in vector-valued H^2, and the map space is globally (C^*)^m x (H^2_0)^m. This yields a proved complete anchored metric and almost-everywhere nontangential limits with radial L^2 convergence in target distance. A fixed standard compact-open metric is nevertheless incomplete, and finite characteristic does not force finite radial variation.\n\nCandidate contribution (theorem; novelty confidence low): For every uniformly logarithmically elliptic toric Kähler form on (C^*)^m, the finite-characteristic disk space admits the explicit product parametrization f -> (f(0), g-g(0)), a complete anchored product metric, and radial L^2 convergence in the Kähler target distance; under the flat logarithmic form the energy is exactly pi times the squared H^2 norm of the centered logarithm." }, { "id": 20000516, "problem_number": "AIM-ANALYSIS-0048", "title": "Levi-flat reduction and a global first-integral obstruction for real-analytic Fatou-Bieberbach boundaries", "statement": "Is there a Fatou-Bieberbach domain with real analytic boundary?", "original_statement": "Is there a Fatou-Bieberbach domain with real analytic boundary?", "clean_statement": "Is there a Fatou-Bieberbach domain with real analytic boundary?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Problems on holomorphic function spaces and complex dynamics*, section *Problems on Holomorphic Function Spaces*, Problem 1.55) asks exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.55\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[47]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a Fatou-Bieberbach domain with real analytic boundary?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0048", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The real-analytic-boundary question remains open. For every C^2-smooth Fatou-Bieberbach domain in C^n, the domain, its complement, and its boundary are unbounded, and the boundary Levi form is semipositive and everywhere degenerate; in C^2 it is Levi-flat. Hence a real-analytic example in C^2 has local Cartan holomorphic flattenings, but these cannot globalize to a nonconstant holomorphic map from the whole domain into a half-plane, and there is no global bounded-above plurisubharmonic defining function. Existence in C^2 propagates to every n at least 2 by taking products.\n\nCandidate contribution (proposition; novelty confidence low): A real-analytic-boundary Fatou-Bieberbach domain in C^2 must have local holomorphic transverse coordinates along its Levi-flat boundary, but no choice of them can globalize to a nonconstant half-plane-valued holomorphic first integral on the entire domain; in particular, no component of {Im G > 0} for a nonconstant entire G can be Fatou-Bieberbach." }, { "id": 20000517, "problem_number": "AIM-ANALYSIS-0049", "title": "Boundary multipliers and Levi resonance for autonomous attracting basins", "statement": "Can region of attractions for non-polynomial automorphisms of $\\mathbb{C}^2$ have $C^1$ boundary?", "original_statement": "Can region of attractions for non-polynomial automorphisms of $\\mathbb{C}^2$ have $C^1$ boundary?", "clean_statement": "Can region of attractions for non-polynomial automorphisms of $\\mathbb{C}^2$ have $C^1$ boundary?", "statement_status": "exact", "statement_verification": "The canonical AIM record (Analysis, problem 1.6, source index 48) reads verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.6\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[48]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can region of attractions for non-polynomial automorphisms of $\\\\mathbb{C}^2$ have $C^1$ boundary?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0049", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any invariant domain of an autonomous holomorphic automorphism of C^2, a periodic point on a C^1 boundary forces the complex tangent line to be invariant and the induced complex-normal quotient multiplier to be a positive real eigenvalue, without any diagonalizability assumption. At C^2 regularity the exact identity (nu-|tau|^2)L_rho=0 holds, so every saddle periodic boundary point is Levi-flat and dense saddle periodic points force global Levi-flatness. The report also proves conjugacy covariance, gives explicit non-polynomial conjugates showing the empty-boundary and coordinate-dependence ambiguities, and proves that every proper Fatou-Bieberbach domain has noncompact boundary. These are rigorous necessary conditions and reductions; the existence question remains open.\n\nCandidate contribution (lemma; novelty confidence low): Candidate synthesis/generalization: at a periodic point of any invariant C^1 domain boundary for an autonomous holomorphic automorphism of C^2, the complex-normal quotient multiplier is positive real; at C^2 regularity it satisfies the Levi resonance (nu-|tau|^2)L=0, yielding a dense-saddle-to-global-Levi-flat reduction." }, { "id": 20000518, "problem_number": "AIM-ANALYSIS-0050", "title": "Logarithmic lattice-packing reduction and necessary character constraints", "statement": "Is there a Fatou-Bieberbach domain contained in $\\mathbb{C}^* \\times \\mathbb{C}^*$?", "original_statement": "Is there a Fatou-Bieberbach domain contained in $\\mathbb{C}^* \\times \\mathbb{C}^*$?", "clean_statement": "Is there a Fatou-Bieberbach domain contained in $\\mathbb{C}^* \\times \\mathbb{C}^*$?", "statement_status": "exact", "statement_verification": "The exact canonical record is AIM-ANALYSIS-0050, Problem 1.65 in the AIM list *Problems on Holomorphic Function Spaces*. The archived AIM page agrees with the record and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.65\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[49]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a Fatou-Bieberbach domain contained in $\\\\mathbb{C}^* \\\\times \\\\mathbb{C}^*$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0050", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A Fatou-Bieberbach domain Omega contained in (C*)^2 exists if and only if there is a Fatou-Bieberbach domain D in C^2 disjoint from every nonzero translate by the deck lattice (2 pi i Z)^2; in that case the translates are exactly the components of the full logarithmic preimage. Any such Omega maps surjectively under every nontrivial algebraic torus character, while every lift D has Euclidean inradius at most pi, imaginary slices of area at most 4 pi^2, infinite real-part projection area, and quadratic radial volume growth. These are proved reductions and necessary conditions, not an existence or nonexistence solution.\n\nCandidate contribution (equivalence_and_obstruction_package; novelty confidence low): Candidate novelty is the combined exact converse and deck-component form of the Rosay-Rudin logarithmic criterion together with surjectivity of every nontrivial torus character and the explicit inradius/infinite-real-projection consequences." }, { "id": 20000519, "problem_number": "AIM-ANALYSIS-0051", "title": "Positive logarithmic capacity does not force a nonconstant H^2 function", "statement": "For a domain $\\Omega \\subset \\mathbb{C}$, define\n\\begin{equation*}\nH^2(\\Omega):=\\{ f \\,\\, \\mbox{holomorphic} \\, \\, \\mbox{on} \\, \\, \\Omega : |f|^2 \\, \\, \\mbox{has a harmonic majorant} \\}.\n\\end{equation*}\nDoes log-capacity$(\\mathbb{C}\\setminus \\Omega) > 0$ imply ${\\rm dim}\\, H^2 (\\Omega) > 1$?", "original_statement": "For a domain $\\Omega \\subset \\mathbb{C}$, define\n\\begin{equation*}\nH^2(\\Omega):=\\{ f \\,\\, \\mbox{holomorphic} \\, \\, \\mbox{on} \\, \\, \\Omega : |f|^2 \\, \\, \\mbox{has a harmonic majorant} \\}.\n\\end{equation*}\nDoes log-capacity$(\\mathbb{C}\\setminus \\Omega) > 0$ imply ${\\rm dim}\\, H^2 (\\Omega) > 1$?", "clean_statement": "For a domain $\\Omega \\subset \\mathbb{C}$, define\n\\begin{equation*}\nH^2(\\Omega):=\\{ f \\,\\, \\mbox{holomorphic} \\, \\, \\mbox{on} \\, \\, \\Omega : |f|^2 \\, \\, \\mbox{has a harmonic majorant} \\}.\n\\end{equation*}\nDoes log-capacity$(\\mathbb{C}\\setminus \\Omega) > 0$ imply ${\\rm dim}\\, H^2 (\\Omega) > 1$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists problem 1.7 from “Problems on Holomorphic Function Spaces,” stored at index 50 of `aim-analysis-notes.json`. The source record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.7\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[50]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For a domain $\\\\Omega \\\\subset \\\\mathbb{C}$, define\\n\\\\begin{equation*}\\nH^2(\\\\Omega):=\\\\{ f \\\\,\\\\, \\\\mbox{holomorphic} \\\\, \\\\, \\\\mbox{on} \\\\, \\\\, \\\\Omega : |f|^2 \\\\, \\\\, \\\\mbox{has a harmonic majorant} \\\\}.\\n\\\\end{equation*}\\nDoes log-capacity$(\\\\mathbb{C}\\\\setminus \\\\Omega) > 0$ imply ${\\\\rm dim}\\\\, H^2 (\\\\Omega) > 1$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0051", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Specializing Hasumi's Theorem 2.5 to the convex function phi(t)=exp(2t) produces a compact totally disconnected set E of positive logarithmic capacity that belongs to the H^2 null class. Applying its defining removability identity with the ambient domain C gives H^2(C minus E)=H^2(C)=C, so Omega=C minus E is a domain with cap(C minus Omega)>0 but dim H^2(Omega)=1. The later correction literature was checked and concerns a separate 1/z witness in Hasumi's strict-classification argument, not the null-set/removability theorem used here.\n\nCandidate contribution (corollary; novelty confidence low): For every kappa>0 and c in C, affine normalization of a Hasumi null set gives a compact totally disconnected E_{kappa,c} with logarithmic capacity exactly kappa, analytic capacity zero, and H^2(C minus E_{kappa,c})=C." }, { "id": 20000520, "problem_number": "AIM-ANALYSIS-0052", "title": "Toeplitz and mode reductions for weighted Szegő projections on the smooth worm", "statement": "Can we choose continuous multiples of surface area on the boundary of the Diederich-Fornaess worm (for example, powers of Levi-form) mitigating the irregularity of the Szeg\\\"{o} projection (defined using surface area/modified measure?)", "original_statement": "Can we choose continuous multiples of surface area on the boundary of the Diederich-Fornaess worm (for example, powers of Levi-form) mitigating the irregularity of the Szeg\\\"{o} projection (defined using surface area/modified measure?)", "clean_statement": "Can we choose continuous multiples of surface area on the boundary of the Diederich-Fornaess worm (for example, powers of Levi-form) mitigating the irregularity of the Szeg\\\"{o} projection (defined using surface area/modified measure?)", "statement_status": "exact", "statement_verification": "The AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.75\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[51]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we choose continuous multiples of surface area on the boundary of the Diederich-Fornaess worm (for example, powers of Levi-form) mitigating the irregularity of the Szeg\\\\\\\"{o} projection (defined using surface area/modified measure?)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0052", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every strictly positive bounded boundary density, the weighted Szegő projection satisfies the proved identity S_w=T_w^{-1}SM_w, with exact difference and adjoint formulas; smooth invertible holomorphic-square weights are exact similarities and therefore cannot mitigate regularity. For a continuous density that vanishes but is positive almost everywhere, the proved surviving facts are the weak equation SM_wS_w=SM_w, an exact Toeplitz coercivity and regularization criterion, a literal Hardy-mass concentration estimate, and an S^1 Fourier-mode reduction. The existence of a concentrating sequence is conditional, the behavior of the individual worm modes is unproved, and whether a degenerate Levi-power weight actually improves the smooth-worm Szegő projection remains open.\n\nCandidate contribution (reduction; novelty confidence low): For a rotation-invariant continuous Levi-power density on the smooth bounded worm, the degenerate weighted problem reduces to modewise noncoercive Toeplitz equations with the exact law ||T_{w+epsilon}^{-1}||=1/(c_w+epsilon), where c_w=inf_j c_{w,j}; any sequence forcing c_w=0 must concentrate its unweighted Hardy mass in arbitrarily small Levi sublevel sets. In the equivalent-weight regime, improvement requires failure of regularity-space Toeplitz invertibility, while weights |g|^2 with g holomorphic and invertible are ruled out by exact similarity.", "research_difficulty_suggested": "L4" }, { "id": 20000521, "problem_number": "AIM-ANALYSIS-0053", "title": "Holomorphic Neumann data as Hardy traces with period constraints", "statement": "Characterize the boundary data space for Neumann boundary problem for holomorphic functions on planar domain. Are there any representation formula for the solution?", "original_statement": "Characterize the boundary data space for Neumann boundary problem for holomorphic functions on planar domain. Are there any representation formula for the solution?", "clean_statement": "Characterize the boundary data space for Neumann boundary problem for holomorphic functions on planar domain. Are there any representation formula for the solution?", "statement_status": "exact", "statement_verification": "The archived AIM page was inspected. It contains exactly this sentence and no attribution, status note, definition of “Neumann,” regularity hypothesis, or choice of boundary norm. Thus the ambiguity is in the source, not an extraction error.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Holomorphic Function Spaces\nSource item: 1.8\nSource URL: http://aimpl.org/scvproblems/1/\nCanonical location: aim-analysis-notes.json notes[52]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Characterize the boundary data space for Neumann boundary problem for holomorphic functions on planar domain. Are there any representation formula for the solution?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/1/", "tags": [ "aim", "AIM-ANALYSIS-0053", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The intended full complex Neumann problem is substantially answered in published work: on bounded finitely connected Lipschitz planar domains its L^p data are exactly those g for which i times the conjugate unit tangent times g is a holomorphic Hardy trace and the flux of g vanishes on each independent inner boundary component; the derivative is its Cauchy integral and the solution is a normalized primitive. A self-contained Holder-class proof and disk logarithmic formula are given here. Separately, the report proves a low-confidence candidate synthesis for the source's alternative real-data reading: pointwise real part is a real-linear bijection from full complex holomorphic Neumann data onto real harmonic Neumann data with zero flux on every component, with inverse phi mapped to phi minus i times the tangential derivative of the normalized harmonic Neumann solution.\n\nCandidate contribution (equivalence_synthesis; novelty confidence low): On a bounded finitely connected C^{2,alpha} planar domain, the real-part map from full complex holomorphic Neumann data to real harmonic Neumann data is a real-linear bijection onto the componentwise zero-flux space, and its inverse is phi mapped to phi-i partial_s u_phi, where u_phi is the normalized harmonic Neumann solution." }, { "id": 20000522, "problem_number": "AIM-ANALYSIS-0054", "title": "Wild realization and a constant-Jacobian packing obstruction", "statement": "Let $F$ be a non-polynomial automorphism of $\\mathbb{C}^2$. What are the possible biholomorphic types of Fatou components of $F$?", "original_statement": "Let $F$ be a non-polynomial automorphism of $\\mathbb{C}^2$. What are the possible biholomorphic types of Fatou components of $F$?", "clean_statement": "Let $F$ be a non-polynomial automorphism of $\\mathbb{C}^2$. What are the possible biholomorphic types of Fatou components of $F$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ANALYSIS-0054, problem 2.1 in the section “Problems on Complex Dynamics” of the AIM list *Problems on holomorphic function spaces and complex dynamics*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Complex Dynamics\nSource item: 2.1\nSource URL: http://aimpl.org/scvproblems/2/\nCanonical location: aim-analysis-notes.json notes[53]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $F$ be a non-polynomial automorphism of $\\\\mathbb{C}^2$. What are the possible biholomorphic types of Fatou components of $F$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/2/", "tags": [ "aim", "AIM-ANALYSIS-0054", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the forward P^2-Fatou convention, no exhaustive classification is currently known. Boc Thaler's realization theorem implies that every bounded convex domain can occur exactly as either an escaping or an oscillating wandering Fatou component; applying it to E_m = {(|z|^2 + |w|^(2m) < 1)} gives an explicit pairwise non-biholomorphic family, and the realizing maps are non-polynomial by the Friedland-Milnor dichotomy together with the polynomial Henon filtration. The candidate novel result is a whole-component packing inequality: if det_C DF is the constant delta and a finite-positive-volume wandering component Omega has entire images F^n(Omega) contained in a finite-volume set K at times S, then Vol_4(K) is at least Vol_4(Omega) times the sum over S of |delta|^(2n), with the analogous inverse-time inequality. This constrains whole component images, not pointwise or compact-subset recurrence.\n\nCandidate contribution (obstruction; novelty confidence low): For a constant-Jacobian automorphism with det_C DF = delta, any finite-positive-volume wandering component Omega and finite-volume measurable K satisfy Vol_4(K) >= Vol_4(Omega) sum_{n: F^n(Omega) subset K} |delta|^(2n), and Vol_4(K) >= Vol_4(Omega) sum_{n: F^(-n)(Omega) subset K} |delta|^(-2n). Thus whole forward component images can be contained in K only finitely often when |delta| >= 1, and whole backward images only finitely often when |delta| <= 1." }, { "id": 20000523, "problem_number": "AIM-ANALYSIS-0055", "title": "Pole-orbit stability for rational Henon recurrences", "statement": "Investigate the dynamics of maps of the form\n\\begin{equation*}(x, y) \\mapsto (p(x)-\\delta y, x),\n\\end{equation*}\nwhere $p(x)$ is a degree $2$ rational map.", "original_statement": "Investigate the dynamics of maps of the form\n\\begin{equation*}(x, y) \\mapsto (p(x)-\\delta y, x),\n\\end{equation*}\nwhere $p(x)$ is a degree $2$ rational map.", "clean_statement": "Investigate the dynamics of maps of the form\n\\begin{equation*}(x, y) \\mapsto (p(x)-\\delta y, x),\n\\end{equation*}\nwhere $p(x)$ is a degree $2$ rational map.", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Complex Dynamics\nSource item: 2.2\nSource URL: http://aimpl.org/scvproblems/2/\nCanonical location: aim-analysis-notes.json notes[54]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate the dynamics of maps of the form\\n\\\\begin{equation*}(x, y) \\\\mapsto (p(x)-\\\\delta y, x),\\n\\\\end{equation*}\\nwhere $p(x)$ is a degree $2$ rational map.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/2/", "tags": [ "aim", "AIM-ANALYSIS-0055", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For nonzero delta, the natural birational extension of (x,y) -> (p(x)-delta y,x) to P^1 x P^1 has exceptional curves exactly over the poles of p. If infinity is not a pole, writing c=p(infinity) and L_delta(t)=c-delta t, the extension is algebraically stable exactly when L_delta^k(a) avoids the pole set for every pole a and every k>=1. In the stable case its bidegrees satisfy the Pell recurrence and its first dynamical degree is 1+sqrt(2). Unless both 0 and c are poles, the nonstable parameters form at most a countable union of finite algebraic sets; for p(x)=1/(x^2-1), stability holds exactly when delta is not a root of unity. The result also proves that delta=1 is never stable on the naive product model and explains why the naive P^2 compactification has different raw degree behavior.\n\nCandidate contribution (criterion_and_generic_parameter_theorem; novelty confidence low): Candidate novelty: the exact reduction of algebraic stability on P^1 x P^1 to avoidance of the pole set under L_delta(t)=p(infinity)-delta t, together with the resulting co-countable parameter theorem, the root-of-unity classification for p(x)=1/(x^2-1), and the product-versus-projective compactification audit." }, { "id": 20000524, "problem_number": "AIM-ANALYSIS-0056", "title": "An exact petal criterion for a diagonal family with a degenerate direction", "statement": "Let $F$ be a germ of automorphism of $\\mathbb{C}^n (n > 1)$ tangent to the identity with a degenerate characteristic direction $v$. When is there a domain of attraction tangent to the origin along $v$?", "original_statement": "Let $F$ be a germ of automorphism of $\\mathbb{C}^n (n > 1)$ tangent to the identity with a degenerate characteristic direction $v$. When is there a domain of attraction tangent to the origin along $v$?", "clean_statement": "Let $F$ be a germ of automorphism of $\\mathbb{C}^n (n > 1)$ tangent to the identity with a degenerate characteristic direction $v$. When is there a domain of attraction tangent to the origin along $v$?", "statement_status": "exact", "statement_verification": "The canonical record is `aim-analysis-notes.json`, record index 55, Problem 2.3 of the AIM list *Problems on holomorphic function spaces and complex dynamics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Complex Dynamics\nSource item: 2.3\nSource URL: http://aimpl.org/scvproblems/2/\nCanonical location: aim-analysis-notes.json notes[55]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $F$ be a germ of automorphism of $\\\\mathbb{C}^n (n > 1)$ tangent to the identity with a degenerate characteristic direction $v$. When is there a domain of attraction tangent to the origin along $v$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/2/", "tags": [ "aim", "AIM-ANALYSIS-0056", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the local biholomorphic germs F(z,w_1,...,w_{n-1})=(z-z^{r+1},(1-b_1z^{t_1})w_1,...,(1-b_{n-1}z^{t_{n-1}})w_{n-1}), where 1 <= t_j < r and b_j is nonzero, the degenerate characteristic direction [1:0:...:0] has a full-dimensional attracting domain if and only if some r-th root of unity zeta satisfies Re(b_j zeta^{t_j})>0 for every j. The proof constructs invariant product wedges and proves the converse for arbitrary open basins. In dimension two, failure occurs exactly when r=2t and b is nonzero purely imaginary; on this boundary generic transverse moduli grow polynomially with log|w_N/w_0|=(|b|^2/(2r))log N+L+o(1).\n\nCandidate contribution (exact criterion and obstruction; novelty confidence low): Candidate novelty: the petal-by-petal if-and-only-if criterion for the all-dimensional diagonal family, including the arbitrary-open-basin converse, the simultaneous-contraction obstruction in dimension three, the exact two-dimensional exceptional arithmetic, and the second-order neutral cocycle asymptotic." }, { "id": 20000525, "problem_number": "AIM-ANALYSIS-0057", "title": "Two-neutral hedgehogs and a rigorous product model", "statement": "Study hedgehogs for gems of automorphism of $\\mathbb{C}^2$ with both elliptic eigenvalues.", "original_statement": "Study hedgehogs for gems of automorphism of $\\mathbb{C}^2$ with both elliptic eigenvalues.", "clean_statement": "Study hedgehogs for gems of automorphism of $\\mathbb{C}^2$ with both elliptic eigenvalues.", "statement_status": "exact", "statement_verification": "The canonical record reads, literally:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Complex Dynamics\nSource item: 2.4\nSource URL: http://aimpl.org/scvproblems/2/\nCanonical location: aim-analysis-notes.json notes[56]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Study hedgehogs for gems of automorphism of $\\\\mathbb{C}^2$ with both elliptic eigenvalues.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/2/", "tags": [ "aim", "AIM-ANALYSIS-0057", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a decoupled germ F=f_1 times f_2 with two irrationally indifferent one-variable factors and admissible product domain, the maximal two-sided invariant set and its component at the origin factor exactly. The resulting compactum K=K_1 times K_2 is compact, connected, completely invariant, touches both boundary faces, has connected complement in C^2, and is polynomially convex; F is holomorphically linearizable exactly when both factors are. If both factors are nonlinearizable then K contains no nonconstant analytic disk, whereas a nonlinearizable factor times an irrational rotation gives an empty-interior K that does contain vertical analytic disks. Independently, any compact completely invariant local set whose ambient interior contains the fixed point forces holomorphic linearization.\n\nCandidate contribution (special_case; novelty confidence low): For products of two irrationally indifferent one-variable germs, the product-domain maximal invariant component factors exactly and is polynomially convex; its analytic disks distinguish the double-Cremer case, where none exist, from the Cremer-times-rotation case, where vertical disks exist despite nonlinearizability." }, { "id": 20000526, "problem_number": "AIM-ANALYSIS-0058", "title": "Exact Green functions and zero leafwise critical measure on the Cayley fiber", "statement": "Let $C$ be the cubic surface defined by\n\\begin{equation*}\nx^2 + y^2 + z^2 - xyz = D,\n\\end{equation*}\nwhere $D$ is a complex parameter. Let $s_x: C \\rightarrow C$ be defined by\n\\begin{equation*}\ns_x(x, y, z) := (yz - x, y, z).\n\\end{equation*}\nThe maps $s_y, s_z$ are defined similarly.\n\nConsider, for example, the map $S:= s_x \\circ s_y \\circ s_z$. It is known that $S$ has non-zero entropy. Adapt the study of critical measure, etc from the study of H\\'{e}non maps to that of $F$.", "original_statement": "Let $C$ be the cubic surface defined by\n\\begin{equation*}\nx^2 + y^2 + z^2 - xyz = D,\n\\end{equation*}\nwhere $D$ is a complex parameter. Let $s_x: C \\rightarrow C$ be defined by\n\\begin{equation*}\ns_x(x, y, z) := (yz - x, y, z).\n\\end{equation*}\nThe maps $s_y, s_z$ are defined similarly.\n\nConsider, for example, the map $S:= s_x \\circ s_y \\circ s_z$. It is known that $S$ has non-zero entropy. Adapt the study of critical measure, etc from the study of H\\'{e}non maps to that of $F$.", "clean_statement": "Let $C$ be the cubic surface defined by\n\\begin{equation*}\nx^2 + y^2 + z^2 - xyz = D,\n\\end{equation*}\nwhere $D$ is a complex parameter. Let $s_x: C \\rightarrow C$ be defined by\n\\begin{equation*}\ns_x(x, y, z) := (yz - x, y, z).\n\\end{equation*}\nThe maps $s_y, s_z$ are defined similarly.\n\nConsider, for example, the map $S:= s_x \\circ s_y \\circ s_z$. It is known that $S$ has non-zero entropy. Adapt the study of critical measure, etc from the study of H\\'{e}non maps to that of $F$.", "statement_status": "exact", "statement_verification": "This is Problem 2.5 in the AIM list *Problems on holomorphic function spaces and complex dynamics*, section “Problems on Complex Dynamics.” The canonical record is `aim-analysis-notes.json`, zero-based index 57. Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Problems on holomorphic function spaces and complex dynamics\nSection: Problems on Complex Dynamics\nSource item: 2.5\nSource URL: http://aimpl.org/scvproblems/2/\nCanonical location: aim-analysis-notes.json notes[57]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $C$ be the cubic surface defined by\\n\\\\begin{equation*}\\nx^2 + y^2 + z^2 - xyz = D,\\n\\\\end{equation*}\\nwhere $D$ is a complex parameter. Let $s_x: C \\\\rightarrow C$ be defined by\\n\\\\begin{equation*}\\ns_x(x, y, z) := (yz - x, y, z).\\n\\\\end{equation*}\\nThe maps $s_y, s_z$ are defined similarly.\\n\\nConsider, for example, the map $S:= s_x \\\\circ s_y \\\\circ s_z$. It is known that $S$ has non-zero entropy. Adapt the study of critical measure, etc from the study of H\\\\'{e}non maps to that of $F$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/scvproblems/2/", "tags": [ "aim", "AIM-ANALYSIS-0058", "aim-domain:analysis", "aim-workshop:scvproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For S = s_x composed with s_y composed with s_z on the Cayley fiber C_4, the finite torus cover conjugates S to the monomial map with exponent matrix [[3,-2],[-2,1]]. With lambda = 2 + sqrt(5) and phi = (1 + sqrt(5))/2, the normalized Green functions pull back to |phi log|u| - log|v||/sqrt(5) and phi|log|u| + phi log|v||/sqrt(5). Their complex derivatives along unstable and stable escape components are nonzero constants, so the corresponding Bedford-Smillie-type leafwise critical loci and critical measures vanish. The bounded full-orbit set is the torus image, its maximal-entropy measure is pushed-forward Haar measure, and its Lyapunov exponents are plus or minus log(lambda).\n\nCandidate contribution (explicit_formula_and_critical_measure_lemma; novelty confidence low): Candidate novelty: for the exact word S = s_x s_y s_z on C_4, the two normalized escape-rate functions have the explicit absolute-value formulas in the artifacts, and these formulas imply empty stable and unstable escape leafwise critical loci and hence zero natural critical measures." }, { "id": 20000527, "problem_number": "AIM-ANALYSIS-0059", "title": "Explicit projective-space solution to item 9 of the 2016 complex Monge-Ampere list", "statement": "1. Monge-Amp` ere masses supported by analytic sets (a) Compact K¨ ahler case (e.g., CP n \\ H) [S. Dinew]\n\n• (A good) definition? Possible criteria: 1. If un ↓ and vn ↓, un, v n ∈ L∞ ∩ P SH and lim un = lim vn,then lim M A (un) = lim M A (vn). 2. Take u and un = max( u, −n). Try M A (u) = M A (un) (if it exists).\n\n• Solvability of M A (u) = μ, where μ = measure supported on an analytic set (not a point), Green's function, possibly with prescribed singularities. (b) Similar questions for the Dirichlet problem on Ω ⊂⊂ Cn.(c) Suppose 6 ∃ KE metric on X, with c1(X) > 0: Consider e.g. the continuity method ψt:= φt − sup φt. Show that a subsequence ψt →\n\nψ whose M A (ψ) is defined and M A (ψ) = μ, μ supported in the multiplier ideal sheaf. (e.g., X = CP 2 with one point blown-up.) (d) Real analogues (from the Toric case, for example). (e) X Fano, D smooth anti-canonical divisor. Let ω[U+000F] satisfy Ric( ωt) = [U+000F]ω t + (1 − [U+000F])[ D].\n\nWhat is the limit of (subsequence) ω[U+000F]? In particular, is its Ricci (in a suitable sense) supported on D? [H. Guenancia] 2. Are there non-product solutions of ( dd cu)n = 0 on M compact (e.g., CP 2), where u is smooth and not necessarily PSH? [Y. Rubinstein] 3. Are there solutions of ( dd cu)n = 0 on Cn, where u is PSH? [W. He] 4. Let Ω be a strongly pseudo-convex domain, ∂Ω ∈ C3,1. Consider the problem ( dd cu)n = f, f ≥ 0, f 1\n\n> n\n\n∈ C1,1 and u|∂Ω = φ, φ ∈ C3,1(∂Ω). Find an analytic (independent from Krylov's) proof of u ∈ C1,1(Ω). 15. Same question as above but with f 1\n\n> n−1\n\n∈ C1,1 (in this case, not covered by Krylov.) (Special case known: Ω = Bn, φ = 0: yes by Pli´ s.) 6. Can one construct a counterexample to the maximal rank question from the example of Ross-Witt-Nystrom of solutions of the HCMA without foliation? [M. Paun] 7. Solving ( π∗ω + i∂∂u )n+1 = 0 on X × A, where A is an annulus, and ω\n\nis possibly degenerate. u|X×{ t=1,e } = φ0, φ 1, φ0 and φ1 ω-psh (with some regularity) and ∫\n\n> X\n\nωn > 0. [E. Di Nezza] 8. Find a PDE proof of Kolodziej's L∞ estimate; find optimal constant for a ball. [Z. Blocki] 9. Find X KE Fano and u ∈ Λ1 such that ∫\n\n> X\n\nu3ωnKE 6 = 0. [H. Macbeth] 10. ( dd cu)n = 1 on Ω, ∆ u ∈ Ln(n−1) =⇒ u ∈ C∞? [T. Collins] 11. Complex version of Pogorelov's estimate. 12. Let p: X → D, X K¨ ahler and KX nef. Study solutions of Ric( ωt[U+000F]) =\n\n−ωt[U+000F] − [U+000F]β t on Xt. [M. Paun] 13. Find an analytic proof of the ACC Conjecture/Theorem. [T. Collins] 2", "original_statement": "1. Monge-Amp` ere masses supported by analytic sets (a) Compact K¨ ahler case (e.g., CP n \\ H) [S. Dinew] \n\n• (A good) definition? Possible criteria: 1. If un ↓ and vn ↓, un, v n ∈ L∞ ∩ P SH and lim un = lim vn,then lim M A (un) = lim M A (vn). 2. Take u and un = max( u, −n). Try M A (u) = M A (un) (if it exists). \n\n• Solvability of M A (u) = μ, where μ = measure supported on an analytic set (not a point), Green's function, possibly with prescribed singularities. (b) Similar questions for the Dirichlet problem on Ω ⊂⊂ Cn.(c) Suppose 6 ∃ KE metric on X, with c1(X) > 0: Consider e.g. the continuity method ψt:= φt − sup φt. Show that a subsequence ψt →\n\nψ whose M A (ψ) is defined and M A (ψ) = μ, μ supported in the multiplier ideal sheaf. (e.g., X = CP 2 with one point blown-up.) (d) Real analogues (from the Toric case, for example). (e) X Fano, D smooth anti-canonical divisor. Let ω\u000f satisfy Ric( ωt) = \u000fω t + (1 − \u000f)[ D].\n\nWhat is the limit of (subsequence) ω\u000f? In particular, is its Ricci (in a suitable sense) supported on D? [H. Guenancia] 2. Are there non-product solutions of ( dd cu)n = 0 on M compact (e.g., CP 2), where u is smooth and not necessarily PSH? [Y. Rubinstein] 3. Are there solutions of ( dd cu)n = 0 on Cn, where u is PSH? [W. He] 4. Let Ω be a strongly pseudo-convex domain, ∂Ω ∈ C3,1. Consider the problem ( dd cu)n = f, f ≥ 0, f 1 \n\n> n\n\n∈ C1,1 and u|∂Ω = φ, φ ∈ C3,1(∂Ω). Find an analytic (independent from Krylov's) proof of u ∈ C1,1(Ω). 15. Same question as above but with f 1 \n\n> n−1\n\n∈ C1,1 (in this case, not covered by Krylov.) (Special case known: Ω = Bn, φ = 0: yes by Pli´ s.) 6. Can one construct a counterexample to the maximal rank question from the example of Ross-Witt-Nystrom of solutions of the HCMA without foliation? [M. Paun] 7. Solving ( π∗ω + i∂∂u )n+1 = 0 on X × A, where A is an annulus, and ω\n\nis possibly degenerate. u|X×{ t=1,e } = φ0, φ 1, φ0 and φ1 ω-psh (with some regularity) and ∫ \n\n> X\n\nωn > 0. [E. Di Nezza] 8. Find a PDE proof of Kolodziej's L∞ estimate; find optimal constant for a ball. [Z. Blocki] 9. Find X KE Fano and u ∈ Λ1 such that ∫ \n\n> X\n\nu3ωnKE 6 = 0. [H. Macbeth] 10. ( dd cu)n = 1 on Ω, ∆ u ∈ Ln(n−1) =⇒ u ∈ C∞? [T. Collins] 11. Complex version of Pogorelov's estimate. 12. Let p: X → D, X K¨ ahler and KX nef. Study solutions of Ric( ωt\u000f) = \n\n−ωt\u000f − \u000fβ t on Xt. [M. Paun] 13. Find an analytic proof of the ACC Conjecture/Theorem. [T. Collins] 2", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record is not one problem. It is the entire two-page list *Complex Monge-Ampère Equation Workshop: Open problems*, edited by M. Dellatorre and dated September 8, 2016. The original PDF and the workshop report were checked directly. The extraction merged all thirteen numbered items into record 1. It also turned item 5 into “15.” by adjoining the page-one footer, split the exponents \\(1/n\\) and \\(1/(n-1)\\), inserted a control character in place of \\(\\varepsilon\\), and appended the page number “2” to item 13.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: The complex Monge-Ampere equation\nSection: \nSource item: 1\nSource URL: https://aimath.org/pastworkshops/mongeampereproblems.pdf\nCanonical location: aim-analysis-notes.json notes[58]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Monge-Amp` ere masses supported by analytic sets (a) Compact K¨ ahler case (e.g., CP n \\\\ H) [S. Dinew] \\n\\n• (A good) definition? Possible criteria: 1. If un ↓ and vn ↓, un, v n ∈ L∞ ∩ P SH and lim un = lim vn,then lim M A (un) = lim M A (vn). 2. Take u and un = max( u, −n). Try M A (u) = M A (un) (if it exists). \\n\\n• Solvability of M A (u) = μ, where μ = measure supported on an analytic set (not a point), Green's function, possibly with prescribed singularities. (b) Similar questions for the Dirichlet problem on Ω ⊂⊂ Cn.(c) Suppose 6 ∃ KE metric on X, with c1(X) > 0: Consider e.g. the continuity method ψt:= φt − sup φt. Show that a subsequence ψt →\\n\\nψ whose M A (ψ) is defined and M A (ψ) = μ, μ supported in the multiplier ideal sheaf. (e.g., X = CP 2 with one point blown-up.) (d) Real analogues (from the Toric case, for example). (e) X Fano, D smooth anti-canonical divisor. Let ω\\u000f satisfy Ric( ωt) = \\u000fω t + (1 − \\u000f)[ D].\\n\\nWhat is the limit of (subsequence) ω\\u000f? In particular, is its Ricci (in a suitable sense) supported on D? [H. Guenancia] 2. Are there non-product solutions of ( dd cu)n = 0 on M compact (e.g., CP 2), where u is smooth and not necessarily PSH? [Y. Rubinstein] 3. Are there solutions of ( dd cu)n = 0 on Cn, where u is PSH? [W. He] 4. Let Ω be a strongly pseudo-convex domain, ∂Ω ∈ C3,1. Consider the problem ( dd cu)n = f, f ≥ 0, f 1 \\n\\n> n\\n\\n∈ C1,1 and u|∂Ω = φ, φ ∈ C3,1(∂Ω). Find an analytic (independent from Krylov's) proof of u ∈ C1,1(Ω). 15. Same question as above but with f 1 \\n\\n> n−1\\n\\n∈ C1,1 (in this case, not covered by Krylov.) (Special case known: Ω = Bn, φ = 0: yes by Pli´ s.) 6. Can one construct a counterexample to the maximal rank question from the example of Ross-Witt-Nystrom of solutions of the HCMA without foliation? [M. Paun] 7. Solving ( π∗ω + i∂∂u )n+1 = 0 on X × A, where A is an annulus, and ω\\n\\nis possibly degenerate. u|X×{ t=1,e } = φ0, φ 1, φ0 and φ1 ω-psh (with some regularity) and ∫ \\n\\n> X\\n\\nωn > 0. [E. Di Nezza] 8. Find a PDE proof of Kolodziej's L∞ estimate; find optimal constant for a ball. [Z. Blocki] 9. Find X KE Fano and u ∈ Λ1 such that ∫ \\n\\n> X\\n\\nu3ωnKE 6 = 0. [H. Macbeth] 10. ( dd cu)n = 1 on Ω, ∆ u ∈ Ln(n−1) =⇒ u ∈ C∞? [T. Collins] 11. Complex version of Pogorelov's estimate. 12. Let p: X → D, X K¨ ahler and KX nef. Study solutions of Ric( ωt\\u000f) = \\n\\n−ωt\\u000f − \\u000fβ t on Xt. [M. Paun] 13. Find an analytic proof of the ACC Conjecture/Theorem. [T. Collins] 2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/mongeampereproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0059", "aim-domain:analysis", "aim-workshop:mongeampereproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The canonical record is the full thirteen-item 2016 AIM workshop list. Under the standard convention that Lambda_1 is the first positive complex-Laplacian eigenspace for Ric(omega)=omega, item 9 has an explicit solution on every complex projective space CP^m with m at least 2: the centered moment-map coordinate u=|Z_0|^2/sum_j|Z_j|^2-1/(m+1) satisfies Delta u=u and has normalized cubic moment 2m(m-1)/((m+1)^3(m+2)(m+3)), which is positive. The report also proves a divisorial truncation formula relevant to item 1 and a proper-exhaustion obstruction relevant to item 3, and gives a conservative status map for all thirteen items.\n\nCandidate contribution (explicit_family; novelty confidence low): For every m at least 2, the centered first-eigenfunction on CP^m above has normalized cubic moment 2m(m-1)/((m+1)^3(m+2)(m+3)); in particular CP^2 gives 1/135." }, { "id": 20000528, "problem_number": "AIM-ANALYSIS-0060", "title": "Finite-range reduction and measurable affine patching for a one-variable vector-field maximal operator", "statement": "Let \\(u:\\mathbb R\\to\\mathbb R\\) be a measurable function. Define the maximal operator along the planar vector field \\((1,u)\\) by\n\\[\nM_u f(x,y):=\\sup_{\\epsilon>0}\\left|\n\\frac{1}{2\\epsilon}\\int_{-\\epsilon}^{\\epsilon}\nf(x-t,y-u(x)t)\\,dt\n\\right|.\n\\]\nDoes \\(M_u\\) satisfy any \\(L^p\\) bound for certain \\(p<\\infty\\)?", "original_statement": "Question 1 (Thiele). Let u: R → R be a measurable function. Define the maximal operator along the planar vector field (1, u ) by \n\nMuf (x, y ):= sup \n\n> \u000f> 0\n\n∣∣∣∣\n\n12\u000f\n\n∫ \u000f\n\n> −\u000f\n\nf (x − t, y − u(x)t)dt \n\n∣∣∣∣. (0.1) \n\nDoes Mu satisfy any Lp bound for certain p < ∞?", "clean_statement": "Let \\(u:\\mathbb R\\to\\mathbb R\\) be a measurable function. Define the maximal operator along the planar vector field \\((1,u)\\) by\n\\[\nM_u f(x,y):=\\sup_{\\epsilon>0}\\left|\n\\frac{1}{2\\epsilon}\\int_{-\\epsilon}^{\\epsilon}\nf(x-t,y-u(x)t)\\,dt\n\\right|.\n\\]\nDoes \\(M_u\\) satisfy any \\(L^p\\) bound for certain \\(p<\\infty\\)?", "statement_status": "corrected_verified", "statement_verification": "Here “the vector field \\((1,u)\\)” means \\(v(x,y)=(1,u(x))\\). The following repairs were made to the corpus OCR, and all were checked against the displayed formula in the PDF: 1. Each control character `\\u000f` is the glyph \\(\\epsilon\\). 2. The broken string `12\\u000f` is the fraction \\(1/(2\\epsilon)\\). 3. The integral limits are \\(-\\epsilon\\) and \\(\\epsilon\\). 4. The vertical bars enclose the whole signed average; the source does not put \\(|f|\\) inside the integral. 5. Fragmented line breaks and the superscripts in \\(\\mathbb R\\) and \\(L^p\\) were restored.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 1\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[59]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 1 (Thiele). Let u: R → R be a measurable function. Define the maximal operator along the planar vector field (1, u ) by \\n\\nMuf (x, y ):= sup \\n\\n> \\u000f> 0\\n\\n∣∣∣∣\\n\\n12\\u000f\\n\\n∫ \\u000f\\n\\n> −\\u000f\\n\\nf (x − t, y − u(x)t)dt \\n\\n∣∣∣∣. (0.1) \\n\\nDoes Mu satisfy any Lp bound for certain p < ∞?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0060", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every 1 < p < infinity, the optimal uniform L^p test-function constant over all measurable slopes u equals the corresponding supremum over finite rational-valued slopes. In addition, if u agrees with N affine functions on an arbitrary measurable partition of the real line, then the full-scale operator satisfies ||M_u f||_p <= C_p N^(1/p) ||f||_p, uniformly in the partition and affine coefficients. The proof uses an exact measure-preserving shear from each affine branch to a polynomial parabolic maximal average. Strong L^p boundedness fails for 0 < p <= 1 already when u = 0.\n\nCandidate contribution (reduction_and_partial_theorem; novelty confidence low): Candidate novelty: the uniform arbitrary-measurable problem is exactly reducible to finite rational-valued slope fields with a bound independent of finite slope complexity; combined with an exact affine-to-parabola conjugacy, this yields a C_p N^(1/p) bound for arbitrary measurable pasting of N affine branches." }, { "id": 20000529, "problem_number": "AIM-ANALYSIS-0061", "title": "Fourier norms of indicator functions and a two-interval splitting benchmark", "statement": "Question 2 (Christ). On Rd, let E ⊂ Rd with |E| = 1. Given q > 2,describe the set E that maximises the quantity ‖̂1E ‖q.\n\nCertain partial progress has been made.\n\nTheorem 0.1 (Christ [3]). 1. For any q > 2, the extremizing set exists. 2. For any dimension d, for any sufficiently large q which is also sufficiently close to 2N, a set E is a extremizer iff E is an ellipsoid. 3. If d = 1, then for any q close to 2N, a set E is a extremizer iff E is an ellipsoid. 4. If d = 2, then for any q close to 4, a set E is a extremizer iff E is an ellipsoid.", "original_statement": "Question 2 (Christ). On Rd, let E ⊂ Rd with |E| = 1. Given q > 2,describe the set E that maximises the quantity ‖̂1E ‖q.\n\nCertain partial progress has been made. \n\nTheorem 0.1 (Christ [3]). 1. For any q > 2, the extremizing set exists. 2. For any dimension d, for any sufficiently large q which is also sufficiently close to 2N, a set E is a extremizer iff E is an ellipsoid. 3. If d = 1, then for any q close to 2N, a set E is a extremizer iff E is an ellipsoid. 4. If d = 2, then for any q close to 4, a set E is a extremizer iff E is an ellipsoid.", "clean_statement": "Question 2 (Christ). On Rd, let E ⊂ Rd with |E| = 1. Given q > 2,describe the set E that maximises the quantity ‖̂1E ‖q.\n\nCertain partial progress has been made.\n\nTheorem 0.1 (Christ [3]). 1. For any q > 2, the extremizing set exists. 2. For any dimension d, for any sufficiently large q which is also sufficiently close to 2N, a set E is a extremizer iff E is an ellipsoid. 3. If d = 1, then for any q close to 2N, a set E is a extremizer iff E is an ellipsoid. 4. If d = 2, then for any q close to 4, a set E is a extremizer iff E is an ellipsoid.", "statement_status": "exact", "statement_verification": "This record is Question 2 from the AIM workshop problem list *Carleson theorems and multilinear operators*. The PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 2\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[60]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 2 (Christ). On Rd, let E ⊂ Rd with |E| = 1. Given q > 2,describe the set E that maximises the quantity ‖̂1E ‖q.\\n\\nCertain partial progress has been made. \\n\\nTheorem 0.1 (Christ [3]). 1. For any q > 2, the extremizing set exists. 2. For any dimension d, for any sufficiently large q which is also sufficiently close to 2N, a set E is a extremizer iff E is an ellipsoid. 3. If d = 1, then for any q close to 2N, a set E is a extremizer iff E is an ellipsoid. 4. If d = 2, then for any q close to 4, a set E is a extremizer iff E is an ellipsoid.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0061", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the one-dimensional unit-measure family E_g formed by two intervals of length one half separated by a gap g, the exact fourth-power Fourier norm is 1/2 + (4/3)(1/2-g)_+^3, and its deficit from the unit interval is delta/2 - delta^2/2 + delta^3/6 where delta = min(2g,1) is the distance to unit intervals in symmetric difference. For two half-intervals whose separation tends to infinity, the ratio of q-th powers to the unit interval tends to c_q = 2 Gamma((q+1)/2)/(sqrt(pi) Gamma((q+2)/2)) < 1 for every q>2, with c_q = 1 - (log 2 - 1/2)(q-2) + O((q-2)^2) near q=2. These are rigorous special-family results; the global classification remains open away from the known neighborhoods of even exponents.\n\nCandidate contribution (explicit_family_and_quantitative_formula; novelty confidence low): Candidate novelty: the exact cubic q=4 gap deficit as a polynomial in symmetric-difference distance, together with the general-q gamma-function splitting constant and its first-order degeneration at q=2, provides a concrete and fully proved dichotomy benchmark for the equal two-interval family." }, { "id": 20000530, "problem_number": "AIM-ANALYSIS-0062", "title": "Sharp decay for bounded-degree chirps along null directions", "statement": "Question 3 (Christ). Let B be the unit ball in R3. Let N be a positive integer. Let {Vj: 1 ≤ j ≤ N } be N different light cones in R3. Prove that\n\n∣∣∣∣∣∣∫\n\n> B\n\neiλx 23\n\n> N\n\n∏\n\n> j=1\n\nfj (x · vj )dx\n\n∣∣∣∣∣∣. λ−[U+000F]N∏\n\n> j=1\n\n‖fj ‖∞, (0.2)\n\nfor certain positive [U+000F], where vj ∈ Vj. If possible, find the optimal [U+000F].\n\n1So far (0.2) has only been proved for N ≤ 5, see [4].", "original_statement": "Question 3 (Christ). Let B be the unit ball in R3. Let N be a positive integer. Let {Vj: 1 ≤ j ≤ N } be N different light cones in R3. Prove that \n\n∣∣∣∣∣∣∫\n\n> B\n\neiλx 23\n\n> N\n\n∏\n\n> j=1\n\nfj (x · vj )dx \n\n∣∣∣∣∣∣. λ−\u000fN∏\n\n> j=1\n\n‖fj ‖∞, (0.2) \n\nfor certain positive \u000f, where vj ∈ Vj. If possible, find the optimal \u000f.\n\n1So far (0.2) has only been proved for N ≤ 5, see [4].", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Question 3 (attributed to Michael Christ) in the AIM open-problem list *Carleson theorems and multilinear operators*. The extraction damage can be repaired directly from the official PDF and its TeX source: the phase is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 3\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[61]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 3 (Christ). Let B be the unit ball in R3. Let N be a positive integer. Let {Vj: 1 ≤ j ≤ N } be N different light cones in R3. Prove that \\n\\n∣∣∣∣∣∣∫\\n\\n> B\\n\\neiλx 23\\n\\n> N\\n\\n∏\\n\\n> j=1\\n\\nfj (x · vj )dx \\n\\n∣∣∣∣∣∣. λ−\\u000fN∏\\n\\n> j=1\\n\\n‖fj ‖∞, (0.2) \\n\\nfor certain positive \\u000f, where vj ∈ Vj. If possible, find the optimal \\u000f.\\n\\n1So far (0.2) has only been proved for N ≤ 5, see [4].\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0062", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After reconstructing the intended data as distinct lines spanned by light-cone vectors, the attempt proves an exact tensor criterion for quadratic phase cancellation and a sharp structured-input theorem: for arbitrary N null directions and arbitrary real polynomial chirps of any fixed degree D, the integral over the unit ball is bounded by C_D |lambda|^{-1/2}, uniformly in N, the directions, and all polynomial coefficients. The same estimate holds for finite-measure mixtures in the product total-variation norm, and exponent 1/2 is optimal by stationary phase with all inputs equal to one.\n\nCandidate contribution (theorem; novelty confidence low): For every fixed degree D and every finite collection of null directions in R^3, polynomial-chirp ridge inputs and their finite-measure mixtures satisfy the sharp |lambda|^{-1/2} decay estimate with a constant independent of the number and placement of directions and of all polynomial coefficients." }, { "id": 20000531, "problem_number": "AIM-ANALYSIS-0063", "title": "An unequal-cardinality trilinear Kakeya estimate in four dimensions", "statement": "Question 4 (Bennett). Suppose we are in R4. Let [U+000F] > 0. Suppose that T1,\n\nT2 and T3 are three transversal families of δ-tubes (short sides δ and long side 1) such that for each j ∈ { 1, 2, 3}, {e(Tj ): Tj ∈ Tj } forms a δ-separated subset of S3. If q ≥ 43 and 1\n\n> p\n\n+ 3\n\n> q\n\n≤ 3, then there exists a constant C[U+000F] > 0\n\nsuch that\n\n∥∥∥∥∥∥\n\n> 3\n\n∏\n\n> j=1\n\n ∑\n\n> Tj∈Tj\n\nχTj\n\n∥∥∥∥∥∥Lq/ 3(R4)\n\n≤ C[U+000F]\n\n> 3\n\n∏\n\n> j=1\n\nδ 4\n\n> q−3\n> p′−[U+000F]\n\n(Tj )1/p. (0.3)\n\nHere e(T ) ∈ S3 denotes the direction of the long side of a tube T.", "original_statement": "Question 4 (Bennett). Suppose we are in R4. Let \u000f > 0. Suppose that T1,\n\nT2 and T3 are three transversal families of δ-tubes (short sides δ and long side 1) such that for each j ∈ { 1, 2, 3}, {e(Tj ): Tj ∈ Tj } forms a δ-separated subset of S3. If q ≥ 43 and 1 \n\n> p\n\n+ 3 \n\n> q\n\n≤ 3, then there exists a constant C\u000f > 0\n\nsuch that \n\n∥∥∥∥∥∥\n\n> 3\n\n∏\n\n> j=1\n\n ∑ \n\n> Tj∈Tj\n\nχTj\n\n∥∥∥∥∥∥Lq/ 3(R4)\n\n≤ C\u000f\n\n> 3\n\n∏\n\n> j=1\n\nδ 4 \n\n> q−3\n> p′−\u000f\n\n(Tj )1/p. (0.3) \n\nHere e(T ) ∈ S3 denotes the direction of the long side of a tube T.", "clean_statement": "Suppose that we are in \\(\\mathbb R^4\\), and let \\(\\varepsilon>0\\). Suppose that \\(\\mathbb T_1,\\mathbb T_2,\\mathbb T_3\\) are three transversal families of \\(\\delta\\)-tubes (three short sides of length \\(\\delta\\) and one long side of length \\(1\\)) such that, for each \\(j\\in\\{1,2,3\\}\\),\n\\[\n\\{e(T):T\\in\\mathbb T_j\\}\n\\]\nis a \\(\\delta\\)-separated subset of \\(S^3\\). If\n\\[\nq\\geq \\frac43,\n\\qquad\n\\frac1p+\\frac3q\\leq 3,\n\\]\nthen is there a constant \\(C_\\varepsilon>0\\) such that\n\\[\n\\left\\|\n\\prod_{j=1}^{3}\n\\left(\\sum_{T\\in\\mathbb T_j}\\chi_T\\right)\n\\right\\|_{L^{q/3}(\\mathbb R^4)}\n\\leq C_\\varepsilon\n\\prod_{j=1}^{3}\n\\left[\n\\delta^{\\,4/q-3/p'-\\varepsilon}\n(\\#\\mathbb T_j)^{1/p}\n\\right]?\n\\tag{AIM}\n\\]\nHere \\(e(T)\\in S^3\\) is the direction of the long side of \\(T\\), and \\(p'\\) is the conjugate exponent.", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Question 4 (Bennett) in the AIM workshop list *Carleson theorems and multilinear operators*. The JSON extraction has several consequential OCR errors: “43” is \\(4/3\\), the control character is \\(\\varepsilon\\), and line breaks obscure both the admissibility condition and the location of the product. Inspection of page 2 of the authoritative AIM PDF gives the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 4\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[62]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 4 (Bennett). Suppose we are in R4. Let \\u000f > 0. Suppose that T1,\\n\\nT2 and T3 are three transversal families of δ-tubes (short sides δ and long side 1) such that for each j ∈ { 1, 2, 3}, {e(Tj ): Tj ∈ Tj } forms a δ-separated subset of S3. If q ≥ 43 and 1 \\n\\n> p\\n\\n+ 3 \\n\\n> q\\n\\n≤ 3, then there exists a constant C\\u000f > 0\\n\\nsuch that \\n\\n∥∥∥∥∥∥\\n\\n> 3\\n\\n∏\\n\\n> j=1\\n\\n ∑ \\n\\n> Tj∈Tj\\n\\nχTj\\n\\n∥∥∥∥∥∥Lq/ 3(R4)\\n\\n≤ C\\u000f\\n\\n> 3\\n\\n∏\\n\\n> j=1\\n\\nδ 4 \\n\\n> q−3\\n> p′−\\u000f\\n\\n(Tj )1/p. (0.3) \\n\\nHere e(T ) ∈ S3 denotes the direction of the long side of a tube T.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0063", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For three uniformly transverse direction-separated families of delta-tubes in R^4, with 0 < N_1 <= N_2 <= N_3, integer replication in the Guth-Zahl polynomial-Wolff estimate proves ||F_1 F_2 F_3||_{L^{13/27}} <= C delta^{99/13-eta} N_1^{10/13} N_2 N_3, equivalently the conjectured endpoint cardinality powers times (N_2 N_3/N_1^2)^{1/13}. Interpolation with the Bennett-Carbery-Tao epsilon endpoint gives the explicit imbalance power theta/13 for 13/9 <= q <= 3/2 and proves the requested range q >= 13/9 for comparable family sizes; two affine-foliation constructions rigorously recover both necessary conditions q >= 4/3 and 1/p + 3/q <= 3.\n\nCandidate contribution (theorem; novelty confidence low): The explicit unequal-cardinality Guth-Zahl corollary has imbalance loss (N_2 N_3/N_1^2)^{1/13} at (p,q)=(13/12,13/9), and loss exponent theta/13 after interpolation, obtained by exact integer multiplicities whose common scale cancels against the K^{1/9} polynomial-Wolff dependence." }, { "id": 20000532, "problem_number": "AIM-ANALYSIS-0064", "title": "A hybrid spectral-complexity reduction for the maximal directional Hilbert transform", "statement": "Question 5 (Di Plinio). Let N ∈ N. Given a collection of N directions\n\n{vj ∈ S1: 1 ≤ j ≤ N }. Does it hold true that\n\n‖ sup\n\n> j∈{ 1,2,...,N }\n\n|Hvj f |‖ L2,∞(R2). √log N ‖f ‖2? (0.4)\n\nHere Hvj f denotes the Hilbert transform along the given direction vj, namely\n\nHvj f (x):=\n\n∫\n\n> R\n\nf (x − tv j ) dt t. (0.5) The maximal variant of the estimate (0.4) has been proved by Katz [8]. Moreover, (0.4) has also been verified for sets of two \"extreme\" structures: the lacunary set and the Vargas set. One typical example of the Vargas set is the set of uniformly distributed directions. See Demeter [5], Demeter and Di Plinio [6].", "original_statement": "Question 5 (Di Plinio). Let N ∈ N. Given a collection of N directions \n\n{vj ∈ S1: 1 ≤ j ≤ N }. Does it hold true that \n\n‖ sup \n\n> j∈{ 1,2,...,N }\n\n|Hvj f |‖ L2,∞(R2). √log N ‖f ‖2? (0.4) \n\nHere Hvj f denotes the Hilbert transform along the given direction vj, namely \n\nHvj f (x):= \n\n∫\n\n> R\n\nf (x − tv j ) dt t. (0.5) The maximal variant of the estimate (0.4) has been proved by Katz [8]. Moreover, (0.4) has also been verified for sets of two \"extreme\" structures: the lacunary set and the Vargas set. One typical example of the Vargas set is the set of uniformly distributed directions. See Demeter [5], Demeter and Di Plinio [6].", "clean_statement": "Question 5 (Di Plinio). Let N ∈ N. Given a collection of N directions\n\n{vj ∈ S1: 1 ≤ j ≤ N }. Does it hold true that\n\n‖ sup\n\n> j∈{ 1,2,...,N }\n\n|Hvj f |‖ L2,∞(R2). √log N ‖f ‖2? (0.4)\n\nHere Hvj f denotes the Hilbert transform along the given direction vj, namely\n\nHvj f (x):=\n\n∫\n\n> R\n\nf (x − tv j ) dt t. (0.5) The maximal variant of the estimate (0.4) has been proved by Katz [8]. Moreover, (0.4) has also been verified for sets of two \"extreme\" structures: the lacunary set and the Vargas set. One typical example of the Vargas set is the set of uniformly distributed directions. See Demeter [5], Demeter and Di Plinio [6].", "statement_status": "exact", "statement_verification": "The canonical record is Question 5 (Di Plinio) in *Carleson theorems and multilinear operators: Open problems*, p. 2 of the AIM PDF. The authoritative display is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 5\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[63]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 5 (Di Plinio). Let N ∈ N. Given a collection of N directions \\n\\n{vj ∈ S1: 1 ≤ j ≤ N }. Does it hold true that \\n\\n‖ sup \\n\\n> j∈{ 1,2,...,N }\\n\\n|Hvj f |‖ L2,∞(R2). √log N ‖f ‖2? (0.4) \\n\\nHere Hvj f denotes the Hilbert transform along the given direction vj, namely \\n\\nHvj f (x):= \\n\\n∫\\n\\n> R\\n\\nf (x − tv j ) dt t. (0.5) The maximal variant of the estimate (0.4) has been proved by Katz [8]. Moreover, (0.4) has also been verified for sets of two \\\"extreme\\\" structures: the lacunary set and the Vargas set. One typical example of the Vargas set is the set of uniformly distributed directions. See Demeter [5], Demeter and Di Plinio [6].\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0064", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite planar direction set with M distinct projective directions, if the Fourier support of an input lies in L dyadic radial annuli and K angular sign cells determined by the directions, then the maximal directional Hilbert transform has weak-L2 norm at most C min(sqrt(L), log(K+1)) times the input L2 norm. More generally, projecting an arbitrary input onto such a region gives the same term for the projection plus C log(M+1) times the L2 norm of the Fourier-energy remainder. Hence a spectrally finite input attaining the conjectured sqrt(log(1+M)) scale must use at least order log(1+M) radial bands and exponentially many in sqrt(log(1+M)) angular cells.\n\nCandidate contribution (theorem/reduction; novelty confidence low): Candidate novelty: the proved stable hybrid radial-angular spectral-complexity estimate and its explicit two-axis obstruction for finite-complexity near-extremizers." }, { "id": 20000533, "problem_number": "AIM-ANALYSIS-0065", "title": "Sharp bounds and an elliptic-radial nullspace for the determinantal trilinear form", "statement": "Question 6 (Street). Prove\n\n∣∣∣∣∫\n\n> R2\n\n∫\n\n> R2\n\nf (x)g(y)h(x + y) 1\n\ndet (x, y ) dxdy\n\n∣∣∣∣. ‖f ‖p‖g‖q‖h‖r, (0.6)\n\nfor certain p, q and r.\n\nThis question has a quite satisfactory answer. See Gressman et al. [7]. 2", "original_statement": "Question 6 (Street). Prove \n\n∣∣∣∣∫\n\n> R2\n\n∫\n\n> R2\n\nf (x)g(y)h(x + y) 1\n\ndet (x, y ) dxdy \n\n∣∣∣∣. ‖f ‖p‖g‖q‖h‖r, (0.6) \n\nfor certain p, q and r.\n\nThis question has a quite satisfactory answer. See Gressman et al. [7]. 2", "clean_statement": "Question 6 (Street). Prove\n\n∣∣∣∣∫\n\n> R2\n\n∫\n\n> R2\n\nf (x)g(y)h(x + y) 1\n\ndet (x, y ) dxdy\n\n∣∣∣∣. ‖f ‖p‖g‖q‖h‖r, (0.6)\n\nfor certain p, q and r.\n\nThis question has a quite satisfactory answer. See Gressman et al. [7]. 2", "statement_status": "exact", "statement_verification": "The canonical record is Question 6 (Street) from the AIM workshop *Carleson theorems and multilinear operators*. The official AIM TeX reads", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 6\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[64]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 6 (Street). Prove \\n\\n∣∣∣∣∫\\n\\n> R2\\n\\n∫\\n\\n> R2\\n\\nf (x)g(y)h(x + y) 1\\n\\ndet (x, y ) dxdy \\n\\n∣∣∣∣. ‖f ‖p‖g‖q‖h‖r, (0.6) \\n\\nfor certain p, q and r.\\n\\nThis question has a quite satisfactory answer. See Gressman et al. [7]. 2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0065", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The intended determinant-symmetric principal value is exactly three times the Gressman--He--Kovač--Street--Thiele--Yung form after reflecting the third input. Their published theorem gives the sharp Banach range 20, the truncated form T_epsilon(f,g,h) vanishes identically if any two inputs are Q-radial; consequently, in the sharp exponent range, Q-radialization yields an explicit bound by the angular defects of either selected pair of inputs." }, { "id": 20000534, "problem_number": "AIM-ANALYSIS-0066", "title": "A transfer principle and the full Banach range for lacunary rotation-coupled circle averages", "statement": "Question 7 (Krause). On the plane R2, let R: S1 → S1 be the rotation by\n\nπ/ 3. Prove\n\n∥∥∥∥ sup\n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x − tR (w)) dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q, (0.7)\n\nfor certain p, q and r.\n\nDuring the workshop, this has been shown to be equivalent to\n\n∥∥∥∥ sup\n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x + tw )dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q. (0.8)", "original_statement": "Question 7 (Krause). On the plane R2, let R: S1 → S1 be the rotation by \n\nπ/ 3. Prove \n\n∥∥∥∥ sup \n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x − tR (w)) dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q, (0.7) \n\nfor certain p, q and r.\n\nDuring the workshop, this has been shown to be equivalent to \n\n∥∥∥∥ sup \n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x + tw )dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q. (0.8)", "clean_statement": "Question 7 (Krause). On the plane R2, let R: S1 → S1 be the rotation by\n\nπ/ 3. Prove\n\n∥∥∥∥ sup\n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x − tR (w)) dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q, (0.7)\n\nfor certain p, q and r.\n\nDuring the workshop, this has been shown to be equivalent to\n\n∥∥∥∥ sup\n\n> t∈R+\n\n∣∣∣∣∫\n\n> S1\n\nf (x − tw )g(x + tw )dσ (w)\n\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q. (0.8)", "statement_status": "exact", "statement_verification": "The record is Question 7 (attributed to Krause) from the AIM workshop *Carleson theorems and multilinear operators*. The OCR in `input.json` breaks several displayed formulas, so the statement was checked against the official AIM TeX source as well as the PDF linked in the record. The TeX source reads as follows (with only notation typeset more compactly here):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 7\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[65]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 7 (Krause). On the plane R2, let R: S1 → S1 be the rotation by \\n\\nπ/ 3. Prove \\n\\n∥∥∥∥ sup \\n\\n> t∈R+\\n\\n∣∣∣∣∫\\n\\n> S1\\n\\nf (x − tw )g(x − tR (w)) dσ (w)\\n\\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q, (0.7) \\n\\nfor certain p, q and r.\\n\\nDuring the workshop, this has been shown to be equivalent to \\n\\n∥∥∥∥ sup \\n\\n> t∈R+\\n\\n∣∣∣∣∫\\n\\n> S1\\n\\nf (x − tw )g(x + tw )dσ (w)\\n\\n∣∣∣∣∥∥∥∥r. ‖f ‖p‖g‖q. (0.8)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0066", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any radius set E, an adjustable-exponent Holder inequality pointwise bounds the rotation-coupled bilinear maximal operator by a product of two ordinary circular maximal operators. This recovers the known continuous-radius bounds on the Holder line for output r>2 and proves that the dyadic-radius AIM operator, uniformly for every rotation including pi/3 and pi, maps L^p(R^2) x L^q(R^2) to L^r(R^2) whenever 1/r=1/p+1/q<1. The continuous-radius range p,q>2 and 11, presented through a linear-threshold transfer packaging; the elementary underlying Holder step itself is not claimed novel." }, { "id": 20000535, "problem_number": "AIM-ANALYSIS-0067", "title": "The solved discrete quadratic Carleson problem and a sharp-scale finite sampling reduction", "statement": "Question 8 (Anderson, Pierce). Generalise Stein and Wainger's polynomial Carleson's theorem to the discrete setting, namely to prove\n\n∥∥∥∥∥sup\n\n> λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.9) Let Λ ⊂ [0, 1]. Define sup\n\n> λ∈Λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣. (0.10) A sufficient condition has been given on the set Λ, to guarantee the l2\n\nboundedness of (0.10). See Krause and Lacey [9].", "original_statement": "Question 8 (Anderson, Pierce). Generalise Stein and Wainger's polynomial Carleson's theorem to the discrete setting, namely to prove \n\n∥∥∥∥∥sup \n\n> λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.9) Let Λ ⊂ [0, 1]. Define sup \n\n> λ∈Λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣. (0.10) A sufficient condition has been given on the set Λ, to guarantee the l2\n\nboundedness of (0.10). See Krause and Lacey [9].", "clean_statement": "Question 8 (Anderson, Pierce). Generalise Stein and Wainger's polynomial Carleson's theorem to the discrete setting, namely to prove\n\n∥∥∥∥∥sup\n\n> λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.9) Let Λ ⊂ [0, 1]. Define sup\n\n> λ∈Λ\n\n∣∣∣∣∣∑\n\n> m∈Z\n\nf (n − m) eiλm 2\n\nm\n\n∣∣∣∣∣. (0.10) A sufficient condition has been given on the set Λ, to guarantee the l2\n\nboundedness of (0.10). See Krause and Lacey [9].", "statement_status": "exact", "statement_verification": "The canonical record is Question 8 (Anderson, Pierce) in *Carleson theorems and multilinear operators: Open problems*, p. 3 of the authoritative AIM PDF. The PDF asks for the discrete analogue of Stein--Wainger's polynomial Carleson theorem:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 8\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[66]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 8 (Anderson, Pierce). Generalise Stein and Wainger's polynomial Carleson's theorem to the discrete setting, namely to prove \\n\\n∥∥∥∥∥sup \\n\\n> λ\\n\\n∣∣∣∣∣∑\\n\\n> m∈Z\\n\\nf (n − m) eiλm 2\\n\\nm\\n\\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.9) Let Λ ⊂ [0, 1]. Define sup \\n\\n> λ∈Λ\\n\\n∣∣∣∣∣∑\\n\\n> m∈Z\\n\\nf (n − m) eiλm 2\\n\\nm\\n\\n∣∣∣∣∣. (0.10) A sufficient condition has been given on the set Λ, to guarantee the l2\\n\\nboundedness of (0.10). See Krause and Lacey [9].\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0067", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM discrete quadratic Carleson estimate is solved in the published literature and now follows from a stronger maximally truncated theorem for every 1 N\n\n∑\n\n> n=1\n\nane2πin dte2πin ·x‖Lp(T2). N 12 − d+1\n\n> p+[U+000F]\n\n(\n\n> N\n\n∑\n\n> n=1\n\n|an|2)1/2. (0.11) This is related to Waring's problem.", "original_statement": "Question 9 (Li). Let d ≥ 3. For p ≥ 2( d + 1), prove \n\n‖\n\n> N\n\n∑\n\n> n=1\n\nane2πin dte2πin ·x‖Lp(T2). N 12 − d+1 \n\n> p+\u000f\n\n(\n\n> N\n\n∑\n\n> n=1\n\n|an|2)1/2. (0.11) This is related to Waring's problem.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON extraction is visibly corrupted: superscripts, the exponent of \\(N\\), the coefficient subscript, and the two torus variables have been split by PDF extraction. The original AIM workshop PDF was checked. Its Question 9 is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 9\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[67]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 9 (Li). Let d ≥ 3. For p ≥ 2( d + 1), prove \\n\\n‖\\n\\n> N\\n\\n∑\\n\\n> n=1\\n\\nane2πin dte2πin ·x‖Lp(T2). N 12 − d+1 \\n\\n> p+\\u000f\\n\\n(\\n\\n> N\\n\\n∑\\n\\n> n=1\\n\\n|an|2)1/2. (0.11) This is related to Waring's problem.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0068", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The full deterministic estimate remains open in part. For s=d+1 and arbitrary coefficients b_n, independent uniform phase rotations by any q-th roots of unity with q>s satisfy E||sum omega_n b_n e(nx+n^d t)||_{2s}^{2s} <= s!||b||_2^{2s}. Thus a deterministic choice from the finite alphabet of (d+2)-nd roots gives the critical estimate with no N^epsilon loss, and outside a fraction lambda^{-2s} of choices the same phase vector gives the conjectured N^{1/2-s/p} power simultaneously for every p>=2s. The attempt also proves the exact endpoint convolution identity and computes the generic full-moment-curve trace loss N^{(d-2)/4}.\n\nCandidate contribution (theorem; novelty confidence low): For every d>=3, N, and prescribed complex coefficient vector b, averaging coefficientwise rotations over the finite alphabet of (d+2)-nd roots of unity yields the exact critical 2(d+1)-moment bound with constant (d+1)!, and one good phase vector simultaneously obeys the conjectured high-p scaling for every real p>=2(d+1)." }, { "id": 20000537, "problem_number": "AIM-ANALYSIS-0069", "title": "Near-sharp trilinear convolution range on transverse patches of the 3-sphere", "statement": "Question 10 (Bez). Let S1, S 2 and S3 be transversal patches of the unit sphere S3 in R4. Let σ1, σ 2 and σ3 be the surface measure separately. Deter-mine the full range of exponents p, q > 0 such that the multi-linear singular convolution estimate\n\n‖g1dσ 1 ∗ g2dσ 2 ∗ g3dσ 3‖q.\n\n> 3\n\n∏\n\n> j=1\n\n‖gj ‖p (0.12)\n\nholds.\n\n3", "original_statement": "Question 10 (Bez). Let S1, S 2 and S3 be transversal patches of the unit sphere S3 in R4. Let σ1, σ 2 and σ3 be the surface measure separately. Deter-mine the full range of exponents p, q > 0 such that the multi-linear singular convolution estimate \n\n‖g1dσ 1 ∗ g2dσ 2 ∗ g3dσ 3‖q.\n\n> 3\n\n∏\n\n> j=1\n\n‖gj ‖p (0.12) \n\nholds. \n\n3", "clean_statement": "Question 10 (Bez). Let S1, S 2 and S3 be transversal patches of the unit sphere S3 in R4. Let σ1, σ 2 and σ3 be the surface measure separately. Deter-mine the full range of exponents p, q > 0 such that the multi-linear singular convolution estimate\n\n‖g1dσ 1 ∗ g2dσ 2 ∗ g3dσ 3‖q.\n\n> 3\n\n∏\n\n> j=1\n\n‖gj ‖p (0.12)\n\nholds.\n\n3", "statement_status": "exact", "statement_verification": "The canonical JSON record is visibly damaged by PDF extraction. I therefore checked the official AIM TeX source as well as the linked PDF. The source reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 10\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[68]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 10 (Bez). Let S1, S 2 and S3 be transversal patches of the unit sphere S3 in R4. Let σ1, σ 2 and σ3 be the surface measure separately. Deter-mine the full range of exponents p, q > 0 such that the multi-linear singular convolution estimate \\n\\n‖g1dσ 1 ∗ g2dσ 2 ∗ g3dσ 3‖q.\\n\\n> 3\\n\\n∏\\n\\n> j=1\\n\\n‖gj ‖p (0.12) \\n\\nholds. \\n\\n3\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0069", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For quantitatively transverse compact patches of the unit sphere S^3 in R^4, the convolution estimate holds exactly for p >= 1 when 0 < q <= 1. When 1 < q <= infinity, it holds throughout the strict region 15/p < 8 + 7/q and fails throughout the exterior region 15/p > 8 + 7/q. Thus the only unresolved exponents are on the strong-type boundary 1 < q <= infinity, 15/p = 8 + 7/q. Sufficiency follows from Bejenaru's 2022 sharp-up-to-endpoint trilinear restriction theorem, an explicit characteristic-function-to-Lorentz argument, and interpolation; necessity follows from a one-cap test and a squashed-cap construction.\n\nCandidate contribution (reduction; novelty confidence low): Combining Bejenaru's k = 3, n = 4 restriction theorem at every s > 14/15 with the proved characteristic-function Fourier L1 estimate and factorized Lorentz summation yields the full strict convolution region 15/p < 8 + 7/q; the matching squashed-cap obstruction reduces the AIM problem to only the boundary 15/p = 8 + 7/q for q > 1." }, { "id": 20000538, "problem_number": "AIM-ANALYSIS-0070", "title": "A diagonal-trace reduction for a bilinear polynomial Carleson operator", "statement": "Question 11 (Muscalu). Let K: R2 → R be a function such that\n\n|∂α ˆK(ξ)|. 1\n\n|ξ||α|, ∀ξ ∈ R2 \\ { 0}, (0.13)\n\nfor sufficiently many multi-indices α. Generalise Stein and Wainger's poly-nomial Carleson's theorem to the multi-linear setting. For example, to prove\n\n‖ sup\n\n> λ∈R\n\n|\n\n∫\n\n> R2\n\nf (x − t)g(x − s)K(t, s )eiλs 2t2\n\ndtds |‖ 2. ‖f ‖4‖g‖4. (0.14) The multi-parameter Carleson's theorem has been proved by Li and Mus-calu [11]: Let K be given as in (0.13). Define\n\nC2(f, g )( x):= sup\n\n> N1,N 2\n\n∣∣∣∣∫\n\n> R2\n\nˆK(ξ1 − N1, ξ 2, N 2) ˆf1(ξ1) ˆf2(ξ2)dξ 1dξ 2\n\n∣∣∣∣, (0.15) then\n\n‖C2(f1, f 2)‖2. ‖f1‖4‖f2‖4. (0.16)", "original_statement": "Question 11 (Muscalu). Let K: R2 → R be a function such that \n\n|∂α ˆK(ξ)|. 1\n\n|ξ||α|, ∀ξ ∈ R2 \\ { 0}, (0.13) \n\nfor sufficiently many multi-indices α. Generalise Stein and Wainger's poly-nomial Carleson's theorem to the multi-linear setting. For example, to prove \n\n‖ sup \n\n> λ∈R\n\n|\n\n∫\n\n> R2\n\nf (x − t)g(x − s)K(t, s )eiλs 2t2\n\ndtds |‖ 2. ‖f ‖4‖g‖4. (0.14) The multi-parameter Carleson's theorem has been proved by Li and Mus-calu [11]: Let K be given as in (0.13). Define \n\nC2(f, g )( x):= sup \n\n> N1,N 2\n\n∣∣∣∣∫\n\n> R2\n\nˆK(ξ1 − N1, ξ 2, N 2) ˆf1(ξ1) ˆf2(ξ2)dξ 1dξ 2\n\n∣∣∣∣, (0.15) then \n\n‖C2(f1, f 2)‖2. ‖f1‖4‖f2‖4. (0.16)", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Question 11 (Camil Muscalu) from the 2015 AIM workshop *Carleson theorems and multilinear operators*. The official AIM PDF gives the following multiplier hypothesis. With $m=\\widehat K$,", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 11\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[69]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 11 (Muscalu). Let K: R2 → R be a function such that \\n\\n|∂α ˆK(ξ)|. 1\\n\\n|ξ||α|, ∀ξ ∈ R2 \\\\ { 0}, (0.13) \\n\\nfor sufficiently many multi-indices α. Generalise Stein and Wainger's poly-nomial Carleson's theorem to the multi-linear setting. For example, to prove \\n\\n‖ sup \\n\\n> λ∈R\\n\\n|\\n\\n∫\\n\\n> R2\\n\\nf (x − t)g(x − s)K(t, s )eiλs 2t2\\n\\ndtds |‖ 2. ‖f ‖4‖g‖4. (0.14) The multi-parameter Carleson's theorem has been proved by Li and Mus-calu [11]: Let K be given as in (0.13). Define \\n\\nC2(f, g )( x):= sup \\n\\n> N1,N 2\\n\\n∣∣∣∣∫\\n\\n> R2\\n\\nˆK(ξ1 − N1, ξ 2, N 2) ˆf1(ξ1) ˆf2(ξ2)dξ 1dξ 2\\n\\n∣∣∣∣, (0.15) then \\n\\n‖C2(f1, f 2)‖2. ‖f1‖4‖f2‖4. (0.16)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0070", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The operator with phase exp(i lambda t^2 s^2) is exactly the diagonal restriction of a two-dimensional linear polynomial Carleson maximal operator acting on the tensor F=f tensor g. Under the standard ambient L2 degree-four polynomial maximal estimate, a fixed-parameter normal trace argument proves an all-scale L2 bound controlled by the normal Sobolev norm with Fourier weight (1+|xi-eta|^2)^sigma for every sigma>1/2. Independently, every dyadic multiplier annulus satisfies the requested L4 x L4 to L2 estimate uniformly over all lambda, and finite or l1-summable collections satisfy the corresponding summed bound.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: if a translation-invariant maximal convolution family on R2 is L2-bounded, then its restriction to the diagonal on tensor input f tensor g is L2-bounded by the normal X^sigma norm for every sigma>1/2; applying this to the ambient degree-four polynomial maximal theorem yields a rigorous Sobolev-input partial theorem for the AIM operator." }, { "id": 20000539, "problem_number": "AIM-ANALYSIS-0071", "title": "The critical logarithmic endpoint model", "statement": "Question 12 (Guo). To prove that there exists a universal constant C > 0\n\nsuch that ∀[U+000F] ∈ (0, 1/2), it holds that\n\n‖ sup\n\n> λ∈R\n\n∫\n\n> R\n\nf (x − t)eiλ |t|[U+000F] dt t ‖2 ≤ C‖f ‖2. (0.17)", "original_statement": "Question 12 (Guo). To prove that there exists a universal constant C > 0\n\nsuch that ∀\u000f ∈ (0, 1/2), it holds that \n\n‖ sup \n\n> λ∈R\n\n∫\n\n> R\n\nf (x − t)eiλ |t|\u000f dt t ‖2 ≤ C‖f ‖2. (0.17)", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The JSON extraction contains damaged occurrences of the exponent and loses some absolute-value and principal-value notation. Question 12 in the AIM workshop problem list, checked against the source PDF and against Guo's definition of the same operator, is the following.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 12\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[70]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 12 (Guo). To prove that there exists a universal constant C > 0\\n\\nsuch that ∀\\u000f ∈ (0, 1/2), it holds that \\n\\n‖ sup \\n\\n> λ∈R\\n\\n∫\\n\\n> R\\n\\nf (x − t)eiλ |t|\\u000f dt t ‖2 ≤ C‖f ‖2. (0.17)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0071", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The universal AIM estimate remains open, but it necessarily contains a nontrivial critical endpoint model. At the allowed scaling lambda=mu/epsilon, removal of the scalar phase exp(i mu/epsilon) makes the operators converge pointwise on C_c^1 to S_mu f(x)=p.v. integral f(x-t)|t|^{i mu}dt/t. Therefore any universal AIM bound implies, with the same constant, the L2 estimate for sup_mu |S_mu f|. The exact multiplier is 2 Gamma(i mu)sinh(pi mu/2) sgn(xi)|xi|^{-i mu}, with its mu=0 value understood as -i pi sgn(xi), and the conjugated kernels converge in L2 operator norm at rate O(epsilon) on every fixed annulus uniformly for bounded mu.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: the blowing-coefficient scaling lambda=mu/epsilon reduces any universal bound in AIM Question 12 to an L2 maximal inequality for the logarithmically modulated Hilbert transforms p.v. integral f(x-t)|t|^{i mu}dt/t; this reduction has a finite-parameter Fatou proof, an exact imaginary-power multiplier formula, and quantitative O(epsilon) convergence on fixed annuli.", "research_difficulty_suggested": "L4" }, { "id": 20000540, "problem_number": "AIM-ANALYSIS-0072", "title": "A resonant obstruction to the displayed physical-space maximal estimate", "statement": "Question 13 (Carbery). On Rn, it is a big open problem whether\n\n∥∥∥∥∥sup\n\n> R\n\n∣∣∣∣∣∫\n\n> |ξ|≤ R\n\nˆf (ξ)e2πixξ dξ\n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.18)\n\nHow about ∥∥∥∥∥sup\n\n> R\n\n∣∣∣∣∣f ∗\n\n(\n\nei|x|\n\n|x| n+1 2\n\n· χ{| x|≤ R}\n\n)∣ ∣∣∣∣∥∥∥∥∥2. ‖f ‖2? (0.19) For detailed discussions, see Carbery et al. [2]. 4", "original_statement": "Question 13 (Carbery). On Rn, it is a big open problem whether \n\n∥∥∥∥∥sup \n\n> R\n\n∣∣∣∣∣∫\n\n> |ξ|≤ R\n\nˆf (ξ)e2πixξ dξ \n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.18) \n\nHow about ∥∥∥∥∥sup \n\n> R\n\n∣∣∣∣∣f ∗\n\n(\n\nei|x|\n\n|x| n+1 2\n\n· χ{| x|≤ R}\n\n)∣ ∣∣∣∣∥∥∥∥∥2. ‖f ‖2? (0.19) For detailed discussions, see Carbery et al. [2]. 4", "clean_statement": "Question 13 (Carbery). On Rn, it is a big open problem whether\n\n∥∥∥∥∥sup\n\n> R\n\n∣∣∣∣∣∫\n\n> |ξ|≤ R\n\nˆf (ξ)e2πixξ dξ\n\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.18)\n\nHow about ∥∥∥∥∥sup\n\n> R\n\n∣∣∣∣∣f ∗\n\n(\n\nei|x|\n\n|x| n+1 2\n\n· χ{| x|≤ R}\n\n)∣ ∣∣∣∣∥∥∥∥∥2. ‖f ‖2? (0.19) For detailed discussions, see Carbery et al. [2]. 4", "statement_status": "exact", "statement_verification": "The canonical JSON record is visibly damaged by PDF text extraction. I checked the official AIM problem-list PDF, *Carleson theorems and multilinear operators: Open problems*, page 4 of the PDF. With the Fourier convention", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 13\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[71]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 13 (Carbery). On Rn, it is a big open problem whether \\n\\n∥∥∥∥∥sup \\n\\n> R\\n\\n∣∣∣∣∣∫\\n\\n> |ξ|≤ R\\n\\nˆf (ξ)e2πixξ dξ \\n\\n∣∣∣∣∣∥∥∥∥∥2. ‖f ‖2. (0.18) \\n\\nHow about ∥∥∥∥∥sup \\n\\n> R\\n\\n∣∣∣∣∣f ∗\\n\\n(\\n\\nei|x|\\n\\n|x| n+1 2\\n\\n· χ{| x|≤ R}\\n\\n)∣ ∣∣∣∣∥∥∥∥∥2. ‖f ‖2? (0.19) For detailed discussions, see Carbery et al. [2]. 4\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0072", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For the literal kernel in (0.19), K_R(y)=e^{i|y|}|y|^{-(n+1)/2} 1_{|y|<=R}, n>=2, the fixed-truncation Fourier multiplier satisfies m_R((2pi)^{-1}e_1)=(2pi)^{(n-1)/2}e^{i pi(n-1)/4} log R+O_n(1). Hence the L2 operator norms are not uniformly bounded and the requested maximal L2 estimate is false. In dimensions n>=4, evaluation at zero frequency gives the stronger lower bound of order R^{(n-3)/2}. This refutes only (0.19); the spherical partial-sum problem (0.18) remains open in dimensions n>=2.\n\nCandidate contribution (obstruction; novelty confidence low): The literal AIM outgoing-wave truncations have L2 operator norm at least c_n log R-C_n for every n>=2, at least c'_n R^{(n-3)/2} for n>=4, and a modified two-branch leading kernel ae^{ir}+be^{-ir} can cancel the resonant logarithm only if ae^{i pi(n-1)/4}+be^{-i pi(n-1)/4}=0." }, { "id": 20000541, "problem_number": "AIM-ANALYSIS-0073", "title": "Normalization and the constant-one multilinear Kakeya question", "statement": "Question 14 (Iliopoulou). In Rn, let Ti, i ∈ { 1, 2,..., n } be a collection of tubes with width one and infinity length. We know that\n\n∫  ∑\n\n> T1∈T 1\n\nχT1\n\n...\n\n( ∑\n\n> Tn∈T n\n\nχTn\n\n)\n\nw(T1) ∧... ∧ w(Tn)\n\n\n\n> 1\n> n−1\n\n≤ Cnn∏\n\n> i=1\n\n(Ti) 1\n\n> n−1,\n\n(0.20)\n\nwhere w(Ti) is the unit vector parallel to the long side of the tube Ti. Could we prove (0.20) with Cn = 1?", "original_statement": "Question 14 (Iliopoulou). In Rn, let Ti, i ∈ { 1, 2,..., n } be a collection of tubes with width one and infinity length. We know that \n\n∫  ∑\n\n> T1∈T 1\n\nχT1\n\n... \n\n( ∑\n\n> Tn∈T n\n\nχTn\n\n)\n\nw(T1) ∧... ∧ w(Tn)\n\n\n\n> 1\n> n−1\n\n≤ Cnn∏\n\n> i=1\n\n(Ti) 1 \n\n> n−1,\n\n(0.20) \n\nwhere w(Ti) is the unit vector parallel to the long side of the tube Ti. Could we prove (0.20) with Cn = 1?", "clean_statement": "Question 14 (Iliopoulou). In Rn, let Ti, i ∈ { 1, 2,..., n } be a collection of tubes with width one and infinity length. We know that\n\n∫  ∑\n\n> T1∈T 1\n\nχT1\n\n...\n\n( ∑\n\n> Tn∈T n\n\nχTn\n\n)\n\nw(T1) ∧... ∧ w(Tn)\n\n\n\n> 1\n> n−1\n\n≤ Cnn∏\n\n> i=1\n\n(Ti) 1\n\n> n−1,\n\n(0.20)\n\nwhere w(Ti) is the unit vector parallel to the long side of the tube Ti. Could we prove (0.20) with Cn = 1?", "statement_status": "exact", "statement_verification": "The canonical record is Question 14 (Marina Iliopoulou) from the 18--22 May 2015 AIM workshop *Carleson theorems and multilinear operators*. The official TeX says that, for collections \\(\\mathcal T_i\\) of doubly infinite tubes of width one in \\(\\mathbb R^n\\),", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Carleson theorems and multilinear operators\nSection: \nSource item: 14\nSource URL: https://aimath.org/pastworkshops/multilinopsproblems.pdf\nCanonical location: aim-analysis-notes.json notes[72]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 14 (Iliopoulou). In Rn, let Ti, i ∈ { 1, 2,..., n } be a collection of tubes with width one and infinity length. We know that \\n\\n∫  ∑\\n\\n> T1∈T 1\\n\\nχT1\\n\\n... \\n\\n( ∑\\n\\n> Tn∈T n\\n\\nχTn\\n\\n)\\n\\nw(T1) ∧... ∧ w(Tn)\\n\\n\\n\\n> 1\\n> n−1\\n\\n≤ Cnn∏\\n\\n> i=1\\n\\n(Ti) 1 \\n\\n> n−1,\\n\\n(0.20) \\n\\nwhere w(Ti) is the unit vector parallel to the long side of the tube Ti. Could we prove (0.20) with Cn = 1?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/multilinopsproblems.pdf", "tags": [ "aim", "AIM-ANALYSIS-0073", "aim-domain:analysis", "aim-workshop:multilinopsproblems", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM display is reconstructed as the n-fold sum of tube intersections weighted by the unsigned determinant of their directions. The numerical constant depends essentially on the undefined meaning of width one: for strips of total width w the exact planar best constant is C_2(w)=w^2, so the Carbery-Valdimarsson radius-one convention gives C_2=4 and refutes constant one under that reading, while total-width-one normalization gives C_2=1. More generally, for fixed-direction-per-family tubes of transverse masses at most m_i, an affine Loomis-Whitney argument proves the sharp bound K_n <= product_i (m_i #T_i)^{1/(n-1)}, with equality on volume-one cubical grids. The transverse-mass-one, varying-direction problem for n >= 3 remains unresolved here.\n\nCandidate contribution (theorem; novelty confidence low): Candidate normalization theorem: the exact planar constant is C_2(w)=w^2, the global radius scaling is C_n(r)=r^n C_n(1), and the sharp fixed-direction subclass constant with transverse mass bounds m_i is product_i m_i^{1/(n-1)}; finite dense lattices of round tubes approach the corresponding mass factor." }, { "id": 20000542, "problem_number": "AIM-ANALYSIS-0074", "title": "Trace-norm matrix signing: positive case solved with sharp exponent", "statement": "QUESTION 1: Does there exist a subset J of $1, ..., m$ such that\n$$\\|\\sum_{j\\in J} a_j - \\sum_{j\\in J^c} a_j\\|< O(\\epsilon^{1/8})\\ ?$$\n\nCOMMENT: The MSS method for Kadison-Singer seems to rely on small norm and rank one, but is it really about rank or is it about trace norm being small? Of course the 1/8 is just something to think about. We know $\\epsilon$ won't work, but maybe $\\sqrt \\epsilon$. This is related to the question raised by in the last section of the MSS paper.\n\nA von Neumann algebra is a weakly closed self adjoint sub algebra of $B(H)$. A factor is a von Neumann algebra with trivial center. Murray and von Neumann classified factors into types I, II, and III. Type I factors all have the form $B(H)$, and the only difference between them is the cardinality of $H$. Types II and III, on the other hand, each display uncountably many distinct variations as subalgebras of $B(H)$ for separable $H$. These have MASA's that are also of different types, and a Kadison-Singer problem arises as follows.", "original_statement": "QUESTION 1: Does there exist a subset J of $1, ..., m$ such that\n$$\\|\\sum_{j\\in J} a_j - \\sum_{j\\in J^c} a_j\\|< O(\\epsilon^{1/8})\\ ?$$\n\nCOMMENT: The MSS method for Kadison-Singer seems to rely on small norm and rank one, but is it really about rank or is it about trace norm being small? Of course the 1/8 is just something to think about. We know $\\epsilon$ won't work, but maybe $\\sqrt \\epsilon$. This is related to the question raised by in the last section of the MSS paper.\n\nA von Neumann algebra is a weakly closed self adjoint sub algebra of $B(H)$. A factor is a von Neumann algebra with trivial center. Murray and von Neumann classified factors into types I, II, and III. Type I factors all have the form $B(H)$, and the only difference between them is the cardinality of $H$. Types II and III, on the other hand, each display uncountably many distinct variations as subalgebras of $B(H)$ for separable $H$. These have MASA's that are also of different types, and a Kadison-Singer problem arises as follows.", "clean_statement": "QUESTION 1: Does there exist a subset J of $1, ..., m$ such that\n$$\\|\\sum_{j\\in J} a_j - \\sum_{j\\in J^c} a_j\\|< O(\\epsilon^{1/8})\\ ?$$\n\nCOMMENT: The MSS method for Kadison-Singer seems to rely on small norm and rank one, but is it really about rank or is it about trace norm being small? Of course the 1/8 is just something to think about. We know $\\epsilon$ won't work, but maybe $\\sqrt \\epsilon$. This is related to the question raised by in the last section of the MSS paper.\n\nA von Neumann algebra is a weakly closed self adjoint sub algebra of $B(H)$. A factor is a von Neumann algebra with trivial center. Murray and von Neumann classified factors into types I, II, and III. Type I factors all have the form $B(H)$, and the only difference between them is the cardinality of $H$. Types II and III, on the other hand, each display uncountably many distinct variations as subalgebras of $B(H)$ for separable $H$. These have MASA's that are also of different types, and a Kadison-Singer problem arises as follows.", "statement_status": "exact", "statement_verification": "The canonical record is source index 73 of aim-analysis-notes.json. Its displayed question omits the hypotheses that immediately precede it in the original workshop document. The original Google document gives the following setup:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: 1\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[73]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"QUESTION 1: Does there exist a subset J of $1, ..., m$ such that\\n$$\\\\|\\\\sum_{j\\\\in J} a_j - \\\\sum_{j\\\\in J^c} a_j\\\\|< O(\\\\epsilon^{1/8})\\\\ ?$$\\n\\nCOMMENT: The MSS method for Kadison-Singer seems to rely on small norm and rank one, but is it really about rank or is it about trace norm being small? Of course the 1/8 is just something to think about. We know $\\\\epsilon$ won't work, but maybe $\\\\sqrt \\\\epsilon$. This is related to the question raised by in the last section of the MSS paper.\\n\\nA von Neumann algebra is a weakly closed self adjoint sub algebra of $B(H)$. A factor is a von Neumann algebra with trivial center. Murray and von Neumann classified factors into types I, II, and III. Type I factors all have the form $B(H)$, and the only difference between them is the cardinality of $H$. Types II and III, on the other hand, each display uncountably many distinct variations as subalgebras of $B(H)$ for separable $H$. These have MASA's that are also of different types, and a Kadison-Singer problem arises as follows.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0074", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source is ambiguous between positive and merely self-adjoint summands. For the positive formulation, Bownik's Theorem 4.1 gives a subset approximation of at most 2 sqrt(epsilon); taking all target coefficients equal to one half yields the requested signing norm at most 4 sqrt(epsilon), for arbitrary-rank positive matrices. The report also proves an exact equiangular-tight-frame signing-energy identity and uses Singer harmonic frames to show that every exponent greater than one half is impossible. The broader self-adjoint formulation remains unresolved without additional absolute-variation control.\n\nCandidate contribution (identity/lemma; novelty confidence low): For a unit-norm ETF u_1,...,u_m in C^n and A_j=(n/m)u_j u_j^*, every signing D_s=sum_j s_j A_j satisfies the exact identity tr(D_s^2)=(n/m)^2[m+((m-n)/(n(m-1)))((sum_j s_j)^2-m)], hence ||D_s|| is at least sqrt((n-1)/(m-1)); Singer ETFs turn this into an explicit all-signings lower family asymptotic to sqrt(epsilon)." }, { "id": 20000543, "problem_number": "AIM-ANALYSIS-0075", "title": "Pure-state extension from a MASA of a factor", "statement": "QUESTION 2: Given a MASA $A$ of a factor $M$, do the pure states of $A$ have unique state extension to $M$?\n\nSOME ANSWERS: MSS showed that for discrete MASAs of $B(H)$, $H$ separable, the answer is \"yes\". As Bernard pointed out in his opening talk, if the MASA has any continuous part, the answer is no (that was shown in the original Kadison-Singer paper).\n\nFor Type II factors on separable $H$ the answer is always NO, for any MASA but for non separable $H$, the answer is sometimes yes and sometimes no with no general results as yet. (Popa, Akemann, Sherman)\n\nFor type III factors there are no results at all for any factors and for any MASAs.", "original_statement": "QUESTION 2: Given a MASA $A$ of a factor $M$, do the pure states of $A$ have unique state extension to $M$?\n\nSOME ANSWERS: MSS showed that for discrete MASAs of $B(H)$, $H$ separable, the answer is \"yes\". As Bernard pointed out in his opening talk, if the MASA has any continuous part, the answer is no (that was shown in the original Kadison-Singer paper). \n\nFor Type II factors on separable $H$ the answer is always NO, for any MASA but for non separable $H$, the answer is sometimes yes and sometimes no with no general results as yet. (Popa, Akemann, Sherman)\n\nFor type III factors there are no results at all for any factors and for any MASAs.", "clean_statement": "QUESTION 2: Given a MASA $A$ of a factor $M$, do the pure states of $A$ have unique state extension to $M$?\n\nSOME ANSWERS: MSS showed that for discrete MASAs of $B(H)$, $H$ separable, the answer is \"yes\". As Bernard pointed out in his opening talk, if the MASA has any continuous part, the answer is no (that was shown in the original Kadison-Singer paper).\n\nFor Type II factors on separable $H$ the answer is always NO, for any MASA but for non separable $H$, the answer is sometimes yes and sometimes no with no general results as yet. (Popa, Akemann, Sherman)\n\nFor type III factors there are no results at all for any factors and for any MASAs.", "statement_status": "exact", "statement_verification": "The record was checked against the plain-text export of the linked AIM Google document. The text is intact; no OCR repair or mathematical reconstruction is needed. The source is the problem compilation for the AIM workshop *Beyond Kadison--Singer: paving and consequences*, held December 1--5, 2014. The compilation is dated November 30, 2014.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: 2\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[74]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"QUESTION 2: Given a MASA $A$ of a factor $M$, do the pure states of $A$ have unique state extension to $M$?\\n\\nSOME ANSWERS: MSS showed that for discrete MASAs of $B(H)$, $H$ separable, the answer is \\\"yes\\\". As Bernard pointed out in his opening talk, if the MASA has any continuous part, the answer is no (that was shown in the original Kadison-Singer paper). \\n\\nFor Type II factors on separable $H$ the answer is always NO, for any MASA but for non separable $H$, the answer is sometimes yes and sometimes no with no general results as yet. (Popa, Akemann, Sherman)\\n\\nFor type III factors there are no results at all for any factors and for any MASAs.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0075", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every MASA A in a von Neumann algebra M with separable predual, the all-pure-state unique extension property holds exactly when M is type I and A is the range of a normal conditional expectation; hence every MASA in a separable-predual type II or type III factor has a singular pure state with nonunique, necessarily singular extensions. The arbitrary-predual factor problem remains unclassified. In addition, if A admits both a normal conditional expectation E and a singular conditional expectation F, then ||E-F||=2 and the supremum, over pure states of A, of the diameters of their state-extension fibers is 2.\n\nCandidate contribution (proposition; novelty confidence low): If an abelian von Neumann subalgebra A of M admits a normal conditional expectation E:M->A and a singular conditional expectation F:M->A, then ||E-F||=2 and sup_{chi in A-hat} diam Ext_M(chi)=2." }, { "id": 20000544, "problem_number": "AIM-ANALYSIS-0076", "title": "Pure-state extensions across a split ideal", "statement": "QUESTION 3: Let a C*-algebra $B$ be the direct sum of an ideal and a MASA, $B = I \\oplus A$. Under what circumstances do the pure states of $A$ have unique extension to states of $B$?\n\n\n* Cynthia Vinzant\n - there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs?\n - Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture\n - Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n\n* Greg Knese\n - understanding intrinsic proofs of Feichtinger's conjecture\n - extend the theory of interlacing families to several variables\n - use a theorem to define interlacing in several variables\n - how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\n\n* Petter Branden\n - combinatorics generally\n - better bounds, more quantitative results in Weaver's result\n - for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.\n - has a witness to a hyperbolic version of the problem\n - more generally, matching lower bounds for the different equivalences (the paving lower bound in the \\R case is 1/\\eps^2 and authors showed 6/\\eps^2)\n - Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.\n - consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?\n - serious about hyperbolic frame theory!\n - extensions to hyperbolic polynomials --> just posted a new paper on the ArXiv\n\n* Marcin Bownik\n - interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?\n - Frames of complex of exponentials on a subset of the torus = exponential frames, derive better/best bounds for Feichtinger's conjecture for these frames.\n - exponential frame L^2([0,1]), consider measurable subset E \\subset [0,1]\n - \\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} are an exponential frame\n\n* Darrin Speegle\n - interested in Feichtinger conjecture, how large the partition has to be to get Riesz basic sequences, especially for complex exponentials, this suggests that there's something about Fourier series that we don't understand.\n - whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?\n - More concretely, can we find E, |E| = \\frac12 such that\n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?\n - there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n\n* Bill Johnson\n -Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$.\n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t.\n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j\n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary;\nMatrix norm inequalities and the relative Dixmier property.\nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$.\n\n - Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell\n - design algorithms to get pavings, especially for stochastic optimization\n - interested in algorithms that are efficient!\n - algorithmic developments that use pavings\n - sampling and RIP constructions for sparse approximation, do these lead to constructions or insight for KS\n\n* Anna Gilbert\n - Fourier sampling questions, RIP for the DFT, Rudelson selection lemma, how to remove extra factors on the bounds for RIP for DFT\n\n* Olaf Mordhorst\n - applications to convex geometry and Banach spaces\n - Restricted invertibility: provide a concrete example of optimality which is proven via random matrices\n\n* Pierre Youssef\n - consequences in convex geometry\n - if you have an identity decamp in \\R^n, BSS approximated the identity with many fewer, albeit weighted, vectors. If you use MSS, you get an identity approximation without weights. Let's add a condition that the vectors are balanced (say they come from a convex body).\n - \\sum_j c_j x_j x_j^T with \\|x_j\\| = 1 and \\sum_j c_j x_j = 0\n - Find S \\subset [m] , |S| \\approx d so that\n C \\sum_{j \\in S} x_j x_j^T \\approx I with a balancing condition \\| \\sum x_j \\| \\leq B\n - this was done by Srivastava but got 4 + \\eps approximation of the identity\n - Given a convex body K in \\R^n, Restricted Invertibility says there exists an E \\subset \\R^n with dim(E) = (1-\\eps)n so that\n distance( K \\cap E, B_1) \\approx \\sqrt{dim(E)}\n - KS says that we can partition K into subspaces like this. Can we improve the Banach-Mazur distance of a convex body to the cube using this partition? The best bound currently is n^{5/6}.\n - Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n\n* Leonid Gurvits\n - there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\n\n* Mirko Visontai\n - understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.\n - can we generate new interlacing families?\n\n* Mihai Putinar\n - perturbation determinant: in interlacing families, we have quotients of such polynomials, they form rational functions in the Nevalinna class of functions that preserve the upper half plane\n - understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n\n* Dan Edidin\n - wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.\n - For KS2, the problem is static: given v, find a good partition. Wants a more dynamic version: look at all vectors together (in a manifold structure), so that you replace the discrete structure with a continuous one. Look over some parametrized space with geometric structure and this may be more helpful to characterize when there are lots of partitions (i.e., count) that satisfies KS2.\n\n* Pablo Parrilo\n - algorithmic questions!\n - To try to understand why the MSS proof works: many existence proofs use probabilistic methods or fixed point methods and they are non-constructive. But MSS proof combines different techniques. Understand how these pieces fit together more generally.\n\n* Rachel Ward\n - see the above on algorithmic!\n\n* Leonid Gurvits\n - Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.\n - Conjecture: vol(Zon) = 1.\n - KS analog: symmetrize Zon and then seek a partition S \\cup T so that\n - \\sum_{i \\in S} \\bar z_i \\subset (1/2 \\pm \\eps) Zon.\n\n* Bernhard Bodmann\n - how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n\n* Adam Marcus\n - one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!\n - There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?\n - Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.\n - There was something about adding rank-2 perturbations rather than rank-1 perturbations.", "original_statement": "QUESTION 3: Let a C*-algebra $B$ be the direct sum of an ideal and a MASA, $B = I \\oplus A$. Under what circumstances do the pure states of $A$ have unique extension to states of $B$?\n\n \n* Cynthia Vinzant\n - there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs? \n - Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture\n - Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n \n* Greg Knese\n - understanding intrinsic proofs of Feichtinger's conjecture\n - extend the theory of interlacing families to several variables\n - use a theorem to define interlacing in several variables\n - how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\n \n* Petter Branden\n - combinatorics generally\n - better bounds, more quantitative results in Weaver's result\n - for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.\n - has a witness to a hyperbolic version of the problem\n - more generally, matching lower bounds for the different equivalences (the paving lower bound in the \\R case is 1/\\eps^2 and authors showed 6/\\eps^2)\n - Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.\n - consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?\n - serious about hyperbolic frame theory!\n - extensions to hyperbolic polynomials --> just posted a new paper on the ArXiv\n \n* Marcin Bownik\n - interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?\n - Frames of complex of exponentials on a subset of the torus = exponential frames, derive better/best bounds for Feichtinger's conjecture for these frames.\n - exponential frame L^2([0,1]), consider measurable subset E \\subset [0,1]\n - \\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} are an exponential frame\n \n* Darrin Speegle\n - interested in Feichtinger conjecture, how large the partition has to be to get Riesz basic sequences, especially for complex exponentials, this suggests that there's something about Fourier series that we don't understand.\n - whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?\n - More concretely, can we find E, |E| = \\frac12 such that \n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?\n - there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n \n* Bill Johnson\n -Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$. \n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t. \n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j \n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary; \nMatrix norm inequalities and the relative Dixmier property. \nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$. \n\n - Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell\n - design algorithms to get pavings, especially for stochastic optimization\n - interested in algorithms that are efficient!\n - algorithmic developments that use pavings\n - sampling and RIP constructions for sparse approximation, do these lead to constructions or insight for KS\n \n* Anna Gilbert\n - Fourier sampling questions, RIP for the DFT, Rudelson selection lemma, how to remove extra factors on the bounds for RIP for DFT\n \n* Olaf Mordhorst\n - applications to convex geometry and Banach spaces\n - Restricted invertibility: provide a concrete example of optimality which is proven via random matrices\n\n* Pierre Youssef\n - consequences in convex geometry\n - if you have an identity decamp in \\R^n, BSS approximated the identity with many fewer, albeit weighted, vectors. If you use MSS, you get an identity approximation without weights. Let's add a condition that the vectors are balanced (say they come from a convex body). \n - \\sum_j c_j x_j x_j^T with \\|x_j\\| = 1 and \\sum_j c_j x_j = 0\n - Find S \\subset [m] , |S| \\approx d so that\n C \\sum_{j \\in S} x_j x_j^T \\approx I with a balancing condition \\| \\sum x_j \\| \\leq B\n - this was done by Srivastava but got 4 + \\eps approximation of the identity\n - Given a convex body K in \\R^n, Restricted Invertibility says there exists an E \\subset \\R^n with dim(E) = (1-\\eps)n so that\n distance( K \\cap E, B_1) \\approx \\sqrt{dim(E)}\n - KS says that we can partition K into subspaces like this. Can we improve the Banach-Mazur distance of a convex body to the cube using this partition? The best bound currently is n^{5/6}.\n - Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n \n* Leonid Gurvits\n - there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\n \n* Mirko Visontai\n - understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.\n - can we generate new interlacing families?\n\n* Mihai Putinar\n - perturbation determinant: in interlacing families, we have quotients of such polynomials, they form rational functions in the Nevalinna class of functions that preserve the upper half plane\n - understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n \n* Dan Edidin\n - wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.\n - For KS2, the problem is static: given v, find a good partition. Wants a more dynamic version: look at all vectors together (in a manifold structure), so that you replace the discrete structure with a continuous one. Look over some parametrized space with geometric structure and this may be more helpful to characterize when there are lots of partitions (i.e., count) that satisfies KS2.\n\n* Pablo Parrilo\n - algorithmic questions!\n - To try to understand why the MSS proof works: many existence proofs use probabilistic methods or fixed point methods and they are non-constructive. But MSS proof combines different techniques. Understand how these pieces fit together more generally.\n \n* Rachel Ward\n - see the above on algorithmic!\n \n* Leonid Gurvits\n - Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.\n - Conjecture: vol(Zon) = 1.\n - KS analog: symmetrize Zon and then seek a partition S \\cup T so that\n - \\sum_{i \\in S} \\bar z_i \\subset (1/2 \\pm \\eps) Zon.\n\n* Bernhard Bodmann\n - how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n \n* Adam Marcus\n - one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!\n - There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?\n - Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.\n - There was something about adding rank-2 perturbations rather than rank-1 perturbations.", "clean_statement": "QUESTION 3: Let a C*-algebra $B$ be the direct sum of an ideal and a MASA, $B = I \\oplus A$. Under what circumstances do the pure states of $A$ have unique extension to states of $B$?\n\n\n* Cynthia Vinzant\n - there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs?\n - Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture\n - Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n\n* Greg Knese\n - understanding intrinsic proofs of Feichtinger's conjecture\n - extend the theory of interlacing families to several variables\n - use a theorem to define interlacing in several variables\n - how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\n\n* Petter Branden\n - combinatorics generally\n - better bounds, more quantitative results in Weaver's result\n - for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.\n - has a witness to a hyperbolic version of the problem\n - more generally, matching lower bounds for the different equivalences (the paving lower bound in the \\R case is 1/\\eps^2 and authors showed 6/\\eps^2)\n - Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.\n - consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?\n - serious about hyperbolic frame theory!\n - extensions to hyperbolic polynomials --> just posted a new paper on the ArXiv\n\n* Marcin Bownik\n - interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?\n - Frames of complex of exponentials on a subset of the torus = exponential frames, derive better/best bounds for Feichtinger's conjecture for these frames.\n - exponential frame L^2([0,1]), consider measurable subset E \\subset [0,1]\n - \\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} are an exponential frame\n\n* Darrin Speegle\n - interested in Feichtinger conjecture, how large the partition has to be to get Riesz basic sequences, especially for complex exponentials, this suggests that there's something about Fourier series that we don't understand.\n - whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?\n - More concretely, can we find E, |E| = \\frac12 such that\n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?\n - there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n\n* Bill Johnson\n -Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$.\n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t.\n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j\n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary;\nMatrix norm inequalities and the relative Dixmier property.\nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$.\n\n - Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell\n - design algorithms to get pavings, especially for stochastic optimization\n - interested in algorithms that are efficient!\n - algorithmic developments that use pavings\n - sampling and RIP constructions for sparse approximation, do these lead to constructions or insight for KS\n\n* Anna Gilbert\n - Fourier sampling questions, RIP for the DFT, Rudelson selection lemma, how to remove extra factors on the bounds for RIP for DFT\n\n* Olaf Mordhorst\n - applications to convex geometry and Banach spaces\n - Restricted invertibility: provide a concrete example of optimality which is proven via random matrices\n\n* Pierre Youssef\n - consequences in convex geometry\n - if you have an identity decamp in \\R^n, BSS approximated the identity with many fewer, albeit weighted, vectors. If you use MSS, you get an identity approximation without weights. Let's add a condition that the vectors are balanced (say they come from a convex body).\n - \\sum_j c_j x_j x_j^T with \\|x_j\\| = 1 and \\sum_j c_j x_j = 0\n - Find S \\subset [m] , |S| \\approx d so that\n C \\sum_{j \\in S} x_j x_j^T \\approx I with a balancing condition \\| \\sum x_j \\| \\leq B\n - this was done by Srivastava but got 4 + \\eps approximation of the identity\n - Given a convex body K in \\R^n, Restricted Invertibility says there exists an E \\subset \\R^n with dim(E) = (1-\\eps)n so that\n distance( K \\cap E, B_1) \\approx \\sqrt{dim(E)}\n - KS says that we can partition K into subspaces like this. Can we improve the Banach-Mazur distance of a convex body to the cube using this partition? The best bound currently is n^{5/6}.\n - Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n\n* Leonid Gurvits\n - there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\n\n* Mirko Visontai\n - understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.\n - can we generate new interlacing families?\n\n* Mihai Putinar\n - perturbation determinant: in interlacing families, we have quotients of such polynomials, they form rational functions in the Nevalinna class of functions that preserve the upper half plane\n - understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n\n* Dan Edidin\n - wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.\n - For KS2, the problem is static: given v, find a good partition. Wants a more dynamic version: look at all vectors together (in a manifold structure), so that you replace the discrete structure with a continuous one. Look over some parametrized space with geometric structure and this may be more helpful to characterize when there are lots of partitions (i.e., count) that satisfies KS2.\n\n* Pablo Parrilo\n - algorithmic questions!\n - To try to understand why the MSS proof works: many existence proofs use probabilistic methods or fixed point methods and they are non-constructive. But MSS proof combines different techniques. Understand how these pieces fit together more generally.\n\n* Rachel Ward\n - see the above on algorithmic!\n\n* Leonid Gurvits\n - Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.\n - Conjecture: vol(Zon) = 1.\n - KS analog: symmetrize Zon and then seek a partition S \\cup T so that\n - \\sum_{i \\in S} \\bar z_i \\subset (1/2 \\pm \\eps) Zon.\n\n* Bernhard Bodmann\n - how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n\n* Adam Marcus\n - one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!\n - There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?\n - Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.\n - There was something about adding rank-2 perturbations rather than rank-1 perturbations.", "statement_status": "exact", "statement_verification": "The canonical record is number 3 in the AIM workshop document *Beyond Kadison--Singer: paving and consequences* (source file `aim-analysis-notes.json`, zero-based index 75). The mathematical question is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: 3\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[75]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"QUESTION 3: Let a C*-algebra $B$ be the direct sum of an ideal and a MASA, $B = I \\\\oplus A$. Under what circumstances do the pure states of $A$ have unique extension to states of $B$?\\n\\n \\n* Cynthia Vinzant\\n - there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs? \\n - Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture\\n - Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\\n \\n* Greg Knese\\n - understanding intrinsic proofs of Feichtinger's conjecture\\n - extend the theory of interlacing families to several variables\\n - use a theorem to define interlacing in several variables\\n - how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\\n \\n* Petter Branden\\n - combinatorics generally\\n - better bounds, more quantitative results in Weaver's result\\n - for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\\\sqrt\\\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.\\n - has a witness to a hyperbolic version of the problem\\n - more generally, matching lower bounds for the different equivalences (the paving lower bound in the \\\\R case is 1/\\\\eps^2 and authors showed 6/\\\\eps^2)\\n - Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.\\n - consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?\\n - serious about hyperbolic frame theory!\\n - extensions to hyperbolic polynomials --> just posted a new paper on the ArXiv\\n \\n* Marcin Bownik\\n - interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\\\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?\\n - Frames of complex of exponentials on a subset of the torus = exponential frames, derive better/best bounds for Feichtinger's conjecture for these frames.\\n - exponential frame L^2([0,1]), consider measurable subset E \\\\subset [0,1]\\n - \\\\{ \\\\econst^{2 \\\\pi \\\\iunit n \\\\ell \\\\chi_E \\\\}_{n \\\\in \\\\Z} are an exponential frame\\n \\n* Darrin Speegle\\n - interested in Feichtinger conjecture, how large the partition has to be to get Riesz basic sequences, especially for complex exponentials, this suggests that there's something about Fourier series that we don't understand.\\n - whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?\\n - More concretely, can we find E, |E| = \\\\frac12 such that \\n\\\\{ \\\\econst^{2 \\\\pi \\\\iunit n \\\\ell \\\\chi_E \\\\}_{n \\\\in \\\\Z} cannot be partitioned into two Riesz sequences?\\n - there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\\n \\n* Bill Johnson\\n -Is paving true for $\\\\ell_p^n$? Specifically:\\n\\nSuppose $1\\\\le p<\\\\infty$. \\n\\n$Pav_p$: Does there exist for every $\\\\epsilon > 0$ a constant $k=k(\\\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\\\|A\\\\|_p = 1$, there is a partition $\\\\sigma(1), \\\\dots, \\\\sigma(k) of $\\\\{1,\\\\dots,n\\\\}$ s.t. \\n\\n$$\\n\\\\|\\\\sum_{j=1}^k P_{\\\\sigma(j)} A P_{\\\\sigma(j)} \\\\|_p = \\\\max_j \\n\\\\| P_{\\\\sigma(j)} A P_{\\\\sigma(j)} \\\\|_p \\\\le \\\\epsilon .\\n$$\\n\\nHere $\\\\|\\\\dot \\\\|_$ is the operator norm for operators on $\\\\ell_p^n$, NOT the Schatten $p$-class norm.\\n\\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\\n\\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary; \\nMatrix norm inequalities and the relative Dixmier property. \\nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\\n\\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\\\ell_p^n$, but they got optimal estimates only for $p=2$. \\n\\n - Take A = n \\\\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\\n\\n* Deanna Needell\\n - design algorithms to get pavings, especially for stochastic optimization\\n - interested in algorithms that are efficient!\\n - algorithmic developments that use pavings\\n - sampling and RIP constructions for sparse approximation, do these lead to constructions or insight for KS\\n \\n* Anna Gilbert\\n - Fourier sampling questions, RIP for the DFT, Rudelson selection lemma, how to remove extra factors on the bounds for RIP for DFT\\n \\n* Olaf Mordhorst\\n - applications to convex geometry and Banach spaces\\n - Restricted invertibility: provide a concrete example of optimality which is proven via random matrices\\n\\n* Pierre Youssef\\n - consequences in convex geometry\\n - if you have an identity decamp in \\\\R^n, BSS approximated the identity with many fewer, albeit weighted, vectors. If you use MSS, you get an identity approximation without weights. Let's add a condition that the vectors are balanced (say they come from a convex body). \\n - \\\\sum_j c_j x_j x_j^T with \\\\|x_j\\\\| = 1 and \\\\sum_j c_j x_j = 0\\n - Find S \\\\subset [m] , |S| \\\\approx d so that\\n C \\\\sum_{j \\\\in S} x_j x_j^T \\\\approx I with a balancing condition \\\\| \\\\sum x_j \\\\| \\\\leq B\\n - this was done by Srivastava but got 4 + \\\\eps approximation of the identity\\n - Given a convex body K in \\\\R^n, Restricted Invertibility says there exists an E \\\\subset \\\\R^n with dim(E) = (1-\\\\eps)n so that\\n distance( K \\\\cap E, B_1) \\\\approx \\\\sqrt{dim(E)}\\n - KS says that we can partition K into subspaces like this. Can we improve the Banach-Mazur distance of a convex body to the cube using this partition? The best bound currently is n^{5/6}.\\n - Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\\n \\n* Leonid Gurvits\\n - there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\\n \\n* Mirko Visontai\\n - understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.\\n - can we generate new interlacing families?\\n\\n* Mihai Putinar\\n - perturbation determinant: in interlacing families, we have quotients of such polynomials, they form rational functions in the Nevalinna class of functions that preserve the upper half plane\\n - understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\\n \\n* Dan Edidin\\n - wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.\\n - For KS2, the problem is static: given v, find a good partition. Wants a more dynamic version: look at all vectors together (in a manifold structure), so that you replace the discrete structure with a continuous one. Look over some parametrized space with geometric structure and this may be more helpful to characterize when there are lots of partitions (i.e., count) that satisfies KS2.\\n\\n* Pablo Parrilo\\n - algorithmic questions!\\n - To try to understand why the MSS proof works: many existence proofs use probabilistic methods or fixed point methods and they are non-constructive. But MSS proof combines different techniques. Understand how these pieces fit together more generally.\\n \\n* Rachel Ward\\n - see the above on algorithmic!\\n \\n* Leonid Gurvits\\n - Is there an analog of KS for zonotopes? Let z_1,\\\\ldots,z_k \\\\in \\\\R^n, k > n and let \\\\bar z_i denote the interval \\\\{ \\\\alpha z_i | 0 \\\\leq \\\\alpha \\\\leq 1\\\\}. Consider the Minkowski sum\\n Zon = \\\\alpha_1 \\\\bar z_1 + \\\\cdots \\\\alpha_k \\\\bar z_k.\\n - Conjecture: vol(Zon) = 1.\\n - KS analog: symmetrize Zon and then seek a partition S \\\\cup T so that\\n - \\\\sum_{i \\\\in S} \\\\bar z_i \\\\subset (1/2 \\\\pm \\\\eps) Zon.\\n\\n* Bernhard Bodmann\\n - how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\\n \\n* Adam Marcus\\n - one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!\\n - There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?\\n - Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.\\n - There was something about adding rank-2 perturbations rather than rank-1 perturbations.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0076", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an internal split B = I dotplus A and a character phi of A, state extensions of phi are parametrized exactly by positive functionals tau on I of norm at most one satisfying tau(ai) = phi(a)tau(i) and tau(ia) = tau(i)phi(a); hence uniqueness is equivalent to vanishing of this eigenfunctional cone. If I is isomorphic to K(H) and A is a MASA, the cone always vanishes. In contrast, an explicit separable unital continuous-field construction gives B = I dotplus A with A a MASA and a character having two distinct pure extensions, so the MASA and splitting hypotheses alone are insufficient.\n\nCandidate contribution (counterexample; novelty confidence low): There is an explicit separable unital C*-algebra B = I dotplus A with I isomorphic to C_0((0,1], K(C plus L^2[0,1])) and A isomorphic to C([0,1]) a MASA, such that evaluation at 1/2 on A has two distinct pure state extensions; the construction uses an eigenvector born only in the endpoint fiber." }, { "id": 20000545, "problem_number": "AIM-ANALYSIS-0077", "title": "Certified constructive descent for interlacing proofs", "statement": "there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs?", "original_statement": "there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs?", "clean_statement": "there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs?", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b4\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[76]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"there are existence proofs for KS, how to make them constructive? Likes stable polynomials and interlacers. How to make constructive proofs?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0077", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any finite interlacing tree, choosing at each level the child with the smallest certified largest-root upper bound yields a leaf whose additive loss is at most the sum of the child-interval widths. A binary degree-one construction forces additive loss h eta/2 for every deterministic local selector given only width-eta child intervals, so linear dependence on depth is unavoidable up to a factor of two in that information model. The artifacts also prove a parent-label obstruction and a greedy operator-discrepancy bound alpha for pairwise commuting rank-one tight frames.\n\nCandidate contribution (theorem; novelty confidence low): Certified child-root interval widths add along interlacing descent, while every deterministic local interval-only selector can be forced to lose h eta/2 on a depth-h binary degree-one interlacing tree; thus linear depth dependence is necessary up to a factor of two." }, { "id": 20000546, "problem_number": "AIM-ANALYSIS-0078", "title": "A deterministic dense unequal-norm Weaver partition algorithm", "statement": "Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture", "original_statement": "Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture", "clean_statement": "Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture", "statement_status": "exact", "statement_verification": "The canonical record is the following question from the AIM workshop *Beyond Kadison--Singer: paving and consequences* (December 2014):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b5\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[77]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Find an algorithm for Weaver's conjecture/equivalence/theorem (or call it the vector partitioning problem), similarly for Paving Conjecture\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0078", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a rational Parseval frame with sum_i v_i v_i^* = I_d and max_i ||v_i||^2 <= 1/(49d), a deterministic greedy algorithm that repeatedly selects the vector of minimum normalized leverage (v_i^* B^{-1} v_i)/||v_i||^2 until the selected trace crosses d/2 returns an unsplit partition S,S^c for which I_d/4 <= sum_{i in S} v_i v_i^* <= 3I_d/4 and the same bounds hold for the complementary sum. The proof gives polynomial Turing bit complexity for exact rational input. This is a dense-regime, constant-error partial theorem only: it does not achieve the general Marcus-Spielman-Srivastava O(sqrt(alpha)) scale or a dimension-free constructive paving theorem, and those general algorithmic targets remain open in the literature checked.\n\nCandidate contribution (theorem; novelty confidence low): The minimum-normalized-leverage rule extends the deterministic dense-regime Weaver construction from equal-norm Parseval frames to arbitrary unequal vector norms, without splitting, duplicating, or reweighting vectors, under max_i ||v_i||^2 <= 1/(49d)." }, { "id": 20000547, "problem_number": "AIM-ANALYSIS-0079", "title": "Exact one-way communication bounds for a frame-partition surrogate", "statement": "Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n\n* Greg Knese", "original_statement": "Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n \n* Greg Knese", "clean_statement": "Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\n\n* Greg Knese", "statement_status": "exact", "statement_verification": "The canonical record is item `b6` from the 2014 AIM workshop *Beyond Kadison--Singer: paving and consequences* (`aim-analysis-notes.json`, zero-based source index 78). Its extracted text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b6\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[78]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Chuck started to describe a cooperative two-player game with partitioning the sets --> could we get an impossibility result via communication complexity?\\n \\n* Greg Knese\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0079", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The original two-player game is unrecovered: the primary AIM source contains only a one-sentence prompt, and the appended Greg Knese text is the next speaker heading. For an explicitly defined surrogate in which Alice knows a perfect even r-uniform hypermatching with d blocks on N=dr labeled indices and Bob must remotely output an exact block-balancing signing, the deterministic one-way communication cost is d log_2(2^r/binom(r,r/2)) plus an additive O(log(dr)); explicit finite lower and upper bounds are proved by exact incidence counting and a probabilistic cover. This does not imply centralized Kadison-Singer hardness: the surrogate has a direct O(dr)-time centralized solution.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: the deterministic one-way remote exact frame-partition game for perfect even r-uniform hypermatchings has communication complexity d log_2(2^r/binom(r,r/2)) + O(log(dr)), with the explicit lower cover ratio binom(dr,dr/2)/binom(r,r/2)^d and an upper cover of size at most ceil((ln H_{d,r}+1)(2^r/binom(r,r/2))^d)." }, { "id": 20000548, "problem_number": "AIM-ANALYSIS-0080", "title": "Directional hyperbolic families and the obstruction to a uniform leaf", "statement": "extend the theory of interlacing families to several variables", "original_statement": "extend the theory of interlacing families to several variables", "clean_statement": "extend the theory of interlacing families to several variables", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b7\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[79]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"extend the theory of interlacing families to several variables\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0080", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For same-degree homogeneous forms positive at a direction e, hyperbolicity of every nonzero nonnegative combination is equivalent to common interlacing of the univariate restrictions on every e-parallel line. This yields MSS leaf selection separately for each fixed slice, but not one leaf uniformly over all slices. Antipodal slices force two-sided spectral containment; for linear forms, a child controls a cone K exactly when its normalized-root difference from the parent lies in the dual cone K*, so all-space control requires normalized equality. A square-free quadratic pair gives a nondegenerate counterexample to uniform selection.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate novelty: the directional-compatibility framework gives an exact slice/common-interlacing equivalence and isolates the synchronization failure of multivariate leaf selection; in degree one, cone-uniform selection holds exactly under the dual-cone certificate r-r_i in K*, while the explicit square-free pair s(s-2z), (s-z)(s+z) shows that even hyperbolic compatibility of every conic combination does not produce one child controlling all slices." }, { "id": 20000549, "problem_number": "AIM-ANALYSIS-0081", "title": "Sharp root motion for a first-order stability preserver", "statement": "how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\n\n* Petter Branden", "original_statement": "how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\n \n* Petter Branden", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is `aim-analysis-notes.json`, zero-based index 80, from the AIM workshop *Beyond Kadison--Singer: paving and consequences*. Its extracted text is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b8\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[80]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"how do operators that preserve stability affect locations of roots (i.e., quantitative versions of MSS)\\n \\n* Petter Branden\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0081", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every degree-n real-rooted polynomial p with largest root A and root diameter W, and every t>0, p-tp' has a unique root above A, which is its largest root; its displacement u lies in an explicit sharp interval [L_{n,t}(W), U_{n,t}(W)], with the lower and upper endpoints attained exactly by the two-point root multisets {A,B,...,B} and {A,...,A,B}. The proof also gives coordinatewise gap monotonicity. Separately, the family I-aD^2 shows that qualitative stability preservation plus fixing constants and linear polynomials cannot yield an operator-independent displacement bound. An exact Rayleigh-deficit barrier identity is included as a standard synthesis connecting the univariate calculation to multivariate MSS methods, not as a novelty claim.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: among degree-n real-rooted polynomials with fixed largest root A, root diameter W, and t>0, the largest-root displacement under I-tD is bounded by the explicit functions L_{n,t}(W) and U_{n,t}(W) in the artifacts; both bounds are sharp, with exact two-point extremizers, and the displacement is monotone decreasing under coordinatewise enlargement of the sorted root-gap vector." }, { "id": 20000550, "problem_number": "AIM-ANALYSIS-0082", "title": "Exact lower witnesses in the complete-edge Weaver family", "statement": "for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.", "original_statement": "for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.", "clean_statement": "for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\sqrt\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.", "statement_status": "exact", "statement_verification": "### Exact canonical record", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b9\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[81]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"for the random MSS, there's a witness for the best bound, is there a conjecture for Weaver's conjecture? Weaver has a construction that \\\\sqrt\\\\eps is necessary but MSS has no constructive lower bound. Pete C. *does* have a constructive lower bound.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0082", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the equal-norm Parseval frame formed by the normalized edge vectors of K_n, every signing has operator norm at least 1/sqrt(n), with the stronger bound 1/sqrt(n-1) for even n. Equality in the first bound is equivalent to the signing matrix being the core of a normalized symmetric conference matrix. Paley signings attain equality for every prime power q congruent to 1 modulo 4, giving the exact discrepancy W(V_q)=1/sqrt(q)=sqrt(alpha_q/2), alpha_q=2/q, and the exact optimal larger two-partition block norm 1/2+1/(2sqrt(q)).\n\nCandidate contribution (theorem; novelty confidence low): The normalized complete-edge Parseval frame has a universal 1/sqrt(n) signing lower bound, its equality signings are exactly normalized symmetric conference-matrix cores, and Paley cores give an infinite explicit family with exact Weaver discrepancy sqrt(alpha/2)." }, { "id": 20000551, "problem_number": "AIM-ANALYSIS-0083", "title": "A spectral multiplicity refinement of the Casazza DFT near-counterexample", "statement": "Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.", "original_statement": "Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "It occurs in the notes for the AIM workshop *Beyond Kadison--Singer: paving and consequences* (December 1--5, 2014), under a list headed by Petter Brändén. The immediately preceding source bullets ask for matching quantitative lower bounds, say that the real paving lower bound has order \\(\\varepsilon^{-2}\\), and say that “Pete C.” has a constructive lower bound. The record itself does not define “Pete,” identify the example, specify a conjecture, or say what “better” means. Thus no unique mathematical statement can be recovered from the record alone.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b10\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[82]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Pete's example: can we make it better? It will affect all of the equivalences! It's almost a counter-example.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0083", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the strongest source-supported reading of the ambiguous workshop bullet, the explicit Casazza--Fickus--Mixon--Tremain two-block DFT construction admits an exact spectral refinement: every s-row subset of the first DFT block with s at least n has Gram compression P[S] = (1/(n+1)) I + K_S with K_S positive semidefinite of rank exactly n-1. Thus the flat eigenvalue 1/(n+1) has multiplicity s-n+1, every bipartition forces at least two near-unpaved modes across the two complementary-projection compression spectra, and one zero-diagonal reflection block has norm at least (n-1)/(n+1). A Weyl-stability corollary and an exact even/odd replication no-amplification lemma are also proved; the separate conference-matrix reading is identified and its known epsilon^{-2} lower bound is derived.\n\nCandidate contribution (lemma; novelty confidence low): For the explicit two-block DFT projection, every first-block principal compression P[S] with |S|=s at least n has eigenvalue 1/(n+1) with exact multiplicity s-n+1; consequently every bipartition forces at least two eigenvalues at least n/(n+1) across the two complementary-projection compression spectra." }, { "id": 20000552, "problem_number": "AIM-ANALYSIS-0084", "title": "Facewise strongly-Rayleigh completion and a sharp marginal packing threshold", "statement": "consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?", "original_statement": "consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?", "clean_statement": "consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?", "statement_status": "exact", "statement_verification": "The canonical record is problem **b11** from the AIM workshop *Beyond Kadison--Singer: paving and consequences*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b11\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[83]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"consequences to matroid theory: can we prove a continuous version of matroid theory using MSS theorem and what consequences to matroid theory does MSS have?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0084", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any weak-half-plane-property matroid with a fixed positive stable basis witness, every point x of the closed base polytope is exactly the marginal vector of an external-field strongly Rayleigh law supported on the bases in the minimal face containing x. Consequently, Anari--Oveis Gharan spectral selection can round x to a basis in that same face, and the matroid has k disjoint bases if and only if it admits such a law with all marginals at most 1/k. The threshold 1/k is universally sharp, even for uniform strongly Rayleigh laws. The marginal-completion theorem also extends to every matroid with Lorentzian in place of strongly Rayleigh.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: stable restriction to the minimal face followed by a relative-entropy external-field projection realizes every closed base-polytope point of a WHPP matroid as exact strongly-Rayleigh marginals on that face; this gives face-preserving spectral rounding and an exact, sharp 1/k strongly-Rayleigh characterization of k-base packing." }, { "id": 20000553, "problem_number": "AIM-ANALYSIS-0085", "title": "Exact finite Feichtinger partition count and a sharp ceiling obstruction", "statement": "interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?", "original_statement": "interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?", "clean_statement": "interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?", "statement_status": "exact", "statement_verification": "The corpus record is item `b12` from the AIM workshop *Beyond Kadison--Singer: paving and consequences*. Its extracted text asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b12\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[84]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"interested in Feichtinger conjecture: suppose you have a Parseval frame and norms of vectors are bounded away from zero, what is the minimum number of partitions can we form that make Riesz basic sequences? Count the number of partitions? How does this number depend on \\\\epsilon, the lower bound (Pete C. also will show some results tomorrow on this question)? How to exhibit/construct such partitions?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0085", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The published Bownik-Casazza-Marcus-Speegle theorem solves the requested asymptotic with N(epsilon) = Theta(epsilon^-2). This attempt proves the sharpened lower bound N(epsilon) >= ceil(epsilon^-2) for the original countable complex Parseval class, proves the exact finite unquantified formula N_fin(epsilon) = ceil(epsilon^-2), and gives both an explicit full-spark Fourier witness for every parameter and an exact-arithmetic polynomial-time construction via linear-matroid partition.\n\nCandidate contribution (theorem; novelty confidence low): For every 0 < epsilon <= 1, a rectangular Fourier Parseval frame yields the all-parameter ceiling obstruction N(epsilon) >= ceil(epsilon^-2); combined with the standard Rado-Horn upper bound, this gives the exact finite formula N_fin(epsilon) = ceil(epsilon^-2) and an optimal finite matroid-partition algorithm." }, { "id": 20000554, "problem_number": "AIM-ANALYSIS-0086", "title": "Equal-measure sets can have different exponential-frame partition data", "statement": "whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?", "original_statement": "whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?", "clean_statement": "whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ANALYSIS-0086, item b13 in the AIM workshop “Beyond Kadison-Singer: paving and consequences.” Its exact extracted problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b13\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[85]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"whether the number of partitions depends on |E| for exponential frames or any other properties of E? Are there any examples of two sets with same measure such that the number of partitions is different (or the constants are different)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0086", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "There exist measurable subsets E_0 and E_* of the torus, both of measure 1/2, for which the restricted integer exponentials have minimum Riesz-sequence partition counts r(E_0)=2 and r(E_*)>2, respectively. The second set is an affine normalization of a gap-shifted Kozma-Nitzan-Olevskii construction, and the obstruction follows from the Enstad-van Velthoven critical-density theorem and complement density. In addition, every rational grid union E_A has exact count ceil(N/|A|), and the explicit equal-measure grid sets [0,1/2) and [0,1/4) union [3/8,5/8) both have count two but provably different unrestricted optimal common two-piece lower Riesz constants.\n\nCandidate contribution (theorem; novelty confidence low): A position-invariant version of the Kozma-Nitzan-Olevskii recursive set can be scaled into the torus with measure 1/2 so that its restricted integer exponential frame cannot be partitioned into two Riesz sequences, while the interval of measure 1/2 can; independently, two explicit rational-grid sets of measure 1/2 and exact count two have different unrestricted optimal common lower Riesz constants." }, { "id": 20000555, "problem_number": "AIM-ANALYSIS-0087", "title": "A half-measure Fourier frame with no two-Riesz partition", "statement": "More concretely, can we find E, |E| = \\frac12 such that\n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?", "original_statement": "More concretely, can we find E, |E| = \\frac12 such that \n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?", "clean_statement": "More concretely, can we find E, |E| = \\frac12 such that\n\\{ \\econst^{2 \\pi \\iunit n \\ell \\chi_E \\}_{n \\in \\Z} cannot be partitioned into two Riesz sequences?", "statement_status": "exact", "statement_verification": "The canonical record is item b14 in the American Institute of Mathematics workshop list *Beyond Kadison--Singer: paving and consequences* (December 1--5, 2014). It appears under Darrin Speegle's name. The extracted formula is visibly damaged:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b14\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[86]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"More concretely, can we find E, |E| = \\\\frac12 such that \\n\\\\{ \\\\econst^{2 \\\\pi \\\\iunit n \\\\ell \\\\chi_E \\\\}_{n \\\\in \\\\Z} cannot be partitioned into two Riesz sequences?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0087", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "There exists a compact measurable set E contained in [0,1), of Lebesgue measure 1/2, such that no partition Z = A disjoint-union B makes both restricted exponential families indexed by A and B Riesz sequences in L^2(E). More strongly, E admits no exponential Riesz sequence, even with arbitrary real frequencies, whose upper Beurling density is 1/2. The construction shifts the recursively refined component in the Kozma–Nitzan–Olevskii no-Riesz-basis example from [2,3] to [3/2,5/2], affinely normalizes the resulting set into one torus period, and combines Enstad–van Velthoven's critical-density theorem with Landau's density inequality and complement counting.\n\nCandidate contribution (theorem; novelty confidence low): The Kozma–Nitzan–Olevskii construction remains a no-exponential-Riesz-basis set after its recursively refined component is shifted from [2,3] to [3/2,5/2]; the shifted set satisfies twice its measure greater than its diameter, so affine normalization plus the Enstad–van Velthoven critical-density theorem yields a compact half-measure torus set whose integer Fourier frame has no partition into two Riesz sequences." }, { "id": 20000556, "problem_number": "AIM-ANALYSIS-0088", "title": "Constructive syndetic partitions via restricted DFT matroids and irrational rotations", "statement": "there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n\n* Bill Johnson", "original_statement": "there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n \n* Bill Johnson", "clean_statement": "there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\n\n* Bill Johnson", "statement_status": "exact", "statement_verification": "The canonical record in `aim-analysis-notes.json`, at zero-based index 87, literally reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b15\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[87]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"there's a connection to synthetic sets and arithmetic progressions (Bourgain's work with Tzafriri): we now know we can partition exponential frames into synthetic sets (MSS says we can do so). How to do this (constructively?)?\\n \\n* Bill Johnson\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0088", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a spectrum formed by s cells of an N-cell rational grid, restricted-DFT fiberization and Edmonds matroid partition construct a globally optimal partition of the integers into ceil(N/s) periodic syndetic Riesz sequences, with exact Riesz bounds given by squared singular values divided by N. For every open spectrum S of normalized measure m, an explicit equal-window irrational-rotation coloring gives a full partition into floor(1/m)+1 Riesz classes, each with gap at most twice that number. A Londner--Olevskii consequence shows that some measurable spectra admit no nonempty periodic Riesz set, so the arbitrary-measurable constructive problem remains unresolved.\n\nCandidate contribution (synthesis; novelty confidence low): Candidate synthesis: for every rational-grid spectrum S_A, the vector matroid of the restricted N-point DFT has arboricity exactly ceil(N/|A|); matroid partition therefore yields a globally optimal periodic syndetic Riesz coloring of all integers, and the color submatrices supply finite singular-value certificates." }, { "id": 20000557, "problem_number": "AIM-ANALYSIS-0089", "title": "Induced ell-p paving endpoints and simultaneous all-p conference paving", "statement": "Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$.\n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t.\n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j\n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary;\nMatrix norm inequalities and the relative Dixmier property.\nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$.", "original_statement": "Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$. \n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t. \n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j \n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary; \nMatrix norm inequalities and the relative Dixmier property. \nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$.", "clean_statement": "Is paving true for $\\ell_p^n$? Specifically:\n\nSuppose $1\\le p<\\infty$.\n\n$Pav_p$: Does there exist for every $\\epsilon > 0$ a constant $k=k(\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\|A\\|_p = 1$, there is a partition $\\sigma(1), \\dots, \\sigma(k) of $\\{1,\\dots,n\\}$ s.t.\n\n$$\n\\|\\sum_{j=1}^k P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p = \\max_j\n\\| P_{\\sigma(j)} A P_{\\sigma(j)} \\|_p \\le \\epsilon .\n$$\n\nHere $\\|\\dot \\|_$ is the operator norm for operators on $\\ell_p^n$, NOT the Schatten $p$-class norm.\n\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\n\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary;\nMatrix norm inequalities and the relative Dixmier property.\nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\n\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\ell_p^n$, but they got optimal estimates only for $p=2$.", "statement_status": "exact", "statement_verification": "The record is Bill Johnson's question from the AIM workshop *Beyond Kadison--Singer: paving and consequences*. The exact Google-document export agrees with `input.json`. It asks, for \\(1\\leq p<\\infty\\):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b16\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[88]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Is paving true for $\\\\ell_p^n$? Specifically:\\n\\nSuppose $1\\\\le p<\\\\infty$. \\n\\n$Pav_p$: Does there exist for every $\\\\epsilon > 0$ a constant $k=k(\\\\epsilon, p)$ s.t. for every $n$ and every $n$ by $n$ zero diagonal matrix $A$ for which $\\\\|A\\\\|_p = 1$, there is a partition $\\\\sigma(1), \\\\dots, \\\\sigma(k) of $\\\\{1,\\\\dots,n\\\\}$ s.t. \\n\\n$$\\n\\\\|\\\\sum_{j=1}^k P_{\\\\sigma(j)} A P_{\\\\sigma(j)} \\\\|_p = \\\\max_j \\n\\\\| P_{\\\\sigma(j)} A P_{\\\\sigma(j)} \\\\|_p \\\\le \\\\epsilon .\\n$$\\n\\nHere $\\\\|\\\\dot \\\\|_$ is the operator norm for operators on $\\\\ell_p^n$, NOT the Schatten $p$-class norm.\\n\\n$Pav_1$, due IIRC to Bourgain has been known for a long time and is relatively easy. It is more-or-less equivalent to $Pav_2$ for matrices that have non-negative entries, a result that is contained in\\n\\nBerman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary; \\nMatrix norm inequalities and the relative Dixmier property. \\nIntegral Equations Operator Theory 11 (1988), no. 1, 28-48.\\n\\nIn their original paper, Bourgain and Tzafriri proved restricted invertibility for $\\\\ell_p^n$, but they got optimal estimates only for $p=2$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0089", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The induced-operator-norm problem is quantitatively true at p=1 and p=2, with R_1(epsilon) at most ceil(2/epsilon), R_2(epsilon) of order epsilon^{-2}, and exact conjugate duality R_p=R_{p'}. No general settlement was located for 1=epsilon^{-min(p,p')}. An explicit exactly 2-pavable rank-one direct-sum family proves that naive endpoint interpolation can nevertheless have unbounded dimension-dependent defect.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem/synthesis: every real symmetric conference matrix has one partition into at most ceil(54 epsilon^{-2}) blocks which is an epsilon-paving simultaneously for all induced ell-p operator norms, 1<=p<=infinity; over the Paley family, every fixed-p uniform paving requires at least epsilon^{-min(p,p')} blocks." }, { "id": 20000558, "problem_number": "AIM-ANALYSIS-0090", "title": "Support-coloring cases and a recursive paving bottleneck", "statement": "Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell", "original_statement": "Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell", "clean_statement": "Take A = n \\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\n\n* Deanna Needell", "statement_status": "exact", "statement_verification": "The canonical record (AIM, *Beyond Kadison--Singer: paving and consequences*, item b17, attributed to Deanna Needell) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b17\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[89]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Take A = n \\\\times n matrix that's a commutator and write it as a commutator of B,C with B diagonal of norm 1 and C with bounded norm, with some minimization. Bill's recent PNAS paper with Ozawa and Schechtmann showed something that needed KS with optimal estimates to improve bound from n^\\\\alpha (which uses only restricted invertibility) to poly(n), and that the existence of a bound independent of n implies KS. Can you improve the result using KS to get a bound independent of n?\\n\\n* Deanna Needell\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0090", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The universal dimension-free operator-norm commutator bound remains open. This attempt proves that every zero-diagonal matrix whose undirected support graph is k-colorable has a diagonal-unitary commutator representation with factor at most (1/2) sum_{d=1}^{k-1} csc(pi d/k), and that the factor is exactly 1/2 for bipartite support. It also proves an explicit dimension-free criterion for a nested (r,q)-paving encoded by separated planar spectral clusters, and shows by disk packing that this architecture necessarily requires q= sqrt(r)/e. A further lemma shows two-sided well-conditioned frame blocks each span the full ambient space. No improvement of the universal exponent is claimed.\n\nCandidate contribution (theorem/synthesis; novelty confidence low): The candidate contribution is the combined block-gluing inequality, its explicit b/a=o(r^(1/3)) quadratic threshold for an o(n^(5/6)) conclusion, and the exact factorial determinant-volume obstruction showing a sqrt(r) global penalty despite distortion-one cube sections.", "research_difficulty_suggested": "L4" }, { "id": 20000561, "problem_number": "AIM-ANALYSIS-0093", "title": "Algorithmic Kadison-Singer beyond weighted BSS sparsification", "statement": "Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n\n* Leonid Gurvits", "original_statement": "Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n \n* Leonid Gurvits", "clean_statement": "Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\n\n* Leonid Gurvits", "statement_status": "exact", "statement_verification": "The canonical AIM record (Analysis, source index 92, item b20 from the 2014 workshop *Beyond Kadison--Singer: paving and consequences*) reads exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b20\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[92]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Algorithmic version of KS: try the Batson-S-S approach to KS? BUT, they had convinced themselves that their approach wouldn't work. Perhaps it is worth to revisit this?\\n \\n* Leonid Gurvits\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0093", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The broad polynomial-time Weaver/Kadison-Singer construction remains open, although BSS-inspired determinant barriers now solve the equal-norm dense regime m >= 49 d^2. This attempt proves a deterministic rounded-state dynamic program for rational PSD decompositions of the identity: in dimension d it returns a signing within additive τ of optimal operator-norm discrepancy using at most (6dm/τ)^(d(d+1)/2) states per layer. It also proves a one-dimensional counterexample to support-only deweighting of a BSS certificate and an α-discrepancy theorem for commuting rank-one summands.\n\nCandidate contribution (algorithm/theorem; novelty confidence low): For rational symmetric PSD matrices A_1,...,A_m summing to the identity, a deterministic rounded-state dynamic program computes a signing with operator-norm discrepancy at most OPT + τ using at most (6dm/τ)^(d(d+1)/2) states per layer and explicit rational bit complexity; consequently the MSS bound is constructive up to additive τ in every fixed dimension." }, { "id": 20000562, "problem_number": "AIM-ANALYSIS-0094", "title": "A mixed-discriminant reconstruction of the five-way spanning problem", "statement": "there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\n\n* Mirko Visontai", "original_statement": "there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\n \n* Mirko Visontai", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is a workshop note from *Beyond Kadison--Singer: paving and consequences*. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b21\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[93]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"there's an algorithm: start with k vectors in n-dimnl space. Can you partition them into 5 sets so that set spans the space. Take each vector, repeat it 5 times and augment those matrices by identity to get dimension k. You'll get k semi-def-pos. matrices. Now you check each matrix for some property xxxx. This is an algorithm to carry out Rado-Horn?\\n \\n* Mirko Visontai\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0094", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The literal workshop record is not a well-posed problem: it omits the property denoted by xxxx, the size hypothesis, and the matrix construction. For the mathematically coherent repaired formulation, a complete theorem is proved. Given indexed vectors v_1,...,v_m in F^n, an integer r with m >= rn, and q = m-rn, define A_i = (I_r tensor v_i v_i^*) direct-sum I_q. Then the mixed discriminant D(A_1,...,A_m) is positive if and only if the vectors contain r pairwise disjoint bases, if and only if they partition into r spanning classes, if and only if |[m] minus J| >= r(n-rank(J)) for every J. The case r=5 reconstructs every surviving phrase of the note. A separate proved tuple-padding theorem recovers classical Rado-Horn independent coloring when m <= rn.\n\nCandidate contribution (synthesis/reduction; novelty confidence low): Candidate novelty is an explicit two-sided identity-padding dictionary: when m > rn, adding a common identity block I_{m-rn} to every colored-lift matrix encodes r spanning classes; when m < rn, appending rn-m identity matrices as tuple arguments encodes r independent classes; at m = rn the encodings coincide. Literal uncolored r-fold copying is proved to collapse to independence or spanning of the original family and therefore cannot be the intended reduction.", "research_difficulty_suggested": "L2" }, { "id": 20000563, "problem_number": "AIM-ANALYSIS-0095", "title": "Quantitative counts of Weaver-good bipartitions", "statement": "understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.", "original_statement": "understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.", "clean_statement": "understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.", "statement_status": "exact", "statement_verification": "The exact canonical problem field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b22\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[94]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"understand limitations of the proof, we don't have any idea of how many good partitions there are? come up with more quantitative versions of the proof.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0095", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every rank-one Parseval frame with m vectors and squared vector norms at most alpha, where 0 < alpha <= 1/4, there are at least m+1 distinct unordered bipartitions satisfying the historical Marcus-Spielman-Srivastava discrepancy bound sqrt(8 alpha)+2 alpha. The proof combines a general weighted Hamming-stability theorem with the sharper Bownik-Casazza-Marcus-Speegle witness and its strict one-flip slack. Further proved results give a matrix-Bernstein count, an exact repeated-coordinate product count whose good fraction vanishes at certified KS thresholds, and an abstract example showing that common interlacing alone can certify only one favorable leaf.\n\nCandidate contribution (counting theorem and synthesis; novelty confidence low): Candidate novelty: the weighted subset-sum/Hamming-ball abundance theorem, when combined with the BCMS improvement, implies the explicit worst-case lower bound of m+1 unordered partitions at the original MSS threshold; the exact repeated-coordinate family simultaneously shows that no dimension-free positive-fraction strengthening is possible at that threshold." }, { "id": 20000564, "problem_number": "AIM-ANALYSIS-0096", "title": "Generating interlacing families by real-rootedness preservers", "statement": "can we generate new interlacing families?\n\n* Mihai Putinar", "original_statement": "can we generate new interlacing families?\n\n* Mihai Putinar", "clean_statement": "can we generate new interlacing families?\n\n* Mihai Putinar", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b23\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[95]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"can we generate new interlacing families?\\n\\n* Mihai Putinar\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0096", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A uniform-degree linear real-rootedness preserver maps every classical positive-leading interlacing family to another one. More generally, any bilinear map that sends every pair of positive-leading real-rooted inputs of the prescribed degrees to a positive-leading real-rooted output combines two interlacing-family trees into a new family under every order-preserving shuffle of their decision coordinates. Symmetric additive finite free convolution is a concrete instance. An exact quadratic example verifies all conditional interlacers while showing that the flattened leaves need not have a global common interlacer, and a degree-one example disproves diagonal pointwise-product closure.\n\nCandidate contribution (closure_theorem; novelty confidence low): Candidate shuffle-product closure theorem: a bilinear real-rootedness preserver sends the Cartesian product of two fixed-degree positive-leading interlacing-family trees to a new interlacing family for every shuffle preserving the internal order of each input's coordinates." }, { "id": 20000565, "problem_number": "AIM-ANALYSIS-0097", "title": "A phase-shift dictionary for Kadison-Singer and a free-phase obstruction", "statement": "understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n\n* Dan Edidin", "original_statement": "understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n \n* Dan Edidin", "clean_statement": "understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\n\n* Dan Edidin", "statement_status": "exact", "statement_verification": "The canonical record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b24\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[96]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"understand KS in terms of the phase shift theory, perhaps getting a proof of the KS in terms that analysts can understand\\n \\n* Dan Edidin\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0097", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The original document shows that the question belongs to Mihai Putinar and concerns spectral/scattering phase shifts arising from perturbation determinants, not phase retrieval. For a positive rank-one update, the report proves an exact equivalence among the normalized perturbation determinant, its Krein phase shift, and the resolvent barrier increment; it also proves that a one-step expected rank-one characteristic polynomial is exactly an effective deterministic rank-one update. Two explicit six-vector Parseval frames show that individual phase shifts relative to zero lose essential eigendirection information, while cumulative relative phase intervals yield the bound 1/r+(1-1/r)delta for every commuting rank-one decomposition of the identity. No full phase-shift proof of the noncommutative MSS/Kadison-Singer theorem is supplied; the multivariate barrier/resolvent step remains the precise gap.\n\nCandidate contribution (synthesis/obstruction; novelty confidence low): Candidate novelty is the explicit KS-oriented boundary theorem combining: exact effective rank-one realization of a one-step expected characteristic polynomial and its phase-moment barrier formula; a pair of equal-norm Parseval frames with identical individual free phase data but different exact signing optima; and sufficiency of cumulative relative phase intervals for the quantitative commuting bound 1/r+(1-1/r)delta." }, { "id": 20000566, "problem_number": "AIM-ANALYSIS-0098", "title": "Fixed-diagonal geometry and an exact 4x4 two-paving profile", "statement": "wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.", "original_statement": "wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.", "clean_statement": "wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b25\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[97]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"wants to geometrize this work! Paving projections is especially interesting. Fix r. Find the conditions on the diagonal of a projection so it can be r-paved? Conditions should be independent of dimension.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0098", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For complex rank-two projections P in M_4 with diagonal (1/2,1/2,1/2,1/2), the optimal nonempty direct-compression two-paving constant is pi_2(P)=1/2+min(|p_12|,|p_13|,|p_14|), and its exact range over the fiber is [1/2, 1/2+1/(2 sqrt(3))]; an explicit projection path realizes every value. Every real projection in the same fiber instead has pi_2=1/2. Thus the diagonal has distinct exact existential and universal quantitative constants and cannot decide (2,3/4)-pavability of an individual complex projection, although this does not separate mere strict two-pavability. The result is scoped to r=2, n=4, rank two, direct projection compression, and nonempty blocks; a separate proved proposition gives the exact diagonal bin-packing characterization in rank one.\n\nCandidate contribution (exact fixed-fiber extremal theorem and quantitative counterexample; novelty confidence low): Candidate novelty: the exact formula and full interval [1/2, 1/2+1/(2 sqrt(3))] for the complex half-diagonal M_4 two-paving profile, together with the real-fiber collapse to 1/2 and the resulting exact existential/universal separation." }, { "id": 20000567, "problem_number": "AIM-ANALYSIS-0099", "title": "A support-discrepancy formulation for zonotope generator partitions", "statement": "Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.", "original_statement": "Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.", "clean_statement": "Is there an analog of KS for zonotopes? Let z_1,\\ldots,z_k \\in \\R^n, k > n and let \\bar z_i denote the interval \\{ \\alpha z_i | 0 \\leq \\alpha \\leq 1\\}. Consider the Minkowski sum\n Zon = \\alpha_1 \\bar z_1 + \\cdots \\alpha_k \\bar z_k.", "statement_status": "exact", "statement_verification": "The canonical record (AIM Problem Lists, workshop notes for *Beyond Kadison--Singer: paving and consequences*) is attributed in the original Google document to Leonid Gurvits. It reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b26\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[98]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Is there an analog of KS for zonotopes? Let z_1,\\\\ldots,z_k \\\\in \\\\R^n, k > n and let \\\\bar z_i denote the interval \\\\{ \\\\alpha z_i | 0 \\\\leq \\\\alpha \\\\leq 1\\\\}. Consider the Minkowski sum\\n Zon = \\\\alpha_1 \\\\bar z_1 + \\\\cdots \\\\alpha_k \\\\bar z_k.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0099", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the centered, consistently weighted repair Z = sum_i [-v_i,v_i] of the ambiguous Gurvits source question, simultaneous approximate-half inclusions for both generator subzonotopes are exactly equivalent to the uniform inequality |sum_i epsilon_i ||| <= 2 eta sum_i ||. If every generator segment is contained in delta Z and d is the span dimension, a proved multiscale Rademacher-net argument gives a partition with relative support discrepancy at most min{1, 12 sqrt(3 delta d)}, hence eta <= min{1/2, 6 sqrt(3 delta d)}. Every partition has discrepancy at least max{0, 2 delta - 1}; an explicit full-dimensional volume-one box has delta = 1, so volume normalization alone cannot imply a nontrivial half-partition. Parallel generator classes admit the sharper proved error eta <= rho/2.\n\nCandidate contribution (partial theorem; novelty confidence low): Candidate contribution: under the directional small-segment condition [-v_i,v_i] subset delta Z, the centered unweighted generator bipartition has sandwich error eta <= min{1/2, 6 sqrt(3 delta d)}; for parallel classes it has eta <= rho/2, while a volume-one example and the bound max{0, 2 delta - 1} show that volume normalization cannot replace directional smallness." }, { "id": 20000568, "problem_number": "AIM-ANALYSIS-0100", "title": "What rank-one perturbation can and cannot eliminate", "statement": "how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n\n* Adam Marcus", "original_statement": "how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n \n* Adam Marcus", "clean_statement": "how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\n\n* Adam Marcus", "statement_status": "exact", "statement_verification": "The canonical record is item `b27` from the AIM workshop *Beyond Kadison--Singer: paving and consequences*. Its exact extracted text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b27\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[99]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"how to eliminate those polynomials?! Borgain-Tzafriri's proof did not have such polynomials, they used rank-1 perturbation theory. Can we reproduce such a proof at the level of multi-variate polynomials?\\n \\n* Adam Marcus\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0100", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a positive-definite determinant pencil, every distinct-index mixed derivative has an exact permutation-cycle expansion in resolvent traces. When every operated covariance slot is rank one, this hierarchy collapses to principal minors of a resolvent Gram matrix, so the full mixed differential operator is exactly a deterministic rank-one matrix shift, with singular intermediate shifts handled by adjugates or regularization. For a fixed positive-definite base matrix and a Hermitian covariance, this deterministic-shift interpretation for every shift parameter holds if and only if the covariance has rank at most one. A determinant-tilted expectation further makes barrier monotonicity equivalent, in the common positive-definite resolvent region, to an explicit covariance-domination inequality. These results eliminate explicit polynomial computation in the rank-one case but do not eliminate the real-stability, root-region, convexity, or interlacing logic needed for the MSS Kadison-Singer bound.\n\nCandidate contribution (equivalence; novelty confidence low): Candidate contribution: the determinant-tilted identity partial_k Phi_q^j = -E_tilde tr(R_X A_k R_X A_j) + Cov_tilde(tr(R_X A_j), tr(R_X A_k)), together with the sharp fixed-base rank-one deterministic-shift boundary, gives a testable separation between computational elimination of mixed determinant polynomials and logical elimination of stability; in the common positive-definite region, barrier monotonicity is exactly the displayed covariance-domination inequality." }, { "id": 20000569, "problem_number": "AIM-ANALYSIS-0101", "title": "A portable tail-certificate principle behind interlacing families", "statement": "one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!", "original_statement": "one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!", "clean_statement": "one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!", "statement_status": "exact", "statement_verification": "The exact canonical record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b28\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[100]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"one of the biggest comments we get is that KS was so popular because it was ubiquitous so it would require some new tool that other people could use in lots of other areas, unfortunately, these polynomial tools are not portable to other areas. If someone *did* prove KS in your area, what would it look like? Let's construct such a tool so that you have a hammer to bang away with in your area! So, please think about what this tool would like!\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0101", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source record is an aspirational Adam Marcus workshop prompt rather than a mathematical proposition. As an extracted portable tool, this attempt proves exact and robust tail-certificate descent on finite recursive outcome trees: affine compatibility, one-sided tail soundness, normalized growth kappa, compatibility error a, and absolute oracle error epsilon yield a leaf whose badness increases by at most (a+2 epsilon)/kappa per level. This is a rigorously scoped abstraction of the selection module, not a solution of Kadison--Singer and not a construction of such certificates in every domain.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: under the stated local guard, approximate affine compatibility, tail-soundness, growth, and oracle axioms, deterministic descent incurs the explicit additive loss (a+2 epsilon)/kappa at each level; a polynomial corollary and two exact examples show that this tail condition is necessary for the mechanism yet strictly weaker than common interlacing.", "research_difficulty_suggested": "L2" }, { "id": 20000570, "problem_number": "AIM-ANALYSIS-0102", "title": "The scalar-and-zero mixed-characteristic extremizer in dimension two", "statement": "There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?", "original_statement": "There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?", "clean_statement": "There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?", "statement_status": "exact", "statement_verification": "The exact canonical record is the following question from the AIM workshop *Beyond Kadison--Singer: paving and consequences* (December 1--5, 2014):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b29\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[101]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"There's a conjecture about repeated copies of Identity with zero padding that this gives you the worse bounds on roots?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0102", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The workshop fragment is identified with the unnumbered Marcus-Spielman-Srivastava conjecture that capped scalar copies of the identity followed by zeros maximize the largest root of the mixed characteristic polynomial. For d=2 and every fixed feasible m (m epsilon >= 2), this attempt proves the conjecture for arbitrary positive semidefinite Hermitian matrices, without assuming commutativity: mu(x)=(x-1)^2-sum_i det(A_i), and the unique maximizer up to permutation and zero padding consists of floor(2/epsilon) copies of (epsilon/2)I_2 and one residual scalar matrix when needed. An exact nonnegative deficit identity separates trace-packing loss from Frobenius anisotropy. The result does not resolve d>=3, whose exact extremizer remains open in the literature checked, and a displayed example shows that the stronger coordinatewise all-roots reading is false.\n\nCandidate contribution (theorem; novelty confidence low): For q=floor(2/epsilon), r=2-q epsilon, and t_i=tr(A_i), every feasible d=2 family satisfies (q epsilon^2+r^2)/4-sum_i det(A_i) = (q epsilon^2+r^2-sum_i t_i^2)/4 + (1/2)sum_i ||A_i-(t_i/2)I_2||_F^2; hence the capped scalar-and-zero family is the unique largest-root maximizer up to permutation and zero padding." }, { "id": 20000571, "problem_number": "AIM-ANALYSIS-0103", "title": "A one-sided endpoint criterion for largest-root selection", "statement": "Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.", "original_statement": "Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.", "clean_statement": "Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: GoogleDoc\nAIM domain: Analysis\nWorkshop: Beyond Kadison-Singer: paving and consequences\nSection: \nSource item: b30\nSource URL: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit\nCanonical location: aim-analysis-notes.json notes[102]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Using interlacing polynomials but if you only want the maximum root, you don't need all these other roots to exist, can we trim our conditions so that the machinery is more usable? Is there some weaker form of interlacing if you don't care about all the roots.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit", "tags": [ "aim", "AIM-ANALYSIS-0103", "aim-domain:analysis", "aim-workshop:edit", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For positive-leading real polynomials f_i, each having a largest real zero r_i, and strictly positive weights, let F be their weighted sum and r_- and r_+ the minimum and maximum of the r_i. The single condition F(r_-) <= 0 implies that F has a largest real zero in [r_-, r_+], so a child with largest zero no greater than the parent's is selectable. Imposing this one-point certificate at each node of a finite positive-mixture tree yields an MSS-style leaf selection theorem without equal degrees or real-rooted node polynomials. A common nonpositive right barrier is an easy structural sufficient condition, and explicit examples show strict separation from common interlacing.\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: the mixture-specific certificate p_v(min_u R(p_u)) <= 0 composes through a finite positive-mixture polynomial tree to select a leaf with largest real zero at most that of the root, even when node degrees vary and node polynomials have nonreal zeros; the report packages this with a common-right-barrier sufficient condition and sharp separating examples." }, { "id": 20000572, "problem_number": "AIM-ANALYSIS-0104", "title": "Linear bi-Lipschitz extension constants and affine perturbative certificates", "statement": "Linear dependence of constants in bi-Lipschitz extension theorems\n\nDo bi-Lipschitz extension theorems admit linear dependence on constants?", "original_statement": "Linear dependence of constants in bi-Lipschitz extension theorems\n\nDo bi-Lipschitz extension theorems admit linear dependence on constants?", "clean_statement": "Linear dependence of constants in bi-Lipschitz extension theorems\n\nDo bi-Lipschitz extension theorems admit linear dependence on constants?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.1, “Extensions,” from the AIM workshop *Mapping theory in metric spaces*. Its exact problem field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Extensions\nSource item: 1.1\nSource URL: http://aimpl.org/mappingmetric/1/\nCanonical location: aim-analysis-notes.json notes[103]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Linear dependence of constants in bi-Lipschitz extension theorems\\n\\nDo bi-Lipschitz extension theorems admit linear dependence on constants?\"\nOriginal remarks: [\"For bi-Lipschitz embeddings of the circle, the answer is yes for extension to the disk, and no for extension to the plane. Reference: Kovalev, L.V., \\\"Optimal extension of Lipschitz embeddings in the plane\\\", Bull. London Math. Soc. 51 (2019), no. 4, 622-632.\"]\nOriginal literature field (JSON string): \"Every $L$-bi-Lipschitz embedding of ${\\\\mathbb R}$ into ${\\\\mathbb R}^2$ extends to a $2000L$-bi-Lipschitz map of ${\\\\mathbb R}^2$. Is something similar for the circle? Daneri--Pratelli proved that every $L$-bi-Lipschitz embedding of ${\\\\mathbb S}^1$ into ${\\\\mathbb R}^2$ extends to a $CL^4$-bi-Lipschitz map of ${\\\\mathbb R}^2$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/1/", "tags": [ "aim", "AIM-ANALYSIS-0104", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The problem splits into inequivalent separate-constant and symmetric one-parameter formulations: published work gives separate linear control for disk extension and refutes it for general global plane extension, whereas the keyhole counterexample does not refute a symmetric O(K) global theorem, which remains apparently open in the literature checked. A proved perturbative theorem shows that if a global (L0, ell0)-bi-Lipschitz baseline G0 differs from boundary data f by a residual of Lipschitz seminorm delta < ell0, then f has a surjective global extension with constants (L0 + delta, ell0 - delta). Applied to affine baselines and Kovalev's keyhole family, this yields the quantitative obstruction m_aff(f_epsilon) <= 4 epsilon/(2 pi - 2 epsilon).\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: the explicit affine-extension-margin formulation, together with the proved deduction that Kovalev's universal keyhole separation bound forces every positive affine residual certificate to have margin at most 4 epsilon/(2 pi - 2 epsilon)." }, { "id": 20000573, "problem_number": "AIM-ANALYSIS-0105", "title": "Sharp scalar parameter stability and a finite square-root obstruction", "statement": "Dependence on parameters\n\nDo canonical Lipschitz extensions exist which depend nicely on parameters?", "original_statement": "Dependence on parameters\n\nDo canonical Lipschitz extensions exist which depend nicely on parameters?", "clean_statement": "Dependence on parameters\n\nDo canonical Lipschitz extensions exist which depend nicely on parameters?", "statement_status": "exact", "statement_verification": "The exact canonical AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Extensions\nSource item: 1.2\nSource URL: http://aimpl.org/mappingmetric/1/\nCanonical location: aim-analysis-notes.json notes[104]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Dependence on parameters\\n\\nDo canonical Lipschitz extensions exist which depend nicely on parameters?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Eva Kopecka (preprint) has constructed a Kirszbraun extension operator which depends continuously on parameters. Other types of dependence on parameters could be considered.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/1/", "tags": [ "aim", "AIM-ANALYSIS-0105", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For scalar-valued data on an arbitrary metric space, the midpoint of the McShane-Whitney extremal extensions is a canonical same-constant extension, is an isometry in boundary-data uniform distance, and obeys the proved joint bound ||B_{d,L}f-B_{d',L}g||_infinity <= delta+L eta for labeled source-metric perturbations, with sharp universal constants. In contrast, on a fixed five-point star metric pair with target R^2, explicit feasible data at uniform distance O(h^2) have uniquely forced hub values at distance h, so no extension selection on the feasible-data space is locally Holder of exponent greater than 1/2 near the limiting datum. The star is a general metric space, not a Hilbert domain, so this is a rigorous partial obstruction rather than a resolution of the Hilbert-domain problem.\n\nCandidate contribution (counterexample; novelty confidence low): Candidate novelty: the explicit fixed five-point star construction gives a selection-independent local obstruction to every Holder exponent alpha>1/2 by moving the four boundary centers exactly sqrt(1+5h^2)-1 in sup distance while positive active-normal weights force the unique extension value to move by h; paired with it is the joint scalar metric-kernel estimate delta+L eta." }, { "id": 20000574, "problem_number": "AIM-ANALYSIS-0106", "title": "Deterministic sharp-order Lee--Naor extension and finite gentle-partition aggregation", "statement": "Deterministic proof of the Lee--Naor Lipschitz extension theorem\n\nThe Lee--Naor theorem states that, given a doubling subset $A$ of a metric space $X$ and an $L$-Lipschitz map from $A$ to a Banach space $Y$, there exists a $C(\\log m)L$-Lipschitz extension $X\\to Y$. Here $m$ is the doubling constant of $A$.\n\nIs there a deterministic proof of the Lee--Naor Lipschitz extension theorem giving the sharp dependence on the doubling constant?", "original_statement": "Deterministic proof of the Lee--Naor Lipschitz extension theorem\n\nThe Lee--Naor theorem states that, given a doubling subset $A$ of a metric space $X$ and an $L$-Lipschitz map from $A$ to a Banach space $Y$, there exists a $C(\\log m)L$-Lipschitz extension $X\\to Y$. Here $m$ is the doubling constant of $A$.\n\nIs there a deterministic proof of the Lee--Naor Lipschitz extension theorem giving the sharp dependence on the doubling constant?", "clean_statement": "Deterministic proof of the Lee--Naor Lipschitz extension theorem\n\nThe Lee--Naor theorem states that, given a doubling subset $A$ of a metric space $X$ and an $L$-Lipschitz map from $A$ to a Banach space $Y$, there exists a $C(\\log m)L$-Lipschitz extension $X\\to Y$. Here $m$ is the doubling constant of $A$.\n\nIs there a deterministic proof of the Lee--Naor Lipschitz extension theorem giving the sharp dependence on the doubling constant?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.3 in the **Extensions** section of the AIM list *Mapping theory in metric spaces*. Its exact mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Extensions\nSource item: 1.3\nSource URL: http://aimpl.org/mappingmetric/1/\nCanonical location: aim-analysis-notes.json notes[105]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Deterministic proof of the Lee--Naor Lipschitz extension theorem\\n\\nThe Lee--Naor theorem states that, given a doubling subset $A$ of a metric space $X$ and an $L$-Lipschitz map from $A$ to a Banach space $Y$, there exists a $C(\\\\log m)L$-Lipschitz extension $X\\\\to Y$. Here $m$ is the doubling constant of $A$.\\n\\nIs there a deterministic proof of the Lee--Naor Lipschitz extension theorem giving the sharp dependence on the doubling constant?\"\nOriginal remarks: [\"There are deterministic proofs of similar results by Lang-Schlichenmaier and Brudnyi-Brudnyi.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/1/", "tags": [ "aim", "AIM-ANALYSIS-0106", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The original AIM question is fully answered: Brudnyi--Brudnyi gave a constructive integral-average proof with O(log M) loss, and Basso's deterministic Whitney-cover construction gives the explicit bound Lip(F) <= 10^5 log(M) Lip(f). Separately, this attempt proves that for every finite nonempty subset A, any K-gentle partition can be aggregated onto the uniform probability space A with at most |A| atoms, without increasing K and while preserving the induced extension operator for every Banach-valued map; it also proves that a single bounded partition cannot be uniformly padded on a connected metric space at a scale below its diameter.\n\nCandidate contribution (lemma; novelty confidence low): Finite selector-fibre aggregation: if A is finite and nonempty, every K-gentle partition relative to A has an operator-preserving realization on the uniform |A|-point probability space, with selector id_A and no increase in K; paired with the connected-space zero-padding obstruction, this gives a precise two-level derandomization principle." }, { "id": 20000575, "problem_number": "AIM-ANALYSIS-0107", "title": "Unordered tuples of metric trees: qualitative ALR versus injectivity", "statement": "Absolute Lipschitz retract constructions\n\n${\\mathbb Q}_Q(Y)$ denotes the space of unordered $Q$-tuples $y=[y_1,\\ldots,y_Q]$ of elements of $Y$ equipped with the metric $d(y,z) = \\min \\{ \\max \\{ d(y_j,z_{\\sigma(j)}) \\} : \\sigma \\in S_Q \\}$, where $S_Q$ denotes the symmetric group on $Q$ letters.\n\nIf $Y$ is an absolute Lipschitz retract, is the same true of the space ${\\mathbb Q}_Q(Y)$?", "original_statement": "Absolute Lipschitz retract constructions\n\n${\\mathbb Q}_Q(Y)$ denotes the space of unordered $Q$-tuples $y=[y_1,\\ldots,y_Q]$ of elements of $Y$ equipped with the metric $d(y,z) = \\min \\{ \\max \\{ d(y_j,z_{\\sigma(j)}) \\} : \\sigma \\in S_Q \\}$, where $S_Q$ denotes the symmetric group on $Q$ letters.\n\nIf $Y$ is an absolute Lipschitz retract, is the same true of the space ${\\mathbb Q}_Q(Y)$?", "clean_statement": "Absolute Lipschitz retract constructions\n\n${\\mathbb Q}_Q(Y)$ denotes the space of unordered $Q$-tuples $y=[y_1,\\ldots,y_Q]$ of elements of $Y$ equipped with the metric $d(y,z) = \\min \\{ \\max \\{ d(y_j,z_{\\sigma(j)}) \\} : \\sigma \\in S_Q \\}$, where $S_Q$ denotes the symmetric group on $Q$ letters.\n\nIf $Y$ is an absolute Lipschitz retract, is the same true of the space ${\\mathbb Q}_Q(Y)$?", "statement_status": "exact", "statement_verification": "This is Problem 1.4, “Absolute Lipschitz retract constructions,” in the “Extensions” section of the AIM workshop list *Mapping theory in metric spaces*. The canonical record (source file `aim-analysis-notes.json`, index 106) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Extensions\nSource item: 1.4\nSource URL: http://aimpl.org/mappingmetric/1/\nCanonical location: aim-analysis-notes.json notes[106]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Absolute Lipschitz retract constructions\\n\\n${\\\\mathbb Q}_Q(Y)$ denotes the space of unordered $Q$-tuples $y=[y_1,\\\\ldots,y_Q]$ of elements of $Y$ equipped with the metric $d(y,z) = \\\\min \\\\{ \\\\max \\\\{ d(y_j,z_{\\\\sigma(j)}) \\\\} : \\\\sigma \\\\in S_Q \\\\}$, where $S_Q$ denotes the symmetric group on $Q$ letters.\\n\\nIf $Y$ is an absolute Lipschitz retract, is the same true of the space ${\\\\mathbb Q}_Q(Y)$?\"\nOriginal remarks: [\"The space ${\\\\mathbb Q}_Q(Y)$ is similar to the $Q$th symmetric product space; similar questions could be of interest for symmetric products.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/1/", "tags": [ "aim", "AIM-ANALYSIS-0107", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted qualitative ALR-inheritance question remains open. For every complete R-tree T and every Q at least 2, Goblet's theorem implies that the bottleneck tuple space Q_Q(T) is an ordinary absolute Lipschitz retract, while the present proof classifies its constant-one behavior: Q_Q(T) is injective if and only if T has no branch point. At any branch point an explicit padded triple has pairwise distance 2r and circumradius exactly 3r/2, so the uniform absolute retraction constant is at least 3/2.\n\nCandidate contribution (theorem; novelty confidence low): For each complete R-tree T and Q at least 2, Q_Q(T) with the bottleneck metric is injective exactly when T is branchless; in the branched case an explicit three-center configuration has normalized circumradius 3/2 and forces every uniform absolute retraction constant to be at least 3/2." }, { "id": 20000576, "problem_number": "AIM-ANALYSIS-0108", "title": "Exact finite localization and summable Hilbert-block extensions for the L2-to-L1 problem", "statement": "Banach space pairs with the Lipschitz extension property\n\nDoes the pair $(L^2,L^1)$ have the Lipschitz extension property?", "original_statement": "Banach space pairs with the Lipschitz extension property\n\nDoes the pair $(L^2,L^1)$ have the Lipschitz extension property?", "clean_statement": "Banach space pairs with the Lipschitz extension property\n\nDoes the pair $(L^2,L^1)$ have the Lipschitz extension property?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem 1.5 in the “Extensions” section of the January 2012 workshop *Mapping theory in metric spaces*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Extensions\nSource item: 1.5\nSource URL: http://aimpl.org/mappingmetric/1/\nCanonical location: aim-analysis-notes.json notes[107]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Banach space pairs with the Lipschitz extension property\\n\\nDoes the pair $(L^2,L^1)$ have the Lipschitz extension property?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This problem has been asked by Naor and coauthors in several papers.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/1/", "tags": [ "aim", "AIM-ANALYSIS-0108", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If a Banach target Y is the range of a norm-one projection from its canonical bidual, then for every metric domain X its global Lipschitz extension modulus equals both the supremum of its finite-boundary moduli and the supremum of its fully finite local extension moduli. Applied separately to Y = L1[0,1] and Y = ell1, this gives the exact no-loss equality e(H,Y) = sup_m e(ell2^m,Y) = sup_{m,n} e_n(ell2^m,Y) for an infinite-dimensional separable Hilbert space H. A second proved criterion extends every map admitting a summable decomposition into finite-dimensional Hilbertian target blocks, with Lipschitz loss bounded by the total block cost. These results reduce but do not solve the open question, because finiteness of the common modulus is not proved and the two L1 targets are not conflated.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novel synthesis: for each target L1[0,1] and ell1 separately, the Hilbert-to-target extension problem is exactly, with no constant loss, the uniform boundedness problem over finite subsets of finite-dimensional Hilbert spaces; maps with finite summable Hilbert-block cost form an explicit globally extendable class." }, { "id": 20000577, "problem_number": "AIM-ANALYSIS-0109", "title": "Quantitative local obstruction and conditional Nagata upper bound for Hadamard ALRs", "statement": "Absolute Lipschitz retracts\n\nAre all Hadamard manifolds absolute Lipschitz retracts?", "original_statement": "Absolute Lipschitz retracts\n\nAre all Hadamard manifolds absolute Lipschitz retracts?", "clean_statement": "Absolute Lipschitz retracts\n\nAre all Hadamard manifolds absolute Lipschitz retracts?", "statement_status": "exact", "statement_verification": "The canonical record is number 1.6, under “Extensions,” in the AIM list *Mapping theory in metric spaces*. Its exact mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Extensions\nSource item: 1.6\nSource URL: http://aimpl.org/mappingmetric/1/\nCanonical location: aim-analysis-notes.json notes[108]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Absolute Lipschitz retracts\\n\\nAre all Hadamard manifolds absolute Lipschitz retracts?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/1/", "tags": [ "aim", "AIM-ANALYSIS-0109", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted question remains open for Hadamard manifolds of dimension at least four (while dimensions at most three are known positively). This attempt proves that every m-dimensional Riemannian manifold has absolute Lipschitz retraction constant at least sqrt(2m/(m+1)), and shows conditionally that a Hadamard manifold satisfying Nagata(n,c) has absolute retraction constant at most 1000(c+1) log_2(n+2) by Basso's extension theorem. Finite Nagata dimension is used only as a sufficient condition and is not claimed necessary.\n\nCandidate contribution (theorem; novelty confidence low): Every m-dimensional smooth Riemannian manifold with its intrinsic distance has absolute Lipschitz retraction constant at least sqrt(2m/(m+1))." }, { "id": 20000578, "problem_number": "AIM-ANALYSIS-0110", "title": "Lipschitz homotopy of Heisenberg groups and an infinite-order Hopf class", "statement": "Lipschitz homotopy groups of the Heisenberg group\n\n${\\mathbb H}^n$ denotes the $n$th Heisenberg group, equipped with the Carnot-Carath\\'eodory metric. $\\pi_k^{\\scriptstyle{Lip}}(X)$ denotes the $k$th Lipschitz homotopy group of a metric space $X$. It is known that $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n) \\ne 0$ (DeJarnette--Hajlasz--Lukyanenko--Tyson) \\cite{arXiv:1109.4641}.\n\nIf $\\pi_k({\\mathbb S}^n)$ is nontrivial, is the same true for $\\pi_k^{\\scriptstyle{Lip}}({\\mathbb H}^n)$? In particular, is $\\pi_3^{\\scriptstyle{Lip}}({\\mathbb H}^2) \\ne 0$?\n\\label{Heis-Lip}", "original_statement": "Lipschitz homotopy groups of the Heisenberg group\n\n${\\mathbb H}^n$ denotes the $n$th Heisenberg group, equipped with the Carnot-Carath\\'eodory metric. $\\pi_k^{\\scriptstyle{Lip}}(X)$ denotes the $k$th Lipschitz homotopy group of a metric space $X$. It is known that $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n) \\ne 0$ (DeJarnette--Hajlasz--Lukyanenko--Tyson) \\cite{arXiv:1109.4641}.\n\nIf $\\pi_k({\\mathbb S}^n)$ is nontrivial, is the same true for $\\pi_k^{\\scriptstyle{Lip}}({\\mathbb H}^n)$? In particular, is $\\pi_3^{\\scriptstyle{Lip}}({\\mathbb H}^2) \\ne 0$?\n\\label{Heis-Lip}", "clean_statement": "Lipschitz homotopy groups of the Heisenberg group\n\n${\\mathbb H}^n$ denotes the $n$th Heisenberg group, equipped with the Carnot-Carath\\'eodory metric. $\\pi_k^{\\scriptstyle{Lip}}(X)$ denotes the $k$th Lipschitz homotopy group of a metric space $X$. It is known that $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n) \\ne 0$ (DeJarnette--Hajlasz--Lukyanenko--Tyson) \\cite{arXiv:1109.4641}.\n\nIf $\\pi_k({\\mathbb S}^n)$ is nontrivial, is the same true for $\\pi_k^{\\scriptstyle{Lip}}({\\mathbb H}^n)$? In particular, is $\\pi_3^{\\scriptstyle{Lip}}({\\mathbb H}^2) \\ne 0$?\n\\label{Heis-Lip}", "statement_status": "exact", "statement_verification": "This is Problem 1.7 in the “Extensions” section of the AIM problem list from the 2012 workshop *Mapping theory in metric spaces*. The canonical source record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Extensions\nSource item: 1.7\nSource URL: http://aimpl.org/mappingmetric/1/\nCanonical location: aim-analysis-notes.json notes[109]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Lipschitz homotopy groups of the Heisenberg group\\n\\n${\\\\mathbb H}^n$ denotes the $n$th Heisenberg group, equipped with the Carnot-Carath\\\\'eodory metric. $\\\\pi_k^{\\\\scriptstyle{Lip}}(X)$ denotes the $k$th Lipschitz homotopy group of a metric space $X$. It is known that $\\\\pi_n^{\\\\scriptstyle{Lip}}({\\\\mathbb H}^n) \\\\ne 0$ (DeJarnette--Hajlasz--Lukyanenko--Tyson) \\\\cite{arXiv:1109.4641}.\\n\\nIf $\\\\pi_k({\\\\mathbb S}^n)$ is nontrivial, is the same true for $\\\\pi_k^{\\\\scriptstyle{Lip}}({\\\\mathbb H}^n)$? In particular, is $\\\\pi_3^{\\\\scriptstyle{Lip}}({\\\\mathbb H}^2) \\\\ne 0$?\\n\\\\label{Heis-Lip}\"\nOriginal remarks: [\"Solved by Hajlasz, Tyson, Schikorra, Wenger, Young... in HOMOTOPY GROUPS OF SPHERES AND LIPSCHITZ HOMOTOPY GROUPS OF HEISENBERG GROUPS as well as \\\"Lipschitz Homotopy Groups of the Heisenberg Groups\\\", to appear in GAFA (Geometric and Functional Analysis) in February 2014.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/1/", "tags": [ "aim", "AIM-ANALYSIS-0110", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The highlighted case is solved affirmatively: pi_3^Lip(H^2) is nonzero by the Hajlasz--Schikorra--Tyson rank-essentiality theorem, while the blanket implication from nonzero pi_k(S^n) to nonzero pi_k^Lip(H^n) remains open in the literature checked through 2026-07-28. In addition, the report proves that every standard HST class represented by a horizontal sphere embedding composed with a map of nonzero classical Hopf invariant has infinite order, not merely that it is nonzero.\n\nCandidate contribution (corollary; novelty confidence low): For every d >= 1, if Phi:S^{2d}->H^{2d} is a smooth horizontal embedding and f:S^{4d-1}->S^{2d} has nonzero classical Hopf invariant, then [Phi composed with f] has infinite order in pi_{4d-1}^Lip(H^{2d}); equivalently, Phi_* has infinite image and is injective on the cyclic subgroup generated by [f]." }, { "id": 20000579, "problem_number": "AIM-ANALYSIS-0111", "title": "A current-valued obstruction to torsion", "statement": "Lipschitz homotopy groups of the Heisenberg group II\n\nDoes $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n)$ contain torsion elements?", "original_statement": "Lipschitz homotopy groups of the Heisenberg group II\n\nDoes $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n)$ contain torsion elements?", "clean_statement": "Lipschitz homotopy groups of the Heisenberg group II\n\nDoes $\\pi_n^{\\scriptstyle{Lip}}({\\mathbb H}^n)$ contain torsion elements?", "statement_status": "exact", "statement_verification": "The neighboring AIM record gives these definitions and cites DHLT for $\\pi_n^{\\mathrm{Lip}}(\\mathbb H^n)\\neq 0$. No corruption or ambiguity in the displayed torsion question was found.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Extensions\nSource item: 1.8\nSource URL: http://aimpl.org/mappingmetric/1/\nCanonical location: aim-analysis-notes.json notes[110]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Lipschitz homotopy groups of the Heisenberg group II\\n\\nDoes $\\\\pi_n^{\\\\scriptstyle{Lip}}({\\\\mathbb H}^n)$ contain torsion elements?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"See problem \\\\ref{Heis-Lip} for definitions.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/1/", "tags": [ "aim", "AIM-ANALYSIS-0111", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The exact question whether pi_n^Lip(H^n) contains torsion remains open in the literature located through 2026-07-28. For every n at least 1, pushforward of the cubical fundamental current defines a well-defined homomorphism C_n from pi_n^Lip(H^n) to the torsion-free group of integral n-cycles in H^n; therefore every finite-order class lies in ker(C_n). Horizontal embedded-sphere classes, and more generally the current-detected exclusive-patch sector, have infinite order. This does not prove global torsion-freeness, and for n=1 it does not rule out current-zero commutator-type behavior or assert that the dilation subgroup is a free nonabelian group.\n\nCandidate contribution (reduction; novelty confidence low): For any complete purely (n+1)-unrectifiable metric space X, the assignment [f] to f_#[[Q^n]] is a natural cycle-valued homomorphism C_X: pi_n^Lip(X) -> Z_n^IC(X), so every finite-order Lipschitz homotopy class lies in its kernel; for H^n this gives a global torsion obstruction that is injective on exclusive-patch embedded-sphere spans." }, { "id": 20000580, "problem_number": "AIM-ANALYSIS-0112", "title": "Deterministic Assouad embeddings and a finite-scale stopping certificate", "statement": "Deterministic proofs of bi-Lipschitz embedding theorems\n\nAssouad's embedding theorem asserts that to each doubling metric space $(X,d)$ and each $\\epsilon \\in (0,1)$, there corresponds $n$ so that the snowflake metric space $(X,d^\\epsilon)$ admits a bi-Lipschitz embedding into ${\\mathbb R}^n$. Naor--Neiman proved that $n$ can be chosen independent of the snowflake parameter $\\epsilon$, for $\\epsilon$ near one (say, $\\epsilon>\\tfrac12$). Their proof is probabilistic and nonconstructive. Can one give an explicit construction of such embedding?\n\nGive a deterministic proof of Naor--Neiman's improved Assouad embedding theorem.\\label{deterministic bi-Lipschitz}", "original_statement": "Deterministic proofs of bi-Lipschitz embedding theorems\n\nAssouad's embedding theorem asserts that to each doubling metric space $(X,d)$ and each $\\epsilon \\in (0,1)$, there corresponds $n$ so that the snowflake metric space $(X,d^\\epsilon)$ admits a bi-Lipschitz embedding into ${\\mathbb R}^n$. Naor--Neiman proved that $n$ can be chosen independent of the snowflake parameter $\\epsilon$, for $\\epsilon$ near one (say, $\\epsilon>\\tfrac12$). Their proof is probabilistic and nonconstructive. Can one give an explicit construction of such embedding?\n\nGive a deterministic proof of Naor--Neiman's improved Assouad embedding theorem.\\label{deterministic bi-Lipschitz}", "clean_statement": "Deterministic proofs of bi-Lipschitz embedding theorems\n\nAssouad's embedding theorem asserts that to each doubling metric space $(X,d)$ and each $\\epsilon \\in (0,1)$, there corresponds $n$ so that the snowflake metric space $(X,d^\\epsilon)$ admits a bi-Lipschitz embedding into ${\\mathbb R}^n$. Naor--Neiman proved that $n$ can be chosen independent of the snowflake parameter $\\epsilon$, for $\\epsilon$ near one (say, $\\epsilon>\\tfrac12$). Their proof is probabilistic and nonconstructive. Can one give an explicit construction of such embedding?\n\nGive a deterministic proof of Naor--Neiman's improved Assouad embedding theorem.\\label{deterministic bi-Lipschitz}", "statement_status": "exact", "statement_verification": "This is Problem 2.1 in the “Embeddings” section of the AIM problem list from the 2012 workshop *Mapping theory in metric spaces*. In the notation of the canonical record, the problem asks for a deterministic, explicit proof of the following assertion:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Embeddings\nSource item: 2.1\nSource URL: http://aimpl.org/mappingmetric/2/\nCanonical location: aim-analysis-notes.json notes[111]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Deterministic proofs of bi-Lipschitz embedding theorems\\n\\nAssouad's embedding theorem asserts that to each doubling metric space $(X,d)$ and each $\\\\epsilon \\\\in (0,1)$, there corresponds $n$ so that the snowflake metric space $(X,d^\\\\epsilon)$ admits a bi-Lipschitz embedding into ${\\\\mathbb R}^n$. Naor--Neiman proved that $n$ can be chosen independent of the snowflake parameter $\\\\epsilon$, for $\\\\epsilon$ near one (say, $\\\\epsilon>\\\\tfrac12$). Their proof is probabilistic and nonconstructive. Can one give an explicit construction of such embedding?\\n\\nGive a deterministic proof of Naor--Neiman's improved Assouad embedding theorem.\\\\label{deterministic bi-Lipschitz}\"\nOriginal remarks: [\"Solved by Guy David and Marie Snipes in \\\"A Non-Probabilistic Proof of the Assouad Embedding Theorem with Bounds on the Dimension.\\\" Analysis and Geometry in Metric Spaces 1 (2013): 36-41.\"]\nOriginal literature field (JSON string): \"Urs Lang remarks that a potential approach to this problem would combine the existence of bi-Lipschitz embeddings of doubling R-trees into finite-dimensional Euclidean spaces with a suitable adaptation of prior results on bi-Lipschitz embeddings of spaces of finite Nagata dimension into products of R-trees. For the former result, see Section 2.3 (and the references therein) in Lee-Naor-Peres, Trees and Markov convexity, GAFA 18 (2009). [In fact, this is a result about the vertex set of a graph-theoretical tree with weighted edges (= of different lengths), but since doubling trees are separable it is easy to promote this to a result about completely general R-trees.]\"\nResearch attempt: 2; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/2/", "tags": [ "aim", "AIM-ANALYSIS-0112", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "David--Snipes fully answer the requested qualitative problem by giving a deterministic, non-probabilistic proof that every alpha-snowflake with 1/2 < alpha < 1 of a doubling metric space embeds bi-Lipschitzly into a Euclidean dimension depending only on the doubling constant. Their quantitative dimension and distortion bounds are weaker than Naor--Neiman's, and their arbitrary unbounded-space extension uses a diagonal compactness limit rather than supplying a finite algorithm. This attempt additionally proves an explicit finite-truncation certificate: on a finite metric, stopping the David--Snipes series once the next scale is below a stated multiple of the minimum separation preserves a positive co-Lipschitz bound without increasing the target dimension.\n\nCandidate contribution (corollary; novelty confidence low): For a finite C0-doubling metric space in the David--Snipes sparse-scale regime alpha > 2/3, with Q = 2 C0^5 blocks and scales r_k = tau^(2k), the partial map through scale K remains bi-Lipschitz whenever r_(K+1) <= delta (tau^3/(64 sqrt(Q)))^(1/alpha), where delta is the minimum nonzero distance; its lower coefficient is tau^5/16, and a logarithmic-in-aspect-ratio number of scales suffices." }, { "id": 20000581, "problem_number": "AIM-ANALYSIS-0113", "title": "SRA curvature and finite Carnot-tangent obstructions", "statement": "Curvature conditions and bi-Lipschitz embeddings\n\nLet $X$ be a doubling metric space. Can one formulate a curvature-type condition which, if satisfied by $X$, ensures that $X$ admits a bi-Lipschitz embedding into some finite-dimensional Euclidean space?", "original_statement": "Curvature conditions and bi-Lipschitz embeddings\n\nLet $X$ be a doubling metric space. Can one formulate a curvature-type condition which, if satisfied by $X$, ensures that $X$ admits a bi-Lipschitz embedding into some finite-dimensional Euclidean space?", "clean_statement": "Curvature conditions and bi-Lipschitz embeddings\n\nLet $X$ be a doubling metric space. Can one formulate a curvature-type condition which, if satisfied by $X$, ensures that $X$ admits a bi-Lipschitz embedding into some finite-dimensional Euclidean space?", "statement_status": "exact", "statement_verification": "This is Problem 2.3 in the **Embeddings** section of the AIM workshop list *Mapping theory in metric spaces*. The AIM page attributes it to Leonid Kovalev. The canonical record and the live AIM page agree, and no reconstruction or OCR repair is needed.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Embeddings\nSource item: 2.3\nSource URL: http://aimpl.org/mappingmetric/2/\nCanonical location: aim-analysis-notes.json notes[112]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Curvature conditions and bi-Lipschitz embeddings\\n\\nLet $X$ be a doubling metric space. Can one formulate a curvature-type condition which, if satisfied by $X$, ensures that $X$ admits a bi-Lipschitz embedding into some finite-dimensional Euclidean space?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Such curvature condition should rule out all nonabelian Carnot groups. Prior results in the spirit of this problem include Lang--Plaut's embedding result for doubling spaces admitting bicombings and geodesic extension properties.\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/2/", "tags": [ "aim", "AIM-ANALYSIS-0113", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal sufficient-condition question has a positive answer: Zolotov's SRA-free condition is a synthetic large-angle packing condition that quantitatively implies a bi-Lipschitz embedding into finite-dimensional Euclidean space, although it is sufficient rather than necessary and is not a characterization. In addition, the attempt proves that for each fixed target dimension N and finite distortion threshold D, any proper pointed Gromov--Hausdorff tangent T that has no D-distortion embedding into R^N yields a finite witness whose distortion into R^N remains greater than D in arbitrarily small balls of the original space. Pansu differentiability applies this to every nonabelian Carnot tangent. The witness depends on T, N, and D, and the proof supplies no effective cardinality, gap, or scale bound.\n\nCandidate contribution (theorem; novelty confidence low): For every fixed N >= 1 and finite D >= 1, a nonabelian Carnot pointed Gromov--Hausdorff tangent supplies a finite metric obstruction with Euclidean distortion c_N > D that transfers to finite subsets in arbitrarily small balls of the original space." }, { "id": 20000582, "problem_number": "AIM-ANALYSIS-0114", "title": "Lacunary orthogonal layers admit nonlinear dimension reduction", "statement": "Bi-Lipschitz embeddings: Hilbert spaces vs. finite-dimensional Euclidean spaces\n\nSuppose that $X$ is a doubling space which admits a bi-Lipschitz embedding into Hilbert space. Does $X$ necessarily admit a bi-Lipschitz embedding into some finite-dimensional Euclidean space?", "original_statement": "Bi-Lipschitz embeddings: Hilbert spaces vs. finite-dimensional Euclidean spaces\n\nSuppose that $X$ is a doubling space which admits a bi-Lipschitz embedding into Hilbert space. Does $X$ necessarily admit a bi-Lipschitz embedding into some finite-dimensional Euclidean space?", "clean_statement": "Bi-Lipschitz embeddings: Hilbert spaces vs. finite-dimensional Euclidean spaces\n\nSuppose that $X$ is a doubling space which admits a bi-Lipschitz embedding into Hilbert space. Does $X$ necessarily admit a bi-Lipschitz embedding into some finite-dimensional Euclidean space?", "statement_status": "exact", "statement_verification": "The canonical record is zero-based index 113 of `aim-analysis-notes.json`, Problem 2.4 in the “Embeddings” section of the AIM workshop list *Mapping theory in metric spaces*. Its exact question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Embeddings\nSource item: 2.4\nSource URL: http://aimpl.org/mappingmetric/2/\nCanonical location: aim-analysis-notes.json notes[113]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Bi-Lipschitz embeddings: Hilbert spaces vs. finite-dimensional Euclidean spaces\\n\\nSuppose that $X$ is a doubling space which admits a bi-Lipschitz embedding into Hilbert space. Does $X$ necessarily admit a bi-Lipschitz embedding into some finite-dimensional Euclidean space?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is a well known open problem, asked for instance by Naor and Neiman.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/2/", "tags": [ "aim", "AIM-ANALYSIS-0114", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general Lang-Plaut problem remains open in the primary literature located through 2026-07-28. A proved structural special case is obtained: if a Hilbert space is an orthogonal sum of layers X_j lying in uniform annuli at scales r_j with r_{j+1} at most lambda times r_j, and the layers have uniform doubling constants and uniform distortion-D embeddings into R^m, then their union with the origin is explicitly doubling and embeds bi-Lipschitzly into R^{m+1}, with constants depending only on the uniform parameters. The compact set consisting of the origin and lacunary points r_j e_j embeds into the line but defeats every bounded finite-rank linear map, showing that infinitely many orthogonal directions alone are not a nonlinear obstruction and that ambient linear projection is insufficient.\n\nCandidate contribution (theorem; novelty confidence low): For uniformly K-doubling subsets X_j of mutually orthogonal Hilbert summands, contained in shells a r_j <= ||x|| <= b r_j with r_{j+1} <= lambda r_j < r_j and admitting uniform distortion-D embeddings into R^m, the union {0} union all X_j is (1+M K^2)-doubling for M=1+ceil(log(2b/a)/log(1/lambda)) and embeds into R^{m+1} via the explicit scale-coordinate map x in X_j maps to (r_j,f_j(x)); a one-point-layer corollary simultaneously exhibits nonlinear line embeddability and failure of every bounded finite-rank linear map." }, { "id": 20000583, "problem_number": "AIM-ANALYSIS-0115", "title": "A strict critical-dimension obstruction for Heisenberg snowflakes", "statement": "Snowflake embeddings of the Heisenberg group into low-dimensional Euclidean spaces\n\nLet $d_{cc}$ denote the Carnot-Carath\\'eodory metric.\n\nDoes there exist $\\epsilon<1$ so that the first Heisenberg group ${\\mathbb H}^1$ equipped with the metric $d_{cc}^\\epsilon$ bi-Lipschitz embeds into ${\\mathbb R}^5$?", "original_statement": "Snowflake embeddings of the Heisenberg group into low-dimensional Euclidean spaces\n\nLet $d_{cc}$ denote the Carnot-Carath\\'eodory metric.\n\nDoes there exist $\\epsilon<1$ so that the first Heisenberg group ${\\mathbb H}^1$ equipped with the metric $d_{cc}^\\epsilon$ bi-Lipschitz embeds into ${\\mathbb R}^5$?", "clean_statement": "Snowflake embeddings of the Heisenberg group into low-dimensional Euclidean spaces\n\nLet $d_{cc}$ denote the Carnot-Carath\\'eodory metric.\n\nDoes there exist $\\epsilon<1$ so that the first Heisenberg group ${\\mathbb H}^1$ equipped with the metric $d_{cc}^\\epsilon$ bi-Lipschitz embeds into ${\\mathbb R}^5$?", "statement_status": "exact", "statement_verification": "The canonical record is problem 2.4 in the **Embeddings** section of the AIM workshop list *Mapping theory in metric spaces* (`aim-analysis-notes.json`, zero-based record index 114). Its question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Embeddings\nSource item: 2.4\nSource URL: http://aimpl.org/mappingmetric/2/\nCanonical location: aim-analysis-notes.json notes[114]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Snowflake embeddings of the Heisenberg group into low-dimensional Euclidean spaces\\n\\nLet $d_{cc}$ denote the Carnot-Carath\\\\'eodory metric.\\n\\nDoes there exist $\\\\epsilon<1$ so that the first Heisenberg group ${\\\\mathbb H}^1$ equipped with the metric $d_{cc}^\\\\epsilon$ bi-Lipschitz embeds into ${\\\\mathbb R}^5$?\"\nOriginal remarks: [\"Instead one could use the Kor\\\\'anyi metric $d_0(p,q) = ||p^{-1}*q||_0$, where $||(z,t)||_0 = (|z|^4+t^2)^{1/4}$.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/2/", "tags": [ "aim", "AIM-ANALYSIS-0115", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The checked literature does not appear to settle whether any snowflake of the first Heisenberg group embeds bi-Lipschitzly into R^5. This attempt proves that every such embedding must have exponent alpha strictly greater than 4/5. Hausdorff dimension excludes alpha<4/5; at alpha=4/5, Ahlfors 5-regularity and a density-point blow-up, combined with exact Heisenberg dilations, would produce a surjective bi-Lipschitz homeomorphism from a space topologically R^3 onto R^5, a contradiction. A companion finite-net estimate gives distortion D at least c(R/delta)^((4-5alpha)/5) on suitable delta-nets at CC scale R. The interval 4/54/5; in particular the dimension-critical exponent alpha=4/5 is impossible by a density-point and self-similar blow-up argument." }, { "id": 20000584, "problem_number": "AIM-ANALYSIS-0116", "title": "Higher-dimensional quantitative bi-Lipschitz parameterization and a scalar-curvature obstruction", "statement": "Bi-Lipschitz parameterization by Euclidean spaces\n\nAre there higher dimensional analogs of the Bonk--Lang parameterization theorem which give quantitative control on bi-Lipschitz constants?", "original_statement": "Bi-Lipschitz parameterization by Euclidean spaces\n\nAre there higher dimensional analogs of the Bonk--Lang parameterization theorem which give quantitative control on bi-Lipschitz constants?", "clean_statement": "Bi-Lipschitz parameterization by Euclidean spaces\n\nAre there higher dimensional analogs of the Bonk--Lang parameterization theorem which give quantitative control on bi-Lipschitz constants?", "statement_status": "exact", "statement_verification": "The canonical record is source index 115 of **aim-analysis-notes.json**. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Uniformization and parameterization\nSource item: 33.1\nSource URL: http://aimpl.org/mappingmetric/3/\nCanonical location: aim-analysis-notes.json notes[115]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Bi-Lipschitz parameterization by Euclidean spaces\\n\\nAre there higher dimensional analogs of the Bonk--Lang parameterization theorem which give quantitative control on bi-Lipschitz constants?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/3/", "tags": [ "aim", "AIM-ANALYSIS-0116", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Known higher-dimensional quantitative analogs of Bonk--Lang exist in special conformal Q-curvature settings, while the unrestricted same-dimensional problem remains open. A proved obstruction shows that raw total L^1 scalar curvature cannot by itself control the global bi-Lipschitz constant: for every n at least 3, epsilon greater than 0, and L_0 greater than 1, there is a complete smooth conformally flat metric on R^n, Euclidean outside a compact set, whose total absolute scalar curvature is below epsilon but whose every bi-Lipschitz parameterization by Euclidean R^n has symmetric constant greater than L_0.\n\nCandidate contribution (counterexample; novelty confidence low): The explicit conformal-well family combines Euclidean packing with the scaling law for total L^1 scalar curvature to make that curvature arbitrarily small while forcing arbitrarily large same-dimensional bi-Lipschitz distortion." }, { "id": 20000585, "problem_number": "AIM-ANALYSIS-0117", "title": "Quasisymmetric uniformization via weak metric doubling measures", "statement": "Quasisymmetric uniformization of metric $2$-spheres\n\nGive a geometric characterization of metric $2$-spheres which are quasisymmetrically equivalent to the standard $2$-sphere.", "original_statement": "Quasisymmetric uniformization of metric $2$-spheres\n\nGive a geometric characterization of metric $2$-spheres which are quasisymmetrically equivalent to the standard $2$-sphere.", "clean_statement": "Quasisymmetric uniformization of metric $2$-spheres\n\nGive a geometric characterization of metric $2$-spheres which are quasisymmetrically equivalent to the standard $2$-sphere.", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ANALYSIS-0117, source file aim-analysis-notes.json, zero-based source index 116, from the AIM workshop *Mapping theory in metric spaces*, section *Uniformization and parameterization*, Problem 33.2. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Uniformization and parameterization\nSource item: 33.2\nSource URL: http://aimpl.org/mappingmetric/3/\nCanonical location: aim-analysis-notes.json notes[116]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Quasisymmetric uniformization of metric $2$-spheres\\n\\nGive a geometric characterization of metric $2$-spheres which are quasisymmetrically equivalent to the standard $2$-sphere.\"\nOriginal remarks: [\"See also: https://link.springer.com/article/10.1007/s00229-012-0555-0\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/3/", "tags": [ "aim", "AIM-ANALYSIS-0117", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The unrestricted AIM problem is solved by the theorem of Lohvansuu, Rajala, and Rasimus: a metric space homeomorphic to the 2-sphere is quasisymmetrically equivalent to the standard sphere if and only if it is linearly locally connected and carries a weak metric doubling measure of dimension 2. The report explains the reduction through the weak chain metric to the Ahlfors 2-regular Bonk-Kleiner theorem and proves an additional quantitative snowflake proposition showing that this chain metric recovers the unsnowflaked geodesic geometry.\n\nCandidate contribution (proposition; novelty confidence low): If (Z,rho,nu) is a compact geodesic Ahlfors 2-regular metric-measure space with regularity constant A and d=rho^alpha for 0= 1/q(x) on an h_f-realizing sequence implies k_f(x) <= 4q(x)h_f(x); in particular, convex selected image balls imply the AIM conclusion." }, { "id": 20000588, "problem_number": "AIM-ANALYSIS-0120", "title": "An axis-trace reduction and an obstruction to the canonical fat-Cantor trace", "statement": "Mapping properties of planar maps of exponentially integrable distortion\n\nDoes there exist a homeomorphism $f$ of the plane with exponentially integrable distortion such that $f$ fixes the real axis and sends a set of positive length onto the $\\tfrac13$ Cantor set?", "original_statement": "Mapping properties of planar maps of exponentially integrable distortion\n\nDoes there exist a homeomorphism $f$ of the plane with exponentially integrable distortion such that $f$ fixes the real axis and sends a set of positive length onto the $\\tfrac13$ Cantor set?", "clean_statement": "Mapping properties of planar maps of exponentially integrable distortion\n\nDoes there exist a homeomorphism $f$ of the plane with exponentially integrable distortion such that $f$ fixes the real axis and sends a set of positive length onto the $\\tfrac13$ Cantor set?", "statement_status": "exact", "statement_verification": "The canonical record is problem 4.2 in the AIM workshop list *Mapping theory in metric spaces* (source file `aim-analysis-notes.json`, zero-based record index 119). The original AIM page was also inspected through its archived page data. It attributes the problem to Pekka Koskela and gives exactly the same text, with no clarifying remark:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Regularity\nSource item: 4.2\nSource URL: http://aimpl.org/mappingmetric/4/\nCanonical location: aim-analysis-notes.json notes[119]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Mapping properties of planar maps of exponentially integrable distortion\\n\\nDoes there exist a homeomorphism $f$ of the plane with exponentially integrable distortion such that $f$ fixes the real axis and sends a set of positive length onto the $\\\\tfrac13$ Cantor set?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/4/", "tags": [ "aim", "AIM-ANALYSIS-0120", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If “fixes the real axis” means pointwise fixation, the requested map is impossible because injectivity forces the source set to be the null middle-thirds Cantor set itself. Under the substantive setwise reconstruction f(R)=R, the problem is exactly equivalent to finding an increasing real-line homeomorphism h with |h^{-1}(C)|>0 that admits a David extension to the upper half-plane; reflection then gives the planar map. For an explicit one-parameter family of canonical address-preserving traces h_a from fat Cantor sets E_a of length 1-a onto C, the symmetric distortion satisfies exponential-in-generation lower bounds on positive-area rectangles, so exp(q rho_{h_a}) is nonintegrable for every q>0. This proves only that the Beurling–Ahlfors extension of this canonical trace is not David; it does not rule out another extension or another trace.\n\nCandidate contribution (obstruction; novelty confidence low): For every a in (0,1), let E_a be the symmetric Cantor set whose generation-n gaps have length a·3^{-n}, and let h_a be the address-preserving homeomorphism with h_a(E_a)=C, affine on complementary gaps. There are 2^{n-1} (x,t)-rectangles of total area (a^2/64)(2/9)^n on which rho_{h_a}(x,t) is at least c_a A^n, with A=3^{log_2(3)-1}>1. Hence the integral of exp(q rho_{h_a}) over (0,1)^2 diverges for every q>0, and the Beurling–Ahlfors extension of h_a cannot have exponentially integrable distortion." }, { "id": 20000589, "problem_number": "AIM-ANALYSIS-0121", "title": "Simultaneous Sobolev approximation for radial planar homeomorphisms", "statement": "Approximation of Sobolev homeomorphisms by diffeomorphisms\n\nLet $f$ be a homeomorphism of the plane such that $f$ and $f^{-1}$ lie in some Sobolev class, e.g., $W^{1,2}$. Can one approximate $f$ by diffeomorphisms $f_j$ so that $f_j$ and $f_j^{-1}$ converge in the Sobolev norm?", "original_statement": "Approximation of Sobolev homeomorphisms by diffeomorphisms\n\nLet $f$ be a homeomorphism of the plane such that $f$ and $f^{-1}$ lie in some Sobolev class, e.g., $W^{1,2}$. Can one approximate $f$ by diffeomorphisms $f_j$ so that $f_j$ and $f_j^{-1}$ converge in the Sobolev norm?", "clean_statement": "Approximation of Sobolev homeomorphisms by diffeomorphisms\n\nLet $f$ be a homeomorphism of the plane such that $f$ and $f^{-1}$ lie in some Sobolev class, e.g., $W^{1,2}$. Can one approximate $f$ by diffeomorphisms $f_j$ so that $f_j$ and $f_j^{-1}$ converge in the Sobolev norm?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, workshop *Mapping theory in metric spaces*, section “Regularity,” Problem 4.3, source file `aim-analysis-notes.json`, record index 120. The live AIM page was checked on 28 July 2026 and still gives the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Regularity\nSource item: 4.3\nSource URL: http://aimpl.org/mappingmetric/4/\nCanonical location: aim-analysis-notes.json notes[120]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Approximation of Sobolev homeomorphisms by diffeomorphisms\\n\\nLet $f$ be a homeomorphism of the plane such that $f$ and $f^{-1}$ lie in some Sobolev class, e.g., $W^{1,2}$. Can one approximate $f$ by diffeomorphisms $f_j$ so that $f_j$ and $f_j^{-1}$ converge in the Sobolev norm?\"\nOriginal remarks: [\"The one-sided approximation, namely $f_j \\\\to f$ without the convergence of inverses, is possible in this case (Iwaniec--Kovalev--Onninen).\", \"Even the one-sided case remains open in the non-reflexive space $W^{1,1}$, or for any Sobolev space in dimension three.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/4/", "tags": [ "aim", "AIM-ANALYSIS-0121", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every orientation-preserving radial self-homeomorphism of the closed unit disk with both the map and inverse in W^{1,2}, a sequence of radial smooth diffeomorphisms is constructed that converges strongly in W^{1,2}, and uniformly, simultaneously with its inverses. The attempt also proves an exact inverse-energy iff criterion for uniformly convergent smooth planar diffeomorphisms and constructs a smooth radial sequence converging one-sidedly to the identity in W^{1,2} whose inverse energies diverge. These are partial results only; no priority or solution of the unrestricted planar problem is claimed.\n\nCandidate contribution (theorem; novelty confidence low): Every orientation-preserving radial bi-W^{1,2} self-homeomorphism of the closed unit disk admits radial C-infinity diffeomorphisms converging strongly in W^{1,2} simultaneously with their inverses." }, { "id": 20000590, "problem_number": "AIM-ANALYSIS-0122", "title": "First-order scaling and the elementary-proof bottleneck", "statement": "Rigidity of $n$-harmonic functions in ${\\mathbb R}^n$\n\nIs there an elementary proof for the fact that every entire $n$-harmonic function in ${\\mathbb R}^n$ with linear growth is affine?", "original_statement": "Rigidity of $n$-harmonic functions in ${\\mathbb R}^n$\n\nIs there an elementary proof for the fact that every entire $n$-harmonic function in ${\\mathbb R}^n$ with linear growth is affine?", "clean_statement": "Rigidity of $n$-harmonic functions in ${\\mathbb R}^n$\n\nIs there an elementary proof for the fact that every entire $n$-harmonic function in ${\\mathbb R}^n$ with linear growth is affine?", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ANALYSIS-0122, from aim-analysis-notes.json at zero-based index 121. The live AIM page *Mapping theory in metric spaces*, Section 5 (“Rigidity”), gives the same text as Problem 5.1 and attributes it to Mario Bonk:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Rigidity\nSource item: 5.1\nSource URL: http://aimpl.org/mappingmetric/5/\nCanonical location: aim-analysis-notes.json notes[121]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Rigidity of $n$-harmonic functions in ${\\\\mathbb R}^n$\\n\\nIs there an elementary proof for the fact that every entire $n$-harmonic function in ${\\\\mathbb R}^n$ with linear growth is affine?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/5/", "tags": [ "aim", "AIM-ANALYSIS-0122", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The scalar rigidity theorem is classical and settled: the scale-invariant interior C^{1,alpha} estimate implies that every entire p-harmonic function on R^d satisfying liminf_{R to infinity} osc_{B_R}(u)/R^{1+alpha}=0 is affine, hence covers linear growth for d=p=n and, more generally, all 1 t^3 has pointwise metric dilatation one but is not a similarity. The intended nonsmooth infinite-dimensional question remains open in the literature checked. As a proved partial theorem, if H and H' are real Hilbert spaces of dimension at least three and f:H->H' is a C3 Frechet diffeomorphism with ball/exterior metric dilatation H_f=1 everywhere, then f is a similarity. The proof identifies Df=lambda U, derives the flat conformal identity for g=log(lambda), shows distributionally that q=1/lambda has Hessian a constant multiple of the identity, and uses global positivity plus Mazur-Ulam to conclude that lambda is constant and f is affine-similar.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: every C3 Frechet diffeomorphism with ball/exterior metric dilatation identically one between arbitrary, not necessarily separable, real Hilbert spaces of dimension at least three is a global similarity." }, { "id": 20000592, "problem_number": "AIM-ANALYSIS-0124", "title": "Solved status and the null G_delta structure of completely nonrectifiable fibers", "statement": "Quasiconformal mappings of the plane which destroy the rectifiability of uncountably many disjoint lines\n\nIs there a quasiconformal map $f$ of the plane and an uncountable set $E\\subset{\\mathbb R}$ so that $f(E\\times{\\mathbb R})$ contains no rectifiable curves?", "original_statement": "Quasiconformal mappings of the plane which destroy the rectifiability of uncountably many disjoint lines\n\nIs there a quasiconformal map $f$ of the plane and an uncountable set $E\\subset{\\mathbb R}$ so that $f(E\\times{\\mathbb R})$ contains no rectifiable curves?", "clean_statement": "Quasiconformal mappings of the plane which destroy the rectifiability of uncountably many disjoint lines\n\nIs there a quasiconformal map $f$ of the plane and an uncountable set $E\\subset{\\mathbb R}$ so that $f(E\\times{\\mathbb R})$ contains no rectifiable curves?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 5.3 in the “Rigidity” section of the AIM list *Mapping theory in metric spaces*. The AIM page was checked on 2026-07-28. It agrees with the repository record and requires no reconstruction:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Rigidity\nSource item: 5.3\nSource URL: http://aimpl.org/mappingmetric/5/\nCanonical location: aim-analysis-notes.json notes[123]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Quasiconformal mappings of the plane which destroy the rectifiability of uncountably many disjoint lines\\n\\nIs there a quasiconformal map $f$ of the plane and an uncountable set $E\\\\subset{\\\\mathbb R}$ so that $f(E\\\\times{\\\\mathbb R})$ contains no rectifiable curves?\"\nOriginal remarks: [\"The same question may also be interesting for mappings of exponentially integrable distortion.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/5/", "tags": [ "aim", "AIM-ANALYSIS-0124", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Bishop, Hakobyan, and Williams, Corollary 1.8, gives a published full affirmative solution: after rotating their construction, one obtains an uncountable E of Hausdorff dimension arbitrarily close to one for which f(E x R) contains no nonconstant rectifiable curve. This attempt additionally proves that for every planar quasiconformal homeomorphism f, the parameters of completely nonrectifiable vertical fibers form a Lebesgue-null G_delta set B_f, that f(E x R) is curve-free exactly when E is contained in B_f, that any uncountable witness has a compact perfect null replacement, and that compact slice lengths satisfy a local weak-L2 estimate.\n\nCandidate contribution (proposition; novelty confidence low): For every planar K-quasiconformal homeomorphism f, the set B_f of vertical fibers containing no nonconstant rectifiable curve is a null G_delta; for every E subset R, f(E x R) contains no such curve if and only if E is contained in B_f; every uncountable witness therefore has a compact perfect null replacement; and for bounded intervals I,J, |{x in I: length(f({x} x J)) > lambda}| is at most K|J||f(I x J)|/lambda^2." }, { "id": 20000593, "problem_number": "AIM-ANALYSIS-0125", "title": "Global permutation rigidity for repeated Carnot factors", "statement": "Quasisymmetric maps of products of irreducible Carnot groups\n\nLet $H$ be an irreducible Carnot group, $G$ a product of several copies of $H$, and $f:G\\to G$ a quasisymmetric map. Must $f$ be a product map (after permutation of the factors)?", "original_statement": "Quasisymmetric maps of products of irreducible Carnot groups\n\nLet $H$ be an irreducible Carnot group, $G$ a product of several copies of $H$, and $f:G\\to G$ a quasisymmetric map. Must $f$ be a product map (after permutation of the factors)?", "clean_statement": "Quasisymmetric maps of products of irreducible Carnot groups\n\nLet $H$ be an irreducible Carnot group, $G$ a product of several copies of $H$, and $f:G\\to G$ a quasisymmetric map. Must $f$ be a product map (after permutation of the factors)?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 5.4, from the 2012 workshop *Mapping theory in metric spaces*. Its question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Rigidity\nSource item: 5.4\nSource URL: http://aimpl.org/mappingmetric/5/\nCanonical location: aim-analysis-notes.json notes[124]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Quasisymmetric maps of products of irreducible Carnot groups\\n\\nLet $H$ be an irreducible Carnot group, $G$ a product of several copies of $H$, and $f:G\\\\to G$ a quasisymmetric map. Must $f$ be a product map (after permutation of the factors)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"A Carnot group is irreducible if it is not the real line and is not the direct sum of two Carnot groups. Assume $G_1$ and $G_2$ are direct sums of at least two irreducible Carnot groups and $f : G_1 \\\\to G_2$ a quasisymmetric map. Using Pansu's differentiability theorem, one can show that the differential of $f$ is a product map (after permutation of the summands). It follows that $f$ is a product map (after permutation) and is bilipschitz, provided $G_1$ has at least two non-isomorphic irreducible summands.\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/5/", "tags": [ "aim", "AIM-ANALYSIS-0125", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM question has an affirmative answer under its stated irreducibility convention: Kleiner--Muller--Xie Theorem 1.1 implies that every quasisymmetric self-homeomorphism of H^m, with H nonabelian and Carnot-indecomposable, is a product of component homeomorphisms after one fixed global permutation. The artifacts also prove a measurable block-calibration proposition: once the splitting is fixed, an almost-everywhere horizontal condition-number bound K forces every component differential into one common interval [c,K^2 c] and yields a global quasisimilarity.\n\nCandidate contribution (proposition; novelty confidence low): For m at least 2, a product homeomorphism of H^m with the standard quasiconformal ACL, null-set, and inverse-chain-rule properties and with L(D_P F) at most K times ell(D_P F) almost everywhere admits one c>0 such that c <= ell(D_P f_i) <= L(D_P f_i) <= K^2 c almost everywhere for every factor; consequently the whole product map is globally (c,K^2 c)-bilipschitz for the orthogonal product Carnot metric." }, { "id": 20000594, "problem_number": "AIM-ANALYSIS-0126", "title": "A five-state area-preserving bi-Lipschitz counterexample to product rigidity", "statement": "Product quasiconformal mappings of the plane\n\nLet $f$ be a quasiconformal map of the plane. Suppose at a.e.\\ point, the differential of $f$ either preserves both the $x$-direction and the $y$-direction, or switches the two directions. Does this imply $f$ is a product map?", "original_statement": "Product quasiconformal mappings of the plane\n\nLet $f$ be a quasiconformal map of the plane. Suppose at a.e.\\ point, the differential of $f$ either preserves both the $x$-direction and the $y$-direction, or switches the two directions. Does this imply $f$ is a product map?", "clean_statement": "Product quasiconformal mappings of the plane\n\nLet $f$ be a quasiconformal map of the plane. Suppose at a.e.\\ point, the differential of $f$ either preserves both the $x$-direction and the $y$-direction, or switches the two directions. Does this imply $f$ is a product map?", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 5.5 in the Rigidity section of the 2012 workshop list *Mapping theory in metric spaces*. It asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Rigidity\nSource item: 5.5\nSource URL: http://aimpl.org/mappingmetric/5/\nCanonical location: aim-analysis-notes.json notes[125]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Product quasiconformal mappings of the plane\\n\\nLet $f$ be a quasiconformal map of the plane. Suppose at a.e.\\\\ point, the differential of $f$ either preserves both the $x$-direction and the $y$-direction, or switches the two directions. Does this imply $f$ is a product map?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Leonid Kovalev has a proof when the distortion of $f$ is small. The same question can be asked in the bi-Lipschitz category.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/5/", "tags": [ "aim", "AIM-ANALYSIS-0126", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Kleiner, Muller, Szekelyhidi, and Xie refuted the AIM question in arXiv:2603.05979v1 by constructing a global orientation-preserving, area-preserving, non-product bi-Lipschitz homeomorphism of the plane whose differential is diagonal or anti-diagonal almost everywhere. This attempt verifies the exact hypothesis match and extracts the quantitative corollary that the counterexample may be chosen 3-bi-Lipschitz and at most 9-quasiconformal, with its differential taking five explicitly listed determinant-one values. It also proves an elementary two-state obstruction: if a Sobolev gradient takes only two fixed values A and B with det(A-B) nonzero, then it is constant almost everywhere; hence one same-orientation diagonal state and one anti-diagonal state cannot produce the required phase mixing.\n\nCandidate contribution (proposition; novelty confidence low): Candidate quantitative proposition: the c=3 endpoint of the five-state construction, periodically tiled using its affine boundary data, yields a global area-preserving 3-bi-Lipschitz and 9-quasiconformal non-product map; additionally, a distributional curl argument proves rigidity for every fixed two-state pair with det(A-B) nonzero." }, { "id": 20000595, "problem_number": "AIM-ANALYSIS-0127", "title": "Conditional cell-mass concentration and a radial no-go theorem for the standard Sierpinski carpet", "statement": "Loewner Sierpi\\'nski carpets\n\nIs the usual $\\tfrac13$ Sierpi\\'nski carpet $S_3$ quasisymmetrically equivalent to a Loewner space?", "original_statement": "Loewner Sierpi\\'nski carpets\n\nIs the usual $\\tfrac13$ Sierpi\\'nski carpet $S_3$ quasisymmetrically equivalent to a Loewner space?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is AIM Problem List 5.6 in the “Rigidity” section of *Mapping theory in metric spaces*. The live AIM page attributes the problem to Hrant Hakobyan and states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Rigidity\nSource item: 5.6\nSource URL: http://aimpl.org/mappingmetric/5/\nCanonical location: aim-analysis-notes.json notes[126]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Loewner Sierpi\\\\'nski carpets\\n\\nIs the usual $\\\\tfrac13$ Sierpi\\\\'nski carpet $S_3$ quasisymmetrically equivalent to a Loewner space?\"\nOriginal remarks: [\"This problem is closely related to the well known open problem of determining the conformal dimension of $S_3$.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/5/", "tags": [ "aim", "AIM-ANALYSIS-0127", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM yes/no question remains open. The Loewner-uniformization problem is equivalent to attainment of the Ahlfors-regular conformal dimension p. Conditional on attainment, every normalized optimal measure mu satisfies 8^n mu(K_{s(x)|n}) -> infinity for mu-almost every x along an explicit Borel-selected nested address; consequently no optimal measure has a uniform upper bound mu(K_w) <= C 8^{-|w|}. Independently, no increasing concave radial metric transform phi composed with the Euclidean distance can be Loewner.\n\nCandidate contribution (theorem; novelty confidence low): Conditional on attainment, normalized optimal cell masses obey 8^n mu(K_{s(x)|n}) -> infinity at mu-almost every point for an explicit overlap-safe Borel address selector s." }, { "id": 20000596, "problem_number": "AIM-ANALYSIS-0128", "title": "Finite-cell obstructions for proper self-embeddings of the standard Sierpinski carpet", "statement": "Bi-Lipschitz embeddings of the Sierpi\\'nski carpet into itself\n\nIs every bi-Lipschitz embedding of the $\\tfrac13$ Sierpi\\'nski carpet $S_3$ into itself the restriction of an affine mapping?", "original_statement": "Bi-Lipschitz embeddings of the Sierpi\\'nski carpet into itself\n\nIs every bi-Lipschitz embedding of the $\\tfrac13$ Sierpi\\'nski carpet $S_3$ into itself the restriction of an affine mapping?", "clean_statement": "Bi-Lipschitz embeddings of the Sierpi\\'nski carpet into itself\n\nIs every bi-Lipschitz embedding of the $\\tfrac13$ Sierpi\\'nski carpet $S_3$ into itself the restriction of an affine mapping?", "statement_status": "exact", "statement_verification": "The record adds that Bonk--Merenkov classify quasisymmetric maps **onto** \\(S_3\\). The word “onto” is essential. Here an embedding means an injective map \\[ f:S_3\\longrightarrow S_3 \\] for which, for some \\(L\\geq 1\\), \\[ L^{-1}|x-y|\\leq |f(x)-f(y)|\\leq L|x-y|\\qquad(x,y\\in S_3), \\] and it need not be surjective. “Restriction of an affine mapping” means \\(f=A|_{S_3}\\) for a plane affine map \\(A(x)=Mx+b\\). The source statement has no apparent OCR corruption or notational ambiguity.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Rigidity\nSource item: 5.7\nSource URL: http://aimpl.org/mappingmetric/5/\nCanonical location: aim-analysis-notes.json notes[127]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Bi-Lipschitz embeddings of the Sierpi\\\\'nski carpet into itself\\n\\nIs every bi-Lipschitz embedding of the $\\\\tfrac13$ Sierpi\\\\'nski carpet $S_3$ into itself the restriction of an affine mapping?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"More generally, one can ask about quasisymmetric mappings. By work of Bonk--Merenkov, every quasisymmetric map of $S_3$ onto itself is of this type.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/5/", "tags": [ "aim", "AIM-ANALYSIS-0128", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The original question for arbitrary bi-Lipschitz embeddings remains apparently open. This attempt proves that an injective map of S_3 which restricts to a plane similarity on every construction cell of any one fixed level must be the restriction of one global similarity. As a corollary, if a quasisymmetric self-embedding maps each of the eight first-generation cells onto a construction cell, even with target depths initially allowed to vary, then all target depths agree, the targets are the eight children of one surviving parent, and the embedding is a level-cell similarity composed with a dihedral symmetry.\n\nCandidate contribution (theorem; novelty confidence low): Fixed-level similarity gluing: if an injective map f:S_3 to R^2 agrees on every surviving cell of one fixed level with a plane similarity, then all local similarities coincide and f is one global similarity; consequently, every first-generation construction-cell-respecting quasisymmetric self-embedding is a level-cell similarity followed by a dihedral symmetry." }, { "id": 20000597, "problem_number": "AIM-ANALYSIS-0129", "title": "Explicit positive-area carpets with non-isometric quasisymmetries", "statement": "Quasisymmetric rigidity and non-rigidity of positive area Sierpi\\'nski carpets\n\nA set $X \\subset {\\mathbb R}^n$ is said to be \\emph{quasisymmetrically rigid} if every quasisymmetric map of $X$ onto itself coincides the restriction to $X$ of an isometry of ${\\mathbb R}^n$. Bonk and Merenkov have shown that the usual $\\tfrac13$ Sierpi\\'snki carpet is quasisymmetrically rigid.\n\nWhich positive area Sierpi\\'nski carpets are quasisymmetrically rigid? Which are nonrigid?", "original_statement": "Quasisymmetric rigidity and non-rigidity of positive area Sierpi\\'nski carpets\n\nA set $X \\subset {\\mathbb R}^n$ is said to be \\emph{quasisymmetrically rigid} if every quasisymmetric map of $X$ onto itself coincides the restriction to $X$ of an isometry of ${\\mathbb R}^n$. Bonk and Merenkov have shown that the usual $\\tfrac13$ Sierpi\\'snki carpet is quasisymmetrically rigid.\n\nWhich positive area Sierpi\\'nski carpets are quasisymmetrically rigid? Which are nonrigid?", "clean_statement": "Quasisymmetric rigidity and non-rigidity of positive area Sierpi\\'nski carpets\n\nA set $X \\subset {\\mathbb R}^n$ is said to be \\emph{quasisymmetrically rigid} if every quasisymmetric map of $X$ onto itself coincides the restriction to $X$ of an isometry of ${\\mathbb R}^n$. Bonk and Merenkov have shown that the usual $\\tfrac13$ Sierpi\\'snki carpet is quasisymmetrically rigid.\n\nWhich positive area Sierpi\\'nski carpets are quasisymmetrically rigid? Which are nonrigid?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 5.8 from the 2012 workshop *Mapping theory in metric spaces*. Apart from the typographical spelling “Sierpi'snki” and the missing word “with” in “coincides the restriction,” its question is unambiguous:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Rigidity\nSource item: 5.8\nSource URL: http://aimpl.org/mappingmetric/5/\nCanonical location: aim-analysis-notes.json notes[128]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Quasisymmetric rigidity and non-rigidity of positive area Sierpi\\\\'nski carpets\\n\\nA set $X \\\\subset {\\\\mathbb R}^n$ is said to be \\\\emph{quasisymmetrically rigid} if every quasisymmetric map of $X$ onto itself coincides the restriction to $X$ of an isometry of ${\\\\mathbb R}^n$. Bonk and Merenkov have shown that the usual $\\\\tfrac13$ Sierpi\\\\'snki carpet is quasisymmetrically rigid.\\n\\nWhich positive area Sierpi\\\\'nski carpets are quasisymmetrically rigid? Which are nonrigid?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/5/", "tags": [ "aim", "AIM-ANALYSIS-0129", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The overall classification of positive-area planar Sierpiński carpets as quasisymmetrically rigid or nonrigid remains open. A proved affine symmetry-breaking criterion nevertheless gives an explicit nonrigid family: if a variable-grid square carpet Y_n is built with odd n_k at least 3 and sum n_k^{-2}0 and lambda not equal to 1, admits the bilipschitz self-involution (x,y) mapped to (lambda y,x/lambda), which changes a pairwise distance and therefore cannot be the restriction of a Euclidean isometry.\n\nCandidate contribution (proposition; novelty confidence low): If a compact positive-area planar set Y is invariant under the Euclidean isometry R(x)=Ux+b and A has invertible linear part L with LUL^{-1} nonorthogonal, then ARA^{-1} restricts to a bilipschitz self-map of A(Y) that is not the restriction of a Euclidean isometry; applying this criterion to coordinate-symmetric positive-area variable-grid carpets yields the explicit family in the main theorem." }, { "id": 20000598, "problem_number": "AIM-ANALYSIS-0130", "title": "Energy-jump rigidity for planar Hopf-Laplace maps", "statement": "Inner variations\n\nConsider the inner variational equation for planar maps\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nLocally, this equation can be reduced to a first-order equation $f_zf_{\\overline{z}} = 1$, which may also be viewed as a differential inclusion $Df \\in M$, with $M$ a certain set of $2\\times 2$ matrices. Natural assumptions on $f$ are the finiteness of the energy and nonnegativity of the Jacobian.\n\nThe equation\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nexpresses the stationarity of the Dirichlet energy $\\int |Df|^2$ under inner variations of $f$ (precomposition with diffeomorphisms).\n\nIs $|Df|$ continuous?", "original_statement": "Inner variations\n\nConsider the inner variational equation for planar maps\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nLocally, this equation can be reduced to a first-order equation $f_zf_{\\overline{z}} = 1$, which may also be viewed as a differential inclusion $Df \\in M$, with $M$ a certain set of $2\\times 2$ matrices. Natural assumptions on $f$ are the finiteness of the energy and nonnegativity of the Jacobian.\n\nThe equation\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nexpresses the stationarity of the Dirichlet energy $\\int |Df|^2$ under inner variations of $f$ (precomposition with diffeomorphisms).\n\nIs $|Df|$ continuous?", "clean_statement": "Inner variations\n\nConsider the inner variational equation for planar maps\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nLocally, this equation can be reduced to a first-order equation $f_zf_{\\overline{z}} = 1$, which may also be viewed as a differential inclusion $Df \\in M$, with $M$ a certain set of $2\\times 2$ matrices. Natural assumptions on $f$ are the finiteness of the energy and nonnegativity of the Jacobian.\n\nThe equation\n$$\n(f_z\\overline{f_{\\overline{z}}})_{\\overline{z}} = 0.\n$$\nexpresses the stationarity of the Dirichlet energy $\\int |Df|^2$ under inner variations of $f$ (precomposition with diffeomorphisms).\n\nIs $|Df|$ continuous?", "statement_status": "exact", "statement_verification": "The exact AIM record asks about a planar Sobolev map \\(f\\) satisfying", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Variational problems\nSource item: 66.1\nSource URL: http://aimpl.org/mappingmetric/6/\nCanonical location: aim-analysis-notes.json notes[129]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Inner variations\\n\\nConsider the inner variational equation for planar maps\\n$$\\n(f_z\\\\overline{f_{\\\\overline{z}}})_{\\\\overline{z}} = 0.\\n$$\\nLocally, this equation can be reduced to a first-order equation $f_zf_{\\\\overline{z}} = 1$, which may also be viewed as a differential inclusion $Df \\\\in M$, with $M$ a certain set of $2\\\\times 2$ matrices. Natural assumptions on $f$ are the finiteness of the energy and nonnegativity of the Jacobian.\\n\\nThe equation\\n$$\\n(f_z\\\\overline{f_{\\\\overline{z}}})_{\\\\overline{z}} = 0.\\n$$\\nexpresses the stationarity of the Dirichlet energy $\\\\int |Df|^2$ under inner variations of $f$ (precomposition with diffeomorphisms).\\n\\nIs $|Df|$ continuous?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/6/", "tags": [ "aim", "AIM-ANALYSIS-0130", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The continuity question remains open in the literature checked, but the orientation-nonnegative Hopf state set has a sharp rank-one rigidity: two distinct rank-one-compatible states with the same Hopf product must both have zero Jacobian and exactly the same Frobenius norm. Consequently, energy traces match across classical interfaces, the energy density has no jump part when Df is BV, and every continuous Hopf-Laplace map that is affine on the cells of a finite conforming polygonal partition has constant |Df|; if one cell has positive Jacobian, or if the Hopf product is zero, the map is globally affine.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate novelty: for orientation-nonnegative planar Hopf states, every distinct rank-one-compatible pair lies on the zero-Jacobian boundary and has equal Frobenius norm; hence no classical interface, BV-gradient jump, or continuous finite-phase piecewise-affine Hopf construction on a finite conforming partition can create an energy-density jump." }, { "id": 20000599, "problem_number": "AIM-ANALYSIS-0131", "title": "Dilatation energies and diameter-constrained obstacles", "statement": "Variational problems for dilatation functionals\n\nStudy extremal variational problems for dilatation functionals of mappings. For instance, find the extremal domain (and possibly also the extremal mappings) associated to the configuration of a doubly connected planar domain including an obstacle of prescribed diameter.", "original_statement": "Variational problems for dilatation functionals\n\nStudy extremal variational problems for dilatation functionals of mappings. For instance, find the extremal domain (and possibly also the extremal mappings) associated to the configuration of a doubly connected planar domain including an obstacle of prescribed diameter.", "clean_statement": "Variational problems for dilatation functionals\n\nStudy extremal variational problems for dilatation functionals of mappings. For instance, find the extremal domain (and possibly also the extremal mappings) associated to the configuration of a doubly connected planar domain including an obstacle of prescribed diameter.", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ANALYSIS-0131, source file `aim-analysis-notes.json`, zero-based index 130. It is Problem 6.2 in the section “Variational problems” of the AIM list *Mapping theory in metric spaces*. The record states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Variational problems\nSource item: 6.2\nSource URL: http://aimpl.org/mappingmetric/6/\nCanonical location: aim-analysis-notes.json notes[130]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Variational problems for dilatation functionals\\n\\nStudy extremal variational problems for dilatation functionals of mappings. For instance, find the extremal domain (and possibly also the extremal mappings) associated to the configuration of a doubly connected planar domain including an obstacle of prescribed diameter.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/6/", "tags": [ "aim", "AIM-ANALYSIS-0131", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The diameter-only reading has no domain extremum: round domains with a fixed-diameter disk obstacle realize every positive ring modulus, while the unmarked least maximal dilatation between two rings is exactly the larger ratio of their moduli. For fixed concentric annuli, a radial homeomorphism measured with logarithmic cylinder area minimizes every nondecreasing convex penalty of linear dilatation at the constant-log-slope power map; strong convexity gives an explicit squared-profile stability remainder. Separately, the literature gives a sharp geodesic-segment extremizer for the modulus-maximizing side of the normalized fixed-hyperbolic-diameter obstacle problem, while the opposite capacity-maximizing side was left open and no later solution was located.\n\nCandidate contribution (lemma; novelty confidence low): For orientation-preserving absolutely continuous radial maps between arbitrary concentric annuli, using logarithmic cylinder measure, every nondecreasing convex penalty Phi of linear dilatation satisfies the sharp power-map lower bound; if Phi is twice differentiable with Phi' nonnegative and Phi'' at least lambda greater than zero, the energy excess is at least pi times lambda times the squared L2 norm of the deviation of the logarithmic radial derivative from its mean." }, { "id": 20000600, "problem_number": "AIM-ANALYSIS-0132", "title": "Calibrated faces in ell_p^n and the solved dimensional ranges", "statement": "Hausdorff measures in $\\ell^p_n$\n\nLet ${\\mathcal H}^m$ denotes the Hausdorff $m$-measure corresponding to the norm in $\\ell^p_n$.\n\nLet $P$ be a polyhedral $m$-cycle in $\\ell^p_n$, $p\\ne 2$, with faces $F_1,\\ldots,F_k$. Is ${\\mathcal H}^m(F_1) \\le \\sum_{j=2}^k {\\mathcal H}^m(F_j)$?", "original_statement": "Hausdorff measures in $\\ell^p_n$\n\nLet ${\\mathcal H}^m$ denotes the Hausdorff $m$-measure corresponding to the norm in $\\ell^p_n$.\n\nLet $P$ be a polyhedral $m$-cycle in $\\ell^p_n$, $p\\ne 2$, with faces $F_1,\\ldots,F_k$. Is ${\\mathcal H}^m(F_1) \\le \\sum_{j=2}^k {\\mathcal H}^m(F_j)$?", "clean_statement": "Hausdorff measures in $\\ell^p_n$\n\nLet ${\\mathcal H}^m$ denotes the Hausdorff $m$-measure corresponding to the norm in $\\ell^p_n$.\n\nLet $P$ be a polyhedral $m$-cycle in $\\ell^p_n$, $p\\ne 2$, with faces $F_1,\\ldots,F_k$. Is ${\\mathcal H}^m(F_1) \\le \\sum_{j=2}^k {\\mathcal H}^m(F_j)$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 6.3 from the workshop *Mapping theory in metric spaces*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Variational problems\nSource item: 6.3\nSource URL: http://aimpl.org/mappingmetric/6/\nCanonical location: aim-analysis-notes.json notes[131]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Hausdorff measures in $\\\\ell^p_n$\\n\\nLet ${\\\\mathcal H}^m$ denotes the Hausdorff $m$-measure corresponding to the norm in $\\\\ell^p_n$.\\n\\nLet $P$ be a polyhedral $m$-cycle in $\\\\ell^p_n$, $p\\\\ne 2$, with faces $F_1,\\\\ldots,F_k$. Is ${\\\\mathcal H}^m(F_1) \\\\le \\\\sum_{j=2}^k {\\\\mathcal H}^m(F_j)$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This question is related to the lower semicontinuity and existence for the Plateau problem.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/6/", "tags": [ "aim", "AIM-ANALYSIS-0132", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For oriented polyhedral cycles with coefficients, the natural weighted Hausdorff face inequality is affirmative in every finite-dimensional normed space for m=1, m=2, and m=n-1, so it is fully affirmative for all m= 2 and 1 <= m < n, a coordinate m-plane P in ell-infinity_n can be enlarged by an explicit closed countable family of off-plane dyadic satellites N accumulating at every point of P. The set S = P union N is closed, countably m-rectifiable, locally H^m-finite, and minimizing because N and every Lipschitz image of N are H^m-null; nevertheless S is not a local Lipschitz graph over any m-plane at any point of P, a full-H^m-measure subset of S. This refutes only the literal non-reduced AIM wording. The intended reduced/coral formulation remains open in the literature checked.\n\nCandidate contribution (counterexample; novelty confidence low): The explicit dyadic null-padding construction gives a counterexample to the exact archived AIM statement in every positive codimension, and the graph-projection collision argument rules out every possible base m-plane and complement, not only the original coordinate plane." }, { "id": 20000602, "problem_number": "AIM-ANALYSIS-0134", "title": "Strong relative isoperimetry from the measure-theoretic boundary", "statement": "Isoperimetric inequalities in metric measure spaces\n\nCan the perimeter measure $P$ on the right hand side of the relative isoperimetric inequality\n$$\n\\min \\{ \\mu (E\\cap B), \\mu (B \\setminus E) \\} \\le C r P(E,\\lambda B),\n$$\nwhere $E \\subset X$ is a Borel set and $B \\subset X$ is a ball of radius $r$, be replaced with the codimension one Hausdorff measure of the part of the measure-theoretic boundary of $E$ inside $B$ even without knowing ahead of time whether $E$ is of finite perimeter?", "original_statement": "Isoperimetric inequalities in metric measure spaces\n\nCan the perimeter measure $P$ on the right hand side of the relative isoperimetric inequality\n$$\n\\min \\{ \\mu (E\\cap B), \\mu (B \\setminus E) \\} \\le C r P(E,\\lambda B),\n$$\nwhere $E \\subset X$ is a Borel set and $B \\subset X$ is a ball of radius $r$, be replaced with the codimension one Hausdorff measure of the part of the measure-theoretic boundary of $E$ inside $B$ even without knowing ahead of time whether $E$ is of finite perimeter?", "clean_statement": "Isoperimetric inequalities in metric measure spaces\n\nCan the perimeter measure $P$ on the right hand side of the relative isoperimetric inequality\n$$\n\\min \\{ \\mu (E\\cap B), \\mu (B \\setminus E) \\} \\le C r P(E,\\lambda B),\n$$\nwhere $E \\subset X$ is a Borel set and $B \\subset X$ is a ball of radius $r$, be replaced with the codimension one Hausdorff measure of the part of the measure-theoretic boundary of $E$ inside $B$ even without knowing ahead of time whether $E$ is of finite perimeter?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 6.5, “Isoperimetric inequalities in metric measure spaces,” attributed on the live AIM page to Nageswari Shanmugalingam. The exact question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Mapping theory in metric spaces\nSection: Variational problems\nSource item: 6.5\nSource URL: http://aimpl.org/mappingmetric/6/\nCanonical location: aim-analysis-notes.json notes[133]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Isoperimetric inequalities in metric measure spaces\\n\\nCan the perimeter measure $P$ on the right hand side of the relative isoperimetric inequality\\n$$\\n\\\\min \\\\{ \\\\mu (E\\\\cap B), \\\\mu (B \\\\setminus E) \\\\} \\\\le C r P(E,\\\\lambda B),\\n$$\\nwhere $E \\\\subset X$ is a Borel set and $B \\\\subset X$ is a ball of radius $r$, be replaced with the codimension one Hausdorff measure of the part of the measure-theoretic boundary of $E$ inside $B$ even without knowing ahead of time whether $E$ is of finite perimeter?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The relative isoperimetric inequality is equivalent to the $(1,1)$-Poincar\\\\'e inequality. Under certain geometric conditions on $X$ (such as a geometric version of Semmes' ``pencil of curves'' condition), one knows that the answer to the above question is yes. Examples of spaces with such curves include the Euclidean spaces and the Heisenberg group. Can the Semmes pencil requirement be removed?\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/mappingmetric/6/", "tags": [ "aim", "AIM-ANALYSIS-0134", "aim-domain:analysis", "aim-workshop:mappingmetric", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For the intended standard formulation, in which the boundary is measured in the same dilated region as the original perimeter term, Lahti's metric Federer theorem gives min{mu(E intersect B), mu(B minus E)} <= 2 C_P C_d r H(partial-star E intersect lambda B) for every measurable E in every complete doubling (1,1)-PI space, with no prior finite-perimeter assumption; hence the geometric Semmes-pencil requirement is unnecessary. If the AIM phrase 'inside B' is read literally as the undilated ball, that stronger statement is false: an explicit compact doubling 1-PI graph-metric interval has a ball containing positive measure from E and its complement while partial-star E misses the ball (and even misses Lambda B through a quantitative range).\n\nCandidate contribution (counterexample; novelty confidence low): Let X=[0,2] with Lebesgue measure and chordal graph metric d(s,t)=sqrt((s-t)^2+(f(s)-f(t))^2), where f(t)=4t on [0,1/2], f(t)=3-2t on [1/2,3/2], and f(t)=0 on [3/2,2]. For E=[0,3/4) and B=B_d(0,8/5), both E intersect B and B minus E have positive measure, but partial-star E intersect Lambda B is empty for every 1 <= Lambda <= 15 sqrt(5)/32. Thus no general undilated-boundary estimate can hold under only complete doubling 1-PI assumptions." }, { "id": 20000603, "problem_number": "AIM-ANALYSIS-0135", "title": "Log-curvature obstructions and exact interpolation for n at most 2", "statement": "Let $\\{c_k\\}_{k=0}^n\\subset\\mathbb{R}$ be given. Does there exist a real rooted polynomial, $p(x)$, with zeros outside $[0,n]$ such that $c_k=p(k)$, $k=0,1,\\ldots,n$?", "original_statement": "Let $\\{c_k\\}_{k=0}^n\\subset\\mathbb{R}$ be given. Does there exist a real rooted polynomial, $p(x)$, with zeros outside $[0,n]$ such that $c_k=p(k)$, $k=0,1,\\ldots,n$?", "clean_statement": "Let $\\{c_k\\}_{k=0}^n\\subset\\mathbb{R}$ be given. Does there exist a real rooted polynomial, $p(x)$, with zeros outside $[0,n]$ such that $c_k=p(k)$, $k=0,1,\\ldots,n$?", "statement_status": "exact", "statement_verification": "The canonical record is problem 13.1 in the section “Multiplier sequences and CZDS” of the 2011 AIM workshop *Stability, hyperbolicity, and zero localization of functions*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Multiplier sequences and CZDS\nSource item: 13.1\nSource URL: http://aimpl.org/hyperbolicpoly/1/\nCanonical location: aim-analysis-notes.json notes[134]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $\\\\{c_k\\\\}_{k=0}^n\\\\subset\\\\mathbb{R}$ be given. Does there exist a real rooted polynomial, $p(x)$, with zeros outside $[0,n]$ such that $c_k=p(k)$, $k=0,1,\\\\ldots,n$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Note that this implies $\\\\{c_k\\\\}_{k=0}^n$ is an n-CZDS.\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/1/", "tags": [ "aim", "AIM-ANALYSIS-0135", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Universal existence for arbitrary real data is refuted, while the general feasibility characterization remains unresolved. If a nonconstant real-rooted polynomial p has every zero strictly outside [0,n], then its samples have one strict sign and, for L_k=log|p(k)| and kappa_k=2L_k-L_{k-1}-L_{k+1}, one has the exact rootwise formula kappa_k=sum_j log((k-r_j)^2/((k-r_j)^2-1))>0 and Delta^(2q) kappa_k>0 for every q>=1 in the exact range 1<=k and k+2q<=n-1. The positive strictly log-concave n=4 vector (1,1,1/2,1/32,1/1024) fails this obstruction. For n=1, feasibility is equivalent to the two values sharing a strict sign; for n=2, feasibility is equivalent either to equal nonzero data or to one strict sign together with c_1^2>c_0c_2, and an explicit outside-root construction is proved.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate novelty: admissible nonconstant sample data satisfy Delta^(2q) kappa_k>0 for every q>=1 and every 1<=k with k+2q<=n-1, and in dimension n=2 the first curvature inequality is also sufficient via an explicit finite product whose roots lie strictly outside [0,2]." }, { "id": 20000604, "problem_number": "AIM-ANALYSIS-0136", "title": "Meromorphic sampling, gauge freedom, and exponential-rational rigidity", "statement": "Let $f(z)$ be a meromorphic function. When is the sequence $\\{f(k)\\}_{k=0}^\\infty$ a multiplier sequence?", "original_statement": "Let $f(z)$ be a meromorphic function. When is the sequence $\\{f(k)\\}_{k=0}^\\infty$ a multiplier sequence?", "clean_statement": "Let $f(z)$ be a meromorphic function. When is the sequence $\\{f(k)\\}_{k=0}^\\infty$ a multiplier sequence?", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Multiplier sequences and CZDS\nSource item: 13.2\nSource URL: http://aimpl.org/hyperbolicpoly/1/\nCanonical location: aim-analysis-notes.json notes[135]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $f(z)$ be a meromorphic function. When is the sequence $\\\\{f(k)\\\\}_{k=0}^\\\\infty$ a multiplier sequence?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/1/", "tags": [ "aim", "AIM-ANALYSIS-0136", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted problem has the classical exact sample-level answer: the exponential generating function of the real finite samples, or its reflection, must be type-I Laguerre-Polya. Beyond this formal criterion, the attempt proves a complete classification for F(z)=c exp((a+i*pi*epsilon)z)p(z)/q(z), with p and q coprime real polynomials and q nonzero on the nonnegative integers: its samples form a multiplier sequence if and only if q is constant and the exponential Newton polynomial N_p(z)=sum_j Delta^j p(0)z^j/j! has only real nonpositive zeros. It also proves that the kernel of sampling meromorphic functions holomorphic on the nonnegative integers is exactly (1/Gamma(-z))M_0, so arbitrary finitely many off-lattice principal parts can be changed without changing the samples.\n\nCandidate contribution (classification; novelty confidence low): Candidate contribution: the exponential-rational family c exp((a+i*pi*epsilon)z)p(z)/q(z) admits the explicit if-and-only-if finite test that q is constant and N_p has only real nonpositive zeros, complemented by the exact sampling-fibre identity ker(S)=(1/Gamma(-z))M_0 and its finite principal-part freedom corollary." }, { "id": 20000605, "problem_number": "AIM-ANALYSIS-0137", "title": "A spectral-projector Hermitian form and the singular Bezoutian obstruction", "statement": "Given $p\\in\\mathbb{C}[x]$, $\\deg(p) = n$, find an $n\\times n$ Hermitian matrix whose inertia $(n_+, n_{-}, n_0)$ counts the zeros of $p$ in $H_+, H_{-}, H_0=\\mathbb{R}$.", "original_statement": "Given $p\\in\\mathbb{C}[x]$, $\\deg(p) = n$, find an $n\\times n$ Hermitian matrix whose inertia $(n_+, n_{-}, n_0)$ counts the zeros of $p$ in $H_+, H_{-}, H_0=\\mathbb{R}$.", "clean_statement": "Given $p\\in\\mathbb{C}[x]$, $\\deg(p) = n$, find an $n\\times n$ Hermitian matrix whose inertia $(n_+, n_{-}, n_0)$ counts the zeros of $p$ in $H_+, H_{-}, H_0=\\mathbb{R}$.", "statement_status": "exact", "statement_verification": "The canonical record is AIM-ANALYSIS-0137, item 2.3 in the “Matrix Theory” section of the AIM workshop list *Stability, hyperbolicity, and zero localization of functions*. Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Matrix Theory\nSource item: 2.3\nSource URL: http://aimpl.org/hyperbolicpoly/2/\nCanonical location: aim-analysis-notes.json notes[136]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given $p\\\\in\\\\mathbb{C}[x]$, $\\\\deg(p) = n$, find an $n\\\\times n$ Hermitian matrix whose inertia $(n_+, n_{-}, n_0)$ counts the zeros of $p$ in $H_+, H_{-}, H_0=\\\\mathbb{R}$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/2/", "tags": [ "aim", "AIM-ANALYSIS-0137", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the Frobenius companion matrix of p, let P_+, P_-, and P_0 be the Riesz projectors onto the generalized eigenspaces for roots in the open upper half-plane, open lower half-plane, and real axis. The single Hermitian matrix H(p)=P_+^*P_+-P_-^*P_- has inertia exactly equal to the three algebraic root counts, including all repeated and defective roots; this fully answers the literal existential wording. An equivalent CRT-idempotent formula is proved after regional factorization. The result is classified as partial because forming the projectors or regional factors already uses spectral separation and therefore does not settle the stronger intended problem of one coefficient-explicit zero-location matrix.\n\nCandidate contribution (construction; novelty confidence low): The candidate contribution is the explicit one-matrix construction H(p)=P_+^*P_+-P_-^*P_-, its arbitrary-Jordan congruence proof and CRT-idempotent form, together with the diagnosis that p(z)=z^2+1 makes the naive Hermite Bezoutian vanish although the required inertia is (1,1,0)." }, { "id": 20000606, "problem_number": "AIM-ANALYSIS-0138", "title": "Exact zeros and stability obstructions for BMV trace polynomials", "statement": "Study the zeros of Bessis-Moussa-Villani polynomials $t\\to\\text{tr}(A+tB)^m$.", "original_statement": "Study the zeros of Bessis-Moussa-Villani polynomials $t\\to\\text{tr}(A+tB)^m$.", "clean_statement": "Study the zeros of Bessis-Moussa-Villani polynomials $t\\to\\text{tr}(A+tB)^m$.", "statement_status": "exact", "statement_verification": "The canonical AIM record says, verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Matrix Theory\nSource item: 2.1\nSource URL: http://aimpl.org/hyperbolicpoly/2/\nCanonical location: aim-analysis-notes.json notes[137]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Study the zeros of Bessis-Moussa-Villani polynomials $t\\\\to\\\\text{tr}(A+tB)^m$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/2/", "tags": [ "aim", "AIM-ANALYSIS-0138", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every commuting positive-definite Hermitian 2-by-2 pair, an exact Moebius formula lists all zeros of tr(A+tB)^m and yields the sharp Hurwitz criterion (ab+cd)/(ad+bc)>cos(pi/m). The explicit family A=diag(1,r), B=diag(r,1) has all simple zeros on the unit circle and, for every m>=3 when r>tan(pi/4+pi/(2m)), a conjugate pair in the open right half-plane despite strictly positive coefficients. The attempt also proves the full m=2 classification for arbitrary positive-definite Hermitian pairs and an exact, noncommutative m=3 Routh-Hurwitz criterion in terms of four traces.\n\nCandidate contribution (classification; novelty confidence low): For A=diag(a,c), B=diag(b,d) with positive entries and ad not equal to bc, all roots are t_j=(c zeta_j-a)/(b-d zeta_j), where zeta_j^m=-1, and Hurwitz stability holds exactly when (ab+cd)/(ad+bc)>cos(pi/m); in particular A=diag(1,r), B=diag(r,1) gives a unit-circle family with a right-half-plane pair above the explicit sharp threshold r>tan(pi/4+pi/(2m))." }, { "id": 20000607, "problem_number": "AIM-ANALYSIS-0139", "title": "BMV after Stahl: reducible measures and the first three-state phase obstruction", "statement": "Find a simple proof of BMV conjecture for $3\\times 3$ matrices (and an explicit formula for the BMV measure for $3\\times 3$ matrices; the existence of the measure is due to recent work of Stahl).", "original_statement": "Find a simple proof of BMV conjecture for $3\\times 3$ matrices (and an explicit formula for the BMV measure for $3\\times 3$ matrices; the existence of the measure is due to recent work of Stahl).", "clean_statement": "Find a simple proof of BMV conjecture for $3\\times 3$ matrices (and an explicit formula for the BMV measure for $3\\times 3$ matrices; the existence of the measure is due to recent work of Stahl).", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Matrix Theory\nSource item: 2.2\nSource URL: http://aimpl.org/hyperbolicpoly/2/\nCanonical location: aim-analysis-notes.json notes[138]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find a simple proof of BMV conjecture for $3\\\\times 3$ matrices (and an explicit formula for the BMV measure for $3\\\\times 3$ matrices; the existence of the measure is due to recent work of Stahl).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/2/", "tags": [ "aim", "AIM-ANALYSIS-0139", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Stahl's theorem has already established the BMV conjecture in every finite dimension. This attempt proves two complementary explicit three-dimensional results: every common-eigenvector (1+2 reducible) pair has the displayed atom-plus-Bessel density, including the repeated-eigenvalue case; and, for distinct eigenvalues of B and A(epsilon)=D+epsilon C, the BMV measure has a total-variation-convergent coupling expansion with an explicit factorial bound, positive quadratic bridge density, and cubic coefficient nu_3=2 Re(c_12 c_23 c_31) H_a, where H_a is an explicitly positive exponential triangular B-spline. The example B=diag(0,1,2), D=0, c_12=c_23=1, c_31=-1 has cubic density -x on (0,1) and -(2-x) on (1,2), obstructing only coefficientwise-in-epsilon positivity, not BMV itself.\n\nCandidate contribution (density_identity; novelty confidence low): For B=diag(b_1,b_2,b_3) with b_1=2 and degree-n targets with leading coefficient (-1)^n, because det(A-lambda I) always has that leading term; equivalently, every monic real polynomial is required as det(lambda I-A). Under this necessary normalization, the assertion is the solved T_n conjecture: the published Garnett-Shader theorem, translated by A=-T^T, gives the exact strict AIM sign pattern in every order n>=2. The artifacts also give a positive-diagonal canonical form, its exact continuant recurrence, complete explicit constructions for n=2 and n=3, and checks covering repeated and nonreal conjugate roots.\n\nCandidate contribution (constructive_bound; novelty confidence low): For every monic cubic x^3+c2*x^2+c1*x+c0 and every prescribed endpoint a>max(0,-c2), the unique canonical AIM-pattern parameters are b=a+c2, S=c1+a*b, q1=(c0+a*S)/(a+b), and q2=(b*S-c0)/(a+b); strict realization is equivalent to positivity of the two displayed numerators, and every a>=4*max(1,|c2|,|c1|,|c0|) is guaranteed to work." }, { "id": 20000609, "problem_number": "AIM-ANALYSIS-0141", "title": "Hurwitz minors as controlled Toeplitz minors and staircase Schur functions", "statement": "Investigate the signs of Hurwitz minors for P\\'olya frequency sequences.", "original_statement": "Investigate the signs of Hurwitz minors for P\\'olya frequency sequences.", "clean_statement": "Investigate the signs of Hurwitz minors for P\\'olya frequency sequences.", "statement_status": "exact", "statement_verification": "The canonical record is problem 2.5 in the “Matrix Theory” section of the AIM workshop *Stability, hyperbolicity, and zero localization of functions*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Matrix Theory\nSource item: 2.5\nSource URL: http://aimpl.org/hyperbolicpoly/2/\nCanonical location: aim-analysis-notes.json notes[140]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate the signs of Hurwitz minors for P\\\\'olya frequency sequences.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/2/", "tags": [ "aim", "AIM-ANALYSIS-0141", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the standard one-sided convention, the augmented Hurwitz matrix is the even-column submatrix of the Toeplitz matrix, so every Hurwitz minor through order r is nonnegative for a PF_r sequence and every finite Hurwitz minor is nonnegative for PF_infinity. For a finite PF_infinity block with reciprocal-root factorization P(z)=c product_rho(z+x_rho), the k-th classical leading determinant is exactly c^k times the staircase Schur polynomial s_(k,k-1,...,1)(x), and it is positive exactly when at least k root parameters are positive. The order bound cannot generally be extended: exact PF_2 examples give third leading determinants of negative, zero, and positive sign.\n\nCandidate contribution (formula; novelty confidence low): For every finite one-sided PF_infinity coefficient block, including fixed trailing-zero padding, Delta_k=c^k s_(k,k-1,...,1)(x_1,...,x_n), with Delta_k>0 if and only if at least k reciprocal-root parameters x_rho are positive; together with explicit PF_2 examples, this sharply separates within-order nonnegativity from beyond-order sign failure." }, { "id": 20000610, "problem_number": "AIM-ANALYSIS-0142", "title": "A three-nonzero-root theorem for Fisk's 3x3 Toeplitz-minor transform", "statement": "The following conjecture of (McNamara et al.) was proved by P. Br\\\"and\\'en. Let $p(x):=\\sum_{k=0}^n a_k x^k$, with $a_{-1}:=a_{n+1}:=0$.\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{cc}\na_k & a_{k+1} \\\\\na_{k-1} & a_{k} \\end{array}\\right|x^k \\in\\mathcal{L}\\text{-}\\mathcal{P}.\\]\nS. Fisk generalized this statement to polynomials with coefficients formed from the determinants of $3\\times 3$ matrices in the conjecture below.\n\n\\label{fidk_conj}\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{ccc}\na_k & a_{k+1} & a_{k+2} \\\\\na_{k-1} & a_k & a_{k+1}\\\\\na_{k-2} & a_{k-1} & a_k\\end{array}\\right|x^k\\in\\mathcal{L}\\text{-}\\mathcal{P}^+.\\]", "original_statement": "The following conjecture of (McNamara et al.) was proved by P. Br\\\"and\\'en. Let $p(x):=\\sum_{k=0}^n a_k x^k$, with $a_{-1}:=a_{n+1}:=0$.\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{cc}\na_k & a_{k+1} \\\\\na_{k-1} & a_{k} \\end{array}\\right|x^k \\in\\mathcal{L}\\text{-}\\mathcal{P}.\\]\nS. Fisk generalized this statement to polynomials with coefficients formed from the determinants of $3\\times 3$ matrices in the conjecture below.\n\n\\label{fidk_conj}\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{ccc}\na_k & a_{k+1} & a_{k+2} \\\\\na_{k-1} & a_k & a_{k+1}\\\\\na_{k-2} & a_{k-1} & a_k\\end{array}\\right|x^k\\in\\mathcal{L}\\text{-}\\mathcal{P}^+.\\]", "clean_statement": "The following conjecture of (McNamara et al.) was proved by P. Br\\\"and\\'en. Let $p(x):=\\sum_{k=0}^n a_k x^k$, with $a_{-1}:=a_{n+1}:=0$.\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{cc}\na_k & a_{k+1} \\\\\na_{k-1} & a_{k} \\end{array}\\right|x^k \\in\\mathcal{L}\\text{-}\\mathcal{P}.\\]\nS. Fisk generalized this statement to polynomials with coefficients formed from the determinants of $3\\times 3$ matrices in the conjecture below.\n\n\\label{fidk_conj}\nIf $p\\in\\mathcal{L}\\text{-}\\mathcal{P}^+$, then\n\\[\\sum_{k=0}^n \\left|\\begin{array}{ccc}\na_k & a_{k+1} & a_{k+2} \\\\\na_{k-1} & a_k & a_{k+1}\\\\\na_{k-2} & a_{k-1} & a_k\\end{array}\\right|x^k\\in\\mathcal{L}\\text{-}\\mathcal{P}^+.\\]", "statement_status": "exact", "statement_verification": "The canonical record is Conjecture 3.2 in the \"Log Concavity\" section of the AIM problem list *Stability, hyperbolicity, and zero localization of functions*. It asks the following. Given", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Log Concavity\nSource item: 3.2\nSource URL: http://aimpl.org/hyperbolicpoly/3/\nCanonical location: aim-analysis-notes.json notes[141]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"The following conjecture of (McNamara et al.) was proved by P. Br\\\\\\\"and\\\\'en. Let $p(x):=\\\\sum_{k=0}^n a_k x^k$, with $a_{-1}:=a_{n+1}:=0$.\\nIf $p\\\\in\\\\mathcal{L}\\\\text{-}\\\\mathcal{P}^+$, then\\n\\\\[\\\\sum_{k=0}^n \\\\left|\\\\begin{array}{cc}\\na_k & a_{k+1} \\\\\\\\\\na_{k-1} & a_{k} \\\\end{array}\\\\right|x^k \\\\in\\\\mathcal{L}\\\\text{-}\\\\mathcal{P}.\\\\]\\nS. Fisk generalized this statement to polynomials with coefficients formed from the determinants of $3\\\\times 3$ matrices in the conjecture below.\\n\\n\\\\label{fidk_conj}\\nIf $p\\\\in\\\\mathcal{L}\\\\text{-}\\\\mathcal{P}^+$, then\\n\\\\[\\\\sum_{k=0}^n \\\\left|\\\\begin{array}{ccc}\\na_k & a_{k+1} & a_{k+2} \\\\\\\\\\na_{k-1} & a_k & a_{k+1}\\\\\\\\\\na_{k-2} & a_{k-1} & a_k\\\\end{array}\\\\right|x^k\\\\in\\\\mathcal{L}\\\\text{-}\\\\mathcal{P}^+.\\\\]\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/3/", "tags": [ "aim", "AIM-ANALYSIS-0142", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "With the coefficient sequence extended by zero, Fisk's 3x3 transform preserves the Laguerre-Polya class with nonnegative coefficients for every input polynomial having at most three nonzero zeros, counted with multiplicity; arbitrary zero-root multiplicity is allowed. In the exactly three-root case the output coefficients satisfy D1^2 >= 10 D0 D2 and D2^2 >= 10 D1 D3, so Hutchinson's criterion gives three simple negative zeros. More generally, if p(x)=a0 product_j(1+rho_j x), then its kth transformed coefficient is a0^3 s_(3^k)(rho), reducing the full conjecture to real-rootedness of a rectangular-Schur generating polynomial.\n\nCandidate contribution (special_case; novelty confidence low): The 3x3 transform preserves the Laguerre-Polya property for every polynomial with at most three nonzero zeros; in the three-root case both internal second quotients are at least 10.", "research_difficulty_suggested": "L4" }, { "id": 20000611, "problem_number": "AIM-ANALYSIS-0143", "title": "A paired-tail reduction and coefficientwise positivity through n=5", "statement": "Let\n\\[Q_n^{(\\alpha, \\beta)}:=\\sum_{k=0}^n\\binom{n}{k}f_kf_{n-k}((x+\\alpha)_k(x+\\beta)_{n-k} - (x+\\alpha+\\beta)_k(x)_{n-k}),\\]\nwhere $\\alpha, \\beta>0$, $f_k^2\\ge f_{k-1}f_{k+1}$, and $(x)_k = x(x+1)\\cdots(x+k-1)$. Then the Maclaurin coefficients of $Q^{(\\,\\alpha,\\,\\beta)}$ are non-negative.", "original_statement": "Let\n\\[Q_n^{(\\alpha, \\beta)}:=\\sum_{k=0}^n\\binom{n}{k}f_kf_{n-k}((x+\\alpha)_k(x+\\beta)_{n-k} - (x+\\alpha+\\beta)_k(x)_{n-k}),\\]\nwhere $\\alpha, \\beta>0$, $f_k^2\\ge f_{k-1}f_{k+1}$, and $(x)_k = x(x+1)\\cdots(x+k-1)$. Then the Maclaurin coefficients of $Q^{(\\,\\alpha,\\,\\beta)}$ are non-negative.", "clean_statement": "Let\n\\[Q_n^{(\\alpha, \\beta)}:=\\sum_{k=0}^n\\binom{n}{k}f_kf_{n-k}((x+\\alpha)_k(x+\\beta)_{n-k} - (x+\\alpha+\\beta)_k(x)_{n-k}),\\]\nwhere $\\alpha, \\beta>0$, $f_k^2\\ge f_{k-1}f_{k+1}$, and $(x)_k = x(x+1)\\cdots(x+k-1)$. Then the Maclaurin coefficients of $Q^{(\\,\\alpha,\\,\\beta)}$ are non-negative.", "statement_status": "exact", "statement_verification": "The canonical AIM record, problem 3.1 in the “Log Concavity” section of the workshop *Stability, hyperbolicity, and zero localization of functions*, asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Log Concavity\nSource item: 3.1\nSource URL: http://aimpl.org/hyperbolicpoly/3/\nCanonical location: aim-analysis-notes.json notes[142]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let\\n\\\\[Q_n^{(\\\\alpha, \\\\beta)}:=\\\\sum_{k=0}^n\\\\binom{n}{k}f_kf_{n-k}((x+\\\\alpha)_k(x+\\\\beta)_{n-k} - (x+\\\\alpha+\\\\beta)_k(x)_{n-k}),\\\\]\\nwhere $\\\\alpha, \\\\beta>0$, $f_k^2\\\\ge f_{k-1}f_{k+1}$, and $(x)_k = x(x+1)\\\\cdots(x+k-1)$. Then the Maclaurin coefficients of $Q^{(\\\\,\\\\alpha,\\\\,\\\\beta)}$ are non-negative.\"\nOriginal remarks: [\"I have proved the case $\\\\alpha=\\\\beta=1$ and formulated a number of generalizations of this conjecture in http://arxiv.org/abs/1203.1482\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/3/", "tags": [ "aim", "AIM-ANALYSIS-0143", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the intended nonnegative log-concave/no-internal-zero convention, Q_n has nonnegative Maclaurin coefficients for every alpha,beta>0 and every n at most 5. For arbitrary n, pairing k with n-k and applying Chu-Vandermonde plus Abel summation gives Q_n as a sum of nonnegative central-product increments f_r f_(n-r)-f_(r-1)f_(n-r+1) times explicit central truncated kernels; every needed kernel through n=5 has a strictly positive coefficient factorization. The abbreviated local inequalities alone are insufficient: exact n=3 examples separately show failure for sign-changing sequences and for nonnegative sequences with internal zeros.\n\nCandidate contribution (finite_case_and_reduction; novelty confidence low): The exact paired-tail identity reduces the general conjecture to coefficientwise positivity of central truncated Chu-Vandermonde kernels, and explicit positive factorizations of those kernels prove the full arbitrary-parameter conjecture for n=2,3,4,5 with a strict/zero classification." }, { "id": 20000612, "problem_number": "AIM-ANALYSIS-0144", "title": "A lattice criterion and a curvature screen for log-concave mixtures", "statement": "Let $\\{\\phi_k(x)\\}_{k=0}^\\infty$ and $\\{f_k\\}_{k=0}^\\infty$ be log-concave. When is $\\sum f_k\\phi_k(x)$ log-concave ?", "original_statement": "Let $\\{\\phi_k(x)\\}_{k=0}^\\infty$ and $\\{f_k\\}_{k=0}^\\infty$ be log-concave. When is $\\sum f_k\\phi_k(x)$ log-concave ?", "clean_statement": "Let $\\{\\phi_k(x)\\}_{k=0}^\\infty$ and $\\{f_k\\}_{k=0}^\\infty$ be log-concave. When is $\\sum f_k\\phi_k(x)$ log-concave ?", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Log Concavity\nSource item: 3.3\nSource URL: http://aimpl.org/hyperbolicpoly/3/\nCanonical location: aim-analysis-notes.json notes[143]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $\\\\{\\\\phi_k(x)\\\\}_{k=0}^\\\\infty$ and $\\\\{f_k\\\\}_{k=0}^\\\\infty$ be log-concave. When is $\\\\sum f_k\\\\phi_k(x)$ log-concave ?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/3/", "tags": [ "aim", "AIM-ANALYSIS-0144", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM wording is ambiguous, but a rigorous positive answer holds under a lattice-adapted hypothesis: if f has interval support and is log-concave, the continuous nonnegative kernel obeys the floor/ceiling midpoint inequality, and the nonnegative series is finite, then the sum is log-concave by the Klartag-Lehec discrete Prekopa-Leindler theorem and monotone truncation. The separate marginal assumptions are insufficient: a smooth positive jointly log-concave, strictly TP2 Gaussian kernel with f=(1,1,0,...) produces e^{-(x+1)^2}+e^{-(x-1)^2}, which fails midpoint log-concavity. For two positive C2 columns u,v, every mixture au+bv is log-concave exactly when |(log u)'-(log v)'| is at most sqrt(-(log u)'')+sqrt(-(log v)'') pointwise.\n\nCandidate contribution (criterion/equivalence; novelty confidence low): For positive C2 log-concave adjacent kernel columns u and v, all positive two-column mixtures au+bv are log-concave if and only if |(log u)'-(log v)'| <= sqrt(-(log u)'')+sqrt(-(log v)'') pointwise; consequently this inequality is a necessary adjacent-column screen for any kernel preserving log-concavity for every finite-support log-concave weight sequence." }, { "id": 20000613, "problem_number": "AIM-ANALYSIS-0145", "title": "A literal counterexample and an exact symmetric-quartic curvature classification", "statement": "Let $p\\in\\mathbb{R}[x]$, $\\deg(p)=n>2$. There are at least $n-1$ points of extreme curvature $\\kappa(p)$ ( more generally where $\\kappa'(p)=0$ ) \\cite{MR2104693}.", "original_statement": "Let $p\\in\\mathbb{R}[x]$, $\\deg(p)=n>2$. There are at least $n-1$ points of extreme curvature $\\kappa(p)$ ( more generally where $\\kappa'(p)=0$ ) \\cite{MR2104693}.", "clean_statement": "Let $p\\in\\mathbb{R}[x]$, $\\deg(p)=n>2$. There are at least $n-1$ points of extreme curvature $\\kappa(p)$ ( more generally where $\\kappa'(p)=0$ ) \\cite{MR2104693}.", "statement_status": "exact", "statement_verification": "The canonical record, copied without correction, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Zeros of derivatives\nSource item: 5.1\nSource URL: http://aimpl.org/hyperbolicpoly/5/\nCanonical location: aim-analysis-notes.json notes[144]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $p\\\\in\\\\mathbb{R}[x]$, $\\\\deg(p)=n>2$. There are at least $n-1$ points of extreme curvature $\\\\kappa(p)$ ( more generally where $\\\\kappa'(p)=0$ ) \\\\cite{MR2104693}.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/5/", "tags": [ "aim", "AIM-ANALYSIS-0145", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal canonical assertion that every degree-n real polynomial has at least n-1 points of extreme curvature is refuted by p(x)=x^4+2x^2: its curvature derivative has exactly one simple real zero, which is a genuine strict maximum, rather than three. The cited Edwards-Gordon problem is recovered as the opposite, at-most-n-1 conjecture; the literature checked indicates that corrected general conjecture remains open. As a proved special case, every reflection-symmetric quartic is classified exactly and has either one or three genuine signed-curvature extrema.\n\nCandidate contribution (special_case; novelty confidence low): For p(x)=A(x-b)^4+B(x-b)^2+C, after multiplying by -1 if needed to normalize A>0, signed curvature has exactly three simple genuine extrema when B<0 or when B>=0 and B^3=0 and B^3>=A, with a triple stationary zero precisely at B^3=A." }, { "id": 20000614, "problem_number": "AIM-ANALYSIS-0146", "title": "A polynomial/full-Laguerre-Polya boundary for zero-spacing under higher derivatives", "statement": "What are the zero spacing effects of higher order differential operators on functions which belong to $\\mathcal{L}\\text{-}\\mathcal{P}$?", "original_statement": "What are the zero spacing effects of higher order differential operators on functions which belong to $\\mathcal{L}\\text{-}\\mathcal{P}$?", "clean_statement": "What are the zero spacing effects of higher order differential operators on functions which belong to $\\mathcal{L}\\text{-}\\mathcal{P}$?", "statement_status": "exact", "statement_verification": "The exact canonical record is AIM-ANALYSIS-0146, source file `aim-analysis-notes.json`, index 145, workshop section “Zeros of derivatives,” Problem 5.2:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Zeros of derivatives\nSource item: 5.2\nSource URL: http://aimpl.org/hyperbolicpoly/5/\nCanonical location: aim-analysis-notes.json notes[145]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What are the zero spacing effects of higher order differential operators on functions which belong to $\\\\mathcal{L}\\\\text{-}\\\\mathcal{P}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/5/", "tags": [ "aim", "AIM-ANALYSIS-0146", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For polynomial inputs, every translation-invariant hyperbolicity preserver, including the supported constant-coefficient operators and D^r, does not decrease additive mesh. This positive theorem does not extend to arbitrary entire Laguerre-Polya inputs: for every fixed r >= 1 and b > 0, f_{a,b}(x) = x(x^2-b^2)exp(-a x^2) has mesh b, whereas the mesh of D^r f_{a,b} divided by b tends to zero as a tends to infinity. After x = y/sqrt(a) and exact normalization, the polynomial factor converges to a nonzero multiple of H_{r+1}(y), so two persistent simple Hermite zeros give an O(a^{-1/2}) output gap.\n\nCandidate contribution (counterexample_family; novelty confidence low): For every fixed derivative order r >= 1, the explicit Laguerre-Polya family x(x^2-b^2)exp(-a x^2) exhibits arbitrarily large relative mesh collapse under D^r, with the collapsing zero cluster governed by H_{r+1} after Gaussian rescaling." }, { "id": 20000615, "problem_number": "AIM-ANALYSIS-0147", "title": "Rational counterexamples and a finite gap-two parameter theorem", "statement": "Let $(L_n^\\alpha)$ denote the generalized Laguerre polynomials. It is known that $L_m^\\alpha$ and $L_n^\\alpha$ can have common zeros if $|m-n|>1$. Is the number of common zeros linked to arithmetic properties of $\\alpha$?\n\nThere are no common zeros between $L_m^\\alpha$ and $L_n^\\alpha$, $|m-n|>1$, for $\\alpha$ rational.", "original_statement": "Let $(L_n^\\alpha)$ denote the generalized Laguerre polynomials. It is known that $L_m^\\alpha$ and $L_n^\\alpha$ can have common zeros if $|m-n|>1$. Is the number of common zeros linked to arithmetic properties of $\\alpha$?\n\nThere are no common zeros between $L_m^\\alpha$ and $L_n^\\alpha$, $|m-n|>1$, for $\\alpha$ rational.", "clean_statement": "Let $(L_n^\\alpha)$ denote the generalized Laguerre polynomials. It is known that $L_m^\\alpha$ and $L_n^\\alpha$ can have common zeros if $|m-n|>1$. Is the number of common zeros linked to arithmetic properties of $\\alpha$?\n\nThere are no common zeros between $L_m^\\alpha$ and $L_n^\\alpha$, $|m-n|>1$, for $\\alpha$ rational.", "statement_status": "exact", "statement_verification": "The canonical record comes from Conjecture 6.1 in the “Orthogonal polynomials” section of the AIM problem list *Stability and hyperbolicity*. The live page was checked on 2026-07-28. It explicitly labels the following as “Conjecture 6.1” and attributes it to K. Driver; it does not display a restriction on the parameter:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Orthogonal polynomials\nSource item: 6.1\nSource URL: http://aimpl.org/hyperbolicpoly/6/\nCanonical location: aim-analysis-notes.json notes[146]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Let $(L_n^\\\\alpha)$ denote the generalized Laguerre polynomials. It is known that $L_m^\\\\alpha$ and $L_n^\\\\alpha$ can have common zeros if $|m-n|>1$. Is the number of common zeros linked to arithmetic properties of $\\\\alpha$?\\n\\nThere are no common zeros between $L_m^\\\\alpha$ and $L_n^\\\\alpha$, $|m-n|>1$, for $\\\\alpha$ rational.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/6/", "tags": [ "aim", "AIM-ANALYSIS-0147", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM conjecture is refuted by the published rational example L_2^(23)(30)=L_4^(23)(30)=0. In addition, this attempt proves that for every m >= 1 and alpha > -1, L_m^(alpha) and L_(m+2)^(alpha) share a zero if and only if P_m(alpha)=L_m^(alpha)(alpha+2m+3)=0; that common zero is uniquely x=alpha+2m+3, and P_m has exact degree floor(m/2) with explicit nonzero leading coefficients in both parities.\n\nCandidate contribution (theorem; novelty confidence low): For fixed m >= 1, the gap-two exceptional-parameter polynomial P_m(alpha)=L_m^(alpha)(alpha+2m+3) has exact degree floor(m/2), with leading coefficient (-1)^q/(2^q q!) for m=2q and (-1)^(q+1)4(2q+3)/(3*2^q q!) for m=2q+1; consequently at most floor(m/2) parameters alpha > -1 can make L_m^(alpha) and L_(m+2)^(alpha) share a zero." }, { "id": 20000616, "problem_number": "AIM-ANALYSIS-0148", "title": "Constraint pivots, canonical orthogonal bases, and a dual-lacunary Padé equivalence", "statement": "Consider a subspace $S$ of $\\mathbb{C}[x]$ of codimension $k$ (defined by $k$ linear constraints, explicitly given). Given a scalar product on $\\mathbb{C}[x]$, find an orthogonal basis of $S$ (ordered by degree); some degrees will be missing.\n\\begin{enumerate}\n\\item Which degrees get skipped?\n\\item Characterize these orthogonal polynomials.\n\\item Does this have anything to do with lacunary Pad\\'e approximants?\n\\end{enumerate}", "original_statement": "Consider a subspace $S$ of $\\mathbb{C}[x]$ of codimension $k$ (defined by $k$ linear constraints, explicitly given). Given a scalar product on $\\mathbb{C}[x]$, find an orthogonal basis of $S$ (ordered by degree); some degrees will be missing.\n\\begin{enumerate}\n\\item Which degrees get skipped?\n\\item Characterize these orthogonal polynomials.\n\\item Does this have anything to do with lacunary Pad\\'e approximants?\n\\end{enumerate}", "clean_statement": "Consider a subspace $S$ of $\\mathbb{C}[x]$ of codimension $k$ (defined by $k$ linear constraints, explicitly given). Given a scalar product on $\\mathbb{C}[x]$, find an orthogonal basis of $S$ (ordered by degree); some degrees will be missing.\n\\begin{enumerate}\n\\item Which degrees get skipped?\n\\item Characterize these orthogonal polynomials.\n\\item Does this have anything to do with lacunary Pad\\'e approximants?\n\\end{enumerate}", "statement_status": "exact", "statement_verification": "The canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Orthogonal polynomials\nSource item: 6.3\nSource URL: http://aimpl.org/hyperbolicpoly/6/\nCanonical location: aim-analysis-notes.json notes[147]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Consider a subspace $S$ of $\\\\mathbb{C}[x]$ of codimension $k$ (defined by $k$ linear constraints, explicitly given). Given a scalar product on $\\\\mathbb{C}[x]$, find an orthogonal basis of $S$ (ordered by degree); some degrees will be missing.\\n\\\\begin{enumerate}\\n\\\\item Which degrees get skipped?\\n\\\\item Characterize these orthogonal polynomials.\\n\\\\item Does this have anything to do with lacunary Pad\\\\'e approximants?\\n\\\\end{enumerate}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/6/", "tags": [ "aim", "AIM-ANALYSIS-0148", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any codimension-k polynomial subspace defined by independent constraints, the skipped degrees are exactly the k rank-jump columns of the truncated constraint matrix, independently of the positive-definite inner product; every present degree has a unique monic vector orthogonal to the preceding filtered subspace, giving a finite Gram-projection construction. Every k-element gap set is realizable, jet/divisibility spaces reduce exactly to a Christoffel-modified measure, and in the coordinate-gap plus moment/Hankel setting the constrained polynomials are precisely denominators with matching coefficient and Laurent-remainder lacunae. A positive non-Hankel coefficient inner product proves that no such one-function Padé interpretation exists for arbitrary scalar products.\n\nCandidate contribution (equivalence; novelty confidence low): For every finite exponent set E, positive moment/Hankel inner product, and degree n not in E, the monic constrained orthogonal polynomial in the coordinate-gap space is exactly the unique degree-n denominator whose coefficients vanish on E intersected with [0,n] and whose Markov-transform remainder has vanishing coefficients at z^(-j-1) for every j in [0,n-1] outside E." }, { "id": 20000617, "problem_number": "AIM-ANALYSIS-0149", "title": "Exact degree-two Jacobi interlacing and its physical interpretation", "statement": "Let $P_n^{\\;\\alpha, \\,\\beta}$ denote the Jacobi polynomial of $\\deg n$ with parameters $\\alpha, \\beta> -1$.\nThen $P_n^{\\;\\alpha', \\,\\beta'}$ and $P_n^{\\;\\alpha, \\,\\beta}$ interlace for $\\alpha\\approx\\alpha'$, and $\\beta\\approx\\beta'$. Does the failure of interlacing correspond to some physical law being ``violated''?", "original_statement": "Let $P_n^{\\;\\alpha, \\,\\beta}$ denote the Jacobi polynomial of $\\deg n$ with parameters $\\alpha, \\beta> -1$.\nThen $P_n^{\\;\\alpha', \\,\\beta'}$ and $P_n^{\\;\\alpha, \\,\\beta}$ interlace for $\\alpha\\approx\\alpha'$, and $\\beta\\approx\\beta'$. Does the failure of interlacing correspond to some physical law being ``violated''?", "clean_statement": "Let $P_n^{\\;\\alpha, \\,\\beta}$ denote the Jacobi polynomial of $\\deg n$ with parameters $\\alpha, \\beta> -1$.\nThen $P_n^{\\;\\alpha', \\,\\beta'}$ and $P_n^{\\;\\alpha, \\,\\beta}$ interlace for $\\alpha\\approx\\alpha'$, and $\\beta\\approx\\beta'$. Does the failure of interlacing correspond to some physical law being ``violated''?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 6.2 in the “Orthogonal polynomials” section of the AIM problem list *Stability, hyperbolicity, and zero localization of functions*. The live page was checked on 2026-07-28. It attributes the problem to K. Driver and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Orthogonal polynomials\nSource item: 6.2\nSource URL: http://aimpl.org/hyperbolicpoly/6/\nCanonical location: aim-analysis-notes.json notes[148]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $P_n^{\\\\;\\\\alpha, \\\\,\\\\beta}$ denote the Jacobi polynomial of $\\\\deg n$ with parameters $\\\\alpha, \\\\beta> -1$.\\nThen $P_n^{\\\\;\\\\alpha', \\\\,\\\\beta'}$ and $P_n^{\\\\;\\\\alpha, \\\\,\\\\beta}$ interlace for $\\\\alpha\\\\approx\\\\alpha'$, and $\\\\beta\\\\approx\\\\beta'$. Does the failure of interlacing correspond to some physical law being ``violated''?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/6/", "tags": [ "aim", "AIM-ANALYSIS-0149", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For degree two, writing the Jacobi zeros as c-r and c+r with c=(beta-alpha)/(alpha+beta+4) and r=2 sqrt((alpha+2)(beta+2))/((alpha+beta+4)sqrt(alpha+beta+3)), two parameter pairs strictly interlace exactly when |r-r'|<|c-c'||r-dot|, while the exact family P_2^(a,a) versus P_2^(a+epsilon,a+epsilon) fails to interlace for every nonzero admissible epsilon, however small. This failure is loss of a cross-parameter geometric ordering, not violation of the canonical fixed-operator Sturm law or loss of the unique strictly convex Stieltjes electrostatic equilibrium.\n\nCandidate contribution (criterion; novelty confidence low): The exact degree-two classification |r-r'|<|c-c'||r-dot| provide a necessary-and-sufficient geometric test for global and infinitesimal strict interlacing, respectively; the concentric symmetric path gives an exact arbitrarily-small obstruction." }, { "id": 20000618, "problem_number": "AIM-ANALYSIS-0150", "title": "Confluent Slater quotients and the first kurtosis obstruction", "statement": "Let $p_k(z)$ be the polynomials orthogonal with respect to $\\Phi$ on $(-\\infty,\\infty)$.\nDefine\n\\[P_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=1}^n}{\\det\\left[z_i^{j-1}\\right]_{i=1,j=1}^n}, \\qquad\\text{and}\\qquad\nQ_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=2}^n}{\\det\\left[z_{i}^{j-2}\\right]_{i,j=2}^{n}}.\\]\n\nInvestigate the zeros of $P_n$ and $Q_n.$", "original_statement": "Let $p_k(z)$ be the polynomials orthogonal with respect to $\\Phi$ on $(-\\infty,\\infty)$.\nDefine\n\\[P_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=1}^n}{\\det\\left[z_i^{j-1}\\right]_{i=1,j=1}^n}, \\qquad\\text{and}\\qquad\nQ_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=2}^n}{\\det\\left[z_{i}^{j-2}\\right]_{i,j=2}^{n}}.\\]\n\nInvestigate the zeros of $P_n$ and $Q_n.$", "clean_statement": "Let $p_k(z)$ be the polynomials orthogonal with respect to $\\Phi$ on $(-\\infty,\\infty)$.\nDefine\n\\[P_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=1}^n}{\\det\\left[z_i^{j-1}\\right]_{i=1,j=1}^n}, \\qquad\\text{and}\\qquad\nQ_n(z):=\\frac{\\det\\left[p_i(z_j)\\right]_{i,j=2}^n}{\\det\\left[z_{i}^{j-2}\\right]_{i,j=2}^{n}}.\\]\n\nInvestigate the zeros of $P_n$ and $Q_n.$", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Riemann $\\Xi$-function\nSource item: 7.1\nSource URL: http://aimpl.org/hyperbolicpoly/7/\nCanonical location: aim-analysis-notes.json notes[149]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $p_k(z)$ be the polynomials orthogonal with respect to $\\\\Phi$ on $(-\\\\infty,\\\\infty)$.\\nDefine\\n\\\\[P_n(z):=\\\\frac{\\\\det\\\\left[p_i(z_j)\\\\right]_{i,j=1}^n}{\\\\det\\\\left[z_i^{j-1}\\\\right]_{i=1,j=1}^n}, \\\\qquad\\\\text{and}\\\\qquad\\nQ_n(z):=\\\\frac{\\\\det\\\\left[p_i(z_j)\\\\right]_{i,j=2}^n}{\\\\det\\\\left[z_{i}^{j-2}\\\\right]_{i,j=2}^{n}}.\\\\]\\n\\nInvestigate the zeros of $P_n$ and $Q_n.$\"\nOriginal remarks: [\"The following generalization of the result mentioned above is true.\\n\\nLet $\\\\mu$ be an even, positive measure on $\\\\mathbb{R}$, and $F(z):=\\\\int_{-\\\\infty}^\\\\infty\\\\cos(zt)d\\\\mu(t)$. Let $\\\\{p_k(z)\\\\}_{k=0}^\\\\infty$ be the orthogonal polynomials with respect to $\\\\mu$ on $(-\\\\infty,\\\\infty)$. Then $F\\\\in\\\\mathcal{L}\\\\text{-}\\\\mathcal{P}$ if and only if the associated polynomial $P_n[p_1,\\\\ldots,p_n]$ has purely imaginary zeros, if and only if $Q_n[p_1,\\\\ldots, p_n]$ has no purely imaginary zeros. (for all $n$)\"]\nOriginal literature field (JSON string): \"The motivation for this problem is the following. If for all $n\\\\in\\\\mathbb{N}$, $P_n(z,z,\\\\ldots,z)$ vanishes only when $z=i\\\\alpha$, with $\\\\alpha\\\\in\\\\mathbb{R}$, then the Riemann hypothesis is true. Alternatively, if for each $n\\\\in\\\\mathbb{N}$, $Q_n(z,z,\\\\ldots,z)$ is non-zero on the imaginary axis, then the Riemann hypothesis is true.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/7/", "tags": [ "aim", "AIM-ANALYSIS-0150", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After fixing monic orthogonal polynomials and the standard confluent Wronskian normalization, the first Slater quotient equals the normalized moment integral m_0^{-1} integral product_j(z_j-t) dmu(t). Writing r=m_2/m_0 and R=m_0 m_4/m_2^2, the diagonal P_1, P_2, and P_3 tests hold automatically, P_4 is purely imaginary-rooted exactly when R is at most 9, Q_2 automatically avoids the imaginary axis, and Q_3 avoids it exactly when R is less than 9. These are separate finite tests, with distinct behavior at R=9; a bounded Favard construction realizes every R>1 and proves sharpness. This is no progress toward proving the all-degree condition or the Riemann hypothesis. The reported numerical value R approximately 2.791102858 for the AIM kernel Phi is explicitly non-certified and is not used in the proof.\n\nCandidate contribution (proposition; novelty confidence low): For every even positive infinite-support measure, the first nonautomatic diagonal Slater tests are governed sharply by R=m_0 m_4/m_2^2: P_4 is imaginary-rooted if and only if R is at most 9, while Q_3 has no imaginary-axis zero if and only if R is less than 9; moreover every R>1 is realized by an even compactly supported infinite-support measure via the bounded Jacobi sequence b_1=1, b_2=R-1, and b_k=1 for k at least 3." }, { "id": 20000619, "problem_number": "AIM-ANALYSIS-0151", "title": "Large-power zero-free disks for powers of the Riemann xi kernel", "statement": "Are there any non-real zeros of\n\\[\\int_{0}^\\infty \\Phi^\\alpha(t)\\cos(zt)dt \\qquad \\text{for}\\qquad \\alpha>0 ?\\]", "original_statement": "Are there any non-real zeros of\n\\[\\int_{0}^\\infty \\Phi^\\alpha(t)\\cos(zt)dt \\qquad \\text{for}\\qquad \\alpha>0 ?\\]", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Riemann $\\Xi$-function\nSource item: 78.2\nSource URL: http://aimpl.org/hyperbolicpoly/7/\nCanonical location: aim-analysis-notes.json notes[150]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Are there any non-real zeros of\\n\\\\[\\\\int_{0}^\\\\infty \\\\Phi^\\\\alpha(t)\\\\cos(zt)dt \\\\qquad \\\\text{for}\\\\qquad \\\\alpha>0 ?\\\\]\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/7/", "tags": [ "aim", "AIM-ANALYSIS-0151", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every alpha > 0 the cosine transform F_alpha is an even entire function with no zeros on the imaginary axis. After normalization by F_alpha(0), it converges locally uniformly to 1 as alpha tends to infinity. More quantitatively, if M_alpha(r) is the normalized moment E_alpha[T^2 exp(rT)] for the probability density proportional to Phi(t)^alpha, then (r^2/2) M_alpha(r) < 1 certifies that F_alpha has no zero in |z| <= r, and the resulting certified radii tend to infinity with alpha.\n\nCandidate contribution (criterion_and_asymptotic_theorem; novelty confidence low): For the AIM pointwise-power family, the explicit Rouche criterion (r^2/2) E_alpha[T^2 exp(rT)] < 1 yields zero-free disks, while weighted concentration at the unique maximum of Phi proves that these certified disks exhaust every fixed compact subset of the complex plane as alpha tends to infinity." }, { "id": 20000620, "problem_number": "AIM-ANALYSIS-0152", "title": "Index tests and quantitative boundary-root pushing for zero-free Mergelyan approximation", "statement": "\\label{nonzero_approx}\nLet be $K\\subset\\mathbb{C}$ be compact, let $\\mathbb{C}\\setminus K$ be connected, and let $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$. Is there a sequence of polynomials $\\{p_n\\}_{n=1}^\\infty$ with no zeros in $K$ such that $p_n\\to f$?", "original_statement": "\\label{nonzero_approx}\nLet be $K\\subset\\mathbb{C}$ be compact, let $\\mathbb{C}\\setminus K$ be connected, and let $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$. Is there a sequence of polynomials $\\{p_n\\}_{n=1}^\\infty$ with no zeros in $K$ such that $p_n\\to f$?", "clean_statement": "\\label{nonzero_approx}\nLet be $K\\subset\\mathbb{C}$ be compact, let $\\mathbb{C}\\setminus K$ be connected, and let $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$. Is there a sequence of polynomials $\\{p_n\\}_{n=1}^\\infty$ with no zeros in $K$ such that $p_n\\to f$?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 9.1 in the “Miscellaneous Problems” section of the AIM problem list *Stability, hyperbolicity, and zero localization of functions*. The live page was checked on 2026-07-28. It attributes the problem to P. Gauthier and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Miscellaneous Problems\nSource item: 9.1\nSource URL: http://aimpl.org/hyperbolicpoly/9/\nCanonical location: aim-analysis-notes.json notes[151]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"\\\\label{nonzero_approx}\\nLet be $K\\\\subset\\\\mathbb{C}$ be compact, let $\\\\mathbb{C}\\\\setminus K$ be connected, and let $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$. Is there a sequence of polynomials $\\\\{p_n\\\\}_{n=1}^\\\\infty$ with no zeros in $K$ such that $p_n\\\\to f$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/9/", "tags": [ "aim", "AIM-ANALYSIS-0152", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The intended convergence is uniform on K, and the unrestricted zero-free Mergelyan problem remains apparently open. This attempt proves that any zero-free polynomial approximation forces a nonnegative winding index on each Jordan curve in K where the limit is nonzero, while connected complement forces the Jordan interior into K and hence makes every admissible index zero; without connected complement, K equal to the unit circle and f(z)=1/z give an exact negative-index counterexample. It also quantifies boundary-root pushing: if q=a product(z-zeta_j) is zero-free on the interior of K, moving each boundary root by less than delta outside K changes q by at most |a| d delta (R+M+delta)^(d-1) on K, roots counted with multiplicity. Consequently every factorization f=hq with h invertible in A(K) and q a nonzero polynomial zero-free on the interior admits zero-free polynomial approximants, with explicit two-stage tolerances.\n\nCandidate contribution (quantitative_reduction; novelty confidence low): For q(z)=a product_{j=1}^d(z-zeta_j), zero-free on K interior, simultaneous displacement by less than delta of every boundary root to the complement gives a zero-free-on-K polynomial q_delta satisfying ||q_delta-q||_K <= |a| d delta (R+M+delta)^(d-1); combined with Mergelyan approximation below the new minimum modulus, this yields a quantitative theorem for every f=hq with h invertible in A(K)." }, { "id": 20000621, "problem_number": "AIM-ANALYSIS-0153", "title": "Upper-density universality and zero-free approximation", "statement": "\\label{conj_universal}\nLet be $K\\subset\\mathbb{C}$ be compact, $\\mathbb{C}\\setminus K$ connected, and $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$.\nSuppose $K\\subseteq\\{z:\\frac{1}{2} 0,\\]\nwhere\n\n\\[\\overline{d}(E):=\\limsup_{T\\to\\infty} \\frac{m(E\\cap[0,T\\,])}{T}.\\]", "original_statement": "\\label{conj_universal}\nLet be $K\\subset\\mathbb{C}$ be compact, $\\mathbb{C}\\setminus K$ connected, and $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$.\nSuppose $K\\subseteq\\{z:\\frac{1}{2} 0,\\]\nwhere\n\n\\[\\overline{d}(E):=\\limsup_{T\\to\\infty} \\frac{m(E\\cap[0,T\\,])}{T}.\\]", "clean_statement": "\\label{conj_universal}\nLet be $K\\subset\\mathbb{C}$ be compact, $\\mathbb{C}\\setminus K$ connected, and $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$.\nSuppose $K\\subseteq\\{z:\\frac{1}{2} 0,\\]\nwhere\n\n\\[\\overline{d}(E):=\\limsup_{T\\to\\infty} \\frac{m(E\\cap[0,T\\,])}{T}.\\]", "statement_status": "exact", "statement_verification": "The canonical record is object 152 (zero-based) of *aim-analysis-notes.json*. Its text is truncated after \\(\\frac12\\) and is not mathematically usable.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Miscellaneous Problems\nSource item: 9.2\nSource URL: http://aimpl.org/hyperbolicpoly/9/\nCanonical location: aim-analysis-notes.json notes[152]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"\\\\label{conj_universal}\\nLet be $K\\\\subset\\\\mathbb{C}$ be compact, $\\\\mathbb{C}\\\\setminus K$ connected, and $f$ be a continuous function on $K$ which is holomorphic and non-zero on the interior of $K$.\\nSuppose $K\\\\subseteq\\\\{z:\\\\frac{1}{2} 0,\\\\]\\nwhere\\n\\n\\\\[\\\\overline{d}(E):=\\\\limsup_{T\\\\to\\\\infty} \\\\frac{m(E\\\\cap[0,T\\\\,])}{T}.\\\\]\"\nOriginal remarks: [\"Andersson has shown that a positive answer to\\nProblem \\\\ref{nonzero_approx} and a confirmation of Conjecture \\\\ref{conj_universal} are equivalent.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/9/", "tags": [ "aim", "AIM-ANALYSIS-0153", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The corrupted live statement is recovered as a positive upper-density universality conjecture in the strip 1/2 < Re(s) < 1. A proved zero-bearing-shift estimate shows that the live upper-density form, Andersson's stronger lower-density form, and zero-free Mergelyan approximation are equivalent. More quantitatively, after affine compression into a disk with left edge sigma_0, good-shift measure of order T^alpha along a subsequence already suffices whenever alpha exceeds 3(1-sigma_0)/(2-sigma_0); an explicit endpoint in the disk centered at 3/4 of radius 1/8 uses T^(9/11)(log T)^5. This is a reduction only and proves neither the general conjecture nor the Riemann hypothesis.\n\nCandidate contribution (equivalence; novelty confidence low): For every fixed alpha in (0,1), choose a disk compactly contained in 1/2 < Re(s) < 1 and sufficiently close to Re(s)=1 that Ingham's exponent theta=3(1-sigma_0)/(2-sigma_0) is less than alpha. Then the global zero-free polynomial approximation conjecture is equivalent to requiring, only for targets in that disk, positive limsup of good-shift measure divided by T^alpha; at the explicit disk |s-3/4| <= 1/8, domination of T^(9/11)(log T)^5 is sufficient and equivalent.", "research_difficulty_suggested": "L4" }, { "id": 20000622, "problem_number": "AIM-ANALYSIS-0154", "title": "Stability in logarithmic moment coordinates", "statement": "Let $\\mu$ be a probability measure on $\\{0,1\\}^n$. Consider\n\\[f(\\vec{z}\\,) = \\int {\\vec{z}}^{\\;\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,)\\]\nand\n\\[g(\\vec{\\lambda}\\,) = \\int e^{\\;\\vec{\\lambda}\\cdot\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,).\\]\nWhat property of $g$ is equivalent to the stability of $f$?", "original_statement": "Let $\\mu$ be a probability measure on $\\{0,1\\}^n$. Consider\n\\[f(\\vec{z}\\,) = \\int {\\vec{z}}^{\\;\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,)\\]\nand\n\\[g(\\vec{\\lambda}\\,) = \\int e^{\\;\\vec{\\lambda}\\cdot\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,).\\]\nWhat property of $g$ is equivalent to the stability of $f$?", "clean_statement": "Let $\\mu$ be a probability measure on $\\{0,1\\}^n$. Consider\n\\[f(\\vec{z}\\,) = \\int {\\vec{z}}^{\\;\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,)\\]\nand\n\\[g(\\vec{\\lambda}\\,) = \\int e^{\\;\\vec{\\lambda}\\cdot\\vec{\\alpha}} d\\mu(\\vec{\\alpha}\\,).\\]\nWhat property of $g$ is equivalent to the stability of $f$?", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analysis\nWorkshop: Stability, hyperbolicity, and zero localization of functions\nSection: Miscellaneous Problems\nSource item: 9.3\nSource URL: http://aimpl.org/hyperbolicpoly/9/\nCanonical location: aim-analysis-notes.json notes[153]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $\\\\mu$ be a probability measure on $\\\\{0,1\\\\}^n$. Consider\\n\\\\[f(\\\\vec{z}\\\\,) = \\\\int {\\\\vec{z}}^{\\\\;\\\\vec{\\\\alpha}} d\\\\mu(\\\\vec{\\\\alpha}\\\\,)\\\\]\\nand\\n\\\\[g(\\\\vec{\\\\lambda}\\\\,) = \\\\int e^{\\\\;\\\\vec{\\\\lambda}\\\\cdot\\\\vec{\\\\alpha}} d\\\\mu(\\\\vec{\\\\alpha}\\\\,).\\\\]\\nWhat property of $g$ is equivalent to the stability of $f$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "http://aimpl.org/hyperbolicpoly/9/", "tags": [ "aim", "AIM-ANALYSIS-0154", "aim-domain:analysis", "aim-workshop:hyperbolicpoly", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Writing g(lambda)=f(exp(lambda_1),...,exp(lambda_n)), upper-half-plane stability of f is exactly zero-freeness of g on the open polystrip 0 J\n\n∑\n\n> j=1\n\nEj ≥ nn + 2 (2 π)2(n/ |Sn−1|| Ω|)2/n J(n+2) /n, J = 1, 2, 3,....\n\nBerezin proved in 1972 that\n\n∑\n\n> j\n\n(E − Ej )σ\n\n> +\n\n≤ |Ω|\n\n(2 π)n\n\n∫\n\n> Rn\n\n(E − | p|2)σ\n\n> +\n\ndp, σ ≥ 1, E > 0.\n\nThe cases 0 ≤ σ < 1 remain open. The P´ olya conjecture is exactly the case σ = 0. The inequality for σ = 1 implies the Li-Yau inequality via the Legendre transform.\n\n> 1Corrections and updates will be gratefully received at rlfrank@math.princeton.edu (for P´ olya and Lieb- Thirring inequalities) and Laugesen@illinois.edu (for Gap and Laplace eigenvalue inequalities).\n> 1\n\n(2) (Timo Weidl) Can one strengthen the Li-Yau result by including a correction term, perhaps involving the surface area of the boundary? This has been done for the discrete Laplacian on domains in a lattice; J. K. Freericks, E. H. Lieb, D. Ueltschi, Phase separation due to quantum mechanical correlations, Phys. Rev. Lett. 88, 106401 1-4 (2002). There is a result by Melas of Li-Yau type with corrections involving moments of iner-tia rather than surface area, see A. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003), 631-636. February 2009: The result of Melas has been strengthened by inclusion of a correction term involving the surface area of the boundary; see H. Kovarik, S. Vugalter, T. Weidl,\n\nTwo-dimensional Berezin-Li-Yau inequalities with a correction term, Comm. Math. Phys. 287 (2009), 959-981. An earlier improvement for γ ≥ 3/2, involving a notion of effective boundary, is due to T. Weidl, Improved Berezin-Li-Yau inequalities with a remainder term,in: Spectral theory of differential operators, T. Suslina and D. Yafaev (eds.), Amer. Math. Soc. Transl. Ser. 2, 225 (2008). For γ < 3/2 further improvements seem possible and desirable. (3) (Timo Weidl) There are analogues of the P´ olya and Li-Yau inequalities under Neumann boundary conditions, with the inequality signs reversed. The P´ olya Conjecture remains open for Neumann boundary conditions for j ≥ 2, except it was recently proved for j =2 in two dimensions by A. Girouard et al., J. Diff. Geometry, to appear. The analogue of Li-Yau was proved by Pawel Kr¨ oger; see P. Kr¨ oger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal. 106 (1992), no. 2, 353-357, and also A. Laptev Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal. 151 (1997), 531-545. Can one strengthen the Kr¨ oger result by including a correction term? (4) (Timo Weidl) The questions raised above are meaningful in the presence of a magnetic field. For more information and some progress see Item (10) below. (5) (Evans Harrell, Joachim Stubbe) For σ ≥ 2 the mapping\n\nrσ: E 7 → E−σ−d/ 2\n\n( |Ω|\n\n(2 π)n\n\n∫\n\n> Rn\n\n(E − | p|2)σdp − ∑\n\n> j\n\n(E − Ej )σ\n\n> +\n\n)\n\nis non-increasing. This was proved in E. M. Harrell and L. Hermi, Differential inequali-ties for Riesz means and Weyl-type bounds for eigenvalues, J. Funct. Analysis 254 (2008), 3173-3191, using the trace identities of E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, and Universal bounds and semiclassical estimates for eigen-values of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. According to Weyl's asymptotic formula rσ(E) tends to zero as E tends to infinity and therefore rσ(E) ≥ 0, which is the Berezin-Li-Yau-inequality. Can one strengthen this\n\n> 2\n\nbound in the trace identity of Harrell-Stubbe to obtain correction terms involving the sur-face area of the boundary? For the Laplacian with periodic boundary conditions a similar monotonicity property holds (for details see E. M. Harrell and J. Stubbe, Trace identities for commutators, with applications to the distribution of eigenvalues, preprint 2009, avail-able as arXiv:0903.0563). In this case the search for the correction term is related to the famous Gauss circle problem (or lattice point problem). (6) (Evans Harrell, Joachim Stubbe) Prove monotonicity results like in Item (5) for higher order operators (e.g. clamped plate problem) and fractional powers of Laplacians (for some results on √−∆ see E. M. Harrell and S. Yıldırım Yolcu, Eigenvalue inequalities for Klein-Gordon Operators, accepted for publication in J. Funct. Analysis) leading to Berezin-Li-Yau inequalities for these operators. (7) (Evans Harrell, Joachim Stubbe) For p > 0 let\n\nMp(J):=\n\n(n + 2 pn\n\n1\n\nJ\n\n> J\n\n∑\n\n> j=1\n\nEpj\n\n) 1\n\n> p\n\n(1) and for p = 0 define\n\nM0(J):= e 2\n\n> n\n\n( J∏\n\n> j=1\n\nEj\n\n) 1\n\n> J. (2) According the the Weyl asymptotic formula, for all p ≥ 0,\n\nMp(J) ∼ (2 π)2(n/ |Sn−1|| Ω|)2/n J2/n\n\nas J → ∞. In E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, it has been shown that\n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2(≥ 0)\n\nand\n\nM1(J) −\n\n√\n\nM 21 (J) − M2(J) ≤ EJ ≤ EJ+1 ≤ M1(J) +\n\n√\n\nM 21 (J) − M2(J).\n\nBoth inequalities are sharp in the Weyl limit. For extensions to other Mp(J) see E. M. Harrell and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. For p > 0\n\nfind an upper bound of the form\n\nM 2pp (J) − M p\n\n> 2p\n\n(J) ≤ C(p, Ω) E2p\n\n> 1\n\nJ2pκ\n\nwith κ < 2/n.\n\n> 3\n\n(8) (Evans Harrell, Joachim Stubbe) With the above notations does\n\nEJ ≤ Mp(J)\n\nhold for all J and all p ≥ 0? Can one find Ω and J such that the inequality\n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2\n\nis saturated? 2. Lieb-Thirring Inequalities\n\nWrite E1 < E 2 ≤ E3 ≤ · · · ≤ 0 for the eigenvalues of −∆ − V on L2(Rn), meaning\n\n(−∆ − V )uj = Ej uj.\n\nThe eigenfunctions uj represent bound states with energies Ej. For simplicity we assume V ≥ 0.Assume n ≥ 1.The Lieb-Thirring inequality can be written as\n\n∑\n\n> j\n\n|Ej |γ ≤ Ln,γ\n\n∫\n\n> Rn\n\nV γ+n/ 2 dx,\n\nThis inequality holds (with a constant Ln,γ independent of V ) iff the parameter γ satisfies γ ≥ 1/2\n\nif n = 1, γ > 0 if n = 2 and γ ≥ 0 if n ≥ 3. The case γ = 0 (counting eigenvalues) is the Cwikel-Lieb-Rozenblum Inequality (CLR). In other words Tr (−∆ − V )γ\n\n> −\n\n≤ Cn,γ\n\n(2 π)n\n\n∫\n\n> Rn\n\n∫\n\n> Rn\n\n(|p|2 − V (x)) γ\n\n> −\n\ndpdx,\n\nwhere\n\nCn,γ = Ln,γ\n\nLcl\n\n> n,γ\n\nand Lcl\n\n> n,γ\n\n= 1(2 π)n\n\n∫\n\n> Rn\n\n(|p|2 − 1) γ\n\n> −\n\ndp.\n\nThe constant Lcl\n\n> n,γ\n\nis called the semiclassical Lieb-Thirring constant. Note that Cn,γ ≥ 1 always, by the Weyl asymptotics, and that Cn,γ is decreasing in γ for each fixed n, by the Aizenman-Lieb monotonicity result. To start with, let us summarize some known results on the constants Cn,γ, along with conjectures about best (smallest) values of Cn,γ. 4n γ Best known Cn,γ Best constant? status last updated\n\n1 12 2 2 known\n\n(12, 32 ) 2* 2\n\n(γ−1/2\n\n> γ+1 /2\n\n)γ−1/2\n\nconjectured\n\n[32, ∞) 1 1 known\n\n2 (0, 12 )?\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 Feb. 2009\n\n[32, ∞) 1 1 known\n\n= 3 [0, 12 ) 6.87 8/√3 ' 4.62 conjectured\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 1 conjectured Feb. 2009\n\n[32, ∞) 1 1 known\n\n≥ 4 [0, 12 ) 10.34 Feb. 2009\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 1 conjectured Feb. 2009\n\n[32, ∞) 1 1 known *better is known for γ ∈ [1, 32 ), e.g. C1,1 ≤ π√3 ' 1.82 via work of Eden-Foias.", "original_statement": "1. P´ olya and Related Inequalities \n\nConsider eigenvalues of the Dirichlet Laplacian on a bounded domain Ω ⊂ Rn:\n\n{\n\n−∆uj = Ej uj in Ω,\n\nuj = 0 on ∂Ω.Assume n ≥ 2.(1) (Michael Loss, Timo Weidl) The P´ olya Conjecture claims that the Weyl asymptotic formula provides a lower bound: \n\nEj ≥ (2 π)2(n/ |Sn−1|| Ω|)2/n j2/n, j = 1, 2, 3,.... \n\nThe conjecture remains open even for j = 3.The best partial result known is with a factor of n/ (n + 2) (which is less than 1) on the right hand side, as one deduces by estimating Ej ≤ EJ in the following inequality due to Li and Yau, \n\n> J\n\n∑\n\n> j=1\n\nEj ≥ nn + 2 (2 π)2(n/ |Sn−1|| Ω|)2/n J(n+2) /n, J = 1, 2, 3,.... \n\nBerezin proved in 1972 that \n\n∑\n\n> j\n\n(E − Ej )σ \n\n> +\n\n≤ |Ω|\n\n(2 π)n\n\n∫\n\n> Rn\n\n(E − | p|2)σ \n\n> +\n\ndp, σ ≥ 1, E > 0.\n\nThe cases 0 ≤ σ < 1 remain open. The P´ olya conjecture is exactly the case σ = 0. The inequality for σ = 1 implies the Li-Yau inequality via the Legendre transform. \n\n> 1Corrections and updates will be gratefully received at rlfrank@math.princeton.edu (for P´ olya and Lieb- Thirring inequalities) and Laugesen@illinois.edu (for Gap and Laplace eigenvalue inequalities).\n> 1\n\n(2) (Timo Weidl) Can one strengthen the Li-Yau result by including a correction term, perhaps involving the surface area of the boundary? This has been done for the discrete Laplacian on domains in a lattice; J. K. Freericks, E. H. Lieb, D. Ueltschi, Phase separation due to quantum mechanical correlations, Phys. Rev. Lett. 88, 106401 1-4 (2002). There is a result by Melas of Li-Yau type with corrections involving moments of iner-tia rather than surface area, see A. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003), 631-636. February 2009: The result of Melas has been strengthened by inclusion of a correction term involving the surface area of the boundary; see H. Kovarik, S. Vugalter, T. Weidl, \n\nTwo-dimensional Berezin-Li-Yau inequalities with a correction term, Comm. Math. Phys. 287 (2009), 959-981. An earlier improvement for γ ≥ 3/2, involving a notion of effective boundary, is due to T. Weidl, Improved Berezin-Li-Yau inequalities with a remainder term,in: Spectral theory of differential operators, T. Suslina and D. Yafaev (eds.), Amer. Math. Soc. Transl. Ser. 2, 225 (2008). For γ < 3/2 further improvements seem possible and desirable. (3) (Timo Weidl) There are analogues of the P´ olya and Li-Yau inequalities under Neumann boundary conditions, with the inequality signs reversed. The P´ olya Conjecture remains open for Neumann boundary conditions for j ≥ 2, except it was recently proved for j =2 in two dimensions by A. Girouard et al., J. Diff. Geometry, to appear. The analogue of Li-Yau was proved by Pawel Kr¨ oger; see P. Kr¨ oger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal. 106 (1992), no. 2, 353-357, and also A. Laptev Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal. 151 (1997), 531-545. Can one strengthen the Kr¨ oger result by including a correction term? (4) (Timo Weidl) The questions raised above are meaningful in the presence of a magnetic field. For more information and some progress see Item (10) below. (5) (Evans Harrell, Joachim Stubbe) For σ ≥ 2 the mapping \n\nrσ: E 7 → E−σ−d/ 2\n\n( |Ω|\n\n(2 π)n\n\n∫\n\n> Rn\n\n(E − | p|2)σdp − ∑\n\n> j\n\n(E − Ej )σ\n\n> +\n\n)\n\nis non-increasing. This was proved in E. M. Harrell and L. Hermi, Differential inequali-ties for Riesz means and Weyl-type bounds for eigenvalues, J. Funct. Analysis 254 (2008), 3173-3191, using the trace identities of E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, and Universal bounds and semiclassical estimates for eigen-values of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. According to Weyl's asymptotic formula rσ(E) tends to zero as E tends to infinity and therefore rσ(E) ≥ 0, which is the Berezin-Li-Yau-inequality. Can one strengthen this \n\n> 2\n\nbound in the trace identity of Harrell-Stubbe to obtain correction terms involving the sur-face area of the boundary? For the Laplacian with periodic boundary conditions a similar monotonicity property holds (for details see E. M. Harrell and J. Stubbe, Trace identities for commutators, with applications to the distribution of eigenvalues, preprint 2009, avail-able as arXiv:0903.0563). In this case the search for the correction term is related to the famous Gauss circle problem (or lattice point problem). (6) (Evans Harrell, Joachim Stubbe) Prove monotonicity results like in Item (5) for higher order operators (e.g. clamped plate problem) and fractional powers of Laplacians (for some results on √−∆ see E. M. Harrell and S. Yıldırım Yolcu, Eigenvalue inequalities for Klein-Gordon Operators, accepted for publication in J. Funct. Analysis) leading to Berezin-Li-Yau inequalities for these operators. (7) (Evans Harrell, Joachim Stubbe) For p > 0 let \n\nMp(J):= \n\n(n + 2 pn\n\n1\n\nJ\n\n> J\n\n∑\n\n> j=1\n\nEpj\n\n) 1\n\n> p\n\n(1) and for p = 0 define \n\nM0(J):= e 2\n\n> n\n\n( J∏\n\n> j=1\n\nEj\n\n) 1\n\n> J. (2) According the the Weyl asymptotic formula, for all p ≥ 0,\n\nMp(J) ∼ (2 π)2(n/ |Sn−1|| Ω|)2/n J2/n \n\nas J → ∞. In E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, it has been shown that \n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2(≥ 0) \n\nand \n\nM1(J) −\n\n√\n\nM 21 (J) − M2(J) ≤ EJ ≤ EJ+1 ≤ M1(J) + \n\n√\n\nM 21 (J) − M2(J).\n\nBoth inequalities are sharp in the Weyl limit. For extensions to other Mp(J) see E. M. Harrell and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. For p > 0\n\nfind an upper bound of the form \n\nM 2pp (J) − M p\n\n> 2p\n\n(J) ≤ C(p, Ω) E2p \n\n> 1\n\nJ2pκ \n\nwith κ < 2/n.\n\n> 3\n\n(8) (Evans Harrell, Joachim Stubbe) With the above notations does \n\nEJ ≤ Mp(J)\n\nhold for all J and all p ≥ 0? Can one find Ω and J such that the inequality \n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2\n\nis saturated? 2. Lieb-Thirring Inequalities \n\nWrite E1 < E 2 ≤ E3 ≤ · · · ≤ 0 for the eigenvalues of −∆ − V on L2(Rn), meaning \n\n(−∆ − V )uj = Ej uj.\n\nThe eigenfunctions uj represent bound states with energies Ej. For simplicity we assume V ≥ 0.Assume n ≥ 1.The Lieb-Thirring inequality can be written as \n\n∑\n\n> j\n\n|Ej |γ ≤ Ln,γ \n\n∫\n\n> Rn\n\nV γ+n/ 2 dx, \n\nThis inequality holds (with a constant Ln,γ independent of V ) iff the parameter γ satisfies γ ≥ 1/2\n\nif n = 1, γ > 0 if n = 2 and γ ≥ 0 if n ≥ 3. The case γ = 0 (counting eigenvalues) is the Cwikel-Lieb-Rozenblum Inequality (CLR). In other words Tr (−∆ − V )γ \n\n> −\n\n≤ Cn,γ \n\n(2 π)n\n\n∫\n\n> Rn\n\n∫\n\n> Rn\n\n(|p|2 − V (x)) γ \n\n> −\n\ndpdx, \n\nwhere \n\nCn,γ = Ln,γ \n\nLcl \n\n> n,γ\n\nand Lcl \n\n> n,γ\n\n= 1(2 π)n\n\n∫\n\n> Rn\n\n(|p|2 − 1) γ \n\n> −\n\ndp. \n\nThe constant Lcl \n\n> n,γ\n\nis called the semiclassical Lieb-Thirring constant. Note that Cn,γ ≥ 1 always, by the Weyl asymptotics, and that Cn,γ is decreasing in γ for each fixed n, by the Aizenman-Lieb monotonicity result. To start with, let us summarize some known results on the constants Cn,γ, along with conjectures about best (smallest) values of Cn,γ. 4n γ Best known Cn,γ Best constant? status last updated \n\n1 12 2 2 known \n\n(12, 32 ) 2* 2\n\n(γ−1/2\n\n> γ+1 /2\n\n)γ−1/2\n\nconjectured \n\n[32, ∞) 1 1 known \n\n2 (0, 12 )?\n\n[12, 1) 3.64 Feb. 2009 \n\n[1, 32 ) 1.82 Feb. 2009 \n\n[32, ∞) 1 1 known \n\n= 3 [0, 12 ) 6.87 8/√3 ' 4.62 conjectured \n\n[12, 1) 3.64 Feb. 2009 \n\n[1, 32 ) 1.82 1 conjectured Feb. 2009 \n\n[32, ∞) 1 1 known \n\n≥ 4 [0, 12 ) 10.34 Feb. 2009 \n\n[12, 1) 3.64 Feb. 2009 \n\n[1, 32 ) 1.82 1 conjectured Feb. 2009 \n\n[32, ∞) 1 1 known *better is known for γ ∈ [1, 32 ), e.g. C1,1 ≤ π√3 ' 1.82 via work of Eden-Foias.", "clean_statement": "1. P´ olya and Related Inequalities\n\nConsider eigenvalues of the Dirichlet Laplacian on a bounded domain Ω ⊂ Rn:\n\n{\n\n−∆uj = Ej uj in Ω,\n\nuj = 0 on ∂Ω.Assume n ≥ 2.(1) (Michael Loss, Timo Weidl) The P´ olya Conjecture claims that the Weyl asymptotic formula provides a lower bound:\n\nEj ≥ (2 π)2(n/ |Sn−1|| Ω|)2/n j2/n, j = 1, 2, 3,....\n\nThe conjecture remains open even for j = 3.The best partial result known is with a factor of n/ (n + 2) (which is less than 1) on the right hand side, as one deduces by estimating Ej ≤ EJ in the following inequality due to Li and Yau,\n\nd/2 J\n\n∑\n\nd/2 j=1\n\nEj ≥ nn + 2 (2 π)2(n/ |Sn−1|| Ω|)2/n J(n+2) /n, J = 1, 2, 3,....\n\nBerezin proved in 1972 that\n\n∑\n\nd/2 j\n\n(E − Ej )σ\n\nd/2 +\n\n≤ |Ω|\n\n(2 π)n\n\n∫\n\nd/2 Rn\n\n(E − | p|2)σ\n\nd/2 +\n\ndp, σ ≥ 1, E d/2 0.\n\nThe cases 0 ≤ σ < 1 remain open. The P´ olya conjecture is exactly the case σ = 0. The inequality for σ = 1 implies the Li-Yau inequality via the Legendre transform.\n\nd/2 1Corrections and updates will be gratefully received at rlfrank@math.princeton.edu (for P´ olya and Lieb- Thirring inequalities) and Laugesen@illinois.edu (for Gap and Laplace eigenvalue inequalities).\nd/2 1\n\n(2) (Timo Weidl) Can one strengthen the Li-Yau result by including a correction term, perhaps involving the surface area of the boundary? This has been done for the discrete Laplacian on domains in a lattice; J. K. Freericks, E. H. Lieb, D. Ueltschi, Phase separation due to quantum mechanical correlations, Phys. Rev. Lett. 88, 106401 1-4 (2002). There is a result by Melas of Li-Yau type with corrections involving moments of iner-tia rather than surface area, see A. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003), 631-636. February 2009: The result of Melas has been strengthened by inclusion of a correction term involving the surface area of the boundary; see H. Kovarik, S. Vugalter, T. Weidl,\n\nTwo-dimensional Berezin-Li-Yau inequalities with a correction term, Comm. Math. Phys. 287 (2009), 959-981. An earlier improvement for γ ≥ 3/2, involving a notion of effective boundary, is due to T. Weidl, Improved Berezin-Li-Yau inequalities with a remainder term,in: Spectral theory of differential operators, T. Suslina and D. Yafaev (eds.), Amer. Math. Soc. Transl. Ser. 2, 225 (2008). For γ < 3/2 further improvements seem possible and desirable. (3) (Timo Weidl) There are analogues of the P´ olya and Li-Yau inequalities under Neumann boundary conditions, with the inequality signs reversed. The P´ olya Conjecture remains open for Neumann boundary conditions for j ≥ 2, except it was recently proved for j =2 in two dimensions by A. Girouard et al., J. Diff. Geometry, to appear. The analogue of Li-Yau was proved by Pawel Kr¨ oger; see P. Kr¨ oger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal. 106 (1992), no. 2, 353-357, and also A. Laptev Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal. 151 (1997), 531-545. Can one strengthen the Kr¨ oger result by including a correction term? (4) (Timo Weidl) The questions raised above are meaningful in the presence of a magnetic field. For more information and some progress see Item (10) below. (5) (Evans Harrell, Joachim Stubbe) For σ ≥ 2 the mapping\n\nrσ: E 7 → E−σ−d/ 2\n\n( |Ω|\n\n(2 π)n\n\n∫\n\nd/2 Rn\n\n(E − | p|2)σdp − ∑\n\nd/2 j\n\n(E − Ej )σ\n\nd/2 +\n\n)\n\nis non-increasing. This was proved in E. M. Harrell and L. Hermi, Differential inequali-ties for Riesz means and Weyl-type bounds for eigenvalues, J. Funct. Analysis 254 (2008), 3173-3191, using the trace identities of E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, and Universal bounds and semiclassical estimates for eigen-values of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. According to Weyl's asymptotic formula rσ(E) tends to zero as E tends to infinity and therefore rσ(E) ≥ 0, which is the Berezin-Li-Yau-inequality. Can one strengthen this\n\nd/2 2\n\nbound in the trace identity of Harrell-Stubbe to obtain correction terms involving the sur-face area of the boundary? For the Laplacian with periodic boundary conditions a similar monotonicity property holds (for details see E. M. Harrell and J. Stubbe, Trace identities for commutators, with applications to the distribution of eigenvalues, preprint 2009, avail-able as arXiv:0903.0563). In this case the search for the correction term is related to the famous Gauss circle problem (or lattice point problem). (6) (Evans Harrell, Joachim Stubbe) Prove monotonicity results like in Item (5) for higher order operators (e.g. clamped plate problem) and fractional powers of Laplacians (for some results on √−∆ see E. M. Harrell and S. Yıldırım Yolcu, Eigenvalue inequalities for Klein-Gordon Operators, accepted for publication in J. Funct. Analysis) leading to Berezin-Li-Yau inequalities for these operators. (7) (Evans Harrell, Joachim Stubbe) For p d/2 0 let\n\nMp(J):=\n\n(n + 2 pn\n\n1\n\nJ\n\nd/2 J\n\n∑\n\nd/2 j=1\n\nEpj\n\n) 1\n\nd/2 p\n\n(1) and for p = 0 define\n\nM0(J):= e 2\n\nd/2 n\n\n( J∏\n\nd/2 j=1\n\nEj\n\n) 1\n\nd/2 J. (2) According the the Weyl asymptotic formula, for all p ≥ 0,\n\nMp(J) ∼ (2 π)2(n/ |Sn−1|| Ω|)2/n J2/n\n\nas J → ∞. In E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, it has been shown that\n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2(≥ 0)\n\nand\n\nM1(J) −\n\n√\n\nM 21 (J) − M2(J) ≤ EJ ≤ EJ+1 ≤ M1(J) +\n\n√\n\nM 21 (J) − M2(J).\n\nBoth inequalities are sharp in the Weyl limit. For extensions to other Mp(J) see E. M. Harrell and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. For p d/2 0\n\nfind an upper bound of the form\n\nM 2pp (J) − M p\n\nd/2 2p\n\n(J) ≤ C(p, Ω) E2p\n\nd/2 1\n\nJ2pκ\n\nwith κ < 2/n.\n\nd/2 3\n\n(8) (Evans Harrell, Joachim Stubbe) With the above notations does\n\nEJ ≤ Mp(J)\n\nhold for all J and all p ≥ 0? Can one find Ω and J such that the inequality\n\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2\n\nis saturated? 2. Lieb-Thirring Inequalities\n\nWrite E1 < E 2 ≤ E3 ≤ · · · ≤ 0 for the eigenvalues of −∆ − V on L2(Rn), meaning\n\n(−∆ − V )uj = Ej uj.\n\nThe eigenfunctions uj represent bound states with energies Ej. For simplicity we assume V ≥ 0.Assume n ≥ 1.The Lieb-Thirring inequality can be written as\n\n∑\n\nd/2 j\n\n|Ej |γ ≤ Ln,γ\n\n∫\n\nd/2 Rn\n\nV γ+n/ 2 dx,\n\nThis inequality holds (with a constant Ln,γ independent of V ) iff the parameter γ satisfies γ ≥ 1/2\n\nif n = 1, γ d/2 0 if n = 2 and γ ≥ 0 if n ≥ 3. The case γ = 0 (counting eigenvalues) is the Cwikel-Lieb-Rozenblum Inequality (CLR). In other words Tr (−∆ − V )γ\n\nd/2 −\n\n≤ Cn,γ\n\n(2 π)n\n\n∫\n\nd/2 Rn\n\n∫\n\nd/2 Rn\n\n(|p|2 − V (x)) γ\n\nd/2 −\n\ndpdx,\n\nwhere\n\nCn,γ = Ln,γ\n\nLcl\n\nd/2 n,γ\n\nand Lcl\n\nd/2 n,γ\n\n= 1(2 π)n\n\n∫\n\nd/2 Rn\n\n(|p|2 − 1) γ\n\nd/2 −\n\ndp.\n\nThe constant Lcl\n\nd/2 n,γ\n\nis called the semiclassical Lieb-Thirring constant. Note that Cn,γ ≥ 1 always, by the Weyl asymptotics, and that Cn,γ is decreasing in γ for each fixed n, by the Aizenman-Lieb monotonicity result. To start with, let us summarize some known results on the constants Cn,γ, along with conjectures about best (smallest) values of Cn,γ. 4n γ Best known Cn,γ Best constant? status last updated\n\n1 12 2 2 known\n\n(12, 32 ) 2* 2\n\n(γ−1/2\n\nd/2 γ+1 /2\n\n)γ−1/2\n\nconjectured\n\n[32, ∞) 1 1 known\n\n2 (0, 12 )?\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 Feb. 2009\n\n[32, ∞) 1 1 known\n\n= 3 [0, 12 ) 6.87 8/√3 ' 4.62 conjectured\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 1 conjectured Feb. 2009\n\n[32, ∞) 1 1 known\n\n≥ 4 [0, 12 ) 10.34 Feb. 2009\n\n[12, 1) 3.64 Feb. 2009\n\n[1, 32 ) 1.82 1 conjectured Feb. 2009\n\n[32, ∞) 1 1 known *better is known for γ ∈ [1, 32 ), e.g. C1,1 ≤ π√3 ' 1.82 via work of Eden-Foias.", "statement_status": "reconstructed_unverified", "statement_verification": "1. In Item P5 the exponent in the Harrell--Stubbe normalized deficit must use the ambient dimension \\(n\\). The extracted `d/2` is inconsistent with the section's notation and the surrounding \\(\\mathbb R^n\\) integral. 2. Several superscripts and subscripts in Items P7--P8 collapse. The displayed source text layer reads like \\(M_1^2-M_2\\), which is dimensionally inconsistent with the stated root-normalized \\(M_p\\). The dimensionally consistent reconstructed quantity is \\[ M_1^2(J)-M_2^2(J), \\] and the corresponding general dispersion is read as \\[ M_p^{2p}(J)-M_{2p}^{2p}(J). \\] This is an explicit reconstruction, not a claim that the PDF's text layer itself is unambiguous. Confirming it against the original TeX or the authors is a useful editorial next step. 3. In the Ovals item the operator is \\[ H_C=-\\frac{d^2}{ds^2}+\\kappa(s)^2 \\] on \\(2\\pi\\)-periodic functions. The extracted run-on...", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Low eigenvalues of Laplace and Schrodinger operators\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/loweigenvalues/loweigenvalues.pdf\nCanonical location: aim-analysis-notes.json notes[156]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. P´ olya and Related Inequalities \\n\\nConsider eigenvalues of the Dirichlet Laplacian on a bounded domain Ω ⊂ Rn:\\n\\n{\\n\\n−∆uj = Ej uj in Ω,\\n\\nuj = 0 on ∂Ω.Assume n ≥ 2.(1) (Michael Loss, Timo Weidl) The P´ olya Conjecture claims that the Weyl asymptotic formula provides a lower bound: \\n\\nEj ≥ (2 π)2(n/ |Sn−1|| Ω|)2/n j2/n, j = 1, 2, 3,.... \\n\\nThe conjecture remains open even for j = 3.The best partial result known is with a factor of n/ (n + 2) (which is less than 1) on the right hand side, as one deduces by estimating Ej ≤ EJ in the following inequality due to Li and Yau, \\n\\n> J\\n\\n∑\\n\\n> j=1\\n\\nEj ≥ nn + 2 (2 π)2(n/ |Sn−1|| Ω|)2/n J(n+2) /n, J = 1, 2, 3,.... \\n\\nBerezin proved in 1972 that \\n\\n∑\\n\\n> j\\n\\n(E − Ej )σ \\n\\n> +\\n\\n≤ |Ω|\\n\\n(2 π)n\\n\\n∫\\n\\n> Rn\\n\\n(E − | p|2)σ \\n\\n> +\\n\\ndp, σ ≥ 1, E > 0.\\n\\nThe cases 0 ≤ σ < 1 remain open. The P´ olya conjecture is exactly the case σ = 0. The inequality for σ = 1 implies the Li-Yau inequality via the Legendre transform. \\n\\n> 1Corrections and updates will be gratefully received at rlfrank@math.princeton.edu (for P´ olya and Lieb- Thirring inequalities) and Laugesen@illinois.edu (for Gap and Laplace eigenvalue inequalities).\\n> 1\\n\\n(2) (Timo Weidl) Can one strengthen the Li-Yau result by including a correction term, perhaps involving the surface area of the boundary? This has been done for the discrete Laplacian on domains in a lattice; J. K. Freericks, E. H. Lieb, D. Ueltschi, Phase separation due to quantum mechanical correlations, Phys. Rev. Lett. 88, 106401 1-4 (2002). There is a result by Melas of Li-Yau type with corrections involving moments of iner-tia rather than surface area, see A. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003), 631-636. February 2009: The result of Melas has been strengthened by inclusion of a correction term involving the surface area of the boundary; see H. Kovarik, S. Vugalter, T. Weidl, \\n\\nTwo-dimensional Berezin-Li-Yau inequalities with a correction term, Comm. Math. Phys. 287 (2009), 959-981. An earlier improvement for γ ≥ 3/2, involving a notion of effective boundary, is due to T. Weidl, Improved Berezin-Li-Yau inequalities with a remainder term,in: Spectral theory of differential operators, T. Suslina and D. Yafaev (eds.), Amer. Math. Soc. Transl. Ser. 2, 225 (2008). For γ < 3/2 further improvements seem possible and desirable. (3) (Timo Weidl) There are analogues of the P´ olya and Li-Yau inequalities under Neumann boundary conditions, with the inequality signs reversed. The P´ olya Conjecture remains open for Neumann boundary conditions for j ≥ 2, except it was recently proved for j =2 in two dimensions by A. Girouard et al., J. Diff. Geometry, to appear. The analogue of Li-Yau was proved by Pawel Kr¨ oger; see P. Kr¨ oger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal. 106 (1992), no. 2, 353-357, and also A. Laptev Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal. 151 (1997), 531-545. Can one strengthen the Kr¨ oger result by including a correction term? (4) (Timo Weidl) The questions raised above are meaningful in the presence of a magnetic field. For more information and some progress see Item (10) below. (5) (Evans Harrell, Joachim Stubbe) For σ ≥ 2 the mapping \\n\\nrσ: E 7 → E−σ−d/ 2\\n\\n( |Ω|\\n\\n(2 π)n\\n\\n∫\\n\\n> Rn\\n\\n(E − | p|2)σdp − ∑\\n\\n> j\\n\\n(E − Ej )σ\\n\\n> +\\n\\n)\\n\\nis non-increasing. This was proved in E. M. Harrell and L. Hermi, Differential inequali-ties for Riesz means and Weyl-type bounds for eigenvalues, J. Funct. Analysis 254 (2008), 3173-3191, using the trace identities of E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, and Universal bounds and semiclassical estimates for eigen-values of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. According to Weyl's asymptotic formula rσ(E) tends to zero as E tends to infinity and therefore rσ(E) ≥ 0, which is the Berezin-Li-Yau-inequality. Can one strengthen this \\n\\n> 2\\n\\nbound in the trace identity of Harrell-Stubbe to obtain correction terms involving the sur-face area of the boundary? For the Laplacian with periodic boundary conditions a similar monotonicity property holds (for details see E. M. Harrell and J. Stubbe, Trace identities for commutators, with applications to the distribution of eigenvalues, preprint 2009, avail-able as arXiv:0903.0563). In this case the search for the correction term is related to the famous Gauss circle problem (or lattice point problem). (6) (Evans Harrell, Joachim Stubbe) Prove monotonicity results like in Item (5) for higher order operators (e.g. clamped plate problem) and fractional powers of Laplacians (for some results on √−∆ see E. M. Harrell and S. Yıldırım Yolcu, Eigenvalue inequalities for Klein-Gordon Operators, accepted for publication in J. Funct. Analysis) leading to Berezin-Li-Yau inequalities for these operators. (7) (Evans Harrell, Joachim Stubbe) For p > 0 let \\n\\nMp(J):= \\n\\n(n + 2 pn\\n\\n1\\n\\nJ\\n\\n> J\\n\\n∑\\n\\n> j=1\\n\\nEpj\\n\\n) 1\\n\\n> p\\n\\n(1) and for p = 0 define \\n\\nM0(J):= e 2\\n\\n> n\\n\\n( J∏\\n\\n> j=1\\n\\nEj\\n\\n) 1\\n\\n> J. (2) According the the Weyl asymptotic formula, for all p ≥ 0,\\n\\nMp(J) ∼ (2 π)2(n/ |Sn−1|| Ω|)2/n J2/n \\n\\nas J → ∞. In E. M. Harrell and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809, it has been shown that \\n\\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2(≥ 0) \\n\\nand \\n\\nM1(J) −\\n\\n√\\n\\nM 21 (J) − M2(J) ≤ EJ ≤ EJ+1 ≤ M1(J) + \\n\\n√\\n\\nM 21 (J) − M2(J).\\n\\nBoth inequalities are sharp in the Weyl limit. For extensions to other Mp(J) see E. M. Harrell and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schr¨ odinger operators, preprint 2008, available as arXiv:0808.1133. For p > 0\\n\\nfind an upper bound of the form \\n\\nM 2pp (J) − M p\\n\\n> 2p\\n\\n(J) ≤ C(p, Ω) E2p \\n\\n> 1\\n\\nJ2pκ \\n\\nwith κ < 2/n.\\n\\n> 3\\n\\n(8) (Evans Harrell, Joachim Stubbe) With the above notations does \\n\\nEJ ≤ Mp(J)\\n\\nhold for all J and all p ≥ 0? Can one find Ω and J such that the inequality \\n\\nM 21 (J) − M2(J) ≥ 14(EJ+1 − EJ )2\\n\\nis saturated? 2. Lieb-Thirring Inequalities \\n\\nWrite E1 < E 2 ≤ E3 ≤ · · · ≤ 0 for the eigenvalues of −∆ − V on L2(Rn), meaning \\n\\n(−∆ − V )uj = Ej uj.\\n\\nThe eigenfunctions uj represent bound states with energies Ej. For simplicity we assume V ≥ 0.Assume n ≥ 1.The Lieb-Thirring inequality can be written as \\n\\n∑\\n\\n> j\\n\\n|Ej |γ ≤ Ln,γ \\n\\n∫\\n\\n> Rn\\n\\nV γ+n/ 2 dx, \\n\\nThis inequality holds (with a constant Ln,γ independent of V ) iff the parameter γ satisfies γ ≥ 1/2\\n\\nif n = 1, γ > 0 if n = 2 and γ ≥ 0 if n ≥ 3. The case γ = 0 (counting eigenvalues) is the Cwikel-Lieb-Rozenblum Inequality (CLR). In other words Tr (−∆ − V )γ \\n\\n> −\\n\\n≤ Cn,γ \\n\\n(2 π)n\\n\\n∫\\n\\n> Rn\\n\\n∫\\n\\n> Rn\\n\\n(|p|2 − V (x)) γ \\n\\n> −\\n\\ndpdx, \\n\\nwhere \\n\\nCn,γ = Ln,γ \\n\\nLcl \\n\\n> n,γ\\n\\nand Lcl \\n\\n> n,γ\\n\\n= 1(2 π)n\\n\\n∫\\n\\n> Rn\\n\\n(|p|2 − 1) γ \\n\\n> −\\n\\ndp. \\n\\nThe constant Lcl \\n\\n> n,γ\\n\\nis called the semiclassical Lieb-Thirring constant. Note that Cn,γ ≥ 1 always, by the Weyl asymptotics, and that Cn,γ is decreasing in γ for each fixed n, by the Aizenman-Lieb monotonicity result. To start with, let us summarize some known results on the constants Cn,γ, along with conjectures about best (smallest) values of Cn,γ. 4n γ Best known Cn,γ Best constant? status last updated \\n\\n1 12 2 2 known \\n\\n(12, 32 ) 2* 2\\n\\n(γ−1/2\\n\\n> γ+1 /2\\n\\n)γ−1/2\\n\\nconjectured \\n\\n[32, ∞) 1 1 known \\n\\n2 (0, 12 )?\\n\\n[12, 1) 3.64 Feb. 2009 \\n\\n[1, 32 ) 1.82 Feb. 2009 \\n\\n[32, ∞) 1 1 known \\n\\n= 3 [0, 12 ) 6.87 8/√3 ' 4.62 conjectured \\n\\n[12, 1) 3.64 Feb. 2009 \\n\\n[1, 32 ) 1.82 1 conjectured Feb. 2009 \\n\\n[32, ∞) 1 1 known \\n\\n≥ 4 [0, 12 ) 10.34 Feb. 2009 \\n\\n[12, 1) 3.64 Feb. 2009 \\n\\n[1, 32 ) 1.82 1 conjectured Feb. 2009 \\n\\n[32, ∞) 1 1 known *better is known for γ ∈ [1, 32 ), e.g. C1,1 ≤ π√3 ' 1.82 via work of Eden-Foias.\"\nOriginal remarks: [\"Remark. References to the results in the table and to many of the questions below can be found in the lecture notes by Michael Loss and Timo Weidl, and in the survey paper by Dirk Hundertmark (which further states some better estimates on Cn,γ for special values of n and γ). For February 2009 updates see J. Dolbeault, A. Laptev, M. Loss, Lieb-Thirring inequalities with improved constants, J. Eur. Math. Soc. 10 (2008), and R. L. Frank, E. H. Lieb, R. Seiringer, \\n\\nNumber of bound states of Schr¨ odinger operators with matrix-valued potentials, Lett. Math. Phys. 82, 107 (2007). Now we state open problems on Lieb-Thirring inequalities. (1) (Richard Laugesen) Must an optimal potential V exist, for those Lieb-Thirring inequalities in which the best constant is not known? In particular this question is open for n = 1 and \\n\\n> 12\\n\\n< γ < 32.A restricted version of the problem asks: within the class of potentials having m bound states (where m ≥ 1 is given), does an optimal potential exist? (2) (Richard Laugesen) If an optimal potential exists, then does it have just a single bound state? (In other words, does −∆ − V have just a single eigenvalue?) When n = 1 and 12 <γ < 32, the natural conjecture is that the optimal potential is the one found by J. B. Keller when he determined the best constant in |E1|γ ≤ C ∫ \\n\\n> R\\n\\nV γ+1 /2 dx (see J. Mathematical Phys. 2:262-266, 1961). 5This \\\"single bound state\\\" conjecture is due to Lieb and Thirring, 1976. In dimension \\n\\nn = 1, the conjecture is known to be true in the endpoint cases γ = 1 /2 (in which case \\n\\nV is a delta function) and γ = 3 /2 (in which case V is a transparent or reflectionless potential). (3) (Eric Carlen) Does there exist a bound of the form ∑ \\n\\n> j\\n\\n|Ej |γ ≤ C|E1|γ? Here the factor C\\n\\ncould depend on n, γ, and on the integrability of a power of V sufficient to guarantee that the lefthand side is finite. (4) (Rafael Benguria) The use of Korteweg-de Vries (KdV) integrable system methods when \\n\\nn = 1, γ = 3 /2, suggests that one might similarly study Lieb-Thirring inequalities for the linear equation associated with the Benjamin-Ono equation (another integrable system). Tomas Ekholm, Rupert Frank and Dirk Hundertmark made progress during the Workshop already, by obtaining the analog of the Aizenman-Lieb \\\"monotonicity toward best con-stants\\\" result. The Lax pair for the Benjamin-Ono equation can be found for example in R.L. Anderson and E. Tafflin, The Benjamin-Ono equation -Recursivity of linearization mapsLax pairs, Letters in Mathematical Physics, 9 (1985), 299-311. See also, D.J. Kaup and Y. Matsuno, The inverse scattering transform for the Benjamin-Ono equation, Studies in applied mathematics 101 (1998), 73-98. (5) (Rupert Frank) The best constant when n = 1, γ = 1, is due to Eden-Foias (see A. Eden and C. Foias, A simple proof of the generalized Lieb-Thirring inequalities of one-space dimension, Journal of mathematical analysis and applications, 162 (1991), 250-254.) More precisely, they proved a Sobolev inequality, which then gives a Lieb-Thirring inequality via the Legendre transform. So a question is: can one find a more direct proof of this Lieb-Thirring inequality? Also, can one sharpen the Eden-Foias bound by including correction terms in their ar-gument? February 2009: An operator-valued version of the Eden-Foias bound has been proved by J. Dolbeault, A. Laptev, M. Loss, Lieb-Thirring inequalities with improved constants, J. Eur. Math. Soc. 10 (2008). By the 'lifting of dimension'-argument this result leads to the best known values for the constants in the Lieb-Thirring inequalities for γ ≥ 1 if n = 1 \\n\\nand for γ ≥ 1/2 if n ≥ 2.(6) (Timo Weidl) Can one find a way to directly estimate the sum of the eigenvalues, without going through the Birman-Schwinger transformation (which counts the eigenvalues rather than summing them)? (7) (Almut Burchard) The Ovals Problem. Consider a smooth closed curve γ of length 2π in \\n\\nR3, and let κ(s) be its curvature as a function of arclength. The curve determines the one-dimensional Schr¨ odinger operator HC = −d2/ds 2 + κ2 acting on 2π-periodic functions. \\n\\n> 6\\n\\nThis operator appears in the equation for the tension of a smooth, elastic, inextensible loop [5], and in connection with a Lieb-Thirring inequality in one dimension [4]; similar Schr¨ odinger operators with quadratic curvature potentials have been studied in connec-tion with quantum mechanics on narrow channels [2], Dirac operators on the sphere [3], and curvature-driven flows describing the motion of interfaces in reaction-diffusion equa-tions [1]. A natural conjecture is that the principal eigenvalue e(γ) is minimal when γ is a circle, where it takes the value 1. This question is open even for planar loops that enclose convex sets ( 'ovals' ). It is known that the value e(γ) = 1 is attained for an entire family of planar curves whose curvature is given by κ(s) = (α2 cos 2 s + α−2 sin 2 s)−1. When α → 0,these curves collapse onto two straight line segments of length π joined at the ends. The inequality e(γ) ≥ 1 has recently been shown for curves in some neighborhood of the family [5], and for curves satisfying additional geometric constraints [6]. The best universal lower bound on e(γ) that is currently known is.6085 [6]. Several participants at the Workshop had worked on this problem previously (includ-ing Benguria, Loss, Burchard, Thomas, and Linde). All agreed that classical Calculus of Variations techniques may be exhausted at this point, and that rearrangement techniques seem to fail. Linde and Burchard claimed that minimizers can be shown to exist, and should be convex, but could conceivably contain one corner, or two corners joined by a straight line segment. Benguria pointed to the family of putative minimizers (which look like ellipses in polar coordinates) as evidence that the problem may have a hidden affine symmetry. Carlen, Mazzeo, and Benguria proposed to search for geometric flows that drive e(γ) towards its minimum. The affine curvature flow [7] was mentioned as a promis-ing candidate. Rapti and Lee proposed to analyze the Euler-Lagrange equation using ODE methods. Laugesen suggested applying the Birman-Schwinger transformation, after which the conjecture becomes that the largest eigenvalue of the operator T = κ(d2/ds 2 + γ)−1κ\\n\\nis larger than 1, for each constant 0 < γ < 1. Equivalently, take γ = 1 and try to show the largest eigenvalue of T is larger than 1, when T acts on functions ψ with κψ orthogonal to \\n\\nsin s and cos s. The hope is that a good choice of trial function (in the variational principle for the largest eigenvalue) might suffice to prove this conjecture.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/loweigenvalues/loweigenvalues.pdf", "tags": [ "aim", "AIM-ANALYSIS-0157", "aim-domain:analysis", "aim-workshop:loweigenvalues", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For q=1+n/2, A=L_{1,n}^{cl}|Omega|, and the uncorrected Weyl saddle Lambda_J=(J/(qA))^{1/(q-1)}, a corrected order-one Riesz bound needed only at that single point, R_1(Lambda_J)<=A Lambda_J^q-Phi_Omega(Lambda_J), implies M_1(J)>=Lambda_J+(n+2)Phi_Omega(Lambda_J)/(nJ). This exact Legendre-duality transfer combines with positive fractional integration, which propagates correction profiles to every higher Riesz order and carries the sharp boundary coefficient exactly. Applied to the proved two-term Dirichlet Riesz asymptotic on the unit square, it yields M_1(J)=4 pi J+(32 sqrt(pi)/3)J^{1/2}+o(J^{1/2}).\n\nCandidate contribution (one_point_correction_transfer; novelty confidence low): Candidate novelty: the exact one-cutoff transfer from a domain-dependent R_1 deficit at the uncorrected Weyl saddle to the source-normalized M_1 correction, packaged with exact higher-order boundary-coefficient propagation and the explicit unit-square coefficient 32 sqrt(pi)/3." }, { "id": 20000626, "problem_number": "AIM-ANALYSIS-0158", "title": "No ensemble-free extreme-root asymptotic and a Kac small-ball transfer theorem", "statement": "Question 1: Wenbo Li Consider random polynomials in one variable (real or complex). Find the asymptotics of the norm of the largest zero as the degree of the random polynomials tends to infinity.", "original_statement": "Question 1: Wenbo Li Consider random polynomials in one variable (real or complex). Find the asymptotics of the norm of the largest zero as the degree of the random polynomials tends to infinity. \u0005", "clean_statement": "Question 1: Wenbo Li Consider random polynomials in one variable (real or complex). Find the asymptotics of the norm of the largest zero as the degree of the random polynomials tends to infinity.", "statement_status": "corrected_verified", "statement_verification": "The control character `U+0005` at the end of the extracted record is not mathematical content. Inspection of the original PDF shows the same end marker after every question. The PDF contains no preceding definition of a random-polynomial ensemble, and the following questions do not supply one.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Random analytic functions\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/randomzeros/arcc1.pdf\nCanonical location: aim-analysis-notes.json notes[157]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 1: Wenbo Li Consider random polynomials in one variable (real or complex). Find the asymptotics of the norm of the largest zero as the degree of the random polynomials tends to infinity. \\u0005\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/randomzeros/arcc1.pdf", "tags": [ "aim", "AIM-ANALYSIS-0158", "aim-domain:analysis", "aim-workshop:arcc1", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM PDF specifies no random-polynomial ensemble, so no universal extreme-root asymptotic exists: Gaussian Kac extremes are tight, while a Vieta bound forces Gaussian Weyl extremes to be at least (e^{-1/2}-o_P(1)) sqrt(n). For i.i.d. real or complex Kac coefficients with P(|xi| <= u) two-sided comparable to u^alpha near zero and E|xi|^alpha finite, the largest-root tail is uniformly comparable to x^{-alpha} at every finite degree and for the limiting power-series law; moments below alpha are uniformly finite and converge, whereas moments at or above alpha diverge.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: a two-sided u^alpha coefficient small-ball law plus a finite alpha moment transfers uniformly in degree to a two-sided x^{-alpha} largest-Kac-root tail, yielding the exact moment threshold and convergence of all subcritical moments." }, { "id": 20000627, "problem_number": "AIM-ANALYSIS-0159", "title": "Conditional-Poisson covariance proxy and Jensen-gap reduction for Ginibre permanental roots", "statement": "Question 2: Yan Fyodorov What is the mean density of permanental polynomials? This is unknown for random matrices of size greater than 5 × 5. [U+0005]", "original_statement": "Question 2: Yan Fyodorov What is the mean density of permanental polynomials? This is unknown for random matrices of size greater than 5 × 5. \u0005", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record comes from the American Institute of Mathematics workshop problem list *Random Analytic Functions*, compiled by Swaminathan Sethuraman and dated April 17, 2006. The original two-page PDF states, verbatim apart from its decorative end marker:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Random analytic functions\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/randomzeros/arcc1.pdf\nCanonical location: aim-analysis-notes.json notes[158]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 2: Yan Fyodorov What is the mean density of permanental polynomials? This is unknown for random matrices of size greater than 5 × 5. \\u0005\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/randomzeros/arcc1.pdf", "tags": [ "aim", "AIM-ANALYSIS-0159", "aim-domain:analysis", "aim-workshop:arcc1", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For p_N(z) = Per(z I_N - Z) with variance-1/N complex Ginibre Z, the covariance kernel is (N!/N^N) S_N(N z conjugate(w)). Its normalized annealed covariance proxy has exact disk mass E[Y]/N and planar density Var(Y)/(pi x), where Y is Poisson(x) conditioned on Y <= N; this proxy converges weakly to the uniform unit-disk law, with pointwise density convergence away from the edge. The actual normalized mean disk count differs exactly by -r J_N'(r)/(2N) almost everywhere and in the radial distributional formulation. Thus an explicit o(N/r) radial Jensen-gap derivative estimate would transfer the proxy limit to the actual mean zero measure. The N=1 calculation proves that the proxy is not the actual finite-N density.\n\nCandidate contribution (reduction; novelty confidence low): The finite-N conditional-Poisson formulas for the normalized covariance proxy, combined with the exact radial Jensen-gap disk-count identity, reduce the actual complex-Ginibre mean-density conjecture to control of r J_N'(r)/N.", "research_difficulty_suggested": "L4" }, { "id": 20000628, "problem_number": "AIM-ANALYSIS-0160", "title": "A CUE critical-point sum rule and the small-exponent Jensen bridge", "statement": "Question 3: Ashkan Nikeghbali What can one tell about the distribution of the zeros of the derivative of characteristic polynomial of random unitary matrix, especially near the boundary of the unit circle.Also what can one tell about E[|f ′|s] as s tends to zero? [U+0005]", "original_statement": "Question 3: Ashkan Nikeghbali What can one tell about the distribution of the zeros of the derivative of characteristic polynomial of random unitary matrix, especially near the boundary of the unit circle.Also what can one tell about E[|f ′|s] as s tends to zero? \u0005", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The record is Question 3 from the AIM workshop *Random analytic functions*. The AIM PDF, *Open Problems at the Random Analytic functions Workshop at AIM* (Swaminathan Sethuraman, April 17, 2006), reads, with only mathematical typesetting and spacing restored:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Random analytic functions\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/randomzeros/arcc1.pdf\nCanonical location: aim-analysis-notes.json notes[159]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 3: Ashkan Nikeghbali What can one tell about the distribution of the zeros of the derivative of characteristic polynomial of random unitary matrix, especially near the boundary of the unit circle.Also what can one tell about E[|f ′|s] as s tends to zero? \\u0005\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/randomzeros/arcc1.pdf", "tags": [ "aim", "AIM-ANALYSIS-0160", "aim-domain:analysis", "aim-workshop:arcc1", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For Haar U_N and Λ_N(z)=det(I-zU_N†), the critical points c_{j,N} satisfy exactly ∏_{j=1}^{N-1}|c_{j,N}|=|Tr U_N|/N. Hence their total logarithmic radial depth D_N=Σ log(1/|c_{j,N}|) obeys D_N−log N⇒W with P(W≤x)=exp(−e^(−2x)), and K_N(r)≤D_N/log(1/r), giving quantitative deep-zero concentration. Under explicit two-sided exponential-integrability hypotheses, the q→0 slope of log E|Λ_N′(r)|^q relative to r=0 equals the integrated expected radial critical-point count; expansion of the Simm–Wei fixed-interior limit gives its first two log-cumulants.\n\nCandidate contribution (theorem; novelty confidence low): The exact CUE critical-radius product/sum rule, its scaled-Gumbel total-depth limit, and the deterministic deep-critical-point bound appear not to have been stated together." }, { "id": 20000629, "problem_number": "AIM-ANALYSIS-0161", "title": "Random supports, Newton polytopes, and local Viro complexity", "statement": "Question 4: Maurice Rojas Investigate the connections between random sparse polygons and Newton polytopes. This should be extended to the case of random Viro diagrams. [U+0005]", "original_statement": "Question 4: Maurice Rojas Investigate the connections between random sparse polygons and Newton polytopes. This should be extended to the case of random Viro diagrams. \u0005", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Question 4 from the AIM workshop *Random analytic functions* (April 2006), attributed to Maurice Rojas. Its printable text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Random analytic functions\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/randomzeros/arcc1.pdf\nCanonical location: aim-analysis-notes.json notes[160]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 4: Maurice Rojas Investigate the connections between random sparse polygons and Newton polytopes. This should be extended to the case of random Viro diagrams. \\u0005\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/randomzeros/arcc1.pdf", "tags": [ "aim", "AIM-ANALYSIS-0161", "aim-domain:analysis", "aim-workshop:arcc1", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under an explicit Bernoulli-support, independent-lifting, fair-sign model, conditional on a pure d-dimensional regular triangulation T with N top simplices and n_r(T) unordered top-simplex pairs sharing r vertices, the active local Viro-chart count Y satisfies E[Y | T] = (1 - 2^{-d})N and Var(Y | T) = N(1 - 2^{-d})2^{-d} + 2^{1-2d} sum_{r=2}^d n_r(T)(2^{r-1} - 1). A vertex-incidence McDiarmid bound and an exact law-of-total-variance transfer to random T are also proved, together with an exact directional support-function law for a Bernoulli Newton polytope.\n\nCandidate contribution (moment_formula; novelty confidence low): Exact overlap-incidence covariance and variance formulas for the number of active top-simplex Viro charts, together with their law-of-total-variance transfer to a random Newton subdivision." }, { "id": 20000630, "problem_number": "AIM-ANALYSIS-0162", "title": "A constant-intensity obstruction to negatively correlated GAF zeros", "statement": "Question 5: Balint Virag Find a Gaussian entire function with negatively correlated zeros. We know that there exists such a function on the unit disc. This is related to the repulsion properties of random polynomials. [U+0005]", "original_statement": "Question 5: Balint Virag Find a Gaussian entire function with negatively correlated zeros. We know that there exists such a function on the unit disc. This is related to the repulsion properties of random polynomials. \u0005", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record comes from Question 5 of the AIM workshop list *Random analytic functions* (compiled by Swaminathan Sethuraman, dated April 17, 2006). The original PDF gives, with line breaks normalized:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Random analytic functions\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/randomzeros/arcc1.pdf\nCanonical location: aim-analysis-notes.json notes[161]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 5: Balint Virag Find a Gaussian entire function with negatively correlated zeros. We know that there exists such a function on the unit disc. This is related to the repulsion properties of random polynomials. \\u0005\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/randomzeros/arcc1.pdf", "tags": [ "aim", "AIM-ANALYSIS-0162", "aim-domain:analysis", "aim-workshop:arcc1", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unqualified wording admits the finite-rank literal example G(z)=e^z(xi_0+xi_1 z), whose single zero is a rank-one projection DPP, but this is explicitly quarantined as degenerate. For the intended infinite-zero problem, every nondegenerate entire GAF with constant zero intensity L/pi has the planar-GAF zero law; its normalized pair correlation exceeds one for t=L|z-w|^2/2 >= 2. In the explicit disks A_L=D(0,1/(4 sqrt L)) and B_L=D(3/sqrt L,1/(4 sqrt L)), the zero-count covariance is at least 1/[128(e^(49/4)-1)]>0. The standard hyperbolic disk family either loses zeros at alpha=1 or converges under positive-intensity scaling to this obstructed planar law. The inhomogeneous infinite-zero case remains open.\n\nCandidate contribution (quantitative_obstruction; novelty confidence low): Every nondegenerate entire GAF of constant zero intensity L/pi has positively correlated zero counts in the explicit scale-normalized disks A_L and B_L, with the uniform lower bound Cov(N(A_L),N(B_L)) >= 1/[128(e^(49/4)-1)]." }, { "id": 20000631, "problem_number": "AIM-ANALYSIS-0163", "title": "Morse reduction and exact Fubini-Study minimization in degrees one and two", "statement": "Question 6: Bernard Shiffman Does the Fubini-Study metric on P1 minimize the expected number of critical points? There are reasons to conjecture that the answer is Yes. [U+0005]", "original_statement": "Question 6: Bernard Shiffman Does the Fubini-Study metric on P1 minimize the expected number of critical points? There are reasons to conjecture that the answer is Yes. \u0005", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The [AIM workshop PDF](https://aimath.org/WWN/randomzeros/arcc1.pdf) gives no degree, probability law, or definition of critical point. Those data can be recovered from the Douglas--Shiffman--Zelditch papers underlying this workshop question. The reading used in this report is therefore the following, explicitly labeled reconstruction.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Random analytic functions\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/randomzeros/arcc1.pdf\nCanonical location: aim-analysis-notes.json notes[162]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 6: Bernard Shiffman Does the Fubini-Study metric on P1 minimize the expected number of critical points? There are reasons to conjecture that the answer is Yes. \\u0005\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/randomzeros/arcc1.pdf", "tags": [ "aim", "AIM-ANALYSIS-0163", "aim-domain:analysis", "aim-workshop:arcc1", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In the recovered Douglas-Shiffman-Zelditch ensemble, Poincare-Hopf gives S-M=N-2 for every section with simple zeros and Morse Chern-critical points, while compactness forces M>=1; hence #Crit=N-2+2M>=N and, for N>=2, the expected-count problem is exactly expected-maxima minimization. The exact Fubini-Study formula attains this universal bound for N=1 and N=2, proving global minimization in those degrees (without uniqueness, because scaling and PGL(2,C) pullbacks give ties). For an exactly area-preserving conformal variation at Fubini-Study, the constrained Calabi Hessian is (1/2) integral (Lf-8f)^2, with kernel precisely the degree-one spherical harmonics, so the leading expected-count Hessian is (1/(27 pi N)) integral (Lf-8f)^2. Exact global minimization for each fixed N>=3 remains unresolved in the literature checked.\n\nCandidate contribution (reduction_and_special_case; novelty confidence low): For Gaussian-almost-every Morse section, S-M=N-2 and #Crit=N-2+2M, so fixed-degree minimization is exactly expected-maxima minimization; this yields the universal bound #Crit>=N, attained by Fubini-Study for N=1,2 and therefore proves exact global minimization in those two degrees." }, { "id": 20000632, "problem_number": "AIM-ANALYSIS-0164", "title": "Covariance, diagonal, and normalized-kernel criteria for Gaussian entire functions", "statement": "Question 7: Steven Evans Is there a necessary and sufficient condition for a given correlation function to be the correlation function of an enire Gaussian function? [U+0005]", "original_statement": "Question 7: Steven Evans Is there a necessary and sufficient condition for a given correlation function to be the correlation function of an enire Gaussian function? \u0005", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Question 7 from the AIM workshop *Random analytic functions* (April 2006), attributed to Steven Evans. The original two-page PDF itself, not merely the extracted JSON, prints:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Random analytic functions\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/randomzeros/arcc1.pdf\nCanonical location: aim-analysis-notes.json notes[163]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 7: Steven Evans Is there a necessary and sufficient condition for a given correlation function to be the correlation function of an enire Gaussian function? \\u0005\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/randomzeros/arcc1.pdf", "tags": [ "aim", "AIM-ANALYSIS-0164", "aim-domain:analysis", "aim-workshop:arcc1", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the standard proper-covariance reading, a kernel K is realizable by a centered proper Gaussian entire function if and only if it is Hermitian positive semidefinite and entire in its first variable and anti-entire in its second, equivalently if and only if all of its finite leading Taylor coefficient matrices are positive semidefinite; the proof gives an almost surely locally uniformly convergent RKHS Gaussian-series realization. The attempt also proves exact diagonal and real-translation-stationary criteria and a base-point harmonic-phase criterion for normalized covariances, while conservatively separating the nonproper and zero-point-process readings.\n\nCandidate contribution (theorem; novelty confidence low): For a continuous Hermitian positive semidefinite unit-diagonal kernel R with a nonvanishing base section r(z)=R(z,z0), R is a normalized covariance of a centered proper Gaussian entire function if and only if C(z,w)=R(z,w)/(r(z) conjugate(r(w))) is sesquientire and a continuous argument of r is harmonic; the rank-one kernel R_*(z,w)=exp(i(|z|^2-|w|^2)) is an explicit sharp phase obstruction." }, { "id": 20000633, "problem_number": "AIM-ANALYSIS-0165", "title": "Exact planar probabilities and sphere witnesses for random Viro diagrams", "statement": "Question 8: Maurice Rojas What is the probability that a random Viro diagram contains no sphere? [U+0005]", "original_statement": "Question 8: Maurice Rojas What is the probability that a random Viro diagram contains no sphere? \u0005", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Question 8 in the AIM workshop list *Random analytic functions*, compiled by Swaminathan Sethuraman and dated April 17, 2006. The original PDF reads, with line breaks normalized:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Random analytic functions\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/randomzeros/arcc1.pdf\nCanonical location: aim-analysis-notes.json notes[164]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 8: Maurice Rojas What is the probability that a random Viro diagram contains no sphere? \\u0005\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/randomzeros/arcc1.pdf", "tags": [ "aim", "AIM-ANALYSIS-0165", "aim-domain:analysis", "aim-workshop:arcc1", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM wording has no source-defined probability space. Under an explicit orthant-glued planar toric model with used triangulation vertices A and iid Bernoulli(p) signs, the exact no-S^1-component probability is 1_{A mod 2 is a singleton}[p^{|A|}+(1-p)^{|A|}]; this averages exactly over any random support/subdivision law, and every primitive triangulation has probability zero. For a one-interior-vertex positive-orthant fan with m boundary vertices, the exact no-local-loop probability is 1-[p(1-p)^m+(1-p)p^m]. In the abstract Bernoulli Viro-subcomplex model, an isolated sign produces an S^{n-1} component, yielding the fair-sign bound (1-2^{-Delta})^{N/(Delta^2+1)}, while a connected closed triangulated surface has exact no-circle probability p^N+(1-p)^N.\n\nCandidate contribution (exact_probability_reduction; novelty confidence low): For a globally glued planar Viro patchwork with used triangulation vertex set A, the no-S^1 event has conditional probability 1_{|A mod 2|=1}[p^{|A|}+(1-p)^{|A|}], so arbitrary random support/subdivision laws reduce exactly to averaging this parity-and-cardinality statistic; primitive triangulations have probability zero." }, { "id": 20000634, "problem_number": "AIM-ANALYSIS-0166", "title": "Exact GUE-zero insertion amplitudes and a coefficient-space obstruction", "statement": "Question 9: Ashkan Nikeghbali What are the natural physical examples of random functions with GUE zeros on the real line? [U+0005]\n\n> ∗Partially supported by NSF grants DMS-0211458, CAREER DMS-0349309, and AIM.\n\n1", "original_statement": "Question 9: Ashkan Nikeghbali What are the natural physical examples of random functions with GUE zeros on the real line? \u0005\n\n> ∗Partially supported by NSF grants DMS-0211458, CAREER DMS-0349309, and AIM.\n\n1", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record contains", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Random analytic functions\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/randomzeros/arcc1.pdf\nCanonical location: aim-analysis-notes.json notes[165]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 9: Ashkan Nikeghbali What are the natural physical examples of random functions with GUE zeros on the real line? \\u0005\\n\\n> ∗Partially supported by NSF grants DMS-0211458, CAREER DMS-0349309, and AIM.\\n\\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/randomzeros/arcc1.pdf", "tags": [ "aim", "AIM-ANALYSIS-0166", "aim-domain:analysis", "aim-workshop:arcc1", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A measured N-fermion harmonic-trap ground state yields a physically sourced random insertion polynomial with exact finite-N GUE zeros: after removal of the nonzero Gaussian envelope, the Slater-determinant ratio is the monic polynomial product over the measured positions. Pushing this exact root law through Vieta gives the coefficient density proportional to the hyperbolicity indicator times exp[-alpha^2(a_1^2-2a_2)] times the square root of the discriminant, which excludes every full-rank jointly Gaussian coefficient vector for N at least 2. Rigorous Sine_2 and Airy_2 entire-function/operator models answer the two scaling-limit readings, while chaotic spectral determinants generally provide only GUE-type universality.\n\nCandidate contribution (coefficient-space characterization; novelty confidence low): For harmonic-trap ground-state fermion positions, the Gaussian-stripped Slater insertion ratio is the product of (z-X_j) and has exact finite-N GUE zeros; its coefficients have density proportional to the hyperbolicity indicator times exp[-alpha^2(a_1^2-2a_2)] times sqrt(Disc(P)), so no full-rank jointly Gaussian coefficient vector can realize exact GUE zeros when N is at least 2." }, { "id": 20000635, "problem_number": "AIM-ANALYSIS-0167", "title": "Which web is meant by a Gaussian free-field zero set?", "statement": "Question 10: Scott Sheffield Consider a function who Fourier transform is white noise on the unit circle. We aim to understand the web like appearance, that is the zero level lines of the Gaussian free field. Zelditch and Schramm make the above question more precise. [U+0005]", "original_statement": "Question 10: Scott Sheffield Consider a function who Fourier transform is white noise on the unit circle. We aim to understand the web like appearance, that is the zero level lines of the Gaussian free field. Zelditch and Schramm make the above question more precise. \u0005", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Question 10 from the April 2006 AIM workshop *Random analytic functions*, attributed to Scott Sheffield. Direct inspection of the two-page source PDF gives the exact text:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Random analytic functions\nSection: \nSource item: 10\nSource URL: https://aimath.org/WWN/randomzeros/arcc1.pdf\nCanonical location: aim-analysis-notes.json notes[166]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 10: Scott Sheffield Consider a function who Fourier transform is white noise on the unit circle. We aim to understand the web like appearance, that is the zero level lines of the Gaussian free field. Zelditch and Schramm make the above question more precise. \\u0005\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/randomzeros/arcc1.pdf", "tags": [ "aim", "AIM-ANALYSIS-0167", "aim-domain:analysis", "aim-workshop:arcc1", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM wording must be separated into a monochromatic Gaussian plane wave, a boundary white-noise model, and the distribution-valued two-dimensional GFF: the first is a real-analytic Helmholtz field with genuine nodal curves, while canonical GFF level lines are local sets with SLE_4/CLE_4 laws rather than pointwise zero sets. For zero-boundary GFF circle averages, B_t=sqrt(2pi) h_{e^{-t}}(0) is Brownian motion and the exact expected number of adjacent-scale sign flips is (1/pi) sum_{n=1}^{N-1} arcsin(1/sqrt(n+1))=(2/pi)sqrt(N)+O(1), with infinitely many flips almost surely. For the explicitly defined boundary logarithmic Fourier cutoff, the exact expected number of zeros is sqrt(2N(N+1)/H_N), asymptotic to sqrt(2)N/sqrt(log N). These formulas quantify cutoff instability but do not imply convergence or nonconvergence of specially selected unlabelled interfaces.\n\nCandidate contribution (exact regularization-instability diagnostic; novelty confidence low): Candidate novelty: the paired finite-N diagnostics consisting of the exact adjacent geometric-scale GFF circle-average sign-flip formula and the exact boundary logarithmic Fourier-cutoff zero-count formula give a concrete quantitative explanation of why a pointwise zero-set interpretation is unstable while canonical local-set interfaces remain meaningful." }, { "id": 20000636, "problem_number": "AIM-ANALYSIS-0168", "title": "Stationary zero-process obstruction to reconstruction from real zeros", "statement": "Question 11: Yan Fyodorov Given a random entire function on order 1, real on real line with given distribution of real zeros, what is the distribution of zeros of f? [U+0005]\n\n2", "original_statement": "Question 11: Yan Fyodorov Given a random entire function on order 1, real on real line with given distribution of real zeros, what is the distribution of zeros of f? \u0005\n\n2", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The source itself has “on order 1”; this is almost certainly a grammatical error for “of order 1.” The control character in the extracted JSON is the end-of-question marker, and the final “2” is the PDF page number. Inspection of the PDF text gives no evidence for a prime on the final \\(f\\).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analysis\nWorkshop: Random analytic functions\nSection: \nSource item: 11\nSource URL: https://aimath.org/WWN/randomzeros/arcc1.pdf\nCanonical location: aim-analysis-notes.json notes[167]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 11: Yan Fyodorov Given a random entire function on order 1, real on real line with given distribution of real zeros, what is the distribution of zeros of f? \\u0005\\n\\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/randomzeros/arcc1.pdf", "tags": [ "aim", "AIM-ANALYSIS-0168", "aim-domain:analysis", "aim-workshop:arcc1", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The real-zero divisor law does not determine the complete complex-zero law, even among random real-entire Cartwright functions of exact order one, fixed type 3π and indicator 3π|sin θ| whose real and complete zero-divisor processes are stationary: F_a(z)=sin π(z−V) sin π(z−U−ia) sin π(z−U+ia) has real zeros V+Z for every a>0 but nonreal zeros U+Z±ia, giving mutually singular laws for distinct a. In addition, finite conjugate-pair polynomial insertion preserves the real divisor, order, type, indicator, and any prescribed finite Taylor jet.\n\nCandidate contribution (counterexample_family; novelty confidence low): An explicit one-parameter Cartwright family has stationary real and complete zero-divisor processes, identical real-zero law, exact order, type, and full indicator, but pairwise mutually singular nonreal-zero laws; a companion conjugate-pair insertion construction is invisible to any prescribed finite Taylor jet." }, { "id": 20000637, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0001", "title": "Near-rational improvements and a linear-phase delta-method bottleneck", "statement": "Obtain cancellations in additive twists of GL(3) Fourier coefficients\n\nLet $\\pi$ be a GL(3) Hecke--Maass cusp form with the n-th Fourier coefficient $\\lambda_\\pi(1,n)$. For $\\alpha\\in\\mathbb{R}$, prove a bound of the form\n$$ \\sum_{n\\ll N} \\lambda_\\pi(1,n) e(n\\alpha) \\ll_\\pi N^{1-\\delta}, $$\nfor some $\\delta>0$ using a delta method. Aim to improve upon Miller's bound.", "original_statement": "Obtain cancellations in additive twists of GL(3) Fourier coefficients\n\nLet $\\pi$ be a GL(3) Hecke--Maass cusp form with the n-th Fourier coefficient $\\lambda_\\pi(1,n)$. For $\\alpha\\in\\mathbb{R}$, prove a bound of the form\n$$ \\sum_{n\\ll N} \\lambda_\\pi(1,n) e(n\\alpha) \\ll_\\pi N^{1-\\delta}, $$\nfor some $\\delta>0$ using a delta method. Aim to improve upon Miller's bound.", "clean_statement": "Obtain cancellations in additive twists of GL(3) Fourier coefficients\n\nLet $\\pi$ be a GL(3) Hecke--Maass cusp form with the n-th Fourier coefficient $\\lambda_\\pi(1,n)$. For $\\alpha\\in\\mathbb{R}$, prove a bound of the form\n$$ \\sum_{n\\ll N} \\lambda_\\pi(1,n) e(n\\alpha) \\ll_\\pi N^{1-\\delta}, $$\nfor some $\\delta>0$ using a delta method. Aim to improve upon Miller's bound.", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.02 from the AIM workshop “Delta symbols and the subconvexity problem” (October 16–20, 2023):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Delta symbols and the subconvexity problem\nSection: Problems\nSource item: 1.02\nSource URL: http://aimpl.org/deltasubconvex2/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Obtain cancellations in additive twists of GL(3) Fourier coefficients\\n\\nLet $\\\\pi$ be a GL(3) Hecke--Maass cusp form with the n-th Fourier coefficient $\\\\lambda_\\\\pi(1,n)$. For $\\\\alpha\\\\in\\\\mathbb{R}$, prove a bound of the form\\n$$ \\\\sum_{n\\\\ll N} \\\\lambda_\\\\pi(1,n) e(n\\\\alpha) \\\\ll_\\\\pi N^{1-\\\\delta}, $$\\nfor some $\\\\delta>0$ using a delta method. Aim to improve upon Miller's bound.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/deltasubconvex2/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0001", "aim-domain:analytic-number-theory", "aim-workshop:deltasubconvex2", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a fixed normalized level-one GL(3) Hecke--Maass cusp form, a reduced fraction a/q, and alpha=a/q+beta with N at least q^3, the report proves the sharp-cutoff bound |S_pi(N,alpha)| << q^(3/4) N^(1/2+theta/2+epsilon)(1+N|beta|), using Jääsaari's rational-twist theorem, Miller's estimate below the intermediate threshold x=q^3, and Abel summation. With theta=5/14, this improves Miller on the explicit arcs q<=N^kappa and |beta|<=N^(-1+lambda) whenever 3kappa/4+lambda<1/14. It also proves an explicitly scoped Poisson-localization lemma showing that, in a standard smooth duplicated-variable delta setup, the linear phase selects at most one rational residue under a stated width condition but supplies no curvature gain; a uniform delta-method improvement remains open.\n\nCandidate contribution (near-rational transfer theorem; novelty confidence low): The exact-rational saving is stable for sharp sums throughout the two-parameter region alpha=a/q+beta, q<=N^kappa, |beta|<=N^(-1+lambda), and 3kappa/4+lambda<1/14, with the explicit exponent 3/4-(1/14-3kappa/4-lambda)+epsilon." }, { "id": 20000638, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0002", "title": "Fixed-shift cancellation and a four-detector residual ledger", "statement": "Shifted convolution sum via various delta methods\n\nFor a holomorphic or a Hecke--Maass cusp form $f$ with the nth Fourier coefficient $\\lambda_f(n)$, obtain cancellations in the summation\n$$ \\sum_{n\\leq N} \\lambda_f(n)\\lambda_f(n+1), $$\nvia\n\\begin{enumerate}\n\\item Duke-Friedlander-Iwaniec's delta method\n\\item Trivial delta method\n\\item GL(2) Petersson trace formula as a delta method\n\\item GL(3) Kuznetsov trace formula as a delta method.\n\\end{enumerate}\n\nNote: We do not yet have an explicit GL(3) Kuznetsov trace formula with arbitrary level and nebentypus.", "original_statement": "Shifted convolution sum via various delta methods\n\nFor a holomorphic or a Hecke--Maass cusp form $f$ with the nth Fourier coefficient $\\lambda_f(n)$, obtain cancellations in the summation\n$$ \\sum_{n\\leq N} \\lambda_f(n)\\lambda_f(n+1), $$\nvia\n\\begin{enumerate}\n\\item Duke-Friedlander-Iwaniec's delta method\n\\item Trivial delta method\n\\item GL(2) Petersson trace formula as a delta method\n\\item GL(3) Kuznetsov trace formula as a delta method.\n\\end{enumerate}\n\nNote: We do not yet have an explicit GL(3) Kuznetsov trace formula with arbitrary level and nebentypus.", "clean_statement": "Shifted convolution sum via various delta methods\n\nFor a holomorphic or a Hecke--Maass cusp form $f$ with the nth Fourier coefficient $\\lambda_f(n)$, obtain cancellations in the summation\n$$ \\sum_{n\\leq N} \\lambda_f(n)\\lambda_f(n+1), $$\nvia\n\\begin{enumerate}\n\\item Duke-Friedlander-Iwaniec's delta method\n\\item Trivial delta method\n\\item GL(2) Petersson trace formula as a delta method\n\\item GL(3) Kuznetsov trace formula as a delta method.\n\\end{enumerate}\n\nNote: We do not yet have an explicit GL(3) Kuznetsov trace formula with arbitrary level and nebentypus.", "statement_status": "exact", "statement_verification": "The canonical source URL is . It timed out during this run, so the wording above was checked against the exact repository record rather than a newly downloaded copy. There is no visible OCR corruption or mathematical ambiguity.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Delta symbols and the subconvexity problem\nSection: Problems\nSource item: 1.04\nSource URL: http://aimpl.org/deltasubconvex2/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Shifted convolution sum via various delta methods\\n\\nFor a holomorphic or a Hecke--Maass cusp form $f$ with the nth Fourier coefficient $\\\\lambda_f(n)$, obtain cancellations in the summation\\n$$ \\\\sum_{n\\\\leq N} \\\\lambda_f(n)\\\\lambda_f(n+1), $$\\nvia\\n\\\\begin{enumerate}\\n\\\\item Duke-Friedlander-Iwaniec's delta method\\n\\\\item Trivial delta method\\n\\\\item GL(2) Petersson trace formula as a delta method\\n\\\\item GL(3) Kuznetsov trace formula as a delta method.\\n\\\\end{enumerate}\\n\\nNote: We do not yet have an explicit GL(3) Kuznetsov trace formula with arbitrary level and nebentypus.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/deltasubconvex2/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0002", "aim-domain:analytic-number-theory", "aim-workshop:deltasubconvex2", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Known work gives square-root cancellation for the fixed smooth level-one shifted convolution target for holomorphic or Hecke--Maass forms, and a sharp X^{2/3+epsilon} bound for fixed level-one holomorphic forms. This attempt proves an exact single-modulus aliasing--conductor dichotomy on an unpartitioned dyadic box and supplies exact Petersson and GL(3) Kuznetsov detector identities retaining the full Kloosterman--Bessel, continuous-spectrum, and Weyl-cell residual terms; it does not complete four independent power-saving proofs.\n\nCandidate contribution (proposition; novelty confidence low): For a single exact congruence modulus q on an unpartitioned smooth dyadic box with R_X=max|m-n-h|, either q>R_X and two separate GL(2) Voronoi transforms have dual length q^2/X comparable to or exceeding the original length, or q<=R_X and every occurring nonzero alias m-n-h=dq must be retained; combined with the exact Petersson and GL(3) residual identities, this yields a testable four-detector conservation ledger." }, { "id": 20000639, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0003", "title": "Prime-level direct delta solution and an exponent-transfer lemma", "statement": "Level aspect subconvexity for GL(2)\n\nUse a delta method to obtain level aspect subconvexity for GL(2) L-functions.", "original_statement": "Level aspect subconvexity for GL(2)\n\nUse a delta method to obtain level aspect subconvexity for GL(2) L-functions.", "clean_statement": "Level aspect subconvexity for GL(2)\n\nUse a delta method to obtain level aspect subconvexity for GL(2) L-functions.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Delta symbols and the subconvexity problem\nSection: Problems\nSource item: 1.06\nSource URL: http://aimpl.org/deltasubconvex2/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Level aspect subconvexity for GL(2)\\n\\nUse a delta method to obtain level aspect subconvexity for GL(2) L-functions.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/deltasubconvex2/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0003", "aim-domain:analytic-number-theory", "aim-workshop:deltasubconvex2", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The standard varying-newform reading is now solved at prime level by the 2024 AKKLLY preprint: their direct delta bound L(1/2,f⊗g)≪p^{1/2−1/524+ε}, specialized to the normalized level-one Eisenstein series with L(s,f⊗E)=L(s,f)^2, gives L(1/2,f)≪p^{1/4−1/1048+ε}. This attempt also proves, conditional on the entire precise Proposition-4.2-shaped engine E_sigma and all its admissibility/error estimates, the transfer savings sigma/[4(7−9sigma)] (degree four) and sigma/[8(7−9sigma)] (standard GL2), with prime-only variants sigma/[2(8−9sigma)] and sigma/[4(8−9sigma)]; arbitrary composite/depth level remains open in the checked sources.\n\nCandidate contribution (conditional lemma; novelty confidence low): For 00$ would imply the result of Holowinsky and Soundararajan (conditionally on non-negativity of the central values $L(1/2, sym^2f)$).\n\nUse a delta method in order to prove the above bound.", "original_statement": "Application to Quantum Unique Ergodicity\n\nHolowinsky and Soundararajan proved the Quantum Unique Ergodicity conjecture in the case of holomorphic Hecke eigenforms by combining two different approaches: By proving a non-trivial bound on certain shifted convolution sum, and by proving (weak) subconvexity bound for certain $L$-functions.\n\nObtaining a bound of the form\n$$ \\sum_{f,g \\in \\mathcal{B}_k} L(1/2, sym^2f) L(1/2, sym^2g)\\big|\\sum_{n\\leq k} \\lambda_f(n)\\lambda_g(n+h)\\big|^2 \\ll k^{4-\\delta},$$\nfor $|h|\\ll 1$ and some $\\delta>0$ would imply the result of Holowinsky and Soundararajan (conditionally on non-negativity of the central values $L(1/2, sym^2f)$).\n\nUse a delta method in order to prove the above bound.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "This is an explicit reconstruction, not text verified on the unavailable AIM page. If \\(\\mathcal B_k\\) instead means an orthonormal Fourier basis, or if harmonic weights are intended, both the scale and the applicable trace formula change.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Delta symbols and the subconvexity problem\nSection: Problems\nSource item: 1.16\nSource URL: http://aimpl.org/deltasubconvex2/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[7]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Application to Quantum Unique Ergodicity\\n\\nHolowinsky and Soundararajan proved the Quantum Unique Ergodicity conjecture in the case of holomorphic Hecke eigenforms by combining two different approaches: By proving a non-trivial bound on certain shifted convolution sum, and by proving (weak) subconvexity bound for certain $L$-functions.\\n\\nObtaining a bound of the form\\n$$ \\\\sum_{f,g \\\\in \\\\mathcal{B}_k} L(1/2, sym^2f) L(1/2, sym^2g)\\\\big|\\\\sum_{n\\\\leq k} \\\\lambda_f(n)\\\\lambda_g(n+h)\\\\big|^2 \\\\ll k^{4-\\\\delta},$$\\nfor $|h|\\\\ll 1$ and some $\\\\delta>0$ would imply the result of Holowinsky and Soundararajan (conditionally on non-negativity of the central values $L(1/2, sym^2f)$).\\n\\nUse a delta method in order to prove the above bound.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/deltasubconvex2/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0008", "aim-domain:analytic-number-theory", "aim-workshop:deltasubconvex2", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the natural fixed-weight interpretation and conditional nonnegativity of W_f=L(1/2,sym^2 f), the proposed double moment has the exact weighted-Gram factorization M_h=Tr(G_0G_h) and satisfies M_h <= N A_W(I_0)A_W(I_h), reducing the requested k^{4-delta} estimate to a power-saving weighted spectral large-sieve norm. Conversely, M_h <= k^{4-delta} plus W_f >= k^{-alpha} yields only max |S_{f,f}| <= k^{2-delta/2+alpha} and at most O(epsilon^{-2}k^{2-delta+2alpha}) forms with |S_{f,f}| >= epsilon k; hence the moment alone gives density-one cancellation only for delta>1+2alpha and uniform o(k) only for delta>2+2alpha. Nonnegativity alone supplies no alpha and permits zero weights, so the source's claim that any delta>0 implies individual QUE is not justified as written.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: the exact weighted-Gram/operator-norm reduction for the AIM moment, paired with the explicit sharp information thresholds delta>1+2alpha for density-one and delta>2+2alpha for uniform shifted cancellation.", "research_difficulty_suggested": "L4" }, { "id": 20000645, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0009", "title": "The solved maximal-parabolic formula and an all-level Borel support reduction", "statement": "$GL(3)$ Kuznetsov trace formula in the level aspect\n\nDevelop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_0(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ \\star & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$ for any integer $N\\geq1$.\n\nFor the purpose of applications, develop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_B(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ 0 & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$.", "original_statement": "$GL(3)$ Kuznetsov trace formula in the level aspect\n\nDevelop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_0(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ \\star & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$ for any integer $N\\geq1$.\n\nFor the purpose of applications, develop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_B(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ 0 & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$.", "clean_statement": "$GL(3)$ Kuznetsov trace formula in the level aspect\n\nDevelop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_0(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ \\star & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$ for any integer $N\\geq1$.\n\nFor the purpose of applications, develop a Kuznetsov trace formula for automorphic forms for the group $\\Gamma_B(N) = \\left\\lbrace \\begin{pmatrix} \\star & \\star & \\star \\\\ 0 & \\star & \\star \\\\ 0 & 0 & \\star \\end{pmatrix} \\bmod N \\right\\rbrace$.", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 1.18, “Delta symbols and the subconvexity problem.” Its exact statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Delta symbols and the subconvexity problem\nSection: Problems\nSource item: 1.18\nSource URL: http://aimpl.org/deltasubconvex2/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[8]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"$GL(3)$ Kuznetsov trace formula in the level aspect\\n\\nDevelop a Kuznetsov trace formula for automorphic forms for the group $\\\\Gamma_0(N) = \\\\left\\\\lbrace \\\\begin{pmatrix} \\\\star & \\\\star & \\\\star \\\\\\\\ \\\\star & \\\\star & \\\\star \\\\\\\\ 0 & 0 & \\\\star \\\\end{pmatrix} \\\\bmod N \\\\right\\\\rbrace$ for any integer $N\\\\geq1$.\\n\\nFor the purpose of applications, develop a Kuznetsov trace formula for automorphic forms for the group $\\\\Gamma_B(N) = \\\\left\\\\lbrace \\\\begin{pmatrix} \\\\star & \\\\star & \\\\star \\\\\\\\ 0 & \\\\star & \\\\star \\\\\\\\ 0 & 0 & \\\\star \\\\end{pmatrix} \\\\bmod N \\\\right\\\\rbrace$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/deltasubconvex2/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0009", "aim-domain:analytic-number-theory", "aim-workshop:deltasubconvex2", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Blomer–Buttcane–Maga Theorem 6 already supplies the requested standard-cusp Kuznetsov formula for Gamma_0(N) for every positive integer N; its later prime-level applications do not restrict the formula. For the remaining Gamma_B(N) problem, this attempt proves that in BBM Pluecker coordinates Gamma_B(N) is characterized by N dividing A_1, B_1, and B_2. Consequently the w_4 support is N|D_2 and ND_2|D_1, the w_5 support is N|D_1 and ND_1|D_2, and the long cell has N|D_1,D_2 with both residue variables B_1,B_2 divisible by N, all for arbitrary composite N. It also proves [Gamma_0(N):Gamma_B(N)]=N product_{p|N}(1+1/p) and identifies the Gamma_0 formula as the finite-cover conditional-expectation compression of the still-missing Borel formula.\n\nCandidate contribution (lemma; novelty confidence low): Candidate all-level Borel Bruhat-support lemma: adding the single Pluecker congruence N|B_2 to the known Gamma_0(N) parametrization yields exactly the strengthened w_4 and w_5 modulus conditions and the symmetric long-cell residue restrictions N|B_1,B_2, while the finite-index conditional expectation shows why the scalar Gamma_0 identity cannot recover the complementary Iwahori directions." }, { "id": 20000646, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0010", "title": "Subconvex prime-twist families for GL4xGL2 and GL3xGL3", "statement": "Subconvexity bound for Rankin-Selberg $L$-functions\n\nObtain a subconvexity bound for a $\\rm GL_n\\times GL_m$ Rankin--Selberg $L$-function when $n$ and $m$ are not coprime.\n\nThe simplest first cases would be obtaining a subconvexity bound (in any aspect) for a $\\rm GL_4\\times GL_2$ $L$-function or a $\\rm GL_3\\times GL_3$ $L$-function.", "original_statement": "Subconvexity bound for Rankin-Selberg $L$-functions\n\nObtain a subconvexity bound for a $\\rm GL_n\\times GL_m$ Rankin--Selberg $L$-function when $n$ and $m$ are not coprime.\n\nThe simplest first cases would be obtaining a subconvexity bound (in any aspect) for a $\\rm GL_4\\times GL_2$ $L$-function or a $\\rm GL_3\\times GL_3$ $L$-function.", "clean_statement": "Subconvexity bound for Rankin-Selberg $L$-functions\n\nObtain a subconvexity bound for a $\\rm GL_n\\times GL_m$ Rankin--Selberg $L$-function when $n$ and $m$ are not coprime.\n\nThe simplest first cases would be obtaining a subconvexity bound (in any aspect) for a $\\rm GL_4\\times GL_2$ $L$-function or a $\\rm GL_3\\times GL_3$ $L$-function.", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.2 in the AIM list *Delta symbols and the subconvexity problem*, source file `aim-analytic-number-theory-notes.json`, zero-based index 9. The exact recorded problem is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Delta symbols and the subconvexity problem\nSection: Problems\nSource item: 1.2\nSource URL: http://aimpl.org/deltasubconvex2/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[9]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Subconvexity bound for Rankin-Selberg $L$-functions\\n\\nObtain a subconvexity bound for a $\\\\rm GL_n\\\\times GL_m$ Rankin--Selberg $L$-function when $n$ and $m$ are not coprime.\\n\\nThe simplest first cases would be obtaining a subconvexity bound (in any aspect) for a $\\\\rm GL_4\\\\times GL_2$ $L$-function or a $\\\\rm GL_3\\\\times GL_3$ $L$-function.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/deltasubconvex2/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0010", "aim-domain:analytic-number-theory", "aim-workshop:deltasubconvex2", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a fixed everywhere-finite-unramified spherical non-solvable-polyhedral GL2 cusp representation pi over Q with trivial central character and any primitive character chi of odd prime conductor q, exact Clebsch-Gordan factorizations and current factor bounds give |L(1/2, Sym^3 pi x (pi tensor chi))| << C^(1/4-1/64+epsilon), with C asymptotic to q^8, and |L(1/2, Sym^2 pi x (Sym^2 pi tensor chi))| << C^(1/4-5/216+epsilon), with C asymptotic to q^9. These are proved special functorial subfamilies of both first rank pairs proposed by AIM, not generic solutions.\n\nCandidate contribution (special_case_theorem; novelty confidence low): Candidate novelty: the explicit paired deductions C^(1/4-1/64+epsilon) for the Sym^3(pi) x (pi tensor chi) GL4xGL2 family and C^(1/4-5/216+epsilon) for the Sym^2(pi) x (Sym^2(pi) tensor chi) GL3xGL3 family, with exact local conductor budgets 8=5+3 and 9=5+3+1." }, { "id": 20000647, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0011", "title": "Spectral large-sieve bounds for GL(3) and a GL(4) conductor window", "statement": "Cancellations in additive twists on average\n\nLet $\\alpha\\in\\mathbb{R}$ and $f_j$ be a Maass form for $\\rm SL(4, \\mathbb{Z})$ with eigenvalue $t_j$ and Fourier coefficients $\\{\\lambda_j(1,1,n)\\}_{n\\geq1}$. For parameters $T$ and $N$, prove a non-trivial bound for the averaged sum\n$$ \\sum_{t_j\\sim T} \\bigg|\\sum_{n\\leq N}\\lambda_j(1,1,n)e(n\\alpha) \\bigg|^2. $$\n\nIf $\\{f_j\\}$ are Maass forms for $\\rm SL(3, \\mathbb{Z})$ improve Miller's bound on average over the spectral parameter $t_j$.", "original_statement": "Cancellations in additive twists on average\n\nLet $\\alpha\\in\\mathbb{R}$ and $f_j$ be a Maass form for $\\rm SL(4, \\mathbb{Z})$ with eigenvalue $t_j$ and Fourier coefficients $\\{\\lambda_j(1,1,n)\\}_{n\\geq1}$. For parameters $T$ and $N$, prove a non-trivial bound for the averaged sum\n$$ \\sum_{t_j\\sim T} \\bigg|\\sum_{n\\leq N}\\lambda_j(1,1,n)e(n\\alpha) \\bigg|^2. $$\n\nIf $\\{f_j\\}$ are Maass forms for $\\rm SL(3, \\mathbb{Z})$ improve Miller's bound on average over the spectral parameter $t_j$.", "clean_statement": "Cancellations in additive twists on average\n\nLet $\\alpha\\in\\mathbb{R}$ and $f_j$ be a Maass form for $\\rm SL(4, \\mathbb{Z})$ with eigenvalue $t_j$ and Fourier coefficients $\\{\\lambda_j(1,1,n)\\}_{n\\geq1}$. For parameters $T$ and $N$, prove a non-trivial bound for the averaged sum\n$$ \\sum_{t_j\\sim T} \\bigg|\\sum_{n\\leq N}\\lambda_j(1,1,n)e(n\\alpha) \\bigg|^2. $$\n\nIf $\\{f_j\\}$ are Maass forms for $\\rm SL(3, \\mathbb{Z})$ improve Miller's bound on average over the spectral parameter $t_j$.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.22, “Cancellations in additive twists on average,” from the AIM list *Delta symbols and the subconvexity problem*. The live AIM page was checked and agrees with the repository record. Its text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Delta symbols and the subconvexity problem\nSection: Problems\nSource item: 1.22\nSource URL: http://aimpl.org/deltasubconvex2/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[10]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Cancellations in additive twists on average\\n\\nLet $\\\\alpha\\\\in\\\\mathbb{R}$ and $f_j$ be a Maass form for $\\\\rm SL(4, \\\\mathbb{Z})$ with eigenvalue $t_j$ and Fourier coefficients $\\\\{\\\\lambda_j(1,1,n)\\\\}_{n\\\\geq1}$. For parameters $T$ and $N$, prove a non-trivial bound for the averaged sum\\n$$ \\\\sum_{t_j\\\\sim T} \\\\bigg|\\\\sum_{n\\\\leq N}\\\\lambda_j(1,1,n)e(n\\\\alpha) \\\\bigg|^2. $$\\n\\nIf $\\\\{f_j\\\\}$ are Maass forms for $\\\\rm SL(3, \\\\mathbb{Z})$ improve Miller's bound on average over the spectral parameter $t_j$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/deltasubconvex2/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0011", "aim-domain:analytic-number-theory", "aim-workshop:deltasubconvex2", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For generic compact GL(3) spectral boxes, the Blomer--Buttcane large sieve gives the phase-uniform moment M_3(T,N,alpha) << (TN)^epsilon(T^5 N+T^2 N^2), whose normalized RMS improves Miller's N^(3/4) scale for N= A^(1/3+eta), the hybrid Weyl bound implies a conductor saving eta/8, while at t=t_f it misses conductor convexity by A^(1/12). After the balanced choice X=1 in the approximate functional equation, a target saving delta requires nontrivial estimates only for K^(1/2-2delta) <= N <= K^(1/2); for general X, the two formal cutoff scales have product K and satisfy the minimax relation." }, { "id": 20000655, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0019", "title": "A power-saving GL(4) short second moment", "statement": "Bound for short second moment of $\\rm GL_4$ $L$-functions\n\nLet $f$ be a fixed Hecke-Maass cusp form for $\\rm GL_4(\\mathbb{Z})$, $T$ and $\\Delta$ be parameters. Prove a non-trivial bound for\n$$\\int_T^{T+\\Delta} |L(\\frac12+it, f)|^2 dt, $$\nfor $\\Delta = T^\\delta$ with some $\\delta<1$.", "original_statement": "Bound for short second moment of $\\rm GL_4$ $L$-functions\n\nLet $f$ be a fixed Hecke-Maass cusp form for $\\rm GL_4(\\mathbb{Z})$, $T$ and $\\Delta$ be parameters. Prove a non-trivial bound for\n$$\\int_T^{T+\\Delta} |L(\\frac12+it, f)|^2 dt, $$\nfor $\\Delta = T^\\delta$ with some $\\delta<1$.", "clean_statement": "Bound for short second moment of $\\rm GL_4$ $L$-functions\n\nLet $f$ be a fixed Hecke-Maass cusp form for $\\rm GL_4(\\mathbb{Z})$, $T$ and $\\Delta$ be parameters. Prove a non-trivial bound for\n$$\\int_T^{T+\\Delta} |L(\\frac12+it, f)|^2 dt, $$\nfor $\\Delta = T^\\delta$ with some $\\delta<1$.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.38 in the AIM list *Delta symbols and the subconvexity problem* (`aim-analytic-number-theory-notes.json`, zero-based index 18). Its mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Delta symbols and the subconvexity problem\nSection: Problems\nSource item: 1.38\nSource URL: http://aimpl.org/deltasubconvex2/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[18]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Bound for short second moment of $\\\\rm GL_4$ $L$-functions\\n\\nLet $f$ be a fixed Hecke-Maass cusp form for $\\\\rm GL_4(\\\\mathbb{Z})$, $T$ and $\\\\Delta$ be parameters. Prove a non-trivial bound for\\n$$\\\\int_T^{T+\\\\Delta} |L(\\\\frac12+it, f)|^2 dt, $$\\nfor $\\\\Delta = T^\\\\delta$ with some $\\\\delta<1$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/deltasubconvex2/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0019", "aim-domain:analytic-number-theory", "aim-workshop:deltasubconvex2", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For a fixed Hecke--Maass cusp form f on GL(4,Z), T >= 2, and 1 <= Delta <= T, one has I_f(T,Delta) <<_{f,epsilon} min(T^{2+epsilon}, Delta T^{2-1/150+epsilon}). Consequently, Delta=T^delta gives a strict power saving below the approximate-functional-equation benchmark T^2 for every fixed delta<1/150. The pointwise branch follows from Yang's explicit preprint bound, while Nelson independently supplies a weaker qualitative completion; the T^2 branch is proved by a moving-weight Montgomery--Vaughan argument.\n\nCandidate contribution (quantitative_corollary; novelty confidence low): The explicit crossover envelope I_f(T,T^delta) << T^{2-(1/150-delta)_+ + epsilon} rigorously combines the best explicit general pointwise GL(4) theorem located with an unconditional length-T^2 mean-value bound, isolating delta=1/150 as the exact crossover of these two mechanisms." }, { "id": 20000656, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0020", "title": "Certified Petersson resolution for finite point counts", "statement": "Applications of trace formula towards point counting\n\nVarious circle methods and $\\delta$-methods have been successfully used to count points on varieties. Could one use Petersson or Kuznetsov trace formula as a $\\delta$-method to count point on varieties?", "original_statement": "Applications of trace formula towards point counting\n\nVarious circle methods and $\\delta$-methods have been successfully used to count points on varieties. Could one use Petersson or Kuznetsov trace formula as a $\\delta$-method to count point on varieties?", "clean_statement": "Applications of trace formula towards point counting\n\nVarious circle methods and $\\delta$-methods have been successfully used to count points on varieties. Could one use Petersson or Kuznetsov trace formula as a $\\delta$-method to count point on varieties?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.4, “Applications of trace formula towards point counting,” from the AIM workshop *Delta symbols and the subconvexity problem*. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Delta symbols and the subconvexity problem\nSection: Problems\nSource item: 1.4\nSource URL: http://aimpl.org/deltasubconvex2/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[19]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Applications of trace formula towards point counting\\n\\nVarious circle methods and $\\\\delta$-methods have been successfully used to count points on varieties. Could one use Petersson or Kuznetsov trace formula as a $\\\\delta$-method to count point on varieties?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/deltasubconvex2/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0020", "aim-domain:analytic-number-theory", "aim-workshop:deltasubconvex2", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For finite nonempty sets A and B with positive integer-valued maps F and G, value height H, and M = |A||B|, the full-space Petersson spectral cross moment T_{k,N} satisfies the proved uniform bound |T_{k,N} - #{F=G}| <= 4 pi M (2 pi e H/(N(k-1)))^(k-1). Hence an explicit level-weight inequality makes the exact point count the unique nearest integer to the spectral moment. Applied to ad-bc=r in the positive box, this gives a certified spectral formula and an exact Hecke-Mobius separation that isolates a shifted divisor-Hecke correlation. This is a proof of principle, not a competitive counting asymptotic.\n\nCandidate contribution (exact_rounding_criterion; novelty confidence low): An explicit universal level-weight rounding criterion certifies exact recovery of any finite positive integer equality count from the Petersson spectral cross moment, with error at most 4 pi M (2 pi e H/(N(k-1)))^(k-1)." }, { "id": 20000657, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0021", "title": "An averaged short-additive-twist criterion for a GL(3) Hardy theorem", "statement": "Hardy-type theorem for $\\rm GL_3$ $L$-functions\n\nUse a $\\delta$-method to prove that there are infinitely many critical zeros for a $\\rm GL_3(\\mathbb{Z})$ $L$-functions.", "original_statement": "Hardy-type theorem for $\\rm GL_3$ $L$-functions\n\nUse a $\\delta$-method to prove that there are infinitely many critical zeros for a $\\rm GL_3(\\mathbb{Z})$ $L$-functions.", "clean_statement": "Hardy-type theorem for $\\rm GL_3$ $L$-functions\n\nUse a $\\delta$-method to prove that there are infinitely many critical zeros for a $\\rm GL_3(\\mathbb{Z})$ $L$-functions.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.42 in the AIM list *Delta symbols and the subconvexity problem*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Delta symbols and the subconvexity problem\nSection: Problems\nSource item: 1.42\nSource URL: http://aimpl.org/deltasubconvex2/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[20]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Hardy-type theorem for $\\\\rm GL_3$ $L$-functions\\n\\nUse a $\\\\delta$-method to prove that there are infinitely many critical zeros for a $\\\\rm GL_3(\\\\mathbb{Z})$ $L$-functions.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: open; rigor: conditional; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/deltasubconvex2/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0021", "aim-domain:analytic-number-theory", "aim-workshop:deltasubconvex2", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let N=(T/2pi)^(3/2), L=N^(2/3), and partition the stationary range for the fixed level-one GL(3) Hardy moment into length-L blocks. If, for some delta>0, the resulting phase-locked linear additive twists satisfy the unproved averaged estimate sum_M |sum_n A_pi(1,n)e(g'(M)n)U_M((n-M)/L)| << N^(5/6-delta+epsilon), where g(x)=-3x^(2/3)/2, then L(s,pi) has infinitely many zeros on Re(s)=1/2. A stronger unproved pointwise estimate of size L^(3/4-eta) implies this criterion with delta=2eta/3. The proof combines a Gaussian L1 mass lower bound with the standard approximate functional equation and stationary phase; it is a conditional reduction, not a solution of the missing additive-twist estimate.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: the known global nonlinear-twist threshold N^(5/6-delta) is refined into a sufficient averaged absolute bound for phase-locked short linear GL(3) additive twists on the maximal linearization scale L=N^(2/3); the pointwise local threshold is exactly L^(3/4), and a saving L^(-eta) yields delta=2eta/3.", "research_difficulty_suggested": "L4" }, { "id": 20000658, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0022", "title": "A quantitative negative-rectangle obstruction to bounded fixed-shift Liouville correlations", "statement": "Let $h \\in \\N$ be fixed. Is there a sequence $X_k \\to \\infty$ such that\n\\begin{equation*}\n \\left| \\sum_{n \\leq X_k} \\lambda(n) \\lambda(n + h) \\right| \\to \\infty\n\\end{equation*}\nas $k \\to \\infty$? We could also consider variants for multiplicative functions $f$ with values in $\\{\\pm 1\\}$. We can handle $f$ pretentious, and can also do it assuming $4$-point Chowla or the existence of arbitrarily long strings of $+1$'s for $\\lambda$. Would a Sudoku-type argument work here? If $h = h_k$ is allowed to vary with $k$, then we can probably show this as well.", "original_statement": "Let $h \\in \\N$ be fixed. Is there a sequence $X_k \\to \\infty$ such that\n\\begin{equation*}\n \\left| \\sum_{n \\leq X_k} \\lambda(n) \\lambda(n + h) \\right| \\to \\infty\n\\end{equation*}\nas $k \\to \\infty$? We could also consider variants for multiplicative functions $f$ with values in $\\{\\pm 1\\}$. We can handle $f$ pretentious, and can also do it assuming $4$-point Chowla or the existence of arbitrarily long strings of $+1$'s for $\\lambda$. Would a Sudoku-type argument work here? If $h = h_k$ is allowed to vary with $k$, then we can probably show this as well.", "clean_statement": "Let $h \\in \\N$ be fixed. Is there a sequence $X_k \\to \\infty$ such that\n\\begin{equation*}\n \\left| \\sum_{n \\leq X_k} \\lambda(n) \\lambda(n + h) \\right| \\to \\infty\n\\end{equation*}\nas $k \\to \\infty$? We could also consider variants for multiplicative functions $f$ with values in $\\{\\pm 1\\}$. We can handle $f$ pretentious, and can also do it assuming $4$-point Chowla or the existence of arbitrarily long strings of $+1$'s for $\\lambda$. Would a Sudoku-type argument work here? If $h = h_k$ is allowed to vary with $k$, then we can probably show this as well.", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List item 1.1 in the workshop list *Sarnak's conjecture*, section “Lower bounds.” It asks, for fixed \\(h\\in\\mathbb N\\), whether there are \\(X_k\\to\\infty\\) such that", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Lower bounds\nSource item: 1.1\nSource URL: http://aimpl.org/sarnakconjecture/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[21]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $h \\\\in \\\\N$ be fixed. Is there a sequence $X_k \\\\to \\\\infty$ such that\\n\\\\begin{equation*}\\n \\\\left| \\\\sum_{n \\\\leq X_k} \\\\lambda(n) \\\\lambda(n + h) \\\\right| \\\\to \\\\infty\\n\\\\end{equation*}\\nas $k \\\\to \\\\infty$? We could also consider variants for multiplicative functions $f$ with values in $\\\\{\\\\pm 1\\\\}$. We can handle $f$ pretentious, and can also do it assuming $4$-point Chowla or the existence of arbitrarily long strings of $+1$'s for $\\\\lambda$. Would a Sudoku-type argument work here? If $h = h_k$ is allowed to vary with $k$, then we can probably show this as well.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0022", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let C_h(X)=sum_{n<=X} lambda(n)lambda(n+h). If |C_h(X)| were bounded globally by an integer B, then for every H>4B^2 a fixed distinct displacement d in {1,...,H-1}\\{h} would have logarithmic four-point rectangle liminf at most -(H-4B^2)/(H(H-1)-2*1_{h 0$ and such that\n\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} \\lambda(n) a(n) \\right| > \\epsilon X\n\\end{equation*}\nfor all $X$ and some fixed $\\epsilon > 0$? Can we say something about the Furstenberg systems of such an $a(n)$? In particular, do they have positive entropy?", "original_statement": "Is there a bounded sequence $a(n)$ such that each $a(n)$ is computable in time $O(\\log^A n)$ for some fixed $A > 0$ and such that\n\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} \\lambda(n) a(n) \\right| > \\epsilon X\n\\end{equation*}\nfor all $X$ and some fixed $\\epsilon > 0$? Can we say something about the Furstenberg systems of such an $a(n)$? In particular, do they have positive entropy?", "clean_statement": "Is there a bounded sequence $a(n)$ such that each $a(n)$ is computable in time $O(\\log^A n)$ for some fixed $A > 0$ and such that\n\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} \\lambda(n) a(n) \\right| > \\epsilon X\n\\end{equation*}\nfor all $X$ and some fixed $\\epsilon > 0$? Can we say something about the Furstenberg systems of such an $a(n)$? In particular, do they have positive entropy?", "statement_status": "exact", "statement_verification": "The canonical record is item 1.2, “Lower bounds,” from the AIM workshop problem list *Sarnak's conjecture* (December 2018). Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Lower bounds\nSource item: 1.2\nSource URL: http://aimpl.org/sarnakconjecture/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[22]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a bounded sequence $a(n)$ such that each $a(n)$ is computable in time $O(\\\\log^A n)$ for some fixed $A > 0$ and such that\\n\\n\\\\begin{equation*}\\n \\\\left| \\\\sum_{n \\\\leq X} \\\\lambda(n) a(n) \\\\right| > \\\\epsilon X\\n\\\\end{equation*}\\nfor all $X$ and some fixed $\\\\epsilon > 0$? Can we say something about the Furstenberg systems of such an $a(n)$? In particular, do they have positive entropy?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0023", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Unconditionally, for a real sequence |a(n)| <= 1, the all-prefix lower bound forces the Liouville correlation sums to have an eventual fixed sign and logarithmic correlation at least epsilon; the Frantzikinakis-Host logarithmic theorem then implies that every logarithmic Furstenberg measure with at most countably many ergodic components has positive entropy, in particular every ergodic one. Conditionally on full Cesaro Chowla, every Cesaro Furstenberg system of any complex |a(n)| <= 1 satisfying the bound has entropy at least epsilon^2/2 nats per shift. The classical deterministic existence question is not claimed solved: a=lambda is available if deterministic factoring is in P, and in standard bounded-error or verified expected quantum polynomial time via Shor.\n\nCandidate contribution (quantitative_entropy_bound; novelty confidence low): Candidate novelty: under full Cesaro Chowla, every Cesaro Furstenberg system of any bounded complex sequence with all-prefix Liouville correlation at least epsilon has Kolmogorov-Sinai entropy at least epsilon^2/2 nats per shift after normalization |a| <= 1, or epsilon^2/(2B^2) when |a| <= B." }, { "id": 20000660, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0024", "title": "Endpoint variation and a correlation-budget criterion for Liouville short sums", "statement": "Can we obtain lower bounds for\n\n\\begin{equation*}\n \\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) \\right| d x?\n\\end{equation*}\nIn particular, can we get a lower bound of the form $XH^{\\frac{1}{2}-\\epsilon}$?", "original_statement": "Can we obtain lower bounds for\n\n\\begin{equation*}\n \\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) \\right| d x?\n\\end{equation*}\nIn particular, can we get a lower bound of the form $XH^{\\frac{1}{2}-\\epsilon}$?", "clean_statement": "Can we obtain lower bounds for\n\n\\begin{equation*}\n \\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) \\right| d x?\n\\end{equation*}\nIn particular, can we get a lower bound of the form $XH^{\\frac{1}{2}-\\epsilon}$?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List item 1.3, in the section “Lower bounds” of the 2018 workshop on Sarnak's conjecture. Its question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Lower bounds\nSource item: 1.3\nSource URL: http://aimpl.org/sarnakconjecture/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[23]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we obtain lower bounds for\\n\\n\\\\begin{equation*}\\n \\\\int_X^{2X} \\\\left| \\\\sum_{x\\\\leq n\\\\leq x+H} \\\\lambda(n) \\\\right| d x?\\n\\\\end{equation*}\\nIn particular, can we get a lower bound of the form $XH^{\\\\frac{1}{2}-\\\\epsilon}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0024", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Unconditionally, for every fixed integer h, the logarithmically weighted first absolute moment satisfies sum_{m<=Y} |sum_{j=1}^h lambda(m+j)|/m >= (1/2-o_h(1)) log Y; consequently limsup_j I(2^j,h)/2^j >= (log 2)/2, and for odd h the elementary bound I(N,h)>=N holds at every integer N. Separately, a proved finite deterministic criterion shows that uniform two-point bounds |C_2|<=eta N/h and four-distinct-point bounds |C_4|<=eta N/h^2 imply I(N,h)>=(1-eta)^(3/2)/sqrt(3+7 eta) times N sqrt(h). The latter correlation budgets are not proved for the Liouville function in a growing-h range, so the AIM target remains open.\n\nCandidate contribution (lemma; novelty confidence low): Candidate correlation-budget lemma: for any sign sequence, the finite bounds |C_2(r,s)|<=eta N/h for distinct pairs and |C_4(r,s,t,u)|<=eta N/h^2 for four distinct shifts force an explicit N sqrt(h) first-moment lower bound; the fourth-moment audit uses the exact count 6h^3-14h^2+8h for tuples with odd-multiplicity support of size two." }, { "id": 20000661, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0025", "title": "Quantitative discrepancy for completely multiplicative signs", "statement": "Can we get a reasonable omega result for\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} f(n) \\right|,\n\\end{equation*}\nwhere $f: \\N \\to \\{\\pm1\\}$ is totally multiplicative? Anything better than $\\log \\log \\log \\log \\log X$ would be interesting.", "original_statement": "Can we get a reasonable omega result for\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} f(n) \\right|,\n\\end{equation*}\nwhere $f: \\N \\to \\{\\pm1\\}$ is totally multiplicative? Anything better than $\\log \\log \\log \\log \\log X$ would be interesting.", "clean_statement": "Can we get a reasonable omega result for\n\\begin{equation*}\n \\left| \\sum_{n \\leq X} f(n) \\right|,\n\\end{equation*}\nwhere $f: \\N \\to \\{\\pm1\\}$ is totally multiplicative? Anything better than $\\log \\log \\log \\log \\log X$ would be interesting.", "statement_status": "exact", "statement_verification": "This is record `AIM-ANALYTIC_NUMBER_THEORY-0025`, zero-based index 24 in `aim-analytic-number-theory-notes.json`, from the AIM workshop list “Sarnak's conjecture,” section “Lower bounds,” Problem 1.4.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Lower bounds\nSource item: 1.4\nSource URL: http://aimpl.org/sarnakconjecture/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[24]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we get a reasonable omega result for\\n\\\\begin{equation*}\\n \\\\left| \\\\sum_{n \\\\leq X} f(n) \\\\right|,\\n\\\\end{equation*}\\nwhere $f: \\\\N \\\\to \\\\{\\\\pm1\\\\}$ is totally multiplicative? Anything better than $\\\\log \\\\log \\\\log \\\\log \\\\log X$ would be interesting.\"\nOriginal remarks: [\"This problem was solved by R. McNamara in his PhD thesis [https://escholarship.org/uc/item/4wr015m0], giving a growth rate of $(\\\\log \\\\log X)^{1/484-o(1)}$. H. Helfgott and M. Radziwiłł outline in [https://arxiv.org/abs/2103.06853] a significant improvement to the exponent of $\\\\log \\\\log X$.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0025", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "McNamara's quantitative Hilbert-sphere Erdős discrepancy theorem, combined with the identity f(jd)=f(j)f(d), proves that for every completely multiplicative f:N→S^1, every fixed 00$.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/4/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0031", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Pilatte's April 2026 preprint lowers the best unconditional local Fourier-uniformity range for the Liouville function to H at least exp((log X)^(2/5+epsilon)), improving the AIM record's 5/8 threshold and Walsh's intervening 1/2 threshold; the conjecture for every H(X) tending to infinity remains open. A proved deterministic transfer inequality converts this maximal Fourier estimate into average decay of every fixed balanced even spectral moment at the new 2/5 range, with the q=2 case giving an explicit mean-square local autocorrelation-energy estimate.\n\nCandidate contribution (corollary; novelty confidence low): For every 1-bounded sequence a, every interval of integer length H, and every integer q at least 2, the local spectral moment E_q^a(x,H)=integral_0^1 |sum_{n in [x,x+H)} a(n)e(n alpha)|^(2q) d alpha is at most H^(2q-2) times the maximal Fourier coefficient on that interval. Applied to Pilatte's theorem, this gives sum_x E_q(x,H)=o(H^(2q-1)X) in the new exponent-2/5 range; for q=2 this is exactly sum_x sum_{|r|0$.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/4/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0032", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The full slowly-growing-H nilsequence conjecture remains open: MRTTZ prove local polynomial and fixed-nilsequence uniformity for every fixed power scale H >= X^theta, and polynomial phases down to exp((log X)^(5/8+epsilon)); later unconditional improvements below that exponent are linear only. This attempt proves a quantitative abelian transference theorem: if A_k is the normalized moving supremum against all degree-k polynomial phases, then for every R >= 2 the normalized correlation with the supremum over all degree-k polynomial torus orbits and the entire fixed Lipschitz ball inside the interval average is at most (4R+1)^d A_k + C_d L/R. Optimization gives O_d((1+L)^(d/(d+1)) A_k^(1/(d+1))), with no additional loss in the admissible H-range.\n\nCandidate contribution (transference_lemma; novelty confidence low): A moving-observable toral transference inequality preserves the interval-by-interval suprema over both polynomial orbits g and all observables with sup norm at most 1 and Lipschitz constant at most L: T_{d,k,L} <= (4R+1)^d A_k + C_d L/R for every integer R >= 2." }, { "id": 20000669, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0033", "title": "Large-field local Fourier uniformity and a higher-order obstruction", "statement": "Can we prove the local Fourier uniformity conjecture in function fields? Would that suffice to obtain the full logarithmic Chowla conjecture?", "original_statement": "Can we prove the local Fourier uniformity conjecture in function fields? Would that suffice to obtain the full logarithmic Chowla conjecture?", "clean_statement": "Can we prove the local Fourier uniformity conjecture in function fields? Would that suffice to obtain the full logarithmic Chowla conjecture?", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Fourier uniformity\nSource item: 4.3\nSource URL: http://aimpl.org/sarnakconjecture/4/\nCanonical location: aim-analytic-number-theory-notes.json notes[32]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we prove the local Fourier uniformity conjecture in function fields? Would that suffice to obtain the full logarithmic Chowla conjecture?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/4/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0033", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an odd prime p and a power q of p satisfying q > 16 p^2 e^2, polynomial Liouville over F_q[T] satisfies center-adaptive local 1-Fourier uniformity, quantitatively limsup_N L_{N,h}(lambda_q) <= (2 q^{-h} - q^{-2h})^{1/4}. The proof transfers the Sawin-Shusterman affine Mobius correlations to Liouville through a truncated square-divisor identity and then applies an exact fourth spectral moment. The analogous characteristic-two fourth-moment bound is only conditional on the needed distinct-shift correlations. Linear local Fourier uniformity does not by itself yield full logarithmic Chowla: higher local Gowers layers are required.\n\nCandidate contribution (partial_theorem; novelty confidence low): The candidate contribution is the explicit large-field center-adaptive bound limsup_N L_{N,h}(lambda_q) <= (2 q^{-h} - q^{-2h})^{1/4}, together with the exact collision counts 2Q^2-Q in odd characteristic and 3Q^2-2Q in characteristic two." }, { "id": 20000670, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0034", "title": "A quantized gap for the Liouville Goldbach correlation", "statement": "Show that\n\\begin{equation*}\n \\left| \\sum_{1 \\leq n < N} \\lambda(n) \\lambda(N-n) \\right| < N-1.\n\\end{equation*}", "original_statement": "Show that\n\\begin{equation*}\n \\left| \\sum_{1 \\leq n < N} \\lambda(n) \\lambda(N-n) \\right| < N-1.\n\\end{equation*}", "clean_statement": "Show that\n\\begin{equation*}\n \\left| \\sum_{1 \\leq n < N} \\lambda(n) \\lambda(N-n) \\right| < N-1.\n\\end{equation*}", "statement_status": "exact", "statement_verification": "The exact extracted AIM record (workshop “Sarnak's conjecture,” section “Special correlations,” Problem 5.1) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Special correlations\nSource item: 5.1\nSource URL: http://aimpl.org/sarnakconjecture/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[33]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Show that\\n\\\\begin{equation*}\\n \\\\left| \\\\sum_{1 \\\\leq n < N} \\\\lambda(n) \\\\lambda(N-n) \\\\right| < N-1.\\n\\\\end{equation*}\"\nOriginal remarks: [\"This has been proved by Sacha Mangerel (https://arxiv.org/abs/2404.12117) for all $N \\\\geq N_0$, where $N_0$ is an ineffective constant.\", \"It has been shown in the published version of the aforementioned paper (https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnae149/7704606?utm_source=advanceaccess&utm_campaign=imrn&utm_medium=email&login=false) that the above bound holds for all $N \\\\geq 11$, and only fails when $N$ is $2,3,5$ or $10$.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0034", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Mangerel's published theorem solves the original AIM problem by proving that |L_lambda(N)| = N-1 exactly for N in {2,3,5,10}, and therefore the requested strict inequality holds for every N >= 11. Dependent on that cited classification, the artifacts prove the stronger corollary |L_lambda(N)| <= N-5 for every N >= 7 with N != 10, using a mod-4 reflection-defect lemma, quantitative divisor propagation, and exact seed values at 8, 12, and 20.\n\nCandidate contribution (corollary; novelty confidence low): For every integer N >= 7 with N != 10, the Liouville Goldbach correlation satisfies |L_lambda(N)| <= N-5; more generally, reflected products of any sign sequence have an exact even-orbit defect and their correlation is congruent to N-1 modulo 4." }, { "id": 20000671, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0035", "title": "Finite-prime parity laws and a covariance reduction", "statement": "Show that\n\\begin{equation*}\n \\left| \\mathbb{E}_{n \\in \\mathbb{N}}^{\\log} \\lambda(n^2+1) \\right| 0$. It would even be interesting to show that both sets $\\{p : \\lambda(p-1) = 1 \\}$ and $\\{p : \\lambda(p-1) = -1 \\}$ are infinite. For the first bound, using the entropy decrement argument, wouldn't it be enough to show that $\\left| \\sum_{p \\approx P} \\sum_{n \\approx X} \\lambda(n^2 +p^2) \\right| = o\\left(X\\frac{P}{\\log P}\\right)$ with $P = \\log \\log \\log X$?", "original_statement": "Show that\n\\begin{equation*}\n \\left| \\mathbb{E}_{n \\in \\mathbb{N}}^{\\log} \\lambda(n^2+1) \\right| 0$. It would even be interesting to show that both sets $\\{p : \\lambda(p-1) = 1 \\}$ and $\\{p : \\lambda(p-1) = -1 \\}$ are infinite. For the first bound, using the entropy decrement argument, wouldn't it be enough to show that $\\left| \\sum_{p \\approx P} \\sum_{n \\approx X} \\lambda(n^2 +p^2) \\right| = o\\left(X\\frac{P}{\\log P}\\right)$ with $P = \\log \\log \\log X$?", "clean_statement": "Show that\n\\begin{equation*}\n \\left| \\mathbb{E}_{n \\in \\mathbb{N}}^{\\log} \\lambda(n^2+1) \\right| 0$. It would even be interesting to show that both sets $\\{p : \\lambda(p-1) = 1 \\}$ and $\\{p : \\lambda(p-1) = -1 \\}$ are infinite. For the first bound, using the entropy decrement argument, wouldn't it be enough to show that $\\left| \\sum_{p \\approx P} \\sum_{n \\approx X} \\lambda(n^2 +p^2) \\right| = o\\left(X\\frac{P}{\\log P}\\right)$ with $P = \\log \\log \\log X$?", "statement_status": "exact", "statement_verification": "The canonical JSON record is visibly truncated. Its `problem` field is preserved verbatim here:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Special correlations\nSource item: 5.2\nSource URL: http://aimpl.org/sarnakconjecture/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[34]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Show that\\n\\\\begin{equation*}\\n \\\\left| \\\\mathbb{E}_{n \\\\in \\\\mathbb{N}}^{\\\\log} \\\\lambda(n^2+1) \\\\right| 0$. It would even be interesting to show that both sets $\\\\{p : \\\\lambda(p-1) = 1 \\\\}$ and $\\\\{p : \\\\lambda(p-1) = -1 \\\\}$ are infinite. For the first bound, using the entropy decrement argument, wouldn't it be enough to show that $\\\\left| \\\\sum_{p \\\\approx P} \\\\sum_{n \\\\approx X} \\\\lambda(n^2 +p^2) \\\\right| = o\\\\left(X\\\\frac{P}{\\\\log P}\\\\right)$ with $P = \\\\log \\\\log \\\\log X$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0035", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The two AIM constant-saving bounds remain open. For every fixed z, this attempt proves exact finite-prime parity laws for lambda(n^2+1) under logarithmic averaging and lambda(p-1) under prime averaging: the respective local biases are A_z = product over primes q <= z, q congruent to 1 mod 4, of (q-3)/(q+1), and B_z = -(1/3) times the product over odd primes q <= z of (1-2q/(q^2-1)); both have magnitude asymptotic in order to (log z)^(-2). It then proves covariance inequalities reducing each original mean to the corresponding explicit local bias plus the covariance between local valuation parity and the complementary tail parity. No nontrivial bound for either covariance is claimed.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: a matched pair of exact finite-prime parity Euler products for n^2+1 and p-1, together with parallel proved covariance reductions that isolate the complementary parity-tail dependence as the precise remaining bottleneck in both halves of AIM Problem 5.2.", "research_difficulty_suggested": "L4" }, { "id": 20000672, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0036", "title": "Moving-cutoff smooth-number correlations and a weighted mean-value consequence", "statement": "What can we say about correlations with itself of\n\\begin{equation*}\n \\mathbb{1}_{p^+(n) \\leq n^\\delta},\n\\end{equation*}\nwith $\\delta \\to 0$? Here $p^+(n) := \\max\\{p: p|n\\}$. Even upper bounds would be welcome. This kind of result would imply sharp mean values for Dirichlet polynomials supported on smooth numbers.", "original_statement": "What can we say about correlations with itself of\n\\begin{equation*}\n \\mathbb{1}_{p^+(n) \\leq n^\\delta},\n\\end{equation*}\nwith $\\delta \\to 0$? Here $p^+(n) := \\max\\{p: p|n\\}$. Even upper bounds would be welcome. This kind of result would imply sharp mean values for Dirichlet polynomials supported on smooth numbers.", "clean_statement": "What can we say about correlations with itself of\n\\begin{equation*}\n \\mathbb{1}_{p^+(n) \\leq n^\\delta},\n\\end{equation*}\nwith $\\delta \\to 0$? Here $p^+(n) := \\max\\{p: p|n\\}$. Even upper bounds would be welcome. This kind of result would imply sharp mean values for Dirichlet polynomials supported on smooth numbers.", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 5.3 in the section “Special correlations” of the AIM list from the workshop *Sarnak's conjecture*. Its exact mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Special correlations\nSource item: 5.3\nSource URL: http://aimpl.org/sarnakconjecture/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[35]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What can we say about correlations with itself of\\n\\\\begin{equation*}\\n \\\\mathbb{1}_{p^+(n) \\\\leq n^\\\\delta},\\n\\\\end{equation*}\\nwith $\\\\delta \\\\to 0$? Here $p^+(n) := \\\\max\\\\{p: p|n\\\\}$. Even upper bounds would be welcome. This kind of result would imply sharp mean values for Dirichlet polynomials supported on smooth numbers.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0036", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every fixed eta > 0 and 0 < epsilon < 5/8, if delta = delta(X) tends to zero and delta log X >= (log log X)^(5/3+eta), then current fixed-threshold smooth-pair technology transfers to the literal AIM indicator: for some kappa > 0, uniformly for 1 <= h <= X^kappa, the correlation over X < n <= 2X is O(X rho(1/delta)^(13/8-epsilon)), while the support has size X rho(1/delta)(1+o(1)). A positive Fejer-kernel identity consequently gives a diagonal asymptotic for the associated weighted Dirichlet-polynomial second moment when T >= 4X^(1-kappa) and T/(X rho(1/delta)^(5/8-epsilon)) tends to infinity. In the ultra-small regime, the correlation is zero if (3X)^delta < 2 and at most one if (3X)^delta < 3.\n\nCandidate contribution (corollary; novelty confidence low): Candidate novelty: combining Jain's affine 13/8-epsilon smooth-pair estimate with a quantified Dickman-stability argument transfers the bound to the moving cutoff P+(n) <= n^delta, and an exact positive Fejer-kernel expansion yields the explicit sharp weighted mean-value threshold T/(X rho(1/delta)^(5/8-epsilon)) -> infinity." }, { "id": 20000673, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0037", "title": "Exact level sets inherit bounded-multiplicative logarithmic disjointness", "statement": "Let $(X, \\mu, T)$ be a totally uniquely ergodic $0$-entropy system. We know from recent work of Frantzikinakis-Host that if $g:\\N \\to \\C$ is a bounded multiplicative function, then for all $f \\in C(X)$ with $\\int_X f d \\mu = 0$ and $x \\in X$ we have\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} g(n) f(T^nx) = 0.\n\\end{equation*}\n\nNow, if $g: \\N \\to \\C$ is a (possibly unbounded) multiplicative function, show that for all $f \\in C(X)$, $x \\in X$ and fixed $c \\in \\C$,\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} f(T^nx) = \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} \\cdot \\int_X f d \\mu.\n\\end{equation*}\nThis can be done for totally ergodic nilsystems.", "original_statement": "Let $(X, \\mu, T)$ be a totally uniquely ergodic $0$-entropy system. We know from recent work of Frantzikinakis-Host that if $g:\\N \\to \\C$ is a bounded multiplicative function, then for all $f \\in C(X)$ with $\\int_X f d \\mu = 0$ and $x \\in X$ we have\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} g(n) f(T^nx) = 0.\n\\end{equation*}\n\nNow, if $g: \\N \\to \\C$ is a (possibly unbounded) multiplicative function, show that for all $f \\in C(X)$, $x \\in X$ and fixed $c \\in \\C$,\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} f(T^nx) = \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} \\cdot \\int_X f d \\mu.\n\\end{equation*}\nThis can be done for totally ergodic nilsystems.", "clean_statement": "Let $(X, \\mu, T)$ be a totally uniquely ergodic $0$-entropy system. We know from recent work of Frantzikinakis-Host that if $g:\\N \\to \\C$ is a bounded multiplicative function, then for all $f \\in C(X)$ with $\\int_X f d \\mu = 0$ and $x \\in X$ we have\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} g(n) f(T^nx) = 0.\n\\end{equation*}\n\nNow, if $g: \\N \\to \\C$ is a (possibly unbounded) multiplicative function, show that for all $f \\in C(X)$, $x \\in X$ and fixed $c \\in \\C$,\n\n\\begin{equation*}\n \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} f(T^nx) = \\mathbb{E}^{\\log}_{n \\in \\mathbb{N}} \\mathbb{1}_{g(n)=c} \\cdot \\int_X f d \\mu.\n\\end{equation*}\nThis can be done for totally ergodic nilsystems.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 6.1 in the “Unbounded multiplicative functions” section of the AIM problem list from the workshop *Sarnak's conjecture*. The archived AIM page was checked and agrees with the record; no correction or reconstruction is needed.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Unbounded multiplicative functions\nSource item: 6.1\nSource URL: http://aimpl.org/sarnakconjecture/6/\nCanonical location: aim-analytic-number-theory-notes.json notes[36]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $(X, \\\\mu, T)$ be a totally uniquely ergodic $0$-entropy system. We know from recent work of Frantzikinakis-Host that if $g:\\\\N \\\\to \\\\C$ is a bounded multiplicative function, then for all $f \\\\in C(X)$ with $\\\\int_X f d \\\\mu = 0$ and $x \\\\in X$ we have\\n\\n\\\\begin{equation*}\\n \\\\mathbb{E}^{\\\\log}_{n \\\\in \\\\mathbb{N}} g(n) f(T^nx) = 0.\\n\\\\end{equation*}\\n\\nNow, if $g: \\\\N \\\\to \\\\C$ is a (possibly unbounded) multiplicative function, show that for all $f \\\\in C(X)$, $x \\\\in X$ and fixed $c \\\\in \\\\C$,\\n\\n\\\\begin{equation*}\\n \\\\mathbb{E}^{\\\\log}_{n \\\\in \\\\mathbb{N}} \\\\mathbb{1}_{g(n)=c} f(T^nx) = \\\\mathbb{E}^{\\\\log}_{n \\\\in \\\\mathbb{N}} \\\\mathbb{1}_{g(n)=c} \\\\cdot \\\\int_X f d \\\\mu.\\n\\\\end{equation*}\\nThis can be done for totally ergodic nilsystems.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/6/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0037", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The requested identity follows for every possibly unbounded complex multiplicative function. For a nonzero level c, Haar Fourier inversion on the compact dual of the discrete group generated by the nonzero values of g expresses the level-set indicator as an integral of unit-disc-valued multiplicative functions; the Frantzikinakis-Host theorem applies to every character twist, and dominated convergence transfers the cancellation. The zero level is the difference of the bounded multiplicative weights 1 and the indicator of g being nonzero. Ruzsa's density theorem supplies existence of the scalar logarithmic density.\n\nCandidate contribution (lemma; novelty confidence low): Compact-dual level-set transfer lemma: any bounded sequence that is logarithmically orthogonal to every unit-disc-valued multiplicative function is logarithmically orthogonal to every exact level-set indicator of every possibly unbounded complex multiplicative function." }, { "id": 20000674, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0038", "title": "Prime-power obstructions and divisor-bounded positive results", "statement": "Can anything be done for correlations of unbounded functions? What is the behavior in almost all intervals of\n\\begin{equation*}\n \\sum_{x \\leq n \\leq x + H} g(n),\n\\end{equation*}\nwith $g$ an unbounded multiplicative function such that $g(p) = O(1)$, and $H = \\log^A x$ for $A>0$ fixed?", "original_statement": "Can anything be done for correlations of unbounded functions? What is the behavior in almost all intervals of\n\\begin{equation*}\n \\sum_{x \\leq n \\leq x + H} g(n),\n\\end{equation*}\nwith $g$ an unbounded multiplicative function such that $g(p) = O(1)$, and $H = \\log^A x$ for $A>0$ fixed?", "clean_statement": "Can anything be done for correlations of unbounded functions? What is the behavior in almost all intervals of\n\\begin{equation*}\n \\sum_{x \\leq n \\leq x + H} g(n),\n\\end{equation*}\nwith $g$ an unbounded multiplicative function such that $g(p) = O(1)$, and $H = \\log^A x$ for $A>0$ fixed?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List item 6.2 in the section “Unbounded multiplicative functions.” Its exact question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Unbounded multiplicative functions\nSource item: 6.2\nSource URL: http://aimpl.org/sarnakconjecture/6/\nCanonical location: aim-analytic-number-theory-notes.json notes[37]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can anything be done for correlations of unbounded functions? What is the behavior in almost all intervals of\\n\\\\begin{equation*}\\n \\\\sum_{x \\\\leq n \\\\leq x + H} g(n),\\n\\\\end{equation*}\\nwith $g$ an unbounded multiplicative function such that $g(p) = O(1)$, and $H = \\\\log^A x$ for $A>0$ fixed?\"\nOriginal remarks: [\"This problem is addressed in the following preprint: https://arxiv.org/abs/2108.11401, assuming $|g(p)|$ is not too sparsely supported, and $g$ is bounded by a suitable (generalized) divisor function. The method is an adaptation of the work of Matomaki-Radziwill in https://arxiv.org/abs/2007.04290. The exponent $A$ in the range $H = (\\\\log x)^A$ depends on the growth of $\\\\sum_{p \\\\leq X} (|g(p)|-1)^2p^{-1}$.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/6/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0038", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "For each fixed A>0, there exists a nonnegative unbounded ordinary multiplicative function g (depending on A), with g(p)=1 for every prime, and a sequence X_j tending to infinity such that H_j=(log X_j)^A and at least X_j/20 centers in [X_j,2X_j] simultaneously have both their normalized short mean of g and their normalized shift-one correlation of g differ from the corresponding dyadic mean by at least five times that dyadic mean. Thus g(p)=O(1) alone cannot imply a universal almost-all short-to-long law; this does not contradict the positive divisor-bounded results summarized in the artifacts.\n\nCandidate contribution (obstruction; novelty confidence low): For each prescribed fixed A>0, a sparse prime-power spike construction gives a multiplicative g with g(p)=1 at every prime and positive-density failures, on a sequence of scales with H=(log X)^A, simultaneously for the first moment and the shift-one correlation." }, { "id": 20000675, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0039", "title": "Sparse-orbit counterexamples and the exact Polynomial Chowla implication", "statement": "We have the following polynomial versions of the Sarnak and Chowla conjectures, where $(X, \\mu, T)$ is a $0$-entropy system, $f \\in C(X)$, $x \\in X$ and $p(t) \\in \\Z[t]$:\n\n\\begin{itemize}\n \\item Polynomial Sarnak Conjecture 1: If $T$ is minimal, then $\\sum_{n\\leq X} \\lambda(n) f(T^{p(n)}x) = o(X)$.\n\n \\item Polynomial Sarnak Conjecture 2: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) f(T^nx) = o(X)$.\n\n \\item Polynomial Chowla Conjecture: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) = o(X)$.\n\\end{itemize}\n\nIs polynomial Sarnak true, and does it follow from polynomial Chowla in any of the two cases?", "original_statement": "We have the following polynomial versions of the Sarnak and Chowla conjectures, where $(X, \\mu, T)$ is a $0$-entropy system, $f \\in C(X)$, $x \\in X$ and $p(t) \\in \\Z[t]$:\n\n\\begin{itemize}\n \\item Polynomial Sarnak Conjecture 1: If $T$ is minimal, then $\\sum_{n\\leq X} \\lambda(n) f(T^{p(n)}x) = o(X)$.\n\n \\item Polynomial Sarnak Conjecture 2: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) f(T^nx) = o(X)$.\n\n \\item Polynomial Chowla Conjecture: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) = o(X)$.\n\\end{itemize}\n\nIs polynomial Sarnak true, and does it follow from polynomial Chowla in any of the two cases?", "clean_statement": "We have the following polynomial versions of the Sarnak and Chowla conjectures, where $(X, \\mu, T)$ is a $0$-entropy system, $f \\in C(X)$, $x \\in X$ and $p(t) \\in \\Z[t]$:\n\n\\begin{itemize}\n \\item Polynomial Sarnak Conjecture 1: If $T$ is minimal, then $\\sum_{n\\leq X} \\lambda(n) f(T^{p(n)}x) = o(X)$.\n\n \\item Polynomial Sarnak Conjecture 2: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) f(T^nx) = o(X)$.\n\n \\item Polynomial Chowla Conjecture: If $p$ is not a constant multiple of the square of a polynomial, then $\\sum_{n \\leq X} \\lambda(p(n)) = o(X)$.\n\\end{itemize}\n\nIs polynomial Sarnak true, and does it follow from polynomial Chowla in any of the two cases?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 7.05 in the “Generalizations” section of the AIM problem list from the workshop *Sarnak's conjecture*. The archived AIM page was checked. Its mathematical text agrees with the canonical record:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Generalizations\nSource item: 7.05\nSource URL: http://aimpl.org/sarnakconjecture/7/\nCanonical location: aim-analytic-number-theory-notes.json notes[38]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"We have the following polynomial versions of the Sarnak and Chowla conjectures, where $(X, \\\\mu, T)$ is a $0$-entropy system, $f \\\\in C(X)$, $x \\\\in X$ and $p(t) \\\\in \\\\Z[t]$:\\n\\n\\\\begin{itemize}\\n \\\\item Polynomial Sarnak Conjecture 1: If $T$ is minimal, then $\\\\sum_{n\\\\leq X} \\\\lambda(n) f(T^{p(n)}x) = o(X)$.\\n\\n \\\\item Polynomial Sarnak Conjecture 2: If $p$ is not a constant multiple of the square of a polynomial, then $\\\\sum_{n \\\\leq X} \\\\lambda(p(n)) f(T^nx) = o(X)$.\\n\\n \\\\item Polynomial Chowla Conjecture: If $p$ is not a constant multiple of the square of a polynomial, then $\\\\sum_{n \\\\leq X} \\\\lambda(p(n)) = o(X)$.\\n\\\\end{itemize}\\n\\nIs polynomial Sarnak true, and does it follow from polynomial Chowla in any of the two cases?\"\nOriginal remarks: [\"A counterexample to Polynomial Sarnak Conjecture 1 has recently\\nbeen given in\\n``Prime Number Theorem for Analytic Skew Products'' by A. Kanigowski, M. Lema\\\\'nczyk and M. Radziwill in the class of strictly ergodic (i.e. minimal and uniquely ergodic) continuous Anzai skew products. Independently and simultaneously, a counterexample has also been given by Z. Liang and R. Shi in\\n``A counter-example for polynomial Sarnak\\nConjecture'' in the class of minimal Toeplitz subshifts.\", \"Polynomial Sarnak Conjecture 2 follows from the Polynomial Chowla Conjecture using the exact same argument that shows that the Chowla conjecture for $\\\\lambda(n)$ implies the Sarnak Conjecture for $\\\\lambda(n)$ (this argument applies to any bounded sequence $a(n)$ with vanishing correlations, in particular it applies to $a(n)=\\\\lambda(p(n))$ if we assume that the polynomial Chowla conjecture holds).\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/7/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0039", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Polynomial Sarnak 1 is false in a maximal sense for every eventually increasing polynomial of degree at least two: Pavlov's sparse-set interpolation theorem gives a minimal zero-entropy subshift whose Liouville correlation along the polynomial times tends to one. For Polynomial Sarnak 2, the globally quantified one-point Polynomial Chowla conjecture is sufficient: a proved translate-product parity lemma shows that every product of distinct translates of a nonsquare polynomial remains nonsquare, so global Polynomial Chowla supplies every finite self-correlation of lambda(p(n)); Bernoulli genericity and the standard Bernoulli/zero-entropy joining argument then imply Polynomial Sarnak 2. The unconditional truth of Polynomial Sarnak 2 remains open because global Polynomial Chowla remains open.\n\nCandidate contribution (lemma; novelty confidence low): Translate-product parity lemma: if p is not a constant multiple of a square and the shifts h_1,...,h_k are distinct, then the product of p(t+h_j) over those shifts is not a constant multiple of a square; consequently, globally quantified one-point Polynomial Chowla yields full Bernoulli block statistics for every sequence lambda(p(n))." }, { "id": 20000676, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0040", "title": "Solved divisibility-parity means and exact correlations in the complementary convergent regime", "statement": "Let $\\mathcal{A} = \\{a_1, a_2, \\dots\\}$ be a sequence of integers with $(a_i, a_j) = 1$ for $i \\not= j$. Let $\\pi(n):= (-1)^{\\#\\{a_j:a_j|n\\}}$. Is it true that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n) = o(X)?\n\\end{equation*}\nCan we find a set $\\mathcal{A}$ such that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n+h_1) \\dots \\pi(n+h_k) = o(X)\n\\end{equation*}\nfor all distinct $h_1, \\dots, h_k$?\n\n\\begin{itemize}\n \\item Remark: Let\n\\[ \\lambda_y(p) = \\begin{cases}\n -1, &\\text{if } p \\leq y \\\\\n +1, &\\text{if } p > y.\n \\end{cases}\n\\]\nIf $y = X^{o(1)}$ then\n\\begin{equation*}\n \\sum_{n\\leq X} \\lambda_y(n+h_1) \\cdots \\lambda_y(n+h_k) = o(X).\n\\end{equation*}\nFor $k=2$ this was done by Daboussi-Sarkar.\n\\end{itemize}", "original_statement": "Let $\\mathcal{A} = \\{a_1, a_2, \\dots\\}$ be a sequence of integers with $(a_i, a_j) = 1$ for $i \\not= j$. Let $\\pi(n):= (-1)^{\\#\\{a_j:a_j|n\\}}$. Is it true that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n) = o(X)?\n\\end{equation*}\nCan we find a set $\\mathcal{A}$ such that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n+h_1) \\dots \\pi(n+h_k) = o(X)\n\\end{equation*}\nfor all distinct $h_1, \\dots, h_k$?\n\n\\begin{itemize}\n \\item Remark: Let\n\\[ \\lambda_y(p) = \\begin{cases}\n -1, &\\text{if } p \\leq y \\\\\n +1, &\\text{if } p > y.\n \\end{cases}\n\\]\nIf $y = X^{o(1)}$ then\n\\begin{equation*}\n \\sum_{n\\leq X} \\lambda_y(n+h_1) \\cdots \\lambda_y(n+h_k) = o(X).\n\\end{equation*}\nFor $k=2$ this was done by Daboussi-Sarkar.\n\\end{itemize}", "clean_statement": "Let $\\mathcal{A} = \\{a_1, a_2, \\dots\\}$ be a sequence of integers with $(a_i, a_j) = 1$ for $i \\not= j$. Let $\\pi(n):= (-1)^{\\#\\{a_j:a_j|n\\}}$. Is it true that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n) = o(X)?\n\\end{equation*}\nCan we find a set $\\mathcal{A}$ such that\n\\begin{equation*}\n \\sum_{n \\leq X} \\pi(n+h_1) \\dots \\pi(n+h_k) = o(X)\n\\end{equation*}\nfor all distinct $h_1, \\dots, h_k$?\n\n\\begin{itemize}\n \\item Remark: Let\n\\[ \\lambda_y(p) = \\begin{cases}\n -1, &\\text{if } p \\leq y \\\\\n +1, &\\text{if } p > y.\n \\end{cases}\n\\]\nIf $y = X^{o(1)}$ then\n\\begin{equation*}\n \\sum_{n\\leq X} \\lambda_y(n+h_1) \\cdots \\lambda_y(n+h_k) = o(X).\n\\end{equation*}\nFor $k=2$ this was done by Daboussi-Sarkar.\n\\end{itemize}", "statement_status": "exact", "statement_verification": "The canonical record is Problem 7.2 in the “Generalizations” section of the AIM list from the workshop *Sarnak's conjecture*. The exact problem field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Generalizations\nSource item: 7.2\nSource URL: http://aimpl.org/sarnakconjecture/7/\nCanonical location: aim-analytic-number-theory-notes.json notes[39]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $\\\\mathcal{A} = \\\\{a_1, a_2, \\\\dots\\\\}$ be a sequence of integers with $(a_i, a_j) = 1$ for $i \\\\not= j$. Let $\\\\pi(n):= (-1)^{\\\\#\\\\{a_j:a_j|n\\\\}}$. Is it true that\\n\\\\begin{equation*}\\n \\\\sum_{n \\\\leq X} \\\\pi(n) = o(X)?\\n\\\\end{equation*}\\nCan we find a set $\\\\mathcal{A}$ such that\\n\\\\begin{equation*}\\n \\\\sum_{n \\\\leq X} \\\\pi(n+h_1) \\\\dots \\\\pi(n+h_k) = o(X)\\n\\\\end{equation*}\\nfor all distinct $h_1, \\\\dots, h_k$?\\n\\n\\\\begin{itemize}\\n \\\\item Remark: Let\\n\\\\[ \\\\lambda_y(p) = \\\\begin{cases}\\n -1, &\\\\text{if } p \\\\leq y \\\\\\\\\\n +1, &\\\\text{if } p > y.\\n \\\\end{cases}\\n\\\\]\\nIf $y = X^{o(1)}$ then\\n\\\\begin{equation*}\\n \\\\sum_{n\\\\leq X} \\\\lambda_y(n+h_1) \\\\cdots \\\\lambda_y(n+h_k) = o(X).\\n\\\\end{equation*}\\nFor $k=2$ this was done by Daboussi-Sarkar.\\n\\\\end{itemize}\"\nOriginal remarks: [\"In the preprint [https://arxiv.org/abs/2304.05344] it is shown that if $\\\\mathcal{A}$ is taken to be a subset of primes satisfying:\\ni) $\\\\frac{1}{\\\\pi(x)} |\\\\mathcal{A} \\\\cap [1,x]| \\\\rightarrow 0$, and\\nii) $\\\\sum_{p \\\\in \\\\mathcal{A}} 1/p = \\\\infty$\\nthen all of the $k$-point correlations of $\\\\pi = \\\\pi_{\\\\mathcal{A}}$ are $0$. This resolves the second problem.\", \"The mean-value problem for $\\\\pi(n)$ was completely solved by Yichen You, a Ph.D student at Durham University, in her recent preprint https://arxiv.org/pdf/2312.06012.pdf.\", \"In this question we assume also that $\\\\sum_j 1/a_j =\\\\infty$, otherwise the conclusion does not hold.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/7/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0040", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM problem is fully resolved with its later editorial qualification. As literally written, the universal mean-zero assertion is false: for the infinite pairwise-coprime set A={p^2:p prime}, the mean is the positive product over p of (1-2/p^2). With the intended hypothesis sum_{a in A}1/a=infinity, Yichen You proved the mean is zero for every mutually coprime A; Klurman-Mangerel-Teravainen constructed relative-density-zero prime sets with divergent reciprocal sum whose ordinary correlations of every order vanish, and You extended this to general mutually coprime A with sparse prime part and affine forms. In addition to citing those deep solved theorems, this attempt proves that in the complementary regime sum 1/a X^\\epsilon$ implies\n\\begin{equation*}\n \\sum_{n \\leq X} \\frac{\\lambda(n)\\lambda(n+1)}{n} = o(\\log X)?\n\\end{equation*}", "original_statement": "Is is possible to show that\n\\begin{equation*}\n \\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) e(n \\alpha) \\right| d x = o(XH)\n\\end{equation*}\nwith $H > X^\\epsilon$ implies\n\\begin{equation*}\n \\sum_{n \\leq X} \\frac{\\lambda(n)\\lambda(n+1)}{n} = o(\\log X)?\n\\end{equation*}", "clean_statement": "Is is possible to show that\n\\begin{equation*}\n \\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} \\lambda(n) e(n \\alpha) \\right| d x = o(XH)\n\\end{equation*}\nwith $H > X^\\epsilon$ implies\n\\begin{equation*}\n \\sum_{n \\leq X} \\frac{\\lambda(n)\\lambda(n+1)}{n} = o(\\log X)?\n\\end{equation*}", "statement_status": "exact", "statement_verification": "The canonical record is Problem 8.1 in the AIM workshop list *Sarnak's conjecture*, section “Proof techniques and examples.” The archived source page was checked directly and agrees with the repository record, including the duplicated word “Is” and the absence of a supremum over the frequency. Its mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Proof techniques and examples\nSource item: 8.1\nSource URL: http://aimpl.org/sarnakconjecture/8/\nCanonical location: aim-analytic-number-theory-notes.json notes[42]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is is possible to show that\\n\\\\begin{equation*}\\n \\\\int_X^{2X} \\\\left| \\\\sum_{x\\\\leq n\\\\leq x+H} \\\\lambda(n) e(n \\\\alpha) \\\\right| d x = o(XH)\\n\\\\end{equation*}\\nwith $H > X^\\\\epsilon$ implies\\n\\\\begin{equation*}\\n \\\\sum_{n \\\\leq X} \\\\frac{\\\\lambda(n)\\\\lambda(n+1)}{n} = o(\\\\log X)?\\n\\\\end{equation*}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/8/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0043", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Tao proved the requested logarithmically averaged lag-one Liouville correlation unconditionally in 2016. The historically relevant implication uses a uniform-in-frequency short-sum estimate at every growing scale together with multiplicativity and entropy decrement; the polynomial-scale estimate printed by AIM does not by itself supply the shorter unspecified scale selected by that argument. A direct Fourier-square extraction loses one factor of H, and a rigorous paired-Rademacher countermodel shows this loss is genuine for general bounded sequences.\n\nCandidate contribution (counterexample; novelty confidence low): There exists a sequence a(n) in {-1,+1} whose supremum-in-frequency exponential sum on every interval of polynomial length H is O(sqrt(H log X)), uniformly for intervals based in [X,2X], while its logarithmically averaged adjacent correlation is (1/2+o(1)) log X." }, { "id": 20000680, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0044", "title": "Uniform Mobius disjointness for horocycles: an AOP obstruction and a moving-basepoint reduction", "statement": "Does the Sarnak conjecture hold uniformly in $x \\in X$ in the case of horocycle flows? It is known that the uniform Sarnak conjecture follows from the Sarnak conjecture, but we don't know that it follows case by case.", "original_statement": "Does the Sarnak conjecture hold uniformly in $x \\in X$ in the case of horocycle flows? It is known that the uniform Sarnak conjecture follows from the Sarnak conjecture, but we don't know that it follows case by case.", "clean_statement": "Does the Sarnak conjecture hold uniformly in $x \\in X$ in the case of horocycle flows? It is known that the uniform Sarnak conjecture follows from the Sarnak conjecture, but we don't know that it follows case by case.", "statement_status": "exact", "statement_verification": "The exact AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Proof techniques and examples\nSource item: 8.2\nSource URL: http://aimpl.org/sarnakconjecture/8/\nCanonical location: aim-analytic-number-theory-notes.json notes[43]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does the Sarnak conjecture hold uniformly in $x \\\\in X$ in the case of horocycle flows? It is known that the uniform Sarnak conjecture follows from the Sarnak conjecture, but we don't know that it follows case by case.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/8/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0044", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The requested full-sequence uniform-in-x Mobius theorem for the classical horocycle time-one map remains open in the literature checked. For compact quotients, two unconditional facts are proved here: the classical map fails AOP maximally via explicit geodesic graph joinings, and its Mobius averages have Lipschitz norm O(N^2), reducing uniformity to checks on o(N^{-2})-nets. The main candidate contribution is a proved conditional reduction: a moving-basepoint multi-joining isolation hypothesis, allowing at most one nonproduct prime-pair marginal in diagonal-start empirical limits, implies uniform Mobius disjointness through the finite Bourgain-Sarnak-Ziegler criterion. The isolation hypothesis itself is not proved.\n\nCandidate contribution (reduction; novelty confidence low): For a compact classical horocycle map, uniform Mobius disjointness follows from the explicit moving-basepoint multi-joining isolation condition that every weak-* limit of finite-prime diagonal-start empirical measures has at most one nonproduct two-coordinate marginal; this separates the fixed-basepoint one-exception BSZ conclusion from the unproved stability needed when the basepoint moves with the averaging horizon.", "research_difficulty_suggested": "L4" }, { "id": 20000681, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0045", "title": "Correlation with a uniquely ergodic system", "statement": "The Chowla conjecture implies that $\\mu$ cannot be orthogonal to all uniquely ergodic systems (allowing non-zero entropy). Can we find a concrete example of a uniquely ergodic system that correlates with $\\mu$? A way to find a uniquely ergodic system (of positive entropy) that does not correlate with $\\lambda$ is known.", "original_statement": "The Chowla conjecture implies that $\\mu$ cannot be orthogonal to all uniquely ergodic systems (allowing non-zero entropy). Can we find a concrete example of a uniquely ergodic system that correlates with $\\mu$? A way to find a uniquely ergodic system (of positive entropy) that does not correlate with $\\lambda$ is known.", "clean_statement": "The Chowla conjecture implies that $\\mu$ cannot be orthogonal to all uniquely ergodic systems (allowing non-zero entropy). Can we find a concrete example of a uniquely ergodic system that correlates with $\\mu$? A way to find a uniquely ergodic system (of positive entropy) that does not correlate with $\\lambda$ is known.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 8.3 in the “Proof techniques and examples” section of the AIM problem list from the workshop *Sarnak's conjecture*. Its exact problem field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Sarnak's conjecture\nSection: Proof techniques and examples\nSource item: 8.3\nSource URL: http://aimpl.org/sarnakconjecture/8/\nCanonical location: aim-analytic-number-theory-notes.json notes[44]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The Chowla conjecture implies that $\\\\mu$ cannot be orthogonal to all uniquely ergodic systems (allowing non-zero entropy). Can we find a concrete example of a uniquely ergodic system that correlates with $\\\\mu$? A way to find a uniquely ergodic system (of positive entropy) that does not correlate with $\\\\lambda$ is known.\"\nOriginal remarks: [\"Since the system of Problem 3.1 is disjoint from all ergodic systems, an example of an ergodic sequence (or uniquely ergodic system) that correlates with the Liouville function would give a negative answer to Problem 3.1.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: open; rigor: conditional; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/sarnakconjecture/8/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0045", "aim-domain:analytic-number-theory", "aim-workshop:sarnakconjecture", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The requested concrete uniquely ergodic system with proved nonzero ordinary Möbius correlation was not located and appears open in the literature checked. Under full Chowla, Weiss's strict-ergodic Besicovitch approximation theorem, as used by Conze--Downarowicz--Serafin, gives abstract conditional existence: any approximant within distance 6/pi^2 of Möbius already correlates. The new conditional theorem proved here gives a quantitative obstruction: if a finite-valued observable of a uniquely ergodic system has Möbius correlation magnitude c, then its entropy is at least c^2/(2q), q=6/pi^2; for a binary observable the sharper lower bound is q[log 2-H_b((1-c/q)/2)].\n\nCandidate contribution (conditional_entropy_bound; novelty confidence low): Under the standard dynamical form of full Chowla, correlation magnitude c between Möbius and a finite-valued continuous observable of a uniquely ergodic system forces h_top at least c^2/(2q), where q=6/pi^2; for a binary observable it forces h_top at least q[log 2-H_b((1-c/q)/2)].", "research_difficulty_suggested": "L4" }, { "id": 20000682, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0046", "title": "Microscopic even moments and a diagonal crossover mechanism for short divisor sums", "statement": "Moments of divisor functions in short intervals\n\nEvaluate $$\\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} d_k(n) \\ - \\ \\mbox{main term}\\right|^{2\\ell}\\,dx$$ for different ranges of $H$.", "original_statement": "Moments of divisor functions in short intervals\n\nEvaluate $$\\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} d_k(n) \\ - \\ \\mbox{main term}\\right|^{2\\ell}\\,dx$$ for different ranges of $H$.", "clean_statement": "Moments of divisor functions in short intervals\n\nEvaluate $$\\int_X^{2X} \\left| \\sum_{x\\leq n\\leq x+H} d_k(n) \\ - \\ \\mbox{main term}\\right|^{2\\ell}\\,dx$$ for different ranges of $H$.", "statement_status": "exact", "statement_verification": "The archived AIM page was checked directly. Problem 1.1, attributed there to S. Lester, asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Divisor sums\nSource item: 1.1\nSource URL: http://aimpl.org/zetamoments/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[45]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Moments of divisor functions in short intervals\\n\\nEvaluate $$\\\\int_X^{2X} \\\\left| \\\\sum_{x\\\\leq n\\\\leq x+H} d_k(n) \\\\ - \\\\ \\\\mbox{main term}\\\\right|^{2\\\\ell}\\\\,dx$$ for different ranges of $H$.\"\nOriginal remarks: [\"Find connections with the paper of Keating, Rodgers, Roditty-Gershon, and Rudnick.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0046", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every fixed k >= 2 and even exponent r = 2 ell, the AIM moment at H = 1 is asymptotic to A_{k,r} X (log X)^{k^r-1}, with an explicit positive Euler-product constant, for the canonical smooth centering and more generally for any centering of size O((log X)^{k-1}). In addition, for every integer 1 <= H <= X, the all-equal-index sub-sum in the exact r-point correlation expansion is asymptotic to A_{k,r} X H (log X)^{k^r-1}. These are proved statements. Comparing the diagonal scale with Gaussian pairing predicts a polylogarithmic crossover, but that prediction is explicitly heuristic and is not claimed as an asymptotic for the full growing-H moment.\n\nCandidate contribution (special_case_and_diagonal_reduction; novelty confidence low): Candidate novelty: a centering-robust closed asymptotic for every even moment at H = 1, together with a uniform evaluation of the all-equal-index diagonal for 1 <= H <= X and the falsifiable crossover exponent beta_{k,ell} = (k^{2ell}-1-ell(k^2-1))/(ell-1) for ell >= 2." }, { "id": 20000683, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0047", "title": "Geometric rigidity and a Hecke obstruction for the apparent Ramanujan coefficients", "statement": "Some coefficients of the Ramanujan tau function appear as coefficients of the piecewise polynomial in the work of Keating, Rodgers, Roditty-Gershon, and Rudnick. Is this a coincidence or not?", "original_statement": "Some coefficients of the Ramanujan tau function appear as coefficients of the piecewise polynomial in the work of Keating, Rodgers, Roditty-Gershon, and Rudnick. Is this a coincidence or not?", "clean_statement": "Some coefficients of the Ramanujan tau function appear as coefficients of the piecewise polynomial in the work of Keating, Rodgers, Roditty-Gershon, and Rudnick. Is this a coincidence or not?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Moments of zeta and correlations of divisor sums*, section “Divisor sums,” Problem 1.2) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Divisor sums\nSource item: 1.2\nSource URL: http://aimpl.org/zetamoments/1/\nCanonical location: aim-analytic-number-theory-notes.json notes[46]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Some coefficients of the Ramanujan tau function appear as coefficients of the piecewise polynomial in the work of Keating, Rodgers, Roditty-Gershon, and Rudnick. Is this a coincidence or not?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/1/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0047", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the KRRGR k=3 middle-chamber polynomial Q, the coefficients 24, -252, and -4830 correspond under the natural reversed indexing to tau(2), tau(3), and tau(5). The full polynomial is uniquely forced by reflection symmetry, fourth-order chamber gluing, and its Selberg mass, giving a non-arithmetic derivation of all coefficients. Moreover, the reversed sequence fails the first weight-12 Hecke prime-square relation by exactly -40, so the literal coefficientwise modular-form explanation is impossible, although a more indirect connection is not excluded.\n\nCandidate contribution (obstruction_and_rigidity_lemma; novelty confidence low): The KRRGR middle polynomial is the unique degree-at-most-eight polynomial satisfying reflection symmetry, C^3 gluing to c^8 at c=1, and middle mass 40/9; its reversed coefficient sequence matches tau at 2, 3, and 5 but has Hecke defect b_4-(b_2^2-2^11)=-40." }, { "id": 20000684, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0048", "title": "Nonsquare dual polynomials in Florea's Poisson formula", "statement": "Analyze Alexandra Florea's formula for $V$ not equal to a square.", "original_statement": "Analyze Alexandra Florea's formula for $V$ not equal to a square.", "clean_statement": "Analyze Alexandra Florea's formula for $V$ not equal to a square.", "statement_status": "exact", "statement_verification": "The canonical record says, verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Function fields\nSource item: 2.4\nSource URL: http://aimpl.org/zetamoments/2/\nCanonical location: aim-analytic-number-theory-notes.json notes[47]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Analyze Alexandra Florea's formula for $V$ not equal to a square.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/2/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0048", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM instruction is identified with high confidence as Florea's published second/third-moment nonsquare-dual calculation, which proves aggregate bounds S_{2g}(V not square) << q^{g(1+epsilon)} and S_{3g}(V not square) << q^{3g(1+epsilon)/2}. Beyond that status result, for every nonconstant squarefree dual polynomial V this attempt proves an exact support theorem for G(V,chi_f), an all-k divisor-weighted Euler factorization F_{k,V}=L(w,chi_V)^k E_{k,V} with E absolutely convergent for |w|deg(V)+1." }, { "id": 20000685, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0049", "title": "A safe-half-plane Euler product for ratios in the hyperelliptic ensemble", "statement": "Evaluate moments of ratios of $L$-functions over function fields.", "original_statement": "Evaluate moments of ratios of $L$-functions over function fields.", "clean_statement": "Evaluate moments of ratios of $L$-functions over function fields.", "statement_status": "exact", "statement_verification": "The exact canonical record, from the 2016 AIM workshop *Moments of zeta and correlations of divisor sums*, section “Function fields,” Problem 2.5, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Function fields\nSource item: 2.5\nSource URL: http://aimpl.org/zetamoments/2/\nCanonical location: aim-analytic-number-theory-notes.json notes[48]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Evaluate moments of ratios of $L$-functions over function fields.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/2/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0049", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The intended primary family is reconstructed as the odd-characteristic hyperelliptic ensemble H_{2g+1,q} with fixed q and g tending to infinity. In the absolutely convergent region Re(alpha), Re(beta)>1/2, the normalized one-over-one ratio has a rigorous large-genus limit equal to an explicit normally convergent Euler product whose local factor is 1 + |P|/(|P|+1) times (|P|^{-1-2alpha}-|P|^{-1-alpha-beta})/(1-|P|^{-1-2alpha}). Its diagonal derivative gives an explicit prime sum for the limiting averaged logarithmic derivative. An exact functional-equation covariance is also proved, while current literature is separated into finite-family identities, fixed-q large-genus partial theorems, and fixed-genus large-q matrix limits.\n\nCandidate contribution (rigorous_special_case; novelty confidence low): For fixed odd q and Re(alpha), Re(beta)>1/2, the one-over-one hyperelliptic ratio average converges to the explicit square-diagonal Euler product in Theorem 1, and differentiating on alpha=beta=r yields the prime-sum identity in Corollary 1 for the averaged logarithmic derivative." }, { "id": 20000686, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0050", "title": "The local inverse-zeta fingerprint of a forced rational point", "statement": "Consider a family of elliptic curves with a known rational point. Evaluate moments of $L$-functions in this family at certain points. How does the fixed rational point analytically affect calculations? E.g. $y^2=x^3+ax+b^2$ with $a,b$ integers having $|a|,|b|\\leq X$ (or in $\\mathbb{F}_q[t]$).", "original_statement": "Consider a family of elliptic curves with a known rational point. Evaluate moments of $L$-functions in this family at certain points. How does the fixed rational point analytically affect calculations? E.g. $y^2=x^3+ax+b^2$ with $a,b$ integers having $|a|,|b|\\leq X$ (or in $\\mathbb{F}_q[t]$).", "clean_statement": "Consider a family of elliptic curves with a known rational point. Evaluate moments of $L$-functions in this family at certain points. How does the fixed rational point analytically affect calculations? E.g. $y^2=x^3+ax+b^2$ with $a,b$ integers having $|a|,|b|\\leq X$ (or in $\\mathbb{F}_q[t]$).", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Moments of zeta and correlations of divisor sums*, section “Function fields,” Problem 2.2) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Function fields\nSource item: 2.2\nSource URL: http://aimpl.org/zetamoments/2/\nCanonical location: aim-analytic-number-theory-notes.json notes[49]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Consider a family of elliptic curves with a known rational point. Evaluate moments of $L$-functions in this family at certain points. How does the fixed rational point analytically affect calculations? E.g. $y^2=x^3+ax+b^2$ with $a,b$ integers having $|a|,|b|\\\\leq X$ (or in $\\\\mathbb{F}_q[t]$).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/2/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0050", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the family y^2=x^3+ax+b^2 over every finite field F_Q of characteristic greater than 3, the exact full-box normalized trace moments are E(lambda_Q)=-Q^{-1/2}+Q^{-3/2} and E(lambda_Q^2)=1-Q^{-1}. After conditioning on good reduction they are E(lambda_Q | good)=-Q^{-1/2}-chi(6)Q^{-3/2} and E(lambda_Q^2 | good)=1-Q^{-2}. The first bias yields one zeta_K(1+alpha)^{-1} factor per shift in the global moment recipe. Over Q, the marked point is non-torsion for density-one integer parameters and then central vanishing is unconditional by modularity and the Gross-Zagier-Kolyvagin rank-zero theorem, without forcing root number -1. Young's full derivative-moment asymptotic remains conjectural.\n\nCandidate contribution (theorem; novelty confidence low): For every finite field F_Q of characteristic greater than 3, the nonsingular subfamily y^2=x^3+ax+b^2 has exact conditioned normalized trace moments -Q^{-1/2}-chi(6)Q^{-3/2} and 1-Q^{-2}; the singular-locus correction to the first raw trace sum is exactly (Q-1)chi(6)." }, { "id": 20000687, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0051", "title": "A parity-projector certificate for fixed-field hyperelliptic moments", "statement": "Develop a heuristic for moments of L-functions over function fields. Can one carry out the analogous Conrey-Keating calculation for high moments (fixed $q$, large degree $g$)?", "original_statement": "Develop a heuristic for moments of L-functions over function fields. Can one carry out the analogous Conrey-Keating calculation for high moments (fixed $q$, large degree $g$)?", "clean_statement": "Develop a heuristic for moments of L-functions over function fields. Can one carry out the analogous Conrey-Keating calculation for high moments (fixed $q$, large degree $g$)?", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Function fields\nSource item: 2.1\nSource URL: http://aimpl.org/zetamoments/2/\nCanonical location: aim-analytic-number-theory-notes.json notes[50]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop a heuristic for moments of L-functions over function fields. Can one carry out the analogous Conrey-Keating calculation for high moments (fixed $q$, large degree $g$)?\"\nOriginal remarks: [\"For this problem, it may be easier to start with low moments first.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/2/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0051", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the square-diagonal Euler product in the fixed-q hyperelliptic CFKRS recipe, an exact parity projector factors the shifted series into precisely k(k+1)/2 pairwise zeta factors times an arithmetic product that converges absolutely and is holomorphic for |Re z_i|<1/4. A one-variable specialization rigorously recovers both the leading and constant coefficients of the complete principal first-moment polynomial. Literature checked through 2026 also shows that the CFKRS polynomial is now proved for every fixed moment when the fixed odd field is sufficiently large relative to that moment, while arbitrary small fixed odd fields remain unresolved in general.\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: the closed parity-projector local factor, its uniform eta<1/4 holomorphy audit after removal of the k(k+1)/2 zeta poles, and the direct one-variable extraction of the full principal k=1 polynomial form a compact three-part certificate for proposed fixed-field CFKRS calculations." }, { "id": 20000688, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0052", "title": "Families and moment heuristics for automorphic L-functions over function fields", "statement": "Develop a heuristic for moments of L-functions of automorphic forms over function fields. What are families in this context?", "original_statement": "Develop a heuristic for moments of L-functions of automorphic forms over function fields. What are families in this context?", "clean_statement": "Develop a heuristic for moments of L-functions of automorphic forms over function fields. What are families in this context?", "statement_status": "exact", "statement_verification": "The canonical AIM record says, verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Function fields\nSource item: 2.3\nSource URL: http://aimpl.org/zetamoments/2/\nCanonical location: aim-analytic-number-theory-notes.json notes[51]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop a heuristic for moments of L-functions of automorphic forms over function fields. What are families in this context?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/2/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0052", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A usable automorphic family must specify the global field, group and L-group representation, global and local constraints, measure, root-number sector, asymptotic aspect, and order of limits. For a generic non-self-dual conductor family this leads to a two-factor prediction: a local Plancherel arithmetic factor times a global unitary characteristic-polynomial moment. As a proved test, if Q is prime of degree d and eta is a fixed nontrivial character of F_q^times, then the characters modulo Q restricting to eta all have L-degree N=d-1 and exact average central second moment d=N+1, exactly matching the Haar U(N) moment.\n\nCandidate contribution (exact finite-family theorem; novelty confidence low): For every q>2, prime Q of degree d, and fixed nontrivial eta on F_q^times, the prime-conductor character fiber with restriction eta has exact central second moment d, equal to the finite-N Haar U(N) moment for N=d-1." }, { "id": 20000689, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0053", "title": "Asymmetric lengths: a safe-range obstruction and conductor-matched boundary cancellation", "statement": "Can one find a version of the recipe using asymmetric approximate functional equations, giving a ``robust'' version of the calculations? Similarly, can one apply the Conrey-Keating method using two asymmetric Dirichlet polynomials\n$$\n\\sum_{m\\leq X} \\frac{\\tau_A(m)}{m^s} \\ \\ \\ \\ \\ \\ \\mbox{and} \\ \\ \\ \\ \\ \\\n\\sum_{n\\leq Y} \\frac{\\tau_B(n)}{n^{1-s}}\n$$\nto get the same answer as before, i.e. when $X=Y$?", "original_statement": "Can one find a version of the recipe using asymmetric approximate functional equations, giving a ``robust'' version of the calculations? Similarly, can one apply the Conrey-Keating method using two asymmetric Dirichlet polynomials\n$$\n\\sum_{m\\leq X} \\frac{\\tau_A(m)}{m^s} \\ \\ \\ \\ \\ \\ \\mbox{and} \\ \\ \\ \\ \\ \\\n\\sum_{n\\leq Y} \\frac{\\tau_B(n)}{n^{1-s}}\n$$\nto get the same answer as before, i.e. when $X=Y$?", "clean_statement": "Can one find a version of the recipe using asymmetric approximate functional equations, giving a ``robust'' version of the calculations? Similarly, can one apply the Conrey-Keating method using two asymmetric Dirichlet polynomials\n$$\n\\sum_{m\\leq X} \\frac{\\tau_A(m)}{m^s} \\ \\ \\ \\ \\ \\ \\mbox{and} \\ \\ \\ \\ \\ \\\n\\sum_{n\\leq Y} \\frac{\\tau_B(n)}{n^{1-s}}\n$$\nto get the same answer as before, i.e. when $X=Y$?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 3.1 in the “Recipe” section of the AIM list *Moments of zeta and correlations of divisor sums*. The live source page was checked and attributes the problem to C. Hughes. It asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: The Recipe\nSource item: 3.1\nSource URL: http://aimpl.org/zetamoments/3/\nCanonical location: aim-analytic-number-theory-notes.json notes[52]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can one find a version of the recipe using asymmetric approximate functional equations, giving a ``robust'' version of the calculations? Similarly, can one apply the Conrey-Keating method using two asymmetric Dirichlet polynomials\\n$$\\n\\\\sum_{m\\\\leq X} \\\\frac{\\\\tau_A(m)}{m^s} \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\mbox{and} \\\\ \\\\ \\\\ \\\\ \\\\ \\\\\\n\\\\sum_{n\\\\leq Y} \\\\frac{\\\\tau_B(n)}{n^{1-s}}\\n$$\\nto get the same answer as before, i.e. when $X=Y$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/3/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0053", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Two complementary facts isolate the meaning of robustness. First, for max(X,Y) at most T^(1-eta), the smoothed mean of the two bare shifted-divisor polynomials equals T times their common truncated diagonal through min(X,Y), with an arbitrarily power-saving error; hence it is generally not invariant under changing the aspect ratio. Second, for the singleton shifted approximate-functional-equation diagonal, the leading dependence on complementary lengths x,y is exactly x^(-delta)((xy/C)^delta-1)/delta, so it cancels when xy equals the local conductor C=t/(2 pi) and measures the defect when conductor matching fails.\n\nCandidate contribution (obstruction_and_boundary_identity; novelty confidence low): The candidate contribution is the combined boundary-flux plus safe-obstruction criterion: below the first off-diagonal threshold the asymmetric bare-polynomial mean necessarily retains min(X,Y), whereas the paired singleton AFE diagonal has the explicit conductor-mismatch correction x^(-delta)((xy/C)^delta-1)/delta, vanishing exactly at xy=C." }, { "id": 20000690, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0054", "title": "A split-dependent diagonal obstruction in the four-factor approximate functional equation", "statement": "Can one prove the fourth moment formula starting from\n$$\n\\int \\prod_{j=1}^4 \\left( \\sum_{n\\leq x_j} \\frac{1}{n^{1/2+it}}+\\chi(s) \\sum_{n\\leq t/x_j} \\frac{1}{n^{1/2-it}}\\right)\\,dt \\ ?\n$$", "original_statement": "Can one prove the fourth moment formula starting from\n$$\n\\int \\prod_{j=1}^4 \\left( \\sum_{n\\leq x_j} \\frac{1}{n^{1/2+it}}+\\chi(s) \\sum_{n\\leq t/x_j} \\frac{1}{n^{1/2-it}}\\right)\\,dt \\ ?\n$$", "clean_statement": "Can one prove the fourth moment formula starting from\n$$\n\\int \\prod_{j=1}^4 \\left( \\sum_{n\\leq x_j} \\frac{1}{n^{1/2+it}}+\\chi(s) \\sum_{n\\leq t/x_j} \\frac{1}{n^{1/2-it}}\\right)\\,dt \\ ?\n$$", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: The Recipe\nSource item: 3.2\nSource URL: http://aimpl.org/zetamoments/3/\nCanonical location: aim-analytic-number-theory-notes.json notes[53]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can one prove the fourth moment formula starting from\\n$$\\n\\\\int \\\\prod_{j=1}^4 \\\\left( \\\\sum_{n\\\\leq x_j} \\\\frac{1}{n^{1/2+it}}+\\\\chi(s) \\\\sum_{n\\\\leq t/x_j} \\\\frac{1}{n^{1/2-it}}\\\\right)\\\\,dt \\\\ ?\\n$$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/3/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0054", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After recovering the intended product as two asymmetric approximants to zeta(1/2+it) and two conjugate approximants, with complementary cutoffs xy=t/(2pi), a rigorous logarithmic-polytope theorem evaluates every unequal-cutoff multiplicative diagonal. For a common split x=T^alpha, the six chi-balanced pure diagonals have coefficient F(alpha)/zeta(2), where F(alpha) is explicit and split-dependent. At alpha=1/2 this is 3/(8pi^2), exactly three quarters of Ingham's 1/(2pi^2), leaving a 1/(8pi^2) leading deficit. Moreover, the h=plus-or-minus-one unbalanced pieces possess exact interior stationary points, so they cannot all be dismissed by nonstationary phase. This is a proved obstruction and reduction, not a complete proof of the fourth moment.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate novelty: the unequal-cutoff diagonal-volume formula D(T^lambda_1,...,T^lambda_4)=V(lambda)/zeta(2) times log^4 T plus O(log^3 T), specialized to the explicit common-split coefficient F(alpha), proves a split-dependent diagonal-only answer and an exact three-quarter deficit at the symmetric split; the exact integer family L=125k, m=a=b=100k, n=64k independently falsifies a uniform nonstationary-phase dismissal of the h=1 pieces." }, { "id": 20000691, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0055", "title": "Quadratic moments, Poisson squares, and swap terms", "statement": "Using the ideas of Conrey and Keating, develop a heuristic for conjecturing asymptotic formulas for high moments of quadratic Dirichlet $L$-functions. What new terms contribute to the main term when $k\\geq 3$?", "original_statement": "Using the ideas of Conrey and Keating, develop a heuristic for conjecturing asymptotic formulas for high moments of quadratic Dirichlet $L$-functions. What new terms contribute to the main term when $k\\geq 3$?", "clean_statement": "Using the ideas of Conrey and Keating, develop a heuristic for conjecturing asymptotic formulas for high moments of quadratic Dirichlet $L$-functions. What new terms contribute to the main term when $k\\geq 3$?", "statement_status": "exact", "statement_verification": "The exact AIM record is Problem 5.1 in the section “Other families of \\(L\\)-functions” of the workshop list *Moments of zeta and correlations of divisor sums*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Other families of $L$-functions\nSource item: 5.1\nSource URL: http://aimpl.org/zetamoments/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[54]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Using the ideas of Conrey and Keating, develop a heuristic for conjecturing asymptotic formulas for high moments of quadratic Dirichlet $L$-functions. What new terms contribute to the main term when $k\\\\geq 3$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0055", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the family of all positive or all negative fundamental discriminants, the square-diagonal local law is uniform over every prime, including p=2: chi_d(p) has limiting probabilities 1/(p+1), p/(2(p+1)), p/(2(p+1)) for 0,+1,-1. This yields an exact Euler factor and extracts exactly k(k+1)/2 zeta poles. Under the same full-zeta extraction, the chi_{8d} and full-family local arithmetic factors at 2 are respectively 2^{-K} and 2^{-K}E_{full,2}, so their ratio is E_{full,2}, with no extra 2^{-K}. In the Conrey-Keating long-polynomial organization, the natural length for the kth moment is D^{k/2}; quadratic Poisson summation and the Conrey-Rodgers length theorem identify the genuinely new principal contributions from k=3 onward as nonzero-square dual frequencies, equivalently 1-swap residues. These belong to X times the principal logarithmic polynomial and are distinct from secondary powers such as the cubic X^{3/4} term.\n\nCandidate contribution (normalization lemma and synthesis; novelty confidence low): Candidate novelty: the all-prime square-diagonal lemma and the explicit p=2 ratio R_{k,2}=1/3+[(1-2^{-1/2})^{-k}+(1+2^{-1/2})^{-k}]/3 between the full fundamental-discriminant and chi_{8d} zero-swap constants under the same full-zeta extraction, together with the statement that this transfer changes neither the moment degree nor the swap threshold." }, { "id": 20000692, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0056", "title": "Fixed-form automorphic diagonals and the missing Type-I mechanism", "statement": "Develop a heuristic for moments of $L$-functions of automorphic forms.", "original_statement": "Develop a heuristic for moments of $L$-functions of automorphic forms.", "clean_statement": "Develop a heuristic for moments of $L$-functions of automorphic forms.", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Moments of zeta and correlations of divisor sums*, section “Other families of \\(L\\)-functions,” problem 5.2) says exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Other families of $L$-functions\nSource item: 5.2\nSource URL: http://aimpl.org/zetamoments/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[55]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop a heuristic for moments of $L$-functions of automorphic forms.\"\nOriginal remarks: [\"One of the problems here is that $\\\\sum a_m a_{m+h}=0$, so there are no Type I terms.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0056", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a fixed primitive holomorphic GL(2) newform f and every fixed k, the Dirichlet series of the squared coefficients of L(s,f)^k factors exactly as L(s,f x conjugate(f))^(k^2) times an Euler product absolutely convergent for Re(s)>1/2. This proves a k^2-order pole, an explicit Rankin-Selberg arithmetic constant, and a safe-length mean-square asymptotic for the associated Dirichlet polynomial. Comparing its two approximate-functional-equation diagonals with the unitary CFKRS constant shows that the sixth moment is the first forced mismatch: the combined non-diagonal strata must supply the normalized leading deficit 42 - 2(3/2)^9 = -8931/256. This is a rigorous diagonal theorem plus an audited heuristic obstruction, not a full high-moment solution.\n\nCandidate contribution (theorem_and_obstruction; novelty confidence low): Candidate fixed-form diagonal/deficit package: the exact pole-normalized Euler product and safe-length k-fold polynomial mean, together with the testable sixth-moment checksum -8931/256 for any proposed automorphic Type-II construction." }, { "id": 20000693, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0057", "title": "Multiplicity-sensitive moments of products of Dirichlet L-functions", "statement": "Develop a heuristic for moments of imprimitive $L$-functions, or if your $L$-function factors into degree $1$ factors, e.g.\n$$\n\\int |L(\\tfrac{1}{2}+it,\\chi_1)\\cdots L(\\tfrac{1}{2}+it,\\chi_k)|^2\\,dt\n$$\n(or different powers).", "original_statement": "Develop a heuristic for moments of imprimitive $L$-functions, or if your $L$-function factors into degree $1$ factors, e.g.\n$$\n\\int |L(\\tfrac{1}{2}+it,\\chi_1)\\cdots L(\\tfrac{1}{2}+it,\\chi_k)|^2\\,dt\n$$\n(or different powers).", "clean_statement": "Develop a heuristic for moments of imprimitive $L$-functions, or if your $L$-function factors into degree $1$ factors, e.g.\n$$\n\\int |L(\\tfrac{1}{2}+it,\\chi_1)\\cdots L(\\tfrac{1}{2}+it,\\chi_k)|^2\\,dt\n$$\n(or different powers).", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, workshop *Moments of zeta and correlations of divisor sums*, section “Other families of \\(L\\)-functions,” Problem 5.3. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Other families of $L$-functions\nSource item: 5.3\nSource URL: http://aimpl.org/zetamoments/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[56]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop a heuristic for moments of imprimitive $L$-functions, or if your $L$-function factors into degree $1$ factors, e.g.\\n$$\\n\\\\int |L(\\\\tfrac{1}{2}+it,\\\\chi_1)\\\\cdots L(\\\\tfrac{1}{2}+it,\\\\chi_k)|^2\\\\,dt\\n$$\\n(or different powers).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0057", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any fixed multiset of possibly imprimitive Dirichlet characters, the coefficient-square Dirichlet series of their product factors into one partial Dirichlet L-function for every ordered pair of primitive inducers, times finitely many bad-prime factors and an Euler product absolutely convergent for real part greater than 1/2. Its pole order at 1 is E = sum of the squared multiplicities of the primitive-character classes, its regularized value is positive, and Selberg--Delange gives the corresponding coefficient and harmonic-coefficient asymptotics. A factorwise CFKRS/approximate-functional-equation argument consequently predicts a global smoothed moment polynomial of degree E, but that global assertion remains conjectural in general.\n\nCandidate contribution (factorization theorem; novelty confidence low): Candidate novelty: an explicit ordered-pair factorization for arbitrary multisets of imprimitive Dirichlet characters, with equality tested at the primitive-inducer level, all bad primes isolated, an auxiliary Euler product absolutely convergent for real part greater than 1/2, a positive regularized constant, and a directed pairing-matrix extension for different integer powers." }, { "id": 20000694, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0058", "title": "Root-number conditioning in quadratic-twist moments", "statement": "Develop a heuristic for quadratic twists of $L$-functions", "original_statement": "Develop a heuristic for quadratic twists of $L$-functions", "clean_statement": "Develop a heuristic for quadratic twists of $L$-functions", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 5.4 from the workshop *Moments of zeta and correlations of divisor sums*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Other families of $L$-functions\nSource item: 5.4\nSource URL: http://aimpl.org/zetamoments/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[57]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop a heuristic for quadratic twists of $L$-functions\"\nOriginal remarks: [\"What role does the root number play?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0058", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For quadratic twists of a fixed primitive holomorphic newform of even weight, squarefree level N>1, and trivial nebentypus, restricted to odd fundamental discriminants of fixed sign and coprime to N, the exact sign law epsilon(f tensor chi_d)=epsilon(f)chi_d(-N) makes root-sign conditioning a single parity constraint on the local signs at primes dividing N. A squarefree-progression argument proves that all fixed good-prime 0,+1,-1 laws, and hence the square-diagonal rule, are unchanged with normalized O(X^{-1/2}) error. The complete change is an explicit finite parity-cube Fourier factor at primes dividing 2N. This supplies a root-aware correction to the CFKRS recipe, while central-value and derivative moment asymptotics at general order remain conjectural.\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: a quantitative parity-cube factorization theorem, including the dyadic coordinate, which expresses root-sign-conditioned finite local averages as the product of unchanged good-prime measures and the bad-prime correction E(G)+kappa E(G H_N), with normalized O(X^{-1/2}) error for fixed local data; for separable bad data it gives an explicit two-term Euler correction." }, { "id": 20000695, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0059", "title": "Exact Petersson--Kloosterman reduction of the fixed-weight symmetric-square second moment", "statement": "Find an exact formula for $\\sum_{f\\in H_k}^h L(\\frac{1}{2},Sym^2f)^2$.", "original_statement": "Find an exact formula for $\\sum_{f\\in H_k}^h L(\\frac{1}{2},Sym^2f)^2$.", "clean_statement": "Find an exact formula for $\\sum_{f\\in H_k}^h L(\\frac{1}{2},Sym^2f)^2$.", "statement_status": "exact", "statement_verification": "The canonical record (AIM workshop “Moments of zeta and correlations of divisor sums,” section “Other families of $L$-functions,” Problem 5.7) says exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Other families of $L$-functions\nSource item: 5.7\nSource URL: http://aimpl.org/zetamoments/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[58]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find an exact formula for $\\\\sum_{f\\\\in H_k}^h L(\\\\frac{1}{2},Sym^2f)^2$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0059", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the standard level-one harmonic normalization, the fixed-weight moment is exactly the sum of an explicit diagonal Mellin integral and an absolutely convergent Kloosterman--Bessel off-diagonal series. The diagonal collapses to the Mellin transform of zeta(1+2u)^3; for the kernel exp(u^2), it equals a fully displayed cubic residue P_k plus O(k^{-2}), with P_k=(log k)^3/3+O((log k)^2). This does not evaluate the off-diagonal and is therefore a rigorous reduction, not a full solution or a moment asymptotic.\n\nCandidate contribution (trace reduction; novelty confidence low): Candidate contribution: a normalization-complete exact fixed-single-weight AFE--Petersson formula whose full diagonal is a one-variable zeta(1+2u)^3 Mellin integral, together with an explicit polygamma/Stieltjes cubic residue and O(k^{-2}) contour remainder for G(u)=exp(u^2).", "research_difficulty_suggested": "L4" }, { "id": 20000696, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0060", "title": "A safe-range bridge between character and t-aspect orthogonality", "statement": "The mechanism for computing moments in families like $\\chi$ (mod $q$), $q\\leq Q$, involve different orthogonality relations than $|m-n|=$small. Yet the answers are the same. Why?", "original_statement": "The mechanism for computing moments in families like $\\chi$ (mod $q$), $q\\leq Q$, involve different orthogonality relations than $|m-n|=$small. Yet the answers are the same. Why?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The conservative reconstruction used here is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Other families of $L$-functions\nSource item: 5.5\nSource URL: http://aimpl.org/zetamoments/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[59]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The mechanism for computing moments in families like $\\\\chi$ (mod $q$), $q\\\\leq Q$, involve different orthogonality relations than $|m-n|=$small. Yet the answers are the same. Why?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0060", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every k and every finite Dirichlet polynomial of length X, the normalized 2k-th moment over all characters modulo a prime q is exactly the same product-diagonal form as a suitably band-limited t-average whenever q>X^k and T log(1+X^{-k})>2 pi. For primitive even characters, q>2X^k gives an explicit rank-one principal-character correction. At first aliasing, the t-aspect direct shift and character-family complementary divisor both have scale Y/C, and the common diagonal Euler product has pole order k^2. These proved statements explain the shared diagonal and shift geometry but do not evaluate the general long-range off-diagonal terms.\n\nCandidate contribution (transfer lemma; novelty confidence low): The candidate contribution is the exact all-k package joining the all-character and band-limited t-aspect projector identities, the primitive-even rank-one correction, and a quantitative first-aliasing scale and logarithmic-linearization bound." }, { "id": 20000697, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0061", "title": "Small moments from defect-stable mollified correlations", "statement": "Is there any way to get lower bounds for small moments (e.g. first moment) in families, even assuming GRH?", "original_statement": "Is there any way to get lower bounds for small moments (e.g. first moment) in families, even assuming GRH?", "clean_statement": "Is there any way to get lower bounds for small moments (e.g. first moment) in families, even assuming GRH?", "statement_status": "exact", "statement_verification": "The source record is Problem 5.6 in the section “Other families of $L$-functions” of the 2016 AIM workshop *Moments of zeta and correlations of divisor sums* (August 29--September 2, 2016). Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Other families of $L$-functions\nSource item: 5.6\nSource URL: http://aimpl.org/zetamoments/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[60]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there any way to get lower bounds for small moments (e.g. first moment) in families, even assuming GRH?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0061", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any finite weighted family, complex values Z, A, B, and 0 c the missing factor tends to exp(c^3). A compatible macroscopic rate function is also proved. This is a rigorous random-matrix benchmark and not a zeta-moment theorem.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: a direct real-k logarithmic error theorem for the exact CUE moment, together with the explicit three-regime crossover k=o(N^(1/3)), k~cN^(1/3), and N^(1/3)< 0 and 0 < a < b, the boundary-layer asymptotic with coefficient -pi(e^{2b lambda}-e^{2a lambda})/(2 lambda) is equivalent to E_{1-lambda/log T}(T^b)-E_{1-lambda/log T}(T^a)=o(log T); the coefficient tends to the rigorously known endpoint coefficient -pi(b-a) as lambda tends to zero. A companion smoothing lemma extracts the complete two-term fixed-sigma formula with O(T^{-1/2}) error from Lau's first-moment estimate for E_sigma." }, { "id": 20000708, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0072", "title": "Fixed rational twists of zeta moments and a prime-exponent response law", "statement": "Develop a heuristic for twisted moments, e.g.\n$$\n\\int_0^T \\left( \\frac{m}{n}\\right)^{it} |\\zeta(\\tfrac{1}{2}+it)|^{2k}\\,dt.\n$$", "original_statement": "Develop a heuristic for twisted moments, e.g.\n$$\n\\int_0^T \\left( \\frac{m}{n}\\right)^{it} |\\zeta(\\tfrac{1}{2}+it)|^{2k}\\,dt.\n$$", "clean_statement": "Develop a heuristic for twisted moments, e.g.\n$$\n\\int_0^T \\left( \\frac{m}{n}\\right)^{it} |\\zeta(\\tfrac{1}{2}+it)|^{2k}\\,dt.\n$$", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 8.2 in the “Miscellaneous” section of the workshop *Moments of zeta and correlations of divisor sums*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Miscellaneous\nSource item: 8.2\nSource URL: http://aimpl.org/zetamoments/8/\nCanonical location: aim-analytic-number-theory-notes.json notes[71]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop a heuristic for twisted moments, e.g.\\n$$\\n\\\\int_0^T \\\\left( \\\\frac{m}{n}\\\\right)^{it} |\\\\zeta(\\\\tfrac{1}{2}+it)|^{2k}\\\\,dt.\\n$$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: heuristic; problem status at run: solved; rigor: heuristic; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/8/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0072", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The Hughes--Young modification of the CFKRS recipe predicts that, for fixed integral moment order r and fixed reduced twist u/v, the leading twisted moment is the ordinary Keating--Snaith/CFKRS leading term multiplied by (uv)^(-1/2) times an explicit finite Euler product B_r(u,v). This agrees with the rigorous twisted second and fourth moments. The report additionally proves that each prime-power local modifier R_{r,1/p}(a) is exactly a degree-(r-1) polynomial in the exponent a with positive binomial-basis coefficients, an explicit leading coefficient, and the bounds 1=2 and a>=1.\n\nCandidate contribution (lemma; novelty confidence low): For every integer r>=1 and 00 and c_{r,0}(x)=1; its leading coefficient and strict divisor-function bounds are explicit. Consequently the conjectural prime-power twist response p^{-a/2}R_{r,1/p}(a) is a finite, directly testable expression." }, { "id": 20000709, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0073", "title": "High moments of a prime Dirichlet polynomial: collision corrections and a finite-height cutoff", "statement": "Develop a heuristic for\n$$\n\\int_T^{2T} \\left| \\sum_{p\\leq X} \\frac{1}{p^{1/2+it}}\\right|^{2k} \\,dt.\n$$\nFind connections with the work of Bogomolny-Keating and with the Ratios Conjectures.", "original_statement": "Develop a heuristic for\n$$\n\\int_T^{2T} \\left| \\sum_{p\\leq X} \\frac{1}{p^{1/2+it}}\\right|^{2k} \\,dt.\n$$\nFind connections with the work of Bogomolny-Keating and with the Ratios Conjectures.", "clean_statement": "Develop a heuristic for\n$$\n\\int_T^{2T} \\left| \\sum_{p\\leq X} \\frac{1}{p^{1/2+it}}\\right|^{2k} \\,dt.\n$$\nFind connections with the work of Bogomolny-Keating and with the Ratios Conjectures.", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Moments of zeta and correlations of divisor sums*, section “Miscellaneous,” problem 8.3) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Miscellaneous\nSource item: 8.3\nSource URL: http://aimpl.org/zetamoments/8/\nCanonical location: aim-analytic-number-theory-notes.json notes[72]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop a heuristic for\\n$$\\n\\\\int_T^{2T} \\\\left| \\\\sum_{p\\\\leq X} \\\\frac{1}{p^{1/2+it}}\\\\right|^{2k} \\\\,dt.\\n$$\\nFind connections with the work of Bogomolny-Keating and with the Ratios Conjectures.\"\nOriginal remarks: [\"A key issue here is to take $k$ large. Unless $X^{k} \\\\geq T$ we don't need a heuristic anyway, we can just apply mean value results for Dirichlet polynomials. But when I suggested the problem I had in mind very large $k$ (e.g. growing with $T$ at some rate), for which this would connect with questions about the maximum size of the zeta function.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/8/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0073", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The moment has an exact Fourier-kernel expansion whose unresolved part is a shifted correlation of the ordered k-prime-product coefficient. Its diagonal is exactly the corresponding Steinhaus prime-sum moment. For every X at least 2 and every positive integer k, a proved multinomial occupancy identity gives the diagonal as k! L^k times 1 - k(k-1)S_2/(4L^2) plus an explicit error bounded by k(k-1)(k-2)S_3/L^3 + 6 binom(k,4)S_2^2/L^4. A separate, explicitly labeled heuristic uses the Bessel cumulant rate function to predict the finite-height transition from the random-model moment to a maximum-dominated moment and gives a concrete generating-function bridge to Bogomolny-Keating shifted correlations and a generalized Ratios recipe.\n\nCandidate contribution (lemma; novelty confidence low): For the diagonal D_{k,X}, the exact identity D_{k,X}/(k!L^k) = E product_p (1/M_p!) with multinomial cell probabilities 1/(pL) yields the uniform first-collision expansion D_{k,X}=k!L^k(1-k(k-1)S_2/(4L^2)+epsilon), with |epsilon| bounded by k(k-1)(k-2)S_3/L^3 + 6 binom(k,4)S_2^2/L^4." }, { "id": 20000710, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0074", "title": "Bessel endpoint averages and a Chebyshev moment reduction", "statement": "Evaluate $$\\int_0^T f(|\\zeta(\\tfrac{1}{2}+it)|)\\,dt,$$ where $f=J_{2k}$, the Bessel function, or $f=T_{2k}$, the Chebyshev polynomial.", "original_statement": "Evaluate $$\\int_0^T f(|\\zeta(\\tfrac{1}{2}+it)|)\\,dt,$$ where $f=J_{2k}$, the Bessel function, or $f=T_{2k}$, the Chebyshev polynomial.", "clean_statement": "Evaluate $$\\int_0^T f(|\\zeta(\\tfrac{1}{2}+it)|)\\,dt,$$ where $f=J_{2k}$, the Bessel function, or $f=T_{2k}$, the Chebyshev polynomial.", "statement_status": "exact", "statement_verification": "The canonical AIM record (source file `aim-analytic-number-theory-notes.json`, zero-based index 73) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Miscellaneous\nSource item: 8.5\nSource URL: http://aimpl.org/zetamoments/8/\nCanonical location: aim-analytic-number-theory-notes.json notes[73]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Evaluate $$\\\\int_0^T f(|\\\\zeta(\\\\tfrac{1}{2}+it)|)\\\\,dt,$$ where $f=J_{2k}$, the Bessel function, or $f=T_{2k}$, the Chebyshev polynomial.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/8/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0074", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every fixed bounded continuous f on [0,infinity) having a limit at infinity, Selberg's central limit theorem implies that the average of f(|zeta(1/2+it)|) over [0,T] is ((f(0)+f(infinity))/2)T+o(T). Hence the standard Bessel transforms satisfy B_0(T)=T/2+o(T) and B_k(T)=o(T) for each fixed k>=1. For the standard first-kind Chebyshev polynomial, an exact coefficient formula expresses C_k(T) as an invertible triangular linear combination of the ordinary even zeta moments M_0(T),...,M_k(T); this gives unconditional formulae for k=1,2 and shows that k>=3 contains the open higher-moment problem. A sharper B_k(T)~T/(2k sqrt(pi log log T)) prediction is derived only under an explicitly stated local-limit hypothesis.\n\nCandidate contribution (theorem; novelty confidence low): Candidate folklore corollary: for every fixed bounded continuous f:[0,infinity)->C with a finite limit at infinity, the limiting normalized zeta-value transform is the equal endpoint mixture, T^{-1} integral_0^T f(|zeta(1/2+it)|) dt -> (f(0)+f(infinity))/2; in particular T^{-1}B_0(T)->1/2 and T^{-1}B_k(T)->0 for fixed k>=1." }, { "id": 20000711, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0075", "title": "Second moments on periodic comb sets", "statement": "Evaluate moments of zeta over sets other than $[0,T]$ or $[T,2T]$.", "original_statement": "Evaluate moments of zeta over sets other than $[0,T]$ or $[T,2T]$.", "clean_statement": "Evaluate moments of zeta over sets other than $[0,T]$ or $[T,2T]$.", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 8.4 in the “Miscellaneous” section of the workshop *Moments of zeta and correlations of divisor sums*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Miscellaneous\nSource item: 8.4\nSource URL: http://aimpl.org/zetamoments/8/\nCanonical location: aim-analytic-number-theory-notes.json notes[74]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Evaluate moments of zeta over sets other than $[0,T]$ or $[T,2T]$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/8/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0075", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a fixed duty cycle 0T^(1515/4816+delta), together with the exact sampled-remainder identity identifying endpoint cancellation as the obstruction below this scale." }, { "id": 20000712, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0076", "title": "A compensated-Laplace window reduction for fractional zeta moments", "statement": "Evaluate fractional moments of zeta.", "original_statement": "Evaluate fractional moments of zeta.", "clean_statement": "Evaluate fractional moments of zeta.", "statement_status": "exact", "statement_verification": "The canonical AIM record says, exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Moments of zeta and correlations of divisor sums\nSection: Miscellaneous\nSource item: 8.7\nSource URL: http://aimpl.org/zetamoments/8/\nCanonical location: aim-analytic-number-theory-notes.json notes[75]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Evaluate fractional moments of zeta.\"\nOriginal remarks: [\"There are conjectures on lower order terms, but there is no analogue yet of Conrey-Keating\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/zetamoments/8/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0076", "aim-domain:analytic-number-theory", "aim-workshop:zetamoments", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every fixed nonintegral k with 0-1/2 and show from the exact k=1 zeta moment that a naive CUE-only first lower-order term misses the arithmetic correction 2 gamma - 1.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: for fixed nonintegral 00 in the standard Hurwitz normalization, the fractional part of H(m) is 1/3 exactly when m=3s^2, is 1/2 exactly when m=4s^2, and is 0 otherwise; these two cases are precisely the residual rigidified CM automorphism strata j=0 and j=1728." }, { "id": 20000723, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0087", "title": "Coupling-weight criteria and formal obstructions for an all-genus series", "statement": "Does $F= \\sum f_g(\\tau)\\lambda^{2g-2}$ have transformation properties with respect to $\\lambda$?", "original_statement": "Does $F= \\sum f_g(\\tau)\\lambda^{2g-2}$ have transformation properties with respect to $\\lambda$?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is problem 4.3, in the “Specific functions” section of the 2013 AIM workshop *Gromov-Witten invariants and number theory*. Its entire mathematical text is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Gromov-Witten invariants and number theory\nSection: Specific functions\nSource item: 4.3\nSource URL: http://aimpl.org/gromwitnumthry/4/\nCanonical location: aim-analytic-number-theory-notes.json notes[86]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does $F= \\\\sum f_g(\\\\tau)\\\\lambda^{2g-2}$ have transformation properties with respect to $\\\\lambda$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/4/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0087", "aim-domain:analytic-number-theory", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the formal all-genus series F(tau,lambda)=sum_g f_g(tau)lambda^(2g-2), a scalar coupled modular law F(gamma tau,lambda/(c tau+d)^r)=chi(gamma)(c tau+d)^s F(tau,lambda) holds if and only if every active coefficient has the common character chi and weight s+r(2g-2). Constant lambda-scalings are classified by the gcd of the active exponents, while literal inversion lambda -> A/lambda forces f_g=0 for g>=3 and f_0=A^2 f_2. Under the plausible elliptic-cover hypothesis f_g in quasimodular weight 6g-6 for g>=2, lambda has modular weight -3: the holomorphic tail transforms by an explicit E_2-translation, and its canonical almost-holomorphic completion is invariant.\n\nCandidate contribution (reduction; novelty confidence low): A single coefficientwise diagnostic separates scalar coupled modularity, quasimodular anomaly, and literal coupling inversion: affine coefficient weights are necessary and sufficient for a coupled lambda-rescaling; in the weight-6g-6 elliptic-cover case the exact anomaly is exp((6c/(pi i(c tau+d))) partial_{E_2}) with lambda scaled by (c tau+d)^(-3); and any nonzero genus-g coefficient with g>=3 obstructs perturbative lambda inversion." }, { "id": 20000724, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0088", "title": "Mirror-quintic periods and a base-change-stable obstruction to classical modularity", "statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?", "original_statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?", "clean_statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Gromov--Witten invariants and number theory*, section “Specific functions,” problem 4.4) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Gromov-Witten invariants and number theory\nSection: Specific functions\nSource item: 4.4\nSource URL: http://aimpl.org/gromwitnumthry/4/\nCanonical location: aim-analytic-number-theory-notes.json notes[87]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is $\\\\sum \\\\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\\nDo other solutions to the Picard-Fuchs equation have modular properties?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/4/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0088", "aim-domain:analytic-number-theory", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After the exact rescaling x=5^5 z, the AIM series is the fundamental mirror-quintic period for the hypergeometric operator theta^4-x(theta+1/5)(theta+2/5)(theta+3/5)(theta+4/5). Its four solutions form a symplectic, monodromy-equivariant period vector and yield a canonical real-analytic Hodge-norm invariant; Movasati's work supplies a seven-generator modular-type differential algebra. However, the conifold monodromy at x=1 is a nontrivial rank-one unipotent transvection. A local Jordan-rank argument proves that, after every finite algebraic base change and every rank-one twist, the rank-four local system is not the third symmetric power of a rank-two local system. This excludes the standard scalar weight-three modular-form/Hauptmodul mechanism without excluding broader vector-valued, thin-group, or period-domain automorphy.\n\nCandidate contribution (obstruction_lemma; novelty confidence low): For every connected finite algebraic cover p:U'->U of the mirror-quintic base, every rank-two complex local system W on U', and every rank-one local system chi on U', the pullback p*V is not isomorphic to chi tensor Sym^3(W); locally, every power of the conifold transvection differs from the identity by rank one, whereas a nontrivial twisted symmetric cube with unipotent spectrum differs from the identity by rank three." }, { "id": 20000725, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0089", "title": "Shadow-controlled functional-equation defects for mixed mock coefficient series", "statement": "Do the L-series of mixed mock modular forms have interesting properties?", "original_statement": "Do the L-series of mixed mock modular forms have interesting properties?", "clean_statement": "Do the L-series of mixed mock modular forms have interesting properties?", "statement_status": "exact", "statement_verification": "The exact canonical record is problem 5.1 in the section “Other problems” of the AIM workshop *Gromov–Witten invariants and number theory*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Gromov-Witten invariants and number theory\nSection: Other problems\nSource item: 5.1\nSource URL: http://aimpl.org/gromwitnumthry/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[88]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Do the L-series of mixed mock modular forms have interesting properties?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0089", "aim-domain:analytic-number-theory", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a depth-one product-type mixed mock modular form F^+=g h^+ of integral total weight whose real-analytic completion has a scalar Fricke law, the compactly supported Fourier-coefficient L-functional satisfies an exact inhomogeneous functional equation: its defect is precisely the anti-invariant Mellin/Laplace functional of the nonholomorphic shadow correction P=g h^-. Fricke-symmetric test kernels give an unregularized central-cancellation identity, and, whenever ordinary Mellin transforms are justified, every nonzero shadow mode contributes an explicit Gauss hypergeometric kernel. This gives a rigorous reduction of the raw coefficient Dirichlet-series question to a concrete shadow double series.\n\nCandidate contribution (explicit_reduction; novelty confidence low): Candidate novelty: the exact compact-support Fricke-defect identity, its symmetric-kernel cancellation law, and the mode formula K_kappa(s;ell,m)=Gamma(s+1-kappa)/(s(4 pi m)^s) times 2F1(s+1-kappa,s;s+1;(m-ell)/(2m)) package the raw mixed-mock coefficient functional's failure of a homogeneous functional equation entirely in terms of shadow coefficients." }, { "id": 20000726, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0090", "title": "A rational j-equation for a mixed mock Kaneko-Zagier solution", "statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?", "original_statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?", "clean_statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?", "statement_status": "exact", "statement_verification": "The canonical record is problem 5.2 in the AIM workshop list *Gromov--Witten invariants and number theory*, section “Other problems”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Gromov-Witten invariants and number theory\nSection: Other problems\nSource item: 5.2\nSource URL: http://aimpl.org/gromwitnumthry/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[89]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Do mixed mock modular forms satisfy differential equations with respect to a modular function?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0090", "aim-domain:analytic-number-theory", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The broad question is partially solved and interpretation-dependent. For Guerzhoy's weight-four mixed mock solution F_4=E_4 h_4, the weight-zero factor satisfies (dh_4/dj)^6=1/(j^8(j-1728)^3) and the linear equation h_4''+(4/(3j)+1/(2(j-1728)))h_4'=0 over C(j). A proved vector-valued monodromy descent theorem also shows that the raw F_4 satisfies some linear ODE over C(j) of order at most 10, and gives rational-coefficient ODEs for the finite-dimensional Mertens-Raum class under explicit weight or normalization hypotheses.\n\nCandidate contribution (proposition; novelty confidence low): For Guerzhoy's normalized function theta h_4=eta^20/E_4^2, one has (dh_4/dj)^6=1/(j^8(j-1728)^3) and hence h_4''+(4/(3j)+1/(2(j-1728)))h_4'=0; moreover the raw mixed mock form E_4 h_4 has a C(j)-linear ODE of order at most 10." }, { "id": 20000727, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0091", "title": "Mock modularity in geometric counting: polar data, wall crossing, and holomorphic ambiguity", "statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.", "original_statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.", "clean_statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.", "statement_status": "exact", "statement_verification": "The canonical record is problem 5.3 in the AIM workshop list *Gromov--Witten invariants and number theory*, section “Other problems.” Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Analytic number theory\nWorkshop: Gromov-Witten invariants and number theory\nSection: Other problems\nSource item: 5.3\nSource URL: http://aimpl.org/gromwitnumthry/5/\nCanonical location: aim-analytic-number-theory-notes.json notes[90]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/5/", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0091", "aim-domain:analytic-number-theory", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The broad AIM problem is partially solved in explicit Donaldson, Vafa-Witten/sheaf, logarithmic Gromov-Witten, DT/BPS, and black-hole families, but there is no universal correspondence and generic OSV remains conjectural. A proved comparison theorem shows that two meromorphic Jacobi partition functions with the same Jacobi data and full torsion-point principal parts have finite parts with identical nonholomorphic anomalies; their difference is precisely a weakly holomorphic Jacobi form. An exact Gaussian wall-flux identity gives the parallel anomaly test for a two-cone indefinite theta series. Thus agreement of shadows is a necessary but insufficient test for equality of geometric invariants.\n\nCandidate contribution (reduction_and_obstruction; novelty confidence low): Candidate principal-part shadow-transfer criterion: after two geometric or BPS partition functions are independently proved to be meromorphic Jacobi forms with the same weight, index, multiplier, and full polar data, their canonical finite parts have the same complete nonholomorphic anomaly, and all remaining coefficient-level uncertainty lies in a bounded-cusp weakly holomorphic Jacobi-form space; matching shadows alone therefore cannot identify invariants." }, { "id": 20000728, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0092", "title": "Weight-only modularity recognition has a finite-data obstruction", "statement": "Problem 1.1. Given a q-series, determine fast methods to find (heuristically) if it is modular.\n\nThe method should require knowledge of the weight, but not require knowledge of the level or group.", "original_statement": "Problem 1.1. Given a q-series, determine fast methods to find (heuristically) if it is modular. \n\nThe method should require knowledge of the weight, but not require knowledge of the level or group.", "clean_statement": "Problem 1.1. Given a q-series, determine fast methods to find (heuristically) if it is modular.\n\nThe method should require knowledge of the weight, but not require knowledge of the level or group.", "statement_status": "exact", "statement_verification": "The source is the two-page AIM workshop problem list *Mock Modular Forms*, edited by Sharon Anne Garthwaite after the workshop “Mock modular forms in combinatorics and arithmetic geometry,” March 8–12, 2010. The PDF was inspected directly. Its text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Mock modular forms in combinatorics and arithmetic geometry\nSection: \nSource item: 1.1\nSource URL: https://aimath.org/WWN/mockmodular/mockmodular.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[91]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.1. Given a q-series, determine fast methods to find (heuristically) if it is modular. \\n\\nThe method should require knowledge of the weight, but not require knowledge of the level or group.\"\nOriginal remarks: [\"Remark. Zagier has a method involving asymptotics at a point.\", \"Remark. It is worth exploring a p-adic method. For example, more coefficients should be divisible by primes than expected by chance.\", \"Remark. One might consider the distribution of ap and compare to Sato-Tate.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/mockmodular/mockmodular.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0092", "aim-domain:analytic-number-theory", "aim-workshop:mockmodular", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In the scalar holomorphic congruence setting, for every even weight k at least 2 and every finite complex coefficient prefix through q^M, an explicit triangular oldform construction gives a modular continuation at level N_k lcm(1,...,M), while adding a later monomial gives a provably nonmodular continuation with the same prefix. Thus weight plus finitely many coefficients cannot decide modularity at unrestricted level. For normalized eigenform candidates, local Hecke defects nevertheless force explicit divisors and lower bounds for any possible level, and multiplicativity tests on disjoint pairs of distinct primes admit an exact binomial false-positive calibration under a stated independent-uniform residue null.\n\nCandidate contribution (obstruction theorem; novelty confidence low): Candidate novelty: every finite complex q-expansion prefix in every even integral weight k at least 2 has both (i) an explicitly constructed holomorphic modular continuation of level N_k lcm(1,...,M), with N_2=11, N_4=5, and N_k=11 for k at least 6, and (ii) an explicit nonmodular continuation obtained by adding q^R for any R greater than M." }, { "id": 20000729, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0093", "title": "A finite-theta certificate for sum-side modularity", "statement": "Problem 1.2. Find a method to prove modularity directly from the sum ( q-hypergeometric) expansion.\n\nFor motivation, consider the Rogers-Ramanujan identities\n\nG(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2\n\n(q; q)n\n\n= 1\n\n(q; q5)∞(q4; q5)∞,\n\nH(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2+n\n\n(q; q)n\n\n= 1\n\n(q2; q5)∞(q3; q5)∞;\n\nit is apparent from the right hand side that q−1/60 G(q) and q11 /60 H(q) are modular. How can we see this from the left hand side?", "original_statement": "Problem 1.2. Find a method to prove modularity directly from the sum ( q-hypergeometric) expansion. \n\nFor motivation, consider the Rogers-Ramanujan identities \n\nG(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2\n\n(q; q)n\n\n= 1\n\n(q; q5)∞(q4; q5)∞,\n\nH(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2+n\n\n(q; q)n\n\n= 1\n\n(q2; q5)∞(q3; q5)∞;\n\nit is apparent from the right hand side that q−1/60 G(q) and q11 /60 H(q) are modular. How can we see this from the left hand side?", "clean_statement": "Problem 1.2. Find a method to prove modularity directly from the sum ( q-hypergeometric) expansion.\n\nFor motivation, consider the Rogers-Ramanujan identities\n\nG(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2\n\n(q; q)n\n\n= 1\n\n(q; q5)∞(q4; q5)∞,\n\nH(q) =\n\n> ∞\n\n∑\n\n> n=0\n\nqn2+n\n\n(q; q)n\n\n= 1\n\n(q2; q5)∞(q3; q5)∞;\n\nit is apparent from the right hand side that q−1/60 G(q) and q11 /60 H(q) are modular. How can we see this from the left hand side?", "statement_status": "exact", "statement_verification": "The source is the AIM workshop problem list *Mock modular forms in combinatorics and arithmetic geometry* (March 8--12, 2010), Problem 1.2. With", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Mock modular forms in combinatorics and arithmetic geometry\nSection: \nSource item: 1.2\nSource URL: https://aimath.org/WWN/mockmodular/mockmodular.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[92]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.2. Find a method to prove modularity directly from the sum ( q-hypergeometric) expansion. \\n\\nFor motivation, consider the Rogers-Ramanujan identities \\n\\nG(q) =\\n\\n> ∞\\n\\n∑\\n\\n> n=0\\n\\nqn2\\n\\n(q; q)n\\n\\n= 1\\n\\n(q; q5)∞(q4; q5)∞,\\n\\nH(q) =\\n\\n> ∞\\n\\n∑\\n\\n> n=0\\n\\nqn2+n\\n\\n(q; q)n\\n\\n= 1\\n\\n(q2; q5)∞(q3; q5)∞;\\n\\nit is apparent from the right hand side that q−1/60 G(q) and q11 /60 H(q) are modular. How can we see this from the left hand side?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/mockmodular/mockmodular.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0093", "aim-domain:analytic-number-theory", "aim-workshop:mockmodular", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Bressoud's terminating Gaussian-polynomial identities give a product-free route from the defining unilateral Rogers-Ramanujan sums to eta-normalized unary theta differences. A compact-open convergence proof and Poisson summation then yield the exact two-dimensional S- and T-transformation matrices for (q^{-1/60}G, q^{11/60}H). More generally, closure of the limiting theta residue coefficient space under the finite quadratic-phase and Fourier matrices is proved to be a sufficient, mechanically checkable certificate of vector-valued modularity.\n\nCandidate contribution (criterion; novelty confidence low): Candidate novelty: recurrence-certified finitization plus compact-open convergence to an eta-denominated unary-theta vector, followed by finite Weil-matrix closure of its residue coefficient space, is an explicit sufficient sum-side modularity certificate; for the Rogers-Ramanujan pair it gives the full S and T matrices without either Rogers-Ramanujan product or Jacobi's triple product." }, { "id": 20000730, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0094", "title": "Torsion continuous q-Hermite expansions as Siegel modular units", "statement": "Problem 1.3. Is there any kind of relations between the modularity of a series and the expansion in terms of q-orthogonal polynomials?", "original_statement": "Problem 1.3. Is there any kind of relations between the modularity of a series and the expansion in terms of q-orthogonal polynomials?", "clean_statement": "Problem 1.3. Is there any kind of relations between the modularity of a series and the expansion in terms of q-orthogonal polynomials?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.3 from the AIM workshop *Mock modular forms in combinatorics and arithmetic geometry* (March 8--12, 2010):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Mock modular forms in combinatorics and arithmetic geometry\nSection: \nSource item: 1.3\nSource URL: https://aimath.org/WWN/mockmodular/mockmodular.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[93]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.3. Is there any kind of relations between the modularity of a series and the expansion in terms of q-orthogonal polynomials?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/mockmodular/mockmodular.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0094", "aim-domain:analytic-number-theory", "aim-workshop:mockmodular", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every rational beta, the continuous q-Hermite generating expansion sum over n of H_n(cos(2 pi beta) | q) q^(n/2)/(q;q)_n, after multiplication by q^(1/24), is an explicit constant times the inverse Siegel function g_(1/2,beta). If M is the denominator of the torsion vector (1/2,beta), its 12M-th power is therefore a modular unit on X(M), and its logarithmic derivative is a proved weight-two modular form with explicit twisted odd-divisor Fourier coefficients. The report also proves that the bare existence of an orthogonal-polynomial expansion cannot carry modularity information without additional arithmetic structure.\n\nCandidate contribution (proved_special_case_and_explicit_modularity_certificate; novelty confidence low): Candidate novelty: q^(1/24) times the torsion-specialized continuous q-Hermite generating series equals -exp(-pi i beta/2) g_(1/2,beta)^(-1), and its logarithmic derivative has coefficient at q^(m/2) equal to one half the sum over odd d dividing m of d times (zeta^(m/d)+zeta^(-m/d))." }, { "id": 20000731, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0095", "title": "A forced mixing invariant for the Rogers-Ramanujan modular matrices", "statement": "Problem 1.4. For vector-valued modular forms, do the matrices involved have interesting digi-talization?\n\nFor example, in the Rogers-Ramanujan example, the transformation z 7 → z+1 leads to a diagonal matrix, where as z 7 → − 1/z does not. Is the diagonalization of this interesting?\n\n> 12EDITED BY SHARON ANNE GARTHWAITE\n\n2. B LOCH GROUP METHOD PROBLEMS\n\nWork by Werner Nahm gives us a way to attack modularity questions with Algebraic K-theory. Let B(F) denote the Bloch group for a field F. For more about the Bloch group, see Nahm's article\n\nConformal Field Theory and Torsion Elements of the Bloch Group, which is available on the arXiv. Let A ∈ Mr×r(Q) be a positive definite symmetric matrix. Let B ∈ Qr and C ∈ Q; define\n\nfA,B,C (τ): =\n\n∑\n\n> n!,..., nr≥0\n\nq1\n\n> 2\n> −→n A −→nt+B−→n+C\n\n(q)n1 · · · (q)nr,\n\nwhere q:= e2πiτ and (q)m:= (1 − q)(1 − q2) · · · (1 − qm), taking (q)0:= 1.In the case r = 1, Zagier proved that this function is only modular for seven choices of A, B, C,such as (2, 0, −1/60) and (2, 1, 11 /60); the proof involves looking at the behavior of the function as q tends to one. Nahm conjectures that for general r, there is an A for which fA,B,C (τ) is modular if and only if the image of A in the Bloch group is torsion. If such an A exists, there may be multiple choices of\n\nB and C for which fA,B,C (τ) is indeed modular. Motivation for this idea comes from the study of the dilogarithm function and other related functions. For more on this, see Don Zagier's chapter ' The Dilogarithm Function in \"Frontiers in number theory, physics, and geometry II.\" The problems throughout this section assume the notation above.", "original_statement": "Problem 1.4. For vector-valued modular forms, do the matrices involved have interesting digi-talization? \n\nFor example, in the Rogers-Ramanujan example, the transformation z 7 → z+1 leads to a diagonal matrix, where as z 7 → − 1/z does not. Is the diagonalization of this interesting? \n\n> 12EDITED BY SHARON ANNE GARTHWAITE\n\n2. B LOCH GROUP METHOD PROBLEMS \n\nWork by Werner Nahm gives us a way to attack modularity questions with Algebraic K-theory. Let B(F) denote the Bloch group for a field F. For more about the Bloch group, see Nahm's article \n\nConformal Field Theory and Torsion Elements of the Bloch Group, which is available on the arXiv. Let A ∈ Mr×r(Q) be a positive definite symmetric matrix. Let B ∈ Qr and C ∈ Q; define \n\nfA,B,C (τ): =\n\n∑ \n\n> n!,..., nr≥0\n\nq1 \n\n> 2\n> −→n A −→nt+B−→n+C\n\n(q)n1 · · · (q)nr,\n\nwhere q:= e2πiτ and (q)m:= (1 − q)(1 − q2) · · · (1 − qm), taking (q)0:= 1.In the case r = 1, Zagier proved that this function is only modular for seven choices of A, B, C,such as (2, 0, −1/60) and (2, 1, 11 /60); the proof involves looking at the behavior of the function as q tends to one. Nahm conjectures that for general r, there is an A for which fA,B,C (τ) is modular if and only if the image of A in the Bloch group is torsion. If such an A exists, there may be multiple choices of \n\nB and C for which fA,B,C (τ) is indeed modular. Motivation for this idea comes from the study of the dilogarithm function and other related functions. For more on this, see Don Zagier's chapter ' The Dilogarithm Function in \"Frontiers in number theory, physics, and geometry II.\" The problems throughout this section assume the notation above.", "clean_statement": "Problem 1.4. For vector-valued modular forms, do the matrices involved have interesting digi-talization?\n\nFor example, in the Rogers-Ramanujan example, the transformation z 7 → z+1 leads to a diagonal matrix, where as z 7 → − 1/z does not. Is the diagonalization of this interesting?\n\n> 12EDITED BY SHARON ANNE GARTHWAITE\n\n2. B LOCH GROUP METHOD PROBLEMS\n\nWork by Werner Nahm gives us a way to attack modularity questions with Algebraic K-theory. Let B(F) denote the Bloch group for a field F. For more about the Bloch group, see Nahm's article\n\nConformal Field Theory and Torsion Elements of the Bloch Group, which is available on the arXiv. Let A ∈ Mr×r(Q) be a positive definite symmetric matrix. Let B ∈ Qr and C ∈ Q; define\n\nfA,B,C (τ): =\n\n∑\n\n> n!,..., nr≥0\n\nq1\n\n> 2\n> −→n A −→nt+B−→n+C\n\n(q)n1 · · · (q)nr,\n\nwhere q:= e2πiτ and (q)m:= (1 − q)(1 − q2) · · · (1 − qm), taking (q)0:= 1.In the case r = 1, Zagier proved that this function is only modular for seven choices of A, B, C,such as (2, 0, −1/60) and (2, 1, 11 /60); the proof involves looking at the behavior of the function as q tends to one. Nahm conjectures that for general r, there is an A for which fA,B,C (τ) is modular if and only if the image of A in the Bloch group is torsion. If such an A exists, there may be multiple choices of\n\nB and C for which fA,B,C (τ) is indeed modular. Motivation for this idea comes from the study of the dilogarithm function and other related functions. For more on this, see Don Zagier's chapter ' The Dilogarithm Function in \"Frontiers in number theory, physics, and geometry II.\" The problems throughout this section assume the notation above.", "statement_status": "exact", "statement_verification": "The source is the American Institute of Mathematics problem list from the March 8--12, 2010 workshop *Mock modular forms in combinatorics and arithmetic geometry*, Problem 1.4 on PDF page 1. The first sentence in the PDF is literally:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Mock modular forms in combinatorics and arithmetic geometry\nSection: \nSource item: 1.4\nSource URL: https://aimath.org/WWN/mockmodular/mockmodular.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[94]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.4. For vector-valued modular forms, do the matrices involved have interesting digi-talization? \\n\\nFor example, in the Rogers-Ramanujan example, the transformation z 7 → z+1 leads to a diagonal matrix, where as z 7 → − 1/z does not. Is the diagonalization of this interesting? \\n\\n> 12EDITED BY SHARON ANNE GARTHWAITE\\n\\n2. B LOCH GROUP METHOD PROBLEMS \\n\\nWork by Werner Nahm gives us a way to attack modularity questions with Algebraic K-theory. Let B(F) denote the Bloch group for a field F. For more about the Bloch group, see Nahm's article \\n\\nConformal Field Theory and Torsion Elements of the Bloch Group, which is available on the arXiv. Let A ∈ Mr×r(Q) be a positive definite symmetric matrix. Let B ∈ Qr and C ∈ Q; define \\n\\nfA,B,C (τ): =\\n\\n∑ \\n\\n> n!,..., nr≥0\\n\\nq1 \\n\\n> 2\\n> −→n A −→nt+B−→n+C\\n\\n(q)n1 · · · (q)nr,\\n\\nwhere q:= e2πiτ and (q)m:= (1 − q)(1 − q2) · · · (1 − qm), taking (q)0:= 1.In the case r = 1, Zagier proved that this function is only modular for seven choices of A, B, C,such as (2, 0, −1/60) and (2, 1, 11 /60); the proof involves looking at the behavior of the function as q tends to one. Nahm conjectures that for general r, there is an A for which fA,B,C (τ) is modular if and only if the image of A in the Bloch group is torsion. If such an A exists, there may be multiple choices of \\n\\nB and C for which fA,B,C (τ) is indeed modular. Motivation for this idea comes from the study of the dilogarithm function and other related functions. For more on this, see Don Zagier's chapter ' The Dilogarithm Function in \\\"Frontiers in number theory, physics, and geometry II.\\\" The problems throughout this section assume the notation above.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/mockmodular/mockmodular.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0095", "aim-domain:analytic-number-theory", "aim-workshop:mockmodular", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every irreducible two-dimensional weight-zero representation of PSL_2(Z) with T = diag(a,b) and a != b, the modular relations force S_12 S_21 = (a^2-ab+b^2)/(a-b)^2 and the conjugacy-invariant identity det(ST-TS) = a^2-ab+b^2. For the Rogers-Ramanujan exponents this gives S_12 S_21 = (5-sqrt(5))/10, forces non-simultaneous diagonalizability, and in the symmetric normalization yields the golden-ratio S-matrix that diagonalizes the Lee-Yang fusion matrix.\n\nCandidate contribution (basis-invariant lemma; novelty confidence low): The explicit formula det(ST-TS) = a^2-ab+b^2, together with S_12 S_21 = (a^2-ab+b^2)/(a-b)^2 in a simple T-eigenbasis, is a testable intrinsic mixing certificate determined solely by the two T-eigenvalues; applying it to Rogers-Ramanujan recovers the exact golden-ratio mixing without evaluating a hypergeometric connection formula." }, { "id": 20000732, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0096", "title": "Finite asymptotic certificates and a cusp-width span bound for fixed-matrix Nahm sums", "statement": "Problem 2.1. Given a particular A, can we bound the number of possible B? Similarly, given a particular A, can we bound the dimension of the vector space determined by the B?\n\nThe bounds may or may not be effective.", "original_statement": "Problem 2.1. Given a particular A, can we bound the number of possible B? Similarly, given a particular A, can we bound the dimension of the vector space determined by the B?\n\nThe bounds may or may not be effective.", "clean_statement": "Problem 2.1. Given a particular A, can we bound the number of possible B? Similarly, given a particular A, can we bound the dimension of the vector space determined by the B?\n\nThe bounds may or may not be effective.", "statement_status": "exact", "statement_verification": "The exact canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Mock modular forms in combinatorics and arithmetic geometry\nSection: \nSource item: 2.1\nSource URL: https://aimath.org/WWN/mockmodular/mockmodular.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[95]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.1. Given a particular A, can we bound the number of possible B? Similarly, given a particular A, can we bound the dimension of the vector space determined by the B?\\n\\nThe bounds may or may not be effective.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/mockmodular/mockmodular.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0096", "aim-domain:analytic-number-theory", "aim-workshop:mockmodular", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For fixed rational positive-definite A, scalar modularity forces B to satisfy the Vlasenko-Zwegers polynomial equations P_p^A(B)=0, with degree at most 2p and C uniquely determined. Any zero-dimensional finite truncation gives an effective quotient-length bound on the number of modular B, and Noetherianity shows that finiteness of the full obstruction locus always has such a finite certificate. Separately, for Nahm sums with fixed A that are scalar weight-zero modular functions with trivial multiplier on one common finite-index group Gamma, the Garoufalidis-Zagier cusp valuation bound and Riemann-Roch give an explicit span-dimension bound in terms of A and the cusp widths of Gamma.\n\nCandidate contribution (reduction; novelty confidence low): Candidate synthesis: the fixed-A modular-B problem admits a finite exact-algebraic zero-dimensionality certificate whose quotient length bounds the number of B, while any prescribed common scalar modular group admits the explicit bound dim(span) <= 1 + sum over cusps a of max(0, floor(-h_a C_0(A))).", "research_difficulty_suggested": "L4" }, { "id": 20000733, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0097", "title": "Radial Laplace obstruction and an exact C-law for Stokes jumps", "statement": "Problem 2.2. Consider the Laplace transform of the q-hypergeometric series above. Note that the Mellin transform will not converge. Look at the properties, in particular the \"jumps;\" what can we say about the dependence on A, B, and C?", "original_statement": "Problem 2.2. Consider the Laplace transform of the q-hypergeometric series above. Note that the Mellin transform will not converge. Look at the properties, in particular the \"jumps;\" what can we say about the dependence on A, B, and C?", "clean_statement": "Problem 2.2. Consider the Laplace transform of the q-hypergeometric series above. Note that the Mellin transform will not converge. Look at the properties, in particular the \"jumps;\" what can we say about the dependence on A, B, and C?", "statement_status": "exact", "statement_verification": "The exact AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Mock modular forms in combinatorics and arithmetic geometry\nSection: \nSource item: 2.2\nSource URL: https://aimath.org/WWN/mockmodular/mockmodular.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[96]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.2. Consider the Laplace transform of the q-hypergeometric series above. Note that the Mellin transform will not converge. Look at the properties, in particular the \\\"jumps;\\\" what can we say about the dependence on A, B, and C?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/mockmodular/mockmodular.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0097", "aim-domain:analytic-number-theory", "aim-workshop:mockmodular", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the explicitly defined radial function R(t)=F_{A,B,C}(e^{-t}), the Garoufalidis-Zagier cusp asymptotic is R(t)=K(A,B) exp(Lambda(A)/t)(1+O(t)) with Lambda(A)>0; hence the ordinary radial Laplace and Mellin transforms both fail local integrability at t=0 for every positive-definite Nahm datum. The dominant action depends only on A and the leading amplitude is the explicit A,B-dependent factor K, independent of C. Under the separately stated Borel-Laplace interpretation, the normalized all-orders series obeys Phi_{A,B,C}(t)=exp(-Ct)Phi_{A,B,0}(t), so whenever lateral sums exist every Stokes jump is exactly exp(-Ct) times the C=0 jump and no nonzero jump direction can depend on C.\n\nCandidate contribution (parameter-dependence proposition; novelty confidence low): Candidate novelty: separating the transform conventions yields a universal obstruction to the literal radial Laplace transform and the exact testable Stokes law Disc_theta(A,B,C)=exp(-Ct) Disc_theta(A,B,0), while Lambda is A-only and K(A,B+Delta B)/K(A,B)=product_i z_i(A)^{Delta B_i}." }, { "id": 20000734, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0098", "title": "Coupled rank-two Nahm sums as modular units and partition series", "statement": "Problem 2.3. Look at explicit examples where r ≥ 2 and fA,B,C (τ) is modular. Can we write these in terms of natural objects? Similarly, can we related them to combinatorial identities; in the case r = 1, the two triples given relate to the Rogers-Ramanujan identities.", "original_statement": "Problem 2.3. Look at explicit examples where r ≥ 2 and fA,B,C (τ) is modular. Can we write these in terms of natural objects? Similarly, can we related them to combinatorial identities; in the case r = 1, the two triples given relate to the Rogers-Ramanujan identities.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Problem 2.3 in the two-page problem list from the AIM workshop *Mock modular forms in combinatorics and arithmetic geometry* (March 8--12, 2010). The record reads, exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Mock modular forms in combinatorics and arithmetic geometry\nSection: \nSource item: 2.3\nSource URL: https://aimath.org/WWN/mockmodular/mockmodular.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[97]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.3. Look at explicit examples where r ≥ 2 and fA,B,C (τ) is modular. Can we write these in terms of natural objects? Similarly, can we related them to combinatorial identities; in the case r = 1, the two triples given relate to the Rogers-Ramanujan identities.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/mockmodular/mockmodular.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0098", "aim-domain:analytic-number-theory", "aim-workshop:mockmodular", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the genuinely coupled matrix A=((4,1),(1,1)), the two known triples B=(0,1/2), C=1/120 and B=(2,1/2), C=49/120 reduce respectively to (-q;q)_infinity G(q^2) and (-q;q)_infinity H(q^2), hence are eta/Siegel modular-unit products and generate partitions avoiding residues {0,4,6} and {0,2,8} modulo 10. In addition to recovering these published identities, this attempt proves an explicit weight- and charge-preserving bijection from each coupled double-sum model to a pair consisting of a distinct partition and an even gap-at-least-four partition, with minimum part 2 or 4 respectively.\n\nCandidate contribution (bijection; novelty confidence low): Candidate novelty: for each epsilon in {0,1}, the explicitly defined map Phi_epsilon is a weight-preserving bijection between triples (i, alpha, beta), where alpha has parts at most i and beta has distinct parts greater than i, and pairs (delta, rho), where delta is distinct and rho has even parts with gaps at least four and minimum part 2+2 epsilon; it also carries i to the number of parts of rho." }, { "id": 20000735, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0099", "title": "A Bailey-Chebyshev-Bloch dictionary and terminal-choice blindness", "statement": "Problem 2.4. Relate Bailey pair techniques with Bloch group techniques. In particular, develop a dictionary. Perhaps this might might help with the computation of torsion. Similarly, connect orthogonal polynomials to Bloch group techniques.", "original_statement": "Problem 2.4. Relate Bailey pair techniques with Bloch group techniques. In particular, develop a dictionary. Perhaps this might might help with the computation of torsion. Similarly, connect orthogonal polynomials to Bloch group techniques.", "clean_statement": "Problem 2.4. Relate Bailey pair techniques with Bloch group techniques. In particular, develop a dictionary. Perhaps this might might help with the computation of torsion. Similarly, connect orthogonal polynomials to Bloch group techniques.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 2.4 from the AIM workshop list *Mock modular forms in combinatorics and arithmetic geometry*. The source file is `aim-analytic-number-theory-notes.json`, record index 98. The exact extracted statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Mock modular forms in combinatorics and arithmetic geometry\nSection: \nSource item: 2.4\nSource URL: https://aimath.org/WWN/mockmodular/mockmodular.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[98]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.4. Relate Bailey pair techniques with Bloch group techniques. In particular, develop a dictionary. Perhaps this might might help with the computation of torsion. Similarly, connect orthogonal polynomials to Bloch group techniques.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/mockmodular/mockmodular.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0099", "aim-domain:analytic-number-theory", "aim-workshop:mockmodular", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the standard rank-r Andrews-Gordon Bailey-chain family, the quadratic Nahm matrix is A_ij = 2 min(i,j), and its unique positive saddle has the explicit Chebyshev cross-ratio form x_i = U_{i-1}(c)U_{i+1}(c)/U_i(c)^2 with c = cos(pi/(2r+3)). The associated sum of Bloch symbols is a torsion element over the totally real field Q(c). For fixed r, all r+1 Andrews-Gordon terminal choices have distinct linear terms and product residue classes but the same saddle and ordinary Bloch class, proving that an ordinary-Bloch-valued Bailey dictionary necessarily loses terminal-choice data.\n\nCandidate contribution (obstruction_and_explicit_dictionary; novelty confidence low): Candidate novelty: fixed Bailey depth r gives r+1 distinct Andrews-Gordon companions but only one ordinary distinguished Bloch class; the report proves this non-injectivity and packages the common class in explicit Chebyshev cross-ratio coordinates." }, { "id": 20000736, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0100", "title": "Mock forms as boundary pieces of twisted Nahm sums", "statement": "Problem 2.5. Determine how mock modular forms fit into this theory.", "original_statement": "Problem 2.5. Determine how mock modular forms fit into this theory.", "clean_statement": "Problem 2.5. Determine how mock modular forms fit into this theory.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Mock modular forms in combinatorics and arithmetic geometry\nSection: \nSource item: 2.5\nSource URL: https://aimath.org/WWN/mockmodular/mockmodular.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[99]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.5. Determine how mock modular forms fit into this theory.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/mockmodular/mockmodular.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0100", "aim-domain:analytic-number-theory", "aim-workshop:mockmodular", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source question is reconstructed as asking for an extension of the Nahm-sum, Bloch-group, and asymptotic program that detects mock modularity and shadows. For twisted multivariate Nahm kernels with denominators (epsilon_i q;q)_{n_i}, we prove an exact q-difference equation whose affine source is (1-epsilon_i) times the kernel on the face n_i=0. In rank one for A>1, an explicit absolutely convergent negative-cone tail cancels this source. At A=2 and epsilon=-1, the cone decomposition gives the fifth-order pairs f_0+2 psi_0 and f_1+2 psi_1; McLaughlin supplies their product evaluations, and Zwegers supplies the compatible normalized holomorphic modular vector. A formal twisted saddle calculation gives 1-epsilon_i Q_i=product_j Q_j^{A_ij}, showing why the ordinary distinguished Bloch class alone cannot encode the sign-twisted mock example.\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: the arbitrary-rank twisted cone-boundary identity U(x)-epsilon_i U(s_i x)=(1-epsilon_i)U_face+q^(A_ii/2)x_i U(r_i x), together with its explicit rank-one source-cancelling negative-cone correction, isolates a finite face-source filtration as a coefficientwise testable Nahm-to-mock diagnostic." }, { "id": 20000737, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0101", "title": "An exact q-dilogarithmic action and finite-q Nahm equation", "statement": "Problem 2.6. How do q-analogues of the dilogarithm figure into the theory?", "original_statement": "Problem 2.6. How do q-analogues of the dilogarithm figure into the theory?", "clean_statement": "Problem 2.6. How do q-analogues of the dilogarithm figure into the theory?", "statement_status": "exact", "statement_verification": "The exact source question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Mock modular forms in combinatorics and arithmetic geometry\nSection: \nSource item: 2.6\nSource URL: https://aimath.org/WWN/mockmodular/mockmodular.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[100]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.6. How do q-analogues of the dilogarithm figure into the theory?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/mockmodular/mockmodular.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0101", "aim-domain:analytic-number-theory", "aim-workshop:mockmodular", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every Nahm summand is exactly the exponential of a compact-q-dilogarithmic action. For real 0 1 and thus we know the con-vexity bound in the s-aspect. Do we know the convexity bound in any aspect for any m Ê 1 and any n Ê 1? No! For instance, when m = n = 2, it is known via L2-theory of automorphic forms. Is there a geometric method which could give the result for any m Ê 1 and any n Ê 1?\n\n• These subconvexity problems were mainly suggested by Michel. Let f\n\nbe a Hecke Maass cusp form of level 1 and Laplacian eigenvalue λf:=\n\n1/4 + i t 2\n\n> f. We want to break the convexity bound for L( f, 1/2 + i t f ) in the spectral aspect namely to find δ > 0 (even microcospic) such that (1.1) L( f, 1/2 + i t f ) ¿ε t 1/4 −δ+ε\n\n> f\n\nfor any ε > 0. About this problem, two questions arose during the dis-cussion.\n\n- Is there a work on this from Luo?\n\n> Date: Version of December 8, 2006. Guillaume.Ricotta@math.u-bordeaux1.fr.\n> 12G. RICOTTA\n\n- Does somebody know an arithmetic application of such unkown convexity bound? Note that this is a case for which the analytic conductor drops since\n\nQ( f, 1/2 + i t f ) = (1 + ∣∣1/2 + i t f − i t f\n\n∣∣) ( 1 + ∣∣1/2 + i t f + i t f\n\n∣∣) ≈ t f.Thus, previous experience suggests that it should be difficult to prove. You can also think of the method of moments to be convinced: you will have to estimate an higher moment if you want to break convexity. An-other example in which the analytic conductor drops is given by (1.2) L( f × g, 1/2 + i t f ) ¿g,ε t 1/2 −δ+ε\n\n> f\n\nfor any ε > 0. Here, g is a fixed Hecke Maass cusp form. Note that if\n\ng is an holomorphic Hecke cusp form of weight 4 and level q, such L-functions already appear in Phillips-Sarnak's deformation theory even if people are more interested in non-vanishing results in this context. Roughly speaking, this theory deals with the deformation of Γ0(q) in the direction given by g. The authors proved that a positive proportion (in a suitable sense) of L( f × g, 1/2 + i t f ) does not vanish which entails that a positive proportion (in the same suitable sense) of f 's is annihilated by such deformation.\n\n• Let f and g be two fixed Hecke Maass forms. We want to break the con-vexity bound for L( f × g, 1/2 + i t ) in the s-aspect namely to find δ > 0such that (1.3) L( f × g, 1/2 + i t ) ¿f,g,ε t 1−δ+ε\n\nfor any ε > 0. Jutila suggested an extra-average over the spectral param-eter t f of f to produce some saving. For instance, he proved with Moto-hashi that\n\nL( f × g, 1/2 + i t ) ¿g,ε\n\n{t 1+ε /√ t f if t 3/2\n\n> f\n\n¿ t ¿ε t 2−ε\n\n> f,\n\n(t + t f\n\n)2/3 +ε if t ¿ t 3/2\n\n> f.How can we extend this range?\n\n• Let us talk about the symmetric-square L-function L(Sym 2 f, s) for any Hecke Maass cusp form of level q f, spectral parameter t f and nebenty-pus χf. This L-function is of degree 3. Thus, the subconvexity problem in the s-aspect is given by (1.4) L(Sym 2 f, 1/2 + i t ) ¿f,ε t 3/4 −δ+ε\n\nfor any ε > 0 and for some δ > 0. About the three spectral parameters at infinity of L(Sym 2 f, s), one is of constant size and the two others are of size t f. Thus, the subconvexity problem in the spectral aspect is given by (1.5) L(Sym 2 f, 1/2 + i t ) ¿q f,t,ε t 1/2 −δ+ε\n\n> f\n\nfor any ε > 0 and for some δ > 0. The subconvexity problem in the level aspect is given by (1.6) L(Sym 2 f, 1/2 + i t ) ¿q f,t,ε qε ×\n\n{q1/2 −δ\n\n> f\n\nif χf trivial or non-quadratic,\n\nq1/4 −δ\n\n> f\n\notherwise PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 3\n\nfor some δ > 0 and for any ε > 0 since\n\nL( f × f, s) = L(χf, s)L(Sym 2 f, s). Michel said that if you know a subconvexity bound for L(Sym 2 f, s) in the\n\ns-aspect then you know (by Cauchy-Schwarz) a subconvexity bound for\n\nL( f × g, s) when f 6 = g and g is fixed in the s-aspect. He also said that the subconvexity problem for symmetric square L-functions could have some link with metaplectic tools on ˜GL 2 via an integral representation of this L-function in which an Eisenstein series on ˜GL 2 occurs.\n\n• Let us talk about triple L-functions. Let f, g, h some Hecke Maass forms,\n\nf being of level q f, spectral parameter t f, weight k f and the same nota-tions for the two others. We could be interested in the following subcon-vexity problems (1.7) L( f × g × h, 1/2 + i t ) ¿f,g,t,qh k2−δ+ε\n\nwhen h is holomorphic. (1.8) L( f × g × h, 1/2 + i t ) ¿f,g,h,ε t 2−δ+ε.(1.9) L( f × f × h, 1/2) ¿q f,h,ε t 1−δ+ε\n\n> f.This last case is again an example of situation in which the conductor drops. Reznikov said that it is easier and doable to prove (1.10) L( f × g × h, 1/2) ¿ε t 2−δ+ε\n\n> f\n\nwhen f, g and h have some comparable but not equal spectral parame-ters at infinity such that the conductor does not drop. 1.2. Universality of convexity breaking exponents.\n\n• One aims at explaining the apparently unrelated occurences of Weyl's subconvexity exponent given by 1/4(1 − 1/3) and Burgess' subconvexity exponent given by 1/4(1 − 1/4). Remember that Weyl's subconvexity ex-ponent appears\n\n- in the subconvexity problem for GL 2 L-functions L( f, s) in the s and spectral aspect,\n\n- in the subconvexity problem for twisted L-functions L( f × χ, s) in the level aspect ( f and χ are of same level),\n\n- in the subconvexity problem for Rankin-Selberg L-functions L( f ×\n\ng, s) in the t f aspect whereas Burgess' subconvexity exponent appears\n\n- in the subconvexity problem for Dirichlet L-functions L(χ, s) in the level aspect,\n\n- in the subconvexity problem for twisted L-functions L( f × χ, s) in the conductor aspect of the character.\n\n• A natural problem is to find particular L-functions for which we know how to prove better exponents than Weyl and Burgess' ones. Soundarara-jan suggested to look at L(χ, s) when χ is of conductor say 3 n and n goes to infinity by taking advantage of Vinogradov's method. We can talk about subconvexity in the depth aspect in such context. A similar ex-ample is L( f × χ, s) by taking advantage of Graham-Ringrose's method when the modulus of χ is a higly divisible square-free integer. 4 G. RICOTTA\n\n• It is also natural to wonder if there exists some applications which need a better subconvexity exponent than Weyl or Burgess' one. Michel sug-gested an application to André Oort conjecture discovered by Edixhoven. This conjecture asserts that a curve contained in X0(1) × X0(1) which does not project to X0(1) itself and contains infinitely many CM points is the modular curve itself (embedded in the product as a graph of Hecke correspondence). Assuming GRH for quadratic imaginary fields, Edix-hoven \"proved\" this conjecture. The main hole for an unconditional proof being that he needs to know that the number of primes less than log 2 (|d|) log 22 (|d|) which are split in the quadratic imaginary field of dis-criminant d tends to infinity with d. Fouvry, applying a result of Linnik and Vinogradov, noticed that the number of primes less than |d|1/4 +ε\n\nwhich are split in the quadratic imaginary field of discriminant d tends to infinity with d. Improving Burgess' bound for character sums will im-prove the previous range. Studying these small split primes is a challeng-ing problem because it may occur when one wants to build an efficient amplifier. For instance, Duke-Friedlander-Iwaniec faced this problem when they tried to prove subconvexity bound for class group L-functions without appealing to the spectral theory of automorphic forms. 2. T UESDAY PROBLEM SESSION\n\n2.1. General definition and interesting examples of periods. Lindenstrauss gave the following general definition of period. Let G be a group and H be a subgroup of G. The general space is given by\n\nX:= G(Q) ∖G(AQ). To any automorphic form f on X (a smooth function on X which belongs to the space of an automorphic representation), we define some periods by\n\n∫\n\n> H(Q)∖H(AQ)\n\nf (h)g (h)d h\n\nfor any automorphic form g on H (Q) ∖H (AQ). Then, people gave fundamental examples of periods.\n\nExample 1: Fourier coefficients of cusp forms on GL 2\n\nHere, G = GL 2 and H is the unipotent subgroup of G namely\n\nH:=\n\n{( 1 x\n\n0 1\n\n), x ∈ R\n\n}.For any g in GL 2(Q) ∖GL 2(AQ), the period\n\n∫\n\n> R\n\nf\n\n(\n\ng\n\n(1 t\n\n0 1\n\n))\n\ne(−nt )d t\n\nis directly linked to the n-th Fourier coefficient of f for any n ∈ Z.\n\nExample 2: Special values of GL 2 L-functions Here, F is a number field, G = GL 2 and H is the torus subgroup of G namely\n\nH:=\n\n{( y 00 1\n\n), y > 0\n\n}.PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 5\n\nThe period ∫\n\n> F×∖A×\n> F\n\nf\n\n(( y 00 1\n\n))\n\nd× y\n\nis directly linked to the special value L( f, 1/2) up to Γ-factors.\n\nExample 3: Triple product formula Here, G = GL 2 × GL 2 and H = GL 2 is a subgroup of G via the diagonal embed-ding.Thus, an automorphic form F on G is a pair of automorphic forms f1 and\n\nf2 on GL 2. If h ∈ H then F (h) = f1(h) f2(h). The period\n\n∫\n\n> GL 2(Q)∖GL 2(AQ)\n\nf1(h) f2(h) f3(h)d h\n\nis linked (up to some factors) to the special value L( f1 × f2 × f3, 1/2). 2.2. A convexity bound for periods? The discussion was about understanding what could be a general bound for period which specializes to (sub)convex bounds for L-functions. Lindenstrauss said that a convex bound for period should be a bound that comes from general harmonic analysis results (it is the case for L-functions). Then, Miller defined an automorphic period which is our previous geometric period up to some factors (which are sometimes special values of L-functions). Thus, bounding these factors turns out to bounding both (automor-phic and geometric) periods. Another problem was trying to understand if there is a canonical way to define a period such that it does not depend on the choice of test vectors in the space of representations that occur. 2.3. List of problems to be discussed into small groups.\n\n• Subconvexity problems when the analytic conductor drops (in particu-lar for the symmetric square L-function).\n\n• Improving Weyl's exponent in various examples. In particular, investi-gate Soundararajan's idea about Dirichlet characters of highly compos-ite moduli or try to prove some explicit spectral decomposition of shifted convolution sums (Harcos' suggestion).\n\n• Develop explicit Good-Motoashi type identities.\n\n• Develop associativity type identities in the conductor aspect.\n\n• Formulate clearly period problems in relation to L-functions and repre-sentation theory.\n\n• Quantitative equidistribution results to prove subconvexity bounds. 3. W EDNESDAY PROBLEM SESSION\n\nHere is an incomplete list of the problems which could be understood in a close future.\n\n• Silberman suggested to try to prove a strong hybrid subconvexity bound for standard L-functions on GL n namely try to find δ > 0 such that\n\nL(π, 1/2 + i t ) ¿ε Q(π, 1/2 + i t )1/4 −δ+ε\n\nfor any ε > 0. According to Garett, the Diaconu-Garrett-Goldfeld ex-tension of Good's method to GL n xGL n−1 may have something to offer in this direction.Venkatesh also suggested to try to find some applica-tions of such subconvexity bound before proving them. For instance, 6 G. RICOTTA\n\nit can be interesting to clarify the links between subconvexity problems and equidistribution problems in higher rank for classical groups. Duke also had in mind to extract explicit information from higher rank Artin\n\nL-functions.\n\n• About periods, a very important point is to find an heuristic way of pre-dicting what is an analogue of convexity bounds and Lindelöf hypoth-esis for periods. Also, prove some bounds for periods which special-ize to subconvexity bounds for L-functions. Silberman mentioned the problem of predicting what could be the expected bound for the infinite norm of general automorphic forms when the spectral parameters go to infinity.\n\n• Jutila suggested to prove some Ω-results about the error term which oc-curs in some moments of families of L-functions. For instance,\n\n∫\n\n> tÉT\n\n∣∣L( f, 1/2 + i t )∣∣2 dt = Main( T ) + Error( T )where Error( T ) = Ω(pT ) is expected to hold for any GL 2-automorphic form. What could be some applications of such Ω-results?\n\n• Venkatesh suggested to find a way to guess when it is possible to prove some asymptotic formula for some moment given by\n\n∑\n\n> f∈F\n\nL( f, 1/2) where as usual the conductor of each L-function of F is of size almost constant say Q(F ) in the logarithmic scale and Q(F ) → +∞. If 4 log |F | > log Q(F )then we generally can prove an asymptotic formula. At the moment, the record is 6 log |F | = log Q(F )in the work of Conrey and Iwaniec on the cubic moment of automorphic\n\nL-functions. Can we do better?\n\n• Soundararajan asked if it is possible to prove some subconvexity bound in the critical strip but outside the critical line and eventually near the edge of the critical strip. One known instance is the ζ function in the\n\ns-aspect near ℜs = 1 via Vinogradov.\n\n• Michel asked about an analogy of (sub)convexity for p-adic L-functions. The answer could be some integrality property (Mazur, Prasad,...). 4. T HURSDAY PROBLEM SESSION\n\n• Venkatesh convinced us that many equidistribution results are useful to prove asymptotic formula for moments of L-functions with a power sav-ing in the error term if the suitable equidistribution results are quantita-tive ones. He illustrated this by two GL 2-examples namely\n\n∫ +T\n\n> −T\n\n∣∣L( f, 1/2 + i t )∣∣2 dt\n\nand ×∑\n\n> χmod ( q)\n\n∫\n\n> R\n\n∣∣L( f × χ, 1/2 + i t )∣∣2 dt.PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 7\n\n• In addition, he mentioned what could be the obstacles to do the same in higher rank cases. On one hand, it is very hard to prove an inte-gral representation (period) of L-functions which occured in higher rank since the archimedean computation may be very far away from obvious. According to Garett, the Diaconu-Garrett-Goldfeld extension of Good's spectral-theory-based method to GL n xGL n−1 illustrates the complica-tions at archimedean places. On the other hand, it is necessary to anal-yse such integral representation via ergodic theory or spectral methods. Two difficult instances are given by On × On−1 and GL n ×GL n−1.\n\n• An application of subconvexity bounds for GL n may be some informa-tion about the 2 n-moment of the Riemann ζ function. For instance, some Motohashi type formula make the link between GL 3 and |ζ|6. We also have to mention the work of Conrey and Iwaniec. 5. F RIDAY PROBLEM SESSION\n\n• There will be a website dedicated to subconvexity for L-functions. Peo-ple agreed that it should contain references to important results, a list of people with their current attempts and previous results also.\n\n• Venkatesh put the stress on the subconvexity problem in higher-rank cases. Few methods are known which have bearing on it. After Venkatesh-Lindenstrauss and Bernstein-Reznikoff treatments of triple products, the exceptions are works in progress mentionned by Garett: Venkatesh's ap-plications of ergodic theoretic ideas coming from Ratner and Clozel, and Diaconu-Garrett-Goldfeld's GL n version of an old method of Good. Venkatesh suggested that a first higher rank example to undertake should be GL 3 × GL 2 with f3 on GL 3 is fixed and f2 is varying. In order to get some insight into that, it would be profitable for everybody that classi-cal analytic number theorists try to understand the case when f3 is an Eisenstein series namely\n\n∑\n\n> fHecke-Maass of level 1 and eigenvalue 1/4 +t2\n> f\n> tfvT\n\n∫\n\n> tvT\n\n∣∣L( f, 1/2 + i t )∣∣6 dt.Note that the size of the family is about T 3 whereas the size of the an-alytic conductor is about T 12 which reveals the level of difficulty. Also, people should understand where the GL 3-theory occurs in the analytic analysis.", "original_statement": "1. Monday problem session 11.1. Unknown cases of (sub)convexity 11.2. Universality of convexity breaking exponents 32. Tuesday problem session 42.1. General definition and interesting examples of periods 42.2. A convexity bound for periods? 52.3. List of problems to be discussed into small groups 53. Wednesday problem session 54. Thursday problem session 65. Friday problem session 71. M ONDAY PROBLEM SESSION \n\n1.1. Unknown cases of (sub)convexity. \n\n• This point was mentioned by Reznikov. Let π be an automorphic cuspi-dal representation of GL m for m Ê 1 and π′ be an automorphic cuspidal representation of GL n for n Ê 1. Do we know the convexity bound for \n\nL(π×π′, s)? It absolutely converges on ℜs > 1 and thus we know the con-vexity bound in the s-aspect. Do we know the convexity bound in any aspect for any m Ê 1 and any n Ê 1? No! For instance, when m = n = 2, it is known via L2-theory of automorphic forms. Is there a geometric method which could give the result for any m Ê 1 and any n Ê 1? \n\n• These subconvexity problems were mainly suggested by Michel. Let f\n\nbe a Hecke Maass cusp form of level 1 and Laplacian eigenvalue λf:=\n\n1/4 + i t 2 \n\n> f. We want to break the convexity bound for L( f, 1/2 + i t f ) in the spectral aspect namely to find δ > 0 (even microcospic) such that (1.1) L( f, 1/2 + i t f ) ¿ε t 1/4 −δ+ε\n\n> f\n\nfor any ε > 0. About this problem, two questions arose during the dis-cussion. \n\n- Is there a work on this from Luo? \n\n> Date: Version of December 8, 2006. Guillaume.Ricotta@math.u-bordeaux1.fr.\n> 12G. RICOTTA\n\n- Does somebody know an arithmetic application of such unkown convexity bound? Note that this is a case for which the analytic conductor drops since \n\nQ( f, 1/2 + i t f ) = (1 + ∣∣1/2 + i t f − i t f\n\n∣∣) ( 1 + ∣∣1/2 + i t f + i t f\n\n∣∣) ≈ t f.Thus, previous experience suggests that it should be difficult to prove. You can also think of the method of moments to be convinced: you will have to estimate an higher moment if you want to break convexity. An-other example in which the analytic conductor drops is given by (1.2) L( f × g, 1/2 + i t f ) ¿g,ε t 1/2 −δ+ε\n\n> f\n\nfor any ε > 0. Here, g is a fixed Hecke Maass cusp form. Note that if \n\ng is an holomorphic Hecke cusp form of weight 4 and level q, such L-functions already appear in Phillips-Sarnak's deformation theory even if people are more interested in non-vanishing results in this context. Roughly speaking, this theory deals with the deformation of Γ0(q) in the direction given by g. The authors proved that a positive proportion (in a suitable sense) of L( f × g, 1/2 + i t f ) does not vanish which entails that a positive proportion (in the same suitable sense) of f 's is annihilated by such deformation. \n\n• Let f and g be two fixed Hecke Maass forms. We want to break the con-vexity bound for L( f × g, 1/2 + i t ) in the s-aspect namely to find δ > 0such that (1.3) L( f × g, 1/2 + i t ) ¿f,g,ε t 1−δ+ε\n\nfor any ε > 0. Jutila suggested an extra-average over the spectral param-eter t f of f to produce some saving. For instance, he proved with Moto-hashi that \n\nL( f × g, 1/2 + i t ) ¿g,ε\n\n{t 1+ε /√ t f if t 3/2 \n\n> f\n\n¿ t ¿ε t 2−ε \n\n> f,\n\n(t + t f\n\n)2/3 +ε if t ¿ t 3/2 \n\n> f.How can we extend this range? \n\n• Let us talk about the symmetric-square L-function L(Sym 2 f, s) for any Hecke Maass cusp form of level q f, spectral parameter t f and nebenty-pus χf. This L-function is of degree 3. Thus, the subconvexity problem in the s-aspect is given by (1.4) L(Sym 2 f, 1/2 + i t ) ¿f,ε t 3/4 −δ+ε\n\nfor any ε > 0 and for some δ > 0. About the three spectral parameters at infinity of L(Sym 2 f, s), one is of constant size and the two others are of size t f. Thus, the subconvexity problem in the spectral aspect is given by (1.5) L(Sym 2 f, 1/2 + i t ) ¿q f,t,ε t 1/2 −δ+ε\n\n> f\n\nfor any ε > 0 and for some δ > 0. The subconvexity problem in the level aspect is given by (1.6) L(Sym 2 f, 1/2 + i t ) ¿q f,t,ε qε ×\n\n{q1/2 −δ \n\n> f\n\nif χf trivial or non-quadratic, \n\nq1/4 −δ \n\n> f\n\notherwise PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 3\n\nfor some δ > 0 and for any ε > 0 since \n\nL( f × f, s) = L(χf, s)L(Sym 2 f, s). Michel said that if you know a subconvexity bound for L(Sym 2 f, s) in the \n\ns-aspect then you know (by Cauchy-Schwarz) a subconvexity bound for \n\nL( f × g, s) when f 6 = g and g is fixed in the s-aspect. He also said that the subconvexity problem for symmetric square L-functions could have some link with metaplectic tools on ˜GL 2 via an integral representation of this L-function in which an Eisenstein series on ˜GL 2 occurs. \n\n• Let us talk about triple L-functions. Let f, g, h some Hecke Maass forms, \n\nf being of level q f, spectral parameter t f, weight k f and the same nota-tions for the two others. We could be interested in the following subcon-vexity problems (1.7) L( f × g × h, 1/2 + i t ) ¿f,g,t,qh k2−δ+ε\n\nwhen h is holomorphic. (1.8) L( f × g × h, 1/2 + i t ) ¿f,g,h,ε t 2−δ+ε.(1.9) L( f × f × h, 1/2) ¿q f,h,ε t 1−δ+ε \n\n> f.This last case is again an example of situation in which the conductor drops. Reznikov said that it is easier and doable to prove (1.10) L( f × g × h, 1/2) ¿ε t 2−δ+ε\n\n> f\n\nwhen f, g and h have some comparable but not equal spectral parame-ters at infinity such that the conductor does not drop. 1.2. Universality of convexity breaking exponents. \n\n• One aims at explaining the apparently unrelated occurences of Weyl's subconvexity exponent given by 1/4(1 − 1/3) and Burgess' subconvexity exponent given by 1/4(1 − 1/4). Remember that Weyl's subconvexity ex-ponent appears \n\n- in the subconvexity problem for GL 2 L-functions L( f, s) in the s and spectral aspect, \n\n- in the subconvexity problem for twisted L-functions L( f × χ, s) in the level aspect ( f and χ are of same level), \n\n- in the subconvexity problem for Rankin-Selberg L-functions L( f ×\n\ng, s) in the t f aspect whereas Burgess' subconvexity exponent appears \n\n- in the subconvexity problem for Dirichlet L-functions L(χ, s) in the level aspect, \n\n- in the subconvexity problem for twisted L-functions L( f × χ, s) in the conductor aspect of the character. \n\n• A natural problem is to find particular L-functions for which we know how to prove better exponents than Weyl and Burgess' ones. Soundarara-jan suggested to look at L(χ, s) when χ is of conductor say 3 n and n goes to infinity by taking advantage of Vinogradov's method. We can talk about subconvexity in the depth aspect in such context. A similar ex-ample is L( f × χ, s) by taking advantage of Graham-Ringrose's method when the modulus of χ is a higly divisible square-free integer. 4 G. RICOTTA \n\n• It is also natural to wonder if there exists some applications which need a better subconvexity exponent than Weyl or Burgess' one. Michel sug-gested an application to André Oort conjecture discovered by Edixhoven. This conjecture asserts that a curve contained in X0(1) × X0(1) which does not project to X0(1) itself and contains infinitely many CM points is the modular curve itself (embedded in the product as a graph of Hecke correspondence). Assuming GRH for quadratic imaginary fields, Edix-hoven \"proved\" this conjecture. The main hole for an unconditional proof being that he needs to know that the number of primes less than log 2 (|d|) log 22 (|d|) which are split in the quadratic imaginary field of dis-criminant d tends to infinity with d. Fouvry, applying a result of Linnik and Vinogradov, noticed that the number of primes less than |d|1/4 +ε\n\nwhich are split in the quadratic imaginary field of discriminant d tends to infinity with d. Improving Burgess' bound for character sums will im-prove the previous range. Studying these small split primes is a challeng-ing problem because it may occur when one wants to build an efficient amplifier. For instance, Duke-Friedlander-Iwaniec faced this problem when they tried to prove subconvexity bound for class group L-functions without appealing to the spectral theory of automorphic forms. 2. T UESDAY PROBLEM SESSION \n\n2.1. General definition and interesting examples of periods. Lindenstrauss gave the following general definition of period. Let G be a group and H be a subgroup of G. The general space is given by \n\nX:= G(Q) ∖G(AQ). To any automorphic form f on X (a smooth function on X which belongs to the space of an automorphic representation), we define some periods by \n\n∫ \n\n> H(Q)∖H(AQ)\n\nf (h)g (h)d h\n\nfor any automorphic form g on H (Q) ∖H (AQ). Then, people gave fundamental examples of periods. \n\nExample 1: Fourier coefficients of cusp forms on GL 2\n\nHere, G = GL 2 and H is the unipotent subgroup of G namely \n\nH:=\n\n{( 1 x\n\n0 1\n\n), x ∈ R\n\n}.For any g in GL 2(Q) ∖GL 2(AQ), the period \n\n∫\n\n> R\n\nf\n\n(\n\ng\n\n(1 t\n\n0 1\n\n)) \n\ne(−nt )d t\n\nis directly linked to the n-th Fourier coefficient of f for any n ∈ Z.\n\nExample 2: Special values of GL 2 L-functions Here, F is a number field, G = GL 2 and H is the torus subgroup of G namely \n\nH:=\n\n{( y 00 1\n\n), y > 0\n\n}.PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 5\n\nThe period ∫ \n\n> F×∖A×\n> F\n\nf\n\n(( y 00 1\n\n)) \n\nd× y\n\nis directly linked to the special value L( f, 1/2) up to Γ-factors. \n\nExample 3: Triple product formula Here, G = GL 2 × GL 2 and H = GL 2 is a subgroup of G via the diagonal embed-ding.Thus, an automorphic form F on G is a pair of automorphic forms f1 and \n\nf2 on GL 2. If h ∈ H then F (h) = f1(h) f2(h). The period \n\n∫\n\n> GL 2(Q)∖GL 2(AQ)\n\nf1(h) f2(h) f3(h)d h\n\nis linked (up to some factors) to the special value L( f1 × f2 × f3, 1/2). 2.2. A convexity bound for periods? The discussion was about understanding what could be a general bound for period which specializes to (sub)convex bounds for L-functions. Lindenstrauss said that a convex bound for period should be a bound that comes from general harmonic analysis results (it is the case for L-functions). Then, Miller defined an automorphic period which is our previous geometric period up to some factors (which are sometimes special values of L-functions). Thus, bounding these factors turns out to bounding both (automor-phic and geometric) periods. Another problem was trying to understand if there is a canonical way to define a period such that it does not depend on the choice of test vectors in the space of representations that occur. 2.3. List of problems to be discussed into small groups. \n\n• Subconvexity problems when the analytic conductor drops (in particu-lar for the symmetric square L-function). \n\n• Improving Weyl's exponent in various examples. In particular, investi-gate Soundararajan's idea about Dirichlet characters of highly compos-ite moduli or try to prove some explicit spectral decomposition of shifted convolution sums (Harcos' suggestion). \n\n• Develop explicit Good-Motoashi type identities. \n\n• Develop associativity type identities in the conductor aspect. \n\n• Formulate clearly period problems in relation to L-functions and repre-sentation theory. \n\n• Quantitative equidistribution results to prove subconvexity bounds. 3. W EDNESDAY PROBLEM SESSION \n\nHere is an incomplete list of the problems which could be understood in a close future. \n\n• Silberman suggested to try to prove a strong hybrid subconvexity bound for standard L-functions on GL n namely try to find δ > 0 such that \n\nL(π, 1/2 + i t ) ¿ε Q(π, 1/2 + i t )1/4 −δ+ε\n\nfor any ε > 0. According to Garett, the Diaconu-Garrett-Goldfeld ex-tension of Good's method to GL n xGL n−1 may have something to offer in this direction.Venkatesh also suggested to try to find some applica-tions of such subconvexity bound before proving them. For instance, 6 G. RICOTTA \n\nit can be interesting to clarify the links between subconvexity problems and equidistribution problems in higher rank for classical groups. Duke also had in mind to extract explicit information from higher rank Artin \n\nL-functions. \n\n• About periods, a very important point is to find an heuristic way of pre-dicting what is an analogue of convexity bounds and Lindelöf hypoth-esis for periods. Also, prove some bounds for periods which special-ize to subconvexity bounds for L-functions. Silberman mentioned the problem of predicting what could be the expected bound for the infinite norm of general automorphic forms when the spectral parameters go to infinity. \n\n• Jutila suggested to prove some Ω-results about the error term which oc-curs in some moments of families of L-functions. For instance, \n\n∫ \n\n> tÉT\n\n∣∣L( f, 1/2 + i t )∣∣2 dt = Main( T ) + Error( T )where Error( T ) = Ω(pT ) is expected to hold for any GL 2-automorphic form. What could be some applications of such Ω-results? \n\n• Venkatesh suggested to find a way to guess when it is possible to prove some asymptotic formula for some moment given by \n\n∑ \n\n> f∈F\n\nL( f, 1/2) where as usual the conductor of each L-function of F is of size almost constant say Q(F ) in the logarithmic scale and Q(F ) → +∞. If 4 log |F | > log Q(F )then we generally can prove an asymptotic formula. At the moment, the record is 6 log |F | = log Q(F )in the work of Conrey and Iwaniec on the cubic moment of automorphic \n\nL-functions. Can we do better? \n\n• Soundararajan asked if it is possible to prove some subconvexity bound in the critical strip but outside the critical line and eventually near the edge of the critical strip. One known instance is the ζ function in the \n\ns-aspect near ℜs = 1 via Vinogradov. \n\n• Michel asked about an analogy of (sub)convexity for p-adic L-functions. The answer could be some integrality property (Mazur, Prasad,...). 4. T HURSDAY PROBLEM SESSION \n\n• Venkatesh convinced us that many equidistribution results are useful to prove asymptotic formula for moments of L-functions with a power sav-ing in the error term if the suitable equidistribution results are quantita-tive ones. He illustrated this by two GL 2-examples namely \n\n∫ +T\n\n> −T\n\n∣∣L( f, 1/2 + i t )∣∣2 dt\n\nand ×∑ \n\n> χmod ( q)\n\n∫\n\n> R\n\n∣∣L( f × χ, 1/2 + i t )∣∣2 dt.PROBLEM SESSIONS: \"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\" 7\n\n• In addition, he mentioned what could be the obstacles to do the same in higher rank cases. On one hand, it is very hard to prove an inte-gral representation (period) of L-functions which occured in higher rank since the archimedean computation may be very far away from obvious. According to Garett, the Diaconu-Garrett-Goldfeld extension of Good's spectral-theory-based method to GL n xGL n−1 illustrates the complica-tions at archimedean places. On the other hand, it is necessary to anal-yse such integral representation via ergodic theory or spectral methods. Two difficult instances are given by On × On−1 and GL n ×GL n−1.\n\n• An application of subconvexity bounds for GL n may be some informa-tion about the 2 n-moment of the Riemann ζ function. For instance, some Motohashi type formula make the link between GL 3 and |ζ|6. We also have to mention the work of Conrey and Iwaniec. 5. F RIDAY PROBLEM SESSION \n\n• There will be a website dedicated to subconvexity for L-functions. Peo-ple agreed that it should contain references to important results, a list of people with their current attempts and previous results also. \n\n• Venkatesh put the stress on the subconvexity problem in higher-rank cases. Few methods are known which have bearing on it. After Venkatesh-Lindenstrauss and Bernstein-Reznikoff treatments of triple products, the exceptions are works in progress mentionned by Garett: Venkatesh's ap-plications of ergodic theoretic ideas coming from Ratner and Clozel, and Diaconu-Garrett-Goldfeld's GL n version of an old method of Good. Venkatesh suggested that a first higher rank example to undertake should be GL 3 × GL 2 with f3 on GL 3 is fixed and f2 is varying. In order to get some insight into that, it would be profitable for everybody that classi-cal analytic number theorists try to understand the case when f3 is an Eisenstein series namely \n\n∑ \n\n> fHecke-Maass of level 1 and eigenvalue 1/4 +t2\n> f\n> tfvT\n\n∫ \n\n> tvT\n\n∣∣L( f, 1/2 + i t )∣∣6 dt.Note that the size of the family is about T 3 whereas the size of the an-alytic conductor is about T 12 which reveals the level of difficulty. Also, people should understand where the GL 3-theory occurs in the analytic analysis.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical object cannot be recovered as one mathematical problem. Its metadata say number “1” and tag “problem,” but its 16,182-character problem field begins with the complete table of contents", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Subconvexity bounds for L-functions\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/subconvexity/subconvexity.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[101]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Monday problem session 11.1. Unknown cases of (sub)convexity 11.2. Universality of convexity breaking exponents 32. Tuesday problem session 42.1. General definition and interesting examples of periods 42.2. A convexity bound for periods? 52.3. List of problems to be discussed into small groups 53. Wednesday problem session 54. Thursday problem session 65. Friday problem session 71. M ONDAY PROBLEM SESSION \\n\\n1.1. Unknown cases of (sub)convexity. \\n\\n• This point was mentioned by Reznikov. Let π be an automorphic cuspi-dal representation of GL m for m Ê 1 and π′ be an automorphic cuspidal representation of GL n for n Ê 1. Do we know the convexity bound for \\n\\nL(π×π′, s)? It absolutely converges on ℜs > 1 and thus we know the con-vexity bound in the s-aspect. Do we know the convexity bound in any aspect for any m Ê 1 and any n Ê 1? No! For instance, when m = n = 2, it is known via L2-theory of automorphic forms. Is there a geometric method which could give the result for any m Ê 1 and any n Ê 1? \\n\\n• These subconvexity problems were mainly suggested by Michel. Let f\\n\\nbe a Hecke Maass cusp form of level 1 and Laplacian eigenvalue λf:=\\n\\n1/4 + i t 2 \\n\\n> f. We want to break the convexity bound for L( f, 1/2 + i t f ) in the spectral aspect namely to find δ > 0 (even microcospic) such that (1.1) L( f, 1/2 + i t f ) ¿ε t 1/4 −δ+ε\\n\\n> f\\n\\nfor any ε > 0. About this problem, two questions arose during the dis-cussion. \\n\\n- Is there a work on this from Luo? \\n\\n> Date: Version of December 8, 2006. Guillaume.Ricotta@math.u-bordeaux1.fr.\\n> 12G. RICOTTA\\n\\n- Does somebody know an arithmetic application of such unkown convexity bound? Note that this is a case for which the analytic conductor drops since \\n\\nQ( f, 1/2 + i t f ) = (1 + ∣∣1/2 + i t f − i t f\\n\\n∣∣) ( 1 + ∣∣1/2 + i t f + i t f\\n\\n∣∣) ≈ t f.Thus, previous experience suggests that it should be difficult to prove. You can also think of the method of moments to be convinced: you will have to estimate an higher moment if you want to break convexity. An-other example in which the analytic conductor drops is given by (1.2) L( f × g, 1/2 + i t f ) ¿g,ε t 1/2 −δ+ε\\n\\n> f\\n\\nfor any ε > 0. Here, g is a fixed Hecke Maass cusp form. Note that if \\n\\ng is an holomorphic Hecke cusp form of weight 4 and level q, such L-functions already appear in Phillips-Sarnak's deformation theory even if people are more interested in non-vanishing results in this context. Roughly speaking, this theory deals with the deformation of Γ0(q) in the direction given by g. The authors proved that a positive proportion (in a suitable sense) of L( f × g, 1/2 + i t f ) does not vanish which entails that a positive proportion (in the same suitable sense) of f 's is annihilated by such deformation. \\n\\n• Let f and g be two fixed Hecke Maass forms. We want to break the con-vexity bound for L( f × g, 1/2 + i t ) in the s-aspect namely to find δ > 0such that (1.3) L( f × g, 1/2 + i t ) ¿f,g,ε t 1−δ+ε\\n\\nfor any ε > 0. Jutila suggested an extra-average over the spectral param-eter t f of f to produce some saving. For instance, he proved with Moto-hashi that \\n\\nL( f × g, 1/2 + i t ) ¿g,ε\\n\\n{t 1+ε /√ t f if t 3/2 \\n\\n> f\\n\\n¿ t ¿ε t 2−ε \\n\\n> f,\\n\\n(t + t f\\n\\n)2/3 +ε if t ¿ t 3/2 \\n\\n> f.How can we extend this range? \\n\\n• Let us talk about the symmetric-square L-function L(Sym 2 f, s) for any Hecke Maass cusp form of level q f, spectral parameter t f and nebenty-pus χf. This L-function is of degree 3. Thus, the subconvexity problem in the s-aspect is given by (1.4) L(Sym 2 f, 1/2 + i t ) ¿f,ε t 3/4 −δ+ε\\n\\nfor any ε > 0 and for some δ > 0. About the three spectral parameters at infinity of L(Sym 2 f, s), one is of constant size and the two others are of size t f. Thus, the subconvexity problem in the spectral aspect is given by (1.5) L(Sym 2 f, 1/2 + i t ) ¿q f,t,ε t 1/2 −δ+ε\\n\\n> f\\n\\nfor any ε > 0 and for some δ > 0. The subconvexity problem in the level aspect is given by (1.6) L(Sym 2 f, 1/2 + i t ) ¿q f,t,ε qε ×\\n\\n{q1/2 −δ \\n\\n> f\\n\\nif χf trivial or non-quadratic, \\n\\nq1/4 −δ \\n\\n> f\\n\\notherwise PROBLEM SESSIONS: \\\"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\\\" 3\\n\\nfor some δ > 0 and for any ε > 0 since \\n\\nL( f × f, s) = L(χf, s)L(Sym 2 f, s). Michel said that if you know a subconvexity bound for L(Sym 2 f, s) in the \\n\\ns-aspect then you know (by Cauchy-Schwarz) a subconvexity bound for \\n\\nL( f × g, s) when f 6 = g and g is fixed in the s-aspect. He also said that the subconvexity problem for symmetric square L-functions could have some link with metaplectic tools on ˜GL 2 via an integral representation of this L-function in which an Eisenstein series on ˜GL 2 occurs. \\n\\n• Let us talk about triple L-functions. Let f, g, h some Hecke Maass forms, \\n\\nf being of level q f, spectral parameter t f, weight k f and the same nota-tions for the two others. We could be interested in the following subcon-vexity problems (1.7) L( f × g × h, 1/2 + i t ) ¿f,g,t,qh k2−δ+ε\\n\\nwhen h is holomorphic. (1.8) L( f × g × h, 1/2 + i t ) ¿f,g,h,ε t 2−δ+ε.(1.9) L( f × f × h, 1/2) ¿q f,h,ε t 1−δ+ε \\n\\n> f.This last case is again an example of situation in which the conductor drops. Reznikov said that it is easier and doable to prove (1.10) L( f × g × h, 1/2) ¿ε t 2−δ+ε\\n\\n> f\\n\\nwhen f, g and h have some comparable but not equal spectral parame-ters at infinity such that the conductor does not drop. 1.2. Universality of convexity breaking exponents. \\n\\n• One aims at explaining the apparently unrelated occurences of Weyl's subconvexity exponent given by 1/4(1 − 1/3) and Burgess' subconvexity exponent given by 1/4(1 − 1/4). Remember that Weyl's subconvexity ex-ponent appears \\n\\n- in the subconvexity problem for GL 2 L-functions L( f, s) in the s and spectral aspect, \\n\\n- in the subconvexity problem for twisted L-functions L( f × χ, s) in the level aspect ( f and χ are of same level), \\n\\n- in the subconvexity problem for Rankin-Selberg L-functions L( f ×\\n\\ng, s) in the t f aspect whereas Burgess' subconvexity exponent appears \\n\\n- in the subconvexity problem for Dirichlet L-functions L(χ, s) in the level aspect, \\n\\n- in the subconvexity problem for twisted L-functions L( f × χ, s) in the conductor aspect of the character. \\n\\n• A natural problem is to find particular L-functions for which we know how to prove better exponents than Weyl and Burgess' ones. Soundarara-jan suggested to look at L(χ, s) when χ is of conductor say 3 n and n goes to infinity by taking advantage of Vinogradov's method. We can talk about subconvexity in the depth aspect in such context. A similar ex-ample is L( f × χ, s) by taking advantage of Graham-Ringrose's method when the modulus of χ is a higly divisible square-free integer. 4 G. RICOTTA \\n\\n• It is also natural to wonder if there exists some applications which need a better subconvexity exponent than Weyl or Burgess' one. Michel sug-gested an application to André Oort conjecture discovered by Edixhoven. This conjecture asserts that a curve contained in X0(1) × X0(1) which does not project to X0(1) itself and contains infinitely many CM points is the modular curve itself (embedded in the product as a graph of Hecke correspondence). Assuming GRH for quadratic imaginary fields, Edix-hoven \\\"proved\\\" this conjecture. The main hole for an unconditional proof being that he needs to know that the number of primes less than log 2 (|d|) log 22 (|d|) which are split in the quadratic imaginary field of dis-criminant d tends to infinity with d. Fouvry, applying a result of Linnik and Vinogradov, noticed that the number of primes less than |d|1/4 +ε\\n\\nwhich are split in the quadratic imaginary field of discriminant d tends to infinity with d. Improving Burgess' bound for character sums will im-prove the previous range. Studying these small split primes is a challeng-ing problem because it may occur when one wants to build an efficient amplifier. For instance, Duke-Friedlander-Iwaniec faced this problem when they tried to prove subconvexity bound for class group L-functions without appealing to the spectral theory of automorphic forms. 2. T UESDAY PROBLEM SESSION \\n\\n2.1. General definition and interesting examples of periods. Lindenstrauss gave the following general definition of period. Let G be a group and H be a subgroup of G. The general space is given by \\n\\nX:= G(Q) ∖G(AQ). To any automorphic form f on X (a smooth function on X which belongs to the space of an automorphic representation), we define some periods by \\n\\n∫ \\n\\n> H(Q)∖H(AQ)\\n\\nf (h)g (h)d h\\n\\nfor any automorphic form g on H (Q) ∖H (AQ). Then, people gave fundamental examples of periods. \\n\\nExample 1: Fourier coefficients of cusp forms on GL 2\\n\\nHere, G = GL 2 and H is the unipotent subgroup of G namely \\n\\nH:=\\n\\n{( 1 x\\n\\n0 1\\n\\n), x ∈ R\\n\\n}.For any g in GL 2(Q) ∖GL 2(AQ), the period \\n\\n∫\\n\\n> R\\n\\nf\\n\\n(\\n\\ng\\n\\n(1 t\\n\\n0 1\\n\\n)) \\n\\ne(−nt )d t\\n\\nis directly linked to the n-th Fourier coefficient of f for any n ∈ Z.\\n\\nExample 2: Special values of GL 2 L-functions Here, F is a number field, G = GL 2 and H is the torus subgroup of G namely \\n\\nH:=\\n\\n{( y 00 1\\n\\n), y > 0\\n\\n}.PROBLEM SESSIONS: \\\"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\\\" 5\\n\\nThe period ∫ \\n\\n> F×∖A×\\n> F\\n\\nf\\n\\n(( y 00 1\\n\\n)) \\n\\nd× y\\n\\nis directly linked to the special value L( f, 1/2) up to Γ-factors. \\n\\nExample 3: Triple product formula Here, G = GL 2 × GL 2 and H = GL 2 is a subgroup of G via the diagonal embed-ding.Thus, an automorphic form F on G is a pair of automorphic forms f1 and \\n\\nf2 on GL 2. If h ∈ H then F (h) = f1(h) f2(h). The period \\n\\n∫\\n\\n> GL 2(Q)∖GL 2(AQ)\\n\\nf1(h) f2(h) f3(h)d h\\n\\nis linked (up to some factors) to the special value L( f1 × f2 × f3, 1/2). 2.2. A convexity bound for periods? The discussion was about understanding what could be a general bound for period which specializes to (sub)convex bounds for L-functions. Lindenstrauss said that a convex bound for period should be a bound that comes from general harmonic analysis results (it is the case for L-functions). Then, Miller defined an automorphic period which is our previous geometric period up to some factors (which are sometimes special values of L-functions). Thus, bounding these factors turns out to bounding both (automor-phic and geometric) periods. Another problem was trying to understand if there is a canonical way to define a period such that it does not depend on the choice of test vectors in the space of representations that occur. 2.3. List of problems to be discussed into small groups. \\n\\n• Subconvexity problems when the analytic conductor drops (in particu-lar for the symmetric square L-function). \\n\\n• Improving Weyl's exponent in various examples. In particular, investi-gate Soundararajan's idea about Dirichlet characters of highly compos-ite moduli or try to prove some explicit spectral decomposition of shifted convolution sums (Harcos' suggestion). \\n\\n• Develop explicit Good-Motoashi type identities. \\n\\n• Develop associativity type identities in the conductor aspect. \\n\\n• Formulate clearly period problems in relation to L-functions and repre-sentation theory. \\n\\n• Quantitative equidistribution results to prove subconvexity bounds. 3. W EDNESDAY PROBLEM SESSION \\n\\nHere is an incomplete list of the problems which could be understood in a close future. \\n\\n• Silberman suggested to try to prove a strong hybrid subconvexity bound for standard L-functions on GL n namely try to find δ > 0 such that \\n\\nL(π, 1/2 + i t ) ¿ε Q(π, 1/2 + i t )1/4 −δ+ε\\n\\nfor any ε > 0. According to Garett, the Diaconu-Garrett-Goldfeld ex-tension of Good's method to GL n xGL n−1 may have something to offer in this direction.Venkatesh also suggested to try to find some applica-tions of such subconvexity bound before proving them. For instance, 6 G. RICOTTA \\n\\nit can be interesting to clarify the links between subconvexity problems and equidistribution problems in higher rank for classical groups. Duke also had in mind to extract explicit information from higher rank Artin \\n\\nL-functions. \\n\\n• About periods, a very important point is to find an heuristic way of pre-dicting what is an analogue of convexity bounds and Lindelöf hypoth-esis for periods. Also, prove some bounds for periods which special-ize to subconvexity bounds for L-functions. Silberman mentioned the problem of predicting what could be the expected bound for the infinite norm of general automorphic forms when the spectral parameters go to infinity. \\n\\n• Jutila suggested to prove some Ω-results about the error term which oc-curs in some moments of families of L-functions. For instance, \\n\\n∫ \\n\\n> tÉT\\n\\n∣∣L( f, 1/2 + i t )∣∣2 dt = Main( T ) + Error( T )where Error( T ) = Ω(pT ) is expected to hold for any GL 2-automorphic form. What could be some applications of such Ω-results? \\n\\n• Venkatesh suggested to find a way to guess when it is possible to prove some asymptotic formula for some moment given by \\n\\n∑ \\n\\n> f∈F\\n\\nL( f, 1/2) where as usual the conductor of each L-function of F is of size almost constant say Q(F ) in the logarithmic scale and Q(F ) → +∞. If 4 log |F | > log Q(F )then we generally can prove an asymptotic formula. At the moment, the record is 6 log |F | = log Q(F )in the work of Conrey and Iwaniec on the cubic moment of automorphic \\n\\nL-functions. Can we do better? \\n\\n• Soundararajan asked if it is possible to prove some subconvexity bound in the critical strip but outside the critical line and eventually near the edge of the critical strip. One known instance is the ζ function in the \\n\\ns-aspect near ℜs = 1 via Vinogradov. \\n\\n• Michel asked about an analogy of (sub)convexity for p-adic L-functions. The answer could be some integrality property (Mazur, Prasad,...). 4. T HURSDAY PROBLEM SESSION \\n\\n• Venkatesh convinced us that many equidistribution results are useful to prove asymptotic formula for moments of L-functions with a power sav-ing in the error term if the suitable equidistribution results are quantita-tive ones. He illustrated this by two GL 2-examples namely \\n\\n∫ +T\\n\\n> −T\\n\\n∣∣L( f, 1/2 + i t )∣∣2 dt\\n\\nand ×∑ \\n\\n> χmod ( q)\\n\\n∫\\n\\n> R\\n\\n∣∣L( f × χ, 1/2 + i t )∣∣2 dt.PROBLEM SESSIONS: \\\"SUBCONVEXITY BOUNDS FOR L-FUNCTIONS\\\" 7\\n\\n• In addition, he mentioned what could be the obstacles to do the same in higher rank cases. On one hand, it is very hard to prove an inte-gral representation (period) of L-functions which occured in higher rank since the archimedean computation may be very far away from obvious. According to Garett, the Diaconu-Garrett-Goldfeld extension of Good's spectral-theory-based method to GL n xGL n−1 illustrates the complica-tions at archimedean places. On the other hand, it is necessary to anal-yse such integral representation via ergodic theory or spectral methods. Two difficult instances are given by On × On−1 and GL n ×GL n−1.\\n\\n• An application of subconvexity bounds for GL n may be some informa-tion about the 2 n-moment of the Riemann ζ function. For instance, some Motohashi type formula make the link between GL 3 and |ζ|6. We also have to mention the work of Conrey and Iwaniec. 5. F RIDAY PROBLEM SESSION \\n\\n• There will be a website dedicated to subconvexity for L-functions. Peo-ple agreed that it should contain references to important results, a list of people with their current attempts and previous results also. \\n\\n• Venkatesh put the stress on the subconvexity problem in higher-rank cases. Few methods are known which have bearing on it. After Venkatesh-Lindenstrauss and Bernstein-Reznikoff treatments of triple products, the exceptions are works in progress mentionned by Garett: Venkatesh's ap-plications of ergodic theoretic ideas coming from Ratner and Clozel, and Diaconu-Garrett-Goldfeld's GL n version of an old method of Good. Venkatesh suggested that a first higher rank example to undertake should be GL 3 × GL 2 with f3 on GL 3 is fixed and f2 is varying. In order to get some insight into that, it would be profitable for everybody that classi-cal analytic number theorists try to understand the case when f3 is an Eisenstein series namely \\n\\n∑ \\n\\n> fHecke-Maass of level 1 and eigenvalue 1/4 +t2\\n> f\\n> tfvT\\n\\n∫ \\n\\n> tvT\\n\\n∣∣L( f, 1/2 + i t )∣∣6 dt.Note that the size of the family is about T 3 whereas the size of the an-alytic conductor is about T 12 which reveals the level of difficulty. Also, people should understand where the GL 3-theory occurs in the analytic analysis.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/subconvexity/subconvexity.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0102", "aim-domain:analytic-number-theory", "aim-workshop:subconvexity", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical object is a malformed whole-document extraction rather than one mathematical problem. For the recurring conductor-dropping theme, this attempt proves that an r-fold tempered spherical GL(2) archimedean tensor product along t_j=a_j T+O(1) and height tau=bT+O(1) has conductor C_infinity asymptotic to T^(2^r-N), where N counts resonant sign vectors. If exactly k slopes are positive, then N is at most 2^(r-k) times binomial(k,floor(k/2)), sharply; a multiscale variant sums the individual gamma-factor exponents.\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: the archimedean conductor exponent on a linear spectral ray is 2^r-N, with resonant signed gamma parameters counted with multiplicity, and the collision count satisfies the sharp bound N <= 2^(r-k) binomial(k,floor(k/2)) when k input slopes grow.", "research_difficulty_suggested": "L2" }, { "id": 20000739, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0103", "title": "Character-phase evidence for Elliott--Halberstam and a sparse twisted-psi diagonal", "statement": "1. Evidence in support of the Elliott-Halberstam conjecture.\n\n• J.B. Conrey asked what evidence exists in support of the Elliott-Halberstam conjecture (EH). He also suggested that the search for further such evidence may shed some light on the conjecture or even lead to the resolution of an \"easy\" case, which in turn would su ffi ce to establish the existence of infinitely many bounded gaps between primes by the GPY method. Similar comments were also made by G. Harcos in his contributed statement.\n\n• R.C. Vaughan observed that the Barban-Davenport-Halberstam theorem is consistent with EH.\n\n• A. Granville noted that the Cram´ er probabilistic model also supports EH.\n\n• In an earlier lecture on the Bombieri-Vinogradov theorem and the large sieve, R.C. Vaughan had remarked that progress toward EH would most likely exploit cancellation among the values of ¯ χ(a) in the formula\n\nψ(x; q, a) = 1\n\nφ(q)\n\n∑\n\n> χmod q\n\n¯χ(a)ψ(x, χ ). (1) J.B. Conrey asked whether known results on the asymptotic of the sum\n\n∑∗\n\n> χmod q\n\nχ(a)|L(1 /2, χ )|2\n\nsuggest that the desired cancellation in (1) may be nonexistent.\n\n• K. Soundararajan noted that the results mentioned by Conrey can be explained via the Euler product of L(s, χ )and that there is no such phenomenon in the case of ψ(x, χ ). Therefore, it seems that Vaughan's remark remains valid until further notice.", "original_statement": "1. Evidence in support of the Elliott-Halberstam conjecture. \n\n• J.B. Conrey asked what evidence exists in support of the Elliott-Halberstam conjecture (EH). He also suggested that the search for further such evidence may shed some light on the conjecture or even lead to the resolution of an \"easy\" case, which in turn would su ffi ce to establish the existence of infinitely many bounded gaps between primes by the GPY method. Similar comments were also made by G. Harcos in his contributed statement. \n\n• R.C. Vaughan observed that the Barban-Davenport-Halberstam theorem is consistent with EH. \n\n• A. Granville noted that the Cram´ er probabilistic model also supports EH. \n\n• In an earlier lecture on the Bombieri-Vinogradov theorem and the large sieve, R.C. Vaughan had remarked that progress toward EH would most likely exploit cancellation among the values of ¯ χ(a) in the formula \n\nψ(x; q, a) = 1\n\nφ(q)\n\n∑ \n\n> χmod q\n\n¯χ(a)ψ(x, χ ). (1) J.B. Conrey asked whether known results on the asymptotic of the sum \n\n∑∗ \n\n> χmod q\n\nχ(a)|L(1 /2, χ )|2\n\nsuggest that the desired cancellation in (1) may be nonexistent. \n\n• K. Soundararajan noted that the results mentioned by Conrey can be explained via the Euler product of L(s, χ )and that there is no such phenomenon in the case of ψ(x, χ ). Therefore, it seems that Vaughan's remark remains valid until further notice.", "clean_statement": "1. Evidence in support of the Elliott-Halberstam conjecture.\n\n• J.B. Conrey asked what evidence exists in support of the Elliott-Halberstam conjecture (EH). He also suggested that the search for further such evidence may shed some light on the conjecture or even lead to the resolution of an \"easy\" case, which in turn would su ffi ce to establish the existence of infinitely many bounded gaps between primes by the GPY method. Similar comments were also made by G. Harcos in his contributed statement.\n\n• R.C. Vaughan observed that the Barban-Davenport-Halberstam theorem is consistent with EH.\n\n• A. Granville noted that the Cram´ er probabilistic model also supports EH.\n\n• In an earlier lecture on the Bombieri-Vinogradov theorem and the large sieve, R.C. Vaughan had remarked that progress toward EH would most likely exploit cancellation among the values of ¯ χ(a) in the formula\n\nψ(x; q, a) = 1\n\nφ(q)\n\n∑\n\n> χmod q\n\n¯χ(a)ψ(x, χ ). (1) J.B. Conrey asked whether known results on the asymptotic of the sum\n\n∑∗\n\n> χmod q\n\nχ(a)|L(1 /2, χ )|2\n\nsuggest that the desired cancellation in (1) may be nonexistent.\n\n• K. Soundararajan noted that the results mentioned by Conrey can be explained via the Euler product of L(s, χ )and that there is no such phenomenon in the case of ψ(x, χ ). Therefore, it seems that Vaughan's remark remains valid until further notice.", "statement_status": "exact", "statement_verification": "The source is Problem 1 in Angel Kumchev's notes from the December 2005 ARCC workshop *Gaps Between Primes*. The original PDF was inspected directly. The extracted record is faithful in substance, but its PDF-to-text conversion split ligatures and displayed sums vertically. With only those typographical defects repaired, the key passage is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[102]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Evidence in support of the Elliott-Halberstam conjecture. \\n\\n• J.B. Conrey asked what evidence exists in support of the Elliott-Halberstam conjecture (EH). He also suggested that the search for further such evidence may shed some light on the conjecture or even lead to the resolution of an \\\"easy\\\" case, which in turn would su ffi ce to establish the existence of infinitely many bounded gaps between primes by the GPY method. Similar comments were also made by G. Harcos in his contributed statement. \\n\\n• R.C. Vaughan observed that the Barban-Davenport-Halberstam theorem is consistent with EH. \\n\\n• A. Granville noted that the Cram´ er probabilistic model also supports EH. \\n\\n• In an earlier lecture on the Bombieri-Vinogradov theorem and the large sieve, R.C. Vaughan had remarked that progress toward EH would most likely exploit cancellation among the values of ¯ χ(a) in the formula \\n\\nψ(x; q, a) = 1\\n\\nφ(q)\\n\\n∑ \\n\\n> χmod q\\n\\n¯χ(a)ψ(x, χ ). (1) J.B. Conrey asked whether known results on the asymptotic of the sum \\n\\n∑∗ \\n\\n> χmod q\\n\\nχ(a)|L(1 /2, χ )|2\\n\\nsuggest that the desired cancellation in (1) may be nonexistent. \\n\\n• K. Soundararajan noted that the results mentioned by Conrey can be explained via the Euler product of L(s, χ )and that there is no such phenomenon in the case of ψ(x, χ ). Therefore, it seems that Vaughan's remark remains valid until further notice.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0103", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted Elliott--Halberstam conjecture remains open, but two rigorous partial results sharpen the workshop discussion. First, an explicit concentration theorem proves that Barban--Davenport--Halberstam character energy yields an Elliott--Halberstam-strength bound with high probability after independent randomization of the nonprincipal character phases, isolating actual phase coherence as the missing deterministic issue. Second, primitive-character orthogonality gives an exact formula for the twisted second moment of psi(x,chi); its zero-shift line n=a m vanishes for every fixed non-prime-power twist a and has an exact logarithmic formula when a is a prime power. This explains why the long Euler-product diagonal in twisted central L-value moments does not transfer to von Mangoldt coefficients, while leaving nonzero shifted correlations and actual EH unresolved.\n\nCandidate contribution (lemma; novelty confidence low): For the least positive fixed twist 2 <= a < q with (a,q)=1, the exact n=a m contribution to the primitive twisted moment sum over primitive chi mod q of chi(a)|psi(x,chi)|^2 is zero unless a=p^k is a prime power; if a=p^k, it equals phi*(q)(log p)^2 max(0,floor(log_p x)-k)." }, { "id": 20000740, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0104", "title": "Exact indicator extremizers for complete-grid large sieve inequalities", "statement": "2. On the large sieve, I.\n\n• H. Helfgott asked whether the standard large sieve inequality for the exponential sum\n\nS (α) =\n\n> M+N\n\n∑\n\n> n=M+1\n\nane(αn)is best possible for coe ffi cient sequences with an ∈ { 0, 1}. He noted that the usual proof that the large sieve is essentially best possible uses a sequence that does not belong to this class.\n\n• R.C. Vaughan said that the answer to this question is most likely in the a ffi rmative.", "original_statement": "2. On the large sieve, I. \n\n• H. Helfgott asked whether the standard large sieve inequality for the exponential sum \n\nS (α) =\n\n> M+N\n\n∑\n\n> n=M+1\n\nane(αn)is best possible for coe ffi cient sequences with an ∈ { 0, 1}. He noted that the usual proof that the large sieve is essentially best possible uses a sequence that does not belong to this class. \n\n• R.C. Vaughan said that the answer to this question is most likely in the a ffi rmative.", "clean_statement": "2. On the large sieve, I.\n\n• H. Helfgott asked whether the standard large sieve inequality for the exponential sum\n\nS (α) =\nM+N\n\n∑\nn=M+1\n\n, and the split best possible for coe ffi cient sequences with an ∈ { 0, 1}. He noted that the usual proof that the large sieve is essentially best possible uses a sequence that does not belong to this class.\n\n• R.C. Vaughan said that the answer to this question is most likely in the a ffi rmative.", "statement_status": "reconstructed_unverified", "statement_verification": "A second plausible reading is the additive arithmetic/Farey form \\[ \\mathcal L_Q(a):= \\sum_{q\\leq Q}\\ \\sum_{\\substack{1\\leq h\\leq q\\\\(h,q)=1}} \\left|S\\!\\left(\\frac hq\\right)\\right|^2 \\leq (N-1+Q^2)\\sum_n|a_n|^2, \\tag{FLS} \\] where \\(h/q\\) is viewed modulo one and \\(q=1,h=1\\) supplies the point \\(0\\). This follows from (LS), since two distinct reduced fractions of denominator at most \\(Q\\) are at circular distance at least \\(Q^{-2}\\). It is often written with \\(N+Q^2\\). Both readings are treated below; their sharpness questions are not identical.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[103]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. On the large sieve, I. \\n\\n• H. Helfgott asked whether the standard large sieve inequality for the exponential sum \\n\\nS (α) =\\n\\n> M+N\\n\\n∑\\n\\n> n=M+1\\n\\nane(αn)is best possible for coe ffi cient sequences with an ∈ { 0, 1}. He noted that the usual proof that the large sieve is essentially best possible uses a sequence that does not belong to this class. \\n\\n• R.C. Vaughan said that the answer to this question is most likely in the a ffi rmative.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0104", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the analytic delta-spaced large sieve, the complete grid alpha_r=r/R has exact operator norm R ceil(N/R) even after coefficients are restricted to {0,1}; an extremizer is the indicator of a fullest residue class modulo R. Hence, when N is congruent to 1 modulo R, a {0,1} sequence attains the sharp factor N+R-1 exactly, and for arbitrary delta the construction comes within less than one additively of N-1+delta^{-1} along infinitely many lengths. For the alternative fixed-Farey interpretation, singleton and all-ones indicators prove the N+Q^2 order only up to an absolute factor, leaving its critical leading constant unresolved.\n\nCandidate contribution (exact_extremal_lemma; novelty confidence low): On the complete rational grid of order R, the supremum of the large-sieve Rayleigh quotient over nonzero {0,1} coefficient vectors is exactly R ceil(N/R), equal to the unrestricted complex-coefficient norm; consequently the restricted analytic constant is within less than one of N-1+delta^{-1} when R=floor(delta^{-1}) and N is 1 modulo R." }, { "id": 20000741, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0105", "title": "A spliced large-sieve/zeta record and a local 1-line zeta certificate", "statement": "3. On the large sieve, II. Y. Motohashi proposed to try to improve the large sieve by restricting the moduli. He mentioned that D. Wolke has results in that direction for prime moduli. 14. |tζ(1 + it )| ≥ 1.\n\n• J. Pintz asked whether it is true that\n\n|tζ(1 + it )| ≥ 1 for all t ∈ R.\n\nHe noted that this inequality holds when |t| ≥ t0 and when |t| ≤ δ. A complete proof would simplify the GPY method at certain points.\n\n• R.C. Vaughan recalled being asked a similar question in the past and that he was able to answer it in the affi rmative. He said that he thinks that he has the solution written somewhere in his files.", "original_statement": "3. On the large sieve, II. Y. Motohashi proposed to try to improve the large sieve by restricting the moduli. He mentioned that D. Wolke has results in that direction for prime moduli. 14. |tζ(1 + it )| ≥ 1.\n\n• J. Pintz asked whether it is true that \n\n|tζ(1 + it )| ≥ 1 for all t ∈ R.\n\nHe noted that this inequality holds when |t| ≥ t0 and when |t| ≤ δ. A complete proof would simplify the GPY method at certain points. \n\n• R.C. Vaughan recalled being asked a similar question in the past and that he was able to answer it in the affi rmative. He said that he thinks that he has the solution written somewhere in his files.", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "The canonical record is not one mathematical problem. It joins two consecutive items from the AIM workshop list *Gaps between primes*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[104]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. On the large sieve, II. Y. Motohashi proposed to try to improve the large sieve by restricting the moduli. He mentioned that D. Wolke has results in that direction for prime moduli. 14. |tζ(1 + it )| ≥ 1.\\n\\n• J. Pintz asked whether it is true that \\n\\n|tζ(1 + it )| ≥ 1 for all t ∈ R.\\n\\nHe noted that this inequality holds when |t| ≥ t0 and when |t| ≤ δ. A complete proof would simplify the GPY method at certain points. \\n\\n• R.C. Vaughan recalled being asked a similar question in the past and that he was able to answer it in the affi rmative. He said that he thinks that he has the solution written somewhere in his files.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0105", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is not a single problem: primary-PDF inspection shows that it splices Problem 3, an underspecified restricted-modulus large-sieve direction, to Problem 4, the inequality |t zeta(1+it)| >= 1; the apparent heading 14 is footer page number 1 fused to heading 4. Recovered Problem 4 is solved by C. Y. Yıldırım's 2026 theorem giving strict inequality for every nonzero real t. Independently, this attempt proves the fourth-order expansion |t zeta(1+it)|^2 = 1 + (gamma^2 + 2 gamma_1)t^2 + (gamma_1^2 - gamma gamma_2 - gamma_3/3)t^4 + O(t^6), proves gamma^2 + 2 gamma_1 > 0 by elementary rational estimates, and derives the explicit range |t| >= 250 from a published reciprocal-zeta bound.\n\nCandidate contribution (lemma; novelty confidence low): Candidate contribution: a table-free rational certificate gamma > 1/2 and gamma_1 > -1/8 proves that t = 0 is a strict local minimum of the continuously extended function |t zeta(1+it)|^2; together with the exact quartic coefficient and an explicit reciprocal bound, it isolates the historical issue to a bounded intermediate range and gives the cutoff |t| >= 250.", "research_difficulty_suggested": "L2" }, { "id": 20000742, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0106", "title": "Local singular series and multiplicative-order correlations for Elliott's shifted-prime ratio", "statement": "5. The multiplicative twin-prime problem of Elliott.\n\n• In 2003 P.D.T.A. Elliott proved that for every fixed integer a > 1, the equation\n\nak = p + 1\n\nq + 1 (2) has infinitely many solutions in primes p, q and integers k with 2 ≤ k ≤ log q. K.B. Ford suggested to try to adapt the GPY method to reduce the range for k in Elliott's result.\n\n• J. Pintz remarked that an upper bound of the form k\n (log q)1/2+[U+000F] seemed a reasonable goal at that stage of our understanding of the GPY method.\n\n• Several participants noted that the adaptation of the GPY method to this problem would require an analogue of Gallagher's result on the average size of the singular series for prime k-tuples. A. Granville noted that the problem of understanding the singular series of (2) may be of independent interest.", "original_statement": "5. The multiplicative twin-prime problem of Elliott. \n\n• In 2003 P.D.T.A. Elliott proved that for every fixed integer a > 1, the equation \n\nak = p + 1\n\nq + 1 (2) has infinitely many solutions in primes p, q and integers k with 2 ≤ k ≤ log q. K.B. Ford suggested to try to adapt the GPY method to reduce the range for k in Elliott's result. \n\n• J. Pintz remarked that an upper bound of the form k \u001c (log q)1/2+\u000f seemed a reasonable goal at that stage of our understanding of the GPY method. \n\n• Several participants noted that the adaptation of the GPY method to this problem would require an analogue of Gallagher's result on the average size of the singular series for prime k-tuples. A. Granville noted that the problem of understanding the singular series of (2) may be of independent interest.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record is visibly damaged by vertical-layout extraction and two control characters. In particular, it contains the lines", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[105]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"5. The multiplicative twin-prime problem of Elliott. \\n\\n• In 2003 P.D.T.A. Elliott proved that for every fixed integer a > 1, the equation \\n\\nak = p + 1\\n\\nq + 1 (2) has infinitely many solutions in primes p, q and integers k with 2 ≤ k ≤ log q. K.B. Ford suggested to try to adapt the GPY method to reduce the range for k in Elliott's result. \\n\\n• J. Pintz remarked that an upper bound of the form k \\u001c (log q)1/2+\\u000f seemed a reasonable goal at that stage of our understanding of the GPY method. \\n\\n• Several participants noted that the adaptation of the GPY method to this problem would require an analogue of Gallagher's result on the average size of the singular series for prime k-tuples. A. Granville noted that the problem of understanding the singular series of (2) may be of independent interest.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0106", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For A = a^k, the two forms n and A n + A - 1 have one forbidden residue modulo a prime ell exactly when ell divides A(A-1), and two otherwise. Hence they are always admissible and their singular series is 2 C_2 times the finite correction product over odd primes ell dividing a(a^k-1) of (ell-1)/(ell-2). More generally, geometric tuples are automatically admissible. For every fixed prime cutoff z, the exponent-average of the truncated pair singular series is given exactly by a subset sum whose denominators are lcms of the orders ord_ell(a); this yields an explicit nonnegative correlation surplus over the naive independent-events model.\n\nCandidate contribution (theorem; novelty confidence low): For fixed a and prime cutoff z, the normalized average over 1 <= k <= K of the truncated singular series equals the exact subset sum sum_D c_D floor(K/r_D)/K, where c_D is the product of 1/(ell-2) and r_D is the lcm of the multiplicative orders ord_ell(a). Its fixed-z limit dominates the independent Euler-product benchmark, and the pairwise covariance is (gcd(r_ell,r_m)-1)/(r_ell r_m)." }, { "id": 20000743, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0107", "title": "A multiplier-repetition obstruction for long prime chains", "statement": "6. Long chains of primes.\n\n• K.B. Ford proposed the following problem: Let x be large and consider a sequence of primes p1,..., pk ≤ x\n\nsuch that for all j = 2,..., k,\n\np j = m j p j−1 + 1 for some m j ∈ N.\n\nObviously, k = O(log x). Prove or disprove that k = o(log x).\n\n• H.L. Montgomery noted that this question reminded him of a problem of Erd ¨ os: Show that there are finitely many n such that n − 2k is prime for all integers k with 2 k < n.\n\n• R.C. Vaughan recalled that Erd¨ os thought that n = 105 is the largest such n.", "original_statement": "6. Long chains of primes. \n\n• K.B. Ford proposed the following problem: Let x be large and consider a sequence of primes p1,..., pk ≤ x\n\nsuch that for all j = 2,..., k,\n\np j = m j p j−1 + 1 for some m j ∈ N.\n\nObviously, k = O(log x). Prove or disprove that k = o(log x). \n\n• H.L. Montgomery noted that this question reminded him of a problem of Erd ¨ os: Show that there are finitely many n such that n − 2k is prime for all integers k with 2 k < n.\n\n• R.C. Vaughan recalled that Erd¨ os thought that n = 105 is the largest such n.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical input is record 106 (zero-based) of `aim-analytic-number-theory-notes.json`. Its primary source is Angel Kumchev's notes from the December 2005 ARCC workshop *Gaps Between Primes* [AIM2005, p. 2 of the PDF]. The source was inspected directly, rather than relying only on the extracted record.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[106]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"6. Long chains of primes. \\n\\n• K.B. Ford proposed the following problem: Let x be large and consider a sequence of primes p1,..., pk ≤ x\\n\\nsuch that for all j = 2,..., k,\\n\\np j = m j p j−1 + 1 for some m j ∈ N.\\n\\nObviously, k = O(log x). Prove or disprove that k = o(log x). \\n\\n• H.L. Montgomery noted that this question reminded him of a problem of Erd ¨ os: Show that there are finitely many n such that n − 2k is prime for all integers k with 2 k < n.\\n\\n• R.C. Vaughan recalled that Erd¨ os thought that n = 105 is the largest such n.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0107", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any prime chain p_j = m_j p_{j-1} + 1 with k at least 3, let n = k - 2 and let R be the maximum multiplicity of one value among m_3,...,m_k. Then p_k > (2n/(eR))^n. Consequently, R = o(k) forces k = o(log p_k), and bounded-multiplicity chains have k at most (1+o(1)) log p_k/log log p_k. Conversely, any family with n at least eta log p_k must reuse one bounded even multiplier on a positive proportion of its tail transitions; it also contains, for every fixed length, infinitely recurring fixed bounded multiplier words after passage to a subsequence.\n\nCandidate contribution (lemma; novelty confidence low): The explicit tail-multiplier inequality p_k > (2n/(eR))^n and its quantified consequence that a linear-logarithmic counterexample must have a single bounded even multiplier occurring with positive density." }, { "id": 20000744, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0108", "title": "PDF-splice repair and a gap-scale reduction for two independent prime questions", "statement": "7. Small gaps between primes in thin sequences.\n\n• A. Kantorovich asked whether it is possible to prove the existence of small gaps between primes from the sequence\n\n{p ≤ x | p = [n log n] for some n ∈ N}.\n\nThis is a \"thin\" set of primes: #{p ≤ x | p = [n log n] for some n ∈ N} ∼ x(log x)−2.\n\n• Several participants noted that if one is interested in this question, then one may also investigate the analogous question for Piatetski-Shapiro primes. 28. Mersenne composites.\n\n• A. Granville asked whether it is possible to use the GPY method to prove (under the assumption of EH, if necessary) that there are infinitely many composites of the form 2 p − 1 (Mersenne composites).\n\n• C. Elsholtz noted that there may exist related work by M.R. Murty under the assumption of Artin's primitive root conjecture.", "original_statement": "7. Small gaps between primes in thin sequences. \n\n• A. Kantorovich asked whether it is possible to prove the existence of small gaps between primes from the sequence \n\n{p ≤ x | p = [n log n] for some n ∈ N}.\n\nThis is a \"thin\" set of primes: #{p ≤ x | p = [n log n] for some n ∈ N} ∼ x(log x)−2.\n\n• Several participants noted that if one is interested in this question, then one may also investigate the analogous question for Piatetski-Shapiro primes. 28. Mersenne composites. \n\n• A. Granville asked whether it is possible to use the GPY method to prove (under the assumption of EH, if necessary) that there are infinitely many composites of the form 2 p − 1 (Mersenne composites). \n\n• C. Elsholtz noted that there may exist related work by M.R. Murty under the assumption of Artin's primitive root conjecture.", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "The canonical record is not one mathematical problem. It is an extraction splice from Angel Kumchev's notes for the December 2005 ARCC workshop *Gaps Between Primes*. The primary five-page PDF and the neighboring canonical records were checked directly.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[107]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"7. Small gaps between primes in thin sequences. \\n\\n• A. Kantorovich asked whether it is possible to prove the existence of small gaps between primes from the sequence \\n\\n{p ≤ x | p = [n log n] for some n ∈ N}.\\n\\nThis is a \\\"thin\\\" set of primes: #{p ≤ x | p = [n log n] for some n ∈ N} ∼ x(log x)−2.\\n\\n• Several participants noted that if one is interested in this question, then one may also investigate the analogous question for Piatetski-Shapiro primes. 28. Mersenne composites. \\n\\n• A. Granville asked whether it is possible to use the GPY method to prove (under the assumption of EH, if necessary) that there are infinitely many composites of the form 2 p − 1 (Mersenne composites). \\n\\n• C. Elsholtz noted that there may exist related work by M.R. Murty under the assumption of Artin's primitive root conjecture.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0108", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is invalid as a single statement: it concatenates Problem 7 on prime values of floor(n log n), PDF footer 2, and Problem 8 on prime-exponent Mersenne composites. After preserving both recovered questions, a proved scale-dictionary reduction shows that if q_j=floor(n_j log n_j) are the prime values and h_j=n_{j+1}-n_j, then liminf (q_{j+1}-q_j)/log q_j equals liminf h_j in the extended sense, while mean-normalized gaps have liminf zero exactly when h_j/log n_j has liminf zero. Thus GPY's usual log-normalized zero target is obstructed by the sequence mesh, and the precise strong target is bounded producing-index gaps. The Mersenne fragment remains open; the report proves the standard divisor-form obstruction and Sophie-Germain conditional route without claiming a solution.\n\nCandidate contribution (reduction; novelty confidence low): For a_n=floor(n log n), prime values q_j=a_{n_j}, and h_j=n_{j+1}-n_j, the exact identities liminf (q_{j+1}-q_j)/log q_j = liminf h_j and [liminf (q_{j+1}-q_j)/(log q_j)^2=0 iff liminf h_j/log n_j=0] give a rigorous scale dictionary and mesh obstruction for the workshop's undefined term 'small gaps'.", "research_difficulty_suggested": "L2" }, { "id": 20000745, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0109", "title": "Signed large and recurrent fixed second differences of consecutive primes", "statement": "9. Second order di ff erences between primes.\n\n• T.D. Wooley proposed the following problem: How often are the second di ff erences\n\npn+2 − 2pn+1 + pn\n\nsmall (large)? He noted that Erd¨ os has proved that\n\n|pn+2 − 2pn+1 + pn| ≥ (1 + δ) log pn (3) infinitely often.\n\n• A. Balog added that he had thought about the problem in the past and had convinced himself that the left side of (3) is infinitely often as large as the largest known gap between consecutive primes.", "original_statement": "9. Second order di ff erences between primes. \n\n• T.D. Wooley proposed the following problem: How often are the second di ff erences \n\npn+2 − 2pn+1 + pn\n\nsmall (large)? He noted that Erd¨ os has proved that \n\n|pn+2 − 2pn+1 + pn| ≥ (1 + δ) log pn (3) infinitely often. \n\n• A. Balog added that he had thought about the problem in the past and had convinced himself that the left side of (3) is infinitely often as large as the largest known gap between consecutive primes.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "This record is Problem 9 in the AIM workshop notes *Gaps Between Primes* (December 2005), recorded by Angel Kumchev. The source PDF gives the following question (typographical spacing and OCR errors have been repaired, but the mathematical wording has not been strengthened):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[108]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"9. Second order di ff erences between primes. \\n\\n• T.D. Wooley proposed the following problem: How often are the second di ff erences \\n\\npn+2 − 2pn+1 + pn\\n\\nsmall (large)? He noted that Erd¨ os has proved that \\n\\n|pn+2 − 2pn+1 + pn| ≥ (1 + δ) log pn (3) infinitely often. \\n\\n• A. Balog added that he had thought about the problem in the past and had convinced himself that the left side of (3) is infinitely often as large as the largest known gap between consecutive primes.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0109", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Writing A_n=p_{n+2}-2p_{n+1}+p_n=d_{n+1}-d_n, the Sun--Pan bounded-cluster/long-gap construction implies A_n is at least a positive constant times L(p_n)=log(p_n)log_2(p_n)log_4(p_n)/log_3(p_n) for at least on the order of log X indices with p_n<=X. A convex/concave specialization of Baker--Freiberg's two-gap chain theorem proves limsup A_n/F(p_n)=+infinity and liminf A_n/F(p_n)=-infinity for F(x)=log(x)log_2(x)log_5(x)/log_3(x). Pintz additionally gives on the order of log X occurrences of each sign on the classical Erdos--Rankin scale, while the fixed-tuple argument of Banks--Freiberg--Turnage-Butterbaugh gives recurring fixed nonzero values of each sign.\n\nCandidate contribution (corollary; novelty confidence low): Retaining the every-sufficiently-large-N uniformity in Sun--Pan yields an explicit Omega(log X) count of positive full-L-scale second differences, while scaled convex and concave 64-point configurations in Baker--Freiberg Theorem 6.4 yield both signed infinite extremal limits after an explicit denominator-transfer check." }, { "id": 20000746, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0110", "title": "Sparse-gap data do not determine Montgomery pair correlation", "statement": "10. Di ff erences between primes and pair correlation.\n\n• Let F(α) be Montgomery's pair correlation function. In 1982 D.R. Heath-Brown proved that, under RH and Montgomery's Pair Correlation Conjecture, lim inf\n\n> n→∞\n\npn+1 − pn\n\nlog pn\n\n= 0.\n\nT.H. Chan asked whether it is possible to prove a converse result (which, of course, would be unconditional post-GPY).\n\n• J. Pintz expressed serious doubt.\n\n• In the same paper, Heath-Brown proved also (under the same assumptions) that\n\npn+1 − pn\n √pn log pn.\n\nJ.B. Conrey asked whether it is possible to use random matrix theory (RMT) to improve further on this result.\n\n• K. Soundararajan felt quite strongly that RMT should not help in this problem.", "original_statement": "10. Di ff erences between primes and pair correlation. \n\n• Let F(α) be Montgomery's pair correlation function. In 1982 D.R. Heath-Brown proved that, under RH and Montgomery's Pair Correlation Conjecture, lim inf \n\n> n→∞\n\npn+1 − pn\n\nlog pn\n\n= 0.\n\nT.H. Chan asked whether it is possible to prove a converse result (which, of course, would be unconditional post-GPY). \n\n• J. Pintz expressed serious doubt. \n\n• In the same paper, Heath-Brown proved also (under the same assumptions) that \n\npn+1 − pn \u001c √pn log pn.\n\nJ.B. Conrey asked whether it is possible to use random matrix theory (RMT) to improve further on this result. \n\n• K. Soundararajan felt quite strongly that RMT should not help in this problem.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The record is Problem 10 in Angel Kumchev's notes from the December 2005 ARCC workshop *Gaps Between Primes*. The official AIM PDF was checked, as was the primary paper of Heath-Brown. The mathematically unambiguous repaired statement is as follows. Write", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 10\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[109]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"10. Di ff erences between primes and pair correlation. \\n\\n• Let F(α) be Montgomery's pair correlation function. In 1982 D.R. Heath-Brown proved that, under RH and Montgomery's Pair Correlation Conjecture, lim inf \\n\\n> n→∞\\n\\npn+1 − pn\\n\\nlog pn\\n\\n= 0.\\n\\nT.H. Chan asked whether it is possible to prove a converse result (which, of course, would be unconditional post-GPY). \\n\\n• J. Pintz expressed serious doubt. \\n\\n• In the same paper, Heath-Brown proved also (under the same assumptions) that \\n\\npn+1 − pn \\u001c √pn log pn.\\n\\nJ.B. Conrey asked whether it is possible to use random matrix theory (RMT) to improve further on this result. \\n\\n• K. Soundararajan felt quite strongly that RMT should not help in this problem.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0110", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The OCR-damaged bound is recovered as g_n << sqrt(p_n log p_n), with the entire product under the radical, and the requested converse is identified as small normalized prime gaps implying pair correlation. Two rigorous obstructions are proved: an optimized single-gap lower bound J_theta(Y,2G/3) >= 4G^3/27 exposes the square-root barrier of a natural L2/positivity argument, and an explicit PNT-density generalized-prime point set has liminf normalized gap zero while its dyadic short-interval variance at h=(log X)^2 is O(X log^2 X), a full logarithm below the pair-correlation scale. This does not refute a converse special to the actual primes, but proves that no model-independent converse can use only PNT density and the scalar liminf statement.\n\nCandidate contribution (countermodel theorem; novelty confidence low): For the explicit recursion a_{m+1}=a_m+log a_m, followed by insertion of lacunary consecutive pairs q_j,q_j+1, the resulting PNT-density point set B satisfies liminf (b_{n+1}-b_n)/log b_n=0 but J_B^*(X,(log X)^2)=O(X log^2 X)=o(X log^2 X log(X/log^2 X))." }, { "id": 20000747, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0111", "title": "Support-gap obstructions and a local-discrepancy converse for zeta", "statement": "Traditionally, analytic information about the zeta-function has been\nused to derive upper bounds for differences between consecutive primes.\nY. Motohashi asked whether this process can be reversed. For example,\nwhat (if anything) can be said about \\(\\zeta(s)\\) in the critical strip\nunder (H)?", "original_statement": "11. Di ff erences between primes and ζ(s). Traditionally, analytic information about the zeta-function has been used to derive upper bounds for the di ff erences between consecutive primes. Y. Motohashi asked whether this process can be reversed. For example, what (if anything) can be said about ζ(s) in the critical strip under the assumption that \n\npn+1 − pn \u001c\u000f p\u000f\n\n> n?", "clean_statement": "Traditionally, analytic information about the zeta-function has been\nused to derive upper bounds for differences between consecutive primes.\nY. Motohashi asked whether this process can be reversed. For example,\nwhat (if anything) can be said about \\(\\zeta(s)\\) in the critical strip\nunder (H)?", "statement_status": "corrected_verified", "statement_verification": "The canonical record is problem 11 from the December 2005 ARCC workshop *Gaps Between Primes*, in notes prepared by Angel Kumchev. The repository extraction ends with corrupted control characters: The official AIM PDF was checked at the displayed formula on its third page. The formula is unambiguously", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 11\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[110]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"11. Di ff erences between primes and ζ(s). Traditionally, analytic information about the zeta-function has been used to derive upper bounds for the di ff erences between consecutive primes. Y. Motohashi asked whether this process can be reversed. For example, what (if anything) can be said about ζ(s) in the critical strip under the assumption that \\n\\npn+1 − pn \\u001c\\u000f p\\u000f\\n\\n> n?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0111", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source hypothesis is exactly p_{n+1}-p_n \\ll_\\epsilon p_n^\\epsilon for every fixed epsilon>0. It yields only one-sided nonemptiness of intervals, not signed prime-counting discrepancy. We prove three complementary results: fixed zero terms are invisible at the guaranteed gap scale; a bounded positive Dirichlet-series logarithmic derivative supported at every integer can realize an arbitrary prescribed conjugate pair of zero signatures; and the genuinely stronger uniform estimate psi(x+x^eta)-psi(x)=x^eta+O(x^{eta-delta}) telescopes to psi(x)=x+O(x^{1-delta}) and hence forces zeta(s) to be zero-free in Re(s)>1-delta.\n\nCandidate contribution (obstruction theorem; novelty confidence low): For every rho=beta+i gamma with 00 supported at every integer m>=2 such that D_rho(s)=sum a_m m^{-s} has residues +1 at 1 and -1 at rho and conjugate rho, while the associated meromorphic function with -Z'/Z=D_rho has a pole at 1 and simple zeros at that prescribed conjugate pair." }, { "id": 20000748, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0112", "title": "Merged GPY and Gaussian-prime questions with a finite-direction occupancy reduction", "statement": "12. Limits of the GPY method.\n\n• H. Helfgott noted that it seems that the GPY method is by nature a lower bound method and will most likely never yield an asymptotic formula for the number of primes with a given spacing. He asked those who were well-versed in the GPY method whether this is indeed so and also whether the GPY method is capable of producing a right-order lower bound.\n\n• J. Pintz replied that an asymptotic formula of any kind appears to be out of the reach of the method. He also noted that the method could potentially yield a \"right-order lower bound\", but that that will involve considerable technical di ffi culties. Some ideas in that direction were sketched in C. Yildirim's talk. It should be noted that in this context, a \"right-order lower bound\" means that one can show that a positive proportion of the k-tuples we consider contain at least two primes. 313. Distances between Gaussian primes. Several members of the audience asked whether the GPY method can be adapted to study small distances between Gaussian primes. T.D. Wooley remarked that such an adaptation should be straightforward.", "original_statement": "12. Limits of the GPY method. \n\n• H. Helfgott noted that it seems that the GPY method is by nature a lower bound method and will most likely never yield an asymptotic formula for the number of primes with a given spacing. He asked those who were well-versed in the GPY method whether this is indeed so and also whether the GPY method is capable of producing a right-order lower bound. \n\n• J. Pintz replied that an asymptotic formula of any kind appears to be out of the reach of the method. He also noted that the method could potentially yield a \"right-order lower bound\", but that that will involve considerable technical di ffi culties. Some ideas in that direction were sketched in C. Yildirim's talk. It should be noted that in this context, a \"right-order lower bound\" means that one can show that a positive proportion of the k-tuples we consider contain at least two primes. 313. Distances between Gaussian primes. Several members of the audience asked whether the GPY method can be adapted to study small distances between Gaussian primes. T.D. Wooley remarked that such an adaptation should be straightforward.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical problem field is preserved here verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 12\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[111]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"12. Limits of the GPY method. \\n\\n• H. Helfgott noted that it seems that the GPY method is by nature a lower bound method and will most likely never yield an asymptotic formula for the number of primes with a given spacing. He asked those who were well-versed in the GPY method whether this is indeed so and also whether the GPY method is capable of producing a right-order lower bound. \\n\\n• J. Pintz replied that an asymptotic formula of any kind appears to be out of the reach of the method. He also noted that the method could potentially yield a \\\"right-order lower bound\\\", but that that will involve considerable technical di ffi culties. Some ideas in that direction were sketched in C. Yildirim's talk. It should be noted that in this context, a \\\"right-order lower bound\\\" means that one can show that a positive proportion of the k-tuples we consider contain at least two primes. 313. Distances between Gaussian primes. Several members of the audience asked whether the GPY method can be adapted to study small distances between Gaussian primes. T.D. Wooley remarked that such an adaptation should be straightforward.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0112", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is malformed: the official AIM PDF shows that its string '313.' is page footer 3 followed by the separate heading 13, so Problems 12 and 13 were merged. For either integer primes or Gaussian primes, a proved finite-union theorem gives max Q_ij <= T_2 <= sum Q_ij and the Bonferroni refinement M_2 - 2 M_3 <= T_2 <= M_2. Consequently, a Hardy-Littlewood-order lower bound for at-least-two-prime occupancy of a fixed finite tuple forces one fixed admissible gap direction to have that order on infinitely many ranges. Maynard resolves a distinct positive-proportion statement over offset patterns via a multidimensional GPY refinement; Castillo et al. unconditionally resolve bounded Gaussian-prime distances, sharpened explicitly by Vatwani to 246, while prescribed-direction right-order pair counts remain open.\n\nCandidate contribution (reduction; novelty confidence low): For every fixed finite pattern in an abelian group, its at-least-two-hit translation count lies between its largest fixed-pair count and the sum of its fixed-pair counts, with M_2 - 2 M_3 <= T_2 <= M_2; hence a natural-scale occupancy liminf forces a single fixed pair-direction natural-scale limsup. Applied to Z and Z[i], this gives a unified obstruction diagnostic, and in Z[i] a two-point direction a+bi is admissible exactly when a and b have the same parity.", "research_difficulty_suggested": "L2" }, { "id": 20000749, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0113", "title": "A quantitative short-interval transfer for small values of ap-q", "statement": "14. Small values of linear forms at prime arguments. A. Balog proposed the following problem: Show that the di ff erence |ap − q|, where p, q are primes and a > 1 is a fixed integer, can be small. More generally, consider the case when a is real, a, 1.", "original_statement": "14. Small values of linear forms at prime arguments. A. Balog proposed the following problem: Show that the di ff erence |ap − q|, where p, q are primes and a > 1 is a fixed integer, can be small. More generally, consider the case when a is real, a, 1.", "clean_statement": "14. Small values of linear forms at prime arguments. A. Balog proposed the following problem: Show that the di ff erence |ap − q|, where p, q are primes and a > 1 is a fixed integer, can be small. More generally, consider the case when a is real, a, 1.", "statement_status": "exact", "statement_verification": "The canonical record is item 14 of the AIM workshop list *Gaps between primes*. Its OCR text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 14\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[112]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"14. Small values of linear forms at prime arguments. A. Balog proposed the following problem: Show that the di ff erence |ap − q|, where p, q are primes and a > 1 is a fixed integer, can be small. More generally, consider the case when a is real, a, 1.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0113", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every fixed positive real a and every eta > 0, every sufficiently large block (P,3P/2) contains at least a positive constant times P/log P primes p for which some prime q satisfies 0 < q-ap < C(a,eta)p^(1/20+eta). This follows rigorously from Jia's published O(X log^(-B) X) exceptional-set theorem and a bounded-multiplicity endpoint map. For rational a=A/B, an exact local calculation shows that the conjectural optimal bounded error is 1/B when AB is even and 2/B when AB is odd, with attainability conditional on Dickson's conjecture.\n\nCandidate contribution (quantitative transfer theorem; novelty confidence low): Jia's quantitative almost-all short-interval theorem implies, for each designated a > 0, a block-by-block lower bound of order P/log P for working prime inputs p and a lower bound of order P^(1+1/20+delta)/(log P)^2 for ordered prime pairs at error scale p^(1/20+delta)." }, { "id": 20000750, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0114", "title": "Alternative GPY weights: exact perturbation geometry and a radial escape", "statement": "15. Alternative weights in the GPY method.\n\n• The GPY method uses the weights\n\nΛR(n; H) = ∑\n\n> d|P(n;H)\n\nλd, λd = μ(d) (log +(R/d)) k+l\n\n(k + l)!,\n\nwhere P(n; H) = (n + h1) · · · (n + hk) and log + x = max(0, log x). These λd's are Selberg-type sieve weights. J.B. Conrey asked whether it is possible to strenghten the method quantitatively by using a di ff erent choice of the\n\nλd's. For the duration of the workshop, there was an ongoing discussion of this question. Much of it focused on weights of the form\n\nλd = μ(d)\n\n{ ∏\n\n> p|d\n\nf\n\n( log p\n\nlog R\n\n) }\n\nQ( log +(R/d)),\n\nwhere f and Q are smooth functions at our disposal.\n\n• R.C. Vaughan noted that it may be worth investigating how the method is a ff ected by changing Selberg's weights to other traditional sieve weights.\n\n• Another suggestion was to try to combine upper and lower sieves a la Chen. K. Soundararajan commented that there will be an issue calculating the negative contribution.", "original_statement": "15. Alternative weights in the GPY method. \n\n• The GPY method uses the weights \n\nΛR(n; H) = ∑\n\n> d|P(n;H)\n\nλd, λd = μ(d) (log +(R/d)) k+l\n\n(k + l)!,\n\nwhere P(n; H) = (n + h1) · · · (n + hk) and log + x = max(0, log x). These λd's are Selberg-type sieve weights. J.B. Conrey asked whether it is possible to strenghten the method quantitatively by using a di ff erent choice of the \n\nλd's. For the duration of the workshop, there was an ongoing discussion of this question. Much of it focused on weights of the form \n\nλd = μ(d)\n\n{ ∏\n\n> p|d\n\nf\n\n( log p\n\nlog R\n\n) } \n\nQ( log +(R/d)),\n\nwhere f and Q are smooth functions at our disposal. \n\n• R.C. Vaughan noted that it may be worth investigating how the method is a ff ected by changing Selberg's weights to other traditional sieve weights. \n\n• Another suggestion was to try to combine upper and lower sieves a la Chen. K. Soundararajan commented that there will be an issue calculating the negative contribution.", "clean_statement": "15. Alternative weights in the GPY method.\n\n• The GPY method uses the weights\n\nΛR(n; H) = ∑\n\n> d|P(n;H)\n\nλd, λd = μ(d) (log +(R/d)) k+l\n\n(k + l)!,\n\nwhere P(n; H) = (n + h1) · · · (n + hk) and log + x = max(0, log x). These λd's are Selberg-type sieve weights. J.B. Conrey asked whether it is possible to strenghten the method quantitatively by using a di ff erent choice of the\n\nλd's. For the duration of the workshop, there was an ongoing discussion of this question. Much of it focused on weights of the form\n\nλd = μ(d)\n\n{ ∏\n\n> p|d\n\nf\n\n( log p\n\nlog R\n\n) }\n\nQ( log +(R/d)),\n\nwhere f and Q are smooth functions at our disposal.\n\n• R.C. Vaughan noted that it may be worth investigating how the method is a ff ected by changing Selberg's weights to other traditional sieve weights.\n\n• Another suggestion was to try to combine upper and lower sieves a la Chen. K. Soundararajan commented that there will be an issue calculating the negative contribution.", "statement_status": "exact", "statement_verification": "This is Problem 15 in the AIM workshop list *Gaps between primes*. The official PDF is the authority for the reconstruction below. The JSON extraction lost fraction layout and line breaks but did not change the mathematical content.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 15\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[113]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"15. Alternative weights in the GPY method. \\n\\n• The GPY method uses the weights \\n\\nΛR(n; H) = ∑\\n\\n> d|P(n;H)\\n\\nλd, λd = μ(d) (log +(R/d)) k+l\\n\\n(k + l)!,\\n\\nwhere P(n; H) = (n + h1) · · · (n + hk) and log + x = max(0, log x). These λd's are Selberg-type sieve weights. J.B. Conrey asked whether it is possible to strenghten the method quantitatively by using a di ff erent choice of the \\n\\nλd's. For the duration of the workshop, there was an ongoing discussion of this question. Much of it focused on weights of the form \\n\\nλd = μ(d)\\n\\n{ ∏\\n\\n> p|d\\n\\nf\\n\\n( log p\\n\\nlog R\\n\\n) } \\n\\nQ( log +(R/d)),\\n\\nwhere f and Q are smooth functions at our disposal. \\n\\n• R.C. Vaughan noted that it may be worth investigating how the method is a ff ected by changing Selberg's weights to other traditional sieve weights. \\n\\n• Another suggestion was to try to combine upper and lower sieves a la Chen. K. Soundararajan commented that there will be an issue calculating the negative contribution.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0114", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The alternative-weight question admits an exact finite-level Gram/Rayleigh reduction: adjoining one candidate divisor-coefficient vector improves a baseline precisely when its orthogonalized direction either couples to the baseline or has larger Rayleigh quotient, with the optimum given by a closed two-by-two eigenvalue formula. In the standard Maynard simplex functional, no nonzero continuous radial profile is an eigenfunction for k at least 2; its Rayleigh residual therefore gives a strict, explicitly quantified two-plane improvement, and at a radial critical profile this escape direction is genuinely nonradial. A separate signed-mixture lemma identifies the correction term needed before an upper-minus-lower sieve expression can support a positivity argument.\n\nCandidate contribution (lemma; novelty confidence low): For every k at least 2, every nonzero continuous radial profile on the standard Maynard simplex has a nonzero Rayleigh residual; the exact two-plane eigenvalue formula yields a strict certified improvement, and for a profile stationary under radial variations the residual is a canonical genuinely nonradial correction." }, { "id": 20000751, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0115", "title": "A smooth-modulus 21/40 small-residue maximum and a weight-quantifier audit", "statement": "16. Possible improvements on the Bombieri-Vinogradov theorem.\n\n• J.B. Friedlander posed several questions in his talk:\n\n◦ Show that for some fixed θ > 1/2 and D = xθ, one has\n\n∑\n\n> d≤D\n> (d,a)=1\n\nμ(d)\n\n(\n\nψ(x; d, a) − x\n\nφ(d)\n\n)\n\n\n x(log x)−A.\n\n◦ Prove anything beyond the Bombieri-Vinogradov theorem for the sum\n\n∑\n\n> d≤D\n\nmax\n\n> (a,d)=1\n> |a|<(log x)2005\n\n∣∣∣∣ψ(x; d, a) − x\n\nφ(d)\n\n∣∣∣∣.\n\n◦ For any \"reasonable\" weights λd, show, for some fixed θ > 1/2 and D = xθ, that\n\n∑\n\n> d≤D\n> (d,a)=1\n\nλd\n\n(\n\nψ(x; d, ¯a) − x\n\nφ(d)\n\n)\n\n\n x(log x)−A.\n\nHere, ¯ a is defined modulo d by a¯a ≡ 1 (mod d). All of these were motivated by the limitations of the Bombieri-Friedlander-Iwaniec method for primes in arith-metic progressions to large moduli. 4• Several participants proposed to investigate theorems of the Bombieri-Vinogradov type for averages of arith-metic functions other than Λ(n) (cf. known results for the divisor functions d(n) and d3(n)).\n\n• A. Granville asked whether it is possible to formulate a reasonable conjecture about zeros of L-functions that would yield EH( θ) with a fixed θ > 1/2.", "original_statement": "16. Possible improvements on the Bombieri-Vinogradov theorem. \n\n• J.B. Friedlander posed several questions in his talk: \n\n◦ Show that for some fixed θ > 1/2 and D = xθ, one has \n\n∑\n\n> d≤D\n> (d,a)=1\n\nμ(d)\n\n(\n\nψ(x; d, a) − x\n\nφ(d)\n\n)\n\n\u001c x(log x)−A.\n\n◦ Prove anything beyond the Bombieri-Vinogradov theorem for the sum \n\n∑\n\n> d≤D\n\nmax \n\n> (a,d)=1\n> |a|<(log x)2005\n\n∣∣∣∣ψ(x; d, a) − x\n\nφ(d)\n\n∣∣∣∣.\n\n◦ For any \"reasonable\" weights λd, show, for some fixed θ > 1/2 and D = xθ, that \n\n∑\n\n> d≤D\n> (d,a)=1\n\nλd\n\n(\n\nψ(x; d, ¯a) − x\n\nφ(d)\n\n)\n\n\u001c x(log x)−A.\n\nHere, ¯ a is defined modulo d by a¯a ≡ 1 (mod d). All of these were motivated by the limitations of the Bombieri-Friedlander-Iwaniec method for primes in arith-metic progressions to large moduli. 4• Several participants proposed to investigate theorems of the Bombieri-Vinogradov type for averages of arith-metic functions other than Λ(n) (cf. known results for the divisor functions d(n) and d3(n)). \n\n• A. Granville asked whether it is possible to formulate a reasonable conjecture about zeros of L-functions that would yield EH( θ) with a fixed θ > 1/2.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The record is Problem 16 in the AIM workshop list *Gaps between primes* (2005). I checked the typeset workshop PDF rather than relying on the damaged extracted text. With \\[ \\psi(x;q,a)=\\sum_{\\substack{n\\le x\\\\n\\equiv a\\pmod q}}\\Lambda(n), \\] the three questions attributed to J. B. Friedlander are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 16\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[114]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"16. Possible improvements on the Bombieri-Vinogradov theorem. \\n\\n• J.B. Friedlander posed several questions in his talk: \\n\\n◦ Show that for some fixed θ > 1/2 and D = xθ, one has \\n\\n∑\\n\\n> d≤D\\n> (d,a)=1\\n\\nμ(d)\\n\\n(\\n\\nψ(x; d, a) − x\\n\\nφ(d)\\n\\n)\\n\\n\\u001c x(log x)−A.\\n\\n◦ Prove anything beyond the Bombieri-Vinogradov theorem for the sum \\n\\n∑\\n\\n> d≤D\\n\\nmax \\n\\n> (a,d)=1\\n> |a|<(log x)2005\\n\\n∣∣∣∣ψ(x; d, a) − x\\n\\nφ(d)\\n\\n∣∣∣∣.\\n\\n◦ For any \\\"reasonable\\\" weights λd, show, for some fixed θ > 1/2 and D = xθ, that \\n\\n∑\\n\\n> d≤D\\n> (d,a)=1\\n\\nλd\\n\\n(\\n\\nψ(x; d, ¯a) − x\\n\\nφ(d)\\n\\n)\\n\\n\\u001c x(log x)−A.\\n\\nHere, ¯ a is defined modulo d by a¯a ≡ 1 (mod d). All of these were motivated by the limitations of the Bombieri-Friedlander-Iwaniec method for primes in arith-metic progressions to large moduli. 4• Several participants proposed to investigate theorems of the Bombieri-Vinogradov type for averages of arith-metic functions other than Λ(n) (cf. known results for the divisor functions d(n) and d3(n)). \\n\\n• A. Granville asked whether it is possible to formulate a reasonable conjecture about zeros of L-functions that would yield EH( θ) with a fixed θ > 1/2.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0115", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every fixed K,A>0 and 00 such that the sum over q<=x^(21/40-epsilon), q dividing P(x^delta), of the maximum over coprime nonzero |a|<(log x)^K of |psi(x;q,a)-x/phi(q)| is O(x/(log x)^A). This gives the workshop's K=2005 maximum genuinely beyond level 1/2 for squarefree smooth moduli. The report also proves an exact duality obstruction for unrestricted 1-bounded weights and rewrites the inverse-residue sum as a prime/divisor correlation on an-1.\n\nCandidate contribution (corollary; novelty confidence low): Stadlmann's residue-uniform prime-counting theorem implies an explicit psi-version with a single maximum over all coprime |a|<(log x)^2005 at level 21/40-epsilon on q dividing P(x^delta), after a moving-scale partial-summation argument and an aggregate divisor-switch bound for prime powers." }, { "id": 20000752, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0116", "title": "A moment ladder and tuple-error barrier for a converse from bounded gaps", "statement": "17. Elliott-Halberstam from bounded gaps.\n\n• During the workshop, much attention focused on how to prove EH( θ) with θ > 1/2, so we can deduce the existence of bounded gaps between primes. V. Blomer asked whether it is possible to go the opposite way and derive an improvement on the Bombieri-Vinogradov theorem from a \"strong\" quantitative result on bounded gaps between primes.\n\n• R.C. Vaughan commented that such an improvement would encode so much information about L-functions that the hypothesis would have to be extremely strong.\n\n• K. Soundararajan added that the \"strong quantitative result\" could be as strong as the Hardy-Littlewood k-tuple conjecture with a sharp error term, say O(N1/2+[U+000F] ).", "original_statement": "17. Elliott-Halberstam from bounded gaps. \n\n• During the workshop, much attention focused on how to prove EH( θ) with θ > 1/2, so we can deduce the existence of bounded gaps between primes. V. Blomer asked whether it is possible to go the opposite way and derive an improvement on the Bombieri-Vinogradov theorem from a \"strong\" quantitative result on bounded gaps between primes. \n\n• R.C. Vaughan commented that such an improvement would encode so much information about L-functions that the hypothesis would have to be extremely strong. \n\n• K. Soundararajan added that the \"strong quantitative result\" could be as strong as the Hardy-Littlewood k-tuple conjecture with a sharp error term, say O(N1/2+\u000f ).", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "This is Problem 17 in the AIM workshop list *Gaps between primes* (December 2005), notes by Angel Kumchev. The exact mathematical prompt is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 17\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[115]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"17. Elliott-Halberstam from bounded gaps. \\n\\n• During the workshop, much attention focused on how to prove EH( θ) with θ > 1/2, so we can deduce the existence of bounded gaps between primes. V. Blomer asked whether it is possible to go the opposite way and derive an improvement on the Bombieri-Vinogradov theorem from a \\\"strong\\\" quantitative result on bounded gaps between primes. \\n\\n• R.C. Vaughan commented that such an improvement would encode so much information about L-functions that the hypothesis would have to be extremely strong. \\n\\n• K. Soundararajan added that the \\\"strong quantitative result\\\" could be as strong as the Hardy-Littlewood k-tuple conjecture with a sharp error term, say O(N1/2+\\u000f ).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: open; rigor: conditional; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0116", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the exact coprime-centered residue discrepancies, an exact character/correlation expansion shows which uniform prime-tuple data would be needed. A Gaussian-scale dyadic 2r-th moment bound implies endpoint Elliott--Halberstam for every theta < r/(r+1). In contrast, separately uniform O(x^{1/2+epsilon}) errors for distinct prime tuples, if used only through the triangle inequality over their shift lattices, control that moment scale only for R >= x^{1-1/(2r)+o(1)}, beyond the useful range when r > 1. A periodic weighted model also proves that total mass plus one sharp bounded-gap correlation need not imply even equidistribution modulo 5.\n\nCandidate contribution (conditional reduction and obstruction; novelty confidence low): Candidate novelty: the dyadic Gaussian 2r-th-moment implication theta < r/(r+1), paired with the incompatible triangle-summed square-root tuple-error threshold R >= x^{1-1/(2r)+o(1)}, isolates aggregate tuple-remainder cancellation as the missing hypothesis in this converse route.", "research_difficulty_suggested": "L4" }, { "id": 20000753, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0117", "title": "Tuple extension requires shifted correlation", "statement": "18. From k-tuples to (k + 1) -tuples.\n\n• J.B. Conrey asked whether it is possible to deduce the existence of prime ( k + 1)-tuples from a version of the Elliott-Halberstam conjecture for prime k-tuples.\n\n• T.D. Wooley and several others pointed out that it is not quite clear what the proper statement of EH for prime\n\nk-tuples is. It was mentioned that there are results by Balog, Kawada, and Mikawa of the form\n\n∑\n\n> q≤Q\n\nmax\n\n> (a,q)=1\n\n∑\n\n> 1≤b≤q\n> (b,q)=1\n\n∣∣∣∣∑\n\n> n≤x/q\n\n$(qn + a)$(qn + b) − MT( x, q; a, b)\n\n∣∣∣∣\n x2(log x)−A.\n\nHere, $(·) is the characteristic function of the primes and MT( x, q; a, b) is a main term.\n\n• R.C. Vaughan suggested that the following estimate is another possible candidate:\n\n∑\n\n> q≤Q\n\nmax\n\n> (a,q)=1\n\n∣∣∣∣∑\n\n> 1≤b≤B\n> (b,q)=1\n\n( ∑\n\n> n≤x/q\n\n$(qn + a)$(qn + b) − MT( x, q; a, b)\n\n)∣ ∣∣∣\n Bx (log x)−A.\n\n• J. Pintz proposed to try to use EH for twin primes to obtain two pairs of twin primes that are close to each other.", "original_statement": "18. From k-tuples to (k + 1) -tuples. \n\n• J.B. Conrey asked whether it is possible to deduce the existence of prime ( k + 1)-tuples from a version of the Elliott-Halberstam conjecture for prime k-tuples. \n\n• T.D. Wooley and several others pointed out that it is not quite clear what the proper statement of EH for prime \n\nk-tuples is. It was mentioned that there are results by Balog, Kawada, and Mikawa of the form \n\n∑\n\n> q≤Q\n\nmax \n\n> (a,q)=1\n\n∑\n\n> 1≤b≤q\n> (b,q)=1\n\n∣∣∣∣∑\n\n> n≤x/q\n\n$(qn + a)$(qn + b) − MT( x, q; a, b)\n\n∣∣∣∣ \u001c x2(log x)−A.\n\nHere, $(·) is the characteristic function of the primes and MT( x, q; a, b) is a main term. \n\n• R.C. Vaughan suggested that the following estimate is another possible candidate: \n\n∑\n\n> q≤Q\n\nmax \n\n> (a,q)=1\n\n∣∣∣∣∑\n\n> 1≤b≤B\n> (b,q)=1\n\n( ∑\n\n> n≤x/q\n\n$(qn + a)$(qn + b) − MT( x, q; a, b)\n\n)∣ ∣∣∣ \u001c Bx (log x)−A.\n\n• J. Pintz proposed to try to use EH for twin primes to obtain two pairs of twin primes that are close to each other.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The source is page 5 of Angel Kumchev's notes from the December 2005 ARCC workshop *Gaps Between Primes*. The heading and first question are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 18\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[116]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"18. From k-tuples to (k + 1) -tuples. \\n\\n• J.B. Conrey asked whether it is possible to deduce the existence of prime ( k + 1)-tuples from a version of the Elliott-Halberstam conjecture for prime k-tuples. \\n\\n• T.D. Wooley and several others pointed out that it is not quite clear what the proper statement of EH for prime \\n\\nk-tuples is. It was mentioned that there are results by Balog, Kawada, and Mikawa of the form \\n\\n∑\\n\\n> q≤Q\\n\\nmax \\n\\n> (a,q)=1\\n\\n∑\\n\\n> 1≤b≤q\\n> (b,q)=1\\n\\n∣∣∣∣∑\\n\\n> n≤x/q\\n\\n$(qn + a)$(qn + b) − MT( x, q; a, b)\\n\\n∣∣∣∣ \\u001c x2(log x)−A.\\n\\nHere, $(·) is the characteristic function of the primes and MT( x, q; a, b) is a main term. \\n\\n• R.C. Vaughan suggested that the following estimate is another possible candidate: \\n\\n∑\\n\\n> q≤Q\\n\\nmax \\n\\n> (a,q)=1\\n\\n∣∣∣∣∑\\n\\n> 1≤b≤B\\n> (b,q)=1\\n\\n( ∑\\n\\n> n≤x/q\\n\\n$(qn + a)$(qn + b) − MT( x, q; a, b)\\n\\n)∣ ∣∣∣ \\u001c Bx (log x)−A.\\n\\n• J. Pintz proposed to try to use EH for twin primes to obtain two pairs of twin primes that are close to each other.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0117", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every admissible k-shift pattern, a simple congruence construction supplies extension shifts for which all one- and two-shift enlargements remain admissible. Exact first- and second-moment identities then give a quantitative conditional lower bound for extendible occurrences, exposing that the needed inputs are aggregate (k+1)- and (k+2)-point correlations rather than marginal Elliott--Halberstam distribution of the base tuple sequence. An abstract probabilistic construction proves the logical obstruction sharply: disjoint prime-density 0-1 sequences can both satisfy marginal progression discrepancies smaller than N/(log N)^C to every fixed level below one. For Pintz's twin-pair target, the fixed four-shift pattern {0,2,d,d+2} is admissible exactly for d >= 6 divisible by 6.\n\nCandidate contribution (transfer_lemma_and_obstruction; novelty confidence low): The candidate contribution is the combined quantifier-exact boundary consisting of an admissible-extension construction, exact Paley--Zygmund moment transfer through correlation orders k+1 and k+2, and an EH-strength sparse disjointness countermodel showing that marginal progression data alone cannot force a tuple extension; the same audit reduces boundedly close twin pairs to a finite disjunction of admissible patterns {0,2,6m,6m+2}." }, { "id": 20000754, "problem_number": "AIM-ANALYTIC_NUMBER_THEORY-0118", "title": "Primitive balanced proper triples of sums of two squares", "statement": "19. Triples in prime-like sequences.\n\n• C. Elsholtz asked whether it is possible to find infinitely many triples sn, sn+1, sn+2, with sn+2 − sn bounded, in sequences that are commonly considered to be similar to but more manageable than the primes (e.g., integers that are sums of two squares).\n\n• K.B. Ford noted that for sums of two squares the problem is trivial, because of triples of the form n2, n2 +1, n2 +4, for example.\n\n• J.B. Friedlander suggested that such trivialities can be avoided by requiring that the two squares be of compara-ble sizes. S.W. Graham and T.D. Wooley proposed another option: produce \"many\" gaps.\n\n• R.C. Vaughan proposed considering the more general question of triples of values of norm-forms. 5", "original_statement": "19. Triples in prime-like sequences. \n\n• C. Elsholtz asked whether it is possible to find infinitely many triples sn, sn+1, sn+2, with sn+2 − sn bounded, in sequences that are commonly considered to be similar to but more manageable than the primes (e.g., integers that are sums of two squares). \n\n• K.B. Ford noted that for sums of two squares the problem is trivial, because of triples of the form n2, n2 +1, n2 +4, for example. \n\n• J.B. Friedlander suggested that such trivialities can be avoided by requiring that the two squares be of compara-ble sizes. S.W. Graham and T.D. Wooley proposed another option: produce \"many\" gaps. \n\n• R.C. Vaughan proposed considering the more general question of triples of values of norm-forms. 5", "clean_statement": "19. Triples in prime-like sequences.\n\n• C. Elsholtz asked whether it is possible to find infinitely many triples sn, sn+1, sn+2, with sn+2 − sn bounded, in sequences that are commonly considered to be similar to but more manageable than the primes (e.g., integers that are sums of two squares).\n\n• K.B. Ford noted that for sums of two squares the problem is trivial, because of triples of the form n2, n2 +1, n2 +4, for example.\n\n• J.B. Friedlander suggested that such trivialities can be avoided by requiring that the two squares be of compara-ble sizes. S.W. Graham and T.D. Wooley proposed another option: produce \"many\" gaps.\n\n• R.C. Vaughan proposed considering the more general question of triples of values of norm-forms. 5", "statement_status": "exact", "statement_verification": "The source is Angel Kumchev's notes from the December 2005 ARCC workshop *Gaps Between Primes*, Problem 19, p. 5. The original PDF was inspected directly. Apart from line wrapping (including the split word “compara-ble”) and the extracted page number, the repository record is accurate. The PDF says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Analytic number theory\nWorkshop: Gaps between primes\nSection: \nSource item: 19\nSource URL: https://aimath.org/WWN/primegaps/WorkshopProblems.pdf\nCanonical location: aim-analytic-number-theory-notes.json notes[117]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"19. Triples in prime-like sequences. \\n\\n• C. Elsholtz asked whether it is possible to find infinitely many triples sn, sn+1, sn+2, with sn+2 − sn bounded, in sequences that are commonly considered to be similar to but more manageable than the primes (e.g., integers that are sums of two squares). \\n\\n• K.B. Ford noted that for sums of two squares the problem is trivial, because of triples of the form n2, n2 +1, n2 +4, for example. \\n\\n• J.B. Friedlander suggested that such trivialities can be avoided by requiring that the two squares be of compara-ble sizes. S.W. Graham and T.D. Wooley proposed another option: produce \\\"many\\\" gaps. \\n\\n• R.C. Vaughan proposed considering the more general question of triples of values of norm-forms. 5\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 1, "set_id": 14, "source_url": "https://aimath.org/WWN/primegaps/WorkshopProblems.pdf", "tags": [ "aim", "AIM-ANALYTIC_NUMBER_THEORY-0118", "aim-domain:analytic-number-theory", "aim-workshop:workshopproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 1, "name": "number_theory", "display_name": "Number Theory", "description": "Properties of integers, prime numbers, Diophantine equations.", "slug": "number-theory", "order_index": 1, "created_at": "2026-07-31T15:26:25.670Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "There is a fixed CRT-defined arithmetic progression of integers n for which 5n^2+5, 5n^2+20, and 5n^2+45 are exact consecutive terms of the full increasing sequence of sums of two squares. Their displayed representations (n+2k)^2+(2n-k)^2 for k=1,2,3 are primitive and 2-comparable, and the construction yields at least a constant times sqrt(X) such proper triplets up to X. An orthogonality argument also gives bounded-diameter consecutive triples for every positive-definite integral binary quadratic form, but not for general higher-degree norm forms.\n\nCandidate contribution (explicit construction; novelty confidence low): A fixed CRT-defined progression produces at least a constant times sqrt(X) proper consecutive sum-of-two-squares triplets up to X with exact gaps 15 and 25, each term carrying the explicit primitive 2-comparable representation (n+2k)^2+(2n-k)^2 for k=1,2,3." }, { "id": 20000755, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0001", "title": "Higher-weight special cycles: known cohomological modularity and a rational non-neat descent criterion", "statement": "Is it possible to define a generating series of special cycles over Shimura varieties related to automorphic representations of higher weight? Do we have similar geometric/arithmetic modularity and Siegel-Weil formulas as in the Kudla program?", "original_statement": "Is it possible to define a generating series of special cycles over Shimura varieties related to automorphic representations of higher weight? Do we have similar geometric/arithmetic modularity and Siegel-Weil formulas as in the Kudla program?", "clean_statement": "Is it possible to define a generating series of special cycles over Shimura varieties related to automorphic representations of higher weight? Do we have similar geometric/arithmetic modularity and Siegel-Weil formulas as in the Kudla program?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.1 in the section **“Special cycles, modularity and arithmetic intersection”** of the AIM workshop *Arithmetic intersection theory on Shimura varieties* (January 8--12, 2024). The exact recorded question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Arithmetic intersection theory on Shimura varieties\nSection: Special cycles, modularity and arithmetic intersection\nSource item: 1.1\nSource URL: http://aimpl.org/intersectshimura/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is it possible to define a generating series of special cycles over Shimura varieties related to automorphic representations of higher weight? Do we have similar geometric/arithmetic modularity and Siegel-Weil formulas as in the Kudla program?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/intersectshimura/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0001", "aim-domain:arithmetic-geometry", "aim-workshop:intersectshimura", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Funke--Millson already solve a precise orthogonal cohomological higher-weight version of the question, while a general coefficient-valued arithmetic Siegel--Weil theory remains unavailable. This attempt proves a rational non-neat-level descent theorem: on the finite tensor-generated decoration space of an orthogonal special cycle, the descendable coefficient is exactly the orientation-twisted Reynolds projection; normalized pushforward from a suitable neat cover realizes this projection. It yields an explicit parity--orientation obstruction in rank one and proves that a scalar higher-weight Siegel--Weil formula requires a dual coefficient insertion under Zariski-density and irreducibility hypotheses.\n\nCandidate contribution (descent criterion and obstruction; novelty confidence low): Candidate novelty: the orientation-twisted Reynolds projector, its exact normalized finite-cover trace formula, the rank-one parity--orientation test, and the dual-pairing obstruction form a finite, falsifiable gate for proposed higher-weight orthogonal special cycles at non-neat level." }, { "id": 20000756, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0002", "title": "Finite-level geometric refinement of Li–Zhang local modularity", "statement": "In recent proof of Kudla-Rapoport conjectures by Li--Zhang, numerical local modularity results for local Kudla--Rapoport divisors have been discovered and proved. Is it possible to formulate local modularity geometrically e.g. using moduli of p-adic Shtukas? Could the method of Feng-Yun-Zhang for global modularity be adapted to prove geometric local modularity?", "original_statement": "In recent proof of Kudla-Rapoport conjectures by Li--Zhang, numerical local modularity results for local Kudla--Rapoport divisors have been discovered and proved. Is it possible to formulate local modularity geometrically e.g. using moduli of p-adic Shtukas? Could the method of Feng-Yun-Zhang for global modularity be adapted to prove geometric local modularity?", "clean_statement": "In recent proof of Kudla-Rapoport conjectures by Li--Zhang, numerical local modularity results for local Kudla--Rapoport divisors have been discovered and proved. Is it possible to formulate local modularity geometrically e.g. using moduli of p-adic Shtukas? Could the method of Feng-Yun-Zhang for global modularity be adapted to prove geometric local modularity?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.3 in the section “Special cycles, modularity and arithmetic intersection” of the AIM list *Arithmetic intersection theory on Shimura varieties*. The live AIM page agrees with the repository record and attributes the question to Q. He and Z. Zhang. The exact statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Arithmetic intersection theory on Shimura varieties\nSection: Special cycles, modularity and arithmetic intersection\nSource item: 1.3\nSource URL: http://aimpl.org/intersectshimura/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In recent proof of Kudla-Rapoport conjectures by Li--Zhang, numerical local modularity results for local Kudla--Rapoport divisors have been discovered and proved. Is it possible to formulate local modularity geometrically e.g. using moduli of p-adic Shtukas? Could the method of Feng-Yun-Zhang for global modularity be adapted to prove geometric local modularity?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/intersectshimura/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0002", "aim-domain:arithmetic-geometry", "aim-workshop:intersectshimura", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Li–Zhang's Theorem 6.4.9 formally assembles into a canonical coefficient-valued Fourier eigenidentity in their finite-dimensional quotient of K_0 by the kernel of the Chern-character/cycle-class realization. For a vertex lattice Lambda of type t=2d+1 this identity is exactly the normalized finite Fourier identity on A_Lambda=Lambda^vee/Lambda, with factor q^{-t} and Weil constant gamma_V=-1 in the nonsplit convention. For raw K_0 representatives, the Fourier defect lies pointwise in ker(cl composed with ch); using the injective divisor cycle-class map invoked in Li–Zhang's Lemma 6.4.5, its degree-one Chern character vanishes, giving an unconditional unquotiented CH^1-valued local modularity identity. A separate conditional trace square specifies the support and realization compatibilities needed for a p-adic-shtuka categorification.\n\nCandidate contribution (finite-level reformulation and obstruction lemma; novelty confidence low): Candidate novelty: the scalar-functional local modularity theorem canonically descends to a normalized finite Fourier eigenvector on Lambda^vee/Lambda; the raw K_0 Fourier defect is pointwise in ker(cl composed with ch), while its divisor-degree Chern character is exactly zero in unquotiented CH^1." }, { "id": 20000757, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0003", "title": "Proper support and clean excess for improper AFL intersections", "statement": "In the formulation of the arithmetic fundamental lemma, arithmetic intersection numbers are only formulated for regular semi-simple orbits because of proper intersections of cycles. Is it possible to define arithmetic intersection numbers in case of improper intersections of cycles (and formulate a similar analytic side)?", "original_statement": "In the formulation of the arithmetic fundamental lemma, arithmetic intersection numbers are only formulated for regular semi-simple orbits because of proper intersections of cycles. Is it possible to define arithmetic intersection numbers in case of improper intersections of cycles (and formulate a similar analytic side)?", "clean_statement": "In the formulation of the arithmetic fundamental lemma, arithmetic intersection numbers are only formulated for regular semi-simple orbits because of proper intersections of cycles. Is it possible to define arithmetic intersection numbers in case of improper intersections of cycles (and formulate a similar analytic side)?", "statement_status": "exact", "statement_verification": "The canonical record is item 1.4 in the section **“Special cycles, modularity and arithmetic intersection”** of the AIM workshop list **“Arithmetic intersection theory on Shimura varieties.”** The exact corpus statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Arithmetic intersection theory on Shimura varieties\nSection: Special cycles, modularity and arithmetic intersection\nSource item: 1.4\nSource URL: http://aimpl.org/intersectshimura/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In the formulation of the arithmetic fundamental lemma, arithmetic intersection numbers are only formulated for regular semi-simple orbits because of proper intersections of cycles. Is it possible to define arithmetic intersection numbers in case of improper intersections of cycles (and formulate a similar analytic side)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/intersectshimura/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0003", "aim-domain:arithmetic-geometry", "aim-workshop:intersectshimura", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For two perfect cycles in a regular formal DVR-space, a derived arithmetic intersection is a finite signed integer whenever their classical intersection is a proper vertical scheme. Under an explicit clean-intersection convention, with excess normal bundle Q = coker(N_{E/Z_2} -> N_{Z_1/X}|_E), the value is chi_W(E, lambda_{-1}(Q^vee)); a trivial line summand of Q forces vanishing. A projective family verifies deformation compatibility, while an affine family has Tor_0 = Tor_1 = k[y] and proves that derived tensor product alone cannot repair nonproper support. The proposed analytic finite-part identity is conditional on a normalized, path-independent germ expansion.\n\nCandidate contribution (reduction; novelty confidence low): Candidate AFL-specific reduction: any numerical extension to a clean non-regular degeneration should first pass the proper-perfect envelope test, then use chi_W(E, lambda_{-1}(Q^vee)); it must vanish when Q has a trivial line, whereas a nonproper Koszul fiber supplies an explicit obstruction to any uncorrected numerical extension. Conditional analytic matching therefore targets a fully normalized constant germ coefficient rather than pointwise evaluation at the singular orbit." }, { "id": 20000758, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0004", "title": "A parahoric Koszul descent criterion at a semistable node", "statement": "Derived special cycles have been defined on integral models of certain Shimura varieties (resp. moduli of Shtukas) in recent works of Madapusi (resp. Feng-Yun-Zhang) in the good reduction case. Is it possible to extend their method and define derived special cycles in the bad reduction cases e.g. for some specific parahoric levels?", "original_statement": "Derived special cycles have been defined on integral models of certain Shimura varieties (resp. moduli of Shtukas) in recent works of Madapusi (resp. Feng-Yun-Zhang) in the good reduction case. Is it possible to extend their method and define derived special cycles in the bad reduction cases e.g. for some specific parahoric levels?", "clean_statement": "Derived special cycles have been defined on integral models of certain Shimura varieties (resp. moduli of Shtukas) in recent works of Madapusi (resp. Feng-Yun-Zhang) in the good reduction case. Is it possible to extend their method and define derived special cycles in the bad reduction cases e.g. for some specific parahoric levels?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.2 in the section “Special cycles, modularity and arithmetic intersection” of the AIM workshop list *Arithmetic intersection theory on Shimura varieties*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Arithmetic intersection theory on Shimura varieties\nSection: Special cycles, modularity and arithmetic intersection\nSource item: 1.2\nSource URL: http://aimpl.org/intersectshimura/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[3]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Derived special cycles have been defined on integral models of certain Shimura varieties (resp. moduli of Shtukas) in recent works of Madapusi (resp. Feng-Yun-Zhang) in the good reduction case. Is it possible to extend their method and define derived special cycles in the bad reduction cases e.g. for some specific parahoric levels?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/intersectshimura/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0004", "aim-domain:arithmetic-geometry", "aim-workshop:intersectshimura", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Assuming a parahoric local-model diagram and a canonical equivariant vector-bundle section on the local model, the section descends through the parahoric torsor to an integral derived zero locus that is quasi-smooth relative to the Shimura model, perfect over it, compatible with arbitrary derived base change, and multiplicative under direct sums. Absolute cotangent perfectness holds if and only if the ambient cotangent complex restricted to the cycle is perfect; the lci hypothesis on the local model is a sufficient condition for absolute quasi-smoothness. On the GL_2 Iwahori node XY=pi, the vertical equation X=0 has derived special-fiber homology H_0=k[Y] and H_1=(Y), displaying the excess class lost by ordinary base change.\n\nCandidate contribution (criterion_and_test_case; novelty confidence low): Candidate novelty: the equations-descent-perfectness criterion, together with the nonperfect ambient-cotangent obstruction and the explicit Iwahori-node unit test requiring the complementary-branch module H_1=(Y), gives a concrete falsifiable gate for proposed parahoric derived-special-cycle constructions.", "research_difficulty_suggested": "L4" }, { "id": 20000759, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0005", "title": "A canonical obstruction to correction-free arithmetic transfer", "statement": "In any arithmetic transfer conjectures, could we find perfect test transfer functions for (derived) orbital integrals to match arithmetic intersection numbers with no correction factors?", "original_statement": "In any arithmetic transfer conjectures, could we find perfect test transfer functions for (derived) orbital integrals to match arithmetic intersection numbers with no correction factors?", "clean_statement": "In any arithmetic transfer conjectures, could we find perfect test transfer functions for (derived) orbital integrals to match arithmetic intersection numbers with no correction factors?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.5 in the section “Special cycles, modularity and arithmetic intersection” of the AIM workshop list *Arithmetic intersection theory on Shimura varieties*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Arithmetic intersection theory on Shimura varieties\nSection: Special cycles, modularity and arithmetic intersection\nSource item: 1.5\nSource URL: http://aimpl.org/intersectshimura/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[4]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In any arithmetic transfer conjectures, could we find perfect test transfer functions for (derived) orbital integrals to match arithmetic intersection numbers with no correction factors?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/intersectshimura/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0005", "aim-domain:arithmetic-geometry", "aim-workshop:intersectshimura", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any fixed linear arithmetic-transfer setup with ordinary-transfer map T and derivative-orbital map D, the class [a-D(f0)] in N/D(ker T) is independent of the chosen transfer f0 and vanishes exactly when a correction-free transfer exists. A weight-translation difference supplies a sufficient lifting of correction orbital data into ker T, as in Li--Mihatsch, while an explicit two-dimensional model proves that ordinary-transfer surjectivity, transfer nonuniqueness, and correction representability alone do not force correction removal.\n\nCandidate contribution (obstruction criterion; novelty confidence low): The candidate contribution is the support/level-filtered profile of canonical obstruction classes o_alpha in N/D(ker(T restricted to V_alpha)); its transition maps prove monotonicity under enlargement of linear test spaces, and finite Hecke/orbit truncations detect its vanishing by an augmented-matrix rank equality.", "research_difficulty_suggested": "L4" }, { "id": 20000760, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0006", "title": "Derived-excess reduction for commutativity of unitary RZ Hecke operators", "statement": "In the recent work of Li--Rapoport--Zhang on arithmetic fundamental lemma conjecture for spherical Hecke algebras, there is a conjecture on commutativity of certain Hecke operators on Grothendieck groups of coherent sheaves on unitary Rapoport--Zink spaces. How to prove this conjecture?", "original_statement": "In the recent work of Li--Rapoport--Zhang on arithmetic fundamental lemma conjecture for spherical Hecke algebras, there is a conjecture on commutativity of certain Hecke operators on Grothendieck groups of coherent sheaves on unitary Rapoport--Zink spaces. How to prove this conjecture?", "clean_statement": "In the recent work of Li--Rapoport--Zhang on arithmetic fundamental lemma conjecture for spherical Hecke algebras, there is a conjecture on commutativity of certain Hecke operators on Grothendieck groups of coherent sheaves on unitary Rapoport--Zink spaces. How to prove this conjecture?", "statement_status": "exact", "statement_verification": "The exact AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Arithmetic intersection theory on Shimura varieties\nSection: Special cycles, modularity and arithmetic intersection\nSource item: 1.6\nSource URL: http://aimpl.org/intersectshimura/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[5]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In the recent work of Li--Rapoport--Zhang on arithmetic fundamental lemma conjecture for spherical Hecke algebras, there is a conjecture on commutativity of certain Hecke operators on Grothendieck groups of coherent sheaves on unitary Rapoport--Zink spaces. How to prove this conjecture?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/intersectshimura/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0006", "aim-domain:arithmetic-geometry", "aim-workshop:intersectshimura", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the two ordered Li--Rapoport--Zhang atomic Hecke compositions, a proved derived-kernel formula identifies the commutator on supported K_0 with the integral-transform action of the difference of the two ordered convolution kernel classes. Under an endpoint-preserving isomorphism of the classical four-step convolution spaces, this kernel difference is exactly the pushforward of the difference of their positive multi-Tor Euler classes. Hence rational equality of those excess classes is a sufficient criterion for the LRZ conjecture, and the obstruction is localized on the vertical non-Tor-independent locus.\n\nCandidate contribution (reduction; novelty confidence low): Conditional on an endpoint-preserving exchange of the two ordered classical LRZ four-step convolution spaces, the commutator kernel is the pushed-forward difference of the positive multi-Tor Euler classes; in the first nontrivial n=4, (t,t')=(2,4) case, commutativity reduces to one explicit rational G_0 identity on the vertical bad locus.", "research_difficulty_suggested": "L4" }, { "id": 20000761, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0007", "title": "A finite-slice criterion for numerical modularity of CM-cycle series", "statement": "Similar to Kudla program, we could also define a generating series of (derived) complex multiplication cycles on Shimura varieties and conjecture its modularity. Is it possible to compute arithmetic intersection of the generating series with special divisors / Hodge bundle and prove its numerical modularity when the Shimura variety has low dimension e.g. on Siegel threefold or Hilbert modular surface?", "original_statement": "Similar to Kudla program, we could also define a generating series of (derived) complex multiplication cycles on Shimura varieties and conjecture its modularity. Is it possible to compute arithmetic intersection of the generating series with special divisors / Hodge bundle and prove its numerical modularity when the Shimura variety has low dimension e.g. on Siegel threefold or Hilbert modular surface?", "clean_statement": "Similar to Kudla program, we could also define a generating series of (derived) complex multiplication cycles on Shimura varieties and conjecture its modularity. Is it possible to compute arithmetic intersection of the generating series with special divisors / Hodge bundle and prove its numerical modularity when the Shimura variety has low dimension e.g. on Siegel threefold or Hilbert modular surface?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 2.1 in the AIM list *Arithmetic intersection theory on Shimura varieties*, section “Arithmetic intersection theory beyond classical unitary Shimura varieties.” The live AIM page was checked on 2026-07-29. It attributes the question to A. Mihatsch and contains no status update or remark. The exact statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Arithmetic intersection theory on Shimura varieties\nSection: Arithmetic intersection theory beyond classical unitary Shimura varieties\nSource item: 2.1\nSource URL: http://aimpl.org/intersectshimura/2/\nCanonical location: aim-arithmetic-geometry-notes.json notes[6]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Similar to Kudla program, we could also define a generating series of (derived) complex multiplication cycles on Shimura varieties and conjecture its modularity. Is it possible to compute arithmetic intersection of the generating series with special divisors / Hodge bundle and prove its numerical modularity when the Shimura variety has low dimension e.g. on Siegel threefold or Hilbert modular surface?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/intersectshimura/2/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0007", "aim-domain:arithmetic-geometry", "aim-workshop:intersectshimura", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the two-variable arithmetic kernel K(m,n) obtained by pairing virtual/refined CM cycles with special divisors, suppose every fixed-CM divisor-index row belongs to one s-dimensional modular-form space M with injective q-expansion. Then s Fourier-coordinate divisor indices, selected solely from M, suffice: all complete CM-indexed intersection series are modular in any fixed linear target space if and only if the s selected complete series are modular. This is a proved finite-dimensional reduction, valid for properly defined derived arithmetic coefficients, not a proof of the still-missing CM-index identities.\n\nCandidate contribution (criterion; novelty confidence low): For any arithmetic CM/special-divisor kernel whose divisor-direction rows lie in a fixed s-dimensional modular-form space, there exist s Fourier-coordinate divisor tests such that modularity of their complete CM-indexed series is equivalent to modularity against every special divisor; the same statement holds for vector-valued rows after selecting Fourier-component coordinates and for derived cycles when refined arithmetic intersections define the coefficients.", "research_difficulty_suggested": "L4" }, { "id": 20000762, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0008", "title": "A mixed-excess obstruction and forced Hodge normalization for U(a,b)", "statement": "Could we formulate and prove arithmetic Siegel--Weil formulas of special cycles on unitary Shimura varieties for U(a,b) (a, b>1) where there are no special divisors?", "original_statement": "Could we formulate and prove arithmetic Siegel--Weil formulas of special cycles on unitary Shimura varieties for U(a,b) (a, b>1) where there are no special divisors?", "clean_statement": "Could we formulate and prove arithmetic Siegel--Weil formulas of special cycles on unitary Shimura varieties for U(a,b) (a, b>1) where there are no special divisors?", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Arithmetic intersection theory on Shimura varieties\nSection: Arithmetic intersection theory beyond classical unitary Shimura varieties\nSource item: 2.2\nSource URL: http://aimpl.org/intersectshimura/2/\nCanonical location: aim-arithmetic-geometry-notes.json notes[7]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Could we formulate and prove arithmetic Siegel--Weil formulas of special cycles on unitary Shimura varieties for U(a,b) (a, b>1) where there are no special divisors?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 3; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/intersectshimura/2/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0008", "aim-domain:arithmetic-geometry", "aim-workshop:intersectshimura", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a positive-definite r-plane P and a negative-definite s-plane N in a Hermitian space of signature (a,b), the associated signed special-cycle components have codimensions rb and sa. They meet only when P is orthogonal to N; then the intersection is clean of excess rs with constant excess bundle Hom(N,V/P^perp). For r,s>0 the ordinary rational Chow product and the marked generic-fibre derived Euler class vanish. Hence a nonzero mixed arithmetic coefficient must be carried by vertical derived and/or archimedean arithmetic data. In rank one, the unique Hodge power making the mixed product degree-valued is (a-1)(b-1).\n\nCandidate contribution (excess-intersection lemma and reduction; novelty confidence low): The mixed signed excess bundle is canonically the constant rank-rs bundle Hom(N,V/P^perp); therefore the marked generic-fibre derived intersection class vanishes for every r,s>0, and the rank-one scalar arithmetic formulation is forced to insert exactly (a-1)(b-1) Hodge classes." }, { "id": 20000763, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0009", "title": "A Hermitian and dual-pair filter for exceptional Siegel--Weil programs", "statement": "Could we formulate and prove arithmetic (resp. higher) Siegel--Weil formulas on Shimura varieties (resp. moduli of Shtukas) for exceptional groups?", "original_statement": "Could we formulate and prove arithmetic (resp. higher) Siegel--Weil formulas on Shimura varieties (resp. moduli of Shtukas) for exceptional groups?", "clean_statement": "Could we formulate and prove arithmetic (resp. higher) Siegel--Weil formulas on Shimura varieties (resp. moduli of Shtukas) for exceptional groups?", "statement_status": "exact", "statement_verification": "The canonical record is item 2.3 in the section “Arithmetic intersection theory beyond classical unitary Shimura varieties” of the AIM problem list attached to the January 8--12, 2024 workshop *Arithmetic intersection theory on Shimura varieties*. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Arithmetic intersection theory on Shimura varieties\nSection: Arithmetic intersection theory beyond classical unitary Shimura varieties\nSource item: 2.3\nSource URL: http://aimpl.org/intersectshimura/2/\nCanonical location: aim-arithmetic-geometry-notes.json notes[8]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Could we formulate and prove arithmetic (resp. higher) Siegel--Weil formulas on Shimura varieties (resp. moduli of Shtukas) for exceptional groups?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/intersectshimura/2/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0009", "aim-domain:arithmetic-geometry", "aim-workshop:intersectshimura", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the explicit hypotheses that the geometric member is a positive-dimensional exceptional Shimura factor and that the theta mechanism uses a nonzero semisimple mutual-centralizer pair in a simple exceptional ambient algebra, the only complex candidates are (E6,A2) in E8 and (E7,A1) in E8. The proof intersects the Shimura special-node classification, which leaves only E6(-14) and E7(-25), with the complete exceptional dual-pair list. A second proved obstruction shows that Z[t]/(f(t,1)) is finite free of rank three only for unit leading coefficient, so the full G2 Fourier series proposed by the AIM workshop must use Delone--Faddeev cubic rings and GL2(Z)-orbit data rather than only monogenic quotients. No arithmetic or higher exceptional Siegel--Weil formula is claimed.\n\nCandidate contribution (classification reduction; novelty confidence low): Candidate novelty: the intersection of the positive-dimensional exceptional Shimura-factor list with the nonzero semisimple mutual-centralizer pairs in simple complex exceptional Lie algebras consists exactly of (E6,A2) in E8 and (E7,A1) in E8." }, { "id": 20000764, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0010", "title": "Central normalization and an explicit quaternionic reduced locus", "statement": "Could we describe the reduced locus of Rapoport--Zink spaces (integral local Shimura varieties) for inner forms of general linear groups? It will be useful for studying arithmetic transfer conjectures in the context of linear arithmetic fundamental lemmas.", "original_statement": "Could we describe the reduced locus of Rapoport--Zink spaces (integral local Shimura varieties) for inner forms of general linear groups? It will be useful for studying arithmetic transfer conjectures in the context of linear arithmetic fundamental lemmas.", "clean_statement": "Could we describe the reduced locus of Rapoport--Zink spaces (integral local Shimura varieties) for inner forms of general linear groups? It will be useful for studying arithmetic transfer conjectures in the context of linear arithmetic fundamental lemmas.", "statement_status": "exact", "statement_verification": "The canonical source is aim-arithmetic-geometry-notes.json, zero-based index 9, problem 2.4 from the AIM workshop *Arithmetic intersection theory on Shimura varieties*. The exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Arithmetic intersection theory on Shimura varieties\nSection: Arithmetic intersection theory beyond classical unitary Shimura varieties\nSource item: 2.4\nSource URL: http://aimpl.org/intersectshimura/2/\nCanonical location: aim-arithmetic-geometry-notes.json notes[9]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Could we describe the reduced locus of Rapoport--Zink spaces (integral local Shimura varieties) for inner forms of general linear groups? It will be useful for studying arithmetic transfer conjectures in the context of linear arithmetic fundamental lemmas.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/intersectshimura/2/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0010", "aim-domain:arithmetic-geometry", "aim-workshop:intersectshimura", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any inner form G = GL_m(D), the perfect reduced locus is the affine Deligne--Lusztig variety attached to the datum. The report proves that simultaneous central translation of b and the ADLV bound leaves this perfect scheme unchanged, and that non-emptiness forces equality of the reduced-norm and cocharacter Kottwitz invariants. It then gives the complete basic quaternionic Drinfeld special fiber as a Bruhat--Tits tree of projective lines and proves explicit finite-radius component, node, boundary, and cohomology formulas. The general ordinary reduced-locus problem, including the degree-four invariant-3/4 arithmetic-transfer space, remains open.\n\nCandidate contribution (explicit truncation lemma; novelty confidence low): After central normalization (b, (a+1,a)) to (b*pi^(-a), (1,0)), the radius-R truncation of the basic quaternionic Drinfeld reduced locus has N_R = 1 + (q+1)(q^R-1)/(q-1) projective-line components, N_R-1 internal nodes, (q+1)q^R boundary nodes, vanishing H^1 of its structure sheaf, and l-adic Euler characteristic N_R+1." }, { "id": 20000765, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0011", "title": "Triality and a middle-dimensional G2 cycle for O(4,4)", "statement": "Study non-algebraic locally symmetric spaces for $O(4,4)$ and natural special cycles on them. What kind of structures do we expect?", "original_statement": "Study non-algebraic locally symmetric spaces for $O(4,4)$ and natural special cycles on them. What kind of structures do we expect?", "clean_statement": "Study non-algebraic locally symmetric spaces for $O(4,4)$ and natural special cycles on them. What kind of structures do we expect?", "statement_status": "exact", "statement_verification": "The record is `AIM-ARITHMETIC_GEOMETRY-0011`, source file `aim-arithmetic-geometry-notes.json`, zero-based source index 10. The neighboring records confirm that the intended setting is an extension of Kudla-style cycle and intersection structures beyond algebraic Shimura varieties. There is no visible OCR corruption. The record does not specify a rational quadratic space, level, connected component, or cohomology theory, so those choices must be made explicitly.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Arithmetic intersection theory on Shimura varieties\nSection: Arithmetic intersection theory beyond classical unitary Shimura varieties\nSource item: 2.5\nSource URL: http://aimpl.org/intersectshimura/2/\nCanonical location: aim-arithmetic-geometry-notes.json notes[10]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Study non-algebraic locally symmetric spaces for $O(4,4)$ and natural special cycles on them. What kind of structures do we expect?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/intersectshimura/2/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0011", "aim-domain:arithmetic-geometry", "aim-workshop:intersectshimura", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the split rational group Spin(4,4), a pinned order-three triality automorphism and any neat triality-stable arithmetic level produce an embedded finite-volume totally geodesic fixed cycle modeled on split G2/SO(4). It has real dimension 8 inside the 16-dimensional ambient quotient, is not a standard rank-two positive-plane orthogonal cycle, and its real rank-8 normal bundle has the canonical complex structure J=(d tau-(d tau)^2)/sqrt(3), so the Thom self-restriction is the top Chern class c4(N). A complementary proved incidence lemma gives codimension 4r, normal bundle E* tensor U, and a positive-Gram-matrix criterion for ordinary rank-r cycles.\n\nCandidate contribution (geometric construction; novelty confidence low): At every neat order-three-triality-stable arithmetic level for split Spin(4,4), the identity-twisted split-G2 fixed locus is a natural exceptional middle-dimensional special cycle, distinct in symmetric-space type from the Kudla-Millson rank-two cycles, with canonical complex rank-4 normal bundle and Thom self-restriction c4(N)." }, { "id": 20000766, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0012", "title": "Positive semidecision and a rank-one geometric decision algorithm", "statement": "Is there an algorithm to compute derived equivalence for K3 surfaces over a finitely generated field?", "original_statement": "Is there an algorithm to compute derived equivalence for K3 surfaces over a finitely generated field?", "clean_statement": "Is there an algorithm to compute derived equivalence for K3 surfaces over a finitely generated field?", "statement_status": "exact", "statement_verification": "The canonical record is AIM problem 1.05 in the section “K3 surfaces” of the workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.05\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[11]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there an algorithm to compute derived equivalence for K3 surfaces over a finitely generated field?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0012", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For explicitly presented K3 surfaces over an effectively presented finitely generated field, ground-field Fourier--Mukai equivalence is positively semidecidable by enumerating perfect kernels and finite certificates for two inverse convolution identities. In addition, for K3 surfaces over a number field with geometric Picard rank one, geometric derived equivalence after algebraic closure is decidable: compute the geometric Neron--Severi Galois modules, construct the finite Hosono--Lian--Oguiso--Yau list of moduli partners, and apply Yasuda's isomorphism algorithm. This does not decide descent of a geometric equivalence to the ground field.\n\nCandidate contribution (algorithmic_reduction; novelty confidence low): For number-field K3 surfaces whose source has geometric Picard rank one and primitive polarization square 2n, geometric derived equivalence is decidable using one candidate isomorphism test when n=1 and at most 2^(omega(n)-1) tests when n>1, by combining the unconditional K3 Neron--Severi algorithm, the complete rank-one Fourier--Mukai-partner classification, effective moduli construction, and decidability of K3 isomorphism with finite automorphism groups." }, { "id": 20000767, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0013", "title": "Uniqueness, counting, and a descent sieve for the degree-eight K3 partner", "statement": "If you fix a degree 2 K3 surface over $\\mathbb{Q}$ with a 2-torsion Brauer class $\\alpha$ (over $\\overline{\\mathbb{Q}}$), van Geeman tells us that we can find a variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and such that $\\alpha$ is the obstruction for the moduli space to be fine.\n\nIs $Y$ unique? How many such $Y$ are there? If $X$ is defined over $\\mathbb{Q}$ and $Y$ is defined over some nontrivial extension of $\\mathbb{Q}$, is it possible that $\\alpha$ is defined over $\\mathbb{Q}$, i.e.~in the image of $\\mathrm{Br}(X) \\to \\mathrm{Br}(\\overline{X})$?", "original_statement": "If you fix a degree 2 K3 surface over $\\mathbb{Q}$ with a 2-torsion Brauer class $\\alpha$ (over $\\overline{\\mathbb{Q}}$), van Geeman tells us that we can find a variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and such that $\\alpha$ is the obstruction for the moduli space to be fine.\n\nIs $Y$ unique? How many such $Y$ are there? If $X$ is defined over $\\mathbb{Q}$ and $Y$ is defined over some nontrivial extension of $\\mathbb{Q}$, is it possible that $\\alpha$ is defined over $\\mathbb{Q}$, i.e.~in the image of $\\mathrm{Br}(X) \\to \\mathrm{Br}(\\overline{X})$?", "clean_statement": "If you fix a degree 2 K3 surface over $\\mathbb{Q}$ with a 2-torsion Brauer class $\\alpha$ (over $\\overline{\\mathbb{Q}}$), van Geeman tells us that we can find a variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and such that $\\alpha$ is the obstruction for the moduli space to be fine.\n\nIs $Y$ unique? How many such $Y$ are there? If $X$ is defined over $\\mathbb{Q}$ and $Y$ is defined over some nontrivial extension of $\\mathbb{Q}$, is it possible that $\\alpha$ is defined over $\\mathbb{Q}$, i.e.~in the image of $\\mathrm{Br}(X) \\to \\mathrm{Br}(\\overline{X})$?", "statement_status": "exact", "statement_verification": "The AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.15\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[12]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If you fix a degree 2 K3 surface over $\\\\mathbb{Q}$ with a 2-torsion Brauer class $\\\\alpha$ (over $\\\\overline{\\\\mathbb{Q}}$), van Geeman tells us that we can find a variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and such that $\\\\alpha$ is the obstruction for the moduli space to be fine.\\n\\nIs $Y$ unique? How many such $Y$ are there? If $X$ is defined over $\\\\mathbb{Q}$ and $Y$ is defined over some nontrivial extension of $\\\\mathbb{Q}$, is it possible that $\\\\alpha$ is defined over $\\\\mathbb{Q}$, i.e.~in the image of $\\\\mathrm{Br}(X) \\\\to \\\\mathrm{Br}(\\\\overline{X})$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0013", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a complex degree-two K3 surface of geometric Picard rank one, a fixed nonzero two-torsion Brauer class has a degree-eight K3 realization exactly in the K3-type/even-theta case, and then the polarized realization is unique; varying the K3-type class yields 524,800 polarized partners. In the arithmetic smooth-net setting, a Galois-invariant class has a Galois-fixed partner and theta characteristic, and either trivial polarized automorphisms or vanishing of the theta characteristic's Picard-Brauer boundary forces the partner to descend. Thus a genuinely non-descending partner attached to a class coming from Br(X) can occur only on an exceptional locus where both obstructions are nontrivial.\n\nCandidate contribution (descent_obstruction_criterion; novelty confidence low): In the rank-one K3-type non-effective-even-theta setting, if the associated polarized degree-eight K3 surface has no model over the ground field, then its polarized automorphism group is nontrivial and the invariant theta characteristic has a nonzero two-torsion Picard-Brauer boundary; moreover the branch sextic has no zero-cycle of odd degree." }, { "id": 20000768, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0014", "title": "Odd and transcendental 3-torsion Brauer--Manin obstructions on K3 surfaces", "statement": "Does there exist a K3 surface $X$ and a Brauer class $\\alpha$ of odd order $n$ defined over a number field $k$ such that $\\alpha$ gives an obstruction to the existence of rational points?\n\nWhat about an example with $n>2$?\n\nWhat about a transcendental example?", "original_statement": "Does there exist a K3 surface $X$ and a Brauer class $\\alpha$ of odd order $n$ defined over a number field $k$ such that $\\alpha$ gives an obstruction to the existence of rational points?\n\nWhat about an example with $n>2$?\n\nWhat about a transcendental example?", "clean_statement": "Does there exist a K3 surface $X$ and a Brauer class $\\alpha$ of odd order $n$ defined over a number field $k$ such that $\\alpha$ gives an obstruction to the existence of rational points?\n\nWhat about an example with $n>2$?\n\nWhat about a transcendental example?", "statement_status": "exact", "statement_verification": "The canonical record is item 1.2 in the K3-surfaces section of the AIM workshop list *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.2\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[13]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does there exist a K3 surface $X$ and a Brauer class $\\\\alpha$ of odd order $n$ defined over a number field $k$ such that $\\\\alpha$ gives an obstruction to the existence of rational points?\\n\\nWhat about an example with $n>2$?\\n\\nWhat about a transcendental example?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0014", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Berg and Várilly-Alvarado constructed a smooth degree-2 K3 surface Y over Q with adelic points and a Q-defined Brauer class alpha of exact order 3 for which the alpha-Brauer set is empty; geometric Picard rank one gives Br_1(Y)=Br_0(Y), and the obstruction makes alpha nonconstant, so alpha is transcendental. Thus one published example answers affirmatively the odd-order, order n>2, and transcendental versions of the AIM question. Corn--Nakahara independently provide an algebraic exact-order-3 example.\n\nCandidate contribution (lemma; novelty confidence low): For an odd-order Brauer class beta on a double cover with covering involution iota and iota^*beta=-beta, evaluation changes sign along iota-orbits and vanishes on rational ramification points; applied to the Berg--Várilly-Alvarado example, the 3-adic invariant image is exactly {1/3, 2/3} and the branch sextic has no Q_3-point." }, { "id": 20000769, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0015", "title": "Derived invariance of the stable reduction profile and Kulikov type of K3 surfaces", "statement": "Say we have a K3 surface over a number field. Are the places of bad reduction of the surface determined by the derived category? What about over $\\mathbb{C}((t))$?", "original_statement": "Say we have a K3 surface over a number field. Are the places of bad reduction of the surface determined by the derived category? What about over $\\mathbb{C}((t))$?", "clean_statement": "Say we have a K3 surface over a number field. Are the places of bad reduction of the surface determined by the derived category? What about over $\\mathbb{C}((t))$?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*, section “K3 surfaces”, Problem 1.25) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.25\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[14]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Say we have a K3 surface over a number field. Are the places of bad reduction of the surface determined by the derived category? What about over $\\\\mathbb{C}((t))$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0015", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For K3 surfaces joined by a ground-field Fourier--Mukai equivalence, the rational Mukai realizations give a noncanonical isomorphism of their second cohomology Galois modules for every auxiliary prime, hence equal Artin and Swan conductors and equal rational monodromy profiles at every finite place. Potential good reduction and good reduction after a finite unramified extension are derived invariants. Over C((t)), the minimal unipotent base-change degree, Kulikov type, and actual good reduction are derived invariants. Over a number-field completion, actual good reduction over the original field remains reduced to comparison of the canonical-reduction Weyl descent classes; geometric derived equivalence without ground-field descent is explicitly shown insufficient by a quadratic double-sextic twist.\n\nCandidate contribution (proposition; novelty confidence low): Candidate stable-inertia proposition: a ground-field Fourier--Mukai equivalence of K3 surfaces forces equality at every finite place, with no residue-prime exclusion in rational cohomology, of the H^2 Artin conductor, Swan conductor, and all rational monodromy ranks; rational Krull--Schmidt cancellation gives a noncanonical H^2 Galois-module isomorphism, and over C((t)) this determines the common minimal unipotent base change and Kulikov type. An explicit double-sextic quadratic twist proves that geometric equivalence alone does not preserve reduction." }, { "id": 20000770, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0016", "title": "Order-three fineness obstructions on degree-two and degree-four K3 surfaces", "statement": "Can we identify 3-torsion elements of the Brauer group of a degree 2 K3 surface as obstructions to the fineness of some moduli space?\n\nCan we do this for K3 surfaces of other degrees? What about degree 4?", "original_statement": "Can we identify 3-torsion elements of the Brauer group of a degree 2 K3 surface as obstructions to the fineness of some moduli space?\n\nCan we do this for K3 surfaces of other degrees? What about degree 4?", "clean_statement": "Can we identify 3-torsion elements of the Brauer group of a degree 2 K3 surface as obstructions to the fineness of some moduli space?\n\nCan we do this for K3 surfaces of other degrees? What about degree 4?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.3 in the K3-surfaces section of the AIM workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.3\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[15]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we identify 3-torsion elements of the Brauer group of a degree 2 K3 surface as obstructions to the fineness of some moduli space?\\n\\nCan we do this for K3 surfaces of other degrees? What about degree 4?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0016", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a complex Picard-rank-one K3 surface S of degree 2d with d=1 or 2, an order-three cyclic Brauer subgroup C is the universal-sheaf obstruction of a two-dimensional moduli space of ordinary sheaves on an untwisted K3 surface if and only if its index-three kernel T_C is of K3 type. Every such subgroup is realized as S = M_X(3,H,3d) with H^2 = 18d; the Mukai vector is primitive isotropic, has divisibility exactly 3, and hence gives an exact-order-three obstruction. Thus degree 2 uses a degree-18 source and degree 4 uses a degree-36 source. A discriminant-group comparison further proves that the order-three class from a general discriminant-18 cubic fourfold is a distinct non-K3-type orbit and cannot be this fineness obstruction.\n\nCandidate contribution (lattice_obstruction; novelty confidence low): The cyclic/noncyclic discriminant-group sieve separates the G2/P degree-18 K3 fineness class (kernel discriminant group Z/18) from the discriminant-18 cubic-fourfold/sextic-del-Pezzo class (kernel discriminant group Z/6 plus Z/3), and, combined with the fixed-class Fourier-Mukai count, identifies the degree-4 analogue as the degree-36 Mukai construction with exactly two polarized sources for each K3-type subgroup." }, { "id": 20000771, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0017", "title": "The universal-sheaf gerbe and a finite descent-spectrum test", "statement": "If $X$ is a non-fine moduli space of sheaves on a K3 surface $Y$, and the dimension of $X$ is greater than 2, can the corresponding Brauer class obstruct the Hasse principle?", "original_statement": "If $X$ is a non-fine moduli space of sheaves on a K3 surface $Y$, and the dimension of $X$ is greater than 2, can the corresponding Brauer class obstruct the Hasse principle?", "clean_statement": "If $X$ is a non-fine moduli space of sheaves on a K3 surface $Y$, and the dimension of $X$ is greater than 2, can the corresponding Brauer class obstruct the Hasse principle?", "statement_status": "exact", "statement_verification": "The archived AIM URL in the record is . It returned an HTTP 502 error during this run, so the wording above is verified from the canonical JSON record and its neighbouring K3 questions, not from a currently accessible copy of the web page. There is no visible OCR corruption.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.35\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[16]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If $X$ is a non-fine moduli space of sheaves on a K3 surface $Y$, and the dimension of $X$ is greater than 2, can the corresponding Brauer class obstruct the Hasse principle?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0017", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general existence question remains open in the literature checked. For any proper stable-sheaf moduli space, evaluation of its universal-sheaf Brauer class at a local coarse point is exactly the gerbe obstruction to representing that point by an actual local sheaf; the Brauer-Manin question therefore reduces to a finite sumset of local descent spectra, and the class cannot obstruct if an actual stable sheaf exists over every completion. In addition, for a geometrically Picard-rank-one polarized K3 surface with H^2=2g-2, the non-fine 2g-fold M_H(0,H,1-g) has obstruction order g-1 for g at least 3 but contains the rational point represented by O_C for every smooth k-rational member C of |H|, so its obstruction cannot violate the Hasse principle. A parallel vector (0,H,0) yields conditional examples including non-fine fourfolds with rational points.\n\nCandidate contribution (special-case theorem and reduction; novelty confidence low): For every g at least 3 and every number-field K3 surface (Y,H) with H primitive base-point-free, H^2=2g-2, and geometric Neron-Severi group ZH, the 2g-dimensional moduli space M_H(0,H,1-g) is geometrically non-fine of obstruction order g-1 yet has a neutral-gerbe k-point [i_*O_C], and hence this class cannot obstruct the Hasse principle; more generally the obstruction question is equivalent to zero membership in the finite Minkowski sum of the local residual-gerbe invariant sets." }, { "id": 20000772, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0018", "title": "Noncyclic discriminant kernels obstruct prescribed fineness classes", "statement": "Given a K3 surface $X$ with a Brauer class, is that Brauer class the obstruction to fineness as a moduli space of sheaves?", "original_statement": "Given a K3 surface $X$ with a Brauer class, is that Brauer class the obstruction to fineness as a moduli space of sheaves?", "clean_statement": "Given a K3 surface $X$ with a Brauer class, is that Brauer class the obstruction to fineness as a moduli space of sheaves?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List item 1.4 in the section “K3 surfaces” of the 2013 workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.4\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[17]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given a K3 surface $X$ with a Brauer class, is that Brauer class the obstruction to fineness as a moduli space of sheaves?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0018", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the standard complex-projective K3-to-K3 interpretation, the universal affirmative answer is false. If X has Picard rank one and degree 2d, then for every n>1 one can choose an exact order-n Brauer class alpha_n whose kernel is U(n) plus U plus two copies of E8(-1) plus the rank-one lattice of square -2d. Its discriminant group is (Z/n)^2 plus Z/(2d), hence noncyclic. Any obstruction class of a compact two-dimensional moduli space of stable sheaves on a K3 surface would, by Caldararu's theorem, have kernel Hodge-isometric to the source transcendental lattice; the source is forced to have Picard rank one and therefore cyclic discriminant group. Thus alpha_n is not realizable.\n\nCandidate contribution (counterexample_family; novelty confidence low): For every integer n>1, including composite n, and every complex projective Picard-rank-one K3 surface X of degree 2d, the hyperbolic-coordinate class alpha_n(ae+bf+w)=a/n mod Z is not a K3 sheaf-moduli fineness obstruction; its kernel has discriminant group (Z/n)^2 plus Z/(2d)." }, { "id": 20000773, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0019", "title": "Brauer lifts and Beauville--Mukai torsors", "statement": "Let $X$ be a K3 surface of degree $2d$. Consider the Hilbert scheme $\\mathscr{H}$ of length $d+1$ zero-dimensional subschemes of $X$. $\\mathrm{Br}(X)=\\mathrm{Br}(\\mathscr{H})$. There is a map $\\mathscr{H} \\to \\mathbb{P}^{d+1}$ with generic fiber an abelian variety. Does every $\\alpha \\in \\mathrm{Br}(X)$ be correspond to a torsor for the relative Jacobian of this vibration in such a way that the order of the torsor associated to $\\alpha$ times the order of $\\alpha$ is the order of the torsor $\\mathscr{H}$? In this question we are using the same correspondence between principal homogeneous spaces and Brauer classes as in Bianca's talk.\n\nIf we consider the moduli space of $\\alpha$-twisted sheaves on $X$ of dimension 0 and length $d+1$, does this realize the torsors of the previous part of this question?", "original_statement": "Let $X$ be a K3 surface of degree $2d$. Consider the Hilbert scheme $\\mathscr{H}$ of length $d+1$ zero-dimensional subschemes of $X$. $\\mathrm{Br}(X)=\\mathrm{Br}(\\mathscr{H})$. There is a map $\\mathscr{H} \\to \\mathbb{P}^{d+1}$ with generic fiber an abelian variety. Does every $\\alpha \\in \\mathrm{Br}(X)$ be correspond to a torsor for the relative Jacobian of this vibration in such a way that the order of the torsor associated to $\\alpha$ times the order of $\\alpha$ is the order of the torsor $\\mathscr{H}$? In this question we are using the same correspondence between principal homogeneous spaces and Brauer classes as in Bianca's talk.\n\nIf we consider the moduli space of $\\alpha$-twisted sheaves on $X$ of dimension 0 and length $d+1$, does this realize the torsors of the previous part of this question?", "clean_statement": "Let $X$ be a K3 surface of degree $2d$. Consider the Hilbert scheme $\\mathscr{H}$ of length $d+1$ zero-dimensional subschemes of $X$. $\\mathrm{Br}(X)=\\mathrm{Br}(\\mathscr{H})$. There is a map $\\mathscr{H} \\to \\mathbb{P}^{d+1}$ with generic fiber an abelian variety. Does every $\\alpha \\in \\mathrm{Br}(X)$ be correspond to a torsor for the relative Jacobian of this vibration in such a way that the order of the torsor associated to $\\alpha$ times the order of $\\alpha$ is the order of the torsor $\\mathscr{H}$? In this question we are using the same correspondence between principal homogeneous spaces and Brauer classes as in Bianca's talk.\n\nIf we consider the moduli space of $\\alpha$-twisted sheaves on $X$ of dimension 0 and length $d+1$, does this realize the torsors of the previous part of this question?", "statement_status": "exact", "statement_verification": "The canonical record (AIM Problem Lists, workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*, item 1.45) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.45\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[18]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X$ be a K3 surface of degree $2d$. Consider the Hilbert scheme $\\\\mathscr{H}$ of length $d+1$ zero-dimensional subschemes of $X$. $\\\\mathrm{Br}(X)=\\\\mathrm{Br}(\\\\mathscr{H})$. There is a map $\\\\mathscr{H} \\\\to \\\\mathbb{P}^{d+1}$ with generic fiber an abelian variety. Does every $\\\\alpha \\\\in \\\\mathrm{Br}(X)$ be correspond to a torsor for the relative Jacobian of this vibration in such a way that the order of the torsor associated to $\\\\alpha$ times the order of $\\\\alpha$ is the order of the torsor $\\\\mathscr{H}$? In this question we are using the same correspondence between principal homogeneous spaces and Brauer classes as in Bianca's talk.\\n\\nIf we consider the moduli space of $\\\\alpha$-twisted sheaves on $X$ of dimension 0 and length $d+1$, does this realize the torsors of the previous part of this question?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0019", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a complex projective K3 surface with a base-point-free polarization h of square 2d, the rational fibration on X^[d+1] is birational to the degree-(d+1) Beauville--Mukai relative Picard torsor. Its generic period divides 2d/gcd(2d,d+1), so the multiplicative order formula recorded in the AIM question cannot hold for every Brauer class. Huybrechts--Mattei's 2026 exact sequence 0 -> Z/mZ -> Sha(X,h) -> Br(X) -> 0 supplies the corrected noncanonical lifting statement; the realizing moduli spaces consist of pure one-dimensional twisted sheaves, while nonzero Brauer classes cannot support rank-one twisted ideal sheaves.\n\nCandidate contribution (obstruction; novelty confidence low): The generic Hilbert/Beauville--Mukai torsor has period dividing q_d = 2d/gcd(2d,d+1); since Br(X) has elements of arbitrarily large prime order, no torsor assignment on all Brauer classes can satisfy per(T_alpha) ord(alpha) = per(H). The exact replacement is ord(x) = ord(alpha) ord(ord(alpha) x) for a lift x in Sha(X,h), with the residual element lying in the cyclic kernel." }, { "id": 20000774, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0020", "title": "A polarized rank-drop bound and the transcendental exponent bottleneck", "statement": "Fix a number field and a degree. Look at all K3 surfaces over the number field of degree $2d$. Look at the Brauer group of the K3 surface modulo the Brauer group of the number field. Is there a uniform bound on the size of the quotient? What about the algebraic part? The transcendental part of the Brauer group? Note: the first two groups are known to be finite.", "original_statement": "Fix a number field and a degree. Look at all K3 surfaces over the number field of degree $2d$. Look at the Brauer group of the K3 surface modulo the Brauer group of the number field. Is there a uniform bound on the size of the quotient? What about the algebraic part? The transcendental part of the Brauer group? Note: the first two groups are known to be finite.", "clean_statement": "Fix a number field and a degree. Look at all K3 surfaces over the number field of degree $2d$. Look at the Brauer group of the K3 surface modulo the Brauer group of the number field. Is there a uniform bound on the size of the quotient? What about the algebraic part? The transcendental part of the Brauer group? Note: the first two groups are known to be finite.", "statement_status": "exact", "statement_verification": "The canonical AIM record (source file `aim-arithmetic-geometry-notes.json`, zero-based index 19) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.5\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[19]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Fix a number field and a degree. Look at all K3 surfaces over the number field of degree $2d$. Look at the Brauer group of the K3 surface modulo the Brauer group of the number field. Is there a uniform bound on the size of the quotient? What about the algebraic part? The transcendental part of the Brauer group? Note: the first two groups are known to be finite.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0020", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a polarized K3 surface (X,H) over a number field with geometric Picard rank r, the algebraic Brauer quotient has exponent dividing M(r-1) and order dividing M(r-1)^(r-1), where M(s) is Minkowski's universal divisibility bound for finite subgroups of GL_s(Z). If e_X is the exponent of Br(X)/Br_1(X), then #Br(X)/Br_0(X) is at most M(r-1)^(r-1)e_X^(22-r). Hence the algebraic part is uniformly bounded by M(19)^19 independently of the field and degree, and full fixed-degree uniformity is equivalent to uniform boundedness of the transcendental exponent, which remains open in general.\n\nCandidate contribution (lemma; novelty confidence low): The fixed positive polarization class makes the finite Galois image on Pic(X_bar) act faithfully on the rank-(r-1) quotient Pic(X_bar)/Z h_0, yielding exp(Br_1/Br_0) | M(r-1), #(Br_1/Br_0) | M(r-1)^(r-1), and the combined bound #(Br/Br_0) <= M(r-1)^(r-1) exp(Br/Br_1)^(22-r)." }, { "id": 20000775, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0021", "title": "Potential descent, odd-class lower bounds, and the Galois-orbit obstruction", "statement": "If we have a K3 surface $X$ defined over some number field $k$, what can we say about the fields of definition of its Fourier-Mukai partners (partners over $\\mathbb{C}$)? What about twisted K3 surfaces? What about Abelian varieties? What about other varieties?\n\nIf we look at real quadratic extensions of $\\mathbb{Q}$, can the order of the odd part of the class group go to $\\infty$? Does this answer the previous question? How does this relate to the previous problem?", "original_statement": "If we have a K3 surface $X$ defined over some number field $k$, what can we say about the fields of definition of its Fourier-Mukai partners (partners over $\\mathbb{C}$)? What about twisted K3 surfaces? What about Abelian varieties? What about other varieties?\n\nIf we look at real quadratic extensions of $\\mathbb{Q}$, can the order of the odd part of the class group go to $\\infty$? Does this answer the previous question? How does this relate to the previous problem?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "where $h^+(p)$ is the narrow class number. Thus the wording “real quadratic extensions” has two plausible readings:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.55\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[20]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If we have a K3 surface $X$ defined over some number field $k$, what can we say about the fields of definition of its Fourier-Mukai partners (partners over $\\\\mathbb{C}$)? What about twisted K3 surfaces? What about Abelian varieties? What about other varieties?\\n\\nIf we look at real quadratic extensions of $\\\\mathbb{Q}$, can the order of the odd part of the class group go to $\\\\infty$? Does this answer the previous question? How does this relate to the previous problem?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0021", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every complex Fourier--Mukai partner of a projective K3 surface over a number field has a model after finite extension, one common finite extension works for all partners, and a Picard-split surface has models of all partners over the base field. For every positive fundamental discriminant D and every complex K3 surface whose Neron--Severi lattice is even hyperbolic of rank two and determinant -D, the Fourier--Mukai number is at least (|Cl^+(D)_odd|+1)/2. Gang Yu's theorem therefore gives unbounded partner counts for arbitrary real-quadratic discriminants, but a separate bound on the number of Galois orbits is necessary to force growing fields of definition.\n\nCandidate contribution (reduction; novelty confidence low): For a positive fundamental discriminant D, the principal genus theorem and the HLOY formula give |FM(X)| >= (|Cl^+(D)_odd|+1)/2 for every rank-two K3 surface with determinant -D; if the Galois action on its FM partners has at most R>1 orbits, some partner has field-of-moduli degree at least ceil((|Cl^+(D)_odd|-1)/(2(R-1))). The Picard-split criterion proves that the orbit hypothesis cannot be omitted." }, { "id": 20000776, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0022", "title": "Explicit addition of fineness obstructions on a Picard-rank-one cyclic Brauer ray", "statement": "How does the group structure of the Brauer group relate to Brauer classes as obstructions to fineness of moduli spaces? Given a K3 surface $X$ and two K3 surfaces $S_1$ and $S_2$ such that $X$ is a moduli space of sheaves on $S_1$ and $S_1$, can we explicitly construct a third K3 surface or variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and the obstruction class to fineness of this moduli space is the sum of the obstruction classes coming from $S_1$ and $S_2$?", "original_statement": "How does the group structure of the Brauer group relate to Brauer classes as obstructions to fineness of moduli spaces? Given a K3 surface $X$ and two K3 surfaces $S_1$ and $S_2$ such that $X$ is a moduli space of sheaves on $S_1$ and $S_1$, can we explicitly construct a third K3 surface or variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and the obstruction class to fineness of this moduli space is the sum of the obstruction classes coming from $S_1$ and $S_2$?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "This report treats that sentence as an explicit reconstruction, not as verified source text. The original record also omits a base field and stability data. We work over $\\mathbb C$, use primitive isotropic Mukai vectors and a generic polarization, and keep the chosen identifications with $X$ as part of the data. These hypotheses are essential: the two obstruction classes live in the same group only after transporting them to $\\operatorname{Br}(X)$.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.6\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[21]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How does the group structure of the Brauer group relate to Brauer classes as obstructions to fineness of moduli spaces? Given a K3 surface $X$ and two K3 surfaces $S_1$ and $S_2$ such that $X$ is a moduli space of sheaves on $S_1$ and $S_1$, can we explicitly construct a third K3 surface or variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and the obstruction class to fineness of this moduli space is the sum of the obstruction classes coming from $S_1$ and $S_2$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0022", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a marked complex K3 surface X with NS(X)=Zh and h^2=2d, the cyclic subgroup C_g of Br(X) consisting of characters supported on the rank-one discriminant direction g is entirely realized by fineness obstructions of sheaf-moduli presentations of X and is closed under Brauer addition. If q=a/n is reduced, a source K3 of degree 2dn^2 and the primitive isotropic vector v=(n,kH,dnk^2), with k a unit chosen from a up to the fixed obstruction-sign convention, give X as the corresponding moduli K3 with obstruction alpha_q. For arbitrary sums, a separate proved reduction says that the twisted algebraic Mukai lattice for B_1+B_2 must contain a primitive hyperbolic plane; the unrestricted AIM question remains open.\n\nCandidate contribution (explicit_construction_and_special_case_theorem; novelty confidence low): Candidate novelty: on a marked Picard-rank-one K3 surface, the full cyclic Brauer ray C_g is made of fineness-obstruction classes and is closed under addition; the class a/n is realized by a degree-2dn^2 source and Mukai vector (n,kH,dnk^2), and a Bezout/discriminant-gluing calculation proves that the target is the original X rather than only a Fourier-Mukai partner." }, { "id": 20000777, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0023", "title": "Presentation gauge, a binary complement, and the two Clifford K3 surfaces", "statement": "Pick three quadrics, $Q_0$, $Q_1$, and $Q_2$. Let $X=V(x_0Q_0+x_1Q_1+x_2Q_2)$. This is a cubic fourfold which contains the plane $V(x_0,x_1,x_2)$. The net of quadrics generated by the $Q_i$ gives us a family of quadric fourfolds over $\\mathbb{P}^2$. Let $\\mathcal{V}$ be the locus of maximal isotropic subspaces of the fibers. If we take the Stein factorization of the map $\\mathcal{V} \\to \\mathbb{P}^2$, this gives a bundle of projective spaces of dimension 3 over a double cover of $\\mathbb{P}^2$. The double cover will be a K3 surface. This give a Brauer class on the K3 surface.\n\nIf we look at 3-planes in $\\mathbb{P}^5$ containing the plane, they will be parametrized by $\\mathbb{P}^2$, and we will get a family of quadric surfaces over the $\\mathbb{P}^2$, namely the residual intersection of $X$ with the 3-plane. We can do the same construction as before with the space of maximal isotropic subspaces of the fibers to get another projective bundle over a K3 surface.\n\nIs there an intrinsic description of the relation between these twisted K3 surfaces?", "original_statement": "Pick three quadrics, $Q_0$, $Q_1$, and $Q_2$. Let $X=V(x_0Q_0+x_1Q_1+x_2Q_2)$. This is a cubic fourfold which contains the plane $V(x_0,x_1,x_2)$. The net of quadrics generated by the $Q_i$ gives us a family of quadric fourfolds over $\\mathbb{P}^2$. Let $\\mathcal{V}$ be the locus of maximal isotropic subspaces of the fibers. If we take the Stein factorization of the map $\\mathcal{V} \\to \\mathbb{P}^2$, this gives a bundle of projective spaces of dimension 3 over a double cover of $\\mathbb{P}^2$. The double cover will be a K3 surface. This give a Brauer class on the K3 surface.\n\nIf we look at 3-planes in $\\mathbb{P}^5$ containing the plane, they will be parametrized by $\\mathbb{P}^2$, and we will get a family of quadric surfaces over the $\\mathbb{P}^2$, namely the residual intersection of $X$ with the 3-plane. We can do the same construction as before with the space of maximal isotropic subspaces of the fibers to get another projective bundle over a K3 surface.\n\nIs there an intrinsic description of the relation between these twisted K3 surfaces?", "clean_statement": "Pick three quadrics, $Q_0$, $Q_1$, and $Q_2$. Let $X=V(x_0Q_0+x_1Q_1+x_2Q_2)$. This is a cubic fourfold which contains the plane $V(x_0,x_1,x_2)$. The net of quadrics generated by the $Q_i$ gives us a family of quadric fourfolds over $\\mathbb{P}^2$. Let $\\mathcal{V}$ be the locus of maximal isotropic subspaces of the fibers. If we take the Stein factorization of the map $\\mathcal{V} \\to \\mathbb{P}^2$, this gives a bundle of projective spaces of dimension 3 over a double cover of $\\mathbb{P}^2$. The double cover will be a K3 surface. This give a Brauer class on the K3 surface.\n\nIf we look at 3-planes in $\\mathbb{P}^5$ containing the plane, they will be parametrized by $\\mathbb{P}^2$, and we will get a family of quadric surfaces over the $\\mathbb{P}^2$, namely the residual intersection of $X$ with the 3-plane. We can do the same construction as before with the space of maximal isotropic subspaces of the fibers to get another projective bundle over a K3 surface.\n\nIs there an intrinsic description of the relation between these twisted K3 surfaces?", "statement_status": "exact", "statement_verification": "This is Problem 1.1 in the “K3 surfaces” section of the AIM workshop list *Brauer groups and obstruction problems: moduli spaces and arithmetic* (25 February--1 March 2013). The exact source record asks the following.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: K3 surfaces\nSource item: 1.1\nSource URL: http://aimpl.org/brauermoduli/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[22]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Pick three quadrics, $Q_0$, $Q_1$, and $Q_2$. Let $X=V(x_0Q_0+x_1Q_1+x_2Q_2)$. This is a cubic fourfold which contains the plane $V(x_0,x_1,x_2)$. The net of quadrics generated by the $Q_i$ gives us a family of quadric fourfolds over $\\\\mathbb{P}^2$. Let $\\\\mathcal{V}$ be the locus of maximal isotropic subspaces of the fibers. If we take the Stein factorization of the map $\\\\mathcal{V} \\\\to \\\\mathbb{P}^2$, this gives a bundle of projective spaces of dimension 3 over a double cover of $\\\\mathbb{P}^2$. The double cover will be a K3 surface. This give a Brauer class on the K3 surface.\\n\\nIf we look at 3-planes in $\\\\mathbb{P}^5$ containing the plane, they will be parametrized by $\\\\mathbb{P}^2$, and we will get a family of quadric surfaces over the $\\\\mathbb{P}^2$, namely the residual intersection of $X$ with the 3-plane. We can do the same construction as before with the space of maximal isotropic subspaces of the fibers to get another projective bundle over a K3 surface.\\n\\nIs there an intrinsic description of the relation between these twisted K3 surfaces?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0023", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The residual rank-four quadric-surface bundle is exactly the restriction of the rank-six net form to the tautological bundle associated with 3-planes through the fixed plane. Every Koszul change of the presentation of the same cubic fixes this restricted form and its Clifford class, while it can change the rank-six net. Over the generic point, a binary orthogonal complement measures the difference of the signed discriminant covers and gives equality of the Clifford classes when it is hyperbolic. For very general complex data, the two twisted K3 surfaces are not Fourier--Mukai partners because their twisted transcendental lattices have inequivalent discriminant forms of cyclic order eight.\n\nCandidate contribution (reduction; novelty confidence low): Candidate gauge-rigidity reduction: Koszul presentation changes fix the residual rank-four quadratic bundle exactly but generally move the rank-six net, and for a fixed presentation the signed discriminant of the binary orthogonal complement is the precise square-class obstruction to equality of the two generic discriminant covers." }, { "id": 20000778, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0024", "title": "Phantoms on projective space and the failure of a surface rationality criterion", "statement": "Consider $D^b(\\mathbb{P}^n)$. Take an admissible thick subcategory, $\\mathcal{A}$, i.e. a triangulated subcategory closed under taking summands and such that the inclusion has a left (equivalently right) adjoint. If $K_0(\\mathcal{A})=0$, is $\\mathcal{A}=0$? Such an $\\mathcal{A}$ is called a phantom. When $n=1$, it is known that there is no such phantom subcategory. This is related to Caldararu's conjecture on the existence of exceptional collections on projective homogeneous varieties. The answer is no if we replace $\\mathbb{P}^n$ by an arbitrary variety (there is a surface counterexample).\n\nIf we replace $\\mathbb{P}^n$ by another regular surface, is the condition of not having phantom subcategories equivalent to rationality? What if we take $K_0(X) \\otimes \\mathbb{Q}$?", "original_statement": "Consider $D^b(\\mathbb{P}^n)$. Take an admissible thick subcategory, $\\mathcal{A}$, i.e. a triangulated subcategory closed under taking summands and such that the inclusion has a left (equivalently right) adjoint. If $K_0(\\mathcal{A})=0$, is $\\mathcal{A}=0$? Such an $\\mathcal{A}$ is called a phantom. When $n=1$, it is known that there is no such phantom subcategory. This is related to Caldararu's conjecture on the existence of exceptional collections on projective homogeneous varieties. The answer is no if we replace $\\mathbb{P}^n$ by an arbitrary variety (there is a surface counterexample).\n\nIf we replace $\\mathbb{P}^n$ by another regular surface, is the condition of not having phantom subcategories equivalent to rationality? What if we take $K_0(X) \\otimes \\mathbb{Q}$?", "clean_statement": "Consider $D^b(\\mathbb{P}^n)$. Take an admissible thick subcategory, $\\mathcal{A}$, i.e. a triangulated subcategory closed under taking summands and such that the inclusion has a left (equivalently right) adjoint. If $K_0(\\mathcal{A})=0$, is $\\mathcal{A}=0$? Such an $\\mathcal{A}$ is called a phantom. When $n=1$, it is known that there is no such phantom subcategory. This is related to Caldararu's conjecture on the existence of exceptional collections on projective homogeneous varieties. The answer is no if we replace $\\mathbb{P}^n$ by an arbitrary variety (there is a surface counterexample).\n\nIf we replace $\\mathbb{P}^n$ by another regular surface, is the condition of not having phantom subcategories equivalent to rationality? What if we take $K_0(X) \\otimes \\mathbb{Q}$?", "statement_status": "exact", "statement_verification": "The canonical record is problem 2.1, “Other,” from the 2013 AIM workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: Other\nSource item: 2.1\nSource URL: http://aimpl.org/brauermoduli/2/\nCanonical location: aim-arithmetic-geometry-notes.json notes[23]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Consider $D^b(\\\\mathbb{P}^n)$. Take an admissible thick subcategory, $\\\\mathcal{A}$, i.e. a triangulated subcategory closed under taking summands and such that the inclusion has a left (equivalently right) adjoint. If $K_0(\\\\mathcal{A})=0$, is $\\\\mathcal{A}=0$? Such an $\\\\mathcal{A}$ is called a phantom. When $n=1$, it is known that there is no such phantom subcategory. This is related to Caldararu's conjecture on the existence of exceptional collections on projective homogeneous varieties. The answer is no if we replace $\\\\mathbb{P}^n$ by an arbitrary variety (there is a surface counterexample).\\n\\nIf we replace $\\\\mathbb{P}^n$ by another regular surface, is the condition of not having phantom subcategories equivalent to rationality? What if we take $K_0(X) \\\\otimes \\\\mathbb{Q}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/2/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0024", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over the complex numbers, the phantom question has an affirmative answer for projective spaces of dimensions one and two, including after rationalizing K_0, while no general resolution for dimension at least three was located. For surfaces, phantom-freeness is not equivalent to rationality: Krah's blow-up of the plane at ten general points is rational and has a universal phantom, whereas a projective K3 surface is nonrational and has no rational phantom. A proved blow-up persistence lemma yields a smooth rational surface of every Picard number at least 11 containing a universal phantom equivalent to Krah's.\n\nCandidate contribution (lemma_and_explicit_family; novelty confidence low): For every integer rho at least 11, an iterated point blow-up of Krah's surface has Picard number rho and contains the same universal phantom up to equivalence; more generally, integral, rational, and universal phantoms persist under blow-up along any smooth center." }, { "id": 20000779, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0025", "title": "Derived equivalence preserves all splitting fields but not the genus-one curve", "statement": "Suppose we have two genus one curves $C$ and $C'$ over a field $F$ which are derived-equivalent. Is $C(F) \\neq \\emptyset$ equivalent to $C'(F) \\neq \\emptyset$? Are $C$ and $C'$ isomorphic over $F$?", "original_statement": "Suppose we have two genus one curves $C$ and $C'$ over a field $F$ which are derived-equivalent. Is $C(F) \\neq \\emptyset$ equivalent to $C'(F) \\neq \\emptyset$? Are $C$ and $C'$ isomorphic over $F$?", "clean_statement": "Suppose we have two genus one curves $C$ and $C'$ over a field $F$ which are derived-equivalent. Is $C(F) \\neq \\emptyset$ equivalent to $C'(F) \\neq \\emptyset$? Are $C$ and $C'$ isomorphic over $F$?", "statement_status": "exact", "statement_verification": "There is no visible corruption or missing notation. Following the paper that arose from the same AIM workshop, “derived-equivalent” is interpreted as an \\(F\\)-linear exact equivalence \\[ D^b(\\operatorname{Coh} C)\\simeq D^b(\\operatorname{Coh} C') \\] of triangulated categories. A genus-one curve means a smooth, projective, geometrically connected curve of genus one. No perfectness hypothesis is imposed: the classification used below is stated over an arbitrary field.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: Other\nSource item: 2.2\nSource URL: http://aimpl.org/brauermoduli/2/\nCanonical location: aim-arithmetic-geometry-notes.json notes[24]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Suppose we have two genus one curves $C$ and $C'$ over a field $F$ which are derived-equivalent. Is $C(F) \\\\neq \\\\emptyset$ equivalent to $C'(F) \\\\neq \\\\emptyset$? Are $C$ and $C'$ isomorphic over $F$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/2/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0025", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For smooth projective geometrically connected genus-one curves over any field F, an F-linear exact derived equivalence identifies their Jacobians and relates their Weil--Chatelet classes by a Jacobian automorphism and multiplication by a unit modulo the period. Consequently C(L) is nonempty if and only if C'(L) is nonempty for every extension L/F, so period, finite splitting extensions, and index agree. Isomorphism need not hold: over Q, torsors with classes alpha and 2 alpha for a period-five class alpha on a non-CM elliptic curve are derived-equivalent but not isomorphic.\n\nCandidate contribution (corollary; novelty confidence low): Derived-equivalent genus-one curves have identical splitting fields over every extension of the base field; moreover, if the common Jacobian E has Aut_F(E)={+/-1}, then a torsor of period n has exactly one curve-isomorphism class of derived partners for n=1,2 and exactly phi(n)/2 classes for n>2, hence categorical rigidity holds exactly for n in {1,2,3,4,6}." }, { "id": 20000780, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0026", "title": "A toric-pencil obstruction to irrational curve subfields", "statement": "Consider $\\mathbb{C}(u,v)=\\mathbb{C}(\\mathbb{P}^2)$. Look at the cyclic algebra $A=(u,v)_n$, defined to be $\\mathbb{C}(u,v)\\langle x,y \\rangle/(x^n-u, y^n-v,xy-\\zeta yx)$ with $\\zeta$ a primitive $n$th root of unity. Does there exist a transcendence degree 1 subfield of $A$ (also a $\\mathbb{C}$-subalgebra) which is not rational? Geometrically, can we find a branched cover $S$ of $\\mathbb{P}^2$ corresponding to a maximal subfield of the cyclic algebra, which has a rational map to an irrational curve?", "original_statement": "Consider $\\mathbb{C}(u,v)=\\mathbb{C}(\\mathbb{P}^2)$. Look at the cyclic algebra $A=(u,v)_n$, defined to be $\\mathbb{C}(u,v)\\langle x,y \\rangle/(x^n-u, y^n-v,xy-\\zeta yx)$ with $\\zeta$ a primitive $n$th root of unity. Does there exist a transcendence degree 1 subfield of $A$ (also a $\\mathbb{C}$-subalgebra) which is not rational? Geometrically, can we find a branched cover $S$ of $\\mathbb{P}^2$ corresponding to a maximal subfield of the cyclic algebra, which has a rational map to an irrational curve?", "clean_statement": "Consider $\\mathbb{C}(u,v)=\\mathbb{C}(\\mathbb{P}^2)$. Look at the cyclic algebra $A=(u,v)_n$, defined to be $\\mathbb{C}(u,v)\\langle x,y \\rangle/(x^n-u, y^n-v,xy-\\zeta yx)$ with $\\zeta$ a primitive $n$th root of unity. Does there exist a transcendence degree 1 subfield of $A$ (also a $\\mathbb{C}$-subalgebra) which is not rational? Geometrically, can we find a branched cover $S$ of $\\mathbb{P}^2$ corresponding to a maximal subfield of the cyclic algebra, which has a rational map to an irrational curve?", "statement_status": "exact", "statement_verification": "The canonical record is problem 2.3 in the “Other” section of the 2013 AIM workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: Other\nSource item: 2.3\nSource URL: http://aimpl.org/brauermoduli/2/\nCanonical location: aim-arithmetic-geometry-notes.json notes[25]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Consider $\\\\mathbb{C}(u,v)=\\\\mathbb{C}(\\\\mathbb{P}^2)$. Look at the cyclic algebra $A=(u,v)_n$, defined to be $\\\\mathbb{C}(u,v)\\\\langle x,y \\\\rangle/(x^n-u, y^n-v,xy-\\\\zeta yx)$ with $\\\\zeta$ a primitive $n$th root of unity. Does there exist a transcendence degree 1 subfield of $A$ (also a $\\\\mathbb{C}$-subalgebra) which is not rational? Geometrically, can we find a branched cover $S$ of $\\\\mathbb{P}^2$ corresponding to a maximal subfield of the cyclic algebra, which has a rational map to an irrational curve?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/2/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0026", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let D=(u,v)_n over C(u,v). For every primitive integer vector (a,b), any commutative transcendence-degree-one C-subfield F of D that contains C(u^a v^b) is rational. More generally, for a symbol-compatible pencil K=k(t,s) with [D]=r[(t,s)_n] and gcd(r,n)=1, every such F containing k(t) is classified as F=k(alpha^d), where alpha^n=t and d divides n. The proof works even when KF is not maximal, using a maximal-subfield index budget and the residue at s=0. This is a rigorous partial obstruction; the unrestricted AIM existence question remains open here.\n\nCandidate contribution (obstruction_and_classification; novelty confidence low): Candidate novelty: all curve subfields of the generic symbol division algebra lying over any primitive toric pencil are necessarily cyclic monomial fields k(alpha^d), alpha^n=u^a v^b, and hence have genus zero; equivalently, a positive-genus fibration of a maximal-subfield cover cannot carry a factored primitive toric coordinate." }, { "id": 20000781, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0027", "title": "No essential transcendental Brauer-Manin obstruction on abelian torsors", "statement": "Does any Abelian variety have a transcendental Brauer class which obstructs the existence of rational points?\n\nA theorem of Manin says that if an Abelian variety has a transcendental Brauer class, and $\\mathrm{Sha}(A)$ is finite, $E$ a torsor for $A$, and $E(\\mathbb{A})^{\\mathrm{Br}_{\\mathrm{alg}}} \\neq \\emptyset$, then $E(K) \\neq \\emptyset$. Can we show that the existence of a transcendental obstruction implies the existence of algebraic obstructions which rule out the same adelic points as coming from global points?", "original_statement": "Does any Abelian variety have a transcendental Brauer class which obstructs the existence of rational points?\n\nA theorem of Manin says that if an Abelian variety has a transcendental Brauer class, and $\\mathrm{Sha}(A)$ is finite, $E$ a torsor for $A$, and $E(\\mathbb{A})^{\\mathrm{Br}_{\\mathrm{alg}}} \\neq \\emptyset$, then $E(K) \\neq \\emptyset$. Can we show that the existence of a transcendental obstruction implies the existence of algebraic obstructions which rule out the same adelic points as coming from global points?", "clean_statement": "Does any Abelian variety have a transcendental Brauer class which obstructs the existence of rational points?\n\nA theorem of Manin says that if an Abelian variety has a transcendental Brauer class, and $\\mathrm{Sha}(A)$ is finite, $E$ a torsor for $A$, and $E(\\mathbb{A})^{\\mathrm{Br}_{\\mathrm{alg}}} \\neq \\emptyset$, then $E(K) \\neq \\emptyset$. Can we show that the existence of a transcendental obstruction implies the existence of algebraic obstructions which rule out the same adelic points as coming from global points?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop “Brauer groups and obstruction problems: moduli spaces and arithmetic,” section “Other,” Problem 2.4) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: Other\nSource item: 2.4\nSource URL: http://aimpl.org/brauermoduli/2/\nCanonical location: aim-arithmetic-geometry-notes.json notes[26]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does any Abelian variety have a transcendental Brauer class which obstructs the existence of rational points?\\n\\nA theorem of Manin says that if an Abelian variety has a transcendental Brauer class, and $\\\\mathrm{Sha}(A)$ is finite, $E$ a torsor for $A$, and $E(\\\\mathbb{A})^{\\\\mathrm{Br}_{\\\\mathrm{alg}}} \\\\neq \\\\emptyset$, then $E(K) \\\\neq \\\\emptyset$. Can we show that the existence of a transcendental obstruction implies the existence of algebraic obstructions which rule out the same adelic points as coming from global points?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/2/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0027", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal pointed question has a negative answer because every abelian variety has its rational identity. For the intended torsor question, Creutz proved unconditionally that the full, algebraic, and Albanese Brauer-Manin sets coincide, and that emptiness is witnessed by one locally constant algebraic class; this explicitly solves AIM Problem 2.4. Finiteness of Tate-Shafarevich groups is needed only to deduce a rational point from nonempty algebraic Brauer set via the Manin-Cassels-Tate pairing, not for equality of Brauer sets. A proved synthesis corollary further shows that an individually obstructing n-torsion class on a locally soluble torsor of period P compresses to a single uniformly obstructing locally constant algebraic l-primary class for some prime l dividing gcd(n,P), with no new prime introduced.\n\nCandidate contribution (synthesis_corollary; novelty confidence low): If X is a locally soluble torsor under an abelian variety A over a number field, with period P, and an order-n class alpha in Br(X) satisfies X(A_k)^alpha empty, then for some prime l dividing gcd(n,P) there is a locally constant algebraic class beta_l, killed by l^(e(A) v_l(n)), whose Brauer-Manin value is one fixed nonzero l-primary element on every adelic point; hence beta_l alone obstructs." }, { "id": 20000782, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0028", "title": "The spin^c obstruction rules out the proposed threefold", "statement": "Does there exist a smooth projective complex threefold \\(X\\) and\n\\(\\gamma\\in H^2(X,\\mathbb Z/2)\\) for which\n\\([\\beta_2(\\gamma^2)]\\ne0\\) in\n\\(\\mathcal Q_{\\beta_2(\\gamma)}(X)\\)?", "original_statement": "Let $X$ be a smooth projective complex threefold and a class $\\gamma \\in H^2(X,\\Z/2)$. Consider the integral Bockstein of its square, $\\beta(\\gamma^2) \\in H^5(X,\\Z)$ (coming from the short exact sequence $0 \\to \\Z \\to \\Z \\to \\Z/2 \\to 0$). Take $H^5(X,\\Z)/(H^2(X,\\Z) \\mathbb{C}up \\beta(\\gamma))$. If $\\beta(\\gamma^2) \\neq 0$ in this quotient, then the period index conjecture would be false, since we would have a Brauer class of period 2, and index at least 8. Is there a threefold satisfying these conditions? The period-index conjecture says that the index of $\\alpha$ divides the period of $\\alpha$ to the $(d-1)$-th power, where $d$ is the dimension of the variety.", "clean_statement": "Does there exist a smooth projective complex threefold \\(X\\) and\n\\(\\gamma\\in H^2(X,\\mathbb Z/2)\\) for which\n\\([\\beta_2(\\gamma^2)]\\ne0\\) in\n\\(\\mathcal Q_{\\beta_2(\\gamma)}(X)\\)?", "statement_status": "corrected_verified", "statement_verification": "The source has two corruptions. 1. The integral Bockstein is the connecting map for \\[ 0\\longrightarrow\\mathbb Z\\xrightarrow{\\times2}\\mathbb Z \\longrightarrow\\mathbb Z/2\\longrightarrow0. \\] The multiplication-by-\\(2\\) label is missing from the AIM text. 2. Comparison with Antieau--Williams, *The topological period-index problem over 6-complexes*, identifies the string “\\(\\mathbb{C}up\\)” as a corrupted cup product.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: Other\nSource item: 2.5\nSource URL: http://aimpl.org/brauermoduli/2/\nCanonical location: aim-arithmetic-geometry-notes.json notes[27]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X$ be a smooth projective complex threefold and a class $\\\\gamma \\\\in H^2(X,\\\\Z/2)$. Consider the integral Bockstein of its square, $\\\\beta(\\\\gamma^2) \\\\in H^5(X,\\\\Z)$ (coming from the short exact sequence $0 \\\\to \\\\Z \\\\to \\\\Z \\\\to \\\\Z/2 \\\\to 0$). Take $H^5(X,\\\\Z)/(H^2(X,\\\\Z) \\\\mathbb{C}up \\\\beta(\\\\gamma))$. If $\\\\beta(\\\\gamma^2) \\\\neq 0$ in this quotient, then the period index conjecture would be false, since we would have a Brauer class of period 2, and index at least 8. Is there a threefold satisfying these conditions? The period-index conjecture says that the index of $\\\\alpha$ divides the period of $\\\\alpha$ to the $(d-1)$-th power, where $d$ is the dimension of the variety.\"\nOriginal remarks: [\"I will fix the Bbb Z. Also, what is the string of symbols after \\\"Take\\\" in the 3rd sentence?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/2/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0028", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "No smooth projective complex threefold satisfies the recovered AIM condition. Crowley and Grant prove that for every closed spin^c 6-manifold N and every gamma in H^2(N,Z/2), the class beta_2(gamma^2) equals beta_2(gamma) cup e_gamma for some integral degree-2 class e_gamma. A smooth projective complex threefold is a closed almost-complex, hence spin^c, 6-manifold, so beta_2(gamma^2) is always zero in H^5(X,Z)/(H^2(X,Z) cup beta_2(gamma)).\n\nCandidate contribution (corollary; novelty confidence low): For smooth projective complex threefolds, raw square-Bockstein vanishing beta_2(delta^2)=0 for every delta in H^2(-,Z/2) is a birational invariant and holds for every threefold birational to a product S x C of a smooth projective surface and a smooth projective curve." }, { "id": 20000783, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0029", "title": "Affine automorphism-orbit finiteness and a two-class torus example", "statement": "Fix a variety $X$ and a Brauer class $\\alpha$. Is the set of $\\beta$ such that the $D^b(X,\\alpha)$ are equivalent finite?", "original_statement": "Fix a variety $X$ and a Brauer class $\\alpha$. Is the set of $\\beta$ such that the $D^b(X,\\alpha)$ are equivalent finite?", "clean_statement": "Fix a variety $X$ and a Brauer class $\\alpha$. Is the set of $\\beta$ such that the $D^b(X,\\alpha)$ are equivalent finite?", "statement_status": "exact", "statement_verification": "The canonical record (AIM workshop *Brauer groups and obstruction problems: moduli spaces and arithmetic*, item 2.6) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Brauer groups and obstruction problems: moduli spaces and arithmetic\nSection: Other\nSource item: 2.6\nSource URL: http://aimpl.org/brauermoduli/2/\nCanonical location: aim-arithmetic-geometry-notes.json notes[28]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Fix a variety $X$ and a Brauer class $\\\\alpha$. Is the set of $\\\\beta$ such that the $D^b(X,\\\\alpha)$ are equivalent finite?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"If $X$ is affine, then the set of such $\\\\beta$ might be a singleton by the theory of Morita-equivalence.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/brauermoduli/2/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0029", "aim-domain:arithmetic-geometry", "aim-workshop:brauermoduli", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth affine complex variety X=Spec(R), the Brauer classes beta for which D^b(X,beta) is C-linearly exactly equivalent to D^b(X,alpha) form precisely the Aut_C(X)-orbit of alpha; hence the set is finite and has cardinality at most |Br(X)[per(alpha)]|. If equivalences are R-linear, the set is the singleton {alpha}. As an explicit worked family, for X=(G_m)^2 and the order-n symbol class alpha_n=(x,y)_n with n>=3, the set is exactly {alpha_n,-alpha_n}, so the tentative affine singleton fails without X-linearity.\n\nCandidate contribution (explicit_worked_family; novelty confidence low): For every integer n>=3, the symbol class alpha_n=(x,y)_n on the smooth affine complex torus (G_m)^2 has exactly the two-element fixed-variety C-linear derived-equivalence set {alpha_n,-alpha_n}." }, { "id": 20000784, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0030", "title": "Factorwise transfer of density-one ordinary reduction", "statement": "Let $A$ be an abelian variety over a number field $K$. Does there exist a finite extension $K'/K$ such that for a set of primes $v$ of $K'$ of density $1$, $A/\\F_v$ is ordinary?", "original_statement": "Let $A$ be an abelian variety over a number field $K$. Does there exist a finite extension $K'/K$ such that for a set of primes $v$ of $K'$ of density $1$, $A/\\F_v$ is ordinary?", "clean_statement": "Let $A$ be an abelian variety over a number field $K$. Does there exist a finite extension $K'/K$ such that for a set of primes $v$ of $K'$ of density $1$, $A/\\F_v$ is ordinary?", "statement_status": "exact", "statement_verification": "The repository record has no visible OCR corruption. Its cited source URL, , timed out during this run, so it could not be compared directly with the live page. The same formulation appears at the beginning of Section 7 of Pink [Pin98] and in Section 1.1 of Cantoral Farfán--Li--Mantovan--Pries--Tang [CFLMPT25]. There is therefore no substantive ambiguity in the recovered question.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Cohomological methods in abelian varieties\nSection: Problems\nSource item: 1.1\nSource URL: http://aimpl.org/cohomabelian/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[29]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $A$ be an abelian variety over a number field $K$. Does there exist a finite extension $K'/K$ such that for a set of primes $v$ of $K'$ of density $1$, $A/\\\\F_v$ is ordinary?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"It is known that if $\\\\dim A \\\\leq 2$. The result is also true for some other varieties, such as $K3$ surfaces or ordinary cubic fourfolds. Look at Ogus, LN 900.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/cohomabelian/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0030", "aim-domain:arithmetic-geometry", "aim-workshop:cohomabelian", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Potential density-one ordinary reduction is stable under geometric isogeny decomposition: if A over K is geometrically isogenous to a finite product of powers of factors B_i and every B_i has density-one ordinary reduction after some finite extension of a field of definition, then A has density-one ordinary reduction after one common finite extension. Quantitatively, the upper Dirichlet density of the defect set over a common field M is at most the sum of [M:F_i] times the factorwise upper defect densities. Consequently, the AIM question has an affirmative answer in arbitrary total dimension whenever every geometrically simple factor has dimension at most two; this includes restrictions of scalars of elliptic curves and abelian surfaces.\n\nCandidate contribution (closure_lemma_and_special_case; novelty confidence low): Candidate novelty: for a common overfield M of factor fields F_i, the nonordinary-or-bad defect density of an abelian variety geometrically isogenous to a product is bounded by the sum of [M:F_i] times the factor defect densities; this yields the explicit arbitrary-dimensional theorem for varieties all of whose geometric simple factors have dimension at most two, and the corresponding restriction-of-scalars family." }, { "id": 20000785, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0031", "title": "A torsion and multiplication sieve for rational multisections", "statement": "Let $X/\\mathbb C$ be a surface of general type. Then Lang's conjecture implies that there exists a constant $N$, depending only on $X$, such that if $C \\subseteq X$ is rational, then $K_X \\cdot C \\leq N$.\n\nWhat happens if $X$ is an elliptic surface? Is there a bound on $K_X \\cdot C$ as $C$ varies through rational curves?", "original_statement": "Let $X/\\mathbb C$ be a surface of general type. Then Lang's conjecture implies that there exists a constant $N$, depending only on $X$, such that if $C \\subseteq X$ is rational, then $K_X \\cdot C \\leq N$.\n\nWhat happens if $X$ is an elliptic surface? Is there a bound on $K_X \\cdot C$ as $C$ varies through rational curves?", "clean_statement": "Let $X/\\mathbb C$ be a surface of general type. Then Lang's conjecture implies that there exists a constant $N$, depending only on $X$, such that if $C \\subseteq X$ is rational, then $K_X \\cdot C \\leq N$.\n\nWhat happens if $X$ is an elliptic surface? Is there a bound on $K_X \\cdot C$ as $C$ varies through rational curves?", "statement_status": "exact", "statement_verification": "The canonical AIM record (source file `aim-arithmetic-geometry-notes.json`, zero-based index 30) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Cohomological methods in abelian varieties\nSection: Problems\nSource item: 1.2\nSource URL: http://aimpl.org/cohomabelian/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[30]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X/\\\\mathbb C$ be a surface of general type. Then Lang's conjecture implies that there exists a constant $N$, depending only on $X$, such that if $C \\\\subseteq X$ is rational, then $K_X \\\\cdot C \\\\leq N$.\\n\\nWhat happens if $X$ is an elliptic surface? Is there a bound on $K_X \\\\cdot C$ as $C$ varies through rational curves?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Ulmer conjectures that there is a bound. However, this result is false over finite fields.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/cohomabelian/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0031", "aim-domain:arithmetic-geometry", "aim-workshop:cohomabelian", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a relatively minimal complex Jacobian elliptic surface over P^1, the canonical-degree problem is nontrivial only in height h >= 3, where K_X.C = (h-2)deg(C) for horizontal curves. Every torsion rational multisection has fiber degree at most 96 (at most 6 for constant j), and the fiberwise m-multiple of any rational multisection is again rational with degree dividing the original degree. Therefore neither torsion nor multiplication orbits can produce unbounded canonical degree; any counterexample must use genuinely new non-torsion rational multisections of unbounded degree. A separate orbifold Riemann-Hurwitz inequality rules out rational multisections when the multiple-fiber contribution is at least 2.\n\nCandidate contribution (reduction; novelty confidence low): Candidate torsion-multiplication sieve: torsion rational multisections on a complex Jacobian elliptic surface have degree at most 96 (at most 6 in the isotrivial case), while every fiberwise multiple has degree dividing the source degree, so any unbounded sequence must consist of non-torsion multisections outside the multiplication closure of every finite set." }, { "id": 20000786, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0032", "title": "CM factors force infinitely many nonordinary primes", "statement": "Let $K$ be a number field, and let $A/K$ be an abelian variety. Do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is not ordinary? More strongly, do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is supersingular?", "original_statement": "Let $K$ be a number field, and let $A/K$ be an abelian variety. Do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is not ordinary? More strongly, do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is supersingular?", "clean_statement": "Let $K$ be a number field, and let $A/K$ be an abelian variety. Do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is not ordinary? More strongly, do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\F_v$ is supersingular?", "statement_status": "exact", "statement_verification": "Here and below, $v$ means a finite place of good reduction and $\\mathbb F_v=\\kappa(v)$ is its residue field. This restriction is implicit in the phrase \"the reduction $A/\\mathbb F_v$.\" The local canonical JSON record is internally consistent with the adjacent workshop problems 1.1, 1.2, and 1.4. The original AIM URL timed out during this run, so the wording above is verified from the canonical repository copy rather than a fresh rendering of the web page. There is no apparent OCR corruption.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Cohomological methods in abelian varieties\nSection: Problems\nSource item: 1.3\nSource URL: http://aimpl.org/cohomabelian/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[31]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $K$ be a number field, and let $A/K$ be an abelian variety. Do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\\\F_v$ is not ordinary? More strongly, do there exist infinitely many places $v$ of $K$ such that the reduction $A/\\\\F_v$ is supersingular?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This result was proven by Elkies for $\\\\dim A = 1$ over number fields $K$ with a real place. Poonen says that the supersingular part is probably false, since the supersingular locus in the moduli space of abelian varieties has codimension $>1$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/cohomabelian/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0032", "aim-domain:arithmetic-geometry", "aim-workshop:cohomabelian", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If an abelian variety A/K has a positive-dimensional geometric CM isogeny factor, let L be the Galois compositum of the CM fields of that factor and let c be complex conjugation. Outside finitely many exceptions, every K-place over a rational prime with Frobenius c in Gal(L/Q) is nonordinary; these rational characteristics have density 1/[L:Q], giving at least (1/[L:Q]+o(1))Li(X^(1/[K:Q])) nonordinary K-places of norm at most X. If A is potentially CM, the same places are supersingular. If L intersected with the normal closure of K lies in L^+, a degree-one subfamily has lower density at least [K:Q]/[L K^n:Q].\n\nCandidate contribution (theorem; novelty confidence low): Candidate CM-factor Chebotarev sieve: a single positive-dimensional CM geometric factor forces infinitely many nonordinary reductions even with an arbitrary non-CM complement, with an explicit norm-count lower bound; for potentially CM A it forces supersingular reductions, and the intersection condition L cap K^n subset L^+ yields an explicit positive-density degree-one subfamily." }, { "id": 20000787, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0033", "title": "Geometric 1-motives versus integral Jacobian splittings", "statement": "(a) Let $K$ be an algebraically closed field, $X/K$ an abelian variety. The Voevodsky motive $F$ is given as a sheaf of groups $F(S) = \\textrm{Mor}_K(S,X)$. Is $F$ geometric (i.e. in the triangulated subcategory generated by smooth projective varieties)?\n\n(b) (This is a stronger variant of (a)): Is every principally polarized abelian variety a direct summand of a Jacobian integrally (as an abelian variety)?", "original_statement": "(a) Let $K$ be an algebraically closed field, $X/K$ an abelian variety. The Voevodsky motive $F$ is given as a sheaf of groups $F(S) = \\textrm{Mor}_K(S,X)$. Is $F$ geometric (i.e. in the triangulated subcategory generated by smooth projective varieties)?\n\n(b) (This is a stronger variant of (a)): Is every principally polarized abelian variety a direct summand of a Jacobian integrally (as an abelian variety)?", "clean_statement": "(a) Let $K$ be an algebraically closed field, $X/K$ an abelian variety. The Voevodsky motive $F$ is given as a sheaf of groups $F(S) = \\textrm{Mor}_K(S,X)$. Is $F$ geometric (i.e. in the triangulated subcategory generated by smooth projective varieties)?\n\n(b) (This is a stronger variant of (a)): Is every principally polarized abelian variety a direct summand of a Jacobian integrally (as an abelian variety)?", "statement_status": "exact", "statement_verification": "The record has no apparent OCR corruption. It does, however, suppress three choices which affect the answer to (a): Nisnevich versus étale motives, the coefficient ring, and the placement of the sheaf as a complex. The standard reconstruction is that \\(F_X\\) is the homotopy-invariant sheaf with transfers represented by the commutative group scheme \\(X\\), placed in cohomological degree zero in \\(DM^{\\mathrm{eff}}_-(K,\\mathbb Z)\\). The transfer along a finite correspondence is induced by the sum/trace map on the abelian variety.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Cohomological methods in abelian varieties\nSection: Problems\nSource item: 1.4\nSource URL: http://aimpl.org/cohomabelian/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[32]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(a) Let $K$ be an algebraically closed field, $X/K$ an abelian variety. The Voevodsky motive $F$ is given as a sheaf of groups $F(S) = \\\\textrm{Mor}_K(S,X)$. Is $F$ geometric (i.e. in the triangulated subcategory generated by smooth projective varieties)?\\n\\n(b) (This is a stronger variant of (a)): Is every principally polarized abelian variety a direct summand of a Jacobian integrally (as an abelian variety)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/cohomabelian/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0033", "aim-domain:arithmetic-geometry", "aim-workshop:cohomabelian", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an abelian variety X over an algebraically closed field of exponential characteristic p, the sheaf with transfers F_X is geometric in effective etale motives with coefficients Z[1/p], and hence integrally so in characteristic zero; it is also geometric in the usual Nisnevich category rationally. In contrast, the 2026 parity theorem of Engel--de Gaay Fortman--Schreieder implies that a very general complex principally polarized abelian variety of dimension at least four is not a direct summand even of a product of Jacobians, refuting part (b). Moreover, every quasi-section q s = [n] of every Jacobian quotient onto such an abelian variety has n even, so no quotient splits over Z_(2). The unqualified integral Nisnevich and positive-characteristic p-primary versions of part (a) are not decided here.\n\nCandidate contribution (lemma; novelty confidence low): If a prime ell divides the degree of every isogeny A x Y to a product of Jacobians, then every relation q composed with s = [n] for a Jacobian quotient q:J(C)->A satisfies ell divides n; equivalently no such quotient splits after tensoring Hom groups with Z_(ell). Applied to very general complex PPAVs of dimension at least four and ell=2, this gives a 2-local nonsplitting obstruction despite integral etale motivic geometricity." }, { "id": 20000788, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0034", "title": "Minimal finite fields and a complete dimension-two analysis", "statement": "Let $p$ be a prime number. For a fixed dimension $g$ and a fixed $p$-rank $f$, what is the smallest field of definition of a simple abelian variety over $\\overline{\\F_p}$ with dimension $g$ and $p$-rank $f$?", "original_statement": "Let $p$ be a prime number. For a fixed dimension $g$ and a fixed $p$-rank $f$, what is the smallest field of definition of a simple abelian variety over $\\overline{\\F_p}$ with dimension $g$ and $p$-rank $f$?", "clean_statement": "Let $p$ be a prime number. For a fixed dimension $g$ and a fixed $p$-rank $f$, what is the smallest field of definition of a simple abelian variety over $\\overline{\\F_p}$ with dimension $g$ and $p$-rank $f$?", "statement_status": "exact", "statement_verification": "The canonical record (AIM workshop *Cohomological methods in abelian varieties*, Problems 1.5) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Cohomological methods in abelian varieties\nSection: Problems\nSource item: 1.5\nSource URL: http://aimpl.org/cohomabelian/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[33]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $p$ be a prime number. For a fixed dimension $g$ and a fixed $p$-rank $f$, what is the smallest field of definition of a simple abelian variety over $\\\\overline{\\\\F_p}$ with dimension $g$ and $p$-rank $f$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/cohomabelian/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0034", "aim-domain:arithmetic-geometry", "aim-workshop:cohomabelian", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Interpreting the AIM wording literally as geometric simplicity and defining the minimum by least extension degree for existence, the complete dimension-two table is N_p(2,0)=infinity and N_p(2,1)=N_p(2,2)=1 for every prime p. Explicit all-prime quartic Weil polynomials realize the p-rank-one case over F_p and also show that the finite-field-simple variant has minimum 1 even at p-rank zero. More generally, every pair except (2,0) occurs over some finite extension; ordinary rank occurs over F_p in every dimension, and current 2025 work implies F_p suffices for every prescribed p-rank in all sufficiently large dimensions at fixed p.\n\nCandidate contribution (explicit_construction; novelty confidence low): The displayed quartic families P_p(T)=T^4+T^3+2T+4 for p=2 and T^4+T^3+pT^2+pT+p^2 for odd p uniformly yield geometrically simple p-rank-one surfaces over F_p; paired supersingular quartics give a complete dimension-two minimum-field table under both geometric and finite-field simplicity." }, { "id": 20000789, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0035", "title": "A fixed-degree moduli obstruction for plane-curve Jacobian quotients", "statement": "Is every abelian variety the quotient of $\\textrm{Jac}(C)$, where $C$ is a smooth plane curve?", "original_statement": "Is every abelian variety the quotient of $\\textrm{Jac}(C)$, where $C$ is a smooth plane curve?", "clean_statement": "Is every abelian variety the quotient of $\\textrm{Jac}(C)$, where $C$ is a smooth plane curve?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Cohomological methods in abelian varieties*, Problems 1.6) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Cohomological methods in abelian varieties\nSection: Problems\nSource item: 1.6\nSource URL: http://aimpl.org/cohomabelian/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[34]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is every abelian variety the quotient of $\\\\textrm{Jac}(C)$, where $C$ is a smooth plane curve?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Ciliberto: Yes, if every surface $S$ of general type is birational to $S' \\\\subseteq \\\\mathbb P^3$, where $S'$ has only isolated singularities.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/cohomabelian/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0035", "aim-domain:arithmetic-geometry", "aim-workshop:cohomabelian", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over the complex numbers, for fixed plane degree d, target dimension n, and polarization type, the locus of polarized abelian n-folds that are unpolarized quotients of Jacobians of smooth degree-d plane curves is contained in a countable union of closed algebraic subsets, each of dimension at most binomial(d+2,2)-9. Hence a very general target can occur only if d is at least ceil((-3+sqrt(4n^2+4n+73))/2), which equals n for n at least 8. The report also proves the elliptic case and an explicit power-pullback construction producing higher-degree smooth plane witnesses from any existing plane witness.\n\nCandidate contribution (moduli-dimension obstruction; novelty confidence low): For fixed d, n, and polarization type, the unpolarized plane-Jacobian quotient locus in the moduli of polarized abelian n-folds has moduli dimension at most binomial(d+2,2)-9; in particular a very general n-fold needs plane degree at least ceil((-3+sqrt(4n^2+4n+73))/2)." }, { "id": 20000790, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0036", "title": "Very general complex ppav's have no Chow-theoretic half theta square", "statement": "Let $X$ be a principally polarized abelian variety of dimension $\\geq 4$. Is $\\theta^2 \\in CH^2(X)/\\textrm{tors}$ divisible by $2$?", "original_statement": "Let $X$ be a principally polarized abelian variety of dimension $\\geq 4$. Is $\\theta^2 \\in CH^2(X)/\\textrm{tors}$ divisible by $2$?", "clean_statement": "Let $X$ be a principally polarized abelian variety of dimension $\\geq 4$. Is $\\theta^2 \\in CH^2(X)/\\textrm{tors}$ divisible by $2$?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Cohomological methods in abelian varieties*, Problems 1.7) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Cohomological methods in abelian varieties\nSection: Problems\nSource item: 1.7\nSource URL: http://aimpl.org/cohomabelian/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[35]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X$ be a principally polarized abelian variety of dimension $\\\\geq 4$. Is $\\\\theta^2 \\\\in CH^2(X)/\\\\textrm{tors}$ divisible by $2$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Voisin thinks the answer may be \\\"no\\\", by using the method of Esnault-Bloch.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/cohomabelian/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0036", "aim-domain:arithmetic-geometry", "aim-workshop:cohomabelian", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Using Engel--de Gaay Fortman--Schreieder, Theorem 1.1 (arXiv:2507.15704v3), a very general principally polarized complex abelian variety of every dimension g at least 4 has theta^2 nondivisible by 2 in CH^2 modulo torsion: a hypothetical half would have Betti class [Theta]^2/2, whereas every codimension-two algebraic class has even coefficient relative to that primitive integral class. In addition, an exact product-factor criterion is proved, yielding both positive products of elliptic curves and decomposable counterexamples in every dimension at least 4.\n\nCandidate contribution (reduction; novelty confidence low): For a finite product of abelian varieties with the product polarization line bundle, theta^2 is divisible by 2 in ordinary CH^2 modulo torsion if and only if the square of every factor polarization is divisible by 2 modulo torsion on that factor." }, { "id": 20000791, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0037", "title": "The polarization-orbit ring detects the full cohomological spectral radius", "statement": "Let $A$ be an abelian variety over a finite field $K$, and $u: A \\to A$ a fixed automorphism. Then $u$ induces $u^*: H^*(\\overline{A}, \\Q_l) \\to H^*(\\overline{A}, \\Q_l)$. Let $\\alpha_i$ be its eigenvalues, and let $|\\alpha_i|$ denote the archimedean size of $\\alpha_i$. Take the subring in $H^*(\\overline{A}, \\Q_l)$ generated by a polarization and stabilized by $u$. Is the maximum archimedean size of the eigenvalues of the subring equal to the maximum archimedean size of the eigenvalues of $H^*(\\overline{A}, \\Q_l)$?", "original_statement": "Let $A$ be an abelian variety over a finite field $K$, and $u: A \\to A$ a fixed automorphism. Then $u$ induces $u^*: H^*(\\overline{A}, \\Q_l) \\to H^*(\\overline{A}, \\Q_l)$. Let $\\alpha_i$ be its eigenvalues, and let $|\\alpha_i|$ denote the archimedean size of $\\alpha_i$. Take the subring in $H^*(\\overline{A}, \\Q_l)$ generated by a polarization and stabilized by $u$. Is the maximum archimedean size of the eigenvalues of the subring equal to the maximum archimedean size of the eigenvalues of $H^*(\\overline{A}, \\Q_l)$?", "clean_statement": "Let $A$ be an abelian variety over a finite field $K$, and $u: A \\to A$ a fixed automorphism. Then $u$ induces $u^*: H^*(\\overline{A}, \\Q_l) \\to H^*(\\overline{A}, \\Q_l)$. Let $\\alpha_i$ be its eigenvalues, and let $|\\alpha_i|$ denote the archimedean size of $\\alpha_i$. Take the subring in $H^*(\\overline{A}, \\Q_l)$ generated by a polarization and stabilized by $u$. Is the maximum archimedean size of the eigenvalues of the subring equal to the maximum archimedean size of the eigenvalues of $H^*(\\overline{A}, \\Q_l)$?", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Cohomological methods in abelian varieties\nSection: Problems\nSource item: 1.8\nSource URL: http://aimpl.org/cohomabelian/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[36]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $A$ be an abelian variety over a finite field $K$, and $u: A \\\\to A$ a fixed automorphism. Then $u$ induces $u^*: H^*(\\\\overline{A}, \\\\Q_l) \\\\to H^*(\\\\overline{A}, \\\\Q_l)$. Let $\\\\alpha_i$ be its eigenvalues, and let $|\\\\alpha_i|$ denote the archimedean size of $\\\\alpha_i$. Take the subring in $H^*(\\\\overline{A}, \\\\Q_l)$ generated by a polarization and stabilized by $u$. Is the maximum archimedean size of the eigenvalues of the subring equal to the maximum archimedean size of the eigenvalues of $H^*(\\\\overline{A}, \\\\Q_l)$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"OK if $\\\\dim A = 2$, or if $K = \\\\C$ and $A$ is a smooth projective variety.\"\nResearch attempt: 2; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/cohomabelian/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0037", "aim-domain:arithmetic-geometry", "aim-workshop:cohomabelian", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the natural interpretation of the AIM subring as the smallest u*-stable graded algebra generated by an ample class theta, the answer is yes. Hu's published equality chi_{2k}(u)=lambda_k(u), combined with intersection-number growth, shows that for every k the cyclic space spanned by u^{n*}(theta^k) already has spectral radius chi_{2k}(u). Hu's conjugate-pair theorem for H^1 and the exterior-algebra description of abelian-variety cohomology show that no odd degree has larger spectral radius than both adjacent even degrees. Therefore the polarization-orbit ring and total l-adic cohomology have equal maximum archimedean eigenvalue modulus. The proof extends to every surjective self-map of an abelian variety over an algebraically closed field of arbitrary characteristic.\n\nCandidate contribution (lemma; novelty confidence low): For every surjective self-map f of an abelian variety in arbitrary characteristic, every ample class theta, and every k, the cyclic space span{f^{n*}(theta^k): n >= 0} has spectral radius chi_{2k}(f); consequently the smallest f*-stable algebra generated by theta detects the spectral radius on total cohomology." }, { "id": 20000792, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0038", "title": "Uniform finite-extension envelopes and valuation invariance", "statement": "If $u(F) < \\infty$, does there exist $B \\in \\N$ such that $u(F') <\n \\infty$ for all $F'/F$ finite? In particular, is $B = 2u(F)$\n sufficient?", "original_statement": "If $u(F) < \\infty$, does there exist $B \\in \\N$ such that $u(F') <\n \\infty$ for all $F'/F$ finite? In particular, is $B = 2u(F)$\n sufficient?", "clean_statement": "If $u(F) < \\infty$, does there exist $B \\in \\N$ such that $u(F') <\n \\infty$ for all $F'/F$ finite? In particular, is $B = 2u(F)$\n sufficient?", "statement_status": "exact", "statement_verification": "The exact canonical record in `aim-arithmetic-geometry-notes.json`, source index 37, reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Deformation theory, patching, quadratic forms, and the Brauer group\nSection: The $u$-invariant problem\nSource item: 1.1\nSource URL: http://aimpl.org/deformationbrauer/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[37]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"If $u(F) < \\\\infty$, does there exist $B \\\\in \\\\N$ such that $u(F') <\\n \\\\infty$ for all $F'/F$ finite? In particular, is $B = 2u(F)$\\n sufficient?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/deformationbrauer/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0038", "aim-domain:arithmetic-geometry", "aim-workshop:deformationbrauer", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The canonical text is malformed because its bound B is unused. Under the only coherent reconstruction, U_fin(F)=sup over finite L/F of u(L), the general uniform question remains open; Leep's bound nevertheless settles literal individual finiteness and proves U_fin over extensions of degree at most three is at most 2u(F). For characteristic not two, purely inseparable extensions preserve u, 2U_fin(F) is at most u(F(t)), and for every complete discretely valued K with non-dyadic residue field k one has the exact identities u(K)=2u(k) and U_fin(K)=2U_fin(k). Thus finiteness and the proposed factor-two inequality hold for K exactly when they hold for k.\n\nCandidate contribution (lemma; novelty confidence low): For a complete discretely valued field K with residue field k of characteristic not two, the ordered pair (u(K), U_fin(K)) equals twice (u(k), U_fin(k)); consequently the normalized gap U_fin/u and the truth or failure of U_fin <= 2u are invariant under a complete non-dyadic discrete valuation step." }, { "id": 20000793, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0039", "title": "A hereditary square-closure screen and a quadratic-closure test family", "statement": "Find interesting classes $\\mathcal F$ of fields such that for $F \\in\n \\mathcal F$,\n\\[u(F) < \\infty \\Longrightarrow u(F(t)) < \\infty.\\]", "original_statement": "Find interesting classes $\\mathcal F$ of fields such that for $F \\in\n \\mathcal F$,\n\\[u(F) < \\infty \\Longrightarrow u(F(t)) < \\infty.\\]", "clean_statement": "Find interesting classes $\\mathcal F$ of fields such that for $F \\in\n \\mathcal F$,\n\\[u(F) < \\infty \\Longrightarrow u(F(t)) < \\infty.\\]", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop section “The $u$-invariant problem,” Problem 1.2) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Deformation theory, patching, quadratic forms, and the Brauer group\nSection: The $u$-invariant problem\nSource item: 1.2\nSource URL: http://aimpl.org/deformationbrauer/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[38]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find interesting classes $\\\\mathcal F$ of fields such that for $F \\\\in\\n \\\\mathcal F$,\\n\\\\[u(F) < \\\\infty \\\\Longrightarrow u(F(t)) < \\\\infty.\\\\]\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Examples to try:\\n\\n\\\\[ \\\\mathcal F = \\\\{F \\\\ \\\\ :\\\\ \\\\ |F^{\\\\times}/F^{\\\\times 2}| < \\\\infty \\\\}\\n\\\\]\\nand\\n\\\\[ \\\\mathcal F' = \\\\{F \\\\ \\\\ : \\\\ \\\\ F^{\\\\times} = F^{\\\\times 2} \\\\}\\\\]\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/deformationbrauer/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0039", "aim-domain:arithmetic-geometry", "aim-workshop:deformationbrauer", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In characteristic different from 2 with Kaplansky's u-invariant, a square-closed field F has u(F(t))=2 exactly when every finite extension of F is square-closed; if a finite extension contains a nonsquare c with minimal polynomial h, the quaternion (t,h(t)) has nonzero residue c and its 4-dimensional norm form is anisotropic. Combining this criterion with Grimm--Leep gives the concrete family F=k_quad for every number field k, with u(F)=1 but u(F(t)) at least 4. The work also identifies finite square-class number plus finite u with finiteness of W(F), supplies exact strong-u Laurent-tower subclasses, and does not claim to settle finiteness in the general AIM classes.\n\nCandidate contribution (worked_family_synthesis; novelty confidence low): For every number field k, its quadratic closure F=k_quad satisfies u(F)=1 while u(F(t))>=4; a finite extension E/F with a nonsquare c, guaranteed by hereditary non-quadratic-closure, yields the explicit anisotropic quaternion norm form <1,-t,-h(t),t h(t)> over F(t), where h is the minimal polynomial of c over F." }, { "id": 20000794, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0040", "title": "Higher u-invariants and a flat-profile criterion", "statement": "Define \\[u_k(F) = \\max \\{ \\dim (q)\\ \\ |\\ \\ q \\in I^k \\}.\\]\nThen $u_0 = u$.\n\nCan we compute $u_k(F)$? How about $u_3(F)$?", "original_statement": "Define \\[u_k(F) = \\max \\{ \\dim (q)\\ \\ |\\ \\ q \\in I^k \\}.\\]\nThen $u_0 = u$.\n\nCan we compute $u_k(F)$? How about $u_3(F)$?", "clean_statement": "Define \\[u_k(F) = \\max \\{ \\dim (q)\\ \\ |\\ \\ q \\in I^k \\}.\\]\nThen $u_0 = u$.\n\nCan we compute $u_k(F)$? How about $u_3(F)$?", "statement_status": "exact", "statement_verification": "The canonical repository record (source index 39 in `aim-arithmetic-geometry-notes.json`) transcribes the 2011 AIM problem as", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Deformation theory, patching, quadratic forms, and the Brauer group\nSection: The $u$-invariant problem\nSource item: 1.3\nSource URL: http://aimpl.org/deformationbrauer/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[39]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Define \\\\[u_k(F) = \\\\max \\\\{ \\\\dim (q)\\\\ \\\\ |\\\\ \\\\ q \\\\in I^k \\\\}.\\\\]\\nThen $u_0 = u$.\\n\\nCan we compute $u_k(F)$? How about $u_3(F)$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Note that forms in\\n$I^3$ have trivial Clifford invariant.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/deformationbrauer/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0040", "aim-domain:arithmetic-geometry", "aim-workshop:deformationbrauer", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After correcting the literal definition to use anisotropic Witt representatives, a field of characteristic different from two with u(F)=2^d and I(F)^d nonzero has I(F)^(d+1)=0 and the exact profile u_k(F)=2^d for 0<=k<=d and u_k(F)=0 for k>d. Applying Tsen-Lang and an explicit valuation proof for the generic d-fold Pfister form computes u_k(k(x_1,...,x_d)) for algebraically closed k: it is 2^d through level d and zero afterward; hence u_3 is zero for d<3 and 2^d for d>=3.\n\nCandidate contribution (lemma; novelty confidence low): Flat-profile criterion: the two testable hypotheses u(F)=2^d and I(F)^d nonzero alone force I(F)^(d+1)=0 and (u_0,...,u_d)=(2^d,...,2^d); combined with the explicit generic Pfister form this yields the full higher-u profile of k(x_1,...,x_d)." }, { "id": 20000795, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0041", "title": "An explicit cyclic cubic counterexample from an A4-extension", "statement": "Let \\[\\nu(F) = \\max \\{n\\ |\\ I^n \\neq 0 \\}. \\]\n\nIf $\\nu(F) < \\infty$, is it true that $\\nu(F') < \\nu(F) +1$ for all\n$F'/F$ finite?", "original_statement": "Let \\[\\nu(F) = \\max \\{n\\ |\\ I^n \\neq 0 \\}. \\]\n\nIf $\\nu(F) < \\infty$, is it true that $\\nu(F') < \\nu(F) +1$ for all\n$F'/F$ finite?", "clean_statement": "Let \\[\\nu(F) = \\max \\{n\\ |\\ I^n \\neq 0 \\}. \\]\n\nIf $\\nu(F) < \\infty$, is it true that $\\nu(F') < \\nu(F) +1$ for all\n$F'/F$ finite?", "statement_status": "exact", "statement_verification": "The canonical record at index 40 of `aim-arithmetic-geometry-notes.json`, from the 2011 AIM workshop *Deformation theory, patching, quadratic forms, and the Brauer group*, asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Deformation theory, patching, quadratic forms, and the Brauer group\nSection: The $u$-invariant problem\nSource item: 1.4\nSource URL: http://aimpl.org/deformationbrauer/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[40]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let \\\\[\\\\nu(F) = \\\\max \\\\{n\\\\ |\\\\ I^n \\\\neq 0 \\\\}. \\\\]\\n\\nIf $\\\\nu(F) < \\\\infty$, is it true that $\\\\nu(F') < \\\\nu(F) +1$ for all\\n$F'/F$ finite?\"\nOriginal remarks: [\"$2^{\\\\nu(F)} < u(F)$\", \"This is related to the computation of Galois groups of\\n quadratically closed fields.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/deformationbrauer/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0041", "aim-domain:arithmetic-geometry", "aim-workshop:deformationbrauer", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The proposed finite-extension bound is false already at nu(F)=0. Let F be the quadratic closure of Q and let N/Q be the A4 splitting field of x^4+2x^3+2x^2+2. Since N is disjoint from F, the V4-fixed field L inside NF is cyclic cubic over F, while NF/L has group V4 and supplies quadratic extensions of L. Thus nu(F)=0 but I(L) is nonzero. Because L contains Q(i) and cd_2(L)<=2, Milnor's conjecture and the Arason-Pfister intersection theorem give I(L)^3=0, so 1<=nu(L)<=2. The failure is therefore finite, separable, Galois, and odd-degree.\n\nCandidate contribution (lemma; novelty confidence low): Finite-group certificate: if a realizable finite Galois group G has no nontrivial 2-group quotient but contains a subgroup H with a C2 quotient, then after base change to the quadratic closure the H-fixed field is a finite non-quadratically-closed extension of a quadratically closed field; (A4,V4) is minimal by group order." }, { "id": 20000796, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0042", "title": "Finite-field counterexamples and a low-u torsion-index obstruction", "statement": "For a quadratic form $q$ over $F$, we define\n\\[\\textrm{splitting degree} = \\min\\{[L:F]\\ :\\ q_{L}\\ \\textrm{is a\n direct sum of hyperbolics}\\}.\\]\nThe \\emph{torsion index} of $F$, denoted $\\tau_F$ is the maximum\nsplitting degree, taken over all even dimensional $q$.\n\nIs $\\tau_F < 2^{(\\frac{u(F)}{2} -1)}$?", "original_statement": "For a quadratic form $q$ over $F$, we define\n\\[\\textrm{splitting degree} = \\min\\{[L:F]\\ :\\ q_{L}\\ \\textrm{is a\n direct sum of hyperbolics}\\}.\\]\nThe \\emph{torsion index} of $F$, denoted $\\tau_F$ is the maximum\nsplitting degree, taken over all even dimensional $q$.\n\nIs $\\tau_F < 2^{(\\frac{u(F)}{2} -1)}$?", "clean_statement": "For a quadratic form $q$ over $F$, we define\n\\[\\textrm{splitting degree} = \\min\\{[L:F]\\ :\\ q_{L}\\ \\textrm{is a\n direct sum of hyperbolics}\\}.\\]\nThe \\emph{torsion index} of $F$, denoted $\\tau_F$ is the maximum\nsplitting degree, taken over all even dimensional $q$.\n\nIs $\\tau_F < 2^{(\\frac{u(F)}{2} -1)}$?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.5 in the AIM list for the workshop *Deformation theory, patching, quadratic forms, and the Brauer group*. It says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Deformation theory, patching, quadratic forms, and the Brauer group\nSection: The $u$-invariant problem\nSource item: 1.5\nSource URL: http://aimpl.org/deformationbrauer/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[41]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For a quadratic form $q$ over $F$, we define\\n\\\\[\\\\textrm{splitting degree} = \\\\min\\\\{[L:F]\\\\ :\\\\ q_{L}\\\\ \\\\textrm{is a\\n direct sum of hyperbolics}\\\\}.\\\\]\\nThe \\\\emph{torsion index} of $F$, denoted $\\\\tau_F$ is the maximum\\nsplitting degree, taken over all even dimensional $q$.\\n\\nIs $\\\\tau_F < 2^{(\\\\frac{u(F)}{2} -1)}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/deformationbrauer/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0042", "aim-domain:arithmetic-geometry", "aim-workshop:deformationbrauer", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal strict inequality is false: for every odd finite field F_q, u(F_q)=2 and tau(F_q)=2, whereas the proposed right side is 1. More generally, in characteristic different from 2 and for finite classical u-invariant n, Witt reduction and diagonal coefficient pairing prove tau(F) <= 2^{floor(n/2)}; moreover tau(F)=1 for u=1, tau(F)=2 for u=2, and tau(F) is either 2 or 4 for u=4, so the proposed strict inequality fails at every one of these low values.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate low-u obstruction package: for fields of characteristic different from 2, tau(F) <= 2^{floor(u(F)/2)}, with tau(F)=1 at u=1, tau(F)=2 at u=2, and tau(F) in {2,4} at u=4; hence the displayed strict estimate fails structurally throughout the low-u range." }, { "id": 20000797, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0043", "title": "Uniform degrees for splitting quadratic forms", "statement": "When is $\\tau_F$ finite?", "original_statement": "When is $\\tau_F$ finite?", "clean_statement": "When is $\\tau_F$ finite?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.6 in the AIM workshop section *The $u$-invariant problem*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Deformation theory, patching, quadratic forms, and the Brauer group\nSection: The $u$-invariant problem\nSource item: 1.6\nSource URL: http://aimpl.org/deformationbrauer/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[42]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"When is $\\\\tau_F$ finite?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/deformationbrauer/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0043", "aim-domain:arithmetic-geometry", "aim-workshop:deformationbrauer", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For nonsingular quadratic forms over fields of characteristic different from 2, the workshop maximum is rigorously interpreted as a supremum of finite individual splitting degrees. If r(F)=dim_F2(F^*/F^{*2}) and u_e(F) is the largest even anisotropic dimension, then tau_F <= 2^{min(r(F),u_e(F)/2)} whenever either parameter is finite; every finite extension E/F gives tau_F <= [E:F] tau_E; and sup{ind(alpha): alpha in Br(F)[2]} <= tau_F. Thus finite square-class rank or finite u-invariant is sufficient, while unbounded Brauer 2-indices force tau_F=infinity. In addition, an exact local-global calculation proves tau_F=4 for every number field F, with a four-dimensional quaternion-norm witness for the lower bound.\n\nCandidate contribution (exact_worked_family; novelty confidence low): For every number field F, the AIM invariant defined as the supremum of minimal degrees of finite extensions making even-dimensional nonsingular forms hyperbolic satisfies tau_F=4; a degree-4 lower witness is q=<1,-a,-b,abd>, where (a,b) is a quaternion division algebra and F(sqrt(d)) is chosen not to split it." }, { "id": 20000798, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0044", "title": "Henselian reduction and exact value-group obstructions for the torsion index", "statement": "When $F = \\R$ , we have $u(F) = \\infty$ but $\\tau_F < \\infty$. Is there\n an example where this holds for $F$ not formally real?", "original_statement": "When $F = \\R$ , we have $u(F) = \\infty$ but $\\tau_F < \\infty$. Is there\n an example where this holds for $F$ not formally real?", "clean_statement": "When $F = \\R$ , we have $u(F) = \\infty$ but $\\tau_F < \\infty$. Is there\n an example where this holds for $F$ not formally real?", "statement_status": "exact", "statement_verification": "The canonical record in `aim-arithmetic-geometry-notes.json`, source index 43, asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Deformation theory, patching, quadratic forms, and the Brauer group\nSection: The $u$-invariant problem\nSource item: 1.7\nSource URL: http://aimpl.org/deformationbrauer/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[43]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"When $F = \\\\R$ , we have $u(F) = \\\\infty$ but $\\\\tau_F < \\\\infty$. Is there\\n an example where this holds for $F$ not formally real?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/deformationbrauer/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0044", "aim-domain:arithmetic-geometry", "aim-workshop:deformationbrauer", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a Henselian valued field K with residue field k of characteristic not two and value group Gamma, if d=dim_F2(Gamma/2Gamma) is finite then tau_k <= tau_K <= 2^d tau_k. If d is infinite, explicit 2m-dimensional diagonal forms supported on 2m independent value classes have splitting degree exactly 2^m, so tau_K is infinite. Thus tau_K is finite exactly when tau_k and d are both finite. This is a partial reduction and obstruction; it does not decide whether the requested non-formally-real example exists.\n\nCandidate contribution (valuation-theoretic reduction and exact witness family; novelty confidence low): In the non-dyadic Henselian setting, tau_k <= tau_K <= 2^d tau_k for finite d=dim_F2(Gamma/2Gamma), while q_m=<1,a_1,...,a_{2m-1}> on independent value classes has exact splitting degree 2^m; consequently finite tau_K is equivalent to finite tau_k and finite d." }, { "id": 20000799, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0045", "title": "A Laurent-rank counterexample to the proposed period-index formula", "statement": "Find reasonable classes $\\mathcal F$ of fields such that $u(F)$ is a\n power of 2 for all $F \\in \\mathcal F$. For these fields, assume $k$ is the\n period-index bound for $l=2$. Does this imply $u(F) = 2^{k+1}$?", "original_statement": "Find reasonable classes $\\mathcal F$ of fields such that $u(F)$ is a\n power of 2 for all $F \\in \\mathcal F$. For these fields, assume $k$ is the\n period-index bound for $l=2$. Does this imply $u(F) = 2^{k+1}$?", "clean_statement": "Find reasonable classes $\\mathcal F$ of fields such that $u(F)$ is a\n power of 2 for all $F \\in \\mathcal F$. For these fields, assume $k$ is the\n period-index bound for $l=2$. Does this imply $u(F) = 2^{k+1}$?", "statement_status": "exact", "statement_verification": "The record is Problem 1.8 in the section *The \\(u\\)-invariant problem* from the AIM workshop *Deformation theory, patching, quadratic forms, and the Brauer group*. Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Deformation theory, patching, quadratic forms, and the Brauer group\nSection: The $u$-invariant problem\nSource item: 1.8\nSource URL: http://aimpl.org/deformationbrauer/1/\nCanonical location: aim-arithmetic-geometry-notes.json notes[44]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find reasonable classes $\\\\mathcal F$ of fields such that $u(F)$ is a\\n power of 2 for all $F \\\\in \\\\mathcal F$. For these fields, assume $k$ is the\\n period-index bound for $l=2$. Does this imply $u(F) = 2^{k+1}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/deformationbrauer/1/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0045", "aim-domain:arithmetic-geometry", "aim-workshop:deformationbrauer", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Let kappa be algebraically closed of characteristic different from 2, let K_n=kappa((t_1))...((t_n)), and let F_n be the class of all finite extensions E/K_n. For every E in F_n, u(E)=2^n, while the minimal stable Brauer 2-dimension, the minimal fieldwise period-index exponent, and the exponent-two index bound all equal floor(n/2). Hence the proposed identity u(E)=2^(k+1) fails for every n at least 3; at n=3 it predicts 4 although u(E)=8. The ratio is exactly 2^(ceil(n/2)-1), so the gap is unbounded.\n\nCandidate contribution (exact counterexample family; novelty confidence low): The finite-extension-closed Laurent families have exact profile (log_2 u, Brd_2)=(n,floor(n/2)), giving a convention-robust AIM equality gap of ceil(n/2)-1 powers of 2 that is unbounded in n." }, { "id": 20000800, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0046", "title": "Classification of an AIM test fixture", "statement": "Name of the problem block\n\nIntro to problem block\n\nThis is the problem statement.", "original_statement": "Name of the problem block\n\nIntro to problem block\n\nThis is the problem statement.", "clean_statement": "Name of the problem block\n\nIntro to problem block\n\nThis is the problem statement.", "statement_status": "exact", "statement_verification": "The exact canonical record is labeled as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Deformation theory, patching, quadratic forms, and the Brauer group\nSection: Testing section\nSource item: 33.1\nSource URL: http://aimpl.org/deformationbrauer/3/\nCanonical location: aim-arithmetic-geometry-notes.json notes[45]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Name of the problem block\\n\\nIntro to problem block\\n\\nThis is the problem statement.\"\nOriginal remarks: [\"This is a remark.\"]\nOriginal literature field (JSON string): \"This is the status statement.\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/deformationbrauer/3/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0046", "aim-domain:arithmetic-geometry", "aim-workshop:deformationbrauer", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The exact record contains no truth-apt mathematical proposition: it supplies no mathematical domain, objects, predicate, quantification, hypotheses, or requested output. Non-identifiability is proved by exhibiting two inequivalent conjectures compatible with every surviving generic field. The explicit section title 'Testing section', sibling record 33.2's 'throw-away problem', isolated 33.x numbering, and the repository parser's one-to-one mapping from name/intro/body/remark/status to the five placeholder sentences establish that this is intentional fixture context rather than a corrupted conjecture. The correct terminal classification is therefore context_only with problem_status_at_run not_a_problem.\n\nCandidate contribution (provenance_well_posedness_criterion; novelty confidence low): A five-step non-recoverability certificate for placeholder-like corpus records is developed and applied: quote all fields, audit domain/object/relation/scope, exhibit two incompatible completions, require two independent fixture-provenance signals, and state a reversal condition. For AIM-ARITHMETIC_GEOMETRY-0046 this certificate reproducibly yields context_only rather than invalid_statement.", "research_difficulty_suggested": "L2" }, { "id": 20000801, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0047", "title": "A self-identifying throw-away fixture", "statement": "Test introduction\n\nthe statement of a throw-away problem", "original_statement": "Test introduction\n\nthe statement of a throw-away problem", "clean_statement": "Test introduction\n\nthe statement of a throw-away problem", "statement_status": "exact", "statement_verification": "The exact canonical metadata are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Arithmetic geometry\nWorkshop: Deformation theory, patching, quadratic forms, and the Brauer group\nSection: Testing section\nSource item: 33.2\nSource URL: http://aimpl.org/deformationbrauer/3/\nCanonical location: aim-arithmetic-geometry-notes.json notes[46]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Test introduction\\n\\nthe statement of a throw-away problem\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "http://aimpl.org/deformationbrauer/3/", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0047", "aim-domain:arithmetic-geometry", "aim-workshop:deformationbrauer", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The exact two-line record fails the D/O/R/Q truth-aptness audit independently: it specifies no mathematical domain, objects, relation, hypotheses, quantifiers, or requested output, and the workshop metadata admit inequivalent possible completions. Its phrase 'throw-away problem' is therefore a self-fixture marker. Together with the consecutive placeholder record 0046, the testing-section provenance certifies the maximal cluster {0046, 0047} as fixture content. A semantic veto tested on the genuine Black-Scholes 'test problem' AIM-COMPUTATION-0087 prevents lexical false positives.\n\nCandidate contribution (methodological_rule; novelty confidence low): Candidate fixture-cluster certification rule: a maximal contiguous same-section, same-source run may be fixture-certified only when every member independently fails D/O/R/Q, at least one member self-identifies as fixture content, an independent structural signal corroborates the reading, and no primary source contradicts it; any concrete D/O/R/Q content semantically vetoes keyword- or cluster-based exclusion.", "research_difficulty_suggested": "L2" }, { "id": 20000802, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0048", "title": "Packed square-fat-point lower bounds for tangent spaces on the smoothable component", "statement": "Problem 1. Describe the singularities of the smoothable component of Hilb d(An). To be more specific,\n\n• How large can the dimension of the Zariski tangent space to this component get?\n\n• Does the maximum occur in the intersection of components? Does it occur in the smoothable component?", "original_statement": "Problem 1. Describe the singularities of the smoothable component of Hilb d(An). To be more specific, \n\n• How large can the dimension of the Zariski tangent space to this component get? \n\n• Does the maximum occur in the intersection of components? Does it occur in the smoothable component?", "clean_statement": "Problem 1. Describe the singularities of the smoothable component of Hilb d(An). To be more specific,\n\n• How large can the dimension of the Zariski tangent space to this component get?\n\n• Does the maximum occur in the intersection of components? Does it occur in the smoothable component?", "statement_status": "exact", "statement_verification": "The final sentence is verified verbatim from the PDF; it is not an extraction error. Read literally, its last question is redundant, since the tangent space “to this component” is based at a point of the component. A plausible intended contrast is between points lying on several components and points lying only on the smoothable component, or between the ambient Hilbert-scheme tangent space along the smoothable component and the tangent space of the reduced component itself. The source does not settle that ambiguity. This report therefore distinguishes \\[ T_{[Z]}\\operatorname{Hilb}^{D}(\\mathbb A^n) \\quad\\text{from}\\quad T_{[Z]}\\mathcal R_D^n, \\] where \\(\\mathcal R_D^n\\) is the reduced smoothable component, and constructs points at which these spaces are equal. It separately determines when the constructed point is known to lie at an intersection of components.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[47]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1. Describe the singularities of the smoothable component of Hilb d(An). To be more specific, \\n\\n• How large can the dimension of the Zariski tangent space to this component get? \\n\\n• Does the maximum occur in the intersection of components? Does it occur in the smoothable component?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0048", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over an algebraically closed characteristic-zero field, write D=r(n+1)+s with 0<=s<=n. A disjoint union of r first infinitesimal neighborhoods defined by squares of maximal ideals and s reduced points lies on the smoothable component of Hilb^D(A^n), and its tangent space to that reduced component equals the full ambient Hilbert-scheme tangent space. Their common dimension is nD+r*n(n+1)(n-2)/2. This gives an explicit uniform lower bound for the maximum asked about. For n>=7 and r>=1, Lee's conical symmetric chart shows that the same point lies at an intersection of components.\n\nCandidate contribution (explicit lower bound and witness family; novelty confidence low): For characteristic zero and n>=2, M_sm(D,n) is at least nD+floor(D/(n+1))*n(n+1)(n-2)/2; this is witnessed by packed square-fat points at which the tangent space of the reduced smoothable component equals the entire ambient Hilbert tangent space, and for n>=7 the witnesses may be chosen at component intersections." }, { "id": 20000803, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0049", "title": "Tangent spaces and prescribed-jet smoothings on the monogenic chart", "statement": "Problem 2. Can you describe the Zariski tangent space to the smoothable component of Hilb d(An)?", "original_statement": "Problem 2. Can you describe the Zariski tangent space to the smoothable component of Hilb d(An)?", "clean_statement": "Problem 2. Can you describe the Zariski tangent space to the smoothable component of Hilb d(An)?", "statement_status": "exact", "statement_verification": "The exact canonical extraction is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[48]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2. Can you describe the Zariski tangent space to the smoothable component of Hilb d(An)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 3; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0049", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over an algebraically closed field of arbitrary characteristic, the locus where 1,x_1,...,x_1^{d-1} is a basis of R/I is an affine nd-space with ideals (f(x_1),x_2-p_2(x_1),...,x_n-p_n(x_1)). It lies in the smoothable component. At every such point the component tangent, whole-Hilbert tangent, and Hom_R(I,R/I) coincide and are isomorphic to (R/I)^n, of dimension nd. Every tangent vector is also the exact first jet of a polynomial family whose generic fiber is d distinct points. After a coordinate change, this includes every curvilinear scheme.\n\nCandidate contribution (lemma; novelty confidence low): Prescribed-first-jet smoothing lemma: at every monogenic length-d point of Hilb^d(A^n), every vector in Hom_R(I,R/I) is realized by a polynomial smoothing curve with that exact first jet; in coefficient space one may take gamma(s)=q+s v+s^2(q_*-q-v) with q_* squarefree." }, { "id": 20000804, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0050", "title": "Gauss-inseparability strata and a Frobenius plane-curve family", "statement": "Problem 3. Fix d, g, r, e > 0. Let Hilb sm d,g (Pr) be the open subscheme of the Hilbert scheme\n\nHilb d,g (Pr) that parameterizes smooth curves. For each point [C] ∈ Hilb sm d,g (Pr), we define the Gauss map C → Gr(1, r ) sending a point of C to the tangent line at that point. Define\n\nZe:= {[C] ∈ Hilb sm d,g (Pr)| the Gauss map of the curve C is inseparable of degree pe}.\n\nThen ∪\n\n> e≥0\n\nZe = Hilb sm d,g (Pr).What can we say about the set Ze? Can we construct exotic components (i.e. components that only exist in characteristic p) using this stratification? Study the action by the Galois group\n\nGal( Fp/Fp).", "original_statement": "Problem 3. Fix d, g, r, e > 0. Let Hilb sm d,g (Pr) be the open subscheme of the Hilbert scheme \n\nHilb d,g (Pr) that parameterizes smooth curves. For each point [C] ∈ Hilb sm d,g (Pr), we define the Gauss map C → Gr(1, r ) sending a point of C to the tangent line at that point. Define \n\nZe:= {[C] ∈ Hilb sm d,g (Pr)| the Gauss map of the curve C is inseparable of degree pe}.\n\nThen ∪ \n\n> e≥0\n\nZe = Hilb sm d,g (Pr).What can we say about the set Ze? Can we construct exotic components (i.e. components that only exist in characteristic p) using this stratification? Study the action by the Galois group \n\nGal( Fp/Fp).", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The primary source is the four-page problem list *Components of Hilbert Schemes*, recorded by Izzet Coskun and edited by Li Li after the AIM workshop of July 19--23, 2010. The PDF typography restores the corrupted extraction as follows.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[49]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3. Fix d, g, r, e > 0. Let Hilb sm d,g (Pr) be the open subscheme of the Hilbert scheme \\n\\nHilb d,g (Pr) that parameterizes smooth curves. For each point [C] ∈ Hilb sm d,g (Pr), we define the Gauss map C → Gr(1, r ) sending a point of C to the tangent line at that point. Define \\n\\nZe:= {[C] ∈ Hilb sm d,g (Pr)| the Gauss map of the curve C is inseparable of degree pe}.\\n\\nThen ∪ \\n\\n> e≥0\\n\\nZe = Hilb sm d,g (Pr).What can we say about the set Ze? Can we construct exotic components (i.e. components that only exist in characteristic p) using this stratification? Study the action by the Galois group \\n\\nGal( Fp/Fp).\"\nOriginal remarks: [\"Remark. There are some other stratifications one may consider, e.g. the one obtained by the topological type of the ramification divisor.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0050", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "On the smooth geometrically integral nondegenerate curve locus, every fixed-inseparable-degree set Z_e is constructible, defined over F_p, and Galois stable; it is empty unless p^e divides 2d+2g-2, and each geometric Hilbert component has a unique generic exponent. For q=p^e, the smooth plane curves F_B=(X^[q])^T B X=0 with B invertible form a closed subvariety W_e isomorphic to PGL_3 inside Z_e for (d,g,r)=(q+1,q(q-1)/2,2). Their Gauss maps are exactly [X] maps to [B^T X^[q]], and W_e is 8-dimensional but lies properly inside the unique horizontal plane-curve Hilbert component, so it does not itself produce an exotic component.\n\nCandidate contribution (explicit classified family and component obstruction; novelty confidence low): For q=p^e, the Frobenius-linear part of the AIM stratum for smooth degree-q+1 plane curves is exactly the closed Hilbert subvariety W_e isomorphic to PGL_3 parametrized by invertible projective matrices B in the equations (X^[q])^T B X=0; arithmetic Frobenius acts by [B] to [B^(p)], and W_e is a proper subvariety rather than a Hilbert component." }, { "id": 20000805, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0051", "title": "Vertical Hilbert components and a Borel-fixed boundary reduction", "statement": "Problem 4. (1) Is there a component of Hilb d(An) that exists only in characteristic p for some p?(2) Same question for the Hilbert schemes of curves in P3.", "original_statement": "Problem 4. (1) Is there a component of Hilb d(An) that exists only in characteristic p for some p?(2) Same question for the Hilbert schemes of curves in P3.", "clean_statement": "Problem 4. (1) Is there a component of Hilb d(An) that exists only in characteristic p for some p?(2) Same question for the Hilbert schemes of curves in P3.", "statement_status": "exact", "statement_verification": "The AIM workshop list *Components of Hilbert Schemes* asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[50]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4. (1) Is there a component of Hilb d(An) that exists only in characteristic p for some p?(2) Same question for the Hilbert schemes of curves in P3.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0051", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Jelisiejew's Corollary 5.2 answers the affine-points clause affirmatively in the strongest relative sense: for every prime p, some Hilb^d(A^16_Z) has a component entirely supported over p. No answer was found for curves in fixed P^3. For that open clause, a proved boundary-incidence theorem shows that every hypothetical p-vertical component of Hilb^P(P^3_Z) must meet a horizontal component at an integrally liftable Borel-fixed monomial ideal; equivalently, a relevant local Hilbert ring must have distinct vertical and horizontal minimal primes.\n\nCandidate contribution (reduction; novelty confidence low): Every p-vertical irreducible component of a projective Hilbert scheme Hilb^P(P^n_Z), after geometric base change, contains a Borel-fixed monomial point that lifts tautologically over Z and therefore also lies on a horizontal component; hence the P^3-curve search reduces to detecting mixed vertical/horizontal minimal primes at Borel-fixed ideals." }, { "id": 20000806, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0052", "title": "Generically nonreduced components of Hilbert schemes of points", "statement": "Problem 5. Is there a nonreduced component of Hilb d(An)? If so, find it.", "original_statement": "Problem 5. Is there a nonreduced component of Hilb d(An)? If so, find it.", "clean_statement": "Problem 5. Is there a nonreduced component of Hilb d(An)? If so, find it.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 5 from the 2010 AIM workshop list *Components of Hilbert schemes*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[51]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 5. Is there a nonreduced component of Hilb d(An)? If so, find it.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0052", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Jelisiejew's 2024 theorem answers the AIM existence problem in characteristic zero: for s in {6,7,8,9}, the very-compressed locus with Hilbert function (1,4,10,s) is the reduced support of a generically nonreduced irreducible component of Hilb^{15+s}(A^4), and disjoint reduced points give such components in every length d >= 21. This report supplies a tangent-excess criterion, derives explicit generic lower-bound excesses 10, 7, 12, and 25 in lengths 21 through 24, and proves the completed-local-ring propagation statement.\n\nCandidate contribution (quantitative lemma package; novelty confidence low): On the dense cubic-generated parts of the four known components, the Hilbert tangent dimension exceeds the reduced component dimension by at least 10, 7, 12, and 25 for lengths 21, 22, 23, and 24; after adjoining m disjoint reduced points on A^4, the completed local ring gains 4m formal variables, so the corresponding tangent-excess lower bound and generic nilpotents persist." }, { "id": 20000807, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0053", "title": "Galois-moving point-Hilbert components via constant-field transfer", "statement": "Problem 6. (1) Give an explicit example (or show it does not happen) of a geometrically irreducible component of Hilb d(An), which is not fixed under the action of Gal( Q/Q).(2) Same question for the Hilbert schemes of curves in P3.", "original_statement": "Problem 6. (1) Give an explicit example (or show it does not happen) of a geometrically irreducible component of Hilb d(An), which is not fixed under the action of Gal( Q/Q).(2) Same question for the Hilbert schemes of curves in P3.", "clean_statement": "Problem 6. (1) Give an explicit example (or show it does not happen) of a geometrically irreducible component of Hilb d(An), which is not fixed under the action of Gal( Q/Q).(2) Same question for the Hilbert schemes of curves in P3.", "statement_status": "exact", "statement_verification": "The typeset AIM source reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[52]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 6. (1) Give an explicit example (or show it does not happen) of a geometrically irreducible component of Hilb d(An), which is not fixed under the action of Gal( Q/Q).(2) Same question for the Hilbert schemes of curves in P3.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0053", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A component-level synthesis of Easton-Vakil's faithful absolute-Galois action on surface-moduli components with Jelisiejew's functorial truncation and TNT-frame retractions proves that, for every nontrivial sigma in Gal(Qbar/Q), some Hilb^d(A^N_Qbar) has a geometric component C with sigma(C) != C. If the source surface is canonically embedded in P^r, the frame construction has N=2r+8 and d=dim(T/J). This is constructive existence but not a compact numerical example. No answer was found for curves in fixed P^3. A second proved theorem shows that every component in either clause contains a Q-rational monomial degeneration, so any moved component must meet all its conjugates at a singular monomial point.\n\nCandidate contribution (construction_and_reduction; novelty confidence low): Nontrivial component constant fields transfer through a section-retraction diagram; applying this at the generic point of an Easton-Vakil moved embedded-surface component and spreading Jelisiejew's TNT frame yields a Galois-moved component of Hilb^d(A^(2r+8)). Independently, any moved point- or projective-Hilbert component must collide with every conjugate at a Q-rational monomial ideal." }, { "id": 20000808, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0054", "title": "Irrational Hilbert-scheme components and a threshold-10 MRC strengthening", "statement": "Problem 7. Does there exist a nonrational component of Hilb d(An)?\n\n> 12RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI", "original_statement": "Problem 7. Does there exist a nonrational component of Hilb d(An)? \n\n> 12RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI", "clean_statement": "Problem 7. Does there exist a nonrational component of Hilb d(An)?\n\n> 12RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI", "statement_status": "exact", "statement_verification": "The primary source is the four-page AIM problem list *Components of Hilbert Schemes*, recorded by Izzet Coskun and edited by Li Li after the workshop of 19--23 July 2010. Its notation paragraph says that \\(\\operatorname{Hilb}^{d}(\\mathbb A^{n})\\) denotes the Hilbert scheme of \\(d\\) points in affine \\(n\\)-space and that “component” means irreducible component. Page 1 gives the exact problem:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[53]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 7. Does there exist a nonrational component of Hilb d(An)? \\n\\n> 12RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0054", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The OCR-contaminated AIM question has a positive answer over every algebraically closed field of characteristic zero. Farkas–Pandharipande–Sammartano constructed non-rationally-connected components for ambient dimension at least 12, and Wu's June 2026 one-variable-frame preprint lowers this to every n at least 10. Combining Wu's freely varying genus source with MRC functoriality and the length-preserving ambient-dimension extension shows that for each g at least 24 one length D_g works for all n at least 10 and yields components whose reduced varieties have MRC dimension at least 3g-3; hence they are neither rational, stably rational, nor unirational.\n\nCandidate contribution (corollary; novelty confidence low): Simultaneous-length threshold-10 MRC corollary: for every g at least 24 there is one D_g such that for every n at least 10, Hilb^{D_g}(A^n) has a reduced underlying component dominating M_g and therefore of MRC dimension at least 3g-3; in particular unbounded MRC dimension extends from the published n at least 12 construction to n=10 and n=11." }, { "id": 20000809, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0055", "title": "A tangent-weight criterion for rational Hilbert components", "statement": "Problem 8. If a component of the Hilbert scheme contains a smooth Borel fixed point, does the component have to be rational?", "original_statement": "Problem 8. If a component of the Hilbert scheme contains a smooth Borel fixed point, does the component have to be rational?", "clean_statement": "Problem 8. If a component of the Hilbert scheme contains a smooth Borel fixed point, does the component have to be rational?", "statement_status": "exact", "statement_verification": "The exact 2010 AIM question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[54]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 8. If a component of the Hilbert scheme contains a smooth Borel fixed point, does the component have to be rational?\"\nOriginal remarks: [\"Remark. If the ideal is a segment ideal ( i.e. a monomial ideal generated in some degree s,where s ≤ its regularity, by the maximal monomials with respect to some term ordering), then the component is known to be rational, c.f. [P. Lella, M. Roggero, Rational components of Hilbert schemes, arXiv:0903.1029]. Note that every segment ideal is a Borel fixed ideal, but the converse is not true.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0055", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over an algebraically closed field of characteristic zero, a smooth Borel-fixed Hilbert point lies on a rational component whenever all characters of its multigraded tangent space Hom_S(J,S/J)_0 lie in one open half-space: the separating cocharacter has a full-dimensional Bialynicki-Birula attracting cell isomorphic to affine space. Gordan's theorem gives an exact finite alternative: either such an integral cocharacter exists, or a nonzero nonnegative rational relation among tangent characters certifies that no diagonal one-parameter subgroup makes the point a source or sink. The general AIM question remains open in the literature checked.\n\nCandidate contribution (criterion and finite obstruction certificate; novelty confidence low): For a smooth Borel-fixed ideal J, the multigraded tangent representation alone gives an auditable dichotomy: a strict separating cocharacter proves rationality of the unique component through J via a dense affine Bialynicki-Birula cell, while a nonnegative dependence among tangent characters exactly certifies failure of every diagonal one-parameter-subgroup source/sink proof at J." }, { "id": 20000810, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0056", "title": "A fixed-polynomial Cartier-divisor bridge for the locally Cohen-Macaulay curve locus", "statement": "Problem 9. Is the Hilbert scheme of local Cohen-Macaulay curves in P3 connected?", "original_statement": "Problem 9. Is the Hilbert scheme of local Cohen-Macaulay curves in P3 connected?", "clean_statement": "Problem 9. Is the Hilbert scheme of local Cohen-Macaulay curves in P3 connected?", "statement_status": "exact", "statement_verification": "The primary AIM workshop PDF (page 2 of the PDF, numbered page 2) literally prints:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[55]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 9. Is the Hilbert scheme of local Cohen-Macaulay curves in P3 connected?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0056", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth geometrically integral surface S in projective three-space and a nontrivial effective divisor class D, the complete linear system |D| gives an irreducible flat family entirely inside one locally Cohen-Macaulay Hilbert locus H_{d,g}, with d=D.H and g=1+(D^2+D.K_S)/2. Consequently, a graph of such linear-system images, with edges witnessed by shared curves, certifies connectedness of every covered connected subgraph without passing through embedded or isolated points or changing the Hilbert polynomial. This yields explicit verified connected families for divisors on smooth quadrics and for complete intersections, but does not settle the global problem.\n\nCandidate contribution (connectedness criterion; novelty confidence low): The Cartier-overlap graph is a sufficient, fixed-polynomial connectedness certificate: vertices are images of complete linear systems on smooth integral surfaces, an edge is witnessed by one common Cartier curve, and any connected subgraph maps to a connected subset of H_{d,g} while all reducible and nonreduced intermediate fibers remain locally Cohen-Macaulay." }, { "id": 20000811, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0057", "title": "Known irreducibility regions and the explicit q=1 component decomposition", "statement": "Problem 10. Let ρ = ( p, q, r, 0, 0,... ) and define Ep,q,r:= Eρ to be the moduli space of finite length graded modules of function ρ. For which p, q, r is Ep,q,r irreducible?", "original_statement": "Problem 10. Let ρ = ( p, q, r, 0, 0,... ) and define Ep,q,r:= Eρ to be the moduli space of finite length graded modules of function ρ. For which p, q, r is Ep,q,r irreducible?", "clean_statement": "Problem 10. Let ρ = ( p, q, r, 0, 0,... ) and define Ep,q,r:= Eρ to be the moduli space of finite length graded modules of function ρ. For which p, q, r is Ep,q,r irreducible?", "statement_status": "exact", "statement_verification": "The AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 10\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[56]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 10. Let ρ = ( p, q, r, 0, 0,... ) and define Ep,q,r:= Eρ to be the moduli space of finite length graded modules of function ρ. For which p, q, r is Ep,q,r irreducible?\"\nOriginal remarks: [\"Remark. It is proved that if 4 q < max(6 p + r, 6r + p) then Ep,q,r is reducible (c.f. [M. Martin-Deschamps, D. Perrin, Courbes gauches et modules de Rao, J. Reine Angew. Math. 439 (1993), 103-145. page 119, Theorem 2.1]). Conjecturally, if 4 q ≥ max(6 p + r, 6r + p) then Ep,q,r is irreducible.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0057", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After recovering E_{p,q,r} as the based affine scheme of four maps in each adjacent degree satisfying six commutativity relations, the report corrects the AIM remark by restoring the exceptional irreducible triple (1,1,1), translates Ginouillac's later results into a sharp known/open region, and proves that (E_{p,1,r})_red is the union of the two zero-map affine subspaces and the rank-at-most-one determinantal variety of (p+r)-by-4 matrices. This gives exactly one, two, or three irreducible components according as p=r=1, exactly one of p,r equals 1, or p,r>1.\n\nCandidate contribution (component_decomposition; novelty confidence low): For every p,r>0 over an algebraically closed field, (E_{p,1,r})_red is {A_bullet=0} union {B_bullet=0} union {rank[A_bullet;B_bullet]<=1}, yielding component count 1, 2, or 3 according as (p,r)=(1,1), exactly one equals 1, or both exceed 1." }, { "id": 20000812, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0058", "title": "Complete reduced component decomposition for Rao functions (p,1,r)", "statement": "Problem 11. Describe the irreducible component of Eρ, the moduli of finite length graded modules of function ρ.", "original_statement": "Problem 11. Describe the irreducible component of Eρ, the moduli of finite length graded modules of function ρ.", "clean_statement": "Problem 11. Describe the irreducible component of Eρ, the moduli of finite length graded modules of function ρ.", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 11\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[57]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 11. Describe the irreducible component of Eρ, the moduli of finite length graded modules of function ρ.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0058", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every p,r >= 1 over an algebraically closed field, the reduced framed scheme of graded k[x1,x2,x3,x4]-module structures with Hilbert function (p,1,r) is the union of the two coordinate loci where one adjacent multiplication vanishes and the determinantal locus where the combined 4 by (p+r) multiplication matrix has rank at most one. After removing contained loci, this gives one component for (1,1), two components when exactly one of p,r is one, and three components when p,r > 1, with dimensions listed and generic module types proved in the artifacts.\n\nCandidate contribution (theorem; novelty confidence low): The radical of the cross-commutation ideal for the middle-value-one family is (X) intersect (Y) intersect I_2([X Y]), yielding an exact one-, two-, or three-component classification of the reduced scheme and an explicit generic one-variable module on the mixed component." }, { "id": 20000813, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0059", "title": "Hyperplane lifting defects and a Gorenstein Rao-module obstruction", "statement": "Problem 12. What do properties of the Rao modules imply about C? For example, if MC is Gorenstein or annihilated by a linear form, does C have any nice properties?", "original_statement": "Problem 12. What do properties of the Rao modules imply about C? For example, if MC is Gorenstein or annihilated by a linear form, does C have any nice properties?", "clean_statement": "Problem 12. What do properties of the Rao modules imply about C? For example, if MC is Gorenstein or annihilated by a linear form, does C have any nice properties?", "statement_status": "exact", "statement_verification": "The primary AIM workshop PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 12\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[58]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 12. What do properties of the Rao modules imply about C? For example, if MC is Gorenstein or annihilated by a linear form, does C have any nice properties?\"\nOriginal remarks: [\"Remark. We might have to require C to be minimal.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0059", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If a linear form ell annihilates the Rao module of a locally Cohen-Macaulay curve and its hyperplane meets the curve properly, then every Rao class is exactly an obstruction to lifting an equation of the hyperplane section; a self-duality M_C congruent to M_C^vee(-s) gives perfect pairings between complementary lifting-defect spaces. Conversely, the explicit cyclic module R/(x,yz,yt,zt,y^4-z^4,y^4-t^4) is an Artinian Gorenstein algebra annihilated by x, but its six minimal first relations violate the four-relation condition for cyclic Rao modules of double-plane curves. Rao's theorem therefore yields an even liaison class, minimal curves included, with both module properties but no double-plane representative.\n\nCandidate contribution (counterexample and cohomological criterion; novelty confidence low): The cyclic Artinian Gorenstein module R/(x,yz,yt,zt,y^4-z^4,y^4-t^4) is linearly annihilated yet its entire even liaison class has no curve in any double plane; separately, for a proper annihilating hyperplane H, chosen Rao self-duality induces perfect pairings Q_m(C,H) x Q_{s+2-m}(C,H) -> k between complementary equation-lifting defects." }, { "id": 20000814, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0060", "title": "A bad-surface and Rao-module reduction for smooth limits of complete intersections", "statement": "Problem 13. Let Ct be a family of curves in P3 such that a general curve in this family is a smooth complete intersection, and the special curve C0 is smooth. Does it imply that C0 is also a complete intersection, assuming that the characteristic is 0?", "original_statement": "Problem 13. Let Ct be a family of curves in P3 such that a general curve in this family is a smooth complete intersection, and the special curve C0 is smooth. Does it imply that C0 is also a complete intersection, assuming that the characteristic is 0?", "clean_statement": "Problem 13. Let Ct be a family of curves in P3 such that a general curve in this family is a smooth complete intersection, and the special curve C0 is smooth. Does it imply that C0 is also a complete intersection, assuming that the characteristic is 0?", "statement_status": "exact", "statement_verification": "The AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 13\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[59]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 13. Let Ct be a family of curves in P3 such that a general curve in this family is a smooth complete intersection, and the special curve C0 is smooth. Does it imply that C0 is also a complete intersection, assuming that the characteristic is 0?\"\nOriginal remarks: [\"Remark. If the characteristic is p > 0, the answer is negative; if n > 3, the answer is negative; if C0 is not smooth, the answer is negative. The reference is [P. Ellia, R. Hartshorne, Smooth specializations of space curves: questions and examples, Commutative algebra and algebraic geometry (Ferrara), 53-79, Lecture Notes in Pure and Appl. Math., 206, Dekker, New York, 1999].\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0060", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a flat trait family in characteristic zero whose generic fiber is a smooth complete intersection of type (a,b) in P3 and whose special fiber is smooth, the special fiber retains the complete-intersection degree, genus, and subcanonical equality, and it lies on a degree-a surface. It is a complete intersection exactly when its Rao module vanishes, and it is necessarily a complete intersection if a smooth degree-a containing surface exists. A direct analysis of planes and all ranks of quadrics proves the original assertion for a <= 2. Thus any counterexample must have a >= 3, nonzero Rao module, and only singular degree-a containing surfaces.\n\nCandidate contribution (reduction; novelty confidence low): Any characteristic-zero smooth non-complete-intersection limit of complete intersections of type (a,b) must simultaneously have a >= 3, nonzero Rao module, omega_C isomorphic to O_C(a+b-4), and every degree-a surface containing C singular; in particular, no such counterexample exists for a <= 2." }, { "id": 20000815, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0061", "title": "Generic points via elementary support factorization", "statement": "Problem 14. Give a geometric algebraic description of generic points of irreducible components of Hilb d(An).", "original_statement": "Problem 14. Give a geometric algebraic description of generic points of irreducible components of Hilb d(An).", "clean_statement": "Problem 14. Give a geometric algebraic description of generic points of irreducible components of Hilb d(An).", "statement_status": "exact", "statement_verification": "The exact extracted record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 14\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[60]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 14. Give a geometric algebraic description of generic points of irreducible components of Hilb d(An).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0061", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every irreducible component of Hilb^d(A^n) has a unique multiset of elementary one-support component factors; a general member is their disjoint union, and component dimension, generic tangent dimension, dimension excess, tangent defect, generic reducedness, and smoothability are determined factorwise. In characteristic zero, concatenating the known elementary blocks with Hilbert functions (1,4,3), (1,5,3), and (1,6,3) gives explicit components C_{a,b,c,s} of dimension nd-7a-4b and the stated coefficient-extraction lower bounds for the number of components.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: the support-tangent fingerprint and its explicit length-8/9/10 concatenation package give uniquely identifiable generic ideals in every length, the formula dim C_{a,b,c,s}=nd-7a-4b, a supportwise smoothability test, and component-count generating functions 1/((1-t)(1-t^8)), 1/((1-t)(1-t^8)(1-t^9)), and 1/((1-t)(1-t^8)(1-t^9)(1-t^10)) in ambient dimensions 4, 5, and at least 6 respectively." }, { "id": 20000816, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0062", "title": "A rank-three flat family where the restricted Gröbner fan jumps", "statement": "Problem 15. Is the Gr¨ obner fan a discrete invariant that distinguishes the components of Hilb d(An)?", "original_statement": "Problem 15. Is the Gr¨ obner fan a discrete invariant that distinguishes the components of Hilb d(An)?", "clean_statement": "Problem 15. Is the Gr¨ obner fan a discrete invariant that distinguishes the components of Hilb d(An)?", "statement_status": "exact", "statement_verification": "The source is Problem 15 in the AIM workshop list *Components of Hilbert Schemes* (2010). The OCR record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 15\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[61]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 15. Is the Gr¨ obner fan a discrete invariant that distinguishes the components of Hilb d(An)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0062", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal raw-invariant reading is false over every field: the flat family J=(x^2-t y,xy,y^2) in k[t,x,y] gives an irreducible curve in Hilb^3(A^2), with the monomial fiber J_0=(x,y)^2 having one maximal restricted Gröbner cone, while every nonzero fiber has exactly two maximal cones separated by w_y=2w_x and monomial initial ideals (x,y)^2 and (y,x^3). The endpoint local algebras have embedding dimensions two and one, respectively, so they are not isomorphic. A separate localization argument proves that an unlabelled fan with its maximal monomial labels is nevertheless constant on a dense open of every irreducible finite-type family, making injectivity of the double-generic component fan the precise surviving question.\n\nCandidate contribution (counterexample; novelty confidence low): The characteristic-free family (x^2-t y,xy,y^2) is an explicit degree-three witness, with a complete hand proof, that the restricted Gröbner fan jumps from one to exactly two maximal cones inside one irreducible Hilbert-scheme curve; the report also proves generic constancy and isolates double-generic injectivity as the corrected formulation." }, { "id": 20000817, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0063", "title": "Monomial signatures distinguish components through affine length eight", "statement": "Problem 16. Does the set of monomial ideals contained in a component of Hilb d(An) determine that component? (Or more generally for Hilb P (Pn) for an arbitrary Hilbert polynomial P.)", "original_statement": "Problem 16. Does the set of monomial ideals contained in a component of Hilb d(An) determine that component? (Or more generally for Hilb P (Pn) for an arbitrary Hilbert polynomial P.)", "clean_statement": "Problem 16. Does the set of monomial ideals contained in a component of Hilb d(An) determine that component? (Or more generally for Hilb P (Pn) for an arbitrary Hilbert polynomial P.)", "statement_status": "exact", "statement_verification": "The AIM record (Problem 16 in *Components of Hilbert schemes*) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 16\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[62]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 16. Does the set of monomial ideals contained in a component of Hilb d(An) determine that component? (Or more generally for Hilb P (Pn) for an arbitrary Hilbert polynomial P.)\"\nOriginal remarks: [\"Remark. For the analogous question for the Hilbert scheme of curves, Richard Liebling might have given a counterexample in his thesis. COMPONENTS OF HILBERT SCHEMES 3\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0063", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an algebraically closed field of characteristic zero, the set of monomial ideals distinguishes every irreducible component of Hilb^d(A^n) for all d at most 8 and all n. More generally, a component containing an ambient-smooth monomial point is determined by its monomial signature, so the smoothable component is always distinguished by the curvilinear ideal (x_1^d,x_2,...,x_n). In the first reducible case, an explicit length-eight monomial ideal lies on both components, showing that overlap of monomial sets does not imply equality. The analogous projective assertion holds for the lexicographic component and, in characteristic zero, whenever there are at most two Borel-fixed ideals.\n\nCandidate contribution (theorem; novelty confidence low): Candidate corollary: in characteristic zero, C maps to its full monomial signature injectively on the components of Hilb^d(A^n) for every d at most 8; nevertheless the two components of Hilb^8(A^4) share the displayed singular monomial collision point." }, { "id": 20000818, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0064", "title": "A sharp uniqueness threshold and conductor-preserving flag criterion", "statement": "Problem 17. Let S ⊂ P3 be an integral surface of degree d with a double curve D of degree e, and triple points, pinch point,.... Conjecture: there exists n0 ∈ Z such that for any smooth curve C ⊂ S of degree n ≥ n0, let HC\n\nbe the irreducible component of the Hilbert scheme containing C, then a general C′ ∈ HC is also contained in a surface S′ ⊂ P3 of degree d and a double curve D′ of degree e.", "original_statement": "Problem 17. Let S ⊂ P3 be an integral surface of degree d with a double curve D of degree e, and triple points, pinch point,.... Conjecture: there exists n0 ∈ Z such that for any smooth curve C ⊂ S of degree n ≥ n0, let HC\n\nbe the irreducible component of the Hilbert scheme containing C, then a general C′ ∈ HC is also contained in a surface S′ ⊂ P3 of degree d and a double curve D′ of degree e.", "clean_statement": "Problem 17. Let S ⊂ P3 be an integral surface of degree d with a double curve D of degree e, and triple points, pinch point,.... Conjecture: there exists n0 ∈ Z such that for any smooth curve C ⊂ S of degree n ≥ n0, let HC\n\nbe the irreducible component of the Hilbert scheme containing C, then a general C′ ∈ HC is also contained in a surface S′ ⊂ P3 of degree d and a double curve D′ of degree e.", "statement_status": "exact", "statement_verification": "The exact extracted record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 17\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[63]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 17. Let S ⊂ P3 be an integral surface of degree d with a double curve D of degree e, and triple points, pinch point,.... Conjecture: there exists n0 ∈ Z such that for any smooth curve C ⊂ S of degree n ≥ n0, let HC\\n\\nbe the irreducible component of the Hilbert scheme containing C, then a general C′ ∈ HC is also contained in a surface S′ ⊂ P3 of degree d and a double curve D′ of degree e.\"\nOriginal remarks: [\"Remark. The conjecture is posed in [P. Ellia, R. Hartshorne, Smooth specializations of space curves: questions and examples, Commutative algebra and algebraic geometry (Ferrara), 53-79, Lecture Notes in Pure and Appl. Math., 206, Dekker, New York, 1999]. It is proved to be true for d = 3 in [J. Brevik, F. Mordasini, Curves on a ruled cubic surface, Collect. Math. 54 (2003), no. 3, 269-281].\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0064", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth degree-n curve C on an integral degree-d surface S in P3, the inequality n>d^2 forces S to be the unique degree-d surface containing C and excludes all lower-degree containing surfaces; d^2 is the optimal uniform threshold. For a smooth conductor-flat stratum B of ordinary surfaces, the image of the restricted flag differential consists exactly of curve deformations xi for which xi(F) is the restriction of a tangent deformation in B. If the curve and flag points are smooth and this explicit tangent containment holds, the flag projection is locally an open immersion, so the general curve in the Hilbert component lies on a unique degree-d surface with double curve degree e.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: above the sharp threshold n>d^2, the AIM conjecture for a specified simultaneous-normalization/conductor stratum is equivalent locally, under smoothness, to the computable containment rho(H^0(N_C)) subset im(T_S B -> H^0(O_C(d))); the quotient Q=rho(H^0(N_C))/(rho(H^0(N_C)) intersect im r) measures the missing conductor-preserving tangent directions, and its nonvanishing at a general flag obstructs dominance." }, { "id": 20000819, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0065", "title": "Grading and resolution-dependence obstructions to higher-dimensional Nakajima operators", "statement": "Problem 18. Is it possible to define analogues of the Nakajima operators for the cohomology of a desingularization of the smoothable component of Hilb d(An)?", "original_statement": "Problem 18. Is it possible to define analogues of the Nakajima operators for the cohomology of a desingularization of the smoothable component of Hilb d(An)?", "clean_statement": "Problem 18. Is it possible to define analogues of the Nakajima operators for the cohomology of a desingularization of the smoothable component of Hilb d(An)?", "statement_status": "exact", "statement_verification": "The AIM source asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 18\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[64]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 18. Is it possible to define analogues of the Nakajima operators for the cohomology of a desingularization of the smoothable component of Hilb d(An)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0065", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a pure expected-dimensional length-r punctual incidence family of complex dimension delta, Borel--Moore push--pull on fixed resolutions gives creation degree p+2[n(r-1)-delta] and transpose-annihilation degree q-2n-2delta. With paired labels p+q=2n, a nonzero scalar Heisenberg commutator therefore requires 2delta=n(r-1). The curvilinear family has delta=(n-1)(r-1), so for every n>2 and r>1 its ordinary fundamental class has commutator degree -2(n-2)(r-1) and cannot satisfy the unshifted Nakajima relation. Independently, iterated point blow-ups above the smoothable singular point [m^2] in Hilb^4(A^3) produce strong projective resolutions with unbounded Betti numbers, proving that full ordinary cohomology is not resolution-independent.\n\nCandidate contribution (obstruction; novelty confidence low): The balance criterion 2delta=n(r-1), its explicit curvilinear defect -2(n-2)(r-1), and the iterated-blow-up ambiguity for Hilb^4(A^3) together give a concrete, testable obstruction to defining literal fundamental-class Nakajima operators on the full cohomology of an unspecified resolution." }, { "id": 20000820, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0066", "title": "A forbidden strip and a determinantal chart for lCM surfaces in P4", "statement": "Problem 19. What is the geography of locally Cohen-Macaulay surfaces in P4? To be more precise, what numerical invariants (degree, sectional genus,... ) occur?", "original_statement": "Problem 19. What is the geography of locally Cohen-Macaulay surfaces in P4? To be more precise, what numerical invariants (degree, sectional genus,... ) occur?", "clean_statement": "Problem 19. What is the geography of locally Cohen-Macaulay surfaces in P4? To be more precise, what numerical invariants (degree, sectional genus,... ) occur?", "statement_status": "exact", "statement_verification": "The AIM workshop list *Components of Hilbert schemes* asks in Problem 19:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 19\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[65]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 19. What is the geography of locally Cohen-Macaulay surfaces in P4? To be more precise, what numerical invariants (degree, sectional genus,... ) occur?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0066", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every locally Cohen-Macaulay surface of degree d in P4 has sectional genus either binomial(d-1,2) or at most binomial(d-2,2), by regular hyperplane section and Hartshorne's lCM space-curve theorem. For the ACM codimension-two subgeography, a Hilbert-Burch resolution yields explicit second-, third-, and fourth-moment formulas for degree, sectional genus, and Euler characteristic. In particular, maximal minors of a general t by (t+1) matrix of degree-e forms realize a two-parameter family of exact triples, and proper complete-intersection linkage transforms any realized triple by proved closed formulas.\n\nCandidate contribution (realizable_family; novelty confidence low): Candidate explicit geography chart: for every positive pair (t,e), there is an ACM surface with d=t(t+1)e^2/2, sectional genus 1+t(t+1)e^2((2t+1)e-6)/6, and Euler characteristic t(t+1)e^2((3t^2+3t+1)e^2-10(2t+1)e+35)/24; sufficiently positive proper complete-intersection links generate further exact triples by the displayed transformation laws." }, { "id": 20000821, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0067", "title": "A support-deficit criterion and the fold-singularity test family", "statement": "Problem 20. Let C be a curve, can Hilb d(C) have a component of dim < d?", "original_statement": "Problem 20. Let C be a curve, can Hilb d(C) have a component of dim < d?", "clean_statement": "Problem 20. Let C be a curve, can Hilb d(C) have a component of dim < d?", "statement_status": "exact", "statement_verification": "The canonical AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 20\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[66]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 20. Let C be a curve, can Hilb d(C) have a component of dim < d?\"\nOriginal remarks: [\"Remark. There exists a non-smoothable component of dim = d > 1.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0067", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an integral curve with isolated singularities, a support-separated stratum with local lengths a_j and punctual-family dimensions q_j has dimension d-sum(a_j)+sum(q_j); it can be a component of dimension below d only if it has a genuine punctual deficit sum(q_j)= 4g-3+dim Stab(C). Thus every positive-genus nonspecial embedding has an actual non-projective Hilbert deformation; in particular no elliptic curve is rigid in any projective space, and the only rigid rational curves are rational normal curves. The same deformation-defect calculation recovers that rational normal curves are the only smooth rigid curves for n <= 3, while the general special higher-genus case in n >= 4 remains open.\n\nCandidate contribution (deformation_obstruction; novelty confidence low): A projectively rigid smooth nondegenerate curve must satisfy the explicit stabilizer-aware speciality inequality (n+1)h^1(O_C(1)) >= 4g-3+dim Stab(C), obtained by combining the Hilbert-component lower bound with Riemann-Roch for the hyperplane bundle." }, { "id": 20000825, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0071", "title": "A Pfaffian-colon reducedness certificate for Hilb^8(A^4)", "statement": "Problem 24. Is Hilb 8(A4) reduced? More generally, develop techniques to prove the reducedness of Hilbert schemes.", "original_statement": "Problem 24. Is Hilb 8(A4) reduced? More generally, develop techniques to prove the reducedness of Hilbert schemes.", "clean_statement": "Problem 24. Is Hilb 8(A4) reduced? More generally, develop techniques to prove the reducedness of Hilbert schemes.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 24 in the AIM workshop list *Components of Hilbert Schemes* (July 2010):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 24\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[70]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 24. Is Hilb 8(A4) reduced? More generally, develop techniques to prove the reducedness of Hilbert schemes.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0071", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In characteristic zero, reducedness of Hilb^8(A^4) admits a finite scheme-theoretic certificate on its monomial-basis cover. Up to coordinate permutations there are 49 chart types: 38 one-component charts whose coordinate rings must be proved radical, and 11 Hilbert-function-(1,4,3) charts. On each of the latter, if f is the lifted Salmon-Turnbull Pfaffian and q_G is the prime of the 25-dimensional component, it suffices to prove A/(f) reduced and (0:f)=q_G. The underlying local algebra theorem proves these two conditions exclude embedded infinitesimal glue. A worked counterexample shows why generic reducedness, reduced components and intersection, and tangent dimensions cannot replace the colon test.\n\nCandidate contribution (finite reduction and local algebra criterion; novelty confidence low): Candidate contribution: the CEVV Pfaffian equation combines with the proved one-equation colon lemma to reduce reducedness of Hilb^8(A^4) to 38 radicality checks and 11 exact quotient-and-colon checks after an exhaustive 684-to-49 coordinate-symmetry reduction; failure of (0:f)=q_G detects embedded nilpotents invisible to component, intersection, and tangent data.", "research_difficulty_suggested": "L4" }, { "id": 20000826, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0072", "title": "Connected toric and square-zero classes in three variables", "statement": "Problem 25. For R = k[x1, x 2, x 3]. Is every multigraded Hilbert scheme connected?", "original_statement": "Problem 25. For R = k[x1, x 2, x 3]. Is every multigraded Hilbert scheme connected?", "clean_statement": "Problem 25. For R = k[x1, x 2, x 3]. Is every multigraded Hilbert scheme connected?", "statement_status": "exact", "statement_verification": "The exact AIM workshop record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 25\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[71]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 25. For R = k[x1, x 2, x 3]. Is every multigraded Hilbert scheme connected?\"\nOriginal remarks: [\"Remark. For R = k[x1,..., x 26 ], there are non-connected examples, c.f. [Francisco Santos, Non-connected toric Hilbert-schemes, MR2181765, arXiv:0204044]. Find examples with fewer variables.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0072", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The full arbitrary-Hilbert-function question remains open in the literature checked. The positive toric subcase in three variables is affirmative: positivity forces lattice rank at most two, rank two is smooth and irreducible by Maclagan--Thomas, rank one has toric Hilbert scheme P^1, and rank zero gives a point. Beyond the toric case, if a positive grading has a Hilbert function supported at degree zero and indecomposable variable degrees, with zero products in positive degree, then its multigraded Hilbert scheme is naturally a product of Grassmannians and is therefore smooth, irreducible, and connected.\n\nCandidate contribution (theorem; novelty confidence low): For any positive abelian grading, every Hilbert function with h(0)=1 and nonzero support only in indecomposable variable degrees represents a square-zero quotient and has multigraded Hilbert scheme canonically isomorphic to the product over those degrees of Grassmannians of the prescribed quotient ranks." }, { "id": 20000827, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0073", "title": "The resolved four-ruling-lines degeneration and an explicit liaison audit", "statement": "Problem 26. (i) Let C be of bidegree (3, 7) on a nonsingular quadric surface in P3. Can C be connected to an extremal curve in Hilb 10,12 (P3)?(ii) Given 4 skew lines C1 on a nonsingular quadric Q1 in P3. Does there exists a family\n\nQt 2H, and a family Ct ⊂ Qt such that C0 ⊂ Q0 = 2 H is locally Cohen-Macaulay? 4 RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI", "original_statement": "Problem 26. (i) Let C be of bidegree (3, 7) on a nonsingular quadric surface in P3. Can C be connected to an extremal curve in Hilb 10,12 (P3)?(ii) Given 4 skew lines C1 on a nonsingular quadric Q1 in P3. Does there exists a family \n\nQt 2H, and a family Ct ⊂ Qt such that C0 ⊂ Q0 = 2 H is locally Cohen-Macaulay? 4 RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI", "clean_statement": "**Problem 26.** (i) Let \\(C\\) be of bidegree \\((3,7)\\) on a nonsingular\nquadric surface in \\(\\mathbf P^3\\). Can \\(C\\) be connected to an extremal\ncurve in \\(\\operatorname{Hilb}_{10,12}(\\mathbf P^3)\\)?\n\n(ii) Given four skew lines \\(C_1\\) on a nonsingular quadric \\(Q_1\\), does\nthere exist a family \\(Q_t\\rightsquigarrow 2H\\), and a family\n\\(C_t\\subset Q_t\\), such that \\(C_0\\subset Q_0=2H\\) is locally\nCohen--Macaulay?\n\n**Remark.** (ii) implies (i).", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record preserves OCR and page-footer damage. Its problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 26\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[72]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 26. (i) Let C be of bidegree (3, 7) on a nonsingular quadric surface in P3. Can C be connected to an extremal curve in Hilb 10,12 (P3)?(ii) Given 4 skew lines C1 on a nonsingular quadric Q1 in P3. Does there exists a family \\n\\nQt 2H, and a family Ct ⊂ Qt such that C0 ⊂ Q0 = 2 H is locally Cohen-Macaulay? 4 RECORDED ON WHITEBOARD BY IZZET COSKUN, AND EDITED BY LI LI\"\nOriginal remarks: [\"Remark. (ii) ⇒(i).\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0073", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Lella and Schlesinger proved both parts positively: for every d at least 3, d disjoint ruling lines on a smooth quadric admit a flat degeneration to an extremal locally Cohen-Macaulay curve in a double plane, and every effective divisor on a smooth quadric lies in the connected component containing extremal curves. In particular every bidegree (3,7) divisor lies in the extremal component of H_{10,12}. This attempt supplies the explicit characteristic-zero parameters {0,1,3,7}, the family Q_t: x^2+t^3 xw-t^5 yz=0, and its central extremal ideal. It also proves that a direct (2,7)-residual of the degree-four extremal limit is not itself H_{10,12}-extremal, because its Rao-module length is 24 rather than 384; the AIM implication therefore genuinely uses biliaison and double-plane connectedness.\n\nCandidate contribution (explicit_specialization_and_liaison_obstruction; novelty confidence low): For the four ruling parameters {0,1,3,7}, the Lella-Schlesinger weight degeneration has central ideal (x^2,xy,y^4,xG-y^3F), with G=(w+z)(w+3z)(w+4z)(w+7z)(w+8z)(w+10z) and F=z^3(2w+11z); moreover any direct (2,7)-residual has Rao length 24 and cannot be an extremal degree-10 genus-12 curve, whose Rao length is 384." }, { "id": 20000828, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0074", "title": "An exact ACM certificate for reverse-lexicographic gins of locally Cohen-Macaulay space curves", "statement": "Problem 27. Are there necessary or sufficient conditions on Borel fixed monomial ideals such that they are the generic initial ideals of local Cohen-Macaulay curves?", "original_statement": "Problem 27. Are there necessary or sufficient conditions on Borel fixed monomial ideals such that they are the generic initial ideals of local Cohen-Macaulay curves?", "clean_statement": "Problem 27. Are there necessary or sufficient conditions on Borel fixed monomial ideals such that they are the generic initial ideals of local Cohen-Macaulay curves?", "statement_status": "exact", "statement_verification": "The canonical record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 27\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[73]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 27. Are there necessary or sufficient conditions on Borel fixed monomial ideals such that they are the generic initial ideals of local Cohen-Macaulay curves?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0074", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over an infinite characteristic-zero field, let J be a saturated strongly stable ideal in k[x0,x1,x2,x3] with a curve Hilbert polynomial, using degree reverse lexicographic order. Then J is the gin of an arithmetically Cohen-Macaulay space curve if and only if no minimal generator of J involves x2 or x3, equivalently J is the extension of a finite-colength strongly stable ideal B in k[x0,x1]. In that case J itself defines an ACM locally Cohen-Macaulay multiple line and gin(J)=J. If an x2-generator occurs, any locally Cohen-Macaulay realization must instead be non-ACM with nonzero Rao module.\n\nCandidate contribution (criterion; novelty confidence low): The report packages a finite exact certificate for the full ACM region of the AIM realizability question: after the baseline saturation and curve-polynomial checks, absence of x2 from the minimal generators is necessary and sufficient for an explicit ACM/LCM realization, while its presence certifies that every possible LCM realization is non-ACM; the degree and genus are reconstructed from the finite Hilbert function of B by d=sum h_i and g=1-d+sum i h_i." }, { "id": 20000829, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0075", "title": "A local nonsmoothability threshold and very-compressed sieve for Hilb^d(A^3)", "statement": "Problem 28. What is the smallest d such that Hilb d(A3) is reducible?", "original_statement": "Problem 28. What is the smallest d such that Hilb d(A3) is reducible?", "clean_statement": "Problem 28. What is the smallest d such that Hilb d(A3) is reducible?", "statement_status": "exact", "statement_verification": "The AIM record (Components of Hilbert Schemes, Problem 28) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 28\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[74]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 28. What is the smallest d such that Hilb d(A3) is reducible?\"\nOriginal remarks: [\"Remark. 10 < d ≤ 78. The bound d > 10 needs the assumption that char= 0 and follows from a recent paper of Sivic.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0075", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over an algebraically closed characteristic-zero field, the smallest d for which Hilb^d(A^3) is reducible is exactly the smallest length of a nonsmoothable local quotient of k[x,y,z]. This follows from a proved componentwise smoothability lemma using idempotent lifting, and also shows reducibility propagates upward in d. Current literature gives 12 <= d <= 78. Any improvement below 78 must be a local non-very-compressed algebra: very-compressed ternary algebras through length 95 are known smoothable, and an independent exact Grassmannian dimension audit shows their locus is smaller than 3d for 2 <= d <= 95 and first exceeds 3d at d=96.\n\nCandidate contribution (reduction and exclusion sieve; novelty confidence low): Candidate contribution: d_min equals the minimum length of a nonsmoothable local k[x,y,z]-quotient, and every candidate improving d_min <= 78 must have a Hilbert function different from the very-compressed function; disconnected witnesses and all very-compressed witnesses of length at most 77 are rigorously excluded.", "research_difficulty_suggested": "L4" }, { "id": 20000830, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0076", "title": "The degree-two branchcurve locus is an explicit connected quotient", "statement": "Problem 29. Fix a Hilbert polynomial P, consider the moduli space BrV P Pn of branchvarieties that are equidimensional and connected in codimension 1. Is BrV P (Pn) connected when non-empty?", "original_statement": "Problem 29. Fix a Hilbert polynomial P, consider the moduli space BrV P Pn of branchvarieties that are equidimensional and connected in codimension 1. Is BrV P (Pn) connected when non-empty?", "clean_statement": "Problem 29. Fix a Hilbert polynomial P, consider the moduli space BrV P Pn of branchvarieties that are equidimensional and connected in codimension 1. Is BrV P (Pn) connected when non-empty?", "statement_status": "exact", "statement_verification": "The canonical record is an OCR rendering of AIM Problem 29:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 29\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[75]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 29. Fix a Hilbert polynomial P, consider the moduli space BrV P Pn of branchvarieties that are equidimensional and connected in codimension 1. Is BrV P (Pn) connected when non-empty?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0076", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Over an algebraically closed field of characteristic not 2, the stack of equidimensional geometrically reduced degree-two branchcurves X to P1 with Hilbert polynomial P(m)=2m+1-g is the quotient [(H0(P1,O(2g+2)) minus 0)/Gm] for the weight-two action. For every g at least 0, every such source is automatically connected in codimension one; the stack is smooth and irreducible, is a mu_2-gerbe over the coarse space P^(2g+2), and any two points are connected by a chain of at most two explicit A1-families with all fibers reduced, connected, and at fixed P. At g=-1 the full stack is B mu_2 but the connected-in-codimension-one locus is empty.\n\nCandidate contribution (special_case_classification_and_deformation_bridge; novelty confidence low): The exact stack-level refinement identifies the weight-two quotient, proves that the connected-in-codimension-one condition is redundant precisely for g>=0, isolates the g=-1 exception, and constructs a two-step affine-line deformation bridge between every pair of geometric points without leaving the fixed-Hilbert-polynomial reduced connected locus." }, { "id": 20000831, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0077", "title": "Exact deformation tangent spaces for radical-square-zero local algebras", "statement": "**Recovered local question.** Over an algebraically closed field \\(k\\), is\nthere a nontrivial finite local \\(k\\)-algebra \\(A\\) with\n\\(T^1_{A/k}=0\\)?", "original_statement": "Problem 30. Is there a rigid local Artinian algebra besides kn?", "clean_statement": "**Recovered local question.** Over an algebraically closed field \\(k\\), is\nthere a nontrivial finite local \\(k\\)-algebra \\(A\\) with\n\\(T^1_{A/k}=0\\)?", "statement_status": "corrected_verified", "statement_verification": "The record comes from Problem 30 in the AIM workshop list *Components of Hilbert Schemes*. Direct inspection of the typeset PDF, including its decompressed page-content stream, shows that the last expression is \\(k^n\\); the string “kn” is an OCR loss of the superscript. Thus the literal source statement is: There is an internal inconsistency: \\(k^n\\) is a product of fields and is not local when \\(n>1\\). The natural local baseline is \\(k\\), whereas \\(k^n\\) is the familiar finite étale baseline when locality is dropped. Jelisiejew’s 2026 survey corroborates this repair. Its Problem XXIII asks whether there is an irreducible finite \\(k\\)-scheme \\(\\Gamma\\), other than \\(\\operatorname{Spec} k\\), such that \\(T^1_\\Gamma=0\\), with characteristic zero assumed for safety.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 30\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[76]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 30. Is there a rigid local Artinian algebra besides kn?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0077", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source's OCR string 'kn' is the typeset expression k^n, which is incompatible with locality for n > 1; the modern open formulation asks for a nontrivial irreducible finite scheme with vanishing T^1. For every field k and every e-dimensional V with e >= 2, the radical-square-zero local algebra A = k plus V has T^1 naturally isomorphic to Hom(Sym^2 V,V) modulo the injective e-dimensional space of maps beta_c(u,v) = c(u)v + c(v)u. Hence dim T^1 = e^2(e+1)/2 - e > 0. The explicit nonzero class psi(x_1,x_1) = x_2 integrates to the finite free family x_1^2 = t x_2. For the minimal embedding, the Hilbert tangent has dimension e^2(e+1)/2 and the ambient-derivation image has dimension e.\n\nCandidate contribution (theorem; novelty confidence low): A characteristic-free exact quotient and dimension formula for T^1 of every radical-square-zero local algebra, together with an explicit integrated nontrivial class and the matching embedded Hilbert tangent dimension." }, { "id": 20000832, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0078", "title": "The generic very-compressed point for Hilbert function (1,4,10,a)", "statement": "Problem 31. Consider 1, 4, 10, a where 6 ≤ a ≤ 10. The general Artinian algebra with this Hilbert function is nonsmoothable. What is the generic point of the component it lies on?", "original_statement": "Problem 31. Consider 1, 4, 10, a where 6 ≤ a ≤ 10. The general Artinian algebra with this Hilbert function is nonsmoothable. What is the generic point of the component it lies on?", "clean_statement": "Problem 31. Consider 1, 4, 10, a where 6 ≤ a ≤ 10. The general Artinian algebra with this Hilbert function is nonsmoothable. What is the generic point of the component it lies on?", "statement_status": "exact", "statement_verification": "The AIM record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 31\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[77]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 31. Consider 1, 4, 10, a where 6 ≤ a ≤ 10. The general Artinian algebra with this Hilbert function is nonsmoothable. What is the generic point of the component it lies on?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0078", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a=6,7,8,9 in characteristic zero, Jelisiejew's 2024 theorem identifies the very-compressed locus B_a = A^4 x Gr(a,20) as the reduced support of a generically nonreduced component of Hilb^{15+a}(A^4). Its topological generic point has residue field K_a=k(B_a) and represents the generic quotient K_a[y_1,...,y_4]/(W_eta+(y)^4), while the Hilbert-scheme local ring there is a nonreduced Artinian thickening of K_a. For a=10, the same formula describes the generic algebra of the 104-dimensional very-compressed locus, but whether that locus is a component remains open. Uniformly for a=6,...,10, the generic quotient is level of type a, the nonlevel locus has codimension at least 4a-9, and all tangent directions normal to the reduced locus have negative weight.\n\nCandidate contribution (lemma and generic-point reduction; novelty confidence low): For every 6 <= a <= 10, the locus of W in Gr(20-a,S_3) for which S/(W+m^4) has socle below degree three has codimension at least 4a-9; moreover Hom_S(I,S/I)_{>0}=0, Hom_S(I,S/I)_0 is canonically Hom_k(W,S_3/W), and translations inject as four degree-minus-one tangents, so every infinitesimal direction normal to A^4 x Gr(a,20) has negative weight." }, { "id": 20000833, "problem_number": "AIM-ARITHMETIC_GEOMETRY-0079", "title": "Exact incidence and complete-intersection interpolation", "statement": "Problem 32. Fix d, g. Let H ⊂ Hilb d,g (P3) be an irreducible component. What is the largest number of points in general position you can make these curves pass through?", "original_statement": "Problem 32. Fix d, g. Let H ⊂ Hilb d,g (P3) be an irreducible component. What is the largest number of points in general position you can make these curves pass through?", "clean_statement": "Problem 32. Fix d, g. Let H ⊂ Hilb d,g (P3) be an irreducible component. What is the largest number of points in general position you can make these curves pass through?", "statement_status": "exact", "statement_verification": "The source states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Components of Hilbert schemes\nSection: \nSource item: 32\nSource URL: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[78]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 32. Fix d, g. Let H ⊂ Hilb d,g (P3) be an irreducible component. What is the largest number of points in general position you can make these curves pass through?\"\nOriginal remarks: [\"Remark. This is a hard problem. If the curves are arithmetically Cohen-Macaulay, it should be treatable.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0079", "aim-domain:arithmetic-geometry", "aim-workshop:hilbertschemes", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth curve at a smooth Hilbert point, dominance of the r-point incidence map is controlled exactly by surjectivity of H0(N_C) to the direct sum of the r normal fibers. Applying this to the complete-intersection component of type (a,b), 1<=a<=b, gives the sharp component-wide formula iota(H_{a,b})=binom(a+3,3)-1 when a g,n, is Rg−1+ k ∼= Q? What is Rg−1+ k generated by? When should we expect a 1-dimensional socle, and what should we expect for R(M≤kg,n )? 4 A LIST OF OPEN PROBLEMS AND QUESTIONS ON THE MODULI SPACE OF CURVES\n\n(11) (Igusa) Are there operations which relate the stable classes on BΓ∞? We have H∗\n\n> spec\n\n(CP )∞−1 ∼=\n\nZ[c1] · u, where u is the Thom class in degree −2. How is this reflected at the level of infinite loop spaces? What are the stable maps CP ∞−1 → CP ∞−1?(12) (Baldwin) Has anyone computed the intersection cohomology of Mg,n (Pr, d )? These can be arbitrarily singular, but this is what intersection cohomology is designed for. Is there a good notion of the tautological ring here? Perhaps the virtual fundamental class plays the usual role of the fundamental class. (13) (Faber) From Ekedahl and van der Geer, λg is 0 on Ag rationally but not integrally. The order in integral cohomology has been computed up to a factor of two. What is it? (Compare to 5 above.) (14) (Bertram) When will Getzler's paper on M1 appear (even just as a preprint)? Conjecture:\n\nt → ∞. Getzler comments that he does not like how this question is phrased. (15) (Ellenberg) Consider Hurwitz space (genus g, degree d). Could the cohomology stabilize as\n\ng → ∞ with d fixed? The reason behind this question is that point counting over finite fields gives exactly the behavior we would expect if we had Harer stability in degree 2. So, could some sort of Harer stability hold for some sort of Hurwitz schemes? Motivation for this question comes from work on number fields/function fields done in the '80s by Darskovksy and Wright. The general philosophy is this: suppose we have a nice sequence of varieties {Xn}n∈N,and lim n→∞\n\npoints on Xn(Fq )\n\nqdim Xn\n\nexists for all q. Is this because of some version of Harer stability at play here? (16) (Tseng) Same question for C → BG, with G a finite group. Tillmann says \"yes\" for G = S1.More precisely, consider\n\nEDiff( Fg, 1) ×Diff( Fg, 1) Map ∂ (Fg, 1, BG )for G connected, such as S1. This stabilizes by gluing in tori and the induced map on homology is an isomorphism in some range. Is this related? (17) (Sullivan) Fix a curve C and look at all unbranched covers of it. This gives points in Mg.Do these become uniformly dense for any C (with respect to the Teichmuller metric) in universally defined regions Ug of Mg for g large? More precisely, given [U+000F] > 0, can we find\n\ng0 such that, for g > g 0 every point of Ug is within distance [U+000F] of an unbranched cover of C\n\nof genus g? This would imply that, given curves C1, C 2, one could find covers ˜C1, ˜C2 which are arbitrarily close in the moduli space, the Siegel-Ehrenpreis problem. (18) (Morita) There are many numerical invariants that can be associated to the moduli space. (a) One may compute the signature of the cohomology ring of Mg,n.(b) Since Mg,n is a rational cohomology manifold, there are Thom's rational Pontrjagin classes, and one may compute the rational L-genus with respect to these. (c) Lastly, Mg,n is an orbifold of a complex manifold, so there are orbifold Chern classes, and hence orbifold Pontrjagin classes. Thus one may talk about the orbifold L-genus. What are these numbers? Do they agree? Probably not, but their disagreement would tell us interesting information about the types of singularities in the moduli space. The differ-ence between the signature and the rational L-genus detects geometric singularities. The difference between the rational L-genus and the orbifold L-genus detects complex analytic singularities. One may similarly ask questions about the signature of the tautological ring, and many other variations on this theme. A LIST OF OPEN PROBLEMS AND QUESTIONS ON THE MODULI SPACE OF CURVES 5\n\n(19) (Sullivan) This question regards the algebraic structure on the homology of the free loop space of a manifold. There are maps\n\nH∗LM ∆\n\n−→ H∗LM ⊗ H∗LM\n\nand\n\nH∗LM ⊗ H∗LM μ\n\n−→ H∗LM\n\nFirst the naive question: what is the algebraic structure here? The spectral sequence converging to H∗LM has E2 term a tensor product of a Hopf algebra (coming from the base\n\nM ) and a Frobenius algebra (coming from the fibre). But the differential does not respect these structures. A perhaps better question is: can we illuminate the situation be reformulating in terms of the category of spaces over M? We have\n\nLM LM ×M LM LM × LM M M M × M\n\n[U+000F]\n\n[U+000F]\n\n/\n\n/\n\n[U+000F]\n\n[U+000F]\n\n/\n\n/\n\n[U+000F]\n\n[U+000F]//\n\n> id\n\n/ / ∆\n\nThe left square corresponds to the Frobenius algebra part, and the right square corresponds to the Hopf algebra part. So, what is the full algebraic structure here?", "original_statement": "Conjecture 1, p. 4). For that matter, prove Morita's conjecture! Perhaps this is the \"right\" way to prove Faber's intersection number conjecture. (10) (Mondello) For M≤krational components \n\n> g,n, is Rg−1+ k ∼= Q? What is Rg−1+ k generated by? When should we expect a 1-dimensional socle, and what should we expect for R(M≤kg,n )? 4 A LIST OF OPEN PROBLEMS AND QUESTIONS ON THE MODULI SPACE OF CURVES \n\n(11) (Igusa) Are there operations which relate the stable classes on BΓ∞? We have H∗\n\n> spec\n\n(CP )∞−1 ∼=\n\nZ[c1] · u, where u is the Thom class in degree −2. How is this reflected at the level of infinite loop spaces? What are the stable maps CP ∞−1 → CP ∞−1?(12) (Baldwin) Has anyone computed the intersection cohomology of Mg,n (Pr, d )? These can be arbitrarily singular, but this is what intersection cohomology is designed for. Is there a good notion of the tautological ring here? Perhaps the virtual fundamental class plays the usual role of the fundamental class. (13) (Faber) From Ekedahl and van der Geer, λg is 0 on Ag rationally but not integrally. The order in integral cohomology has been computed up to a factor of two. What is it? (Compare to 5 above.) (14) (Bertram) When will Getzler's paper on M1 appear (even just as a preprint)? Conjecture: \n\nt → ∞. Getzler comments that he does not like how this question is phrased. (15) (Ellenberg) Consider Hurwitz space (genus g, degree d). Could the cohomology stabilize as \n\ng → ∞ with d fixed? The reason behind this question is that point counting over finite fields gives exactly the behavior we would expect if we had Harer stability in degree 2. So, could some sort of Harer stability hold for some sort of Hurwitz schemes? Motivation for this question comes from work on number fields/function fields done in the '80s by Darskovksy and Wright. The general philosophy is this: suppose we have a nice sequence of varieties {Xn}n∈N,and lim n→∞ \n\npoints on Xn(Fq )\n\nqdim Xn\n\nexists for all q. Is this because of some version of Harer stability at play here? (16) (Tseng) Same question for C → BG, with G a finite group. Tillmann says \"yes\" for G = S1.More precisely, consider \n\nEDiff( Fg, 1) ×Diff( Fg, 1) Map ∂ (Fg, 1, BG )for G connected, such as S1. This stabilizes by gluing in tori and the induced map on homology is an isomorphism in some range. Is this related? (17) (Sullivan) Fix a curve C and look at all unbranched covers of it. This gives points in Mg.Do these become uniformly dense for any C (with respect to the Teichmuller metric) in universally defined regions Ug of Mg for g large? More precisely, given \u000f > 0, can we find \n\ng0 such that, for g > g 0 every point of Ug is within distance \u000f of an unbranched cover of C\n\nof genus g? This would imply that, given curves C1, C 2, one could find covers ˜C1, ˜C2 which are arbitrarily close in the moduli space, the Siegel-Ehrenpreis problem. (18) (Morita) There are many numerical invariants that can be associated to the moduli space. (a) One may compute the signature of the cohomology ring of Mg,n.(b) Since Mg,n is a rational cohomology manifold, there are Thom's rational Pontrjagin classes, and one may compute the rational L-genus with respect to these. (c) Lastly, Mg,n is an orbifold of a complex manifold, so there are orbifold Chern classes, and hence orbifold Pontrjagin classes. Thus one may talk about the orbifold L-genus. What are these numbers? Do they agree? Probably not, but their disagreement would tell us interesting information about the types of singularities in the moduli space. The differ-ence between the signature and the rational L-genus detects geometric singularities. The difference between the rational L-genus and the orbifold L-genus detects complex analytic singularities. One may similarly ask questions about the signature of the tautological ring, and many other variations on this theme. A LIST OF OPEN PROBLEMS AND QUESTIONS ON THE MODULI SPACE OF CURVES 5\n\n(19) (Sullivan) This question regards the algebraic structure on the homology of the free loop space of a manifold. There are maps \n\nH∗LM ∆\n\n−→ H∗LM ⊗ H∗LM \n\nand \n\nH∗LM ⊗ H∗LM μ\n\n−→ H∗LM \n\nFirst the naive question: what is the algebraic structure here? The spectral sequence converging to H∗LM has E2 term a tensor product of a Hopf algebra (coming from the base \n\nM ) and a Frobenius algebra (coming from the fibre). But the differential does not respect these structures. A perhaps better question is: can we illuminate the situation be reformulating in terms of the category of spaces over M? We have \n\nLM LM ×M LM LM × LM M M M × M\n\n\u000f\n\n\u000f\n\n/\n\n/\n\n\u000f\n\n\u000f\n\n/\n\n/\n\n\u000f\n\n\u000f// \n\n> id\n\n/ / ∆\n\nThe left square corresponds to the Frobenius algebra part, and the right square corresponds to the Hopf algebra part. So, what is the full algebraic structure here?", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "The canonical JSON object is not one problem. Comparison with the five-page AIM source shows that it starts in the middle of numbered Problem (9), at the parenthetical citation ``[Mor05] (Conjecture 1, p. 4),'' and then concatenates the end of Problem (9) with all of Problems (10)--(19). The JSON fields `number: \"1\"` and `tag: \"conjecture\"` were evidently inferred from the words ``Conjecture 1'' inside that citation, not from a heading for a new problem. Page headers, broken formulae, and control characters were also incorporated into the object. Consequently there is no single proposition whose truth could constitute a solution of the record as stored.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Arithmetic geometry\nWorkshop: Topology and geometry of the moduli space of curves\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/modspacecurves/modspacecurves.pdf\nCanonical location: aim-arithmetic-geometry-notes.json notes[80]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 1, p. 4). For that matter, prove Morita's conjecture! Perhaps this is the \\\"right\\\" way to prove Faber's intersection number conjecture. (10) (Mondello) For M≤krational components \\n\\n> g,n, is Rg−1+ k ∼= Q? What is Rg−1+ k generated by? When should we expect a 1-dimensional socle, and what should we expect for R(M≤kg,n )? 4 A LIST OF OPEN PROBLEMS AND QUESTIONS ON THE MODULI SPACE OF CURVES \\n\\n(11) (Igusa) Are there operations which relate the stable classes on BΓ∞? We have H∗\\n\\n> spec\\n\\n(CP )∞−1 ∼=\\n\\nZ[c1] · u, where u is the Thom class in degree −2. How is this reflected at the level of infinite loop spaces? What are the stable maps CP ∞−1 → CP ∞−1?(12) (Baldwin) Has anyone computed the intersection cohomology of Mg,n (Pr, d )? These can be arbitrarily singular, but this is what intersection cohomology is designed for. Is there a good notion of the tautological ring here? Perhaps the virtual fundamental class plays the usual role of the fundamental class. (13) (Faber) From Ekedahl and van der Geer, λg is 0 on Ag rationally but not integrally. The order in integral cohomology has been computed up to a factor of two. What is it? (Compare to 5 above.) (14) (Bertram) When will Getzler's paper on M1 appear (even just as a preprint)? Conjecture: \\n\\nt → ∞. Getzler comments that he does not like how this question is phrased. (15) (Ellenberg) Consider Hurwitz space (genus g, degree d). Could the cohomology stabilize as \\n\\ng → ∞ with d fixed? The reason behind this question is that point counting over finite fields gives exactly the behavior we would expect if we had Harer stability in degree 2. So, could some sort of Harer stability hold for some sort of Hurwitz schemes? Motivation for this question comes from work on number fields/function fields done in the '80s by Darskovksy and Wright. The general philosophy is this: suppose we have a nice sequence of varieties {Xn}n∈N,and lim n→∞ \\n\\npoints on Xn(Fq )\\n\\nqdim Xn\\n\\nexists for all q. Is this because of some version of Harer stability at play here? (16) (Tseng) Same question for C → BG, with G a finite group. Tillmann says \\\"yes\\\" for G = S1.More precisely, consider \\n\\nEDiff( Fg, 1) ×Diff( Fg, 1) Map ∂ (Fg, 1, BG )for G connected, such as S1. This stabilizes by gluing in tori and the induced map on homology is an isomorphism in some range. Is this related? (17) (Sullivan) Fix a curve C and look at all unbranched covers of it. This gives points in Mg.Do these become uniformly dense for any C (with respect to the Teichmuller metric) in universally defined regions Ug of Mg for g large? More precisely, given \\u000f > 0, can we find \\n\\ng0 such that, for g > g 0 every point of Ug is within distance \\u000f of an unbranched cover of C\\n\\nof genus g? This would imply that, given curves C1, C 2, one could find covers ˜C1, ˜C2 which are arbitrarily close in the moduli space, the Siegel-Ehrenpreis problem. (18) (Morita) There are many numerical invariants that can be associated to the moduli space. (a) One may compute the signature of the cohomology ring of Mg,n.(b) Since Mg,n is a rational cohomology manifold, there are Thom's rational Pontrjagin classes, and one may compute the rational L-genus with respect to these. (c) Lastly, Mg,n is an orbifold of a complex manifold, so there are orbifold Chern classes, and hence orbifold Pontrjagin classes. Thus one may talk about the orbifold L-genus. What are these numbers? Do they agree? Probably not, but their disagreement would tell us interesting information about the types of singularities in the moduli space. The differ-ence between the signature and the rational L-genus detects geometric singularities. The difference between the rational L-genus and the orbifold L-genus detects complex analytic singularities. One may similarly ask questions about the signature of the tautological ring, and many other variations on this theme. A LIST OF OPEN PROBLEMS AND QUESTIONS ON THE MODULI SPACE OF CURVES 5\\n\\n(19) (Sullivan) This question regards the algebraic structure on the homology of the free loop space of a manifold. There are maps \\n\\nH∗LM ∆\\n\\n−→ H∗LM ⊗ H∗LM \\n\\nand \\n\\nH∗LM ⊗ H∗LM μ\\n\\n−→ H∗LM \\n\\nFirst the naive question: what is the algebraic structure here? The spectral sequence converging to H∗LM has E2 term a tensor product of a Hopf algebra (coming from the base \\n\\nM ) and a Frobenius algebra (coming from the fibre). But the differential does not respect these structures. A perhaps better question is: can we illuminate the situation be reformulating in terms of the category of spaces over M? We have \\n\\nLM LM ×M LM LM × LM M M M × M\\n\\n\\u000f\\n\\n\\u000f\\n\\n/\\n\\n/\\n\\n\\u000f\\n\\n\\u000f\\n\\n/\\n\\n/\\n\\n\\u000f\\n\\n\\u000f// \\n\\n> id\\n\\n/ / ∆\\n\\nThe left square corresponds to the Frobenius algebra part, and the right square corresponds to the Hopf algebra part. So, what is the full algebraic structure here?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 5, "set_id": 14, "source_url": "https://aimath.org/WWN/modspacecurves/modspacecurves.pdf", "tags": [ "aim", "AIM-ARITHMETIC_GEOMETRY-0081", "aim-domain:arithmetic-geometry", "aim-workshop:modspacecurves", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 5, "name": "algebraic_geometry", "display_name": "Algebraic Geometry", "description": "Geometric objects defined by polynomial equations.", "slug": "algebraic-geometry", "order_index": 5, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The stored object is not one mathematical problem: it begins inside the citation at the end of source Problem (9) and concatenates Problems (10)--(19). For the intact Sullivan Problem (17), a proved exact genus-spectrum theorem shows that a fixed base curve of genus h has connected unbranched covers only in genera 1+d(h-1) when h is at least two (with the separate spectra {0} and {1} in genera zero and one). Thus the literal all-sufficiently-large-g quantifier is impossible for every base genus other than two whenever the regions U_g are eventually nonempty; for two hyperbolic bases, common cover genera are exactly 1 modulo lcm(h_1-1,h_2-1).\n\nCandidate contribution (quantifier_obstruction; novelty confidence low): The exact cover-genus spectrum gives a testable repair of Sullivan's AIM wording: restrict g to 1+d(h-1); genus two is the unique hyperbolic base for which every genus at least two is arithmetically admissible, and two bases have a common cover genus precisely when g-1 is divisible by both h_i-1.", "research_difficulty_suggested": "L2" }, { "id": 20000836, "problem_number": "AIM-BIOLOGY-0001", "title": "Identifiability of stage-resolved retinal vascular change in diabetes", "statement": "How does vasculature in eye change with diabetes?", "original_statement": "How does vasculature in eye change with diabetes?", "clean_statement": "How does vasculature in eye change with diabetes?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Modeling the eye as a window on the body\nSection: Diabetes\nSource item: 1.1\nSource URL: http://aimpl.org/eyewindow/1/\nCanonical location: aim-biology-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How does vasculature in eye change with diabetes?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/eyewindow/1/", "tags": [ "aim", "AIM-BIOLOGY-0001", "aim-domain:biology", "aim-workshop:eyewindow", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Diabetic ocular vascular change is stage-, compartment-, modality-, and treatment-dependent rather than a single monotone vessel-density change. For an idealized layer-specific OCTA model, one threshold cannot distinguish anatomical or durable capillary loss from retained segments whose positive flow falls below detection, even when area and skeleton density are both measured. The complete calibrated threshold curves uniquely determine the positive-flow centerline-length measure and its width-weighted counterpart, with flow-band apparent caliber equal to the ratio of their tail differences, but no such curve can distinguish an absent segment from an anatomically present zero-flow segment. A proved finite-error bound shows that recovery is unstable when a flow band contains too little detected length.\n\nCandidate contribution (identifiability theorem; novelty confidence low): An exact threshold-curve identifiability theorem, zero-flow impossibility result, and finite-error band-caliber bound for paired OCTA skeleton-length and additive-area measurements, proposed as a falsifiable diabetic-retinopathy imaging design criterion." }, { "id": 20000837, "problem_number": "AIM-BIOLOGY-0002", "title": "A conditional shunt-steal criterion for retinal oxygenation", "statement": "How does change in vasculature can change oxygenation in the eye?", "original_statement": "How does change in vasculature can change oxygenation in the eye?", "clean_statement": "How does change in vasculature can change oxygenation in the eye?", "statement_status": "exact", "statement_verification": "The exact canonical record is AIM Problem Lists workshop *Modeling the eye as a window on the body*, section “Diabetes,” problem 1.2:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Modeling the eye as a window on the body\nSection: Diabetes\nSource item: 1.2\nSource URL: http://aimpl.org/eyewindow/1/\nCanonical location: aim-biology-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How does change in vasculature can change oxygenation in the eye?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/eyewindow/1/", "tags": [ "aim", "AIM-BIOLOGY-0002", "aim-domain:biology", "aim-workshop:eyewindow", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In a reduced retinal circuit with a shared upstream resistance and parallel nutritive and bypass conductances, the exact first-order condition for a vascular perturbation to raise nutritive flow is (1-theta*phi) eta_n > theta(1-phi) eta_s. Coupling this circuit to a finite-exchange, monotone tissue oxygen balance proves that the same inequality determines the sign of tissue pO2. Pure preferential shunt dilation can therefore increase total retinal flow and mixed venous oxygen content while decreasing nutritive flow, retinal uptake, tissue pO2, and the arteriovenous oxygen difference. A separate comparison theorem shows that spatial hypoxic burden is controlled by the effective oxygen source field, not vessel density alone.\n\nCandidate contribution (conditional theorem; novelty confidence low): The candidate contribution is the closed-form shared-resistance conductance threshold, coupled to a finite-exchange proof, that yields the falsifiable sign quartet total flow up, tissue pO2 down, venous oxygen content up, and arteriovenous difference down under pure preferential shunt dilation." }, { "id": 20000838, "problem_number": "AIM-BIOLOGY-0003", "title": "Repeatability-calibrated sequential warning from retinal microvascular monitoring", "statement": "Can we identify any early warning signs by monitoring microvasculature?", "original_statement": "Can we identify any early warning signs by monitoring microvasculature?", "clean_statement": "Can we identify any early warning signs by monitoring microvasculature?", "statement_status": "exact", "statement_verification": "The exact canonical question is: “Can we identify any early warning signs by monitoring microvasculature?” It appears as Diabetes Problem 1.3 in the AIM workshop “Modeling the eye as a window on the body.” The official workshop summary explains that ocular vascular and structural changes can be measured noninvasively, that one problem group studied early imaging warnings and diabetes progression, and that the group proposed following vascular measurements over a long period. The summary specifically mentions color-Doppler waveform parameters in retrobulbar vessels; those are upstream flow measurements rather than direct retinal capillary measurements. The present record itself says “microvasculature,” so OCT angiography (OCTA) is the most direct candidate modality, with Doppler indices treated as optional auxiliary observations.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Modeling the eye as a window on the body\nSection: Diabetes\nSource item: 1.3\nSource URL: http://aimpl.org/eyewindow/1/\nCanonical location: aim-biology-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we identify any early warning signs by monitoring microvasculature?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/eyewindow/1/", "tags": [ "aim", "AIM-BIOLOGY-0003", "aim-domain:biology", "aim-workshop:eyewindow", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Prospective OCTA cohorts support prognostic association but do not yet provide a universally validated longitudinal warning rule. Under a bounded device/physiology drift model and conservative sub-Gaussian repeatability bounds, a proved alpha-spending threshold controls the per-patient probability of any false alert across multiple biomarkers and visits by alpha without assuming independent errors. A companion power bound gives a sufficient detectable biological change of 2B + v[sqrt(2 log(1/alpha_tj)) + sqrt(2 log(1/beta))], while an identifiability proposition proves that unrestricted drift is observationally indistinguishable from biological change.\n\nCandidate contribution (sequential_detection_bound; novelty confidence low): A bias-aware conversion from same-protocol repeatability to a multimetric sequential retinal warning threshold and detectable-change bound, valid under arbitrary shared-baseline dependence, together with a proof that an unbounded drift term makes true microvascular change unidentifiable." }, { "id": 20000839, "problem_number": "AIM-BIOLOGY-0004", "title": "Separating retinal barrier influx from delivery, volume, and clearance", "statement": "Can we characterize and quantify the leakage factors from a clinical and modeling perspective in diabetes?", "original_statement": "Can we characterize and quantify the leakage factors from a clinical and modeling perspective in diabetes?", "clean_statement": "Can we characterize and quantify the leakage factors from a clinical and modeling perspective in diabetes?", "statement_status": "exact", "statement_verification": "The canonical source is the AIM workshop *Modeling the eye as a window on the body*, section “Diabetes,” problem 1.4. The exact record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Modeling the eye as a window on the body\nSection: Diabetes\nSource item: 1.4\nSource URL: http://aimpl.org/eyewindow/1/\nCanonical location: aim-biology-notes.json notes[3]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we characterize and quantify the leakage factors from a clinical and modeling perspective in diabetes?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/eyewindow/1/", "tags": [ "aim", "AIM-BIOLOGY-0004", "aim-domain:biology", "aim-workshop:eyewindow", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a calibrated one-compartment retinal tracer model, a known local free-plasma input and nondegenerate dynamic extravascular amount curve structurally identify the ROI permeability-surface product and effective clearance, whereas a single late endpoint admits a continuum of indistinguishable influx-clearance pairs. Unknown input scale and the factorization PS create further exact nonidentifiabilities. An irreversible finite-window estimate underestimates PS by a relative amount bounded above by 1-exp(-kT), itself at most kT; if imaging measures concentration while edema volume changes, the logarithmic volume-growth rate is added to apparent clearance.\n\nCandidate contribution (identifiability theorem and bias bound; novelty confidence low): Candidate contribution: a unified uptake-washout audit for diabetic retinal leakage combines exact nonidentifiability families, the clearance-neglect bias bound 0 <= 1-PS_hat/PS <= 1-exp(-kT) <= kT, and the result that unmeasured edema-volume growth adds dot(Ve)/Ve to apparent concentration clearance.", "research_difficulty_suggested": "L4" }, { "id": 20000840, "problem_number": "AIM-BIOLOGY-0005", "title": "A deployable landmark model for imaging-based diabetic-retinopathy progression warnings", "statement": "How can we develop a predictive model for early imaging related warnings and diabetes progression?", "original_statement": "How can we develop a predictive model for early imaging related warnings and diabetes progression?", "clean_statement": "How can we develop a predictive model for early imaging related warnings and diabetes progression?", "statement_status": "exact", "statement_verification": "The canonical AIM record (Biology, workshop *Modeling the eye as a window on the body*, Diabetes problem 1.5) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Modeling the eye as a window on the body\nSection: Diabetes\nSource item: 1.5\nSource URL: http://aimpl.org/eyewindow/1/\nCanonical location: aim-biology-notes.json notes[4]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How can we develop a predictive model for early imaging related warnings and diabetes progression?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/eyewindow/1/", "tags": [ "aim", "AIM-BIOLOGY-0005", "aim-domain:biology", "aim-workshop:eyewindow", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The underspecified AIM prompt is converted into a 24-month landmark cumulative-incidence prognosis under a declared treatment and surveillance policy. For a locked prediction measurable at the landmark, a proved three-term inverse-censoring loss has expectation equal to the binary horizon Brier risk while scoring death as a competing non-progression event. Exact counterexamples and bounds additionally show that post-landmark image leakage can create perfect but useless performance, untreated risk is not identified under deterministic warning-triggered treatment, and person-level risk cannot be recovered from two eye marginals without dependence assumptions.\n\nCandidate contribution (theorem; novelty confidence low): Candidate contribution: a problem-specific four-part acceptance protocol combining landmark measurability, a proved competing-event inverse-censoring Brier identity, an explicit prevailing-policy treatment estimand, and a sharp two-eye Frechet audit; together these yield observable failure tests for leakage, censoring/death mishandling, policy nonidentification, and invalid patient-level aggregation." }, { "id": 20000841, "problem_number": "AIM-BIOLOGY-0006", "title": "A compensation and identifiability framework for aging, IOP, and ocular structure", "statement": "How does aging influence intraocular pressure(IOP)of an eye? How does aging influence the structure of the eye?", "original_statement": "How does aging influence intraocular pressure(IOP)of an eye? How does aging influence the structure of the eye?", "clean_statement": "How does aging influence intraocular pressure(IOP)of an eye? How does aging influence the structure of the eye?", "statement_status": "exact", "statement_verification": "The canonical AIM record, from the workshop *Modeling the eye as a window on the body*, section “Aging” (source URL: http://aimpl.org/eyewindow/2/), asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Modeling the eye as a window on the body\nSection: Aging\nSource item: 2.1\nSource URL: http://aimpl.org/eyewindow/2/\nCanonical location: aim-biology-notes.json notes[5]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How does aging influence intraocular pressure(IOP)of an eye? How does aging influence the structure of the eye?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/eyewindow/2/", "tags": [ "aim", "AIM-BIOLOGY-0006", "aim-domain:biology", "aim-workshop:eyewindow", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "There is no population-independent signed age effect on true intraocular pressure: in the modified Goldmann balance, the trans-episcleral pressure gradient rises, stays fixed, or falls according as the proportional change in net conventional-flow demand exceeds, equals, or falls below the proportional change in conventional outflow facility. A single cornea-dependent tonometric response cannot identify true pressure drift, while two calibrated responses with distinct biomechanical sensitivities yield an offset-invariant longitudinal pressure estimator and an explicit noise-conditioning bound. This separates true pressure, measurement bias, and structural aging.\n\nCandidate contribution (reduction; novelty confidence low): The exact finite-age aqueous-compensation threshold, paired offset-invariant longitudinal pressure estimator, and its conditioning bound form a single prospective identifiability criterion for aging-eye studies.", "research_difficulty_suggested": "L4" }, { "id": 20000842, "problem_number": "AIM-BIOLOGY-0007", "title": "Flow certification for healthy-age retrobulbar Doppler comparisons", "statement": "How does velocity profile change in young vs old healthy population?", "original_statement": "How does velocity profile change in young vs old healthy population?", "clean_statement": "How does velocity profile change in young vs old healthy population?", "statement_status": "exact", "statement_verification": "The exact canonical statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Modeling the eye as a window on the body\nSection: Aging\nSource item: 2.2\nSource URL: http://aimpl.org/eyewindow/2/\nCanonical location: aim-biology-notes.json notes[6]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How does velocity profile change in young vs old healthy population?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/eyewindow/2/", "tags": [ "aim", "AIM-BIOLOGY-0007", "aim-domain:biology", "aim-workshop:eyewindow", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The workshop context identifies the target as color-Doppler cardiac waveforms in the retrobulbar ophthalmic, central retinal, and posterior ciliary arteries. For participant-level measurements, or group geometric means, true old-to-young volume flow equals the observed velocity ratio times the squared lumen-radius ratio, the inverse profile-factor ratio, and the relative angle-correction gain. This gives an exact uncertainty interval that certifies a flow decrease or increase only when the full interval lies below or above one. Without radius and profile/angle information, velocity does not identify flow direction. An explicit Poiseuille counterexample shows that mean velocity can rise while flow falls, and a normalized-waveform result proves that RI, PI, and waveform shape are invariant to constant multiplicative angle/device gain.\n\nCandidate contribution (uncertainty-certification theorem; novelty confidence low): The candidate contribution is a vessel-specific old-versus-young flow-certification interval combining measured Doppler velocity, squared caliber ratio, profile/sampling uncertainty, and a rigorous angular error bound; it supplies a testable decision rule for when an age-associated velocity change does or does not support a volumetric-flow claim." }, { "id": 20000843, "problem_number": "AIM-BIOLOGY-0008", "title": "Supply-demand thresholds for retinal oxygenation in light and darkness", "statement": "Can we quantify and characterize the factors that determine the oxygenation in the eye? (e.g. Light + dark ).", "original_statement": "Can we quantify and characterize the factors that determine the oxygenation in the eye? (e.g. Light + dark ).", "clean_statement": "Can we quantify and characterize the factors that determine the oxygenation in the eye? (e.g. Light + dark ).", "statement_status": "exact", "statement_verification": "The canonical AIM record, in the “Basic Physiology” section of the workshop *Modeling the eye as a window on the body*, asks exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Modeling the eye as a window on the body\nSection: Basic Physiology\nSource item: 3.1\nSource URL: http://aimpl.org/eyewindow/3/\nCanonical location: aim-biology-notes.json notes[7]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we quantify and characterize the factors that determine the oxygenation in the eye? (e.g. Light + dark ).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/eyewindow/3/", "tags": [ "aim", "AIM-BIOLOGY-0008", "aim-domain:biology", "aim-workshop:eyewindow", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A baseline retinal oxygenation question is reduced to separate outer- and inner-retinal supply-demand balances. A proved layered comparison theorem shows that greater dark consumption under fixed supply cannot raise tissue oxygen. In a homogeneous capillary-free slab, an exact light-dark identity partitions midpoint change into boundary-supply compensation minus a quadratic demand penalty; the dimensionless demand number Theta reaches hypoxia at exactly 8 and gives an explicit below-threshold width. A Green-function counterexample proves that even perfect flow and arterial/venous oxygen measurements identify total extraction but not the local tissue oxygen profile.\n\nCandidate contribution (theorem; novelty confidence low): Candidate contribution: a light-dark oxygen-reserve acceptance test combining a variable-layer comparison theorem, the exact compensation identity, the threshold Theta=8 with hypoxic-core width, and a Green-function proof that vascular extraction alone cannot identify local retinal oxygen." }, { "id": 20000844, "problem_number": "AIM-BIOLOGY-0009", "title": "A two-number epithelial transport reduction with an aqueous-production identifiability test", "statement": "Can we develop kidney inspired models for aqueous humor production?", "original_statement": "Can we develop kidney inspired models for aqueous humor production?", "clean_statement": "Can we develop kidney inspired models for aqueous humor production?", "statement_status": "exact", "statement_verification": "The canonical AIM record is from the workshop *Modeling the eye as a window on the body*, section “Aqueous Humor Formation,” item 4.2. It asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Modeling the eye as a window on the body\nSection: Aqueous Humor Formation\nSource item: 4.2\nSource URL: http://aimpl.org/eyewindow/4/\nCanonical location: aim-biology-notes.json notes[8]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we develop kidney inspired models for aqueous humor production?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/eyewindow/4/", "tags": [ "aim", "AIM-BIOLOGY-0009", "aim-domain:biology", "aim-workshop:eyewindow", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Kidney-inspired aqueous-production models are already feasible when renal epithelial conservation, electroneutrality, osmotic water coupling, paracellular leak, and supply limitation are rewired for the PE-NPE ciliary bilayer rather than copied as nephron anatomy. In a minimal local-osmosis reduction, forward production is controlled by two dimensionless groups, has a sharp net-source inhibition threshold, and is not identifiable from one steady production measurement. Four fixed-source pressure clamps yield a linear effective-parameter reconstruction and the overidentifying invariant b = a k, whose failure falsifies the one-domain model.\n\nCandidate contribution (identifiability theorem; novelty confidence low): For the stated ciliary local-osmosis reduction, the dimensionless secretion law has an exact source threshold, every single positive steady state has an explicitly parameterized continuum of hydraulic/leak realizations, and four full-rank pressure clamps identify effective parameters while testing the necessary and sufficient invariant b = a k." }, { "id": 20000845, "problem_number": "AIM-BIOLOGY-0010", "title": "A rate-based criterion for targeting plastic tumor states", "statement": "Characterizing and modeling tumor heterogeneity", "original_statement": "Characterizing and modeling tumor heterogeneity", "clean_statement": "Characterizing and modeling tumor heterogeneity", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.1, “Characterizing and modeling tumor heterogeneity,” from the AIM workshop *Systems approaches to drug discovery and development in oncology*. Its main questions are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Top four problem areas\nSource item: 1.1\nSource URL: http://aimpl.org/systemsoncology/1/\nCanonical location: aim-biology-notes.json notes[9]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Characterizing and modeling tumor heterogeneity\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Main questions:\\n\\nHow do we characterize and account for heterogeneity? Are we targeting the right cell populations? Do cancer stem cells exist? If so, do they matter?\\n\\nNotes:\\n\\n(a) John Lowengrub (UC-Irvine) and Heiko Enderling (Tufts University) have studied math models about stem cells, and their results suggest that the presence of stem cells makes a big difference in outcomes.\\n\\n(b) Whether there is agreement about what the cells might be called; if there is agreement about their function, that is a starting point.\\n\\n(c) Stem cells might develop mutations and cause cancer, and non-stem cells may develop mutations and cause cancer.\\n\\n(d) There exist viruses that are too virulent for their own good, and thus die out while less virulent viruses remain. Is there a corresponding scenario for tumor cells? Can we characterize the prognostic threat from different cells in the tumor? For example, are some cells more cloaked from the immune system than others?\\n\\n(e) Research underway studying single cell heterogeneity in tumors, e.g., from Stephen Quake's lab at Stanford (Dalerba et al. 2011).\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/1/", "tags": [ "aim", "AIM-BIOLOGY-0010", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a constant-rate two-state tumor model with reversible transitions from a stem-like state S to a differentiated state D at rate alpha and back at rate beta, differentiated-state-only killing can robustly clear both states only if the untreated S diagonal rate a0 is negative; in that case the exact threshold is kD > d0 + alpha beta / |a0|. The additional term is a plasticity tax for the D-to-S-to-D feedback loop. Perron eigenvalue sensitivities rank targets by reproductive-value-weighted occupancy rather than abundance alone, and an explicit construction proves that even a stable state composition cannot identify the growth exponent without rate data.\n\nCandidate contribution (theorem; novelty confidence low): Candidate model theorem: differentiated-state-only therapy has the sharp clearance threshold kD > d0 + alpha beta / |a0| when a0 < 0 and cannot produce robust exponential clearance for any finite kD when a0 >= 0; this sanctuary criterion is paired with a constructive proof that stable state fractions do not identify tumor threat." }, { "id": 20000846, "problem_number": "AIM-BIOLOGY-0011", "title": "A mechanism gate for resistance-driven drug scheduling", "statement": "Systems approaches to drug resistance", "original_statement": "Systems approaches to drug resistance", "clean_statement": "Systems approaches to drug resistance", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 1.2, **“Systems approaches to drug resistance,”** from the workshop *Systems approaches to drug discovery and development in oncology*. Its accompanying questions are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Top four problem areas\nSource item: 1.2\nSource URL: http://aimpl.org/systemsoncology/1/\nCanonical location: aim-biology-notes.json notes[10]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Systems approaches to drug resistance\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"What mechanisms are driving resistance? How do we predict better drug combinations to combat resistance? What kind of data/studies do we need to better understand resistance?\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/1/", "tags": [ "aim", "AIM-BIOLOGY-0011", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In a two-state constant-exposure model, treatment-induced sensitive-to-tolerant switching and selection of a pre-existing non-switching resistant subpopulation can produce exactly the same total-burden trajectory for all times. Nevertheless, when drug A creates a tolerant state and drug B selectively kills that state, equal-exposure schedules satisfy N_BA - N_AB = (1-exp(-b tau_B)) (k S_0/(d+k)) (1-exp(-(d+k) tau_A)) >= 0, with strict benefit for A followed by B in every nondegenerate case; the observationally equivalent pre-existing diagonal model has zero order effect. Thus a monotherapy bulk curve cannot identify the schedule decision, and lineage/state/washout/exposure-resolved perturbations are required.\n\nCandidate contribution (theorem; novelty confidence low): Candidate mechanism-gated schedule non-identifiability theorem: the same exact monotherapy bulk trajectory can arise from a pre-existing-resistance model with commuting drug actions and zero order effect or from an induced-tolerance model with the explicit positive order gap (1-exp(-b tau_B)) (k S_0/(d+k)) (1-exp(-(d+k) tau_A))." }, { "id": 20000847, "problem_number": "AIM-BIOLOGY-0012", "title": "A closure test from signaling dynamics to tumor growth", "statement": "Linking signaling models to phenotype (e.g., tumor growth)", "original_statement": "Linking signaling models to phenotype (e.g., tumor growth)", "clean_statement": "Linking signaling models to phenotype (e.g., tumor growth)", "statement_status": "exact", "statement_verification": "The exact canonical problem is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Top four problem areas\nSource item: 1.3\nSource URL: http://aimpl.org/systemsoncology/1/\nCanonical location: aim-biology-notes.json notes[11]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Linking signaling models to phenotype (e.g., tumor growth)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Consortium between pharma industry and academics? Multi-scale modeling from pathway to cell to whole tissue?\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/1/", "tags": [ "aim", "AIM-BIOLOGY-0012", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A mean pathway signal is an exact population-growth closure for every signaling distribution only when the signal-to-growth map is affine; with bounded curvature, its error is bounded by one half the curvature bound times the signal variance. For a causal two-state signaling decoder with state retention across division, the true population exponent equals the isolated-cell occupancy-weighted growth rate plus an exact nonnegative lineage-selection correction. At fixed stationary signaling occupancy, the exponent strictly decreases as switching is accelerated, converging from the faster state-specific growth rate to the occupancy average. A worked example shows that the occupancy average can predict clearance while the actual population grows.\n\nCandidate contribution (theorem; novelty confidence low): Candidate bridge theorem: in the constant-rate two-state signaling model, the tumor-growth exponent is the isolated-cell occupancy average plus (sqrt(C^2+4V)-C)/2, a strictly positive lineage-selection correction when both states communicate and have unequal causal growth rates; holding occupancy fixed while varying switching speed yields a monotone, falsifiable growth prediction." }, { "id": 20000848, "problem_number": "AIM-BIOLOGY-0013", "title": "A blinded finite-portfolio audit for preclinical-to-human prediction", "statement": "Translating pre-clinical models to human", "original_statement": "Translating pre-clinical models to human", "clean_statement": "Translating pre-clinical models to human", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 1.4, **“Translating pre-clinical models to human,”** in the workshop *Systems approaches to drug discovery and development in oncology*. Its accompanying questions are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Top four problem areas\nSource item: 1.4\nSource URL: http://aimpl.org/systemsoncology/1/\nCanonical location: aim-biology-notes.json notes[12]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Translating pre-clinical models to human\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"How do we test and interpret predictability of pre-clinical animal models? How might an industrial consortium conduct blind pre-clinical studies on marketed and failed drugs?\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/1/", "tags": [ "aim", "AIM-BIOLOGY-0013", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a registered finite portfolio of drug-indication pairs, a retrospective collection selected for preclinical-human concordance can have perfect apparent Brier loss, calibration, accuracy, and AUC even when the predictor is independent of the human target in the full portfolio. If only fraction q is audited, bounded portfolio loss is sharply identified only within [q times observed loss, q times observed loss plus 1-q]. In contrast, when each pair has a positive known inclusion probability fixed before unblinding and the prediction pipeline is frozen, the inverse-probability Horvitz-Thompson loss is design-unbiased for finite-portfolio risk, with an explicit unbiased variance estimator under independent inclusion. A separate conditional affine transport envelope gives sharp human-effect and threshold-decision bounds, preventing statistical prediction from being mistaken for a mechanistic transport guarantee.\n\nCandidate contribution (theorem; novelty confidence low): Candidate selection-coverage-audit certificate for blinded preclinical oncology benchmarking: combine an explicit concordance-selection counterexample that manufactures perfect apparent calibration and discrimination, a sharp bounded-loss interval for incomplete portfolio coverage, and a probability-sampled outcome-blind finite-portfolio estimator with an unbiased design variance estimate; couple these to a distinct sharp transport envelope rather than treating retrospective agreement as mechanistic validity." }, { "id": 20000849, "problem_number": "AIM-BIOLOGY-0014", "title": "A feedback-aware clearance threshold for metabolism and signaling", "statement": "The role of cellular metabolism in cancer", "original_statement": "The role of cellular metabolism in cancer", "clean_statement": "The role of cellular metabolism in cancer", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 2.1, **“The role of cellular metabolism in cancer,”** from the workshop *Systems approaches to drug discovery and development in oncology*. The record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Other biology-centric problem areas\nSource item: 2.1\nSource URL: http://aimpl.org/systemsoncology/2/\nCanonical location: aim-biology-notes.json notes[13]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The role of cellular metabolism in cancer\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Main questions:\\n\\nWhat is the connection between cancer cell metabolism and growth factor signaling? Is there potential for drugs targeting cell metabolism or metabolism and signaling simultaneously?\\n\\nNotes:\\n\\n(a) There exist strict diet regimens that can be used to affect patients' metabolic activities, which have beneficial anti-cancer outcomes but are very difficult to manage. This nutrition approach is currently in use for epilepsy to control seizures. Nutrition as a cancer therapy, or therapy booster.\\n\\n(b) For some context about the current questions in cancer metabolism, see the agenda from the Keystone Symposium on 'Cancer and Metabolism': http://www.keystonesymposia.org/meetings/viewMeetings.cfm?MeetingID=1136\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/2/", "tags": [ "aim", "AIM-BIOLOGY-0014", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a positive two-node signaling-metabolism feedback model with effective deactivation rates A = mu + u and B = nu + v, the quasi-steady tumor exponent is negative exactly when H(A,B) = delta A B - p a B - q c A - (delta beta gamma + p beta c + q gamma a) is positive. This yields exact, possibly degenerate sanctuary requirements A > p a/delta and B > q c/delta, the boundary Bcrit(A) = (q c A + C)/(delta A - p a), and an if-and-only-if feasibility test H(mu + U, nu + V) > 0 under tolerated effective-rate caps. With Kb = q c p a + delta C, Bcrit strictly decreases and has a lower-end vertical asymptote only when Kb > 0; when Kb = 0 it is identically q c/delta. A linked snapshot lemma and saturable-uptake construction give the corresponding identifiability warnings.\n\nCandidate contribution (theorem; novelty confidence low): Candidate feedback-aware clearance theorem: the two-node model has an exact clearance isobole with possibly degenerate dual sanctuary thresholds, a complete Kb-positive versus Kb-zero boundary classification, and an admissible treatment inside rectangular tolerated effective-rate caps if and only if the maximally tolerated pair makes H positive; the threshold is paired with exact snapshot and diet-to-flux counterexamples." }, { "id": 20000850, "problem_number": "AIM-BIOLOGY-0015", "title": "Geometry-specific transport thresholds for 2D and 3D tumor assays", "statement": "Spatial effects on cancer cells", "original_statement": "Spatial effects on cancer cells", "clean_statement": "Spatial effects on cancer cells", "statement_status": "exact", "statement_verification": "The canonical AIM record is “Spatial effects on cancer cells,” problem 2.2 in the “Other biology-centric problem areas” from the workshop *Systems approaches to drug discovery and development in oncology*. Its exact main questions are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Other biology-centric problem areas\nSource item: 2.2\nSource URL: http://aimpl.org/systemsoncology/2/\nCanonical location: aim-biology-notes.json notes[14]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Spatial effects on cancer cells\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Main questions:\\n\\nWhat are the pros and cons of 2D vs. 3D experiments and modeling? Is one or both necessary? What are the effects of mechanical stress, or of a tumor's necrotic core?\\n\\nNotes:\\n\\n(a) Notion from the literature that microenvironment can be stressful on tumors, and possible connections with why some therapies might be more effective in a 3D environment - are cells in a 3D environment under more stress (e.g., undergoing necrosis) that makes them more sensitive to drug?\\n\\n(b) There are studies emerging from the lab of Doug Lauffenburger (MIT) comparing cell behaviors in 2D vs. 3D environments (e.g., Hughes-Alford and Lauffenburger, 2012).\\n\\n(c) 2D vs. 3D cell behaviors are also likely cell line dependent.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/2/", "tags": [ "aim", "AIM-BIOLOGY-0015", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a homogeneous constant-consumption spherical spheroid, necrosis begins at radius sqrt(6 D_O (c_b-c_N)/q), and the post-onset nonconsuming core satisfies an exact unique free-boundary equation. Matching the sphere's pre-necrotic central oxygen concentration with a planar slab requires half-thickness R/sqrt(3). In contrast, a linearly depleted drug with inverse penetration length kappa requires the agent-specific half-thickness kappa^{-1} arcosh(sinh(kappa R)/(kappa R)), whose ratio to R changes from 1/sqrt(3) in weak depletion toward 1 in strong depletion. Thus no molecule-independent effective planar thickness can preserve central exposure across oxygen and drugs. A separate normalization proposition shows why a necrotic core can change reported potency without changing killing of viable cells.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate geometry-kinetics obstruction: the planar thickness that center-matches a spherical tumor is R/sqrt(3) for zero-order oxygen consumption but is kappa^{-1} arcosh(sinh(kappa R)/(kappa R)) for a first-order depleted agent, so a single 2D-to-3D effective thickness cannot preserve exposure across molecules with different uptake strengths." }, { "id": 20000851, "problem_number": "AIM-BIOLOGY-0016", "title": "A size-phase audit for receptor, signaling, and drug-response variation", "statement": "Cell cycle-dependent variation in tumor cells.", "original_statement": "Cell cycle-dependent variation in tumor cells.", "clean_statement": "Cell cycle-dependent variation in tumor cells.", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 2.3, **“Cell cycle-dependent variation in tumor cells.”** Its full accompanying text asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Other biology-centric problem areas\nSource item: 2.3\nSource URL: http://aimpl.org/systemsoncology/2/\nCanonical location: aim-biology-notes.json notes[15]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Cell cycle-dependent variation in tumor cells.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Main questions:\\n\\nHow might such cell cycle-specific changes affect signaling activity and therapeutic efficacy? How does cell receptor number vary with cell cycle?\\n\\nNotes:\\n\\n(a) There are published studies about the variability in growth factor receptor expression as a function of stage in the cell cycle. For example, see Dong et al. (1991), or Urdiales et al. (1998).\\n\\n(b) There is quite a bit of variability in receptor expression per cell.\\n\\n(c) How much of observed cell-cell variability in protein expression is due to variation in cell size?\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/2/", "tags": [ "aim", "AIM-BIOLOGY-0016", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A receptor-versus-cell-cycle study is not identified by phase means alone. The exact identity E[R|C]/E[A|C] = E[D|C] + Cov(A,D|C)/E[A|C] shows that mean count and mean area do not determine mean surface density, with an explicit two-cell counterexample. For paired data, a two-input Shapley allocation gives a nonnegative, exhaustive predictive partition of receptor variance between correlated size and phase plus residual variation. A gate-confusion matrix identifies true phase-specific means exactly when it has full column rank under nondifferential error. In a deterministic balanced asynchronous population, phase weights follow the stable age density 2 lambda exp(-lambda a), not duration fractions; using the wrong phase mixture changes short-pulse efficacy by at most total variation times the range of phase-conditioned survival, and this bound is sharp. A curvature bound separately quantifies the error of predicting nonlinear signaling from mean density alone.\n\nCandidate contribution (identifiability framework; novelty confidence low): Candidate paired size-phase audit certificate: for a receptor and therapy fixed in advance, jointly require the paired density correction, the nonnegative order-averaged size/phase variance fractions, the phase-gate rank diagnostic, and the stable-age composition correction with its sharp short-pulse response bound before interpreting receptor variation as cell-cycle-specific." }, { "id": 20000852, "problem_number": "AIM-BIOLOGY-0017", "title": "Landmark selection can reverse metastatic and adjuvant drug rankings", "statement": "Accounting for and modeling metastasis", "original_statement": "Accounting for and modeling metastasis", "clean_statement": "Accounting for and modeling metastasis", "statement_status": "exact", "statement_verification": "The canonical record is AIM-BIOLOGY-0017, source file `aim-biology-notes.json`, zero-based source index 16, from the AIM workshop *Systems approaches to drug discovery and development in oncology*. Its short problem is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Other modeling-centric problem areas\nSource item: 3.1\nSource URL: http://aimpl.org/systemsoncology/3/\nCanonical location: aim-biology-notes.json notes[16]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Accounting for and modeling metastasis\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Main questions:\\n\\nHow do we consider modeling the metastatic tumor cells as opposed to the primary tumor cells, and/or the transition from primary to metastatic cells (known as the epithelial-mesenchymal transition (EMT))?\\n\\nNotes:\\n\\n(a) Issue of understanding what about the primary tumor will cause it to spread, versus studying already-spread metastatic tumor cells.\\n\\n(b) Making sure we are testing and developing models for the relevant state, if we give drugs to patients who already have metastatic cancer.\\n\\n(c) Clinical trials likely occur in metastatic patients, but if that drug is successful, it will likely be used on non-metastatic patients. Does that affect drug effectiveness?\\n\\n(d) For an example of a systems approach to studying the epithelial-mesenchymal transition, see Kim et al. (2011).\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/3/", "tags": [ "aim", "AIM-BIOLOGY-0017", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A stage-resolved metastasis model must separate primary growth and epithelial-mesenchymal plasticity, dissemination, occult survival/dormancy and colonization, and response of established lesions. In a proved two-phenotype landmark reduction, early metastatic cases weight phenotype i by 1-exp(-lambda_i tau), while clinically negative patients weight it by exp(-lambda_i tau). These opposite weights can reverse two drug rankings even when phenotype-specific intrinsic activity is assumed to transfer perfectly between occult and established disease. The same formulas prove that the metastatic phenotype mixture does not identify the landmark-negative occult reservoir, and an explicit competing-death formula distinguishes metastasis incidence from overall survival.\n\nCandidate contribution (obstruction; novelty confidence low): Candidate landmark-selection transfer obstruction: conditioning on clinical metastasis by a landmark and conditioning on remaining clinically negative generate opposite time-to-detection weights, which can reverse therapy rankings under perfect assumed transfer of phenotype-specific activity; moreover, fixed metastatic mixture odds are compatible with occult-reservoir odds approaching zero or infinity." }, { "id": 20000853, "problem_number": "AIM-BIOLOGY-0018", "title": "Nuisance-adjusted intervention criteria for testing biological model structures", "statement": "Uncertainty in model structure", "original_statement": "Uncertainty in model structure", "clean_statement": "Uncertainty in model structure", "statement_status": "exact", "statement_verification": "The record points to Ciaccio et al. (2010) and Morris et al. (2011) as examples of methods for exploring structures. The first reference can be identified unambiguously as the microwestern-array study of EGF-receptor signaling [Ciaccio2010]. The year, topic, and workshop context identify the second as the constrained-fuzzy-logic study of inflammatory signaling [Morris2011]. No corruption of the short source statement was found.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Other modeling-centric problem areas\nSource item: 3.2\nSource URL: http://aimpl.org/systemsoncology/3/\nCanonical location: aim-biology-notes.json notes[17]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Uncertainty in model structure\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"What kind of methods/experiments are available, or should be available, to efficiently test model structures?\\n\\nNotes:\\n\\n(a) For examples of methods to explore model structures, see Ciaccio et al. (2010) and Morris et al. (2011).\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/3/", "tags": [ "aim", "AIM-BIOLOGY-0018", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For two locally affine, whitened prediction families S_i(E) = b_i(E) + col X_i(E), their nuisance-adjusted separation is Delta_E = ||(I - A_E A_E^dagger)d_E||, where A_E = [X_0(E), -X_1(E)] and d_E = b_1(E) - b_0(E); noiseless structural discrimination is equivalent to an augmented-rank increase, and nearest-family classification is uniformly correct under bounded error ||epsilon|| <= eta when Delta_E > 2 eta, with the factor two sharp. An added intervention strictly improves separation exactly when no nuisance contrast minimizing the old discrepancy also fits the new block. A separate state-space similarity proposition proves that passive input-output data cannot identify labeled hidden mechanisms beyond admissible realization equivalence.\n\nCandidate contribution (theorem; novelty confidence low): Candidate sequential intervention theorem: for affine localizations of rival biological structures with shared and experiment-specific nuisance effects, a new experiment improves worst-case whitened separation strictly if and only if it destroys every old best-fitting nuisance contrast; pairing this equivalence-set test with the sharp Delta > 2 eta bounded-error certificate yields a concrete experiment-screening and stopping rule." }, { "id": 20000854, "problem_number": "AIM-BIOLOGY-0019", "title": "A coordinate-aware audit for uncertain model parameters", "statement": "Uncertainty in model parameters", "original_statement": "Uncertainty in model parameters", "clean_statement": "Uncertainty in model parameters", "statement_status": "exact", "statement_verification": "The canonical AIM record comes from the workshop *Systems approaches to drug discovery and development in oncology*, section “Other modeling-centric problem areas,” problem 3.3. Its title is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Other modeling-centric problem areas\nSource item: 3.3\nSource URL: http://aimpl.org/systemsoncology/3/\nCanonical location: aim-biology-notes.json notes[18]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Uncertainty in model parameters\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Main questions:\\n\\nCan we create a 'best practices' approach for estimating model parameters? Can we better define local vs. global parameter fits?\\n\\nNotes:\\n\\n(a) What do people mean by 'global' (e.g., global sensitivity analysis), given that there are not well established/widely accepted methods for this purpose.\\n\\n(b) If we are doing parameter estimation, how do we efficiently crawl through the search space, assuming the problem is not convex.\\n\\n(c) This topic was partially discussed in a break out group. They thought that a balance must be established between simple and understandable methods, and methods that are easily parallelizable. For example, Kalman filtering is not parallelizable, but particle filtering is parallelizable, even though it is more complicated to do Particle filtering. Thus it may be useful to look for tools that are inherently parallelizable.\\n\\n(d) There are methods that can be useful for moderately sized problems, but fail for higher dimensional problems.\\n\\n(e) Must work with experimentalists to get a solid understanding of what are valid and reasonable estimates of parameter values a priori.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/3/", "tags": [ "aim", "AIM-BIOLOGY-0019", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A rigorous certificate ladder separates local curvature, global optimization, global sensitivity, and multistart coverage. At a stationary fit, generalized Hessian eigenvalues relative to a transported biological scale metric are invariant under smooth reparameterization, unlike raw Hessian eigenvalues. For a pilot-frozen incumbent and N fresh independent starts from a declared measure, zero delta-better endpoints gives the exact upper confidence bound p_delta <= 1 - alpha^(1/N) on the measure of algorithm-accessible better basins; this bound is coordinate invariant only when the measure and optimizer are transported together. A smooth bump counterexample proves that no finite convergence log can certify a global optimum without additional assumptions, while a Lipschitz grid bound supplies a genuine approximate certificate at exponential dimension cost.\n\nCandidate contribution (theorem; novelty confidence low): Candidate two-stage multistart audit: freeze an incumbent from pilot fits, use independent validation starts to report an exact zero-improvement bound on the declared start-measure mass of algorithm-accessible delta-better basins, require joint pushforward of start measure and optimizer for coordinate invariance, and state the finite-query impossibility result alongside the bound so it cannot be misreported as global optimality." }, { "id": 20000855, "problem_number": "AIM-BIOLOGY-0020", "title": "Bounded disclosure delay, participation, and milestone thresholds", "statement": "Challenges: Time lines, focus, intellectual property", "original_statement": "Challenges: Time lines, focus, intellectual property", "clean_statement": "Challenges: Time lines, focus, intellectual property", "statement_status": "exact", "statement_verification": "The canonical record is AIM-BIOLOGY-0020, source file aim-biology-notes.json, zero-based source index 19, from the AIM workshop *Systems approaches to drug discovery and development in oncology*. Its complete problem field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Mechanisms of industrial-academic collaborations\nSource item: 4.1\nSource URL: http://aimpl.org/systemsoncology/4/\nCanonical location: aim-biology-notes.json notes[19]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Challenges: Time lines, focus, intellectual property\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/4/", "tags": [ "aim", "AIM-BIOLOGY-0020", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source item is a discussion heading, not a formal unresolved problem. For a clearly labeled surrogate model, joint surplus has the unique optimum x*=min{1,b/a} and d*=min{dbar,[log(Vk/c)]_+/k}, while bilateral participation is possible exactly for transfers between the academic cost and sponsor gross-value endpoints. In the corrected milestone model, with statewise intrinsic relative utilities h_theta=u_theta(M)-u_theta(R) and fixed payment difference Delta=t_M-t_R, the desired strict choices M in state M and R in state R are uniquely implemented exactly when -h_Mh_R; non-implementation is conditional rather than universal, and all endpoint and equality cases are characterized.\n\nCandidate contribution (threshold characterization; novelty confidence low): Candidate novelty: a cap-compensate-threshold audit pairing the closed-form optimal bounded disclosure delay, the exact bilateral participation-transfer interval, and the necessary-and-sufficient fixed-milestone incentive interval (-h_M,-h_R), including restricted feasible payment sets and every tie boundary.", "research_difficulty_suggested": "L2" }, { "id": 20000856, "problem_number": "AIM-BIOLOGY-0021", "title": "A robust allocation certificate for student research clinics", "statement": "Opportunities: Student clinics, Research Experience for Undergraduates (REU) programs, internships", "original_statement": "Opportunities: Student clinics, Research Experience for Undergraduates (REU) programs, internships", "clean_statement": "Opportunities: Student clinics, Research Experience for Undergraduates (REU) programs, internships", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Mechanisms of industrial-academic collaborations\nSource item: 4.2\nSource URL: http://aimpl.org/systemsoncology/4/\nCanonical location: aim-biology-notes.json notes[20]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Opportunities: Student clinics, Research Experience for Undergraduates (REU) programs, internships\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/4/", "tags": [ "aim", "AIM-BIOLOGY-0021", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source record is an opportunity label under mechanisms of industrial-academic collaboration, not a posed mathematical or empirical problem. As a useful formalization, this attempt proves an exact robust allocation theorem for a fixed set of clinic, REU, or internship projects. If project p must retain b_p students after any a student withdrawals, its inflated initial demand is d_p=b_p+a. A robust allocation exists exactly when every inflated demand fits the project cap, every preassigned mentor can carry the sum of demands, and every project subset Q has at least the sum of its demands in collectively eligible students. The proof uses project-slot cloning and Hall's theorem. A positive Hall deficit is a quantitative infeasibility certificate; a two-project example disproves naive first-come assignment; and a binomial-tail plus union bound quantifies continuity risk under explicitly independent attrition.\n\nCandidate contribution (theorem; novelty confidence low): Candidate robust clinic certificate: before offers, translate a declared tolerance of a withdrawals into project demands b_p+a, either publish a matching satisfying all project-subset Hall inequalities and mentor caps or publish an explicit positive Hall-deficit subset, and optimize preferences only after feasibility; accompany this worst-case certificate with a separately labeled independent-attrition sensitivity bound.", "research_difficulty_suggested": "L2" }, { "id": 20000857, "problem_number": "AIM-BIOLOGY-0022", "title": "A dated funding pointer and a staged-funding decision rule", "statement": "Funding for industry/academic interface: US National Institutes of Health (1) R01 \"Interface of the Life and Physical Sciences\" grants (PAR-10-141, PAR-10-142), (2) Small Business Innovation research (SBIR) and Small Business Technology Transfer (STTR) grants", "original_statement": "Funding for industry/academic interface: US National Institutes of Health (1) R01 \"Interface of the Life and Physical Sciences\" grants (PAR-10-141, PAR-10-142), (2) Small Business Innovation research (SBIR) and Small Business Technology Transfer (STTR) grants", "clean_statement": "Funding for industry/academic interface: US National Institutes of Health (1) R01 \"Interface of the Life and Physical Sciences\" grants (PAR-10-141, PAR-10-142), (2) Small Business Innovation research (SBIR) and Small Business Technology Transfer (STTR) grants", "statement_status": "exact", "statement_verification": "The exact corpus record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Biology\nWorkshop: Systems approaches to drug discovery and development in oncology\nSection: Mechanisms of industrial-academic collaborations\nSource item: 4.3\nSource URL: http://aimpl.org/systemsoncology/4/\nCanonical location: aim-biology-notes.json notes[21]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Funding for industry/academic interface: US National Institutes of Health (1) R01 \\\"Interface of the Life and Physical Sciences\\\" grants (PAR-10-141, PAR-10-142), (2) Small Business Innovation research (SBIR) and Small Business Technology Transfer (STTR) grants\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "http://aimpl.org/systemsoncology/4/", "tags": [ "aim", "AIM-BIOLOGY-0022", "aim-domain:biology", "aim-workshop:systemsoncology", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The corpus entry is a dated funding-resource list rather than an open research problem: PAR-10-141 and PAR-10-142 both expired on May 19, 2012; after the April 13, 2026 SBIR/STTR reauthorization, NIH posted PA-27-100 and PA-27-102 with an August 5, 2026 open date, so on the July 29 run date they were posted but not yet open for submission. A proved decision rule combines a verified notice-and-eligibility gate, review-delay/runway survival, and the option to stop after a first-stage signal; its exact advantage over forced continuation is the discounted expected negative part of later-stage value.\n\nCandidate contribution (decision-rule proposition; novelty confidence low): For an industry-academic funding route with feasibility gate a, cost c, award event F, delay T, runway tau, first-stage value A, and learned continuation value X, the staged score is -c+a E[1_F exp(-rT)1_{T<=tau}(A+X_+)]; its advantage over forced continuation is exactly a E[1_F exp(-rT)1_{T<=tau}X_-], and an expired or not-yet-open notice fails the gate before any success-probability calculation.", "research_difficulty_suggested": "L2" }, { "id": 20000858, "problem_number": "AIM-BIOLOGY-0023", "title": "What aggregate V1 suppression cannot identify", "statement": "Chapter A: How does human vision make good perceptual guesses about objects?\n\nDaniel Kersten Talk Summary\n\nThe speaker provided an explanation of how human vision makes good perceptual guesses about objects using bayesian influence graphs. To this end, he decomposed an object S and its image I into world/object properties and features.The resulting probability distribution was then represented using a graph where nodes represent random variables and links represent the influences. He then illustrated this approach using examples of discounting and cue integration. In the second half of his talk the speaker concentrated on V1 and LOC mechanisms involved in perception. His experiments revealed that perceptual organization correlated with reduced V1 activity and increases LOC activity. He showed that V1 activity can predict percept on the time-scale of behaviour. The decrease in V1 activity could mean two things 1) Predictive Coding 2) Sparsification. In predictive Coding high-level object models project back predictions of the incoming data. In this case a good fit implies a low activity at the lower areas due to subtraction. (The \"shut-up\" theory). In sparsification, a good high-level fit tells the lower areas to 'stop gossiping\". This essentially amplifies the activity for features belong to the object and suppress the rest. Since predictive coding and sparsification have the same empirical fMRI observation, the experiments were inconclusive in deciding which of the two mechanisms (shut-up or stop gossiping) reduces the V1 activity when a higher level (LOC) has a good explanation for an object.\n\nQuestions During the Presentation:\n\n• Bill: Could you explain the term Discounting?\n\n• Ans: Discounting is throwing away information that is not required for perception. For eg: lighting\n\n• Jeff: (Regarding shape cues explaining away pinkishness as white paper and pink light)Are these things learnt?\n\n• Ans: Don't know yet. We are in the process of esigning and experiment to find that out.\n\nDiscussion after the Presentation:\n\n• Grossberg: In non-stationary environments priors don't exist. Also the bayesian framework breaks apart for some Kanisza illusions.\n\n• Bruno: Nothing in the Bayesian framework is inconsistent with Kanizsa. Bayesian framework is taking out the Kanizsa illusions and explaining those. The correct choice of priors is the trick.\n\n• Dan: For eg: one prior for natural vision is that motion tends to be slow.\n\n• Bill Softky: You showed us data showing suppression of activity in V1. Is there any data for suppression of thalamus?\n\n• Dan: In an fMRI experiment, localizing the thalamus and especially the LGN is not easy. Hence we don't have any data for that.\n\n• This was followed by Steve explaining a Kanizsa illusion. 4", "original_statement": "Chapter A: How does human vision make good perceptual guesses about objects? \n\nDaniel Kersten Talk Summary \n\nThe speaker provided an explanation of how human vision makes good perceptual guesses about objects using bayesian influence graphs. To this end, he decomposed an object S and its image I into world/object properties and features.The resulting probability distribution was then represented using a graph where nodes represent random variables and links represent the influences. He then illustrated this approach using examples of discounting and cue integration. In the second half of his talk the speaker concentrated on V1 and LOC mechanisms involved in perception. His experiments revealed that perceptual organization correlated with reduced V1 activity and increases LOC activity. He showed that V1 activity can predict percept on the time-scale of behaviour. The decrease in V1 activity could mean two things 1) Predictive Coding 2) Sparsification. In predictive Coding high-level object models project back predictions of the incoming data. In this case a good fit implies a low activity at the lower areas due to subtraction. (The \"shut-up\" theory). In sparsification, a good high-level fit tells the lower areas to 'stop gossiping\". This essentially amplifies the activity for features belong to the object and suppress the rest. Since predictive coding and sparsification have the same empirical fMRI observation, the experiments were inconclusive in deciding which of the two mechanisms (shut-up or stop gossiping) reduces the V1 activity when a higher level (LOC) has a good explanation for an object. \n\nQuestions During the Presentation: \n\n• Bill: Could you explain the term Discounting? \n\n• Ans: Discounting is throwing away information that is not required for perception. For eg: lighting \n\n• Jeff: (Regarding shape cues explaining away pinkishness as white paper and pink light)Are these things learnt? \n\n• Ans: Don't know yet. We are in the process of esigning and experiment to find that out. \n\nDiscussion after the Presentation: \n\n• Grossberg: In non-stationary environments priors don't exist. Also the bayesian framework breaks apart for some Kanisza illusions. \n\n• Bruno: Nothing in the Bayesian framework is inconsistent with Kanizsa. Bayesian framework is taking out the Kanizsa illusions and explaining those. The correct choice of priors is the trick. \n\n• Dan: For eg: one prior for natural vision is that motion tends to be slow. \n\n• Bill Softky: You showed us data showing suppression of activity in V1. Is there any data for suppression of thalamus? \n\n• Dan: In an fMRI experiment, localizing the thalamus and especially the LGN is not easy. Hence we don't have any data for that. \n\n• This was followed by Steve explaining a Kanizsa illusion. 4", "clean_statement": "Chapter A: How does human vision make good perceptual guesses about objects?\n\nDaniel Kersten Talk Summary\n\nThe speaker provided an explanation of how human vision makes good perceptual guesses about objects using bayesian influence graphs. To this end, he decomposed an object S and its image I into world/object properties and features.The resulting probability distribution was then represented using a graph where nodes represent random variables and links represent the influences. He then illustrated this approach using examples of discounting and cue integration. In the second half of his talk the speaker concentrated on V1 and LOC mechanisms involved in perception. His experiments revealed that perceptual organization correlated with reduced V1 activity and increases LOC activity. He showed that V1 activity can predict percept on the time-scale of behaviour. The decrease in V1 activity could mean two things 1) Predictive Coding 2) Sparsification. In predictive Coding high-level object models project back predictions of the incoming data. In this case a good fit implies a low activity at the lower areas due to subtraction. (The \"shut-up\" theory). In sparsification, a good high-level fit tells the lower areas to 'stop gossiping\". This essentially amplifies the activity for features belong to the object and suppress the rest. Since predictive coding and sparsification have the same empirical fMRI observation, the experiments were inconclusive in deciding which of the two mechanisms (shut-up or stop gossiping) reduces the V1 activity when a higher level (LOC) has a good explanation for an object.\n\nQuestions During the Presentation:\n\n• Bill: Could you explain the term Discounting?\n\n• Ans: Discounting is throwing away information that is not required for perception. For eg: lighting\n\n• Jeff: (Regarding shape cues explaining away pinkishness as white paper and pink light)Are these things learnt?\n\n• Ans: Don't know yet. We are in the process of esigning and experiment to find that out.\n\nDiscussion after the Presentation:\n\n• Grossberg: In non-stationary environments priors don't exist. Also the bayesian framework breaks apart for some Kanisza illusions.\n\n• Bruno: Nothing in the Bayesian framework is inconsistent with Kanizsa. Bayesian framework is taking out the Kanizsa illusions and explaining those. The correct choice of priors is the trick.\n\n• Dan: For eg: one prior for natural vision is that motion tends to be slow.\n\n• Bill Softky: You showed us data showing suppression of activity in V1. Is there any data for suppression of thalamus?\n\n• Dan: In an fMRI experiment, localizing the thalamus and especially the LGN is not easy. Hence we don't have any data for that.\n\n• This was followed by Steve explaining a Kanizsa illusion. 4", "statement_status": "exact", "statement_verification": "The canonical record is Chapter A of the 2003 AIM workshop notes *Inference and Prediction in Neocortical Circuits*, a talk summary for Daniel Kersten [AIM2003]. The official PDF confirms the following central passage:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Inference and prediction in neocortical circuits\nSection: \nSource item: Chapter A\nSource URL: https://aimath.org/WWN/brain/brain.pdf\nCanonical location: aim-biology-notes.json notes[22]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"Chapter A: How does human vision make good perceptual guesses about objects? \\n\\nDaniel Kersten Talk Summary \\n\\nThe speaker provided an explanation of how human vision makes good perceptual guesses about objects using bayesian influence graphs. To this end, he decomposed an object S and its image I into world/object properties and features.The resulting probability distribution was then represented using a graph where nodes represent random variables and links represent the influences. He then illustrated this approach using examples of discounting and cue integration. In the second half of his talk the speaker concentrated on V1 and LOC mechanisms involved in perception. His experiments revealed that perceptual organization correlated with reduced V1 activity and increases LOC activity. He showed that V1 activity can predict percept on the time-scale of behaviour. The decrease in V1 activity could mean two things 1) Predictive Coding 2) Sparsification. In predictive Coding high-level object models project back predictions of the incoming data. In this case a good fit implies a low activity at the lower areas due to subtraction. (The \\\"shut-up\\\" theory). In sparsification, a good high-level fit tells the lower areas to 'stop gossiping\\\". This essentially amplifies the activity for features belong to the object and suppress the rest. Since predictive coding and sparsification have the same empirical fMRI observation, the experiments were inconclusive in deciding which of the two mechanisms (shut-up or stop gossiping) reduces the V1 activity when a higher level (LOC) has a good explanation for an object. \\n\\nQuestions During the Presentation: \\n\\n• Bill: Could you explain the term Discounting? \\n\\n• Ans: Discounting is throwing away information that is not required for perception. For eg: lighting \\n\\n• Jeff: (Regarding shape cues explaining away pinkishness as white paper and pink light)Are these things learnt? \\n\\n• Ans: Don't know yet. We are in the process of esigning and experiment to find that out. \\n\\nDiscussion after the Presentation: \\n\\n• Grossberg: In non-stationary environments priors don't exist. Also the bayesian framework breaks apart for some Kanisza illusions. \\n\\n• Bruno: Nothing in the Bayesian framework is inconsistent with Kanizsa. Bayesian framework is taking out the Kanizsa illusions and explaining those. The correct choice of priors is the trick. \\n\\n• Dan: For eg: one prior for natural vision is that motion tends to be slow. \\n\\n• Bill Softky: You showed us data showing suppression of activity in V1. Is there any data for suppression of thalamus? \\n\\n• Dan: In an fMRI experiment, localizing the thalamus and especially the LGN is not easy. Hence we don't have any data for that. \\n\\n• This was followed by Steve explaining a Kanizsa illusion. 4\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/brain/brain.pdf", "tags": [ "aim", "AIM-BIOLOGY-0023", "aim-domain:biology", "aim-workshop:brain", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The section record contains a genuine unresolved identification problem despite being a talk summary. Under a declared linearized observation map y=H Delta x, predictive-cancellation and sparsification implementations are observationally equivalent exactly when their latent contrast difference delta lies in ker H; a scalar regional mean leaves an (n-1)-dimensional ambiguity for an n-dimensional neural state. For a fixed ambiguous pair, one added scalar measurement is minimal and sufficient exactly when it is not orthogonal to delta. For a k-dimensional unresolved mechanism subspace, at least k independent scalar rows are necessary and sufficient. An explicit same-mean example gives opposite signs along an independently localized feature-tuning direction, a deterministic margin bound audits sign robustness, and a feedback-by-organization difference-in-differences turns the direction into a falsifiable experiment. This resolves the observation-design question under stated assumptions, not which biological mechanism operates in V1.\n\nCandidate contribution (identifiability theorem; novelty confidence low): Candidate observation-rank certificate for the AIM ambiguity: express both mechanisms in a common latent neural variable, state the measurement operator, verify that the current mechanistic difference lies in its null space, add the minimum measurement rows whose restriction to the prespecified ambiguity subspace is injective, certify the feature-alignment sign with an explicit error margin, and combine that contrast with a causal feedback perturbation." }, { "id": 20000859, "problem_number": "AIM-BIOLOGY-0024", "title": "A space-time exclusion certificate for visual-cortical route attribution", "statement": "Chapter B: Anatomical Substrates for Functional Responses of Neurons in the Primate Visual Cerebral Cortex\n\nAlessandra Angelucci\n\nAlessandra's talk addressed specific brain circuits and perception. Specifically, it re-ported results on the study \"How does the spatial scale of classical receptive fields and and surround modulation compare to the spatial scale of horizontal, feedforward and feedback connections? The study consisted of injecting tracers and correlating the connection spread to the receptive field size at various levels (V1,..., ). The speaker noted that receptive field sizes depend on the measurement technique. Typically encountered measurements of receptive fields are (1) the minimum response field is denoted by mRF, (2) the summation receptive field at high contract denoted by hsRF and (3) the summation receptive field at low contract denoted by hsRF. Her study showed that horizontal connections are commensurate with the low contrast sRF size of V1 cells and might mediate expansion of the sRF at low contrast and collinear facilitation effects. Moreover, the extent of feedback connections were found to be commen-surate with the whole range of V1 cells center surround field sizes.These connections might mediate center-surround interactions and global-to-local signal integration. She also reported the following; (1)Feedback connections are 10 times faster than hor-izontal connections, and as fast as feedorward connections. (2)Feedback connections are patchy (3)Feedback connections to V1 are specific (4)Horizontal and feedback connections arise from excitatory neurons (5)There is no such thing as \"long-range inhibitory connec-tions\" in visual cortex. (6)Horizontal and feedback axons contact excitatory (approx 80%) as well as inhibitory (approx 20%) neurons\n\nDiscussion:Question: Our impression was that feedback is diffused and not orientation specific? Ans: The results presented here say the opposite. It says that feeback is patchy and orientation specific. Qn: Connections from V1 to LGN? Ans: We don't have any data on that. Qn: Did you say that 98% of the feedback connections go to excitatory neurons? Ans: No, the division is 80/20", "original_statement": "Chapter B: Anatomical Substrates for Functional Responses of Neurons in the Primate Visual Cerebral Cortex \n\nAlessandra Angelucci \n\nAlessandra's talk addressed specific brain circuits and perception. Specifically, it re-ported results on the study \"How does the spatial scale of classical receptive fields and and surround modulation compare to the spatial scale of horizontal, feedforward and feedback connections? The study consisted of injecting tracers and correlating the connection spread to the receptive field size at various levels (V1,..., ). The speaker noted that receptive field sizes depend on the measurement technique. Typically encountered measurements of receptive fields are (1) the minimum response field is denoted by mRF, (2) the summation receptive field at high contract denoted by hsRF and (3) the summation receptive field at low contract denoted by hsRF. Her study showed that horizontal connections are commensurate with the low contrast sRF size of V1 cells and might mediate expansion of the sRF at low contrast and collinear facilitation effects. Moreover, the extent of feedback connections were found to be commen-surate with the whole range of V1 cells center surround field sizes.These connections might mediate center-surround interactions and global-to-local signal integration. She also reported the following; (1)Feedback connections are 10 times faster than hor-izontal connections, and as fast as feedorward connections. (2)Feedback connections are patchy (3)Feedback connections to V1 are specific (4)Horizontal and feedback connections arise from excitatory neurons (5)There is no such thing as \"long-range inhibitory connec-tions\" in visual cortex. (6)Horizontal and feedback axons contact excitatory (approx 80%) as well as inhibitory (approx 20%) neurons \n\nDiscussion:Question: Our impression was that feedback is diffused and not orientation specific? Ans: The results presented here say the opposite. It says that feeback is patchy and orientation specific. Qn: Connections from V1 to LGN? Ans: We don't have any data on that. Qn: Did you say that 98% of the feedback connections go to excitatory neurons? Ans: No, the division is 80/20", "clean_statement": "Chapter B: Anatomical Substrates for Functional Responses of Neurons in the Primate Visual Cerebral Cortex\n\nAlessandra Angelucci\n\nAlessandra's talk addressed specific brain circuits and perception. Specifically, it re-ported results on the study \"How does the spatial scale of classical receptive fields and and surround modulation compare to the spatial scale of horizontal, feedforward and feedback connections? The study consisted of injecting tracers and correlating the connection spread to the receptive field size at various levels (V1,..., ). The speaker noted that receptive field sizes depend on the measurement technique. Typically encountered measurements of receptive fields are (1) the minimum response field is denoted by mRF, (2) the summation receptive field at high contract denoted by hsRF and (3) the summation receptive field at low contract denoted by hsRF. Her study showed that horizontal connections are commensurate with the low contrast sRF size of V1 cells and might mediate expansion of the sRF at low contrast and collinear facilitation effects. Moreover, the extent of feedback connections were found to be commen-surate with the whole range of V1 cells center surround field sizes.These connections might mediate center-surround interactions and global-to-local signal integration. She also reported the following; (1)Feedback connections are 10 times faster than hor-izontal connections, and as fast as feedorward connections. (2)Feedback connections are patchy (3)Feedback connections to V1 are specific (4)Horizontal and feedback connections arise from excitatory neurons (5)There is no such thing as \"long-range inhibitory connec-tions\" in visual cortex. (6)Horizontal and feedback axons contact excitatory (approx 80%) as well as inhibitory (approx 20%) neurons\n\nDiscussion:Question: Our impression was that feedback is diffused and not orientation specific? Ans: The results presented here say the opposite. It says that feeback is patchy and orientation specific. Qn: Connections from V1 to LGN? Ans: We don't have any data on that. Qn: Did you say that 98% of the feedback connections go to excitatory neurons? Ans: No, the division is 80/20", "statement_status": "exact", "statement_verification": "This canonical record is tagged `section`, but it contains a genuine research question rather than only a heading. Chapter B of the AIM workshop notes summarizes Alessandra Angelucci's question:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Inference and prediction in neocortical circuits\nSection: \nSource item: Chapter B\nSource URL: https://aimath.org/WWN/brain/brain.pdf\nCanonical location: aim-biology-notes.json notes[23]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"Chapter B: Anatomical Substrates for Functional Responses of Neurons in the Primate Visual Cerebral Cortex \\n\\nAlessandra Angelucci \\n\\nAlessandra's talk addressed specific brain circuits and perception. Specifically, it re-ported results on the study \\\"How does the spatial scale of classical receptive fields and and surround modulation compare to the spatial scale of horizontal, feedforward and feedback connections? The study consisted of injecting tracers and correlating the connection spread to the receptive field size at various levels (V1,..., ). The speaker noted that receptive field sizes depend on the measurement technique. Typically encountered measurements of receptive fields are (1) the minimum response field is denoted by mRF, (2) the summation receptive field at high contract denoted by hsRF and (3) the summation receptive field at low contract denoted by hsRF. Her study showed that horizontal connections are commensurate with the low contrast sRF size of V1 cells and might mediate expansion of the sRF at low contrast and collinear facilitation effects. Moreover, the extent of feedback connections were found to be commen-surate with the whole range of V1 cells center surround field sizes.These connections might mediate center-surround interactions and global-to-local signal integration. She also reported the following; (1)Feedback connections are 10 times faster than hor-izontal connections, and as fast as feedorward connections. (2)Feedback connections are patchy (3)Feedback connections to V1 are specific (4)Horizontal and feedback connections arise from excitatory neurons (5)There is no such thing as \\\"long-range inhibitory connec-tions\\\" in visual cortex. (6)Horizontal and feedback axons contact excitatory (approx 80%) as well as inhibitory (approx 20%) neurons \\n\\nDiscussion:Question: Our impression was that feedback is diffused and not orientation specific? Ans: The results presented here say the opposite. It says that feeback is patchy and orientation specific. Qn: Connections from V1 to LGN? Ans: We don't have any data on that. Qn: Did you say that 98% of the feedback connections go to excitatory neurons? Ans: No, the division is 80/20\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/brain/brain.pdf", "tags": [ "aim", "AIM-BIOLOGY-0024", "aim-domain:biology", "aim-workshop:brain", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The section record contains a genuine route-identification question. The official AIM PDF itself duplicates hsRF, while the primary paper establishes that the intended third measurement is the low-contrast summation field. For a specified direct monosynaptic horizontal route, define a lower cortical-distance bound d_-, an upper one-way reach R_H^+, an upper velocity v_H^+, lower source and fixed-delay bounds, and an observed-onset interval. The route is rigorously excluded if d_->R_H^+ or if even the latest plausible observed onset precedes the earliest possible arrival t_src^-+tau_H^-+d_-/v_H^+. This is a necessary-condition test and cannot prove feedback. A second proved proposition shows that, around every nonnegative radial baseline having finite positive radial mass, finite radial third-weighted integral, and a positive local lower bound inside a sampling annulus, there are two nonnegative profiles with identical finitely sampled disk-summation responses and identical total radial mass but different normalized radial second-moment scales. Current causal literature supports feedback contributions to contextual modulation, but horizontal, feedback, feedforward, recurrent, and indirect routes remain condition-dependent and potentially concurrent.\n\nCandidate contribution (exclusion theorem and identifiability obstruction; novelty confidence low): Candidate favorable-bounds reach-latency certificate for the AIM anatomy-receptive-field comparison: convert declared visual separation to a lower cortical-distance bound, normalize all spread measures to one-way reach, and exclude a direct horizontal mechanism only when its full uncertainty set misses a reach-truncated latency cone; pair this with a constructive proof that finite disk-summation samples and total radial mass do not identify the normalized radial second-moment scale around any qualifying baseline." }, { "id": 20000860, "problem_number": "AIM-BIOLOGY-0025", "title": "A rate-delay abstraction certificate for filtered spiking circuits", "statement": "Chapter C: Breakthroughs in Brain Computing\n\nSteve Grossberg Discussion Following the Talk\n\nQ1: Why do you choose very specific kind of laminar circuits? What is the particular reason for ur circuits? Any predictions for us? Answer: I would like you to refer to my paper with Razada. There we stimulate in layer 6 and observe what happens in layer 4. The reason for Laminar architecture and how we came to it: We started working on grouping. What are the units of perception? To get at the grouping properties and analog coherence required certain ordering of mechanisms or else the data would collapse. Needed something which is robust and not parameter sensitive. We could see by analyzing the anatomy that it provided this required ordering of mechanisms in a compact way. We 5\n\ntried knocking down parts of it to see whether this breaks down and we found that the entire thing was necessary. What we basically have is a minimal mechanis for the required ordering property. Qn:Do you insist that the lateral connections add superlinearly? Ans: Not superlinearly but faster. Comment: The normalization property with just one inhibitory neron might not be sufficient for all. By variations of the simple circuit you can get a variety of effects from bipole to modulatory. Qn: What is your prediction for an experiment where there is a stimulus in the center and the flanks are detected? Qn: Does your model need spikes? Answer: No you don't need spikes. Spikes are important in some situations (self syncrhonizing nets, order preserving limit cycles) but not required for other. Qn: Why are spikes there? Ans: So that we have non attenuated signal transmission over long distance. Also with spikes, we can compensate for axonal delay by increasing the diameter of axons. Also I talked about a balance between excitation and inhibition. If these balances doesn't occur due to development then the system becomes to unstable. If the system is designed to 'stabilize' using spikes, then spikes become important. These systems show that they love to resynchronize these spikes for stability reasons. Qn: Does your model do computer vision tasks that are not done yet? For this the speaker answered with a list of computer vision tasks his algorithms have been employed on. Qn: The neurophysiological detail is that there are spikes. A: Why don't u put in all the channels? Jeff (Comment): Analogy between computers and quantum mechanics. Although the transistors operate based on principles of quantum mechanics, an understanding of quantum mechanics is not required to understand the working of computers. Speaker(Comment): My equations are mean values of stochastic differential equations. Speaker(Comment): An example where spikes are not important: Phonemic Restora-tion. BB noise colored by eel is on the wheel wagon peel orange heel is on the shoes. Pentti(Comment): Syncrhonous computing might not need spikes. Need not be the case for asynchronous operations.", "original_statement": "Chapter C: Breakthroughs in Brain Computing \n\nSteve Grossberg Discussion Following the Talk \n\nQ1: Why do you choose very specific kind of laminar circuits? What is the particular reason for ur circuits? Any predictions for us? Answer: I would like you to refer to my paper with Razada. There we stimulate in layer 6 and observe what happens in layer 4. The reason for Laminar architecture and how we came to it: We started working on grouping. What are the units of perception? To get at the grouping properties and analog coherence required certain ordering of mechanisms or else the data would collapse. Needed something which is robust and not parameter sensitive. We could see by analyzing the anatomy that it provided this required ordering of mechanisms in a compact way. We 5\n\ntried knocking down parts of it to see whether this breaks down and we found that the entire thing was necessary. What we basically have is a minimal mechanis for the required ordering property. Qn:Do you insist that the lateral connections add superlinearly? Ans: Not superlinearly but faster. Comment: The normalization property with just one inhibitory neron might not be sufficient for all. By variations of the simple circuit you can get a variety of effects from bipole to modulatory. Qn: What is your prediction for an experiment where there is a stimulus in the center and the flanks are detected? Qn: Does your model need spikes? Answer: No you don't need spikes. Spikes are important in some situations (self syncrhonizing nets, order preserving limit cycles) but not required for other. Qn: Why are spikes there? Ans: So that we have non attenuated signal transmission over long distance. Also with spikes, we can compensate for axonal delay by increasing the diameter of axons. Also I talked about a balance between excitation and inhibition. If these balances doesn't occur due to development then the system becomes to unstable. If the system is designed to 'stabilize' using spikes, then spikes become important. These systems show that they love to resynchronize these spikes for stability reasons. Qn: Does your model do computer vision tasks that are not done yet? For this the speaker answered with a list of computer vision tasks his algorithms have been employed on. Qn: The neurophysiological detail is that there are spikes. A: Why don't u put in all the channels? Jeff (Comment): Analogy between computers and quantum mechanics. Although the transistors operate based on principles of quantum mechanics, an understanding of quantum mechanics is not required to understand the working of computers. Speaker(Comment): My equations are mean values of stochastic differential equations. Speaker(Comment): An example where spikes are not important: Phonemic Restora-tion. BB noise colored by eel is on the wheel wagon peel orange heel is on the shoes. Pentti(Comment): Syncrhonous computing might not need spikes. Need not be the case for asynchronous operations.", "clean_statement": "Chapter C: Breakthroughs in Brain Computing\n\nSteve Grossberg Discussion Following the Talk\n\nQ1: Why do you choose very specific kind of laminar circuits? What is the particular reason for ur circuits? Any predictions for us? Answer: I would like you to refer to my paper with Razada. There we stimulate in layer 6 and observe what happens in layer 4. The reason for Laminar architecture and how we came to it: We started working on grouping. What are the units of perception? To get at the grouping properties and analog coherence required certain ordering of mechanisms or else the data would collapse. Needed something which is robust and not parameter sensitive. We could see by analyzing the anatomy that it provided this required ordering of mechanisms in a compact way. We 5\n\ntried knocking down parts of it to see whether this breaks down and we found that the entire thing was necessary. What we basically have is a minimal mechanis for the required ordering property. Qn:Do you insist that the lateral connections add superlinearly? Ans: Not superlinearly but faster. Comment: The normalization property with just one inhibitory neron might not be sufficient for all. By variations of the simple circuit you can get a variety of effects from bipole to modulatory. Qn: What is your prediction for an experiment where there is a stimulus in the center and the flanks are detected? Qn: Does your model need spikes? Answer: No you don't need spikes. Spikes are important in some situations (self syncrhonizing nets, order preserving limit cycles) but not required for other. Qn: Why are spikes there? Ans: So that we have non attenuated signal transmission over long distance. Also with spikes, we can compensate for axonal delay by increasing the diameter of axons. Also I talked about a balance between excitation and inhibition. If these balances doesn't occur due to development then the system becomes to unstable. If the system is designed to 'stabilize' using spikes, then spikes become important. These systems show that they love to resynchronize these spikes for stability reasons. Qn: Does your model do computer vision tasks that are not done yet? For this the speaker answered with a list of computer vision tasks his algorithms have been employed on. Qn: The neurophysiological detail is that there are spikes. A: Why don't u put in all the channels? Jeff (Comment): Analogy between computers and quantum mechanics. Although the transistors operate based on principles of quantum mechanics, an understanding of quantum mechanics is not required to understand the working of computers. Speaker(Comment): My equations are mean values of stochastic differential equations. Speaker(Comment): An example where spikes are not important: Phonemic Restora-tion. BB noise colored by eel is on the wheel wagon peel orange heel is on the shoes. Pentti(Comment): Syncrhonous computing might not need spikes. Need not be the case for asynchronous operations.", "statement_status": "exact", "statement_verification": "The canonical record is AIM-BIOLOGY-0025, zero-based index 24 of aim-biology-notes.json. It is tagged section and headed “Chapter C: Breakthroughs in Brain Computing.” The official source is the report of the AIM workshop *Inference and Prediction in Neocortical Circuits*, held 21--24 September 2003 [AIM2003]. The passage contains discussion after a talk by Steve Grossberg, not a stated open problem.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Inference and prediction in neocortical circuits\nSection: \nSource item: Chapter C\nSource URL: https://aimath.org/WWN/brain/brain.pdf\nCanonical location: aim-biology-notes.json notes[24]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"Chapter C: Breakthroughs in Brain Computing \\n\\nSteve Grossberg Discussion Following the Talk \\n\\nQ1: Why do you choose very specific kind of laminar circuits? What is the particular reason for ur circuits? Any predictions for us? Answer: I would like you to refer to my paper with Razada. There we stimulate in layer 6 and observe what happens in layer 4. The reason for Laminar architecture and how we came to it: We started working on grouping. What are the units of perception? To get at the grouping properties and analog coherence required certain ordering of mechanisms or else the data would collapse. Needed something which is robust and not parameter sensitive. We could see by analyzing the anatomy that it provided this required ordering of mechanisms in a compact way. We 5\\n\\ntried knocking down parts of it to see whether this breaks down and we found that the entire thing was necessary. What we basically have is a minimal mechanis for the required ordering property. Qn:Do you insist that the lateral connections add superlinearly? Ans: Not superlinearly but faster. Comment: The normalization property with just one inhibitory neron might not be sufficient for all. By variations of the simple circuit you can get a variety of effects from bipole to modulatory. Qn: What is your prediction for an experiment where there is a stimulus in the center and the flanks are detected? Qn: Does your model need spikes? Answer: No you don't need spikes. Spikes are important in some situations (self syncrhonizing nets, order preserving limit cycles) but not required for other. Qn: Why are spikes there? Ans: So that we have non attenuated signal transmission over long distance. Also with spikes, we can compensate for axonal delay by increasing the diameter of axons. Also I talked about a balance between excitation and inhibition. If these balances doesn't occur due to development then the system becomes to unstable. If the system is designed to 'stabilize' using spikes, then spikes become important. These systems show that they love to resynchronize these spikes for stability reasons. Qn: Does your model do computer vision tasks that are not done yet? For this the speaker answered with a list of computer vision tasks his algorithms have been employed on. Qn: The neurophysiological detail is that there are spikes. A: Why don't u put in all the channels? Jeff (Comment): Analogy between computers and quantum mechanics. Although the transistors operate based on principles of quantum mechanics, an understanding of quantum mechanics is not required to understand the working of computers. Speaker(Comment): My equations are mean values of stochastic differential equations. Speaker(Comment): An example where spikes are not important: Phonemic Restora-tion. BB noise colored by eel is on the wheel wagon peel orange heel is on the shoes. Pentti(Comment): Syncrhonous computing might not need spikes. Need not be the case for asynchronous operations.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/brain/brain.pdf", "tags": [ "aim", "AIM-BIOLOGY-0025", "aim-domain:biology", "aim-workshop:brain", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source is a 2003 discussion transcript rather than a formal problem. For a clearly stated surrogate, the report proves that the average of N independent Poisson spike trains after linear filtering has exactly the rate-filtered mean and variance N^{-1} times the single-train variance; for an exponential filter and rate bounded by R, the variance is at most R/(2 N tau_s), with a corresponding Lipschitz-readout bound, while a square nonlinearity has an exact positive variance bias. Separately, the scalar delayed negative-feedback rate equation T z'(t)=-z(t)-g z(t-D) is stable for every finite delay when 0<=g<=1, and for g>1 is stable exactly when D=2 distinct point centers and bounded circular displacement J>0, worst-case unique nearest-center decoding holds if and only if d_min>2J, implying K0 and g_I>0, a single holding-potential current map is locally noninjective along an explicit null direction; boundary pairs are not covered and may be uniquely pinned by nonnegativity in special sign/current cases. Two distinct ideal holding potentials yield a nonsingular two-conductance system under known-reversal and stationarity assumptions.\n\nCandidate contribution (theorem_and_identifiability_obstruction; novelty confidence low): Candidate novelty: a joint audit tailored to the AIM exchange proving a nonnegative train-versus-held-out peak-window optimism gap, identifying the exact estimand of the two-fold cross-fit, and coupling it to a spike-output obstruction plus local single-holding-potential E/I nonidentifiability at every interior feasible conductance pair, with an explicit two-holding-potential recovery condition.", "research_difficulty_suggested": "L2" }, { "id": 20000867, "problem_number": "AIM-BIOLOGY-0032", "title": "A robust target-offset test for late 3D pop-out signals", "statement": "Chapter J: Resonance Prediction and Priors\n\nTai Sing Lee\n\nQ: Can you tell us anything about the time course of the response between V1 and V2? A: This is hard to detect in our experiments. Q: Is there a connection between predictive ideas and pop out? A: Here top down predictions are more like a prior Qn: What is the attention effect in V2 neurons? A: The V2 neurons don't get attenuated even when you are attending to other locations. Q: Did you try turning off the target? If you turn the target off after 100ms. then probably you would see something interesting. A: We didn't try that. But this looks like an interesting experiment to do. Q: Did you do experiments where the monkey doesnt know where the pop out ball is? Q: The latency of the response to the popout seems to be very long to me comapred to the poput of bars. A: Here the pop out is working on 3D surfaces. Infering this 3D takes longer time. 11", "original_statement": "Chapter J: Resonance Prediction and Priors \n\nTai Sing Lee \n\nQ: Can you tell us anything about the time course of the response between V1 and V2? A: This is hard to detect in our experiments. Q: Is there a connection between predictive ideas and pop out? A: Here top down predictions are more like a prior Qn: What is the attention effect in V2 neurons? A: The V2 neurons don't get attenuated even when you are attending to other locations. Q: Did you try turning off the target? If you turn the target off after 100ms. then probably you would see something interesting. A: We didn't try that. But this looks like an interesting experiment to do. Q: Did you do experiments where the monkey doesnt know where the pop out ball is? Q: The latency of the response to the popout seems to be very long to me comapred to the poput of bars. A: Here the pop out is working on 3D surfaces. Infering this 3D takes longer time. 11", "clean_statement": "Chapter J: Resonance Prediction and Priors\n\nTai Sing Lee\n\nQ: Can you tell us anything about the time course of the response between V1 and V2? A: This is hard to detect in our experiments. Q: Is there a connection between predictive ideas and pop out? A: Here top down predictions are more like a prior Qn: What is the attention effect in V2 neurons? A: The V2 neurons don't get attenuated even when you are attending to other locations. Q: Did you try turning off the target? If you turn the target off after 100ms. then probably you would see something interesting. A: We didn't try that. But this looks like an interesting experiment to do. Q: Did you do experiments where the monkey doesnt know where the pop out ball is? Q: The latency of the response to the popout seems to be very long to me comapred to the poput of bars. A: Here the pop out is working on 3D surfaces. Infering this 3D takes longer time. 11", "statement_status": "exact", "statement_verification": "This canonical record is Chapter J of the discussion notes from the AIM workshop *Inference and Prediction in Neocortical Circuits*. It is tagged `section`, has no separate remarks or bibliography, and is not a formal problem statement. The official PDF was checked directly. On the tenth physical PDF page (the page carrying the printed footer 11), it reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Inference and prediction in neocortical circuits\nSection: \nSource item: Chapter J\nSource URL: https://aimath.org/WWN/brain/brain.pdf\nCanonical location: aim-biology-notes.json notes[31]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"Chapter J: Resonance Prediction and Priors \\n\\nTai Sing Lee \\n\\nQ: Can you tell us anything about the time course of the response between V1 and V2? A: This is hard to detect in our experiments. Q: Is there a connection between predictive ideas and pop out? A: Here top down predictions are more like a prior Qn: What is the attention effect in V2 neurons? A: The V2 neurons don't get attenuated even when you are attending to other locations. Q: Did you try turning off the target? If you turn the target off after 100ms. then probably you would see something interesting. A: We didn't try that. But this looks like an interesting experiment to do. Q: Did you do experiments where the monkey doesnt know where the pop out ball is? Q: The latency of the response to the popout seems to be very long to me comapred to the poput of bars. A: Here the pop out is working on 3D surfaces. Infering this 3D takes longer time. 11\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/brain/brain.pdf", "tags": [ "aim", "AIM-BIOLOGY-0032", "aim-domain:biology", "aim-workshop:brain", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The AIM record is a discussion transcript rather than a determinate open problem, and it is identifiable with high confidence as discussion of Lee et al.'s 2002 macaque shape-from-shading pop-out study. This attempt proves a transcript-specific experimental-design result: after an oddball target is replaced by a uniform distractor, a late V1 or V2 history contrast exceeding the sum of a simultaneous confidence radius and a certified transition-artifact bound rejects a declared bounded stimulus-memory null and yields an explicit lower bound on the residual history signal. Randomized, validated V2-to-V1 terminal perturbation then identifies only its causal effect on that contrast. Separate lemmas show that uncertain V1/V2 onset estimates cannot establish order unless their gap exceeds the sum of timing errors, that even certified order does not identify a causal arrow, that posterior-like response scores do not identify a prior-versus-prediction decomposition, and that a 3D-specific latency residual requires subtracting both timing and non-3D nuisance bounds.\n\nCandidate contribution (experimental_design_theorem; novelty confidence low): Candidate contribution: an explicit offset-perturbation certificate for the transcript's unperformed 100 ms experiment, combining oddball-to-uniform replacement, an exactly matched post-offset display, bounded transition artifacts, a declared memory and timing margin, simultaneous confidence bounds, and a randomized V2-to-V1 difference-in-differences with strictly one-way causal interpretation.", "research_difficulty_suggested": "L2" }, { "id": 20000868, "problem_number": "AIM-BIOLOGY-0033", "title": "A model-discrimination and reporting certificate for Breakout Session I", "statement": "Chapter K: Notes from Breakout Session I\n\nOn the second day of the workshop the participants broke out into four different groups based on their research oreintation. The groups were (1) Pyschology and Psychophysics, (2) Anatomists, and (3) Theorists. Due to the large number of theorists, the theory group was split into two (Theory 1 and Theory 2). The groups met separately for 45 minutes. Each group was supposed to come up with a list of three things that they want from the other groups. At the end of 45 minutes all the groups met at one place and a discussion ensued on the requirements. What follows is a compilation of suggestions from different groups.\n\nSuggestions From Psychology/Psychophysics group\n\nFor theorists\n\n• models that can act on real stimuli real images, moving images\n\n• Are there associative memory models running on the hardware that we have? (spiking models)\n\n• Theory of mid-level vision that is understandable to us. For anatomists\n\n• more quantitative descriptions organized into ciruits. proportions of neurons in one area compared to another area are they myelinated or not\n\n• More oranized quantitative information.\n\n• Timing\n\n• Thalamus\n\n• Anatomists comments on theory slides.\n\nFrom Theory Group I\n\nFor Anatomists\n\n• Naturalize Stimulii\n\n• Make the raw data available.\n\n• Fast-forward development of multi-electrode recordings.\n\n• Specify data on excitatory/inhibitory..which layer they are in. For Psychologists\n\n• Is the BOLD signal reactive or predictive? Need explanation.\n\n• Request for more mathematical/theoretical training.\n\n• Theoretically driven experiments.\n\n• Theories posited before data.\n\nFrom Anatomists\n\nFor Theorists\n\n• Falsifiable predictions crucial tests. - test on how bayesian inference occurs in brain. 12\n\n• Rigorous comparisons between models. - surface representation (biologically plausible) - Where does the model fail?\n\n• Different feedback systems. - predictions from these models. To Psychologists\n\n• Ken: What approach, what end point?\n\n• More natural stimulii.\n\nFrom Theory group II\n\nFor Both Anatomists and Psychologists\n\n• Better communication - theoretical training (academic curriculum) - tutorials or invited talks @ conferences.\n\n• Better Data A) centralized database, systematic connectivity diagram B) Reporting (availability/completeness) C) Natural stimulii/ multi-electrode recordings.\n\n• Hire us!!", "original_statement": "Chapter K: Notes from Breakout Session I \n\nOn the second day of the workshop the participants broke out into four different groups based on their research oreintation. The groups were (1) Pyschology and Psychophysics, (2) Anatomists, and (3) Theorists. Due to the large number of theorists, the theory group was split into two (Theory 1 and Theory 2). The groups met separately for 45 minutes. Each group was supposed to come up with a list of three things that they want from the other groups. At the end of 45 minutes all the groups met at one place and a discussion ensued on the requirements. What follows is a compilation of suggestions from different groups. \n\nSuggestions From Psychology/Psychophysics group \n\nFor theorists \n\n• models that can act on real stimuli real images, moving images \n\n• Are there associative memory models running on the hardware that we have? (spiking models) \n\n• Theory of mid-level vision that is understandable to us. For anatomists \n\n• more quantitative descriptions organized into ciruits. proportions of neurons in one area compared to another area are they myelinated or not \n\n• More oranized quantitative information. \n\n• Timing \n\n• Thalamus \n\n• Anatomists comments on theory slides. \n\nFrom Theory Group I \n\nFor Anatomists \n\n• Naturalize Stimulii \n\n• Make the raw data available. \n\n• Fast-forward development of multi-electrode recordings. \n\n• Specify data on excitatory/inhibitory..which layer they are in. For Psychologists \n\n• Is the BOLD signal reactive or predictive? Need explanation. \n\n• Request for more mathematical/theoretical training. \n\n• Theoretically driven experiments. \n\n• Theories posited before data. \n\nFrom Anatomists \n\nFor Theorists \n\n• Falsifiable predictions crucial tests. - test on how bayesian inference occurs in brain. 12 \n\n• Rigorous comparisons between models. - surface representation (biologically plausible) - Where does the model fail? \n\n• Different feedback systems. - predictions from these models. To Psychologists \n\n• Ken: What approach, what end point? \n\n• More natural stimulii. \n\nFrom Theory group II \n\nFor Both Anatomists and Psychologists \n\n• Better communication - theoretical training (academic curriculum) - tutorials or invited talks @ conferences. \n\n• Better Data A) centralized database, systematic connectivity diagram B) Reporting (availability/completeness) C) Natural stimulii/ multi-electrode recordings. \n\n• Hire us!!", "clean_statement": "Chapter K: Notes from Breakout Session I\n\nOn the second day of the workshop the participants broke out into four different groups based on their research oreintation. The groups were (1) Pyschology and Psychophysics, (2) Anatomists, and (3) Theorists. Due to the large number of theorists, the theory group was split into two (Theory 1 and Theory 2). The groups met separately for 45 minutes. Each group was supposed to come up with a list of three things that they want from the other groups. At the end of 45 minutes all the groups met at one place and a discussion ensued on the requirements. What follows is a compilation of suggestions from different groups.\n\nSuggestions From Psychology/Psychophysics group\n\nFor theorists\n\n• models that can act on real stimuli real images, moving images\n\n• Are there associative memory models running on the hardware that we have? (spiking models)\n\n• Theory of mid-level vision that is understandable to us. For anatomists\n\n• more quantitative descriptions organized into ciruits. proportions of neurons in one area compared to another area are they myelinated or not\n\n• More oranized quantitative information.\n\n• Timing\n\n• Thalamus\n\n• Anatomists comments on theory slides.\n\nFrom Theory Group I\n\nFor Anatomists\n\n• Naturalize Stimulii\n\n• Make the raw data available.\n\n• Fast-forward development of multi-electrode recordings.\n\n• Specify data on excitatory/inhibitory..which layer they are in. For Psychologists\n\n• Is the BOLD signal reactive or predictive? Need explanation.\n\n• Request for more mathematical/theoretical training.\n\n• Theoretically driven experiments.\n\n• Theories posited before data.\n\nFrom Anatomists\n\nFor Theorists\n\n• Falsifiable predictions crucial tests. - test on how bayesian inference occurs in brain. 12\n\n• Rigorous comparisons between models. - surface representation (biologically plausible) - Where does the model fail?\n\n• Different feedback systems. - predictions from these models. To Psychologists\n\n• Ken: What approach, what end point?\n\n• More natural stimulii.\n\nFrom Theory group II\n\nFor Both Anatomists and Psychologists\n\n• Better communication - theoretical training (academic curriculum) - tutorials or invited talks @ conferences.\n\n• Better Data A) centralized database, systematic connectivity diagram B) Reporting (availability/completeness) C) Natural stimulii/ multi-electrode recordings.\n\n• Hire us!!", "statement_status": "exact", "statement_verification": "This record is Chapter K, “Notes from Breakout Session I,” in the official AIM workshop notes *Inference and Prediction in Neocortical Circuits*, version dated 24 October 2003. It is a compilation of requests exchanged by workshop groups, not a single open problem.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Inference and prediction in neocortical circuits\nSection: \nSource item: Chapter K\nSource URL: https://aimath.org/WWN/brain/brain.pdf\nCanonical location: aim-biology-notes.json notes[32]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"Chapter K: Notes from Breakout Session I \\n\\nOn the second day of the workshop the participants broke out into four different groups based on their research oreintation. The groups were (1) Pyschology and Psychophysics, (2) Anatomists, and (3) Theorists. Due to the large number of theorists, the theory group was split into two (Theory 1 and Theory 2). The groups met separately for 45 minutes. Each group was supposed to come up with a list of three things that they want from the other groups. At the end of 45 minutes all the groups met at one place and a discussion ensued on the requirements. What follows is a compilation of suggestions from different groups. \\n\\nSuggestions From Psychology/Psychophysics group \\n\\nFor theorists \\n\\n• models that can act on real stimuli real images, moving images \\n\\n• Are there associative memory models running on the hardware that we have? (spiking models) \\n\\n• Theory of mid-level vision that is understandable to us. For anatomists \\n\\n• more quantitative descriptions organized into ciruits. proportions of neurons in one area compared to another area are they myelinated or not \\n\\n• More oranized quantitative information. \\n\\n• Timing \\n\\n• Thalamus \\n\\n• Anatomists comments on theory slides. \\n\\nFrom Theory Group I \\n\\nFor Anatomists \\n\\n• Naturalize Stimulii \\n\\n• Make the raw data available. \\n\\n• Fast-forward development of multi-electrode recordings. \\n\\n• Specify data on excitatory/inhibitory..which layer they are in. For Psychologists \\n\\n• Is the BOLD signal reactive or predictive? Need explanation. \\n\\n• Request for more mathematical/theoretical training. \\n\\n• Theoretically driven experiments. \\n\\n• Theories posited before data. \\n\\nFrom Anatomists \\n\\nFor Theorists \\n\\n• Falsifiable predictions crucial tests. - test on how bayesian inference occurs in brain. 12 \\n\\n• Rigorous comparisons between models. - surface representation (biologically plausible) - Where does the model fail? \\n\\n• Different feedback systems. - predictions from these models. To Psychologists \\n\\n• Ken: What approach, what end point? \\n\\n• More natural stimulii. \\n\\nFrom Theory group II \\n\\nFor Both Anatomists and Psychologists \\n\\n• Better communication - theoretical training (academic curriculum) - tutorials or invited talks @ conferences. \\n\\n• Better Data A) centralized database, systematic connectivity diagram B) Reporting (availability/completeness) C) Natural stimulii/ multi-electrode recordings. \\n\\n• Hire us!!\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/brain/brain.pdf", "tags": [ "aim", "AIM-BIOLOGY-0033", "aim-domain:biology", "aim-workshop:brain", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The record is a context-only list of interdisciplinary requests, not an open problem. For two fixed predictive models on finite raw observations, the minimum equal-prior classification error from a released summary is one half times one minus the summarized total-variation distance; deterministic summaries contract total variation, with equality exactly when the model-probability difference has constant sign in each summary fiber. A separate pure-delay construction shows that uncertain BOLD time and haemodynamic delay identify only a neural-time interval, so reactive versus predictive timing is justified only when that entire interval lies on one side of task time zero.\n\nCandidate contribution (identifiability_certificate; novelty confidence low): A report is sufficient for binary discrimination of the fixed model pair exactly when p minus q has no sign change inside any report fiber; paired with the feasible BOLD neural-time interval [b_L-d_max, b_U-d_min], this gives a concrete two-part audit of the breakout requests for rigorous model comparison, complete reporting, and reactive-versus-predictive BOLD interpretation.", "research_difficulty_suggested": "L2" }, { "id": 20000869, "problem_number": "AIM-BIOLOGY-0034", "title": "Shared-input model discrimination and recording-overlap identifiability", "statement": "Chapter L: Notes from Breakout Session II\n\nThe four groups met separately to prepare responses to the comments from the previous day.\n\nReply From Psychologists\n\nThe theorists wanted to be hired. The primate experiments would go much better if they did psychophysical experiments. Giving away raw data looks like giving away ones own work. Alessandra: Once something is published it is public domain and so the data should me bade public. Judith: It could be that you have one sort of analysis and more analysis might be waiting. you just don't want to give away the data. We should be able to publish the data\n\nFrom Theory group I.\n\n• Are there associative spiking models with spiking neurons? (Yes). Fritz will give a presentation on that.\n\n• Theoreticians will become aware of training in experimentation. This will give us more intuition. But we need more help in doing that. We would also like the data from you!\n\n• Russ: Rigorous model comparisons between different models. One can apply the same inputs to models and get the results. Should document when the models fail.\n\n• Falsifiable predictions?: We alread do.\n\n• The job of theorists is not just to make testable hypothesis. 13\n\nFrom Anatomists\n\n• Now more labs are producing less descriptive and more quantitative data. There is some kind of questions that take too much time to address. (For eg: number of synapses). How can computer science help us for this?\n\n• There are computer vision techniques and tools available that could be used. Multi-electrode recording? Its a lot of effort. If the cells cannot be activated at the same time, it amounts to recording one after another.\n\n• Natural stimuli. We have acquired multichannel recording systems and we are racing to do multi electrode recording.\n\nFrom Theory II\n\n• Bayesian Framework.: Bayesian framework is a principled way of looking at the computational problems faced by the brian given the ambiguous Priors can be implemented as lateral connectivity or synaptic weights or top-down predictions. Predictions for testing the bayesian framework: Rajesh Rao's work\n\n• Different forms of feedback: We really don't know how these forms of feedback are. The diffused feedback that is topographic could be doing an spatial priming and the patchy ones are doing feature priming.\n\n• Tony: Natural stimulii. Adaptive models: Natural stimulii. Yes. In the case of hardwired models: it will give tighter prediction to the experimentalists.", "original_statement": "Chapter L: Notes from Breakout Session II \n\nThe four groups met separately to prepare responses to the comments from the previous day. \n\nReply From Psychologists \n\nThe theorists wanted to be hired. The primate experiments would go much better if they did psychophysical experiments. Giving away raw data looks like giving away ones own work. Alessandra: Once something is published it is public domain and so the data should me bade public. Judith: It could be that you have one sort of analysis and more analysis might be waiting. you just don't want to give away the data. We should be able to publish the data \n\nFrom Theory group I. \n\n• Are there associative spiking models with spiking neurons? (Yes). Fritz will give a presentation on that. \n\n• Theoreticians will become aware of training in experimentation. This will give us more intuition. But we need more help in doing that. We would also like the data from you! \n\n• Russ: Rigorous model comparisons between different models. One can apply the same inputs to models and get the results. Should document when the models fail. \n\n• Falsifiable predictions?: We alread do. \n\n• The job of theorists is not just to make testable hypothesis. 13 \n\nFrom Anatomists \n\n• Now more labs are producing less descriptive and more quantitative data. There is some kind of questions that take too much time to address. (For eg: number of synapses). How can computer science help us for this? \n\n• There are computer vision techniques and tools available that could be used. Multi-electrode recording? Its a lot of effort. If the cells cannot be activated at the same time, it amounts to recording one after another. \n\n• Natural stimuli. We have acquired multichannel recording systems and we are racing to do multi electrode recording. \n\nFrom Theory II \n\n• Bayesian Framework.: Bayesian framework is a principled way of looking at the computational problems faced by the brian given the ambiguous Priors can be implemented as lateral connectivity or synaptic weights or top-down predictions. Predictions for testing the bayesian framework: Rajesh Rao's work \n\n• Different forms of feedback: We really don't know how these forms of feedback are. The diffused feedback that is topographic could be doing an spatial priming and the patchy ones are doing feature priming. \n\n• Tony: Natural stimulii. Adaptive models: Natural stimulii. Yes. In the case of hardwired models: it will give tighter prediction to the experimentalists.", "clean_statement": "Chapter L: Notes from Breakout Session II\n\nThe four groups met separately to prepare responses to the comments from the previous day.\n\nReply From Psychologists\n\nThe theorists wanted to be hired. The primate experiments would go much better if they did psychophysical experiments. Giving away raw data looks like giving away ones own work. Alessandra: Once something is published it is public domain and so the data should me bade public. Judith: It could be that you have one sort of analysis and more analysis might be waiting. you just don't want to give away the data. We should be able to publish the data\n\nFrom Theory group I.\n\n• Are there associative spiking models with spiking neurons? (Yes). Fritz will give a presentation on that.\n\n• Theoreticians will become aware of training in experimentation. This will give us more intuition. But we need more help in doing that. We would also like the data from you!\n\n• Russ: Rigorous model comparisons between different models. One can apply the same inputs to models and get the results. Should document when the models fail.\n\n• Falsifiable predictions?: We alread do.\n\n• The job of theorists is not just to make testable hypothesis. 13\n\nFrom Anatomists\n\n• Now more labs are producing less descriptive and more quantitative data. There is some kind of questions that take too much time to address. (For eg: number of synapses). How can computer science help us for this?\n\n• There are computer vision techniques and tools available that could be used. Multi-electrode recording? Its a lot of effort. If the cells cannot be activated at the same time, it amounts to recording one after another.\n\n• Natural stimuli. We have acquired multichannel recording systems and we are racing to do multi electrode recording.\n\nFrom Theory II\n\n• Bayesian Framework.: Bayesian framework is a principled way of looking at the computational problems faced by the brian given the ambiguous Priors can be implemented as lateral connectivity or synaptic weights or top-down predictions. Predictions for testing the bayesian framework: Rajesh Rao's work\n\n• Different forms of feedback: We really don't know how these forms of feedback are. The diffused feedback that is topographic could be doing an spatial priming and the patchy ones are doing feature priming.\n\n• Tony: Natural stimulii. Adaptive models: Natural stimulii. Yes. In the case of hardwired models: it will give tighter prediction to the experimentalists.", "statement_status": "exact", "statement_verification": "This canonical record is Chapter L, “Notes from Breakout Session II,” in the American Institute of Mathematics workshop notes *Inference and Prediction in Neocortical Circuits* (version dated 24 October 2003). It is a discussion summary, tagged `section`, rather than a formal mathematical problem.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Inference and prediction in neocortical circuits\nSection: \nSource item: Chapter L\nSource URL: https://aimath.org/WWN/brain/brain.pdf\nCanonical location: aim-biology-notes.json notes[33]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"Chapter L: Notes from Breakout Session II \\n\\nThe four groups met separately to prepare responses to the comments from the previous day. \\n\\nReply From Psychologists \\n\\nThe theorists wanted to be hired. The primate experiments would go much better if they did psychophysical experiments. Giving away raw data looks like giving away ones own work. Alessandra: Once something is published it is public domain and so the data should me bade public. Judith: It could be that you have one sort of analysis and more analysis might be waiting. you just don't want to give away the data. We should be able to publish the data \\n\\nFrom Theory group I. \\n\\n• Are there associative spiking models with spiking neurons? (Yes). Fritz will give a presentation on that. \\n\\n• Theoreticians will become aware of training in experimentation. This will give us more intuition. But we need more help in doing that. We would also like the data from you! \\n\\n• Russ: Rigorous model comparisons between different models. One can apply the same inputs to models and get the results. Should document when the models fail. \\n\\n• Falsifiable predictions?: We alread do. \\n\\n• The job of theorists is not just to make testable hypothesis. 13 \\n\\nFrom Anatomists \\n\\n• Now more labs are producing less descriptive and more quantitative data. There is some kind of questions that take too much time to address. (For eg: number of synapses). How can computer science help us for this? \\n\\n• There are computer vision techniques and tools available that could be used. Multi-electrode recording? Its a lot of effort. If the cells cannot be activated at the same time, it amounts to recording one after another. \\n\\n• Natural stimuli. We have acquired multichannel recording systems and we are racing to do multi electrode recording. \\n\\nFrom Theory II \\n\\n• Bayesian Framework.: Bayesian framework is a principled way of looking at the computational problems faced by the brian given the ambiguous Priors can be implemented as lateral connectivity or synaptic weights or top-down predictions. Predictions for testing the bayesian framework: Rajesh Rao's work \\n\\n• Different forms of feedback: We really don't know how these forms of feedback are. The diffused feedback that is topographic could be doing an spatial priming and the patchy ones are doing feature priming. \\n\\n• Tony: Natural stimulii. Adaptive models: Natural stimulii. Yes. In the case of hardwired models: it will give tighter prediction to the experimentalists.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/brain/brain.pdf", "tags": [ "aim", "AIM-BIOLOGY-0034", "aim-domain:biology", "aim-workshop:brain", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "This source record is a workshop discussion, not a formal problem. It yields two proved, sharply scoped results. For two fully specified scalar Gaussian response models with common known variance, freely repeatable inputs, and a fixed trial budget, repeating an input with maximum absolute prediction gap Blackwell-dominates every adaptive input policy; its equal-prior Bayes error is Phi(-sqrt(n)D/(2 sigma)). This prescription fails for composite models, as an explicit two-input counterexample shows. Separately, under arbitrary additive common epoch drift, a cell-response contrast is identifiable exactly when the two cell vertices lie in the same connected component of the cell-epoch observation graph; purely sequential single-cell epochs identify no between-cell contrasts without extra drift assumptions.\n\nCandidate contribution (theorem_and_identifiability_criterion; novelty confidence low): Candidate novelty: a two-certificate audit tailored to the AIM exchange, combining a constructive Blackwell-dominance proof for maximum-gap fixed stimulation against all fixed-horizon adaptive policies in the two-simple-Gaussian setting with an if-and-only-if recording-overlap graph criterion for drift-robust cell contrasts, plus a composite-model counterexample that sharply limits the first theorem.", "research_difficulty_suggested": "L2" }, { "id": 20000870, "problem_number": "AIM-BIOLOGY-0035", "title": "Provenance-safe multidisciplinary priority setting", "statement": "A.1 Introduction to Discussion Session Goals\n\nThis document presents an outline of some of content of the afternoon discussions. In the interest of producing these notes as quickly as possible, no attempt has been made to track down every reference or attribute every comment to a participant. Also, as the scribe, I summarized what I understood from the discussion, so attributed comments are, in some cases, paraphrases. (Thus, you may wish to check with the attributed before making any consequent attribution.) Each afternoon session was centered around some theme. The purpose of the ses-sions was to try to identify problems that people might want to pursue, and identify which problems are most important to work on. Also, with participants from many different aca-demic communities (biology, combinatorics, topology, statistics), it was important for each community to hear perspectives from the others.", "original_statement": "A.1 Introduction to Discussion Session Goals \n\nThis document presents an outline of some of content of the afternoon discussions. In the interest of producing these notes as quickly as possible, no attempt has been made to track down every reference or attribute every comment to a participant. Also, as the scribe, I summarized what I understood from the discussion, so attributed comments are, in some cases, paraphrases. (Thus, you may wish to check with the attributed before making any consequent attribution.) Each afternoon session was centered around some theme. The purpose of the ses-sions was to try to identify problems that people might want to pursue, and identify which problems are most important to work on. Also, with participants from many different aca-demic communities (biology, combinatorics, topology, statistics), it was important for each community to hear perspectives from the others.", "clean_statement": "A.1 Introduction to Discussion Session Goals\n\nThis document presents an outline of some of content of the afternoon discussions. In the interest of producing these notes as quickly as possible, no attempt has been made to track down every reference or attribute every comment to a participant. Also, as the scribe, I summarized what I understood from the discussion, so attributed comments are, in some cases, paraphrases. (Thus, you may wish to check with the attributed before making any consequent attribution.) Each afternoon session was centered around some theme. The purpose of the ses-sions was to try to identify problems that people might want to pursue, and identify which problems are most important to work on. Also, with participants from many different aca-demic communities (biology, combinatorics, topology, statistics), it was important for each community to hear perspectives from the others.", "statement_status": "exact", "statement_verification": "This canonical record is item A.1, tagged as a section rather than as a question, in the AIM workshop notes *Geometric models of biological phenomena*. The source record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Geometric models of biological phenomena\nSection: \nSource item: A.1\nSource URL: https://aimath.org/WWN/geombio/geombio.pdf\nCanonical location: aim-biology-notes.json notes[34]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.1 Introduction to Discussion Session Goals \\n\\nThis document presents an outline of some of content of the afternoon discussions. In the interest of producing these notes as quickly as possible, no attempt has been made to track down every reference or attribute every comment to a participant. Also, as the scribe, I summarized what I understood from the discussion, so attributed comments are, in some cases, paraphrases. (Thus, you may wish to check with the attributed before making any consequent attribution.) Each afternoon session was centered around some theme. The purpose of the ses-sions was to try to identify problems that people might want to pursue, and identify which problems are most important to work on. Also, with participants from many different aca-demic communities (biology, combinatorics, topology, statistics), it was important for each community to hear perspectives from the others.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/geombio/geombio.pdf", "tags": [ "aim", "AIM-BIOLOGY-0035", "aim-domain:biology", "aim-workshop:geombio", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "A.1 is a methodological section rather than an open problem. Its warnings can be made mathematically actionable by a sound set-valued provenance theorem—universal consequences of conservatively propagated paraphrase sets are safe—and by an exact balanced-priority test: under closed Cartesian community-score intervals and weight floors alpha_g, problem p robustly strictly outranks q exactly when sum_g alpha_g(L_g(p)-U_g(q)) + (1-sum_g alpha_g) min_g(L_g(p)-U_g(q)) is positive.\n\nCandidate contribution (protocol_and_lemma; novelty confidence low): The candidate contribution is a paraphrase-safe balanced-priority certificate coupling universal provenance propagation to the explicit floor-simplex margin test for multidisciplinary problem rankings; it states that individual attribution is safe exactly when the claimed agent occurs in every globally admissible world, that exact quotation additionally requires the claimed exact wording and verbatim status in every such world, and that a pairwise ranking is certified exactly when the displayed margin is positive.", "research_difficulty_suggested": "L2" }, { "id": 20000871, "problem_number": "AIM-BIOLOGY-0036", "title": "Majority topology and the 0.66-or-1 branch-length question", "statement": "A.2 Sunday Discussion on Biological Issues\n\nThe discussion was moderated by John Luecke. Susan Holmes opened the discussion by suggesting that we might ask the biologists: what mathematical or biological questions related to phylogenetic trees are most important to biologists? She also invited clarifications about things that have been addressed in talks earlier in the day. Q. What properties of phylogenetic trees make them different from random trees? Is there some kind of structure that makes them different? (Epstein)\n\n• Phylogenetic trees don't look like trees generated by random branching processes. (Huelsenbeck) Q. What are the characteristics that a \"distance between trees\" should have that biologists would want? (Luecke)\n\n• Luecke: For instance, the Billara-Holmes-Vogtmann (BHV) metric measures dis-tances along geodesics in tree space, and the trees change as you move along the geodesic, but does this have a reasonable biological intepretation?\n\n• Sober: that does not have biological meaning. Biologists just want to know how to tell if trees are similar or not.\n\n• Felsenstein: you are assuming we (biologists) want a notion of distance between trees? (Scribe: this appears to have been in contrast to just a notion of similarity. This elicited remarks from mathematicians about the fruitfulness of using a metric to tell how similar two trees are.)\n\n• Huelsenbeck pointed out several ways to measure how different trees are, e.g., involv-ing the contraction/expansion of edges to get from one to the other, or differences in length of edges, or squared branch length, etc. Some discussion of the Robinson-Foulds distance ensued.\n\n• Luecke: so computational simplicity would be one important consideration for a distance.\n\n• Huelsenbeck: Any distance notion on phylogenetic trees should have a good theoret-ical foundation. 4\n\nQ. Is there a difference between some real distribution of trees on tree space rather than some random distribution? (Holmes)\n\n• Evans: this will help us to construct meaningful confidence sets on trees.\n\n• Felsenstein: Tree spaces are weird. For a cloud of trees, we want to characterize where in this space those trees are. Q. Problem: make notions of distances accessible to biologists, so they might use them. (Huelsenbeck)\n\n• Some comments were made about the use of methods of Kuhner-Felsenstein, weighted Robinson-Foulds, in biology.\n\n• Diaconis pointed out that in his work on non-standard data structures, the notions of distances are very useful and basic. In doing exploratory analyses, they can be useful in statistical tasks. So, what do biologists think of these distances?\n\n• Holmes pointed out that how fast MCMC methods converge depends on the geometry of the space, and the metric chosen.\n\n• Huelsenbeck suggested that a useful task for mathematicians would be to make some sense of the uses of these distances to biologists? (such as in an article pitched to biologists). More comments about tree space topology and branch lengths:\n\n• Huelsenbeck: are there versions of tree space that biologists might not be interested in, due to technical conditions? For instance, if you give a program an alignment, the branches have to be short enough so that you can align them.\n\n• Felsenstein: In many cases in biology we are interested in grappling with, such as origins of mammals, many of the orders seem to have popped up rather quickly. We could be interested in (a) the branch length and (b) the topology. When the orders diverged, important things were happening very quickly, such as morphological changes. But the molecules you are studying may not have been involved. If you are interested in those morphological changes, it is important to know what order the branching occurred. On the other hand, in some other questions, topology many not be as important as branch length. It really depends on the question you are asking.\n\n• Sober: A third issue is important: the character states of the interior nodes of trees. You might want to know what is the sequence of changes that occurred on some branch? for instance in some calculation, you may want to integrate over all possible states of interior nodes. Felsenstein pointed out that these states on interior nodes are sometimes overinterpreted- they are often viewed as actual states, rather than estimates.\n\n• Penny: in biology, knowing the \"true\" tree may only be a starting point of the real investigation- some other aspect that the biologist is truly interested in. Huelsen-beck gave an example of an evolution biologist interested in sexual selection. Al-though she made a phylogeny (what she thought was the best tree), she wasn't primarily interested in the phylogeny, but in further questions about sexual selec-tion. Q. How to pick a tree (or tree average) from a set of trees resulting from data? 5\n\n• Epstein: in bacteria dna fragments, each one gives a tree. Which one to use? Need a tree distance to analyze the resulting 22 different trees. The distance he used was the BHV metric, but he suspects one obtains much the same answer with many different tree metrics.\n\n• Evans: this is similar to Mallow's model in statistics, where you have a distribution on permutations, centered on a permutation, dropping off in some radial sense. Here, you have a central tree, and then the probability that you observe some other tree dies off as you move farther away.\n\n• Penny: one approach to identify \"bad\" portions of a tree is to try to \"identify the guilty taxon\", by succedssively removing taxa and then see whether this stabilizes the tree significantly. Q. How to understand or deal with residuals, such as non-tree like data?\n\n• Holmes: gave an example in which a friend has a reference tree, and 8 different plumage trees. In tree space, if the plumage tree differs from the reference dna tree in some direction, is that present in the way these things actually evolved? In statistics, when doing regression, you compare data points to a fitted line, and ignore the ones that are way off. Here, in tree space, there's a similar question for non-tree like data: how much did I have to bend the data to make it into a tree?\n\n• Sober: how do you decide how much off is too much, before you are worried? (Scribe's note: this seems to require a notion of distance not just on tree space, but some larger space- such as on the space of DNA sequences- in which tree space is embedded. We discussed embedding questions on Wednesday.) More comments on distances:\n\n• Diaconis told a story where distances fit data remarkably well. Perception psycholo-gist Roger Shepard studied the visual system, by showing subjects a configuration of blocks, then some other configuration, and then asked them: are they are the same? He found that the time it took people to decide was the geodesic in the three di-mensional rotation group. The data is quite remarkable, and gave straight line plots. We know what it means to measure the distance in the rotation group (but why the brain should know about this is some fascinating subject in itself.)\n\n• Similarly, does a distance between trees have to have some interpretation in terms of evolution, in order to be the \"right\" notion of distance?\n\n• Epstein: For instance, given dna or amino acid sequences, one for each taxon, this data produces a tree. As sequences change or evolve one nucleotide at a time, the trees will change. What path does this trace out in tree space? (Scribe's note: this appears to require a space of trees that includes trees with many different numbers of leaves, so that one can speak of how a tree evolves as species split off from one another.)\n\n• In response to a question, Felsenstein mentioned a few sources of variation in trees: (a) statistical error. (b) coalescence: take a gene copy in three species, and think of copies ancestral to these. The copies do not come together instantly, but have some stochastic chance of mixing, and they may come in some random order that conflicts with a species tree. (c) horizontal gene transfer. (Scribe's note: a more detailed discussion of this topic can be found in Tuesday's discussion.) 6\n\nQ. Which is better: concatenating DNA sequences first or averaging trees later?\n\n• Holmes: empirical studies show that if you take all the data and compute one tree, you generally do less well at estimating the tree than than if you take the fragments, use them, then average them. This comes from the CAT(0) property. The intuition behind it is if you average in a negatively curved space, you converge much faster than you should.\n\n• St. John: pointed out an example with 20 simulated DNA sequences: if a is one part of the genome, and b is another part, then the trees obtained from using a combined with b do not overlap, and are somewhat in between, the trees obtained from a and from b alone. Felsenstein commented that the stochastic effects pushed a and b in different directions in tree space. She also noted that in studying hybridization (e.g., sunflowers), she was surprised that she didn't get more overlap, and often the right answer was with part of the data, not all the data.\n\n• Felsenstein: Biologists are in disagreement about whether concatenating or averaging is better. Statistically, which is better?\n\n• It was pointed out that a paper by Cunningham (1997) does comparisons. Also, work by Amit, and statistical literature on boosting and bagging.\n\n• Billera: there's a notion of equivalent metrics in topology, and any of these will do. (In doing geometry, where issues like curvature come into play, the choice of metrics is important.) It may be the case that even though certain metrics are good enough for some purposes, it doesn't mean that other metrics aren't valid. Q. What is the right notion of an \"average\" of trees?\n\n• The biologists present agreed that this was a very interesting question for biologists.\n\n• Vert: Two important ideas emerge when working with decision trees: (1) average decrease some variance, (2) averaging can leave the space. Boosting will leave the space, if we don't leave the space it's really not boosting. The averaging we are discussing here doesn't leave the space of trees.\n\n• More discussion on averaging took place here. Holmes made a comment about the \"non-associativity\" of trees (related to considering trees of trees, when building trees one character at a time). Billera made some remarks about how averaging may be better than concatenation, but we have no guarantee that it is any good.\n\n• Penny discussed the notion of a median tree: the tree that is closest on average to all the others.\n\n• Felsenstein: other examples are the consensus tree, and majority rule consensus tree. Q. What are good properties of averaging? (Luecke)\n\n• Penny: would like a fully resolved binary tree, though he points out that some others might sacrifice that in favor of other features.\n\n• Felsenstein posed a problem about averaging properties: given a bunch of trees from different genes, to study a question like: are chimps closer to humans than gorillas? Say 2/3 of the trees trees show humans with chimps, others show chimps with gorillas. Suppose in trees that group humans with chimps, the average branch length is 1, but the others don't have it. When you average... do you want it to be length.66 or length 1? 7\n\nThere was a question about how different are trees that result from diffrent averaging methods. It was pointed out (Huelsenbeck) that differences between methods for a single gene is much smaller than the differences across genes. Q. What would be the distribution on trees that gives majority rule consensus as its average? (Holmes)\n\n• Holmes: would like any tree average to be an expected value with respect to some distribution on tree space.", "original_statement": "A.2 Sunday Discussion on Biological Issues \n\nThe discussion was moderated by John Luecke. Susan Holmes opened the discussion by suggesting that we might ask the biologists: what mathematical or biological questions related to phylogenetic trees are most important to biologists? She also invited clarifications about things that have been addressed in talks earlier in the day. Q. What properties of phylogenetic trees make them different from random trees? Is there some kind of structure that makes them different? (Epstein) \n\n• Phylogenetic trees don't look like trees generated by random branching processes. (Huelsenbeck) Q. What are the characteristics that a \"distance between trees\" should have that biologists would want? (Luecke) \n\n• Luecke: For instance, the Billara-Holmes-Vogtmann (BHV) metric measures dis-tances along geodesics in tree space, and the trees change as you move along the geodesic, but does this have a reasonable biological intepretation? \n\n• Sober: that does not have biological meaning. Biologists just want to know how to tell if trees are similar or not. \n\n• Felsenstein: you are assuming we (biologists) want a notion of distance between trees? (Scribe: this appears to have been in contrast to just a notion of similarity. This elicited remarks from mathematicians about the fruitfulness of using a metric to tell how similar two trees are.) \n\n• Huelsenbeck pointed out several ways to measure how different trees are, e.g., involv-ing the contraction/expansion of edges to get from one to the other, or differences in length of edges, or squared branch length, etc. Some discussion of the Robinson-Foulds distance ensued. \n\n• Luecke: so computational simplicity would be one important consideration for a distance. \n\n• Huelsenbeck: Any distance notion on phylogenetic trees should have a good theoret-ical foundation. 4\n\nQ. Is there a difference between some real distribution of trees on tree space rather than some random distribution? (Holmes) \n\n• Evans: this will help us to construct meaningful confidence sets on trees. \n\n• Felsenstein: Tree spaces are weird. For a cloud of trees, we want to characterize where in this space those trees are. Q. Problem: make notions of distances accessible to biologists, so they might use them. (Huelsenbeck) \n\n• Some comments were made about the use of methods of Kuhner-Felsenstein, weighted Robinson-Foulds, in biology. \n\n• Diaconis pointed out that in his work on non-standard data structures, the notions of distances are very useful and basic. In doing exploratory analyses, they can be useful in statistical tasks. So, what do biologists think of these distances? \n\n• Holmes pointed out that how fast MCMC methods converge depends on the geometry of the space, and the metric chosen. \n\n• Huelsenbeck suggested that a useful task for mathematicians would be to make some sense of the uses of these distances to biologists? (such as in an article pitched to biologists). More comments about tree space topology and branch lengths: \n\n• Huelsenbeck: are there versions of tree space that biologists might not be interested in, due to technical conditions? For instance, if you give a program an alignment, the branches have to be short enough so that you can align them. \n\n• Felsenstein: In many cases in biology we are interested in grappling with, such as origins of mammals, many of the orders seem to have popped up rather quickly. We could be interested in (a) the branch length and (b) the topology. When the orders diverged, important things were happening very quickly, such as morphological changes. But the molecules you are studying may not have been involved. If you are interested in those morphological changes, it is important to know what order the branching occurred. On the other hand, in some other questions, topology many not be as important as branch length. It really depends on the question you are asking. \n\n• Sober: A third issue is important: the character states of the interior nodes of trees. You might want to know what is the sequence of changes that occurred on some branch? for instance in some calculation, you may want to integrate over all possible states of interior nodes. Felsenstein pointed out that these states on interior nodes are sometimes overinterpreted- they are often viewed as actual states, rather than estimates. \n\n• Penny: in biology, knowing the \"true\" tree may only be a starting point of the real investigation- some other aspect that the biologist is truly interested in. Huelsen-beck gave an example of an evolution biologist interested in sexual selection. Al-though she made a phylogeny (what she thought was the best tree), she wasn't primarily interested in the phylogeny, but in further questions about sexual selec-tion. Q. How to pick a tree (or tree average) from a set of trees resulting from data? 5\n\n• Epstein: in bacteria dna fragments, each one gives a tree. Which one to use? Need a tree distance to analyze the resulting 22 different trees. The distance he used was the BHV metric, but he suspects one obtains much the same answer with many different tree metrics. \n\n• Evans: this is similar to Mallow's model in statistics, where you have a distribution on permutations, centered on a permutation, dropping off in some radial sense. Here, you have a central tree, and then the probability that you observe some other tree dies off as you move farther away. \n\n• Penny: one approach to identify \"bad\" portions of a tree is to try to \"identify the guilty taxon\", by succedssively removing taxa and then see whether this stabilizes the tree significantly. Q. How to understand or deal with residuals, such as non-tree like data? \n\n• Holmes: gave an example in which a friend has a reference tree, and 8 different plumage trees. In tree space, if the plumage tree differs from the reference dna tree in some direction, is that present in the way these things actually evolved? In statistics, when doing regression, you compare data points to a fitted line, and ignore the ones that are way off. Here, in tree space, there's a similar question for non-tree like data: how much did I have to bend the data to make it into a tree? \n\n• Sober: how do you decide how much off is too much, before you are worried? (Scribe's note: this seems to require a notion of distance not just on tree space, but some larger space- such as on the space of DNA sequences- in which tree space is embedded. We discussed embedding questions on Wednesday.) More comments on distances: \n\n• Diaconis told a story where distances fit data remarkably well. Perception psycholo-gist Roger Shepard studied the visual system, by showing subjects a configuration of blocks, then some other configuration, and then asked them: are they are the same? He found that the time it took people to decide was the geodesic in the three di-mensional rotation group. The data is quite remarkable, and gave straight line plots. We know what it means to measure the distance in the rotation group (but why the brain should know about this is some fascinating subject in itself.) \n\n• Similarly, does a distance between trees have to have some interpretation in terms of evolution, in order to be the \"right\" notion of distance? \n\n• Epstein: For instance, given dna or amino acid sequences, one for each taxon, this data produces a tree. As sequences change or evolve one nucleotide at a time, the trees will change. What path does this trace out in tree space? (Scribe's note: this appears to require a space of trees that includes trees with many different numbers of leaves, so that one can speak of how a tree evolves as species split off from one another.) \n\n• In response to a question, Felsenstein mentioned a few sources of variation in trees: (a) statistical error. (b) coalescence: take a gene copy in three species, and think of copies ancestral to these. The copies do not come together instantly, but have some stochastic chance of mixing, and they may come in some random order that conflicts with a species tree. (c) horizontal gene transfer. (Scribe's note: a more detailed discussion of this topic can be found in Tuesday's discussion.) 6\n\nQ. Which is better: concatenating DNA sequences first or averaging trees later? \n\n• Holmes: empirical studies show that if you take all the data and compute one tree, you generally do less well at estimating the tree than than if you take the fragments, use them, then average them. This comes from the CAT(0) property. The intuition behind it is if you average in a negatively curved space, you converge much faster than you should. \n\n• St. John: pointed out an example with 20 simulated DNA sequences: if a is one part of the genome, and b is another part, then the trees obtained from using a combined with b do not overlap, and are somewhat in between, the trees obtained from a and from b alone. Felsenstein commented that the stochastic effects pushed a and b in different directions in tree space. She also noted that in studying hybridization (e.g., sunflowers), she was surprised that she didn't get more overlap, and often the right answer was with part of the data, not all the data. \n\n• Felsenstein: Biologists are in disagreement about whether concatenating or averaging is better. Statistically, which is better? \n\n• It was pointed out that a paper by Cunningham (1997) does comparisons. Also, work by Amit, and statistical literature on boosting and bagging. \n\n• Billera: there's a notion of equivalent metrics in topology, and any of these will do. (In doing geometry, where issues like curvature come into play, the choice of metrics is important.) It may be the case that even though certain metrics are good enough for some purposes, it doesn't mean that other metrics aren't valid. Q. What is the right notion of an \"average\" of trees? \n\n• The biologists present agreed that this was a very interesting question for biologists. \n\n• Vert: Two important ideas emerge when working with decision trees: (1) average decrease some variance, (2) averaging can leave the space. Boosting will leave the space, if we don't leave the space it's really not boosting. The averaging we are discussing here doesn't leave the space of trees. \n\n• More discussion on averaging took place here. Holmes made a comment about the \"non-associativity\" of trees (related to considering trees of trees, when building trees one character at a time). Billera made some remarks about how averaging may be better than concatenation, but we have no guarantee that it is any good. \n\n• Penny discussed the notion of a median tree: the tree that is closest on average to all the others. \n\n• Felsenstein: other examples are the consensus tree, and majority rule consensus tree. Q. What are good properties of averaging? (Luecke) \n\n• Penny: would like a fully resolved binary tree, though he points out that some others might sacrifice that in favor of other features. \n\n• Felsenstein posed a problem about averaging properties: given a bunch of trees from different genes, to study a question like: are chimps closer to humans than gorillas? Say 2/3 of the trees trees show humans with chimps, others show chimps with gorillas. Suppose in trees that group humans with chimps, the average branch length is 1, but the others don't have it. When you average... do you want it to be length.66 or length 1? 7\n\nThere was a question about how different are trees that result from diffrent averaging methods. It was pointed out (Huelsenbeck) that differences between methods for a single gene is much smaller than the differences across genes. Q. What would be the distribution on trees that gives majority rule consensus as its average? (Holmes) \n\n• Holmes: would like any tree average to be an expected value with respect to some distribution on tree space.", "clean_statement": "A.2 Sunday Discussion on Biological Issues\n\nThe discussion was moderated by John Luecke. Susan Holmes opened the discussion by suggesting that we might ask the biologists: what mathematical or biological questions related to phylogenetic trees are most important to biologists? She also invited clarifications about things that have been addressed in talks earlier in the day. Q. What properties of phylogenetic trees make them different from random trees? Is there some kind of structure that makes them different? (Epstein)\n\n• Phylogenetic trees don't look like trees generated by random branching processes. (Huelsenbeck) Q. What are the characteristics that a \"distance between trees\" should have that biologists would want? (Luecke)\n\n• Luecke: For instance, the Billara-Holmes-Vogtmann (BHV) metric measures dis-tances along geodesics in tree space, and the trees change as you move along the geodesic, but does this have a reasonable biological intepretation?\n\n• Sober: that does not have biological meaning. Biologists just want to know how to tell if trees are similar or not.\n\n• Felsenstein: you are assuming we (biologists) want a notion of distance between trees? (Scribe: this appears to have been in contrast to just a notion of similarity. This elicited remarks from mathematicians about the fruitfulness of using a metric to tell how similar two trees are.)\n\n• Huelsenbeck pointed out several ways to measure how different trees are, e.g., involv-ing the contraction/expansion of edges to get from one to the other, or differences in length of edges, or squared branch length, etc. Some discussion of the Robinson-Foulds distance ensued.\n\n• Luecke: so computational simplicity would be one important consideration for a distance.\n\n• Huelsenbeck: Any distance notion on phylogenetic trees should have a good theoret-ical foundation. 4\n\nQ. Is there a difference between some real distribution of trees on tree space rather than some random distribution? (Holmes)\n\n• Evans: this will help us to construct meaningful confidence sets on trees.\n\n• Felsenstein: Tree spaces are weird. For a cloud of trees, we want to characterize where in this space those trees are. Q. Problem: make notions of distances accessible to biologists, so they might use them. (Huelsenbeck)\n\n• Some comments were made about the use of methods of Kuhner-Felsenstein, weighted Robinson-Foulds, in biology.\n\n• Diaconis pointed out that in his work on non-standard data structures, the notions of distances are very useful and basic. In doing exploratory analyses, they can be useful in statistical tasks. So, what do biologists think of these distances?\n\n• Holmes pointed out that how fast MCMC methods converge depends on the geometry of the space, and the metric chosen.\n\n• Huelsenbeck suggested that a useful task for mathematicians would be to make some sense of the uses of these distances to biologists? (such as in an article pitched to biologists). More comments about tree space topology and branch lengths:\n\n• Huelsenbeck: are there versions of tree space that biologists might not be interested in, due to technical conditions? For instance, if you give a program an alignment, the branches have to be short enough so that you can align them.\n\n• Felsenstein: In many cases in biology we are interested in grappling with, such as origins of mammals, many of the orders seem to have popped up rather quickly. We could be interested in (a) the branch length and (b) the topology. When the orders diverged, important things were happening very quickly, such as morphological changes. But the molecules you are studying may not have been involved. If you are interested in those morphological changes, it is important to know what order the branching occurred. On the other hand, in some other questions, topology many not be as important as branch length. It really depends on the question you are asking.\n\n• Sober: A third issue is important: the character states of the interior nodes of trees. You might want to know what is the sequence of changes that occurred on some branch? for instance in some calculation, you may want to integrate over all possible states of interior nodes. Felsenstein pointed out that these states on interior nodes are sometimes overinterpreted- they are often viewed as actual states, rather than estimates.\n\n• Penny: in biology, knowing the \"true\" tree may only be a starting point of the real investigation- some other aspect that the biologist is truly interested in. Huelsen-beck gave an example of an evolution biologist interested in sexual selection. Al-though she made a phylogeny (what she thought was the best tree), she wasn't primarily interested in the phylogeny, but in further questions about sexual selec-tion. Q. How to pick a tree (or tree average) from a set of trees resulting from data? 5\n\n• Epstein: in bacteria dna fragments, each one gives a tree. Which one to use? Need a tree distance to analyze the resulting 22 different trees. The distance he used was the BHV metric, but he suspects one obtains much the same answer with many different tree metrics.\n\n• Evans: this is similar to Mallow's model in statistics, where you have a distribution on permutations, centered on a permutation, dropping off in some radial sense. Here, you have a central tree, and then the probability that you observe some other tree dies off as you move farther away.\n\n• Penny: one approach to identify \"bad\" portions of a tree is to try to \"identify the guilty taxon\", by succedssively removing taxa and then see whether this stabilizes the tree significantly. Q. How to understand or deal with residuals, such as non-tree like data?\n\n• Holmes: gave an example in which a friend has a reference tree, and 8 different plumage trees. In tree space, if the plumage tree differs from the reference dna tree in some direction, is that present in the way these things actually evolved? In statistics, when doing regression, you compare data points to a fitted line, and ignore the ones that are way off. Here, in tree space, there's a similar question for non-tree like data: how much did I have to bend the data to make it into a tree?\n\n• Sober: how do you decide how much off is too much, before you are worried? (Scribe's note: this seems to require a notion of distance not just on tree space, but some larger space- such as on the space of DNA sequences- in which tree space is embedded. We discussed embedding questions on Wednesday.) More comments on distances:\n\n• Diaconis told a story where distances fit data remarkably well. Perception psycholo-gist Roger Shepard studied the visual system, by showing subjects a configuration of blocks, then some other configuration, and then asked them: are they are the same? He found that the time it took people to decide was the geodesic in the three di-mensional rotation group. The data is quite remarkable, and gave straight line plots. We know what it means to measure the distance in the rotation group (but why the brain should know about this is some fascinating subject in itself.)\n\n• Similarly, does a distance between trees have to have some interpretation in terms of evolution, in order to be the \"right\" notion of distance?\n\n• Epstein: For instance, given dna or amino acid sequences, one for each taxon, this data produces a tree. As sequences change or evolve one nucleotide at a time, the trees will change. What path does this trace out in tree space? (Scribe's note: this appears to require a space of trees that includes trees with many different numbers of leaves, so that one can speak of how a tree evolves as species split off from one another.)\n\n• In response to a question, Felsenstein mentioned a few sources of variation in trees: (a) statistical error. (b) coalescence: take a gene copy in three species, and think of copies ancestral to these. The copies do not come together instantly, but have some stochastic chance of mixing, and they may come in some random order that conflicts with a species tree. (c) horizontal gene transfer. (Scribe's note: a more detailed discussion of this topic can be found in Tuesday's discussion.) 6\n\nQ. Which is better: concatenating DNA sequences first or averaging trees later?\n\n• Holmes: empirical studies show that if you take all the data and compute one tree, you generally do less well at estimating the tree than than if you take the fragments, use them, then average them. This comes from the CAT(0) property. The intuition behind it is if you average in a negatively curved space, you converge much faster than you should.\n\n• St. John: pointed out an example with 20 simulated DNA sequences: if a is one part of the genome, and b is another part, then the trees obtained from using a combined with b do not overlap, and are somewhat in between, the trees obtained from a and from b alone. Felsenstein commented that the stochastic effects pushed a and b in different directions in tree space. She also noted that in studying hybridization (e.g., sunflowers), she was surprised that she didn't get more overlap, and often the right answer was with part of the data, not all the data.\n\n• Felsenstein: Biologists are in disagreement about whether concatenating or averaging is better. Statistically, which is better?\n\n• It was pointed out that a paper by Cunningham (1997) does comparisons. Also, work by Amit, and statistical literature on boosting and bagging.\n\n• Billera: there's a notion of equivalent metrics in topology, and any of these will do. (In doing geometry, where issues like curvature come into play, the choice of metrics is important.) It may be the case that even though certain metrics are good enough for some purposes, it doesn't mean that other metrics aren't valid. Q. What is the right notion of an \"average\" of trees?\n\n• The biologists present agreed that this was a very interesting question for biologists.\n\n• Vert: Two important ideas emerge when working with decision trees: (1) average decrease some variance, (2) averaging can leave the space. Boosting will leave the space, if we don't leave the space it's really not boosting. The averaging we are discussing here doesn't leave the space of trees.\n\n• More discussion on averaging took place here. Holmes made a comment about the \"non-associativity\" of trees (related to considering trees of trees, when building trees one character at a time). Billera made some remarks about how averaging may be better than concatenation, but we have no guarantee that it is any good.\n\n• Penny discussed the notion of a median tree: the tree that is closest on average to all the others.\n\n• Felsenstein: other examples are the consensus tree, and majority rule consensus tree. Q. What are good properties of averaging? (Luecke)\n\n• Penny: would like a fully resolved binary tree, though he points out that some others might sacrifice that in favor of other features.\n\n• Felsenstein posed a problem about averaging properties: given a bunch of trees from different genes, to study a question like: are chimps closer to humans than gorillas? Say 2/3 of the trees trees show humans with chimps, others show chimps with gorillas. Suppose in trees that group humans with chimps, the average branch length is 1, but the others don't have it. When you average... do you want it to be length.66 or length 1? 7\n\nThere was a question about how different are trees that result from diffrent averaging methods. It was pointed out (Huelsenbeck) that differences between methods for a single gene is much smaller than the differences across genes. Q. What would be the distribution on trees that gives majority rule consensus as its average? (Holmes)\n\n• Holmes: would like any tree average to be an expected value with respect to some distribution on tree space.", "statement_status": "exact", "statement_verification": "This record is Section A.2, “Sunday Discussion on Biological Issues,” in the official AIM workshop notes *Geometric Models of Biological Phenomena*, version dated 18 June 2003. It is a moderated, multi-question discussion rather than one narrowly stated conjecture. It nevertheless contains genuine research questions about biologically meaningful tree metrics, confidence sets and distributions on tree space, topology versus branch length and ancestral states, non-tree-like residuals, concatenating sequence data versus combining gene trees, and the correct notion of a tree average.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Geometric models of biological phenomena\nSection: \nSource item: A.2\nSource URL: https://aimath.org/WWN/geombio/geombio.pdf\nCanonical location: aim-biology-notes.json notes[35]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.2 Sunday Discussion on Biological Issues \\n\\nThe discussion was moderated by John Luecke. Susan Holmes opened the discussion by suggesting that we might ask the biologists: what mathematical or biological questions related to phylogenetic trees are most important to biologists? She also invited clarifications about things that have been addressed in talks earlier in the day. Q. What properties of phylogenetic trees make them different from random trees? Is there some kind of structure that makes them different? (Epstein) \\n\\n• Phylogenetic trees don't look like trees generated by random branching processes. (Huelsenbeck) Q. What are the characteristics that a \\\"distance between trees\\\" should have that biologists would want? (Luecke) \\n\\n• Luecke: For instance, the Billara-Holmes-Vogtmann (BHV) metric measures dis-tances along geodesics in tree space, and the trees change as you move along the geodesic, but does this have a reasonable biological intepretation? \\n\\n• Sober: that does not have biological meaning. Biologists just want to know how to tell if trees are similar or not. \\n\\n• Felsenstein: you are assuming we (biologists) want a notion of distance between trees? (Scribe: this appears to have been in contrast to just a notion of similarity. This elicited remarks from mathematicians about the fruitfulness of using a metric to tell how similar two trees are.) \\n\\n• Huelsenbeck pointed out several ways to measure how different trees are, e.g., involv-ing the contraction/expansion of edges to get from one to the other, or differences in length of edges, or squared branch length, etc. Some discussion of the Robinson-Foulds distance ensued. \\n\\n• Luecke: so computational simplicity would be one important consideration for a distance. \\n\\n• Huelsenbeck: Any distance notion on phylogenetic trees should have a good theoret-ical foundation. 4\\n\\nQ. Is there a difference between some real distribution of trees on tree space rather than some random distribution? (Holmes) \\n\\n• Evans: this will help us to construct meaningful confidence sets on trees. \\n\\n• Felsenstein: Tree spaces are weird. For a cloud of trees, we want to characterize where in this space those trees are. Q. Problem: make notions of distances accessible to biologists, so they might use them. (Huelsenbeck) \\n\\n• Some comments were made about the use of methods of Kuhner-Felsenstein, weighted Robinson-Foulds, in biology. \\n\\n• Diaconis pointed out that in his work on non-standard data structures, the notions of distances are very useful and basic. In doing exploratory analyses, they can be useful in statistical tasks. So, what do biologists think of these distances? \\n\\n• Holmes pointed out that how fast MCMC methods converge depends on the geometry of the space, and the metric chosen. \\n\\n• Huelsenbeck suggested that a useful task for mathematicians would be to make some sense of the uses of these distances to biologists? (such as in an article pitched to biologists). More comments about tree space topology and branch lengths: \\n\\n• Huelsenbeck: are there versions of tree space that biologists might not be interested in, due to technical conditions? For instance, if you give a program an alignment, the branches have to be short enough so that you can align them. \\n\\n• Felsenstein: In many cases in biology we are interested in grappling with, such as origins of mammals, many of the orders seem to have popped up rather quickly. We could be interested in (a) the branch length and (b) the topology. When the orders diverged, important things were happening very quickly, such as morphological changes. But the molecules you are studying may not have been involved. If you are interested in those morphological changes, it is important to know what order the branching occurred. On the other hand, in some other questions, topology many not be as important as branch length. It really depends on the question you are asking. \\n\\n• Sober: A third issue is important: the character states of the interior nodes of trees. You might want to know what is the sequence of changes that occurred on some branch? for instance in some calculation, you may want to integrate over all possible states of interior nodes. Felsenstein pointed out that these states on interior nodes are sometimes overinterpreted- they are often viewed as actual states, rather than estimates. \\n\\n• Penny: in biology, knowing the \\\"true\\\" tree may only be a starting point of the real investigation- some other aspect that the biologist is truly interested in. Huelsen-beck gave an example of an evolution biologist interested in sexual selection. Al-though she made a phylogeny (what she thought was the best tree), she wasn't primarily interested in the phylogeny, but in further questions about sexual selec-tion. Q. How to pick a tree (or tree average) from a set of trees resulting from data? 5\\n\\n• Epstein: in bacteria dna fragments, each one gives a tree. Which one to use? Need a tree distance to analyze the resulting 22 different trees. The distance he used was the BHV metric, but he suspects one obtains much the same answer with many different tree metrics. \\n\\n• Evans: this is similar to Mallow's model in statistics, where you have a distribution on permutations, centered on a permutation, dropping off in some radial sense. Here, you have a central tree, and then the probability that you observe some other tree dies off as you move farther away. \\n\\n• Penny: one approach to identify \\\"bad\\\" portions of a tree is to try to \\\"identify the guilty taxon\\\", by succedssively removing taxa and then see whether this stabilizes the tree significantly. Q. How to understand or deal with residuals, such as non-tree like data? \\n\\n• Holmes: gave an example in which a friend has a reference tree, and 8 different plumage trees. In tree space, if the plumage tree differs from the reference dna tree in some direction, is that present in the way these things actually evolved? In statistics, when doing regression, you compare data points to a fitted line, and ignore the ones that are way off. Here, in tree space, there's a similar question for non-tree like data: how much did I have to bend the data to make it into a tree? \\n\\n• Sober: how do you decide how much off is too much, before you are worried? (Scribe's note: this seems to require a notion of distance not just on tree space, but some larger space- such as on the space of DNA sequences- in which tree space is embedded. We discussed embedding questions on Wednesday.) More comments on distances: \\n\\n• Diaconis told a story where distances fit data remarkably well. Perception psycholo-gist Roger Shepard studied the visual system, by showing subjects a configuration of blocks, then some other configuration, and then asked them: are they are the same? He found that the time it took people to decide was the geodesic in the three di-mensional rotation group. The data is quite remarkable, and gave straight line plots. We know what it means to measure the distance in the rotation group (but why the brain should know about this is some fascinating subject in itself.) \\n\\n• Similarly, does a distance between trees have to have some interpretation in terms of evolution, in order to be the \\\"right\\\" notion of distance? \\n\\n• Epstein: For instance, given dna or amino acid sequences, one for each taxon, this data produces a tree. As sequences change or evolve one nucleotide at a time, the trees will change. What path does this trace out in tree space? (Scribe's note: this appears to require a space of trees that includes trees with many different numbers of leaves, so that one can speak of how a tree evolves as species split off from one another.) \\n\\n• In response to a question, Felsenstein mentioned a few sources of variation in trees: (a) statistical error. (b) coalescence: take a gene copy in three species, and think of copies ancestral to these. The copies do not come together instantly, but have some stochastic chance of mixing, and they may come in some random order that conflicts with a species tree. (c) horizontal gene transfer. (Scribe's note: a more detailed discussion of this topic can be found in Tuesday's discussion.) 6\\n\\nQ. Which is better: concatenating DNA sequences first or averaging trees later? \\n\\n• Holmes: empirical studies show that if you take all the data and compute one tree, you generally do less well at estimating the tree than than if you take the fragments, use them, then average them. This comes from the CAT(0) property. The intuition behind it is if you average in a negatively curved space, you converge much faster than you should. \\n\\n• St. John: pointed out an example with 20 simulated DNA sequences: if a is one part of the genome, and b is another part, then the trees obtained from using a combined with b do not overlap, and are somewhat in between, the trees obtained from a and from b alone. Felsenstein commented that the stochastic effects pushed a and b in different directions in tree space. She also noted that in studying hybridization (e.g., sunflowers), she was surprised that she didn't get more overlap, and often the right answer was with part of the data, not all the data. \\n\\n• Felsenstein: Biologists are in disagreement about whether concatenating or averaging is better. Statistically, which is better? \\n\\n• It was pointed out that a paper by Cunningham (1997) does comparisons. Also, work by Amit, and statistical literature on boosting and bagging. \\n\\n• Billera: there's a notion of equivalent metrics in topology, and any of these will do. (In doing geometry, where issues like curvature come into play, the choice of metrics is important.) It may be the case that even though certain metrics are good enough for some purposes, it doesn't mean that other metrics aren't valid. Q. What is the right notion of an \\\"average\\\" of trees? \\n\\n• The biologists present agreed that this was a very interesting question for biologists. \\n\\n• Vert: Two important ideas emerge when working with decision trees: (1) average decrease some variance, (2) averaging can leave the space. Boosting will leave the space, if we don't leave the space it's really not boosting. The averaging we are discussing here doesn't leave the space of trees. \\n\\n• More discussion on averaging took place here. Holmes made a comment about the \\\"non-associativity\\\" of trees (related to considering trees of trees, when building trees one character at a time). Billera made some remarks about how averaging may be better than concatenation, but we have no guarantee that it is any good. \\n\\n• Penny discussed the notion of a median tree: the tree that is closest on average to all the others. \\n\\n• Felsenstein: other examples are the consensus tree, and majority rule consensus tree. Q. What are good properties of averaging? (Luecke) \\n\\n• Penny: would like a fully resolved binary tree, though he points out that some others might sacrifice that in favor of other features. \\n\\n• Felsenstein posed a problem about averaging properties: given a bunch of trees from different genes, to study a question like: are chimps closer to humans than gorillas? Say 2/3 of the trees trees show humans with chimps, others show chimps with gorillas. Suppose in trees that group humans with chimps, the average branch length is 1, but the others don't have it. When you average... do you want it to be length.66 or length 1? 7\\n\\nThere was a question about how different are trees that result from diffrent averaging methods. It was pointed out (Huelsenbeck) that differences between methods for a single gene is much smaller than the differences across genes. Q. What would be the distribution on trees that gives majority rule consensus as its average? (Holmes) \\n\\n• Holmes: would like any tree average to be an expected value with respect to some distribution on tree space.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/geombio/geombio.pdf", "tags": [ "aim", "AIM-BIOLOGY-0036", "aim-domain:biology", "aim-workshop:geombio", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a distribution on fixed-leaf trees, expected unnormalized Robinson-Foulds loss equals a constant plus the sum over selected splits of 1 minus twice their occurrence probability; strict-majority splits are automatically compatible and give the RF median topology, with precisely characterized half-frequency ties. Conditional on a selected split, the absence-penalized quadratic loss E[I(x-L)^2 + lambda(1-I)x^2] has unique minimizer p mu divided by p + lambda(1-p). Consequently, Felsenstein's p=2/3 and conditional-mean-one example gives length 2/3 when absence is coded as zero and length 1 when absence is ignored. A separate real-line counterexample proves that CAT(0) geometry alone cannot imply biological consistency or decide concatenation versus gene-tree aggregation.\n\nCandidate contribution (decision_theoretic_loss_family; novelty confidence low): The explicit absence-penalty continuum x_lambda = p mu / [p + lambda(1-p)], together with the RF majority-median decomposition, gives a testable resolution of the workshop's exact 0.66-versus-1 branch-length example: the answer is determined by the declared loss assigned to split absence." }, { "id": 20000872, "problem_number": "AIM-BIOLOGY-0037", "title": "Exact counting with prescribed phylogenetic splits", "statement": "A.3 Monday Discussion on Combinatorial Issues\n\nThe disussion was moderated by Michelle Wachs. She started off the discussion by asking: Q. What combinatorial questions arise in biology that might be of interest to biologists? Or, what problems motivated by biology would be of interest to mathematicians, to give us some interesting problems?\n\n• Huelsenbeck: In putting priors on the set of all possible trees, we sometimes need to count trees with particular characteristics, such as ones with a particular kind of edge (split, bipartition), or a list of either this or that kind of edge. The constraints may be multiple or conflicting. The count is needed so that we can sum probabilities over these trees.\n\n• Holmes pointed out that inclusion/exclusion issues are involved here. There was some discussion about NP-hard problems, but Billera pointed out that complexity issues are not that relevant for the basic question of how to count these things. Q. Given a set S of k edges, and a subset J of S, how many leaf labeled binary trees have some subset J of S but not the edges in S − J? Do this for all J.\n\n• Evans asked why we can't just use MCMC to solve this problem? Holmes pointed out that the combinatorics may give some insight.\n\n• Another example of a question that arises: the number of trees that are distance d\n\nfrom a particular set of trees (e.g., using a distance such as nearest-neighbor inter-change)? Q. What are good codings for trees?\n\n• Wachs described a bijection between phylogenetic trees and type B permutations whose left-to-right minimum is unbarred. (Type B permutations are permutations in which each number can be barred or unbarred.) This can be used as a coding for trees.\n\n• There was a question about Penny's method for coding trees (in which each time you add an edge there are 2 n − 1 places to add it, so that you get (2 n − 1)!! of them). Billera said that this is the same as what was on the board.\n\n• Penny mentioned the existence of another scheme for coding each tree by a unique number. Holmes pointed out that the biologists use a Newick notation (New Hamp-shire format) that uses lots of parentheses. Another coding is a matrix where the columns are edges, and the rows are vertices, and you put a 1 if the edge is an ancestor of the vertex. (Brooks used it in biology, and Graham-Winkler used for addressing a network). 8\n\n• Holmes described a matching representation is one due to Diaconis-Holmes. When\n\nn = 2 there is one matching, when n = 4 there are 3 matchings, when n = 6 there are 15 matchings, corresponding to 4 leaf trees. The matchings are the sibling pairs. Example: n = 6. Call the numbers from 1 to 4 available numbers, and the other numbers unavailable. Given a matching 16 − 42 − 53, pick the first available pair of numbers. By pigeonhole argument, there is some pair, that's a sibling pair in a tree. Then the smallest non-available number (5, in this case) is the parent, and the other member of the pair (3) is the sibling of 5. Continue.\n\n• Diaconis pointed out their matching method is related to Billera's method using Prufer codes. Q. Is there an efficient representation for coding trees? (Billera) Q. Is there a nice neat notation for trees that is continuous when doing a random walk on trees? In other words, do small changes in notation correspond to close trees? (Diaconis)\n\n• The type B permutation coding might be very amenable to doing a random walk. Bridson pointed out that this representation is related to subsets of the Cayley graph of the Weyl group. Q. Are there questions that combinatorialists might have for biologists? Combinatorial structures that biologists might be interesting? (Wachs) Q. If we took trees and instead of putting real numbers of them, putting discrete values, like 0-1, would that be of interest in biologists? (Diaconis)\n\n• Huelsenbeck: this maybe useful in character-mapping, mapping characters that are on or off. Perhaps some substitutions are more likely when characters are on.\n\n• Epstein: it certainly seems reasonable to associate to each edge a vector, rather than a real number.\n\n• Evans: as in Penny's talk, characters of each of the vertices can be thought of as elements of the Klein 4-group... the edges can represent differences of elements at the vertices.\n\n• Wachs: what about k-ary trees?\n\n• Penny: people find cycles useful for representing uncertainty. Referred to some who tried to define how tree-like the data was, something tree-like would have only small cycles.\n\n• Holmes referred to a program called splitstree that makes such diagrams.\n\n• Penny: biologist would like structures that represents distances accurately.\n\n• Shareshian: described a space that arose in his work, a space in which ( n − 2)! cycles correspond to associahedra.", "original_statement": "A.3 Monday Discussion on Combinatorial Issues \n\nThe disussion was moderated by Michelle Wachs. She started off the discussion by asking: Q. What combinatorial questions arise in biology that might be of interest to biologists? Or, what problems motivated by biology would be of interest to mathematicians, to give us some interesting problems? \n\n• Huelsenbeck: In putting priors on the set of all possible trees, we sometimes need to count trees with particular characteristics, such as ones with a particular kind of edge (split, bipartition), or a list of either this or that kind of edge. The constraints may be multiple or conflicting. The count is needed so that we can sum probabilities over these trees. \n\n• Holmes pointed out that inclusion/exclusion issues are involved here. There was some discussion about NP-hard problems, but Billera pointed out that complexity issues are not that relevant for the basic question of how to count these things. Q. Given a set S of k edges, and a subset J of S, how many leaf labeled binary trees have some subset J of S but not the edges in S − J? Do this for all J.\n\n• Evans asked why we can't just use MCMC to solve this problem? Holmes pointed out that the combinatorics may give some insight. \n\n• Another example of a question that arises: the number of trees that are distance d\n\nfrom a particular set of trees (e.g., using a distance such as nearest-neighbor inter-change)? Q. What are good codings for trees? \n\n• Wachs described a bijection between phylogenetic trees and type B permutations whose left-to-right minimum is unbarred. (Type B permutations are permutations in which each number can be barred or unbarred.) This can be used as a coding for trees. \n\n• There was a question about Penny's method for coding trees (in which each time you add an edge there are 2 n − 1 places to add it, so that you get (2 n − 1)!! of them). Billera said that this is the same as what was on the board. \n\n• Penny mentioned the existence of another scheme for coding each tree by a unique number. Holmes pointed out that the biologists use a Newick notation (New Hamp-shire format) that uses lots of parentheses. Another coding is a matrix where the columns are edges, and the rows are vertices, and you put a 1 if the edge is an ancestor of the vertex. (Brooks used it in biology, and Graham-Winkler used for addressing a network). 8\n\n• Holmes described a matching representation is one due to Diaconis-Holmes. When \n\nn = 2 there is one matching, when n = 4 there are 3 matchings, when n = 6 there are 15 matchings, corresponding to 4 leaf trees. The matchings are the sibling pairs. Example: n = 6. Call the numbers from 1 to 4 available numbers, and the other numbers unavailable. Given a matching 16 − 42 − 53, pick the first available pair of numbers. By pigeonhole argument, there is some pair, that's a sibling pair in a tree. Then the smallest non-available number (5, in this case) is the parent, and the other member of the pair (3) is the sibling of 5. Continue. \n\n• Diaconis pointed out their matching method is related to Billera's method using Prufer codes. Q. Is there an efficient representation for coding trees? (Billera) Q. Is there a nice neat notation for trees that is continuous when doing a random walk on trees? In other words, do small changes in notation correspond to close trees? (Diaconis) \n\n• The type B permutation coding might be very amenable to doing a random walk. Bridson pointed out that this representation is related to subsets of the Cayley graph of the Weyl group. Q. Are there questions that combinatorialists might have for biologists? Combinatorial structures that biologists might be interesting? (Wachs) Q. If we took trees and instead of putting real numbers of them, putting discrete values, like 0-1, would that be of interest in biologists? (Diaconis) \n\n• Huelsenbeck: this maybe useful in character-mapping, mapping characters that are on or off. Perhaps some substitutions are more likely when characters are on. \n\n• Epstein: it certainly seems reasonable to associate to each edge a vector, rather than a real number. \n\n• Evans: as in Penny's talk, characters of each of the vertices can be thought of as elements of the Klein 4-group... the edges can represent differences of elements at the vertices. \n\n• Wachs: what about k-ary trees? \n\n• Penny: people find cycles useful for representing uncertainty. Referred to some who tried to define how tree-like the data was, something tree-like would have only small cycles. \n\n• Holmes referred to a program called splitstree that makes such diagrams. \n\n• Penny: biologist would like structures that represents distances accurately. \n\n• Shareshian: described a space that arose in his work, a space in which ( n − 2)! cycles correspond to associahedra.", "clean_statement": "A.3 Monday Discussion on Combinatorial Issues\n\nThe disussion was moderated by Michelle Wachs. She started off the discussion by asking: Q. What combinatorial questions arise in biology that might be of interest to biologists? Or, what problems motivated by biology would be of interest to mathematicians, to give us some interesting problems?\n\n• Huelsenbeck: In putting priors on the set of all possible trees, we sometimes need to count trees with particular characteristics, such as ones with a particular kind of edge (split, bipartition), or a list of either this or that kind of edge. The constraints may be multiple or conflicting. The count is needed so that we can sum probabilities over these trees.\n\n• Holmes pointed out that inclusion/exclusion issues are involved here. There was some discussion about NP-hard problems, but Billera pointed out that complexity issues are not that relevant for the basic question of how to count these things. Q. Given a set S of k edges, and a subset J of S, how many leaf labeled binary trees have some subset J of S but not the edges in S − J? Do this for all J.\n\n• Evans asked why we can't just use MCMC to solve this problem? Holmes pointed out that the combinatorics may give some insight.\n\n• Another example of a question that arises: the number of trees that are distance d\n\nfrom a particular set of trees (e.g., using a distance such as nearest-neighbor inter-change)? Q. What are good codings for trees?\n\n• Wachs described a bijection between phylogenetic trees and type B permutations whose left-to-right minimum is unbarred. (Type B permutations are permutations in which each number can be barred or unbarred.) This can be used as a coding for trees.\n\n• There was a question about Penny's method for coding trees (in which each time you add an edge there are 2 n − 1 places to add it, so that you get (2 n − 1)!! of them). Billera said that this is the same as what was on the board.\n\n• Penny mentioned the existence of another scheme for coding each tree by a unique number. Holmes pointed out that the biologists use a Newick notation (New Hamp-shire format) that uses lots of parentheses. Another coding is a matrix where the columns are edges, and the rows are vertices, and you put a 1 if the edge is an ancestor of the vertex. (Brooks used it in biology, and Graham-Winkler used for addressing a network). 8\n\n• Holmes described a matching representation is one due to Diaconis-Holmes. When\n\nn = 2 there is one matching, when n = 4 there are 3 matchings, when n = 6 there are 15 matchings, corresponding to 4 leaf trees. The matchings are the sibling pairs. Example: n = 6. Call the numbers from 1 to 4 available numbers, and the other numbers unavailable. Given a matching 16 − 42 − 53, pick the first available pair of numbers. By pigeonhole argument, there is some pair, that's a sibling pair in a tree. Then the smallest non-available number (5, in this case) is the parent, and the other member of the pair (3) is the sibling of 5. Continue.\n\n• Diaconis pointed out their matching method is related to Billera's method using Prufer codes. Q. Is there an efficient representation for coding trees? (Billera) Q. Is there a nice neat notation for trees that is continuous when doing a random walk on trees? In other words, do small changes in notation correspond to close trees? (Diaconis)\n\n• The type B permutation coding might be very amenable to doing a random walk. Bridson pointed out that this representation is related to subsets of the Cayley graph of the Weyl group. Q. Are there questions that combinatorialists might have for biologists? Combinatorial structures that biologists might be interesting? (Wachs) Q. If we took trees and instead of putting real numbers of them, putting discrete values, like 0-1, would that be of interest in biologists? (Diaconis)\n\n• Huelsenbeck: this maybe useful in character-mapping, mapping characters that are on or off. Perhaps some substitutions are more likely when characters are on.\n\n• Epstein: it certainly seems reasonable to associate to each edge a vector, rather than a real number.\n\n• Evans: as in Penny's talk, characters of each of the vertices can be thought of as elements of the Klein 4-group... the edges can represent differences of elements at the vertices.\n\n• Wachs: what about k-ary trees?\n\n• Penny: people find cycles useful for representing uncertainty. Referred to some who tried to define how tree-like the data was, something tree-like would have only small cycles.\n\n• Holmes referred to a program called splitstree that makes such diagrams.\n\n• Penny: biologist would like structures that represents distances accurately.\n\n• Shareshian: described a space that arose in his work, a space in which ( n − 2)! cycles correspond to associahedra.", "statement_status": "exact", "statement_verification": "This record is Section A.3, “Monday Discussion on Combinatorial Issues,” in the American Institute of Mathematics workshop notes *Geometric Models of Biological Phenomena*. Unlike a purely contextual section, it contains a definite counting problem:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Geometric models of biological phenomena\nSection: \nSource item: A.3\nSource URL: https://aimath.org/WWN/geombio/geombio.pdf\nCanonical location: aim-biology-notes.json notes[36]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.3 Monday Discussion on Combinatorial Issues \\n\\nThe disussion was moderated by Michelle Wachs. She started off the discussion by asking: Q. What combinatorial questions arise in biology that might be of interest to biologists? Or, what problems motivated by biology would be of interest to mathematicians, to give us some interesting problems? \\n\\n• Huelsenbeck: In putting priors on the set of all possible trees, we sometimes need to count trees with particular characteristics, such as ones with a particular kind of edge (split, bipartition), or a list of either this or that kind of edge. The constraints may be multiple or conflicting. The count is needed so that we can sum probabilities over these trees. \\n\\n• Holmes pointed out that inclusion/exclusion issues are involved here. There was some discussion about NP-hard problems, but Billera pointed out that complexity issues are not that relevant for the basic question of how to count these things. Q. Given a set S of k edges, and a subset J of S, how many leaf labeled binary trees have some subset J of S but not the edges in S − J? Do this for all J.\\n\\n• Evans asked why we can't just use MCMC to solve this problem? Holmes pointed out that the combinatorics may give some insight. \\n\\n• Another example of a question that arises: the number of trees that are distance d\\n\\nfrom a particular set of trees (e.g., using a distance such as nearest-neighbor inter-change)? Q. What are good codings for trees? \\n\\n• Wachs described a bijection between phylogenetic trees and type B permutations whose left-to-right minimum is unbarred. (Type B permutations are permutations in which each number can be barred or unbarred.) This can be used as a coding for trees. \\n\\n• There was a question about Penny's method for coding trees (in which each time you add an edge there are 2 n − 1 places to add it, so that you get (2 n − 1)!! of them). Billera said that this is the same as what was on the board. \\n\\n• Penny mentioned the existence of another scheme for coding each tree by a unique number. Holmes pointed out that the biologists use a Newick notation (New Hamp-shire format) that uses lots of parentheses. Another coding is a matrix where the columns are edges, and the rows are vertices, and you put a 1 if the edge is an ancestor of the vertex. (Brooks used it in biology, and Graham-Winkler used for addressing a network). 8\\n\\n• Holmes described a matching representation is one due to Diaconis-Holmes. When \\n\\nn = 2 there is one matching, when n = 4 there are 3 matchings, when n = 6 there are 15 matchings, corresponding to 4 leaf trees. The matchings are the sibling pairs. Example: n = 6. Call the numbers from 1 to 4 available numbers, and the other numbers unavailable. Given a matching 16 − 42 − 53, pick the first available pair of numbers. By pigeonhole argument, there is some pair, that's a sibling pair in a tree. Then the smallest non-available number (5, in this case) is the parent, and the other member of the pair (3) is the sibling of 5. Continue. \\n\\n• Diaconis pointed out their matching method is related to Billera's method using Prufer codes. Q. Is there an efficient representation for coding trees? (Billera) Q. Is there a nice neat notation for trees that is continuous when doing a random walk on trees? In other words, do small changes in notation correspond to close trees? (Diaconis) \\n\\n• The type B permutation coding might be very amenable to doing a random walk. Bridson pointed out that this representation is related to subsets of the Cayley graph of the Weyl group. Q. Are there questions that combinatorialists might have for biologists? Combinatorial structures that biologists might be interesting? (Wachs) Q. If we took trees and instead of putting real numbers of them, putting discrete values, like 0-1, would that be of interest in biologists? (Diaconis) \\n\\n• Huelsenbeck: this maybe useful in character-mapping, mapping characters that are on or off. Perhaps some substitutions are more likely when characters are on. \\n\\n• Epstein: it certainly seems reasonable to associate to each edge a vector, rather than a real number. \\n\\n• Evans: as in Penny's talk, characters of each of the vertices can be thought of as elements of the Klein 4-group... the edges can represent differences of elements at the vertices. \\n\\n• Wachs: what about k-ary trees? \\n\\n• Penny: people find cycles useful for representing uncertainty. Referred to some who tried to define how tree-like the data was, something tree-like would have only small cycles. \\n\\n• Holmes referred to a program called splitstree that makes such diagrams. \\n\\n• Penny: biologist would like structures that represents distances accurately. \\n\\n• Shareshian: described a space that arose in his work, a space in which ( n − 2)! cycles correspond to associahedra.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/geombio/geombio.pdf", "tags": [ "aim", "AIM-BIOLOGY-0037", "aim-domain:biology", "aim-workshop:geombio", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under an explicit unrooted, nonplanar binary phylogenetic-tree convention on a fixed n-leaf label set, the prescribed-split subproblem has an exact solution. A required family F has zero containing trees if it is incompatible; otherwise its unique least-resolved tree T_F has exactly the product over internal vertices v of (2 deg(v)-5)!! binary refinements. For distinct nontrivial candidate splits S, the number whose intersection with S is exactly J equals the Boolean Mobius inversion sum over K containing J of (-1)^(|K|-|J|) R(K). A multivariate generating polynomial packages all exact patterns. The artifacts also give a rooted outgroup translation, forced-trivial and duplicate-split rules, and four- and five-leaf checks.\n\nCandidate contribution (enumeration_formula_and_reduction; novelty confidence low): Candidate novelty: a convention-complete all-pattern enumerator for the AIM question that combines compatibility-pruned local refinement products with Boolean Mobius inversion, explicitly preprocesses duplicate and trivial splits, and transfers rooted clade constraints to the same formula by adjoining an outgroup leaf." }, { "id": 20000873, "problem_number": "AIM-BIOLOGY-0038", "title": "Exact split-residual certificates in BHV tree space", "statement": "A.4 Tuesday Discussion on Statistical Issues\n\nThis discussion was moderated by Ruth Charney. Q. Diaconis suggested making a list of the kinds of noise, or sources of variation, that are implicit in estimating trees? Participant discussion led to the following list:\n\n• alignment\n\n• errors in sequencing, (though Epstein pointed out that the quality is steadily improv-ing) 9\n\n• misspecification of the model (process of going from the data to a tree): mutation rates, independence between sites, change of nucleotide composition\n\n• variation within species (1 in 1000 genes)\n\n• bias in corrections for distances (for distance based models)\n\n• variation between fragments of same DNA (bias created by choice of fragments)\n\n• selection varies in different parts of the genome\n\n• gene identification (problems created by gene duplication and gene loss)\n\n• hybridization (branching comes back together, structure is not a tree) and horizontal gene transfer\n\n• optimization\n\n• not enough data- length of sequences, number of taxa Q. are these problems worth working on incrementally or all at once? Which ones are most important? (Diaconis)\n\n• Penny: differences in nucleotide composition? Q. Can we define the geometry (distance) on tree space to behave well with respect to a particular model?\n\n• Vert: what are the implications of the noise in the definition of the tree space? Instead of defining the geometry of tree space beforehand, and then asking what properties it satisfies, would it be possible to define a tree space in terms in such a way that it has good properties (is stable, averages are at least as good as the trees themselves, etc.)\n\n• Diaconis: this is perhaps related to statistical geometry or information geometry- using the Fisher information to define a Riemannian metric on the space of distribu-tions.\n\n• Epstein: an average is a summary statistic, but doesn't summarize everything we might want.\n\n• Evans: information geometry example: using the upper half plane to parametrize normals on a line, the geometry that best reflects closeness of normals is hyperbolic geometry. Q. What are residuals for trees, and how do we estimate the residuals? (Residuals measure how far each data point is from fitted tree.) Are there graphical methods for detecting departure from the model? (Penny)\n\n• Penny also asked why there are so many definitions of maximum likelihood?\n\n• Diaconis suggested that one topic probabalists and biologists may benefit from is work of Aldous and students, on analyzing rates of convergence of random walks on phylogenetic trees. Aldous responded by noting that the case they can analyze is not necessarily so useful to biologists (one where leaf is taken off and pushed somewhere else). For an n-leaf tree, the random walk needs n2 random steps. A more realistic example is to cut somewhere, and attach whole subtree in a different place. It is believed that about n3/2 steps is needed to mix this kind of chain. This could be useful in MCMC algorithms. Q. Random walks on sets of trees (using tree rotations)- how fast does it converge?\n\n• Once again, the question of how to measure distance in the space of trees emerged. Q. What are appropriate meeasures on tree space? 10\n\n• Glenn asked if there a need for a non-parametric likelihood on trees? Diaconis said that empirical likelihood related to the boostrap, so it could contribute. Q. If exact distances aren't easy to compute, can we obtain any upper and lower bounds for distances (such as the BHV metric)? (Su)", "original_statement": "A.4 Tuesday Discussion on Statistical Issues \n\nThis discussion was moderated by Ruth Charney. Q. Diaconis suggested making a list of the kinds of noise, or sources of variation, that are implicit in estimating trees? Participant discussion led to the following list: \n\n• alignment \n\n• errors in sequencing, (though Epstein pointed out that the quality is steadily improv-ing) 9\n\n• misspecification of the model (process of going from the data to a tree): mutation rates, independence between sites, change of nucleotide composition \n\n• variation within species (1 in 1000 genes) \n\n• bias in corrections for distances (for distance based models) \n\n• variation between fragments of same DNA (bias created by choice of fragments) \n\n• selection varies in different parts of the genome \n\n• gene identification (problems created by gene duplication and gene loss) \n\n• hybridization (branching comes back together, structure is not a tree) and horizontal gene transfer \n\n• optimization \n\n• not enough data- length of sequences, number of taxa Q. are these problems worth working on incrementally or all at once? Which ones are most important? (Diaconis) \n\n• Penny: differences in nucleotide composition? Q. Can we define the geometry (distance) on tree space to behave well with respect to a particular model? \n\n• Vert: what are the implications of the noise in the definition of the tree space? Instead of defining the geometry of tree space beforehand, and then asking what properties it satisfies, would it be possible to define a tree space in terms in such a way that it has good properties (is stable, averages are at least as good as the trees themselves, etc.) \n\n• Diaconis: this is perhaps related to statistical geometry or information geometry- using the Fisher information to define a Riemannian metric on the space of distribu-tions. \n\n• Epstein: an average is a summary statistic, but doesn't summarize everything we might want. \n\n• Evans: information geometry example: using the upper half plane to parametrize normals on a line, the geometry that best reflects closeness of normals is hyperbolic geometry. Q. What are residuals for trees, and how do we estimate the residuals? (Residuals measure how far each data point is from fitted tree.) Are there graphical methods for detecting departure from the model? (Penny) \n\n• Penny also asked why there are so many definitions of maximum likelihood? \n\n• Diaconis suggested that one topic probabalists and biologists may benefit from is work of Aldous and students, on analyzing rates of convergence of random walks on phylogenetic trees. Aldous responded by noting that the case they can analyze is not necessarily so useful to biologists (one where leaf is taken off and pushed somewhere else). For an n-leaf tree, the random walk needs n2 random steps. A more realistic example is to cut somewhere, and attach whole subtree in a different place. It is believed that about n3/2 steps is needed to mix this kind of chain. This could be useful in MCMC algorithms. Q. Random walks on sets of trees (using tree rotations)- how fast does it converge? \n\n• Once again, the question of how to measure distance in the space of trees emerged. Q. What are appropriate meeasures on tree space? 10 \n\n• Glenn asked if there a need for a non-parametric likelihood on trees? Diaconis said that empirical likelihood related to the boostrap, so it could contribute. Q. If exact distances aren't easy to compute, can we obtain any upper and lower bounds for distances (such as the BHV metric)? (Su)", "clean_statement": "A.4 Tuesday Discussion on Statistical Issues\n\nThis discussion was moderated by Ruth Charney. Q. Diaconis suggested making a list of the kinds of noise, or sources of variation, that are implicit in estimating trees? Participant discussion led to the following list:\n\n• alignment\n\n• errors in sequencing, (though Epstein pointed out that the quality is steadily improv-ing) 9\n\n• misspecification of the model (process of going from the data to a tree): mutation rates, independence between sites, change of nucleotide composition\n\n• variation within species (1 in 1000 genes)\n\n• bias in corrections for distances (for distance based models)\n\n• variation between fragments of same DNA (bias created by choice of fragments)\n\n• selection varies in different parts of the genome\n\n• gene identification (problems created by gene duplication and gene loss)\n\n• hybridization (branching comes back together, structure is not a tree) and horizontal gene transfer\n\n• optimization\n\n• not enough data- length of sequences, number of taxa Q. are these problems worth working on incrementally or all at once? Which ones are most important? (Diaconis)\n\n• Penny: differences in nucleotide composition? Q. Can we define the geometry (distance) on tree space to behave well with respect to a particular model?\n\n• Vert: what are the implications of the noise in the definition of the tree space? Instead of defining the geometry of tree space beforehand, and then asking what properties it satisfies, would it be possible to define a tree space in terms in such a way that it has good properties (is stable, averages are at least as good as the trees themselves, etc.)\n\n• Diaconis: this is perhaps related to statistical geometry or information geometry- using the Fisher information to define a Riemannian metric on the space of distribu-tions.\n\n• Epstein: an average is a summary statistic, but doesn't summarize everything we might want.\n\n• Evans: information geometry example: using the upper half plane to parametrize normals on a line, the geometry that best reflects closeness of normals is hyperbolic geometry. Q. What are residuals for trees, and how do we estimate the residuals? (Residuals measure how far each data point is from fitted tree.) Are there graphical methods for detecting departure from the model? (Penny)\n\n• Penny also asked why there are so many definitions of maximum likelihood?\n\n• Diaconis suggested that one topic probabalists and biologists may benefit from is work of Aldous and students, on analyzing rates of convergence of random walks on phylogenetic trees. Aldous responded by noting that the case they can analyze is not necessarily so useful to biologists (one where leaf is taken off and pushed somewhere else). For an n-leaf tree, the random walk needs n2 random steps. A more realistic example is to cut somewhere, and attach whole subtree in a different place. It is believed that about n3/2 steps is needed to mix this kind of chain. This could be useful in MCMC algorithms. Q. Random walks on sets of trees (using tree rotations)- how fast does it converge?\n\n• Once again, the question of how to measure distance in the space of trees emerged. Q. What are appropriate meeasures on tree space? 10\n\n• Glenn asked if there a need for a non-parametric likelihood on trees? Diaconis said that empirical likelihood related to the boostrap, so it could contribute. Q. If exact distances aren't easy to compute, can we obtain any upper and lower bounds for distances (such as the BHV metric)? (Su)", "statement_status": "exact", "statement_verification": "The canonical record is section A.4, “Tuesday Discussion on Statistical Issues,” in the American Institute of Mathematics workshop notes *Geometric models of biological phenomena*. The official PDF was inspected directly. It identifies the American Institute of Mathematics, gives version time “Wed Jun 18 16:51:55 2003,” and credits the discussion-session notes to Francis Su at the chapter level. A.4 occupies PDF pages 8–10 (printed pages 9–10) and says the session was moderated by Ruth Charney.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Geometric models of biological phenomena\nSection: \nSource item: A.4\nSource URL: https://aimath.org/WWN/geombio/geombio.pdf\nCanonical location: aim-biology-notes.json notes[37]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.4 Tuesday Discussion on Statistical Issues \\n\\nThis discussion was moderated by Ruth Charney. Q. Diaconis suggested making a list of the kinds of noise, or sources of variation, that are implicit in estimating trees? Participant discussion led to the following list: \\n\\n• alignment \\n\\n• errors in sequencing, (though Epstein pointed out that the quality is steadily improv-ing) 9\\n\\n• misspecification of the model (process of going from the data to a tree): mutation rates, independence between sites, change of nucleotide composition \\n\\n• variation within species (1 in 1000 genes) \\n\\n• bias in corrections for distances (for distance based models) \\n\\n• variation between fragments of same DNA (bias created by choice of fragments) \\n\\n• selection varies in different parts of the genome \\n\\n• gene identification (problems created by gene duplication and gene loss) \\n\\n• hybridization (branching comes back together, structure is not a tree) and horizontal gene transfer \\n\\n• optimization \\n\\n• not enough data- length of sequences, number of taxa Q. are these problems worth working on incrementally or all at once? Which ones are most important? (Diaconis) \\n\\n• Penny: differences in nucleotide composition? Q. Can we define the geometry (distance) on tree space to behave well with respect to a particular model? \\n\\n• Vert: what are the implications of the noise in the definition of the tree space? Instead of defining the geometry of tree space beforehand, and then asking what properties it satisfies, would it be possible to define a tree space in terms in such a way that it has good properties (is stable, averages are at least as good as the trees themselves, etc.) \\n\\n• Diaconis: this is perhaps related to statistical geometry or information geometry- using the Fisher information to define a Riemannian metric on the space of distribu-tions. \\n\\n• Epstein: an average is a summary statistic, but doesn't summarize everything we might want. \\n\\n• Evans: information geometry example: using the upper half plane to parametrize normals on a line, the geometry that best reflects closeness of normals is hyperbolic geometry. Q. What are residuals for trees, and how do we estimate the residuals? (Residuals measure how far each data point is from fitted tree.) Are there graphical methods for detecting departure from the model? (Penny) \\n\\n• Penny also asked why there are so many definitions of maximum likelihood? \\n\\n• Diaconis suggested that one topic probabalists and biologists may benefit from is work of Aldous and students, on analyzing rates of convergence of random walks on phylogenetic trees. Aldous responded by noting that the case they can analyze is not necessarily so useful to biologists (one where leaf is taken off and pushed somewhere else). For an n-leaf tree, the random walk needs n2 random steps. A more realistic example is to cut somewhere, and attach whole subtree in a different place. It is believed that about n3/2 steps is needed to mix this kind of chain. This could be useful in MCMC algorithms. Q. Random walks on sets of trees (using tree rotations)- how fast does it converge? \\n\\n• Once again, the question of how to measure distance in the space of trees emerged. Q. What are appropriate meeasures on tree space? 10 \\n\\n• Glenn asked if there a need for a non-parametric likelihood on trees? Diaconis said that empirical likelihood related to the boostrap, so it could contribute. Q. If exact distances aren't easy to compute, can we obtain any upper and lower bounds for distances (such as the BHV metric)? (Su)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/geombio/geombio.pdf", "tags": [ "aim", "AIM-BIOLOGY-0038", "aim-domain:biology", "aim-workshop:geombio", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For fixed-leaf BHV tree space and any declared set A of allowed splits, let M_A contain every compatible metric tree whose positive splits lie in A. The BHV distance from a tree T to M_A is exactly the Euclidean norm of the lengths of T's forbidden splits, and coordinate deletion is the unique nearest tree. This yields an exact per-split residual decomposition, a 1-Lipschitz finite-error misspecification certificate, and a dataset least-squares lower bound. The report also recovers the known sharp square-root-of-two BHV two-tree bracket without claiming it as new.\n\nCandidate contribution (exact_residual_certificate; novelty confidence low): The candidate contribution packages the exact identity dist_BHV(T,M_A)^2 = sum over forbidden positive splits sigma of x_sigma(T)^2 with uniqueness of the coordinate-deletion projection, and proves the testable robustness rule that any target within BHV error epsilon of an estimate has residual at least max(0, observed residual minus epsilon)." }, { "id": 20000874, "problem_number": "AIM-BIOLOGY-0039", "title": "Sharp scale-bounded comparison of BHV and Robinson-Foulds distance", "statement": "A.5 Wednesday Discussion on Geometric Issues\n\nThis session was moderated by John Shareshian. The discussion started off with some comments about Riemannian metrics and whether the set of metrics on tree space would be useful to study.\n\n• Vogtmann asked if this space may be too large to study?\n\n• Forman noted that when geometers look at the space of all Riemannian metrics on a given space, they do it to try to find the \"best\" metric in some sense (e.g., such as one with constant curvature). Q. Are there many useful or interesting metrics on tree space? Q. Is the BHV metric the only one (up to scalars) of non-positive curvature? (Bridson)\n\n• Charney: metrics are slightly different, but they should all be similar. If there is no reason that biologists suggest that one is better than another, then it makes sense to choose one that has useful properties, such as metrics of nonpositive curvature.\n\n• Billera: for biologists, what metrics do you want to have on the orthants? Q. What metrics \"should\" be used on the subspace obtained by fixing the tree type?\n\n• St. John: uses the L1 metric, though Felsenstein uses L2.\n\n• Penny: we use L1 because then lengths scale with time.\n\n• Charney: since the data is not completely in tree space, maybe we should be consid-ering non-intrinsic metrics.\n\n• Forman: want to choose metrics so that the statistical methods we are using are continuous.\n\n• Flath: perhaps some edges are more important than others?\n\n• Penny: we are not surprised when we get edges near leaves are accurate, but we are really interested in getting deep internal edges right.\n\n• Penny: taxonomic studies, just care about the branch order (weight 1 on edges), but when we consider time studies, we do want the lengths.\n\n• Forman: measuring the residuals seem to imply that embedding tree space in some larger space and considering an extrinsic metric would be important.\n\n• Vogtmann noted that Felsenstein was talking about embedding tree space in Eu-clidean space (using the Robinson-Foulds metric). Q. How does the Robinson-Foulds metric compare with the BHV metric? (Diaconis) What's the Lipshitz constant? (Bridson)\n\n• Vert: Another distance (referred to by Felsenstein in his talk) is Kullback-Leibler distance (relative entropy) between probabilities. Fix a model. Each tree defines a probability on the set of assignments of letters (A,G,C,T) to leaves. Maximum likelihood corresponds to projection of the empirical measure of a data point to tree space according to this distance. 11\n\nThere was some explanation and discussion of this embedding. Diaconis noted that while there are many metrics on probability distributions to consider here, the Kullback-Liebler separation is good for maximum likelihood. Q. Should the metric on tree space come from an intrinsic metric, or an extrinsic metric on a larger space in which tree space is embedded? Which is a more natural way to view tree space?\n\n• Forman: studying the residuals seems to be very important. This larger space is part of what you are given in the data, and should be useful. Charney rephrased the question as: what is the right ambient space to embed tree space? While Bridson remarked that he viewed tree space as God-given, Forman asserted that he views the data as God-given.\n\n• Vert remarked that the \"average\" of trees with high likelihood may not be high likelihood with the intrinsic metric. Q. How do you deliver geometry to biologists? (Billera)\n\n• Billera pointed out that interesting mathematics in biology seems to happen by serendipity- cases in which someone happened to know some mathematics or knew someone who did.\n\n• Penny: notes that there is a role for mathematics to play. Also new majors in mathematical biology (at some schools) have arisen that will train students to think in both disciplines.\n\n• Diaconis notes that as a result of this conference, and ensuing discussions, people might write an article for Science. This may bring biologists to mathematics.\n\n• Snel suggests advocating a notion of average, and communicate why it is better. For instance, articles on concatenated alignments have appeared in Science.\n\n• Huelsenbeck: write review articles that explain things clearly. Write for Systematics Biology - that's where stuff on trees would appear.\n\n• Billera asked again how one spreads mathematical knowledge in the biological com-munity, noting that courses offered at most institutions are either at too low or too high a level for biologists.", "original_statement": "A.5 Wednesday Discussion on Geometric Issues \n\nThis session was moderated by John Shareshian. The discussion started off with some comments about Riemannian metrics and whether the set of metrics on tree space would be useful to study. \n\n• Vogtmann asked if this space may be too large to study? \n\n• Forman noted that when geometers look at the space of all Riemannian metrics on a given space, they do it to try to find the \"best\" metric in some sense (e.g., such as one with constant curvature). Q. Are there many useful or interesting metrics on tree space? Q. Is the BHV metric the only one (up to scalars) of non-positive curvature? (Bridson) \n\n• Charney: metrics are slightly different, but they should all be similar. If there is no reason that biologists suggest that one is better than another, then it makes sense to choose one that has useful properties, such as metrics of nonpositive curvature. \n\n• Billera: for biologists, what metrics do you want to have on the orthants? Q. What metrics \"should\" be used on the subspace obtained by fixing the tree type? \n\n• St. John: uses the L1 metric, though Felsenstein uses L2.\n\n• Penny: we use L1 because then lengths scale with time. \n\n• Charney: since the data is not completely in tree space, maybe we should be consid-ering non-intrinsic metrics. \n\n• Forman: want to choose metrics so that the statistical methods we are using are continuous. \n\n• Flath: perhaps some edges are more important than others? \n\n• Penny: we are not surprised when we get edges near leaves are accurate, but we are really interested in getting deep internal edges right. \n\n• Penny: taxonomic studies, just care about the branch order (weight 1 on edges), but when we consider time studies, we do want the lengths. \n\n• Forman: measuring the residuals seem to imply that embedding tree space in some larger space and considering an extrinsic metric would be important. \n\n• Vogtmann noted that Felsenstein was talking about embedding tree space in Eu-clidean space (using the Robinson-Foulds metric). Q. How does the Robinson-Foulds metric compare with the BHV metric? (Diaconis) What's the Lipshitz constant? (Bridson) \n\n• Vert: Another distance (referred to by Felsenstein in his talk) is Kullback-Leibler distance (relative entropy) between probabilities. Fix a model. Each tree defines a probability on the set of assignments of letters (A,G,C,T) to leaves. Maximum likelihood corresponds to projection of the empirical measure of a data point to tree space according to this distance. 11 \n\nThere was some explanation and discussion of this embedding. Diaconis noted that while there are many metrics on probability distributions to consider here, the Kullback-Liebler separation is good for maximum likelihood. Q. Should the metric on tree space come from an intrinsic metric, or an extrinsic metric on a larger space in which tree space is embedded? Which is a more natural way to view tree space? \n\n• Forman: studying the residuals seems to be very important. This larger space is part of what you are given in the data, and should be useful. Charney rephrased the question as: what is the right ambient space to embed tree space? While Bridson remarked that he viewed tree space as God-given, Forman asserted that he views the data as God-given. \n\n• Vert remarked that the \"average\" of trees with high likelihood may not be high likelihood with the intrinsic metric. Q. How do you deliver geometry to biologists? (Billera) \n\n• Billera pointed out that interesting mathematics in biology seems to happen by serendipity- cases in which someone happened to know some mathematics or knew someone who did. \n\n• Penny: notes that there is a role for mathematics to play. Also new majors in mathematical biology (at some schools) have arisen that will train students to think in both disciplines. \n\n• Diaconis notes that as a result of this conference, and ensuing discussions, people might write an article for Science. This may bring biologists to mathematics. \n\n• Snel suggests advocating a notion of average, and communicate why it is better. For instance, articles on concatenated alignments have appeared in Science. \n\n• Huelsenbeck: write review articles that explain things clearly. Write for Systematics Biology - that's where stuff on trees would appear. \n\n• Billera asked again how one spreads mathematical knowledge in the biological com-munity, noting that courses offered at most institutions are either at too low or too high a level for biologists.", "clean_statement": "A.5 Wednesday Discussion on Geometric Issues\n\nThis session was moderated by John Shareshian. The discussion started off with some comments about Riemannian metrics and whether the set of metrics on tree space would be useful to study.\n\n• Vogtmann asked if this space may be too large to study?\n\n• Forman noted that when geometers look at the space of all Riemannian metrics on a given space, they do it to try to find the \"best\" metric in some sense (e.g., such as one with constant curvature). Q. Are there many useful or interesting metrics on tree space? Q. Is the BHV metric the only one (up to scalars) of non-positive curvature? (Bridson)\n\n• Charney: metrics are slightly different, but they should all be similar. If there is no reason that biologists suggest that one is better than another, then it makes sense to choose one that has useful properties, such as metrics of nonpositive curvature.\n\n• Billera: for biologists, what metrics do you want to have on the orthants? Q. What metrics \"should\" be used on the subspace obtained by fixing the tree type?\n\n• St. John: uses the L1 metric, though Felsenstein uses L2.\n\n• Penny: we use L1 because then lengths scale with time.\n\n• Charney: since the data is not completely in tree space, maybe we should be consid-ering non-intrinsic metrics.\n\n• Forman: want to choose metrics so that the statistical methods we are using are continuous.\n\n• Flath: perhaps some edges are more important than others?\n\n• Penny: we are not surprised when we get edges near leaves are accurate, but we are really interested in getting deep internal edges right.\n\n• Penny: taxonomic studies, just care about the branch order (weight 1 on edges), but when we consider time studies, we do want the lengths.\n\n• Forman: measuring the residuals seem to imply that embedding tree space in some larger space and considering an extrinsic metric would be important.\n\n• Vogtmann noted that Felsenstein was talking about embedding tree space in Eu-clidean space (using the Robinson-Foulds metric). Q. How does the Robinson-Foulds metric compare with the BHV metric? (Diaconis) What's the Lipshitz constant? (Bridson)\n\n• Vert: Another distance (referred to by Felsenstein in his talk) is Kullback-Leibler distance (relative entropy) between probabilities. Fix a model. Each tree defines a probability on the set of assignments of letters (A,G,C,T) to leaves. Maximum likelihood corresponds to projection of the empirical measure of a data point to tree space according to this distance. 11\n\nThere was some explanation and discussion of this embedding. Diaconis noted that while there are many metrics on probability distributions to consider here, the Kullback-Liebler separation is good for maximum likelihood. Q. Should the metric on tree space come from an intrinsic metric, or an extrinsic metric on a larger space in which tree space is embedded? Which is a more natural way to view tree space?\n\n• Forman: studying the residuals seems to be very important. This larger space is part of what you are given in the data, and should be useful. Charney rephrased the question as: what is the right ambient space to embed tree space? While Bridson remarked that he viewed tree space as God-given, Forman asserted that he views the data as God-given.\n\n• Vert remarked that the \"average\" of trees with high likelihood may not be high likelihood with the intrinsic metric. Q. How do you deliver geometry to biologists? (Billera)\n\n• Billera pointed out that interesting mathematics in biology seems to happen by serendipity- cases in which someone happened to know some mathematics or knew someone who did.\n\n• Penny: notes that there is a role for mathematics to play. Also new majors in mathematical biology (at some schools) have arisen that will train students to think in both disciplines.\n\n• Diaconis notes that as a result of this conference, and ensuing discussions, people might write an article for Science. This may bring biologists to mathematics.\n\n• Snel suggests advocating a notion of average, and communicate why it is better. For instance, articles on concatenated alignments have appeared in Science.\n\n• Huelsenbeck: write review articles that explain things clearly. Write for Systematics Biology - that's where stuff on trees would appear.\n\n• Billera asked again how one spreads mathematical knowledge in the biological com-munity, noting that courses offered at most institutions are either at too low or too high a level for biologists.", "statement_status": "exact", "statement_verification": "This record is Section A.5, “Wednesday Discussion on Geometric Issues,” in the official AIM workshop notes *Geometric Models of Biological Phenomena*, version dated 18 June 2003. John Shareshian moderated the session. The record contains genuine questions:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Geometric models of biological phenomena\nSection: \nSource item: A.5\nSource URL: https://aimath.org/WWN/geombio/geombio.pdf\nCanonical location: aim-biology-notes.json notes[38]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.5 Wednesday Discussion on Geometric Issues \\n\\nThis session was moderated by John Shareshian. The discussion started off with some comments about Riemannian metrics and whether the set of metrics on tree space would be useful to study. \\n\\n• Vogtmann asked if this space may be too large to study? \\n\\n• Forman noted that when geometers look at the space of all Riemannian metrics on a given space, they do it to try to find the \\\"best\\\" metric in some sense (e.g., such as one with constant curvature). Q. Are there many useful or interesting metrics on tree space? Q. Is the BHV metric the only one (up to scalars) of non-positive curvature? (Bridson) \\n\\n• Charney: metrics are slightly different, but they should all be similar. If there is no reason that biologists suggest that one is better than another, then it makes sense to choose one that has useful properties, such as metrics of nonpositive curvature. \\n\\n• Billera: for biologists, what metrics do you want to have on the orthants? Q. What metrics \\\"should\\\" be used on the subspace obtained by fixing the tree type? \\n\\n• St. John: uses the L1 metric, though Felsenstein uses L2.\\n\\n• Penny: we use L1 because then lengths scale with time. \\n\\n• Charney: since the data is not completely in tree space, maybe we should be consid-ering non-intrinsic metrics. \\n\\n• Forman: want to choose metrics so that the statistical methods we are using are continuous. \\n\\n• Flath: perhaps some edges are more important than others? \\n\\n• Penny: we are not surprised when we get edges near leaves are accurate, but we are really interested in getting deep internal edges right. \\n\\n• Penny: taxonomic studies, just care about the branch order (weight 1 on edges), but when we consider time studies, we do want the lengths. \\n\\n• Forman: measuring the residuals seem to imply that embedding tree space in some larger space and considering an extrinsic metric would be important. \\n\\n• Vogtmann noted that Felsenstein was talking about embedding tree space in Eu-clidean space (using the Robinson-Foulds metric). Q. How does the Robinson-Foulds metric compare with the BHV metric? (Diaconis) What's the Lipshitz constant? (Bridson) \\n\\n• Vert: Another distance (referred to by Felsenstein in his talk) is Kullback-Leibler distance (relative entropy) between probabilities. Fix a model. Each tree defines a probability on the set of assignments of letters (A,G,C,T) to leaves. Maximum likelihood corresponds to projection of the empirical measure of a data point to tree space according to this distance. 11 \\n\\nThere was some explanation and discussion of this embedding. Diaconis noted that while there are many metrics on probability distributions to consider here, the Kullback-Liebler separation is good for maximum likelihood. Q. Should the metric on tree space come from an intrinsic metric, or an extrinsic metric on a larger space in which tree space is embedded? Which is a more natural way to view tree space? \\n\\n• Forman: studying the residuals seems to be very important. This larger space is part of what you are given in the data, and should be useful. Charney rephrased the question as: what is the right ambient space to embed tree space? While Bridson remarked that he viewed tree space as God-given, Forman asserted that he views the data as God-given. \\n\\n• Vert remarked that the \\\"average\\\" of trees with high likelihood may not be high likelihood with the intrinsic metric. Q. How do you deliver geometry to biologists? (Billera) \\n\\n• Billera pointed out that interesting mathematics in biology seems to happen by serendipity- cases in which someone happened to know some mathematics or knew someone who did. \\n\\n• Penny: notes that there is a role for mathematics to play. Also new majors in mathematical biology (at some schools) have arisen that will train students to think in both disciplines. \\n\\n• Diaconis notes that as a result of this conference, and ensuing discussions, people might write an article for Science. This may bring biologists to mathematics. \\n\\n• Snel suggests advocating a notion of average, and communicate why it is better. For instance, articles on concatenated alignments have appeared in Science. \\n\\n• Huelsenbeck: write review articles that explain things clearly. Write for Systematics Biology - that's where stuff on trees would appear. \\n\\n• Billera asked again how one spreads mathematical knowledge in the biological com-munity, noting that courses offered at most institutions are either at too low or too high a level for biologists.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/geombio/geombio.pdf", "tags": [ "aim", "AIM-BIOLOGY-0039", "aim-domain:biology", "aim-workshop:geombio", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For fixed-leaf metric trees, let r be unnormalized topology-only Robinson-Foulds distance, let D_C be the Euclidean length discrepancy on common splits, and suppose every unmatched positive edge has length in [delta,M]. Then sqrt(D_C^2 + delta^2 r) is at most the BHV distance, which is at most sqrt(D_C^2 + 2 M^2 r), and all constants are sharp. Without a lower unmatched-edge bound, RF cannot be Lipschitz-controlled by BHV; without tracking common-edge lengths or imposing an upper unmatched-edge bound, BHV cannot be Lipschitz-controlled by RF. Positive split-consistent diagonal weights also yield a CAT(0) orthant metric isometric to BHV by coordinate rescaling, while nonconstant weights are not scalar multiples on fixed physical length coordinates.\n\nCandidate contribution (sharp_metric_comparison_certificate; novelty confidence low): The sharp certificate sqrt(D_C^2 + delta^2 d_RF) <= d_BHV <= sqrt(D_C^2 + 2 M^2 d_RF), together with exact star/refinement, same-orthant, and incompatible-quartet witnesses, states precisely which scale and common-length data are necessary to make the workshop's RF-versus-BHV Lipschitz question well posed." }, { "id": 20000875, "problem_number": "AIM-BIOLOGY-0040", "title": "When concatenation is score averaging", "statement": "A.6 Thursday Open Discussion\n\nThis discussion was moderated by Lou Billera. Billera noted that, as it was the final discussion session, anything is a valid topic today. The discussion started off with some comments about the usefulness of the workshop:\n\n• Billera began by remarking that the organizers worried that having 4 different groups of people at this workshop from different backgrounds could have been disastrous. But he noted that it didn't appear to be... and had found it interesting, and was pleased with how much discussion had taken place.\n\n• Diaconis emphasized the usefulness of this workshop for generating problems to work on, and cited an example of residuals for discrete data analysis, as something worth following up, as well as the discussion in the geometry session about what are natural measures on tree space. He noted that the workshop was valuable because he couldn't think of another setting in which all these ideas from different arenas could all come together. 12\n\nQ. Concatenation as \"averaging\"?\n\n• Billera noted that in the discussion about averaging, he was struck with the idea that concatenation was an average in sequence space, an idea that he hadn't appreciated until this workshop. One question He noted that the conference had touched on many ways to go from sequences to trees, and one of the big questions was whether these methods were coherent with concatenation-averaging.\n\n• Penny: it is worrisome that there are many different ways of concatenating, and how to handle the number of parameters? Q. Likelihood contours?\n\n• There was further discussion about confidence sets and likelihood contours, and how these might intersect transition points where the tree space branches in several di-rections. For instance, if a maximum likelihood process produces a tree with very short edge, then it would be near a transition. There was also discussion of whether the notion of curvature on tree space would be helpful. Q. Multivariate \"metrics\"?\n\n• Holmes: in the date that arises for trees, you get multivariate distances. We may want to speak of directions? is there anything like that in metric topology?\n\n• Diaconis: there are some distances that take values in partially ordered sets, etc., distances as a vector, etc. Q. Alternate representations (not trees)?\n\n• Epstein asked if are there other notions besides trees that would be helpful?\n\n• St. John: Networks? Trees with several non-tree portions. Q. Random walks on trees (generate letters, prodcue new trees)\n\n• Vert: One can imagine a random walks in this space of trees, by starting a tree, generate data, estimate a tree (by maximum likelihood). Do this number of times, get a random walk? Might be interesting to study this walk? Q. Would it be useful to study 2-colored trees to model orthology and paralogy? (Shareshian) During the session, Diaconis suggested that, as a follow up to this conference, people can now get together in small groups and communicate and start to work on a problem, whereas this wasn't true before the beginning of the week. AIM could facilitate such follow-up meetings. There was some brief discussion about holding a follow-up conference to this one, since all the participants now have a solid foundation on which to try to work on some joint problems. ARCC and the organizers (Lou Billera, Susan Holmes, Karen Vogtmann) were all thanked enthusiastically for a stimulating week of talks and discussions. The workshop was concluded over evening refreshments.", "original_statement": "A.6 Thursday Open Discussion \n\nThis discussion was moderated by Lou Billera. Billera noted that, as it was the final discussion session, anything is a valid topic today. The discussion started off with some comments about the usefulness of the workshop: \n\n• Billera began by remarking that the organizers worried that having 4 different groups of people at this workshop from different backgrounds could have been disastrous. But he noted that it didn't appear to be... and had found it interesting, and was pleased with how much discussion had taken place. \n\n• Diaconis emphasized the usefulness of this workshop for generating problems to work on, and cited an example of residuals for discrete data analysis, as something worth following up, as well as the discussion in the geometry session about what are natural measures on tree space. He noted that the workshop was valuable because he couldn't think of another setting in which all these ideas from different arenas could all come together. 12 \n\nQ. Concatenation as \"averaging\"? \n\n• Billera noted that in the discussion about averaging, he was struck with the idea that concatenation was an average in sequence space, an idea that he hadn't appreciated until this workshop. One question He noted that the conference had touched on many ways to go from sequences to trees, and one of the big questions was whether these methods were coherent with concatenation-averaging. \n\n• Penny: it is worrisome that there are many different ways of concatenating, and how to handle the number of parameters? Q. Likelihood contours? \n\n• There was further discussion about confidence sets and likelihood contours, and how these might intersect transition points where the tree space branches in several di-rections. For instance, if a maximum likelihood process produces a tree with very short edge, then it would be near a transition. There was also discussion of whether the notion of curvature on tree space would be helpful. Q. Multivariate \"metrics\"? \n\n• Holmes: in the date that arises for trees, you get multivariate distances. We may want to speak of directions? is there anything like that in metric topology? \n\n• Diaconis: there are some distances that take values in partially ordered sets, etc., distances as a vector, etc. Q. Alternate representations (not trees)? \n\n• Epstein asked if are there other notions besides trees that would be helpful? \n\n• St. John: Networks? Trees with several non-tree portions. Q. Random walks on trees (generate letters, prodcue new trees) \n\n• Vert: One can imagine a random walks in this space of trees, by starting a tree, generate data, estimate a tree (by maximum likelihood). Do this number of times, get a random walk? Might be interesting to study this walk? Q. Would it be useful to study 2-colored trees to model orthology and paralogy? (Shareshian) During the session, Diaconis suggested that, as a follow up to this conference, people can now get together in small groups and communicate and start to work on a problem, whereas this wasn't true before the beginning of the week. AIM could facilitate such follow-up meetings. There was some brief discussion about holding a follow-up conference to this one, since all the participants now have a solid foundation on which to try to work on some joint problems. ARCC and the organizers (Lou Billera, Susan Holmes, Karen Vogtmann) were all thanked enthusiastically for a stimulating week of talks and discussions. The workshop was concluded over evening refreshments.", "clean_statement": "A.6 Thursday Open Discussion\n\nThis discussion was moderated by Lou Billera. Billera noted that, as it was the final discussion session, anything is a valid topic today. The discussion started off with some comments about the usefulness of the workshop:\n\n• Billera began by remarking that the organizers worried that having 4 different groups of people at this workshop from different backgrounds could have been disastrous. But he noted that it didn't appear to be... and had found it interesting, and was pleased with how much discussion had taken place.\n\n• Diaconis emphasized the usefulness of this workshop for generating problems to work on, and cited an example of residuals for discrete data analysis, as something worth following up, as well as the discussion in the geometry session about what are natural measures on tree space. He noted that the workshop was valuable because he couldn't think of another setting in which all these ideas from different arenas could all come together. 12\n\nQ. Concatenation as \"averaging\"?\n\n• Billera noted that in the discussion about averaging, he was struck with the idea that concatenation was an average in sequence space, an idea that he hadn't appreciated until this workshop. One question He noted that the conference had touched on many ways to go from sequences to trees, and one of the big questions was whether these methods were coherent with concatenation-averaging.\n\n• Penny: it is worrisome that there are many different ways of concatenating, and how to handle the number of parameters? Q. Likelihood contours?\n\n• There was further discussion about confidence sets and likelihood contours, and how these might intersect transition points where the tree space branches in several di-rections. For instance, if a maximum likelihood process produces a tree with very short edge, then it would be near a transition. There was also discussion of whether the notion of curvature on tree space would be helpful. Q. Multivariate \"metrics\"?\n\n• Holmes: in the date that arises for trees, you get multivariate distances. We may want to speak of directions? is there anything like that in metric topology?\n\n• Diaconis: there are some distances that take values in partially ordered sets, etc., distances as a vector, etc. Q. Alternate representations (not trees)?\n\n• Epstein asked if are there other notions besides trees that would be helpful?\n\n• St. John: Networks? Trees with several non-tree portions. Q. Random walks on trees (generate letters, prodcue new trees)\n\n• Vert: One can imagine a random walks in this space of trees, by starting a tree, generate data, estimate a tree (by maximum likelihood). Do this number of times, get a random walk? Might be interesting to study this walk? Q. Would it be useful to study 2-colored trees to model orthology and paralogy? (Shareshian) During the session, Diaconis suggested that, as a follow up to this conference, people can now get together in small groups and communicate and start to work on a problem, whereas this wasn't true before the beginning of the week. AIM could facilitate such follow-up meetings. There was some brief discussion about holding a follow-up conference to this one, since all the participants now have a solid foundation on which to try to work on some joint problems. ARCC and the organizers (Lou Billera, Susan Holmes, Karen Vogtmann) were all thanked enthusiastically for a stimulating week of talks and discussions. The workshop was concluded over evening refreshments.", "statement_status": "exact", "statement_verification": "This record is section A.6, “Thursday Open Discussion,” in the AIM workshop report *Geometric models of biological phenomena*. It is a collection of discussion prompts, not a single formal open problem. The source PDF was checked against the extracted JSON. The isolated “12” in the JSON is the printed page number. The phrases “One question He noted,” “date,” and “prodcue” occur in the PDF itself and are retained as source defects; the mathematics below does not depend on reconstructing them.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Biology\nWorkshop: Geometric models of biological phenomena\nSection: \nSource item: A.6\nSource URL: https://aimath.org/WWN/geombio/geombio.pdf\nCanonical location: aim-biology-notes.json notes[39]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.6 Thursday Open Discussion \\n\\nThis discussion was moderated by Lou Billera. Billera noted that, as it was the final discussion session, anything is a valid topic today. The discussion started off with some comments about the usefulness of the workshop: \\n\\n• Billera began by remarking that the organizers worried that having 4 different groups of people at this workshop from different backgrounds could have been disastrous. But he noted that it didn't appear to be... and had found it interesting, and was pleased with how much discussion had taken place. \\n\\n• Diaconis emphasized the usefulness of this workshop for generating problems to work on, and cited an example of residuals for discrete data analysis, as something worth following up, as well as the discussion in the geometry session about what are natural measures on tree space. He noted that the workshop was valuable because he couldn't think of another setting in which all these ideas from different arenas could all come together. 12 \\n\\nQ. Concatenation as \\\"averaging\\\"? \\n\\n• Billera noted that in the discussion about averaging, he was struck with the idea that concatenation was an average in sequence space, an idea that he hadn't appreciated until this workshop. One question He noted that the conference had touched on many ways to go from sequences to trees, and one of the big questions was whether these methods were coherent with concatenation-averaging. \\n\\n• Penny: it is worrisome that there are many different ways of concatenating, and how to handle the number of parameters? Q. Likelihood contours? \\n\\n• There was further discussion about confidence sets and likelihood contours, and how these might intersect transition points where the tree space branches in several di-rections. For instance, if a maximum likelihood process produces a tree with very short edge, then it would be near a transition. There was also discussion of whether the notion of curvature on tree space would be helpful. Q. Multivariate \\\"metrics\\\"? \\n\\n• Holmes: in the date that arises for trees, you get multivariate distances. We may want to speak of directions? is there anything like that in metric topology? \\n\\n• Diaconis: there are some distances that take values in partially ordered sets, etc., distances as a vector, etc. Q. Alternate representations (not trees)? \\n\\n• Epstein asked if are there other notions besides trees that would be helpful? \\n\\n• St. John: Networks? Trees with several non-tree portions. Q. Random walks on trees (generate letters, prodcue new trees) \\n\\n• Vert: One can imagine a random walks in this space of trees, by starting a tree, generate data, estimate a tree (by maximum likelihood). Do this number of times, get a random walk? Might be interesting to study this walk? Q. Would it be useful to study 2-colored trees to model orthology and paralogy? (Shareshian) During the session, Diaconis suggested that, as a follow up to this conference, people can now get together in small groups and communicate and start to work on a problem, whereas this wasn't true before the beginning of the week. AIM could facilitate such follow-up meetings. There was some brief discussion about holding a follow-up conference to this one, since all the participants now have a solid foundation on which to try to work on some joint problems. ARCC and the organizers (Lou Billera, Susan Holmes, Karen Vogtmann) were all thanked enthusiastically for a stimulating week of talks and discussions. The workshop was concluded over evening refreshments.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 20, "set_id": 14, "source_url": "https://aimath.org/WWN/geombio/geombio.pdf", "tags": [ "aim", "AIM-BIOLOGY-0040", "aim-domain:biology", "aim-workshop:geombio", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 20, "name": "miscellaneous", "display_name": "Miscellaneous", "description": "Problems whose source classification does not fit the main mathematical categories.", "slug": "miscellaneous", "order_index": 20, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "For a finite candidate set, fully partitioned profiling makes the concatenated score exactly the weighted sum of block profile scores. A candidate's winning weights therefore form an explicit closed convex polyhedral cone cut out by pairwise score margins, and it wins for every nonnegative weighting exactly when it wins every block. No rule retaining only the block-winning topologies can recover this additive winner in general. If nuisance parameters are instead shared, the shared score is at most the partitioned score, with equality exactly when all positively weighted block optimizer sets intersect; an explicit two-block score example shows that linking parameters can reverse the selected candidate.\n\nCandidate contribution (criterion; novelty confidence low): The combined score-coherence certificate consists of explicit winner cones in block-weight space, a proof that this certificate cannot factor through per-block winner labels, and an exact optimizer-intersection criterion for when nuisance-parameter linking leaves the partitioned score unchanged.", "research_difficulty_suggested": "L2" }, { "id": 20000876, "problem_number": "AIM-COMBINATORICS-0001", "title": "Sharp lower bounds for entropic subspace approximation", "statement": "Entropy Freiman-Ruzsa lower bounds\n\nTheorem. Let $X$ be any $\\mathbb{F}_2^n$-valued random variable such that $d[X,X]\\leq \\log K$, where $d$ is the entropic Ruzsa distance. Then there is a subspace $H\\subset \\mathbb{F}_2^n$ such that $d[X,U_H]\\leq 11\\log K$, where $U_H$ is uniform on $H$.\n\nCan we give lower bounds for $d[X,U_H]$?", "original_statement": "Entropy Freiman-Ruzsa lower bounds\n\nTheorem. Let $X$ be any $\\mathbb{F}_2^n$-valued random variable such that $d[X,X]\\leq \\log K$, where $d$ is the entropic Ruzsa distance. Then there is a subspace $H\\subset \\mathbb{F}_2^n$ such that $d[X,U_H]\\leq 11\\log K$, where $U_H$ is uniform on $H$.\n\nCan we give lower bounds for $d[X,U_H]$?", "clean_statement": "Entropy Freiman-Ruzsa lower bounds\n\nTheorem. Let $X$ be any $\\mathbb{F}_2^n$-valued random variable such that $d[X,X]\\leq \\log K$, where $d$ is the entropic Ruzsa distance. Then there is a subspace $H\\subset \\mathbb{F}_2^n$ such that $d[X,U_H]\\leq 11\\log K$, where $U_H$ is uniform on $H$.\n\nCan we give lower bounds for $d[X,U_H]$?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.1, “Entropy Freiman-Ruzsa lower bounds,” from the AIM workshop *High-dimensional phenomena in discrete analysis*, section “Entropic methods.” Its mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Entropic methods\nSource item: 1.1\nSource URL: http://aimpl.org/highdimdiscrete/1/\nCanonical location: aim-combinatorics-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Entropy Freiman-Ruzsa lower bounds\\n\\nTheorem. Let $X$ be any $\\\\mathbb{F}_2^n$-valued random variable such that $d[X,X]\\\\leq \\\\log K$, where $d$ is the entropic Ruzsa distance. Then there is a subspace $H\\\\subset \\\\mathbb{F}_2^n$ such that $d[X,U_H]\\\\leq 11\\\\log K$, where $U_H$ is uniform on $H$.\\n\\nCan we give lower bounds for $d[X,U_H]$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/1/", "tags": [ "aim", "AIM-COMBINATORICS-0001", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every subspace H of F_2^n, d[X;U_H] is exactly one half of the sum of the quotient entropy H(pi_H X) and the conditional entropy deficit log|H|-H(X|pi_H X), and the entropic triangle inequality gives d[X;U_H] >= d[X;X]/2. This factor 1/2 is universally sharp even at any prescribed positive self-distance. An exact optimization for X=U_V+B_p v gives R(X)=min(h(p),log 2-h(p))/2; the balanced choice h(p_0)=log 2/2 yields the explicit obstruction C_* >= 1.1707587808... for the best universal entropic inverse-theorem constant, while Liao's current checked theorem gives C_* <= 5.\n\nCandidate contribution (sharpness construction and explicit constant obstruction; novelty confidence low): Tensorized rare Bernoulli laws make the lower factor 1/2 sharp while d[X;X] tends to any prescribed t>0, and the balanced one-bit law h(p_0)=log 2/2 gives C_* >= (log 2/4)/(h(2p_0(1-p_0))-log 2/2) = 1.1707587808... ." }, { "id": 20000877, "problem_number": "AIM-COMBINATORICS-0002", "title": "A finite entropic coupling calculus with forest and even-cycle certificates", "statement": "Entropy proof of Sidorenko\n\nThere is a proof system for graph homomorphism inequalities using flag algebras.\n\nDefine a proof system for graph homomorphism inequalities using entropy.", "original_statement": "Entropy proof of Sidorenko\n\nThere is a proof system for graph homomorphism inequalities using flag algebras.\n\nDefine a proof system for graph homomorphism inequalities using entropy.", "clean_statement": "Entropy proof of Sidorenko\n\nThere is a proof system for graph homomorphism inequalities using flag algebras.\n\nDefine a proof system for graph homomorphism inequalities using entropy.", "statement_status": "exact", "statement_verification": "The canonical AIM record is workshop *High-dimensional phenomena in discrete analysis*, section “Entropic methods,” Problem 1.2. Its problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Entropic methods\nSource item: 1.2\nSource URL: http://aimpl.org/highdimdiscrete/1/\nCanonical location: aim-combinatorics-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Entropy proof of Sidorenko\\n\\nThere is a proof system for graph homomorphism inequalities using flag algebras.\\n\\nDefine a proof system for graph homomorphism inequalities using entropy.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"For example, prove that \\\"$H$ is Sidorenko\\\" is not provable in this proof system, for $H=K_{5,5}\\\\setminus C_{10}$.\\n\\nCan all known cases of Sidorenko be entropized?\\n\\nCan the proof that even girth graphs are locally Sidorenko be entropized?\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/1/", "tags": [ "aim", "AIM-COMBINATORICS-0002", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A sound finite Shannon entropic-coupling calculus is specified using typed supported distribution terms, exact-marginal relatively independent joins, structural law-equality proofs, Shannon elemental inequalities, and support/cardinality rules. The calculus gives uniform Sidorenko certificates for every forest and every even cycle. In the restricted fragment that only attaches a fresh edge along one old vertex, the constructible source graphs are exactly forests, so K_{5,5} minus C_{10} is unprovable in that fragment, but no nonprovability result is claimed for the full calculus.\n\nCandidate contribution (proof_system_and_limitation_lemma; novelty confidence low): Candidate contribution: the typed certificate discipline separates supported-law construction, exact separator-law equality, and rational entropy arithmetic; within it, a two-copy endpoint join gives a uniform certificate for all even cycles, while the one-vertex fresh-edge fragment has an exact forest-only closure invariant." }, { "id": 20000878, "problem_number": "AIM-COMBINATORICS-0003", "title": "The Renyi-Shannon bottleneck in entropic inverse arguments", "statement": "Inverse theorems with polynomial bounds\n\nGiven $f:\\mathbb{F}_2^n\\to \\mathbb{F}_2^m$ with $\\mathbb{P}_{x+y=z+w}(f(x)+f(y)=f(z)+f(w))\\geq \\epsilon$, then Polynomial Freiman-Ruzsa implies that $f$ correlates with a linear function.\n\nSet $D_hf(x)=f(x+h)-f(x)$. If $\\mathbb{P}_{x,h_1,h_2,h_3}(D_{h_1}D_{h_2}D_{h_3}f(x)=0)\\geq \\epsilon$, then PFR implies that $f$ correlates with a quadratic function.\n\nIs there a direct entropic proof of the above results with polynomial bounds?", "original_statement": "Inverse theorems with polynomial bounds\n\nGiven $f:\\mathbb{F}_2^n\\to \\mathbb{F}_2^m$ with $\\mathbb{P}_{x+y=z+w}(f(x)+f(y)=f(z)+f(w))\\geq \\epsilon$, then Polynomial Freiman-Ruzsa implies that $f$ correlates with a linear function.\n\nSet $D_hf(x)=f(x+h)-f(x)$. If $\\mathbb{P}_{x,h_1,h_2,h_3}(D_{h_1}D_{h_2}D_{h_3}f(x)=0)\\geq \\epsilon$, then PFR implies that $f$ correlates with a quadratic function.\n\nIs there a direct entropic proof of the above results with polynomial bounds?", "clean_statement": "Inverse theorems with polynomial bounds\n\nGiven $f:\\mathbb{F}_2^n\\to \\mathbb{F}_2^m$ with $\\mathbb{P}_{x+y=z+w}(f(x)+f(y)=f(z)+f(w))\\geq \\epsilon$, then Polynomial Freiman-Ruzsa implies that $f$ correlates with a linear function.\n\nSet $D_hf(x)=f(x+h)-f(x)$. If $\\mathbb{P}_{x,h_1,h_2,h_3}(D_{h_1}D_{h_2}D_{h_3}f(x)=0)\\geq \\epsilon$, then PFR implies that $f$ correlates with a quadratic function.\n\nIs there a direct entropic proof of the above results with polynomial bounds?", "statement_status": "exact", "statement_verification": "This is Problem 1.3, “Inverse theorems with polynomial bounds,” from the May 2024 AIM workshop *High-dimensional phenomena in discrete analysis*, in the section “Entropic methods.” The live AimPL problem URL returned an access error during this run, so the exact canonical JSON record and the neighboring record 1.1 were used. Neighbor 1.1 states the entropic Polynomial Freiman–Ruzsa theorem with constant 11, confirming that Problem 1.3 asks for a more direct entropic derivation of inverse consequences, not for a proof of PFR itself.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Entropic methods\nSource item: 1.3\nSource URL: http://aimpl.org/highdimdiscrete/1/\nCanonical location: aim-combinatorics-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Inverse theorems with polynomial bounds\\n\\nGiven $f:\\\\mathbb{F}_2^n\\\\to \\\\mathbb{F}_2^m$ with $\\\\mathbb{P}_{x+y=z+w}(f(x)+f(y)=f(z)+f(w))\\\\geq \\\\epsilon$, then Polynomial Freiman-Ruzsa implies that $f$ correlates with a linear function.\\n\\nSet $D_hf(x)=f(x+h)-f(x)$. If $\\\\mathbb{P}_{x,h_1,h_2,h_3}(D_{h_1}D_{h_2}D_{h_3}f(x)=0)\\\\geq \\\\epsilon$, then PFR implies that $f$ correlates with a quadratic function.\\n\\nIs there a direct entropic proof of the above results with polynomial bounds?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The second result is known with bounds $1/\\\\exp(\\\\exp(1/\\\\epsilon))$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/1/", "tags": [ "aim", "AIM-COMBINATORICS-0003", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the four-point test, the exact graph-sum identity is alpha_2(f)=2^n Pr(S=S'), so high acceptance controls collision (Renyi-2) entropy rather than Shannon entropic Ruzsa distance. An explicit family f_n:F_2^n to F_2^n, equal to zero on a codimension-three subspace and x^3 off it, has alpha_2(f_n) at least 2^-9 but Shannon Ruzsa self-distance at least (17/32)n-3. Thus the naive direct step from high energy to small Shannon entropic doubling is impossible; a genuine conditioning or entropic Balog-Szemeredi-Gowers step is necessary. A separate exact character identity expresses the vector third-derivative acceptance as the average of scalar U^3 eighth powers and records the zero-character baseline.\n\nCandidate contribution (counterexample; novelty confidence low): The explicit cubic-on-a-complement graph family has fixed four-point acceptance at least 2^-9 and graph-sum collision entropy n+O(1), while its Shannon entropic Ruzsa self-distance grows at least as (17/32)n-O(1), disproving any dimension-free direct conversion from the test probability to small Shannon Ruzsa distance." }, { "id": 20000879, "problem_number": "AIM-COMBINATORICS-0004", "title": "A reduced-degree masking counterexample to nonlinear Roth", "statement": "Non-linear Roth\n\nLet $P$ be any (non-linear) polynomial, $A\\subset \\mathbb{F}_p$ with $|A|>p^{0.99}$. Does there always exist distinct $x,y$ such that\n$$x,y,y+P(x)-P(y)\\in A?$$", "original_statement": "Non-linear Roth\n\nLet $P$ be any (non-linear) polynomial, $A\\subset \\mathbb{F}_p$ with $|A|>p^{0.99}$. Does there always exist distinct $x,y$ such that\n$$x,y,y+P(x)-P(y)\\in A?$$", "clean_statement": "Non-linear Roth\n\nLet $P$ be any (non-linear) polynomial, $A\\subset \\mathbb{F}_p$ with $|A|>p^{0.99}$. Does there always exist distinct $x,y$ such that\n$$x,y,y+P(x)-P(y)\\in A?$$", "statement_status": "exact", "statement_verification": "The canonical record is problem 2.1, “Non-linear Roth,” from the additive-combinatorics section of the May 2024 AIM workshop *High-dimensional phenomena in discrete analysis*. Its exact mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Additive combinatorics\nSource item: 2.1\nSource URL: http://aimpl.org/highdimdiscrete/2/\nCanonical location: aim-combinatorics-notes.json notes[3]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Non-linear Roth\\n\\nLet $P$ be any (non-linear) polynomial, $A\\\\subset \\\\mathbb{F}_p$ with $|A|>p^{0.99}$. Does there always exist distinct $x,y$ such that\\n$$x,y,y+P(x)-P(y)\\\\in A?$$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"When $P(x)=2x$, this recovers Roth's Theorem on 3-APs.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/2/", "tags": [ "aim", "AIM-COMBINATORICS-0004", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal AIM statement is false for every sufficiently large odd prime. Choose a 3-AP-free set A inside [1,floor((p-1)/3)] with |A|>p^0.99 using the Behrend-Elkin construction, and set P_A(T)=2T+product_{a in A}(T-a). Then 2<=deg P_A=|A|(p+1)/2.\n\nCandidate contribution (counterexample construction; novelty confidence low): For every sufficiently large odd prime p, a Behrend-Elkin 3-AP-free set A of size greater than p^0.99 and the reduced polynomial P_A(T)=2T+product_{a in A}(T-a) satisfy deg P_A=|A|0$ and sufficiently large $N$, if $A\\subset [N]$ with $|A|\\geq N/\\log^C N$, then there exists $x$ and $p$ prime such that\n$$x,x+p-1,x+2(p-1)\\in A.$$", "original_statement": "Roth with common difference $p-1$\n\nFor all $C>0$ and sufficiently large $N$, if $A\\subset [N]$ with $|A|\\geq N/\\log^C N$, then there exists $x$ and $p$ prime such that\n$$x,x+p-1,x+2(p-1)\\in A.$$", "clean_statement": "Roth with common difference $p-1$\n\nFor all $C>0$ and sufficiently large $N$, if $A\\subset [N]$ with $|A|\\geq N/\\log^C N$, then there exists $x$ and $p$ prime such that\n$$x,x+p-1,x+2(p-1)\\in A.$$", "statement_status": "exact", "statement_verification": "The canonical AIM record is workshop *High-dimensional phenomena in discrete analysis*, section “Additive combinatorics,” Conjecture 2.2. Its exact main statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Additive combinatorics\nSource item: 2.2\nSource URL: http://aimpl.org/highdimdiscrete/2/\nCanonical location: aim-combinatorics-notes.json notes[4]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Roth with common difference $p-1$\\n\\nFor all $C>0$ and sufficiently large $N$, if $A\\\\subset [N]$ with $|A|\\\\geq N/\\\\log^C N$, then there exists $x$ and $p$ prime such that\\n$$x,x+p-1,x+2(p-1)\\\\in A.$$\"\nOriginal remarks: [\"What if we replace the set {p-1 : p prime} with the set of positive integers that can be written as a sum of two squares?\"]\nOriginal literature field (JSON string): \"A hard result of Green says that if $|A|>N^{1-c}$, then we can find $x,x+p-1\\\\in A$, so we don't reasonably expect to get power-saving bounds for the above conjecture.\\n\\nA model problem is the following. Let $W=\\\\prod_{p\\\\text{ prime}, p\\\\leq w}p$ and $S$ the set of positive integers coprime to $W$. Then find in $A$ a 3-AP of the form $x,x+y,x+2y$ with $y\\\\in S$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/2/", "tags": [ "aim", "AIM-COMBINATORICS-0005", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The proposed coprime-to-W model is false even at constant density: whenever 3 divides W, the maximum density of a periodic set avoiding every three-term progression with difference coprime to W is exactly 2/3. In contrast, every nonempty periodic set contains a progression with difference p-1 and one with sum-of-two-squares difference. Exact Möbius, residue-fiber, and cyclic Fourier identities then isolate progression-count balance, rather than residue-cardinality balance, as a sufficient repair of the model.\n\nCandidate contribution (obstruction_and_counting_reduction; novelty confidence low): Candidate novel package: a sharp 2/3 classification of periodic obstructions to the literal coprime-W model, a proof that neither shifted-prime nor sum-of-two-squares differences admit any nonempty periodic obstruction, and exact AP-divisor and Ramanujan-multiplier criteria specifying the additional progression-count balance a repaired model must control.", "research_difficulty_suggested": "L4" }, { "id": 20000881, "problem_number": "AIM-COMBINATORICS-0006", "title": "Tensor growth rate and exact one-dimensional obstructions for constrained differences", "statement": "Roth with constrained differences\n\nFor which $0\\in D\\subseteq \\mathbb{F}_p$ is it true that any $A\\subset \\mathbb{F}_p^n$ avoiding\n$$x,x+y,x+2y\\in A\\quad \\text{with } y\\in D^n\\setminus\\{0\\}$$\nhas size $\\leq C^n$ for some $C < p$?", "original_statement": "Roth with constrained differences\n\nFor which $0\\in D\\subseteq \\mathbb{F}_p$ is it true that any $A\\subset \\mathbb{F}_p^n$ avoiding\n$$x,x+y,x+2y\\in A\\quad \\text{with } y\\in D^n\\setminus\\{0\\}$$\nhas size $\\leq C^n$ for some $C < p$?", "clean_statement": "Roth with constrained differences\n\nFor which $0\\in D\\subseteq \\mathbb{F}_p$ is it true that any $A\\subset \\mathbb{F}_p^n$ avoiding\n$$x,x+y,x+2y\\in A\\quad \\text{with } y\\in D^n\\setminus\\{0\\}$$\nhas size $\\leq C^n$ for some $C < p$?", "statement_status": "exact", "statement_verification": "The source is Problem 2.3 in the “Additive combinatorics” section of the AIM workshop list *High-dimensional phenomena in discrete analysis*. The exact mathematical question in the supplied record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Additive combinatorics\nSource item: 2.3\nSource URL: http://aimpl.org/highdimdiscrete/2/\nCanonical location: aim-combinatorics-notes.json notes[5]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Roth with constrained differences\\n\\nFor which $0\\\\in D\\\\subseteq \\\\mathbb{F}_p$ is it true that any $A\\\\subset \\\\mathbb{F}_p^n$ avoiding\\n$$x,x+y,x+2y\\\\in A\\\\quad \\\\text{with } y\\\\in D^n\\\\setminus\\\\{0\\\\}$$\\nhas size $\\\\leq C^n$ for some $C < p$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Slice rank + \\\"multilinear trick\\\" implies the problem is true for $|D|>p/2$.\\n\\nFor $D=\\\\{0,1,2\\\\}$, it is known that we must have $|A|\\\\leq p^n/(\\\\log\\\\log\\\\log n)^c$.\\n\\nWhen $|D|=2$, Hales-Jewett gives $|A|=o(p^n)$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/2/", "tags": [ "aim", "AIM-COMBINATORICS-0006", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the constrained-difference extremal function M_D(n), Cartesian products imply supermultiplicativity and hence an exact rate rho_D = lim M_D(n)^(1/n) = sup M_D(n)^(1/n); the AIM exponential-bound property is equivalent to rho_D < p. The rate is monotone under enlarging D and invariant under nonzero scaling. For every odd p and every two-element D, M_D(1) = floor(2p/3), giving rho_D at least floor(2p/3). Nevertheless, the explicit supermultiplicative profile p^n 2^(-sqrt(n)) has vanishing density but nth-root limit p, proving that density Hales-Jewett plus tensor closure cannot by themselves settle the exponential question. The literal p=2 formulation is also completely classified.\n\nCandidate contribution (reduction; novelty confidence low): The constrained-difference problem is exactly the strict tensor-rate gap rho_D < p; for two-element D the exact one-dimensional tensor obstruction is floor(2p/3)^n, while an explicit supermultiplicative vanishing-density profile proves that the known qualitative density conclusion and product closure are logically insufficient to force this gap." }, { "id": 20000882, "problem_number": "AIM-COMBINATORICS-0007", "title": "A root-system reduction for the spherical triangle form", "statement": "Inverse theorem for equilateral triangles on the sphere\n\nDenote by $S^n\\subset \\mathbb{R}^{n+1}$ the unit sphere. For a function $f:S^n\\to \\mathbb{R}$, define the quantity\n$$Tf:= \\mathbb{E}_{x,y,z\\in S^n,x+y+z=0} f(x)f(y)f(z).$$\n\nIf $f$ is 1-bounded and $|Tf|>\\delta$, does $f$ correlate with a function that depends on $O_{\\delta}(1)$ coordinates (with respect to some basis)?", "original_statement": "Inverse theorem for equilateral triangles on the sphere\n\nDenote by $S^n\\subset \\mathbb{R}^{n+1}$ the unit sphere. For a function $f:S^n\\to \\mathbb{R}$, define the quantity\n$$Tf:= \\mathbb{E}_{x,y,z\\in S^n,x+y+z=0} f(x)f(y)f(z).$$\n\nIf $f$ is 1-bounded and $|Tf|>\\delta$, does $f$ correlate with a function that depends on $O_{\\delta}(1)$ coordinates (with respect to some basis)?", "clean_statement": "Inverse theorem for equilateral triangles on the sphere\n\nDenote by $S^n\\subset \\mathbb{R}^{n+1}$ the unit sphere. For a function $f:S^n\\to \\mathbb{R}$, define the quantity\n$$Tf:= \\mathbb{E}_{x,y,z\\in S^n,x+y+z=0} f(x)f(y)f(z).$$\n\nIf $f$ is 1-bounded and $|Tf|>\\delta$, does $f$ correlate with a function that depends on $O_{\\delta}(1)$ coordinates (with respect to some basis)?", "statement_status": "exact", "statement_verification": "The source record asks the following.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Additive combinatorics\nSource item: 2.4\nSource URL: http://aimpl.org/highdimdiscrete/2/\nCanonical location: aim-combinatorics-notes.json notes[6]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Inverse theorem for equilateral triangles on the sphere\\n\\nDenote by $S^n\\\\subset \\\\mathbb{R}^{n+1}$ the unit sphere. For a function $f:S^n\\\\to \\\\mathbb{R}$, define the quantity\\n$$Tf:= \\\\mathbb{E}_{x,y,z\\\\in S^n,x+y+z=0} f(x)f(y)f(z).$$\\n\\nIf $f$ is 1-bounded and $|Tf|>\\\\delta$, does $f$ correlate with a function that depends on $O_{\\\\delta}(1)$ coordinates (with respect to some basis)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is motivated by the following conjecture: the largest subset of the sphere avoiding $x+y+z=0$ is a hemisphere.\\n\\nLet $T_{\\\\alpha}f=\\\\mathbb{E}_{x,y\\\\in S^n,\\\\langle x,y\\\\rangle=\\\\alpha} f(x)f(y)$. It is known that if $|T_{\\\\alpha}f|>\\\\delta$, then $f$ correlates with a low-degree harmonic function.\\n\\nAn analogous problem can be posed on the plane. For example, given $f:[0,10]^2\\\\to\\\\mathbb{C}$, if we instead define $Tf=\\\\mathbb{E}_{x,y,z\\\\text{ unit equilateral triangle}} f(x)f(y)f(z)$, can we prove an inverse theorem in this setting? What about for unit equilateral triangles in $\\\\mathbb{F}_p^2$?\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/2/", "tags": [ "aim", "AIM-COMBINATORICS-0007", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "With the constrained expectation defined by Haar measure on the orthogonal orbit of an equilateral frame, the form Tf is exactly the rotation-average of the directed-triangle density of weighted type-A_d root samples. Consequently, |Tf| > delta forces a sampled root matrix with operator norm greater than delta(d-1), and such a certificate occurs on a positive-measure set of rotations at half the threshold. In addition, a dimension-uniform Sobolev hypothesis forces low spherical-harmonic degree mass, and the inverse conclusion is proved outright for affine functions.\n\nCandidate contribution (reduction; novelty confidence low): For N=d+1 and v_ij=(e_i-e_j)/sqrt(2) identified isometrically with roots in R^d, Tf equals the Haar rotation-average of tr(W_R^3)/(N(N-1)(N-2)); hence |Tf|>delta implies ||W_R||_op>delta(N-2) for some rotation, with a quantitative positive-measure variant." }, { "id": 20000883, "problem_number": "AIM-COMBINATORICS-0008", "title": "Reciprocal constants and exact entropy-to-set bounds for easier arithmetic Kakeya", "statement": "Easier arithmetic Kakeya\n\nLet $S\\subset\\mathbb{Z}^n$ be such that for all $(d_1,\\ldots,d_r)\\in ([N]^n)^r$, $S$ contains the GAP (for some translate $a_0$)\n$$\\{a_0+i_1d_1+\\cdots +i_rd_r : i_1,\\ldots,i_r\\in \\{0,1,\\ldots,k-1\\}\\}.$$\nThen $|S|\\geq N^{c_kn+o_{k\\to\\infty}(1)}$.", "original_statement": "Easier arithmetic Kakeya\n\nLet $S\\subset\\mathbb{Z}^n$ be such that for all $(d_1,\\ldots,d_r)\\in ([N]^n)^r$, $S$ contains the GAP (for some translate $a_0$)\n$$\\{a_0+i_1d_1+\\cdots +i_rd_r : i_1,\\ldots,i_r\\in \\{0,1,\\ldots,k-1\\}\\}.$$\nThen $|S|\\geq N^{c_kn+o_{k\\to\\infty}(1)}$.", "clean_statement": "Easier arithmetic Kakeya\n\nLet $S\\subset\\mathbb{Z}^n$ be such that for all $(d_1,\\ldots,d_r)\\in ([N]^n)^r$, $S$ contains the GAP (for some translate $a_0$)\n$$\\{a_0+i_1d_1+\\cdots +i_rd_r : i_1,\\ldots,i_r\\in \\{0,1,\\ldots,k-1\\}\\}.$$\nThen $|S|\\geq N^{c_kn+o_{k\\to\\infty}(1)}$.", "statement_status": "exact", "statement_verification": "The supplied AIM record, Problem 2.5 in the “Additive combinatorics” section of *High-dimensional phenomena in discrete analysis*, states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Additive combinatorics\nSource item: 2.5\nSource URL: http://aimpl.org/highdimdiscrete/2/\nCanonical location: aim-combinatorics-notes.json notes[7]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Easier arithmetic Kakeya\\n\\nLet $S\\\\subset\\\\mathbb{Z}^n$ be such that for all $(d_1,\\\\ldots,d_r)\\\\in ([N]^n)^r$, $S$ contains the GAP (for some translate $a_0$)\\n$$\\\\{a_0+i_1d_1+\\\\cdots +i_rd_r : i_1,\\\\ldots,i_r\\\\in \\\\{0,1,\\\\ldots,k-1\\\\}\\\\}.$$\\nThen $|S|\\\\geq N^{c_kn+o_{k\\\\to\\\\infty}(1)}$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The case $r=1$ is a version of arithmetic Kakeya, and is known with bound $N^{0.59n+o_k(1)}$.\\n\\nThere is an equivalent entropic formulation. Let $X,Y_1,\\\\ldots,Y_r$ be $\\\\mathbb{Z}$-valued random variables such that\\n$$H(X+a_1Y_1+\\\\ldots+a_rY_r)\\\\leq h\\\\quad \\\\forall a_1,\\\\ldots,a_r\\\\in \\\\{0,1,\\\\ldots,k-1\\\\}.$$\\nThen $H(Y_1,\\\\ldots,Y_r)\\\\leq rhc_k$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/2/", "tags": [ "aim", "AIM-COMBINATORICS-0008", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A finite canonical selector and a simultaneous additive encoding from Z^n to Z give an exact entropy-to-set dictionary: an entropy inequality H(Y_1,...,Y_r) <= C max_a H(X+sum a_jY_j) implies |S| >= N^(rn/C). Thus the source's multiplier r c_k produces set exponent 1/c_k, not c_k; the literal same-constant reading is disproved already at k=2. Combining the dictionary with the Pohoata-Zakharov product theorem gives |S| >= N^((1-((beta_k-1)/beta_k)^r)n), including N^((1-2^(-r))n) for k=2. Independently, a labeled difference-star map proves |S| >= N^(rn/(r+1)).\n\nCandidate contribution (reduction; novelty confidence low): For the exact AIM labeled multi-GAP condition, a measurable finite selector and one simultaneous Freiman encoding yield the no-loss pointwise implication |S| >= N^(rn/beta(R)); this fixes the source's reciprocal mismatch and transfers the generalized entropy product theorem to the explicit exponent 1-((beta_k-1)/beta_k)^r.", "research_difficulty_suggested": "L4" }, { "id": 20000884, "problem_number": "AIM-COMBINATORICS-0009", "title": "Popular square differences: a radix obstruction and a sharp characteristic-two no-wrap model", "statement": "Popular Furstenberg-Sárközy\n\n(Furstenberg-Sárközy) For all $\\epsilon>0$, there exists $N_0=N_0(\\epsilon)$ such that for all $N\\geq N_0$, $A\\subseteq [N]$ with $|A|\\geq \\delta N$, there exists $y\\neq 0$ such that\n$$|\\{x : x,x+y^2\\in A\\}|\\geq (\\delta^2-\\epsilon)N,$$\nwith $N_0(\\epsilon)\\leq e^{1/\\epsilon^c}$.\n\n(1) Prove lower bounds of the form $N_0(\\epsilon)>superpoly(1/\\epsilon)$, or even just $N_0(\\epsilon)>1/\\epsilon^{10}$.\n\n(2) What happens in the function field case $\\mathbb{F}_p[t]_{< n}$? Can we prove upper/lower bounds?", "original_statement": "Popular Furstenberg-Sárközy\n\n(Furstenberg-Sárközy) For all $\\epsilon>0$, there exists $N_0=N_0(\\epsilon)$ such that for all $N\\geq N_0$, $A\\subseteq [N]$ with $|A|\\geq \\delta N$, there exists $y\\neq 0$ such that\n$$|\\{x : x,x+y^2\\in A\\}|\\geq (\\delta^2-\\epsilon)N,$$\nwith $N_0(\\epsilon)\\leq e^{1/\\epsilon^c}$.\n\n(1) Prove lower bounds of the form $N_0(\\epsilon)>superpoly(1/\\epsilon)$, or even just $N_0(\\epsilon)>1/\\epsilon^{10}$.\n\n(2) What happens in the function field case $\\mathbb{F}_p[t]_{< n}$? Can we prove upper/lower bounds?", "clean_statement": "Popular Furstenberg-Sárközy\n\n(Furstenberg-Sárközy) For all $\\epsilon>0$, there exists $N_0=N_0(\\epsilon)$ such that for all $N\\geq N_0$, $A\\subseteq [N]$ with $|A|\\geq \\delta N$, there exists $y\\neq 0$ such that\n$$|\\{x : x,x+y^2\\in A\\}|\\geq (\\delta^2-\\epsilon)N,$$\nwith $N_0(\\epsilon)\\leq e^{1/\\epsilon^c}$.\n\n(1) Prove lower bounds of the form $N_0(\\epsilon)>superpoly(1/\\epsilon)$, or even just $N_0(\\epsilon)>1/\\epsilon^{10}$.\n\n(2) What happens in the function field case $\\mathbb{F}_p[t]_{< n}$? Can we prove upper/lower bounds?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 2.6 in the additive-combinatorics section of the AIM workshop *High-dimensional phenomena in discrete analysis*. Its main paragraph reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Additive combinatorics\nSource item: 2.6\nSource URL: http://aimpl.org/highdimdiscrete/2/\nCanonical location: aim-combinatorics-notes.json notes[8]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Popular Furstenberg-Sárközy\\n\\n(Furstenberg-Sárközy) For all $\\\\epsilon>0$, there exists $N_0=N_0(\\\\epsilon)$ such that for all $N\\\\geq N_0$, $A\\\\subseteq [N]$ with $|A|\\\\geq \\\\delta N$, there exists $y\\\\neq 0$ such that\\n$$|\\\\{x : x,x+y^2\\\\in A\\\\}|\\\\geq (\\\\delta^2-\\\\epsilon)N,$$\\nwith $N_0(\\\\epsilon)\\\\leq e^{1/\\\\epsilon^c}$.\\n\\n(1) Prove lower bounds of the form $N_0(\\\\epsilon)>superpoly(1/\\\\epsilon)$, or even just $N_0(\\\\epsilon)>1/\\\\epsilon^{10}$.\\n\\n(2) What happens in the function field case $\\\\mathbb{F}_p[t]_{< n}$? Can we prove upper/lower bounds?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/2/", "tags": [ "aim", "AIM-COMBINATORICS-0009", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the least integer threshold, an explicit Beigel--Gasarch radix family gives N_Z(epsilon) > c epsilon^{-gamma} for all sufficiently small epsilon, where gamma = log(205)/log(205/12) = 1.8755518603..., and every admissible nonzero square shift has count zero while the requested target is positive. This does not reach exponent 10. In the explicitly defined no-wrap model V_n = F_2[t]_{=rho^m n^v, hence s(F)=m. Moreover inj(F,G)>=rho^m n^v-binom(v,2)n^(v-1); if e(G)=pn^3 and (6p)^m n>=2 binom(v,2), then inj(F,G)>=p^m n^v. The all-density injective reading is false, while the known loose-triangle exponent s=4 forces every universal C to be at least 4/3.\n\nCandidate contribution (finite_size_embedding_transfer; novelty confidence low): For every 3-uniform loose forest F with m>=1 edges and v vertices, every n-vertex simple 3-graph G with pn^3 edges satisfies inj(F,G)>=(6p)^m n^v-binom(v,2)n^(v-1), and therefore inj(F,G)>=p^m n^v whenever (6p)^m n>=2 binom(v,2).", "research_difficulty_suggested": "L4" }, { "id": 20000886, "problem_number": "AIM-COMBINATORICS-0011", "title": "Expansion can hide only global Sidorenko obstructions", "statement": "Expansion of non-Sidorenko graphs\n\nGiven a $k$-graph $F$, define the $r$-expansion of $F$ to be the $r$-graph obtained by enlarging each $k$-edge of $F$ with a set of $r-k$ vertices of degree one.\n\nDoes there exist a non-Sidorenko (hyper)graph $H$ whose $r$-expansion is Sidorenko?", "original_statement": "Expansion of non-Sidorenko graphs\n\nGiven a $k$-graph $F$, define the $r$-expansion of $F$ to be the $r$-graph obtained by enlarging each $k$-edge of $F$ with a set of $r-k$ vertices of degree one.\n\nDoes there exist a non-Sidorenko (hyper)graph $H$ whose $r$-expansion is Sidorenko?", "clean_statement": "Expansion of non-Sidorenko graphs\n\nGiven a $k$-graph $F$, define the $r$-expansion of $F$ to be the $r$-graph obtained by enlarging each $k$-edge of $F$ with a set of $r-k$ vertices of degree one.\n\nDoes there exist a non-Sidorenko (hyper)graph $H$ whose $r$-expansion is Sidorenko?", "statement_status": "exact", "statement_verification": "The record is problem 3.2, “Expansion of non-Sidorenko graphs,” from the AIM workshop *High-dimensional phenomena in discrete analysis*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Extremal graph theory\nSource item: 3.2\nSource URL: http://aimpl.org/highdimdiscrete/3/\nCanonical location: aim-combinatorics-notes.json notes[10]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Expansion of non-Sidorenko graphs\\n\\nGiven a $k$-graph $F$, define the $r$-expansion of $F$ to be the $r$-graph obtained by enlarging each $k$-edge of $F$ with a set of $r-k$ vertices of degree one.\\n\\nDoes there exist a non-Sidorenko (hyper)graph $H$ whose $r$-expansion is Sidorenko?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"It is known that an expansion of a Sidorenko graph is Sidorenko.\\n\\nIt is known that $C_5$ and its 3-expansion are not Sidorenko. More generally, if the girth (length of the shortest loose cycle) of a hypergraph $F$ is odd, then $F$ is not Sidorenko.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/3/", "tags": [ "aim", "AIM-COMBINATORICS-0011", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the strong r-expansion, homomorphism density is exactly t_{E^r(F)}(W)=t_F(M_{r to k}W), where M is the k-coordinate marginal. A bounded symmetric Hoeffding/ANOVA right inverse of M has operator norm at most C_{k,r}=sum_{j=0}^k binom(r,j)2^j. Therefore every sufficiently small L-infinity non-Sidorenko perturbation of an interior constant kernel lifts to a counterexample for every expansion. In particular, a Sidorenko expansion must have a locally Sidorenko core. Expansion transitivity and a strong-coloring obstruction further show that any positive example must occur at r at least the strong chromatic number and would persist at all higher uniformities.\n\nCandidate contribution (lemma; novelty confidence low): The coordinate-marginal map governing strong hypergraph expansion admits an explicit bounded symmetric Hoeffding/ANOVA right inverse, yielding the quantitative local-reflection implication: if E^r(F) is Sidorenko, then F has no arbitrarily small L-infinity Sidorenko counterexamples around an interior constant kernel." }, { "id": 20000887, "problem_number": "AIM-COMBINATORICS-0012", "title": "Induced packing versus dynamic editing for C5", "statement": "Induced packing and covering\n\nLet $H$ be a fixed graph, $G$ an $n$-vertex graph. If the minimum number of edges needed to add or remove from $G$ in order to remove all induced copies of $H$ is $o(n^2)$, then the regularity lemma implies that the maximum number of induced copies of $H$, with each pair intersecting in at most one vertex, is $o(n^2)$.\n\nCan we get better bounds, even for the case $H=C_5$?", "original_statement": "Induced packing and covering\n\nLet $H$ be a fixed graph, $G$ an $n$-vertex graph. If the minimum number of edges needed to add or remove from $G$ in order to remove all induced copies of $H$ is $o(n^2)$, then the regularity lemma implies that the maximum number of induced copies of $H$, with each pair intersecting in at most one vertex, is $o(n^2)$.\n\nCan we get better bounds, even for the case $H=C_5$?", "clean_statement": "Induced packing and covering\n\nLet $H$ be a fixed graph, $G$ an $n$-vertex graph. If the minimum number of edges needed to add or remove from $G$ in order to remove all induced copies of $H$ is $o(n^2)$, then the regularity lemma implies that the maximum number of induced copies of $H$, with each pair intersecting in at most one vertex, is $o(n^2)$.\n\nCan we get better bounds, even for the case $H=C_5$?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 3.3 in the extremal-graph-theory section of the AIM workshop *High-dimensional phenomena in discrete analysis*. The live AIM page, which attributes the problem to Jacob Fox, has exactly the following introduction and question:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Extremal graph theory\nSource item: 3.3\nSource URL: http://aimpl.org/highdimdiscrete/3/\nCanonical location: aim-combinatorics-notes.json notes[11]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Induced packing and covering\\n\\nLet $H$ be a fixed graph, $G$ an $n$-vertex graph. If the minimum number of edges needed to add or remove from $G$ in order to remove all induced copies of $H$ is $o(n^2)$, then the regularity lemma implies that the maximum number of induced copies of $H$, with each pair intersecting in at most one vertex, is $o(n^2)$.\\n\\nCan we get better bounds, even for the case $H=C_5$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/3/", "tags": [ "aim", "AIM-COMBINATORICS-0012", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "With nu_H(G) denoting the largest family of induced H-copies whose vertex sets pairwise meet in at most one vertex, rho_H(G) the static pair-transversal of current copies, and d_H(G) the true induced-H edit distance, one always has nu_H <= rho_H <= d_H and rho_H <= binom(h,2) nu_H. Thus the implication literally printed in the AIM source, d_H=o(n^2) implies nu_H=o(n^2), is immediate; induced removal is needed for the converse, which follows from N_H^ind(G) <= binom(h,2) nu_H(G) binom(n-2,h-2). The new structured-host result is d_H(G) <= h Delta(G) nu_H(G) whenever H has no isolated vertex. For C5, self-complementarity gives d_C5(G) <= 5 min(Delta(G),Delta(complement G)) nu_C5(G). An explicit nine-vertex gadget, and all its disjoint unions, has rho_C5=nu_C5 but d_C5=2 nu_C5, proving that the static transversal and dynamic edit distance genuinely differ.\n\nCandidate contribution (structured_host_theorem_and_obstruction; novelty confidence low): Candidate novelty: every graph G satisfies d_C5(G) <= 5 min{Delta(G), Delta(complement G)} nu_C5(G), via an explicit isolation/universalization repair that cannot create new induced C5s; moreover, there is an explicit bounded-degree family J^{disjoint union q} with rho_C5=nu_C5=q and d_C5=2q." }, { "id": 20000888, "problem_number": "AIM-COMBINATORICS-0013", "title": "Quantifier-sensitive random barriers for dense linear tripartite patterns", "statement": "Turan exponents of hypergraphs\n\nThere exists $c>0$ such that the following holds. If $H$ is an $n$-vertex, 3-uniform hypergraph with at least $n^{3-c/d}$ edges, then $H$ contains some/every 3-uniform, 3-partite linear hypergraph on $d$ vertices with at least $d^2/100$ edges.", "original_statement": "Turan exponents of hypergraphs\n\nThere exists $c>0$ such that the following holds. If $H$ is an $n$-vertex, 3-uniform hypergraph with at least $n^{3-c/d}$ edges, then $H$ contains some/every 3-uniform, 3-partite linear hypergraph on $d$ vertices with at least $d^2/100$ edges.", "clean_statement": "Turan exponents of hypergraphs\n\nThere exists $c>0$ such that the following holds. If $H$ is an $n$-vertex, 3-uniform hypergraph with at least $n^{3-c/d}$ edges, then $H$ contains some/every 3-uniform, 3-partite linear hypergraph on $d$ vertices with at least $d^2/100$ edges.", "statement_status": "exact", "statement_verification": "The AIM record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Extremal graph theory\nSource item: 3.4\nSource URL: http://aimpl.org/highdimdiscrete/3/\nCanonical location: aim-combinatorics-notes.json notes[12]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Turan exponents of hypergraphs\\n\\nThere exists $c>0$ such that the following holds. If $H$ is an $n$-vertex, 3-uniform hypergraph with at least $n^{3-c/d}$ edges, then $H$ contains some/every 3-uniform, 3-partite linear hypergraph on $d$ vertices with at least $d^2/100$ edges.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"It is known that if $H$ has $\\\\geq n^{3-c/d^2}$ edges, then it contains $K_{d,d,d}^{(3)}$.\\n\\nInteresting cases are when $H$ is the set of triangles in some graph $G$, or a $p$-random subset of triangles in $G$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/3/", "tags": [ "aim", "AIM-COMBINATORICS-0013", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source's 'some/every' alternatives are different family-Turan problems. The maximum number of edges in a d-vertex 3-partite linear 3-graph is exactly floor(d/3) floor((d+1)/3), so the requested family is feasible after rounding. In the complete-base binomial model G^(3)(n,12n^(-c/d)), every isomorphism type of d-vertex 3-partite linear 3-graph occurs simultaneously with high probability for c<9, whereas for c>9 a cyclic Latin-square pattern with d^2/9 edges is absent for a suitable d; hence every deterministic constant for the 'every' reading is at most 9. A separate random-deletion construction shows that every constant for the 'some' reading is at most 100. For triangle hosts, F is contained in the triangle hypergraph of G exactly when its 2-shadow is contained in G.\n\nCandidate contribution (quantifier_sensitive_random_barrier; novelty confidence low): At probability 12n^(-c/d), simultaneous containment of all d-vertex 3-partite linear isomorphism types has sharp exponent constant 9 in the complete-base random model, while avoidance of every type with at least ceil(d^2/100) edges has the finite-d deletion barrier d(d-3)/(ceil(d^2/100)-1), tending to 100.", "research_difficulty_suggested": "L4" }, { "id": 20000889, "problem_number": "AIM-COMBINATORICS-0014", "title": "A holonomy reduction to equiangular lines", "statement": "Equi-isoclinic subspaces\n\nGiven $k$-dimensional linear subspaces $U,V\\leq \\mathbb{R}^d$, we say that $U,V$ are equi-isoclinic with angle $\\alpha$ if every unit vector in $U$ projects down onto $V$ a vector of length $\\alpha$.\n\nLet $N_\\alpha^k(d)$ be the maximum number of $k$-dimensional subspaces of $\\mathbb{R}^d$, every pair of which is equi-isoclinic with angle $\\alpha$. Then\n\n(1) $N_{\\alpha}^k(d)=O_{\\alpha,k}(d)$.\n\n(2) $N_{\\alpha}^k(d)=O_{k}(d)+o_{\\alpha,k}(d)$.", "original_statement": "Equi-isoclinic subspaces\n\nGiven $k$-dimensional linear subspaces $U,V\\leq \\mathbb{R}^d$, we say that $U,V$ are equi-isoclinic with angle $\\alpha$ if every unit vector in $U$ projects down onto $V$ a vector of length $\\alpha$.\n\nLet $N_\\alpha^k(d)$ be the maximum number of $k$-dimensional subspaces of $\\mathbb{R}^d$, every pair of which is equi-isoclinic with angle $\\alpha$. Then\n\n(1) $N_{\\alpha}^k(d)=O_{\\alpha,k}(d)$.\n\n(2) $N_{\\alpha}^k(d)=O_{k}(d)+o_{\\alpha,k}(d)$.", "clean_statement": "Equi-isoclinic subspaces\n\nGiven $k$-dimensional linear subspaces $U,V\\leq \\mathbb{R}^d$, we say that $U,V$ are equi-isoclinic with angle $\\alpha$ if every unit vector in $U$ projects down onto $V$ a vector of length $\\alpha$.\n\nLet $N_\\alpha^k(d)$ be the maximum number of $k$-dimensional subspaces of $\\mathbb{R}^d$, every pair of which is equi-isoclinic with angle $\\alpha$. Then\n\n(1) $N_{\\alpha}^k(d)=O_{\\alpha,k}(d)$.\n\n(2) $N_{\\alpha}^k(d)=O_{k}(d)+o_{\\alpha,k}(d)$.", "statement_status": "exact", "statement_verification": "The AIM record, problem 4.1 from the workshop *High-dimensional phenomena in discrete analysis*, states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: High-dimensional phenomena in discrete analysis\nSection: Discrete geometry\nSource item: 4.1\nSource URL: http://aimpl.org/highdimdiscrete/4/\nCanonical location: aim-combinatorics-notes.json notes[13]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Equi-isoclinic subspaces\\n\\nGiven $k$-dimensional linear subspaces $U,V\\\\leq \\\\mathbb{R}^d$, we say that $U,V$ are equi-isoclinic with angle $\\\\alpha$ if every unit vector in $U$ projects down onto $V$ a vector of length $\\\\alpha$.\\n\\nLet $N_\\\\alpha^k(d)$ be the maximum number of $k$-dimensional subspaces of $\\\\mathbb{R}^d$, every pair of which is equi-isoclinic with angle $\\\\alpha$. Then\\n\\n(1) $N_{\\\\alpha}^k(d)=O_{\\\\alpha,k}(d)$.\\n\\n(2) $N_{\\\\alpha}^k(d)=O_{k}(d)+o_{\\\\alpha,k}(d)$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The case $k=1$ gives equiangular lines, which is known, using Ramsey Theory.\\n\\n$k=2$ generalizes \\\"complex equiangular lines\\\". Conjecture 1 is known for complex equiangular lines.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/highdimdiscrete/4/", "tags": [ "aim", "AIM-COMBINATORICS-0014", "aim-domain:combinatorics", "aim-workshop:highdimdiscrete", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For 02$\n\nSuppose $A\\subset \\mathbb{F}_p[x]$ is of finite size, and every element of $A$ is a square. Show that $|A+A|\\geq |A|^{1+\\epsilon}$ for some constant $\\epsilon$.", "original_statement": "Rudin's problem for $\\mathbb{F}_p[x], p>2$\n\nSuppose $A\\subset \\mathbb{F}_p[x]$ is of finite size, and every element of $A$ is a square. Show that $|A+A|\\geq |A|^{1+\\epsilon}$ for some constant $\\epsilon$.", "clean_statement": "Rudin's problem for $\\mathbb{F}_p[x], p>2$\n\nSuppose $A\\subset \\mathbb{F}_p[x]$ is of finite size, and every element of $A$ is a square. Show that $|A+A|\\geq |A|^{1+\\epsilon}$ for some constant $\\epsilon$.", "statement_status": "exact", "statement_verification": "The canonical record is from the 2018 AIM workshop *Additive combinatorics and its applications*, section “Polynomial method,” problem 1.1. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Polynomial method\nSource item: 1.1\nSource URL: http://aimpl.org/addcombapp/1/\nCanonical location: aim-combinatorics-notes.json notes[16]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Rudin's problem for $\\\\mathbb{F}_p[x], p>2$\\n\\nSuppose $A\\\\subset \\\\mathbb{F}_p[x]$ is of finite size, and every element of $A$ is a square. Show that $|A+A|\\\\geq |A|^{1+\\\\epsilon}$ for some constant $\\\\epsilon$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"We know that Ruzsa's modeling theorem, and $4AP$ conjecture, implies the given problem.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/1/", "tags": [ "aim", "AIM-COMBINATORICS-0017", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source is ambiguous about dependence on p. A uniform exponent is false: the constant squares Q_p have (p+1)/2 elements and Q_p+Q_p=F_p. For the natural fixed-p reading, if S is any choice of one square root for each element of A, then E(A) is at most n^2+(p-1) times the ordered sum of tau(r^2-u^2) over distinct r,u in S, where tau counts monic divisors; hence |A+A| is at least n^4 divided by that quantity. This yields an explicit average-divisor criterion for power growth, exact Sidon theorems for degree/valuation-separated sets and nondegenerate affine root lines, and a scalar-degeneracy theorem for four square polynomials in arithmetic progression when p is at least 5.\n\nCandidate contribution (reduction; novelty confidence low): For every finite square set A in F_p[x] and every root section S, |A+A| >= n^4/(n^2+(p-1)D(S)), where D(S)=sum_{r!=u} tau(r^2-u^2) over ordered root pairs and tau is the monic-divisor function; in particular, an average bound tau_bar <= C n^(1-2 eta) implies |A+A| >= n^(1+eta) above an explicit threshold." }, { "id": 20000893, "problem_number": "AIM-COMBINATORICS-0018", "title": "Uniform-prime counterexample and a fixed-characteristic polynomial reduction", "statement": "3APs in sets of small growth\n\nObtain a proof of the following theorem using the polynomial method.\n\nTheorem. There is an absolute constant $c>0$ that the following holds. If $A\\subset \\mathbb{F}_p^n$ and $|A+A|\\leq |A|^{1+c}$ then $A$ contains a 3AP.", "original_statement": "3APs in sets of small growth\n\nObtain a proof of the following theorem using the polynomial method.\n\nTheorem. There is an absolute constant $c>0$ that the following holds. If $A\\subset \\mathbb{F}_p^n$ and $|A+A|\\leq |A|^{1+c}$ then $A$ contains a 3AP.", "clean_statement": "3APs in sets of small growth\n\nObtain a proof of the following theorem using the polynomial method.\n\nTheorem. There is an absolute constant $c>0$ that the following holds. If $A\\subset \\mathbb{F}_p^n$ and $|A+A|\\leq |A|^{1+c}$ then $A$ contains a 3AP.", "statement_status": "exact", "statement_verification": "The AIM record, from the workshop “Additive combinatorics and its applications,” Polynomial method, Problem 1.2, says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Polynomial method\nSource item: 1.2\nSource URL: http://aimpl.org/addcombapp/1/\nCanonical location: aim-combinatorics-notes.json notes[17]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3APs in sets of small growth\\n\\nObtain a proof of the following theorem using the polynomial method.\\n\\nTheorem. There is an absolute constant $c>0$ that the following holds. If $A\\\\subset \\\\mathbb{F}_p^n$ and $|A+A|\\\\leq |A|^{1+c}$ then $A$ contains a 3AP.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The current proof uses Ruzsa's modeling lemma.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/1/", "tags": [ "aim", "AIM-COMBINATORICS-0018", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The theorem as literally stated has no constant uniform over all primes: for every c>0, a sufficiently large Behrend set embedded without wraparound in F_p gives a proper-3AP-free A with |A+A| at most |A|^(1+c). For every fixed odd p, however, random linear Freiman compression followed by the Ellenberg-Gijswijt polynomial bound proves |A+A|/|A| at least C_p |A|^eta_p, where rho_p=log_p Gamma_p, eta_p=(1-rho_p)/(4 rho_p), and C_p=(3(2p)^rho_p)^(-1/(4 rho_p)). This yields a p-dependent exponent c_p>0 after treating finitely many small cardinalities separately.\n\nCandidate contribution (reduction; novelty confidence low): The candidate contribution is an explicit packaged compression-to-cap inequality K >= (3(2p)^rho_p)^(-1/(4 rho_p)) |A|^((1-rho_p)/(4 rho_p)), paired with a Behrend obstruction to any prime-uniform exponent and a proof that the support-only progression-tensor decomposition yields only |A| <= |A+A|." }, { "id": 20000894, "problem_number": "AIM-COMBINATORICS-0019", "title": "A normalization counterexample for analytic-rank tensorization", "statement": "Polynomial method via tensorizing\n\nThe purpose of this problem is to extend the current polynomial method techniques to handle 4APs and more complicated patterns. It's about suggesting a notion of tensor rank that tensorizes. Let $\\mathbb{F}$ be a field and $A\\in \\mathbb{F}^{n\\times \\cdots \\times n}$ be a $d$-dimensional tensor. We can think of $A$ as a multilinear polynomial $f:(\\mathbb{F}^n)^d\\rightarrow \\mathbb{F}$ on variables $X_1,\\cdots,X_d \\in \\mathbb{F}^n$ where $A$ specifies the coefficients of the monomials. In the following we introduce several notions of rank.\nLet $rank(A)$ be the minimum number of rank-one tensors that sum up to $A$. We need to specify the definition of a rank-one tensor. There are several definitions of a rank-1 tensor $f$.\n\n$rank$-1:\n$$f(X_1,\\cdots,X_d) = g_1(X_1)\\cdots g_d(X_d)$$\nwhere each $g_i:\\mathbb{F}^n\\rightarrow\\mathbb{F}$ is linear\n\n$srank$-1: There is some $i\\in [d]$ so that\n$$f(X_1,\\cdots,X_d) = g(X_i)f'(X_1,\\cdots,X_d)$$\nwhere $g$ is linear and $f'$ does not depend on $X_i$.\n\n$prank$-1: There is a subset $\\phi\\neq S\\subset[d]$ so that\n$$f(X_1,\\cdots,X_d) = g(X_i: i\\in S) h(X_i: i\\notin S)$$\nNote that $prank(\\cdot)\\leq srank(\\cdot)\\leq rank(\\cdot)$.\nWe introduce another notion of rank (only for $\\mathbb{F}_p$, where $p$ is prime) called analytic rank.\nLet $$Bias (f) = \\mathbb{E}_{X_1,\\cdots,X_d} \\omega^{f(X_1,\\cdots,X_d)}$$\nwhere $\\omega$ is a $p$th root of unity. Also note that $Bias(f)\\in [0,1]$.\nDefine $$arank(f) = -\\log Bias(f).$$\nWe have that $arank(\\cdot)\\leq prank(\\cdot)$.\n\nNow, let's see how these notions of rank are used to solve, say the capset problem. The idea is that the identity tensor has high rank for all definitions of $arank,srank,prank$, but the capset tensor has low rank. The capset tensor is $C\\in \\mathbb{F}_3^{n\\times n \\times n}$ where $n= 3^k$ for some $k$. Given $x,y,z\\in \\mathbb{F}_3^k$ we have $C(x,y,z)= 0 $ if $x+y+z\\neq 0$ and is 1 otherwise.\nIf $S\\subset \\mathbb{F}_3^k$ is $AP$-free, then $C_{|S}$ ($C$ restricted to the set $S$) is identity.\n\nLet $\\mathbf{r}$ be either $arank$ or $prank$. Let $A$ be a tensor, and $Id$ is the identity tensor on the same space as $A$. Suppose $\\mathbf{r}(A)< \\mathbf{r}(Id)$. Is it true that $\\mathbf{r}(A^{\\otimes m}) < \\mathbf{r}(Id)^m c^m$ for some constant $c<1$?", "original_statement": "Polynomial method via tensorizing\n\nThe purpose of this problem is to extend the current polynomial method techniques to handle 4APs and more complicated patterns. It's about suggesting a notion of tensor rank that tensorizes. Let $\\mathbb{F}$ be a field and $A\\in \\mathbb{F}^{n\\times \\cdots \\times n}$ be a $d$-dimensional tensor. We can think of $A$ as a multilinear polynomial $f:(\\mathbb{F}^n)^d\\rightarrow \\mathbb{F}$ on variables $X_1,\\cdots,X_d \\in \\mathbb{F}^n$ where $A$ specifies the coefficients of the monomials. In the following we introduce several notions of rank.\nLet $rank(A)$ be the minimum number of rank-one tensors that sum up to $A$. We need to specify the definition of a rank-one tensor. There are several definitions of a rank-1 tensor $f$.\n\n$rank$-1:\n$$f(X_1,\\cdots,X_d) = g_1(X_1)\\cdots g_d(X_d)$$\nwhere each $g_i:\\mathbb{F}^n\\rightarrow\\mathbb{F}$ is linear\n\n$srank$-1: There is some $i\\in [d]$ so that\n$$f(X_1,\\cdots,X_d) = g(X_i)f'(X_1,\\cdots,X_d)$$\nwhere $g$ is linear and $f'$ does not depend on $X_i$.\n\n$prank$-1: There is a subset $\\phi\\neq S\\subset[d]$ so that\n$$f(X_1,\\cdots,X_d) = g(X_i: i\\in S) h(X_i: i\\notin S)$$\nNote that $prank(\\cdot)\\leq srank(\\cdot)\\leq rank(\\cdot)$.\nWe introduce another notion of rank (only for $\\mathbb{F}_p$, where $p$ is prime) called analytic rank.\nLet $$Bias (f) = \\mathbb{E}_{X_1,\\cdots,X_d} \\omega^{f(X_1,\\cdots,X_d)}$$\nwhere $\\omega$ is a $p$th root of unity. Also note that $Bias(f)\\in [0,1]$.\nDefine $$arank(f) = -\\log Bias(f).$$\nWe have that $arank(\\cdot)\\leq prank(\\cdot)$.\n\nNow, let's see how these notions of rank are used to solve, say the capset problem. The idea is that the identity tensor has high rank for all definitions of $arank,srank,prank$, but the capset tensor has low rank. The capset tensor is $C\\in \\mathbb{F}_3^{n\\times n \\times n}$ where $n= 3^k$ for some $k$. Given $x,y,z\\in \\mathbb{F}_3^k$ we have $C(x,y,z)= 0 $ if $x+y+z\\neq 0$ and is 1 otherwise.\nIf $S\\subset \\mathbb{F}_3^k$ is $AP$-free, then $C_{|S}$ ($C$ restricted to the set $S$) is identity.\n\nLet $\\mathbf{r}$ be either $arank$ or $prank$. Let $A$ be a tensor, and $Id$ is the identity tensor on the same space as $A$. Suppose $\\mathbf{r}(A)< \\mathbf{r}(Id)$. Is it true that $\\mathbf{r}(A^{\\otimes m}) < \\mathbf{r}(Id)^m c^m$ for some constant $c<1$?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "1. The printed `\\(\\phi\\neq S\\subset[d]\\)` is almost certainly an extraction/OCR error for the standard condition \\(\\varnothing\\ne S\\subsetneq[d]\\). Both factors in a partition-rank-one decomposition must use nonempty complementary sets of modes. 2. \\(\\omega\\) must be a **nontrivial** additive character, i.e. a primitive \\(p\\)-th root in the prime-field notation. If \\(\\omega=1\\), every bias is one. 3. Standard analytic rank is \\[ \\operatorname{arank}_p(T)=-\\log_p\\operatorname{bias}(T). \\] The missing logarithm base matters critically when a numerical rank is raised to the \\(m\\)-th power. Base \\(p\\) is also the normalization for which matrix analytic rank equals matrix rank and Lovett's inequality \\(\\operatorname{arank}\\le\\operatorname{prank}\\) has constant one. 4. The identity must be the order-\\(d\\), side-\\(n\\) diagonal tensor \\[ I_{n,d}=\\sum_{i=1}^n e_i^*\\otimes\\cdots\\otimes e_i^*. \\...", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Polynomial method\nSource item: 1.3\nSource URL: http://aimpl.org/addcombapp/1/\nCanonical location: aim-combinatorics-notes.json notes[18]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Polynomial method via tensorizing\\n\\nThe purpose of this problem is to extend the current polynomial method techniques to handle 4APs and more complicated patterns. It's about suggesting a notion of tensor rank that tensorizes. Let $\\\\mathbb{F}$ be a field and $A\\\\in \\\\mathbb{F}^{n\\\\times \\\\cdots \\\\times n}$ be a $d$-dimensional tensor. We can think of $A$ as a multilinear polynomial $f:(\\\\mathbb{F}^n)^d\\\\rightarrow \\\\mathbb{F}$ on variables $X_1,\\\\cdots,X_d \\\\in \\\\mathbb{F}^n$ where $A$ specifies the coefficients of the monomials. In the following we introduce several notions of rank.\\nLet $rank(A)$ be the minimum number of rank-one tensors that sum up to $A$. We need to specify the definition of a rank-one tensor. There are several definitions of a rank-1 tensor $f$.\\n\\n$rank$-1:\\n$$f(X_1,\\\\cdots,X_d) = g_1(X_1)\\\\cdots g_d(X_d)$$\\nwhere each $g_i:\\\\mathbb{F}^n\\\\rightarrow\\\\mathbb{F}$ is linear\\n\\n$srank$-1: There is some $i\\\\in [d]$ so that\\n$$f(X_1,\\\\cdots,X_d) = g(X_i)f'(X_1,\\\\cdots,X_d)$$\\nwhere $g$ is linear and $f'$ does not depend on $X_i$.\\n\\n$prank$-1: There is a subset $\\\\phi\\\\neq S\\\\subset[d]$ so that\\n$$f(X_1,\\\\cdots,X_d) = g(X_i: i\\\\in S) h(X_i: i\\\\notin S)$$\\nNote that $prank(\\\\cdot)\\\\leq srank(\\\\cdot)\\\\leq rank(\\\\cdot)$.\\nWe introduce another notion of rank (only for $\\\\mathbb{F}_p$, where $p$ is prime) called analytic rank.\\nLet $$Bias (f) = \\\\mathbb{E}_{X_1,\\\\cdots,X_d} \\\\omega^{f(X_1,\\\\cdots,X_d)}$$\\nwhere $\\\\omega$ is a $p$th root of unity. Also note that $Bias(f)\\\\in [0,1]$.\\nDefine $$arank(f) = -\\\\log Bias(f).$$\\nWe have that $arank(\\\\cdot)\\\\leq prank(\\\\cdot)$.\\n\\nNow, let's see how these notions of rank are used to solve, say the capset problem. The idea is that the identity tensor has high rank for all definitions of $arank,srank,prank$, but the capset tensor has low rank. The capset tensor is $C\\\\in \\\\mathbb{F}_3^{n\\\\times n \\\\times n}$ where $n= 3^k$ for some $k$. Given $x,y,z\\\\in \\\\mathbb{F}_3^k$ we have $C(x,y,z)= 0 $ if $x+y+z\\\\neq 0$ and is 1 otherwise.\\nIf $S\\\\subset \\\\mathbb{F}_3^k$ is $AP$-free, then $C_{|S}$ ($C$ restricted to the set $S$) is identity.\\n\\nLet $\\\\mathbf{r}$ be either $arank$ or $prank$. Let $A$ be a tensor, and $Id$ is the identity tensor on the same space as $A$. Suppose $\\\\mathbf{r}(A)< \\\\mathbf{r}(Id)$. Is it true that $\\\\mathbf{r}(A^{\\\\otimes m}) < \\\\mathbf{r}(Id)^m c^m$ for some constant $c<1$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is known if $\\\\mathbf{r}$ is chosen to be $srank$.\\nAlso, this problem, if true, has several consequences such as exponential upper bounds on size of $4AP$-free sets.\\nMore generally, it would be great to find any useful notion of rank that tensorizes.\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/1/", "tags": [ "aim", "AIM-COMBINATORICS-0019", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "With standard base-p analytic rank and the fixed-order Kronecker product, the analytic-rank branch of the printed AIM implication is false. For p=2, d=3, n=2, a nonzero pure tensor A has analytic rank a=log_2(4/3), the identity has rank 2a, and every Kronecker power of A still has rank a; because 2a<1, the proposed upper bound (2ac)^m tends to zero for every fixed c<1. Replacing r(Id)^m by r(Id^{boxtimes m}) repairs scale invariance, and on every k-supported diagonal tensor the corrected analytic-, partition-, and slice-rank gap is exactly (k/n)^m.\n\nCandidate contribution (counterexample; novelty confidence low): The pure tensor e_1^{* tensor 3} over F_2 with ambient side length two is an explicit counterexample to the literal analytic-rank tensor-power formula, while the powered-identity normalization gives an exact (k/n)^m ratio on every k-supported diagonal family." }, { "id": 20000895, "problem_number": "AIM-COMBINATORICS-0020", "title": "Arithmetic removal in cyclic groups: corrected quantifiers, matching distances, and exact CRT-compatible lifts", "statement": "Arithmetic removal lemma for cyclic groups\n\nThe arithmetic removal lemma is the following. $\\forall\\varepsilon>0, \\exists\\delta>0$ such that if $G$ is an abelian group and $A,B,C\\subset G$, and suppose there are at least $\\delta|G|^2$ solutions to $a+b+c=0, a\\in A, b\\in B, c\\in C$. Then you can remove $\\varepsilon|G|$ elements from each of the sets $A,B,C$ so that there are no solutions to $a+b+c=0$. In the case of $G = \\mathbb{F}_p^n$ we know $\\delta = \\varepsilon^{c_p+o(1)}$ where $c_p$ is a known absolute constant and this is sharp. In the case of cyclic groups, we know that because of Behrend's construction $\\delta$ can not be a polynomial in $\\varepsilon$. The known bound is however is of tower type.\n\nObtain improved bound for arithmetic regularity lemma for cyclic groups.", "original_statement": "Arithmetic removal lemma for cyclic groups\n\nThe arithmetic removal lemma is the following. $\\forall\\varepsilon>0, \\exists\\delta>0$ such that if $G$ is an abelian group and $A,B,C\\subset G$, and suppose there are at least $\\delta|G|^2$ solutions to $a+b+c=0, a\\in A, b\\in B, c\\in C$. Then you can remove $\\varepsilon|G|$ elements from each of the sets $A,B,C$ so that there are no solutions to $a+b+c=0$. In the case of $G = \\mathbb{F}_p^n$ we know $\\delta = \\varepsilon^{c_p+o(1)}$ where $c_p$ is a known absolute constant and this is sharp. In the case of cyclic groups, we know that because of Behrend's construction $\\delta$ can not be a polynomial in $\\varepsilon$. The known bound is however is of tower type.\n\nObtain improved bound for arithmetic regularity lemma for cyclic groups.", "clean_statement": "Arithmetic removal lemma for cyclic groups\n\nThe arithmetic removal lemma is the following. $\\forall\\varepsilon>0, \\exists\\delta>0$ such that if $G$ is an abelian group and $A,B,C\\subset G$, and suppose there are at least $\\delta|G|^2$ solutions to $a+b+c=0, a\\in A, b\\in B, c\\in C$. Then you can remove $\\varepsilon|G|$ elements from each of the sets $A,B,C$ so that there are no solutions to $a+b+c=0$. In the case of $G = \\mathbb{F}_p^n$ we know $\\delta = \\varepsilon^{c_p+o(1)}$ where $c_p$ is a known absolute constant and this is sharp. In the case of cyclic groups, we know that because of Behrend's construction $\\delta$ can not be a polynomial in $\\varepsilon$. The known bound is however is of tower type.\n\nObtain improved bound for arithmetic regularity lemma for cyclic groups.", "statement_status": "exact", "statement_verification": "The AIM record is titled **“Arithmetic removal lemma for cyclic groups.”** Its displayed formulation says that if there are *at least* \\(\\delta |G|^2\\) solutions of \\[ a+b+c=0,\\qquad (a,b,c)\\in A\\times B\\times C, \\] then one may remove \\(\\varepsilon |G|\\) elements from each set and destroy all solutions. It then contrasts the sharp finite-vector-space bound with a tower-type bound for cyclic groups, mentions Behrend's obstruction to polynomial dependence, and ends with “Obtain improved bound for arithmetic regularity lemma for cyclic groups.” The accompanying literature note points to tri-colored sum-free sets.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Polynomial method\nSource item: 1.4\nSource URL: http://aimpl.org/addcombapp/1/\nCanonical location: aim-combinatorics-notes.json notes[19]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Arithmetic removal lemma for cyclic groups\\n\\nThe arithmetic removal lemma is the following. $\\\\forall\\\\varepsilon>0, \\\\exists\\\\delta>0$ such that if $G$ is an abelian group and $A,B,C\\\\subset G$, and suppose there are at least $\\\\delta|G|^2$ solutions to $a+b+c=0, a\\\\in A, b\\\\in B, c\\\\in C$. Then you can remove $\\\\varepsilon|G|$ elements from each of the sets $A,B,C$ so that there are no solutions to $a+b+c=0$. In the case of $G = \\\\mathbb{F}_p^n$ we know $\\\\delta = \\\\varepsilon^{c_p+o(1)}$ where $c_p$ is a known absolute constant and this is sharp. In the case of cyclic groups, we know that because of Behrend's construction $\\\\delta$ can not be a polynomial in $\\\\varepsilon$. The known bound is however is of tower type.\\n\\nObtain improved bound for arithmetic regularity lemma for cyclic groups.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Suppose we have three sets $A=\\\\{a_i: 1\\\\leq i\\\\leq n\\\\},B=\\\\{b_i: 1\\\\leq i\\\\leq n\\\\},C= \\\\{c_i: 1\\\\leq i\\\\leq n\\\\}\\\\subset G$. This triple is called tri-colored sumfree if $$a_i+b_j+c_k=0 \\\\iff i=j=k.$$\\nObtaining strong upper bounds on the size of tri-colored sum-free sets when $G$ is a cyclic group would possibly result in an improved arithmetic removal lemma.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/1/", "tags": [ "aim", "AIM-COMBINATORICS-0020", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM statement has its removal hypothesis reversed: the literal 'at least' form is false, while the corrected cyclic quantitative problem remains open. For the corrected problem, this attempt proves exact convention-aware relationships among triangle matchings and total/per-color deletion distance, including an exact product-lift identity: for full lifts of X,Y,Z from a finite abelian group H to H x K, the triangle count is multiplied by |K|^2 and the minimum total colored deletion distance is multiplied exactly by |K|. The complementary-slice proof yields CRT propagation of normalized obstructions from C_M to C_{MQ} when gcd(M,Q)=1, with at most a factor-three loss for the source's per-color convention. A Behrend construction is also normalized explicitly to recover the known superpolynomial obstruction.\n\nCandidate contribution (lemma; novelty confidence low): For full product lifts, tau_1(X x K,Y x K,Z x K)=|K| tau_1(X,Y,Z) exactly; consequently normalized total deletion distance and triangle density are invariant, and cyclic obstructions propagate to coprime multiples via CRT, while per-color distance is preserved within a factor of three." }, { "id": 20000896, "problem_number": "AIM-COMBINATORICS-0021", "title": "Parity, polynomial rank, and an exact prime-distance case", "statement": "A polynomial method proof of Frankl-Rodl theorem\n\nThe Frankl-Rodl theorem is the following: let $A\\subset \\mathbb{F}_2^n$ be so that $A+A$ does not contain any element with hamming weight $k$ for some $k$ satisfying $C\\sqrt[]{\\log \\alpha^{-1}n}\\leq k\\leq n-C\\sqrt[]{\\log \\alpha^{-1}n}$. Then $A$ has density at most $\\alpha$.\n\nFind a proof of Frankl-Rodl theorem using the polynomial method, in particular Croot-Lev-Pach method.", "original_statement": "A polynomial method proof of Frankl-Rodl theorem\n\nThe Frankl-Rodl theorem is the following: let $A\\subset \\mathbb{F}_2^n$ be so that $A+A$ does not contain any element with hamming weight $k$ for some $k$ satisfying $C\\sqrt[]{\\log \\alpha^{-1}n}\\leq k\\leq n-C\\sqrt[]{\\log \\alpha^{-1}n}$. Then $A$ has density at most $\\alpha$.\n\nFind a proof of Frankl-Rodl theorem using the polynomial method, in particular Croot-Lev-Pach method.", "clean_statement": "A polynomial method proof of Frankl-Rodl theorem\n\nThe Frankl-Rodl theorem is the following: let $A\\subset \\mathbb{F}_2^n$ be so that $A+A$ does not contain any element with hamming weight $k$ for some $k$ satisfying $C\\sqrt[]{\\log \\alpha^{-1}n}\\leq k\\leq n-C\\sqrt[]{\\log \\alpha^{-1}n}$. Then $A$ has density at most $\\alpha$.\n\nFind a proof of Frankl-Rodl theorem using the polynomial method, in particular Croot-Lev-Pach method.", "statement_status": "exact", "statement_verification": "The canonical `problem` field is reproduced verbatim below. This preserves the missing parity hypothesis and the ambiguous expression under the square root.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Polynomial method\nSource item: 1.5\nSource URL: http://aimpl.org/addcombapp/1/\nCanonical location: aim-combinatorics-notes.json notes[20]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A polynomial method proof of Frankl-Rodl theorem\\n\\nThe Frankl-Rodl theorem is the following: let $A\\\\subset \\\\mathbb{F}_2^n$ be so that $A+A$ does not contain any element with hamming weight $k$ for some $k$ satisfying $C\\\\sqrt[]{\\\\log \\\\alpha^{-1}n}\\\\leq k\\\\leq n-C\\\\sqrt[]{\\\\log \\\\alpha^{-1}n}$. Then $A$ has density at most $\\\\alpha$.\\n\\nFind a proof of Frankl-Rodl theorem using the polynomial method, in particular Croot-Lev-Pach method.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/1/", "tags": [ "aim", "AIM-COMBINATORICS-0021", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literal source statement is false for odd k, as witnessed by the density-1/2 even-weight subspace. After restoring the necessary even-distance condition and the strongly supported sqrt(n log(1/alpha)) scale, a direct two-variable CLP-style argument proves that if p is an odd prime and A in F_2^n avoids distance 2p, then, for N=min(n,4p-1), its density is at most 2(M(N,p-1)+M(N,p-2))/2^N. This is below 4 exp(-(n-2p)^2/(2n)) when 2p<=n<=4p-1 and below 4 exp(-p/2) when n>=4p. Exact Krawtchouk and multilinear-degree calculations also explain why the ordinary one-sphere Hoffman bound and the naive radial CLP kernel cannot establish the full theorem.\n\nCandidate contribution (special_case; novelty confidence low): For every odd prime p and n>=2p, splitting each (4p-1)-coordinate fiber into its two parity classes converts the Fermat divisibility kernel into an identity matrix and gives the exact single-distance bound mu(A)<=2(M(N,p-1)+M(N,p-2))/2^N, with explicit upper-endpoint and long-cube entropy regimes." }, { "id": 20000897, "problem_number": "AIM-COMBINATORICS-0022", "title": "Popular-direction planarization for no-five-coplanar sumsets", "statement": "Doubling of generic set of points\n\nThere is a result of Stancescu as follows: If $A\\subset \\mathbb{R}^2$ is a finite set with no three points on a line, then $$|A+A|\\geq |A|(\\log |A|)^c $$ for some $c>0$.\nAlso we know that the lower bound of $|A|^{1-\\varepsilon}$ is false because of the Behrend construction. We would like to generalize this statement to higher dimensions.\n\nLet $A\\subset \\mathbb{R}^3$ be a finite set with no five points on a plane. What is the best inequality $$|A+A|\\geq |A| f(A)$$\nthat we can have?", "original_statement": "Doubling of generic set of points\n\nThere is a result of Stancescu as follows: If $A\\subset \\mathbb{R}^2$ is a finite set with no three points on a line, then $$|A+A|\\geq |A|(\\log |A|)^c $$ for some $c>0$.\nAlso we know that the lower bound of $|A|^{1-\\varepsilon}$ is false because of the Behrend construction. We would like to generalize this statement to higher dimensions.\n\nLet $A\\subset \\mathbb{R}^3$ be a finite set with no five points on a plane. What is the best inequality $$|A+A|\\geq |A| f(A)$$\nthat we can have?", "clean_statement": "Doubling of generic set of points\n\nThere is a result of Stancescu as follows: If $A\\subset \\mathbb{R}^2$ is a finite set with no three points on a line, then $$|A+A|\\geq |A|(\\log |A|)^c $$ for some $c>0$.\nAlso we know that the lower bound of $|A|^{1-\\varepsilon}$ is false because of the Behrend construction. We would like to generalize this statement to higher dimensions.\n\nLet $A\\subset \\mathbb{R}^3$ be a finite set with no five points on a plane. What is the best inequality $$|A+A|\\geq |A| f(A)$$\nthat we can have?", "statement_status": "exact", "statement_verification": "The exact AIM record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Additive combinatorics\nSource item: 2.05\nSource URL: http://aimpl.org/addcombapp/2/\nCanonical location: aim-combinatorics-notes.json notes[21]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Doubling of generic set of points\\n\\nThere is a result of Stancescu as follows: If $A\\\\subset \\\\mathbb{R}^2$ is a finite set with no three points on a line, then $$|A+A|\\\\geq |A|(\\\\log |A|)^c $$ for some $c>0$.\\nAlso we know that the lower bound of $|A|^{1-\\\\varepsilon}$ is false because of the Behrend construction. We would like to generalize this statement to higher dimensions.\\n\\nLet $A\\\\subset \\\\mathbb{R}^3$ be a finite set with no five points on a plane. What is the best inequality $$|A+A|\\\\geq |A| f(A)$$\\nthat we can have?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"It's known that $f(A)\\\\geq \\\\exp{\\\\log^c|A|}$. Also here, the Behrend construction does not imply $f(A)\\\\leq |A|^c$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/2/", "tags": [ "aim", "AIM-COMBINATORICS-0022", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If A is an n-point subset of R^3 with no five distinct points in an affine plane and S=|A+A|n^2/(4S) and |X+X|<=S. Combining this proved reduction with the Ruzsa-Stanchescu planar bound and the May 2026 Roth bound r_3(N)<=N exp(-c(log N)^(1/6)/log log N) yields |A+A|>=C n exp(c(log n)^(1/6)/log log n) for sufficiently large n. The optimal factor remains open; the earlier 1/9 Roth estimate gives the corresponding independent fallback.\n\nCandidate contribution (lemma; novelty confidence low): For every no-five-coplanar A with |A+A|<|A|^2/2, there is a nonzero popular difference v such that the quotient along v contains a no-three-collinear set X of size greater than |A|^2/(4|A+A|), with X+X contained in the quotient image of A+A; consequently any eventual planar growth factor g transfers through F^2 >= (1/4)g(|A|/(4F)), where F=|A+A|/|A|." }, { "id": 20000898, "problem_number": "AIM-COMBINATORICS-0023", "title": "Off-diagonal van der Waerden numbers: refuted quadratic conjecture and a difference-sensitive transversal criterion", "statement": "A question about 2-coloring of integers\n\nLet $W(k,\\ell)$ be the minimum $n$ so that any red/blue coloring of $\\{1,\\cdots, n\\}$ has either a red $k$-AP or a blue $\\ell$-AP.\nThe following conjecture is supported by numerical evidence.\n\n$W(3,\\ell)= \\ell^{2+o(1)}$.", "original_statement": "A question about 2-coloring of integers\n\nLet $W(k,\\ell)$ be the minimum $n$ so that any red/blue coloring of $\\{1,\\cdots, n\\}$ has either a red $k$-AP or a blue $\\ell$-AP.\nThe following conjecture is supported by numerical evidence.\n\n$W(3,\\ell)= \\ell^{2+o(1)}$.", "clean_statement": "A question about 2-coloring of integers\n\nLet $W(k,\\ell)$ be the minimum $n$ so that any red/blue coloring of $\\{1,\\cdots, n\\}$ has either a red $k$-AP or a blue $\\ell$-AP.\nThe following conjecture is supported by numerical evidence.\n\n$W(3,\\ell)= \\ell^{2+o(1)}$.", "statement_status": "exact", "statement_verification": "The exact AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Additive combinatorics\nSource item: 2.1\nSource URL: http://aimpl.org/addcombapp/2/\nCanonical location: aim-combinatorics-notes.json notes[22]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A question about 2-coloring of integers\\n\\nLet $W(k,\\\\ell)$ be the minimum $n$ so that any red/blue coloring of $\\\\{1,\\\\cdots, n\\\\}$ has either a red $k$-AP or a blue $\\\\ell$-AP.\\nThe following conjecture is supported by numerical evidence.\\n\\n$W(3,\\\\ell)= \\\\ell^{2+o(1)}$.\"\nOriginal remarks: [\"Progress on finding the true bounds by Ben Green:\\nhttps://arxiv.org/abs/2102.01543\"]\nOriginal literature field (JSON string): \"Using Roth's theorem one can get $W(3,\\\\ell)\\\\leq 2^{\\\\ell^{1+o(1)}}$.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/2/", "tags": [ "aim", "AIM-COMBINATORICS-0023", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The conjecture W(3,l)=l^{2+o(1)} is refuted: Green proved a superpolynomial lower bound and Hunter strengthened it to exp(c(log l)^2/loglog l). Combining the exact 3AP-free-transversal formulation with Raghavan's March 2026 preprint bound for r_3(N) gives the current inferred upper bound exp(C(log l)^6(loglog l)^6), subject to the preprint caveat. This attempt additionally proves an exact equivalence between W(3,l)>N and the existence of a 3AP-free hitting set for every l-AP, an exact SAT formulation, and a difference-sensitive Lovasz-local-lemma certificate e max{p^3,(1-p)^l} l(N-1+l|Delta|)<=1, including fully audited dependency counts and a constructive interpretation.\n\nCandidate contribution (lemma; novelty confidence low): For admissible long-progression differences Delta, if e max{p^3,(1-p)^l} l(N-1+l|Delta|)<=1 for some p in (0,1), then [N] admits a coloring with no red nonconstant 3AP and no blue l-AP having difference in Delta; the criterion follows from the exact bound of N-1+l|Delta| relevant bad events through each integer and supports Moser-Tardos resampling under slack." }, { "id": 20000899, "problem_number": "AIM-COMBINATORICS-0024", "title": "A variational energy core for stronger BSG", "statement": "A stronger BSG theorem\n\nThe standard BSG theorem is as follows: let $E(A) = |\\{(a,b,c,d)\\in A^4: a+b=c+d\\}|$. If $\\frac{E(A)}{|A|^3}\\geq \\frac{1}{K}$ then there is a subset $A'\\subset A$ so that $|A'|\\geq \\frac{|A|}{K^{100}}$ and $|A'+A'|\\leq K^{100}|A'|$.\nOne of the drawbacks of BSG theorem is that we lose control on $E(A')$. Can we have a stronger BSG theorem as follows?\n\nLet $A\\subset G$, and suppose $E(A)\\geq \\frac{|A|^3}{K}$. Then there is a set $A'\\subset A$ so that $$|A'|\\geq \\frac{|A|}{K^{100}},$$\n$$|A'+A'|\\leq K^{100}|A'|,$$\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3}$$\nCan we moreover obtain the following?\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3} \\left(\\frac{|A|}{|A'|}\\right)^{\\frac{1}{100}}$$", "original_statement": "A stronger BSG theorem\n\nThe standard BSG theorem is as follows: let $E(A) = |\\{(a,b,c,d)\\in A^4: a+b=c+d\\}|$. If $\\frac{E(A)}{|A|^3}\\geq \\frac{1}{K}$ then there is a subset $A'\\subset A$ so that $|A'|\\geq \\frac{|A|}{K^{100}}$ and $|A'+A'|\\leq K^{100}|A'|$.\nOne of the drawbacks of BSG theorem is that we lose control on $E(A')$. Can we have a stronger BSG theorem as follows?\n\nLet $A\\subset G$, and suppose $E(A)\\geq \\frac{|A|^3}{K}$. Then there is a set $A'\\subset A$ so that $$|A'|\\geq \\frac{|A|}{K^{100}},$$\n$$|A'+A'|\\leq K^{100}|A'|,$$\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3}$$\nCan we moreover obtain the following?\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3} \\left(\\frac{|A|}{|A'|}\\right)^{\\frac{1}{100}}$$", "clean_statement": "A stronger BSG theorem\n\nThe standard BSG theorem is as follows: let $E(A) = |\\{(a,b,c,d)\\in A^4: a+b=c+d\\}|$. If $\\frac{E(A)}{|A|^3}\\geq \\frac{1}{K}$ then there is a subset $A'\\subset A$ so that $|A'|\\geq \\frac{|A|}{K^{100}}$ and $|A'+A'|\\leq K^{100}|A'|$.\nOne of the drawbacks of BSG theorem is that we lose control on $E(A')$. Can we have a stronger BSG theorem as follows?\n\nLet $A\\subset G$, and suppose $E(A)\\geq \\frac{|A|^3}{K}$. Then there is a set $A'\\subset A$ so that $$|A'|\\geq \\frac{|A|}{K^{100}},$$\n$$|A'+A'|\\leq K^{100}|A'|,$$\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3}$$\nCan we moreover obtain the following?\n$$\\frac{E(A')}{|A'|^3}\\geq \\frac{E(A)}{|A|^3} \\left(\\frac{|A|}{|A'|}\\right)^{\\frac{1}{100}}$$", "statement_status": "exact", "statement_verification": "The canonical record is Problem 2.15, “A stronger BSG theorem,” from the AIM list *Additive combinatorics and its applications*, section “Additive combinatorics.” The stored source record is uncorrupted and asks the following.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Additive combinatorics\nSource item: 2.15\nSource URL: http://aimpl.org/addcombapp/2/\nCanonical location: aim-combinatorics-notes.json notes[23]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A stronger BSG theorem\\n\\nThe standard BSG theorem is as follows: let $E(A) = |\\\\{(a,b,c,d)\\\\in A^4: a+b=c+d\\\\}|$. If $\\\\frac{E(A)}{|A|^3}\\\\geq \\\\frac{1}{K}$ then there is a subset $A'\\\\subset A$ so that $|A'|\\\\geq \\\\frac{|A|}{K^{100}}$ and $|A'+A'|\\\\leq K^{100}|A'|$.\\nOne of the drawbacks of BSG theorem is that we lose control on $E(A')$. Can we have a stronger BSG theorem as follows?\\n\\nLet $A\\\\subset G$, and suppose $E(A)\\\\geq \\\\frac{|A|^3}{K}$. Then there is a set $A'\\\\subset A$ so that $$|A'|\\\\geq \\\\frac{|A|}{K^{100}},$$\\n$$|A'+A'|\\\\leq K^{100}|A'|,$$\\n$$\\\\frac{E(A')}{|A'|^3}\\\\geq \\\\frac{E(A)}{|A|^3}$$\\nCan we moreover obtain the following?\\n$$\\\\frac{E(A')}{|A'|^3}\\\\geq \\\\frac{E(A)}{|A|^3} \\\\left(\\\\frac{|A|}{|A'|}\\\\right)^{\\\\frac{1}{100}}$$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/2/", "tags": [ "aim", "AIM-COMBINATORICS-0024", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every finite nonempty A and every 0 < alpha <= 2, a subset B maximizing E(B)/|B|^(3-alpha) satisfies |B| >= e(A)^(1/alpha)|A| and e(B) >= e(A)(|A|/|B|)^alpha. At alpha = 1/100 and e(A) >= 1/K, this proves simultaneously the requested |B| >= |A|/K^100 and the stronger normalized-energy inequality. Maximality also gives an exact all-subset energy-removal inequality and a minimum-degree popular-sum graph on all of B, reducing the unresolved problem to proving ordinary K^100-small doubling for such an energy core.\n\nCandidate contribution (reduction; novelty confidence low): The candidate contribution is the exact variational-core package: maximizing E(X)/|X|^(3-alpha) yields the exponent-matched size and energy-amplification bounds, the removal inequality E(B)-E(B\\S) >= E(B)[1-(1-|S|/|B|)^(3-alpha)], and a whole-core popular-sum graph with minimum degree at least e(B)|B|/8 and restricted sumset size at most 8|B|/e(B).", "research_difficulty_suggested": "L4" }, { "id": 20000900, "problem_number": "AIM-COMBINATORICS-0025", "title": "Extremal sumset examples: Hamming balls and sharp no-carry digit cubes", "statement": "Finding examples.\n\nFind counter examples to the polynomial Bogolyubov Ruzsa theorem.\n\nFind interesting examples of subset $A\\subset \\mathbb{F}_2^n$ so that $A+A$ is far from all of $\\mathbb{F}_2^n$.\n\nFind interesting examples of $A\\subset [N]$ of density $\\alpha$ so that $4A$ does not contain long APs.", "original_statement": "Finding examples.\n\nFind counter examples to the polynomial Bogolyubov Ruzsa theorem.\n\nFind interesting examples of subset $A\\subset \\mathbb{F}_2^n$ so that $A+A$ is far from all of $\\mathbb{F}_2^n$.\n\nFind interesting examples of $A\\subset [N]$ of density $\\alpha$ so that $4A$ does not contain long APs.", "clean_statement": "Finding examples.\n\nFind counter examples to the polynomial Bogolyubov Ruzsa theorem.\n\nFind interesting examples of subset $A\\subset \\mathbb{F}_2^n$ so that $A+A$ is far from all of $\\mathbb{F}_2^n$.\n\nFind interesting examples of $A\\subset [N]$ of density $\\alpha$ so that $4A$ does not contain long APs.", "statement_status": "exact", "statement_verification": "The AIM record, under the heading “Finding examples,” contains three requests:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Additive combinatorics\nSource item: 2.2\nSource URL: http://aimpl.org/addcombapp/2/\nCanonical location: aim-combinatorics-notes.json notes[24]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Finding examples.\\n\\nFind counter examples to the polynomial Bogolyubov Ruzsa theorem.\\n\\nFind interesting examples of subset $A\\\\subset \\\\mathbb{F}_2^n$ so that $A+A$ is far from all of $\\\\mathbb{F}_2^n$.\\n\\nFind interesting examples of $A\\\\subset [N]$ of density $\\\\alpha$ so that $4A$ does not contain long APs.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/2/", "tags": [ "aim", "AIM-COMBINATORICS-0025", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source-era counterexample program must now be separated from the proved characteristic-2 Polynomial Freiman-Ruzsa theorem and the still-stronger fixed-summand Polynomial Bogolyubov conjecture. This attempt gives two rigorous extremal families. For a Hamming ball A=B_n(r) in F_2^n, every iterated sumset is exactly jA=B_n(min{jr,n}); A+A can miss 1-o(1) of the cube, A is exponentially far in symmetric difference from every affine subspace of comparable size, and the largest subspace in jA has exactly dimension jr before saturation. For integers, a base-B binary digit cube with B>=2s+1 has longest nonconstant arithmetic progression in sA exactly s+1. In particular, a density N^{-(1-log_9 2)} subset of [N] along N=9^n has a fourfold sumset containing a 5-AP but no 6-AP.\n\nCandidate contribution (lemma; novelty confidence low): For all integers s,n>=1 and B>=2s+1, the set D={1+sum_{i=0}^{n-1} epsilon_i B^i: epsilon_i in {0,1}} has longest nonconstant arithmetic progression in sD exactly s+1." }, { "id": 20000901, "problem_number": "AIM-COMBINATORICS-0026", "title": "Local Hilbertian variance and obstructions to p-tightness", "statement": "Approximate Caratheodory theorem\n\nThere is the following important theorem:\nLet $A\\subset \\mathbb{R}^d$ be a finite set of points. Also assume that $A$ is contained in the $L^p$ unit Hamming ball for $p\\geq 2$.\nThen any arbitrary point $a$ in the convex hull of $A$ can be approximated by a convex combination of only $O(p\\varepsilon^{-2})$ many points of $A$.\n\nQ. First, assuming some assumptions on the point set $A$, find a better dependence (sub-linear) on $p$.\n\nQ. Characterize configurations for which the bound $O(p\\varepsilon^{-2})$ is tight.\nstatus", "original_statement": "Approximate Caratheodory theorem\n\nThere is the following important theorem:\nLet $A\\subset \\mathbb{R}^d$ be a finite set of points. Also assume that $A$ is contained in the $L^p$ unit Hamming ball for $p\\geq 2$.\nThen any arbitrary point $a$ in the convex hull of $A$ can be approximated by a convex combination of only $O(p\\varepsilon^{-2})$ many points of $A$.\n\nQ. First, assuming some assumptions on the point set $A$, find a better dependence (sub-linear) on $p$.\n\nQ. Characterize configurations for which the bound $O(p\\varepsilon^{-2})$ is tight.\nstatus", "clean_statement": "Approximate Caratheodory theorem\n\nThere is the following important theorem:\nLet $A\\subset \\mathbb{R}^d$ be a finite set of points. Also assume that $A$ is contained in the $L^p$ unit Hamming ball for $p\\geq 2$.\nThen any arbitrary point $a$ in the convex hull of $A$ can be approximated by a convex combination of only $O(p\\varepsilon^{-2})$ many points of $A$.\n\nQ. First, assuming some assumptions on the point set $A$, find a better dependence (sub-linear) on $p$.\n\nQ. Characterize configurations for which the bound $O(p\\varepsilon^{-2})$ is tight.\nstatus", "statement_status": "exact", "statement_verification": "The canonical `problem` field is reproduced verbatim below. In particular, this preserves the phrases “unit Hamming ball,” the omitted approximation norm, and the trailing word `status`.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Additive combinatorics\nSource item: 2.25\nSource URL: http://aimpl.org/addcombapp/2/\nCanonical location: aim-combinatorics-notes.json notes[25]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Approximate Caratheodory theorem\\n\\nThere is the following important theorem:\\nLet $A\\\\subset \\\\mathbb{R}^d$ be a finite set of points. Also assume that $A$ is contained in the $L^p$ unit Hamming ball for $p\\\\geq 2$.\\nThen any arbitrary point $a$ in the convex hull of $A$ can be approximated by a convex combination of only $O(p\\\\varepsilon^{-2})$ many points of $A$.\\n\\nQ. First, assuming some assumptions on the point set $A$, find a better dependence (sub-linear) on $p$.\\n\\nQ. Characterize configurations for which the bound $O(p\\\\varepsilon^{-2})$ is tight.\\nstatus\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/2/", "tags": [ "aim", "AIM-COMBINATORICS-0026", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The corrupted source is strongly reconstructed as the standard unit-l_p-ball theorem with approximation measured in l_p; the general O(p/epsilon^2) worst-case bound is known to be tight. For an individual target a, this attempt proves the sharper certificate e_{p,m}(A,a)<=sqrt(V_T(lambda)/m) for every barycentric law lambda and every Hilbert majorant T of l_p on span(A-a). In particular e_{p,m}<=sqrt(V_2(A,a)/m), giving p-independent sparsification when the minimum Euclidean barycentric variance is small. A constant-factor sqrt(p/m) lower curve therefore forces V_2(A,a)=Omega(p), and the Reis-Rothvoss theorem simultaneously forces ambient dimension exponential in p. The coordinate simplex is solved exactly and shown not to be p-tight.\n\nCandidate contribution (obstruction; novelty confidence low): If a unit-l_p configuration has e_{p,m}(A,a)>=c sqrt(p/m), then every Hilbert-majorant barycentric representation has energy at least c^2 p; in particular V_2(A,a)>=c^2 p, while the Reis-Rothvoss discrepancy bound forces d>=(m/2) exp((c/C_RR)^2 p). Thus large target-local Hilbertian energy and exponential dimension are simultaneous necessary conditions for p-tightness." }, { "id": 20000902, "problem_number": "AIM-COMBINATORICS-0027", "title": "Good Freiman models from polynomial coset covers", "statement": "Finding good modeling\n\nLet $A,B$ be subset of abelian groups $G, H$ respectively. A map $\\phi: A\\rightarrow B$ is called a Freiman isomorphism if\n$$a+b=c+d \\iff \\phi(a)+\\phi(b)=\\phi(c)+\\phi(d).$$\nWe have the following modeling lemmas.\n\n1. Let $A\\subset \\mathbb{F}_2^n$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of $\\mathbb{F}_2^m$ where $|\\mathbb{F}_2^m|\\leq K^{O(1)}|A|$.\n\n2. Let $A\\subset \\mathbb{Z}^d$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of a group $H$ where $|H|\\leq K^{O(1)}|A|$.\n\nMoreover, this is false for arbitrary groups instead of $\\mathbb{Z}^d$ or $\\mathbb{F}_2^n$.\n\nSuppose $A\\subset G$, and $|A+A|\\leq K |A|$. Can we find a subset $A'\\subset A$ with $|A'|\\geq K^{-O(1)}|A|$ and a group $H$ so that $A'$ is Freiman isomorphic to a subset of $H$ and $|H|\\leq K^{O(1)}|A|$?", "original_statement": "Finding good modeling\n\nLet $A,B$ be subset of abelian groups $G, H$ respectively. A map $\\phi: A\\rightarrow B$ is called a Freiman isomorphism if\n$$a+b=c+d \\iff \\phi(a)+\\phi(b)=\\phi(c)+\\phi(d).$$\nWe have the following modeling lemmas.\n\n1. Let $A\\subset \\mathbb{F}_2^n$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of $\\mathbb{F}_2^m$ where $|\\mathbb{F}_2^m|\\leq K^{O(1)}|A|$.\n\n2. Let $A\\subset \\mathbb{Z}^d$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of a group $H$ where $|H|\\leq K^{O(1)}|A|$.\n\nMoreover, this is false for arbitrary groups instead of $\\mathbb{Z}^d$ or $\\mathbb{F}_2^n$.\n\nSuppose $A\\subset G$, and $|A+A|\\leq K |A|$. Can we find a subset $A'\\subset A$ with $|A'|\\geq K^{-O(1)}|A|$ and a group $H$ so that $A'$ is Freiman isomorphic to a subset of $H$ and $|H|\\leq K^{O(1)}|A|$?", "clean_statement": "Finding good modeling\n\nLet $A,B$ be subset of abelian groups $G, H$ respectively. A map $\\phi: A\\rightarrow B$ is called a Freiman isomorphism if\n$$a+b=c+d \\iff \\phi(a)+\\phi(b)=\\phi(c)+\\phi(d).$$\nWe have the following modeling lemmas.\n\n1. Let $A\\subset \\mathbb{F}_2^n$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of $\\mathbb{F}_2^m$ where $|\\mathbb{F}_2^m|\\leq K^{O(1)}|A|$.\n\n2. Let $A\\subset \\mathbb{Z}^d$ and $|A+A|\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of a group $H$ where $|H|\\leq K^{O(1)}|A|$.\n\nMoreover, this is false for arbitrary groups instead of $\\mathbb{Z}^d$ or $\\mathbb{F}_2^n$.\n\nSuppose $A\\subset G$, and $|A+A|\\leq K |A|$. Can we find a subset $A'\\subset A$ with $|A'|\\geq K^{-O(1)}|A|$ and a group $H$ so that $A'$ is Freiman isomorphic to a subset of $H$ and $|H|\\leq K^{O(1)}|A|$?", "statement_status": "exact", "statement_verification": "The canonical record (AIM workshop *Additive combinatorics and its applications*, problem 2.3) asks the following. If \\(A\\) is a finite subset of an abelian group \\(G\\) and", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Additive combinatorics\nSource item: 2.3\nSource URL: http://aimpl.org/addcombapp/2/\nCanonical location: aim-combinatorics-notes.json notes[26]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Finding good modeling\\n\\nLet $A,B$ be subset of abelian groups $G, H$ respectively. A map $\\\\phi: A\\\\rightarrow B$ is called a Freiman isomorphism if\\n$$a+b=c+d \\\\iff \\\\phi(a)+\\\\phi(b)=\\\\phi(c)+\\\\phi(d).$$\\nWe have the following modeling lemmas.\\n\\n1. Let $A\\\\subset \\\\mathbb{F}_2^n$ and $|A+A|\\\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of $\\\\mathbb{F}_2^m$ where $|\\\\mathbb{F}_2^m|\\\\leq K^{O(1)}|A|$.\\n\\n2. Let $A\\\\subset \\\\mathbb{Z}^d$ and $|A+A|\\\\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of a group $H$ where $|H|\\\\leq K^{O(1)}|A|$.\\n\\nMoreover, this is false for arbitrary groups instead of $\\\\mathbb{Z}^d$ or $\\\\mathbb{F}_2^n$.\\n\\nSuppose $A\\\\subset G$, and $|A+A|\\\\leq K |A|$. Can we find a subset $A'\\\\subset A$ with $|A'|\\\\geq K^{-O(1)}|A|$ and a group $H$ so that $A'$ is Freiman isomorphic to a subset of $H$ and $|H|\\\\leq K^{O(1)}|A|$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/2/", "tags": [ "aim", "AIM-COMBINATORICS-0027", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If A is covered by M cosets of a finite subgroup V with |V| at most L|A|, then a densest coset slice A' has size at least |A|/M and translation into V is an injective Freiman 2-isomorphism, with |V| at most L|A| and at most LM|A'|. Consequently the 2025 characteristic-two PFR theorem and its bounded-exponent extension affirm the modeling question for fixed exponent m, with density (2K)^(-O(m^3)) and |V| at most |A|; the dependence on m prevents a uniform arbitrary-abelian-group solution. A quotient q:G to G/L is a Freiman 2-isomorphism on A exactly when L intersects 2A-2A only at zero, so quotienting by the PFR covering subgroup fails on every non-singleton slice contained in one of its cosets.\n\nCandidate contribution (lemma; novelty confidence low): Candidate cover-to-model-and-kernel principle: an M-coset cover by V yields, by translation of a densest slice, a Freiman 2-model A' in V with |A'| at least |A|/M and |V| at most LM|A'|, while quotienting by V is valid exactly under the incompatible condition V intersect (2A'-2A') equals {0}." }, { "id": 20000903, "problem_number": "AIM-COMBINATORICS-0028", "title": "A sparse interval obstruction and corrected product-factor tradeoff", "statement": "Product sets in iterated sumsets\n\nObserve that the interval $[N]$ contains $[N^c][N^c]$ for $c=0.5$.\n\nThere exists $n,\\varepsilon,\\delta$ for which the following holds.\nLet $A\\subset \\mathbb{F}_p$ and suppose $|A+A|\\leq |A|^{1+\\varepsilon}$. Then $nA-nA$ contains a product set $XY$ where $|X|,|Y|\\geq |A|^{1+\\varepsilon}$.\n\nIf $X$ is polynomially large, how big can you make $Y$?", "original_statement": "Product sets in iterated sumsets\n\nObserve that the interval $[N]$ contains $[N^c][N^c]$ for $c=0.5$.\n\nThere exists $n,\\varepsilon,\\delta$ for which the following holds.\nLet $A\\subset \\mathbb{F}_p$ and suppose $|A+A|\\leq |A|^{1+\\varepsilon}$. Then $nA-nA$ contains a product set $XY$ where $|X|,|Y|\\geq |A|^{1+\\varepsilon}$.\n\nIf $X$ is polynomially large, how big can you make $Y$?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The most plausible repair is to replace both occurrences of \\(|A|^{1+\\varepsilon}\\) in the conclusion by \\(|A|^\\delta\\), or more generally to ask for \\[ |X|\\ge |A|^\\alpha,\\qquad |Y|\\ge |A|^\\beta, \\tag{1} \\] under an explicit density window such as \\(|A|\\le p^{1-\\eta}\\), with \\(n,\\varepsilon,\\alpha,\\beta,\\eta\\) absolute. This is a reconstruction, not verified source text.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Additive combinatorics\nSource item: 2.35\nSource URL: http://aimpl.org/addcombapp/2/\nCanonical location: aim-combinatorics-notes.json notes[27]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Product sets in iterated sumsets\\n\\nObserve that the interval $[N]$ contains $[N^c][N^c]$ for $c=0.5$.\\n\\nThere exists $n,\\\\varepsilon,\\\\delta$ for which the following holds.\\nLet $A\\\\subset \\\\mathbb{F}_p$ and suppose $|A+A|\\\\leq |A|^{1+\\\\varepsilon}$. Then $nA-nA$ contains a product set $XY$ where $|X|,|Y|\\\\geq |A|^{1+\\\\varepsilon}$.\\n\\nIf $X$ is polynomially large, how big can you make $Y$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/2/", "tags": [ "aim", "AIM-COMBINATORICS-0028", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal AIM assertion is false for every fixed n >= 1 and epsilon > 0: for arbitrarily large M and a prime p larger than both 2nM+1 and M^(1+epsilon), the interval A={1,...,M} satisfies |A+A| <= |A|^(1+epsilon), while |nA-nA|=2n(M-1)+1 < M^(1+epsilon); since |XY| >= |X| whenever |Y|>1, nA-nA cannot contain factors X,Y of the requested sizes. For the coherent corrected interval problem, an arithmetic progression A of length M has X=d{1,...,L}, Y={1,...,R} with XY contained in nA-nA whenever LR <= n(M-1), giving the exponent tradeoff |X| about M^alpha and |Y| about M^(1-alpha).\n\nCandidate contribution (counterexample; novelty confidence low): For every fixed integer n >= 1 and real epsilon > 0, sparse no-wraparound intervals in sufficiently large prime fields refute the literal factor bounds |X|,|Y| >= |A|^(1+epsilon), even when the ambient field is large enough to hold such factors; the same family supplies a testable corrected alpha versus 1-alpha factor-size tradeoff and separates from the Cauchy-Davenport saturation regime." }, { "id": 20000904, "problem_number": "AIM-COMBINATORICS-0029", "title": "Riemann-profile transference for discrete entropy monotonicity", "statement": "Entropy of sum of random variables.\n\nLet $G$ be a torsion free abelian group. Suppose $X$ is a random variable on $G$ and $S_n$ is sum of $n$ many $i.i.d$ copies of $X$. Let $H(\\cdot)$ be the discrete entropy function.\n\nWe have the following theorem by T. Tao:\n$$H(S_2)\\geq H(S_1)+\\log_2 \\sqrt[]{2}-o(1) $$ where the $o(1)$ term goes to zero as $H(X)$ grows to infinity.\n\nShow $$H(S_n)\\geq H(S_{n-1})+\\log_2 \\sqrt[]{\\frac{n}{n-1}} - o(1).$$", "original_statement": "Entropy of sum of random variables.\n\nLet $G$ be a torsion free abelian group. Suppose $X$ is a random variable on $G$ and $S_n$ is sum of $n$ many $i.i.d$ copies of $X$. Let $H(\\cdot)$ be the discrete entropy function.\n\nWe have the following theorem by T. Tao:\n$$H(S_2)\\geq H(S_1)+\\log_2 \\sqrt[]{2}-o(1) $$ where the $o(1)$ term goes to zero as $H(X)$ grows to infinity.\n\nShow $$H(S_n)\\geq H(S_{n-1})+\\log_2 \\sqrt[]{\\frac{n}{n-1}} - o(1).$$", "clean_statement": "Entropy of sum of random variables.\n\nLet $G$ be a torsion free abelian group. Suppose $X$ is a random variable on $G$ and $S_n$ is sum of $n$ many $i.i.d$ copies of $X$. Let $H(\\cdot)$ be the discrete entropy function.\n\nWe have the following theorem by T. Tao:\n$$H(S_2)\\geq H(S_1)+\\log_2 \\sqrt[]{2}-o(1) $$ where the $o(1)$ term goes to zero as $H(X)$ grows to infinity.\n\nShow $$H(S_n)\\geq H(S_{n-1})+\\log_2 \\sqrt[]{\\frac{n}{n-1}} - o(1).$$", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Additive combinatorics and its applications*, section 2.4) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Additive combinatorics\nSource item: 2.4\nSource URL: http://aimpl.org/addcombapp/2/\nCanonical location: aim-combinatorics-notes.json notes[28]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Entropy of sum of random variables.\\n\\nLet $G$ be a torsion free abelian group. Suppose $X$ is a random variable on $G$ and $S_n$ is sum of $n$ many $i.i.d$ copies of $X$. Let $H(\\\\cdot)$ be the discrete entropy function.\\n\\nWe have the following theorem by T. Tao:\\n$$H(S_2)\\\\geq H(S_1)+\\\\log_2 \\\\sqrt[]{2}-o(1) $$ where the $o(1)$ term goes to zero as $H(X)$ grows to infinity.\\n\\nShow $$H(S_n)\\\\geq H(S_{n-1})+\\\\log_2 \\\\sqrt[]{\\\\frac{n}{n-1}} - o(1).$$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is not even known for $n=3$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/2/", "tags": [ "aim", "AIM-COMBINATORICS-0029", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general fixed-n conjecture remains open, including arbitrary n=3, although it is known for log-concave and other regular classes. This attempt proves that for every continuous compactly supported probability density f, the integer law p_M(j) proportional to f(j/M) satisfies H_2(S_{k,M})=log_2 M+h_2(f^{*k})+o(1) for every fixed k. Continuous entropy monotonicity then gives Tao's desired increment for every fixed n along the whole high-resolution family. An explicit bimodal polynomial profile yields a non-log-concave worked family. The attempt also proves exact mutual-information and finite-support compression-to-Z reductions.\n\nCandidate contribution (special_case; novelty confidence low): For p_M(j)=f(j/M)/(M a_M), where f is any continuous compactly supported probability density and a_M=M^{-1}sum_j f(j/M), every fixed convolution satisfies H_2(p_M^{*k})=log_2 M+h_2(f^{*k})+o(1); hence H_2(p_M^{*n})-H_2(p_M^{*(n-1)}) is at least (1/2)log_2(n/(n-1))-o(1)." }, { "id": 20000905, "problem_number": "AIM-COMBINATORICS-0030", "title": "Quantitative transfer from PFR covers to dense Bogolyubov containment", "statement": "Polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.\n\nProve that polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.", "original_statement": "Polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.\n\nProve that polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.", "clean_statement": "Polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.\n\nProve that polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.", "statement_status": "exact", "statement_verification": "The canonical record (AIM problem 2.45, workshop *Additive combinatorics and its applications*) says exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Additive combinatorics\nSource item: 2.45\nSource URL: http://aimpl.org/addcombapp/2/\nCanonical location: aim-combinatorics-notes.json notes[29]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.\\n\\nProve that polynomial Freiman Ruzsa conjecture implies Polynomial Bogolyubov Ruzsa conjecture.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/2/", "tags": [ "aim", "AIM-COMBINATORICS-0030", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The requested implication from polynomial Freiman-Ruzsa to fixed-fourfold polynomial Bogolyubov-Ruzsa remains unresolved even though PFR is now a theorem. This attempt proves that a cover by M cosets of a subgroup H with |H| <= L|A| transfers any dense t-fold containment modulus D_t to an affine subspace in tA of size at least M^{-1} 2^{-D_t((LM)^{-1})}|A|, making explicit that logarithmic dense codimension is exactly what yields a polynomial conclusion. It also proves an elementary codimension-2 delta^{-2} Fourier bound and a linearly independent product-padding reduction whose evident PFR-form cover has q <= 4K_A cosets but whose fourfold containment recovers an arbitrary dense Bogolyubov instance with only polynomial loss.\n\nCandidate contribution (reduction; novelty confidence low): For every density-delta set B in an F_2-vector space, the paired constructions give (i) the exact cover-to-dense cardinality loss M^{-1} 2^{-D_t((LM)^{-1})}, and (ii) a padded set A=B x S with q asymptotic to delta^{-1}, actual doubling q/4 <= K_A <= q^2, a cover by q <= 4K_A cosets of a subgroup no larger than A, and the property that any subspace W contained in 4A projects to a subspace in 4B of size at least |W|/q^4." }, { "id": 20000906, "problem_number": "AIM-COMBINATORICS-0031", "title": "Centered affine-fibre density increments for almost containment in A+A", "statement": "Finding almost subspace inside A+A\n\nLet $A\\subset \\mathbb{F}_2^n$ be a set of density $\\alpha$. Then there is a subspace $V$ of codimension $\\log^{O(1)}(\\varepsilon^{-1}\\alpha^{-1})$ so that $$|V\\cap (A+A)|\\geq (1-\\varepsilon)|V|$$", "original_statement": "Finding almost subspace inside A+A\n\nLet $A\\subset \\mathbb{F}_2^n$ be a set of density $\\alpha$. Then there is a subspace $V$ of codimension $\\log^{O(1)}(\\varepsilon^{-1}\\alpha^{-1})$ so that $$|V\\cap (A+A)|\\geq (1-\\varepsilon)|V|$$", "clean_statement": "Finding almost subspace inside A+A\n\nLet $A\\subset \\mathbb{F}_2^n$ be a set of density $\\alpha$. Then there is a subspace $V$ of codimension $\\log^{O(1)}(\\varepsilon^{-1}\\alpha^{-1})$ so that $$|V\\cap (A+A)|\\geq (1-\\varepsilon)|V|$$", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, workshop *Additive combinatorics and its applications*, section *Additive combinatorics*, Problem 2.5, “Finding almost subspace inside \\(A+A\\).” Its text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Additive combinatorics\nSource item: 2.5\nSource URL: http://aimpl.org/addcombapp/2/\nCanonical location: aim-combinatorics-notes.json notes[30]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Finding almost subspace inside A+A\\n\\nLet $A\\\\subset \\\\mathbb{F}_2^n$ be a set of density $\\\\alpha$. Then there is a subspace $V$ of codimension $\\\\log^{O(1)}(\\\\varepsilon^{-1}\\\\alpha^{-1})$ so that $$|V\\\\cap (A+A)|\\\\geq (1-\\\\varepsilon)|V|$$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/2/", "tags": [ "aim", "AIM-COMBINATORICS-0031", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The conjectured polylogarithmic codimension bound remains open. This attempt proves that every nonempty density-alpha set A in F_2^n and every 00\\) and \\(c'=c'(n,\\varepsilon)>0\\) such that the following holds uniformly for every finite abelian group \\(G\\), every nonempty \\(A\\subseteq G\\), and every \\(K>1\\). If\n\\[\n\\rho_{2n+2}(A)^{(n-1)/n}\\le K\\rho_{2n}(A),\n\\]\nthen there is a subset \\(H\\subseteq(n-1)A\\) such that\n\\[\n|H|\\ge \\rho_{2n}(A)^{1/(n-1)}K^{-c},\n\\qquad\nE_4(H)\\ge\n\\frac{|H|^3}{|A|^\\varepsilon K^{c'}}.\n\\]", "statement_status": "reconstructed_unverified", "statement_verification": "The higher-energy conjecture must quantify \\(n\\). A conservative reconstruction is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Additive combinatorics and its applications\nSection: Additive combinatorics\nSource item: 2.6\nSource URL: http://aimpl.org/addcombapp/2/\nCanonical location: aim-combinatorics-notes.json notes[32]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A conjecture about generalized additive energy\\n\\nLet $A\\\\subset G$ and define $$E_{2n}(A) = |\\\\{(a_1,\\\\cdots,a_n)\\\\in A^{2n}: a_1+\\\\cdots+a_n = a_{n+1}+\\\\cdots+a_{2n}\\\\}|.$$\\nTrivial bounds on additive energy are $$|A|^n\\\\leq E_{2n}(A)\\\\leq |A|^{2n-1}.$$\\nDefine $$\\\\rho_{2n}(A)= \\\\frac{E_{2n}(A)}{|A|^n}.$$\\n\\nGiven this, one can compare various $\\\\rho_k$'s. For example by an applications of Holder's inequality one can get\\n$$\\\\rho_8(A) \\\\geq \\\\rho_4^3(A).$$\\nMoreover, there is the following theorem.\\n\\nTheorem. Fix some $0<\\\\varepsilon<1$. Suppose $\\\\rho_8(A)\\\\leq K\\\\rho_4^3(A)$. then there is a subset $H\\\\subset A$, so that\\n$$|H|\\\\geq \\\\rho_4(A)/K^c$$ and\\n$$E_4(H)\\\\geq \\\\frac{|H|^3}{|A|^{\\\\varepsilon}K^{c'}}.$$\\n\\nNow, using Holder's inequality one can show\\n$$\\\\rho_{2n}\\\\leq \\\\rho_{2n+2}^{\\\\frac{n-1}{n}}.$$\\n\\nSuppose $$\\\\rho_{2n+2}^{\\\\frac{n-1}{n}}(A)< K\\\\rho_{2n}(A)$$. Then there is $H\\\\subset (n-1)A, |H|\\\\geq \\\\frac{\\\\rho_{2n}^{\\\\frac{1}{n-1}}}{K^c}$ so that $$E_4(H)\\\\geq \\\\frac{|H|^3}{|A|^{\\\\varepsilon}K^{c'}}.$$\"\nOriginal remarks: [\"The problem is currently misstated because of an exaggeration of independence of constants. So for instance in the statement of the theorem, the inequality should have an extra factor of $|A|^{\\\\epsilon}$ in the denominator\\n$E_4(H) \\\\geq {|H|^3 \\\\over |A|^{\\\\epsilon} K^{c'}}$\\nSimilarly in the statement of the problem.\\n\\nIt is in fact an open problem whether those factors are really needed. But that isn't the most important open\\nproblem. Even with the extra factors, the connection to Polynomial Freiman Ruzsa is pretty strong.\"]\nOriginal literature field (JSON string): \"This conjecture has applications to PFR. In fact, given a set $A$ , we can list all the energies for various $n$, and separate to two cases.\\n\\n1. Energy always increases (Smoothing.)\\n\\n2. There is one point that the assumption of the conjecture is satisfied.(Non-smoothing) . The conjecture will give us some structure.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/addcombapp/2/", "tags": [ "aim", "AIM-COMBINATORICS-0033", "aim-domain:combinatorics", "aim-workshop:addcombapp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After correcting the definition to use 2n variables and making the dependence on n and epsilon explicit, the consecutive higher-energy conjecture remains open. Writing X(chi)=|hat(1_A)(chi)|^2/|A| under uniform measure on the dual gives rho_{2j}=E[X^j], which verifies every displayed Holder exponent. This attempt proves a quantitative spectral stability theorem for the defect Q_n=rho_{2n+2}^{(n-1)/n}/rho_{2n}, classifies Q_n=1 exactly as cosets of subgroups, and proves the conjectured physical-space conclusion with H=(n-1)A whenever its further growth is at most |A|^epsilon times a polynomial in K; in particular this holds, without the |A|^epsilon loss, when the separate doubling constant D(A) is polynomial in K.\n\nCandidate contribution (lemma; novelty confidence low): Let s=rho_{2n}(A)^{1/(n-1)} and bias the dual group by lambda_n(chi)=X(chi)^n/(|G| rho_{2n}(A)). If Q_n(A)<=K, then the exact quantitative estimate E_{lambda_n}[(sqrt(X/s)-sqrt(s/X))^2] <= K^{n/(n-1)}-1 holds, yielding an explicit multiplicative-window tail bound; zero defect forces the two-level Fourier law and, after using the trivial coefficient, triangle equality, Parseval, and annihilator cardinality, is equivalent to A being a coset of a subgroup." }, { "id": 20000909, "problem_number": "AIM-COMBINATORICS-0034", "title": "Kernel saturation and rank-sensitive Komlos discrepancy", "statement": "Koml\\'{o}s conjecture\n\nThere is a universal constant $c>0$ s.t. for every $v_1,v_2,...,v_n \\in \\mathbb{B_2}^d$, there exist signs $\\epsilon_1,...,\\epsilon_n\\in \\{-1,1\\}$ s.t. $|| \\sum_{i=1}^n \\epsilon_i v_i||_{\\infty} \\le c$", "original_statement": "Koml\\'{o}s conjecture\n\nThere is a universal constant $c>0$ s.t. for every $v_1,v_2,...,v_n \\in \\mathbb{B_2}^d$, there exist signs $\\epsilon_1,...,\\epsilon_n\\in \\{-1,1\\}$ s.t. $|| \\sum_{i=1}^n \\epsilon_i v_i||_{\\infty} \\le c$", "clean_statement": "Koml\\'{o}s conjecture\n\nThere is a universal constant $c>0$ s.t. for every $v_1,v_2,...,v_n \\in \\mathbb{B_2}^d$, there exist signs $\\epsilon_1,...,\\epsilon_n\\in \\{-1,1\\}$ s.t. $|| \\sum_{i=1}^n \\epsilon_i v_i||_{\\infty} \\le c$", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, workshop *Hereditary discrepancy and factorization norms*, section *Open problems*, Problem 1.05, “Komlós conjecture.” Its exact mathematical content is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.05\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[33]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Koml\\\\'{o}s conjecture\\n\\nThere is a universal constant $c>0$ s.t. for every $v_1,v_2,...,v_n \\\\in \\\\mathbb{B_2}^d$, there exist signs $\\\\epsilon_1,...,\\\\epsilon_n\\\\in \\\\{-1,1\\\\}$ s.t. $|| \\\\sum_{i=1}^n \\\\epsilon_i v_i||_{\\\\infty} \\\\le c$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"$c=O(\\\\sqrt{\\\\log min\\\\{d,n\\\\}})$ was given by [Banaszczyk'98] and [Naor'05]\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0034", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The Komlos conjecture remains open; the current general worst-case bound is the randomized polynomial-time tilde-O(log(n)^(1/4)) theorem of Bansal and Jiang. This attempt proves a deterministic rank-sensitive partial theorem: if A has unit Euclidean norm columns and rank r, a kernel coloring can be saturated to at most r fractional coordinates and rounded to signs epsilon with ||A epsilon||_infinity <= ||A epsilon||_2 <= sqrt(Q_A(x)) <= sqrt(r), where Q_A(x)=sum_{i fractional}(1-x_i^2)||v_i||_2^2. The same bound holds hereditarily for every column submatrix. Identity matrices show the stronger Euclidean estimate is sharp while infinity discrepancy is only 1, isolating the coordinatewise obstruction to upgrading this quadratic route to a dimension-free bound.\n\nCandidate contribution (lemma; novelty confidence low): For every unit-column matrix A and every kernel coloring x saturated to at most rank(A) fractional coordinates, biased rounding can be deterministically derandomized to produce signs epsilon satisfying ||A epsilon||_infinity <= sqrt(sum_{i fractional}(1-x_i^2)||v_i||_2^2) <= sqrt(rank(A)); consequently herdisc(A) <= sqrt(rank(A))." }, { "id": 20000910, "problem_number": "AIM-COMBINATORICS-0035", "title": "Constructive Komlos and deterministic balancing with bounded direction overlap", "statement": "Constructive Koml\\'{o}s\n\nFind a polynomial time algorithm to find the signs above.", "original_statement": "Constructive Koml\\'{o}s\n\nFind a polynomial time algorithm to find the signs above.", "clean_statement": "Constructive Koml\\'{o}s\n\nFind a polynomial time algorithm to find the signs above.", "statement_status": "exact", "statement_verification": "The exact extracted record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.1\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[34]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Constructive Koml\\\\'{o}s\\n\\nFind a polynomial time algorithm to find the signs above.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"$O(\\\\log min\\\\{d,n\\\\})$ [Bansal'10, Lovett-Meka'14, Rothvoss'15]\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0035", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a rational Komlos matrix V, partition its nonzero columns into projective direction classes and define beta(V) as the maximum, over coordinates q, of the sum over classes C of max_{i in C}|v_i(q)|. Exact scalar-greedy cancellation within every class produces signs deterministically in polynomial bit complexity with ||V epsilon||_infinity <= beta(V) <= rho(V), where rho is the maximum number of direction classes meeting one coordinate. A supplied approximate-direction decomposition obeys an analogous bound with an explicit rowwise l1 error term. This gives a constant constructive theorem whenever rho (or beta) is bounded, but does not solve the general constant Komlos conjecture.\n\nCandidate contribution (algorithmic_special_case; novelty confidence low): Candidate contribution: an exact rational-input, bit-polynomial projective-direction algorithm with the computable coordinate-sensitive certificate beta(V), together with a robust supplied-decomposition lemma whose perturbation loss is exactly bounded by the maximum rowwise l1 error." }, { "id": 20000911, "problem_number": "AIM-COMBINATORICS-0036", "title": "Constructive Beck-Fiala status and an exact pseudoforest incidence theorem", "statement": "Beck-Fiala conjecture and making it constructive\n\nGiven a set system $(V,\\mathcal{S})$ with $V=[n]$, $\\mathcal{S}=S_1,...,S_m$. Each $i\\in V$ occurs in at most $t$ sets from $\\mathcal{S}$. Then there is a coloring to the elements of $V$ achieving a discrepancy of $O(\\sqrt{t})$. Koml\\'{o}s conjecture implies this by treating the incidence vector of each $i\\in V$ as a vector and scaling it down by $\\sqrt{t}$.", "original_statement": "Beck-Fiala conjecture and making it constructive\n\nGiven a set system $(V,\\mathcal{S})$ with $V=[n]$, $\\mathcal{S}=S_1,...,S_m$. Each $i\\in V$ occurs in at most $t$ sets from $\\mathcal{S}$. Then there is a coloring to the elements of $V$ achieving a discrepancy of $O(\\sqrt{t})$. Koml\\'{o}s conjecture implies this by treating the incidence vector of each $i\\in V$ as a vector and scaling it down by $\\sqrt{t}$.", "clean_statement": "Beck-Fiala conjecture and making it constructive\n\nGiven a set system $(V,\\mathcal{S})$ with $V=[n]$, $\\mathcal{S}=S_1,...,S_m$. Each $i\\in V$ occurs in at most $t$ sets from $\\mathcal{S}$. Then there is a coloring to the elements of $V$ achieving a discrepancy of $O(\\sqrt{t})$. Koml\\'{o}s conjecture implies this by treating the incidence vector of each $i\\in V$ as a vector and scaling it down by $\\sqrt{t}$.", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.15, “Beck-Fiala conjecture and making it constructive,” from the AIM workshop *Hereditary discrepancy and factorization norms*. Its mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.15\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[35]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Beck-Fiala conjecture and making it constructive\\n\\nGiven a set system $(V,\\\\mathcal{S})$ with $V=[n]$, $\\\\mathcal{S}=S_1,...,S_m$. Each $i\\\\in V$ occurs in at most $t$ sets from $\\\\mathcal{S}$. Then there is a coloring to the elements of $V$ achieving a discrepancy of $O(\\\\sqrt{t})$. Koml\\\\'{o}s conjecture implies this by treating the incidence vector of each $i\\\\in V$ as a vector and scaling it down by $\\\\sqrt{t}$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"A $2t-1$ guarantee which is also algorithmic was given by Beck-Fiala. Recently, [Bukh] gave a guarantee of $2t-\\\\log^*t$. Banaszczyk's result implies a bound of $O(\\\\sqrt{t\\\\log m})$ but is non-constructive. Other algorithmic guarantees are $O(\\\\sqrt{t}\\\\log n)$ [Bansal'10, Lovett-Meka'12, Rothvoss'15] and $O(\\\\sqrt{t\\\\log n\\\\log|S_j|})$ [Bansal-Garg'15].\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0036", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The full all-t Beck-Fiala conjecture remains open, although Bansal-Jiang prove the exact O(sqrt(t)) bound by a randomized polynomial-time algorithm for t=Omega(log^2 n), and a July 2026 unrefereed preprint claims the same bound down to t at least C log n (log log n)^(2+eta). As a proved structural special case, every set system whose bipartite incidence graph is a pseudoforest has a deterministic linear-time coloring of discrepancy at most 2. It has discrepancy at most 1 unless some unicyclic component has an odd number of cycle set-vertices and every such set-vertex has full degree 2; in that exceptional case its discrepancy is exactly 2.\n\nCandidate contribution (theorem; novelty confidence low): For pseudoforest bipartite incidence graphs, full discrepancy is exactly 2 precisely when there is a unicyclic component whose cycle has an odd number of set-vertices all of full degree 2; otherwise full discrepancy is at most 1, and the coloring is constructible in deterministic linear time." }, { "id": 20000912, "problem_number": "AIM-COMBINATORICS-0037", "title": "Random Beck-Fiala systems and the hereditary saturation obstruction", "statement": "Beck-Fiala for random set system\n\nEach element $i\\in [n]$ lies in $t$ randomly selected sets from $\\mathcal{S}$.", "original_statement": "Beck-Fiala for random set system\n\nEach element $i\\in [n]$ lies in $t$ randomly selected sets from $\\mathcal{S}$.", "clean_statement": "Beck-Fiala for random set system\n\nEach element $i\\in [n]$ lies in $t$ randomly selected sets from $\\mathcal{S}$.", "statement_status": "exact", "statement_verification": "The exact corpus record is problem 1.2 from the AIM workshop *Hereditary discrepancy and factorization norms*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.2\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[36]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Beck-Fiala for random set system\\n\\nEach element $i\\\\in [n]$ lies in $t$ randomly selected sets from $\\\\mathcal{S}$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"They show that when $m\\\\ge n$, hereditary discrepancy of the set system is $O(\\\\sqrt{t\\\\log t})$ whp. When $n\\\\ge m^t$, hereditary discrepancy is $O(1)$. An open question is to explore this setting for the remaining range of parameters i.e. when $n^{1/t} < m < n$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0037", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In the verified Ezra-Lovett model, columns are independent uniform t-subsets of [m]. For every repeated-column instance A, hereditary discrepancy equals the maximum ordinary discrepancy over simple subfamilies of its distinct column support; after all binomial(m,t) types are collected, it therefore equals the worst discrepancy of a simple t-uniform system. For projective-plane parameters m=q^2+q+1 and t=q+1, already at n=ceil(2 binomial(m,t) log m), which lies in the AIM intermediate range n^(1/t)= sqrt(t-1) with probability at least 1-1/m at n=ceil(2 binomial(m,t) log m) in the stated intermediate range for every prime-power projective-plane parameter q." }, { "id": 20000913, "problem_number": "AIM-COMBINATORICS-0038", "title": "Random sub-neighborhood discrepancy on the Boolean cube", "statement": "Hypercube set system\n\nGiven a set system where the elements are the vertices of the hypercube $\\{0,1\\}^n$. Corresponding to each vertex $v$, there is a set $S_v$ which is generated by picking a random subset of the neighbours of $v$ in the hypercube. What is the discrepancy of this set system?", "original_statement": "Hypercube set system\n\nGiven a set system where the elements are the vertices of the hypercube $\\{0,1\\}^n$. Corresponding to each vertex $v$, there is a set $S_v$ which is generated by picking a random subset of the neighbours of $v$ in the hypercube. What is the discrepancy of this set system?", "clean_statement": "Hypercube set system\n\nGiven a set system where the elements are the vertices of the hypercube $\\{0,1\\}^n$. Corresponding to each vertex $v$, there is a set $S_v$ which is generated by picking a random subset of the neighbours of $v$ in the hypercube. What is the discrepancy of this set system?", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.25\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[37]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Hypercube set system\\n\\nGiven a set system where the elements are the vertices of the hypercube $\\\\{0,1\\\\}^n$. Corresponding to each vertex $v$, there is a set $S_v$ which is generated by picking a random subset of the neighbours of $v$ in the hypercube. What is the discrepancy of this set system?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"If instead of picking a random subset of the neighbours, $S_v$ was the full set of neighbours of $v$, then there is an easy solution giving discrepancy 1. Color each vertex depending on the parity of its first $n/2$ bits.\"\nResearch attempt: 1; result status: conditional_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0038", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM statement omits the random-subset density and dependence model. Under either natural Bernoulli-half interpretation--independent ordered incidences or a symmetric random cube subgraph--the optimized discrepancy is at least sqrt(n)/64 with high probability; explicitly, for n >= 1024 the failure probabilities are at most (3/16)^(2^n) and (3/8)^(2^(n-1)), respectively. Every deterministic realization S_v subset N(v) has discrepancy at most sqrt(2n log(2e(n^2+1))) by the Lovasz local lemma. The full-neighborhood discrepancy is exactly 0 for even n and 1 for odd n, while the suggested parity coloring has maximum row imbalance between ceil(n/2)-1 and ceil(n/2) with high probability after half-sampling. Exact Bernoulli-p and fixed-size-r one-row laws are also derived.\n\nCandidate contribution (model_specific_lower_bound; novelty confidence low): Candidate contribution: a uniform Littlewood-Offord union-bound proof that both independent directed half-neighborhood sampling and reciprocal undirected half-edge sampling on Q_n have discrepancy Omega(sqrt(n)) with high probability, together with an explicit distribution-free O(sqrt(n log n)) upper bound and a proof that the AIM parity benchmark becomes (1/2-o(1))n after half-sampling.", "research_difficulty_suggested": "L4" }, { "id": 20000914, "problem_number": "AIM-COMBINATORICS-0039", "title": "A high-probability lower bound for random permutation discrepancy", "statement": "Discrepancy of permutations\n\na) discrepancy of k-permutations\n\nFor $k=3$, $[NNN'12]$ showed a lower bound of $\\Omega(\\log n)$ where $n$ is the number of elements in the ground set. For general $k$, we know that the answer lies in $[\\sqrt{k}+\\log n,\\sqrt{k}\\log n]$. An open question is to clsoe this gap.\n\nb) discrepancy of k-random permutations\n\nA lower bound of $\\Omega(\\sqrt{k})$ is known.", "original_statement": "Discrepancy of permutations\n\na) discrepancy of k-permutations\n\nFor $k=3$, $[NNN'12]$ showed a lower bound of $\\Omega(\\log n)$ where $n$ is the number of elements in the ground set. For general $k$, we know that the answer lies in $[\\sqrt{k}+\\log n,\\sqrt{k}\\log n]$. An open question is to clsoe this gap.\n\nb) discrepancy of k-random permutations\n\nA lower bound of $\\Omega(\\sqrt{k})$ is known.", "clean_statement": "Discrepancy of permutations\n\na) discrepancy of k-permutations\n\nFor $k=3$, $[NNN'12]$ showed a lower bound of $\\Omega(\\log n)$ where $n$ is the number of elements in the ground set. For general $k$, we know that the answer lies in $[\\sqrt{k}+\\log n,\\sqrt{k}\\log n]$. An open question is to clsoe this gap.\n\nb) discrepancy of k-random permutations\n\nA lower bound of $\\Omega(\\sqrt{k})$ is known.", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Hereditary discrepancy and factorization norms*, Open Problem 1.3) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.3\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[38]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Discrepancy of permutations\\n\\na) discrepancy of k-permutations\\n\\nFor $k=3$, $[NNN'12]$ showed a lower bound of $\\\\Omega(\\\\log n)$ where $n$ is the number of elements in the ground set. For general $k$, we know that the answer lies in $[\\\\sqrt{k}+\\\\log n,\\\\sqrt{k}\\\\log n]$. An open question is to clsoe this gap.\\n\\nb) discrepancy of k-random permutations\\n\\nA lower bound of $\\\\Omega(\\\\sqrt{k})$ is known.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0039", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For k independent uniform permutations of [n], the probability that optimized prefix discrepancy is below sqrt(n)/8 is at most exp(n log 2-k/100). Thus for k at least 200(log 2)n, prefix and interval discrepancy are at least sqrt(n)/8 with failure probability at most 2^{-n}; combined carefully with the dense Spencer-Srinivasan-Tetali interval upper bound, both discrepancies are Theta(sqrt(n)) throughout every fixed linear window 200(log 2)n <= k <= A n.\n\nCandidate contribution (theorem; novelty confidence low): An explicit tail bound, uniform over the coloring chosen after observing the input, holds for iid uniform permutations: Pr[disc_pref < sqrt(n)/8] <= exp(n log 2-k/100)." }, { "id": 20000915, "problem_number": "AIM-COMBINATORICS-0040", "title": "Matrix Spencer: exact dilation bookkeeping and a sharp Frobenius-energy case", "statement": "Matrix Spencer\n\nGiven matrices $A_1,...,A_n \\in\\mathbb{R}^{n\\times n}$ satisfying $||A_i||_{op} \\le 1$ where $||.||_{op}$ is the operator norm of matrices. Do there exists signs $\\epsilon_1,...,\\epsilon_n\\in\\{-1,1\\}$ s.t. $||\\sum_{i=1}^n\\epsilon_i A_i || =O(\\sqrt{n})$.", "original_statement": "Matrix Spencer\n\nGiven matrices $A_1,...,A_n \\in\\mathbb{R}^{n\\times n}$ satisfying $||A_i||_{op} \\le 1$ where $||.||_{op}$ is the operator norm of matrices. Do there exists signs $\\epsilon_1,...,\\epsilon_n\\in\\{-1,1\\}$ s.t. $||\\sum_{i=1}^n\\epsilon_i A_i || =O(\\sqrt{n})$.", "clean_statement": "Matrix Spencer\n\nGiven matrices $A_1,...,A_n \\in\\mathbb{R}^{n\\times n}$ satisfying $||A_i||_{op} \\le 1$ where $||.||_{op}$ is the operator norm of matrices. Do there exists signs $\\epsilon_1,...,\\epsilon_n\\in\\{-1,1\\}$ s.t. $||\\sum_{i=1}^n\\epsilon_i A_i || =O(\\sqrt{n})$.", "statement_status": "exact", "statement_verification": "Source metadata: `aim-combinatorics-notes.json`, zero-based record index 39, AIM problem ID `AIM-COMBINATORICS-0040`, workshop *Hereditary discrepancy and factorization norms*, source URL `http://aimpl.org/hereddiscrep/1/`.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.35\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[39]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Matrix Spencer\\n\\nGiven matrices $A_1,...,A_n \\\\in\\\\mathbb{R}^{n\\\\times n}$ satisfying $||A_i||_{op} \\\\le 1$ where $||.||_{op}$ is the operator norm of matrices. Do there exists signs $\\\\epsilon_1,...,\\\\epsilon_n\\\\in\\\\{-1,1\\\\}$ s.t. $||\\\\sum_{i=1}^n\\\\epsilon_i A_i || =O(\\\\sqrt{n})$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"We can assume that the matrices are symmetric. Matrix chernoff gives a bound of $O(\\\\sqrt{n\\\\log n})$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0040", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general Matrix Spencer conjecture remains open. This attempt proves that its square symmetric formulation implies the arbitrary real square formulation with only a factor sqrt(2), by dilating n matrices to size 2n and appending n zeros; transfers the Bansal-Jiang-Meka low-Frobenius theorem to arbitrary real matrices under ||A_i||_F^2 <= n/(2 log^3 n); and gives a deterministic greedy signing with operator norm at most (sum_i ||A_i||_F^2)^{1/2}. A normalized symmetric star family has norm exactly sqrt((n-1)/2) for every signing, so the last bound is sharp in order when ||A_i||_F <= 1.\n\nCandidate contribution (lemma; novelty confidence low): Candidate contribution: the exact dilation-and-zero-padding reduction for the equal-count/equal-dimension source formulation, its factor-two Frobenius transfer of Bansal-Jiang-Meka to nonsymmetric inputs, and the greedy total-energy certificate paired with an exact normalized-star lower family." }, { "id": 20000916, "problem_number": "AIM-COMBINATORICS-0041", "title": "The Euclidean Steinitz conjecture and a sharp ell_1 obstruction", "statement": "Steinitz conjecture\n\nGiven vectors $v_1,...,v_n\\in\\mathbb{R}^d$ and a norm $X$ satisfying $\\sum_{i=1}^n v_i=0$ and $||v_i||_X\\le 1$ for all $i$. Define $S_n(X,d)=min_{\\pi\\in S^n}max_{k=1,...,n}|| \\sum_{i=1}^k v_{\\pi(i)}||_X$ where $S^n$ is the set of all permutations of $[n]$. Let $S(X,d)=sup_n S_n(X,d)$. Then, $S(X,d) =O(\\sqrt{d})$.", "original_statement": "Steinitz conjecture\n\nGiven vectors $v_1,...,v_n\\in\\mathbb{R}^d$ and a norm $X$ satisfying $\\sum_{i=1}^n v_i=0$ and $||v_i||_X\\le 1$ for all $i$. Define $S_n(X,d)=min_{\\pi\\in S^n}max_{k=1,...,n}|| \\sum_{i=1}^k v_{\\pi(i)}||_X$ where $S^n$ is the set of all permutations of $[n]$. Let $S(X,d)=sup_n S_n(X,d)$. Then, $S(X,d) =O(\\sqrt{d})$.", "clean_statement": "Steinitz conjecture\n\nGiven vectors $v_1,...,v_n\\in\\mathbb{R}^d$ and a norm $X$ satisfying $\\sum_{i=1}^n v_i=0$ and $||v_i||_X\\le 1$ for all $i$. Define $S_n(X,d)=min_{\\pi\\in S^n}max_{k=1,...,n}|| \\sum_{i=1}^k v_{\\pi(i)}||_X$ where $S^n$ is the set of all permutations of $[n]$. Let $S(X,d)=sup_n S_n(X,d)$. Then, $S(X,d) =O(\\sqrt{d})$.", "statement_status": "exact", "statement_verification": "The AIM record is Problem 1.4, “Steinitz conjecture,” from the workshop list *Hereditary discrepancy and factorization norms*. Its displayed text says that, for zero-sum vectors \\(v_1,\\ldots,v_n\\in\\mathbb R^d\\) of norm at most one in a norm \\(X\\), \\[ \\min_{\\pi}\\max_{1\\le k\\le n} \\left\\|\\sum_{i=1}^k v_{\\pi(i)}\\right\\|_X =O(\\sqrt d), \\] after taking a supremum over \\(n\\). As written, this is not a consistent definition or conjecture:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.4\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[40]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Steinitz conjecture\\n\\nGiven vectors $v_1,...,v_n\\\\in\\\\mathbb{R}^d$ and a norm $X$ satisfying $\\\\sum_{i=1}^n v_i=0$ and $||v_i||_X\\\\le 1$ for all $i$. Define $S_n(X,d)=min_{\\\\pi\\\\in S^n}max_{k=1,...,n}|| \\\\sum_{i=1}^k v_{\\\\pi(i)}||_X$ where $S^n$ is the set of all permutations of $[n]$. Let $S(X,d)=sup_n S_n(X,d)$. Then, $S(X,d) =O(\\\\sqrt{d})$.\"\nOriginal remarks: [\"An algorithmic guarantee of $\\\\sqrt{d}log\\\\,n$ was worked upon in the workshop.\"]\nOriginal literature field (JSON string): \"i) Sevastiyanov showed $S(X,d) \\\\le d$. For $X$ a symmetric norm, Banaszczyk improved this to $d-\\\\frac{1}{d}$\\n\\nii) If $X=l_p$ for $1< p < 2$, $S(X,d) =\\\\Omega(d)$\\n\\niii) For $X=l_2$, Banaszczyk'13 gave a bound of $O(\\\\sqrt{d}+\\\\sqrt{\\\\log n})$. A lower bound of $\\\\sqrt{d}$ is known here.\\n\\niv) It is known that $S(X,d) \\\\ge \\\\alpha(X,X,d)$ [Chobanyan] (refer to vector balancing problem below for definition of $\\\\alpha$) . Given in Barany's epic document. Moreover, for a given set of vectors $v_1,..,v_n$ , given a signing of it achieving discrepancy $\\\\alpha$, we can recover a permutation achieving a guarantee of $\\\\alpha$ in the Steinitz setting [Harvey].\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0041", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM record conflates the arbitrary-norm Steinitz lemma with the open Euclidean O(sqrt(d)) conjecture and also contains a corrupted ell_p lower-bound line. For the admissible 2d-vector family consisting of e_1,...,e_d and d copies of -(1/d) times the all-ones vector, every ordering has a forced length-d prefix of ell_p norm at least d^(1/p)/2. In ell_1 the exact optimal maximum prefix norm of this family is d/2 for even d and d/2+1/(2d) for odd d, attained by the alternating order. Moreover, for 1 <= p <= 2, S_p(d) <= d^(1/p-1/2) S_2(d), so the Euclidean conjecture would make the d^(1/p) scale sharp. The recovered Euclidean conjecture remains open as of 2026-07-30.\n\nCandidate contribution (exact_example_and_transfer_lemma; novelty confidence low): For the classical family {e_1,...,e_d} together with d indexed copies of -(1/d)1, its exact ell_1 minimax prefix cost is d/2 in even dimension and d/2+1/(2d) in odd dimension; the same forced length-d prefix yields an explicit ell_p obstruction, and norm comparison gives S_p(d) <= d^(1/p-1/2)S_2(d)." }, { "id": 20000917, "problem_number": "AIM-COMBINATORICS-0042", "title": "Corrected volumetric characterization of vector balancing", "statement": "Vector Balancing\n\nLet $X,Y$ be norms on $\\mathbb{R}^d$. Given vectors $v_1,...,v_n\\in \\mathbb{R}^d$ satisfying $||v_i||_X\\le 1$ for all $i$. We want to find signs $\\epsilon_i \\in \\{-1,1\\}$ to minimize $|| \\sum_{i=1}^n\\epsilon_iv_i ||_Y$. Call this the discrepancy of the vectors. Define $\\alpha_n(X,Y,d)$ as the maximum of this discrepancy over all choice of $n$ vectors satisfying $||v_i||_X\\le 1$ and $\\alpha(X,Y,d) =sup_n \\alpha_n(X,Y,d)$. Banaszczyk proved $\\alpha(X,Y,d) \\ge \\sqrt{d}(\\frac{vol(B_X)}{vol(B_Y)})^{1/d}$ where $B_X,B_Y$ are the unit balls of the corresponding norms.\n\nIs this lower bound also an upper bound for $\\alpha(X,Y,d)$ upto $\\log d$ factors?", "original_statement": "Vector Balancing\n\nLet $X,Y$ be norms on $\\mathbb{R}^d$. Given vectors $v_1,...,v_n\\in \\mathbb{R}^d$ satisfying $||v_i||_X\\le 1$ for all $i$. We want to find signs $\\epsilon_i \\in \\{-1,1\\}$ to minimize $|| \\sum_{i=1}^n\\epsilon_iv_i ||_Y$. Call this the discrepancy of the vectors. Define $\\alpha_n(X,Y,d)$ as the maximum of this discrepancy over all choice of $n$ vectors satisfying $||v_i||_X\\le 1$ and $\\alpha(X,Y,d) =sup_n \\alpha_n(X,Y,d)$. Banaszczyk proved $\\alpha(X,Y,d) \\ge \\sqrt{d}(\\frac{vol(B_X)}{vol(B_Y)})^{1/d}$ where $B_X,B_Y$ are the unit balls of the corresponding norms.\n\nIs this lower bound also an upper bound for $\\alpha(X,Y,d)$ upto $\\log d$ factors?", "clean_statement": "Vector Balancing\n\nLet $X,Y$ be norms on $\\mathbb{R}^d$. Given vectors $v_1,...,v_n\\in \\mathbb{R}^d$ satisfying $||v_i||_X\\le 1$ for all $i$. We want to find signs $\\epsilon_i \\in \\{-1,1\\}$ to minimize $|| \\sum_{i=1}^n\\epsilon_iv_i ||_Y$. Call this the discrepancy of the vectors. Define $\\alpha_n(X,Y,d)$ as the maximum of this discrepancy over all choice of $n$ vectors satisfying $||v_i||_X\\le 1$ and $\\alpha(X,Y,d) =sup_n \\alpha_n(X,Y,d)$. Banaszczyk proved $\\alpha(X,Y,d) \\ge \\sqrt{d}(\\frac{vol(B_X)}{vol(B_Y)})^{1/d}$ where $B_X,B_Y$ are the unit balls of the corresponding norms.\n\nIs this lower bound also an upper bound for $\\alpha(X,Y,d)$ upto $\\log d$ factors?", "statement_status": "exact", "statement_verification": "Let $X,Y$ be norms on $\\mathbb R^d$, with unit balls $C=B_X$ and $K=B_Y$. For $v_1,\\ldots,v_N\\in C$, the source defines \\[ \\operatorname{disc}_Y(v_1,\\ldots,v_N) =\\min_{\\varepsilon\\in\\{-1,1\\}^N} \\left\\|\\sum_{i=1}^N\\varepsilon_i v_i\\right\\|_Y, \\] then takes the maximum over all such $N$-tuples to obtain $\\alpha_N(X,Y,d)$ and finally $\\alpha(X,Y,d)=\\sup_N\\alpha_N(X,Y,d)$.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.5\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[41]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Vector Balancing\\n\\nLet $X,Y$ be norms on $\\\\mathbb{R}^d$. Given vectors $v_1,...,v_n\\\\in \\\\mathbb{R}^d$ satisfying $||v_i||_X\\\\le 1$ for all $i$. We want to find signs $\\\\epsilon_i \\\\in \\\\{-1,1\\\\}$ to minimize $|| \\\\sum_{i=1}^n\\\\epsilon_iv_i ||_Y$. Call this the discrepancy of the vectors. Define $\\\\alpha_n(X,Y,d)$ as the maximum of this discrepancy over all choice of $n$ vectors satisfying $||v_i||_X\\\\le 1$ and $\\\\alpha(X,Y,d) =sup_n \\\\alpha_n(X,Y,d)$. Banaszczyk proved $\\\\alpha(X,Y,d) \\\\ge \\\\sqrt{d}(\\\\frac{vol(B_X)}{vol(B_Y)})^{1/d}$ where $B_X,B_Y$ are the unit balls of the corresponding norms.\\n\\nIs this lower bound also an upper bound for $\\\\alpha(X,Y,d)$ upto $\\\\log d$ factors?\"\nOriginal remarks: [\"In fact, this lower bound can be too small by a factor of $\\\\sqrt{d}$, for example when $X = Y = \\\\ell_1^d$. However, the following stronger lower bound may in fact also be an upper bound, up to logarithmic factors: $\\\\max_{k = 1}^{d}\\\\max_{W: \\\\mathrm{dim}(W) = k} \\\\min_{B_Y \\\\cap W \\\\subseteq E} \\\\sqrt{d} \\\\frac{vol(E)^{1/d}}{vol(B_X \\\\cap W)^{1/d}}$, where the maximum is over subspaces $W$ of dimension $k$ and the minimum is over ellipsoids $E$ containing $B_Y \\\\cap W$.\", \"The formula above should be $\\\\max_{k = 1}^d \\\\max_{W: \\\\mathrm{dim}(W) = k} \\\\min_{E: B_X \\\\cap W \\\\subseteq E} \\\\sqrt{k} \\\\frac{vol(E)^{1/k}}{vol(B_Y \\\\cap W)^{1/k}}$.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0042", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal full-dimensional volume-ratio conjecture is false, as shown by X=Y=ell_1^d, where the vector-balancing discrepancy equals d while the displayed lower bound has order sqrt(d). The corrected AIM functional Phi, formed by maximizing over k-dimensional sections and the minimum-volume ellipsoid containing B_X intersected with the section, does characterize alpha up to an O(log d) factor. This follows from the 2018 theorem of Dadush, Nikolov, Talwar, and Tomczak-Jaegermann together with a proved constant-factor equivalence between Phi and their determinant/preimage-volume lower bound.\n\nCandidate contribution (equivalence; novelty confidence low): For every pair of origin-symmetric convex bodies C,K in R^d, the corrected AIM Löwner-section functional Phi(C,K) is within universal constant factors of the determinant/preimage-volume functional V(C,K); explicitly, c Phi(C,K) <= V(C,K) <= C Phi(C,K)." }, { "id": 20000918, "problem_number": "AIM-COMBINATORICS-0043", "title": "Tusnady's problem: canonical models and monotone-chain certificates", "statement": "Tusnady's problem\n\nLet $d_n$ be the discrepancy of $n$ points in $\\mathbb{R}^2$ wrt all alix-aligned rectangles i.e. the set system comprises of all axis-aligned rectangles in $\\mathbb{R}^2$ as sets, maximised over all choices of $n$ points. Tusnady's problem then asks for the exact value of $d_n$.\n\nWe can also ask for the same question in $\\mathbb{R}^d$.", "original_statement": "Tusnady's problem\n\nLet $d_n$ be the discrepancy of $n$ points in $\\mathbb{R}^2$ wrt all alix-aligned rectangles i.e. the set system comprises of all axis-aligned rectangles in $\\mathbb{R}^2$ as sets, maximised over all choices of $n$ points. Tusnady's problem then asks for the exact value of $d_n$.\n\nWe can also ask for the same question in $\\mathbb{R}^d$.", "clean_statement": "Tusnady's problem\n\nLet $d_n$ be the discrepancy of $n$ points in $\\mathbb{R}^2$ wrt all alix-aligned rectangles i.e. the set system comprises of all axis-aligned rectangles in $\\mathbb{R}^2$ as sets, maximised over all choices of $n$ points. Tusnady's problem then asks for the exact value of $d_n$.\n\nWe can also ask for the same question in $\\mathbb{R}^d$.", "statement_status": "exact", "statement_verification": "Source metadata: `aim-combinatorics-notes.json`, zero-based index 42, problem ID `AIM-COMBINATORICS-0043`, workshop *Hereditary discrepancy and factorization norms*, problem number 1.65, source URL `http://aimpl.org/hereddiscrep/1/`.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.65\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[42]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Tusnady's problem\\n\\nLet $d_n$ be the discrepancy of $n$ points in $\\\\mathbb{R}^2$ wrt all alix-aligned rectangles i.e. the set system comprises of all axis-aligned rectangles in $\\\\mathbb{R}^2$ as sets, maximised over all choices of $n$ points. Tusnady's problem then asks for the exact value of $d_n$.\\n\\nWe can also ask for the same question in $\\\\mathbb{R}^d$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"In $\\\\mathbb{R^2}$, we know a lower bound of $O(\\\\log n)$ on $d_n$ and an upper bound of $O(\\\\log^{2.5}n)$. In $\\\\mathbb{R}^d$, we know that $d_n\\\\in[\\\\log^{d-1}n,log^{d+1/2}n]$\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0043", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The valid fixed-dimensional status is a lower bound c_d (log n)^{d-1} for infinitely many n and an upper bound C_d (log n)^{d-1/2} for all n, leaving the planar gap log n versus (log n)^{3/2}; a purported 2021 improvement was withdrawn in 2022 for an unfixable proof flaw. This attempt proves exact finite reductions between all boxes and anchored orthants (within 2^d), to general position, and in the plane to permutation rectangles. Its main new structural certificate is disc(P) <= mu(pi), where mu(pi) is the minimum number of increasing-or-decreasing subsequences partitioning the associated permutation, witnessed by alternating signs on every chain. It also proves that every nonempty full Cartesian grid has exact box discrepancy one.\n\nCandidate contribution (structural_lemma; novelty confidence low): Candidate contribution: for a planar general-position point set with permutation pi, its all-rectangle discrepancy is at most mu(pi), the least number of increasing-or-decreasing subsequences partitioning pi; alternating signs independently on the chains is an explicit witness. In particular LDS(pi) <= k implies discrepancy at most k, and 321-avoiding configurations have discrepancy at most two." }, { "id": 20000919, "problem_number": "AIM-COMBINATORICS-0044", "title": "Linear Beck--Fiala systems and an incidence-forest reduction", "statement": "Beating LLL\n\nConsider a matrix where both columns are rows are t-sparse. LLL gives a discrepancy bound of $O(\\sqrt{t\\log t})$ here.\nIf we assume the stronger property that every pair of rows intersect in at most one element, can we get a $O(\\sqrt{t})$?", "original_statement": "Beating LLL\n\nConsider a matrix where both columns are rows are t-sparse. LLL gives a discrepancy bound of $O(\\sqrt{t\\log t})$ here.\nIf we assume the stronger property that every pair of rows intersect in at most one element, can we get a $O(\\sqrt{t})$?", "clean_statement": "Beating LLL\n\nConsider a matrix where both columns are rows are t-sparse. LLL gives a discrepancy bound of $O(\\sqrt{t\\log t})$ here.\nIf we assume the stronger property that every pair of rows intersect in at most one element, can we get a $O(\\sqrt{t})$?", "statement_status": "exact", "statement_verification": "The AIM entry is Problem 1.7, “Beating LLL,” from the workshop *Hereditary discrepancy and factorization norms*. The stored text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.7\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[43]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Beating LLL\\n\\nConsider a matrix where both columns are rows are t-sparse. LLL gives a discrepancy bound of $O(\\\\sqrt{t\\\\log t})$ here.\\nIf we assume the stronger property that every pair of rows intersect in at most one element, can we get a $O(\\\\sqrt{t})$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0044", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Interpreting the problem as discrepancy of bounded-rank, bounded-degree linear hypergraphs, the class is closed under column restriction, so its uniform ordinary and hereditary O(sqrt(t)) conjectures are equivalent. For every finite hypergraph H, define phi(H) as the least possible maximum deleted-incidence load at a row among incidence-edge sets whose removal makes the Levi graph a forest. A direct rooted-forest coloring proves herdisc(H) <= 1 + phi(H). For projective planes of order q, disc(H) >= sqrt(q) while phi(H) = q = t-1, showing both that the requested order is optimal and that the feedback reduction cannot resolve dense linear cores. The maximum hereditary discrepancy in the class is exactly 2 when t=2.\n\nCandidate contribution (reduction; novelty confidence low): For every finite hypergraph H, herdisc(H) <= 1 + phi(H), where phi is the minimum maximum per-row incidence load required to delete to make the Levi graph a forest; moreover phi equals t-1 for a projective plane with line size t." }, { "id": 20000920, "problem_number": "AIM-COMBINATORICS-0045", "title": "Normalization-safe hardness below constructive Komlos bounds", "statement": "Hardness of Komlos\n\nCan we prove some hardness statement about finding a coloring achieving Komlos guarantee or the Banaszczyk discrepancy guarantee?", "original_statement": "Hardness of Komlos\n\nCan we prove some hardness statement about finding a coloring achieving Komlos guarantee or the Banaszczyk discrepancy guarantee?", "clean_statement": "Hardness of Komlos\n\nCan we prove some hardness statement about finding a coloring achieving Komlos guarantee or the Banaszczyk discrepancy guarantee?", "statement_status": "exact", "statement_verification": "The canonical record is problem 1.75, “Hardness of Komlos,” from the AIM workshop *Hereditary discrepancy and factorization norms*. Its complete question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.75\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[44]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Hardness of Komlos\\n\\nCan we prove some hardness statement about finding a coloring achieving Komlos guarantee or the Banaszczyk discrepancy guarantee?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0045", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "There is a universal constant delta > 0 for which it is NP-hard to distinguish rational matrices with column Euclidean norms at most one and discrepancy zero from such matrices with discrepancy at least delta. The reduction scales the Charikar-Newman-Nikolov 0-1 hard instances by the exactly encodable integer ceil(sqrt(M)), preserving a constant gap when M=O(N). This is threshold/additive hardness, not hardness against every unspecified O(1) Komlos guarantee. In contrast, one-row exact zero discrepancy is weakly NP-complete while a greedy algorithm always finds discrepancy at most one, and the Banaszczyk cube guarantee is now constructively attainable.\n\nCandidate contribution (reduction; novelty confidence low): The Charikar-Newman-Nikolov zero-versus-Omega(sqrt(N)) incidence-matrix gap transfers to a rational zero-versus-constant Komlos promise gap by setting B=A/ceil(sqrt(M)); the transfer preserves unit column norms, has O(log M)-bit entries, and explicitly identifies why it does not imply hardness against every absolute constant guarantee." }, { "id": 20000921, "problem_number": "AIM-COMBINATORICS-0046", "title": "Reverse Banaszczyk minimization and the ellipsoidal regime", "statement": "Reverse Banaszczyk-type problems\n\nSay that a convex body $K$ has property B if for any set of vectors of length no more than $c$ (some small constant), there exists a signing of them s.t. this signed sum lies inside $K$.\n\nWhat is the convex body with the smallest gaussian volume which has property $B$?", "original_statement": "Reverse Banaszczyk-type problems\n\nSay that a convex body $K$ has property B if for any set of vectors of length no more than $c$ (some small constant), there exists a signing of them s.t. this signed sum lies inside $K$.\n\nWhat is the convex body with the smallest gaussian volume which has property $B$?", "clean_statement": "Reverse Banaszczyk-type problems\n\nSay that a convex body $K$ has property B if for any set of vectors of length no more than $c$ (some small constant), there exists a signing of them s.t. this signed sum lies inside $K$.\n\nWhat is the convex body with the smallest gaussian volume which has property $B$?", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 1.8, “Reverse Banaszczyk-type problems,” from the workshop *Hereditary discrepancy and factorization norms*. Its stored text says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hereditary discrepancy and factorization norms\nSection: Open problems\nSource item: 1.8\nSource URL: http://aimpl.org/hereddiscrep/1/\nCanonical location: aim-combinatorics-notes.json notes[45]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Reverse Banaszczyk-type problems\\n\\nSay that a convex body $K$ has property B if for any set of vectors of length no more than $c$ (some small constant), there exists a signing of them s.t. this signed sum lies inside $K$.\\n\\nWhat is the convex body with the smallest gaussian volume which has property $B$?\"\nOriginal remarks: [\"During the workshop it was shown that if $K$ has property $B$, then $\\\\mathbb{E} \\\\|G\\\\|_K = (\\\\log n)^O(1)$, where $G$ is a standard Gaussian random vector.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hereddiscrep/1/", "tags": [ "aim", "AIM-COMBINATORICS-0046", "aim-domain:combinatorics", "aim-workshop:hereddiscrep", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After fixing the intended normalization as arbitrary finite Euclidean-norm-bounded sequences with a universal constant c, the general Gaussian-measure minimization remains open. A proved special-case theorem determines its exponential scale for origin-symmetric ellipsoids: property B_c forces c^2 sum_j a_j^{-2} <= 1 in principal-axis coordinates, hence forces a rotated copy of cB_infinity^n inside the ellipsoid and Gaussian measure at least (2Phi(c)-1)^n. Conversely, c sqrt(n) B_2^n has B_c by floating-variable reduction and randomized rounding, and for fixed 0(2/3)sqrt(N), the explicit bound r_3(P_3,K_t)<=ceil(9t^2/4), and an explicit P_3 obstruction in the Cooper-Mubayi grid. The asymptotic logarithmic Ramsey gap remains open.\n\nCandidate contribution (structural_theorem; novelty confidence low): A source-P_3-free simple 3-graph is characterized exactly by all vertex links being disjoint unions of stars and triangles; consequently e(H)<=N(N-1)/3, and this constant is attained for every N=4^d by taking all triples on affine F_4-lines." }, { "id": 20000924, "problem_number": "AIM-COMBINATORICS-0049", "title": "The minimal empty-core tight path and the 2026 uniformity-four resolution", "statement": "Question: Is it true that for fixed $k$, and $s$ sufficiently large (in terms of $k$), one has $r_k(P_s^{(k)},K_t^{(k)}) = \\Theta(t^{k-1})$.", "original_statement": "Question: Is it true that for fixed $k$, and $s$ sufficiently large (in terms of $k$), one has $r_k(P_s^{(k)},K_t^{(k)}) = \\Theta(t^{k-1})$.", "clean_statement": "Question: Is it true that for fixed $k$, and $s$ sufficiently large (in terms of $k$), one has $r_k(P_s^{(k)},K_t^{(k)}) = \\Theta(t^{k-1})$.", "statement_status": "exact", "statement_verification": "The canonical record (AIM Graph Ramsey theory workshop, section “Tight paths versus cliques,” question 2) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Tight paths versus cliques\nSource item: 2\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[48]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question: Is it true that for fixed $k$, and $s$ sufficiently large (in terms of $k$), one has $r_k(P_s^{(k)},K_t^{(k)}) = \\\\Theta(t^{k-1})$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0049", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the AIM convention that P_s^(k) has s edges and copies are unordered, its common edge intersection has size max(k-s+1,0), so it is a nontrivial tight k-tree exactly for s at least k+1. Nie's 2026 theorem therefore settles the question affirmatively for k=4 and s at least 5 (and recovers k=3, s at least 4), while k at least 5 remains open. Containment and a proved explicit random-deletion upper bound show that a sharp lower bound for the single minimal empty-core path P_(k+1)^(k) would imply Theta(t^(k-1)) for every longer fixed path.\n\nCandidate contribution (reduction; novelty confidence low): In the workshop's unordered edge-length convention, the sharp Ramsey lower-bound program throughout the nontrivial range s >= k+1 reduces to the single sufficient target P_(k+1)^(k): P_(k+1)^(k) is the minimal empty-core path, r_k(P_(k+1)^(k),K_t^(k)) <= r_k(P_s^(k),K_t^(k)) for all s >= k+1, and every fixed path has an unconditional O(t^(k-1)) upper bound with the explicit constant proved in the artifacts." }, { "id": 20000925, "problem_number": "AIM-COMBINATORICS-0050", "title": "The exceptional three-color path problem: star-forest and triangle-density reductions", "statement": "Paths in 3-colorings of 6-chromatic graphs\n\nAndr\\'as Gy\\'arf\\'as\n\n$P_k$ denotes a path of length $k$, i.e., with $k$ edges\n\nFact: $r(P_3,P_3,P_3) = 6$\n\nQuestion: Is it true that in any 3-edge-coloring of a graph\nof chromatic number at least 6, there is a monochromatic path of\nlength 3?\n\nKnown: (Garrison) For $k \\ne 3$, if we $k$-edge-color an\n$\\ell$-chromatic graph and $\\ell \\ge r(\\underbrace{P_3,\\dots,P_3}_{k\n \\text{ times}})$, then we get a monochromatic $P_3$. (That is, $k=3$\nis the only unknown case.)\n\nComment: For $k\\ne 3$, this follows from\n$\\operatorname{ex}(n,P_3)\\le n$ (look at the majority color).", "original_statement": "Paths in 3-colorings of 6-chromatic graphs\n\nAndr\\'as Gy\\'arf\\'as\n\n$P_k$ denotes a path of length $k$, i.e., with $k$ edges\n\nFact: $r(P_3,P_3,P_3) = 6$\n\nQuestion: Is it true that in any 3-edge-coloring of a graph\nof chromatic number at least 6, there is a monochromatic path of\nlength 3?\n\nKnown: (Garrison) For $k \\ne 3$, if we $k$-edge-color an\n$\\ell$-chromatic graph and $\\ell \\ge r(\\underbrace{P_3,\\dots,P_3}_{k\n \\text{ times}})$, then we get a monochromatic $P_3$. (That is, $k=3$\nis the only unknown case.)\n\nComment: For $k\\ne 3$, this follows from\n$\\operatorname{ex}(n,P_3)\\le n$ (look at the majority color).", "clean_statement": "Paths in 3-colorings of 6-chromatic graphs\n\nAndr\\'as Gy\\'arf\\'as\n\n$P_k$ denotes a path of length $k$, i.e., with $k$ edges\n\nFact: $r(P_3,P_3,P_3) = 6$\n\nQuestion: Is it true that in any 3-edge-coloring of a graph\nof chromatic number at least 6, there is a monochromatic path of\nlength 3?\n\nKnown: (Garrison) For $k \\ne 3$, if we $k$-edge-color an\n$\\ell$-chromatic graph and $\\ell \\ge r(\\underbrace{P_3,\\dots,P_3}_{k\n \\text{ times}})$, then we get a monochromatic $P_3$. (That is, $k=3$\nis the only unknown case.)\n\nComment: For $k\\ne 3$, this follows from\n$\\operatorname{ex}(n,P_3)\\le n$ (look at the majority color).", "statement_status": "exact", "statement_verification": "The source record states that \\(P_k\\) means a path of **length** \\(k\\), hence with \\(k\\) edges. It records \\[ r(P_3,P_3,P_3)=6 \\] and asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Paths in 3-colorings of 6-chromatic graphs\nSource item: 3\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[49]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Paths in 3-colorings of 6-chromatic graphs\\n\\nAndr\\\\'as Gy\\\\'arf\\\\'as\\n\\n$P_k$ denotes a path of length $k$, i.e., with $k$ edges\\n\\nFact: $r(P_3,P_3,P_3) = 6$\\n\\nQuestion: Is it true that in any 3-edge-coloring of a graph\\nof chromatic number at least 6, there is a monochromatic path of\\nlength 3?\\n\\nKnown: (Garrison) For $k \\\\ne 3$, if we $k$-edge-color an\\n$\\\\ell$-chromatic graph and $\\\\ell \\\\ge r(\\\\underbrace{P_3,\\\\dots,P_3}_{k\\n \\\\text{ times}})$, then we get a monochromatic $P_3$. (That is, $k=3$\\nis the only unknown case.)\\n\\nComment: For $k\\\\ne 3$, this follows from\\n$\\\\operatorname{ex}(n,P_3)\\\\le n$ (look at the majority color).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0050", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The question remains apparently open, with the standard three-edge path denoted P4 in the literature and known chromatic Ramsey interval 6 <= chi(P4,3) <= 7. This attempt proves that any graph whose edges partition into k star forests is (2k-1)-colorable for k>=2, so every three-coloring of a triangle-free 6-chromatic host contains a monochromatic three-edge path. More generally, in every induced 6-critical counterexample at least half the vertices lie in monochromatic triangles in at least two colors, forcing at least |V|/3 monochromatic triangle components; in the all-triangle dual reduction, a perfect matching would imply a five-edge-coloring and hence cannot exist.\n\nCandidate contribution (structural_theorem; novelty confidence low): Candidate contribution: unions of k star forests are (2k-1)-colorable for k>=2; consequently the AIM assertion holds for triangle-free hosts. For an unrestricted 6-critical counterexample, a canonical star/triangle orientation forces at least half the vertices to be in triangles of at least two colors and at least |V|/3 monochromatic triangle components, while an all-triangle counterexample's cubic tripartite linear dual must lack a perfect matching." }, { "id": 20000926, "problem_number": "AIM-COMBINATORICS-0051", "title": "A saturated-list kernel for one-point extensions of four-color Ramsey colorings", "statement": "4-color Ramsey number of triangles\n\nFan Chung\n\nProblem: Find $r(3,3,3,3)$. Is it 51?\n\nKnown: $51 \\le r(3,3,3,3) \\le 62$. Lower bound due to (Chung 1973),\nand upper bound due to (Fettes--Kramer--Radziszowski 2004).", "original_statement": "4-color Ramsey number of triangles\n\nFan Chung\n\nProblem: Find $r(3,3,3,3)$. Is it 51?\n\nKnown: $51 \\le r(3,3,3,3) \\le 62$. Lower bound due to (Chung 1973),\nand upper bound due to (Fettes--Kramer--Radziszowski 2004).", "clean_statement": "4-color Ramsey number of triangles\n\nFan Chung\n\nProblem: Find $r(3,3,3,3)$. Is it 51?\n\nKnown: $51 \\le r(3,3,3,3) \\le 62$. Lower bound due to (Chung 1973),\nand upper bound due to (Fettes--Kramer--Radziszowski 2004).", "statement_status": "exact", "statement_verification": "The canonical AIM record (Graph Ramsey theory workshop, section “4-color Ramsey number of triangles,” problem 4, attributed to Fan Chung) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: 4-color Ramsey number of triangles\nSource item: 4\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[50]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"4-color Ramsey number of triangles\\n\\nFan Chung\\n\\nProblem: Find $r(3,3,3,3)$. Is it 51?\\n\\nKnown: $51 \\\\le r(3,3,3,3) \\\\le 62$. Lower bound due to (Chung 1973),\\nand upper bound due to (Fettes--Kramer--Radziszowski 2004).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0051", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "One-point extension of any good four-coloring is exactly a four-label partition problem: the vertices assigned label i must be independent in color i. If the raw edge-clause system is infeasible, it becomes feasible after deletion of any vertex, via the canonical pivot assignment. Every color degree is at most 16; a degree-16 color is logically forbidden as an extension label, and its 16-neighborhood has each of the other three color graphs 5-regular. For a good K_50, two endpoints joined in color i, each with degree pattern 1 in color i and 16 in the other colors, give a concrete nonextendability certificate. These results reduce and constrain the extension problem but do not determine R_4(3).\n\nCandidate contribution (structural reduction; novelty confidence low): Candidate novelty: the one-point extension problem admits a two-layer exact kernel in which every infeasible raw edge-clause instance is vertex-minimal infeasible, while saturated degree-16 neighborhoods give logically implied label deletions, three 5-regular link graphs, and a degree-only forced-pair certificate for nonextendability of a good K_50." }, { "id": 20000927, "problem_number": "AIM-COMBINATORICS-0052", "title": "The hypergraph size-Ramsey equality is false, with critical-edge structure for minimal witnesses", "statement": "Hypergraph size Ramsey numbers\n\nAndrzej Dudek\n\nThe size Ramsey number $\\hat r(n,n)$ is the minimum number $m$\nfor which there exists a graph with $m$ edges such that every\n2-edge-coloring of it produces a monochromatic $K_n$. Similarly define\n$\\hat r_3(n,n)$ for the corresponding 3-uniform size Ramsey number.\n\nFor graphs, $\\hat r (n,n) = \\binom{r(n,n)}{2}$.\n\nProblem: (Dudek--R\\\"odl) Is $\\hat r_3(n,n) = \\binom{r_3(n,n)}{3}$?\n\nIn particular,", "original_statement": "Hypergraph size Ramsey numbers\n\nAndrzej Dudek\n\nThe size Ramsey number $\\hat r(n,n)$ is the minimum number $m$\nfor which there exists a graph with $m$ edges such that every\n2-edge-coloring of it produces a monochromatic $K_n$. Similarly define\n$\\hat r_3(n,n)$ for the corresponding 3-uniform size Ramsey number.\n\nFor graphs, $\\hat r (n,n) = \\binom{r(n,n)}{2}$.\n\nProblem: (Dudek--R\\\"odl) Is $\\hat r_3(n,n) = \\binom{r_3(n,n)}{3}$?\n\nIn particular,", "clean_statement": "Hypergraph size Ramsey numbers\n\nAndrzej Dudek\n\nThe size Ramsey number $\\hat r(n,n)$ is the minimum number $m$\nfor which there exists a graph with $m$ edges such that every\n2-edge-coloring of it produces a monochromatic $K_n$. Similarly define\n$\\hat r_3(n,n)$ for the corresponding 3-uniform size Ramsey number.\n\nFor graphs, $\\hat r (n,n) = \\binom{r(n,n)}{2}$.\n\nProblem: (Dudek--R\\\"odl) Is $\\hat r_3(n,n) = \\binom{r_3(n,n)}{3}$?\n\nIn particular,", "statement_status": "exact", "statement_verification": "The canonical AIM Graph Ramsey theory workshop record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Hypergraph size Ramsey numbers\nSource item: 5\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[51]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Hypergraph size Ramsey numbers\\n\\nAndrzej Dudek\\n\\nThe size Ramsey number $\\\\hat r(n,n)$ is the minimum number $m$\\nfor which there exists a graph with $m$ edges such that every\\n2-edge-coloring of it produces a monochromatic $K_n$. Similarly define\\n$\\\\hat r_3(n,n)$ for the corresponding 3-uniform size Ramsey number.\\n\\nFor graphs, $\\\\hat r (n,n) = \\\\binom{r(n,n)}{2}$.\\n\\nProblem: (Dudek--R\\\\\\\"odl) Is $\\\\hat r_3(n,n) = \\\\binom{r_3(n,n)}{3}$?\\n\\nIn particular,\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0052", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The proposed identity is false: using r_3(4,4)=13 and McKay's verified arrowing K_13^(3)-e -> (K_4^(3))_2 gives hat r_3(4,4) <= 285 < 286. In addition, an edge-minimal 3-uniform Ramsey host has, at every edge e, red and blue witness copies of K_n^(3) obtained from one avoiding coloring of H-e whose vertex sets intersect exactly in e. This yields minimum supported-vertex degree at least 2*binom(n-1,2)-1, positive pair-codegree at least 2n-5, and a corresponding elementary lower bound for every size-optimal host.\n\nCandidate contribution (structural_lemma; novelty confidence low): For every edge e of an edge-minimal 3-graph H arrowing K_n^(3) in two colors, one avoiding coloring of H-e produces a red witness vertex set R_e and a blue witness vertex set B_e with R_e intersect B_e exactly e; consequently every supported vertex has degree at least 2*binom(n-1,2)-1 and every positive pair has codegree at least 2n-5." }, { "id": 20000928, "problem_number": "AIM-COMBINATORICS-0053", "title": "Endpoint-deck gluing for one-edge deletion at a 3-uniform Ramsey threshold", "statement": "Problem: (Dudek--R\\\"odl) Let $N = r_3(n,n)$. Is it true that every\n2-edge-coloring of $K_N^{(3)}-e$ contains a monochromatic $K_n^{(3)}$? Here $K_N^{(3)} - e$ denotes $K_N^{(3)}$ with one edge deleted.\n\nLower bound: (Dudek--La Fleur--Mubayi--R\\\"odl) $\\hat r_3(n,n) \\ge\n\\text{poly}(n) r_3(n,n)^2$\n\nComment: $r_3(4,4) = 13$ ($\\ge$ Seymour; $\\le$\nRadziszowski). So does it work for $n=4$? This is a matter of computational verification.", "original_statement": "Problem: (Dudek--R\\\"odl) Let $N = r_3(n,n)$. Is it true that every\n2-edge-coloring of $K_N^{(3)}-e$ contains a monochromatic $K_n^{(3)}$? Here $K_N^{(3)} - e$ denotes $K_N^{(3)}$ with one edge deleted.\n\nLower bound: (Dudek--La Fleur--Mubayi--R\\\"odl) $\\hat r_3(n,n) \\ge\n\\text{poly}(n) r_3(n,n)^2$\n\nComment: $r_3(4,4) = 13$ ($\\ge$ Seymour; $\\le$\nRadziszowski). So does it work for $n=4$? This is a matter of computational verification.", "clean_statement": "Problem: (Dudek--R\\\"odl) Let $N = r_3(n,n)$. Is it true that every\n2-edge-coloring of $K_N^{(3)}-e$ contains a monochromatic $K_n^{(3)}$? Here $K_N^{(3)} - e$ denotes $K_N^{(3)}$ with one edge deleted.\n\nLower bound: (Dudek--La Fleur--Mubayi--R\\\"odl) $\\hat r_3(n,n) \\ge\n\\text{poly}(n) r_3(n,n)^2$\n\nComment: $r_3(4,4) = 13$ ($\\ge$ Seymour; $\\le$\nRadziszowski). So does it work for $n=4$? This is a matter of computational verification.", "statement_status": "exact", "statement_verification": "The source record (AIM problem list, workshop *Graph Ramsey theory*, section *Hypergraph size Ramsey numbers*) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Hypergraph size Ramsey numbers\nSource item: 6\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[52]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem: (Dudek--R\\\\\\\"odl) Let $N = r_3(n,n)$. Is it true that every\\n2-edge-coloring of $K_N^{(3)}-e$ contains a monochromatic $K_n^{(3)}$? Here $K_N^{(3)} - e$ denotes $K_N^{(3)}$ with one edge deleted.\\n\\nLower bound: (Dudek--La Fleur--Mubayi--R\\\\\\\"odl) $\\\\hat r_3(n,n) \\\\ge\\n\\\\text{poly}(n) r_3(n,n)^2$\\n\\nComment: $r_3(4,4) = 13$ ($\\\\ge$ Seymour; $\\\\le$\\nRadziszowski). So does it work for $n=4$? This is a matter of computational verification.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0053", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "McKay's published exhaustive computation settles the requested n=4 case affirmatively: K_13^(3)-e arrows K_4^(3), so the corresponding size-Ramsey number is at most 285. For the intended general range n>=5, no resolution was found. Independently, a good coloring of K_N^(q)-e is proved equivalent to a compatible q-tuple of good colorings on the complete (N-1)-vertex endpoint-deleted decks; at N=r_q(n,n), any countercoloring also forces oppositely colored rooted almost-K_n^(q) witnesses whose vertex sets intersect exactly in e. The unrestricted literal wording has the trivial counterexample n=3.\n\nCandidate contribution (equivalence; novelty confidence low): For e={x_1,...,x_q}, K_V^(q)-e admits a red-blue coloring with no monochromatic K_n^(q) if and only if there are pairwise-compatible (n,n;q)-good colorings of K_(V\\{x_i})^(q) for every endpoint x_i; moreover, at N=r_q(n,n), every such countercoloring contains red and blue rooted almost-cliques meeting exactly in e.", "research_difficulty_suggested": "L4" }, { "id": 20000929, "problem_number": "AIM-COMBINATORICS-0054", "title": "Solved diagonal online-Ramsey lower bound and a common-core obstruction to greedy Painter", "statement": "Online Ramsey numbers\n\nJacob Fox\n\nThe online Ramsey number $\\tilde r(s,n)$ is defined via a game as\nfollows. Initially there are infinitely many vertices and no\nedges. Each turn, the builder adds an edge to the graph, and\nthe painter colors the edge blue or red. We define $\\tilde\nr(s,n)$ to be the minimum $m$ such that the builder can force a win in\n$m$ moves, i.e., force a red $K_s$ or a blue $K_n$.\n\nNote: $\\tilde r(s,n) \\le \\hat r (s,n) = \\binom{r(s,n)}{2}$.\n\n(Conlon) For infinitely many $n$, $\\tilde r(n,n) \\le (0.99)^n\n\\binom{r(n,n)}{2}$. The constant $0.99$ can probably be made much\nsmaller.\n\nNote that if $\\tilde r(n,n) \\ge (1.999)^n$ for all $n$, then $r(n,n)\n\\ge (\\sqrt 2 + \\delta)^n$ for some $\\delta > 0$. This is a possible\nstrategy for improving lower bounds on diagonal Ramsey numbers. More\nmodestly, we can ask:\n\nProblem: Show that $\\tilde r(n,n) \\ge (\\sqrt 2 + \\epsilon)^n$\nfor some $\\epsilon > 0$.\n\nRelated: (Conlon--Fox--Grinshpun) $\\tilde r(3,n) = \\tilde \\Theta(n^3)$, where\n$\\tilde\\Theta$ means up to a $\\text{poly\\,log}(n)$ factor. The lower bound is\nproved via a lopsided local lemma.\n\n\\section{Erd\\H{o}s--Hajnal for tournaments}\n\nMaria Chudnovsky\n\nLet $T^*$ denote the following tournament on $6$ vertices labeled by\n$\\{1, 2, \\dots, 6\\}$: orient $6 \\to 1, 6 \\to 3, 5 \\to 2, 4 \\to 1$, and\norient all remaining pairs $i \\to j$ where $i < j$.", "original_statement": "Online Ramsey numbers\n\nJacob Fox\n\nThe online Ramsey number $\\tilde r(s,n)$ is defined via a game as\nfollows. Initially there are infinitely many vertices and no\nedges. Each turn, the builder adds an edge to the graph, and\nthe painter colors the edge blue or red. We define $\\tilde\nr(s,n)$ to be the minimum $m$ such that the builder can force a win in\n$m$ moves, i.e., force a red $K_s$ or a blue $K_n$.\n\nNote: $\\tilde r(s,n) \\le \\hat r (s,n) = \\binom{r(s,n)}{2}$.\n\n(Conlon) For infinitely many $n$, $\\tilde r(n,n) \\le (0.99)^n\n\\binom{r(n,n)}{2}$. The constant $0.99$ can probably be made much\nsmaller.\n\nNote that if $\\tilde r(n,n) \\ge (1.999)^n$ for all $n$, then $r(n,n)\n\\ge (\\sqrt 2 + \\delta)^n$ for some $\\delta > 0$. This is a possible\nstrategy for improving lower bounds on diagonal Ramsey numbers. More\nmodestly, we can ask:\n\nProblem: Show that $\\tilde r(n,n) \\ge (\\sqrt 2 + \\epsilon)^n$\nfor some $\\epsilon > 0$.\n\nRelated: (Conlon--Fox--Grinshpun) $\\tilde r(3,n) = \\tilde \\Theta(n^3)$, where\n$\\tilde\\Theta$ means up to a $\\text{poly\\,log}(n)$ factor. The lower bound is\nproved via a lopsided local lemma.\n\n\\section{Erd\\H{o}s--Hajnal for tournaments}\n\nMaria Chudnovsky\n\nLet $T^*$ denote the following tournament on $6$ vertices labeled by\n$\\{1, 2, \\dots, 6\\}$: orient $6 \\to 1, 6 \\to 3, 5 \\to 2, 4 \\to 1$, and\norient all remaining pairs $i \\to j$ where $i < j$.", "clean_statement": "Online Ramsey numbers\n\nJacob Fox\n\nThe online Ramsey number $\\tilde r(s,n)$ is defined via a game as\nfollows. Initially there are infinitely many vertices and no\nedges. Each turn, the builder adds an edge to the graph, and\nthe painter colors the edge blue or red. We define $\\tilde\nr(s,n)$ to be the minimum $m$ such that the builder can force a win in\n$m$ moves, i.e., force a red $K_s$ or a blue $K_n$.\n\nNote: $\\tilde r(s,n) \\le \\hat r (s,n) = \\binom{r(s,n)}{2}$.\n\n(Conlon) For infinitely many $n$, $\\tilde r(n,n) \\le (0.99)^n\n\\binom{r(n,n)}{2}$. The constant $0.99$ can probably be made much\nsmaller.\n\nNote that if $\\tilde r(n,n) \\ge (1.999)^n$ for all $n$, then $r(n,n)\n\\ge (\\sqrt 2 + \\delta)^n$ for some $\\delta > 0$. This is a possible\nstrategy for improving lower bounds on diagonal Ramsey numbers. More\nmodestly, we can ask:\n\nProblem: Show that $\\tilde r(n,n) \\ge (\\sqrt 2 + \\epsilon)^n$\nfor some $\\epsilon > 0$.\n\nRelated: (Conlon--Fox--Grinshpun) $\\tilde r(3,n) = \\tilde \\Theta(n^3)$, where\n$\\tilde\\Theta$ means up to a $\\text{poly\\,log}(n)$ factor. The lower bound is\nproved via a lopsided local lemma.\n\n\\section{Erd\\H{o}s--Hajnal for tournaments}\n\nMaria Chudnovsky\n\nLet $T^*$ denote the following tournament on $6$ vertices labeled by\n$\\{1, 2, \\dots, 6\\}$: orient $6 \\to 1, 6 \\to 3, 5 \\to 2, 4 \\to 1$, and\norient all remaining pairs $i \\to j$ where $i < j$.", "statement_status": "exact", "statement_verification": "The online-Ramsey part of the verified AIM record defines the game on infinitely many initially isolated vertices: in each move Builder exposes one edge and Painter immediately colors it red or blue. It then asks", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Online Ramsey numbers\nSource item: 7\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[53]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Online Ramsey numbers\\n\\nJacob Fox\\n\\nThe online Ramsey number $\\\\tilde r(s,n)$ is defined via a game as\\nfollows. Initially there are infinitely many vertices and no\\nedges. Each turn, the builder adds an edge to the graph, and\\nthe painter colors the edge blue or red. We define $\\\\tilde\\nr(s,n)$ to be the minimum $m$ such that the builder can force a win in\\n$m$ moves, i.e., force a red $K_s$ or a blue $K_n$.\\n\\nNote: $\\\\tilde r(s,n) \\\\le \\\\hat r (s,n) = \\\\binom{r(s,n)}{2}$.\\n\\n(Conlon) For infinitely many $n$, $\\\\tilde r(n,n) \\\\le (0.99)^n\\n\\\\binom{r(n,n)}{2}$. The constant $0.99$ can probably be made much\\nsmaller.\\n\\nNote that if $\\\\tilde r(n,n) \\\\ge (1.999)^n$ for all $n$, then $r(n,n)\\n\\\\ge (\\\\sqrt 2 + \\\\delta)^n$ for some $\\\\delta > 0$. This is a possible\\nstrategy for improving lower bounds on diagonal Ramsey numbers. More\\nmodestly, we can ask:\\n\\nProblem: Show that $\\\\tilde r(n,n) \\\\ge (\\\\sqrt 2 + \\\\epsilon)^n$\\nfor some $\\\\epsilon > 0$.\\n\\nRelated: (Conlon--Fox--Grinshpun) $\\\\tilde r(3,n) = \\\\tilde \\\\Theta(n^3)$, where\\n$\\\\tilde\\\\Theta$ means up to a $\\\\text{poly\\\\,log}(n)$ factor. The lower bound is\\nproved via a lopsided local lemma.\\n\\n\\\\section{Erd\\\\H{o}s--Hajnal for tournaments}\\n\\nMaria Chudnovsky\\n\\nLet $T^*$ denote the following tournament on $6$ vertices labeled by\\n$\\\\{1, 2, \\\\dots, 6\\\\}$: orient $6 \\\\to 1, 6 \\\\to 3, 5 \\\\to 2, 4 \\\\to 1$, and\\norient all remaining pairs $i \\\\to j$ where $i < j$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0054", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the necessary asymptotic reading, the AIM problem is solved by the published bound of Conlon, Fox, Grinshpun, and He: tilde r(n,n) is at least 2^((2-sqrt(2))n-O(1)), whose exponential base 2^(2-sqrt(2)) exceeds sqrt(2). The report also proves a quantified subsequence-to-ordinary-Ramsey transfer and an explicit common-core obstruction: the Painter rule that chooses red unless red creates K_s is defeated on s+n-2 vertices in binom(s-2,2)+(s-2)n+binom(n,2) moves, or 2n^2-5n+3 moves diagonally.\n\nCandidate contribution (obstruction; novelty confidence low): For the one-sided greedy rule that colors an edge red unless this would create a red K_s, every forced blue edge xy has a red K_s-minus-xy witness whose auxiliary vertices avoid all other vertices of a terminal blue K_n; nevertheless a fixed common red core of size s-2 lets Builder force a blue K_n in binom(s-2,2)+(s-2)n+binom(n,2) moves on s+n-2 vertices." }, { "id": 20000930, "problem_number": "AIM-COMBINATORICS-0055", "title": "The Erdős-Hajnal property for the Paley seven-vertex tournament with one vertex removed", "statement": "Question: Does there exist $\\epsilon > 0$ such that if $T_n$\nis an $n$-vertex tournament not containing $T^*$ as a subtournament,\nthen $T_n$ contains a transitive tournament with $\\ge n^\\epsilon$\nvertices?\n\nRemark: (Berger--Choromanski--Chudnovsky) This is the smallest\ntournament for which this conclusion is not known.\n\nRemark: Such a statement for all tournaments is equivalent to\nthe Erd\\H os--Hajnal conjecture.", "original_statement": "Question: Does there exist $\\epsilon > 0$ such that if $T_n$\nis an $n$-vertex tournament not containing $T^*$ as a subtournament,\nthen $T_n$ contains a transitive tournament with $\\ge n^\\epsilon$\nvertices?\n\nRemark: (Berger--Choromanski--Chudnovsky) This is the smallest\ntournament for which this conclusion is not known.\n\nRemark: Such a statement for all tournaments is equivalent to\nthe Erd\\H os--Hajnal conjecture.", "clean_statement": "Question: Does there exist $\\epsilon > 0$ such that if $T_n$\nis an $n$-vertex tournament not containing $T^*$ as a subtournament,\nthen $T_n$ contains a transitive tournament with $\\ge n^\\epsilon$\nvertices?\n\nRemark: (Berger--Choromanski--Chudnovsky) This is the smallest\ntournament for which this conclusion is not known.\n\nRemark: Such a statement for all tournaments is equivalent to\nthe Erd\\H os--Hajnal conjecture.", "statement_status": "exact", "statement_verification": "The canonical record asks whether there is an \\(\\epsilon>0\\) such that every \\(n\\)-vertex tournament with no subtournament isomorphic to \\(T^*\\) contains a transitive subtournament on at least \\(n^\\epsilon\\) vertices. Its historical remark says that Berger--Choromanski--Chudnovsky had reduced the six-vertex case to this one tournament.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Online Ramsey numbers\nSource item: 8\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[54]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question: Does there exist $\\\\epsilon > 0$ such that if $T_n$\\nis an $n$-vertex tournament not containing $T^*$ as a subtournament,\\nthen $T_n$ contains a transitive tournament with $\\\\ge n^\\\\epsilon$\\nvertices?\\n\\nRemark: (Berger--Choromanski--Chudnovsky) This is the smallest\\ntournament for which this conclusion is not known.\\n\\nRemark: Such a statement for all tournaments is equivalent to\\nthe Erd\\\\H os--Hajnal conjecture.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0055", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The recovered T* is the exceptional six-vertex tournament K6 of Berger-Choromanski-Chudnovsky, equivalently the Paley tournament P7 with vertex 0 removed. It is two-colourable because {1,2,3} and {4,5,6} are transitive. Nguyen, Scott, and Seymour proved in 2025 that every two-colourable tournament has the Erdős-Hajnal property, so the workshop question has an affirmative answer. In addition, every tournament recursively built by substitutions through templates of order at most five is proved T*-free and to contain a transitive subtournament of order at least n^(log_5 2).\n\nCandidate contribution (quantitative special case; novelty confidence low): If a tournament G is recursively constructed by substituting tournaments into arbitrary tournament templates of order at most five, then G is T*-free and tr(G) >= |G|^(log_5 2), without any balance hypothesis on the substitution blocks." }, { "id": 20000931, "problem_number": "AIM-COMBINATORICS-0056", "title": "Local triangle trees and K4-minor support girth", "statement": "Local trees of triangles\n\nVojt\\v{e}ch R\\\"odl\n\nA tree of triangles is any graph built recursively by starting with a triangle, and then adding two edges at a time that forms a single new triangle with an existing edge.\n\nKnown: Given any tree $T$ of triangles, there exists $k$ such\nthat if $G \\xrightarrow{k} K_3$ (the $k$ above the arrow means with $k$\ncolors), then $G$ contains $T$.\n\nQuestion: For every $\\ell$, does there exist a graph $G\n\\xrightarrow{2} K_3$ which on every $\\ell$ vertices induces a subgraph\nof a tree of triangles.\n\nThe property in the conclusion of the question will be referred to as\nlocally a tree of triangles.", "original_statement": "Local trees of triangles\n\nVojt\\v{e}ch R\\\"odl\n\nA tree of triangles is any graph built recursively by starting with a triangle, and then adding two edges at a time that forms a single new triangle with an existing edge.\n\nKnown: Given any tree $T$ of triangles, there exists $k$ such\nthat if $G \\xrightarrow{k} K_3$ (the $k$ above the arrow means with $k$\ncolors), then $G$ contains $T$.\n\nQuestion: For every $\\ell$, does there exist a graph $G\n\\xrightarrow{2} K_3$ which on every $\\ell$ vertices induces a subgraph\nof a tree of triangles.\n\nThe property in the conclusion of the question will be referred to as\nlocally a tree of triangles.", "clean_statement": "Local trees of triangles\n\nVojt\\v{e}ch R\\\"odl\n\nA tree of triangles is any graph built recursively by starting with a triangle, and then adding two edges at a time that forms a single new triangle with an existing edge.\n\nKnown: Given any tree $T$ of triangles, there exists $k$ such\nthat if $G \\xrightarrow{k} K_3$ (the $k$ above the arrow means with $k$\ncolors), then $G$ contains $T$.\n\nQuestion: For every $\\ell$, does there exist a graph $G\n\\xrightarrow{2} K_3$ which on every $\\ell$ vertices induces a subgraph\nof a tree of triangles.\n\nThe property in the conclusion of the question will be referred to as\nlocally a tree of triangles.", "statement_status": "exact", "statement_verification": "The record comes from the January 2015 AIM workshop *Graph Ramsey theory* and attributes the question to Vojtěch Rödl. In normalized notation it asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Local trees of triangles\nSource item: 9\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[55]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Local trees of triangles\\n\\nVojt\\\\v{e}ch R\\\\\\\"odl\\n\\nA tree of triangles is any graph built recursively by starting with a triangle, and then adding two edges at a time that forms a single new triangle with an existing edge.\\n\\nKnown: Given any tree $T$ of triangles, there exists $k$ such\\nthat if $G \\\\xrightarrow{k} K_3$ (the $k$ above the arrow means with $k$\\ncolors), then $G$ contains $T$.\\n\\nQuestion: For every $\\\\ell$, does there exist a graph $G\\n\\\\xrightarrow{2} K_3$ which on every $\\\\ell$ vertices induces a subgraph\\nof a tree of triangles.\\n\\nThe property in the conclusion of the question will be referred to as\\nlocally a tree of triangles.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0056", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The question has an affirmative answer as a direct specialization of Reiher and Rodl's 2023 Girth Ramsey Theorem, Theorem 1.6: with F=K_3, r=2, and n=ell it produces a triangle-Ramsey graph whose induced subgraphs on at most ell vertices are partial forests of triangle copies. An explicit width-two tree decomposition proves that every such partial forest is an ordinary subgraph of an AIM tree of triangles. Thus for every ell there is a graph G arrowing K_3 in two colors that is locally a tree of triangles through ell.\n\nCandidate contribution (structural equivalence; novelty confidence low): Candidate novelty: if sigma_4(G) is the minimum size of a vertex set X for which G[X] contains a K_4 minor, then G is locally a tree of triangles through ell exactly when sigma_4(G)>ell. Hence the solved AIM problem is equivalently the unboundedness of K_4-minor support number among triangle-Ramsey graphs; the terminology bridge is witnessed by a one-bag-per-triangle width-two decomposition." }, { "id": 20000932, "problem_number": "AIM-COMBINATORICS-0057", "title": "Local trees of triangles force quadratic order", "statement": "Question Does there exist any finite graph where every edge\nis in at least two triangles and the graph is locally a tree of triangles? (Solved by J. Verstraete)\n\n(Ne\\v{s}et\\v{r}il--R\\\"odl) Partial results", "original_statement": "Question Does there exist any finite graph where every edge\nis in at least two triangles and the graph is locally a tree of triangles? (Solved by J. Verstraete)\n\n(Ne\\v{s}et\\v{r}il--R\\\"odl) Partial results", "clean_statement": "Question Does there exist any finite graph where every edge\nis in at least two triangles and the graph is locally a tree of triangles? (Solved by J. Verstraete)\n\n(Ne\\v{s}et\\v{r}il--R\\\"odl) Partial results", "statement_status": "exact", "statement_verification": "The extracted source record reads exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Local trees of triangles\nSource item: 10\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[56]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question Does there exist any finite graph where every edge\\nis in at least two triangles and the graph is locally a tree of triangles? (Solved by J. Verstraete)\\n\\n(Ne\\\\v{s}et\\\\v{r}il--R\\\\\\\"odl) Partial results\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 3; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0057", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the recovered AIM formulation, for every ell at least 3, each connected component containing an edge in an ell-local witness has minimum degree at least ell. Averaging the edge bound for ell-vertex partial 2-trees then forces component order n at least 1+ceil(ell^2(ell-1)/(2(2ell-3))). This exact lower bound is attained for ell=3 by K4 and for ell=4 by K_{2,2,2}; for ell=1,2 the exact minimum nontrivial component order is four. The result is a necessary obstruction and small-case solution, not a reconstruction of Verstraete's credited general existence proof.\n\nCandidate contribution (bound; novelty confidence low): For ell at least 3, the link-cycle and K4-minor obstruction forces delta(H) at least ell in every component H containing an edge, and consequently |V(H)| is at least 1+ceil(ell^2(ell-1)/(2(2ell-3))); the bound is sharp for ell=3 and ell=4." }, { "id": 20000933, "problem_number": "AIM-COMBINATORICS-0058", "title": "A downgrade criterion for clique-order monotonicity", "statement": "Ramsey minimal graphs\n\nTibor Szab\\'o\n\nLet $G \\xrightarrow{r} K_k$ be minimal, i.e., $G - e\n\\not\\xrightarrow{r} K_3$ for any $e \\in E(G)$.\nLet $\\mathcal{M}_r(k)$ denote the family of such graphs $G$. Let\n\\[\ns_r(k) = \\min\\{\\delta(G) : G \\in \\mathcal{M}_r(k)\\}\n\\]\nwhere $\\delta(G)$ is the minimum degree of $G$.\n\nConjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o)\n$s_r(k) \\le s_r(k+1)$ for all $k$.\n\nResults: $s_2(k) = (k-1)^2$ (Erd\\H os--Burr--Lov\\'asz 1976)\n\n$\\Omega(r^2 \\log r) \\le s_r(3) \\le O(r^2(\\log r)^2 )$\n\n$\\Omega\\left(\\frac{r^2 \\log r }{ \\log\\log r}\\right) \\le s_r(k) \\le\nO\\left(r^2 (\\log r)^{8(k-1)^2}\\right)$ for $k \\ge 4$", "original_statement": "Ramsey minimal graphs\n\nTibor Szab\\'o\n\nLet $G \\xrightarrow{r} K_k$ be minimal, i.e., $G - e\n\\not\\xrightarrow{r} K_3$ for any $e \\in E(G)$.\nLet $\\mathcal{M}_r(k)$ denote the family of such graphs $G$. Let\n\\[\ns_r(k) = \\min\\{\\delta(G) : G \\in \\mathcal{M}_r(k)\\}\n\\]\nwhere $\\delta(G)$ is the minimum degree of $G$.\n\nConjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o)\n$s_r(k) \\le s_r(k+1)$ for all $k$.\n\nResults: $s_2(k) = (k-1)^2$ (Erd\\H os--Burr--Lov\\'asz 1976)\n\n$\\Omega(r^2 \\log r) \\le s_r(3) \\le O(r^2(\\log r)^2 )$\n\n$\\Omega\\left(\\frac{r^2 \\log r }{ \\log\\log r}\\right) \\le s_r(k) \\le\nO\\left(r^2 (\\log r)^{8(k-1)^2}\\right)$ for $k \\ge 4$", "clean_statement": "Ramsey minimal graphs\n\nTibor Szab\\'o\n\nLet $G \\xrightarrow{r} K_k$ be minimal, i.e., $G - e\n\\not\\xrightarrow{r} K_3$ for any $e \\in E(G)$.\nLet $\\mathcal{M}_r(k)$ denote the family of such graphs $G$. Let\n\\[\ns_r(k) = \\min\\{\\delta(G) : G \\in \\mathcal{M}_r(k)\\}\n\\]\nwhere $\\delta(G)$ is the minimum degree of $G$.\n\nConjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o)\n$s_r(k) \\le s_r(k+1)$ for all $k$.\n\nResults: $s_2(k) = (k-1)^2$ (Erd\\H os--Burr--Lov\\'asz 1976)\n\n$\\Omega(r^2 \\log r) \\le s_r(3) \\le O(r^2(\\log r)^2 )$\n\n$\\Omega\\left(\\frac{r^2 \\log r }{ \\log\\log r}\\right) \\le s_r(k) \\le\nO\\left(r^2 (\\log r)^{8(k-1)^2}\\right)$ for $k \\ge 4$", "statement_status": "exact", "statement_verification": "The source record is from the AIM workshop *Graph Ramsey theory*, section “Ramsey minimal graphs,” problem 11, attributed to Tibor Szabó. The displayed source text contains", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Ramsey minimal graphs\nSource item: 11\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[57]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Ramsey minimal graphs\\n\\nTibor Szab\\\\'o\\n\\nLet $G \\\\xrightarrow{r} K_k$ be minimal, i.e., $G - e\\n\\\\not\\\\xrightarrow{r} K_3$ for any $e \\\\in E(G)$.\\nLet $\\\\mathcal{M}_r(k)$ denote the family of such graphs $G$. Let\\n\\\\[\\ns_r(k) = \\\\min\\\\{\\\\delta(G) : G \\\\in \\\\mathcal{M}_r(k)\\\\}\\n\\\\]\\nwhere $\\\\delta(G)$ is the minimum degree of $G$.\\n\\nConjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\\\'o)\\n$s_r(k) \\\\le s_r(k+1)$ for all $k$.\\n\\nResults: $s_2(k) = (k-1)^2$ (Erd\\\\H os--Burr--Lov\\\\'asz 1976)\\n\\n$\\\\Omega(r^2 \\\\log r) \\\\le s_r(3) \\\\le O(r^2(\\\\log r)^2 )$\\n\\n$\\\\Omega\\\\left(\\\\frac{r^2 \\\\log r }{ \\\\log\\\\log r}\\\\right) \\\\le s_r(k) \\\\le\\nO\\\\left(r^2 (\\\\log r)^{8(k-1)^2}\\\\right)$ for $k \\\\ge 4$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0058", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After correcting the source typo from K_3 to K_k, the conjecture is equivalent to P_r(k-1) <= P_r(k). A proved local downgrade criterion shows this inequality whenever one optimal P_r(k) packing has, in each colour graph, a K_k-free spanning subgraph retaining a K_{k-1} inside every original K_k; an exact vertex K_k-transversal is a sufficient certificate. Independently, monotonicity in the number of colours and published explicit bounds give an unconditional comparison s_r(K_k) <= s_r(K_l) for sufficiently polynomially separated clique sizes. The adjacent conjecture for general r >= 3 remains open in the literature checked.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: a local downgrade certificate on an optimal clique-packing witness implies the adjacent inequality P_r(k-1) <= P_r(k), with exact vertex clique transversals as a concrete sufficient condition; combining colour-monotonicity with explicit published estimates also yields the stated separated-size comparison.", "research_difficulty_suggested": "L4" }, { "id": 20000934, "problem_number": "AIM-COMBINATORICS-0059", "title": "Sharp lower bound from the resolved Erdős--Rogers scale", "statement": "Conjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o) $s_r(k)\n= \\Theta(r \\cdot f_{k,k+1}(r)^2)$ where $f_{k,k+1}$ is the\nErd\\H{o}s-Rogers function, to be discussed in Dudek's talk.", "original_statement": "Conjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o) $s_r(k)\n= \\Theta(r \\cdot f_{k,k+1}(r)^2)$ where $f_{k,k+1}$ is the\nErd\\H{o}s-Rogers function, to be discussed in Dudek's talk.", "clean_statement": "Conjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\'o) $s_r(k)\n= \\Theta(r \\cdot f_{k,k+1}(r)^2)$ where $f_{k,k+1}$ is the\nErd\\H{o}s-Rogers function, to be discussed in Dudek's talk.", "statement_status": "exact", "statement_verification": "The exact AIM record (Graph Ramsey Theory workshop, section \"Ramsey minimal graphs\", problem 12) says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Ramsey minimal graphs\nSource item: 12\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[58]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture: (Fox--Grinshpun--Liebenau--Person--Szab\\\\'o) $s_r(k)\\n= \\\\Theta(r \\\\cdot f_{k,k+1}(r)^2)$ where $f_{k,k+1}$ is the\\nErd\\\\H{o}s-Rogers function, to be discussed in Dudek's talk.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0059", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM record has a one-step index mismatch: the primary FGLPS conjecture is s_r(K_k)=Theta_k(r f_{k-1,k}(r)^2), equivalently s_r(K_{q+1})=Theta_q(r f_{q,q+1}(r)^2). Morris--Sahasrabudhe--Verstraëte's July 2026 theorem f_{q,q+1}(n)=Theta_q(sqrt(n log n)), combined with the FGLPS recurrence for P_r(q)=s_r(K_{q+1}), rigorously yields s_r(K_k)=Omega_k(r^2 log r) for every fixed k>=3. At n=Theta_q(r^2 log r), the existence of one required Erdős--Rogers graph is now known; the matching upper bound for k>=4 reduces to packing r such graphs pairwise edge-disjointly, which remains open.\n\nCandidate contribution (new_status_corollary_and_reduction; novelty confidence low): Candidate novelty: for every fixed k>=4, the July 2026 Erdős--Rogers theorem and the FGLPS packing recurrence imply the sharp lower bound s_r(K_k)=Omega_k(r^2 log r), removing the former log-log loss; moreover, the complementary upper bound is isolated as a simultaneous edge-disjoint packing problem at n=Theta_k(r^2 log r), since the single-graph input is now available.", "research_difficulty_suggested": "L4" }, { "id": 20000935, "problem_number": "AIM-COMBINATORICS-0060", "title": "Solved by semi-random triangle-free graph packings", "statement": "Conjecture: $s_r(3) = \\Theta(r^2 \\log r)$\n\nRemark: This would follow from the extension of the triangle-free\nprocess to $r$ colors.", "original_statement": "Conjecture: $s_r(3) = \\Theta(r^2 \\log r)$\n\nRemark: This would follow from the extension of the triangle-free\nprocess to $r$ colors.", "clean_statement": "Conjecture: $s_r(3) = \\Theta(r^2 \\log r)$\n\nRemark: This would follow from the extension of the triangle-free\nprocess to $r$ colors.", "statement_status": "exact", "statement_verification": "The exact AIM record (Graph Ramsey Theory workshop, section \"Ramsey minimal graphs\", problem 13) states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Ramsey minimal graphs\nSource item: 13\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[59]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture: $s_r(3) = \\\\Theta(r^2 \\\\log r)$\\n\\nRemark: This would follow from the extension of the triangle-free\\nprocess to $r$ colors.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0060", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM conjecture s_r(3)=Theta(r^2 log r), recovered as s_r(K_3)=Theta(r^2 log r), was fully resolved by Guo and Warnke. FGLPS proved the matching lower order and the identity s_r(K_3)=P_r(2); Guo--Warnke constructed Theta(sqrt(n/log n)) pairwise edge-disjoint triangle-free graphs on n common vertices, each with independence number O(sqrt(n log n)). Taking n=Theta(r^2 log r) supplies r layers with alphasum_i alpha(G_i). Consequently, for every fixed eta in (0,1), Guo--Warnke yields an O_eta(r^2 log r)-vertex certificate whose strong-coloring property survives deletion of every set of fewer than eta times |V| vertices." }, { "id": 20000936, "problem_number": "AIM-COMBINATORICS-0061", "title": "A linear worst-case lower bound for Ramsey numbers of rectilinear knots", "statement": "Ramsey numbers of knots\n\nDavid Conlon\n\nTheorem. (Negami) Fix a knot $K$. If $n$ if large enough, any\nlinear spatial embedding of $K_n$ contains $K$.\n\nLinear spatial embedding of a graph means placing the vertices of the graph in $\\mathbb{R}^3$ and drawing edges as straight lines between the vertices.\n\n``Contains'' would mean either\n\n(i) a cycle homeomorphic to $K$, or\n\n(ii) a cycle isotopic to $K$\n\nThe distinction is that (ii) does not allow reflections.\n\nKnown: Any\nlinear spatial embedding of $K_7$ contains a homeomorphic copy of a\ntrefoil knot.\n\nQuestion: If $K$ has $k$ crossings, how big does $n = n(k)$\nhave to be?\n\nKnown:\n(Negami) at most 5-fold exponential\n\n(Conlon--Fox) $n(k) \\le 2^{2^{ck}}$", "original_statement": "Ramsey numbers of knots\n\nDavid Conlon\n\nTheorem. (Negami) Fix a knot $K$. If $n$ if large enough, any\nlinear spatial embedding of $K_n$ contains $K$.\n\nLinear spatial embedding of a graph means placing the vertices of the graph in $\\mathbb{R}^3$ and drawing edges as straight lines between the vertices.\n\n``Contains'' would mean either \n\n(i) a cycle homeomorphic to $K$, or \n\n(ii) a cycle isotopic to $K$ \n\nThe distinction is that (ii) does not allow reflections.\n\nKnown: Any\nlinear spatial embedding of $K_7$ contains a homeomorphic copy of a\ntrefoil knot.\n\nQuestion: If $K$ has $k$ crossings, how big does $n = n(k)$\nhave to be?\n\nKnown:\n(Negami) at most 5-fold exponential\n\n(Conlon--Fox) $n(k) \\le 2^{2^{ck}}$", "clean_statement": "Ramsey numbers of knots\n\nDavid Conlon\n\nTheorem. (Negami) Fix a knot $K$. If $n$ if large enough, any\nlinear spatial embedding of $K_n$ contains $K$.\n\nLinear spatial embedding of a graph means placing the vertices of the graph in $\\mathbb{R}^3$ and drawing edges as straight lines between the vertices.\n\n``Contains'' would mean either\n\n(i) a cycle homeomorphic to $K$, or\n\n(ii) a cycle isotopic to $K$\n\nThe distinction is that (ii) does not allow reflections.\n\nKnown: Any\nlinear spatial embedding of $K_7$ contains a homeomorphic copy of a\ntrefoil knot.\n\nQuestion: If $K$ has $k$ crossings, how big does $n = n(k)$\nhave to be?\n\nKnown:\n(Negami) at most 5-fold exponential\n\n(Conlon--Fox) $n(k) \\le 2^{2^{ck}}$", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Ramsey numbers of knots\nSource item: 14\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[60]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Ramsey numbers of knots\\n\\nDavid Conlon\\n\\nTheorem. (Negami) Fix a knot $K$. If $n$ if large enough, any\\nlinear spatial embedding of $K_n$ contains $K$.\\n\\nLinear spatial embedding of a graph means placing the vertices of the graph in $\\\\mathbb{R}^3$ and drawing edges as straight lines between the vertices.\\n\\n``Contains'' would mean either \\n\\n(i) a cycle homeomorphic to $K$, or \\n\\n(ii) a cycle isotopic to $K$ \\n\\nThe distinction is that (ii) does not allow reflections.\\n\\nKnown: Any\\nlinear spatial embedding of $K_7$ contains a homeomorphic copy of a\\ntrefoil knot.\\n\\nQuestion: If $K$ has $k$ crossings, how big does $n = n(k)$\\nhave to be?\\n\\nKnown:\\n(Negami) at most 5-fold exponential\\n\\n(Conlon--Fox) $n(k) \\\\le 2^{2^{ck}}$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0061", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After separating reflection-allowing homeomorphism containment from fixed-handed ambient-isotopy containment, and exact from at-most crossing profiles, every one of the four worst-case profiles satisfies W(k) >= k + floor(k/3) + 3 for every k >= 3. The witness B_k is an explicit connected sum of q=floor(k/3) alternating bridge-two knots with crossing totals adjusted by a figure-eight or 5_2 factor. Johnson's cyclic-polytope obstruction R(K) >= arc-index(K) + bridge(K), together with arc-index(B_k)=k+2 and bridge(B_k)=q+1, proves the claim. The broad gap to the workshop's double-exponential upper bound remains open.\n\nCandidate contribution (explicit_profile_corollary; novelty confidence low): Candidate novelty: for every integer k >= 3, the exact-crossing and at-most-crossing worst-case knot Ramsey profiles, under both homeomorphism-up-to-reflection and fixed-handed ambient-isotopy containment, are at least k + floor(k/3) + 3." }, { "id": 20000937, "problem_number": "AIM-COMBINATORICS-0062", "title": "Polynomial bounds for rectilinear knot Ramsey numbers", "statement": "Question:\nIs it polynomial in $k$?\n\nLower bound: $n(K) \\ge \\sqrt{\\nu(K)}$, where $\\nu(K)$ is the crossing\nnumber of a knot.\n\nWe can also ask the same question for links.\nFor the simplest nontrivial link of two unknots (i.e., Hopf link), the answer is 6.", "original_statement": "Question:\nIs it polynomial in $k$?\n\nLower bound: $n(K) \\ge \\sqrt{\\nu(K)}$, where $\\nu(K)$ is the crossing\nnumber of a knot.\n\nWe can also ask the same question for links.\nFor the simplest nontrivial link of two unknots (i.e., Hopf link), the answer is 6.", "clean_statement": "Question:\nIs it polynomial in $k$?\n\nLower bound: $n(K) \\ge \\sqrt{\\nu(K)}$, where $\\nu(K)$ is the crossing\nnumber of a knot.\n\nWe can also ask the same question for links.\nFor the simplest nontrivial link of two unknots (i.e., Hopf link), the answer is 6.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Ramsey numbers of knots\nSource item: 15\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[61]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question:\\nIs it polynomial in $k$?\\n\\nLower bound: $n(K) \\\\ge \\\\sqrt{\\\\nu(K)}$, where $\\\\nu(K)$ is the crossing\\nnumber of a knot.\\n\\nWe can also ask the same question for links.\\nFor the simplest nontrivial link of two unknots (i.e., Hopf link), the answer is 6.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0062", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted link extension cannot be bounded by crossing number alone: every link L satisfies R(L) >= s(L) >= 3 mu(L), so the nontrivial split links consisting of a Hopf link and mu-2 split unknots have crossing number 2 but Ramsey number at least 3 mu. For knots, the published arc-index and bridge-number inequality gives the infinite linear lower family R(T(2,q)) >= q+4 for odd q >= 3, while the uniform polynomial upper-bound question remains open in the literature checked.\n\nCandidate contribution (obstruction; novelty confidence low): There is no function F of crossing number alone with R(L) <= F(c(L)) for all links of unrestricted component number; even the nontrivial links H disjoint-union U_(mu-2) have c=2 and R >= 3 mu." }, { "id": 20000938, "problem_number": "AIM-COMBINATORICS-0063", "title": "Fixed-polynomial semi-algebraic Ramsey bounds and unbounded-degree universality", "statement": "Semi-algebraic Ramsey numbers\n\nAndrew Suk\n\nSuppose that that for each $n$ we have a $k$-uniform hypergraph $H_n$\nwhere $V(H_n)$ is a set of $n$ points in the plane, say $V(H_n) =\n\\{v_1, \\dots, v_n\\} \\subseteq \\mathbb{R}^2$, and the edges are defined\nby a $2k$-variable polynomial $f$, namely\n\\[\n\\{v_{i_1}, \\dots, v_{i_k}\\} \\in E(H_n) \\Leftrightarrow\nf(v_{i_1},\\dots, v_{i_k}) \\ge 0, \\qquad i_1 < i_2 < \\dots < i_k.\n\\]\n\nQuestion: What is the size of the largest\nhomogeneous set (i.e., clique or independent set) that is guaranteed to\nexist in every such\n$H_n$?\n\nLet $r_f(n) = \\min_{H_n} \\max\\{\\alpha(H_n),\n\\omega(H_n)\\}$ denote the above quantity.\n\nKnown: From bounds on Ramsey numbers we have $r_f(n) \\ge c\n\\underbrace{\\log \\dots \\log}_{k-1 \\text{ times}} n$.", "original_statement": "Semi-algebraic Ramsey numbers\n\nAndrew Suk\n\nSuppose that that for each $n$ we have a $k$-uniform hypergraph $H_n$\nwhere $V(H_n)$ is a set of $n$ points in the plane, say $V(H_n) =\n\\{v_1, \\dots, v_n\\} \\subseteq \\mathbb{R}^2$, and the edges are defined\nby a $2k$-variable polynomial $f$, namely\n\\[\n\\{v_{i_1}, \\dots, v_{i_k}\\} \\in E(H_n) \\Leftrightarrow\nf(v_{i_1},\\dots, v_{i_k}) \\ge 0, \\qquad i_1 < i_2 < \\dots < i_k.\n\\]\n\nQuestion: What is the size of the largest\nhomogeneous set (i.e., clique or independent set) that is guaranteed to\nexist in every such\n$H_n$?\n\nLet $r_f(n) = \\min_{H_n} \\max\\{\\alpha(H_n),\n\\omega(H_n)\\}$ denote the above quantity.\n\nKnown: From bounds on Ramsey numbers we have $r_f(n) \\ge c\n\\underbrace{\\log \\dots \\log}_{k-1 \\text{ times}} n$.", "clean_statement": "Semi-algebraic Ramsey numbers\n\nAndrew Suk\n\nSuppose that that for each $n$ we have a $k$-uniform hypergraph $H_n$\nwhere $V(H_n)$ is a set of $n$ points in the plane, say $V(H_n) =\n\\{v_1, \\dots, v_n\\} \\subseteq \\mathbb{R}^2$, and the edges are defined\nby a $2k$-variable polynomial $f$, namely\n\\[\n\\{v_{i_1}, \\dots, v_{i_k}\\} \\in E(H_n) \\Leftrightarrow\nf(v_{i_1},\\dots, v_{i_k}) \\ge 0, \\qquad i_1 < i_2 < \\dots < i_k.\n\\]\n\nQuestion: What is the size of the largest\nhomogeneous set (i.e., clique or independent set) that is guaranteed to\nexist in every such\n$H_n$?\n\nLet $r_f(n) = \\min_{H_n} \\max\\{\\alpha(H_n),\n\\omega(H_n)\\}$ denote the above quantity.\n\nKnown: From bounds on Ramsey numbers we have $r_f(n) \\ge c\n\\underbrace{\\log \\dots \\log}_{k-1 \\text{ times}} n$.", "statement_status": "exact", "statement_verification": "The source asks the following question, attributed to Andrew Suk. For each \\(n\\), let \\(H_n\\) be a \\(k\\)-uniform hypergraph whose vertices are labelled points \\[ P=(v_1,\\ldots,v_n)\\in(\\mathbb R^2)^n, \\] and, for \\(i_1<\\cdots n^{(log n)^{1/3-o(1)}}, but their theorem does not settle the literal question for one fixed planar polynomial f. A proved universal parameter-elimination lemma shows that every complexity-(d,D,m) ordered semialgebraic hypergraph embeds into one fixed description in dimension d + m*binom(rd+D,D) + 3^m, using m + 3^m fixed polynomials of degree at most D+1. Hence the Jin-Tomon lower bound does transfer to one fixed higher-dimensional multi-polynomial Boolean description for all n, while the planar single-inequality problem remains open. The 4-uniform motivation is made exact by an explicit degree-eight polynomial whose (5,n) Ramsey number on general-position point sets equals the Erdos-Szekeres number ES(n).\n\nCandidate contribution (lemma; novelty confidence low): For every fixed (r,d,D,m), there is one explicit fixed semialgebraic description Lambda_univ in dimension d + m*binom(rd+D,D) + 3^m, with m + 3^m polynomials of degree at most D+1, that realizes every ordered complexity-(d,D,m) hypergraph on a constant-parameter slice and therefore satisfies R_r^{Lambda_univ}(s,n) >= R_r^{d,D,m}(s,n)." }, { "id": 20000941, "problem_number": "AIM-COMBINATORICS-0066", "title": "Quantifier-resolved off-diagonal semi-algebraic graph Ramsey numbers", "statement": "Question: (Fox) Is $R_f^{(2)}(3,n) = n^{1 + o(1)}$?", "original_statement": "Question: (Fox) Is $R_f^{(2)}(3,n) = n^{1 + o(1)}$?", "clean_statement": "Question: (Fox) Is $R_f^{(2)}(3,n) = n^{1 + o(1)}$?", "statement_status": "exact", "statement_verification": "The complete extracted record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Semi-algebraic Ramsey numbers\nSource item: 19\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[65]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question: (Fox) Is $R_f^{(2)}(3,n) = n^{1 + o(1)}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0066", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The exponent-one assertion has different answers under the relevant quantifiers. Tomon's theorem gives R(3,n) between n and n times a fixed polylogarithm for every fixed-complexity semilinear class, hence n^{1+o(1)}. In the full bounded-complexity semi-algebraic class it is false: the Suk-Tomon Hasse construction can be written using one fixed symmetric Boolean description in R^4 with degree-two tests and gives R(3,n)=Omega(n^{4/3}). This does not automatically settle the narrower planar single-polynomial formulation. Independently, for d>=1 and every fixed additive threshold f(x,y)=u(x)+u(y)-tau on R^d, every triangle-free realization is a star plus isolates, so R_f(3,n)<=n+1, and a strict two-value realizability condition gives the exact value n+1.\n\nCandidate contribution (theorem; novelty confidence low): Let d>=1, let u:R^d->R be continuous, let tau be real, and put f(x,y)=u(x)+u(y)-tau. Every triangle-free finite f-realization has all edges incident with a single maximum-weight vertex, hence R_f(3,n)<=n+1. If w=u-tau/2 has points c,q with w(q)<0=2." }, { "id": 20000942, "problem_number": "AIM-COMBINATORICS-0067", "title": "A defect-graph certificate for large crossing families", "statement": "Large cliques in straight edge intersection graphs\n\nJ\\'anos Pach\n\nQuestion: If $K_n$ is drawn in the plane with straight line\nedges, can we always find $\\Omega(n)$ pairwise crossing edges?\n\nKnown: (Aronov--Erd\\H{os}--Kleitman--Pach) Can get\n$\\Omega(\\sqrt{n})$.\n\nResults: (Csaba--Fox--Pach) For some $\\epsilon > 0$, if edges\nare curves and each pair crosses $O(1)$ times, then we get\n$\\Omega(n^{\\epsilon})$ pairwise crossing curves.", "original_statement": "Large cliques in straight edge intersection graphs\n\nJ\\'anos Pach\n\nQuestion: If $K_n$ is drawn in the plane with straight line\nedges, can we always find $\\Omega(n)$ pairwise crossing edges?\n\nKnown: (Aronov--Erd\\H{os}--Kleitman--Pach) Can get\n$\\Omega(\\sqrt{n})$.\n\nResults: (Csaba--Fox--Pach) For some $\\epsilon > 0$, if edges\nare curves and each pair crosses $O(1)$ times, then we get\n$\\Omega(n^{\\epsilon})$ pairwise crossing curves.", "clean_statement": "Large cliques in straight edge intersection graphs\n\nJ\\'anos Pach\n\nQuestion: If $K_n$ is drawn in the plane with straight line\nedges, can we always find $\\Omega(n)$ pairwise crossing edges?\n\nKnown: (Aronov--Erd\\H{os}--Kleitman--Pach) Can get\n$\\Omega(\\sqrt{n})$.\n\nResults: (Csaba--Fox--Pach) For some $\\epsilon > 0$, if edges\nare curves and each pair crosses $O(1)$ times, then we get\n$\\Omega(n^{\\epsilon})$ pairwise crossing curves.", "statement_status": "exact", "statement_verification": "The canonical record is item 20 of the AIM workshop list “Graph Ramsey theory,” attributed there to János Pach:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Large cliques in straight edge intersection graphs\nSource item: 20\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[66]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Large cliques in straight edge intersection graphs\\n\\nJ\\\\'anos Pach\\n\\nQuestion: If $K_n$ is drawn in the plane with straight line\\nedges, can we always find $\\\\Omega(n)$ pairwise crossing edges?\\n\\nKnown: (Aronov--Erd\\\\H{os}--Kleitman--Pach) Can get\\n$\\\\Omega(\\\\sqrt{n})$.\\n\\nResults: (Csaba--Fox--Pach) For some $\\\\epsilon > 0$, if edges\\nare curves and each pair crosses $O(1)$ times, then we get\\n$\\\\Omega(n^{\\\\epsilon})$ pairwise crossing curves.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0067", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For disjoint point blocks A and B, form a defect graph on A by joining two points when their full line meets conv(B), and define the symmetric graph on B. The maximum crossing-family size is at least the minimum of the two independence numbers. Caro--Wei therefore gives an explicit bound in terms of the two defect edge counts; in particular, two linear-size blocks with bounded average defect degree force a linear crossing family. A secondary proposition proves a linear bound for point sets of bounded convex-hull onion depth.\n\nCandidate contribution (reduction; novelty confidence low): The explicit two-sided defect-graph inequality T(P) >= min(alpha(D_A(B)), alpha(D_B(A))) and its Caro--Wei edge-count consequence provide a checkable approximate-mutual-avoidance certificate for crossing families." }, { "id": 20000943, "problem_number": "AIM-COMBINATORICS-0068", "title": "Hall ratio three: current coloring bound and a nested critical-core certificate", "statement": "Chromatic number of graphs with everywhere high independence number\n\nJacque Verstraete\n\nQuestion: Is there an $\\epsilon > 0$ so that $\\chi(G) \\le\nd^{1-\\epsilon}$ for every graph $G$ of maximum degree $d$ such that\n$\\alpha(H) \\geq \\frac13 |V(H)|$ for every subgraph $H \\subseteq G$?\n\nMotivation: Kneser graphs $KG_{3n,n}$ satisfies the constraint\nand has chromatic number $\\Theta(\\log d)$.\n\nNote that any graph satisfying the constraint is $K_4$-free, and it is\nknown (Shearer) that $\\alpha(G) \\ge \\frac{cn\\log d}{d \\log\\log d}$ if\n$G$ is $K_4$-free, and hence $\\chi(G) \\le \\frac{n}{\\alpha(G)} \\le\n\\frac{d \\log\\log d}{c \\log d}$.\n\n\\section{Burr--Erd\\H{o}s for hypergraphs}\n\nJacob Fox\n\nA graph $G$ is $d$-degenerate if every $H \\subseteq G$ has\n$\\delta(H)\\le d$.", "original_statement": "Chromatic number of graphs with everywhere high independence number\n\nJacque Verstraete\n\nQuestion: Is there an $\\epsilon > 0$ so that $\\chi(G) \\le\nd^{1-\\epsilon}$ for every graph $G$ of maximum degree $d$ such that\n$\\alpha(H) \\geq \\frac13 |V(H)|$ for every subgraph $H \\subseteq G$?\n\nMotivation: Kneser graphs $KG_{3n,n}$ satisfies the constraint\nand has chromatic number $\\Theta(\\log d)$.\n\nNote that any graph satisfying the constraint is $K_4$-free, and it is\nknown (Shearer) that $\\alpha(G) \\ge \\frac{cn\\log d}{d \\log\\log d}$ if\n$G$ is $K_4$-free, and hence $\\chi(G) \\le \\frac{n}{\\alpha(G)} \\le\n\\frac{d \\log\\log d}{c \\log d}$.\n\n\\section{Burr--Erd\\H{o}s for hypergraphs}\n\nJacob Fox\n\nA graph $G$ is $d$-degenerate if every $H \\subseteq G$ has\n$\\delta(H)\\le d$.", "clean_statement": "Chromatic number of graphs with everywhere high independence number\n\nJacque Verstraete\n\nQuestion: Is there an $\\epsilon > 0$ so that $\\chi(G) \\le\nd^{1-\\epsilon}$ for every graph $G$ of maximum degree $d$ such that\n$\\alpha(H) \\geq \\frac13 |V(H)|$ for every subgraph $H \\subseteq G$?\n\nMotivation: Kneser graphs $KG_{3n,n}$ satisfies the constraint\nand has chromatic number $\\Theta(\\log d)$.\n\nNote that any graph satisfying the constraint is $K_4$-free, and it is\nknown (Shearer) that $\\alpha(G) \\ge \\frac{cn\\log d}{d \\log\\log d}$ if\n$G$ is $K_4$-free, and hence $\\chi(G) \\le \\frac{n}{\\alpha(G)} \\le\n\\frac{d \\log\\log d}{c \\log d}$.\n\n\\section{Burr--Erd\\H{o}s for hypergraphs}\n\nJacob Fox\n\nA graph $G$ is $d$-degenerate if every $H \\subseteq G$ has\n$\\delta(H)\\le d$.", "statement_status": "exact", "statement_verification": "The mathematical question in the source record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Chromatic number of graphs with everywhere high independence number\nSource item: 21\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[67]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Chromatic number of graphs with everywhere high independence number\\n\\nJacque Verstraete\\n\\nQuestion: Is there an $\\\\epsilon > 0$ so that $\\\\chi(G) \\\\le\\nd^{1-\\\\epsilon}$ for every graph $G$ of maximum degree $d$ such that\\n$\\\\alpha(H) \\\\geq \\\\frac13 |V(H)|$ for every subgraph $H \\\\subseteq G$?\\n\\nMotivation: Kneser graphs $KG_{3n,n}$ satisfies the constraint\\nand has chromatic number $\\\\Theta(\\\\log d)$.\\n\\nNote that any graph satisfying the constraint is $K_4$-free, and it is\\nknown (Shearer) that $\\\\alpha(G) \\\\ge \\\\frac{cn\\\\log d}{d \\\\log\\\\log d}$ if\\n$G$ is $K_4$-free, and hence $\\\\chi(G) \\\\le \\\\frac{n}{\\\\alpha(G)} \\\\le\\n\\\\frac{d \\\\log\\\\log d}{c \\\\log d}$.\\n\\n\\\\section{Burr--Erd\\\\H{o}s for hypergraphs}\\n\\nJacob Fox\\n\\nA graph $G$ is $d$-degenerate if every $H \\\\subseteq G$ has\\n$\\\\delta(H)\\\\le d$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0068", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The hereditary independence condition is exactly Hall ratio at most 3. Davies--Kang--Pirot--Sereni therefore give the currently strongest directly applicable bound chi(G) <= (K(3)+o(1)) Delta/log Delta, which does not yield a fixed power saving. In addition, every k-chromatic graph under the hypothesis contains a nested chain of j-critical subgraphs F_j with delta(F_j) >= j-1 and |F_{j-1}| <= floor(2|F_j|/3); consequently its critical witness has at least a_k vertices, where a_1=1 and a_j=ceil(3a_{j-1}/2), hence at least (3/2)^(k-1) vertices.\n\nCandidate contribution (reduction; novelty confidence low): Every k-chromatic graph of Hall ratio at most 3 admits an exact nested critical chain F_1 subset ... subset F_k in which F_j is j-critical, delta(F_j) >= j-1, and a maximum independent-set deletion leaves a (j-1)-chromatic graph containing F_{j-1}, with |F_{j-1}| <= floor(2|F_j|/3); thus |F_k| >= a_k for a_1=1 and a_j=ceil(3a_{j-1}/2)." }, { "id": 20000944, "problem_number": "AIM-COMBINATORICS-0069", "title": "Burr--Erdos, skeletal degeneracy, and a sharp hereditary edge envelope", "statement": "Conjecture: (Burr--Erd\\H{o}s) $r(G,G) \\le c(d) n$ if $G$ is\nan $n$-vertex $d$-degenerate graph.\n\nKnown: (Kostochka--Sudakov) $r(G,G) \\le n^{1+o_d(1)}$.\n\nIf $G$ is a 3-uniform $d$-degenerate hypergraph, the statement can be\nfalse. There is a 1-degenerate 4-uniform $G$ such that $r_4(G,G) \\ge\n2^{\\Omega(n^{1/3})}$.\n\nPerhaps we can redefine degeneracy for $k$-uniform hypergraphs: call a\n$k$-uniform hypergraph $d$-degenerate if the 2-graph obtained by\nreplacing every edge by a graph clique $K_k^{(2)}$ is $d$-degenerate\nas a graph.", "original_statement": "Conjecture: (Burr--Erd\\H{o}s) $r(G,G) \\le c(d) n$ if $G$ is\nan $n$-vertex $d$-degenerate graph.\n\nKnown: (Kostochka--Sudakov) $r(G,G) \\le n^{1+o_d(1)}$.\n\nIf $G$ is a 3-uniform $d$-degenerate hypergraph, the statement can be\nfalse. There is a 1-degenerate 4-uniform $G$ such that $r_4(G,G) \\ge\n2^{\\Omega(n^{1/3})}$.\n\nPerhaps we can redefine degeneracy for $k$-uniform hypergraphs: call a\n$k$-uniform hypergraph $d$-degenerate if the 2-graph obtained by\nreplacing every edge by a graph clique $K_k^{(2)}$ is $d$-degenerate\nas a graph.", "clean_statement": "Conjecture: (Burr--Erd\\H{o}s) $r(G,G) \\le c(d) n$ if $G$ is\nan $n$-vertex $d$-degenerate graph.\n\nKnown: (Kostochka--Sudakov) $r(G,G) \\le n^{1+o_d(1)}$.\n\nIf $G$ is a 3-uniform $d$-degenerate hypergraph, the statement can be\nfalse. There is a 1-degenerate 4-uniform $G$ such that $r_4(G,G) \\ge\n2^{\\Omega(n^{1/3})}$.\n\nPerhaps we can redefine degeneracy for $k$-uniform hypergraphs: call a\n$k$-uniform hypergraph $d$-degenerate if the 2-graph obtained by\nreplacing every edge by a graph clique $K_k^{(2)}$ is $d$-degenerate\nas a graph.", "statement_status": "exact", "statement_verification": "The canonical AIM record (Graph Ramsey theory workshop, item 22, zero-based source index 68) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Chromatic number of graphs with everywhere high independence number\nSource item: 22\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[68]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture: (Burr--Erd\\\\H{o}s) $r(G,G) \\\\le c(d) n$ if $G$ is\\nan $n$-vertex $d$-degenerate graph.\\n\\nKnown: (Kostochka--Sudakov) $r(G,G) \\\\le n^{1+o_d(1)}$.\\n\\nIf $G$ is a 3-uniform $d$-degenerate hypergraph, the statement can be\\nfalse. There is a 1-degenerate 4-uniform $G$ such that $r_4(G,G) \\\\ge\\n2^{\\\\Omega(n^{1/3})}$.\\n\\nPerhaps we can redefine degeneracy for $k$-uniform hypergraphs: call a\\n$k$-uniform hypergraph $d$-degenerate if the 2-graph obtained by\\nreplacing every edge by a graph clique $K_k^{(2)}$ is $d$-degenerate\\nas a graph.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0069", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Lee's 2017 theorem solves the graph Burr--Erdos conjecture, and Fox--Sankar--Simkin--Tidor--Zhou's Theorem 1.5 proves the hypergraph analogue for exactly the clique-shadow notion proposed by AIM, now called skeletal degeneracy. As a self-contained proved contribution, if a nonempty simple k-uniform hypergraph has skeletal degeneracy s, then every m-vertex induced subhypergraph has at most binom(s,k)+(m-s)binom(s,k-1) edges for m at least s (and at most binom(m,k) for m 0$?\n\nFurthermore,\nis it true that for any graph $G$, edges labeled $1, 2, \\dots,\n|E(G)|$, we always have $\\sum_{v \\in V(G)} t(v) \\ge |E(G)|$?\n\nKnown: (Calderbank--Chung--Sturtevant 1984).\nThere exists a labeling with $\\max_v t(v) \\le n/2$.\n\nKnown: (Graham--Kleitman 1973) $\\max_v t(v) \\ge (1+o(1))\n\\sqrt{n}$ for $K_n$\n\nRecently claimed to be improved to $\\ge (\\sqrt{2} + o(1)) \\sqrt{n}$ by a\ngroup of graduate students.", "original_statement": "Increasing paths in edge labelings\n\nRon Graham\n\nLabel the edges of $K_n$ with numbers $1, 2, \\dots,\n\\binom{n}{2}$. For each $v \\in V(K_n)$, let $t(v)$ be the length of\nthe longest increasing path from $v$.\n\nQuestion: Is it true that $\\max_v t(v) \\ge cn$ for some $c > 0$?\n\nFurthermore,\nis it true that for any graph $G$, edges labeled $1, 2, \\dots,\n|E(G)|$, we always have $\\sum_{v \\in V(G)} t(v) \\ge |E(G)|$?\n\nKnown: (Calderbank--Chung--Sturtevant 1984).\nThere exists a labeling with $\\max_v t(v) \\le n/2$.\n\nKnown: (Graham--Kleitman 1973) $\\max_v t(v) \\ge (1+o(1))\n\\sqrt{n}$ for $K_n$\n\nRecently claimed to be improved to $\\ge (\\sqrt{2} + o(1)) \\sqrt{n}$ by a\ngroup of graduate students.", "clean_statement": "Increasing paths in edge labelings\n\nRon Graham\n\nLabel the edges of $K_n$ with numbers $1, 2, \\dots,\n\\binom{n}{2}$. For each $v \\in V(K_n)$, let $t(v)$ be the length of\nthe longest increasing path from $v$.\n\nQuestion: Is it true that $\\max_v t(v) \\ge cn$ for some $c > 0$?\n\nFurthermore,\nis it true that for any graph $G$, edges labeled $1, 2, \\dots,\n|E(G)|$, we always have $\\sum_{v \\in V(G)} t(v) \\ge |E(G)|$?\n\nKnown: (Calderbank--Chung--Sturtevant 1984).\nThere exists a labeling with $\\max_v t(v) \\le n/2$.\n\nKnown: (Graham--Kleitman 1973) $\\max_v t(v) \\ge (1+o(1))\n\\sqrt{n}$ for $K_n$\n\nRecently claimed to be improved to $\\ge (\\sqrt{2} + o(1)) \\sqrt{n}$ by a\ngroup of graduate students.", "statement_status": "exact", "statement_verification": "The AIM record, attributed to Ron Graham, gives every edge of a graph a distinct label. For the complete graph it asks whether there is an absolute constant \\(c>0\\) such that \\[ \\max_{v\\in V(K_n)}t(v)\\ge cn, \\] where \\(t(v)\\) is the maximum length of an increasing path starting at \\(v\\). It then asks, for every edge-ordered graph \\(G\\), whether \\[ \\sum_{v\\in V(G)}t(v)\\ge |E(G)|. \\tag{1.1} \\]", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Increasing paths in edge labelings\nSource item: 29\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[75]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Increasing paths in edge labelings\\n\\nRon Graham\\n\\nLabel the edges of $K_n$ with numbers $1, 2, \\\\dots,\\n\\\\binom{n}{2}$. For each $v \\\\in V(K_n)$, let $t(v)$ be the length of\\nthe longest increasing path from $v$.\\n\\nQuestion: Is it true that $\\\\max_v t(v) \\\\ge cn$ for some $c > 0$?\\n\\nFurthermore,\\nis it true that for any graph $G$, edges labeled $1, 2, \\\\dots,\\n|E(G)|$, we always have $\\\\sum_{v \\\\in V(G)} t(v) \\\\ge |E(G)|$?\\n\\nKnown: (Calderbank--Chung--Sturtevant 1984).\\nThere exists a labeling with $\\\\max_v t(v) \\\\le n/2$.\\n\\nKnown: (Graham--Kleitman 1973) $\\\\max_v t(v) \\\\ge (1+o(1))\\n\\\\sqrt{n}$ for $K_n$\\n\\nRecently claimed to be improved to $\\\\ge (\\\\sqrt{2} + o(1)) \\\\sqrt{n}$ by a\\ngroup of graduate students.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0076", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For simple paths whose length counts edges, the current best published complete-graph lower bound is f(K_n) >= n/2^{O(sqrt(log n log log n))}=n^{1-o(1)}, while a constant-linear bound and the stronger sum_v t(v) >= |E(G)| question remain open. Two proved certificates give partial progress: S(G) >= 2a_lambda(G), where a_lambda counts edges contained in no increasing closed trail, and S(G) >= n_+ + sum_P binom(length(P),2) for every vertex-disjoint family of increasing paths. Hence the sum conjecture holds if at least half the edges are temporally acyclic or if the path-packing credit is at least |E|-n_+; in particular every forest satisfies the sharp stronger bound S(G) >= 2|E(G)|.\n\nCandidate contribution (lemma; novelty confidence low): In every finite simple edge-ordered graph, chronological loop erasure of the standard token-switching trails proves sum_v t(v) >= 2a_lambda(G), with a_lambda(G) the number of edges lying in no increasing closed trail; independently, every vertex-disjoint increasing-path packing P proves sum_v t(v) >= n_+ + sum_{P in P} binom(length(P),2)." }, { "id": 20000952, "problem_number": "AIM-COMBINATORICS-0077", "title": "A reciprocity bound for long directed paths", "statement": "Directed paths in Eulerian digraphs\n\nJacques Verstraete\n\nThe following is a special case of a conjecture of Bollob\\'{a}s and Scott on weighted paths in directed graphs (see: A proof of a conjecture of Bondy concerning paths in weighted digraphs. J.\\ Combin.\\ Theory Ser.\\ B 66 (1996), no. 2, 283--292):\n\nQuestion: Does every Eulerian digraph of average degree $d$\nhave a directed path of length $\\Omega(d)$?\n\nGetting $\\Omega(\\sqrt{d})$ is not too hard.", "original_statement": "Directed paths in Eulerian digraphs\n\nJacques Verstraete\n\nThe following is a special case of a conjecture of Bollob\\'{a}s and Scott on weighted paths in directed graphs (see: A proof of a conjecture of Bondy concerning paths in weighted digraphs. J.\\ Combin.\\ Theory Ser.\\ B 66 (1996), no. 2, 283--292):\n\nQuestion: Does every Eulerian digraph of average degree $d$\nhave a directed path of length $\\Omega(d)$?\n\nGetting $\\Omega(\\sqrt{d})$ is not too hard.", "clean_statement": "Directed paths in Eulerian digraphs\n\nJacques Verstraete\n\nThe following is a special case of a conjecture of Bollob\\'{a}s and Scott on weighted paths in directed graphs (see: A proof of a conjecture of Bondy concerning paths in weighted digraphs. J.\\ Combin.\\ Theory Ser.\\ B 66 (1996), no. 2, 283--292):\n\nQuestion: Does every Eulerian digraph of average degree $d$\nhave a directed path of length $\\Omega(d)$?\n\nGetting $\\Omega(\\sqrt{d})$ is not too hard.", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Directed paths in Eulerian digraphs\nSource item: 30\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[76]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Directed paths in Eulerian digraphs\\n\\nJacques Verstraete\\n\\nThe following is a special case of a conjecture of Bollob\\\\'{a}s and Scott on weighted paths in directed graphs (see: A proof of a conjecture of Bondy concerning paths in weighted digraphs. J.\\\\ Combin.\\\\ Theory Ser.\\\\ B 66 (1996), no. 2, 283--292):\\n\\nQuestion: Does every Eulerian digraph of average degree $d$\\nhave a directed path of length $\\\\Omega(d)$?\\n\\nGetting $\\\\Omega(\\\\sqrt{d})$ is not too hard.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0077", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite simple loopless digraph D, let b(D) be the number of unordered vertex pairs supporting both opposite arcs, and let rho(D)=2b(D)/|A(D)|. The longest directed vertex-simple path satisfies ell(D) >= 2b(D)/|V(D)| = rho(D)d, where d=|A(D)|/|V(D)|. Thus every Eulerian family with reciprocal-arc density bounded below satisfies the requested linear bound, every bidirected case satisfies ell(D) >= d, and any asymptotic counterexample family must have rho(D)=o(1). The proof uses the Erdos-Gallai undirected path theorem on the reciprocal graph. The general linear conjecture remains open; the best general bound located is d/(log d+1).\n\nCandidate contribution (structural bound and reduction; novelty confidence low): The exact reciprocity-parameter inequality ell(D) >= rho(D)d reduces every possible asymptotic counterexample to an asymptotically oriented Eulerian family, meaning that only o(|A(D)|) arcs may lie in directed 2-cycles." }, { "id": 20000953, "problem_number": "AIM-COMBINATORICS-0078", "title": "Free-cut decomposition for ordered matchings versus triangles", "statement": "Ordered Ramsey numbers\n\nDavid Conlon, Jacob Fox, Choongbum Lee, and Benny Sudakov\n\nAn ordered graph on $N$ vertices is a graph whose vertices have been labeled with $\\{1,\\dots,N\\}$. We say that an ordered graph $G$ on $\\{1,\\dots,N\\}$ contains another ordered graph $H$ on $\\{1,\\dots,n\\}$ if $H$ occurs as a subgraph of $G$ with vertices appearing in the correct order.\n\nLet $r_<(H)$ denote the minimum $N$ such that every 2-edge coloring of $K_N$ contains a monochromatic ordered copy of $H$. Similarly for $r_<(H_1, H_2)$.\n\nQuestion: Does there exist $\\epsilon > 0$ such that $r_<(K_3,M) = O(n^{2-\\epsilon})$ for every ordered matching $M$ on $n$ vertices.\n\nKnown: $r_<(K_3, M) \\le r_< (K_3, K_n) = \\Theta(n^2/\\log n)$.\n\nKnown: There exists an ordered matching $M$ on $n$ vertices such that $r_<(K_3, M) = n^{4/3 + o(1)}$.", "original_statement": "Ordered Ramsey numbers\n\nDavid Conlon, Jacob Fox, Choongbum Lee, and Benny Sudakov\n\nAn ordered graph on $N$ vertices is a graph whose vertices have been labeled with $\\{1,\\dots,N\\}$. We say that an ordered graph $G$ on $\\{1,\\dots,N\\}$ contains another ordered graph $H$ on $\\{1,\\dots,n\\}$ if $H$ occurs as a subgraph of $G$ with vertices appearing in the correct order.\n\nLet $r_<(H)$ denote the minimum $N$ such that every 2-edge coloring of $K_N$ contains a monochromatic ordered copy of $H$. Similarly for $r_<(H_1, H_2)$.\n\nQuestion: Does there exist $\\epsilon > 0$ such that $r_<(K_3,M) = O(n^{2-\\epsilon})$ for every ordered matching $M$ on $n$ vertices.\n\nKnown: $r_<(K_3, M) \\le r_< (K_3, K_n) = \\Theta(n^2/\\log n)$.\n\nKnown: There exists an ordered matching $M$ on $n$ vertices such that $r_<(K_3, M) = n^{4/3 + o(1)}$.", "clean_statement": "Ordered Ramsey numbers\n\nDavid Conlon, Jacob Fox, Choongbum Lee, and Benny Sudakov\n\nAn ordered graph on $N$ vertices is a graph whose vertices have been labeled with $\\{1,\\dots,N\\}$. We say that an ordered graph $G$ on $\\{1,\\dots,N\\}$ contains another ordered graph $H$ on $\\{1,\\dots,n\\}$ if $H$ occurs as a subgraph of $G$ with vertices appearing in the correct order.\n\nLet $r_<(H)$ denote the minimum $N$ such that every 2-edge coloring of $K_N$ contains a monochromatic ordered copy of $H$. Similarly for $r_<(H_1, H_2)$.\n\nQuestion: Does there exist $\\epsilon > 0$ such that $r_<(K_3,M) = O(n^{2-\\epsilon})$ for every ordered matching $M$ on $n$ vertices.\n\nKnown: $r_<(K_3, M) \\le r_< (K_3, K_n) = \\Theta(n^2/\\log n)$.\n\nKnown: There exists an ordered matching $M$ on $n$ vertices such that $r_<(K_3, M) = n^{4/3 + o(1)}$.", "statement_status": "exact", "statement_verification": "The canonical record is Question 31 in the AIM workshop list *Graph Ramsey theory*, section “Ordered Ramsey numbers,” attributed to David Conlon, Jacob Fox, Choongbum Lee, and Benny Sudakov. It asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Ordered Ramsey numbers\nSource item: 31\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[77]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Ordered Ramsey numbers\\n\\nDavid Conlon, Jacob Fox, Choongbum Lee, and Benny Sudakov\\n\\nAn ordered graph on $N$ vertices is a graph whose vertices have been labeled with $\\\\{1,\\\\dots,N\\\\}$. We say that an ordered graph $G$ on $\\\\{1,\\\\dots,N\\\\}$ contains another ordered graph $H$ on $\\\\{1,\\\\dots,n\\\\}$ if $H$ occurs as a subgraph of $G$ with vertices appearing in the correct order.\\n\\nLet $r_<(H)$ denote the minimum $N$ such that every 2-edge coloring of $K_N$ contains a monochromatic ordered copy of $H$. Similarly for $r_<(H_1, H_2)$.\\n\\nQuestion: Does there exist $\\\\epsilon > 0$ such that $r_<(K_3,M) = O(n^{2-\\\\epsilon})$ for every ordered matching $M$ on $n$ vertices.\\n\\nKnown: $r_<(K_3, M) \\\\le r_< (K_3, K_n) = \\\\Theta(n^2/\\\\log n)$.\\n\\nKnown: There exists an ordered matching $M$ on $n$ vertices such that $r_<(K_3, M) = n^{4/3 + o(1)}$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0078", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general uniform subquadratic question remains open in the primary literature checked. For an ordered perfect matching M on n vertices, decompose M at every cut crossed by no matching edge, and let v_i be the resulting block sizes and b=max v_i. Ordered-sum subadditivity and the classical triangle-versus-clique estimate give the proved bound R(M) <= C sum_i v_i^2/log(v_i+1) <= C n b/log(b+1). Hence the AIM conclusion holds uniformly whenever b <= n^(1-delta), with exponent 2-delta. An exact matched-block obstruction certificate is also proved for every avoiding coloring.\n\nCandidate contribution (structural_reduction; novelty confidence low): If M has free-cut block sizes v_1,...,v_t and largest block b, then R(M) <= C sum_i v_i^2/log(v_i+1) <= C n b/log(b+1); moreover, in every red-triangle-free, blue-M-free coloring and every partition into n nonempty consecutive host blocks, some pattern edge indexes a completely red pair of blocks, each of which is internally a blue clique." }, { "id": 20000954, "problem_number": "AIM-COMBINATORICS-0079", "title": "Order-robust ordered Ramsey numbers of regular graphs", "statement": "Question: Does there exist a constant $c > 0$ such that for every sufficiently large $n$ there exists a 100-regular graph $H$ on $n$ vertices such that $r_<(H) \\ge n^{1 + c}$ for every ordering of $H$?\n\n\\section{Ramsey meets Erd\\H{o}s--Szekeres}\n\nDhruv Mubayi\n\nLet $g(s,n)$ be the minimum $N$ such that for every set of $N$ points in general position in the plane, and every red/blue coloring of the pairs, there are $s$ points in convex position, with all pairs red, or $n$ points in convex position with all pairs blue.\n\nKnown: $4^n < g(n,n) < 2^{O( n^2 \\log n)}$\n\n$2^{n-1} < g(s,n) < r(4^s, 4^n) < (4^n)^{4^s}$", "original_statement": "Question: Does there exist a constant $c > 0$ such that for every sufficiently large $n$ there exists a 100-regular graph $H$ on $n$ vertices such that $r_<(H) \\ge n^{1 + c}$ for every ordering of $H$?\n\n\\section{Ramsey meets Erd\\H{o}s--Szekeres}\n\nDhruv Mubayi\n\nLet $g(s,n)$ be the minimum $N$ such that for every set of $N$ points in general position in the plane, and every red/blue coloring of the pairs, there are $s$ points in convex position, with all pairs red, or $n$ points in convex position with all pairs blue.\n\nKnown: $4^n < g(n,n) < 2^{O( n^2 \\log n)}$\n\n$2^{n-1} < g(s,n) < r(4^s, 4^n) < (4^n)^{4^s}$", "clean_statement": "Question: Does there exist a constant $c > 0$ such that for every sufficiently large $n$ there exists a 100-regular graph $H$ on $n$ vertices such that $r_<(H) \\ge n^{1 + c}$ for every ordering of $H$?\n\n\\section{Ramsey meets Erd\\H{o}s--Szekeres}\n\nDhruv Mubayi\n\nLet $g(s,n)$ be the minimum $N$ such that for every set of $N$ points in general position in the plane, and every red/blue coloring of the pairs, there are $s$ points in convex position, with all pairs red, or $n$ points in convex position with all pairs blue.\n\nKnown: $4^n < g(n,n) < 2^{O( n^2 \\log n)}$\n\n$2^{n-1} < g(s,n) < r(4^s, 4^n) < (4^n)^{4^s}$", "statement_status": "exact", "statement_verification": "The canonical record begins in the section “Ordered Ramsey numbers” and asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Ordered Ramsey numbers\nSource item: 32\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[78]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question: Does there exist a constant $c > 0$ such that for every sufficiently large $n$ there exists a 100-regular graph $H$ on $n$ vertices such that $r_<(H) \\\\ge n^{1 + c}$ for every ordering of $H$?\\n\\n\\\\section{Ramsey meets Erd\\\\H{o}s--Szekeres}\\n\\nDhruv Mubayi\\n\\nLet $g(s,n)$ be the minimum $N$ such that for every set of $N$ points in general position in the plane, and every red/blue coloring of the pairs, there are $s$ points in convex position, with all pairs red, or $n$ points in convex position with all pairs blue.\\n\\nKnown: $4^n < g(n,n) < 2^{O( n^2 \\\\log n)}$\\n\\n$2^{n-1} < g(s,n) < r(4^s, 4^n) < (4^n)^{4^s}$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0079", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Balko, Jelinek, and Valtr proved that almost every fixed-degree regular graph has ordered Ramsey number at least n^(3/2-1/d)/(4 log n log log n) simultaneously for every vertex ordering. At d=100 this gives one 100-regular graph for every sufficiently large n with r_<(H) at least n^(1+c) under every ordering for every fixed c<49/100, so the AIM question is answered affirmatively (for example, c=12/25). As an additional contribution, a same-parity regular-completion theorem transfers any order-robust lower bound from degree d to degree D without changing the vertex set.\n\nCandidate contribution (theorem; novelty confidence low): If 0<=d<=D, D-d is even, and n>=max(D+2,2d+2), then every d-regular n-vertex graph G has a spanning D-regular supergraph H satisfying min_prec r_<((H,prec)) >= min_prec r_<((G,prec)); in particular, robust degree-4 examples lift to degree 100 for n>=102." }, { "id": 20000955, "problem_number": "AIM-COMBINATORICS-0080", "title": "Off-diagonal convex geometric Ramsey bounds", "statement": "Problem: Improve these bounds.", "original_statement": "Problem: Improve these bounds.", "clean_statement": "Problem: Improve these bounds.", "statement_status": "exact", "statement_verification": "The canonical record consists only of", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Ordered Ramsey numbers\nSource item: 33\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[79]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem: Improve these bounds.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0080", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The context-dependent record is the off-diagonal convex geometric Ramsey problem g(s,n). Writing ES(k) for the Erdos-Szekeres number and T=(s-1)^2+(n-1)^2+1, this attempt proves (ES(s)-1)(ES(n)-1)+1 <= g(s,n) <= min{r(ES(s),ES(n)), (T+1)2^T T!}. The lower bound uses an order-type-preserving geometric cluster product; the factorial upper bound optimizes the asymmetric thresholds in the Mubayi-Suk canonical-selection proof. Consequently log_2 g(n,n) <= (4+o(1))n^2 log_2 n, and for fixed s >= 3, g(s,n) <= 2^{(ES(s)-1)n+O_s(sqrt(n log n))}/n^{ES(s)-2}. In particular, 2^{n-1}+1 <= g(3,n) <= 2^{2n+O(sqrt(n log n))}/n. The main diagonal exponential gap remains open.\n\nCandidate contribution (product construction and optimized structural bound; novelty confidence low): The exact candidate contribution is the combined bound (ES(s)-1)(ES(n)-1)+1 <= g(s,n) <= (T+1)2^T T!, where T=(s-1)^2+(n-1)^2+1; it includes an asymmetric order-type-preserving lexicographic lower construction and improves the explicit leading constant from the published canonical-selection upper bound to log_2 g(n,n) <= (4+o(1))n^2 log_2 n." }, { "id": 20000956, "problem_number": "AIM-COMBINATORICS-0081", "title": "A four-point Hamming barrier in Euclidean Ramsey theory", "statement": "Euclidean Ramsey sets\n\nRon Graham\n\nLet $X$ be a finite set in $\\mathbb{E}^n$ ($n$-dimensional Euclidean\nspace). We say that $X$ is Ramsey if for all positive integers\n$r$ there exists $N = N(X,r)$ such that if one colors the points of\n$\\mathbb{E}^N$ with $r$ colors, then there exists a monochromatic\nset congruent to $X$.\n\nFor example, if $X$ consist of two points,\nthen $X$ is Ramsey. (Consider an $r$-coloring of the $2r+1$ vertices of\na unit simplex simplex in $\\mathbb{E}^{2r}$).\n\nA set is spherical if it lies on some sphere.\n\nTheorem. (Erd\\H{o}s--Graham--Montgomery--Rothschild--Spencer--Straus). Every Ramsey set is spherical.\n\nConjecture. (\\$1000) Every finite spherical set is Ramsey.\n\nA ``warm up'':", "original_statement": "Euclidean Ramsey sets\n\nRon Graham\n\nLet $X$ be a finite set in $\\mathbb{E}^n$ ($n$-dimensional Euclidean\nspace). We say that $X$ is Ramsey if for all positive integers\n$r$ there exists $N = N(X,r)$ such that if one colors the points of\n$\\mathbb{E}^N$ with $r$ colors, then there exists a monochromatic\nset congruent to $X$.\n\nFor example, if $X$ consist of two points,\nthen $X$ is Ramsey. (Consider an $r$-coloring of the $2r+1$ vertices of\na unit simplex simplex in $\\mathbb{E}^{2r}$).\n\nA set is spherical if it lies on some sphere.\n\nTheorem. (Erd\\H{o}s--Graham--Montgomery--Rothschild--Spencer--Straus). Every Ramsey set is spherical.\n\nConjecture. (\\$1000) Every finite spherical set is Ramsey.\n\nA ``warm up'':", "clean_statement": "Euclidean Ramsey sets\n\nRon Graham\n\nLet $X$ be a finite set in $\\mathbb{E}^n$ ($n$-dimensional Euclidean\nspace). We say that $X$ is Ramsey if for all positive integers\n$r$ there exists $N = N(X,r)$ such that if one colors the points of\n$\\mathbb{E}^N$ with $r$ colors, then there exists a monochromatic\nset congruent to $X$.\n\nFor example, if $X$ consist of two points,\nthen $X$ is Ramsey. (Consider an $r$-coloring of the $2r+1$ vertices of\na unit simplex simplex in $\\mathbb{E}^{2r}$).\n\nA set is spherical if it lies on some sphere.\n\nTheorem. (Erd\\H{o}s--Graham--Montgomery--Rothschild--Spencer--Straus). Every Ramsey set is spherical.\n\nConjecture. (\\$1000) Every finite spherical set is Ramsey.\n\nA ``warm up'':", "statement_status": "exact", "statement_verification": "The canonical record is Conjecture 34 in the AIM workshop list *Graph Ramsey theory*, section “Euclidean Ramsey sets,” attributed to Ron Graham. Its exact mathematical claim is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Euclidean Ramsey sets\nSource item: 34\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[80]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Euclidean Ramsey sets\\n\\nRon Graham\\n\\nLet $X$ be a finite set in $\\\\mathbb{E}^n$ ($n$-dimensional Euclidean\\nspace). We say that $X$ is Ramsey if for all positive integers\\n$r$ there exists $N = N(X,r)$ such that if one colors the points of\\n$\\\\mathbb{E}^N$ with $r$ colors, then there exists a monochromatic\\nset congruent to $X$.\\n\\nFor example, if $X$ consist of two points,\\nthen $X$ is Ramsey. (Consider an $r$-coloring of the $2r+1$ vertices of\\na unit simplex simplex in $\\\\mathbb{E}^{2r}$).\\n\\nA set is spherical if it lies on some sphere.\\n\\nTheorem. (Erd\\\\H{o}s--Graham--Montgomery--Rothschild--Spencer--Straus). Every Ramsey set is spherical.\\n\\nConjecture. (\\\\$1000) Every finite spherical set is Ramsey.\\n\\nA ``warm up'':\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0081", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Graham's conjecture that every finite spherical set is Ramsey remains open. This attempt proves that a finite Euclidean configuration embeds in a rectangular-box vertex set exactly when its squared-distance vector lies in the cut cone. For four points this is equivalent to every constituent triangle being nonobtuse, via an explicit decomposition using at most six cuts. Consequently four distinct coplanar points can occur in a box, or in a product of regular simplices, only when they form a rectangle. This is a rigorous obstruction to a natural product-based proof of the four-point cyclic warm-up, not an obstruction to Ramsey-ness itself.\n\nCandidate contribution (method_obstruction; novelty confidence low): A four-point Euclidean configuration is box-embeddable if and only if every one of its constituent triangles is nonobtuse, with an explicit six-cut certificate; hence the only four-point planar configurations contained in rectangular boxes or products of regular simplices are rectangles.", "research_difficulty_suggested": "L4" }, { "id": 20000957, "problem_number": "AIM-COMBINATORICS-0082", "title": "Four concyclic points and bounded Euclidean Ramsey witnesses", "statement": "Conjecture. (\\$100) Every 4-point subset of a circle is\nRamsey.\n\nA rival conjecture:", "original_statement": "Conjecture. (\\$100) Every 4-point subset of a circle is\nRamsey.\n\nA rival conjecture:", "clean_statement": "Conjecture. (\\$100) Every 4-point subset of a circle is\nRamsey.\n\nA rival conjecture:", "statement_status": "exact", "statement_verification": "The canonical record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Euclidean Ramsey sets\nSource item: 35\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[81]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture. (\\\\$100) Every 4-point subset of a circle is\\nRamsey.\\n\\nA rival conjecture:\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0082", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The four-point circle conjecture remains open. A proved lower-semicontinuity theorem shows that the minimum cardinality w_r(X) of a finite r-Ramsey witness satisfies w_r(X) <= liminf w_r(X_j) whenever fixed-size Euclidean configurations X_j converge to a distinct-point configuration X in labeled distance matrices. Since rational-arc cyclic quadrilaterals are dense and are Ramsey as subsets of regular polygons, the AIM conjecture reduces to bounding their finite witness sizes locally; any counterexample would force those sizes to diverge for some fixed number of colors along every rational-arc approximation.\n\nCandidate contribution (reduction_theorem; novelty confidence low): For fixed s and r, the finite Euclidean Ramsey witness number w_r is lower semicontinuous under convergence of labeled s-point distance matrices to a distinct-point configuration; consequently, any non-Ramsey cyclic quadrilateral forces w_r to tend to infinity along every sequence of rational-arc regular-polygon approximants for some fixed r.", "research_difficulty_suggested": "L4" }, { "id": 20000958, "problem_number": "AIM-COMBINATORICS-0083", "title": "A cut-cone certificate for binary transitive hosts", "statement": "Conjecture. (Leader--Russell--Walters) A finite set is Ramsey\nif and only if it is a subset of some set with a transitive symmetry group.\n\nFor 3-point sets:", "original_statement": "Conjecture. (Leader--Russell--Walters) A finite set is Ramsey\nif and only if it is a subset of some set with a transitive symmetry group.\n\nFor 3-point sets:", "clean_statement": "Conjecture. (Leader--Russell--Walters) A finite set is Ramsey\nif and only if it is a subset of some set with a transitive symmetry group.\n\nFor 3-point sets:", "statement_status": "exact", "statement_verification": "The canonical record is number 36 in the section “Euclidean Ramsey sets” of the AIM workshop list “Graph Ramsey theory.” Its mathematical content is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Euclidean Ramsey sets\nSource item: 36\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[82]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture. (Leader--Russell--Walters) A finite set is Ramsey\\nif and only if it is a subset of some set with a transitive symmetry group.\\n\\nFor 3-point sets:\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0083", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite Euclidean set X, embedding into a transitive orbit of an elementary abelian 2-group is equivalent to embedding into a rectangular-box vertex set, and equivalent to the squared-distance vector of X lying in the cut cone. This is a finite linear-feasibility certificate and, by Kříž's theorem, implies that X is Ramsey. For a nondegenerate triangle the certificate holds exactly when the triangle is non-obtuse; the minimal box dimension is three in the acute case and two in the right case.\n\nCandidate contribution (equivalence_and_special_case; novelty confidence low): Candidate synthesis: binary subtransitivity is exactly cut-cone membership of the squared-distance matrix; for triangles this gives unique cut coefficients, an if-and-only-if non-obtuse criterion, and the sharp minimal binary-host dimension.", "research_difficulty_suggested": "L4" }, { "id": 20000959, "problem_number": "AIM-COMBINATORICS-0084", "title": "A finite certificate for three-stripe triangle avoidance", "statement": "Conjecture. Let $T$ be a set of three points in\n$\\mathbb{E}^2$. Then there exists a 3-coloring of $\\mathbb{E}^2$ with no\nmonochromatic copy of $T$.", "original_statement": "Conjecture. Let $T$ be a set of three points in\n$\\mathbb{E}^2$. Then there exists a 3-coloring of $\\mathbb{E}^2$ with no\nmonochromatic copy of $T$.", "clean_statement": "Conjecture. Let $T$ be a set of three points in\n$\\mathbb{E}^2$. Then there exists a 3-coloring of $\\mathbb{E}^2$ with no\nmonochromatic copy of $T$.", "statement_status": "exact", "statement_verification": "The exact canonical record is Conjecture 37 in the AIM workshop list *Graph Ramsey theory*, section “Euclidean Ramsey sets”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Euclidean Ramsey sets\nSource item: 37\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[83]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture. Let $T$ be a set of three points in\\n$\\\\mathbb{E}^2$. Then there exists a 3-coloring of $\\\\mathbb{E}^2$ with no\\nmonochromatic copy of $T$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0084", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The conjecture that every three-point subset of the Euclidean plane can be avoided by a coloring with three colors remains open. This attempt gives an exact finite certificate for the natural cyclic equal-width three-stripe family: a monochromatic congruent copy exists exactly when a coherent 3w-shifted triple of vertical projections has residual range strictly below the strip width w. Only finitely many strip offsets and at most twelve critical angles per offset pair must be checked. It also supplies a boundary-safe proof that the stripe coloring avoids every nondegenerate triangle whose smallest angle is at least 30 degrees, and proves why line-constant and one-side proper-coloring strategies cannot settle the full conjecture.\n\nCandidate contribution (finite_reduction; novelty confidence low): For any prescribed three-point set and strip width w, avoidance by the cyclic three-stripe coloring is exactly decidable using at most 12(2K+1)^2 explicit trigonometric evaluations, where K=ceil((diam(T)+w)/(3w)); the certificate accounts exactly for translations, rotations, reflections, coherent strip offsets, and half-open boundaries.", "research_difficulty_suggested": "L4" }, { "id": 20000960, "problem_number": "AIM-COMBINATORICS-0085", "title": "Two-colored planes and the exact stripe spectrum", "statement": "Conjecture. In any 2-coloring of $\\mathbb{E}^2$, every\n3-point set $X$ occurs monochromatically (as some congruent copy), except\npossibly for the three points that form some specific equilateral triangle.\n\n(Example coloring: half-open horizontal stripes of height $\\sqrt{3}/2$ in\nalternating colors will stop the three vertices of a unit equilateral triangle\nfrom occurring monochromatically).\n\nKnown: If $T$ is a set of 3 collinear points, then there exists\na 16-coloring of $\\mathbb{E}^N$ with no monochromatic copy of $T$.", "original_statement": "Conjecture. In any 2-coloring of $\\mathbb{E}^2$, every\n3-point set $X$ occurs monochromatically (as some congruent copy), except\npossibly for the three points that form some specific equilateral triangle.\n\n(Example coloring: half-open horizontal stripes of height $\\sqrt{3}/2$ in\nalternating colors will stop the three vertices of a unit equilateral triangle\nfrom occurring monochromatically).\n\nKnown: If $T$ is a set of 3 collinear points, then there exists\na 16-coloring of $\\mathbb{E}^N$ with no monochromatic copy of $T$.", "clean_statement": "Conjecture. In any 2-coloring of $\\mathbb{E}^2$, every\n3-point set $X$ occurs monochromatically (as some congruent copy), except\npossibly for the three points that form some specific equilateral triangle.\n\n(Example coloring: half-open horizontal stripes of height $\\sqrt{3}/2$ in\nalternating colors will stop the three vertices of a unit equilateral triangle\nfrom occurring monochromatically).\n\nKnown: If $T$ is a set of 3 collinear points, then there exists\na 16-coloring of $\\mathbb{E}^N$ with no monochromatic copy of $T$.", "statement_status": "exact", "statement_verification": "The exact source record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Euclidean Ramsey sets\nSource item: 38\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[84]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture. In any 2-coloring of $\\\\mathbb{E}^2$, every\\n3-point set $X$ occurs monochromatically (as some congruent copy), except\\npossibly for the three points that form some specific equilateral triangle.\\n\\n(Example coloring: half-open horizontal stripes of height $\\\\sqrt{3}/2$ in\\nalternating colors will stop the three vertices of a unit equilateral triangle\\nfrom occurring monochromatically).\\n\\nKnown: If $T$ is a set of 3 collinear points, then there exists\\na 16-coloring of $\\\\mathbb{E}^N$ with no monochromatic copy of $T$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0085", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the alternating half-open slab coloring chi_{h,u}(x) = floor(/h) mod 2, the exceptional equilateral spectrum is exactly {2h/sqrt(3)}: an equilateral triangle is avoided if and only if its altitude equals the stripe width. The proof uses a strict circular-residue gap criterion that handles boundary points. Using the EGMRSS side-length theorem, the original nonequilateral-triple conjecture is also shown equivalent to saying that every two-coloring has at most one exceptional equilateral side length.\n\nCandidate contribution (parameter_characterization; novelty confidence low): The standard alternating half-open stripe coloring of width h avoids exactly the equilateral side length 2h/sqrt(3), and no other equilateral side length.", "research_difficulty_suggested": "L4" }, { "id": 20000961, "problem_number": "AIM-COMBINATORICS-0086", "title": "Uniform color bounds for avoiding three-point collinear configurations", "statement": "Question: What is the minimum number of colors needed above?", "original_statement": "Question: What is the minimum number of colors needed above?", "clean_statement": "Question: What is the minimum number of colors needed above?", "statement_status": "exact", "statement_verification": "The canonical record says only:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Euclidean Ramsey sets\nSource item: 39\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[85]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question: What is the minimum number of colors needed above?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0086", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The ambiguous prompt asks for the least universal number q_* of colors such that, for every nondegenerate collinear triple T and every dimension n, some coloring of Euclidean n-space avoids a monochromatic congruent T. Current verified bounds are 3 <= q_* <= 16: the upper bound is Straus's classical theorem, while Currier--Moore--Yip's 2024 plane theorem for equally spaced triples gives the updated lower bound. In addition, chi_1(T)=2 for every T, chi_n(T) is nondecreasing in n, and if T has reduced consecutive-gap ratio p:q then the explicit radial coloring floor(||x||^2/s^2) modulo (p+q)(pq+1) avoids T in every dimension. This yields 4 colors for equal gaps and 9 for ratio 1:2.\n\nCandidate contribution (explicit coloring lemma; novelty confidence low): For a collinear triple with consecutive gaps ps and qs, where p:q is reduced, the coloring c(x)=floor(||x||^2/s^2) mod ((p+q)(pq+1)) avoids every monochromatic congruent copy in every finite Euclidean dimension; in particular, ratio 1:2 has an explicit dimension-uniform 9-color avoidance coloring." }, { "id": 20000962, "problem_number": "AIM-COMBINATORICS-0087", "title": "An explicit 94-vertex cubic-residue candidate and certificate reduction", "statement": "Folkman graphs\n\nRon Graham\n\nA Folkman graph is a $K_4$-free graph such that every\n2-edge-coloring contains a monochromatic triangle.\n\nFolkman showed that such a graph exists, but he required an enormous\nnumber of vertices. Proofs of existence of smaller Folkman graphs were\ngiven by Frankl--R\\\"odl, Spencer, Lu, Dudek--R\\\"odl,\nLange--Radziszowski--Xu.\n\nChallenge. (\\$100) Find a Folkman graph with fewer than 100\nvertices.\n\nCandidate construction: (Exoo) take $\\mathbb{Z}_{127}$ as vertices, and join\n$i$ with $j$ if $i-j$ is a cubic residue mod 127, and then delete 33\nvertices (three independent sets of size 11).", "original_statement": "Folkman graphs\n\nRon Graham\n\nA Folkman graph is a $K_4$-free graph such that every\n2-edge-coloring contains a monochromatic triangle.\n\nFolkman showed that such a graph exists, but he required an enormous\nnumber of vertices. Proofs of existence of smaller Folkman graphs were\ngiven by Frankl--R\\\"odl, Spencer, Lu, Dudek--R\\\"odl,\nLange--Radziszowski--Xu.\n\nChallenge. (\\$100) Find a Folkman graph with fewer than 100\nvertices.\n\nCandidate construction: (Exoo) take $\\mathbb{Z}_{127}$ as vertices, and join\n$i$ with $j$ if $i-j$ is a cubic residue mod 127, and then delete 33\nvertices (three independent sets of size 11).", "clean_statement": "Folkman graphs\n\nRon Graham\n\nA Folkman graph is a $K_4$-free graph such that every\n2-edge-coloring contains a monochromatic triangle.\n\nFolkman showed that such a graph exists, but he required an enormous\nnumber of vertices. Proofs of existence of smaller Folkman graphs were\ngiven by Frankl--R\\\"odl, Spencer, Lu, Dudek--R\\\"odl,\nLange--Radziszowski--Xu.\n\nChallenge. (\\$100) Find a Folkman graph with fewer than 100\nvertices.\n\nCandidate construction: (Exoo) take $\\mathbb{Z}_{127}$ as vertices, and join\n$i$ with $j$ if $i-j$ is a cubic residue mod 127, and then delete 33\nvertices (three independent sets of size 11).", "statement_status": "exact", "statement_verification": "The source record is challenge 40 from the AIM workshop section “Folkman graphs”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: Folkman graphs\nSource item: 40\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[86]\nCanonical tag: challenge\nOriginal extracted problem text (JSON string): \"Folkman graphs\\n\\nRon Graham\\n\\nA Folkman graph is a $K_4$-free graph such that every\\n2-edge-coloring contains a monochromatic triangle.\\n\\nFolkman showed that such a graph exists, but he required an enormous\\nnumber of vertices. Proofs of existence of smaller Folkman graphs were\\ngiven by Frankl--R\\\\\\\"odl, Spencer, Lu, Dudek--R\\\\\\\"odl,\\nLange--Radziszowski--Xu.\\n\\nChallenge. (\\\\$100) Find a Folkman graph with fewer than 100\\nvertices.\\n\\nCandidate construction: (Exoo) take $\\\\mathbb{Z}_{127}$ as vertices, and join\\n$i$ with $j$ if $i-j$ is a cubic residue mod 127, and then delete 33\\nvertices (three independent sets of size 11).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0087", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:challenge" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The fewer-than-100-vertex Folkman challenge remains open, with the May 2026 literature still giving 21 <= F_e(3,3;4) <= 786. This attempt makes the source's underspecified deletion candidate exact: for the cubic-residue Cayley graph on F_127, let D=H intersect (1+H) and delete D, 2D, and 1+D, the common-neighbor sets of the three edges of the triangle {0,1,2}. These are pairwise disjoint independent 11-sets. The residual graph is a proved K4-free graph on 94 vertices with 1440 edges and 3750 triangles, and its arrowing property is equivalent to unsatisfiability of an explicit 1440-variable, 7500-clause 3-CNF. No unsatisfiability certificate is supplied, so this is a rigorous finite reduction rather than a full solution.\n\nCandidate contribution (finite_reduction; novelty confidence low): The verbal three-independent-set deletion is canonically instantiated as U=N(0,1) union N(0,2) union N(1,2)=D union 2D union (1+D), producing an explicit 94-vertex, 1440-edge, 3750-triangle K4-free graph whose Folkman property is exactly a 1440-variable, 7500-clause UNSAT certificate problem.", "research_difficulty_suggested": "L4" }, { "id": 20000963, "problem_number": "AIM-COMBINATORICS-0088", "title": "van der Waerden numbers and sparse forcing sets", "statement": "van der Waerden numbers\n\nRon Graham\n\nLet $W(n,m)$ denote the least $N$ so that any red/blue coloring of $\\{1,\n\\dots, N\\}$ contains either a red $n$-term arithmetic progression or a\nblue $m$-term arithmetic progression.\n\nLet $W(n) = W(n,n)$.\n\nSome data:\n{c|ccccc}\n $n$ & 2 & 3 & 4 & 5 & 6\n\n\n \\hline\n $W(n)$ & 3 & 9 & 35 & 178 & 1132\n\nKnown: (Berlekamp) $W(n+1) \\ge n 2^n$ for $n$ prime.\n\n(Gowers) $W(n) \\le 2^{2^{2^{2^{2^{n+9}}}}}$.\n\nConjecture: (\\$1000) $W(n) \\le 2^{n^2}$.\n\nLet $W^*(n)$ denote the size of the smallest subset $S$ of\n$\\mathbb{N}$ such that any 2-coloring of $S$ has a monochromatic\n$n$-term arithmetic progression.\n\nKnown: (Elkies) $W^*(3) = W(3).$\n\n$W^*(4) \\le 27 < W(4) = 35.$", "original_statement": "van der Waerden numbers\n\nRon Graham\n\nLet $W(n,m)$ denote the least $N$ so that any red/blue coloring of $\\{1,\n\\dots, N\\}$ contains either a red $n$-term arithmetic progression or a\nblue $m$-term arithmetic progression.\n\nLet $W(n) = W(n,n)$.\n\nSome data:\n{c|ccccc}\n $n$ & 2 & 3 & 4 & 5 & 6\n \n\n \\hline\n $W(n)$ & 3 & 9 & 35 & 178 & 1132\n\nKnown: (Berlekamp) $W(n+1) \\ge n 2^n$ for $n$ prime.\n\n(Gowers) $W(n) \\le 2^{2^{2^{2^{2^{n+9}}}}}$.\n\nConjecture: (\\$1000) $W(n) \\le 2^{n^2}$.\n\nLet $W^*(n)$ denote the size of the smallest subset $S$ of\n$\\mathbb{N}$ such that any 2-coloring of $S$ has a monochromatic\n$n$-term arithmetic progression.\n\nKnown: (Elkies) $W^*(3) = W(3).$\n\n$W^*(4) \\le 27 < W(4) = 35.$", "clean_statement": "van der Waerden numbers\n\nRon Graham\n\nLet $W(n,m)$ denote the least $N$ so that any red/blue coloring of $\\{1,\n\\dots, N\\}$ contains either a red $n$-term arithmetic progression or a\nblue $m$-term arithmetic progression.\n\nLet $W(n) = W(n,n)$.\n\nSome data:\n{c|ccccc}\n $n$ & 2 & 3 & 4 & 5 & 6\n\n\n \\hline\n $W(n)$ & 3 & 9 & 35 & 178 & 1132\n\nKnown: (Berlekamp) $W(n+1) \\ge n 2^n$ for $n$ prime.\n\n(Gowers) $W(n) \\le 2^{2^{2^{2^{2^{n+9}}}}}$.\n\nConjecture: (\\$1000) $W(n) \\le 2^{n^2}$.\n\nLet $W^*(n)$ denote the size of the smallest subset $S$ of\n$\\mathbb{N}$ such that any 2-coloring of $S$ has a monochromatic\n$n$-term arithmetic progression.\n\nKnown: (Elkies) $W^*(3) = W(3).$\n\n$W^*(4) \\le 27 < W(4) = 35.$", "statement_status": "exact", "statement_verification": "The source record, attributed to Ron Graham, defines \\(W(n,m)\\) to be the least \\(N\\) such that every red/blue coloring of \\([N]=\\{1,\\ldots,N\\}\\) contains either a red \\(n\\)-term arithmetic progression or a blue \\(m\\)-term arithmetic progression, and puts \\(W(n)=W(n,n)\\). It records", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: van der Waerden numbers\nSource item: 41\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[87]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"van der Waerden numbers\\n\\nRon Graham\\n\\nLet $W(n,m)$ denote the least $N$ so that any red/blue coloring of $\\\\{1,\\n\\\\dots, N\\\\}$ contains either a red $n$-term arithmetic progression or a\\nblue $m$-term arithmetic progression.\\n\\nLet $W(n) = W(n,n)$.\\n\\nSome data:\\n{c|ccccc}\\n $n$ & 2 & 3 & 4 & 5 & 6\\n \\n\\n \\\\hline\\n $W(n)$ & 3 & 9 & 35 & 178 & 1132\\n\\nKnown: (Berlekamp) $W(n+1) \\\\ge n 2^n$ for $n$ prime.\\n\\n(Gowers) $W(n) \\\\le 2^{2^{2^{2^{2^{n+9}}}}}$.\\n\\nConjecture: (\\\\$1000) $W(n) \\\\le 2^{n^2}$.\\n\\nLet $W^*(n)$ denote the size of the smallest subset $S$ of\\n$\\\\mathbb{N}$ such that any 2-coloring of $S$ has a monochromatic\\n$n$-term arithmetic progression.\\n\\nKnown: (Elkies) $W^*(3) = W(3).$\\n\\n$W^*(4) \\\\le 27 < W(4) = 35.$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0088", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let m(k) be the minimum number of edges in a non-two-colorable k-uniform hypergraph and let P_k(v) be the number of k-term arithmetic progressions in [v]. Klotz's extremal progression theorem implies W*(k) is at least the smallest v for which P_k(v) >= m(k). In particular, m(4)=23 and P_4(13)=22 give W*(4) >= 14, so the source's upper bound becomes 14 <= W*(4) <= 27. The final Grill-Linzmayer bounds similarly give W*(5) >= 18, W*(6) >= 28, W*(7) >= 43, W*(8) >= 65, and W*(9) >= 97. The Radhakrishnan-Srinivasan Property B bound yields W*(k) = Omega(2^(k/2) k^(3/4)/(log k)^(1/4)). Graham's W(k) <= 2^(k^2) conjecture remains open.\n\nCandidate contribution (reduction_and_lower_bound; novelty confidence low): For every k >= 2, W*(k) >= min{v >= k : P_k(v) >= m(k)}; consequently W*(4) >= 14, the corrected small bounds for k=5,...,9 are 18, 28, 43, 65, 97, and W*(k) = Omega(2^(k/2) k^(3/4)/(log k)^(1/4)).", "research_difficulty_suggested": "L4" }, { "id": 20000964, "problem_number": "AIM-COMBINATORICS-0089", "title": "Sparse witnesses and off-diagonal van der Waerden numbers", "statement": "Question: Does $W(n) - W^*(n) \\to \\infty$?\n\n\\medskip\n\nOff diagonal van der Waerden numbers $W(k,3)$\n\nFrom computed values, $W(k,3)$ seems to grow quadratically in $k$.\n\nKnown: $k^{2-o(1)} < W(k) < e^{O(k \\log^5 k)}$.", "original_statement": "Question: Does $W(n) - W^*(n) \\to \\infty$?\n\n\\medskip\n\nOff diagonal van der Waerden numbers $W(k,3)$\n\nFrom computed values, $W(k,3)$ seems to grow quadratically in $k$.\n\nKnown: $k^{2-o(1)} < W(k) < e^{O(k \\log^5 k)}$.", "clean_statement": "Question: Does $W(n)-W^*(n)\\to\\infty$?\n\nOff diagonal van der Waerden numbers $W(k,3)$\n\nFrom computed values, $W(k,3)$ seems to grow quadratically in $k$.\n\nKnown: $k^{2-o(1)}=2k, then W(3,k)<=N. This transfers published density bounds to W(3,k)<=exp(O((log k)^12)) and the March 2026 Raghavan preprint to W(3,k)<=exp(O((log k)^6(log log k)^6)).\n\nCandidate contribution (structural certificate and explicit lower bound; novelty confidence low): Every vertex of a cardinality-minimal W*(n) witness is the unique intersection of two forcing n-term arithmetic progressions, and the witness contains at least 2^(n-1) such progressions; together these give an explicit lower bound and a local pruning certificate for searches." }, { "id": 20000965, "problem_number": "AIM-COMBINATORICS-0090", "title": "Polynomial growth is false for both diagonal and off-diagonal readings", "statement": "Question: Does $W(k)$ grow polynomially in $k$?", "original_statement": "Question: Does $W(k)$ grow polynomially in $k$?", "clean_statement": "Question: Does $W(k)$ grow polynomially in $k$?", "statement_status": "exact", "statement_verification": "The exact source record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: OverleafTeX\nAIM domain: Combinatorics\nWorkshop: Graph Ramsey theory\nSection: van der Waerden numbers\nSource item: 43\nSource URL: https://www.overleaf.com/read/mnvcscjjysvg\nCanonical location: aim-combinatorics-notes.json notes[89]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question: Does $W(k)$ grow polynomially in $k$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://www.overleaf.com/read/mnvcscjjysvg", "tags": [ "aim", "AIM-COMBINATORICS-0090", "aim-domain:combinatorics", "aim-workshop:mnvcscjjysvg", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The notation is ambiguous, but the answer is no under both readings. Literally, the earlier definition W(k)=W(k,k) is incompatible with polynomial growth by Berlekamp's bound W(p+1)>=p*2^p along primes. Contextually, the intended function is the off-diagonal w(3,k)=W(k,3), and Hunter's peer-reviewed bound w(3,k)>=exp(c(log k)^2/log log k) proves superpolynomial growth. The current broad off-diagonal window is exp(c(log k)^2/log log k)<=w(3,k)<=exp(C(log k)^9). Beyond the published resolution, this attempt proves an exact 3-AP-free-transversal formulation and a two-sided quasipolynomial window for the consecutive-block certificate threshold Q(k).\n\nCandidate contribution (exact_reduction; novelty confidence low): A coloring avoiding a blue 3-AP and red k-AP is exactly a 3-AP-free transversal of the k-AP hypergraph; moreover the consecutive-block certificate threshold Q(k)=min{N:r_3(N)=3 incidence tree. Using all k-1 modes with positive weights gives kernel exactly the constants on every incidence-connected hypergraph and finite one-sink reduced cyclotomic modules; asymmetric conjugate-mode weights retain oriented cyclic information.\n\nCandidate contribution (theorem; novelty confidence low): The exact incidence-tree mode-count formula dim ker L = 1+(k-1-|S|)|E| and rank of the degree-zero cyclotomic cokernel = (k-1-|S|)|E| yield a sharp obstruction: all k-1 nontrivial Fourier modes are necessary for a finite graph-style critical module uniformly on k-uniform incidence trees, while positive weights on all modes are sufficient on every incidence-connected hypergraph and can preserve cyclic orientation." }, { "id": 20000967, "problem_number": "AIM-COMBINATORICS-0092", "title": "Regular-matroid Jacobian sampling is solved; torsion obstructs the unrestricted simplicial version", "statement": "(2) Jeremy Martin: \"We can efficiently sample random spanning trees of a graph in a uniform way by sampling the Jacobian group randomly. Can this approach be made to work when I change the graph to something more general like a simplicial complex?\" (a) Farbod Shokrieh: \"The class of regular matroids is a good place to start looking for such a generalization. In this case we can define the Jacobian and we know the size is correct (that is, is equal to the size of a basis of the matroid). The problem is finding an efficient bijection. We have no analog of Dhar's burning algorithm at this level.\"\n\n> 1The wokshop's website is aimath.org/WWN/chipfiring/.\n> 12PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\n(b) Jeremy Martin: \"I can state the problem more concretely as follows. Let D be an n × m matrix over Z of rank n, and let L: Zn → Zn be given by L:= DD t. Define\n\nJ(D):= Zn/im L = coker( L).\n\nWe can uniformly sample from J(D). But can we sample uni-formly from the column bases of D?\" (c) Matt Baker: \"If D is unimodular, then the usual Cauchy-Binet proof of Matrix-Tree theorem works and shows that J(D) is in bijection with a column basis of D. We are really asking: is this bijection efficiently computable?\" (d) Jeremy Martin: \"Actually, I believe you do not need unimodu-larity; you just need the determinants of all the maximal minors to be ±1.\"", "original_statement": "(2) Jeremy Martin: \"We can efficiently sample random spanning trees of a graph in a uniform way by sampling the Jacobian group randomly. Can this approach be made to work when I change the graph to something more general like a simplicial complex?\" (a) Farbod Shokrieh: \"The class of regular matroids is a good place to start looking for such a generalization. In this case we can define the Jacobian and we know the size is correct (that is, is equal to the size of a basis of the matroid). The problem is finding an efficient bijection. We have no analog of Dhar's burning algorithm at this level.\" \n\n> 1The wokshop's website is aimath.org/WWN/chipfiring/.\n> 12PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\n(b) Jeremy Martin: \"I can state the problem more concretely as follows. Let D be an n × m matrix over Z of rank n, and let L: Zn → Zn be given by L:= DD t. Define \n\nJ(D):= Zn/im L = coker( L).\n\nWe can uniformly sample from J(D). But can we sample uni-formly from the column bases of D?\" (c) Matt Baker: \"If D is unimodular, then the usual Cauchy-Binet proof of Matrix-Tree theorem works and shows that J(D) is in bijection with a column basis of D. We are really asking: is this bijection efficiently computable?\" (d) Jeremy Martin: \"Actually, I believe you do not need unimodu-larity; you just need the determinants of all the maximal minors to be ±1.\"", "clean_statement": "(2) Jeremy Martin: \"We can efficiently sample random spanning trees of a graph in a uniform way by sampling the Jacobian group randomly. Can this approach be made to work when I change the graph to something more general like a simplicial complex?\" (a) Farbod Shokrieh: \"The class of regular matroids is a good place to start looking for such a generalization. In this case we can define the Jacobian and we know the size is correct (that is, is equal to the size of a basis of the matroid). The problem is finding an efficient bijection. We have no analog of Dhar's burning algorithm at this level.\"\n\n> 1The wokshop's website is aimath.org/WWN/chipfiring/.\n> 12PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\n(b) Jeremy Martin: \"I can state the problem more concretely as follows. Let D be an n × m matrix over Z of rank n, and let L: Zn → Zn be given by L:= DD t. Define\n\nJ(D):= Zn/im L = coker( L).\n\nWe can uniformly sample from J(D). But can we sample uni-formly from the column bases of D?\" (c) Matt Baker: \"If D is unimodular, then the usual Cauchy-Binet proof of Matrix-Tree theorem works and shows that J(D) is in bijection with a column basis of D. We are really asking: is this bijection efficiently computable?\" (d) Jeremy Martin: \"Actually, I believe you do not need unimodu-larity; you just need the determinants of all the maximal minors to be ±1.\"", "statement_status": "exact", "statement_verification": "This is Problem 2 in the problem list from the July 8--12, 2013 AIM workshop *Generalizations of chip-firing and the critical group*. The source PDF is the authoritative text [AIM13]. The corpus extraction contains three OCR/layout errors that matter:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 2\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[91]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(2) Jeremy Martin: \\\"We can efficiently sample random spanning trees of a graph in a uniform way by sampling the Jacobian group randomly. Can this approach be made to work when I change the graph to something more general like a simplicial complex?\\\" (a) Farbod Shokrieh: \\\"The class of regular matroids is a good place to start looking for such a generalization. In this case we can define the Jacobian and we know the size is correct (that is, is equal to the size of a basis of the matroid). The problem is finding an efficient bijection. We have no analog of Dhar's burning algorithm at this level.\\\" \\n\\n> 1The wokshop's website is aimath.org/WWN/chipfiring/.\\n> 12PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\\n\\n(b) Jeremy Martin: \\\"I can state the problem more concretely as follows. Let D be an n × m matrix over Z of rank n, and let L: Zn → Zn be given by L:= DD t. Define \\n\\nJ(D):= Zn/im L = coker( L).\\n\\nWe can uniformly sample from J(D). But can we sample uni-formly from the column bases of D?\\\" (c) Matt Baker: \\\"If D is unimodular, then the usual Cauchy-Binet proof of Matrix-Tree theorem works and shows that J(D) is in bijection with a column basis of D. We are really asking: is this bijection efficiently computable?\\\" (d) Jeremy Martin: \\\"Actually, I believe you do not need unimodu-larity; you just need the determinants of all the maximal minors to be ±1.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0092", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Backman--Baker--Yuen (2019) solved the concrete regular-matroid problem by explicit Jacobian torsors and polynomial-time basis bijections. For the exact matrix hypothesis in the AIM note, this report proves that any full-row-rank integer D whose nonzero maximal minors are all plus or minus one can be changed by an explicit integral row operation into a totally unimodular representation A while the map [x] to [Ux] identifies coker(DD^T) with coker(AA^T), so the published BBY algorithm applies to the original group. Independently, the projection kernel D^T(DD^T)^{-1}D yields an exact sequential sampler with basis probability det(D_B)^2/det(DD^T), which is uniform precisely in this unimodular regime. General simplicial complexes retain torsion-squared weights, so the unrestricted unweighted formulation is only partially solved.\n\nCandidate contribution (reduction_and_exact_sampler; novelty confidence low): Candidate synthesis: maximal-minor unimodularity of the displayed AIM matrix is converted by U=D_B0^{-1} into the totally unimodular representation needed by BBY, the same U explicitly transports the AIM cokernel, and a Schur-complement projection-kernel algorithm gives an exact uniform sampler directly on the unnormalized D; for D=[I|v] with v in {0,plus or minus 1}^n, [v] explicitly generates J(D) and gives a linear-time bijection to the s+1 bases." }, { "id": 20000968, "problem_number": "AIM-COMBINATORICS-0093", "title": "A q-Eulerian last-site law for cycle IDLA", "statement": "(3) Persi Diaconis: \"Say you play the explorers game [16] on a circle with n vertices labeled cyclically 0 through n − 1 and you repeatedly explore from 0. The probability that the last point occupied before the circle is filled is j is given by A(n, j )/n!, where A(n, j ) are the Eulerian numbers. 2 This fact was shown to me by Andrea Sportiello during the lunch break. What is interesting is that the situation is very different from a random walk, in which the distribution is uniform. It may be worth looking at the probability distribution of last occupied spot in the explorers game for other sets (for example in higher dimensions or on circulant graphs).\" (a) Lionel Levine: \"Jim Propp made this observation about the Eulerian numbers a while ago.\" (b) James Propp: \"There are connections here to other models such as P´ olya's urn and Bernard Friedman's urn. Are these connec-tions useful?\" (c) Andrea Sportiello: \"If you change the graph to something like Z2,I would guess that things become ugly very fast.\" (d) Lionel Levine: \"It is not clear how to generalize to higher di-mensions because so much is going on, but sticking to the one-dimensional case and putting weights on the edges may give some new q-analog.\"", "original_statement": "(3) Persi Diaconis: \"Say you play the explorers game [16] on a circle with n vertices labeled cyclically 0 through n − 1 and you repeatedly explore from 0. The probability that the last point occupied before the circle is filled is j is given by A(n, j )/n!, where A(n, j ) are the Eulerian numbers. 2 This fact was shown to me by Andrea Sportiello during the lunch break. What is interesting is that the situation is very different from a random walk, in which the distribution is uniform. It may be worth looking at the probability distribution of last occupied spot in the explorers game for other sets (for example in higher dimensions or on circulant graphs).\" (a) Lionel Levine: \"Jim Propp made this observation about the Eulerian numbers a while ago.\" (b) James Propp: \"There are connections here to other models such as P´ olya's urn and Bernard Friedman's urn. Are these connec-tions useful?\" (c) Andrea Sportiello: \"If you change the graph to something like Z2,I would guess that things become ugly very fast.\" (d) Lionel Levine: \"It is not clear how to generalize to higher di-mensions because so much is going on, but sticking to the one-dimensional case and putting weights on the edges may give some new q-analog.\"", "clean_statement": "(3) Persi Diaconis: \"Say you play the explorers game [16] on a circle with n vertices labeled cyclically 0 through n − 1 and you repeatedly explore from 0. The probability that the last point occupied before the circle is filled is j is given by A(n, j )/n!, where A(n, j ) are the Eulerian numbers. 2 This fact was shown to me by Andrea Sportiello during the lunch break. What is interesting is that the situation is very different from a random walk, in which the distribution is uniform. It may be worth looking at the probability distribution of last occupied spot in the explorers game for other sets (for example in higher dimensions or on circulant graphs).\" (a) Lionel Levine: \"Jim Propp made this observation about the Eulerian numbers a while ago.\" (b) James Propp: \"There are connections here to other models such as P´ olya's urn and Bernard Friedman's urn. Are these connec-tions useful?\" (c) Andrea Sportiello: \"If you change the graph to something like Z2,I would guess that things become ugly very fast.\" (d) Lionel Levine: \"It is not clear how to generalize to higher di-mensions because so much is going on, but sticking to the one-dimensional case and putting weights on the edges may give some new q-analog.\"", "statement_status": "exact", "statement_verification": "The official AIM PDF is the problem-session record from the July 2013 workshop “Generalizations of chip-firing and the critical group.” Its introduction warns that the quotations are summaries rather than direct transcriptions.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 3\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[92]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(3) Persi Diaconis: \\\"Say you play the explorers game [16] on a circle with n vertices labeled cyclically 0 through n − 1 and you repeatedly explore from 0. The probability that the last point occupied before the circle is filled is j is given by A(n, j )/n!, where A(n, j ) are the Eulerian numbers. 2 This fact was shown to me by Andrea Sportiello during the lunch break. What is interesting is that the situation is very different from a random walk, in which the distribution is uniform. It may be worth looking at the probability distribution of last occupied spot in the explorers game for other sets (for example in higher dimensions or on circulant graphs).\\\" (a) Lionel Levine: \\\"Jim Propp made this observation about the Eulerian numbers a while ago.\\\" (b) James Propp: \\\"There are connections here to other models such as P´ olya's urn and Bernard Friedman's urn. Are these connec-tions useful?\\\" (c) Andrea Sportiello: \\\"If you change the graph to something like Z2,I would guess that things become ugly very fast.\\\" (d) Lionel Levine: \\\"It is not clear how to generalize to higher di-mensions because so much is going on, but sticking to the one-dimensional case and putting weights on the edges may give some new q-analog.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0093", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The explorers game is sequential internal DLA, and the workshop's displayed cycle law has a one-unit indexing inconsistency: on C_n with source 0 initially occupied, the fair last-site law is Eulerian(n-1,j-1)/(n-1)! for 1<=j<=n-1. For arbitrary positive undirected edge conductances, an exact two-index recurrence is derived from cumulative resistances. For a homogeneous biased walk with clockwise probability p and rho=(1-p)/p, the exact law is P(L=j)=A_{n-1,n-j-1}(rho)/[n-1]_rho!, where A is Carlitz's descent-major-index q-Eulerian number. A general finite-graph subset recurrence and a source-stabilizer symmetry principle are also proved.\n\nCandidate contribution (exact q-analogue theorem; novelty confidence low): Biased nearest-neighbour sequential IDLA on the n-cycle has last-site distribution equal to the normalized Carlitz descent-major-index q-Eulerian row, with q=(1-p)/p." }, { "id": 20000969, "problem_number": "AIM-COMBINATORICS-0094", "title": "A quotient-root formula for every parking/toppling Betti number", "statement": "(4) Sam Hopkins: \"This question concerns a recent preprint of Venkatesh and Viswanath [34], which shows that for a finite graph G on n ver-tices, the dimension of the root space associated to the root that is the sum of all simple roots in the Kac-Moody algebra having Cartan matrix 2 In − A(G) is the number of maximal G-parking functions. (Here A(G) is the adjacency matrix of G.) Can this be exploited further? Are all G-parking functions present somewhere? Does the sandpile group act on some such root space?\"\n\n> 2For a definition of Eulerian numbers, see en.wikipedia.org/wiki/Eulerian number. PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 3\n\n(a) Sam Hopkins: \"I should also mention that there is a natural labeling of a basis of this root space by maximal parking func-tions.\" (b) Anton Dochtermann: \"The number of maximal parking func-tions is also the number of last syzygies in a minimal free reso-lution of the toppling ideal, and also the number of chambers of a certain slice of the graphic arrangement. Can we connect the Lie theory to commutative algebra or hyperlane arrangements?\"", "original_statement": "(4) Sam Hopkins: \"This question concerns a recent preprint of Venkatesh and Viswanath [34], which shows that for a finite graph G on n ver-tices, the dimension of the root space associated to the root that is the sum of all simple roots in the Kac-Moody algebra having Cartan matrix 2 In − A(G) is the number of maximal G-parking functions. (Here A(G) is the adjacency matrix of G.) Can this be exploited further? Are all G-parking functions present somewhere? Does the sandpile group act on some such root space?\" \n\n> 2For a definition of Eulerian numbers, see en.wikipedia.org/wiki/Eulerian number. PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 3\n\n(a) Sam Hopkins: \"I should also mention that there is a natural labeling of a basis of this root space by maximal parking func-tions.\" (b) Anton Dochtermann: \"The number of maximal parking func-tions is also the number of last syzygies in a minimal free reso-lution of the toppling ideal, and also the number of chambers of a certain slice of the graphic arrangement. Can we connect the Lie theory to commutative algebra or hyperlane arrangements?\"", "clean_statement": "(4) Sam Hopkins: \"This question concerns a recent preprint of Venkatesh and Viswanath [34], which shows that for a finite graph G on n ver-tices, the dimension of the root space associated to the root that is the sum of all simple roots in the Kac-Moody algebra having Cartan matrix 2 In − A(G) is the number of maximal G-parking functions. (Here A(G) is the adjacency matrix of G.) Can this be exploited further? Are all G-parking functions present somewhere? Does the sandpile group act on some such root space?\"\n\n> 2For a definition of Eulerian numbers, see en.wikipedia.org/wiki/Eulerian number. PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 3\n\n(a) Sam Hopkins: \"I should also mention that there is a natural labeling of a basis of this root space by maximal parking func-tions.\" (b) Anton Dochtermann: \"The number of maximal parking func-tions is also the number of last syzygies in a minimal free reso-lution of the toppling ideal, and also the number of chambers of a certain slice of the graphic arrangement. Can we connect the Lie theory to commutative algebra or hyperlane arrangements?\"", "statement_status": "exact", "statement_verification": "The canonical JSON record is an OCR extraction of problem (4) from the AIM workshop list *Generalizations of chip-firing and the critical group*. The extraction splits words at line endings, renders the identity matrix incorrectly, misspells “hyperplane,” and inserts a page header and an Eulerian-number footnote belonging to the preceding problem. Inspection of pages 2–3 of the original PDF gives the following recovered text (with only mathematical typesetting normalized):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 4\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[93]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(4) Sam Hopkins: \\\"This question concerns a recent preprint of Venkatesh and Viswanath [34], which shows that for a finite graph G on n ver-tices, the dimension of the root space associated to the root that is the sum of all simple roots in the Kac-Moody algebra having Cartan matrix 2 In − A(G) is the number of maximal G-parking functions. (Here A(G) is the adjacency matrix of G.) Can this be exploited further? Are all G-parking functions present somewhere? Does the sandpile group act on some such root space?\\\" \\n\\n> 2For a definition of Eulerian numbers, see en.wikipedia.org/wiki/Eulerian number. PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 3\\n\\n(a) Sam Hopkins: \\\"I should also mention that there is a natural labeling of a basis of this root space by maximal parking func-tions.\\\" (b) Anton Dochtermann: \\\"The number of maximal parking func-tions is also the number of last syzygies in a minimal free reso-lution of the toppling ideal, and also the number of chambers of a certain slice of the graphic arrangement. Can we connect the Lie theory to commutative algebra or hyperlane arrangements?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0094", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a connected finite simple graph G, every total Betti number of the parking-function and toppling quotients equals a sum of full-support Kac-Moody root multiplicities: beta_{r-1}(S/M_G)=beta_{r-1}(S/I_G)=sum over connected r-partitions Pi of dim(g(2I-A(Q_Pi)))_{delta_Pi}, where Q_Pi is the simple support of the quotient graph. This extends the AIM last-syzygy/root-space coincidence to every homological degree. All parking functions occur as the standard-monomial basis and as a sandpile torsor, but the standard sandpile action preserves the maximal parking functions only for trees.\n\nCandidate contribution (theorem; novelty confidence low): The all-degree Betti-root formula obtained by attaching a full-support Kac-Moody root space to every connected quotient partition, together with the proof that the usual sandpile torsor action cannot restrict to the maximal-parking-function root basis for any connected graph with a cycle." }, { "id": 20000970, "problem_number": "AIM-COMBINATORICS-0095", "title": "Exact block-product sampling of maximal G-parking functions", "statement": "(5) Lionel Levine: \"How can we uniformly and efficiently sample from the set of maximal G-parking functions?\" (a) Persi Diacons: \"There may be a connection to my paper with Christos Athanasiadis [5].\"", "original_statement": "(5) Lionel Levine: \"How can we uniformly and efficiently sample from the set of maximal G-parking functions?\" (a) Persi Diacons: \"There may be a connection to my paper with Christos Athanasiadis [5].\"", "clean_statement": "(5) Lionel Levine: \"How can we uniformly and efficiently sample from the set of maximal G-parking functions?\" (a) Persi Diacons: \"There may be a connection to my paper with Christos Athanasiadis [5].\"", "statement_status": "exact", "statement_verification": "The source is the AIM workshop problem list *Generalizations of chip-firing and the critical group*, compiled after the workshop of 8--12 July 2013. The printed item is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 5\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[94]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(5) Lionel Levine: \\\"How can we uniformly and efficiently sample from the set of maximal G-parking functions?\\\" (a) Persi Diacons: \\\"There may be a connection to my paper with Christos Athanasiadis [5].\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0095", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Using the Benson--Chakrabarty--Tetali bijection, the target is exactly the set of acyclic orientations with prescribed unique source q. Restriction to the rooted blocks gives a Cartesian-product bijection over locally unique-source orientations. Consequently, if every block is a bridge, a cycle, or a clique, independent explicit local choices give an exactly uniform maximal G-parking function in O(|V|+|E|) time and space, with count equal to the product of (r-1) over cycle blocks and (r-1)! over clique blocks. The general sampling problem remains open in the literature checked through 30 July 2026.\n\nCandidate contribution (exact_sampling_algorithm; novelty confidence low): For every connected simple graph rooted at q whose blocks are bridges, cycles, or cliques, the rooted block-cut product algorithm proved in the artifacts samples maximal G-parking functions exactly uniformly in O(|V|+|E|) time and space and yields an explicit product formula for their number." }, { "id": 20000971, "problem_number": "AIM-COMBINATORICS-0096", "title": "Volume and resistance crossover for the proposed parabolic IDLA domains", "statement": "(6) Persi Diaconis: \"When we play the explorers game and repeatedly explore from the origin on a square lattice in any dimension, the lim-iting shape is round. In dimensions d ≥ 3, the fluctuations after n\n\nsteps from this ball of radius n1/d are tiny, on the order of √log( n). For d = 2, the fluctuations are of size log( n), and for d = 1 the fluctuations are of size √n. These results in the higher dimensional cases have only been obtained recently; see [22], [4], and [21]. Are there other growth models where the fluctuations take on some in-termediate value, say between √n and log( n)?\" (a) Andrea Sportiello: \"If we consider a wedge of the two dimen-sional lattice Z2, is it clear that the limiting shape is the sector of the circle and does not depend on the angles?\" (b) Charles Smart: \"Away from the border we should be fine.\" (c) Andrea Sportiello: \"Then if you take the part of the lattice inside a parabola of exponent γ (that is, where |y| ≤ x1/γ ), which is a further generalization of a wedge, this interpolates between one and two dimensions and is thus a candidate for intermediate growth of fluctuations.\" (d) Igor Pak: \"Yes, in general if we consider the part of the lattice given by |y| ≤ f (x), we should be able to interpolate however we want between √n and log( n) by choosing the appropriate function f.\"", "original_statement": "(6) Persi Diaconis: \"When we play the explorers game and repeatedly explore from the origin on a square lattice in any dimension, the lim-iting shape is round. In dimensions d ≥ 3, the fluctuations after n\n\nsteps from this ball of radius n1/d are tiny, on the order of √log( n). For d = 2, the fluctuations are of size log( n), and for d = 1 the fluctuations are of size √n. These results in the higher dimensional cases have only been obtained recently; see [22], [4], and [21]. Are there other growth models where the fluctuations take on some in-termediate value, say between √n and log( n)?\" (a) Andrea Sportiello: \"If we consider a wedge of the two dimen-sional lattice Z2, is it clear that the limiting shape is the sector of the circle and does not depend on the angles?\" (b) Charles Smart: \"Away from the border we should be fine.\" (c) Andrea Sportiello: \"Then if you take the part of the lattice inside a parabola of exponent γ (that is, where |y| ≤ x1/γ ), which is a further generalization of a wedge, this interpolates between one and two dimensions and is thus a candidate for intermediate growth of fluctuations.\" (d) Igor Pak: \"Yes, in general if we consider the part of the lattice given by |y| ≤ f (x), we should be able to interpolate however we want between √n and log( n) by choosing the appropriate function f.\"", "clean_statement": "(6) Persi Diaconis: \"When we play the explorers game and repeatedly explore from the origin on a square lattice in any dimension, the lim-iting shape is round. In dimensions d ≥ 3, the fluctuations after n\n\nsteps from this ball of radius n1/d are tiny, on the order of √log( n). For d = 2, the fluctuations are of size log( n), and for d = 1 the fluctuations are of size √n. These results in the higher dimensional cases have only been obtained recently; see [22], [4], and [21]. Are there other growth models where the fluctuations take on some in-termediate value, say between √n and log( n)?\" (a) Andrea Sportiello: \"If we consider a wedge of the two dimen-sional lattice Z2, is it clear that the limiting shape is the sector of the circle and does not depend on the angles?\" (b) Charles Smart: \"Away from the border we should be fine.\" (c) Andrea Sportiello: \"Then if you take the part of the lattice inside a parabola of exponent γ (that is, where |y| ≤ x1/γ ), which is a further generalization of a wedge, this interpolates between one and two dimensions and is thus a candidate for intermediate growth of fluctuations.\" (d) Igor Pak: \"Yes, in general if we consider the part of the lattice given by |y| ≤ f (x), we should be able to interpolate however we want between √n and log( n) by choosing the appropriate function f.\"", "statement_status": "exact", "statement_verification": "This is Problem 6 in the AIM workshop problem list *Generalizations of chip-firing and the critical group* (workshop held 8--12 July 2013). The introduction to the list says that the displayed comments are recorder's summaries rather than necessarily verbatim quotations. The source is the official AIM PDF [1].", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 6\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[95]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(6) Persi Diaconis: \\\"When we play the explorers game and repeatedly explore from the origin on a square lattice in any dimension, the lim-iting shape is round. In dimensions d ≥ 3, the fluctuations after n\\n\\nsteps from this ball of radius n1/d are tiny, on the order of √log( n). For d = 2, the fluctuations are of size log( n), and for d = 1 the fluctuations are of size √n. These results in the higher dimensional cases have only been obtained recently; see [22], [4], and [21]. Are there other growth models where the fluctuations take on some in-termediate value, say between √n and log( n)?\\\" (a) Andrea Sportiello: \\\"If we consider a wedge of the two dimen-sional lattice Z2, is it clear that the limiting shape is the sector of the circle and does not depend on the angles?\\\" (b) Charles Smart: \\\"Away from the border we should be fine.\\\" (c) Andrea Sportiello: \\\"Then if you take the part of the lattice inside a parabola of exponent γ (that is, where |y| ≤ x1/γ ), which is a further generalization of a wedge, this interpolates between one and two dimensions and is thus a candidate for intermediate growth of fluctuations.\\\" (d) Igor Pak: \\\"Yes, in general if we consider the part of the lattice given by |y| ≤ f (x), we should be able to interpolate however we want between √n and log( n) by choosing the appropriate function f.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0096", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the induced lattice graph G_alpha = {(k,j): k >= 0, |j| <= floor(k^alpha)} with 0 < alpha < 1, intrinsic balls satisfy |B(R)| = 2 R^(1+alpha)/(1+alpha) + O(R + R^(2 alpha)), and the effective resistance from the origin to the wired column k=R is asymptotic to R^(1-alpha)/(2(1-alpha)). An explicit Neumann continuum odometer also shows that the deterministic cone benchmark is a circular sector, with angle affecting the mass-radius normalization. These proved facts supply the correct normalization and a potential-theoretic obstruction for the AIM interpolation proposal, but do not prove an IDLA limit shape or fluctuation exponent.\n\nCandidate contribution (theorem; novelty confidence low): For every fixed 0 < alpha < 1, R_eff(o <-> L_R) is asymptotic to R^(1-alpha)/(2(1-alpha)) on the AIM parabolic induced graph; the upper bound is realized by a unit flow whose energy exceeds the Nash-Williams cutset sum by only O(log R), and this is paired with the exact leading intrinsic-volume asymptotic." }, { "id": 20000972, "problem_number": "AIM-COMBINATORICS-0097", "title": "A fixed Laplacian quotient and hereditary firing staircases", "statement": "(7) Matt Macauley: \"Consider the equivalence relation on the set of all acyclic orientations of a graph G of converting sources to sinks. This relation encodes the chip-firing game on G among a special class of divisors. In this special case, there is a nice geometric way to view chip-firing via passing from the graphic arrangement (whose regions are in bijection with all acyclic orientations) to the toric graphic arrangement. Can we find a similar geometric way of viewing chip-firing equivalence for other (larger) classes of divisors?\" (a) Andrea Sportiello: \"I think we need to look at multi-toppling, or what has also been called hereditary chip-firing, to accomplish this.\" 4 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\n(b) James Propp: \"Does multi-toppling have the abelian property?\" (c) Andrea Sportiello: \"It has the abelian property precisely when it is hereditary, when we fire clusters and then sub-clusters and so on.\" (d) Lionel Levine: \"A good source for hereditary chip-firing is a paper of Spencer Backman [6].\"", "original_statement": "(7) Matt Macauley: \"Consider the equivalence relation on the set of all acyclic orientations of a graph G of converting sources to sinks. This relation encodes the chip-firing game on G among a special class of divisors. In this special case, there is a nice geometric way to view chip-firing via passing from the graphic arrangement (whose regions are in bijection with all acyclic orientations) to the toric graphic arrangement. Can we find a similar geometric way of viewing chip-firing equivalence for other (larger) classes of divisors?\" (a) Andrea Sportiello: \"I think we need to look at multi-toppling, or what has also been called hereditary chip-firing, to accomplish this.\" 4 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP \n\n(b) James Propp: \"Does multi-toppling have the abelian property?\" (c) Andrea Sportiello: \"It has the abelian property precisely when it is hereditary, when we fire clusters and then sub-clusters and so on.\" (d) Lionel Levine: \"A good source for hereditary chip-firing is a paper of Spencer Backman [6].\"", "clean_statement": "(7) Matt Macauley: \"Consider the equivalence relation on the set of all acyclic orientations of a graph G of converting sources to sinks. This relation encodes the chip-firing game on G among a special class of divisors. In this special case, there is a nice geometric way to view chip-firing via passing from the graphic arrangement (whose regions are in bijection with all acyclic orientations) to the toric graphic arrangement. Can we find a similar geometric way of viewing chip-firing equivalence for other (larger) classes of divisors?\" (a) Andrea Sportiello: \"I think we need to look at multi-toppling, or what has also been called hereditary chip-firing, to accomplish this.\" 4 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\n(b) James Propp: \"Does multi-toppling have the abelian property?\" (c) Andrea Sportiello: \"It has the abelian property precisely when it is hereditary, when we fire clusters and then sub-clusters and so on.\" (d) Lionel Levine: \"A good source for hereditary chip-firing is a paper of Spencer Backman [6].\"", "statement_status": "exact", "statement_verification": "This is Problem 7 in the official AIM list *Generalizations of chip-firing and the critical group*, produced after the workshop of 8--12 July 2013 [1]. The PDF introduction warns that the displayed comments are recorder's summaries and not necessarily verbatim quotations.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 7\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[96]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(7) Matt Macauley: \\\"Consider the equivalence relation on the set of all acyclic orientations of a graph G of converting sources to sinks. This relation encodes the chip-firing game on G among a special class of divisors. In this special case, there is a nice geometric way to view chip-firing via passing from the graphic arrangement (whose regions are in bijection with all acyclic orientations) to the toric graphic arrangement. Can we find a similar geometric way of viewing chip-firing equivalence for other (larger) classes of divisors?\\\" (a) Andrea Sportiello: \\\"I think we need to look at multi-toppling, or what has also been called hereditary chip-firing, to accomplish this.\\\" 4 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP \\n\\n(b) James Propp: \\\"Does multi-toppling have the abelian property?\\\" (c) Andrea Sportiello: \\\"It has the abelian property precisely when it is hereditary, when we fire clusters and then sub-clusters and so on.\\\" (d) Lionel Levine: \\\"A good source for hereditary chip-firing is a paper of Spencer Backman [6].\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0097", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every connected loopless finite multigraph with sink and every hereditary cover of the nonsink vertices, the legal cluster thresholds define a bounded monomial staircase whose lattice exponent vectors are exactly the stable configurations, while the lattice generated by all allowed cluster firings is always the full reduced-Laplacian lattice. Thus the finite quotient is independent of the hereditary rule, whereas the stable staircase and Backman's recurrent section retain the rule. On a cycle C_m, source-to-sink classes of acyclic orientations map bijectively to all nonzero classes of Pic^0(C_m), so adjoining the single missing identity class gives a precise degree-zero completion.\n\nCandidate contribution (theorem; novelty confidence low): The candidate contribution is the combined fixed-quotient/moving-staircase theorem for hereditary cluster firing, together with the explicit one-point completion of cycle source-to-sink classes to Pic^0(C_m)." }, { "id": 20000973, "problem_number": "AIM-COMBINATORICS-0098", "title": "Naturality as an affine automorphism action", "statement": "(8) Jordan Ellenberg: \"By the Matrix-Tree theorem, the number of spanning trees of a graph G equals the size of the critical group of G. When are the spanning trees a torsor for the critical group? Making G into a ribbon graph, and given choice of sink, we do get a torsor structure from the rotor-router model [20]. But under what conditions can we get a natural torsor action without this additional data?\" (a) Jordan Ellenberg: \"If you do not know what a torsor is, I am asking when there is a simply-transitive action of the critical group on the set of spanning trees. Simply-transitive means that for any pair of spanning trees T and T ′, there is a unique group element taking T to T ′. When I say I want this action to be 'natural', I mean at least that I want it to respect the automorphism group of G.\" (b) Farbod Shokrieh: \"David Wagner has an argument (although not a proof) that we cannot expect to get a torsor in general.\" (c) Matt Baker: \"A recent paper of mine and several coauthors [2] almost achieves this through a polyhedral decomposition of the real g-dimensional torus (where g is the genus of the graph). The cells of this decomposition are canonically in bijection with the spanning trees, the vertices are canonically in bijection with Pic g(G), and Pic g(G) is canonically a torsor for the critical group. The only thing that is not canonical here is the bijection between vertices and cells of the decomposition.\" (d) Lionel Levine: \"Why is the genus special here?\" (e) Matt Baker: \"The set of reduced divisors depends on the choice of sink vertex q. It turns out that there is another nice set of coset representatives (well not really coset representatives because they do not form a group, but representatives of the set of degree g divisors modulo chip-firing equivalence) that can be defined in a canonical way.\" (f) Melody Chan: \"In my recent paper with Church and Gro-chow [14], we look at when the choice of sink is needed in the rotor-router model. We show that the resulting torsor is in-dependent of the choice of sink if and only if the graph G is planar.\"", "original_statement": "(8) Jordan Ellenberg: \"By the Matrix-Tree theorem, the number of spanning trees of a graph G equals the size of the critical group of G. When are the spanning trees a torsor for the critical group? Making G into a ribbon graph, and given choice of sink, we do get a torsor structure from the rotor-router model [20]. But under what conditions can we get a natural torsor action without this additional data?\" (a) Jordan Ellenberg: \"If you do not know what a torsor is, I am asking when there is a simply-transitive action of the critical group on the set of spanning trees. Simply-transitive means that for any pair of spanning trees T and T ′, there is a unique group element taking T to T ′. When I say I want this action to be 'natural', I mean at least that I want it to respect the automorphism group of G.\" (b) Farbod Shokrieh: \"David Wagner has an argument (although not a proof) that we cannot expect to get a torsor in general.\" (c) Matt Baker: \"A recent paper of mine and several coauthors [2] almost achieves this through a polyhedral decomposition of the real g-dimensional torus (where g is the genus of the graph). The cells of this decomposition are canonically in bijection with the spanning trees, the vertices are canonically in bijection with Pic g(G), and Pic g(G) is canonically a torsor for the critical group. The only thing that is not canonical here is the bijection between vertices and cells of the decomposition.\" (d) Lionel Levine: \"Why is the genus special here?\" (e) Matt Baker: \"The set of reduced divisors depends on the choice of sink vertex q. It turns out that there is another nice set of coset representatives (well not really coset representatives because they do not form a group, but representatives of the set of degree g divisors modulo chip-firing equivalence) that can be defined in a canonical way.\" (f) Melody Chan: \"In my recent paper with Church and Gro-chow [14], we look at when the choice of sink is needed in the rotor-router model. We show that the resulting torsor is in-dependent of the choice of sink if and only if the graph G is planar.\"", "clean_statement": "(8) Jordan Ellenberg: \"By the Matrix-Tree theorem, the number of spanning trees of a graph G equals the size of the critical group of G. When are the spanning trees a torsor for the critical group? Making G into a ribbon graph, and given choice of sink, we do get a torsor structure from the rotor-router model [20]. But under what conditions can we get a natural torsor action without this additional data?\" (a) Jordan Ellenberg: \"If you do not know what a torsor is, I am asking when there is a simply-transitive action of the critical group on the set of spanning trees. Simply-transitive means that for any pair of spanning trees T and T ′, there is a unique group element taking T to T ′. When I say I want this action to be 'natural', I mean at least that I want it to respect the automorphism group of G.\" (b) Farbod Shokrieh: \"David Wagner has an argument (although not a proof) that we cannot expect to get a torsor in general.\" (c) Matt Baker: \"A recent paper of mine and several coauthors [2] almost achieves this through a polyhedral decomposition of the real g-dimensional torus (where g is the genus of the graph). The cells of this decomposition are canonically in bijection with the spanning trees, the vertices are canonically in bijection with Pic g(G), and Pic g(G) is canonically a torsor for the critical group. The only thing that is not canonical here is the bijection between vertices and cells of the decomposition.\" (d) Lionel Levine: \"Why is the genus special here?\" (e) Matt Baker: \"The set of reduced divisors depends on the choice of sink vertex q. It turns out that there is another nice set of coset representatives (well not really coset representatives because they do not form a group, but representatives of the set of degree g divisors modulo chip-firing equivalence) that can be defined in a canonical way.\" (f) Melody Chan: \"In my recent paper with Church and Gro-chow [14], we look at when the choice of sink is needed in the rotor-router model. We show that the resulting torsor is in-dependent of the choice of sink if and only if the graph G is planar.\"", "statement_status": "exact", "statement_verification": "This is Problem 8 from the AIM workshop list *Generalizations of chip-firing and the critical group*. Jordan Ellenberg asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 8\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[97]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(8) Jordan Ellenberg: \\\"By the Matrix-Tree theorem, the number of spanning trees of a graph G equals the size of the critical group of G. When are the spanning trees a torsor for the critical group? Making G into a ribbon graph, and given choice of sink, we do get a torsor structure from the rotor-router model [20]. But under what conditions can we get a natural torsor action without this additional data?\\\" (a) Jordan Ellenberg: \\\"If you do not know what a torsor is, I am asking when there is a simply-transitive action of the critical group on the set of spanning trees. Simply-transitive means that for any pair of spanning trees T and T ′, there is a unique group element taking T to T ′. When I say I want this action to be 'natural', I mean at least that I want it to respect the automorphism group of G.\\\" (b) Farbod Shokrieh: \\\"David Wagner has an argument (although not a proof) that we cannot expect to get a torsor in general.\\\" (c) Matt Baker: \\\"A recent paper of mine and several coauthors [2] almost achieves this through a polyhedral decomposition of the real g-dimensional torus (where g is the genus of the graph). The cells of this decomposition are canonically in bijection with the spanning trees, the vertices are canonically in bijection with Pic g(G), and Pic g(G) is canonically a torsor for the critical group. The only thing that is not canonical here is the bijection between vertices and cells of the decomposition.\\\" (d) Lionel Levine: \\\"Why is the genus special here?\\\" (e) Matt Baker: \\\"The set of reduced divisors depends on the choice of sink vertex q. It turns out that there is another nice set of coset representatives (well not really coset representatives because they do not form a group, but representatives of the set of degree g divisors modulo chip-firing equivalence) that can be defined in a canonical way.\\\" (f) Melody Chan: \\\"In my recent paper with Church and Gro-chow [14], we look at when the choice of sink is needed in the rotor-router model. We show that the resulting torsor is in-dependent of the choice of sink if and only if the graph G is planar.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0098", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "An automorphism-equivariant critical-group torsor is equivalent to an affine automorphism action on the Jacobian determined by a 1-cocycle. Hence, for every automorphism subgroup H, the H-fixed spanning trees are either absent or number exactly the H-fixed Jacobian elements. This rules out every complete graph K_n for n at least 4: a transposition fixes (n-2)^(n-3) trees but n^(n-3) Jacobian elements. In the positive direction, every connected finite simple unicyclic graph has a choice-free automorphism-equivariant torsor, constructed by identifying its Jacobian and its omitted-cycle-edge tree set with Z/m in an orientation-independent way.\n\nCandidate contribution (theorem; novelty confidence low): For every n >= 4, no Aut(K_n)-equivariant Jac(K_n)-torsor exists on the spanning trees, even after allowing arbitrary affine cocycles; in contrast, every connected finite simple unicyclic graph has the explicit choice-free Aut(G)-equivariant torsor given by the cycle-coordinate formula in the artifacts." }, { "id": 20000974, "problem_number": "AIM-COMBINATORICS-0099", "title": "Complexity and an exact two-vertex formula for divisor rank", "statement": "(9) Lionel Levine: \"Is there an efficient way to compute the rank (in the sense of Baker and Norine [7]) of a divisor?\" PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 5", "original_statement": "(9) Lionel Levine: \"Is there an efficient way to compute the rank (in the sense of Baker and Norine [7]) of a divisor?\" PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 5", "clean_statement": "(9) Lionel Levine: \"Is there an efficient way to compute the rank (in the sense of Baker and Norine [7]) of a divisor?\" PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 5", "statement_status": "exact", "statement_verification": "The canonical JSON record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 9\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[98]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(9) Lionel Levine: \\\"Is there an efficient way to compute the rank (in the sense of Baker and Norine [7]) of a divisor?\\\" PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 5\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0099", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The broad polynomial-time exact-rank question has a negative complexity-theoretic answer: exact Baker--Norine rank is NP-hard even on simple graphs, although reduced-divisor effectivity, fixed thresholds, fixed vertex count, and several graph classes remain tractable. As a proved special-family contribution, for the multigraph B_m with two vertices joined by m parallel edges and D=(x,y), if a is the least nonnegative residue of x modulo m and b=x+y-a, then r(D)=min{a+max(b-m+1,0), max(b,-1)}. This handles arbitrary signed, binary-encoded inputs in polynomial bit time.\n\nCandidate contribution (explicit_formula_and_algorithmic_specialization; novelty confidence low): After one modular normalization, the exact rank of every signed divisor on the binary-encoded two-vertex multigraph B_m is the minimum of two explicit positive-part expressions; a direct proof shows that exactly two obstruction sectors suffice and yields a bit-polynomial algorithm independent of the numerical chip count." }, { "id": 20000975, "problem_number": "AIM-COMBINATORICS-0100", "title": "Semisimple character partitions and a rational cycle obstruction", "statement": "(10) Lionel Levine: \"This question is inspired by Criel Merino's work [25] relating the Tutte polynomial to chip-firing. The Tutte polynomial is the most general invariant of a graph that has a deletion-contraction recurrence. Is there a deletion-contraction recurrence of some sort for critical groups? It would have to be at the level of group algebras rather than groups themselves.\" (a) Jeremy Martin: \"Start by considering just a cycle graph.\" (b) Lionel Levine: \"For hyperplane arrangement, there is a nice deletion-contraction rule that gives an exact sequence of alge-bras; is there a corresponding exact sequence for critical groups?\"", "original_statement": "(10) Lionel Levine: \"This question is inspired by Criel Merino's work [25] relating the Tutte polynomial to chip-firing. The Tutte polynomial is the most general invariant of a graph that has a deletion-contraction recurrence. Is there a deletion-contraction recurrence of some sort for critical groups? It would have to be at the level of group algebras rather than groups themselves.\" (a) Jeremy Martin: \"Start by considering just a cycle graph.\" (b) Lionel Levine: \"For hyperplane arrangement, there is a nice deletion-contraction rule that gives an exact sequence of alge-bras; is there a corresponding exact sequence for critical groups?\"", "clean_statement": "(10) Lionel Levine: \"This question is inspired by Criel Merino's work [25] relating the Tutte polynomial to chip-firing. The Tutte polynomial is the most general invariant of a graph that has a deletion-contraction recurrence. Is there a deletion-contraction recurrence of some sort for critical groups? It would have to be at the level of group algebras rather than groups themselves.\" (a) Jeremy Martin: \"Start by considering just a cycle graph.\" (b) Lionel Levine: \"For hyperplane arrangement, there is a nice deletion-contraction rule that gives an exact sequence of alge-bras; is there a corresponding exact sequence for critical groups?\"", "statement_status": "exact", "statement_verification": "The canonical record is problem 10 from the AIM workshop *Generalizations of chip-firing and the critical group*. The official PDF was checked directly (page 5 of the printed report, PDF page 4). Apart from the line-break OCR error “alge-bras,” the corpus record is accurate:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 10\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[99]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(10) Lionel Levine: \\\"This question is inspired by Criel Merino's work [25] relating the Tutte polynomial to chip-firing. The Tutte polynomial is the most general invariant of a graph that has a deletion-contraction recurrence. Is there a deletion-contraction recurrence of some sort for critical groups? It would have to be at the level of group algebras rather than groups themselves.\\\" (a) Jeremy Martin: \\\"Start by considering just a cycle graph.\\\" (b) Lionel Levine: \\\"For hyperplane arrangement, there is a nice deletion-contraction rule that gives an exact sequence of alge-bras; is there a corresponding exact sequence for critical groups?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0100", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Manjunath's 2022 critical-module sequences give a substantial positive answer to the broad deletion-contraction request. For the literal group-algebra formulation, this attempt proves that over an algebraically closed characteristic-zero field a split exact sequence with contraction algebra as ideal and deletion algebra as quotient always exists, but such sequences are exactly arbitrary partitions and bijections of character sets and therefore encode no critical-group structure without a naturality rule. It gives an explicit Fourier-idempotent sequence for cycles and unicyclic graphs, proves the sequence is neither group-induced nor Hopf-exact, and proves that the triangle admits no analogous sequence over Q or Z in either endpoint order.\n\nCandidate contribution (theorem; novelty confidence low): The candidate contribution is the character-partition classification of all semisimple deletion-contraction group-algebra sequences, including its equivariant naturality criterion, combined with the explicit cycle/unicyclic Fourier sequence and the two-direction rational obstruction for the triangle." }, { "id": 20000976, "problem_number": "AIM-COMBINATORICS-0101", "title": "Periodic Laplacian certificates for robustness and explosion", "statement": "(11) Lionel Levine: \"Suppose I give you a stable, periodic sandpile con-figuration c on the square grid Z2. Is it decidable whether there is an integer n such that c + nδ 0 (i.e. the addition of n grains of sand to the origin) is not stabilizable?\" (a) Matt Baker: \"What do the sets of n such that c + nδ 0 is not stabilizable look like when there is such an n?\" (b) Lionel Levine: \"In that case there is a minimal such n.\" (c) Andrea Sportiello: \"This minimal n must depend on the period; what else could it depend on?\" (d) Lionel Levine: \"Well that would certainly make the problem decidable.\"", "original_statement": "(11) Lionel Levine: \"Suppose I give you a stable, periodic sandpile con-figuration c on the square grid Z2. Is it decidable whether there is an integer n such that c + nδ 0 (i.e. the addition of n grains of sand to the origin) is not stabilizable?\" (a) Matt Baker: \"What do the sets of n such that c + nδ 0 is not stabilizable look like when there is such an n?\" (b) Lionel Levine: \"In that case there is a minimal such n.\" (c) Andrea Sportiello: \"This minimal n must depend on the period; what else could it depend on?\" (d) Lionel Levine: \"Well that would certainly make the problem decidable.\"", "clean_statement": "(11) Lionel Levine: \"Suppose I give you a stable, periodic sandpile con-figuration c on the square grid Z2. Is it decidable whether there is an integer n such that c + nδ 0 (i.e. the addition of n grains of sand to the origin) is not stabilizable?\" (a) Matt Baker: \"What do the sets of n such that c + nδ 0 is not stabilizable look like when there is such an n?\" (b) Lionel Levine: \"In that case there is a minimal such n.\" (c) Andrea Sportiello: \"This minimal n must depend on the period; what else could it depend on?\" (d) Lionel Levine: \"Well that would certainly make the problem decidable.\"", "statement_status": "exact", "statement_verification": "The original AIM PDF, *Problems from the AIM Chip-Firing Workshop*, was checked directly. Problem 11 reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 11\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[100]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(11) Lionel Levine: \\\"Suppose I give you a stable, periodic sandpile con-figuration c on the square grid Z2. Is it decidable whether there is an integer n such that c + nδ 0 (i.e. the addition of n grains of sand to the origin) is not stabilizable?\\\" (a) Matt Baker: \\\"What do the sets of n such that c + nδ 0 is not stabilizable look like when there is such an n?\\\" (b) Lionel Levine: \\\"In that case there is a minimal such n.\\\" (c) Andrea Sportiello: \\\"This minimal n must depend on the period; what else could it depend on?\\\" (d) Lionel Levine: \\\"Well that would certainly make the problem decidable.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0101", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a stable periodic background c, the nonstabilizable source sizes form either the empty set or an upper ray. More strongly, the entire threshold set is invariant when c is changed by the Laplacian of a bounded integer periodic potential, provided both backgrounds remain nonnegative. This yields a finite integer-linear robustness certificate c + Delta p <= 2 and a complementary finite explosion certificate: if c is periodically Laplacian-equivalent to a {2,3}-valued cell having a 3 in every horizontal and vertical residue class, then its explosion threshold is bounded by a computable function of the declared periods. These results do not decide every periodic cell, and the exact two-dimensional problem remains open in the literature checked.\n\nCandidate contribution (theorem; novelty confidence low): The complete one-source stabilizability threshold set E(c), including its minimal explosion threshold, is invariant under bounded periodic integer-Laplacian changes of the background; this invariance gives finite robust and explosive certificates, with a computable period-only threshold bound for the face-saturated explosive subclass." }, { "id": 20000977, "problem_number": "AIM-COMBINATORICS-0102", "title": "Abelian hard-object growth and a greedy obstruction", "statement": "(12) Persi Diaconis: \"Is there an abelian growth model for non-overlapping discs growing in the plane?\" (a) James Propp: \"There is such an abelian model in the case of\n\nnon-nesting discs; firing means if you have a disc that is totally contained in another one, you move it in some fixed way.\" (b) Anton Dochtermann: \"We could even ask if there is an abelian model of the one-dimensional case of non-overlapping intervals on a line.\"", "original_statement": "(12) Persi Diaconis: \"Is there an abelian growth model for non-overlapping discs growing in the plane?\" (a) James Propp: \"There is such an abelian model in the case of \n\nnon-nesting discs; firing means if you have a disc that is totally contained in another one, you move it in some fixed way.\" (b) Anton Dochtermann: \"We could even ask if there is an abelian model of the one-dimensional case of non-overlapping intervals on a line.\"", "clean_statement": "(12) Persi Diaconis: \"Is there an abelian growth model for non-overlapping discs growing in the plane?\" (a) James Propp: \"There is such an abelian model in the case of\n\nnon-nesting discs; firing means if you have a disc that is totally contained in another one, you move it in some fixed way.\" (b) Anton Dochtermann: \"We could even ask if there is an abelian model of the one-dimensional case of non-overlapping intervals on a line.\"", "statement_status": "exact", "statement_verification": "The official AIM problem list states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 12\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[101]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(12) Persi Diaconis: \\\"Is there an abelian growth model for non-overlapping discs growing in the plane?\\\" (a) James Propp: \\\"There is such an abelian model in the case of \\n\\nnon-nesting discs; firing means if you have a disc that is totally contained in another one, you move it in some fixed way.\\\" (b) Anton Dochtermann: \\\"We could even ask if there is an abelian model of the one-dimensional case of non-overlapping intervals on a line.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0102", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Because the AIM prompt leaves growth, overlap, firing, termination, and abelianity unspecified, it has no single formal status. Under an explicit discrete accretion interpretation, a local balanced-stack rule on Z^d is proved to terminate on every finite input and to have a unique site odometer, final state, and hard-object packing; in dimension one it gives non-overlapping intervals and in dimension two lattice-centered non-overlapping discs, with commuting additions. Conversely, the natural rule that grows a chosen fixed-center interval or disc maximally until its target or first contact is proved nonconfluent on three collinear centers.\n\nCandidate contribution (explicit_model_and_obstruction; novelty confidence low): Balanced nearest-neighbor instruction stacks in which every direction occurs infinitely often yield a finite-input odometer-abelian hard-interval/hard-disc accretion model, while fixed-center irreversible maximal-radius growth has two distinct terminal states for centers 0, 2, 3 and target radius 3/4." }, { "id": 20000978, "problem_number": "AIM-COMBINATORICS-0103", "title": "A residual-diamond model for abelian dynamics with shared memory", "statement": "(13) Lionel Levine: \"What is the right definition of an abelian network with shared memory?\" (a) Lionel Levine: \"I have unified some subset of the class of models that behave like chip-firing does, but a lot of models with an abelian property still fall outside this definition and seem to re-quire some kind of shared memory. An example of shared mem-ory is cluster-firing, as investigated by Spencer Backman [6], who in particular studied when the abelian property is pre-served while allowing firing of multiple vertices at once.\" (b) Jordan Ellenberg: \"This seems formally similar to Persi Diaco-nis's question on non-overlapping discs.\"", "original_statement": "(13) Lionel Levine: \"What is the right definition of an abelian network with shared memory?\" (a) Lionel Levine: \"I have unified some subset of the class of models that behave like chip-firing does, but a lot of models with an abelian property still fall outside this definition and seem to re-quire some kind of shared memory. An example of shared mem-ory is cluster-firing, as investigated by Spencer Backman [6], who in particular studied when the abelian property is pre-served while allowing firing of multiple vertices at once.\" (b) Jordan Ellenberg: \"This seems formally similar to Persi Diaco-nis's question on non-overlapping discs.\"", "clean_statement": "(13) Lionel Levine: \"What is the right definition of an abelian network with shared memory?\" (a) Lionel Levine: \"I have unified some subset of the class of models that behave like chip-firing does, but a lot of models with an abelian property still fall outside this definition and seem to re-quire some kind of shared memory. An example of shared mem-ory is cluster-firing, as investigated by Spencer Backman [6], who in particular studied when the abelian property is pre-served while allowing firing of multiple vertices at once.\" (b) Jordan Ellenberg: \"This seems formally similar to Persi Diaco-nis's question on non-overlapping discs.\"", "statement_status": "exact", "statement_verification": "The canonical record is Problem 13 from the AIM workshop *Generalizations of chip-firing and the critical group*. Its extracted text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 13\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[102]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(13) Lionel Levine: \\\"What is the right definition of an abelian network with shared memory?\\\" (a) Lionel Levine: \\\"I have unified some subset of the class of models that behave like chip-firing does, but a lot of models with an abelian property still fall outside this definition and seem to re-quire some kind of shared memory. An example of shared mem-ory is cluster-firing, as investigated by Spencer Backman [6], who in particular studied when the abelian property is pre-served while allowing firing of multiple vertices at once.\\\" (b) Jordan Ellenberg: \\\"This seems formally similar to Persi Diaco-nis's question on non-overlapping discs.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0103", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A shared-memory event system has the full least-action and odometer property when its lifted handlers commute, equivalently when its memory transitions commute and its state-dependent outputs satisfy an explicit cocycle identity. For the weaker batch setting, commuting primitive actions plus difference closure and residual legality give a local diamond; termination then gives a unique normal form, and faithfulness gives a unique atomic odometer even though batch histories can differ. Hereditary cluster-firing satisfies this criterion, terminates by a reduced-Laplacian potential, and admits an explicit two-vertex example with unique vertex odometer but different cluster histories.\n\nCandidate contribution (framework_and_theorem; novelty confidence low): Candidate shared-memory criterion: combine the exact memory-output cocycle for individual handlers with a residual-legality axiom for guarded batches; every terminating difference-closed system satisfying this axiom is confluent, and if its primitive action is faithful then its atomic odometer is unique even when its batch odometer is not." }, { "id": 20000979, "problem_number": "AIM-COMBINATORICS-0104", "title": "The DFS-burning solution and a chordwise Hamilton-path refinement", "statement": "(14) David Perkinson: \"This question is due to Richard Stanley; it is problem 4 of chapter 6 in his notes on hyperplane arrangements [32]. Is there a natural bijection φ between labeled trees T on {0, 1, 2,..., n }6 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\nrooted at 0 and parking functions of length n such that deg( φ(T )) = g − inv( T ),\n\nwhere g is the genus of the complete graph Kn+1 and inv( T ) is the number of inversions of T? I believe such a bijection would extend to an arbitrary graph by replacing the inversion statistic with the κ\n\nstatistic of Ira Gessel [19]. No bijection I have tried works (and there are many bijections between spanning trees and parking functions in the literature).\" (a) Recorder's note: this problem was resolved in [30].", "original_statement": "(14) David Perkinson: \"This question is due to Richard Stanley; it is problem 4 of chapter 6 in his notes on hyperplane arrangements [32]. Is there a natural bijection φ between labeled trees T on {0, 1, 2,..., n }6 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP \n\nrooted at 0 and parking functions of length n such that deg( φ(T )) = g − inv( T ),\n\nwhere g is the genus of the complete graph Kn+1 and inv( T ) is the number of inversions of T? I believe such a bijection would extend to an arbitrary graph by replacing the inversion statistic with the κ\n\nstatistic of Ira Gessel [19]. No bijection I have tried works (and there are many bijections between spanning trees and parking functions in the literature).\" (a) Recorder's note: this problem was resolved in [30].", "clean_statement": "(14) David Perkinson: \"This question is due to Richard Stanley; it is problem 4 of chapter 6 in his notes on hyperplane arrangements [32]. Is there a natural bijection φ between labeled trees T on {0, 1, 2,..., n }6 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\n\nrooted at 0 and parking functions of length n such that deg( φ(T )) = g − inv( T ),\n\nwhere g is the genus of the complete graph Kn+1 and inv( T ) is the number of inversions of T? I believe such a bijection would extend to an arbitrary graph by replacing the inversion statistic with the κ\n\nstatistic of Ira Gessel [19]. No bijection I have tried works (and there are many bijections between spanning trees and parking functions in the literature).\" (a) Recorder's note: this problem was resolved in [30].", "statement_status": "exact", "statement_verification": "The canonical record is David Perkinson's Problem (14) in the AIM chip-firing problem list. The extracted text has only a page-break OCR artifact. Comparison with the linked AIM PDF recovers the question as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 14\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[103]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(14) David Perkinson: \\\"This question is due to Richard Stanley; it is problem 4 of chapter 6 in his notes on hyperplane arrangements [32]. Is there a natural bijection φ between labeled trees T on {0, 1, 2,..., n }6 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP \\n\\nrooted at 0 and parking functions of length n such that deg( φ(T )) = g − inv( T ),\\n\\nwhere g is the genus of the complete graph Kn+1 and inv( T ) is the number of inversions of T? I believe such a bijection would extend to an arbitrary graph by replacing the inversion statistic with the κ\\n\\nstatistic of Ira Gessel [19]. No bijection I have tried works (and there are many bijections between spanning trees and parking functions in the literature).\\\" (a) Recorder's note: this problem was resolved in [30].\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0104", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Perkinson, Yang, and Yu's DFS-burning bijection is the resolution cited as [30]: for every connected simple graph G it satisfies deg(P)=g(G)-kappa(G,T), and for the complete graph kappa is ordinary tree inversion, exactly answering Stanley's question. In addition, this attempt proves a static refinement for any rooted Hamilton-path spanning tree 0-a_1-...-a_n: the inverse parking coordinate at a_k counts precisely the chords {a_j,a_k} with j at most k-2 and a_k>a_{j+1}; every remaining chord corresponds uniquely to the kappa-inversion (a_{j+1},a_k). For the complete graph this restricts the DFS inverse to an explicit bijection between Hamilton paths and functions P with 0<=P(v)<=v-1, inverted by sequential insertion.\n\nCandidate contribution (special-case bijection and edgewise refinement; novelty confidence low): For a rooted Hamilton-path spanning tree 0-a_1-...-a_n, the PYY inverse partitions every cotree chord {a_j,a_k}, j<=k-2, edge by edge: it contributes to P(a_k) iff a_k>a_{j+1}, and otherwise gives the unique kappa-inversion (a_{j+1},a_k); on K_{n+1} the map is exactly the bijection P(a_k)=#{ell T(K_i) -> coker(L_i^ud) -> Hom(ker_Z partial_{i+1}^T,Z) -> 0; when H_i(X;Q)=0, T(K_i)=K_i. A separate proved lemma shows that coordinatewise effectiveness cannot be invariant under independent higher-cell orientation reversals without becoming the whole chain lattice.\n\nCandidate contribution (comparison_theorem; novelty confidence low): The candidate contribution is the explicit coordinate-free comparison identifying the Kim--Perkinson realizable multidegree lattice with Hom(ker_Z L_i^ud,Z), together with the orientation no-go lemma explaining why their fixed-orientation effective cone cannot be promoted unchanged to an orientation-independent cone." }, { "id": 20000987, "problem_number": "AIM-COMBINATORICS-0112", "title": "A dissipative ridge-routing sandpile from a fitting orientation", "statement": "(22) Caroline Klivans: \"Can we develop a chip-firing model for directed rooted forests (as defined, for example, by Olivier Bernardi [10]) in higher dimension?\"", "original_statement": "(22) Caroline Klivans: \"Can we develop a chip-firing model for directed rooted forests (as defined, for example, by Olivier Bernardi [10]) in higher dimension?\"", "clean_statement": "(22) Caroline Klivans: \"Can we develop a chip-firing model for directed rooted forests (as defined, for example, by Olivier Bernardi [10]) in higher dimension?\"", "statement_status": "exact", "statement_verification": "The exact canonical statement is Problem (22) from the 2013 AIM workshop *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 22\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[111]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(22) Caroline Klivans: \\\"Can we develop a chip-firing model for directed rooted forests (as defined, for example, by Olivier Bernardi [10]) in higher dimension?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0112", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Bernardi-Klivans fitting orientations already give the accepted higher-dimensional directed rooted forests. For every such fitting orientation on a finite simplicial complex, the proposed ridge-routing rule sends chips from a nonroot ridge through its matched facet to all other ridges. A proof using forest independence and the chain identity partial squared equals zero shows every nonroot ridge reaches the root, even when the fitting orientation has closed V-paths. Therefore the reduced toppling matrix is a nonsingular M-matrix with finite order-independent stabilization; its determinant counts auxiliary routing arborescences, lies between 1 and d to the power |F|, and attains the upper bound exactly for discrete gradient orientations. In dimension one it is exactly firing toward the roots of the ordinary forest.\n\nCandidate contribution (model_and_theorem; novelty confidence low): Every Bernardi-Klivans fitting orientation canonically induces a dissipative nonnegative ridge-routing sandpile; its critical-group order is the number of routing arborescences and equals d^{|F|} if and only if the fitting orientation has no closed V-path." }, { "id": 20000988, "problem_number": "AIM-COMBINATORICS-0113", "title": "The full toppling Gröbner fan in the tree case", "statement": "(23) Farbod Shokrieh: \"What is the full Gr¨ obner fan of the toppling ideal (as in [26])?\"", "original_statement": "(23) Farbod Shokrieh: \"What is the full Gr¨ obner fan of the toppling ideal (as in [26])?\"", "clean_statement": "(23) Farbod Shokrieh: \"What is the full Gr¨ obner fan of the toppling ideal (as in [26])?\"", "statement_status": "exact", "statement_verification": "The source record is problem 23 from the AIM workshop *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 23\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[112]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(23) Farbod Shokrieh: \\\"What is the full Gr¨ obner fan of the toppling ideal (as in [26])?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0113", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every tree T on n at least 2 vertices, over every field, the homogeneous toppling ideal is the diagonal ideal generated by all x_i-x_j. If S is the set of minimum coordinates of a weight w (using maximum-weight initial forms), then in_w(I_T) is generated by x_i for i outside S and x_s-x_s0 for s in S other than a chosen s0. Hence the complete fan, including all boundary cones, is the normal fan of the reflected (n-1)-simplex: it has 2^n-1 cones, n maximal cones, and lineality R times the all-ones vector. The corresponding nonnegative sink-dehomogenized slice is also determined explicitly.\n\nCandidate contribution (theorem; novelty confidence low): The full homogeneous Gröbner fan of the toppling ideal of every tree is the reflected simplex fan, with the exact boundary formula in_w(I_T)=+ for S=argmin(w), together with an explicit formula for every cell in the nonnegative sink-dehomogenized slice." }, { "id": 20000989, "problem_number": "AIM-COMBINATORICS-0114", "title": "Exact cellular sandpile dynamics on simplex-boundary spheres", "statement": "(24) Art Duval: \"Can we figure out the abelian sandpile dynamics in higher dimensions?\" (a) Matt Baker: \"Do we have pretty three-dimensional pictures?\" (b) Caroline Klivans: \"I have code for this.\"", "original_statement": "(24) Art Duval: \"Can we figure out the abelian sandpile dynamics in higher dimensions?\" (a) Matt Baker: \"Do we have pretty three-dimensional pictures?\" (b) Caroline Klivans: \"I have code for this.\"", "clean_statement": "(24) Art Duval: \"Can we figure out the abelian sandpile dynamics in higher dimensions?\" (a) Matt Baker: \"Do we have pretty three-dimensional pictures?\" (b) Caroline Klivans: \"I have code for this.\"", "statement_status": "exact", "statement_verification": "The source is Problem 24 from the AIM workshop *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 24\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[113]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(24) Art Duval: \\\"Can we figure out the abelian sandpile dynamics in higher dimensions?\\\" (a) Matt Baker: \\\"Do we have pretty three-dimensional pictures?\\\" (b) Caroline Klivans: \\\"I have code for this.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0114", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For X equal to the boundary of a (d+1)-simplex with d at least 2, a codimension-one reduced up-down Laplacian associated with a dual spanning-tree root can be made into an M-matrix by reorienting retained ridges if and only if the dual tree is a path. For every path root the Laplacian is A_{d+1}; stabilization terminates and is confluent, the recurrent configurations are the all-ones vector and the vectors with exactly one zero, the superstables are zero and the unit vectors, and the critical group is cyclic of order d+2. Repeated addition at any ridge has an exact modular moving-vacancy formula, period, and closed Green-kernel odometer in every dimension.\n\nCandidate contribution (theorem; novelty confidence low): On the codimension-one cellular sandpile of every simplex-boundary sphere, a dual spanning-tree root admits graph-style nonnegative (L,L) dynamics after ridge reorientation exactly when the dual tree is a path; for those and only those roots in this orientation-only sense, all recurrent and superstable states and the repeated-addition vacancy orbit and Green-kernel odometer are given explicitly and uniformly in the dimension." }, { "id": 20000990, "problem_number": "AIM-COMBINATORICS-0115", "title": "Explorer aggregation on oriented codimension-one faces", "statement": "(25) Andrea Sportiello: \"What is the higher-dimensional version of the explorer model [16]? Is this model more amenable to generalization the abelian sandpile model?\"", "original_statement": "(25) Andrea Sportiello: \"What is the higher-dimensional version of the explorer model [16]? Is this model more amenable to generalization the abelian sandpile model?\"", "clean_statement": "(25) Andrea Sportiello: \"What is the higher-dimensional version of the explorer model [16]? Is this model more amenable to generalization the abelian sandpile model?\"", "statement_status": "exact", "statement_verification": "The exact corpus record is Andrea Sportiello's Problem (25) from the 2013 AIM workshop *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 25\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[114]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(25) Andrea Sportiello: \\\"What is the higher-dimensional version of the explorer model [16]? Is this model more amenable to generalization the abelian sandpile model?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0115", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Reference [16] is the Diaconis-Fulton explorer model, now called multiple-source internal aggregation or IDLA. Combining its stack stabilization with the Parzanchevski-Rosenthal p-lazy walk on oriented codimension-one faces of a finite pure simplicial complex gives an order-independent capacity-one explorer process; it terminates for fair stacks when every active state reaches a root, conserves particles up to root absorption, and specializes exactly in dimension one to the original graph explorer model. However, giving opposite orientations a single occupancy slot makes the process factor exactly through ordinary IDLA on the unsigned coface-adjacency graph. The filled triangle and its boundary cycle then both give the C3 explorer process despite having first homology 0 and Z, respectively, proving a concrete topology-loss obstruction.\n\nCandidate contribution (model_and_obstruction_theorem; novelty confidence low): Diaconis-Fulton stabilization along the Parzanchevski-Rosenthal oriented ridge walk is abelian and root-terminating under explicit fair-stack hypotheses, while its one-slot-per-unoriented-ridge quotient factors through unsigned upper-adjacency IDLA and can erase homology, as witnessed by the filled triangle versus its boundary cycle." }, { "id": 20000991, "problem_number": "AIM-COMBINATORICS-0116", "title": "Uniform versus torsion-squared higher spanning trees", "statement": "(26) Caroline Klivans: \"Is there an efficient algorithm to generate a higher dimensional spanning tree uniformly at random?\"", "original_statement": "(26) Caroline Klivans: \"Is there an efficient algorithm to generate a higher dimensional spanning tree uniformly at random?\"", "clean_statement": "(26) Caroline Klivans: \"Is there an efficient algorithm to generate a higher dimensional spanning tree uniformly at random?\"", "statement_status": "exact", "statement_verification": "This is Problem 26 from the AIM workshop *Generalizations of chip-firing and the critical group*. The exact source record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 26\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[115]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(26) Caroline Klivans: \\\"Is there an efficient algorithm to generate a higher dimensional spanning tree uniformly at random?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0116", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an integer boundary matrix B, determinant or projection-DPP sampling weights each basis S by the square t(S)^2 of its integral lattice index, equivalently the square of the torsion order of the associated higher spanning forest. These samples become exactly uniform if one accepts a proposal S with probability 1/t(S)^2; the expected proposal count is exactly E_U[t(S)^2]. Thus this is an expected-polynomial exact sampler whenever the uniform torsion second moment is polynomially bounded, and it has no rejection for totally unimodular or, more generally, constant-torsion representations. Independently, known matroid basis walks give polynomial-time epsilon-total-variation uniform sampling for every explicit rational boundary matrix.\n\nCandidate contribution (algorithmic theorem; novelty confidence low): Accepting a basis drawn from the torsion-squared determinant law with probability equal to the reciprocal of its squared torsion order yields an exact uniform basis, and its expected number of proposals is exactly the uniform second moment of the torsion order." }, { "id": 20000992, "problem_number": "AIM-COMBINATORICS-0117", "title": "A planar analytic certificate for metric chip-firing", "statement": "(27) Adrien Kassel: \"How does chip-firing on planar metric graphs relate to questions in analysis?\"", "original_statement": "(27) Adrien Kassel: \"How does chip-firing on planar metric graphs relate to questions in analysis?\"", "clean_statement": "(27) Adrien Kassel: \"How does chip-firing on planar metric graphs relate to questions in analysis?\"", "statement_status": "exact", "statement_verification": "The canonical AIM record is the following broad question from the workshop *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 27\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[116]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(27) Adrien Kassel: \\\"How does chip-firing on planar metric graphs relate to questions in analysis?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0117", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a fixed plane model of a compact metric graph, the weak Poisson solution for a degree-zero atomic source is the minimum-energy real relaxation of chip-firing. If q is any routing of the source, then its difference from the Green-current solution is the gradient of a scalar face potential solving a Poisson equation on the reciprocal-length planar dual, with an exact Pythagorean energy identity. For an integral divisor and integral q, the divisor is principal exactly when this dual gradient is integral. An additional proved H^{-1} estimate controls the error from moving divisor atoms to nearby model vertices.\n\nCandidate contribution (duality theorem and integrality criterion; novelty confidence low): The reciprocal-length dual Poisson equation gives an exact face-potential certificate for metric divisor principality: the energy excess of an integral routing is precisely the dual Dirichlet energy, and the divisor is principal if and only if the resulting dual gradient is integral." }, { "id": 20000993, "problem_number": "AIM-COMBINATORICS-0118", "title": "Tate defects and character splitting for critical groups of abelian graph covers", "statement": "(28) Vic Reiner: \"We know results about the homomorphism between critical groups for two graphs that are related by a Galois covering map (see, e.g., [31]). How do these results relate to class field theory, when the Galois group of the covering is abelian?\" 8 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP", "original_statement": "(28) Vic Reiner: \"We know results about the homomorphism between critical groups for two graphs that are related by a Galois covering map (see, e.g., [31]). How do these results relate to class field theory, when the Galois group of the covering is abelian?\" 8 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP", "clean_statement": "(28) Vic Reiner: \"We know results about the homomorphism between critical groups for two graphs that are related by a Galois covering map (see, e.g., [31]). How do these results relate to class field theory, when the Galois group of the covering is abelian?\" 8 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP", "statement_status": "exact", "statement_verification": "The canonical record is problem (28), attributed to Vic Reiner, from the AIM workshop *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 28\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[117]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(28) Vic Reiner: \\\"We know results about the homomorphism between critical groups for two graphs that are related by a Galois covering map (see, e.g., [31]). How do these results relate to class field theory, when the Galois group of the covering is abelian?\\\" 8 PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0118", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a connected regular abelian graph cover Y to X with deck group A of order n, the ambiguous critical classes J(Y)^A/p^*J(X) and the Prym coinvariant defect ker(p_*)/I_AJ(Y) are canonically the Tate groups H-hat^0(A,J(Y)) and H-hat^{-1}(A,J(Y)); both are killed by n. At every prime not dividing n, the transfer idempotent gives a canonical base/Prym splitting, and over a splitting unramified coefficient ring this refines to the full abelian-character decomposition. For the cyclic cover C_{mr} to C_m, both Tate defects are Z/r, the norm exact sequence splits exactly when gcd(m,r)=1, and the existence of these covers for arbitrary r proves that critical-group quotients cannot classify abelian graph covers.\n\nCandidate contribution (theorem_and_obstruction; novelty confidence low): The paired Tate-cohomological identification of the ambiguous and Prym defects, together with its degree-support theorem and complete calculation for C_{mr} to C_m, gives an explicit integral bridge to class-field norm formalism and an exact obstruction to naive critical-group reciprocity." }, { "id": 20000994, "problem_number": "AIM-COMBINATORICS-0119", "title": "The full Tutte polynomial from chip-firing data", "statement": "(29) Matt Baker: \"Is there a relation between the full Tutte polynomial (as opposed to just TG(1, y ) from Merino's theorem [25]) and chip-firing? We made to use some additional geometric structure.\"", "original_statement": "(29) Matt Baker: \"Is there a relation between the full Tutte polynomial (as opposed to just TG(1, y ) from Merino's theorem [25]) and chip-firing? We made to use some additional geometric structure.\"", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The last sentence is ungrammatical. A highly plausible reconstruction is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 29\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[118]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(29) Matt Baker: \\\"Is there a relation between the full Tutte polynomial (as opposed to just TG(1, y ) from Merino's theorem [25]) and chip-firing? We made to use some additional geometric structure.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0119", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The broad AIM question has a known affirmative answer with auxiliary choices: Bernardi's rooted-embedding activity expansion and his level-preserving bijection between spanning trees and recurrent sandpiles express T_G(x,y) as the sum, over recurrent configurations c, of x raised to the transported internal activity times y raised to the level of c; for plane graphs, x is the level of the complementary dual-tree sandpile and y is the primal level. The new contribution proved here is a canonicity obstruction and explicit repair: on rooted cycles no automorphism-invariant positive monomial x-statistic can recover the full Tutte polynomial, while choosing a cycle orientation gives a positional statistic; this formula extends over arbitrary attached bridges to all connected simple unicyclic graphs with sink on the cycle.\n\nCandidate contribution (obstruction and explicit family theorem; novelty confidence low): For every rooted cycle C_n with n at least 3, reflection symmetry prevents any rooted-automorphism-invariant statistic J from recovering T_Cn(x,y) by summing the positive monomial x raised to J(c) times y raised to the level of c over recurrent configurations c. After orienting the cycle, assigning exponent j to the recurrent configuration whose unique zero is at the j-th nonsink vertex and exponent 0 to the all-one configuration gives the full Tutte polynomial. For a connected simple unicyclic graph, adding the number of bridges to this exponent yields the corresponding full formula." }, { "id": 20000995, "problem_number": "AIM-COMBINATORICS-0120", "title": "Drift splits generalized IDLA scaling", "statement": "(30) Persi Diaconis: \"What does a limiting shape theory for generalized internal DLA (see [22], [4], and [21]) look like?\"", "original_statement": "(30) Persi Diaconis: \"What does a limiting shape theory for generalized internal DLA (see [22], [4], and [21]) look like?\"", "clean_statement": "(30) Persi Diaconis: \"What does a limiting shape theory for generalized internal DLA (see [22], [4], and [21]) look like?\"", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 30\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[119]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(30) Persi Diaconis: \\\"What does a limiting shape theory for generalized internal DLA (see [22], [4], and [21]) look like?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0120", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For one-dimensional internal DLA in which every particle starts at zero and takes nearest-neighbor steps right with probability p and left with probability q, where p>q>0, the upstream occupation length L_N satisfies L_N/log_{p/q}(N) -> 1 almost surely. The proof gives explicit finite-N upper and lower tails by uniformly sandwiching the changing boundary waiting times with biased gambler's-ruin probabilities. A complementary generator expansion explains why centered kernels select elliptic N^{1/d} scaling while fixed drift selects transverse N^{1/(d+1)} and longitudinal N^{2/(d+1)} parabolic scaling under suitable shape-theorem hypotheses.\n\nCandidate contribution (sharp asymptotic boundary law; novelty confidence low): In fixed-bias nearest-neighbor IDLA on Z, the upstream length has the sharp almost-sure asymptotic L_N ~ log_{p/q}(N), with the two explicit finite-size tail inequalities stated in Theorem 5.1." }, { "id": 20000996, "problem_number": "AIM-COMBINATORICS-0121", "title": "A diagonal-lattice reduction for the tropical Picard group of a product of graphs", "statement": "(31) Dustin Cartwright: \"What is the (tropical) Picard group (in the sense of [13]) of the product of graphs? I have a guess if one of the graphs is a tree.\"", "original_statement": "(31) Dustin Cartwright: \"What is the (tropical) Picard group (in the sense of [13]) of the product of graphs? I have a guess if one of the graphs is a tree.\"", "clean_statement": "(31) Dustin Cartwright: \"What is the (tropical) Picard group (in the sense of [13]) of the product of graphs? I have a guess if one of the graphs is a tree.\"", "statement_status": "exact", "statement_verification": "The canonical record is AIM problem 31 from the workshop *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 31\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[120]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(31) Dustin Cartwright: \\\"What is the (tropical) Picard group (in the sense of [13]) of the product of graphs? I have a guess if one of the graphs is a tree.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0121", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Lazar proved Cartwright's tree-factor conjecture in 2017: the natural factor map Pic(G) x Pic(H) -> Pic_ridge(Gamma) is injective and is an isomorphism when either factor is a tree. Thus Pic_ridge(G x T) is Pic(G) x Z, equivalently Z^2 plus the critical group K(G). For arbitrary connected loopless multigraphs, this attempt proves a canonical exact sequence whose quotient is the lattice of diagonal coefficients of Cartier divisors modulo the principal diagonal lattice. The latter is the image of a signed Kronecker product of the two incidence matrices and is primitive. Hence the cokernel of the factor map is always free abelian and all torsion in Pic_ridge(Gamma) is exactly K(G) x K(H); Lazar's stronger conjecture is reduced to computing the remaining free rank.\n\nCandidate contribution (exact_sequence; novelty confidence low): For every diagonal triangulation Gamma of the cubical product of finite connected loopless multigraphs G and H, Pic_ridge(Gamma)/im(gamma) is Lambda_Gamma/Q_Gamma, where Lambda_Gamma is the diagonal restriction of the integral Cartier lattice and Q_Gamma is the primitive image of a row-signed mixed incidence matrix B_G tensor B_H. Consequently this quotient is free abelian and Tor Pic_ridge(Gamma) is canonically K(G) x K(H)." }, { "id": 20000997, "problem_number": "AIM-COMBINATORICS-0122", "title": "Rotor antiparticles and inverse topplings: a common quotient but different full dynamics", "statement": "(32) Jim Propp: \"Is Andrea Sportiello's model of inverse topplings [12] the same as the particle creation and deletion model?\"", "original_statement": "(32) Jim Propp: \"Is Andrea Sportiello's model of inverse topplings [12] the same as the particle creation and deletion model?\"", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "There is a second plausible reading. CPS explicitly identify one of their Markov chains with the Karmakar--Manna particle--hole protocol. On that reading the answer is tautologically yes, but it does not answer the rotor-routing question described by the workshop summary. Both readings are separated below.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 32\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[121]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(32) Jim Propp: \\\"Is Andrea Sportiello's model of inverse topplings [12] the same as the particle creation and deletion model?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0122", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The official AIM workshop summary shows that the likely intended comparison is between rotor-routing particles/antiparticles and Caracciolo-Paoletti-Sportiello addition/removal operators, not merely the Karmakar-Manna chain. For finite undirected sinked graphs, both systems factor onto signed translations by the same generators of the critical group: CPS addition/removal act by plus or minus [e_v] modulo the toppling lattice, while rotor particle/antiparticle addition act by the same elements on rotor-equivalence classes, a critical-group torsor. This quotient agreement does not lift in general to a conjugacy of full state spaces. On an explicit five-edge simple graph with twelve stable height states and twelve rotor states, the center CPS addition map has image size nine whereas the center rotor-particle map has image size eight.\n\nCandidate contribution (obstruction and quotient comparison; novelty confidence low): For the simple graph with nonsinks c,l,r, sink t, and edges ct, cl, cr, lt, rt, with rotor orders c:(t,l,r), l:(t,c), r:(t,c), the CPS center addition map and the rotor center-particle map act on equal twelve-element state spaces but have fiber-size multisets {0,0,0,1,1,1,1,1,1,2,2,2} and {0,0,0,0,1,1,1,1,1,2,2,3}, respectively. Hence no state-space bijection conjugates these labeled maps, although both factor through translation by [e_c] on the same order-eight critical group." }, { "id": 20000998, "problem_number": "AIM-COMBINATORICS-0123", "title": "A positive-density obstruction for the prop-agator limit", "statement": "(33) Lionel Levine: \"What is the infinite volume limit of Andrea's prop-agator model [12] with a finite density of propagators?\"", "original_statement": "(33) Lionel Levine: \"What is the infinite volume limit of Andrea's prop-agator model [12] with a finite density of propagators?\"", "clean_statement": "(33) Lionel Levine: \"What is the infinite volume limit of Andrea's prop-agator model [12] with a finite density of propagators?\"", "statement_status": "exact", "statement_verification": "The AIM list asks, verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 33\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[122]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(33) Lionel Levine: \\\"What is the infinite volume limit of Andrea's prop-agator model [12] with a finite density of propagators?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0123", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the documented reconstruction of the prop-agator as the idempotent Pi_x = a_x^dagger a_x, run to the common absorbing state on a prescribed active set I, the maximal BTW initial state on a wired box has at least |I|/(2d+1) height defects and dissipates at least that many grains to the sink. Hence active density p forces defect density at least p/(2d+1) in every local weak limit. Moreover, no positive-density stationary active set admits a translation-covariant conservative infinite-volume pair-stable realization with a finite odometer at every site; a random-walk telescoping proof gives this without assuming the odometer is integrable.\n\nCandidate contribution (obstruction; novelty confidence low): For the active-set absorbing reconstruction, I is contained in the closed nearest-neighbourhood of the final defect set, so |D| and wired-boundary loss are at least |I|/(2d+1); the corresponding positive-density state cannot arise from a stationary conservative stabilization having finitely many topplings per site." }, { "id": 20000999, "problem_number": "AIM-COMBINATORICS-0124", "title": "An arrival-gated interpolation from IDLA to the abelian sandpile", "statement": "(34) Jordan Ellenberg: \"What does a stochastic sandpile model that in-terpolates between DLA [16] and the sandpile model look like?\"", "original_statement": "(34) Jordan Ellenberg: \"What does a stochastic sandpile model that in-terpolates between DLA [16] and the sandpile model look like?\"", "clean_statement": "(34) Jordan Ellenberg: \"What does a stochastic sandpile model that in-terpolates between DLA [16] and the sandpile model look like?\"", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 34\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[123]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(34) Jordan Ellenberg: \\\"What does a stochastic sandpile model that in-terpolates between DLA [16] and the sandpile model look like?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0124", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Reference [16] is Diaconis-Fulton internal DLA, not external DLA. On any finite directed multigraph with a global sink and nonsink outdegree at least two, an arrival-gated unary network that independently chooses a first-vacancy explorer gate with probability 1-theta or a full degree-threshold sandpile gate with probability theta is pathwise abelian, conservative up to sink loss, and almost surely terminating. Its theta=0 law is exactly IDLA, its theta=1 stabilization is exactly the classical abelian sandpile, and every fixed finite-input terminal law is continuous in theta in total variation. On the one-vertex d-arc family, the stationary law is explicit and shows that the long-time limit and theta down to zero do not commute for d at least three.\n\nCandidate contribution (abelian interpolation theorem and exact stationary obstruction; novelty confidence low): The arrival-gated processor family has exact Diaconis-Fulton IDLA and classical abelian-sandpile endpoints, pathwise order independence, almost-sure finite-volume termination, and total-variation continuity; on one vertex with d sink arcs its stationary probabilities are theta/(theta+d-1) at height zero and 1/(theta+d-1) at every positive height, yielding a noncommuting endpoint/long-time limit for d at least three." }, { "id": 20001000, "problem_number": "AIM-COMBINATORICS-0125", "title": "Small-density i.i.d. sandpiles stabilize", "statement": "(35) Leonardo Rolla: \"If you start with an i.i.d. number of chips with some small density μ in Zd, is this configuration stabilizable with high probability?\"", "original_statement": "(35) Leonardo Rolla: \"If you start with an i.i.d. number of chips with some small density μ in Zd, is this configuration stabilizable with high probability?\"", "clean_statement": "the **ordinary fixed-energy abelian sandpile**. A configuration is \\(\\eta:\\mathbb Z^d\\to\\mathbb Z_{\\ge0}\\); a site with at least \\(2d\\) chips may topple, losing \\(2d\\) chips and sending one to each nearest neighbour. Randomness is only in the i.i.d. initial heights. The configuration is stabilizable if a legal toppling procedure reaches heights at most \\(2d-1\\) while toppling every site only finitely often. This matches the terminology, workshop setting, and the precise infinite-volume problem studied by Fey--Meester--Redig.", "statement_status": "reconstructed_unverified", "statement_verification": "Throughout this report, \\(\\mu=\\mathbb E\\eta(0)\\) denotes mean chip density. For a product law, the stabilization event is translation invariant and the law is ergodic. Its probability is therefore exactly zero or one. Thus “with high probability” in infinite volume should be read as “almost surely.” On a finite wired box every configuration stabilizes, so a literal finite-volume stabilization probability would be identically one and would not express the intended question.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 35\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[124]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(35) Leonardo Rolla: \\\"If you start with an i.i.d. number of chips with some small density μ in Zd, is this configuration stabilizable with high probability?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0125", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the ordinary fixed-energy abelian-sandpile reconstruction, the qualitative AIM question is already answered affirmatively: every i.i.d. nonnegative integer height field on Z^d of mean less than d stabilizes almost surely, and the product zero-one law upgrades 'high probability' to probability one. In addition, if q(t)=4d^2 exp(-t) E[exp(t eta(0))] is less than one for some t>0, then stabilization holds and the toppled component at the origin has tail at most (4d^2)^(-1) q(t)^n/(1-q(t)). For Poisson mean mu this optimizes to q_d(mu)=4d^2 mu exp(1-mu), giving a closed-form quantitative localization certificate.\n\nCandidate contribution (criterion; novelty confidence low): The explicit sufficient condition 4d^2 exp(-t) E[exp(t eta(0))] < 1 yields the stated exponential toppled-cluster tail; for Poisson input it becomes 4d^2 mu exp(1-mu) < 1, and hence mu < 1/(4 e d^2) is a simple sufficient condition." }, { "id": 20001001, "problem_number": "AIM-COMBINATORICS-0126", "title": "Cartwright local matrices as link walks with potential", "statement": "(36) Lionel Levine: \"The graph Laplacian is the generator of a random walk on a graph. Does the Laplacian of a simplicial complex generate some kind of stochastic process on this higher-dimensional object?\" (a) Caroline Klivans: \"Look up the work of Russel Lyons [24].\" (b) Lionel Levine: \"I have and it is interesting but not what I want.\" (c) Andrea Sportiello: \"Dustin Cartwright's model [13] looks more like a random walk with the way it assigns weights.\" (d) Matt Baker: \"So the concrete question is: does Dustin's model generate some stochastic process?\"", "original_statement": "(36) Lionel Levine: \"The graph Laplacian is the generator of a random walk on a graph. Does the Laplacian of a simplicial complex generate some kind of stochastic process on this higher-dimensional object?\" (a) Caroline Klivans: \"Look up the work of Russel Lyons [24].\" (b) Lionel Levine: \"I have and it is interesting but not what I want.\" (c) Andrea Sportiello: \"Dustin Cartwright's model [13] looks more like a random walk with the way it assigns weights.\" (d) Matt Baker: \"So the concrete question is: does Dustin's model generate some stochastic process?\"", "clean_statement": "(36) Lionel Levine: \"The graph Laplacian is the generator of a random walk on a graph. Does the Laplacian of a simplicial complex generate some kind of stochastic process on this higher-dimensional object?\" (a) Caroline Klivans: \"Look up the work of Russel Lyons [24].\" (b) Lionel Levine: \"I have and it is interesting but not what I want.\" (c) Andrea Sportiello: \"Dustin Cartwright's model [13] looks more like a random walk with the way it assigns weights.\" (d) Matt Baker: \"So the concrete question is: does Dustin's model generate some stochastic process?\"", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 36 from the workshop *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 36\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[125]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(36) Lionel Levine: \\\"The graph Laplacian is the generator of a random walk on a graph. Does the Laplacian of a simplicial complex generate some kind of stochastic process on this higher-dimensional object?\\\" (a) Caroline Klivans: \\\"Look up the work of Russel Lyons [24].\\\" (b) Lionel Levine: \\\"I have and it is interesting but not what I want.\\\" (c) Andrea Sportiello: \\\"Dustin Cartwright's model [13] looks more like a random walk with the way it assigns weights.\\\" (d) Matt Baker: \\\"So the concrete question is: does Dustin's model generate some stochastic process?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0126", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every codimension-two face q of a weak tropical complex, Cartwright's local intersection matrix has the proved decomposition M_q = Q_link(q) + diag(deg_link(q) - alpha), where Q_link(q) is the conservative continuous-time random-walk generator on the local link. Hence M_q is itself a Markov generator exactly when alpha equals local link degree at every state, and a single scalar normalization makes its semigroup Markov exactly when the discrepancy is constant. For general simplicial Hodge Laplacians, an explicit conservative Markov chain on a doubled signed-cell state space represents the heat semigroup through a signed expectation. The filled-triangle calculation shows why the full Hodge Laplacian need not directly move among cells.\n\nCandidate contribution (lemma; novelty confidence low): Cartwright's local intersection matrix satisfies M_q = Q_link(q) + diag(deg_link(q) - alpha), yielding an exact local Feynman--Kac representation and the necessary-and-sufficient conservation criterion alpha = deg_link(q)." }, { "id": 20001002, "problem_number": "AIM-COMBINATORICS-0127", "title": "Scaled intertwining, time change, and lumpability for harmonic Markov maps", "statement": "(37) James Propp: \"What is known about harmonic maps (in the sense of Baker and Norine [8], Cory [15], etc.) between Markov chains?\" (a) Matt Baker: \"The paper of Urakawa's [33] that we cite in our original article already is aware of this connection to random walks. I have not followed it up since then. We also have a notion of harmonic maps for metric graphs.\"", "original_statement": "(37) James Propp: \"What is known about harmonic maps (in the sense of Baker and Norine [8], Cory [15], etc.) between Markov chains?\" (a) Matt Baker: \"The paper of Urakawa's [33] that we cite in our original article already is aware of this connection to random walks. I have not followed it up since then. We also have a notion of harmonic maps for metric graphs.\"", "clean_statement": "(37) James Propp: \"What is known about harmonic maps (in the sense of Baker and Norine [8], Cory [15], etc.) between Markov chains?\" (a) Matt Baker: \"The paper of Urakawa's [33] that we cite in our original article already is aware of this connection to random walks. I have not followed it up since then. We also have a notion of harmonic maps for metric graphs.\"", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 37 from the workshop *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 37\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[126]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(37) James Propp: \\\"What is known about harmonic maps (in the sense of Baker and Norine [8], Cory [15], etc.) between Markov chains?\\\" (a) Matt Baker: \\\"The paper of Urakawa's [33] that we cite in our original article already is aware of this connection to random walks. I have not followed it up since then. We also have a notion of harmonic maps for metric graphs.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0127", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For finite Markov kernels, pullback of all functions harmonic at a target state is equivalent to a scaled block-intertwining identity p_x=(1-a(x))delta_y+a(x)Q(y,·). For irreducible zero-holding target chains, this makes the successive distinct image states an exact target chain; the raw image has one target-state transition kernel valid for every initial law exactly when a(x) is constant on each fiber, and it equals the target kernel exactly when a=1. For a Baker--Norine harmonic graph morphism the scale is a(x)=m_phi(x)deg_H(phi(x))/deg_G(x)=1-v_phi(x)/deg_G(x), yielding a ramification--lumpability trichotomy, a stationary pushforward formula, and a precise multigraph converse obstruction.\n\nCandidate contribution (theorem_and_counterexample; novelty confidence low): Candidate contribution: the scaled local-harmonicity identity, random-time-change theorem, and strong-lumpability criterion combine to give the explicit graph trichotomy a=1-v/deg; in the fiberwise-constant case the pushed-forward degree measure is proportional to deg_H(y)/a_y, while a two-vertex parallel-edge example shows that vertex-chain data cannot recover Baker--Norine edgewise harmonicity for multigraph targets." }, { "id": 20001003, "problem_number": "AIM-COMBINATORICS-0128", "title": "Recognizing undirected reduced Laplacian lattices", "statement": "(38) David Perkinson: \"If you have an n × n integer matrix of full rank, is that the Laplacian of a graph? Well, of course not in general, but if you are allowed to do row operations you can always get a graph (in fact an infinite number of them) with this property. This is due to an algorithm of Wilmes [29]. But to accomplish this, you need to allow the resulting graph to be directed. (a) Spencer Backman: \"The question is related to trying to under-stand when a directed graph has the Riemann-Roch property. Amini and Manjunath [1] studied when orthogonal lattices have the Riemann-Roch property, and via the algorithm of Wilmes, Arash Asadi and I [3] were able to reduce this to a question of directed graphs. So a solution to Dave's question would help PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 9\n\nwith understanding Riemann-Roch on directed graphs (which appears to be a very delicate question).\"", "original_statement": "(38) David Perkinson: \"If you have an n × n integer matrix of full rank, is that the Laplacian of a graph? Well, of course not in general, but if you are allowed to do row operations you can always get a graph (in fact an infinite number of them) with this property. This is due to an algorithm of Wilmes [29]. But to accomplish this, you need to allow the resulting graph to be directed. (a) Spencer Backman: \"The question is related to trying to under-stand when a directed graph has the Riemann-Roch property. Amini and Manjunath [1] studied when orthogonal lattices have the Riemann-Roch property, and via the algorithm of Wilmes, Arash Asadi and I [3] were able to reduce this to a question of directed graphs. So a solution to Dave's question would help PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 9\n\nwith understanding Riemann-Roch on directed graphs (which appears to be a very delicate question).\"", "clean_statement": "(38) David Perkinson: \"If you have an n × n integer matrix of full rank, is that the Laplacian of a graph? Well, of course not in general, but if you are allowed to do row operations you can always get a graph (in fact an infinite number of them) with this property. This is due to an algorithm of Wilmes [29]. But to accomplish this, you need to allow the resulting graph to be directed. (a) Spencer Backman: \"The question is related to trying to under-stand when a directed graph has the Riemann-Roch property. Amini and Manjunath [1] studied when orthogonal lattices have the Riemann-Roch property, and via the algorithm of Wilmes, Arash Asadi and I [3] were able to reduce this to a question of directed graphs. So a solution to Dave's question would help PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 9\n\nwith understanding Riemann-Roch on directed graphs (which appears to be a very delicate question).\"", "statement_status": "exact", "statement_verification": "The AIM workshop list records David Perkinson's setup:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 38\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[127]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(38) David Perkinson: \\\"If you have an n × n integer matrix of full rank, is that the Laplacian of a graph? Well, of course not in general, but if you are allowed to do row operations you can always get a graph (in fact an infinite number of them) with this property. This is due to an algorithm of Wilmes [29]. But to accomplish this, you need to allow the resulting graph to be directed. (a) Spencer Backman: \\\"The question is related to trying to under-stand when a directed graph has the Riemann-Roch property. Amini and Manjunath [1] studied when orthogonal lattices have the Riemann-Roch property, and via the algorithm of Wilmes, Arash Asadi and I [3] were able to reduce this to a question of directed graphs. So a solution to Dave's question would help PROBLEMS FROM THE AIM CHIP-FIRING WORKSHOP 9\\n\\nwith understanding Riemann-Roch on directed graphs (which appears to be a very delicate question).\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0128", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The truncated AIM entry is recovered as Perkinson--Perlman--Wilmes Question 4.15: characterize full-rank sublattices of Z^n that are reduced Laplacian lattices of undirected multigraphs. For a lattice of index D, such a realization exists exactly when a nonnegative integral root-Gram Laplacian with every edge weight at most D has all rows in the lattice and determinant D; this gives a finite exact recognition certificate. In rank two the criterion is: find a,b,c between 0 and D with ab+ac+bc=D and rows (a+c,-c),(-c,b+c) in the lattice. Every index-at-most-three rank-two lattice passes, while row-span((2,0),(1,2)) is a proved index-four obstruction. A graphical index-four lattice with the same Smith invariants shows that the abstract critical group is insufficient.\n\nCandidate contribution (criterion_and_obstruction; novelty confidence low): Candidate contribution: every undirected realization of an index-D embedded lattice has each edge multiplicity at most D, so bounded root-Gram enumeration plus row membership and determinant D is an exact certificate; in rank two this yields the explicit ab+ac+bc test, a smallest-index obstruction at D=4, and a paired counterexample showing Smith normal form does not decide graphicality." }, { "id": 20001004, "problem_number": "AIM-COMBINATORICS-0129", "title": "Fixed-graph recognition and a rotation certificate for maximal harmonic actions", "statement": "(39) Gregg Musiker: \"Can we characterize which genus g graphs have a maximal harmonic group (in the sense of Corry [15]) of size 6( g−1)?\" (a) Scott Corry: \"If you know the maximal graph group, I can make a graph on which it acts harmonically by a variation of a Cayley graph construction.\" (b) Gregg Musiker: \"But if I hand you a graph, can you tell me if it has a group of size 6( g − 1) that acts harmonically on it?\"", "original_statement": "(39) Gregg Musiker: \"Can we characterize which genus g graphs have a maximal harmonic group (in the sense of Corry [15]) of size 6( g−1)?\" (a) Scott Corry: \"If you know the maximal graph group, I can make a graph on which it acts harmonically by a variation of a Cayley graph construction.\" (b) Gregg Musiker: \"But if I hand you a graph, can you tell me if it has a group of size 6( g − 1) that acts harmonically on it?\"", "clean_statement": "(39) Gregg Musiker: \"Can we characterize which genus g graphs have a maximal harmonic group (in the sense of Corry [15]) of size 6( g−1)?\" (a) Scott Corry: \"If you know the maximal graph group, I can make a graph on which it acts harmonically by a variation of a Cayley graph construction.\" (b) Gregg Musiker: \"But if I hand you a graph, can you tell me if it has a group of size 6( g − 1) that acts harmonically on it?\"", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 39 from the 2013 workshop *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 39\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[128]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(39) Gregg Musiker: \\\"Can we characterize which genus g graphs have a maximal harmonic group (in the sense of Corry [15]) of size 6( g−1)?\\\" (a) Scott Corry: \\\"If you know the maximal graph group, I can make a graph on which it acts harmonically by a variation of a Cayley graph construction.\\\" (b) Gregg Musiker: \\\"But if I hand you a graph, can you tell me if it has a group of size 6( g − 1) that acts harmonically on it?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0129", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every finite connected loopless multigraph Y of Betti genus g at least 2, maximality is decided exactly by whether Aut(Y) contains a subgroup of order 6(g-1) acting semiregularly on darts. If Y has minimum degree at least two, Corry's uniform-subdivision theorem sharpens this: suppressing degree-two vertices must give a loopless cubic graph C with uniform subdivision lengths, and C is admissible exactly when some choice of local cyclic orders gives a dart permutation group generated by vertex rotation and dart reversal of order |D(C)|. A necessary arithmetic obstruction valid even with leaves is |E(Y)| = 3k(g-1).\n\nCandidate contribution (equivalence; novelty confidence low): Candidate fixed-graph certificate: a leafless graph is maximal exactly when it is a uniform subdivision of a loopless cubic graph C for which a choice of cyclic order at each vertex makes the monodromy group generated by the local rotation permutation and dart reversal have order |D(C)|; equivalently its centralizer is a dart-regular automorphism group. Additionally every maximal graph satisfies 3(g-1) divides |E|." }, { "id": 20001005, "problem_number": "AIM-COMBINATORICS-0130", "title": "Cone-stack moduli and stabilizer criteria for harmonic metric-graph quotients", "statement": "(40) Jordan Ellenberg: \"Is there a Deligne-Mumford stack on graphs?\" (a) Matt Baker: \"We have a paper on metric graphs that accom-plishes some of this.\" (b) Jordan: \"But there is still a question of what are harmonic group actions on metric graphs?\"", "original_statement": "(40) Jordan Ellenberg: \"Is there a Deligne-Mumford stack on graphs?\" (a) Matt Baker: \"We have a paper on metric graphs that accom-plishes some of this.\" (b) Jordan: \"But there is still a question of what are harmonic group actions on metric graphs?\"", "clean_statement": "(40) Jordan Ellenberg: \"Is there a Deligne-Mumford stack on graphs?\" (a) Matt Baker: \"We have a paper on metric graphs that accom-plishes some of this.\" (b) Jordan: \"But there is still a question of what are harmonic group actions on metric graphs?\"", "statement_status": "exact", "statement_verification": "The exact canonical record is Problem 40 from the 2013 AIM workshop *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 40\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[129]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(40) Jordan Ellenberg: \\\"Is there a Deligne-Mumford stack on graphs?\\\" (a) Matt Baker: \\\"We have a paper on metric graphs that accom-plishes some of this.\\\" (b) Jordan: \\\"But there is still a question of what are harmonic group actions on metric graphs?\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0130", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The stable form of the requested Deligne--Mumford analogue is now supplied by the cone stack of tropical curves, whose cells are finite quotients [R_{>0}^{E(G)}/Aut(G)]. For a finite isometric action K on a connected metric graph, after resolving edge inversions, the standard quotient metric l(bar e)=|K_e|l(e) makes the orbit map harmonic with edge slope |K_e|, local degree |K_x|, and global degree |K|. The additional proved criterion is that the unscaled path-metric orbit map is harmonic if and only if the generic edge-stabilizer order |K_e| is constant over all edges; its degree is then |K|/|K_e|.\n\nCandidate contribution (criterion_and_obstruction; novelty confidence low): Candidate contribution: after equivariantly resolving inversions, the ordinary path-metric quotient of a connected metric graph by a finite isometry group is harmonic exactly when all generic edge stabilizers have the same order; a C2-action on a Y-graph with one fixed edge and two exchanged edges is the sharp smallest obstruction. The criterion is subdivision invariant and explains precisely when the coarse path metric differs essentially from the stabilizer-weighted harmonic-Galois metric." }, { "id": 20001006, "problem_number": "AIM-COMBINATORICS-0131", "title": "What the critical group sees of a harmonic action", "statement": "(41) Lionel Levine: \"If you have a graph with a harmonic action, then its critical group comes with some extra structure, namely a group of automorphisms of the graph. That extra object might be worth studying.\"", "original_statement": "(41) Lionel Levine: \"If you have a graph with a harmonic action, then its critical group comes with some extra structure, namely a group of automorphisms of the graph. That extra object might be worth studying.\"", "clean_statement": "(41) Lionel Levine: \"If you have a graph with a harmonic action, then its critical group comes with some extra structure, namely a group of automorphisms of the graph. That extra object might be worth studying.\"", "statement_status": "exact", "statement_verification": "This is item (41), attributed to Lionel Levine, in the AIM workshop list *Generalizations of chip-firing and the critical group*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 41\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[130]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(41) Lionel Levine: \\\"If you have a graph with a harmonic action, then its critical group comes with some extra structure, namely a group of automorphisms of the graph. That extra object might be worth studying.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0131", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Levine's item is a research suggestion rather than a fixed problem. For a finite connected bridgeless loopless multigraph X with n vertices, let K be the kernel of Aut(X) acting on Jac(X), and let P be the subgroup fixing every vertex. The vertex-independent displacement d(sigma)=[sigma(v)-v] gives an injection K/P into Jac(X)[n]. Hence for simple bridgeless graphs the invisible automorphism group is abelian n-torsion. Combining this with the 2025 semiregular faithfulness theorem of Estelyi--Karabas--Mednykh--Nedela proves that the full automorphism group of every simple connected 3-edge-connected graph acts faithfully on its critical group. In contrast, every finite group has a faithful harmonic action on a banana multigraph that is completely invisible on the critical group.\n\nCandidate contribution (theorem; novelty confidence low): The candidate contribution is the canonical kernel-displacement injection ker(Aut(X) to Aut(Jac(X)))/ker(Aut(X) to Sym(V(X))) into Jac(X)[|V(X)|] for bridgeless multigraphs, together with its consequence that Aut(X) acts faithfully on Jac(X) for every simple connected 3-edge-connected graph.", "research_difficulty_suggested": "L2" }, { "id": 20001007, "problem_number": "AIM-COMBINATORICS-0132", "title": "The pentagon lift and a periodic obstruction on general graphs", "statement": "(42) Matt Baker: \"Peter Winkler [35] gives an interesting solution to a famous math Olympiad problem involving numbers on a pentagon in his book on math puzzles. His solution takes infinite covers of the pentagon. I have not tried to generalize this proof to the usual chip-firing game; Farbod and I [9] have a solution that uses potential theory. But maybe we could apply Winkler's method to general graphs?\" (a) Matt Macauley: \"Is this pentagon game the same as numbers game for Coxeter groups?\" (b) Matt Baker \"In fact it is a special case. The numbers game is a different generalization of chip-firing on a cycle than chip-firing on a general graph is.\"", "original_statement": "(42) Matt Baker: \"Peter Winkler [35] gives an interesting solution to a famous math Olympiad problem involving numbers on a pentagon in his book on math puzzles. His solution takes infinite covers of the pentagon. I have not tried to generalize this proof to the usual chip-firing game; Farbod and I [9] have a solution that uses potential theory. But maybe we could apply Winkler's method to general graphs?\" (a) Matt Macauley: \"Is this pentagon game the same as numbers game for Coxeter groups?\" (b) Matt Baker \"In fact it is a special case. The numbers game is a different generalization of chip-firing on a cycle than chip-firing on a general graph is.\"", "clean_statement": "(42) Matt Baker: \"Peter Winkler [35] gives an interesting solution to a famous math Olympiad problem involving numbers on a pentagon in his book on math puzzles. His solution takes infinite covers of the pentagon. I have not tried to generalize this proof to the usual chip-firing game; Farbod and I [9] have a solution that uses potential theory. But maybe we could apply Winkler's method to general graphs?\" (a) Matt Macauley: \"Is this pentagon game the same as numbers game for Coxeter groups?\" (b) Matt Baker \"In fact it is a special case. The numbers game is a different generalization of chip-firing on a cycle than chip-firing on a general graph is.\"", "statement_status": "exact", "statement_verification": "The AIM workshop record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Generalizations of chip-firing and the critical group\nSection: \nSource item: 42\nSource URL: https://aimath.org/pastworkshops/chipfiringproblems.pdf\nCanonical location: aim-combinatorics-notes.json notes[131]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(42) Matt Baker: \\\"Peter Winkler [35] gives an interesting solution to a famous math Olympiad problem involving numbers on a pentagon in his book on math puzzles. His solution takes infinite covers of the pentagon. I have not tried to generalize this proof to the usual chip-firing game; Farbod and I [9] have a solution that uses potential theory. But maybe we could apply Winkler's method to general graphs?\\\" (a) Matt Macauley: \\\"Is this pentagon game the same as numbers game for Coxeter groups?\\\" (b) Matt Baker \\\"In fact it is a special case. The numbers game is a different generalization of chip-firing on a cycle than chip-firing on a general graph is.\\\"\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0132", "aim-domain:combinatorics", "aim-workshop:chipfiringproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "Winkler's periodic partial-sum lift gives a corrected inversion-orbit proof of termination on every cycle, but the natural positive-total general-graph extension is false. For every m at least 3, unit borrowing and batch debt-clearing on K_{2,m}, started with value -1 on the two-vertex part and +1 on the m-vertex part, follow a legal period of length m+2 and have positive total m-2. In addition, a batch move at a vertex of degree d and debt y changes the Baker-Shokrieh summed energy by n(d-2)y^2+2sy, exactly isolating the degree-two mechanism.\n\nCandidate contribution (counterexample family and energy identity; novelty confidence low): For every graph genus g at least 2, K_{2,g+1} has an explicit forced periodic borrowing orbit of degree g-1, and the same orbit is a periodic batch debt-clearing orbit; paired with the exact summed-energy change n(d-2)y^2+2sy, this supplies a concrete obstruction to extending the cycle sorting proof verbatim." }, { "id": 20001008, "problem_number": "AIM-COMBINATORICS-0133", "title": "Composition-resolved LL fibers and a finite-flatness obstruction", "statement": "A more complete introduction to this problem is given in V. Ripoll's paper \\htmladdnormallink{Lyashoko-Looijenga Morphisms and Submaximal Factorizations of a Coxeter Element}{http://arxiv.org/pdf/1012.3825v3.pdf} \\cite{arXiv:1012.3825}.\n\nLet $W$ be a well-generated irreducible complex reflection group. One of the major open problems in Catalan combinatorics that V. Reiner outlined was to give a uniform proof that noncrossing objects are counted by $$Cat^p(W) = \\prod_{i=1}^n \\frac{ph+d_i}{d_i}.$$\n\nThis is a suggestion for how to uniformly prove this formula for such $W$ (note that in the non-well-generated case, $Cat^p(W)$ is not even an integer). Such a uniform proof would be new, even for $p=1$.\n\nWe know case-by-case that $Cat^p(W)$ counts multichains $w_1 \\leq w_2 \\leq \\cdots \\leq w_p \\leq c$ of length $p$ in $NC(W,c)$. Using standard zeta polynomial computations, we can relate multichains in $[1,c]_T$ to strictly increasing chains in $[1,c]_T$, which we then interpret as reduced factorizations of the Coxeter element into nontrivial factors. It is interesting that even though the formula for multichains has a beautiful uniform expression, the formula for strict chains does not seem nice in general.\n\nA $k$-block factorization of $c$ is a factorization of c into $k$ factors $$c=u_1 u_2 \\cdots u_k,$$ such that $\\sum_{i=1}^k \\ell_T(u_i) = \\ell_T(c) = n$ and $u_i \\neq 1$.\n\nLet $Fact_k(c)$ denote the number of $k$-block factorizations of $c$.\n\nUsing the relation between multichains and strict chains, we obtain the identity $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c).$$\n\nDespite the fact that $Fact_k (c)$ does not seem to have a nice uniform formula in general, we do understand it geometrically---$Fact_k (c)$ counts the cardinality of fibers of the Lyashko-Looijenga map. In particular, D. Bessis showed case-by-case that the generic fibers of this map correspond to $n$-block factorizations, or reduced decompositions in the reflections $T$ \\cite{arXiv:math/0610777}. Even though it was done case-by-case, we have the uniform formula $$Fact_n(c) = \\frac{n! h^n}{|W|} = \\frac{(h)(2h)(3h)\\cdots(nh)}{d_1 d_2 \\cdots d_n}.$$ V. Ripoll gave a similar formula for the almost generic fibers.\n\nUniformly prove $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c)$$ by interpreting the right-hand side as fibers of the Lyashko-Looijenga map.", "original_statement": "A more complete introduction to this problem is given in V. Ripoll's paper \\htmladdnormallink{Lyashoko-Looijenga Morphisms and Submaximal Factorizations of a Coxeter Element}{http://arxiv.org/pdf/1012.3825v3.pdf} \\cite{arXiv:1012.3825}.\n\nLet $W$ be a well-generated irreducible complex reflection group. One of the major open problems in Catalan combinatorics that V. Reiner outlined was to give a uniform proof that noncrossing objects are counted by $$Cat^p(W) = \\prod_{i=1}^n \\frac{ph+d_i}{d_i}.$$\n\nThis is a suggestion for how to uniformly prove this formula for such $W$ (note that in the non-well-generated case, $Cat^p(W)$ is not even an integer). Such a uniform proof would be new, even for $p=1$.\n\nWe know case-by-case that $Cat^p(W)$ counts multichains $w_1 \\leq w_2 \\leq \\cdots \\leq w_p \\leq c$ of length $p$ in $NC(W,c)$. Using standard zeta polynomial computations, we can relate multichains in $[1,c]_T$ to strictly increasing chains in $[1,c]_T$, which we then interpret as reduced factorizations of the Coxeter element into nontrivial factors. It is interesting that even though the formula for multichains has a beautiful uniform expression, the formula for strict chains does not seem nice in general.\n\nA $k$-block factorization of $c$ is a factorization of c into $k$ factors $$c=u_1 u_2 \\cdots u_k,$$ such that $\\sum_{i=1}^k \\ell_T(u_i) = \\ell_T(c) = n$ and $u_i \\neq 1$.\n\nLet $Fact_k(c)$ denote the number of $k$-block factorizations of $c$.\n\nUsing the relation between multichains and strict chains, we obtain the identity $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c).$$\n\nDespite the fact that $Fact_k (c)$ does not seem to have a nice uniform formula in general, we do understand it geometrically---$Fact_k (c)$ counts the cardinality of fibers of the Lyashko-Looijenga map. In particular, D. Bessis showed case-by-case that the generic fibers of this map correspond to $n$-block factorizations, or reduced decompositions in the reflections $T$ \\cite{arXiv:math/0610777}. Even though it was done case-by-case, we have the uniform formula $$Fact_n(c) = \\frac{n! h^n}{|W|} = \\frac{(h)(2h)(3h)\\cdots(nh)}{d_1 d_2 \\cdots d_n}.$$ V. Ripoll gave a similar formula for the almost generic fibers.\n\nUniformly prove $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c)$$ by interpreting the right-hand side as fibers of the Lyashko-Looijenga map.", "clean_statement": "A more complete introduction to this problem is given in V. Ripoll's paper \\htmladdnormallink{Lyashoko-Looijenga Morphisms and Submaximal Factorizations of a Coxeter Element}{http://arxiv.org/pdf/1012.3825v3.pdf} \\cite{arXiv:1012.3825}.\n\nLet $W$ be a well-generated irreducible complex reflection group. One of the major open problems in Catalan combinatorics that V. Reiner outlined was to give a uniform proof that noncrossing objects are counted by $$Cat^p(W) = \\prod_{i=1}^n \\frac{ph+d_i}{d_i}.$$\n\nThis is a suggestion for how to uniformly prove this formula for such $W$ (note that in the non-well-generated case, $Cat^p(W)$ is not even an integer). Such a uniform proof would be new, even for $p=1$.\n\nWe know case-by-case that $Cat^p(W)$ counts multichains $w_1 \\leq w_2 \\leq \\cdots \\leq w_p \\leq c$ of length $p$ in $NC(W,c)$. Using standard zeta polynomial computations, we can relate multichains in $[1,c]_T$ to strictly increasing chains in $[1,c]_T$, which we then interpret as reduced factorizations of the Coxeter element into nontrivial factors. It is interesting that even though the formula for multichains has a beautiful uniform expression, the formula for strict chains does not seem nice in general.\n\nA $k$-block factorization of $c$ is a factorization of c into $k$ factors $$c=u_1 u_2 \\cdots u_k,$$ such that $\\sum_{i=1}^k \\ell_T(u_i) = \\ell_T(c) = n$ and $u_i \\neq 1$.\n\nLet $Fact_k(c)$ denote the number of $k$-block factorizations of $c$.\n\nUsing the relation between multichains and strict chains, we obtain the identity $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c).$$\n\nDespite the fact that $Fact_k (c)$ does not seem to have a nice uniform formula in general, we do understand it geometrically---$Fact_k (c)$ counts the cardinality of fibers of the Lyashko-Looijenga map. In particular, D. Bessis showed case-by-case that the generic fibers of this map correspond to $n$-block factorizations, or reduced decompositions in the reflections $T$ \\cite{arXiv:math/0610777}. Even though it was done case-by-case, we have the uniform formula $$Fact_n(c) = \\frac{n! h^n}{|W|} = \\frac{(h)(2h)(3h)\\cdots(nh)}{d_1 d_2 \\cdots d_n}.$$ V. Ripoll gave a similar formula for the almost generic fibers.\n\nUniformly prove $$\\prod_{i=1}^n \\frac{ph+d_i}{d_i} = \\sum_{k=1}^n \\binom{p+1}{k} Fact_k (c)$$ by interpreting the right-hand side as fibers of the Lyashko-Looijenga map.", "statement_status": "exact", "statement_verification": "The canonical record is `aim-combinatorics-notes.json`, zero-based index 132, from the AIM workshop *Rational Catalan combinatorics*, section “Catalan Combinatorics and Reflection Groups,” Problem 1.1. The supplied source URL is . It returned an HTTP 502 error during this run (30 July 2026), so the text below is preserved exactly from `input.json`. In particular, the misspelling “Lyashoko-Looijenga” is in the source; the standard spelling is “Lyashko--Looijenga.”", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Rational Catalan combinatorics\nSection: Catalan Combinatorics and Reflection Groups\nSource item: 1.1\nSource URL: http://aimpl.org/rationalcatalan/1/\nCanonical location: aim-combinatorics-notes.json notes[132]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A more complete introduction to this problem is given in V. Ripoll's paper \\\\htmladdnormallink{Lyashoko-Looijenga Morphisms and Submaximal Factorizations of a Coxeter Element}{http://arxiv.org/pdf/1012.3825v3.pdf} \\\\cite{arXiv:1012.3825}.\\n\\nLet $W$ be a well-generated irreducible complex reflection group. One of the major open problems in Catalan combinatorics that V. Reiner outlined was to give a uniform proof that noncrossing objects are counted by $$Cat^p(W) = \\\\prod_{i=1}^n \\\\frac{ph+d_i}{d_i}.$$\\n\\nThis is a suggestion for how to uniformly prove this formula for such $W$ (note that in the non-well-generated case, $Cat^p(W)$ is not even an integer). Such a uniform proof would be new, even for $p=1$.\\n\\nWe know case-by-case that $Cat^p(W)$ counts multichains $w_1 \\\\leq w_2 \\\\leq \\\\cdots \\\\leq w_p \\\\leq c$ of length $p$ in $NC(W,c)$. Using standard zeta polynomial computations, we can relate multichains in $[1,c]_T$ to strictly increasing chains in $[1,c]_T$, which we then interpret as reduced factorizations of the Coxeter element into nontrivial factors. It is interesting that even though the formula for multichains has a beautiful uniform expression, the formula for strict chains does not seem nice in general.\\n\\nA $k$-block factorization of $c$ is a factorization of c into $k$ factors $$c=u_1 u_2 \\\\cdots u_k,$$ such that $\\\\sum_{i=1}^k \\\\ell_T(u_i) = \\\\ell_T(c) = n$ and $u_i \\\\neq 1$.\\n\\nLet $Fact_k(c)$ denote the number of $k$-block factorizations of $c$.\\n\\nUsing the relation between multichains and strict chains, we obtain the identity $$\\\\prod_{i=1}^n \\\\frac{ph+d_i}{d_i} = \\\\sum_{k=1}^n \\\\binom{p+1}{k} Fact_k (c).$$\\n\\nDespite the fact that $Fact_k (c)$ does not seem to have a nice uniform formula in general, we do understand it geometrically---$Fact_k (c)$ counts the cardinality of fibers of the Lyashko-Looijenga map. In particular, D. Bessis showed case-by-case that the generic fibers of this map correspond to $n$-block factorizations, or reduced decompositions in the reflections $T$ \\\\cite{arXiv:math/0610777}. Even though it was done case-by-case, we have the uniform formula $$Fact_n(c) = \\\\frac{n! h^n}{|W|} = \\\\frac{(h)(2h)(3h)\\\\cdots(nh)}{d_1 d_2 \\\\cdots d_n}.$$ V. Ripoll gave a similar formula for the almost generic fibers.\\n\\nUniformly prove $$\\\\prod_{i=1}^n \\\\frac{ph+d_i}{d_i} = \\\\sum_{k=1}^n \\\\binom{p+1}{k} Fact_k (c)$$ by interpreting the right-hand side as fibers of the Lyashko-Looijenga map.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/rationalcatalan/1/", "tags": [ "aim", "AIM-COMBINATORICS-0133", "aim-domain:combinatorics", "aim-workshop:rationalcatalan", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The all-complex classification-free Lyashko--Looijenga proof requested by the source remains open, although type-independent Fuss-chain formulas are now known for real reflection groups. This attempt proves a precise reduction: weak p-chains are block factorizations with nonidentity blocks placed in p+1 separator slots; each k-block total is a sum of reduced LL fibers indexed by ordered compositions of n; and the desired product is equivalent to explicit Newton finite-difference formulas for those totals. It also proves that the unchanged LL map is finite flat of rank n!h^n/|W|, so no single one of its fibers can itself realize the unbounded Fuss--Catalan polynomial. The decoration or aggregation requested by the problem is therefore essential, not cosmetic.\n\nCandidate contribution (obstruction_and_reduction; novelty confidence low): Candidate contribution: a single fiber of the original unchanged LL morphism cannot realize the Fuss--Catalan polynomial for all p because LL is finite flat of fixed rank; the exact viable replacement is the binomially decorated sum of composition-specific reduced LL fibers, whose aggregate cardinalities must equal the explicit Newton differences Delta^k[P(x-1)] at x=0.", "research_difficulty_suggested": "L4" }, { "id": 20001009, "problem_number": "AIM-COMBINATORICS-0134", "title": "Quarantining a test placeholder by an item-identifiability criterion", "statement": "This is just a test!", "original_statement": "This is just a test!", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "**Recovered statement.** No mathematical statement is recoverable at item level from the canonical record. The record is an AIMPL test/placeholder entry, or the residue of an overwritten entry; the available evidence does not distinguish those possibilities. The status used here is `invalid_statement`, with the underlying record classified as `not_a_problem`.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Rational Catalan combinatorics\nSection: Symmetric Functions, Macdonald Polynomials, and Diagonal Harmonics\nSource item: 2.1\nSource URL: http://aimpl.org/rationalcatalan/2/\nCanonical location: aim-combinatorics-notes.json notes[133]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"This is just a test!\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/rationalcatalan/2/", "tags": [ "aim", "AIM-COMBINATORICS-0134", "aim-domain:combinatorics", "aim-workshop:rationalcatalan", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The exact body, \"This is just a test!\", is metatext rather than a mathematical statement. A 2013 workshop outline recovers four distinct problems in the same subsection, but no checked source associates any one of them with AIMPL endpoint /rationalcatalan/2/; moreover, the adjacent endpoint /1/ corresponds to outline Problem 3, refuting ordinal matching. A proved evidence-equivariance obstruction therefore rules out selecting a unique contextual replacement, so this item must remain invalid_statement unless an item-level snapshot or database export is recovered. Separately, a proved finite-fiber criterion characterizes when two statistics admit a swapping involution, providing a sharply formulated synthesis of the section's q,t-symmetry theme without imputing it to this record.\n\nCandidate contribution (recovery criterion; novelty confidence low): For a corpus item containing explicit placeholder metatext, if no item-identifying witness is recoverable and at least two distinct section-level candidates remain, evidence-equivariance forbids a unique supported selection; the item must be quarantined rather than repaired by order or section membership.", "research_difficulty_suggested": "L2" }, { "id": 20001010, "problem_number": "AIM-COMBINATORICS-0135", "title": "Calibrating curvature measurements on empirical networks", "statement": "How do we measure the curvature of real networks?", "original_statement": "How do we measure the curvature of real networks?", "clean_statement": "How do we measure the curvature of real networks?", "statement_status": "exact", "statement_verification": "The exact AIM Problem List record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Geometry of large networks\nSection: Problem session\nSource item: 11.05\nSource URL: http://aimpl.org/largenetworks/1/\nCanonical location: aim-combinatorics-notes.json notes[134]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How do we measure the curvature of real networks?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/largenetworks/1/", "tags": [ "aim", "AIM-COMBINATORICS-0135", "aim-domain:combinatorics", "aim-workshop:largenetworks", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "There is no canonical curvature of a bare empirical graph: the metric, transition kernel, generator, face structure, conductance semantics, and scale select different measurements. A proved calibration theorem shows that on a girth-at-least-six edge of degrees p and q, lazy Ollivier curvature is determined by the reciprocal-degree statistic 1/p+1/q, whereas Forman curvature is 4-p-q. Explicit finite double-star trees therefore have equal Forman curvature but opposite-sign Lin-Lu-Yau curvature, or equal Lin-Lu-Yau curvature but different Forman curvature. A second proposition bounds fixed-topology Ollivier-curvature error by local total-variation and normalized cost errors.\n\nCandidate contribution (comparison theorem and stability certificate; novelty confidence low): The explicit double-star witnesses (degree pairs (1,5) versus (2,4), and (2,6) versus (3,3)) paired with the fixed-topology bound |delta kappa| <= 2(TV_x+TV_y)+eta form a testable calibration certificate: raw Forman/Ollivier correlation cannot establish interchangeability, while transition-probability error has a computable curvature bound when both costs keep the central edge length normalized to one." }, { "id": 20001011, "problem_number": "AIM-COMBINATORICS-0136", "title": "Variable-curvature distortion bounds for embedded neighborhood graphs", "statement": "Need ways to compute geodesics on spaces where curvature is variable (using embedded graphs).", "original_statement": "Need ways to compute geodesics on spaces where curvature is variable (using embedded graphs).", "clean_statement": "Need ways to compute geodesics on spaces where curvature is variable (using embedded graphs).", "statement_status": "exact", "statement_verification": "The canonical repository record is AIM-COMBINATORICS-0136, item 11.1 in the “Problem session” of the AIM workshop *Geometry of large networks*. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Geometry of large networks\nSection: Problem session\nSource item: 11.1\nSource URL: http://aimpl.org/largenetworks/1/\nCanonical location: aim-combinatorics-notes.json notes[135]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Need ways to compute geodesics on spaces where curvature is variable (using embedded graphs).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/largenetworks/1/", "tags": [ "aim", "AIM-COMBINATORICS-0136", "aim-domain:combinatorics", "aim-workshop:largenetworks", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an intrinsic h-net on a complete C2 embedded manifold, an embedded graph that contains all sample pairs within coverage radius r_- and has no intrinsically nonlocal edges satisfies a two-sided geodesic-distance estimate. With ambient chord weights, the lower relative loss is at most eta^2/24, where eta is the largest edgewise product of extrinsic geodesic curvature and intrinsic edge length, while the upper sampling detour is at most 2h/(r_- - 2h). This yields an explicit epsilon-accuracy rule and separates variable-curvature bias, sampling density, and global fold-shortcut control.\n\nCandidate contribution (theorem; novelty confidence low): Candidate finite-sample theorem: separate graph coverage and locality radii and impose the edgewise variable-curvature budget sup_e K_e s_e; then (1-eta^2/24)d_M <= d_G <= (1+2h/(r_- - 2h))d_M. In the uniform exact-radius case, epsilon accuracy is feasible when Kh <= sqrt(6) epsilon^(3/2)/(1+epsilon)." }, { "id": 20001012, "problem_number": "AIM-COMBINATORICS-0137", "title": "Operational meaning and identifiability limits of network curvature", "statement": "Why measure it? (What are the real-world applications?)", "original_statement": "Why measure it? (What are the real-world applications?)", "clean_statement": "Why measure it? (What are the real-world applications?)", "statement_status": "exact", "statement_verification": "The exact canonical record, zero-based index 136 of `aim-combinatorics-notes.json`, asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Geometry of large networks\nSection: Problem session\nSource item: 11.15\nSource URL: http://aimpl.org/largenetworks/1/\nCanonical location: aim-combinatorics-notes.json notes[136]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Why measure it? (What are the real-world applications?)\"\nOriginal remarks: [\"Wasn't that your result, that the internet has curvature $-1$ and dimension $2$?\", \"So, yes, we need more examples like that, showing the significance of curvature.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/largenetworks/1/", "tags": [ "aim", "AIM-COMBINATORICS-0137", "aim-domain:combinatorics", "aim-workshop:largenetworks", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The 2011 remark about the Internet having curvature -1 and dimension 2 is best reconstructed as a statement about a fitted two-dimensional latent hyperbolic model, not an intrinsic graph-curvature measurement. Curvature has rigorous operational value only after its notion and task are fixed: additive target-distance error below 1/2 certifies shortest-path greedy routing; every n-vertex tree has centroid transit load at least floor((n-1)^2/4), yet a star and clique have the same vertex four-point hyperbolicity delta_4=0 and opposite congestion; and hyperbolic volume growth identifies only (d-1)kappa, not dimension and curvature separately. These results motivate a three-gate certificate requiring identification, task sufficiency, and stability.\n\nCandidate contribution (operational criterion and theorem package; novelty confidence low): Candidate contribution: use a three-gate operational identifiability certificate for network-curvature applications, with explicit sharp witnesses—uniform routing-score error below 1/2 guarantees success and stretch 1; (n, delta_4) cannot identify point congestion because S_n and K_n both have delta_4=0 but respective maximum internal loads binomial(n-1,2) and 0; hyperbolic growth determines only (d-1)kappa; and same-node metric perturbation by epsilon changes delta_4 by at most 2epsilon." }, { "id": 20001013, "problem_number": "AIM-COMBINATORICS-0138", "title": "A hybrid distance-to-tree and sparse separating families", "statement": "Find (different) notions of distance on the space of graphs, or perhaps measures on this space. In particular: how far is a given graph from being a tree?", "original_statement": "Find (different) notions of distance on the space of graphs, or perhaps measures on this space. In particular: how far is a given graph from being a tree?", "clean_statement": "Find (different) notions of distance on the space of graphs, or perhaps measures on this space. In particular: how far is a given graph from being a tree?", "statement_status": "exact", "statement_verification": "The canonical record is item 11.2 in the problem session of the 2011 AIM workshop *Geometry of large networks*. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Geometry of large networks\nSection: Problem session\nSource item: 11.2\nSource URL: http://aimpl.org/largenetworks/1/\nCanonical location: aim-combinatorics-notes.json notes[137]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find (different) notions of distance on the space of graphs, or perhaps measures on this space. In particular: how far is a given graph from being a tree?\"\nOriginal remarks: [\"perhaps need to look at non-dense graphs; I have the impression that the problem is solved for dense ones.\", \"How about tree-width?\", \"Tree-width is not good enough. There is a relation to tree-length.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/largenetworks/1/", "tags": [ "aim", "AIM-COMBINATORICS-0138", "aim-domain:combinatorics", "aim-workshop:largenetworks", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "On connected graphs with a common labeled vertex set, the sum of edge symmetric-difference and lambda times sup-norm shortest-path distortion is a genuine metric and descends to an unlabeled quotient metric. Its distance to the set of trees is at least the cyclomatic number plus lambda for every non-tree. Long cycles and cliques prove that edit distance, tree-width, tree-length, and additive Steiner tree-metric defect rank tree-likeness in opposite ways; additionally, dense cut distance collapses every o(n^2)-edge sequence to the zero graphon.\n\nCandidate contribution (theorem; novelty confidence low): Candidate hybrid formulation: D_lambda(G,H)=|E(G) symmetric-difference E(H)|+lambda||d_G-d_H||_infinity is a metric whose tree distance combines exact cycle deletion and global path distortion. The paired families satisfy A_infinity(C_{12k}) >= 3k/2 with beta=1 and tree-width 2, whereas A_infinity(K_n)=0 and tree-length 1 with beta=(n-1)(n-2)/2 and tree-width n-1; dense cut distance equals 2m/n^2 from the zero graphon." }, { "id": 20001014, "problem_number": "AIM-COMBINATORICS-0139", "title": "Curvature versus clustering in regular networks", "statement": "Is curvature related to clustering?", "original_statement": "Is curvature related to clustering?", "clean_statement": "Is curvature related to clustering?", "statement_status": "exact", "statement_verification": "The canonical record is problem 11.25 from the AIM workshop **Geometry of large networks**:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Geometry of large networks\nSection: Problem session\nSource item: 11.25\nSource URL: http://aimpl.org/largenetworks/1/\nCanonical location: aim-combinatorics-notes.json notes[138]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is curvature related to clustering?\"\nOriginal remarks: [\"What is meant by clustering?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/largenetworks/1/", "tags": [ "aim", "AIM-COMBINATORICS-0139", "aim-domain:combinatorics", "aim-workshop:largenetworks", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After fixing non-lazy Ollivier curvature and Watts--Strogatz triangle clustering, every simple unweighted d-regular graph satisfies -2+4/d+((3d-5)/d)C(x) <= K_0(x) <= ((d-1)/d)C(x). The cube Q_3 and Heawood graph are finite connected 3-regular bipartite graphs with C=0 everywhere, yet their constant LLY edge curvatures are respectively +2/3 and -2/3 (and their non-lazy curvatures are 0 and -2/3). Thus clustering quantitatively bounds a specified curvature but does not determine it or even determine the sign after degree is fixed.\n\nCandidate contribution (theorem_and_counterexample; novelty confidence low): Candidate novelty: the unconditional endpoint-chord scalar bound -2+4/d+((3d-5)/d)C(x) <= K_0(x), requiring no same-branch hypothesis on incident triangle counts, paired with the explicit Q_3/Heawood degree-controlled LLY sign-reversal certificate." }, { "id": 20001015, "problem_number": "AIM-COMBINATORICS-0140", "title": "A typed and scale-aware definition protocol for network curvature", "statement": "What is the curvature of a network? How to define it?", "original_statement": "What is the curvature of a network? How to define it?", "clean_statement": "What is the curvature of a network? How to define it?", "statement_status": "exact", "statement_verification": "The exact canonical AIM Problem Lists record, zero-based index 139 of `aim-combinatorics-notes.json`, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Geometry of large networks\nSection: Problem session\nSource item: 11.3\nSource URL: http://aimpl.org/largenetworks/1/\nCanonical location: aim-combinatorics-notes.json notes[139]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the curvature of a network? How to define it?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/largenetworks/1/", "tags": [ "aim", "AIM-COMBINATORICS-0140", "aim-domain:combinatorics", "aim-workshop:largenetworks", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "There is no unqualified scalar curvature of a bare network that simultaneously captures the established local and global comparison principles. A curvature statement should instead specify an enhanced object, output carrier, scale, comparison axiom, and normalization. This necessity is proved in two directions: every fixed-radius edge functional is constant on all sufficiently long cycles although delta_4(C_{4m})=m, while a symmetric local Gauss–Bonnet allocation on a declared polygonal complex uniquely forces K(v)=1-deg(v)/2+sum_{f containing v}1/|boundary f|, yet the same K4 adjacency has vertex curvature -1/2 as a graph and +1/2 as a tetrahedral sphere. An exact formula for nonlazy Ollivier curvature on tree edges further separates local branching from global hyperbolicity.\n\nCandidate contribution (axiomatic compatibility and impossibility theorem package; novelty confidence low): Candidate contribution: a typed-curvature minimality certificate requiring both (i) the enhancement needed by the comparison axiom and (ii) an explicit scale or global descriptor. It is witnessed sharply by the uniqueness of the symmetric local Gauss–Bonnet coefficients A=1, B=-1/2, C_k=1/k once polygonal faces are supplied; the K4 sign reversal under two fillings; the equality of every fixed-radius local edge value on C_{4m} while delta_4=m grows without bound; and the exact tree formula kappa_0(x,y)=-2 max(0,1-1/deg(x)-1/deg(y))." }, { "id": 20001016, "problem_number": "AIM-COMBINATORICS-0141", "title": "Exact degree-sequence obstruction to hyperbolicity indicators", "statement": "For random graphs, is there an idea for a degree of heterogeneity that could indicate hyperbolicity", "original_statement": "For random graphs, is there an idea for a degree of heterogeneity that could indicate hyperbolicity", "clean_statement": "For random graphs, is there an idea for a degree of heterogeneity that could indicate hyperbolicity", "statement_status": "exact", "statement_verification": "The source URL in the record did not yield a browsable copy during this run. The statement above is therefore reproduced from the exact canonical repository record; it contains no apparent OCR corruption.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Geometry of large networks\nSection: Problem session\nSource item: 11.35\nSource URL: http://aimpl.org/largenetworks/1/\nCanonical location: aim-combinatorics-notes.json notes[140]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For random graphs, is there an idea for a degree of heterogeneity that could indicate hyperbolicity\"\nOriginal remarks: [\"Similar to the question of frequency of hyperbolic groups among finitely generated/presented groups: could ask --\\n\\n* What is a set of conditions ensuring hyperbolicity of an ensemble of random graphs (i.e., the set of hyperbolic graphs in the ensemble has probability one)?\\n\\n* (Mahoney) important to emphasize randomness.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/largenetworks/1/", "tags": [ "aim", "AIM-COMBINATORICS-0141", "aim-domain:combinatorics", "aim-workshop:largenetworks", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every m,k >= 1 there are two connected simple unicyclic graphs on m+4k vertices with exactly the same degree multiset (m+2, 2 repeated 4k-1 times, 1 repeated m times), but four-point vertex hyperbolicities 0 and k. Uniform random relabeling gives vertex-exchangeable ensembles with identical complete degree data almost surely but opposite fixed-C and normalized hyperbolicity behavior. For random cactus graphs, a complementary positive result shows that hyperbolicity is tight in probability exactly when the longest cycle length is tight in probability.\n\nCandidate contribution (theorem; novelty confidence low): Candidate exact-degree exchangeable obstruction: the displayed unicyclic degree sequence has realizations S_{m,k} and L_{m,k} with delta(S_{m,k})=0 and delta(L_{m,k})=k; after uniform relabeling, no degree-only statistic can distinguish an almost-surely 0-hyperbolic ensemble from one with delta/diameter tending to 1/2." }, { "id": 20001017, "problem_number": "AIM-COMBINATORICS-0142", "title": "Positive graph curvature requires a scale audit", "statement": "What would be meant by a graph having *positive* curvature? (Grids don't have it; trees don't; how about spheres?)", "original_statement": "What would be meant by a graph having *positive* curvature? (Grids don't have it; trees don't; how about spheres?)", "clean_statement": "What would be meant by a graph having *positive* curvature? (Grids don't have it; trees don't; how about spheres?)", "statement_status": "exact", "statement_verification": "The canonical record is problem 11.4 from the AIM workshop *Geometry of large networks*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Geometry of large networks\nSection: Problem session\nSource item: 11.4\nSource URL: http://aimpl.org/largenetworks/1/\nCanonical location: aim-combinatorics-notes.json notes[141]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What would be meant by a graph having *positive* curvature? (Grids don't have it; trees don't; how about spheres?)\"\nOriginal remarks: [\"That depends on a choice of scale.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/largenetworks/1/", "tags": [ "aim", "AIM-COMBINATORICS-0142", "aim-domain:combinatorics", "aim-workshop:largenetworks", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For r-step half-lazy Ollivier curvature, every finite connected graph becomes uniformly positively curved after sufficiently many steps, with an explicit spectral lower bound, whereas every edge of the infinite grid remains exactly flat at all diffusion times. Locally, the central edge of a finite double-star has curvature 2/d-1<0 but eventually becomes positive, every octahedral edge has curvature 1/2, and uniform L-subdivision forces at least the fraction 1-2/L of all edges to have exactly zero one-step curvature. These results rigorously show that positivity is incomplete without specifying diffusion time, metric normalization, and behavior under refinement.\n\nCandidate contribution (theorem; novelty confidence low): The candidate contribution is a paired scale-audit certificate for one fixed curvature convention: an explicit spectral estimate proves eventual post-mixing positivity on every finite connected graph, while an exact subdivision theorem proves fixed-hop flattening on at least a 1-2/L fraction of refined edges; the double-star supplies an exact negative-to-positive sign flip between these regimes." }, { "id": 20001018, "problem_number": "AIM-COMBINATORICS-0143", "title": "Curvature, cores, capacity, and the price of detours", "statement": "What precise notions of negative curvature imply congestion?", "original_statement": "What precise notions of negative curvature imply congestion?", "clean_statement": "What precise notions of negative curvature imply congestion?", "statement_status": "exact", "statement_verification": "The exact AIM record, from the workshop *Geometry of large networks*, Problem 11.45, asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Geometry of large networks\nSection: Problem session\nSource item: 11.45\nSource URL: http://aimpl.org/largenetworks/1/\nCanonical location: aim-combinatorics-notes.json notes[142]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What precise notions of negative curvature imply congestion?\"\nOriginal remarks: [\"What price are you willing to pay to avoid congestion?\", \"*Where* does congestion occur? (something analogous to convex core?)\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/largenetworks/1/", "tags": [ "aim", "AIM-COMBINATORICS-0143", "aim-domain:combinatorics", "aim-workshop:largenetworks", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Global thin-triangle hyperbolicity, together with uniform all-pairs geodesic demand and bounded core capacity, yields a quantitative utilization lower bound: the Chepoi-Dragan-Vaxes ball B(m,4 delta) forces at least half the pair demand, so maximum utilization in the ball is at least binom(n,2)/(2 C(B)). More generally, for any proposed core S and additive route allowance a, deleting S defines a forced-demand spectrum D_S(a), and every such routing satisfies max_{v in S} L(v)/c(v) >= D_S(a)/C(S). A Morse-lemma enlargement gives the analogous conclusion for local quasi-geodesics. Conversely, fixed-degree high-girth vertex-transitive expanders have negative non-lazy Ollivier and Forman curvature at every edge but no fixed-radius core capturing a positive fraction under symmetric geodesic routing.\n\nCandidate contribution (theorem; novelty confidence low): For arbitrary split demand, positive vertex capacities, a proposed core S, and additive allowance a, let D_S(a) be the total demand whose distance increases by more than a after deleting S. Every routing supported on paths of length at most d(s,t)+a obeys max_{v in S} L(v)/c(v) >= D_S(a)/C(S), with an explicit endpoint correction for internal-transit load; combining this with the hyperbolic half-pair core and Morse stability separates additive detour price, quasi-geodesic degradation, and installed capacity." }, { "id": 20001019, "problem_number": "AIM-COMBINATORICS-0144", "title": "Typed identifiability and minimal data for congestion", "statement": "What are the data required to define congestion?", "original_statement": "What are the data required to define congestion?", "clean_statement": "What are the data required to define congestion?", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 11.5 from the problem session of the workshop *Geometry of large networks*. It is record AIM-COMBINATORICS-0144 at zero-based index 143 of aim-combinatorics-notes.json. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Geometry of large networks\nSection: Problem session\nSource item: 11.5\nSource URL: http://aimpl.org/largenetworks/1/\nCanonical location: aim-combinatorics-notes.json notes[143]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What are the data required to define congestion?\"\nOriginal remarks: [\"Do weights on the network or some such affect this?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/largenetworks/1/", "tags": [ "aim", "AIM-COMBINATORICS-0144", "aim-domain:combinatorics", "aim-workshop:largenetworks", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For finite static splittable multicommodity peak utilization, route-edge incidence with path-to-commodity typing, capacities, and demands is an exact sufficient statistic. Its optimum equals the maximum demand-weighted allowed-path distance over nonnegative edge lengths with capacity-weighted budget one. Scaling laws, a cut obstruction, and explicit parallel-link, P4, Pigou, temporal, objective, and failure witnesses prove that omitting capacity semantics, demand, route admissibility, assignment behavior, objective, or temporal/scenario data makes congestion non-identifiable or can reverse the reported ranking.\n\nCandidate contribution (theorem; novelty confidence low): Candidate typed-identifiability package: the tuple (A,c,d), with path-to-commodity typing, is sufficient for static splittable min-max congestion and has the stated capacity-budgeted shortest-path dual; a field-by-field incompatible-completion test suite and an exact P4 betweenness-versus-utilization reversal certify why the named semantic fields cannot be silently omitted." }, { "id": 20001020, "problem_number": "AIM-COMBINATORICS-0145", "title": "A manifold scale needs calibration and identifiability", "statement": "Is there a scale at which the structure of a (real) network ``looks like'' a manifold? (e.g., Krioukov's result)", "original_statement": "Is there a scale at which the structure of a (real) network ``looks like'' a manifold? (e.g., Krioukov's result)", "clean_statement": "Is there a scale at which the structure of a (real) network ``looks like'' a manifold? (e.g., Krioukov's result)", "statement_status": "exact", "statement_verification": "The exact canonical record, item 11.55 in the problem session of the AIM workshop *Geometry of large networks*, asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Geometry of large networks\nSection: Problem session\nSource item: 11.55\nSource URL: http://aimpl.org/largenetworks/1/\nCanonical location: aim-combinatorics-notes.json notes[144]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there a scale at which the structure of a (real) network ``looks like'' a manifold? (e.g., Krioukov's result)\"\nOriginal remarks: [\"This relates to the (previous) questions about how to define hyperbolicity or curvature for a {\\\\it single} finite graph.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/largenetworks/1/", "tags": [ "aim", "AIM-COMBINATORICS-0145", "aim-domain:combinatorics", "aim-workshop:largenetworks", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The Krioukov antecedent is a latent hyperbolic model of the Internet, which is logically distinct from intrinsic manifold reconstruction. Every finite graph has an exact high-dimensional spherical and hyperbolic distance-threshold realization, and simultaneous metric/threshold rescaling leaves its adjacency unchanged while changing curvature magnitude. Conversely, for an intrinsic epsilon-net of a closed Riemannian manifold with a known radius-s connection rule, the rescaled hop metric is within (s+2 epsilon Q)/2 in Gromov-Hausdorff distance, where Q=ceil(diameter/(s-2 epsilon)); Latschev's theorem then certifies graph-power scales whose clique complexes recover the manifold. Stars show that exact latent manifold fit can coexist with no nontrivial pre-saturation intrinsic manifold scale.\n\nCandidate contribution (theorem; novelty confidence low): The candidate contribution is a paired latent-versus-intrinsic identifiability certificate: an explicit Gram construction realizes every graph as both a spherical and hyperbolic threshold graph; a scale-gauge proposition proves absolute curvature magnitude is unidentifiable from adjacency; an explicit radius-graph Gromov-Hausdorff bound yields a graph-power Latschev reconstruction window; and the star family separates latent realizability from every pre-saturation intrinsic manifold scale." }, { "id": 20001021, "problem_number": "AIM-COMBINATORICS-0146", "title": "Unique balance and a codegree-variance obstruction for the tetrahedron conjecture", "statement": "$\\pi(K_4^3)=\\frac59$.\\label{K34}", "original_statement": "$\\pi(K_4^3)=\\frac59$.\\label{K34}", "clean_statement": "$\\pi(K_4^3)=\\frac59$.\\label{K34}", "statement_status": "exact", "statement_verification": "The canonical record is Conjecture 11.1 in the AIM list *Hypergraph Turán problem*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Complete hypergraphs\nSource item: 11.1\nSource URL: http://aimpl.org/hypergraphturan/1/\nCanonical location: aim-combinatorics-notes.json notes[145]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"$\\\\pi(K_4^3)=\\\\frac59$.\\\\label{K34}\"\nOriginal remarks: [\"Fon-der-Flaass~\\\\cite{fonderflaass:88} presented a construction of\\n$K_4^3$-free graphs from digraphs. A weakening of Conjecture~\\\\ref{cj:K34}\\nis that Fon-der-Flaass' construction cannot beat $\\\\frac59$; some\\nprogress in this direction was made by Razborov~\\\\cite{razborov:flaass}.\", \"Kalai~\\\\cite{kalai:85:gc} (see also~\\\\cite[Section~11]{keevash:survey})\\npresented an interesting approach to $\\\\pi(K_4^3)$.\"]\nOriginal literature field (JSON string): \"There are many different constructions that achieve the lower bound\\n(see Brown~\\\\cite{brown:83}, Kostochka~\\\\cite{kostochka:82}, and\\nFon-der-Flaass~\\\\cite{fonderflaass:88}), which is one of the reasons\\nwhy this problem is so difficult. Successively better upper bounds were proved by\\nde Caen~\\\\cite{decaen:88}, Giraud (see~\\\\cite{chung+lu:01}), and\\nChung and Lu~\\\\cite{chung+lu:01}. Razborov's~\\\\cite{razborov:10} flag algebra\\napproach suggests that $\\\\pi(K_4^3)\\\\le 0.561...$ (and, if needed, this can be\\nconverted into a rigorous proof).\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/1/", "tags": [ "aim", "AIM-COMBINATORICS-0146", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the canonical cyclic three-part K_4^3-free construction, the balanced proportions are the unique global maximizer and the exact density deficit is 5/9-d=Q+(3/2)(Delta-R), implying a quadratic local stability bound. Independently, the solved l2-codegree theorem implies that any K_4^3-free sequence of density 5/9+eta must have normalized pair-codegree variance at most 2/81-(10/9)eta-eta^2, a compression of at least (10/9)eta+eta^2 relative to the balanced cyclic construction.\n\nCandidate contribution (proposition; novelty confidence low): The explicit combination of the exact cyclic-template imbalance identity with the universal variance-compression inequality rules out two concrete counterexample mechanisms: changing only the three part proportions, and retaining the standard construction's two-level codegree variance while increasing edge density.", "research_difficulty_suggested": "L4" }, { "id": 20001022, "problem_number": "AIM-COMBINATORICS-0147", "title": "An alternating flat family for the complete 3-graph lower bound", "statement": "$\\pi(K_m^3)=1-\\left(\\frac{2}{m-1}\\right)^2$.\n\\label{Km3}", "original_statement": "$\\pi(K_m^3)=1-\\left(\\frac{2}{m-1}\\right)^2$.\n\\label{Km3}", "clean_statement": "$\\pi(K_m^3)=1-\\left(\\frac{2}{m-1}\\right)^2$.\n\\label{Km3}", "statement_status": "exact", "statement_verification": "This is attempt 1 for canonical record AIM-COMBINATORICS-0147, zero-based record 146 of aim-combinatorics-notes.json. Its exact mathematical assertion is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Complete hypergraphs\nSource item: 11.2\nSource URL: http://aimpl.org/hypergraphturan/1/\nCanonical location: aim-combinatorics-notes.json notes[146]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"$\\\\pi(K_m^3)=1-\\\\left(\\\\frac{2}{m-1}\\\\right)^2$.\\n\\\\label{Km3}\"\nOriginal remarks: [\"A construction that achieves the lower bound can be found\\nin~\\\\cite[Section~7]{sidorenko:95}. Mubayi and Keevash\\n(see~\\\\cite[Section 9]{keevash:survey}) found a different construction\\n(via digraphs).\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/1/", "tags": [ "aim", "AIM-COMBINATORICS-0147", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For m >= 5 with m != 6, an explicit one-parameter alternating reweighting of the complementary directed-cycle construction gives K_m^3-free 3-graphs at the conjectured density 1-4/(m-1)^2. Every vertex has the same limiting normalized degree, while the empirical pair-codegree distribution recovers the parameter; distinct fixed parameters are therefore separated by Omega(n^3) edge edits. This is an unconditional lower-bound construction and does not prove the conjectured upper bound.\n\nCandidate contribution (theorem; novelty confidence low): The directed-cycle lower-bound construction has an explicit alternating-weight flat family, for every m >= 5 except m = 6, whose members are asymptotically vertex-degree-regular but whose pair-codegree distributions certify Omega(n^3) edit separation.", "research_difficulty_suggested": "L4" }, { "id": 20001023, "problem_number": "AIM-COMBINATORICS-0148", "title": "An editorial placeholder and a two-axis non-recovery certificate", "statement": "Intro to this problem ...\n\nThis problem is just a placeholder. Replace it bu a real problem,\nor we can jsut delete it later.", "original_statement": "Intro to this problem ...\n\nThis problem is just a placeholder. Replace it bu a real problem,\nor we can jsut delete it later.", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "**Recovered statement:** no mathematical statement is recoverable from this record. The text is an editorial instruction to replace or delete a stub.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Complete hypergraphs\nSource item: 11.3\nSource URL: http://aimpl.org/hypergraphturan/1/\nCanonical location: aim-combinatorics-notes.json notes[147]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Intro to this problem ...\\n\\nThis problem is just a placeholder. Replace it bu a real problem,\\nor we can jsut delete it later.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/1/", "tags": [ "aim", "AIM-COMBINATORICS-0148", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical text is an editorial stub rather than a mathematical proposition, and archived official AIMPL snapshots from 2015 and 2021 contain the same placeholder wording under the same database object identifier. A proved local audit shows that the neighboring three-uniform row and varying-uniformity diagonal of complete-hypergraph Turan densities intersect only at (3,4), while the elementary balanced (m-1)-partite construction has density (m-2)(m-3)/(m-1)^2, differing from the neighboring row conjecture by 3(m-3)/(m-1)^2. Thus no neighboring statement can be imported as a unique repair of this record.\n\nCandidate contribution (lemma; novelty confidence low): Candidate two-axis non-recovery certificate: for this archived object, duplicate numbering cannot identify a replacement; the parameter loci R={(3,m):m>=4} and D={(k,k+1):k>=3} meet only at (3,4); and the ordinary partite construction differs from the neighboring row formula by exactly 3(m-3)/(m-1)^2.", "research_difficulty_suggested": "L2" }, { "id": 20001024, "problem_number": "AIM-COMBINATORICS-0149", "title": "The refuted de Caen clique-covering conjecture and an exact rational recursion", "statement": "Let us mention here another very interesting question for whose solution de\nCaen~\\cite[Page~190]{decaen:94} offered 500 Canadian dollars.\n\nDoes $k(1-\\pi(K_{k+1}^k))$ tend to $\\infty$\nas $k\\to\\infty$?", "original_statement": "Let us mention here another very interesting question for whose solution de\nCaen~\\cite[Page~190]{decaen:94} offered 500 Canadian dollars.\n\nDoes $k(1-\\pi(K_{k+1}^k))$ tend to $\\infty$\nas $k\\to\\infty$?", "clean_statement": "Let us mention here another very interesting question for whose solution de\nCaen~\\cite[Page~190]{decaen:94} offered 500 Canadian dollars.\n\nDoes $k(1-\\pi(K_{k+1}^k))$ tend to $\\infty$\nas $k\\to\\infty$?", "statement_status": "exact", "statement_verification": "The AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Complete hypergraphs\nSource item: 11.3\nSource URL: http://aimpl.org/hypergraphturan/1/\nCanonical location: aim-combinatorics-notes.json notes[148]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let us mention here another very interesting question for whose solution de\\nCaen~\\\\cite[Page~190]{decaen:94} offered 500 Canadian dollars.\\n\\nDoes $k(1-\\\\pi(K_{k+1}^k))$ tend to $\\\\infty$\\nas $k\\\\to\\\\infty$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/1/", "tags": [ "aim", "AIM-COMBINATORICS-0149", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The finite complement identity gives 1-pi(K_{k+1}^k)=t(k+1,k). Pikhurko's 2025 theorem proves t(k+1,k)<=4.911/(k+1) for all sufficiently large k, so the proposed divergence is false. A self-contained specialization of the published recursive architecture, using beta=39/50, c=14/5, and mu=25/4, proves the uniform finite bound T(n,k+1,k)<=25 binom(n,k)/(4(k+1)); an exact calculation also shows why the crude independent-sampling plus one-repair-charge-per-uncovered-set certificate tends to density 1 when n tends to infinity first.\n\nCandidate contribution (proposition; novelty confidence low): The exact rational parameter certificate (39/50, 14/5, 25/4) yields a hand-verifiable uniform 25/[4(k+1)] covering bound, and its pairing with the sharp minimum 1-k(n-k)^(-1/k)/(k+1) for the naive one-shot alteration certificate isolates a precise limit-order obstruction." }, { "id": 20001025, "problem_number": "AIM-COMBINATORICS-0150", "title": "Exact recursion and local profile stability for the broken tetrahedron construction", "statement": "$K_4^3$ minus an edge\n\nLet $K_4^-$ be obtained from $K_4^3$ by removing one edge.\n\n$\\pi(K_4^-)=\\frac27$", "original_statement": "$K_4^3$ minus an edge\n\nLet $K_4^-$ be obtained from $K_4^3$ by removing one edge.\n\n$\\pi(K_4^-)=\\frac27$", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is `aim-combinatorics-notes.json`, zero-based index 149, from the AIM workshop “Hypergraph Turan problem,” section “Complete hypergraphs,” number 11.4. Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Complete hypergraphs\nSource item: 11.4\nSource URL: http://aimpl.org/hypergraphturan/1/\nCanonical location: aim-combinatorics-notes.json notes[149]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"$K_4^3$ minus an edge\\n\\nLet $K_4^-$ be obtained from $K_4^3$ by removing one edge.\\n\\n$\\\\pi(K_4^-)=\\\\frac27$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The lower bound comes by partitioning $[n]$ into six parts, taking\\ncertain 10 complete 3-partite 3-graphs, and then recursively repeating\\nthe same construction inside each part, see \\\\cite[Page\\n323]{frankl+furedi:84}.\\n\\nThe best known upper bounds come from flag algebra computations: Baber\\nand Talbot~\\\\cite{baber+talbot} (by using the method of\\nRazborov~\\\\cite{razborov:10} and generating a larger SDP program than\\nthat in~\\\\cite{razborov:10}) showed that $\\\\pi(K_4^-)\\\\le 0.2871$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/1/", "tags": [ "aim", "AIM-COMBINATORICS-0150", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The classical conjecture remains open with the primary-source bracket 2/7 <= pi(K_4^-) <= 0.286889, improving the canonical record's obsolete 0.2871 upper endpoint. The Frankl-Furedi iterated H_6 construction is reconstructed explicitly and proved K_4^--free via triangle-free links; on n=6^k vertices it has exactly (n^3-n)/21 edges. For arbitrary fixed recursive part profile x, its density is 6P(x)/(1-sum x_i^3), and an exact degree/codegree expansion proves that the balanced profile is the unique local maximizer in the explicit ball max_i |x_i-1/6| < 1/12, with a quantitative quadratic gap.\n\nCandidate contribution (lemma; novelty confidence low): Candidate local profile-stability lemma: writing x=(1/6,...,1/6)+y and M=max |y_i|, the exact deficit is 1-S(x)-21P(x)=3 sum y_i^2-(sum y_i^3+21P(y)), the cubic remainder has absolute value at most 36M sum y_i^2, and hence every non-balanced fixed profile with M<1/12 has recursive density strictly below 2/7; for M<=1/24 the loss is at least (3/7) sum (x_i-1/6)^2.", "research_difficulty_suggested": "L4" }, { "id": 20001026, "problem_number": "AIM-COMBINATORICS-0151", "title": "Explicit residue formulas for the solved generalized-triangle problem", "statement": "Frankl and F\\\"uredi~\\cite{frankl+furedi:89} determined\n$\\pi(B_{k,2})$ for $k=5,6$; in both cases the lower bounds comes by\nblowing up a small design. The following question is still open (see\nFrankl and F\\\"uredi~\\cite[Conjecture~1.5]{frankl+furedi:89}):\n\nDetermine $\\operatorname{ex}(n,B_{5,2})$ and $\\operatorname{ex}(n,B_{6,2})$ exactly for all\nlarge $n$.", "original_statement": "Frankl and F\\\"uredi~\\cite{frankl+furedi:89} determined\n$\\pi(B_{k,2})$ for $k=5,6$; in both cases the lower bounds comes by\nblowing up a small design. The following question is still open (see\nFrankl and F\\\"uredi~\\cite[Conjecture~1.5]{frankl+furedi:89}):\n\nDetermine $\\operatorname{ex}(n,B_{5,2})$ and $\\operatorname{ex}(n,B_{6,2})$ exactly for all\nlarge $n$.", "clean_statement": "Frankl and F\\\"uredi~\\cite{frankl+furedi:89} determined\n$\\pi(B_{k,2})$ for $k=5,6$; in both cases the lower bounds comes by\nblowing up a small design. The following question is still open (see\nFrankl and F\\\"uredi~\\cite[Conjecture~1.5]{frankl+furedi:89}):\n\nDetermine $\\operatorname{ex}(n,B_{5,2})$ and $\\operatorname{ex}(n,B_{6,2})$ exactly for all\nlarge $n$.", "statement_status": "exact", "statement_verification": "The canonical record is AIM-COMBINATORICS-0151, zero-based record 150 of aim-combinatorics-notes.json, attempt 1. It asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Turán functions for books\nSource item: 22.1\nSource URL: http://aimpl.org/hypergraphturan/2/\nCanonical location: aim-combinatorics-notes.json notes[150]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Frankl and F\\\\\\\"uredi~\\\\cite{frankl+furedi:89} determined\\n$\\\\pi(B_{k,2})$ for $k=5,6$; in both cases the lower bounds comes by\\nblowing up a small design. The following question is still open (see\\nFrankl and F\\\\\\\"uredi~\\\\cite[Conjecture~1.5]{frankl+furedi:89}):\\n\\nDetermine $\\\\operatorname{ex}(n,B_{5,2})$ and $\\\\operatorname{ex}(n,B_{6,2})$ exactly for all\\nlarge $n$.\"\nOriginal remarks: [\"One difficulty for this problem is that it is not\\nclear how to prove \\\\emph{the stability property}, that is, that all\\nalmost extremal graphs have similar structure.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/2/", "tags": [ "aim", "AIM-COMBINATORICS-0151", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM problem was solved by Norin and Yepremyan in 2017: B_{k,2} is the generalized triangle T_k, and for k=5,6 every sufficiently large extremal graph is a blow-up of the unique S(4,5,11) or S(5,6,12) Witt design, with the required global stability. Building on that theorem, this attempt proves that every maximizing integer blow-up is eventually balanced and gives explicit exact edge-count polynomials for each residue class modulo 11 or 12.\n\nCandidate contribution (theorem; novelty confidence low): For a uniquely uniformly maximizing S(r-1,r,m) Steiner template, every sufficiently large integer blow-up maximizer has class sizes differing by at most one; applying this to the two Witt systems yields explicit residue constants M5=(0,0,0,0,0,1,1,3,8,18,36) and M6=(0,0,0,0,0,0,1,1,4,12,30,66) in the exact B_{5,2} and B_{6,2} formulas." }, { "id": 20001027, "problem_number": "AIM-COMBINATORICS-0152", "title": "Exact profiles for the odd book construction", "statement": "\\[\n \\pi(B_{5,5})=\\frac{40}{81},\n\\]\nand\n\\[\n \\pi(B_{6,6})=\\frac12.\n\\]", "original_statement": "\\[\n \\pi(B_{5,5})=\\frac{40}{81},\n\\]\nand\n\\[\n \\pi(B_{6,6})=\\frac12.\n\\]", "clean_statement": "\\[\n \\pi(B_{5,5})=\\frac{40}{81},\n\\]\nand\n\\[\n \\pi(B_{6,6})=\\frac12.\n\\]", "statement_status": "exact", "statement_verification": "The AIM record, from the workshop list “Hypergraph Turán problem,” section “Turán functions for books,” conjectures", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Turán functions for books\nSource item: 22.2\nSource URL: http://aimpl.org/hypergraphturan/2/\nCanonical location: aim-combinatorics-notes.json notes[151]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"\\\\[\\n \\\\pi(B_{5,5})=\\\\frac{40}{81},\\n\\\\]\\nand\\n\\\\[\\n \\\\pi(B_{6,6})=\\\\frac12.\\n\\\\]\"\nOriginal remarks: [\"The lower bounds come from a ``bipartite'' construction. It was\\nproved in \\\\cite{bohman+frieze+mubayi+pikhurko:10} that\\n$\\\\pi(B_{5,5})\\\\le 0.534...$ and that the bipartite construction is not optimal for $\\\\pi(B_{k,k})$ when $k\\\\ge 7$.\", \"The Tur\\\\'an density is unknown for $B_{5,3}$ and $B_{5,4}$ which is an\\ninteresting (and perhaps tractable) open problem.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/2/", "tags": [ "aim", "AIM-COMBINATORICS-0152", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The standard odd construction is proved directly to be B_{k,k}-free and optimized for every k. For the two open cases its exact density deficits are derived: 1/2-p_6(y)=32(y-1/2)^6, while 40/81-p_5(y)=(5/81)(3y-1)^2(8-33y+54y^2-27y^3). The latter yields the sharp global quadratic coefficient (10/9)(1-sqrt(3)/9), whereas the former proves that no positive quadratic part-imbalance bound can hold near the balanced k=6 optimizer. These results are confined to the conjectured construction family and do not prove either open upper bound.\n\nCandidate contribution (sharp construction-profile theorem; novelty confidence low): Within the odd construction family, the k=5 density deficit has the stated exact factorization and optimal global quadratic stability constant, while the k=6 deficit is exactly sixth order and therefore obstructs every positive local quadratic part-imbalance estimate.", "research_difficulty_suggested": "L4" }, { "id": 20001028, "problem_number": "AIM-COMBINATORICS-0153", "title": "Exact deficit identity for recursive tight-cycle constructions", "statement": "Mubayi and R\\\"odl~\\cite{mubayi+rodl:02} have given bounds on\n$\\pi(C_5^3)$, where $C_5^3$ is the tight $3$-graph $5$-cycle:\n $$\n C_5^3=\\{123,234,345,451,512\\}.\n $$\n In particular, the lower bound $\\pi(C_5^3)\\ge 2\\sqrt3-3$ comes from the\nfollowing construction: partition the vertex set into two parts $A$ and $B$, take all triples that intersect $A$ precisely in $2$ vertices, and\nrecursively repeat this construction within $B$. Finding the optimal ratio\nbetween $|A|$ and $|B|$ gives the required.\nRazborov's~\\cite{razborov:10} flag algebra computations showed that\n$\\pi(C_5^3)< 0.4683$ (note that $2\\sqrt3-3=0.4641...$). This makes the following\nconjecture plausible.\n\n$\\pi(C_5^3)=2\\sqrt3-3$", "original_statement": "Mubayi and R\\\"odl~\\cite{mubayi+rodl:02} have given bounds on\n$\\pi(C_5^3)$, where $C_5^3$ is the tight $3$-graph $5$-cycle:\n $$\n C_5^3=\\{123,234,345,451,512\\}.\n $$\n In particular, the lower bound $\\pi(C_5^3)\\ge 2\\sqrt3-3$ comes from the\nfollowing construction: partition the vertex set into two parts $A$ and $B$, take all triples that intersect $A$ precisely in $2$ vertices, and\nrecursively repeat this construction within $B$. Finding the optimal ratio\nbetween $|A|$ and $|B|$ gives the required.\nRazborov's~\\cite{razborov:10} flag algebra computations showed that\n$\\pi(C_5^3)< 0.4683$ (note that $2\\sqrt3-3=0.4641...$). This makes the following\nconjecture plausible.\n\n$\\pi(C_5^3)=2\\sqrt3-3$", "clean_statement": "Mubayi and R\\\"odl~\\cite{mubayi+rodl:02} have given bounds on\n$\\pi(C_5^3)$, where $C_5^3$ is the tight $3$-graph $5$-cycle:\n $$\n C_5^3=\\{123,234,345,451,512\\}.\n $$\n In particular, the lower bound $\\pi(C_5^3)\\ge 2\\sqrt3-3$ comes from the\nfollowing construction: partition the vertex set into two parts $A$ and $B$, take all triples that intersect $A$ precisely in $2$ vertices, and\nrecursively repeat this construction within $B$. Finding the optimal ratio\nbetween $|A|$ and $|B|$ gives the required.\nRazborov's~\\cite{razborov:10} flag algebra computations showed that\n$\\pi(C_5^3)< 0.4683$ (note that $2\\sqrt3-3=0.4641...$). This makes the following\nconjecture plausible.\n\n$\\pi(C_5^3)=2\\sqrt3-3$", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 33.1 in the workshop list *Hypergraph Turán problem*, section “Tight \\(5\\)-Cycle,” at http://aimpl.org/hypergraphturan/3/ . It defines \\[ C_5^3=\\{123,234,345,451,512\\} \\] and asks for the classical edge-density Turán value. Its conjecture is \\[ \\boxed{\\pi(C_5^3)=2\\sqrt3-3.} \\] Here \\[ \\pi(F)=\\lim_{n\\to\\infty}\\frac{\\operatorname{ex}(n,F)}{\\binom n3}, \\] where copies are ordinary (not necessarily induced) injective copies of a 3-uniform hypergraph.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Tight $5$-Cycle\nSource item: 33.1\nSource URL: http://aimpl.org/hypergraphturan/3/\nCanonical location: aim-combinatorics-notes.json notes[152]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Mubayi and R\\\\\\\"odl~\\\\cite{mubayi+rodl:02} have given bounds on\\n$\\\\pi(C_5^3)$, where $C_5^3$ is the tight $3$-graph $5$-cycle:\\n $$\\n C_5^3=\\\\{123,234,345,451,512\\\\}.\\n $$\\n In particular, the lower bound $\\\\pi(C_5^3)\\\\ge 2\\\\sqrt3-3$ comes from the\\nfollowing construction: partition the vertex set into two parts $A$ and $B$, take all triples that intersect $A$ precisely in $2$ vertices, and\\nrecursively repeat this construction within $B$. Finding the optimal ratio\\nbetween $|A|$ and $|B|$ gives the required.\\nRazborov's~\\\\cite{razborov:10} flag algebra computations showed that\\n$\\\\pi(C_5^3)< 0.4683$ (note that $2\\\\sqrt3-3=0.4641...$). This makes the following\\nconjecture plausible.\\n\\n$\\\\pi(C_5^3)=2\\\\sqrt3-3$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/3/", "tags": [ "aim", "AIM-COMBINATORICS-0153", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the full Mubayi--Rodl one-branch recursive C_5^3-free construction, allowing an arbitrary split ratio b_j at every level, the density deficit from alpha=2sqrt(3)-3 is exactly alpha q_infinity + 2sqrt(3) sum_j q_j(1-b_j)(b_j-beta)^2, where beta=(sqrt(3)-1)/2 and q_j is the surviving triple weight. This proves global optimality and quantitative weighted stability throughout the entire level-dependent recursive family. The optimized finite recursive edge count satisfies an exact recurrence and equals alpha binom(n,3)+O(n^2). This does not prove the upper bound for arbitrary C_5^3-free 3-graphs.\n\nCandidate contribution (exact deficit identity; novelty confidence low): The exact telescoping Bellman sum-of-squares formula for every level-dependent split profile, together with its weighted near-equality bounds and the finite recurrence R(n)=max_m{m binom(n-m,2)+R(m)} with O(n^2) asymptotic error, is a concrete candidate contribution.", "research_difficulty_suggested": "L4" }, { "id": 20001029, "problem_number": "AIM-COMBINATORICS-0154", "title": "Tight 5-cycle minus an edge: solved density and a recursive deficit identity", "statement": "Tight $5$-Cycle Minus an Edge\n\nLet the $3$-graph $C_5^-$ be obtained from $C_5^3$ by removing one\nedge. An example of a $C_5^-$-free $3$-graph can be obtained by taking\na complete $3$-partite $3$-graph and repeating this construction\nrecursively within each of the three parts. This gives density $1/4$\nin the limit.\n\n$\\pi(C_5^-)=1/4$.", "original_statement": "Tight $5$-Cycle Minus an Edge\n\nLet the $3$-graph $C_5^-$ be obtained from $C_5^3$ by removing one\nedge. An example of a $C_5^-$-free $3$-graph can be obtained by taking\na complete $3$-partite $3$-graph and repeating this construction\nrecursively within each of the three parts. This gives density $1/4$\nin the limit.\n\n$\\pi(C_5^-)=1/4$.", "clean_statement": "Tight $5$-Cycle Minus an Edge\n\nLet the $3$-graph $C_5^-$ be obtained from $C_5^3$ by removing one\nedge. An example of a $C_5^-$-free $3$-graph can be obtained by taking\na complete $3$-partite $3$-graph and repeating this construction\nrecursively within each of the three parts. This gives density $1/4$\nin the limit.\n\n$\\pi(C_5^-)=1/4$.", "statement_status": "exact", "statement_verification": "The canonical record is zero-based index 153 of aim-combinatorics-notes.json, problem 33.2 in the AIM workshop list *Hypergraph Turan problem*, section “Tight \\(5\\)-Cycle.” It states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Tight $5$-Cycle\nSource item: 33.2\nSource URL: http://aimpl.org/hypergraphturan/3/\nCanonical location: aim-combinatorics-notes.json notes[153]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Tight $5$-Cycle Minus an Edge\\n\\nLet the $3$-graph $C_5^-$ be obtained from $C_5^3$ by removing one\\nedge. An example of a $C_5^-$-free $3$-graph can be obtained by taking\\na complete $3$-partite $3$-graph and repeating this construction\\nrecursively within each of the three parts. This gives density $1/4$\\nin the limit.\\n\\n$\\\\pi(C_5^-)=1/4$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/3/", "tags": [ "aim", "AIM-COMBINATORICS-0154", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM conjecture is solved: for C_5^- with edge set {123,234,345,451}, independent computer-assisted flag-algebra works of Lidicky--Mattes--Pfender and Bodnar--Leon--Liu--Pikhurko prove pi(C_5^-)=1/4. In addition, this attempt proves an exact elementary telescoping identity for every finite pure ternary recursive construction: (n^3-n)/24-e(T) is the sum of a nonnegative AM--GM imbalance deficit over all internal nodes plus 1/4 for every two-vertex leaf.\n\nCandidate contribution (identity; novelty confidence low): For any finite ternary recursive construction T, if an internal node v has positive child sizes a_v,b_v,c_v and m_v=a_v+b_v+c_v, then (n^3-n)/24-e(T) equals the sum over v of ((a_v+b_v)(b_v+c_v)(c_v+a_v)-8a_vb_vc_v)/8 plus L_2(T)/4, where L_2(T) is the number of two-vertex leaves." }, { "id": 20001030, "problem_number": "AIM-COMBINATORICS-0155", "title": "A 4/9 quadratic upper bound for the first Brown--Erdos--Sos case", "statement": "For any $r\\ge 3$ and $s\\ge 4$ we have\n $$\n f^r(n,s(r-2)+3,s)=o(n^2).\n $$", "original_statement": "For any $r\\ge 3$ and $s\\ge 4$ we have\n $$\n f^r(n,s(r-2)+3,s)=o(n^2).\n $$", "clean_statement": "For any $r\\ge 3$ and $s\\ge 4$ we have\n $$\n f^r(n,s(r-2)+3,s)=o(n^2).\n $$", "statement_status": "exact", "statement_verification": "The canonical AIM record (Hypergraph Turan problem workshop, Section 44.1) states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Ruzsa-Szemer\\'edi Theorem and Relatives\nSource item: 44.1\nSource URL: http://aimpl.org/hypergraphturan/4/\nCanonical location: aim-combinatorics-notes.json notes[154]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"For any $r\\\\ge 3$ and $s\\\\ge 4$ we have\\n $$\\n f^r(n,s(r-2)+3,s)=o(n^2).\\n $$\"\nOriginal remarks: [\"In~\\\\cite{sarkozy+selkow:05} it is proved that\\n$f^r(n,s(r-2)+[{\\\\log_2s}],s)=o(n^2)$. The first remaining open case is to\\nprove the conjecture for $f^3(n,7,4)$ (probably very hard).\", \"One possible direction here is to look at multiple hypergraphs (when\\nthe same $r$-tuple can appear a multiple number of times) and ask for\\n$F^r(n,p,s)$ maximum size of an $r$-multi-hypergraph such that every\\n$s$-set spans at most $p$ edges. See F\\\\\\\"uredi and\\nK\\\\\\\"undgen~\\\\cite{furedi+kundgen:02} for results in the graph case\\n($r=2$).\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/4/", "tags": [ "aim", "AIM-COMBINATORICS-0155", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every simple 3-uniform hypergraph on n vertices with no four edges spanning at most seven vertices has fewer than (4/9)n^2+2n edges. The proof six-colors the two-vertex-intersection conflict graph using Brooks' theorem, then applies the June 2026 Gishboliner--Solymosi finite density theorem to each linear color class. In addition, for C_{r,e}=1+binom(r,2)(e-2), the exact reduction f_r(n,e(r-2)+3,e) <= C_{r,e} f_3(n,e+3,e) is proved, as is constant-factor equivalence with the natural repeated-edge multihypergraph version. These results do not prove the conjectured o(n^2) bound.\n\nCandidate contribution (explicit upper bound; novelty confidence low): The conflict-graph/Brooks decomposition combined with the June 2026 dense-linear theorem gives the concrete general-hypergraph estimate f_3(n,7,4)<(4/9)n^2+2n.", "research_difficulty_suggested": "L4" }, { "id": 20001031, "problem_number": "AIM-COMBINATORICS-0156", "title": "A sharp three-edge-color counterexample to the literal conjecture", "statement": "A related question is as follows. Let $A$, $B$, and $C$ be disjoint\nsets each of size $n$. Let $M_1,\\dots ,M_l$, be matchings, where each\nedge of $M_i$ has one point in each of $A$, $B$, and $C$. The\nforbidden configuration is: three edges $abc$, $a'b'c'$, $a''b''c''$\nall in some $M_i$ and one edge of the form $ab'c''$ in some other\n$M_j$ (that is, the edge from $M_j$ crosses the three edges of $M_i$).\nAdditionally, we require that the union of all matchings $M_i$ makes a\nsimple (linear) $3$-graph, call it $M$.\n\n$$\n |M| = o(n^2).\n $$", "original_statement": "A related question is as follows. Let $A$, $B$, and $C$ be disjoint\nsets each of size $n$. Let $M_1,\\dots ,M_l$, be matchings, where each\nedge of $M_i$ has one point in each of $A$, $B$, and $C$. The\nforbidden configuration is: three edges $abc$, $a'b'c'$, $a''b''c''$\nall in some $M_i$ and one edge of the form $ab'c''$ in some other\n$M_j$ (that is, the edge from $M_j$ crosses the three edges of $M_i$).\nAdditionally, we require that the union of all matchings $M_i$ makes a\nsimple (linear) $3$-graph, call it $M$.\n\n$$\n |M| = o(n^2).\n $$", "clean_statement": "A related question is as follows. Let $A$, $B$, and $C$ be disjoint\nsets each of size $n$. Let $M_1,\\dots ,M_l$, be matchings, where each\nedge of $M_i$ has one point in each of $A$, $B$, and $C$. The\nforbidden configuration is: three edges $abc$, $a'b'c'$, $a''b''c''$\nall in some $M_i$ and one edge of the form $ab'c''$ in some other\n$M_j$ (that is, the edge from $M_j$ crosses the three edges of $M_i$).\nAdditionally, we require that the union of all matchings $M_i$ makes a\nsimple (linear) $3$-graph, call it $M$.\n\n$$\n |M| = o(n^2).\n $$", "statement_status": "exact", "statement_verification": "The exact AIM record is Conjecture 10 in the 2011 workshop list, in the section “Ruzsa–Szemerédi Theorem and Relatives.” It takes three disjoint sets \\(A,B,C\\), each of size \\(n\\), and matchings \\(M_1,\\ldots,M_\\ell\\) of triples with one vertex in each part. The forbidden configuration consists of three edges", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Ruzsa-Szemer\\'edi Theorem and Relatives\nSource item: 44.2\nSource URL: http://aimpl.org/hypergraphturan/4/\nCanonical location: aim-combinatorics-notes.json notes[155]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"A related question is as follows. Let $A$, $B$, and $C$ be disjoint\\nsets each of size $n$. Let $M_1,\\\\dots ,M_l$, be matchings, where each\\nedge of $M_i$ has one point in each of $A$, $B$, and $C$. The\\nforbidden configuration is: three edges $abc$, $a'b'c'$, $a''b''c''$\\nall in some $M_i$ and one edge of the form $ab'c''$ in some other\\n$M_j$ (that is, the edge from $M_j$ crosses the three edges of $M_i$).\\nAdditionally, we require that the union of all matchings $M_i$ makes a\\nsimple (linear) $3$-graph, call it $M$.\\n\\n$$\\n |M| = o(n^2).\\n $$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"If true, this implies Roth's Theorem (every set of integers of\\npositive upper density has a $3$-term arithmetic progression). An\\nobvious generalization is to consider the $k$-graph version where we\\nhave parts $A_1, ..., A_k$, each of size $n$, and instead of three\\nedges in $M_i$ we take $k$ edges in $M_i$ and another crossing edge as\\nthe forbidden configuration; as before, we require that the union $M$\\nis a linear $k$-graph. Again, we believe that $|M| = o(n^2)$ and, if\\ntrue, this would imply Szemer\\\\'edi's Theorem. The case $k=2$ is\\nequivalent to the Ruzsa--Szemer\\\\'edi Theorem that $f^3(n,6,3)=o(n^2)$.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/4/", "tags": [ "aim", "AIM-COMBINATORICS-0156", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM conjecture as literally quantified is false. For every n=3m with odd m at least 5, the cyclic Latin-square linear tripartite 3-graph with n^2 edges can be partitioned into pairwise edge-disjoint matchings of exactly three edges such that no ordered crossing of any color class is an edge of the entire union. Since every linear tripartite 3-graph with n vertices per part has at most n^2 edges, this is a sharp counterexample. The same model identifies a crossing exactly with x+y=2z and thereby explains the intended Roth connection before the natural perfect-matching colors are refined.\n\nCandidate contribution (explicit extremal counterexample; novelty confidence low): For every odd m at least 5 and n=3m, the classes M_{t,r}={e_x^t:x in {3r,3r+1,3r+5}} partition a maximum-size n^2-edge linear tripartite 3-graph into matchings of exactly three edges, and every possible crossing of each class is absent from the whole graph." }, { "id": 20001032, "problem_number": "AIM-COMBINATORICS-0157", "title": "The sparse-removal conjecture fails literally and at the true extremal exponent", "statement": "The following conjecture seems to be related.\n\nLet $F$ be a graph and\n$\\alpha>1$ be such that $\\operatorname{ex}(n,F)=\\Omega(n^{\\alpha}$). Then for any\n$\\epsilon > 0$ there is $n_0$ so that if $n > n_0$ and a\ngraph $H$ is the edge-disjoint union of $m=\\lceil \\epsilon\nn^{\\alpha}\\rceil$ copies of $F$, then $H$ contains another copy of $F$\n(i.e.\\ has at least $m+1$ copies of $F$).", "original_statement": "The following conjecture seems to be related.\n\nLet $F$ be a graph and\n$\\alpha>1$ be such that $\\operatorname{ex}(n,F)=\\Omega(n^{\\alpha}$). Then for any\n$\\epsilon > 0$ there is $n_0$ so that if $n > n_0$ and a\ngraph $H$ is the edge-disjoint union of $m=\\lceil \\epsilon\nn^{\\alpha}\\rceil$ copies of $F$, then $H$ contains another copy of $F$\n(i.e.\\ has at least $m+1$ copies of $F$).", "clean_statement": "The following conjecture seems to be related.\n\nLet $F$ be a graph and\n$\\alpha>1$ be such that $\\operatorname{ex}(n,F)=\\Omega(n^{\\alpha}$). Then for any\n$\\epsilon > 0$ there is $n_0$ so that if $n > n_0$ and a\ngraph $H$ is the edge-disjoint union of $m=\\lceil \\epsilon\nn^{\\alpha}\\rceil$ copies of $F$, then $H$ contains another copy of $F$\n(i.e.\\ has at least $m+1$ copies of $F$).", "statement_status": "exact", "statement_verification": "The AIM record (Hypergraph Turan problem workshop, Section “Ruzsa--Szemerédi Theorem and Relatives,” item 44.3) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Combinatorics\nWorkshop: Hypergraph Turan problem\nSection: Ruzsa-Szemer\\'edi Theorem and Relatives\nSource item: 44.3\nSource URL: http://aimpl.org/hypergraphturan/4/\nCanonical location: aim-combinatorics-notes.json notes[156]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"The following conjecture seems to be related.\\n\\nLet $F$ be a graph and\\n$\\\\alpha>1$ be such that $\\\\operatorname{ex}(n,F)=\\\\Omega(n^{\\\\alpha}$). Then for any\\n$\\\\epsilon > 0$ there is $n_0$ so that if $n > n_0$ and a\\ngraph $H$ is the edge-disjoint union of $m=\\\\lceil \\\\epsilon\\nn^{\\\\alpha}\\\\rceil$ copies of $F$, then $H$ contains another copy of $F$\\n(i.e.\\\\ has at least $m+1$ copies of $F$).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "http://aimpl.org/hypergraphturan/4/", "tags": [ "aim", "AIM-COMBINATORICS-0157", "aim-domain:combinatorics", "aim-workshop:hypergraphturan", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "After repairing the missing TeX brace and making the copy-counting quantifiers explicit, the AIM conjecture is false in two senses. Literally, K_3 refutes it for every chosen 1=5 and every sufficiently large prime p congruent to 1 modulo 4 in the Timmons-Verstraete construction, with n=2kp^2, every integer m satisfying 1<=m<=p^2(p-1) is realized by an n-vertex graph whose edge set is the pairwise edge-disjoint union of exactly m copies of H_k and which contains no other H_k; in particular this realizes m=ceil(epsilon_k n^{3/2}) for epsilon_k=1/[4(2k)^{3/2}]." }, { "id": 20001033, "problem_number": "AIM-COMBINATORICS-0158", "title": "Exact trace certificates and a regular counterexample normal form for Caccetta--Haggkvist", "statement": "Conjecture 2.\n1. (L. Caccetta, R. H¨ aggkvist [5]) Every simple n-vertex digraph with minimum out-degree at least r has a cycle with length at most d n\n\n> r\n\ne.\n\nThis can be restated in the following way: Let A be an n × n 0-1 matrix such that aij = 1 implies\n\naji 6 = 1 for all i 6 = j. Let aii = 1 for all i. If the sum of every row in A is at least r + 1, Adn/r e\n\nhas trace greater than n.\n\n#2.1 Partial Results\n\nThe C-H conjecture has been proved for:\n\n• r = 2 by Caccetta and H¨ aggkvist [5]\n\n• r = 3 by Hamidoune [17]\n\n• r = 4 and r = 5 by Ho´ ang and Reed [19]\n\n• r ≤ √n/ 2 by Shen [30]. For the exact statement of his result, see Theorem 5.1. This shows that for any given r, the number of counterexamples to the conjecture (if any) is finite.\n\n• Cayley graphs (which implies all vertex transitive graphs using coset representations) by Hamidoune [15]. This proof uses a lemma of Kemperman [21] (Lemma 5.9). Also, Shen [32] proved that if deg +(u) + deg +(v) ≥ 4 for all ( u, v ) ∈ E(G), then g ≤ d n/ 2e,where g denotes the girth of G. This is an average local outdegree version for the r = 2 case of the Caccetta-H¨ aggkvist conjecture.\n\n#2.2 Approximate Results I - Additive Constant\n\nAnother approach is to show that if δ+\n\n> G\n\n≥ r, then there is a cycle of length at most n\n\n> r\n\n+ c for some small c. This has been proved for some values of c, as follows:\n\n• c = 2500 by Chv´ atal and Szemer´ edi [9]\n\n• c = 304 by Nishimura [27]\n\n• c = 73 by Shen [31]. 22.3 Approximate Results II - Special Case n/3\n\nThe case r = n/ 2 is trivial, but r = n/ 3 has received much attention. Research has sought the minimum constant c such that δ+\n\n> G\n\n≥ cn in an n-vertex simple digraph G forces a directed cycle of length at most 3. The conjecture is that c = 1 /3, and the current results are:\n\n• c ≤ (3 − √5) /2 = 0.382 by Caccetta and H¨ aggkvist [5]\n\n• c ≤ (2 √6 − 3) /5 = 0.3797 by Bondy in a neat subgraph counting argument [4]\n\n• c ≤ 3 − √7 = 0.3542 by [29] Similarly, Seymour, Graaf, and Schrijver [12] asked for the minimum value of β so that when the minimum inand out-degrees of G are at least βn, G has a directed cycle of length at most 3. They proved that β ≤ 0.3487 and gave a formula relating β and c. Shen applied this formula to his 1998 result to get a slight improvement to β ≤ 0.3477 [29].\n\n3 Seymour's Second Neighborhood Conjecture\n\nThis conjecture implies the special case of Caccetta-H¨ aggkvist when both inand out-degrees are at least n/ 3, and has received much attention of its own.", "original_statement": "Conjecture 2.\n1. (L. Caccetta, R. H¨ aggkvist [5]) Every simple n-vertex digraph with minimum out-degree at least r has a cycle with length at most d n \n\n> r\n\ne.\n\nThis can be restated in the following way: Let A be an n × n 0-1 matrix such that aij = 1 implies \n\naji 6 = 1 for all i 6 = j. Let aii = 1 for all i. If the sum of every row in A is at least r + 1, Adn/r e\n\nhas trace greater than n.\n\n#2.1 Partial Results \n\nThe C-H conjecture has been proved for: \n\n• r = 2 by Caccetta and H¨ aggkvist [5] \n\n• r = 3 by Hamidoune [17] \n\n• r = 4 and r = 5 by Ho´ ang and Reed [19] \n\n• r ≤ √n/ 2 by Shen [30]. For the exact statement of his result, see Theorem 5.1. This shows that for any given r, the number of counterexamples to the conjecture (if any) is finite. \n\n• Cayley graphs (which implies all vertex transitive graphs using coset representations) by Hamidoune [15]. This proof uses a lemma of Kemperman [21] (Lemma 5.9). Also, Shen [32] proved that if deg +(u) + deg +(v) ≥ 4 for all ( u, v ) ∈ E(G), then g ≤ d n/ 2e,where g denotes the girth of G. This is an average local outdegree version for the r = 2 case of the Caccetta-H¨ aggkvist conjecture. \n\n#2.2 Approximate Results I - Additive Constant \n\nAnother approach is to show that if δ+ \n\n> G\n\n≥ r, then there is a cycle of length at most n \n\n> r\n\n+ c for some small c. This has been proved for some values of c, as follows: \n\n• c = 2500 by Chv´ atal and Szemer´ edi [9] \n\n• c = 304 by Nishimura [27] \n\n• c = 73 by Shen [31]. 22.3 Approximate Results II - Special Case n/3 \n\nThe case r = n/ 2 is trivial, but r = n/ 3 has received much attention. Research has sought the minimum constant c such that δ+ \n\n> G\n\n≥ cn in an n-vertex simple digraph G forces a directed cycle of length at most 3. The conjecture is that c = 1 /3, and the current results are: \n\n• c ≤ (3 − √5) /2 = 0.382 by Caccetta and H¨ aggkvist [5] \n\n• c ≤ (2 √6 − 3) /5 = 0.3797 by Bondy in a neat subgraph counting argument [4] \n\n• c ≤ 3 − √7 = 0.3542 by [29] Similarly, Seymour, Graaf, and Schrijver [12] asked for the minimum value of β so that when the minimum inand out-degrees of G are at least βn, G has a directed cycle of length at most 3. They proved that β ≤ 0.3487 and gave a formula relating β and c. Shen applied this formula to his 1998 result to get a slight improvement to β ≤ 0.3477 [29]. \n\n3 Seymour's Second Neighborhood Conjecture \n\nThis conjecture implies the special case of Caccetta-H¨ aggkvist when both inand out-degrees are at least n/ 3, and has received much attention of its own.", "clean_statement": "Conjecture 2.\n1. (L. Caccetta, R. H¨ aggkvist [5]) Every simple n-vertex digraph with minimum out-degree at least r has a cycle with length at most d n\n\n> r\n\ne.\n\nThis can be restated in the following way: Let A be an n × n 0-1 matrix such that aij = 1 implies\n\naji 6 = 1 for all i 6 = j. Let aii = 1 for all i. If the sum of every row in A is at least r + 1, Adn/r e\n\nhas trace greater than n.\n\n#2.1 Partial Results\n\nThe C-H conjecture has been proved for:\n\n• r = 2 by Caccetta and H¨ aggkvist [5]\n\n• r = 3 by Hamidoune [17]\n\n• r = 4 and r = 5 by Ho´ ang and Reed [19]\n\n• r ≤ √n/ 2 by Shen [30]. For the exact statement of his result, see Theorem 5.1. This shows that for any given r, the number of counterexamples to the conjecture (if any) is finite.\n\n• Cayley graphs (which implies all vertex transitive graphs using coset representations) by Hamidoune [15]. This proof uses a lemma of Kemperman [21] (Lemma 5.9). Also, Shen [32] proved that if deg +(u) + deg +(v) ≥ 4 for all ( u, v ) ∈ E(G), then g ≤ d n/ 2e,where g denotes the girth of G. This is an average local outdegree version for the r = 2 case of the Caccetta-H¨ aggkvist conjecture.\n\n#2.2 Approximate Results I - Additive Constant\n\nAnother approach is to show that if δ+\n\n> G\n\n≥ r, then there is a cycle of length at most n\n\n> r\n\n+ c for some small c. This has been proved for some values of c, as follows:\n\n• c = 2500 by Chv´ atal and Szemer´ edi [9]\n\n• c = 304 by Nishimura [27]\n\n• c = 73 by Shen [31]. 22.3 Approximate Results II - Special Case n/3\n\nThe case r = n/ 2 is trivial, but r = n/ 3 has received much attention. Research has sought the minimum constant c such that δ+\n\n> G\n\n≥ cn in an n-vertex simple digraph G forces a directed cycle of length at most 3. The conjecture is that c = 1 /3, and the current results are:\n\n• c ≤ (3 − √5) /2 = 0.382 by Caccetta and H¨ aggkvist [5]\n\n• c ≤ (2 √6 − 3) /5 = 0.3797 by Bondy in a neat subgraph counting argument [4]\n\n• c ≤ 3 − √7 = 0.3542 by [29] Similarly, Seymour, Graaf, and Schrijver [12] asked for the minimum value of β so that when the minimum inand out-degrees of G are at least βn, G has a directed cycle of length at most 3. They proved that β ≤ 0.3487 and gave a formula relating β and c. Shen applied this formula to his 1998 result to get a slight improvement to β ≤ 0.3477 [29].\n\n3 Seymour's Second Neighborhood Conjecture\n\nThis conjecture implies the special case of Caccetta-H¨ aggkvist when both inand out-degrees are at least n/ 3, and has received much attention of its own.", "statement_status": "exact", "statement_verification": "The source is Blair D. Sullivan's 14 April 2006 AIM workshop survey, *A Summary of Results and Problems Related to the Caccetta--Häggkvist Conjecture*. The canonical `input.json` preserves the full OCR extraction. Inspection of the original PDF recovers Conjecture 2.1 as follows.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[157]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 2.\\n1. (L. Caccetta, R. H¨ aggkvist [5]) Every simple n-vertex digraph with minimum out-degree at least r has a cycle with length at most d n \\n\\n> r\\n\\ne.\\n\\nThis can be restated in the following way: Let A be an n × n 0-1 matrix such that aij = 1 implies \\n\\naji 6 = 1 for all i 6 = j. Let aii = 1 for all i. If the sum of every row in A is at least r + 1, Adn/r e\\n\\nhas trace greater than n.\\n\\n#2.1 Partial Results \\n\\nThe C-H conjecture has been proved for: \\n\\n• r = 2 by Caccetta and H¨ aggkvist [5] \\n\\n• r = 3 by Hamidoune [17] \\n\\n• r = 4 and r = 5 by Ho´ ang and Reed [19] \\n\\n• r ≤ √n/ 2 by Shen [30]. For the exact statement of his result, see Theorem 5.1. This shows that for any given r, the number of counterexamples to the conjecture (if any) is finite. \\n\\n• Cayley graphs (which implies all vertex transitive graphs using coset representations) by Hamidoune [15]. This proof uses a lemma of Kemperman [21] (Lemma 5.9). Also, Shen [32] proved that if deg +(u) + deg +(v) ≥ 4 for all ( u, v ) ∈ E(G), then g ≤ d n/ 2e,where g denotes the girth of G. This is an average local outdegree version for the r = 2 case of the Caccetta-H¨ aggkvist conjecture. \\n\\n#2.2 Approximate Results I - Additive Constant \\n\\nAnother approach is to show that if δ+ \\n\\n> G\\n\\n≥ r, then there is a cycle of length at most n \\n\\n> r\\n\\n+ c for some small c. This has been proved for some values of c, as follows: \\n\\n• c = 2500 by Chv´ atal and Szemer´ edi [9] \\n\\n• c = 304 by Nishimura [27] \\n\\n• c = 73 by Shen [31]. 22.3 Approximate Results II - Special Case n/3 \\n\\nThe case r = n/ 2 is trivial, but r = n/ 3 has received much attention. Research has sought the minimum constant c such that δ+ \\n\\n> G\\n\\n≥ cn in an n-vertex simple digraph G forces a directed cycle of length at most 3. The conjecture is that c = 1 /3, and the current results are: \\n\\n• c ≤ (3 − √5) /2 = 0.382 by Caccetta and H¨ aggkvist [5] \\n\\n• c ≤ (2 √6 − 3) /5 = 0.3797 by Bondy in a neat subgraph counting argument [4] \\n\\n• c ≤ 3 − √7 = 0.3542 by [29] Similarly, Seymour, Graaf, and Schrijver [12] asked for the minimum value of β so that when the minimum inand out-degrees of G are at least βn, G has a directed cycle of length at most 3. They proved that β ≤ 0.3487 and gave a formula relating β and c. Shen applied this formula to his 1998 result to get a slight improvement to β ≤ 0.3477 [29]. \\n\\n3 Seymour's Second Neighborhood Conjecture \\n\\nThis conjecture implies the special case of Caccetta-H¨ aggkvist when both inand out-degrees are at least n/ 3, and has received much attention of its own.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0158", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every loopless digraph with adjacency matrix B and A=I+B, tr(A^t)=n+sum_{k=2}^t binom(t,k)tr(B^k), so tr(A^t)>n is exactly equivalent to the existence of a simple directed cycle of length at most t; if the directed girth is g, the excess is at least g binom(t,g) times the number of g-cycles. Any hypothetical Caccetta--Haggkvist counterexample reduces to a strongly connected exactly r-outregular oriented counterexample. For the sharp circulant C(n,r), the first trace jump at L=ceil(n/r) is computed exactly as n times the number of bounded compositions of n into L positive parts. These results clarify and sharpen the matrix formulation but do not prove the conjecture.\n\nCandidate contribution (exact matrix-walk equivalence and sharp family analysis; novelty confidence low): The packaged diagonal-padding trace identity, irreducible constant-row-sum counterexample reduction, and coefficient formula tr((I+B)^L)=n+n[x^n](x+...+x^r)^L for the sharp circulant family form a concrete candidate contribution.", "research_difficulty_suggested": "L4" }, { "id": 20001034, "problem_number": "AIM-COMBINATORICS-0159", "title": "A feedback-set bound for second neighborhoods", "statement": "Conjecture 3.\n1. (Seymour) Any simple digraph with no loops or digons has a vertex v whose second neighborhood is at least as big as its first neighborhood, i.e. |N +2 (v)| ≥ | N +(v)|.\n\nThe following is known for Seymour's second neighborhood conjecture:\n\n• When G is a tournament, this is Dean's conjecture, and was proved by Fisher [13] using probabilistic methods. There is also a combinatorial proof by Havet and Thomass´ e [18].\n\n• It was proved for digraphs with minimum outdegree ≤ 6 by Kaneko and Locke [20].\n\n• There is a vertex v where |N +2 (v)| ≥ γ|N +(v)| and γ = 0.657298... is the unique real root of 2 x3 + x2 − 1 =\n0. (Note the conjecture is that γ = 1) by Chen, Shen, and Yuster [7]. They also claim a slight improvement to γ = 0.67815 (proof unpublished).\n\n• Godbole, Cole, and Wright [14] showed that the conjecture holds for almost all digraphs.\n\n4 r-Regular Digraphs\n\nA digraph G is r-regular if every vertex v has δ+\n\n> G\n\n(v) = δ−\n\n> G\n\n(v) = r.", "original_statement": "Conjecture 3.\n1. (Seymour) Any simple digraph with no loops or digons has a vertex v whose second neighborhood is at least as big as its first neighborhood, i.e. |N +2 (v)| ≥ | N +(v)|.\n\nThe following is known for Seymour's second neighborhood conjecture: \n\n• When G is a tournament, this is Dean's conjecture, and was proved by Fisher [13] using probabilistic methods. There is also a combinatorial proof by Havet and Thomass´ e [18]. \n\n• It was proved for digraphs with minimum outdegree ≤ 6 by Kaneko and Locke [20]. \n\n• There is a vertex v where |N +2 (v)| ≥ γ|N +(v)| and γ = 0.657298... is the unique real root of 2 x3 + x2 − 1 = \n0. (Note the conjecture is that γ = 1) by Chen, Shen, and Yuster [7]. They also claim a slight improvement to γ = 0.67815 (proof unpublished). \n\n• Godbole, Cole, and Wright [14] showed that the conjecture holds for almost all digraphs. \n\n4 r-Regular Digraphs \n\nA digraph G is r-regular if every vertex v has δ+ \n\n> G\n\n(v) = δ− \n\n> G\n\n(v) = r.", "clean_statement": "Conjecture 3.\n1. (Seymour) Any simple digraph with no loops or digons has a vertex v whose second neighborhood is at least as big as its first neighborhood, i.e. |N +2 (v)| ≥ | N +(v)|.\n\nThe following is known for Seymour's second neighborhood conjecture:\n\n• When G is a tournament, this is Dean's conjecture, and was proved by Fisher [13] using probabilistic methods. There is also a combinatorial proof by Havet and Thomass´ e [18].\n\n• It was proved for digraphs with minimum outdegree ≤ 6 by Kaneko and Locke [20].\n\n• There is a vertex v where |N +2 (v)| ≥ γ|N +(v)| and γ = 0.657298... is the unique real root of 2 x3 + x2 − 1 =\n0. (Note the conjecture is that γ = 1) by Chen, Shen, and Yuster [7]. They also claim a slight improvement to γ = 0.67815 (proof unpublished).\n\n• Godbole, Cole, and Wright [14] showed that the conjecture holds for almost all digraphs.\n\n4 r-Regular Digraphs\n\nA digraph G is r-regular if every vertex v has δ+\n\n> G\n\n(v) = δ−\n\n> G\n\n(v) = r.", "statement_status": "exact", "statement_verification": "The source is the AIM workshop problem list *The Caccetta--Haggkvist conjecture*, Conjecture 3.1 (Seymour). After repairing the PDF extraction, the mathematical statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[158]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 3.\\n1. (Seymour) Any simple digraph with no loops or digons has a vertex v whose second neighborhood is at least as big as its first neighborhood, i.e. |N +2 (v)| ≥ | N +(v)|.\\n\\nThe following is known for Seymour's second neighborhood conjecture: \\n\\n• When G is a tournament, this is Dean's conjecture, and was proved by Fisher [13] using probabilistic methods. There is also a combinatorial proof by Havet and Thomass´ e [18]. \\n\\n• It was proved for digraphs with minimum outdegree ≤ 6 by Kaneko and Locke [20]. \\n\\n• There is a vertex v where |N +2 (v)| ≥ γ|N +(v)| and γ = 0.657298... is the unique real root of 2 x3 + x2 − 1 = \\n0. (Note the conjecture is that γ = 1) by Chen, Shen, and Yuster [7]. They also claim a slight improvement to γ = 0.67815 (proof unpublished). \\n\\n• Godbole, Cole, and Wright [14] showed that the conjecture holds for almost all digraphs. \\n\\n4 r-Regular Digraphs \\n\\nA digraph G is r-regular if every vertex v has δ+ \\n\\n> G\\n\\n(v) = δ− \\n\\n> G\\n\\n(v) = r.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0159", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every vertex v of a finite oriented graph D, the exact second out-neighborhood has size at least the minimum out-degree of D minus the directed feedback vertex number of the subdigraph induced by the first out-neighborhood of v. Hence the second-neighborhood deficit of a minimum-outdegree vertex is at most that local feedback number. An explicit eight-vertex tournament attains equality with out-degree three, second-neighborhood size two, and local feedback number one, while also showing that an arbitrary minimum-outdegree vertex need not be a Seymour vertex.\n\nCandidate contribution (structural inequality with sharp example; novelty confidence low): The candidate contribution is the quantitative inequality |N_2^+(v)| >= delta^+(D) - tau(D[N_1^+(v)]) for every vertex v, together with an eight-vertex equality tournament.", "research_difficulty_suggested": "L4" }, { "id": 20001035, "problem_number": "AIM-COMBINATORICS-0160", "title": "Exact anatomy of the BCW circular construction", "statement": "Conjecture 4.\n1. (Behzad, Chartrand, Wall [2]) The minimum number of vertices in an r-regular digraph G with girth g is r(g − 1) + 1.\n\nBehzad, Chartrand and Wall give an example achieving this by placing r(g − 1) + 1 vertices on a circle with each vertex having edges to the next r vertices in clockwise order. The Caccetta-H¨ aggkvist Conjecture is a generalization of this earlier conjecture. The Behzad-Chartrand-Wall conjecture was proved for the following special cases: 3• r = 2 by Behzad [1]\n\n• r = 3 by Bermond [3]\n\n• Vertex-transitive graphs by Hamidoune [16]\n\n• If δ+\n\n> G\n\n≥ r, then g ≤ 3d n\n\n> r\n\nln( 2+ √7\n\n> 3\n\n)e ≈ 1.312 n\n\n> r\n\nby Shen [31].\n\n5 Related Results\n\nTheorem 5.\n1. (Shen [30]) For a digraph G on n vertices, if δ+\n\n> G\n\n≥ r, and n ≥ 2r2 − 3r + 1,then G has a cycle of length at most d n\n\n> r\n\ne.\n\nIn a graph G, for u, v ∈ V (G), κ(u, v ) denotes the maximum number of internally disjoint paths between u and v. If G is a digraph, κ counts the maximum number of internally disjoint directed paths from u to v.In a graph G, for u, v ∈ V (G), λ(u, v ) denotes the maximum number of edge-disjoint paths between u and v. If G is a digraph, λ counts the maximum number of edge-disjoint directed paths from u to v.\n\nTheorem 5.\n2. (Thomassen [34]) For all positive integers r, there is a digraph D without digons with δ+\n\n> D\n\n≥ r and δ−\n\n> D\n\n≥ r such that: i. no vertex v ∈ V (D) is contained in three openly disjoint circuits (that is, three circuits which pairwise share only v)ii. no edge (x, y ) ∈ E(D) has κ(y, x ) ≥ 3.\n\nTheorem 5.\n3. (Mader [25]) For every integer k ≥ 0, If G has |V (G)| > k 2(k + 1) and at most\n\nk2(k + 1) vertices of G have out-degree at most k3(k + 1), then there are vertices x 6 = y so that\n\nκ(x, y ) > k.\n\n#5.1 Undirected Graph Theorems\n\nTheorem 5.\n4. (Mader [22]) Every graph G with δG ≥ r contains vertices x, y with κ(x, y ) ≥ r\n\nwhen r ≥ 1.\n\nTheorem 5.\n5. (Mader [23]) Every graph G with δG ≥ r contains r + 1 vertices v1,..., v r+1\n\nwith λ(vi, v j ) ≥ r for all i 6 = j.\n\n#5.2 Additive Number Theory Results\n\nGiven an additive group Γ, and sets A, B ⊆ Γ, let A + B:= {a + b | a ∈ A, b ∈ B}, and A ˆ+B:= {a + b | a ∈ A, b ∈ B, a 6 = b}. Finally, for a positive integer r, let rB:= {b1 + · · · + bh| all\n\nbi ∈ B, not necessarily distinct }.\n\nTheorem 5.\n6. (Cauchy [6] and Davenport [10], [11]) Let p be a prime, and A, B ⊆ Z/p Z be nonempty. Then |A + B| ≥ min( p, |A| + |B| − 1).\n\n4Theorem 5.\n7. (I. Chowla [8]) Let m be a positive integer, and A, B ⊆ Z/m Z such that 0 ∈ B\n\nand gcd (b, m ) = 1 for all nonzero b ∈ B. Then |A + B| ≥ min( m, |A| + |B| − 1).\n\nTheorem 5.\n8. (Dias de Silva and Hamidoune [33]) The Erd¨ os-Heilbronn Conjecture: Let\n\nA ⊆ Z/p Z, with p prime. Then |A ˆ+A| ≥ min(2 |A| − 3, p ).\n\nFor a multiplicative group Γ and sets A, B ⊆ Γ, let AB:= {ab | a ∈ A, b ∈ B}.\n\nLemma 5.\n9. (Kemperman [21]) Given a group Γ and finite non-empty subsets A, B ⊆ Γ, if\n\n1 ∈ A, B but (1, 1) is the only pair (a, b ) with a ∈ A, b ∈ B such that ab = 1, then |AB | ≥ |A| + |B| − 1.\n\nWe say G is a layered digraph if G is a digraph with V (G) = ∪hi=0 Vi with Vi 6 = ∅, and Vi ∩ Vj = ∅\n\nfor all i 6 = j, and ( u, v ) ∈ E(G) implies u ∈ Vi−1, v ∈ Vi for some i ∈ { 1,..., h }.A Pl¨ unnecke graph is a layered digraph G with the following two properties: 1. If u, v, w 1,..., w k are vertices of G with ( u, v ), (v, w 1),..., (v, w k) ∈ E(G), then there are distinct vertices v1,..., v k so that ( u, v i), (vi, w i) ∈ E(G) for i = 1,..., k.2. If v, w, u 1,..., u k are vertices of G with ( v, w ), (u1, v ),..., (uk, v ) ∈ E(G), then there are distinct vertices v1,..., v k so that ( ui, v i), (vi, w ) ∈ E(G) for i = 1,..., k.Let G be a digraph, and X, Y nonempty subsets of V (G). Then\n\nIm(X,Y):= {y ∈ Y | there is a directed path from X to y}. The magnification ratio D(X,Y) is\n\nD(X, Y ):= min\n\n> Z⊆X,Z 6=∅\n\n{|Im (Z, Y )|\n\n|Z|\n\n}.\n\nTheorem 5.\n1\n0. (Pl¨ unnecke [28]) In a Pl¨ unnecke graph, let Di = D(V0, V i). Then\n\nD1 ≥ D 1\n\n> 2\n> 2\n\n≥ · · · ≥ D 1\n\n> h\n> h.\n\nThe following are consequences of applying Pl¨ unnecke's Inequalities to a special graph created from subsets A, B of a group:\n\nTheorem 5.11. For sets A, B ⊂ Γ:1. |iB | 1\n\n> i\n\n≥ | hB | 1\n\n> h\n\nfor all 0 ≤ i < h.2. If |B| = k, and |B + B| ≤ ck, then |hB | ≤ chk.3. If |A| = n, and |A + B| < cn then for all k, ` ∈ Z+, we have that |kB − `B | ≤ ck+`n where\n\nkB − `B denotes the set of all elements expressable as (b1 + · · · + bk) − (b′\n\n> 1\n\n+ · · · + b′\n\n> `\n\n) where all bi, b ′\n\n> i\n\nare in B.\n\nTheorem 5.12. (generalization of Erd¨ os-Heilbronn (Thm. 5.8)) Let A, B ⊆ Z/p Z, with p prime and |A| 6 = |B|. Let C = A ˆ+B. Then |C| ≥ min( |A| + |B| − 2, p ).\n\n56 Open Problems and Conjectures\n\n#6.1 Rainbow Conjectures\n\n6.1.1 A Colored Generalization of Seymour's Second Neighborhood\n\nGiven a digraph G = ( V, E ) with each edge e ∈ E having a set Se of labels in {1, 2,..., k G}, a\n\nrainbow structure H in G (such as a path or cycle) means one in which there is a way to assign each edge e ∈ E(H) a label `(e) ∈ Se so that ell (e) 6 = `(f ) for all edges e 6 = f in E(H).", "original_statement": "Conjecture 4.\n1. (Behzad, Chartrand, Wall [2]) The minimum number of vertices in an r-regular digraph G with girth g is r(g − 1) + 1.\n\nBehzad, Chartrand and Wall give an example achieving this by placing r(g − 1) + 1 vertices on a circle with each vertex having edges to the next r vertices in clockwise order. The Caccetta-H¨ aggkvist Conjecture is a generalization of this earlier conjecture. The Behzad-Chartrand-Wall conjecture was proved for the following special cases: 3• r = 2 by Behzad [1] \n\n• r = 3 by Bermond [3] \n\n• Vertex-transitive graphs by Hamidoune [16] \n\n• If δ+ \n\n> G\n\n≥ r, then g ≤ 3d n \n\n> r\n\nln( 2+ √7 \n\n> 3\n\n)e ≈ 1.312 n \n\n> r\n\nby Shen [31]. \n\n5 Related Results \n\nTheorem 5.\n1. (Shen [30]) For a digraph G on n vertices, if δ+ \n\n> G\n\n≥ r, and n ≥ 2r2 − 3r + 1,then G has a cycle of length at most d n \n\n> r\n\ne.\n\nIn a graph G, for u, v ∈ V (G), κ(u, v ) denotes the maximum number of internally disjoint paths between u and v. If G is a digraph, κ counts the maximum number of internally disjoint directed paths from u to v.In a graph G, for u, v ∈ V (G), λ(u, v ) denotes the maximum number of edge-disjoint paths between u and v. If G is a digraph, λ counts the maximum number of edge-disjoint directed paths from u to v.\n\nTheorem 5.\n2. (Thomassen [34]) For all positive integers r, there is a digraph D without digons with δ+ \n\n> D\n\n≥ r and δ− \n\n> D\n\n≥ r such that: i. no vertex v ∈ V (D) is contained in three openly disjoint circuits (that is, three circuits which pairwise share only v)ii. no edge (x, y ) ∈ E(D) has κ(y, x ) ≥ 3.\n\nTheorem 5.\n3. (Mader [25]) For every integer k ≥ 0, If G has |V (G)| > k 2(k + 1) and at most \n\nk2(k + 1) vertices of G have out-degree at most k3(k + 1), then there are vertices x 6 = y so that \n\nκ(x, y ) > k.\n\n#5.1 Undirected Graph Theorems \n\nTheorem 5.\n4. (Mader [22]) Every graph G with δG ≥ r contains vertices x, y with κ(x, y ) ≥ r\n\nwhen r ≥ 1.\n\nTheorem 5.\n5. (Mader [23]) Every graph G with δG ≥ r contains r + 1 vertices v1,..., v r+1 \n\nwith λ(vi, v j ) ≥ r for all i 6 = j.\n\n#5.2 Additive Number Theory Results \n\nGiven an additive group Γ, and sets A, B ⊆ Γ, let A + B:= {a + b | a ∈ A, b ∈ B}, and A ˆ+B:= {a + b | a ∈ A, b ∈ B, a 6 = b}. Finally, for a positive integer r, let rB:= {b1 + · · · + bh| all \n\nbi ∈ B, not necessarily distinct }.\n\nTheorem 5.\n6. (Cauchy [6] and Davenport [10], [11]) Let p be a prime, and A, B ⊆ Z/p Z be nonempty. Then |A + B| ≥ min( p, |A| + |B| − 1).\n\n4Theorem 5.\n7. (I. Chowla [8]) Let m be a positive integer, and A, B ⊆ Z/m Z such that 0 ∈ B\n\nand gcd (b, m ) = 1 for all nonzero b ∈ B. Then |A + B| ≥ min( m, |A| + |B| − 1).\n\nTheorem 5.\n8. (Dias de Silva and Hamidoune [33]) The Erd¨ os-Heilbronn Conjecture: Let \n\nA ⊆ Z/p Z, with p prime. Then |A ˆ+A| ≥ min(2 |A| − 3, p ).\n\nFor a multiplicative group Γ and sets A, B ⊆ Γ, let AB:= {ab | a ∈ A, b ∈ B}.\n\nLemma 5.\n9. (Kemperman [21]) Given a group Γ and finite non-empty subsets A, B ⊆ Γ, if \n\n1 ∈ A, B but (1, 1) is the only pair (a, b ) with a ∈ A, b ∈ B such that ab = 1, then |AB | ≥ |A| + |B| − 1.\n\nWe say G is a layered digraph if G is a digraph with V (G) = ∪hi=0 Vi with Vi 6 = ∅, and Vi ∩ Vj = ∅\n\nfor all i 6 = j, and ( u, v ) ∈ E(G) implies u ∈ Vi−1, v ∈ Vi for some i ∈ { 1,..., h }.A Pl¨ unnecke graph is a layered digraph G with the following two properties: 1. If u, v, w 1,..., w k are vertices of G with ( u, v ), (v, w 1),..., (v, w k) ∈ E(G), then there are distinct vertices v1,..., v k so that ( u, v i), (vi, w i) ∈ E(G) for i = 1,..., k.2. If v, w, u 1,..., u k are vertices of G with ( v, w ), (u1, v ),..., (uk, v ) ∈ E(G), then there are distinct vertices v1,..., v k so that ( ui, v i), (vi, w ) ∈ E(G) for i = 1,..., k.Let G be a digraph, and X, Y nonempty subsets of V (G). Then \n\nIm(X,Y):= {y ∈ Y | there is a directed path from X to y}. The magnification ratio D(X,Y) is \n\nD(X, Y ):= min \n\n> Z⊆X,Z 6=∅\n\n{|Im (Z, Y )|\n\n|Z|\n\n}.\n\nTheorem 5.\n1\n0. (Pl¨ unnecke [28]) In a Pl¨ unnecke graph, let Di = D(V0, V i). Then \n\nD1 ≥ D 1 \n\n> 2\n> 2\n\n≥ · · · ≥ D 1 \n\n> h\n> h.\n\nThe following are consequences of applying Pl¨ unnecke's Inequalities to a special graph created from subsets A, B of a group: \n\nTheorem 5.11. For sets A, B ⊂ Γ:1. |iB | 1 \n\n> i\n\n≥ | hB | 1 \n\n> h\n\nfor all 0 ≤ i < h.2. If |B| = k, and |B + B| ≤ ck, then |hB | ≤ chk.3. If |A| = n, and |A + B| < cn then for all k, ` ∈ Z+, we have that |kB − `B | ≤ ck+`n where \n\nkB − `B denotes the set of all elements expressable as (b1 + · · · + bk) − (b′ \n\n> 1\n\n+ · · · + b′\n\n> `\n\n) where all bi, b ′ \n\n> i\n\nare in B.\n\nTheorem 5.12. (generalization of Erd¨ os-Heilbronn (Thm. 5.8)) Let A, B ⊆ Z/p Z, with p prime and |A| 6 = |B|. Let C = A ˆ+B. Then |C| ≥ min( |A| + |B| − 2, p ).\n\n56 Open Problems and Conjectures \n\n#6.1 Rainbow Conjectures \n\n6.1.1 A Colored Generalization of Seymour's Second Neighborhood \n\nGiven a digraph G = ( V, E ) with each edge e ∈ E having a set Se of labels in {1, 2,..., k G}, a \n\nrainbow structure H in G (such as a path or cycle) means one in which there is a way to assign each edge e ∈ E(H) a label `(e) ∈ Se so that ell (e) 6 = `(f ) for all edges e 6 = f in E(H).", "clean_statement": "Conjecture 4.\n1. (Behzad, Chartrand, Wall [2]) The minimum number of vertices in an r-regular digraph G with girth g is r(g − 1) + 1.\n\nBehzad, Chartrand and Wall give an example achieving this by placing r(g − 1) + 1 vertices on a circle with each vertex having edges to the next r vertices in clockwise order. The Caccetta-H¨ aggkvist Conjecture is a generalization of this earlier conjecture. The Behzad-Chartrand-Wall conjecture was proved for the following special cases: 3• r = 2 by Behzad [1]\n\n• r = 3 by Bermond [3]\n\n• Vertex-transitive graphs by Hamidoune [16]\n\n• If δ+\n\n> G\n\n≥ r, then g ≤ 3d n\n\n> r\n\nln( 2+ √7\n\n> 3\n\n)e ≈ 1.312 n\n\n> r\n\nby Shen [31].\n\n5 Related Results\n\nTheorem 5.\n1. (Shen [30]) For a digraph G on n vertices, if δ+\n\n> G\n\n≥ r, and n ≥ 2r2 − 3r + 1,then G has a cycle of length at most d n\n\n> r\n\ne.\n\nIn a graph G, for u, v ∈ V (G), κ(u, v ) denotes the maximum number of internally disjoint paths between u and v. If G is a digraph, κ counts the maximum number of internally disjoint directed paths from u to v.In a graph G, for u, v ∈ V (G), λ(u, v ) denotes the maximum number of edge-disjoint paths between u and v. If G is a digraph, λ counts the maximum number of edge-disjoint directed paths from u to v.\n\nTheorem 5.\n2. (Thomassen [34]) For all positive integers r, there is a digraph D without digons with δ+\n\n> D\n\n≥ r and δ−\n\n> D\n\n≥ r such that: i. no vertex v ∈ V (D) is contained in three openly disjoint circuits (that is, three circuits which pairwise share only v)ii. no edge (x, y ) ∈ E(D) has κ(y, x ) ≥ 3.\n\nTheorem 5.\n3. (Mader [25]) For every integer k ≥ 0, If G has |V (G)| > k 2(k + 1) and at most\n\nk2(k + 1) vertices of G have out-degree at most k3(k + 1), then there are vertices x 6 = y so that\n\nκ(x, y ) > k.\n\n#5.1 Undirected Graph Theorems\n\nTheorem 5.\n4. (Mader [22]) Every graph G with δG ≥ r contains vertices x, y with κ(x, y ) ≥ r\n\nwhen r ≥ 1.\n\nTheorem 5.\n5. (Mader [23]) Every graph G with δG ≥ r contains r + 1 vertices v1,..., v r+1\n\nwith λ(vi, v j ) ≥ r for all i 6 = j.\n\n#5.2 Additive Number Theory Results\n\nGiven an additive group Γ, and sets A, B ⊆ Γ, let A + B:= {a + b | a ∈ A, b ∈ B}, and A ˆ+B:= {a + b | a ∈ A, b ∈ B, a 6 = b}. Finally, for a positive integer r, let rB:= {b1 + · · · + bh| all\n\nbi ∈ B, not necessarily distinct }.\n\nTheorem 5.\n6. (Cauchy [6] and Davenport [10], [11]) Let p be a prime, and A, B ⊆ Z/p Z be nonempty. Then |A + B| ≥ min( p, |A| + |B| − 1).\n\n4Theorem 5.\n7. (I. Chowla [8]) Let m be a positive integer, and A, B ⊆ Z/m Z such that 0 ∈ B\n\nand gcd (b, m ) = 1 for all nonzero b ∈ B. Then |A + B| ≥ min( m, |A| + |B| − 1).\n\nTheorem 5.\n8. (Dias de Silva and Hamidoune [33]) The Erd¨ os-Heilbronn Conjecture: Let\n\nA ⊆ Z/p Z, with p prime. Then |A ˆ+A| ≥ min(2 |A| − 3, p ).\n\nFor a multiplicative group Γ and sets A, B ⊆ Γ, let AB:= {ab | a ∈ A, b ∈ B}.\n\nLemma 5.\n9. (Kemperman [21]) Given a group Γ and finite non-empty subsets A, B ⊆ Γ, if\n\n1 ∈ A, B but (1, 1) is the only pair (a, b ) with a ∈ A, b ∈ B such that ab = 1, then |AB | ≥ |A| + |B| − 1.\n\nWe say G is a layered digraph if G is a digraph with V (G) = ∪hi=0 Vi with Vi 6 = ∅, and Vi ∩ Vj = ∅\n\nfor all i 6 = j, and ( u, v ) ∈ E(G) implies u ∈ Vi−1, v ∈ Vi for some i ∈ { 1,..., h }.A Pl¨ unnecke graph is a layered digraph G with the following two properties: 1. If u, v, w 1,..., w k are vertices of G with ( u, v ), (v, w 1),..., (v, w k) ∈ E(G), then there are distinct vertices v1,..., v k so that ( u, v i), (vi, w i) ∈ E(G) for i = 1,..., k.2. If v, w, u 1,..., u k are vertices of G with ( v, w ), (u1, v ),..., (uk, v ) ∈ E(G), then there are distinct vertices v1,..., v k so that ( ui, v i), (vi, w ) ∈ E(G) for i = 1,..., k.Let G be a digraph, and X, Y nonempty subsets of V (G). Then\n\nIm(X,Y):= {y ∈ Y | there is a directed path from X to y}. The magnification ratio D(X,Y) is\n\nD(X, Y ):= min\n\n> Z⊆X,Z 6=∅\n\n{|Im (Z, Y )|\n\n|Z|\n\n}.\n\nTheorem 5.\n1\n0. (Pl¨ unnecke [28]) In a Pl¨ unnecke graph, let Di = D(V0, V i). Then\n\nD1 ≥ D 1\n\n> 2\n> 2\n\n≥ · · · ≥ D 1\n\n> h\n> h.\n\nThe following are consequences of applying Pl¨ unnecke's Inequalities to a special graph created from subsets A, B of a group:\n\nTheorem 5.11. For sets A, B ⊂ Γ:1. |iB | 1\n\n> i\n\n≥ | hB | 1\n\n> h\n\nfor all 0 ≤ i < h.2. If |B| = k, and |B + B| ≤ ck, then |hB | ≤ chk.3. If |A| = n, and |A + B| < cn then for all k, ` ∈ Z+, we have that |kB − `B | ≤ ck+`n where\n\nkB − `B denotes the set of all elements expressable as (b1 + · · · + bk) − (b′\n\n> 1\n\n+ · · · + b′\n\n> `\n\n) where all bi, b ′\n\n> i\n\nare in B.\n\nTheorem 5.12. (generalization of Erd¨ os-Heilbronn (Thm. 5.8)) Let A, B ⊆ Z/p Z, with p prime and |A| 6 = |B|. Let C = A ˆ+B. Then |C| ≥ min( |A| + |B| − 2, p ).\n\n56 Open Problems and Conjectures\n\n#6.1 Rainbow Conjectures\n\n6.1.1 A Colored Generalization of Seymour's Second Neighborhood\n\nGiven a digraph G = ( V, E ) with each edge e ∈ E having a set Se of labels in {1, 2,..., k G}, a\n\nrainbow structure H in G (such as a path or cycle) means one in which there is a way to assign each edge e ∈ E(H) a label `(e) ∈ Se so that ell (e) 6 = `(f ) for all edges e 6 = f in E(H).", "statement_status": "exact", "statement_verification": "The source is Blair D. Sullivan's 2006 AIM workshop survey *A Summary of Results and Problems Related to the Caccetta--Häggkvist Conjecture*. Section 4 defines an \\(r\\)-regular digraph by \\[ d^+(v)=d^-(v)=r\\qquad\\text{for every vertex }v. \\] After repairing the extraction, Conjecture 4.1 reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[159]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 4.\\n1. (Behzad, Chartrand, Wall [2]) The minimum number of vertices in an r-regular digraph G with girth g is r(g − 1) + 1.\\n\\nBehzad, Chartrand and Wall give an example achieving this by placing r(g − 1) + 1 vertices on a circle with each vertex having edges to the next r vertices in clockwise order. The Caccetta-H¨ aggkvist Conjecture is a generalization of this earlier conjecture. The Behzad-Chartrand-Wall conjecture was proved for the following special cases: 3• r = 2 by Behzad [1] \\n\\n• r = 3 by Bermond [3] \\n\\n• Vertex-transitive graphs by Hamidoune [16] \\n\\n• If δ+ \\n\\n> G\\n\\n≥ r, then g ≤ 3d n \\n\\n> r\\n\\nln( 2+ √7 \\n\\n> 3\\n\\n)e ≈ 1.312 n \\n\\n> r\\n\\nby Shen [31]. \\n\\n5 Related Results \\n\\nTheorem 5.\\n1. (Shen [30]) For a digraph G on n vertices, if δ+ \\n\\n> G\\n\\n≥ r, and n ≥ 2r2 − 3r + 1,then G has a cycle of length at most d n \\n\\n> r\\n\\ne.\\n\\nIn a graph G, for u, v ∈ V (G), κ(u, v ) denotes the maximum number of internally disjoint paths between u and v. If G is a digraph, κ counts the maximum number of internally disjoint directed paths from u to v.In a graph G, for u, v ∈ V (G), λ(u, v ) denotes the maximum number of edge-disjoint paths between u and v. If G is a digraph, λ counts the maximum number of edge-disjoint directed paths from u to v.\\n\\nTheorem 5.\\n2. (Thomassen [34]) For all positive integers r, there is a digraph D without digons with δ+ \\n\\n> D\\n\\n≥ r and δ− \\n\\n> D\\n\\n≥ r such that: i. no vertex v ∈ V (D) is contained in three openly disjoint circuits (that is, three circuits which pairwise share only v)ii. no edge (x, y ) ∈ E(D) has κ(y, x ) ≥ 3.\\n\\nTheorem 5.\\n3. (Mader [25]) For every integer k ≥ 0, If G has |V (G)| > k 2(k + 1) and at most \\n\\nk2(k + 1) vertices of G have out-degree at most k3(k + 1), then there are vertices x 6 = y so that \\n\\nκ(x, y ) > k.\\n\\n#5.1 Undirected Graph Theorems \\n\\nTheorem 5.\\n4. (Mader [22]) Every graph G with δG ≥ r contains vertices x, y with κ(x, y ) ≥ r\\n\\nwhen r ≥ 1.\\n\\nTheorem 5.\\n5. (Mader [23]) Every graph G with δG ≥ r contains r + 1 vertices v1,..., v r+1 \\n\\nwith λ(vi, v j ) ≥ r for all i 6 = j.\\n\\n#5.2 Additive Number Theory Results \\n\\nGiven an additive group Γ, and sets A, B ⊆ Γ, let A + B:= {a + b | a ∈ A, b ∈ B}, and A ˆ+B:= {a + b | a ∈ A, b ∈ B, a 6 = b}. Finally, for a positive integer r, let rB:= {b1 + · · · + bh| all \\n\\nbi ∈ B, not necessarily distinct }.\\n\\nTheorem 5.\\n6. (Cauchy [6] and Davenport [10], [11]) Let p be a prime, and A, B ⊆ Z/p Z be nonempty. Then |A + B| ≥ min( p, |A| + |B| − 1).\\n\\n4Theorem 5.\\n7. (I. Chowla [8]) Let m be a positive integer, and A, B ⊆ Z/m Z such that 0 ∈ B\\n\\nand gcd (b, m ) = 1 for all nonzero b ∈ B. Then |A + B| ≥ min( m, |A| + |B| − 1).\\n\\nTheorem 5.\\n8. (Dias de Silva and Hamidoune [33]) The Erd¨ os-Heilbronn Conjecture: Let \\n\\nA ⊆ Z/p Z, with p prime. Then |A ˆ+A| ≥ min(2 |A| − 3, p ).\\n\\nFor a multiplicative group Γ and sets A, B ⊆ Γ, let AB:= {ab | a ∈ A, b ∈ B}.\\n\\nLemma 5.\\n9. (Kemperman [21]) Given a group Γ and finite non-empty subsets A, B ⊆ Γ, if \\n\\n1 ∈ A, B but (1, 1) is the only pair (a, b ) with a ∈ A, b ∈ B such that ab = 1, then |AB | ≥ |A| + |B| − 1.\\n\\nWe say G is a layered digraph if G is a digraph with V (G) = ∪hi=0 Vi with Vi 6 = ∅, and Vi ∩ Vj = ∅\\n\\nfor all i 6 = j, and ( u, v ) ∈ E(G) implies u ∈ Vi−1, v ∈ Vi for some i ∈ { 1,..., h }.A Pl¨ unnecke graph is a layered digraph G with the following two properties: 1. If u, v, w 1,..., w k are vertices of G with ( u, v ), (v, w 1),..., (v, w k) ∈ E(G), then there are distinct vertices v1,..., v k so that ( u, v i), (vi, w i) ∈ E(G) for i = 1,..., k.2. If v, w, u 1,..., u k are vertices of G with ( v, w ), (u1, v ),..., (uk, v ) ∈ E(G), then there are distinct vertices v1,..., v k so that ( ui, v i), (vi, w ) ∈ E(G) for i = 1,..., k.Let G be a digraph, and X, Y nonempty subsets of V (G). Then \\n\\nIm(X,Y):= {y ∈ Y | there is a directed path from X to y}. The magnification ratio D(X,Y) is \\n\\nD(X, Y ):= min \\n\\n> Z⊆X,Z 6=∅\\n\\n{|Im (Z, Y )|\\n\\n|Z|\\n\\n}.\\n\\nTheorem 5.\\n1\\n0. (Pl¨ unnecke [28]) In a Pl¨ unnecke graph, let Di = D(V0, V i). Then \\n\\nD1 ≥ D 1 \\n\\n> 2\\n> 2\\n\\n≥ · · · ≥ D 1 \\n\\n> h\\n> h.\\n\\nThe following are consequences of applying Pl¨ unnecke's Inequalities to a special graph created from subsets A, B of a group: \\n\\nTheorem 5.11. For sets A, B ⊂ Γ:1. |iB | 1 \\n\\n> i\\n\\n≥ | hB | 1 \\n\\n> h\\n\\nfor all 0 ≤ i < h.2. If |B| = k, and |B + B| ≤ ck, then |hB | ≤ chk.3. If |A| = n, and |A + B| < cn then for all k, ` ∈ Z+, we have that |kB − `B | ≤ ck+`n where \\n\\nkB − `B denotes the set of all elements expressable as (b1 + · · · + bk) − (b′ \\n\\n> 1\\n\\n+ · · · + b′\\n\\n> `\\n\\n) where all bi, b ′ \\n\\n> i\\n\\nare in B.\\n\\nTheorem 5.12. (generalization of Erd¨ os-Heilbronn (Thm. 5.8)) Let A, B ⊆ Z/p Z, with p prime and |A| 6 = |B|. Let C = A ˆ+B. Then |C| ≥ min( |A| + |B| − 2, p ).\\n\\n56 Open Problems and Conjectures \\n\\n#6.1 Rainbow Conjectures \\n\\n6.1.1 A Colored Generalization of Seymour's Second Neighborhood \\n\\nGiven a digraph G = ( V, E ) with each edge e ∈ E having a set Se of labels in {1, 2,..., k G}, a \\n\\nrainbow structure H in G (such as a path or cycle) means one in which there is a way to assign each edge e ∈ E(H) a label `(e) ∈ Se so that ell (e) 6 = `(f ) for all edges e 6 = f in E(H).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0160", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the interval circulant C(n,r) on Z_n with arcs x to x+s for 1 <= s <= r, the in- and out-degrees are both r, the directed girth is exactly ceil(n/r), and the digraph is oriented exactly when n > 2r. Its shortest directed cycles admit an exact coefficient count. At the Behzad--Chartrand--Wall order n=r(g-1)+1, the number of unrooted directed g-cycles is ((r(g-1)+1)/g) times binomial(r+g-2,g-1). In addition, the minimum cages of girth two and three are classified respectively as the complete bidirected digraph and all regular tournaments of the relevant orders.\n\nCandidate contribution (exact enumeration and extremal classification; novelty confidence low): The candidate contribution is an exact shortest-cycle census for every interval circulant C(n,r), specializing at the BCW extremal order to ((r(g-1)+1)/g) binomial(r+g-2,g-1), together with a minimal-zero-sum proof and complete equality classifications for girths two and three.", "research_difficulty_suggested": "L4" }, { "id": 20001036, "problem_number": "AIM-COMBINATORICS-0161", "title": "Rainbow reachability and the multi-label core obstruction", "statement": "Conjecture 6.\n1. (Seymour, Sullivan) Let G be a simple digraph on the vertex set V, and\n\nE1,..., E k ⊆ E(G). Say an edge e ∈ E(G) has label set Se ⊆ { 1,..., k } where i ∈ Se if and only if e ∈ Ei. Finally, let Gi = ( V, E i).i. There exists a rainbow (di)cycle in G or ii. There exists a vertex v such that |{ w | there exists a rainbow path from v to w}| ≥ ∑ki=1 δ+\n\n> Gi\n\n(v).Notes: This is false if you require that the colors appear in an increasing order (cyclic on the cycle). We have been able to show that this conjecture holds when G1,..., G k are Cayley graphs on a common group Γ (using induction on Lemma 5.9), and when δ+\n\n> Gi\n\n(v) ≤ 1 for all v and all i\n\nexcept i = 1, where we allow the outdegrees to be unbounded (but finite).\n\n6.1.2 Implications of", "original_statement": "Conjecture 6.\n1. (Seymour, Sullivan) Let G be a simple digraph on the vertex set V, and \n\nE1,..., E k ⊆ E(G). Say an edge e ∈ E(G) has label set Se ⊆ { 1,..., k } where i ∈ Se if and only if e ∈ Ei. Finally, let Gi = ( V, E i).i. There exists a rainbow (di)cycle in G or ii. There exists a vertex v such that |{ w | there exists a rainbow path from v to w}| ≥ ∑ki=1 δ+ \n\n> Gi\n\n(v).Notes: This is false if you require that the colors appear in an increasing order (cyclic on the cycle). We have been able to show that this conjecture holds when G1,..., G k are Cayley graphs on a common group Γ (using induction on Lemma 5.9), and when δ+ \n\n> Gi\n\n(v) ≤ 1 for all v and all i\n\nexcept i = 1, where we allow the outdegrees to be unbounded (but finite). \n\n6.1.2 Implications of", "clean_statement": "Conjecture 6.\n1. (Seymour, Sullivan) Let G be a simple digraph on the vertex set V, and\n\nE1,..., E k ⊆ E(G). Say an edge e ∈ E(G) has label set Se ⊆ { 1,..., k } where i ∈ Se if and only if e ∈ Ei. Finally, let Gi = ( V, E i).i. There exists a rainbow (di)cycle in G or ii. There exists a vertex v such that |{ w | there exists a rainbow path from v to w}| ≥ ∑ki=1 δ+\n\n> Gi\n\n(v).Notes: This is false if you require that the colors appear in an increasing order (cyclic on the cycle). We have been able to show that this conjecture holds when G1,..., G k are Cayley graphs on a common group Γ (using induction on Lemma 5.9), and when δ+\n\n> Gi\n\n(v) ≤ 1 for all v and all i\n\nexcept i = 1, where we allow the outdegrees to be unbounded (but finite).\n\n6.1.2 Implications of", "statement_status": "exact", "statement_verification": "The record comes from Blair D. Sullivan's 2006 AIM workshop summary *A Summary of Problems and Results Related to the Caccetta--Häggkvist Conjecture*, Section 6.1.1, “A Colored Generalization of Seymour's Second Neighborhood.” The corpus extraction runs into the next heading (“6.1.2 Implications of”); that phrase is not part of the conjecture.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[160]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n1. (Seymour, Sullivan) Let G be a simple digraph on the vertex set V, and \\n\\nE1,..., E k ⊆ E(G). Say an edge e ∈ E(G) has label set Se ⊆ { 1,..., k } where i ∈ Se if and only if e ∈ Ei. Finally, let Gi = ( V, E i).i. There exists a rainbow (di)cycle in G or ii. There exists a vertex v such that |{ w | there exists a rainbow path from v to w}| ≥ ∑ki=1 δ+ \\n\\n> Gi\\n\\n(v).Notes: This is false if you require that the colors appear in an increasing order (cyclic on the cycle). We have been able to show that this conjecture holds when G1,..., G k are Cayley graphs on a common group Γ (using induction on Lemma 5.9), and when δ+ \\n\\n> Gi\\n\\n(v) ≤ 1 for all v and all i\\n\\nexcept i = 1, where we allow the outdegrees to be unbounded (but finite). \\n\\n6.1.2 Implications of\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0161", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The original PDF is recovered with delta^+_{G_i}(v) meaning the outdegree of the particular vertex and with positive-length rainbow paths, so v is excluded from its own reachability set. Rainbow means that arc label sets admit distinct representatives, equivalently transversal-matroid independence. For each vertex v, a proved local bound separates its distinct labelled outneighbors from the surplus caused by multiple labels and certifies the conjectural reachability alternative when new rainbow second targets cover that surplus. Consequently the full conjecture holds whenever the subdigraph of arcs carrying at least two labels is acyclic. Any counterexample must have positive minimum outdegree in that multi-label core, fail the local surplus bound everywhere, and contain a directed core cycle with a Hall-deficient label subfamily of size at least three. A directed core triangle must use one common two-label palette on all three arcs.\n\nCandidate contribution (reduction; novelty confidence low): Let M consist of arcs e with |S_e|>=2, let sigma(v)=sum_{vx}(|S_{vx}|-1) over labelled outgoing arcs, and let B(v) be the new vertices reached by rainbow two-arc paths from v. If |B(v)|>=sigma(v), the reachability alternative holds at v; hence an instance with neither alternative must satisfy d_M^+(v)>=1 and |B(v)|<=sigma(v)-1 at every vertex and must contain a directed cycle C with a Hall-deficient subfamily Q of at least three arcs. In particular the conjecture holds when M is acyclic, and every directed triangle of M in a counterexample has the same two-element label set on all three arcs.", "research_difficulty_suggested": "L4" }, { "id": 20001037, "problem_number": "AIM-COMBINATORICS-0162", "title": "The doubled-label slice is exactly Seymour's second-neighborhood conjecture", "statement": "Conjecture 6.2. Seymour's Second Neighborhood Conjecture Notes: To see this, for a digraph H, let k = 2, G = H, and E1 = E2 = E(H).", "original_statement": "Conjecture 6.2. Seymour's Second Neighborhood Conjecture Notes: To see this, for a digraph H, let k = 2, G = H, and E1 = E2 = E(H).", "clean_statement": "Conjecture 6.2. Seymour's Second Neighborhood Conjecture Notes: To see this, for a digraph H, let k = 2, G = H, and E1 = E2 = E(H).", "statement_status": "exact", "statement_verification": "The canonical record is the following extracted text from Blair D. Sullivan's AIM survey:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.2\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[161]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.2. Seymour's Second Neighborhood Conjecture Notes: To see this, for a digraph H, let k = 2, G = H, and E1 = E2 = E(H).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0162", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source item is both a title-only repeat of Seymour's Second Neighborhood Conjecture and an assertion that colored Conjecture 6.1 implies it. Under the source's intended positive-length path convention, the uniform two-label restriction k=2 and E1=E2=E(D) is logically equivalent to Seymour's conjecture: rainbow-reachable vertices from v are exactly N_1^+(v) disjoint union N_2^+(v), the colored degree sum is 2d^+(v), and rainbow cycles are precisely loops or digons. More generally, the uniform m-label restriction is equivalent to the alternative that D has a directed cycle of length at most m or some v satisfies sum_{j=1}^m |N_j^+(v)| >= m d^+(v). Counting trivial paths weakens the m=2 conclusion by exactly one.\n\nCandidate contribution (equivalence and convention-sensitive reduction; novelty confidence low): For every m >= 1, the uniform m-label restriction of the Seymour--Sullivan colored conjecture is exactly equivalent to the directed-distance layer-mass alternative; at m=2 this is a two-way equivalence with Seymour's conjecture, while counting the trivial path produces the exact additive defect |N_2^+(v)| >= d^+(v)-1 and excluding rainbow digons makes the restriction fail on the two-vertex digon.", "research_difficulty_suggested": "L4" }, { "id": 20001038, "problem_number": "AIM-COMBINATORICS-0163", "title": "The repeated-label reduction and its sharp threshold", "statement": "Conjecture 6.3. Caccetta-H¨ aggkvist Conjecture (general case) Notes: For a digraph H, take G = H, k = d n\n\n> δ+\n> H\n\ne, and E1 = · · · = Ek = E(H). We must get a rainbow cycle because the sum of the outdegrees at each vertex is ≥ n. This corresponds to a dicycle of length at most d n\n\n> δ+\n> H\n\ne in H, as desired.", "original_statement": "Conjecture 6.3. Caccetta-H¨ aggkvist Conjecture (general case) Notes: For a digraph H, take G = H, k = d n\n\n> δ+\n> H\n\ne, and E1 = · · · = Ek = E(H). We must get a rainbow cycle because the sum of the outdegrees at each vertex is ≥ n. This corresponds to a dicycle of length at most d n\n\n> δ+\n> H\n\ne in H, as desired.", "clean_statement": "Conjecture 6.3. Caccetta-H¨ aggkvist Conjecture (general case) Notes: For a digraph H, take G = H, k = d n\n\n> δ+\n> H\n\ne, and E1 = · · · = Ek = E(H). We must get a rainbow cycle because the sum of the outdegrees at each vertex is ≥ n. This corresponds to a dicycle of length at most d n\n\n> δ+\n> H\n\ne in H, as desired.", "statement_status": "exact", "statement_verification": "The source record is Conjecture 6.3 in the AIM notes from the 2006 workshop “The Caccetta--Haggkvist conjecture.” Its displayed text is an implication from the preceding Seymour--Sullivan rainbow conjecture, rather than a new independent formulation:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.3\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[162]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.3. Caccetta-H¨ aggkvist Conjecture (general case) Notes: For a digraph H, take G = H, k = d n\\n\\n> δ+\\n> H\\n\\ne, and E1 = · · · = Ek = E(H). We must get a rainbow cycle because the sum of the outdegrees at each vertex is ≥ n. This corresponds to a dicycle of length at most d n\\n\\n> δ+\\n> H\\n\\ne in H, as desired.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0163", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The Seymour--Sullivan rainbow reachability conjecture, interpreted using positive-length reachability, implies the Caccetta--Haggkvist bound by the complete-list lift with k = ceil(n/delta). The proof uses only a load-threshold rainbow-cycle principle, whose restriction to complete-list lifts is exactly equivalent to Caccetta--Haggkvist. The threshold n is sharp, is unconditionally sufficient for arbitrary label lists on functional digraphs, and counting the trivial path weakens the source argument by one precisely when delta divides n.\n\nCandidate contribution (sharp threshold theorem and convention-sensitive reduction; novelty confidence low): For n-vertex list-labelled functional digraphs, total labelled outdegree at least n at every vertex forces a rainbow directed cycle, and n is best possible; in the repeated-label reduction, allowing the trivial path causes a one-unit ceiling loss exactly when the minimum outdegree divides n.", "research_difficulty_suggested": "L4" }, { "id": 20001039, "problem_number": "AIM-COMBINATORICS-0164", "title": "A corrected colored reduction and a Sullivan-2 counterexample normal form", "statement": "Conjecture 6.4. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2|N −(v)|.Notes: Recall, for comparison, SSN can be written as |N +2 (v)| + |N +(v)| ≥ 2|N +(v)|. To see how", "original_statement": "Conjecture 6.4. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2|N −(v)|.Notes: Recall, for comparison, SSN can be written as |N +2 (v)| + |N +(v)| ≥ 2|N +(v)|. To see how", "clean_statement": "Conjecture 6.4. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2|N −(v)|.Notes: Recall, for comparison, SSN can be written as |N +2 (v)| + |N +(v)| ≥ 2|N +(v)|. To see how", "statement_status": "exact", "statement_verification": "The canonical record is visibly truncated after “To see how.” Its exact extracted text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.4\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[163]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.4. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\\n\\n|N+(v)| ≥ 2|N −(v)|.Notes: Recall, for comparison, SSN can be written as |N +2 (v)| + |N +(v)| ≥ 2|N +(v)|. To see how\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0164", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The truncated record is Sullivan's second conjecture d^{++}(v)+d^+(v) >= 2d^-(v). Its printed proof note has a genuine ambient-edge defect: the third label class contains nonedges of H although the note sets G'=H. Replacing G' by the explicit completion with arcs F_H={(u,x): x is not rainbow-reachable to u in the doubled-label H} repairs the proof and gives a counterexample-lifting map from Sullivan-2 to colored Conjecture 6.1. Unconditionally, any Sullivan-2 counterexample contains a strong induced counterexample C of order at least 6 with minimum indegree at least 2, at least 2|V(C)| arcs, directed diameter at least 3, and sum_v d_C^{++}(v) <= |A(C)|-|V(C)|. Hence all source strong components of diameter at most 2 form a proved class.\n\nCandidate contribution (corrected reduction and counterexample normal form; novelty confidence low): The explicit completion \\hat H=(V(H),A(H) union F_H) makes Sullivan's intended colored implication rigorous and lifts every Sullivan-2 counterexample to a counterexample of colored Conjecture 6.1; moreover every Sullivan-2 counterexample reduces to a strong induced graph C with |V(C)|>=6, minimum indegree at least 2, directed diameter at least 3, and total exact second-neighborhood mass at most |A(C)|-|V(C)|.", "research_difficulty_suggested": "L4" }, { "id": 20001040, "problem_number": "AIM-COMBINATORICS-0165", "title": "Repairing the safe-completion reduction from rainbow reachability to Sullivan-2", "statement": "Conjecture 6.1 implies 6.4, take a simple digraph H with no loops or digons, and set G = H and\n\nE1 = E2 = E(H). G cannot have a rainbow cycle by definition of H. Define N +∗\n\n> G\n\n(u) = {vertices you can reach by a rainbow path in G from u}, and N −∗\n\n> G\n\n(u) = {vertices that have a rainbow path in G to u}. Let E3 be the edges {(u, v ) | v is not in the set N −∗\n\n> G\n\n(u)}. Let G′ = H, and have subsets E1, E 2, E 3 ⊆ E(G′) giving rise to label sets S′\n\n> e\n\n⊆ { 1, 2, 3} for e ∈ G′. We can see from these definitions that for any u in V (G′),\n\n> 3\n\n∑\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) = 2 |N +(u)| + (( n − 1) − | N −(u)| − | N −\n\n> 2\n\n(u)|),\n\n6where all neighborhoods referenced on the RHS are in H. We can rewrite this as:\n\n> 3\n\n∑\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) = (( n − 1) − (|N −(u)| + |N −\n\n> 2\n\n(u)| − 2|N +(u)|)).\n\nThen ∑3\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) ≥ n whenever |N −(u)| + |N −\n\n> 2\n\n(u)| < 2|N +(u)|. If this were true for all vertices\n\nu, then no vertex could have |N +∗\n\n> G′\n\n(u)| ≥ ∑3\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) ≥ n, so we must have a rainbow cycle in G′,by", "original_statement": "Conjecture 6.1 implies 6.4, take a simple digraph H with no loops or digons, and set G = H and \n\nE1 = E2 = E(H). G cannot have a rainbow cycle by definition of H. Define N +∗ \n\n> G\n\n(u) = {vertices you can reach by a rainbow path in G from u}, and N −∗ \n\n> G\n\n(u) = {vertices that have a rainbow path in G to u}. Let E3 be the edges {(u, v ) | v is not in the set N −∗ \n\n> G\n\n(u)}. Let G′ = H, and have subsets E1, E 2, E 3 ⊆ E(G′) giving rise to label sets S′ \n\n> e\n\n⊆ { 1, 2, 3} for e ∈ G′. We can see from these definitions that for any u in V (G′), \n\n> 3\n\n∑\n\n> i=1\n\nδ+ \n\n> Gi\n\n(u) = 2 |N +(u)| + (( n − 1) − | N −(u)| − | N − \n\n> 2\n\n(u)|),\n\n6where all neighborhoods referenced on the RHS are in H. We can rewrite this as: \n\n> 3\n\n∑\n\n> i=1\n\nδ+ \n\n> Gi\n\n(u) = (( n − 1) − (|N −(u)| + |N − \n\n> 2\n\n(u)| − 2|N +(u)|)).\n\nThen ∑3 \n\n> i=1\n\nδ+ \n\n> Gi\n\n(u) ≥ n whenever |N −(u)| + |N − \n\n> 2\n\n(u)| < 2|N +(u)|. If this were true for all vertices \n\nu, then no vertex could have |N +∗ \n\n> G′\n\n(u)| ≥ ∑3 \n\n> i=1\n\nδ+ \n\n> Gi\n\n(u) ≥ n, so we must have a rainbow cycle in G′,by", "clean_statement": "Conjecture 6.1 implies 6.4, take a simple digraph H with no loops or digons, and set G = H and\n\nE1 = E2 = E(H). G cannot have a rainbow cycle by definition of H. Define N +∗\n\n> G\n\n(u) = {vertices you can reach by a rainbow path in G from u}, and N −∗\n\n> G\n\n(u) = {vertices that have a rainbow path in G to u}. Let E3 be the edges {(u, v ) | v is not in the set N −∗\n\n> G\n\n(u)}. Let G′ = H, and have subsets E1, E 2, E 3 ⊆ E(G′) giving rise to label sets S′\n\n> e\n\n⊆ { 1, 2, 3} for e ∈ G′. We can see from these definitions that for any u in V (G′),\n\n> 3\n\n∑\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) = 2 |N +(u)| + (( n − 1) − | N −(u)| − | N −\n\n> 2\n\n(u)|),\n\n6where all neighborhoods referenced on the RHS are in H. We can rewrite this as:\n\n> 3\n\n∑\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) = (( n − 1) − (|N −(u)| + |N −\n\n> 2\n\n(u)| − 2|N +(u)|)).\n\nThen ∑3\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) ≥ n whenever |N −(u)| + |N −\n\n> 2\n\n(u)| < 2|N +(u)|. If this were true for all vertices\n\nu, then no vertex could have |N +∗\n\n> G′\n\n(u)| ≥ ∑3\n\n> i=1\n\nδ+\n\n> Gi\n\n(u) ≥ n, so we must have a rainbow cycle in G′,by", "statement_status": "exact", "statement_verification": "This canonical record is not an independent conjecture. It is the middle of the proof, split across records AIM-COMBINATORICS-0164 through AIM-COMBINATORICS-0166, that the rainbow-reachability Conjecture 6.1 of Seymour and Sullivan implies Sullivan's neighborhood Conjecture 6.4. The exact source is Blair D. Sullivan's 2006 AIM workshop summary *A Summary of Problems and Results Related to the Caccetta--Häggkvist Conjecture*, pages 5--6. The original arXiv TeX source was also inspected.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.1\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[164]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.1 implies 6.4, take a simple digraph H with no loops or digons, and set G = H and \\n\\nE1 = E2 = E(H). G cannot have a rainbow cycle by definition of H. Define N +∗ \\n\\n> G\\n\\n(u) = {vertices you can reach by a rainbow path in G from u}, and N −∗ \\n\\n> G\\n\\n(u) = {vertices that have a rainbow path in G to u}. Let E3 be the edges {(u, v ) | v is not in the set N −∗ \\n\\n> G\\n\\n(u)}. Let G′ = H, and have subsets E1, E 2, E 3 ⊆ E(G′) giving rise to label sets S′ \\n\\n> e\\n\\n⊆ { 1, 2, 3} for e ∈ G′. We can see from these definitions that for any u in V (G′), \\n\\n> 3\\n\\n∑\\n\\n> i=1\\n\\nδ+ \\n\\n> Gi\\n\\n(u) = 2 |N +(u)| + (( n − 1) − | N −(u)| − | N − \\n\\n> 2\\n\\n(u)|),\\n\\n6where all neighborhoods referenced on the RHS are in H. We can rewrite this as: \\n\\n> 3\\n\\n∑\\n\\n> i=1\\n\\nδ+ \\n\\n> Gi\\n\\n(u) = (( n − 1) − (|N −(u)| + |N − \\n\\n> 2\\n\\n(u)| − 2|N +(u)|)).\\n\\nThen ∑3 \\n\\n> i=1\\n\\nδ+ \\n\\n> Gi\\n\\n(u) ≥ n whenever |N −(u)| + |N − \\n\\n> 2\\n\\n(u)| < 2|N +(u)|. If this were true for all vertices \\n\\nu, then no vertex could have |N +∗ \\n\\n> G′\\n\\n(u)| ≥ ∑3 \\n\\n> i=1\\n\\nδ+ \\n\\n> Gi\\n\\n(u) ≥ n, so we must have a rainbow cycle in G′,by\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0165", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The assigned record is a truncated proof fragment rather than an independent problem. Its literal declaration that the augmented host remains H is inconsistent with its full-complement degree count. Adjoining exactly the missing fresh-label arcs repairs the proof: this safe completion creates no rainbow directed cycle. More generally, assuming Conjecture 6.1, every labelled simple digraph G without a rainbow directed cycle has a vertex u satisfying |R_G^-(u)| >= sum_i d^+_{G_i}(u); duplicating two labels on an oriented graph and reversing all arcs recovers the intended Sullivan-2 implication.\n\nCandidate contribution (reduction; novelty confidence low): For any finite labelled simple digraph G with no rainbow directed cycle, adjoining a fresh-labelled arc u->v exactly when v is not old-label rainbow-in-reachable to u preserves the absence of rainbow directed cycles; consequently, Conjecture 6.1 implies the conditional dual inequality |R_G^-(u)| >= sum_i d^+_{G_i}(u) for some u.", "research_difficulty_suggested": "L2" }, { "id": 20001041, "problem_number": "AIM-COMBINATORICS-0166", "title": "The exact fresh-color safety criterion behind the Sullivan-2 reduction", "statement": "Conjecture 6.1. However, by construction, since G has no rainbow cycle, G′ has no rainbow cycle. Thus there is a vertex v ∈ V (H) so that |N −(u)| + |N −\n\n> 2\n\n(u)| ≥ 2|N +(u)|. If we reverse all edges in H, this gives |N +(u)| + |N +2 (u)| ≥ 2|N −(u)|, as claimed.", "original_statement": "Conjecture 6.1. However, by construction, since G has no rainbow cycle, G′ has no rainbow cycle. Thus there is a vertex v ∈ V (H) so that |N −(u)| + |N − \n\n> 2\n\n(u)| ≥ 2|N +(u)|. If we reverse all edges in H, this gives |N +(u)| + |N +2 (u)| ≥ 2|N −(u)|, as claimed.", "clean_statement": "Conjecture 6.1. However, by construction, since G has no rainbow cycle, G′ has no rainbow cycle. Thus there is a vertex v ∈ V (H) so that |N −(u)| + |N −\n\n> 2\n\n(u)| ≥ 2|N +(u)|. If we reverse all edges in H, this gives |N +(u)| + |N +2 (u)| ≥ 2|N −(u)|, as claimed.", "statement_status": "exact", "statement_verification": "The assigned record is not an independent conjecture. It is the final OCR-split fragment of the notes following Conjecture 6.4 in Blair D. Sullivan's 2006 AIM workshop summary *A Summary of Problems and Results Related to the Caccetta--Häggkvist Conjecture*. The exact PDF and original arXiv TeX were inspected. The source passage reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.1\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[165]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.1. However, by construction, since G has no rainbow cycle, G′ has no rainbow cycle. Thus there is a vertex v ∈ V (H) so that |N −(u)| + |N − \\n\\n> 2\\n\\n(u)| ≥ 2|N +(u)|. If we reverse all edges in H, this gives |N +(u)| + |N +2 (u)| ≥ 2|N −(u)|, as claimed.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0166", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The assigned record is the final fragment of a conditional proof, not an independent conjecture. For any labelled simple digraph G and any ordered-pair set F carrying one common fresh label, the augmented graph has a rainbow directed cycle exactly when G already has one or some fresh arc u->v closes an old rainbow path v->u. Hence, when G has no rainbow cycle, the source's reachability complement is the unique maximum safe fresh-label set and pointwise maximizes fresh-label outdegree. This validates the phrase 'by construction' after correcting the host to adjoin the required nonedges, and the audited strict-negation and exact-distance reversal steps recover the intended conditional Sullivan-2 implication.\n\nCandidate contribution (characterization; novelty confidence low): A set F of arcs carrying one common fresh label is safe for a rainbow-cycle-free labelled digraph G if and only if every u->v in F satisfies v not in R_G^-(u); consequently F_max(G)={(u,v):u!=v and v not in R_G^-(u)} is the unique maximum safe set, and every safe F obeys d_F^+(u)<=n-1-|R_G^-(u)| with equality at all vertices exactly for F=F_max(G).", "research_difficulty_suggested": "L2" }, { "id": 20001042, "problem_number": "AIM-COMBINATORICS-0167", "title": "A sharp two-label frontier for Seymour's load conjecture", "statement": "Conjecture 6.\n5. (Seymour) Under the hypotheses of", "original_statement": "Conjecture 6.\n5. (Seymour) Under the hypotheses of", "clean_statement": "Conjecture 6.\n5. (Seymour) Under the hypotheses of", "statement_status": "exact", "statement_verification": "The canonical record is severely truncated:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[166]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n5. (Seymour) Under the hypotheses of\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0167", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The truncated record is the genuine Seymour load conjecture asserting that an n-vertex list-labelled digraph with pointwise total labelled outdegree at least n has a rainbow directed cycle. Unconditionally, the conjecture holds for at most two labels: every rainbow-cycle-free two-label instance has total load at most n(n-1), with equality attainable, so average load greater than n-1 forces a rainbow digon. The source's claimed failure on d+1 vertices is witnessed sharply by a directed (d+1)-cycle whose arcs all have the common d-label list. The complete-list three-label restriction is exactly the directed-triangle Caccetta--Haggkvist problem.\n\nCandidate contribution (sharp threshold theorem; novelty confidence low): For every n-vertex loopless simple digraph with arc lists contained in a two-element label set, absence of a rainbow directed cycle implies total label load at most n(n-1); this bound is exact even when both labels occur.", "research_difficulty_suggested": "L4" }, { "id": 20001043, "problem_number": "AIM-COMBINATORICS-0168", "title": "Rigidity of the deficit-one rainbow-cycle obstruction", "statement": "Conjecture 6.1, if |V | = d and ∑ki=1 δGi (v) ≥\n\nd for all v, G must have a rainbow cycle. Note: This conjecture is false if |V | = d is replaced by |V | = d + 1.\n\n6.1.3 Other Conjectures Inspired by (or related to)", "original_statement": "Conjecture 6.1, if |V | = d and ∑ki=1 δGi (v) ≥\n\nd for all v, G must have a rainbow cycle. Note: This conjecture is false if |V | = d is replaced by |V | = d + 1. \n\n6.1.3 Other Conjectures Inspired by (or related to)", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is the second half of Conjecture 6.5 in Blair Sullivan's summary of the 2006 AIM workshop *The Caccetta--Haggkvist conjecture*. The preceding canonical record, AIM-COMBINATORICS-0167, contains only the truncated prefix “Conjecture 6.5. (Seymour) Under the hypotheses of”. Reading the two records against page 6 of the source PDF recovers the complete statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.1\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[167]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.1, if |V | = d and ∑ki=1 δGi (v) ≥\\n\\nd for all v, G must have a rainbow cycle. Note: This conjecture is false if |V | = d is replaced by |V | = d + 1. \\n\\n6.1.3 Other Conjectures Inspired by (or related to)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0168", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source's d+1-vertex sharpness note is verified for every d >= 1 by the directed (d+1)-cycle with common palette [d]. More strongly, every n-vertex functional list-labelled digraph with total labelled outdegree at least n-1 at each vertex either has a rainbow directed cycle or is exactly a spanning directed n-cycle whose arc lists are all one common (n-1)-element palette; the converse also holds. Thus the source example is the unique functional deficit-one obstruction up to relabelling, and one fresh label on any arc repairs it.\n\nCandidate contribution (classification theorem; novelty confidence low): Among n-vertex functional list-labelled digraphs with load at least n-1 at every vertex, absence of a rainbow directed cycle forces the whole digraph to be a directed n-cycle and forces every arc list to equal one common set of exactly n-1 labels.", "research_difficulty_suggested": "L4" }, { "id": 20001044, "problem_number": "AIM-COMBINATORICS-0169", "title": "A surplus lattice for Seymour and Sullivan neighborhood inequalities", "statement": "Conjecture 6.1\n\nIf we believe Seymour's second neighborhood conjecture and", "original_statement": "Conjecture 6.1 \n\nIf we believe Seymour's second neighborhood conjecture and", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The assigned record is an extremely truncated transition, not a conjecture by itself. Comparison with the original arXiv TeX and AIM PDF recovers the complete sentence:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.1\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[168]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.1 \\n\\nIf we believe Seymour's second neighborhood conjecture and\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0169", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The assigned record is a truncated transition into the compromise conjectures, not an independent problem. For a vertex with a=d^+, b=d^-, and c=d^{++}, the Seymour, Sullivan-1, and Sullivan-2 surpluses s=c-a, u=c-b, and t=c+a-2b satisfy 2u=s+t. The Sullivan-3 surplus m=c+a-2min(a,b) equals max(s,t)=u+|a-b|. Thus Sullivan-3 is exactly the pointwise disjunction of Seymour and Sullivan-2, while the sharp thresholds u>=-|a-b|, u>=0, and u>=|a-b| characterize their disjunction, Sullivan-1, and their conjunction. A depth-two oriented gadget realizes the sharp local nonimplications, and all four properties coincide at degree-balanced vertices.\n\nCandidate contribution (equivalence; novelty confidence low): At every vertex, 2 sigma_1=sigma_S+sigma_2 and sigma_3=max(sigma_S,sigma_2)=sigma_1+|d^+-d^-|; consequently S or S2, S1, and S and S2 are exactly the three thresholds sigma_1>=-|d^+-d^-|, sigma_1>=0, and sigma_1>=|d^+-d^-|. Every missing individual implication is realized at a distinguished vertex of an explicit finite oriented depth-two gadget.", "research_difficulty_suggested": "L2" }, { "id": 20001045, "problem_number": "AIM-COMBINATORICS-0170", "title": "The exact interpolation among second-neighborhood inequalities", "statement": "Conjecture 6.1 (specifically 6.4), we might be led to ask if the following holds:", "original_statement": "Conjecture 6.1 (specifically 6.4), we might be led to ask if the following holds:", "clean_statement": "Conjecture 6.1 (specifically 6.4), we might be led to ask if the following holds:", "statement_status": "exact", "statement_verification": "The canonical record is not a complete conjecture. It is the second half of a transition sentence split across two extracted records:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.1\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[169]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.1 (specifically 6.4), we might be led to ask if the following holds:\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0170", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is the second half of a transition sentence, not an independent conjecture. Its surrounding context distinguishes Seymour's inequality, Sullivan--1 (the AIM Compromise Conjecture), Sullivan--2 (AIM Conjecture 6.4), and Sullivan--3 (the minimum inequality). Pointwise, the Sullivan--3 slack equals d++ - d- + |d+ - d-|, so it is exactly Seymour's predicate when d+ <= d- and Sullivan--2 when d- <= d+. This yields the complete implication lattice, four small oriented separation examples, and a global necessary defect budget for any Sullivan--3 counterexample.\n\nCandidate contribution (equivalence and obstruction; novelty confidence low): If an n-vertex oriented graph with m arcs has no Sullivan--3 vertex, then every vertex satisfies d- >= d++ + |d+ - d-| + 1 and globally sum_v d++(v) + sum_v |d+(v)-d-(v)| <= m-n; moreover, all invalid pointwise implications among Seymour and Sullivan--1/2/3 are separated by explicit oriented examples.", "research_difficulty_suggested": "L2" }, { "id": 20001046, "problem_number": "AIM-COMBINATORICS-0171", "title": "An arc--nonedge defect identity for Sullivan-1", "statement": "Conjecture 6.6. (\"Compromise Conjecture\") Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| ≥ | N −(v)|.", "original_statement": "Conjecture 6.6. (\"Compromise Conjecture\") Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| ≥ | N −(v)|.", "clean_statement": "Conjecture 6.6. (\"Compromise Conjecture\") Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| ≥ | N −(v)|.", "statement_status": "exact", "statement_verification": "The exact corpus record is Conjecture 6.6 from the AIM workshop list on the Caccetta--Haggkvist conjecture:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[170]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.6. (\\\"Compromise Conjecture\\\") Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| ≥ | N −(v)|.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0171", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every vertex v of an oriented graph, the Sullivan-1 deficit d^-(v)-d^{++}(v) equals |U(v)|-|R(v)|, where U(v) indexes incoming arcs at v lying in no directed cyclic triangle and R(v) is the set of nonadjacent exact second out-neighbors. Summing gives q(D)-rho(D), so q(D)-rho(D)<|V(D)| is sufficient. Any counterexample reduces to a strong induced source component H of order at least six with minimum indegree at least two, no directed 2-king, an untriangulated incoming arc at every vertex, and q(H)>=rho(H)+|V(H)|.\n\nCandidate contribution (exact_identity_and_reduction; novelty confidence low): The exact local identity d^-(v)-d^{++}(v)=|U(v)|-|R(v)| recasts Sullivan-1 as a balance between untriangulated incoming arcs and reachable nonedges; with the source-component reduction it yields the testable counterexample certificate |U(v)|>=|R(v)|+1 at every vertex and q(H)>=rho(H)+|V(H)| globally.", "research_difficulty_suggested": "L4" }, { "id": 20001047, "problem_number": "AIM-COMBINATORICS-0172", "title": "One-sided lifting and polarized minimal counterexamples for Sullivan-3", "statement": "Conjecture 6.7. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2 min( |N −(v)|, |N +(v)|).", "original_statement": "Conjecture 6.7. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2 min( |N −(v)|, |N +(v)|).", "clean_statement": "Conjecture 6.7. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\n\n|N+(v)| ≥ 2 min( |N −(v)|, |N +(v)|).", "statement_status": "exact", "statement_verification": "The original arXiv TeX and AIM PDF give the following unambiguous statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.7\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[171]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.7. Any simple digraph with no loops or digons has a vertex v such that |N +2 (v)| +\\n\\n|N+(v)| ≥ 2 min( |N −(v)|, |N +(v)|).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0172", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every initial induced subgraph, Sullivan-1 and Sullivan-2 witnesses lift to the ambient oriented graph; for every terminal induced subgraph, Seymour witnesses lift. Consequently, every proper initial set of a vertex-minimal Sullivan-3 counterexample is Sullivan-1/2-free but contains a Seymour-only witness with d^+ <= d^{++} < d^-, whereas every proper terminal set is Seymour-free but contains a Sullivan-2-only witness with d^- < d^+ and 2d^- - d^+ <= d^{++} < d^+. This yields weak connectivity, source/sink strong-component lower bounds, and attachment-closed extension classes from known local-tournament and quasi-transitive results.\n\nCandidate contribution (reduction; novelty confidence low): Initial/terminal polarity: Sullivan-1/2 witnesses lift through arbitrary outward attachments to an initial induced subgraph, Seymour witnesses lift through arbitrary inward attachments to a terminal induced subgraph, and hence every proper one-sided cut of a smallest Sullivan-3 counterexample has the explicit opposite-witness inequalities proved in Theorem 5.2.", "research_difficulty_suggested": "L4" }, { "id": 20001048, "problem_number": "AIM-COMBINATORICS-0173", "title": "A tight overlap-core reduction for Conjecture 6.8", "statement": "Conjecture 6.8. Under the hypotheses of", "original_statement": "Conjecture 6.8. Under the hypotheses of", "clean_statement": "Conjecture 6.8. Under the hypotheses of", "statement_status": "exact", "statement_verification": "The canonical record is split across two consecutive extraction records. The exact text in the assigned record is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.8\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[172]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.8. Under the hypotheses of\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0173", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every counterexample to the recovered directed overlapping-list rainbow-cycle conjecture can be reduced, by deleting label incidences and iterating sink strongly connected components, to a strongly connected tight instance whose exact layer outdegrees sum to its number of vertices. In that instance the subdigraph of arcs carrying at least two labels has minimum outdegree at least one; every directed cycle in this overlap core contains a Hall-deficient subset of at least three arcs. Consequently the core has no digon, and every directed core triangle has three identical two-element label lists. The threshold n is also shown sharp by a directed-cycle family at total degree n-1.\n\nCandidate contribution (structural reduction; novelty confidence low): Any counterexample admits a tight strongly connected normalization whose forced multi-label overlap core has minimum outdegree one, no digons, uniform-pair labels on every directed triangle, and a Hall-deficient subset of at least three arcs on every directed core cycle.", "research_difficulty_suggested": "L4" }, { "id": 20001049, "problem_number": "AIM-COMBINATORICS-0174", "title": "The exact two-label boundary for directed rainbow cycles", "statement": "Conjecture 6.1, if δ+\n\n> Gi\n\n≥ ri, and ∑ti=1 ri ≥ | V |, there is a rainbow cycle in G.", "original_statement": "Conjecture 6.1, if δ+ \n\n> Gi\n\n≥ ri, and ∑ti=1 ri ≥ | V |, there is a rainbow cycle in G.", "clean_statement": "Conjecture 6.1, if δ+\n\n> Gi\n\n≥ ri, and ∑ti=1 ri ≥ | V |, there is a rainbow cycle in G.", "statement_status": "exact", "statement_verification": "The canonical record is the second half of a sentence split across two extraction records. Its literal text is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.1\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[173]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.1, if δ+ \\n\\n> Gi\\n\\n≥ ri, and ∑ti=1 ri ≥ | V |, there is a rainbow cycle in G.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0174", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any two distinct labels i and j in the recovered directed overlapping-list model, if their minimum layer outdegrees satisfy delta_i + delta_j >= n, a labeled loop or an i,j-rainbow digon is forced. This proves Conjecture 6.8 for one and two labels. At the sharp boundary delta_i + delta_j = n-1, every obstruction is classified exactly: both layers are diregular and E_j is the complement of the reversal of E_i. Conversely every diregular layer and its reverse complement gives a two-label obstruction at n-1. A general incidence bound also shows that sum_i |E_i| <= k binom(n,2) whenever there is no labeled loop or rainbow digon.\n\nCandidate contribution (special_case_and_extremal_classification; novelty confidence low): Every two-label obstruction at total minimum outdegree n-1 is exactly a diregular digraph F together with the reverse-complement layer Omega minus F^R, while total minimum outdegree n forces a rainbow digon; the same reversal packing supplies a pairwise criterion inside any larger palette.", "research_difficulty_suggested": "L4" }, { "id": 20001050, "problem_number": "AIM-COMBINATORICS-0175", "title": "A cut-critical kernel for the vertexwise rainbow-cycle conjecture", "statement": "Conjecture 6.9. Under the hypotheses of", "original_statement": "Conjecture 6.9. Under the hypotheses of", "clean_statement": "Conjecture 6.9. Under the hypotheses of", "statement_status": "exact", "statement_verification": "The assigned canonical record is an extraction fragment:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.9\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[174]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.9. Under the hypotheses of\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0175", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every counterexample among nonempty finite simple loopless digraphs to the recovered vertexwise-total-load Conjecture 6.9 contains a nonempty deletion subinstance on m >= 2 vertices with exact outgoing label load m at every vertex and a strict cut certificate: for every nonempty proper X, some vertex outside X sends at least |X|+1 label incidences into X. Consequently the subdigraph of arcs carrying at least two labels is strongly connected. Every cycle in that overlap core also contains a Hall-deficient subset of at least three arcs. A separate directed-cycle family proves that the vertexwise hypothesis is strictly more general than the sum-of-layer-minima hypothesis of Conjecture 6.8, and the threshold m is sharp against m-1.\n\nCandidate contribution (cut-critical reduction; novelty confidence low): Every counterexample contains a nonempty exact-load deletion kernel on at least two vertices satisfying max_{v outside X} sum_{x in X}|S_{vx}| >= |X|+1 for every nontrivial vertex set X; in particular, its multi-label overlap core is strongly connected.", "research_difficulty_suggested": "L4" }, { "id": 20001051, "problem_number": "AIM-COMBINATORICS-0176", "title": "Palette transversals and a rooted Hall obstruction", "statement": "Conjecture 6.1, if ∑ti=1 δ+\n\n> Gi\n\n(v) ≥ | V | for all vertices\n\nv, there is a rainbow cycle in G.", "original_statement": "Conjecture 6.1, if ∑ti=1 δ+ \n\n> Gi\n\n(v) ≥ | V | for all vertices \n\nv, there is a rainbow cycle in G.", "clean_statement": "Conjecture 6.1, if ∑ti=1 δ+\n\n> Gi\n\n(v) ≥ | V | for all vertices\n\nv, there is a rainbow cycle in G.", "statement_status": "exact", "statement_verification": "The canonical record is an OCR fragment:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.1\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[175]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.1, if ∑ti=1 δ+ \\n\\n> Gi\\n\\n(v) ≥ | V | for all vertices \\n\\nv, there is a rainbow cycle in G.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0176", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a nonempty finite directed list-labeled graph, Hall's condition on the active outgoing label palettes P(v) forces a rainbow directed cycle: match a distinct available label to every tail, choose one corresponding outgoing arc per vertex, and take a cycle in the resulting functional digraph. Therefore every counterexample satisfying the recovered vertexwise load condition has an inclusion-minimal deficient palette set X of size at least three with |P(X)|=|X|-1. Deleting any root makes the full palettes matchable, but the internally realizable palettes remain Hall-deficient; every vertex of X also has one layer outdegree at least ceil(n/(|X|-1)). A separate balanced-support theorem proves the conjecture when the minimum palette size is at least the maximum number of tails supporting any one label.\n\nCandidate contribution (palette_transversal_obstruction; novelty confidence low): In any counterexample, every inclusion-minimal Hall-deficient tail-palette set X has size at least three and exact deficiency one; for every root x, the full palettes on X minus x have an SDR onto P(X), while the palettes realizable by arcs internal to X minus x violate Hall. Moreover each vertex of X has some layer outdegree at least ceil(n/(|X|-1)).", "research_difficulty_suggested": "L4" }, { "id": 20001052, "problem_number": "AIM-COMBINATORICS-0177", "title": "Resolving the average in DeVos's ordered-label conjecture", "statement": "Conjecture 6.\n1\n0. (Devos) Under the hypotheses of", "original_statement": "Conjecture 6.\n1\n0. (Devos) Under the hypotheses of", "clean_statement": "Conjecture 6.\n1\n0. (Devos) Under the hypotheses of", "statement_status": "exact", "statement_verification": "The assigned canonical record is only the first fragment", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[176]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n1\\n0. (Devos) Under the hypotheses of\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0177", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The printed Conjecture 6.10 does not specify the averaging variable. Under the coherent uniform-start-vertex reading, increasing reachability is exactly the endpoint relation of a time-expanded acyclic digraph, or equivalently the support of the Boolean product (I or A_1)...(I or A_k). The conjectured bound is exactly a global collision-deficit inequality: endpoints first lost through overlaps among one-step layer neighborhoods must be compensated by new endpoints requiring at least two labels. A five-vertex two-label example with reachability sizes (3,4,4,5,5) and minimum degrees (2,1) has no increasing cycle, disproving the tempting pointwise reading while satisfying the vertex-average bound. A capacitated-Hall neighborhood condition proves the stronger pointwise bound for an explicit arbitrary-label class.\n\nCandidate contribution (equivalence_and_counterexample; novelty confidence low): On the uniform start-vertex interpretation, DeVos's conjecture is equivalent to the time-expanded endpoint-density and collision-deficit inequalities proved in Theorem 5.1; the explicit five-vertex construction separates that average statement from the false pointwise reading, and the capacitated-Hall condition gives a testable sufficient class.", "research_difficulty_suggested": "L4" }, { "id": 20001053, "problem_number": "AIM-COMBINATORICS-0178", "title": "Two-label extremality and the meaning of average reachability", "statement": "Conjecture 6.1, if there is no rainbow cycle in G with strictly increasing edge labels, then the average number of vertices reachable from a fixed vertex v by label-increasing (possibly trivial) paths is at least 1 + ∑ki=1 δ+\n\n> Gi.Note: This can be proved when Gi = Cayley(Γ, A i) for some group Γ, using Lemma 5.9\n\n#6.2 Second & Kth Neighborhood Conjectures", "original_statement": "Conjecture 6.1, if there is no rainbow cycle in G with strictly increasing edge labels, then the average number of vertices reachable from a fixed vertex v by label-increasing (possibly trivial) paths is at least 1 + ∑ki=1 δ+ \n\n> Gi.Note: This can be proved when Gi = Cayley(Γ, A i) for some group Γ, using Lemma 5.9 \n\n#6.2 Second & Kth Neighborhood Conjectures", "clean_statement": "Conjecture 6.1, if there is no rainbow cycle in G with strictly increasing edge labels, then the average number of vertices reachable from a fixed vertex v by label-increasing (possibly trivial) paths is at least 1 + ∑ki=1 δ+\n\n> Gi.Note: This can be proved when Gi = Cayley(Γ, A i) for some group Γ, using Lemma 5.9\n\n#6.2 Second & Kth Neighborhood Conjectures", "statement_status": "exact", "statement_verification": "The assigned record is the second part of a split extraction. The preceding source fragment supplies only the heading “Conjecture 6.10. (Devos) Under the hypotheses of”; the assigned record and the original PDF/TeX supply the rest. The exact TeX source reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.1\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[177]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.1, if there is no rainbow cycle in G with strictly increasing edge labels, then the average number of vertices reachable from a fixed vertex v by label-increasing (possibly trivial) paths is at least 1 + ∑ki=1 δ+ \\n\\n> Gi.Note: This can be proved when Gi = Cayley(Γ, A i) for some group Γ, using Lemma 5.9 \\n\\n#6.2 Second & Kth Neighborhood Conjectures\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0178", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the most defensible reconstruction of Conjecture 6.10, the average is over nonempty starting vertices, labels use their natural order, trivial paths count the start, and the right side uses global layer minimum outdegrees. For two labels, absence of an increasing rainbow cycle implies delta_1+delta_2 <= n-1; at equality every vertex reaches all n vertices by an increasing path, proving the conjectured bound with equality. A verified five-vertex example with minima (2,1) has reachable-set sizes (5,3,4,5,5), so it refutes the plausible pointwise interpretation while satisfying the average bound. An exact layered-DAG double-count identity is also given.\n\nCandidate contribution (sharp special case and obstruction; novelty confidence low): In every nonempty two-label simple loopless instance with no increasing rainbow cycle, delta_1+delta_2 <= n-1, and equality forces R(v)=V for every vertex; below equality, the explicit five-vertex instance in the artifacts disproves the pointwise lower bound |R(v)| >= 1+delta_1+delta_2.", "research_difficulty_suggested": "L4" }, { "id": 20001054, "problem_number": "AIM-COMBINATORICS-0179", "title": "Finite-boundary transfer for infinite second neighborhoods", "statement": "Conjecture 6.11. Is Seymour's Second Neighborhood true for locally finite digraphs? What if we just require the outdegrees to be finite?", "original_statement": "Conjecture 6.11. Is Seymour's Second Neighborhood true for locally finite digraphs? What if we just require the outdegrees to be finite?", "clean_statement": "Conjecture 6.11. Is Seymour's Second Neighborhood true for locally finite digraphs? What if we just require the outdegrees to be finite?", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.11\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[178]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.11. Is Seymour's Second Neighborhood true for locally finite digraphs? What if we just require the outdegrees to be finite?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0179", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every 0 <= lambda <= 1 and every induced-hereditary class of oriented graphs, if every nonempty finite member has a vertex v with |N_2^+(v)| >= lambda |N_1^+(v)|, then every arbitrary-cardinality member with finite outdegrees has such a vertex. Consequently, both exact infinite variants in the source are equivalent to the still-open finite Seymour conjecture, while the finite Huang-Peng factor gamma = 0.715538... transfers unconditionally to every finite-outdegree infinite oriented graph. The proof uses a minimum finite outer boundary and requires neither finite indegrees, countability, connectivity, an exhaustion, nor Konig's infinity lemma.\n\nCandidate contribution (transfer_theorem; novelty confidence low): Candidate novelty: the finite-boundary argument transfers every multiplicative lambda-Seymour theorem for 0 <= lambda <= 1 within any induced-hereditary class from finite graphs to arbitrary-cardinality graphs with finite outdegrees; in particular it yields the explicit gamma = 0.715538... infinite finite-outdegree bound.", "research_difficulty_suggested": "L4" }, { "id": 20001055, "problem_number": "AIM-COMBINATORICS-0180", "title": "Local edge budgets for Thomassé's nonneighbor conjecture", "statement": "Conjecture 6.\n1\n2. (Thomass´ e) Let G be a digraph with no directed cycle of length at most three. Then there is a vertex v ∈ V (G) with δ+(v) at most the number of non-neighbors of v.\n\nThe following generalization of second neighborhood to kth neighborhood was taken from a (unpublished) paper of Serge Burckel: 7", "original_statement": "Conjecture 6.\n1\n2. (Thomass´ e) Let G be a digraph with no directed cycle of length at most three. Then there is a vertex v ∈ V (G) with δ+(v) at most the number of non-neighbors of v.\n\nThe following generalization of second neighborhood to kth neighborhood was taken from a (unpublished) paper of Serge Burckel: 7", "clean_statement": "Conjecture 6.\n1\n2. (Thomass´ e) Let G be a digraph with no directed cycle of length at most three. Then there is a vertex v ∈ V (G) with δ+(v) at most the number of non-neighbors of v.\n\nThe following generalization of second neighborhood to kth neighborhood was taken from a (unpublished) paper of Serge Burckel: 7", "statement_status": "exact", "statement_verification": "The canonical JSON has two extraction defects: the displayed number is split as `6.\\n1\\n2`, and the conjecture is followed by the opening sentence of the next item. Inspection of the TeX source behind the AIM survey recovers the item as **Conjecture 6.12** and gives the statement", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[179]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n1\\n2. (Thomass´ e) Let G be a digraph with no directed cycle of length at most three. Then there is a vertex v ∈ V (G) with δ+(v) at most the number of non-neighbors of v.\\n\\nThe following generalization of second neighborhood to kth neighborhood was taken from a (unpublished) paper of Serge Burckel: 7\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0180", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a minimum-outdegree vertex v of a finite digraph with no directed cycle of length at most three, with δ=d+(v) and q(v) nonneighbors, δ² ≤ e(G[N+(v)])+e(N+(v),U(v)) ≤ binom(δ,2)+δq(v), hence q(v) ≥ ceil((δ+1)/2). Equality analysis proves Thomassé's conjecture directly when the minimum outdegree is at most 3. The artifacts also prove the exact global identity sum_v(q(v)-d+(v))=n(n-1)-3m, reduce the conjecture to Seymour's Second Neighborhood Conjecture, settle the vertex-transitive case, and give a circular family attaining equality both in d+(v)≤q(v) and in the density certificate m=n(n-1)/3.\n\nCandidate contribution (local counting lemma and rigidity argument; novelty confidence low): At every minimum-outdegree vertex v in a 3-free finite digraph, the explicit edge-budget inequality δ² ≤ e(G[N+(v)])+e(N+(v),U(v)) ≤ binom(δ,2)+δq(v) holds; when δ=3, the sole numerical obstruction q(v)=2 forces a transitive three-vertex tournament whose sink has total outdegree 2, a contradiction.", "research_difficulty_suggested": "L4" }, { "id": 20001056, "problem_number": "AIM-COMBINATORICS-0181", "title": "Frontier Hall obstructions for exact distance shells", "statement": "Conjecture 6.13. Any simple digraph with no directed cycles of length at most k has a vertex\n\nv such that |N +\n\n> k\n\n(v)| ≥ | N +\n\n> k−1\n\n(v)|.\n\nSerge Burckel also asked the following structural question:", "original_statement": "Conjecture 6.13. Any simple digraph with no directed cycles of length at most k has a vertex \n\nv such that |N + \n\n> k\n\n(v)| ≥ | N +\n\n> k−1\n\n(v)|.\n\nSerge Burckel also asked the following structural question:", "clean_statement": "Conjecture 6.13. Any simple digraph with no directed cycles of length at most k has a vertex\n\nv such that |N +\n\n> k\n\n(v)| ≥ | N +\n\n> k−1\n\n(v)|.\n\nSerge Burckel also asked the following structural question:", "statement_status": "exact", "statement_verification": "The canonical record is visibly damaged by line-oriented PDF extraction:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.13\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[180]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.13. Any simple digraph with no directed cycles of length at most k has a vertex \\n\\nv such that |N + \\n\\n> k\\n\\n(v)| ≥ | N +\\n\\n> k−1\\n\\n(v)|.\\n\\nSerge Burckel also asked the following structural question:\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0181", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For k at least 2, let D be a finite simple sinkless digraph of directed girth greater than k. If a root v violates |N_k^+(v)| >= |N_{k-1}^+(v)|, its bipartite last-frontier graph has an inclusion-minimal Hall-deficient set of exact deficiency one. A singleton witness forces a cross-branch shortcut and an underlying cycle of length at most 2k-1; a larger witness forces two coalescing length-k geodesics and an underlying cycle of length at most 2k. Hence the recovered Burckel conjecture holds when the underlying undirected girth exceeds 2k. A two-branch construction shows the 2k threshold is sharp for guaranteeing that each sinkless root is good.\n\nCandidate contribution (rooted_frontier_hall_obstruction; novelty confidence low): Every bad root in a finite simple sinkless digraph of directed girth greater than k has an exact-deficiency-one Hall witness between its exact distance-(k-1) and distance-k shells; that witness certifies an underlying cycle of length at most 2k inside the radius-k out-ball, with a locally sharp family at underlying girth 2k.", "research_difficulty_suggested": "L4" }, { "id": 20001057, "problem_number": "AIM-COMBINATORICS-0182", "title": "A two-vertex digon counterexample and an oriented local descent theorem", "statement": "Conjecture 6.14. For any k and any digraph G, define G∗ to be those vertices with |N +2 (v)| ≥ |N+(v)|. Then any vertex of out-degree k is at distance at most k from a vertex in G∗.Notes: He motivates this with the following remark: \"If a vertex x has one successor y, then if y has no successors, y ∈ G∗, otherwise x ∈ G∗. This property seems to generalize for any out-degree, and if it is true, is optimal considering 'pyramids' where any vertex of out-degree k\n\nis at distance exactly k from the (unique) solution.\" His 'pyramids' are formed by placing one vertex, then two in the row beneath it, and so forth ( i vertices in row i) to form a triangle. The vertices in row i are then completely joined to all vertices in row i − 1 for i ≥ 2.", "original_statement": "Conjecture 6.14. For any k and any digraph G, define G∗ to be those vertices with |N +2 (v)| ≥ |N+(v)|. Then any vertex of out-degree k is at distance at most k from a vertex in G∗.Notes: He motivates this with the following remark: \"If a vertex x has one successor y, then if y has no successors, y ∈ G∗, otherwise x ∈ G∗. This property seems to generalize for any out-degree, and if it is true, is optimal considering 'pyramids' where any vertex of out-degree k\n\nis at distance exactly k from the (unique) solution.\" His 'pyramids' are formed by placing one vertex, then two in the row beneath it, and so forth ( i vertices in row i) to form a triangle. The vertices in row i are then completely joined to all vertices in row i − 1 for i ≥ 2.", "clean_statement": "Conjecture 6.14. For any k and any digraph G, define G∗ to be those vertices with |N +2 (v)| ≥ |N+(v)|. Then any vertex of out-degree k is at distance at most k from a vertex in G∗.Notes: He motivates this with the following remark: \"If a vertex x has one successor y, then if y has no successors, y ∈ G∗, otherwise x ∈ G∗. This property seems to generalize for any out-degree, and if it is true, is optimal considering 'pyramids' where any vertex of out-degree k\n\nis at distance exactly k from the (unique) solution.\" His 'pyramids' are formed by placing one vertex, then two in the row beneath it, and so forth ( i vertices in row i) to form a triangle. The vertices in row i are then completely joined to all vertices in row i − 1 for i ≥ 2.", "statement_status": "exact", "statement_verification": "The source is Conjecture 6.14 in the AIM workshop problem list *The Caccetta--Häggkvist conjecture*. The source list declares at the outset that digraphs are finite unless explicitly stated otherwise, and defines \\(N_j^+(v)\\) to be the vertices at directed distance exactly \\(j\\) from \\(v\\). With \\[ G^*=\\{v\\in V(G): |N_2^+(v)|\\geq |N^+(v)|\\}, \\] the displayed conjecture says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.14\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[181]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.14. For any k and any digraph G, define G∗ to be those vertices with |N +2 (v)| ≥ |N+(v)|. Then any vertex of out-degree k is at distance at most k from a vertex in G∗.Notes: He motivates this with the following remark: \\\"If a vertex x has one successor y, then if y has no successors, y ∈ G∗, otherwise x ∈ G∗. This property seems to generalize for any out-degree, and if it is true, is optimal considering 'pyramids' where any vertex of out-degree k\\n\\nis at distance exactly k from the (unique) solution.\\\" His 'pyramids' are formed by placing one vertex, then two in the row beneath it, and so forth ( i vertices in row i) to form a triangle. The vertices in row i are then completely joined to all vertices in row i − 1 for i ≥ 2.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0182", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The exact canonical statement for any digraph is refuted by the two-vertex digon a->b->a: each vertex has one first out-neighbor but no exact second out-neighbor, so G* is empty and neither degree-one vertex is within distance one of G*. The natural repaired formulation for finite oriented graphs remains open. For that repair, the distance bound is proved whenever every nonempty out-neighborhood induces a graph with a sink, with an equality certificate N^+(v_{i+1}) = N_2^+(v_i), and it is also proved unconditionally for every starting vertex of out-degree at most three.\n\nCandidate contribution (theorem_and_equality_certificate; novelty confidence low): In any finite oriented graph where each nonempty out-neighborhood induces a sink, every degree-k vertex has a path to a good vertex along which out-degree drops by at least one per step; moreover, if its distance to the good set is exactly k, every descent step satisfies N^+(v_{i+1}) = N_2^+(v_i) and |N_2^+(v_i)| = k-i-1." }, { "id": 20001058, "problem_number": "AIM-COMBINATORICS-0183", "title": "A collision budget for Eulerian second neighborhoods", "statement": "Conjecture 6.\n1\n5. (Seymour &/or Jackson) If G is an Eulerian digraph with no loops or digons, then ∑\n\n> v∈V(G)\n\n|N+2 (v)| ≥ ∑\n\n> v∈V(G)\n\n|N+(v)|.", "original_statement": "Conjecture 6.\n1\n5. (Seymour &/or Jackson) If G is an Eulerian digraph with no loops or digons, then ∑ \n\n> v∈V(G)\n\n|N+2 (v)| ≥ ∑ \n\n> v∈V(G)\n\n|N+(v)|.", "clean_statement": "Conjecture 6.\n1\n5. (Seymour &/or Jackson) If G is an Eulerian digraph with no loops or digons, then ∑\n\n> v∈V(G)\n\n|N+2 (v)| ≥ ∑\n\n> v∈V(G)\n\n|N+(v)|.", "statement_status": "exact", "statement_verification": "The canonical JSON breaks the conjecture number across lines as “6. 1 5” and inserts `>` extraction debris before the summation indices. The primary AIM PDF gives the unambiguous statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[182]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n1\\n5. (Seymour &/or Jackson) If G is an Eulerian digraph with no loops or digons, then ∑ \\n\\n> v∈V(G)\\n\\n|N+2 (v)| ≥ ∑ \\n\\n> v∈V(G)\\n\\n|N+(v)|.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0183", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every finite balanced loopless digon-free digraph, the second-neighborhood surplus has the exact decomposition M-m=S-tau-C, where S is the ordered degree-pair budget, tau counts transitive triangles, and C is two-path collision excess. Since tau is at most S/2, C at most S/2 is sufficient for the conjecture. A local nonnegative refinement proves the conjecture, with an equality characterization, whenever the maximum outdegree is at most two; the same framework proves it when the underlying undirected graph is C4-free and gives a bounded-multiplicity criterion.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: the exact collision-triangle surplus identity M-m=S-tau-C, together with the local formula M-m=sum_{d(x)=2}(2-t_x-c_x), proves Conjecture 6.15 for balanced loopless digon-free digraphs of maximum outdegree two and characterizes equality; it also yields a C4-free strengthening.", "research_difficulty_suggested": "L4" }, { "id": 20001059, "problem_number": "AIM-COMBINATORICS-0184", "title": "A literal parallel-arc counterexample and a collision obstruction for the simple repair", "statement": "Conjecture 6.\n1\n6. (Thomass´ e & Kral) Let G be an Eulerian digraph on n vertices, so that\n\n|E(G)| ≥ n2/3. Then G has a directed cycle of length at most three.\n\n#6.3 Matrices", "original_statement": "Conjecture 6.\n1\n6. (Thomass´ e & Kral) Let G be an Eulerian digraph on n vertices, so that \n\n|E(G)| ≥ n2/3. Then G has a directed cycle of length at most three. \n\n#6.3 Matrices", "clean_statement": "Conjecture 6.\n1\n6. (Thomass´ e & Kral) Let G be an Eulerian digraph on n vertices, so that\n\n|E(G)| ≥ n2/3. Then G has a directed cycle of length at most three.\n\n#6.3 Matrices", "statement_status": "exact", "statement_verification": "The canonical JSON record is damaged by line breaks and OCR. The source PDF gives the following statement on page 7:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[183]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n1\\n6. (Thomass´ e & Kral) Let G be an Eulerian digraph on n vertices, so that \\n\\n|E(G)| ≥ n2/3. Then G has a directed cycle of length at most three. \\n\\n#6.3 Matrices\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0184", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the source PDF's explicit convention that a digraph may have parallel arcs, Conjecture 6.16 is false: for every n at least 4, replacing each arc of a directed n-cycle by ceil(n/3) parallel copies gives a connected Eulerian multidigraph with at least n^2/3 arcs and directed girth n. For the evidently intended simple formulation, an exact proved identity expresses the global second-neighborhood defect as T+2W-C, where T counts transitive triangles, W counts open in-wedges, and C is excess multiplicity of directed two-paths to nonneighbors. Any simple counterexample at the stated density must satisfy C at least T+2W+n.\n\nCandidate contribution (identity_and_obstruction; novelty confidence low): For every finite simple Eulerian digraph with no directed cycle of length at most three, sum_v(|N_2^+(v)|-d(v))=T+2W-C; consequently any graph in this class with at least n^2/3 arcs must have C at least T+2W+n. The standard circular family at m=n(n-1)/3 satisfies C=T and W=0, giving equality in the identity." }, { "id": 20001060, "problem_number": "AIM-COMBINATORICS-0185", "title": "Low temporal repetition forces excess trace", "statement": "Conjecture 6.17. The following were all presented by Seymour, with no other attributions given: For the following questions, matrices are assumed to be n × n 0-1 matrices such that aij = 1\n\nimplies aji 6 = 1, and all diagonal elements equal to 1.1. Let A1, A 2,..., A dn/r e be matrices (not necessarily distinct) so that the row sums of Ai are at least r + 1 for all i. Does A1A2 · · · Adn/r e have trace > n? This is a special case of", "original_statement": "Conjecture 6.17. The following were all presented by Seymour, with no other attributions given: For the following questions, matrices are assumed to be n × n 0-1 matrices such that aij = 1 \n\nimplies aji 6 = 1, and all diagonal elements equal to 1.1. Let A1, A 2,..., A dn/r e be matrices (not necessarily distinct) so that the row sums of Ai are at least r + 1 for all i. Does A1A2 · · · Adn/r e have trace > n? This is a special case of", "clean_statement": "Conjecture 6.17. The following were all presented by Seymour, with no other attributions given: For the following questions, matrices are assumed to be n × n 0-1 matrices such that aij = 1\n\nimplies aji 6 = 1, and all diagonal elements equal to 1.1. Let A1, A 2,..., A dn/r e be matrices (not necessarily distinct) so that the row sums of Ai are at least r + 1 for all i. Does A1A2 · · · Adn/r e have trace > n? This is a special case of", "statement_status": "exact", "statement_verification": "The canonical record is an OCR-truncated extraction of item 1 in AIM Conjecture 6.17. It renders a ceiling as “dn/r e,” omits the off-diagonal qualification from the asymmetry condition, joins the item number to the preceding sentence, and stops after “This is a special case of.”", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.17\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[184]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.17. The following were all presented by Seymour, with no other attributions given: For the following questions, matrices are assumed to be n × n 0-1 matrices such that aij = 1 \\n\\nimplies aji 6 = 1, and all diagonal elements equal to 1.1. Let A1, A 2,..., A dn/r e be matrices (not necessarily distinct) so that the row sums of Ai are at least r + 1 for all i. Does A1A2 · · · Adn/r e have trace > n? This is a special case of\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0185", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an ordered family A_1,...,A_t with diagonal ones and asymmetric off-diagonal supports, let L be the total number of off-diagonal incidences and suppose each directed entry occurs in at most q layers. Then tr(A_1...A_t) is at least n + q max(0, L - q binom(n,2)). Under Seymour's row bounds and t=ceil(n/r), directed-entry multiplicity at most two therefore forces the sharp bound tr(A_1...A_t) >= 3n. Any counterexample, whose trace would equal n, must orient every unordered pair consistently across time and repeat some ordered arc in at least ceil(2tr/(n-1)) >= 3 layers.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: the explicit temporal-multiplicity lower bound tr(A_1...A_t) >= n + q max(0, L - q binom(n,2)); in particular, maximum directed-entry multiplicity two implies the sharp estimate tr(A_1...A_t) >= 3n, while trace n requires triple persistence of some ordered arc.", "research_difficulty_suggested": "L4" }, { "id": 20001061, "problem_number": "AIM-COMBINATORICS-0186", "title": "Temporal trace semantics and a bounded-repetition special case", "statement": "The matrices \\(A_1,\\ldots,A_t\\) are \\(n\\times n\\) \\(0\\)-\\(1\\) matrices, \\(a^{(i)}_{uv}=1\\) implies \\(a^{(i)}_{vu}\\ne1\\) for \\(u\\ne v\\), and all diagonal entries equal \\(1\\). They need not be distinct. If every row sum of \\(A_i\\) is at least \\(r_i+1\\) and\n\\[\n\\sum_{i=1}^t r_i\\ge n,\n\\]\nmust ordinary matrix multiplication satisfy\n\\[\n\\operatorname{tr}(A_1A_2\\cdots A_t)>n?\n\\]", "original_statement": "Conjecture 6.5. 2. Let A1, A 2,..., A t be matrices (not necessarily distinct) so Ai has row sums at least ri + 1 \n\nand ∑ti=1 ri ≥ n. Does A1A2 · · · At have trace > n? This is equivalent to", "clean_statement": "The matrices \\(A_1,\\ldots,A_t\\) are \\(n\\times n\\) \\(0\\)-\\(1\\) matrices, \\(a^{(i)}_{uv}=1\\) implies \\(a^{(i)}_{vu}\\ne1\\) for \\(u\\ne v\\), and all diagonal entries equal \\(1\\). They need not be distinct. If every row sum of \\(A_i\\) is at least \\(r_i+1\\) and\n\\[\n\\sum_{i=1}^t r_i\\ge n,\n\\]\nmust ordinary matrix multiplication satisfy\n\\[\n\\operatorname{tr}(A_1A_2\\cdots A_t)>n?\n\\]", "statement_status": "corrected_verified", "statement_verification": "This fragment was split in the middle of an item. Inspection of page 7 of the primary source recovers it as item 2 under **Conjecture 6.17**, not as a new item numbered “Conjecture 6.5.2.” The shared preamble and continuation give the complete recovered statement: The missing words after the owned fragment are “Conjecture 6.5.” Items 3 and 4, printed immediately afterward, give the intended layered-path and increasing-color-cycle pictures. In item 3 there are \\(t+1\\) copies of the vertex set; an \\(A_i\\)-arc goes from layer \\(i-1\\) to layer \\(i\\), and diagonal entries are the horizontal waiting edges. The question is whether some vertex has a non-horizontal path back to its copy in the last layer. Item 4 identifies this with a nontrivial rainbow directed cycle whose colors occur in increasing order.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.5\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[185]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.5. 2. Let A1, A 2,..., A t be matrices (not necessarily distinct) so Ai has row sums at least ri + 1 \\n\\nand ∑ti=1 ri ≥ n. Does A1A2 · · · At have trace > n? This is equivalent to\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0186", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The split record is item 2 of Conjecture 6.17: its trace counts closed time-ordered walks, so trace greater than n is exactly equivalent to a nonconstant layered closed walk and an increasing-color directed cycle. A proved support-counting theorem shows that trace equal to n forces all off-diagonal supports into one common orientation, D at most q(n-1)/2, and repeated-support excess X at least nD-binomial(n,2), where D is the sum of actual minimum off-diagonal row degrees and q is maximum cross-layer arc multiplicity. Hence the conjecture holds whenever q is at most 2 under D at least n. Repeated regular tournaments show sharpness at the subcritical boundary D=n-1.\n\nCandidate contribution (theorem_and_special_case; novelty confidence low): If a family satisfying the recovered matrix conventions has trace(A1...At)=n, then its union support is oriented, D <= q(n-1)/2, and X >= nD-binomial(n,2); moreover trace(A1...At)-n is at least the sum over i2, every counterexample has at least ceil(n/(t-2)) distinct arcs of multiplicity at least three.", "research_difficulty_suggested": "L4" }, { "id": 20001063, "problem_number": "AIM-COMBINATORICS-0188", "title": "Charbit's spectral short-cycle conjecture: the sharp digon case and a Perron-period obstruction", "statement": "Conjecture 6.\n1\n8. (Charbit) Let A be the adjacency matrix of a digraph G with zero on the diagonal and aij = 1 if and only if the edge (i, j ) ∈ E(G). If the spectral radius of A is at least\n\nn/k, then G has a cycle of length ≤ k.\n\n#6.4 Disjoint Cycles", "original_statement": "Conjecture 6.\n1\n8. (Charbit) Let A be the adjacency matrix of a digraph G with zero on the diagonal and aij = 1 if and only if the edge (i, j ) ∈ E(G). If the spectral radius of A is at least \n\nn/k, then G has a cycle of length ≤ k.\n\n#6.4 Disjoint Cycles", "clean_statement": "Conjecture 6.\n1\n8. (Charbit) Let A be the adjacency matrix of a digraph G with zero on the diagonal and aij = 1 if and only if the edge (i, j ) ∈ E(G). If the spectral radius of A is at least\n\nn/k, then G has a cycle of length ≤ k.\n\n#6.4 Disjoint Cycles", "statement_status": "exact", "statement_verification": "The canonical record is an OCR extraction of Conjecture 6.18 in Blair Sullivan's report from the 2006 AIM workshop on the Caccetta--Haggkvist conjecture. The split number “6. 1 8” is an OCR artifact, and the attached heading “6.4 Disjoint Cycles” begins the next section. The recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[187]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n1\\n8. (Charbit) Let A be the adjacency matrix of a digraph G with zero on the diagonal and aij = 1 if and only if the edge (i, j ) ∈ E(G). If the spectral radius of A is at least \\n\\nn/k, then G has a cycle of length ≤ k.\\n\\n#6.4 Disjoint Cycles\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0188", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every digraph whose loopless simple support has no directed 2-cycle, its zero-one support adjacency matrix satisfies rho(A) <= (n-1)/2 < n/2, proving Charbit's conjecture for k=2; equality in the non-strict bound occurs exactly when the support is a regular tournament. More generally, if an m-vertex spectral-radius-attaining support component has period h and cyclic-class sizes m_i, then rho(A) <= (product_i m_i)^(1/h) <= m/h, with equality for h >= 2 exactly when that support component is a balanced complete cyclic blow-up. Hence rho(A) >= n/k forces h <= mk/n <= k for every Perron component.\n\nCandidate contribution (reduction; novelty confidence low): If H is an m-vertex Perron component of the loopless simple support of a digraph G, with period h and cyclic-class sizes m_0,...,m_{h-1}, then rho(G) <= (product_i m_i)^(1/h) <= m/h; equality for h >= 2 holds exactly when the support H is a balanced complete cyclic blow-up, and Charbit's threshold rho(G) >= n/k consequently imposes the ambient-size obstruction h <= mk/n.", "research_difficulty_suggested": "L4" }, { "id": 20001064, "problem_number": "AIM-COMBINATORICS-0189", "title": "A multigraph counterexample and a low-exposure reduction", "statement": "Conjecture 6.\n1\n9. (Bermond-Thomassen) In a digraph D with δ+\n\n> D\n\n≥ 2k − 1, there are k vertex disjoint cycles. Notes: Open for k ≥ 3, and the proof for k = 2 by Thomassen is not intuitive.", "original_statement": "Conjecture 6.\n1\n9. (Bermond-Thomassen) In a digraph D with δ+ \n\n> D\n\n≥ 2k − 1, there are k vertex disjoint cycles. Notes: Open for k ≥ 3, and the proof for k = 2 by Thomassen is not intuitive.", "clean_statement": "Conjecture 6.\n1\n9. (Bermond-Thomassen) In a digraph D with δ+\n\n> D\n\n≥ 2k − 1, there are k vertex disjoint cycles. Notes: Open for k ≥ 3, and the proof for k = 2 by Thomassen is not intuitive.", "statement_status": "exact", "statement_verification": "The corpus record is Conjecture 6.19 in Blair D. Sullivan's report from the 2006 AIM workshop on the Caccetta--Haggkvist conjecture. After repairing only the PDF line breaks, its statement is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[188]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n1\\n9. (Bermond-Thomassen) In a digraph D with δ+ \\n\\n> D\\n\\n≥ 2k − 1, there are k vertex disjoint cycles. Notes: Open for k ≥ 3, and the proof for k = 2 by Thomassen is not intuitive.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0189", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the AIM report's explicit allowance of parallel arcs and the standard multiplicity-counted outdegree convention, the printed statement is false for every k at least 2: two vertices with 2k-1 parallel arcs in each direction have the required minimum outdegree but cycle-packing number at most one. Under the intended no-parallel-arcs formulation, which remains open for k at least 4, a proved low-exposure extension lemma shows that s prepacked cycles extend to k cycles whenever their union costs each outside vertex at most 2s out-neighbors and the Bermond-Thomassen conjecture is known for k-s. The known k-s=3 case implies that any k=4 counterexample is digon-free and every nonspanning cycle has an outside vertex with at least three out-neighbors on it.\n\nCandidate contribution (reduction; novelty confidence low): In the standard finite loopless simple-digraph setting, if a digraph of minimum outdegree at least 2k-1 has s disjoint cycles whose union is a proper vertex set receiving at most 2s out-neighbors from every outside vertex, and the Bermond-Thomassen assertion holds for k-s, then those cycles extend to k disjoint cycles; consequently every k=4 counterexample is digon-free and every directed triangle is completely out-dominated by some outside vertex." }, { "id": 20001065, "problem_number": "AIM-COMBINATORICS-0190", "title": "A convention-sensitive counterexample and a maximal-family obstruction certificate", "statement": "Conjecture 6.\n2\n0. (Ho´ ang & Reed) If G is a digraph with minimum outdegree r, then there are directed cycles C1,..., C r such that for all `,\n\n|V (C`) ∩ (∪`−1\n\n> i=1\n\nV (Ci)) | ≤ 1.\n\nNotes: The related conjecture that given minimum outdegree at least r, there should be a vertex\n\nv with r cycles through v which are otherwise vertex-disjoint is false. A counterexample for r = 3 was given by Thomassen in 1985 (see Theorem 5.2 for his complete result). Adding the condition that the minimum indegree is also at least r does not improve the veracity of the conjecture, though it is still open when indegree and outdegree are identically r everywhere.\n\n#6.5 Connectivity", "original_statement": "Conjecture 6.\n2\n0. (Ho´ ang & Reed) If G is a digraph with minimum outdegree r, then there are directed cycles C1,..., C r such that for all `,\n\n|V (C`) ∩ (∪`−1 \n\n> i=1\n\nV (Ci)) | ≤ 1.\n\nNotes: The related conjecture that given minimum outdegree at least r, there should be a vertex \n\nv with r cycles through v which are otherwise vertex-disjoint is false. A counterexample for r = 3 was given by Thomassen in 1985 (see Theorem 5.2 for his complete result). Adding the condition that the minimum indegree is also at least r does not improve the veracity of the conjecture, though it is still open when indegree and outdegree are identically r everywhere. \n\n#6.5 Connectivity", "clean_statement": "Conjecture 6.\n2\n0. (Ho´ ang & Reed) If G is a digraph with minimum outdegree r, then there are directed cycles C1,..., C r such that for all `,\n\n|V (C`) ∩ (∪`−1\n\n> i=1\n\nV (Ci)) | ≤ 1.\n\nNotes: The related conjecture that given minimum outdegree at least r, there should be a vertex\n\nv with r cycles through v which are otherwise vertex-disjoint is false. A counterexample for r = 3 was given by Thomassen in 1985 (see Theorem 5.2 for his complete result). Adding the condition that the minimum indegree is also at least r does not improve the veracity of the conjecture, though it is still open when indegree and outdegree are identically r everywhere.\n\n#6.5 Connectivity", "statement_status": "exact", "statement_verification": "The canonical JSON record is an OCR-damaged extraction of Conjecture 6.20 in Blair D. Sullivan's 2006 AIM survey. The PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[189]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n2\\n0. (Ho´ ang & Reed) If G is a digraph with minimum outdegree r, then there are directed cycles C1,..., C r such that for all `,\\n\\n|V (C`) ∩ (∪`−1 \\n\\n> i=1\\n\\nV (Ci)) | ≤ 1.\\n\\nNotes: The related conjecture that given minimum outdegree at least r, there should be a vertex \\n\\nv with r cycles through v which are otherwise vertex-disjoint is false. A counterexample for r = 3 was given by Thomassen in 1985 (see Theorem 5.2 for his complete result). Adding the condition that the minimum indegree is also at least r does not improve the veracity of the conjecture, though it is still open when indegree and outdegree are identically r everywhere. \\n\\n#6.5 Connectivity\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0190", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "If the parallel arcs permitted by the AIM survey are counted with multiplicity in outdegree, then for every r at least 2 and n at least 3 the r-fold parallel directed n-cycle is loopless, digon-free, strongly connected, exactly r-in/r-out-regular, and has no admissible pair of directed cycles because every cycle uses all n vertices. This does not refute the intended simple or distinct-outneighbor formulation, which is known for r at most 3 and for tournaments but remains open in general. For the intended simple setting, a cycle family is proved admissible exactly when its overlap-incidence graph is a forest; a nonextendable family has a union U meeting every directed cycle at least twice, each deletion D minus (U minus {u}) is acyclic, and |U| is at least r+1.\n\nCandidate contribution (counterexample; novelty confidence low): Under the natural incidence-counted interpretation of degree for the multidigraphs allowed by Sullivan's AIM survey, the r-fold parallel directed n-cycle is, for every r at least 2 and n at least 3, a loopless, digon-free, strongly connected, exactly r-in/r-out-regular counterexample to that reading of Conjecture 6.20.", "research_difficulty_suggested": "L4" }, { "id": 20001066, "problem_number": "AIM-COMBINATORICS-0191", "title": "A convention-sensitive counterexample to Hamidoune's reverse local-connectivity conjecture", "statement": "Conjecture 6.\n2\n1. (Hamidoune, 1981) Let D be a digraph with δ+\n\n> D\n\n≥ r, δ−\n\n> D\n\n≥ r, and r ≥ 1.Then there is an edge (x, y ) such that κ(y, x ) ≥ r.\n\nNote that a counterexample to the above conjecture for all r ≥ 3 is given by Thomassen in Theorem 5.2.", "original_statement": "Conjecture 6.\n2\n1. (Hamidoune, 1981) Let D be a digraph with δ+ \n\n> D\n\n≥ r, δ− \n\n> D\n\n≥ r, and r ≥ 1.Then there is an edge (x, y ) such that κ(y, x ) ≥ r.\n\nNote that a counterexample to the above conjecture for all r ≥ 3 is given by Thomassen in Theorem 5.2.", "clean_statement": "Conjecture 6.\n2\n1. (Hamidoune, 1981) Let D be a digraph with δ+\n\n> D\n\n≥ r, δ−\n\n> D\n\n≥ r, and r ≥ 1.Then there is an edge (x, y ) such that κ(y, x ) ≥ r.\n\nNote that a counterexample to the above conjecture for all r ≥ 3 is given by Thomassen in Theorem 5.2.", "statement_status": "exact", "statement_verification": "The OCR in `input.json` breaks the subscripts and the conjecture number. The AIM source gives the following statement as Conjecture 6.21:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[190]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n2\\n1. (Hamidoune, 1981) Let D be a digraph with δ+ \\n\\n> D\\n\\n≥ r, δ− \\n\\n> D\\n\\n≥ r, and r ≥ 1.Then there is an edge (x, y ) such that κ(y, x ) ≥ r.\\n\\nNote that a counterexample to the above conjecture for all r ≥ 3 is given by Thomassen in Theorem 5.2.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0191", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the AIM source's explicitly permitted parallel-arc convention and the standard convention that degree counts arcs with multiplicity, the conjecture fails for every r >= 2: replace each arc of a directed cycle of length at least three by r parallel copies. Every vertex then has indegree and outdegree r, while every reverse path for any selected arc follows the unique support route and all such paths share an internal vertex, so kappa(y,x)=1. The r=1 case is true; replacing multiplicity degree by support degree is exactly equivalent to the simple-digraph form; and the repaired r=2 statement is proved here for bidirected supports but was not resolved in the literature checked.\n\nCandidate contribution (counterexample; novelty confidence low): For every m >= 3 and r >= 2, the r-fold parallel directed m-cycle has minimum arc indegree and outdegree r but reverse local vertex-connectivity exactly one at every arc; moreover, the support-degree repair is logically equivalent to the simple-digraph conjecture." }, { "id": 20001067, "problem_number": "AIM-COMBINATORICS-0192", "title": "Cut covers and small cases of Mader's local-connectivity conjecture", "statement": "Conjecture 6.\n2\n2. (Mader) If a digraph D has δ+\n\n> D\n\n≥ r, then there are vertices x 6 = y such that\n\nλ(x, y ) ≥ r. Also, there is an edge (x, y ) so that λ(x, y ) ≥ r.\n\nThe first part of this conjecture was proven by Mader [24] for λ(x, y ) ≥ r − 1.\n\n#6.6 Weighted Versions", "original_statement": "Conjecture 6.\n2\n2. (Mader) If a digraph D has δ+ \n\n> D\n\n≥ r, then there are vertices x 6 = y such that \n\nλ(x, y ) ≥ r. Also, there is an edge (x, y ) so that λ(x, y ) ≥ r.\n\nThe first part of this conjecture was proven by Mader [24] for λ(x, y ) ≥ r − 1. \n\n#6.6 Weighted Versions", "clean_statement": "Conjecture 6.\n2\n2. (Mader) If a digraph D has δ+\n\n> D\n\n≥ r, then there are vertices x 6 = y such that\n\nλ(x, y ) ≥ r. Also, there is an edge (x, y ) so that λ(x, y ) ≥ r.\n\nThe first part of this conjecture was proven by Mader [24] for λ(x, y ) ≥ r − 1.\n\n#6.6 Weighted Versions", "statement_status": "exact", "statement_verification": "The corpus record is Conjecture 6.22 in Section 6.5 (“Connectivity”) of Blair D. Sullivan's report from the 2006 AIM workshop on the Caccetta--Haggkvist conjecture. The primary PDF reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[191]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n2\\n2. (Mader) If a digraph D has δ+ \\n\\n> D\\n\\n≥ r, then there are vertices x 6 = y such that \\n\\nλ(x, y ) ≥ r. Also, there is an edge (x, y ) so that λ(x, y ) ≥ r.\\n\\nThe first part of this conjecture was proven by Mader [24] for λ(x, y ) ≥ r − 1. \\n\\n#6.6 Weighted Versions\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0192", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The sharp conjecture remains open in the literature checked, but the stronger adjacent-endpoint clause is proved here for every finite loopless directed multigraph when r is at most 2 and for every finite loopless simple digraph of order at most r+2. Directed Menger's theorem also gives an exact counterexample certificate: every arc must lie in an outgoing cut of size at most r-1; in the simple case, every such cut has at least r+1 vertices on its tail side. Thus a simple counterexample has at least r+3 vertices.\n\nCandidate contribution (special_case; novelty confidence low): The adjacent-endpoint clause holds for loopless directed multigraphs for r<=2 and for loopless simple digraphs on at most r+2 vertices; moreover, every simple counterexample has at least r+3 vertices and admits an all-arc cover by outgoing cuts of size at most r-1 whose tail sides each have at least r+1 vertices.", "research_difficulty_suggested": "L4" }, { "id": 20001068, "problem_number": "AIM-COMBINATORICS-0193", "title": "A counterexample to the unqualified weak-weight formulation", "statement": "Conjecture 6.\n2\n3. (Bollob´ as & Scott) Let p: E(G) → [0, 1]. If ∑\n\n> v∈N+\n> G(u)\n\np(uv ) ≥ 1 and\n\n∑\n\n> v∈N−\n> G(u)\n\np(vu ) ≥ 1 for all u ∈ V (G), there is a directed cycle in G of total weight ≥ 1.Note: There is a nice proof that there is a dipath of total weight at least 1.", "original_statement": "Conjecture 6.\n2\n3. (Bollob´ as & Scott) Let p: E(G) → [0, 1]. If ∑ \n\n> v∈N+\n> G(u)\n\np(uv ) ≥ 1 and \n\n∑ \n\n> v∈N−\n> G(u)\n\np(vu ) ≥ 1 for all u ∈ V (G), there is a directed cycle in G of total weight ≥ 1.Note: There is a nice proof that there is a dipath of total weight at least 1.", "clean_statement": "Conjecture 6.\n2\n3. (Bollob´ as & Scott) Let p: E(G) → [0, 1]. If ∑\n\n> v∈N+\n> G(u)\n\np(uv ) ≥ 1 and\n\n∑\n\n> v∈N−\n> G(u)\n\np(vu ) ≥ 1 for all u ∈ V (G), there is a directed cycle in G of total weight ≥ 1.Note: There is a nice proof that there is a dipath of total weight at least 1.", "statement_status": "exact", "statement_verification": "The assigned record is Conjecture 6.23 in Section 6.6 (“Weighted Versions”) of Blair D. Sullivan's 2006 AIM report on the Caccetta--Haggkvist conjecture. The primary PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[192]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n2\\n3. (Bollob´ as & Scott) Let p: E(G) → [0, 1]. If ∑ \\n\\n> v∈N+\\n> G(u)\\n\\np(uv ) ≥ 1 and \\n\\n∑ \\n\\n> v∈N−\\n> G(u)\\n\\np(vu ) ≥ 1 for all u ∈ V (G), there is a directed cycle in G of total weight ≥ 1.Note: There is a nice proof that there is a dipath of total weight at least 1.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0193", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "AIM Conjecture 6.23 is false as written. An explicit finite loopless simple weighted digraph on 26 vertices and 77 arcs is constructed with minimum weighted indegree and outdegree both 39/38, yet every simple directed cycle has total weight exactly 39/40. A general one-way reverse-gluing lemma explains the obstruction: the AIM report combined weak inequalities with no connectivity hypothesis, whereas Bollobas and Scott proposed those inequalities only for strongly connected digraphs.\n\nCandidate contribution (counterexample; novelty confidence low): There is a 26-vertex, 77-arc loopless simple digraph with all arc weights in [0,1], minimum weighted indegree and outdegree 39/38, and maximum directed-cycle weight 39/40; more generally, any outweight-only counterexample can be converted into an unqualified two-sided weak-inequality counterexample by one-way reverse gluing." }, { "id": 20001069, "problem_number": "AIM-COMBINATORICS-0194", "title": "Reciprocal-weight cycles and a winding-number obstruction", "statement": "Conjecture 6.\n2\n4. (Zhang) Let G be a digraph on n vertices, and f: E(G) → { 0, 1,... }. If ∑\n\n> e∈E+(v)\n\nf (e) ≥ n/k for all v ∈ V (G), then there is a directed cycle C such that ∑\n\n> e∈C\n> 1\n> f(e)\n\n≤ k.Counterexample: (Charbit) Take the directed Cayley graph G with group G = Z8, and generators\n\n{1, 2}. Let the weight function f be 1 on 2-edges, and 2 on 1-edges (where r-edges are those coming from the generator r). Now let k = 3. We can calculate ∑\n\n> e∈E+(v)\n\nf (e) = 4 ≥ 8/3 for all vertices v, but there is no directed cycle with ∑\n\n> e∈C\n> 1\n> f(e)\n\n≤ 3. 96.7 Averaged Outdegree Conditions", "original_statement": "Conjecture 6.\n2\n4. (Zhang) Let G be a digraph on n vertices, and f: E(G) → { 0, 1,... }. If ∑ \n\n> e∈E+(v)\n\nf (e) ≥ n/k for all v ∈ V (G), then there is a directed cycle C such that ∑ \n\n> e∈C\n> 1\n> f(e)\n\n≤ k.Counterexample: (Charbit) Take the directed Cayley graph G with group G = Z8, and generators \n\n{1, 2}. Let the weight function f be 1 on 2-edges, and 2 on 1-edges (where r-edges are those coming from the generator r). Now let k = 3. We can calculate ∑ \n\n> e∈E+(v)\n\nf (e) = 4 ≥ 8/3 for all vertices v, but there is no directed cycle with ∑ \n\n> e∈C\n> 1\n> f(e)\n\n≤ 3. 96.7 Averaged Outdegree Conditions", "clean_statement": "Conjecture 6.\n2\n4. (Zhang) Let G be a digraph on n vertices, and f: E(G) → { 0, 1,... }. If ∑\n\n> e∈E+(v)\n\nf (e) ≥ n/k for all v ∈ V (G), then there is a directed cycle C such that ∑\n\n> e∈C\n> 1\n> f(e)\n\n≤ k.Counterexample: (Charbit) Take the directed Cayley graph G with group G = Z8, and generators\n\n{1, 2}. Let the weight function f be 1 on 2-edges, and 2 on 1-edges (where r-edges are those coming from the generator r). Now let k = 3. We can calculate ∑\n\n> e∈E+(v)\n\nf (e) = 4 ≥ 8/3 for all vertices v, but there is no directed cycle with ∑\n\n> e∈C\n> 1\n> f(e)\n\n≤ 3. 96.7 Averaged Outdegree Conditions", "statement_status": "exact", "statement_verification": "The OCR in `input.json` splits the conjecture number and appends the next page number and section heading. The primary PDF gives the following display as Conjecture 6.24:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[193]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n2\\n4. (Zhang) Let G be a digraph on n vertices, and f: E(G) → { 0, 1,... }. If ∑ \\n\\n> e∈E+(v)\\n\\nf (e) ≥ n/k for all v ∈ V (G), then there is a directed cycle C such that ∑ \\n\\n> e∈C\\n> 1\\n> f(e)\\n\\n≤ k.Counterexample: (Charbit) Take the directed Cayley graph G with group G = Z8, and generators \\n\\n{1, 2}. Let the weight function f be 1 on 2-edges, and 2 on 1-edges (where r-edges are those coming from the generator r). Now let k = 3. We can calculate ∑ \\n\\n> e∈E+(v)\\n\\nf (e) = 4 ≥ 8/3 for all vertices v, but there is no directed cycle with ∑ \\n\\n> e∈C\\n> 1\\n> f(e)\\n\\n≤ 3. 96.7 Averaged Outdegree Conditions\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0194", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The source's Charbit example is valid but contains an arithmetic typo: its outgoing-weight sum is 2+1=3, not 4, and 3 is still at least 8/3. More generally, for every integer k >= 2, the positively weighted simple digraph Cay(Z_{2(k+1)},{1,2}), with weight 2 on 1-arcs and weight 1 on 2-arcs, has outgoing-weight sum 3 >= 2(k+1)/k while every directed cycle has reciprocal cost exactly k+1. A general winding-number theorem explains this obstruction, and a sharp positive version is proved for digraphs with exactly one outgoing arc at every vertex.\n\nCandidate contribution (counterexample_family; novelty confidence low): If S is a set of positive generators modulo n and a is divisible by every s in S, assigning weight a/s to every s-arc makes the reciprocal cost of each closed walk equal to qn/a for a positive integer winding number q. In particular, the two-generator specialization gives a positive-integer counterexample for every k >= 2, and on Cay(Z_{2t},{1,2}) every simple directed cycle has cost exactly t with sharp failure interval 2t/3 <= k < t." }, { "id": 20001070, "problem_number": "AIM-COMBINATORICS-0195", "title": "A parametric audit of a refuted logarithmic outdegree condition", "statement": "Conjecture 6.25. If D is a digraph on n vertices with\n\n∑\n\n> v∈V(D)\n\nlog(1 + 1\n\nδ+\n\n> D\n\n(v)) ≥ n log(1 + 1\n\nr ),\n\nthen D has a cycle of length at most dn/r e.Counterexample: Take a transitive tournament of size n − 1, and replace the edge from the vertex of out-degree n − 2 to the vertex of out-degree zero with a path of length 2 in the opposite direction (thus increasing the number of vertices to n).", "original_statement": "Conjecture 6.25. If D is a digraph on n vertices with \n\n∑ \n\n> v∈V(D)\n\nlog(1 + 1\n\nδ+ \n\n> D\n\n(v)) ≥ n log(1 + 1\n\nr ),\n\nthen D has a cycle of length at most dn/r e.Counterexample: Take a transitive tournament of size n − 1, and replace the edge from the vertex of out-degree n − 2 to the vertex of out-degree zero with a path of length 2 in the opposite direction (thus increasing the number of vertices to n).", "clean_statement": "Conjecture 6.25. If D is a digraph on n vertices with\n\n∑\n\n> v∈V(D)\n\nlog(1 + 1\n\nδ+\n\n> D\n\n(v)) ≥ n log(1 + 1\n\nr ),\n\nthen D has a cycle of length at most dn/r e.Counterexample: Take a transitive tournament of size n − 1, and replace the edge from the vertex of out-degree n − 2 to the vertex of out-degree zero with a path of length 2 in the opposite direction (thus increasing the number of vertices to n).", "statement_status": "exact", "statement_verification": "The canonical record comes from Conjecture 6.25 in Blair D. Sullivan's 2006 AIM survey. Inspection of page 10 of the PDF confirms that the final bound uses a ceiling, not a floor. In modern notation, the displayed statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6.25\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[194]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.25. If D is a digraph on n vertices with \\n\\n∑ \\n\\n> v∈V(D)\\n\\nlog(1 + 1\\n\\nδ+ \\n\\n> D\\n\\n(v)) ≥ n log(1 + 1\\n\\nr ),\\n\\nthen D has a cycle of length at most dn/r e.Counterexample: Take a transitive tournament of size n − 1, and replace the edge from the vertex of out-degree n − 2 to the vertex of out-degree zero with a path of length 2 in the opposite direction (thus increasing the number of vertices to n).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0195", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The transitive-tournament construction printed with Conjecture 6.25 is verified exactly: for every n at least 4 it has degree product 4(n-2)^2/(n-3), directed cycle spectrum {4,...,n}, and refutes the claimed implication for precisely the positive integers r at least ceil(n/3). More generally, replacing the extreme tournament arc by an oppositely oriented q-arc path gives an explicit two-parameter family with a closed degree product, exact cycle spectrum, and exact two-threshold counterexample criterion. The report also isolates the zero-outdegree domain defect and derives a valid factor-2 logarithmic replacement from a 2023 nonuniform-degree theorem.\n\nCandidate contribution (counterexample_family; novelty confidence low): For the reverse-path family T_{m,q} on N=m+q-1 vertices, the degree product is 2^q(m-1)^2/(m-2), the directed cycle lengths are exactly q+2 through N, and it refutes the source implication exactly for integers r at least the maximum of ceil(N/(q+1)) and ceil(1/((2^q(m-1)^2/(m-2))^(1/N)-1))." }, { "id": 20001071, "problem_number": "AIM-COMBINATORICS-0196", "title": "Local average outdegree and short directed cycles", "statement": "Conjecture 6.\n2\n6. (Shen [32]) Let G be a digraph with n vertices and minimum outdegree at least one. If δ+\n\n> G\n\n(v) + δ+\n\n> G\n\n(u) ≥ 2r for every edge (u, v ) in G, then the girth g of G is at most\n\ndn/r e.Note: This was proved by Shen for r = 2 in [32].\n\n#6.8 UnCategorized\n\nThe following conjecture would imply that in a counterexample for Caccetta-H¨ aggkvist for r =\n\nn/ 3, one can order the vertices so that at least 75% of the edges go from left to right:", "original_statement": "Conjecture 6.\n2\n6. (Shen [32]) Let G be a digraph with n vertices and minimum outdegree at least one. If δ+ \n\n> G\n\n(v) + δ+ \n\n> G\n\n(u) ≥ 2r for every edge (u, v ) in G, then the girth g of G is at most \n\ndn/r e.Note: This was proved by Shen for r = 2 in [32]. \n\n#6.8 UnCategorized \n\nThe following conjecture would imply that in a counterexample for Caccetta-H¨ aggkvist for r =\n\nn/ 3, one can order the vertices so that at least 75% of the edges go from left to right:", "clean_statement": "Conjecture 6.\n2\n6. (Shen [32]) Let G be a digraph with n vertices and minimum outdegree at least one. If δ+\n\n> G\n\n(v) + δ+\n\n> G\n\n(u) ≥ 2r for every edge (u, v ) in G, then the girth g of G is at most\n\ndn/r e.Note: This was proved by Shen for r = 2 in [32].\n\n#6.8 UnCategorized\n\nThe following conjecture would imply that in a counterexample for Caccetta-H¨ aggkvist for r =\n\nn/ 3, one can order the vertices so that at least 75% of the edges go from left to right:", "statement_status": "exact", "statement_verification": "The primary AIM PDF gives the following statement. Its bracket glyphs are a ceiling, not a floor:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[195]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n2\\n6. (Shen [32]) Let G be a digraph with n vertices and minimum outdegree at least one. If δ+ \\n\\n> G\\n\\n(v) + δ+ \\n\\n> G\\n\\n(u) ≥ 2r for every edge (u, v ) in G, then the girth g of G is at most \\n\\ndn/r e.Note: This was proved by Shen for r = 2 in [32]. \\n\\n#6.8 UnCategorized \\n\\nThe following conjecture would imply that in a counterexample for Caccetta-H¨ aggkvist for r =\\n\\nn/ 3, one can order the vertices so that at least 75% of the edges go from left to right:\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0196", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For Shen's intended no-parallel-arcs convention, the conjecture is known for r=1 and r=2 and remains open in the literature checked for general r at least 3. This attempt proves the conjectured bound sharply for every cyclic complete blow-up, classifies equality as uniform layers for odd cycle length and alternating layer sizes r-s, r+s for even cycle length, and proves a sufficient condition based on pairwise-disjoint outneighborhoods along a shortest cycle. It also shows that the literal AIM multiarc reading, when outdegree counts arc multiplicity, is false for every r at least 2 via an alternating-multiplicity directed cycle.\n\nCandidate contribution (special_case_and_obstruction; novelty confidence low): In a cyclic complete blow-up satisfying the local endpoint outdegree-sum condition, n is at least rm and equality forces all layer sizes to equal r when m is odd, while for even m they alternate r-s and r+s with |s| at most r-1; moreover, allowing parallel arcs counted with multiplicity produces an explicit counterexample for every r at least 2.", "research_difficulty_suggested": "L4" }, { "id": 20001072, "problem_number": "AIM-COMBINATORICS-0197", "title": "Exact feedback sets in cyclic tournament substitutions", "statement": "Conjecture 6.\n2\n7. (Chudnovsky, Seymour, Sullivan) Let G be a simple digraph with k non-edges (unordered pairs {u, v } where both uv and vu are not in E(G)). If G has no directed cycle of length at most 3, one can delete at most k/ 2 edges from G and obtain a graph with no directed cycle. Notes: We know that there are tight examples for this conjecture (transitive tournaments, C4,and products of these). It is also known that a minimal counterexample has no source or sink vertex, and no directed cut. Kostochka recently proved all vertices in a minimal counterexample have at least 3 and at most ( n − 1) /2 non-neighbors. He has an argument using these facts to show the conjecture for all k ≤ 14.", "original_statement": "Conjecture 6.\n2\n7. (Chudnovsky, Seymour, Sullivan) Let G be a simple digraph with k non-edges (unordered pairs {u, v } where both uv and vu are not in E(G)). If G has no directed cycle of length at most 3, one can delete at most k/ 2 edges from G and obtain a graph with no directed cycle. Notes: We know that there are tight examples for this conjecture (transitive tournaments, C4,and products of these). It is also known that a minimal counterexample has no source or sink vertex, and no directed cut. Kostochka recently proved all vertices in a minimal counterexample have at least 3 and at most ( n − 1) /2 non-neighbors. He has an argument using these facts to show the conjecture for all k ≤ 14.", "clean_statement": "Conjecture 6.\n2\n7. (Chudnovsky, Seymour, Sullivan) Let G be a simple digraph with k non-edges (unordered pairs {u, v } where both uv and vu are not in E(G)). If G has no directed cycle of length at most 3, one can delete at most k/ 2 edges from G and obtain a graph with no directed cycle. Notes: We know that there are tight examples for this conjecture (transitive tournaments, C4,and products of these). It is also known that a minimal counterexample has no source or sink vertex, and no directed cut. Kostochka recently proved all vertices in a minimal counterexample have at least 3 and at most ( n − 1) /2 non-neighbors. He has an argument using these facts to show the conjecture for all k ≤ 14.", "statement_status": "exact", "statement_verification": "The canonical record is Conjecture 6.27 in Blair D. Sullivan's 2006 AIM survey. The primary PDF gives the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[196]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n2\\n7. (Chudnovsky, Seymour, Sullivan) Let G be a simple digraph with k non-edges (unordered pairs {u, v } where both uv and vu are not in E(G)). If G has no directed cycle of length at most 3, one can delete at most k/ 2 edges from G and obtain a graph with no directed cycle. Notes: We know that there are tight examples for this conjecture (transitive tournaments, C4,and products of these). It is also known that a minimal counterexample has no source or sink vertex, and no directed cut. Kostochka recently proved all vertices in a minimal counterexample have at least 3 and at most ( n − 1) /2 non-neighbors. He has an argument using these facts to show the conjecture for all k ≤ 14.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0197", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The Chudnovsky-Seymour-Sullivan conjecture remains open, with the strongest explicit global coefficient located being 0.8616 and no verified universal exact extension of the AIM workshop range beyond 14 missing pairs. This attempt proves an exact special-family theorem: replacing the vertices of a directed cycle of length ell at least 4 by transitive tournaments of positive sizes a_i gives feedback arc number min_i a_i a_{i+1}, with an explicit missing-pair formula. For four bags of sizes a,b,c,d this becomes beta=min{ab,bc,cd,da} and gamma=ac+bd, and all equality cases in the integral bound beta=floor(gamma/2) are classified. A separate product theorem shows that 3-freeness and CSS validity are preserved by lexicographic products.\n\nCandidate contribution (exact_special_class_theorem; novelty confidence low): For cyclic substitutions B_ell(a_0,...,a_{ell-1}) by transitive tournaments, beta(B_ell)=min_i a_i a_{i+1}. For ell=4, beta=floor(gamma/2) holds exactly, up to cyclic rotation, for (p,q,p,q) with |p-q| at most 1 and for (2,1,1,1); real equality beta=gamma/2 occurs exactly for four equal bags.", "research_difficulty_suggested": "L4" }, { "id": 20001073, "problem_number": "AIM-COMBINATORICS-0198", "title": "LP obstructions and cyclic blow-ups for DeVos's distribution conjecture", "statement": "Conjecture 6.\n2\n8. (Devos) For any digraph G with no directed cycles of length at most three, there is a probability distribution p on V (G) such that at every vertex v, p(N +(v)) ≥ p(N −(v))\n\nand p(N −\n\n> 2\n\n(v)) ≥ p(N −(v)), where p(S):= ∑\n\n> s∈S\n\np(s) for a set S ⊆ V (G).Notes: For tournaments, such a distribution exists, and is unique. Its existence for a general digraph would imply Seymour's Second Neighborhood Conjecture, and actually it would suffice to just have a probability distribution p so that p(N −\n\n> 2\n\n(v)) ≥ p(N −(v)) on average in G.Given a digraph G = ( V, E ), we say F ⊆ E(G) is a feedback arc set if the digraph G′ = ( V, E −F )has no directed cycles.", "original_statement": "Conjecture 6.\n2\n8. (Devos) For any digraph G with no directed cycles of length at most three, there is a probability distribution p on V (G) such that at every vertex v, p(N +(v)) ≥ p(N −(v)) \n\nand p(N − \n\n> 2\n\n(v)) ≥ p(N −(v)), where p(S):= ∑ \n\n> s∈S\n\np(s) for a set S ⊆ V (G).Notes: For tournaments, such a distribution exists, and is unique. Its existence for a general digraph would imply Seymour's Second Neighborhood Conjecture, and actually it would suffice to just have a probability distribution p so that p(N − \n\n> 2\n\n(v)) ≥ p(N −(v)) on average in G.Given a digraph G = ( V, E ), we say F ⊆ E(G) is a feedback arc set if the digraph G′ = ( V, E −F )has no directed cycles.", "clean_statement": "Conjecture 6.\n2\n8. (Devos) For any digraph G with no directed cycles of length at most three, there is a probability distribution p on V (G) such that at every vertex v, p(N +(v)) ≥ p(N −(v))\n\nand p(N −\n\n> 2\n\n(v)) ≥ p(N −(v)), where p(S):= ∑\n\n> s∈S\n\np(s) for a set S ⊆ V (G).Notes: For tournaments, such a distribution exists, and is unique. Its existence for a general digraph would imply Seymour's Second Neighborhood Conjecture, and actually it would suffice to just have a probability distribution p so that p(N −\n\n> 2\n\n(v)) ≥ p(N −(v)) on average in G.Given a digraph G = ( V, E ), we say F ⊆ E(G) is a feedback arc set if the digraph G′ = ( V, E −F )has no directed cycles.", "statement_status": "exact", "statement_verification": "The primary source is Sullivan's AIM problem-list article on the Caccetta--Häggkvist conjecture. The corpus extraction split the subscript in the second inequality and appended text from the next problem. Using the definitions earlier in that source, the recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[197]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n2\\n8. (Devos) For any digraph G with no directed cycles of length at most three, there is a probability distribution p on V (G) such that at every vertex v, p(N +(v)) ≥ p(N −(v)) \\n\\nand p(N − \\n\\n> 2\\n\\n(v)) ≥ p(N −(v)), where p(S):= ∑ \\n\\n> s∈S\\n\\np(s) for a set S ⊆ V (G).Notes: For tournaments, such a distribution exists, and is unique. Its existence for a general digraph would imply Seymour's Second Neighborhood Conjecture, and actually it would suffice to just have a probability distribution p so that p(N − \\n\\n> 2\\n\\n(v)) ≥ p(N −(v)) on average in G.Given a digraph G = ( V, E ), we say F ⊆ E(G) is a feedback arc set if the digraph G′ = ( V, E −F )has no directed cycles.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0198", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unweighted-over-v interpretation of the AIM source's averaged weakening is proved exactly equivalent to Seymour's Second Neighborhood Conjecture by a double-counting identity. Joint pointwise feasibility is characterized by an explicit two-weight Gordan--Stiemke obstruction, whose second-neighborhood weight must be nonzero in every obstruction for a digon-free graph. Feasibility lifts from sink strong components, and for every terminal complete cyclic blow-up of length at least four the full feasible polytope consists exactly of distributions assigning mass 1/k to each cyclic part.\n\nCandidate contribution (equivalence_and_classification; novelty confidence low): Candidate contribution: the combined exact averaged equivalence, explicit two-weight alternative with nonzero second-neighborhood component in any digon-free obstruction, and complete feasible-distribution classification for terminal cyclic blow-ups.", "research_difficulty_suggested": "L4" }, { "id": 20001074, "problem_number": "AIM-COMBINATORICS-0199", "title": "Minimal feedback arc sets and long paths", "statement": "Conjecture 6.\n2\n9. (Lichiardopol's Conjecture) Every digraph D has some minimal feedback arc set F which contains a path of length δ+\n\n> D.\n\n10 Notes: This implies Ho´ ang & Reed (6.20), Caccetta-H¨ aggkvist (2.1), Bermond-Thomassen (6.19), and Thomass´ e (6.34).", "original_statement": "Conjecture 6.\n2\n9. (Lichiardopol's Conjecture) Every digraph D has some minimal feedback arc set F which contains a path of length δ+\n\n> D.\n\n10 Notes: This implies Ho´ ang & Reed (6.20), Caccetta-H¨ aggkvist (2.1), Bermond-Thomassen (6.19), and Thomass´ e (6.34).", "clean_statement": "Conjecture 6.\n2\n9. (Lichiardopol's Conjecture) Every digraph D has some minimal feedback arc set F which contains a path of length δ+\n\n> D.\n\n10 Notes: This implies Ho´ ang & Reed (6.20), Caccetta-H¨ aggkvist (2.1), Bermond-Thomassen (6.19), and Thomass´ e (6.34).", "statement_status": "exact", "statement_verification": "The primary AIM PDF defines a feedback arc set immediately before the conjecture: if \\(D=(V,E)\\), then \\(F\\subseteq E\\) is a feedback arc set (FAS) when \\((V,E\\setminus F)\\) has no directed cycle. The exact statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[198]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n2\\n9. (Lichiardopol's Conjecture) Every digraph D has some minimal feedback arc set F which contains a path of length δ+\\n\\n> D.\\n\\n10 Notes: This implies Ho´ ang & Reed (6.20), Caccetta-H¨ aggkvist (2.1), Bermond-Thomassen (6.19), and Thomass´ e (6.34).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0199", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Lichiardopol's conjecture is refuted. For every integer k at least 3, the Botler--Schneider construction gives a finite simple loopless digon-free strongly connected digraph of minimum outdegree k in which every inclusion-minimal feedback arc set has longest directed path at most two. This attempt proves the obstruction through a sharp two-sided degree-core certificate: every internal vertex of a path in a minimal feedback arc set has both indegree and outdegree at least two. It also classifies all minimal feedback arc sets of an r-fold parallel directed cycle, showing that the literal AIM multiplicity-degree convention fails for every r at least 2.\n\nCandidate contribution (structural_obstruction; novelty confidence low): If K(D) is the set of vertices having both indegree and outdegree at least two, then every directed path in every inclusion-minimal feedback arc set F has length at most rho(D[K(D)])+1, where rho is maximum path order; this bound is sharp for every value of rho." }, { "id": 20001075, "problem_number": "AIM-COMBINATORICS-0200", "title": "A sharp lower-bound certificate for the first open transitive tournament case", "statement": "Conjecture 6.\n3\n0. (Mader, 1985) For all k ∈ Z+, there exists r ∈ Z+ such that every r-out-regular digraph contains a subdivision of the transitive tournament on k vertices. Notes: This conjecture is known for k = 3 (where r = 2), and k = 4 ( r = 3 proven by Mader in 1996 [26]). The existence of r for k = 5 is still not known, though r = 6 has been conjectured.", "original_statement": "Conjecture 6.\n3\n0. (Mader, 1985) For all k ∈ Z+, there exists r ∈ Z+ such that every r-out-regular digraph contains a subdivision of the transitive tournament on k vertices. Notes: This conjecture is known for k = 3 (where r = 2), and k = 4 ( r = 3 proven by Mader in 1996 [26]). The existence of r for k = 5 is still not known, though r = 6 has been conjectured.", "clean_statement": "Conjecture 6.\n3\n0. (Mader, 1985) For all k ∈ Z+, there exists r ∈ Z+ such that every r-out-regular digraph contains a subdivision of the transitive tournament on k vertices. Notes: This conjecture is known for k = 3 (where r = 2), and k = 4 ( r = 3 proven by Mader in 1996 [26]). The existence of r for k = 5 is still not known, though r = 6 has been conjectured.", "statement_status": "exact", "statement_verification": "The canonical record is Conjecture 6.30 from the AIM workshop list *The Caccetta--Haggkvist conjecture*. The OCR has split the number “30” across lines, but the mathematical text recovers unambiguously as follows.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[199]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n3\\n0. (Mader, 1985) For all k ∈ Z+, there exists r ∈ Z+ such that every r-out-regular digraph contains a subdivision of the transitive tournament on k vertices. Notes: This conjecture is known for k = 3 (where r = 2), and k = 4 ( r = 3 proven by Mader in 1996 [26]). The existence of r for k = 5 is still not known, though r = 6 has been conjectured.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0200", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The exact r-out-regular formulation in the AIM record is equivalent, for every fixed r, to Mader's modern minimum-outdegree formulation by independently thinning each outneighbourhood. Moreover, the symmetric lift of any undirected topological-K_k-free graph is TT_k-subdivision-free. Applying this to the 5-regular planar icosahedral graph gives an explicit exactly 5-out-regular digraph with no TT_5-subdivision. Hence any TT_5 threshold is at least 6, so the historically conjectured value r=6 is provably sharp on the lower-bound side, while its sufficiency remains open.\n\nCandidate contribution (reduction_and_structural_obstruction; novelty confidence low): Exact outdegree and minimum outdegree are equivalent forcing conditions at each fixed r, and symmetric lifting transfers every undirected topological-K_k obstruction to TT_k; in particular, the symmetric lift of the icosahedral graph is a twelve-vertex exact 5-out-regular certificate proving tau(5) at least 6.", "research_difficulty_suggested": "L4" }, { "id": 20001076, "problem_number": "AIM-COMBINATORICS-0201", "title": "An order-statistic form of Shen's deficit conjecture", "statement": "Conjecture 6.\n3\n1. (Shen) For a digraph G on n vertices of girth g, define\n\nt(G, r ) = ∑\n\n> u:δ+\n> G(u) G\n\n(u)).\n\nIf δ+\n\n> G\n\n≥ 1, then n ≥ r(g − 1) + 1 − t(G, r ).", "original_statement": "Conjecture 6.\n3\n1. (Shen) For a digraph G on n vertices of girth g, define \n\nt(G, r ) = ∑ \n\n> u:δ+\n> G(u) G\n\n(u)).\n\nIf δ+ \n\n> G\n\n≥ 1, then n ≥ r(g − 1) + 1 − t(G, r ).", "clean_statement": "Conjecture 6.\n3\n1. (Shen) For a digraph G on n vertices of girth g, define\n\nt(G, r ) = ∑\n\n> u:δ+\n> G(u) G\n\n(u)).\n\nIf δ+\n\n> G\n\n≥ 1, then n ≥ r(g − 1) + 1 − t(G, r ).", "statement_status": "exact", "statement_verification": "The primary AIM PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[200]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n3\\n1. (Shen) For a digraph G on n vertices of girth g, define \\n\\nt(G, r ) = ∑ \\n\\n> u:δ+\\n> G(u) G\\n\\n(u)).\\n\\nIf δ+ \\n\\n> G\\n\\n≥ 1, then n ≥ r(g − 1) + 1 − t(G, r ).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0201", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite sinkless digraph without parallel arcs and with girth g at least 2, write its outdegrees as d_1 <= ... <= d_n. Maximizing Shen's expression r(g-1)-t(G,r) over positive integers r gives exactly the sum d_1+...+d_{g-1}, attained at r=d_{g-1}. Thus the full parameterized conjecture is equivalent to the single inequality d_1+...+d_{g-1} <= n-1. This inequality is proved here for girth at most 3 and for complete cyclic blow-ups; equality is classified in those cases, and a literal parallel-arc reading of the AIM source is refuted by parallel directed cycles.\n\nCandidate contribution (equivalence; novelty confidence low): Candidate novelty: Shen's entire family of deficit inequalities is exactly equivalent to the order-statistic inequality that the sum of the g-1 smallest outdegrees is at most n-1; the associated discrete derivative identifies the degree quantile containing every maximizing threshold. The report also develops sharp proofs and equality classifications for girth at most 3 and complete cyclic blow-ups.", "research_difficulty_suggested": "L4" }, { "id": 20001077, "problem_number": "AIM-COMBINATORICS-0202", "title": "Exact identities and rounding loss for induced directed paths", "statement": "Conjecture 6.\n3\n2. (Thomass´ e) If G is a loopless digon-free digraph the maximum number of induced directed 2-edge paths is n3/15 + O(n2).Notes: First, note that n3/15 + O(n2) can be obtained by substituting C4's inside C4's. Next, given a digraph G, let τ be the number of induced directed 2-paths, η the number of induced edges, and θ the number of cyclic triangles. Then if |V (G)| = n,\n\nτ + η/ 3 + 2 θ = n3/12 + O(n2) − 1\n\n6\n\n∑\n\n> v∈V(G)\n\n(\n\n(δ+(v) − δ−(v)) 2 + ( n\n\n2 − δ+(v)) 2 + ( n\n\n2 − δ−(v)) 2\n\n).\n\nF¨ uredi rewrote the terms on the right hand side in terms of τ, η, and θ, and showed\n\nτ ≤ n3\n\n12 + O(n2).\n\nBondy has a slight improvement of this result, proving that\n\nτ ≤ 2n3\n\n25.\n\nDefine an α/β-digraph to be a digraph D on vertex set V and edges defined by σ1,..., σ β\n\npermutations (linear orders) on V where the edge ( i, j ) ∈ E ⇐⇒ i < j in at least α of the σi.We say G is a majority digraph if α\n\n> β\n\n> 2/3.", "original_statement": "Conjecture 6.\n3\n2. (Thomass´ e) If G is a loopless digon-free digraph the maximum number of induced directed 2-edge paths is n3/15 + O(n2).Notes: First, note that n3/15 + O(n2) can be obtained by substituting C4's inside C4's. Next, given a digraph G, let τ be the number of induced directed 2-paths, η the number of induced edges, and θ the number of cyclic triangles. Then if |V (G)| = n,\n\nτ + η/ 3 + 2 θ = n3/12 + O(n2) − 1\n\n6\n\n∑ \n\n> v∈V(G)\n\n(\n\n(δ+(v) − δ−(v)) 2 + ( n\n\n2 − δ+(v)) 2 + ( n\n\n2 − δ−(v)) 2\n\n).\n\nF¨ uredi rewrote the terms on the right hand side in terms of τ, η, and θ, and showed \n\nτ ≤ n3\n\n12 + O(n2).\n\nBondy has a slight improvement of this result, proving that \n\nτ ≤ 2n3\n\n25.\n\nDefine an α/β-digraph to be a digraph D on vertex set V and edges defined by σ1,..., σ β\n\npermutations (linear orders) on V where the edge ( i, j ) ∈ E ⇐⇒ i < j in at least α of the σi.We say G is a majority digraph if α \n\n> β\n\n> 2/3.", "clean_statement": "Conjecture 6.\n3\n2. (Thomass´ e) If G is a loopless digon-free digraph the maximum number of induced directed 2-edge paths is n3/15 + O(n2).Notes: First, note that n3/15 + O(n2) can be obtained by substituting C4's inside C4's. Next, given a digraph G, let τ be the number of induced directed 2-paths, η the number of induced edges, and θ the number of cyclic triangles. Then if |V (G)| = n,\n\nτ + η/ 3 + 2 θ = n3/12 + O(n2) − 1\n\n6\n\n∑\n\n> v∈V(G)\n\n(\n\n(δ+(v) − δ−(v)) 2 + ( n\n\n2 − δ+(v)) 2 + ( n\n\n2 − δ−(v)) 2\n\n).\n\nF¨ uredi rewrote the terms on the right hand side in terms of τ, η, and θ, and showed\n\nτ ≤ n3\n\n12 + O(n2).\n\nBondy has a slight improvement of this result, proving that\n\nτ ≤ 2n3\n\n25.\n\nDefine an α/β-digraph to be a digraph D on vertex set V and edges defined by σ1,..., σ β\n\npermutations (linear orders) on V where the edge ( i, j ) ∈ E ⇐⇒ i < j in at least α of the σi.We say G is a majority digraph if α\n\n> β\n\n> 2/3.", "statement_status": "exact", "statement_verification": "The primary AIM PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[201]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n3\\n2. (Thomass´ e) If G is a loopless digon-free digraph the maximum number of induced directed 2-edge paths is n3/15 + O(n2).Notes: First, note that n3/15 + O(n2) can be obtained by substituting C4's inside C4's. Next, given a digraph G, let τ be the number of induced directed 2-paths, η the number of induced edges, and θ the number of cyclic triangles. Then if |V (G)| = n,\\n\\nτ + η/ 3 + 2 θ = n3/12 + O(n2) − 1\\n\\n6\\n\\n∑ \\n\\n> v∈V(G)\\n\\n(\\n\\n(δ+(v) − δ−(v)) 2 + ( n\\n\\n2 − δ+(v)) 2 + ( n\\n\\n2 − δ−(v)) 2\\n\\n).\\n\\nF¨ uredi rewrote the terms on the right hand side in terms of τ, η, and θ, and showed \\n\\nτ ≤ n3\\n\\n12 + O(n2).\\n\\nBondy has a slight improvement of this result, proving that \\n\\nτ ≤ 2n3\\n\\n25.\\n\\nDefine an α/β-digraph to be a digraph D on vertex set V and edges defined by σ1,..., σ β\\n\\npermutations (linear orders) on V where the edge ( i, j ) ∈ E ⇐⇒ i < j in at least α of the σi.We say G is a majority digraph if α \\n\\n> β\\n\\n> 2/3.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0202", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every finite simple oriented graph, the degree-square formula in the AIM note is an exact identity (with no asymptotic error) when eta counts three-sets inducing one arc and an isolated vertex. For the balanced iterated directed-C4 construction B_n, the induced directed two-edge-path count satisfies an exact four-way recurrence. Its defect from (n^3-n)/15 is nonnegative and at most (n/5) ceiling(log_4 n), so the construction has n^3/15-O(n log n) copies for every n and exactly (n^3-n)/15 when n is a power of four. This is a construction theorem and does not prove the conjectured general upper bound.\n\nCandidate contribution (exact recurrence and quantitative construction bound; novelty confidence low): The balanced recursive directed-C4 construction has the explicit base-four defect recurrence displayed in the artifacts and loses at most (n/5) ceiling(log_4 n) induced directed paths relative to (n^3-n)/15 for every n.", "research_difficulty_suggested": "L4" }, { "id": 20001078, "problem_number": "AIM-COMBINATORICS-0203", "title": "Defect feedback sets and a fractional obstruction for four-order majority digraphs", "statement": "Conjecture 6.\n3\n3. (Thomass´ e & Charbit) Caccetta-H¨ aggkvist holds for 3/4-digraphs. Notes: Majority digraphs have no cycles of length at most three. The class of 3/4-digraphs is stable under substitution, and contains the circular interval graph on 3 k + 1 vertices with outdegree k going clockwise. The class includes all known extremal examples for C-H, yet this class of 3/4-digraphs seems manageably small. It is still open whether or not 3/4 digraphs must have vertices x of outdegree less than n/ 3. One can more generally ask if Caccetta-H¨ aggkvist holds for larger classes of majority digraphs. 11", "original_statement": "Conjecture 6.\n3\n3. (Thomass´ e & Charbit) Caccetta-H¨ aggkvist holds for 3/4-digraphs. Notes: Majority digraphs have no cycles of length at most three. The class of 3/4-digraphs is stable under substitution, and contains the circular interval graph on 3 k + 1 vertices with outdegree k going clockwise. The class includes all known extremal examples for C-H, yet this class of 3/4-digraphs seems manageably small. It is still open whether or not 3/4 digraphs must have vertices x of outdegree less than n/ 3. One can more generally ask if Caccetta-H¨ aggkvist holds for larger classes of majority digraphs. 11", "clean_statement": "Conjecture 6.\n3\n3. (Thomass´ e & Charbit) Caccetta-H¨ aggkvist holds for 3/4-digraphs. Notes: Majority digraphs have no cycles of length at most three. The class of 3/4-digraphs is stable under substitution, and contains the circular interval graph on 3 k + 1 vertices with outdegree k going clockwise. The class includes all known extremal examples for C-H, yet this class of 3/4-digraphs seems manageably small. It is still open whether or not 3/4 digraphs must have vertices x of outdegree less than n/ 3. One can more generally ask if Caccetta-H¨ aggkvist holds for larger classes of majority digraphs. 11", "statement_status": "exact", "statement_verification": "The canonical AIM record reads, after repairing line-break and accent OCR:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[202]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n3\\n3. (Thomass´ e & Charbit) Caccetta-H¨ aggkvist holds for 3/4-digraphs. Notes: Majority digraphs have no cycles of length at most three. The class of 3/4-digraphs is stable under substitution, and contains the circular interval graph on 3 k + 1 vertices with outdegree k going clockwise. The class includes all known extremal examples for C-H, yet this class of 3/4-digraphs seems manageably small. It is still open whether or not 3/4 digraphs must have vertices x of outdegree less than n/ 3. One can more generally ask if Caccetta-H¨ aggkvist holds for larger classes of majority digraphs. 11\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0203", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a digraph represented by four orders at threshold three, the four unique-dissenting-order arc classes are disjoint feedback arc sets, so every directed cycle uses all four classes and every directed 4-cycle is rainbow. The class is closed under synchronized substitution with an exact outdegree formula, and the conjectured strict n/3 outdegree bound is equivalent to excluding a supported probability vector that gives every support vertex outneighbourhood mass at least 1/3. Representations containing an equal or reversed pair of orders are acyclic.\n\nCandidate contribution (structural reduction; novelty confidence low): The combined defect-color and blow-up argument gives an exact, testable fractional obstruction: the four-order conjecture fails if and only if some four-order digraph has a supported probability vector p with p(N+(v)) at least 1/3 for every v in its support, while each of the four defect-color classes is a feedback arc set and every directed 4-cycle is rainbow.", "research_difficulty_suggested": "L4" }, { "id": 20001079, "problem_number": "AIM-COMBINATORICS-0204", "title": "Thomasse's path-girth conjecture: refutation and exact cyclic blow-ups", "statement": "Conjecture 6.\n3\n4. (Thomass´ e) Every digraph D has a path of length δ+\n\n> D\n\n(g − 1), where g is the girth of D.Note: This is open for g = 3, and implies Caccetta-H¨ aggkvist.", "original_statement": "Conjecture 6.\n3\n4. (Thomass´ e) Every digraph D has a path of length δ+\n\n> D\n\n(g − 1), where g is the girth of D.Note: This is open for g = 3, and implies Caccetta-H¨ aggkvist.", "clean_statement": "Conjecture 6.\n3\n4. (Thomass´ e) Every digraph D has a path of length δ+\n\n> D\n\n(g − 1), where g is the girth of D.Note: This is open for g = 3, and implies Caccetta-H¨ aggkvist.", "statement_status": "exact", "statement_verification": "The canonical JSON is visibly damaged by PDF extraction: the conjecture number is split across lines as 6.34, and the superscript and subscript in the degree symbol are detached. Page 11 of the primary AIM PDF reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[203]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n3\\n4. (Thomass´ e) Every digraph D has a path of length δ+\\n\\n> D\\n\\n(g − 1), where g is the girth of D.Note: This is open for g = 3, and implies Caccetta-H¨ aggkvist.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0204", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The OCR-damaged AIM statement is the arc-length inequality ell(D) >= delta^+(D)(g(D)-1). The conjecture is now known to be false for every girth g >= 4, while the g=3 oriented-graph case remains open. This run gives a self-contained simple oriented girth-four family X_d with minimum outdegree d and exact longest-path length 2d+2 < 3d for every d >= 3. It also proves an exact positive theorem: a complete nonuniform blow-up of a directed m-cycle with layer sizes a_i, minimum s, and longest cyclic run R of layers larger than s has longest-path length sm+R-1, so it satisfies Thomasse's target with equality only for the bare cycle.\n\nCandidate contribution (exact_formula; novelty confidence low): Candidate novelty: for every complete nonuniform cyclic blow-up B(a_0,...,a_{m-1}), if s=min_i a_i and R is the maximum cyclically consecutive run of layers with size at least s+1, then ell(B)=sm+R-1; consequently equality in Thomasse's inequality inside this family occurs exactly when all a_i=1." }, { "id": 20001080, "problem_number": "AIM-COMBINATORICS-0205", "title": "Convention obstructions and an oriented-pseudoforest theorem", "statement": "Conjecture 6.\n3\n5. (Thomass´ e) In a digraph D,\n\n∑\n\n> v∈V(D)\n\n|δ+\n\n> D\n\n(v) − δ−\n\n> D\n\n(v)| + |{ (u, v )|d(u, v ) ≤ 2}| ≥ 2|E(D)| + |{ v|δ+\n\n> D\n\n(v) > δ −\n\n> D\n\n(v)}|.\n\nNote: This is exact for transitive tournaments.", "original_statement": "Conjecture 6.\n3\n5. (Thomass´ e) In a digraph D,\n\n∑ \n\n> v∈V(D)\n\n|δ+\n\n> D\n\n(v) − δ− \n\n> D\n\n(v)| + |{ (u, v )|d(u, v ) ≤ 2}| ≥ 2|E(D)| + |{ v|δ+\n\n> D\n\n(v) > δ − \n\n> D\n\n(v)}|.\n\nNote: This is exact for transitive tournaments.", "clean_statement": "Conjecture 6.\n3\n5. (Thomass´ e) In a digraph D,\n\n∑\n\n> v∈V(D)\n\n|δ+\n\n> D\n\n(v) − δ−\n\n> D\n\n(v)| + |{ (u, v )|d(u, v ) ≤ 2}| ≥ 2|E(D)| + |{ v|δ+\n\n> D\n\n(v) > δ −\n\n> D\n\n(v)}|.\n\nNote: This is exact for transitive tournaments.", "statement_status": "exact", "statement_verification": "The primary AIM PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[204]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n3\\n5. (Thomass´ e) In a digraph D,\\n\\n∑ \\n\\n> v∈V(D)\\n\\n|δ+\\n\\n> D\\n\\n(v) − δ− \\n\\n> D\\n\\n(v)| + |{ (u, v )|d(u, v ) ≤ 2}| ≥ 2|E(D)| + |{ v|δ+\\n\\n> D\\n\\n(v) > δ − \\n\\n> D\\n\\n(v)}|.\\n\\nNote: This is exact for transitive tournaments.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0205", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the general digraph convention explicitly defined in the AIM source, Conjecture 6.35 is false: a two-vertex digon violates it, and a doubled directed triangle shows that forbidding digons without requiring simplicity is still insufficient. The transitive-tournament equality note forces the distance count to use ordered pairs of distinct vertices. For the likely repaired class of finite simple oriented graphs, the inequality is equivalent to q(D)+S(D) >= m+p(D). An exact nonnegative pendant-vertex recurrence proves this repaired inequality for every oriented pseudoforest and gives a recursive equality classification. No general proof for arbitrary oriented graphs is claimed.\n\nCandidate contribution (special-family theorem and equality classification; novelty confidence low): For every finite simple oriented pseudoforest, the repaired Thomasse inequality holds; its slack changes under a pendant attachment by the exact piecewise formulas in the artifacts, and equality is characterized by zero-slack isolated or cycle cores together with zero-increment pendant attachments." }, { "id": 20001081, "problem_number": "AIM-COMBINATORICS-0206", "title": "The prime maximal Z15 boundary counterexample", "statement": "Conjecture 6.\n3\n6. (Thomass´ e) Let G be a digraph on n vertices with minimum outdegree at least 4n/ 15 so that G is maximal with no cycles of length at most three, and has no homogeneous set (in other words, G cannot be obtained by substitution). Then G is a Cayley graph on 3k + 1\n\nvertices with 1,..., k as generators for some k (i.e. the circular interval graph with everyone joined to next k clockwise). Note: Z/15 Z with generators S = {1, 2, 4, 8} gives exactly 4 n/ 15.\n\n#Acknowledgements\n\nThis research was performed while on appointment as a U.S. Department of Homeland Security (DHS) Fellow under the DHS Scholarship and Fellowship Program, a program administered by the Oak Ridge Institute for Science and Education (ORISE) for DHS through an interagency agreement with the U.S Department of Energy (DOE). ORISE is managed by Oak Ridge Asso-ciated Universities under DOE contract number DE-AC05-06OR23100. All opinions expressed in this paper are the author's and do not necessarily reflect the policies and views of DHS, DOE, or ORISE.", "original_statement": "Conjecture 6.\n3\n6. (Thomass´ e) Let G be a digraph on n vertices with minimum outdegree at least 4n/ 15 so that G is maximal with no cycles of length at most three, and has no homogeneous set (in other words, G cannot be obtained by substitution). Then G is a Cayley graph on 3k + 1 \n\nvertices with 1,..., k as generators for some k (i.e. the circular interval graph with everyone joined to next k clockwise). Note: Z/15 Z with generators S = {1, 2, 4, 8} gives exactly 4 n/ 15. \n\n#Acknowledgements \n\nThis research was performed while on appointment as a U.S. Department of Homeland Security (DHS) Fellow under the DHS Scholarship and Fellowship Program, a program administered by the Oak Ridge Institute for Science and Education (ORISE) for DHS through an interagency agreement with the U.S Department of Energy (DOE). ORISE is managed by Oak Ridge Asso-ciated Universities under DOE contract number DE-AC05-06OR23100. All opinions expressed in this paper are the author's and do not necessarily reflect the policies and views of DHS, DOE, or ORISE.", "clean_statement": "Conjecture 6.\n3\n6. (Thomass´ e) Let G be a digraph on n vertices with minimum outdegree at least 4n/ 15 so that G is maximal with no cycles of length at most three, and has no homogeneous set (in other words, G cannot be obtained by substitution). Then G is a Cayley graph on 3k + 1\n\nvertices with 1,..., k as generators for some k (i.e. the circular interval graph with everyone joined to next k clockwise). Note: Z/15 Z with generators S = {1, 2, 4, 8} gives exactly 4 n/ 15.\n\n#Acknowledgements\n\nThis research was performed while on appointment as a U.S. Department of Homeland Security (DHS) Fellow under the DHS Scholarship and Fellowship Program, a program administered by the Oak Ridge Institute for Science and Education (ORISE) for DHS through an interagency agreement with the U.S Department of Energy (DOE). ORISE is managed by Oak Ridge Asso-ciated Universities under DOE contract number DE-AC05-06OR23100. All opinions expressed in this paper are the author's and do not necessarily reflect the policies and views of DHS, DOE, or ORISE.", "statement_status": "exact", "statement_verification": "The primary source is page 11 of Blair D. Sullivan's 2006 AIM report. It prints the following statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Caccetta-Haggkvist conjecture\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/caccetta/caccetta.pdf\nCanonical location: aim-combinatorics-notes.json notes[205]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 6.\\n3\\n6. (Thomass´ e) Let G be a digraph on n vertices with minimum outdegree at least 4n/ 15 so that G is maximal with no cycles of length at most three, and has no homogeneous set (in other words, G cannot be obtained by substitution). Then G is a Cayley graph on 3k + 1 \\n\\nvertices with 1,..., k as generators for some k (i.e. the circular interval graph with everyone joined to next k clockwise). Note: Z/15 Z with generators S = {1, 2, 4, 8} gives exactly 4 n/ 15. \\n\\n#Acknowledgements \\n\\nThis research was performed while on appointment as a U.S. Department of Homeland Security (DHS) Fellow under the DHS Scholarship and Fellowship Program, a program administered by the Oak Ridge Institute for Science and Education (ORISE) for DHS through an interagency agreement with the U.S Department of Energy (DOE). ORISE is managed by Oak Ridge Asso-ciated Universities under DOE contract number DE-AC05-06OR23100. All opinions expressed in this paper are the author's and do not necessarily reflect the policies and views of DHS, DOE, or ORISE.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/caccetta/caccetta.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0206", "aim-domain:combinatorics", "aim-workshop:caccetta", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The Cayley digraph Cay(Z/15Z,{1,2,4,8}) satisfies every hypothesis of Conjecture 6.36 as literally printed: it is 4-regular, has no directed cycle of length at most three, is globally arc-maximal 3-free, and has no nontrivial directed module. Since its order 15 is not 3k+1 for any integer k, it refutes the printed at-least-4n/15 formulation. Replacing the threshold by strict inequality excludes this example and changes the integral hypothesis only when 15 divides n; that repaired conjecture remains open.\n\nCandidate contribution (counterexample certificate; novelty confidence low): The source's noted boundary graph is upgraded to a complete falsification certificate: exact sumset identities prove global arc-maximality, and a three-orbit forced-signature closure table proves that Cay(Z/15Z,{1,2,4,8}) is prime; moreover the difference between the printed non-strict threshold and the likely strict repair occurs exactly at orders divisible by 15." }, { "id": 20001082, "problem_number": "AIM-COMBINATORICS-0207", "title": "Sharp threshold and finite torsion certificates for torus difference sets", "statement": "Problem 1.1 (Y. Katznelson). Given a constant a > 0, does there exist d0 = d0(a)such that for any integer d > d 0 and any closed subset A of the d-dimensional torus group with the measure μ(A) > a, the difference set A − A contains a subgroup?", "original_statement": "Problem 1.1 (Y. Katznelson). Given a constant a > 0, does there exist d0 = d0(a)such that for any integer d > d 0 and any closed subset A of the d-dimensional torus group with the measure μ(A) > a, the difference set A − A contains a subgroup?", "clean_statement": "Problem 1.1 (Y. Katznelson). Given a constant a > 0, does there exist d0 = d0(a)such that for any integer d > d 0 and any closed subset A of the d-dimensional torus group with the measure μ(A) > a, the difference set A − A contains a subgroup?", "statement_status": "exact", "statement_verification": "The source record is Katznelson's Problem 1.1 from the AIM workshop notes on additive combinatorics:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.1\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[206]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.1 (Y. Katznelson). Given a constant a > 0, does there exist d0 = d0(a)such that for any integer d > d 0 and any closed subset A of the d-dimensional torus group with the measure μ(A) > a, the difference set A − A contains a subgroup?\"\nOriginal remarks: [\"Remark(s). For a > 0.5 this is trivial as in this case A − A is the whole group, the case a < 0.5 is open. Katznelson adds: Bourgain observes that the answer is no. The key is a construction of Ruzsa's [Arithmetic progressions in sumsets. Acta Arith. 60 (1991), no. 2, 191-202] which produces, for arbitrary ε > 0 and prime p > p 0(ε), sequences in Zp, of density \\n\\n> 1/2 − ε, such that A − A contains no arithmetic progression of length exp((log p) 2 \\n\\n> 3+ ε\\n\\n). Now given any d, one can take large p and roll Zp into Td properly, replace the points in Ruzsa's set by appropriate cubes, and obtain a set Ω In Td of measure close to 1 /2and such that Ω − Ω contains no infinite subgroup. The (still open) \\\"real problem\\\" that motivated me [the presenter] to raise the, now answered, question is the following. Given Λ ⊂ N, denote by χ(Λ) = χ (ZΛ) the chromatic number of the Cayley graph \\n\\nZΛ. Is it true that χ(Λ) = ∞ if, and only if, Λ is arithmetically rich enough to satisfy Dirichlet's theorem? In terms of recurrence the question is: Is topological recurrence equivalent to Bohr recurrence (recurrence for rigid translations on tori)? For background see [Y. Katznelson, Chromatic numbers of Cayley graphs on Z and recurrence, Combinatorica, 21:211-219, 2001.] Can be seen also at http://math.stanford.edu/˜katznel/erdosvol.pdf\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0207", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the intended infinite- or one-dimensional-subgroup interpretation, Bourgain's result as recorded by the AIM notes and Green gives the exact answer: d_0(a) exists exactly when a is at least 1/2; below 1/2 there are counterexamples in every dimension, while measure greater than 1/2 forces A-A to equal the whole torus. In addition, an exact proved finite-to-torus reduction shows that cubically thickening E in F_p^d gives a closed set of measure rho^d|E|/p^d whose p-torsion difference trace is exactly E-E; if that finite difference set has no nonzero line, the torus difference set has no infinite subgroup. A sharp 2^{-d} theorem is also proved for Cartesian-product sets.\n\nCandidate contribution (reduction; novelty confidence low): For every prime p, E subset F_p^d, and 0 δ |G0|,then there exist equitable partitions\n\nA = A1 ∪ · · · ∪ As, B = B1 ∪ · · · ∪ Bt,\n\nwhere s, t < C (≤, δ ), such that for (1 − ≤)st of the pairs ( i, j ) the restriction of G to\n\nAi × Bj is ≤-regular relative to G0.\n\n> 12COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\n\nThe presenter remarks that despite being a generalization of the already rather diffi-cult hypergraph regularity lemmas, if done correctly the proof of such a result may be\n\neasier than that of the existing regularity lemmas. This is because the induction used to prove such lemmas may be cleaner.", "original_statement": "Problem 1.2 (T. Tao). Is there a hypergraph regularity lemma for subsets of pseu-dorandom sparse hypergraphs of large density? If so, it would give a new proof that there are arbitrarily long arithmetic progressions among the primes, which may possi-bly extend to a more general situation. The following analogue for graphs is known: If \n\n|A| = |B| = N, G0 ⊆ A × B is \"sparsely c(≤, δ )-quasirandom\", G ⊂ G0, |G| > δ |G0|,then there exist equitable partitions \n\nA = A1 ∪ · · · ∪ As, B = B1 ∪ · · · ∪ Bt,\n\nwhere s, t < C (≤, δ ), such that for (1 − ≤)st of the pairs ( i, j ) the restriction of G to \n\nAi × Bj is ≤-regular relative to G0. \n\n> 12COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\n\nThe presenter remarks that despite being a generalization of the already rather diffi-cult hypergraph regularity lemmas, if done correctly the proof of such a result may be \n\neasier than that of the existing regularity lemmas. This is because the induction used to prove such lemmas may be cleaner.", "clean_statement": "Problem 1.2 (T. Tao). Is there a hypergraph regularity lemma for subsets of pseu-dorandom sparse hypergraphs of large density? If so, it would give a new proof that there are arbitrarily long arithmetic progressions among the primes, which may possi-bly extend to a more general situation. The following analogue for graphs is known: If\n\n|A| = |B| = N, G0 ⊆ A × B is \"sparsely c(≤, δ )-quasirandom\", G ⊂ G0, |G| > δ |G0|,then there exist equitable partitions\n\nA = A1 ∪ · · · ∪ As, B = B1 ∪ · · · ∪ Bt,\n\nwhere s, t < C (≤, δ ), such that for (1 − ≤)st of the pairs ( i, j ) the restriction of G to\n\nAi × Bj is ≤-regular relative to G0.\n\n> 12COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\n\nThe presenter remarks that despite being a generalization of the already rather diffi-cult hypergraph regularity lemmas, if done correctly the proof of such a result may be\n\neasier than that of the existing regularity lemmas. This is because the induction used to prove such lemmas may be cleaner.", "statement_status": "exact", "statement_verification": "This is Problem 1.2, attributed to T. Tao, in the AIM workshop list *Recent trends in additive combinatorics* (collected by Ernie Croot and Vsevolod F. Lev). It asks whether a subset of large relative density in a pseudorandom sparse hypergraph satisfies a hypergraph regularity lemma, and observes that such a result could reprove the existence of arbitrarily long arithmetic progressions in the primes.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.2\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[207]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.2 (T. Tao). Is there a hypergraph regularity lemma for subsets of pseu-dorandom sparse hypergraphs of large density? If so, it would give a new proof that there are arbitrarily long arithmetic progressions among the primes, which may possi-bly extend to a more general situation. The following analogue for graphs is known: If \\n\\n|A| = |B| = N, G0 ⊆ A × B is \\\"sparsely c(≤, δ )-quasirandom\\\", G ⊂ G0, |G| > δ |G0|,then there exist equitable partitions \\n\\nA = A1 ∪ · · · ∪ As, B = B1 ∪ · · · ∪ Bt,\\n\\nwhere s, t < C (≤, δ ), such that for (1 − ≤)st of the pairs ( i, j ) the restriction of G to \\n\\nAi × Bj is ≤-regular relative to G0. \\n\\n> 12COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\\n\\nThe presenter remarks that despite being a generalization of the already rather diffi-cult hypergraph regularity lemmas, if done correctly the proof of such a result may be \\n\\neasier than that of the existing regularity lemmas. This is because the induction used to prove such lemmas may be cleaner.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0208", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The intended arithmetic application is known under explicit linear-forms hypotheses, but the AIM wording is under-specified for arbitrary pseudorandom hosts. Independently of raw host sparsity, a normalized host-mass energy increment gives an edge-space partition into at most 2^{ceil(epsilon^{-4})} cells with irregular cells of total host mass at most epsilon. This cannot by itself yield sparse counting: the density-q^{-1} host x+y+z=0 over F_q has exactly uniform conditional expectations on every proper coordinate set but has a nontrivial three-character correlation and zero nondegenerate cherry count, versus 1-q^{-1} in the dense benchmark.\n\nCandidate contribution (lemma_and_obstruction; novelty confidence low): Candidate normalization-and-obstruction boundary: arbitrary normalized sparse hosts admit an epsilon-weak edge-space regularization with at most 2^{ceil(epsilon^{-4})} cells when exceptional mass is measured by the host, while the hosts q 1_{x+y+z=0} have perfect proper-coordinate marginals but fail a nondegenerate two-edge count maximally." }, { "id": 20001084, "problem_number": "AIM-COMBINATORICS-0209", "title": "Sliding progression-free windows via gap words", "statement": "Problem 1.3 (G. Freiman). Fix an integer s ≥ 3 and suppose that A = ( a1, a 2,... ) is a strictly increasing sequence of integers such that no segment of this sequence of the form ( ai+1, a i+2,..., a i+s) ( i = 0, 1,... ) contains a three-term arithmetic progression. How large can the density of A be under this assumption? Describe all extremal sets. Examples: 1) For s = 4 one can take A = (0, 1, 3, 4, 6, 7, 9, 10,... ) (all non-negative integers congruent to any of 0, 1, 3, and 4 modulo six) with the density 2 /3; 2) for s = 8 one can take A = (0, 1, 3, 4, 9, 10, 12, 13, 18, 19, 21,... ) (all non-negative integers congruent to any of 0, 1, 3, 4, 9, 10, 12, and 13 modulo eighteen) with the density 4 /9.", "original_statement": "Problem 1.3 (G. Freiman). Fix an integer s ≥ 3 and suppose that A = ( a1, a 2,... ) is a strictly increasing sequence of integers such that no segment of this sequence of the form ( ai+1, a i+2,..., a i+s) ( i = 0, 1,... ) contains a three-term arithmetic progression. How large can the density of A be under this assumption? Describe all extremal sets. Examples: 1) For s = 4 one can take A = (0, 1, 3, 4, 6, 7, 9, 10,... ) (all non-negative integers congruent to any of 0, 1, 3, and 4 modulo six) with the density 2 /3; 2) for s = 8 one can take A = (0, 1, 3, 4, 9, 10, 12, 13, 18, 19, 21,... ) (all non-negative integers congruent to any of 0, 1, 3, 4, 9, 10, 12, and 13 modulo eighteen) with the density 4 /9.", "clean_statement": "Problem 1.3 (G. Freiman). Fix an integer s ≥ 3 and suppose that A = ( a1, a 2,... ) is a strictly increasing sequence of integers such that no segment of this sequence of the form ( ai+1, a i+2,..., a i+s) ( i = 0, 1,... ) contains a three-term arithmetic progression. How large can the density of A be under this assumption? Describe all extremal sets. Examples: 1) For s = 4 one can take A = (0, 1, 3, 4, 6, 7, 9, 10,... ) (all non-negative integers congruent to any of 0, 1, 3, and 4 modulo six) with the density 2 /3; 2) for s = 8 one can take A = (0, 1, 3, 4, 9, 10, 12, 13, 18, 19, 21,... ) (all non-negative integers congruent to any of 0, 1, 3, 4, 9, 10, 12, and 13 modulo eighteen) with the density 4 /9.", "statement_status": "exact", "statement_verification": "The primary AIM workshop PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.3\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[208]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.3 (G. Freiman). Fix an integer s ≥ 3 and suppose that A = ( a1, a 2,... ) is a strictly increasing sequence of integers such that no segment of this sequence of the form ( ai+1, a i+2,..., a i+s) ( i = 0, 1,... ) contains a three-term arithmetic progression. How large can the density of A be under this assumption? Describe all extremal sets. Examples: 1) For s = 4 one can take A = (0, 1, 3, 4, 6, 7, 9, 10,... ) (all non-negative integers congruent to any of 0, 1, 3, and 4 modulo six) with the density 2 /3; 2) for s = 8 one can take A = (0, 1, 3, 4, 9, 10, 12, 13, 18, 19, 21,... ) (all non-negative integers congruent to any of 0, 1, 3, 4, 9, 10, 12, and 13 modulo eighteen) with the density 4 /9.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0209", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For upper asymptotic density, the problem is exactly equivalent to minimizing the lower mean of a positive-integer gap word in which no two adjacent nonempty subwords of combined length at most s-1 have equal sums. This gives the sharp maximum 2/3 for s=3 and s=4 and classifies periodic extremizers there. More generally, any r-element 3AP-free set B contained in [0,D], repeated modulo M>2D, is valid through window size 2r and has density r/M, explaining both AIM examples. For every fixed gap cap M, the exact optimal density is the reciprocal of a finite graph's minimum cycle mean and is attained periodically.\n\nCandidate contribution (construction theorem and finite-state reduction; novelty confidence low): If B is an r-element 3AP-free subset of [0,D] and M>2D, then B+M times the nonnegative integers is s-admissible for every s at most 2r, with density r/M; in parallel, every fixed-M bounded-gap version is exactly a minimum-cycle-mean problem on at most M^(s-2) legal suffix states." }, { "id": 20001085, "problem_number": "AIM-COMBINATORICS-0210", "title": "Exact quadratic-residue half-set counts and a quantitative Varnavides transfer", "statement": "Problem 1.4 (B. Green). For a prime p, what is the least number of three-term arithmetic progressions that a subset A ⊂ Fp with |A| = ( p − 1) /2 can have? What happens if |A| = δp where δ < 0.5?", "original_statement": "Problem 1.4 (B. Green). For a prime p, what is the least number of three-term arithmetic progressions that a subset A ⊂ Fp with |A| = ( p − 1) /2 can have? What happens if |A| = δp where δ < 0.5?", "clean_statement": "Problem 1.4 (B. Green). For a prime p, what is the least number of three-term arithmetic progressions that a subset A ⊂ Fp with |A| = ( p − 1) /2 can have? What happens if |A| = δp where δ < 0.5?", "statement_status": "exact", "statement_verification": "The AIM list *Recent trends in additive combinatorics*, Problem 1.4 (attributed to B. Green), asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.4\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[209]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.4 (B. Green). For a prime p, what is the least number of three-term arithmetic progressions that a subset A ⊂ Fp with |A| = ( p − 1) /2 can have? What happens if |A| = δp where δ < 0.5?\"\nOriginal remarks: [\"Remark(s). As it follows from a result by Varnavides, this number is at least cp 2\\n\\nwith some c = c(δ), and Croot has recently shown that it is in fact cp 2(1 + o(1)) as \\n\\np → ∞. It seems to be a difficult problem to determine the rough order of magnitude of the constant c as δ → 0.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0210", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "With ordered progressions (x,d), including d=0, the nonzero quadratic residues Q in F_p have exactly T_p(Q)=(p-1)(p-2-2 chi(2)-chi(-1))/8 progressions. This yields four explicit mod-8 formulas and beats the exact fixed-cardinality random benchmark for p congruent to 1 or 7 modulo 8, giving a deterministic half-density upper-bound family without claiming optimality. The report also proves the exact complement identity and an explicit Varnavides incidence lemma which, combined with the current 2026 Roth bound, gives a quantitative small-density lower bound.\n\nCandidate contribution (explicit_family; novelty confidence low): The exact mod-8 quadratic-residue progression formula, paired with the exact hypergeometric half-density benchmark, gives an explicit deterministic construction strictly below that random benchmark for every prime p congruent to 1 or 7 modulo 8 (p at least 7)." }, { "id": 20001086, "problem_number": "AIM-COMBINATORICS-0211", "title": "A missing threshold and an explicit Freiman-model repair", "statement": "Problem 1.5 (N. Katz). Fix integer k ≥ 3 and real C ≥ 2. Given that A is a finite set of integers with |A + A| < C |A|, is it true that the number of k-term arithmetic progressions in A is at least c|A|2 with a positive constant c depending on k and C\n\nonly? B. Green comments: The answer to this is surely \"yes\". A (or a large subset of it) is Freiman isomorphic to a dense subset of Z/p Z by a lemma of Ruzsa. Now apply Szemeredi's theorem. J. Solymosi mentions that Katz's question was part of an induction step in his alter-native proof of the Balog-Szemeredi theorem.", "original_statement": "Problem 1.5 (N. Katz). Fix integer k ≥ 3 and real C ≥ 2. Given that A is a finite set of integers with |A + A| < C |A|, is it true that the number of k-term arithmetic progressions in A is at least c|A|2 with a positive constant c depending on k and C\n\nonly? B. Green comments: The answer to this is surely \"yes\". A (or a large subset of it) is Freiman isomorphic to a dense subset of Z/p Z by a lemma of Ruzsa. Now apply Szemeredi's theorem. J. Solymosi mentions that Katz's question was part of an induction step in his alter-native proof of the Balog-Szemeredi theorem.", "clean_statement": "Problem 1.5 (N. Katz). Fix integer k ≥ 3 and real C ≥ 2. Given that A is a finite set of integers with |A + A| < C |A|, is it true that the number of k-term arithmetic progressions in A is at least c|A|2 with a positive constant c depending on k and C\n\nonly? B. Green comments: The answer to this is surely \"yes\". A (or a large subset of it) is Freiman isomorphic to a dense subset of Z/p Z by a lemma of Ruzsa. Now apply Szemeredi's theorem. J. Solymosi mentions that Katz's question was part of an induction step in his alter-native proof of the Balog-Szemeredi theorem.", "statement_status": "exact", "statement_verification": "The primary AIM workshop sheet states (with its mathematical typography restored):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.5\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[210]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.5 (N. Katz). Fix integer k ≥ 3 and real C ≥ 2. Given that A is a finite set of integers with |A + A| < C |A|, is it true that the number of k-term arithmetic progressions in A is at least c|A|2 with a positive constant c depending on k and C\\n\\nonly? B. Green comments: The answer to this is surely \\\"yes\\\". A (or a large subset of it) is Freiman isomorphic to a dense subset of Z/p Z by a lemma of Ruzsa. Now apply Szemeredi's theorem. J. Solymosi mentions that Katz's question was part of an induction step in his alter-native proof of the Balog-Szemeredi theorem.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0211", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the standard convention that a k-term arithmetic progression has nonzero common difference, the literal all-finite-set statement is false: A={0,...,k-2} has |A+A|=2k-3<2|A| but no k-term progression. The intended asymptotic assertion is true. If L is a Szemeredi threshold at density 1/(8C^4), then every A subset of the integers with |A| at least 2L and |A+A| at most C|A| contains at least |A|^2/(128 C^4 L^2) increasing k-term progressions. The proof models at least half of A densely in a prime cyclic group, obtains a quadratic cyclic count by Varnavides averaging, and transfers it using only Freiman order 2.\n\nCandidate contribution (quantitative transfer lemma and quantifier audit; novelty confidence low): Let L=S_k(1/(8C^4)). Ruzsa's order-2 model plus an explicit affine-interval averaging argument yields the testable constants N_0=2L and c=1/(128 C^4 L^2); every cyclic k-progression, including one that wraps modulo the model prime, pulls back because its k-2 consecutive second-difference identities are two-sum relations." }, { "id": 20001087, "problem_number": "AIM-COMBINATORICS-0212", "title": "A two-sided transfer principle for the least doubling of 3-AP-free sets", "statement": "Problem 1.6 (G. Freiman). Given that A is an n-element integer set free of three-term arithmetic progressions, how small can |2A| be?", "original_statement": "Problem 1.6 (G. Freiman). Given that A is an n-element integer set free of three-term arithmetic progressions, how small can |2A| be?", "clean_statement": "Problem 1.6 (G. Freiman). Given that A is an n-element integer set free of three-term arithmetic progressions, how small can |2A| be?", "statement_status": "exact", "statement_verification": "Problem 1.6 of the AIM list *Recent trends in additive combinatorics*, attributed to G. Freiman, asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.6\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[211]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.6 (G. Freiman). Given that A is an n-element integer set free of three-term arithmetic progressions, how small can |2A| be?\"\nOriginal remarks: [\"Remark(s). Behrend's construction yields a set A with |2A| ∼ ne c√log n, where c is an absolute constant. Freiman proved that |2A|/n tends to infinity and Ruzsa proved that this quotient is at least ( n/r 3(n)) 1/4, where r3(n) is the size of the largest progression-free subset of [ n].\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0212", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Writing D_3(n) for the minimum of |A+A| over n-element 3-AP-free integer sets, r_3(N) for the interval extremal function, and rho_3(n)=min{N:r_3(N)>=n}, the report proves the explicit transfer bounds 20^{-1/4} n(n/r_3(n))^{1/4} <= D_3(n) <= 2 rho_3(n)-1. It also rigorously inverts Behrend-type construction bounds without changing the sign or leading square-root constant. Combining the lower transfer with Raghavan's March 2026 preprint, including its odd-modulus restriction, gives the announced current lower bound D_3(n) >= n exp(c (log n)^{1/6}/log log n); the best construction cited gives D_3(n) <= n exp((2 sqrt(log(24/7))+o(1)) sqrt(log n)). The wide gap remains open.\n\nCandidate contribution (transfer_theorem; novelty confidence low): The explicit constant-tracked two-sided dictionary 20^{-1/4} n(n/r_3(n))^{1/4} <= D_3(n) <= 2 rho_3(n)-1, together with an audited inversion lemma and the parity-safe transfer of the March 2026 odd-N estimate, is a concrete candidate synthesis not found verbatim in the literature checked." }, { "id": 20001088, "problem_number": "AIM-COMBINATORICS-0213", "title": "Cap-set status and a sharp quadratic-graph dimension barrier", "statement": "Problem 1.7 (Brought by T. Tao). For a positive integer r, what is the largest possible size of a subset A ⊆ Fr\n\n> 3\n\ncontaining no three points on a line?", "original_statement": "Problem 1.7 (Brought by T. Tao). For a positive integer r, what is the largest possible size of a subset A ⊆ Fr \n\n> 3\n\ncontaining no three points on a line?", "clean_statement": "Problem 1.7 (Brought by T. Tao). For a positive integer r, what is the largest possible size of a subset A ⊆ Fr\n\n> 3\n\ncontaining no three points on a line?", "statement_status": "exact", "statement_verification": "The AIM workshop list *Recent trend in additive combinatorics*, Problem 1.7 (brought by T. Tao), asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.7\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[212]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.7 (Brought by T. Tao). For a positive integer r, what is the largest possible size of a subset A ⊆ Fr \\n\\n> 3\\n\\ncontaining no three points on a line?\"\nOriginal remarks: [\"Remark(s). Meshulam has shown that this size is O(3 r/r ); on the other hand, it is easy to construct a set A with the property in question such that |A| = 2 r: just fix PALO ALTO PROBLEMS 3\\n\\narbitrarily a basis {e1,..., e r} of Fr \\n\\n> 3\\n\\nover F3 and let A:= {≤1e1 + · · · + ≤rer: ≤1,..., ≤ r ∈{0, 1}}. The best known construction is due to Edel who has constructed sets in A ⊆ Fr\\n\\n> 3\\n\\nof size (2.217... )n containing no three points on a line by finding a particular example in rather large dimension and then taking products of several copies of it.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0213", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a polynomial map q:F_3^n to F_3^m of coordinate degree at most two, with homogeneous quadratic part Q, its full graph is a cap exactly when Q has no nonzero common zero. A self-contained Chevalley-Warning argument shows that this is possible exactly when n is at most 2m, and paired forms x_(2j-1)^2+x_(2j)^2 attain the boundary. Therefore the exact largest full quadratic cap graph in ambient dimension r has 3^floor(2r/3) points. This is a proved restricted-class theorem, not a solution of the global cap-set problem.\n\nCandidate contribution (restricted_class_theorem; novelty confidence low): Full quadratic cap graphs F_3^n to F_3^m exist if and only if n<=2m; after optimizing over all domain/codomain splittings of F_3^r, their exact maximum size is 3^floor(2r/3)." }, { "id": 20001089, "problem_number": "AIM-COMBINATORICS-0214", "title": "An explicit local-lemma certificate and bounded high-overlap neighbourhood for van der Waerden hypergraphs", "statement": "Problem 1.8 (R. Graham). Define W (k) to be the least integer such that in any 2-coloring of the integers {1, 2,..., W (k)} there must always exist a monochromatic k-term arithmetic progression. What is the true order of growth of W (k)? The presenter offers $1000 for the proof that W (k) ≤ 2k2.", "original_statement": "Problem 1.8 (R. Graham). Define W (k) to be the least integer such that in any 2-coloring of the integers {1, 2,..., W (k)} there must always exist a monochromatic k-term arithmetic progression. What is the true order of growth of W (k)? The presenter offers $1000 for the proof that W (k) ≤ 2k2.", "clean_statement": "Problem 1.8 (R. Graham). Define W (k) to be the least integer such that in any 2-coloring of the integers {1, 2,..., W (k)} there must always exist a monochromatic k-term arithmetic progression. What is the true order of growth of W (k)? The presenter offers $1000 for the proof that W (k) ≤ 2k2.", "statement_status": "exact", "statement_verification": "The exact canonical record assigned to this attempt is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.8\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[213]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.8 (R. Graham). Define W (k) to be the least integer such that in any 2-coloring of the integers {1, 2,..., W (k)} there must always exist a monochromatic k-term arithmetic progression. What is the true order of growth of W (k)? The presenter offers $1000 for the proof that W (k) ≤ 2k2.\"\nOriginal remarks: [\"Remark(s). It is known from the work of Gowers that \\n\\nW (k) ≤ 22222k+9,\\n\\nand Berlekamp proved that \\n\\nW (p + 1) ≥ p2p.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0214", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every k at least 3, an elementary symmetric Lovasz local lemma argument proves W(k)>1+floor((k-1)2^(k-1)/(e k^2)). More structurally, for any k-term arithmetic progression P in [N], at most binom(k,2)^2-binom(k,2) other k-term progressions share two or more points with P, independently of N. Thus all but polynomially many events in the dependency neighbourhood of P arise from one-point intersections. This does not improve the known Kozik-Shabanov exponential lower bound, but it gives a proved quantitative decomposition of the near-simple progression hypergraph and identifies the structure discarded by the elementary benchmark.\n\nCandidate contribution (lemma; novelty confidence low): For every edge P of the interval k-term-progression hypergraph, the number of distinct other edges Q with |P intersect Q| at least 2 is at most binom(k,2)^2-binom(k,2); paired with the explicit symmetric-local-lemma certificate, this separates a polynomial-size high-overlap obstruction from the much larger one-point dependency neighbourhood." }, { "id": 20001090, "problem_number": "AIM-COMBINATORICS-0215", "title": "A diameter-free Property B bound for arbitrary van der Waerden sets", "statement": "Problem 1.9 (R. Graham). Define W ∗(k) to be the size of the smallest set X ⊆ Z such that any 2-coloring of X always has a monochromatic k-term arithmetic progression. Then, W ∗(k) ≤ W (k). For $100: Is W (k) − W ∗(k) unbounded as k → ∞? Does lim\n\n> k→∞\n\nW ∗(k)\n\nW (k) = 1?", "original_statement": "Problem 1.9 (R. Graham). Define W ∗(k) to be the size of the smallest set X ⊆ Z such that any 2-coloring of X always has a monochromatic k-term arithmetic progression. Then, W ∗(k) ≤ W (k). For $100: Is W (k) − W ∗(k) unbounded as k → ∞? Does lim \n\n> k→∞\n\nW ∗(k)\n\nW (k) = 1?", "clean_statement": "Problem 1.9 (R. Graham). Define W ∗(k) to be the size of the smallest set X ⊆ Z such that any 2-coloring of X always has a monochromatic k-term arithmetic progression. Then, W ∗(k) ≤ W (k). For $100: Is W (k) − W ∗(k) unbounded as k → ∞? Does lim\n\n> k→∞\n\nW ∗(k)\n\nW (k) = 1?", "statement_status": "exact", "statement_verification": "The stored `problem` field is reproduced verbatim below. It is visibly damaged by PDF extraction around the displayed limit.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.9\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[214]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.9 (R. Graham). Define W ∗(k) to be the size of the smallest set X ⊆ Z such that any 2-coloring of X always has a monochromatic k-term arithmetic progression. Then, W ∗(k) ≤ W (k). For $100: Is W (k) − W ∗(k) unbounded as k → ∞? Does lim \\n\\n> k→∞\\n\\nW ∗(k)\\n\\nW (k) = 1?\"\nOriginal remarks: [\"Remark(s). W ∗(3) = W (3) = 9, W ∗(4) ≤ 27, W (4) = 35.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0215", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The OCR-damaged source is recovered as asking whether W(k)-W*(k) is unbounded and whether W*(k)/W(k) tends to 1; neither question was resolved in the literature checked through 2026-07-30. For the k-AP hypergraph of an arbitrary n-point integer set X, the report proves the diameter-free vertex-degree estimate deg(x) <= k(n-1)/2. The symmetric Lovasz local lemma then gives W*(k) > 2^k/(e k^2), with the r-color extension W*_r(k) > 2 r^(k-1)/(e k^2). A separate balanced-partition argument proves W*(k) >= 2k+1 for k >= 4, and a Freiman-2-isomorphism lemma identifies exactly which transformations preserve the full progression hypergraph.\n\nCandidate contribution (local_lemma_bound; novelty confidence low): The explicit package deg_{H_k(X)}(x) <= k(|X|-1)/2 and W*(k) > 2^k/(e k^2), together with the balanced lower bound W*(k) >= 2k+1 and the Freiman-2-invariance audit, is a concrete candidate contribution for Graham's arbitrary-set parameter." }, { "id": 20001091, "problem_number": "AIM-COMBINATORICS-0216", "title": "Graham's reciprocal-mass square conjecture: a dyadic extremal reduction", "statement": "Problem 1.10 (R. Graham). Let A ⊆ Z × Z satisfy\n\n∑\n\n> (x,y )∈A\n\n1\n\nx2 + y2 = ∞.\n\nConjecture ($1000): A contains the four vertices of a square, i.e. four points of the form (x, y ), (x + a, y ), (x, y + a), and ( x + a, y + a). (More generally, it should be true that\n\nA contains a k × k square grid.)", "original_statement": "Problem 1.10 (R. Graham). Let A ⊆ Z × Z satisfy \n\n∑\n\n> (x,y )∈A\n\n1\n\nx2 + y2 = ∞.\n\nConjecture ($1000): A contains the four vertices of a square, i.e. four points of the form (x, y ), (x + a, y ), (x, y + a), and ( x + a, y + a). (More generally, it should be true that \n\nA contains a k × k square grid.)", "clean_statement": "Problem 1.10 (R. Graham). Let A ⊆ Z × Z satisfy\n\n∑\n\n> (x,y )∈A\n\n1\n\nx2 + y2 = ∞.\n\nConjecture ($1000): A contains the four vertices of a square, i.e. four points of the form (x, y ), (x + a, y ), (x, y + a), and ( x + a, y + a). (More generally, it should be true that\n\nA contains a k × k square grid.)", "statement_status": "exact", "statement_verification": "The original AIM workshop PDF was checked directly. Problem 1.10, presented by R. Graham, reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.10\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[215]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.10 (R. Graham). Let A ⊆ Z × Z satisfy \\n\\n∑\\n\\n> (x,y )∈A\\n\\n1\\n\\nx2 + y2 = ∞.\\n\\nConjecture ($1000): A contains the four vertices of a square, i.e. four points of the form (x, y ), (x + a, y ), (x, y + a), and ( x + a, y + a). (More generally, it should be true that \\n\\nA contains a k × k square grid.)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0216", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After repairing the source's missing exclusion of the origin and zero side length, the reciprocal-mass hypothesis is exactly equivalent, up to absolute constants, to divergence of the sum of normalized dyadic-shell densities. Any square-free counterexample must have shell densities tending to zero but with divergent sum. A summable dyadic bound for the finite square-free extremal function, in particular r_square(N) = O(N^2/(log N)^(1+epsilon)), would prove the conjecture. An explicit zero-upper-density set with divergent reciprocal mass shows why positive-density theorems alone do not suffice, and the full k-by-k conclusion is proved for coordinatewise down-sets.\n\nCandidate contribution (reduction; novelty confidence low): Candidate dyadic Dini bridge: every counterexample has vanishing but nonsummable normalized shell densities, while summability of r_square(2^j)/4^j rules out all counterexamples; the N^2/(log N)^(1+epsilon) finite threshold, explicit zero-density divergent model, and all-k down-set theorem are proved as a single obstruction package." }, { "id": 20001092, "problem_number": "AIM-COMBINATORICS-0217", "title": "Ceiling-aware bounds and a carry-free density Hales-Jewett construction", "statement": "Problem 1.11 (R. Graham). Given real α ∈ (0, 1) and integer k ≥ 3, estimate the size of the smallest set Sα,k with the properties that 1) If X ⊆ Sα,k satisfies |X| ≥ α|Sα,k |, then X has a k-term arithmetic progression; 2) Sα,k has no ( k + 1)-term arithmetic progression.", "original_statement": "Problem 1.11 (R. Graham). Given real α ∈ (0, 1) and integer k ≥ 3, estimate the size of the smallest set Sα,k with the properties that 1) If X ⊆ Sα,k satisfies |X| ≥ α|Sα,k |, then X has a k-term arithmetic progression; 2) Sα,k has no ( k + 1)-term arithmetic progression.", "clean_statement": "Problem 1.11 (R. Graham). Given real α ∈ (0, 1) and integer k ≥ 3, estimate the size of the smallest set Sα,k with the properties that 1) If X ⊆ Sα,k satisfies |X| ≥ α|Sα,k |, then X has a k-term arithmetic progression; 2) Sα,k has no ( k + 1)-term arithmetic progression.", "statement_status": "exact", "statement_verification": "The exact canonical record assigned to this attempt is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.11\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[216]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.11 (R. Graham). Given real α ∈ (0, 1) and integer k ≥ 3, estimate the size of the smallest set Sα,k with the properties that 1) If X ⊆ Sα,k satisfies |X| ≥ α|Sα,k |, then X has a k-term arithmetic progression; 2) Sα,k has no ( k + 1)-term arithmetic progression.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0217", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let m_A(alpha,k) be the minimum witness cardinality over finite integer or real sets, with the threshold interpreted as ceil(alpha|S|). A universal alteration argument proves m_A(alpha,k)>max{(k-1)/alpha, 1+(2/k)(1-1/k)^(k-1)alpha^(-(k-1))}. A fully audited base-(2k-1) digit embedding of a density Hales-Jewett cube proves m_A(alpha,k)<=k^D(k,alpha), where D(k,alpha) is the corresponding density Hales-Jewett dimension; every combinatorial line becomes a k-AP and signed-digit uniqueness forbids all nontrivial (k+1)-APs. In addition, m_A(alpha,k)=k exactly when alpha>(k-1)/k. For k=3 the new lower baseline is m_A(alpha,3)>1+8/(27alpha^2), while a published container theorem yields an exponential-in-alpha inverse upper estimate with implicit constants.\n\nCandidate contribution (theorem; novelty confidence low): With the source's exact ceiling convention, the combined explicit bounds m_A(alpha,k)>1+(2/k)(1-1/k)^(k-1)alpha^(-(k-1)) and m_A(alpha,k)<=k^D(k,alpha) hold for both integer and real ambient classes, using a base-(2k-1) witness that is proved free of every nontrivial (k+1)-AP; moreover m_A(alpha,k)=k if and only if alpha>(k-1)/k." }, { "id": 20001093, "problem_number": "AIM-COMBINATORICS-0218", "title": "Multiplicity and an exact weighted-cut identity for three-term progressions", "statement": "Problem 1.12 (R. Graham). Let S be a set of homogeneous linear equations which is partition regular, i.e. in any r-coloring of Z there is always a non-trivial monochromatic solution to S. What is the minimum number of monochromatic solutions to S which can occur in some r-coloring of {1, 2,..., n } as a function of n and r?For example, with r = 2 and S is the single equation x + y = z, the correct answer is n2(1 + o(1)) /22 (Schoen, Roberts-Zeilberger). A random 2-coloring of {1, 2,..., n }\n\nwould give ∼ n2/16 such solutions. What happens for the equation x+y = 2 z? Are random 2-colorings best in this case? 4 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV", "original_statement": "Problem 1.12 (R. Graham). Let S be a set of homogeneous linear equations which is partition regular, i.e. in any r-coloring of Z there is always a non-trivial monochromatic solution to S. What is the minimum number of monochromatic solutions to S which can occur in some r-coloring of {1, 2,..., n } as a function of n and r?For example, with r = 2 and S is the single equation x + y = z, the correct answer is n2(1 + o(1)) /22 (Schoen, Roberts-Zeilberger). A random 2-coloring of {1, 2,..., n }\n\nwould give ∼ n2/16 such solutions. What happens for the equation x+y = 2 z? Are random 2-colorings best in this case? 4 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV", "clean_statement": "Problem 1.12 (R. Graham). Let S be a set of homogeneous linear equations which is partition regular, i.e. in any r-coloring of Z there is always a non-trivial monochromatic solution to S. What is the minimum number of monochromatic solutions to S which can occur in some r-coloring of {1, 2,..., n } as a function of n and r?For example, with r = 2 and S is the single equation x + y = z, the correct answer is n2(1 + o(1)) /22 (Schoen, Roberts-Zeilberger). A random 2-coloring of {1, 2,..., n }\n\nwould give ∼ n2/16 such solutions. What happens for the equation x+y = 2 z? Are random 2-colorings best in this case? 4 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV", "statement_status": "exact", "statement_verification": "The stored record is OCR text from Problem 1.12 of the AIM list *Recent trends in additive combinatorics*. Inspection of the official PDF confirms that the lost superscripts and fractions are \\[ \\frac{n^2(1+o(1))}{22}\\qquad\\hbox{and}\\qquad \\frac{n^2}{16}. \\] The words beginning “COLLECTED BY” are a page footer, not part of the problem. The source also says “Roberts-Zeilberger”; the cited paper is by Aaron Robertson and Doron Zeilberger.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.12\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[217]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.12 (R. Graham). Let S be a set of homogeneous linear equations which is partition regular, i.e. in any r-coloring of Z there is always a non-trivial monochromatic solution to S. What is the minimum number of monochromatic solutions to S which can occur in some r-coloring of {1, 2,..., n } as a function of n and r?For example, with r = 2 and S is the single equation x + y = z, the correct answer is n2(1 + o(1)) /22 (Schoen, Roberts-Zeilberger). A random 2-coloring of {1, 2,..., n }\\n\\nwould give ∼ n2/16 such solutions. What happens for the equation x+y = 2 z? Are random 2-colorings best in this case? 4 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0218", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "With nonconstant three-term arithmetic progressions counted once up to endpoint reversal, the highlighted equation x+y=2z is known not to be minimized by random two-colorings: the best published bounds located are (1675/32768)n^2(1+o(1)) <= V(n) <= (117/2192)n^2(1+o(1)), while random colorings give n^2/16+O(n); the exact constant remains open. This attempt also proves the exact finite-n identity M_r=T-C_r/2+R_r/2, where C_r is an explicitly weighted bichromatic pair cut and R_r is the number of rainbow progressions. For two colors the problem is exactly a weighted Max-Cut problem.\n\nCandidate contribution (reduction; novelty confidence low): For every r-coloring of [n], the number M_r of nonconstant monochromatic three-term progressions satisfies M_r=T-C_r/2+R_r/2 with edge weights w_n(i,j)=1_{2j-i<=n}+1_{2i-j>=1}+1_{2 divides (j-i)}; in particular, for r=2 minimizing M_2 is exactly weighted Max-Cut, and for r>=3 the sole correction is half the rainbow count." }, { "id": 20001094, "problem_number": "AIM-COMBINATORICS-0219", "title": "Quantitative Hales-Jewett bounds and exact binary calibration", "statement": "Problem 1.13 (R. Graham). Obtain \"reasonable\" bounds for the Hales-Jewett theo-rem and for the density version of it.", "original_statement": "Problem 1.13 (R. Graham). Obtain \"reasonable\" bounds for the Hales-Jewett theo-rem and for the density version of it.", "clean_statement": "Problem 1.13 (R. Graham). Obtain \"reasonable\" bounds for the Hales-Jewett theo-rem and for the density version of it.", "statement_status": "exact", "statement_verification": "The exact extracted AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.13\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[218]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.13 (R. Graham). Obtain \\\"reasonable\\\" bounds for the Hales-Jewett theo-rem and for the density version of it.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0219", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literature now gives fixed-height or better 'reasonable' bounds for both versions at alphabet size three: Conlon's direct coloring bound is HJ(3,r) at most double exponential in r, while Bhangale-Khot-Liu-Minzer give a fourfold-exponential upper bound for dhj(3,delta) after inversion. Comparable general-alphabet density bounds remain open. Independently, the report proves an extremal-chromatic threshold calculus and the exact binary formulas HJ(2,r)=r and dhj(2,delta)=min{n: 2^{-n} binom(n,floor(n/2))=2, primes q>2m, and t>=1, the shifted set of t-digit base-q numbers with digits 0,...,m-1 has size m^t and its unrestricted two-fold sumset has longest arithmetic progression exactly 2m-1. Consequently F_theta(N) is bounded by a theta-dependent constant for every fixed theta<1; more sharply, F_theta(N)=3 for all sufficiently large N whenever theta=k when k is odd and theta>1-1/(k-1).\n\nCandidate contribution (construction; novelty confidence low): Candidate explicit synthesis: the no-carry base-q family has progression length exactly 2m-1 at every digit depth; its base-5, digit-{0,1} specialization yields the exact extremal equality F_theta(N)=3 for every fixed theta s\n\n∑\n\n> i=1\n\naixi = 0, where\n\n> s\n\n∑\n\n> i=1\n\nai = 0 and |ai| < A?What about the same question, but with\n\n> t\n\n∑\n\n> i=1\n\naix2\n\n> i\n\n= 0? N. Alon comments: the answer to the question is \"No\", if, for example, a1 = a2 = 1 and a3 = a4 = −1, then S cannot have size bigger than (1 + o(1)) √n (it is a Sidon set). For the quadratic version the answer is also \"no\"; if say,\n\ns = 100, a 1 = a2 = · · · = a50 = 1, a 51 = −1,..., a 100 = −1,\n\nthen by the pigeonhole principle S cannot of size bigger than O(n1/25 ). For the linear case the Behrend construction easily generalizes if only one ai is positive and the others are negative (even simultaneously for all such sets ai). There are also extensions to some other cases that appear in papers of Ruzsa in Acta Arithmetica in 93 or so.", "original_statement": "Problem 1.16 (T. Wooley). Can one generalize Behrend's construction to produce large subsets S ⊂ [N ] such that S does not admit solutions to \n\n> s\n\n∑\n\n> i=1\n\naixi = 0, where \n\n> s\n\n∑\n\n> i=1\n\nai = 0 and |ai| < A?What about the same question, but with \n\n> t\n\n∑\n\n> i=1\n\naix2 \n\n> i\n\n= 0? N. Alon comments: the answer to the question is \"No\", if, for example, a1 = a2 = 1 and a3 = a4 = −1, then S cannot have size bigger than (1 + o(1)) √n (it is a Sidon set). For the quadratic version the answer is also \"no\"; if say, \n\ns = 100, a 1 = a2 = · · · = a50 = 1, a 51 = −1,..., a 100 = −1,\n\nthen by the pigeonhole principle S cannot of size bigger than O(n1/25 ). For the linear case the Behrend construction easily generalizes if only one ai is positive and the others are negative (even simultaneously for all such sets ai). There are also extensions to some other cases that appear in papers of Ruzsa in Acta Arithmetica in 93 or so.", "clean_statement": "Problem 1.16 (T. Wooley). Can one generalize Behrend's construction to produce large subsets S ⊂ [N ] such that S does not admit solutions to\n\n> s\n\n∑\n\n> i=1\n\naixi = 0, where\n\n> s\n\n∑\n\n> i=1\n\nai = 0 and |ai| < A?What about the same question, but with\n\n> t\n\n∑\n\n> i=1\n\naix2\n\n> i\n\n= 0? N. Alon comments: the answer to the question is \"No\", if, for example, a1 = a2 = 1 and a3 = a4 = −1, then S cannot have size bigger than (1 + o(1)) √n (it is a Sidon set). For the quadratic version the answer is also \"no\"; if say,\n\ns = 100, a 1 = a2 = · · · = a50 = 1, a 51 = −1,..., a 100 = −1,\n\nthen by the pigeonhole principle S cannot of size bigger than O(n1/25 ). For the linear case the Behrend construction easily generalizes if only one ai is positive and the others are negative (even simultaneously for all such sets ai). There are also extensions to some other cases that appear in papers of Ruzsa in Acta Arithmetica in 93 or so.", "statement_status": "exact", "statement_verification": "The official AIM workshop PDF, *Recent trends in additive combinatorics*, Problem 1.16 (T. Wooley), asks whether Behrend's construction can be generalized to give large sets \\(S\\subset [N]\\) having no solutions of \\[ \\sum_{i=1}^{s}a_i x_i=0, \\qquad \\sum_{i=1}^{s}a_i=0, \\qquad |a_i| s\\n\\n∑\\n\\n> i=1\\n\\naixi = 0, where \\n\\n> s\\n\\n∑\\n\\n> i=1\\n\\nai = 0 and |ai| < A?What about the same question, but with \\n\\n> t\\n\\n∑\\n\\n> i=1\\n\\naix2 \\n\\n> i\\n\\n= 0? N. Alon comments: the answer to the question is \\\"No\\\", if, for example, a1 = a2 = 1 and a3 = a4 = −1, then S cannot have size bigger than (1 + o(1)) √n (it is a Sidon set). For the quadratic version the answer is also \\\"no\\\"; if say, \\n\\ns = 100, a 1 = a2 = · · · = a50 = 1, a 51 = −1,..., a 100 = −1,\\n\\nthen by the pigeonhole principle S cannot of size bigger than O(n1/25 ). For the linear case the Behrend construction easily generalizes if only one ai is positive and the others are negative (even simultaneously for all such sets ai). There are also extensions to some other cases that appear in papers of Ruzsa in Acta Arithmetica in 93 or so.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0222", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under Ruzsa's equality-class convention, avoiding nontrivial solutions to the balanced power equation x_1^d+...+x_h^d=x_{h+1}^d+...+x_{2h}^d is exactly the strong B_h property for the set of dth powers of S. Consequently binom(|S|+h-1,h) is at most h(N^d-1)+1 and |S| is O_h(N^{d/h}); at h=50 and d=2 this rigorously gives the AIM note's O(N^{1/25}) obstruction. A carry-free sphere proof also records the known simultaneous Behrend-scale construction for every bounded invariant one-sign coefficient vector.\n\nCandidate contribution (lemma; novelty confidence low): For every h,d,N, a subset S of [N] avoiding Ruzsa-nontrivial solutions to the balanced degree-d power equation satisfies the explicit finite inequality binom(|S|+h-1,h) <= h(N^d-1)+1; equivalently, its dth-power image is a strong B_h set with repetitions allowed." }, { "id": 20001098, "problem_number": "AIM-COMBINATORICS-0223", "title": "Recognizing self-sumsets over finite fields", "statement": "Problem 1.17 (A. Granville). Given a set in a finite field, how to determine (in rea-sonable time) whether it is a sumset of yet another set?", "original_statement": "Problem 1.17 (A. Granville). Given a set in a finite field, how to determine (in rea-sonable time) whether it is a sumset of yet another set?", "clean_statement": "Problem 1.17 (A. Granville). Given a set in a finite field, how to determine (in rea-sonable time) whether it is a sumset of yet another set?", "statement_status": "exact", "statement_verification": "The stored record reads verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.17\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[222]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.17 (A. Granville). Given a set in a finite field, how to determine (in rea-sonable time) whether it is a sumset of yet another set?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0223", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Primary literature now resolves Granville's intended A+A recognition problem: it is NP-complete over the integers and, for every prime p, over growing vector spaces F_p^d. This report makes the input model and characteristic dependence explicit, proves that roots lie in (1/2)S in odd characteristic and can be translated to satisfy 0 in A subset S in characteristic two, and derives an exact support-sensitive enumeration algorithm. It also completely classifies targets of size at most four: in characteristic two the nonempty possibilities are {0}, {0,s}, and two-dimensional F_2-subspaces, while in odd characteristic they are singletons and nondegenerate three-term arithmetic progressions.\n\nCandidate contribution (exact_small_support_classification; novelty confidence low): Candidate novelty: a uniform proof that across all finite fields the self-sumsets of cardinality at most four are exactly the characteristic-two sets {0}, {0,s}, and two-dimensional F_2-subspaces, or in odd characteristic the singletons and nondegenerate three-term arithmetic progressions; in particular sizes three and four give sharp characteristic-dependent obstructions." }, { "id": 20001099, "problem_number": "AIM-COMBINATORICS-0224", "title": "An odd-order five-coset counterexample above the one-third threshold", "statement": "Problem 1.18 (V. Lev). Let A be a finite non-empty subset of an abelian group G, and write D:= A−A. If any d ∈ D has strictly more than |A|/2 representations of the form\n\nd = a′ − a′′ with a′, a ′′ ∈ A, then D is a subgroup: indeed, by the pigeonhole principle for any d1, d 2 ∈ D there exists a pair of representations d1 = a′\n\n> 1\n\n− a′′\n\n> 1, d 2 = a′\n\n> 2\n\n− a′′\n\n> 2\n\nsuch that a′′\n\n> 1\n\n= a′′\n\n> 2, and it follows that d1 − d2 = a′\n\n> 1\n\n− a′\n\n> 2\n\n∈ D.Assume now that any d ∈ D is only guaranteed to have at least |A|/2 representations as d = a′ −a′′ with a′, a ′′ ∈ A. In this case the argument above doesn't work, and in fact, the conclusion is not true either. To see this, consider the set A:= H ∪ (g + H), where PALO ALTO PROBLEMS 5\n\nH < G is a finite subgroup and g ∈ G is so chosen that the order of g in the quotient group G/H is at least five. Then D = ( −g + H) ∪ H ∪ (g + H) is not a subgroup, but a union of three cosets. At the same time, it is easily seen that any d ∈ D has at least\n\n|H| = |A|/2 representations of the form d = a′ − a′′.Is this example unique? In other words, given that any d ∈ D has at least |A|/2representations as d = a′ − a′′, is it necessarily true that D is either a subgroup or a union of three cosets? For practical applications one should go somewhat beyond the |A|/2 bound. Problem: assuming that any d ∈ D:= A − A has more than |A|/3representations of the form d = a′ − a′′ with a′, a ′′ ∈ A, is it necessarily true that D is either a subgroup or a union of three cosets?", "original_statement": "Problem 1.18 (V. Lev). Let A be a finite non-empty subset of an abelian group G, and write D:= A−A. If any d ∈ D has strictly more than |A|/2 representations of the form \n\nd = a′ − a′′ with a′, a ′′ ∈ A, then D is a subgroup: indeed, by the pigeonhole principle for any d1, d 2 ∈ D there exists a pair of representations d1 = a′ \n\n> 1\n\n− a′′ \n\n> 1, d 2 = a′ \n\n> 2\n\n− a′′ \n\n> 2\n\nsuch that a′′ \n\n> 1\n\n= a′′ \n\n> 2, and it follows that d1 − d2 = a′ \n\n> 1\n\n− a′ \n\n> 2\n\n∈ D.Assume now that any d ∈ D is only guaranteed to have at least |A|/2 representations as d = a′ −a′′ with a′, a ′′ ∈ A. In this case the argument above doesn't work, and in fact, the conclusion is not true either. To see this, consider the set A:= H ∪ (g + H), where PALO ALTO PROBLEMS 5\n\nH < G is a finite subgroup and g ∈ G is so chosen that the order of g in the quotient group G/H is at least five. Then D = ( −g + H) ∪ H ∪ (g + H) is not a subgroup, but a union of three cosets. At the same time, it is easily seen that any d ∈ D has at least \n\n|H| = |A|/2 representations of the form d = a′ − a′′.Is this example unique? In other words, given that any d ∈ D has at least |A|/2representations as d = a′ − a′′, is it necessarily true that D is either a subgroup or a union of three cosets? For practical applications one should go somewhat beyond the |A|/2 bound. Problem: assuming that any d ∈ D:= A − A has more than |A|/3representations of the form d = a′ − a′′ with a′, a ′′ ∈ A, is it necessarily true that D is either a subgroup or a union of three cosets?", "clean_statement": "Problem 1.18 (V. Lev). Let A be a finite non-empty subset of an abelian group G, and write D:= A−A. If any d ∈ D has strictly more than |A|/2 representations of the form\n\nd = a′ − a′′ with a′, a ′′ ∈ A, then D is a subgroup: indeed, by the pigeonhole principle for any d1, d 2 ∈ D there exists a pair of representations d1 = a′\n\n> 1\n\n− a′′\n\n> 1, d 2 = a′\n\n> 2\n\n− a′′\n\n> 2\n\nsuch that a′′\n\n> 1\n\n= a′′\n\n> 2, and it follows that d1 − d2 = a′\n\n> 1\n\n− a′\n\n> 2\n\n∈ D.Assume now that any d ∈ D is only guaranteed to have at least |A|/2 representations as d = a′ −a′′ with a′, a ′′ ∈ A. In this case the argument above doesn't work, and in fact, the conclusion is not true either. To see this, consider the set A:= H ∪ (g + H), where PALO ALTO PROBLEMS 5\n\nH < G is a finite subgroup and g ∈ G is so chosen that the order of g in the quotient group G/H is at least five. Then D = ( −g + H) ∪ H ∪ (g + H) is not a subgroup, but a union of three cosets. At the same time, it is easily seen that any d ∈ D has at least\n\n|H| = |A|/2 representations of the form d = a′ − a′′.Is this example unique? In other words, given that any d ∈ D has at least |A|/2representations as d = a′ − a′′, is it necessarily true that D is either a subgroup or a union of three cosets? For practical applications one should go somewhat beyond the |A|/2 bound. Problem: assuming that any d ∈ D:= A − A has more than |A|/3representations of the form d = a′ − a′′ with a′, a ′′ ∈ A, is it necessarily true that D is either a subgroup or a union of three cosets?", "statement_status": "exact", "statement_verification": "The canonical input remains unchanged in input.json. Its problem field is reproduced verbatim here, including extraction line breaks and the page-footer intrusion:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.18\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[223]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.18 (V. Lev). Let A be a finite non-empty subset of an abelian group G, and write D:= A−A. If any d ∈ D has strictly more than |A|/2 representations of the form \\n\\nd = a′ − a′′ with a′, a ′′ ∈ A, then D is a subgroup: indeed, by the pigeonhole principle for any d1, d 2 ∈ D there exists a pair of representations d1 = a′ \\n\\n> 1\\n\\n− a′′ \\n\\n> 1, d 2 = a′ \\n\\n> 2\\n\\n− a′′ \\n\\n> 2\\n\\nsuch that a′′ \\n\\n> 1\\n\\n= a′′ \\n\\n> 2, and it follows that d1 − d2 = a′ \\n\\n> 1\\n\\n− a′ \\n\\n> 2\\n\\n∈ D.Assume now that any d ∈ D is only guaranteed to have at least |A|/2 representations as d = a′ −a′′ with a′, a ′′ ∈ A. In this case the argument above doesn't work, and in fact, the conclusion is not true either. To see this, consider the set A:= H ∪ (g + H), where PALO ALTO PROBLEMS 5\\n\\nH < G is a finite subgroup and g ∈ G is so chosen that the order of g in the quotient group G/H is at least five. Then D = ( −g + H) ∪ H ∪ (g + H) is not a subgroup, but a union of three cosets. At the same time, it is easily seen that any d ∈ D has at least \\n\\n|H| = |A|/2 representations of the form d = a′ − a′′.Is this example unique? In other words, given that any d ∈ D has at least |A|/2representations as d = a′ − a′′, is it necessarily true that D is either a subgroup or a union of three cosets? For practical applications one should go somewhat beyond the |A|/2 bound. Problem: assuming that any d ∈ D:= A − A has more than |A|/3representations of the form d = a′ − a′′ with a′, a ′′ ∈ A, is it necessarily true that D is either a subgroup or a union of three cosets?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0224", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "In the odd cyclic group C_3 x C_7, the eight-element set consisting of full C_3-fibres at quotient heights 0 and 2 and the two-point fibre {0,1} at height 1 has difference set C_3 x {0,+/-1,+/-2}. Its ordered difference multiplicities are 3 on the extreme layers, 4 on the adjacent layers, and 8,7,7 on the central layer, so every represented difference has more than |A|/3 representations. The 15-element difference set is neither a subgroup nor a union of at most three cosets of any common subgroup. Thus the AIM one-third conclusion is false even in a finite odd-order, two-torsion-free group.\n\nCandidate contribution (counterexample; novelty confidence low): The explicit set A=(C_3 x {0}) union ({0,1} x {1}) union (C_3 x {2}) in C_3 x C_7 refutes the strict one-third subgroup/three-coset conclusion in odd order; more generally, the three-layer construction has exact relative minimum min(h,2m)/(2h+m), maximized at 2/5." }, { "id": 20001100, "problem_number": "AIM-COMBINATORICS-0225", "title": "Bourgain's arbitrary-modulus theorem answers Chang's sum-product question", "statement": "Problem 1.19 (M.-C. Chang). Is it true that for any ≤ > 0 there exists ≤′ > 0 with the following property: if A ⊆ Z/q Z (with a sufficiently large integer q) satisfies |A +\n\nA| + |A · A| < q ≤, then either |A| > q 1−≤′\n\nor there exists q1 | q, q 1 > 1 such that the canonical image of A in Z/q 1Z has at most q≤′\n\nelements? 2. Sumsets Problem Session", "original_statement": "Problem 1.19 (M.-C. Chang). Is it true that for any ≤ > 0 there exists ≤′ > 0 with the following property: if A ⊆ Z/q Z (with a sufficiently large integer q) satisfies |A +\n\nA| + |A · A| < q ≤, then either |A| > q 1−≤′\n\nor there exists q1 | q, q 1 > 1 such that the canonical image of A in Z/q 1Z has at most q≤′\n\nelements? 2. Sumsets Problem Session", "clean_statement": "Problem 1.19 (M.-C. Chang). Is it true that for any ≤ > 0 there exists ≤′ > 0 with the following property: if A ⊆ Z/q Z (with a sufficiently large integer q) satisfies |A +\n\nA| + |A · A| < q ≤, then either |A| > q 1−≤′\n\nor there exists q1 | q, q 1 > 1 such that the canonical image of A in Z/q 1Z has at most q≤′\n\nelements? 2. Sumsets Problem Session", "statement_status": "exact", "statement_verification": "The exact stored record is visibly corrupted by OCR:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 1.19\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[224]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.19 (M.-C. Chang). Is it true that for any ≤ > 0 there exists ≤′ > 0 with the following property: if A ⊆ Z/q Z (with a sufficiently large integer q) satisfies |A +\\n\\nA| + |A · A| < q ≤, then either |A| > q 1−≤′\\n\\nor there exists q1 | q, q 1 > 1 such that the canonical image of A in Z/q 1Z has at most q≤′\\n\\nelements? 2. Sumsets Problem Session\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0225", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The corrupted source statement is recovered with epsilon and epsilon-prime exponents. Bourgain's arbitrary-modulus sum-product theorem proves the AIM assertion, and the report gives the complete parameter transfer with the explicit choice epsilon-prime=epsilon: failure of both structural alternatives implies |A|>q^epsilon via the allowed identity projection modulo q, so Bourgain's relative lower bound q^delta|A| exceeds q^(epsilon+delta). A stronger trichotomy is recorded. In addition, if tau_*(epsilon) is the infimum of admissible conclusion exponents, then epsilon/2<=tau_*(epsilon)<=epsilon for 0 k\n\n≤ rk + rk+1 + rk+2 + · · · (∗)6 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\n\nfor any k ≥ 1. Problem: what are the general properties shared by the functions ν for all finite sets A, B ⊆ Z, other than that reflected by ( ∗)? Notice that for any t ∈ N, the length of the tth row of the above described diagram (counting the rows from the bottom) is Nt:= {n: ν(n) ≥ t}. From a well-known result of Pollard it follows that N1 + · · · + Nt ≥ t(|A| + |B| − t) for any t ≤ min {| A|, |B|}, and this can be derived also as a corollary of ( ∗).", "original_statement": "Problem 2.2 (V. Lev). Given two finite integer sets A and B, write \n\nν(n):= {(a, b ) ∈ A × B: a + b = n}; n ∈ Z.\n\nThe spectrum of ν defines a partition of the integer |A|| B| which can be visualized using a Ferrers diagram; that is, an arrangement of |A|| B| square boxes in bottom-aligned columns such that the height of the leftmost column is the largest value attained by ν,the height of next column is the second largest value of ν, and so on. It is not difficult to show that if rk denotes the height of the kth column of the diagram (that is, the kth largest value attained by ν), then \n\nr2 \n\n> k\n\n≤ rk + rk+1 + rk+2 + · · · (∗)6 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV \n\nfor any k ≥ 1. Problem: what are the general properties shared by the functions ν for all finite sets A, B ⊆ Z, other than that reflected by ( ∗)? Notice that for any t ∈ N, the length of the tth row of the above described diagram (counting the rows from the bottom) is Nt:= {n: ν(n) ≥ t}. From a well-known result of Pollard it follows that N1 + · · · + Nt ≥ t(|A| + |B| − t) for any t ≤ min {| A|, |B|}, and this can be derived also as a corollary of ( ∗).", "clean_statement": "Problem 2.2 (V. Lev). Given two finite integer sets A and B, write\n\nν(n):= {(a, b ) ∈ A × B: a + b = n}; n ∈ Z.\n\nThe spectrum of ν defines a partition of the integer |A|| B| which can be visualized using a Ferrers diagram; that is, an arrangement of |A|| B| square boxes in bottom-aligned columns such that the height of the leftmost column is the largest value attained by ν,the height of next column is the second largest value of ν, and so on. It is not difficult to show that if rk denotes the height of the kth column of the diagram (that is, the kth largest value attained by ν), then\n\nr2\n\n> k\n\n≤ rk + rk+1 + rk+2 + · · · (∗)6 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\n\nfor any k ≥ 1. Problem: what are the general properties shared by the functions ν for all finite sets A, B ⊆ Z, other than that reflected by ( ∗)? Notice that for any t ∈ N, the length of the tth row of the above described diagram (counting the rows from the bottom) is Nt:= {n: ν(n) ≥ t}. From a well-known result of Pollard it follows that N1 + · · · + Nt ≥ t(|A| + |B| − t) for any t ≤ min {| A|, |B|}, and this can be derived also as a corollary of ( ∗).", "statement_status": "exact", "statement_verification": "The raw corpus record defines, for finite integer sets \\(A,B\\), \\[ \\nu(n):=\\#\\{(a,b)\\in A\\times B:a+b=n\\},\\qquad n\\in\\mathbb Z, \\] then sorts the positive values of \\(\\nu\\) as column heights \\(r_1\\ge r_2\\ge\\cdots\\) of a Ferrers diagram. The extracted display is corrupted across a page boundary as `r2 > k <= rk + rk+1 + ...`.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 2.2\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[226]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.2 (V. Lev). Given two finite integer sets A and B, write \\n\\nν(n):= {(a, b ) ∈ A × B: a + b = n}; n ∈ Z.\\n\\nThe spectrum of ν defines a partition of the integer |A|| B| which can be visualized using a Ferrers diagram; that is, an arrangement of |A|| B| square boxes in bottom-aligned columns such that the height of the leftmost column is the largest value attained by ν,the height of next column is the second largest value of ν, and so on. It is not difficult to show that if rk denotes the height of the kth column of the diagram (that is, the kth largest value attained by ν), then \\n\\nr2 \\n\\n> k\\n\\n≤ rk + rk+1 + rk+2 + · · · (∗)6 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV \\n\\nfor any k ≥ 1. Problem: what are the general properties shared by the functions ν for all finite sets A, B ⊆ Z, other than that reflected by ( ∗)? Notice that for any t ∈ N, the length of the tth row of the above described diagram (counting the rows from the bottom) is Nt:= {n: ν(n) ≥ t}. From a well-known result of Pollard it follows that N1 + · · · + Nt ≥ t(|A| + |B| − t) for any t ≤ min {| A|, |B|}, and this can be derived also as a corollary of ( ∗).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0227", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The corrupted source formula is recovered as r_k^2 <= r_k+r_{k+1}+r_{k+2}+... . Beyond this sorted-spectrum condition, every finite integer-set convolution nu=1_A*1_B satisfies the location-sensitive inequality sum_{d>=1} min{nu(n-d),nu(n+d)} >= binomial(nu(n),2). More precisely, the positive-difference multiplicities of the representation fiber R_n=A intersection (n-B) inject into representations at both mirrored points n-d and n+d. The inequality is sharp for every fiber size and every difference profile via A=R, B=n-R, and it excludes spatial arrangements whose sorted spectrum satisfies the AIM inequality.\n\nCandidate contribution (inequality; novelty confidence low): Candidate novel contribution: for every finite A,B subset of the integers and every n, sum_{d>=1} min{nu(n-d),nu(n+d)} is at least binomial(nu(n),2), with the stronger fiberwise lower bounds nu(n-d),nu(n+d) >= q_n(d); this is sharp for all central multiplicities and is not implied by the sorted Ferrers spectrum condition." }, { "id": 20001103, "problem_number": "AIM-COMBINATORICS-0228", "title": "Small sumsets with no three collinear", "statement": "Problem 2.3 (G. Freiman). Suppose A ⊆ Z2 is finite set no three points of which are on a line. What is the smallest size of 2 A, given that |A| = n?", "original_statement": "Problem 2.3 (G. Freiman). Suppose A ⊆ Z2 is finite set no three points of which are on a line. What is the smallest size of 2 A, given that |A| = n?", "clean_statement": "Problem 2.3 (G. Freiman). Suppose A ⊆ Z2 is finite set no three points of which are on a line. What is the smallest size of 2 A, given that |A| = n?", "statement_status": "exact", "statement_verification": "The canonical JSON lost superscripts and comparison symbols. The AIM workshop PDF gives the following problem.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 2.3\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[227]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.3 (G. Freiman). Suppose A ⊆ Z2 is finite set no three points of which are on a line. What is the smallest size of 2 A, given that |A| = n?\"\nOriginal remarks: [\"Remark(s). Stanchescu has shown that (i) |2A| ¿ n(log n)1/8, and (ii) there is no positive constant ≤ such that the inequality |2A| ¿ n1+ ≤ holds for every finite set A ⊆ Z2\\n\\ncontaining no three points on a line.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0228", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every prescribed integer n >= 3, there is an n-point set A in Z^2 with no three collinear and |A+A| <= n exp(C sqrt(log n)); this is proved by selecting a large equal-norm layer in a high-dimensional integer box, taking exactly n points, and applying a generic integral projection that preserves all triple determinants and cannot enlarge the sumset. In the other direction, Stanchescu's inequality combined with Raghavan's March 2026 preprint gives the conditional current bound |A+A| >= (n/2) exp(c (log n)^(1/6)/log log n) for sufficiently large n.\n\nCandidate contribution (construction; novelty confidence low): For every integer n >= 3, rather than only for arbitrarily large cardinalities along an unspecified sequence, there exists A subset Z^2 with |A| = n, no three collinear, and |A+A| <= n exp(C sqrt(log n)); an explicit rounding argument and generic integral projection lemma prove this all-n quantifier." }, { "id": 20001104, "problem_number": "AIM-COMBINATORICS-0229", "title": "An anchor-container reduction for imbalanced restricted sumsets", "statement": "Problem 2.4 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, where m > n, and suppose that G is a subset of A × B having size at least δmn. Further, suppose that\n\n|{ a + b: a ∈ A, b ∈ B, (a, b ) ∈ G}| < Cm.\n\nDoes this imply anything about the structure of A and B? In particular, must there exist A′ ⊂ A, B′ ⊂ B, |A′| ≥ cm, |B′| ≥ cn such that |A′ + B′| ≤ Km, where c and K\n\ndepend on δ and C?", "original_statement": "Problem 2.4 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, where m > n, and suppose that G is a subset of A × B having size at least δmn. Further, suppose that \n\n|{ a + b: a ∈ A, b ∈ B, (a, b ) ∈ G}| < Cm. \n\nDoes this imply anything about the structure of A and B? In particular, must there exist A′ ⊂ A, B′ ⊂ B, |A′| ≥ cm, |B′| ≥ cn such that |A′ + B′| ≤ Km, where c and K\n\ndepend on δ and C?", "clean_statement": "Problem 2.4 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, where m > n, and suppose that G is a subset of A × B having size at least δmn. Further, suppose that\n\n|{ a + b: a ∈ A, b ∈ B, (a, b ) ∈ G}| < Cm.\n\nDoes this imply anything about the structure of A and B? In particular, must there exist A′ ⊂ A, B′ ⊂ B, |A′| ≥ cm, |B′| ≥ cn such that |A′ + B′| ≤ Km, where c and K\n\ndepend on δ and C?", "statement_status": "exact", "statement_verification": "No corruption of the mathematical statement was found. The line breaks in the canonical JSON are extraction artifacts only.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 2.4\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[228]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.4 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\\n\\nand |B| = n, where m > n, and suppose that G is a subset of A × B having size at least δmn. Further, suppose that \\n\\n|{ a + b: a ∈ A, b ∈ B, (a, b ) ∈ G}| < Cm. \\n\\nDoes this imply anything about the structure of A and B? In particular, must there exist A′ ⊂ A, B′ ⊂ B, |A′| ≥ cm, |B′| ≥ cn such that |A′ + B′| ≤ Km, where c and K\\n\\ndepend on δ and C?\"\nOriginal remarks: [\"Remark(s). The case m = n is the Balog-Szemeredi's theorem.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0229", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If G is a delta-dense bipartite graph on A x B and S is its restricted sumset, there are subsets A' and B' with |A'| at least delta^2|A| and |B'| at least delta^2|B|/(2-delta^2) such that the entire sumset A'+B' lies in S+S-S. Hence Tao's requested ratio-free conclusion holds whenever S has bounded threefold growth, and in particular when S lies in a bounded-rank generalized arithmetic progression of size O(|A|). Independently, the hypotheses imply mixed energy at least (delta^2/C)|A||B|^2, so Tao-Vu's asymmetric BSG theorem gives the known near-answer with arbitrarily small powers of |A|/|B| lost.\n\nCandidate contribution (reduction; novelty confidence low): A single codegree anchor retains at least delta^2 of the larger vertex class and delta^2/(2-delta^2) of the smaller class while putting their full sumset inside the exact threefold restricted-label container S+S-S." }, { "id": 20001105, "problem_number": "AIM-COMBINATORICS-0230", "title": "Lacunary counterexamples to the asymmetric Freiman container question", "statement": "Problem 2.5 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, m > n, satisfying |A+B| < Km. Must there exist a generalized arithmetic progression P of rank c(K) containing B such that A ⊂ P + X and |P + X| ≤ c(K)|A|?", "original_statement": "Problem 2.5 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, m > n, satisfying |A+B| < Km. Must there exist a generalized arithmetic progression P of rank c(K) containing B such that A ⊂ P + X and |P + X| ≤ c(K)|A|?", "clean_statement": "Problem 2.5 (T. Tao). Suppose that A and B are finite sets of integers with |A| = m\n\nand |B| = n, m > n, satisfying |A+B| < Km. Must there exist a generalized arithmetic progression P of rank c(K) containing B such that A ⊂ P + X and |P + X| ≤ c(K)|A|?", "statement_status": "exact", "statement_verification": "The canonical record asks the following question of T. Tao. Let \\(A,B\\subseteq\\mathbb Z\\) be finite, with \\[ |A|=m>|B|=n, \\qquad |A+B| n, satisfying |A+B| < Km. Must there exist a generalized arithmetic progression P of rank c(K) containing B such that A ⊂ P + X and |P + X| ≤ c(K)|A|?\"\nOriginal remarks: [\"Remark(s). This can be done by Plunnecke's inequality if one weakens the hypotheses on P to |P + P | ≤ c(K, ≤ )m≤|P |. In this weakened version, P is no longer a progression, but merely a set with somewhat small sumset. Notice that the case m = n is Freiman's theorem.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0230", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For every fixed K>1 and every proposed rank bound d and size multiplier C, there are finite integer sets A,B with |A|>|B| and |A+B|1 and every proposed constants d,C, an explicit counterexample to the distinct-summand Freiman container conclusion after r,k,h,N are chosen in the stated order." }, { "id": 20001106, "problem_number": "AIM-COMBINATORICS-0231", "title": "Minimum difference sets in fixed dimension", "statement": "Problem 2.6 (Y. Stanchescu). Suppose that A is a finite subset of Zd, not contained in a hyperplane of dimension smaller than d. Determine the smallest possible value of\n\n|A − A| as a function of |A|.", "original_statement": "Problem 2.6 (Y. Stanchescu). Suppose that A is a finite subset of Zd, not contained in a hyperplane of dimension smaller than d. Determine the smallest possible value of \n\n|A − A| as a function of |A|.", "clean_statement": "Problem 2.6 (Y. Stanchescu). Suppose that A is a finite subset of Zd, not contained in a hyperplane of dimension smaller than d. Determine the smallest possible value of\n\n|A − A| as a function of |A|.", "statement_status": "exact", "statement_verification": "The canonical JSON contains line-break and fraction damage. The official AIM workshop PDF gives:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 2.6\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[230]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.6 (Y. Stanchescu). Suppose that A is a finite subset of Zd, not contained in a hyperplane of dimension smaller than d. Determine the smallest possible value of \\n\\n|A − A| as a function of |A|.\"\nOriginal remarks: [\"Remark(s). For every d ≥ 1 Freiman, Heppes, and Uhrin proved that |A − A| ≥ \\n\\n(d + 1) |A| − 1 \\n\\n> 2\\n\\nd(d + 1), and this inequality is best possible for d = 1, 2. In the case d = 3 the presenter has shown that a best possible result is |A − A| ≥ 4.5|A| − 9. For d ≥ 4the presenter conjectures that \\n\\n|A − A| ≥\\n\\n(\\n\\n2d − 2 + 1\\n\\nd − 1\\n\\n)\\n\\n|A| − Cd,\\n\\nfor some constant Cd.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0231", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Conlon and Lim proved the sharp Stanchescu inequality |A-A| >= (2d-2+1/(d-1))|A|-(2d^2-4d+3) for every sufficiently large full-dimensional finite A in R^d, resolving the intended large-set extremal problem. Beyond this status update, the artifacts prove an exact difference-count formula for a two-height simplex-cylinder family and derive an explicit residue-sensitive O_d(1)-width window for the literal pointwise minimum g_d(n); the bounds coincide for every sufficiently large n divisible by 2(d-1).\n\nCandidate contribution (construction; novelty confidence low): For q=d-1 and B_{a,b}=(U+P_a) union (e_d-U+P_b), the exact formula |B_{a,b}-B_{a,b}|=(q^2-q+1)(2 max(a,b)-1)+q(q+1)(a+b-1), combined with balanced heights, odd-cardinality rounding, and dimension-preserving deletion, yields an explicit all-residue pointwise window for g_d(n)." }, { "id": 20001107, "problem_number": "AIM-COMBINATORICS-0232", "title": "Dense nonsumsets and cyclic-interval square roots", "statement": "Problem 2.7 (B. Green). What is the size of the largest subset of Fp which is not a sumset B + B?PALO ALTO PROBLEMS 7", "original_statement": "Problem 2.7 (B. Green). What is the size of the largest subset of Fp which is not a sumset B + B?PALO ALTO PROBLEMS 7", "clean_statement": "Problem 2.7 (B. Green). What is the size of the largest subset of Fp which is not a sumset B + B?PALO ALTO PROBLEMS 7", "statement_status": "exact", "statement_verification": "The official AIM PDF gives the following question.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 2.7\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[231]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.7 (B. Green). What is the size of the largest subset of Fp which is not a sumset B + B?PALO ALTO PROBLEMS 7\"\nOriginal remarks: [\"Remark(s). Denoting this size by \\n\\nf (p):= max \\n\\n> A⊆FpA6=B+B\\n\\n|A|,\\n\\nthe presenter can prove that \\n\\np − p2/3+ ≤ < f (p) < p − log p\\n\\n9for any fixed ≤ > 0 and p large enough.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0232", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Writing D(p) for the least complement size of a nonsumset and M(p)=p-D(p) for the AIM maximum, the current checked literature gives Omega(sqrt(p/log p)) <= D(p) <= O(p^{3/5}(log p)^{19/10}); the upper exponent 3/5 is the 2025 Alon-Pham improvement. In addition, an exact structured classification is proved: the complement of a t-term cyclic arithmetic progression in F_p has an arithmetic-progression square root B with B+B equal to that complement if and only if t is even, with an explicit root.\n\nCandidate contribution (classification; novelty confidence low): For every odd prime p, nonzero step d, and 1 <= t <= p-1, F_p minus I_d(a,t) is B+B for an arithmetic progression B if and only if t is even; for even t one may take B=I_d((a+td)/2,(p-t+1)/2)." }, { "id": 20001108, "problem_number": "AIM-COMBINATORICS-0233", "title": "Order lifting for sparse additive bases of polynomial values", "statement": "Problem 2.8 (T. Wooley). Suppose A is a subset of the naturals. We say that A is an additive basis of order h for a polynomial sequence {f (n): n = 1, 2,... } if hA contains this sequence. If f is linear, and A is any order h basis for f (n), then |A∩[n]| ≥ n1/h. If d =deg( f ) ≥\n\n2, then one can trivially deduce that |A ∩ [n]| ≥ n1/hd. Can one get a substantially sharper lower bound in the case d ≥ 2?", "original_statement": "Problem 2.8 (T. Wooley). Suppose A is a subset of the naturals. We say that A is an additive basis of order h for a polynomial sequence {f (n): n = 1, 2,... } if hA contains this sequence. If f is linear, and A is any order h basis for f (n), then |A∩[n]| ≥ n1/h. If d =deg( f ) ≥\n\n2, then one can trivially deduce that |A ∩ [n]| ≥ n1/hd. Can one get a substantially sharper lower bound in the case d ≥ 2?", "clean_statement": "Problem 2.8 (T. Wooley). Suppose A is a subset of the naturals. We say that A is an additive basis of order h for a polynomial sequence {f (n): n = 1, 2,... } if hA contains this sequence. If f is linear, and A is any order h basis for f (n), then |A∩[n]| ≥ n1/h. If d =deg( f ) ≥\n\n2, then one can trivially deduce that |A ∩ [n]| ≥ n1/hd. Can one get a substantially sharper lower bound in the case d ≥ 2?", "statement_status": "exact", "statement_verification": "This is Problem 2.8, attributed to T. Wooley, in the AIM workshop list *Recent trends in additive combinatorics*. The repository extraction has lost superscript formatting: its strings `n1/h` and `n1/hd` mean, respectively, $n^{1/h}$ and $n^{1/(hd)}$. This was checked against the source PDF.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 2.8\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[232]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.8 (T. Wooley). Suppose A is a subset of the naturals. We say that A is an additive basis of order h for a polynomial sequence {f (n): n = 1, 2,... } if hA contains this sequence. If f is linear, and A is any order h basis for f (n), then |A∩[n]| ≥ n1/h. If d =deg( f ) ≥\\n\\n2, then one can trivially deduce that |A ∩ [n]| ≥ n1/hd. Can one get a substantially sharper lower bound in the case d ≥ 2?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0233", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let lambda_2(F_N) be the minimum size of a two-basis for the first N values of an eventually increasing degree-d polynomial f. If A is an order-h basis for these values and s=ceil(h/2), then A(f(N)) is at least (lambda_2(F_N)/2)^(1/s). Hence every finite two-basis exponent gamma transfers to A(x) >= x^(gamma/(d s)-o(1)). For powers, published two-basis estimates give gamma_2=2/3 and gamma_d=max(1/2, 3/4-1/(2 sqrt(d))-1/(2(d-1))) for d>=3. In particular, for squares A(x) >= x^(1/(3 ceil(h/2))-o(1)), improving the trivial bound for even h and odd h>=5.\n\nCandidate contribution (reduction; novelty confidence low): Splitting every h-term representation into blocks of sizes floor(h/2) and ceil(h/2) proves the explicit transfer A(f(N)) >= (lambda_2(F_N)/2)^(1/ceil(h/2)); combined with known two-basis bounds, this yields explicit all-order lower bounds for bases of power sequences." }, { "id": 20001109, "problem_number": "AIM-COMBINATORICS-0234", "title": "Seven-corner completion sets", "statement": "Problem 2.9 (T. Gowers). Suppose A ⊆ Z, |A| = n. Let\n\nS = {x + a + b + c: x, x + a, x + b, x + c, x + a + b, x + b + c, x + a + c ∈ A}.\n\nIf |S| < cn, then can one deduce anything about the structure of A?B. Green comments: The answer to this is \"no\" as it stands. For example S could be a dissociated set. Tim, Terry and I [Green] tried to formulate a decent question along these lines but couldn't come up with anything we liked.", "original_statement": "Problem 2.9 (T. Gowers). Suppose A ⊆ Z, |A| = n. Let \n\nS = {x + a + b + c: x, x + a, x + b, x + c, x + a + b, x + b + c, x + a + c ∈ A}.\n\nIf |S| < cn, then can one deduce anything about the structure of A?B. Green comments: The answer to this is \"no\" as it stands. For example S could be a dissociated set. Tim, Terry and I [Green] tried to formulate a decent question along these lines but couldn't come up with anything we liked.", "clean_statement": "Problem 2.9 (T. Gowers). Suppose A ⊆ Z, |A| = n. Let\n\nS = {x + a + b + c: x, x + a, x + b, x + c, x + a + b, x + b + c, x + a + c ∈ A}.\n\nIf |S| < cn, then can one deduce anything about the structure of A?B. Green comments: The answer to this is \"no\" as it stands. For example S could be a dissociated set. Tim, Terry and I [Green] tried to formulate a decent question along these lines but couldn't come up with anything we liked.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 2.9 from the AIM workshop problem list *Recent Trends in Additive Combinatorics*. The official PDF was checked directly. Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 2.9\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[233]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.9 (T. Gowers). Suppose A ⊆ Z, |A| = n. Let \\n\\nS = {x + a + b + c: x, x + a, x + b, x + c, x + a + b, x + b + c, x + a + c ∈ A}.\\n\\nIf |S| < cn, then can one deduce anything about the structure of A?B. Green comments: The answer to this is \\\"no\\\" as it stands. For example S could be a dissociated set. Tim, Terry and I [Green] tried to formulate a decent question along these lines but couldn't come up with anything we liked.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0234", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal AIM question is refuted: A is always contained in its completion set S(A), while every Sidon set satisfies S(A)=A; the explicit sets {1,3,...,3^(n-1)} therefore have |S(A)|=n but maximal |A+A|=n(n+1)/2. Beyond this known obstruction, the artifacts prove the multiplicity correction C_x(A) <= |A|[E(A)-(2|A|^2-|A|)] for seven-corners with all steps nonzero, yielding a linear-size small-difference subset by Balog-Szemeredi-Gowers when the average nonzero-step completion multiplicity is of order n^3. They also compute S({0,...,N}) exactly as the integer interval from -floor(N/2) to floor(3N/2).\n\nCandidate contribution (lemma; novelty confidence low): If C_x(A) counts ordered seven-corners (x,a,b,c) with abc nonzero, then C_x(A) <= |A| times the off-diagonal additive energy E(A)-(2|A|^2-|A|); consequently, order-|A|^3 average nonzero-step multiplicity over S(A) forces a linearly large subset with small difference set." }, { "id": 20001110, "problem_number": "AIM-COMBINATORICS-0235", "title": "Minimal order-two bases and quantitative thinning", "statement": "Problem 2.10. Suppose A is a subset of the naturals, and is an additive basis of N of order 2. Does there exist a proper subset B of A, where B is an additive basis of the naturals of order 2? What is the slowest growing\n\nB(x):= |{ b ∈ B: b ≤ x}|?", "original_statement": "Problem 2.10. Suppose A is a subset of the naturals, and is an additive basis of N of order 2. Does there exist a proper subset B of A, where B is an additive basis of the naturals of order 2? What is the slowest growing \n\nB(x):= |{ b ∈ B: b ≤ x}|?", "clean_statement": "Problem 2.10. Suppose A is a subset of the naturals, and is an additive basis of N of order 2. Does there exist a proper subset B of A, where B is an additive basis of the naturals of order 2? What is the slowest growing\n\nB(x):= |{ b ∈ B: b ≤ x}|?", "statement_status": "exact", "statement_verification": "The record is Problem 2.10 in the AIM workshop list *Recent trends in additive combinatorics*. The official PDF and the corpus record agree on the wording:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 2.10\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[234]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.10. Suppose A is a subset of the naturals, and is an additive basis of N of order 2. Does there exist a proper subset B of A, where B is an additive basis of the naturals of order 2? What is the slowest growing \\n\\nB(x):= |{ b ∈ B: b ≤ x}|?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0235", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal universal assertion is false under both standard conventions: minimal asymptotic order-two bases are classical, including thin strongly minimal examples, while a finite-representation compactness argument shows that every exact order-two basis contains a minimal exact subbasis. Pair counting forces every order-two basis to satisfy B(x) >= sqrt(2x)-O(1), and thin constructions attain Theta(sqrt(x)); thus this is the optimal global scale, although a prescribed minimal A has no proper basis subset at all. A proved representation-surplus lemma further shows that r_A(n) > |F intersect [0,n]| eventually guarantees that A minus F remains an asymptotic basis, yielding an infinite removable F whenever r_A(n) tends to infinity.\n\nCandidate contribution (lemma; novelty confidence low): For the unordered order-two representation function, if r_A(n) > |F intersect [0,n]| for all sufficiently large n, then A minus F is an asymptotic basis; in particular every asymptotic basis with r_A(n) tending to infinity admits an infinite removable subset F constructed by delaying its j-th element past a threshold where r_A(n) > j." }, { "id": 20001111, "problem_number": "AIM-COMBINATORICS-0236", "title": "A parity-enhanced shell inequality for the Erdős–Turán additive-basis problem", "statement": "Problem 2.11 (Brought by V. Vu, originally stated by Erd˝ os and Turan). Suppose\n\nA ⊆ N is an additive basis of order n. Let r(m) be the number of pairs ( a1, a 2) ∈ A × A\n\nsuch that m = a1 + a2. Must lim sup m→∞ r(m) = ∞?", "original_statement": "Problem 2.11 (Brought by V. Vu, originally stated by Erd˝ os and Turan). Suppose \n\nA ⊆ N is an additive basis of order n. Let r(m) be the number of pairs ( a1, a 2) ∈ A × A\n\nsuch that m = a1 + a2. Must lim sup m→∞ r(m) = ∞?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The official AIM PDF prints:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 2.11\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[235]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.11 (Brought by V. Vu, originally stated by Erd˝ os and Turan). Suppose \\n\\nA ⊆ N is an additive basis of order n. Let r(m) be the number of pairs ( a1, a 2) ∈ A × A\\n\\nsuch that m = a1 + a2. Must lim sup m→∞ r(m) = ∞?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0236", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an asymptotic additive basis A of order 2, fixed 0 < alpha < 1/2, q = floor(alpha N), and Delta = A(N) - A(q), the ordered representation maximum satisfies R(2N) >= (9(1-2 alpha)^2/32 + o(1)) N/Delta^2. Hence the Erdős–Turán conclusion holds whenever Delta = o(sqrt(N)) along an unbounded sequence. Conversely, any hypothetical eventual bound r_A(n) <= K forces every such shell to contain at least (3(1-2 alpha)/(4 sqrt(2K)) + o(1)) sqrt(N) basis elements.\n\nCandidate contribution (inequality; novelty confidence low): The parity-enhanced annular amplification inequality R(2N) >= (9(1-2 alpha)^2/32 + o(1)) N/[A(N)-A(floor(alpha N))]^2, together with its sparse-shell sufficient criterion and explicit necessary shell-thickness bound for a bounded counterexample." }, { "id": 20001112, "problem_number": "AIM-COMBINATORICS-0237", "title": "Simultaneously avoiding bounded weighted averages", "statement": "Problem 2.12 (Y. Stanchescu). Fix an integer t ≥ 1 and suppose that A ⊆ [N ] is a set such that none of the t2 equations mx + ny = ( m + n)z with 1 ≤ m, n ≤ t\n\nhas a non-trivial solution in the variables x, y, z ∈ A. How large can A be under this assumption?", "original_statement": "Problem 2.12 (Y. Stanchescu). Fix an integer t ≥ 1 and suppose that A ⊆ [N ] is a set such that none of the t2 equations mx + ny = ( m + n)z with 1 ≤ m, n ≤ t\n\nhas a non-trivial solution in the variables x, y, z ∈ A. How large can A be under this assumption?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record comes from Problem 2.12 of the AIM workshop list *Recent trends in additive combinatorics*. Direct inspection of the official PDF gives the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 2.12\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[236]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2.12 (Y. Stanchescu). Fix an integer t ≥ 1 and suppose that A ⊆ [N ] is a set such that none of the t2 equations mx + ny = ( m + n)z with 1 ≤ m, n ≤ t\\n\\nhas a non-trivial solution in the variables x, y, z ∈ A. How large can A be under this assumption?\"\nOriginal remarks: [\"Remark(s). Certainly, one has |A| ≤ r3(N ), where r3(N ) is the size of any largest subset of [ N ] containing no three-term arithmetic progressions. The presenter has shown that there is no positive constant ≤ such that |2A| ¿ | A|1+ ≤ holds true for all such sets. 3. Sum-product estimates Problem Session\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0237", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Writing F_t(N) for the extremal size, the problem remains open because F_1(N)=r_3(N). The equation family admits an exact reduced-gap formulation: a sorted triple is forbidden precisely when both reduced adjacent gaps are at most t, and the t^2 nominal equations collapse to sum_{k<=t} phi(k) unoriented constraints. This gives F_t(N) <= min(r_3(N), 2 ceil(N/(t+2))) and the exact endpoint F_t(N)=2 for N>=3 and t>=N-2. A carry-free sphere construction proves F_t(N) >= q^d/(d(q-1)^2+1) whenever (2tq)^d<=N, recovering the fixed-t Behrend scale with explicit t-dependence.\n\nCandidate contribution (theorem; novelty confidence low): For every d,q>=2 with (2tq)^d<=N, one has F_t(N)>=q^d/(d(q-1)^2+1); paired with the reduced-gap packing bound F_t(N)<=2 ceil(N/(t+2)), this yields a uniform growing-t description and the exact value F_t(N)=2 for t>=N-2." }, { "id": 20001113, "problem_number": "AIM-COMBINATORICS-0238", "title": "An explicit finite-field two-source expander", "statement": "Problem 3.1 (J. Bourgain). Find explicitly a function f: Fp × Fp → Fp such that for every A, B ⊆ Fp with |A|, |B| ∼ p1/2 we have\n\n|f (A × B)| ≥ p1/2+ ≤.8 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV", "original_statement": "Problem 3.1 (J. Bourgain). Find explicitly a function f: Fp × Fp → Fp such that for every A, B ⊆ Fp with |A|, |B| ∼ p1/2 we have \n\n|f (A × B)| ≥ p1/2+ ≤.8 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV", "clean_statement": "Problem 3.1 (J. Bourgain). Find explicitly a function f: Fp × Fp → Fp such that for every A, B ⊆ Fp with |A|, |B| ∼ p1/2 we have\n\n|f (A × B)| ≥ p1/2+ ≤.8 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV", "statement_status": "exact", "statement_verification": "The canonical record is Problem 3.1 in the AIM workshop problem list *Recent Trends in Additive Combinatorics*. The official PDF was checked directly. It states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.1\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[237]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.1 (J. Bourgain). Find explicitly a function f: Fp × Fp → Fp such that for every A, B ⊆ Fp with |A|, |B| ∼ p1/2 we have \\n\\n|f (A × B)| ≥ p1/2+ ≤.8 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0238", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The fixed polynomial f_p(x,y)=x^2+xy solves the recovered AIM problem. A collision-to-Cartesian-incidence argument using the Stevens--de Zeeuw theorem proves |f_p(A,B)| >> N^{5/4} for equal sources of size N with N^3 << p^2; pruning comparable sources of size asymptotic to p^{1/2} gives |f_p(A,B)| >> p^{5/8}, hence the literal AIM inequality with any epsilon<1/8, for example epsilon=1/9, for sufficiently large p. This is Bourgain's published construction, not a new solution.\n\nCandidate contribution (corollary; novelty confidence low): For independent uniform X and Y on equal N-element subsets in the incidence range, Z=X(X+Y) is within total variation tau of a distribution with min-entropy at least (5/4)log_2 N-log_2(C/(tau(1-tau))) for every 0 p\n\nand |H| > p δ, what is the smallest k\n\nwhich is guaranteed to satisfy kH (= H + · · · + H) = Fp? It is known that this holds provided that log k > (1 /δ )C, but can one do better?", "original_statement": "Problem 3.2 (J. Bourgain). Given that H ≤ F∗ \n\n> p\n\nand |H| > p δ, what is the smallest k\n\nwhich is guaranteed to satisfy kH (= H + · · · + H) = Fp? It is known that this holds provided that log k > (1 /δ )C, but can one do better?", "clean_statement": "Problem 3.2 (J. Bourgain). Given that H ≤ F∗\n\n> p\n\nand |H| > p δ, what is the smallest k\n\nwhich is guaranteed to satisfy kH (= H + · · · + H) = Fp? It is known that this holds provided that log k > (1 /δ )C, but can one do better?", "statement_status": "exact", "statement_verification": "The corpus record is visibly damaged by PDF extraction: the subgroup notation, the exponent on \\(p\\), and the exponent on \\(1/\\delta\\) were separated from their surrounding text. The AIM workshop PDF and the expanded problem list of Croot--Lev give the following unambiguous statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.2\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[238]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.2 (J. Bourgain). Given that H ≤ F∗ \\n\\n> p\\n\\nand |H| > p δ, what is the smallest k\\n\\nwhich is guaranteed to satisfy kH (= H + · · · + H) = Fp? It is known that this holds provided that log k > (1 /δ )C, but can one do better?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0239", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Writing K(delta) for the least exact sumset order that works uniformly for every prime p and subgroup H <= F_p^* with |H| > p^delta, the published Glibichuk-Konyagin theorem gives K(delta) <= C 4^(1/delta). This attempt proves the explicit subgroup lower bound K(delta) >= floor(2^(1/delta-1)) for 0 < delta <= 1/2 by choosing a prime p < 2^(1/delta) and H = {+1,-1}, whose exact additive basis order is p-1. Hence log K(delta) = Theta(1/delta), with the exponential constant lying between log 2 and log 4. It also proves an energy-aware Fourier criterion: p rho(A)^(k-2s) T_s(A) < |A|^(2s) implies kA = F_p.\n\nCandidate contribution (lower_bound; novelty confidence low): For the subgroup-only uniform function, K(delta) >= floor(2^(1/delta-1)) for every 0 < delta <= 1/2; together with the published upper bound this gives an explicit finite-delta bracket and proves log K(delta) = Theta(1/delta)." }, { "id": 20001115, "problem_number": "AIM-COMBINATORICS-0240", "title": "Cancellation thresholds and a prime-repunit obstruction for multiplicative subgroups", "statement": "Problem 3.3 (J. Bourgain). Suppose that H ≤ F∗\n\n> p. How large must H be in order that ∣∣∣∣∣∑\n\n> x∈H\n\nep(ax )\n\n∣∣∣∣∣ = o(|H|)to hold for all a 6 = 0?", "original_statement": "Problem 3.3 (J. Bourgain). Suppose that H ≤ F∗\n\n> p. How large must H be in order that ∣∣∣∣∣∑\n\n> x∈H\n\nep(ax )\n\n∣∣∣∣∣ = o(|H|)to hold for all a 6 = 0?", "clean_statement": "Problem 3.3 (J. Bourgain). Suppose that H ≤ F∗\n\n> p. How large must H be in order that ∣∣∣∣∣∑\n\n> x∈H\n\nep(ax )\n\n∣∣∣∣∣ = o(|H|)to hold for all a 6 = 0?", "statement_status": "exact", "statement_verification": "The repository record is visibly damaged by PDF extraction: subscripts, superscripts, the absolute-value bars, and the not-equal sign have been split across lines. The official AIM PDF, page 8 of the printed document (PDF page 7), displays Problem 3.3 as follows.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.3\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[239]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.3 (J. Bourgain). Suppose that H ≤ F∗\\n\\n> p. How large must H be in order that ∣∣∣∣∣∑\\n\\n> x∈H\\n\\nep(ax )\\n\\n∣∣∣∣∣ = o(|H|)to hold for all a 6 = 0?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0240", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The OCR-damaged question is recovered as uniform additive-character cancellation over H <= F_p^*. Published work gives the sufficient condition |H| > exp(57 log p/log log p), while the conjectured |H|/log p -> infinity threshold remains open. This attempt proves exact multiplicative-coset Fourier moment identities and the mixing bound |r_k(t)-|H|^k/p| <= M(H)^(k-2)|H|(1-|H|/p), explicitly connecting spectral cancellation to kH=F_p. Its strongest new obstruction is that whenever p=(g^q-1)/(g-1) is prime with q prime, H= has order q and |sum_{x in H} e_p(x)-q| <= 2 pi; hence any infinite fixed-base family of such prime repunits has logarithmic-size subgroups with M(H)/|H| -> 1.\n\nCandidate contribution (obstruction; novelty confidence low): Prime-repunit phase alignment: if g >= 2, q is prime, and p=(g^q-1)/(g-1) is prime, then the order-q subgroup H= of F_p^* satisfies |sum_{x in H} e_p(x)-q| <= 2 pi. Consequently, infinitely many such primes in one fixed base would give a logarithmic-scale family with essentially no cancellation." }, { "id": 20001116, "problem_number": "AIM-COMBINATORICS-0241", "title": "Two-step product growth in SL_2(F_p)", "statement": "Problem 3.4 (Brought by B. Green, originally stated by A. Venkatesh). Let A ⊆\n\nSL 2(Fp) satisfy |A| ∼ p5/2. Does it follow that\n\n|A · A| > p5/2+ δ?", "original_statement": "Problem 3.4 (Brought by B. Green, originally stated by A. Venkatesh). Let A ⊆\n\nSL 2(Fp) satisfy |A| ∼ p5/2. Does it follow that \n\n|A · A| > p5/2+ δ?", "clean_statement": "Problem 3.4 (Brought by B. Green, originally stated by A. Venkatesh). Let A ⊆\n\nSL 2(Fp) satisfy |A| ∼ p5/2. Does it follow that\n\n|A · A| > p5/2+ δ?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 3.4 in the AIM workshop list *Recent Trends in Additive Combinatorics*. The official PDF was inspected directly. Its mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.4\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[240]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.4 (Brought by B. Green, originally stated by A. Venkatesh). Let A ⊆\\n\\nSL 2(Fp) satisfy |A| ∼ p5/2. Does it follow that \\n\\n|A · A| > p5/2+ δ?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0241", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM problem is solved by the Babai--Nikolov--Pyber two-step product-mixing theorem. For G=SL_2(F_p), every nontrivial complex representation has dimension at least (p-1)/2, so their product bound gives |AA| >> p^3 whenever |A| is comparable to p^(5/2). Hence the requested strict inequality holds for every fixed delta<1/2, for example delta=1/4, for all sufficiently large primes. The artifacts include a self-contained noncommutative Fourier proof and an elementary unipotent-character proof of the representation-degree bound.\n\nCandidate contribution (quantitative_refinement; novelty confidence low): If G has order n and every nontrivial irreducible complex representation has dimension at least d, then every nonempty A,B subset G satisfies |AB| >= n/[1 + (n^2/(d|A||B|))(1-|A|/n)(1-|B|/n)]." }, { "id": 20001117, "problem_number": "AIM-COMBINATORICS-0242", "title": "Small doubling, tripling, and balanced quotients below two", "statement": "Problem 3.5 (T. Tao). Let A be a finite subset of a (not necessarily abelian) group. Given that A · A is small, is A · A · A necessarily small, too? T. Tao comments: it was pointed out to me by Ben Green and Mei-Chu Chang that the problem as stated is false, as it follows by considering A = H ∪ { x} where H is a non-normal subgroup and x is not in the normalizer of H. However, what does appear to be true is that there is a large subset A′ of A such that A′A′A′ is small. A more ambitious problem would be to attempt an inverse theorem; for instance, if |AA | < 2|A|,what can one say about A?", "original_statement": "Problem 3.5 (T. Tao). Let A be a finite subset of a (not necessarily abelian) group. Given that A · A is small, is A · A · A necessarily small, too? T. Tao comments: it was pointed out to me by Ben Green and Mei-Chu Chang that the problem as stated is false, as it follows by considering A = H ∪ { x} where H is a non-normal subgroup and x is not in the normalizer of H. However, what does appear to be true is that there is a large subset A′ of A such that A′A′A′ is small. A more ambitious problem would be to attempt an inverse theorem; for instance, if |AA | < 2|A|,what can one say about A?", "clean_statement": "Problem 3.5 (T. Tao). Let A be a finite subset of a (not necessarily abelian) group. Given that A · A is small, is A · A · A necessarily small, too? T. Tao comments: it was pointed out to me by Ben Green and Mei-Chu Chang that the problem as stated is false, as it follows by considering A = H ∪ { x} where H is a non-normal subgroup and x is not in the normalizer of H. However, what does appear to be true is that there is a large subset A′ of A such that A′A′A′ is small. A more ambitious problem would be to attempt an inverse theorem; for instance, if |AA | < 2|A|,what can one say about A?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 3.5 in *Problems Presented at the Workshop on Recent Trend in Additive Combinatorics*, collected by Ernie Croot and Vsevolod F. Lev. The official AIM PDF, page 8 (PDF page index 7), says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.5\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[241]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.5 (T. Tao). Let A be a finite subset of a (not necessarily abelian) group. Given that A · A is small, is A · A · A necessarily small, too? T. Tao comments: it was pointed out to me by Ben Green and Mei-Chu Chang that the problem as stated is false, as it follows by considering A = H ∪ { x} where H is a non-normal subgroup and x is not in the normalizer of H. However, what does appear to be true is that there is a large subset A′ of A such that A′A′A′ is small. A more ambitious problem would be to attempt an inverse theorem; for instance, if |AA | < 2|A|,what can one say about A?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0242", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal small-doubling-to-small-tripling implication is quantitatively false, but the corrected large-subset assertion holds with explicit bounds: if |A^2| <= K|A|, some A' subset A has |A'| >= |A|/K and |(A')^3| <= K^3|A'|. In the strict threshold-two regime, a direct new synthesis proves that if d = 2|A|-|A^2| > 0, then A^{-1}A = AA^{-1}; every element of this common quotient set has at least d representations in each orientation, so its size is at most |A|^2/d. Later weak-Kneser and critical-pair theorems provide the broader inverse structure requested by the source.\n\nCandidate contribution (lemma; novelty confidence low): Candidate balanced-quotient lemma: every finite nonempty subset A of any group with d=2|A|-|A^2|>0 satisfies A^{-1}A=AA^{-1}, each element has at least d representations as both a^{-1}b and ab^{-1}, and the common set has size at most |A|^2/d; the strict threshold is sharp." }, { "id": 20001118, "problem_number": "AIM-COMBINATORICS-0243", "title": "Square-difference sets, Paley cliques, and an exact external-defect variance identity", "statement": "Problem 3.6 (Brought by V. Lev). Given a prime p, how large can a set A ⊆ Fp be given that the difference between any two elements of A is a quadratic residue modulo\n\np?", "original_statement": "Problem 3.6 (Brought by V. Lev). Given a prime p, how large can a set A ⊆ Fp be given that the difference between any two elements of A is a quadratic residue modulo \n\np?", "clean_statement": "Problem 3.6 (Brought by V. Lev). Given a prime p, how large can a set A ⊆ Fp be given that the difference between any two elements of A is a quadratic residue modulo\n\np?", "statement_status": "exact", "statement_verification": "The AIM workshop PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.6\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[242]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.6 (Brought by V. Lev). Given a prime p, how large can a set A ⊆ Fp be given that the difference between any two elements of A is a quadratic residue modulo \\n\\np?\"\nOriginal remarks: [\"Remark(s). This is actually an old problem on which nothing is known beyond the estimate |A| < √p. A simple elementary proof is as follows. Suppose that |A| > √p.Then for any x ∈ Fp there exist a1, b 1, a 2, b 2 ∈ A such that a1x + b1 = a2x + b2 and \\n\\na1 6 = a2. Consequently, x = ( b1 −b2)/(a2 −a1) and since any x ∈ Fp has a representation of this form, the set of all non-zero elements of A−A is not contained in a multiplicative subgroup of Fp.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0243", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source remark shows that all ordered nonzero differences are required. Accordingly the maximum is 2 for p=2, is 1 for odd p congruent to 3 modulo 4, and equals the Paley graph clique number for p congruent to 1 modulo 4. In the nontrivial case, Hanson-Petridis gives the current closed-form upper bound n(n-1) <= (p-1)/2, hence n <= (1+sqrt(2p-1))/2. This attempt additionally proves exact first and second moments, and an exact variance formula, for the number of nonsquare differences from each external vertex to any Paley clique.\n\nCandidate contribution (moment_identity; novelty confidence low): If A is a Paley clique of size n in F_p and nu_A(x) counts nonsquare differences x-a for x outside A, then sum nu_A(x)=n(p-1)/2, sum nu_A(x)^2=n(n+1)(p-1)/4, and the sum of squared deviations from n(p-1)/(2(p-n)) equals n(p-1)(p-n^2)/(4(p-n))." }, { "id": 20001119, "problem_number": "AIM-COMBINATORICS-0244", "title": "A divisor-profile lower bound and a polynomial-height near-Sidon theorem for squares", "statement": "Problem 3.7 (Brought by B. Green and T. Tao). Take a finite subset of the squares of integers, |A| = N. What lower bound can you get on A + A? One can get better than cN via Freiman's theorem. B. Green comments: I am not sure who posed this. It is certainly implicit in a paper of Chang on Rudin's problem (are the squares a Λ( p) set?)", "original_statement": "Problem 3.7 (Brought by B. Green and T. Tao). Take a finite subset of the squares of integers, |A| = N. What lower bound can you get on A + A? One can get better than cN via Freiman's theorem. B. Green comments: I am not sure who posed this. It is certainly implicit in a paper of Chang on Rudin's problem (are the squares a Λ( p) set?)", "clean_statement": "Problem 3.7 (Brought by B. Green and T. Tao). Take a finite subset of the squares of integers, |A| = N. What lower bound can you get on A + A? One can get better than cN via Freiman's theorem. B. Green comments: I am not sure who posed this. It is certainly implicit in a paper of Chang on Rudin's problem (are the squares a Λ( p) set?)", "statement_status": "exact", "statement_verification": "The canonical record is Problem 3.7 from the AIM workshop *Recent trends in additive combinatorics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.7\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[243]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.7 (Brought by B. Green and T. Tao). Take a finite subset of the squares of integers, |A| = N. What lower bound can you get on A + A? One can get better than cN via Freiman's theorem. B. Green comments: I am not sure who posed this. It is certainly implicit in a paper of Chang on Rudin's problem (are the squares a Λ( p) set?)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0244", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For A=X^2 with X a finite set of N nonnegative integers, the additive energy satisfies E(A) <= N^2 + sum_{x != z} tau(|x^2-z^2|), and therefore |A+A| >= N^4/(N^2 + sum_{x != z} tau(|x^2-z^2|)). If all roots are at most H, this gives |A+A| >= N^2/(1 + max_{m <= H^2} tau(m)); hence H <= N^C for fixed C implies |A+A| >= N^{2-o_C(1)}. Conversely, |A+A| <= N^{1+c} forces the average divisor count of |x^2-z^2| over ordered distinct root pairs to be at least N(N^{1-c}-1)/(N-1).\n\nCandidate contribution (lemma; novelty confidence low): Candidate divisor-profile certificate: |X^2+X^2| >= |X|^4/(|X|^2 + sum_{x != z} tau(|x^2-z^2|)), with the explicit contrapositive that any N-element square set satisfying |A+A| <= N^{1+c} has average divisor profile at least N(N^{1-c}-1)/(N-1)." }, { "id": 20001120, "problem_number": "AIM-COMBINATORICS-0245", "title": "The finite-field Kakeya problem in dimension three", "statement": "Problem 3.8 (T. Tao). Take F to be a finite field, and suppose that E ⊆ F × F × F,where E is a Besicovich set; i.e. E contains a line in every direction. It is known from the work of Wolf that |E| ≥ | F|5/2; prove the better lower bound |E| ≥ | F|5/2+ ≤.", "original_statement": "Problem 3.8 (T. Tao). Take F to be a finite field, and suppose that E ⊆ F × F × F,where E is a Besicovich set; i.e. E contains a line in every direction. It is known from the work of Wolf that |E| ≥ | F|5/2; prove the better lower bound |E| ≥ | F|5/2+ ≤.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Problem 3.8 in *Problems Presented at the Workshop on Recent Trend in Additive Combinatorics*, collected by Ernie Croot and Vsevolod F. Lev. The official AIM PDF, page 8 (PDF page index 7), reads, after repairing its text layer:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.8\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[244]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.8 (T. Tao). Take F to be a finite field, and suppose that E ⊆ F × F × F,where E is a Besicovich set; i.e. E contains a line in every direction. It is known from the work of Wolf that |E| ≥ | F|5/2; prove the better lower bound |E| ≥ | F|5/2+ ≤.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0245", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The damaged source is the finite-field Kakeya problem asking to improve the Wolff exponent 5/2 to 5/2+epsilon. Dvir's polynomial method solves it with the optimal exponent: every Kakeya set E in F_q^3 has at least binom(q+2,3) points, hence at least q^3/6. A self-contained proof is given. The current sharper result of Bukh and Chao is |E| >= (q^3+q)/4, asymptotically matching constructions. As a developed extension, if E contains full lines in |S| non-horizontal directions (u,v,1) and |S|>mq for 0<=m<=q-1, then |E|>=binom(m+3,3).\n\nCandidate contribution (partial-direction theorem; novelty confidence low): Candidate exact partial-direction interpolation: if E in F_q^3 contains a full affine line in every direction (u,v,1) indexed by S subset F_q^2, and |S|>mq for an integer 0<=m<=q-1, then |E| is at least binom(m+3,3); equivalently, with r=ceil(|S|/q), |E| is at least binom(r+2,3)." }, { "id": 20001121, "problem_number": "AIM-COMBINATORICS-0246", "title": "Direction-complete finite-field sets and quadratic secant lifting", "statement": "Problem 3.9 (A. Granville). Given a finite field F and an integer n ≥ 1, find the smallest size of a subset E ⊂ F n which determines all directions in Fn.PALO ALTO PROBLEMS 9", "original_statement": "Problem 3.9 (A. Granville). Given a finite field F and an integer n ≥ 1, find the smallest size of a subset E ⊂ F n which determines all directions in Fn.PALO ALTO PROBLEMS 9", "clean_statement": "Problem 3.9 (A. Granville). Given a finite field F and an integer n ≥ 1, find the smallest size of a subset E ⊂ F n which determines all directions in Fn.PALO ALTO PROBLEMS 9", "statement_status": "exact", "statement_verification": "The canonical record is an OCR extraction of Problem 3.9 from the AIM workshop list *Problems presented at the workshop: Additive Combinatorics*. The PDF reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.9\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[245]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.9 (A. Granville). Given a finite field F and an integer n ≥ 1, find the smallest size of a subset E ⊂ F n which determines all directions in Fn.PALO ALTO PROBLEMS 9\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0246", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The corrupted source is recovered as the secant-direction problem in F_q^n, not a Kakeya or affine blocking-set problem. Croot and Lev's later publication records Konyagin's order-sharp reduction to difference bases: the exact pair-counting lower bound and the Kozma-Lev construction imply M_q(n)=Theta(q^{(n-1)/2}) with absolute constants, while the general exact value remains unresolved. This attempt proves an exact two-layer equivalence and an odd-characteristic quadratic secant-lifting lemma yielding M_q(a+b) <= q^a + M_q(b) - 1, together with the exact obstruction to that graph-alone mechanism in characteristic two.\n\nCandidate contribution (explicit_construction; novelty confidence low): For odd q, a >= b >= 1, every surjective F_q-linear map pi:F_{q^a}->F_q^b has the property that the graph {(x,pi(x^2))} has difference set containing every (u,w) with u nonzero; adjoining a vertical direction-complete set proves M_q(a+b) <= q^a + M_q(b) - 1. In characteristic two, a chord with first difference u has fixed second difference pi(u^2), so the graph-alone assertion fails whenever pi is nonzero." }, { "id": 20001122, "problem_number": "AIM-COMBINATORICS-0247", "title": "Finite-field point-line incidences and a projective no-wrap criterion", "statement": "Problem 3.10 (T. Tao). Find an analogue for the Szemer´ edi-Trotter theorem for\n\nFp × Fp. More precisely, suppose we have a system of n points and l lines in Fp × Fp.Does the number i of point-line incidences necessarily satisfy\n\ni ø (nl )2/3 + n + l?In particular, if both n and l are about log p, is it true that i = O(( nl )2/3)?", "original_statement": "Problem 3.10 (T. Tao). Find an analogue for the Szemer´ edi-Trotter theorem for \n\nFp × Fp. More precisely, suppose we have a system of n points and l lines in Fp × Fp.Does the number i of point-line incidences necessarily satisfy \n\ni ø (nl )2/3 + n + l?In particular, if both n and l are about log p, is it true that i = O(( nl )2/3)?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The record is Problem 3.10, attributed to T. Tao, in the AIM workshop list *Recent trends in additive combinatorics*. Direct inspection of the official PDF gives the following statement (notation normalized but not changed mathematically):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.10\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[246]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.10 (T. Tao). Find an analogue for the Szemer´ edi-Trotter theorem for \\n\\nFp × Fp. More precisely, suppose we have a system of n points and l lines in Fp × Fp.Does the number i of point-line incidences necessarily satisfy \\n\\ni ø (nl )2/3 + n + l?In particular, if both n and l are about log p, is it true that i = O(( nl )2/3)?\"\nOriginal remarks: [\"Remark(s). For n = l = p a recent paper by Bourgain, Katz, and the presenter shows that the trivial bound ( nl )3/2 can be improved to ( nl )3/2−≤ for some explicit but very small ≤ > 0. The presenter indicates that if n and l are both large then i ø (nl )2/3 +n+l\\n\\nmay fail: for n = l = p2 one can get p3 incidences.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0247", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted proposed Szemeredi-Trotter inequality is false in the dense regime: all p^2 affine points and p^2 affine lines give p^3 incidences. The arbitrary balanced N asymptotic to log p case was not found resolved. A proved structured special case is obtained by projective rectification: if selected point lifts in Z^3 have height H_P, selected line-normal lifts have height H_L, and 3 H_P H_L < p, then reduction modulo p preserves the complete selected incidence graph over the real projective plane, so the real Szemeredi-Trotter bound applies.\n\nCandidate contribution (theorem; novelty confidence low): Candidate projective no-wrap theorem: a finite point-line configuration in P^2(F_p) admitting integer homogeneous point lifts of height H_P and integer line-normal lifts of height H_L with 3 H_P H_L < p satisfies I(P,L) << (|P||L|)^(2/3) + |P| + |L|, because its entire selected incidence graph embeds unchanged in P^2(R)." }, { "id": 20001123, "problem_number": "AIM-COMBINATORICS-0248", "title": "Small sums and products near the square-root scale", "statement": "Problem 3.11 (J. Solymosi). Do there exist sets A, B, C ⊆ Fp with |A|, |B|, |C| ≈ √p\n\nand |A + B| + |AC | < p?", "original_statement": "Problem 3.11 (J. Solymosi). Do there exist sets A, B, C ⊆ Fp with |A|, |B|, |C| ≈ √p\n\nand |A + B| + |AC | < p?", "clean_statement": "Problem 3.11 (J. Solymosi). Do there exist sets A, B, C ⊆ Fp with |A|, |B|, |C| ≈ √p\n\nand |A + B| + |AC | < p?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 3.11, attributed to J. Solymosi, from the AIM workshop list *Problems Presented at the Workshop on Recent Trend in Additive Combinatorics*. Its literal question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.11\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[247]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.11 (J. Solymosi). Do there exist sets A, B, C ⊆ Fp with |A|, |B|, |C| ≈ √p\\n\\nand |A + B| + |AC | < p?\"\nOriginal remarks: [\"Remark(s). The presenter suspected that there can be a counterexample, and in-deed this was noticed by T. Tao and N. Alon. Tao suggests taking A = B = C =[1, b√p/ 100 c]. Alon comments: take \\n\\nA = B = C = {1, 2, 3,.., k = √p}.\\n\\nThen, |A + B| is about size 2 k = 2 √p and |AC | is the number of distinct elements in the multiplication table of size k by k, which is, as is well known, (and as follows easily from the fact that almost all numbers between 1 and k have about log log k prime divisors) o(k2) = o(p). Tao observes that the problem becomes non-trivial if one replaces |A|, |B|, |C| ≈ √p\\n\\nby |A|, |B|, |C| ≈ p1−≤ with ≤ < 0.5, or |A + B| + |AC | < p by |A + B| + |AC | = o(p).\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0248", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal existence question and both printed strengthenings have affirmative answers. The interval [1,floor(sqrt(p))] has sumset size 2 floor(sqrt(p))-1 and product-set size equal to the integer multiplication-table function M(floor(sqrt(p)))=o(p). More generally, for every fixed epsilon>0 and N=floor(p^(1-epsilon)), there is an N-point set X in F_p such that |X+X|+|XX| is at most 4 ceil(sqrt(pN))-2=O(p^(1-epsilon/2))=o(p); taking A=B=C=X solves the larger-set variant for 0= 2 and n <= s <= n^2 with s dividing n^2, there is an explicit configuration of exactly n^2 lines with plane cap s and n selected points per line supported on at most 4 n^(7/2) s^(-1/2) points; the proof also controls planes other than the designated parallel block planes." }, { "id": 20001125, "problem_number": "AIM-COMBINATORICS-0250", "title": "The joints theorem and fixed-direction stability", "statement": "Problem 3.13 (T. Tao). Take n lines in R3. Define a joint to be a point with three lines passing through it that are not coplanar. How many joints can there be?", "original_statement": "Problem 3.13 (T. Tao). Take n lines in R3. Define a joint to be a point with three lines passing through it that are not coplanar. How many joints can there be?", "clean_statement": "Problem 3.13 (T. Tao). Take n lines in R3. Define a joint to be a point with three lines passing through it that are not coplanar. How many joints can there be?", "statement_status": "exact", "statement_verification": "The record is Problem 3.13, attributed to T. Tao, in the AIM workshop list *Recent trends in additive combinatorics*. Direct inspection of the official PDF recovers the statement as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.13\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[249]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.13 (T. Tao). Take n lines in R3. Define a joint to be a point with three lines passing through it that are not coplanar. How many joints can there be?\"\nOriginal remarks: [\"Remark(s). It is known that there are at least n3/2, and a trivial upper bound is n2.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0250", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM question is solved: N distinct lines in R^3 determine Theta(N^(3/2)) joints. A self-contained pruning and polynomial derivative-descent proof gives the explicit upper bound |J| < 5 N^(3/2), while grid and generic-plane constructions give the matching order. Current sharp work gives the exact generalized-binomial bound. In addition, a proved fixed-direction result gives a constant-one weighted Loomis-Whitney inequality and explicit near-extremal balance and active-line estimates.\n\nCandidate contribution (stability_bound; novelty confidence low): Candidate fixed-direction stability statement: if three line families in independent fixed directions have total size N and at least (1-epsilon)(N/3)^(3/2) common joints, then every family size N_i satisfies |N_i-N/3| <= (N/3)sqrt(8 epsilon), and at least a (1-epsilon)^2 fraction of every family consists of lines containing a joint; this follows from a sharp weighted constant-one incidence form of Loomis-Whitney." }, { "id": 20001126, "problem_number": "AIM-COMBINATORICS-0251", "title": "Joints of lines over finite fields", "statement": "Problem 3.14 (T. Tao). Same problem, but over finite fields.", "original_statement": "Problem 3.14 (T. Tao). Same problem, but over finite fields.", "clean_statement": "Problem 3.14 (T. Tao). Same problem, but over finite fields.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.14\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[250]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.14 (T. Tao). Same problem, but over finite fields.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0251", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The finite-field joints problem has the same sharp L^{3/2} order as the real problem, with no characteristic or field-size hypothesis. A self-contained minimal-degree polynomial and deletion argument proves J <= (6^{1/3}+1)^{3/2} L^{3/2}; the positive-characteristic zero-derivative case is closed by taking a p-th root over the perfect finite field. The current exact arbitrary-field theorem of Chao and Yu yields J <= L(sqrt(8L+1)-3)/6.\n\nCandidate contribution (finite_field_bound_synthesis; novelty confidence low): For L distinct affine lines in F_q^3, the explicit three-ceiling envelope J <= min{q^3, qL/3, L(sqrt(8L+1)-3)/6} holds; the axis-grid family simultaneously attains the first two ceilings at L=3q^2, while an explicit Vandermonde-plane family attains the third whenever L=binom(M,2) with M <= q." }, { "id": 20001127, "problem_number": "AIM-COMBINATORICS-0252", "title": "Literal normalization obstruction and the solved polynomial Freiman-Ruzsa theorem", "statement": "Problem 3.15 (Brought by T. Tao, originally stated by I. Ruzsa). Choose A ⊆ Fn\n\n> 2\n\nsuch that\n\n|A + A| ≤ k|A|.\n\nDoes there exist a subspace V ⊆ Fn\n\n> 2\n\nsuch that |V | < k c|A|, and\n\n|A ∩ V | ≥ k−c|A|?", "original_statement": "Problem 3.15 (Brought by T. Tao, originally stated by I. Ruzsa). Choose A ⊆ Fn\n\n> 2\n\nsuch that \n\n|A + A| ≤ k|A|.\n\nDoes there exist a subspace V ⊆ Fn \n\n> 2\n\nsuch that |V | < k c|A|, and \n\n|A ∩ V | ≥ k−c|A|?", "clean_statement": "Problem 3.15 (Brought by T. Tao, originally stated by I. Ruzsa). Choose A ⊆ Fn\n\n> 2\n\nsuch that\n\n|A + A| ≤ k|A|.\n\nDoes there exist a subspace V ⊆ Fn\n\n> 2\n\nsuch that |V | < k c|A|, and\n\n|A ∩ V | ≥ k−c|A|?", "statement_status": "exact", "statement_verification": "The official AIM PDF gives the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.15\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[251]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.15 (Brought by T. Tao, originally stated by I. Ruzsa). Choose A ⊆ Fn\\n\\n> 2\\n\\nsuch that \\n\\n|A + A| ≤ k|A|.\\n\\nDoes there exist a subspace V ⊆ Fn \\n\\n> 2\\n\\nsuch that |V | < k c|A|, and \\n\\n|A ∩ V | ≥ k−c|A|?\"\nOriginal remarks: [\"Remark(s). An equivalent reformulation due to Ruzsa is as follows. Suppose that \\n\\nf: Fm \\n\\n> 2\\n\\n→ F∞\\n\\n> 2\\n\\nand consider \\n\\nS:= {f (x + y) − f (x) − f (y): x, y ∈ Fm \\n\\n> 2\\n\\n}.10 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV \\n\\nCan f be written as f = g + h, where g is linear and the image of h has size polynomial in |S|?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0252", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The exact AIM wording is false: a non-linear affine coset A=a+H has doubling constant 1 but cannot have the required large intersection with a linear subspace of size less than |A|, and the family A=a+(H minus {0}) gives a robust obstruction for every fixed exponent as the doubling constant tends to 1. The intended PFR statement is nevertheless solved: Liao's 2024 exponent-nine covering theorem yields the exact AIM inequalities with c=10 for K>=2, and Gowers-Green-Manners-Tao Corollary 1.4 directly proves Ruzsa's approximate-homomorphism formulation.\n\nCandidate contribution (counterexample_and_quantitative_corollary; novelty confidence low): For every fixed c>0, the explicit family A=a+(H minus {0}) with |H|=q has K=q/(q-1) tending to 1 and defeats both pure K^{+c} and K^{-c} linear-subspace bounds; in contrast, Liao's 2K^9 coset cover implies the precise AIM intersection bounds with the single exponent c=10 whenever K>=2." }, { "id": 20001128, "problem_number": "AIM-COMBINATORICS-0253", "title": "Uniform iterated-sumset growth at the first doubling threshold", "statement": "Problem 3.16 (T. Tao). Suppose A ⊆ Z. If |2A| < k |A|, describe the sumset |nA | as\n\nn tends to infinity. Find f (n, k ), which is the smallest number such that\n\n|nA | ≤ f (n, k )|A|\n\nfor all A such that |2A| < k |A|.", "original_statement": "Problem 3.16 (T. Tao). Suppose A ⊆ Z. If |2A| < k |A|, describe the sumset |nA | as \n\nn tends to infinity. Find f (n, k ), which is the smallest number such that \n\n|nA | ≤ f (n, k )|A|\n\nfor all A such that |2A| < k |A|.", "clean_statement": "Problem 3.16 (T. Tao). Suppose A ⊆ Z. If |2A| < k |A|, describe the sumset |nA | as\n\nn tends to infinity. Find f (n, k ), which is the smallest number such that\n\n|nA | ≤ f (n, k )|A|\n\nfor all A such that |2A| < k |A|.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 3.16 from the AIM workshop *Recent trends in additive combinatorics*, attributed to T. Tao:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.16\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[252]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.16 (T. Tao). Suppose A ⊆ Z. If |2A| < k |A|, describe the sumset |nA | as \\n\\nn tends to infinity. Find f (n, k ), which is the smallest number such that \\n\\n|nA | ≤ f (n, k )|A|\\n\\nfor all A such that |2A| < k |A|.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0253", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general exact extremal function remains open, but the first threshold range is resolved in growth order. For every n >= 1, f(n,2)=n exactly. For every real 2=2, a base construction forces uniform degree at least ceil(2k)-3.\n\nCandidate contribution (threshold_theorem; novelty confidence low): Candidate uniform threshold theorem: f(n,2)=n, while for every real 2=1 one has (n+1)(n+2)/6 <= f(n,k) <= (2n^2+2n+1)/5; hence the uniform growth exponent jumps discontinuously from one to two at k=2." }, { "id": 20001129, "problem_number": "AIM-COMBINATORICS-0254", "title": "A singleton obstruction to the Katz critical-set question", "statement": "Problem 3.17 (N. Katz). Does there exist a finite subset A ⊂ Z, |2A| = k|A|, so that for the function f (n, k ) defined in the previous problem every proper A′ ⊂ A satisfies\n\n|nA ′| ≥ f (n, k )|A′|?", "original_statement": "Problem 3.17 (N. Katz). Does there exist a finite subset A ⊂ Z, |2A| = k|A|, so that for the function f (n, k ) defined in the previous problem every proper A′ ⊂ A satisfies \n\n|nA ′| ≥ f (n, k )|A′|?", "clean_statement": "Problem 3.17 (N. Katz). Does there exist a finite subset A ⊂ Z, |2A| = k|A|, so that for the function f (n, k ) defined in the previous problem every proper A′ ⊂ A satisfies\n\n|nA ′| ≥ f (n, k )|A′|?", "statement_status": "exact", "statement_verification": "The official AIM PDF confirms all three potentially surprising features: Problem 3.16 uses the strict inequality \\(<\\), Problem 3.17 puts \\(A\\) on the equality boundary \\(=\\), and the required inequality for every proper subset is \\(\\geq\\). These are not extraction errors.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.17\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[253]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.17 (N. Katz). Does there exist a finite subset A ⊂ Z, |2A| = k|A|, so that for the function f (n, k ) defined in the previous problem every proper A′ ⊂ A satisfies \\n\\n|nA ′| ≥ f (n, k )|A′|?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0254", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the natural literal reading with fixed integer n >= 2 and real k > 1, define f(n,k) using the strict doubling condition from Problem 3.16. A nontrivial finite A satisfying |2A|=k|A| and |nA'| >= f(n,k)|A'| for every proper subset exists if and only if k=3/2, and then exactly the two-element sets work. Thus the answer is uniformly negative if k was intended to be an integer at least 2. The classification follows because singleton proper subsets force f(n,k)=1, which occurs exactly for 1 < k <= 3/2.\n\nCandidate contribution (obstruction; novelty confidence low): For any finite A in an abelian group with |A| > r >= 1, min_{A' proper subset of A, |A'| >= r} |nA'|/|A'| <= binom(n+r-1,r-1)/r. Hence any cardinality-cutoff repair of the Katz condition has this universal cap; r=1 recovers the singleton collapse." }, { "id": 20001130, "problem_number": "AIM-COMBINATORICS-0255", "title": "Exact horizontal Plancherel identities for the diagonal Diffie--Hellman sum", "statement": "Problem 3.18 (J. Bourgain). Find a good upper bound for the absolute value of the exponential sum ∑\n\n> x∈Fp\n\nep(aθ x + bθ x2\n\n),\n\nwhere θ is a generator for F∗\n\n> p.", "original_statement": "Problem 3.18 (J. Bourgain). Find a good upper bound for the absolute value of the exponential sum ∑\n\n> x∈Fp\n\nep(aθ x + bθ x2\n\n),\n\nwhere θ is a generator for F∗\n\n> p.", "clean_statement": "Problem 3.18 (J. Bourgain). Find a good upper bound for the absolute value of the exponential sum ∑\n\n> x∈Fp\n\nep(aθ x + bθ x2\n\n),\n\nwhere θ is a generator for F∗\n\n> p.", "statement_status": "exact", "statement_verification": "The corpus OCR reads", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.18\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[254]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.18 (J. Bourgain). Find a good upper bound for the absolute value of the exponential sum ∑\\n\\n> x∈Fp\\n\\nep(aθ x + bθ x2\\n\\n),\\n\\nwhere θ is a generator for F∗\\n\\n> p.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0255", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After repairing the source's noncanonical indexing, the canonical full-period sum T(a,b)=sum_{x mod p-1} e_p(a theta^x+b theta^{x^2}) and the literal representative sum S over x=0,...,p-1 satisfy S=T+e_p(a+b). For every fixed b, their exact second moments over a are respectively p(p-1) and p(p+2). Hence both have square-root size for all but an explicitly bounded exceptional set of first coefficients. The coefficient line b=0 is also evaluated exactly. This is an average-in-a partial result, not the requested worst-case pointwise bound.\n\nCandidate contribution (exact_identity_and_reduction; novelty confidence low): The endpoint-aware reduction S(a,b)=T(a,b)+e_p(a+b), together with the exact identities sum_a |T(a,b)|^2=p(p-1) and sum_a |S(a,b)|^2=p(p+2), gives a uniform-in-b, explicit square-root exceptional-set theorem while reconciling the AIM formula with the canonical exponent-period formulation." }, { "id": 20001131, "problem_number": "AIM-COMBINATORICS-0256", "title": "Compact Freiman models and relation peeling", "statement": "Problem 3.19 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nFor any integer n ≥ 2 the set {0, 1, 2, 4,..., 2n−2} is \"linear\" (has Freiman rank one) and not contained in an arithmetic progression with difference larger than one, hence it is not isomorphic to a set of integers of length smaller than 2 n−2. Is this the extremal case? That is, is it true that in any class of isomorphic n-element sets there is a set of length at most 2 n−2? Conjecture: for n ≥ 7 any n-element set of integers is isomorphic (in Freiman's sense) to a subset of [0, 2n−2].", "original_statement": "Problem 3.19 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nFor any integer n ≥ 2 the set {0, 1, 2, 4,..., 2n−2} is \"linear\" (has Freiman rank one) and not contained in an arithmetic progression with difference larger than one, hence it is not isomorphic to a set of integers of length smaller than 2 n−2. Is this the extremal case? That is, is it true that in any class of isomorphic n-element sets there is a set of length at most 2 n−2? Conjecture: for n ≥ 7 any n-element set of integers is isomorphic (in Freiman's sense) to a subset of [0, 2n−2].", "clean_statement": "Problem 3.19 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nFor any integer n ≥ 2 the set {0, 1, 2, 4,..., 2n−2} is \"linear\" (has Freiman rank one) and not contained in an arithmetic progression with difference larger than one, hence it is not isomorphic to a set of integers of length smaller than 2 n−2. Is this the extremal case? That is, is it true that in any class of isomorphic n-element sets there is a set of length at most 2 n−2? Conjecture: for n ≥ 7 any n-element set of integers is isomorphic (in Freiman's sense) to a subset of [0, 2n−2].", "statement_status": "exact", "statement_verification": "The canonical JSON has lost superscripts. Inspection of the official AIM PDF and the original Konyagin--Lev formulation recovers the statement as follows.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 3.19\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[255]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3.19 (Brought by V. Lev, originally stated by Konyagin and the presenter).\\n\\nFor any integer n ≥ 2 the set {0, 1, 2, 4,..., 2n−2} is \\\"linear\\\" (has Freiman rank one) and not contained in an arithmetic progression with difference larger than one, hence it is not isomorphic to a set of integers of length smaller than 2 n−2. Is this the extremal case? That is, is it true that in any class of isomorphic n-element sets there is a set of length at most 2 n−2? Conjecture: for n ≥ 7 any n-element set of integers is isomorphic (in Freiman's sense) to a subset of [0, 2n−2].\"\nOriginal remarks: [\"Remark(s). All Sidon sets of the same cardinality are isomorphic to each other, and it is well-known that for N large enough the interval [0, N ] contains a Sidon set of cardinality about √N. Thus, any n-element Sidon set is isomorphic to a subset of [0, n 2(1 + o(1))]. For n ≤ 6, however, this n2(1 + o(1)) turns out to be larger than 2n−2: that is, [0, 2n−2] contains no n-element Sidon set. This is the only reason for the restriction n ≥ 7 in the problem above. 4. Erd˝ os Distance and Kakea Problem Session\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0256", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The OCR-corrected Konyagin--Lev conjecture asks for a Freiman-isomorphic model in [0,2^{n-2}], not [0,2n-2]. The sharp powers-of-two lower example is verified directly. A new explicit special case is proved: if the elements can be ordered so that each new element participates in a pair-sum relation with three earlier elements, recursive normalization gives an integral affine Freiman model whose diameter at most doubles at each step, hence is at most 2^{n-2}.\n\nCandidate contribution (special_case_and_equality_obstruction; novelty confidence low): For every relation-peelable n-element integer set, the recursive normalized Freiman model has diameter at most 2^{n-2}; equality holds exactly when every step adjoins 2u-v or 2v-u from the old extrema u,v, with the defining relation using the same endpoint twice and subtracting the opposite endpoint. Thus the extremal peelable models are endpoint-reflection chains." }, { "id": 20001132, "problem_number": "AIM-COMBINATORICS-0257", "title": "A linear unit-distance bound for algebraically concentrated p-point sets", "statement": "Problem 4.1 (T. Tao). Suppose that S ⊆ F2\n\n> p\n\nwith |S| = p. How many pairs of points of distance one apart can there be? That is, how large can #{(x, y ) ∈ S × S: ( x1 − x2)2 + ( y1 − y2)2 = 1 }\n\nbe?", "original_statement": "Problem 4.1 (T. Tao). Suppose that S ⊆ F2 \n\n> p\n\nwith |S| = p. How many pairs of points of distance one apart can there be? That is, how large can #{(x, y ) ∈ S × S: ( x1 − x2)2 + ( y1 − y2)2 = 1 }\n\nbe?", "clean_statement": "Problem 4.1 (T. Tao). Suppose that S ⊆ F2\n\n> p\n\nwith |S| = p. How many pairs of points of distance one apart can there be? That is, how large can #{(x, y ) ∈ S × S: ( x1 − x2)2 + ( y1 − y2)2 = 1 }\n\nbe?", "statement_status": "exact", "statement_verification": "Inspection of the original AIM PDF recovers the statement as follows.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 4.1\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[256]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.1 (T. Tao). Suppose that S ⊆ F2 \\n\\n> p\\n\\nwith |S| = p. How many pairs of points of distance one apart can there be? That is, how large can #{(x, y ) ∈ S × S: ( x1 − x2)2 + ( y1 − y2)2 = 1 }\\n\\nbe?\"\nOriginal remarks: [\"Remark(s). The trivial upper bound is p3/2. The best lower bound example is size p.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0257", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For odd p, if a p-point set S in F_p^2 is contained in a geometrically irreducible plane curve of degree d, then it determines at most 2dp ordered unit pairs (dp unordered), by a uniform Bezout argument. A full affine line is classified exactly and attains 2p ordered pairs when its direction has nonzero square norm. The ambient finite Euclidean graph is also analyzed exactly: it has degree p-chi(-1), and two distinct vertices at quadratic distance t have zero common unit neighbors when t=0 and 1+chi(t(4-t)) otherwise. These results give a structural linear regime but do not improve the p^(3/2) bound for arbitrary sets.\n\nCandidate contribution (partial_theorem; novelty confidence low): Every p-point subset of F_p^2 contained in a geometrically irreducible degree-d plane curve has at most 2dp ordered unit-distance pairs; consequently any family with a superlinear number of unit pairs must escape all geometrically irreducible curves of bounded degree." }, { "id": 20001133, "problem_number": "AIM-COMBINATORICS-0258", "title": "A global density theorem below the 3k-4 threshold", "statement": "Problem 4.2 (V. S´ os). Let A be strictly increasing infinite sequence of integers, and denote by An the initial n-element segment of A. Suppose that\n\n|An + An| < Cn.\n\nWhat can be said about the structure of A?PALO ALTO PROBLEMS 11", "original_statement": "Problem 4.2 (V. S´ os). Let A be strictly increasing infinite sequence of integers, and denote by An the initial n-element segment of A. Suppose that \n\n|An + An| < Cn. \n\nWhat can be said about the structure of A?PALO ALTO PROBLEMS 11", "clean_statement": "Problem 4.2 (V. S´ os). Let A be strictly increasing infinite sequence of integers, and denote by An the initial n-element segment of A. Suppose that\n\n|An + An| < Cn.\n\nWhat can be said about the structure of A?PALO ALTO PROBLEMS 11", "statement_status": "exact", "statement_verification": "The AIM PDF gives the following problem (subscripts and the accent in the presenter's name have been restored from the typeset PDF):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 4.2\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[257]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.2 (V. S´ os). Let A be strictly increasing infinite sequence of integers, and denote by An the initial n-element segment of A. Suppose that \\n\\n|An + An| < Cn. \\n\\nWhat can be said about the structure of A?PALO ALTO PROBLEMS 11\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0258", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let q_n=|A_n+A_n| and K=limsup q_n/n. If K<3, the nested prefix gcds d_n=gcd(a_2-a_1,...,a_n-a_1) stabilize to the global gcd d, and Lev-Smeliansky's reduced-length theorem yields limsup (a_n-a_1)/(dn) <= K-1. Hence the normalized set (A-a_1)/d has lower natural density at least 1/(K-1). In particular, a uniform bound q_n= 6, and the real optimizer expansion rho_d = sqrt(d/2) - 5/8 + O(d^{-1/2}). Combined with the 2024 classification, this supplies an exact factorial formula for the global maximum in its sufficiently-large-d range.\n\nCandidate contribution (lemma; novelty confidence low): For the Crane--Kedlaya families F_I^x in A_d, the exact ratio R_d(k)=((k+1)/k)((d-2k)(d-2k-1)/((d-k)(d-k-1))) strictly decreases; there is no adjacent tie for any d >= 6, the unique optimizer is k_d=ceil(rho_d), and rho_d=sqrt(d/2)-5/8+O(d^{-1/2})." }, { "id": 20001135, "problem_number": "AIM-COMBINATORICS-0260", "title": "A chain-deficit theorem and exact multidimensional constructions for restricted sums", "statement": "Problem 4.4 (G. Freiman). Let A ⊆ Z with |A| = n and let G be a subset of A × A.Write\n\nG1 = {ai + aj: (ai, a j ) ∈ G}\n\nand\n\nG2 = {ai − aj: (ai, a j ) ∈ G}.\n\nGiven |G1|, estimate |G2| from above and describe those sets A with largest possible\n\n|G2|.Examples: 1) If |G1| = 1, then |G2| = n;2) If |G1| = 2, then |G2| = 2 n − 1, and A is an arithmetic progression; 3) If |G1| = 4, then |G2| = 4 n − c√n, and A is isomorphic to the set of interior points of some convex set; 4) If |G1| = 8, then |G2| = 8 n − cn 2/3, and A is near a three-dimensional convex body.", "original_statement": "Problem 4.4 (G. Freiman). Let A ⊆ Z with |A| = n and let G be a subset of A × A.Write \n\nG1 = {ai + aj: (ai, a j ) ∈ G}\n\nand \n\nG2 = {ai − aj: (ai, a j ) ∈ G}.\n\nGiven |G1|, estimate |G2| from above and describe those sets A with largest possible \n\n|G2|.Examples: 1) If |G1| = 1, then |G2| = n;2) If |G1| = 2, then |G2| = 2 n − 1, and A is an arithmetic progression; 3) If |G1| = 4, then |G2| = 4 n − c√n, and A is isomorphic to the set of interior points of some convex set; 4) If |G1| = 8, then |G2| = 8 n − cn 2/3, and A is near a three-dimensional convex body.", "clean_statement": "Problem 4.4 (G. Freiman). Let A ⊆ Z with |A| = n and let G be a subset of A × A.Write\n\nG1 = {ai + aj: (ai, a j ) ∈ G}\n\nand\n\nG2 = {ai − aj: (ai, a j ) ∈ G}.\n\nGiven |G1|, estimate |G2| from above and describe those sets A with largest possible\n\n|G2|.Examples: 1) If |G1| = 1, then |G2| = n;2) If |G1| = 2, then |G2| = 2 n − 1, and A is an arithmetic progression; 3) If |G1| = 4, then |G2| = 4 n − c√n, and A is isomorphic to the set of interior points of some convex set; 4) If |G1| = 8, then |G2| = 8 n − cn 2/3, and A is near a three-dimensional convex body.", "statement_status": "exact", "statement_verification": "Inspection of the original AIM PDF gives the following exact typography.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 4.4\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[259]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.4 (G. Freiman). Let A ⊆ Z with |A| = n and let G be a subset of A × A.Write \\n\\nG1 = {ai + aj: (ai, a j ) ∈ G}\\n\\nand \\n\\nG2 = {ai − aj: (ai, a j ) ∈ G}.\\n\\nGiven |G1|, estimate |G2| from above and describe those sets A with largest possible \\n\\n|G2|.Examples: 1) If |G1| = 1, then |G2| = n;2) If |G1| = 2, then |G2| = 2 n − 1, and A is an arithmetic progression; 3) If |G1| = 4, then |G2| = 4 n − c√n, and A is isomorphic to the set of interior points of some convex set; 4) If |G1| = 8, then |G2| = 8 n − cn 2/3, and A is near a three-dimensional convex body.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0260", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source examples must be read extremally rather than as implications for every graph. For exactly two restricted sum labels s,t, if c_{s-t}(A) is the number of arithmetic-chain components of A under steps s-t, then |G2| is at most 2n-c_{s-t}(A). Thus a deficit at most r from 2n forces A to be a union of at most r common-difference progressions, and equality 2n-1 forces an arithmetic progression, proving F_2(n)=2n-1. In addition, for every d and L an explicit Freiman-embedded d-box has |A|=L^d, |G1|=2^d, and |G2|=(2L-1)^d; for d=3 this is exactly 8n-12n^(2/3)+6n^(1/3)-1.\n\nCandidate contribution (stability_theorem; novelty confidence low): For two restricted sum labels s and t, the deficit from the trivial 2n bound is at least the number of connected arithmetic chains of A under translation by s-t; consequently |G2| at least 2n-r forces A to be a union of at most r arithmetic progressions of common difference |s-t|." }, { "id": 20001136, "problem_number": "AIM-COMBINATORICS-0261", "title": "Cantor-direction Kakeya sets and a Hölder graph obstruction", "statement": "Problem 4.5 (N. Katz). Let C be a triadic Cantor set. Does there exist E ⊂ R2 with the following properties: 1) E is the union of a 1-D family of unit line segments whose slopes are in C;2) the Lebesgue measure of E is 0. 3) the union of the doubles of the above line segments has positive Lebesgue mea-sure.", "original_statement": "Problem 4.5 (N. Katz). Let C be a triadic Cantor set. Does there exist E ⊂ R2 with the following properties: 1) E is the union of a 1-D family of unit line segments whose slopes are in C;2) the Lebesgue measure of E is 0. 3) the union of the doubles of the above line segments has positive Lebesgue mea-sure.", "clean_statement": "Problem 4.5 (N. Katz). Let C be a triadic Cantor set. Does there exist E ⊂ R2 with the following properties: 1) E is the union of a 1-D family of unit line segments whose slopes are in C;2) the Lebesgue measure of E is 0. 3) the union of the doubles of the above line segments has positive Lebesgue mea-sure.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 4.5, attributed to N. Katz, in the AIM workshop list *Recent trends in additive combinatorics*. The official PDF reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 4.5\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[260]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.5 (N. Katz). Let C be a triadic Cantor set. Does there exist E ⊂ R2 with the following properties: 1) E is the union of a 1-D family of unit line segments whose slopes are in C;2) the Lebesgue measure of E is 0. 3) the union of the doubles of the above line segments has positive Lebesgue mea-sure.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0261", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Bateman and Katz affirmatively settled the standard finite-scale Kakeya-type interpretation by constructing N=3^n Cantor-slope tubes whose axially doubled-to-original union area ratio grows at least as a constant times log log N. The literal fixed-set wording remains under-specified and is not a formal consequence of the displayed finite-scale estimates alone. This attempt additionally proves that, for one exact unit segment per slope m in the standard Cantor set with a beta-Hölder placement map, every fixed axial dilation has upper box dimension at most 1+(log 2/log 3)/beta and hence remains planar-null whenever beta>log 2/log 3.\n\nCandidate contribution (obstruction theorem; novelty confidence low): For a one-segment-per-slope graph over the standard middle-thirds Cantor set, a beta-Hölder placement with beta>log 2/log 3 forces the union of every fixed centered or endpoint-anchored axial dilation to have planar Lebesgue measure zero." }, { "id": 20001137, "problem_number": "AIM-COMBINATORICS-0262", "title": "A two-sided rich core in an extremal point-line arrangement", "statement": "Problem 4.6 (T. Tao). Given n lines and n points in R2, the number of point-line incidences is O(n4/3). Suppose that this number of incidences is indeed of this order. What can be said about the structure of our configuration of points and lines?", "original_statement": "Problem 4.6 (T. Tao). Given n lines and n points in R2, the number of point-line incidences is O(n4/3). Suppose that this number of incidences is indeed of this order. What can be said about the structure of our configuration of points and lines?", "clean_statement": "Problem 4.6 (T. Tao). Given n lines and n points in R2, the number of point-line incidences is O(n4/3). Suppose that this number of incidences is indeed of this order. What can be said about the structure of our configuration of points and lines?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 4.6, attributed to T. Tao, from the AIM workshop list Recent trends in additive combinatorics. The extracted text reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 4.6\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[261]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.6 (T. Tao). Given n lines and n points in R2, the number of point-line incidences is O(n4/3). Suppose that this number of incidences is indeed of this order. What can be said about the structure of our configuration of points and lines?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0262", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If n distinct points and n distinct real affine lines determine at least c n^{4/3} incidences, then, with gamma=min(c,K) for a Szemeredi-Trotter constant K and all sufficiently large n, they contain an induced core on at least delta n vertices in each part, where delta=(gamma/(2K))^{3/2}, with minimum degree at least (gamma/8)n^{1/3} and at least (3gamma/4)n^{4/3} retained incidences. Both the distinct line-pair shadow represented by core points and the distinct point-pair shadow represented by core lines have size at least (delta gamma^2/256)n^{5/3}. An exact balanced family with n=k^3 has 3k^4/4-k^3/2-k^2/4 incidences.\n\nCandidate contribution (lemma; novelty confidence low): Every balanced Szemeredi-Trotter-scale arrangement has a constant-explicit induced two-sided rich core of linear size and n^{1/3} minimum degree, simultaneously supporting Omega(n^{5/3}) distinct represented pairs on each geometric side." }, { "id": 20001138, "problem_number": "AIM-COMBINATORICS-0263", "title": "Exact subgroup self-sumsets and a robust quotient-character leakage bound", "statement": "Problem 4.7 (A. Granville and T. Tao). Suppose that G is a proper subgroup of the multiplicative group of Fp. Let A ⊂ Fp such that A + A = G or A + A is slightly larger than G. Do such A exist, and do they necessarily have structure? Another version of this question is as follows: given that |A| > p ≤, can A + A be contained in a proper subgroup of F∗\n\n> p?", "original_statement": "Problem 4.7 (A. Granville and T. Tao). Suppose that G is a proper subgroup of the multiplicative group of Fp. Let A ⊂ Fp such that A + A = G or A + A is slightly larger than G. Do such A exist, and do they necessarily have structure? Another version of this question is as follows: given that |A| > p ≤, can A + A be contained in a proper subgroup of F∗\n\n> p?", "clean_statement": "Problem 4.7 (A. Granville and T. Tao). Suppose that G is a proper subgroup of the multiplicative group of Fp. Let A ⊂ Fp such that A + A = G or A + A is slightly larger than G. Do such A exist, and do they necessarily have structure? Another version of this question is as follows: given that |A| > p ≤, can A + A be contained in a proper subgroup of F∗\n\n> p?", "statement_status": "exact", "statement_verification": "The canonical JSON extraction is visibly damaged: it prints `p ≤` where an exponent should occur, loses the subscript and star in \\(\\mathbb F_p^*\\), and ends the remark after “very close to”. The official AIM workshop PDF restores the text as follows (notation normalized only from the PDF typography):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 4.7\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[262]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.7 (A. Granville and T. Tao). Suppose that G is a proper subgroup of the multiplicative group of Fp. Let A ⊂ Fp such that A + A = G or A + A is slightly larger than G. Do such A exist, and do they necessarily have structure? Another version of this question is as follows: given that |A| > p ≤, can A + A be contained in a proper subgroup of F∗\\n\\n> p?\"\nOriginal remarks: [\"Remark(s). Probably a very hard question, at least for small ≤ > 0. Negative answer would imply Vinogradov's conjecture that the least quadratic non-residue modulo p is smaller than p≤, for p sufficiently large. This problem is very close to\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0263", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The corrupted AIM statement is recovered with |A| > p^epsilon and a proper subgroup of F_p^*, with the remark continuing to Problem 3.6. The exact equality A+A=G is now known to be impossible for every nontrivial proper multiplicative subgroup of a prime field (Yip 2024), while the p^epsilon containment and informal near-equality questions remain open. A proved candidate lemma gives an index-sensitive robust obstruction: if d=[F_p^*:G], n=|A|, M counts ordered pairs (a,b) with a+b outside G (zero included), and z=|A intersect (-A)|, then dM >= (d-1)(n^2-n(p-n)/sqrt(p))+z. Hence the number t of distinct exceptional sum values is at least ((d-1)/d)(n-(p-n)/sqrt(p))_+.\n\nCandidate contribution (lemma; novelty confidence low): For every proper multiplicative subgroup G of index d in an odd prime field and every nonempty A, the ordered leakage M satisfies dM >= (d-1)(|A|^2-|A|(p-|A|)/sqrt(p))+|A intersect (-A)|, yielding the explicit distinct-value leakage bound |(A+A) setminus G| >= ((d-1)/d)(|A|-(p-|A|)/sqrt(p))_+." }, { "id": 20001139, "problem_number": "AIM-COMBINATORICS-0264", "title": "Transcendental dilate sumsets and an exact digit-box construction", "statement": "**Problem 4.8 (I. Łaba).** Suppose that \\(\\alpha\\) is transcendental, and \\(|A|=n\\). What is the best lower bound for \\(|A+\\alpha A|\\)?\n\n**Remark(s).** Konyagin and Łaba have shown that this cardinality is\n\\[\n\\gg \\frac{n\\log n}{\\log\\log n}.\n\\]\nThe best example (lowest known cardinality) is\n\\[\nn e^{c\\sqrt{\\log n}}.\n\\]", "original_statement": "Problem 4.8 (I. √ Laba). Suppose that α is transcendental, and |A| = n. What is the best lower bound for |A + αA |?12 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV", "clean_statement": "**Problem 4.8 (I. Łaba).** Suppose that \\(\\alpha\\) is transcendental, and \\(|A|=n\\). What is the best lower bound for \\(|A+\\alpha A|\\)?\n\n**Remark(s).** Konyagin and Łaba have shown that this cardinality is\n\\[\n\\gg \\frac{n\\log n}{\\log\\log n}.\n\\]\nThe best example (lowest known cardinality) is\n\\[\nn e^{c\\sqrt{\\log n}}.\n\\]", "statement_status": "corrected_verified", "statement_verification": "The canonical JSON has three OCR defects: the author is Izabella Łaba, not “I. √ Laba”; a page footer was inserted into the problem; and the exponent in the example was flattened. The official AIM PDF places the problem at the bottom of printed page 11 and the remark at the top of printed page 12. With mathematical typography restored, it reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 4.8\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[263]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.8 (I. √ Laba). Suppose that α is transcendental, and |A| = n. What is the best lower bound for |A + αA |?12 COLLECTED BY ERNIE CROOT AND VSEVOLOD F. LEV\"\nOriginal remarks: [\"Remark(s). Konyagin and √ Laba have shown that this cardinality is & n log n/ (log log n). The best example (lowest known cardinality) is ne c√log n.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0264", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Conlon and Lim proved that every finite A contained in the real numbers and every real transcendental alpha satisfy |A+alpha A| at least |A| exp(c sqrt(log |A|)); the Konyagin--Laba construction gives the matching upper scale, so log(f_alpha(n)/n)=Theta(sqrt(log n)). Thus the growth scale is settled, while the optimal exponential constant and finer behavior remain open. This attempt also proves the exact identity |B_{h,m}+alpha B_{h,m}|=h^2(2h-1)^{m-1} for power-basis digit boxes and optimizes it to give, for every sufficiently large N, an N-element example with upper bound N exp((2 sqrt(log 2)+o(1)) sqrt(log N)).\n\nCandidate contribution (exact construction; novelty confidence low): For B_{h,m}={sum_{i=0}^{m-1} a_i alpha^i : 0<=a_i=2, the set exponent is 2 log(q)/log(2q-1), the entropy ratio is 2 log(q)/h_q with h_q given explicitly, and the underlying q by q square is maximal for injectivity of (x,y) maps to bx+ay.\n\nCandidate contribution (counterexample; novelty confidence low): For every coprime nonzero integer pair a,b with a not equal to b and q=max(|a|,|b|)>=2, the q by q grid certificate is maximal among square grids for injectivity of bx+ay and simultaneously proves failure of historical SD(0,-b/a,infinity; alpha) for alpha at most 2 log(q)/log(2q-1), while its uniform distribution gives the exact entropy ratio 2 log(q)/h_q.", "research_difficulty_suggested": "L2" }, { "id": 20001141, "problem_number": "AIM-COMBINATORICS-0266", "title": "The arithmetic Kakeya entropy constant and an exact bounded-support case", "statement": "**Problem 4.10 (T. Tao).** What is the best \\(\\varepsilon\\) for which there\nexist real numbers \\(r_1,\\ldots,r_n\\) with the following property: given any\ntwo random values \\(x,y\\) taking finitely many real values, and obeying the\nentropy bound \\(H(x+r_jy)<\\log N\\) for all \\(j=1,\\ldots,N\\), one necessarily\nhas \\(H(x-y)<(1+\\varepsilon)\\log N\\).", "original_statement": "Problem 4.10 (T. Tao). What is the best ≤ for which there exist real numbers r1,..., r n\n\nwith the following property: given any two random values x, y taking finitely many real values, and obeying the entropy bound H(x + rj y) < log N for all j = 1,..., N, one necessarily has H(x − y) < (1 + ≤) log N.", "clean_statement": "**Problem 4.10 (T. Tao).** What is the best \\(\\varepsilon\\) for which there\nexist real numbers \\(r_1,\\ldots,r_n\\) with the following property: given any\ntwo random values \\(x,y\\) taking finitely many real values, and obeying the\nentropy bound \\(H(x+r_jy)<\\log N\\) for all \\(j=1,\\ldots,N\\), one necessarily\nhas \\(H(x-y)<(1+\\varepsilon)\\log N\\).", "statement_status": "corrected_verified", "statement_verification": "The record comes from Problem 4.10 of the AIM workshop *Recent Trends in Additive Combinatorics* (September 9--12, 2004). Inspection of the official PDF, rather than the OCR record alone, recovers the missing symbol as \\(\\varepsilon\\). The PDF literally states: There are genuine defects in the printed statement, not merely OCR defects.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 4.10\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[265]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.10 (T. Tao). What is the best ≤ for which there exist real numbers r1,..., r n\\n\\nwith the following property: given any two random values x, y taking finitely many real values, and obeying the entropy bound H(x + rj y) < log N for all j = 1,..., N, one necessarily has H(x − y) < (1 + ≤) log N.\"\nOriginal remarks: [\"Remark(s). Best known ≤ = 0.67512....\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0266", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The official source is recovered and conservatively repaired as the arithmetic Kakeya entropy conjecture: the n/N mismatch and omission of r_j != -1 are genuine printing defects, and the current best global upper bound is epsilon <= 0.675130870566646..., while epsilon = 0 remains conjectural. A proved special-case theorem shows that for every joint law of (X,Y) supported on at most m points, any binomial(m,2)+1 distinct admissible real slopes contain a projection that is injective on the joint support; consequently H(X-Y) <= max_r H(X+rY), with no independence assumption. The pair-collision count is attained by an explicit parabola family, and for m=2 the two-slope count is genuinely necessary.\n\nCandidate contribution (special_case_theorem; novelty confidence low): For every m >= 1, the explicit slope set {0,1,...,binomial(m,2)} gives entropy exponent 1 for every arbitrarily dependent real-valued joint law supported on at most m atoms; the binomial(m,2) pair-collision bound is attained by the points (2^(2i),2^i), and two slopes are necessary when m=2." }, { "id": 20001142, "problem_number": "AIM-COMBINATORICS-0267", "title": "Favard length and an exact tube formula for the four-corner Cantor set", "statement": "Problem 4.11 (Brought by B. Green, implicit in a paper of Solomyak and Peres).\n\nGive an estimate for the size of the δ-thickened Kahane Besicovich set C4 × C 4, where\n\nC4 =\n\n{ ∞∑\n\n> n=0\n\nan\n\n4n: an ∈ { 0, 1}\n\n}.", "original_statement": "Problem 4.11 (Brought by B. Green, implicit in a paper of Solomyak and Peres).\n\nGive an estimate for the size of the δ-thickened Kahane Besicovich set C4 × C 4, where \n\nC4 =\n\n{ ∞∑\n\n> n=0\n\nan\n\n4n: an ∈ { 0, 1}\n\n}.", "clean_statement": "Problem 4.11 (Brought by B. Green, implicit in a paper of Solomyak and Peres).\n\nGive an estimate for the size of the δ-thickened Kahane Besicovich set C4 × C 4, where\n\nC4 =\n\n{ ∞∑\n\n> n=0\n\nan\n\n4n: an ∈ { 0, 1}\n\n}.", "statement_status": "exact", "statement_verification": "The official AIM workshop PDF, on its final page and in the section headed “Erdős Distance and Kakea Problem Session,” prints:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 4.11\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[266]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.11 (Brought by B. Green, implicit in a paper of Solomyak and Peres).\\n\\nGive an estimate for the size of the δ-thickened Kahane Besicovich set C4 × C 4, where \\n\\nC4 =\\n\\n{ ∞∑\\n\\n> n=0\\n\\nan\\n\\n4n: an ∈ { 0, 1}\\n\\n}.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0267", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The cited Peres--Solomyak context shows that the intended size is the Favard length of the Euclidean delta-neighborhood of the four-corner Cantor set, not its planar area. For every p<1/6, the conservatively verified published bounds transfer to every sufficiently small delta as c log log(1/delta)/log(1/delta) <= Fav(E_delta) <= C_p (log(1/delta))^{-p}; the exact deterministic order remains open. For the literal planar-area reading, this attempt proves an exact every-radius max-norm tube formula and shows liminf V_infinity(r)/r=32/3 and limsup V_infinity(r)/r=12, together with explicit Euclidean bounds (16 sqrt(2)/3)r <= V_2(r) <= 12r.\n\nCandidate contribution (exact tube formula; novelty confidence low): If C={sum_{n>=0} a_n 4^{-n}: a_n in {0,1}}, E=C x C, and (1/3)4^{-(k+1)} <= r <= (1/3)4^{-k}, then the max-norm r-neighborhood has exact area [(4/3)2^{-(k+1)}+2^{k+2}r]^2; equivalently V_infinity(r)/r=(2+4t)^2/(3t) for t=3*4^k*r in [1/4,1], so its exact oscillation interval is [32/3,12]." }, { "id": 20001143, "problem_number": "AIM-COMBINATORICS-0268", "title": "Complete slope classification for the four-corner Cantor projection", "statement": "Problem 4.12 (T. Tao). For which real numbers α the set C4 + αC4 has zero Lebesgue measure, where C4 is as in the previous problem?", "original_statement": "Problem 4.12 (T. Tao). For which real numbers α the set C4 + αC4 has zero Lebesgue measure, where C4 is as in the previous problem?", "clean_statement": "Problem 4.12 (T. Tao). For which real numbers α the set C4 + αC4 has zero Lebesgue measure, where C4 is as in the previous problem?", "statement_status": "exact", "statement_verification": "The official AIM TeX source gives Problem 4.11 as \\[ \\mathcal C_4= \\left\\{ \\sum_{n=0}^{\\infty}\\frac{a_n}{4^n}:a_n\\in\\{0,1\\} \\right\\}, \\] and Problem 4.12, attributed to T. Tao, reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 4.12\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[267]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.12 (T. Tao). For which real numbers α the set C4 + αC4 has zero Lebesgue measure, where C4 is as in the previous problem?\"\nOriginal remarks: [\"Remark(s). This is known to hold for almost all α.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0268", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "With the AIM normalization C_4={sum from n=0 to infinity of a_n/4^n:a_n in {0,1}}, the sum C_4+alpha C_4 has zero Lebesgue measure exactly when alpha=0, when alpha is irrational, or when alpha=p/q is a nonzero reduced rational whose 2-adic valuation gap v_2(|p|)-v_2(q) is even. Irrational cases have Hausdorff dimension 1; rational zero-measure cases have an exact overlap and dimension below 1. A rational parameter with odd valuation gap instead gives positive measure and a nondegenerate interval. The report transfers the published classification to AIM's indexing and supplies an elementary Fourier/residue proof of the rational measure dichotomy.\n\nCandidate contribution (alternative_proof; novelty confidence low): After exact reductions by sign, inversion, and alpha maps to 4 alpha, the rational classification follows from a paired elementary certificate: an odd valuation gap yields a complete four-digit residue system modulo 4 whose natural measure pushes forward to Haar measure modulo 1, while an even gap yields a nonzero Fourier coefficient repeated along 4^N. In the normalized positive case p=2u with u,q odd, the argument also proves Leb(C_4+(p/q)C_4)>=4/q." }, { "id": 20001144, "problem_number": "AIM-COMBINATORICS-0269", "title": "A weighted product of chordal distances", "statement": "Problem 4.13 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nSuppose we are given r ≥ 1 points z1,..., z r on the unit circle and corresponding non-negative weights p1,..., p r, normalized by the condition p1 + · · · + pr = r. We want to find yet another point z on the circle which should be as far as possible from all points\n\nzj in the sense that the product ∏rj=1 |z − zj |pj is to be maximized. Conjecture: for any points zj and weights pj as above, there exists z such that\n\n> r\n\n∏\n\n> j=1\n\n|z − zj |pj ≥ 2.", "original_statement": "Problem 4.13 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nSuppose we are given r ≥ 1 points z1,..., z r on the unit circle and corresponding non-negative weights p1,..., p r, normalized by the condition p1 + · · · + pr = r. We want to find yet another point z on the circle which should be as far as possible from all points \n\nzj in the sense that the product ∏rj=1 |z − zj |pj is to be maximized. Conjecture: for any points zj and weights pj as above, there exists z such that \n\n> r\n\n∏\n\n> j=1\n\n|z − zj |pj ≥ 2.", "clean_statement": "Problem 4.13 (Brought by V. Lev, originally stated by Konyagin and the presenter).\n\nSuppose we are given r ≥ 1 points z1,..., z r on the unit circle and corresponding non-negative weights p1,..., p r, normalized by the condition p1 + · · · + pr = r. We want to find yet another point z on the circle which should be as far as possible from all points\n\nzj in the sense that the product ∏rj=1 |z − zj |pj is to be maximized. Conjecture: for any points zj and weights pj as above, there exists z such that\n\n> r\n\n∏\n\n> j=1\n\n|z − zj |pj ≥ 2.", "statement_status": "exact", "statement_verification": "The source is Problem 4.13 in the AIM workshop list *Recent trends in additive combinatorics*. Its mathematical statement, with the typography restored but without changing its content, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: Recent trends in additive combinatorics\nSection: \nSource item: 4.13\nSource URL: https://aimath.org/WWN/additivecomb/additivecomb.pdf\nCanonical location: aim-combinatorics-notes.json notes[268]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4.13 (Brought by V. Lev, originally stated by Konyagin and the presenter).\\n\\nSuppose we are given r ≥ 1 points z1,..., z r on the unit circle and corresponding non-negative weights p1,..., p r, normalized by the condition p1 + · · · + pr = r. We want to find yet another point z on the circle which should be as far as possible from all points \\n\\nzj in the sense that the product ∏rj=1 |z − zj |pj is to be maximized. Conjecture: for any points zj and weights pj as above, there exists z such that \\n\\n> r\\n\\n∏\\n\\n> j=1\\n\\n|z − zj |pj ≥ 2.\"\nOriginal remarks: [\"Remark(s). The constant two in the right-hand side is easily seen to be best possible. This conjecture has been established in a number of special cases: in particular if all weights pj equal each other or if zj are equally spaced on the unit circle. It can be re-stated as a conjecture about the maximum possible value of a polynomial on the unit circle.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0269", "aim-domain:combinatorics", "aim-workshop:additivecomb", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "After combining repeated positive-weight locations, let m be the number of distinct active points and n the number of source entries. Dubinin's sharp hedgehog-capacity inequality, through the published weighted conformal-map correspondence, gives max_{|z|=1} product_j |z-z_j|^{p_j} >= 2^{n/m} >= 2, proving the AIM conjecture and identifying the regular-polygon/equal-weight equality cases. The report also derives the monic-polynomial form ||P||_T >= 2^{d/m}.\n\nCandidate contribution (quantitative_bound; novelty confidence low): If the active support is contained in a closed arc of angular length L and has m distinct locations, then the maximum M satisfies M >= max{2^{n/m}, [2 cos(L/4)]^n}; in the complete n=2 audit, M=2 occurs exactly for antipodal points of weights 1 and 1." }, { "id": 20001145, "problem_number": "AIM-COMBINATORICS-0270", "title": "Malformed perfect-graph workshop mega-record and an NP-certificate bridge", "statement": "Problem 4. NP Characterization of Perfect Graphs 5. Recognition Algorithm Given the List of Maximal Cliques a. Berge Graphs with Poly-bounded Number of Max Cliques 6. TDI Matrices 7. Fixed Parameter Algorithms 8. Clique Joins 9. Polynomial Size Decomposition Tree B. Structural Characterization of Perfect Graphs............... 6C. Coloring Perfect Graphs......................... 71. Uniquely colorable perfect graphs D. Optimization on Perfect Graphs..................... 71. New Optimization Problems on Perfect Graphs a. A Possible New Problem E. Skew-Partitions............................ 71. Extending a Skew -Partition 2. Graphs Without Skew-Partitions 3. Graphs Without Star Cutsets 4. Finding Skew-Partitions in Berge Graphs 5. Interaction Between Different Skew-Partitions in a Graph 6. Skew -Partitions of Balanced Size 7. Recognizing Balanced Skew-Partitions 8. Even-Pair Skew-Partition F. Even Pairs in Berge Graphs....................... 91. Coloring Berge Graphs Using Even Pairs 2. Recognizing Even Pairs 3. Quasi-Parity and Strict Quasi-Parity Graphs a. Forbidden Subgraphs for The Class of Strict Quasi-Parity Graphs b. Recognition of Quasi-Parity and Strict Quasi-Parity Graphs 4. Perfectly Contractile Graphs a. Perfectly Contractile Graphs and the Decomposition Method 5. Possible Structure Theorem for Berge Graphs 6. Odd holes and odd walks G. Forbidding Holes and Antiholes.................... 12 1. 2-divisible Graphs 2. Clique Coloring of Perfect Graphs 3. Recognition of Odd-Hole-Free Graphs 4. Even-Hole-Free Graphs 5. Even-hole-free circulants, 6. beta-perfect graphs H. Partitionable Graphs......................... 14 1. Perfect, Partitionable, and Kernel-Solvable Graphs 3\n\n2. Partitionable graphs and odd holes 3. A Property of Partitionable Graphs 4. Small Transversals in Partitionable Graphs I. The Imperfection Ratio........................ 16 J. Integer Programming......................... 18 1. Partitionable Graphs as Cutting Planes for Packing Problems? 2. Feasibility/Membership Problem For the Theta Body K. Balanced Graphs........................... 18 1. Balanced circulants L. P4-structure and Its Relatives..................... 19 4\n\nChapter A: Recognition of Perfect Graphs\n\nCan one decide in polynomial time if a graph is perfect?\n\nA.1 Polynomial Recognition Algorithm Found\n\nA polynomial algorithm to test whether a graph is Berge was found in November 2002. A paper summarizing the work of two groups- Chudnovsky and Seymour and Cornuejols, Liu and Vuscovic- is due to appear in Combinatorica. The algorithm is independent of the proof of the strong perfect graph conjecture.\n\nA.2 Interaction Between Skew-Partitions and 2-joins\n\nOne can think of algorithms for testing for odd holes if you use only 2-joins to decom-pose a graph, or if you use only skew partitions. However, the interaction between skew partition steps and 2-join steps adds another level of difficulty. Can one argue that this can be reduced only to testing for holes as you decompose using skew partitions only and testing for holes using 2-joins only? Contributed by Jeremy Spinrad Similar approach worked in algorithms for recognizing even-hole-free graphs: first the graph is decomposed via vertex cut-sets, and only then via 2-joins. The 2-join decomposition blocks are defined in such a way that no new vertex cutset is introduced. Contributed by Kristina Vuskoic\n\nA.3 The Perfect-Graph Robust Algorithm Problem\n\nAn algorithm, which for any easily recognizable input, A, finds either an easily rec-ognizable B or an easily recognizable C, is sometimes called a \"robust algorithm\" - to be distinguished from a non-robust algorithm, which, for any A without a B, finds a C. Either provides a proof of the \"existentially polytime (EP) theorem\": For any A, there exists a B or a C. In other words: For any A without a B, there is a C. In [92i:68043 ], Jack Edmonds and Kathie Cameron advocated seeking a robust al-gorithm which, for any graph G, finds either a clique and colouring the same size or else finds an easily recognizable combinatorial obstruction to G being perfect. The obstruction might be specified to be a \"alpha-omega partitioned subgraph\", or it might be specified more particularly to be an odd hole or odd antihole. Such an algorithm might be simpler than an algorithm for recognizing whether or not a graph is perfect, in view of precedents, and since what it would do is incomparable with perfect-graph recognition. Such an algorithm could end up giving a clique and colouring the same size in a non-perfect graph. Here are two examples of similar problems which have been solved. Edmonds has given a simple robust algorithm which, for any graph G, either finds an odd cycle > 3 with at most one chord (a defining obstruction to G being Meyniel) or else finds a clique and colouring the same size. This is an improvement on the non-robust algorithms of Hoang and Hertz which, assuming a graph is Meyniel, find a clique and colouring the same size. Edmonds' algorithm is much simpler than the Burlet-Fonlupt decomposition algorithm for recognizing Meyniel graphs, which was motivated by an interest in optimizing in Meyniel graphs, and which is used by Hoang and Hertz. 5\n\nConforti and Cornuejols give a complicated decomposition algorithm for recognizing whether or not a matrix is balanced. At about the same time, to motivate the advocacy of a robust algorithm for either node-colouring a graph or recognizing it to be not perfect, Cameron and Edmonds presented a simple algorithm which, for any 0-1 matrix M, either finds, where x is the largest number of ones in any row, an x-colouring of the columns so that the 1's of any row are in different coloured columns, or else finds \"an odd hole\" in M (the defining obstruction to M being balanced). This introduced the \"EP - robust algorithm\" paradigm which is followed in Edmonds' Meyniel-related algorithm, and is related to the Conforti-Cornuejols-Rao treatment of balanced matrices in the same way that Edmonds' is related to the Burlet-Fonlupt treatment of Meyniel graphs. Following the same paradigm we expect there to be a robust algorithm proving the SPCG, related in the same way to the Chudnovsky-Robertson-Seymour-Thomas decomposition of Berge graphs. In conclusion, we know the following EP Theorem 1: For any graph, there is either a clique and a colouring of the same size, or there is an alpha-omega partitioned subgraph (or both). EP Theorem 2 (SPGT): For any graph, there is either a clique and a colouring of the same size, or there is a odd hole or odd antihole (or both). So: Give a combinatorial polytime algorithm to find what the EP theorem asserts to exist. Contributed by Kathie Cameron and Jack Edmonds\n\nA.4 NP Description of Perfect Graphs\n\nGive an NP description of perfect graphs. Contributed by Jack Edmonds.\n\nA.5 Recognition Algorithm Given the List of Maximal Cliques\n\nIs there a polytime recognition algorithm for perfect graphs where the input is the list of all maximal cliques in the graph? This probelm has been resolved, since a perfect graph can itself now be recognized. Contributed by Bruce Shepherd\n\nA.5.a Berge Graphs with Poly-bounded Number of Max Cliques. Give a poly-nomial time recognition algorithm for Berge graphs with polynomially bounded number of maximal cliques. Contributed by Jeremy Spinrad.\n\nA.6 TDI Matrices\n\n1. Given an m × n 0 − 1 matrix A and an m-dimensional vector b, decide whether the system\n\nAx ≤ b is totally dual integral (TDI), that is, is there an integer dual solution for every objective function for which the dual optimum exists.\n\nThis is the 0 − 1 special case of the well-known problem of TDI system recognition. Let\n\nP:= Ax ≤ b, x ≥ 0. Let A(u) be the matrix whose rows are the normal vectors of facets of\n\nP containing u, each row is integer and the gcd of its entries is 1. There are several known relations between TDI and unimodular systems. As Serkan Ho¸ sten pointed out, 'nondegenerate' TDI matrices are exactly those in which A(u) is an\n\nn × n matrix having determinant 1 for every vertex u.The following problem involves unimodularity in perfectness test: 6\n\nINPUT: m × n 0-1 matrix A and an m-dimensional positive vector b,QUESTION: Is the matrix A(u) for every vertex u of P a square matrix of determinant 1?\n\n2. Can this problem be solved in polynomial time?\n\nFor reducing the test for the perfectness of matrix A to this problem define b with a 'lexicographic perturbation' from the all 1 objective function. Contributed by Andr´ as Seb˝ o\n\nA.7 Fixed Parameter Algorithms\n\nIn the context of fixed parameter algorithms, which was recently introduced by Downey and Fellows, it would be interesting to design an algorithm with running time O(f (k)|V |c)where k is the size of the maximum clique, c is a small constant independent of k and f (k)is a (exponential) function of k. Such an algorithm can work for small k even for large n.Contributed by Mohammad Taghi Hajiaghayi\n\nA.8 Clique Joins\n\nA k-clique-join of G = ( V, E ) is a set of pairs {(A0, B 0), (A1, B 1),..., (Ak, B k)}, where\n\n{A0, B 0} is a partition of V, both A0 and B0 contain at least one ω-clique, and Ai ⊆ A0,\n\nBi ⊆ B0 (i = 1,..., k ) (not necessarily disjoint), moreover (i) If x ∈ Ai and y ∈ Bi, then xy ∈ E\n\n(ii) If K is an ω-clique of G that meets both A0 and B0, then there exists i so that\n\nK ⊆ Ai ∪ Bi.A partitionable graph does not contain a k-clique-join for k < 2( ω − 1), on the other hand odd holes, odd antiholes do all contain 2( ω − 1)-clique-joins.\n\nCould the minimum of k for which a k-clique-join exists be computed (or well-characterized)? For Berge-graphs? Is there a variant of this operation that would allow to compose perfect graphs and keep perfectness?\n\nContributed by Andr´ as Seb˝ o\n\nA.9 Polynomial Size Decomposition Tree\n\nThe problem with using skew-paritions (or star cutsets) for recognition algorithms is that the decomposition tree they induce does not have polynomial size. The following questions have been suggested during the workshop: 1. Suggest different endblocks of the decompostition (other than basic perfect graphs), that can be recognized in polynomial time and yet make the decomposition tree polynomial (Bruce Reed) 2. Normally every skew-partition has 4 decomposition blocks. What if we could prove that it is enough to consider only two blocks for each skew-partition. Would that imply a polynomial size decomposition tree? Possibly introducing new endblocks or using cleaning? (Kristina Vuskovic). 3.Does decomposition of C4-free graphs via star-cutsets induce a polynomial size de-composition tree? Possibly using cleaning? (Kristina Vuskovic). 7\n\nChapter B: Structural Characterization of Perfect Graphs\n\nPossible structural characterization of perfect graphs. Give explicit constructions for subclasses of Berge graphs. Contributed by Paul Seymour\n\nChapter C: Coloring Perfect Graphs\n\nCan one find an efficient algorithms to color a perfect graph?\n\nC.1 Uniquely colorable perfect graphs\n\nUniquely colorable perfect graphs (in which there is a unique partition into ω stable-sets) are closely related to minimal imperfect graphs: according to a result of Padberg (Perfect zero-one matrices, Math Programming, 6, (1974)) if G is minimal imperfect then for all of its vertices v, the graph G − v is uniquely colorable. There is also a combinatorial good characterization theorem for unique colorability of perfect graphs, and a polynomial algorithm for testing the property using the ellipsoid method (IPCO 1, Kannan, Pulleyblank eds, Waterloo Univ. Press, 1990). In other words\n\nUNIQUE COLORABILITY is a tractable property for perfect graphs, closely related to min-imal imperfect graphs. Yet, some simple conjectures related to the SPGT, resist through the years. The following one arises both by specializing more general conjectures occurring in various papers, and does not seem to trivially follow from the SPGT:\n\nIf G is perfect and uniquely colorable, does there exist two ω-cliques which meet in ω −1\n\npoints?\n\nLet us call the two vertices in the symmetric difference of two such cliques forced.\n\nIs it true that every known uniquely colorable perfect graph collapses to an ω-clique by successive identification of forced vertices?\n\nIt can be simply proved that a minimal imperfect graph with three forced vertices in particular positions is an odd hole or an odd antihole. A simpler proof of the following statement would shortcut the proof of the SPGT:\n\nIf G is minimal imperfect, there is a vertex v so that N (v) is uniquely colorable.\n\nContributed by Jean Fonlupt and Andr´ as Seb˝ o\n\nChapter D: Optimization on Perfect Graphs\n\nOptimization on perfect graphs without using the ellipsoid method.\n\nD.1 New Optimization Problems on Perfect Graphs\n\nAre there any new optimization problems (other that coloring and finding the size of the max clique) that are easier to solve for a perfect graph that for a general graph? Can we use any of the existing (future) recognition algorithms in order to do that? Contributed by Mohammad Hajiaghayi\n\nD.1.a A Possible New Problem. Solve in a perfect graph: do two given vertices belong to an induced hole? Contributed by Bruce Reed 8\n\nChapter E: Skew-Partitions\n\nE.1 Extending a Skew -Partition\n\nWhen can a skew partition of an induced subgraph be extended to a skew partition of G? Algorithmically this is answered by the algorithm of de Figueiredo, Klein, Kohayakawa and Reed [2001j:05114], but what about a theorem? Is there some theorem that says \"either the skew partition is extendable, or there is a reason why not (an obstruction)\"? Contributed by Paul Seymour\n\nE.2 Graphs Without Skew-Partitions\n\nIs there a structure theorem for such graphs? Can they all be constructed somehow? Maybe by starting with a small one, and adding little bits so that at each stage there is no skew partition? Contributed by Paul Seymour\n\nE.3 Graphs Without Star Cutsets\n\nIs there a structure theorem for such graphs? Can they all be constructed somehow? Maybe by starting with a small one, and adding little bits so that at each stage there is no star cutset? Contributed by Bruce Reed\n\nConjecture If neither G nor Gc has a star cutset then the disk-structure of G is connected (A disk is a hole or an antihole. Two disks are adjacent in the disk structure if they share at least 2 vertices). Contributed by Ryan Hayward\n\nE.4 Finding Skew-Partitions in Berge Graphs\n\nIs it easier to detect skew partitions in Berge graphs than in general ones? Contributed by Paul Seymour\n\nE.5 Interaction Between Different Skew-Partitions in a Graph\n\nFor 1 ≤ i ≤ n, let ( Ai, B i, C i, D i) be a skew partition of G, where there are no edges between Ai and Bi, and Ci is complete to Di. For each i, choose one of Ai, B i, C i, D i, say\n\nXi, and let X be the union of all these Xi. Call G \\ X a chunk. If there is an odd hole or antihole in G, then it belongs to the chunk (for some choice of the Xi's), so to check Bergeness of G, it is enough to check Bergeness of all the chunks. But even if n is linear in the size of G, the number of chunks can be exponential. Maybe there is a way around this. With decomposition theorems that come up from excluded minors, using separations instead of skew partitions, the same exponential blowup happens, but it can be avoided by using separations that are pairwise noncrossing - then you only get linearly many pieces. Is there an analogous nice way for skew partitions to fit together, so that we only get linearly many (or polynomially many) chunks? Contributed by Paul Seymour 9\n\nE.6 Skew -Partitions of Balanced Size\n\nThe problem with recursive skew decomposition is that at least at first glance, you get exponential behavior. This would not occur if you were always able to find a decomposition in which each of A,B,C,D have at least n/c vertices for some c. Call this a skew partition of balanced size. a) Can you find a skew partition of balanced in polynomial time, if one exists? b) Will always looking for a balanced skew partition if possible lead to a polynomial size decomposition tree for perfect graphs (ie always use the skew partition which maximizes the size of the smallest set) Contributed by Jeremy Spinrad Answer: There exists a graph admitting no skew-partition of balanced size: take a clique and for some edges e1,..., e k of it add a vertices v1,..., v k s.t. each vi has degree 2 and is adjacent to both ends of ei.\n\nE.7 Recognizing Balanced Skew-Partitions\n\nGiven a skew-partition, can one check in polynomial time whether it is balanced. Contributed by Jeremy Spinrad\n\nE.8 Even-Pair Skew-Partition\n\nAn even-pair skew-partition is a partition of the vertex set of a graph G into four sets\n\nA, B, C, D s.t. A is complete to B and C is anti-complete to D, and any two non-adjacent vertices in A or B are an even pair.", "original_statement": "Problem 4. NP Characterization of Perfect Graphs 5. Recognition Algorithm Given the List of Maximal Cliques a. Berge Graphs with Poly-bounded Number of Max Cliques 6. TDI Matrices 7. Fixed Parameter Algorithms 8. Clique Joins 9. Polynomial Size Decomposition Tree B. Structural Characterization of Perfect Graphs............... 6C. Coloring Perfect Graphs......................... 71. Uniquely colorable perfect graphs D. Optimization on Perfect Graphs..................... 71. New Optimization Problems on Perfect Graphs a. A Possible New Problem E. Skew-Partitions............................ 71. Extending a Skew -Partition 2. Graphs Without Skew-Partitions 3. Graphs Without Star Cutsets 4. Finding Skew-Partitions in Berge Graphs 5. Interaction Between Different Skew-Partitions in a Graph 6. Skew -Partitions of Balanced Size 7. Recognizing Balanced Skew-Partitions 8. Even-Pair Skew-Partition F. Even Pairs in Berge Graphs....................... 91. Coloring Berge Graphs Using Even Pairs 2. Recognizing Even Pairs 3. Quasi-Parity and Strict Quasi-Parity Graphs a. Forbidden Subgraphs for The Class of Strict Quasi-Parity Graphs b. Recognition of Quasi-Parity and Strict Quasi-Parity Graphs 4. Perfectly Contractile Graphs a. Perfectly Contractile Graphs and the Decomposition Method 5. Possible Structure Theorem for Berge Graphs 6. Odd holes and odd walks G. Forbidding Holes and Antiholes.................... 12 1. 2-divisible Graphs 2. Clique Coloring of Perfect Graphs 3. Recognition of Odd-Hole-Free Graphs 4. Even-Hole-Free Graphs 5. Even-hole-free circulants, 6. beta-perfect graphs H. Partitionable Graphs......................... 14 1. Perfect, Partitionable, and Kernel-Solvable Graphs 3\n\n2. Partitionable graphs and odd holes 3. A Property of Partitionable Graphs 4. Small Transversals in Partitionable Graphs I. The Imperfection Ratio........................ 16 J. Integer Programming......................... 18 1. Partitionable Graphs as Cutting Planes for Packing Problems? 2. Feasibility/Membership Problem For the Theta Body K. Balanced Graphs........................... 18 1. Balanced circulants L. P4-structure and Its Relatives..................... 19 4\n\nChapter A: Recognition of Perfect Graphs \n\nCan one decide in polynomial time if a graph is perfect? \n\nA.1 Polynomial Recognition Algorithm Found \n\nA polynomial algorithm to test whether a graph is Berge was found in November 2002. A paper summarizing the work of two groups- Chudnovsky and Seymour and Cornuejols, Liu and Vuscovic- is due to appear in Combinatorica. The algorithm is independent of the proof of the strong perfect graph conjecture. \n\nA.2 Interaction Between Skew-Partitions and 2-joins \n\nOne can think of algorithms for testing for odd holes if you use only 2-joins to decom-pose a graph, or if you use only skew partitions. However, the interaction between skew partition steps and 2-join steps adds another level of difficulty. Can one argue that this can be reduced only to testing for holes as you decompose using skew partitions only and testing for holes using 2-joins only? Contributed by Jeremy Spinrad Similar approach worked in algorithms for recognizing even-hole-free graphs: first the graph is decomposed via vertex cut-sets, and only then via 2-joins. The 2-join decomposition blocks are defined in such a way that no new vertex cutset is introduced. Contributed by Kristina Vuskoic \n\nA.3 The Perfect-Graph Robust Algorithm Problem \n\nAn algorithm, which for any easily recognizable input, A, finds either an easily rec-ognizable B or an easily recognizable C, is sometimes called a \"robust algorithm\" - to be distinguished from a non-robust algorithm, which, for any A without a B, finds a C. Either provides a proof of the \"existentially polytime (EP) theorem\": For any A, there exists a B or a C. In other words: For any A without a B, there is a C. In [92i:68043 ], Jack Edmonds and Kathie Cameron advocated seeking a robust al-gorithm which, for any graph G, finds either a clique and colouring the same size or else finds an easily recognizable combinatorial obstruction to G being perfect. The obstruction might be specified to be a \"alpha-omega partitioned subgraph\", or it might be specified more particularly to be an odd hole or odd antihole. Such an algorithm might be simpler than an algorithm for recognizing whether or not a graph is perfect, in view of precedents, and since what it would do is incomparable with perfect-graph recognition. Such an algorithm could end up giving a clique and colouring the same size in a non-perfect graph. Here are two examples of similar problems which have been solved. Edmonds has given a simple robust algorithm which, for any graph G, either finds an odd cycle > 3 with at most one chord (a defining obstruction to G being Meyniel) or else finds a clique and colouring the same size. This is an improvement on the non-robust algorithms of Hoang and Hertz which, assuming a graph is Meyniel, find a clique and colouring the same size. Edmonds' algorithm is much simpler than the Burlet-Fonlupt decomposition algorithm for recognizing Meyniel graphs, which was motivated by an interest in optimizing in Meyniel graphs, and which is used by Hoang and Hertz. 5\n\nConforti and Cornuejols give a complicated decomposition algorithm for recognizing whether or not a matrix is balanced. At about the same time, to motivate the advocacy of a robust algorithm for either node-colouring a graph or recognizing it to be not perfect, Cameron and Edmonds presented a simple algorithm which, for any 0-1 matrix M, either finds, where x is the largest number of ones in any row, an x-colouring of the columns so that the 1's of any row are in different coloured columns, or else finds \"an odd hole\" in M (the defining obstruction to M being balanced). This introduced the \"EP - robust algorithm\" paradigm which is followed in Edmonds' Meyniel-related algorithm, and is related to the Conforti-Cornuejols-Rao treatment of balanced matrices in the same way that Edmonds' is related to the Burlet-Fonlupt treatment of Meyniel graphs. Following the same paradigm we expect there to be a robust algorithm proving the SPCG, related in the same way to the Chudnovsky-Robertson-Seymour-Thomas decomposition of Berge graphs. In conclusion, we know the following EP Theorem 1: For any graph, there is either a clique and a colouring of the same size, or there is an alpha-omega partitioned subgraph (or both). EP Theorem 2 (SPGT): For any graph, there is either a clique and a colouring of the same size, or there is a odd hole or odd antihole (or both). So: Give a combinatorial polytime algorithm to find what the EP theorem asserts to exist. Contributed by Kathie Cameron and Jack Edmonds \n\nA.4 NP Description of Perfect Graphs \n\nGive an NP description of perfect graphs. Contributed by Jack Edmonds. \n\nA.5 Recognition Algorithm Given the List of Maximal Cliques \n\nIs there a polytime recognition algorithm for perfect graphs where the input is the list of all maximal cliques in the graph? This probelm has been resolved, since a perfect graph can itself now be recognized. Contributed by Bruce Shepherd \n\nA.5.a Berge Graphs with Poly-bounded Number of Max Cliques. Give a poly-nomial time recognition algorithm for Berge graphs with polynomially bounded number of maximal cliques. Contributed by Jeremy Spinrad. \n\nA.6 TDI Matrices \n\n1. Given an m × n 0 − 1 matrix A and an m-dimensional vector b, decide whether the system \n\nAx ≤ b is totally dual integral (TDI), that is, is there an integer dual solution for every objective function for which the dual optimum exists. \n\nThis is the 0 − 1 special case of the well-known problem of TDI system recognition. Let \n\nP:= Ax ≤ b, x ≥ 0. Let A(u) be the matrix whose rows are the normal vectors of facets of \n\nP containing u, each row is integer and the gcd of its entries is 1. There are several known relations between TDI and unimodular systems. As Serkan Ho¸ sten pointed out, 'nondegenerate' TDI matrices are exactly those in which A(u) is an \n\nn × n matrix having determinant 1 for every vertex u.The following problem involves unimodularity in perfectness test: 6\n\nINPUT: m × n 0-1 matrix A and an m-dimensional positive vector b,QUESTION: Is the matrix A(u) for every vertex u of P a square matrix of determinant 1? \n\n2. Can this problem be solved in polynomial time? \n\nFor reducing the test for the perfectness of matrix A to this problem define b with a 'lexicographic perturbation' from the all 1 objective function. Contributed by Andr´ as Seb˝ o\n\nA.7 Fixed Parameter Algorithms \n\nIn the context of fixed parameter algorithms, which was recently introduced by Downey and Fellows, it would be interesting to design an algorithm with running time O(f (k)|V |c)where k is the size of the maximum clique, c is a small constant independent of k and f (k)is a (exponential) function of k. Such an algorithm can work for small k even for large n.Contributed by Mohammad Taghi Hajiaghayi \n\nA.8 Clique Joins \n\nA k-clique-join of G = ( V, E ) is a set of pairs {(A0, B 0), (A1, B 1),..., (Ak, B k)}, where \n\n{A0, B 0} is a partition of V, both A0 and B0 contain at least one ω-clique, and Ai ⊆ A0,\n\nBi ⊆ B0 (i = 1,..., k ) (not necessarily disjoint), moreover (i) If x ∈ Ai and y ∈ Bi, then xy ∈ E\n\n(ii) If K is an ω-clique of G that meets both A0 and B0, then there exists i so that \n\nK ⊆ Ai ∪ Bi.A partitionable graph does not contain a k-clique-join for k < 2( ω − 1), on the other hand odd holes, odd antiholes do all contain 2( ω − 1)-clique-joins. \n\nCould the minimum of k for which a k-clique-join exists be computed (or well-characterized)? For Berge-graphs? Is there a variant of this operation that would allow to compose perfect graphs and keep perfectness? \n\nContributed by Andr´ as Seb˝ o\n\nA.9 Polynomial Size Decomposition Tree \n\nThe problem with using skew-paritions (or star cutsets) for recognition algorithms is that the decomposition tree they induce does not have polynomial size. The following questions have been suggested during the workshop: 1. Suggest different endblocks of the decompostition (other than basic perfect graphs), that can be recognized in polynomial time and yet make the decomposition tree polynomial (Bruce Reed) 2. Normally every skew-partition has 4 decomposition blocks. What if we could prove that it is enough to consider only two blocks for each skew-partition. Would that imply a polynomial size decomposition tree? Possibly introducing new endblocks or using cleaning? (Kristina Vuskovic). 3.Does decomposition of C4-free graphs via star-cutsets induce a polynomial size de-composition tree? Possibly using cleaning? (Kristina Vuskovic). 7\n\nChapter B: Structural Characterization of Perfect Graphs \n\nPossible structural characterization of perfect graphs. Give explicit constructions for subclasses of Berge graphs. Contributed by Paul Seymour \n\nChapter C: Coloring Perfect Graphs \n\nCan one find an efficient algorithms to color a perfect graph? \n\nC.1 Uniquely colorable perfect graphs \n\nUniquely colorable perfect graphs (in which there is a unique partition into ω stable-sets) are closely related to minimal imperfect graphs: according to a result of Padberg (Perfect zero-one matrices, Math Programming, 6, (1974)) if G is minimal imperfect then for all of its vertices v, the graph G − v is uniquely colorable. There is also a combinatorial good characterization theorem for unique colorability of perfect graphs, and a polynomial algorithm for testing the property using the ellipsoid method (IPCO 1, Kannan, Pulleyblank eds, Waterloo Univ. Press, 1990). In other words \n\nUNIQUE COLORABILITY is a tractable property for perfect graphs, closely related to min-imal imperfect graphs. Yet, some simple conjectures related to the SPGT, resist through the years. The following one arises both by specializing more general conjectures occurring in various papers, and does not seem to trivially follow from the SPGT: \n\nIf G is perfect and uniquely colorable, does there exist two ω-cliques which meet in ω −1\n\npoints? \n\nLet us call the two vertices in the symmetric difference of two such cliques forced.\n\nIs it true that every known uniquely colorable perfect graph collapses to an ω-clique by successive identification of forced vertices? \n\nIt can be simply proved that a minimal imperfect graph with three forced vertices in particular positions is an odd hole or an odd antihole. A simpler proof of the following statement would shortcut the proof of the SPGT: \n\nIf G is minimal imperfect, there is a vertex v so that N (v) is uniquely colorable. \n\nContributed by Jean Fonlupt and Andr´ as Seb˝ o\n\nChapter D: Optimization on Perfect Graphs \n\nOptimization on perfect graphs without using the ellipsoid method. \n\nD.1 New Optimization Problems on Perfect Graphs \n\nAre there any new optimization problems (other that coloring and finding the size of the max clique) that are easier to solve for a perfect graph that for a general graph? Can we use any of the existing (future) recognition algorithms in order to do that? Contributed by Mohammad Hajiaghayi \n\nD.1.a A Possible New Problem. Solve in a perfect graph: do two given vertices belong to an induced hole? Contributed by Bruce Reed 8\n\nChapter E: Skew-Partitions \n\nE.1 Extending a Skew -Partition \n\nWhen can a skew partition of an induced subgraph be extended to a skew partition of G? Algorithmically this is answered by the algorithm of de Figueiredo, Klein, Kohayakawa and Reed [2001j:05114], but what about a theorem? Is there some theorem that says \"either the skew partition is extendable, or there is a reason why not (an obstruction)\"? Contributed by Paul Seymour \n\nE.2 Graphs Without Skew-Partitions \n\nIs there a structure theorem for such graphs? Can they all be constructed somehow? Maybe by starting with a small one, and adding little bits so that at each stage there is no skew partition? Contributed by Paul Seymour \n\nE.3 Graphs Without Star Cutsets \n\nIs there a structure theorem for such graphs? Can they all be constructed somehow? Maybe by starting with a small one, and adding little bits so that at each stage there is no star cutset? Contributed by Bruce Reed \n\nConjecture If neither G nor Gc has a star cutset then the disk-structure of G is connected (A disk is a hole or an antihole. Two disks are adjacent in the disk structure if they share at least 2 vertices). Contributed by Ryan Hayward \n\nE.4 Finding Skew-Partitions in Berge Graphs \n\nIs it easier to detect skew partitions in Berge graphs than in general ones? Contributed by Paul Seymour \n\nE.5 Interaction Between Different Skew-Partitions in a Graph \n\nFor 1 ≤ i ≤ n, let ( Ai, B i, C i, D i) be a skew partition of G, where there are no edges between Ai and Bi, and Ci is complete to Di. For each i, choose one of Ai, B i, C i, D i, say \n\nXi, and let X be the union of all these Xi. Call G \\ X a chunk. If there is an odd hole or antihole in G, then it belongs to the chunk (for some choice of the Xi's), so to check Bergeness of G, it is enough to check Bergeness of all the chunks. But even if n is linear in the size of G, the number of chunks can be exponential. Maybe there is a way around this. With decomposition theorems that come up from excluded minors, using separations instead of skew partitions, the same exponential blowup happens, but it can be avoided by using separations that are pairwise noncrossing - then you only get linearly many pieces. Is there an analogous nice way for skew partitions to fit together, so that we only get linearly many (or polynomially many) chunks? Contributed by Paul Seymour 9\n\nE.6 Skew -Partitions of Balanced Size \n\nThe problem with recursive skew decomposition is that at least at first glance, you get exponential behavior. This would not occur if you were always able to find a decomposition in which each of A,B,C,D have at least n/c vertices for some c. Call this a skew partition of balanced size. a) Can you find a skew partition of balanced in polynomial time, if one exists? b) Will always looking for a balanced skew partition if possible lead to a polynomial size decomposition tree for perfect graphs (ie always use the skew partition which maximizes the size of the smallest set) Contributed by Jeremy Spinrad Answer: There exists a graph admitting no skew-partition of balanced size: take a clique and for some edges e1,..., e k of it add a vertices v1,..., v k s.t. each vi has degree 2 and is adjacent to both ends of ei.\n\nE.7 Recognizing Balanced Skew-Partitions \n\nGiven a skew-partition, can one check in polynomial time whether it is balanced. Contributed by Jeremy Spinrad \n\nE.8 Even-Pair Skew-Partition \n\nAn even-pair skew-partition is a partition of the vertex set of a graph G into four sets \n\nA, B, C, D s.t. A is complete to B and C is anti-complete to D, and any two non-adjacent vertices in A or B are an even pair.", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "The canonical input is not one mathematical problem. It is a malformed extraction from the 21-page AIM workshop document *Perfect Graphs* (version dated 24 August 2004). The input begins", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[269]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4. NP Characterization of Perfect Graphs 5. Recognition Algorithm Given the List of Maximal Cliques a. Berge Graphs with Poly-bounded Number of Max Cliques 6. TDI Matrices 7. Fixed Parameter Algorithms 8. Clique Joins 9. Polynomial Size Decomposition Tree B. Structural Characterization of Perfect Graphs............... 6C. Coloring Perfect Graphs......................... 71. Uniquely colorable perfect graphs D. Optimization on Perfect Graphs..................... 71. New Optimization Problems on Perfect Graphs a. A Possible New Problem E. Skew-Partitions............................ 71. Extending a Skew -Partition 2. Graphs Without Skew-Partitions 3. Graphs Without Star Cutsets 4. Finding Skew-Partitions in Berge Graphs 5. Interaction Between Different Skew-Partitions in a Graph 6. Skew -Partitions of Balanced Size 7. Recognizing Balanced Skew-Partitions 8. Even-Pair Skew-Partition F. Even Pairs in Berge Graphs....................... 91. Coloring Berge Graphs Using Even Pairs 2. Recognizing Even Pairs 3. Quasi-Parity and Strict Quasi-Parity Graphs a. Forbidden Subgraphs for The Class of Strict Quasi-Parity Graphs b. Recognition of Quasi-Parity and Strict Quasi-Parity Graphs 4. Perfectly Contractile Graphs a. Perfectly Contractile Graphs and the Decomposition Method 5. Possible Structure Theorem for Berge Graphs 6. Odd holes and odd walks G. Forbidding Holes and Antiholes.................... 12 1. 2-divisible Graphs 2. Clique Coloring of Perfect Graphs 3. Recognition of Odd-Hole-Free Graphs 4. Even-Hole-Free Graphs 5. Even-hole-free circulants, 6. beta-perfect graphs H. Partitionable Graphs......................... 14 1. Perfect, Partitionable, and Kernel-Solvable Graphs 3\\n\\n2. Partitionable graphs and odd holes 3. A Property of Partitionable Graphs 4. Small Transversals in Partitionable Graphs I. The Imperfection Ratio........................ 16 J. Integer Programming......................... 18 1. Partitionable Graphs as Cutting Planes for Packing Problems? 2. Feasibility/Membership Problem For the Theta Body K. Balanced Graphs........................... 18 1. Balanced circulants L. P4-structure and Its Relatives..................... 19 4\\n\\nChapter A: Recognition of Perfect Graphs \\n\\nCan one decide in polynomial time if a graph is perfect? \\n\\nA.1 Polynomial Recognition Algorithm Found \\n\\nA polynomial algorithm to test whether a graph is Berge was found in November 2002. A paper summarizing the work of two groups- Chudnovsky and Seymour and Cornuejols, Liu and Vuscovic- is due to appear in Combinatorica. The algorithm is independent of the proof of the strong perfect graph conjecture. \\n\\nA.2 Interaction Between Skew-Partitions and 2-joins \\n\\nOne can think of algorithms for testing for odd holes if you use only 2-joins to decom-pose a graph, or if you use only skew partitions. However, the interaction between skew partition steps and 2-join steps adds another level of difficulty. Can one argue that this can be reduced only to testing for holes as you decompose using skew partitions only and testing for holes using 2-joins only? Contributed by Jeremy Spinrad Similar approach worked in algorithms for recognizing even-hole-free graphs: first the graph is decomposed via vertex cut-sets, and only then via 2-joins. The 2-join decomposition blocks are defined in such a way that no new vertex cutset is introduced. Contributed by Kristina Vuskoic \\n\\nA.3 The Perfect-Graph Robust Algorithm Problem \\n\\nAn algorithm, which for any easily recognizable input, A, finds either an easily rec-ognizable B or an easily recognizable C, is sometimes called a \\\"robust algorithm\\\" - to be distinguished from a non-robust algorithm, which, for any A without a B, finds a C. Either provides a proof of the \\\"existentially polytime (EP) theorem\\\": For any A, there exists a B or a C. In other words: For any A without a B, there is a C. In [92i:68043 ], Jack Edmonds and Kathie Cameron advocated seeking a robust al-gorithm which, for any graph G, finds either a clique and colouring the same size or else finds an easily recognizable combinatorial obstruction to G being perfect. The obstruction might be specified to be a \\\"alpha-omega partitioned subgraph\\\", or it might be specified more particularly to be an odd hole or odd antihole. Such an algorithm might be simpler than an algorithm for recognizing whether or not a graph is perfect, in view of precedents, and since what it would do is incomparable with perfect-graph recognition. Such an algorithm could end up giving a clique and colouring the same size in a non-perfect graph. Here are two examples of similar problems which have been solved. Edmonds has given a simple robust algorithm which, for any graph G, either finds an odd cycle > 3 with at most one chord (a defining obstruction to G being Meyniel) or else finds a clique and colouring the same size. This is an improvement on the non-robust algorithms of Hoang and Hertz which, assuming a graph is Meyniel, find a clique and colouring the same size. Edmonds' algorithm is much simpler than the Burlet-Fonlupt decomposition algorithm for recognizing Meyniel graphs, which was motivated by an interest in optimizing in Meyniel graphs, and which is used by Hoang and Hertz. 5\\n\\nConforti and Cornuejols give a complicated decomposition algorithm for recognizing whether or not a matrix is balanced. At about the same time, to motivate the advocacy of a robust algorithm for either node-colouring a graph or recognizing it to be not perfect, Cameron and Edmonds presented a simple algorithm which, for any 0-1 matrix M, either finds, where x is the largest number of ones in any row, an x-colouring of the columns so that the 1's of any row are in different coloured columns, or else finds \\\"an odd hole\\\" in M (the defining obstruction to M being balanced). This introduced the \\\"EP - robust algorithm\\\" paradigm which is followed in Edmonds' Meyniel-related algorithm, and is related to the Conforti-Cornuejols-Rao treatment of balanced matrices in the same way that Edmonds' is related to the Burlet-Fonlupt treatment of Meyniel graphs. Following the same paradigm we expect there to be a robust algorithm proving the SPCG, related in the same way to the Chudnovsky-Robertson-Seymour-Thomas decomposition of Berge graphs. In conclusion, we know the following EP Theorem 1: For any graph, there is either a clique and a colouring of the same size, or there is an alpha-omega partitioned subgraph (or both). EP Theorem 2 (SPGT): For any graph, there is either a clique and a colouring of the same size, or there is a odd hole or odd antihole (or both). So: Give a combinatorial polytime algorithm to find what the EP theorem asserts to exist. Contributed by Kathie Cameron and Jack Edmonds \\n\\nA.4 NP Description of Perfect Graphs \\n\\nGive an NP description of perfect graphs. Contributed by Jack Edmonds. \\n\\nA.5 Recognition Algorithm Given the List of Maximal Cliques \\n\\nIs there a polytime recognition algorithm for perfect graphs where the input is the list of all maximal cliques in the graph? This probelm has been resolved, since a perfect graph can itself now be recognized. Contributed by Bruce Shepherd \\n\\nA.5.a Berge Graphs with Poly-bounded Number of Max Cliques. Give a poly-nomial time recognition algorithm for Berge graphs with polynomially bounded number of maximal cliques. Contributed by Jeremy Spinrad. \\n\\nA.6 TDI Matrices \\n\\n1. Given an m × n 0 − 1 matrix A and an m-dimensional vector b, decide whether the system \\n\\nAx ≤ b is totally dual integral (TDI), that is, is there an integer dual solution for every objective function for which the dual optimum exists. \\n\\nThis is the 0 − 1 special case of the well-known problem of TDI system recognition. Let \\n\\nP:= Ax ≤ b, x ≥ 0. Let A(u) be the matrix whose rows are the normal vectors of facets of \\n\\nP containing u, each row is integer and the gcd of its entries is 1. There are several known relations between TDI and unimodular systems. As Serkan Ho¸ sten pointed out, 'nondegenerate' TDI matrices are exactly those in which A(u) is an \\n\\nn × n matrix having determinant 1 for every vertex u.The following problem involves unimodularity in perfectness test: 6\\n\\nINPUT: m × n 0-1 matrix A and an m-dimensional positive vector b,QUESTION: Is the matrix A(u) for every vertex u of P a square matrix of determinant 1? \\n\\n2. Can this problem be solved in polynomial time? \\n\\nFor reducing the test for the perfectness of matrix A to this problem define b with a 'lexicographic perturbation' from the all 1 objective function. Contributed by Andr´ as Seb˝ o\\n\\nA.7 Fixed Parameter Algorithms \\n\\nIn the context of fixed parameter algorithms, which was recently introduced by Downey and Fellows, it would be interesting to design an algorithm with running time O(f (k)|V |c)where k is the size of the maximum clique, c is a small constant independent of k and f (k)is a (exponential) function of k. Such an algorithm can work for small k even for large n.Contributed by Mohammad Taghi Hajiaghayi \\n\\nA.8 Clique Joins \\n\\nA k-clique-join of G = ( V, E ) is a set of pairs {(A0, B 0), (A1, B 1),..., (Ak, B k)}, where \\n\\n{A0, B 0} is a partition of V, both A0 and B0 contain at least one ω-clique, and Ai ⊆ A0,\\n\\nBi ⊆ B0 (i = 1,..., k ) (not necessarily disjoint), moreover (i) If x ∈ Ai and y ∈ Bi, then xy ∈ E\\n\\n(ii) If K is an ω-clique of G that meets both A0 and B0, then there exists i so that \\n\\nK ⊆ Ai ∪ Bi.A partitionable graph does not contain a k-clique-join for k < 2( ω − 1), on the other hand odd holes, odd antiholes do all contain 2( ω − 1)-clique-joins. \\n\\nCould the minimum of k for which a k-clique-join exists be computed (or well-characterized)? For Berge-graphs? Is there a variant of this operation that would allow to compose perfect graphs and keep perfectness? \\n\\nContributed by Andr´ as Seb˝ o\\n\\nA.9 Polynomial Size Decomposition Tree \\n\\nThe problem with using skew-paritions (or star cutsets) for recognition algorithms is that the decomposition tree they induce does not have polynomial size. The following questions have been suggested during the workshop: 1. Suggest different endblocks of the decompostition (other than basic perfect graphs), that can be recognized in polynomial time and yet make the decomposition tree polynomial (Bruce Reed) 2. Normally every skew-partition has 4 decomposition blocks. What if we could prove that it is enough to consider only two blocks for each skew-partition. Would that imply a polynomial size decomposition tree? Possibly introducing new endblocks or using cleaning? (Kristina Vuskovic). 3.Does decomposition of C4-free graphs via star-cutsets induce a polynomial size de-composition tree? Possibly using cleaning? (Kristina Vuskovic). 7\\n\\nChapter B: Structural Characterization of Perfect Graphs \\n\\nPossible structural characterization of perfect graphs. Give explicit constructions for subclasses of Berge graphs. Contributed by Paul Seymour \\n\\nChapter C: Coloring Perfect Graphs \\n\\nCan one find an efficient algorithms to color a perfect graph? \\n\\nC.1 Uniquely colorable perfect graphs \\n\\nUniquely colorable perfect graphs (in which there is a unique partition into ω stable-sets) are closely related to minimal imperfect graphs: according to a result of Padberg (Perfect zero-one matrices, Math Programming, 6, (1974)) if G is minimal imperfect then for all of its vertices v, the graph G − v is uniquely colorable. There is also a combinatorial good characterization theorem for unique colorability of perfect graphs, and a polynomial algorithm for testing the property using the ellipsoid method (IPCO 1, Kannan, Pulleyblank eds, Waterloo Univ. Press, 1990). In other words \\n\\nUNIQUE COLORABILITY is a tractable property for perfect graphs, closely related to min-imal imperfect graphs. Yet, some simple conjectures related to the SPGT, resist through the years. The following one arises both by specializing more general conjectures occurring in various papers, and does not seem to trivially follow from the SPGT: \\n\\nIf G is perfect and uniquely colorable, does there exist two ω-cliques which meet in ω −1\\n\\npoints? \\n\\nLet us call the two vertices in the symmetric difference of two such cliques forced.\\n\\nIs it true that every known uniquely colorable perfect graph collapses to an ω-clique by successive identification of forced vertices? \\n\\nIt can be simply proved that a minimal imperfect graph with three forced vertices in particular positions is an odd hole or an odd antihole. A simpler proof of the following statement would shortcut the proof of the SPGT: \\n\\nIf G is minimal imperfect, there is a vertex v so that N (v) is uniquely colorable. \\n\\nContributed by Jean Fonlupt and Andr´ as Seb˝ o\\n\\nChapter D: Optimization on Perfect Graphs \\n\\nOptimization on perfect graphs without using the ellipsoid method. \\n\\nD.1 New Optimization Problems on Perfect Graphs \\n\\nAre there any new optimization problems (other that coloring and finding the size of the max clique) that are easier to solve for a perfect graph that for a general graph? Can we use any of the existing (future) recognition algorithms in order to do that? Contributed by Mohammad Hajiaghayi \\n\\nD.1.a A Possible New Problem. Solve in a perfect graph: do two given vertices belong to an induced hole? Contributed by Bruce Reed 8\\n\\nChapter E: Skew-Partitions \\n\\nE.1 Extending a Skew -Partition \\n\\nWhen can a skew partition of an induced subgraph be extended to a skew partition of G? Algorithmically this is answered by the algorithm of de Figueiredo, Klein, Kohayakawa and Reed [2001j:05114], but what about a theorem? Is there some theorem that says \\\"either the skew partition is extendable, or there is a reason why not (an obstruction)\\\"? Contributed by Paul Seymour \\n\\nE.2 Graphs Without Skew-Partitions \\n\\nIs there a structure theorem for such graphs? Can they all be constructed somehow? Maybe by starting with a small one, and adding little bits so that at each stage there is no skew partition? Contributed by Paul Seymour \\n\\nE.3 Graphs Without Star Cutsets \\n\\nIs there a structure theorem for such graphs? Can they all be constructed somehow? Maybe by starting with a small one, and adding little bits so that at each stage there is no star cutset? Contributed by Bruce Reed \\n\\nConjecture If neither G nor Gc has a star cutset then the disk-structure of G is connected (A disk is a hole or an antihole. Two disks are adjacent in the disk structure if they share at least 2 vertices). Contributed by Ryan Hayward \\n\\nE.4 Finding Skew-Partitions in Berge Graphs \\n\\nIs it easier to detect skew partitions in Berge graphs than in general ones? Contributed by Paul Seymour \\n\\nE.5 Interaction Between Different Skew-Partitions in a Graph \\n\\nFor 1 ≤ i ≤ n, let ( Ai, B i, C i, D i) be a skew partition of G, where there are no edges between Ai and Bi, and Ci is complete to Di. For each i, choose one of Ai, B i, C i, D i, say \\n\\nXi, and let X be the union of all these Xi. Call G \\\\ X a chunk. If there is an odd hole or antihole in G, then it belongs to the chunk (for some choice of the Xi's), so to check Bergeness of G, it is enough to check Bergeness of all the chunks. But even if n is linear in the size of G, the number of chunks can be exponential. Maybe there is a way around this. With decomposition theorems that come up from excluded minors, using separations instead of skew partitions, the same exponential blowup happens, but it can be avoided by using separations that are pairwise noncrossing - then you only get linearly many pieces. Is there an analogous nice way for skew partitions to fit together, so that we only get linearly many (or polynomially many) chunks? Contributed by Paul Seymour 9\\n\\nE.6 Skew -Partitions of Balanced Size \\n\\nThe problem with recursive skew decomposition is that at least at first glance, you get exponential behavior. This would not occur if you were always able to find a decomposition in which each of A,B,C,D have at least n/c vertices for some c. Call this a skew partition of balanced size. a) Can you find a skew partition of balanced in polynomial time, if one exists? b) Will always looking for a balanced skew partition if possible lead to a polynomial size decomposition tree for perfect graphs (ie always use the skew partition which maximizes the size of the smallest set) Contributed by Jeremy Spinrad Answer: There exists a graph admitting no skew-partition of balanced size: take a clique and for some edges e1,..., e k of it add a vertices v1,..., v k s.t. each vi has degree 2 and is adjacent to both ends of ei.\\n\\nE.7 Recognizing Balanced Skew-Partitions \\n\\nGiven a skew-partition, can one check in polynomial time whether it is balanced. Contributed by Jeremy Spinrad \\n\\nE.8 Even-Pair Skew-Partition \\n\\nAn even-pair skew-partition is a partition of the vertex set of a graph G into four sets \\n\\nA, B, C, D s.t. A is complete to B and C is anti-complete to D, and any two non-adjacent vertices in A or B are an even pair.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0270", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is invalid as a single statement: it begins inside the source PDF's table of contents, repeats Chapters A through E, and ends just before E.8 Question 1, while the next records split at generic Question 1 and Question 2 labels. For the clearly delimited A.4 subproblem, perfect-graph recognition is in P by O(n^9) Berge recognition plus the Strong Perfect Graph Theorem, so the literal NP request is solved. As a developed structural special case, a perfect elimination ordering gives one polynomial certificate for all induced-subgraph equalities, and the E.6 clique-plus-degree-two family has the explicit ordering consisting of all added vertices followed by the original clique.\n\nCandidate contribution (certificate_synthesis; novelty confidence low): The E.6 clique-plus-edge-vertices family embedded later in this malformed record has the explicit perfect-elimination certificate 'all added degree-two vertices, then the original clique'; the same certificate yields an explicit equal-size clique and coloring, with the necessary m=2 exception where the optimum is three.", "research_difficulty_suggested": "L2" }, { "id": 20001146, "problem_number": "AIM-COMBINATORICS-0271", "title": "A 21-vertex counterexample via the complement of the WBGKSF", "statement": "Question 1 Is it true that every Berge graph is either basic or has a 2-join or has an even-pair skew-partition?", "original_statement": "Question 1 Is it true that every Berge graph is either basic or has a 2-join or has an even-pair skew-partition?", "clean_statement": "Question 1 Is it true that every Berge graph is either basic or has a 2-join or has an even-pair skew-partition?", "statement_status": "exact", "statement_verification": "The canonical record comes from the AIM workshop *The Perfect Graph Conjecture* (official PDF version dated 24 August 2004). The source defines an **even-pair skew-partition** as a partition of \\(V(G)\\) into four sets \\(A,B,C,D\\) such that \\(A\\) is complete to \\(B\\), \\(C\\) is anticomplete to \\(D\\), and every two nonadjacent vertices lying together in \\(A\\) or together in \\(B\\) form an even pair. As usual for a split of a skew partition, all four sets are nonempty. An even pair is a nonadjacent pair for which every induced path between its vertices has even length.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[270]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 1 Is it true that every Berge graph is either basic or has a 2-join or has an even-pair skew-partition?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0271", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Let W be the 21-vertex Worst Berge Graph Known So Far displayed in Trotignon's survey and let H be its complement. The survey states that W is perfect, nonbasic, has neither a 2-join nor a complement 2-join, and that every edge of W is the middle edge of an induced P4; hence H is a nonbasic Berge graph with no 2-join and no even pair. In any even-pair-free graph, an even-pair skew-partition exists exactly when there is a clique cutset of size at least two. An exhaustive check of all 3,229 cliques of H from the reproduced adjacency list finds no clique cutset. Therefore H has no even-pair skew-partition and refutes the AIM question.\n\nCandidate contribution (counterexample; novelty confidence low): Candidate novelty: an even-pair-free graph has an even-pair skew-partition if and only if it has a clique cutset of size at least two; applying this equivalence and a complete no-clique-cutset certificate to the complement of the 21-vertex WBGKSF yields an explicit counterexample to AIM-COMBINATORICS-0271." }, { "id": 20001147, "problem_number": "AIM-COMBINATORICS-0272", "title": "Even-pair skew partitions and a natural two-block localization theorem", "statement": "Question 2 Is even-pair skew-partition a composition? Contributed by Bruce Reed\n\nChapter F: Even Pairs in Berge Graphs\n\nAn even pair is a pair of vertices such that each chordless path between them has even length. Results of Fonlupt and Uhry, Meyniel, and also Bertschi and Reed imply that no minimal imperfect graph contains an even pair. A graph G is called quasi-parity (QP) if, for every induced subgraph H of G on at least two vertices, either H or its complement has an even pair. A graph G is called strict quasi-parity (SQP) if every induced subgraph H of G either H is aclique or has an even pair. In the past 20 years many classical families of perfect graphs were proven to be SQP, which shows the interest of this class. However there are perfect graphs with no even pairs, e.g., all the line-graphs of 3-connected bipartite graphs. None of the following questions or conjectures has been settled in full. Solutions are known only for special subclasses of graphs, e.g., planar graphs, claw-free graphs, bull-free graphs, etc. Contributed by Fr´ ed´ eric Maffray 10\n\nF.1 Coloring Berge Graphs Using Even Pairs\n\nIt is known (from Fonlupt and Uhry) that contracting an even pair in a perfect graph yields a perfect graph with the same chromaticnumber. This idea can be used as the basis for a conceptually simple coloring algorithm. Contributed by Fr´ ed´ eric Maffray\n\nF.2 Recognizing Even Pairs\n\nCan one decide in polynomial time if a given Berge graph has an even pair? (The general problem, i.e., not restricted to Berge graphs, is known to be co-NP-complete.) Contributed by Fr´ ed´ eric Maffray Can one find even pairs using balanced skew-partitions? (Is there always an even pair in the cutset of a balanced skew-partition?) Contributed by Bruce Ree\n\nF.3 Quasi-Parity and Strict Quasi-Parity Graphs\n\nA graph G is called quasi-parity (QP) if, for every induced subgraph H of G on at least two vertices, either H or its complement has an even pair. A graph G is called strict quasi-parity (SQP) if every induced subgraph H of G either H is a clique or has an even pair. In the past 20 years many classical families of perfect graphs were proven to be SQP, which shows the interest of this class. Contributed by Fr´ ed´ eric Maffray\n\nF.3.a Forbidden Subgraphs for The Class of Strict Quasi-Parity Graphs. Hougardy conjectured that the minimal forbidden induced subgraphs for the class SQP are odd hole, antiholes, and some line-graphs of bipartite graphs (not determined explicitly). Contributed by Fr´ ed´ eric Maffray\n\nF.3.b Recognition of Quasi-Parity and Strict Quasi-Parity Graphs. Can one decide in polynomial time if a given Berge graph is in the class QP, or SQP? Contributed by Fr´ ed´ eric Maffray\n\nF.4 Perfectly Contractile Graphs\n\nBertschi called a graph G even-contractile if there exists a sequence of even-pair con-tractions that turn G into a clique, and he called a graph G perfectly contractile (PC) if every induced subgraph of G is even-contractile. Many classical families of graphs (Meyniel graphs, weakly chordal graphs, perfectly orderable graphs, etc) are perfectly contractile, and for some of them (Meyniel graphs, weakly chordal graphs) the coloring algorithm based on even-pair contractions is the most efficient that is known so far. Everett and Reed conjectured that a graph is PC if and only if it contains no odd hole, no antihole, and no odd prism (two disjoint triangles with three disjoint chordless odd paths between them). Maffray and Trotignon proved a weaker form of this conjecture, also due to Everett and Reed: if a graph contains no odd hole, no antihole, and no prism, and it is not a clique, then the graph admits an even pair whose contraction yields a graph with no odd hole, no 11\n\nantihole, and no prism. The proof is an algorithm to find such a pair. Here is a link to a preprint 1.The same authors found a polynomial-time algorithm to decide if a graph belongs to that class (the class of graphs with no odd hole, no antihole, and no prism). Here is a link to a preprint 2.Contributed by Fr´ ed´ eric Maffray\n\nF.4.a Perfectly Contractile Graphs and the Decomposition Method. The following conjecture, due to Everett and Reed attempts to characterize perfectly contractile graphs.\n\nPerfectly Contractile Graph Conjecture (PCGC) A graph is perfectly contractile if and only if it does not contain an odd hole, an antihole nor an odd prism.\n\nWe propose to investigate the PCGC and a possible construction of a polynomial-time recognition algorithm for perfectly contractile graphs, through the decomposition method. The decomposition method is based on a decomposition theorem of the following form, for the class of graphs C we want to analyse.\n\nDecomposition Theorem If G ∈ C, then G is either basic or it contains certain types of cutsets.\n\nBasic stands for a certain \"simple\" subclass of C.The idea of a decomposition based recognition algorithm for the class C is as follows. In a connected graph G, a node set (or an edge set or a combination of the two) is a\n\ncutset if its removal disconnects G into two or more connected components. From these components blocks of decomposition are constructed by adding some more nodes and edges. A decomposition is C-preserving if it satisfies the following: G belongs to C if and only if all the blocks of decomposition belong to C. A decomposition based recognition algorithm takes an input graph G and decomposes it using C-preserving decompositions into a polynomial number of basic blocks, which are then checked, in polynomial time, whether they belong to\n\nC.Such a construction of blocks works nicely for clique cutsets. A node set S is a star cutset of a graph G if its removal disconnects G and S contains a node that is adjacent to all the other nodes of S. With the usual construction of blocks for the node cutsets, the star cutset decomposition is not preserving for the class of perfectly contractile graphs. A generalization of star cutsets is obtained as follows. 1-Amalgams are defined and used in for the construction of a recognition algorithm for Meyniel graphs. A graph G has a\n\n1-amalgam if its vertex set can be partitioned into sets V1, V2 and K (where K is possibly empty) in such a way that:\n\n• for i = 1, 2, |Vi| ≥ 2 and Vi contains a nonempty set Ai;\n\n• every node of A1 is adjacent to every node of A2 and these are the only adjacencies between the nodes of V1 and the nodes of V2; and\n\n• if K 6 = ∅, then it induces a clique, and every node of K is adjacent to every node of\n\nA1 ∪ A2.A graph G has a 2-join if its node set can be partitioned into sets V1 and V2 so that for i = 1, 2, Vi contains disjoint nonempty sets Ai and Bi, and the following properties hold:\n\n> 1http://www-leibniz.imag.fr/LesCahiers/2002/Cahier67/ResumCahier67.html\n> 2http://www-leibniz.imag.fr/NEWLEIBNIZ/LesCahiers/Cahier106/ResumCahier106.html 12\n\n• every node of A1 (resp. B1) is adjacent to every node of A2 (resp. B2), and these are the only adjacencies between the nodes of V1 and the nodes of V2;\n\n• for i = 1, 2, let Pi be the set of all chordless paths in G[Vi] with one endnode in Ai,the other endnode in Bi, and no intermediate node in Ai ∪ Bi. For i = 1, 2, Pi 6 = ∅\n\nand G[Vi] is not isomorphic to a path in Pi.Let Cpc denote the class of perfectly contractile graphs. We have the following conjectures.", "original_statement": "Question 2 Is even-pair skew-partition a composition? Contributed by Bruce Reed \n\nChapter F: Even Pairs in Berge Graphs \n\nAn even pair is a pair of vertices such that each chordless path between them has even length. Results of Fonlupt and Uhry, Meyniel, and also Bertschi and Reed imply that no minimal imperfect graph contains an even pair. A graph G is called quasi-parity (QP) if, for every induced subgraph H of G on at least two vertices, either H or its complement has an even pair. A graph G is called strict quasi-parity (SQP) if every induced subgraph H of G either H is aclique or has an even pair. In the past 20 years many classical families of perfect graphs were proven to be SQP, which shows the interest of this class. However there are perfect graphs with no even pairs, e.g., all the line-graphs of 3-connected bipartite graphs. None of the following questions or conjectures has been settled in full. Solutions are known only for special subclasses of graphs, e.g., planar graphs, claw-free graphs, bull-free graphs, etc. Contributed by Fr´ ed´ eric Maffray 10 \n\nF.1 Coloring Berge Graphs Using Even Pairs \n\nIt is known (from Fonlupt and Uhry) that contracting an even pair in a perfect graph yields a perfect graph with the same chromaticnumber. This idea can be used as the basis for a conceptually simple coloring algorithm. Contributed by Fr´ ed´ eric Maffray \n\nF.2 Recognizing Even Pairs \n\nCan one decide in polynomial time if a given Berge graph has an even pair? (The general problem, i.e., not restricted to Berge graphs, is known to be co-NP-complete.) Contributed by Fr´ ed´ eric Maffray Can one find even pairs using balanced skew-partitions? (Is there always an even pair in the cutset of a balanced skew-partition?) Contributed by Bruce Ree \n\nF.3 Quasi-Parity and Strict Quasi-Parity Graphs \n\nA graph G is called quasi-parity (QP) if, for every induced subgraph H of G on at least two vertices, either H or its complement has an even pair. A graph G is called strict quasi-parity (SQP) if every induced subgraph H of G either H is a clique or has an even pair. In the past 20 years many classical families of perfect graphs were proven to be SQP, which shows the interest of this class. Contributed by Fr´ ed´ eric Maffray \n\nF.3.a Forbidden Subgraphs for The Class of Strict Quasi-Parity Graphs. Hougardy conjectured that the minimal forbidden induced subgraphs for the class SQP are odd hole, antiholes, and some line-graphs of bipartite graphs (not determined explicitly). Contributed by Fr´ ed´ eric Maffray \n\nF.3.b Recognition of Quasi-Parity and Strict Quasi-Parity Graphs. Can one decide in polynomial time if a given Berge graph is in the class QP, or SQP? Contributed by Fr´ ed´ eric Maffray \n\nF.4 Perfectly Contractile Graphs \n\nBertschi called a graph G even-contractile if there exists a sequence of even-pair con-tractions that turn G into a clique, and he called a graph G perfectly contractile (PC) if every induced subgraph of G is even-contractile. Many classical families of graphs (Meyniel graphs, weakly chordal graphs, perfectly orderable graphs, etc) are perfectly contractile, and for some of them (Meyniel graphs, weakly chordal graphs) the coloring algorithm based on even-pair contractions is the most efficient that is known so far. Everett and Reed conjectured that a graph is PC if and only if it contains no odd hole, no antihole, and no odd prism (two disjoint triangles with three disjoint chordless odd paths between them). Maffray and Trotignon proved a weaker form of this conjecture, also due to Everett and Reed: if a graph contains no odd hole, no antihole, and no prism, and it is not a clique, then the graph admits an even pair whose contraction yields a graph with no odd hole, no 11 \n\nantihole, and no prism. The proof is an algorithm to find such a pair. Here is a link to a preprint 1.The same authors found a polynomial-time algorithm to decide if a graph belongs to that class (the class of graphs with no odd hole, no antihole, and no prism). Here is a link to a preprint 2.Contributed by Fr´ ed´ eric Maffray \n\nF.4.a Perfectly Contractile Graphs and the Decomposition Method. The following conjecture, due to Everett and Reed attempts to characterize perfectly contractile graphs. \n\nPerfectly Contractile Graph Conjecture (PCGC) A graph is perfectly contractile if and only if it does not contain an odd hole, an antihole nor an odd prism. \n\nWe propose to investigate the PCGC and a possible construction of a polynomial-time recognition algorithm for perfectly contractile graphs, through the decomposition method. The decomposition method is based on a decomposition theorem of the following form, for the class of graphs C we want to analyse. \n\nDecomposition Theorem If G ∈ C, then G is either basic or it contains certain types of cutsets. \n\nBasic stands for a certain \"simple\" subclass of C.The idea of a decomposition based recognition algorithm for the class C is as follows. In a connected graph G, a node set (or an edge set or a combination of the two) is a \n\ncutset if its removal disconnects G into two or more connected components. From these components blocks of decomposition are constructed by adding some more nodes and edges. A decomposition is C-preserving if it satisfies the following: G belongs to C if and only if all the blocks of decomposition belong to C. A decomposition based recognition algorithm takes an input graph G and decomposes it using C-preserving decompositions into a polynomial number of basic blocks, which are then checked, in polynomial time, whether they belong to \n\nC.Such a construction of blocks works nicely for clique cutsets. A node set S is a star cutset of a graph G if its removal disconnects G and S contains a node that is adjacent to all the other nodes of S. With the usual construction of blocks for the node cutsets, the star cutset decomposition is not preserving for the class of perfectly contractile graphs. A generalization of star cutsets is obtained as follows. 1-Amalgams are defined and used in for the construction of a recognition algorithm for Meyniel graphs. A graph G has a \n\n1-amalgam if its vertex set can be partitioned into sets V1, V2 and K (where K is possibly empty) in such a way that: \n\n• for i = 1, 2, |Vi| ≥ 2 and Vi contains a nonempty set Ai;\n\n• every node of A1 is adjacent to every node of A2 and these are the only adjacencies between the nodes of V1 and the nodes of V2; and \n\n• if K 6 = ∅, then it induces a clique, and every node of K is adjacent to every node of \n\nA1 ∪ A2.A graph G has a 2-join if its node set can be partitioned into sets V1 and V2 so that for i = 1, 2, Vi contains disjoint nonempty sets Ai and Bi, and the following properties hold: \n\n> 1http://www-leibniz.imag.fr/LesCahiers/2002/Cahier67/ResumCahier67.html\n> 2http://www-leibniz.imag.fr/NEWLEIBNIZ/LesCahiers/Cahier106/ResumCahier106.html 12\n\n• every node of A1 (resp. B1) is adjacent to every node of A2 (resp. B2), and these are the only adjacencies between the nodes of V1 and the nodes of V2;\n\n• for i = 1, 2, let Pi be the set of all chordless paths in G[Vi] with one endnode in Ai,the other endnode in Bi, and no intermediate node in Ai ∪ Bi. For i = 1, 2, Pi 6 = ∅\n\nand G[Vi] is not isomorphic to a path in Pi.Let Cpc denote the class of perfectly contractile graphs. We have the following conjectures.", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "The canonical input is another extraction mega-record, not one mathematical question. In the official 21-page AIM document *Perfect Graphs* (version 24 August 2004), the record begins at the second question of Section E.8 on PDF page index 8:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[271]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 2 Is even-pair skew-partition a composition? Contributed by Bruce Reed \\n\\nChapter F: Even Pairs in Berge Graphs \\n\\nAn even pair is a pair of vertices such that each chordless path between them has even length. Results of Fonlupt and Uhry, Meyniel, and also Bertschi and Reed imply that no minimal imperfect graph contains an even pair. A graph G is called quasi-parity (QP) if, for every induced subgraph H of G on at least two vertices, either H or its complement has an even pair. A graph G is called strict quasi-parity (SQP) if every induced subgraph H of G either H is aclique or has an even pair. In the past 20 years many classical families of perfect graphs were proven to be SQP, which shows the interest of this class. However there are perfect graphs with no even pairs, e.g., all the line-graphs of 3-connected bipartite graphs. None of the following questions or conjectures has been settled in full. Solutions are known only for special subclasses of graphs, e.g., planar graphs, claw-free graphs, bull-free graphs, etc. Contributed by Fr´ ed´ eric Maffray 10 \\n\\nF.1 Coloring Berge Graphs Using Even Pairs \\n\\nIt is known (from Fonlupt and Uhry) that contracting an even pair in a perfect graph yields a perfect graph with the same chromaticnumber. This idea can be used as the basis for a conceptually simple coloring algorithm. Contributed by Fr´ ed´ eric Maffray \\n\\nF.2 Recognizing Even Pairs \\n\\nCan one decide in polynomial time if a given Berge graph has an even pair? (The general problem, i.e., not restricted to Berge graphs, is known to be co-NP-complete.) Contributed by Fr´ ed´ eric Maffray Can one find even pairs using balanced skew-partitions? (Is there always an even pair in the cutset of a balanced skew-partition?) Contributed by Bruce Ree \\n\\nF.3 Quasi-Parity and Strict Quasi-Parity Graphs \\n\\nA graph G is called quasi-parity (QP) if, for every induced subgraph H of G on at least two vertices, either H or its complement has an even pair. A graph G is called strict quasi-parity (SQP) if every induced subgraph H of G either H is a clique or has an even pair. In the past 20 years many classical families of perfect graphs were proven to be SQP, which shows the interest of this class. Contributed by Fr´ ed´ eric Maffray \\n\\nF.3.a Forbidden Subgraphs for The Class of Strict Quasi-Parity Graphs. Hougardy conjectured that the minimal forbidden induced subgraphs for the class SQP are odd hole, antiholes, and some line-graphs of bipartite graphs (not determined explicitly). Contributed by Fr´ ed´ eric Maffray \\n\\nF.3.b Recognition of Quasi-Parity and Strict Quasi-Parity Graphs. Can one decide in polynomial time if a given Berge graph is in the class QP, or SQP? Contributed by Fr´ ed´ eric Maffray \\n\\nF.4 Perfectly Contractile Graphs \\n\\nBertschi called a graph G even-contractile if there exists a sequence of even-pair con-tractions that turn G into a clique, and he called a graph G perfectly contractile (PC) if every induced subgraph of G is even-contractile. Many classical families of graphs (Meyniel graphs, weakly chordal graphs, perfectly orderable graphs, etc) are perfectly contractile, and for some of them (Meyniel graphs, weakly chordal graphs) the coloring algorithm based on even-pair contractions is the most efficient that is known so far. Everett and Reed conjectured that a graph is PC if and only if it contains no odd hole, no antihole, and no odd prism (two disjoint triangles with three disjoint chordless odd paths between them). Maffray and Trotignon proved a weaker form of this conjecture, also due to Everett and Reed: if a graph contains no odd hole, no antihole, and no prism, and it is not a clique, then the graph admits an even pair whose contraction yields a graph with no odd hole, no 11 \\n\\nantihole, and no prism. The proof is an algorithm to find such a pair. Here is a link to a preprint 1.The same authors found a polynomial-time algorithm to decide if a graph belongs to that class (the class of graphs with no odd hole, no antihole, and no prism). Here is a link to a preprint 2.Contributed by Fr´ ed´ eric Maffray \\n\\nF.4.a Perfectly Contractile Graphs and the Decomposition Method. The following conjecture, due to Everett and Reed attempts to characterize perfectly contractile graphs. \\n\\nPerfectly Contractile Graph Conjecture (PCGC) A graph is perfectly contractile if and only if it does not contain an odd hole, an antihole nor an odd prism. \\n\\nWe propose to investigate the PCGC and a possible construction of a polynomial-time recognition algorithm for perfectly contractile graphs, through the decomposition method. The decomposition method is based on a decomposition theorem of the following form, for the class of graphs C we want to analyse. \\n\\nDecomposition Theorem If G ∈ C, then G is either basic or it contains certain types of cutsets. \\n\\nBasic stands for a certain \\\"simple\\\" subclass of C.The idea of a decomposition based recognition algorithm for the class C is as follows. In a connected graph G, a node set (or an edge set or a combination of the two) is a \\n\\ncutset if its removal disconnects G into two or more connected components. From these components blocks of decomposition are constructed by adding some more nodes and edges. A decomposition is C-preserving if it satisfies the following: G belongs to C if and only if all the blocks of decomposition belong to C. A decomposition based recognition algorithm takes an input graph G and decomposes it using C-preserving decompositions into a polynomial number of basic blocks, which are then checked, in polynomial time, whether they belong to \\n\\nC.Such a construction of blocks works nicely for clique cutsets. A node set S is a star cutset of a graph G if its removal disconnects G and S contains a node that is adjacent to all the other nodes of S. With the usual construction of blocks for the node cutsets, the star cutset decomposition is not preserving for the class of perfectly contractile graphs. A generalization of star cutsets is obtained as follows. 1-Amalgams are defined and used in for the construction of a recognition algorithm for Meyniel graphs. A graph G has a \\n\\n1-amalgam if its vertex set can be partitioned into sets V1, V2 and K (where K is possibly empty) in such a way that: \\n\\n• for i = 1, 2, |Vi| ≥ 2 and Vi contains a nonempty set Ai;\\n\\n• every node of A1 is adjacent to every node of A2 and these are the only adjacencies between the nodes of V1 and the nodes of V2; and \\n\\n• if K 6 = ∅, then it induces a clique, and every node of K is adjacent to every node of \\n\\nA1 ∪ A2.A graph G has a 2-join if its node set can be partitioned into sets V1 and V2 so that for i = 1, 2, Vi contains disjoint nonempty sets Ai and Bi, and the following properties hold: \\n\\n> 1http://www-leibniz.imag.fr/LesCahiers/2002/Cahier67/ResumCahier67.html\\n> 2http://www-leibniz.imag.fr/NEWLEIBNIZ/LesCahiers/Cahier106/ResumCahier106.html 12\\n\\n• every node of A1 (resp. B1) is adjacent to every node of A2 (resp. B2), and these are the only adjacencies between the nodes of V1 and the nodes of V2;\\n\\n• for i = 1, 2, let Pi be the set of all chordless paths in G[Vi] with one endnode in Ai,the other endnode in Bi, and no intermediate node in Ai ∪ Bi. For i = 1, 2, Pi 6 = ∅\\n\\nand G[Vi] is not isomorphic to a path in Pi.Let Cpc denote the class of perfectly contractile graphs. We have the following conjectures.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0272", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is malformed: it starts at E.8 Question 2, absorbs Chapter F through most of F.4.a, and ends immediately before the conjecture that starts the next record. For the delimited E.8 question, the historical word 'composition' is not formally defined. A precise necessary composition property is proved: if (A,B,C,D) is an even-pair skew split, every odd hole and odd antihole lies in one of the induced overlapping blocks G[A union B union C] or G[A union B union D]. Consequently G is Berge if and only if both blocks are Berge, and by the Strong Perfect Graph Theorem the same equivalence holds for perfectness. A fully checked six-vertex example shows the reduction fails when the even-pair condition is dropped.\n\nCandidate contribution (localization_lemma; novelty confidence low): For an even-pair skew split (A,B,C,D), odd holes and odd antiholes cannot meet both C and D, so Berge and perfect membership localize exactly to the two natural induced blocks; the even-pair hypothesis is sharp for this statement, as witnessed by the explicit six-vertex graph with edge set {xc,cy,yd,de,ex,bx,by,be}.", "research_difficulty_suggested": "L2" }, { "id": 20001148, "problem_number": "AIM-COMBINATORICS-0273", "title": "Perfect contractility and 1-amalgam blocks", "statement": "Conjecture 1-Amalgam decomposition is Cpc -preserving.", "original_statement": "Conjecture 1-Amalgam decomposition is Cpc -preserving.", "clean_statement": "Conjecture 1-Amalgam decomposition is Cpc -preserving.", "statement_status": "exact", "statement_verification": "The canonical record contains only:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[272]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 1-Amalgam decomposition is Cpc -preserving.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0273", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the standard marker-block convention, perfect contractility of a 1-amalgam implies perfect contractility of both blocks for arbitrary K. The converse is proved when K is empty and both attachments are singletons, and when both attachments fill their sides with arbitrary K. For K empty, induced-path parity between two same-side vertices agrees exactly with the corresponding marker block, and every marker-free block contraction sequence lifts to the original graph.\n\nCandidate contribution (special_case_and_reduction; novelty confidence low): For a 1-amalgam with K empty, every induced path between two vertices of one side either stays in that side or makes a two-edge excursion through a single opposite attachment vertex; replacing that vertex by the marker preserves inducedness and parity, and the correspondence survives every marker-free side contraction. This yields full preservation for singleton attachments, while a separate join argument yields full preservation for full attachments with arbitrary K.", "research_difficulty_suggested": "L4" }, { "id": 20001149, "problem_number": "AIM-COMBINATORICS-0274", "title": "What a 2-join block must remember for perfect contractility", "statement": "Conjecture 2-Join decomposition is Cpc -preserving.\n\nConjecture No minimal non perfectly contractile graph has a star cutset. Note that the PCGC implies all three of these conjectures. Contributed by Claudia Linhares Sales.\n\nF.5 Possible Structure Theorem for Berge Graphs\n\nConjecture For every even-pairfree Berge graph, either it or its complement is the line graph of a bipartite graph, or has a 2-join. A direct proof (if there is one) might give a shorter proof of the SPGC. Contributed by Robin Thomas A general even pair is not a compositiona non-Berge graph may become Berge by contracting an even pair. How much more do we need in order to be able to find a construction for Berge graphs using even pairs?\n\nQuestion Give a sufficient condition such that if x, y is an even pair satisfying the condition then G is Berge if and only if the graph obtained from G by contracting x, y is Berge. Contributed by Paul Seymour\n\nF.6 Odd holes and odd walks\n\nGiven a graph G = ( V, E ) and two vertices a, b can the following problem be solved in polynomial time?\n\nFind a triangle-free odd (a, b )-walk in G, where a walk can also contain repetitions of edges, and triangle-free means that the vertex-set of the walk does not contain any triangle (but can contain an odd hole).\n\nA polynomial algorithm for this problem would specialize to a polynomial algorithm for finding odd holes (and even pairs in odd-hole-free graphs). Bienstock proved that it is NP-hard to find odd holes containing a given a ∈ V. However, a triangle-free odd ( a, a )-walk exists in a 2-connected graph if and only if there exists an odd hole in G (not necessarily containing a). Contributed by Andr´ as Seb˝ o and Nicolas Trotignon 13\n\nChapter G: Forbidding Holes and Antiholes\n\nG.1 2-divisible Graphs\n\nA 2 division of a graph is a partition of its vertex set into two parts neither of which contains a maximum clique. Hoang and McDiarmid call a graph 2-divisible if all of its induced subgraphs permit 2-division. Every perfect graph is 2-divisible, and an odd hole has no 2-division. Thus 2-divisible graphs are odd-hole-free. Hoang and McDiarmid made the following conjectures: (1) G is 2-divisible iff G is odd-hole-free Contributed by Bruce Reed\n\nG.2 Clique Coloring of Perfect Graphs\n\nIt has been asked by Duffus et al [92e:06009] whether the clique hypergraph of perfect graphs is colorable with a constant number of colors. Several results followed showing that for some classes of perfect graphs this constant is actually 2 or 3: They proved it for comparability and cocomparability graphs, Bacs´ o et al proved the same for some more perfect graphs, and noticed that 'almost all' perfect graphs are 3-clique-colorable (applying Pr¨ omel and Steger's result (Probability and Computation, 1, 1992)); they ask whether this bound holds for all perfect graphs:\n\nCan the vertex-set of any perfect graph be partitioned into three classes, so that no (inclusionwise) maximal clique (of size > 1) is included in any of these? Is the same true already for odd hole free graphs?\n\nIf the clique-hypergraph of a graph and of all of its subgraphs can be colored with k\n\ncolors, that is, there exists a partition of the vertex-set into k parts so that none of them contains an (inclusionwise) maximal clique, then Ho` ang and McDiarmid [2002j:05110] say the graph is strongly k-divisible. This is indeed a sharpening of k-divisibility where 'maximal' is replaced by 'maximum' (cardinality). In these terms the above conjecture states that perfect graphs are strongly 3-divisible. We formulate another problem in this language:\n\nCan strong 2-divisibility be decided in polytime?\n\nA major difficulty with the coloration of the maximal clique hypergraph is that it is NP-hard to decide whether a partition of the vertices is a clique-coloration, and even in very particular classes of perfect graphs. Contributed by Myriam Preissmann and Andr´ as Seb˝ o\n\nG.3 Recognition of Odd-Hole-Free Graphs\n\nFind a polynomial algorithm to recognize odd-hole-free graphs. Contributed by Chinh Hoang\n\nG.4 Even-Hole-Free Graphs\n\nA k-division of a graph G is a partition of its vertex-set into sets V1,..., V k such that no Vi contains a largest clique of G. A graph is k-divisible if each of its induced subgraphs with at least one edge has a k-division. Conforti, Cornu´ ejols, Kapoor and Vuˇ skovi´ c designed a polynomial algorithm to recog-nize even-hole-free graphs. 14", "original_statement": "Conjecture 2-Join decomposition is Cpc -preserving. \n\nConjecture No minimal non perfectly contractile graph has a star cutset. Note that the PCGC implies all three of these conjectures. Contributed by Claudia Linhares Sales. \n\nF.5 Possible Structure Theorem for Berge Graphs \n\nConjecture For every even-pairfree Berge graph, either it or its complement is the line graph of a bipartite graph, or has a 2-join. A direct proof (if there is one) might give a shorter proof of the SPGC. Contributed by Robin Thomas A general even pair is not a compositiona non-Berge graph may become Berge by contracting an even pair. How much more do we need in order to be able to find a construction for Berge graphs using even pairs? \n\nQuestion Give a sufficient condition such that if x, y is an even pair satisfying the condition then G is Berge if and only if the graph obtained from G by contracting x, y is Berge. Contributed by Paul Seymour \n\nF.6 Odd holes and odd walks \n\nGiven a graph G = ( V, E ) and two vertices a, b can the following problem be solved in polynomial time? \n\nFind a triangle-free odd (a, b )-walk in G, where a walk can also contain repetitions of edges, and triangle-free means that the vertex-set of the walk does not contain any triangle (but can contain an odd hole). \n\nA polynomial algorithm for this problem would specialize to a polynomial algorithm for finding odd holes (and even pairs in odd-hole-free graphs). Bienstock proved that it is NP-hard to find odd holes containing a given a ∈ V. However, a triangle-free odd ( a, a )-walk exists in a 2-connected graph if and only if there exists an odd hole in G (not necessarily containing a). Contributed by Andr´ as Seb˝ o and Nicolas Trotignon 13 \n\nChapter G: Forbidding Holes and Antiholes \n\nG.1 2-divisible Graphs \n\nA 2 division of a graph is a partition of its vertex set into two parts neither of which contains a maximum clique. Hoang and McDiarmid call a graph 2-divisible if all of its induced subgraphs permit 2-division. Every perfect graph is 2-divisible, and an odd hole has no 2-division. Thus 2-divisible graphs are odd-hole-free. Hoang and McDiarmid made the following conjectures: (1) G is 2-divisible iff G is odd-hole-free Contributed by Bruce Reed \n\nG.2 Clique Coloring of Perfect Graphs \n\nIt has been asked by Duffus et al [92e:06009] whether the clique hypergraph of perfect graphs is colorable with a constant number of colors. Several results followed showing that for some classes of perfect graphs this constant is actually 2 or 3: They proved it for comparability and cocomparability graphs, Bacs´ o et al proved the same for some more perfect graphs, and noticed that 'almost all' perfect graphs are 3-clique-colorable (applying Pr¨ omel and Steger's result (Probability and Computation, 1, 1992)); they ask whether this bound holds for all perfect graphs: \n\nCan the vertex-set of any perfect graph be partitioned into three classes, so that no (inclusionwise) maximal clique (of size > 1) is included in any of these? Is the same true already for odd hole free graphs? \n\nIf the clique-hypergraph of a graph and of all of its subgraphs can be colored with k\n\ncolors, that is, there exists a partition of the vertex-set into k parts so that none of them contains an (inclusionwise) maximal clique, then Ho` ang and McDiarmid [2002j:05110] say the graph is strongly k-divisible. This is indeed a sharpening of k-divisibility where 'maximal' is replaced by 'maximum' (cardinality). In these terms the above conjecture states that perfect graphs are strongly 3-divisible. We formulate another problem in this language: \n\nCan strong 2-divisibility be decided in polytime? \n\nA major difficulty with the coloration of the maximal clique hypergraph is that it is NP-hard to decide whether a partition of the vertices is a clique-coloration, and even in very particular classes of perfect graphs. Contributed by Myriam Preissmann and Andr´ as Seb˝ o\n\nG.3 Recognition of Odd-Hole-Free Graphs \n\nFind a polynomial algorithm to recognize odd-hole-free graphs. Contributed by Chinh Hoang \n\nG.4 Even-Hole-Free Graphs \n\nA k-division of a graph G is a partition of its vertex-set into sets V1,..., V k such that no Vi contains a largest clique of G. A graph is k-divisible if each of its induced subgraphs with at least one edge has a k-division. Conforti, Cornu´ ejols, Kapoor and Vuˇ skovi´ c designed a polynomial algorithm to recog-nize even-hole-free graphs. 14", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record begins with", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[273]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 2-Join decomposition is Cpc -preserving. \\n\\nConjecture No minimal non perfectly contractile graph has a star cutset. Note that the PCGC implies all three of these conjectures. Contributed by Claudia Linhares Sales. \\n\\nF.5 Possible Structure Theorem for Berge Graphs \\n\\nConjecture For every even-pairfree Berge graph, either it or its complement is the line graph of a bipartite graph, or has a 2-join. A direct proof (if there is one) might give a shorter proof of the SPGC. Contributed by Robin Thomas A general even pair is not a compositiona non-Berge graph may become Berge by contracting an even pair. How much more do we need in order to be able to find a construction for Berge graphs using even pairs? \\n\\nQuestion Give a sufficient condition such that if x, y is an even pair satisfying the condition then G is Berge if and only if the graph obtained from G by contracting x, y is Berge. Contributed by Paul Seymour \\n\\nF.6 Odd holes and odd walks \\n\\nGiven a graph G = ( V, E ) and two vertices a, b can the following problem be solved in polynomial time? \\n\\nFind a triangle-free odd (a, b )-walk in G, where a walk can also contain repetitions of edges, and triangle-free means that the vertex-set of the walk does not contain any triangle (but can contain an odd hole). \\n\\nA polynomial algorithm for this problem would specialize to a polynomial algorithm for finding odd holes (and even pairs in odd-hole-free graphs). Bienstock proved that it is NP-hard to find odd holes containing a given a ∈ V. However, a triangle-free odd ( a, a )-walk exists in a 2-connected graph if and only if there exists an odd hole in G (not necessarily containing a). Contributed by Andr´ as Seb˝ o and Nicolas Trotignon 13 \\n\\nChapter G: Forbidding Holes and Antiholes \\n\\nG.1 2-divisible Graphs \\n\\nA 2 division of a graph is a partition of its vertex set into two parts neither of which contains a maximum clique. Hoang and McDiarmid call a graph 2-divisible if all of its induced subgraphs permit 2-division. Every perfect graph is 2-divisible, and an odd hole has no 2-division. Thus 2-divisible graphs are odd-hole-free. Hoang and McDiarmid made the following conjectures: (1) G is 2-divisible iff G is odd-hole-free Contributed by Bruce Reed \\n\\nG.2 Clique Coloring of Perfect Graphs \\n\\nIt has been asked by Duffus et al [92e:06009] whether the clique hypergraph of perfect graphs is colorable with a constant number of colors. Several results followed showing that for some classes of perfect graphs this constant is actually 2 or 3: They proved it for comparability and cocomparability graphs, Bacs´ o et al proved the same for some more perfect graphs, and noticed that 'almost all' perfect graphs are 3-clique-colorable (applying Pr¨ omel and Steger's result (Probability and Computation, 1, 1992)); they ask whether this bound holds for all perfect graphs: \\n\\nCan the vertex-set of any perfect graph be partitioned into three classes, so that no (inclusionwise) maximal clique (of size > 1) is included in any of these? Is the same true already for odd hole free graphs? \\n\\nIf the clique-hypergraph of a graph and of all of its subgraphs can be colored with k\\n\\ncolors, that is, there exists a partition of the vertex-set into k parts so that none of them contains an (inclusionwise) maximal clique, then Ho` ang and McDiarmid [2002j:05110] say the graph is strongly k-divisible. This is indeed a sharpening of k-divisibility where 'maximal' is replaced by 'maximum' (cardinality). In these terms the above conjecture states that perfect graphs are strongly 3-divisible. We formulate another problem in this language: \\n\\nCan strong 2-divisibility be decided in polytime? \\n\\nA major difficulty with the coloration of the maximal clique hypergraph is that it is NP-hard to decide whether a partition of the vertices is a clique-coloration, and even in very particular classes of perfect graphs. Contributed by Myriam Preissmann and Andr´ as Seb˝ o\\n\\nG.3 Recognition of Odd-Hole-Free Graphs \\n\\nFind a polynomial algorithm to recognize odd-hole-free graphs. Contributed by Chinh Hoang \\n\\nG.4 Even-Hole-Free Graphs \\n\\nA k-division of a graph G is a partition of its vertex-set into sets V1,..., V k such that no Vi contains a largest clique of G. A graph is k-divisible if each of its induced subgraphs with at least one edge has a k-division. Conforti, Cornu´ ejols, Kapoor and Vuˇ skovi´ c designed a polynomial algorithm to recog-nize even-hole-free graphs. 14\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0274", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is malformed and its exact opening conjecture is under-specified: the AIM source defines the non-path 2-join split but never defines the blocks whose C_pc-preservation is asserted. For a precise repair, an honest block obtained by retaining one side and one actual attachment path from the other is proved to be a proper induced subgraph, so G perfectly contractile implies every honest block perfectly contractile. Attachment paths are also proved to generate canonical crossing cycles whose parity forces parity-preserving markers in odd-hole-free graphs. Finally, a fully explicit seven-vertex non-path 2-join has PC side graphs K3 and C4 but is not PC because it contains an induced complement of C6 with no even pair; this refutes only the marker-free side-only reading, not the standard marker-path conjecture.\n\nCandidate contribution (obstruction_and_reduction_lemma; novelty confidence low): For the AIM 2-join definition, actual-path blocks are proper induced subgraphs and hence preserve perfect contractility in the forward direction; parity is the datum required to compress canonical crossing cycles, while omission of the marker fails on the explicit graph with vertices {x1,x2,x3,y1,y2,y3,z} and edges consisting of the two x/y triangles, the three matching rungs, and zx1,zy1.", "research_difficulty_suggested": "L2" }, { "id": 20001150, "problem_number": "AIM-COMBINATORICS-0275", "title": "Three-way divisibility and a list strengthening for even-hole-free graphs", "statement": "Conjecture 2 (Ho` ang). Every even-hole-free graph is 3-divisible.", "original_statement": "Conjecture 2 (Ho` ang). Every even-hole-free graph is 3-divisible.", "clean_statement": "Conjecture 2 (Ho` ang). Every even-hole-free graph is 3-divisible.", "statement_status": "exact", "statement_verification": "The canonical record is in Chapter G.4, “Even-Hole-Free Graphs,” of the AIM workshop list *The Perfect Graph Conjecture* (official PDF version dated 24 August 2004). The nearby source text gives the needed definition:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[274]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 2 (Ho` ang). Every even-hole-free graph is 3-divisible.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0275", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The conjecture is true. Chudnovsky and Seymour proved in 2023 that every nonempty even-hole-free graph has a bisimplicial vertex, and Chen, Lan, and Zhong explicitly confirmed Hoang's 3-divisibility conjecture in a May 2026 preprint. A direct proof is given here: every induced subgraph H has a vertex of degree at most 2(omega(H)-1), which yields a 3-division. More strongly, for arbitrary three-element color lists on V(H), the vertices can be colored from their lists so that no maximum clique of H is monochromatic. The cycle C5 shows that three colors are necessary.\n\nCandidate contribution (strengthening; novelty confidence low): Candidate novelty: if every induced subgraph J with an edge has a vertex v satisfying d_J(v) <= q(omega(J)-1), then the maximum-clique hypergraph of every such J is (q+1)-choosable; consequently, maximum-clique hypergraphs of even-hole-free graphs are hereditarily 3-choosable." }, { "id": 20001151, "problem_number": "AIM-COMBINATORICS-0276", "title": "List coloring and equality structure for even-hole-free graphs", "statement": "Conjecture 3 (Ho` ang). If G is an even-hole-free graph then χ(G) ≤ 2ω(G) − 1. Hayward and Reed proposed the following.", "original_statement": "Conjecture 3 (Ho` ang). If G is an even-hole-free graph then χ(G) ≤ 2ω(G) − 1. Hayward and Reed proposed the following.", "clean_statement": "Conjecture 3 (Ho` ang). If G is an even-hole-free graph then χ(G) ≤ 2ω(G) − 1. Hayward and Reed proposed the following.", "statement_status": "exact", "statement_verification": "The canonical record is in Chapter G.4, “Even-Hole-Free Graphs,” of the AIM workshop list *The Perfect Graph Conjecture* (official PDF version dated 24 August 2004). The exact record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[275]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 3 (Ho` ang). If G is an even-hole-free graph then χ(G) ≤ 2ω(G) − 1. Hayward and Reed proposed the following.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0276", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The conjecture is true. Chudnovsky and Seymour proved in 2023 that every non-null even-hole-free graph has a bisimplicial vertex. Applying this theorem in every induced suffix gives an ordering in which vertex v_i has at most 2 omega(G_i)-2 later neighbors, where G_i is the suffix beginning at v_i. Reverse greedy coloring proves the stronger adaptive list statement and in particular chi_l(G) <= 2 omega(G)-1, hence chi(G) <= 2 omega(G)-1. Odd holes show sharpness at clique number two.\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: if an even-hole-free graph G with omega(G)=w>=2 satisfies chi(G)=2w-1, then every induced-minimal subgraph H with chi(H)=chi(G) has omega(H)=w and contains a vertex v with degree 2w-2 whose neighborhood partitions into two (w-1)-cliques, while chi(H-v)=2w-2 and omega(H-v)=w." }, { "id": 20001152, "problem_number": "AIM-COMBINATORICS-0277", "title": "Bisimplicial vertices and protected elimination in even-hole-free graphs", "statement": "Conjecture 4 (Hayward and Reed). An even-hole-free graph contains a vertex whose neighbourhood can be partitioned into two cliques.", "original_statement": "Conjecture 4 (Hayward and Reed). An even-hole-free graph contains a vertex whose neighbourhood can be partitioned into two cliques.", "clean_statement": "Conjecture 4 (Hayward and Reed). An even-hole-free graph contains a vertex whose neighbourhood can be partitioned into two cliques.", "statement_status": "exact", "statement_verification": "The canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[276]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 4 (Hayward and Reed). An even-hole-free graph contains a vertex whose neighbourhood can be partitioned into two cliques.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0277", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Chudnovsky and Seymour proved in Journal of Combinatorial Theory, Series B 161 (2023), 331--381 that every non-null even-hole-free graph has a bisimplicial vertex, resolving the recovered Hayward--Reed conjecture; their stronger theorem locates one outside the closed neighborhood of any nondominating clique of size at most two. Iterating that rooted theorem yields a protected elimination ordering, a full acyclic orientation whose out-neighborhoods are unions of two cliques, and an exact edge identity outside the protected core.\n\nCandidate contribution (corollary_and_certificate; novelty confidence low): For every even-hole-free graph G and clique K of size at most two, all vertices M outside the closed neighborhood N[K] admit a bisimplicial elimination order while N[K] remains fixed. Appending an ordinary elimination of the core gives a full acyclic orientation with every out-neighborhood covered by two cliques, and the rooted prefix certifies that the number of edges not internal to N[K] equals the sum of the successive degrees of the eliminated vertices." }, { "id": 20001153, "problem_number": "AIM-COMBINATORICS-0278", "title": "The chromatic-to-divisibility implication hidden in a source fragment", "statement": "Conjecture 3 which in turn implies", "original_statement": "Conjecture 3 which in turn implies", "clean_statement": "Conjecture 3 which in turn implies", "statement_status": "exact", "statement_verification": "The exact canonical record is the five-word fragment", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[277]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 3 which in turn implies\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0278", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The five-word record is not an independent conjecture: the official source sentence is ‘Conjecture 4 implies Conjecture 3 which in turn implies Conjecture 2.’ Its encoded implication is formalized and proved with exact rounding. For every graph F with an edge, the least number d(F) of parts needed to avoid a maximum clique satisfies d(F) <= ceil(chi(F)/(omega(F)-1)); equality holds for every triangle-free F. More generally, a hereditary chi-bound f transfers to the uniform divisibility constant sup over attained integers w>=2 of ceil(f(w)/(w-1)). Thus chi(H)<=2omega(H)-1 for every induced H implies 3-divisibility, and odd cycles show the constant three is sharp among even-hole-free graphs. Chudnovsky and Seymour's 2023 bisimplicial-vertex theorem makes the historical even-hole-free implication chain unconditional.\n\nCandidate contribution (hereditary_transfer_lemma; novelty confidence low): For a hereditary class with chi(H)<=f(omega(H)) on every induced H with an edge, the uniform maximum-clique divisibility constant is at most sup over attained integers w>=2 of ceil(f(w)/(w-1)); the underlying one-graph bound d(F)<=ceil(chi(F)/(omega(F)-1)) is exact for all triangle-free graphs, and odd cycles make the value three sharp for the even-hole-free specialization f(w)=2w-1.", "research_difficulty_suggested": "L2" }, { "id": 20001154, "problem_number": "AIM-COMBINATORICS-0279", "title": "An extraction fragment and an exact triangle-free list-coloring corollary", "statement": "Conjecture 2. Let G be an even-hole-free graph.", "original_statement": "Conjecture 2. Let G be an even-hole-free graph.", "clean_statement": "Conjecture 2. Let G be an even-hole-free graph.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[278]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 2. Let G be an even-hole-free graph.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0279", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical text is not an independent Conjecture 2: the original PDF shows that it splices the final words of the sentence saying Conjecture 4 implies Conjecture 3, which implies Conjecture 2, to the opening sentence of the ensuing proof. As a substantive result motivated by that proof, every finite triangle-free even-hole-free graph is 2-degenerate and has χ equal to its list-chromatic number: 0 for the null graph, 1 for a non-null edgeless graph, 2 for a forest with an edge, and 3 for a cyclic graph.\n\nCandidate contribution (corollary; novelty confidence low): Candidate exact list-coloring corollary: every finite triangle-free even-hole-free graph has ordinary and list chromatic numbers equal, with the exact null/edgeless/forest/cyclic values 0, 1, 2, and 3 respectively; equivalently, the bipartite members are precisely the forests.", "research_difficulty_suggested": "L2" }, { "id": 20001155, "problem_number": "AIM-COMBINATORICS-0280", "title": "Exact structure and coloring of the proposed even-hole-free circulants", "statement": "Conjecture 4 implies that each induced subgraph H of G has a vertex of degree at most 2 ω(H) − 2, and therefore χ(G) ≤ 2ω(G) − 1. Since any graph F is\n\n> χ(F)\n> ω(F)−1\n\n-divisible, G is 3-divisible. Contributed by Chinh Hoang.\n\nG.5 Even-hole-free circulants,\n\nGiven interer k ≥ 1 and m ≥ 0, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {1,..., n }, where n = k(2 m + 1), and ( i, j ) ∈ E iff\n\ni − j + t (2 m + 1) = 0 or + 1 or − 1 ( mod n )for some integer t. (For convenience, the loops i = j are included.) E.g. if k = 5, m = 2 then n = 25 and ( i, j ) ∈ E iff i − j(mod 25) ∈ { 4, 5, 6; 9, 10, 11; 14, 15, 16; 19, 20, 21; 24, 0, 1}.\n\nIt is not difficult to check that G(k, m ) has no even holes, (in fact, it can only have holes of length 2 m + 1); furthermore,\n\nω(G(k, m )) = 2 k, 2k + dk/m e ≤ χ(G(k, m )) ≤ 2k + dk/m e + 1, and G(k, m ) satisfies Conjectures 2,3,4 from the section \"Even-Hole-Free Graphs\".\n\nConjecture. Every non-empty even-hole-free circulant is isomorphic to a G(k, m ). Contributed by Diogo Andrade, Endre Boros, and Vladimir Gurvich\n\nG.6 beta-perfect graphs\n\nDefinition β(G) = max G′⊆G(mindeg (G′) + 1) (the maximum is taken over all induced subgraphs). Note that β(G) ≥ χ(G). A graph G is called β-perfect if β(G′) = χ(G′) for all induced subgraphs G′ of G.\n\nQuestion Characterize β-perfect graphs. Even holes and graphs obtained from odd holes by replacing every vertex by two adjacent vertices preserving the adjacencies in the hole (so every edge is replaced by a K4)are known not to be β-perfect. So a β-perfect graph has no induced subgraph of those types. Contributed by Bruce Reed\n\nChapter H: Partitionable Graphs\n\nH.1 Perfect, Partitionable, and Kernel-Solvable Graphs\n\nGiven a graph G = (V,E), assign to every its edge e = (u,v) either the directed arc [u,v), or [v,u), or both. The obtained directed multi-graph D = (V,A) is called an orientation of G. 15\n\nA vertex-subset K of V is called a KERNEL if K is (i) independent and (ii) absorbant, that is for each u from V K there is an arc [u,v) in A such that v in K. Orientation D is called clique acyclic if every clique of G has a kernel in D. Orientation\n\nD is called kernel-less if it has no kernel. Graph G is called kernel-solvable if every its clique-acyclic orientation has a kernel. Berge and Duchet (1983) conjectured that (BD1) Perfect graphs are kernel-solvable, and (BD2) Kernel-solvable graphs are perfect. BD1 was proved by Boros and Gurvich (1996) and by Holzman and Aharoni (1998), BD2 follows from the SPGC but no independent proof is known. An orientation D of a PARTITIONABLE graph G is called UNIFORM if D is (0) kernel-less and clique acyclic; (a) for each maximum stable set S there exists a unique unabsorbed vertex v(S); (b) v(S) belongs to the vis-a-vis clique C(S) of S; (c) for each vertex v there exists a unique maximal stable set S(v) which does not absorb v. Sebo (1998) proved that every kernel-less and clique-acyclic orientation of a minimal imperfect graph is uniform. Conjecture. Each partitionable graph has a uniform orientation. This, if true, implies BD2 Contributed by Boros and Gurvich\n\nH.2 Partitionable graphs and odd holes\n\nLet G = ( V, E ) be a graph, and α, ω arbitrary natural numbers. Assume that a, v, b ∈\n\nV, av ∈ E, vb / ∈ E are such that G − a, G − v have a partition of size α into ω-cliques, and\n\nG − v, G − b have a partition of size ω into α-stable sets. It is easy to show then that G is not perfect.\n\nGiven the four partitions, find an odd hole or an odd antihole.\n\nThis would imply SPGC. This contains the following:\n\nGiven a partitionable graph (with all the partitions), find an odd hole or an odd antihole. Does the fact that the partitions are given make the task easier?\n\nContributed by Andr´ as Seb˝ o\n\nH.3 A Property of Partitionable Graphs\n\nWe say that a graph satisfies the \"no-week-pair\" property if each pair of vertices of a graph is either in a maximum clique or in a maximum stable set Conjecture: If a partitionable graph satisfies the \"no-week-pair\" property then the graph is an odd hole or an odd anti-hole. The following graph is a counter-example to this conjecture: take a 17-gon and and add all 34- and 5-chords. (This 17-vertex graph is the only known partitionable graph without a small transver-sal.) It's still interesting if there are other such graphs (i.e. partitionable graph satisfying \"no-week-pair\" property). If there are then it would be nice to characterize them. 16\n\nContributed by Ara Markosian\n\nH.4 Small Transversals in Partitionable Graphs\n\nFollowing Bland, Huang, and Trotter [80g:05034];[86e:05075] a graph is called parti-tionable if, for some r and s, it has rs + 1 vertices and, no matter which vertex is removed, the set of the remaining rs vertices can be partitioned into r pairwise disjoint cliques of size\n\ns and also into s pairwise disjoint stable sets of size r. Odd holes and odd antiholes are partitionable; many additional partitionable graphs have been constructed by V. Chvtal, R. L. Graham, A. F. Perold, and S. H. Whitesides [81b:05044]. A small transversal in a graph G is a set of α(G) + ω(G) − 1 vertices which meets all cliques of size ω(G) and all stable sets of size α(G). The following problem is an easier variation on a conjecture contributed to the 1993 workshop on perfect graphs 3 by Gurvich and Temkin and on two conjectures proposed by Bacso, Boros, Gurvich, Maffray, and Preissmann [2000h:05116].\n\nConjecture. Every partitionable graph G with α(G) > 2 and ω(G) > 2 has a small transversal or else contains a hole of length five. One of the milestones in the development of our understanding of perfect graphs was the theorem of Lovasz [46 #8885], asserting that every minimal imperfect graph G has precisely α(G)ω(G) +1 vertices. This theorem implies that every minimal imperfect graph is partitionable and that - as pointed out by Chvatal [86h:05091] - no minimal imperfect graph contains a small transversal. It follows that a proof of the conjecture would provide another proof of the Strong Perfect Graph Theorem. A partitionable graph without a small transversal has been constructed by by Chvatal, Graham, Perold, and Whitesides (op.cit.). Its vertices are 0, 1,..., 16; vertices i and j are adjacent if and only if |i − j| mod 17 is one of 1, 3, 4, 5, 12, 13, 14, 16. Ara Markosian claims here 4 that this is the only known partitionable graph without a small transversal. One of the many holes of length five in this graph is 1 − 4 − 8 − 12 − 15 − 1Additional information on related results and problems can be found here 5\n\nContributed by Vasek Chvatal\n\nChapter I: The Imperfection Ratio\n\nA demand vector for a graph G with node set V is a non-negative vector of integers indexed by nodes of G. Given a graph G and a demand vector x = ( xv: v ∈ V (G)) a coloring of the pair ( G.x ) is an assignment of a set of xv colors to each node v of G such that two adjacet nodes receive disjoint sets of colors. Coloring the pair ( G, x ) corresponds exactly to usual proper coloring of the replicated graph Gx. Let G be a graph.Define the\n\nimperfection ratio of G by setting\n\nimp (G) = max x{χf (Gx)\n\nω(Gx) }\n\n> 3http://dimacs.rutgers.edu/\"\n> 4http://www.aimath.org/WWN/perfectgraph/articles/html/46a/\n> 5http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#partitionable 17\n\nwhere the maximum is over all non-zero integral demand vectors x and χF (Gx) and\n\nω(Gx) are the fractional chromatic number and the clique number of the replicated graph Gx\n\nrespectively. (The ratios on the right-hand side above do indeed attain a maximum value). Observe that imp (G) ≥ 1. The next result establishes the connection of the imperfection ratio with perfection.\n\nProposition For any graph G, imp (G) = 1 iff G is perfect. One of the motivations for studying graph imperfection is its connection to frequency assignment. With this motivation in mind one is particularly interested in bounding the imperfection ratio for graphs of relevant graph classes. One relevant graph class are for example unit disk graph, that is graphs the node set of which can be represented by unit size disks such that two nodes are adjacent if and only if the corresponding disks intersect. It is known that imp (G) ≤ 2.155 for any unit disk graph G, and that there exists a unit disk graph G with imp (G) arbitrarily close to 3/2.\n\nConjecture: For any unit disk graph G, imp (G) ≤ 3/2. A subclass of unit disk graphs are the induced subgraphs of the triangular lattice. These graphs are of importance for channel assignment, since a pattern of omni-directional transmitters in two dimensions laid out like nodes of the triangular lattice in the plane give good coverage. Let us now consider an induced subgraph G of the triangular lattice T. Such a graph has a natural 3-coloring. It is possible to have ω(Gx) = 3 and χ(Gx)=4 for such a graph G.There is a polynomial-time coloring algorithm (by McDiarmid and Reed) which shows that for such a graph G we always have\n\nχ(Gx) ≤ 4ω(Gx) + 1\n\n3Thus imp (G) ≤ 4\n\n> 3\n\nfor any finite induced subgraph G of the triangular lattice T. The 9-cycle C9 is an induced subgraph of the triangular lattice T. For any integer k the graph obtained from C9 by replicating each of its nodes k times has clique number 2 k and chromatic number d 9k\n\n> 4\n\ne Is the ratio 9\n\n> 8\n\nof chromatic number to clique number asymptotically the worst (greatest) possible with large demands? This questions may be rephrased in terms of imp (G)as follows:\n\nConjecture For any induced subgraph G of the triangular lattice T, we have imp (G) ≤\n\n> 9\n> 8\n\nThis would imply the following weaker and perhaps more tractable conjecture:\n\nConjecture If G is a trianlgle-free induced subgraph of the triangular lattice then\n\n|V (G)| ≤ 9\n\n> 4\n\nα(G) (where α(G) is the size of the maximum stable set in G), and indeed\n\nχf (G) ≤ 9\n\n> 4\n\nTo get a feeling for the behavior of the imperfection ratio, we mention the following elementary decomposition result: if G is composed of two parts G1 and G2 that are either disjoint or overlap in a clique, then\n\nimp (G) = max {imp (G1), imp (G2)}\n\nThe following is another property of imperfection which is desirable for any graph invariant related to perfection: 18\n\nProposition For any graph G imp (G) = imp (Gc) where Gc denotes the completemt of G.Another graph class of interest are planar graphs. It follows from the 4-colour theorem that imp (G) ≤ 2 for any planar graph G. It is known that one can improve a little on this but we conjecture that the true value is 3 /2.\n\nConjecture: For any planar graph G, imp (G) ≤ 3/2. Another area of interest are complexity issues concerning imperfection. It is known that it is NP hard to determine the imperfection ratio. One open question is whether for a fixed k one can determine in polynomial time whether a graph G satisfies imp (G) ≤ k. For the special case of k = 1, this is the recognition problem for perfect graphs. Another open question is how hard is it to approximate the imperfection ratio of a graph. Initial contribution by Bruce Reed, extended by Stefanie Gerke\n\nChapter J: Integer Programming\n\nJ.1 Partitionable Graphs as Cutting Planes for Packing Problems?\n\nAs is well known, the strong perfect graph conjecture has been of interest to the integer programming community as well as the combinatorics community. Now that the SPGC has been established I wanted to mention another problem which may shed light on cutting plane approaches to packing problems. Sewell (and later Bram Verweij and Aardel) gave successful cutting plane codes for solving maximum stable set problems in sparse graphs by adding odd hole inequalities as they are violated by fractional solutions. Moura studied this approach for some problems arising in design theory, but met with much less success since the graph instances were much more dense (and hence odd hole inequalities were unlikely to be violated). Can we extend the class of odd cycle inequalities to the class of partitionable graph inequalities\n\n∑\n\n> v∈I\n\nxv ≤ α(I)for each partitionable subgraph. I.e., can we develop algorithms to solve the separa-tion problem for this class of inequalities. One positive result is that partitionable graphs themselves can be recognized in polynomial time (another problem is to find a combinatorial algorithm to recognize partitionable graphs). Contributed by Bruce Shepherd.\n\nJ.2 Feasibility/Membership Problem For the Theta Body\n\nFind a polynomial time algorithm to solve the (exact) feasibility/membership problem for the theta body. Contributed by Bruce Shepherd\n\nChapter K: Balanced Graphs\n\nDefinition A graph is balanced if every induced cycle has length 0( mod 4). Clearly balanced graphs are bipartite. 19\n\nA balanced graph is basic if all its vertices on one side of the bipartition have degree at most 2 or G contains a hole H such that the vertices of G \\ H induce a complete bipartite graph. Here are two conjectures concerning balanced graphs.", "original_statement": "Conjecture 4 implies that each induced subgraph H of G has a vertex of degree at most 2 ω(H) − 2, and therefore χ(G) ≤ 2ω(G) − 1. Since any graph F is \n\n> χ(F)\n> ω(F)−1\n\n-divisible, G is 3-divisible. Contributed by Chinh Hoang. \n\nG.5 Even-hole-free circulants, \n\nGiven interer k ≥ 1 and m ≥ 0, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {1,..., n }, where n = k(2 m + 1), and ( i, j ) ∈ E iff \n\ni − j + t (2 m + 1) = 0 or + 1 or − 1 ( mod n )for some integer t. (For convenience, the loops i = j are included.) E.g. if k = 5, m = 2 then n = 25 and ( i, j ) ∈ E iff i − j(mod 25) ∈ { 4, 5, 6; 9, 10, 11; 14, 15, 16; 19, 20, 21; 24, 0, 1}.\n\nIt is not difficult to check that G(k, m ) has no even holes, (in fact, it can only have holes of length 2 m + 1); furthermore, \n\nω(G(k, m )) = 2 k, 2k + dk/m e ≤ χ(G(k, m )) ≤ 2k + dk/m e + 1, and G(k, m ) satisfies Conjectures 2,3,4 from the section \"Even-Hole-Free Graphs\". \n\nConjecture. Every non-empty even-hole-free circulant is isomorphic to a G(k, m ). Contributed by Diogo Andrade, Endre Boros, and Vladimir Gurvich \n\nG.6 beta-perfect graphs \n\nDefinition β(G) = max G′⊆G(mindeg (G′) + 1) (the maximum is taken over all induced subgraphs). Note that β(G) ≥ χ(G). A graph G is called β-perfect if β(G′) = χ(G′) for all induced subgraphs G′ of G.\n\nQuestion Characterize β-perfect graphs. Even holes and graphs obtained from odd holes by replacing every vertex by two adjacent vertices preserving the adjacencies in the hole (so every edge is replaced by a K4)are known not to be β-perfect. So a β-perfect graph has no induced subgraph of those types. Contributed by Bruce Reed \n\nChapter H: Partitionable Graphs \n\nH.1 Perfect, Partitionable, and Kernel-Solvable Graphs \n\nGiven a graph G = (V,E), assign to every its edge e = (u,v) either the directed arc [u,v), or [v,u), or both. The obtained directed multi-graph D = (V,A) is called an orientation of G. 15 \n\nA vertex-subset K of V is called a KERNEL if K is (i) independent and (ii) absorbant, that is for each u from V K there is an arc [u,v) in A such that v in K. Orientation D is called clique acyclic if every clique of G has a kernel in D. Orientation \n\nD is called kernel-less if it has no kernel. Graph G is called kernel-solvable if every its clique-acyclic orientation has a kernel. Berge and Duchet (1983) conjectured that (BD1) Perfect graphs are kernel-solvable, and (BD2) Kernel-solvable graphs are perfect. BD1 was proved by Boros and Gurvich (1996) and by Holzman and Aharoni (1998), BD2 follows from the SPGC but no independent proof is known. An orientation D of a PARTITIONABLE graph G is called UNIFORM if D is (0) kernel-less and clique acyclic; (a) for each maximum stable set S there exists a unique unabsorbed vertex v(S); (b) v(S) belongs to the vis-a-vis clique C(S) of S; (c) for each vertex v there exists a unique maximal stable set S(v) which does not absorb v. Sebo (1998) proved that every kernel-less and clique-acyclic orientation of a minimal imperfect graph is uniform. Conjecture. Each partitionable graph has a uniform orientation. This, if true, implies BD2 Contributed by Boros and Gurvich \n\nH.2 Partitionable graphs and odd holes \n\nLet G = ( V, E ) be a graph, and α, ω arbitrary natural numbers. Assume that a, v, b ∈\n\nV, av ∈ E, vb / ∈ E are such that G − a, G − v have a partition of size α into ω-cliques, and \n\nG − v, G − b have a partition of size ω into α-stable sets. It is easy to show then that G is not perfect. \n\nGiven the four partitions, find an odd hole or an odd antihole. \n\nThis would imply SPGC. This contains the following: \n\nGiven a partitionable graph (with all the partitions), find an odd hole or an odd antihole. Does the fact that the partitions are given make the task easier? \n\nContributed by Andr´ as Seb˝ o\n\nH.3 A Property of Partitionable Graphs \n\nWe say that a graph satisfies the \"no-week-pair\" property if each pair of vertices of a graph is either in a maximum clique or in a maximum stable set Conjecture: If a partitionable graph satisfies the \"no-week-pair\" property then the graph is an odd hole or an odd anti-hole. The following graph is a counter-example to this conjecture: take a 17-gon and and add all 34- and 5-chords. (This 17-vertex graph is the only known partitionable graph without a small transver-sal.) It's still interesting if there are other such graphs (i.e. partitionable graph satisfying \"no-week-pair\" property). If there are then it would be nice to characterize them. 16 \n\nContributed by Ara Markosian \n\nH.4 Small Transversals in Partitionable Graphs \n\nFollowing Bland, Huang, and Trotter [80g:05034];[86e:05075] a graph is called parti-tionable if, for some r and s, it has rs + 1 vertices and, no matter which vertex is removed, the set of the remaining rs vertices can be partitioned into r pairwise disjoint cliques of size \n\ns and also into s pairwise disjoint stable sets of size r. Odd holes and odd antiholes are partitionable; many additional partitionable graphs have been constructed by V. Chvtal, R. L. Graham, A. F. Perold, and S. H. Whitesides [81b:05044]. A small transversal in a graph G is a set of α(G) + ω(G) − 1 vertices which meets all cliques of size ω(G) and all stable sets of size α(G). The following problem is an easier variation on a conjecture contributed to the 1993 workshop on perfect graphs 3 by Gurvich and Temkin and on two conjectures proposed by Bacso, Boros, Gurvich, Maffray, and Preissmann [2000h:05116]. \n\nConjecture. Every partitionable graph G with α(G) > 2 and ω(G) > 2 has a small transversal or else contains a hole of length five. One of the milestones in the development of our understanding of perfect graphs was the theorem of Lovasz [46 #8885], asserting that every minimal imperfect graph G has precisely α(G)ω(G) +1 vertices. This theorem implies that every minimal imperfect graph is partitionable and that - as pointed out by Chvatal [86h:05091] - no minimal imperfect graph contains a small transversal. It follows that a proof of the conjecture would provide another proof of the Strong Perfect Graph Theorem. A partitionable graph without a small transversal has been constructed by by Chvatal, Graham, Perold, and Whitesides (op.cit.). Its vertices are 0, 1,..., 16; vertices i and j are adjacent if and only if |i − j| mod 17 is one of 1, 3, 4, 5, 12, 13, 14, 16. Ara Markosian claims here 4 that this is the only known partitionable graph without a small transversal. One of the many holes of length five in this graph is 1 − 4 − 8 − 12 − 15 − 1Additional information on related results and problems can be found here 5\n\nContributed by Vasek Chvatal \n\nChapter I: The Imperfection Ratio \n\nA demand vector for a graph G with node set V is a non-negative vector of integers indexed by nodes of G. Given a graph G and a demand vector x = ( xv: v ∈ V (G)) a coloring of the pair ( G.x ) is an assignment of a set of xv colors to each node v of G such that two adjacet nodes receive disjoint sets of colors. Coloring the pair ( G, x ) corresponds exactly to usual proper coloring of the replicated graph Gx. Let G be a graph.Define the \n\nimperfection ratio of G by setting \n\nimp (G) = max x{χf (Gx)\n\nω(Gx) }\n\n> 3http://dimacs.rutgers.edu/\"\n> 4http://www.aimath.org/WWN/perfectgraph/articles/html/46a/\n> 5http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#partitionable 17\n\nwhere the maximum is over all non-zero integral demand vectors x and χF (Gx) and \n\nω(Gx) are the fractional chromatic number and the clique number of the replicated graph Gx\n\nrespectively. (The ratios on the right-hand side above do indeed attain a maximum value). Observe that imp (G) ≥ 1. The next result establishes the connection of the imperfection ratio with perfection. \n\nProposition For any graph G, imp (G) = 1 iff G is perfect. One of the motivations for studying graph imperfection is its connection to frequency assignment. With this motivation in mind one is particularly interested in bounding the imperfection ratio for graphs of relevant graph classes. One relevant graph class are for example unit disk graph, that is graphs the node set of which can be represented by unit size disks such that two nodes are adjacent if and only if the corresponding disks intersect. It is known that imp (G) ≤ 2.155 for any unit disk graph G, and that there exists a unit disk graph G with imp (G) arbitrarily close to 3/2. \n\nConjecture: For any unit disk graph G, imp (G) ≤ 3/2. A subclass of unit disk graphs are the induced subgraphs of the triangular lattice. These graphs are of importance for channel assignment, since a pattern of omni-directional transmitters in two dimensions laid out like nodes of the triangular lattice in the plane give good coverage. Let us now consider an induced subgraph G of the triangular lattice T. Such a graph has a natural 3-coloring. It is possible to have ω(Gx) = 3 and χ(Gx)=4 for such a graph G.There is a polynomial-time coloring algorithm (by McDiarmid and Reed) which shows that for such a graph G we always have \n\nχ(Gx) ≤ 4ω(Gx) + 1 \n\n3Thus imp (G) ≤ 4 \n\n> 3\n\nfor any finite induced subgraph G of the triangular lattice T. The 9-cycle C9 is an induced subgraph of the triangular lattice T. For any integer k the graph obtained from C9 by replicating each of its nodes k times has clique number 2 k and chromatic number d 9k \n\n> 4\n\ne Is the ratio 9 \n\n> 8\n\nof chromatic number to clique number asymptotically the worst (greatest) possible with large demands? This questions may be rephrased in terms of imp (G)as follows: \n\nConjecture For any induced subgraph G of the triangular lattice T, we have imp (G) ≤\n\n> 9\n> 8\n\nThis would imply the following weaker and perhaps more tractable conjecture: \n\nConjecture If G is a trianlgle-free induced subgraph of the triangular lattice then \n\n|V (G)| ≤ 9 \n\n> 4\n\nα(G) (where α(G) is the size of the maximum stable set in G), and indeed \n\nχf (G) ≤ 9\n\n> 4\n\nTo get a feeling for the behavior of the imperfection ratio, we mention the following elementary decomposition result: if G is composed of two parts G1 and G2 that are either disjoint or overlap in a clique, then \n\nimp (G) = max {imp (G1), imp (G2)}\n\nThe following is another property of imperfection which is desirable for any graph invariant related to perfection: 18 \n\nProposition For any graph G imp (G) = imp (Gc) where Gc denotes the completemt of G.Another graph class of interest are planar graphs. It follows from the 4-colour theorem that imp (G) ≤ 2 for any planar graph G. It is known that one can improve a little on this but we conjecture that the true value is 3 /2.\n\nConjecture: For any planar graph G, imp (G) ≤ 3/2. Another area of interest are complexity issues concerning imperfection. It is known that it is NP hard to determine the imperfection ratio. One open question is whether for a fixed k one can determine in polynomial time whether a graph G satisfies imp (G) ≤ k. For the special case of k = 1, this is the recognition problem for perfect graphs. Another open question is how hard is it to approximate the imperfection ratio of a graph. Initial contribution by Bruce Reed, extended by Stefanie Gerke \n\nChapter J: Integer Programming \n\nJ.1 Partitionable Graphs as Cutting Planes for Packing Problems? \n\nAs is well known, the strong perfect graph conjecture has been of interest to the integer programming community as well as the combinatorics community. Now that the SPGC has been established I wanted to mention another problem which may shed light on cutting plane approaches to packing problems. Sewell (and later Bram Verweij and Aardel) gave successful cutting plane codes for solving maximum stable set problems in sparse graphs by adding odd hole inequalities as they are violated by fractional solutions. Moura studied this approach for some problems arising in design theory, but met with much less success since the graph instances were much more dense (and hence odd hole inequalities were unlikely to be violated). Can we extend the class of odd cycle inequalities to the class of partitionable graph inequalities \n\n∑\n\n> v∈I\n\nxv ≤ α(I)for each partitionable subgraph. I.e., can we develop algorithms to solve the separa-tion problem for this class of inequalities. One positive result is that partitionable graphs themselves can be recognized in polynomial time (another problem is to find a combinatorial algorithm to recognize partitionable graphs). Contributed by Bruce Shepherd. \n\nJ.2 Feasibility/Membership Problem For the Theta Body \n\nFind a polynomial time algorithm to solve the (exact) feasibility/membership problem for the theta body. Contributed by Bruce Shepherd \n\nChapter K: Balanced Graphs \n\nDefinition A graph is balanced if every induced cycle has length 0( mod 4). Clearly balanced graphs are bipartite. 19 \n\nA balanced graph is basic if all its vertices on one side of the bipartition have degree at most 2 or G contains a hole H such that the vertices of G \\ H induce a complete bipartite graph. Here are two conjectures concerning balanced graphs.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical **problem** field is not one mathematical record. It begins in the middle of the proof of the implications among Conjectures 2, 3, and 4 in Section G.4, contains all of Section G.5, and then absorbs G.6, Chapters H, I, and J, and the beginning of Chapter K. The exact 13,034-character field is preserved in **input.json**. Thus the canonical record, as a record, has status `invalid_statement`: it has no single set of hypotheses or conclusion.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[279]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 4 implies that each induced subgraph H of G has a vertex of degree at most 2 ω(H) − 2, and therefore χ(G) ≤ 2ω(G) − 1. Since any graph F is \\n\\n> χ(F)\\n> ω(F)−1\\n\\n-divisible, G is 3-divisible. Contributed by Chinh Hoang. \\n\\nG.5 Even-hole-free circulants, \\n\\nGiven interer k ≥ 1 and m ≥ 0, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {1,..., n }, where n = k(2 m + 1), and ( i, j ) ∈ E iff \\n\\ni − j + t (2 m + 1) = 0 or + 1 or − 1 ( mod n )for some integer t. (For convenience, the loops i = j are included.) E.g. if k = 5, m = 2 then n = 25 and ( i, j ) ∈ E iff i − j(mod 25) ∈ { 4, 5, 6; 9, 10, 11; 14, 15, 16; 19, 20, 21; 24, 0, 1}.\\n\\nIt is not difficult to check that G(k, m ) has no even holes, (in fact, it can only have holes of length 2 m + 1); furthermore, \\n\\nω(G(k, m )) = 2 k, 2k + dk/m e ≤ χ(G(k, m )) ≤ 2k + dk/m e + 1, and G(k, m ) satisfies Conjectures 2,3,4 from the section \\\"Even-Hole-Free Graphs\\\". \\n\\nConjecture. Every non-empty even-hole-free circulant is isomorphic to a G(k, m ). Contributed by Diogo Andrade, Endre Boros, and Vladimir Gurvich \\n\\nG.6 beta-perfect graphs \\n\\nDefinition β(G) = max G′⊆G(mindeg (G′) + 1) (the maximum is taken over all induced subgraphs). Note that β(G) ≥ χ(G). A graph G is called β-perfect if β(G′) = χ(G′) for all induced subgraphs G′ of G.\\n\\nQuestion Characterize β-perfect graphs. Even holes and graphs obtained from odd holes by replacing every vertex by two adjacent vertices preserving the adjacencies in the hole (so every edge is replaced by a K4)are known not to be β-perfect. So a β-perfect graph has no induced subgraph of those types. Contributed by Bruce Reed \\n\\nChapter H: Partitionable Graphs \\n\\nH.1 Perfect, Partitionable, and Kernel-Solvable Graphs \\n\\nGiven a graph G = (V,E), assign to every its edge e = (u,v) either the directed arc [u,v), or [v,u), or both. The obtained directed multi-graph D = (V,A) is called an orientation of G. 15 \\n\\nA vertex-subset K of V is called a KERNEL if K is (i) independent and (ii) absorbant, that is for each u from V K there is an arc [u,v) in A such that v in K. Orientation D is called clique acyclic if every clique of G has a kernel in D. Orientation \\n\\nD is called kernel-less if it has no kernel. Graph G is called kernel-solvable if every its clique-acyclic orientation has a kernel. Berge and Duchet (1983) conjectured that (BD1) Perfect graphs are kernel-solvable, and (BD2) Kernel-solvable graphs are perfect. BD1 was proved by Boros and Gurvich (1996) and by Holzman and Aharoni (1998), BD2 follows from the SPGC but no independent proof is known. An orientation D of a PARTITIONABLE graph G is called UNIFORM if D is (0) kernel-less and clique acyclic; (a) for each maximum stable set S there exists a unique unabsorbed vertex v(S); (b) v(S) belongs to the vis-a-vis clique C(S) of S; (c) for each vertex v there exists a unique maximal stable set S(v) which does not absorb v. Sebo (1998) proved that every kernel-less and clique-acyclic orientation of a minimal imperfect graph is uniform. Conjecture. Each partitionable graph has a uniform orientation. This, if true, implies BD2 Contributed by Boros and Gurvich \\n\\nH.2 Partitionable graphs and odd holes \\n\\nLet G = ( V, E ) be a graph, and α, ω arbitrary natural numbers. Assume that a, v, b ∈\\n\\nV, av ∈ E, vb / ∈ E are such that G − a, G − v have a partition of size α into ω-cliques, and \\n\\nG − v, G − b have a partition of size ω into α-stable sets. It is easy to show then that G is not perfect. \\n\\nGiven the four partitions, find an odd hole or an odd antihole. \\n\\nThis would imply SPGC. This contains the following: \\n\\nGiven a partitionable graph (with all the partitions), find an odd hole or an odd antihole. Does the fact that the partitions are given make the task easier? \\n\\nContributed by Andr´ as Seb˝ o\\n\\nH.3 A Property of Partitionable Graphs \\n\\nWe say that a graph satisfies the \\\"no-week-pair\\\" property if each pair of vertices of a graph is either in a maximum clique or in a maximum stable set Conjecture: If a partitionable graph satisfies the \\\"no-week-pair\\\" property then the graph is an odd hole or an odd anti-hole. The following graph is a counter-example to this conjecture: take a 17-gon and and add all 34- and 5-chords. (This 17-vertex graph is the only known partitionable graph without a small transver-sal.) It's still interesting if there are other such graphs (i.e. partitionable graph satisfying \\\"no-week-pair\\\" property). If there are then it would be nice to characterize them. 16 \\n\\nContributed by Ara Markosian \\n\\nH.4 Small Transversals in Partitionable Graphs \\n\\nFollowing Bland, Huang, and Trotter [80g:05034];[86e:05075] a graph is called parti-tionable if, for some r and s, it has rs + 1 vertices and, no matter which vertex is removed, the set of the remaining rs vertices can be partitioned into r pairwise disjoint cliques of size \\n\\ns and also into s pairwise disjoint stable sets of size r. Odd holes and odd antiholes are partitionable; many additional partitionable graphs have been constructed by V. Chvtal, R. L. Graham, A. F. Perold, and S. H. Whitesides [81b:05044]. A small transversal in a graph G is a set of α(G) + ω(G) − 1 vertices which meets all cliques of size ω(G) and all stable sets of size α(G). The following problem is an easier variation on a conjecture contributed to the 1993 workshop on perfect graphs 3 by Gurvich and Temkin and on two conjectures proposed by Bacso, Boros, Gurvich, Maffray, and Preissmann [2000h:05116]. \\n\\nConjecture. Every partitionable graph G with α(G) > 2 and ω(G) > 2 has a small transversal or else contains a hole of length five. One of the milestones in the development of our understanding of perfect graphs was the theorem of Lovasz [46 #8885], asserting that every minimal imperfect graph G has precisely α(G)ω(G) +1 vertices. This theorem implies that every minimal imperfect graph is partitionable and that - as pointed out by Chvatal [86h:05091] - no minimal imperfect graph contains a small transversal. It follows that a proof of the conjecture would provide another proof of the Strong Perfect Graph Theorem. A partitionable graph without a small transversal has been constructed by by Chvatal, Graham, Perold, and Whitesides (op.cit.). Its vertices are 0, 1,..., 16; vertices i and j are adjacent if and only if |i − j| mod 17 is one of 1, 3, 4, 5, 12, 13, 14, 16. Ara Markosian claims here 4 that this is the only known partitionable graph without a small transversal. One of the many holes of length five in this graph is 1 − 4 − 8 − 12 − 15 − 1Additional information on related results and problems can be found here 5\\n\\nContributed by Vasek Chvatal \\n\\nChapter I: The Imperfection Ratio \\n\\nA demand vector for a graph G with node set V is a non-negative vector of integers indexed by nodes of G. Given a graph G and a demand vector x = ( xv: v ∈ V (G)) a coloring of the pair ( G.x ) is an assignment of a set of xv colors to each node v of G such that two adjacet nodes receive disjoint sets of colors. Coloring the pair ( G, x ) corresponds exactly to usual proper coloring of the replicated graph Gx. Let G be a graph.Define the \\n\\nimperfection ratio of G by setting \\n\\nimp (G) = max x{χf (Gx)\\n\\nω(Gx) }\\n\\n> 3http://dimacs.rutgers.edu/\\\"\\n> 4http://www.aimath.org/WWN/perfectgraph/articles/html/46a/\\n> 5http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#partitionable 17\\n\\nwhere the maximum is over all non-zero integral demand vectors x and χF (Gx) and \\n\\nω(Gx) are the fractional chromatic number and the clique number of the replicated graph Gx\\n\\nrespectively. (The ratios on the right-hand side above do indeed attain a maximum value). Observe that imp (G) ≥ 1. The next result establishes the connection of the imperfection ratio with perfection. \\n\\nProposition For any graph G, imp (G) = 1 iff G is perfect. One of the motivations for studying graph imperfection is its connection to frequency assignment. With this motivation in mind one is particularly interested in bounding the imperfection ratio for graphs of relevant graph classes. One relevant graph class are for example unit disk graph, that is graphs the node set of which can be represented by unit size disks such that two nodes are adjacent if and only if the corresponding disks intersect. It is known that imp (G) ≤ 2.155 for any unit disk graph G, and that there exists a unit disk graph G with imp (G) arbitrarily close to 3/2. \\n\\nConjecture: For any unit disk graph G, imp (G) ≤ 3/2. A subclass of unit disk graphs are the induced subgraphs of the triangular lattice. These graphs are of importance for channel assignment, since a pattern of omni-directional transmitters in two dimensions laid out like nodes of the triangular lattice in the plane give good coverage. Let us now consider an induced subgraph G of the triangular lattice T. Such a graph has a natural 3-coloring. It is possible to have ω(Gx) = 3 and χ(Gx)=4 for such a graph G.There is a polynomial-time coloring algorithm (by McDiarmid and Reed) which shows that for such a graph G we always have \\n\\nχ(Gx) ≤ 4ω(Gx) + 1 \\n\\n3Thus imp (G) ≤ 4 \\n\\n> 3\\n\\nfor any finite induced subgraph G of the triangular lattice T. The 9-cycle C9 is an induced subgraph of the triangular lattice T. For any integer k the graph obtained from C9 by replicating each of its nodes k times has clique number 2 k and chromatic number d 9k \\n\\n> 4\\n\\ne Is the ratio 9 \\n\\n> 8\\n\\nof chromatic number to clique number asymptotically the worst (greatest) possible with large demands? This questions may be rephrased in terms of imp (G)as follows: \\n\\nConjecture For any induced subgraph G of the triangular lattice T, we have imp (G) ≤\\n\\n> 9\\n> 8\\n\\nThis would imply the following weaker and perhaps more tractable conjecture: \\n\\nConjecture If G is a trianlgle-free induced subgraph of the triangular lattice then \\n\\n|V (G)| ≤ 9 \\n\\n> 4\\n\\nα(G) (where α(G) is the size of the maximum stable set in G), and indeed \\n\\nχf (G) ≤ 9\\n\\n> 4\\n\\nTo get a feeling for the behavior of the imperfection ratio, we mention the following elementary decomposition result: if G is composed of two parts G1 and G2 that are either disjoint or overlap in a clique, then \\n\\nimp (G) = max {imp (G1), imp (G2)}\\n\\nThe following is another property of imperfection which is desirable for any graph invariant related to perfection: 18 \\n\\nProposition For any graph G imp (G) = imp (Gc) where Gc denotes the completemt of G.Another graph class of interest are planar graphs. It follows from the 4-colour theorem that imp (G) ≤ 2 for any planar graph G. It is known that one can improve a little on this but we conjecture that the true value is 3 /2.\\n\\nConjecture: For any planar graph G, imp (G) ≤ 3/2. Another area of interest are complexity issues concerning imperfection. It is known that it is NP hard to determine the imperfection ratio. One open question is whether for a fixed k one can determine in polynomial time whether a graph G satisfies imp (G) ≤ k. For the special case of k = 1, this is the recognition problem for perfect graphs. Another open question is how hard is it to approximate the imperfection ratio of a graph. Initial contribution by Bruce Reed, extended by Stefanie Gerke \\n\\nChapter J: Integer Programming \\n\\nJ.1 Partitionable Graphs as Cutting Planes for Packing Problems? \\n\\nAs is well known, the strong perfect graph conjecture has been of interest to the integer programming community as well as the combinatorics community. Now that the SPGC has been established I wanted to mention another problem which may shed light on cutting plane approaches to packing problems. Sewell (and later Bram Verweij and Aardel) gave successful cutting plane codes for solving maximum stable set problems in sparse graphs by adding odd hole inequalities as they are violated by fractional solutions. Moura studied this approach for some problems arising in design theory, but met with much less success since the graph instances were much more dense (and hence odd hole inequalities were unlikely to be violated). Can we extend the class of odd cycle inequalities to the class of partitionable graph inequalities \\n\\n∑\\n\\n> v∈I\\n\\nxv ≤ α(I)for each partitionable subgraph. I.e., can we develop algorithms to solve the separa-tion problem for this class of inequalities. One positive result is that partitionable graphs themselves can be recognized in polynomial time (another problem is to find a combinatorial algorithm to recognize partitionable graphs). Contributed by Bruce Shepherd. \\n\\nJ.2 Feasibility/Membership Problem For the Theta Body \\n\\nFind a polynomial time algorithm to solve the (exact) feasibility/membership problem for the theta body. Contributed by Bruce Shepherd \\n\\nChapter K: Balanced Graphs \\n\\nDefinition A graph is balanced if every induced cycle has length 0( mod 4). Clearly balanced graphs are bipartite. 19 \\n\\nA balanced graph is basic if all its vertices on one side of the bipartition have degree at most 2 or G contains a hole H such that the vertices of G \\\\ H induce a complete bipartite graph. Here are two conjectures concerning balanced graphs.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0280", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical field is an invalid over-merge of many unrelated sections. Isolating only the first coherent new problem, G.5, the underlying simple graph defined there satisfies G(k,0)=K_k, G(k,1)=K_{3k}, and G(k,m)=C_{2m+1}[K_k] for m>=2. Consequently every hole for m>=2 has length 2m+1, omega=2k, and for every m>=1 the exact chromatic number is 2k+ceil(k/m), closing the source's one-color gap. A fully explicit cyclic-interval coloring proves the upper bound. The unrestricted disconnected wording is false because Cay(Z_6,{2,4})=2K_3, while every G(k,m) is connected; the authors' intended connected classification remains unresolved in the literature checked.\n\nCandidate contribution (explicit_coloring_and_source_obstruction; novelty confidence low): For q=2k+c, c=ceil(k/m), and delta=mc-k, advancing the first 2m cyclic k-palette starts by k+1 exactly delta times and by k otherwise makes the last start q-k and yields an explicit optimal q-coloring of G(k,m); combined with the residue-fiber proof, this simultaneously closes the printed chromatic gap, identifies the m=0 and m=1 degeneracies, and separates the false disconnected reading via the exact counterexample 2K_3.", "research_difficulty_suggested": "L2" }, { "id": 20001156, "problem_number": "AIM-COMBINATORICS-0281", "title": "The balanced-graph trichotomy for C4-free and subcubic graphs", "statement": "Conjecture 1 (Conforti, Cornuej´ ols,Rao) Every balanced graph is either basic or has a 2-join or a skew-partition.", "original_statement": "Conjecture 1 (Conforti, Cornuej´ ols,Rao) Every balanced graph is either basic or has a 2-join or a skew-partition.", "clean_statement": "Conjecture 1 (Conforti, Cornuej´ ols,Rao) Every balanced graph is either basic or has a 2-join or a skew-partition.", "statement_status": "exact", "statement_verification": "The canonical record states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[280]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 1 (Conforti, Cornuej´ ols,Rao) Every balanced graph is either basic or has a 2-join or a skew-partition.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0281", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every finite balanced graph that is either induced-C4-free or has maximum degree at most three satisfies the exact AIM conclusion: it is basic, has a proper 2-join, or has a skew partition with four nonempty parts. The proof derives the connected case from the published decomposition theorem for balanceable graphs, excludes 6-joins and R10 by induced six-cycles, converts every remaining star cutset to a skew partition apart from the basic graph P3, and then lifts the result to disconnected graphs. The unrestricted AIM conjecture was not resolved, and its present status could not be verified from the primary literature checked.\n\nCandidate contribution (special_case; novelty confidence low): The exact AIM trichotomy, with the proper 2-join convention and four nonempty skew-partition parts, holds for every possibly disconnected balanced graph that is C4-free or subcubic.", "research_difficulty_suggested": "L4" }, { "id": 20001157, "problem_number": "AIM-COMBINATORICS-0282", "title": "Safe edge deletion, modular unichord certificates, and the 3-core", "statement": "Conjecture 2 (Conforti, Rao) In every balanced graph, there exists an edge that can be deleted, so that the resulting graph remains balanced. Contributed by Gerard Cornuej` ols\n\nK.1 Balanced circulants\n\nGiven interer k ≥ 1 and m ≥ 1, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {0, 1,..., n − 1}, where n = 4 km, and ( i, j ) ∈ E iff\n\ni − j + 4 mt = +1 or − 1 ( mod n )for some integer t. E.g. if k = 5, m = 2 then n = 40 and ( i, j ) ∈ E iff i − j(mod 40) ∈{7, 9; 15, 17; 23, 25; 31, 33; 39, 1}.\n\nIt is not difficult to check that G(k, m ) is balanced, i.e. it has no odd cycles, nor (4 i + 2)-holes. In fact, it may only contain holes of length 4 and 4 m. Moreover, every (4 i + 2)-cycle has at least two chords. Hence, G(k, m ) − e is still balanced for any edge e, in agreement with", "original_statement": "Conjecture 2 (Conforti, Rao) In every balanced graph, there exists an edge that can be deleted, so that the resulting graph remains balanced. Contributed by Gerard Cornuej` ols \n\nK.1 Balanced circulants \n\nGiven interer k ≥ 1 and m ≥ 1, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {0, 1,..., n − 1}, where n = 4 km, and ( i, j ) ∈ E iff \n\ni − j + 4 mt = +1 or − 1 ( mod n )for some integer t. E.g. if k = 5, m = 2 then n = 40 and ( i, j ) ∈ E iff i − j(mod 40) ∈{7, 9; 15, 17; 23, 25; 31, 33; 39, 1}.\n\nIt is not difficult to check that G(k, m ) is balanced, i.e. it has no odd cycles, nor (4 i + 2)-holes. In fact, it may only contain holes of length 4 and 4 m. Moreover, every (4 i + 2)-cycle has at least two chords. Hence, G(k, m ) − e is still balanced for any edge e, in agreement with", "clean_statement": "Conjecture 2 (Conforti, Rao) In every balanced graph, there exists an edge that can be deleted, so that the resulting graph remains balanced. Contributed by Gerard Cornuej` ols\n\nK.1 Balanced circulants\n\nGiven interer k ≥ 1 and m ≥ 1, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {0, 1,..., n − 1}, where n = 4 km, and ( i, j ) ∈ E iff\n\ni − j + 4 mt = +1 or − 1 ( mod n )for some integer t. E.g. if k = 5, m = 2 then n = 40 and ( i, j ) ∈ E iff i − j(mod 40) ∈{7, 9; 15, 17; 23, 25; 31, 33; 39, 1}.\n\nIt is not difficult to check that G(k, m ) is balanced, i.e. it has no odd cycles, nor (4 i + 2)-holes. In fact, it may only contain holes of length 4 and 4 m. Moreover, every (4 i + 2)-cycle has at least two chords. Hence, G(k, m ) − e is still balanced for any edge e, in agreement with", "statement_status": "exact", "statement_verification": "The exact leading conjecture in the canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[281]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 2 (Conforti, Rao) In every balanced graph, there exists an edge that can be deleted, so that the resulting graph remains balanced. Contributed by Gerard Cornuej` ols \\n\\nK.1 Balanced circulants \\n\\nGiven interer k ≥ 1 and m ≥ 1, let us introduce a graph G(k, m ) = ( V, E ) with circular symmetry as follows: V = Zn = {0, 1,..., n − 1}, where n = 4 km, and ( i, j ) ∈ E iff \\n\\ni − j + 4 mt = +1 or − 1 ( mod n )for some integer t. E.g. if k = 5, m = 2 then n = 40 and ( i, j ) ∈ E iff i − j(mod 40) ∈{7, 9; 15, 17; 23, 25; 31, 33; 39, 1}.\\n\\nIt is not difficult to check that G(k, m ) is balanced, i.e. it has no odd cycles, nor (4 i + 2)-holes. In fact, it may only contain holes of length 4 and 4 m. Moreover, every (4 i + 2)-cycle has at least two chords. Hence, G(k, m ) − e is still balanced for any edge e, in agreement with\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0282", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a balanced graph G, an edge e=uv is unsafe to delete if and only if it is the one-edge branch of an induced configuration consisting of e and two internally disjoint u-v paths, each of length 3 modulo 4. Consequently every edge incident with a vertex of degree at most 2 is safe, and repeated such deletions preserve balancedness and remove exactly all edges outside the 3-core. In particular every balanced 2-degenerate graph has a complete balancedness-preserving edge-deletion ordering. The global Conforti-Rao conjecture remains open.\n\nCandidate contribution (structural_reduction_and_constructive_special_case; novelty confidence low): Unsafe edges have an exact induced 1,3,3 modulo 4 short-branch-theta certificate; iterating the resulting low-degree safe-deletion rule yields a balancedness-preserving edge ordering that stops exactly at the 3-core and deletes every edge of a balanced 2-degenerate graph.", "research_difficulty_suggested": "L4" }, { "id": 20001158, "problem_number": "AIM-COMBINATORICS-0283", "title": "Block-local safe edges for the Conforti--Rao conjecture", "statement": "Conjecture 2 from the section \"Balanced Graphs\".", "original_statement": "Conjecture 2 from the section \"Balanced Graphs\".", "clean_statement": "Conjecture 2 from the section \"Balanced Graphs\".", "statement_status": "exact", "statement_verification": "The exact canonical record is only the fragment", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[282]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 2 from the section \\\"Balanced Graphs\\\".\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0283", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical text is a spillover cross-reference, not a separate conjecture: the official AIM PDF shows that it points to the Conforti--Rao conjecture that every nonempty balanced graph has an edge whose deletion preserves balancedness. For that recovered open problem, this attempt proves that unsafe status localizes exactly to the unique block containing an edge. It also proves that an edge bisimplicial inside its block is globally safe even when it is not bisimplicial in the whole graph. Consequently, any vertex-minimal counterexample is 2-connected, has minimum degree at least three, has no bisimplicial edge, and gives each edge two induced detours of lengths 3 modulo 4; published special cases further force maximum degree at least four and an induced C4.\n\nCandidate contribution (reduction; novelty confidence low): For every block B of a finite balanced graph G, the unsafe edges satisfy U(G) intersect E(B) equals U(B); hence an edge bisimplicial relative to B is safe in G, even if extra neighbors at a cut vertex make it non-bisimplicial in G.", "research_difficulty_suggested": "L2" }, { "id": 20001159, "problem_number": "AIM-COMBINATORICS-0284", "title": "A merged record and the corrected balanced-circulant classification", "statement": "Conjecture 1 of the same section also holds for G(k, m ). Indeed, if k > 1 then S = {0; 4 mi + 1, 4mi − 1 | i = 1,..., k }\n\nis a star cutset: 0 is an isolated vertex in ¯G[S], while 4 mj is an isolated vertex in G[V \\ S]for every j = 1,..., k; and if k = 1 then G(k, m ) is 4 m-cycle, that is a basic graph CONJECTURE. Every non-empty balanced circulant is isomorphic to a G(k, m ). Contributed by Diogo Andrade, Endre Boros, and Vladimir Gurvich\n\nChapter L: P4-structure and Its Relatives\n\nThe P4-structure of a graph G is the 4-uniform hypergraph whose vertex-set V is the vertex-set of G and whose hyperedges are the subsets of V that induce P4's (chordless paths on four vertices) in G. The graph property of being Berge can be formulated directly in terms of P4-structure: a graph is Berge if and only if its P4-structure contains no induced\n\nodd ring, meaning a 4-uniform hypergraph with vertices\n\nu0, u 1,..., u k−1,\n\nwhere k is odd and at least five, and with the k hyperedges\n\n{ui+1, u i+2, u i+3, u i+4 },\n\nwhere the subscripts are taken modulo k. (The \"if\" part is trivial and the \"only if\" part is easy, even though a little tedious; see [MR 86j:05119] for a sketch of the argument.) Can the Chudnovsky-Robertson-Seymour-Thomas decomposition theorem for Berge graphs be reformulated directly in terms of P4-structure? The five classes of basic graphs featured in the theorem lend themselves nicely to such reformulations: there are classes C of 4-uniform hypergraphs such that 20\n\n• the P4-structure of every basic graph belongs to C,\n\n• no 4-uniform hypergraph in C contains an induced odd ring,\n\n• membership in C can be tested in polynomial time. One such class is defined in terms of a certain directed graph, D6(H), associated with every 4-uniform hypergraph H: C consists of all 4-uniform hypergraphs H such that\n\n• H contains no induced ring with five vertices and\n\n• all strongly connected components of D6(H) are bipartite. The vertices of D6(H) are all the ordered 6-tuples (u1, u 2, u 3, u 4, u 5, u 6)of distinct vertices of H such that the sub-hypergraph of H induced by the set\n\n{u1, u 2, u 3, u 4, u 5, u 6}\n\nconsists of the three hyperedges\n\n{u1, u 2, u 3, u 4}, {u2, u 3, u 4, u 5}, {u3, u 4, u 5, u 6};there is a directed edge from vertex (u1, u 2, u 3, u 4, u 5, u 6)of D6(H) to vertex (v1, v 2, v 3, v 4, v 5, v 6)of D6(H) if and only if\n\nv1 = u2, v 2 = u3, v 3 = u4, v 4 = u5, v 5 = u6.\n\nTrivially, membership in C can be tested in polynomial time; trivially, no 4-uniform hyper-graph in C contains an induced odd ring; a proof that the P4-structure of every basic graph belongs to C is easy, even though a little tedious (here 6 is a sketch of the argument). The four kinds of structural faults featured in the decomposition theorem suggest the following three problems.", "original_statement": "Conjecture 1 of the same section also holds for G(k, m ). Indeed, if k > 1 then S = {0; 4 mi + 1, 4mi − 1 | i = 1,..., k }\n\nis a star cutset: 0 is an isolated vertex in ¯G[S], while 4 mj is an isolated vertex in G[V \\ S]for every j = 1,..., k; and if k = 1 then G(k, m ) is 4 m-cycle, that is a basic graph CONJECTURE. Every non-empty balanced circulant is isomorphic to a G(k, m ). Contributed by Diogo Andrade, Endre Boros, and Vladimir Gurvich \n\nChapter L: P4-structure and Its Relatives \n\nThe P4-structure of a graph G is the 4-uniform hypergraph whose vertex-set V is the vertex-set of G and whose hyperedges are the subsets of V that induce P4's (chordless paths on four vertices) in G. The graph property of being Berge can be formulated directly in terms of P4-structure: a graph is Berge if and only if its P4-structure contains no induced \n\nodd ring, meaning a 4-uniform hypergraph with vertices \n\nu0, u 1,..., u k−1,\n\nwhere k is odd and at least five, and with the k hyperedges \n\n{ui+1, u i+2, u i+3, u i+4 },\n\nwhere the subscripts are taken modulo k. (The \"if\" part is trivial and the \"only if\" part is easy, even though a little tedious; see [MR 86j:05119] for a sketch of the argument.) Can the Chudnovsky-Robertson-Seymour-Thomas decomposition theorem for Berge graphs be reformulated directly in terms of P4-structure? The five classes of basic graphs featured in the theorem lend themselves nicely to such reformulations: there are classes C of 4-uniform hypergraphs such that 20 \n\n• the P4-structure of every basic graph belongs to C,\n\n• no 4-uniform hypergraph in C contains an induced odd ring, \n\n• membership in C can be tested in polynomial time. One such class is defined in terms of a certain directed graph, D6(H), associated with every 4-uniform hypergraph H: C consists of all 4-uniform hypergraphs H such that \n\n• H contains no induced ring with five vertices and \n\n• all strongly connected components of D6(H) are bipartite. The vertices of D6(H) are all the ordered 6-tuples (u1, u 2, u 3, u 4, u 5, u 6)of distinct vertices of H such that the sub-hypergraph of H induced by the set \n\n{u1, u 2, u 3, u 4, u 5, u 6}\n\nconsists of the three hyperedges \n\n{u1, u 2, u 3, u 4}, {u2, u 3, u 4, u 5}, {u3, u 4, u 5, u 6};there is a directed edge from vertex (u1, u 2, u 3, u 4, u 5, u 6)of D6(H) to vertex (v1, v 2, v 3, v 4, v 5, v 6)of D6(H) if and only if \n\nv1 = u2, v 2 = u3, v 3 = u4, v 4 = u5, v 5 = u6.\n\nTrivially, membership in C can be tested in polynomial time; trivially, no 4-uniform hyper-graph in C contains an induced odd ring; a proof that the P4-structure of every basic graph belongs to C is easy, even though a little tedious (here 6 is a sketch of the argument). The four kinds of structural faults featured in the decomposition theorem suggest the following three problems.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is not one coherent problem. It begins in the middle of the balanced-circulant item K.1, gives a conjecture, and then runs through the heading and introductory material of Chapter L on P4-structure. The latter material sets up later, separately indexed problems and is not part of the K.1 conjecture. Consequently the record is classified as `invalid_statement` as a unit.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[283]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 1 of the same section also holds for G(k, m ). Indeed, if k > 1 then S = {0; 4 mi + 1, 4mi − 1 | i = 1,..., k }\\n\\nis a star cutset: 0 is an isolated vertex in ¯G[S], while 4 mj is an isolated vertex in G[V \\\\ S]for every j = 1,..., k; and if k = 1 then G(k, m ) is 4 m-cycle, that is a basic graph CONJECTURE. Every non-empty balanced circulant is isomorphic to a G(k, m ). Contributed by Diogo Andrade, Endre Boros, and Vladimir Gurvich \\n\\nChapter L: P4-structure and Its Relatives \\n\\nThe P4-structure of a graph G is the 4-uniform hypergraph whose vertex-set V is the vertex-set of G and whose hyperedges are the subsets of V that induce P4's (chordless paths on four vertices) in G. The graph property of being Berge can be formulated directly in terms of P4-structure: a graph is Berge if and only if its P4-structure contains no induced \\n\\nodd ring, meaning a 4-uniform hypergraph with vertices \\n\\nu0, u 1,..., u k−1,\\n\\nwhere k is odd and at least five, and with the k hyperedges \\n\\n{ui+1, u i+2, u i+3, u i+4 },\\n\\nwhere the subscripts are taken modulo k. (The \\\"if\\\" part is trivial and the \\\"only if\\\" part is easy, even though a little tedious; see [MR 86j:05119] for a sketch of the argument.) Can the Chudnovsky-Robertson-Seymour-Thomas decomposition theorem for Berge graphs be reformulated directly in terms of P4-structure? The five classes of basic graphs featured in the theorem lend themselves nicely to such reformulations: there are classes C of 4-uniform hypergraphs such that 20 \\n\\n• the P4-structure of every basic graph belongs to C,\\n\\n• no 4-uniform hypergraph in C contains an induced odd ring, \\n\\n• membership in C can be tested in polynomial time. One such class is defined in terms of a certain directed graph, D6(H), associated with every 4-uniform hypergraph H: C consists of all 4-uniform hypergraphs H such that \\n\\n• H contains no induced ring with five vertices and \\n\\n• all strongly connected components of D6(H) are bipartite. The vertices of D6(H) are all the ordered 6-tuples (u1, u 2, u 3, u 4, u 5, u 6)of distinct vertices of H such that the sub-hypergraph of H induced by the set \\n\\n{u1, u 2, u 3, u 4, u 5, u 6}\\n\\nconsists of the three hyperedges \\n\\n{u1, u 2, u 3, u 4}, {u2, u 3, u 4, u 5}, {u3, u 4, u 5, u 6};there is a directed edge from vertex (u1, u 2, u 3, u 4, u 5, u 6)of D6(H) to vertex (v1, v 2, v 3, v 4, v 5, v 6)of D6(H) if and only if \\n\\nv1 = u2, v 2 = u3, v 3 = u4, v 4 = u5, v 5 = u6.\\n\\nTrivially, membership in C can be tested in polynomial time; trivially, no 4-uniform hyper-graph in C contains an induced odd ring; a proof that the P4-structure of every basic graph belongs to C is easy, even though a little tedious (here 6 is a sketch of the argument). The four kinds of structural faults featured in the decomposition theorem suggest the following three problems.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0284", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record overmerges the end of the K.1 balanced-circulant item with the beginning of an unrelated P4-structure chapter. The coherent K.1 conjecture is false as written: K_{3,3}=Cay(Z_6,{1,3,5}) is a connected non-empty balanced circulant but cannot be any G(k,m), whose order is divisible by four. Using the published Morris-Spiga-Webb classification and the proved identity G(k,m)=C_{4m}[Kbar_k], the exact repair is that connected balanced circulants are G(k,m) for m at least 2 or K_{t,t}; relative to the original AIM family, exactly K_{t,t} with odd t are missing. The report also proves the family’s hole, two-chord, edge-deletion, and star-cutset properties and corrects an endpoint error in the printed cutset argument.\n\nCandidate contribution (classification_correction; novelty confidence low): Within the published connected classification, the exact set-theoretic gap in the AIM family is K_{2r+1,2r+1}: even-part complete bipartite graphs equal G(r,1), while all non-complete-bipartite cases equal G(k,m) with m at least 2.", "research_difficulty_suggested": "L2" }, { "id": 20001160, "problem_number": "AIM-COMBINATORICS-0285", "title": "A literal NP class for the P4-structure of proper 2-joins", "statement": "Problem 1: Find a class C1 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with a 2-join belongs to C1,\n\n• no odd ring belongs to C1,\n\n• C1 belongs to NP.", "original_statement": "Problem 1: Find a class C1 of 4-uniform hypergraphs such that \n\n• the P4-structure of every graph with a 2-join belongs to C1,\n\n• no odd ring belongs to C1,\n\n• C1 belongs to NP.", "clean_statement": "Problem 1: Find a class C1 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with a 2-join belongs to C1,\n\n• no odd ring belongs to C1,\n\n• C1 belongs to NP.", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[284]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1: Find a class C1 of 4-uniform hypergraphs such that \\n\\n• the P4-structure of every graph with a 2-join belongs to C1,\\n\\n• no odd ring belongs to C1,\\n\\n• C1 belongs to NP.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0285", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "Under the connected non-path 2-join convention explicitly stated in the source, the class of 4-uniform hypergraphs admitting a graph realization with such a 2-join satisfies all three printed bullets and is in NP via an O(n^4)-time certificate verifier. Odd rings are excluded using Chvatal's classification of their realizations together with direct proofs that neither an odd hole nor an odd antihole has a source-proper 2-join. The literal formulation also admits the intrinsic polynomial-time class of all hypergraphs not isomorphic to an odd ring.\n\nCandidate contribution (theorem; novelty confidence low): The explicit realization-plus-proper-2-join witness language is an NP solution of the literal AIM bullets; its odd-ring gate has a direct hole/antihole proof, and the accompanying semantic audit proves both that the maximal non-ring class is already a P solution and that strengthening exact ring exclusion to induced-ring exclusion makes the universal first bullet inconsistent." }, { "id": 20001161, "problem_number": "AIM-COMBINATORICS-0286", "title": "An existential NP class for P4-structures of M-join graphs", "statement": "Problem 2: Find a class C2 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with an M-join belongs to C2,\n\n• no odd ring belongs to C2,\n\n• C2 belongs to NP.", "original_statement": "Problem 2: Find a class C2 of 4-uniform hypergraphs such that \n\n• the P4-structure of every graph with an M-join belongs to C2,\n\n• no odd ring belongs to C2,\n\n• C2 belongs to NP.", "clean_statement": "Problem 2: Find a class C2 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with an M-join belongs to C2,\n\n• no odd ring belongs to C2,\n\n• C2 belongs to NP.", "statement_status": "exact", "statement_verification": "The official AIM PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[285]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2: Find a class C2 of 4-uniform hypergraphs such that \\n\\n• the P4-structure of every graph with an M-join belongs to C2,\\n\\n• no odd ring belongs to C2,\\n\\n• C2 belongs to NP.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0286", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the contemporaneous six-nonempty-set M-join convention and the literal language-membership reading of the AIM bullets, define C2 as the four-uniform hypergraphs admitting a graph realization with an M-join. An adjacency matrix and six-set partition give an O(n^4)-verifiable NP certificate. No odd ring belongs to this class: odd-ring realizations are, by Chvatal's uniqueness theorem, odd holes or their complements; an odd hole cannot have an M-join because a vertex in A would have distinct neighbors in B, C, and F, and complement invariance excludes antiholes. This is a direct literal answer but may not meet the intended hypergraph-intrinsic reformulation. An explicit M-join graph with an induced C5 in A proves that the class fails the stronger condition that no member contain an induced odd ring.\n\nCandidate contribution (construction; novelty confidence low): The existential realization language C2^exists satisfies all three literal AIM bullets in NP, while an explicit six-part M-join construction separates the literal condition 'no odd ring belongs' from the stronger condition 'no member contains an induced odd ring.'", "research_difficulty_suggested": "L4" }, { "id": 20001162, "problem_number": "AIM-COMBINATORICS-0287", "title": "A scope trichotomy for the balanced-skew-partition class problem", "statement": "Problem 3: Find a class C3 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with a balanced skew partition belongs to C3,\n\n• no odd ring belongs to C3,\n\n• C3 belongs to NP.\n\n> 6http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#basic 21\n\nCh´ ınh Ho` ang introduced additional graph functions that are invariant under comple-mentation and determine whether or not a graph is Berge: these are the co-paw-structure\n\n([MR 2001a:05065]), the co-C4-structure (Discrete Math. 252 (2002), 141-159), and the\n\nco-P3-structure (to appear in SIAM J. Discrete Math.). Can the decomposition theorem be reformulated directly in terms of one of Ho` ang's invariants? Contributed by Vaˇ sek Chv´ atal", "original_statement": "Problem 3: Find a class C3 of 4-uniform hypergraphs such that \n\n• the P4-structure of every graph with a balanced skew partition belongs to C3,\n\n• no odd ring belongs to C3,\n\n• C3 belongs to NP.\n\n> 6http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#basic 21\n\nCh´ ınh Ho` ang introduced additional graph functions that are invariant under comple-mentation and determine whether or not a graph is Berge: these are the co-paw-structure \n\n([MR 2001a:05065]), the co-C4-structure (Discrete Math. 252 (2002), 141-159), and the \n\nco-P3-structure (to appear in SIAM J. Discrete Math.). Can the decomposition theorem be reformulated directly in terms of one of Ho` ang's invariants? Contributed by Vaˇ sek Chv´ atal", "clean_statement": "Problem 3: Find a class C3 of 4-uniform hypergraphs such that\n\n• the P4-structure of every graph with a balanced skew partition belongs to C3,\n\n• no odd ring belongs to C3,\n\n• C3 belongs to NP.\n\n> 6http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#basic 21\n\nCh´ ınh Ho` ang introduced additional graph functions that are invariant under comple-mentation and determine whether or not a graph is Berge: these are the co-paw-structure\n\n([MR 2001a:05065]), the co-C4-structure (Discrete Math. 252 (2002), 141-159), and the\n\nco-P3-structure (to appear in SIAM J. Discrete Math.). Can the decomposition theorem be reformulated directly in terms of one of Ho` ang's invariants? Contributed by Vaˇ sek Chv´ atal", "statement_status": "exact", "statement_verification": "The official AIM page states the numbered item as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Combinatorics\nWorkshop: The Perfect Graph Conjecture\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/perfectgraph/perfectgraph.pdf\nCanonical location: aim-combinatorics-notes.json notes[286]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3: Find a class C3 of 4-uniform hypergraphs such that \\n\\n• the P4-structure of every graph with a balanced skew partition belongs to C3,\\n\\n• no odd ring belongs to C3,\\n\\n• C3 belongs to NP.\\n\\n> 6http://www.cs.rutgers.edu/ ∼chvatal/perfect/problems.html#basic 21\\n\\nCh´ ınh Ho` ang introduced additional graph functions that are invariant under comple-mentation and determine whether or not a graph is Berge: these are the co-paw-structure \\n\\n([MR 2001a:05065]), the co-C4-structure (Discrete Math. 252 (2002), 141-159), and the \\n\\nco-P3-structure (to appear in SIAM J. Discrete Math.). Can the decomposition theorem be reformulated directly in terms of one of Ho` ang's invariants? Contributed by Vaˇ sek Chv´ atal\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 2, "set_id": 14, "source_url": "https://aimath.org/WWN/perfectgraph/perfectgraph.pdf", "tags": [ "aim", "AIM-COMBINATORICS-0287", "aim-domain:combinatorics", "aim-workshop:perfectgraph", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 2, "name": "combinatorics", "display_name": "Combinatorics", "description": "Counting problems, graph theory, discrete structures.", "slug": "combinatorics", "order_index": 2, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "For the clean numbered Problem 3 under its literal whole-hypergraph membership wording, the class of realizable P4-structures that are not themselves odd rings satisfies all three requested properties and is in P. The proof combines polynomial P4-realizability recognition with Chvatal's uniqueness theorem and a direct proof that odd holes and antiholes have no skew partition. Under the contextual Berge reading, an explicit realization-and-partition certificate gives an NP class with induced-odd-ring exclusion. The stronger all-graphs plus induced-ring-free reading is impossible, witnessed by the connected family K_2 join (C_k disjoint union K_1) for odd k at least 5. The natural exact existential class is proved only to lie in Sigma_2^P; its NP membership remains unverified.\n\nCandidate contribution (obstruction_and_scope_reduction; novelty confidence low): The three plausible quantifier scopes separate explicitly: the literal class is in P, the Berge-scoped induced-ring-free class is in NP, and the all-graphs induced-ring-free version is impossible via the connected family K_2 join (C_{2r+1} disjoint union K_1)." }, { "id": 20001163, "problem_number": "AIM-COMPUTATION-0001", "title": "Overmerged factorization questions and a batched splitting criterion", "statement": "1. Is there a deterministic algorithm that is polynomial time in n and log( q)? State of the art algorithm should be seen on May 16th as a talk. 2. How quickly can we factor x2 −a without hypotheses (Qi Cheng's question) State of the art algorithm Burgess O(p 1\n\n> 2e\n\n)\n\n1.2 Open Question\n\nWhat is the exponent of the probabilistic complexity (for q = 2)?\n\n1.3 Open Question\n\nGiven a set of polynomials with very low degree (2 or 3), decide if all of them factor into linear factors. Is it faster to do this by factoring their product or each one individually? (Tanja Lange's question)\n\n2 Sparse Polynomials in Fq[x]\n\n2.1 Open Question\n\nDecide whether a trinomial xα + ax β + b with 0 < β < α ≤ q − 1, a, b ∈ Fq has a root in Fq, in polynomial time (in log( q)). (Erich Kaltofen's Question)\n\n2.2 Open Questions", "original_statement": "1. Is there a deterministic algorithm that is polynomial time in n and log( q)? State of the art algorithm should be seen on May 16th as a talk. 2. How quickly can we factor x2 −a without hypotheses (Qi Cheng's question) State of the art algorithm Burgess O(p 1 \n\n> 2e\n\n)\n\n1.2 Open Question \n\nWhat is the exponent of the probabilistic complexity (for q = 2)? \n\n1.3 Open Question \n\nGiven a set of polynomials with very low degree (2 or 3), decide if all of them factor into linear factors. Is it faster to do this by factoring their product or each one individually? (Tanja Lange's question) \n\n2 Sparse Polynomials in Fq[x]\n\n2.1 Open Question \n\nDecide whether a trinomial xα + ax β + b with 0 < β < α ≤ q − 1, a, b ∈ Fq has a root in Fq, in polynomial time (in log( q)). (Erich Kaltofen's Question) \n\n2.2 Open Questions", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "The canonical record is not one mathematical problem. The official three-page AIM PDF verifies that extraction joined the following distinct items from page 1:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: The computational complexity of polynomial factorization\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/polyfactor/polyfactor.pdf\nCanonical location: aim-computation-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Is there a deterministic algorithm that is polynomial time in n and log( q)? State of the art algorithm should be seen on May 16th as a talk. 2. How quickly can we factor x2 −a without hypotheses (Qi Cheng's question) State of the art algorithm Burgess O(p 1 \\n\\n> 2e\\n\\n)\\n\\n1.2 Open Question \\n\\nWhat is the exponent of the probabilistic complexity (for q = 2)? \\n\\n1.3 Open Question \\n\\nGiven a set of polynomials with very low degree (2 or 3), decide if all of them factor into linear factors. Is it faster to do this by factoring their product or each one individually? (Tanja Lange's question) \\n\\n2 Sparse Polynomials in Fq[x]\\n\\n2.1 Open Question \\n\\nDecide whether a trinomial xα + ax β + b with 0 < β < α ≤ q − 1, a, b ∈ Fq has a root in Fq, in polynomial time (in log( q)). (Erich Kaltofen's Question) \\n\\n2.2 Open Questions\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/polyfactor/polyfactor.pdf", "tags": [ "aim", "AIM-COMPUTATION-0001", "aim-domain:computation", "aim-workshop:polyfactor", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is not a single problem: the official PDF shows that it merges five distinct questions from sections 1.1, 1.2, 1.3, and 2.1, two section headings, and a corrupted displayed bound. For the clearly recovered section 1.3 component, this attempt proves that nonzero polynomials f_i over F_q all split completely into linear factors, with multiplicity allowed, if and only if the radical of their product divides X^q-X, equivalently X^q is congruent to X modulo that radical. This gives a deterministic decision test without producing factors and precisely identifies why the product-versus-individual speed comparison needs a specified arithmetic and output model.\n\nCandidate contribution (equivalence; novelty confidence low): For a batch of nonzero finite-field polynomials, complete splitting of every input is exactly one radical divisibility test rad(product f_i) divides X^q-X; the formulation remains valid with repeated factors and derivative-zero inputs and separates Boolean splitting tests from factor output.", "research_difficulty_suggested": "L2" }, { "id": 20001164, "problem_number": "AIM-COMPUTATION-0002", "title": "A malformed composite record and an Artin--Schreier certificate family", "statement": "1. Finding better certificates for irreducibility (a) Are there certificates for irreducibility that one can verify faster than the existing irreducibility tests? (Victor Miller's Question) (b) Can you exhibit families of sparse polynomials for which you can find shorter certificates? 2. Find a polynomial-time algorithm in log( q), where q = pn, that solves\n\n∑n−1\n\n> i,j =0\n\nai,j xpi +pj\n\n+ ∑n−1\n\n> i=0\n\nbixpi\n\n+ c in Fq (or shows no solutions) and works for 1\n\n> poly\n\nof the inputs, and never lies. (We know that this is NP-complete.) (Jintai Ding's Question) 13 Factoring over Q[x]\n\n3.1 Open\nQuestion", "original_statement": "1. Finding better certificates for irreducibility (a) Are there certificates for irreducibility that one can verify faster than the existing irreducibility tests? (Victor Miller's Question) (b) Can you exhibit families of sparse polynomials for which you can find shorter certificates? 2. Find a polynomial-time algorithm in log( q), where q = pn, that solves \n\n∑n−1 \n\n> i,j =0\n\nai,j xpi +pj\n\n+ ∑n−1 \n\n> i=0\n\nbixpi\n\n+ c in Fq (or shows no solutions) and works for 1 \n\n> poly\n\nof the inputs, and never lies. (We know that this is NP-complete.) (Jintai Ding's Question) 13 Factoring over Q[x]\n\n3.1 Open \nQuestion", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "The canonical JSON record is not one mathematical problem. It is a damaged extraction of page 1 of the AIM workshop notes *The computational complexity of polynomial factorization*. Comparison with the source PDF recovers the following text in Section 2.2, “Open Questions” (line breaks normalized but wording retained):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: The computational complexity of polynomial factorization\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/polyfactor/polyfactor.pdf\nCanonical location: aim-computation-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Finding better certificates for irreducibility (a) Are there certificates for irreducibility that one can verify faster than the existing irreducibility tests? (Victor Miller's Question) (b) Can you exhibit families of sparse polynomials for which you can find shorter certificates? 2. Find a polynomial-time algorithm in log( q), where q = pn, that solves \\n\\n∑n−1 \\n\\n> i,j =0\\n\\nai,j xpi +pj\\n\\n+ ∑n−1 \\n\\n> i=0\\n\\nbixpi\\n\\n+ c in Fq (or shows no solutions) and works for 1 \\n\\n> poly\\n\\nof the inputs, and never lies. (We know that this is NP-complete.) (Jintai Ding's Question) 13 Factoring over Q[x]\\n\\n3.1 Open \\nQuestion\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/polyfactor/polyfactor.pdf", "tags": [ "aim", "AIM-COMPUTATION-0002", "aim-domain:computation", "aim-workshop:polyfactor", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is not one well-formed problem: it overmerges two independent Section 2.2 questions, a page number, and the truncated next section heading, while leaving the success semantics of the equation-solving question ambiguous. For the clearly recovered sparse-certificate subquestion, the report proves that T^p-T-a over F_{p^n} is irreducible exactly when the absolute trace of a is nonzero. In a trusted trace-normalized normal basis, this gives a one-base-field-element certificate checked with n-1 base-field additions plus equality and nonzero tests.\n\nCandidate contribution (certificate packaging; novelty confidence low): For the three-term family T^p-T-a over F_{p^n}, express a in a pre-established trace-normalized normal basis and use its nonzero coordinate sum as a one-F_p-element irreducibility certificate; a sound verifier recomputes that sum using exactly n-1 F_p additions, then performs equality and nonzero checks.", "research_difficulty_suggested": "L2" }, { "id": 20001165, "problem_number": "AIM-COMPUTATION-0003", "title": "Source repair and a sharp quadratic specialization theorem", "statement": "1. How long does the gradual feeding (or some modification) algorithm actu-ally take to solve one approximate linear equation (with the ci's bounded by kn bits)? (Mark van Hoeij's Question) 2. (a) Can we do lattice basis reduction in essentially linear time in the total number of digits of the input? (Dan Bernstein's Question) (b) Is there a softly linear time reduction from linear system solving to basis reduction in lattices? (Joachim von zur Gathen's question)\n\n3.2 Open Question\n\nCan we use Niederreiter's equation to find a faster algorithm to factor dense polynomials in Q[x]? (Shuhong Gao's Question) The main possible advantage of such a technique would be to use linear algebra over Fp instead of lattice basis reduction.\n\n4 Bivariate\n\n4.1 Open Question\n\nFor Dense Bivariate polynomials over Fp with total degree, t: Can we improve on the probabilistic complexity ˜O( d3) + Factorization Complexity of Univariate, degree d, polynomial. Lecerf believes ˜O( dω ) is possible. The bottleneck is solving d2 equations in d variables.\n\n4.2 Open Question\n\nCan we find low-degree ( ≤ n) factors of a polynomial with t terms and degree\n\nd (t\n d2, n\n d)\n\n4.3 Open Question\n\nFind an Absolute Factorization (or test for irreducibility) over fields with small (p < d 2) characteristic, in ˜O( d3).\n\n5 Intermezzo\n\nGiven f ∈ Q[x] irreducible, and f (α) = 0. Factor f over Q[α]. State of Art: Belabas 26 Numerical\n\n6.1 Open Question\n\nChallenge Problem #1 in Kaltofen's JSC 2000 paper.\n\n6.2 Open Question\n\nGiven f ∈ C[x, y ] find the nearest polynomial that factors into linear factors in\n\nC[x, y ]. Find/Bound the distance to the nearest polynomial that factors.\n\n6.3 Open Question\n\nGiven f ∈ Q[x, y, z ], for which values of z ∈ Q does f factor (completely) in\n\nC[x, y ]? What is the degree of a given z?\n\n6.4 Open Question\n\nHow to compute/approximate the structured condition number of a Ruppert Matrix. 3", "original_statement": "1. How long does the gradual feeding (or some modification) algorithm actu-ally take to solve one approximate linear equation (with the ci's bounded by kn bits)? (Mark van Hoeij's Question) 2. (a) Can we do lattice basis reduction in essentially linear time in the total number of digits of the input? (Dan Bernstein's Question) (b) Is there a softly linear time reduction from linear system solving to basis reduction in lattices? (Joachim von zur Gathen's question) \n\n3.2 Open Question \n\nCan we use Niederreiter's equation to find a faster algorithm to factor dense polynomials in Q[x]? (Shuhong Gao's Question) The main possible advantage of such a technique would be to use linear algebra over Fp instead of lattice basis reduction. \n\n4 Bivariate \n\n4.1 Open Question \n\nFor Dense Bivariate polynomials over Fp with total degree, t: Can we improve on the probabilistic complexity ˜O( d3) + Factorization Complexity of Univariate, degree d, polynomial. Lecerf believes ˜O( dω ) is possible. The bottleneck is solving d2 equations in d variables. \n\n4.2 Open Question \n\nCan we find low-degree ( ≤ n) factors of a polynomial with t terms and degree \n\nd (t \u001c d2, n \u001c d)\n\n4.3 Open Question \n\nFind an Absolute Factorization (or test for irreducibility) over fields with small (p < d 2) characteristic, in ˜O( d3). \n\n5 Intermezzo \n\nGiven f ∈ Q[x] irreducible, and f (α) = 0. Factor f over Q[α]. State of Art: Belabas 26 Numerical \n\n6.1 Open Question \n\nChallenge Problem #1 in Kaltofen's JSC 2000 paper. \n\n6.2 Open Question \n\nGiven f ∈ C[x, y ] find the nearest polynomial that factors into linear factors in \n\nC[x, y ]. Find/Bound the distance to the nearest polynomial that factors. \n\n6.3 Open Question \n\nGiven f ∈ Q[x, y, z ], for which values of z ∈ Q does f factor (completely) in \n\nC[x, y ]? What is the degree of a given z?\n\n6.4 Open Question \n\nHow to compute/approximate the structured condition number of a Ruppert Matrix. 3", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "The canonical `problem` field is preserved verbatim in `input.json`. It is not a single mathematical question. Comparison with the official three-page AIM PDF shows that the extraction joined almost all of pages 2 and 3 of a workshop problem list into one record. More precisely, the record contains:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: The computational complexity of polynomial factorization\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/polyfactor/polyfactor.pdf\nCanonical location: aim-computation-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. How long does the gradual feeding (or some modification) algorithm actu-ally take to solve one approximate linear equation (with the ci's bounded by kn bits)? (Mark van Hoeij's Question) 2. (a) Can we do lattice basis reduction in essentially linear time in the total number of digits of the input? (Dan Bernstein's Question) (b) Is there a softly linear time reduction from linear system solving to basis reduction in lattices? (Joachim von zur Gathen's question) \\n\\n3.2 Open Question \\n\\nCan we use Niederreiter's equation to find a faster algorithm to factor dense polynomials in Q[x]? (Shuhong Gao's Question) The main possible advantage of such a technique would be to use linear algebra over Fp instead of lattice basis reduction. \\n\\n4 Bivariate \\n\\n4.1 Open Question \\n\\nFor Dense Bivariate polynomials over Fp with total degree, t: Can we improve on the probabilistic complexity ˜O( d3) + Factorization Complexity of Univariate, degree d, polynomial. Lecerf believes ˜O( dω ) is possible. The bottleneck is solving d2 equations in d variables. \\n\\n4.2 Open Question \\n\\nCan we find low-degree ( ≤ n) factors of a polynomial with t terms and degree \\n\\nd (t \\u001c d2, n \\u001c d)\\n\\n4.3 Open Question \\n\\nFind an Absolute Factorization (or test for irreducibility) over fields with small (p < d 2) characteristic, in ˜O( d3). \\n\\n5 Intermezzo \\n\\nGiven f ∈ Q[x] irreducible, and f (α) = 0. Factor f over Q[α]. State of Art: Belabas 26 Numerical \\n\\n6.1 Open Question \\n\\nChallenge Problem #1 in Kaltofen's JSC 2000 paper. \\n\\n6.2 Open Question \\n\\nGiven f ∈ C[x, y ] find the nearest polynomial that factors into linear factors in \\n\\nC[x, y ]. Find/Bound the distance to the nearest polynomial that factors. \\n\\n6.3 Open Question \\n\\nGiven f ∈ Q[x, y, z ], for which values of z ∈ Q does f factor (completely) in \\n\\nC[x, y ]? What is the degree of a given z?\\n\\n6.4 Open Question \\n\\nHow to compute/approximate the structured condition number of a Ruppert Matrix. 3\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/polyfactor/polyfactor.pdf", "tags": [ "aim", "AIM-COMPUTATION-0003", "aim-domain:computation", "aim-workshop:polyfactor", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is invalidly segmented: it concatenates sections 3.1 through 6.4 of the AIM workshop PDF, includes a lost asymptotic glyph, and contains an ambiguous question about the degree of a rational parameter. For the clearly recovered section 6.3, a proved special-case theorem gives an exact certificate when f is quadratic in x and y: a nonzero fiber at zeta splits completely over C if and only if Delta_f(zeta)=0 for Delta_f=4ach+bde-ae^2-cd^2-hb^2. If f has total degree at most D, then deg Delta_f is at most 3D-4; this bounds both the number and algebraic degree of exceptional parameters, and the bound is sharp over Q for every D at least 2.\n\nCandidate contribution (theorem; novelty confidence low): For total-degree-D families quadratic in x and y, the explicit polynomial Delta_f=4ach+bde-ae^2-cd^2-hb^2 detects every nonzero completely split fiber, has degree at most 3D-4, and the resulting parameter-count and algebraic-degree bound is attained by an explicit integral family for every D at least 2.", "research_difficulty_suggested": "L2" }, { "id": 20001166, "problem_number": "AIM-COMPUTATION-0004", "title": "Exact ridge reduction for Lévy PIDEs", "statement": "A.1 Arisawa, Mariko\n\nWe would like to introduce a viscosity solution's approach to a class of second-order partial-integro-differential equations (PIDE, in short) arising in the mathematical finance. The usual brownian models for the processes of the asset prices are replaced by the jump-diffusion processes. The derived PIDEs contain in the integral term, various Le'vy measures which correspond to the frequencies and the sizes of the jumps. In addition, according to the financial settings such as the American option, the look back option, the problem with transaction costs, a variety of nonlinear PIDEs are engendered. We shall introduce a new definition of the viscosity solution for such problems, and shall give the comparison, the existence, and the regularity of the solutions of viscosity solutions in this framework. As far as the high-dimensional numerical analysis, a very few is known at this stage, while from the practical view point a lot of important problems should be explored, such as the modelization of the high-dimensional Le'vy measures arising in the mathematical finances. This is the problem which I would like to discuss with the participants of the workshop.", "original_statement": "A.1 Arisawa, Mariko \n\nWe would like to introduce a viscosity solution's approach to a class of second-order partial-integro-differential equations (PIDE, in short) arising in the mathematical finance. The usual brownian models for the processes of the asset prices are replaced by the jump-diffusion processes. The derived PIDEs contain in the integral term, various Le'vy measures which correspond to the frequencies and the sizes of the jumps. In addition, according to the financial settings such as the American option, the look back option, the problem with transaction costs, a variety of nonlinear PIDEs are engendered. We shall introduce a new definition of the viscosity solution for such problems, and shall give the comparison, the existence, and the regularity of the solutions of viscosity solutions in this framework. As far as the high-dimensional numerical analysis, a very few is known at this stage, while from the practical view point a lot of important problems should be explored, such as the modelization of the high-dimensional Le'vy measures arising in the mathematical finances. This is the problem which I would like to discuss with the participants of the workshop.", "clean_statement": "A.1 Arisawa, Mariko\n\nWe would like to introduce a viscosity solution's approach to a class of second-order partial-integro-differential equations (PIDE, in short) arising in the mathematical finance. The usual brownian models for the processes of the asset prices are replaced by the jump-diffusion processes. The derived PIDEs contain in the integral term, various Le'vy measures which correspond to the frequencies and the sizes of the jumps. In addition, according to the financial settings such as the American option, the look back option, the problem with transaction costs, a variety of nonlinear PIDEs are engendered. We shall introduce a new definition of the viscosity solution for such problems, and shall give the comparison, the existence, and the regularity of the solutions of viscosity solutions in this framework. As far as the high-dimensional numerical analysis, a very few is known at this stage, while from the practical view point a lot of important problems should be explored, such as the modelization of the high-dimensional Le'vy measures arising in the mathematical finances. This is the problem which I would like to discuss with the participants of the workshop.", "statement_status": "exact", "statement_verification": "The canonical item is Section A.1, “Arisawa, Mariko,” in the participant contributions to the AIM workshop *Numerical methods for optimal control in high dimensions*. The official PDF and the extracted record agree. The entry says that second-order partial integro-differential equations (PIDEs) arising from jump-diffusion models in mathematical finance involve a variety of Lévy measures and nonlinearities associated with American options, lookback options, and transaction costs. It proposes a viscosity-solution framework with comparison, existence, and regularity results. With the PDF's malformed accent in “Lévy” normalized, the entry closes with:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Numerical methods for optimal control in high dimensions\nSection: \nSource item: A.1\nSource URL: https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf\nCanonical location: aim-computation-notes.json notes[3]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.1 Arisawa, Mariko \\n\\nWe would like to introduce a viscosity solution's approach to a class of second-order partial-integro-differential equations (PIDE, in short) arising in the mathematical finance. The usual brownian models for the processes of the asset prices are replaced by the jump-diffusion processes. The derived PIDEs contain in the integral term, various Le'vy measures which correspond to the frequencies and the sizes of the jumps. In addition, according to the financial settings such as the American option, the look back option, the problem with transaction costs, a variety of nonlinear PIDEs are engendered. We shall introduce a new definition of the viscosity solution for such problems, and shall give the comparison, the existence, and the regularity of the solutions of viscosity solutions in this framework. As far as the high-dimensional numerical analysis, a very few is known at this stage, while from the practical view point a lot of important problems should be explored, such as the modelization of the high-dimensional Le'vy measures arising in the mathematical finances. This is the problem which I would like to discuss with the participants of the workshop.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf", "tags": [ "aim", "AIM-COMPUTATION-0004", "aim-domain:computation", "aim-workshop:optimalcontrol", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source is a participant research agenda rather than a quantified open problem. A rigorous truncation-aware reduction is proved for its high-dimensional Lévy-modeling theme: for u(x)=v(Wx), the d-dimensional Lévy generator equals the r-dimensional generator whose measure is the pushforward away from zero, whose covariance is WQW^T, and whose drift is Wb plus the explicit correction integral of h_r(Wz)-W h_d(z). The projected triplet is also necessary and sufficient for equality on every Fourier ridge observable. A semigroup reduction follows for bounded ridge payoffs, while viscosity, obstacle, boundary, state-dependent-kernel, and numerical-convergence extensions are explicitly left conditional.\n\nCandidate contribution (exact_reduction_and_observability_criterion; novelty confidence low): For translation-invariant Lévy PIDEs observed only through a fixed linear feature map W, the truncation-corrected projected triplet is both sufficient and necessary for equality on all Fourier ridge observables; an explicit nonsymmetric cutoff-crossing example proves that omitting the drift correction changes the operator.", "research_difficulty_suggested": "L2" }, { "id": 20001167, "problem_number": "AIM-COMPUTATION-0005", "title": "Nuisance-adjusted information for semiparametric MRS design", "statement": "A.2 Capobianco, Enrico\n\nPart 1\n\nSystems Biology: On Dimensionality Reduction and Feature Se-lection in Genomics (work carried out at Boston University, Biomedical Engineering Department) Gene networks offer a wealth of data; this is mainly due to the genomic dimensionality rather than the samples, as the latter usually come from measurements obtained under only a few experimental conditions or time points. It is therefore a challenging task to design suitable statistical models and to develop effective reverse engineering algorithms. Network inference and reverse engineering deal basically with an inverse problem that is stimulating a great deal of biomedical research in systems biology: reconstructing gene-gene interactions from measurements obtained under specific experimental conditions. Some of the open problems calling for solutions are now listed. First, the signature of noise is pervasive in genetic networks. For instance, in pertur-bation experiments only a few genes change expression value, while most genes show either noisy or constant patterns. The measurements are usually taken at an initial time and then regularly recorded until the genes reach a steady state. During this time interval, the gene temporal patterns are subject to several kinds of fluctuations, and only some of them are induced by the perturbations, directly or indirectly as a cascade effect. Noise is thus a strong conditioning factor for the gene temporal patterns; the ones that appear smooth are usually of interest compared to those characterized by a random signature. Consequently, sharp time-localized fluctuations lead to discard many patterns and focus just on subsets of genes. An important tool is thresholding, which deals with the separation of signal from noise. It identifies gene groups by selecting the genes that are differentially expressed with regard to 4\n\nnoise-dependent patterns. Furthermore, a gene can be considered outlying if its expression value is significantly different (overor under-expressed) compared to the average expression value computed for all the genes.\n\nMethodological aspects\n\nStatistically speaking, if we are able to isolate a certain number of modes or components which represent in a compact way the systems dynamics underlying the gene network under examination, we might also try to compute statistics based on these modes through the gene sets that have been identified by them. Due to the inherent biological characterization of the estimated components, the thresh-olding step can validate or contradict the hypothesis that the observed expression values of the genes are significantly different from specific test statistics. The idea is that for each available condition (a sample point where a measurement is taken or an experiment is conducted), a genomic profile X is considered a mixture (via a certain matrix A) of unknown biological influences (modes or components) S that may have more or less impact over each specific condition. Then, feature sets are identified by linear combinations of genes which are delivered by each identified and estimated component. More formally, consider at time t an ≤− noise genomic system such as:\n\nX = AS + ≤ (1) The dimensions of the model (for k = 1) are S ∈ Rm, X ∈ Rn, and A ∈ Rn×m, and the gene expression matrix X has rows representing the different genes involved in the regulatory network under exam, while the columns list the measurements at successive conditions. The mixture signals in X represent k−dimensional vectors xi, i = 1,..., n, and the components S represent k−dimensional vectors sj, j = 1,..., m. The latter are mixed up linearly (or non-linearly) through the mixing matrix A. Both S and A are unknown variables and must be estimated from the only available information set X.This gene selection goal implies that a genetic regulatory network is a redundant system because inherently noisy, but also owing to the high-dimensionality and dependence between genes, only in part justified by biological reasons. Equivalently, the underlying system of gene dynamics is sparse; in other words, the gene-gene interaction matrix is only partially active, as some of the supposed links among genes are not supported by significant evidence. Exploiting sparsity when designing statistical inference procedures for gene network analysis as a key goal is to target just subsets of genes when measuring the effects of the experiments. And sparsity refers to dimensionality too, through the possibility of approxi-mating the truly intrinsic dimension of the input space. In turn, this may bring some valuable experimental insight too, by suggesting for instance how to calibrate the perturbations and selecting the genes expected to become targets. In order to explore these two aspects, redundancy and sparsity, clustering has been widely employed by computational biologists, but several limitations have appeared; there-fore, independent and principal component analysis (ICA and PCA, respectively) can also 5\n\nbe proposed (and compared) as examples of flexible approximation tools targeted to dimen-sionality reduction and gene feature selection.\n\nPart 2\n\nSemi-Parametric Estimation in In Vivo MR Spectroscopy (ongoing joint work with: H. Ratiney, C. Cudalbu, S. Cavassila, D. Graveron-Demilly RMN, CNRS UMR 5012, Univ. Claude Bernard LYON I-CPE, France R. de Beer, D. van Ormondt, Applied Physics, TU Delft, The Netherlands). Magnetic Resonance Spectroscopy (MRS) is a unique tool for non-invasive in vivo detection and quantitation of metabolites. This makes MRS an indispensable tool for com-bating major diseases. Although modern MRS-methods are increasingly capable of detecting and quantifying metabolites, perturbation by signals from so-called macromolecules and un-wanted metabolites poses problems. Three procedures for alleviating the problem exist: 1) Tuning of the scanner to a metabolite of interest - 'spectral editing' - can clean up the MRS signal significantly. 2) Separate (approximate) measurement of the macromolecule signal and subsequent subtraction from the MRS signal. 3) Semi-parametric estimation of the model parameters of interest from the MRS signal. Our strategy is to process directly in the measurement domain; in MRS, this is the time domain. Alternatively, one can process in the frequency or spectral domain. A main disadvantage of the latter is that it starts with estimation of the spectrum which is not trivial in the case of missing samples or non-Cartesian sampling.\n\nMethodological aspects\n\nIn semi-parametric estimation, the task is to disentangle the parametric and non-parametric parts of a signal. In MRS, the parametric part pertains to the metabolites, the non-parametric part to the macromolecules. In the measurement domain, the metabolite signal persists over time much longer than the macromolecule signal. On the other hand, during the initial period, the latter strongly dominates the former. This property is exploited to bring about the disentanglement. Through simulations, we have successfully investigated: 1) Iteration of the disentanglement procedure. 2) Derivation of Cramer-Rao Bounds (CRB) for the metabolite concentrations, taking the non-parametric macromolecule signal into account. This contributes to experimental design. Furthermore: 1) Our previous contributions did not report on iteration of the disentanglement pro-cedure. Now, we include cases where such iteration clearly helps. 2) We have devised a novel way (in MRS, at least) to estimate CRBs for the parametric part, taking the non-parametric part into account. 3) Only information about the point of time where the non-parametric part 'decays into the noise' is needed. We thus have shown that through semiparametric models it is possible to augment the applicability of MRS to Medicine, while in order to have the best possible impact we need to optimize the accuracy of the estimates for the parameters of interest. 6\n\nIn particular, one seeks an improved reliability of the lower bounds computed on pa-rameters to be estimated with a planned scan, from which it depends in turn an improved reliability of the 'experimental design', or, in other words, a better prediction of feasibility of expensive MRS-scans in the clinic. Then, one would like to address: Awhat is the predicted minimum detectable concentration of each of the 40 or so metabolites of interest with a given scanner measurement protocol? BIs the try justifiable?", "original_statement": "A.2 Capobianco, Enrico \n\nPart 1 \n\nSystems Biology: On Dimensionality Reduction and Feature Se-lection in Genomics (work carried out at Boston University, Biomedical Engineering Department) Gene networks offer a wealth of data; this is mainly due to the genomic dimensionality rather than the samples, as the latter usually come from measurements obtained under only a few experimental conditions or time points. It is therefore a challenging task to design suitable statistical models and to develop effective reverse engineering algorithms. Network inference and reverse engineering deal basically with an inverse problem that is stimulating a great deal of biomedical research in systems biology: reconstructing gene-gene interactions from measurements obtained under specific experimental conditions. Some of the open problems calling for solutions are now listed. First, the signature of noise is pervasive in genetic networks. For instance, in pertur-bation experiments only a few genes change expression value, while most genes show either noisy or constant patterns. The measurements are usually taken at an initial time and then regularly recorded until the genes reach a steady state. During this time interval, the gene temporal patterns are subject to several kinds of fluctuations, and only some of them are induced by the perturbations, directly or indirectly as a cascade effect. Noise is thus a strong conditioning factor for the gene temporal patterns; the ones that appear smooth are usually of interest compared to those characterized by a random signature. Consequently, sharp time-localized fluctuations lead to discard many patterns and focus just on subsets of genes. An important tool is thresholding, which deals with the separation of signal from noise. It identifies gene groups by selecting the genes that are differentially expressed with regard to 4\n\nnoise-dependent patterns. Furthermore, a gene can be considered outlying if its expression value is significantly different (overor under-expressed) compared to the average expression value computed for all the genes. \n\nMethodological aspects \n\nStatistically speaking, if we are able to isolate a certain number of modes or components which represent in a compact way the systems dynamics underlying the gene network under examination, we might also try to compute statistics based on these modes through the gene sets that have been identified by them. Due to the inherent biological characterization of the estimated components, the thresh-olding step can validate or contradict the hypothesis that the observed expression values of the genes are significantly different from specific test statistics. The idea is that for each available condition (a sample point where a measurement is taken or an experiment is conducted), a genomic profile X is considered a mixture (via a certain matrix A) of unknown biological influences (modes or components) S that may have more or less impact over each specific condition. Then, feature sets are identified by linear combinations of genes which are delivered by each identified and estimated component. More formally, consider at time t an ≤− noise genomic system such as: \n\nX = AS + ≤ (1) The dimensions of the model (for k = 1) are S ∈ Rm, X ∈ Rn, and A ∈ Rn×m, and the gene expression matrix X has rows representing the different genes involved in the regulatory network under exam, while the columns list the measurements at successive conditions. The mixture signals in X represent k−dimensional vectors xi, i = 1,..., n, and the components S represent k−dimensional vectors sj, j = 1,..., m. The latter are mixed up linearly (or non-linearly) through the mixing matrix A. Both S and A are unknown variables and must be estimated from the only available information set X.This gene selection goal implies that a genetic regulatory network is a redundant system because inherently noisy, but also owing to the high-dimensionality and dependence between genes, only in part justified by biological reasons. Equivalently, the underlying system of gene dynamics is sparse; in other words, the gene-gene interaction matrix is only partially active, as some of the supposed links among genes are not supported by significant evidence. Exploiting sparsity when designing statistical inference procedures for gene network analysis as a key goal is to target just subsets of genes when measuring the effects of the experiments. And sparsity refers to dimensionality too, through the possibility of approxi-mating the truly intrinsic dimension of the input space. In turn, this may bring some valuable experimental insight too, by suggesting for instance how to calibrate the perturbations and selecting the genes expected to become targets. In order to explore these two aspects, redundancy and sparsity, clustering has been widely employed by computational biologists, but several limitations have appeared; there-fore, independent and principal component analysis (ICA and PCA, respectively) can also 5\n\nbe proposed (and compared) as examples of flexible approximation tools targeted to dimen-sionality reduction and gene feature selection. \n\nPart 2 \n\nSemi-Parametric Estimation in In Vivo MR Spectroscopy (ongoing joint work with: H. Ratiney, C. Cudalbu, S. Cavassila, D. Graveron-Demilly RMN, CNRS UMR 5012, Univ. Claude Bernard LYON I-CPE, France R. de Beer, D. van Ormondt, Applied Physics, TU Delft, The Netherlands). Magnetic Resonance Spectroscopy (MRS) is a unique tool for non-invasive in vivo detection and quantitation of metabolites. This makes MRS an indispensable tool for com-bating major diseases. Although modern MRS-methods are increasingly capable of detecting and quantifying metabolites, perturbation by signals from so-called macromolecules and un-wanted metabolites poses problems. Three procedures for alleviating the problem exist: 1) Tuning of the scanner to a metabolite of interest - 'spectral editing' - can clean up the MRS signal significantly. 2) Separate (approximate) measurement of the macromolecule signal and subsequent subtraction from the MRS signal. 3) Semi-parametric estimation of the model parameters of interest from the MRS signal. Our strategy is to process directly in the measurement domain; in MRS, this is the time domain. Alternatively, one can process in the frequency or spectral domain. A main disadvantage of the latter is that it starts with estimation of the spectrum which is not trivial in the case of missing samples or non-Cartesian sampling. \n\nMethodological aspects \n\nIn semi-parametric estimation, the task is to disentangle the parametric and non-parametric parts of a signal. In MRS, the parametric part pertains to the metabolites, the non-parametric part to the macromolecules. In the measurement domain, the metabolite signal persists over time much longer than the macromolecule signal. On the other hand, during the initial period, the latter strongly dominates the former. This property is exploited to bring about the disentanglement. Through simulations, we have successfully investigated: 1) Iteration of the disentanglement procedure. 2) Derivation of Cramer-Rao Bounds (CRB) for the metabolite concentrations, taking the non-parametric macromolecule signal into account. This contributes to experimental design. Furthermore: 1) Our previous contributions did not report on iteration of the disentanglement pro-cedure. Now, we include cases where such iteration clearly helps. 2) We have devised a novel way (in MRS, at least) to estimate CRBs for the parametric part, taking the non-parametric part into account. 3) Only information about the point of time where the non-parametric part 'decays into the noise' is needed. We thus have shown that through semiparametric models it is possible to augment the applicability of MRS to Medicine, while in order to have the best possible impact we need to optimize the accuracy of the estimates for the parameters of interest. 6\n\nIn particular, one seeks an improved reliability of the lower bounds computed on pa-rameters to be estimated with a planned scan, from which it depends in turn an improved reliability of the 'experimental design', or, in other words, a better prediction of feasibility of expensive MRS-scans in the clinic. Then, one would like to address: Awhat is the predicted minimum detectable concentration of each of the 40 or so metabolites of interest with a given scanner measurement protocol? BIs the try justifiable?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The source is Section A.2, “Capobianco, Enrico,” of the AIM workshop document *Numerical Methods for Optimal Control in High Dimensions*, version dated 19 August 2005. The table of contents identifies Appendix A as “Participant Contributions.” Thus this canonical `tag: section` record is an agenda contribution, not one theorem-like open problem.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Numerical methods for optimal control in high dimensions\nSection: \nSource item: A.2\nSource URL: https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf\nCanonical location: aim-computation-notes.json notes[4]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.2 Capobianco, Enrico \\n\\nPart 1 \\n\\nSystems Biology: On Dimensionality Reduction and Feature Se-lection in Genomics (work carried out at Boston University, Biomedical Engineering Department) Gene networks offer a wealth of data; this is mainly due to the genomic dimensionality rather than the samples, as the latter usually come from measurements obtained under only a few experimental conditions or time points. It is therefore a challenging task to design suitable statistical models and to develop effective reverse engineering algorithms. Network inference and reverse engineering deal basically with an inverse problem that is stimulating a great deal of biomedical research in systems biology: reconstructing gene-gene interactions from measurements obtained under specific experimental conditions. Some of the open problems calling for solutions are now listed. First, the signature of noise is pervasive in genetic networks. For instance, in pertur-bation experiments only a few genes change expression value, while most genes show either noisy or constant patterns. The measurements are usually taken at an initial time and then regularly recorded until the genes reach a steady state. During this time interval, the gene temporal patterns are subject to several kinds of fluctuations, and only some of them are induced by the perturbations, directly or indirectly as a cascade effect. Noise is thus a strong conditioning factor for the gene temporal patterns; the ones that appear smooth are usually of interest compared to those characterized by a random signature. Consequently, sharp time-localized fluctuations lead to discard many patterns and focus just on subsets of genes. An important tool is thresholding, which deals with the separation of signal from noise. It identifies gene groups by selecting the genes that are differentially expressed with regard to 4\\n\\nnoise-dependent patterns. Furthermore, a gene can be considered outlying if its expression value is significantly different (overor under-expressed) compared to the average expression value computed for all the genes. \\n\\nMethodological aspects \\n\\nStatistically speaking, if we are able to isolate a certain number of modes or components which represent in a compact way the systems dynamics underlying the gene network under examination, we might also try to compute statistics based on these modes through the gene sets that have been identified by them. Due to the inherent biological characterization of the estimated components, the thresh-olding step can validate or contradict the hypothesis that the observed expression values of the genes are significantly different from specific test statistics. The idea is that for each available condition (a sample point where a measurement is taken or an experiment is conducted), a genomic profile X is considered a mixture (via a certain matrix A) of unknown biological influences (modes or components) S that may have more or less impact over each specific condition. Then, feature sets are identified by linear combinations of genes which are delivered by each identified and estimated component. More formally, consider at time t an ≤− noise genomic system such as: \\n\\nX = AS + ≤ (1) The dimensions of the model (for k = 1) are S ∈ Rm, X ∈ Rn, and A ∈ Rn×m, and the gene expression matrix X has rows representing the different genes involved in the regulatory network under exam, while the columns list the measurements at successive conditions. The mixture signals in X represent k−dimensional vectors xi, i = 1,..., n, and the components S represent k−dimensional vectors sj, j = 1,..., m. The latter are mixed up linearly (or non-linearly) through the mixing matrix A. Both S and A are unknown variables and must be estimated from the only available information set X.This gene selection goal implies that a genetic regulatory network is a redundant system because inherently noisy, but also owing to the high-dimensionality and dependence between genes, only in part justified by biological reasons. Equivalently, the underlying system of gene dynamics is sparse; in other words, the gene-gene interaction matrix is only partially active, as some of the supposed links among genes are not supported by significant evidence. Exploiting sparsity when designing statistical inference procedures for gene network analysis as a key goal is to target just subsets of genes when measuring the effects of the experiments. And sparsity refers to dimensionality too, through the possibility of approxi-mating the truly intrinsic dimension of the input space. In turn, this may bring some valuable experimental insight too, by suggesting for instance how to calibrate the perturbations and selecting the genes expected to become targets. In order to explore these two aspects, redundancy and sparsity, clustering has been widely employed by computational biologists, but several limitations have appeared; there-fore, independent and principal component analysis (ICA and PCA, respectively) can also 5\\n\\nbe proposed (and compared) as examples of flexible approximation tools targeted to dimen-sionality reduction and gene feature selection. \\n\\nPart 2 \\n\\nSemi-Parametric Estimation in In Vivo MR Spectroscopy (ongoing joint work with: H. Ratiney, C. Cudalbu, S. Cavassila, D. Graveron-Demilly RMN, CNRS UMR 5012, Univ. Claude Bernard LYON I-CPE, France R. de Beer, D. van Ormondt, Applied Physics, TU Delft, The Netherlands). Magnetic Resonance Spectroscopy (MRS) is a unique tool for non-invasive in vivo detection and quantitation of metabolites. This makes MRS an indispensable tool for com-bating major diseases. Although modern MRS-methods are increasingly capable of detecting and quantifying metabolites, perturbation by signals from so-called macromolecules and un-wanted metabolites poses problems. Three procedures for alleviating the problem exist: 1) Tuning of the scanner to a metabolite of interest - 'spectral editing' - can clean up the MRS signal significantly. 2) Separate (approximate) measurement of the macromolecule signal and subsequent subtraction from the MRS signal. 3) Semi-parametric estimation of the model parameters of interest from the MRS signal. Our strategy is to process directly in the measurement domain; in MRS, this is the time domain. Alternatively, one can process in the frequency or spectral domain. A main disadvantage of the latter is that it starts with estimation of the spectrum which is not trivial in the case of missing samples or non-Cartesian sampling. \\n\\nMethodological aspects \\n\\nIn semi-parametric estimation, the task is to disentangle the parametric and non-parametric parts of a signal. In MRS, the parametric part pertains to the metabolites, the non-parametric part to the macromolecules. In the measurement domain, the metabolite signal persists over time much longer than the macromolecule signal. On the other hand, during the initial period, the latter strongly dominates the former. This property is exploited to bring about the disentanglement. Through simulations, we have successfully investigated: 1) Iteration of the disentanglement procedure. 2) Derivation of Cramer-Rao Bounds (CRB) for the metabolite concentrations, taking the non-parametric macromolecule signal into account. This contributes to experimental design. Furthermore: 1) Our previous contributions did not report on iteration of the disentanglement pro-cedure. Now, we include cases where such iteration clearly helps. 2) We have devised a novel way (in MRS, at least) to estimate CRBs for the parametric part, taking the non-parametric part into account. 3) Only information about the point of time where the non-parametric part 'decays into the noise' is needed. We thus have shown that through semiparametric models it is possible to augment the applicability of MRS to Medicine, while in order to have the best possible impact we need to optimize the accuracy of the estimates for the parameters of interest. 6\\n\\nIn particular, one seeks an improved reliability of the lower bounds computed on pa-rameters to be estimated with a planned scan, from which it depends in turn an improved reliability of the 'experimental design', or, in other words, a better prediction of feasibility of expensive MRS-scans in the clinic. Then, one would like to address: Awhat is the predicted minimum detectable concentration of each of the 40 or so metabolites of interest with a given scanner measurement protocol? BIs the try justifiable?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf", "tags": [ "aim", "AIM-COMPUTATION-0005", "aim-domain:computation", "aim-workshop:optimalcontrol", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is a 2005 participant agenda spanning genomics and in vivo MR spectroscopy, not one sharply quantified problem. For a rigorous extraction of its MRS design question, the report proves that nuisance-adjusted metabolite Fisher information is the weighted residual projection J-tilde^T(I-Q_Z)J-tilde, gives the exact identifiability and singular-contrast criteria, and shows that early-time samples contribute exactly R_E^T R_E after residualization against a macromolecular nuisance space. It derives zero and strict CRB-gain conditions and a one-sided, level-and-power-dependent minimum detectable concentration formula under a known-covariance linear Gaussian model.\n\nCandidate contribution (projection criterion and experimental-design synthesis; novelty confidence low): For a prespecified MRS macromolecular decay cutoff, the residualized early metabolite Jacobian R_E=(I-Q_ZE)J_E completely determines the information retained from early samples: the increment is R_E^T R_E, it vanishes exactly when col(J_E) is contained in col(Z_E), its directional gain is the squared residual norm, and relative to an identifiable late block A the CRB of contrast c improves strictly exactly when R_E^T R_E A^{-1}c is nonzero.", "research_difficulty_suggested": "L2" }, { "id": 20001168, "problem_number": "AIM-COMPUTATION-0006", "title": "Bellman residuals, policy performance, and coverage", "statement": "A.3 de Farias, Daniela\n\nIn this workshop, I am interested in learning more about the main open issues and approaches across the different subfields related to high-dimensional optimal control. In addition, I also see the workshop as an opportunity to discuss some of the open issues and explore potential directions in my own subfield. My current research interests lie in approximate methods for large-scale stochastic dynamic programming, with particular focus on value function approximation. I would hope to see at least some activities and discussions pertaining to that topic. Some of the directions in approximate dynamic programming that I consider worth exploring are as follows:\n\n• Approaches for validation of approximate DP methods. Interest in approximate DP has been fuelled by successful application in a variety of areas ranging from finance to games and helicopter control. Yet, it is still difficult to know a priori when, how and why a given method will be successful in tackling a particular problem. In my view, there is a need for more systematic approaches to validation of algorithms beyond illustration via application to one or two specific problems. These systematic approaches could be of many different kinds, which could be discussed in more or less technical detail at the workshop. For instance, one could pursue the development theoretical guarantees of performance and errors, or a more systematic empirical comparison based on a benchmark of large-scale DP problems.\n\n• Comparison and unification of different algorithms. Approximate DP is currently represented by a conglomeration of ideas and algorithms, and it is often difficult to know how they relate to one another and which is most appropriate for any class of applications. A streamlined comparison among the approaches and potentially unifying theory would be useful developments to the field.\n\n• Systematic and/or adaptive design of approximation architectures. A central issue in approximate dynamic programming that remains largely open is how to choose the approximation architecture, which remains a problem-specific task, and approximate DP algorithms typically start with the premise that an approximation architecture has been pre-specified. More streamlined methods of approximation architecture selection together with theory characterizing optimality and convergence rates would be desirable. Alternatively, it may be worth trying to explore whether a canonical approximation architecture could be developed at least for some specific class of problems.\n\n• Performance-Oriented Approximate DP. The design, analysis and comparison of value function approximation algorithms is often based on approximation or Bellman er-rors. Both situations entail the choice of a norm whereby to measure and trade off approximation or Bellman errors across states. Little work has been done to develop 7\n\na better understanding of how different criteria capture the quality of the policies being generated. Theoretical analysis is required to identify which criteria - Bell-man or approximation error, which norms? - best reflect policy performance; at the same time, algorithms that take such criteria into account must be designed.\n\n• Alternative paradigms and connections with stochastic programming. As a more open-ended direction, one could consider whether extensions of the Markov deci-sion process paradigm, possibly combining it with stochastic programming, and/or a completely different approach could be more suitable for tackling certain classes of real-world problems without running into the curse of dimensionality.", "original_statement": "A.3 de Farias, Daniela \n\nIn this workshop, I am interested in learning more about the main open issues and approaches across the different subfields related to high-dimensional optimal control. In addition, I also see the workshop as an opportunity to discuss some of the open issues and explore potential directions in my own subfield. My current research interests lie in approximate methods for large-scale stochastic dynamic programming, with particular focus on value function approximation. I would hope to see at least some activities and discussions pertaining to that topic. Some of the directions in approximate dynamic programming that I consider worth exploring are as follows: \n\n• Approaches for validation of approximate DP methods. Interest in approximate DP has been fuelled by successful application in a variety of areas ranging from finance to games and helicopter control. Yet, it is still difficult to know a priori when, how and why a given method will be successful in tackling a particular problem. In my view, there is a need for more systematic approaches to validation of algorithms beyond illustration via application to one or two specific problems. These systematic approaches could be of many different kinds, which could be discussed in more or less technical detail at the workshop. For instance, one could pursue the development theoretical guarantees of performance and errors, or a more systematic empirical comparison based on a benchmark of large-scale DP problems. \n\n• Comparison and unification of different algorithms. Approximate DP is currently represented by a conglomeration of ideas and algorithms, and it is often difficult to know how they relate to one another and which is most appropriate for any class of applications. A streamlined comparison among the approaches and potentially unifying theory would be useful developments to the field. \n\n• Systematic and/or adaptive design of approximation architectures. A central issue in approximate dynamic programming that remains largely open is how to choose the approximation architecture, which remains a problem-specific task, and approximate DP algorithms typically start with the premise that an approximation architecture has been pre-specified. More streamlined methods of approximation architecture selection together with theory characterizing optimality and convergence rates would be desirable. Alternatively, it may be worth trying to explore whether a canonical approximation architecture could be developed at least for some specific class of problems. \n\n• Performance-Oriented Approximate DP. The design, analysis and comparison of value function approximation algorithms is often based on approximation or Bellman er-rors. Both situations entail the choice of a norm whereby to measure and trade off approximation or Bellman errors across states. Little work has been done to develop 7\n\na better understanding of how different criteria capture the quality of the policies being generated. Theoretical analysis is required to identify which criteria - Bell-man or approximation error, which norms? - best reflect policy performance; at the same time, algorithms that take such criteria into account must be designed. \n\n• Alternative paradigms and connections with stochastic programming. As a more open-ended direction, one could consider whether extensions of the Markov deci-sion process paradigm, possibly combining it with stochastic programming, and/or a completely different approach could be more suitable for tackling certain classes of real-world problems without running into the curse of dimensionality.", "clean_statement": "A.3 de Farias, Daniela\n\nIn this workshop, I am interested in learning more about the main open issues and approaches across the different subfields related to high-dimensional optimal control. In addition, I also see the workshop as an opportunity to discuss some of the open issues and explore potential directions in my own subfield. My current research interests lie in approximate methods for large-scale stochastic dynamic programming, with particular focus on value function approximation. I would hope to see at least some activities and discussions pertaining to that topic. Some of the directions in approximate dynamic programming that I consider worth exploring are as follows:\n\n• Approaches for validation of approximate DP methods. Interest in approximate DP has been fuelled by successful application in a variety of areas ranging from finance to games and helicopter control. Yet, it is still difficult to know a priori when, how and why a given method will be successful in tackling a particular problem. In my view, there is a need for more systematic approaches to validation of algorithms beyond illustration via application to one or two specific problems. These systematic approaches could be of many different kinds, which could be discussed in more or less technical detail at the workshop. For instance, one could pursue the development theoretical guarantees of performance and errors, or a more systematic empirical comparison based on a benchmark of large-scale DP problems.\n\n• Comparison and unification of different algorithms. Approximate DP is currently represented by a conglomeration of ideas and algorithms, and it is often difficult to know how they relate to one another and which is most appropriate for any class of applications. A streamlined comparison among the approaches and potentially unifying theory would be useful developments to the field.\n\n• Systematic and/or adaptive design of approximation architectures. A central issue in approximate dynamic programming that remains largely open is how to choose the approximation architecture, which remains a problem-specific task, and approximate DP algorithms typically start with the premise that an approximation architecture has been pre-specified. More streamlined methods of approximation architecture selection together with theory characterizing optimality and convergence rates would be desirable. Alternatively, it may be worth trying to explore whether a canonical approximation architecture could be developed at least for some specific class of problems.\n\n• Performance-Oriented Approximate DP. The design, analysis and comparison of value function approximation algorithms is often based on approximation or Bellman er-rors. Both situations entail the choice of a norm whereby to measure and trade off approximation or Bellman errors across states. Little work has been done to develop 7\n\na better understanding of how different criteria capture the quality of the policies being generated. Theoretical analysis is required to identify which criteria - Bell-man or approximation error, which norms? - best reflect policy performance; at the same time, algorithms that take such criteria into account must be designed.\n\n• Alternative paradigms and connections with stochastic programming. As a more open-ended direction, one could consider whether extensions of the Markov deci-sion process paradigm, possibly combining it with stochastic programming, and/or a completely different approach could be more suitable for tackling certain classes of real-world problems without running into the curse of dimensionality.", "statement_status": "exact", "statement_verification": "This record is item A.3, Daniela de Farias's participant contribution to the AIM workshop *Numerical methods for optimal control in high dimensions* (August 29--September 2, 2005). It is a research agenda rather than a single quantified open problem. The contribution asks for progress on five directions in approximate dynamic programming (ADP):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Numerical methods for optimal control in high dimensions\nSection: \nSource item: A.3\nSource URL: https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf\nCanonical location: aim-computation-notes.json notes[5]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.3 de Farias, Daniela \\n\\nIn this workshop, I am interested in learning more about the main open issues and approaches across the different subfields related to high-dimensional optimal control. In addition, I also see the workshop as an opportunity to discuss some of the open issues and explore potential directions in my own subfield. My current research interests lie in approximate methods for large-scale stochastic dynamic programming, with particular focus on value function approximation. I would hope to see at least some activities and discussions pertaining to that topic. Some of the directions in approximate dynamic programming that I consider worth exploring are as follows: \\n\\n• Approaches for validation of approximate DP methods. Interest in approximate DP has been fuelled by successful application in a variety of areas ranging from finance to games and helicopter control. Yet, it is still difficult to know a priori when, how and why a given method will be successful in tackling a particular problem. In my view, there is a need for more systematic approaches to validation of algorithms beyond illustration via application to one or two specific problems. These systematic approaches could be of many different kinds, which could be discussed in more or less technical detail at the workshop. For instance, one could pursue the development theoretical guarantees of performance and errors, or a more systematic empirical comparison based on a benchmark of large-scale DP problems. \\n\\n• Comparison and unification of different algorithms. Approximate DP is currently represented by a conglomeration of ideas and algorithms, and it is often difficult to know how they relate to one another and which is most appropriate for any class of applications. A streamlined comparison among the approaches and potentially unifying theory would be useful developments to the field. \\n\\n• Systematic and/or adaptive design of approximation architectures. A central issue in approximate dynamic programming that remains largely open is how to choose the approximation architecture, which remains a problem-specific task, and approximate DP algorithms typically start with the premise that an approximation architecture has been pre-specified. More streamlined methods of approximation architecture selection together with theory characterizing optimality and convergence rates would be desirable. Alternatively, it may be worth trying to explore whether a canonical approximation architecture could be developed at least for some specific class of problems. \\n\\n• Performance-Oriented Approximate DP. The design, analysis and comparison of value function approximation algorithms is often based on approximation or Bellman er-rors. Both situations entail the choice of a norm whereby to measure and trade off approximation or Bellman errors across states. Little work has been done to develop 7\\n\\na better understanding of how different criteria capture the quality of the policies being generated. Theoretical analysis is required to identify which criteria - Bell-man or approximation error, which norms? - best reflect policy performance; at the same time, algorithms that take such criteria into account must be designed. \\n\\n• Alternative paradigms and connections with stochastic programming. As a more open-ended direction, one could consider whether extensions of the Markov deci-sion process paradigm, possibly combining it with stochastic programming, and/or a completely different approach could be more suitable for tackling certain classes of real-world problems without running into the curse of dimensionality.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf", "tags": [ "aim", "AIM-COMPUTATION-0006", "aim-domain:computation", "aim-workshop:optimalcontrol", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source is a five-part participant research agenda rather than a quantified open problem. For its performance-oriented approximate-DP direction, a proved two-occupancy certificate bounds the loss of a value-greedy policy by the absolute Bellman residual under the normalized discounted occupancies of the optimal and returned policies. Weighted Holder then exposes exact density-ratio dependence. A proved two-state family with bounded rewards and approximate values, strict greediness, and ergodicity under every stationary policy shows that an arbitrary finite-p sampling norm can tend to zero while policy loss remains fixed; its density ratio has the necessary inverse-power scaling, and the occupancy certificate is asymptotically sharp on the family.\n\nCandidate contribution (worked_counterexample_and_certificate; novelty confidence low): A single exact two-state family simultaneously has bounded rewards and approximate values, a unique erroneous greedy action, positive transition probabilities making every stationary-policy chain irreducible and aperiodic, residual norm eta^(1/p) under a full-support sampling distribution, the matching eta^(-1/p) density-ratio requirement, explicit normalized occupancies, and asymptotic sharpness of the paired optimal/returned-occupancy residual certificate.", "research_difficulty_suggested": "L2" }, { "id": 20001169, "problem_number": "AIM-COMPUTATION-0007", "title": "A finite-sample Rao-Blackwell diagnostic between Kalman and particle filtering", "statement": "A.4 Haykin, Simon\n\nFor my talk, I will do the following: 1. Provide a brief background on the classical Kalman filtering algorithm, emphasizing its virtues and limitations. 2. Describe particle filtering, rooted in Bayesian estimation and Monte Carlo simula-tion. Here again, I will highlight the virtues and limitations of this second approach. 3. The background would then be set for describing a new approach that has the potential for making a significant difference to the literature. In particular, I will pose a nonlinear recursive problem that is currently the stumbling block. As such, to progress further with this approach, we have to solve this problem. It could be that the solution I am looking for will pop up in the supplemental technical breakout session following my talk on the second day.", "original_statement": "A.4 Haykin, Simon \n\nFor my talk, I will do the following: 1. Provide a brief background on the classical Kalman filtering algorithm, emphasizing its virtues and limitations. 2. Describe particle filtering, rooted in Bayesian estimation and Monte Carlo simula-tion. Here again, I will highlight the virtues and limitations of this second approach. 3. The background would then be set for describing a new approach that has the potential for making a significant difference to the literature. In particular, I will pose a nonlinear recursive problem that is currently the stumbling block. As such, to progress further with this approach, we have to solve this problem. It could be that the solution I am looking for will pop up in the supplemental technical breakout session following my talk on the second day.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The source is Appendix A.4, “Haykin, Simon,” in the AIM document *Numerical Methods for Optimal Control in High Dimensions*, version dated 19 August 2005. Appendix A is explicitly a collection of participant contributions. The complete A.4 text, with line wrapping normalized, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Numerical methods for optimal control in high dimensions\nSection: \nSource item: A.4\nSource URL: https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf\nCanonical location: aim-computation-notes.json notes[6]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.4 Haykin, Simon \\n\\nFor my talk, I will do the following: 1. Provide a brief background on the classical Kalman filtering algorithm, emphasizing its virtues and limitations. 2. Describe particle filtering, rooted in Bayesian estimation and Monte Carlo simula-tion. Here again, I will highlight the virtues and limitations of this second approach. 3. The background would then be set for describing a new approach that has the potential for making a significant difference to the literature. In particular, I will pose a nonlinear recursive problem that is currently the stumbling block. As such, to progress further with this approach, we have to solve this problem. It could be that the solution I am looking for will pop up in the supplemental technical breakout session following my talk on the second day.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf", "tags": [ "aim", "AIM-COMPUTATION-0007", "aim-domain:computation", "aim-workshop:optimalcontrol", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The official PDF confirms that A.4 is only a talk agenda: it promises that a nonlinear recursion will be posed orally but does not state it, and A.5 begins immediately afterward. For a rigorous connection between the named Kalman and particle paradigms, the report proves a finite-N matched-outer-particle Rao-Blackwell identity. Conditioning on observations, nonlinear particles, normalized weights, resampling ancestry, and exact per-particle Kalman moments, analytic integration of the linear Gaussian substate preserves finite-N expectation and bias while removing exactly the conditional inner-sampling variance. For a linear target the realized MSE gain is sum_i W_i^2 b_i^T P_i b_i, with exact equality, strictness, effective-sample-size, unbiased exact-posterior, and consistency statements.\n\nCandidate contribution (finite-sample variance diagnostic; novelty confidence low): For a matched outer particle system whose normalized weights do not depend on subsequently sampled conditionally Gaussian substates, Rao-Blackwellizing a linear target removes exactly D_N=sum_i W_i^2 b_i^T P_i b_i conditional MSE; D_N is zero exactly when every active P_i^{1/2}b_i vanishes and equals v/ESS_2 when the conditional variances are constant.", "research_difficulty_suggested": "L2" }, { "id": 20001170, "problem_number": "AIM-COMPUTATION-0008", "title": "Exact support and trace-controlled ellipsoidal reachability", "statement": "A.5 Kurzhanski, Alexander\n\nAmong the central topics of modern control theory are problems of control synthesis for complex systems. These include problems of control under uncertainty and conflict as well as those under complex constraints. The solutions to such problems are well formalizable within Hamiltonian techniques and Dynamic Programming ideas, being as a rule reduced to the solution of HJB and HJBI partial differential equations or/and variational inequalities. An example where such methods are successfully applied are problems of forward and backward reachability for uncertain systems with further passage to control synthesis, safety verification, measurement feedback and related issues. The advent of HJB methods for continuous systems is due to the introduction of appropriate theories for generalized viscosity-type solutions and their equivalents. Another important topic is the solution to problems in dynamics and control for set-valued systems in terms of Hamiltonian formalism and Dynamic Programming. At present there is a sharp necessity for a breakthrough in numerical methods for solving equations and variational inequalities of the HJB-HJBI types motivated by problems of the above. There is also an interest in numerical methods for the comparison principle in HJB theory, which allows to calculate upper and lower bounds to exact solutions of HJB equations. Another perspective is the calculation of set-valued solutions to problems in evolution dynamics, estimation and control. Together with my colleagues we have developed an ellipsoidal calculus aimed at such problems and closely connected to HJB theory. The calculus is applicable to systems with original linear structure, but its methods allow effective calculations in high dimensions as well as computer animation in high dimensions through computer windows. For nonlinear systems some comparison methods were indicated. 8\n\nThe discussion of such methods as well as those for stochastic dynamics are within my primary interests at the AIM workshop.", "original_statement": "A.5 Kurzhanski, Alexander \n\nAmong the central topics of modern control theory are problems of control synthesis for complex systems. These include problems of control under uncertainty and conflict as well as those under complex constraints. The solutions to such problems are well formalizable within Hamiltonian techniques and Dynamic Programming ideas, being as a rule reduced to the solution of HJB and HJBI partial differential equations or/and variational inequalities. An example where such methods are successfully applied are problems of forward and backward reachability for uncertain systems with further passage to control synthesis, safety verification, measurement feedback and related issues. The advent of HJB methods for continuous systems is due to the introduction of appropriate theories for generalized viscosity-type solutions and their equivalents. Another important topic is the solution to problems in dynamics and control for set-valued systems in terms of Hamiltonian formalism and Dynamic Programming. At present there is a sharp necessity for a breakthrough in numerical methods for solving equations and variational inequalities of the HJB-HJBI types motivated by problems of the above. There is also an interest in numerical methods for the comparison principle in HJB theory, which allows to calculate upper and lower bounds to exact solutions of HJB equations. Another perspective is the calculation of set-valued solutions to problems in evolution dynamics, estimation and control. Together with my colleagues we have developed an ellipsoidal calculus aimed at such problems and closely connected to HJB theory. The calculus is applicable to systems with original linear structure, but its methods allow effective calculations in high dimensions as well as computer animation in high dimensions through computer windows. For nonlinear systems some comparison methods were indicated. 8\n\nThe discussion of such methods as well as those for stochastic dynamics are within my primary interests at the AIM workshop.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Section A.5, “Kurzhanski, Alexander,” in the participant contributions to the AIM workshop *Numerical methods for optimal control in high dimensions*. It is a research agenda, not a numbered conjecture. It identifies:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Numerical methods for optimal control in high dimensions\nSection: \nSource item: A.5\nSource URL: https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf\nCanonical location: aim-computation-notes.json notes[7]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.5 Kurzhanski, Alexander \\n\\nAmong the central topics of modern control theory are problems of control synthesis for complex systems. These include problems of control under uncertainty and conflict as well as those under complex constraints. The solutions to such problems are well formalizable within Hamiltonian techniques and Dynamic Programming ideas, being as a rule reduced to the solution of HJB and HJBI partial differential equations or/and variational inequalities. An example where such methods are successfully applied are problems of forward and backward reachability for uncertain systems with further passage to control synthesis, safety verification, measurement feedback and related issues. The advent of HJB methods for continuous systems is due to the introduction of appropriate theories for generalized viscosity-type solutions and their equivalents. Another important topic is the solution to problems in dynamics and control for set-valued systems in terms of Hamiltonian formalism and Dynamic Programming. At present there is a sharp necessity for a breakthrough in numerical methods for solving equations and variational inequalities of the HJB-HJBI types motivated by problems of the above. There is also an interest in numerical methods for the comparison principle in HJB theory, which allows to calculate upper and lower bounds to exact solutions of HJB equations. Another perspective is the calculation of set-valued solutions to problems in evolution dynamics, estimation and control. Together with my colleagues we have developed an ellipsoidal calculus aimed at such problems and closely connected to HJB theory. The calculus is applicable to systems with original linear structure, but its methods allow effective calculations in high dimensions as well as computer animation in high dimensions through computer windows. For nonlinear systems some comparison methods were indicated. 8\\n\\nThe discussion of such methods as well as those for stochastic dynamics are within my primary interests at the AIM workshop.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf", "tags": [ "aim", "AIM-COMPUTATION-0008", "aim-domain:computation", "aim-workshop:optimalcontrol", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source is a participant research agenda rather than a quantified problem; the isolated numeral 8 in the extraction is the printed page-8 footer. For a linear time-varying system with a possibly singular ellipsoidal initial set and measurable pointwise ellipsoidal inputs, the terminal reachable set is proved compact and convex and its support is exactly l·c_T+||F_0^T l||+∫||G(s)^T l||ds. Weighted Cauchy-Schwarz gives a continuum family of computable outer ellipsoids, with the trace-minimizing weight in that declared family proportional to the square root of the propagated shape trace. Exact tight directions, the standard two-ellipsoid formula, polyhedral safety certificates, and a Lipschitz midpoint-quadrature safety margin are proved.\n\nCandidate contribution (support_to_safety_synthesis; novelty confidence low): Under measurable time-varying data and singular shape matrices, the exact reachable support, the trace-minimizing member of an explicit continuous weighted-Cauchy outer-ellipsoid family, an if-and-only-if directionwise tightness criterion, and a certified halfspace quadrature margin form one rigorously proved support-to-safety package.", "research_difficulty_suggested": "L2" }, { "id": 20001171, "problem_number": "AIM-COMPUTATION-0009", "title": "From an approximate HJB value to implemented control", "statement": "A.6 Kushner, Harold\n\nI have been interested in numerical methods in stochastic (and deterministic) control for several decades, and have developed the current methods of choice for the stochastic problem (which is often the best method for determinstic problems as well, as seen in applications to complex variational problems in out book). These cover virtually all of the cost functions and systems of current interest. But, in applications, there is a serious dimensioanlity problem, beyond 4 dimensions. Various approximation schemes (say, Q-learning or neural nets) have been hyped, but with llittle to show, outside of some very special cases. Additionally, solving the Bellman or HJ-Bellman equation is not enough. One wants the control, and to investigate the effects of implemetation of reasonable approximations. The visualization problem, even in 3 dimensions, is formidable.", "original_statement": "A.6 Kushner, Harold \n\nI have been interested in numerical methods in stochastic (and deterministic) control for several decades, and have developed the current methods of choice for the stochastic problem (which is often the best method for determinstic problems as well, as seen in applications to complex variational problems in out book). These cover virtually all of the cost functions and systems of current interest. But, in applications, there is a serious dimensioanlity problem, beyond 4 dimensions. Various approximation schemes (say, Q-learning or neural nets) have been hyped, but with llittle to show, outside of some very special cases. Additionally, solving the Bellman or HJ-Bellman equation is not enough. One wants the control, and to investigate the effects of implemetation of reasonable approximations. The visualization problem, even in 3 dimensions, is formidable.", "clean_statement": "A.6 Kushner, Harold\n\nI have been interested in numerical methods in stochastic (and deterministic) control for several decades, and have developed the current methods of choice for the stochastic problem (which is often the best method for determinstic problems as well, as seen in applications to complex variational problems in out book). These cover virtually all of the cost functions and systems of current interest. But, in applications, there is a serious dimensioanlity problem, beyond 4 dimensions. Various approximation schemes (say, Q-learning or neural nets) have been hyped, but with llittle to show, outside of some very special cases. Additionally, solving the Bellman or HJ-Bellman equation is not enough. One wants the control, and to investigate the effects of implemetation of reasonable approximations. The visualization problem, even in 3 dimensions, is formidable.", "statement_status": "exact", "statement_verification": "This record is item A.6, Harold Kushner's participant contribution to the AIM workshop *Numerical methods for optimal control in high dimensions* (August 29--September 2, 2005). The official PDF, version dated August 19, 2005, contains a short position statement rather than a single quantified conjecture. Kushner says that established numerical methods cover broad stochastic and deterministic control classes but face a severe dimensionality barrier beyond four dimensions; he is skeptical of then-current claims for Q-learning and neural-network approximations; and he stresses that solving a Bellman or Hamilton--Jacobi--Bellman (HJB) equation is insufficient because one must extract a control and understand what happens when an approximation is implemented. He also flags visualization in three dimensions as formidable.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Numerical methods for optimal control in high dimensions\nSection: \nSource item: A.6\nSource URL: https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf\nCanonical location: aim-computation-notes.json notes[8]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.6 Kushner, Harold \\n\\nI have been interested in numerical methods in stochastic (and deterministic) control for several decades, and have developed the current methods of choice for the stochastic problem (which is often the best method for determinstic problems as well, as seen in applications to complex variational problems in out book). These cover virtually all of the cost functions and systems of current interest. But, in applications, there is a serious dimensioanlity problem, beyond 4 dimensions. Various approximation schemes (say, Q-learning or neural nets) have been hyped, but with llittle to show, outside of some very special cases. Additionally, solving the Bellman or HJ-Bellman equation is not enough. One wants the control, and to investigate the effects of implemetation of reasonable approximations. The visualization problem, even in 3 dimensions, is formidable.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf", "tags": [ "aim", "AIM-COMPUTATION-0009", "aim-domain:computation", "aim-workshop:optimalcontrol", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source is a historical participant position statement, not a quantified open problem. For a discounted control-affine diffusion with control-independent noise and unconstrained strictly quadratic control cost, and under explicit classical verification, admissibility, nonexplosion, integrability, and transversality assumptions, the loss of the feedback extracted from a smooth approximate HJB value equals exactly one half of the discounted energy of its actuation-projected gradient error. In normalized implemented-policy occupancy form the loss is the projected gradient energy divided by twice the discount, and it also equals the initial value error plus the occupancy-mean approximate-HJB residual divided by the discount. A scalar linear-quadratic family verifies sharpness, while a high-frequency perturbation proves that uniform value accuracy alone does not ensure pointwise feedback accuracy.\n\nCandidate contribution (exact_identity_and_diagnostic; novelty confidence low): The exact three-way implementation balance packages policy suboptimality, normalized implemented-occupancy actuation-projected value-gradient energy, and initial value error plus closed-loop HJB residual mean into one equality; together with constant-shift cancellation, underactuated null directions, a local uniform-value obstruction, and a sharp scalar LQ calibration, it yields an explicit falsifiable implementation audit.", "research_difficulty_suggested": "L2" }, { "id": 20001172, "problem_number": "AIM-COMPUTATION-0010", "title": "Certified weak-coupling tubes for structured reachability", "statement": "A.7 Mitchell, Ian\n\nThe solution to the Hamilton-Jacobi(-Bellman)(-Isaacs) partial differential equation (HJ PDE) provides the optimal value function (eg cost to go) for bounded input optimal control or zero-sum differential game problems respectively. These equations arise in many disparate fields, and my personal interest lies in how they can be used to approximate the reachable sets of continuous dynamic systems in order to verify and/or synthesize provably safe control policies. For these purposes, it may be possible to relax the goal of a convergent approximation of the solution of the PDE. Instead, we seek guaranteed under or over approximations of the solutions; for example, an underapproximation of the solution might overapproximate the unsafe reachable set, which might thereby generate false negative verifications but no false positives. With or without such relaxations, is it possible to take advantage of structure in the system dynamics to reduce the typically exponential growth in computational cost with system dimension? For example, is it possible to take advantage of weak or linear coupling between certain components to reduce the degrees of freedom needed by an approximation, perhaps through partial parameterization? Or another approach: Can sampling methods be applied to produce quantifiable probabilistic verifications? In addition to these questions related to the application area of verification, I am inter-ested in the class of algorithms designed to approximate solutions of HJ PDEs - often broadly called \"level set methods.\" This interest includes both the time-dependent and stationary (or static) versions of the equation, as well as issues related to the use of such numerical approximations in the control of robotic systems.", "original_statement": "A.7 Mitchell, Ian \n\nThe solution to the Hamilton-Jacobi(-Bellman)(-Isaacs) partial differential equation (HJ PDE) provides the optimal value function (eg cost to go) for bounded input optimal control or zero-sum differential game problems respectively. These equations arise in many disparate fields, and my personal interest lies in how they can be used to approximate the reachable sets of continuous dynamic systems in order to verify and/or synthesize provably safe control policies. For these purposes, it may be possible to relax the goal of a convergent approximation of the solution of the PDE. Instead, we seek guaranteed under or over approximations of the solutions; for example, an underapproximation of the solution might overapproximate the unsafe reachable set, which might thereby generate false negative verifications but no false positives. With or without such relaxations, is it possible to take advantage of structure in the system dynamics to reduce the typically exponential growth in computational cost with system dimension? For example, is it possible to take advantage of weak or linear coupling between certain components to reduce the degrees of freedom needed by an approximation, perhaps through partial parameterization? Or another approach: Can sampling methods be applied to produce quantifiable probabilistic verifications? In addition to these questions related to the application area of verification, I am inter-ested in the class of algorithms designed to approximate solutions of HJ PDEs - often broadly called \"level set methods.\" This interest includes both the time-dependent and stationary (or static) versions of the equation, as well as issues related to the use of such numerical approximations in the control of robotic systems.", "clean_statement": "A.7 Mitchell, Ian\n\nThe solution to the Hamilton-Jacobi(-Bellman)(-Isaacs) partial differential equation (HJ PDE) provides the optimal value function (eg cost to go) for bounded input optimal control or zero-sum differential game problems respectively. These equations arise in many disparate fields, and my personal interest lies in how they can be used to approximate the reachable sets of continuous dynamic systems in order to verify and/or synthesize provably safe control policies. For these purposes, it may be possible to relax the goal of a convergent approximation of the solution of the PDE. Instead, we seek guaranteed under or over approximations of the solutions; for example, an underapproximation of the solution might overapproximate the unsafe reachable set, which might thereby generate false negative verifications but no false positives. With or without such relaxations, is it possible to take advantage of structure in the system dynamics to reduce the typically exponential growth in computational cost with system dimension? For example, is it possible to take advantage of weak or linear coupling between certain components to reduce the degrees of freedom needed by an approximation, perhaps through partial parameterization? Or another approach: Can sampling methods be applied to produce quantifiable probabilistic verifications? In addition to these questions related to the application area of verification, I am inter-ested in the class of algorithms designed to approximate solutions of HJ PDEs - often broadly called \"level set methods.\" This interest includes both the time-dependent and stationary (or static) versions of the equation, as well as issues related to the use of such numerical approximations in the control of robotic systems.", "statement_status": "exact", "statement_verification": "This record is item A.7, attributed to Ian Mitchell, in the 2005 AIM workshop list *Numerical methods for optimal control in high dimensions*. The source is a research agenda rather than a single formally quantified problem. The official PDF was checked directly; A.7 occupies page 7 and ends immediately before A.8. The JSON's `inter-ested` is only line-break hyphenation. The source's `eg` is retained in the extraction and is naturally read as “e.g.”", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Numerical methods for optimal control in high dimensions\nSection: \nSource item: A.7\nSource URL: https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf\nCanonical location: aim-computation-notes.json notes[9]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.7 Mitchell, Ian \\n\\nThe solution to the Hamilton-Jacobi(-Bellman)(-Isaacs) partial differential equation (HJ PDE) provides the optimal value function (eg cost to go) for bounded input optimal control or zero-sum differential game problems respectively. These equations arise in many disparate fields, and my personal interest lies in how they can be used to approximate the reachable sets of continuous dynamic systems in order to verify and/or synthesize provably safe control policies. For these purposes, it may be possible to relax the goal of a convergent approximation of the solution of the PDE. Instead, we seek guaranteed under or over approximations of the solutions; for example, an underapproximation of the solution might overapproximate the unsafe reachable set, which might thereby generate false negative verifications but no false positives. With or without such relaxations, is it possible to take advantage of structure in the system dynamics to reduce the typically exponential growth in computational cost with system dimension? For example, is it possible to take advantage of weak or linear coupling between certain components to reduce the degrees of freedom needed by an approximation, perhaps through partial parameterization? Or another approach: Can sampling methods be applied to produce quantifiable probabilistic verifications? In addition to these questions related to the application area of verification, I am inter-ested in the class of algorithms designed to approximate solutions of HJ PDEs - often broadly called \\\"level set methods.\\\" This interest includes both the time-dependent and stationary (or static) versions of the equation, as well as issues related to the use of such numerical approximations in the control of robotic systems.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf", "tags": [ "aim", "AIM-COMPUTATION-0010", "aim-domain:computation", "aim-workshop:optimalcontrol", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Although A.7 is a multi-question research agenda rather than a single proposition, this attempt proves a concrete partial theorem addressing its weak-coupling and one-sided-verification branch. If the blockwise mismatch between a true system and a structured surrogate is bounded by a nonnegative matrix comparison inequality, then an m-dimensional positive linear ODE gives a guaranteed block-error tube. The true reachable set lies in the surrogate reachable set plus that block tube, and a block-Lipschitz or exact dual-norm margin turns the inclusion into a sound safety certificate.\n\nCandidate contribution (comparison theorem and safety certificate; novelty confidence low): Candidate contribution: the explicit synthesis of an m-dimensional positive block-error comparison ODE, a time-indexed Minkowski outer reachable-set inclusion, and an exact dual-block-norm halfspace safety margin converts independently computed structured surrogate reachable sets into a no-false-safe certificate without reconstructing the full HJ value function.", "research_difficulty_suggested": "L2" }, { "id": 20001173, "problem_number": "AIM-COMPUTATION-0011", "title": "Finite-switch implementation of chattering controls", "statement": "A.8 Oberman, Adam\n\nI am interested in finite difference methods for high dimensional control and stochastic control problems. The types of general questions I am interested in are in 2 groups: - we know in theory that we can solve the PDE for the value function. How can we do this in practice for medium dimensional (3-4-5) problems? - Given an approximate numerical solution of the PDE for 9\n\nthe value function, it isn't trivial to implement a near-optimal control strategy. How do we best do this and can we understand how far from optimal these approximations are? I am curious about how much of a gap there is between theory and practice for nonlinear control problems. Given a control problem, we can write down the PDE for the value function, and we know that this PDE has unique viscosity solutions. From solutions of the PDE, we can then obtain the control. What I would like to know is the following: Suppose we have an approximate solution of the PDE, and we use this approximation to derive a control. How far from optimal will the resulting solution be? Also: suppose the optimal solution swtiches strategy infinitely many times. (This happens even in well-known examples, like stabilizing the linear/nonliner pendulum). Even supposing we had the exact solution, how far off are we from optimal if we approximate the control by one which only switches finitely many times, (which is all that we can do in practice).", "original_statement": "A.8 Oberman, Adam \n\nI am interested in finite difference methods for high dimensional control and stochastic control problems. The types of general questions I am interested in are in 2 groups: - we know in theory that we can solve the PDE for the value function. How can we do this in practice for medium dimensional (3-4-5) problems? - Given an approximate numerical solution of the PDE for 9\n\nthe value function, it isn't trivial to implement a near-optimal control strategy. How do we best do this and can we understand how far from optimal these approximations are? I am curious about how much of a gap there is between theory and practice for nonlinear control problems. Given a control problem, we can write down the PDE for the value function, and we know that this PDE has unique viscosity solutions. From solutions of the PDE, we can then obtain the control. What I would like to know is the following: Suppose we have an approximate solution of the PDE, and we use this approximation to derive a control. How far from optimal will the resulting solution be? Also: suppose the optimal solution swtiches strategy infinitely many times. (This happens even in well-known examples, like stabilizing the linear/nonliner pendulum). Even supposing we had the exact solution, how far off are we from optimal if we approximate the control by one which only switches finitely many times, (which is all that we can do in practice).", "clean_statement": "A.8 Oberman, Adam\n\nI am interested in finite difference methods for high dimensional control and stochastic control problems. The types of general questions I am interested in are in 2 groups: - we know in theory that we can solve the PDE for the value function. How can we do this in practice for medium dimensional (3-4-5) problems? - Given an approximate numerical solution of the PDE for 9\n\nthe value function, it isn't trivial to implement a near-optimal control strategy. How do we best do this and can we understand how far from optimal these approximations are? I am curious about how much of a gap there is between theory and practice for nonlinear control problems. Given a control problem, we can write down the PDE for the value function, and we know that this PDE has unique viscosity solutions. From solutions of the PDE, we can then obtain the control. What I would like to know is the following: Suppose we have an approximate solution of the PDE, and we use this approximation to derive a control. How far from optimal will the resulting solution be? Also: suppose the optimal solution swtiches strategy infinitely many times. (This happens even in well-known examples, like stabilizing the linear/nonliner pendulum). Even supposing we had the exact solution, how far off are we from optimal if we approximate the control by one which only switches finitely many times, (which is all that we can do in practice).", "statement_status": "exact", "statement_verification": "This is item A.8, contributed by Adam Oberman to the AIM workshop *Numerical methods for optimal control in high dimensions*. The record asks three broad questions rather than posing a single theorem:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Numerical methods for optimal control in high dimensions\nSection: \nSource item: A.8\nSource URL: https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf\nCanonical location: aim-computation-notes.json notes[10]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.8 Oberman, Adam \\n\\nI am interested in finite difference methods for high dimensional control and stochastic control problems. The types of general questions I am interested in are in 2 groups: - we know in theory that we can solve the PDE for the value function. How can we do this in practice for medium dimensional (3-4-5) problems? - Given an approximate numerical solution of the PDE for 9\\n\\nthe value function, it isn't trivial to implement a near-optimal control strategy. How do we best do this and can we understand how far from optimal these approximations are? I am curious about how much of a gap there is between theory and practice for nonlinear control problems. Given a control problem, we can write down the PDE for the value function, and we know that this PDE has unique viscosity solutions. From solutions of the PDE, we can then obtain the control. What I would like to know is the following: Suppose we have an approximate solution of the PDE, and we use this approximation to derive a control. How far from optimal will the resulting solution be? Also: suppose the optimal solution swtiches strategy infinitely many times. (This happens even in well-known examples, like stabilizing the linear/nonliner pendulum). Even supposing we had the exact solution, how far off are we from optimal if we approximate the control by one which only switches finitely many times, (which is all that we can do in practice).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf", "tags": [ "aim", "AIM-COMPUTATION-0011", "aim-domain:computation", "aim-workshop:optimalcontrol", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For finite-action deterministic finite-horizon ODE control with uniformly Lipschitz dynamics and costs, replacing any measurable control u by v on a time set of measure eta changes the state by at most M_f exp(L_f H) eta and the Bolza cost by at most C_H eta, where C_H=M_l+(H L_l+L_g)M_f exp(L_f H). Finite-switch controls are dense in this mismatch metric, so their cost infimum equals the unrestricted value. Conversely, on a fixed state-free two-action system, an explicit family of bounded increasingly oscillatory time-dependent costs forces every N-switch control to lose at least 1/2-(N+1)/(4K), proving that no uniform switch-count-only rate exists over merely measurable cost data. A smooth HJB residual bound composes additively with the implementation term.\n\nCandidate contribution (performance_bound_and_obstruction; novelty confidence low): Candidate contribution: the explicit certificate J(v)-V <= policy-extraction loss + [M_l+(H L_l+L_g)M_f exp(L_f H)] eta(u,v), paired with the alternating-cost lower bound inf_{v with at most N switches} J_K(v) >= 1/2-(N+1)/(4K), identifies mismatch measure as a sufficient implementation modulus and proves switch count alone is not a uniform substitute." }, { "id": 20001174, "problem_number": "AIM-COMPUTATION-0012", "title": "An intrinsic-dimension certificate for projected Hamilton--Jacobi equations", "statement": "A.9 Osher, Stanley\n\nI am interested in solving high dimensional Hamilton-Jacobi equations numerically, especially those arising in control theory and involving the level set method.", "original_statement": "A.9 Osher, Stanley \n\nI am interested in solving high dimensional Hamilton-Jacobi equations numerically, especially those arising in control theory and involving the level set method.", "clean_statement": "A.9 Osher, Stanley\n\nI am interested in solving high dimensional Hamilton-Jacobi equations numerically, especially those arising in control theory and involving the level set method.", "statement_status": "exact", "statement_verification": "The canonical record is item A.9 of the AIM workshop notes *Numerical methods for optimal control in high dimensions* (Stanley Osher):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Numerical methods for optimal control in high dimensions\nSection: \nSource item: A.9\nSource URL: https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf\nCanonical location: aim-computation-notes.json notes[11]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.9 Osher, Stanley \\n\\nI am interested in solving high dimensional Hamilton-Jacobi equations numerically, especially those arising in control theory and involving the level set method.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf", "tags": [ "aim", "AIM-COMPUTATION-0012", "aim-domain:computation", "aim-workshop:optimalcontrol", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Although AIM item A.9 is a broad research agenda rather than a formal problem, it admits a rigorous structural contribution: for a row-orthonormal projection P, the lift of a k-dimensional viscosity solution approximates the n-dimensional Hamilton--Jacobi solution within epsilon times t, where epsilon tests the full and reduced Hamiltonians only on covectors P^T q actually sampled by the lift. This yields exact reduction when epsilon is zero, certified level-set inclusions, additive reduced-scheme error, and the control bound epsilon <= L delta_f + delta_ell. A scalar fiberwise minimax lemma proves that half the Hamiltonian oscillation on each projection fiber is the smallest possible pointwise reduced-model defect.\n\nCandidate contribution (stability theorem and reduction certificate; novelty confidence low): For projected first-order Hamilton--Jacobi equations, the restricted-covector defect gives the dimension-independent bound ||u-v composed with P||_infinity <= epsilon t; the scalar midpoint of the Hamiltonian range on each fiber minimizes the pointwise defect, and adding a certified reduced numerical error gives explicit non-strict level-set enclosures.", "research_difficulty_suggested": "L2" }, { "id": 20001175, "problem_number": "AIM-COMPUTATION-0013", "title": "Coordinate-covariant orientation laws for no-trade regions", "statement": "A.10 Ostrov, Daniel\n\nMy primary interest in optimal control in multidimensions has come from applications in financial mathematics. For example, I have recently been working (with Jonathan Good-man) on how to determine the best strategy for trading a number of different stocks so that the expected \"utility\" of the stocks is maximized. (The term \"utility\" refers to a specified balance between the desires for a higher return from the stocks and a low total variance.) It is known that in the presence of transactions costs (that is, a charge for buying or selling stocks), there is a \"hold region\" in the multidimensional space of stock prices, and within this region, one should not trade stocks, but on the boundary of the hold region, one trades so that the stocks do not leave this hold region. It is believed, but not known for certain, that in the asymptotic limit as the transaction costs get small, the hold region becomes a multidimensional parallelogram. However, even assuming that this is true, it is not clear what the orientation of the parallelogram should be.", "original_statement": "A.10 Ostrov, Daniel \n\nMy primary interest in optimal control in multidimensions has come from applications in financial mathematics. For example, I have recently been working (with Jonathan Good-man) on how to determine the best strategy for trading a number of different stocks so that the expected \"utility\" of the stocks is maximized. (The term \"utility\" refers to a specified balance between the desires for a higher return from the stocks and a low total variance.) It is known that in the presence of transactions costs (that is, a charge for buying or selling stocks), there is a \"hold region\" in the multidimensional space of stock prices, and within this region, one should not trade stocks, but on the boundary of the hold region, one trades so that the stocks do not leave this hold region. It is believed, but not known for certain, that in the asymptotic limit as the transaction costs get small, the hold region becomes a multidimensional parallelogram. However, even assuming that this is true, it is not clear what the orientation of the parallelogram should be.", "clean_statement": "A.10 Ostrov, Daniel\n\nMy primary interest in optimal control in multidimensions has come from applications in financial mathematics. For example, I have recently been working (with Jonathan Good-man) on how to determine the best strategy for trading a number of different stocks so that the expected \"utility\" of the stocks is maximized. (The term \"utility\" refers to a specified balance between the desires for a higher return from the stocks and a low total variance.) It is known that in the presence of transactions costs (that is, a charge for buying or selling stocks), there is a \"hold region\" in the multidimensional space of stock prices, and within this region, one should not trade stocks, but on the boundary of the hold region, one trades so that the stocks do not leave this hold region. It is believed, but not known for certain, that in the asymptotic limit as the transaction costs get small, the hold region becomes a multidimensional parallelogram. However, even assuming that this is true, it is not clear what the orientation of the parallelogram should be.", "statement_status": "exact", "statement_verification": "This record is item A.10, attributed to Daniel Ostrov, in the AIM workshop list *Numerical methods for optimal control in high dimensions*. The official nine-page PDF was checked directly. A.10 occupies page 9 of the displayed document (PDF page index 8), lines 274--285 in the extracted text. The name printed across a line break as `Good-man` is Jonathan **Goodman**; the hyphen is not part of his surname.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Numerical methods for optimal control in high dimensions\nSection: \nSource item: A.10\nSource URL: https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf\nCanonical location: aim-computation-notes.json notes[12]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.10 Ostrov, Daniel \\n\\nMy primary interest in optimal control in multidimensions has come from applications in financial mathematics. For example, I have recently been working (with Jonathan Good-man) on how to determine the best strategy for trading a number of different stocks so that the expected \\\"utility\\\" of the stocks is maximized. (The term \\\"utility\\\" refers to a specified balance between the desires for a higher return from the stocks and a low total variance.) It is known that in the presence of transactions costs (that is, a charge for buying or selling stocks), there is a \\\"hold region\\\" in the multidimensional space of stock prices, and within this region, one should not trade stocks, but on the boundary of the hold region, one trades so that the stocks do not leave this hold region. It is believed, but not known for certain, that in the asymptotic limit as the transaction costs get small, the hold region becomes a multidimensional parallelogram. However, even assuming that this is true, it is not clear what the orientation of the parallelogram should be.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/optimalcontrol/optimalcontrol.pdf", "tags": [ "aim", "AIM-COMPUTATION-0013", "aim-domain:computation", "aim-workshop:optimalcontrol", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The continuous-time question is only partially resolved: rigorous small-cost theory gives an order-epsilon-to-the-one-third scale and a multidimensional corrector problem, while Goodman and Ostrov obtain a leading-order box for uncorrelated returns but no universal correlated-assets parallelotope. In a separately declared static local model, this attempt proves that the no-trade condition is the dual inclusion gradient U(x) in lambda K, that the rescaled local no-trade set converges in Hausdorff distance to -Q^{-1}K, and that the formula is exact for quadratic utility. It also proves the coordinate transformation law and constructs a three-channel transaction gauge whose exact quadratic no-trade region is a hexagon, showing that orientation and even parallelotope shape depend jointly on the opportunity-loss Hessian, transaction channels, and coordinates.\n\nCandidate contribution (local asymptotic theorem and geometric obstruction; novelty confidence low): Candidate contribution: for a smooth strictly concave one-step objective and a convex positively homogeneous transaction gauge, the local rescaled no-trade set obeys the coordinate-covariant dual-ball law lambda^{-1}(N_lambda-x_star) converging to -Q^{-1}partial g(0); together with an explicit three-channel hexagon, this gives a testable obstruction to treating a parallelogram or its principal-axis orientation as universal." }, { "id": 20001176, "problem_number": "AIM-COMPUTATION-0014", "title": "When frequency selectivity can and cannot improve multi-user capacity", "statement": "1. Spectral efficiency AP: One can consider spectral efficiency (bps/Hz) for (a) a single user (b) multiple users within the same cell (c) multiple cells AP: HDB cannot improve capacity. (This fact has not been proven for the multi-user case, although it is believed to be true.) BH: In the multi-user case, HDB provides frequency diversity. 2. Coverage (SNR + fade margins) AP: SF does not buy coverage. HDB only buys coverage through di-versity. GP: There is a trade-off between coverage and transmission rate. 3. Reliability 14 AP: Reliability is a measure on the statistics of the link and HDB helps, but there is no benefit from SF. AP: Tx processing does not gain over Rx processing. CT: The capacity of a SIMO channel is the same as the capacity of a MISO channel (if CSI is available at Tx), but the SIMO channel does not have SF. 4. Channel estimation AP: For the same amount of power, you have to estimate a lot more parameters in a HDB channel than in a non-HDB channel. This is problematic, especially for the weaker taps. AP: What is the interaction with SF? This relates to iterative TR. 5. Signaling overhead AP: Current systems have about 25-30 % signaling overhead, which eats up spectral efficiency. In these scenarios, iterative TR would not be worth its cost in delay. However, channel estimation is especially important if there is fading. 6. Low probability of intercept AP: In this case, SF is very important. In the context of TR, it can be achieved even with 1 transmit antenna. CDMA technology is not a competitive contender in this regard, because pseudo-random sequences are well-known. LPI applications are probably prepared to throw away bandwidth. In UWB applications, e.g., cable replacement applications, one is not interested in spectral efficiency. Also, the transmit power is limited. GP: In each LPI transmission, the power delivered can be very low, but the power from several transmissions will add up. GP: Are there commercial applications for LPI? Probably not, because they wouldn't tolerate the high rate back-off. In these cases, se-curity is achieved with higher layer mechanisms. BH: You don't have to do a complete rate back-off, but only to the point which the equalizer can handle the channel. 15 VI. Further Discussion on Topics from Open Discussion Session on Thursday", "original_statement": "1. Spectral efficiency AP: One can consider spectral efficiency (bps/Hz) for (a) a single user (b) multiple users within the same cell (c) multiple cells AP: HDB cannot improve capacity. (This fact has not been proven for the multi-user case, although it is believed to be true.) BH: In the multi-user case, HDB provides frequency diversity. 2. Coverage (SNR + fade margins) AP: SF does not buy coverage. HDB only buys coverage through di-versity. GP: There is a trade-off between coverage and transmission rate. 3. Reliability 14 AP: Reliability is a measure on the statistics of the link and HDB helps, but there is no benefit from SF. AP: Tx processing does not gain over Rx processing. CT: The capacity of a SIMO channel is the same as the capacity of a MISO channel (if CSI is available at Tx), but the SIMO channel does not have SF. 4. Channel estimation AP: For the same amount of power, you have to estimate a lot more parameters in a HDB channel than in a non-HDB channel. This is problematic, especially for the weaker taps. AP: What is the interaction with SF? This relates to iterative TR. 5. Signaling overhead AP: Current systems have about 25-30 % signaling overhead, which eats up spectral efficiency. In these scenarios, iterative TR would not be worth its cost in delay. However, channel estimation is especially important if there is fading. 6. Low probability of intercept AP: In this case, SF is very important. In the context of TR, it can be achieved even with 1 transmit antenna. CDMA technology is not a competitive contender in this regard, because pseudo-random sequences are well-known. LPI applications are probably prepared to throw away bandwidth. In UWB applications, e.g., cable replacement applications, one is not interested in spectral efficiency. Also, the transmit power is limited. GP: In each LPI transmission, the power delivered can be very low, but the power from several transmissions will add up. GP: Are there commercial applications for LPI? Probably not, because they wouldn't tolerate the high rate back-off. In these cases, se-curity is achieved with higher layer mechanisms. BH: You don't have to do a complete rate back-off, but only to the point which the equalizer can handle the channel. 15 VI. Further Discussion on Topics from Open Discussion Session on Thursday", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The source is not a single numbered theorem. It is Section V, “Open Discussion,” in the Friday session of the 2004 AIM workshop *Time Reversal Communications in Richly Scattering Environments*. The extraction merged all six discussion headings into one record:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Time-reversal communications in richly scattering environments\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/timerev/timerev.pdf\nCanonical location: aim-computation-notes.json notes[13]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Spectral efficiency AP: One can consider spectral efficiency (bps/Hz) for (a) a single user (b) multiple users within the same cell (c) multiple cells AP: HDB cannot improve capacity. (This fact has not been proven for the multi-user case, although it is believed to be true.) BH: In the multi-user case, HDB provides frequency diversity. 2. Coverage (SNR + fade margins) AP: SF does not buy coverage. HDB only buys coverage through di-versity. GP: There is a trade-off between coverage and transmission rate. 3. Reliability 14 AP: Reliability is a measure on the statistics of the link and HDB helps, but there is no benefit from SF. AP: Tx processing does not gain over Rx processing. CT: The capacity of a SIMO channel is the same as the capacity of a MISO channel (if CSI is available at Tx), but the SIMO channel does not have SF. 4. Channel estimation AP: For the same amount of power, you have to estimate a lot more parameters in a HDB channel than in a non-HDB channel. This is problematic, especially for the weaker taps. AP: What is the interaction with SF? This relates to iterative TR. 5. Signaling overhead AP: Current systems have about 25-30 % signaling overhead, which eats up spectral efficiency. In these scenarios, iterative TR would not be worth its cost in delay. However, channel estimation is especially important if there is fading. 6. Low probability of intercept AP: In this case, SF is very important. In the context of TR, it can be achieved even with 1 transmit antenna. CDMA technology is not a competitive contender in this regard, because pseudo-random sequences are well-known. LPI applications are probably prepared to throw away bandwidth. In UWB applications, e.g., cable replacement applications, one is not interested in spectral efficiency. Also, the transmit power is limited. GP: In each LPI transmission, the power delivered can be very low, but the power from several transmissions will add up. GP: Are there commercial applications for LPI? Probably not, because they wouldn't tolerate the high rate back-off. In these cases, se-curity is achieved with higher layer mechanisms. BH: You don't have to do a complete rate back-off, but only to the point which the equalizer can handle the channel. 15 VI. Further Discussion on Topics from Open Discussion Session on Thursday\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/timerev/timerev.pdf", "tags": [ "aim", "AIM-COMPUTATION-0014", "aim-domain:computation", "aim-workshop:timerev", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source is an overmerged six-branch discussion rather than one formal problem. For a K-user, M-tone scalar Gaussian multiple-access channel with receiver CSI, joint decoding, fixed tone-independent per-user spectral masks p_k, and a flat comparator preserving each user's average power gain, every subset constraint of the frequency-selective capacity region is no larger than the corresponding flat constraint by Jensen's inequality; hence the whole selective region is contained in the flat region. This does not survive a standard change of model: with per-user total power P and transmitter frequency CSI, two users with complementary two-tap channels having power gains (2,0) and (0,2) achieve normalized sum capacity log2(1+2P), strictly above the log2(1+P) flat-channel optimum despite identical average gain and impulse-response energy.\n\nCandidate contribution (theorem_and_counterexample; novelty confidence low): Candidate contribution: subset-wise Jensen containment proves full MAC-region dominance, not merely a sum-rate inequality, under fixed white power, while the explicit equal-energy two-user/two-tone FIR construction reverses the sum-capacity comparison for every P>0 under transmitter CSI and per-user total-power allocation; together they isolate frequency power mobility as an assumption that flips the AIM HDB claim." }, { "id": 20001177, "problem_number": "AIM-COMPUTATION-0015", "title": "What is actually exponential in single-carrier equalization?", "statement": "1. Equalization complexity AP: MC modulation, e.g., OFDM, can handle the equalization task easily. However, for HDB channels, one would need long symbols, and problems with channel variability and channel estimation arise. Equalization complexity in MC modulation is O(N log N ), whereas in SC modulation, the complexity is exponential in N (N\n\nis the channel length). AP: Linear vs. nonlinear equalization Linear equalization algorithms are not efficient. It appears that nonlinear techniques, e.g., THP, DPC, perform better, but their interactions with SF have not been investigated. AP: Special equalization techniques The problem frequently reduces to that of equalizing a long but sparse channel. These channels are encountered in TV systems, and are handled in DVB-T with OFDM-like techniques. 2. OFDM AP: OFDM is the system of choice for 3.5G, 4G and Wi-Fi systems. The delay spread and the Doppler define the cyclic prefix, symbol length and FFT size for an OFDM system. BH: Coding in OFDM with fine frequency spacing, such as in HDB channels that are very frequency selective, becomes more complex. AP: Spatial focusing OFDM and SF have not been studied. We expect some perfor-mance and complexity trade-offs. 3. Waveform Selection TS: The waveform selection problem is not only relevant to HDB chan-nels. AP: In UWB systems, OFDM and PPM are contenders. 4. Open vs. closed loop techniques 16 AP: What happens when you don't know the channel exactly? What happens when you estimate the channel on the uplink and use this knowledge on the downlink? 17 A Acronyms Acronyms\n\nCDMA Code Division Multiple Access CSI Channel State Information DFE Decision Feedback Equalizer DPC Dirty Paper Coding DVB-T Digital Video Broadcasting - Terrestrial FDD Frequency Division Duplexing FFT Fast Fourier Transform FIR Finite Impulse Response HDB High Delay spread Bandwidth HSDPA High Speed Data Packet Access IIR Infinite Impulse Response ISI Inter Symbol Interference ITU International Telecommunication Union LE Linear Equalizer LMS Least Mean Square LPI Low Probability of Intercept MC Multi Carrier MIMO Multiple Input Multiple Output MISO Multiple Input Single Output ML Maximum Likelihood 18 OFDM Orthogonal Frequency Division Multiplexing PDP Power Delay Profile PPM Pulse Position Modulation QAM Quadrature Amplitude Modulation Rx Receiver SC Single Carrier SDMA Space Division Multiple Access SF Spatial Focusing SIMO Single Input Multiple Output SINR Signal to Interference plus Noise Ratio SNR Signal to Noise Ratio SONAR Sonic Radar TDD Time Division Duplexing THP Tomlinson Harashima Precoding TR Time Reversal TV Television Tx Transmitter UWB Ultra Wideband 19 B Participants Abbreviations of Names of Participants\n\nAA Agissilaos Athanassoulis AF Albert Fannjiang AK Arnold Kim AP Arogyaswami Paulraj BH Babak Hassibi CT Chrysoula Tsogka DB Deborah Berebichez FL Frederick Lee ES Erik Stauffer GD Gregoire Derveaux GL Gabriel Lerosey GP George Papanicolaou HB Harry Bims HS Heechun Song JF Jean-Pierre Fouque JH Jan Hansen JS James Frederick Sifferlen KS Knut Solna LB Liliana Borcea MF Mathias Fink 20 MV Mai Vu MC Mohamad Charafeddine ME Majid Emami OO Ozgur Oyman PK Persefoni Kyritsi RC Robert Calderbank RD Robert Clark Daniels RH Robert Heath RM Raghuraman Mudumbai RN Rohit Nabar ST Stephan ten Brink TS Thomas Strohmer UM Upamanyu Madhow 21", "original_statement": "1. Equalization complexity AP: MC modulation, e.g., OFDM, can handle the equalization task easily. However, for HDB channels, one would need long symbols, and problems with channel variability and channel estimation arise. Equalization complexity in MC modulation is O(N log N ), whereas in SC modulation, the complexity is exponential in N (N\n\nis the channel length). AP: Linear vs. nonlinear equalization Linear equalization algorithms are not efficient. It appears that nonlinear techniques, e.g., THP, DPC, perform better, but their interactions with SF have not been investigated. AP: Special equalization techniques The problem frequently reduces to that of equalizing a long but sparse channel. These channels are encountered in TV systems, and are handled in DVB-T with OFDM-like techniques. 2. OFDM AP: OFDM is the system of choice for 3.5G, 4G and Wi-Fi systems. The delay spread and the Doppler define the cyclic prefix, symbol length and FFT size for an OFDM system. BH: Coding in OFDM with fine frequency spacing, such as in HDB channels that are very frequency selective, becomes more complex. AP: Spatial focusing OFDM and SF have not been studied. We expect some perfor-mance and complexity trade-offs. 3. Waveform Selection TS: The waveform selection problem is not only relevant to HDB chan-nels. AP: In UWB systems, OFDM and PPM are contenders. 4. Open vs. closed loop techniques 16 AP: What happens when you don't know the channel exactly? What happens when you estimate the channel on the uplink and use this knowledge on the downlink? 17 A Acronyms Acronyms \n\nCDMA Code Division Multiple Access CSI Channel State Information DFE Decision Feedback Equalizer DPC Dirty Paper Coding DVB-T Digital Video Broadcasting - Terrestrial FDD Frequency Division Duplexing FFT Fast Fourier Transform FIR Finite Impulse Response HDB High Delay spread Bandwidth HSDPA High Speed Data Packet Access IIR Infinite Impulse Response ISI Inter Symbol Interference ITU International Telecommunication Union LE Linear Equalizer LMS Least Mean Square LPI Low Probability of Intercept MC Multi Carrier MIMO Multiple Input Multiple Output MISO Multiple Input Single Output ML Maximum Likelihood 18 OFDM Orthogonal Frequency Division Multiplexing PDP Power Delay Profile PPM Pulse Position Modulation QAM Quadrature Amplitude Modulation Rx Receiver SC Single Carrier SDMA Space Division Multiple Access SF Spatial Focusing SIMO Single Input Multiple Output SINR Signal to Interference plus Noise Ratio SNR Signal to Noise Ratio SONAR Sonic Radar TDD Time Division Duplexing THP Tomlinson Harashima Precoding TR Time Reversal TV Television Tx Transmitter UWB Ultra Wideband 19 B Participants Abbreviations of Names of Participants \n\nAA Agissilaos Athanassoulis AF Albert Fannjiang AK Arnold Kim AP Arogyaswami Paulraj BH Babak Hassibi CT Chrysoula Tsogka DB Deborah Berebichez FL Frederick Lee ES Erik Stauffer GD Gregoire Derveaux GL Gabriel Lerosey GP George Papanicolaou HB Harry Bims HS Heechun Song JF Jean-Pierre Fouque JH Jan Hansen JS James Frederick Sifferlen KS Knut Solna LB Liliana Borcea MF Mathias Fink 20 MV Mai Vu MC Mohamad Charafeddine ME Majid Emami OO Ozgur Oyman PK Persefoni Kyritsi RC Robert Calderbank RD Robert Clark Daniels RH Robert Heath RM Raghuraman Mudumbai RN Rohit Nabar ST Stephan ten Brink TS Thomas Strohmer UM Upamanyu Madhow 21", "clean_statement": "1. Equalization complexity AP: MC modulation, e.g., OFDM, can handle the equalization task easily. However, for HDB channels, one would need long symbols, and problems with channel variability and channel estimation arise. Equalization complexity in MC modulation is O(N log N ), whereas in SC modulation, the complexity is exponential in N (N\n\nis the channel length). AP: Linear vs. nonlinear equalization Linear equalization algorithms are not efficient. It appears that nonlinear techniques, e.g., THP, DPC, perform better, but their interactions with SF have not been investigated. AP: Special equalization techniques The problem frequently reduces to that of equalizing a long but sparse channel. These channels are encountered in TV systems, and are handled in DVB-T with OFDM-like techniques. 2. OFDM AP: OFDM is the system of choice for 3.5G, 4G and Wi-Fi systems. The delay spread and the Doppler define the cyclic prefix, symbol length and FFT size for an OFDM system. BH: Coding in OFDM with fine frequency spacing, such as in HDB channels that are very frequency selective, becomes more complex. AP: Spatial focusing OFDM and SF have not been studied. We expect some perfor-mance and complexity trade-offs. 3. Waveform Selection TS: The waveform selection problem is not only relevant to HDB chan-nels. AP: In UWB systems, OFDM and PPM are contenders. 4. Open vs. closed loop techniques 16 AP: What happens when you don't know the channel exactly? What happens when you estimate the channel on the uplink and use this knowledge on the downlink? 17 A Acronyms Acronyms\n\nCDMA Code Division Multiple Access CSI Channel State Information DFE Decision Feedback Equalizer DPC Dirty Paper Coding DVB-T Digital Video Broadcasting - Terrestrial FDD Frequency Division Duplexing FFT Fast Fourier Transform FIR Finite Impulse Response HDB High Delay spread Bandwidth HSDPA High Speed Data Packet Access IIR Infinite Impulse Response ISI Inter Symbol Interference ITU International Telecommunication Union LE Linear Equalizer LMS Least Mean Square LPI Low Probability of Intercept MC Multi Carrier MIMO Multiple Input Multiple Output MISO Multiple Input Single Output ML Maximum Likelihood 18 OFDM Orthogonal Frequency Division Multiplexing PDP Power Delay Profile PPM Pulse Position Modulation QAM Quadrature Amplitude Modulation Rx Receiver SC Single Carrier SDMA Space Division Multiple Access SF Spatial Focusing SIMO Single Input Multiple Output SINR Signal to Interference plus Noise Ratio SNR Signal to Noise Ratio SONAR Sonic Radar TDD Time Division Duplexing THP Tomlinson Harashima Precoding TR Time Reversal TV Television Tx Transmitter UWB Ultra Wideband 19 B Participants Abbreviations of Names of Participants\n\nAA Agissilaos Athanassoulis AF Albert Fannjiang AK Arnold Kim AP Arogyaswami Paulraj BH Babak Hassibi CT Chrysoula Tsogka DB Deborah Berebichez FL Frederick Lee ES Erik Stauffer GD Gregoire Derveaux GL Gabriel Lerosey GP George Papanicolaou HB Harry Bims HS Heechun Song JF Jean-Pierre Fouque JH Jan Hansen JS James Frederick Sifferlen KS Knut Solna LB Liliana Borcea MF Mathias Fink 20 MV Mai Vu MC Mohamad Charafeddine ME Majid Emami OO Ozgur Oyman PK Persefoni Kyritsi RC Robert Calderbank RD Robert Clark Daniels RH Robert Heath RM Raghuraman Mudumbai RN Rohit Nabar ST Stephan ten Brink TS Thomas Strohmer UM Upamanyu Madhow 21", "statement_status": "exact", "statement_verification": "The canonical record comes from the final open-discussion section of the AIM workshop notes *Time-reversal communications in richly scattering environments* (18--22 October 2004). The official PDF was inspected directly. The mathematical discussion occupies PDF pages 16--17 (PDF indices 15--16), and has four branches:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Time-reversal communications in richly scattering environments\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/timerev/timerev.pdf\nCanonical location: aim-computation-notes.json notes[14]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Equalization complexity AP: MC modulation, e.g., OFDM, can handle the equalization task easily. However, for HDB channels, one would need long symbols, and problems with channel variability and channel estimation arise. Equalization complexity in MC modulation is O(N log N ), whereas in SC modulation, the complexity is exponential in N (N\\n\\nis the channel length). AP: Linear vs. nonlinear equalization Linear equalization algorithms are not efficient. It appears that nonlinear techniques, e.g., THP, DPC, perform better, but their interactions with SF have not been investigated. AP: Special equalization techniques The problem frequently reduces to that of equalizing a long but sparse channel. These channels are encountered in TV systems, and are handled in DVB-T with OFDM-like techniques. 2. OFDM AP: OFDM is the system of choice for 3.5G, 4G and Wi-Fi systems. The delay spread and the Doppler define the cyclic prefix, symbol length and FFT size for an OFDM system. BH: Coding in OFDM with fine frequency spacing, such as in HDB channels that are very frequency selective, becomes more complex. AP: Spatial focusing OFDM and SF have not been studied. We expect some perfor-mance and complexity trade-offs. 3. Waveform Selection TS: The waveform selection problem is not only relevant to HDB chan-nels. AP: In UWB systems, OFDM and PPM are contenders. 4. Open vs. closed loop techniques 16 AP: What happens when you don't know the channel exactly? What happens when you estimate the channel on the uplink and use this knowledge on the downlink? 17 A Acronyms Acronyms \\n\\nCDMA Code Division Multiple Access CSI Channel State Information DFE Decision Feedback Equalizer DPC Dirty Paper Coding DVB-T Digital Video Broadcasting - Terrestrial FDD Frequency Division Duplexing FFT Fast Fourier Transform FIR Finite Impulse Response HDB High Delay spread Bandwidth HSDPA High Speed Data Packet Access IIR Infinite Impulse Response ISI Inter Symbol Interference ITU International Telecommunication Union LE Linear Equalizer LMS Least Mean Square LPI Low Probability of Intercept MC Multi Carrier MIMO Multiple Input Multiple Output MISO Multiple Input Single Output ML Maximum Likelihood 18 OFDM Orthogonal Frequency Division Multiplexing PDP Power Delay Profile PPM Pulse Position Modulation QAM Quadrature Amplitude Modulation Rx Receiver SC Single Carrier SDMA Space Division Multiple Access SF Spatial Focusing SIMO Single Input Multiple Output SINR Signal to Interference plus Noise Ratio SNR Signal to Noise Ratio SONAR Sonic Radar TDD Time Division Duplexing THP Tomlinson Harashima Precoding TR Time Reversal TV Television Tx Transmitter UWB Ultra Wideband 19 B Participants Abbreviations of Names of Participants \\n\\nAA Agissilaos Athanassoulis AF Albert Fannjiang AK Arnold Kim AP Arogyaswami Paulraj BH Babak Hassibi CT Chrysoula Tsogka DB Deborah Berebichez FL Frederick Lee ES Erik Stauffer GD Gregoire Derveaux GL Gabriel Lerosey GP George Papanicolaou HB Harry Bims HS Heechun Song JF Jean-Pierre Fouque JH Jan Hansen JS James Frederick Sifferlen KS Knut Solna LB Liliana Borcea MF Mathias Fink 20 MV Mai Vu MC Mohamad Charafeddine ME Majid Emami OO Ozgur Oyman PK Persefoni Kyritsi RC Robert Calderbank RD Robert Clark Daniels RH Robert Heath RM Raghuraman Mudumbai RN Rohit Nabar ST Stephan ten Brink TS Thomas Strohmer UM Upamanyu Madhow 21\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/timerev/timerev.pdf", "tags": [ "aim", "AIM-COMPUTATION-0015", "aim-domain:computation", "aim-workshop:timerev", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source's blanket assertion that single-carrier equalization is exponential in channel length is false without an objective and complexity model. For a cyclic-prefix single-carrier block of length B with channel memory nu and nu+1 <= B, linear ZF or LMMSE frequency-domain equalization costs O(B log B). For exact finite-alphabet ML sequence detection, a channel supported on delays in gZ splits into g independent memory-nu/g trellises; in particular, the long two-path channel with delays {0,nu} has exact ML complexity O((B+nu) M^2), not exponential in nu. For a nearly lattice-supported channel h=h0+e, a proved observation-dependent best-versus-second-best metric-margin test certifies when the cheap polyphase decision is also the unique exact full-channel ML decision.\n\nCandidate contribution (robustness certificate; novelty confidence low): Let h0 be lattice-supported, let rho=||h-h0||_1, and compute the unique h0-ML winner and its best-to-second-best gap gamma0 by a two-best polyphase Viterbi recursion. With C=A sqrt(B) and Delta=2(||y||_2+||h0||_1 C) rho C+rho^2 C^2, the strict test gamma0>2 Delta certifies that the h0 winner is the unique exact ML sequence for the full channel h." }, { "id": 20001178, "problem_number": "AIM-COMPUTATION-0016", "title": "Degree-two-and-three Markov basis for the Birkhoff ranking model", "statement": "Problem 1. Explicitly describe a Markov basis for tables with fixed first-order summary. That is, describe a set of moves which will connect any two surveys that have the same first order summary. In a purely mathematical language: describe a generating set for the toric ideal given by the p2 × p! matrix whose columns are the p! p × p permutation matrices.\n\nContingency tables with quadratic statistics.", "original_statement": "Problem 1. Explicitly describe a Markov basis for tables with fixed first-order summary. That is, describe a set of moves which will connect any two surveys that have the same first order summary. In a purely mathematical language: describe a generating set for the toric ideal given by the p2 × p! matrix whose columns are the p! p × p permutation matrices. \n\nContingency tables with quadratic statistics.", "clean_statement": "Problem 1. Explicitly describe a Markov basis for tables with fixed first-order summary. That is, describe a set of moves which will connect any two surveys that have the same first order summary. In a purely mathematical language: describe a generating set for the toric ideal given by the p2 × p! matrix whose columns are the p! p × p permutation matrices.\n\nContingency tables with quadratic statistics.", "statement_status": "exact", "statement_verification": "The official AIM PDF *Computational Algebraic Statistics*, version 27 March 2004, was checked directly. On displayed page 3 (PDF page index 2), under Persi Diaconis's heading “Fixed First-Order Summaries,” it defines a survey as a frequency function $f:S_p\\to\\mathbb N$. Its first-order summary is the $p\\times p$ table whose $(i,j)$ entry counts voters who put candidate $i$ in position $j$. The problem is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[15]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1. Explicitly describe a Markov basis for tables with fixed first-order summary. That is, describe a set of moves which will connect any two surveys that have the same first order summary. In a purely mathematical language: describe a generating set for the toric ideal given by the p2 × p! matrix whose columns are the p! p × p permutation matrices. \\n\\nContingency tables with quadratic statistics.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0016", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM problem is the complete-ranking Birkhoff model. For p at least 3, all balanced replacements of two or three rankings form a finite Markov basis, and degree three is both sufficient and necessary; for p equal to 1 or 2 the toric ideal is zero. The report translates the published degree-three theorem into the requested survey moves and proves a cycle-cube description of every quadratic fiber, a block-lifting lemma, an explicit indispensable cubic, and a deterministic fiber-component compression recipe.\n\nCandidate contribution (fiber-structure lemma and compressed-basis recipe; novelty confidence low): Candidate contribution: a degree-two summary whose associated two-regular bipartite multigraph has c nontrivial cycle components has one survey when c is zero and exactly 2^(c-1) surveys when c is positive, with a cycle-cube spanning basis; this structure combines with a proved block-diagonal fiber embedding and a component-tree construction to give a deterministic compressed Markov basis from the published degree-three bound." }, { "id": 20001179, "problem_number": "AIM-COMPUTATION-0017", "title": "Indispensable cubics and a five-move Markov basis on the zero-middle-row face", "statement": "Problem 2. Find a Markov basis for studying the set X of 2-way tables with a fixed quadratic statistic:\n\nX = {T | ∑\n\n> j\n\nTij = Ri, ∑\n\n> i\n\nTij = Cj and ∑\n\n> i,j\n\n(i − 2)( j − 2) Tij = Q}.\n\nFor 3 × 4 tables, the matrix that computes the sufficient statistics (i.e. the matrix which represents the log-linear model) is the following:\n\n\n\n1 1 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 0 0 0 00 0 0 0 0 0 0 0 1 1 1 11 0 0 0 1 0 0 0 1 0 0 00 1 0 0 0 1 0 0 0 1 0 00 0 1 0 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1 0 0 0 11 0 −1 −2 0 0 0 0 −1 0 1 2\n\n.\n\nPolyhedral Cones and Testing.\n\nObserve\n\nXij = {10|P (Xij |Xij ) = Z−1(θ)i)exij θj }4\n\nTest θi = θ∀i against A · θ ≥ 0 with A fixed.\n\nExample 3. Possible conditions enforced by A · · · θ ≥ 0 are\n\nθ1 ≥ θ2 ≥ · · · ≥ θI\n\nor\n\nθi ≥ θI ∀i.\n\nThere should be a nice interaction between statistics and polyhedral combinatorics in this problem.\n\nCompare \"Competing\" Techniques.\n\nA. Classical Asymptotics B. Edgeworth Corrections C. StatXact D. Sequential Importance Sampling E. Others How do these different techniques for estimating p-values relate to each other? With respect to sequential importance sampling: what are possible multi-way gener-alizations of the Gale-Ryser theorem which gives necessary and sufficient conditions for the existence of a 0 /1 matrix with fixed row and column sums?\n\nStephen Fienberg\n\nQuestions about Odds-Ratios.", "original_statement": "Problem 2. Find a Markov basis for studying the set X of 2-way tables with a fixed quadratic statistic: \n\nX = {T | ∑\n\n> j\n\nTij = Ri, ∑\n\n> i\n\nTij = Cj and ∑\n\n> i,j\n\n(i − 2)( j − 2) Tij = Q}.\n\nFor 3 × 4 tables, the matrix that computes the sufficient statistics (i.e. the matrix which represents the log-linear model) is the following: \n\n\n\n1 1 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 0 0 0 00 0 0 0 0 0 0 0 1 1 1 11 0 0 0 1 0 0 0 1 0 0 00 1 0 0 0 1 0 0 0 1 0 00 0 1 0 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1 0 0 0 11 0 −1 −2 0 0 0 0 −1 0 1 2\n\n.\n\nPolyhedral Cones and Testing. \n\nObserve \n\nXij = {10|P (Xij |Xij ) = Z−1(θ)i)exij θj }4\n\nTest θi = θ∀i against A · θ ≥ 0 with A fixed. \n\nExample 3. Possible conditions enforced by A · · · θ ≥ 0 are \n\nθ1 ≥ θ2 ≥ · · · ≥ θI\n\nor \n\nθi ≥ θI ∀i. \n\nThere should be a nice interaction between statistics and polyhedral combinatorics in this problem. \n\nCompare \"Competing\" Techniques. \n\nA. Classical Asymptotics B. Edgeworth Corrections C. StatXact D. Sequential Importance Sampling E. Others How do these different techniques for estimating p-values relate to each other? With respect to sequential importance sampling: what are possible multi-way gener-alizations of the Gale-Ryser theorem which gives necessary and sufficient conditions for the existence of a 0 /1 matrix with fixed row and column sums? \n\nStephen Fienberg \n\nQuestions about Odds-Ratios.", "clean_statement": "Problem 2. Find a Markov basis for studying the set X of 2-way tables with a fixed quadratic statistic:\n\nX = {T | ∑\n\n> j\n\nTij = Ri, ∑\n\n> i\n\nTij = Cj and ∑\n\n> i,j\n\n(i − 2)( j − 2) Tij = Q}.\n\nFor 3 × 4 tables, the matrix that computes the sufficient statistics (i.e. the matrix which represents the log-linear model) is the following:\n\n\n\n1 1 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 0 0 0 00 0 0 0 0 0 0 0 1 1 1 11 0 0 0 1 0 0 0 1 0 0 00 1 0 0 0 1 0 0 0 1 0 00 0 1 0 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1 0 0 0 11 0 −1 −2 0 0 0 0 −1 0 1 2\n\n.\n\nPolyhedral Cones and Testing.\n\nObserve\n\nXij = {10|P (Xij |Xij ) = Z−1(θ)i)exij θj }4\n\nTest θi = θ∀i against A · θ ≥ 0 with A fixed.\n\nExample 3. Possible conditions enforced by A · · · θ ≥ 0 are\n\nθ1 ≥ θ2 ≥ · · · ≥ θI\n\nor\n\nθi ≥ θI ∀i.\n\nThere should be a nice interaction between statistics and polyhedral combinatorics in this problem.\n\nCompare \"Competing\" Techniques.\n\nA. Classical Asymptotics B. Edgeworth Corrections C. StatXact D. Sequential Importance Sampling E. Others How do these different techniques for estimating p-values relate to each other? With respect to sequential importance sampling: what are possible multi-way gener-alizations of the Gale-Ryser theorem which gives necessary and sufficient conditions for the existence of a 0 /1 matrix with fixed row and column sums?\n\nStephen Fienberg\n\nQuestions about Odds-Ratios.", "statement_status": "exact", "statement_verification": "The canonical JSON record contains extraction spillover. The authoritative AIM PDF, *Computational Algebraic Statistics* (version dated 27 March 2004), gives the following problem and then ends it immediately after the displayed matrix:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[16]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2. Find a Markov basis for studying the set X of 2-way tables with a fixed quadratic statistic: \\n\\nX = {T | ∑\\n\\n> j\\n\\nTij = Ri, ∑\\n\\n> i\\n\\nTij = Cj and ∑\\n\\n> i,j\\n\\n(i − 2)( j − 2) Tij = Q}.\\n\\nFor 3 × 4 tables, the matrix that computes the sufficient statistics (i.e. the matrix which represents the log-linear model) is the following: \\n\\n\\n\\n1 1 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 0 0 0 00 0 0 0 0 0 0 0 1 1 1 11 0 0 0 1 0 0 0 1 0 0 00 1 0 0 0 1 0 0 0 1 0 00 0 1 0 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1 0 0 0 11 0 −1 −2 0 0 0 0 −1 0 1 2\\n\\n.\\n\\nPolyhedral Cones and Testing. \\n\\nObserve \\n\\nXij = {10|P (Xij |Xij ) = Z−1(θ)i)exij θj }4\\n\\nTest θi = θ∀i against A · θ ≥ 0 with A fixed. \\n\\nExample 3. Possible conditions enforced by A · · · θ ≥ 0 are \\n\\nθ1 ≥ θ2 ≥ · · · ≥ θI\\n\\nor \\n\\nθi ≥ θI ∀i. \\n\\nThere should be a nice interaction between statistics and polyhedral combinatorics in this problem. \\n\\nCompare \\\"Competing\\\" Techniques. \\n\\nA. Classical Asymptotics B. Edgeworth Corrections C. StatXact D. Sequential Importance Sampling E. Others How do these different techniques for estimating p-values relate to each other? With respect to sequential importance sampling: what are possible multi-way gener-alizations of the Gale-Ryser theorem which gives necessary and sufficient conditions for the existence of a 0 /1 matrix with fixed row and column sums? \\n\\nStephen Fienberg \\n\\nQuestions about Odds-Ratios.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0017", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the exact AIM 3-by-4 fixed-margin, fixed-quadratic-statistic model, no nonzero degree-two move exists and two explicit consecutive-column degree-three moves are indispensable, as witnessed by two-point permutation fibers. On the entire face R_2=0, five explicit lifted moves form a genuine Markov basis for every nonnegative fiber; the proof identifies them as the full Graver basis of the four-position total-and-first-moment configuration and uses conformal decomposition to preserve all coordinate bounds.\n\nCandidate contribution (special_case_theorem; novelty confidence low): The exact 3-by-4 model has no quadratic moves and has the two displayed indispensable consecutive-column cubics; moreover, the five displayed lifts of (1,-2,1,0), (1,-1,-1,1), (0,1,-2,1), (2,-3,0,1), and (1,0,-3,2) connect every nonnegative fiber on the face R_2=0." }, { "id": 20001180, "problem_number": "AIM-COMPUTATION-0018", "title": "Two fixed odds ratios form a line segment, with a complete boundary audit", "statement": "Question 4. For a 2 × 2 table of probabilities\n\n( p00 p01\n\np10 p11\n\n): does it makes sense to fix values of two of the odds ratios α = p00 p11\n\n> p01 p10\n\nand α∗ = p00 p01\n\n> p11 p10\n\nand look at the locus of possible values for the third odds-ratio α∗∗ = p00 p10\n\n> p01 p11?Describe the curve (in the probability simplex) obtained by intersecting two of these quadric surfaces.", "original_statement": "Question 4. For a 2 × 2 table of probabilities \n\n( p00 p01 \n\np10 p11 \n\n): does it makes sense to fix values of two of the odds ratios α = p00 p11 \n\n> p01 p10\n\nand α∗ = p00 p01 \n\n> p11 p10\n\nand look at the locus of possible values for the third odds-ratio α∗∗ = p00 p10 \n\n> p01 p11?Describe the curve (in the probability simplex) obtained by intersecting two of these quadric surfaces.", "clean_statement": "Question 4. For a 2 × 2 table of probabilities\n\n( p00 p01\n\np10 p11\n\n): does it makes sense to fix values of two of the odds ratios α = p00 p11\n\n> p01 p10\n\nand α∗ = p00 p01\n\n> p11 p10\n\nand look at the locus of possible values for the third odds-ratio α∗∗ = p00 p10\n\n> p01 p11?Describe the curve (in the probability simplex) obtained by intersecting two of these quadric surfaces.", "statement_status": "exact", "statement_verification": "Write", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[17]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 4. For a 2 × 2 table of probabilities \\n\\n( p00 p01 \\n\\np10 p11 \\n\\n): does it makes sense to fix values of two of the odds ratios α = p00 p11 \\n\\n> p01 p10\\n\\nand α∗ = p00 p01 \\n\\n> p11 p10\\n\\nand look at the locus of possible values for the third odds-ratio α∗∗ = p00 p10 \\n\\n> p01 p11?Describe the curve (in the probability simplex) obtained by intersecting two of these quadric surfaces.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0018", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For fixed positive A=alpha and B=alpha*, put r=sqrt(AB) and s=sqrt(A/B). The exact positive locus is P(t)=(r,t,1,st)/(1+r+(1+s)t), t>0, a single smooth connected open affine segment, and the third ratio is alpha**=B/t^2, ranging bijectively over all positive values. Every positive triple has the unique normalized table (sqrt(ABC),sqrt(B),sqrt(C),sqrt(A)). The closed-simplex audit distinguishes this rational locus from its two-endpoint closure and from the larger cleared-denominator intersection: the latter is a four-line projective curve whose nonnegative part consists of the genuine connector plus two spurious coordinate edges, attached at two explicitly verified nodes. Finite-zero, projective zero/infinity, and degenerate cleared cases are also classified.\n\nCandidate contribution (boundary classification; novelty confidence low): Candidate novelty: an integrated, explicit boundary dictionary for the precise pair (alpha, alpha*) that separates the rational odds-ratio locus, its topological closure, and the cleared-denominator variety; identifies all four linear components, the two spurious nonnegative edges, and the endpoint Jacobian nodes; and classifies finite-zero, projective zero/infinity, and degenerate cleared strata." }, { "id": 20001181, "problem_number": "AIM-COMPUTATION-0019", "title": "Linear collapse, binomial hypersurfaces, and the image of local-odds fibers", "statement": "Question 5. What is the algebro-geometric structure of passing from local to global odds ratios? Note that local odds ratio is obtained by looking at a 2 k subtable of a given table whereas a global odds ratio is obtained by looking at the odds-ratio in a 2 k collapsing of a given table.", "original_statement": "Question 5. What is the algebro-geometric structure of passing from local to global odds ratios? Note that local odds ratio is obtained by looking at a 2 k subtable of a given table whereas a global odds ratio is obtained by looking at the odds-ratio in a 2 k collapsing of a given table.", "clean_statement": "Question 5. What is the algebro-geometric structure of passing from local to global odds ratios? Note that local odds ratio is obtained by looking at a 2 k subtable of a given table whereas a global odds ratio is obtained by looking at the odds-ratio in a 2 k collapsing of a given table.", "statement_status": "exact", "statement_verification": "The canonical record reads “\\(2 k\\)” twice because superscripts were lost during extraction. The official AIM HTML source preserves the mathematical alternative text and resolves both occurrences as \\(2^k\\). The verified statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[18]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 5. What is the algebro-geometric structure of passing from local to global odds ratios? Note that local odds ratio is obtained by looking at a 2 k subtable of a given table whereas a global odds ratio is obtained by looking at the odds-ratio in a 2 k collapsing of a given table.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0019", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The corrupted source notation is 2^k. For any k-way dichotomizing collapse, a fixed finite nonzero global odds ratio is the denominator-open part of an irreducible degree-2^{k-1} binomial hypersurface pulled back by the linear block-sum map, with an exact even-block/odd-block boundary decomposition. For two-way tables the collapsed ratio is an explicit convex combination of cross-block local ratios. Complete local ratios fix a table only up to row-column scaling; on a one-sided collapse fiber, the global ratio is linear-fractional, has positive image exactly the open interval between the reference-row local ratios (or a singleton if they coincide), and has Laurent-hyperplane level sets.\n\nCandidate contribution (special_case_theorem; novelty confidence low): On a positive complete local-odds-ratio fiber of an (m+1)-by-2 table, collapsing m rows against one reference row restricts the global odds ratio to a linear-fractional map whose positive image is exactly the relative interior of the convex hull of the m reference-row local ratios, and whose fixed-value fibers are the displayed Laurent hyperplanes." }, { "id": 20001182, "problem_number": "AIM-COMPUTATION-0020", "title": "Compatible and complete odds-ratio specifications", "statement": "Question 6. For multi-way tables, what are the complete and partial specifications that come from specifying sets of odds-ratios?\n\nExistence of Maximum Likelihood Estimates.", "original_statement": "Question 6. For multi-way tables, what are the complete and partial specifications that come from specifying sets of odds-ratios? \n\nExistence of Maximum Likelihood Estimates.", "clean_statement": "Question 6. For multi-way tables, what are the complete and partial specifications that come from specifying sets of odds-ratios?\n\nExistence of Maximum Likelihood Estimates.", "statement_status": "exact", "statement_verification": "The official AIM PDF *Computational Algebraic Statistics*, version 27 March 2004, was inspected directly. On displayed page 4 (PDF page index 3), Stephen Fienberg's subsection is headed “Questions about Odds-Ratios.” It first asks about three multiplicative contrasts in a positive \\(2\\times2\\) probability table (Question 4), then distinguishes local odds ratios on \\(2^k\\) subtables from global odds ratios formed after a \\(2^k\\) collapsing (Question 5). The assigned statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[19]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 6. For multi-way tables, what are the complete and partial specifications that come from specifying sets of odds-ratios? \\n\\nExistence of Maximum Likelihood Estimates.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0020", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a positive m-cell table and a k by m integer matrix C of homogeneous multiplicative cell contrasts, target odds ratios q are compatible exactly when log q lies in the image of C; a compatible fiber is real-analytically diffeomorphic to R^(m-1-rank(C)), and the specification is complete exactly when rank(C)=m-1. For positive binary d-way tables, all overlap constraints among the full arrays of conditional pairwise odds ratios are explicit Walsh-transform equalities; the arrays contain 2^d-d-1 independent association coordinates, leave a d-dimensional fiber, and become complete after d one-factor product ratios are supplied. Global collapsed odds ratios remain outside this linear classification.\n\nCandidate contribution (compatibility theorem and completion formula; novelty confidence low): Candidate contribution: for every positive binary d-way table, proposed conditional pairwise odds-ratio arrays are jointly compatible if and only if their normalized Walsh transforms agree whenever two pairs extract the same higher interaction; these equations have exactly binomial(d,2) times 2^(d-2) minus (2^d-d-1) independent constraints, and adding the d singleton product ratios yields the explicit unique table." }, { "id": 20001183, "problem_number": "AIM-COMPUTATION-0021", "title": "Zero patterns, facial sets, and exact support repair in chordal models", "statement": "Question 7. What conditions on the structures of zeros in a table of counts will guarantee that the maximum likelihood estimates do not exist? In this setting, a maximum likelihood estimate is required to have all strictly positive probabilities. 5\n\nAs Alessandro Rinaldo, later pointed out in his talk, this is equivalent to giving a combinatorial description of the facets of the cone of all feasible margins. This polyhedral cone is obtained by taking of the positive hull of the columns of the matrices AG which appear in the working questions on graphical models by Chris Meek and Serkan Ho¸ sten.", "original_statement": "Question 7. What conditions on the structures of zeros in a table of counts will guarantee that the maximum likelihood estimates do not exist? In this setting, a maximum likelihood estimate is required to have all strictly positive probabilities. 5\n\nAs Alessandro Rinaldo, later pointed out in his talk, this is equivalent to giving a combinatorial description of the facets of the cone of all feasible margins. This polyhedral cone is obtained by taking of the positive hull of the columns of the matrices AG which appear in the working questions on graphical models by Chris Meek and Serkan Ho¸ sten.", "clean_statement": "Question 7. What conditions on the structures of zeros in a table of counts will guarantee that the maximum likelihood estimates do not exist? In this setting, a maximum likelihood estimate is required to have all strictly positive probabilities. 5\n\nAs Alessandro Rinaldo, later pointed out in his talk, this is equivalent to giving a combinatorial description of the facets of the cone of all feasible margins. This polyhedral cone is obtained by taking of the positive hull of the columns of the matrices AG which appear in the working questions on graphical models by Chris Meek and Serkan Ho¸ sten.", "statement_status": "exact", "statement_verification": "The source is Question 7 in the AIM workshop notes *Computational algebraic statistics* (version dated 27 March 2004). The extracted record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[20]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 7. What conditions on the structures of zeros in a table of counts will guarantee that the maximum likelihood estimates do not exist? In this setting, a maximum likelihood estimate is required to have all strictly positive probabilities. 5\\n\\nAs Alessandro Rinaldo, later pointed out in his talk, this is equivalent to giving a combinatorial description of the facets of the cone of all feasible margins. This polyhedral cone is obtained by taking of the positive hull of the columns of the matrices AG which appear in the working questions on graphical models by Chris Meek and Serkan Ho¸ sten.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0021", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The known general answer is that the ordinary strictly positive log-linear MLE exists exactly when the observed margin lies in the relative interior of the marginal cone, equivalently when the observed support is not contained in a proper facial set. For a full Cartesian chordal graphical model, this becomes the explicit proved criterion that every maximal-clique margin cell is positive. Building on it, the attempt proves an exact set-cover formulation for the minimum support repair and a closed formula for models with exactly two maximal cliques: the repair number is the sum over separator states of the larger of the two numbers of missing clique configurations in that state.\n\nCandidate contribution (theorem; novelty confidence low): For a full two-clique decomposable graphical model with separator R, the minimum number of distinct new positive support cells required for ordinary-MLE existence is sum over s in I_R of max(m_1(s), m_2(s)), where m_k(s) counts missing configurations of clique C_k with separator state s; for arbitrary chordal models the repair number is exactly the stated binary set-cover program." }, { "id": 20001184, "problem_number": "AIM-COMPUTATION-0022", "title": "A binary-collapse-invisible likelihood face", "statement": "Conjecture 8. Maximum likelihood estimates for an I × J × K table under the no 3-way interaction model do not exist if and only if they do not exist for all collapsings of the table to a 2 × 2 × 2 table.\n\nLater in the week, Nicholas Eriksson showed that this conjecture is false already for 3 × 3 × 3 tables. Mathias Drton asked \"How can we incorporate structural zeroes into this problem?\" Stephen Roehrig asked \"Is it possible to relate the MLE solutions for decomposable models to decomposable network flow problems?\"\n\nAkimichi Takemura", "original_statement": "Conjecture 8. Maximum likelihood estimates for an I × J × K table under the no 3-way interaction model do not exist if and only if they do not exist for all collapsings of the table to a 2 × 2 × 2 table. \n\nLater in the week, Nicholas Eriksson showed that this conjecture is false already for 3 × 3 × 3 tables. Mathias Drton asked \"How can we incorporate structural zeroes into this problem?\" Stephen Roehrig asked \"Is it possible to relate the MLE solutions for decomposable models to decomposable network flow problems?\" \n\nAkimichi Takemura", "clean_statement": "Conjecture 8. Maximum likelihood estimates for an I × J × K table under the no 3-way interaction model do not exist if and only if they do not exist for all collapsings of the table to a 2 × 2 × 2 table.\n\nLater in the week, Nicholas Eriksson showed that this conjecture is false already for 3 × 3 × 3 tables. Mathias Drton asked \"How can we incorporate structural zeroes into this problem?\" Stephen Roehrig asked \"Is it possible to relate the MLE solutions for decomposable models to decomposable network flow problems?\"\n\nAkimichi Takemura", "statement_status": "exact", "statement_verification": "The source is Conjecture 8 in the AIM workshop notes *Computational algebraic statistics* (version dated 27 March 2004). The official PDF says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[21]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 8. Maximum likelihood estimates for an I × J × K table under the no 3-way interaction model do not exist if and only if they do not exist for all collapsings of the table to a 2 × 2 × 2 table. \\n\\nLater in the week, Nicholas Eriksson showed that this conjecture is false already for 3 × 3 × 3 tables. Mathias Drton asked \\\"How can we incorporate structural zeroes into this problem?\\\" Stephen Roehrig asked \\\"Is it possible to relate the MLE solutions for decomposable models to decomposable network flow problems?\\\" \\n\\nAkimichi Takemura\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0022", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal universal wording in the AIM source is already false for a 2 x 2 x 3 table, while the substantive existential binary-detection conjecture has an explicit 3 x 3 x 3 counterexample. For the displayed nine-cell zero set, three pair-potential matrices sum to its indicator, proving that the original sufficient statistic lies on a proper likelihood face; nevertheless, an exhaustive transparent classification of all 27 binary level partitions shows that every collapsed 2 x 2 x 2 table has at most one zero cell and therefore has an ordinary MLE. More generally, a collapsed MLE exists exactly when its empty-block zero set is independent in the binary cube graph augmented by antipodal edges.\n\nCandidate contribution (criterion; novelty confidence low): For any full three-way table with sampling-zero set Z and positive counts elsewhere, a specified binary collapse has an ordinary no-three-factor-effect MLE if and only if the set of fully empty binary blocks is independent in the graph on the binary cube whose edges join pairs at Hamming distance one or three; the explicit 3 x 3 x 3 zero set in the report passes this test for all 27 partitions while its indicator is pair-additive and exposes a proper face upstairs." }, { "id": 20001185, "problem_number": "AIM-COMPUTATION-0023", "title": "Unique minimal Markov bases, bipartite cycles, and padding monotonicity", "statement": "Question 9. Do I × J × K tables under the no 3-way interaction model always have a unique minimal Markov basis? Bernd Sturmfels asked, \"What about problems for 3-way tables that have structural zeros coming from a very regular design?\" Henry Wynn asked, \"Does the group structure of circuits play a role in the solution of this problem?\"\n\nEmily Gamundi", "original_statement": "Question 9. Do I × J × K tables under the no 3-way interaction model always have a unique minimal Markov basis? Bernd Sturmfels asked, \"What about problems for 3-way tables that have structural zeros coming from a very regular design?\" Henry Wynn asked, \"Does the group structure of circuits play a role in the solution of this problem?\" \n\nEmily Gamundi", "clean_statement": "Question 9. Do I × J × K tables under the no 3-way interaction model always have a unique minimal Markov basis? Bernd Sturmfels asked, \"What about problems for 3-way tables that have structural zeros coming from a very regular design?\" Henry Wynn asked, \"Does the group structure of circuits play a role in the solution of this problem?\"\n\nEmily Gamundi", "statement_status": "exact", "statement_verification": "Question 9 in the AIM workshop notes *Computational algebraic statistics* (version dated 27 March 2004) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[22]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 9. Do I × J × K tables under the no 3-way interaction model always have a unique minimal Markov basis? Bernd Sturmfels asked, \\\"What about problems for 3-way tables that have structural zeros coming from a very regular design?\\\" Henry Wynn asked, \\\"Does the group structure of circuits play a role in the solution of this problem?\\\" \\n\\nEmily Gamundi\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0023", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In the primary and recent sources checked, universal uniqueness for the no-three-factor-interaction model remains an open conjecture, with no verified counterexample. This attempt gives a self-contained proof of the full 2 by J by K family: the unique minimal basis consists of indispensable lifts of all simple even cycles of K_{J,K}, with an exact degree-by-degree count and one symmetry orbit per cycle length. It also proves that any failure of uniqueness at one format persists at every componentwise larger full format by embedding the exact lower-degree-disconnected witness fiber with zero outside margins.\n\nCandidate contribution (theorem; novelty confidence low): If the full no-three-factor-interaction model for format (I0,J0,K0) has nonunique minimal Markov bases, then every full componentwise larger format also has nonunique minimal Markov bases; equivalently, uniqueness descends to every coordinate subtable format. Consequently, using the established special families, the first sorted unresolved search frontiers are 3 by 5 by 5 and 4 by 4 by 5." }, { "id": 20001186, "problem_number": "AIM-COMPUTATION-0024", "title": "Exact endpoint and stability census for the symmetric triangle", "statement": "Question 10. Are there always only 0, 1, or 2 statistically interesting solutions to the blocking probability equations in network tomography?\n\nRuriko Yoshida\n\nRuriko Yoshida discussed short encodings of the set of points in polytopes by means of rational generating functions, and the applications of rational generating functions for giving short encodings of Markov bases. Ian Dinwoodie asked, \"Is there a way to extract (random) monomials from the gener-ating functions?\" Akimichi Takemura asked, \"Is there a way to extract only the absolutely essential Markov basis elements that are needed to connect a given fiber of the form {Ax = b|x ∈ Nd}?Ruriko answered,\"We are still attempting to solve these problems.\" Henry Wynn asked, \" Can these techniques be used to approximate the number of integral points inside an arbitrary convex set?\"\n\nMathias Drton\n\nMathias Drton discussed multimodality of the likelihood function for the model of seemingly unrelated regression (SUR). These can be thought of as conditional independence models induced by the chain graph with two sets of nodes, ( X1,..., X n) and ( Y1,..., Y n)and directed edges from Xi → Yi and undirected edges between any pair Yi − Yj.6", "original_statement": "Question 10. Are there always only 0, 1, or 2 statistically interesting solutions to the blocking probability equations in network tomography? \n\nRuriko Yoshida \n\nRuriko Yoshida discussed short encodings of the set of points in polytopes by means of rational generating functions, and the applications of rational generating functions for giving short encodings of Markov bases. Ian Dinwoodie asked, \"Is there a way to extract (random) monomials from the gener-ating functions?\" Akimichi Takemura asked, \"Is there a way to extract only the absolutely essential Markov basis elements that are needed to connect a given fiber of the form {Ax = b|x ∈ Nd}?Ruriko answered,\"We are still attempting to solve these problems.\" Henry Wynn asked, \" Can these techniques be used to approximate the number of integral points inside an arbitrary convex set?\" \n\nMathias Drton \n\nMathias Drton discussed multimodality of the likelihood function for the model of seemingly unrelated regression (SUR). These can be thought of as conditional independence models induced by the chain graph with two sets of nodes, ( X1,..., X n) and ( Y1,..., Y n)and directed edges from Xi → Yi and undirected edges between any pair Yi − Yj.6", "clean_statement": "Question 10. Are there always only 0, 1, or 2 statistically interesting solutions to the blocking probability equations in network tomography?\n\nRuriko Yoshida\n\nRuriko Yoshida discussed short encodings of the set of points in polytopes by means of rational generating functions, and the applications of rational generating functions for giving short encodings of Markov bases. Ian Dinwoodie asked, \"Is there a way to extract (random) monomials from the gener-ating functions?\" Akimichi Takemura asked, \"Is there a way to extract only the absolutely essential Markov basis elements that are needed to connect a given fiber of the form {Ax = b|x ∈ Nd}?Ruriko answered,\"We are still attempting to solve these problems.\" Henry Wynn asked, \" Can these techniques be used to approximate the number of integral points inside an arbitrary convex set?\"\n\nMathias Drton\n\nMathias Drton discussed multimodality of the likelihood function for the model of seemingly unrelated regression (SUR). These can be thought of as conditional independence models induced by the chain graph with two sets of nodes, ( X1,..., X n) and ( Y1,..., Y n)and directed edges from Xi → Yi and undirected edges between any pair Yi − Yj.6", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 10\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[23]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 10. Are there always only 0, 1, or 2 statistically interesting solutions to the blocking probability equations in network tomography? \\n\\nRuriko Yoshida \\n\\nRuriko Yoshida discussed short encodings of the set of points in polytopes by means of rational generating functions, and the applications of rational generating functions for giving short encodings of Markov bases. Ian Dinwoodie asked, \\\"Is there a way to extract (random) monomials from the gener-ating functions?\\\" Akimichi Takemura asked, \\\"Is there a way to extract only the absolutely essential Markov basis elements that are needed to connect a given fiber of the form {Ax = b|x ∈ Nd}?Ruriko answered,\\\"We are still attempting to solve these problems.\\\" Henry Wynn asked, \\\" Can these techniques be used to approximate the number of integral points inside an arbitrary convex set?\\\" \\n\\nMathias Drton \\n\\nMathias Drton discussed multimodality of the likelihood function for the model of seemingly unrelated regression (SUR). These can be thought of as conditional independence models induced by the chain graph with two sets of nodes, ( X1,..., X n) and ( Y1,..., Y n)and directed edges from Xi → Yi and undirected edges between any pair Yi − Yj.6\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0024", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the verified symmetric-triangle large-capacity blocking map T_c(b)=max{0,1-c/[2(1+2b(1-b))]}, statistically interesting solutions are fixed points b in (0,1). With b_*=(4-sqrt(10))/6 and c_*=(26+10sqrt(10))/27, their distinct count is one for 0c_*. The boundary point b=0 is fixed exactly for c>=2. The smaller positive branch is repelling, the larger branch is attracting, and the fold has multiplier +1.\n\nCandidate contribution (proposition; novelty confidence low): Candidate endpoint-complete refinement: the exact symmetric-triangle census includes both transition parameters, proves the fold root has multiplicity two, separates open-interval probability roots from the boundary fixed point, and proves the iteration multiplier on every branch." }, { "id": 20001187, "problem_number": "AIM-COMPUTATION-0025", "title": "An algebraic one-mode certificate for MANOVA", "statement": "Question 11. Can algebraic techniques be used to see the unimodality of the likelihood function in a MANOVA model. These models correspond to the chain graphs as described above but with the extra edges Xi → Yj for each i and j.Akimichi Takemura asked,\"As the number of samples grows, how quickly does the likelihood converge to a unimodal function in the bivariate SUR model?\"\n\nAleksandra Slavkovic\n\nAleksandra Slavkovic spoke about disclosure limitation problems associated with the releasse of marginals and conditionals. Since fixing conditionals is a linear constraint, the theory of Markov bases can be applied to the problem of deciding how many tables have the given fixed conditionals.", "original_statement": "Question 11. Can algebraic techniques be used to see the unimodality of the likelihood function in a MANOVA model. These models correspond to the chain graphs as described above but with the extra edges Xi → Yj for each i and j.Akimichi Takemura asked,\"As the number of samples grows, how quickly does the likelihood converge to a unimodal function in the bivariate SUR model?\" \n\nAleksandra Slavkovic \n\nAleksandra Slavkovic spoke about disclosure limitation problems associated with the releasse of marginals and conditionals. Since fixing conditionals is a linear constraint, the theory of Markov bases can be applied to the problem of deciding how many tables have the given fixed conditionals.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The source is Question 11 in the AIM workshop notes *Computational algebraic statistics*. The relevant text in the official PDF is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 11\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[24]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 11. Can algebraic techniques be used to see the unimodality of the likelihood function in a MANOVA model. These models correspond to the chain graphs as described above but with the extra edges Xi → Yj for each i and j.Akimichi Takemura asked,\\\"As the number of samples grows, how quickly does the likelihood converge to a unimodal function in the bivariate SUR model?\\\" \\n\\nAleksandra Slavkovic \\n\\nAleksandra Slavkovic spoke about disclosure limitation problems associated with the releasse of marginals and conditionals. Since fixing conditionals is a linear constraint, the theory of Markov bases can be applied to the problem of deciding how many tables have the given fixed conditionals.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0025", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For Gaussian common-design MANOVA with rank(X)=p and positive-definite least-squares residual cross-product S0, the full likelihood has exactly one local/global mode, (Bhat,S0/n). After maximizing over the covariance, the exact log-likelihood gap is (n/2) log det(I+C^T C), where C=(X^T X)^{1/2}(B-Bhat)S0^{-1/2}; Cauchy-Binet writes the determinant as one plus the sum of squares of every nonempty minor of C, and this yields an explicit global mode-isolation bound. The result extends exactly to SUR systems whose equation design column spaces are identical. Drton and Richardson's published bivariate crossed-SUR theorem gives almost-sure eventual uniqueness but no explicit finite-sample rate.\n\nCandidate contribution (algebraic_certificate; novelty confidence low): The profiled MANOVA likelihood ratio admits the explicit all-minors certificate det(I+C^T C)=1+sum_{k,I,J} det(C_{I,J})^2, giving both a global quantitative separation from the unique mode and, through a common-basis reparametrization, an exact finite-sample unimodality criterion for all equal-column-space SUR systems." }, { "id": 20001188, "problem_number": "AIM-COMPUTATION-0026", "title": "Exact slack lifts for Markov bases under rounded summaries", "statement": "Question 12. What can be done with Markov bases when their are floating point approxi-mations made in marginals or conditionals?", "original_statement": "Question 12. What can be done with Markov bases when their are floating point approxi-mations made in marginals or conditionals?", "clean_statement": "Question 12. What can be done with Markov bases when their are floating point approxi-mations made in marginals or conditionals?", "statement_status": "exact", "statement_verification": "The exact extracted record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 12\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[25]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 12. What can be done with Markov bases when their are floating point approxi-mations made in marginals or conditionals?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0026", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an integral summary map A and a componentwise floating-point tolerance report, exact ceiling/floor tightening converts the report to integer bounds L <= Ax <= U. The map x -> (x, Ax-L, U-Ax) is a bijection to one ordinary nonnegative integer fiber of an explicit two-slack augmented matrix. Therefore every Markov basis of the augmented matrix projects faithfully to moves connecting the entire tolerance fiber, with no nonzero slack-only move. The attempt also proves a sharp total-interval feasibility criterion for unrestricted two-way row and column margin intervals and identifies the necessary target correction when margins vary.\n\nCandidate contribution (theorem; novelty confidence low): For every finite componentwise tolerance fiber of integral linear summaries, integer tightening followed by the explicit two-slack lift is bijective to an exact fiber; any augmented Markov basis projects to a connecting move set with no slack-only moves, and the same bijection transports any specified target distribution for a correct Metropolis-Hastings chain." }, { "id": 20001189, "problem_number": "AIM-COMPUTATION-0027", "title": "Sufficient releases, conditional fibers, and incompatible Gibbs updates", "statement": "Question 13. What is the family of distributions that has margins and conditionals as sufficient statistics? What should the stationary distribution be for MCMC? Jon Forster asked, \"Is it safe to release full conditionals that have been perturbed?\" Aleksandra replied, \"Probably not.\"\n\nLuis Garcia\n\nLuis Garcia spoke about the algebraic structure of the probability distributions which arise from graphical models with hidden variables. Chris Meek asked, \"What is the algebraic characterization of the Verma constraints? (non-independence constraints)\" Mathias Drton asked, \"Are these the same constraints described by Richardson and Spirtes (Annals of Statistics 2002)?\" Henry Wynn asked, \"How do inequalities play a role in this problem?\"\n\nHenry Wynn\n\nHenry Wynn spoke about formulae relating cumulants to moments.", "original_statement": "Question 13. What is the family of distributions that has margins and conditionals as sufficient statistics? What should the stationary distribution be for MCMC? Jon Forster asked, \"Is it safe to release full conditionals that have been perturbed?\" Aleksandra replied, \"Probably not.\" \n\nLuis Garcia \n\nLuis Garcia spoke about the algebraic structure of the probability distributions which arise from graphical models with hidden variables. Chris Meek asked, \"What is the algebraic characterization of the Verma constraints? (non-independence constraints)\" Mathias Drton asked, \"Are these the same constraints described by Richardson and Spirtes (Annals of Statistics 2002)?\" Henry Wynn asked, \"How do inequalities play a role in this problem?\" \n\nHenry Wynn \n\nHenry Wynn spoke about formulae relating cumulants to moments.", "clean_statement": "Question 13. What is the family of distributions that has margins and conditionals as sufficient statistics? What should the stationary distribution be for MCMC? Jon Forster asked, \"Is it safe to release full conditionals that have been perturbed?\" Aleksandra replied, \"Probably not.\"\n\nLuis Garcia\n\nLuis Garcia spoke about the algebraic structure of the probability distributions which arise from graphical models with hidden variables. Chris Meek asked, \"What is the algebraic characterization of the Verma constraints? (non-independence constraints)\" Mathias Drton asked, \"Are these the same constraints described by Richardson and Spirtes (Annals of Statistics 2002)?\" Henry Wynn asked, \"How do inequalities play a role in this problem?\"\n\nHenry Wynn\n\nHenry Wynn spoke about formulae relating cumulants to moments.", "statement_status": "exact", "statement_verification": "This record comes from the AIM workshop *Computational algebraic statistics*, Question 13. The canonical JSON extraction contains later speaker material, but inspection of the workshop PDF shows that the question ends before the heading “Luis Garcia.” The recovered statement is therefore:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 13\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[26]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 13. What is the family of distributions that has margins and conditionals as sufficient statistics? What should the stationary distribution be for MCMC? Jon Forster asked, \\\"Is it safe to release full conditionals that have been perturbed?\\\" Aleksandra replied, \\\"Probably not.\\\" \\n\\nLuis Garcia \\n\\nLuis Garcia spoke about the algebraic structure of the probability distributions which arise from graphical models with hidden variables. Chris Meek asked, \\\"What is the algebraic characterization of the Verma constraints? (non-independence constraints)\\\" Mathias Drton asked, \\\"Are these the same constraints described by Richardson and Spirtes (Annals of Statistics 2002)?\\\" Henry Wynn asked, \\\"How do inequalities play a role in this problem?\\\" \\n\\nHenry Wynn \\n\\nHenry Wynn spoke about formulae relating cumulants to moments.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0027", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a countable dominated table family, margins and conditional frequencies S are sufficient exactly for families factoring as p_theta(n)=h(n)g_theta(S(n)); the exact conditional MCMC target on a positive-probability fiber is proportional to h, and for log-linear count models it is typically proportional to 1/prod_j n_j!, not uniform. Positive finite full conditionals are compatible exactly when all coordinate-edge ratio products around closed walks equal one. In the two-block case, the unique random-scan stationary laws at two distinct interior scan weights coincide if and only if the conditionals are compatible; hence incompatible perturbed conditionals have no scan-weight-independent Gibbs target.\n\nCandidate contribution (theorem; novelty confidence low): Candidate two-scan-weight compatibility certificate: for strictly positive kernels on a finite two-block product space, equality of the unique invariant laws for any two distinct random-scan weights alpha,beta in (0,1) is equivalent to compatibility of the two full conditionals; an exact binary odds calculation supplies a concrete perturbation obstruction and scan-sensitive stationary family." }, { "id": 20001190, "problem_number": "AIM-COMPUTATION-0028", "title": "Binary independence and hierarchical models in cumulant coordinates", "statement": "Problem 14. Use the transformations from probabilities to moments to cumulants to de-scribe independence models/ hierarchical models in terms of polynomial functions in the cumulants.", "original_statement": "Problem 14. Use the transformations from probabilities to moments to cumulants to de-scribe independence models/ hierarchical models in terms of polynomial functions in the cumulants.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The exact extracted record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 14\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[27]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 14. Use the transformations from probabilities to moments to cumulants to de-scribe independence models/ hierarchical models in terms of polynomial functions in the cumulants.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0028", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The normalized binary probability-to-moment-to-square-free-cumulant transformation is an integral triangular polynomial automorphism, so every hierarchical toric ideal can be transported exactly to cumulant coordinates. For the hierarchical complex containing every proper subset of [n] but omitting the full interaction, the parity binomial becomes a monic polynomial of exact degree 2^(n-1)-1 in the top cumulant k_[n] when all proper cumulants are fixed. A positive three-variable table is also exhibited that has no three-factor log-linear interaction but has k_123 = -2304/95^3, proving that forbidden log-linear interactions cannot naively be replaced by cumulant vanishing.\n\nCandidate contribution (theorem; novelty confidence low): For n at least 2, the binary no-full-n-factor hierarchical model has, in square-free cumulant coordinates, a defining parity equation that is monic of exact degree 2^(n-1)-1 in k_[n] with all proper cumulants fixed; the admissible positive roots are exactly those also satisfying every reconstructed affine cell inequality." }, { "id": 20001191, "problem_number": "AIM-COMPUTATION-0029", "title": "Exact cumulant ideal of the normalized binary four-cycle", "statement": "Question 15. What is the cumulant ideal of the binary four-cycle model?\n\nRussell Steele\n\nRussell Steele spoke about mixture models, model selection, and the Bayesian Infor-mation Criterion (BIC).", "original_statement": "Question 15. What is the cumulant ideal of the binary four-cycle model? \n\nRussell Steele \n\nRussell Steele spoke about mixture models, model selection, and the Bayesian Infor-mation Criterion (BIC).", "clean_statement": "Question 15. What is the cumulant ideal of the binary four-cycle model?\n\nRussell Steele\n\nRussell Steele spoke about mixture models, model selection, and the Bayesian Infor-mation Criterion (BIC).", "statement_status": "exact", "statement_verification": "The source is the AIM workshop list *Computational Algebraic Statistics* (PDF version dated 27 March 2004). The recovered question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 15\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[28]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 15. What is the cumulant ideal of the binary four-cycle model? \\n\\nRussell Steele \\n\\nRussell Steele spoke about mixture models, model selection, and the Bayesian Infor-mation Criterion (BIC).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0029", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "On the normalized affine probability chart, the probability-to-squarefree-cumulant transformation is a polynomial automorphism. Pulling the known prime binary C4 toric ideal through this automorphism gives an exact cumulant generating set: the pullbacks of eight conditional-independence quadrics and eight quartics. Equivalently, the prime cumulant ideal is the saturation of the transformed CI ideal by the product of all sixteen reconstructed cell-probability polynomials. The report also proves a two-equation description of the equal-coupling zero-field Ising slice in a separately defined plus/minus-one cumulant coding.\n\nCandidate contribution (special_case; novelty confidence low): For the plus/minus-one, zero-field, equal-edge-coupling Ising model on C4, if a is an edge second cumulant, d a diagonal second cumulant, and h the fourth joint cumulant, then 2a^2=d+d^2 and h=-2d^2; over the rationals the Zariski closure of the rational high-temperature parameterization has ideal generated by these two relations." }, { "id": 20001192, "problem_number": "AIM-COMPUTATION-0030", "title": "Holonomic Bayesian normalizers with certified posterior moments", "statement": "Question 16. Computing normalization constants and posterior distributions for continuous random variables involves the computation of integrals involving rational and transcendental funtions. Can algebraic techniques be used to give exact expressions for these integrals or help give approximations for them? 7", "original_statement": "Question 16. Computing normalization constants and posterior distributions for continuous random variables involves the computation of integrals involving rational and transcendental funtions. Can algebraic techniques be used to give exact expressions for these integrals or help give approximations for them? 7", "clean_statement": "Question 16. Computing normalization constants and posterior distributions for continuous random variables involves the computation of integrals involving rational and transcendental funtions. Can algebraic techniques be used to give exact expressions for these integrals or help give approximations for them? 7", "statement_status": "exact", "statement_verification": "The record is Question 16 in the AIM workshop problem list *Computational algebraic statistics* (version dated March 27, 2004). It occurs under the speaker heading “Russell Steele,” after a sentence about mixture models, model selection, and BIC. Inspection of the official PDF gives the source text:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 16\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[29]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 16. Computing normalization constants and posterior distributions for continuous random variables involves the computation of integrals involving rational and transcendental funtions. Can algebraic techniques be used to give exact expressions for these integrals or help give approximations for them? 7\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0030", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the exponentially tilted beta posterior, the normalizer is the classical Kummer function B(a,b) 1F1(a;a+b;t), satisfies Kummer's differential equation, and its derivatives recover all raw posterior moments through an exact three-term recurrence. In addition, finite sums of beta values have explicit Taylor-remainder radii that rigorously enclose the normalizer and every unnormalized moment for all real tilts; when the normalizer interval excludes zero, explicit quotient intervals certify all posterior moments and the variance. A fully rational worked certificate is given for (a,b,t,N)=(2,3,1,4), and a scoped undecidability reduction explains why no comparable canonical exact procedure can cover arbitrary transcendental input classes.\n\nCandidate contribution (certificate; novelty confidence low): For every a,b>0, real t, and k,N in the nonnegative integers, the finite beta sum P_{k,N}=sum_{n=0}^N t^n B(a+k+n,b)/n! encloses the k-th unnormalized tilted-beta moment with radius exp(max(t,0)) |t|^(N+1) B(a+k+N+1,b)/(N+1)!; these simultaneous enclosures yield explicit certified intervals for every raw posterior moment and the variance, while a geometric positive-tail bound sharpens the certificate for t>=0." }, { "id": 20001193, "problem_number": "AIM-COMPUTATION-0031", "title": "BIC evidence, mixture singularities, and an exact quotient diagnostic", "statement": "Question 17. Why does the BIC work for model fitting with mixture models? What is being approximated (and why is it good to approximate these things?) This question raised some controversy during the discussion following the talk which was later resolved. The main issue that is interesting in this situation is that the likelihood function for mixture models does not satisfy the necessary regularity conditions to apply the classical results about the BIC.", "original_statement": "Question 17. Why does the BIC work for model fitting with mixture models? What is being approximated (and why is it good to approximate these things?) This question raised some controversy during the discussion following the talk which was later resolved. The main issue that is interesting in this situation is that the likelihood function for mixture models does not satisfy the necessary regularity conditions to apply the classical results about the BIC.", "clean_statement": "Question 17. Why does the BIC work for model fitting with mixture models? What is being approximated (and why is it good to approximate these things?) This question raised some controversy during the discussion following the talk which was later resolved. The main issue that is interesting in this situation is that the likelihood function for mixture models does not satisfy the necessary regularity conditions to apply the classical results about the BIC.", "statement_status": "exact", "statement_verification": "The canonical record is Question 17 from the AIM workshop *Computational algebraic statistics*. The source PDF was inspected directly. The preceding heading says that Russell Steele spoke about mixture models, model selection, and the Bayesian Information Criterion (BIC). The exact question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 17\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[30]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 17. Why does the BIC work for model fitting with mixture models? What is being approximated (and why is it good to approximate these things?) This question raised some controversy during the discussion following the talk which was later resolved. The main issue that is interesting in this situation is that the likelihood function for mixture models does not satisfy the necessary regularity conditions to apply the classical results about the BIC.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0031", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Ordinary BIC approximates negative twice the Bayesian marginal log likelihood by regular Laplace expansion, but that dimension-based expansion is generally wrong on mixture singularities; mixture-order consistency under Keribin-type assumptions is a separate property. A proved quotient-Laplace proposition and exact two-component Bernoulli-mixture calculation show that a nominally three-parameter mixture has interior evidence penalty (1/2) log n rather than (3/2) log n, while a Beta(a,b) quotient prior gives exact boundary coefficients a and b.\n\nCandidate contribution (proposition; novelty confidence low): For the labelled two-component Bernoulli mixture q = alpha p1 + (1-alpha) p2, the evidence coefficient is exactly 1/2 at an interior truth for any proper prior with positive continuous quotient density, but is a at q0 = 0 and b at q0 = 1 for a Beta(a,b) quotient prior; finite label switching can affect only constants and cannot explain these coefficient changes." }, { "id": 20001194, "problem_number": "AIM-COMPUTATION-0032", "title": "Transcendence-aware algebraic localization of symmetric Gaussian-mixture modes", "statement": "Question 18. Can algebra be used to better approximate the location of the modes in a mixture?", "original_statement": "Question 18. Can algebra be used to better approximate the location of the modes in a mixture?", "clean_statement": "Question 18. Can algebra be used to better approximate the location of the modes in a mixture?", "statement_status": "exact", "statement_verification": "The exact AIM question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 18\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[31]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 18. Can algebra be used to better approximate the location of the modes in a mixture?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0032", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the equal-weight Gaussian mixture with means plus and minus mu and one positive-definite common covariance Sigma, all stationary points lie on the line through mu and reduce to z=tanh(lambda z), where lambda=mu^T Sigma^{-1}mu. The origin is the unique strict global mode for lambda at most one, including the Hessian-degenerate threshold; for lambda greater than one the only global modes are plus and minus z_* mu. When algebraic lambda is greater than one, z_* is transcendental, but explicit polynomial equations define nested algebraic lower and upper bounds for z_*^2, with proved monotone convergence and a posteriori error bounds.\n\nCandidate contribution (certified_bound; novelty confidence low): For the symmetric common-covariance two-Gaussian family, the nonzero mode scalar z_* is transcendental whenever algebraic lambda is greater than one, while roots b_m and c_m of explicit polynomial truncation equations satisfy b_m increasing to z_*^2 and c_m decreasing to z_*^2, with computable algebraic error bounds." }, { "id": 20001195, "problem_number": "AIM-COMPUTATION-0033", "title": "Algebraic fibers and sharp sets for missing categorical data", "statement": "Question 19. Can algebraic techniques be used to better imputing missing categorical variable?", "original_statement": "Question 19. Can algebraic techniques be used to better imputing missing categorical variable?", "clean_statement": "Question 19. Can algebraic techniques be used to better imputing missing categorical variable?", "statement_status": "exact", "statement_verification": "The record is Question 19 in the AIM workshop problem list *Computational algebraic statistics*, version dated March 27, 2004. Inspection of the official PDF places it under Russell Steele, immediately after Question 18 about locating modes in a mixture and immediately before Question 20 about collapsing levels of categorical variables for multiple imputation. The source says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 19\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[32]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 19. Can algebraic techniques be used to better imputing missing categorical variable?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0033", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For one fully observed categorical variable X and one partly observed categorical variable Y, the complete-data fiber over respondent cells a_ij and missing row masses m_i is exactly a product of scaled simplices. MAR gives a unique rational completion in every positive-response stratum but leaves zero-response strata unidentified; MCAR with positive overall response is compatible exactly when the 2-by-2 minors of the response-versus-X margin matrix vanish and then gives a unique completion. Intersecting the unrestricted fiber with the rank-one substantive model X independent of Y yields a lower-bounded simplex: with w_i=P(X=i) and L_j=max_{i:w_i>0} a_ij/w_i, feasibility is equivalent to sum_j L_j<=1, all compatible laws are p_ij=w_i theta_j with theta_j>=L_j, and explicit endpoint formulas give sharp intervals for every missing-case imputation probability.\n\nCandidate contribution (theorem; novelty confidence low): In the response-margin observation model with X fully observed, Y partly observed, and complete-data independence X independent of Y, the compatible full laws are exactly the lower-bounded simplex {theta in Delta: theta_j>=max_{i:w_i>0} a_ij/w_i}; consequently sum_j max_i a_ij/w_i<=1 is necessary and sufficient for feasibility, and for every row with m_i>0 the sharp missing-case bounds are (w_i L_j-a_ij)/m_i <= P(Y=j|X=i,R=0) <= (w_i[1-sum_{k!=j}L_k]-a_ij)/m_i." }, { "id": 20001196, "problem_number": "AIM-COMPUTATION-0034", "title": "Determinantal certificates for safe categorical collapse before multiple imputation", "statement": "Question 20. Can algebraic techniques be used to collapse levels of categorical variables in order to improve the quality of multiple imputation without destroying the quality of inference?\n\nFrantisek Matus\n\nFrantisek Matus spoke about the representability of conditional independence (CI) structures.", "original_statement": "Question 20. Can algebraic techniques be used to collapse levels of categorical variables in order to improve the quality of multiple imputation without destroying the quality of inference? \n\nFrantisek Matus \n\nFrantisek Matus spoke about the representability of conditional independence (CI) structures.", "clean_statement": "Question 20. Can algebraic techniques be used to collapse levels of categorical variables in order to improve the quality of multiple imputation without destroying the quality of inference?\n\nFrantisek Matus\n\nFrantisek Matus spoke about the representability of conditional independence (CI) structures.", "statement_status": "exact", "statement_verification": "The canonical record is Question 20 from the AIM workshop list *Computational algebraic statistics*. Direct inspection of the official PDF and its neighboring text shows the following layout:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 20\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[33]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 20. Can algebraic techniques be used to collapse levels of categorical variables in order to improve the quality of multiple imputation without destroying the quality of inference? \\n\\nFrantisek Matus \\n\\nFrantisek Matus spoke about the representability of conditional independence (CI) structures.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0034", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a fully observed fine categorical predictor X, a proposed collapse Z=g(X), fully observed finite covariates W, and a missing categorical outcome Y satisfying fine-level MAR and positivity, collapsing before saturated imputation preserves the target law P(Y|Z,W) exactly if and only if explicit fiberwise 2-by-2 determinant polynomials vanish. The same calculation gives the exact target-score bias Cov_pi(rho_i,m_i)/rho_bar, a sharp range bound, universal robustness criteria, and a complete two-level factorization. Retaining X through imputation and collapsing afterward avoids this additional determinant condition for coarsened targets. Counterexamples establish loss of fine-level targets and nonidentifiability under unrestricted MNAR.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: the MI-specific safe-collapse ideal generated by the fiberwise determinants, paired with the exact covariance bias identity, the universal robustness dichotomy, and the two-level factorization." }, { "id": 20001197, "problem_number": "AIM-COMPUTATION-0035", "title": "Exact CI rationalization through rational local charts", "statement": "Conjecture 21. Every p-representable CI-structure can be p-represented by a random vector living on a finite probability space endowed with the uniform distribution.", "original_statement": "Conjecture 21. Every p-representable CI-structure can be p-represented by a random vector living on a finite probability space endowed with the uniform distribution.", "clean_statement": "Conjecture 21. Every p-representable CI-structure can be p-represented by a random vector living on a finite probability space endowed with the uniform distribution.", "statement_status": "exact", "statement_verification": "The canonical record is Conjecture 21 from the AIM workshop list *Computational algebraic statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 21\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[34]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 21. Every p-representable CI-structure can be p-represented by a random vector living on a finite probability space endowed with the uniform distribution.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0035", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Conjecture 21 remains open, with Matúš proving rationality through four variables and Boege explicitly recording an equivalent formulation as open in 2024. For fixed finite alphabets, a proved exact-pattern rational-chart lemma shows that any p-representation lying on a Q-rationally parametrized local branch contained in the asserted-CI locus admits a rational representation, provided rational parameters are dense near the representing point and probability-domain conditions persist: finitely many selected nonzero CI minors preserve every absent CI, and a common-denominator refinement then yields a uniform finite probability space. A second proved proposition derives the equivalence with rational counterexamples to every invalid finite CI implication by coordinatewise independent products.\n\nCandidate contribution (lemma; novelty confidence low): Exact-pattern rational-chart lemma: on fixed finite alphabets, a Q-rational chart whose local image remains in the asserted-CI locus and probability simplex, whose denominators stay nonzero, and whose rational parameter points are dense near an exact representing point contains a rational probability table with precisely the same complete CI structure; one chosen nonzero polynomial minor for each absent CI proves exact-pattern preservation.", "research_difficulty_suggested": "L4" }, { "id": 20001198, "problem_number": "AIM-COMPUTATION-0036", "title": "Componentwise entropy rigidity and a reduced search for a non-multilinear matroid", "statement": "Conjecture 22. There exists a connected matroid whose CI-structure is p-representable but not multilinear.\n\nSuppose independence vatieties are properly defined, also for the CI constraints involv-ing functional independences, over the field of complex numbers.", "original_statement": "Conjecture 22. There exists a connected matroid whose CI-structure is p-representable but not multilinear. \n\nSuppose independence vatieties are properly defined, also for the CI constraints involv-ing functional independences, over the field of complex numbers.", "clean_statement": "Conjecture 22. There exists a connected matroid whose CI-structure is p-representable but not multilinear.\n\nSuppose independence vatieties are properly defined, also for the CI constraints involv-ing functional independences, over the field of complex numbers.", "statement_status": "exact", "statement_verification": "The exact conjecture in the official AIM *Computational Algebraic Statistics* workshop report is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 22\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[35]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 22. There exists a connected matroid whose CI-structure is p-representable but not multilinear. \\n\\nSuppose independence vatieties are properly defined, also for the CI constraints involv-ing functional independences, over the field of complex numbers.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0036", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite matroid with connected components E_j, any finite discrete random vector satisfying all independence and functional-dependence equalities of the augmented matroid CI structure has entropy H(X_A)=sum_j c_j r(A intersect E_j), with nonnegative component scales and positive scales on every positive-rank component for an exact representation; conversely, any entropic function of this form with positive relevant scales has exactly the augmented matroid CI zero pattern. Hence, for connected matroids, p-representability is equivalent to entropic representability, and the AIM conjecture is precisely the still-open request for a connected entropic matroid that is not multilinear. Any witness must have rank and corank at least three, cannot have partition degree two or three, cannot be a rank-at-least-three Dowling geometry, and an entropic Ingleton violator would immediately suffice.\n\nCandidate contribution (reduction; novelty confidence low): The componentwise entropy-rigidity formula H(X_A)=sum_j c_j r(A intersect E_j), including the exact-zero-pattern converse, gives a direct decomposition theorem for probabilistic realizations of augmented matroid CI structures and yields a four-part certified filter for any proposed entropic non-multilinear witness.", "research_difficulty_suggested": "L4" }, { "id": 20001199, "problem_number": "AIM-COMPUTATION-0037", "title": "A zero-dimensional parity independence variety", "statement": "Conjecture 23. When P belongs to such a zero dimensional independence variety then P\n\nis the distribution of a random vector with its CI-structure equal to the CI-structure of a matroid.\n\nDonald Richards\n\nDonald Richards spoke on the problem of determining the number of solutions to generic maximum likelihood problems and the use of computational algebra, homotopy con-tinuation, and mixed volume in these calculations.", "original_statement": "Conjecture 23. When P belongs to such a zero dimensional independence variety then P\n\nis the distribution of a random vector with its CI-structure equal to the CI-structure of a matroid. \n\nDonald Richards \n\nDonald Richards spoke on the problem of determining the number of solutions to generic maximum likelihood problems and the use of computational algebra, homotopy con-tinuation, and mixed volume in these calculations.", "clean_statement": "Conjecture 23. When P belongs to such a zero dimensional independence variety then P\n\nis the distribution of a random vector with its CI-structure equal to the CI-structure of a matroid.\n\nDonald Richards\n\nDonald Richards spoke on the problem of determining the number of solutions to generic maximum likelihood problems and the use of computational algebra, homotopy con-tinuation, and mixed volume in these calculations.", "statement_status": "exact", "statement_verification": "The canonical record contains:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 23\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[36]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Conjecture 23. When P belongs to such a zero dimensional independence variety then P\\n\\nis the distribution of a random vector with its CI-structure equal to the CI-structure of a matroid. \\n\\nDonald Richards \\n\\nDonald Richards spoke on the problem of determining the number of solutions to generic maximum likelihood problems and the use of computational algebra, homotopy con-tinuation, and mixed volume in these calculations.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0037", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For three binary variables on the fixed functional-dependence component Z=X xor Y, the three pairwise conditional-independence determinants generate a homogeneous ideal whose projective complex variety is eight reduced points: four coordinate vertices and four torus sign points. Probability normalization leaves exactly four point masses and the uniform parity law; projective torus saturation retains the four sign points, while normalization followed by full-support saturation retains only the uniform law. Every simplex point has exact augmented matroid CI structure, namely U_{0,3} at the vertices and U_{2,3} at the uniform point. More generally, any nondegenerate three-binary fixed-map model with full (X,Y) input support and all three pairwise independences is XOR or XNOR with the uniform law. Ordinary CI equations alone cannot define a zero-dimensional normalized model because they contain the positive-dimensional Segre family of product laws.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: the explicit comparison of the eight-point projective parity scheme, five-point normalized scheme, four-point projective torus saturation, and one-point normalized full-support saturation, together with exact matroid identification of every probability point and the binary fixed-map rigidity proposition.", "research_difficulty_suggested": "L4" }, { "id": 20001200, "problem_number": "AIM-COMPUTATION-0038", "title": "Behrens–Fisher likelihood equations and an equal-scatter bifurcation", "statement": "Problem 24 (Behrens-Fischer Problem). Solve the maximum likelihood equations for the mixture of two multivariate Gaussians with the same mean and different covariance matrices.", "original_statement": "Problem 24 (Behrens-Fischer Problem). Solve the maximum likelihood equations for the mixture of two multivariate Gaussians with the same mean and different covariance matrices.", "clean_statement": "Problem 24 (Behrens-Fischer Problem). Solve the maximum likelihood equations for the mixture of two multivariate Gaussians with the same mean and different covariance matrices.", "statement_status": "exact", "statement_verification": "The canonical AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 24\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[37]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 24 (Behrens-Fischer Problem). Solve the maximum likelihood equations for the mixture of two multivariate Gaussians with the same mean and different covariance matrices.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0038", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM wording is resolved as the labelled two-sample Behrens–Fisher model, not an unlabelled finite mixture: Buot, Hoşten, and Richards proved in 2007 that its generic saturated likelihood system has ML degree 2p+1, and Belloni and Didier supplied a globally convergent maximization algorithm in 2008. This attempt rederives the exact profiled rational equations with the 1/n maximum-likelihood scatter convention and proves a global special-family theorem: for equal sample sizes and equal positive-definite within-group scatters, the Mahalanobis separation D=2 is a sharp bifurcation, with a unique midpoint MLE for D≤2 (quartically flat at equality) and exactly two explicit global MLE means plus one nonmaximizing midpoint stationary point for D>2. Under the literal latent-mixture reading, an elementary covariance-collapse proof shows the unconstrained likelihood is unbounded.\n\nCandidate contribution (theorem; novelty confidence low): For the labelled multivariate Behrens–Fisher model with n1=n2=n>p and equal positive-definite within-group ML scatters S, let D be the S^{-1}-Mahalanobis distance between sample means. If D≤2, their midpoint is the unique global common-mean MLE, with quartic flatness at D=2; if D>2, the only stationary means are the midpoint and m ± sqrt(D^2/4-1)(xbar2-xbar1)/D, and the latter two are precisely the global MLEs." }, { "id": 20001201, "problem_number": "AIM-COMPUTATION-0039", "title": "Gaussian observed-likelihood equations and a rank-certified monotone solution", "statement": "Problem 25. Solve the maximum likelihood equations for a multivariate guassian with arbitrary missing data patterns.", "original_statement": "Problem 25. Solve the maximum likelihood equations for a multivariate guassian with arbitrary missing data patterns.", "clean_statement": "Problem 25. Solve the maximum likelihood equations for a multivariate guassian with arbitrary missing data patterns.", "statement_status": "exact", "statement_verification": "The canonical record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 25\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[38]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 25. Solve the maximum likelihood equations for a multivariate guassian with arbitrary missing data patterns.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0039", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For arbitrary fixed ignorable observation patterns, the report derives the exact observed-data score equations for an unrestricted Gaussian mean and positive-definite covariance. The Gaussian parameter is structurally identifiable from the population marginals of the occurring patterns exactly when every coordinate is observed and every pair is jointly observed. For the two-block monotone family, where X is observed in all N rows and Y in n complete rows, the likelihood factors into a marginal Gaussian likelihood for X and a conditional multivariate regression for Y given X. An interior global MLE exists exactly when the centered all-row X scatter and minimized complete-case regression residual scatter are positive definite; it is unique exactly when the augmented complete-case design [1,X] also has full column rank. Explicit rational formulas are proved, while rank-deficient design gives a coefficient family and singular scatter makes the likelihood unbounded at the covariance boundary.\n\nCandidate contribution (special_case; novelty confidence low): The vertex-and-pair pattern-coverage criterion and the two-block monotone three-rank certificate together give a testable algebraic preconditioner that separates structural identifiability, a unique rational likelihood solution, a positive-dimensional coefficient family, and covariance-boundary blow-up." }, { "id": 20001202, "problem_number": "AIM-COMPUTATION-0040", "title": "Semialgebraic stratification of maximum likelihood", "statement": "Question 26. What are the possible implications of semi-algebraic geometry to maximum likelihood estimation?\n\nSeth Sullivant\n\nSeth Sullivant spoke about Markov bases for hierarchical models and theoretical results about the structure of minimal Markov basis elements. 8", "original_statement": "Question 26. What are the possible implications of semi-algebraic geometry to maximum likelihood estimation? \n\nSeth Sullivant \n\nSeth Sullivant spoke about Markov bases for hierarchical models and theoretical results about the structure of minimal Markov basis elements. 8", "clean_statement": "Question 26. What are the possible implications of semi-algebraic geometry to maximum likelihood estimation?\n\nSeth Sullivant\n\nSeth Sullivant spoke about Markov bases for hierarchical models and theoretical results about the structure of minimal Markov basis elements. 8", "statement_status": "exact", "statement_verification": "The canonical record contains Question 26 followed by material from the next speaker entry. The official AIM workshop PDF shows that the recovered statement is exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 26\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[39]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 26. What are the possible implications of semi-algebraic geometry to maximum likelihood estimation? \\n\\nSeth Sullivant \\n\\nSeth Sullivant spoke about Markov bases for hierarchical models and theoretical results about the structure of minimal Markov basis elements. 8\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0040", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the folded Bernoulli model q(theta)=theta(1-theta) on 0= max(a,b,c) and r(a+b+c-2) < abc, proving that matrix flattening minors are vacuous while the three-way hidden-class ideal is nonzero, together with an explicit three-state quartic witness whose cleared commutator has (1,1)-entry 1." }, { "id": 20001207, "problem_number": "AIM-COMPUTATION-0045", "title": "The stochastic image of a binary phylogenetic tripod", "statement": "Problem 31. Determine the image of the stochastic parameterization (as opposed to the complex parameterization).", "original_statement": "Problem 31. Determine the image of the stochastic parameterization (as opposed to the complex parameterization).", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "Three historically plausible readings must be distinguished.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 31\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[44]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 31. Determine the image of the stochastic parameterization (as opposed to the complex parameterization).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0045", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After documenting that the source leaves the phylogenetic tree, state space, and substitution model unspecified, the artifacts prove an endpoint-complete semialgebraic membership theorem for the uniform-root discrete-time binary symmetric tripod. In three Fourier correlation coordinates, the theorem distinguishes the raw complex image, its Zariski closure, the closed nonnegative stochastic image, the strictly positive transition image, and the continuous-time CFN subimage; it also recovers generic and singular parameter fibers, detects deterministic edges, and gives an all-positive observed distribution that nevertheless forces a boundary transition matrix. A separate general proposition proves that the closed finite-state general-Markov image is the Euclidean closure of the strictly positive image.\n\nCandidate contribution (theorem; novelty confidence low): The exact combined closed/strict/complex membership and parameter-fiber stratification for the uniform-root discrete-time binary symmetric tripod, including the positive-observed distribution that forces a deterministic edge, is a candidate new endpoint-complete synthesis." }, { "id": 20001208, "problem_number": "AIM-COMPUTATION-0046", "title": "Generator-invariant local efficacy for phylogenetic invariants", "statement": "Problem 32. \"Finding good invariants\": Do some of the invariants have more of a statis-tical/biological significance than others?", "original_statement": "Problem 32. \"Finding good invariants\": Do some of the invariants have more of a statis-tical/biological significance than others?", "clean_statement": "Problem 32. \"Finding good invariants\": Do some of the invariants have more of a statis-tical/biological significance than others?", "statement_status": "exact", "statement_verification": "This is Problem 32 in Elizabeth Allman's contribution to the AIM workshop *Computational algebraic statistics*. The exact canonical record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 32\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[45]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 32. \\\"Finding good invariants\\\": Do some of the invariants have more of a statis-tical/biological significance than others?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0046", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "At a regular interior site-pattern distribution p, let f be a vector of polynomial invariant residuals, J=Df(p), and V=J(diag(p)-pp^T)J^T positive definite. The statistic N f(phat)^T V^{-1}f(phat) has an asymptotic chi-square null law and a noncentral chi-square law under p+h/sqrt(N), with noncentrality (Jh)^T V^{-1}(Jh). The statistic and local efficacy are exactly invariant under invertible constant recombinations or rescalings of the residual basis. Among nonzero scalar combinations, the optimal combination for a specified local alternative is proportional to V^{-1}Jh; if Jh=0, the chosen residual span has no root-N first-order power. This supplies a task-specific statistical meaning of a good invariant while proving that raw algebraic generator status cannot intrinsically rank residuals.\n\nCandidate contribution (theorem; novelty confidence low): The candidate contribution packages multinomial null calibration, local noncentrality, exact invariance under constant changes of invariant basis, the optimal scalar residual combination for a specified biological alternative, and the Jh=0 first-order-power obstruction into one proved criterion." }, { "id": 20001209, "problem_number": "AIM-COMPUTATION-0047", "title": "A joint tree and JC69/K80 selector with a finite-sample certificate", "statement": "Question 33. How can we efficiently decide which trees fit the data best? Which model of mutation is most accurate?", "original_statement": "Question 33. How can we efficiently decide which trees fit the data best? Which model of mutation is most accurate?", "clean_statement": "Question 33. How can we efficiently decide which trees fit the data best? Which model of mutation is most accurate?", "statement_status": "exact", "statement_verification": "The canonical record is Question 33 in the American Institute of Mathematics list *Computational Algebraic Statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 33\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[46]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 33. How can we efficiently decide which trees fit the data best? Which model of mutation is most accurate?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0047", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For N iid DNA sites generated on one n-taxon unrooted binary tree by a homogeneous stationary-uniform JC69 or K80 process with bounded positive rates and edge lengths, a pairwise-only selector jointly recovers the tree topology and the minimal adequate member of {JC69,K80}. It uses LogDet four-point splits and a normalized transition-versus-transversion contrast. With rho=4^-4 exp(-12R(n-1)t_max), Gamma=24r t_min, lambda_0=(delta t_min/4)exp(-4R(n-1)t_max), and epsilon_N=sqrt(log(32 binom(n,2)/eta)/(2N)), simultaneous recovery has probability at least 1-eta when epsilon_N4 epsilon, all quartets are declared correctly and uniquely recover the tree; N>8 kappa^4 gamma^{-2} log(2 binom(n,4) kappa^4/delta) suffices.\n\nCandidate contribution (theorem; novelty confidence low): The candidate contribution is a proved safe-abstention SVD-rank quartet rule with simultaneous dependence-robust familywise correctness, a compatible partial-tree confidence set, a decisive positive-weight assembly guarantee, and an explicit exact-recovery sample bound." }, { "id": 20001212, "problem_number": "AIM-COMPUTATION-0050", "title": "Root obstruction and outgroup-oriented algebraic clade certificates", "statement": "Question 36. Can we develop algebraic invariant techniques for identifying clades?\n\nShmuel Onn\n\nShmuel Onn spoke about the spectra of hierarchical models. The spectra is defined to be the set of all sets of possible cell entries in for a multiway nonnegative integral table with fixed marginal totals.", "original_statement": "Question 36. Can we develop algebraic invariant techniques for identifying clades? \n\nShmuel Onn \n\nShmuel Onn spoke about the spectra of hierarchical models. The spectra is defined to be the set of all sets of possible cell entries in for a multiway nonnegative integral table with fixed marginal totals.", "clean_statement": "Question 36. Can we develop algebraic invariant techniques for identifying clades?\n\nShmuel Onn\n\nShmuel Onn spoke about the spectra of hierarchical models. The spectra is defined to be the set of all sets of possible cell entries in for a multiway nonnegative integral table with fixed marginal totals.", "statement_status": "exact", "statement_verification": "The official AIM PDF resolves the apparent topic change. On PDF page 7, Questions 29--36 are listed under the heading **Elizabeth Allman**. Immediately after Question 36, **Shmuel Onn** appears as a new speaker heading; the paragraph about spectra follows it, and Problems 37--38 concern those spectra. Thus the last two paragraphs in the canonical record are extraction bleed from the next speaker section. They are preserved verbatim in `input.json`, but they are not part of the phylogenetic question and are not used below.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 36\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[49]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 36. Can we develop algebraic invariant techniques for identifying clades? \\n\\nShmuel Onn \\n\\nShmuel Onn spoke about the spectra of hierarchical models. The spectra is defined to be the set of all sets of possible cell entries in for a multiway nonnegative integral table with fixed marginal totals.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0050", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The canonical record contains extraction bleed: the verified Question 36 is only the clade question, while the Shmuel Onn table-spectra paragraphs begin the next speaker section. For site-pattern data, a common-stationary-distribution reversible model is exactly invariant under rerooting at every stochastic parameter point, so no population invariant can orient an edge split into a clade. Under the k-state general Markov model, an edge-split flattening has rank at most k for all parameters, while a nontrivial nonedge split has generic nondegenerate rank at least k^2. If the root is explicitly constrained to a declared outgroup's pendant edge, the side of a detected split not containing the outgroup is a clade. The tail singular-value score obeys D_k(hat p)<=||hat p-p||_2 and, for N independent sites, P[D_k>1/sqrt(N alpha)]<=alpha for every true split; this gives a conservative, potentially vacuous, one-sided clade-rejection certificate.\n\nCandidate contribution (theorem; novelty confidence low): A root-aware clade-identifiability trichotomy is proved: stationary reversible leaf data cannot orient a split; GM edge minors generically identify the unordered nontrivial split; an explicit root constraint on the outgroup side converts that split into a clade. The synthesis includes the distribution-free finite-sample rejection threshold 1/sqrt(N alpha) and proves transpose-based orientation blindness of every edge-minor or singular-value score." }, { "id": 20001213, "problem_number": "AIM-COMPUTATION-0051", "title": "A gap-move bound and the exact decomposable-model spectrum", "statement": "Problem 37. Find the spectra of various models and classes of models.", "original_statement": "Problem 37. Find the spectra of various models and classes of models.", "clean_statement": "Problem 37. Find the spectra of various models and classes of models.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 37 from the 2004 AIM workshop list *Computational algebraic statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 37\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[50]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 37. Find the spectra of various models and classes of models.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0051", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite nonnegative integer fiber and a Markov basis B, every difference between consecutive attainable values of coordinate q is at most max_{g in B}|g_q|; when that maximum is zero the coordinate range is a singleton. Consequently any model with a {0,+1,-1}-valued Markov basis has interval cell ranges. Dobra's primitive-data-swap theorem then implies that the spectrum of the class of all decomposable graphical contingency-table models, over arbitrary finite formats, is exactly the collection of nonempty finite integer intervals. A direct integral-flow proof also gives the exact Frechet interval for every cell in the two-clique model [AS][SB].\n\nCandidate contribution (theorem_and_classification; novelty confidence low): Candidate gap-move spectrum principle: for every finite fiber, each run of missing q-coordinate values has length at most max_{g in B}|g_q|-1 for any Markov basis B with positive q-width (and zero width forces a singleton); hence, using primitive data swaps, the union-over-formats spectrum of decomposable graphical models is exactly all nonempty finite integer intervals." }, { "id": 20001214, "problem_number": "AIM-COMPUTATION-0052", "title": "A precise probabilistic repair and algebraic criteria for interval cell spectra", "statement": "Question 38. With respect to some distribution is the set of tables whose range of cell entries are intervals dense among all tables?\n\nJoseph Landsberg Ilias Kotsireas\n\nChallenge problem in Gr¨ obner bases of polynomial ideals.\n\nThe system of polynomial equations given in 3 different formats at http://www.cargo.wlu.ca/hi/had7.cocoa http://www.cargo.wlu.ca/hi/had7.maple http://www.cargo.wlu.ca/hi/had7.reduce has 41 equations in the 27 variables a1,..., a27.Prove that this system of polynomial equations, does not have any solutions. This would constitute an algebraic proof of the combinatorial result of L. D. Baumert:\n\nthere are no Hadamard matrices of order 28 with one circulant core.9\n\nMoreover, it is of interest to write down the effective Nullstellensatz for this ideal, i.e. write down explicitly a linear combination of (some of) the generators of the ideal, which is equal to 1. Similar algebraic proofs could then be devised for Hadamard matrices with one circulant core of bigger order. Reference: Ilias Kotsireas, Christos Koukouvinos, Jennifer Seberry, Hadamard ideals and Hadamard matrices with one circulant core. Submitted, November 2003. Ideas that were suggested during the workshop:\n\n• (Bernd Sturmfels) Devise some kind of incremental process to write down the effective Nullstellensatz. e.g. break the system into pieces, write the effective Nullstellensatz for each piece separately and then combine the results to obtain the effective Null-stellensatz for the whole system;\n\n• (Beth Arnold) Use CoCoA and her own mod p Gr¨ obner implementation, to see if the Gr¨ obner basis can be computed.\n\n• (Nick Eriksson) Use 4ti2, to see if the Grobner basis can be computed.\n\nDesign efficient heuristics in binary trees.\n\nGiven sets of binary vectors in a set of binary trees with given depths, design efficient heuristics to predict the positions of the analogous vectors in binary trees of bigger depth. The adjective \"analogous\" can take on various meanings. In our context, it designates solu-tions of the associated polynomial system. Ideas that were suggested during the workshop: (Persi Diaconis) first of all obtain graphical visualizations and then analyze the data with DFT and other transforms. Work in progress.\n\nIndicator function approach for Hadamard Equivalence.\n\nCheck data sets of Hadamard matrices of big orders to identify the inequivalent Hadamard matrices. Ideas that were suggested during the workshop: (Maria Piera Rogantin) A JSPI paper by Fontana-Pistone-Rogantin contains a description of the indicator function approach to check Hadamard equivalence. In general, one needs to compute the indicator function which has exponentially many monomials. However, this approach is especially efficient for Hadamard matrices with some structure, e.g. with one circulant core. The main reason for this is that the exponentially many monomials fall naturally in a few categories and we only need to check equality of the coefficients of one representative from these categories. Work in progress.\n\nFirst Open Problem Session", "original_statement": "Question 38. With respect to some distribution is the set of tables whose range of cell entries are intervals dense among all tables? \n\nJoseph Landsberg Ilias Kotsireas \n\nChallenge problem in Gr¨ obner bases of polynomial ideals. \n\nThe system of polynomial equations given in 3 different formats at http://www.cargo.wlu.ca/hi/had7.cocoa http://www.cargo.wlu.ca/hi/had7.maple http://www.cargo.wlu.ca/hi/had7.reduce has 41 equations in the 27 variables a1,..., a27.Prove that this system of polynomial equations, does not have any solutions. This would constitute an algebraic proof of the combinatorial result of L. D. Baumert: \n\nthere are no Hadamard matrices of order 28 with one circulant core.9\n\nMoreover, it is of interest to write down the effective Nullstellensatz for this ideal, i.e. write down explicitly a linear combination of (some of) the generators of the ideal, which is equal to 1. Similar algebraic proofs could then be devised for Hadamard matrices with one circulant core of bigger order. Reference: Ilias Kotsireas, Christos Koukouvinos, Jennifer Seberry, Hadamard ideals and Hadamard matrices with one circulant core. Submitted, November 2003. Ideas that were suggested during the workshop: \n\n• (Bernd Sturmfels) Devise some kind of incremental process to write down the effective Nullstellensatz. e.g. break the system into pieces, write the effective Nullstellensatz for each piece separately and then combine the results to obtain the effective Null-stellensatz for the whole system; \n\n• (Beth Arnold) Use CoCoA and her own mod p Gr¨ obner implementation, to see if the Gr¨ obner basis can be computed. \n\n• (Nick Eriksson) Use 4ti2, to see if the Grobner basis can be computed. \n\nDesign efficient heuristics in binary trees. \n\nGiven sets of binary vectors in a set of binary trees with given depths, design efficient heuristics to predict the positions of the analogous vectors in binary trees of bigger depth. The adjective \"analogous\" can take on various meanings. In our context, it designates solu-tions of the associated polynomial system. Ideas that were suggested during the workshop: (Persi Diaconis) first of all obtain graphical visualizations and then analyze the data with DFT and other transforms. Work in progress. \n\nIndicator function approach for Hadamard Equivalence. \n\nCheck data sets of Hadamard matrices of big orders to identify the inequivalent Hadamard matrices. Ideas that were suggested during the workshop: (Maria Piera Rogantin) A JSPI paper by Fontana-Pistone-Rogantin contains a description of the indicator function approach to check Hadamard equivalence. In general, one needs to compute the indicator function which has exponentially many monomials. However, this approach is especially efficient for Hadamard matrices with some structure, e.g. with one circulant core. The main reason for this is that the exponentially many monomials fall naturally in a few categories and we only need to check equality of the coefficients of one representative from these categories. Work in progress. \n\nFirst Open Problem Session", "clean_statement": "Question 38. With respect to some distribution is the set of tables whose range of cell entries are intervals dense among all tables?\n\nJoseph Landsberg Ilias Kotsireas\n\nChallenge problem in Gr¨ obner bases of polynomial ideals.\n\nThe system of polynomial equations given in 3 different formats at http://www.cargo.wlu.ca/hi/had7.cocoa http://www.cargo.wlu.ca/hi/had7.maple http://www.cargo.wlu.ca/hi/had7.reduce has 41 equations in the 27 variables a1,..., a27.Prove that this system of polynomial equations, does not have any solutions. This would constitute an algebraic proof of the combinatorial result of L. D. Baumert:\n\nthere are no Hadamard matrices of order 28 with one circulant core.9\n\nMoreover, it is of interest to write down the effective Nullstellensatz for this ideal, i.e. write down explicitly a linear combination of (some of) the generators of the ideal, which is equal to 1. Similar algebraic proofs could then be devised for Hadamard matrices with one circulant core of bigger order. Reference: Ilias Kotsireas, Christos Koukouvinos, Jennifer Seberry, Hadamard ideals and Hadamard matrices with one circulant core. Submitted, November 2003. Ideas that were suggested during the workshop:\n\n• (Bernd Sturmfels) Devise some kind of incremental process to write down the effective Nullstellensatz. e.g. break the system into pieces, write the effective Nullstellensatz for each piece separately and then combine the results to obtain the effective Null-stellensatz for the whole system;\n\n• (Beth Arnold) Use CoCoA and her own mod p Gr¨ obner implementation, to see if the Gr¨ obner basis can be computed.\n\n• (Nick Eriksson) Use 4ti2, to see if the Grobner basis can be computed.\n\nDesign efficient heuristics in binary trees.\n\nGiven sets of binary vectors in a set of binary trees with given depths, design efficient heuristics to predict the positions of the analogous vectors in binary trees of bigger depth. The adjective \"analogous\" can take on various meanings. In our context, it designates solu-tions of the associated polynomial system. Ideas that were suggested during the workshop: (Persi Diaconis) first of all obtain graphical visualizations and then analyze the data with DFT and other transforms. Work in progress.\n\nIndicator function approach for Hadamard Equivalence.\n\nCheck data sets of Hadamard matrices of big orders to identify the inequivalent Hadamard matrices. Ideas that were suggested during the workshop: (Maria Piera Rogantin) A JSPI paper by Fontana-Pistone-Rogantin contains a description of the indicator function approach to check Hadamard equivalence. In general, one needs to compute the indicator function which has exponentially many monomials. However, this approach is especially efficient for Hadamard matrices with some structure, e.g. with one circulant core. The main reason for this is that the exponentially many monomials fall naturally in a few categories and we only need to check equality of the coefficients of one representative from these categories. Work in progress.\n\nFirst Open Problem Session", "statement_status": "exact", "statement_verification": "The canonical JSON record contains a genuine question followed by unrelated text about Hadamard matrices. The official AIM workshop page separates these items by speaker. Under **Shmuel Onn** it says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 38\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[51]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 38. With respect to some distribution is the set of tables whose range of cell entries are intervals dense among all tables? \\n\\nJoseph Landsberg Ilias Kotsireas \\n\\nChallenge problem in Gr¨ obner bases of polynomial ideals. \\n\\nThe system of polynomial equations given in 3 different formats at http://www.cargo.wlu.ca/hi/had7.cocoa http://www.cargo.wlu.ca/hi/had7.maple http://www.cargo.wlu.ca/hi/had7.reduce has 41 equations in the 27 variables a1,..., a27.Prove that this system of polynomial equations, does not have any solutions. This would constitute an algebraic proof of the combinatorial result of L. D. Baumert: \\n\\nthere are no Hadamard matrices of order 28 with one circulant core.9\\n\\nMoreover, it is of interest to write down the effective Nullstellensatz for this ideal, i.e. write down explicitly a linear combination of (some of) the generators of the ideal, which is equal to 1. Similar algebraic proofs could then be devised for Hadamard matrices with one circulant core of bigger order. Reference: Ilias Kotsireas, Christos Koukouvinos, Jennifer Seberry, Hadamard ideals and Hadamard matrices with one circulant core. Submitted, November 2003. Ideas that were suggested during the workshop: \\n\\n• (Bernd Sturmfels) Devise some kind of incremental process to write down the effective Nullstellensatz. e.g. break the system into pieces, write the effective Nullstellensatz for each piece separately and then combine the results to obtain the effective Null-stellensatz for the whole system; \\n\\n• (Beth Arnold) Use CoCoA and her own mod p Gr¨ obner implementation, to see if the Gr¨ obner basis can be computed. \\n\\n• (Nick Eriksson) Use 4ti2, to see if the Grobner basis can be computed. \\n\\nDesign efficient heuristics in binary trees. \\n\\nGiven sets of binary vectors in a set of binary trees with given depths, design efficient heuristics to predict the positions of the analogous vectors in binary trees of bigger depth. The adjective \\\"analogous\\\" can take on various meanings. In our context, it designates solu-tions of the associated polynomial system. Ideas that were suggested during the workshop: (Persi Diaconis) first of all obtain graphical visualizations and then analyze the data with DFT and other transforms. Work in progress. \\n\\nIndicator function approach for Hadamard Equivalence. \\n\\nCheck data sets of Hadamard matrices of big orders to identify the inequivalent Hadamard matrices. Ideas that were suggested during the workshop: (Maria Piera Rogantin) A JSPI paper by Fontana-Pistone-Rogantin contains a description of the indicator function approach to check Hadamard equivalence. In general, one needs to compute the indicator function which has exponentially many monomials. However, this approach is especially efficient for Hadamard matrices with some structure, e.g. with one circulant core. The main reason for this is that the exponentially many monomials fall naturally in a few categories and we only need to check equality of the coefficients of one representative from these categories. Work in progress. \\n\\nFirst Open Problem Session\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0052", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM question is not a well-posed probability statement until a law and a meaning of density are supplied. For a fixed integer marginal matrix A, the cell spectrum has the exact representation S_j(b)={k>=0: b-k a_j is in N A_{-j}}. If the deleted-column semigroup is normal relative to its lattice and a_j lies in that lattice, then every such spectrum is an interval; if this holds for all cells, every distribution gives interval-good tables probability one. Independently, total unimodularity of A also forces all spectra to be intervals. The order of a_j in Z A/Z A_{-j} supplies a congruence obstruction, and a toy multinomial model realizes repaired asymptotic density zero.\n\nCandidate contribution (criterion; novelty confidence low): Deleted-column normality N A_{-j}=cone(A_{-j}) intersect Z A_{-j}, together with a_j in Z A_{-j}, is a checkable sufficient condition for the j-th cell spectrum to be an interval for every attainable margin; imposed for every j, it yields probability exactly one under any table or margin distribution." }, { "id": 20001215, "problem_number": "AIM-COMPUTATION-0053", "title": "One observable mode can lift to infinitely many hidden-parameter maxima", "statement": "Question 39 (Sturmfels). In a graphical model with hidden variables how many (nonneg-ative) real modes might exist in solutions to the likelihood equations?", "original_statement": "Question 39 (Sturmfels). In a graphical model with hidden variables how many (nonneg-ative) real modes might exist in solutions to the likelihood equations?", "clean_statement": "Question 39 (Sturmfels). In a graphical model with hidden variables how many (nonneg-ative) real modes might exist in solutions to the likelihood equations?", "statement_status": "exact", "statement_verification": "The canonical record is Question 39 from the AIM workshop notes *Computational algebraic statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 39\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[52]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 39 (Sturmfels). In a graphical model with hidden variables how many (nonneg-ative) real modes might exist in solutions to the likelihood equations?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0053", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the latent-class graph X<-H->Y with r,c>=2 observed states, h>=min(r,c) hidden classes, and strictly positive table counts, the observable model is the full r-by-c probability simplex and its likelihood has the unique strict global mode p_hat=u/N. Nevertheless, the fiber over p_hat contains an uncountable fully interior curve of positive hidden parameters, every point of which is a global maximizer and a constrained score-equation solution. The curve is exhibited by S_t=(1-rt)I+t11^T, A_t=diag(q)S_t, and B_t=S_t^{-1}B (after transposition if cmin(r,c). Finite hidden-label switching does not remove the continuum. Thus the same model has one observable mode but infinitely many non-strict parameter maximizers, proving that mode counts require a declared space, quotient, strictness convention, and boundary treatment.\n\nCandidate contribution (theorem; novelty confidence low): An explicit stochastic change-of-factor curve proves a mode-counting separation for every positive r-by-c distribution when h>=min(r,c): exactly one strict observable likelihood mode for positive data, but uncountably many fully interior non-strict parameter-space global maximizers satisfying the likelihood score equations, even modulo the finite hidden-label action." }, { "id": 20001216, "problem_number": "AIM-COMPUTATION-0054", "title": "An ancestral-observation certificate for likelihood unimodality", "statement": "Question 40 (Wynn). Which nodes being hidden imply the multimodality/ unimodality of the likelihood? Given a particular graph, what nodes must be observed to ensure unimodal-ity?", "original_statement": "Question 40 (Wynn). Which nodes being hidden imply the multimodality/ unimodality of the likelihood? Given a particular graph, what nodes must be observed to ensure unimodal-ity?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Question 40 from the 2004 AIM workshop list *Computational algebraic statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 40\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[53]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 40 (Wynn). Which nodes being hidden imply the multimodality/ unimodality of the likelihood? Given a particular graph, what nodes must be observed to ensure unimodal-ity?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0054", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite discrete DAG, if the observed node set is ancestral, then all hidden nodes are barren relative to the observations and sum out in reverse topological order. The observed likelihood becomes a sum of complete-data multinomial log likelihoods: every local maximum is global, the exact maximum set fixes each positive-parent-count observed CPD block to its empirical conditional distribution while leaving zero-parent-count and hidden blocks flat, and boundary and identifiability behavior follow explicitly. For required targets T, An_G(T) is the unique minimal observation set certified by this theorem. A fixed-star example proves that graph and hidden-node placement alone are insufficient: the binary-hidden star H->X,H->Y is saturated for binary observables but has three distinct global observable modes for four-state observables with the 100-Swiss-Franc data.\n\nCandidate contribution (theorem_and_obstruction; novelty confidence low): Candidate ancestral-observation certificate: observing the unique minimal ancestral closure An_G(T) makes the discrete-DAG likelihood quotient-unimodal and gives its exact connected maximum stratum, including boundary and nonidentifiable blocks; moreover, one fixed hidden-star placement changes from a unique observable MLE to three global observable modes when observable cardinalities and data change." }, { "id": 20001217, "problem_number": "AIM-COMPUTATION-0055", "title": "ML degree, feasibility, and a Gaussian likelihood-root trichotomy", "statement": "Question 41 (Ho¸ sten). How many complex roots to the likelihood equations can there be? What is the statistical significance of complex roots/ negative real roots or other roots that do not yield a valid statistical model (e.g. covariance matrices that are not positive definite)?", "original_statement": "Question 41 (Ho¸ sten). How many complex roots to the likelihood equations can there be? What is the statistical significance of complex roots/ negative real roots or other roots that do not yield a valid statistical model (e.g. covariance matrices that are not positive definite)?", "clean_statement": "Question 41 (Ho¸ sten). How many complex roots to the likelihood equations can there be? What is the statistical significance of complex roots/ negative real roots or other roots that do not yield a valid statistical model (e.g. covariance matrices that are not positive definite)?", "statement_status": "exact", "statement_verification": "The official AIM workshop page gives the following standalone item in the first open-problem session:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 41\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[54]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 41 (Ho¸ sten). How many complex roots to the likelihood equations can there be? What is the statistical significance of complex roots/ negative real roots or other roots that do not yield a valid statistical model (e.g. covariance matrices that are not positive definite)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0055", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The modern generic answer to the complex-root count is the maximum-likelihood degree of the saturated observable likelihood correspondence, while real feasibility and boundary maxima require separate analysis. For the identifiable zero-mean bivariate Gaussian family with covariance [[1,rho],[rho,1]] and positive-definite sample scatter, the saturated likelihood equation is the cubic rho^3-b rho^2+(a-1)rho-b=0 and has ML degree three. Off its discriminant, its roots have exactly one of the statistical budgets (valid real, invalid real, nonreal conjugate pairs)=(3,0,0), (1,2,0), or (1,0,1). Exact positive-definite datasets realize all three cases, and the discriminant exhibits a complex pair continuing into two valid likelihood modes.\n\nCandidate contribution (worked_family; novelty confidence low): For the fixed-unit-variance bivariate Gaussian correlation model and positive-definite scatter data off the score discriminant, the saturated ML-degree-three roots obey the exhaustive trichotomy (P,I,C)=(3,0,0), (1,2,0), or (1,0,1); the explicit discriminant and exact datasets in the artifacts realize every case and separate determinant poles from statistical infeasibility." }, { "id": 20001218, "problem_number": "AIM-COMPUTATION-0056", "title": "Complex count solved, with an exact real-root cusp slice", "statement": "Question 42 (Richards). What can be said about the number and structure of solutions to the likelihood equations which arise in the Behrens-Fischer problem? 10", "original_statement": "Question 42 (Richards). What can be said about the number and structure of solutions to the likelihood equations which arise in the Behrens-Fischer problem? 10", "clean_statement": "Question 42 (Richards). What can be said about the number and structure of solutions to the likelihood equations which arise in the Behrens-Fischer problem? 10", "statement_status": "exact", "statement_verification": "The canonical record quotes Question 42 from the AIM workshop notes *Computational algebraic statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 42\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[55]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 42 (Richards). What can be said about the number and structure of solutions to the likelihood equations which arise in the Behrens-Fischer problem? 10\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0056", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The trailing 10 in the canonical record is a page-number extraction artifact, and AIM Problem 24 identifies Question 42 with two labeled multivariate normal samples having a common mean and different unknown covariance matrices. The generic complex count is known: Buot, Hosten, and Richards proved that for dimension p and N1,N2>p there are almost surely exactly 2p+1 denominator-saturated complex likelihood solutions. For the univariate equal-sample-size slice with sample means -a,+a and positive ML-divisor residual variances vX,vY, this attempt proves that profiling the variances reduces the real score equation to mu^3+A mu+B=0, where A=(vX+vY-2a^2)/2, B=a(vY-vX)/2, and discriminant Delta=[2(2a^2-vX-vY)^3-27a^2(vY-vX)^2]/4. Delta<0 gives one strict global maximum; Delta>0 gives two outer local maxima separated by a profile minimum that is a full-likelihood saddle; and the discriminant-zero cases are classified. When vX=vY=v, the equation factors as mu(mu^2+v-a^2)=0, with two equal global maxima exactly when a^2>v and a quartically flat strict maximum at a^2=v.\n\nCandidate contribution (theorem; novelty confidence low): On the equal-sample-size, symmetric-mean Behrens-Fisher slice, the real likelihood equation is put in the explicit depressed-cubic cusp normal form mu^3+A mu+B=0 with discriminant [2(2a^2-vX-vY)^3-27a^2(vY-vX)^2]/4, and every repeated/simple real root is classified in both profile and full parameter space. At the symmetric triple-root locus vX=vY=a^2, qX(mu)qY(mu)=mu^4+4a^4 proves that the zero-curvature stationary point is nevertheless a strict quartically flat global maximum." }, { "id": 20001219, "problem_number": "AIM-COMPUTATION-0057", "title": "A support-aware exchange between levels and binary variables", "statement": "Question 43 (Sturmfels). Which is more important to solve: fixing the model and increasing the number of levels in the model, or fixing binary random variables and letting the number of random variables get large? Meek: It depends on the type of problems you are interested in studying. Roehrig: You can exchange a solution to problems on binary random variables to a problem on nonbinary random variables.", "original_statement": "Question 43 (Sturmfels). Which is more important to solve: fixing the model and increasing the number of levels in the model, or fixing binary random variables and letting the number of random variables get large? Meek: It depends on the type of problems you are interested in studying. Roehrig: You can exchange a solution to problems on binary random variables to a problem on nonbinary random variables.", "clean_statement": "Question 43 (Sturmfels). Which is more important to solve: fixing the model and increasing the number of levels in the model, or fixing binary random variables and letting the number of random variables get large? Meek: It depends on the type of problems you are interested in studying. Roehrig: You can exchange a solution to problems on binary random variables to a problem on nonbinary random variables.", "statement_status": "exact", "statement_verification": "The canonical record is Question 43 from the 2004 AIM workshop list *Computational algebraic statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 43\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[56]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 43 (Sturmfels). Which is more important to solve: fixing the model and increasing the number of levels in the model, or fixing binary random variables and letting the number of random variables get large? Meek: It depends on the type of problems you are interested in studying. Roehrig: You can exchange a solution to problems on binary random variables to a problem on nonbinary random variables.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0057", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A q_i-state variable can be encoded by ceil(log2 q_i) bits with invalid codewords treated as structural zeros. Coordinate relabeling on the valid support exactly preserves statistical models, likelihoods, normalizers, MLEs, and support-relative critical equations. Adding identity sufficient-statistic rows for invalid cells separately preserves integer fibers, lattice kernels, toric relations, and Markov bases, including move degree and support. This formalizes Roehrig's exchange, but the structural-zero face is essential and graphical locality is not preserved: a factor on S expands to all code bits in those blocks, while collapsing n bits into one variable creates 2^n levels.\n\nCandidate contribution (equivalence_theorem_and_obstruction; novelty confidence low): Candidate support-aware exchange theorem: blockwise binary coding induces an exact likelihood/fiber/toric/Markov equivalence once probability support restriction and identity-row fiber augmentation are kept distinct, with the sharp cell bound Q <= 2^M < 2^d Q and explicit q=3 and q=4 counterexamples to support-free or locality-preserving interpretations." }, { "id": 20001220, "problem_number": "AIM-COMPUTATION-0058", "title": "When flattening and symmetry-aware collapse are informative", "statement": "Question 44 (Landsberg). Why is flattening used so frequently in statistics? Might other linear transformations on categorical data be more useful? Why not project onto the irre-ducible representations of the symmetric group? What about other forms of collapsing data that takes into account group actions?", "original_statement": "Question 44 (Landsberg). Why is flattening used so frequently in statistics? Might other linear transformations on categorical data be more useful? Why not project onto the irre-ducible representations of the symmetric group? What about other forms of collapsing data that takes into account group actions?", "clean_statement": "Question 44 (Landsberg). Why is flattening used so frequently in statistics? Might other linear transformations on categorical data be more useful? Why not project onto the irre-ducible representations of the symmetric group? What about other forms of collapsing data that takes into account group actions?", "statement_status": "exact", "statement_verification": "The official AIM workshop page for *Computational algebraic statistics* gives the following question:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 44\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[57]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 44 (Landsberg). Why is flattening used so frequently in statistics? Might other linear transformations on categorical data be more useful? Why not project onto the irre-ducible representations of the symmetric group? What about other forms of collapsing data that takes into account group actions?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0058", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Flattening is canonical for a chosen bipartition and is equivariant under modewise linear changes of coordinates; invertible modewise Fourier or irreducible coordinate changes therefore preserve its rank information. For categorical multinomial models that are pointwise invariant under a finite group, orbit totals are sufficient and preserve all within-model likelihood ratios and maximum-likelihood estimation. In contrast, projection onto selected irreducible components is generally lossy: a two-category S_2 example shows that even a setwise group-stable model can lose all likelihood discrimination when its sign component is discarded.\n\nCandidate contribution (synthesis_and_losslessness_criterion; novelty confidence low): Candidate synthesis: retaining all irreducible blocks is a lossless reparametrization, whereas a noninvertible categorical group collapse is guaranteed to preserve multinomial likelihood when the model is pointwise invariant on every collapsed orbit; setwise model symmetry alone is insufficient, as witnessed by an explicit two-point Bernoulli model." }, { "id": 20001221, "problem_number": "AIM-COMPUTATION-0059", "title": "Exact Bayesian evidence and prediction for finite latent discrete models", "statement": "Problem 45 (Fienberg). Computing the full distributions for observables using Bayesian extensions. Predictive distributions. Dinwoodie: How might you use algebra to pursue Bayesian directions in model fit criteria? Sturmfels: How to you compute this in Macaulay 2? What is Bayesian algebraic geometry? Fienberg: Taking mixtures of things the different models we have discussed at this conference. Meek: Finding the modes of the posterior distributions could be an interesting research problem. Pistone: You could use moment generating functions and cumulant generating func-tions for these problems. Fienberg: Identifiability is a big issue in the Bayesian context. Wynn: How may Edgeworth corrections play a role in this problem?", "original_statement": "Problem 45 (Fienberg). Computing the full distributions for observables using Bayesian extensions. Predictive distributions. Dinwoodie: How might you use algebra to pursue Bayesian directions in model fit criteria? Sturmfels: How to you compute this in Macaulay 2? What is Bayesian algebraic geometry? Fienberg: Taking mixtures of things the different models we have discussed at this conference. Meek: Finding the modes of the posterior distributions could be an interesting research problem. Pistone: You could use moment generating functions and cumulant generating func-tions for these problems. Fienberg: Identifiability is a big issue in the Bayesian context. Wynn: How may Edgeworth corrections play a role in this problem?", "clean_statement": "Problem 45 (Fienberg). Computing the full distributions for observables using Bayesian extensions. Predictive distributions. Dinwoodie: How might you use algebra to pursue Bayesian directions in model fit criteria? Sturmfels: How to you compute this in Macaulay 2? What is Bayesian algebraic geometry? Fienberg: Taking mixtures of things the different models we have discussed at this conference. Meek: Finding the modes of the posterior distributions could be an interesting research problem. Pistone: You could use moment generating functions and cumulant generating func-tions for these problems. Fienberg: Identifiability is a big issue in the Bayesian context. Wynn: How may Edgeworth corrections play a role in this problem?", "statement_status": "exact", "statement_verification": "The source is Problem 45 from the AIM workshop list *Computational algebraic statistics*. The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 45\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[58]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 45 (Fienberg). Computing the full distributions for observables using Bayesian extensions. Predictive distributions. Dinwoodie: How might you use algebra to pursue Bayesian directions in model fit criteria? Sturmfels: How to you compute this in Macaulay 2? What is Bayesian algebraic geometry? Fienberg: Taking mixtures of things the different models we have discussed at this conference. Meek: Finding the modes of the posterior distributions could be an interesting research problem. Pistone: You could use moment generating functions and cumulant generating func-tions for these problems. Fienberg: Identifiability is a big issue in the Bayesian context. Wynn: How may Edgeworth corrections play a role in this problem?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0059", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any finite discrete Bayesian network with an arbitrary observed-hidden split and independent positive Dirichlet priors on its conditional probability-table rows, the ordered-data marginal likelihood is an exact finite positive sum over hidden completion tables, with explicit multinomial coefficients and products of multivariate beta ratios. Shifted evidence ratios give every finite posterior predictive law and every mixed moment of the observable cell-probability vector; a partition formula gives all joint cumulants. Under an invariant finite hidden-label group, the evidence sum compresses by orbit-stabilizer factors while any isolated parameter-space posterior mode is replicated around its label orbit.\n\nCandidate contribution (theorem; novelty confidence low): The coefficient-exact, convention-aware package consisting of the general hidden-completion evidence formula, ordered-versus-count evidence correction, all predictive distributions and cumulants as shifted evidence ratios, and exact hidden-label orbit compression applies to every finite latent discrete Bayesian network with independent Dirichlet CPT rows." }, { "id": 20001222, "problem_number": "AIM-COMPUTATION-0060", "title": "Algebraic certificates for nonparametric mixture likelihood", "statement": "Question 46 (Steele). What might be the application of these techniques to nonparametric models?", "original_statement": "Question 46 (Steele). What might be the application of these techniques to nonparametric models?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "That reading is faithful to the surrounding discussion of mixtures and likelihood, but it is not asserted to be the only intended reading.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 46\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[59]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 46 (Steele). What might be the application of these techniques to nonparametric models?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0060", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a compact nonparametric mixture model observed at s distinct values, the likelihood has a unique fitted vector and an optimal discrete mixing distribution supported on at most min{s, affdim(conv(v(Theta)))+1} contact points. Global optimality is equivalent to a pointwise dual contact inequality. For polynomial kernels, or rational kernels with strictly positive denominators, that inequality becomes nonnegativity of a contact polynomial on a compact semialgebraic parameter set, allowing stratified critical equations to generate candidate atoms and moment/SOS methods to certify them. An explicit symmetric family proves that the s-point term is sharp.\n\nCandidate contribution (reduction; novelty confidence low): Combine the rank-aware support bound q <= min{s, affdim(conv(v(Theta)))+1} with a cleared contact polynomial for polynomial or positive-denominator rational kernels: generate atoms from stratified critical ideals, reconstruct their weights, and accept the candidate only after globally certifying contact-polynomial nonnegativity; the supplied symmetric family shows the s term is sharp." }, { "id": 20001223, "problem_number": "AIM-COMPUTATION-0061", "title": "A four-point counterexample and the border-basis repair", "statement": "Question 47. [Pistone] Take a design of N distinct rational vectors D = {v1,..., v N } ⊆ Qd\n\nand xα1,..., x αN are monomials in Q[x1,..., x n] which are linearly independent modulo the design ideal I(D). Further suppose that these monomials are equal to the set of standard monomials of some monomial ideal M.Suppose that M = 〈m1,..., m r〉. Since the monomials xα1,..., x αN are a basis for a quotient space, there is a unique representation of the mi as linear combinations of the xαj.This produces r polynomials f1,..., f r which belong to I(D). Question: are the only points which vanish on these polynomials the design points; that is, does V (f1,..., f r) = D?", "original_statement": "Question 47. [Pistone] Take a design of N distinct rational vectors D = {v1,..., v N } ⊆ Qd\n\nand xα1,..., x αN are monomials in Q[x1,..., x n] which are linearly independent modulo the design ideal I(D). Further suppose that these monomials are equal to the set of standard monomials of some monomial ideal M.Suppose that M = 〈m1,..., m r〉. Since the monomials xα1,..., x αN are a basis for a quotient space, there is a unique representation of the mi as linear combinations of the xαj.This produces r polynomials f1,..., f r which belong to I(D). Question: are the only points which vanish on these polynomials the design points; that is, does V (f1,..., f r) = D?", "clean_statement": "Question 47. [Pistone] Take a design of N distinct rational vectors D = {v1,..., v N } ⊆ Qd\n\nand xα1,..., x αN are monomials in Q[x1,..., x n] which are linearly independent modulo the design ideal I(D). Further suppose that these monomials are equal to the set of standard monomials of some monomial ideal M.Suppose that M = 〈m1,..., m r〉. Since the monomials xα1,..., x αN are a basis for a quotient space, there is a unique representation of the mi as linear combinations of the xαj.This produces r polynomials f1,..., f r which belong to I(D). Question: are the only points which vanish on these polynomials the design points; that is, does V (f1,..., f r) = D?", "statement_status": "exact", "statement_verification": "The official AIM page for the 2003 workshop *Computational algebraic statistics* records the following question (Question 47, attributed to Pistone). Let \\[ D=\\{v_1,\\ldots,v_N\\}\\subseteq\\mathbb Q^d \\] be a design of \\(N\\) distinct rational points. Let \\[ B=\\{x^{\\alpha_1},\\ldots,x^{\\alpha_N}\\} \\] be monomials that are linearly independent modulo the design ideal \\(I(D)\\), and suppose \\(B\\) is the set of standard monomials of a zero-dimensional monomial ideal \\[ M=\\langle m_1,\\ldots,m_r\\rangle. \\] Because \\(|B|=N=\\dim_{\\mathbb Q}\\mathbb Q[x]/I(D)\\), \\(B\\) is a quotient basis. Interpolate each \\(m_i\\) uniquely in the basis \\(B\\): \\[ m_i\\equiv h_i\\pmod {I(D)},\\qquad h_i\\in\\operatorname{span}_{\\mathbb Q}B, \\] and put \\(f_i=m_i-h_i\\in I(D)\\). Must \\[ V(f_1,\\ldots,f_r)=D? \\]", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 47\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[60]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 47. [Pistone] Take a design of N distinct rational vectors D = {v1,..., v N } ⊆ Qd\\n\\nand xα1,..., x αN are monomials in Q[x1,..., x n] which are linearly independent modulo the design ideal I(D). Further suppose that these monomials are equal to the set of standard monomials of some monomial ideal M.Suppose that M = 〈m1,..., m r〉. Since the monomials xα1,..., x αN are a basis for a quotient space, there is a unique representation of the mi as linear combinations of the xαj.This produces r polynomials f1,..., f r which belong to I(D). Question: are the only points which vanish on these polynomials the design points; that is, does V (f1,..., f r) = D?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0061", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For D={(0,0),(1,1),(2,2),(0,1)}, B={1,x,y,xy}, and M= in Q[x,y], the evaluation determinant is -2, so B is a quotient basis and is the standard-monomial set of M. Interpolation for all minimal generators gives f_x=x^2-xy and f_y=y^2-xy-y+x, but their ideal is intersect , so their common zero set is the entire diagonal together with (0,1), strictly larger than D. If instead M is an actual initial ideal of I(D), the interpolated corner relations are exactly the reduced Groebner basis and cut out D; for an arbitrary supported order ideal, using every border relation gives the correct commuting border basis.\n\nCandidate contribution (explicit_counterexample_and_repair; novelty confidence low): Candidate contribution: the four-point rational construction D={(0,0),(1,1),(2,2),(0,1)}, B={1,x,y,xy}, M= yields the radical corner ideal = intersect ; its failure is certified by contradictory term-order inequalities and repaired explicitly by adjoining the two missing border relations x^2y+2x-3xy and xy^2+2x-3xy." }, { "id": 20001224, "problem_number": "AIM-COMPUTATION-0062", "title": "An exact witness and certificate for a recorded negative answer", "statement": "Question 47 was answered in the negative by Bernd Sturmfels at the conference.", "original_statement": "Question 47 was answered in the negative by Bernd Sturmfels at the conference.", "clean_statement": "Question 47 was answered in the negative by Bernd Sturmfels at the conference.", "statement_status": "exact", "statement_verification": "The canonical record is not a new question. It says, in full:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 47\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[61]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 47 was answered in the negative by Bernd Sturmfels at the conference.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0062", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical record is a context sentence recording that the preceding Question 47 has a negative answer. An independently constructed exact witness is D={(0,0),(1,-1),(2,-2),(-1,-2)} with B={1,x,y,xy}: its two interpolating corner relations factor as (x+y)(x+1) and (x+y)(y+2), so their common complex zero set contains the entire line x+y=0. Complementarily, for every such design construction with corner ideal J contained in I(D), the exact sequence 0→I(D)/J→R/J→R/I(D)→0 proves that dim_Q(R/J)=N if and only if J=I(D); a full B-border basis with commuting multiplication matrices certifies this only when all full border relations are verified to belong to the corner ideal.\n\nCandidate contribution (counterexample_and_certificate; novelty confidence low): The displayed four-point rational counterexample, together with the explicit length-N and full-border-membership certificate, supplies a hand-verifiable negative witness and exact repair for the AIM status sentence: the full interpolating border matrices commute, but the two noncorner border relations are not consequences of the corner ideal.", "research_difficulty_suggested": "L2" }, { "id": 20001225, "problem_number": "AIM-COMPUTATION-0063", "title": "Markov-basis complexity depends on representation", "statement": "Question 48 (Meek). Is there a polynomial time algorithm for describing the Markov basis of an arbitrary hierarchical model? Can you randomly choose elements from the Markov basis in polynomial time? Sullivant: Is the number of symmetry classes a polynomial in the size of the problem?", "original_statement": "Question 48 (Meek). Is there a polynomial time algorithm for describing the Markov basis of an arbitrary hierarchical model? Can you randomly choose elements from the Markov basis in polynomial time? Sullivant: Is the number of symmetry classes a polynomial in the size of the problem?", "clean_statement": "Question 48 (Meek). Is there a polynomial time algorithm for describing the Markov basis of an arbitrary hierarchical model? Can you randomly choose elements from the Markov basis in polynomial time? Sullivant: Is the number of symmetry classes a polynomial in the size of the problem?", "statement_status": "exact", "statement_verification": "The exact source is the AIM workshop *Computational algebraic statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 48\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[62]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 48 (Meek). Is there a polynomial time algorithm for describing the Markov basis of an arbitrary hierarchical model? Can you randomly choose elements from the Markov basis in polynomial time? Sullivant: Is the number of symmetry classes a polynomial in the size of the problem?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0063", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The original polynomial-time and random-sampling questions have no representation-independent answer. For A=[1,1,1], the three inclusion-minimal Markov bases are the spanning trees on the three cells, so sampling from 'the' minimal basis is undefined until a basis and distribution are selected. For the I by J independence model, all 2 by 2 rectangle moves form the unique minimal basis up to signs, number binom(I,2)binom(J,2), form one S_I by S_J orbit, and admit an exact uniform sparse sampler with expected O(log I+log J) fair random bits. Yet explicit listing is exponentially long when I,J are binary encoded, while an orbit template is succinct and listing is polynomial at the explicit-cell input scale.\n\nCandidate contribution (obstruction; novelty confidence low): A minimal specification audit combines a three-cell noncanonicity obstruction with an encoding trichotomy for I by J independence: one symmetry orbit and expected-polynomial exact sparse move sampling coexist with an exponential explicit-output lower bound under binary level encoding." }, { "id": 20001226, "problem_number": "AIM-COMPUTATION-0064", "title": "Localization and Fourier coordinates beyond polynomial models", "statement": "Question 49 (Wynn). Why polynomials? What can be said about algebraic methods when applied to other rings, like rings of rational functions, or rings where Fourier analysis can be used? 11", "original_statement": "Question 49 (Wynn). Why polynomials? What can be said about algebraic methods when applied to other rings, like rings of rational functions, or rings where Fourier analysis can be used? 11", "clean_statement": "Question 49 (Wynn). Why polynomials? What can be said about algebraic methods when applied to other rings, like rings of rational functions, or rings where Fourier analysis can be used? 11", "statement_status": "exact", "statement_verification": "The corpus record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 49\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[63]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 49 (Wynn). Why polynomials? What can be said about algebraic methods when applied to other rings, like rings of rational functions, or rings where Fourier analysis can be used? 11\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0064", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Rational equations are handled faithfully on their domain by localization: if numerator ideal I is extended to R_s, its contraction is I:s^infinity, the rational zero set is V(I) intersect D(s), and its affine closure is V(I:s^infinity); the equation xy/x=0 shows exactly how naive clearing adds the pole component V(x). For a finite abelian group over a splitting field of nonmodular characteristic, the full Fourier transform is an algebra isomorphism from convolution to coordinatewise multiplication. Applied to a uniform-root Z/2 four-leaf star, this makes permitted joint Fourier coordinates monomials in edge parameters, forces odd-parity coordinates to zero and the trivial coordinate to one, and yields an explicit toric binomial relation.\n\nCandidate contribution (combined_domain_and_spectrum_preservation_rule; novelty confidence low): Candidate synthesis: algebraic use of rational equations is faithful only after all forbidden denominators are inverted, the localized ideal is contracted by saturation, and the pole inequation is retained; algebraic use of Fourier coordinates is faithful only when the full character spectrum is retained over a semisimple splitting field, after which model-forced frequencies may be removed and convolutional parameters become monomial." }, { "id": 20001227, "problem_number": "AIM-COMPUTATION-0065", "title": "Gaussian conditional independence as an algebraic minor", "statement": "Question 50 (Kuhnt). What about adding continuous random variables into these prob-lems? Might an algebraic framework be useful? Dinwoodie: Can we try to apply algebraic techniques to microarray data? Meek: Gaussian (or similar) assumptions on the random variables might be possible to handle with algebraic techniques.", "original_statement": "Question 50 (Kuhnt). What about adding continuous random variables into these prob-lems? Might an algebraic framework be useful? Dinwoodie: Can we try to apply algebraic techniques to microarray data? Meek: Gaussian (or similar) assumptions on the random variables might be possible to handle with algebraic techniques.", "clean_statement": "Question 50 (Kuhnt). What about adding continuous random variables into these prob-lems? Might an algebraic framework be useful? Dinwoodie: Can we try to apply algebraic techniques to microarray data? Meek: Gaussian (or similar) assumptions on the random variables might be possible to handle with algebraic techniques.", "statement_status": "exact", "statement_verification": "The source is Question 50 from the AIM workshop list *Computational algebraic statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 50\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[64]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 50 (Kuhnt). What about adding continuous random variables into these prob-lems? Might an algebraic framework be useful? Dinwoodie: Can we try to apply algebraic techniques to microarray data? Meek: Gaussian (or similar) assumptions on the random variables might be possible to handle with algebraic techniques.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0065", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a regular multivariate Gaussian distribution, Xi is conditionally independent of Xj given X_S exactly when the almost-principal covariance minor det(Sigma_{iS,jS}) vanishes; the determinant equals det(Sigma_SS) times conditional covariance, its normalized ratio with two principal minors is partial correlation, and conditioning on all other variables specializes to a zero precision-matrix entry. The report also proves a high-dimensional obstruction: in a centered covariance from n observations, every such minor with |S| at least n-1 vanishes automatically by rank and therefore carries no conditional-independence evidence.\n\nCandidate contribution (quantitative_certificate_and_obstruction; novelty confidence low): The explicit three-part screening guardrail combines the scale-free principal/almost-principal determinant ratio for partial correlation, a spectral sandwich converting raw polynomial residuals into quantitative partial-correlation bounds, and the exact sample-rank threshold |S|>=n-1 beyond which empirical minor vanishing is structurally automatic." }, { "id": 20001228, "problem_number": "AIM-COMPUTATION-0066", "title": "Equality constraints do not characterize hidden-variable models", "statement": "Question 51 (Meek). Is there a polynomial time algorithm for describing the constraints on the joint probability distribution on the observed variables in a graphical model with hidden variables?", "original_statement": "Question 51 (Meek). Is there a polynomial time algorithm for describing the constraints on the joint probability distribution on the observed variables in a graphical model with hidden variables?", "clean_statement": "Question 51 (Meek). Is there a polynomial time algorithm for describing the constraints on the joint probability distribution on the observed variables in a graphical model with hidden variables?", "statement_status": "exact", "statement_verification": "The exact AIM workshop question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 51\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[65]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 51 (Meek). Is there a polynomial time algorithm for describing the constraints on the joint probability distribution on the observed variables in a graphical model with hidden variables?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0066", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every fixed finite discrete latent DAG, the observed marginal model is compact semialgebraic, so a finite quantifier-free constraint description exists in principle. For the hidden common-cause model H→X,H→Y with r states, stochastic membership is exactly nonnegative rank at most r. Ordinary determinantal constraints suffice when r≤2, but for every r≥3 the explicit rational probability matrix P_r=(S direct-sum I_{r-3})/(r+5), where S is the 4×4 eight-cycle support matrix, has ordinary rank r and nonnegative rank r+1. It therefore satisfies every polynomial equality of the determinantal/Zariski closure while lying outside the r-state stochastic model. With unbounded common-cause cardinality, the observed model saturates the simplex.\n\nCandidate contribution (counterexample_family; novelty confidence low): The block-cycle family P_r=(S direct-sum I_{r-3})/(r+5), for every r≥3, gives a uniform rational family in the full equality/Zariski model that provably requires r+1 hidden classes, with nonnegative-rank additivity established directly by support rectangles." }, { "id": 20001229, "problem_number": "AIM-COMPUTATION-0067", "title": "Parameter tying and exact homogeneous binary-star invariants", "statement": "Problem 52 (Pachter). Try applying the algebraic techniques to more restrictive families of models. For example, models with homogeneous transition matrices (that is, the same set of parameters is used for more than one edge/ clique in the graph), or genomic models.", "original_statement": "Problem 52 (Pachter). Try applying the algebraic techniques to more restrictive families of models. For example, models with homogeneous transition matrices (that is, the same set of parameters is used for more than one edge/ clique in the graph), or genomic models.", "clean_statement": "Problem 52 (Pachter). Try applying the algebraic techniques to more restrictive families of models. For example, models with homogeneous transition matrices (that is, the same set of parameters is used for more than one edge/ clique in the graph), or genomic models.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 52 from the AIM workshop notes on computational algebraic statistics:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 52\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[66]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 52 (Pachter). Try applying the algebraic techniques to more restrictive families of models. For example, models with homogeneous transition matrices (that is, the same set of parameters is used for more than one edge/ clique in the graph), or genomic models.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0067", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any monomial model with exponent matrix A, tying parameter rows according to an incidence matrix C replaces A by CA, preserves toricity, enlarges the prime binomial ideal from I_A to I_CA, and makes all new constraints lattice relations of the aggregated exponents. Applied to the uniform hidden-root binary n-leaf star with the same symmetric transition matrix on every edge, the complete Fourier ideal sets odd-weight coordinates to zero, equates coordinates of each even weight, and imposes q_{T_r}=z^r for a weight-2 coordinate z. Its stochastic image is exactly 0 <= z <= 1; the discrete-time parameter is generically two-to-one under theta <-> 1-theta, while the continuous-time branch is identifiable.\n\nCandidate contribution (explicit ideal and identifiability theorem; novelty confidence low): Candidate novelty: the complete all-n Fourier ideal, exact stochastic interval, and parameter-fiber audit for the leaf-observed, hidden-root, identical-edge binary star, organized as a direct consequence of exponent-row aggregation under parameter tying." }, { "id": 20001230, "problem_number": "AIM-COMPUTATION-0068", "title": "Exact output-sensitive scaling for decomposable discrete models", "statement": "Question 53 (Fienberg). How can we make the algorithms and existing theoretical results apply to problems of sizes that are of practical interest? Scaling up to 5 10 tables is a hopeful goal.", "original_statement": "Question 53 (Fienberg). How can we make the algorithms and existing theoretical results apply to problems of sizes that are of practical interest? Scaling up to 5 10 tables is a hopeful goal.", "clean_statement": "Question 53 (Fienberg). How can we make the algorithms and existing theoretical results apply to problems of sizes that are of practical interest? Scaling up to 5 10 tables is a hopeful goal.", "statement_status": "exact", "statement_verification": "The canonical corpus record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 53\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[67]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 53 (Fienberg). How can we make the algorithms and existing theoretical results apply to problems of sizes that are of practical interest? Scaling up to 5 10 tables is a hopeful goal.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0068", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a known chordal graph with positive empirical maximal-clique cells, the unique interior multinomial MLE is the empirical clique-product/separator-quotient. A separated junction-tree likelihood proves maximum likelihood, and leaf elimination proves normalization and clique-marginal recovery. At treewidth w with at most r states per variable, its ratio factors occupy O(p r^(w+1)) entries, construction from N cases costs O(N m(w+1)+p r^(w+1)), a joint-cell query costs O(m), and an exact junction-tree marginal query costs O(m r^(w+1)); dense input or output nevertheless costs Omega(r^p). For the official AIM 5^10 benchmark instantiated as a ten-node five-state chain, the 9,765,625-cell ambient table has a 265-entry clique/separator representation or a 225-entry optimized rooted conditional representation, while the fully directed representation uses 230 entries; all have 184 free parameters.\n\nCandidate contribution (output-sensitive scaling certificate; novelty confidence low): The candidate contribution is an explicit, proved certificate that jointly separates factor-construction, factor-storage, joint-cell-query, exact-marginal-query, and unavoidable dense-I/O costs for Fienberg's verified 5^10 target, with audited chain constants of 265 ratio entries, 225 optimized rooted conditional entries, 230 fully directed entries, and 184 free parameters." }, { "id": 20001231, "problem_number": "AIM-COMPUTATION-0069", "title": "Two-sided novelty certificates for sparse binomial output", "statement": "Question 54 (Fienberg). How can we make it easier to identify \"new\" interesting details among long complicated computer output? Karr: can we take advantage of sparse representations of problems?\n\nSecond Open Problem Session", "original_statement": "Question 54 (Fienberg). How can we make it easier to identify \"new\" interesting details among long complicated computer output? Karr: can we take advantage of sparse representations of problems? \n\nSecond Open Problem Session", "clean_statement": "Question 54 (Fienberg). How can we make it easier to identify \"new\" interesting details among long complicated computer output? Karr: can we take advantage of sparse representations of problems?\n\nSecond Open Problem Session", "statement_status": "exact", "statement_verification": "The repository record contains:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 54\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[68]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 54 (Fienberg). How can we make it easier to identify \\\"new\\\" interesting details among long complicated computer output? Karr: can we take advantage of sparse representations of problems? \\n\\nSecond Open Problem Session\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0069", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For pure-difference Laurent binomials, encode each relation x^u-x^v by d=u-v. A candidate x^d-1 is in the ideal generated by earlier relations exactly when d lies in their integer exponent lattice. Smith normal form gives either a compressed integer membership certificate Da=d or, over an algebraically closed characteristic-zero field, an explicit torus point satisfying every old relation and violating the candidate. The same data distinguishes dimension-changing rank novelty from finite-index novelty and quantifies the latter. In characteristic p, set-theoretic implication is instead governed by p-saturation; in an affine polynomial ring, the lattice criterion applies after saturation by the coordinate product.\n\nCandidate contribution (certificate protocol; novelty confidence low): Candidate novelty: a two-sided sparse-output record that attaches to every pure-difference binomial either a compressed exponent-lattice derivation or an explicit separating torus countermodel, together with an exact rank-versus-finite-index novelty label and characteristic/saturation warnings." }, { "id": 20001232, "problem_number": "AIM-COMPUTATION-0070", "title": "Arbitrary logistic cell gaps with positive margins and ordinary-margin obstructions", "statement": "Problem 55 (Karr). Find a set of strictly positive (or strictly > 5) margins for which the set of tables which has these margins has an arbitrarily large gap in a cell entry. Dinwoodie: Are there examples of this phenomenon in logistic regression?", "original_statement": "Problem 55 (Karr). Find a set of strictly positive (or strictly > 5) margins for which the set of tables which has these margins has an arbitrarily large gap in a cell entry. Dinwoodie: Are there examples of this phenomenon in logistic regression?", "clean_statement": "Problem 55 (Karr). Find a set of strictly positive (or strictly > 5) margins for which the set of tables which has these margins has an arbitrarily large gap in a cell entry. Dinwoodie: Are there examples of this phenomenon in logistic regression?", "statement_status": "exact", "statement_verification": "The canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 55\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[69]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 55 (Karr). Find a set of strictly positive (or strictly > 5) margins for which the set of tables which has these margins has an arbitrarily large gap in a cell entry. Dinwoodie: Are there examples of this phenomenon in logistic regression?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0070", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every L>=2 and q>=6, grouped univariate logistic regression at covariate values (0,1,L), with response margins (q(L-1),qL,q) and conditioned total and weighted successes both qL, has exactly the tables with success row ((L-1)t,L(q-t),t), t=0,...,q. Every ordinary row and group margin is strictly greater than 5, while the middle success-cell spectrum is {0,L,2L,...,qL}, leaving L-1 missing integers between consecutive values; this gives arbitrarily large logistic-regression gaps and answers Dinwoodie's clause. A general first-path-edge lemma proves that coordinate gaps are bounded by the largest coordinate of a connecting move set. It implies interval spectra for positive-two-margin 2xJxK no-three-way-interaction fibers via Rapallo-Yoshida connectivity, and gap size at most one for equally spaced positive-response bivariate logistic fibers via Hara-Takemura-Yoshida. De Loera-Onn universality uses zero margins, so Karr's ordinary strictly-positive two-margin construction remains unresolved in the literature checked.\n\nCandidate contribution (explicit family; novelty confidence low): The candidate contribution is the exact three-group logistic family with covariates (0,1,L), response margins (q(L-1),qL,q), and full fiber parameterization X(t), whose middle success-cell spectrum is L times {0,...,q}; it has gap size L-1 while all ordinary row and group margins exceed 5. The accompanying bounded-coordinate connectivity lemma gives a sharp L=2 comparison and a positive-margin 2xJxK interval obstruction." }, { "id": 20001233, "problem_number": "AIM-COMPUTATION-0071", "title": "Exact chain gluing and a sharp cyclic incompatibility certificate", "statement": "Problem 56 (Fienberg). Consider 3-way table missing data problem. This consists of a 3-way table with holes and 2-way margins which also have holes.. How can you estimate the likelihood and \"glue\" this information together into a good global picture of the whole table? Can this problem be formulated as an algebraic variety?", "original_statement": "Problem 56 (Fienberg). Consider 3-way table missing data problem. This consists of a 3-way table with holes and 2-way margins which also have holes.. How can you estimate the likelihood and \"glue\" this information together into a good global picture of the whole table? Can this problem be formulated as an algebraic variety?", "clean_statement": "Problem 56 (Fienberg). Consider 3-way table missing data problem. This consists of a 3-way table with holes and 2-way margins which also have holes.. How can you estimate the likelihood and \"glue\" this information together into a good global picture of the whole table? Can this problem be formulated as an algebraic variety?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 56 from the AIM workshop *Computational algebraic statistics*. The repository record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 56\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[70]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 56 (Fienberg). Consider 3-way table missing data problem. This consists of a 3-way table with holes and 2-way margins which also have holes.. How can you estimate the likelihood and \\\"glue\\\" this information together into a good global picture of the whole table? Can this problem be formulated as an algebraic variety?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0071", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For strictly positive separator-consistent XY and YZ probability margins, p_xyz=p_xy p_yz/p_y is a normalized completion with the prescribed margins, the unique X-independent-of-Z-given-Y completion, the unique maximum-entropy completion among all completions, and the unique algebraic solution of the linear margin equations plus conditional-independence slice minors. In contrast, prescribing the symmetric binary margin M_epsilon=(1/2)[[epsilon,1-epsilon],[1-epsilon,epsilon]] on all three pairs is globally compatible exactly when epsilon is at least 1/3; its distance to the compatible set in the maximum of the three pairwise total-variation distances is max(0,1/3-epsilon).\n\nCandidate contribution (sharp_robust_obstruction; novelty confidence low): For 00 at every monotone dropout level give a directly observable necessary-and-sufficient certificate for a unique positive-definite joint Gaussian MLE, unifying design identifiability and residual positive definiteness in one Gram test per block." }, { "id": 20001235, "problem_number": "AIM-COMPUTATION-0073", "title": "Nondefectivity and low-degree obstructions for the two-component S4 Birkhoff model", "statement": "Problem 58 (Sturmfels). Compute a mixture of the S4 model from Diaconis' talk. What are the equations defining this secant variety? Is the likelihood multimodal?\n\nMLE Project\n\nOn Thursday afternoon, Bernd Sturmfels, Serkan Ho¸ sten, and Mathias Drton led a section on algebraic approaches to maximum likelihood estimations. The purpose was to introduce some algebraic techniques and possible small instances of statistical models to which one could apply these techniques. The eventual goal of this project is to try to come 12\n\nto a better understanding of the maximum likelihood estimation problem using algebraic means.\n\nAlgebraic Methods for Optimization:\n\nA. Singular: Gr¨ obner bases and numerical eigenvalues for companion matrices or lexi-cographic solving B. Lagrange multipliers 1. Transformations out of the simplex? 2. Exponential representation for problems C. Resultants D. Homotopy Methods (e.g. PHCpack) E. Sum of Squares (e.g. SOStools)\n\nSmall Instances of Statistical Models\n\nA. σ3(P2 × P2 × P2), that is, the hidden variable naive Bayes model with three ob-served variables, where all the random variables have three states. This model has 20 parameters. B. Seemingly unrelated regression (SUR) model with three random variables. See ques-tions related to Mathias Drton's talk. This model has 3 parameters. C. Hidden Markov model with two hidden nodes. Hidden variables have 2 state and observed variables have 3 states. This model has 6 parameters. D. An unconditional independence model for 4 binary random variables. X1⊥X2 and\n\nX3⊥X4. This model has 13 parameters. E. Perfect tables on 3 random variables. X1⊥X2, X2⊥X3, and X1⊥X3.\n\nQuestions from the Working Sessions\n\nGraphical Models", "original_statement": "Problem 58 (Sturmfels). Compute a mixture of the S4 model from Diaconis' talk. What are the equations defining this secant variety? Is the likelihood multimodal? \n\nMLE Project \n\nOn Thursday afternoon, Bernd Sturmfels, Serkan Ho¸ sten, and Mathias Drton led a section on algebraic approaches to maximum likelihood estimations. The purpose was to introduce some algebraic techniques and possible small instances of statistical models to which one could apply these techniques. The eventual goal of this project is to try to come 12 \n\nto a better understanding of the maximum likelihood estimation problem using algebraic means. \n\nAlgebraic Methods for Optimization: \n\nA. Singular: Gr¨ obner bases and numerical eigenvalues for companion matrices or lexi-cographic solving B. Lagrange multipliers 1. Transformations out of the simplex? 2. Exponential representation for problems C. Resultants D. Homotopy Methods (e.g. PHCpack) E. Sum of Squares (e.g. SOStools) \n\nSmall Instances of Statistical Models \n\nA. σ3(P2 × P2 × P2), that is, the hidden variable naive Bayes model with three ob-served variables, where all the random variables have three states. This model has 20 parameters. B. Seemingly unrelated regression (SUR) model with three random variables. See ques-tions related to Mathias Drton's talk. This model has 3 parameters. C. Hidden Markov model with two hidden nodes. Hidden variables have 2 state and observed variables have 3 states. This model has 6 parameters. D. An unconditional independence model for 4 binary random variables. X1⊥X2 and \n\nX3⊥X4. This model has 13 parameters. E. Perfect tables on 3 random variables. X1⊥X2, X2⊥X3, and X1⊥X3.\n\nQuestions from the Working Sessions \n\nGraphical Models", "clean_statement": "Problem 58 (Sturmfels). Compute a mixture of the S4 model from Diaconis' talk. What are the equations defining this secant variety? Is the likelihood multimodal?\n\nMLE Project\n\nOn Thursday afternoon, Bernd Sturmfels, Serkan Ho¸ sten, and Mathias Drton led a section on algebraic approaches to maximum likelihood estimations. The purpose was to introduce some algebraic techniques and possible small instances of statistical models to which one could apply these techniques. The eventual goal of this project is to try to come 12\n\nto a better understanding of the maximum likelihood estimation problem using algebraic means.\n\nAlgebraic Methods for Optimization:\n\nA. Singular: Gr¨ obner bases and numerical eigenvalues for companion matrices or lexi-cographic solving B. Lagrange multipliers 1. Transformations out of the simplex? 2. Exponential representation for problems C. Resultants D. Homotopy Methods (e.g. PHCpack) E. Sum of Squares (e.g. SOStools)\n\nSmall Instances of Statistical Models\n\nA. σ3(P2 × P2 × P2), that is, the hidden variable naive Bayes model with three ob-served variables, where all the random variables have three states. This model has 20 parameters. B. Seemingly unrelated regression (SUR) model with three random variables. See ques-tions related to Mathias Drton's talk. This model has 3 parameters. C. Hidden Markov model with two hidden nodes. Hidden variables have 2 state and observed variables have 3 states. This model has 6 parameters. D. An unconditional independence model for 4 binary random variables. X1⊥X2 and\n\nX3⊥X4. This model has 13 parameters. E. Perfect tables on 3 random variables. X1⊥X2, X2⊥X3, and X1⊥X3.\n\nQuestions from the Working Sessions\n\nGraphical Models", "statement_status": "exact", "statement_verification": "The source record begins:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 58\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[72]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 58 (Sturmfels). Compute a mixture of the S4 model from Diaconis' talk. What are the equations defining this secant variety? Is the likelihood multimodal? \\n\\nMLE Project \\n\\nOn Thursday afternoon, Bernd Sturmfels, Serkan Ho¸ sten, and Mathias Drton led a section on algebraic approaches to maximum likelihood estimations. The purpose was to introduce some algebraic techniques and possible small instances of statistical models to which one could apply these techniques. The eventual goal of this project is to try to come 12 \\n\\nto a better understanding of the maximum likelihood estimation problem using algebraic means. \\n\\nAlgebraic Methods for Optimization: \\n\\nA. Singular: Gr¨ obner bases and numerical eigenvalues for companion matrices or lexi-cographic solving B. Lagrange multipliers 1. Transformations out of the simplex? 2. Exponential representation for problems C. Resultants D. Homotopy Methods (e.g. PHCpack) E. Sum of Squares (e.g. SOStools) \\n\\nSmall Instances of Statistical Models \\n\\nA. σ3(P2 × P2 × P2), that is, the hidden variable naive Bayes model with three ob-served variables, where all the random variables have three states. This model has 20 parameters. B. Seemingly unrelated regression (SUR) model with three random variables. See ques-tions related to Mathias Drton's talk. This model has 3 parameters. C. Hidden Markov model with two hidden nodes. Hidden variables have 2 state and observed variables have 3 states. This model has 6 parameters. D. An unconditional independence model for 4 binary random variables. X1⊥X2 and \\n\\nX3⊥X4. This model has 13 parameters. E. Perfect tables on 3 random variables. X1⊥X2, X2⊥X3, and X1⊥X3.\\n\\nQuestions from the Working Sessions \\n\\nGraphical Models\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0073", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the first-order S4 permutation (Birkhoff) toric variety X_B4 in P^23, an explicit positive tangent-space certificate with determinant 3456 proves that its two-component secant has the expected projective dimension 19 and codimension 4. Hence the abstract complex join map is generically finite, though uniqueness is not proved. The secant has no nonzero linear or quadratic equations; ordinary tensor flattening minors do not apply because even an unmixed all-ones point has flattening ranks 4 and 6. Label switching gives paired parameter maxima but does not decide genuine likelihood multimodality modulo labels.\n\nCandidate contribution (explicit dimension certificate; novelty confidence low): With columns 1, f_11, ..., f_33, q, q f_11, ..., q f_33 for q_sigma = 2^fix(sigma), and lexicographic permutation rows omitting 3412, 4213, 4312, and 4321, the resulting 20 by 20 tangent minor has determinant 3456; this gives a reproducible elementary proof that dim sigma_2(X_B4) = 19." }, { "id": 20001236, "problem_number": "AIM-COMPUTATION-0074", "title": "An exact MLE fiber for the smallest binary hidden-variable model", "statement": "Question 59 (Ho¸ sten). What can algebra say about MLE's for models with hidden vari-ables?", "original_statement": "Question 59 (Ho¸ sten). What can algebra say about MLE's for models with hidden vari-ables?", "clean_statement": "Question 59 (Ho¸ sten). What can algebra say about MLE's for models with hidden vari-ables?", "statement_status": "exact", "statement_verification": "The exact extracted record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 59\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[73]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 59 (Ho¸ sten). What can algebra say about MLE's for models with hidden vari-ables?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0074", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a binary hidden variable making two observed binary variables conditionally independent, the observed model is the entire 2 by 2 probability simplex, so positive multinomial data have the empirical table as their unique observed-distribution MLE. For a nonindependent empirical table, however, all interior hidden-parameter MLEs form an explicit nonempty two-dimensional smooth semialgebraic fiber. Two displayed continuous tangent directions span exactly the nullspace of both the parameter Hessian and Fisher information. For an independent table the fiber factors as u v = 0; at the collapsed representation u = v = 0 the Jacobian rank drops from 3 to 2 and the curvature nullity jumps from 2 to 3.\n\nCandidate contribution (exact_special_case_and_singularity_certificate; novelty confidence low): The moment-coordinate chart for the generic two-dimensional parameter MLE fiber, together with its two explicit tangent directions, gives an exact equality between the MLE-fiber tangent and the Hessian/Fisher nullspace; at independence, the crossing equation u v = 0 yields a directly testable nullity jump from 2 to 3 at the collapsed hidden-class representation." }, { "id": 20001237, "problem_number": "AIM-COMPUTATION-0075", "title": "Saturation-aware exceptional likelihood fibers and the 2x2 independence locus", "statement": "Question 60 (Sullivant). For a log-linear model, what can be said about the structure of the set of b with Ax = b for some b and the solution space to the maximum likelihood equations is not zero-dimensional? Ho¸ sten: Look at the leading coefficients of polynomials in the elimination ideals (with parameters) and determine when these are zero.", "original_statement": "Question 60 (Sullivant). For a log-linear model, what can be said about the structure of the set of b with Ax = b for some b and the solution space to the maximum likelihood equations is not zero-dimensional? Ho¸ sten: Look at the leading coefficients of polynomials in the elimination ideals (with parameters) and determine when these are zero.", "clean_statement": "Question 60 (Sullivant). For a log-linear model, what can be said about the structure of the set of b with Ax = b for some b and the solution space to the maximum likelihood equations is not zero-dimensional? Ho¸ sten: Look at the leading coefficients of polynomials in the elimination ideals (with parameters) and determine when these are zero.", "statement_status": "exact", "statement_verification": "The canonical record is Question 60 from the AIM workshop *Computational algebraic statistics*. The exact record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 60\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[74]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 60 (Sullivant). For a log-linear model, what can be said about the structure of the set of b with Ax = b for some b and the solution space to the maximum likelihood equations is not zero-dimensional? Ho¸ sten: Look at the leading coefficients of polynomials in the elimination ideals (with parameters) and determine when these are zero.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0075", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any finite-type parameterized likelihood system with forbidden factors encoded by a Rabinowitsch equation zD-1, the locus of parameters with positive-dimensional valid complex fiber is constructible and is exactly a union of comprehensive Groebner strata; generic leading coefficients give a closed outer certificate, while properness or a fiberwise-compatible projective closure makes the exact jump locus closed. For the 2x2 independence expected-count equations with total N and first row and column margins r1,c1, elimination gives N t-r1 c1. Hence the affine fiber is one reduced point for N nonzero, empty for N=0 and r1 c1 nonzero, and an affine line for N=0 and r1 c1=0. The positive-dimensional locus is exactly V(N,r1 c1), survives saturation by the four cell coordinates, disappears when nonzero normalization N is also inverted, and meets the nonnegative compatible marginal cone only at the zero table.\n\nCandidate contribution (explicit classification and reduction; novelty confidence low): The candidate contribution is the paired saturation-aware comprehensive-Groebner criterion and exact 2x2 marginal classification E_{>=1}=V(N,r1 c1): the leading-coefficient divisor V(N) strictly overestimates the positive-dimensional locus because its complementary stratum N=0, r1 c1 nonzero has empty affine fiber; cell-torus saturation preserves the exceptional lines, whereas inverting N removes them, and the nonnegative cone intersects them only at zero." }, { "id": 20001238, "problem_number": "AIM-COMPUTATION-0076", "title": "Covariance-whitened polynomial invariants for model goodness of fit", "statement": "Question 61 (From graphical models notes). Is there any useful statistical measure that can be obtained from evaluating the polynomials that vanish on probabilities coming from a given model at an empirical distribution? What about\n\n∑ |f (̂p)|?\n\nLinear Polynomial Models\n\nA design D is a finite subset of Rd. The design ideal I(D) is the vanishing ideal of polynomial functions vanishing on D. The set of standard monomials modulo I(D) given a term order τ is denoted Est τ. A linear polynomial model L is a list of monomials. 13", "original_statement": "Question 61 (From graphical models notes). Is there any useful statistical measure that can be obtained from evaluating the polynomials that vanish on probabilities coming from a given model at an empirical distribution? What about \n\n∑ |f (̂p)|?\n\nLinear Polynomial Models \n\nA design D is a finite subset of Rd. The design ideal I(D) is the vanishing ideal of polynomial functions vanishing on D. The set of standard monomials modulo I(D) given a term order τ is denoted Est τ. A linear polynomial model L is a list of monomials. 13", "clean_statement": "Question 61 (From graphical models notes). Is there any useful statistical measure that can be obtained from evaluating the polynomials that vanish on probabilities coming from a given model at an empirical distribution? What about\n\n∑ |f (̂p)|?\n\nLinear Polynomial Models\n\nA design D is a finite subset of Rd. The design ideal I(D) is the vanishing ideal of polynomial functions vanishing on D. The set of standard monomials modulo I(D) given a term order τ is denoted Est τ. A linear polynomial model L is a list of monomials. 13", "statement_status": "exact", "statement_verification": "The recovered AIM question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 61\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[75]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 61 (From graphical models notes). Is there any useful statistical measure that can be obtained from evaluating the polynomials that vanish on probabilities coming from a given model at an empirical distribution? What about \\n\\n∑ |f (̂p)|?\\n\\nLinear Polynomial Models \\n\\nA design D is a finite subset of Rd. The design ideal I(D) is the vanishing ideal of polynomial functions vanishing on D. The set of standard monomials modulo I(D) given a term order τ is denoted Est τ. A linear polynomial model L is a list of monomials. 13\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0076", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The raw score sum_i |f_i(p-hat)| is not intrinsic because nonsingular rescaling or generator replacement preserves the model ideal while changing the score arbitrarily. For i.i.d. multinomial data at an interior regular model point, the covariance-whitened statistic N F(p-hat)^T [J(p-tilde) Sigma(p-tilde) J(p-tilde)^T]^{-1} F(p-hat) converges to chi-square with degrees equal to the number of independent invariant gradients. It is exactly invariant under constant nonsingular changes of the invariant basis and asymptotically invariant under smooth locally nonsingular reformulations. For 2 by 2 independence, whitening the determinant invariant with the fitted-margin covariance gives exactly Pearson's one-degree-of-freedom statistic as a finite-sample algebraic identity, while its chi-square calibration remains asymptotic.\n\nCandidate contribution (generator-invariant synthesis and exact worked equivalence; novelty confidence low): For regular multinomial algebraic models, covariance whitening yields an exactly GL(k)-invariant statistic on any independent local invariant basis, with a stable-rank pseudoinverse extension for redundant generators; in the 2 by 2 independence model the fitted-covariance whitened determinant formula is identically equal to Pearson's statistic for every table with positive fitted margins." }, { "id": 20001239, "problem_number": "AIM-COMPUTATION-0077", "title": "A scale-normalized robustness criterion for algebraic design fans", "statement": "Question 62 (Riccomagno). Is there an algebraic method to decide which Est τ factor is best when working with perturbed designs?", "original_statement": "Question 62 (Riccomagno). Is there an algebraic method to decide which Est τ factor is best when working with perturbed designs?", "clean_statement": "Question 62 (Riccomagno). Is there an algebraic method to decide which Est τ factor is best when working with perturbed designs?", "statement_status": "exact", "statement_verification": "The exact source record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 62\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[76]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 62 (Riccomagno). Is there an algebraic method to decide which Est τ factor is best when working with perturbed designs?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0077", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every coherent standard-monomial basis B of a finite design, the cardinal Lagrange matrix L_B(D') = V_B(D')V_B(D)^{-1} gives a basis-invariant robustness certificate: if e = rho G_B(rho) is less than one, then all allowed point perturbations preserve estimability and kappa_2(L_B(D')) is at most (1+e)/(1-e). Maximizing the resulting certified radius over the finite Groebner fan is therefore an algebraic decision method once physical coordinates and a perturbation metric are fixed. An exact three-point family proves that two genuine fan leaves can have sharply different perturbation robustness, including finite loss of estimability for one leaf while the other remains unchanged.\n\nCandidate contribution (certified robustness criterion and exact separating family; novelty confidence low): The candidate contribution is to rank coherent Est_tau bases by the largest perturbation radius certified through the Lagrange-normalized matrix V_B(D')V_B(D)^{-1}, together with an exact family D_epsilon where the bases {1,x,x^2} and {1,y,y^2} are distinct Groebner-fan leaves and the latter loses rank under a perturbation that leaves the former evaluation matrix fixed." }, { "id": 20001240, "problem_number": "AIM-COMPUTATION-0078", "title": "Base-encoded monomial-curve designs", "statement": "Problem 63 (Riccomagno). Given a model L find a design D with special statistical prop-erties such that the monomials in L are linearly independent modulo I(D).", "original_statement": "Problem 63 (Riccomagno). Given a model L find a design D with special statistical prop-erties such that the monomials in L are linearly independent modulo I(D).", "clean_statement": "Problem 63 (Riccomagno). Given a model L find a design D with special statistical prop-erties such that the monomials in L are linearly independent modulo I(D).", "statement_status": "exact", "statement_verification": "The canonical record is Problem 63 from the AIM workshop list *Computational Algebraic Statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 63\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[77]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 63 (Riccomagno). Given a model L find a design D with special statistical prop-erties such that the monomials in L are linearly independent modulo I(D).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0078", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every finite set L of n distinct ordinary monomials, positional base encoding of their exponent vectors gives an explicit positive monomial curve such that any n distinct parameter values yield a saturated design whose model matrix is a nonsingular generalized Vandermonde matrix; hence the monomials are linearly independent modulo the design ideal. On every compact positive parameter interval, a determinant-maximizing n-run design exists on this curve, every maximizer has distinct nodes, and it is exact D-optimal within the stated curve-restricted equal-replication class.\n\nCandidate contribution (constructive theorem; novelty confidence low): The explicit synthesis of base-B exponent encoding with generalized-Vandermonde identifiability shows that every n-node positive design on one constructed monomial curve is saturated for L, and compact determinant maximization on the same curve supplies an identifiable exact D-optimal design within that constrained class." }, { "id": 20001241, "problem_number": "AIM-COMPUTATION-0079", "title": "Removing term order without hiding the extrapolation choice", "statement": "Question 64 (Laubenbacher). Choosing models modulo the design ideal depends on the term order chosen. How can we eliminate this choice of term order from the problem of finding a model that fits the data? Richards: Could you use random search or genetic algorithms to solve this problem?", "original_statement": "Question 64 (Laubenbacher). Choosing models modulo the design ideal depends on the term order chosen. How can we eliminate this choice of term order from the problem of finding a model that fits the data? Richards: Could you use random search or genetic algorithms to solve this problem?", "clean_statement": "Question 64 (Laubenbacher). Choosing models modulo the design ideal depends on the term order chosen. How can we eliminate this choice of term order from the problem of finding a model that fits the data? Richards: Could you use random search or genetic algorithms to solve this problem?", "statement_status": "exact", "statement_verification": "The canonical record is Question 64 from the AIM workshop problem list *Computational Algebraic Statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 64\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[78]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 64 (Laubenbacher). Choosing models modulo the design ideal depends on the term order chosen. How can we eliminate this choice of term order from the problem of finding a model that fits the data? Richards: Could you use random search or genetic algorithms to solve this problem?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0079", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite real design D, evaluation canonically identifies R[x]/I(D) with the functions on D, so every response vector defines one term-order-independent quotient class and different standard-monomial normal forms are only polynomial sections of that class. No criterion depending only on on-design values can choose an off-design representative: multiplying squared distances to all design points gives an element of I(D) that can change any prescribed outside value arbitrarily. Once a finite dictionary and positive-definite polynomial norm are declared, the unique minimum-norm or ridge representative has an explicit weighted pseudoinverse formula that is invariant under arbitrary feature-basis changes when the Gram matrix transforms by congruence. A two-point design exactly separates opposite-term-order representatives x and y from the declared symmetric-norm solution (x+y)/2.\n\nCandidate contribution (no-go theorem and basis-covariant repair audit; novelty confidence low): The candidate contribution is a four-part term-order-elimination audit: return the canonical quotient class, use the explicit product of squared distances as an unbounded off-design nonidentifiability certificate, require the penalty congruence H -> T^T H T under every feature-basis change, and test the method on D={(0,0),(1,1)}, where opposite term orders return x and y for data (0,1) but the fixed symmetric norm returns (x+y)/2." }, { "id": 20001242, "problem_number": "AIM-COMPUTATION-0080", "title": "Evaluation-rank explanation for the value of knockout time series", "statement": "Question 65 (Laubenbacher). The model selection problem does not seem to be very effec-tive when only one time series is used but does seem to work well when multiple knockout time series are used in combination. Why?", "original_statement": "Question 65 (Laubenbacher). The model selection problem does not seem to be very effec-tive when only one time series is used but does seem to work well when multiple knockout time series are used in combination. Why?", "clean_statement": "Question 65 (Laubenbacher). The model selection problem does not seem to be very effec-tive when only one time series is used but does seem to work well when multiple knockout time series are used in combination. Why?", "statement_status": "exact", "statement_verification": "This is Question 65 in the AIM workshop list *Computational Algebraic Statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 65\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[79]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 65 (Laubenbacher). The model selection problem does not seem to be very effec-tive when only one time series is used but does seem to work well when multiple knockout time series are used in combination. Why?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0080", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a fixed r-dimensional function space over F_q, consistent pooled transition data leave exactly q^(r-rank(E)) coefficient models, where E is the evaluation matrix on distinct predecessor states; each additional knockout trajectory contributes exactly its row-space rank increment. A single deterministic finite-state trajectory eventually cycles and then adds no new evaluation rows. As an explicit certificate, the all-on state together with every single-coordinate knockout is a nonsingular design for affine rules over every field, with coefficients recovered by response differences.\n\nCandidate contribution (identifiability theorem; novelty confidence low): The candidate contribution packages an exact ambiguity count q^(r-rank(E)), a per-knockout marginal information measure equal to the evaluation-rank increment, and a characteristic-free determinant certificate showing that baseline plus all single knockouts identifies every affine rule." }, { "id": 20001243, "problem_number": "AIM-COMPUTATION-0081", "title": "Finite-field least squares as statistical decoding", "statement": "Question 66 (Laubenbacher). The model selection problem over finite fields is very sensitive to noise. Is there some sort of \"least squares\" over finite fields to deal with noisy data?", "original_statement": "Question 66 (Laubenbacher). The model selection problem over finite fields is very sensitive to noise. Is there some sort of \"least squares\" over finite fields to deal with noisy data?", "clean_statement": "Question 66 (Laubenbacher). The model selection problem over finite fields is very sensitive to noise. Is there some sort of \"least squares\" over finite fields to deal with noisy data?", "statement_status": "exact", "statement_verification": "The exact source record is Question 66 in the AIM workshop list *Computational Algebraic Statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 66\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[80]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 66 (Laubenbacher). The model selection problem over finite fields is very sensitive to noise. Is there some sort of \\\"least squares\\\" over finite fields to deal with noisy data?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0081", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A field-valued sum of squared residuals has no intrinsic minimization over a finite field. Under an explicit independent q-ary symmetric error model, maximum-likelihood polynomial fitting is instead nearest-codeword decoding in Hamming distance for the evaluation code ev_S(V). If that code has minimum distance d_min, every error pattern of size at most floor((d_min-1)/2) is corrected uniquely, while no larger uniform correct-recovery radius is possible. Coordinate-dependent error rates yield weighted Hamming MAP fitting, and affine Boolean functions on all of F_2^k form a [2^k,k+1,2^(k-1)] first-order Reed-Muller code.\n\nCandidate contribution (experimental-design diagnostic; novelty confidence low): For an injective evaluation design S and an added sampling block B, d_min(S union B) is strictly larger than d_min(S) if and only if B contains a point where every current minimum-weight nonzero model difference h is nonzero." }, { "id": 20001244, "problem_number": "AIM-COMPUTATION-0082", "title": "Rank-profile compression for finite-field time-series models", "statement": "Question 67 (Laubenbacher). What is the complexity of the algorithms used for time series modelling? How can we make the algorithms faster?", "original_statement": "Question 67 (Laubenbacher). What is the complexity of the algorithms used for time series modelling? How can we make the algorithms faster?", "clean_statement": "Question 67 (Laubenbacher). What is the complexity of the algorithms used for time series modelling? How can we make the algorithms faster?", "statement_status": "exact", "statement_verification": "The canonical AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 67\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[81]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 67 (Laubenbacher). What is the complexity of the algorithms used for time series modelling? How can we make the algorithms faster?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0082", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For fixed candidate functions evaluated on finite-field time-series predecessor states, all output coordinates share one feature matrix E. A row basis E_R gives an exact dichotomy: the full multi-output system EC=Y is consistent exactly when every output row obeys the same row dependencies as E; in that case E_R C=Y_R preserves the complete affine model family, while a failure yields an inconsistency certificate supported on at most rank(E)+1 transitions. After deterministic-state deduplication, dense streaming elimination uses u times the feature-evaluation cost plus O(u(r+d)(rank(E)+1)+d rank(E)^2) field operations and O(rank(E)(r+d)) storage, with one factorization reused for all d outputs.\n\nCandidate contribution (theorem and algorithmic synthesis; novelty confidence low): In the recovered polynomial-dynamical-system setting, one shared rank profile simultaneously compresses every exact coordinate-model equation to at most rank(E) transitions, preserves the entire affine multi-output model family, produces a rank(E)+1-transition inconsistency certificate when the response dependencies fail, and supports an explicit rank-sensitive streaming/reusable-factorization bound." }, { "id": 20001245, "problem_number": "AIM-COMPUTATION-0083", "title": "Exact sketches versus genuine dynamical reduction", "statement": "Question 68 (Sturmfels). In many of the problems for analyzing time series there are very few data points in a high dimensional space. How can techniques for reducing the dimensionality be used?\n\nSoftware for Algebraic Statistics", "original_statement": "Question 68 (Sturmfels). In many of the problems for analyzing time series there are very few data points in a high dimensional space. How can techniques for reducing the dimensionality be used? \n\nSoftware for Algebraic Statistics", "clean_statement": "Question 68 (Sturmfels). In many of the problems for analyzing time series there are very few data points in a high dimensional space. How can techniques for reducing the dimensionality be used?\n\nSoftware for Algebraic Statistics", "statement_status": "exact", "statement_verification": "The canonical record from `aim-computation-notes.json` is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 68\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[82]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 68 (Sturmfels). In many of the problems for analyzing time series there are very few data points in a high dimensional space. How can techniques for reducing the dimensionality be used? \\n\\nSoftware for Algebraic Statistics\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0083", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For m observed states in F_q^n, a uniform k-dimensional linear sketch is injective on the sample with probability at least 1 - binom(m,2)q^{-k}, and any collision-free sketch permits explicit reduced polynomial interpolation of arbitrary observed responses. In contrast, a response factors globally through the sketch if and only if it is invariant under every translation in the sketch kernel; for dynamics, a closed projected update exists if and only if AF has this invariance. An explicit family over F_2 shows that exact sample compression can hold while even the projected dynamics fails to close.\n\nCandidate contribution (diagnostic theorem package; novelty confidence low): The candidate contribution is the paired injective-sketch certificate and kernel-invariance audit for sparse finite-field time-series reduction, together with a minimal all-dimensions counterexample proving that the audit cannot be replaced by perfect sampled-transition fit." }, { "id": 20001246, "problem_number": "AIM-COMPUTATION-0084", "title": "Specialized algebra kernels and certified interoperability", "statement": "Question 69 (Roehrig). Why are there so many different systems for computational alge-bra? Couldn't they be add-ins to other systems? Stillman: Not likely to work since those other systems (Mathematica, Maple, etc.) do not give access to their kernels.", "original_statement": "Question 69 (Roehrig). Why are there so many different systems for computational alge-bra? Couldn't they be add-ins to other systems? Stillman: Not likely to work since those other systems (Mathematica, Maple, etc.) do not give access to their kernels.", "clean_statement": "Question 69 (Roehrig). Why are there so many different systems for computational alge-bra? Couldn't they be add-ins to other systems? Stillman: Not likely to work since those other systems (Mathematica, Maple, etc.) do not give access to their kernels.", "statement_status": "exact", "statement_verification": "The source is Question 69 in the AIM workshop list *Computational Algebraic Statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 69\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[83]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 69 (Roehrig). Why are there so many different systems for computational alge-bra? Couldn't they be add-ins to other systems? Stillman: Not likely to work since those other systems (Mathematica, Maple, etc.) do not give access to their kernels.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0084", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Modern library, interpreter, subprocess, and RPC interfaces show that computational algebra systems can be exposed through common front ends without private kernel access, although specialized kernels persist for substantive algorithmic and systems reasons. More rigorously, for an exactly serialized polynomial ring over an effective field, an opaque engine's candidate G can be certified as a Gröbner basis of the input ideal by host-verified two-way ideal-membership identities and leading-monomial-bounded standard representations of every Buchberger S-pair.\n\nCandidate contribution (certified interoperability contract; novelty confidence low): A versioned CertifiedGroebnerBasis response binding an exact ring/order descriptor and input digest to G, two-way ideal-membership identities, and standard-representation certificates for every S-pair permits a host with exact polynomial arithmetic to accept opaque-kernel output without trusting or accessing the producer's kernel." }, { "id": 20001247, "problem_number": "AIM-COMPUTATION-0085", "title": "Faithful interchange between computational algebra systems", "statement": "Question 70 (Slavkovic). Are there scripts which convert between input formats for the different computational algebra systems?", "original_statement": "Question 70 (Slavkovic). Are there scripts which convert between input formats for the different computational algebra systems?", "clean_statement": "Question 70 (Slavkovic). Are there scripts which convert between input formats for the different computational algebra systems?", "statement_status": "exact", "statement_verification": "The canonical record is Question 70 from the AIM workshop list *Computational algebraic statistics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 70\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[84]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 70 (Slavkovic). Are there scripts which convert between input formats for the different computational algebra systems?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0085", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Current command bridges, object adapters, semantic formats, and remote-call protocols give a qualified affirmative answer, but no untyped universal converter can preserve semantics. Formally, partial parsers and faithful serializers yield a semantics-preserving converter exactly on their shared representable domain; if an untagged target representation identifies two distinct source semantics, no faithful decoder or round trip exists. A concrete exact-polynomial profile specifies the coefficient domain, variables, monomial order, canonical sparse terms, aggregate semantics, operation contract, versions, and loss policy needed to avoid such collisions.\n\nCandidate contribution (theorem_and_conformance_profile; novelty confidence low): Candidate novelty: the typed round-trip and collision theorem is coupled to an Exact Polynomial Interchange Contract and mandatory negative conformance tests, giving a falsifiable criterion that distinguishes a faithful converter on an advertised polynomial profile from a syntax rewriter." }, { "id": 20001248, "problem_number": "AIM-COMPUTATION-0086", "title": "A certificate-first algebraic layer for R", "statement": "Question 71 (Sturmfels). What should be the first algebraic functions available in R? Dinwoodie: Toric Markov bases Riccomagno: Ideal of points Pachter: A general R interface with algebra packages Riccomagno: Gr¨ obner bases over fraction fields, with algebraic numbers and parame-ters. Pachter: When do we find out when these things have been implemented? Allman: Improve the documentation!", "original_statement": "Question 71 (Sturmfels). What should be the first algebraic functions available in R? Dinwoodie: Toric Markov bases Riccomagno: Ideal of points Pachter: A general R interface with algebra packages Riccomagno: Gr¨ obner bases over fraction fields, with algebraic numbers and parame-ters. Pachter: When do we find out when these things have been implemented? Allman: Improve the documentation!", "clean_statement": "Question 71 (Sturmfels). What should be the first algebraic functions available in R? Dinwoodie: Toric Markov bases Riccomagno: Ideal of points Pachter: A general R interface with algebra packages Riccomagno: Gr¨ obner bases over fraction fields, with algebraic numbers and parame-ters. Pachter: When do we find out when these things have been implemented? Allman: Improve the documentation!", "statement_status": "exact", "statement_verification": "The canonical record, preserving its extraction defects, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Computational algebraic statistics\nSection: \nSource item: 71\nSource URL: https://aimath.org/WWN/compalgstat/compalgstat.pdf\nCanonical location: aim-computation-notes.json notes[85]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 71 (Sturmfels). What should be the first algebraic functions available in R? Dinwoodie: Toric Markov bases Riccomagno: Ideal of points Pachter: A general R interface with algebra packages Riccomagno: Gr¨ obner bases over fraction fields, with algebraic numbers and parame-ters. Pachter: When do we find out when these things have been implemented? Allman: Improve the documentation!\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/compalgstat/compalgstat.pdf", "tags": [ "aim", "AIM-COMPUTATION-0086", "aim-domain:computation", "aim-workshop:compalgstat", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite set D of rational points over an effective field and a linearly independent finite polynomial dictionary M, the exact kernel of its evaluation matrix is canonically isomorphic to I(D) intersected with span(M); evaluation-matrix column independence is equivalent to independence of the corresponding residue classes in K[x]/I(D), and every consistent interpolation problem has solution set c0+ker(E). Thus exact evaluation, certificate-returning rank/kernel, and exact solve form a mathematically complete first R layer for the chosen dictionary. Separately, standard 2-by-2 moves constructively connect every fiber of nonnegative 2-by-J tables with fixed margins, giving a small exact conformance test for Markov-basis backends.\n\nCandidate contribution (conformance-first interface synthesis; novelty confidence low): Require two backend-neutral gates before an R algebra interface claims algebraic-statistics readiness: an exact finite-design evaluation/kernel/solve gate satisfying the proved quotient-ring identities, and a toric gate whose moves preserve margins and pass the proved constructive 2-by-J connectivity test." }, { "id": 20001249, "problem_number": "AIM-COMPUTATION-0087", "title": "Four distinct reductions in the American min-put benchmark", "statement": "3. A test problem 4. Reduction of the dimension 5. References 3\n\nChapter A: Open problems\n\nA.1 High-dimensional Problems in Finance and extension\n\nA.1.a Some stochastic control problems in finance. Several examples were considered.\n\nOptimal Stopping and free boundary problems.\n\nLet's consider the following financial market containing a non-risky asset S0\n\n> t\n\n= ert, and\n\nd risky assets (e.g. stocks) with prices modeled by a d-dimensional diffusion process S with dynamics\n\ndS t = rS tdt + diag( St)σ(t, S t)dW t,\n\nwhere W is a standard Brownian motion. An important problem in finance is the pricing of American options. Given a reward function g, an American option is a contract that provides to the owner the right to receive (from the seller) the amount g(St) at any time t if exercised before some fixed maturity T.This right can be exercised only once during the period [0, T ]. The price of this option at time t can be expressed as the value function associated to the optimal stopping problem\n\nv(t, S t) = sup\n\n> τ∈T [t,T ]\n\nE[e−r(τ −t)g(St) | St] (1.1.1) where T[t,T ] is the set of all stopping times with values in [ t, T ]. The associated value function v is the solution of the free boundary problem min {rv − vt − L v; v − g} = 0 on [0, T ) × [0, ∞)d\n\nv(T,. ) = g(.) on [0, ∞)d\n\nwhere L is the generator of the diffusion S.\n\nOptimal Investment and Hamilton-Jacobi-Bellman equations.\n\nAn other important issue in finance is that of optimal investment. Denoting by νt the number of stocks held by a given financial agent at time t, the associated wealth-process X ν\n\nhas dynamics given by\n\ndX t = νtdS t + ( Xt − ν∗\n\n> t\n\nSt)dS 0\n\n> t\n\nwhere ∗ stands for transposition, and S may have a general dynamics of the form\n\ndS t = μ(t, S t, ν t)dt + σ(t, S t, ν t)dW t,\n\nin order to take into account a possible influence of the agent's financial strategy on the dynamics of the risky assets. Given a concave (utility) function V, the agent tries to maximize the expected utility of terminal wealth max E[V (XνT )] (1.1.2) over a set of admissible financial strategies ( νt) with values in some subset U of Rd. The associated value function v is the solution of Hamilton-Jacobi-Bellman equation\n\n−vt − sup\n\n> ν∈U\n\nLν v = 0 on [0, T ) × [0, ∞)d × Rv(T, s, x ) = V (x) for ( s, x ) ∈ [0, ∞)d × R\n\nwhere Lν is the generator of the diffusion ( S, X ν ). 4\n\nA.1.b Extensions and BSDE's. The value function of Problem (1.1.1) can be reformu-lated in terms of the Y component of the solution ( Y, Z, A ) of the Backward Stochastic Differential Equation\n\nYt = g(ST ) −\n\n∫ Tt\n\nZ∗\n\n> t\n\ndW t + AT − At\n\nYt ≥ g(St)where A is a non-decreasing process satisfying\n\n∫ T\n\n> 0\n\n(Yt − g(St)) dA t = 0,\n\nsee e.g. [KKPPQ]. This leads us to consider the more general problem of approximating the solution of Reflected Forward - Backward Stochastic Differential Equations of the form\n\nXt = X0 +\n\n∫ Tt\n\nb(t, X t, Y t, Z t)dt +\n\n∫ Tt\n\na(t, X t, Y t, Z t)dW t\n\nYt = g(T, X T ) +\n\n∫ Tt\n\nf (t, X t, Y t, Z t)dt −\n\n∫ Tt\n\nZ∗\n\n> t\n\ndW t + AT − At\n\nYt ≥ g(t, X t)where A is a non-decreasing process satisfying\n\n∫ T\n\n> 0\n\n(Yt − g(t, X t)) dA t = 0.\n\nThis framework partially includes control problems associated to HJB equations. It is related to non-linear PDE's of the form min {− vt(t, x ) − 1\n\n2Trace[ vxx (t, x )aa ∗(t, x, v (t, x ), θ (t, x ))]\n\n−b(t, x, v (t, x ), θ (t, x )) ∗vx(t, x ) − f (t, x, v (t, x ), θ (t, x )); v(t, x ) − g(t, x )} = 0\n\nv(T,. ) = g(T,. )where θ(t, x ) solves\n\nvx(t, x )a(t, x, v (t, x ), θ (t, x )) = θ(t, x ),\n\nthrough the relations\n\nv(t, X t) = Yt and θ(t, X t)∗ = Zt.\n\nSee e.g. [MY].\n\nA.2 Some New Methodologies\n\nA.2.a The case where b and a does not depend on Y and Z. Several methods were considered.\n\nPure Monte Carlo Methods.\n\nPure Monte-Carlo methods are based on a discrete time approximation of the forward-backward equation. Once discretized the forward process can be simulated. These sim-ulations are then used to compute the conditional expectations involved in the backward 5\n\ndiscrete time approximation of ( Y, Z ). Two different methods can be used to compute these conditional expectations. a. The Longstaff-Schwartz (Carriere) approach consists in approximating the conditional expectations by regressions on a given basis of functions. This provides a very powerful, and, easy to implement, algorithm for which convergence has been shown to hold in the case of the American option pricing problem. However, no rate of convergence is given and the choice of the basis is a quite difficult problem. It has been used successfully in up to 20 factors models. See [C], [LS], [CLP]. b. The Malliavin approach consists in re-writing the conditional expectations as the ratio of two unconditional expectations that can be estimated by standard Monte-Carlo methods. Upper bounds for the rate of convergence are proved. Contrary to the previous approach, this algorithm also provides good approximations for the greeks, i.e. the gradient of the associated value function (which is related to the Z, see above). So far, the existing algorithm has a too important complexity, which explains why it has only been tested in small dimensions (up to 5). Some numerical improvements have been proposed, reducing the complexity from\n\nN 2 to N ln( N )d, where N is the number of simulated paths. This work is still in progress, the aim being to develop an algorithm such that the major part of the work is done before a reward function is specified, so as to reduce as much as possible the effective computation time once a particular payoff is defined. See [BET], [BT].\n\nGrid approximations.\n\nThe Quantization approach consists in approximating the original forward process by a discrete time process which evolves on some finite grid. The grid is constructed so has to provide the best Lp approximation. It was first applied to the pricing of American options in dimension up to 10. The construction of the optimal grid (and the computation of the associated transition probabilities) is very time consuming, but it can be done once for all. Once the grid, which is independent of the payoff function, is constructed, it provides a very quick algorithm for pricing American options on different payoffs, whenever they are written on the same assets. As in the Malliavin and Longstaff-Schwartz approach, the use of a good control variable is required. Extensions to non-linear filtering, optimal control and Asian-type options have also been studied. See [BP1], [BP2], [PP1], [PP2].\n\nDual formulation.\n\nThis algorithm is based on a dual formulation for problem (1.1.1):\n\nv(0, S 0) = inf E\n\n[\n\nsup\n\n> t≤T\n\n(e−rt g(St) − Mt)\n\n]\n\nwhere the inf is taken over a well suited set of martingales M. This algorithm consists in providing a upper bound for v by choosing some martingale M ∗ and computing\n\nE\n\n[\n\nsup\n\n> t≤T\n\n(e−rt g(St) − M ∗\n\n> t\n\n)\n\n]\n\nIn cases where a good martingale ˆM can be found, typically when\n\nE[e−r(T −t)g(ST )|St] =: ˆMt6\n\nis known as a function of S, this provides a quite sharp upper bound for the price of the American option. Numerical experiments, up to dimension 15, have been performed. See [R].\n\nCubature on Wiener spaces.\n\nCubature formula on Wiener spaces have been developed in [VL] in order to construct probability measures with finite support which approximate the Wiener measure in the sense that the expectation of iterated Stratonovich integrals under the approximating measure and the Wiener measure are close. In a sense, this approach is similar to that of the quantization since it allows to reduce to a finite dimensional setting. So far, it has been used to develop high order numerical schemes for high dimensional SDE's and semi-elliptic PDE's.\n\nA.2.b The case where b or a depends on Y and Z. In that case, it is no more possible to approximate (or simulate) the forward process X since its dynamic depends on the solu-tion. In such situations, two different solutions have been proposed: a - Construct an a priori grid for X, possibly based on some a priori on the dynamics of Y\n\nand Z.b - Given an a priori solution Y0 and Z0, approximate (or simulate) the corresponding forward process X0 and use the above methodologies to construct the corresponding solution ( Y1, Z 1)of the BSDE. Then, use this solution ( Y1, Z 1) to approximate (or simulate) the corresponding forward process X1 and go on iterating this procedure. Under some mild assumptions, this algorithm should be convergent. However, it seems to be quite heavy to implement. Solution a. has already been applied in the quantization approach but, so far, does not provide very good results. See [PP1].\n\nA.3 A test problem\n\nA test problem has been proposed to compare the performance of the different methods. It corresponds to the computation of a d dimensional at-the-money American min-put option in the Black-Scholes model. The assets are uncorrelated, have the same volatility and the same initial value:\n\nSit = 100 exp(( r − σ2/2) t − σW it ) for each i = 1,..., d\n\nwhere W = ( W 1,..., W d) is a standard Brownian motion. The payoff function is\n\ng(x) = [100 − min {x1,..., x d}]+.\n\nNumerical tests will be performed for different dimensions with σ = 20% and r = 5%. Results will be collected and compared by J. Cvitanic.\n\nA.4 Reduction of the dimension\n\nDifferent ways of reducing the dimension were proposed. aThe first consists in using Principal Component Analysis in order to \"aggregate\" a large number of random variables in a small number of principal directions. Also it is used in 7\n\npractice, it does not really solve the problem of the dimension, but only surrounds it, replac-ing the initial model by a more tractable, but different, one. bInstead of relying on PCA, dimension could be reduced by considering models where a few number of economic fundamentals could explain the behavior of many financial assets. cIn some cases, it is possible to reduce the dimension without changing the initial model. This is the case in P. Carr's Canadization approach for the computation of American options, where the randomization of the maturity allows to reduce to a time homogeneous problem. Extensions to Markovian control problems have been discussed. See [Carr] and [BKT].\n\nA.5 References\n\n[BP1] Bally V. and G. Pages: A quantization algorithm for solving discrete time multidi-mensional optimal stopping problems, Bernoulli, (9) 6, 1-47, 2003. [BP2] Bally V. and G. Pages: Error analysis of the quantization algorithm for obstacle problems, Stochastic Processes and their Applications,106 (1), 1-40, 2003. [BT] Bouchard B. and N. Touzi: Discrete Time Approximation and Monte-Carlo Simulation of Backward Stochastic Differential Equations, Stochastic Processes and their Applications, to appear. [BET] Bouchard B., I. Ekeland and N. Touzi: On the Malliavin approach to Monte Carlo approximation of conditional expectations, Finance and Stochastics, 8 (1), 2004. [BKT] Bouchard B., N. El Karoui and N. Touzi: Maturity randomization for stochastic control problems, in preparation. [Carr] Carr P.: Randomization and the American Put, The Review of Financial Studies, 11, 597-626, 1997. [C] Carriere E.: Valuation of the Early-Exercise Price for Options using Simulations and Nonparametric Regression, Insurance: mathematics and Economics, 19, 19-30, 1996. [CLP] Cl´ ement E., D. Lamberton, and P. Protter: An analysis of a least squares regression method for American option pricing, Finance and Stochastics, 6, 449-472, 2002. [KKPPQ] El Karoui N., C. Kapoudjan, E. Pardoux, S. Peng and M.C. Quenez: Reflected solutions of backward stochastic differential equations and related obstacle problems for PDE's, Annals of Probability, 25, 702-737, 1997. [LS] Longstaff F.A. and R.S. Schwartz: Valuing American Options By Simulation: A simple Least-Square Approach, Review of Financial Studies, 14, 113-147, 2001. [MY] Ma J. and J. Yong: Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math., 1702, Springer, 1999. [PP1] Pages G. and H. Pham: A quantization algorithm for multidimensional stochastic control problems, preprint LPMA n ◦697, 2001. [PP2] Pages G. and H. Pham: Optimal quantization methods for nonlinear filtering with discrete-time observations, preprint LPMA n ◦778, 2002. 8\n\n[R] Rogers L.C.G.: Monte Carlo valuation of American options, Mathematical Finance, 12, 271-286, 2002. [VL] Victoir N. and T. Lyons: Cubature on Wiener space, Proc. R. Soc. Lond. A, 460, 169-198, 2004.", "original_statement": "3. A test problem 4. Reduction of the dimension 5. References 3\n\nChapter A: Open problems \n\nA.1 High-dimensional Problems in Finance and extension \n\nA.1.a Some stochastic control problems in finance. Several examples were considered. \n\nOptimal Stopping and free boundary problems. \n\nLet's consider the following financial market containing a non-risky asset S0 \n\n> t\n\n= ert, and \n\nd risky assets (e.g. stocks) with prices modeled by a d-dimensional diffusion process S with dynamics \n\ndS t = rS tdt + diag( St)σ(t, S t)dW t,\n\nwhere W is a standard Brownian motion. An important problem in finance is the pricing of American options. Given a reward function g, an American option is a contract that provides to the owner the right to receive (from the seller) the amount g(St) at any time t if exercised before some fixed maturity T.This right can be exercised only once during the period [0, T ]. The price of this option at time t can be expressed as the value function associated to the optimal stopping problem \n\nv(t, S t) = sup \n\n> τ∈T [t,T ]\n\nE[e−r(τ −t)g(St) | St] (1.1.1) where T[t,T ] is the set of all stopping times with values in [ t, T ]. The associated value function v is the solution of the free boundary problem min {rv − vt − L v; v − g} = 0 on [0, T ) × [0, ∞)d\n\nv(T,. ) = g(.) on [0, ∞)d\n\nwhere L is the generator of the diffusion S.\n\nOptimal Investment and Hamilton-Jacobi-Bellman equations. \n\nAn other important issue in finance is that of optimal investment. Denoting by νt the number of stocks held by a given financial agent at time t, the associated wealth-process X ν\n\nhas dynamics given by \n\ndX t = νtdS t + ( Xt − ν∗ \n\n> t\n\nSt)dS 0\n\n> t\n\nwhere ∗ stands for transposition, and S may have a general dynamics of the form \n\ndS t = μ(t, S t, ν t)dt + σ(t, S t, ν t)dW t,\n\nin order to take into account a possible influence of the agent's financial strategy on the dynamics of the risky assets. Given a concave (utility) function V, the agent tries to maximize the expected utility of terminal wealth max E[V (XνT )] (1.1.2) over a set of admissible financial strategies ( νt) with values in some subset U of Rd. The associated value function v is the solution of Hamilton-Jacobi-Bellman equation \n\n−vt − sup \n\n> ν∈U\n\nLν v = 0 on [0, T ) × [0, ∞)d × Rv(T, s, x ) = V (x) for ( s, x ) ∈ [0, ∞)d × R\n\nwhere Lν is the generator of the diffusion ( S, X ν ). 4\n\nA.1.b Extensions and BSDE's. The value function of Problem (1.1.1) can be reformu-lated in terms of the Y component of the solution ( Y, Z, A ) of the Backward Stochastic Differential Equation \n\nYt = g(ST ) −\n\n∫ Tt\n\nZ∗ \n\n> t\n\ndW t + AT − At\n\nYt ≥ g(St)where A is a non-decreasing process satisfying \n\n∫ T\n\n> 0\n\n(Yt − g(St)) dA t = 0,\n\nsee e.g. [KKPPQ]. This leads us to consider the more general problem of approximating the solution of Reflected Forward - Backward Stochastic Differential Equations of the form \n\nXt = X0 +\n\n∫ Tt\n\nb(t, X t, Y t, Z t)dt +\n\n∫ Tt\n\na(t, X t, Y t, Z t)dW t\n\nYt = g(T, X T ) + \n\n∫ Tt\n\nf (t, X t, Y t, Z t)dt −\n\n∫ Tt\n\nZ∗ \n\n> t\n\ndW t + AT − At\n\nYt ≥ g(t, X t)where A is a non-decreasing process satisfying \n\n∫ T\n\n> 0\n\n(Yt − g(t, X t)) dA t = 0.\n\nThis framework partially includes control problems associated to HJB equations. It is related to non-linear PDE's of the form min {− vt(t, x ) − 1\n\n2Trace[ vxx (t, x )aa ∗(t, x, v (t, x ), θ (t, x ))] \n\n−b(t, x, v (t, x ), θ (t, x )) ∗vx(t, x ) − f (t, x, v (t, x ), θ (t, x )); v(t, x ) − g(t, x )} = 0 \n\nv(T,. ) = g(T,. )where θ(t, x ) solves \n\nvx(t, x )a(t, x, v (t, x ), θ (t, x )) = θ(t, x ),\n\nthrough the relations \n\nv(t, X t) = Yt and θ(t, X t)∗ = Zt.\n\nSee e.g. [MY]. \n\nA.2 Some New Methodologies \n\nA.2.a The case where b and a does not depend on Y and Z. Several methods were considered. \n\nPure Monte Carlo Methods. \n\nPure Monte-Carlo methods are based on a discrete time approximation of the forward-backward equation. Once discretized the forward process can be simulated. These sim-ulations are then used to compute the conditional expectations involved in the backward 5\n\ndiscrete time approximation of ( Y, Z ). Two different methods can be used to compute these conditional expectations. a. The Longstaff-Schwartz (Carriere) approach consists in approximating the conditional expectations by regressions on a given basis of functions. This provides a very powerful, and, easy to implement, algorithm for which convergence has been shown to hold in the case of the American option pricing problem. However, no rate of convergence is given and the choice of the basis is a quite difficult problem. It has been used successfully in up to 20 factors models. See [C], [LS], [CLP]. b. The Malliavin approach consists in re-writing the conditional expectations as the ratio of two unconditional expectations that can be estimated by standard Monte-Carlo methods. Upper bounds for the rate of convergence are proved. Contrary to the previous approach, this algorithm also provides good approximations for the greeks, i.e. the gradient of the associated value function (which is related to the Z, see above). So far, the existing algorithm has a too important complexity, which explains why it has only been tested in small dimensions (up to 5). Some numerical improvements have been proposed, reducing the complexity from \n\nN 2 to N ln( N )d, where N is the number of simulated paths. This work is still in progress, the aim being to develop an algorithm such that the major part of the work is done before a reward function is specified, so as to reduce as much as possible the effective computation time once a particular payoff is defined. See [BET], [BT]. \n\nGrid approximations. \n\nThe Quantization approach consists in approximating the original forward process by a discrete time process which evolves on some finite grid. The grid is constructed so has to provide the best Lp approximation. It was first applied to the pricing of American options in dimension up to 10. The construction of the optimal grid (and the computation of the associated transition probabilities) is very time consuming, but it can be done once for all. Once the grid, which is independent of the payoff function, is constructed, it provides a very quick algorithm for pricing American options on different payoffs, whenever they are written on the same assets. As in the Malliavin and Longstaff-Schwartz approach, the use of a good control variable is required. Extensions to non-linear filtering, optimal control and Asian-type options have also been studied. See [BP1], [BP2], [PP1], [PP2]. \n\nDual formulation. \n\nThis algorithm is based on a dual formulation for problem (1.1.1): \n\nv(0, S 0) = inf E\n\n[\n\nsup \n\n> t≤T\n\n(e−rt g(St) − Mt)\n\n]\n\nwhere the inf is taken over a well suited set of martingales M. This algorithm consists in providing a upper bound for v by choosing some martingale M ∗ and computing \n\nE\n\n[\n\nsup \n\n> t≤T\n\n(e−rt g(St) − M ∗ \n\n> t\n\n)\n\n]\n\nIn cases where a good martingale ˆM can be found, typically when \n\nE[e−r(T −t)g(ST )|St] =: ˆMt6\n\nis known as a function of S, this provides a quite sharp upper bound for the price of the American option. Numerical experiments, up to dimension 15, have been performed. See [R]. \n\nCubature on Wiener spaces. \n\nCubature formula on Wiener spaces have been developed in [VL] in order to construct probability measures with finite support which approximate the Wiener measure in the sense that the expectation of iterated Stratonovich integrals under the approximating measure and the Wiener measure are close. In a sense, this approach is similar to that of the quantization since it allows to reduce to a finite dimensional setting. So far, it has been used to develop high order numerical schemes for high dimensional SDE's and semi-elliptic PDE's. \n\nA.2.b The case where b or a depends on Y and Z. In that case, it is no more possible to approximate (or simulate) the forward process X since its dynamic depends on the solu-tion. In such situations, two different solutions have been proposed: a - Construct an a priori grid for X, possibly based on some a priori on the dynamics of Y\n\nand Z.b - Given an a priori solution Y0 and Z0, approximate (or simulate) the corresponding forward process X0 and use the above methodologies to construct the corresponding solution ( Y1, Z 1)of the BSDE. Then, use this solution ( Y1, Z 1) to approximate (or simulate) the corresponding forward process X1 and go on iterating this procedure. Under some mild assumptions, this algorithm should be convergent. However, it seems to be quite heavy to implement. Solution a. has already been applied in the quantization approach but, so far, does not provide very good results. See [PP1]. \n\nA.3 A test problem \n\nA test problem has been proposed to compare the performance of the different methods. It corresponds to the computation of a d dimensional at-the-money American min-put option in the Black-Scholes model. The assets are uncorrelated, have the same volatility and the same initial value: \n\nSit = 100 exp(( r − σ2/2) t − σW it ) for each i = 1,..., d \n\nwhere W = ( W 1,..., W d) is a standard Brownian motion. The payoff function is \n\ng(x) = [100 − min {x1,..., x d}]+.\n\nNumerical tests will be performed for different dimensions with σ = 20% and r = 5%. Results will be collected and compared by J. Cvitanic. \n\nA.4 Reduction of the dimension \n\nDifferent ways of reducing the dimension were proposed. aThe first consists in using Principal Component Analysis in order to \"aggregate\" a large number of random variables in a small number of principal directions. Also it is used in 7\n\npractice, it does not really solve the problem of the dimension, but only surrounds it, replac-ing the initial model by a more tractable, but different, one. bInstead of relying on PCA, dimension could be reduced by considering models where a few number of economic fundamentals could explain the behavior of many financial assets. cIn some cases, it is possible to reduce the dimension without changing the initial model. This is the case in P. Carr's Canadization approach for the computation of American options, where the randomization of the maturity allows to reduce to a time homogeneous problem. Extensions to Markovian control problems have been discussed. See [Carr] and [BKT]. \n\nA.5 References \n\n[BP1] Bally V. and G. Pages: A quantization algorithm for solving discrete time multidi-mensional optimal stopping problems, Bernoulli, (9) 6, 1-47, 2003. [BP2] Bally V. and G. Pages: Error analysis of the quantization algorithm for obstacle problems, Stochastic Processes and their Applications,106 (1), 1-40, 2003. [BT] Bouchard B. and N. Touzi: Discrete Time Approximation and Monte-Carlo Simulation of Backward Stochastic Differential Equations, Stochastic Processes and their Applications, to appear. [BET] Bouchard B., I. Ekeland and N. Touzi: On the Malliavin approach to Monte Carlo approximation of conditional expectations, Finance and Stochastics, 8 (1), 2004. [BKT] Bouchard B., N. El Karoui and N. Touzi: Maturity randomization for stochastic control problems, in preparation. [Carr] Carr P.: Randomization and the American Put, The Review of Financial Studies, 11, 597-626, 1997. [C] Carriere E.: Valuation of the Early-Exercise Price for Options using Simulations and Nonparametric Regression, Insurance: mathematics and Economics, 19, 19-30, 1996. [CLP] Cl´ ement E., D. Lamberton, and P. Protter: An analysis of a least squares regression method for American option pricing, Finance and Stochastics, 6, 449-472, 2002. [KKPPQ] El Karoui N., C. Kapoudjan, E. Pardoux, S. Peng and M.C. Quenez: Reflected solutions of backward stochastic differential equations and related obstacle problems for PDE's, Annals of Probability, 25, 702-737, 1997. [LS] Longstaff F.A. and R.S. Schwartz: Valuing American Options By Simulation: A simple Least-Square Approach, Review of Financial Studies, 14, 113-147, 2001. [MY] Ma J. and J. Yong: Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math., 1702, Springer, 1999. [PP1] Pages G. and H. Pham: A quantization algorithm for multidimensional stochastic control problems, preprint LPMA n ◦697, 2001. [PP2] Pages G. and H. Pham: Optimal quantization methods for nonlinear filtering with discrete-time observations, preprint LPMA n ◦778, 2002. 8\n\n[R] Rogers L.C.G.: Monte Carlo valuation of American options, Mathematical Finance, 12, 271-286, 2002. [VL] Victoir N. and T. Lyons: Cubature on Wiener space, Proc. R. Soc. Lond. A, 460, 169-198, 2004.", "clean_statement": "3. A test problem 4. Reduction of the dimension 5. References 3\n\nChapter A: Open problems\n\nA.1 High-dimensional Problems in Finance and extension\n\nA.1.a Some stochastic control problems in finance. Several examples were considered.\n\nOptimal Stopping and free boundary problems.\n\nLet's consider the following financial market containing a non-risky asset S0\n\n> t\n\n= ert, and\n\nd risky assets (e.g. stocks) with prices modeled by a d-dimensional diffusion process S with dynamics\n\ndS t = rS tdt + diag( St)σ(t, S t)dW t,\n\nwhere W is a standard Brownian motion. An important problem in finance is the pricing of American options. Given a reward function g, an American option is a contract that provides to the owner the right to receive (from the seller) the amount g(St) at any time t if exercised before some fixed maturity T.This right can be exercised only once during the period [0, T ]. The price of this option at time t can be expressed as the value function associated to the optimal stopping problem\n\nv(t, S t) = sup\n\n> τ∈T [t,T ]\n\nE[e−r(τ −t)g(St) | St] (1.1.1) where T[t,T ] is the set of all stopping times with values in [ t, T ]. The associated value function v is the solution of the free boundary problem min {rv − vt − L v; v − g} = 0 on [0, T ) × [0, ∞)d\n\nv(T,. ) = g(.) on [0, ∞)d\n\nwhere L is the generator of the diffusion S.\n\nOptimal Investment and Hamilton-Jacobi-Bellman equations.\n\nAn other important issue in finance is that of optimal investment. Denoting by νt the number of stocks held by a given financial agent at time t, the associated wealth-process X ν\n\nhas dynamics given by\n\ndX t = νtdS t + ( Xt − ν∗\n\n> t\n\nSt)dS 0\n\n> t\n\nwhere ∗ stands for transposition, and S may have a general dynamics of the form\n\ndS t = μ(t, S t, ν t)dt + σ(t, S t, ν t)dW t,\n\nin order to take into account a possible influence of the agent's financial strategy on the dynamics of the risky assets. Given a concave (utility) function V, the agent tries to maximize the expected utility of terminal wealth max E[V (XνT )] (1.1.2) over a set of admissible financial strategies ( νt) with values in some subset U of Rd. The associated value function v is the solution of Hamilton-Jacobi-Bellman equation\n\n−vt − sup\n\n> ν∈U\n\nLν v = 0 on [0, T ) × [0, ∞)d × Rv(T, s, x ) = V (x) for ( s, x ) ∈ [0, ∞)d × R\n\nwhere Lν is the generator of the diffusion ( S, X ν ). 4\n\nA.1.b Extensions and BSDE's. The value function of Problem (1.1.1) can be reformu-lated in terms of the Y component of the solution ( Y, Z, A ) of the Backward Stochastic Differential Equation\n\nYt = g(ST ) −\n\n∫ Tt\n\nZ∗\n\n> t\n\ndW t + AT − At\n\nYt ≥ g(St)where A is a non-decreasing process satisfying\n\n∫ T\n\n> 0\n\n(Yt − g(St)) dA t = 0,\n\nsee e.g. [KKPPQ]. This leads us to consider the more general problem of approximating the solution of Reflected Forward - Backward Stochastic Differential Equations of the form\n\nXt = X0 +\n\n∫ Tt\n\nb(t, X t, Y t, Z t)dt +\n\n∫ Tt\n\na(t, X t, Y t, Z t)dW t\n\nYt = g(T, X T ) +\n\n∫ Tt\n\nf (t, X t, Y t, Z t)dt −\n\n∫ Tt\n\nZ∗\n\n> t\n\ndW t + AT − At\n\nYt ≥ g(t, X t)where A is a non-decreasing process satisfying\n\n∫ T\n\n> 0\n\n(Yt − g(t, X t)) dA t = 0.\n\nThis framework partially includes control problems associated to HJB equations. It is related to non-linear PDE's of the form min {− vt(t, x ) − 1\n\n2Trace[ vxx (t, x )aa ∗(t, x, v (t, x ), θ (t, x ))]\n\n−b(t, x, v (t, x ), θ (t, x )) ∗vx(t, x ) − f (t, x, v (t, x ), θ (t, x )); v(t, x ) − g(t, x )} = 0\n\nv(T,. ) = g(T,. )where θ(t, x ) solves\n\nvx(t, x )a(t, x, v (t, x ), θ (t, x )) = θ(t, x ),\n\nthrough the relations\n\nv(t, X t) = Yt and θ(t, X t)∗ = Zt.\n\nSee e.g. [MY].\n\nA.2 Some New Methodologies\n\nA.2.a The case where b and a does not depend on Y and Z. Several methods were considered.\n\nPure Monte Carlo Methods.\n\nPure Monte-Carlo methods are based on a discrete time approximation of the forward-backward equation. Once discretized the forward process can be simulated. These sim-ulations are then used to compute the conditional expectations involved in the backward 5\n\ndiscrete time approximation of ( Y, Z ). Two different methods can be used to compute these conditional expectations. a. The Longstaff-Schwartz (Carriere) approach consists in approximating the conditional expectations by regressions on a given basis of functions. This provides a very powerful, and, easy to implement, algorithm for which convergence has been shown to hold in the case of the American option pricing problem. However, no rate of convergence is given and the choice of the basis is a quite difficult problem. It has been used successfully in up to 20 factors models. See [C], [LS], [CLP]. b. The Malliavin approach consists in re-writing the conditional expectations as the ratio of two unconditional expectations that can be estimated by standard Monte-Carlo methods. Upper bounds for the rate of convergence are proved. Contrary to the previous approach, this algorithm also provides good approximations for the greeks, i.e. the gradient of the associated value function (which is related to the Z, see above). So far, the existing algorithm has a too important complexity, which explains why it has only been tested in small dimensions (up to 5). Some numerical improvements have been proposed, reducing the complexity from\n\nN 2 to N ln( N )d, where N is the number of simulated paths. This work is still in progress, the aim being to develop an algorithm such that the major part of the work is done before a reward function is specified, so as to reduce as much as possible the effective computation time once a particular payoff is defined. See [BET], [BT].\n\nGrid approximations.\n\nThe Quantization approach consists in approximating the original forward process by a discrete time process which evolves on some finite grid. The grid is constructed so has to provide the best Lp approximation. It was first applied to the pricing of American options in dimension up to 10. The construction of the optimal grid (and the computation of the associated transition probabilities) is very time consuming, but it can be done once for all. Once the grid, which is independent of the payoff function, is constructed, it provides a very quick algorithm for pricing American options on different payoffs, whenever they are written on the same assets. As in the Malliavin and Longstaff-Schwartz approach, the use of a good control variable is required. Extensions to non-linear filtering, optimal control and Asian-type options have also been studied. See [BP1], [BP2], [PP1], [PP2].\n\nDual formulation.\n\nThis algorithm is based on a dual formulation for problem (1.1.1):\n\nv(0, S 0) = inf E\n\n[\n\nsup\n\n> t≤T\n\n(e−rt g(St) − Mt)\n\n]\n\nwhere the inf is taken over a well suited set of martingales M. This algorithm consists in providing a upper bound for v by choosing some martingale M ∗ and computing\n\nE\n\n[\n\nsup\n\n> t≤T\n\n(e−rt g(St) − M ∗\n\n> t\n\n)\n\n]\n\nIn cases where a good martingale ˆM can be found, typically when\n\nE[e−r(T −t)g(ST )|St] =: ˆMt6\n\nis known as a function of S, this provides a quite sharp upper bound for the price of the American option. Numerical experiments, up to dimension 15, have been performed. See [R].\n\nCubature on Wiener spaces.\n\nCubature formula on Wiener spaces have been developed in [VL] in order to construct probability measures with finite support which approximate the Wiener measure in the sense that the expectation of iterated Stratonovich integrals under the approximating measure and the Wiener measure are close. In a sense, this approach is similar to that of the quantization since it allows to reduce to a finite dimensional setting. So far, it has been used to develop high order numerical schemes for high dimensional SDE's and semi-elliptic PDE's.\n\nA.2.b The case where b or a depends on Y and Z. In that case, it is no more possible to approximate (or simulate) the forward process X since its dynamic depends on the solu-tion. In such situations, two different solutions have been proposed: a - Construct an a priori grid for X, possibly based on some a priori on the dynamics of Y\n\nand Z.b - Given an a priori solution Y0 and Z0, approximate (or simulate) the corresponding forward process X0 and use the above methodologies to construct the corresponding solution ( Y1, Z 1)of the BSDE. Then, use this solution ( Y1, Z 1) to approximate (or simulate) the corresponding forward process X1 and go on iterating this procedure. Under some mild assumptions, this algorithm should be convergent. However, it seems to be quite heavy to implement. Solution a. has already been applied in the quantization approach but, so far, does not provide very good results. See [PP1].\n\nA.3 A test problem\n\nA test problem has been proposed to compare the performance of the different methods. It corresponds to the computation of a d dimensional at-the-money American min-put option in the Black-Scholes model. The assets are uncorrelated, have the same volatility and the same initial value:\n\nSit = 100 exp(( r − σ2/2) t − σW it ) for each i = 1,..., d\n\nwhere W = ( W 1,..., W d) is a standard Brownian motion. The payoff function is\n\ng(x) = [100 − min {x1,..., x d}]+.\n\nNumerical tests will be performed for different dimensions with σ = 20% and r = 5%. Results will be collected and compared by J. Cvitanic.\n\nA.4 Reduction of the dimension\n\nDifferent ways of reducing the dimension were proposed. aThe first consists in using Principal Component Analysis in order to \"aggregate\" a large number of random variables in a small number of principal directions. Also it is used in 7\n\npractice, it does not really solve the problem of the dimension, but only surrounds it, replac-ing the initial model by a more tractable, but different, one. bInstead of relying on PCA, dimension could be reduced by considering models where a few number of economic fundamentals could explain the behavior of many financial assets. cIn some cases, it is possible to reduce the dimension without changing the initial model. This is the case in P. Carr's Canadization approach for the computation of American options, where the randomization of the maturity allows to reduce to a time homogeneous problem. Extensions to Markovian control problems have been discussed. See [Carr] and [BKT].\n\nA.5 References\n\n[BP1] Bally V. and G. Pages: A quantization algorithm for solving discrete time multidi-mensional optimal stopping problems, Bernoulli, (9) 6, 1-47, 2003. [BP2] Bally V. and G. Pages: Error analysis of the quantization algorithm for obstacle problems, Stochastic Processes and their Applications,106 (1), 1-40, 2003. [BT] Bouchard B. and N. Touzi: Discrete Time Approximation and Monte-Carlo Simulation of Backward Stochastic Differential Equations, Stochastic Processes and their Applications, to appear. [BET] Bouchard B., I. Ekeland and N. Touzi: On the Malliavin approach to Monte Carlo approximation of conditional expectations, Finance and Stochastics, 8 (1), 2004. [BKT] Bouchard B., N. El Karoui and N. Touzi: Maturity randomization for stochastic control problems, in preparation. [Carr] Carr P.: Randomization and the American Put, The Review of Financial Studies, 11, 597-626, 1997. [C] Carriere E.: Valuation of the Early-Exercise Price for Options using Simulations and Nonparametric Regression, Insurance: mathematics and Economics, 19, 19-30, 1996. [CLP] Cl´ ement E., D. Lamberton, and P. Protter: An analysis of a least squares regression method for American option pricing, Finance and Stochastics, 6, 449-472, 2002. [KKPPQ] El Karoui N., C. Kapoudjan, E. Pardoux, S. Peng and M.C. Quenez: Reflected solutions of backward stochastic differential equations and related obstacle problems for PDE's, Annals of Probability, 25, 702-737, 1997. [LS] Longstaff F.A. and R.S. Schwartz: Valuing American Options By Simulation: A simple Least-Square Approach, Review of Financial Studies, 14, 113-147, 2001. [MY] Ma J. and J. Yong: Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math., 1702, Springer, 1999. [PP1] Pages G. and H. Pham: A quantization algorithm for multidimensional stochastic control problems, preprint LPMA n ◦697, 2001. [PP2] Pages G. and H. Pham: Optimal quantization methods for nonlinear filtering with discrete-time observations, preprint LPMA n ◦778, 2002. 8\n\n[R] Rogers L.C.G.: Monte Carlo valuation of American options, Mathematical Finance, 12, 271-286, 2002. [VL] Victoir N. and T. Lyons: Cubature on Wiener space, Proc. R. Soc. Lond. A, 460, 169-198, 2004.", "statement_status": "exact", "statement_verification": "The canonical `problem` field is 12,517 characters long. It begins with a table-of-contents fragment and then reproduces almost all of “Chapter A: Open problems” from the AIM workshop list *Numerical probabilistic methods for high-dimensional problems in finance* [1]. It includes:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Numerical probabilistic methods for high-dimensional problems in finance\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/highdimfinance/highdimfinance.pdf\nCanonical location: aim-computation-notes.json notes[86]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. A test problem 4. Reduction of the dimension 5. References 3\\n\\nChapter A: Open problems \\n\\nA.1 High-dimensional Problems in Finance and extension \\n\\nA.1.a Some stochastic control problems in finance. Several examples were considered. \\n\\nOptimal Stopping and free boundary problems. \\n\\nLet's consider the following financial market containing a non-risky asset S0 \\n\\n> t\\n\\n= ert, and \\n\\nd risky assets (e.g. stocks) with prices modeled by a d-dimensional diffusion process S with dynamics \\n\\ndS t = rS tdt + diag( St)σ(t, S t)dW t,\\n\\nwhere W is a standard Brownian motion. An important problem in finance is the pricing of American options. Given a reward function g, an American option is a contract that provides to the owner the right to receive (from the seller) the amount g(St) at any time t if exercised before some fixed maturity T.This right can be exercised only once during the period [0, T ]. The price of this option at time t can be expressed as the value function associated to the optimal stopping problem \\n\\nv(t, S t) = sup \\n\\n> τ∈T [t,T ]\\n\\nE[e−r(τ −t)g(St) | St] (1.1.1) where T[t,T ] is the set of all stopping times with values in [ t, T ]. The associated value function v is the solution of the free boundary problem min {rv − vt − L v; v − g} = 0 on [0, T ) × [0, ∞)d\\n\\nv(T,. ) = g(.) on [0, ∞)d\\n\\nwhere L is the generator of the diffusion S.\\n\\nOptimal Investment and Hamilton-Jacobi-Bellman equations. \\n\\nAn other important issue in finance is that of optimal investment. Denoting by νt the number of stocks held by a given financial agent at time t, the associated wealth-process X ν\\n\\nhas dynamics given by \\n\\ndX t = νtdS t + ( Xt − ν∗ \\n\\n> t\\n\\nSt)dS 0\\n\\n> t\\n\\nwhere ∗ stands for transposition, and S may have a general dynamics of the form \\n\\ndS t = μ(t, S t, ν t)dt + σ(t, S t, ν t)dW t,\\n\\nin order to take into account a possible influence of the agent's financial strategy on the dynamics of the risky assets. Given a concave (utility) function V, the agent tries to maximize the expected utility of terminal wealth max E[V (XνT )] (1.1.2) over a set of admissible financial strategies ( νt) with values in some subset U of Rd. The associated value function v is the solution of Hamilton-Jacobi-Bellman equation \\n\\n−vt − sup \\n\\n> ν∈U\\n\\nLν v = 0 on [0, T ) × [0, ∞)d × Rv(T, s, x ) = V (x) for ( s, x ) ∈ [0, ∞)d × R\\n\\nwhere Lν is the generator of the diffusion ( S, X ν ). 4\\n\\nA.1.b Extensions and BSDE's. The value function of Problem (1.1.1) can be reformu-lated in terms of the Y component of the solution ( Y, Z, A ) of the Backward Stochastic Differential Equation \\n\\nYt = g(ST ) −\\n\\n∫ Tt\\n\\nZ∗ \\n\\n> t\\n\\ndW t + AT − At\\n\\nYt ≥ g(St)where A is a non-decreasing process satisfying \\n\\n∫ T\\n\\n> 0\\n\\n(Yt − g(St)) dA t = 0,\\n\\nsee e.g. [KKPPQ]. This leads us to consider the more general problem of approximating the solution of Reflected Forward - Backward Stochastic Differential Equations of the form \\n\\nXt = X0 +\\n\\n∫ Tt\\n\\nb(t, X t, Y t, Z t)dt +\\n\\n∫ Tt\\n\\na(t, X t, Y t, Z t)dW t\\n\\nYt = g(T, X T ) + \\n\\n∫ Tt\\n\\nf (t, X t, Y t, Z t)dt −\\n\\n∫ Tt\\n\\nZ∗ \\n\\n> t\\n\\ndW t + AT − At\\n\\nYt ≥ g(t, X t)where A is a non-decreasing process satisfying \\n\\n∫ T\\n\\n> 0\\n\\n(Yt − g(t, X t)) dA t = 0.\\n\\nThis framework partially includes control problems associated to HJB equations. It is related to non-linear PDE's of the form min {− vt(t, x ) − 1\\n\\n2Trace[ vxx (t, x )aa ∗(t, x, v (t, x ), θ (t, x ))] \\n\\n−b(t, x, v (t, x ), θ (t, x )) ∗vx(t, x ) − f (t, x, v (t, x ), θ (t, x )); v(t, x ) − g(t, x )} = 0 \\n\\nv(T,. ) = g(T,. )where θ(t, x ) solves \\n\\nvx(t, x )a(t, x, v (t, x ), θ (t, x )) = θ(t, x ),\\n\\nthrough the relations \\n\\nv(t, X t) = Yt and θ(t, X t)∗ = Zt.\\n\\nSee e.g. [MY]. \\n\\nA.2 Some New Methodologies \\n\\nA.2.a The case where b and a does not depend on Y and Z. Several methods were considered. \\n\\nPure Monte Carlo Methods. \\n\\nPure Monte-Carlo methods are based on a discrete time approximation of the forward-backward equation. Once discretized the forward process can be simulated. These sim-ulations are then used to compute the conditional expectations involved in the backward 5\\n\\ndiscrete time approximation of ( Y, Z ). Two different methods can be used to compute these conditional expectations. a. The Longstaff-Schwartz (Carriere) approach consists in approximating the conditional expectations by regressions on a given basis of functions. This provides a very powerful, and, easy to implement, algorithm for which convergence has been shown to hold in the case of the American option pricing problem. However, no rate of convergence is given and the choice of the basis is a quite difficult problem. It has been used successfully in up to 20 factors models. See [C], [LS], [CLP]. b. The Malliavin approach consists in re-writing the conditional expectations as the ratio of two unconditional expectations that can be estimated by standard Monte-Carlo methods. Upper bounds for the rate of convergence are proved. Contrary to the previous approach, this algorithm also provides good approximations for the greeks, i.e. the gradient of the associated value function (which is related to the Z, see above). So far, the existing algorithm has a too important complexity, which explains why it has only been tested in small dimensions (up to 5). Some numerical improvements have been proposed, reducing the complexity from \\n\\nN 2 to N ln( N )d, where N is the number of simulated paths. This work is still in progress, the aim being to develop an algorithm such that the major part of the work is done before a reward function is specified, so as to reduce as much as possible the effective computation time once a particular payoff is defined. See [BET], [BT]. \\n\\nGrid approximations. \\n\\nThe Quantization approach consists in approximating the original forward process by a discrete time process which evolves on some finite grid. The grid is constructed so has to provide the best Lp approximation. It was first applied to the pricing of American options in dimension up to 10. The construction of the optimal grid (and the computation of the associated transition probabilities) is very time consuming, but it can be done once for all. Once the grid, which is independent of the payoff function, is constructed, it provides a very quick algorithm for pricing American options on different payoffs, whenever they are written on the same assets. As in the Malliavin and Longstaff-Schwartz approach, the use of a good control variable is required. Extensions to non-linear filtering, optimal control and Asian-type options have also been studied. See [BP1], [BP2], [PP1], [PP2]. \\n\\nDual formulation. \\n\\nThis algorithm is based on a dual formulation for problem (1.1.1): \\n\\nv(0, S 0) = inf E\\n\\n[\\n\\nsup \\n\\n> t≤T\\n\\n(e−rt g(St) − Mt)\\n\\n]\\n\\nwhere the inf is taken over a well suited set of martingales M. This algorithm consists in providing a upper bound for v by choosing some martingale M ∗ and computing \\n\\nE\\n\\n[\\n\\nsup \\n\\n> t≤T\\n\\n(e−rt g(St) − M ∗ \\n\\n> t\\n\\n)\\n\\n]\\n\\nIn cases where a good martingale ˆM can be found, typically when \\n\\nE[e−r(T −t)g(ST )|St] =: ˆMt6\\n\\nis known as a function of S, this provides a quite sharp upper bound for the price of the American option. Numerical experiments, up to dimension 15, have been performed. See [R]. \\n\\nCubature on Wiener spaces. \\n\\nCubature formula on Wiener spaces have been developed in [VL] in order to construct probability measures with finite support which approximate the Wiener measure in the sense that the expectation of iterated Stratonovich integrals under the approximating measure and the Wiener measure are close. In a sense, this approach is similar to that of the quantization since it allows to reduce to a finite dimensional setting. So far, it has been used to develop high order numerical schemes for high dimensional SDE's and semi-elliptic PDE's. \\n\\nA.2.b The case where b or a depends on Y and Z. In that case, it is no more possible to approximate (or simulate) the forward process X since its dynamic depends on the solu-tion. In such situations, two different solutions have been proposed: a - Construct an a priori grid for X, possibly based on some a priori on the dynamics of Y\\n\\nand Z.b - Given an a priori solution Y0 and Z0, approximate (or simulate) the corresponding forward process X0 and use the above methodologies to construct the corresponding solution ( Y1, Z 1)of the BSDE. Then, use this solution ( Y1, Z 1) to approximate (or simulate) the corresponding forward process X1 and go on iterating this procedure. Under some mild assumptions, this algorithm should be convergent. However, it seems to be quite heavy to implement. Solution a. has already been applied in the quantization approach but, so far, does not provide very good results. See [PP1]. \\n\\nA.3 A test problem \\n\\nA test problem has been proposed to compare the performance of the different methods. It corresponds to the computation of a d dimensional at-the-money American min-put option in the Black-Scholes model. The assets are uncorrelated, have the same volatility and the same initial value: \\n\\nSit = 100 exp(( r − σ2/2) t − σW it ) for each i = 1,..., d \\n\\nwhere W = ( W 1,..., W d) is a standard Brownian motion. The payoff function is \\n\\ng(x) = [100 − min {x1,..., x d}]+.\\n\\nNumerical tests will be performed for different dimensions with σ = 20% and r = 5%. Results will be collected and compared by J. Cvitanic. \\n\\nA.4 Reduction of the dimension \\n\\nDifferent ways of reducing the dimension were proposed. aThe first consists in using Principal Component Analysis in order to \\\"aggregate\\\" a large number of random variables in a small number of principal directions. Also it is used in 7\\n\\npractice, it does not really solve the problem of the dimension, but only surrounds it, replac-ing the initial model by a more tractable, but different, one. bInstead of relying on PCA, dimension could be reduced by considering models where a few number of economic fundamentals could explain the behavior of many financial assets. cIn some cases, it is possible to reduce the dimension without changing the initial model. This is the case in P. Carr's Canadization approach for the computation of American options, where the randomization of the maturity allows to reduce to a time homogeneous problem. Extensions to Markovian control problems have been discussed. See [Carr] and [BKT]. \\n\\nA.5 References \\n\\n[BP1] Bally V. and G. Pages: A quantization algorithm for solving discrete time multidi-mensional optimal stopping problems, Bernoulli, (9) 6, 1-47, 2003. [BP2] Bally V. and G. Pages: Error analysis of the quantization algorithm for obstacle problems, Stochastic Processes and their Applications,106 (1), 1-40, 2003. [BT] Bouchard B. and N. Touzi: Discrete Time Approximation and Monte-Carlo Simulation of Backward Stochastic Differential Equations, Stochastic Processes and their Applications, to appear. [BET] Bouchard B., I. Ekeland and N. Touzi: On the Malliavin approach to Monte Carlo approximation of conditional expectations, Finance and Stochastics, 8 (1), 2004. [BKT] Bouchard B., N. El Karoui and N. Touzi: Maturity randomization for stochastic control problems, in preparation. [Carr] Carr P.: Randomization and the American Put, The Review of Financial Studies, 11, 597-626, 1997. [C] Carriere E.: Valuation of the Early-Exercise Price for Options using Simulations and Nonparametric Regression, Insurance: mathematics and Economics, 19, 19-30, 1996. [CLP] Cl´ ement E., D. Lamberton, and P. Protter: An analysis of a least squares regression method for American option pricing, Finance and Stochastics, 6, 449-472, 2002. [KKPPQ] El Karoui N., C. Kapoudjan, E. Pardoux, S. Peng and M.C. Quenez: Reflected solutions of backward stochastic differential equations and related obstacle problems for PDE's, Annals of Probability, 25, 702-737, 1997. [LS] Longstaff F.A. and R.S. Schwartz: Valuing American Options By Simulation: A simple Least-Square Approach, Review of Financial Studies, 14, 113-147, 2001. [MY] Ma J. and J. Yong: Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math., 1702, Springer, 1999. [PP1] Pages G. and H. Pham: A quantization algorithm for multidimensional stochastic control problems, preprint LPMA n ◦697, 2001. [PP2] Pages G. and H. Pham: Optimal quantization methods for nonlinear filtering with discrete-time observations, preprint LPMA n ◦778, 2002. 8\\n\\n[R] Rogers L.C.G.: Monte Carlo valuation of American options, Mathematical Finance, 12, 271-286, 2002. [VL] Victoir N. and T. Lyons: Cubature on Wiener space, Proc. R. Soc. Lond. A, 460, 169-198, 2004.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/highdimfinance/highdimfinance.pdf", "tags": [ "aim", "AIM-COMPUTATION-0087", "aim-domain:computation", "aim-workshop:highdimfinance", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the independent equal-volatility Black-Scholes min-put benchmark, the European terminal value has an exact one-dimensional layer-cake integral involving a product of lognormal survival probabilities, but it is strictly decreasing in every current asset coordinate. Same-minimum states therefore have different two-date Bermudan values, proving that the current minimum is not an exact reduced Markov state. Population log-return covariance is sigma^2 tau I_d, so rank-k PCA captures exactly k/d of total variance and no k k k 1≤a≤8(log n)7/2\\n\\n(X − a)ea: ea ≥ 0\\n\\n}.\\n\\nObserve that every m ∈ I is introspective for every g(X) ∈ T.Let t be the order of the group generated by n and p in ( Z/r Z)∗. There are > t elements in I less than or equal to n2√t.Furthermore, there are > n 2√t distinct polynomials of degree < t which are distinct modulo p and h(X), where h(X) is an irreducible factor of the rth cyclotomic polynomial in Fp. We show this as follows. The number of polynomials in T of degree < t is at least 2 min {t, 8(log n)7/2} by simply considering products of distinct linear factors in T. Assume that t > 4(log n)2 and t < \\n\\n16(log n)\\n5. (We will come back to this: we can choose the value of r to obtain it.) Then \\n\\nn2√t = 2 2√t log n < 2t\\n\\nand \\n\\nn2√t < n 8(log n)5/2\\n\\n= 2 8(log n)7/2.\\n\\nLet F = Fp[X]/(h(X)), and let g1(X), g 2(X) be of degree < t in T and g1(X) 6 = g2(X). Suppose g1(X) = g2(X) (mod p, h (X)). Then \\n\\ng1(Xm) ≡ g1(X)m ≡ g2(X)m ≡ g2(Xm) (mod p, h (X)) 5\\n\\nfor all m ∈ I. Therefore Xm is a root in F of the polynomial g1(Y ) − g2(Y ). This is a contradiction, as each of the elements Xm are distinct (they are roots of unity in the field) for the t representatives of the group generated by n and p, but the degree of g1(Y ) − g2(Y )is < t.Let m1 ≡ m2 (mod r), m1 6 = m2 ∈ I and m1, m 2 ≤ n2√t; this is possible as t is the order of the group generated by n and p in Z/r Z. Let g(X) ∈ T be of degree < t. Then \\n\\ng(X)m1 ≡ g(Xm1 ) ≡ g(Xm2 ) ≡ g(X)m2 (mod p, h (X)) so g(X) (mod h(X), p ) is a root in F of the polynomial Y m1 −Y m2. Since g was an arbitrary element of T, each g(X) is a root of this polynomial with degree ≤ n2√t but there are more than n2√t of them, contradiction. Finally, we show that t > 4(log n)2 and t < 16(log n)5. Suppose the order of n is \\n\\n≤ 4(log n)2 in ( Z/r Z)∗. Then r | ∏ \\n\\n> d≤4(log n)2\\n\\n(nd − 1) < 216(log n)5. Now use the fact that the least common multiple of the first k numbers is at least 2 k.It is easy to see that this algorithm has runtime (log n)10.5.\\n\\n#A.2 Agrawal: Finding Quadratic Nonresidues \\n\\nLet p − 1 = 2 `s, s odd. To find a quadratic non-residue, compute continuously −1, (−1) 1/2, ( −1) 1/4,..., ( −1) 1/2`−1. One can quickly compute ( −1) 1/2 from an algorithm due to Schoof, but one gets stuck at ( −1) 1/8.The idea: in primality testing, we want to know if Z/n Z is a field, so we embed it into (Z/n Z)[ X]/(Xr − 1); this ring has enough structure to pull out a nice algorithm. Assume that ` ≥ 2, and we try only to compute √−1. Now we embed in Fp[X]/(X2 + 1). Consider \\n\\ng(X) = (1 − X)s. Observe that ( g(X)) 2`\\n\\n= 1 in Fp[X]/(X2 + 1), so g(X) is a 2 `th root of unity. Assume that ` = 2. In this case, g(X) is a fourth root of unity. But X is also a fourth root, so g(X) = Xk (mod X − ω) where ω is the 'real' fourth root of unity. Consider \\n\\ng(X) mod ( X2 + 1): observe that g(X) 6 = Xk (mod X2 + 1) for any k. For suppose (1 − X)s ≡ Xk (mod X2 + 1); then (1 − 1/X )s = 1 /X k (mod 1 /X 2 + 1),\\n\\nso (1 − X)s = −(Xs/X k) (mod X2 + 1),\\n\\nhence −(Xs/X k) = Xk (mod X2 + 1), a contradiction because s is odd. So compute gcd( g(X) − Xk, X 2 + 1), for each k, one of them will factor X2 + 1. If ` > 2, then you cannot argue g(X) ≡ Xk (mod X − ω). If ` > 2, then it is possible that g(X) is an eighth root or a sixteenth root or so on. Suppose that g(X) modulo X 2 + 1 is an eighth root, for example. Then g(X2) = (1 − X2)s ≡ Xk (mod X − ζ) for some factor \\n\\nX − ζ of X4 + 1 and k odd. But (1 − X2)s is even, so it cannot be an odd power, so gcd((1 − X2)s − Xk, X 4 + 1) will give either a linear factor (in which case we are done) or a product of quadratic factors ( X − ζ2)( X + ζ2) or the products similar to ( X − ζ)( X − ζ3)and ( X − ζ)( X + ζ3). 6\\n\\nLet h(X) be a quadratic factor of X4 + 1. Now if (1 − X)s 6 ≡ Xk (mod h(X)), then \\n\\nh(X) can be factored and we are done. Suppose (1 − X)s ≡ Xk (mod ( X − ζ)( X − ζ3)) and (1 − X)s ≡ Xk′\\n\\n(mod ( X + ζ)( X + ζ3)).\\n\\nIf you replace X by X3, then you get (1 − X3)s = X3k (mod ( X3 − ζ)( X3 − ζ3)) so (1 − X3)s ≡ X3k = (1 − X)3s (mod ( X − ζ)( X − ζ3)).\\n\\nThat means that 3 is introspective for (1 − X)s. The same argument applies to the other congruence, so one obtains that 3 is introspective for (1 − X)s mod X4 + 1. Now try all over again now with X − a replacing 1 − X; the bad case will be when we have 3 is introspective for ( a − X)s mod X4 + 1 for a large number of a. Here we get stuck, but this should be impossible.\", \"Remarks. \\n\\nA. There was a solution due to Lehmer which says for any fixed ` that you can solve this problem? (Cohen) But that assumes the existence of a nonresidue to begin with, which is exactly our problem. (Bernstein) But also this doesn't seem to scale well. (Cohen) There is a strategy to deal with this, and we always work modulo a degree four polynomial. (Agrawal) B. The fact that s > 1 means that the same techniques as in the primality test do not seem to apply. Can you solve the problem if s = 3, or for other small s? (Lenstra) The case of a Fermat prime is trivial (3 is a nonresidue, and there probably aren't any above 65537). (Lenstra, Elkies) It seems as though if s is bounded, there are only a finite number of problematic a. For example, ( a − X3)s ≡ (a − X)3 (mod h(X)) does not hold for 'many' a\\n\\n(Pomerance). C. One strategy to solve this problem: translate this problem into rings and stare at it. (Lenstra) D. Is there a strategy to deal with cases beyond eighth roots (where we have the special situation that every odd integer has square 1)? (Elkies) Yes, but we need to solve this problem first. (Agrawal) E. How many values of a do you need? (Voloch) It seems unlikely you can generate the whole group with the ( X − a)s without the GRH. (Bernstein) \\n\\n#A.3 Bernstein: Proving Primality After Agrawal-Kayal-Saxena \\n\\nThe lecture notes are available on the speaker's website: the talk (and more) 1, related problems 2, older paper on AKS 3, and putting AKS into context 4.\\n\\n> 1http://cr.yp.to/papers.html#quartic\\n> 2http://cr.yp.to/papers.html#abccong\\n> 3http://cr.yp.to/papers.html#aks\\n> 4http://cr.yp.to/primetests.html 7\\n\\n#A.4 Edixhoven: About Point Counting over Arbitrary Finite Fields \\n\\nConsider a system of equations f1(x1,..., x n) = 0,..., fm(x1,..., x n) = 0 given by polynomials fi ∈ Fq[x1,..., x n]. This is not essentially different than the case of a single hypersurface (every variety is birational to a hypersurface, or, also, one can use the inclusion-exclusion principle). We let q = pr.\\n\\nQuestion. For fixed n, is there an algorithm that computes the number of solutions in Fq in time polynomial in the quantities: log q (or r and log p), d = max i deg fi, and m?If you fix p, and m = 1, then the answer is yes (Lauder-Wan) using p-adic methods. We discuss the case where p is not fixed. If p is not fixed, then one knows that the answer is yes for elliptic curves (Schoof) using `-torsion points and curves of a given genus via the \\n\\n`-torsion of their jacobian (Pila). Conceptually, all methods use Lefschetz fixed points formula: #X(Fq) = ∑2 dim Xi=0 (−1) i Tr(Frob q |Hic(X)) where we denote Hic(X) cohomology with compact support. This is true for X a scheme of finite type which is separated over Fq. For these cohomology groups, one can take a p-adic approach using a de Rham-type cohomology, lifting X to a p-adic ring R and take the hypercohomology of the de Rham sequence, quite explicit and computable but the complexity is worse than linear in p. Instead, one can also use mod ` methods ( ` 6 = p); here one takes the groups Hic(XFq,et, F`) which is a lot less explicit. The Hi derive from injective resolutions on the etale topology. In this setup, there is the advantage that you can choose `. For an elliptic curve, one has \\n\\nE(Fq)[ `]∨ = H1(EFq,et, F`).\\n\\nWhat is the simplest interesting case where we want to but cannot yet compute an H i with \\n\\ni ≥ 2? We think of surfaces, or modular forms of weight ≥ 3 (to generalize the case of elliptic curves, which correspond to eigenforms of weight 2). We assume that there is a cohomology group of dimension ≥ 2 (if it is of dimension 1, Frobenius acts as a power of the cyclotomic character). We consider as an example the modular form ∆ = q ∏\\n\\n> n≥1\\n\\n(1 − qn)24 = ∑ \\n\\n> n≥1\\n\\nτ (n)qn is an eigenform of weight 12, viewed as a function on the upper half-plane. Then ∆( dq/q )⊗6\\n\\nis an SL 2(Z)-invariant on H, so it descends to H/SL 2(Z). The variety we work with is the ten-fold product of the universal elliptic curve E; we find that H 11 (E10 ) has dimension 2. For all p, and ` 6 = p, τ (p) mod ` is the trace of Frob p on H11 (E10 \\n\\n> Fp,et, F`), which is also the trace of Frob p on H11 (E10 \\n\\n> Q,et, F`) with Gal( Q/Q) acting on it: it is the two-dimensional Galois representation (modulo `) associated to ∆. The action factors through Gal( K∆,` /Q) acting faithfully where K∆,` /Q is a finite extension. Compute explicitly the extension K∆,`. One gets as a byproduct a computation of the actual representation, which we cannot easily compute now. A bit of work yields that K∆,` \\n\\nis the Galois closure of the field definition of a suitable element x ∈ J1(`)( Q)[ `], where J1(`)is the jacobian of X1(`), if X1(`)( C) = H/Γ1(`) together with te cusps, and Γ 1(`) is the set of matrices \\n\\n(a bc d\\n\\n)\\n\\nwith a ≡ 1 (mod `), c ≡ 0 (mod `). Still: need to compute this efficiently. This is not so easy because the genus g of X1(`) is quadratic in `.8\\n\\nWe have the following strategy for finding Q(x) (based a suggestion of Jean-Marc Couveignes). We have a surjection \\n\\nX1(`)( C)g → J1(`)( C) = Cg/Λwhere Λ = H1(X1(`)( C), Z). We have Cg/Λ ⊃ (1 /` )Λ /Λ 3 x. (For r prime, Tr · x =\\n\\nτ (r) · x.) This map on complex points is given by ( Q1,..., Q g) ∈ X1(`)( C) maps to the point [ Q1 + · · · + Qg − gP 0] = ∑gi=1 \\n\\n∫ Qi \\n\\n> P0\\n\\n(ω1,..., ω g) ∈ Cg/Λ. Here, for P0 one can choose a \\n\\nQ-rational cusp. Generically, there is a unique point ( Q1,..., Q g) up to permutation which gives the point x, namely, α = ∑ \\n\\n> i\\n\\nj(Qi) ∈ Q(x). Now one estimates the height of α, approximates α\\n\\nin C by lifting (numerically) the straight line path from 0 to x (possible because the map \\n\\nX1(`)g → J1(`) is generically unramified). It will probably be a good idea to replace the divisor gP 0 by a sum of g distinct points P1,..., P g, with small height, defined over a small and solvable extension of Q. As x and the Pi determine the Qi (up to permutation), and as x is a torsion point (N´ eron-Tate height zero) one expects that the height of α is not much bigger than that of the Pi. So one hopes that the required number of correct digits of the approximation of α grows polynomially in `. The tool to be used for estimating the height of α is Arakelov geometry. A good indication that this proposed strategy works is that the height estimate works well in the function field case (a nice application of the Grothendieck-Riemann-Roch theorem). \\n\\n#A.5 Gao: Factoring Polynomials under GRH \\n\\nWe consider the following problem: Given a prime p and f ∈ Fp[x], where deg f = n, f\\n\\nis separable, and f splits completely, find a proper factor of f (in deterministic polynomial time). Berlekamp algorithm reduces general polynomials in Fq[x] ( q a power of p) to polyno-mials of the above type. Without GRH, we are stuck already at x2 − a. So throughout we assume GRH. Ronyai (1988) shows that this can be done in time ( nn log p)O(1), or more precisely (nr log p)O(1) whenever r | n, r > 1, so in particular if n is even f can be split in deter-ministic polynomial time. Bach, von zur Gathen and Lenstra (2001) give an algorithm with polynomial time if φk(p) is smooth for some k where φk(x) is the kth cyclotomic polynomial. Evdokimov (1994) shows for any n and p, f can be factored in time ( nlog n log p)O(1). We dis-cuss work in Cheng and Huang (2000), and Gao (2001) plus some unpublished results. GRH will be needed only to compute an rth nonresidue in Fp or in its extensions, for 1 ≤ r ≤ n.\\n\\nDefinition. An algebra R/ Fp is called elementary if R ∼= ( Fp)⊕ m for some m.Let R be elementary over Fp. Then we write R = Fp≤1 + · · · + Fp≤m where the ≤i are primitive idempotents, which are unique in R.\\n\\nFact. \\n\\nA. If f, g ∈ R[x], then gcd( f, g ) can be defined properly and can be computed in deterministic polynomial time for any elementary algebra R.B. If f ∈ R[x] is monic, separable (i.e. (f, f ′) = (1)), and f splits completely, then R1 =\\n\\nR[x]/(f (x)) is also elementary. Given a zerodivisor in R1, one can compute a proper factor of f or a zerodivisor of R.9\\n\\nC. Given a nontrivial ring endomorphism of R1 over R, one can find a proper factor of f or a zerodivisor of R. (Need GRH to get an rth nonresidue, for 1 ≤ r ≤ n, where n = deg f.) D. Given a quadratic nonresidue in Fp, there is a deterministic polynomial time algorithm for computing square roots in R. More precisely we have a function σ: R → R, such that (i) \\n\\nσ(A) is a square root of A if A is a square in R, (ii) if A = ∑mi=1 ai≤i ∈ R, ai ∈ Fp, then \\n\\nσ(A) = \\n\\n> m\\n\\n∑\\n\\n> i=1\\n\\nσ(ai)≤i,\\n\\nand (iii) for a ∈ Fp, σ(a2) = ±a. For example, if p ≡ 3 (mod 4), then we can take \\n\\nσ(A) = A(p+1) /4, and for a ∈ Fp,\\n\\nσ(a2) = \\n\\n{\\n\\na, if a is a square,\\n\\n−a, otherwise.\\n\\nNow let f ∈ Fp[x] be separable, so that \\n\\nf =\\n\\n> n\\n\\n∏\\n\\n> i=1\\n\\n(x − ai), ai ∈ Fp.\\n\\nTo factor f, define \\n\\nR2 = Fp[z1, z 2]/(f (z1), f (z2)),\\n\\nwhich is the tensor product of Fp[z]/(f (z)) with itself. Then R2 is an elementary algebra. In the following, we will identity z1 and z2 with their images in R2. Let \\n\\n≤i =\\n\\n∏ \\n\\n> j6=i\\n\\n(z1 − aj )\\n\\n∏ \\n\\n> j6=i\\n\\n(ai − aj ), 1 ≤ i ≤ n, \\n\\nand \\n\\nηi =\\n\\n∏ \\n\\n> j6=i\\n\\n(z2 − aj )\\n\\n∏ \\n\\n> j6=i\\n\\n(ai − aj ), 1 ≤ i ≤ n. \\n\\nThen ≤iηj, 1 ≤ i, j ≤ n, are all the primitive idempotents of R2. They have the following properties: n∑\\n\\n> i=1\\n\\n≤i = 1,\\n\\n> n\\n\\n∑\\n\\n> j=1\\n\\nηj = 1,\\n\\nand \\n\\nz1 = a1≤1 + · · · + an≤n = ∑\\n\\n> i,j\\n\\nai≤iηj,z2 = a1η1 + · · · + anηn = ∑\\n\\n> i,j\\n\\naj ≤iηj.\\n\\nLet \\n\\nA = 1\\n\\n2(z1 + z2 + σ(( z1 − z2)2) ∈ R2,\\n\\nwhich can be computed in deterministic polynomial time. Then \\n\\nA = ∑\\n\\n> i,j\\n\\n1\\n\\n2(ai + aj + σ(( ai − aj )2)≤iηj10 \\n\\nwhere 1\\n\\n2(ai + aj + σ(( ai − aj )2)) = \\n\\n{\\n\\nai, if σ(( ai − aj )2) = ai − aj\\n\\naj, if σ(( ai − aj )2) = aj − ai.\\n\\nHence A encodes information about the \\\"squareness\\\" of the differences of the roots of \\n\\nf. By using characteristic polynomials and gcd technique, one can extract the factors of f.\\n\\nDefinition. For 1 ≤ i ≤ n, define ∆i = {1 ≤ j ≤ n: j 6 = i, σ (( ai − aj )2) = −(ai − aj )}.\\n\\nTheorem. We can always find a proper factor of f in Fp[x] except when \\n\\n#∆ i = ( n − 1) /2, 1 ≤ i ≤ n, (1) \\n\\nand in this exception case f can be factored over Fp[z1] as \\n\\nf (x) = ( x − z1)f0f1\\n\\nwhere \\n\\nf0 =\\n\\n> n\\n\\n∑\\n\\n> i=1\\n\\n∏\\n\\n> j∈∆i\\n\\n(x − aj )≤i, f1 =\\n\\n> n\\n\\n∑\\n\\n> i=1\\n\\n∏ \\n\\n> j / ∈∆i,j 6=i\\n\\n(x − aj )≤i.\\n\\nTo further factor f0 over R1 = Fp[z1]/(f (z1)), we compute in the ring \\n\\nR3 = R1[z2, z 3]/(f0(z2), f 0(z3)),\\n\\nand similarly for f1.\\n\\nTheorem. We can always split f0 or f1 except when \\n\\n#(∆ i ∩ ∆j ) = ( n − 3) /4, 1 ≤ i < j ≤ n, (2) \\n\\nand in this exception case, f0 is factored over R1[z2]/(f0(z2)) as \\n\\nf0(x) = ( x − z2)f00 f01 \\n\\nwhere f00, f 01 ∈ R1[z2][ x] both of degree (n − 3) /4; similarly over the ring R1[z2]/(f1(z2)),\\n\\nf1(x) = ( x − z2)f10 f11.\\n\\nA system of subsets satisfying (1) and (2) is called an Hadamard design. When n ≡\\n\\n3 mod 4 is a prime, there is always an Hadamard design. To further split f00, let R2 = R1[z2]/(f0(z2)), and we compute in the ring \\n\\nR4 = R2[z3, z 4]/(f00 (z3), f 00 (z4)).\\n\\nSimilarly for f01, f10, and f11.\\n\\nTheorem. We can always split f00, f01, f10, or f11 except when \\n\\n#(∆ i ∩ ∆j ∩ ∆k) = ( n − 7) /8, 1 ≤ i < j < k ≤ n. (3) 11 \\n\\nIt can be proved, however, that (3) is impossible. It remains open how to explore this information to obtain a proper factor of f in Fp[x]!\\n\\n#A.6 Kedlaya: Counting Points using p-adic Cohomology \\n\\nWe introduce a different p-adic setup for counting points (or equivalently computing zeta functions). If X is a variety over Fq, we want to count points using 'de Rham' cohomol-ogy. We will demonstrate Monsky-Washnitzer cohomology (a kind of rigid cohomology for smooth affine varieties). We restrict to the case where X is an affine curve, since in higher dimensions other methods will be faster. Then X = Spec Fq[t1,..., t n]/(f1,..., f m). Let W = W (Fq) be the Witt vectors, and let K = W [1 /p ] be the fraction field of W.Define \\n\\nW 〈t1,..., t n〉† = {∑ \\n\\n> I\\n\\ncI tI: I = ( i1,..., i n) ∈ Zn\\n\\n> ≥0, c I ∈ W, \\n\\nconvergent for ti ∈ K, |ti| ≤ 1 + ≤ for some ≤ > 0}.\\n\\nIn other words, vp(cI ) ≥ r≤ I − c for some r, c with r > 0. Modulo p, W 〈t1,..., t n〉† reduces to the polynomial ring Fq[t1,..., t n], since all but finitely many coefficients are divisible by \\n\\np. We define K〈t1,..., t n〉† = W 〈t1,..., t n〉†[1 /p ]. Let \\n\\nAint = W 〈˜t1,..., ˜tn〉†/( ˜f1,..., ˜fn)where ˜fi is a lift of fi. Then \\n\\nAint [1 /p ] = K〈˜t1,..., ˜tn〉†/( ˜f1,..., ˜fn).\\n\\nThere exists a lift so that Aint is flat over W.Let Ω 1 \\n\\n> A\\n\\nbe the A-module generated by symbols dt 1,..., dt n modulo the submodule generated by d ˜f1,..., d ˜fm. Then there is a K-linear derivation d: A → Ω1\\n\\n> A. Letting ΩiA = ∧iA ΩiA; you get the de Rham complex \\n\\nA = Ω 0\\n\\n> Ad\\n\\n−→ Ω1\\n\\n> Ad\\n\\n−→... \\n\\nand you 'define' \\n\\nHiM W (X) = ker(Ω iA → Ωi+1 \\n\\n> A\\n\\n)\\n\\nimg(Ω i−1 \\n\\n> A\\n\\n→ ΩiA).\\n\\nIt turns out that HqM W (X) is independent of choices ( A is unique up to noncanonical iso-morphism, funny automorphisms are homotopic to the identity) and given X → Y, there is a map AY → AX and the induced maps HiM W (Y ) → HiM W (X) also do not depend on choices. The spaces Hi(X) are finite-dimensional, but it is not obvious; it relies upon relating this cohomology to rigid cohomology for proper varieties, namely, crystalline cohomology which we know is finite-dimensional for other reasons. Moreover, they satisfy the Lefschetz trace formula: if F: X → X is the q-power Frobenius, then (Monsky) #X(Fqi ) = ∑ \\n\\n> j\\n\\n(−1) j Tr(( qF −1)i|Hj (X)).\\n\\nThe idea: try to compute Hi(X) and the map induced by F (find Aint → Aint lifting \\n\\nq-power Frobenius). 12 \\n\\nExample. Look at X = Spec Fq[x, y, z ]/(y2 − f (x), yz − 1) with char Fq = p odd. Let deg f = 2 g + 1, f monic. Lift it to A = K〈x, y, z 〉†/(y2 − P (x), yz − 1), where P (x) is monic, degree 2 g + 1 over W. It is easy to compute that H0(X) is one-dimensional. Now \\n\\nH1(X) is generated by xi dx/y for i = 0,..., 2g − 1, and xi dx/y 2, i = 0,..., 2g. Note H1(X)splits under y 7 → − y into plus and minus eigenspaces. You need to find relations in H1(X) that d(xi/y j ) = \\n0. (This is a special situation: all relations are 'algebraic'.) Lift the p-power Frobenius by W → W by the Witt vector Frobenius, x 7 → xp, and y 7 → yp√F (P (x)) /P (X)p. Compose this map with itself n times to get a q-power Frobenius lift, and this allows us to compute the zeta function of a genus g\\n\\nhyperelliptic curve over Fpn in time ˜O(g4n3p). \\n\\n#A.7 Lauder: Counting Solutions to Equations in Many Variables over Finite Fields \\n\\nWe present an algorithm which allows us to count solutions to a homogeneous equation \\n\\nf (X1,..., X n) ∈ Fq[X1,..., X n] of degree d (for simplicity we assume d ≥ 2, n ≥ 2) with running time which does not increase exponentially in number of variables. In other words, we are interested in computing the number of projective solutions \\n\\nNk = {(x1: · · ·: xn) ∈ Pn−1 \\n\\n> Fqk: f (x1,..., x n) = 0 }\\n\\nfor every k ≥ 1. We encode these numbers in the generating function \\n\\nZ(f, T ) = exp( ∑\\n\\n> k\\n\\nNkT k/k ) ∈ Q[[ T ]] which, by a theorem of Dwork, is in fact a rational function. We assume that f is nonsingular, i.e. f and ∂f /∂x i for i = 1,..., n have no common projective solution. In this situation, we know that \\n\\nZ(f, T ) = P (T )(−1) n+1 \\n\\n(1 − T )(1 − qT )... (1 − qn−2T )where deg P = (1 /d )(( d − 1) n + ( −1) n(d − 1)). If we compute Nk naively for k = 1,..., deg P then we can of course recover the polynomial P (T ); the time required to do this, however, requires ( qdeg P )n ≈ 2dn−1 log q eval-uations of f. The input is given by (d+n−1\\n\\n> n−1\\n\\n) ≤ dn−1 terms of size log q, and the output size is approximately ( dn−1 log q)O(1). We would like the running time to be polynomial in this quantity. If n = 2, we are counting solutions of a univariate polynomial, and this can be done in time ( d log q)O(1). For n = 3, we have an algorithm of Schoof-Pila for curves which has run time (log q)∆ where ∆ depends on d exponentially. In general, there is an algorithm (due to L. and Wan) which runs in time ( pd n log q)O(n). Notice the n in the exponent-we would like to lose this dependence. The new result: If f is 'sufficiently generic' (we exclude a Zariski closed set which is efficiently computable), p 6 = 2, and p - d, then we can find P (T ) using ( pd n log q)) O(1) bit operations. As a corollary, we see that if f ∈ Z[X1,..., X n] is sufficiently generic, then there exists an algorithm which takes as input a prime p, outputs the number of solutions \\n\\nf mod p = 0, and has run time O(p2+ ≤). Recall that P (T ) = det( I − T Frob q |Hn−2(X)), where we write X for the projective variety defined by the equation f = 0. The action of Frob q can be represented by a matrix 13 \\n\\nwith entries in a field of characteristic zero, and we find that \\n\\nNk = ( −1) n Tr(Frob kq |Hn−2(X)) + 1 + qk + · · · + ( qk)n−2.\\n\\nFor curves, for example, the dimension is n − 2 = 1, and H1(X) is a Z`-module, for ` 6 = p.Instead, we work with the p-adic theory, where Hn−2(X) is a R-module for a ring R ⊃\\n\\nZp. We compute instead Frob q = Frob log p(q) \\n\\n> p, and compute the matrix of Frob p |Hn−2(X). (Specifically, R = Qq(π) where Qq is the unramified extension of Qp of degree log p q and \\n\\nπp−1 = −p. Also our Hn−2(X) is actually the primitive part of the cohomology space.) Consider the family \\n\\nfY =\\n\\n> n\\n\\n∑\\n\\n> i=1\\n\\naiXdi + Y h (X1,..., X n),\\n\\nand assume that a1... a n 6 = 0. Then f1 = f and f0 is a diagonal form, for which it is easy to count the number of solutions. Now Frob p(Y ) is a p-adic analytic function with the property that Frob p(Y ) evaluated at a Teichmuller lift of y1/p is exactly the Frob p associated to fy.We see that Frob p(Y ) = C(Y p)−1 Frob p(0) Cτ −1(Y ) where τ mod p: α 7 → αp, and C(Y )is a matrix of power series around the origin satisfying the differential equation dC/dY =\\n\\nC(Y )B(Y ) with initial condition C(0) = I, where B(Y ) is easily computed. This gives a way to compute Frob p(Y ) in a radius around the origin, and we need to extend this to the closed disc of radius 1. The entries of the matrices are p-adic holomorphic functions, so to compute these modulo a power of p we find rational functions with denominators corresponding to the values of Y where the variety becomes singular and then recover the numerator from the power series. Using overconvergence we get a bound on the degree of these rational functions. We evaluate the rational functions at Y = 1. One gets nice complexity because we work with univariate power series, with decay in the coefficients on the order of 1 /p, and one needs to take the power of p approximately on the order of dn log q.In the old algorithm, we would work in Hn−2(X), the ring of power series in X1,..., X n\\n\\nover an R = Q(m) \\n\\n> p\\n\\n(π) where Q(m) \\n\\n> p\\n\\nis the unramified extension of Qp of degree m (if q = pm)and πp−1 = −p modulo an infinite subspace; the power series we must work with have on the order of ( pd n log q)n terms. \\n\\nProblem. Find an algorithm which counts the number of points on a curve in time (d log q)O(1).\\n\\n#A.8 Lenstra: Primality Testing with Pseudofields \\n\\nThis is joint work with Carl Pomerance. If f, g are real-valued functions on a set X and g > 0, then we say f = ˜O(g) if there exists c ∈ R>0 such that for all x ∈ X, |f (x)| ≤ g(x) max {2, log g(x)}c.\\n\\nTheorem. There is a deterministic algorithm that given n ∈ Z, n > 1, correctly decides whether or not n is prime and that has runtime ˜O((log n)6).\\n\\nThe theorem of AKS begins (in brief) as follows. Let n ∈ Z>1 be a positive integer, r\\n\\na prime number with r - n, and (X + a)n ≡ Xn + a (mod n, X r − 1) 14 \\n\\nfor a small set of a, and the multiplicative order of n modulo r is at least c(log n)2. Together with other conditions, we conclude n is a prime. This has runtime ˜O(r3/2(log n)3). For \\n\\nr = O((log n)5), we get ˜O((log n)21 /2) with effective constant. For an ineffective constant, one can get ˜O((log n)15 /2). It would be optimal to have ˜O((log n)6), which would require \\n\\nr = ˜O((log n)2) but here we run into Artin's conjecture on primes with prescribed primitive root. To get around this, we generalize the situation slightly to give more parameters. In fact, we replace Xr − 1 by ( Xr − 1) /(X − 1) = Xr−1 + · · · + X + 1; the proof remains essentially unchanged. Instead of considering congruences, we instead think of equalities in the ring \\n\\nA = Z[X]/(n, X r−1 + · · · + X + 1) ⊃ Z/n Z.\\n\\nThis is a Galois extension of Z/n Z of degree r − 1. We now replace the polynomial X r−1 +\\n\\n· · · + X + 1 with other polynomials f (X) for which the same proof techniques apply. In this case, A is a field if and only if n is a prime number and n is a primitive root modulo r. We are led to introduce rings that 'try very hard' to be fields. \\n\\nDefinition. A pseudofield is a pair A, α where A is a ring (commutative with 1) and α ∈ A\\n\\nsuch that there exists n ∈ Z>1 and d ∈ Z>0 satisfying: \\n\\n• Z/n Z is a subring of A with A ≤ nd, and there exists σ ∈ Aut A with: \\n\\n- σdα = α;\\n\\n- for all q | d prime, σd/q (α) − α ∈ A∗; and \\n\\n- σα = αn;\\n\\n• Equivalently, A ∼= ( Z/n Z)[ X]/(f ) as a ring with X 7 → α for some monic polynomial f ∈\\n\\n(Z/n Z)[ X] of degree d satisfying: \\n\\n- f (X) | f (Xn); \\n\\n- f (X) | (Xnd\\n\\n− X); and \\n\\n- for all primes q | d, ( f, X nd/q \\n\\n− X) = (1). Also equivalently, a pseudofield is characterized by the conditions that A ⊃ Z/n Z be Galois with cyclic group generated by σ, α generates A as a ring, and σα = αn.\\n\\nTheorem. There is a deterministic algorithm that given n ∈ Z>1 and f ∈ (Z/n Z)[ X] decides if (Z/n Z)[ X]/(f ) is a pseudofield in time \\n\\n˜O(( d + log n)d log n).\\n\\nIt is routine to verify this using the second set of conditions. \\n\\nExample. \\n\\nA. If r is a prime with r - n, then A = ( Z/n Z)[ X]/(Xr−1 + · · · + 1) is a pseudofield if and only if the order of n modulo r is r − 1. B. If n is prime, then A, α is a pseudofield if and only if A is a field and A = Fn[α]. C. If A = Z/n Z (so that d = 1) and α = ( a mod n), then A, α is a pseudofield if and only if \\n\\nan ≡ a (mod n)-in other words, n is a pseudoprime to base a.\\n\\nTheorem. Let A, α be a pseudofield of characteristic n and degree d such that d > (log n/ log 2) 2,\\n\\nn has no prime factor ≤ k = b√d(log n/ log 2) c, and such that \\n\\n(α + a)n = an + a15 \\n\\nfor a = 1, 2,..., k (mod n). Then n is a power of a prime number. \\n\\nThe proof uses: for each prime p | n, there exists a unique τ ∈ 〈 σ〉 such that for all \\n\\nβ ∈ A, τ (β) ≡ βp (mod pA ). This is a result coming from Galois theory for rings. This leads to a deterministic primality test with runtime equal to the time to construct the pseudofield plus the time to check the conditions; the latter takes time ˜O(d3/2(log n)3). In the context of primality testing, there is a procedure which converts any (honest) method for constructing finite fields to a method for checking primality. If the algorithm on input n crashes, then n was not prime; if it returns a polynomial f, then one checks (efficiently) if this gives rise to a pseudofield, which then verifies that n is prime, and otherwise produces a proof that n is not prime. Therefore we look for algorithms for constructing finite fields. Our construction relies on the following theorem: \\n\\nTheorem (Kummer 1846). For r prime and q | (r − 1), put \\n\\nfq,r = ∏ \\n\\n> i∈Fr\\n> iq=1\\n\\n(\\n\\nX − ∑ \\n\\n> j(r−1) /q =i\\n\\nζjr\\n\\n)\\n\\n∈ Z[X]\\n\\nwhere ζr is a primitive rth root of unity in C. The polynomial f is monic and irreducible of degree q. If p is prime, p 6 = r, then (fq,r mod p) ∈ Fp[X] is irreducible if and only if the order of p(r−1) /q modulo r is equal to q.Fact. Let Ai, α i be a pseudofield of characteristic n and degree di > 1 for i = 1, 2 such that gcd( d1, d 2) = 1; then A1 ⊗Z A2, α 1 ⊗ α2 is a pseudofield of characteristic n and degree d1d2.\\n\\nTheorem. There exists an effective computable constant c such that there is a deterministic algorithm that given n ∈ Z>1 finds a finite sequence of pairs (r1, q 1),..., (rk, q k) such that: \\n\\n• qi > 1, qi pairwise coprime; \\n\\n• ri prime, qi | (ri − 1);\\n\\n• The order of n(ri−1) /q i modulo ri is qi; and \\n\\n• d = ∏ qi satisfies \\n\\n(log n/ log 2) 2 < d < c (log n/ log 2) 2\\n\\nand max ri < d.This algorithm runs in time ˜O((log n)24 /11 ), with an effective constant. \\n\\n#A.9 Pomerance and Bleichenbacher: Constructing Finite Fields \\n\\nConsider the following problem: Given a prime p and an integer d > 1, find an irre-ducible polynomial f ∈ Fp[X] of degree d. And do so in time polynomial in d and log p.There is a randomized algorithm which attacks this problem by picking a polynomial at random (approximately 1 out of every d polynomials will be irreducible), and testing each for irreducibility (which is fast), continuing this procedure until you find one, then stop. But we are interested here in a deterministic algorithm. Already for d = 2, this is a difficult problem, equivalent to finding a quadratic nonresidue modulo p.Assuming the ERH, Adleman and Lenstra have a solution to this problem. Uncondi-tionally, they also find an irreducible polynomial of degree d′ with d ≤ d′ < cd log p, where c\\n\\nis an effectively computable number. Letting d = (log n)2 and n = p (where you do not know 16 \\n\\na priori if n is prime), the polynomial produced has degree O(log 3 n), and the runtime of the AKS algorithm becomes ˜O((deg f )3/2 log 3 n) = ˜O((log n)15 /2). We improve this theorem to the following: \\n\\nTheorem. Such a polynomial can be produced with d ≤ d′ ≤ 4d for p sufficiently large and \\n\\nd ≥ (log p)11 /6+ ≤.\\n\\nThe bound for the 'sufficiently large' part depends effectively on the choice of ≤.\\n\\nTheorem. There is an effectively computable function N≤ and a deterministic algorithm such that if ≤ > 0, n > N ≤, and D > (log n)11 /6+ ≤, the algorithm produces pairs (q1, r 1),..., (qk, r k)\\n\\nsuch that for each i, ri is prime, ri < D, qi | ri − 1, the order of n(ri−1) /q i modulo ri is qi.Further, the q1,..., q k are pairwise coprime, and D ≤ ∏ \\n\\n> i\\n\\nqi ≤ 4D. This algorithm runs in time ˜Oeff (D12 /11 ).\\n\\nLet ηi be the Gaussian period of degree qi in Q(ζri ) (the trace of ζri into the unique subfield of degree qi over Q). The element η = η1... η k has degree q1... q k over the rationals (by coprimality). If n is prime, and if f (x) is the minimal polynomial of η then f mod n\\n\\nis irreducible over Fn. Checking if f (X) | f (Xn) in ( Z/n Z)[ X] (together with some other conditions), we see that f gives rise to a pseudofield. Let x = D6/11 −≤/ 4. Throughout, we assume that n is 'sufficiently large' with the bound being effectively computable, depending only on the choice of ≤.\\n\\nProposition. All but O(x/ log 3 x) primes r ≤ x have a prime q | (r−1) with q > x 1/(log log x)2\\n\\nand the order of n(r−1) /q modulo r is equal to q.\\n\\nTherefore up to x, almost all of the primes are useful in the context of our theorem. This proposition is a natural extension of the argument in the original AKS paper, together with an added ingredient about the distribution of primes r such that r − 1 is smooth due to Pomerance and Shperlinski. \\n\\nProposition. Let Q be a set of primes q with x1/(log log x)2\\n\\n< q ≤ x1/2 and ∑ \\n\\n> q∈Q\\n\\n1/(q − 1) <\\n\\n(3 − ≤)/11. Then there are > δx/ log 2 x primes r ≤ x such that r − 1 is free of primes from \\n\\nQ ∪ (x1/2, x ).\\n\\nThis proposition follows from a method of Balog, together with some effective estimates on the distribution of primes in residue classes. Together, these two propositions give the corollary: \\n\\nCorollary. Let Q be the set of primes q in the first proposition satisfying q ≤ √x. Then \\n\\n∑\\n\\n> q∈Q\\n\\n1\\n\\nq − 1 ≥ 3 − ≤\\n\\n11.\\n\\nProof. If this inequality did not hold, then by the first and second propositions, there must be primes r ≤ x having the properties of both propositions. By the first proposition, r − 1has a prime factor q > x 1/) log log x)2\\n\\nand the order of n(r−1) /q modulo r is equal to q. By one of the properties in the second proposition, q ≤ x1/2. Then q ∈ Q. This contradicts the second proposition. §\\n\\nProposition. There exists a subset of Q in the corollary with product in the interval [D, 4D].17 \\n\\nThe proof of this theorem relies upon combinatorial number theory (essentially, you can solve a bin packing problem using the primes q). It relies upon: \\n\\nTheorem (Continuous Frobenius theorem). If S is an open subset of R>0, S is closed under addition, and 1 6 ∈ S, then for any t, 0 < t ≤ 1, the du/u measure of S ∩ (0, t ) is ≤ t, i.e. \\n\\n∫ t\\n\\n> 0\\n\\nχS (u) du \\n\\nu ≤ t\\n\\nwhere χS (u) is the characteristic function of S.\\n\\nNow a remark about effectivity. It was proven by de la Vall´ ee Poussin in 1896 that \\n\\nπ(x, k, a ) = {p ≤ x: x ≡ a (mod k)} ∼ π(x)/φ (k)as x → ∞. The Siegel-Walfisz theorem states that this is true for k < (log x)A for any fixed \\n\\nA. This theorem is inherently ineffective because it depends on the existence or nonexistence of Siegel zeros. The Siegel-Walfisz theorem is ubiquitous in analytic number theory, being used in the Bombieri-Vinogradov theorem, Fouvry's theorem, and much else. The analytic number theory we use is an effective version of the Bombieri-Vinogradov theorem that does not rely on the Siegel-Walfisz theorem, and we replace the Fouvry theorem by a (weaker) result of Deshouillers-Iwaniec. Now we give an outline of the proof of the Frobenius theorem. First, it is sufficient to prove the case where St = S ∩ (0, t ) = ⋃ni=1 (ai, b i) (i.e. St contains only finitely many intervals). Second, since 1 6 ∈ S, for all ( h1,..., h n) ∈ Nn \\n\\n> ≥0\\n\\neither ∑ni=1 hiai ≥ 1 or ∑ni=1 hibi ≤\\n\\n1 (*). Now fix b1,..., b n such that b1 > b 2 > · · · > b n and consider all sets ⋃ni=1 (ai, b i)satisfying b1 ≥ a1 ≥ b2 ≥ · · · ≥ bn ≥ an (**) as well as condition (*). Under these conditions, there exists a maximum to ∑ni=1 (log( bi)−log( ai)). We may thus assume that St = ⋃ni=1 (ai, b i)is a maximum. We show in the paper that we can assume that b1 > a 1 > b 2 > · · · > b n > a n.Let U = {h ∈ Nn \\n\\n> ≥0: ha = 1 } with a = ( a1,..., a n). Let an+1 = bn+1 = 0. For all \\n\\nh = ( h1,..., h n) ∈ U and 1 ≤ k ≤ n,\\n\\nhk\\n\\n( n∑\\n\\n> i=1\\n\\n(bi − ai)hi\\n\\n)\\n\\n≤ hk(bk − bk+1 ).\\n\\nThis is trivial if hk = 0, and otherwise, ∑ni=1 aihi = 1 which implies \\n\\n> n\\n\\n∑\\n\\n> i=1\\n\\naihi − ak + ak+1 < 1.\\n\\nand therefore by assumption (*) \\n\\n> n\\n\\n∑\\n\\n> i=1\\n\\nbihi − bk + bk+1 ≤ 1.\\n\\nLet v = ( v1,..., v n) ∈ Rn such that vh ≥ 0 for all h ∈ U. Then there exists ≤ > 0 such that for all 0 ≤ x ≤ ≤, n⋃\\n\\n> i=1\\n\\n(ai + vix, b i)18 \\n\\nsatisfies (*) and (**). By assumption \\n\\n> n\\n\\n∑\\n\\n> i=1\\n\\n(log bi − log( ai + vix)) is maximal for x = 0, so ∑ni=1 vi/a i ≥ 0. Then a theorem by Farkas (or the dual theorem of linear programming) implies that there exists pj ≥ 0 such that \\n\\n> `\\n\\n∑\\n\\n> i=1\\n\\nhij pj = 1\\n\\nai\\n\\nwhere U = {h1,..., h `} and hj = ( h1j,..., h nj ). Now multiply the equation \\n\\nhk\\n\\n( n∑\\n\\n> i=1\\n\\n(bi − ai)hi\\n\\n)\\n\\n≤ hk(bk − bk+1 ).\\n\\nby akpj and sum up \\n\\n> n\\n\\n∑\\n\\n> k=1\\n> `\\n\\n∑\\n\\n> j=1\\n\\nakpj hkj \\n\\n( n∑\\n\\n> i=1\\n\\n(bi − ai)hij \\n\\n)\\n\\n≤\\n\\n> n\\n\\n∑\\n\\n> k=1\\n> `\\n\\n∑\\n\\n> j=1\\n\\nakpj hkj (bk − bk+1 ).\\n\\nAfter some simple arithmetic, we find that \\n\\n> n\\n\\n∑\\n\\n> i=1\\n\\n(log bi − log ai) ≤ t. \\n\\n#A.10 Silverberg: Applications of Algebraic Tori to Crytography \\n\\nIn this lecture we discuss 'torus-based cryptography', and counterexamples to con-jectures in an article entitled Looking Beyond XTR, and compare TBC with Lucas-based cryptosystems and XTR, and understand LUC, XTR, and Beyond in terms of algebraic tori. This is joint work with Karl Rubin, and inspired by XTR. The cryptosystem XTR (due to A. Lenstra and E. Verheul) concerns the extension \\n\\nFp6 /Fp2: they consider the subgroup of F∗ \\n\\n> p6\\n\\nof order p2 − p + 1 with generator g; the public knowledge is Tr Fp6 /Fp2 (g), what is shared is Tr Fp6 /Fp2 (gab ), where Tr( ga) and Tr( gb) are trans-mitted. In this setup, you get the security of F∗ \\n\\n> pn\\n\\nwhile transmitting only φ(n) elements of \\n\\nFp.Let L/k be a finite cyclic extension with intermediate field F. Let g ∈ L \\\\ F, and denote by Cg the Gal( L/F )-conjugacy class of g, so the characteristic polynomial of g over \\n\\nF is ∏ \\n\\n> h∈Cg\\n\\n(X − h). For L = Fp6, F = Fp2, k = Fp, the polynomial is x3 − s1x2 + s2x − s3,where s1 = Tr L/F (g), s3 = NL/F (g), and s2 = Tr L/F (gg σ), where 〈σ〉 = Gal( L/F ). If g is in the subgroup of L∗ of order p2 − p + 1, then s3 = 1 and s2 = Tr L/F (g)p. Therefore knowing Tr L/F (g) is equivalent to knowing all the symmetric polynomials on Cg which is equivalent to knowing Cg as a set, so you know Cga and this is equivalent to knowing Tr L/F (ga). In other words, you can exponentiate, but you cannot multiply: Tr( g) and Tr( h) do not determine Tr( gh ); i.e. knowing Cg and Ch does not allow you to know Cgh.Bosma-Hutton-Verheul conjecture that for all n, there exists a divisor d | n such that \\n\\nd | φ(n) and for L = Fpn and F = Fpd, you can recover all the coefficients s1,..., s n/d of the 19 \\n\\ncharacteristic polynomial of g over F from the first φ(n)/d of them for all g in the subgroup of L∗ of order Φ n(p) and not in any proper subfield. We show that this is in fact false. In particular, when n = 30, it is false; for p = 7, \\n\\nd = 1, no 10 symmetric polynomials determine all of them, and no 8 determine any of the others (except the ones determined by the symmetry of the characteristic polynomial). For \\n\\np = 7, d = 2, no 4 symmetric polynomials determine all of them. \\n\\nFact. The order Φ n(p) subgroup of F∗ \\n\\n> pn\\n\\nis \\n\\n{α ∈ F∗ \\n\\n> pn: NFpn /M (α) = 1 for all M ( F pn }.\\n\\nLet L/k be an abelian degree n extension of fields. Let \\n\\nTL/k = ker \\n\\n(\\n\\nRes L/k (Gm) ⊕NL/M \\n\\n−−−−→ ⊕ \\n\\n> k⊂M(L\\n\\nRes M/k (Gm)\\n\\n);recall that Gm(k) = k∗, and (Res L/k Gm)( k) = L∗. If L/k is not cyclic, dim( TL/k ) = 0. If \\n\\nL/k is cyclic, then TL/k is an algebraic torus over k of dimension φ(n), i.e. TL/k is isomorphic over k to Gφ(n) \\n\\n> m. Here, TL/k ∼=L Gφ(n) \\n\\n> m.Assume from now on that L/k is cyclic. Then \\n\\nTL/k (k) = {α ∈ L∗: NL/M (α) = 1 for all k ⊂ M ( L}.\\n\\nConjecture (Voskresenskii). TL/k is rational, i.e. there exists a birational map TL/k →\\n\\nAφ(n).\\n\\nThis is true if n = pa or paqb (Klyachko 1988). It is not known for n = pqr. Let us look at the case when n = 2, say, char k 6 = 2. Then L = k(√d), and TL/k = ker( NL/k ) which is a conic which can be parameterized, and we obtain \\n\\nψ: P1 ∼\\n\\n−→ TL/k \\n\\na 7 → (a + √d)/(a − √d)\\n\\n∞ 7 → 1We have ψ(a)ψ(b) = ψ(( ab + d)/(a + b)). (This is also just Hilbert's theorem 90.) Writing \\n\\nTn for TL/k when k = Fq, this induces a way to do the multiplication in T2 in P1(k). We also give an explicit example when n = 6 for char( k) 6 = 3, [ k(ζ9): k] = 6 (e.g. k = Fq\\n\\nwith q ≡ 2, 5 (mod 9)). TL/k is dimension 2 and contained in Res M/k (TL/M ) = T ′ ∼k A3\\n\\nwhere M/k is the subextension of degree 3. But TL/M = ker( NL/M ) is dimension 1, and \\n\\nNL/F = 1 defines a hypersurface in T ′ ∼ A3. Therefore T6 ∼ A2, so we can use the multiplication in T6 but represent elements of T6 by 2 elements of F8. This gives rise to the cryptosystem CEILIDH. Open problems: A. Improve the efficiency of multiplication and exponentiation for the system CEILIDH. B. Repeat this analysis for n = 30, i.e. 1. Find explicit birational isomorphisms between T30 and A8,2. Find prime powers q of size 1024 /30 ≈ 35 bits such that Φ 30 (q) has a 160-bit prime factor, 3. Are there special attacks on DL in F∗ \\n\\n> q30?20 \\n\\nNow we look to understand LUC, XTR, and Beyond in terms of algebraic tori. For H\\n\\na subgroup of Gal( L/k ) = G which is a direct factor, write Σ H for the group of permutations of H. Then Σ H acts on ⊕\\n\\n> σ∈G\\n\\nA1 ∼\\n\\n−→L Res L/k A1 ⊃ Res L/k (Gm).\\n\\nLet \\n\\nXF = img (TL/k → Res L/k (Gm)/ΣGal( L/F )\\n\\n).\\n\\nFor LUC and XTR, you look at \\n\\n{Tr L/F (α): α ∈ TL/k (k)}\\n\\nwhich is the image of TL/k (k) under TL/k → XF → Res F/k (A1), where the latter map is a birational isomorphism. \\n\\nTL/k \\n\\n> ≤\\n> ≤\\n> ≤\\n> ≤\\n> Tr L/F\\n> *\\n> *\\n> UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU\\n\\nXF \\n\\n> Tr L/F //\\n\\nRes F/k (A1) ∼ / / F\\n\\nAssume that n is squarefree. Write Gal( L/F ) = H1 × · · · × Ht where Hi are cyclic of prime order. \\n\\nTheorem. The action of ΣHi on Res L/k (Gm) preserves TL/k and XF is birational to TL/k /(Σ H1 ×· · · × ΣHt ).\\n\\nIn XTR, we obtain XF birational to T6/S 3; in the cases n = 30 and d = 1, 2, one gets \\n\\nT30 /(S2 × S3 × S5), T30 /(S3 × S5), which are not groups. We have maps L → F for every symmetric function s1,..., s [L:F ]. We have a surjection \\n\\nTL/k → XF and an injection XF ↪→ F [L:F ] by the direct sum of these functions. A BHV conjecture implies that for the subset consisting of the first dφ(n)/d e functions (where d =[F: k]), the map remains injective. Further, a BHF conjecture implies that every n has a divisor d so that d also divides φ(n), and the map XF → Aφ(n) (induced by the first φ(n)/d \\n\\nsymmetric functions) is a birational isomorphism. This is true for ( n, d ) = (1, 1) (DH), (2, 1) (LUC), (6, 2) (XTR), and ( `, 1) and (2 `, 2) where ` is prime (see Doing more with fewer bits,by Brouwer-Pellikaan-Verheul), but: \\n\\nTheorem. This is false for n = 30 (d = 1, 2) if char( k) lies outside a finite set. \\n\\nTo prove this, we first do a computer search for 2 elements of T30 (F7) with the same image a ∈ (F7)8 but different images in XF. Using Hensel's Lemma, every lift of a to Z87\\n\\nhas at least 2 inverse images in XF (Q7). Therefore the map is not generically one-to-one over Q7, so it is not generically one-to-one over Q, hence over any field of characteristic 0. Then reduce modulo p to get it over Fp and therefore all fields of characteristic p (outside of a finite set). \\n\\n#A.11 Stein: Modular Forms Database \\n\\nThe lecture notes 5 are available on the speaker's website. The tables 6 are also available there. \\n\\n> 5http://modular.fas.harvard.edu/mfd/talks/mfd1/\\n> 6http://modular.fas.harvard.edu/Tables/ 21\\n\\n#A.12 Voloch: Multiplicative Subgroups of a Finite Field \\n\\nThe lecture notes 7 are available on the speaker's website. \\n\\n#A.13 Wan: Partial Counting of Rational Points over Finite Fields \\n\\nWe are motivated by the following problem. Let Fqd = Fq[α]/h (α), where h is irre-ducible of degree d > 1 over Fq. We look at the group \\n\\nG = 〈a − α: a ∈ Fq〉 = ( F∗ \\n\\n> qd\\n\\n)I ⊂ F∗ \\n\\n> qd,\\n\\nwhere I = [ F∗ \\n\\n> qd: G]. When does I = 1, for example? Let D | (qd − 1), and let \\n\\nN = {(x, y ): x − α = yD, x ∈ Fq, y ∈ Fqd }.\\n\\nBy a character sum argument counting, you can write this as \\n\\nN = ∑ \\n\\n> φ:F∗\\n> qd→C∗\\n> φD=1\\n\\n∑\\n\\n> x∈Fq\\n\\nφ(x − α).\\n\\nBy the Riemann hypothesis (Weil), we have |N − q| ≤ (D − 1)( d − 1) √q. Therefore we have seen: \\n\\nProposition. Let S ⊂ Fq. Let GS = 〈a − α: a ∈ S〉 = ( F∗ \\n\\n> qd\\n\\n)IS. Then \\n\\n(S)IS ≤ N ≤ q + ( IS − 1)( d − 1) √q. \\n\\nIf (S) > (d − 1) √q, then \\n\\nIS ≤ q − (d − 1) √q\\n\\n(S) − (d − 1) √q.\\n\\nIn particular, if (S) = q, then IS = 1 and G = F∗ \\n\\n> qd.\\n\\nTherefore we consider the problem: Can we compute N in time polynomial in d, D,and log q?The general setup: Let f (x1,..., x n) ∈ Fq[x1,..., x n], and d1,..., d n ≥ 1. We want to count \\n\\nNd1,...,d n (f ) = {(x1,..., x n): f (x1,..., x n) = 0, x i ∈ Fdi \\n\\n> q\\n\\n}.\\n\\nCan we compute Nd1,...,d n (f ), or at least estimate it? How does this quantity vary when the \\n\\ndi vary? For example, we consider the Artin-Schreier hypersurface. Let \\n\\nf (x1,..., x n, y 1,..., y n′ ) ∈ Fq[x1,..., x n, y 1,..., y n′ ],\\n\\nwhere n, n ′ ≥ 1. For each d ≥ 1, we consider \\n\\nNd(f ) = {(x0,..., x n, y 1,..., y n′ ): xp \\n\\n> 0\\n\\n− x0 = f (x1,..., x n, y 1,..., y n′ ),xi ∈ Fqd, y j ∈ Fq}.\\n\\nHeuristically (for suitable f ), we expect \\n\\nNd(f ) = qdn +n′\\n\\n+ O(q(dn +n′)/2)where the constant depends on p, f, and d.\\n\\n> 7http://www.ma.utexas.edu/users/voloch/preprint.html 22\\n\\nTheorem (Deligne). Write f = fm + fm−1 + · · · + f0, where fi are homogeneous of degree \\n\\ni. Assume fm defines a smooth projective hypersurface in Pn+n′−1 \\n\\n> Fq, and that p - m, d = 1.Then \\n\\n|N1(f ) − qn+n′\\n\\n| ≤ (p − 1)( m − 1) n+n′\\n\\nq(n+n′)/2.\\n\\nWhat about d > 1? \\n\\nDefinition. If d ≥ 1, we define the dth fibred sum of f to be \\n\\n⊕dyf = f (x11,..., x 1n, y 1,..., y n′ ) + · · · + f (xd1,..., x dn, y 1,..., y n′ ).\\n\\nTheorem (Fu-W). Write f = fm + · · · + f0, and assume that ⊕dy fm is smooth in Pdn +n′−1\\n\\n> Fq\\n\\nand p - m. Then \\n\\n|Nd(f ) − qdn +n′\\n\\n| ≤ (p − 1)( m − 1) dn +n′\\n\\nq(dn +n′)/2.\\n\\nExample. In the case that we can write \\n\\nf (x, y ) = f1m(x1,..., x n) + f2m(y1,..., y n′ ) + f≤m−1(x, y ),\\n\\nand f1m is smooth in Pn−1 \\n\\n> Fq, f2m is smooth in Pn′−1 \\n\\n> Fq. Then ⊕dy fm is smooth in Pdn +n′−1 \\n\\n> Fq\\n\\nif and only if p - d.Since the condition that the fibred sum be smooth is Zariski open, we have shown it is nonempty if p - d and therefore there exist many examples of such f to which the theorem applies. \\n\\nDefinition. Let Md be the set of f over Fq such that ⊕dy fm is smooth. Then Md is Zariski open in the set of all f over Fq with deg f ≤ m.\\n\\nTheorem (Gao-W). Md is Zariski dense if and only if p - d. In fact, \\n\\n> ∞\\n\\n⋂\\n\\n> d=1\\n> p-d\\n\\nMd =\\n\\n> p(m−1) n\\n\\n⋂\\n\\n> d=1\\n> p-d\\n\\nMd\\n\\nand this intersection is Zariski open and dense. \\n\\nProblem. What about Kummer hypersurfaces \\n\\nxD \\n\\n> 0\\n\\n= f (x1,..., x n, y 1,..., y ′\\n\\n> n\\n\\n)\\n\\nwhere xi ∈ Fqd and yj ∈ Fq?\", \"Remark. We expect |Nd − qdn +n′\\n\\n| = O(q(dn +n′)/2), but one can get the weaker estimate \\n\\nO(qdn/ 2+ n′−1/2) in many cases (Katz). Now we consider partial zeta functions over Fq. Let f (x1,..., x n) ∈ Fq[x1,..., x n], \\n\\nd1,..., d n ≥ 1. Define \\n\\nZd1,...,d n (f, T ) = exp \\n\\n( ∞∑\\n\\n> k=1\\n\\nNd1,...,d n,k \\n\\nT k\\n\\nk\\n\\n)\\n\\nwhere \\n\\nNd1,...,d n,k = {(x1,..., x n): f (x1,..., x n) = 0, x i ∈ Fqdik }.\\n\\nWithout loss of generality, we may assume gcd( d1,..., d n) = 1, since otherwise we can just enlarge the ground field Fq.23 \\n\\nProposition. \\n\\nA. If d1 | · · · | dn, then Zd1,...,d n (f, T ) ∈ Q(T ).\\n\\nB. (Faltings) Zd1,...,d n (f, T ) = ∏di=1 Pi(T )ζid, where d = lcm( d1,..., d n) and ζd is a primitive dth root of unity, Pi(T ) ∈ Q(T ), and Pi(0) = 1.\\n\\nBy exponentiation to a root of unity, we mean the formal binomial expansion. From a counting point of view, this is 'as good as rational'. \\n\\nTheorem. In all cases, Zd1,...,d n (f, T ) ∈ Q(T ).Proof. Let Xd = X × · · · × X︸ ︷︷ ︸\\n\\n> d. We have a map \\n\\nσ: Xd → Xd\\n\\n(x(1),..., x (d)) 7 → (x(d), x (1),..., x (d−1) )Faltings constructs a large subvariety Yd1,...,d n ↪→ Xd which is stable under σ. Then \\n\\nNd1,...,d n (f ) = #Fix( σ ◦ Frob q |Y (Fq)) = ∑(−1) i Tr( σ ◦ Frob |Hic(Y )).\\n\\nNote σ ◦ Frob q = Frob q ◦σ. This implies Faltings' 'near' rationality as in the proposition. Now Zd1,...,d n (f, T ) ∈ 1 + T Q[[ T ]]. We refine the above argument as follows. First, for gcd( a, d ) = 1, you have \\n\\nNd1,...,d n (f ) = Fix( σa ◦ Frob q |Y (Fq)).\\n\\nWe consider Y → Y /G, where G = 〈σ〉 ∼= Z/d Z. We have a character χ: G → C∗, and define the L-function \\n\\nL(χ, T ) = exp \\n\\n( ∞∑\\n\\n> k=1\\n\\nT k\\n\\nk\\n\\n(\\n\\n1\\n\\nd\\n\\n∑ \\n\\n> τ∈G\\n\\nχ(τ −1)Fix( τ ◦ Frob kq |Y (Fq)) \\n\\n)).\\n\\nBy Grothendieck, L(χ, T ) ∈ Q(ζd)( T ). This implies that \\n\\nZd1,...,d n (f, T )φ(d) = ∏ \\n\\n> χ∈̂G\\n\\nL(χ, T )∑ \\n\\n> gcd( a,d )=1 χ(σ)a\\n\\n∈ Q(ζd)( T )so Zd1,...,d n (f, T ) ∈ Q(T ) (essentially by unique factorization). §\\n\\nOpen problem: can you bound the total degree of Zd1,...,d n (f, T )? The best bound we have is 3 · 2d+1 (3 + dm )d1+··· +dn+1, m = deg f. Can this be improved to O(d)O(1)? Yes, if \\n\\nd1 = · · · = dr = d and dr+1 = · · · = dn = 1 (Fu-W). \\n\\nChapter B: Problems \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0088", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The canonical record is a parser aggregate: the leading 3 is the printed page number, and the actual target is the unnumbered conjecture that the congruences (1+X)^n = 1+X^n modulo (n,X^r-1) for every r up to log^2 n certify primality. For this exact reading, each test is equivalent to r cyclic binomial-sum congruences. If d divides r, the r-test implies the d-test; consequently all tests through an integer cutoff L are equivalent to only the upper-half tests L/20, and S ⊂ Fp with S = d. Let h(X) ∈\n\nFp[X] with h(s) 6 = 0 for all s ∈ S. Let G be the group generated by X − s for s ∈ S. In the original AKS paper, we have G ≥ 2d. There are techniques for getting G ≥ (5.82) d\n\n(using lattice point counting); there seems to be much more room for improvement (perhaps using ABC). (Bernstein, Voloch) Perhaps (log G)/d → ∞\n\nas d → ∞ uniformly over h.", "original_statement": "Question 2. Consider Fp and d ∈ Z>0, and S ⊂ Fp with S = d. Let h(X) ∈\n\nFp[X] with h(s) 6 = 0 for all s ∈ S. Let G be the group generated by X − s for s ∈ S. In the original AKS paper, we have G ≥ 2d. There are techniques for getting G ≥ (5.82) d\n\n(using lattice point counting); there seems to be much more room for improvement (perhaps using ABC). (Bernstein, Voloch) Perhaps (log G)/d → ∞ \n\nas d → ∞ uniformly over h.", "clean_statement": "Question 2. Consider Fp and d ∈ Z>0, and S ⊂ Fp with S = d. Let h(X) ∈\n\nFp[X] with h(s) 6 = 0 for all s ∈ S. Let G be the group generated by X − s for s ∈ S. In the original AKS paper, we have G ≥ 2d. There are techniques for getting G ≥ (5.82) d\n\n(using lattice point counting); there seems to be much more room for improvement (perhaps using ABC). (Bernstein, Voloch) Perhaps (log G)/d → ∞\n\nas d → ∞ uniformly over h.", "statement_status": "exact", "statement_verification": "The canonical record comes from the AIM workshop list *Future directions in algorithmic number theory*, Question 2. The mathematical notation lost in the JSON extraction is recoverable from the official PDF and the official HTML transcription. The printed question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[89]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 2. Consider Fp and d ∈ Z>0, and S ⊂ Fp with S = d. Let h(X) ∈\\n\\nFp[X] with h(s) 6 = 0 for all s ∈ S. Let G be the group generated by X − s for s ∈ S. In the original AKS paper, we have G ≥ 2d. There are techniques for getting G ≥ (5.82) d\\n\\n(using lattice point counting); there seems to be much more room for improvement (perhaps using ABC). (Bernstein, Voloch) Perhaps (log G)/d → ∞ \\n\\nas d → ∞ uniformly over h.\"\nOriginal remarks: [\"Remarks. \\n\\n(i) It is hard to find examples with G smaller than pd. Every improvement speeds up by a constant factor the AKS by a factor related to square of the base of the improvement. (Bernstein) (ii) Does it help to know if S is an interval? (Voloch, Lenstra) Not used thus far. (iii) For Kummer extensions, the extensions will look like multiplicative cosets of a root of unity times a single number-can this be used? (Bernstein) (iv) Is this problem independent of h? (Cohen) For example, h(X) = X d −1 with group generated by X has small order. (Bernstein) 25 \\n\\n(v) Instead of looking at Fp[X]/(h), look at an elliptic curve E over Fq and the subgroup of \\n\\nE(Fq) generated by simple x coordinates. (Elkies) Using Weil restriction of scalars, you are looking at a curve C inside of an abelian variety over Fp and look at the subgroup of points generated by C(Fp). This is then amenable to class field theory techniques. (Voloch) The interesting case is the analogous case with AKS: q = pd and S = d, dim A = d.\\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0090", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the quotient-ring reconstruction G_h= in (F_p[X]/(h))^*, the literal uniform conjecture is false: for S=F_p\\{a} and h=X-a one has |G_h|=p-1=d and log|G_h|/d tends to zero. For arbitrary nonconstant admissible h of degree m, all nonnegative exponent products of total degree below m are distinct, so |G_h| is at least binomial(d+m-1,d); this proves the proposed divergence uniformly whenever m/d tends to infinity. Moreover, replacing h by its radical changes G_h only by a p-group kernel of size at most p^(deg h-deg rad(h)) and exponent dividing p^ceil(log_p(max e_i)), so the prime-to-p part of |G_h| depends only on rad(h).\n\nCandidate contribution (counterexample_and_structural_theorem; novelty confidence low): The combined degree-ratio/radical audit gives a testable repair of the under-specified AIM question: it supplies a literal linear-modulus counterexample, proves the exact lower bound |G_h| >= binomial(d+deg(h)-1,d) and hence uniform divergence for deg(h)/d -> infinity, and proves that factor multiplicities in h can contribute only a quantitatively bounded p-primary kernel while leaving the prime-to-p part invariant." }, { "id": 20001253, "problem_number": "AIM-COMPUTATION-0091", "title": "Perfect-number recognition and simultaneous polynomial divisors", "statement": "Question 3. Find a deterministic polynomial time algorithm for recognizing perfect numbers. (Lenstra) Given two monic polynomials f, g ∈ Z[X] with no common irreducible factor, and two integers n, m ∈ Z>0, find in polynomial time all x ∈ Z such that f (x) | m and g(x) | n, with\n\n|x| < H (f, g, m, n ) for some function H.", "original_statement": "Question 3. Find a deterministic polynomial time algorithm for recognizing perfect numbers. (Lenstra) Given two monic polynomials f, g ∈ Z[X] with no common irreducible factor, and two integers n, m ∈ Z>0, find in polynomial time all x ∈ Z such that f (x) | m and g(x) | n, with \n\n|x| < H (f, g, m, n ) for some function H.", "clean_statement": "Question 3. Find a deterministic polynomial time algorithm for recognizing perfect numbers. (Lenstra) Given two monic polynomials f, g ∈ Z[X] with no common irreducible factor, and two integers n, m ∈ Z>0, find in polynomial time all x ∈ Z such that f (x) | m and g(x) | n, with\n\n|x| < H (f, g, m, n ) for some function H.", "statement_status": "exact", "statement_verification": "The canonical record is Question 3 from the AIM workshop *Future directions in algorithmic number theory*. The official AIM HTML and the PDF (printed page 25) agree that one numbered question contains two linked tasks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[90]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 3. Find a deterministic polynomial time algorithm for recognizing perfect numbers. (Lenstra) Given two monic polynomials f, g ∈ Z[X] with no common irreducible factor, and two integers n, m ∈ Z>0, find in polynomial time all x ∈ Z such that f (x) | m and g(x) | n, with \\n\\n|x| < H (f, g, m, n ) for some function H.\"\nOriginal remarks: [\"Remarks. \\n\\n(i) The first problem would be more interesting than perfect numbers themselves. You are given the numbers m, n in binary. (Lenstra) A solution to the second problem gives a solution to the first: If x is prime and xk ‖ n, and n is perfect, then 1 + x + · · · + xk | 2n.(ii) There is one solution for very small H which is to just try out the necessary values of x.(iii) There are some results on when m | f (x), which has typographical similarity to our problem. (Coppersmith) (iv) If n is a perfect number, there is only one choice of xk where the factor 2 is irrelevant; therefore you are reduced to the case where m = n. (Pomerance) (v) By Gary Miller's thesis, if you are given a multiple of φ(n), you can factor n using a ran-domized algorithm or deterministically under the GRH. (Pomerance) You can replace φ(n)by σ(n). (Lenstra) (vi) If you allow randomization, the first problem should be doable. (Lenstra) There is a paper of Bach-Shallit. (vii) This algorithm will recognize perfect numbers but it may not recognize imperfect numbers. (Lenstra) (viii) There is a more general notion ( multiply perfect ) where σ(n)/n = k has small height. Does this affect the problem? (Elkies) No, because you try out each k one at a time, and for fixed \\n\\nk this is virtually identical to the original problem. (ix) A different problem is to just consider f = g, or to look at rational functions which are integer-valued. (x) Are there heuristics for the number of such x which are related to heuristics for the largest prime divisor of f (x)? (Pomerance) (xi) Look at f (x) = ( x4 + x5 + · · · + x8) | n. Choose h ∈ R and g ≈ e√8 log n log h. Then you can find the set of all x with |x| ≤ h such that gcd( f (x), n ) > g. This can be done 'reasonably fast'. (Bernstein) This is the state of the art due to LLL, but it completely fails to solve the problem. \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0091", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The full deterministic recognition problem remains open, but the even part has an unconditional deterministic polynomial-time test via Euclid--Euler and AKS, while complete factorizations give polynomial-size certificates for both perfectness and imperfection. For the linked polynomial-divisor task, every solution for positive-degree coprime monic f and g has gcd(|f(x)|,|g(x)|) dividing gcd(m,n,|Res(f,g)|) and lies in an explicit Cauchy height envelope. That envelope need not be bit-polynomial to enumerate, and the literal allowance of the monic constant polynomial 1 yields a fixed pair with superpolynomially many explicit outputs in a natural full window.\n\nCandidate contribution (obstruction; novelty confidence low): For the exact AIM simultaneous-divisor formulation, the resultant-supported overlap bound and intrinsic Cauchy height envelope combine with the allowed fixed pair (f,g)=(X,1) to show that a natural all-solutions window can require superpolynomial explicit output; therefore a complete polynomial-time formulation must restrict degrees or specify compressed/output-sensitive output and a quantitative target for H." }, { "id": 20001254, "problem_number": "AIM-COMPUTATION-0092", "title": "Prime-factor advice in power-residue quotients", "statement": "These questions are motivated by the questions posed by AKS concerning\nfinding quadratic nonresidues modulo a prime \\(p\\).\n\n(a) It is known that finding a single quadratic nonresidue for a given prime\nis polynomially equivalent to solving all quadratic equations. How far can\none do this for finding a single bit of data (or few bits of data) for higher\ndegrees?\n\n(b) We do not know yet that there is a deterministic polynomial time\nalgorithm for finding quadratic nonresidues. Is there a subexponential\nalgorithm?", "original_statement": "Question 4. These questions are motivated by the questions posed by AKS concerning finding quadratic nonresidues modulo a prime p.(a) It is known that finding a single quadratic nonresidue for a given prime is polynomially equivalent to solving all quadratic equations. How far can one do this for finding a single bit of data (or few bits of data) for higher degrees? (Elkies) 26 \n\n(b) We do not know yet that there is a deterministic polynomial time algorithm for finding quadratic nonresidues. Is there a subexponential algorithm?", "clean_statement": "These questions are motivated by the questions posed by AKS concerning\nfinding quadratic nonresidues modulo a prime \\(p\\).\n\n(a) It is known that finding a single quadratic nonresidue for a given prime\nis polynomially equivalent to solving all quadratic equations. How far can\none do this for finding a single bit of data (or few bits of data) for higher\ndegrees?\n\n(b) We do not know yet that there is a deterministic polynomial time\nalgorithm for finding quadratic nonresidues. Is there a subexponential\nalgorithm?", "statement_status": "corrected_verified", "statement_verification": "The source is the AIM workshop list *Future directions in algorithmic number theory*, Question 4. The recovered statement is: This wording and the accompanying remarks were checked against the AIM PDF. The OCR fragment \\(p1/(4\\sqrt e)\\) in the JSON means \\(p^{1/(4\\sqrt e)}\\), and the fragment \\(X p-X\\) means \\(X^p-X\\). The trailing `Problem/` is an extraction artifact rather than part of the question. The source does not specify the computational model for “subexponential”; that ambiguity matters below.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[91]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 4. These questions are motivated by the questions posed by AKS concerning finding quadratic nonresidues modulo a prime p.(a) It is known that finding a single quadratic nonresidue for a given prime is polynomially equivalent to solving all quadratic equations. How far can one do this for finding a single bit of data (or few bits of data) for higher degrees? (Elkies) 26 \\n\\n(b) We do not know yet that there is a deterministic polynomial time algorithm for finding quadratic nonresidues. Is there a subexponential algorithm?\"\nOriginal remarks: [\"Remarks. \\n\\n(i) The best known deterministic algorithm is due to Burgess and Vinogradov which runs in time p1/(4 √e). (Pomerance) This is sheer enumeration. If you allow exponential time, is there something better than enumeration? (Bernstein) (ii) For cubic extensions, you can use Cardano's formula. (Gao) Is it equivalent to finding a quadratic nonresidue in the cubic extension given by a cubic nonresidue? (Elkies) If cubics includes quadratics, then you can make a quadratic extension. Then there is a trick due to Berlekamp which allows you to solve cubics in the extension by solving them in the ground field. (Lenstra) (iii) If you know a kth nonresidue in the appropriate extension, then you can factor polynomials up to degree six (unpublished). (Gao) The Galois group is cyclic, so solvability by radicals applies. (iv) Given a quadratic nonresidue, any even degree polynomial f with f | (X p − X) can be split nontrivially deterministically in polynomial time, due to Ronyai. (Lenstra) But this does not determine all solutions. You can specify quadratic conditions that must be satisfied by the factors. (Gao) There is a certain combinatorial structure on systems of roots which must be attended to, and you run into difficulties at degree 7. Conjecturally, you should be able to go higher. (v) If you have GRH then you can deterministically in polynomial time solve quadratic exten-sions. Can you solve higher degree equations? (Elkies) On the GRH, you can construct appropriate nonresidues (since you can construct finite fields). (Lenstra) You can do it for fixed degree in time nlog n(log p)O(1) with the GRH due to Ronyai, Evdokimov. (Cheng) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0092", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For k dividing p-1, supplied elements a_i=g^{j_i} generate the full quotient F_p^*/(F_p^*)^k exactly when lcm_i(k/gcd(k,j_i))=k, equivalently when for every prime ell dividing k at least one supplied element is an ell-th power nonresidue. Hence any nonresidue suffices at the quotient-generation level when k is prime, while an arbitrary nonresidue can fail for composite k. Separately, the Burgess scan p^{1/(4 sqrt(e))+o(1)} remains exponential in the binary input length; genuine subexponential input time would require p^{o(1)}.\n\nCandidate contribution (structural criterion; novelty confidence low): The candidate contribution is an explicit prime-factor coverage criterion for higher-power advice: a set of supplied residue classes gives the complete k-power residue quotient precisely when it escapes every prime-index maximal-subgroup obstruction, equivalently when it contains an ell-th power nonresidue for each prime ell dividing k; this is paired with an input-length translation that separates a small exponent of p from subexponential bit complexity." }, { "id": 20001255, "problem_number": "AIM-COMPUTATION-0093", "title": "Adjacent nonisogenous fibers in the Legendre family", "statement": "Question 5. Suppose that you have a nonconstant family of elliptic curves Eλ\n\nover Fp (e.g. the Frey curves), λ ∈ P1(Fp). Can you find deterministically λ such that Eλ\n\nand Eλ+1 are not isogenous (i.e. Eλ(Fp) 6 = Eλ+1 (Fp)). (Elkies)", "original_statement": "Question 5. Suppose that you have a nonconstant family of elliptic curves Eλ\n\nover Fp (e.g. the Frey curves), λ ∈ P1(Fp). Can you find deterministically λ such that Eλ\n\nand Eλ+1 are not isogenous (i.e. Eλ(Fp) 6 = Eλ+1 (Fp)). (Elkies)", "clean_statement": "Question 5. Suppose that you have a nonconstant family of elliptic curves Eλ\n\nover Fp (e.g. the Frey curves), λ ∈ P1(Fp). Can you find deterministically λ such that Eλ\n\nand Eλ+1 are not isogenous (i.e. Eλ(Fp) 6 = Eλ+1 (Fp)). (Elkies)", "statement_status": "exact", "statement_verification": "The canonical record is Question 5 in the AIM workshop list *Future directions in algorithmic number theory*. The official PDF and HTML transcription restore the notation lost by extraction:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[92]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 5. Suppose that you have a nonconstant family of elliptic curves Eλ\\n\\nover Fp (e.g. the Frey curves), λ ∈ P1(Fp). Can you find deterministically λ such that Eλ\\n\\nand Eλ+1 are not isogenous (i.e. Eλ(Fp) 6 = Eλ+1 (Fp)). (Elkies)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) You want p sufficiently large and the degree small. (Elkies) If you can do this, then you should also be able to do it for λ and 1 − λ, and then you can 'separate them apart' by applying Schoof's algorithm and obtain a deterministic square root. (ii) Over Q, there are only finitely many isogeny classes, so you just need to check that λ is not a root of a finite list of polynomial equations. (Elkies) (iii) Can you deterministically find λ, μ such that Eλ, E μ are not isogeneous? (Coppersmith) What happens to Schoof's algorithm if you just run it with λ a variable? (Lenstra) (iv) There are bounds p1/2+ ≤ on the number of isogeny classes (Hasse interval) over Fp.(v) You can also ask the question for the family of all elliptic curves over Fp, deterministically. (Pomerance) This you can do by looking at if −1 and −2 are both squares. (Elkies) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0093", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the Legendre family E_lambda:y^2=x(x-1)(x-lambda) over every odd prime field, if A_lambda=sum_x chi(x(x-1)(x-lambda)), then C(0)=sum_lambda A_lambda^2=p^2-2p-1, C(1)=sum_lambda A_lambda A_(lambda+1)=-1-p-p chi(-1), and therefore sum_lambda(A_(lambda+1)-A_lambda)^2=2p(p-1+chi(-1)). After explicitly bounding the three parameters {-1,0,1} where one adjacent fiber is singular, this proves that for every odd p>=5 some two smooth adjacent Legendre fibers have unequal point counts and hence are not F_p-isogenous; moreover at least p/8-O(1) parameters work. Exhaustive Schoof scanning is deterministic in O(p poly(log p)) time, not polynomial in the input length log p.\n\nCandidate contribution (exact_identity_and_special_case_theorem; novelty confidence low): The exact adjacent Legendre trace identity sum_lambda(A_(lambda+1)-A_lambda)^2=2p(p-1+chi(-1)), together with a sharp separation of the three singular adjacent parameters, forces a smooth nonisogenous adjacent pair for every odd prime p>=5 and quantitatively at least p/8-O(1) such parameters." }, { "id": 20001256, "problem_number": "AIM-COMPUTATION-0094", "title": "Irreducible line sections: exact convention-sensitive counts and monodromy status", "statement": "Question 6. Let f (X, Y ) ∈ Fq[X, Y ] be irreducible. Let g(Y ) = f (aY + b, Y ). Count (or estimate) the number of pairs a, b over Fq such that g is irreducible over Fq. (Gao)", "original_statement": "Question 6. Let f (X, Y ) ∈ Fq[X, Y ] be irreducible. Let g(Y ) = f (aY + b, Y ). Count (or estimate) the number of pairs a, b over Fq such that g is irreducible over Fq. (Gao)", "clean_statement": "Question 6. Let f (X, Y ) ∈ Fq[X, Y ] be irreducible. Let g(Y ) = f (aY + b, Y ). Count (or estimate) the number of pairs a, b over Fq such that g is irreducible over Fq. (Gao)", "statement_status": "exact", "statement_verification": "The canonical record is Question 6 from the AIM workshop *Future directions in algorithmic number theory*. The official AIM HTML and workshop PDF give the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[93]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 6. Let f (X, Y ) ∈ Fq[X, Y ] be irreducible. Let g(Y ) = f (aY + b, Y ). Count (or estimate) the number of pairs a, b over Fq such that g is irreducible over Fq. (Gao)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) Is it > 0 when q > d 4, where d is the total degree of f? (Gao) Or perhaps some other bound on q?27 \\n\\n(ii) It is equivalent to count the number of reducible ones, so use the Schoof-Pila algorithm. (Elkies) But it is slightly different because you are counting points on a surface. (iii) The polynomial f (X, Y ) = Xq − X + Y is always divisible by Y under such a substitution. (Lenstra) Therefore we need q > d.(iv) Do you need to assume that f is irreducible over Fq? (Pomerance) (v) What does Hilbert's irreducibility theorem say in this case? (Lenstra) (vi) Applying the Chebotarev density theorem, this number is rq 2 + O(q3/2), where r ∈ Q≥0.(Wan) How big is the constant? (vii) View g as a polynomial in 3 variables, Y, a, b. The surface g = 0 is a cover of the affine plane given by the variables a, b. What is the Galois group of this cover (over Fq(a, b ))? Is it possibly the full symmetric group? (Lenstra) Then r(G) is equal to the number of elements in the Galois group that are a full d-cycle. Note this extension is separable whenever f is irreducible. (Edixhoven) (viii) If r = 0 then all elements of the set are reducible, since any one irreducible element will have Frobenius which is a full d cycle. (Lenstra) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0094", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The ordinary and full-generic-degree interpretations of the AIM question differ. For the irreducible family f(X,Y)=X^2-Y, the ordinary/full counts are ((q^2+1)/2,(q-1)^2/2) for odd q and ((q^2+q)/2,q(q-1)/2) for even q. For the AIM obstruction f(X,Y)=X^q-X+Y, the exact counts are q and 0, respectively: its q ordinary irreducible specializations are all the degree-drop polynomial Y. A degree-d irreducible polynomial with d>1 never restricts identically to zero on such a line, and at most d slopes cause top-degree loss. Existing Chebotarev and hyperplane-section results prove q^2/d+O_d(q^{3/2}) under additional geometric and characteristic hypotheses, but the bare q>d^4 question remains unresolved by the literature checked.\n\nCandidate contribution (exact_count_and_convention_audit; novelty confidence low): For every prime power q, the AIM source family X^q-X+Y has exactly q ordinary irreducible line specializations but no irreducible specialization of its generic degree, while X^2-Y has the explicit ordinary/full count pairs ((q^2+1)/2,(q-1)^2/2) in odd characteristic and ((q^2+q)/2,q(q-1)/2) in characteristic 2." }, { "id": 20001257, "problem_number": "AIM-COMPUTATION-0095", "title": "Binomial windows for prescribed-degree factors", "statement": "Question 7. Let m ≥ n ∈ Z. Prove there is a polynomial g ∈ Fq[x] of degree\n\n≤ 2 log n so that xm + g(x) has an irreducible factor of degree n. (Gao)", "original_statement": "Question 7. Let m ≥ n ∈ Z. Prove there is a polynomial g ∈ Fq[x] of degree \n\n≤ 2 log n so that xm + g(x) has an irreducible factor of degree n. (Gao)", "clean_statement": "Question 7. Let m ≥ n ∈ Z. Prove there is a polynomial g ∈ Fq[x] of degree\n\n≤ 2 log n so that xm + g(x) has an irreducible factor of degree n. (Gao)", "statement_status": "exact", "statement_verification": "The record is Question 7 in the AIM workshop list *Future directions in algorithmic number theory*. The official PDF and its HTML version give the statement", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[94]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 7. Let m ≥ n ∈ Z. Prove there is a polynomial g ∈ Fq[x] of degree \\n\\n≤ 2 log n so that xm + g(x) has an irreducible factor of degree n. (Gao)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) If q is large compared to fixed m, n, this seems provable. (Gao) (ii) For q = 2, there is an application to computing discrete logs in F∗ \\n\\n> 2n. (Coppersmith) (iii) If m is the smallest power of q which is ≥ n, then α is a root of the irreducible factor, then the order of α is at least ≥ n(log n)/(log log n), so this group has large order. (Gao) (iv) For all q, and m = n, it is proven if deg g ≤ n/ 2. (v) Consider the variant where instead of restricting the degree consider restricting the number of nonzero terms of xm + g(x) to a fixed number (such as 7) (Bernstein), or at least ≤ 2 log n\\n\\n(Pomerance). (vi) Alternatively, find a trinomial of degree m ≤ 2n over Fq with a primitive irreducible factor of degree n. (Gao) For small q this may not be possible. (vii) For fixed n, q, what is the sparsest polynomial of degree ≤ 2n with a primitive and irre-ducible factor of degree n? (Pomerance) This should be uniform in q, surely 5 but perhaps \\n7. (Bernstein) Trivially, sparsity n works by taking an irreducible polynomial of degree n\\n\\n(excluding ( n, q ) = (2, 2)). (Lenstra) Sparsity (1 − ≤)n should be possible. (Wan) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0095", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Three rigorous partial results are proved for an arbitrary integer cutoff L. First, if s is a positive multiple of n and m=q^s+r with r>=0, then every degree-n monic irreducible over F_q divides X^m-X^{r+1}; hence g=-X^{r+1} solves the AIM problem whenever r+1<=L, and the binomial even contains a primitive degree-n factor. Second, if t divides q^n-1, ord_t(q)=n, and m has a residue r in [0,L] with r=0, the explicit binomial X^{q^s+r}-X^{r+1} simultaneously contains every monic irreducible polynomial of degree n over F_q, including a primitive one; combined with a multiplicative-order residue-window theorem and an exact quotient-algebra incidence equivalence, this gives a testable package of solved fibers and isolates the remaining obstruction in AIM Question 7." }, { "id": 20001258, "problem_number": "AIM-COMPUTATION-0096", "title": "Information and conditioning audit for zeta-zero primality and factoring proposals", "statement": "Question 8. Look at ∑ nρ over the zeros ρ of the Riemann zeta function with the imaginary part of ρ the interval [ T, 2T ]. This is ≈ T / (2 π)Λ( n). Can you make a primality test out of this? (Conrey) The sum ∑ nρζ′(ρ)/ζ ′′ (ρ) ≈ T / (2 π) log p log q if n = pq. Can you make a factoring algorithm out of this?\n\nProblem/", "original_statement": "Question 8. Look at ∑ nρ over the zeros ρ of the Riemann zeta function with the imaginary part of ρ the interval [ T, 2T ]. This is ≈ T / (2 π)Λ( n). Can you make a primality test out of this? (Conrey) The sum ∑ nρζ′(ρ)/ζ ′′ (ρ) ≈ T / (2 π) log p log q if n = pq. Can you make a factoring algorithm out of this? \n\nProblem/", "clean_statement": "Question 8. Look at ∑ nρ over the zeros ρ of the Riemann zeta function with the imaginary part of ρ the interval [ T, 2T ]. This is ≈ T / (2 π)Λ( n). Can you make a primality test out of this? (Conrey) The sum ∑ nρζ′(ρ)/ζ ′′ (ρ) ≈ T / (2 π) log p log q if n = pq. Can you make a factoring algorithm out of this?\n\nProblem/", "statement_status": "exact", "statement_verification": "The canonical record is Question 8 in the AIM workshop list *Future directions in algorithmic number theory*. The official PDF and HTML restore the extracted superscripts and derivatives as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[95]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 8. Look at ∑ nρ over the zeros ρ of the Riemann zeta function with the imaginary part of ρ the interval [ T, 2T ]. This is ≈ T / (2 π)Λ( n). Can you make a primality test out of this? (Conrey) The sum ∑ nρζ′(ρ)/ζ ′′ (ρ) ≈ T / (2 π) log p log q if n = pq. Can you make a factoring algorithm out of this? \\n\\nProblem/\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0096", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The first displayed formula in the AIM source has the opposite sign from the classical Landau formula: for fixed integer n > 1, summing nontrivial zeros with multiplicity gives sum_{T < Im rho <= 2T} n^rho = -(T/(2 pi)) Lambda(n) + O_n(log T). This is a prime-power detector rather than a direct primality detector, although an exact value plus deterministic perfect-power testing decides primality. For n = pq, exact knowledge of S = log(p)log(q), together with L = log(n), determines log(p) and log(q) as the roots of z^2-Lz+S. If p < q, Delta = log(q/p), and an approximation S-hat satisfies |S-hat-S| <= min(Delta^2/8, Delta/(8p)), then exponentiating the smaller reconstructed root and rounding recovers p. A complementary bound quantifies how ordinate errors propagate into the unweighted zero sum. These proved information and conditioning statements do not establish either proposed algorithm: the weighted zeta'/zeta'' asymptotic remains unverified and under-specified, and no uniform effective, certified, poly(log n)-time method for evaluating either sum was found.\n\nCandidate contribution (conditioning theorem and information audit; novelty confidence low): Candidate novelty: a single explicit audit combines (i) the prime-power ambiguity of the Landau signal, (ii) exact semiprime recovery from log(p)log(q), (iii) the sufficient certified recovery threshold |S-hat-S| <= min(log(q/p)^2/8, log(q/p)/(8p)), and (iv) a zero-ordinate perturbation bound, while isolating the independent zeta'' denominator obstruction in the printed weighted proposal." }, { "id": 20001259, "problem_number": "AIM-COMPUTATION-0097", "title": "Three zeta-like products, degree-count inversion, and the factorization boundary", "statement": "Question 9. Compute the zeta function of Spec Z/n Z, namely\n\nζ(s) = ∏\n\n> p|n\n\n1\n\n1 − p−s,\n\nwithout knowing the prime factorization of n. Computing the special value s = 1 gives you\n\nφ(n) which is enough to factor n. (Wan) 28\n\nReplace n by a polynomial f (x) ∈ Fp[x], so given\n\nζ(s) = ∏\n\n> g|fgmonic,irreducible\n\n1\n\n1 − g\n\ncan you recover the factorization of f in Fq[x] in deterministic polynomial time? (Wan)", "original_statement": "Question 9. Compute the zeta function of Spec Z/n Z, namely \n\nζ(s) = ∏\n\n> p|n\n\n1\n\n1 − p−s,\n\nwithout knowing the prime factorization of n. Computing the special value s = 1 gives you \n\nφ(n) which is enough to factor n. (Wan) 28 \n\nReplace n by a polynomial f (x) ∈ Fp[x], so given \n\nζ(s) = ∏ \n\n> g|fgmonic,irreducible\n\n1\n\n1 − g\n\ncan you recover the factorization of f in Fq[x] in deterministic polynomial time? (Wan)", "clean_statement": "Question 9. Compute the zeta function of Spec Z/n Z, namely\n\nζ(s) = ∏\n\n> p|n\n\n1\n\n1 − p−s,\n\nwithout knowing the prime factorization of n. Computing the special value s = 1 gives you\n\nφ(n) which is enough to factor n. (Wan) 28\n\nReplace n by a polynomial f (x) ∈ Fp[x], so given\n\nζ(s) = ∏\n\n> g|fgmonic,irreducible\n\n1\n\n1 − g\n\ncan you recover the factorization of f in Fq[x] in deterministic polynomial time? (Wan)", "statement_status": "exact", "statement_verification": "This is Question 9 from the AIM workshop *Future directions in algorithmic number theory*. The official AIM HTML and the workshop PDF agree on the following formulas.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[96]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 9. Compute the zeta function of Spec Z/n Z, namely \\n\\nζ(s) = ∏\\n\\n> p|n\\n\\n1\\n\\n1 − p−s,\\n\\nwithout knowing the prime factorization of n. Computing the special value s = 1 gives you \\n\\nφ(n) which is enough to factor n. (Wan) 28 \\n\\nReplace n by a polynomial f (x) ∈ Fp[x], so given \\n\\nζ(s) = ∏ \\n\\n> g|fgmonic,irreducible\\n\\n1\\n\\n1 − g\\n\\ncan you recover the factorization of f in Fq[x] in deterministic polynomial time? (Wan)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) Can you compute the latter using Drinfeld modules? (Lenstra) Take A = Fp[x]/(f ). Define an Fp-linear map T: A → A where T (α) = αp − xα which defines the Carlitz module, and makes A into an Fp[T ]-module. If you write f = ∏ \\n\\n> g\\n\\ngei \\n\\n> i, and define φ(f ) = ∏ \\n\\n> i\\n\\ngei−1 \\n\\n> i\\n\\n(gi −\\n\\n1), then φ(f )( T ) kills A. But also f ∏\\n\\n> i\\n\\n(gi − 1) also kills A. Mimic the factorization of integers using upper bounds on φ(n) to factor f probabilistically using this 'exponent' of the multiplicative group. \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0097", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM polynomial display, the standard Hasse-Weil zeta, and a Goss/Carlitz zeta must be distinguished. For X_f=Spec(F_q[x]/(f)), the standard zeta is the product of (1-U^deg(g))^-1 over distinct irreducible divisors g and is exactly equivalent to the counts c_d of distinct degree-d factors: if log Z_f(U)=sum b_m U^m/m, then b_m=sum_{d|m} d c_d and c_d=(1/d)sum_{e|d} mu(d/e)b_e. These counts reach deterministic distinct-degree factorization but not equal-degree splitting. The literal AIM product retains polynomial coefficients yet has an expanded-product collision over F_2. The printed Carlitz minus-sign annihilator is false in odd characteristic; the standard plus action or a monic sign twist repairs it. Also, Z_n(1)=n/phi(n), not phi(n).\n\nCandidate contribution (representation_sensitive_reduction_and_counterexample; novelty confidence low): The combined audit proves the exact Hasse-Weil degree-count inversion, identifies equal-degree splitting as the remaining algorithmic step, exhibits R_{x(x+1)}=R_{x^2+x+1}=1/(x^2+x) for the literal expanded AIM product over F_2, and disproves the literal minus-sign Carlitz annihilator over F_3 with f=x-1 while giving the sign-twisted repair." }, { "id": 20001260, "problem_number": "AIM-COMPUTATION-0098", "title": "A local-to-global screen for number-field ECM torsion", "statement": "Question 10. To speed up the elliptic curve factorization algorithm, pick E/ Q\n\nwith large torsion group so that its reduction modulo n is more likely to be smooth. Mazur's results show that this torsion subgroup can be no larger than 16. There is a result of Kamienny-Mazur-Merel: there is a bound on E(K)tors in terms of [ K: Q]. So try to find a number field K where p splits completely and such that E(K)tors large. (Pomerance)", "original_statement": "Question 10. To speed up the elliptic curve factorization algorithm, pick E/ Q\n\nwith large torsion group so that its reduction modulo n is more likely to be smooth. Mazur's results show that this torsion subgroup can be no larger than 16. There is a result of Kamienny-Mazur-Merel: there is a bound on E(K)tors in terms of [ K: Q]. So try to find a number field K where p splits completely and such that E(K)tors large. (Pomerance)", "clean_statement": "Question 10. To speed up the elliptic curve factorization algorithm, pick E/ Q\n\nwith large torsion group so that its reduction modulo n is more likely to be smooth. Mazur's results show that this torsion subgroup can be no larger than 16. There is a result of Kamienny-Mazur-Merel: there is a bound on E(K)tors in terms of [ K: Q]. So try to find a number field K where p splits completely and such that E(K)tors large. (Pomerance)", "statement_status": "exact", "statement_verification": "The source is Question 10 in the AIM workshop list *Future directions in algorithmic number theory*. The extracted record is faithful to the workshop PDF apart from lost typography. With notation restored, the question reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 10\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[97]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 10. To speed up the elliptic curve factorization algorithm, pick E/ Q\\n\\nwith large torsion group so that its reduction modulo n is more likely to be smooth. Mazur's results show that this torsion subgroup can be no larger than 16. There is a result of Kamienny-Mazur-Merel: there is a bound on E(K)tors in terms of [ K: Q]. So try to find a number field K where p splits completely and such that E(K)tors large. (Pomerance)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) Whatever advantage you gain might be swamped by the expense of working in the larger number field. (Pomerance) (ii) One problem is that for any given number field, there are only finitely many curves over that number field with a fixed torsion subgroup (since then modular curves have genus ≥ 2). (iii) If n = p2q, choose a discriminant D such that D is a nonsquare modulo q, and find an elliptic curve E/ Q(√D). Then E(Q(√D)) tors injects into E(Fq2 ). (Bleichenbacher) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0098", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an elliptic curve E over a number field K and a rational prime p that splits completely in K, good reduction injects the prime-to-p K-rational torsion into E(F_p). If its order is t_p, then t_p divides #E(F_p), the Hasse interval must meet t_p Z, and, when t_p is B-smooth, full-group B-smoothness is equivalent to B-smoothness of the exact cofactor #E(F_p)/t_p. The eligible completely split primes have density 1/[L:Q], where L is the Galois closure of K, not generally 1/[K:Q]. The same argument rigorously recovers the prime-to-q quadratic inert-prime injection into E(F_{q^2}) and supplies its Hasse screen.\n\nCandidate contribution (reduction; novelty confidence low): Candidate torsion-divisor/Hasse/cofactor/splitting-density certificate: a fixed-number-field ECM family is locally feasible at p only if dist(p+1,t_p Z) <= 2 sqrt(p), its full reduced group is B-smooth exactly when the forced torsion divisor and residual cofactor are B-smooth, and the completely split eligible primes have density 1/[L:Q]; for K=Q(sqrt(D)) and inert q the corresponding screen is dist(q^2+1,t_q Z) <= 2q." }, { "id": 20001261, "problem_number": "AIM-COMPUTATION-0099", "title": "Geometry, residues, and scale thresholds for prescribed-prefix prime products", "statement": "Question 11. Given a (random) number n of 10000 digits, it may be impossible to find 5000 digit primes p, q such that n = pq. Much easier: find some 10000 digit primes\n\np, q such that n is the first 10000 digits of pq. Instead, find 7500 digit primes p, q such that\n\nn is the first 10000 digits of pq. (Coppersmith) Therefore, do this for 6000 digit primes. (Bernstein)", "original_statement": "Question 11. Given a (random) number n of 10000 digits, it may be impossible to find 5000 digit primes p, q such that n = pq. Much easier: find some 10000 digit primes \n\np, q such that n is the first 10000 digits of pq. Instead, find 7500 digit primes p, q such that \n\nn is the first 10000 digits of pq. (Coppersmith) Therefore, do this for 6000 digit primes. (Bernstein)", "clean_statement": "Question 11. Given a (random) number n of 10000 digits, it may be impossible to find 5000 digit primes p, q such that n = pq. Much easier: find some 10000 digit primes\n\np, q such that n is the first 10000 digits of pq. Instead, find 7500 digit primes p, q such that\n\nn is the first 10000 digits of pq. (Coppersmith) Therefore, do this for 6000 digit primes. (Bernstein)", "statement_status": "exact", "statement_verification": "This is Question 11 in the AIM problem list produced by the March 2003 workshop *Future directions in algorithmic number theory*. The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 11\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[98]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 11. Given a (random) number n of 10000 digits, it may be impossible to find 5000 digit primes p, q such that n = pq. Much easier: find some 10000 digit primes \\n\\np, q such that n is the first 10000 digits of pq. Instead, find 7500 digit primes p, q such that \\n\\nn is the first 10000 digits of pq. (Coppersmith) Therefore, do this for 6000 digit primes. (Bernstein)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) Coppersmith's algorithm works as follows. Pick at random p0 of 7500 digits. Pick q0 =\\n\\nb(n/p 0)10 5000 c. Add x to p0 and y to q0 where x, y have 2500 digits a piece to fix this up. We want (p0 + x)( q0 + y) = p0q0 + p0y + q0x + xy ≈ n10 5000 \\n\\nso use lattice reduction. At the end, check to make sure p0 and q0 are prime. (ii) This has applications in cryptography. (Bernstein) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0099", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a D-digit prefix N and A-digit factors with D/2 < A < D, put B=2A-D, W=10^B, U=10^A, and M=NW. The exact prefix condition is M <= pq < M+W. The continuous hyperbolic strip in [10^(A-1),U)^2 has area integral_M^(M+W) log(U^2/t) dt; exactly 2*10^(D-A)-1 top prefixes are excluded by the maximum product (U-1)^2; and, apart from at most one boundary row, each fixed p has a candidate q exactly when the least residue (-M mod p) is below W. The Coppersmith perturbation satisfies the exact inequality r <= q0*x+p0*y+x*y < r+W. Within the standard symmetric small-correction scale requirements U/H <= W and H^2 <= W, a compatible H exists exactly when A >= 3D/4. Thus 7500/10000 is the critical two-variable linearization scale, while at 6000/10000 the natural H=10^4000 gives H^2/W=10^6000. This obstructs that ansatz, not all algorithms. Prime abundance for A>D/2 remains heuristic, and no balanced two-prime 5/6-leading-prefix polynomial-time algorithm was found.\n\nCandidate contribution (geometry-and-scale-threshold synthesis; novelty confidence low): Candidate novelty: an endpoint-aware theorem combining the exact continuous prefix-strip area, the exact count of finite-box-impossible top prefixes, and the row-residue characterization, together with a two-threshold audit separating the heuristic A=D/2 abundance boundary from the exact A=3D/4 compatibility boundary of the standard two-variable linearization." }, { "id": 20001262, "problem_number": "AIM-COMPUTATION-0100", "title": "A nonresidue-to-witness reduction and exact witness count", "statement": "Question 12. Let p be prime, write p − 1 = 2 `m where m is odd, and assume\n\n` ≥ 3. Find in deterministic polynomial time x ∈ Fp such that x2`−1\n\n+ 1 is not a 2 `−1th power. (Kedlaya)", "original_statement": "Question 12. Let p be prime, write p − 1 = 2 `m where m is odd, and assume \n\n` ≥ 3. Find in deterministic polynomial time x ∈ Fp such that x2`−1\n\n+ 1 is not a 2 `−1th power. (Kedlaya)", "clean_statement": "Question 12. Let p be prime, write p − 1 = 2 `m where m is odd, and assume\n\n` ≥ 3. Find in deterministic polynomial time x ∈ Fp such that x2`−1\n\n+ 1 is not a 2 `−1th power. (Kedlaya)", "statement_status": "exact", "statement_verification": "The OCR in the canonical record lost the superscripts and the letter \\(\\ell\\). The official AIM HTML transcription and workshop PDF agree on the following statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 12\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[99]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 12. Let p be prime, write p − 1 = 2 `m where m is odd, and assume \\n\\n` ≥ 3. Find in deterministic polynomial time x ∈ Fp such that x2`−1\\n\\n+ 1 is not a 2 `−1th power. (Kedlaya)\"\nOriginal remarks: [\"Remarks. 29 \\n\\n(i) This should be a sufficient condition for the deterministic nonresidue algorithm of Agrawal to work. \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0100", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Given any supplied quadratic nonresidue modulo p, a non-monotone bisection using membership in H=(F_p^*)^k locates an additive boundary; negation orients it, and an explicit exponentiation formula extracts a k-th root, yielding the requested witness deterministically in time polynomial in log p. This proves a conditional all-primes algorithm under ERH/GRH, an unconditional Las Vegas algorithm, and unconditional deterministic algorithms whenever a nonresidue is known, including p congruent to 17 modulo 24. Independently, the report proves the sharp universal bound #W >= k and an exact Jacobi-sum formula for #W.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty: one supplied quadratic nonresidue is converted to the special-form witness with O(log p) subgroup-membership tests by label-only bisection, orientation reversal using -1 in H, and an explicit k-th-root formula; the same analysis gives the sharp bound #W >= k and an exact Jacobi-sum count.", "research_difficulty_suggested": "L4" }, { "id": 20001263, "problem_number": "AIM-COMPUTATION-0101", "title": "Deformation transport, Fesenko transforms, and a Riemann--Roch size barrier", "statement": "Question 13.\n\n(a) Do the p-adic point counting methods of Lauder which allow you to go from one curve from another in a family apply to p-adic methods without the linear factor? (Edixhoven) (b) Do Fesenko's methods for proving good properties of Hasse-Weil L-functions of curves over number fields provide anything useful for computations? (Edixhoven)", "original_statement": "Question 13. \n\n(a) Do the p-adic point counting methods of Lauder which allow you to go from one curve from another in a family apply to p-adic methods without the linear factor? (Edixhoven) (b) Do Fesenko's methods for proving good properties of Hasse-Weil L-functions of curves over number fields provide anything useful for computations? (Edixhoven)", "clean_statement": "Question 13.\n\n(a) Do the p-adic point counting methods of Lauder which allow you to go from one curve from another in a family apply to p-adic methods without the linear factor? (Edixhoven) (b) Do Fesenko's methods for proving good properties of Hasse-Weil L-functions of curves over number fields provide anything useful for computations? (Edixhoven)", "statement_status": "exact", "statement_verification": "This record is Question 13 in the problem section of the American Institute of Mathematics workshop notes *Future directions in algorithmic number theory*. The workshop was held at AIM on 24--28 March 2003, and the PDF identifies itself as the version of 30 April 2003.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 13\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[100]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 13. \\n\\n(a) Do the p-adic point counting methods of Lauder which allow you to go from one curve from another in a family apply to p-adic methods without the linear factor? (Edixhoven) (b) Do Fesenko's methods for proving good properties of Hasse-Weil L-functions of curves over number fields provide anything useful for computations? (Edixhoven)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) Does Riemann-Roch provide anything useful? (Apparently not; involves linear algebra over matrices of size the number of points.) (Wan) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0101", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Lauder's one-parameter deformation is a plug-in transport mechanism for any base-fibre p-adic routine that returns a compatible Frobenius matrix: the horizontal Frobenius identity is derived with an explicit conservative p-adic precision-loss bound and a singular-fibre/convergence obstruction. Later work confirms this compatibility for hyperelliptic families, but a sublinear-in-p easy-fibre routine does not by itself make every stage of the family algorithm sublinear in p. Fesenko's published Bessel formulas are shown to be a rapidly truncatable forward transform from already-known zeta coefficients to a boundary function, useful for numerical post-processing but not, as presented, an independent curve-to-Euler-factor algorithm. Wan's Riemann--Roch remark is formalized as an Omega(q^2 log q) bit-size obstruction for a dense representation indexed by all rational points at fixed genus.\n\nCandidate contribution (reduction; novelty confidence low): Candidate reduction/obstruction package: compatible Frobenius transport loses at most max(gamma+beta, alpha+beta, alpha+gamma) digits when the three factors are independently certified, with an additional conservative 2 beta inverse-conditioning loss when the inverse is derived; on each compact x-interval away from zero and infinity, Fesenko's displayed coefficient-to-boundary transform has a conditional Gaussian tail O(M^(A+10) exp(-2 pi epsilon^2 M^2)); and a dense Riemann--Roch construction based on the divisor consisting of all rational points has quadratic-in-the-point-count representation size." }, { "id": 20001264, "problem_number": "AIM-COMPUTATION-0102", "title": "A format-sensitive genus barrier for Schoof-Pila", "statement": "Question 14. Pila's method for computing roots of unity is exponential in g. Is there any reason it can't be made linear? (Elkies)", "original_statement": "Question 14. Pila's method for computing roots of unity is exponential in g. Is there any reason it can't be made linear? (Elkies)", "clean_statement": "Question 14. Pila's method for computing roots of unity is exponential in g. Is there any reason it can't be made linear? (Elkies)", "statement_status": "exact", "statement_verification": "The official AIM workshop PDF and HTML both give the question literally as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 14\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[101]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 14. Pila's method for computing roots of unity is exponential in g. Is there any reason it can't be made linear? (Elkies)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) Huang has a randomized algorithm (depends on factoring polynomials) with exponent gO(1).(Lauder) They avoid using a full projective model. (Pila) (ii) For any prime `, you work in a group of size `2g, so shouldn't be much worse. See also Edixhoven's talk. The calculation there (on modular curves) can also be done for Drinfeld modular curves. (Elkies) Is there an analogue of the point-counting problem for Drinfeld modules? (Kedlaya) (iii) Has anyone tried doing Schoof-Pila in genus 2? (Kedlaya) Gaudry and Schost have applied AGM. (Couveignes) They also did small torsion. (Edixhoven) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0102", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The intended target is most naturally a linear-in-g exponent in a Schoof-Pila complexity bound. For an auxiliary prime ell distinct from the characteristic, any implementation that explicitly lists A[ell] or emits a basis-expanded coordinate algebra must perform at least ell^(2g) output operations. In a worst-case CRT implementation whose auxiliary primes reach Theta(g log q), this forces an Omega(g) exponent in log q, so a linear exponent is the natural optimal scale for that representation, not something ruled out by torsion size. The Frobenius polynomial itself is exponentially more compact, so this is not a lower bound for compressed algorithms or for the underlying roots-of-unity problem. Later work attains a linear-in-g exponent for large-characteristic hyperelliptic zeta computation under Las Vegas and uniformity restrictions, but does not uniformly settle Pila's full Fermat-Jacobian root-recovery pipeline.\n\nCandidate contribution (theorem; novelty confidence low): A format-sensitive output theorem separates explicit torsion points, basis-expanded torsion coordinate algebras, and compressed Frobenius invariants: the first two require at least ell^(2g) records, yielding an Omega(g) log-q exponent for full-torsion-explicit CRT implementations reaching ell = Theta(g log q), whereas the Frobenius polynomial requires only O(g log ell) bits modulo ell and O(g^2(1+log q)) bits integrally." }, { "id": 20001265, "problem_number": "AIM-COMPUTATION-0103", "title": "Finite certificate interfaces for finite-field zeta functions", "statement": "Question 15. Given a variety over a finite field, can you verify that a given function is its zeta function (i.e., is the question in NP)? (Lenstra, Edixhoven)", "original_statement": "Question 15. Given a variety over a finite field, can you verify that a given function is its zeta function (i.e., is the question in NP)? (Lenstra, Edixhoven)", "clean_statement": "Question 15. Given a variety over a finite field, can you verify that a given function is its zeta function (i.e., is the question in NP)? (Lenstra, Edixhoven)", "statement_status": "exact", "statement_verification": "The exact canonical record is Question 15 in *Future directions in algorithmic number theory*, source file `aim-computation-notes.json`, record index 102:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 15\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[102]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 15. Given a variety over a finite field, can you verify that a given function is its zeta function (i.e., is the question in NP)? (Lenstra, Edixhoven)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) Yes (for g fixed), because you can verify the orders of the Jacobian over the first g extension fields. (Elkies) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0103", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a smooth projective geometrically connected genus-g curve, the first g curve point counts determine the zeta numerator by Newton identities and the functional equation, and the resulting transcript has O(g^2 log q + g log g) bits. More generally, normalized rational functions with numerator and denominator degree bounds a and b are determined by their first a+b series coefficients. These give deterministic, proved reductions from zeta equality to exact point-count certification, but only a conditional NP transfer: they do not supply NP certificates for the counts. The source's different claim about the first g Jacobian orders is explicitly left unproved; current variable-genus work gives an AM intersection coAM protocol rather than NP.\n\nCandidate contribution (reduction; novelty confidence low): A precise certificate-interface reduction: under polynomial numerator and denominator degree bounds, candidate zeta equality reduces to a polynomial-size bounded transcript of exact extension-field point counts; for smooth projective curves, the known denominator and functional equation sharpen the transcript to exactly the first g curve point counts, with total length O(g^2 log q + g log g), while exposing exact count certification as the remaining complexity bottleneck." }, { "id": 20001266, "problem_number": "AIM-COMPUTATION-0104", "title": "Fields, orbits, and explicit torsion points", "statement": "Question 16. How can you compute with points as in Edixhoven's talk? In that application, you can avoid writing down explicit `-torsion points (only the fields of definition), but can you write them down for other transformation? (Edixhoven)", "original_statement": "Question 16. How can you compute with points as in Edixhoven's talk? In that application, you can avoid writing down explicit `-torsion points (only the fields of definition), but can you write them down for other transformation? (Edixhoven)", "clean_statement": "Question 16. How can you compute with points as in Edixhoven's talk? In that application, you can avoid writing down explicit `-torsion points (only the fields of definition), but can you write them down for other transformation? (Edixhoven)", "statement_status": "exact", "statement_verification": "The canonical record has lost several mathematical glyphs. The official AIM HTML transcription and the workshop PDF agree on the following restoration.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 16\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[103]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 16. How can you compute with points as in Edixhoven's talk? In that application, you can avoid writing down explicit `-torsion points (only the fields of definition), but can you write them down for other transformation? (Edixhoven)\"\nOriginal remarks: [\"Remarks. \\n\\n(i) Can one work out instances of the passage from H1 to H2? E.g., K3 surface of N´ eron-Severi rank 19 (the rank 20 case is standard)? (Elkies) (ii) More comments on `-adic computation of zeta functions? (Pila) \\n\\nProblem/\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0104", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an abelian variety A/F_q of dimension g and a prime ell different from the characteristic, dense enumeration of A[ell] necessarily contains ell^(2g) point records, while a Frobenius matrix is polynomial-size. A geometric torsion point v has field degree min{d >= 1 : F^d v = v}, and the number of points of exact degree d is the Mobius sum over e dividing d of ell^(dim ker(F^e-I)); closed orbits number E_d/d. If the Frobenius characteristic polynomial is squarefree, these dimensions equal deg gcd(chi_F,T^e-1), even when ell divides e. An identity-versus-unipotent example proves that the characteristic polynomial alone fails without squarefreeness. Separately, the report proves the integral rank-two symmetric-square Frobenius cubic relevant to a hypothesized rank-three H^2 realization.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: under the explicit squarefree-characteristic-polynomial hypothesis, exact fields-of-definition statistics for prime-to-characteristic torsion are recovered by E_d = sum_{e|d} mu(d/e) ell^(deg gcd(chi_F,T^e-1)); the package pairs this with the sharp ell^(2g) dense-output barrier, a counterexample showing why squarefreeness matters, and an integral Sym^2 transfer polynomial." }, { "id": 20001267, "problem_number": "AIM-COMPUTATION-0105", "title": "Deformation, specialization, and plane-curve genus", "statement": "Question 17.\n\n(a) Can deformation theory be applied `-adically? (Various) (b) Under what circumstances do Betti numbers stay the same under reduction modulo p?(Various) (c) Is finding the genus of a plane curve polynomial time (in the degree)? (Wan)\n\nProblem/", "original_statement": "Question 17. \n\n(a) Can deformation theory be applied `-adically? (Various) (b) Under what circumstances do Betti numbers stay the same under reduction modulo p?(Various) (c) Is finding the genus of a plane curve polynomial time (in the degree)? (Wan) \n\nProblem/", "clean_statement": "Question 17.\n\n(a) Can deformation theory be applied `-adically? (Various) (b) Under what circumstances do Betti numbers stay the same under reduction modulo p?(Various) (c) Is finding the genus of a plane curve polynomial time (in the degree)? (Wan)\n\nProblem/", "statement_status": "exact", "statement_verification": "This is Question 17 in the AIM workshop list *Future directions in algorithmic number theory* (workshop held March 24--28, 2003; source PDF version dated April 30, 2003). The source record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 17\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[104]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 17. \\n\\n(a) Can deformation theory be applied `-adically? (Various) (b) Under what circumstances do Betti numbers stay the same under reduction modulo p?(Various) (c) Is finding the genus of a plane curve polynomial time (in the degree)? (Wan) \\n\\nProblem/\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0105", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The three-part question separates into established abstract l-adic deformation theory, smooth-proper prime-to-p base-change invariance, and model-dependent genus complexity. A proved synthesis treats a flat plane-curve degeneration with smooth generic and integral special fiber: the first l-adic Betti drop is the sum over singular points of 2 delta_P-r_P+1, and for ordinary m_P-fold points it is exactly the sum of (m_P-1)^2. The same locally certifiable multiplicities give a polynomial-field-operation promised-input algorithm for the normalization genus.\n\nCandidate contribution (proposition; novelty confidence low): For an integral degree-d plane-curve degeneration with smooth generic fiber and only ordinary m_P-fold singularities on the special fiber, the exact prime-to-p first-Betti loss is sum_P (m_P-1)^2; the same tangent-cone certificates compute the normalization genus in polynomially many field operations when the complete singular list is supplied." }, { "id": 20001268, "problem_number": "AIM-COMPUTATION-0106", "title": "Deterministic roots of X^3-2: a density-one-eighteenth residual envelope", "statement": "Question 18. Can you compute roots modulo p of a fixed polynomial (` a la Schoof-Pila) in polynomial time, like x3 − 2? (Schoof) 30", "original_statement": "Question 18. Can you compute roots modulo p of a fixed polynomial (` a la Schoof-Pila) in polynomial time, like x3 − 2? (Schoof) 30", "clean_statement": "Question 18. Can you compute roots modulo p of a fixed polynomial (` a la Schoof-Pila) in polynomial time, like x3 − 2? (Schoof) 30", "statement_status": "exact", "statement_verification": "The canonical record comes from Question 18 of the AIM workshop notes *Future directions in algorithmic number theory*. The web version and the typeset PDF give the intended text as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Computation\nWorkshop: Future directions in algorithmic number theory\nSection: \nSource item: 18\nSource URL: https://aimath.org/WWN/primesinp/primesinp.pdf\nCanonical location: aim-computation-notes.json notes[105]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 18. Can you compute roots modulo p of a fixed polynomial (` a la Schoof-Pila) in polynomial time, like x3 − 2? (Schoof) 30\"\nOriginal remarks: [\"Remarks. \\n\\n(i) A problem is realizing a given polynomial as the characteristic polynomial of an endomor-phism, if its roots do not lie in a CM field. Does it help to consider modular forms? (Pila) \\n\\n#B.1\", \"Remarks on Agrawal's Conjecture \\n\\nThese notes concern Agrawal's conjecture, the first problem in the problem session: \\n\\nConjecture. Let n and r be two coprime positive integers. If \\n\\n(X − 1) n ≡ Xn − 1 (mod n, X r − 1) \\n\\nthen either n is prime or \\n\\nn2 ≡ 1 (mod r).\\n\\n(If Agrawal's conjecture were true, this would improve the polynomial time complexity of the AKS primality testing algorithm from ˜O((log n)7.5) to ˜O((log n)3).) The contents are due to Lenstra and Pomerance and suggest strongly that this conjec-ture is false. \\n\\nProposition (Lenstra). Let p1,..., p k be k pairwise distinct prime integers, and let n =\\n\\np1... p k. Suppose that: \\n\\n(i) k ≡ 1 (mod 4);\\n\\n(ii) pi ≡ 3 (mod 80) for all i;\\n\\n(iii) ( pi − 1) | (n − 1) for all i; and \\n\\n(iv) ( pi + 1) | (n + 1) for all i.Then \\n\\n(X − 1) n ≡ Xn − 1 (mod n, X 5 − 1) \\n\\nand n2 6 ≡ 1 (mod 5).\", \"Remark. This result is also true for k ≡ 3 (mod 4). \\n\\nProof. By assumption we get n = 3 k ≡ 3 (mod 80) because 3 4 ≡ 1 (mod 80). So n ≡ 3(mod 5) and then n2 6 ≡ 1 (mod 5). We also have the following identity: (X − 1, X 4 + X3 + X2 + X + 1, n ) = (1) in the polynomial ring Z[X]. Hence, in order to prove the identity (X − 1) n ≡ Xn − 1 (mod n, X 5 − 1) it suffices to prove that (X − 1) n ≡ Xn − 1 (mod n, X 4 + · · · + X + 1).\\n\\nThe Chinese remainder theorem gives the following isomorphism: \\n\\nZ[X]/(n, X 4 + · · · + X + 1) ∼=\\n\\n> k\\n\\n∏\\n\\n> i=1\\n\\nFpi [X]/(X4 + · · · + X + 1).\\n\\nEach ring factor Ri = Fpi [X]/(X4 + · · · + X + 1) is actually a field since each prime pi is prime to 5 and the 5th cyclotomic polynomial is irreducible in Fp[X] so that Ri is nothing but the splitting field of Fpi [ζ5] for a primitive 5th root of unity ζ5.31 \\n\\nIt therefore suffices to prove that each prime pi = p satisfies (ζ5 − 1) n = ζn \\n\\n> 5\\n\\n− 1in the field Fp[ζ5]. We see from (ii) that (ζ5 − 1) p2\\n\\n= ζp2 \\n\\n> 5\\n\\n− 1 = ζ−15 − 1(since p ≡ 3 (mod 5), we have p2 ≡ − 1 (mod 5)). Thus (ζ5 − 1) p2\\n\\n= −ζ−15 (ζ5 − 1).\\n\\nHence the order of ( ζ5 − 1) in Fp[ζ5] divides 10( p2 − 1). It remains to check the residue class of n modulo 10( p2 − 1); more precisely, it suffices to show that \\n\\nn ≡ p (mod 10( p2 − 1)).\\n\\nWe can factor 10( p2 − 1) into 4 pairwise coprime factors: 10( p2 − 1) = 5(2 4)\\n\\n(p − 1\\n\\n2\\n\\n) ( p + 1 \\n\\n4\\n\\n)\\n\\nso it suffices to verify this modulo each factor. Since n, p ≡ 3 (mod 80) by assumption, the first follows. Assumption (iii) implies that \\n\\nn ≡ 1 (mod ( p − 1) /2) and so \\n\\nn = p (mod ( p − 1) /2) since p ≡ 1 (mod ( p − 1) /2), and \\n\\nn ≡ p (mod ( p + 1) /4) similarly. This completes the proof. §\\n\\nBy this proposition, we have a heuristic which suggests the existence of many counterex-amples to the Agrawal conjecture. This argument taken from analytic number theory is very similar to the one already used by Pomerance to find counterexamples to the Baillie-PSW pri-mality testing algorithm which can be found at http://www.pseudoprime.com/dopo.pdf.Fix some arbitrarily large integer m and let T be very large. Let P = Pm(T ) denote the set of primes p in the interval [ T, T m] such that: A. p ≡ 3 (mod 80); B. ( p − 1) /2 is squarefree and divisible only by primes q ≤ T with q ≡ 3 (mod 4); C. ( p + 1) /4 is squarefree and divisible only by primes r ≤ T with r ≡ 1 (mod 4). Both smoothness conditions (2) and (3) are rather restrictive: heuristically, the cardinality of the set P is asymptotically ( T → ∞ )#P ∼ cm\\n\\nT m\\n\\n(log T m)2\\n\\nfor some positive constant cm that depends on the choice of m. In particular, we can take a sufficiently large integer T such that #P > T m\\n\\n(log T m)3\\n\\nwhich we assume from now on. 32 \\n\\nAlso choose an odd integer k ≡ 1 (mod 4) such that k < T 2/(log T m). We consider the squarefree numbers n that run over products of k distinct primes of the set P. Obviously such an integer n satisfies n < e T 2. The number of choices for n is exactly given by the binomial coefficient (#Pk\\n\\n), and we get the lower bound: \\n\\n(#Pk\\n\\n)\\n\\n≥\\n\\n( T m\\n\\n(log T m)3(T 2/ log T m)\\n\\n)(T 2/ log T m)−4\\n\\n> (T m−3)(T 2/ log T m)−4 = e(1 −3/m )T 2−4( m−3) log T\\n\\n> e 1−(4 /m )T 2.\\n\\nfor large T and fixed m.Let Q denote the product of primes q ≤ T with q ≡ 3 (mod 4), and let R denote the product of primes r ≤ T with r ≡ 1 (mod 4). Then Q and R are coprime and asymptotically the product QR equals e(1+ o(1)) T as T → ∞, so that QR < e 2T for some large T. Thus, the number of choices for the numbers n that satisfy in addition n ≡ 1 (mod Q) and n ≡ − 1(mod R) should be asymptotically \\n\\ne(1 −4/m )T 2\\n\\ne−2T > e T 2(1 −5/m ).\\n\\nBut any such n is a counterexample to Agrawal's conjecture by Lenstra's proposition. We see therefore that for fixed m and for all large T, there should be at least eT 2(1 −5/m )\\n\\ncounterexamples to Agrawal's conjecture below eT 2. That is, if we let x = eT 2, this argument implies that the number of counterexamples ≤ x is expected to be ¿ x1−≤ for any ≤ > 0.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 15, "set_id": 14, "source_url": "https://aimath.org/WWN/primesinp/primesinp.pdf", "tags": [ "aim", "AIM-COMPUTATION-0106", "aim-domain:computation", "aim-workshop:primesinp", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 15, "name": "computer_science", "display_name": "Computer Science", "description": "Computational complexity, algorithms, and theoretical CS.", "slug": "computer-science", "order_index": 15, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For X^3-2, an explicit unconditional deterministic poly(log p) algorithm either constructs a root or proves there is none for a density-17/18 set of primes. The elementary residual envelope consists exactly of primes p congruent to 1 modulo 9 for which 2 is a cubic residue; it is the complete-splitting set of Q(cuberoot(2), zeta_9) and has Chebotarev density 1/18. At every split prime, a supplied cubic nonresidue completes deterministic extraction in poly(log p) time through base-3 digit lifting in the 3-Sylow subgroup. The unconditional worst-case single-prime problem on the residual set remains open; Rónyai gives an all-primes conditional answer under GRH.\n\nCandidate contribution (explicit_algorithm_and_reduction; novelty confidence low): Candidate novelty: the elementary exponent algorithm resolves X^3-2 on a density-17/18 set of primes, its exact untreated envelope is the density-1/18 complete-splitting set for Q(cuberoot(2), zeta_9), and one supplied cubic nonresidue yields a verified poly(log p) extraction algorithm on every split prime." }, { "id": 20001269, "problem_number": "AIM-CONVEX_GEOMETRY-0001", "title": "Sparse-normal formulas for perimeters and related cube-section functionals", "statement": "1. Find the central cross section u⊥:= {x ∈ Rn: x · u = 0 } of Bn\n\n> ∞\n\n=\n\n{x ∈ Rn: |xi| ≤ 1} with the largest perimeter. 2. What is the central cross section of Bn\n\n> ∞\n\nwith smallest perimeter? 3. What if instead of perimeter we take the mean width or intrinsic volume? 4. What about other codimensions? 5. What about other measures such as gaussian measures? 6. What about perimeter of projections?\n\n#2 Bounded Projection Inequality\n\nProposed by Mathieu Meyer\n\nLet Knos be the collection of all origin symmetric convex bodies in Rn.Find an asymptotic bound for the quantity\n\nmax\n\n> K∈Knos\n\nmin\n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥), (1) where Pv,u ⊥ (K) is the projection of K on u⊥ in the direction v ∈ Sn−1.One should note that 2n is a rough upper bound for (1) because for every hyperplane H in Rn, there is a projection PH so that ‖PH ‖ ≤ 2.This would imply PH K ⊆ 2( K ∩ H) and hence ‖PH K‖ ≤ 2n−1.One should also note that\n\nmin\n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥) (2) is affine invariant. If K is an ellipsoid then the quantity in (2) is 1 so (1) is always greater than or equal to 1.\n\n#3 Preservation of Convexity Under the Intersec-tion Body Operator\n\nProposed by Maria Alfonseca\n\n2Suppose K is an origin-symmetric starbody in Rn. let ρK (u):= max {a:\n\nau ∈ K} and define the intersection body IK of K as the body defined by\n\nρIK (u) = Vol n−1(K ∩ u⊥) ∀u ∈ Sn−1. (3) Define the convex kernel of K as the intersection of all convex bodies M\n\nsuch that m ⊆ K. What conditions on the convex kernel of K would imply\n\nIK is convex?\n\n#4 Cotype and Volume Ration of `np ⊗π `nq ⊗π `nr\n\nProposed by Carsten Schütt\n\nFind the cotype and volume ration of `np ⊗π `nq ⊗π `nr.\n\n#5 Banach-Mazur Distance Problem\n\nProposed by Mark Rudelson\n\nLet Knos denote the space of origin-symmetric, convex bodies in Rn and let dBM is the Banach-Mazur distance. Let α ∈ (0, 1) and n ∈ N. Suppose K, L ∈ Knos are such that there exists b > 0 so\n\ndBM (K ∩ E, L ∩ E) ≤ b ∀E ∈ Kαn os. (4) Does there exists a function f so that\n\ndBM (K, L ) ≤ f (b, α )? (5) It's worth noting that M. Rudelson showed during one of the problem sessions that this question is trivially true if dBM is replaced with the geometric distance dg.\n\n#6 Injective Operators which Correspond with the Intersection Body Operator\n\nProposed by Richard Gardner\n\nLet Kno denote the collection of convex bodies (i.e., compact convex sets with nonempty interior) in Rn and let Knos denote the origin-symmetric members of Kno. Is there an F: Kno → Knos such that F is continuous, affine invariant, injective (modulo translations and reflections in the origin), and\n\nF |Kn = I, where I is the intersection body operator. 37 Discrete Tomography\n\nProposed by Richard Gardner\n\n(Discrete Aleksandrov projection theorem.) If two n-dimensional cen-trally symmetric convex lattice sets in Zn are such that the cardinality of their projections on any lattice hyperplane is the same, are they equal up to translation? This question appears with the restriction n ≥ 3 in [GGZ]. For n = 2, the answer is negative, but only an isolated counterexample is known, consisting of three non-congruent origin-symmetric convex lattice sets, each with 11 points. See Figure 4 in the above paper. It is possible that the answer is affirmative when n = 2 for sets containing sufficiently many points.\n\n#8 Geometric Problems on Sections of Convex Bod-ies\n\nProposed by Richard Gardner\n\nLet G (n, k ) denote the set of k-dimensional subspaces of Rn, where\n\n2 ≤ k ≤ n−1 is fixed in the following problems. If K is a set let int K denote the interior of K. Also let o denote the origin in Rn and if S ∈ G (n, k ), let\n\nK|S denote the projection of K on S.", "original_statement": "1. Find the central cross section u⊥:= {x ∈ Rn: x · u = 0 } of Bn \n\n> ∞\n\n=\n\n{x ∈ Rn: |xi| ≤ 1} with the largest perimeter. 2. What is the central cross section of Bn \n\n> ∞\n\nwith smallest perimeter? 3. What if instead of perimeter we take the mean width or intrinsic volume? 4. What about other codimensions? 5. What about other measures such as gaussian measures? 6. What about perimeter of projections? \n\n#2 Bounded Projection Inequality \n\nProposed by Mathieu Meyer \n\nLet Knos be the collection of all origin symmetric convex bodies in Rn.Find an asymptotic bound for the quantity \n\nmax \n\n> K∈Knos\n\nmin \n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥), (1) where Pv,u ⊥ (K) is the projection of K on u⊥ in the direction v ∈ Sn−1.One should note that 2n is a rough upper bound for (1) because for every hyperplane H in Rn, there is a projection PH so that ‖PH ‖ ≤ 2.This would imply PH K ⊆ 2( K ∩ H) and hence ‖PH K‖ ≤ 2n−1.One should also note that \n\nmin \n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥) (2) is affine invariant. If K is an ellipsoid then the quantity in (2) is 1 so (1) is always greater than or equal to 1.\n\n#3 Preservation of Convexity Under the Intersec-tion Body Operator \n\nProposed by Maria Alfonseca \n\n2Suppose K is an origin-symmetric starbody in Rn. let ρK (u):= max {a:\n\nau ∈ K} and define the intersection body IK of K as the body defined by \n\nρIK (u) = Vol n−1(K ∩ u⊥) ∀u ∈ Sn−1. (3) Define the convex kernel of K as the intersection of all convex bodies M\n\nsuch that m ⊆ K. What conditions on the convex kernel of K would imply \n\nIK is convex? \n\n#4 Cotype and Volume Ration of `np ⊗π `nq ⊗π `nr\n\nProposed by Carsten Schütt \n\nFind the cotype and volume ration of `np ⊗π `nq ⊗π `nr.\n\n#5 Banach-Mazur Distance Problem \n\nProposed by Mark Rudelson \n\nLet Knos denote the space of origin-symmetric, convex bodies in Rn and let dBM is the Banach-Mazur distance. Let α ∈ (0, 1) and n ∈ N. Suppose K, L ∈ Knos are such that there exists b > 0 so \n\ndBM (K ∩ E, L ∩ E) ≤ b ∀E ∈ Kαn os. (4) Does there exists a function f so that \n\ndBM (K, L ) ≤ f (b, α )? (5) It's worth noting that M. Rudelson showed during one of the problem sessions that this question is trivially true if dBM is replaced with the geometric distance dg.\n\n#6 Injective Operators which Correspond with the Intersection Body Operator \n\nProposed by Richard Gardner \n\nLet Kno denote the collection of convex bodies (i.e., compact convex sets with nonempty interior) in Rn and let Knos denote the origin-symmetric members of Kno. Is there an F: Kno → Knos such that F is continuous, affine invariant, injective (modulo translations and reflections in the origin), and \n\nF |Kn = I, where I is the intersection body operator. 37 Discrete Tomography \n\nProposed by Richard Gardner \n\n(Discrete Aleksandrov projection theorem.) If two n-dimensional cen-trally symmetric convex lattice sets in Zn are such that the cardinality of their projections on any lattice hyperplane is the same, are they equal up to translation? This question appears with the restriction n ≥ 3 in [GGZ]. For n = 2, the answer is negative, but only an isolated counterexample is known, consisting of three non-congruent origin-symmetric convex lattice sets, each with 11 points. See Figure 4 in the above paper. It is possible that the answer is affirmative when n = 2 for sets containing sufficiently many points. \n\n#8 Geometric Problems on Sections of Convex Bod-ies \n\nProposed by Richard Gardner \n\nLet G (n, k ) denote the set of k-dimensional subspaces of Rn, where \n\n2 ≤ k ≤ n−1 is fixed in the following problems. If K is a set let int K denote the interior of K. Also let o denote the origin in Rn and if S ∈ G (n, k ), let \n\nK|S denote the projection of K on S.", "clean_statement": "1. Find the central cross section u⊥:= {x ∈ Rn: x · u = 0 } of Bn\n\n> ∞\n\n=\n\n{x ∈ Rn: |xi| ≤ 1} with the largest perimeter. 2. What is the central cross section of Bn\n\n> ∞\n\nwith smallest perimeter? 3. What if instead of perimeter we take the mean width or intrinsic volume? 4. What about other codimensions? 5. What about other measures such as gaussian measures? 6. What about perimeter of projections?\n\n#2 Bounded Projection Inequality\n\nProposed by Mathieu Meyer\n\nLet Knos be the collection of all origin symmetric convex bodies in Rn.Find an asymptotic bound for the quantity\n\nmax\n\n> K∈Knos\n\nmin\n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥), (1) where Pv,u ⊥ (K) is the projection of K on u⊥ in the direction v ∈ Sn−1.One should note that 2n is a rough upper bound for (1) because for every hyperplane H in Rn, there is a projection PH so that ‖PH ‖ ≤ 2.This would imply PH K ⊆ 2( K ∩ H) and hence ‖PH K‖ ≤ 2n−1.One should also note that\n\nmin\n\n> u,v ∈Sn−1\n\nPv,u ⊥ (K)\n\nvol n−1(K ∩ u⊥) (2) is affine invariant. If K is an ellipsoid then the quantity in (2) is 1 so (1) is always greater than or equal to 1.\n\n#3 Preservation of Convexity Under the Intersec-tion Body Operator\n\nProposed by Maria Alfonseca\n\n2Suppose K is an origin-symmetric starbody in Rn. let ρK (u):= max {a:\n\nau ∈ K} and define the intersection body IK of K as the body defined by\n\nρIK (u) = Vol n−1(K ∩ u⊥) ∀u ∈ Sn−1. (3) Define the convex kernel of K as the intersection of all convex bodies M\n\nsuch that m ⊆ K. What conditions on the convex kernel of K would imply\n\nIK is convex?\n\n#4 Cotype and Volume Ration of `np ⊗π `nq ⊗π `nr\n\nProposed by Carsten Schütt\n\nFind the cotype and volume ration of `np ⊗π `nq ⊗π `nr.\n\n#5 Banach-Mazur Distance Problem\n\nProposed by Mark Rudelson\n\nLet Knos denote the space of origin-symmetric, convex bodies in Rn and let dBM is the Banach-Mazur distance. Let α ∈ (0, 1) and n ∈ N. Suppose K, L ∈ Knos are such that there exists b > 0 so\n\ndBM (K ∩ E, L ∩ E) ≤ b ∀E ∈ Kαn os. (4) Does there exists a function f so that\n\ndBM (K, L ) ≤ f (b, α )? (5) It's worth noting that M. Rudelson showed during one of the problem sessions that this question is trivially true if dBM is replaced with the geometric distance dg.\n\n#6 Injective Operators which Correspond with the Intersection Body Operator\n\nProposed by Richard Gardner\n\nLet Kno denote the collection of convex bodies (i.e., compact convex sets with nonempty interior) in Rn and let Knos denote the origin-symmetric members of Kno. Is there an F: Kno → Knos such that F is continuous, affine invariant, injective (modulo translations and reflections in the origin), and\n\nF |Kn = I, where I is the intersection body operator. 37 Discrete Tomography\n\nProposed by Richard Gardner\n\n(Discrete Aleksandrov projection theorem.) If two n-dimensional cen-trally symmetric convex lattice sets in Zn are such that the cardinality of their projections on any lattice hyperplane is the same, are they equal up to translation? This question appears with the restriction n ≥ 3 in [GGZ]. For n = 2, the answer is negative, but only an isolated counterexample is known, consisting of three non-congruent origin-symmetric convex lattice sets, each with 11 points. See Figure 4 in the above paper. It is possible that the answer is affirmative when n = 2 for sets containing sufficiently many points.\n\n#8 Geometric Problems on Sections of Convex Bod-ies\n\nProposed by Richard Gardner\n\nLet G (n, k ) denote the set of k-dimensional subspaces of Rn, where\n\n2 ≤ k ≤ n−1 is fixed in the following problems. If K is a set let int K denote the interior of K. Also let o denote the origin in Rn and if S ∈ G (n, k ), let\n\nK|S denote the projection of K on S.", "statement_status": "exact", "statement_verification": "The canonical JSON extraction runs past the end of Problem 1 and includes later problems from the same PDF. The official AIM source was therefore checked directly. On page 2 it gives the title **“Max/min Perimeter of central cross-sections,”** proposed by Hermann König, and asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Sections of convex bodies\nSection: \nSource item: 1\nSource URL: https://aimath.org/pastworkshops/sectionsconvexproblems.pdf\nCanonical location: aim-convex-geometry-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Find the central cross section u⊥:= {x ∈ Rn: x · u = 0 } of Bn \\n\\n> ∞\\n\\n=\\n\\n{x ∈ Rn: |xi| ≤ 1} with the largest perimeter. 2. What is the central cross section of Bn \\n\\n> ∞\\n\\nwith smallest perimeter? 3. What if instead of perimeter we take the mean width or intrinsic volume? 4. What about other codimensions? 5. What about other measures such as gaussian measures? 6. What about perimeter of projections? \\n\\n#2 Bounded Projection Inequality \\n\\nProposed by Mathieu Meyer \\n\\nLet Knos be the collection of all origin symmetric convex bodies in Rn.Find an asymptotic bound for the quantity \\n\\nmax \\n\\n> K∈Knos\\n\\nmin \\n\\n> u,v ∈Sn−1\\n\\nPv,u ⊥ (K)\\n\\nvol n−1(K ∩ u⊥), (1) where Pv,u ⊥ (K) is the projection of K on u⊥ in the direction v ∈ Sn−1.One should note that 2n is a rough upper bound for (1) because for every hyperplane H in Rn, there is a projection PH so that ‖PH ‖ ≤ 2.This would imply PH K ⊆ 2( K ∩ H) and hence ‖PH K‖ ≤ 2n−1.One should also note that \\n\\nmin \\n\\n> u,v ∈Sn−1\\n\\nPv,u ⊥ (K)\\n\\nvol n−1(K ∩ u⊥) (2) is affine invariant. If K is an ellipsoid then the quantity in (2) is 1 so (1) is always greater than or equal to 1.\\n\\n#3 Preservation of Convexity Under the Intersec-tion Body Operator \\n\\nProposed by Maria Alfonseca \\n\\n2Suppose K is an origin-symmetric starbody in Rn. let ρK (u):= max {a:\\n\\nau ∈ K} and define the intersection body IK of K as the body defined by \\n\\nρIK (u) = Vol n−1(K ∩ u⊥) ∀u ∈ Sn−1. (3) Define the convex kernel of K as the intersection of all convex bodies M\\n\\nsuch that m ⊆ K. What conditions on the convex kernel of K would imply \\n\\nIK is convex? \\n\\n#4 Cotype and Volume Ration of `np ⊗π `nq ⊗π `nr\\n\\nProposed by Carsten Schütt \\n\\nFind the cotype and volume ration of `np ⊗π `nq ⊗π `nr.\\n\\n#5 Banach-Mazur Distance Problem \\n\\nProposed by Mark Rudelson \\n\\nLet Knos denote the space of origin-symmetric, convex bodies in Rn and let dBM is the Banach-Mazur distance. Let α ∈ (0, 1) and n ∈ N. Suppose K, L ∈ Knos are such that there exists b > 0 so \\n\\ndBM (K ∩ E, L ∩ E) ≤ b ∀E ∈ Kαn os. (4) Does there exists a function f so that \\n\\ndBM (K, L ) ≤ f (b, α )? (5) It's worth noting that M. Rudelson showed during one of the problem sessions that this question is trivially true if dBM is replaced with the geometric distance dg.\\n\\n#6 Injective Operators which Correspond with the Intersection Body Operator \\n\\nProposed by Richard Gardner \\n\\nLet Kno denote the collection of convex bodies (i.e., compact convex sets with nonempty interior) in Rn and let Knos denote the origin-symmetric members of Kno. Is there an F: Kno → Knos such that F is continuous, affine invariant, injective (modulo translations and reflections in the origin), and \\n\\nF |Kn = I, where I is the intersection body operator. 37 Discrete Tomography \\n\\nProposed by Richard Gardner \\n\\n(Discrete Aleksandrov projection theorem.) If two n-dimensional cen-trally symmetric convex lattice sets in Zn are such that the cardinality of their projections on any lattice hyperplane is the same, are they equal up to translation? This question appears with the restriction n ≥ 3 in [GGZ]. For n = 2, the answer is negative, but only an isolated counterexample is known, consisting of three non-congruent origin-symmetric convex lattice sets, each with 11 points. See Figure 4 in the above paper. It is possible that the answer is affirmative when n = 2 for sets containing sufficiently many points. \\n\\n#8 Geometric Problems on Sections of Convex Bod-ies \\n\\nProposed by Richard Gardner \\n\\nLet G (n, k ) denote the set of k-dimensional subspaces of Rn, where \\n\\n2 ≤ k ≤ n−1 is fixed in the following problems. If K is a set let int K denote the interior of K. Also let o denote the origin in Rn and if S ∈ G (n, k ), let \\n\\nK|S denote the projection of K on S.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/sectionsconvexproblems.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0001", "aim-domain:convex-geometry", "aim-workshop:sectionsconvexproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The global maximum-perimeter section is the known König–Koldobsky balanced two-coordinate section, with perimeter 2^{n-1}(1+(n-2)sqrt(2)) for [-1,1]^n. Beyond this status result, the attempt proves a simultaneous explicit ledger for every intrinsic volume, mean width, induced Gaussian measure, and projection perimeter for all two-coordinate normals; gives an exact equal three-coordinate perimeter benchmark; and combines Vaaler's theorem with the Euclidean isoperimetric inequality to obtain a universal dimensionally exact but nonsharp perimeter lower bound. The coordinate-section global minimum remains an open natural conjecture in the literature checked.\n\nCandidate contribution (explicit_family; novelty confidence low): For u=(a,b,0,...,0), a>=b>=0 and a^2+b^2=1, the section and projection intrinsic-volume generating polynomials are respectively (1+2t/a)(1+2t)^{n-2} and (1+2(a+b)t)(1+2t)^{n-2}; these yield simultaneous exact monotonicity formulas for all intrinsic volumes, mean width, perimeter, one induced-Gaussian-volume interpretation, and projection perimeter, supplemented by the exact three-sparse perimeter 2^{n-3}(6sqrt(2)+3sqrt(3)(n-3)) and a hybrid universal lower bound." }, { "id": 20001270, "problem_number": "AIM-CONVEX_GEOMETRY-0002", "title": "Affinely equivalent central sections and an ellipsoidal anchor", "statement": "1. (See [G,\nProblem 7.4 and Note 7.2].) Suppose o ∈ int K and all sections K ∩ S, S ∈ G (n, k ), are affinely equivalent. Does it follow that K is an ellipsoid? Here K is a star body but the problem is also open when K is convex.", "original_statement": "1. (See [G, \nProblem 7.4 and Note 7.2].) Suppose o ∈ int K and all sections K ∩ S, S ∈ G (n, k ), are affinely equivalent. Does it follow that K is an ellipsoid? Here K is a star body but the problem is also open when K is convex.", "clean_statement": "1. (See [G,\nProblem 7.4 and Note 7.2].) Suppose o ∈ int K and all sections K ∩ S, S ∈ G (n, k ), are affinely equivalent. Does it follow that K is an ellipsoid? Here K is a star body but the problem is also open when K is convex.", "statement_status": "exact", "statement_verification": "The problem is Problem 1 in Section 8, “Geometric Problems on Sections of Convex Bodies,” of the AIM list *Sections of Convex Bodies* (August 2013). The section's introductory paragraph is part of the statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Sections of convex bodies\nSection: \nSource item: 1\nSource URL: https://aimath.org/pastworkshops/sectionsconvexproblems.pdf\nCanonical location: aim-convex-geometry-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (See [G, \\nProblem 7.4 and Note 7.2].) Suppose o ∈ int K and all sections K ∩ S, S ∈ G (n, k ), are affinely equivalent. Does it follow that K is an ellipsoid? Here K is a star body but the problem is also open when K is convex.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/sectionsconvexproblems.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0002", "aim-domain:convex-geometry", "aim-workshop:sectionsconvexproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The original nonsymmetric star-body problem remains open, although many symmetric convex Banach-space cases are settled and a 2025-2026 preprint claims the remaining finite-dimensional cases without yet supplying a verified proof. A proved partial theorem shows that an origin-symmetric star body is an ellipsoid if it has a 2-covering family of ellipsoidal k-sections. Consequently, if all central k-sections are affinely equivalent and one section is an ellipsoid, then the whole star body is an origin-centered ellipsoid; boundedness makes section centers unique, so all affine equivalences automatically fix the origin and become linear.\n\nCandidate contribution (theorem; novelty confidence low): Let K be an origin-symmetric star body in R^n and 2 <= k <= n-1. If there is a family of ellipsoidal central k-sections such that every two-plane is contained in a member of the family, then K is an ellipsoid. In particular, mutual affine equivalence of all k-sections plus one ellipsoidal section forces K to be an ellipsoid." }, { "id": 20001271, "problem_number": "AIM-CONVEX_GEOMETRY-0003", "title": "Affinely equivalent projections and an ellipsoidal 2-cover criterion", "statement": "2. (See [G,\nProblem 3.3 and Note 3.2].) Same problem as in (1) but for projections, i.e. if all projections K|S, S ∈ G (n, k ), are affinely equivalent, is K is an ellipsoid? Here K is convex.", "original_statement": "2. (See [G, \nProblem 3.3 and Note 3.2].) Same problem as in (1) but for projections, i.e. if all projections K|S, S ∈ G (n, k ), are affinely equivalent, is K is an ellipsoid? Here K is convex.", "clean_statement": "2. (See [G,\nProblem 3.3 and Note 3.2].) Same problem as in (1) but for projections, i.e. if all projections K|S, S ∈ G (n, k ), are affinely equivalent, is K is an ellipsoid? Here K is convex.", "statement_status": "exact", "statement_verification": "The official 2013 AIM problem list first fixes the conventions", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Sections of convex bodies\nSection: \nSource item: 2\nSource URL: https://aimath.org/pastworkshops/sectionsconvexproblems.pdf\nCanonical location: aim-convex-geometry-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (See [G, \\nProblem 3.3 and Note 3.2].) Same problem as in (1) but for projections, i.e. if all projections K|S, S ∈ G (n, k ), are affinely equivalent, is K is an ellipsoid? Here K is convex.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/sectionsconvexproblems.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0003", "aim-domain:convex-geometry", "aim-workshop:sectionsconvexproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The original projection problem is known affirmatively for projection dimensions k congruent to 0, 1, or 2 modulo 4, with possible exception k=133, while the recorded k congruent to 3 modulo 4 and k=133 cases remain open. A proved geometric endgame is supplied: if a family of subspaces contains a member above every 2-plane and the projection of K onto every member is an ellipsoid, then K is an ellipsoid, without assuming K is centrally symmetric. Consequently, under the AIM affine-equivalence hypothesis, one ellipsoidal projection forces K itself to be an ellipsoid. The proof glues projected affine centers using the odd support function and then applies the parallelogram law to the squared centered support function.\n\nCandidate contribution (theorem; novelty confidence low): Candidate novelty: a convex body K is an ellipsoid whenever its ellipsoidal projections occur on any 2-covering family of subspaces, and this holds without assumed central symmetry because the odd part of h_K forces all planar projection centers to be projections of one global center." }, { "id": 20001272, "problem_number": "AIM-CONVEX_GEOMETRY-0004", "title": "Canonical homothety fields and rigidity of centered sections", "statement": "3. (See [G,\nProblem 7.1 and Note 7.1].) Suppose K and L are star bodies and that K ∩ S is homethetic to L ∩ S for every G (n, k ). Does it follow that K is homothetic to L? The answer is affirmative if K\n\nand L are convex and o ∈ int K ∩ int L; see [G, Theorem 7.1.1].", "original_statement": "3. (See [G, \nProblem 7.1 and Note 7.1].) Suppose K and L are star bodies and that K ∩ S is homethetic to L ∩ S for every G (n, k ). Does it follow that K is homothetic to L? The answer is affirmative if K\n\nand L are convex and o ∈ int K ∩ int L; see [G, Theorem 7.1.1].", "clean_statement": "3. (See [G,\nProblem 7.1 and Note 7.1].) Suppose K and L are star bodies and that K ∩ S is homethetic to L ∩ S for every G (n, k ). Does it follow that K is homothetic to L? The answer is affirmative if K\n\nand L are convex and o ∈ int K ∩ int L; see [G, Theorem 7.1.1].", "statement_status": "exact", "statement_verification": "The source is Richard Gardner's problem list for the 2013 AIM workshop *Sections of Convex Bodies*. The introductory paragraph fixes an integer \\[ 2\\leq k\\leq n-1 \\] and writes \\(G(n,k)\\) for the Grassmannian of \\(k\\)-dimensional linear subspaces of \\(\\mathbb R^n\\). The extracted record has a grammatical omission: “for every \\(G(n,k)\\)” must read “for every \\(S\\in G(n,k)\\).” The official PDF confirms the surrounding notation and this reconstruction.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Sections of convex bodies\nSection: \nSource item: 3\nSource URL: https://aimath.org/pastworkshops/sectionsconvexproblems.pdf\nCanonical location: aim-convex-geometry-notes.json notes[3]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (See [G, \\nProblem 7.1 and Note 7.1].) Suppose K and L are star bodies and that K ∩ S is homethetic to L ∩ S for every G (n, k ). Does it follow that K is homothetic to L? The answer is affirmative if K\\n\\nand L are convex and o ∈ int K ∩ int L; see [G, Theorem 7.1.1].\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/sectionsconvexproblems.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0004", "aim-domain:convex-geometry", "aim-workshop:sectionsconvexproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For direct homotheties K∩S=a(S)+lambda(S)(L∩S) of corresponding k-dimensional central sections of star bodies, the scale and translation are uniquely recovered from section volumes and centroids and vary continuously on the Grassmannian. If the canonical translation field vanishes, then the radial quotient is globally constant, so K is a positive dilate of L. This proves the problem for centroid-zero sections and, in particular, for arbitrary origin-symmetric star bodies, without convexity.\n\nCandidate contribution (lemma; novelty confidence low): Direct section homotheties of star bodies have canonical continuous parameters lambda(S)=[V_k(K∩S)/V_k(L∩S)]^(1/k) and a(S)=g(K∩S)-lambda(S)g(L∩S); vanishing of a(S) for every S forces one global dilation, while translated homotheties cannot generally be patched by restriction to plane intersections." }, { "id": 20001273, "problem_number": "AIM-CONVEX_GEOMETRY-0005", "title": "Refutation by congruent hyperplane sections and rigidity for off-center balls", "statement": "Let \\(G(n,k)\\) be the Grassmannian of \\(k\\)-dimensional linear subspaces of\n\\(\\mathbb R^n\\), where \\(2\\leq k\\leq n-1\\) is fixed. Suppose \\(K\\) and\n\\(L\\) are star bodies and, for every \\(S\\in G(n,k)\\), the sections\n\\(K\\cap S\\) and \\(L\\cap S\\) are congruent. Must \\(K=\\pm L\\)?\n\nThe source adds that the answer is affirmative for \\(k=2\\) when\n“congruent” is replaced by “a rotation of,” citing [R].", "original_statement": "4. (See [G, \nProblem 7.3 and Note 7.1].) Suppose K and L are star bodies and that K ∩ S is congruent to L ∩ S for every G (n, k ). Does it follow that K = ±L? It has been shown that the answer is affirmative for \n\nk = 2 when \"congruent to\" is replaced by \"a rotation of\". For more on this see [R]. \n4", "clean_statement": "Let \\(G(n,k)\\) be the Grassmannian of \\(k\\)-dimensional linear subspaces of\n\\(\\mathbb R^n\\), where \\(2\\leq k\\leq n-1\\) is fixed. Suppose \\(K\\) and\n\\(L\\) are star bodies and, for every \\(S\\in G(n,k)\\), the sections\n\\(K\\cap S\\) and \\(L\\cap S\\) are congruent. Must \\(K=\\pm L\\)?\n\nThe source adds that the answer is affirmative for \\(k=2\\) when\n“congruent” is replaced by “a rotation of,” citing [R].", "statement_status": "corrected_verified", "statement_verification": "The exact source record in `input.json` is preserved, including two extraction defects. The notation and dimension range are inherited from the beginning of Section 8 of the official AIM PDF, and the final isolated `4` is the printed page number captured by OCR. The recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Sections of convex bodies\nSection: \nSource item: 4\nSource URL: https://aimath.org/pastworkshops/sectionsconvexproblems.pdf\nCanonical location: aim-convex-geometry-notes.json notes[4]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"4. (See [G, \\nProblem 7.3 and Note 7.1].) Suppose K and L are star bodies and that K ∩ S is congruent to L ∩ S for every G (n, k ). Does it follow that K = ±L? It has been shown that the answer is affirmative for \\n\\nk = 2 when \\\"congruent to\\\" is replaced by \\\"a rotation of\\\". For more on this see [R]. \\n4\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/sectionsconvexproblems.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0005", "aim-domain:convex-geometry", "aim-workshop:sectionsconvexproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The inherited quantifier is every S in G(n,k), with fixed 2<=k<=n-1, and congruence permits an arbitrary section-wise orthogonal map followed by a translation. Ning Zhang's 2018 convex-body construction refutes the unrestricted AIM question: for every n>=3 there are K not equal to L or -L whose central hyperplane sections are congruent by origin-fixing orthogonal maps; for n>=4 even direct rotations suffice. As a new positive special case, this attempt proves that two possibly off-center Euclidean balls containing the origin are determined up to ambient antipodal reflection by congruence of all k-dimensional central sections, for every allowed k and with no coherence or origin-fixing assumption on the section motions.\n\nCandidate contribution (special_case; novelty confidence low): If K=B(c,R) and L=B(d,Q) in R^n, n>=3, with |c|=3 there are convex bodies not equal up to translation and reflection in the origin whose projections onto every hyperplane are congruent by origin-fixing orthogonal maps. As a candidate positive special case, this attempt proves that if congruent k-projections are accompanied by equality DK=DL of the difference bodies and every k-projection of the common difference body has orthogonal symmetry group exactly {I,-I}, then K is a translate of L or -L. The proof turns each local orthogonal part into a sign and applies Myroshnychenko's established sign-and-linear-term gluing theorem to the support functions.\n\nCandidate contribution (special_case_and_obstruction; novelty confidence low): Let 2<=k<=n-1. If K|S and L|S are congruent for every S in G(n,k), DK=DL=:D, and Sym_S(D|S)={I_S,-I_S} for every S, then K is a translate of L or -L. Equivalently, under the explicit assumption DK=DL, any counterexample must have at least one projected difference body with an additional orthogonal symmetry." }, { "id": 20001275, "problem_number": "AIM-CONVEX_GEOMETRY-0007", "title": "The sharp symmetric 2n-vertex case of the Euclidean-ball vertex-index conjecture", "statement": "1. Prove that the vertex index of the n-dimensional Euclidean ball Bn\n\n> 2\n\nis\n\n2n3/2, where the vertex index of a centrally-symmetric convex body\n\nK = −K ⊂ Rn is defined as\n\nvein( K) = inf\n\n> {∑\n> i\n\n‖pi‖K | K ⊂ conv {pi}\n\n> }.", "original_statement": "1. Prove that the vertex index of the n-dimensional Euclidean ball Bn \n\n> 2\n\nis \n\n2n3/2, where the vertex index of a centrally-symmetric convex body \n\nK = −K ⊂ Rn is defined as \n\nvein( K) = inf \n\n> {∑\n> i\n\n‖pi‖K | K ⊂ conv {pi}\n\n> }.", "clean_statement": "1. Prove that the vertex index of the n-dimensional Euclidean ball Bn\n\n> 2\n\nis\n\n2n3/2, where the vertex index of a centrally-symmetric convex body\n\nK = −K ⊂ Rn is defined as\n\nvein( K) = inf\n\n> {∑\n> i\n\n‖pi‖K | K ⊂ conv {pi}\n\n> }.", "statement_status": "exact", "statement_verification": "The canonical JSON is damaged by PDF extraction. The official AIM problem PDF, Section 9 (“Vertex Index Problems”), Problem 1, has the following notation and exponent:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Sections of convex bodies\nSection: \nSource item: 1\nSource URL: https://aimath.org/pastworkshops/sectionsconvexproblems.pdf\nCanonical location: aim-convex-geometry-notes.json notes[6]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Prove that the vertex index of the n-dimensional Euclidean ball Bn \\n\\n> 2\\n\\nis \\n\\n2n3/2, where the vertex index of a centrally-symmetric convex body \\n\\nK = −K ⊂ Rn is defined as \\n\\nvein( K) = inf \\n\\n> {∑\\n> i\\n\\n‖pi‖K | K ⊂ conv {pi}\\n\\n> }.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/sectionsconvexproblems.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0007", "aim-domain:convex-geometry", "aim-workshop:sectionsconvexproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If a centrally symmetric polytope with exactly 2n vertices contains the Euclidean unit ball in R^n, then the sum of the Euclidean norms of its vertices is at least 2n^(3/2). Equality holds exactly for rotated regular crosspolytopes with vertices plus or minus sqrt(n) times an orthonormal basis. The proof converts containment into an inverse operator-norm bound, averages over sign vectors to control the reciprocal singular values, and applies the nuclear norm. This does not control configurations with more vertices or nonsymmetric configurations.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: among all centrally symmetric enclosing polytopes with exactly n antipodal vertex pairs, the unique minimizers of total Euclidean vertex cost, up to orthogonal transformations and relabeling, are the regular crosspolytopes sqrt(n) Q B_1^n, and the minimum cost is 2n^(3/2)." }, { "id": 20001276, "problem_number": "AIM-CONVEX_GEOMETRY-0008", "title": "Three-dimensional vertex index and the choice of center", "statement": "2. Find the best possible upper bound for the vertex index of 3-dimensional centrally-symmetric convex body. 3. Provide an upper bound for the vertex index of a convex body K ⊂\n\nRn\n\nvein( K) = inf\n\n> {∑\n> i\n\n‖pi‖K−a | a ∈ K, K − a ⊂ conv {pi}\n\n> }.\n\n#10 Reconstruction of Polytopes with Few Facets (or Few Vertices)\n\nThese questions are motivated by the problem of efficiently estimating poly-topes with few facets or few vertices from random points.", "original_statement": "2. Find the best possible upper bound for the vertex index of 3-dimensional centrally-symmetric convex body. 3. Provide an upper bound for the vertex index of a convex body K ⊂\n\nRn\n\nvein( K) = inf \n\n> {∑\n> i\n\n‖pi‖K−a | a ∈ K, K − a ⊂ conv {pi}\n\n> }.\n\n#10 Reconstruction of Polytopes with Few Facets (or Few Vertices) \n\nThese questions are motivated by the problem of efficiently estimating poly-topes with few facets or few vertices from random points.", "clean_statement": "2. Find the best possible upper bound for the vertex index of 3-dimensional centrally-symmetric convex body. 3. Provide an upper bound for the vertex index of a convex body K ⊂\n\nRn\n\nvein( K) = inf\n\n> {∑\n> i\n\n‖pi‖K−a | a ∈ K, K − a ⊂ conv {pi}\n\n> }.\n\n#10 Reconstruction of Polytopes with Few Facets (or Few Vertices)\n\nThese questions are motivated by the problem of efficiently estimating poly-topes with few facets or few vertices from random points.", "statement_status": "exact", "statement_verification": "The canonical extraction merges two consecutive problems from Section 9 of the official 2013 AIM list and then spills into Section 10. The official PDF gives the following partition.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Sections of convex bodies\nSection: \nSource item: 2\nSource URL: https://aimath.org/pastworkshops/sectionsconvexproblems.pdf\nCanonical location: aim-convex-geometry-notes.json notes[7]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. Find the best possible upper bound for the vertex index of 3-dimensional centrally-symmetric convex body. 3. Provide an upper bound for the vertex index of a convex body K ⊂\\n\\nRn\\n\\nvein( K) = inf \\n\\n> {∑\\n> i\\n\\n‖pi‖K−a | a ∈ K, K − a ⊂ conv {pi}\\n\\n> }.\\n\\n#10 Reconstruction of Polytopes with Few Facets (or Few Vertices) \\n\\nThese questions are motivated by the problem of efficiently estimating poly-topes with few facets or few vertices from random points.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/sectionsconvexproblems.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0008", "aim-domain:convex-geometry", "aim-workshop:sectionsconvexproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The three-dimensional symmetric problem remains open with the currently justified bounds 6 sqrt(3) <= M_3 <= 18 and conjectured upper bound 12. For the standard nonsymmetric definition using interior centers, the attempt proves affine invariance, attainment for every fixed center with a finite cardinality reduction, the universal sharp lower bound n+1, and exact value n+1 for every simplex and every interior center. For the PDF-faithful boundary quantifier, an extended-gauge simplex construction has cost n, showing that allowing boundary centers substantively changes the invariant.\n\nCandidate contribution (structural lemma and diagnostic example; novelty confidence low): For every convex body and fixed interior center, the vertex-index infimum is attained after reduction to finitely many outer-polytope vertices; every interior-centered n-simplex has exact index n+1, whereas the literal boundary-allowed extended-gauge reading has an explicit simplex configuration of cost n." }, { "id": 20001277, "problem_number": "AIM-CONVEX_GEOMETRY-0009", "title": "A necessary moment order and a scale obstruction", "statement": "1. (Identifiability from moments) Let the k-th moment tensor of a con-vex body be the k-th moment tensor of the uniform distribution on it. For any positive integer t, is there k = k(t) (independent of d) so that the following map is injective: The map that takes a d-dimensional polytope with dt facets and maps it to its first k moment tensors. Known: 5(a) [GLPR] Can recover a d-dimensional polytope with n vertices from O(dn ) moments along d random directions. (b) [FJK] Can estimate a parallelepiped efficiently from first 4 mo-ment tensors. (c) [AGR] Can estimate any simplex efficiently from first 3 moment tensors. In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n whose input are the first k moment tensors of any given polytope P as before, and that outputs a polytope within 1/10 Hausdorff distance of P? What if the tensors are not known exactly but come from a sample approximation from a sample of size polynomial in n for any fixed t?", "original_statement": "1. (Identifiability from moments) Let the k-th moment tensor of a con-vex body be the k-th moment tensor of the uniform distribution on it. For any positive integer t, is there k = k(t) (independent of d) so that the following map is injective: The map that takes a d-dimensional polytope with dt facets and maps it to its first k moment tensors. Known: 5(a) [GLPR] Can recover a d-dimensional polytope with n vertices from O(dn ) moments along d random directions. (b) [FJK] Can estimate a parallelepiped efficiently from first 4 mo-ment tensors. (c) [AGR] Can estimate any simplex efficiently from first 3 moment tensors. In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n whose input are the first k moment tensors of any given polytope P as before, and that outputs a polytope within 1/10 Hausdorff distance of P? What if the tensors are not known exactly but come from a sample approximation from a sample of size polynomial in n for any fixed t?", "clean_statement": "1. (Identifiability from moments) Let the k-th moment tensor of a con-vex body be the k-th moment tensor of the uniform distribution on it. For any positive integer t, is there k = k(t) (independent of d) so that the following map is injective: The map that takes a d-dimensional polytope with dt facets and maps it to its first k moment tensors. Known: 5(a) [GLPR] Can recover a d-dimensional polytope with n vertices from O(dn ) moments along d random directions. (b) [FJK] Can estimate a parallelepiped efficiently from first 4 mo-ment tensors. (c) [AGR] Can estimate any simplex efficiently from first 3 moment tensors. In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n whose input are the first k moment tensors of any given polytope P as before, and that outputs a polytope within 1/10 Hausdorff distance of P? What if the tensors are not known exactly but come from a sample approximation from a sample of size polynomial in n for any fixed t?", "statement_status": "exact", "statement_verification": "The canonical record has lost superscript formatting and contains a variable inconsistency. The official AIM PDF, Section 10 (“Reconstruction of Polytopes with Few Facets (or Few Vertices)”), Problem 1, asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Sections of convex bodies\nSection: \nSource item: 1\nSource URL: https://aimath.org/pastworkshops/sectionsconvexproblems.pdf\nCanonical location: aim-convex-geometry-notes.json notes[8]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (Identifiability from moments) Let the k-th moment tensor of a con-vex body be the k-th moment tensor of the uniform distribution on it. For any positive integer t, is there k = k(t) (independent of d) so that the following map is injective: The map that takes a d-dimensional polytope with dt facets and maps it to its first k moment tensors. Known: 5(a) [GLPR] Can recover a d-dimensional polytope with n vertices from O(dn ) moments along d random directions. (b) [FJK] Can estimate a parallelepiped efficiently from first 4 mo-ment tensors. (c) [AGR] Can estimate any simplex efficiently from first 3 moment tensors. In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n whose input are the first k moment tensors of any given polytope P as before, and that outputs a polytope within 1/10 Hausdorff distance of P? What if the tensors are not known exactly but come from a sample approximation from a sample of size polynomial in n for any fixed t?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/sectionsconvexproblems.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0009", "aim-domain:convex-geometry", "aim-workshop:sectionsconvexproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If the first k normalized moment tensors are injective on all full-dimensional d-polytopes with exactly m facets, then binom(d+k,k)-1 must be at least md. This follows from an md-dimensional canonically labeled chart of simple polytopes and invariance of domain. Consequently, for the AIM class with m=d^t and fixed t at least 2, every dimension-independent cutoff must satisfy k(t) at least t+2. Separately, nested planar rectangles prove that no sample bound depending only on dimension and facet count can guarantee absolute Hausdorff error 1/10 over the unnormalized class.\n\nCandidate contribution (theorem; novelty confidence low): The exact topological necessity binom(d+k,k)-1 >= md yields k(t) >= t+2, including failure at the leading-order boundary k=t+1; a Le Cam argument on nested rectangles also rules out the printed scale-free sampling guarantee." }, { "id": 20001278, "problem_number": "AIM-CONVEX_GEOMETRY-0010", "title": "Explicit conditioned stability for relative central sections", "statement": "2. (Stability of reconstruction from relative central sectional areas) Is it true that for any positive integer t there is a positive integer t′ = t′(t)\n\nsuch that for any two isotropic centrally symmetric d-dimensional polytopes P, Q with at most dt facets we have: If for all θ ∈ Sd−1\n\n∣∣∣∣∣\n\nVol n−1(P ∩ θ⊥)\n\nVol (P ) − Vol n−1(Q ∩ θ⊥)\n\nVol (Q)\n\n∣∣∣∣∣ ≤ 1\n\ndt′,\n\nthen the Hausdorff distance between P and Q is at most 1/10?In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n that when given access to the value of relative central sectional areas of any given polytope P as before, to within additive error 1/d t′, it outputs a polytope within 1/10 Hausdorff distance of P?\n\n#References\n\n> [AGR] Joseph Anderson, Navin Goyal, and Luis Rademacher, Efficient learning of simplices (2012). [FJK] Alan Frieze, Mark Jerrum, and Ravi Kannan, Learning linear transformations,Foundations of computer science, 1996. proceedings., 37th annual symposium on, 1996, pp. 359-368. [G] Richard J Gardner, Geometric tomography, Vol. 58, Cambridge University Press Cambridge, 1995. [GGZ] Richard J Gardner, Paolo Gronchi, and Chuanming Zong, Sums, projections, and sections of lattice sets, and the discrete covariogram, Discrete & Com-putational Geometry 34 (2005), no. 3, 391-409.\n\n6[GLPR] Nick Gravin, Jean Lasserre, Dmitrii V Pasechnik, and Sinai Robins, The inverse moment problem for convex polytopes, Discrete & Computational Geome-try 48 (2012), no. 3, 596-621. [R] Dmitry Ryabogin, On the continual rubik's cube, Advances in Mathematics 231\n\n(2012), no. 6, 3429-3444.\n\n7", "original_statement": "2. (Stability of reconstruction from relative central sectional areas) Is it true that for any positive integer t there is a positive integer t′ = t′(t)\n\nsuch that for any two isotropic centrally symmetric d-dimensional polytopes P, Q with at most dt facets we have: If for all θ ∈ Sd−1\n\n∣∣∣∣∣\n\nVol n−1(P ∩ θ⊥)\n\nVol (P ) − Vol n−1(Q ∩ θ⊥)\n\nVol (Q)\n\n∣∣∣∣∣ ≤ 1\n\ndt′,\n\nthen the Hausdorff distance between P and Q is at most 1/10?In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n that when given access to the value of relative central sectional areas of any given polytope P as before, to within additive error 1/d t′, it outputs a polytope within 1/10 Hausdorff distance of P?\n\n#References \n\n> [AGR] Joseph Anderson, Navin Goyal, and Luis Rademacher, Efficient learning of simplices (2012). [FJK] Alan Frieze, Mark Jerrum, and Ravi Kannan, Learning linear transformations,Foundations of computer science, 1996. proceedings., 37th annual symposium on, 1996, pp. 359-368. [G] Richard J Gardner, Geometric tomography, Vol. 58, Cambridge University Press Cambridge, 1995. [GGZ] Richard J Gardner, Paolo Gronchi, and Chuanming Zong, Sums, projections, and sections of lattice sets, and the discrete covariogram, Discrete & Com-putational Geometry 34 (2005), no. 3, 391-409.\n\n6[GLPR] Nick Gravin, Jean Lasserre, Dmitrii V Pasechnik, and Sinai Robins, The inverse moment problem for convex polytopes, Discrete & Computational Geome-try 48 (2012), no. 3, 596-621. [R] Dmitry Ryabogin, On the continual rubik's cube, Advances in Mathematics 231 \n\n(2012), no. 6, 3429-3444. \n\n7", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The source literally says “running in time polynomial in \\(n\\)” in the last paragraph, although Problem 2 defines only \\(d\\). This is almost certainly a notation carryover: Problem 1 immediately before it uses \\(d\\) for dimension and \\(n\\) for the number of vertices. For Problem 2, “polynomial in \\(d\\)” is the natural reading, but this reconstruction is explicitly marked as an interpretation rather than a correction to the source.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Sections of convex bodies\nSection: \nSource item: 2\nSource URL: https://aimath.org/pastworkshops/sectionsconvexproblems.pdf\nCanonical location: aim-convex-geometry-notes.json notes[9]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (Stability of reconstruction from relative central sectional areas) Is it true that for any positive integer t there is a positive integer t′ = t′(t)\\n\\nsuch that for any two isotropic centrally symmetric d-dimensional polytopes P, Q with at most dt facets we have: If for all θ ∈ Sd−1\\n\\n∣∣∣∣∣\\n\\nVol n−1(P ∩ θ⊥)\\n\\nVol (P ) − Vol n−1(Q ∩ θ⊥)\\n\\nVol (Q)\\n\\n∣∣∣∣∣ ≤ 1\\n\\ndt′,\\n\\nthen the Hausdorff distance between P and Q is at most 1/10?In the affirmative case, is there an efficient reconstruction algorithm? Namely, is there an algorithm running in time polynomial in n that when given access to the value of relative central sectional areas of any given polytope P as before, to within additive error 1/d t′, it outputs a polytope within 1/10 Hausdorff distance of P?\\n\\n#References \\n\\n> [AGR] Joseph Anderson, Navin Goyal, and Luis Rademacher, Efficient learning of simplices (2012). [FJK] Alan Frieze, Mark Jerrum, and Ravi Kannan, Learning linear transformations,Foundations of computer science, 1996. proceedings., 37th annual symposium on, 1996, pp. 359-368. [G] Richard J Gardner, Geometric tomography, Vol. 58, Cambridge University Press Cambridge, 1995. [GGZ] Richard J Gardner, Paolo Gronchi, and Chuanming Zong, Sums, projections, and sections of lattice sets, and the discrete covariogram, Discrete & Com-putational Geometry 34 (2005), no. 3, 391-409.\\n\\n6[GLPR] Nick Gravin, Jean Lasserre, Dmitrii V Pasechnik, and Sinai Robins, The inverse moment problem for convex polytopes, Discrete & Computational Geome-try 48 (2012), no. 3, 596-621. [R] Dmitry Ryabogin, On the continual rubik's cube, Advances in Mathematics 231 \\n\\n(2012), no. 6, 3429-3444. \\n\\n7\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/sectionsconvexproblems.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0010", "aim-domain:convex-geometry", "aim-workshop:sectionsconvexproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Exact relative central-section data determine an origin-symmetric star body, including its scale, through a normalized Funk transform and explicit volume/radius formulas. On differences of normalized radial powers supported in even harmonic degrees at most 2m, the inverse admits an explicit sup-norm stability constant; under explicit two-sided radial-power bounds this yields a Hausdorff estimate. A compactness argument also gives qualitative uniform stability in fixed dimension under fixed inner and outer radii. These results do not settle the dimension-uniform polynomial estimate for d^t-facet isotropic polytopes or the requested reconstruction algorithm.\n\nCandidate contribution (partial theorem and conditional obstruction; novelty confidence low): For origin-symmetric star bodies whose normalized radial-power difference is supported in even harmonic degrees through 2m, the report proves an explicit inverse-Funk sup-norm bound with constant (d-1)sqrt(N_{d,m})/(|S^{d-2}| Lambda_{d,m}), and converts it to an explicit Hausdorff bound when a <= q_K,q_L <= b; it also proves that the AIM statement fails under any scale-free covariance-scalar interpretation of isotropic, via scaled cubes." }, { "id": 20001279, "problem_number": "AIM-CONVEX_GEOMETRY-0011", "title": "Sections, projections, polarity, and an ellipsoid obstruction", "statement": "1. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then K ⊂ L?2. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then vol( K) 6 vol( L)? (True if n 6 4) 3. Is it true that for any convex (symmetric) body K there exists a direction u ∈ Sn−1\n\nsuch that vol(St u(K)∗) 6 Cvol( K∗)? 4. What about an average version of the previous question, i.e.:\n\n∫\n\n> Sn−1\n\nvol(St u(K)∗)du 6 Cvol( K∗)? 5. Is the average of P(K) for symmetric polytopes K greater than or equal to P(Bn\n\n> ∞\n\n)? More precisely, let K be a random symmetric polytope of a fixed number of vertices, where each vertex is independent Gaussian random vector is it true that EP(K) >\n\nP(Bn\n\n> ∞\n\n)? 6. Given even log-concave probability measure μ, estimate μ(K)μ(K∗) from above. (It is known that μ(K)μ(K∗) 6 μ(Bn\n\n> 2\n\n) if μ is unconditional) 7. In the above question, characterize the equality case and find lower bound for the un-conditional case. 18. Consider μ∗ in the sense of Artstein-Milman and answer to the same questions about\n\nμ(K)μ∗(K). 9. Let K and L be symmetric convex bodies in Rn and denote by N (K, L ) the covering number of K by L, the minimum number of the translates of L needed to cover K. Is it true that\n\nN (K, L ) 6 [N (L∗, cK ∗]δ,\n\nwhere c, δ are absolute constants? 10. Given random vectors X and Y uniformly distributed in K and K∗ respectively, look at 〈X, Y 〉, up to a \"100\"-moment. Are the first \"100\" moments the same (1 + ε) as those for Gaussian random vectors? 11. Prove Mahler's conjecture for finite-dimensional normed spaces that embed in Lp,\n\np < 1 and, in particular, for convex intersection bodies (which correspond to the case\n\np = −1). 12. What if 1 < p < 2? Is it true that P(BX ) > P(Bnp ) if X is embedded into Lp, 1 < p < 2and dimX = n?\n1", "original_statement": "1. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then K ⊂ L?2. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then vol( K) 6 vol( L)? (True if n 6 4) 3. Is it true that for any convex (symmetric) body K there exists a direction u ∈ Sn−1\n\nsuch that vol(St u(K)∗) 6 Cvol( K∗)? 4. What about an average version of the previous question, i.e.: \n\n∫\n\n> Sn−1\n\nvol(St u(K)∗)du 6 Cvol( K∗)? 5. Is the average of P(K) for symmetric polytopes K greater than or equal to P(Bn\n\n> ∞\n\n)? More precisely, let K be a random symmetric polytope of a fixed number of vertices, where each vertex is independent Gaussian random vector is it true that EP(K) >\n\nP(Bn\n\n> ∞\n\n)? 6. Given even log-concave probability measure μ, estimate μ(K)μ(K∗) from above. (It is known that μ(K)μ(K∗) 6 μ(Bn \n\n> 2\n\n) if μ is unconditional) 7. In the above question, characterize the equality case and find lower bound for the un-conditional case. 18. Consider μ∗ in the sense of Artstein-Milman and answer to the same questions about \n\nμ(K)μ∗(K). 9. Let K and L be symmetric convex bodies in Rn and denote by N (K, L ) the covering number of K by L, the minimum number of the translates of L needed to cover K. Is it true that \n\nN (K, L ) 6 [N (L∗, cK ∗]δ,\n\nwhere c, δ are absolute constants? 10. Given random vectors X and Y uniformly distributed in K and K∗ respectively, look at 〈X, Y 〉, up to a \"100\"-moment. Are the first \"100\" moments the same (1 + ε) as those for Gaussian random vectors? 11. Prove Mahler's conjecture for finite-dimensional normed spaces that embed in Lp,\n\np < 1 and, in particular, for convex intersection bodies (which correspond to the case \n\np = −1). 12. What if 1 < p < 2? Is it true that P(BX ) > P(Bnp ) if X is embedded into Lp, 1 < p < 2and dimX = n?\n1", "clean_statement": "1. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then K ⊂ L?2. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then vol( K) 6 vol( L)? (True if n 6 4) 3. Is it true that for any convex (symmetric) body K there exists a direction u ∈ Sn−1\n\nsuch that vol(St u(K)∗) 6 Cvol( K∗)? 4. What about an average version of the previous question, i.e.:\n\n∫\n\n> Sn−1\n\nvol(St u(K)∗)du 6 Cvol( K∗)? 5. Is the average of P(K) for symmetric polytopes K greater than or equal to P(Bn\n\n> ∞\n\n)? More precisely, let K be a random symmetric polytope of a fixed number of vertices, where each vertex is independent Gaussian random vector is it true that EP(K) >\n\nP(Bn\n\n> ∞\n\n)? 6. Given even log-concave probability measure μ, estimate μ(K)μ(K∗) from above. (It is known that μ(K)μ(K∗) 6 μ(Bn\n\n> 2\n\n) if μ is unconditional) 7. In the above question, characterize the equality case and find lower bound for the un-conditional case. 18. Consider μ∗ in the sense of Artstein-Milman and answer to the same questions about\n\nμ(K)μ∗(K). 9. Let K and L be symmetric convex bodies in Rn and denote by N (K, L ) the covering number of K by L, the minimum number of the translates of L needed to cover K. Is it true that\n\nN (K, L ) 6 [N (L∗, cK ∗]δ,\n\nwhere c, δ are absolute constants? 10. Given random vectors X and Y uniformly distributed in K and K∗ respectively, look at 〈X, Y 〉, up to a \"100\"-moment. Are the first \"100\" moments the same (1 + ε) as those for Gaussian random vectors? 11. Prove Mahler's conjecture for finite-dimensional normed spaces that embed in Lp,\n\np < 1 and, in particular, for convex intersection bodies (which correspond to the case\n\np = −1). 12. What if 1 < p < 2? Is it true that P(BX ) > P(Bnp ) if X is embedded into Lp, 1 < p < 2and dimX = n?\n1", "statement_status": "exact", "statement_verification": "The canonical record is the first extracted record from the AIM workshop *Mahler's conjecture and duality in convex geometry* (August 9--13, 2010), notes by Jaegil Kim. The extraction merged Questions 1--12 from the first two pages of the official four-page PDF. The unmodified OCR record is preserved in input.json.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Mahler's conjecture and duality in convex geometry\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/mahlerduality/mahlerduality.pdf\nCanonical location: aim-convex-geometry-notes.json notes[10]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then K ⊂ L?2. Is it true that if IK ⊂ IL and ΠK ⊂ ΠL then vol( K) 6 vol( L)? (True if n 6 4) 3. Is it true that for any convex (symmetric) body K there exists a direction u ∈ Sn−1\\n\\nsuch that vol(St u(K)∗) 6 Cvol( K∗)? 4. What about an average version of the previous question, i.e.: \\n\\n∫\\n\\n> Sn−1\\n\\nvol(St u(K)∗)du 6 Cvol( K∗)? 5. Is the average of P(K) for symmetric polytopes K greater than or equal to P(Bn\\n\\n> ∞\\n\\n)? More precisely, let K be a random symmetric polytope of a fixed number of vertices, where each vertex is independent Gaussian random vector is it true that EP(K) >\\n\\nP(Bn\\n\\n> ∞\\n\\n)? 6. Given even log-concave probability measure μ, estimate μ(K)μ(K∗) from above. (It is known that μ(K)μ(K∗) 6 μ(Bn \\n\\n> 2\\n\\n) if μ is unconditional) 7. In the above question, characterize the equality case and find lower bound for the un-conditional case. 18. Consider μ∗ in the sense of Artstein-Milman and answer to the same questions about \\n\\nμ(K)μ∗(K). 9. Let K and L be symmetric convex bodies in Rn and denote by N (K, L ) the covering number of K by L, the minimum number of the translates of L needed to cover K. Is it true that \\n\\nN (K, L ) 6 [N (L∗, cK ∗]δ,\\n\\nwhere c, δ are absolute constants? 10. Given random vectors X and Y uniformly distributed in K and K∗ respectively, look at 〈X, Y 〉, up to a \\\"100\\\"-moment. Are the first \\\"100\\\" moments the same (1 + ε) as those for Gaussian random vectors? 11. Prove Mahler's conjecture for finite-dimensional normed spaces that embed in Lp,\\n\\np < 1 and, in particular, for convex intersection bodies (which correspond to the case \\n\\np = −1). 12. What if 1 < p < 2? Is it true that P(BX ) > P(Bnp ) if X is embedded into Lp, 1 < p < 2and dimX = n?\\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/mahlerduality/mahlerduality.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0011", "aim-domain:convex-geometry", "aim-workshop:mahlerduality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The merged record contains twelve separate questions. Under the standard origin-symmetric interpretation, Question 1 is answered completely for n >= 2: it is true in dimension two and false in every dimension n >= 3. The counterexample is the explicit ellipsoid K = diag(R,s,...,s) B_2^n versus L = B_2^n, with R > 1 and s = R^{-1/(n-2)}. More generally, for origin-centered ellipsoids E and F, intersection-body inclusion and projection-body inclusion coincide and are equivalent to (|E|/|F|)F being contained in E; this forces |E| <= |F| and proves Question 2 for all ellipsoid pairs in every dimension.\n\nCandidate contribution (counterexample and special-case classification; novelty confidence low): For origin-centered ellipsoids E,F, the conditions I(E) subset I(F) and Pi(E) subset Pi(F) are each equivalent to (|E|/|F|)F subset E. This yields an explicit counterexample to AIM Question 1 in every n >= 3, while the planar identity IK = Pi K = 2RK proves the implication in n = 2." }, { "id": 20001280, "problem_number": "AIM-CONVEX_GEOMETRY-0012", "title": "Four recovered problems and a normalization obstruction", "statement": "3. (T. Tao) Let K be a symmetric convex body in Rn. Consider the covering radius of\n\nZn for K∗, the smallest r > 0 such that\n\n⋃\n\n> v∈Zn\\K\n\nrK ∗ + v = Rn.\n\nIs it true that K∗ has the largest covering radius when K is a cube, among all sym-metric convex bodies in Rn?14. Does IK = K imply K = cB n\n\n> 2\n\nwhen n > 3? Notice that if ΠK is considered instead of IK, it is not true because the projection body of a cube is a dilate of a cube. It is a special case of\nProblem 8.7 in [G]. There are a few comments about the more general problem in [G, Note 8.6]. Moreover the analogous more general question for projection bodies is [G,\nProblem 4.5], and in [G, Note 4.6] it is stated that Weil solved this for polytopes. 15. For n > 3, is it true that ImK → Bn\n\n> 2\n\nin the Banach-Mazur distance as m → ∞?216. Let n > 5. Construct an example of a polytope K which is an intersection body, not a polar body of a zonotope.\n1", "original_statement": "3. (T. Tao) Let K be a symmetric convex body in Rn. Consider the covering radius of \n\nZn for K∗, the smallest r > 0 such that \n\n⋃\n\n> v∈Zn\\K\n\nrK ∗ + v = Rn.\n\nIs it true that K∗ has the largest covering radius when K is a cube, among all sym-metric convex bodies in Rn?14. Does IK = K imply K = cB n \n\n> 2\n\nwhen n > 3? Notice that if ΠK is considered instead of IK, it is not true because the projection body of a cube is a dilate of a cube. It is a special case of \nProblem 8.7 in [G]. There are a few comments about the more general problem in [G, Note 8.6]. Moreover the analogous more general question for projection bodies is [G, \nProblem 4.5], and in [G, Note 4.6] it is stated that Weil solved this for polytopes. 15. For n > 3, is it true that ImK → Bn \n\n> 2\n\nin the Banach-Mazur distance as m → ∞?216. Let n > 5. Construct an example of a polytope K which is an intersection body, not a polar body of a zonotope. \n1", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is not one problem numbered 3. It is an extraction accident that concatenates Problems 13--16 from page 2 of the official AIM workshop PDF, *Mahler's conjecture and duality in convex geometry* (2010). The preceding canonical record contains Problems 1--12 and the following record starts with Problem 17. The four recovered statements are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Mahler's conjecture and duality in convex geometry\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/mahlerduality/mahlerduality.pdf\nCanonical location: aim-convex-geometry-notes.json notes[11]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (T. Tao) Let K be a symmetric convex body in Rn. Consider the covering radius of \\n\\nZn for K∗, the smallest r > 0 such that \\n\\n⋃\\n\\n> v∈Zn\\\\K\\n\\nrK ∗ + v = Rn.\\n\\nIs it true that K∗ has the largest covering radius when K is a cube, among all sym-metric convex bodies in Rn?14. Does IK = K imply K = cB n \\n\\n> 2\\n\\nwhen n > 3? Notice that if ΠK is considered instead of IK, it is not true because the projection body of a cube is a dilate of a cube. It is a special case of \\nProblem 8.7 in [G]. There are a few comments about the more general problem in [G, Note 8.6]. Moreover the analogous more general question for projection bodies is [G, \\nProblem 4.5], and in [G, Note 4.6] it is stated that Weil solved this for polytopes. 15. For n > 3, is it true that ImK → Bn \\n\\n> 2\\n\\nin the Banach-Mazur distance as m → ∞?216. Let n > 5. Construct an example of a polytope K which is an intersection body, not a polar body of a zonotope. \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/mahlerduality/mahlerduality.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0012", "aim-domain:convex-geometry", "aim-workshop:mahlerduality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source record merges AIM Problems 13-16. For Problem 13 as literally printed, the punctured-center covering radius is finite for every fixed symmetric convex body K, but if a Euclidean ball of radius a lies in K then the radius for lambda K is at least lambda^2 a^2. Under the standard all-lattice-center definition, the covering radius of (lambda K)^circ scales linearly in lambda. Even fixing volume does not repair either reading: determinant-one boxes K_R have exact standard radius (R+R^{-1}+n-2)/2 and literal radius at least R/2. Problem 14 was solved affirmatively in 2025; Problems 15 and 16 remain globally open on the literature checked.\n\nCandidate contribution (obstruction; novelty confidence low): For the exact punctured-center AIM formula, finite deletion never makes the covering radius infinite, yet dilation forces at least quadratic growth; moreover a determinant-one rectangular-box family proves that both the literal and standard readings remain unbounded at fixed volume, with an exact formula for the standard radius." }, { "id": 20001281, "problem_number": "AIM-CONVEX_GEOMETRY-0013", "title": "Dual random inner products: status split and Hanner closure", "statement": "7. (G. Kuperberg) Let K be a symmetric convex body in Rn. For random vectors X and\n\nY distributed in K and K∗ respectively, consider the random variable 〈X, Y 〉. Then the conjecture is that 〈X, Y 〉 has the largest variance when K is an ellipsoid. One may consider a weaker conjecture: the integral of |〈 X, Y 〉| 2 over K × K∗ is maximized when\n\nK is an ellipsoid. More generally, G. Paouris asked the same questions for the quantity\n\n|〈 X, Y 〉| p, (p > 0), instead of |〈 X, Y 〉| 2.Furthermore two other variations can be considered: Given 0 < c < 1, consider either the probability that 〈X, Y 〉 = c, or the volume of the region in which it is so. One may again conjecture that either the probability or at least the volume is maximized when\n\nK is an ellipsoid.\n1", "original_statement": "7. (G. Kuperberg) Let K be a symmetric convex body in Rn. For random vectors X and \n\nY distributed in K and K∗ respectively, consider the random variable 〈X, Y 〉. Then the conjecture is that 〈X, Y 〉 has the largest variance when K is an ellipsoid. One may consider a weaker conjecture: the integral of |〈 X, Y 〉| 2 over K × K∗ is maximized when \n\nK is an ellipsoid. More generally, G. Paouris asked the same questions for the quantity \n\n|〈 X, Y 〉| p, (p > 0), instead of |〈 X, Y 〉| 2.Furthermore two other variations can be considered: Given 0 < c < 1, consider either the probability that 〈X, Y 〉 = c, or the volume of the region in which it is so. One may again conjecture that either the probability or at least the volume is maximized when \n\nK is an ellipsoid. \n1", "clean_statement": "7. (G. Kuperberg) Let K be a symmetric convex body in Rn. For random vectors X and\n\nY distributed in K and K∗ respectively, consider the random variable 〈X, Y 〉. Then the conjecture is that 〈X, Y 〉 has the largest variance when K is an ellipsoid. One may consider a weaker conjecture: the integral of |〈 X, Y 〉| 2 over K × K∗ is maximized when\n\nK is an ellipsoid. More generally, G. Paouris asked the same questions for the quantity\n\n|〈 X, Y 〉| p, (p > 0), instead of |〈 X, Y 〉| 2.Furthermore two other variations can be considered: Given 0 < c < 1, consider either the probability that 〈X, Y 〉 = c, or the volume of the region in which it is so. One may again conjecture that either the probability or at least the volume is maximized when\n\nK is an ellipsoid.\n1", "statement_status": "exact", "statement_verification": "The canonical input preserves an OCR extraction beginning with “7.” Direct inspection of the official AIM PDF shows that this is **Problem 17** in the 2010 workshop list *Mahler's conjecture and duality in convex geometry*. The leading 1 was lost in extraction. The PDF uses \\(K^*\\) for the polar body in \\(\\mathbb R^n\\), and the exponents are \\(2\\) and \\(p>0\\).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Mahler's conjecture and duality in convex geometry\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/mahlerduality/mahlerduality.pdf\nCanonical location: aim-convex-geometry-notes.json notes[12]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"7. (G. Kuperberg) Let K be a symmetric convex body in Rn. For random vectors X and \\n\\nY distributed in K and K∗ respectively, consider the random variable 〈X, Y 〉. Then the conjecture is that 〈X, Y 〉 has the largest variance when K is an ellipsoid. One may consider a weaker conjecture: the integral of |〈 X, Y 〉| 2 over K × K∗ is maximized when \\n\\nK is an ellipsoid. More generally, G. Paouris asked the same questions for the quantity \\n\\n|〈 X, Y 〉| p, (p > 0), instead of |〈 X, Y 〉| 2.Furthermore two other variations can be considered: Given 0 < c < 1, consider either the probability that 〈X, Y 〉 = c, or the volume of the region in which it is so. One may again conjecture that either the probability or at least the volume is maximized when \\n\\nK is an ellipsoid. \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/mahlerduality/mahlerduality.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0013", "aim-domain:convex-geometry", "aim-workshop:mahlerduality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM record has a mixed present-day resolution. Klartag's 2018 construction refutes the normalized variance conjecture in sufficiently high dimension, and the layer-cake identity also forces failure of Kuperberg's intended normalized tail comparison at some threshold. By contrast, Böröczky, Patsalos, and Saroglou's May 2026 preprint proves the unnormalized quadratic integral inequality with equality only for ellipsoids. The report additionally proves exact sum/product covariance and volume recursions and deduces M_2(H)=2n/[3(n+1)(n+2)] and J_2(H)=(4^n/n!)M_2(H) for every n-dimensional Hanner polytope. The general p-moment and unnormalized pointwise tail-volume questions remain unresolved.\n\nCandidate contribution (closure theorem and explicit family; novelty confidence low): For symmetric A in R^a and B in R^b, both their l_1-sum and Cartesian product satisfy M_2(A sum B)=alpha_a M_2(A)+alpha_b M_2(B), with alpha_a=(a+1)(a+2)/[(a+b+1)(a+b+2)] and the analogous alpha_b; their volume products obey S(A sum B)=a!b!/(a+b)! S(A)S(B). Consequently every n-dimensional Hanner polytope has the decomposition-independent values M_2=2n/[3(n+1)(n+2)] and J_2=(4^n/n!)2n/[3(n+1)(n+2)]." }, { "id": 20001282, "problem_number": "AIM-CONVEX_GEOMETRY-0014", "title": "A sharp product-prism family for Kuperberg's fixed-combinatorial-type question", "statement": "8. (G. Kuperberg) In R3, consider a class A of centrally symmetric polytopes of a given combinatorial type, excluding the type of the cube and the octahedron. Then the con-jecture is that the volume product does not have a local minimum in A. Furthermore, one may ask if each such A has a unique critical point up to affine transformations, and it is a local maximum.\n1", "original_statement": "8. (G. Kuperberg) In R3, consider a class A of centrally symmetric polytopes of a given combinatorial type, excluding the type of the cube and the octahedron. Then the con-jecture is that the volume product does not have a local minimum in A. Furthermore, one may ask if each such A has a unique critical point up to affine transformations, and it is a local maximum. \n1", "clean_statement": "8. (G. Kuperberg) In R3, consider a class A of centrally symmetric polytopes of a given combinatorial type, excluding the type of the cube and the octahedron. Then the con-jecture is that the volume product does not have a local minimum in A. Furthermore, one may ask if each such A has a unique critical point up to affine transformations, and it is a local maximum.\n1", "statement_status": "exact", "statement_verification": "The canonical JSON extraction reads “8.” and ends with a stray “1”. Inspection of the official AIM workshop PDF shows that the leading digit was lost and that the terminal digit is a page-number spill. The official statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Mahler's conjecture and duality in convex geometry\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/mahlerduality/mahlerduality.pdf\nCanonical location: aim-convex-geometry-notes.json notes[13]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"8. (G. Kuperberg) In R3, consider a class A of centrally symmetric polytopes of a given combinatorial type, excluding the type of the cube and the octahedron. Then the con-jecture is that the volume product does not have a local minimum in A. Furthermore, one may ask if each such A has a unique critical point up to affine transformations, and it is a local maximum. \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/mahlerduality/mahlerduality.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0014", "aim-domain:convex-geometry", "aim-workshop:mahlerduality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every marked centrally symmetric planar hexagon has an affine normal form H_{p,q}=conv{+-(p,q), +-(0,1), +-(-1,0)} on p,q>0, p+q>1, |p-q|<1, with exact volume product (p+q+1)(4-(p+q-1)^2/(pq)). This product lies in (8,9], has no local minimum, and has one critical affine orbit, the affine-regular hexagon, which is a strict maximum. The identity P(H x [-a,a])=(4/3)P(H) then proves that every centrally symmetric geometric product hexagonal prism, and every polar product-dual hexagonal bipyramid, is not a local minimum even in its full fixed-face-lattice realization stratum; critical-orbit uniqueness and maximality are asserted only within the product submoduli.\n\nCandidate contribution (explicit_family_theorem; novelty confidence low): The exact two-parameter hexagon formula and prism-bipyramid lift give a same-face-lattice lowering curve at every centrally symmetric product hexagonal prism and every polar product-dual bipyramid, with sharp product-submoduli range (32/3,12] and a unique critical affine orbit within that submoduli." }, { "id": 20001283, "problem_number": "AIM-CONVEX_GEOMETRY-0015", "title": "Exact radial gluing and symmetric star-body rigidity", "statement": "9. (R.J. Gardner) Does the following theorem hold for star bodies, or perhaps more generally still? It comes from [G,\nProblem 7.1] and there are some relevant comments in [G, Note 7.1].\n\nTheorem (Rogers, 1965.) Suppose that 2 6 k 6 n − 1 and that K and L are compact convex sets in Rn containing the origin in their relative interiors. If all k-sections K ∩S\n\nand L ∩ S of K and L are homothetic (or translates), then K and L are homothetic (or translates, respectively).\n\n2", "original_statement": "9. (R.J. Gardner) Does the following theorem hold for star bodies, or perhaps more generally still? It comes from [G, \nProblem 7.1] and there are some relevant comments in [G, Note 7.1]. \n\nTheorem (Rogers, 1965.) Suppose that 2 6 k 6 n − 1 and that K and L are compact convex sets in Rn containing the origin in their relative interiors. If all k-sections K ∩S\n\nand L ∩ S of K and L are homothetic (or translates), then K and L are homothetic (or translates, respectively). \n\n2", "clean_statement": "9. (R.J. Gardner) Does the following theorem hold for star bodies, or perhaps more generally still? It comes from [G,\nProblem 7.1] and there are some relevant comments in [G, Note 7.1].\n\nTheorem (Rogers, 1965.) Suppose that 2 6 k 6 n − 1 and that K and L are compact convex sets in Rn containing the origin in their relative interiors. If all k-sections K ∩S\n\nand L ∩ S of K and L are homothetic (or translates), then K and L are homothetic (or translates, respectively).\n\n2", "statement_status": "exact", "statement_verification": "The canonical record is an OCR extraction from the official problem list for the 2010 AIM workshop *Mahler's conjecture and duality in convex geometry*. Three features of the extraction require correction:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Mahler's conjecture and duality in convex geometry\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/mahlerduality/mahlerduality.pdf\nCanonical location: aim-convex-geometry-notes.json notes[14]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"9. (R.J. Gardner) Does the following theorem hold for star bodies, or perhaps more generally still? It comes from [G, \\nProblem 7.1] and there are some relevant comments in [G, Note 7.1]. \\n\\nTheorem (Rogers, 1965.) Suppose that 2 6 k 6 n − 1 and that K and L are compact convex sets in Rn containing the origin in their relative interiors. If all k-sections K ∩S\\n\\nand L ∩ S of K and L are homothetic (or translates), then K and L are homothetic (or translates, respectively). \\n\\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/mahlerduality/mahlerduality.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0015", "aim-domain:convex-geometry", "aim-workshop:mahlerduality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any two star bodies and every 2 <= k <= n, the optimal global uniform log-radial error modulo centered dilation equals the maximum of the corresponding optimal errors over all k-dimensional central sections. Consequently, centered section dilations glue exactly and approximately with no loss in the error constant. For origin-symmetric star bodies, uniqueness of the symmetry center forces arbitrary section homotheties to be centered, proving the homothety branch for 2 <= k <= n-1; section-wise translates force equality and this translation conclusion remains true for k = 1.\n\nCandidate contribution (stability_identity; novelty confidence low): If D(K,L)=inf_c ||log rho_K-log rho_L-c||_infinity and D_S is its restriction to S, then D(K,L)=max_{S in G(n,k)} D_S(K,L) for every k >= 2; hence section-wise multiplicative radial error epsilon after optimally chosen centered dilations implies one global centered dilation with the same error epsilon." }, { "id": 20001284, "problem_number": "AIM-CONVEX_GEOMETRY-0016", "title": "Finite-dimensional local rigidity from central-section perimeters", "statement": "0. (R.J. Gardner)[G,\nProblem 7.6 ] If K and L are origin-symmetric star bodies in R3\n\nwhose sections by every plane through the origin have equal perimeters, is K = L? In particular, This was proved by R. Howard, F. Nazarov, D. Ryabogin and A. Zvavitch for bodies of revolution and by V. Yaskin for polytopes.\n2", "original_statement": "0. (R.J. Gardner)[G, \nProblem 7.6 ] If K and L are origin-symmetric star bodies in R3\n\nwhose sections by every plane through the origin have equal perimeters, is K = L? In particular, This was proved by R. Howard, F. Nazarov, D. Ryabogin and A. Zvavitch for bodies of revolution and by V. Yaskin for polytopes. \n2", "clean_statement": "0. (R.J. Gardner)[G,\nProblem 7.6 ] If K and L are origin-symmetric star bodies in R3\n\nwhose sections by every plane through the origin have equal perimeters, is K = L? In particular, This was proved by R. Howard, F. Nazarov, D. Ryabogin and A. Zvavitch for bodies of revolution and by V. Yaskin for polytopes.\n2", "statement_status": "exact", "statement_verification": "The canonical record is source index 15 of aim-convex-geometry-notes.json. Its number field is \"0\" and its text has stray page-number fragments. The official AIM PDF resolves the extraction error: this is **Problem 20**, not Problem 0, in *Problems from the workshop \"Mahler's conjecture and duality in convex geometry\"*, AIM, August 9--13, 2010, notes by Jaegil Kim.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Mahler's conjecture and duality in convex geometry\nSection: \nSource item: 0\nSource URL: https://aimath.org/WWN/mahlerduality/mahlerduality.pdf\nCanonical location: aim-convex-geometry-notes.json notes[15]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"0. (R.J. Gardner)[G, \\nProblem 7.6 ] If K and L are origin-symmetric star bodies in R3\\n\\nwhose sections by every plane through the origin have equal perimeters, is K = L? In particular, This was proved by R. Howard, F. Nazarov, D. Ryabogin and A. Zvavitch for bodies of revolution and by V. Yaskin for polytopes. \\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/mahlerduality/mahlerduality.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0016", "aim-domain:convex-geometry", "aim-workshop:mahlerduality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every radius R and every fixed even spherical-harmonic band through degree 2m, there are exactly (m+1)(2m+1) central-plane normals such that the corresponding nonlinear section-perimeter measurements are locally injective near the radius-R ball within that bandlimited radial chart. The derivative at the ball is the Funk transform, and the Hessian is the great-circle integral of tangential derivative products divided by R.\n\nCandidate contribution (local uniqueness theorem; novelty confidence low): Exactly dim(V_m) = (m+1)(2m+1) suitably selected central-plane perimeter measurements form a locally invertible nonlinear coordinate system near a ball on the even bandlimited radial model R+V_m." }, { "id": 20001285, "problem_number": "AIM-CONVEX_GEOMETRY-0017", "title": "Few-vertex Mahler bounds and pyramid cores", "statement": "1. (S. Reisner) Improve the following theorem by finding the number of vertices N (n) >n + 3.\n\nTheorem (M. Meyer, S. Reisner)[MR] Let K be a convex polytope in Rn, with at most\n\nn + 3 vertices (or facets) and non-empty interior. Then\n\nP(K) > (n + 1) n+1\n\n(n!) 2\n\n3with equality if and only if K is an n-dimensional simplex.\n\n2", "original_statement": "1. (S. Reisner) Improve the following theorem by finding the number of vertices N (n) >n + 3. \n\nTheorem (M. Meyer, S. Reisner)[MR] Let K be a convex polytope in Rn, with at most \n\nn + 3 vertices (or facets) and non-empty interior. Then \n\nP(K) > (n + 1) n+1 \n\n(n!) 2\n\n3with equality if and only if K is an n-dimensional simplex. \n\n2", "clean_statement": "1. (S. Reisner) Improve the following theorem by finding the number of vertices N (n) >n + 3.\n\nTheorem (M. Meyer, S. Reisner)[MR] Let K be a convex polytope in Rn, with at most\n\nn + 3 vertices (or facets) and non-empty interior. Then\n\nP(K) > (n + 1) n+1\n\n(n!) 2\n\n3with equality if and only if K is an n-dimensional simplex.\n\n2", "statement_status": "exact", "statement_verification": "This record is Problem 21 in the AIM list from the workshop “Mahler's conjecture and duality in convex geometry.” The PDF extraction has lost superscripts and changed a non-strict inequality into a strict one. With \\[ K^z=\\{y\\in{\\mathbb R}^n:\\langle y,x-z\\rangle\\leq 1 \\text{ for every }x\\in K\\} \\] and \\[ {\\cal P}(K)=\\min_{z\\in\\operatorname{int}K}|K|\\,|K^z|, \\] the recovered problem is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Mahler's conjecture and duality in convex geometry\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/mahlerduality/mahlerduality.pdf\nCanonical location: aim-convex-geometry-notes.json notes[16]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (S. Reisner) Improve the following theorem by finding the number of vertices N (n) >n + 3. \\n\\nTheorem (M. Meyer, S. Reisner)[MR] Let K be a convex polytope in Rn, with at most \\n\\nn + 3 vertices (or facets) and non-empty interior. Then \\n\\nP(K) > (n + 1) n+1 \\n\\n(n!) 2\\n\\n3with equality if and only if K is an n-dimensional simplex. \\n\\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/mahlerduality/mahlerduality.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0017", "aim-domain:convex-geometry", "aim-workshop:mahlerduality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a (d-1)-dimensional convex body B normalized by s(B)=0, the unit-height pyramid P has Santaló point (0,1/(d+1)) and satisfies P(P)=((d+1)^(d+1)/d^(d+2))P(B). Since this multiplier is exactly the ratio of consecutive simplex products, normalized Mahler deficit is invariant under iterated pyramiding. Consequently, every n+4-vertex pyramid chain with a planar nonpyramidal core satisfies the conjectured bound unconditionally, and every such chain with a three-dimensional core does so subject to the June 2026 dimension-three preprint; any still-unresolved chain has core dimension at least four.\n\nCandidate contribution (reduction; novelty confidence low): For the n+4-vertex extension in AIM Problem 21, stripping pyramid apices preserves the exact normalized ratio P(K)/M_dim(K); therefore the pyramid-chain part reduces to nonpyramidal fixed-excess cores, with planar cores settled unconditionally, three-dimensional cores settled subject to arXiv:2605.09334v3, and the unresolved range beginning at core dimension four." }, { "id": 20001286, "problem_number": "AIM-CONVEX_GEOMETRY-0018", "title": "An exact quantitative shadow-system model", "statement": "2. (S. Reisner) Quantify the shadow movement theorem (see [MR]).\n2", "original_statement": "2. (S. Reisner) Quantify the shadow movement theorem (see [MR]). \n2", "clean_statement": "2. (S. Reisner) Quantify the shadow movement theorem (see [MR]).\n2", "statement_status": "exact", "statement_verification": "The canonical record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Mahler's conjecture and duality in convex geometry\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/mahlerduality/mahlerduality.pdf\nCanonical location: aim-convex-geometry-notes.json notes[17]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (S. Reisner) Quantify the shadow movement theorem (see [MR]). \\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/mahlerduality/mahlerduality.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0018", "aim-domain:convex-geometry", "aim-workshop:mahlerduality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the explicit unit-speed planar shadow system K_t = conv{+/- (1+t,1), +/- (0,1), +/- (-1,0)}, -1 0 ∀y ∈ Q}\n\nand define its volume product by P(Q) = vol( Q)vol( Q◦). For any ∆ n ⊂ Sn−1 which is determined by distinct n hyperplanes passing through the origin, is it true P(∆ n) 6\n\nP(Rn\n\n> +\n\n∩ Sn−1)? In particular, if n = 3, it is true.", "original_statement": "3. (H. Koenig) For Q ⊂ Sn−1, define its dual Q◦ on the sphere Sn−1 by \n\nQ◦ = {x ∈ Sn−1: 〈x, y 〉 > 0 ∀y ∈ Q}\n\nand define its volume product by P(Q) = vol( Q)vol( Q◦). For any ∆ n ⊂ Sn−1 which is determined by distinct n hyperplanes passing through the origin, is it true P(∆ n) 6\n\nP(Rn \n\n> +\n\n∩ Sn−1)? In particular, if n = 3, it is true.", "clean_statement": "3. (H. Koenig) For Q ⊂ Sn−1, define its dual Q◦ on the sphere Sn−1 by\n\nQ◦ = {x ∈ Sn−1: 〈x, y 〉 > 0 ∀y ∈ Q}\n\nand define its volume product by P(Q) = vol( Q)vol( Q◦). For any ∆ n ⊂ Sn−1 which is determined by distinct n hyperplanes passing through the origin, is it true P(∆ n) 6\n\nP(Rn\n\n> +\n\n∩ Sn−1)? In particular, if n = 3, it is true.", "statement_status": "exact", "statement_verification": "The corpus record is an OCR-damaged extraction from the AIM workshop list *Mahler's conjecture and duality in convex geometry*. Direct inspection of the source PDF recovers the entry as Problem 23, attributed to H. Koenig:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Mahler's conjecture and duality in convex geometry\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/mahlerduality/mahlerduality.pdf\nCanonical location: aim-convex-geometry-notes.json notes[18]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (H. Koenig) For Q ⊂ Sn−1, define its dual Q◦ on the sphere Sn−1 by \\n\\nQ◦ = {x ∈ Sn−1: 〈x, y 〉 > 0 ∀y ∈ Q}\\n\\nand define its volume product by P(Q) = vol( Q)vol( Q◦). For any ∆ n ⊂ Sn−1 which is determined by distinct n hyperplanes passing through the origin, is it true P(∆ n) 6\\n\\nP(Rn \\n\\n> +\\n\\n∩ Sn−1)? In particular, if n = 3, it is true.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/mahlerduality/mahlerduality.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0019", "aim-domain:convex-geometry", "aim-workshop:mahlerduality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every n >= 2 and 0 < alpha < pi, the simplicial cone C_alpha = W_alpha x R_+^{n-2}, where W_alpha is a planar wedge of opening alpha, satisfies the conjectured spherical volume-product inequality. Its normalized product is exactly alpha(pi-alpha)/(2^{2n-2}pi^2) = 4^{-n} - (alpha-pi/2)^2/(4^{n-1}pi^2), so equality holds exactly at the orthant alpha = pi/2 and the deficit is an exact quadratic. More generally, normalized cone volume products multiply under orthogonal Cartesian products, which lifts any verified lower-dimensional instance by positive-orthant factors.\n\nCandidate contribution (exact structured-family theorem; novelty confidence low): The exact all-dimensional wedge-orthant formula, including its equality characterization and quadratic deficit, and the product-lifting permanence principle constitute a concrete candidate partial result toward AIM Problem 23." }, { "id": 20001288, "problem_number": "AIM-CONVEX_GEOMETRY-0020", "title": "Sample-only volume estimation and a sharp box threshold", "statement": "Problem 1 (Santosh Vempala) Let K ⊂ Rn be a convex body. How many uniformly randomly chosen points from K are needed to estimate the volume of K within a factor of\n\n2.", "original_statement": "Problem 1 (Santosh Vempala) Let K ⊂ Rn be a convex body. How many uniformly randomly chosen points from K are needed to estimate the volume of K within a factor of \n\n2.", "clean_statement": "Problem 1 (Santosh Vempala) Let K ⊂ Rn be a convex body. How many uniformly randomly chosen points from K are needed to estimate the volume of K within a factor of\n\n2.", "statement_status": "exact", "statement_verification": "The canonical record contains the line break", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Algorithmic convex geometry\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf\nCanonical location: aim-convex-geometry-notes.json notes[19]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1 (Santosh Vempala) Let K ⊂ Rn be a convex body. How many uniformly randomly chosen points from K are needed to estimate the volume of K within a factor of \\n\\n2.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/convexgeometry/convexgeometry.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0020", "aim-domain:convex-geometry", "aim-workshop:convexgeometry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Eldan's 2011 theorem gives an information-theoretic superpolynomial lower bound for the random-point-only interpretation of the AIM problem. As an exact calibrated submodel, for an unknown axis-aligned box the sample-bounding-box volume divided by true volume is a product of n independent Beta(m-1,2) ranges. If m/n tends to c, this ratio tends in probability to exp(-2/c), so the estimator's factor-2 success probability tends to 1 for c>2/log 2 and to 0 for c<2/log 2. An exact finite-sample Chernoff bound and a dimension-free known-homothetic-shape lemma are also proved.\n\nCandidate contribution (exact_asymptotic_threshold; novelty confidence low): For the natural sample-bounding-box estimator on unknown axis-aligned boxes, the estimated-to-true volume ratio converges to exp(-2/c) when m/n tends to c, yielding the sharp strict factor-2 threshold c*=2/log 2, with the exact transform bound Pr(failure) <= 2^(-s)[m(m-1)/((m-s)(m-1-s))]^n for 0 μ\n\n(P (x, y )is convex) ≥ 1 − [U+000F],\n\nthen S is O(n[U+000F] )-convex.", "original_statement": "Problem 2 (Santosh Vempala) Let S ∈ Rn be compact, and let C ∈ argmin C∈K vol(∆( S, C )) \n\nbe a convex body closest to S (K stands for the set of compact convex sets with nonempty interior, and the empty set, and ∆( ·, ·) stands for the symmetric difference). We say that \n\nS is \u000f-convex if vol(∆( S, C )) ≤ \u000fvol( S). Assume that the center of gravity of S is at the origin. For a pair of points x, y 6 = 0 ∈ Rn, let the subspace spanned by them be H(x, y ) and define P (x, y ):= S ∩ H(x, y ).Let μ be the distribution on 2-dimensional sections P (x, y ) obtained by picking x and y\n\nuniformly at random from S. If \n\nPr \n\n> μ\n\n(P (x, y )is convex) ≥ 1 − \u000f, \n\nthen S is O(n\u000f )-convex.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Problem 2 from the AIM workshop “Algorithmic convex geometry.” Its extracted text contains the control character U+000F in place of a Greek letter and several damaged mathematical relations. Inspection of the official PDF and of its embedded TeX font encoding gives the following repairs.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Algorithmic convex geometry\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf\nCanonical location: aim-convex-geometry-notes.json notes[20]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 2 (Santosh Vempala) Let S ∈ Rn be compact, and let C ∈ argmin C∈K vol(∆( S, C )) \\n\\nbe a convex body closest to S (K stands for the set of compact convex sets with nonempty interior, and the empty set, and ∆( ·, ·) stands for the symmetric difference). We say that \\n\\nS is \\u000f-convex if vol(∆( S, C )) ≤ \\u000fvol( S). Assume that the center of gravity of S is at the origin. For a pair of points x, y 6 = 0 ∈ Rn, let the subspace spanned by them be H(x, y ) and define P (x, y ):= S ∩ H(x, y ).Let μ be the distribution on 2-dimensional sections P (x, y ) obtained by picking x and y\\n\\nuniformly at random from S. If \\n\\nPr \\n\\n> μ\\n\\n(P (x, y )is convex) ≥ 1 − \\u000f, \\n\\nthen S is O(n\\u000f )-convex.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/convexgeometry/convexgeometry.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0021", "aim-domain:convex-geometry", "aim-workshop:convexgeometry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the exact two-sample central-plane law in AIM Problem 2, the pushforward distribution on G(n,2) has density proportional to W_S(H)=∫_{(S∩H)^2}[x,y]^(n-2) dx dy, not the Haar law. If a missing chord has endpoint balls and an excluded midpoint ball of common radius rho, then the probability of a nonconvex sampled section is at least (kappa_n rho^n/|S|)^2. Consequently zero section-failure probability forces convexity for regular-closed full-dimensional compact sets, and in dimension two the failure probability is exactly zero or one according as S is convex or not.\n\nCandidate contribution (reduction; novelty confidence low): Under the AIM sampling law, bad central planes admit an exact determinant-weighted Grassmannian representation, and any missing chord with robustness radius rho contributes failure probability at least (kappa_n rho^n/|S|)^2; this gives a topologically precise zero-error theorem for regular-closed full-dimensional sets." }, { "id": 20001290, "problem_number": "AIM-CONVEX_GEOMETRY-0022", "title": "The planar resolution and a one-dimensional obstruction", "statement": "Problem 3 (Van Vu) Let K ⊂ Rd (d fixed dimension) be a convex body. Let x1,..., x n\n\nbe uniformly random points from K, and let X:= vol(conv( x1,..., x n)). Does X satisfy the central limit theorem?", "original_statement": "Problem 3 (Van Vu) Let K ⊂ Rd (d fixed dimension) be a convex body. Let x1,..., x n\n\nbe uniformly random points from K, and let X:= vol(conv( x1,..., x n)). Does X satisfy the central limit theorem?", "clean_statement": "Problem 3 (Van Vu) Let K ⊂ Rd (d fixed dimension) be a convex body. Let x1,..., x n\n\nbe uniformly random points from K, and let X:= vol(conv( x1,..., x n)). Does X satisfy the central limit theorem?", "statement_status": "exact", "statement_verification": "The official AIM PDF, from the November 5--9, 2007 workshop *Algorithmic Convex Geometry*, gives the following as Problem 3, attributed to Van Vu:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Algorithmic convex geometry\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf\nCanonical location: aim-convex-geometry-notes.json notes[21]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3 (Van Vu) Let K ⊂ Rd (d fixed dimension) be a convex body. Let x1,..., x n\\n\\nbe uniformly random points from K, and let X:= vol(conv( x1,..., x n)). Does X satisfy the central limit theorem?\"\nOriginal remarks: [\"Remark. The answer is known in the affirmative for the case when K is a smooth body or a polytope. The problem is open for general K even for d = 2.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/convexgeometry/convexgeometry.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0022", "aim-domain:convex-geometry", "aim-workshop:convexgeometry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Pardon's 2011 Poisson theorem and 2012 de-Poissonization theorem resolve the intended binomial volume CLT for every planar convex body, making the AIM source's planar-open remark obsolete; no unrestricted fixed-dimensional theorem for arbitrary bodies in d >= 3 was located. Independently, the literal formulation allowing d = 1 is false: for any interval the normalized missing length is exactly Beta(2,n-1), and the centered variance-standardized hull length converges to (2-G)/sqrt(2) with G Gamma(2,1), not to a Gaussian. Hull volume and missing volume are exact reflections after standardization, and Efron's identity is proved but shown to be only a first-moment relation.\n\nCandidate contribution (exact special-family obstruction; novelty confidence low): For the literal d = 1 case, the exact Beta(2,n-1) missing-length law yields a reflected-Gamma standardized limit and the quantitative Kolmogorov lower bound liminf d_K >= 1-Phi(sqrt(2)); hence d >= 2 is an essential silent hypothesis in the intended AIM statement." }, { "id": 20001291, "problem_number": "AIM-CONVEX_GEOMETRY-0023", "title": "Rigidity and a sharp halfspace benchmark for Gaussian cylinders", "statement": "Problem 4 (Ryan O'Donnnell) Let K ⊂ Rn be a convex set, and let X be Gaussian random variable conditioned to lie in K. Does Var( Xθ) = 1 imply that K is a cylinder in direction θ? If Var( Xθ) = 1 − [U+000F] then does it mean that K has a small symmetric difference with a cylinder in direction θ?\n\n1Remarks. K is not necessarily symmetric; when it is symmetric, the solution is given by Sidak's lemma. It is known that Var( Xθ) ≤ 1 in every direction θ, where Xθ is the projection of X in direction θ. This can be proved in a number of ways, e.g., using Brascamp-Lieb inequality.", "original_statement": "Problem 4 (Ryan O'Donnnell) Let K ⊂ Rn be a convex set, and let X be Gaussian random variable conditioned to lie in K. Does Var( Xθ) = 1 imply that K is a cylinder in direction θ? If Var( Xθ) = 1 − \u000f then does it mean that K has a small symmetric difference with a cylinder in direction θ?\n\n1Remarks. K is not necessarily symmetric; when it is symmetric, the solution is given by Sidak's lemma. It is known that Var( Xθ) ≤ 1 in every direction θ, where Xθ is the projection of X in direction θ. This can be proved in a number of ways, e.g., using Brascamp-Lieb inequality.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Problem 4 in the AIM workshop list *Algorithmic Convex Geometry* (workshop held November 5--9, 2007). The PDF asks, with notation restored,", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Algorithmic convex geometry\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf\nCanonical location: aim-convex-geometry-notes.json notes[22]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 4 (Ryan O'Donnnell) Let K ⊂ Rn be a convex set, and let X be Gaussian random variable conditioned to lie in K. Does Var( Xθ) = 1 imply that K is a cylinder in direction θ? If Var( Xθ) = 1 − \\u000f then does it mean that K has a small symmetric difference with a cylinder in direction θ?\\n\\n1Remarks. K is not necessarily symmetric; when it is symmetric, the solution is given by Sidak's lemma. It is known that Var( Xθ) ≤ 1 in every direction θ, where Xθ is the projection of X in direction θ. This can be proved in a number of ways, e.g., using Brascamp-Lieb inequality.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/convexgeometry/convexgeometry.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0023", "aim-domain:convex-geometry", "aim-workshop:convexgeometry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Vempala's 2010 normal-subspace theorem affirmatively resolves the exact rigidity question: unit conditional variance in direction theta is equivalent to the convex set being invariant in that direction. For the still-quantitative part, a centered halfspace with normal u and rho= has variance deficit epsilon=(2/pi)rho^2, while its exact minimum standard-Gaussian symmetric-difference distance from any measurable theta-cylinder is arcsin(|rho|)/pi, attained by a convex projected-halfspace cylinder. Hence the distance equals arcsin(sqrt(pi epsilon/2))/pi and is asymptotic to sqrt(epsilon/(2pi)); no uniform power-law cylinder-stability estimate can have exponent greater than 1/2.\n\nCandidate contribution (sharp worked family; novelty confidence low): For centered tilted halfspaces of Gaussian measure 1/2, the optimal Gaussian symmetric-difference distance to all measurable cylinders in a prescribed direction is exactly arcsin(sqrt(pi epsilon/2))/pi, where epsilon is the directional conditional-variance deficit; in particular exponent 1/2 is a necessary upper limit for any uniform power-law stability theorem." }, { "id": 20001292, "problem_number": "AIM-CONVEX_GEOMETRY-0024", "title": "Exact half-volume graph theorem for symmetric planar rectangles", "statement": "Problem 5 (David Jerison) For a symmetric convex body K, let S be a least area surface that divides the volume of K into two halves (or any other fixed proportion). Is the surface a graph? (The surface in question may not be a hyperplane in general.)", "original_statement": "Problem 5 (David Jerison) For a symmetric convex body K, let S be a least area surface that divides the volume of K into two halves (or any other fixed proportion). Is the surface a graph? (The surface in question may not be a hyperplane in general.)", "clean_statement": "Problem 5 (David Jerison) For a symmetric convex body K, let S be a least area surface that divides the volume of K into two halves (or any other fixed proportion). Is the surface a graph? (The surface in question may not be a hyperplane in general.)", "statement_status": "exact", "statement_verification": "The canonical record is Problem 5, attributed to David Jerison, in the AIM workshop list *Problems from the Workshop on Algorithmic Convex Geometry* (version dated 31 October 2007). Its statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Algorithmic convex geometry\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf\nCanonical location: aim-convex-geometry-notes.json notes[23]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 5 (David Jerison) For a symmetric convex body K, let S be a least area surface that divides the volume of K into two halves (or any other fixed proportion). Is the surface a graph? (The surface in question may not be a hyperplane in general.)\"\nOriginal remarks: [\"Remarks. Simons' cone in Rn is defined by \\n\\nx21 + x22 + x23 + x24 = x25 + x26 + x27 + x28.\\n\\nSimons' cone has the minimum volume among the surfaces that meet the ball in the same boundary, it's not a graph, and the bisecting plane has smaller area. As n → ∞, the area tends to that of the bisecting plane. This is very strange: Since in the Gaussian space this ties (as n → ∞ ) with the half-spaces, does it mean that Borell's theorem (for finite n, half-spaces are unique minimizers) does not hold in the limit?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/convexgeometry/convexgeometry.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0024", "aim-domain:convex-geometry", "aim-workshop:convexgeometry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the centrally symmetric rectangle R=(-a,a) x (-b,b), with a >= b > 0, every finite-perimeter half-area subset E satisfies P(E;R) >= 2b. If a>b, equality holds only for the left and right half-rectangles; for a square, the top and bottom halves are the only additional cases. Thus every half-volume relative-isoperimetric interface in this family is a central segment and a global affine graph after rotation. The proof is fully BV-compatible and reflects E evenly to a half-area set on the 4a-by-4b flat torus, classifies every torus equality set, and imposes the fold symmetries.\n\nCandidate contribution (special_case; novelty confidence low): The exact BV half-volume relative-isoperimetric profile of every planar rectangle R=(-a,a) x (-b,b), a >= b, is 2b, with equality only for central shortest-direction half-rectangle cuts (and the orthogonal tie in the square); the supplied proof includes the seam-free even-reflection identity and derives all surviving fold-invariant torus equality bands." }, { "id": 20001293, "problem_number": "AIM-CONVEX_GEOMETRY-0025", "title": "An explicit iid-uniform counterexample to scalar entropy-power supermodularity", "statement": "Problem 6 (Mokshay Madiman) Let X1, X 2, X 3 be real-valued independent random vari-ables with densities. Is the following inequality true?\n\ne2H( X1+X2+X3) + e2H( X2)?\n\n≥ e2H( X1+X2) + e2H( X2+X3). (1)", "original_statement": "Problem 6 (Mokshay Madiman) Let X1, X 2, X 3 be real-valued independent random vari-ables with densities. Is the following inequality true? \n\ne2H( X1+X2+X3) + e2H( X2)?\n\n≥ e2H( X1+X2) + e2H( X2+X3). (1)", "clean_statement": "Problem 6 (Mokshay Madiman) Let X1, X 2, X 3 be real-valued independent random vari-ables with densities. Is the following inequality true?\n\ne2H( X1+X2+X3) + e2H( X2)?\n\n≥ e2H( X1+X2) + e2H( X2+X3). (1)", "statement_status": "exact", "statement_verification": "The AIM list records Problem 6, attributed to Mokshay Madiman. In the source's notation, \\(H\\) is differential entropy and the variables are independent, real-valued random variables with densities. The question is whether", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Algorithmic convex geometry\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf\nCanonical location: aim-convex-geometry-notes.json notes[24]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 6 (Mokshay Madiman) Let X1, X 2, X 3 be real-valued independent random vari-ables with densities. Is the following inequality true? \\n\\ne2H( X1+X2+X3) + e2H( X2)?\\n\\n≥ e2H( X1+X2) + e2H( X2+X3). (1)\"\nOriginal remarks: [\"Remarks. One gets an equality if Xi are Gaussians even with different variances. If true, (1) implies the following: \\n\\ne2H( X1+... +Xn) ≥ ∑\\n\\n> S⊆[n]\\n\\nβS e2H( XS ),\\n\\nwhere {βS } is a fractional covering of [ n]:= {1,..., n }; that is, for all i ∈ [n] we have ∑ \\n\\n> S:i∈S\\n\\nβS ≥ 1, and βS ≥ 0 for all S ⊆ [n]. The above inequality is known to be true when the underlying hypergraph given by sets \\n\\nS consists of all ( n − 1)-subsets of [ n] [Artstein, Ball, Naor]. It was generalized to regular hypergraphs (hypergraphs for which |{ S: i ∈ S}| is the same for all i ∈ [n]) [Barron, Madiman].\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/convexgeometry/convexgeometry.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0025", "aim-domain:convex-geometry", "aim-workshop:convexgeometry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For independent iid Uniform[0,1] variables, h(X1)=0, h(X1+X2)=1/2, and h(X1+X2+X3)=7/6+log(2)-(sqrt(3)/2)log(2+sqrt(3)). Consequently the AIM supermodularity deficit is 4e^(7/3)(2+sqrt(3))^(-sqrt(3))+1-2e<0. The strict sign is certified by rational series bounds, and entropy continuity shows that the failure persists after sufficiently small independent Gaussian smoothing, producing smooth strictly positive log-concave counterexamples.\n\nCandidate contribution (explicit_counterexample; novelty confidence low): The iid Uniform[0,1] triple gives an explicit scalar compactly supported log-concave counterexample with a closed-form entropy deficit and a rational strict-sign certificate; small Gaussian convolutions give a robust smooth family." }, { "id": 20001294, "problem_number": "AIM-CONVEX_GEOMETRY-0026", "title": "The cube-projection core of the zonoid psi_2 problem", "statement": "Problem 7 (Grigoris Paouris) Which bodies are ψ2-bodies? Are zonoids ψ2-bodies? Are projections of ψ2-bodies ψ2-bodies?\n\n2Remark. It is known that the Bpn for p ≥ 2 is ψ2.", "original_statement": "Problem 7 (Grigoris Paouris) Which bodies are ψ2-bodies? Are zonoids ψ2-bodies? Are projections of ψ2-bodies ψ2-bodies? \n\n2Remark. It is known that the Bpn for p ≥ 2 is ψ2.", "clean_statement": "Problem 7 (Grigoris Paouris) Which bodies are ψ2-bodies? Are zonoids ψ2-bodies? Are projections of ψ2-bodies ψ2-bodies?\n\n2Remark. It is known that the Bpn for p ≥ 2 is ψ2.", "statement_status": "exact", "statement_verification": "The canonical record is Problem 7 in the AIM workshop list *Algorithmic Convex Geometry*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Algorithmic convex geometry\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf\nCanonical location: aim-convex-geometry-notes.json notes[25]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 7 (Grigoris Paouris) Which bodies are ψ2-bodies? Are zonoids ψ2-bodies? Are projections of ψ2-bodies ψ2-bodies? \\n\\n2Remark. It is known that the Bpn for p ≥ 2 is ψ2.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/convexgeometry/convexgeometry.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0026", "aim-domain:convex-geometry", "aim-workshop:convexgeometry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "With b_2(K) defined as the supremum over linear functionals of the psi_2 Orlicz norm divided by the L_2 norm under uniform measure on K, the supremum of b_2 over all zonoids in all dimensions equals the supremum over all orthogonal projections P_E[-1,1]^N of cubes. Every zonotope A[-1,1]^N factors as T(P_E[-1,1]^N), where E=(ker A)^perp and T=A|_E is invertible; b_2 is affine invariant, and the same bound passes to full-dimensional zonoid Hausdorff limits. Thus the zonoid question is exactly the cube-restricted core of the projection question, equivalently a deweighting problem from cube box-spline marginals to uniform measure on their supports.\n\nCandidate contribution (reduction; novelty confidence low): Uniform dimension-free psi_2 bounds for all zonoids are quantitatively equivalent, with no loss in the psi_2/L_2 constant, to such bounds for all orthogonal projections of cubes; equivalently, the open step is a uniform deweighting theorem from cube fiber-volume densities to constant density on their zonotope supports.", "research_difficulty_suggested": "L4" }, { "id": 20001295, "problem_number": "AIM-CONVEX_GEOMETRY-0027", "title": "Super-Gaussian directions and the compact-set ambiguity", "statement": "Problem 8 (Grigoris Paouris) Is it true that every compact set is anti-ψ2 in some di-rection?", "original_statement": "Problem 8 (Grigoris Paouris) Is it true that every compact set is anti-ψ2 in some di-rection?", "clean_statement": "Problem 8 (Grigoris Paouris) Is it true that every compact set is anti-ψ2 in some di-rection?", "statement_status": "exact", "statement_verification": "The AIM source states, verbatim apart from joining a line-broken word:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Algorithmic convex geometry\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf\nCanonical location: aim-convex-geometry-notes.json notes[26]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 8 (Grigoris Paouris) Is it true that every compact set is anti-ψ2 in some di-rection?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/convexgeometry/convexgeometry.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0027", "aim-domain:convex-geometry", "aim-workshop:convexgeometry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "The AIM sentence is undefined without specifying a measure and normalization. Klartag's theorem affirmatively repairs it for every density with median normalization, and standard log-concave scale comparison yields the historically intended L1-normalized convex-body result. In contrast, the natural L1-normalized reading for arbitrary positive-volume compact sets is false: an explicit centered rotational ball-plus-distant-shell family has shell mass exactly epsilon and tail probability at the first-moment threshold at most epsilon in every direction, defeating any universal anti-psi2 lower bound already at t=1.\n\nCandidate contribution (counterexample; novelty confidence low): For every dimension n and every epsilon in (0,1/2), there is a compact, positive-volume, full-dimensional, centered, rotationally invariant set whose normalized Lebesgue measure satisfies P(|| >= E||) <= epsilon simultaneously for every unit direction theta." }, { "id": 20001296, "problem_number": "AIM-CONVEX_GEOMETRY-0028", "title": "The half-mass hitting-time endpoint for lazy reversible chains", "statement": "Problem 9 (Yuval Peres) Consider lazy random walks on graphs or reversible Markov chains. Define the mixing time τTV (1 /2) as minimum t such that the total variation distance between pt(x, ·) and μ (stationary distribution) is at most 1/2.Does there exist c > 0 such that\n\nτTV (1 /2) ≤ c max\n\n> x,A:μ(A)≥1/2\n\nE TA,x,\n\nwhere TA,x is the hitting time for hitting A starting from x.", "original_statement": "Problem 9 (Yuval Peres) Consider lazy random walks on graphs or reversible Markov chains. Define the mixing time τTV (1 /2) as minimum t such that the total variation distance between pt(x, ·) and μ (stationary distribution) is at most 1/2.Does there exist c > 0 such that \n\nτTV (1 /2) ≤ c max \n\n> x,A:μ(A)≥1/2\n\nE TA,x,\n\nwhere TA,x is the hitting time for hitting A starting from x.", "clean_statement": "Problem 9 (Yuval Peres) Consider lazy random walks on graphs or reversible Markov chains. Define the mixing time τTV (1 /2) as minimum t such that the total variation distance between pt(x, ·) and μ (stationary distribution) is at most 1/2.Does there exist c > 0 such that\n\nτTV (1 /2) ≤ c max\n\n> x,A:μ(A)≥1/2\n\nE TA,x,\n\nwhere TA,x is the hitting time for hitting A starting from x.", "statement_status": "exact", "statement_verification": "The source is Problem 9, attributed to Yuval Peres, in *Problems from the AIM Workshop on Algorithmic Convex Geometry* (2007). The PDF asks about lazy random walks on graphs or reversible Markov chains. Its intended worst-case total-variation mixing time is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Algorithmic convex geometry\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf\nCanonical location: aim-convex-geometry-notes.json notes[27]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 9 (Yuval Peres) Consider lazy random walks on graphs or reversible Markov chains. Define the mixing time τTV (1 /2) as minimum t such that the total variation distance between pt(x, ·) and μ (stationary distribution) is at most 1/2.Does there exist c > 0 such that \\n\\nτTV (1 /2) ≤ c max \\n\\n> x,A:μ(A)≥1/2\\n\\nE TA,x,\\n\\nwhere TA,x is the hitting time for hitting A starting from x.\"\nOriginal remarks: [\"Remark. It was proved by Aldous that there exists a c such that \\n\\nτTV (1 /2) ≤ cμ (A) max \\n\\n> x,A\\n\\nETA,x,\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/convexgeometry/convexgeometry.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0028", "aim-domain:convex-geometry", "aim-workshop:convexgeometry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM question has an affirmative published solution: Griffiths, Kang, Oliveira, and Patel proved that T(1/2)=max_{x,A:mu(A)>=1/2} E_x T_A is universally equivalent to the standard t_mix(1/4) for every finite irreducible reversible discrete-time chain with P(x,x)>=1/2. Hence tau_TV(1/2)<=c T(1/2). Their sharp endpoint comparison gives (1/2)T(1/2)<=t_prod<=T(1/2), where t_prod=max mu(A)E_xT_A, and Aldous's theorem supplies the mixing comparison. The report also proves that every lazy two-state chain satisfies the sharper optimal bound tau_TV(1/2)<=T(1/2), and gives an explicit failure for every target threshold beta>1/2.\n\nCandidate contribution (sharp_special_case; novelty confidence low): For every irreducible lazy two-state chain P=[[1-a,a],[b,1-b]], the exact worst-case distance and half-mass hitting parameter imply tau_TV(1/2)<=T(1/2) with optimal uniform constant 1; the family b=1/2-epsilon, a=epsilon^2 makes the ratio tend to 1." }, { "id": 20001297, "problem_number": "AIM-CONVEX_GEOMETRY-0029", "title": "Vertex-weight perturbations do not preserve lazy mixing time", "statement": "Problem 10 (Yuval Peres) Consider random walks on undirected graphs where each node\n\nv is given a positive weight Wv (the probability of going from node u to node v is given by\n\n> WvP\n> w∈N(u)Ww, where N (u) is the set of neighbors of u in the graph). If we change the weights by a bounded factor, does the mixing time also change by a bounded factor?", "original_statement": "Problem 10 (Yuval Peres) Consider random walks on undirected graphs where each node \n\nv is given a positive weight Wv (the probability of going from node u to node v is given by \n\n> WvP\n> w∈N(u)Ww, where N (u) is the set of neighbors of u in the graph). If we change the weights by a bounded factor, does the mixing time also change by a bounded factor?", "clean_statement": "Problem 10 (Yuval Peres) Consider random walks on undirected graphs where each node\n\nv is given a positive weight Wv (the probability of going from node u to node v is given by\n\n> WvP\n> w∈N(u)Ww, where N (u) is the set of neighbors of u in the graph). If we change the weights by a bounded factor, does the mixing time also change by a bounded factor?", "statement_status": "exact", "statement_verification": "The source is Problem 10, attributed to Yuval Peres, in the AIM workshop list *Problems from the AIM Workshop on Algorithmic Convex Geometry*. The corpus transcription has lost the fraction bar. The PDF gives the following transition rule: on a finite undirected graph \\(G=(V,E)\\), give each vertex \\(v\\) a positive weight \\(W_v\\), and set", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Algorithmic convex geometry\nSection: \nSource item: 10\nSource URL: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf\nCanonical location: aim-convex-geometry-notes.json notes[28]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 10 (Yuval Peres) Consider random walks on undirected graphs where each node \\n\\nv is given a positive weight Wv (the probability of going from node u to node v is given by \\n\\n> WvP\\n> w∈N(u)Ww, where N (u) is the set of neighbors of u in the graph). If we change the weights by a bounded factor, does the mixing time also change by a bounded factor?\"\nOriginal remarks: [\"Remark. This is true if the mixing time is replaced by spetral gap.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/convexgeometry/convexgeometry.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0029", "aim-domain:convex-geometry", "aim-workshop:convexgeometry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the standard 1/2-lazy total-variation convention, the answer is negative. Subdividing every edge e of an arbitrary conductance network by a midpoint m_e and assigning weights W(u)=1 on original vertices and W(m_e)=c_e makes the two-step trace on original vertices exactly the original lazy conductance walk. A large-set hitting-time argument proves that the lazy mixing times before and after this lift are comparable by universal constants. Applying this to the Ding-Peres bounded-degree networks converts their edge factors in [1,2] into vertex-weight factors in [1,2] and retains an Omega(log n/log log n) mixing-time ratio.\n\nCandidate contribution (reduction; novelty confidence low): For every finite connected positive-conductance network, the once-subdivided vertex-weight lift W(u)=1 and W(m_e)=c_e has lazy total-variation mixing time comparable, by universal constants, to that of the original lazy conductance walk; consequently arbitrary bounded edge-conductance sensitivity transfers factor-for-factor in order to bounded vertex-weight sensitivity." }, { "id": 20001298, "problem_number": "AIM-CONVEX_GEOMETRY-0030", "title": "The sharp Gaussian perimeter bound for ellipsoids", "statement": "Problem 11 (Ryan O'Donnell) Is it true that every ellipsoid has Gaussian surface area bounded by a universal constant?", "original_statement": "Problem 11 (Ryan O'Donnell) Is it true that every ellipsoid has Gaussian surface area bounded by a universal constant?", "clean_statement": "Problem 11 (Ryan O'Donnell) Is it true that every ellipsoid has Gaussian surface area bounded by a universal constant?", "statement_status": "exact", "statement_verification": "The official AIM PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Algorithmic convex geometry\nSection: \nSource item: 11\nSource URL: https://aimath.org/WWN/convexgeometry/convexgeometry.pdf\nCanonical location: aim-convex-geometry-notes.json notes[29]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 11 (Ryan O'Donnell) Is it true that every ellipsoid has Gaussian surface area bounded by a universal constant?\"\nOriginal remarks: [\"Remark. Keith Ball has shown that for a general convex body the answer is O(n1/4); this bound is attained by intersection of random half-spaces. 3\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/convexgeometry/convexgeometry.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0030", "aim-domain:convex-geometry", "aim-workshop:convexgeometry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the standard Gaussian density (2pi)^(-n/2) exp(-|x|^2/2), every full-dimensional bounded ellipsoid in R^n, including every translated ellipsoid, has Gaussian perimeter strictly less than sqrt(2/pi). The constant is optimal: in every fixed dimension the supremum over such ellipsoids is sqrt(2/pi), approached by intervals shrinking to the origin in dimension one and, in higher dimensions, by long ellipsoids whose perimeters first converge to those of fixed-width slabs and whose slab half-widths then tend to zero. Kane's degree-d polynomial-threshold theorem supplied the historical affirmative solution; a specialized Gaussian Crofton identity gives a self-contained quadratic proof and strict nonattainment.\n\nCandidate contribution (sharp_bound; novelty confidence low): For every fixed n at least 1, the supremum of standard Gaussian perimeter over full-dimensional bounded ellipsoids in R^n is exactly sqrt(2/pi), and no such ellipsoid attains the supremum." }, { "id": 20001299, "problem_number": "AIM-CONVEX_GEOMETRY-0031", "title": "An exact one-harmonic threshold for polar-zonoid candidates", "statement": "1. (R. Schneider) Let Zn ⊂ Rn be a zonoid whose polar Zon is also a zonoid. Let d\n\ndenote the Banach-Mazur distance. Does d(Zn, B n\n\n> 2\n\n) converge to 1 as the dimension\n\nn tends to infinity?", "original_statement": "1. (R. Schneider) Let Zn ⊂ Rn be a zonoid whose polar Zon is also a zonoid. Let d\n\ndenote the Banach-Mazur distance. Does d(Zn, B n \n\n> 2\n\n) converge to 1 as the dimension \n\nn tends to infinity?", "clean_statement": "1. (R. Schneider) Let Zn ⊂ Rn be a zonoid whose polar Zon is also a zonoid. Let d\n\ndenote the Banach-Mazur distance. Does d(Zn, B n\n\n> 2\n\n) converge to 1 as the dimension\n\nn tends to infinity?", "statement_status": "exact", "statement_verification": "The canonical JSON record is visibly damaged by PDF extraction: `Zn`, `Zon`, and `B n > 2` have lost their subscript/superscript placement. The official AIM PDF gives the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[30]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (R. Schneider) Let Zn ⊂ Rn be a zonoid whose polar Zon is also a zonoid. Let d\\n\\ndenote the Banach-Mazur distance. Does d(Zn, B n \\n\\n> 2\\n\\n) converge to 1 as the dimension \\n\\nn tends to infinity?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0031", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The full asymptotic problem remains open in the literature checked. For every fixed m and n at least 2m, the balanced even harmonic Y_{m,n}(u)=(2m)^m product_{j=1}^{2m}u_j has range [-1,1], and h=1+epsilon Y_{m,n} is a zonoid support function if and only if |epsilon| is at most a_{m,n}=(2m-3)!!/product_{j=1}^m(n+2j-1). A smooth-polarity and inverse-cosine-transform continuity argument proves that, for each fixed (m,n), some nonzero interval |epsilon| 12", "original_statement": "2. (R. Schneider) For an origin symmetric convex body K ∈ Rn, let λ(K) be the smallest number λ ≥ 1 such that there exists a zonoid Z with K ⊂ Z ⊂ λK. For example, for the cross-polytope Cn we have \n\nλ(Cn) = n\n\n2n−1\n\n((n−1) [( n−1) /2] \n\n)\n\n≈\n\n√2nπ\n\nas n → ∞. Is λ(K) maximal for the cross-polytope? \n\n> 12", "clean_statement": "2. (R. Schneider) For an origin symmetric convex body K ∈ Rn, let λ(K) be the smallest number λ ≥ 1 such that there exists a zonoid Z with K ⊂ Z ⊂ λK. For example, for the cross-polytope Cn we have\n\nλ(Cn) = n\n\n2n−1\n\n((n−1) [( n−1) /2]\n\n)\n\n≈\n\n√2nπ\n\nas n → ∞. Is λ(K) maximal for the cross-polytope?\n\n> 12", "statement_status": "exact", "statement_verification": "The canonical dataset record is item 2, attributed to R. Schneider, in the AIM problem list *Fourier analytic methods in convex geometry*. Its OCR text splits the displayed fraction across lines, renders the binomial coefficient as `((n-1) [(n-1)/2])`, collapses the square root to `sqrt(2n pi)`, and appends `12` after the problem.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[31]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (R. Schneider) For an origin symmetric convex body K ∈ Rn, let λ(K) be the smallest number λ ≥ 1 such that there exists a zonoid Z with K ⊂ Z ⊂ λK. For example, for the cross-polytope Cn we have \\n\\nλ(Cn) = n\\n\\n2n−1\\n\\n((n−1) [( n−1) /2] \\n\\n)\\n\\n≈\\n\\n√2nπ\\n\\nas n → ∞. Is λ(K) maximal for the cross-polytope? \\n\\n> 12\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0032", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The global cross-polytope extremality question remains open in the literature checked, but Schneider's sandwich parameter admits an exact attained cosine-transform minimax formulation. An optimizer may be centered, the parameter is affine invariant, and an optimizer may be averaged over every compact symmetry group of K; after orthogonalizing the group, its generating measure may also be invariant. Moreover, lambda(K) is at most sqrt(n), equals 1 in dimensions at most two, and the Rademacher zonotope together with Schneider's sharp sign-sum inequality recovers the stated exact cross-polytope value.\n\nCandidate contribution (reduction; novelty confidence low): For every origin-symmetric convex body K, lambda(K) is the attained minimum over finite even positive measures mu with positive minimum ratio of max(C mu/h_K) divided by min(C mu/h_K), and for every compact symmetry group of K an optimizer can be chosen invariant, with an invariant generating measure in orthogonal coordinates." }, { "id": 20001301, "problem_number": "AIM-CONVEX_GEOMETRY-0033", "title": "Radial harmonic universality and linear-size zonotope approximation", "statement": "3. Let Kn denote the class of convex bodies in Rn. A Minkowski class M is a subset of\n\nKn that is closed in the Hausdorff metric, under Minkowski linear combinations and translations. A body K is a generalized M-body if there are two bodies M1, M 2 ∈ M\n\nsuch that K + M1 = M2.Let G be a subgroup of GL (n). We say that M is G-invariant if whenever K ∈ M\n\nand g ∈ G, we have gK ∈ M. Given B ∈ K n, G ⊂ GL (n), we define MB,G as the smallest Minkowski class containing B that is G-invariant.\n\nTheorem (Schneider, F. Schuster) Let B ∈ K n be non-symmetric. Every neighborhood of B contains an affine image B′ of B such that the generalized\n\nMB′,SO (n) bodies are dense in Kn.\n\nProblem: (R. Gardner) Dualize the theorem. If we replace generalized zonoids by \"generalized intersection bodies\", Minkowski sums by radial sums, etc, does a similar theorem hold? 4. In Rn, how many segments do we need to approximate a zonoid by zonotopes? Let\n\nK be a zonoid and Z a zonotope that is the sum of M segments. If d(Z, K ) ≤ 1 + [U+000F],then M is of the order C([U+000F])n log n (Talagrand). If K is the Euclidean ball, this can be improved to C([U+000F])n. Is the extra log n in Talagrand's result necessary?", "original_statement": "3. Let Kn denote the class of convex bodies in Rn. A Minkowski class M is a subset of \n\nKn that is closed in the Hausdorff metric, under Minkowski linear combinations and translations. A body K is a generalized M-body if there are two bodies M1, M 2 ∈ M \n\nsuch that K + M1 = M2.Let G be a subgroup of GL (n). We say that M is G-invariant if whenever K ∈ M\n\nand g ∈ G, we have gK ∈ M. Given B ∈ K n, G ⊂ GL (n), we define MB,G as the smallest Minkowski class containing B that is G-invariant. \n\nTheorem (Schneider, F. Schuster) Let B ∈ K n be non-symmetric. Every neighborhood of B contains an affine image B′ of B such that the generalized \n\nMB′,SO (n) bodies are dense in Kn.\n\nProblem: (R. Gardner) Dualize the theorem. If we replace generalized zonoids by \"generalized intersection bodies\", Minkowski sums by radial sums, etc, does a similar theorem hold? 4. In Rn, how many segments do we need to approximate a zonoid by zonotopes? Let \n\nK be a zonoid and Z a zonotope that is the sum of M segments. If d(Z, K ) ≤ 1 + \u000f,then M is of the order C(\u000f)n log n (Talagrand). If K is the Euclidean ball, this can be improved to C(\u000f)n. Is the extra log n in Talagrand's result necessary?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record has accidentally fused two consecutive problems from the AIM workshop list *Fourier analytic methods in convex geometry*. Inspection of the linked PDF separates them as follows.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[32]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. Let Kn denote the class of convex bodies in Rn. A Minkowski class M is a subset of \\n\\nKn that is closed in the Hausdorff metric, under Minkowski linear combinations and translations. A body K is a generalized M-body if there are two bodies M1, M 2 ∈ M \\n\\nsuch that K + M1 = M2.Let G be a subgroup of GL (n). We say that M is G-invariant if whenever K ∈ M\\n\\nand g ∈ G, we have gK ∈ M. Given B ∈ K n, G ⊂ GL (n), we define MB,G as the smallest Minkowski class containing B that is G-invariant. \\n\\nTheorem (Schneider, F. Schuster) Let B ∈ K n be non-symmetric. Every neighborhood of B contains an affine image B′ of B such that the generalized \\n\\nMB′,SO (n) bodies are dense in Kn.\\n\\nProblem: (R. Gardner) Dualize the theorem. If we replace generalized zonoids by \\\"generalized intersection bodies\\\", Minkowski sums by radial sums, etc, does a similar theorem hold? 4. In Rn, how many segments do we need to approximate a zonoid by zonotopes? Let \\n\\nK be a zonoid and Z a zonotope that is the sum of M segments. If d(Z, K ) ≤ 1 + \\u000f,then M is of the order C(\\u000f)n log n (Talagrand). If K is the Euclidean ball, this can be improved to C(\\u000f)n. Is the extra log n in Talagrand's result necessary?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0033", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The canonical record fuses two AIM problems. For the dualization problem, an exact uniform radial-harmonic criterion is proved: generalized bodies from the rotation-generated radial class of B are dense in all star bodies exactly when every harmonic component of rho_B is nonzero, with the even-degree version in the origin-symmetric category. Standard intersection bodies have even radial functions, giving a quantitative obstruction to nonsymmetric density; nevertheless, arbitrarily small spheroidal perturbations of the Euclidean ball generate a generalized radial class dense among symmetric star bodies. For the zonoid problem, Reis and Rothvoss (arXiv:2606.28147, June 2026) have removed Talagrand's log n factor, proving O(n epsilon^{-2} log(1/epsilon)) segments suffice. An independent elementary O_{r,epsilon}(n) approximation theorem is also proved for direct sums of zonoid blocks of dimension at most r.\n\nCandidate contribution (theorem; novelty confidence low): Candidate theorem: in the uniform radial metric, the generalized rotation-generated radial class of B is dense exactly when rho_B has a nonzero projection in every spherical-harmonic degree (every even degree for symmetric targets); moreover, a generic arbitrarily small spheroidal perturbation of the ball satisfies the even criterion, while standard intersection-body classes have a sharp parity obstruction outside the symmetric category." }, { "id": 20001302, "problem_number": "AIM-CONVEX_GEOMETRY-0034", "title": "Optimal vector balancing extends from zonotopes to all zonoids", "statement": "5. (G. Schechtman) A theorem of Spencer states that if {Xi}ni=1 are in Bn\n\n> ∞\n\nand [U+000F]i = ±1then min\n\n> [U+000F]i\n\n‖\n\n> n\n\n∑\n\n> i=1\n\n[U+000F]iXi‖∞ ≤ C√n.\n\nDoes the same hold for any n-dimensional zonoid? i.e., does there exist a universal constant C such that for all n-dimensional centered zonoid Z, if {Xi}ni=1 ∈ Z then there are signs {[U+000F]i}ni=1 with ∑ni=1 [U+000F]iXi ∈ C√nZ?(If one can remove the log factor in problem 4 then the answer here is positive. As is, the best known substitute for C√n is C√n log log n.) 3", "original_statement": "5. (G. Schechtman) A theorem of Spencer states that if {Xi}ni=1 are in Bn \n\n> ∞\n\nand \u000fi = ±1then min \n\n> \u000fi\n\n‖\n\n> n\n\n∑\n\n> i=1\n\n\u000fiXi‖∞ ≤ C√n. \n\nDoes the same hold for any n-dimensional zonoid? i.e., does there exist a universal constant C such that for all n-dimensional centered zonoid Z, if {Xi}ni=1 ∈ Z then there are signs {\u000fi}ni=1 with ∑ni=1 \u000fiXi ∈ C√nZ?(If one can remove the log factor in problem 4 then the answer here is positive. As is, the best known substitute for C√n is C√n log log n.) 3", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record contains flattened mathematical layout, U+000F control characters, and a trailing page number. The official AIM PDF verifies the following statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[33]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"5. (G. Schechtman) A theorem of Spencer states that if {Xi}ni=1 are in Bn \\n\\n> ∞\\n\\nand \\u000fi = ±1then min \\n\\n> \\u000fi\\n\\n‖\\n\\n> n\\n\\n∑\\n\\n> i=1\\n\\n\\u000fiXi‖∞ ≤ C√n. \\n\\nDoes the same hold for any n-dimensional zonoid? i.e., does there exist a universal constant C such that for all n-dimensional centered zonoid Z, if {Xi}ni=1 ∈ Z then there are signs {\\u000fi}ni=1 with ∑ni=1 \\u000fiXi ∈ C√nZ?(If one can remove the log factor in problem 4 then the answer here is positive. As is, the best known substitute for C√n is C√n log log n.) 3\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0034", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Victor Reis's May 2026 preprint proves that every d-dimensional zonotope P and every r<=d vectors in P admit signs whose sum lies in C sqrt(r log(2d/r)) P. The report proves that the finite vector-balancing functional beta_r is multiplicatively continuous: if aK is contained in L and L is contained in bK, then beta_r(L) differs from beta_r(K) by at most the factor b/a. Hausdorff approximation of a full-dimensional centered zonoid by centered zonotopes gives such sandwiches with b/a tending to 1. Reis's estimate therefore extends with no constant loss to every centered zonoid. Taking r=d=n yields the requested C sqrt(n) bound and answers the exact AIM question affirmatively.\n\nCandidate contribution (closure_extension_lemma; novelty confidence low): For full-dimensional origin-symmetric convex bodies satisfying aK subset L subset bK, the finite vector-balancing functional obeys (a/b) beta_r(K) <= beta_r(L) <= (b/a) beta_r(K); hence it is Hausdorff-continuous and every uniform zonotope balancing theorem passes unchanged to all zonoids." }, { "id": 20001303, "problem_number": "AIM-CONVEX_GEOMETRY-0035", "title": "A polylogarithmic nonsymmetric MM* bound and exact simplex calculation", "statement": "6. (M. Rudelson) Let K be a symmetric convex body. There exists a linear operator\n\nT such that\n\n∫\n\n> Sn−1\n\nhT K (θ) dσ (θ) ·\n\n∫\n\n> Sn−1\n\nh(T K )o (θ) dσ (θ) ≤ C log n.\n\nWhat is the upper bound for the above quantity for non-symmetric convex bodies\n\nK? (in this case we need to take T affine).", "original_statement": "6. (M. Rudelson) Let K be a symmetric convex body. There exists a linear operator \n\nT such that \n\n∫\n\n> Sn−1\n\nhT K (θ) dσ (θ) ·\n\n∫\n\n> Sn−1\n\nh(T K )o (θ) dσ (θ) ≤ C log n. \n\nWhat is the upper bound for the above quantity for non-symmetric convex bodies \n\nK? (in this case we need to take T affine).", "clean_statement": "6. (M. Rudelson) Let K be a symmetric convex body. There exists a linear operator\n\nT such that\n\n∫\n\n> Sn−1\n\nhT K (θ) dσ (θ) ·\n\n∫\n\n> Sn−1\n\nh(T K )o (θ) dσ (θ) ≤ C log n.\n\nWhat is the upper bound for the above quantity for non-symmetric convex bodies\n\nK? (in this case we need to take T affine).", "statement_status": "exact", "statement_verification": "The canonical record is problem 6, attributed to M. Rudelson, in the AIM list *Fourier analytic methods in convex geometry*. Inspection of the official PDF repairs the OCR and gives:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[34]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"6. (M. Rudelson) Let K be a symmetric convex body. There exists a linear operator \\n\\nT such that \\n\\n∫\\n\\n> Sn−1\\n\\nhT K (θ) dσ (θ) ·\\n\\n∫\\n\\n> Sn−1\\n\\nh(T K )o (θ) dσ (θ) ≤ C log n. \\n\\nWhat is the upper bound for the above quantity for non-symmetric convex bodies \\n\\nK? (in this case we need to take T affine).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0035", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Bizeul and Klartag's July 2026 preprint proves that every convex body has an affine isotropic position in which the two AIM support-function integrals have product at most C log^3(n+1), superseding Rudelson's n^(1/3)-times-polylogarithmic bound; the exact logarithmic exponent remains open. Independently, this attempt proves the universal lower bound 1, an asymmetry-sensitive difference-body reduction Phi(K) <= (1+s(K))Phi((K-K)/2)/2, and an exact Gaussian formula showing that the centered regular simplex has product asymptotic to 2 log n.\n\nCandidate contribution (explicit_family; novelty confidence low): For the centered regular simplex Delta_n with unit vertices and pairwise inner products -1/n, Delta_n polar equals -n Delta_n and its MM* product is exactly ((n+1)/(E|G_n|)^2)(E max_{1<=i<=n+1} g_i)^2, which is asymptotic to 2 log n; moreover Phi(K) <= (1+s(K))Phi((K-K)/2)/2 for every convex body K." }, { "id": 20001304, "problem_number": "AIM-CONVEX_GEOMETRY-0036", "title": "Uniform point configurations and exact Fourier certificates for convexity", "statement": "7. (S. Robins) Given N, can we find explicitly N points on Sn−1 that are \"uniformly\" arranged? Are they the same as the vertices of the polytope with N vertices and maximum volume that is contained in Sn−1?8. Let f be a function on Sn−1. Can we decide from the Fourier transform of f if f is the support function of a convex body? Or if f is the radial function of a convex body?", "original_statement": "7. (S. Robins) Given N, can we find explicitly N points on Sn−1 that are \"uniformly\" arranged? Are they the same as the vertices of the polytope with N vertices and maximum volume that is contained in Sn−1?8. Let f be a function on Sn−1. Can we decide from the Fourier transform of f if f is the support function of a convex body? Or if f is the radial function of a convex body?", "clean_statement": "7. (S. Robins) Given N, can we find explicitly N points on Sn−1 that are \"uniformly\" arranged? Are they the same as the vertices of the polytope with N vertices and maximum volume that is contained in Sn−1?8. Let f be a function on Sn−1. Can we decide from the Fourier transform of f if f is the support function of a convex body? Or if f is the radial function of a convex body?", "statement_status": "exact", "statement_verification": "The canonical record is not one problem. It concatenates Problems 7 and 8 of the official 2007 AIM workshop list *Fourier analytic methods in convex geometry*. The PDF (printed page 4) reads, with its numbering restored:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[35]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"7. (S. Robins) Given N, can we find explicitly N points on Sn−1 that are \\\"uniformly\\\" arranged? Are they the same as the vertices of the polytope with N vertices and maximum volume that is contained in Sn−1?8. Let f be a function on Sn−1. Can we decide from the Fourier transform of f if f is the support function of a convex body? Or if f is the radial function of a convex body?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0036", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The canonical record fuses AIM Problems 7 and 8. For Problem 7, the report proves that regular polygons and regular simplices simultaneously maximize inscribed volume, form spherical designs, and optimize separation, but also proves a sharp mismatch: when n is odd and N=n+2 no equal-weight spherical 2-design exists, whereas the Horvath-Langi maximum-volume join of two orthogonal regular simplices is an explicit weighted 2-design. For Problem 8, it proves an exact nonsmooth characterization: a continuous f on S^{n-1} is a support function exactly when every Toeplitz matrix built from the coefficients (1-k^2)c_k of every great-circle restriction is positive semidefinite; a positive f is radial exactly when the same holds for 1/f. A single high odd spherical harmonic perturbation of 1 proves that no fixed finite harmonic cutoff can decide either property.\n\nCandidate contribution (fourier_slice_characterization; novelty confidence low): For every continuous f on S^{n-1}, support-function recognition is equivalent to positive semidefiniteness of all great-circle Toeplitz matrices [(1-(j-k)^2)c_{j-k}]_{j,k=0}^m; applying the same criterion to 1/f characterizes positive radial functions, and for every finite harmonic cutoff L an explicit positive perturbation f=1+2(n-1)Y_m/[m(m+n-2)] with odd m>L has the same low modes as 1 but is neither support nor radial." }, { "id": 20001305, "problem_number": "AIM-CONVEX_GEOMETRY-0037", "title": "Rotation-equivariant Minkowski-additive operators", "statement": "9. (R. Schneider) Classify all continuous T: Kn → K n with the following properties:\n\n• hT (K+L) = hT K + hT L\n\n• T (θK ) = θT K for every rotation θ.\n1", "original_statement": "9. (R. Schneider) Classify all continuous T: Kn → K n with the following properties: \n\n• hT (K+L) = hT K + hT L \n\n• T (θK ) = θT K for every rotation θ.\n1", "clean_statement": "9. (R. Schneider) Classify all continuous T: Kn → K n with the following properties:\n\n• hT (K+L) = hT K + hT L\n\n• T (θK ) = θT K for every rotation θ.\n1", "statement_status": "exact", "statement_verification": "The canonical record is Problem 9, attributed to Rolf Schneider, in the AIM workshop notes *Fourier Analytic Methods in Convexity*. The official PDF gives the following statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[36]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"9. (R. Schneider) Classify all continuous T: Kn → K n with the following properties: \\n\\n• hT (K+L) = hT K + hT L \\n\\n• T (θK ) = θT K for every rotation θ.\\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0037", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every operator under the exact AIM hypotheses has a unique decomposition T(K)=Phi(K)+{A s(K)}, where Phi is a translation-invariant Minkowski endomorphism and A commutes with SO(n); thus A=aI for n at least 3 and A=aI+bJ in dimension 2. The normalized part has a unique centered zonal signed generating measure subject to the nontrivial requirement that convolution preserve every support function. This gives a full planar and weakly-monotone classification, while the intrinsic higher-dimensional generator cone remains open. A harmonic audit proves that the first multiplier vanishes, the zeroth is nonnegative, and zeroth multiplier zero forces Phi=0.\n\nCandidate contribution (theorem; novelty confidence low): Under exactly the unnormalized AIM assumptions, the action on singleton bodies is the entire translation defect and yields the unique formula T(K)=Phi(K)+{A s(K)}; proper rotations permit the exceptional A=aI+bJ in dimension two, and the normalized convolution representation forces m_1=0 and either m_0>0 or Phi=0." }, { "id": 20001306, "problem_number": "AIM-CONVEX_GEOMETRY-0038", "title": "Closedness, dimensional correction, and a dual certificate for polar-zonoid intersection bodies", "statement": "0. (R. Schneider) Show that for most origin-symmetric convex K, the intersection body of K is not the polar of a zonoid. (\"Most\" means in the Baire category sense).\n1", "original_statement": "0. (R. Schneider) Show that for most origin-symmetric convex K, the intersection body of K is not the polar of a zonoid. (\"Most\" means in the Baire category sense). \n1", "clean_statement": "0. (R. Schneider) Show that for most origin-symmetric convex K, the intersection body of K is not the polar of a zonoid. (\"Most\" means in the Baire category sense).\n1", "statement_status": "exact", "statement_verification": "The canonical record reads", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 0\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[37]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"0. (R. Schneider) Show that for most origin-symmetric convex K, the intersection body of K is not the polar of a zonoid. (\\\"Most\\\" means in the Baire category sense). \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0038", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For fixed n at least 3, in the Hausdorff Baire space of full-dimensional origin-symmetric convex bodies, the exceptional set of K for which (IK)^circ is a zonoid is closed. Therefore Schneider's residual non-zonoid assertion is exactly the unresolved density assertion for its open complement; the same affine-invariant reduction applies in the Banach--Mazur compactum. In dimension two the literal claim is false because IK is a rotated and scaled copy of K and every symmetric planar body is a zonoid. An exact Hahn--Banach certificate is also proved: writing f_K=(n-1)/R(rho_K^(n-1))=h_((IK)^circ), non-zonoidality is equivalent to an even signed measure nu with C nu nonnegative pointwise but integral f_K d nu negative, and every strict witness persists under small Hausdorff perturbations.\n\nCandidate contribution (dual obstruction lemma; novelty confidence low): For an origin-symmetric convex K in dimension n at least 3, (IK)^circ is not a zonoid if and only if there is a finite even signed measure nu such that C nu is pointwise nonnegative while (n-1) times the integral of 1/R(rho_K^(n-1)) against nu is strictly negative; a strict witness certifies an entire Hausdorff neighborhood of non-zonoid examples." }, { "id": 20001307, "problem_number": "AIM-CONVEX_GEOMETRY-0039", "title": "Small Fourier data do not force simplicial facets without local normalization", "statement": "1. (S. Robins) (a) Let K be a polytope and consider the expansion of hK in spherical harmonics. If all of the coefficients have small absolute value, are all the facets of K simplices or close to simplices? (b) If, for each integer frequency, the Fourier transform of the indicator function 1 K\n\nis small, does the same conclusion hold? (This conjecture is made by analogy with pseudorandom number sequences which exhibit this behavior in dimension 1, and the intuition here is that thegiven hypothesis on the coefficients should exhibit a \"random\" polytope in some sense.) 4\n\n1", "original_statement": "1. (S. Robins) (a) Let K be a polytope and consider the expansion of hK in spherical harmonics. If all of the coefficients have small absolute value, are all the facets of K simplices or close to simplices? (b) If, for each integer frequency, the Fourier transform of the indicator function 1 K\n\nis small, does the same conclusion hold? (This conjecture is made by analogy with pseudorandom number sequences which exhibit this behavior in dimension 1, and the intuition here is that thegiven hypothesis on the coefficients should exhibit a \"random\" polytope in some sense.) 4\n\n1", "clean_statement": "1. (S. Robins) (a) Let K be a polytope and consider the expansion of hK in spherical harmonics. If all of the coefficients have small absolute value, are all the facets of K simplices or close to simplices? (b) If, for each integer frequency, the Fourier transform of the indicator function 1 K\n\nis small, does the same conclusion hold? (This conjecture is made by analogy with pseudorandom number sequences which exhibit this behavior in dimension 1, and the intuition here is that thegiven hypothesis on the coefficients should exhibit a \"random\" polytope in some sense.) 4\n\n1", "statement_status": "exact", "statement_verification": "The canonical JSON record is an OCR extraction from the AIM workshop compilation *Fourier analytic methods in convex geometry*. Comparison with the original PDF shows that this is Problem **11**, proposed by S. Robins, rather than Problem 1. The initial `1` in the JSON is the second digit of `11`; the trailing `4` is the printed page number, and the final `1` is extraction debris. The source also prints `thegiven`, which should be read as “the given.”", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[38]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (S. Robins) (a) Let K be a polytope and consider the expansion of hK in spherical harmonics. If all of the coefficients have small absolute value, are all the facets of K simplices or close to simplices? (b) If, for each integer frequency, the Fourier transform of the indicator function 1 K\\n\\nis small, does the same conclusion hold? (This conjecture is made by analogy with pseudorandom number sequences which exhibit this behavior in dimension 1, and the intuition here is that thegiven hypothesis on the coefficients should exhibit a \\\"random\\\" polytope in some sense.) 4\\n\\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0039", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For dimensions n at least 3, the literal statement fails by scaling. More strongly, under a natural centered and mean-support-normalized reading of part (a), there are polytopes converging uniformly in support function to the Euclidean ball whose every nonconstant spherical-harmonic coefficient tends uniformly to zero, yet each has a facet of a fixed nonsimplicial diameter-normalized shape. Under the natural nonzero integer-lattice reading of part (b), the centered volume-one cube is isotropic up to scalar, has nonsimplicial facets, and has Fourier transform exactly zero at every nonzero integer lattice point. These are counterexamples only to the literal and explicitly stated natural normalized readings, not to every possible intended conjecture.\n\nCandidate contribution (obstruction theorem; novelty confidence low): A paired normalization audit proves that Steiner centering and mean-support normalization do not rescue support-function coefficient smallness, because near-ball polytopes can retain a shrinking facet with a fixed positive simplex defect, while volume, centroid, and covariance normalization do not rescue integer-lattice Fourier sampling, because the cube has an exactly zero nonzero lattice spectrum." }, { "id": 20001308, "problem_number": "AIM-CONVEX_GEOMETRY-0040", "title": "Zonoid ratio in proportional-dimensional sections", "statement": "2. (Y. Gordon) Let K be a symmetric convex body. We define the zonoid ratio as\n\nzr (K) = min\n\n> Z∈Z,Z ⊂K\n\n|K|1/n\n\n|Z|1/n.\n\nThe volume ratio is defined as\n\nvr (K) = min\n\n> E ellipsoid, E ⊂K\n\n|K|1/n\n\n|E|1/n.\n\nClearly, vr (K) ≥ zr (K). Y. Gordon and M. Junge proved that there is a constant\n\nC0 > 0 such that every centrally symmetric convex body K in Rn has a section L\n\nof dimension [ n/ 2] such that zr (L) ≥ C0 vr (K). Since by K. Ball the local unconditional constant χu(L) is ≥ zr (L) for every convex L, and these parameters of the local theory are always hard to compute, it follows that C0 vr (K) yields an easy computational lower bound for some [ n/ 2] dimensional section L of K.Question: Suppose 0 < λ < 1 is given, what is the the greatest value f (λ) > 0such that for every n ≥ 3, every n-dimensional convex body K will have a section\n\nL of dimension [ λn ] for which zr (L) ≥ f (λ)vr (K)?\n1", "original_statement": "2. (Y. Gordon) Let K be a symmetric convex body. We define the zonoid ratio as \n\nzr (K) = min \n\n> Z∈Z,Z ⊂K\n\n|K|1/n \n\n|Z|1/n.\n\nThe volume ratio is defined as \n\nvr (K) = min \n\n> E ellipsoid, E ⊂K\n\n|K|1/n \n\n|E|1/n.\n\nClearly, vr (K) ≥ zr (K). Y. Gordon and M. Junge proved that there is a constant \n\nC0 > 0 such that every centrally symmetric convex body K in Rn has a section L\n\nof dimension [ n/ 2] such that zr (L) ≥ C0 vr (K). Since by K. Ball the local unconditional constant χu(L) is ≥ zr (L) for every convex L, and these parameters of the local theory are always hard to compute, it follows that C0 vr (K) yields an easy computational lower bound for some [ n/ 2] dimensional section L of K.Question: Suppose 0 < λ < 1 is given, what is the the greatest value f (λ) > 0such that for every n ≥ 3, every n-dimensional convex body K will have a section \n\nL of dimension [ λn ] for which zr (L) ≥ f (λ)vr (K)? \n1", "clean_statement": "2. (Y. Gordon) Let K be a symmetric convex body. We define the zonoid ratio as\n\nzr (K) = min\n\n> Z∈Z,Z ⊂K\n\n|K|1/n\n\n|Z|1/n.\n\nThe volume ratio is defined as\n\nvr (K) = min\n\n> E ellipsoid, E ⊂K\n\n|K|1/n\n\n|E|1/n.\n\nClearly, vr (K) ≥ zr (K). Y. Gordon and M. Junge proved that there is a constant\n\nC0 > 0 such that every centrally symmetric convex body K in Rn has a section L\n\nof dimension [ n/ 2] such that zr (L) ≥ C0 vr (K). Since by K. Ball the local unconditional constant χu(L) is ≥ zr (L) for every convex L, and these parameters of the local theory are always hard to compute, it follows that C0 vr (K) yields an easy computational lower bound for some [ n/ 2] dimensional section L of K.Question: Suppose 0 < λ < 1 is given, what is the the greatest value f (λ) > 0such that for every n ≥ 3, every n-dimensional convex body K will have a section\n\nL of dimension [ λn ] for which zr (L) ≥ f (λ)vr (K)?\n1", "statement_status": "exact", "statement_verification": "The canonical JSON labels this record as Problem 2, but the official AIM PDF places it on PDF page 4 as **Problem 12**, attributed to Y. Gordon. The printed statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[39]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (Y. Gordon) Let K be a symmetric convex body. We define the zonoid ratio as \\n\\nzr (K) = min \\n\\n> Z∈Z,Z ⊂K\\n\\n|K|1/n \\n\\n|Z|1/n.\\n\\nThe volume ratio is defined as \\n\\nvr (K) = min \\n\\n> E ellipsoid, E ⊂K\\n\\n|K|1/n \\n\\n|E|1/n.\\n\\nClearly, vr (K) ≥ zr (K). Y. Gordon and M. Junge proved that there is a constant \\n\\nC0 > 0 such that every centrally symmetric convex body K in Rn has a section L\\n\\nof dimension [ n/ 2] such that zr (L) ≥ C0 vr (K). Since by K. Ball the local unconditional constant χu(L) is ≥ zr (L) for every convex L, and these parameters of the local theory are always hard to compute, it follows that C0 vr (K) yields an easy computational lower bound for some [ n/ 2] dimensional section L of K.Question: Suppose 0 < λ < 1 is given, what is the the greatest value f (λ) > 0such that for every n ≥ 3, every n-dimensional convex body K will have a section \\n\\nL of dimension [ λn ] for which zr (L) ≥ f (λ)vr (K)? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0040", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literal AIM question needs a positive-dimensional rounding convention and, under its intended Banach-space reading, centrally symmetric bodies and central linear sections. For the exact-dimensional extremal value alpha_{n,m}, a cube and John's theorem give the rigorous bound alpha_{n,m} <= sqrt(m) kappa_n^(1/n)/2. Consequently the asymptotic optimal constant is at most min(1, sqrt(pi e/2) sqrt(lambda)), while the corrected all-n constant has the additional three-dimensional cap (4 pi/3)^(1/3)/2. The sharp lower dependence remains open.\n\nCandidate contribution (obstruction; novelty confidence low): For the rounding-corrected extremal formulation of the AIM question, alpha_{n,m} <= sqrt(m) kappa_n^(1/n)/2; hence F_infinity(lambda) <= sqrt(pi e/2) sqrt(lambda), and the all-n version is additionally capped by (4 pi/3)^(1/3)/2." }, { "id": 20001309, "problem_number": "AIM-CONVEX_GEOMETRY-0041", "title": "Lattice facet-count constraints and nonsymmetric projection reconstruction", "statement": "3. (S. Robins) Minkowski's Theorem in Zd. Given a finite number of vectors { ~ni} ⊂\n\nZd and integers αi ∈ Z, is there a polytope such that { ~ni} are normal to the facets and the number of integer points on the ith-facet is αi? Do we need some extra constraints? 14. Which projection or/and section data are needed to determine non-symmetric con-vex bodies? Several known cases: A non-symmetric convex body is uniquely determined if we know:\n\n• The volume of sections passing through a given interior point and the centers of gravity of those sections.\n\n• The mean width and the Steiner points.\n\n• The brightness function and the illumination function. 5\n\nQuestions:\n\n(a) (Schneider) Do the areas of projections plus the centers of gravity determine a body? (b) What is the dual version of the third known case (brightness + illumination)?\n1", "original_statement": "3. (S. Robins) Minkowski's Theorem in Zd. Given a finite number of vectors { ~ni} ⊂ \n\nZd and integers αi ∈ Z, is there a polytope such that { ~ni} are normal to the facets and the number of integer points on the ith-facet is αi? Do we need some extra constraints? 14. Which projection or/and section data are needed to determine non-symmetric con-vex bodies? Several known cases: A non-symmetric convex body is uniquely determined if we know: \n\n• The volume of sections passing through a given interior point and the centers of gravity of those sections. \n\n• The mean width and the Steiner points. \n\n• The brightness function and the illumination function. 5\n\nQuestions: \n\n(a) (Schneider) Do the areas of projections plus the centers of gravity determine a body? (b) What is the dual version of the third known case (brightness + illumination)? \n1", "clean_statement": "3. (S. Robins) Minkowski's Theorem in Zd. Given a finite number of vectors { ~ni} ⊂\n\nZd and integers αi ∈ Z, is there a polytope such that { ~ni} are normal to the facets and the number of integer points on the ith-facet is αi? Do we need some extra constraints? 14. Which projection or/and section data are needed to determine non-symmetric con-vex bodies? Several known cases: A non-symmetric convex body is uniquely determined if we know:\n\n• The volume of sections passing through a given interior point and the centers of gravity of those sections.\n\n• The mean width and the Steiner points.\n\n• The brightness function and the illumination function. 5\n\nQuestions:\n\n(a) (Schneider) Do the areas of projections plus the centers of gravity determine a body? (b) What is the dual version of the third known case (brightness + illumination)?\n1", "statement_status": "exact", "statement_verification": "The canonical record begins with `3. (S. Robins) Minkowski's Theorem in Zd` and then runs directly into `14. Which projection or/and section data...`. Inspection of the official AIM PDF shows that one extracted record has fused two independent consecutive problems. The official boundaries and repaired notation are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[40]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (S. Robins) Minkowski's Theorem in Zd. Given a finite number of vectors { ~ni} ⊂ \\n\\nZd and integers αi ∈ Z, is there a polytope such that { ~ni} are normal to the facets and the number of integer points on the ith-facet is αi? Do we need some extra constraints? 14. Which projection or/and section data are needed to determine non-symmetric con-vex bodies? Several known cases: A non-symmetric convex body is uniquely determined if we know: \\n\\n• The volume of sections passing through a given interior point and the centers of gravity of those sections. \\n\\n• The mean width and the Steiner points. \\n\\n• The brightness function and the illumination function. 5\\n\\nQuestions: \\n\\n(a) (Schneider) Do the areas of projections plus the centers of gravity determine a body? (b) What is the dual version of the third known case (brightness + illumination)? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0041", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The OCR record is proved to fuse official Problems 13 and 14. Under the lattice-polytope reading of Problem 13, a complete planar classification is obtained: primitive distinct cyclic outer normals u_i and edge counts alpha_i are realizable exactly when alpha_i is at least 2, the normals span, and the sum of (alpha_i-1)u_i is zero. In dimension three, every solution must admit integers q_i in [alpha_i-2, 2alpha_i-5] with sum q_i u_i=0; this rules out the positively spanning tetrahedral-normal data with counts (3,3,3,4). For Problem 14(a), projection areas and projection centroids reconstruct every planar body, and their full linearization is injective at the Euclidean ball in every dimension at least 3. For an explicit interpretation of Problem 14(b), central section volumes together with a directed hemispherical transform of rho^(n-1) are proved to reconstruct every star body about the origin.\n\nCandidate contribution (reconstruction theorem; novelty confidence low): For every star body L about the origin in dimension n at least 2, the pair consisting of all central hyperplane-section volumes and the directed radial hemispherical data j_L(u)=integral H()rho_L(v)^(n-1) dS(v) determines L uniquely: the Funk data recover the even part of rho_L^(n-1), while the hemispherical difference recovers its odd part." }, { "id": 20001310, "problem_number": "AIM-CONVEX_GEOMETRY-0042", "title": "Six recovered convex-geometric problems and an inradius lattice-point sandwich", "statement": "5. (A. Zvavitch) Consider the Gaussian measure of sections of the cube Bn\n\n> ∞. It is known that if n ≥ 3,\n\nγn−1\n\n(Bn\n\n> ∞\n\n∩ θ⊥) ≤ γn−1\n\n(√ nn − 1Bn−1\n\n> ∞\n\n).\n\nIs this the best upper bound? If we introduce a dilation factor r > 0, for which\n\nθ = θ(r) is γn−1\n\n(rB n\n\n> ∞\n\n∩ θ⊥) maximal? 16. Minimum of slabs of the cube Bn\n\n> ∞\n\nGiven t ≤ 2√2 − 2, the minimal slab is in the direction of (1, 0,..., 0). Conjecture: there are numbers t1 and t2 such that for 2√2 − 2 < t < t 1, the minimum is in the direction of (1, 1, 0,..., 0); for t1 < t < t 2,the minimum is in the direccion of (1, 1, 1, 0,..., 0) and for t2 < t, the minimum is in the direction of (1, 1,..., 1). 17. Given n + k points x1,..., x n+k on Sn−1, with k ≤ n, we want to cover the sphere with caps of radius r = r(n, k ) centered at those points. What is the best possible position for the points that will minimize r? How does this minimal r behave as function of n and k? 18. Given n + 1 points x1,..., x n+1 on Sn−1, find the configuration such that the con-vex hull of x1,..., x n+1 has the largest mean width. Is it maximal for the regular simplex? 19. Let K ⊂ Rn be a non symmetric set with centroid 0.\n\nvol (K ∩ (−K)) ≥ 2−nvol (K).\n\nIs the simplex the extremal case? If not, what is it? 20. Find the smallest radius R = R(n) such that if a convex body K contains a ball of radius R, then the number of integer points in K is equivalent to the volume of K,up to a multiplicative factor (a polynomial of n). 6\n\n2", "original_statement": "5. (A. Zvavitch) Consider the Gaussian measure of sections of the cube Bn\n\n> ∞. It is known that if n ≥ 3, \n\nγn−1\n\n(Bn \n\n> ∞\n\n∩ θ⊥) ≤ γn−1\n\n(√ nn − 1Bn−1\n\n> ∞\n\n).\n\nIs this the best upper bound? If we introduce a dilation factor r > 0, for which \n\nθ = θ(r) is γn−1\n\n(rB n \n\n> ∞\n\n∩ θ⊥) maximal? 16. Minimum of slabs of the cube Bn \n\n> ∞\n\nGiven t ≤ 2√2 − 2, the minimal slab is in the direction of (1, 0,..., 0). Conjecture: there are numbers t1 and t2 such that for 2√2 − 2 < t < t 1, the minimum is in the direction of (1, 1, 0,..., 0); for t1 < t < t 2,the minimum is in the direccion of (1, 1, 1, 0,..., 0) and for t2 < t, the minimum is in the direction of (1, 1,..., 1). 17. Given n + k points x1,..., x n+k on Sn−1, with k ≤ n, we want to cover the sphere with caps of radius r = r(n, k ) centered at those points. What is the best possible position for the points that will minimize r? How does this minimal r behave as function of n and k? 18. Given n + 1 points x1,..., x n+1 on Sn−1, find the configuration such that the con-vex hull of x1,..., x n+1 has the largest mean width. Is it maximal for the regular simplex? 19. Let K ⊂ Rn be a non symmetric set with centroid 0. \n\nvol (K ∩ (−K)) ≥ 2−nvol (K).\n\nIs the simplex the extremal case? If not, what is it? 20. Find the smallest radius R = R(n) such that if a convex body K contains a ball of radius R, then the number of integer points in K is equivalent to the volume of K,up to a multiplicative factor (a polynomial of n). 6\n\n2", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is an extraction accident: its field numbered 5 starts in the middle of official Problem 15 and then contains official Problems 16--20. The source is the AIM workshop list, Fourier analytic methods in convex geometry. Inspection of the PDF repairs the missing initial digit, display fractions, superscripts, and page debris, but does not otherwise rewrite the questions.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[41]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"5. (A. Zvavitch) Consider the Gaussian measure of sections of the cube Bn\\n\\n> ∞. It is known that if n ≥ 3, \\n\\nγn−1\\n\\n(Bn \\n\\n> ∞\\n\\n∩ θ⊥) ≤ γn−1\\n\\n(√ nn − 1Bn−1\\n\\n> ∞\\n\\n).\\n\\nIs this the best upper bound? If we introduce a dilation factor r > 0, for which \\n\\nθ = θ(r) is γn−1\\n\\n(rB n \\n\\n> ∞\\n\\n∩ θ⊥) maximal? 16. Minimum of slabs of the cube Bn \\n\\n> ∞\\n\\nGiven t ≤ 2√2 − 2, the minimal slab is in the direction of (1, 0,..., 0). Conjecture: there are numbers t1 and t2 such that for 2√2 − 2 < t < t 1, the minimum is in the direction of (1, 1, 0,..., 0); for t1 < t < t 2,the minimum is in the direccion of (1, 1, 1, 0,..., 0) and for t2 < t, the minimum is in the direction of (1, 1,..., 1). 17. Given n + k points x1,..., x n+k on Sn−1, with k ≤ n, we want to cover the sphere with caps of radius r = r(n, k ) centered at those points. What is the best possible position for the points that will minimize r? How does this minimal r behave as function of n and k? 18. Given n + 1 points x1,..., x n+1 on Sn−1, find the configuration such that the con-vex hull of x1,..., x n+1 has the largest mean width. Is it maximal for the regular simplex? 19. Let K ⊂ Rn be a non symmetric set with centroid 0. \\n\\nvol (K ∩ (−K)) ≥ 2−nvol (K).\\n\\nIs the simplex the extremal case? If not, what is it? 20. Find the smallest radius R = R(n) such that if a convex body K contains a ball of radius R, then the number of integer points in K is equivalent to the volume of K,up to a multiplicative factor (a polynomial of n). 6\\n\\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0042", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The fused source record contains official Problems 15--20. The strongest new result proved here is the unit-cell estimate (1-sqrt(n)/(2R))^n vol(K) <= #(K intersect Z^n) <= (1+sqrt(n)/(2R))^n vol(K) for every convex body with Euclidean inradius R>sqrt(n)/2. It yields polynomial comparison once R >= n^(3/2)/log(n+1), while a half-cell-centered empty ball proves that no positive lower comparison is possible below sqrt(n)/2. The artifacts also prove Gaussian small- and large-radius endpoint formulas, an exact restricted slab crossing, the exact n+1-center spherical covering radius, the planar mean-width case, and a centered-simplex intersection formula.\n\nCandidate contribution (theorem; novelty confidence low): For every convex body K containing a Euclidean ball of radius R>sqrt(n)/2, the lattice enumerator satisfies (1-sqrt(n)/(2R))^n vol(K) <= #(K intersect Z^n) <= (1+sqrt(n)/(2R))^n vol(K); consequently R >= n^(3/2)/log(n+1) gives vol(K)/(n+1) <= #(K intersect Z^n) <= sqrt(n+1) vol(K), and the lower threshold cannot be below sqrt(n)/2." }, { "id": 20001311, "problem_number": "AIM-CONVEX_GEOMETRY-0043", "title": "Projection-body eigenbodies: exact planar classification and product formulas", "statement": "1. (W. Weil) Let Π K be the projection body of K. It is known that if K is a polytope and ΠΠ K is homothetic to K, then K is a direct sum of planar polygons (in even dimensions) or a direct sum of planar polygons and a segment (in odd dimensions). If\n\nK is an arbitrary convex body and ΠΠ K is homothetic to K, what can we conclude about K?\n2", "original_statement": "1. (W. Weil) Let Π K be the projection body of K. It is known that if K is a polytope and ΠΠ K is homothetic to K, then K is a direct sum of planar polygons (in even dimensions) or a direct sum of planar polygons and a segment (in odd dimensions). If \n\nK is an arbitrary convex body and ΠΠ K is homothetic to K, what can we conclude about K?\n2", "clean_statement": "1. (W. Weil) Let Π K be the projection body of K. It is known that if K is a polytope and ΠΠ K is homothetic to K, then K is a direct sum of planar polygons (in even dimensions) or a direct sum of planar polygons and a segment (in odd dimensions). If\n\nK is an arbitrary convex body and ΠΠ K is homothetic to K, what can we conclude about K?\n2", "statement_status": "exact", "statement_verification": "The JSON record has two extraction defects: its number is stored as `1` rather than `21`, and a terminal page marker `2` was appended to the problem text. The official AIM workshop PDF confirms the following statement as **Problem 21**:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[42]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (W. Weil) Let Π K be the projection body of K. It is known that if K is a polytope and ΠΠ K is homothetic to K, then K is a direct sum of planar polygons (in even dimensions) or a direct sum of planar polygons and a segment (in odd dimensions). If \\n\\nK is an arbitrary convex body and ΠΠ K is homothetic to K, what can we conclude about K?\\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0043", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Every solution of Pi^2 K = cK + x is a translate of an origin-symmetric zonoid, with the translation forced by its center. In dimension two the equation is classified exactly: Pi^2 K = 2(K+(-K)), so the solutions are precisely centrally symmetric bodies up to translation and the centered multiplier is 4. In every dimension, a linear image of a Cartesian product of origin-symmetric one- and two-dimensional factors is an eigenbody; for the orthogonal centered product the exact formula is Pi^2 K = 2^n V(K)^(n-2) K. The global classification in dimensions at least three remains open, while published local smooth results force ellipsoids near the ball.\n\nCandidate contribution (product identity; novelty confidence low): For K = K_1 x ... x K_r in mutually orthogonal complementary subspaces, Pi_n K is the product of (V(K)/V(K_i)) Pi_{d_i} K_i; if all d_i are one or two and the planar factors are centrally symmetric, iterating gives the explicit multiplier Pi_n^2 K = 2^n V(K)^(n-2) K and V(Pi_n K) = 2^n V(K)^(n-1), including arbitrary nonpolytopal factors." }, { "id": 20001312, "problem_number": "AIM-CONVEX_GEOMETRY-0044", "title": "Polynomial ridge completeness and switching obstructions for the four AIM directions", "statement": "2. (A. Daurat and R. Gardner) We say that a measurable function f: R2 → R is a ridge function if it is constant in one direction, i.e., f (x) = g(〈x, v 〉) for some\n\ng: R → R, and v ∈ S1. Given n ridge functions f1,..., f n with directions v1,..., v n\n\nrespectively, the set E = {x ∈ R2: ( f1 + · · ·, f n)( x) ≥ 0} is uniquely determined among all measurable sets by the X-rays in the directions of v1,..., v n.It is known that there are sets of four directions such that every planar convex body can be determined by X-rays in those directions (for example, the directions of v1 = (1, 0), v2 = (0, 1), v3 = (2, 1) v4 = ( −1, 2) work). However, no set of three directions is enough. Is it possible to represent any convex body K ⊂ R2 in the form K = {x ∈ R2: ( f1 + f2 + f3 + f4)( x) ≥ 0} where f1,..., f 4 are ridge functions in the above four directions? 23. For t > 0, a ∈ Sn−1, consider the slab\n\nSl (a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, |〈 x, a 〉| ≤ t}\n\nand the section\n\nS(a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, 〈x, a 〉 = t}.\n\nDenote V (a, t ) = vol n(Sl (a, t )) and A(a, t ) = vol n−1(S(a, t )). Let fm = (1,..., 1, 0,..., 0) /√m ∈ Sn−1, where 1 ≤ m ≤ n and there are exactly\n\nm ones in fm. It is known (Hensley, Ball;Oleskiewicz-Pelcynski) that 1 ≤ A(a, 0) ≤√2, with equality in the left attained for a = f1, and on the right for a = f", "original_statement": "2. (A. Daurat and R. Gardner) We say that a measurable function f: R2 → R is a ridge function if it is constant in one direction, i.e., f (x) = g(〈x, v 〉) for some \n\ng: R → R, and v ∈ S1. Given n ridge functions f1,..., f n with directions v1,..., v n\n\nrespectively, the set E = {x ∈ R2: ( f1 + · · ·, f n)( x) ≥ 0} is uniquely determined among all measurable sets by the X-rays in the directions of v1,..., v n.It is known that there are sets of four directions such that every planar convex body can be determined by X-rays in those directions (for example, the directions of v1 = (1, 0), v2 = (0, 1), v3 = (2, 1) v4 = ( −1, 2) work). However, no set of three directions is enough. Is it possible to represent any convex body K ⊂ R2 in the form K = {x ∈ R2: ( f1 + f2 + f3 + f4)( x) ≥ 0} where f1,..., f 4 are ridge functions in the above four directions? 23. For t > 0, a ∈ Sn−1, consider the slab \n\nSl (a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, |〈 x, a 〉| ≤ t}\n\nand the section \n\nS(a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, 〈x, a 〉 = t}.\n\nDenote V (a, t ) = vol n(Sl (a, t )) and A(a, t ) = vol n−1(S(a, t )). Let fm = (1,..., 1, 0,..., 0) /√m ∈ Sn−1, where 1 ≤ m ≤ n and there are exactly \n\nm ones in fm. It is known (Hensley, Ball;Oleskiewicz-Pelcynski) that 1 ≤ A(a, 0) ≤√2, with equality in the left attained for a = f1, and on the right for a = f", "clean_statement": "2. (A. Daurat and R. Gardner) We say that a measurable function f: R2 → R is a ridge function if it is constant in one direction, i.e., f (x) = g(〈x, v 〉) for some\n\ng: R → R, and v ∈ S1. Given n ridge functions f1,..., f n with directions v1,..., v n\n\nrespectively, the set E = {x ∈ R2: ( f1 + · · ·, f n)( x) ≥ 0} is uniquely determined among all measurable sets by the X-rays in the directions of v1,..., v n.It is known that there are sets of four directions such that every planar convex body can be determined by X-rays in those directions (for example, the directions of v1 = (1, 0), v2 = (0, 1), v3 = (2, 1) v4 = ( −1, 2) work). However, no set of three directions is enough. Is it possible to represent any convex body K ⊂ R2 in the form K = {x ∈ R2: ( f1 + f2 + f3 + f4)( x) ≥ 0} where f1,..., f 4 are ridge functions in the above four directions? 23. For t > 0, a ∈ Sn−1, consider the slab\n\nSl (a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, |〈 x, a 〉| ≤ t}\n\nand the section\n\nS(a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, 〈x, a 〉 = t}.\n\nDenote V (a, t ) = vol n(Sl (a, t )) and A(a, t ) = vol n−1(S(a, t )). Let fm = (1,..., 1, 0,..., 0) /√m ∈ Sn−1, where 1 ≤ m ≤ n and there are exactly\n\nm ones in fm. It is known (Hensley, Ball;Oleskiewicz-Pelcynski) that 1 ≤ A(a, 0) ≤√2, with equality in the left attained for a = f1, and on the right for a = f", "statement_status": "exact", "statement_verification": "The canonical record accidentally concatenates all of Problem 22 with the beginning of Problem 23. The official AIM PDF shows that the part assigned here is Problem 22 and ends after the ridge-function question. In the PDF, a comma is also missing between the displayed vectors $v_3$ and $v_4$, and OCR has inserted a comma into the sum $f_1+\\cdots+f_n$. With those typographical repairs, the recovered problem is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[43]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (A. Daurat and R. Gardner) We say that a measurable function f: R2 → R is a ridge function if it is constant in one direction, i.e., f (x) = g(〈x, v 〉) for some \\n\\ng: R → R, and v ∈ S1. Given n ridge functions f1,..., f n with directions v1,..., v n\\n\\nrespectively, the set E = {x ∈ R2: ( f1 + · · ·, f n)( x) ≥ 0} is uniquely determined among all measurable sets by the X-rays in the directions of v1,..., v n.It is known that there are sets of four directions such that every planar convex body can be determined by X-rays in those directions (for example, the directions of v1 = (1, 0), v2 = (0, 1), v3 = (2, 1) v4 = ( −1, 2) work). However, no set of three directions is enough. Is it possible to represent any convex body K ⊂ R2 in the form K = {x ∈ R2: ( f1 + f2 + f3 + f4)( x) ≥ 0} where f1,..., f 4 are ridge functions in the above four directions? 23. For t > 0, a ∈ Sn−1, consider the slab \\n\\nSl (a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, |〈 x, a 〉| ≤ t}\\n\\nand the section \\n\\nS(a, t ) = {x ∈ Rn: ‖x‖∞ ≤ 1, 〈x, a 〉 = t}.\\n\\nDenote V (a, t ) = vol n(Sl (a, t )) and A(a, t ) = vol n−1(S(a, t )). Let fm = (1,..., 1, 0,..., 0) /√m ∈ Sn−1, where 1 ≤ m ≤ n and there are exactly \\n\\nm ones in fm. It is known (Hensley, Ball;Oleskiewicz-Pelcynski) that 1 ≤ A(a, 0) ≤√2, with equality in the left attained for a = f1, and on the right for a = f\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0044", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the fixed linear forms x, y, 2x+y, and -x+2y, every bivariate polynomial of total degree at most three is a sum of four univariate ridge polynomials, and every quadratic needs only the first three forms. Consequently every planar ellipse has an explicit exact three-ridge threshold representation. A complementary finite signed zero-marginal criterion is proved: any configuration whose two signs are separated by membership in a set rules out a literal four-ridge threshold representation; an explicit nonzero 16-term switching family is supplied for the AIM directions.\n\nCandidate contribution (explicit special-case theorem; novelty confidence low): For the exact AIM direction tuple, every bivariate polynomial of degree at most three is a sum of four univariate ridge polynomials, while every quadratic uses only the first three; in particular, the report gives an explicit exact three-ridge formula for every planar ellipse." }, { "id": 20001313, "problem_number": "AIM-CONVEX_GEOMETRY-0045", "title": "Extremal slabs, spherical negative moments, and cross-polytope formulas", "statement": "2. (In the complex case, the result is 1 ≤ A(a, 0) ≤ 2). (a) (V. Milman) Is the minimum of V (a, t ) for fixed t > 0 and a ∈ Sn−1 (variable) always attained for a standard vector a = fi?7\n\n(b) If t ≤ 2√2 − 2, is min\n\n> a∈Sn−1\n\nV (a, t ) = V (f1, t )? This is true if t ≤ 3/4 (F. Barthe, A. Koldobsky). In the complex case, it is known to be true for t ≤ 4/5 (H. Koenig). Is it true for t ≤ t0 ≈ 0.867? Here t0\n\nis the solution of\n\nπt 2 = t(t2 − 1) √2 − t2 + (2 t2 − 1) arcsin( t√2 − t2).\n\n(c) (H. Koenig) Determine the best constants in the Khintchine-type inequality\n\nc−1\n\n> p\n\n|a| ≤ ‖\n\n> n\n\n∑\n\n> j=1\n\naj Xj ‖Lp\n\nwhere Xj are i. unif. dist. on Sm−1, −m + 1 < p < 0. (d) (H. Koenig) Find an integral formula for the slab-volume in Bn\n\n> 1\n\nand for the non-central sections of Bn\n\n> 1. Find the extremal directions for the slabs in Bn\n\n> 1.", "original_statement": "2. (In the complex case, the result is 1 ≤ A(a, 0) ≤ 2). (a) (V. Milman) Is the minimum of V (a, t ) for fixed t > 0 and a ∈ Sn−1 (variable) always attained for a standard vector a = fi?7\n\n(b) If t ≤ 2√2 − 2, is min \n\n> a∈Sn−1\n\nV (a, t ) = V (f1, t )? This is true if t ≤ 3/4 (F. Barthe, A. Koldobsky). In the complex case, it is known to be true for t ≤ 4/5 (H. Koenig). Is it true for t ≤ t0 ≈ 0.867? Here t0\n\nis the solution of \n\nπt 2 = t(t2 − 1) √2 − t2 + (2 t2 − 1) arcsin( t√2 − t2).\n\n(c) (H. Koenig) Determine the best constants in the Khintchine-type inequality \n\nc−1 \n\n> p\n\n|a| ≤ ‖ \n\n> n\n\n∑\n\n> j=1\n\naj Xj ‖Lp\n\nwhere Xj are i. unif. dist. on Sm−1, −m + 1 < p < 0. (d) (H. Koenig) Find an integral formula for the slab-volume in Bn \n\n> 1\n\nand for the non-central sections of Bn \n\n> 1. Find the extremal directions for the slabs in Bn \n\n> 1.", "clean_statement": "2. (In the complex case, the result is 1 ≤ A(a, 0) ≤ 2). (a) (V. Milman) Is the minimum of V (a, t ) for fixed t > 0 and a ∈ Sn−1 (variable) always attained for a standard vector a = fi?7\n\n(b) If t ≤ 2√2 − 2, is min\n\n> a∈Sn−1\n\nV (a, t ) = V (f1, t )? This is true if t ≤ 3/4 (F. Barthe, A. Koldobsky). In the complex case, it is known to be true for t ≤ 4/5 (H. Koenig). Is it true for t ≤ t0 ≈ 0.867? Here t0\n\nis the solution of\n\nπt 2 = t(t2 − 1) √2 − t2 + (2 t2 − 1) arcsin( t√2 − t2).\n\n(c) (H. Koenig) Determine the best constants in the Khintchine-type inequality\n\nc−1\n\n> p\n\n|a| ≤ ‖\n\n> n\n\n∑\n\n> j=1\n\naj Xj ‖Lp\n\nwhere Xj are i. unif. dist. on Sm−1, −m + 1 < p < 0. (d) (H. Koenig) Find an integral formula for the slab-volume in Bn\n\n> 1\n\nand for the non-central sections of Bn\n\n> 1. Find the extremal directions for the slabs in Bn\n\n> 1.", "statement_status": "exact", "statement_verification": "This record is the continuation of Problem 23 in the AIM workshop list *Fourier analytic methods in convex geometry*. The extraction starts in the middle of the preceding sentence and labels the record “2”; the official PDF shows that the number is 23 and that the isolated “7” after part (a) is a page-number artifact.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Convex geometry\nWorkshop: Fourier analytic methods in convex geometry\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf\nCanonical location: aim-convex-geometry-notes.json notes[44]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (In the complex case, the result is 1 ≤ A(a, 0) ≤ 2). (a) (V. Milman) Is the minimum of V (a, t ) for fixed t > 0 and a ∈ Sn−1 (variable) always attained for a standard vector a = fi?7\\n\\n(b) If t ≤ 2√2 − 2, is min \\n\\n> a∈Sn−1\\n\\nV (a, t ) = V (f1, t )? This is true if t ≤ 3/4 (F. Barthe, A. Koldobsky). In the complex case, it is known to be true for t ≤ 4/5 (H. Koenig). Is it true for t ≤ t0 ≈ 0.867? Here t0\\n\\nis the solution of \\n\\nπt 2 = t(t2 − 1) √2 − t2 + (2 t2 − 1) arcsin( t√2 − t2).\\n\\n(c) (H. Koenig) Determine the best constants in the Khintchine-type inequality \\n\\nc−1 \\n\\n> p\\n\\n|a| ≤ ‖ \\n\\n> n\\n\\n∑\\n\\n> j=1\\n\\naj Xj ‖Lp\\n\\nwhere Xj are i. unif. dist. on Sm−1, −m + 1 < p < 0. (d) (H. Koenig) Find an integral formula for the slab-volume in Bn \\n\\n> 1\\n\\nand for the non-central sections of Bn \\n\\n> 1. Find the extremal directions for the slabs in Bn \\n\\n> 1.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 6, "set_id": 14, "source_url": "https://aimath.org/WWN/fourierconvex/fourierconvex.pdf", "tags": [ "aim", "AIM-CONVEX_GEOMETRY-0045", "aim-domain:convex-geometry", "aim-workshop:fourierconvex", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 6, "name": "geometry", "display_name": "Geometry", "description": "Euclidean and non-Euclidean geometry, geometric structures.", "slug": "geometry", "order_index": 6, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After repairing the source normalization and OCR, this attempt gives exact Irwin--Hall formulas for every standard cube-slab competitor, two explicit necessary sharp benchmarks for the negative spherical Khintchine constant, an exact signed-simplex divided-difference and Fourier formula for every noncentral section and symmetric slab of B_1^n, and a complete extremal theorem in dimension two: diagonal normals minimize for 0=n and unipotent block products F_m=A^{k_1}B^{l_1}...A^{k_m}B^{l_m}, every entry has individual degree at most n-1 and total degree at most 2m(n-1), with 2m exponent variables and output rank at most n^2-1; however, the unconstrained F_p system is not length-faithful, because restricting an exponent to 0,...,L for L

is exactly the descending intersection of the factorial congruence tubes H+[N!]E(Q). Each fixed tube is positive-existentially definable over Q, with P fixed, by a finite union of cosets [r]P+[m]E(Q), but every finite positive combination of tubes still contains a rank-two subgroup. Independently, no rank-one subgroup of Z^2 plus finite torsion is positive-existentially definable in one variable in the pure group language, even with parameters; any possible positive answer to the intended ring-language question must therefore use genuinely field-arithmetic witnesses.\n\nCandidate contribution (theorem; novelty confidence low): The cyclic subgroup

is an exact factorial intersection of uniformly Diophantine fixed-divisibility tubes, while every finite positive combination of these tubes retains rank two; moreover positive-existential pure-group formulas cannot define rank one." }, { "id": 20002358, "problem_number": "AIM-LOGIC-0134", "title": "Density-zero localizations do not force rank-small integral points", "statement": "Question 21 (Shlapentokh). Let E be an elliptic curve over Q of rank 2. Can we find a subset S of (infinitely many) primes such that the subgroup generated by E(Z[S−1]) has rank one? If S is finite, the Siegel-Mahler theorem states that E(Z[S−1]) is finite. Suppose S is infinite, but of density 0. Is E(Z[S−1]) still \"small\"?", "original_statement": "Question 21 (Shlapentokh). Let E be an elliptic curve over Q of rank 2. Can we find a subset S of (infinitely many) primes such that the subgroup generated by E(Z[S−1]) has rank one? If S is finite, the Siegel-Mahler theorem states that E(Z[S−1]) is finite. Suppose S is infinite, but of density 0. Is E(Z[S−1]) still \"small\"?", "clean_statement": "Question 21 (Shlapentokh). Let E be an elliptic curve over Q of rank 2. Can we find a subset S of (infinitely many) primes such that the subgroup generated by E(Z[S−1]) has rank one? If S is finite, the Siegel-Mahler theorem states that E(Z[S−1]) is finite. Suppose S is infinite, but of density 0. Is E(Z[S−1]) still \"small\"?", "statement_status": "exact", "statement_verification": "Question 21 (Shlapentokh) in the 2005 AIM list *Extensions of Hilbert's tenth problem* reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 21\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[133]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 21 (Shlapentokh). Let E be an elliptic curve over Q of rank 2. Can we find a subset S of (infinitely many) primes such that the subgroup generated by E(Z[S−1]) has rank one? If S is finite, the Siegel-Mahler theorem states that E(Z[S−1]) is finite. Suppose S is infinite, but of density 0. Is E(Z[S−1]) still \\\"small\\\"?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0134", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any fixed integral Weierstrass model of a rank-r elliptic curve over Q, there is an infinite recursive prime set S of both natural and Dirichlet density zero for which the subgroup generated by E(Z[S^{-1}]) has full rank r: include the finite denominator supports of a Mordell-Weil basis and adjoin the sparse set of primes with indices 2^n. Moreover, on the global minimal model 389a1, y^2+y=x^3+x^2-2x, the integral points (0,0) and (1,0) are independent Mordell-Weil generators, so no prime set S at all yields generated rank one. This refutes the universal fixed-model reading while leaving refined positive cases open.\n\nCandidate contribution (reduction; novelty confidence low): The finite-seed/density dichotomy and transverse-support hypergraph criterion show that density zero cannot control generated Mordell-Weil rank and that isolating a primitive rank-one line is exactly a hitting-set problem for denominator supports of all points transverse to that line; the standard minimal model 389a1 supplies an explicit integral-basis obstruction." }, { "id": 20002359, "problem_number": "AIM-LOGIC-0135", "title": "A Prym criterion for rank preservation under Denef base change", "statement": "Question 22 (Zahidi). Look at the Denef curve\n\nE: f (t)Y 2 = f (X)\n\nwhere f is a cubic. If we choose the curve in a good way, then E(k(t)) has rank 1.Define\n\nEu: f (u)Y 2 = f (X)\n\nTry to give conditions on u ∈ k(t) such that Eu(k(t)) also has rank 1.", "original_statement": "Question 22 (Zahidi). Look at the Denef curve \n\nE: f (t)Y 2 = f (X)\n\nwhere f is a cubic. If we choose the curve in a good way, then E(k(t)) has rank 1.Define \n\nEu: f (u)Y 2 = f (X)\n\nTry to give conditions on u ∈ k(t) such that Eu(k(t)) also has rank 1.", "clean_statement": "Question 22 (Zahidi). Look at the Denef curve\n\nE: f (t)Y 2 = f (X)\n\nwhere f is a cubic. If we choose the curve in a good way, then E(k(t)) has rank 1.Define\n\nEu: f (u)Y 2 = f (X)\n\nTry to give conditions on u ∈ k(t) such that Eu(k(t)) also has rank 1.", "statement_status": "exact", "statement_verification": "The canonical JSON record has lost superscript and subscript formatting, but the official AIM workshop PDF gives the following unambiguous statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 22\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[134]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 22 (Zahidi). Look at the Denef curve \\n\\nE: f (t)Y 2 = f (X)\\n\\nwhere f is a cubic. If we choose the curve in a good way, then E(k(t)) has rank 1.Define \\n\\nEu: f (u)Y 2 = f (X)\\n\\nTry to give conditions on u ∈ k(t) such that Eu(k(t)) also has rank 1.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0135", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a characteristic-zero field k, a separable cubic f, E0:y^2=f(x), and nonconstant u in k(t), let C_u be the normalization of s^2=f(u(t)) and P_u the connected kernel of J(C_u)→E0 induced by (t,s)↦(u(t),s). Then rank E_u(k(t)) equals rank End_k(E0) plus rank Hom_k(P_u,E0). Thus, when End_k(E0)=Z, rank one is equivalent to the Prym having no k-homomorphism to E0. For u=t^2 with f(0) nonzero, J(C_u) is k-isogenous to E0 times D_f, where D_f:w^2=z f(z), so rank E_{t^2}(k(t))=1+rank Hom_k(D_f,E0), and rank one holds exactly when D_f is not k-isogenous to E0.\n\nCandidate contribution (theorem; novelty confidence low): For u=t^2 and f(0) nonzero, the rank-preservation question is exactly the k-isogeny test between E0:y^2=f(x) and the explicit elliptic quotient D_f:w^2=z f(z); non-isogeny gives rank one, while isogeny certifies a rank jump." }, { "id": 20002360, "problem_number": "AIM-LOGIC-0136", "title": "Certificate escape for CM j-invariants in the complex rational function field", "statement": "Question 23 (Pheidas). Consider the elliptic curve\n\nE: Y 2 = X3 + aX + b\n\nThe following statement is Diophantine: \" End( E)/(2 End( E)) has more than 2 elements\". Because End( E) is a free finitely generated Z-module, this is equivalent with \" End( E) 6 = Z\". So, we can existentially define the following set in C(Z):\n\n{j ∈ C | j is the j-invariant of a CM elliptic curve }\n\nCan we do anything with this set?", "original_statement": "Question 23 (Pheidas). Consider the elliptic curve \n\nE: Y 2 = X3 + aX + b\n\nThe following statement is Diophantine: \" End( E)/(2 End( E)) has more than 2 elements\". Because End( E) is a free finitely generated Z-module, this is equivalent with \" End( E) 6 = Z\". So, we can existentially define the following set in C(Z):\n\n{j ∈ C | j is the j-invariant of a CM elliptic curve }\n\nCan we do anything with this set?", "clean_statement": "Question 23 (Pheidas). Consider the elliptic curve\n\nE: Y 2 = X3 + aX + b\n\nThe following statement is Diophantine: \" End( E)/(2 End( E)) has more than 2 elements\". Because End( E) is a free finitely generated Z-module, this is equivalent with \" End( E) 6 = Z\". So, we can existentially define the following set in C(Z):\n\n{j ∈ C | j is the j-invariant of a CM elliptic curve }\n\nCan we do anything with this set?", "statement_status": "exact", "statement_verification": "The canonical record is Question 23 in the AIM workshop list *Extensions of Hilbert's tenth problem*. Direct inspection of page 5 of the official PDF gives:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 23\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[135]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 23 (Pheidas). Consider the elliptic curve \\n\\nE: Y 2 = X3 + aX + b\\n\\nThe following statement is Diophantine: \\\" End( E)/(2 End( E)) has more than 2 elements\\\". Because End( E) is a free finitely generated Z-module, this is equivalent with \\\" End( E) 6 = Z\\\". So, we can existentially define the following set in C(Z):\\n\\n{j ∈ C | j is the j-invariant of a CM elliptic curve }\\n\\nCan we do anything with this set?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0136", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The recovered domain is the rational-function field C(Z). For a CM elliptic curve with endomorphism order of discriminant D<0, every noninteger endomorphism has degree at least |D|/4, so bounded-degree non-scalar endomorphisms certify only finitely many singular moduli. More generally, for any fixed Diophantine formula over C(Z) defining exactly the CM j-invariants among the constants, the CM parameters admitting witnesses of Z-degree at most d form a finite set for every d. Thus witness complexity must be unbounded, and the unary predicate alone does not yet supply a Diophantine interpretation of integer arithmetic.\n\nCandidate contribution (obstruction; novelty confidence low): For every fixed Diophantine definition of the singular moduli inside C(Z), each bounded rational-function witness-degree stratum is finite; combined with the explicit bound deg(alpha) >= |disc End(E)|/4 for non-scalar CM endomorphisms, this forces unbounded certificate complexity along the CM locus." }, { "id": 20002361, "problem_number": "AIM-LOGIC-0137", "title": "Bounded square-class width and sign-symmetric factorization in C(Z)", "statement": "Question 24 (Pheidas). If x ∈ C(Z), then\n\nord Z=0\n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= ord Z=∞\n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= 0\n\nCan every f ∈ C(Z) with ord Z=0 (f ) = ord Z=∞(f ) even be written as (obviously, the number 1000 can be changed to any other integer)\n\nf = u21000 ∏\n\n> i=1\n\n1 + Zx 2\n\n> i\n\n1 − Zx 2\n\n> i\n\nWeaker version: is this true at least for f ∈ Q(Z), with u, x i ∈ C(Z)?This would imply that the existential theory of C(Z) is undecidable.", "original_statement": "Question 24 (Pheidas). If x ∈ C(Z), then \n\nord Z=0 \n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= ord Z=∞\n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= 0 \n\nCan every f ∈ C(Z) with ord Z=0 (f ) = ord Z=∞(f ) even be written as (obviously, the number 1000 can be changed to any other integer) \n\nf = u21000 ∏\n\n> i=1\n\n1 + Zx 2\n\n> i\n\n1 − Zx 2\n\n> i\n\nWeaker version: is this true at least for f ∈ Q(Z), with u, x i ∈ C(Z)?This would imply that the existential theory of C(Z) is undecidable.", "clean_statement": "Question 24 (Pheidas). If x ∈ C(Z), then\n\nord Z=0\n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= ord Z=∞\n\n( 1 + Zx 2\n\n1 − Zx 2\n\n)\n\n= 0\n\nCan every f ∈ C(Z) with ord Z=0 (f ) = ord Z=∞(f ) even be written as (obviously, the number 1000 can be changed to any other integer)\n\nf = u21000 ∏\n\n> i=1\n\n1 + Zx 2\n\n> i\n\n1 − Zx 2\n\n> i\n\nWeaker version: is this true at least for f ∈ Q(Z), with u, x i ∈ C(Z)?This would imply that the existential theory of C(Z) is undecidable.", "statement_status": "exact", "statement_verification": "This is Question 24 in the problem list from the March 2005 AIM workshop *Extensions of Hilbert's Tenth Problem*. The canonical JSON record has a layout/OCR corruption: `u21000` is not an exponent, and the `1000` belongs above a product sign. Inspection of the official PDF gives the following normalized statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 24\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[136]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 24 (Pheidas). If x ∈ C(Z), then \\n\\nord Z=0 \\n\\n( 1 + Zx 2\\n\\n1 − Zx 2\\n\\n)\\n\\n= ord Z=∞\\n\\n( 1 + Zx 2\\n\\n1 − Zx 2\\n\\n)\\n\\n= 0 \\n\\nCan every f ∈ C(Z) with ord Z=0 (f ) = ord Z=∞(f ) even be written as (obviously, the number 1000 can be changed to any other integer) \\n\\nf = u21000 ∏\\n\\n> i=1\\n\\n1 + Zx 2\\n\\n> i\\n\\n1 − Zx 2\\n\\n> i\\n\\nWeaker version: is this true at least for f ∈ Q(Z), with u, x i ∈ C(Z)?This would imply that the existential theory of C(Z) is undecidable.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0137", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM factorization question is exactly the assertion that the subgroup of square classes with even valuations at zero and infinity has word width at most 1000 with respect to [(1+Zx^2)/(1-Zx^2)]=[1-Z^2x^4]. If the odd divisor support of f is invariant under a↦-a and consists of m≤1000 sign-pairs, then the requested factorization is proved constructively using constant x-values. Every generator is a norm from C(s), s^2=Z, but that norm map is surjective and gives no obstruction. If witnesses are instead required in Q(Z), the residue at zero must be a rational square up to sign, so f=2 is a counterexample to that stronger, non-source reading.\n\nCandidate contribution (reduction; novelty confidence low): The question admits an exact uniform word-width formulation in C(Z)^×/C(Z)^{×2}, and every class supported on at most 1000 sign-pairs {a,-a} has an explicit requested factorization; additionally, the local residue square-class obstruction sharply separates complex witnesses from a tempting rational-witness strengthening." }, { "id": 20002362, "problem_number": "AIM-LOGIC-0138", "title": "A dense Diophantine shadow of the Z-adic valuation ring", "statement": "Question 25 (Pheidas). Is {f ∈ C(Z) | ord Z=0 (f ) ≥ 0} (existentially) definable in C(Z),where there is a symbol for Z in the language? 6 MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER", "original_statement": "Question 25 (Pheidas). Is {f ∈ C(Z) | ord Z=0 (f ) ≥ 0} (existentially) definable in C(Z),where there is a symbol for Z in the language? 6 MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER", "clean_statement": "Question 25 (Pheidas). Is {f ∈ C(Z) | ord Z=0 (f ) ≥ 0} (existentially) definable in C(Z),where there is a symbol for Z in the language? 6 MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER", "statement_status": "exact", "statement_verification": "The official AIM workshop PDF states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 25\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[137]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 25 (Pheidas). Is {f ∈ C(Z) | ord Z=0 (f ) ≥ 0} (existentially) definable in C(Z),where there is a symbol for Z in the language? 6 MODERATED BY B. POONEN AND T. SCANLON, NOTES BY J. DEMEYER\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0138", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In C(Z), the one-equation existential set S={x: there exists y with y^2=1+Zx^2} has the exact parametrization x=2r/(1-Zr^2). It is a proper subset of the valuation ring O_0=C[Z]_(Z), since 1 is not in S, yet for every N it maps surjectively to O_0/Z^N O_0 and contains an element of every nonnegative valuation. Thus the Hensel formula that exactly defines C[[Z]] in C((Z)) gives a proper Z-adically dense rational shadow in C(Z); the missing condition is global rationality, invisible to every finite jet. A hypothetical existential definition of O_0 would also make its complement existential and, by Daans's Proposition 5.1, imply undecidability of the existential theory of (C(Z);Z).\n\nCandidate contribution (theorem; novelty confidence low): The existential conic predicate y^2=1+Zx^2 defines a proper subset of C[Z]_(Z) that nevertheless surjects onto C[Z]_(Z)/Z^N C[Z]_(Z) for every N, with exact rational parametrization x=2r/(1-Zr^2)." }, { "id": 20002363, "problem_number": "AIM-LOGIC-0139", "title": "Definable valuations on the anisotropic real conic", "statement": "Question 26 (Moret-Bailly). Is there a nontrivial valuation ring\n\nR ⊂ Frac R[x, y ]\n\n(x2 + y2 + 1)\n\nwhich is definable? Same question for \"semi-local ring\" (finite intersection of valuation rings) instead of \"valuation ring\"? This is equivalent with the problem for valuation rings.", "original_statement": "Question 26 (Moret-Bailly). Is there a nontrivial valuation ring \n\nR ⊂ Frac R[x, y ]\n\n(x2 + y2 + 1) \n\nwhich is definable? Same question for \"semi-local ring\" (finite intersection of valuation rings) instead of \"valuation ring\"? This is equivalent with the problem for valuation rings.", "clean_statement": "Question 26 (Moret-Bailly). Is there a nontrivial valuation ring\n\nR ⊂ Frac R[x, y ]\n\n(x2 + y2 + 1)\n\nwhich is definable? Same question for \"semi-local ring\" (finite intersection of valuation rings) instead of \"valuation ring\"? This is equivalent with the problem for valuation rings.", "statement_status": "exact", "statement_verification": "The canonical JSON record has lost a typographical distinction and displays both the sought valuation ring and the real constant field as `R`. The official AIM PDF was downloaded and its page-6 PDF content stream was inspected. The first symbol is set in the ordinary math-italic font CMMI12, whereas the numerator uses the blackboard-bold font MSBM10. Thus the recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 26\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[138]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 26 (Moret-Bailly). Is there a nontrivial valuation ring \\n\\nR ⊂ Frac R[x, y ]\\n\\n(x2 + y2 + 1) \\n\\nwhich is definable? Same question for \\\"semi-local ring\\\" (finite intersection of valuation rings) instead of \\\"valuation ring\\\"? This is equivalent with the problem for valuation rings.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0139", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The official PDF recovers the field as F = Frac(RR[x,y]/(x^2+y^2+1)). For geometric valuations (those trivial on the real constant field), a finite intersection A = intersection_j O_{P_j} existentially defines each chosen O_{P_i} after adjoining a weak-approximation parameter pi_i: x lies in O_{P_i} iff there exist s,u,w in A with sx in A and us+w pi_i = 1. Conversely a valuation ring is a one-member intersection, proving the source's equivalence even for existential definitions. In addition, the transitive SO_3(RR)-action proves that no such valuation ring or nonempty finite intersection is definable using only real parameters. The arbitrary-F-parameter problem remains open.\n\nCandidate contribution (reduction; novelty confidence low): For the exact anisotropic-conic field, the displayed positive-existential Bezout formula extracts a chosen geometric DVR from any definable finite holomorphy ring using one weak-approximation parameter, while SO_3(RR)-symmetry rules out definitions of any such DVR or finite holomorphy ring over real parameters alone.", "research_difficulty_suggested": "L4" }, { "id": 20002364, "problem_number": "AIM-LOGIC-0140", "title": "Symmetry and finite-extension descent for definable valuation rings", "statement": "Question 27 (Shlapentokh). Can one find an algebraically closed field K and a nontrivial valuation ring R ⊂ K(Z) (or a finite extension), which is definable in K(Z)?", "original_statement": "Question 27 (Shlapentokh). Can one find an algebraically closed field K and a nontrivial valuation ring R ⊂ K(Z) (or a finite extension), which is definable in K(Z)?", "clean_statement": "Question 27 (Shlapentokh). Can one find an algebraically closed field K and a nontrivial valuation ring R ⊂ K(Z) (or a finite extension), which is definable in K(Z)?", "statement_status": "exact", "statement_verification": "The exact source is the AIM problem list *Problems related to “Extensions of Hilbert's Tenth Problem”*, moderated by B. Poonen and T. Scanlon, with notes by J. Demeyer. The source PDF was checked directly. Question 27 reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 27\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[139]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 27 (Shlapentokh). Can one find an algebraically closed field K and a nontrivial valuation ring R ⊂ K(Z) (or a finite extension), which is definable in K(Z)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0140", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For algebraically closed K and F=K(Z), no nontrivial valuation ring of F/K is definable in the pure ring language using only parameters from K, because PGL_2(K) moves every place. Moreover, for any finite L/F, a definable nontrivial valuation ring O_w of L contracts, through the standard finite-dimensional interpretation, to a nontrivial definable valuation ring O_w intersect F of F; existentiality is preserved. Hence a positive definition must name a nonconstant element, and the finite-extension option does not bypass the parameter-allowing base-field problem.\n\nCandidate contribution (reduction; novelty confidence low): The paired parameter-cost/descent reduction: a K-trivial valuation of K(Z) cannot be defined from constant parameters alone, while every finite-extension example descends to K(Z) through the standard interpretation, preserving existentiality.", "research_difficulty_suggested": "L4" }, { "id": 20002365, "problem_number": "AIM-LOGIC-0141", "title": "A corrected algebraic p-adic field with definable valuation and algebraically closed residue", "statement": "Question 28 (Shlapentokh). Is there an algebraic extension K of Q and a nontrivial valu-ation ring R ⊂ K, such that the residue field of R is algebraically closed and R is definable over K?Answer: YES. Inside Qalg\n\n> p\n\n= Q ∩ Qp ⊆ Qp, the ring Zalg\n\n> p\n\nis definable.\n\nFact 29 (Pheidas). C[[ Z]] is definable in C(( Z)):\n\nx ∈ C[[ Z]] ⇐⇒ (∃y)(1 + Zx 2 = y2)\n\nProven using Hensel's lemma.", "original_statement": "Question 28 (Shlapentokh). Is there an algebraic extension K of Q and a nontrivial valu-ation ring R ⊂ K, such that the residue field of R is algebraically closed and R is definable over K?Answer: YES. Inside Qalg \n\n> p\n\n= Q ∩ Qp ⊆ Qp, the ring Zalg \n\n> p\n\nis definable. \n\nFact 29 (Pheidas). C[[ Z]] is definable in C(( Z)):\n\nx ∈ C[[ Z]] ⇐⇒ (∃y)(1 + Zx 2 = y2)\n\nProven using Hensel's lemma.", "clean_statement": "Question 28 (Shlapentokh). Is there an algebraic extension K of Q and a nontrivial valu-ation ring R ⊂ K, such that the residue field of R is algebraically closed and R is definable over K?Answer: YES. Inside Qalg\n\n> p\n\n= Q ∩ Qp ⊆ Qp, the ring Zalg\n\n> p\n\nis definable.\n\nFact 29 (Pheidas). C[[ Z]] is definable in C(( Z)):\n\nx ∈ C[[ Z]] ⇐⇒ (∃y)(1 + Zx 2 = y2)\n\nProven using Hensel's lemma.", "statement_status": "exact", "statement_verification": "The canonical record combines two consecutive items from the 2005 AIM workshop list *Extensions of Hilbert's Tenth Problem*. Visual inspection of the official PDF shows the boundary clearly.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 28\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[140]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 28 (Shlapentokh). Is there an algebraic extension K of Q and a nontrivial valu-ation ring R ⊂ K, such that the residue field of R is algebraically closed and R is definable over K?Answer: YES. Inside Qalg \\n\\n> p\\n\\n= Q ∩ Qp ⊆ Qp, the ring Zalg \\n\\n> p\\n\\nis definable. \\n\\nFact 29 (Pheidas). C[[ Z]] is definable in C(( Z)):\\n\\nx ∈ C[[ Z]] ⇐⇒ (∃y)(1 + Zx 2 = y2)\\n\\nProven using Hensel's lemma.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0141", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal AIM example K_0=Qbar∩Q_p has residue field F_p and therefore does not satisfy Question 28. A corrected complete example is K_p=Qbar∩Q_p^nr, the algebraic-number part of the maximal unramified p-adic extension. Its restricted valuation is henselian with value group Z and residue field Fpbar, and its nontrivial valuation ring is parameter-free existentially definable by (∃y)(y^n=1+p x^n) for every n>1 with p not dividing n. For this field the exponent condition is sharp: n=1 defines the whole field, while p|n makes the formula fail at the integral element x=1.\n\nCandidate contribution (construction; novelty confidence low): Replacing the literal AIM intersection Qbar∩Q_p by Qbar∩Q_p^nr repairs the residue-field defect while preserving a one-existential pure-ring definition, and the Hensel predicate (∃y)(y^n=1+p x^n) defines the corrected valuation ring if and only if n>1 and p∤n; the adjacent C((Z)) predicate likewise defines C[[Z]] exactly when n>1." }, { "id": 20002366, "problem_number": "AIM-LOGIC-0142", "title": "Diophantine archimedean balls after the all-number-field breakthrough", "statement": "Question 30 (Shlapentokh). Let K be a number field and OK its ring of integers. Fix an embedding K ↪ → C, with K 6 ⊆ R. Is {α ∈ O K | | α| ≤ 1} Diophantine in OK?If this is true for all K, then Hilbert's Tenth Problem is undecidable for all OK.", "original_statement": "Question 30 (Shlapentokh). Let K be a number field and OK its ring of integers. Fix an embedding K ↪ → C, with K 6 ⊆ R. Is {α ∈ O K | | α| ≤ 1} Diophantine in OK?If this is true for all K, then Hilbert's Tenth Problem is undecidable for all OK.", "clean_statement": "Question 30 (Shlapentokh). Let K be a number field and OK its ring of integers. Fix an embedding K ↪ → C, with K 6 ⊆ R. Is {α ∈ O K | | α| ≤ 1} Diophantine in OK?If this is true for all K, then Hilbert's Tenth Problem is undecidable for all OK.", "statement_status": "exact", "statement_verification": "The exact source is the AIM problem list *Problems related to “Extensions of Hilbert's Tenth Problem”*, moderated by B. Poonen and T. Scanlon, with notes by J. Demeyer. The source PDF was checked directly. Question 30 reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 30\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[141]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 30 (Shlapentokh). Let K be a number field and OK its ring of integers. Fix an embedding K ↪ → C, with K 6 ⊆ R. Is {α ∈ O K | | α| ≤ 1} Diophantine in OK?If this is true for all K, then Hilbert's Tenth Problem is undecidable for all OK.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0142", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Koymans-Pagano, arXiv:2412.01768v3, Theorem 1.2 proves that Z is Diophantine in O_K for every number field. With an integral basis, the coordinate tuples of S_sigma={alpha in O_K: |sigma(alpha)|<=1} form a recursive subset of Z^[K:Q], since signs of real algebraic numbers are decidable exactly; MRDP and existential restriction of all coordinates and witnesses to the Diophantine copy of Z therefore define S_sigma over O_K. Thus Question 30 has a positive answer for every K. In the imaginary-quadratic case S_sigma={0} union O_K^times is defined parameter-free by exists y, x(xy-1)=0; otherwise the Dirichlet log lattice gives a unit epsilon with |sigma(epsilon)|<1 and S_sigma intersect ={epsilon^n:n>=0}.\n\nCandidate contribution (corollary; novelty confidence low): The explicit effective-archimedean transfer and unit-rank dichotomy: every fixed recursive archimedean predicate on O_K is Diophantine after the 2025 all-number-field theorem; for this ball the imaginary-quadratic one-equation formula and higher-rank cyclic-ray description give a sharp signature split." }, { "id": 20002367, "problem_number": "AIM-LOGIC-0143", "title": "Division-ample sets outside finitely many unit orbits", "statement": "Question 31 (Cornelissen). Let K be a number field and OK its ring of integers. A set\n\nA ⊆ O K is said to be division-ample if\n\n• It is Diophantine over OK.\n\n• Any x ∈ O K divides some a ∈ A.\n\n• There exists a positive integer l such that for any a ∈ A, there exists ˜a ∈ Z with ˜a|a\n\nand N (a) ≤ | ˜a|l.Observe that if A ⊆ Z, then one can dispose of the last condition by choosing ˜a = a\n\nand l = [ K: Q].Question: give an example of such A where for any finite S ⊆ O K, A is not a subset of O∗\n\n> K\n\n· (Z ∪ S).Cornelissen-Pheidas-Zahidi have shown that HTP( OK ) has a negative answer if such\n\nA exists and there exists an elliptic curve of rank one over K.", "original_statement": "Question 31 (Cornelissen). Let K be a number field and OK its ring of integers. A set \n\nA ⊆ O K is said to be division-ample if \n\n• It is Diophantine over OK.\n\n• Any x ∈ O K divides some a ∈ A.\n\n• There exists a positive integer l such that for any a ∈ A, there exists ˜a ∈ Z with ˜a|a\n\nand N (a) ≤ | ˜a|l.Observe that if A ⊆ Z, then one can dispose of the last condition by choosing ˜a = a\n\nand l = [ K: Q].Question: give an example of such A where for any finite S ⊆ O K, A is not a subset of O∗ \n\n> K\n\n· (Z ∪ S).Cornelissen-Pheidas-Zahidi have shown that HTP( OK ) has a negative answer if such \n\nA exists and there exists an elliptic curve of rank one over K.", "clean_statement": "Question 31 (Cornelissen). Let K be a number field and OK its ring of integers. A set\n\nA ⊆ O K is said to be division-ample if\n\n• It is Diophantine over OK.\n\n• Any x ∈ O K divides some a ∈ A.\n\n• There exists a positive integer l such that for any a ∈ A, there exists ˜a ∈ Z with ˜a|a\n\nand N (a) ≤ | ˜a|l.Observe that if A ⊆ Z, then one can dispose of the last condition by choosing ˜a = a\n\nand l = [ K: Q].Question: give an example of such A where for any finite S ⊆ O K, A is not a subset of O∗\n\n> K\n\n· (Z ∪ S).Cornelissen-Pheidas-Zahidi have shown that HTP( OK ) has a negative answer if such\n\nA exists and there exists an elliptic curve of rank one over K.", "statement_status": "exact", "statement_verification": "The official AIM PDF was checked at the level of its page-6 content stream. The symbols lost or displaced by OCR are \\(\\widetilde a\\), \\(\\mathbb Z\\), and the superscript star in \\(\\mathcal O_K^*\\). The recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 31\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[142]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 31 (Cornelissen). Let K be a number field and OK its ring of integers. A set \\n\\nA ⊆ O K is said to be division-ample if \\n\\n• It is Diophantine over OK.\\n\\n• Any x ∈ O K divides some a ∈ A.\\n\\n• There exists a positive integer l such that for any a ∈ A, there exists ˜a ∈ Z with ˜a|a\\n\\nand N (a) ≤ | ˜a|l.Observe that if A ⊆ Z, then one can dispose of the last condition by choosing ˜a = a\\n\\nand l = [ K: Q].Question: give an example of such A where for any finite S ⊆ O K, A is not a subset of O∗ \\n\\n> K\\n\\n· (Z ∪ S).Cornelissen-Pheidas-Zahidi have shown that HTP( OK ) has a negative answer if such \\n\\nA exists and there exists an elliptic curve of rank one over K.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0143", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Assuming the now-proved theorem that Z is Diophantine over O_K for every number field K, the AIM question has a complete classification. For every number field K of degree d>1, choose x_0 outside O_K^* Z, enumerate O_K as (b_j), put n_j=max(1,|N(b_j)|), choose distinct increasing m_j at least |N(x_0)|, and take A={n_j b_j} union {m_j x_0}. Every computably enumerable subset of O_K is Diophantine by MRDP plus the Diophantine definition of Z; this A is divisibility-dense, satisfies the absolute norm bound with exponent d+1, and escapes O_K^*(Z union S) for every finite S. For K=Q the requested nonconcentration condition is impossible.\n\nCandidate contribution (construction; novelty confidence low): The explicit two-family construction gives a full post-2025 classification of AIM Question 31: the requested set exists for every K not equal to Q with exponent [K:Q]+1, and Q is the unique impossible field." }, { "id": 20002368, "problem_number": "AIM-LOGIC-0144", "title": "Proper models transfer exact Mordell-Weil stability", "statement": "Question 32 (Poonen). Is is true that for all number fields K, there exists a variety X\n\n(scheme of finite type) over Z such that\n\n(1) X(Z) is infinite.\n\n(2) X(OK ) = X(Z).", "original_statement": "Question 32 (Poonen). Is is true that for all number fields K, there exists a variety X\n\n(scheme of finite type) over Z such that \n\n(1) X(Z) is infinite. \n\n(2) X(OK ) = X(Z).", "clean_statement": "**Question 32 (Poonen).** Is it true that for all number fields $K$, there exists a variety $X$ (scheme of finite type) over $\\mathbb Z$ such that\n\n1. $X(\\mathbb Z)$ is infinite.\n2. $X(\\mathcal O_K)=X(\\mathbb Z)$.", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON extraction reads “Is is true” and writes the ring of integers as `OK`. Inspection of the AIM source identifies these as extraction/OCR defects. The recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 32\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[143]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 32 (Poonen). Is is true that for all number fields K, there exists a variety X\\n\\n(scheme of finite type) over Z such that \\n\\n(1) X(Z) is infinite. \\n\\n(2) X(OK ) = X(Z).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: open; rigor: conditional; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0144", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any number field K, if there exists a positive-rank elliptic curve E/Q with the exact equality E(K)=E(Q), then every proper finite-presentation Z-model of E solves Question 32: its Z-sections are naturally E(Q), its O_K-sections are naturally E(K), and singular bad fibers cause no difficulty. The 2025/2026 rank-stability theorems checked do not supply this hypothesis: equality of ranks yields only a finite quotient E(K)/E(Q), and every nonzero coset survives as an unwanted O_K-section of every proper model.\n\nCandidate contribution (reduction; novelty confidence low): For every proper finite-presentation Z-model of E, the gap between its O_K-sections and Z-sections is exactly the Mordell-Weil quotient E(K)/E(Q); under equal ranks, exact equality is equivalent to saturation of E(Q) in E(K), and changing singular bad fibers within the proper category cannot remove extra points.", "research_difficulty_suggested": "L4" }, { "id": 20002369, "problem_number": "AIM-LOGIC-0145", "title": "Endpoint robustness and a tower criterion for Julia Robinson attainment", "statement": "Question 33 (Videla). Let K ⊆ Qtot. real ⊆ Q. Define\n\nAK:= {s ∈ R>0 | There exist infinitely many α ∈ O K\n\nsuch that α and its conjugates are all in [0, s ]}\n\nQuestion of Julia Robinson: Is the infimum of AK an element of AK? If so, the first order theory of OK is undecidable. For K = Qtot. real, inf( AK ) = 4 ∈ AK.PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 7", "original_statement": "Question 33 (Videla). Let K ⊆ Qtot. real ⊆ Q. Define \n\nAK:= {s ∈ R>0 | There exist infinitely many α ∈ O K\n\nsuch that α and its conjugates are all in [0, s ]}\n\nQuestion of Julia Robinson: Is the infimum of AK an element of AK? If so, the first order theory of OK is undecidable. For K = Qtot. real, inf( AK ) = 4 ∈ AK.PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 7", "clean_statement": "Question 33 (Videla). Let K ⊆ Qtot. real ⊆ Q. Define\n\nAK:= {s ∈ R>0 | There exist infinitely many α ∈ O K\n\nsuch that α and its conjugates are all in [0, s ]}\n\nQuestion of Julia Robinson: Is the infimum of AK an element of AK? If so, the first order theory of OK is undecidable. For K = Qtot. real, inf( AK ) = 4 ∈ AK.PROBLEMS RELATED TO \"EXTENSIONS OF HILBERT'S TENTH PROBLEM\" 7", "statement_status": "exact", "statement_verification": "The source is the AIM problem list *Problems related to “Extensions of Hilbert's Tenth Problem”*, moderated by B. Poonen and T. Scanlon, with notes by J. Demeyer. The typeset PDF gives the following statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 33\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[144]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 33 (Videla). Let K ⊆ Qtot. real ⊆ Q. Define \\n\\nAK:= {s ∈ R>0 | There exist infinitely many α ∈ O K\\n\\nsuch that α and its conjugates are all in [0, s ]}\\n\\nQuestion of Julia Robinson: Is the infimum of AK an element of AK? If so, the first order theory of OK is undecidable. For K = Qtot. real, inf( AK ) = 4 ∈ AK.PROBLEMS RELATED TO \\\"EXTENSIONS OF HILBERT'S TENTH PROBLEM\\\" 7\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0145", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the recovered AIM closed-interval formulation, replacing [0,s] by (0,s) changes only finitely many algebraic integers at each finite threshold. Number fields have empty literal A_K; for K contained in L one has B_K(s)=B_L(s) intersect O_K, hence threshold monotonicity and upward propagation of attainment when thresholds agree; and for every nested number-field exhaustion K=union K_n, attainment at a finite lambda_K is equivalent to unboundedness of the finite counts |B_{K_n}(lambda_K)|. A Kronecker argument also proves the universal lower bound lambda_K>=4 for finite thresholds and shows that infinitely many cyclotomic elements 2+zeta_n+zeta_n^{-1} force lambda_K=4 attained, verifying the printed case of the maximal totally real field.\n\nCandidate contribution (criterion; novelty confidence low): The endpoint-robust exhaustion criterion packages two exact, testable equivalences: the AIM closed-box and modern open-box conventions have identical infinitude behavior at every finite threshold, and for any nested number-field exhaustion K=union K_n, a finite threshold lambda_K is attained exactly when |B_{K_n}(lambda_K)| is unbounded; moreover, attainment propagates upward along inclusions whose numerical thresholds agree." }, { "id": 20002370, "problem_number": "AIM-LOGIC-0146", "title": "Exact alternation depth and variable bounds for two real rational-function fields", "statement": "Question 34 (Zahidi). Let Ralg:= Q ∩ R. It is known that Ralg ≡ R (elementary equiv-alence), but that Ralg (t) 6 ≡ R(t). On the other hand, the existential theories of Ralg (t) and\n\nR(t) are the same. What is the minimal quantifier complexity for which Ralg (t) and R(t)\n\nhave different theories? Another question is the minimal number of variables one needs.", "original_statement": "Question 34 (Zahidi). Let Ralg:= Q ∩ R. It is known that Ralg ≡ R (elementary equiv-alence), but that Ralg (t) 6 ≡ R(t). On the other hand, the existential theories of Ralg (t) and \n\nR(t) are the same. What is the minimal quantifier complexity for which Ralg (t) and R(t)\n\nhave different theories? Another question is the minimal number of variables one needs.", "clean_statement": "Question 34 (Zahidi). Let Ralg:= Q ∩ R. It is known that Ralg ≡ R (elementary equiv-alence), but that Ralg (t) 6 ≡ R(t). On the other hand, the existential theories of Ralg (t) and\n\nR(t) are the same. What is the minimal quantifier complexity for which Ralg (t) and R(t)\n\nhave different theories? Another question is the minimal number of variables one needs.", "statement_status": "exact", "statement_verification": "The OCR record has lost a conjugation bar and has turned `\\(\\not\\equiv\\)` into `6 equiv`. Page 7 of the official AIM PDF gives the following recovered statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 34\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[145]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 34 (Zahidi). Let Ralg:= Q ∩ R. It is known that Ralg ≡ R (elementary equiv-alence), but that Ralg (t) 6 ≡ R(t). On the other hand, the existential theories of Ralg (t) and \\n\\nR(t) are the same. What is the minimal quantifier complexity for which Ralg (t) and R(t)\\n\\nhave different theories? Another question is the minimal number of variables one needs.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0146", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In the parameter-free pure language of rings, R_alg(t) and R(t) agree on existential, universal, forall_1-exists, and exists_1-forall sentences, but a published Liouville-cut construction gives a pure prenex forall_3-exists separator. Therefore their minimum quantifier-block alternation depth is exactly one, while the minimum size of the leading universal block in a forall_n-exists separator is either two or three. At least two distinct object-variable symbols are necessary, but the exact total-variable minimum remains open.\n\nCandidate contribution (bound; novelty confidence low): For the exact AIM pair, combining one-universal-variable preservation with the parameter-free Liouville-cut separator proves an exact alternation minimum of one, a leading-universal-block bound 2 <= n_min <= 3, and a total finite-variable lower bound of two." }, { "id": 20002371, "problem_number": "AIM-LOGIC-0147", "title": "A complete Brody-hyperbolicity algorithm for algebraic curves", "statement": "Question 35 (Pheidas). Let X be a variety over Q. Call X hyperbolic iff there is no nonconstant holomorphic map C → X(C). Is there an algorithm which can decide whether a variety X/ Q over hyperbolic?", "original_statement": "Question 35 (Pheidas). Let X be a variety over Q. Call X hyperbolic iff there is no nonconstant holomorphic map C → X(C). Is there an algorithm which can decide whether a variety X/ Q over hyperbolic?", "clean_statement": "**Is there an algorithm which can decide whether a variety\n\\(X/\\mathbb Q\\) is hyperbolic?**", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record comes from Question 35 of the AIM problem list *Problems related to “Extensions of Hilbert's Tenth Problem”*, moderated by B. Poonen and T. Scanlon, with notes by J. Demeyer. The PDF text reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 35\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[146]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 35 (Pheidas). Let X be a variety over Q. Call X hyperbolic iff there is no nonconstant holomorphic map C → X(C). Is there an algorithm which can decide whether a variety X/ Q over hyperbolic?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0147", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the explicit promise class of locally closed schemes X=V(I)\\V(J) in projective space over Q whose geometric components have dimension at most one, Brody hyperbolicity is decidable exactly. After reduction and geometric irreducible decomposition, let each one-dimensional component have smooth projective normalized completion of genus g and r distinct geometric boundary points. Then X is Brody hyperbolic if and only if 2g-2+r>0 for every component. Standard exact decomposition, normalization, genus, and zero-dimensional radical algorithms compute these invariants and return finite certificates; the only obstruction pairs are (0,0), (0,1), (0,2), and (1,0). Smooth projective curves specialize to the genus-at-least-two test. The unrestricted all-dimensional AIM question is not resolved.\n\nCandidate contribution (algorithmic criterion; novelty confidence low): The normalization--log-Euler package gives one exact terminating decision procedure and finite certificate format for every represented dimension-at-most-one input, simultaneously covering nonproper, reducible, singular, and nonreduced curves: hyperbolicity is equivalent componentwise to positivity of 2g-2+r, and failure is certified by exactly one of four pairs." }, { "id": 20002372, "problem_number": "AIM-LOGIC-0148", "title": "Finite homogeneous maps and a sharp fiber bound", "statement": "**Question 36 (Jarden).** Given $f_1,\\ldots,f_n\\in\\mathbb C[x_1,\\ldots,x_m]$ which are homogeneous of degree $d$, assume that their only common zero is $(0,\\ldots,0)$. Prove that\n\\[\nV\\bigl(f_1(\\vec x)=b_1,\\ldots,f_n(\\vec x)=b_n\\bigr)\n\\]\nis finite for all $b_1,\\ldots,b_n\\in\\mathbb C$.", "original_statement": "Question 36 (Jarden). Given f1,..., f n ∈ C[x1,..., x m] which are homogeneous of degree \n\nd. Assume that the only common zero of the fi is (0,..., 0). Prove that \n\nV (f1(~x) = b1,..., f n(~x) = bn)\n\nis finite, for all b1,..., b n ∈ C.Solution: If it were infinite, then the variety in Pm defined by the homogenizations of the equations would be positive-dimensional, and then it would have to intersect the hyperplane at infinity, which would mean that the fi have a common zero.", "clean_statement": "**Question 36 (Jarden).** Given $f_1,\\ldots,f_n\\in\\mathbb C[x_1,\\ldots,x_m]$ which are homogeneous of degree $d$, assume that their only common zero is $(0,\\ldots,0)$. Prove that\n\\[\nV\\bigl(f_1(\\vec x)=b_1,\\ldots,f_n(\\vec x)=b_n\\bigr)\n\\]\nis finite for all $b_1,\\ldots,b_n\\in\\mathbb C$.", "statement_status": "corrected_verified", "statement_verification": "The canonical JSON preserves the source record but contains extraction artifacts: `f n`, `x m`, `C.Solution`, and `f_i(~x)`. The official AIM PDF confirms the intended subscripts, spacing, and vector notation. The recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Extensions of Hilbert's tenth problem\nSection: \nSource item: 36\nSource URL: https://aimath.org/WWN/hilberts10th/hilberts10th.pdf\nCanonical location: aim-logic-notes.json notes[147]\nCanonical tag: question\nOriginal extracted problem text (JSON string): \"Question 36 (Jarden). Given f1,..., f n ∈ C[x1,..., x m] which are homogeneous of degree \\n\\nd. Assume that the only common zero of the fi is (0,..., 0). Prove that \\n\\nV (f1(~x) = b1,..., f n(~x) = bn)\\n\\nis finite, for all b1,..., b n ∈ C.Solution: If it were infinite, then the variety in Pm defined by the homogenizations of the equations would be positive-dimensional, and then it would have to intersect the hyperplane at infinity, which would mean that the fi have a common zero.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/hilberts10th/hilberts10th.pdf", "tags": [ "aim", "AIM-LOGIC-0148", "aim-domain:logic", "aim-workshop:hilberts10th", "aim-source-tag:question" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The projective homogenization argument supplied in the AIM source is correct and proves Jarden's finiteness statement. More strongly, over any algebraically closed field, homogeneous forms f_1,...,f_n of common degree d with common zero set {0} define a finite morphism F:A^m→A^n. If I=(f_1,...,f_n), every scheme fiber has length at most dim k[x_1,...,x_m]/I, which is at most d^m; hence every set-theoretic fiber has at most d^m points. The scheme-length bound is sharp in every characteristic, and the point bound is sharp when the characteristic does not divide d.\n\nCandidate contribution (theorem; novelty confidence low): The qualitative AIM solution admits the explicit characteristic-free refinement length(F^{-1}(b)) ≤ dim_k k[x_1,...,x_m]/(f_1,...,f_n) ≤ d^m for every b, with F finite and the d^m scheme-length bound sharp." }, { "id": 20002373, "problem_number": "AIM-LOGIC-0149", "title": "A branch-profile audit of the CBH participant note", "statement": "A.1 Aspero, David\n\nMost of Hugh Woodin's deep work on core model theory remains unpublished. I believe this workshop will be an excellent opportunity for me to learn about the Cofinal Branches Hypothesis (about which I don't know much) and about Woodin's refutation of it.", "original_statement": "A.1 Aspero, David \n\nMost of Hugh Woodin's deep work on core model theory remains unpublished. I believe this workshop will be an excellent opportunity for me to learn about the Cofinal Branches Hypothesis (about which I don't know much) and about Woodin's refutation of it.", "clean_statement": "A.1 Aspero, David\n\nMost of Hugh Woodin's deep work on core model theory remains unpublished. I believe this workshop will be an excellent opportunity for me to learn about the Cofinal Branches Hypothesis (about which I don't know much) and about Woodin's refutation of it.", "statement_status": "exact", "statement_verification": "The canonical record is source index 148 of **aim-logic-notes.json**. Its source is the official AIM PDF *Recent Advances in Core Model Theory*, version dated 29 November 2004. The PDF table of contents calls Chapter A “Participant Contributions”; page 3 begins with the same heading and then lists the participants by name. The entries immediately following A.1 are also first-person descriptions of what participants hoped to learn at the workshop.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.1\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[148]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.1 Aspero, David \\n\\nMost of Hugh Woodin's deep work on core model theory remains unpublished. I believe this workshop will be an excellent opportunity for me to learn about the Cofinal Branches Hypothesis (about which I don't know much) and about Woodin's refutation of it.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0149", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The official 2004 AIM source identifies A.1 as a participant contribution and the text is a learning-interest statement, not an open problem. As a concrete mathematical synthesis, for an iteration tree T define c(T) as the number of cofinal branches and w(T) as the number whose direct limits are well-founded. Local CBH is w(T) >= 1 and local UBH is w(T) <= 1, so one tree cannot refute both. Under the explicit Neeman-Steel assumptions (A1)-(A2), their published Theorem 2.1 has profile (c,w)=(2,2), while their distinct Theorem 3.4 has profile (c,w)=(1,0).\n\nCandidate contribution (expository lemma; novelty confidence low): The two-coordinate branch profile (c(T),w(T)) yields a witness-separation lemma: a CBH counterexample has w=0 and locally satisfies UBH, a UBH counterexample has w>=2 and locally satisfies CBH, and simultaneous global failure necessarily requires distinct trees; the Neeman-Steel witnesses realize the orthogonal profiles (1,0) and (2,2).", "research_difficulty_suggested": "L2" }, { "id": 20002374, "problem_number": "AIM-LOGIC-0150", "title": "A forcing diagnostic for CH and Omega-logic", "statement": "A.2 Brooke-Taylor, Andrew\n\nI find progress on the continuum hypothesis very interesting, and so learning what the ramifications of this work are for Woodin's Omega conjecture is particularly enticing. Also, my current area of study is at the level of I3, so it will be good to know more about the difficulties in trying to build models with superstrong cardinals, with an eye to continuing the ascent.", "original_statement": "A.2 Brooke-Taylor, Andrew \n\nI find progress on the continuum hypothesis very interesting, and so learning what the ramifications of this work are for Woodin's Omega conjecture is particularly enticing. Also, my current area of study is at the level of I3, so it will be good to know more about the difficulties in trying to build models with superstrong cardinals, with an eye to continuing the ascent.", "clean_statement": "A.2 Brooke-Taylor, Andrew\n\nI find progress on the continuum hypothesis very interesting, and so learning what the ramifications of this work are for Woodin's Omega conjecture is particularly enticing. Also, my current area of study is at the level of I3, so it will be good to know more about the difficulties in trying to build models with superstrong cardinals, with an eye to continuing the ascent.", "statement_status": "exact", "statement_verification": "The source is the four-page AIM document *Recent Advances in Core Model Theory*, version 29 November 2004. Its table of contents labels Chapter A “Participant Contributions.” On page 3, item A.2 occurs between the analogous personal statements A.1 and A.3. The exact recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.2\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[149]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.2 Brooke-Taylor, Andrew \\n\\nI find progress on the continuum hypothesis very interesting, and so learning what the ramifications of this work are for Woodin's Omega conjecture is particularly enticing. Also, my current area of study is at the level of I3, so it will be good to know more about the difficulties in trying to build models with superstrong cardinals, with an eye to continuing the ascent.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0150", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The official source confirms that AIM-LOGIC-0150 is a participant-interest statement rather than a formal problem. As a mathematically developed synthesis, the report proves that any forcing-persistent theory with a countable transitive model has set-generic extensions satisfying CH and not-CH, so neither ordinary consequence nor any generically sound consequence relation can select a CH truth value from generic stability alone. This identifies finite Omega-star-axiomatizability of the actual theory of H(omega_2) as an indispensable additional hypothesis in Woodin's published conditional anti-CH statement, and separately proves the domain-level obstruction that an I3 map V_lambda to V_lambda is not itself a global superstrong witness or an iteration strategy.\n\nCandidate contribution (lemma; novelty confidence low): For the exact AIM context, the two explicit forcing extensions provide a diagnostic: every proposed derivation of a CH truth value from a forcing-persistent large-cardinal background must use additional non-generic structural content; generic stability alone cannot perform the selection.", "research_difficulty_suggested": "L2" }, { "id": 20002375, "problem_number": "AIM-LOGIC-0151", "title": "A parameter-exact sharp-to-perfect-set bridge", "statement": "A.3 Brown, Elizabeth\n\nCore Model Theory as such is largely a new topic to me; what I know about CMT specifically I have learnt in preparation for this workshop. I am particularly interested in CMT as part of the general study of large cardinals and their equivalencies, and in the implications of CMT for analysis.", "original_statement": "A.3 Brown, Elizabeth \n\nCore Model Theory as such is largely a new topic to me; what I know about CMT specifically I have learnt in preparation for this workshop. I am particularly interested in CMT as part of the general study of large cardinals and their equivalencies, and in the implications of CMT for analysis.", "clean_statement": "A.3 Brown, Elizabeth\n\nCore Model Theory as such is largely a new topic to me; what I know about CMT specifically I have learnt in preparation for this workshop. I am particularly interested in CMT as part of the general study of large cardinals and their equivalencies, and in the implications of CMT for analysis.", "statement_status": "exact", "statement_verification": "The canonical record is item A.3 in the AIM workshop document *Recent advances in core model theory*. The source PDF labels its appendix “Participant Contributions”; A.1, A.2, A.3, and A.4 are individual participants' statements. The text of A.3 is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.3\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[150]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.3 Brown, Elizabeth \\n\\nCore Model Theory as such is largely a new topic to me; what I know about CMT specifically I have learnt in preparation for this workshop. I am particularly interested in CMT as part of the general study of large cardinals and their equivalencies, and in the implications of CMT for analysis.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0151", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source record is a participant-interest statement rather than a mathematical problem. As a rigorous synthesis prompted by its stated interest, for every real a the cardinal gap omega_1^{L[a]} < omega_1 implies that every Sigma^1_2(a) set has the perfect set property; the existence of a-sharp is a sufficient condition for this gap. Consequently, if a-sharp exists for every real a, every boldface Sigma^1_2 set has the perfect set property.\n\nCandidate contribution (synthesis; novelty confidence low): A parameter-audited bridge certificate factors the conclusion into the explicit cardinal-gap premise omega_1^{L[a]} < omega_1 and the relativized Mansfield-Solovay alternative A subseteq L[a] or A contains a perfect subset, while isolating a-sharp as only a sufficient certificate for the gap and exposing the all-real quantifier needed for the boldface conclusion.", "research_difficulty_suggested": "L2" }, { "id": 20002376, "problem_number": "AIM-LOGIC-0152", "title": "Four core-model questions and a modular audit of branch-hypothesis counterexamples", "statement": "Some of my goals/questions for the workshop:\n\nA. I would like a better understanding of the relationship between models of the form \\(\\mathrm{HOD}^M\\) and the classical \\(L[\\vec E]\\) models.\n\nB. Very strong axioms are used in the refutations of UBH and CBH (at least in the versions of these results which I have heard about). How exactly are they used? Are they necessary?\n\nC. I would like to know more about Woodin's recent work on promising extender sequences.\n\nD. I would like to get a picture of the status and significance of the \\(\\Omega\\) conjecture.", "original_statement": "A.4 Cummings, James \n\nSome of my goals/questions for the workshop: A. I would like a better understanding of the relationship between models of the form HOD M and the classical L[ ~E] models. B. Very strong axioms are used in the refutations of UBH and CBH (at least in the versions of these results which I have heard about). How exactly are they used? Are they necessary? C. I would like to know more about Woodin's recent work on promising extender se-quences. D. I would like to get a picture of the status and significance of the Ω conjecture.", "clean_statement": "Some of my goals/questions for the workshop:\n\nA. I would like a better understanding of the relationship between models of the form \\(\\mathrm{HOD}^M\\) and the classical \\(L[\\vec E]\\) models.\n\nB. Very strong axioms are used in the refutations of UBH and CBH (at least in the versions of these results which I have heard about). How exactly are they used? Are they necessary?\n\nC. I would like to know more about Woodin's recent work on promising extender sequences.\n\nD. I would like to get a picture of the status and significance of the \\(\\Omega\\) conjecture.", "statement_status": "corrected_verified", "statement_verification": "The record is item A.4, James Cummings's contribution to the AIM workshop *Recent advances in core model theory*. The canonical extraction is substantially readable, but it loses two pieces of mathematical typography and introduces a line-break hyphen. Inspection of the official AIM PDF verifies the following recovered statement: The PDF content stream places the \\(M\\) as a superscript on HOD and places a vector accent over \\(E\\); thus the intended expressions are \\(\\mathrm{HOD}^M\\) and \\(L[\\vec E]\\), not “HOD M” and a literal \\(L[\\widetilde E]\\). “se-quences” is only a line-break artifact. These corrections are source-verified rather than silent rewrites.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.4\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[151]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.4 Cummings, James \\n\\nSome of my goals/questions for the workshop: A. I would like a better understanding of the relationship between models of the form HOD M and the classical L[ ~E] models. B. Very strong axioms are used in the refutations of UBH and CBH (at least in the versions of these results which I have heard about). How exactly are they used? Are they necessary? C. I would like to know more about Woodin's recent work on promising extender se-quences. D. I would like to get a picture of the status and significance of the Ω conjecture.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0152", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Neeman--Steel's branch-hypothesis construction separates into a local-Woodin collision seed, an overlap-lift and ultrapower-identification module, a Martin--Steel well-foundedness input, and a CBH amplification module. Under their Lemma 3.2 hypotheses, equality of the two transported extenders makes both branch models isomorphic to the same ultrapower, so well-foundedness of either branch transfers to the other. Under the countable-closure hypotheses of their Proposition 2.6, the entire set of well-founded branches is invariant when the lower tree is lifted from V to Ult(V,F); hence that specific overlap-lift cannot manufacture a UBH counterexample from a tree having at most one well-founded branch. This is a method-specific obstruction, not a proof of UBH for all countably closed trees and not a proof that the original large-cardinal assumptions are necessary.\n\nCandidate contribution (lemma; novelty confidence low): The collision-transfer and closure no-go proposition packages Neeman--Steel Lemma 3.2 and Proposition 2.6 into an exact implication: transported-extender equality transfers one branch's well-foundedness to the other, while countable closure preserves the lower tree's well-founded-branch set exactly and therefore blocks this overlap-lift mechanism." }, { "id": 20002377, "problem_number": "AIM-LOGIC-0153", "title": "Games, distributivity, and stationarity preservation", "statement": "A.5 Dobrinen, Natasha\n\nI am delighted to participate in the ARCC Workshop. My general goal is, naturally, to gain a deeper and better understanding of the state of the art in core model research and find some open problems to work on. In particular, my research with games related to distributive laws in Boolean algebras is leading me to look at large cardinals. My hope is to learn techniques for working with large cardinals which will help in settling questions about relationships between games, distributive laws, and stationary subsets of Pκλ.", "original_statement": "A.5 Dobrinen, Natasha \n\nI am delighted to participate in the ARCC Workshop. My general goal is, naturally, to gain a deeper and better understanding of the state of the art in core model research and find some open problems to work on. In particular, my research with games related to distributive laws in Boolean algebras is leading me to look at large cardinals. My hope is to learn techniques for working with large cardinals which will help in settling questions about relationships between games, distributive laws, and stationary subsets of Pκλ.", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "This is a research-interest statement, not a mathematical question with specified hypotheses or a requested conclusion. Thus the correct status is `context_only`, with `problem_status_at_run` equal to `not_a_problem`.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.5\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[152]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.5 Dobrinen, Natasha \\n\\nI am delighted to participate in the ARCC Workshop. My general goal is, naturally, to gain a deeper and better understanding of the state of the art in core model research and find some open problems to work on. In particular, my research with games related to distributive laws in Boolean algebras is leading me to look at large cardinals. My hope is to learn techniques for working with large cardinals which will help in settling questions about relationships between games, distributive laws, and stationary subsets of Pκλ.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0153", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "AIM item A.5 is a verified participant-interest statement rather than a posed problem, and Dobrinen's later primary papers substantially realized its proposed research program. As a mathematically developed synthesis, the artifacts prove that less-than-kappa closure implies a win in an explicitly defined dense-set fusion game, which implies less-than-kappa distributivity, while the converse route to stationarity preservation fails sharply: for every lambda at least omega_1, a countably distributive club-shooting Boolean algebra destroys a ground-model stationary subset of P_{omega_1}(lambda).\n\nCandidate contribution (counterexample; novelty confidence low): For every lambda at least omega_1, lifting a stationary A subset of omega_1 to the set of countable x subset of lambda whose intersection with omega_1 is an ordinal in A gives a stationary subset of P_{omega_1}(lambda) that is destroyed by the countably distributive regular-open completion of club shooting through omega_1 minus A; combined with the explicit fusion-game lemma, this yields a strict closure-to-game-to-distributivity boundary that does not extend to stationarity preservation.", "research_difficulty_suggested": "L2" }, { "id": 20002378, "problem_number": "AIM-LOGIC-0154", "title": "From Jensen covering to a singular-cardinal obstruction", "statement": "A.6 Dzamonja, Mirna\n\nThe primary interest of my research is combinatorial set theory. The fields of core model theory and combinatorial set theory may look rather distant at a first glance, but in fact the work in the last ten or so years have shown that there is a large overlap. I am very 4\n\npleased to participate in the workshop with the idea of learning more methods that have been invented within the core model theory and understanding their combinatorial nature.", "original_statement": "A.6 Dzamonja, Mirna \n\nThe primary interest of my research is combinatorial set theory. The fields of core model theory and combinatorial set theory may look rather distant at a first glance, but in fact the work in the last ten or so years have shown that there is a large overlap. I am very 4\n\npleased to participate in the workshop with the idea of learning more methods that have been invented within the core model theory and understanding their combinatorial nature.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is item A.6 of the AIM workshop document *Recent advances in core model theory*. The PDF places A.6 in Chapter A, “Participant Contributions.” After correcting only a page-layout artifact, the recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.6\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[153]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.6 Dzamonja, Mirna \\n\\nThe primary interest of my research is combinatorial set theory. The fields of core model theory and combinatorial set theory may look rather distant at a first glance, but in fact the work in the last ten or so years have shown that there is a large overlap. I am very 4\\n\\npleased to participate in the workshop with the idea of learning more methods that have been invented within the core model theory and understanding their combinatorial nature.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0154", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source record is a participant-interest statement, and the isolated 4 in its extraction is the printed page number at a PDF page break. As a rigorous synthesis prompted by the statement, if 0-sharp does not exist, Jensen covering yields an explicit family of at most kappa-plus constructible cells covering every cf(kappa)-sized subset of each singular strong-limit kappa; each cell contributes fewer than kappa subsets. Graph coding, Konig's theorem, and bounded-initial-segment coding then give 2^kappa = kappa-plus. Consequently, failure of SCH at a singular strong-limit cardinal implies that 0-sharp exists.\n\nCandidate contribution (synthesis; novelty confidence low): An explicit bounded-cell certificate C = {Y in P(kappa) intersect L : the ambient size of Y is at most max(aleph_1, cf(kappa))} separates the classical covering-to-SCH proof into three testable budgets: at most kappa-plus cells, fewer than kappa relevant subsets per cell, and a kappa-plus graph-coding/Konig lower bound, with a separate aleph_1-sized enlargement in the countable-cofinality case.", "research_difficulty_suggested": "L2" }, { "id": 20002379, "problem_number": "AIM-LOGIC-0155", "title": "A limit-gap obstruction for partial iteration strategies", "statement": "A.7 Fuchs, Gunter\n\nOf the scope of the conference, two topics are of main importance to me. Firstly, getting familiar with the method used to refute the CBH is essential. I am looking forward to learning about this. Secondly, the theory of inner models constructed relative to a sequence of extenders together with partial iteration strategies seems to get more and more important. This is an intriguing area, and I hope to be able to do some research here, using a very widely applicable form of fine structure theory.", "original_statement": "A.7 Fuchs, Gunter \n\nOf the scope of the conference, two topics are of main importance to me. Firstly, getting familiar with the method used to refute the CBH is essential. I am looking forward to learning about this. Secondly, the theory of inner models constructed relative to a sequence of extenders together with partial iteration strategies seems to get more and more important. This is an intriguing area, and I hope to be able to do some research here, using a very widely applicable form of fine structure theory.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The official AIM PDF confirms that A.7 is Gunter Fuchs's participant contribution to the 2004 workshop *Recent advances in core model theory*. The recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.7\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[154]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.7 Fuchs, Gunter \\n\\nOf the scope of the conference, two topics are of main importance to me. Firstly, getting familiar with the method used to refute the CBH is essential. I am looking forward to learning about this. Secondly, the theory of inner models constructed relative to a sequence of extenders together with partial iteration strategies seems to get more and more important. This is an intriguing area, and I hope to be able to do some research here, using a very widely applicable form of fine structure theory.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0155", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source is a participant-interest note, not a formal problem. Its two themes admit a rigorous synthesis: for a limit-length CBH counterexample whose proper limit-stage models are well-founded, the actual continuation branches form a branch-correct partial strategy on every proper limit restriction, but no branch-correct extension can add the union tree. Initial-segment closure makes the admitted restrictions downward closed, so a missing restriction forces omission of every longer restriction; adding directed-union closure therefore forces a cofinal tail gap. A faithful copy with a branch bijection and membership-preserving branch-model embeddings remains a CBH counterexample because every external descending sequence is transported. For the Neeman--Steel tree of length omega times omega, the proper limit stages are exactly the block boundaries omega n, Corollary 3.8 and the Martin--Steel input verify their well-founded continuations, and the final critical-point sequence gives the ill-founded union.\n\nCandidate contribution (lemma; novelty confidence low): The limit-gap and copied-gap theorem gives an exact domain-profile dichotomy: along a locally well-founded CBH counterexample, a prefix-closed branch-correct partial-strategy domain either contains every proper limit restriction and has a single final gap, or omits a cofinal tail; directed-union closure eliminates the single-gap alternative, and faithful copying cannot erase the obstruction.", "research_difficulty_suggested": "L2" }, { "id": 20002380, "problem_number": "AIM-LOGIC-0156", "title": "The relativized Church-Kleene boundary as a fine-structure bridge", "statement": "A.8 Greenberg, Noam\n\nMy mathematical training is mainly in recursion theory on admissible ordinals, in which fine-structural notions come into play. I am also in general interested in set theory; inner model theory in particular has some flavor of recursion theory, and so I hope to gain understanding of the field. As I am a beginner I'm afraid I don't have much to contribute in the way of suggesting problems or issues for consideration.", "original_statement": "A.8 Greenberg, Noam \n\nMy mathematical training is mainly in recursion theory on admissible ordinals, in which fine-structural notions come into play. I am also in general interested in set theory; inner model theory in particular has some flavor of recursion theory, and so I hope to gain understanding of the field. As I am a beginner I'm afraid I don't have much to contribute in the way of suggesting problems or issues for consideration.", "clean_statement": "A.8 Greenberg, Noam\n\nMy mathematical training is mainly in recursion theory on admissible ordinals, in which fine-structural notions come into play. I am also in general interested in set theory; inner model theory in particular has some flavor of recursion theory, and so I hope to gain understanding of the field. As I am a beginner I'm afraid I don't have much to contribute in the way of suggesting problems or issues for consideration.", "statement_status": "exact", "statement_verification": "The official AIM PDF *Recent Advances in Core Model Theory* was checked directly. Item A.8 occurs in Chapter A, “Participant Contributions,” and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.8\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[155]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.8 Greenberg, Noam \\n\\nMy mathematical training is mainly in recursion theory on admissible ordinals, in which fine-structural notions come into play. I am also in general interested in set theory; inner model theory in particular has some flavor of recursion theory, and so I hope to gain understanding of the field. As I am a beginner I'm afraid I don't have much to contribute in the way of suggesting problems or issues for consideration.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0156", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The official AIM item A.8 is explicitly a participant background statement that proposes no problem. As a rigorous bridge matching its theme, the artifacts establish for every real x that omega_1^x is the least x-admissible ordinal, L_{omega_1^x}[x] is the least transitive admissible set containing x, and its reals are exactly Delta^1_1(x). They also prove the exact rank diagnostic: an ordinal beta is below omega_1^x exactly when it has an x-recursive well-order code, equivalently an x-hyperarithmetic well-order code, while omega_1^x itself is externally countable but has neither kind of code.\n\nCandidate contribution (diagnostic_synthesis; novelty confidence low): For each real oracle x, the staged theorem packages a four-way parameter audit: x-recursive and x-hyperarithmetic well-order codes have exactly the same rank spectrum below omega_1^x; L_{omega_1^x}[x] is the first x-admissible KP level and least admissible set over x; and mere external countability does not imply an x-effective presentation, as witnessed by omega_1^x itself.", "research_difficulty_suggested": "L2" }, { "id": 20002381, "problem_number": "AIM-LOGIC-0157", "title": "An internal coding criterion for HOD to satisfy V = HOD", "statement": "A.9 Koellner, Peter\n\nI am interested in Woodin's HOD -analysis. In particular, I would like to come away from the workshop with an understanding of the analysis of HOD L(R) (under the assumption of AD L(R)) and the proof that HOD L(R) thinks V = HOD.", "original_statement": "A.9 Koellner, Peter \n\nI am interested in Woodin's HOD -analysis. In particular, I would like to come away from the workshop with an understanding of the analysis of HOD L(R) (under the assumption of AD L(R)) and the proof that HOD L(R) thinks V = HOD.", "clean_statement": "A.9 Koellner, Peter\n\nI am interested in Woodin's HOD -analysis. In particular, I would like to come away from the workshop with an understanding of the analysis of HOD L(R) (under the assumption of AD L(R)) and the proof that HOD L(R) thinks V = HOD.", "statement_status": "exact", "statement_verification": "This record is item A.9 in the appendix “Participant Contributions” of the AIM workshop report *Recent advances in core model theory*. The canonical JSON lost superscripts. Inspection of the official PDF typography gives the following recovered statement (the superscripts are the only reconstruction):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.9\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[156]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.9 Koellner, Peter \\n\\nI am interested in Woodin's HOD -analysis. In particular, I would like to come away from the workshop with an understanding of the analysis of HOD L(R) (under the assumption of AD L(R)) and the proof that HOD L(R) thinks V = HOD.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0157", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The recovered source is a participant learning goal, not an open problem. For every transitive model M of ZF, the attempt proves that M satisfies V = HOD if and only if M has a set-like global well-order definable from one fixed finite tuple of ordinals, if and only if M has a pure-language ordinal-definable surjection from Ord onto M. It also proves a predicate-elimination corollary and uses a Cohen extension to show that a strategy- or predicate-relative well-order alone is insufficient. Applied to HOD of L(R), the conclusion is deliberately conditional on the deep missing input that the Woodin-Steel strategy presentation is internally recoverable in the pure membership language.\n\nCandidate contribution (diagnostic criterion; novelty confidence low): A strategy-relative presentation of HOD passes the internal V = HOD test precisely once it yields a pure-language ordinal-definable surjection from Ord onto the presented model; definability only in an expanded structure does not pass, as witnessed by the canonical c-relative well-order of L[c] for a Cohen real c.", "research_difficulty_suggested": "L2" }, { "id": 20002382, "problem_number": "AIM-LOGIC-0158", "title": "Local branch hypotheses as uniqueness and gluing principles for strategies", "statement": "A.10 Sargsyan, Grigor\n\nI am a second year graduate student at UC, Berkeley, and have taken inner model theory seminars with Professor Steel and Professor Woodin. Having learnt the basics of inner model theory, I hope to learn, during this workshop, more advanced topics in core model theory.", "original_statement": "A.10 Sargsyan, Grigor \n\nI am a second year graduate student at UC, Berkeley, and have taken inner model theory seminars with Professor Steel and Professor Woodin. Having learnt the basics of inner model theory, I hope to learn, during this workshop, more advanced topics in core model theory.", "clean_statement": "A.10 Sargsyan, Grigor\n\nI am a second year graduate student at UC, Berkeley, and have taken inner model theory seminars with Professor Steel and Professor Woodin. Having learnt the basics of inner model theory, I hope to learn, during this workshop, more advanced topics in core model theory.", "statement_status": "exact", "statement_verification": "The official AIM PDF verifies the canonical A.10 entry as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.10\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[157]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.10 Sargsyan, Grigor \\n\\nI am a second year graduate student at UC, Berkeley, and have taken inner model theory seminars with Professor Steel and Professor Woodin. Having learnt the basics of inner model theory, I hope to learn, during this workshop, more advanced topics in core model theory.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0158", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The official AIM source confirms that A.10 is a participant biography and learning goal, not a mathematical problem. As a developed synthesis of its stated topic, the report proves that local unique-branch hypotheses force minimal branch-correct partial iteration strategies to agree on overlaps and permit their union, while a least strategy disagreement localizes a unique-branch-hypothesis failure. A trace-realizable converse is proved, and the Neeman--Steel two-branch tree gives an exact sharpness example.\n\nCandidate contribution (lemma; novelty confidence low): In the minimal branch-correct selector framework, local UBH forces partial strategies to agree and glue on overlaps; a least disagreement is a localized UBH witness; and any trace-realizable two-well-founded-branch tree produces two strategies whose first disagreement occurs at that tree.", "research_difficulty_suggested": "L2" }, { "id": 20002383, "problem_number": "AIM-LOGIC-0159", "title": "A delegated problem list and a ZFC envelope for Theta of L(R)", "statement": "A.11 Schimmerling, Ernest\n\nPlease see http://www.math.cmu.edu/ ∼eschimme/AIM/Index.html", "original_statement": "A.11 Schimmerling, Ernest \n\nPlease see http://www.math.cmu.edu/ ∼eschimme/AIM/Index.html", "clean_statement": "A.11 Schimmerling, Ernest\n\nPlease see http://www.math.cmu.edu/ ∼eschimme/AIM/Index.html", "statement_status": "exact", "statement_verification": "The canonical record is preserved verbatim in `input.json`:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.11\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[158]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.11 Schimmerling, Ernest \\n\\nPlease see http://www.math.cmu.edu/ ∼eschimme/AIM/Index.html\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0159", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The corrupted A.11 URL is recovered as the live Schimmerling core-model workshop index, which delegates to a list of twenty open problems; malformed HTML hides the twentieth entry, but the linked rough PDF numbers Problems 1-20. Addressing exactly Problem 3, the attempt proves in ZFC that omega_1^V < Theta^{L(R)} <= ((2^{aleph_0})^+)^V and that every alpha below Theta has ambient cardinality at most the continuum. Hence omega_3^V < Theta requires 2^{aleph_0} >= omega_3^V and is impossible when the continuum is at most omega_2. It also proves that transitive same-ordinal, same-real models have identical L(R) and identical Theta, so no-new-real forcing preserves the comparison whenever it preserves the actual ordinal omega_3.\n\nCandidate contribution (invariance and obstruction lemma; novelty confidence low): For the recovered AIM Problem 3, any genuine forcing increase of Theta^{L(R)} requires new reals, while a no-new-real change in omega_3 < Theta can only arise by changing the ambient ordinal labeled omega_3; moreover omega_3 < Theta necessarily implies continuum at least omega_3." }, { "id": 20002384, "problem_number": "AIM-LOGIC-0160", "title": "A necessary HOD diagnostic inside determinacy models", "statement": "A.12 Yoshinobu, Yasuo\n\nI am working in core model theory. My interests are exactly those described by the organizers in their detailed conference description.", "original_statement": "A.12 Yoshinobu, Yasuo \n\nI am working in core model theory. My interests are exactly those described by the organizers in their detailed conference description.", "clean_statement": "A.12 Yoshinobu, Yasuo\n\nI am working in core model theory. My interests are exactly those described by the organizers in their detailed conference description.", "statement_status": "exact", "statement_verification": "The canonical record is item A.12 in Chapter A, “Participant Contributions,” of the official AIM PDF *Recent Advances in Core Model Theory*. The PDF was checked directly and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Logic\nWorkshop: Recent advances in core model theory\nSection: \nSource item: A.12\nSource URL: https://aimath.org/WWN/coremodel/coremodel.pdf\nCanonical location: aim-logic-notes.json notes[159]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.12 Yoshinobu, Yasuo \\n\\nI am working in core model theory. My interests are exactly those described by the organizers in their detailed conference description.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 18, "set_id": 14, "source_url": "https://aimath.org/WWN/coremodel/coremodel.pdf", "tags": [ "aim", "AIM-LOGIC-0160", "aim-domain:logic", "aim-workshop:coremodel", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 18, "name": "logic", "display_name": "Logic", "description": "Problems in mathematical logic, model theory, proof theory, and finite model theory.", "slug": "logic", "order_index": 18, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "AIM item A.12 is a participant-interest statement, and its reference to the organizers' detailed description is recovered from the matching official AIM announcement as including Woodin's analysis of HOD inside AD+ models via extenders and iteration strategies. The artifacts prove an elementary but exact interface theorem: for any transitive M satisfying ZF, HOD^M is an inner model of ZFC with all M-ordinals, and its M-reals form an M-well-orderable set; if M cannot well-order its ambient reals, in particular under AD or AD+, then the HOD reals are a proper subset and HOD^M is proper in M. This yields three necessary falsification tests for any proposed full extender/strategy identification of HOD^M.\n\nCandidate contribution (diagnostic_reduction; novelty confidence low): For a transitive M satisfying ZF+AD+, any proposed full presentation N=HOD^M must simultaneously satisfy ZFC and contain all M-ordinals, have an M-well-orderable set of reals, and omit at least one ambient M-real; failure of any one condition refutes the proposed identification, while passing them is explicitly not sufficient.", "research_difficulty_suggested": "L2" }, { "id": 20002385, "problem_number": "AIM-OPTIMIZATION-0001", "title": "An explicit lifted-LMI cone that is not hyperbolic", "statement": "**Open Problem 1.10.1.** Characterize all \\(d\\)-dimensional, pointed, closed, convex cones in \\(\\mathbb R^d\\) which admit a lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones.", "original_statement": "Problem 1.10.1. Characterize all d-dimensional, pointed, closed, convex cones in \n\nRd which admit a lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones. \n\nOpen", "clean_statement": "**Open Problem 1.10.1.** Characterize all \\(d\\)-dimensional, pointed, closed, convex cones in \\(\\mathbb R^d\\) which admit a lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones.", "statement_status": "corrected_verified", "statement_verification": "The official AIM PDF, *Theory and Algorithms of Linear Matrix Inequalities*, pp. 10--11, places this contribution in Levent Tunçel's section 1.10, “Representation Theory for LMIs.” The PDF first gives the following definition, with notation normalized but mathematical content preserved: The JSON text “Rd” is OCR loss for \\(\\mathbb R^d\\); it also omits the preceding definition. The source's phrase “strictly contain” is reproduced rather than silently corrected. “\\(d\\)-dimensional in \\(\\mathbb R^d\\)” means full-dimensional. Importantly, the source defines the representation through the **strictly feasible interior**, not merely by a weak inequality on all boundary points.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Optimization\nWorkshop: Theory and algorithms of linear matrix inequalities\nSection: \nSource item: 1.10\nSource URL: https://aimath.org/WWN/matrixineq/matrixineq.pdf\nCanonical location: aim-optimization-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.10.1. Characterize all d-dimensional, pointed, closed, convex cones in \\n\\nRd which admit a lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones. \\n\\nOpen\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/matrixineq/matrixineq.pdf", "tags": [ "aim", "AIM-OPTIMIZATION-0001", "aim-domain:optimization", "aim-workshop:matrixineq", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every a>1, the closed pointed full-dimensional cone C_a=Q_3+R_+(1,a,0) has an exact one-variable 3-by-3 lifted LMI in the source's strict-interior convention, but two tangent points in its compact base induce nonexposed extreme rays, so C_a is not a hyperbolicity cone. Its dual is the spectrahedral hyperbolicity cone Q_3 intersected with the halfspace s+au>=0, defined by the product hyperbolic polynomial (s+au)(s^2-u^2-v^2). This proves that lifted-LMI cones are not contained in hyperbolicity cones, while carefully leaving open the converse inclusion required for the source's full strict-containment belief.\n\nCandidate contribution (explicit_family; novelty confidence low): The one-parameter family C_a=Q_3+R_+(1,a,0), a>1, combines an exact size-three strict lifted LMI, explicit nonexposed extreme rays excluding hyperbolicity, and a polar that is the hyperbolicity cone of (s+au)(s^2-u^2-v^2)." }, { "id": 20002386, "problem_number": "AIM-OPTIMIZATION-0002", "title": "Homogenization, exact PSD lift size, and hyperbolic cone obstructions", "statement": "Problem 1.10.2. Characterize all d-dimensional, pointed, closed, convex cones in\n\nRd which admit a poly-time, lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones.\n\nMost specifically:\n\nOpen", "original_statement": "Problem 1.10.2. Characterize all d-dimensional, pointed, closed, convex cones in \n\nRd which admit a poly-time, lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones. \n\nMost specifically: \n\nOpen", "clean_statement": "Problem 1.10.2. Characterize all d-dimensional, pointed, closed, convex cones in\n\nRd which admit a poly-time, lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones.\n\nMost specifically:\n\nOpen", "statement_status": "exact", "statement_verification": "The canonical JSON record has two extraction defects. The string `Rd` is \\(\\mathbb R^d\\), and the trailing words “Most specifically: Open” do not belong to Problem 1.10.2. In the official AIM PDF, “Most specifically:” is a transition to the separately numbered Problem 1.10.3. The recovered statement of the assigned problem is therefore:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Optimization\nWorkshop: Theory and algorithms of linear matrix inequalities\nSection: \nSource item: 1.10\nSource URL: https://aimath.org/WWN/matrixineq/matrixineq.pdf\nCanonical location: aim-optimization-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.10.2. Characterize all d-dimensional, pointed, closed, convex cones in \\n\\nRd which admit a poly-time, lifted-LMI representation. I believe that this set of cones strictly contain the hyperbolic cones. \\n\\nMost specifically: \\n\\nOpen\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/matrixineq/matrixineq.pdf", "tags": [ "aim", "AIM-OPTIMIZATION-0002", "aim-domain:optimization", "aim-workshop:matrixineq", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every nonempty bounded full-dimensional polytope P, its homogenization C(P) is a closed pointed full-dimensional cone and xc_psd(P) <= xc_psd(C(P)) <= xc_psd(P)+1, with no increase in auxiliary variables in the upper construction. A separate double-polar lemma proves that a size-r strict-interior lift in the AIM source convention yields an exact closed PSD lift of matrix size at most r and at most r(r+1)/2 auxiliary variables. Applying Lee--Raghavendra--Steurer's superpolynomial lower bound for the explicit correlation polytopes gives explicit homogenized polyhedral cones, hence hyperbolicity cones, with no lift polynomial in ambient dimension. Thus, under the only nonvacuous family-size reading of the source, the hoped-for inclusion of all hyperbolicity cones is false.\n\nCandidate contribution (theorem; novelty confidence low): The source-faithful package proves the exact homogenization inequality, a quantitative strict-interior-to-exact lift conversion with matrix size at most r and auxiliary dimension at most r(r+1)/2, and transfers the LRS correlation-polytope bound to a family of pointed hyperbolicity cones outside the source-polynomial lifted-LMI class." }, { "id": 20002387, "problem_number": "AIM-OPTIMIZATION-0003", "title": "Encoding and convexity obstructions in the stated hyperbolic-feasibility problem", "statement": "Problem 1.10.3. Are all Hyperbolic Feasibility Problems polynomial-time equivalent to LMI problems?\n\nThis last question needs some definitions and clarifications.\n\nDefinition 1.10.2. Let p1, p 2,..., p m: Rd → R be given polynomials. Then the problem \"does there exist x ∈ Rd such that pi(x) ≥ 0, ∀i ∈ { 1, 2,..., m } is a Hyperbolic Feasibility Problem (HFP) if every pi is a hyperbolic polynomial.\n\nNext, we define the size( HF P ). The \"size\" should involve the basic complexity measures needed to bound the amount of computational effort required (in the Blum-Shub-Smale real computation model) to \"solve\" HFP to [U+000F] ∈ (0, 1) accuracy using some general class of well-established algorithms. For instance, we can define size( HF P ):= max {m, ln(1 /[U+000F] ), ln( R)},\n\nwhere R > 1 denotes the volume of a given ellipsoid E0 which determines the region in which we will decide the solvability of HFP. I.e., our problem is to find ¯ x ∈ E0 satisfying all the inequalities. We require that after poly(size( HLP )) operations the algorithm either outputs ¯ x ∈ Rd such that pi(¯ x) ≥ 0, ∀i ∈ { 1, 2,..., m } or it outputs \"there does not exist a ball of volume at least [U+000F] which is contained in\n\nE0 ∩ {x ∈ Rd: pi(x) ≥ 0, ∀i ∈ { 1, 2,..., m }}.′′\n\nIn this context, when we say HFP is polynomial-time equivalent to LMI we mean that for every HFP (with m, R and a given [U+000F] ∈ (0, 1)), we can explicitly describe an LMI such that 11 • the formulated LMI can be solved to [U+000F] accuracy in time poly (size( HF P )),\n\n• solving the LMI within accuracy [U+000F], solves the original HFP. This notion of poly-time equivalence is quite important in optimization theory. A problem analogous to", "original_statement": "Problem 1.10.3. Are all Hyperbolic Feasibility Problems polynomial-time equivalent to LMI problems? \n\nThis last question needs some definitions and clarifications. \n\nDefinition 1.10.2. Let p1, p 2,..., p m: Rd → R be given polynomials. Then the problem \"does there exist x ∈ Rd such that pi(x) ≥ 0, ∀i ∈ { 1, 2,..., m } is a Hyperbolic Feasibility Problem (HFP) if every pi is a hyperbolic polynomial. \n\nNext, we define the size( HF P ). The \"size\" should involve the basic complexity measures needed to bound the amount of computational effort required (in the Blum-Shub-Smale real computation model) to \"solve\" HFP to \u000f ∈ (0, 1) accuracy using some general class of well-established algorithms. For instance, we can define size( HF P ):= max {m, ln(1 /\u000f ), ln( R)},\n\nwhere R > 1 denotes the volume of a given ellipsoid E0 which determines the region in which we will decide the solvability of HFP. I.e., our problem is to find ¯ x ∈ E0 satisfying all the inequalities. We require that after poly(size( HLP )) operations the algorithm either outputs ¯ x ∈ Rd such that pi(¯ x) ≥ 0, ∀i ∈ { 1, 2,..., m } or it outputs \"there does not exist a ball of volume at least \u000f which is contained in \n\nE0 ∩ {x ∈ Rd: pi(x) ≥ 0, ∀i ∈ { 1, 2,..., m }}.′′ \n\nIn this context, when we say HFP is polynomial-time equivalent to LMI we mean that for every HFP (with m, R and a given \u000f ∈ (0, 1)), we can explicitly describe an LMI such that 11 • the formulated LMI can be solved to \u000f accuracy in time poly (size( HF P )), \n\n• solving the LMI within accuracy \u000f, solves the original HFP. This notion of poly-time equivalence is quite important in optimization theory. A problem analogous to", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record comes from the AIM workshop notes *Theory and Algorithms of Linear Matrix Inequalities*, Open Problem 1.10.3 (printed pp. 10--11; PDF pages 11--12), version dated March 12, 2006. The mathematical core is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Optimization\nWorkshop: Theory and algorithms of linear matrix inequalities\nSection: \nSource item: 1.10\nSource URL: https://aimath.org/WWN/matrixineq/matrixineq.pdf\nCanonical location: aim-optimization-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.10.3. Are all Hyperbolic Feasibility Problems polynomial-time equivalent to LMI problems? \\n\\nThis last question needs some definitions and clarifications. \\n\\nDefinition 1.10.2. Let p1, p 2,..., p m: Rd → R be given polynomials. Then the problem \\\"does there exist x ∈ Rd such that pi(x) ≥ 0, ∀i ∈ { 1, 2,..., m } is a Hyperbolic Feasibility Problem (HFP) if every pi is a hyperbolic polynomial. \\n\\nNext, we define the size( HF P ). The \\\"size\\\" should involve the basic complexity measures needed to bound the amount of computational effort required (in the Blum-Shub-Smale real computation model) to \\\"solve\\\" HFP to \\u000f ∈ (0, 1) accuracy using some general class of well-established algorithms. For instance, we can define size( HF P ):= max {m, ln(1 /\\u000f ), ln( R)},\\n\\nwhere R > 1 denotes the volume of a given ellipsoid E0 which determines the region in which we will decide the solvability of HFP. I.e., our problem is to find ¯ x ∈ E0 satisfying all the inequalities. We require that after poly(size( HLP )) operations the algorithm either outputs ¯ x ∈ Rd such that pi(¯ x) ≥ 0, ∀i ∈ { 1, 2,..., m } or it outputs \\\"there does not exist a ball of volume at least \\u000f which is contained in \\n\\nE0 ∩ {x ∈ Rd: pi(x) ≥ 0, ∀i ∈ { 1, 2,..., m }}.′′ \\n\\nIn this context, when we say HFP is polynomial-time equivalent to LMI we mean that for every HFP (with m, R and a given \\u000f ∈ (0, 1)), we can explicitly describe an LMI such that 11 • the formulated LMI can be solved to \\u000f accuracy in time poly (size( HF P )), \\n\\n• solving the LMI within accuracy \\u000f, solves the original HFP. This notion of poly-time equivalence is quite important in optimization theory. A problem analogous to\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/matrixineq/matrixineq.pdf", "tags": [ "aim", "AIM-OPTIMIZATION-0003", "aim-domain:optimization", "aim-workshop:matrixineq", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The official 2006 formulation is not a well-defined polynomial-time problem: its size measure omits dimension and all polynomial/ellipsoid encoding, and a fixed-(m,R,epsilon) family forces an explicit output of Omega(d) coordinates while the stated size remains constant. Its raw constraints p_i(x)>=0 also need not select convex hyperbolicity components; p(x,y)=xy gives a nonconvex feasible set that cannot itself be an LMI projection. After repairing the input to include chosen directions/components, complete encoding and conditioning data, and explicit polynomial-total-size spectrahedral lifts, the component intersection and encoded ellipsoid reduce exactly to one polynomial-size block-diagonal LMI.\n\nCandidate contribution (obstruction_and_reduction; novelty confidence low): A fixed-(m,R,epsilon), growing-dimension hyperbolic family disproves the proposed output bound under standard explicit-output BSS conventions; independently, xy>=0 supplies a nonconvexity obstruction to the literal LMI-projection reading, while supplied polynomial-size lifts of chosen components yield an exact block-LMI repair.", "research_difficulty_suggested": "L2" }, { "id": 20002388, "problem_number": "AIM-OPTIMIZATION-0004", "title": "Monotone Galerkin approximation of finite Schur complements", "statement": "**1.11 Hugo Woerdeman.** One of the questions I am interested in is how to approximate numerically the Schur complement of a positive definite operator supported on a finite subspace. Of course there is the formula that for a block matrix\n\\[\n\\begin{pmatrix}A&B\\\\ C&D\\end{pmatrix}\n\\]\nthe Schur complement is \\(A-BD^{-1}C\\), but this requires determining the inverse of the infinite operator \\(D\\). The way this question arose is through attempts to develop multivariable analogs of the Gohberg--Semencul formula. One way to prove the Gohberg--Semencul formula is by determining the Schur complement of the inverse of a Toeplitz operator whose symbol is the reciprocal of a positive trigonometric polynomial. In several variables the analogous attempt runs into difficulties. In order to at least get a decent numerical approximation of the inverse of a doubly Toeplitz matrix, one may try to find reasonable numerical methods to determine finitely based Schur complements of infinite operators.", "original_statement": "Problem 1.10.3 was solved in [5] by showing that Second Order Cone Programming is poly.-time equivalent to Linear Programming. \n\n1.11 Hugo Woerdeman \n\nOne of the questions I am interested in is how to approximate numerically the Schur com-plement of a positive definite operator supported on a finite subspace. Of course there is the formula that for a block matrix [A B; C D] the Schur complement is A B inv(D) C, but this requires determining the inverse of the infinite operator D. The way this question arose is through attempts to develop multivariable analogs of the Gohberg-Semencul formula. One way to prove the Gohberg-Semencul formula is by determining the Schur complement of the inverse of a Toeplitz operator whose symbol is the reciprocal of a positive trigonometric poly-nomial. In several variables the analogous attempt runs into difficulties. In order to at least get a decent numerical approximation of the inverse of a doubly Toeplitz matrix, one may try to find reasonable numerical methods to determine finitely based Schur complements of infinite operators. 12 Chapter 2 Ideas for Teaching \n\nMihai: OPEN FOR ADDITIONS, CORRECTIONS, REARRANGEMENTS ToDo \n\nA sketch of a plan: 1. General convexity (Hahn Banach, Minkowski separation theorem, Caratheodory's the-orem on generators of convex hulls) 2. Weighted sums of squares in free *-algebras 3. The spectral theorem for commuting self-adjoint operators. Note the spectral measure in physical terms is just the power spectral density. 4. Multivariate moment problems and their dual: weighted SOS decompositions of poly-nomials 5. Applications (optimization, Lyapunov functions,...) 6. Real algebra, logic and the full Positivestellensatz 7. More optimization (see Tuncel, Henrion, Lasserre) 13 Chapter 3 Other", "clean_statement": "**1.11 Hugo Woerdeman.** One of the questions I am interested in is how to approximate numerically the Schur complement of a positive definite operator supported on a finite subspace. Of course there is the formula that for a block matrix\n\\[\n\\begin{pmatrix}A&B\\\\ C&D\\end{pmatrix}\n\\]\nthe Schur complement is \\(A-BD^{-1}C\\), but this requires determining the inverse of the infinite operator \\(D\\). The way this question arose is through attempts to develop multivariable analogs of the Gohberg--Semencul formula. One way to prove the Gohberg--Semencul formula is by determining the Schur complement of the inverse of a Toeplitz operator whose symbol is the reciprocal of a positive trigonometric polynomial. In several variables the analogous attempt runs into difficulties. In order to at least get a decent numerical approximation of the inverse of a doubly Toeplitz matrix, one may try to find reasonable numerical methods to determine finitely based Schur complements of infinite operators.", "statement_status": "corrected_verified", "statement_verification": "This canonical record is a genuine extraction-boundary collision. The exact record in the assigned input contains all of the following, and none is silently discarded: The official AIM PDF fixes the boundaries exactly. On PDF p. 12, the sentence", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Optimization\nWorkshop: Theory and algorithms of linear matrix inequalities\nSection: \nSource item: 1.10\nSource URL: https://aimath.org/WWN/matrixineq/matrixineq.pdf\nCanonical location: aim-optimization-notes.json notes[3]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1.10.3 was solved in [5] by showing that Second Order Cone Programming is poly.-time equivalent to Linear Programming. \\n\\n1.11 Hugo Woerdeman \\n\\nOne of the questions I am interested in is how to approximate numerically the Schur com-plement of a positive definite operator supported on a finite subspace. Of course there is the formula that for a block matrix [A B; C D] the Schur complement is A B inv(D) C, but this requires determining the inverse of the infinite operator D. The way this question arose is through attempts to develop multivariable analogs of the Gohberg-Semencul formula. One way to prove the Gohberg-Semencul formula is by determining the Schur complement of the inverse of a Toeplitz operator whose symbol is the reciprocal of a positive trigonometric poly-nomial. In several variables the analogous attempt runs into difficulties. In order to at least get a decent numerical approximation of the inverse of a doubly Toeplitz matrix, one may try to find reasonable numerical methods to determine finitely based Schur complements of infinite operators. 12 Chapter 2 Ideas for Teaching \\n\\nMihai: OPEN FOR ADDITIONS, CORRECTIONS, REARRANGEMENTS ToDo \\n\\nA sketch of a plan: 1. General convexity (Hahn Banach, Minkowski separation theorem, Caratheodory's the-orem on generators of convex hulls) 2. Weighted sums of squares in free *-algebras 3. The spectral theorem for commuting self-adjoint operators. Note the spectral measure in physical terms is just the power spectral density. 4. Multivariate moment problems and their dual: weighted SOS decompositions of poly-nomials 5. Applications (optimization, Lyapunov functions,...) 6. Real algebra, logic and the full Positivestellensatz 7. More optimization (see Tuncel, Henrion, Lasserre) 13 Chapter 3 Other\"\nOriginal remarks: [\"Remarks \\n\\n3.1 Leonid Gurvits \\n\\nThe van der Waerden conjecture states that the permanent of n×n doubly stochastic matrix \\n\\nA satisfies the inequality P er (A) ≥ n! \\n\\n> nn\\n\\n(VDW bound) and was finally proven (independently) by D.I. Falikman and G.P. Egorychev in 1981. They both shared Delbert Ray Fulkerson prize in 1982. It was for more than XX years the most important conjecture about permanents. The VDW bound is the simplest and most powerful bound on permanents and therefore among the simplest and most powerful general purpose bounds in combinatorics. We introduce and prove a vast generalization of the VDW conjecture: \\n\\nConsider a homogeneous polynomial p(z1,..., z n) of degree n in n complex variables. As-sume that this polynomial satisfies the property: \\n\\n|p(z1,..., z n)| ≥ ∏\\n\\n> 1≤i≤n\\n\\nRe (zi) on the domain {(z1,..., z n): Re (zi) ≥ 0, 1 ≤ i ≤ n}.\\n\\nWe prove that | ∂n \\n\\n> ∂z 1...∂z n\\n\\np| ≥ n! \\n\\n> nn.\\n\\nOur generalization not only affects the world of permanents, but also has important implications concerning PDEs, stability and control theory, complexity theory. Besides, our proof is much shorter and conceptually simpler than original proofs as of the van der Waerden conjecture for permanents as well of the Bapat's conjecture on mixed discriminants, proved by the author. The paper with the proof is available at \\n\\nhttp://lanl.arxiv.org/abs/math.CO/0504397,see also the paper at \\n\\nhttp://xxx.lanl.gov/abs/math.CO/0404474.14 3.2 Alexandre Megretski \\n\\nSee separate paper \\\"Optimal Model Order Reduction for Finite Length Segments of LTI System Unit Sample Response\\\". \\n\\n3.3 Jiawang Nie \\n\\nSome topics in polynomial optimization: 1. Using gradients in SOS for approximating minimizing polynomials 2. Convergence rate of Lasserre's method for solving constrained polynomial optimzation. 3. Practical sos/moment methods for Maximum-Likelyhood estimations \\n\\n3.4 Frank Vallentin \\n\\nI am a beginner (or more precisely a user) in the theory of linear matrix inequalities and my main interest in the workshop is to learn about applications and possibilities of LMIs as well as about open problems in this theory. In the past years I was a user of LMIs for problems in classical lattice geometry. In my Ph.D. thesis I developed and implemented algorithms for solving lattice packing and covering problems which are based on semidefinite programming. Currently I am interested in LMIs and convex optimization problems which have many symmetries and usually come from discrete geometry or combinatorics. \\n\\n3.5 Jan Willems \\n\\nI am especially interested in obtaining new insights concerning the relations between LMI's and dissipative systems. In particular, I would like to learn how the recent methods involv-ing multivariable polynomials lead to the construction of storage functions and Lyapunov functions for systems described in terms of differential equations using polynomial matrices or matrices of rational functions. 15 Bibliography \\n\\n[1] N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis,Oliver and Boyd, Edinburgh, 1965. [2] B. Fuglede, The multidimensional moment problem, Expo. Math. 1(1983), 47-65. [3] C. Berg and C. Thill, Rotation invariant moment problems, Acta Math. 167(1991), 207-227. \\n\\nFrom Levent Tun¸ cel: \\n\\n[4] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization Analysis, Algorithms, and Engineering Applications, MPS-SIAM Series in Optimization, SIAM, Philadelphia, PA, USA, 2001. [5] A. Ben-Tal and A. Nemirovski, On polyhedral approximations of the second-order cone, \\n\\nMath. Oper. Res. 26 (2001) 93-205. [6] C. B. Chua and L. Tun¸ cel, Invariance and efficiency of convex representations, Research Report 2004-18, Dept. of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Ontario, Canada, June 2004. [7] J. W. Helton and V. Vinnikov, Linear matrix inequality representation of sets, Technical Report, 2002. [8] M. Kojima and A. Takeda, Complexity analysis of successive convex relaxation of non-convex sets, Math. Oper. Res. 26 (2001) 519-542. [9] M. Kojima and L. Tun¸ cel, Cones of matrices and successive convex relaxations of non-convex sets, SIAM J. Optimization 10 (2000) 750-778. [10] M. Kojima and L. Tun¸ cel, Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization problems, Math. Prog. A 89 (2000) 79- 111. [11] J. B. Lasserre, Polynomials nonnegative on a grid and discrete optimization, Trans. Amer. Math. Soc. 354 (2002) 631-649. 16 [12] J. B. Lasserre, An explicit equivalent positive semidefinite program for nonlinear 0-1 programs, SIAM J. Optim. 12 (2002) 756-769. [13] A. S. Lewis, P. A. Parrilo and M. V. Ramana, The Lax conjecture is true, Technical Report, April 2003. [14] Yu. E. Nesterov and A. S. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming, SIAM, Philadelphia, PA, USA, 1994. [15] P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems. Alge-braic and geometric methods in discrete optimization, Math. Program. Ser. B 96 (2003) 293-320. [16] P. A. Parrilo and B. Sturmfels, Minimizing polynomial functions. Algorithmic and quan-titative real algebraic geometry (Piscataway, NJ, 2001), 83-99, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 60, Amer. Math. Soc., Providence, RI, 2003. [17] M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J. 42 (1993) 969-984. [18] J. Renegar, Hyperbolic Programs and Their Derivative Relaxations, Technical Report,March 2004. [19] K. Schm¨ udgen, The K-moment problem for compact semi-algebraic sets, Math. Ann. \\n\\n289 (1991) 203-206. [20] L. Tun¸ cel and S. Xu, Complexity analyses of discretized successive convex relaxation methods. Research Report 99-37, Department of Combinatorics and Optimization, Uni-versity of Waterloo, Waterloo, Ontario, Canada, September 1999. [21] V. Vinnikov, Self-adjoint determinantal representations of real plane curves, Mathema-tische Annalen 296 (1993) 453-479. \\n\\nFrom Antonis Papachristodoulou: \\n\\n[22] P. Apkarian and H. D. Tuan. Parameterized LMIs in control theory. SIAM J. Control Optim., 38(4):1241-1264, 2000. [23] G. Chesi, A. Garulli, A. Tesi, and A. Vicino. Polynomially parameter-dependent Lya-punov functions for robust stability in polytopic systems: An LMI approach. IEEE Transactions on Automatic Control, 50(3):365-370, 2005. [24] K. Gatermann and P. A. Parrilo. Symmetry groups, semidefinite programs, and sums of squares. Journal of Pure and Appl. Algebra, 192(1-3):95-128, 2004. [25] C. W. J. Hol and C. W. Scherer. Sum of squares relaxations for polynomial semidefinite programming. In Proc. of the 16th International Symposium on MTNS, 2004. 17 [26] A. Papachristodoulou, M. Peet, and S. Lall. Constructing Lyapunov-Krasovksii func-tionals for linear time delay systems. In Proceedings of the American Control Conference,2005. [27] A. Papachristodoulou and S. Prajna. Analysis of non-polynomial systems using the sum of squares decomposition. To appear in Positive Polynomials in Control, Springer-Verlag, 2004. [28] F. Wu and S. Prajna. A new solution approach to polynomial LPV system analysis and synthesis. In Proceedings of the American Control Conference, 2004. 18\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 8, "set_id": 14, "source_url": "https://aimath.org/WWN/matrixineq/matrixineq.pdf", "tags": [ "aim", "AIM-OPTIMIZATION-0004", "aim-domain:optimization", "aim-workshop:matrixineq", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 8, "name": "analysis", "display_name": "Analysis", "description": "Limits, continuity, calculus, and function theory.", "slug": "analysis", "order_index": 8, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After recovering Hugo Woerdeman's section 1.11 from a canonical record contaminated by adjacent chapters, this attempt proves that conforming Galerkin compression of a bounded uniformly positive self-adjoint block operator gives stable finite Schur complements S_n that decrease to S in Loewner order and in operator norm when the retained space is finite-dimensional. The exact identities S_n-S=(T-T_n)^*D(T-T_n) and S_n-S_{n+1}=(T_{n+1}-T_n)^*D(T_{n+1}-T_n), together with the squared best-approximation bound ||S_n-S|| <= ||D|| ||(I-P_n)D^{-1}B^*||^2, provide rigorous error certification. Explicit counterexamples delimit uniform coercivity, finite-target, density, nestedness, and self-adjointness assumptions.\n\nCandidate contribution (theorem; novelty confidence low): For a finite retained block and nested conforming Galerkin spaces, the two exact energy-Gram identities for the total Schur error and successive Schur decrement can be combined with a squared operator best-approximation bound to certify stable monotone operator-norm convergence." }, { "id": 20002389, "problem_number": "AIM-PDES-0001", "title": "Horizontal enstrophy budgets and a Couette orientation obstruction", "statement": "Nonlinear enhanced dissipation for the 2D Euler equation with horizontal viscosity", "original_statement": "Nonlinear enhanced dissipation for the 2D Euler equation with horizontal viscosity", "clean_statement": "Nonlinear enhanced dissipation for the 2D Euler equation with horizontal viscosity", "statement_status": "exact", "statement_verification": "The canonical record contains only the title. The original AimPL page is no longer available at its live URL, but the Internet Archive snapshot dated 2024-08-28 recovers the page. Its complete mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Small scale dynamics in incompressible fluid flows\nSection: \nSource item: 1\nSource URL: http://aimpl.org/smallscalefluid/1/\nCanonical location: aim-pdes-notes.json notes[0]\nCanonical tag: section-problem\nOriginal extracted problem text (JSON string): \"Nonlinear enhanced dissipation for the 2D Euler equation with horizontal viscosity\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/smallscalefluid/1/", "tags": [ "aim", "AIM-PDES-0001", "aim-domain:pdes", "aim-workshop:smallscalefluid", "aim-source-tag:section-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the natural smooth, unforced, impermeable-channel formulation, the horizontal-viscosity vorticity equation has an exact enstrophy budget that bounds late dissipation in time-average and large-measure senses but does not exclude isolated spikes. Exact nonlinear Laplace eigenmodes show that the unnormalized instantaneous dissipation supremum is infinite and quantify the regularity cost of order-one dissipation. Linearization about Couette flow has the exact x-mean-free semigroup norm exp(-nu kappa_1^2 t), so horizontal viscosity alone yields no Couette enhanced-dissipation time scale faster than nu^{-1}; the x-independent mode is undamped.\n\nCandidate contribution (identity_obstruction_and_exact_family; novelty confidence low): The candidate contribution is a single formulation audit combining: a nonlinear enstrophy budget with precise tail-average and superlevel-time bounds; an exact nonlinear eigenmode family exposing both literal unboundedness and the Sobolev cost of normalized order-one dissipation; and an exact x-mean-free Couette semigroup calculation showing that the orientation of horizontal viscosity prevents the usual Couette enhancement mechanism." }, { "id": 20002390, "problem_number": "AIM-PDES-0002", "title": "A viscosity-uniform drift-shear family", "statement": "2D stationary Navier-Stokes with viscosity-independent forcing", "original_statement": "2D stationary Navier-Stokes with viscosity-independent forcing", "clean_statement": "2D stationary Navier-Stokes with viscosity-independent forcing", "statement_status": "exact", "statement_verification": "The canonical record contains only the title “2D stationary Navier--Stokes with viscosity-independent forcing.” The archived official AIM Problem Lists page, captured on 28 August 2024, gives the following statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Small scale dynamics in incompressible fluid flows\nSection: \nSource item: 2\nSource URL: http://aimpl.org/smallscalefluid/2/\nCanonical location: aim-pdes-notes.json notes[1]\nCanonical tag: section-problem\nOriginal extracted problem text (JSON string): \"2D stationary Navier-Stokes with viscosity-independent forcing\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/smallscalefluid/2/", "tags": [ "aim", "AIM-PDES-0002", "aim-domain:pdes", "aim-workshop:smallscalefluid", "aim-source-tag:section-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the standard stationary Navier-Stokes normalization on the two-torus, and allowing nonzero mean velocity, every nonzero mean-zero shear force f(x,y)=(0,F(x)) admits an exact branch u^nu=(c,v^nu(x)) for c nonzero, where the Fourier coefficients are F_hat(k)/(nu k^2+i c k). This branch is uniformly bounded for every viscosity nu>0, satisfies a uniform one-derivative Sobolev gain and an O(nu) inviscid-limit estimate, and includes the explicit example f=(0,sin x), u^nu=(1,(nu sin x-cos x)/(1+nu^2)) with squared normalized L2 norm at most 3/2. The archived AIM wording omits a zero-mean velocity condition; the construction does not settle that customary stronger formulation.\n\nCandidate contribution (explicit construction and uniform estimate; novelty confidence low): The drift-shear multiplier formula yields a nonzero viscosity-independent force with an exact stationary branch uniformly bounded for all positive viscosities, together with the estimates ||v^nu||_{Hdot^{s+1}} <= |c|^{-1}||F||_{Hdot^s} and ||v^nu-v^0||_{Hdot^s} <= nu c^{-2}||F||_{Hdot^s}; an explicit sine-force instance has squared normalized L2 norm at most 3/2." }, { "id": 20002391, "problem_number": "AIM-PDES-0003", "title": "Arbitrary-large one-phase solutions for 2D biharmonic Burgers", "statement": "Large data global wellposedness for 2D Burgers with hyper-dissipation", "original_statement": "Large data global wellposedness for 2D Burgers with hyper-dissipation", "clean_statement": "Large data global wellposedness for 2D Burgers with hyper-dissipation", "statement_status": "exact", "statement_verification": "The canonical record says only:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Small scale dynamics in incompressible fluid flows\nSection: \nSource item: 3\nSource URL: http://aimpl.org/smallscalefluid/3/\nCanonical location: aim-pdes-notes.json notes[2]\nCanonical tag: section-problem\nOriginal extracted problem text (JSON string): \"Large data global wellposedness for 2D Burgers with hyper-dissipation\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/smallscalefluid/3/", "tags": [ "aim", "AIM-PDES-0003", "aim-domain:pdes", "aim-workshop:smallscalefluid", "aim-source-tag:section-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the explicitly labeled standard periodic reconstruction u_t+(u dot grad)u=-nu Delta^2 u on T^2, every smooth arbitrary-amplitude one-phase datum u_0(x)=U_0(k dot x), k in Z^2 nonzero, generates a global smooth solution. The invariant profile system decomposes triangularly into scalar one-dimensional biharmonic Burgers for q=k dot U and a linear fourth-order equation for the transverse component. In addition, the exact positive-divergence energy identity and an explicit H^3 estimate show that any finite-time strong breakdown would require the time integral of the L-infinity norm of the positive divergence to diverge.\n\nCandidate contribution (special_case; novelty confidence low): For each nonzero k in Z^2, the full one-phase manifold M_k={u(x)=U(k dot x)} is invariant and globally regular at arbitrary amplitude, with the exact triangular decomposition q_t+q q'= -nu|k|^4 q'''' and V_t+qV'= -nu|k|^4 V''''; therefore a possible singularity cannot remain one-phase." }, { "id": 20002392, "problem_number": "AIM-PDES-0004", "title": "Backward self-similarity in a half-space and a parasitic-profile audit", "statement": "Existence/nonexistence of self-similar solutions for 3D Navier-Stokes", "original_statement": "Existence/nonexistence of self-similar solutions for 3D Navier-Stokes", "clean_statement": "Existence/nonexistence of self-similar solutions for 3D Navier-Stokes", "statement_status": "exact", "statement_verification": "The canonical record has the title", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Small scale dynamics in incompressible fluid flows\nSection: \nSource item: 4\nSource URL: http://aimpl.org/smallscalefluid/4/\nCanonical location: aim-pdes-notes.json notes[3]\nCanonical tag: section-problem\nOriginal extracted problem text (JSON string): \"Existence/nonexistence of self-similar solutions for 3D Navier-Stokes\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/smallscalefluid/4/", "tags": [ "aim", "AIM-PDES-0004", "aim-domain:pdes", "aim-workshop:smallscalefluid", "aim-source-tag:section-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The archived AimPL statement is underdetermined, while the workshop report identifies the intended open target as nonexistence of no-slip backward Leray profiles in the half-space under finite-local-energy or Tsai-type hypotheses. In a rigorously classified special class, every bounded plane-parallel backward profile has affine tangential pressure P(y)=g dot y'+C and velocity V(z)=2g[exp(z^2/4) erfc(z/2)-1]. These give exact classical continuously backward self-similar parasitic solutions, but their physical pressure grows as g dot x'/(-t)^(3/2), their velocity energy on every compact set away from the wall grows as (-t)^(-1), and they lie outside all finite L^q and weak-L^3 profile classes. Pressure normalization or velocity decay makes only this plane-parallel class trivial; the intended finite-local-energy problem remains open.\n\nCandidate contribution (exact_classification_and_formulation_obstruction; novelty confidence low): The candidate contribution is a complete classification of bounded plane-parallel backward Leray profiles in the no-slip half-space, including the unique error-function velocity profile for each affine tangential pressure gradient, together with the exact compact-away-from-the-wall local-energy divergence rate that excludes every nontrivial member from the intended admissible class." }, { "id": 20002393, "problem_number": "AIM-PDES-0005", "title": "Exact SQG sector and sharp interior-PV boundary drift", "statement": "Extending illposedness for SQG to QG", "original_statement": "Extending illposedness for SQG to QG", "clean_statement": "Extending illposedness for SQG to QG", "statement_status": "exact", "statement_verification": "The canonical record gives only the title “Extending illposedness for SQG to QG.” The archived official AIM Problem Lists page, captured on 28 August 2024, contains exactly one mathematical sentence:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Small scale dynamics in incompressible fluid flows\nSection: \nSource item: 5\nSource URL: http://aimpl.org/smallscalefluid/5/\nCanonical location: aim-pdes-notes.json notes[4]\nCanonical tag: section-problem\nOriginal extracted problem text (JSON string): \"Extending illposedness for SQG to QG\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/smallscalefluid/5/", "tags": [ "aim", "AIM-PDES-0005", "aim-domain:pdes", "aim-workshop:smallscalefluid", "aim-source-tag:section-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For one explicitly labeled plausible reconstruction of inviscid three-dimensional QG on the periodic half-space, the harmonic zero-interior-PV sector is an exact invariant copy of SQG. The Poisson lift is an isometry from the SQG homogeneous H^s norm to an explicit trace-bulk QG norm, so the Jeong-Kim smooth critical H^2 norm-inflation family transfers rigorously to that classical QG sector. For nonzero interior PV, the boundary velocity is the SQG Riesz velocity plus an explicitly derived divergence-free drift satisfying a sharp mode-by-mode half-horizontal-derivative estimate with constant 1/sqrt(2). This is a reduction and partial result, not a solution of the underspecified AIM prompt or a nonzero-PV ill-posedness theorem.\n\nCandidate contribution (invariant reduction and sharp estimate; novelty confidence low): The candidate contribution is the combined exact trace-bulk isometry for the harmonic QG lift, its conservative transfer of smooth critical SQG norm inflation to the invariant zero-PV QG sector, and the sharp 1/sqrt(2) half-derivative estimate for the boundary drift generated by interior PV.", "research_difficulty_suggested": "L4" }, { "id": 20002394, "problem_number": "AIM-PDES-0006", "title": "A coherent-stretching diagnostic for exponential material-line growth", "statement": "Exponential growth of the length of Lagrangian flow map for 2D Euler", "original_statement": "Exponential growth of the length of Lagrangian flow map for 2D Euler", "clean_statement": "Exponential growth of the length of Lagrangian flow map for 2D Euler", "statement_status": "exact", "statement_verification": "The canonical corpus record contains only the title", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Small scale dynamics in incompressible fluid flows\nSection: \nSource item: 6\nSource URL: http://aimpl.org/smallscalefluid/6/\nCanonical location: aim-pdes-notes.json notes[5]\nCanonical tag: section-problem\nOriginal extracted problem text (JSON string): \"Exponential growth of the length of Lagrangian flow map for 2D Euler\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/smallscalefluid/6/", "tags": [ "aim", "AIM-PDES-0006", "aim-domain:pdes", "aim-workshop:smallscalefluid", "aim-source-tag:section-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every uniformly Lipschitz incompressible two-dimensional flow, exponential growth of a fixed material curve forces an explicit lower bound on the measure of initial labels whose tangential stretch exceeds any smaller exponential rate. On every regular initial vorticity level curve, that same-label tangential stretch is exactly the ratio of the transported to initial vorticity-gradient magnitudes. These proved identities quantify why known time-dependent L-infinity vorticity-gradient maximizers do not yet provide the one fixed curve required by the AIM problem.\n\nCandidate contribution (quantitative lemma; novelty confidence low): The combined coherent-stretching diagnostic gives |E_mu(t)| >= (C exp(lambda t)-L0 exp(mu t))/(exp(Mt)-exp(mu t)) whenever the denominator and numerator are positive, and identifies the stretch on a regular vorticity level with the same-material-label gradient-amplification ratio. This is a concrete test distinguishing total fixed-curve growth from a drifting pointwise maximizer." }, { "id": 20002395, "problem_number": "AIM-PDES-0007", "title": "Weighted convection cancellation for topographic geostrophic profiles", "statement": "The QG equation with Ekman layer with topography", "original_statement": "The QG equation with Ekman layer with topography", "clean_statement": "The QG equation with Ekman layer with topography", "statement_status": "exact", "statement_verification": "The canonical AIM record says, verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Small scale dynamics in incompressible fluid flows\nSection: \nSource item: 7\nSource URL: http://aimpl.org/smallscalefluid/7/\nCanonical location: aim-pdes-notes.json notes[6]\nCanonical tag: section-problem\nOriginal extracted problem text (JSON string): \"The QG equation with Ekman layer with topography\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/smallscalefluid/7/", "tags": [ "aim", "AIM-PDES-0007", "aim-domain:pdes", "aim-workshop:smallscalefluid", "aim-source-tag:section-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The one-line AIM item is too underspecified to identify a unique QG model. For the closely matching topographic fast-rotation formulation of Chemin, Fanelli, and Gallagher, the exact special-case damping rate is recorded and a self-contained partial result is proved: if u=a(h) grad-perp h and v=b(h) grad-perp h, then the weighted pairing of (u dot grad)u with v has an exact boundary-plus-streamline-weight defect formula. In particular, for any depth weight w(h) and vanishing boundary flux, the quadratic convection is orthogonal to every field in the same topographic geostrophic space. This is projection/energy orthogonality, not pointwise vanishing of convection.\n\nCandidate contribution (lemma; novelty confidence low): For isobath-tangent fields u=a(h) grad-perp h and v=b(h) grad-perp h, the arbitrary spatial weight q satisfies an exact defect identity whose only interior term is -(1/2) integral a(h)^2 b(h) |grad h|^2 grad-perp h dot grad q; hence any streamline-invariant weight q=w(h), including a variable fluid depth, gives zero projected convection when the boundary flux vanishes." }, { "id": 20002396, "problem_number": "AIM-PDES-0008", "title": "Deformation and invariant barriers for low-Sobolev 2D Euler norm inflation", "statement": "2D Euler norm inflation", "original_statement": "2D Euler norm inflation", "clean_statement": "2D Euler norm inflation", "statement_status": "exact", "statement_verification": "The AIM workshop report independently confirms the intended range by naming the working-group problem “Norm inflation for 2D Euler, for \\(u\\in H^s\\), \\(s\\in(0,1)\\).” It says that the difficulty is precisely the low velocity regularity between the two conserved endpoint norms and contrasts the question with the then-recent result for \\(s\\in(1,2)\\). Thus \\(02 and p<2. These are necessary conditions and a reduction, not the norm-inflating construction requested by AIM.\n\nCandidate contribution (quantitative obstruction and reduction; novelty confidence low): The candidate contribution is a unified deformation-and-invariant barrier for the exact AIM range 00. The other algebraic wavenumber always violates the positive-amplitude condition. Consequently a spatially constant nonzero wake is compatible with the pulled double root exactly on alpha+beta=0. This is a proved necessary condition and certification-ready boundary-data filter, not an existence or stability theorem.\n\nCandidate contribution (necessary-condition proposition; novelty confidence low): The parameter-uniform pulled-wake filter includes an explicit proof that exactly one quadratic wake branch has positive amplitude and that imposing a constant wake is valid if and only if alpha+beta=0." }, { "id": 20002399, "problem_number": "AIM-PDES-0011", "title": "Exact pushed-front benchmark and pulled spectral obstruction", "statement": "A starting point for the above problem would be\n\nProve existence and stability for pushed or pulled fronts for a toy model using a computer assisted proof.", "original_statement": "A starting point for the above problem would be\n\nProve existence and stability for pushed or pulled fronts for a toy model using a computer assisted proof.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The official section heading is exactly \"Existence and marginally stability of pushed and pulled fronts.\" The grammatical defect is present on the source page and is therefore not an OCR error. The intended phrase is almost surely \"existence and marginal stability,\" but this reconstruction is not silently substituted for the source.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Computer assisted proofs for stability analysis of nonlinear waves\nSection: Existence and marginally stability of pushed and pulled fronts\nSource item: 2.2\nSource URL: http://aimpl.org/compproofstability/2/\nCanonical location: aim-pdes-notes.json notes[10]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A starting point for the above problem would be\\n\\nProve existence and stability for pushed or pulled fronts for a toy model using a computer assisted proof.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/compproofstability/2/", "tags": [ "aim", "AIM-PDES-0011", "aim-domain:pdes", "aim-workshop:compproofstability", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the scalar toy model u_t=u_xx+u(1-u)(1+a u), setting a=2k^2 with k>1 gives the exact pushed minimal front U=(1+exp(k xi))^{-1} at speed c=k+k^{-1}. A variational equality weight proves minimality, while conjugation by exp(c xi/2) and an exact ground-state identity prove weighted spectral nonpositivity, a simple isolated translational zero, and essential-spectrum distance (k-k^{-1})^2/4 from zero. The a=2 transition loses integrability and localization, and the genuinely pulled Fisher-KPP endpoint is gapless. Concrete contraction and interval-Evans inequalities specify an a-posteriori computer-assisted proof, but no interval computation or nonlinear-stability proof is claimed.\n\nCandidate contribution (benchmark synthesis and spectral obstruction; novelty confidence low): The candidate contribution is a testable analytic-to-validated pushed/pulled benchmark packet: the exact equality weight for pushed-speed minimality, the exact weighted spectral margin (including 9/16 at a=8), and paired profile-contraction and Evans/Rouche pass-fail criteria whose ordinary spectral-gap form provably breaks down at the pulled threshold." }, { "id": 20002400, "problem_number": "AIM-PDES-0012", "title": "A two-radius core-tail certificate for infinite-dimensional spatial dynamics", "statement": "Use computer assisted methods to extend spatial dynamics in infinite\ndimensions from the perturbative regime.", "original_statement": "Use computer assisted methods to extend spatial dynamics in infinite\ndimensions from the perturbative regime.", "clean_statement": "Use computer assisted methods to extend spatial dynamics in infinite\ndimensions from the perturbative regime.", "statement_status": "exact", "statement_verification": "The newline between “infinite” and “dimensions” in `input.json` is a source line wrap, not an OCR error. The archived page contains no model equation, hypotheses, boundary conditions, or indication of whether the desired object is a center manifold, periodic orbit, front, pulse, or other bounded spatial trajectory. Accordingly, this report does **not** silently choose a particular PDE or claim a model-specific existence result.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Computer assisted proofs for stability analysis of nonlinear waves\nSection: Spatial dynamics\nSource item: 3.1\nSource URL: http://aimpl.org/compproofstability/3/\nCanonical location: aim-pdes-notes.json notes[11]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Use computer assisted methods to extend spatial dynamics in infinite\\ndimensions from the perturbative regime.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/compproofstability/3/", "tags": [ "aim", "AIM-PDES-0012", "aim-domain:pdes", "aim-workshop:compproofstability", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a spatial object encoded by a C1 operator F:D subset X to Y on a Banach splitting X=X_core direct-sum X_tail, a bounded injective preconditioner A and rigorous core-tail residual and derivative-block bounds reduce existence and local uniqueness of a full infinite-dimensional zero to two rectangular invariance inequalities and an explicit weighted row-sum contraction bound. A corollary shows how an analytic inverse for the tail linear operator supplies the tail residual and derivative estimates, allowing a large resolved core when, and only when, the uniform full-ball bounds pass.\n\nCandidate contribution (a_posteriori_reduction; novelty confidence low): The explicit anisotropic two-radius core-tail feasibility test is proposed as a reusable validation gate for beyond-perturbation spatial dynamics: interval-certified nonlinear core data are coupled to analytic infinite-tail estimates while an ill-posed spatial initial-value formulation is avoided.", "research_difficulty_suggested": "L4" }, { "id": 20002401, "problem_number": "AIM-PDES-0013", "title": "A two-level validated certificate for Maslov indices of planar elliptic problems", "statement": "Develop rigorous computer assisted methods for computing the Maslov index in two dimensions.", "original_statement": "Develop rigorous computer assisted methods for computing the Maslov index in two dimensions.", "clean_statement": "Develop rigorous computer assisted methods for computing the Maslov index in two dimensions.", "statement_status": "exact", "statement_verification": "The canonical AIM record says exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Computer assisted proofs for stability analysis of nonlinear waves\nSection: Computing Maslov indices and Evans functions in 2 dimensions\nSource item: 4.1\nSource URL: http://aimpl.org/compproofstability/4/\nCanonical location: aim-pdes-notes.json notes[12]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop rigorous computer assisted methods for computing the Maslov index in two dimensions.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/compproofstability/4/", "tags": [ "aim", "AIM-PDES-0013", "aim-domain:pdes", "aim-workshop:compproofstability", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the physical-two-dimensional interpretation, a Fredholm-Lagrangian Cauchy-data path in H^{1/2}(boundary Omega) plus H^{-1/2}(boundary Omega) admits a rigorous finite certificate whenever each chart has a uniformly coercive infinite tail, rigorously enclosed cross-coupling, exhaustive interval crossing boxes, transverse endpoints, and verified chart overlaps. Exact Schur elimination proves that both crossing kernels and crossing-form signatures pass from the infinite operator to a finite matrix; a second scalar Schur complement with derivative bounded away from zero isolates each simple crossing and certifies its sign. The final Maslov index is the sum of these certified signs. An exact rectangular Dirichlet family supplies a physical-2D spectral-flow benchmark and tail audit.\n\nCandidate contribution (validation-certificate theorem; novelty confidence low): A nested infinite-tail and scalar-crossing Schur certificate preserves crossing-form signatures exactly and yields an auditable integer Maslov index while explicitly requiring tail coercivity, cross-coupling control, chart transversality, endpoint transversality, and no-crossing exhaustion.", "research_difficulty_suggested": "L4" }, { "id": 20002402, "problem_number": "AIM-PDES-0014", "title": "A mode-complete certificate for two-dimensional Evans computations", "statement": "Develop rigorous computer assisted methods for computing the Evans function in two dimensions.", "original_statement": "Develop rigorous computer assisted methods for computing the Evans function in two dimensions.", "clean_statement": "Develop rigorous computer assisted methods for computing the Evans function in two dimensions.", "statement_status": "exact", "statement_verification": "The recovered statement is AIM Problem 4.2 in the section “Computing Maslov indices and Evans functions in 2 dimensions” of the 2023 workshop *Computer assisted proofs for stability analysis of nonlinear waves*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Computer assisted proofs for stability analysis of nonlinear waves\nSection: Computing Maslov indices and Evans functions in 2 dimensions\nSource item: 4.2\nSource URL: http://aimpl.org/compproofstability/4/\nCanonical location: aim-pdes-notes.json notes[13]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop rigorous computer assisted methods for computing the Evans function in two dimensions.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/compproofstability/4/", "tags": [ "aim", "AIM-PDES-0014", "aim-domain:pdes", "aim-workshop:compproofstability", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For planar waves with transverse translation invariance, a rigorous two-dimensional Evans computation reduces to finitely many validated one-dimensional fiber computations only after an analytic high-transverse-frequency exclusion. On a periodic strip, a uniform Rouche bound for every retained mode gives the exact full algebraic eigenvalue count after summing the separate positive and negative mode multiplicities; for a continuous transverse parameter, joint contour nonvanishing makes the fiber winding constant on connected parameter intervals. A scalar reaction-diffusion shift identity supplies an exact normalization-aware regression benchmark.\n\nCandidate contribution (mode-complete validation reduction; novelty confidence low): The candidate contribution is a falsifiable end-to-end certificate combining an analytic transverse-tail gate, interval Evans and Rouche validation for every retained mode, a connected-interval winding homotopy for continuous transverse frequency, correct modal multiplicity accounting, and the exact normalized reaction-diffusion identity D_k(lambda)=D_0(lambda+eta_k^2) as a regression test.", "research_difficulty_suggested": "L4" }, { "id": 20002403, "problem_number": "AIM-PDES-0015", "title": "Endpoint correction and sharp acoustic conditioning for planar Gross-Pitaevskii waves", "statement": "Prove existence of a traveling wave for Gross-Pitaevskii equation in\n\\(\\mathbb{R}^2\\) for all wave speeds \\(c \\in [0, \\sqrt{2}]\\).", "original_statement": "Prove existence of a traveling wave for Gross-Pitaevskii equation in\n\\(\\mathbb{R}^2\\) for all wave speeds \\(c \\in [0, \\sqrt{2}]\\).", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record in `aim-pdes-notes.json` (zero-based index 14) states exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Computer assisted proofs for stability analysis of nonlinear waves\nSection: Computing Maslov indices and Evans functions in 2 dimensions\nSource item: 5.1\nSource URL: http://aimpl.org/compproofstability/5/\nCanonical location: aim-pdes-notes.json notes[14]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Prove existence of a traveling wave for Gross-Pitaevskii equation in\\n\\\\(\\\\mathbb{R}^2\\\\) for all wave speeds \\\\(c \\\\in [0, \\\\sqrt{2}]\\\\).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: invalid_statement; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/compproofstability/5/", "tags": [ "aim", "AIM-PDES-0015", "aim-domain:pdes", "aim-workshop:compproofstability", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The exact AIM wording is mathematically vacuous because constant profiles of modulus one solve the standard Gross-Pitaevskii traveling-wave equation at every speed. Under the only substantive reconstruction—nonconstant finite-energy waves—the stated closed interval is false at both endpoints: a two-dimensional Pohozaev argument proves that c=0 admits only constants, and Gravejat's theorem proves the same at c=sqrt(2). The corrected every-prescribed-speed problem on (0,sqrt(2)) remains only partially solved in the literature checked. Independently, after fixing the phase gauge, the acoustic Schur multiplier has sharp L2 coercivity 2-c^2 and exact inverse norm 1/(2-c^2), with a normalized longitudinal low-frequency cone family proving sonic degeneration.\n\nCandidate contribution (sharp_obstruction; novelty confidence low): After explicit phase-gauge handling, the far-field acoustic Schur block for planar Gross-Pitaevskii traveling waves has exact L2 inverse norm (2-c^2)^{-1}; real normalized Fourier test functions supported in shrinking longitudinal annular cones saturate the bound, proving that any unscaled validation estimate explicitly using this inverse must lose conditioning at least at that rate as c approaches sqrt(2).", "research_difficulty_suggested": "L2" }, { "id": 20002404, "problem_number": "AIM-PDES-0016", "title": "A co-periodic constrained-positivity certificate for generalized KdV waves", "statement": "Prove that the spectrum for spatially periodic solutions to\n generalised KdV lies entirely on the imaginary axis.", "original_statement": "Prove that the spectrum for spatially periodic solutions to\n generalised KdV lies entirely on the imaginary axis.", "clean_statement": "Prove that the spectrum for spatially periodic solutions to\n generalised KdV lies entirely on the imaginary axis.", "statement_status": "exact", "statement_verification": "The original AIM page was checked. It contains the same sentence, attributes the item to Jared Bronski, and supplies no equation, nonlinearity, wave family, period, or perturbation class. Thus “generalised” is the source's British spelling, not an OCR error, but the mathematical statement is under-specified. In particular, it does not say whether “the spectrum” means:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Computer assisted proofs for stability analysis of nonlinear waves\nSection: Stability of spatially periodic solution to generalised KdV or other non-integrable dispersive PDE\nSource item: 6.1\nSource URL: http://aimpl.org/compproofstability/6/\nCanonical location: aim-pdes-notes.json notes[15]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Prove that the spectrum for spatially periodic solutions to\\n generalised KdV lies entirely on the imaginary axis.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/compproofstability/6/", "tags": [ "aim", "AIM-PDES-0016", "aim-domain:pdes", "aim-workshop:compproofstability", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The unqualified AIM assertion is false: Bronski and Johnson prove modulationally unstable periodic power-law gKdV waves for f(u)=u^(p+1), c>0, and p>4 near the solitary-wave limit. For the precise co-periodic problem, a new self-contained reduction is proved: if the Hill operator L has one negative eigenvalue, kernel spanned by the translation mode, and <0, then L is nonnegative on the mean-zero subspace with only the translation mode in its constrained kernel, and the entire co-periodic spectrum of partial_x L lies on the imaginary axis.\n\nCandidate contribution (reduction; novelty confidence low): A rigorous co-periodic computer-assisted proof can be reduced to three self-adjoint validations—n(L)=1, a simple translation kernel, and the single interval sign <0—without directly enclosing the spectrum of the non-self-adjoint third-order linearization." }, { "id": 20002405, "problem_number": "AIM-PDES-0017", "title": "Certified mass bounds for basins in finite-dimensional gradient flows", "statement": "Estimate size of the basin of attraction for each equilibria.", "original_statement": "Estimate size of the basin of attraction for each equilibria.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The archived AIM page gives Problem 7.1 in the section “Finite-dimensional gradient flows with many stable equilibria,” attributed to Keith Promislow:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Computer assisted proofs for stability analysis of nonlinear waves\nSection: Finite-dimensional gradient flows with many stable equilibria\nSource item: 7.1\nSource URL: http://aimpl.org/compproofstability/7/\nCanonical location: aim-pdes-notes.json notes[16]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Estimate size of the basin of attraction for each equilibria.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/compproofstability/7/", "tags": [ "aim", "AIM-PDES-0017", "aim-domain:pdes", "aim-workshop:compproofstability", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source's phrase 'each equilibria' is a genuine grammatical defect and the question has no numerical answer until a flow, reference region, and initial-condition measure are specified. For a Morse gradient flow on a compact positively invariant region, a validated enclosure N of all stable manifolds of nonminimum critical points makes the basin label constant on every certified connected component of the complement. If L_i is the union of components validated to converge to minimum i, then L_i is contained in B_i, B_i is contained in L_i union N, and mu(L_i) <= mu(B_i) <= mu(L_i)+mu(N). Explicit Hessian bounds yield local inner-ball volumes, and a separable double-well family gives an exact benchmark with 2^d stable equilibria.\n\nCandidate contribution (validated basin-mass sandwich; novelty confidence low): The candidate contribution is an end-to-end deterministic certificate in which complete separatrix enclosure and component topology reduce every basin-probability uncertainty to the measured mass of a single enclosure N, augmented by explicit Hessian-ball labeling cores and an exact 2^d-well product regression family.", "research_difficulty_suggested": "L4" }, { "id": 20002406, "problem_number": "AIM-PDES-0018", "title": "A precise one-dimensional Swift-Hohenberg quasipattern reduction", "statement": "A starting point would be the following problem.\n\nProve existence of quasiperiodic patters for the 1d Swift-Hogenberg.", "original_statement": "A starting point would be the following problem.\n\nProve existence of quasiperiodic patters for the 1d Swift-Hogenberg.", "clean_statement": "A starting point would be the following problem.\n\nProve existence of pattern - for the 1d Swift-Hogenberg.", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record states exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Computer assisted proofs for stability analysis of nonlinear waves\nSection: Existence of quasiperiodic patterns\nSource item: 8.1\nSource URL: http://aimpl.org/compproofstability/8/\nCanonical location: aim-pdes-notes.json notes[17]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A starting point would be the following problem.\\n\\nProve existence of quasiperiodic patters for the 1d Swift-Hogenberg.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/compproofstability/8/", "tags": [ "aim", "AIM-PDES-0018", "aim-domain:pdes", "aim-workshop:compproofstability", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source certainly contains the typos 'patters' for 'patterns' and 'Swift-Hogenberg' for 'Swift-Hohenberg', but it does not specify whether quasiperiodicity is spatial or temporal, whether the target is stationary or evolving, the equation coefficients, parameter regime, frequency vector, or function space. For the explicit stationary real model u_t = mu u - (1+partial_x^2)^2 u + nu u^2 - u^3 with u(x)=U(omega x), this attempt proves that every nonconstant exact solution has infinite Fourier support: any finite real Fourier core has an exactly nonzero residual -a_{k_*}^3 at an exposed outer index 3k_*. It also proves that the torus-lift linear operator is non-Fredholm for mu >= 0 because its diagonal symbols accumulate at zero, while for mu < 0 it has an H^s inverse bound 1/|mu| and a quantitative small-ball nonexistence estimate.\n\nCandidate contribution (obstruction; novelty confidence low): For every nonconstant finite real Fourier core of the explicitly reconstructed stationary quadratic-cubic one-dimensional Swift-Hohenberg equation, a generic exposing functional selects k_* such that the exact residual coefficient at 3k_* is -a_{k_*}^3; hence every zero-tail approximation has weighted residual at least w_{3k_*}|a_{k_*}|^3 and cannot itself be an exact pattern.", "research_difficulty_suggested": "L2" }, { "id": 20002407, "problem_number": "AIM-PDES-0019", "title": "A determinant-line phase certificate for higher-dimensional stability counts", "statement": "Develop a geometric approach to stability that generalizes to higher dimensions.", "original_statement": "Develop a geometric approach to stability that generalizes to higher dimensions.", "clean_statement": "Develop a geometric approach to stability that generalizes to higher dimensions.", "statement_status": "exact", "statement_verification": "The original AIM page was checked. It contains exactly this sentence, attributes it to Christopher Jones, and gives no introduction, remarks, equations, or status. This is not an OCR error, but it is not a mathematically closed proposition. The workshop list separately contains a section titled “Computing Maslov indices and Evans functions in 2 dimensions,” which makes an Evans/Maslov interpretation plausible but does not settle what “dimensions” means here.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Computer assisted proofs for stability analysis of nonlinear waves\nSection: Geometric phase\nSource item: 9.1\nSource URL: http://aimpl.org/compproofstability/9/\nCanonical location: aim-pdes-notes.json notes[18]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop a geometric approach to stability that generalizes to higher dimensions.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/compproofstability/9/", "tags": [ "aim", "AIM-PDES-0019", "aim-domain:pdes", "aim-workshop:compproofstability", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The finite-system interpretation was substantially answered by Grudzien, Bridges, and Jones, who proved that relative Hopf phase equals the Evans/Chern eigenvalue count and extended it by exterior powers to arbitrary finite-dimensional ODE systems. For higher spatial dimension, this attempt proves a validation-ready survival theorem: a jointly continuous analytic Fredholm/Schatten characteristic determinant that is nonzero on the whole boundary cylinder Gamma times K has constant spectral winding on every compact connected transverse-parameter block K. For planar reaction-diffusion waves, an explicit energy estimate excludes all modes with |eta|>sqrt(beta_plus/d_star), reducing transverse spectral exclusion to one reference count and a uniform boundary-cylinder validation. Loss of Fredholmness or essential-spectrum contact is identified as the failure gate for this isolated-eigenvalue phase count.\n\nCandidate contribution (reduction; novelty confidence low): A determinant-line/Fredholm survival-and-failure certificate reduces planar-wave transverse stability counts to three verifiable gates: nonvanishing of the characteristic determinant on Gamma times a connected compact parameter block, one reference winding count, and the explicit high-frequency cutoff |eta|>sqrt(beta_plus/d_star); the certificate stops at loss of Fredholmness on the boundary cylinder." }, { "id": 20002408, "problem_number": "AIM-PDES-0020", "title": "A full-line Maslov certificate without pulse reversibility", "statement": "The Maslow index allow you to prove stability results for symmetric\npulses in Hamiltonian systems. Can this be done also for non-symmetric\npulses?", "original_statement": "The Maslow index allow you to prove stability results for symmetric\npulses in Hamiltonian systems. Can this be done also for non-symmetric\npulses?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The archived AIM page gives Problem 10.1 in the section “Maslov index for an asymmetric pulse,” attributed to Jonathan Jaquette:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Computer assisted proofs for stability analysis of nonlinear waves\nSection: Maslov index for an asymmetric pulse\nSource item: 10.1\nSource URL: http://aimpl.org/compproofstability/10/\nCanonical location: aim-pdes-notes.json notes[19]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The Maslow index allow you to prove stability results for symmetric\\npulses in Hamiltonian systems. Can this be done also for non-symmetric\\npulses?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/compproofstability/10/", "tags": [ "aim", "AIM-PDES-0020", "aim-domain:pdes", "aim-workshop:compproofstability", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a selfadjoint Hamiltonian Fredholm pencil with hyperbolic limits, a non-reversible pulse can be treated by independently constructing its left unstable and right stable Lagrangian bundles and taking their relative Maslov index in doubled symplectic space. With transverse endpoints and regular crossing forms of one definite sign, the absolute index equals the full eigenvalue multiplicity on the interval. The report proves match-point invariance, fixes the sign of the Hörmander correction required by a fixed reference plane, and gives a finite-window certificate that requires rigorous bounds on both tails. Symmetry is therefore unnecessary for the count, although without symmetry or model-specific information the index need not be nonzero and stability does not follow automatically.\n\nCandidate contribution (validation protocol; novelty confidence low): A candidate symmetry-free computer-assisted validation packet is obtained by combining the doubled-space stable/unstable pair index, certified two-tail enclosures, the explicitly signed endpoint Hörmander correction, separate treatment of translation, and a compactly supported symplectic-gauge deformation of an exactly solvable Schrödinger pencil as a genuinely non-reversible regression benchmark.", "research_difficulty_suggested": "L4" }, { "id": 20002409, "problem_number": "AIM-PDES-0021", "title": "A normalized a posteriori center-graph and chart-exit certificate", "statement": "Develop rigorous computer assisted methods for handling center\nmanifolds.", "original_statement": "Develop rigorous computer assisted methods for handling center\nmanifolds.", "clean_statement": "Develop rigorous computer assisted methods for handling center\nmanifolds.", "statement_status": "exact", "statement_verification": "The canonical record is problem 11.1 in the AIM workshop section “Center manifolds”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Computer assisted proofs for stability analysis of nonlinear waves\nSection: Center manifolds\nSource item: 11.1\nSource URL: http://aimpl.org/compproofstability/11/\nCanonical location: aim-pdes-notes.json notes[20]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Develop rigorous computer assisted methods for handling center\\nmanifolds.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/compproofstability/11/", "tags": [ "aim", "AIM-PDES-0021", "aim-domain:pdes", "aim-workshop:compproofstability", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a normalized graph invariance operator between Banach spaces, an injective bounded preconditioner together with rigorous residual, linear-defect, and derivative-Lipschitz bounds satisfying an explicit radii inequality yields one exact invariant graph and reduced vector field in the selected validation ball, with error Y_0/(1-q). Certified norm embeddings and a Gronwall estimate then give an explicit full-state trajectory error valid through a computable chart-exit margin. The exact family h_A(x)=A exp(1/x) for x<0 and zero for x>=0 in x'=x^2, y'=-y proves that uniqueness cannot be extended beyond the selected normalized function-space ball and that local invariance does not imply an all-time trajectory claim.\n\nCandidate contribution (reduction; novelty confidence low): Candidate residual-to-reduced-dynamics exit certificate: the auditable constants (Y_0, Z_0, Z_1, rho, C_h, C_r, L, M_K, d_c, delta_h), together with preconditioner injectivity and a stated center-manifold selection rule, certify a normalized invariant graph, approximation error, reduced-flow error, and finite chart-validity horizon.", "research_difficulty_suggested": "L4" }, { "id": 20002410, "problem_number": "AIM-PDES-0022", "title": "Landau equation: context recovery and a structure-preserving kernel audit", "statement": "Landau equation", "original_statement": "Landau equation", "clean_statement": "Landau equation", "statement_status": "exact", "statement_verification": "The exact canonical record says only", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlocal differential equations in collective behavior\nSection: \nSource item: 5\nSource URL: http://aimpl.org/nonlocalde/5/\nCanonical location: aim-pdes-notes.json notes[21]\nCanonical tag: section-problem\nOriginal extracted problem text (JSON string): \"Landau equation\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlocalde/5/", "tags": [ "aim", "AIM-PDES-0022", "aim-domain:pdes", "aim-workshop:nonlocalde", "aim-source-tag:section-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The exact record is a section heading rather than a mathematical question. For the spatially homogeneous Landau-form operator with the stated sign convention, an even symmetric positive-semidefinite pair kernel gives a symmetrized weak identity, mass and momentum conservation, and entropy decay; the additional pointwise condition B(z)z=0 is exactly the pair-flux tangency certificate for kinetic-energy conservation. The report also proves a quantitative energy-defect bound and shows that denominator smoothing of the projector produces a strictly positive energy slope on centered Maxwellian data, while smoothing only the radial scalar factor preserves the certificate.\n\nCandidate contribution (structural_lemma_and_counterexample; novelty confidence low): Candidate synthesis: the pointwise if-and-only-if pair-flux test B(z)z=0, the explicit bound |dE/dt| <= (1/2) double-integral ff_* |X_f| |B(z)z|, and the strict Gaussian energy-defect formula together form a directly testable audit for proposed Landau-kernel regularizations.", "research_difficulty_suggested": "L2" }, { "id": 20002411, "problem_number": "AIM-PDES-0023", "title": "A virial clock for homogeneous aggregation-diffusion", "statement": "Nonlocal aggregation-diffusion equations", "original_statement": "Nonlocal aggregation-diffusion equations", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlocal differential equations in collective behavior\nSection: \nSource item: 6\nSource URL: http://aimpl.org/nonlocalde/6/\nCanonical location: aim-pdes-notes.json notes[22]\nCanonical tag: section-problem\nOriginal extracted problem text (JSON string): \"Nonlocal aggregation-diffusion equations\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlocalde/6/", "tags": [ "aim", "AIM-PDES-0023", "aim-domain:pdes", "aim-workshop:nonlocalde", "aim-source-tag:section-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Archive recovery proves that the canonical text is a section heading whose actual AIM page contains eight distinct child problems, so it has no single proposition to solve. As a substantive section-level contribution, the report proves for the convention partial_t rho = Delta(rho^m) + div(rho grad(W*rho)) that an even k-homogeneous interaction satisfies I' = 2d U - k V. The sharp dilation balance is d(m-1)+k=0, where I'=2d(m-1)E. In the attractive singular porous-medium range, negative dissipated energy forces failure of the assumed finite-energy, finite-second-moment continuation class by an explicit time; a profile-dependent mass threshold and exact PDE scaling covariance are also derived.\n\nCandidate contribution (virial diagnostic; novelty confidence low): The candidate contribution is an operational packet combining the explicit continuation deadline T_* = I_0/[-2d(m-1)E_0], the unit-shape mass threshold M_*(f) = [2U_f/((m-1)(-V_f))]^(1/(2-m)), and the covariance T_*[(rho_0)_lambda] = lambda^[-(2+d(m-1))] T_*[rho_0], formulated in a symmetrized weak identity that is safe at a singular interaction diagonal.", "research_difficulty_suggested": "L2" }, { "id": 20002412, "problem_number": "AIM-PDES-0024", "title": "Applied Models heading recovery and a gang-model spectral benchmark", "statement": "Applied Models", "original_statement": "Applied Models", "clean_statement": "Applied Models", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlocal differential equations in collective behavior\nSection: \nSource item: 8\nSource URL: http://aimpl.org/nonlocalde/8/\nCanonical location: aim-pdes-notes.json notes[23]\nCanonical tag: section-problem\nOriginal extracted problem text (JSON string): \"Applied Models\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlocalde/8/", "tags": [ "aim", "AIM-PDES-0024", "aim-domain:pdes", "aim-workshop:nonlocalde", "aim-source-tag:section-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical text 'Applied Models' is a section heading, not a mathematical proposition: archived AIM pages verify that it was visibly section 4, while its metadata carried global list position 8, and that it contained two omitted child prompts which this attempt does not claim. As a source-tied benchmark for the archived four-field gang/graffiti PDE on the normalized periodic two-torus, the report proves agent-mass conservation, exact graffiti-mean relaxation, conditional positivity for classical solutions, and an exact factorization of every Fourier-mode characteristic polynomial. The homogeneous fixed-mass state is strictly linearly stable exactly when 2 c beta sqrt(m_A m_B)<1. Above threshold every nonzero mode has one positive real eigenvalue; that eigenvalue increases strictly with squared wavenumber and converges to the bounded limit 2 c beta sqrt(m_A m_B)-1, so the linearized model has no finite selected wavelength.\n\nCandidate contribution (benchmark_corollary; novelty confidence low): Candidate structure-preserving spectral benchmark: for the exact archived four-field periodic gang model, a numerical implementation should preserve both agent masses and nonnegativity, reproduce exponential graffiti-mean relaxation and the two-quadratic Fourier factorization, and in the supercritical regime reproduce a positive growth rate that is strictly increasing in q=|k|^2 and saturates at 2 c beta sqrt(m_A m_B)-1 rather than selecting a spurious finite wavelength.", "research_difficulty_suggested": "L2" }, { "id": 20002413, "problem_number": "AIM-PDES-0025", "title": "Complex Monge-Ampere symmetry: a literal counterexample and a proper repair", "statement": "Let $u$ plurisubharmonic satisfy\n\\[\n\\mathrm{det}(u_{j \\overline k}) = f(u)\n\\] on $B_1(0)$ with Dirichlet boundary conditions, then $u$ is rotationally symmetric", "original_statement": "Let $u$ plurisubharmonic satisfy\n\\[\n\\mathrm{det}(u_{j \\overline k}) = f(u)\n\\] on $B_1(0)$ with Dirichlet boundary conditions, then $u$ is rotationally symmetric", "clean_statement": "Let $u$ plurisubharmonic satisfy\n\\[\n\\mathrm{det}(u_{j \\overline k}) = f(u)\n\\] on $B_1(0)$ with Dirichlet boundary conditions, then $u$ is rotationally symmetric", "statement_status": "exact", "statement_verification": "The exact canonical statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Monge-Ampère equations\nSource item: 1.05\nSource URL: http://aimpl.org/nonlinpdegeom/1/\nCanonical location: aim-pdes-notes.json notes[24]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $u$ plurisubharmonic satisfy\\n\\\\[\\n\\\\mathrm{det}(u_{j \\\\overline k}) = f(u)\\n\\\\] on $B_1(0)$ with Dirichlet boundary conditions, then $u$ is rotationally symmetric\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/1/", "tags": [ "aim", "AIM-PDES-0025", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM statement is false under its literal unspecified-Dirichlet-data wording: for every real pluriharmonic h, u(z)=|z|^2+h(z) is smooth strictly plurisubharmonic with det(u_{j bar k})=1=f(u), yet is nonradial whenever h is non-unitarily invariant. Under the explicit repaired hypotheses of constant boundary data, a positive C^1 nondecreasing f, and a C^2 strictly plurisubharmonic solution on the closed complex unit ball, linearized log-determinant comparison proves uniqueness; unitary covariance then forces U(n)-invariance. The radial complex Hessian and its exact divergence-form ODE are also derived.\n\nCandidate contribution (counterexample_family_and_partial_theorem; novelty confidence low): Candidate source-specific contribution: the infinite-dimensional pluriharmonic family u=|z|^2+h sharply refutes the literal boundary-unspecified symmetry claim, while the paired uniqueness theorem identifies constant boundary data and the properness condition f'>=0 as a rigorous sufficient repair and supplies the exact radial identity det=(g')^(n-1)(g'+s g'')=[n s^(n-1)]^(-1)(s^n(g')^n)'." }, { "id": 20002414, "problem_number": "AIM-PDES-0026", "title": "Sharp radial regularity and the ambiguity of smooth cone decomposition", "statement": "Let $u$ strictly convex with Dirichlet boundary conditions solve\n\\[\\mathrm{det}(u_{ij}) = 1 + \\delta_{x_0}, \\]\nthen $u=\\varphi+w$ for $\\varphi$ smooth uniformly convex cone, and $w$ smooth.", "original_statement": "Let $u$ strictly convex with Dirichlet boundary conditions solve\n\\[\\mathrm{det}(u_{ij}) = 1 + \\delta_{x_0}, \\]\nthen $u=\\varphi+w$ for $\\varphi$ smooth uniformly convex cone, and $w$ smooth.", "clean_statement": "Let $u$ strictly convex with Dirichlet boundary conditions solve\n\\[\\mathrm{det}(u_{ij}) = 1 + \\delta_{x_0}, \\]\nthen $u=\\varphi+w$ for $\\varphi$ smooth uniformly convex cone, and $w$ smooth.", "statement_status": "exact", "statement_verification": "The archived AIM page reproduces exactly this wording and attributes the problem to Xu-Jia Wang. Thus the omissions are not an OCR error: the page does not specify the domain, dimension, boundary values, solution concept, normalization of the Dirac mass, or the meanings of “smooth uniformly convex cone” and “smooth.”", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Monge-Ampère equations\nSource item: 1.1\nSource URL: http://aimpl.org/nonlinpdegeom/1/\nCanonical location: aim-pdes-notes.json notes[25]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $u$ strictly convex with Dirichlet boundary conditions solve\\n\\\\[\\\\mathrm{det}(u_{ij}) = 1 + \\\\delta_{x_0}, \\\\]\\nthen $u=\\\\varphi+w$ for $\\\\varphi$ smooth uniformly convex cone, and $w$ smooth.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/1/", "tags": [ "aim", "AIM-PDES-0026", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the strictly convex Alexandrov solution on a ball with a centered atom of mass mu and constant Dirichlet data, u(x)=g_R-integral_{|x|}^R(s^n+mu/omega_n)^{1/n} ds, the tangent cone is a|x| with a=(mu/omega_n)^{1/n}. After subtracting it, the remainder is exactly C^n but not C^{n+1} for even n, exactly C^{2n} but not C^{2n+1} for odd n at least 3, and smooth for n=1. Any one-homogeneous cone that leaves a C^1 remainder equals a|x| modulo a linear form. Hence this example refutes the literal C-infinity reading for n at least 2, but positively satisfies the likely intended C^{2,alpha} formulation; the general strict one-atom bounded-domain problem remains unresolved in the literature checked.\n\nCandidate contribution (sharp_special_case; novelty confidence low): For every atom mass mu>0 in the centered radial ball model, the remainder after subtracting any admissible one-homogeneous cone has the parity-sharp regularity C^n without C^{n+1} in even dimension, C^{2n} without C^{2n+1} in odd dimension n>=3, and C-infinity in dimension one." }, { "id": 20002415, "problem_number": "AIM-PDES-0027", "title": "Affine-normal arrival-time models and sharp center regularity", "statement": "For $u$ strictly convex, determine the existence and regularity of solutions to \\[\\mathrm{det}(D^2 u) u^{ij}u_i u_j = 1.\\]", "original_statement": "For $u$ strictly convex, determine the existence and regularity of solutions to \\[\\mathrm{det}(D^2 u) u^{ij}u_i u_j = 1.\\]", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Problem 1.15 in the “Monge-Ampère equations” section of the AIM problem list *Nonlinear PDEs in real and complex geometry*. The current AIM page attributes it to Tristan Collins and displays:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Monge-Ampère equations\nSource item: 1.15\nSource URL: http://aimpl.org/nonlinpdegeom/1/\nCanonical location: aim-pdes-notes.json notes[26]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For $u$ strictly convex, determine the existence and regularity of solutions to \\\\[\\\\mathrm{det}(D^2 u) u^{ij}u_i u_j = 1.\\\\]\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/1/", "tags": [ "aim", "AIM-PDES-0027", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the standard inverse-Hessian convention, the equation is the arrival-time equation for affine normal flow. For every n at least 2 and invertible A, the report proves an explicit affine-normalized ellipsoidal family of classical strictly convex solutions on punctured space, with a sharp C^{1,(n-1)/(n+1)} extension at the puncture and no better gradient Hölder regularity. It also proves radial rigidity, nonexistence in dimension one, exclusion of all classical interior critical points and constant-boundary solutions on bounded convex domains, and a Legendre-dual reduction.\n\nCandidate contribution (explicit_solution_family_and_obstruction; novelty confidence low): The affine-covariant punctured family u=C+(n+1)/(2n)|det A|^{-2/(n+1)}|A(x-x0)|^{2n/(n+1)}, packaged with its sharp center exponent C^{1,(n-1)/(n+1)} and critical-point, one-dimensional, and constant-Dirichlet obstructions, is a concrete benchmark for every proposed regularity formulation of this AIM problem." }, { "id": 20002416, "problem_number": "AIM-PDES-0028", "title": "The refuted C1 conjecture and radial rigidity", "statement": "A function $u:\\Omega \\to \\mathbb R$ satisfies the special Lagrangian equation if \\[\\sum_i \\mathrm{arctan}(\\lambda_i)= c\\] for a constant $c$, where $\\lambda_i$ are the eigenvalues of $D^2u$.\n\n(\\textit{Conjecture of Nadirashvili-Vl\\u{a}du\\c{t}}) Any viscosity solution of the special Lagrangian equation on the unit ball is $C^1(B)$ for sufficiently smooth boundary data.", "original_statement": "A function $u:\\Omega \\to \\mathbb R$ satisfies the special Lagrangian equation if \\[\\sum_i \\mathrm{arctan}(\\lambda_i)= c\\] for a constant $c$, where $\\lambda_i$ are the eigenvalues of $D^2u$.\n\n(\\textit{Conjecture of Nadirashvili-Vl\\u{a}du\\c{t}}) Any viscosity solution of the special Lagrangian equation on the unit ball is $C^1(B)$ for sufficiently smooth boundary data.", "clean_statement": "A function $u:\\Omega \\to \\mathbb R$ satisfies the special Lagrangian equation if \\[\\sum_i \\mathrm{arctan}(\\lambda_i)= c\\] for a constant $c$, where $\\lambda_i$ are the eigenvalues of $D^2u$.\n\n(\\textit{Conjecture of Nadirashvili-Vl\\u{a}du\\c{t}}) Any viscosity solution of the special Lagrangian equation on the unit ball is $C^1(B)$ for sufficiently smooth boundary data.", "statement_status": "exact", "statement_verification": "The canonical AIM record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Monge-Ampère equations\nSource item: 1.2\nSource URL: http://aimpl.org/nonlinpdegeom/1/\nCanonical location: aim-pdes-notes.json notes[27]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A function $u:\\\\Omega \\\\to \\\\mathbb R$ satisfies the special Lagrangian equation if \\\\[\\\\sum_i \\\\mathrm{arctan}(\\\\lambda_i)= c\\\\] for a constant $c$, where $\\\\lambda_i$ are the eigenvalues of $D^2u$.\\n\\n(\\\\textit{Conjecture of Nadirashvili-Vl\\\\u{a}du\\\\c{t}}) Any viscosity solution of the special Lagrangian equation on the unit ball is $C^1(B)$ for sufficiently smooth boundary data.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"There exist known examples which are not $C^2$.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/1/", "tags": [ "aim", "AIM-PDES-0028", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Mooney and Savin's 2024 theorem gives the published negative answer to the Nadirashvili-Vladut C1 conjecture by constructing a three-dimensional Lipschitz viscosity solution of the special Lagrangian equation with discontinuous gradient and smooth behavior near the boundary. In addition, this attempt proves two positive restricted results: every one-dimensional viscosity solution is quadratic, and every radial solution that is C1 at the center and C2/classical off the center is exactly u(x)=a+(1/2)tan(c/n)|x|^2.\n\nCandidate contribution (radial_rigidity_proposition; novelty confidence low): For a radial punctured-classical special-Lagrangian potential u(x)=phi(|x|), the quantity Im(exp(-ic)(r+i phi'(r))^n) is conserved; if u is C1 at the center this constant vanishes and forces u(x)=a+(1/2)tan(c/n)|x|^2." }, { "id": 20002417, "problem_number": "AIM-PDES-0029", "title": "Missing quantifiers and exact Sobolev thresholds for a divergence-form model", "statement": "(\\textit{Conjecture of Nadirashvili-Tkachev-Vl\\u{a}du\\c{t}}) A uniformly elliptic equation in divergence form admits a solution in $W^{1,p}$ for some $p>1$.", "original_statement": "(\\textit{Conjecture of Nadirashvili-Tkachev-Vl\\u{a}du\\c{t}}) A uniformly elliptic equation in divergence form admits a solution in $W^{1,p}$ for some $p>1$.", "clean_statement": "If $u\\in W^{1,2}_{\\mathrm{loc}}(\\Omega)$ solves $\\operatorname{div}(A(x)\\nabla u)=0$, where $A$ is bounded measurable and uniformly elliptic with fixed constants, does $\\nabla u$ belong locally to $L^{2+\\varepsilon}$ for some $\\varepsilon>0$ depending only on dimension and ellipticity?", "statement_status": "reconstructed_unverified", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Monge-Ampère equations\nSource item: 1.25\nSource URL: http://aimpl.org/nonlinpdegeom/1/\nCanonical location: aim-pdes-notes.json notes[28]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(\\\\textit{Conjecture of Nadirashvili-Tkachev-Vl\\\\u{a}du\\\\c{t}}) A uniformly elliptic equation in divergence form admits a solution in $W^{1,p}$ for some $p>1$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/1/", "tags": [ "aim", "AIM-PDES-0029", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The archived AIM source confirms that the displayed conjecture is source-level underquantified: under its literal homogeneous reading constants make it vacuous, under the standard energy reading p=2 is automatic, and under the meaningful p>2 reading it is classical Meyers higher integrability. No primary Nadirashvili-Tkachev-Vladut source matching the divergence-form W1p attribution was found. A proved paired example for one uniformly elliptic coefficient field A_kappa shows that u_+=r^kappa cos(theta) is the energy solution with gradient in Lp exactly for p<2/(1-kappa), while u_-=r^{-kappa} cos(theta) is a non-energy distributional solution with the same smooth boundary trace and gradient in Lp exactly for p<2/(1+kappa). Thus the intended conjecture cannot be recovered, and both ellipticity dependence above energy and solution-class dependence below energy are explicit.\n\nCandidate contribution (paired_threshold_example; novelty confidence low): For A_kappa=e_r tensor e_r+kappa^2 e_theta tensor e_theta on the disk, the two distributional solutions r^{plus or minus kappa} cos(theta) have the same smooth boundary trace and complementary exact gradient thresholds p_+=2/(1-kappa) and p_-=2/(1+kappa), giving sub-energy nonuniqueness and a super-energy Meyers obstruction in one model.", "research_difficulty_suggested": "L2" }, { "id": 20002418, "problem_number": "AIM-PDES-0030", "title": "Literal torus obstruction and background-Hessian reconstruction", "statement": "Let $u$ be a solution to \\[u_{tt}+\\mathrm{det}(u_{ij}) = 0\\] on $T^n\\times[0,1]$ with convex boundary data. Prove a $C^2$ estimate for $u(\\cdot,t)$ for $t$ fixed.", "original_statement": "Let $u$ be a solution to \\[u_{tt}+\\mathrm{det}(u_{ij}) = 0\\] on $T^n\\times[0,1]$ with convex boundary data. Prove a $C^2$ estimate for $u(\\cdot,t)$ for $t$ fixed.", "clean_statement": "Let $u$ be a solution to \\[u_{tt}+\\mathrm{det}(u_{ij}) = 0\\] on $T^n\\times[0,1]$ with convex boundary data. Prove a $C^2$ estimate for $u(\\cdot,t)$ for $t$ fixed.", "statement_status": "exact", "statement_verification": "Thus the repository transcription is faithful: this is not an OCR error introduced by the corpus. The statement on AIM itself is missing definitions essential to a mathematical boundary-value problem:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Monge-Ampère equations\nSource item: 1.3\nSource URL: http://aimpl.org/nonlinpdegeom/1/\nCanonical location: aim-pdes-notes.json notes[29]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $u$ be a solution to \\\\[u_{tt}+\\\\mathrm{det}(u_{ij}) = 0\\\\] on $T^n\\\\times[0,1]$ with convex boundary data. Prove a $C^2$ estimate for $u(\\\\cdot,t)$ for $t$ fixed.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"A solution would imply the maximal rank conjecture for complex tori.\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/1/", "tags": [ "aim", "AIM-PDES-0030", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The live AIM page confirms the corpus text, but the problem is malformed literally: an ordinary convex periodic C2 function is constant, and the Hessian determinant is a null Lagrangian whose torus integral vanishes, so every smooth literal solution has spatial mean affine in time. A primary-source comparison strongly identifies the likely intended equation as the Guan-Phong partial-Legendre equation phi_tt+det(A+D_x^2 phi)=0 with a positive background. For that reconstruction the report proves the exact determinant-mass and mean laws in all dimensions and gives the complete one-dimensional harmonic formula, boundary C2 estimates, preservation of the background convexity constant, and explicit fixed-slice interior smoothing.\n\nCandidate contribution (source_reconstruction_and_compatibility_lemma; novelty confidence low): For every constant symmetric background A and periodic phi, the exact identity integral det(A+D^2 phi)=det A forces the mean of any reconstructed solution to equal linear interpolation of the endpoint means plus (det A)t(1-t)/2; in dimension one this combines with an exact harmonic reduction to preserve the boundary lower bound a+phi_xx at no loss and yield explicit C2 estimates.", "research_difficulty_suggested": "L2" }, { "id": 20002419, "problem_number": "AIM-PDES-0031", "title": "Unified radial C2 criterion and thin-shell obstruction", "statement": "Find conditions that allow for $C^2$ regularity for degenerate Monge-Ampère equation (both real and complex).", "original_statement": "Find conditions that allow for $C^2$ regularity for degenerate Monge-Ampère equation (both real and complex).", "clean_statement": "Find conditions that allow for $C^2$ regularity for degenerate Monge-Ampère equation (both real and complex).", "statement_status": "exact", "statement_verification": "The exact canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Monge-Ampère equations\nSource item: 1.35\nSource URL: http://aimpl.org/nonlinpdegeom/1/\nCanonical location: aim-pdes-notes.json notes[30]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find conditions that allow for $C^2$ regularity for degenerate Monge-Ampère equation (both real and complex).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/1/", "tags": [ "aim", "AIM-PDES-0031", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For normalized radial real and complex Monge-Ampere equations, the tangential Hessian eigenvalue is the nth root of the normalized centered density average A, while the radial eigenvalue is H/A^((n-1)/n). Consequently, C2 extension at the center is equivalent to these two eigenvalues having the same finite limit. A common continuous vanishing-density thin-shell construction shows that, in dimensions n at least 2, continuity and H(0)=0 alone do not even ensure C1,1 regularity.\n\nCandidate contribution (radial_criterion_and_counterexample; novelty confidence low): The same centered-average/pointwise-ratio criterion exactly characterizes C2 center extension in the real radial equation det D2u=H(|x|) and the complex radial equation det(u_jbar_k)=H(|z|^2), and one explicit continuous thin-shell density makes the radial eigenvalue unbounded in both theories for every n at least 2." }, { "id": 20002420, "problem_number": "AIM-PDES-0032", "title": "Quadratic-growth complex Monge-Ampere rigidity with a completeness gap", "statement": "Let $u$ be plurisubharmonic solution to $\\mathrm{det}(u_{\\overline k j}) =1$ on $\\mathbb C^n$ satisfying\n\\[ C^{-1} (|z|^2+1) \\leq u \\leq C (|z|^2+1)\n\\] then $u$ is quadratic.", "original_statement": "Let $u$ be plurisubharmonic solution to $\\mathrm{det}(u_{\\overline k j}) =1$ on $\\mathbb C^n$ satisfying\n\\[ C^{-1} (|z|^2+1) \\leq u \\leq C (|z|^2+1)\n\\] then $u$ is quadratic.", "clean_statement": "Let $u$ be plurisubharmonic solution to $\\mathrm{det}(u_{\\overline k j}) =1$ on $\\mathbb C^n$ satisfying\n\\[ C^{-1} (|z|^2+1) \\leq u \\leq C (|z|^2+1)\n\\] then $u$ is quadratic.", "statement_status": "exact", "statement_verification": "This display agrees with the archived AIM formulation and with Question 1.1 quoted by Li--Sheng; there is no apparent OCR corruption. The reversal \\(u_{\\bar k j}\\) versus the more usual \\(u_{j\\bar k}\\) is harmless. There are, however, three source-level ambiguities or omissions:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Monge-Ampère equations\nSource item: 1.4\nSource URL: http://aimpl.org/nonlinpdegeom/1/\nCanonical location: aim-pdes-notes.json notes[31]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $u$ be plurisubharmonic solution to $\\\\mathrm{det}(u_{\\\\overline k j}) =1$ on $\\\\mathbb C^n$ satisfying\\n\\\\[ C^{-1} (|z|^2+1) \\\\leq u \\\\leq C (|z|^2+1)\\n\\\\] then $u$ is quadratic.\"\nOriginal remarks: [\"The Problem solved recently by AN-MIN LI and LI SHENG the paper in\\narxiv.org/pdf/1809.00824.pdf\"]\nOriginal literature field (JSON string): \"(Y. Wang) Answer is yes, if $u=|z|^2+o(1)$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/1/", "tags": [ "aim", "AIM-PDES-0032", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Li-Sheng's published theorem proves the AIM conclusion for smooth solutions whose associated Kahler metric is complete, but the literal AIM record omits both the solution class and completeness, so the cited paper does not by itself resolve every reading. Independently, the literal statement is proved here in complex dimension one even distributionally: u=|z|^2+Re(az^2+bz+d), with |a|<1 forced by quadratic coercivity. In all dimensions every quadratic solution is classified as u=z^*Hz+Re(z^TSz+b^Tz+d), where H is positive Hermitian with det H=1, and its homogeneous real quadratic part is positive definite exactly when the operator norm of (H^{-1/2})^T S H^{-1/2} is less than one. For a global AIM lower bound, the full affine quadratic must additionally have a strictly positive global minimum.\n\nCandidate contribution (normal_form_and_sharp_coercivity_criterion; novelty confidence low): For every positive Hermitian H with det H=1, the pluriharmonic quadratic perturbations preserving positive-definite real quadratic growth are exactly the symmetric matrices S satisfying ||(H^{-1/2})^T S H^{-1/2}||_op<1; together with harmonic growth this yields the full literal weak theorem in complex dimension one." }, { "id": 20002421, "problem_number": "AIM-PDES-0033", "title": "Separated-variable counterexamples and sharp boundary blow-up", "statement": "Let $u:\\mathbb R^2 \\to \\mathbb R$ be a solution of\n\\[\nu_{11} u_{22} = 1.\n\\]\nIf $u(x) \\leq C(1 + |x|^2)$, then $u$ must be quadratic.\n\nCan the growth condition be removed?\n\nIf one considers instead solutions $u:\\Omega \\to \\mathbb R$ for $\\Omega \\subseteq \\mathbb R^2$ convex, and bounded, and $u$ with Dirichlet boundary conditions, can one obtain a $C^2$ boundary estimate?", "original_statement": "Let $u:\\mathbb R^2 \\to \\mathbb R$ be a solution of\n\\[\nu_{11} u_{22} = 1.\n\\]\nIf $u(x) \\leq C(1 + |x|^2)$, then $u$ must be quadratic.\n\nCan the growth condition be removed?\n\nIf one considers instead solutions $u:\\Omega \\to \\mathbb R$ for $\\Omega \\subseteq \\mathbb R^2$ convex, and bounded, and $u$ with Dirichlet boundary conditions, can one obtain a $C^2$ boundary estimate?", "clean_statement": "Let $u:\\mathbb R^2 \\to \\mathbb R$ be a solution of\n\\[\nu_{11} u_{22} = 1.\n\\]\nIf $u(x) \\leq C(1 + |x|^2)$, then $u$ must be quadratic.\n\nCan the growth condition be removed?\n\nIf one considers instead solutions $u:\\Omega \\to \\mathbb R$ for $\\Omega \\subseteq \\mathbb R^2$ convex, and bounded, and $u$ with Dirichlet boundary conditions, can one obtain a $C^2$ boundary estimate?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem List 1.45 in the section “Monge–Ampère equations” of the workshop *Nonlinear PDEs in real and complex geometry*. The live AIM page was checked on 2026-08-10. It agrees with the corpus and attributes the problem to **Xiangwen Zhang**. The displayed line break before \\(u_{11}u_{22}=1\\) is formatting, not a mathematical symbol.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Monge-Ampère equations\nSource item: 1.45\nSource URL: http://aimpl.org/nonlinpdegeom/1/\nCanonical location: aim-pdes-notes.json notes[32]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $u:\\\\mathbb R^2 \\\\to \\\\mathbb R$ be a solution of\\n\\\\[\\nu_{11} u_{22} = 1.\\n\\\\]\\nIf $u(x) \\\\leq C(1 + |x|^2)$, then $u$ must be quadratic.\\n\\nCan the growth condition be removed?\\n\\nIf one considers instead solutions $u:\\\\Omega \\\\to \\\\mathbb R$ for $\\\\Omega \\\\subseteq \\\\mathbb R^2$ convex, and bounded, and $u$ with Dirichlet boundary conditions, can one obtain a $C^2$ boundary estimate?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/1/", "tags": [ "aim", "AIM-PDES-0033", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "Mooney and Savin answered the growth-removal question negatively by constructing an analytic nonquadratic coordinate-convex entire product solution. This attempt verifies and extends the calculation: every nowhere-zero product solution reduces to pp''=a and qq''=a^{-1}; the canonical positive factor has f(s) asymptotic to sqrt(2)|s|sqrt(log|s|). It also proves that the AIM statement requires the positive elliptic branch, bridges its one-sided growth bound to the published two-sided bound, and derives sharp derivative blow-up rates for a zero-Dirichlet coordinate-convex solution on the square. Thus boundary C2 control fails for a merely bounded convex domain, while the smooth uniformly coordinate-convex boundary problem remains unresolved in the literature checked.\n\nCandidate contribution (asymptotic_obstruction_and_separable_reduction; novelty confidence low): For every nowhere-zero product solution u=p(x)q(y), the PDE is equivalent to pp''=a and qq''=a^{-1}; for the zero-Dirichlet coordinate-convex square solution V=-h(x)h(y), along a fixed interior-height slice at distance d from a flat side, V_11 is asymptotic to h(y)/(d sqrt(2 log(1/d))), V_22 to d sqrt(2 log(1/d))/h(y), and generically |V_12| to |h'(y)| sqrt(2 log(1/d))." }, { "id": 20002422, "problem_number": "AIM-PDES-0034", "title": "A radial phase diagram for a singular complex Monge-Ampere density", "statement": "Let $\\mathrm{det}(u_{\\overline k j}) = \\frac {f(z)}{|z|^\\alpha}$ for $f(z)$ smooth. What is the optimal regularity of $u$?", "original_statement": "Let $\\mathrm{det}(u_{\\overline k j}) = \\frac {f(z)}{|z|^\\alpha}$ for $f(z)$ smooth. What is the optimal regularity of $u$?", "clean_statement": "Let $\\mathrm{det}(u_{\\overline k j}) = \\frac {f(z)}{|z|^\\alpha}$ for $f(z)$ smooth. What is the optimal regularity of $u$?", "statement_status": "exact", "statement_verification": "The exact canonical AIM record, from the workshop *Nonlinear PDEs in real and complex geometry*, Monge--Ampère equations, Problem 1.5, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Monge-Ampère equations\nSource item: 1.5\nSource URL: http://aimpl.org/nonlinpdegeom/1/\nCanonical location: aim-pdes-notes.json notes[33]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $\\\\mathrm{det}(u_{\\\\overline k j}) = \\\\frac {f(z)}{|z|^\\\\alpha}$ for $f(z)$ smooth. What is the optimal regularity of $u$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/1/", "tags": [ "aim", "AIM-PDES-0034", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM question has no unique answer without specifying dimension, domain and origin, alpha range, positivity and vanishing of f, solution class, boundary data, and normalization. For the natural locally bounded radial Bedford-Taylor benchmark with f=c>0 and alpha<2n, the unique branch is u=A|z|^(2-alpha/n)+B, where A=c^(1/n)(1-alpha/(2n))^(-1-1/n). This gives sharp radial transitions: C^(0,2-alpha/n) for n0. If f(0)>0 and alpha>=2n, no locally bounded Bedford-Taylor solution can have the stated measure because the right-hand side is not locally finite.\n\nCandidate contribution (radial_phase_diagram_and_scaling_obstruction; novelty confidence low): For constant positive f, the zero-atom locally bounded radial branch has the unique scale-invariant homogeneity 2-alpha/n; this single exponent yields the exact Holder and C1 transitions, the sharp Hessian and gradient Sobolev thresholds, and a universal obstruction to any stronger regularity theorem covering general smooth positive f." }, { "id": 20002423, "problem_number": "AIM-PDES-0035", "title": "Two quantitative gates for toric Chen-Weber compactness", "statement": "Find a toric proof of the toric case of the orbifold compactness theorem proved by Chen-Weber.", "original_statement": "Find a toric proof of the toric case of the orbifold compactness theorem proved by Chen-Weber.", "clean_statement": "Find a toric proof of the toric case of the orbifold compactness theorem proved by Chen-Weber.", "statement_status": "exact", "statement_verification": "The exact canonical prompt is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Complex geometry\nSource item: 2.1\nSource URL: http://aimpl.org/nonlinpdegeom/2/\nCanonical location: aim-pdes-notes.json notes[34]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find a toric proof of the toric case of the orbifold compactness theorem proved by Chen-Weber.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/2/", "tags": [ "aim", "AIM-PDES-0035", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The Chen-Weber theorem behind the prompt is identified precisely, and two rigorous components of a toric proof are isolated. On a fixed labeled polytope, the labeled Abreu flux identity plus a uniform stability constant gives explicit boundary L1 and interior C0 bounds for normalized symplectic potentials, hence local convex precompactness. Independently, a critical Sobolev constant C_S bounds every local orbifold group by (C_S/S_n)^(n/2), and therefore bounds each toric vertex determinant. The missing step is to derive uniform polytope/stability or M-condition control and tensorial regularity directly from the original Chen-Weber hypotheses; no full toric proof is claimed.\n\nCandidate contribution (quantitative_reduction; novelty confidence low): The paired explicit estimates form a two-gate diagnostic for the requested toric proof: labeled uniform stability bounds normalized potential height, while the critical Sobolev constant bounds toric cone determinants with exponent n/2.", "research_difficulty_suggested": "L4" }, { "id": 20002424, "problem_number": "AIM-PDES-0036", "title": "Goldberg conjecture: corrected statement and counterexample sieve", "statement": "A complex manifold is called \\textit{almost K\\\"ahler} if it admits a complex structure $J$, not necessarily integrable, and a Hermitian metric $g$ such that $g(\\cdot,J\\cdot)$ defines a closed 2-form. One says that $g$ and $J$ are \\textit{compatible}.\n\n(\\textit{Goldberg Conjecture}) Is a compact, almost K\\\"ahler, Einstein manifold necessarily K\\\"ahler?", "original_statement": "A complex manifold is called \\textit{almost K\\\"ahler} if it admits a complex structure $J$, not necessarily integrable, and a Hermitian metric $g$ such that $g(\\cdot,J\\cdot)$ defines a closed 2-form. One says that $g$ and $J$ are \\textit{compatible}.\n\n(\\textit{Goldberg Conjecture}) Is a compact, almost K\\\"ahler, Einstein manifold necessarily K\\\"ahler?", "clean_statement": "A complex manifold is called \\textit{almost K\\\"ahler} if it admits a complex structure $J$, not necessarily integrable, and a Hermitian metric $g$ such that $g(\\cdot,J\\cdot)$ defines a closed 2-form. One says that $g$ and $J$ are \\textit{compatible}.\n\n(\\textit{Goldberg Conjecture}) Is a compact, almost K\\\"ahler, Einstein manifold necessarily K\\\"ahler?", "statement_status": "exact", "statement_verification": "The canonical AIM record (PDEs, workshop *Nonlinear PDEs in real and complex geometry*, Section 2, Problem 2.2) says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Complex geometry\nSource item: 2.2\nSource URL: http://aimpl.org/nonlinpdegeom/2/\nCanonical location: aim-pdes-notes.json notes[35]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A complex manifold is called \\\\textit{almost K\\\\\\\"ahler} if it admits a complex structure $J$, not necessarily integrable, and a Hermitian metric $g$ such that $g(\\\\cdot,J\\\\cdot)$ defines a closed 2-form. One says that $g$ and $J$ are \\\\textit{compatible}.\\n\\n(\\\\textit{Goldberg Conjecture}) Is a compact, almost K\\\\\\\"ahler, Einstein manifold necessarily K\\\\\\\"ahler?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Solved by Sekigawa for non-negative Einstein constant.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/2/", "tags": [ "aim", "AIM-PDES-0036", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After correcting the source's 'complex structure, not necessarily integrable' to 'almost-complex structure,' the compact negative-Einstein case remains open. A proved sieve shows that real dimension two is automatically Kähler without compactness or Einstein assumptions; in every dimension the compatible form is harmonic of constant squared norm n and represents a nonzero cohomology class on a compact manifold; and in real dimension four the exact identity -s/3=|nabla omega|^2/2 holds. Thus any compact counterexample has real dimension at least four, negative Einstein constant, b2 at least one, and nonzero Kähler defect; in dimension four it also has c1·[omega]<0 and positive Weyl defect somewhere.\n\nCandidate contribution (reduction; novelty confidence low): Candidate novelty is a convention-explicit, directly testable counterexample sieve combining dimension at least four, negative Einstein constant, b2 at least one, nonparallel fundamental form, and in dimension four c1·[omega]<0 together with -s/3=|nabla omega|^2/2>0 somewhere." }, { "id": 20002425, "problem_number": "AIM-PDES-0037", "title": "A source-defect diagnosis, forced Aubin--Mabuchi zero mode, and cscK linearization", "statement": "(Question of Donaldson) Find smooth solutions to the equation\n\\[\\ddot{\\varphi}- \\frac 12 |\\nabla \\dot \\varphi|^2_{\\omega_\\varphi}=-\\lambda R_{\\omega_\\varphi}\\] for $\\lambda>0$ on $M \\times [0,1]$ with $\\varphi|_{M\\times\\{0\\}}=0$, $\\varphi|_{M\\times\\{0\\}}=\\varphi_0$.", "original_statement": "(Question of Donaldson) Find smooth solutions to the equation\n\\[\\ddot{\\varphi}- \\frac 12 |\\nabla \\dot \\varphi|^2_{\\omega_\\varphi}=-\\lambda R_{\\omega_\\varphi}\\] for $\\lambda>0$ on $M \\times [0,1]$ with $\\varphi|_{M\\times\\{0\\}}=0$, $\\varphi|_{M\\times\\{0\\}}=\\varphi_0$.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "No corrected primary or archived formulation was located in the searches described below. The following natural **reconstruction is therefore an inference, not verified source text**: Accordingly, the literal AIM problem has status `invalid_statement`. The rest of this report gives rigorous consequences and a linearized solution theory for the explicitly labeled reconstruction (P); it does not claim nonlinear existence for (P).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Complex geometry\nSource item: 2.3\nSource URL: http://aimpl.org/nonlinpdegeom/2/\nCanonical location: aim-pdes-notes.json notes[36]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(Question of Donaldson) Find smooth solutions to the equation\\n\\\\[\\\\ddot{\\\\varphi}- \\\\frac 12 |\\\\nabla \\\\dot \\\\varphi|^2_{\\\\omega_\\\\varphi}=-\\\\lambda R_{\\\\omega_\\\\varphi}\\\\] for $\\\\lambda>0$ on $M \\\\times [0,1]$ with $\\\\varphi|_{M\\\\times\\\\{0\\\\}}=0$, $\\\\varphi|_{M\\\\times\\\\{0\\\\}}=\\\\varphi_0$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"It was pointed out by Donaldson that such a solution would prove the uniqueness of cscK metrics\"\nResearch attempt: 2; result status: invalid_statement; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/2/", "tags": [ "aim", "AIM-PDES-0037", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The literal AIM record is defective: it imposes both endpoint values at t=0, hence is contradictory when the stated phi_0 is nonzero and fails to specify a two-endpoint problem when phi_0=0. For the explicitly unverified reconstruction with endpoints at t=0 and t=1, every smooth solution obeys the exact normalized Aubin--Mabuchi law E(phi_t)=t E(phi_0)+(lambda Rbar/2)t(1-t); spatially constant endpoint data over a cscK metric give an exact solution; and linearization about that forced constant path is u_tt-lambda L u=0, with hyperbolic-sine interpolation on positive Lichnerowicz modes and linear interpolation on its automorphism kernel, so there is no Dirichlet resonance.\n\nCandidate contribution (lemma; novelty confidence low): Candidate novelty: the unshifted scalar-curvature force prescribes an exact quadratic Aubin--Mabuchi zero mode, while after extracting that constant drift the cscK fixed-endpoint linearization has nonvanishing hyperbolic-sine denominators on every positive Lichnerowicz mode and linear interpolation on all kernel modes.", "research_difficulty_suggested": "L2" }, { "id": 20002426, "problem_number": "AIM-PDES-0038", "title": "Mabuchi distances in a big class and their homothety calibration", "statement": "$M$ K\\\"ahler, $\\theta$ big cohomology class. Can we define a family of distances generalizing the Finsler geometry of K\\\"ahler potentials and compatible with weak Mabuchi geodesics?", "original_statement": "$M$ K\\\"ahler, $\\theta$ big cohomology class. Can we define a family of distances generalizing the Finsler geometry of K\\\"ahler potentials and compatible with weak Mabuchi geodesics?", "clean_statement": "$M$ K\\\"ahler, $\\theta$ big cohomology class. Can we define a family of distances generalizing the Finsler geometry of K\\\"ahler potentials and compatible with weak Mabuchi geodesics?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Complex geometry\nSource item: 2.4\nSource URL: http://aimpl.org/nonlinpdegeom/2/\nCanonical location: aim-pdes-notes.json notes[37]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"$M$ K\\\\\\\"ahler, $\\\\theta$ big cohomology class. Can we define a family of distances generalizing the Finsler geometry of K\\\\\\\"ahler potentials and compatible with weak Mabuchi geodesics?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/2/", "tags": [ "aim", "AIM-PDES-0038", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The natural minimal-reference reading of the AIM question is solved: for every compact Kähler manifold, smooth closed real (1,1)-form theta representing a big class, and p at least 1, Gupta's metric makes the full non-pluripolar mass space E^p(X,theta) a complete geodesic metric space whose weak Mabuchi segments are constant-speed metric geodesics. The broader wording remains only partially resolved because it does not specify the singularity type or metric family: Xia's arbitrary positive-mass model slices carry complete metrics, while the cited general weak-segment geodesic theorem covers the minimal/full-mass space and Gupta's prescribed analytic-singularity setting, not every arbitrary model slice. As a proved calibration, scaling theta and potentials by a>0 multiplies the raw distance by a^(1+n/p), while volume normalization changes this factor to a; constant shifts have raw distance |c| Vol(theta)^(1/p) and normalized distance |c|.\n\nCandidate contribution (lemma; novelty confidence low): For the arbitrary-big-class Gupta-Darvas metric, the class homothety u -> au from E^p(X,theta) to E^p(X,a theta) scales the unnormalized distance by a^(1+n/p) and the volume-normalized distance by a; moreover, every constant-direction weak segment u_t=u+tc has unnormalized length |c| Vol(theta)^(1/p) and normalized length |c|." }, { "id": 20002427, "problem_number": "AIM-PDES-0039", "title": "Sharp block-power obstruction for complex Monge-Ampere regularity", "statement": "Let $u$ be a plurisubharmonic function satisfying\n\\[\n(i \\partial \\overline \\partial u)^n = \\psi\n\\]\nwith $\\psi>0$ and smooth. Assume $u$ is $C^{1,\\alpha}$ for $\\alpha > 1 - \\frac 2 n$, then $u\\in C^\\infty$.", "original_statement": "Let $u$ be a plurisubharmonic function satisfying\n\\[\n(i \\partial \\overline \\partial u)^n = \\psi\n\\]\nwith $\\psi>0$ and smooth. Assume $u$ is $C^{1,\\alpha}$ for $\\alpha > 1 - \\frac 2 n$, then $u\\in C^\\infty$.", "clean_statement": "Let $u$ be a plurisubharmonic function satisfying\n\\[\n(i \\partial \\overline \\partial u)^n = \\psi\n\\]\nwith $\\psi>0$ and smooth. Assume $u$ is $C^{1,\\alpha}$ for $\\alpha > 1 - \\frac 2 n$, then $u\\in C^\\infty$.", "statement_status": "exact", "statement_verification": "The exact canonical text is declarative:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Complex geometry\nSource item: 2.6\nSource URL: http://aimpl.org/nonlinpdegeom/2/\nCanonical location: aim-pdes-notes.json notes[38]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $u$ be a plurisubharmonic function satisfying\\n\\\\[\\n(i \\\\partial \\\\overline \\\\partial u)^n = \\\\psi\\n\\\\]\\nwith $\\\\psi>0$ and smooth. Assume $u$ is $C^{1,\\\\alpha}$ for $\\\\alpha > 1 - \\\\frac 2 n$, then $u\\\\in C^\\\\infty$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Solved for $\\\\alpha>1 - \\\\frac 1 n$ by Li-Li-Zheng.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/2/", "tags": [ "aim", "AIM-PDES-0039", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The cited partial theorem is corrected and identified as Chao Li, Jiayu Li, and Xi Zhang: smooth positive density and C^{1,beta} regularity with beta>1-1/n imply local smoothness, while the proposed beta>1-2/n range remains open in the primary literature checked through 2026. New mathematical work proves the conjecture in complex dimension one and for U(n)-invariant weak solutions, and derives the exact formula det(D_C^2(|z|^{2a}A(w)))=a^{p+1}A^n|z|^{2(na-p)}det(ddbar log A). Smooth strict positivity across the axis forces a=p/n and exact gradient exponent 1-2k/n; for k=1 this proves endpoint sharpness but not supercritical regularity.\n\nCandidate contribution (model_rigidity_lemma; novelty confidence low): Within the local product-power ansatz u(z,w)=|z|^{2a}A(w), where z is in C^p, w is in C^k, p+k=n, and log A is strictly plurisubharmonic, a Monge-Ampere density that extends smoothly and strictly positively across z=0 forces a=p/n and therefore forces the exact gradient Holder exponent 1-2k/n." }, { "id": 20002428, "problem_number": "AIM-PDES-0040", "title": "Weak-closure reduction for the maximal-rank counterexample problem", "statement": "Find a counterexample to the maximal rank conjecture.\n\n\\textbf{\\emph{Conjecture}} (Maximal Rank Conjecture)\n\nLet $A\\subseteq\\mathbb C$ be the annulus $A = \\{10$ independent of $\\delta$.", "original_statement": "Find a counterexample to the maximal rank conjecture.\n\n\\textbf{\\emph{Conjecture}} (Maximal Rank Conjecture)\n\nLet $A\\subseteq\\mathbb C$ be the annulus $A = \\{10$ independent of $\\delta$.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is visibly corrupted. It ends with", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Complex geometry\nSource item: 2.5\nSource URL: http://aimpl.org/nonlinpdegeom/2/\nCanonical location: aim-pdes-notes.json notes[39]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Find a counterexample to the maximal rank conjecture.\\n\\n\\\\textbf{\\\\emph{Conjecture}} (Maximal Rank Conjecture)\\n\\nLet $A\\\\subseteq\\\\mathbb C$ be the annulus $A = \\\\{10$ independent of $\\\\delta$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"May try using the counterexample of Ross-Witt Nystr\\\\\\\"om from Hele-Shaw flow to produce such a counterexample.\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/2/", "tags": [ "aim", "AIM-PDES-0040", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The corrupted AIM statement is recovered as the annular fixed-endpoint maximal-rank conjecture, with an explicit remaining ambiguity in its boundary labels. For any smooth positive forms converging weakly as currents, a uniform lower bound on every vertical fiber passes to the limit when tested against all strongly positive vertical test forms. Consequently, an open vertical rank-zero region in a fixed-endpoint limiting geodesic forces the least vertical eigenvalue of every positive smooth regularization to approach zero somewhere on every compact product subregion. This reduces the AIM counterexample problem to constructing smooth fixed endpoints with such limiting degeneration; the Ross-Witt Nystrom example supplies the degeneration mechanism but not the required fixed boundary geometry.\n\nCandidate contribution (lemma; novelty confidence low): Vertical fiber lower bounds are weakly closed under positive-current convergence, and an open vertical rank-zero set forces quantitative minimum-eigenvalue collapse in every positive smooth approximating family.", "research_difficulty_suggested": "L4" }, { "id": 20002429, "problem_number": "AIM-PDES-0041", "title": "The solved J-equation criterion and a sharp chamber on the one-point blow-up", "statement": "Let $(M,\\chi)$ compact K\\\"ahler, and $\\omega$ another K\\\"ahler metric on $M$. There are necessary and sufficient conditions for solving\n\\[\n\\mathrm{tr}_{\\omega_\\varphi} \\chi = c\n\\]\nwhere $c = n \\frac {[\\omega]^{n-1} \\cap [\\chi]} {[\\omega]^n}$ is a constant (Song-Weinkove) in terms of a positivity condition. However, this may be hard to check in general.\n\n(Conjecture of Lejmi-Sz\\'ekelyhidi) There exists a solution if and only if\n\\[\n\\int_V{(c \\omega^p - p\\omega^{p-1}\\wedge \\chi)}>0\n\\]\nfor all proper subvarieties $V \\subset M$, where $p = \\mathrm{dim}(V)$", "original_statement": "Let $(M,\\chi)$ compact K\\\"ahler, and $\\omega$ another K\\\"ahler metric on $M$. There are necessary and sufficient conditions for solving\n\\[\n\\mathrm{tr}_{\\omega_\\varphi} \\chi = c\n\\]\nwhere $c = n \\frac {[\\omega]^{n-1} \\cap [\\chi]} {[\\omega]^n}$ is a constant (Song-Weinkove) in terms of a positivity condition. However, this may be hard to check in general.\n\n(Conjecture of Lejmi-Sz\\'ekelyhidi) There exists a solution if and only if\n\\[\n\\int_V{(c \\omega^p - p\\omega^{p-1}\\wedge \\chi)}>0\n\\]\nfor all proper subvarieties $V \\subset M$, where $p = \\mathrm{dim}(V)$", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical AIM record is problem 2.7 in the “Complex geometry” section of the workshop *Nonlinear PDEs in real and complex geometry*. Its wording has minor extraction defects: the phrase “Let \\((M,\\chi)\\) compact Kähler” is missing “be,” and the cap symbol denotes a cohomological intersection. The displayed mathematics agrees with Lejmi--Székelyhidi's original formulation.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Complex geometry\nSource item: 2.7\nSource URL: http://aimpl.org/nonlinpdegeom/2/\nCanonical location: aim-pdes-notes.json notes[40]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $(M,\\\\chi)$ compact K\\\\\\\"ahler, and $\\\\omega$ another K\\\\\\\"ahler metric on $M$. There are necessary and sufficient conditions for solving\\n\\\\[\\n\\\\mathrm{tr}_{\\\\omega_\\\\varphi} \\\\chi = c\\n\\\\]\\nwhere $c = n \\\\frac {[\\\\omega]^{n-1} \\\\cap [\\\\chi]} {[\\\\omega]^n}$ is a constant (Song-Weinkove) in terms of a positivity condition. However, this may be hard to check in general.\\n\\n(Conjecture of Lejmi-Sz\\\\'ekelyhidi) There exists a solution if and only if\\n\\\\[\\n\\\\int_V{(c \\\\omega^p - p\\\\omega^{p-1}\\\\wedge \\\\chi)}>0\\n\\\\]\\nfor all proper subvarieties $V \\\\subset M$, where $p = \\\\mathrm{dim}(V)$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Condition is necessary, by Donaldson. Solved by Collins-Sz\\\\'ekelyhidi in the toric case.\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/2/", "tags": [ "aim", "AIM-PDES-0041", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Jian Song's Corollary 1.2 in arXiv:2012.07956v1 proves the exact Lejmi--Szekelyhidi criterion for arbitrary smooth compact Kahler manifolds and arbitrary Kahler classes, including nonprojective and transcendental classes. Independently, the attempt proves the complete surface reduction and computes that on Bl_p(P^2), for [omega]=aH-bE and [chi]=xH-yE with a>b>0 and x>y>0, smooth solvability is equivalent to the single sharp inequality 2abx>y(a^2+b^2); the inequality on H-E is automatic.\n\nCandidate contribution (explicit_chamber; novelty confidence low): For arbitrary real Kahler classes alpha=aH-bE and beta=xH-yE on the first Hirzebruch surface, the entire Lejmi--Szekelyhidi numerical criterion reduces to the exceptional-curve wall 2abx>y(a^2+b^2), while (c alpha-beta).(H-E)=(a-b)(x+y)/(a+b)>0 automatically." }, { "id": 20002430, "problem_number": "AIM-PDES-0042", "title": "Regularity of weak geodesics on singular Kähler varieties", "statement": "Can one establish regularity properties of geodesics in the space of K\\\"ahler potentials for singular K\\\"ahler varieties?", "original_statement": "Can one establish regularity properties of geodesics in the space of K\\\"ahler potentials for singular K\\\"ahler varieties?", "clean_statement": "Can one establish regularity properties of geodesics in the space of K\\\"ahler potentials for singular K\\\"ahler varieties?", "statement_status": "exact", "statement_verification": "The exact AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Complex geometry\nSource item: 2.8\nSource URL: http://aimpl.org/nonlinpdegeom/2/\nCanonical location: aim-pdes-notes.json notes[41]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can one establish regularity properties of geodesics in the space of K\\\\\\\"ahler potentials for singular K\\\\\\\"ahler varieties?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/2/", "tags": [ "aim", "AIM-PDES-0042", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The natural smooth-endpoint, regular-locus reading of the AIM question has an affirmative answer through Chu--McCleerey's local C^{1,1} theorem, while the broadly worded question remains only partially solved for arbitrary singularities and endpoint classes. As a proved special case across quotient points, for a compact Kähler manifold with a finite group action and invariant smooth Kähler endpoints, the pluripotential Perron geodesic commutes exactly with passage to the quotient, is unique by lifting, and has C^{1,1} regularity with no loss in the lift-defined orbifold norm. An explicit cyclic A_{m-1} quotient shows that this orbifold regularity need not imply even coarse-coordinate C^1 regularity.\n\nCandidate contribution (theorem; novelty confidence low): For finite global Kähler quotients with invariant smooth endpoints, the full Perron envelope commutes exactly with quotient descent, uniqueness reduces to the upstairs comparison principle, and the lift-defined orbifold C^{1,1} norm has no group-order loss; paired with a cyclic A_{m-1} example, this gives a convention-explicit obstruction to converting orbifold regularity into coarse-coordinate C^1 regularity." }, { "id": 20002431, "problem_number": "AIM-PDES-0043", "title": "Zero-distance conjugate Kähler-Einstein manifolds need not be deformation equivalent", "statement": "Let $(M,\\omega)$ and $(M',\\omega')$ be compact K\\\"ahler-Einstein manifolds which are $\\varepsilon$-close in Gromov-Hausdorff distance. Is there a holomorphic family $M_t$ with $M_0 = M$ and $M_1=M'$?", "original_statement": "Let $(M,\\omega)$ and $(M',\\omega')$ be compact K\\\"ahler-Einstein manifolds which are $\\varepsilon$-close in Gromov-Hausdorff distance. Is there a holomorphic family $M_t$ with $M_0 = M$ and $M_1=M'$?", "clean_statement": "Let $(M,\\omega)$ and $(M',\\omega')$ be compact K\\\"ahler-Einstein manifolds which are $\\varepsilon$-close in Gromov-Hausdorff distance. Is there a holomorphic family $M_t$ with $M_0 = M$ and $M_1=M'$?", "statement_status": "exact", "statement_verification": "The canonical record, AIM Problem List 2.9, reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Complex geometry\nSource item: 2.9\nSource URL: http://aimpl.org/nonlinpdegeom/2/\nCanonical location: aim-pdes-notes.json notes[42]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $(M,\\\\omega)$ and $(M',\\\\omega')$ be compact K\\\\\\\"ahler-Einstein manifolds which are $\\\\varepsilon$-close in Gromov-Hausdorff distance. Is there a holomorphic family $M_t$ with $M_0 = M$ and $M_1=M'$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"By Colding, $M$ and $M'$ are necessarily diffeomorphic for $\\\\varepsilon>0$ sufficiently small.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/2/", "tags": [ "aim", "AIM-PDES-0043", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM implication is false even in fixed real dimension four with the common normalization Ric(g)=-g and Gromov-Hausdorff distance zero. Kharlamov and Kulikov construct a projective surface X with ample canonical bundle such that no self-homeomorphism reverses c1(K_X). Its Aubin-Yau metric g is normalized negative Kähler-Einstein. On the conjugate complex surface (X,-J), the Kähler form is -omega but the real metric g is identical. A proper holomorphic submersion containing X and its conjugate would, by Ehresmann transport and the relative canonical bundle, induce a self-homeomorphism reversing c1(K_X), a contradiction.\n\nCandidate contribution (obstruction; novelty confidence low): If a compact Kähler-Einstein manifold has no self-homeomorphism reversing its canonical class, then it and its conjugate are zero Gromov-Hausdorff distance apart but cannot occur in one smooth proper holomorphic family." }, { "id": 20002432, "problem_number": "AIM-PDES-0044", "title": "Low-regularity flexibility and smooth instability for embedded torus metrics", "statement": "(Problem originally proposed by Fang-Hua Lin) Let $(T^2,g_0) \\subseteq (\\mathbb R^3, g_{\\mathrm{eucl}})$ be an isometric embedding and $g$ be a metric on $T^2$ which is $C^0$ close to $g_0$. Is there an isometric embedding of $(T^2,g)$ in $\\mathbb R^3$?", "original_statement": "(Problem originally proposed by Fang-Hua Lin) Let $(T^2,g_0) \\subseteq (\\mathbb R^3, g_{\\mathrm{eucl}})$ be an isometric embedding and $g$ be a metric on $T^2$ which is $C^0$ close to $g_0$. Is there an isometric embedding of $(T^2,g)$ in $\\mathbb R^3$?", "clean_statement": "(Problem originally proposed by Fang-Hua Lin) Let $(T^2,g_0) \\subseteq (\\mathbb R^3, g_{\\mathrm{eucl}})$ be an isometric embedding and $g$ be a metric on $T^2$ which is $C^0$ close to $g_0$. Is there an isometric embedding of $(T^2,g)$ in $\\mathbb R^3$?", "statement_status": "exact", "statement_verification": "The canonical AIM record (PDEs, workshop *Nonlinear PDEs in real and complex geometry*, Geometry 3.1) says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Geometry\nSource item: 3.1\nSource URL: http://aimpl.org/nonlinpdegeom/3/\nCanonical location: aim-pdes-notes.json notes[43]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(Problem originally proposed by Fang-Hua Lin) Let $(T^2,g_0) \\\\subseteq (\\\\mathbb R^3, g_{\\\\mathrm{eucl}})$ be an isometric embedding and $g$ be a metric on $T^2$ which is $C^0$ close to $g_0$. Is there an isometric embedding of $(T^2,g)$ in $\\\\mathbb R^3$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/3/", "tags": [ "aim", "AIM-PDES-0044", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The answer bifurcates by regularity. For every C1 metric g on the compact torus, scaling the given embedding F0 by any sufficiently small positive factor produces a strictly short embedding; Nash--Kuiper and the global Cao--Szekelyhidi theorem then give isometric embeddings of class C1 and C1,theta for every theta<1/5, so C0 closeness is unnecessary in those categories. In contrast, for every standard torus metric (a+cos t)^2 dx^2+dt^2 with a>1, an explicit cone of arbitrarily C-infinity-small one-sided bump perturbations violates the Han--Lin closure condition and therefore admits no smooth isometric embedding in R3. The C2 status of this bump family is left unresolved.\n\nCandidate contribution (explicit_obstruction_family; novelty confidence low): For every a>1 and every nonzero nonnegative smooth bump h supported compactly in (-pi/2,pi/2), the family f_epsilon=a+cos(t)+epsilon h has Han--Lin closure-defect derivative I_+'(0)=-integral sec^2(t)h(t)dt<0 while the companion integral is exactly unchanged; hence every sufficiently small positive epsilon gives an explicitly certified smoothly nonembeddable metric." }, { "id": 20002433, "problem_number": "AIM-PDES-0045", "title": "Chern--Ricci flow on the standard Hopf surface", "statement": "The \\textit{Chern-Ricci curvature} of a Hermitian metric $g$ is defined by\n\\[\nR^{\\mathcal{Ch}}_{\\overline k j} = -\\partial_j \\partial_{\\overline k} \\mathrm{log det}(g).\n\\]\nA family of Hermitian metrics satisfies the Chern-Ricci flow if\n\\[\\frac \\partial {\\partial t} g_{\\overline k j} = -R^{\\mathcal{Ch}}_{\\overline k j}.\\]\n\nWhat is the behavior of the Chern-Ricci flow on the simplest Hopf surface, $\\mathbb (\\mathbb C^2 \\setminus \\{ 0\\})/((z_1,z_2) \\sim (2z_1,2z_2))$?", "original_statement": "The \\textit{Chern-Ricci curvature} of a Hermitian metric $g$ is defined by\n\\[\nR^{\\mathcal{Ch}}_{\\overline k j} = -\\partial_j \\partial_{\\overline k} \\mathrm{log det}(g).\n\\]\nA family of Hermitian metrics satisfies the Chern-Ricci flow if\n\\[\\frac \\partial {\\partial t} g_{\\overline k j} = -R^{\\mathcal{Ch}}_{\\overline k j}.\\]\n\nWhat is the behavior of the Chern-Ricci flow on the simplest Hopf surface, $\\mathbb (\\mathbb C^2 \\setminus \\{ 0\\})/((z_1,z_2) \\sim (2z_1,2z_2))$?", "clean_statement": "The \\textit{Chern-Ricci curvature} of a Hermitian metric $g$ is defined by\n\\[\nR^{\\mathcal{Ch}}_{\\overline k j} = -\\partial_j \\partial_{\\overline k} \\mathrm{log det}(g).\n\\]\nA family of Hermitian metrics satisfies the Chern-Ricci flow if\n\\[\\frac \\partial {\\partial t} g_{\\overline k j} = -R^{\\mathcal{Ch}}_{\\overline k j}.\\]\n\nWhat is the behavior of the Chern-Ricci flow on the simplest Hopf surface, $\\mathbb (\\mathbb C^2 \\setminus \\{ 0\\})/((z_1,z_2) \\sim (2z_1,2z_2))$?", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Geometric flows\nSource item: 4.1\nSource URL: http://aimpl.org/nonlinpdegeom/4/\nCanonical location: aim-pdes-notes.json notes[44]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The \\\\textit{Chern-Ricci curvature} of a Hermitian metric $g$ is defined by\\n\\\\[\\nR^{\\\\mathcal{Ch}}_{\\\\overline k j} = -\\\\partial_j \\\\partial_{\\\\overline k} \\\\mathrm{log det}(g).\\n\\\\]\\nA family of Hermitian metrics satisfies the Chern-Ricci flow if\\n\\\\[\\\\frac \\\\partial {\\\\partial t} g_{\\\\overline k j} = -R^{\\\\mathcal{Ch}}_{\\\\overline k j}.\\\\]\\n\\nWhat is the behavior of the Chern-Ricci flow on the simplest Hopf surface, $\\\\mathbb (\\\\mathbb C^2 \\\\setminus \\\\{ 0\\\\})/((z_1,z_2) \\\\sim (2z_1,2z_2))$?\"\nOriginal remarks: [\"Chern-Ricci flow is equivalent to a parabolic scalar PDE \\\\[\\\\dot \\\\varphi = \\\\mathrm{log}\\\\frac {\\\\mathrm{det}(g_{\\\\overline k j} - t R^{\\\\mathcal{Ch}}_{\\\\overline k j}+\\\\varphi_{\\\\overline k j})}{\\\\mathrm{det}(g_{\\\\overline k j})}\\\\] for $g_{\\\\overline k j}$ initial Hermitian metric.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/4/", "tags": [ "aim", "AIM-PDES-0045", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The malformed surface name is recovered, by explicit inference from the AIM workshop report, as H=(C^2\\{0})/. For the standard Hopf metric the maximal solution is g(t)=r^{-2}(P_rad+(1-2t)P_hor), 0<=t<1/2; its volume is 8 pi^2 log(2)(1-2t), it collapses in Gromov--Hausdorff topology to the round circle of radius log(2)/(sqrt(2) pi), and the ambient fiber diameter is Theta(sqrt(1-2t)). The Chern scalar and Chern--Ricci norm equal 2/(1-2t), while the unsquared full Chern-curvature norm is Theta((1-2t)^{-2}). For arbitrary Gauduchon initial data only finite maximal time and volume collapse are known in this generality, and arbitrary-initial circle convergence remains open in the literature checked.\n\nCandidate contribution (quantitative_theorem; novelty confidence low): For the standard Hopf solution, the ambient diameter of every F-fiber is Theta(sqrt(1-2t)), the Gromov--Hausdorff distance to the explicitly normalized limit circle is O(sqrt(1-2t)), and the unsquared full Chern-curvature norm is Theta((1-2t)^{-2}); this is paired with the exact spatially constant scalar-potential normalization and all real-metric circle constants." }, { "id": 20002434, "problem_number": "AIM-PDES-0046", "title": "Boundary-audited toric Calabi dissipation and a global product family", "statement": "What is the behavior of $g_{j \\overline k}(t)$ solving the Calabi flow,\n\\[\\frac \\partial {\\partial t} g_{j \\overline k} = \\partial_j \\partial_{\\overline k} R?\\]\nCan long time existence be established for toric surfaces?", "original_statement": "What is the behavior of $g_{j \\overline k}(t)$ solving the Calabi flow,\n\\[\\frac \\partial {\\partial t} g_{j \\overline k} = \\partial_j \\partial_{\\overline k} R?\\]\nCan long time existence be established for toric surfaces?", "clean_statement": "What is the behavior of $g_{j \\overline k}(t)$ solving the Calabi flow,\n\\[\\frac \\partial {\\partial t} g_{j \\overline k} = \\partial_j \\partial_{\\overline k} R?\\]\nCan long time existence be established for toric surfaces?", "statement_status": "exact", "statement_verification": "The repository text has no visible OCR corruption. The live source URL was unavailable during this run, so the wording was checked against the exact canonical record, not silently altered. The mathematical question is nevertheless under-specified in several important ways. The following is the recovered smooth invariant reading used below.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Geometric flows\nSource item: 4.2\nSource URL: http://aimpl.org/nonlinpdegeom/4/\nCanonical location: aim-pdes-notes.json notes[45]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What is the behavior of $g_{j \\\\overline k}(t)$ solving the Calabi flow,\\n\\\\[\\\\frac \\\\partial {\\\\partial t} g_{j \\\\overline k} = \\\\partial_j \\\\partial_{\\\\overline k} R?\\\\]\\nCan long time existence be established for toric surfaces?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/4/", "tags": [ "aim", "AIM-PDES-0046", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Unrestricted smooth long-time existence on toric surfaces remains open through the primary literature checked in August 2026. For a smooth torus-invariant flow with symplectic potential u, the prompt's sign convention gives u_t=bar(S)-S, and the reduced Calabi energy satisfies d/dt integral_P(S-bar(S))^2=-2 integral_P u^{ia}u^{jb} f_{ij}f_{ab}; both Guillemin facet fluxes are shown to vanish. On a rectangular polytope, separable potentials are preserved, the flow splits into two one-dimensional Calabi flows, and the reduced energy factors exactly. Chen's Riemann-surface theorem therefore gives an unconditional global smooth convergent family on CP^1 x CP^1 for product classes and separable toric initial metrics.\n\nCandidate contribution (special_case_theorem; novelty confidence low): For every separable smooth Guillemin potential u_0(x_1,x_2)=u_{1,0}(x_1)+u_{2,0}(x_2) on a rectangle, the CP^1 x CP^1 Calabi flow stays separable, exists smoothly for all time, converges to the product cscK metric, and obeys C_P=|I_2|C_1+|I_1|C_2; the accompanying polytope dissipation identity is proved with both facet boundary terms explicitly controlled." }, { "id": 20002435, "problem_number": "AIM-PDES-0047", "title": "Recovered HCF+ and a complete curve-case continuation and entropy theorem", "statement": "Can the Streets--Tian continuation criterion for \\(HCF_+\\) be reduced to\ncontrol of fewer geometric quantities, and does this exact flow possess\nPerelman-type \\(\\mathcal F\\) and \\(\\mathcal W\\) functionals?", "original_statement": "Consider the flow of inverse Hermitian metrics on a compact complex manifold:\n\\[\n\\frac \\partial {\\partial t} g^{j \\overline k} = g^{m \\overline n} \\partial_m \\partial_{\\overline n} g^{j \\overline k} - \\partial_m g^{j \\overline k} \\partial_{\\overline n} g^{m \\overline k}.\n\\]\nOne has short time existence, and it is known that at the maximal time of existence\n\\[\n| \\mathrm{Rm}^{\\mathcal{Ch}} | + |T| + |\\nabla T| \\to \\infty.\n\\]\nCan this be improved? Can one find analogues of Perelman's $\\mathcal F$ and $\\mathcal W$ functionals?", "clean_statement": "Can the Streets--Tian continuation criterion for \\(HCF_+\\) be reduced to\ncontrol of fewer geometric quantities, and does this exact flow possess\nPerelman-type \\(\\mathcal F\\) and \\(\\mathcal W\\) functionals?", "statement_status": "corrected_verified", "statement_verification": "This is source corruption, not merely corruption introduced into the JSON: the archived AIM page from 14 December 2019 contains exactly the same bad indices and attributes Problem 4.3 to Yuri Ustinovskiy. The intended equation is independently verified by Proposition 3.9 and Corollary 3.10 of Ustinovskiy's 2018 Princeton thesis. The recovered inverse-metric equation is \\[ \\boxed{\\quad \\frac{\\partial}{\\partial t}g^{i\\bar j} =g^{m\\bar n}\\partial_m\\partial_{\\bar n}g^{i\\bar j} -\\bigl(\\partial_m g^{i\\bar n}\\bigr) \\bigl(\\partial_{\\bar n}g^{m\\bar j}\\bigr). \\quad} \\tag{HCF\\(_+^{-1}\\)} \\] Thus the first \\(g^{j\\bar k}\\) in the bad quadratic term must be \\(g^{j\\bar n}\\). All contracted and free indices then occur correctly. Invariantly, this is Ustinovskiy's distinguished Hermitian curvature flow, now often called the **positive Hermitian curvature flow** \\(HCF_+\\): \\[ \\frac{\\partial}{\\partial t...", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Geometric flows\nSource item: 4.3\nSource URL: http://aimpl.org/nonlinpdegeom/4/\nCanonical location: aim-pdes-notes.json notes[46]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Consider the flow of inverse Hermitian metrics on a compact complex manifold:\\n\\\\[\\n\\\\frac \\\\partial {\\\\partial t} g^{j \\\\overline k} = g^{m \\\\overline n} \\\\partial_m \\\\partial_{\\\\overline n} g^{j \\\\overline k} - \\\\partial_m g^{j \\\\overline k} \\\\partial_{\\\\overline n} g^{m \\\\overline k}.\\n\\\\]\\nOne has short time existence, and it is known that at the maximal time of existence\\n\\\\[\\n| \\\\mathrm{Rm}^{\\\\mathcal{Ch}} | + |T| + |\\\\nabla T| \\\\to \\\\infty.\\n\\\\]\\nCan this be improved? Can one find analogues of Perelman's $\\\\mathcal F$ and $\\\\mathcal W$ functionals?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/4/", "tags": [ "aim", "AIM-PDES-0047", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM display is index-corrupt in the archived source. The intended equation is Ustinovskiy's positive Hermitian curvature flow, with inverse-coordinate quadratic term -(partial_m g^{i bar n})(partial_bar n g^{m bar j}). For every compact complex curve this recovered flow is exactly complex-geometric Kahler-Ricci flow: torsion vanishes, h_t = h h_{z bar z} - h_z h_{bar z} = h^2 (log h)_{z bar z}, and u_t = (log u)_{z bar z} for u = h^{-1}. With G=2 Re(u dz tensor dbar z), a direct conformal-curvature calculation gives G_t=-Ric(G), so G-hat(s)=G(2s), s=t/2, is standard real Ricci flow. Consequently a bounded scalar Chern curvature extends the solution, every finite-time singularity has scalar Chern-curvature blow-up, and Perelman's original F and W formulas give genuine monotone functionals after this exact time change. No analogous general improvement or entropy is claimed in complex dimension at least two.\n\nCandidate contribution (regularity_reduction; novelty confidence low): For the source-corrected AIM-PDES-0047 flow on any compact complex curve, the inverse-variable identity reduces HCF+ exactly to complex-geometric Kahler-Ricci flow. In the explicit convention G=2 Re(u dz tensor dbar z), it satisfies G_t=-Ric and becomes standard real Ricci flow under s=t/2. Thus its three-term continuation obstruction collapses to scalar Chern curvature alone, its Perelman F/W analogues are the time-rescaled original surface Ricci-flow functionals, and the constant-curvature family G(t)=(1-K_0 t)G_0 supplies an exact sign and normalization check." }, { "id": 20002436, "problem_number": "AIM-PDES-0048", "title": "Finite-time Kähler–Ricci diameter: status trichotomy and an entropy obstruction", "statement": "Is the diameter bounded at finite time singularities of the K\\\"ahler-Ricci flow on compact K\\\"ahler manifolds?", "original_statement": "Is the diameter bounded at finite time singularities of the K\\\"ahler-Ricci flow on compact K\\\"ahler manifolds?", "clean_statement": "Is the diameter bounded at finite time singularities of the K\\\"ahler-Ricci flow on compact K\\\"ahler manifolds?", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: PDEs\nWorkshop: Nonlinear PDEs in real and complex geometry\nSection: Geometric flows\nSource item: 4.4\nSource URL: http://aimpl.org/nonlinpdegeom/4/\nCanonical location: aim-pdes-notes.json notes[47]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is the diameter bounded at finite time singularities of the K\\\\\\\"ahler-Ricci flow on compact K\\\\\\\"ahler manifolds?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "http://aimpl.org/nonlinpdegeom/4/", "tags": [ "aim", "AIM-PDES-0048", "aim-domain:pdes", "aim-workshop:nonlinpdegeom", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the unnormalized Kähler–Ricci flow, published results give bounded diameter when the limiting nef class has positive top self-intersection and give the sharper O(sqrt(T-t)) decay under total extinction; the only general gap is a nonzero limiting class with zero top self-intersection. In that intermediate collapsing regime, Vu's 2026 variable-class theorem yields a rigorous reduction: a uniform L^p bound, for one p greater than the complex dimension, on the normalized logarithmic volume density implies uniformly bounded diameter. Hence every unbounded-diameter sequence must make this entropy diverge for every p greater than the complex dimension. An exact Kähler–Einstein times Ricci-flat product family verifies the normalization, exhibits bounded intermediate collapse, and has constant normalized density.\n\nCandidate contribution (reduction; novelty confidence low): Along any finite-time unnormalized Kähler–Ricci flow, the cohomological mass hypothesis in Vu's variable-class diameter theorem is automatic; therefore an unbounded-diameter sequence forces, for every p greater than the complex dimension, divergence of the L^p norm of the centered parabolic Monge–Ampère density dot(phi)-log V+log(Omega/eta^n)." }, { "id": 20002437, "problem_number": "AIM-PDES-0049", "title": "Stationary spherical shocks: annular existence and center compatibility", "statement": "Open Problem:Do there exist stationary spherical solutions to the above problem (the idea being that stationary spherical solutions should be the simplest multi-D objects to study). It is not clear how the source terms s and f\n\nmust be chosen in order to have a viscous or inviscid shock. Could f or s be identically zero? What is the role of curvature in the stability of the front and existence of nearby perturbed spherical fronts? Note that in the short-time analysis of Majda, curvature does not play a role. A possible scenario is that as the radius of the spherical shock gets bigger, stability is more likely, and there is likely to be a transition from stability to instability as the radius decreases to a critical radius r∗. Can one prove or disprove this?", "original_statement": "Open Problem:Do there exist stationary spherical solutions to the above problem (the idea being that stationary spherical solutions should be the simplest multi-D objects to study). It is not clear how the source terms s and f\n\nmust be chosen in order to have a viscous or inviscid shock. Could f or s be identically zero? What is the role of curvature in the stability of the front and existence of nearby perturbed spherical fronts? Note that in the short-time analysis of Majda, curvature does not play a role. A possible scenario is that as the radius of the spherical shock gets bigger, stability is more likely, and there is likely to be a transition from stability to instability as the radius decreases to a critical radius r∗. Can one prove or disprove this?", "clean_statement": "Open Problem:Do there exist stationary spherical solutions to the above problem (the idea being that stationary spherical solutions should be the simplest multi-D objects to study). It is not clear how the source terms s and f\n\nmust be chosen in order to have a viscous or inviscid shock. Could f or s be identically zero? What is the role of curvature in the stability of the front and existence of nearby perturbed spherical fronts? Note that in the short-time analysis of Majda, curvature does not play a role. A possible scenario is that as the radius of the spherical shock gets bigger, stability is more likely, and there is likely to be a transition from stability to instability as the radius decreases to a critical radius r∗. Can one prove or disprove this?", "statement_status": "exact", "statement_verification": "The record comes from the problem list *Open Problems in Multidimensional Stability of Waves and Patterns*, compiled by N. Costanzino after the AIM workshop of May 16--20, 2005. The PDF supplies equations that were omitted from the extracted record. In space dimension \\(d=2\\) or \\(3\\), put \\(m=d-1\\), write the density as \\(\\rho(r,t)\\), and take the velocity vector to be \\(u(r,t)x/r\\). The displayed barotropic system is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[48]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:Do there exist stationary spherical solutions to the above problem (the idea being that stationary spherical solutions should be the simplest multi-D objects to study). It is not clear how the source terms s and f\\n\\nmust be chosen in order to have a viscous or inviscid shock. Could f or s be identically zero? What is the role of curvature in the stability of the front and existence of nearby perturbed spherical fronts? Note that in the short-time analysis of Majda, curvature does not play a role. A possible scenario is that as the radius of the spherical shock gets bigger, stability is more likely, and there is likely to be a transition from stability to instability as the radius decreases to a critical radius r∗. Can one prove or disprove this?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0049", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The post-workshop literature constructs source-free stationary spherical Euler shocks and Navier-Stokes profiles on bounded annuli, settling the existence question in that geometry but not the spherical stability question. Independently, the stationary equations imply q=rho u=J r^{-m} when s=0 and (c^2-u^2)rho'=m rho u^2/r when also F=0 in the inviscid case. Any nonsonic planar stationary Lax Rankine-Hugoniot pair therefore extends locally to a spherical shock at every prescribed R>0 after setting J=R^m j. In contrast, a regular center enforces q(r)=r^{-m} integral_0^r tau^m s(tau) dt, so a nonzero-flux front on a full ball requires integral_0^R tau^m s(tau) dt=R^m j and cannot occur with s=0.\n\nCandidate contribution (proposition; novelty confidence low): A fixed nonsonic stationary planar Lax shock pair can be realized locally at any positive spherical radius by the flux covariance J=R^m j, whereas a regular-centered ball obeys the exact source compatibility integral_0^R tau^m s(tau) dt=R^m j and hence admits no nonzero-flux source-free stationary front.", "research_difficulty_suggested": "L4" }, { "id": 20002438, "problem_number": "AIM-PDES-0050", "title": "Frozen Lopatinski symbols and exact spherical capillary mode corrections", "statement": "Open Problem:Determine the relationship between the Lopatinski determinant for stability and recent work on stability of spherical waves using spherical harmonics (c.f. C.C. Wu & P. H. Roberts, Bubble shape instability and sonoluminescence, Phys. Lett. A 250, 131 (1998)).", "original_statement": "Open Problem:Determine the relationship between the Lopatinski determinant for stability and recent work on stability of spherical waves using spherical harmonics (c.f. C.C. Wu & P. H. Roberts, Bubble shape instability and sonoluminescence, Phys. Lett. A 250, 131 (1998)).", "clean_statement": "Open Problem:Determine the relationship between the Lopatinski determinant for stability and recent work on stability of spherical waves using spherical harmonics (c.f. C.C. Wu & P. H. Roberts, Bubble shape instability and sonoluminescence, Phys. Lett. A 250, 131 (1998)).", "statement_status": "exact", "statement_verification": "The spacing after “Problem:” is compressed, but comparison with the original seven-page AIM PDF shows no substantive OCR error in this sentence. The canonical record does, however, omit the setup immediately preceding it. The PDF places the problem under S. Benzoni-Gavage, H. K. Jenssen, and M. Williams, “Existence and Stability of Spherical Fronts.” They propose studying curved shock, reactive, or viscous fronts in the simplest curved geometry. In space dimension two or three they reduce barotropic gas dynamics to radial variables: \\[ \\rho_t+(\\rho u)_r+\\frac{(d-1)\\rho u}{r}=s(r,t), \\tag{1} \\] \\[ (\\rho u)_t+(\\rho u^2+P(\\rho))_r +\\frac{(d-1)\\rho u^2}{r} =\\nu\\left(u_r+\\frac{(d-1)u}{r}\\right)_r+F(r,t). \\tag{2} \\] The first open problem asks for stationary spherical fronts and for the role of curvature and the sources. The present record is the second open problem. The third asks about no...", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[49]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:Determine the relationship between the Lopatinski determinant for stability and recent work on stability of spherical waves using spherical harmonics (c.f. C.C. Wu & P. H. Roberts, Bubble shape instability and sonoluminescence, Phys. Lett. A 250, 131 (1998)).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0050", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM sentence omits an essential source setup and compares two different spectral objects. For an isotropic inviscid planar Lax shock, the frozen principal Lopatinski multiplier on a sphere acts on degree-l harmonics by evaluating the planar symbol at k_l=sqrt(l(l+1))/R, but lower-order radial, source, and base-state terms prevent an exact identity with a full spherical shock determinant. Separately, for a static three-dimensional inviscid incompressible two-fluid capillary sphere, the rigorously derived modal/dispersion determinant is D_sph=lambda^2 R(rho_-/l+rho_+/(l+1))+sigma(l-1)(l+2)/R^2. After multiplication by k_l for l>=1, its exact difference from the planar capillary determinant separates Dirichlet-to-Neumann and mean-curvature corrections and tends to zero only in the joint regime l/R tending to a positive planar frequency. This capillary determinant is not a shock Lopatinski determinant or an Evans function.\n\nCandidate contribution (determinant_identity; novelty confidence low): For l>=1, multiplication of the static spherical capillary modal determinant by k_l=sqrt(l(l+1))/R gives exactly lambda^2[rho_-sqrt((l+1)/l)+rho_+sqrt(l/(l+1))]+sigma(k_l^3-2k_l/R^2); subtracting the normalized planar capillary dispersion determinant yields the explicit density-dependent Dirichlet-to-Neumann correction plus the curvature term -2 sigma k_l/R^2. In the gas-bubble limit this predicts omega_l^2/(sigma k_l^3/rho_+)=1+1/(2l)+O(l^-2).", "research_difficulty_suggested": "L4" }, { "id": 20002439, "problem_number": "AIM-PDES-0051", "title": "Focusing spherical shocks: similarity norms and gauge modes", "statement": "Open Problem:A classical and apparently very hard problem is the analysis of non stationary spherical waves, e.g. a focusing inviscid shock (c.f. R. Courant & K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, (1948) or L.D. Landau & E.M. Lifshitz Fluid Mechanics, Pergamon Press, Oxford (1959)) Are such waves stable? 1ARCC Workshop on Multidimensional Stability of Waves and Patterns 2\n\nP. Szmolyan: (Kinetic and Boltzmann Equations)\n\nSetup:Consider the kinetic equation\n\nft + v · fxQ(f, f )where f = f (x, v, t ), x ∈ Rd. Profiles solve (v − c)f ′ = Q(f, f )", "original_statement": "Open Problem:A classical and apparently very hard problem is the analysis of non stationary spherical waves, e.g. a focusing inviscid shock (c.f. R. Courant & K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, (1948) or L.D. Landau & E.M. Lifshitz Fluid Mechanics, Pergamon Press, Oxford (1959)) Are such waves stable? 1ARCC Workshop on Multidimensional Stability of Waves and Patterns 2\n\nP. Szmolyan: (Kinetic and Boltzmann Equations) \n\nSetup:Consider the kinetic equation \n\nft + v · fxQ(f, f )where f = f (x, v, t ), x ∈ Rd. Profiles solve (v − c)f ′ = Q(f, f )", "clean_statement": "Open Problem:A classical and apparently very hard problem is the analysis of non stationary spherical waves, e.g. a focusing inviscid shock (c.f. R. Courant & K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, (1948) or L.D. Landau & E.M. Lifshitz Fluid Mechanics, Pergamon Press, Oxford (1959)) Are such waves stable? 1ARCC Workshop on Multidimensional Stability of Waves and Patterns 2\n\nP. Szmolyan: (Kinetic and Boltzmann Equations)\n\nSetup:Consider the kinetic equation\n\nft + v · fxQ(f, f )where f = f (x, v, t ), x ∈ Rd. Profiles solve (v − c)f ′ = Q(f, f )", "statement_status": "exact", "statement_verification": "The canonical OCR record must be preserved, but it joins three different pieces of the source PDF. The original first page contains the following complete problem under S. Benzoni-Gavage, H. K. Jenssen, and M. Williams:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[50]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:A classical and apparently very hard problem is the analysis of non stationary spherical waves, e.g. a focusing inviscid shock (c.f. R. Courant & K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, (1948) or L.D. Landau & E.M. Lifshitz Fluid Mechanics, Pergamon Press, Oxford (1959)) Are such waves stable? 1ARCC Workshop on Multidimensional Stability of Waves and Patterns 2\\n\\nP. Szmolyan: (Kinetic and Boltzmann Equations) \\n\\nSetup:Consider the kinetic equation \\n\\nft + v · fxQ(f, f )where f = f (x, v, t ), x ∈ Rd. Profiles solve (v − c)f ′ = Q(f, f )\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0051", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The OCR record contains page-header and next-presenter Boltzmann spillover; the recovered AIM question is only whether nonstationary spherical waves such as a focusing inviscid shock are stable. Rigorous ideal-gas Guderley existence is now known, radial linear and nonspherical modal analyses exist, but no general nonlinear nonspherical stability theorem was located. For a self-similar front R=A(T-t)^alpha, a relative mode exp(s tau) has absolute displacement proportional to (T-t)^(alpha-s), so relative decay requires Re(s)<0 while absolute displacement decay requires Re(s) x\n\n+ 2 ηxΦxGΦ − (GΦ) 2\n\n1 + η2\n\n> x\n\n+ gη = 0 where ̂ G(k) = k tanh( k). The equation admits travelling wave solutions ( ηc, Φc) = ( η(x − ct ), Φc(x − ct )) for a variety of wave speeds.", "original_statement": "Open Problem:Under what conditions do there exist profiles for this equation? Are they stable to multidimensional pertur-bations? What does the Evans function look like. (As a first step, one might start with considering discrete velocity models.) \n\nY. Li: (Stability of travelling waves of the full water wave problem near the critical case) \n\nSetup:Consider the water wave equations \n\nηt = GΦΦt + Φ2 \n\n> x\n\n+ 2 ηxΦxGΦ − (GΦ) 2\n\n1 + η2\n\n> x\n\n+ gη = 0 where ̂ G(k) = k tanh( k). The equation admits travelling wave solutions ( ηc, Φc) = ( η(x − ct ), Φc(x − ct )) for a variety of wave speeds.", "clean_statement": "1. under what collision, end-state, and speed conditions does (1.3) have a\n positive heteroclinic profile;\n2. when is that planar profile stable to perturbations depending on transverse\n spatial variables; and\n3. how should an Evans function be constructed for the resulting kinetic\n spectral problem?", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record is visibly spliced across a page/presenter boundary. It begins with", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[51]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:Under what conditions do there exist profiles for this equation? Are they stable to multidimensional pertur-bations? What does the Evans function look like. (As a first step, one might start with considering discrete velocity models.) \\n\\nY. Li: (Stability of travelling waves of the full water wave problem near the critical case) \\n\\nSetup:Consider the water wave equations \\n\\nηt = GΦΦt + Φ2 \\n\\n> x\\n\\n+ 2 ηxΦxGΦ − (GΦ) 2\\n\\n1 + η2\\n\\n> x\\n\\n+ gη = 0 where ̂ G(k) = k tanh( k). The equation admits travelling wave solutions ( ηc, Φc) = ( η(x − ct ), Φc(x − ct )) for a variety of wave speeds.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0052", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The corrupted source is recovered as P. Szmolyan's kinetic/Boltzmann question, with the subsequent Y. Li water-wave material excluded. For a concrete two-dimensional four-velocity Broadwell collision, the report proves an exact strictly positive logistic shock profile with rational end states, all collision-invariant Rankine–Hugoniot constraints, and one-dimensional unstable-to-stable dynamics on the invariant leaf. For every real transverse frequency and Re(lambda)>0, a detailed-balance symmetrizer proves a 3-left-decaying plus 1-right-decaying spatial splitting and yields an exact 4-by-4 Evans determinant and rank-one far-field dispersion formula. The three collision invariants put lambda=0 in essential spectrum while the localized translation mode also lies there, so a naive simple Evans zero at the origin is invalid without a weighted, integrated, or renormalized construction.\n\nCandidate contribution (explicit_model; novelty confidence low): For the specified four-velocity Broadwell collision at speed c=1/2, the rational end states f_-=(1/2,2,1,1) and f_+=(3/8,49/24,7/8,7/8) are joined by the exact positive profile f=f_-+(-2,2/3,-2,-2)/[16(1+exp(-z/3))]; its transverse spectral ODE has a rigorously proved 3+1 decaying split throughout Re(lambda)>0, an explicit Evans determinant, and a three-branch essential threshold containing translation at (lambda,xi)=(0,0).", "research_difficulty_suggested": "L4" }, { "id": 20002441, "problem_number": "AIM-PDES-0053", "title": "Near-critical solitary gravity waves: recovered status and a two-term KP-II dispersion audit", "statement": "Open Problem:How we determine the stability of such waves near the critical speed c∗.\n\nS. Benzoni-Gavage: (Shocks with Capillarity)\n\nSetup:Consider the equations\n\n∂t% + ∇ · (%u) = 0\n\n∂tu + ( u · ∇ )u + ∇P = ∇\n\n(\n\nK(%)∇% + 12\n\n∂K (%)\n\n∂% |∇ %|2\n\n)\n\nwhere u ∈ R3 is the velocity, % > 0 is the density of the fluid, and P = P (%) is the pressure. The equations admits a planar profile ( %, u). It can be shown that the spectrum of the resulting linear operator obtained by linearization of the equations about the planar profile must lie on the imaginary axis.", "original_statement": "Open Problem:How we determine the stability of such waves near the critical speed c∗.\n\nS. Benzoni-Gavage: (Shocks with Capillarity) \n\nSetup:Consider the equations \n\n∂t% + ∇ · (%u) = 0 \n\n∂tu + ( u · ∇ )u + ∇P = ∇\n\n(\n\nK(%)∇% + 12\n\n∂K (%)\n\n∂% |∇ %|2\n\n)\n\nwhere u ∈ R3 is the velocity, % > 0 is the density of the fluid, and P = P (%) is the pressure. The equations admits a planar profile ( %, u). It can be shown that the spectrum of the resulting linear operator obtained by linearization of the equations about the planar profile must lie on the imaginary axis.", "clean_statement": "Open Problem:How we determine the stability of such waves near the critical speed c∗.\n\nS. Benzoni-Gavage: (Shocks with Capillarity)\n\nSetup:Consider the equations\n\n∂t% + ∇ · (%u) = 0\n\n∂tu + ( u · ∇ )u + ∇P = ∇\n\n(\n\nK(%)∇% + 12\n\n∂K (%)\n\n∂% |∇ %|2\n\n)\n\nwhere u ∈ R3 is the velocity, % > 0 is the density of the fluid, and P = P (%) is the pressure. The equations admits a planar profile ( %, u). It can be shown that the spectrum of the resulting linear operator obtained by linearization of the equations about the planar profile must lie on the imaginary axis.", "statement_status": "exact", "statement_verification": "The exact canonical record is preserved in `input.json`, but it crosses a presenter boundary. The original AIM PDF puts the following material consecutively:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[52]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:How we determine the stability of such waves near the critical speed c∗.\\n\\nS. Benzoni-Gavage: (Shocks with Capillarity) \\n\\nSetup:Consider the equations \\n\\n∂t% + ∇ · (%u) = 0 \\n\\n∂tu + ( u · ∇ )u + ∇P = ∇\\n\\n(\\n\\nK(%)∇% + 12\\n\\n∂K (%)\\n\\n∂% |∇ %|2\\n\\n)\\n\\nwhere u ∈ R3 is the velocity, % > 0 is the density of the fluid, and P = P (%) is the pressure. The equations admits a planar profile ( %, u). It can be shown that the spectrum of the resulting linear operator obtained by linearization of the equations about the planar profile must lie on the imaginary axis.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0053", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The OCR-crossed AIM record is Y. Li's finite-depth pure-gravity solitary-water-wave stability question; the following Benzoni-Gavage capillarity-shock setup is a separate problem. Unit depth gives the critical speed c_*=sqrt(g). Pego-Sun (2016) resolved longitudinal linear asymptotic and spectral stability for small solitary waves, and Rousset-Sun (2025) resolved transverse linear asymptotic and spectral stability for small line solitary waves, while nonlinear stability of the full system remains open. Independently, a uniform exact-symbol calculation in the right-going KP sector k=epsilon K, ell=epsilon^2 L and c=sqrt(g)/(sqrt(1-epsilon^2)) yields (omega-ck)/sqrt(g)=epsilon^3[L^2/(2K)-K/2-K^3/6]+epsilon^5[-L^4/(8K^3)-KL^2/4+19K^5/360-3K/8]+O(epsilon^7), fixing the KP-II sign and the leading neutral curve L^2=K^2+K^4/3.\n\nCandidate contribution (asymptotic_lemma; novelty confidence low): On every compact right-going KP sector 00\\) and velocity \\(u\\in\\mathbb R^3\\). The dimensionally consistent recovery is \\[ \\rho_t+\\nabla\\!\\cdot(\\rho u)=0, \\qquad u_t+(u\\!\\cdot\\!\\nabla)u+\\nabla P(\\rho) =\\nabla\\!\\left(K(\\rho)\\Delta\\rho+ \\frac12K'(\\rho)|\\nabla\\rho|^2\\right). \\tag{EK} \\] The source's glyph before \\(\\rho\\) was extracted as \\(\\nabla\\), but that would add a vector to a scalar inside the outer gradient. The standard Euler--Korteweg formula in Benzoni-Gavage--Danchin--Descombes--Jamet (2005, equation (1.3)) confirms that the intended glyph is \\(\\Delta\\). The workshop calls \\(P\\) “pressure”; in this velocity formula...", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[53]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:Are there any eigenvalues on the imaginary axis? Is the planar profile spectrally stable or unstable? Does linear information tell the whole story? That is, can one prove full nonlinear stability from the linear information. ARCC Workshop on Multidimensional Stability of Waves and Patterns 3\\n\\nR. Pego: (Stability of Toda Lattice Solitons)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0054", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The corrupted workshop statement is recovered as the Euler--Korteweg capillarity system with K(rho) Delta rho. Translation invariance rigorously forces lambda=0 to be an L2 point eigenvalue of the zero-transverse-frequency fiber when the profile derivative lies in the operator domain. The one-dimensional heteroclinic nonlinear question was answered by a constrained-Hamiltonian orbital-stability theorem in 2005, but this neither classifies the nonzero-transverse-frequency Evans roots nor proves multidimensional nonlinear stability. Later transverse instability results apply to homoclinic solitary waves, not the heteroclinic phase boundary.\n\nCandidate contribution (reduction; novelty confidence low): Candidate contribution: a sign- and scope-audited decomposition of the AIM question into a forced zero-frequency point eigenvalue, a solved one-dimensional heteroclinic orbital-stability subproblem, and a remaining nonzero-transverse/multidimensional problem, together with a Fredholm zero-chain diagnostic explicitly restricted to the separate solitary-wave branch.", "research_difficulty_suggested": "L4" }, { "id": 20002443, "problem_number": "AIM-PDES-0055", "title": "Toda one-soliton spectrum, sharp weight threshold, and Floquet convention audit", "statement": "Open Problem:The Toda lattice ¨qk = exp( qk+1 − qk) − exp( qk − qk−1)has the well known solutions\n\nqk = log\n\n( cosh( β(k − ct + 1)) cosh ( β(k − ct ))\n\n), c = sinh ββ\n\nWhat are their spectral stability properties?\n\nM. Williams: (Two Interacting Shocks in 1D)\n\nSetup:Consider the inviscid conservation law\n\n∂tu + ∂xf (u) = 0 (A) and the viscous regularization\n\n∂tuε + ∂xf (uε) = ε∆uε (B) where for simplicity we may take u ∈ R2 and x ∈ R. Take a two-shock solution to the viscous conservation law of the form below:\n\nFigure 1: The Two Shock Setup", "original_statement": "Open Problem:The Toda lattice ¨qk = exp( qk+1 − qk) − exp( qk − qk−1)has the well known solutions \n\nqk = log \n\n( cosh( β(k − ct + 1)) cosh ( β(k − ct )) \n\n), c = sinh ββ\n\nWhat are their spectral stability properties? \n\nM. Williams: (Two Interacting Shocks in 1D) \n\nSetup:Consider the inviscid conservation law \n\n∂tu + ∂xf (u) = 0 (A) and the viscous regularization \n\n∂tuε + ∂xf (uε) = ε∆uε (B) where for simplicity we may take u ∈ R2 and x ∈ R. Take a two-shock solution to the viscous conservation law of the form below: \n\nFigure 1: The Two Shock Setup", "clean_statement": "Open Problem:The Toda lattice ¨qk = exp( qk+1 − qk) − exp( qk − qk−1)has the well known solutions\n\nqk = log\n\n( cosh( β(k − ct + 1)) cosh ( β(k − ct ))\n\n), c = sinh ββ\n\nWhat are their spectral stability properties?\n\nM. Williams: (Two Interacting Shocks in 1D)\n\nSetup:Consider the inviscid conservation law\n\n∂tu + ∂xf (u) = 0 (A) and the viscous regularization\n\n∂tuε + ∂xf (uε) = ε∆uε (B) where for simplicity we may take u ∈ R2 and x ∈ R. Take a two-shock solution to the viscous conservation law of the form below:\n\nFigure 1: The Two Shock Setup", "statement_status": "exact", "statement_verification": "The canonical JSON has lost a fraction bar and then runs into the next speaker's problem. Inspection of page 3 of the original AIM workshop PDF recovers R. Pego's item as follows:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[54]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:The Toda lattice ¨qk = exp( qk+1 − qk) − exp( qk − qk−1)has the well known solutions \\n\\nqk = log \\n\\n( cosh( β(k − ct + 1)) cosh ( β(k − ct )) \\n\\n), c = sinh ββ\\n\\nWhat are their spectral stability properties? \\n\\nM. Williams: (Two Interacting Shocks in 1D) \\n\\nSetup:Consider the inviscid conservation law \\n\\n∂tu + ∂xf (u) = 0 (A) and the viscous regularization \\n\\n∂tuε + ∂xf (uε) = ε∆uε (B) where for simplicity we may take u ∈ R2 and x ∈ R. Take a two-shock solution to the viscous conservation law of the form below: \\n\\nFigure 1: The Two Shock Setup\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0055", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The OCR-damaged AIM item is R. Pego's Toda one-soliton question with q_k=log(cosh(beta(k-ct+1))/cosh(beta(k-ct))) and c=sinh(beta)/beta; the appended M. Williams shock text is a separate problem. Mizumachi and Pego solved the historical question by proving weighted asymptotic stability and the complete traveling-Floquet spectrum. Independently in the AIM sign convention, this attempt proves that the weighted far-field curves are lambda_±=c(-a+i xi)±2 sinh((-a+i xi)/2), with spectral abscissa -ca+2 sinh(a/2), and that the sharp condition 0 0. Can we show that strong shocks for the p-system are stable or unstable? What role, if any, does symmetrizability play in the onset of instability?\n\nK. Promislow: (MultiD front dynamics in optical resonance)\n\nA model for pattern formation in an optical cavity near resonance is given by\n\niϕ t − 12 ∆lϕ + |ϕ|2ϕ + ( i − a)ϕ − γϕ ∗ = 0 where ∆ l = ∂2\n\n> x\n\n+ l−1∂2\n\n> y\n\nand a, γ are real constants. The equation admits a planar front solution. Consider a perturbation of the planar front. What is the Evans function for the perturbed front? What are the dynamics of the front?\n\nB. Sandstede: (Dynamical interpretation of the roots of the D(λ) embedded in the absolute spectra)\n\nSetup:In many circumstances, the Evans function can be extended into branch points of the linear dispersion relation. It is natural to ask what role, if any, roots of the Evans function at branch points play for the temporal dynamics of the linear or nonlinear evolution. It has been shown by Murata [Tohoku Math J 37 (1985) 151-195] that temporal decay rates of scalar linear heat equations depend very much on the presence of these roots.", "original_statement": "Open Problem:Can we construct explicit solutions to the viscous problem (B), say of the form uε(t, x, t/ε, x/ε ), which converge in a reasonable sense to solutions u(t, x ) of the inviscid problem (A) as ε → 0? Note: this question has been answered abstractly (c.f. the C.I.M.E. notes by A. Bressan ), but our goal is to explicitly resolve the dynamics at the corner where Ul and Ur meet in the xt -plane. ARCC Workshop on Multidimensional Stability of Waves and Patterns 4\n\nJ. Humphreys: (Stability for strong viscous shocks of the p-system) \n\nConsider the p-system \n\nvt − ux = 0 \n\nut + p(v)x = ( b(v)u)x\n\nwhere p′ < 0, p′′ > 0. Can we show that strong shocks for the p-system are stable or unstable? What role, if any, does symmetrizability play in the onset of instability? \n\nK. Promislow: (MultiD front dynamics in optical resonance) \n\nA model for pattern formation in an optical cavity near resonance is given by \n\niϕ t − 12 ∆lϕ + |ϕ|2ϕ + ( i − a)ϕ − γϕ ∗ = 0 where ∆ l = ∂2 \n\n> x\n\n+ l−1∂2 \n\n> y\n\nand a, γ are real constants. The equation admits a planar front solution. Consider a perturbation of the planar front. What is the Evans function for the perturbed front? What are the dynamics of the front? \n\nB. Sandstede: (Dynamical interpretation of the roots of the D(λ) embedded in the absolute spectra) \n\nSetup:In many circumstances, the Evans function can be extended into branch points of the linear dispersion relation. It is natural to ask what role, if any, roots of the Evans function at branch points play for the temporal dynamics of the linear or nonlinear evolution. It has been shown by Murata [Tohoku Math J 37 (1985) 151-195] that temporal decay rates of scalar linear heat equations depend very much on the presence of these roots.", "clean_statement": "Open Problem:Can we construct explicit solutions to the viscous problem (B), say of the form uε(t, x, t/ε, x/ε ), which converge in a reasonable sense to solutions u(t, x ) of the inviscid problem (A) as ε → 0? Note: this question has been answered abstractly (c.f. the C.I.M.E. notes by A. Bressan ), but our goal is to explicitly resolve the dynamics at the corner where Ul and Ur meet in the xt -plane. ARCC Workshop on Multidimensional Stability of Waves and Patterns 4\n\nJ. Humphreys: (Stability for strong viscous shocks of the p-system)\n\nConsider the p-system\n\nvt − ux = 0\n\nut + p(v)x = ( b(v)u)x\n\nwhere p′ < 0, p′′ > 0. Can we show that strong shocks for the p-system are stable or unstable? What role, if any, does symmetrizability play in the onset of instability?\n\nK. Promislow: (MultiD front dynamics in optical resonance)\n\nA model for pattern formation in an optical cavity near resonance is given by\n\niϕ t − 12 ∆lϕ + |ϕ|2ϕ + ( i − a)ϕ − γϕ ∗ = 0 where ∆ l = ∂2\n\n> x\n\n+ l−1∂2\n\n> y\n\nand a, γ are real constants. The equation admits a planar front solution. Consider a perturbation of the planar front. What is the Evans function for the perturbed front? What are the dynamics of the front?\n\nB. Sandstede: (Dynamical interpretation of the roots of the D(λ) embedded in the absolute spectra)\n\nSetup:In many circumstances, the Evans function can be extended into branch points of the linear dispersion relation. It is natural to ask what role, if any, roots of the Evans function at branch points play for the temporal dynamics of the linear or nonlinear evolution. It has been shown by Murata [Tohoku Math J 37 (1985) 151-195] that temporal decay rates of scalar linear heat equations depend very much on the presence of these roots.", "statement_status": "exact", "statement_verification": "The source is M. Williams, “Two Interacting Shocks in 1D,” in the AIM workshop list *Open Problems in Multidimensional Stability of Waves and Patterns* (2005). The setup is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[55]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:Can we construct explicit solutions to the viscous problem (B), say of the form uε(t, x, t/ε, x/ε ), which converge in a reasonable sense to solutions u(t, x ) of the inviscid problem (A) as ε → 0? Note: this question has been answered abstractly (c.f. the C.I.M.E. notes by A. Bressan ), but our goal is to explicitly resolve the dynamics at the corner where Ul and Ur meet in the xt -plane. ARCC Workshop on Multidimensional Stability of Waves and Patterns 4\\n\\nJ. Humphreys: (Stability for strong viscous shocks of the p-system) \\n\\nConsider the p-system \\n\\nvt − ux = 0 \\n\\nut + p(v)x = ( b(v)u)x\\n\\nwhere p′ < 0, p′′ > 0. Can we show that strong shocks for the p-system are stable or unstable? What role, if any, does symmetrizability play in the onset of instability? \\n\\nK. Promislow: (MultiD front dynamics in optical resonance) \\n\\nA model for pattern formation in an optical cavity near resonance is given by \\n\\niϕ t − 12 ∆lϕ + |ϕ|2ϕ + ( i − a)ϕ − γϕ ∗ = 0 where ∆ l = ∂2 \\n\\n> x\\n\\n+ l−1∂2 \\n\\n> y\\n\\nand a, γ are real constants. The equation admits a planar front solution. Consider a perturbation of the planar front. What is the Evans function for the perturbed front? What are the dynamics of the front? \\n\\nB. Sandstede: (Dynamical interpretation of the roots of the D(λ) embedded in the absolute spectra) \\n\\nSetup:In many circumstances, the Evans function can be extended into branch points of the linear dispersion relation. It is natural to ask what role, if any, roots of the Evans function at branch points play for the temporal dynamics of the linear or nonlinear evolution. It has been shown by Murata [Tohoku Math J 37 (1985) 151-195] that temporal decay rates of scalar linear heat equations depend very much on the presence of these roots.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0056", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every constant affine conjugate of the diagonal 2x2 Burgers system, two scalar viscous Lax profiles in distinct fields give an exact global fast-variable corner layer. When the shock speeds satisfy s_1/ and lambda_4=-/, derives the corresponding Evans derivative formulas, proves that the constant conservation adjoint is inadmissible because its pairing with U' is zero, and gives the exact equal-diffusion check lambda(k)=-k^2.\n\nCandidate contribution (diagnostic equivalence; novelty confidence low): Candidate novel synthesis: combine the fourth-order Evans translation-branch formula with the exact boundary-jump test excluding the cubic system's constant conservation adjoint, the exact delta=1 spectral-shift benchmark, and a domain-robust Bloch-Evans criterion that distinguishes a secondary spectral long scale from nonlinear biscale mode mixing.", "research_difficulty_suggested": "L4" }, { "id": 20002447, "problem_number": "AIM-PDES-0059", "title": "Arbitrary-period stability exchange on the local Zaitsev branch", "statement": "Open Problem:What is the stability of the KP-I soliton solution with periodic structure?\n\nJ. Albert: (Benjamin-Ono type equations)\n\nSetup:Consider the Benjamin-Ono equation\n\nut + uu x − K ux = 0 where ̂ Ku(k) = |k|ˆu(k). This equation supports travelling waves of the form φ(z) = 41+ z2. The resulting eigenvalue problem is (cv + Kv − φv )z = λv (A) where v is the perturbation v:= u − φ.", "original_statement": "Open Problem:What is the stability of the KP-I soliton solution with periodic structure? \n\nJ. Albert: (Benjamin-Ono type equations) \n\nSetup:Consider the Benjamin-Ono equation \n\nut + uu x − K ux = 0 where ̂ Ku(k) = |k|ˆu(k). This equation supports travelling waves of the form φ(z) = 41+ z2. The resulting eigenvalue problem is (cv + Kv − φv )z = λv (A) where v is the perturbation v:= u − φ.", "clean_statement": "Open Problem:What is the stability of the KP-I soliton solution with periodic structure?\n\nJ. Albert: (Benjamin-Ono type equations)\n\nSetup:Consider the Benjamin-Ono equation\n\nut + uu x − K ux = 0 where ̂ Ku(k) = |k|ˆu(k). This equation supports travelling waves of the form φ(z) = 41+ z2. The resulting eigenvalue problem is (cv + Kv − φv )z = λv (A) where v is the perturbation v:= u − φ.", "statement_status": "exact", "statement_verification": "The canonical JSON record is damaged by a record-boundary error. The AIM PDF puts the following material under M. Haragus, “Stability for KP-I profiles with periodic structure”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 11\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[58]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:What is the stability of the KP-I soliton solution with periodic structure? \\n\\nJ. Albert: (Benjamin-Ono type equations) \\n\\nSetup:Consider the Benjamin-Ono equation \\n\\nut + uu x − K ux = 0 where ̂ Ku(k) = |k|ˆu(k). This equation supports travelling waves of the form φ(z) = 41+ z2. The resulting eigenvalue problem is (cv + Kv − φv )z = λv (A) where v is the perturbation v:= u − φ.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0059", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The corrupted record is recovered as the stability problem for the x-localized, transversely periodic Zaitsev family, excluding the following Benjamin-Ono entry. Yamazaki's local orbital-stability theorem and the Rousset-Tzvetkov line-soliton threshold, conjugated by the exact KP scaling, imply that on every transverse cylinder of period L the critical line speed is 8 pi/(sqrt(3) L), the sufficiently small Zaitsev branch is orbitally stable, and every nonzero wave on that stable branch has the same speed as an orbitally unstable line soliton. The exact speed excess is kappa a^2(2+a^2)/(sqrt(3)(1-a^2)), with kappa=2 pi/L. The full branch toward the stable planar lump remains unresolved.\n\nCandidate contribution (corollary; novelty confidence low): For every fixed transverse period L and every sufficiently small nonzero Zaitsev parameter a, a stable y-periodic Zaitsev soliton and an unstable y-independent KdV line soliton coexist on the same cylinder at exactly the same traveling speed; the report gives the exact speed gap and explicit arbitrary-period bifurcation-kernel modes.", "research_difficulty_suggested": "L4" }, { "id": 20002448, "problem_number": "AIM-PDES-0060", "title": "The differentiated Benjamin--Ono spectral problem", "statement": "Open Problem:What is the spectrum of (A) and how does it relate to the spectrum of\n\ncv + Kv − φv = λv\n\nwhich is well known. A possible approach would be to treat the eigenvalue equation as a two dimensional problem.\n\nK. Zumbrun: (Strong Shocks in one or multiD of viscous conservation laws)\n\nThe Setup:Consider the system of viscous conservation laws in one or several space dimensions d,\n\nut +\n\n> d\n\n∑\n\n> j=1\n\nf j (u)xj =\n\n> d\n\n∑\n\n> j,k =1\n\n(Bjk (u)uxk )xj\n\nwhere u ∈ Rn, f is a smooth mapping from Rn to Rn and Bj,k is a smooth mapping from Rn to Rn×n.The stability of weak shocks in one spatial dimension has been investigated recently by H. Freisthler & P. Szmolyan (Spectral stability of small shock waves, I, Arch. Rat. Mech. Analysis, 164, (2002), 287-309) and R. Plaza & K. Zumbrun (An Evans function approach to spectral stability of small-amplitude shock profiles, J. Discrete and Continuous Dynamical Systems 10, 2004, no. 4, 885-924). The weak shock assumption induces a fast-slow structure (with the small parameter being the strength of the shock) that can be exploited in the calculations. It is not immediately apparent that there is fast-slow structure hidden somewhere for strong shock profiles that can exploited.", "original_statement": "Open Problem:What is the spectrum of (A) and how does it relate to the spectrum of \n\ncv + Kv − φv = λv \n\nwhich is well known. A possible approach would be to treat the eigenvalue equation as a two dimensional problem. \n\nK. Zumbrun: (Strong Shocks in one or multiD of viscous conservation laws) \n\nThe Setup:Consider the system of viscous conservation laws in one or several space dimensions d,\n\nut +\n\n> d\n\n∑\n\n> j=1\n\nf j (u)xj =\n\n> d\n\n∑\n\n> j,k =1\n\n(Bjk (u)uxk )xj\n\nwhere u ∈ Rn, f is a smooth mapping from Rn to Rn and Bj,k is a smooth mapping from Rn to Rn×n.The stability of weak shocks in one spatial dimension has been investigated recently by H. Freisthler & P. Szmolyan (Spectral stability of small shock waves, I, Arch. Rat. Mech. Analysis, 164, (2002), 287-309) and R. Plaza & K. Zumbrun (An Evans function approach to spectral stability of small-amplitude shock profiles, J. Discrete and Continuous Dynamical Systems 10, 2004, no. 4, 885-924). The weak shock assumption induces a fast-slow structure (with the small parameter being the strength of the shock) that can be exploited in the calculations. It is not immediately apparent that there is fast-slow structure hidden somewhere for strong shock profiles that can exploited.", "clean_statement": "Open Problem:What is the spectrum of (A) and how does it relate to the spectrum of\n\ncv + Kv − φv = λv\n\nwhich is well known. A possible approach would be to treat the eigenvalue equation as a two dimensional problem.\n\nK. Zumbrun: (Strong Shocks in one or multiD of viscous conservation laws)\n\nThe Setup:Consider the system of viscous conservation laws in one or several space dimensions d,\n\nut +\n\n> d\n\n∑\n\n> j=1\n\nf j (u)xj =\n\n> d\n\n∑\n\n> j,k =1\n\n(Bjk (u)uxk )xj\n\nwhere u ∈ Rn, f is a smooth mapping from Rn to Rn and Bj,k is a smooth mapping from Rn to Rn×n.The stability of weak shocks in one spatial dimension has been investigated recently by H. Freisthler & P. Szmolyan (Spectral stability of small shock waves, I, Arch. Rat. Mech. Analysis, 164, (2002), 287-309) and R. Plaza & K. Zumbrun (An Evans function approach to spectral stability of small-amplitude shock profiles, J. Discrete and Continuous Dynamical Systems 10, 2004, no. 4, 885-924). The weak shock assumption induces a fast-slow structure (with the small parameter being the strength of the shock) that can be exploited in the calculations. It is not immediately apparent that there is fast-slow structure hidden somewhere for strong shock profiles that can exploited.", "statement_status": "exact", "statement_verification": "The canonical JSON record crosses a presenter boundary. Inspection of the original AIM workshop PDF and the preceding canonical record AIM-PDES-0059 recovers the following J. Albert entry.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 12\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[59]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:What is the spectrum of (A) and how does it relate to the spectrum of \\n\\ncv + Kv − φv = λv \\n\\nwhich is well known. A possible approach would be to treat the eigenvalue equation as a two dimensional problem. \\n\\nK. Zumbrun: (Strong Shocks in one or multiD of viscous conservation laws) \\n\\nThe Setup:Consider the system of viscous conservation laws in one or several space dimensions d,\\n\\nut +\\n\\n> d\\n\\n∑\\n\\n> j=1\\n\\nf j (u)xj =\\n\\n> d\\n\\n∑\\n\\n> j,k =1\\n\\n(Bjk (u)uxk )xj\\n\\nwhere u ∈ Rn, f is a smooth mapping from Rn to Rn and Bj,k is a smooth mapping from Rn to Rn×n.The stability of weak shocks in one spatial dimension has been investigated recently by H. Freisthler & P. Szmolyan (Spectral stability of small shock waves, I, Arch. Rat. Mech. Analysis, 164, (2002), 287-309) and R. Plaza & K. Zumbrun (An Evans function approach to spectral stability of small-amplitude shock profiles, J. Discrete and Continuous Dynamical Systems 10, 2004, no. 4, 885-924). The weak shock assumption induces a fast-slow structure (with the small parameter being the strength of the shock) that can be exploited in the calculations. It is not immediately apparent that there is fast-slow structure hidden somewhere for strong shock profiles that can exploited.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0060", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the maximal unweighted L2 realization A_c = partial_x(|D|+c-phi_c) with domain H2 and phi_c(x)=4c/(1+c^2x^2), the Fredholm essential spectrum and full spectral set are both iR. A Fourier zero-frequency reduction and constrained positivity of the known selfadjoint Hessian exclude every off-axis eigenvalue. The embedded zero has geometric multiplicity one and an exact two-dimensional generalized root space spanned by the translation and speed modes; because zero is embedded, no Riesz algebraic multiplicity is claimed. The selfadjoint and differentiated problems are related exactly by Robin and tangentially differentiated Robin boundary conditions for the harmonic extension.\n\nCandidate contribution (lemma; novelty confidence low): The candidate contribution is a short cancellation-aware spectral correspondence: every off-axis eigenfunction automatically has an L2 antiderivative by an explicit Fourier multiplier formula, the known one-negative-direction spectrum of L_c then forces spectral stability, and a 4pi Fredholm obstruction proves that the translation/speed Jordan chain cannot be extended beyond length two.", "research_difficulty_suggested": "L4" }, { "id": 20002449, "problem_number": "AIM-PDES-0061", "title": "Strong viscous shocks and an amplitude-independent scalar transverse certificate", "statement": "Open Problem:Determine the stability properties of strong shocks for viscous conservation laws in one or several space dimensions. ARCC Workshop on Multidimensional Stability of Waves and Patterns 6\n\nM. Wechselberger: (Loss of hyperbolicity, algebraic decay and the Evans function)\n\nSetup:For certain values of p, the generalized KdV equation\n\nut − uxxx + up x = 0 admits standing waves φ(x) which decay exponentially to zero as |x| → ∞. However, the eigenfunctions of the operator obtained by linearizing about φ need not decay exponentially and depending on the power p\n\n(say p=5) may only decay algebraically as |x| → ∞.", "original_statement": "Open Problem:Determine the stability properties of strong shocks for viscous conservation laws in one or several space dimensions. ARCC Workshop on Multidimensional Stability of Waves and Patterns 6\n\nM. Wechselberger: (Loss of hyperbolicity, algebraic decay and the Evans function) \n\nSetup:For certain values of p, the generalized KdV equation \n\nut − uxxx + up x = 0 admits standing waves φ(x) which decay exponentially to zero as |x| → ∞. However, the eigenfunctions of the operator obtained by linearizing about φ need not decay exponentially and depending on the power p\n\n(say p=5) may only decay algebraically as |x| → ∞.", "clean_statement": "Open Problem:Determine the stability properties of strong shocks for viscous conservation laws in one or several space dimensions. ARCC Workshop on Multidimensional Stability of Waves and Patterns 6\n\nM. Wechselberger: (Loss of hyperbolicity, algebraic decay and the Evans function)\n\nSetup:For certain values of p, the generalized KdV equation\n\nut − uxxx + up x = 0 admits standing waves φ(x) which decay exponentially to zero as |x| → ∞. However, the eigenfunctions of the operator obtained by linearizing about φ need not decay exponentially and depending on the power p\n\n(say p=5) may only decay algebraically as |x| → ∞.", "statement_status": "exact", "statement_verification": "The canonical record is preserved verbatim in input.json, but its problem field straddles two speakers. The original seven-page AIM document was checked directly. On PDF page 5 (printed workshop page 5) the entry is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 13\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[60]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:Determine the stability properties of strong shocks for viscous conservation laws in one or several space dimensions. ARCC Workshop on Multidimensional Stability of Waves and Patterns 6\\n\\nM. Wechselberger: (Loss of hyperbolicity, algebraic decay and the Evans function) \\n\\nSetup:For certain values of p, the generalized KdV equation \\n\\nut − uxxx + up x = 0 admits standing waves φ(x) which decay exponentially to zero as |x| → ∞. However, the eigenfunctions of the operator obtained by linearizing about φ need not decay exponentially and depending on the power p\\n\\n(say p=5) may only decay algebraically as |x| → ∞.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0061", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The broad strong-shock problem is model-dependent and only partially solved: rigorous arbitrary-amplitude results exist for some one-dimensional physical gas models, while multidimensional MHD supplies transverse instability regimes. For a proved strong-amplitude subclass, consider a scalar convex planar shock with constant block-diagonal diffusion beta in the normal direction and C in the transverse block, C positive definite, and affine transverse flux of slope c. Every transverse Fourier fiber is exactly L_eta=L_0-eta^T C eta-i(c dot eta). An integrated energy identity proves that the only point eigenvalue on or to the right of Re lambda=-eta^T C eta is the phase branch lambda=-eta^T C eta-i(c dot eta), with eigenfunction U prime and no L2 Jordan vector. A far-field root count separately places the standard Fredholm essential-spectrum right boundary on the same paraboloid, so the unweighted zero-frequency spectrum still touches the origin.\n\nCandidate contribution (spectral certificate; novelty confidence low): For arbitrary-strength scalar convex shocks with constant block-diagonal normal/transverse diffusion and affine transverse flux, the combined integrated-energy, exact Evans-fiber shift, and embedded-zero audit gives a testable amplitude-independent certificate: all fibers collapse under mu=lambda+eta^T C eta+i(c dot eta), the only point eigenvalue with Re lambda at least -eta^T C eta is the phase root, and no L2 Jordan vector occurs there.", "research_difficulty_suggested": "L4" }, { "id": 20002450, "problem_number": "AIM-PDES-0062", "title": "Evans functions at an algebraic threshold", "statement": "Open Problem:Can one construct an Evans function for this problem? If so, how does it behave? Note that for this problem the absolute spectrum touches the essential spectrum so there is no exponential dichotomy that we can exploit.\n\nP. Howard: (Combination structures in viscous conservation laws)", "original_statement": "Open Problem:Can one construct an Evans function for this problem? If so, how does it behave? Note that for this problem the absolute spectrum touches the essential spectrum so there is no exponential dichotomy that we can exploit. \n\nP. Howard: (Combination structures in viscous conservation laws)", "clean_statement": "Open Problem:Can one construct an Evans function for this problem? If so, how does it behave? Note that for this problem the absolute spectrum touches the essential spectrum so there is no exponential dichotomy that we can exploit.\n\nP. Howard: (Combination structures in viscous conservation laws)", "statement_status": "exact", "statement_verification": "The canonical record is an OCR extraction from the AIM workshop list “Stability criteria for multi-dimensional waves and patterns.” As stored, it reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 14\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[61]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:Can one construct an Evans function for this problem? If so, how does it behave? Note that for this problem the absolute spectrum touches the essential spectrum so there is no exponential dichotomy that we can exploit. \\n\\nP. Howard: (Combination structures in viscous conservation laws)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0062", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The printed stationary p=5 gKdV formulation is inconsistent: its first integral rules out a nonzero smooth zero-background homoclinic, while the standard repaired co-moving gKdV linearization has exponentially decaying L2 eigenfunctions even on essential spectrum and at lambda zero. For a genuinely algebraic rank-one threshold channel, an explicit cutoff renormalization is proved; in the critical inverse-linear case its exponent also gives the sharp L2-versus-resonance boundary.\n\nCandidate contribution (obstruction_and_renormalization_theorem; novelty confidence low): For a rank-one threshold channel z'=(-mu-a(1+s)^(-beta)+r)z with integrable analytic remainder, the classical cutoff determinant has the exact removable factor exp(sum a_sigma H_beta_sigma(R)); for beta=1 the renormalized threshold mode is in L2 exactly when Re(a_sigma)>1/2. Combined with the far-field cubic, this proves that the standard co-moving gKdV reading cannot realize the prompt's merely algebraic L2 eigenfunction.", "research_difficulty_suggested": "L4" }, { "id": 20002451, "problem_number": "AIM-PDES-0063", "title": "Exact state chamber and integrable interaction for a separating thin-film shock pair", "statement": "Open Problem:The thin film equation\n\nut + ( u2 − u3)x = −ε(u3uxxx )x\n\nsupports solutions that are comprised of a Lax shock moving to the left with speed s1 and an undercom-pressive shock moving to the right with speed s2 as in Figure 2. Can one prove the stability or instability of such a structure?\n\nFigure 2: The Combination Structure\n\nC. Jones: (Stability of energized states of NLS)\n\nConsider\n\niu t = urr = n − 1\n\nr ur + f (|u|)u + ωu\n\nIs there really no spectrum off of iR? What is the meaning of this for stability? Does the linear information give us full stability? Opposite Krein signature eigenvalues for NLS. ARCC Workshop on Multidimensional Stability of Waves and Patterns 7\n\nT. Kapitula: (Transient dynamics for vortex patterns)\n\nSetup: Start with the Gross-Pitaevskii equation\n\niq t + ∆ q ± | q|2q = V (x)q\n\nNumerical investigations show that if one starts with a \"nice\" vortex pattern with a lot of symmetry, it evolves to a \"ugly\" vortex pattern with little symmetry or structure, and then evolves further to to a \"nice\" pattern again.", "original_statement": "Open Problem:The thin film equation \n\nut + ( u2 − u3)x = −ε(u3uxxx )x\n\nsupports solutions that are comprised of a Lax shock moving to the left with speed s1 and an undercom-pressive shock moving to the right with speed s2 as in Figure 2. Can one prove the stability or instability of such a structure? \n\nFigure 2: The Combination Structure \n\nC. Jones: (Stability of energized states of NLS) \n\nConsider \n\niu t = urr = n − 1\n\nr ur + f (|u|)u + ωu \n\nIs there really no spectrum off of iR? What is the meaning of this for stability? Does the linear information give us full stability? Opposite Krein signature eigenvalues for NLS. ARCC Workshop on Multidimensional Stability of Waves and Patterns 7\n\nT. Kapitula: (Transient dynamics for vortex patterns) \n\nSetup: Start with the Gross-Pitaevskii equation \n\niq t + ∆ q ± | q|2q = V (x)q\n\nNumerical investigations show that if one starts with a \"nice\" vortex pattern with a lot of symmetry, it evolves to a \"ugly\" vortex pattern with little symmetry or structure, and then evolves further to to a \"nice\" pattern again.", "clean_statement": "Open Problem:The thin film equation\n\nut + ( u2 − u3)x = −ε(u3uxxx )x\n\nsupports solutions that are comprised of a Lax shock moving to the left with speed s1 and an undercom-pressive shock moving to the right with speed s2 as in Figure 2. Can one prove the stability or instability of such a structure?\n\nFigure 2: The Combination Structure\n\nC. Jones: (Stability of energized states of NLS)\n\nConsider\n\niu t = urr = n − 1\n\nr ur + f (|u|)u + ωu\n\nIs there really no spectrum off of iR? What is the meaning of this for stability? Does the linear information give us full stability? Opposite Krein signature eigenvalues for NLS. ARCC Workshop on Multidimensional Stability of Waves and Patterns 7\n\nT. Kapitula: (Transient dynamics for vortex patterns)\n\nSetup: Start with the Gross-Pitaevskii equation\n\niq t + ∆ q ± | q|2q = V (x)q\n\nNumerical investigations show that if one starts with a \"nice\" vortex pattern with a lot of symmetry, it evolves to a \"ugly\" vortex pattern with little symmetry or structure, and then evolves further to to a \"nice\" pattern again.", "statement_status": "exact", "statement_verification": "The canonical record is an OCR extraction from the AIM workshop report *Open Problems in Multidimensional Stability of Waves and Patterns*. Inspection of the original PDF confirms that P. Howard's problem is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 15\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[62]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:The thin film equation \\n\\nut + ( u2 − u3)x = −ε(u3uxxx )x\\n\\nsupports solutions that are comprised of a Lax shock moving to the left with speed s1 and an undercom-pressive shock moving to the right with speed s2 as in Figure 2. Can one prove the stability or instability of such a structure? \\n\\nFigure 2: The Combination Structure \\n\\nC. Jones: (Stability of energized states of NLS) \\n\\nConsider \\n\\niu t = urr = n − 1\\n\\nr ur + f (|u|)u + ωu \\n\\nIs there really no spectrum off of iR? What is the meaning of this for stability? Does the linear information give us full stability? Opposite Krein signature eigenvalues for NLS. ARCC Workshop on Multidimensional Stability of Waves and Patterns 7\\n\\nT. Kapitula: (Transient dynamics for vortex patterns) \\n\\nSetup: Start with the Gross-Pitaevskii equation \\n\\niq t + ∆ q ± | q|2q = V (x)q\\n\\nNumerical investigations show that if one starts with a \\\"nice\\\" vortex pattern with a lot of symmetry, it evolves to a \\\"ugly\\\" vortex pattern with little symmetry or structure, and then evolves further to to a \\\"nice\\\" pattern again.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0063", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For physically ordered states 0 0\n\nf (s) ds > 0 for some t > 0", "original_statement": "Problem: By what mechanism does this happen and how can one capture the general dynamics? \n\nH. Warchall: (Stability of Encapsulated-Vortex Solutions) \n\nSetup:Consider the equations \n\nJu t = ∆ u + g(u) (NLS) \n\nutt = ∆ u + g(u) (NLKG) where u ∈ RN +1 → RM, g: RM → RM continuous satisfying g(y) = h(|y|2)ˆ y, with h: [0, ∞) → R and ˆy ≡ y/ |y|. J is an invertible M × M skew symmetric matrix. Consider standing-wave solutions to (NLS) and (NLKG) of the form \n\nu(x, t ) = eμKt ˆψ(ˆ x)w(r)with μ ∈ R a constant, and K a real skew-symmetric M × M matrix with \n\nK = J−1 for (NLS) \n\nK2 = −I for (NLKG) where w: [0, ∞) → R and ˆψ: SN −1 → SM −1. Here ˆψ is a unit vector valued eigenfunction of the Laplacian on the sphere SN −1 ⊂ RN, with ∆S ˆψ = −l(l + N − 2) ˆψ\n\nWe remark that the possible values of l are limited by the dimension M of range space (see J. Iaia & H.A. Warchall, Encapsulated-vortex solutions to equivariant wave equations: existence, SIAM J. Math. Anal., \n\n30 (1999), 118-139). If u satisfies (NLS) or (NLKG) then the spatial profile w satisfies \n\nw′′ + N − 1\n\nr w′ − l(l + N − 2) \n\nr2 w + f (w) = 0 where f (y) = g(y) + ωy with \n\nω =\n\n{ μ for (NLS) \n\nμ2 for (NLKG) Note: Traveling waves are generated by Galilean or Lorentz boosts. The idea is to generalize, e.g. standing wave solutions u(x, t ) = eiωt eimθ w(r) of −iu t − ∆u = g(u) whose stability was analysed by R.L. Pego & H.A.Warchall (Spectrally stable encapsulated vortices for nonlinear Schr¨ odinger equations, J.Nonlinear Sci., 12 (2002), 347-394). Under appropriate conditions on f there exist smooth exponentially localized solutions w to the profile ODE. One essentially needs \n\nf ′(0) < 0 F (t) = \n\n∫ t\n\n> 0\n\nf (s) ds > 0 for some t > 0", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is split across an extraction boundary. The exact record in `input.json` starts with the final Kapitula question and then incorrectly includes the beginning of H. Warchall's next, unrelated entry. Comparing the preceding canonical record, AIM-PDES-0063, with page 7 of the original AIM workshop PDF recovers the Kapitula entry as follows (typographical spacing is normalized, but the sign is not changed):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 16\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[63]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem: By what mechanism does this happen and how can one capture the general dynamics? \\n\\nH. Warchall: (Stability of Encapsulated-Vortex Solutions) \\n\\nSetup:Consider the equations \\n\\nJu t = ∆ u + g(u) (NLS) \\n\\nutt = ∆ u + g(u) (NLKG) where u ∈ RN +1 → RM, g: RM → RM continuous satisfying g(y) = h(|y|2)ˆ y, with h: [0, ∞) → R and ˆy ≡ y/ |y|. J is an invertible M × M skew symmetric matrix. Consider standing-wave solutions to (NLS) and (NLKG) of the form \\n\\nu(x, t ) = eμKt ˆψ(ˆ x)w(r)with μ ∈ R a constant, and K a real skew-symmetric M × M matrix with \\n\\nK = J−1 for (NLS) \\n\\nK2 = −I for (NLKG) where w: [0, ∞) → R and ˆψ: SN −1 → SM −1. Here ˆψ is a unit vector valued eigenfunction of the Laplacian on the sphere SN −1 ⊂ RN, with ∆S ˆψ = −l(l + N − 2) ˆψ\\n\\nWe remark that the possible values of l are limited by the dimension M of range space (see J. Iaia & H.A. Warchall, Encapsulated-vortex solutions to equivariant wave equations: existence, SIAM J. Math. Anal., \\n\\n30 (1999), 118-139). If u satisfies (NLS) or (NLKG) then the spatial profile w satisfies \\n\\nw′′ + N − 1\\n\\nr w′ − l(l + N − 2) \\n\\nr2 w + f (w) = 0 where f (y) = g(y) + ωy with \\n\\nω =\\n\\n{ μ for (NLS) \\n\\nμ2 for (NLKG) Note: Traveling waves are generated by Galilean or Lorentz boosts. The idea is to generalize, e.g. standing wave solutions u(x, t ) = eiωt eimθ w(r) of −iu t − ∆u = g(u) whose stability was analysed by R.L. Pego & H.A.Warchall (Spectrally stable encapsulated vortices for nonlinear Schr¨ odinger equations, J.Nonlinear Sci., 12 (2002), 347-394). Under appropriate conditions on f there exist smooth exponentially localized solutions w to the profile ODE. One essentially needs \\n\\nf ′(0) < 0 F (t) = \\n\\n∫ t\\n\\n> 0\\n\\nf (s) ds > 0 for some t > 0\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0064", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After repairing the OCR/extraction boundary, the Kapitula problem is shown to be underdetermined but admits a rigorous diagnostic theorem: exact projective isotropy is preserved by any unique equivariant Gross-Pitaevskii flow, so literal exact symmetry cannot disappear and return; a detuned Hermitian two-mode normal form produces an exact nice-to-defect-to-nice excursion with return time pi/Omega and maximum defect kappa^2/Omega^2; and a Duhamel bound controls degradation of that return by unresolved modes. Charge-preserving quotient distances separate true field return from rotation, gauge, vortex permutation, or density-only resemblance, while an energy-norm bound distinguishes coherent beating from nonnormal Hamiltonian amplification.\n\nCandidate contribution (diagnostic theorem; novelty confidence low): Candidate novelty is the combined, falsifiable protocol comprising projective-isotropy persistence, three charge- and symmetry-aware return distances, the detuned two-mode predictions T=pi/Omega and D_max=kappa^2/Omega^2, and an explicit leakage-integral bound; it also proves that inequivalent exact-symmetry modes cannot have the proposed linear coupling." }, { "id": 20002453, "problem_number": "AIM-PDES-0065", "title": "An angular-isotypic stability filter for higher-dimensional encapsulated vortices", "statement": "Open Problem:Under what conditions are the encapsulated-vortex solutions u(x, t ) = eμKt ˆψ(ˆ x)w(r) of (NLS) or (NLKG) with N ≥ 3 stable or unstable?", "original_statement": "Open Problem:Under what conditions are the encapsulated-vortex solutions u(x, t ) = eμKt ˆψ(ˆ x)w(r) of (NLS) or (NLKG) with N ≥ 3 stable or unstable?", "clean_statement": "Open Problem:Under what conditions are the encapsulated-vortex solutions u(x, t ) = eμKt ˆψ(ˆ x)w(r) of (NLS) or (NLKG) with N ≥ 3 stable or unstable?", "statement_status": "exact", "statement_verification": "The canonical record contains only H. Warchall's final open question. Its setup was lost across the record boundary and appears at the end of AIM-PDES-0064. Page 7 of the original AIM workshop PDF verifies the following reconstruction. Let \\[ u:\\mathbb R^{N+1}\\longrightarrow\\mathbb R^M \\] solve either \\[ J u_t=\\Delta u+g(u) \\quad\\text{(NLS)},\\qquad u_{tt}=\\Delta u+g(u) \\quad\\text{(NLKG)}, \\tag{1.1} \\] where \\(J\\) is an invertible skew-symmetric \\(M\\times M\\) matrix and \\[ g(y)=h(|y|^2)\\widehat y,\\qquad \\widehat y=y/|y|. \\tag{1.2} \\] The source assumes \\(g\\) continuous. It considers \\[ u(x,t)=e^{\\mu Kt}\\widehat\\psi(\\widehat x)w(r),\\qquad r=|x|,\\quad \\widehat x=x/r, \\tag{1.3} \\] where \\(K=J^{-1}\\) for NLS, \\(K^2=-I\\) for NLKG, and \\(K\\) is real skew-symmetric. The angular map \\[ \\widehat\\psi:S^{N-1}\\to S^{M-1},\\qquad \\Delta_S\\widehat\\psi=-l(l+N-2)\\widehat\\psi \\tag{1.4} \\] is a unit-vect...", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stability criteria for multi-dimensional waves and patterns\nSection: \nSource item: 17\nSource URL: https://aimath.org/WWN/multidimwaves/multidimwaves.pdf\nCanonical location: aim-pdes-notes.json notes[64]\nCanonical tag: open problem\nOriginal extracted problem text (JSON string): \"Open Problem:Under what conditions are the encapsulated-vortex solutions u(x, t ) = eμKt ˆψ(ˆ x)w(r) of (NLS) or (NLKG) with N ≥ 3 stable or unstable?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/multidimwaves/multidimwaves.pdf", "tags": [ "aim", "AIM-PDES-0065", "aim-domain:pdes", "aim-workshop:multidimwaves", "aim-source-tag:open-problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The full Warchall setup is recovered and an internal NLS sign inconsistency is identified: the literal equation, K=J^{-1}, and e^{+mu Kt} require omega=-mu in the profile equation, while NLKG consistently gives omega=mu^2. Under explicit C1 radial-gradient, localization, far-field-gap, and combined-equivariance hypotheses, the co-rotating Hessian L=-Delta-Dg(Phi)-omega is proved coercive on every combined spatial/target isotypic sector whose angular floor Lambda_sigma exceeds C=ess sup r^2 max(lambda_max(Dg(Phi)-Dg(0)),0). This yields conserved positive quadratic energy and excludes finite-energy exponential growth for both the first-order NLS and gyroscopic NLKG linearizations; when Lambda_sigma tends to infinity, only finitely many low combined angular sectors remain for detailed instability analysis.\n\nCandidate contribution (sectorwise coercivity theorem; novelty confidence low): Candidate novelty is the explicit, symmetry-correct criterion Lambda_sigma>C that reduces possible NLS or NLKG instability of Warchall's vector spherical-eigenmap vortices to low combined spatial/target isotypic sectors, while retaining angular multiplicities and allowing Dg(w psi-hat) to couple scalar spherical-harmonic degrees.", "research_difficulty_suggested": "L4" }, { "id": 20002454, "problem_number": "AIM-PDES-0066", "title": "Branch-aware well balancing and a resonance obstruction for oscillatory forcing", "statement": "A.1 Amadori, Debora\n\nI would be interested in the numerical approximation of the scalar equation\n\nut + f (u)x = 1\n\nε h\n\n(x\n\nε\n\n)\n\n(1) with h continuous and 1-periodic, with zero average; f ∈ C1(R), u · f ′(u) > 0, f (u) → +∞\n\nas |u| → +∞, and with 1-periodic initial data. I can contribute some recent results on the pointwise behavior of the oscillation, as\n\nε → 0, of the solutions to (1); this analysis could be of help in the numerical approximation of (1). More generally, I am interested in the analysis of problems that exibit resonant phe-nomena.", "original_statement": "A.1 Amadori, Debora \n\nI would be interested in the numerical approximation of the scalar equation \n\nut + f (u)x = 1\n\nε h\n\n(x\n\nε\n\n)\n\n(1) with h continuous and 1-periodic, with zero average; f ∈ C1(R), u · f ′(u) > 0, f (u) → +∞\n\nas |u| → +∞, and with 1-periodic initial data. I can contribute some recent results on the pointwise behavior of the oscillation, as \n\nε → 0, of the solutions to (1); this analysis could be of help in the numerical approximation of (1). More generally, I am interested in the analysis of problems that exibit resonant phe-nomena.", "clean_statement": "A.1 Amadori, Debora\n\nI would be interested in the numerical approximation of the scalar equation\n\nut + f (u)x = 1\n\nε h\n\n(x\n\nε\n\n)\n\n(1) with h continuous and 1-periodic, with zero average; f ∈ C1(R), u · f ′(u) > 0, f (u) → +∞\n\nas |u| → +∞, and with 1-periodic initial data. I can contribute some recent results on the pointwise behavior of the oscillation, as\n\nε → 0, of the solutions to (1); this analysis could be of help in the numerical approximation of (1). More generally, I am interested in the analysis of problems that exibit resonant phe-nomena.", "statement_status": "exact", "statement_verification": "The original AIM PDF, *Stiff Sources and Numerical Methods for Conservation Laws*, version dated April 1, 2005, contains the following participant contribution by Debora Amadori: \\[ u_t+\\partial_x f(u)=\\frac1\\varepsilon h\\!\\left(\\frac{x}{\\varepsilon}\\right). \\tag{1.1} \\] It asks for numerical approximation when \\(h\\) is continuous, 1-periodic, and has zero average; \\(f\\in C^1(\\mathbb R)\\), \\(u f'(u)>0\\), and \\(f(u)\\to+\\infty\\) as \\(|u|\\to\\infty\\); and the initial data are 1-periodic. It mentions recent results on the pointwise behavior of the oscillations as \\(\\varepsilon\\to0\\), and a broader interest in resonance.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stiff sources and numerical methods for conservation laws\nSection: \nSource item: A.1\nSource URL: https://aimath.org/WWN/balancelaws/balancelaws.pdf\nCanonical location: aim-pdes-notes.json notes[65]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.1 Amadori, Debora \\n\\nI would be interested in the numerical approximation of the scalar equation \\n\\nut + f (u)x = 1\\n\\nε h\\n\\n(x\\n\\nε\\n\\n)\\n\\n(1) with h continuous and 1-periodic, with zero average; f ∈ C1(R), u · f ′(u) > 0, f (u) → +∞\\n\\nas |u| → +∞, and with 1-periodic initial data. I can contribute some recent results on the pointwise behavior of the oscillation, as \\n\\nε → 0, of the solutions to (1); this analysis could be of help in the numerical approximation of (1). More generally, I am interested in the analysis of problems that exibit resonant phe-nomena.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/balancelaws/balancelaws.pdf", "tags": [ "aim", "AIM-PDES-0066", "aim-domain:pdes", "aim-workshop:balancelaws", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After verifying the equation as u_t+(f(u))_x=epsilon^{-1}h(x/epsilon), this attempt uses a normalized periodic primitive H'=h to rewrite it exactly as u_t+partial_x(f(u)-H(x/epsilon))=0. On the unit torus, where N=epsilon^{-1} must be an integer for a nonconstant forcing of minimal period one, it proves that every branch-separated periodic BV stationary entropy solution is one of the two pure inverse-flux branches phi_+(C+H(Nx)) or phi_-(C+H(Nx)); their oscillation is independent of N and they converge only weakly, not strongly, to their mean. It gives an exact equilibrium-cell-average finite-volume reconstruction that preserves either family on an arbitrarily underresolved mesh. At branch contact it proves an obstruction: for f(u)=u^2/2 and H(y)=1-cos(2 pi y), there are 2^N Lipschitz stationary entropy states, including binomial(N,N/2) distinct mean-zero states for even N, so mass and invariant flux cannot select the microscopic branch pattern.\n\nCandidate contribution (well-balanced reconstruction and obstruction; novelty confidence low): Candidate novel synthesis: exact cell-average reconstruction preserves both separated inverse-flux equilibrium branches without resolving epsilon, while an explicit critical quadratic example has binomial(N,N/2) different mean-zero equilibria with the same invariant flux, proving that an additional microscopic branch variable is indispensable at resonance." }, { "id": 20002455, "problem_number": "AIM-PDES-0067", "title": "A viscous duality benchmark for particle methods with point sources", "statement": "A.2 Chertock, Alina\n\nOne of the projects I am working on is aimed at developing a hybrid finite-volume-particle method for systems of conservation or balance laws coupled with a nonlinear trans-port equation. Solutions of such systems are usually nonsmooth: they may contain shocks, rarefaction waves and contact discontinuities. The presence of a stiff source term adds another level of complexity to the model. Such problems arise, for instance, in modeling transport of a passive pollutant in shallow water (in which case the source of the pollutant may be even a point-source modeled by a delta-function) or compressible inviscid reacting gases (in which case the source is usually stiff, since the reaction is fast and the time scale as-sociated with the reaction is much smaller than that associated with the fluid advection). It is well known that numerical dissipation present in shock-capturing methods may not only seriously degrade the quality of the computed solution but may also lead to nonphysical states, which in turn, may completely destroy the numerical solution. The core idea of the new method is to use a finite-volume method to numerically integrate a system of conservation (balance) laws and a particle method to solve transport equations coupled with the system. This way the specific advantages of each scheme are utilized at the right place. Particle methods applied to transport equations, can ameliorate most of the problems posed by the presence of numerical viscosity since particles provide a non-dissipative approximation of the convection. In these methods, the solution is sought in the form of a linear combination of the delta-functions, whose positions and coefficients represent locations and weights of the particles, respectively. The locations and weights of the particles are then evolved in time according to a system of ODEs, obtained from the weak formulation of the transport equations. We have successfully implemented the finite-volume-particle method for the above as well as some other (inviscid) models and we plan to apply the method to more realistic advection-diffusion-reaction models. This extension is not straightforward since it will in-volve the treatment of diffusion and reaction terms that may appear in the equation. There was a number of attempts in the past to use particle methods for approximating solutions of convection-diffusion models, but each of them has its own drawback, associated primarily with the reconstruction of the point values of the computed solution from its particle distri-bution. Most of known recovering procedures, suitable for smooth functions, typically fail 4\n\nto produce reasonable results in the nonsmooth case. In the purely convective case, we were able to overcome this difficulty using the concept of the dual equation, but it is not clear whether this approach can be generalized for the viscous case. Another (theoretical) difficulty one may encounter while implementing the finite-volume-particle method is related to the lack of smoothness in the right-hand side of the ODE system that describes the evolution of particles and their weights. While the existence of a general-ized solution is guaranteed by the theory of Filippov, the uniqueness can only be obtained via a proper regularization. The presence of a point-source term makes the problem even more challenging and a theoretical justification of the particle method in this case is a wide open problem.", "original_statement": "A.2 Chertock, Alina \n\nOne of the projects I am working on is aimed at developing a hybrid finite-volume-particle method for systems of conservation or balance laws coupled with a nonlinear trans-port equation. Solutions of such systems are usually nonsmooth: they may contain shocks, rarefaction waves and contact discontinuities. The presence of a stiff source term adds another level of complexity to the model. Such problems arise, for instance, in modeling transport of a passive pollutant in shallow water (in which case the source of the pollutant may be even a point-source modeled by a delta-function) or compressible inviscid reacting gases (in which case the source is usually stiff, since the reaction is fast and the time scale as-sociated with the reaction is much smaller than that associated with the fluid advection). It is well known that numerical dissipation present in shock-capturing methods may not only seriously degrade the quality of the computed solution but may also lead to nonphysical states, which in turn, may completely destroy the numerical solution. The core idea of the new method is to use a finite-volume method to numerically integrate a system of conservation (balance) laws and a particle method to solve transport equations coupled with the system. This way the specific advantages of each scheme are utilized at the right place. Particle methods applied to transport equations, can ameliorate most of the problems posed by the presence of numerical viscosity since particles provide a non-dissipative approximation of the convection. In these methods, the solution is sought in the form of a linear combination of the delta-functions, whose positions and coefficients represent locations and weights of the particles, respectively. The locations and weights of the particles are then evolved in time according to a system of ODEs, obtained from the weak formulation of the transport equations. We have successfully implemented the finite-volume-particle method for the above as well as some other (inviscid) models and we plan to apply the method to more realistic advection-diffusion-reaction models. This extension is not straightforward since it will in-volve the treatment of diffusion and reaction terms that may appear in the equation. There was a number of attempts in the past to use particle methods for approximating solutions of convection-diffusion models, but each of them has its own drawback, associated primarily with the reconstruction of the point values of the computed solution from its particle distri-bution. Most of known recovering procedures, suitable for smooth functions, typically fail 4\n\nto produce reasonable results in the nonsmooth case. In the purely convective case, we were able to overcome this difficulty using the concept of the dual equation, but it is not clear whether this approach can be generalized for the viscous case. Another (theoretical) difficulty one may encounter while implementing the finite-volume-particle method is related to the lack of smoothness in the right-hand side of the ODE system that describes the evolution of particles and their weights. While the existence of a general-ized solution is guaranteed by the theory of Filippov, the uniqueness can only be obtained via a proper regularization. The presence of a point-source term makes the problem even more challenging and a theoretical justification of the particle method in this case is a wide open problem.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is a participant contribution in the AIM workshop report *Stiff Sources and Numerical Methods for Conservation Laws*, version dated April 1, 2005. Its exact stored problem field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stiff sources and numerical methods for conservation laws\nSection: \nSource item: A.2\nSource URL: https://aimath.org/WWN/balancelaws/balancelaws.pdf\nCanonical location: aim-pdes-notes.json notes[66]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.2 Chertock, Alina \\n\\nOne of the projects I am working on is aimed at developing a hybrid finite-volume-particle method for systems of conservation or balance laws coupled with a nonlinear trans-port equation. Solutions of such systems are usually nonsmooth: they may contain shocks, rarefaction waves and contact discontinuities. The presence of a stiff source term adds another level of complexity to the model. Such problems arise, for instance, in modeling transport of a passive pollutant in shallow water (in which case the source of the pollutant may be even a point-source modeled by a delta-function) or compressible inviscid reacting gases (in which case the source is usually stiff, since the reaction is fast and the time scale as-sociated with the reaction is much smaller than that associated with the fluid advection). It is well known that numerical dissipation present in shock-capturing methods may not only seriously degrade the quality of the computed solution but may also lead to nonphysical states, which in turn, may completely destroy the numerical solution. The core idea of the new method is to use a finite-volume method to numerically integrate a system of conservation (balance) laws and a particle method to solve transport equations coupled with the system. This way the specific advantages of each scheme are utilized at the right place. Particle methods applied to transport equations, can ameliorate most of the problems posed by the presence of numerical viscosity since particles provide a non-dissipative approximation of the convection. In these methods, the solution is sought in the form of a linear combination of the delta-functions, whose positions and coefficients represent locations and weights of the particles, respectively. The locations and weights of the particles are then evolved in time according to a system of ODEs, obtained from the weak formulation of the transport equations. We have successfully implemented the finite-volume-particle method for the above as well as some other (inviscid) models and we plan to apply the method to more realistic advection-diffusion-reaction models. This extension is not straightforward since it will in-volve the treatment of diffusion and reaction terms that may appear in the equation. There was a number of attempts in the past to use particle methods for approximating solutions of convection-diffusion models, but each of them has its own drawback, associated primarily with the reconstruction of the point values of the computed solution from its particle distri-bution. Most of known recovering procedures, suitable for smooth functions, typically fail 4\\n\\nto produce reasonable results in the nonsmooth case. In the purely convective case, we were able to overcome this difficulty using the concept of the dual equation, but it is not clear whether this approach can be generalized for the viscous case. Another (theoretical) difficulty one may encounter while implementing the finite-volume-particle method is related to the lack of smoothness in the right-hand side of the ODE system that describes the evolution of particles and their weights. While the existence of a general-ized solution is guaranteed by the theory of Filippov, the uniqueness can only be obtained via a proper regularization. The presence of a point-source term makes the problem even more challenging and a theoretical justification of the particle method in this case is a wide open problem.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/balancelaws/balancelaws.pdf", "tags": [ "aim", "AIM-PDES-0067", "aim-domain:pdes", "aim-workshop:balancelaws", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a prescribed smooth advection field and bounded linear reaction on the torus, the advection-diffusion equation with finite-measure initial data and a weak-star measurable measure source, including a continuously injected moving Dirac source, is stable in H^{-s} by a viscosity-uniform backward-duality estimate. Regularizing particle data at scale epsilon gives an explicit O(epsilon^alpha) error in that norm. Complementary proved obstructions show that a finite sum of raw moving Dirac masses cannot exactly represent positive viscosity and that weak or W_1 particle convergence alone cannot justify pointwise reconstruction without a particle/kernel scale relation.\n\nCandidate contribution (stability_theorem_and_obstruction; novelty confidence low): Under s>d/2+alpha, prescribed b and r in L1_t W^{s,infinity}_x, and measure initial/source data, the viscous dual equation yields an H^{-s} particle/blob error bounded by the particle data error plus C epsilon^alpha times total source mass, uniformly for a moving Dirac source and uniformly in viscosity. At the same time, raw ODE-evolved Dirac sums cannot solve a positive-viscosity equation with an order-zero source, and point reconstruction requires an explicit scale condition such as W_1=o(epsilon^{d+1})." }, { "id": 20002456, "problem_number": "AIM-PDES-0068", "title": "Uniform-time matrix-viscosity estimates at a fully damped equilibrium", "statement": "A.3 Christoforou, Cleopatra\n\nMy area of interest is the theory of hyperbolic conservation laws. My current research is an application of the method of vanishing viscosity: The aim is to construct solutions of hyperbolic systems of balance laws with dissipative source terms\n\nut + ( f (u)) x + g(u) = 0 as limits of solutions of parabolic systems\n\nu≤t + ( f (u≤)) x + g(u≤) = ≤u ≤xx\n\nwith viscosity ≤ tending to zero. The analysis of the vanishing viscosity method of Bianchini and Bressan [BiB] is extended to this class of systems. Because of the presence of the dissipative source terms, supplementary Lyapunov functionals are constructed and additional techniques are employed to those already devised in [BiB]. Moreover, an exponential decay of the total variation of vanishing viscosity approximations is established. I am interested in applying these techniques to other systems of conservation laws. A very challenging question would be the case of systems with physical viscosity. Finally, I am very interested in increasing my knowledge in mathematical biology. It is a challenging, fast-growing area that together with numerical methods will introduce me to new tools and assist me to improve my ability to work with physical problems.", "original_statement": "A.3 Christoforou, Cleopatra \n\nMy area of interest is the theory of hyperbolic conservation laws. My current research is an application of the method of vanishing viscosity: The aim is to construct solutions of hyperbolic systems of balance laws with dissipative source terms \n\nut + ( f (u)) x + g(u) = 0 as limits of solutions of parabolic systems \n\nu≤t + ( f (u≤)) x + g(u≤) = ≤u ≤xx \n\nwith viscosity ≤ tending to zero. The analysis of the vanishing viscosity method of Bianchini and Bressan [BiB] is extended to this class of systems. Because of the presence of the dissipative source terms, supplementary Lyapunov functionals are constructed and additional techniques are employed to those already devised in [BiB]. Moreover, an exponential decay of the total variation of vanishing viscosity approximations is established. I am interested in applying these techniques to other systems of conservation laws. A very challenging question would be the case of systems with physical viscosity. Finally, I am very interested in increasing my knowledge in mathematical biology. It is a challenging, fast-growing area that together with numerical methods will introduce me to new tools and assist me to improve my ability to work with physical problems.", "clean_statement": "A.3 Christoforou, Cleopatra\n\nMy area of interest is the theory of hyperbolic conservation laws. My current research is an application of the method of vanishing viscosity: The aim is to construct solutions of hyperbolic systems of balance laws with dissipative source terms\n\nut + ( f (u)) x + g(u) = 0 as limits of solutions of parabolic systems\n\nu≤t + ( f (u≤)) x + g(u≤) = ≤u ≤xx\n\nwith viscosity ≤ tending to zero. The analysis of the vanishing viscosity method of Bianchini and Bressan [BiB] is extended to this class of systems. Because of the presence of the dissipative source terms, supplementary Lyapunov functionals are constructed and additional techniques are employed to those already devised in [BiB]. Moreover, an exponential decay of the total variation of vanishing viscosity approximations is established. I am interested in applying these techniques to other systems of conservation laws. A very challenging question would be the case of systems with physical viscosity. Finally, I am very interested in increasing my knowledge in mathematical biology. It is a challenging, fast-growing area that together with numerical methods will introduce me to new tools and assist me to improve my ability to work with physical problems.", "statement_status": "exact", "statement_verification": "This record is item A.3 in the AIM workshop document *Stiff Sources and Numerical Methods for Conservation Laws*. The canonical JSON has an OCR substitution in which every occurrence of the Greek letter epsilon became the symbol `<=`. The original PDF was checked directly. Its equations are", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stiff sources and numerical methods for conservation laws\nSection: \nSource item: A.3\nSource URL: https://aimath.org/WWN/balancelaws/balancelaws.pdf\nCanonical location: aim-pdes-notes.json notes[67]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.3 Christoforou, Cleopatra \\n\\nMy area of interest is the theory of hyperbolic conservation laws. My current research is an application of the method of vanishing viscosity: The aim is to construct solutions of hyperbolic systems of balance laws with dissipative source terms \\n\\nut + ( f (u)) x + g(u) = 0 as limits of solutions of parabolic systems \\n\\nu≤t + ( f (u≤)) x + g(u≤) = ≤u ≤xx \\n\\nwith viscosity ≤ tending to zero. The analysis of the vanishing viscosity method of Bianchini and Bressan [BiB] is extended to this class of systems. Because of the presence of the dissipative source terms, supplementary Lyapunov functionals are constructed and additional techniques are employed to those already devised in [BiB]. Moreover, an exponential decay of the total variation of vanishing viscosity approximations is established. I am interested in applying these techniques to other systems of conservation laws. A very challenging question would be the case of systems with physical viscosity. Finally, I am very interested in increasing my knowledge in mathematical biology. It is a challenging, fast-growing area that together with numerical methods will introduce me to new tools and assist me to improve my ability to work with physical problems.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/balancelaws/balancelaws.pdf", "tags": [ "aim", "AIM-PDES-0068", "aim-domain:pdes", "aim-workshop:balancelaws", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After correcting the OCR symbol to epsilon and interpreting physical viscosity as epsilon times partial_x(B(u)u_x), this attempt proves a uniform-in-time O(epsilon) vanishing-viscosity estimate for periodic constant-coefficient fully damped symmetrizable balance laws. If SA is symmetric, SD+D^T S is at least 2 gamma S, and SB+B^T S is nonnegative, then the estimate permits singular B and does not require AB=BA. It also proves an O(epsilon) Lipschitz comparison between two entropy-compatible viscosity matrices, and supplies explicit anti-diffusion and uncoupled-mode examples showing why entropy positivity and full damping or Kawashima coupling are essential.\n\nCandidate contribution (uniform matrix-viscosity comparison estimate; novelty confidence low): Candidate novel diagnostic lemma: for a periodic constant-coefficient symmetrizable balance-law linearization with direct source damping rate gamma, two entropy-compatible viscosity matrices B1 and B2, even singular and noncommuting with the transport matrix, produce solutions whose S-norm difference is at most epsilon K_S(B1-B2) t exp(-gamma t) times the initial H^2 seminorm, hence at most epsilon K_S(B1-B2)/(e gamma) times that seminorm uniformly for all time." }, { "id": 20002457, "problem_number": "AIM-PDES-0069", "title": "Energy/AP audit and a kinetic-cone counterexample for a stiff two-moment model", "statement": "A.4 Despres, Bruno\n\nMy interests go in two directions. The first one is not directly related to the subject of the Workshop. It is about the theory of convergence of Finite Volume schemes by means of the old consistency+stability-implies-convergence approach. It helps to get a linear approach of the convegrence of these methods and it is possible to prove quite accurate results even for a non linear scalar con-servation law. The other one is directly related to the subject of Workshop. With a colleague (Christophe Buet) we are currently working on the numerical approximation of the model problem { ut + 1\n\n> ε\n\nvx = 0,vt + 1\n\n> ε\n\nf (u, v ) = − σ\n\n> ε2\n\nv.\n\nOur idea is that we absolutely need an implicite solver for time step requirements of the diffusion limit. Various stability criteria are possible. If the model is the moment modeliza-tion of some kinetic equation, then |v|\n\n> u\n\n≤ 1 is natural. We have develop a 1D solver for this 5\n\nsystem: the solver is implicit (we only solve a linear system with ad-hoc frozen coefficients), stable ( |v|\n\n> u\n\n≤ 1), and has the correct diffusion limit. I will be happy to compare this approach with others.", "original_statement": "A.4 Despres, Bruno \n\nMy interests go in two directions. The first one is not directly related to the subject of the Workshop. It is about the theory of convergence of Finite Volume schemes by means of the old consistency+stability-implies-convergence approach. It helps to get a linear approach of the convegrence of these methods and it is possible to prove quite accurate results even for a non linear scalar con-servation law. The other one is directly related to the subject of Workshop. With a colleague (Christophe Buet) we are currently working on the numerical approximation of the model problem { ut + 1 \n\n> ε\n\nvx = 0,vt + 1 \n\n> ε\n\nf (u, v ) = − σ \n\n> ε2\n\nv. \n\nOur idea is that we absolutely need an implicite solver for time step requirements of the diffusion limit. Various stability criteria are possible. If the model is the moment modeliza-tion of some kinetic equation, then |v| \n\n> u\n\n≤ 1 is natural. We have develop a 1D solver for this 5\n\nsystem: the solver is implicit (we only solve a linear system with ad-hoc frozen coefficients), stable ( |v| \n\n> u\n\n≤ 1), and has the correct diffusion limit. I will be happy to compare this approach with others.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The extracted second equation lacks a spatial derivative. That literal reading cannot yield the claimed diffusion limit because relaxation of $v$ supplies no spatial constitutive law. The balance-law context and the later two-moment radiation equations of Buet and Després both put a spatial derivative on the pressure/second-moment flux. We therefore analyze the explicitly labeled reconstruction", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stiff sources and numerical methods for conservation laws\nSection: \nSource item: A.4\nSource URL: https://aimath.org/WWN/balancelaws/balancelaws.pdf\nCanonical location: aim-pdes-notes.json notes[68]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.4 Despres, Bruno \\n\\nMy interests go in two directions. The first one is not directly related to the subject of the Workshop. It is about the theory of convergence of Finite Volume schemes by means of the old consistency+stability-implies-convergence approach. It helps to get a linear approach of the convegrence of these methods and it is possible to prove quite accurate results even for a non linear scalar con-servation law. The other one is directly related to the subject of Workshop. With a colleague (Christophe Buet) we are currently working on the numerical approximation of the model problem { ut + 1 \\n\\n> ε\\n\\nvx = 0,vt + 1 \\n\\n> ε\\n\\nf (u, v ) = − σ \\n\\n> ε2\\n\\nv. \\n\\nOur idea is that we absolutely need an implicite solver for time step requirements of the diffusion limit. Various stability criteria are possible. If the model is the moment modeliza-tion of some kinetic equation, then |v| \\n\\n> u\\n\\n≤ 1 is natural. We have develop a 1D solver for this 5\\n\\nsystem: the solver is implicit (we only solve a linear system with ad-hoc frozen coefficients), stable ( |v| \\n\\n> u\\n\\n≤ 1), and has the correct diffusion limit. I will be happy to compare this approach with others.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/balancelaws/balancelaws.pdf", "tags": [ "aim", "AIM-PDES-0069", "aim-domain:pdes", "aim-workshop:balancelaws", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The participant note is not an open problem and its second flux derivative is corrupted or omitted in extraction. For the explicitly reconstructed linear telegraph specialization, a fully implicit staggered scheme has an exact epsilon-uniform discrete L2 energy identity. At fixed grid and time step, if u_epsilon^0 converges to a fixed-grid limit and v_epsilon^0 is uniformly bounded, the scheme converges to the standard three-point backward-Euler diffusion sequence starting from that limit; no v_epsilon^0=O(epsilon) well-preparedness is needed. This does not imply the kinetic cone: an exact four-cell periodic example shows that the collocated centered implicit analogue sends cone-admissible data to a state with u < |v|.\n\nCandidate contribution (proposition; novelty confidence low): Candidate audit proposition: the specified staggered backward-Euler telegraph discretization has the exact epsilon-uniform energy law and standard three-point AP diffusion limit proved in the artifacts, while the exact four-cell rational update supplied there disproves cone preservation for its collocated centered analogue.", "research_difficulty_suggested": "L2" }, { "id": 20002458, "problem_number": "AIM-PDES-0070", "title": "Well-balanced reconstruction for hyperbolic chemotaxis", "statement": "A.5 Filbet, Francis\n\nApproximation of Hyperbolic Models for Chemosensitive Movement Numerical methods with different orders of accuracy are proposed to approximate hy-perbolic models for chemosensitive movements. On the one hand, first and second order well-balanced finite volume schemes are presented. This approach provides an exact con-servation of the steady state solutions. On the other hand, a high order finite difference weighted essentially non-oscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme in order to exactly preserve steady states and to retain high order accuracy. Numerical simulations are performed to verify accuracy and the well-balanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments. Keywords: chemotaxis, hyperbolic systems, finite volume methods, finite difference methods, WENO schemes, well-balanced schemes. This work is in collaboration with Chi-Wang Shu, Brown University", "original_statement": "A.5 Filbet, Francis \n\nApproximation of Hyperbolic Models for Chemosensitive Movement Numerical methods with different orders of accuracy are proposed to approximate hy-perbolic models for chemosensitive movements. On the one hand, first and second order well-balanced finite volume schemes are presented. This approach provides an exact con-servation of the steady state solutions. On the other hand, a high order finite difference weighted essentially non-oscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme in order to exactly preserve steady states and to retain high order accuracy. Numerical simulations are performed to verify accuracy and the well-balanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments. Keywords: chemotaxis, hyperbolic systems, finite volume methods, finite difference methods, WENO schemes, well-balanced schemes. This work is in collaboration with Chi-Wang Shu, Brown University", "clean_statement": "A.5 Filbet, Francis\n\nApproximation of Hyperbolic Models for Chemosensitive Movement Numerical methods with different orders of accuracy are proposed to approximate hy-perbolic models for chemosensitive movements. On the one hand, first and second order well-balanced finite volume schemes are presented. This approach provides an exact con-servation of the steady state solutions. On the other hand, a high order finite difference weighted essentially non-oscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme in order to exactly preserve steady states and to retain high order accuracy. Numerical simulations are performed to verify accuracy and the well-balanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments. Keywords: chemotaxis, hyperbolic systems, finite volume methods, finite difference methods, WENO schemes, well-balanced schemes. This work is in collaboration with Chi-Wang Shu, Brown University", "statement_status": "exact", "statement_verification": "The canonical record is tagged `section`, and inspection of the original AIM workshop PDF confirms that it is a contributed-talk abstract, not an open-problem question. The record begins “A.5 Filbet, Francis” and summarizes numerical methods developed with Chi-Wang Shu under the title *Approximation of Hyperbolic Models for Chemosensitive Movement*. The two strings `hy-perbolic` and `con-servation` in the canonical JSON are line-break OCR artifacts. With only those artifacts repaired, the mathematical content says that first- and second-order well-balanced finite-volume schemes and a high-order finite-difference WENO scheme are proposed for hyperbolic chemotaxis, with tests of accuracy, preservation of steady states, and network formation.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stiff sources and numerical methods for conservation laws\nSection: \nSource item: A.5\nSource URL: https://aimath.org/WWN/balancelaws/balancelaws.pdf\nCanonical location: aim-pdes-notes.json notes[69]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.5 Filbet, Francis \\n\\nApproximation of Hyperbolic Models for Chemosensitive Movement Numerical methods with different orders of accuracy are proposed to approximate hy-perbolic models for chemosensitive movements. On the one hand, first and second order well-balanced finite volume schemes are presented. This approach provides an exact con-servation of the steady state solutions. On the other hand, a high order finite difference weighted essentially non-oscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme in order to exactly preserve steady states and to retain high order accuracy. Numerical simulations are performed to verify accuracy and the well-balanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments. Keywords: chemotaxis, hyperbolic systems, finite volume methods, finite difference methods, WENO schemes, well-balanced schemes. This work is in collaboration with Chi-Wang Shu, Brown University\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/balancelaws/balancelaws.pdf", "tags": [ "aim", "AIM-PDES-0070", "aim-domain:pdes", "aim-workshop:balancelaws", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The canonical item is a contributed-talk abstract, not an open problem: comparison with the AIM PDF and the published Filbet--Shu paper recovers the one-dimensional positive rest equilibrium H(n)-chi(c)=K and corrects the scope of the 2005 WENO claim. A proved structural result shows that shared-interface enthalpy reconstruction exactly balances any consistent flux on these equilibria; in the isothermal local multiplicative class, matching every positive equilibrium plus constant-state consistency uniquely forces the exponential Filbet--Shu factor exp(theta-chi(c)).\n\nCandidate contribution (reconstruction_rigidity_lemma; novelty confidence low): For positive isothermal rest equilibria, any local multiplicative reconstruction n*=n R(theta,chi(c)) using a shared interface parameter, matching every equilibrium pair and satisfying R(z,z)=1, must have R(theta,z)=exp(theta-z); for a rest-state-detecting numerical flux, failure of interface density matching necessarily creates nonzero face mass flux.", "research_difficulty_suggested": "L2" }, { "id": 20002459, "problem_number": "AIM-PDES-0071", "title": "A smooth moment-realizability obstruction for non-Gaussian kinetic closures", "statement": "A.6 Gamba, Irene\n\nNon-equilibrium time dependent reactive kinetic-Poisson systems appear in the mod-eling of such diverse areas as electron transport in solids, biological transport, granular and energy dissipative flows. These non-conservative systems exhibit a common feature: their steady or self similar states are given by statistical Stationary Non-equilibrium States (SNS), meaning they are far deviated from Gaussian probability distributions. They entice approx-imating hydrodynamic models modeling \"source\" representing momentum and energy gain or dissipation due to strong friction or forcing scales. When these non-equilibrium regimes take over, classical hydrodynamic models (based on Gaussians/Maxwellian closures) do not apply and need to be corrected to account for the non-equilibrium statistics. I am interested in issues related to the mathematical properties of these models such as existence, uniqueness, self-similarity, stability and to analyze their higher order moment equations (hydro-dynamical corrections) and corresponding boundary value problems; as well as to investigate optimal numerical simulation methods for corresponding quantum, kinetic and macroscopic (hydrodynamic) models. I will present a very recent work related to item (1.i) below: \"Deterministic solvers to transient Boltzmann-Poisson equations\" Abstract: The Boltzmann-Poisson system is the most reliable model for the flow of charged particles in semiconductors devices. Real device models have not already been simulated by deterministic computations due to its high computational cost, although is very well known and general practice to solve these models by Monte-Carlo (DSMC) methods. We focus in a rather easy and fast deterministic solver for a channel flow: one and two-dimension and three-velocity dimension. The system of equations reduces to a linear kinetic (non-local) equation solved by WENO methods coupled with the Poisson equation for the force field acting on the particles accounting for long range interactions. We will 6\n\nfocus on the development of the method, simulation results for diodes and MESFET as well as comparisons to other classical models in the field. In particular we compute, determin-istically, the evolution probability density function with its first three moments. Boundary singularities for 2-space dimensions models are accurately computed. This work has been done in collaboration with J.A. Carrillo, A. Majorana and C.-W. Shu. Finally, I will present work in progress on computations by DG schemes of linear Boltzmann equations, work in collaboration with Jennifer Proft and Ross Heath. Some other issues I am and have been studying, and I am interested in learning more, are: 1)Self-consistent models of kinetic charged transport. Perturbations of Stationary Non-Equilibrium States (SNS). 1.i) Numerical implementation of deterministic kinetic-Poisson systems and compar-isons to DSMC simulations by WENO schemes, and more recently, developing Discontinuous Galerkin (DG) schemes. 1.ii)Boundary value problems, existence and hydrodynamics limits for strong force fields. Coupling of hyperbolic (SNS) and diffusion (SES) scaling regimes by kinetic layers. 1.iii)Biological transport of charged molecules and Chemotaxis kinetic transport. 2) Quantum Trajectory Models (QTM) for charged transport and Quantum hydrody-namics (QHD) from a semi-classical picture: thermalization and Bose-Einstein condensates models. Existence and non-existence to dispersion/diffusion models. Applications and com-putations. 2.i) Strong force field (Chapman-Enskog) expansion to the semi-classical Wigner trans-port equation 2.ii) Finite time flow up for the QHD equations under high velocity data 2.iii)Numerical calculations for quantum states. 3) The Boltzmann equation for energy dissipative flows, such as inelastic collisions in the modeling of rapid granular flows or elastic collisions in mixtures. 3.i) Energy dissipative Maxwell model type-solutions with power like tails-Levy distri-butions. Trends to equilibrium for energy dissipative Pseudo Maxwell models. 3.ii) Point-wise upper bounds for variable hard spheres, both in the elastic and inelastic case. Boundary value problems. Space inhomogeneous equation. 3.iii) Numerical implementations by spectral methods References can be found at www.ma.utexas.edu/users/gamba/research.html", "original_statement": "A.6 Gamba, Irene \n\nNon-equilibrium time dependent reactive kinetic-Poisson systems appear in the mod-eling of such diverse areas as electron transport in solids, biological transport, granular and energy dissipative flows. These non-conservative systems exhibit a common feature: their steady or self similar states are given by statistical Stationary Non-equilibrium States (SNS), meaning they are far deviated from Gaussian probability distributions. They entice approx-imating hydrodynamic models modeling \"source\" representing momentum and energy gain or dissipation due to strong friction or forcing scales. When these non-equilibrium regimes take over, classical hydrodynamic models (based on Gaussians/Maxwellian closures) do not apply and need to be corrected to account for the non-equilibrium statistics. I am interested in issues related to the mathematical properties of these models such as existence, uniqueness, self-similarity, stability and to analyze their higher order moment equations (hydro-dynamical corrections) and corresponding boundary value problems; as well as to investigate optimal numerical simulation methods for corresponding quantum, kinetic and macroscopic (hydrodynamic) models. I will present a very recent work related to item (1.i) below: \"Deterministic solvers to transient Boltzmann-Poisson equations\" Abstract: The Boltzmann-Poisson system is the most reliable model for the flow of charged particles in semiconductors devices. Real device models have not already been simulated by deterministic computations due to its high computational cost, although is very well known and general practice to solve these models by Monte-Carlo (DSMC) methods. We focus in a rather easy and fast deterministic solver for a channel flow: one and two-dimension and three-velocity dimension. The system of equations reduces to a linear kinetic (non-local) equation solved by WENO methods coupled with the Poisson equation for the force field acting on the particles accounting for long range interactions. We will 6\n\nfocus on the development of the method, simulation results for diodes and MESFET as well as comparisons to other classical models in the field. In particular we compute, determin-istically, the evolution probability density function with its first three moments. Boundary singularities for 2-space dimensions models are accurately computed. This work has been done in collaboration with J.A. Carrillo, A. Majorana and C.-W. Shu. Finally, I will present work in progress on computations by DG schemes of linear Boltzmann equations, work in collaboration with Jennifer Proft and Ross Heath. Some other issues I am and have been studying, and I am interested in learning more, are: 1)Self-consistent models of kinetic charged transport. Perturbations of Stationary Non-Equilibrium States (SNS). 1.i) Numerical implementation of deterministic kinetic-Poisson systems and compar-isons to DSMC simulations by WENO schemes, and more recently, developing Discontinuous Galerkin (DG) schemes. 1.ii)Boundary value problems, existence and hydrodynamics limits for strong force fields. Coupling of hyperbolic (SNS) and diffusion (SES) scaling regimes by kinetic layers. 1.iii)Biological transport of charged molecules and Chemotaxis kinetic transport. 2) Quantum Trajectory Models (QTM) for charged transport and Quantum hydrody-namics (QHD) from a semi-classical picture: thermalization and Bose-Einstein condensates models. Existence and non-existence to dispersion/diffusion models. Applications and com-putations. 2.i) Strong force field (Chapman-Enskog) expansion to the semi-classical Wigner trans-port equation 2.ii) Finite time flow up for the QHD equations under high velocity data 2.iii)Numerical calculations for quantum states. 3) The Boltzmann equation for energy dissipative flows, such as inelastic collisions in the modeling of rapid granular flows or elastic collisions in mixtures. 3.i) Energy dissipative Maxwell model type-solutions with power like tails-Levy distri-butions. Trends to equilibrium for energy dissipative Pseudo Maxwell models. 3.ii) Point-wise upper bounds for variable hard spheres, both in the elastic and inelastic case. Boundary value problems. Space inhomogeneous equation. 3.iii) Numerical implementations by spectral methods References can be found at www.ma.utexas.edu/users/gamba/research.html", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is item A.6, “Gamba, Irene,” in the 1 April 2005 AIM workshop document *Stiff Sources and Numerical Methods for Conservation Laws*. The original PDF was inspected on pages 5--6 of the PDF (printed pages 6--7).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stiff sources and numerical methods for conservation laws\nSection: \nSource item: A.6\nSource URL: https://aimath.org/WWN/balancelaws/balancelaws.pdf\nCanonical location: aim-pdes-notes.json notes[70]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.6 Gamba, Irene \\n\\nNon-equilibrium time dependent reactive kinetic-Poisson systems appear in the mod-eling of such diverse areas as electron transport in solids, biological transport, granular and energy dissipative flows. These non-conservative systems exhibit a common feature: their steady or self similar states are given by statistical Stationary Non-equilibrium States (SNS), meaning they are far deviated from Gaussian probability distributions. They entice approx-imating hydrodynamic models modeling \\\"source\\\" representing momentum and energy gain or dissipation due to strong friction or forcing scales. When these non-equilibrium regimes take over, classical hydrodynamic models (based on Gaussians/Maxwellian closures) do not apply and need to be corrected to account for the non-equilibrium statistics. I am interested in issues related to the mathematical properties of these models such as existence, uniqueness, self-similarity, stability and to analyze their higher order moment equations (hydro-dynamical corrections) and corresponding boundary value problems; as well as to investigate optimal numerical simulation methods for corresponding quantum, kinetic and macroscopic (hydrodynamic) models. I will present a very recent work related to item (1.i) below: \\\"Deterministic solvers to transient Boltzmann-Poisson equations\\\" Abstract: The Boltzmann-Poisson system is the most reliable model for the flow of charged particles in semiconductors devices. Real device models have not already been simulated by deterministic computations due to its high computational cost, although is very well known and general practice to solve these models by Monte-Carlo (DSMC) methods. We focus in a rather easy and fast deterministic solver for a channel flow: one and two-dimension and three-velocity dimension. The system of equations reduces to a linear kinetic (non-local) equation solved by WENO methods coupled with the Poisson equation for the force field acting on the particles accounting for long range interactions. We will 6\\n\\nfocus on the development of the method, simulation results for diodes and MESFET as well as comparisons to other classical models in the field. In particular we compute, determin-istically, the evolution probability density function with its first three moments. Boundary singularities for 2-space dimensions models are accurately computed. This work has been done in collaboration with J.A. Carrillo, A. Majorana and C.-W. Shu. Finally, I will present work in progress on computations by DG schemes of linear Boltzmann equations, work in collaboration with Jennifer Proft and Ross Heath. Some other issues I am and have been studying, and I am interested in learning more, are: 1)Self-consistent models of kinetic charged transport. Perturbations of Stationary Non-Equilibrium States (SNS). 1.i) Numerical implementation of deterministic kinetic-Poisson systems and compar-isons to DSMC simulations by WENO schemes, and more recently, developing Discontinuous Galerkin (DG) schemes. 1.ii)Boundary value problems, existence and hydrodynamics limits for strong force fields. Coupling of hyperbolic (SNS) and diffusion (SES) scaling regimes by kinetic layers. 1.iii)Biological transport of charged molecules and Chemotaxis kinetic transport. 2) Quantum Trajectory Models (QTM) for charged transport and Quantum hydrody-namics (QHD) from a semi-classical picture: thermalization and Bose-Einstein condensates models. Existence and non-existence to dispersion/diffusion models. Applications and com-putations. 2.i) Strong force field (Chapman-Enskog) expansion to the semi-classical Wigner trans-port equation 2.ii) Finite time flow up for the QHD equations under high velocity data 2.iii)Numerical calculations for quantum states. 3) The Boltzmann equation for energy dissipative flows, such as inelastic collisions in the modeling of rapid granular flows or elastic collisions in mixtures. 3.i) Energy dissipative Maxwell model type-solutions with power like tails-Levy distri-butions. Trends to equilibrium for energy dissipative Pseudo Maxwell models. 3.ii) Point-wise upper bounds for variable hard spheres, both in the elastic and inelastic case. Boundary value problems. Space inhomogeneous equation. 3.iii) Numerical implementations by spectral methods References can be found at www.ma.utexas.edu/users/gamba/research.html\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/balancelaws/balancelaws.pdf", "tags": [ "aim", "AIM-PDES-0071", "aim-domain:pdes", "aim-workshop:balancelaws", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The AIM item is a 2005 participant research agenda rather than a posed problem. Focusing on its connection between non-Gaussian stationary nonequilibrium states, deterministic probability-density output, and corrected hydrodynamic moments, this attempt constructs smooth strictly positive densities on three-dimensional velocity space with identical mass, zero momentum, and identical full pressure tensor but energy flux spanning all real values. A second smooth family fixes the longitudinal moments through degree three and zero energy flux while its fourth moment is unbounded; a power-law variant shows the fourth moment may be infinite. Thus low moments alone neither determine nor bound the next kinetic flux without an additional closure principle or tail condition.\n\nCandidate contribution (explicit moment-realizability obstruction and closure test family; novelty confidence low): Candidate novel diagnostic: an exactly normalized Gaussian-mixture family of smooth positive three-velocity densities preserves density, momentum, and the entire pressure tensor while its energy-flux component spans the real line, and a paired symmetric family preserves those data plus zero cubic flux while its longitudinal fourth moment tends to infinity.", "research_difficulty_suggested": "L2" }, { "id": 20002460, "problem_number": "AIM-PDES-0072", "title": "A mass-drift certificate for spherical shallow-water patch coupling", "statement": "A.7 Gelb, Anne\n\nI am most interested in shallow water equations as they pertain to environmental fluid dynamics. Specifically I am interested in global and local methods as they can be applied to spheres. I am also interested in problems of long term simulations.", "original_statement": "A.7 Gelb, Anne \n\nI am most interested in shallow water equations as they pertain to environmental fluid dynamics. Specifically I am interested in global and local methods as they can be applied to spheres. I am also interested in problems of long term simulations.", "clean_statement": "A.7 Gelb, Anne\n\nI am most interested in shallow water equations as they pertain to environmental fluid dynamics. Specifically I am interested in global and local methods as they can be applied to spheres. I am also interested in problems of long term simulations.", "statement_status": "exact", "statement_verification": "The exact canonical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stiff sources and numerical methods for conservation laws\nSection: \nSource item: A.7\nSource URL: https://aimath.org/WWN/balancelaws/balancelaws.pdf\nCanonical location: aim-pdes-notes.json notes[71]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.7 Gelb, Anne \\n\\nI am most interested in shallow water equations as they pertain to environmental fluid dynamics. Specifically I am interested in global and local methods as they can be applied to spheres. I am also interested in problems of long term simulations.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/balancelaws/balancelaws.pdf", "tags": [ "aim", "AIM-PDES-0072", "aim-domain:pdes", "aim-workshop:balancelaws", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The AIM item is a research-interest paragraph rather than a posed problem. For a nonoverlapping finite-volume partition of the closed sphere, the global discrete water-volume change equals exactly the negative accumulated mismatch between the two outward time-integrated flux registers at every global/local or coarse/fine interface. This yields a sharp long-time drift bound and shows that refluxing gives exact algebraic mass conservation under Runge--Kutta stages and subcycling. A shared max-bed hydrostatic reconstruction additionally gives identical nonnegative face depths and zero lake-at-rest mass flux, including wet/dry interfaces.\n\nCandidate contribution (space_time_flux_register_identity; novelty confidence low): On a closed spherical finite-volume mesh, the signed sum of coarse/fine time-integrated mass-flux register defects is exactly the observed global mass drift at every synchronization time; combined with a shared max-bed hydrostatic face reconstruction, every possibly dry lake-at-rest interface contributes zero defect.", "research_difficulty_suggested": "L2" }, { "id": 20002461, "problem_number": "AIM-PDES-0073", "title": "A commutator and equilibrium-projection obstruction for stiff splitting", "statement": "A.8 Gerritsen, Margot\n\nJoint contribution with Rami Younis. Our general area of interest is the efficient numerical solution of flow and transport equations in reservoirs. At the moment, we focus primarily on the design of accurate meth-ods for simulation of miscible gas injection and in-situ combustion (or fire-flooding), which 7\n\nare important Enhanced Oil Recovery processes. These processes are inherently multi-scale. They are generally modeled by two sets of equations. One describing the general flow and energy (if non isothermal) in the reservoir, which either parabolic in character. The sec-ond set models transport of components present in the oil, gas and water in the reservoir. The resulting equations are (weakly) hyperbolic, and very strongly nonlinear. In in-situ combustion processes additional reaction terms render the system also very stiff. We are ex-ploring Euler-Lagrangian type methods for gas injection processes, and splitting techniques for treatment of the equations governing in-situ combustion.", "original_statement": "A.8 Gerritsen, Margot \n\nJoint contribution with Rami Younis. Our general area of interest is the efficient numerical solution of flow and transport equations in reservoirs. At the moment, we focus primarily on the design of accurate meth-ods for simulation of miscible gas injection and in-situ combustion (or fire-flooding), which 7\n\nare important Enhanced Oil Recovery processes. These processes are inherently multi-scale. They are generally modeled by two sets of equations. One describing the general flow and energy (if non isothermal) in the reservoir, which either parabolic in character. The sec-ond set models transport of components present in the oil, gas and water in the reservoir. The resulting equations are (weakly) hyperbolic, and very strongly nonlinear. In in-situ combustion processes additional reaction terms render the system also very stiff. We are ex-ploring Euler-Lagrangian type methods for gas injection processes, and splitting techniques for treatment of the equations governing in-situ combustion.", "clean_statement": "A.8 Gerritsen, Margot\n\nJoint contribution with Rami Younis. Our general area of interest is the efficient numerical solution of flow and transport equations in reservoirs. At the moment, we focus primarily on the design of accurate meth-ods for simulation of miscible gas injection and in-situ combustion (or fire-flooding), which 7\n\nare important Enhanced Oil Recovery processes. These processes are inherently multi-scale. They are generally modeled by two sets of equations. One describing the general flow and energy (if non isothermal) in the reservoir, which either parabolic in character. The sec-ond set models transport of components present in the oil, gas and water in the reservoir. The resulting equations are (weakly) hyperbolic, and very strongly nonlinear. In in-situ combustion processes additional reaction terms render the system also very stiff. We are ex-ploring Euler-Lagrangian type methods for gas injection processes, and splitting techniques for treatment of the equations governing in-situ combustion.", "statement_status": "exact", "statement_verification": "This record is participant statement A.8 by Margot Gerritsen, jointly with Rami Younis, in the AIM workshop document *Stiff Sources and Numerical Methods for Conservation Laws*, version dated 1 April 2005. It is a research-interest statement, not an explicit open problem.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stiff sources and numerical methods for conservation laws\nSection: \nSource item: A.8\nSource URL: https://aimath.org/WWN/balancelaws/balancelaws.pdf\nCanonical location: aim-pdes-notes.json notes[72]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.8 Gerritsen, Margot \\n\\nJoint contribution with Rami Younis. Our general area of interest is the efficient numerical solution of flow and transport equations in reservoirs. At the moment, we focus primarily on the design of accurate meth-ods for simulation of miscible gas injection and in-situ combustion (or fire-flooding), which 7\\n\\nare important Enhanced Oil Recovery processes. These processes are inherently multi-scale. They are generally modeled by two sets of equations. One describing the general flow and energy (if non isothermal) in the reservoir, which either parabolic in character. The sec-ond set models transport of components present in the oil, gas and water in the reservoir. The resulting equations are (weakly) hyperbolic, and very strongly nonlinear. In in-situ combustion processes additional reaction terms render the system also very stiff. We are ex-ploring Euler-Lagrangian type methods for gas injection processes, and splitting techniques for treatment of the equations governing in-situ combustion.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/balancelaws/balancelaws.pdf", "tags": [ "aim", "AIM-PDES-0073", "aim-domain:pdes", "aim-workshop:balancelaws", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The source is a research-interest statement, not an open problem, and its phrase 'which either parabolic in character' is already incomplete in the primary PDF. For the linear periodic model U_t+A U_x=epsilon^{-1}R U with symmetric A and symmetric negative-semidefinite R, reaction-last Lie splitting has a rigorously proved local L2 defect bounded by h^2||[R,A]||||U_x||/(2 epsilon), while fixed-mode Lie and Strang expansions expose 1/epsilon and 1/epsilon^2 commutators. If P projects onto ker R, the projected stiff-limit transport step equals the exact reduced step for every small mode-step product if and only if [P,A]=0. An exact 2-by-2 Fourier example shows that when this fails, Strang's slow stiff-limit error is first order globally, 1-cos(h)^(T/h)=Th/2+O(h^2), despite its nominal second order.\n\nCandidate contribution (criterion_and_counterexample; novelty confidence low): Candidate diagnostic: for symmetric A, P exp(-izA) P equals exp(-izPAP) P for every sufficiently small real z if and only if [P,A]=0; combined with the supplied exact 2-by-2 example, this gives a testable equilibrium-projection criterion and witness for stiff-limit Strang order reduction.", "research_difficulty_suggested": "L2" }, { "id": 20002462, "problem_number": "AIM-PDES-0074", "title": "Exact interface balance and a raw-density current diagnostic for semiconductor drift-diffusion", "statement": "A.9 Hauck, Cory\n\nMy interest in this workshop stems from work on hydrodynamic models of electron transport in semiconductors. These models are balance laws that approximate the evolu-tion of a kinetic distribution of electrons by tracking a given set of velocity and/or energy moments. Hydrodynamic equations contain conservative terms, relaxation terms that arise from collisions, and drift terms derived from a electrical potential that satifies a Poisson equation. Hydrodynamic equations suffer from several difficulties. First of all, most electronic devices contain some type of abrupt material interface which make it difficult to devise a numerical scheme that properly captures the balance of forces found at the differential level. For high-field, transition-regime devices, numerical results are characterized by current oscillations that pollute the solution and can even cause breakdown via negative temperatures and densities. It would very helpful to know if this issue can be overcome with a well-balanced numerical scheme or whether it is a model defect. Second, in the low-field, high-density limit, the hydrodynamic equations recover the well-known drift-diffusion model. Hydrodynamic equations become stiff in this limit, and this includes stiff flux terms. Thus, with the restrictions given by a CFL condition, it is not straightforward how to implement an implicit scheme in an efficient way. Finally, I am interested in finding reasonable approximations to relaxation terms and, in particular, relaxation times. In gas dynamics there is an issue of obtaining the correct transport coefficients predicted by the kinetic model in the fluid limit. More specifically, one would like to recover an appropriate ratio of thermal conductivity to viscosity. In electron transport, the drift-diffusion equations contain only one transport coefficient - the mobility - and therefore this is not an issue. However, the behavior of a numerical solution, especially the current and the temperature, is still drastically affecting by the choice of transport coefficients. Although this is more of a physical modeling issue, I think it is important to understand the effects of relaxation times on numerical solutions.", "original_statement": "A.9 Hauck, Cory \n\nMy interest in this workshop stems from work on hydrodynamic models of electron transport in semiconductors. These models are balance laws that approximate the evolu-tion of a kinetic distribution of electrons by tracking a given set of velocity and/or energy moments. Hydrodynamic equations contain conservative terms, relaxation terms that arise from collisions, and drift terms derived from a electrical potential that satifies a Poisson equation. Hydrodynamic equations suffer from several difficulties. First of all, most electronic devices contain some type of abrupt material interface which make it difficult to devise a numerical scheme that properly captures the balance of forces found at the differential level. For high-field, transition-regime devices, numerical results are characterized by current oscillations that pollute the solution and can even cause breakdown via negative temperatures and densities. It would very helpful to know if this issue can be overcome with a well-balanced numerical scheme or whether it is a model defect. Second, in the low-field, high-density limit, the hydrodynamic equations recover the well-known drift-diffusion model. Hydrodynamic equations become stiff in this limit, and this includes stiff flux terms. Thus, with the restrictions given by a CFL condition, it is not straightforward how to implement an implicit scheme in an efficient way. Finally, I am interested in finding reasonable approximations to relaxation terms and, in particular, relaxation times. In gas dynamics there is an issue of obtaining the correct transport coefficients predicted by the kinetic model in the fluid limit. More specifically, one would like to recover an appropriate ratio of thermal conductivity to viscosity. In electron transport, the drift-diffusion equations contain only one transport coefficient - the mobility - and therefore this is not an issue. However, the behavior of a numerical solution, especially the current and the temperature, is still drastically affecting by the choice of transport coefficients. Although this is more of a physical modeling issue, I think it is important to understand the effects of relaxation times on numerical solutions.", "clean_statement": "A.9 Hauck, Cory\n\nMy interest in this workshop stems from work on hydrodynamic models of electron transport in semiconductors. These models are balance laws that approximate the evolu-tion of a kinetic distribution of electrons by tracking a given set of velocity and/or energy moments. Hydrodynamic equations contain conservative terms, relaxation terms that arise from collisions, and drift terms derived from a electrical potential that satifies a Poisson equation. Hydrodynamic equations suffer from several difficulties. First of all, most electronic devices contain some type of abrupt material interface which make it difficult to devise a numerical scheme that properly captures the balance of forces found at the differential level. For high-field, transition-regime devices, numerical results are characterized by current oscillations that pollute the solution and can even cause breakdown via negative temperatures and densities. It would very helpful to know if this issue can be overcome with a well-balanced numerical scheme or whether it is a model defect. Second, in the low-field, high-density limit, the hydrodynamic equations recover the well-known drift-diffusion model. Hydrodynamic equations become stiff in this limit, and this includes stiff flux terms. Thus, with the restrictions given by a CFL condition, it is not straightforward how to implement an implicit scheme in an efficient way. Finally, I am interested in finding reasonable approximations to relaxation terms and, in particular, relaxation times. In gas dynamics there is an issue of obtaining the correct transport coefficients predicted by the kinetic model in the fluid limit. More specifically, one would like to recover an appropriate ratio of thermal conductivity to viscosity. In electron transport, the drift-diffusion equations contain only one transport coefficient - the mobility - and therefore this is not an issue. However, the behavior of a numerical solution, especially the current and the temperature, is still drastically affecting by the choice of transport coefficients. Although this is more of a physical modeling issue, I think it is important to understand the effects of relaxation times on numerical solutions.", "statement_status": "exact", "statement_verification": "This record is participant statement A.9 by Cory Hauck in the AIM workshop document *Stiff Sources and Numerical Methods for Conservation Laws* (version dated April 1, 2005). It is a research agenda, not a single formally quantified problem. It asks about three related issues for hydrodynamic electron-transport models:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stiff sources and numerical methods for conservation laws\nSection: \nSource item: A.9\nSource URL: https://aimath.org/WWN/balancelaws/balancelaws.pdf\nCanonical location: aim-pdes-notes.json notes[73]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.9 Hauck, Cory \\n\\nMy interest in this workshop stems from work on hydrodynamic models of electron transport in semiconductors. These models are balance laws that approximate the evolu-tion of a kinetic distribution of electrons by tracking a given set of velocity and/or energy moments. Hydrodynamic equations contain conservative terms, relaxation terms that arise from collisions, and drift terms derived from a electrical potential that satifies a Poisson equation. Hydrodynamic equations suffer from several difficulties. First of all, most electronic devices contain some type of abrupt material interface which make it difficult to devise a numerical scheme that properly captures the balance of forces found at the differential level. For high-field, transition-regime devices, numerical results are characterized by current oscillations that pollute the solution and can even cause breakdown via negative temperatures and densities. It would very helpful to know if this issue can be overcome with a well-balanced numerical scheme or whether it is a model defect. Second, in the low-field, high-density limit, the hydrodynamic equations recover the well-known drift-diffusion model. Hydrodynamic equations become stiff in this limit, and this includes stiff flux terms. Thus, with the restrictions given by a CFL condition, it is not straightforward how to implement an implicit scheme in an efficient way. Finally, I am interested in finding reasonable approximations to relaxation terms and, in particular, relaxation times. In gas dynamics there is an issue of obtaining the correct transport coefficients predicted by the kinetic model in the fluid limit. More specifically, one would like to recover an appropriate ratio of thermal conductivity to viscosity. In electron transport, the drift-diffusion equations contain only one transport coefficient - the mobility - and therefore this is not an issue. However, the behavior of a numerical solution, especially the current and the temperature, is still drastically affecting by the choice of transport coefficients. Although this is more of a physical modeling issue, I think it is important to understand the effects of relaxation times on numerical solutions.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/balancelaws/balancelaws.pdf", "tags": [ "aim", "AIM-PDES-0074", "aim-domain:pdes", "aim-workshop:balancelaws", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The participant entry is a three-part research agenda rather than one formally stated problem. In the explicitly scoped one-dimensional, nondegenerate, isothermal drift-diffusion model with ideal interface transmission, zero flux is equivalent to constant electrochemical activity and gives the exact density jump n_R/n_L=(N_C,R/N_C,L) exp(-(U_R-U_L)/(k_B T)). A raw-density dissipative detector produces an explicit nonzero equilibrium residual across every nonzero potential jump, whereas logarithmic-mean reconstruction to a shared face potential is exactly the Scharfetter-Gummel flux, preserves that jump, has nonnegative reconstructed traces, generates a positive conservative semidiscretization, and dissipates relative entropy. This proves a narrow discretization-induced equilibrium-current mechanism but does not classify general driven high-field oscillations or solve the stiff hydrodynamic and relaxation-time questions.\n\nCandidate contribution (diagnostic theorem; novelty confidence low): Candidate interface diagnostic: for arbitrary finite prescribed potential jumps under ideal transmission, zero exponentially fitted flux, equal electrochemical activity, and equality of the two positive logarithmic-mean shared-potential traces are equivalent; at the same Boltzmann state a strict raw-density detector has the explicit residual D C (exp(-V_i)-exp(-V_{i+1}))/h, while the fitted semidiscretization is conservative, positivity preserving, and entropy dissipating.", "research_difficulty_suggested": "L2" }, { "id": 20002463, "problem_number": "AIM-PDES-0075", "title": "Flux-conservative Liouville scattering at a potential step", "statement": "A.10 Jin, Shi\n\nMy recent interests include numerical methods for physical problems involving multiple scales. In particular I am interested in the transition from quantum to classical mechanics, from kinetic theory to hydrodynamics, and its numerical relavance. In the workshop I will present recent results on numerical methods for Liouville equations with singular Hamiltoni-ans (which arise either from a discontinuous potential or a discontinuous local wave speed). 8", "original_statement": "A.10 Jin, Shi \n\nMy recent interests include numerical methods for physical problems involving multiple scales. In particular I am interested in the transition from quantum to classical mechanics, from kinetic theory to hydrodynamics, and its numerical relavance. In the workshop I will present recent results on numerical methods for Liouville equations with singular Hamiltoni-ans (which arise either from a discontinuous potential or a discontinuous local wave speed). 8", "clean_statement": "A.10 Jin, Shi\n\nMy recent interests include numerical methods for physical problems involving multiple scales. In particular I am interested in the transition from quantum to classical mechanics, from kinetic theory to hydrodynamics, and its numerical relavance. In the workshop I will present recent results on numerical methods for Liouville equations with singular Hamiltoni-ans (which arise either from a discontinuous potential or a discontinuous local wave speed). 8", "statement_status": "exact", "statement_verification": "The canonical extraction is preserved here verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stiff sources and numerical methods for conservation laws\nSection: \nSource item: A.10\nSource URL: https://aimath.org/WWN/balancelaws/balancelaws.pdf\nCanonical location: aim-pdes-notes.json notes[74]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.10 Jin, Shi \\n\\nMy recent interests include numerical methods for physical problems involving multiple scales. In particular I am interested in the transition from quantum to classical mechanics, from kinetic theory to hydrodynamics, and its numerical relavance. In the workshop I will present recent results on numerical methods for Liouville equations with singular Hamiltoni-ans (which arise either from a discontinuous potential or a discontinuous local wave speed). 8\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/balancelaws/balancelaws.pdf", "tags": [ "aim", "AIM-PDES-0075", "aim-domain:pdes", "aim-workshop:balancelaws", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The AIM record is a participant-interest paragraph rather than an open problem. Motivated by it, this attempt proves that one-dimensional energy-preserving interface scattering transports the normal flux measure |v|f dp: the energy-map momentum Jacobian exactly cancels the velocity ratio, yielding conservation of every integrable Hamiltonian-weighted interface flux. It also proves that a nonzero atomic incident flux exactly at an uphill threshold cannot be transmitted into a finite zero-speed momentum measure without reflection or an explicit interface-storage balance.\n\nCandidate contribution (flux-measure identity and threshold obstruction; novelty confidence low): For general even kinetic energies strictly increasing in momentum magnitude, the flux-measure pushforward formulation simultaneously conserves all integrable Hamiltonian-weighted interface moments, while a threshold atom exposes the necessity of reflection or an interface-storage variable.", "research_difficulty_suggested": "L2" }, { "id": 20002464, "problem_number": "AIM-PDES-0076", "title": "A boundary-work concentration diagnostic for shear-band blow-up", "statement": "A.11 Katsaounis, Theodoros\n\nBalance laws appear as mathematical models in a great number of applications areas such as gas dynamics, mechanics, geophysics, biology. In recent years there has been enor-mous activity on developing numerical methods for capturing correctly the properties and features of the analytical solutions. I am particularly interested in developing numerical schemes for balance laws using relaxation approximation. The starting point of our approach is the class of relaxation schemes, introduced in [JX], which are based on the relaxation approximation to the nonlinear conservation law, that has a linear convection term and needs neither a Riemann solver nor the characteristic decomposition and thus enjoys great simplicity in the expense of increasing the number of unknowns. The stabilization mechanisms are the regularization by wave operators. The idea is to use a local relaxation approximation to construct linear hyperbolic system with a stiff lower order term that approximates the original nonlinear system with a small dissipative correction. Relaxation is a flux approximation and relaxation linearizes the Riemann prob-lem. This simplicity can be of great significance when one has to solve large-scale engineering problems. The numerical schemes are based on finite volume and finite element discretizations of the relaxation models. In [DK1], [DK2] the finite volume(difference) method is used to descritize the relaxation approximation of the shallow water equations in one and two space dimensions respectively. The source term is treated in two different ways. The numerical schemes are of first or second order in space and time, do not need Riemann solvers, they are able to treat the dry bed case(vacuum case) with no extra effort, and satisfy the steady states, an important feature of the analytical solution, within the accuracy of the relaxation parameter ≤.In [DK3] the numerical schemes presented in [DK1], [DK2] are used to compute the transport and diffusion of a passive pollutant by a water flow. The flow is modeled by the well-known shallow water equations and the pollutant propagation is described by a transport equation. It's worth mentioning that that no special treatment is needed for the transport equation in order to obtain accurate results. The relaxation approximation of conservation laws provide a natural setting for apply-ing the finite element method. We apply the standard finite element method combined with appropriate Runge-Kutta methods for the time discretization. Adaptive mesh refinement strategies based on a-posteriori indicators and inverse inequalities are employed for resolv-ing accurately regions with shocks. The resulting schemes have a regularization mechanism with finite speed of propagation, do not need the solution of approximate local Riemann problems, can be formulated as low order or high order schemes, or even a combination of them (h-p methods), and can be extended in multi-dimensions by using the finite element framework, [AKM], [KM], [GM]. Some properties of these schemes, concerning stability and convergence are presented in [AMT]. Simulating a shear band(a narrow layer of intense shearing in a material, not a crack though) is another topic of interest. The mathematical model consists of a system of con-servation laws, close related to that of elastodynamics. The highly nonlinear model has a internal diffusion mechanism which collapses on the shear band. The temperature and the strain rate grow (blow up?) while the velocity develops a δ-function behavior. It is an 9\n\nopen question whether the temperature and the strain rate blow up in finite or infinite time. Numerical simulation of this singular behavior is a challenge, [BKT].", "original_statement": "A.11 Katsaounis, Theodoros \n\nBalance laws appear as mathematical models in a great number of applications areas such as gas dynamics, mechanics, geophysics, biology. In recent years there has been enor-mous activity on developing numerical methods for capturing correctly the properties and features of the analytical solutions. I am particularly interested in developing numerical schemes for balance laws using relaxation approximation. The starting point of our approach is the class of relaxation schemes, introduced in [JX], which are based on the relaxation approximation to the nonlinear conservation law, that has a linear convection term and needs neither a Riemann solver nor the characteristic decomposition and thus enjoys great simplicity in the expense of increasing the number of unknowns. The stabilization mechanisms are the regularization by wave operators. The idea is to use a local relaxation approximation to construct linear hyperbolic system with a stiff lower order term that approximates the original nonlinear system with a small dissipative correction. Relaxation is a flux approximation and relaxation linearizes the Riemann prob-lem. This simplicity can be of great significance when one has to solve large-scale engineering problems. The numerical schemes are based on finite volume and finite element discretizations of the relaxation models. In [DK1], [DK2] the finite volume(difference) method is used to descritize the relaxation approximation of the shallow water equations in one and two space dimensions respectively. The source term is treated in two different ways. The numerical schemes are of first or second order in space and time, do not need Riemann solvers, they are able to treat the dry bed case(vacuum case) with no extra effort, and satisfy the steady states, an important feature of the analytical solution, within the accuracy of the relaxation parameter ≤.In [DK3] the numerical schemes presented in [DK1], [DK2] are used to compute the transport and diffusion of a passive pollutant by a water flow. The flow is modeled by the well-known shallow water equations and the pollutant propagation is described by a transport equation. It's worth mentioning that that no special treatment is needed for the transport equation in order to obtain accurate results. The relaxation approximation of conservation laws provide a natural setting for apply-ing the finite element method. We apply the standard finite element method combined with appropriate Runge-Kutta methods for the time discretization. Adaptive mesh refinement strategies based on a-posteriori indicators and inverse inequalities are employed for resolv-ing accurately regions with shocks. The resulting schemes have a regularization mechanism with finite speed of propagation, do not need the solution of approximate local Riemann problems, can be formulated as low order or high order schemes, or even a combination of them (h-p methods), and can be extended in multi-dimensions by using the finite element framework, [AKM], [KM], [GM]. Some properties of these schemes, concerning stability and convergence are presented in [AMT]. Simulating a shear band(a narrow layer of intense shearing in a material, not a crack though) is another topic of interest. The mathematical model consists of a system of con-servation laws, close related to that of elastodynamics. The highly nonlinear model has a internal diffusion mechanism which collapses on the shear band. The temperature and the strain rate grow (blow up?) while the velocity develops a δ-function behavior. It is an 9\n\nopen question whether the temperature and the strain rate blow up in finite or infinite time. Numerical simulation of this singular behavior is a challenge, [BKT].", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is participant contribution A.11 by Theodoros Katsaounis in the AIM workshop document *Stiff Sources and Numerical Methods for Conservation Laws* (version dated 1 April 2005). Most of the record surveys relaxation schemes. Its final paragraph contains a genuine research question:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: Stiff sources and numerical methods for conservation laws\nSection: \nSource item: A.11\nSource URL: https://aimath.org/WWN/balancelaws/balancelaws.pdf\nCanonical location: aim-pdes-notes.json notes[75]\nCanonical tag: section\nOriginal extracted problem text (JSON string): \"A.11 Katsaounis, Theodoros \\n\\nBalance laws appear as mathematical models in a great number of applications areas such as gas dynamics, mechanics, geophysics, biology. In recent years there has been enor-mous activity on developing numerical methods for capturing correctly the properties and features of the analytical solutions. I am particularly interested in developing numerical schemes for balance laws using relaxation approximation. The starting point of our approach is the class of relaxation schemes, introduced in [JX], which are based on the relaxation approximation to the nonlinear conservation law, that has a linear convection term and needs neither a Riemann solver nor the characteristic decomposition and thus enjoys great simplicity in the expense of increasing the number of unknowns. The stabilization mechanisms are the regularization by wave operators. The idea is to use a local relaxation approximation to construct linear hyperbolic system with a stiff lower order term that approximates the original nonlinear system with a small dissipative correction. Relaxation is a flux approximation and relaxation linearizes the Riemann prob-lem. This simplicity can be of great significance when one has to solve large-scale engineering problems. The numerical schemes are based on finite volume and finite element discretizations of the relaxation models. In [DK1], [DK2] the finite volume(difference) method is used to descritize the relaxation approximation of the shallow water equations in one and two space dimensions respectively. The source term is treated in two different ways. The numerical schemes are of first or second order in space and time, do not need Riemann solvers, they are able to treat the dry bed case(vacuum case) with no extra effort, and satisfy the steady states, an important feature of the analytical solution, within the accuracy of the relaxation parameter ≤.In [DK3] the numerical schemes presented in [DK1], [DK2] are used to compute the transport and diffusion of a passive pollutant by a water flow. The flow is modeled by the well-known shallow water equations and the pollutant propagation is described by a transport equation. It's worth mentioning that that no special treatment is needed for the transport equation in order to obtain accurate results. The relaxation approximation of conservation laws provide a natural setting for apply-ing the finite element method. We apply the standard finite element method combined with appropriate Runge-Kutta methods for the time discretization. Adaptive mesh refinement strategies based on a-posteriori indicators and inverse inequalities are employed for resolv-ing accurately regions with shocks. The resulting schemes have a regularization mechanism with finite speed of propagation, do not need the solution of approximate local Riemann problems, can be formulated as low order or high order schemes, or even a combination of them (h-p methods), and can be extended in multi-dimensions by using the finite element framework, [AKM], [KM], [GM]. Some properties of these schemes, concerning stability and convergence are presented in [AMT]. Simulating a shear band(a narrow layer of intense shearing in a material, not a crack though) is another topic of interest. The mathematical model consists of a system of con-servation laws, close related to that of elastodynamics. The highly nonlinear model has a internal diffusion mechanism which collapses on the shear band. The temperature and the strain rate grow (blow up?) while the velocity develops a δ-function behavior. It is an 9\\n\\nopen question whether the temperature and the strain rate blow up in finite or infinite time. Numerical simulation of this singular behavior is a challenge, [BKT].\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/balancelaws/balancelaws.pdf", "tags": [ "aim", "AIM-PDES-0076", "aim-domain:pdes", "aim-workshop:balancelaws", "aim-source-tag:section" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source contains an explicit shear-band blow-up question, but its literal claim that velocity develops a delta profile conflicts with the cited BKT paper, which reports delta-like behavior of the strain rate v_x and a step-like velocity. For the normalized thermoviscoplastic power-law system, the exact uniform-shear solution has temperature growing only at infinite time while the tangent momentum diffusivity collapses whenever thermal softening dominates strain hardening. For general classical solutions, the proved boundary-work energy identity shows that finite cumulative boundary work uniformly bounds velocity in L2 and temperature in L1: a Dirac mass in velocity is impossible, a Dirac mass in strain rate remains possible, and any finite-time temperature blow-up must concentrate on shrinking superlevel sets.\n\nCandidate contribution (obstruction_and_exact_benchmark; novelty confidence low): Candidate boundary-work diagnostic: for the thermoviscoplastic shear model, bounded cumulative boundary work up to a finite time rules out an atomic distributional limit of velocity and gives |{theta >= H}| <= C/H, while allowing delta concentration of v_x and a step limit of v; combined with the exact uniform-shear formula, it separates diffusion collapse from finite-time blow-up." }, { "id": 20002465, "problem_number": "AIM-PDES-0077", "title": "Quadratic-chain characterization of the Aubry set", "statement": "1. (Albert Fathi) Let f be a smooth vector field on a compact manifold M and let\n\nφtf: M → M be its flow. There is a standard way to include the flow φtf in the flow of a positive definite Lagrangian system. Take a Riemannian metric on M and let\n\n‖ ‖ be the corresponding norm. Define a Lagrangian L: T M → R by\n\nL(x, v ) = 1\n\n2‖v − f (x)‖2.\n\nThen the zero section M ⊂ T M of the tangent bundle is an invariant manifold for the Lagrangian flow. The Aubry set A0 ⊂ T M corresponding to the zero cohomology class 0 ∈ H1(M, R) is contained in M.Problem: give a characterization of A0 in terms of the dynamics of the flow φtf. In particular, does A0 contain the chain recurrent set of the flow φtf?2. Let H: T ∗M → R,\n\nH(x, p ) = 1\n\n2‖p‖2 − f (x) · p\n\nbe the Hamiltonian corresponding to L. Consider the Hamilton-Jacobi equation\n\nH(x, Du (x)) = 1\n\n2‖Du (x)‖2 − f (x) · Du (x) = 0.\n\nQuestion: Under what conditions is u = 0 the unique viscosity solution?", "original_statement": "1. (Albert Fathi) Let f be a smooth vector field on a compact manifold M and let \n\nφtf: M → M be its flow. There is a standard way to include the flow φtf in the flow of a positive definite Lagrangian system. Take a Riemannian metric on M and let \n\n‖ ‖ be the corresponding norm. Define a Lagrangian L: T M → R by \n\nL(x, v ) = 1\n\n2‖v − f (x)‖2.\n\nThen the zero section M ⊂ T M of the tangent bundle is an invariant manifold for the Lagrangian flow. The Aubry set A0 ⊂ T M corresponding to the zero cohomology class 0 ∈ H1(M, R) is contained in M.Problem: give a characterization of A0 in terms of the dynamics of the flow φtf. In particular, does A0 contain the chain recurrent set of the flow φtf?2. Let H: T ∗M → R,\n\nH(x, p ) = 1\n\n2‖p‖2 − f (x) · p\n\nbe the Hamiltonian corresponding to L. Consider the Hamilton-Jacobi equation \n\nH(x, Du (x)) = 1\n\n2‖Du (x)‖2 − f (x) · Du (x) = 0.\n\nQuestion: Under what conditions is u = 0 the unique viscosity solution?", "clean_statement": "1. (Albert Fathi) Let f be a smooth vector field on a compact manifold M and let\n\nφtf: M → M be its flow. There is a standard way to include the flow φtf in the flow of a positive definite Lagrangian system. Take a Riemannian metric on M and let\n\n‖ ‖ be the corresponding norm. Define a Lagrangian L: T M → R by\n\nL(x, v ) = 1\n\n2‖v − f (x)‖2.\n\nThen the zero section M ⊂ T M of the tangent bundle is an invariant manifold for the Lagrangian flow. The Aubry set A0 ⊂ T M corresponding to the zero cohomology class 0 ∈ H1(M, R) is contained in M.Problem: give a characterization of A0 in terms of the dynamics of the flow φtf. In particular, does A0 contain the chain recurrent set of the flow φtf?2. Let H: T ∗M → R,\n\nH(x, p ) = 1\n\n2‖p‖2 − f (x) · p\n\nbe the Hamiltonian corresponding to L. Consider the Hamilton-Jacobi equation\n\nH(x, Du (x)) = 1\n\n2‖Du (x)‖2 − f (x) · Du (x) = 0.\n\nQuestion: Under what conditions is u = 0 the unique viscosity solution?", "statement_status": "exact", "statement_verification": "The primary AIM PDF was checked directly. With typographical layout normalized, it asks about \\[ L_f(x,v)=\\frac12\\lVert v-f(x)\\rVert_x^2 \\] for a smooth vector field \\(f\\) on a compact Riemannian manifold \\(M\\), the relation between its zero-class Aubry set and recurrence of the flow \\(\\phi^t\\) of \\(f\\), and uniqueness of the zero solution of a stationary Hamilton--Jacobi equation. The PDF literally says that the zero section in \\(TM\\) is invariant and literally prints \\[ H_-(x,p)=\\frac12\\lVert p\\rVert_x^2-p(f(x)). \\]", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[76]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (Albert Fathi) Let f be a smooth vector field on a compact manifold M and let \\n\\nφtf: M → M be its flow. There is a standard way to include the flow φtf in the flow of a positive definite Lagrangian system. Take a Riemannian metric on M and let \\n\\n‖ ‖ be the corresponding norm. Define a Lagrangian L: T M → R by \\n\\nL(x, v ) = 1\\n\\n2‖v − f (x)‖2.\\n\\nThen the zero section M ⊂ T M of the tangent bundle is an invariant manifold for the Lagrangian flow. The Aubry set A0 ⊂ T M corresponding to the zero cohomology class 0 ∈ H1(M, R) is contained in M.Problem: give a characterization of A0 in terms of the dynamics of the flow φtf. In particular, does A0 contain the chain recurrent set of the flow φtf?2. Let H: T ∗M → R,\\n\\nH(x, p ) = 1\\n\\n2‖p‖2 − f (x) · p\\n\\nbe the Hamiltonian corresponding to L. Consider the Hamilton-Jacobi equation \\n\\nH(x, Du (x)) = 1\\n\\n2‖Du (x)‖2 − f (x) · Du (x) = 0.\\n\\nQuestion: Under what conditions is u = 0 the unique viscosity solution?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0077", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After correcting the source's tangent/cotangent and Legendre-sign defects, the projected Aubry set of L_f(x,v)=1/2||v-f(x)||^2 is proved to equal the quadratic-chain recurrent set R_2(f), defined by returning chains with arbitrarily large minimum flight time and arbitrarily small sum of squared jump lengths. Hence SCR(f) is contained in R_2(f)=A_f, which is contained in CR(f), and normalized Hamilton-Jacobi uniqueness is equivalent to R_2(f)=M. Under the published Fathi-Figalli-Rifford Mather-disconnectedness hypotheses, including every smooth f in dimension at most three, R_2(f)=CR(f), but the unrestricted higher-dimensional reverse inclusion is not claimed.\n\nCandidate contribution (theorem; novelty confidence low): For a smooth vector field f on a compact connected Riemannian manifold, the projected zero-class Aubry set of the quadratic Mane Lagrangian is exactly the set of points admitting, for every T and epsilon, a returning flow chain with all flight times at least T and total squared jump cost below epsilon." }, { "id": 20002466, "problem_number": "AIM-PDES-0078", "title": "Exact flat-torus Mather quotient and current status", "statement": "3. (John Mather) Let M be a compact manifold and L: T M ×T → R a C∞ Lagrangian satisfying the usual hypotheses of the Mather theory - convexity, superlinearity and completeness. Let Ac ⊂ M be the Aubry set corresponding to the cohomology class\n\nc ∈ H1(M, R). Define a pseudo metric ρc on Ac as follows. Modify the Lagrangian by subtracting a closed 1-form from the cohomology class c and adding a constant\n\nα(c) so that for the new Lc we have inf\n\n{∫\n\nLc dμ | μ is an invariant probablility measure on T M × T\n\n}\n\n= 0.\n\nThen set\n\nhnc (x, y ) = inf\n\n{∫ n\n\n> 0\n\nLc(γ(t), ˙γ(t), t ) dt | γ: [0, n ] → M connects x and y\n\n}\n\nand\n\nρc(x, y ) = lim inf\n\n> n→∞\n\nhnc (x, y ) + lim inf\n\n> n→∞\n\nhnc (y, x ).\n\nDefine an equivalence relation on Ac by x ∼ y iff ρc(x, y ) = 0. Let ¯Ac be the corresponding quotient space. Question: Is it true that ¯Ac is totally disconnected? Does ¯Ac have zero Hausdorff dimension?", "original_statement": "3. (John Mather) Let M be a compact manifold and L: T M ×T → R a C∞ Lagrangian satisfying the usual hypotheses of the Mather theory - convexity, superlinearity and completeness. Let Ac ⊂ M be the Aubry set corresponding to the cohomology class \n\nc ∈ H1(M, R). Define a pseudo metric ρc on Ac as follows. Modify the Lagrangian by subtracting a closed 1-form from the cohomology class c and adding a constant \n\nα(c) so that for the new Lc we have inf \n\n{∫ \n\nLc dμ | μ is an invariant probablility measure on T M × T\n\n}\n\n= 0.\n\nThen set \n\nhnc (x, y ) = inf \n\n{∫ n\n\n> 0\n\nLc(γ(t), ˙γ(t), t ) dt | γ: [0, n ] → M connects x and y\n\n}\n\nand \n\nρc(x, y ) = lim inf \n\n> n→∞\n\nhnc (x, y ) + lim inf \n\n> n→∞\n\nhnc (y, x ).\n\nDefine an equivalence relation on Ac by x ∼ y iff ρc(x, y ) = 0. Let ¯Ac be the corresponding quotient space. Question: Is it true that ¯Ac is totally disconnected? Does ¯Ac have zero Hausdorff dimension?", "clean_statement": "3. (John Mather) Let M be a compact manifold and L: T M ×T → R a C∞ Lagrangian satisfying the usual hypotheses of the Mather theory - convexity, superlinearity and completeness. Let Ac ⊂ M be the Aubry set corresponding to the cohomology class\n\nc ∈ H1(M, R). Define a pseudo metric ρc on Ac as follows. Modify the Lagrangian by subtracting a closed 1-form from the cohomology class c and adding a constant\n\nα(c) so that for the new Lc we have inf\n\n{∫\n\nLc dμ | μ is an invariant probablility measure on T M × T\n\n}\n\n= 0.\n\nThen set\n\nhnc (x, y ) = inf\n\n{∫ n\n\n> 0\n\nLc(γ(t), ˙γ(t), t ) dt | γ: [0, n ] → M connects x and y\n\n}\n\nand\n\nρc(x, y ) = lim inf\n\n> n→∞\n\nhnc (x, y ) + lim inf\n\n> n→∞\n\nhnc (y, x ).\n\nDefine an equivalence relation on Ac by x ∼ y iff ρc(x, y ) = 0. Let ¯Ac be the corresponding quotient space. Question: Is it true that ¯Ac is totally disconnected? Does ¯Ac have zero Hausdorff dimension?", "statement_status": "exact", "statement_verification": "The canonical extraction, preserved without silent correction, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[77]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (John Mather) Let M be a compact manifold and L: T M ×T → R a C∞ Lagrangian satisfying the usual hypotheses of the Mather theory - convexity, superlinearity and completeness. Let Ac ⊂ M be the Aubry set corresponding to the cohomology class \\n\\nc ∈ H1(M, R). Define a pseudo metric ρc on Ac as follows. Modify the Lagrangian by subtracting a closed 1-form from the cohomology class c and adding a constant \\n\\nα(c) so that for the new Lc we have inf \\n\\n{∫ \\n\\nLc dμ | μ is an invariant probablility measure on T M × T\\n\\n}\\n\\n= 0.\\n\\nThen set \\n\\nhnc (x, y ) = inf \\n\\n{∫ n\\n\\n> 0\\n\\nLc(γ(t), ˙γ(t), t ) dt | γ: [0, n ] → M connects x and y\\n\\n}\\n\\nand \\n\\nρc(x, y ) = lim inf \\n\\n> n→∞\\n\\nhnc (x, y ) + lim inf \\n\\n> n→∞\\n\\nhnc (y, x ).\\n\\nDefine an equivalence relation on Ac by x ∼ y iff ρc(x, y ) = 0. Let ¯Ac be the corresponding quotient space. Question: Is it true that ¯Ac is totally disconnected? Does ¯Ac have zero Hausdorff dimension?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0078", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every constant positive-definite kinetic Lagrangian on a flat torus, in every dimension and every cohomology class, the normalized finite-time action is an exact nearest-lattice quadratic expression bounded uniformly by lambda_max(A)d/(8n). Consequently the projected Aubry set is the whole torus while its Mather quotient is a singleton. The report also separates the known autonomous and low-dimensional results from Mather's still-open arbitrary-dimensional smooth time-periodic questions.\n\nCandidate contribution (explicit special-family theorem; novelty confidence low): The normalized n-step action for a constant positive-definite kinetic tensor is given by an exact all-winding lattice-distance formula with a uniform O(1/n) bound for every cohomology class, without any rationality hypothesis on the drift." }, { "id": 20002467, "problem_number": "AIM-PDES-0079", "title": "Static-class criterion for generic Hamilton-Jacobi uniqueness", "statement": "4. (Albert Fathi) Is it true that for generic L ∈ C∞ there exists only a one-parameter family of viscosity solutions of the corresponding Hamilton-Jacobi equation? 4", "original_statement": "4. (Albert Fathi) Is it true that for generic L ∈ C∞ there exists only a one-parameter family of viscosity solutions of the corresponding Hamilton-Jacobi equation? 4", "clean_statement": "4. (Albert Fathi) Is it true that for generic L ∈ C∞ there exists only a one-parameter family of viscosity solutions of the corresponding Hamilton-Jacobi equation? 4", "statement_status": "exact", "statement_verification": "The AIM list *New connections between dynamical systems and PDE's* (version dated 15 August 2003) prints:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[78]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"4. (Albert Fathi) Is it true that for generic L ∈ C∞ there exists only a one-parameter family of viscosity solutions of the corresponding Hamilton-Jacobi equation? 4\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0079", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After reconstructing the intended fixed-cohomology Tonelli weak-KAM setting, the report proves that critical viscosity solutions are unique modulo constants exactly when the quotient Aubry set has one static class. More quantitatively, elementary solutions based at Aubry points a and b have uniform distance modulo constants equal to one half of the static distance rho_c(a,b). Mañe's generic uniqueness of the minimizing measure therefore answers the standard fixed-class, generic-potential reading affirmatively, and a countable Baire intersection gives simultaneous uniqueness for any prescribed countable set of cohomology classes.\n\nCandidate contribution (lemma; novelty confidence low): The map from the quotient Aubry set to elementary critical viscosity solutions modulo constants, [a] -> [h_c(a,.)], is an isometric embedding when the source carries rho_c/2 and the target carries the uniform quotient metric; equivalently, d_infinity([h_c(a,.)],[h_c(b,.)]) = rho_c(a,b)/2." }, { "id": 20002468, "problem_number": "AIM-PDES-0080", "title": "Regular Mather functions, convex duality, and integrability", "statement": "5. (Patrick Bernard) Consider an analytic positive definite Lagrangian L: Tn × Rn ×\n\nT → R, L = L(x, v, t ), satisfying Mather's conditions. Let\n\nαL(c) = − inf\n\n> μ\n\n∫\n\n(L(x, v, t ) − c · v) dμ\n\nbe Mather's function αL: Rn → R.Question: Is it true that if αL is analytic with positive definite second derivative, then the Lagrangian system is completely integrable?", "original_statement": "5. (Patrick Bernard) Consider an analytic positive definite Lagrangian L: Tn × Rn ×\n\nT → R, L = L(x, v, t ), satisfying Mather's conditions. Let \n\nαL(c) = − inf \n\n> μ\n\n∫\n\n(L(x, v, t ) − c · v) dμ \n\nbe Mather's function αL: Rn → R.Question: Is it true that if αL is analytic with positive definite second derivative, then the Lagrangian system is completely integrable?", "clean_statement": "5. (Patrick Bernard) Consider an analytic positive definite Lagrangian L: Tn × Rn ×\n\nT → R, L = L(x, v, t ), satisfying Mather's conditions. Let\n\nαL(c) = − inf\n\n> μ\n\n∫\n\n(L(x, v, t ) − c · v) dμ\n\nbe Mather's function αL: Rn → R.Question: Is it true that if αL is analytic with positive definite second derivative, then the Lagrangian system is completely integrable?", "statement_status": "exact", "statement_verification": "The AIM problem list *New connections between dynamical systems and PDE's* (version dated 15 August 2003) states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[79]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"5. (Patrick Bernard) Consider an analytic positive definite Lagrangian L: Tn × Rn ×\\n\\nT → R, L = L(x, v, t ), satisfying Mather's conditions. Let \\n\\nαL(c) = − inf \\n\\n> μ\\n\\n∫\\n\\n(L(x, v, t ) − c · v) dμ \\n\\nbe Mather's function αL: Rn → R.Question: Is it true that if αL is analytic with positive definite second derivative, then the Lagrangian system is completely integrable?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0080", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a time-periodic Tonelli Lagrangian whose Mather alpha function is analytic with everywhere positive-definite Hessian, the gradient of alpha is a global analytic diffeomorphism, the dual beta function is analytic with inverse Hessian, and every minimizing measure at a fixed cohomology has the same rotation vector and action, although its measure and support need not be unique. This proves C0 integrability in the autonomous two-torus subclass via Massart-Sorrentino. An explicit analytic spatially homogeneous time-periodic family is smoothly integrable and has alpha(c)=c^T(mean B)c/2+(mean a)·c, showing exactly how alpha loses time-resolved coefficient information.\n\nCandidate contribution (explicit_family; novelty confidence low): For L_{B,a}(v,t)=(v-a(t))^T B(t)^{-1}(v-a(t))/2 with analytic periodic B(t)>0 and a(t), the entire Mather function is alpha(c)=c^T(mean B)c/2+(mean a)·c; hence the infinite-dimensional coefficient history collapses to the two averages (mean B, mean a), and all coefficient pairs with those same averages have identical alpha functions." }, { "id": 20002469, "problem_number": "AIM-PDES-0081", "title": "Full-projection Aubry sets under a regular Mather alpha function", "statement": "6. (Gonzalo Contreras) Is it true that under the same condition on αL, each Aubry set\n\nAc covers the whole configuration space Tn?", "original_statement": "6. (Gonzalo Contreras) Is it true that under the same condition on αL, each Aubry set \n\nAc covers the whole configuration space Tn?", "clean_statement": "6. (Gonzalo Contreras) Is it true that under the same condition on αL, each Aubry set\n\nAc covers the whole configuration space Tn?", "statement_status": "exact", "statement_verification": "The canonical extraction is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[80]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"6. (Gonzalo Contreras) Is it true that under the same condition on αL, each Aubry set \\n\\nAc covers the whole configuration space Tn?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0081", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general analytic time-periodic implication remains open in the literature checked. This attempt proves three scoped results: positive-definite Hessian makes the alpha gradient a global analytic diffeomorphism and beta analytic; Massart-Sorrentino's theorem then answers the autonomous two-torus case affirmatively; and any lifted phase-space intersection of Aubry sets for distinct cohomology classes would force an affine segment of alpha. It also gives a genuinely time-dependent analytic drift-and-gauge family with an exact alpha function, exact finite-time Peierls formula, full Aubry projection in every class, and an explicit cotangent foliation.\n\nCandidate contribution (explicit special-family theorem; novelty confidence low): For L(x,v,t)=(v-b(t))^T A(v-b(t))/2 plus the spacetime total derivative of an arbitrary analytic periodic phi, alpha(c)=c dot bar(b)+c^T A^{-1}c/2, the normalized integer-time action is an exact all-winding lattice-distance formula with its gauge endpoint term, every projected Aubry set is the full torus, and the dual Aubry graphs are p=c+D_x phi." }, { "id": 20002470, "problem_number": "AIM-PDES-0082", "title": "A dense analytic rational skeleton for an analytic Aubry foliation", "statement": "7. (John Mather) Let f: T × R → T × R be an analytic twist map. Suppose that\n\nT × R is foliated by invariant curves Γ c (which are Aubry-Mather sets). Are these curves Γ c necessarily analytic?", "original_statement": "7. (John Mather) Let f: T × R → T × R be an analytic twist map. Suppose that \n\nT × R is foliated by invariant curves Γ c (which are Aubry-Mather sets). Are these curves Γ c necessarily analytic?", "clean_statement": "7. (John Mather) Let f: T × R → T × R be an analytic twist map. Suppose that\n\nT × R is foliated by invariant curves Γ c (which are Aubry-Mather sets). Are these curves Γ c necessarily analytic?", "statement_status": "exact", "statement_verification": "The AIM workshop list, Problem 7 attributed to John Mather, states:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[81]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"7. (John Mather) Let f: T × R → T × R be an analytic twist map. Suppose that \\n\\nT × R is foliated by invariant curves Γ c (which are Aubry-Mather sets). Are these curves Γ c necessarily analytic?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0082", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the standard interpretation of the source as an analytic exact symplectic positive twist map continuously foliated by whole Aubry graphs, every rational-rotation leaf is pointwise periodic by Aubry chain recurrence and hence real analytic by the no-conjugate-points analytic implicit-function argument. Such rational leaves are dense by continuity of the cohomology-to-rotation map and the fact that its plateaux can only have rational value. Thus any counterexample must be an irrational leaf approached by analytic rational leaves with degenerating complex-analytic control.\n\nCandidate contribution (reduction; novelty confidence low): An analytic exact positive twist foliation by whole Aubry sets has a dense analytic rational skeleton; consequently Mather's question reduces to uniform complex extension estimates at irrational leaves, or to constructing degeneration of those estimates along rational approximants." }, { "id": 20002471, "problem_number": "AIM-PDES-0083", "title": "Boundary integrability and the explicit confocal-caustic dictionary", "statement": "8. (Vadim Kaloshin) Consider a billiard inside a region bounded by a closed convex curve Γ ⊂ R2. Suppose that a neighborhood of Γ is foliated by caustics for the billiard. Is it true that then Γ is an ellipse and hence the caustics are analytic curves? The billiard in Γ is described by a twist map f, and caustics correspond to invariant curves of f.Hence this questions is a particular case of the previous one.", "original_statement": "8. (Vadim Kaloshin) Consider a billiard inside a region bounded by a closed convex curve Γ ⊂ R2. Suppose that a neighborhood of Γ is foliated by caustics for the billiard. Is it true that then Γ is an ellipse and hence the caustics are analytic curves? The billiard in Γ is described by a twist map f, and caustics correspond to invariant curves of f.Hence this questions is a particular case of the previous one.", "clean_statement": "8. (Vadim Kaloshin) Consider a billiard inside a region bounded by a closed convex curve Γ ⊂ R2. Suppose that a neighborhood of Γ is foliated by caustics for the billiard. Is it true that then Γ is an ellipse and hence the caustics are analytic curves? The billiard in Γ is described by a twist map f, and caustics correspond to invariant curves of f.Hence this questions is a particular case of the previous one.", "statement_status": "exact", "statement_verification": "The AIM list *New connections between dynamical systems and PDE's*, Problem 8 (attributed to Vadim Kaloshin), asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[82]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"8. (Vadim Kaloshin) Consider a billiard inside a region bounded by a closed convex curve Γ ⊂ R2. Suppose that a neighborhood of Γ is foliated by caustics for the billiard. Is it true that then Γ is an ellipse and hence the caustics are analytic curves? The billiard in Γ is described by a twist map f, and caustics correspond to invariant curves of f.Hence this questions is a particular case of the previous one.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0083", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general boundary-integrability implication is a still-open form of the Birkhoff conjecture, with known local and symmetric rigidity regimes. For an elliptical table E_{a,b}, this attempt proves directly that the Joachimsthal quantity J= is billiard-invariant, that every segment is tangent to the confocal conic with parameter lambda=a^2 b^2 J^2, and that the associated phase-space graphs satisfy sin(phi_lambda(x))=sqrt(lambda)/(ab|Ax|). Hence the inner confocal ellipses form an analytic physical collar, while the derivation also shows why a boundary foliation is distinct from both a whole-phase-cylinder foliation and a Lazutkin Cantor family.\n\nCandidate contribution (explicit_equivalence; novelty confidence low): The exact normalization lambda=a^2 b^2^2 together with sin(phi_lambda(x))=sqrt(lambda)/(ab|Ax|) gives a directly verified dictionary between each physical confocal caustic and its paired oriented invariant graphs, and the accompanying hypothesis audit separates open collar integrability from global phase foliation and Cantor-family integrability." }, { "id": 20002472, "problem_number": "AIM-PDES-0084", "title": "A full-support certificate for excluding Denjoy dynamics on invariant twist-map curves", "statement": "9. (John Mather) Does there exist a C3 twist map f: T × R → T × R which has an invariant curve Γ with irrational rotation number and such that f |Γ is not conjugate to a rotation? By the Denjoy example there exist C1 maps of a circle with irrational rotation number not conjugate to a rotation. Can this happen for invariant curves of twist maps?\n1", "original_statement": "9. (John Mather) Does there exist a C3 twist map f: T × R → T × R which has an invariant curve Γ with irrational rotation number and such that f |Γ is not conjugate to a rotation? By the Denjoy example there exist C1 maps of a circle with irrational rotation number not conjugate to a rotation. Can this happen for invariant curves of twist maps? \n1", "clean_statement": "9. (John Mather) Does there exist a C3 twist map f: T × R → T × R which has an invariant curve Γ with irrational rotation number and such that f |Γ is not conjugate to a rotation? By the Denjoy example there exist C1 maps of a circle with irrational rotation number not conjugate to a rotation. Can this happen for invariant curves of twist maps?\n1", "statement_status": "exact", "statement_verification": "The canonical record is item 9, attributed to John Mather, in the AIM workshop list *New connections between dynamical systems and PDE's*. Its exact extracted problem field is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[83]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"9. (John Mather) Does there exist a C3 twist map f: T × R → T × R which has an invariant curve Γ with irrational rotation number and such that f |Γ is not conjugate to a rotation? By the Denjoy example there exist C1 maps of a circle with irrational rotation number not conjugate to a rotation. Can this happen for invariant curves of twist maps? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0084", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an irrational orientation-preserving circle homeomorphism, topological conjugacy to the rigid rotation is equivalent to existence of an invariant probability with full support. Applied to an invariant twist-map graph, any counterexample to Mather's C^3 question must contain wandering open arcs outside the proper Cantor support of its invariant probability, and the graph cannot itself be the support of an invariant minimizing probability. In contrast, every irrational circle homeomorphism, including a Denjoy example, is chain recurrent at every point, so chain recurrence cannot exclude the pathology. A C^2 invariant graph is also ruled out by Denjoy's theorem.\n\nCandidate contribution (reduction; novelty confidence low): Any C^3 twist-map counterexample must exhibit a support gap: its invariant curve strictly contains the proper Cantor support of its invariant probability, with the complement made of wandering arcs; full-support measure is a valid certificate against Denjoy behavior, whereas chain recurrence is automatic and is not such a certificate." }, { "id": 20002473, "problem_number": "AIM-PDES-0085", "title": "Hamilton-Jacobi graph regularity versus KAM conjugacy", "statement": "0. (Patrick Bernard) The existence of KAM invariant tori of a Hamiltonian system on\n\nTn × Rn is equivalent to the existence of regular solutions of the Hamilton-Jacobi equation H(x, c + Du (x)) = α(c). Hence KAM theory can be regarded as regularity result for viscosity solutions. Question: Is it possible to obtain proofs of KAM results using this connection?\n1", "original_statement": "0. (Patrick Bernard) The existence of KAM invariant tori of a Hamiltonian system on \n\nTn × Rn is equivalent to the existence of regular solutions of the Hamilton-Jacobi equation H(x, c + Du (x)) = α(c). Hence KAM theory can be regarded as regularity result for viscosity solutions. Question: Is it possible to obtain proofs of KAM results using this connection? \n1", "clean_statement": "0. (Patrick Bernard) The existence of KAM invariant tori of a Hamiltonian system on\n\nTn × Rn is equivalent to the existence of regular solutions of the Hamilton-Jacobi equation H(x, c + Du (x)) = α(c). Hence KAM theory can be regarded as regularity result for viscosity solutions. Question: Is it possible to obtain proofs of KAM results using this connection?\n1", "statement_status": "exact", "statement_verification": "The original AIM PDF, version dated 15 August 2003, contains the following as Problem 10 (attributed to Patrick Bernard):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 0\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[84]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"0. (Patrick Bernard) The existence of KAM invariant tori of a Hamiltonian system on \\n\\nTn × Rn is equivalent to the existence of regular solutions of the Hamilton-Jacobi equation H(x, c + Du (x)) = α(c). Hence KAM theory can be regarded as regularity result for viscosity solutions. Question: Is it possible to obtain proofs of KAM results using this connection? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0085", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Hamilton-Jacobi and configuration-space KAM proofs exist, so the broad methodological question has an affirmative answer, while a generic viscosity-to-smooth KAM bootstrap was not found in the literature checked. Mathematically, this attempt proves that a C^2 solution H(x,c+Du)=constant is equivalent to invariance of the closed Lagrangian graph p=c+Du, but KAM dynamics additionally requires a torus conjugacy DV(x)H_p(x,c+Du)=omega. It proves an arbitrary-dynamics realization lemma: for any smooth graph q=c+Du_0 and any smooth torus vector field b, the uniformly Tonelli Hamiltonian H=|p-q|^2/2+b·(p-q) has critical value alpha(c)=0 and restricts on that smooth HJ graph to xdot=b. Thus scalar HJ regularity alone cannot force quasiperiodicity. A first-order perturbative calculation also recovers the Diophantine cohomological equation and its analytic strip-loss estimate.\n\nCandidate contribution (obstruction_and_realization_lemma; novelty confidence low): Every smooth vector field b on the torus can be realized exactly as the characteristic dynamics on a smooth critical Hamilton-Jacobi graph of the uniformly fiberwise-convex Hamiltonian H_{b,q}(x,p)=|p-q(x)|^2/2+, where q=c+Du_0; hence regularity of the scalar cell solution alone imposes no KAM-type dynamical restriction, and the missing condition is the explicit conjugacy equation DV·H_p=omega." }, { "id": 20002474, "problem_number": "AIM-PDES-0086", "title": "Critical-level surgery for nonconvex Aubry problems", "statement": "1. (Albert Fathi) For a Hamiltonian H(x, p ) on Tn×Rn, which is superlinear and strictly convex in p, we have a fairly good description both in dynamical terms (Mather theory) and in PDE terms (viscosity solutions) of the Aubry set. It would be nice to understand the situation where H is still superlinear but not necessarily convex:\n\n• Does there exist an Aubry set from the PDE point of view (for example as a uniqueness set for viscosity solutions)?\n\n• Does there exist an Aubry set from the point of view of dynamics, i.e. acanonical graph invariant under the Hamiltonian flow?\n\n• Is there a relationship between the two sets if they exist? 12. Let c0 be the infimum of the c's such that the Hamilton-Jacobi equation H(x, Du (x)) =\n\nc with superlinear H admits a global viscosity subsolution u: Tn → R. Let SS be the set of global viscosity subsolutions of H(x, Du (x)) = c0. Define a function S on\n\nTn × Tn by\n\nS(x, y ) = sup {u(x) − u(y) | u ∈ SS }.5\n\nThen for fixed x, the function S(x, ·) is a viscosity subsolution of H(x, Du (x)) = c0\n\non Tn, and a viscosity solution on Tn \\ { 0}. When H is convex in p, the Aubry set\n\nA0 is the set of x ∈ Tn such that S(x, ·) is a global viscosity solution on the whole\n\nTn.Question: If H is not necessarily convex, does there exist x ∈ Tn such that S(x, ·) is a viscosity solution on the whole Tn?\n1", "original_statement": "1. (Albert Fathi) For a Hamiltonian H(x, p ) on Tn×Rn, which is superlinear and strictly convex in p, we have a fairly good description both in dynamical terms (Mather theory) and in PDE terms (viscosity solutions) of the Aubry set. It would be nice to understand the situation where H is still superlinear but not necessarily convex: \n\n• Does there exist an Aubry set from the PDE point of view (for example as a uniqueness set for viscosity solutions)? \n\n• Does there exist an Aubry set from the point of view of dynamics, i.e. acanonical graph invariant under the Hamiltonian flow? \n\n• Is there a relationship between the two sets if they exist? 12. Let c0 be the infimum of the c's such that the Hamilton-Jacobi equation H(x, Du (x)) = \n\nc with superlinear H admits a global viscosity subsolution u: Tn → R. Let SS be the set of global viscosity subsolutions of H(x, Du (x)) = c0. Define a function S on \n\nTn × Tn by \n\nS(x, y ) = sup {u(x) − u(y) | u ∈ SS }.5\n\nThen for fixed x, the function S(x, ·) is a viscosity subsolution of H(x, Du (x)) = c0\n\non Tn, and a viscosity solution on Tn \\ { 0}. When H is convex in p, the Aubry set \n\nA0 is the set of x ∈ Tn such that S(x, ·) is a global viscosity solution on the whole \n\nTn.Question: If H is not necessarily convex, does there exist x ∈ Tn such that S(x, ·) is a viscosity solution on the whole Tn?\n1", "clean_statement": "1. (Albert Fathi) For a Hamiltonian H(x, p ) on Tn×Rn, which is superlinear and strictly convex in p, we have a fairly good description both in dynamical terms (Mather theory) and in PDE terms (viscosity solutions) of the Aubry set. It would be nice to understand the situation where H is still superlinear but not necessarily convex:\n\n• Does there exist an Aubry set from the PDE point of view (for example as a uniqueness set for viscosity solutions)?\n\n• Does there exist an Aubry set from the point of view of dynamics, i.e. acanonical graph invariant under the Hamiltonian flow?\n\n• Is there a relationship between the two sets if they exist? 12. Let c0 be the infimum of the c's such that the Hamilton-Jacobi equation H(x, Du (x)) =\n\nc with superlinear H admits a global viscosity subsolution u: Tn → R. Let SS be the set of global viscosity subsolutions of H(x, Du (x)) = c0. Define a function S on\n\nTn × Tn by\n\nS(x, y ) = sup {u(x) − u(y) | u ∈ SS }.5\n\nThen for fixed x, the function S(x, ·) is a viscosity subsolution of H(x, Du (x)) = c0\n\non Tn, and a viscosity solution on Tn \\ { 0}. When H is convex in p, the Aubry set\n\nA0 is the set of x ∈ Tn such that S(x, ·) is a global viscosity solution on the whole\n\nTn.Question: If H is not necessarily convex, does there exist x ∈ Tn such that S(x, ·) is a viscosity solution on the whole Tn?\n1", "statement_status": "exact", "statement_verification": "The canonical record is an extraction of two consecutive questions from the 2003 AIM list. It must remain one corpus job, but mathematically it separates as follows.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[85]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (Albert Fathi) For a Hamiltonian H(x, p ) on Tn×Rn, which is superlinear and strictly convex in p, we have a fairly good description both in dynamical terms (Mather theory) and in PDE terms (viscosity solutions) of the Aubry set. It would be nice to understand the situation where H is still superlinear but not necessarily convex: \\n\\n• Does there exist an Aubry set from the PDE point of view (for example as a uniqueness set for viscosity solutions)? \\n\\n• Does there exist an Aubry set from the point of view of dynamics, i.e. acanonical graph invariant under the Hamiltonian flow? \\n\\n• Is there a relationship between the two sets if they exist? 12. Let c0 be the infimum of the c's such that the Hamilton-Jacobi equation H(x, Du (x)) = \\n\\nc with superlinear H admits a global viscosity subsolution u: Tn → R. Let SS be the set of global viscosity subsolutions of H(x, Du (x)) = c0. Define a function S on \\n\\nTn × Tn by \\n\\nS(x, y ) = sup {u(x) − u(y) | u ∈ SS }.5\\n\\nThen for fixed x, the function S(x, ·) is a viscosity subsolution of H(x, Du (x)) = c0\\n\\non Tn, and a viscosity solution on Tn \\\\ { 0}. When H is convex in p, the Aubry set \\n\\nA0 is the set of x ∈ Tn such that S(x, ·) is a global viscosity solution on the whole \\n\\nTn.Question: If H is not necessarily convex, does there exist x ∈ Tn such that S(x, ·) is a viscosity solution on the whole Tn?\\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0086", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The canonical record merges AIM Problems 11 and 12 and contains source-level variable errors. Gomes-Mitake-Tran refuted the corrected maximal-subsolution question in 2018 with a coercive one-dimensional example. This attempt proves that their linear tail can be replaced outside the critical momentum range by a quadratic tail, producing a literal superlinear nonconvex Hamiltonian whose maximal-subsolution Aubry vertex set is still empty. It also proves that positive bounded phase-space rescalings of a Tonelli Hamiltonian about its critical level can be genuinely nonconvex while preserving all critical viscosity solutions, the projected PDE Aubry set, and the invariant lifted Aubry graph up to positive time reparametrization.\n\nCandidate contribution (counterexample_extension; novelty confidence low): The Gomes-Mitake-Tran empty-Aubry counterexample can be made superlinear by leaving its kinetic energy unchanged for momentum radius at most 3 and replacing the outer linear tail by 2+(r-3)+(r-3)^2; the critical value, critical subsolutions, maximal subsolution, and failure of every vertex supersolution test remain unchanged." }, { "id": 20002475, "problem_number": "AIM-PDES-0087", "title": "Frequency arithmetic does not determine Mather-set regularity", "statement": "3. (Walter Craig) Consider a positive definite Lagrangian L: T M × T → R. Let\n\nα: H1(M, R) → R be Mather's α-function. Problem: Relate the regularity of the Mather set Mc and the arithmetic properties of the frequency set ω = Dα (c) ⊂ H1(M, R).\n1", "original_statement": "3. (Walter Craig) Consider a positive definite Lagrangian L: T M × T → R. Let \n\nα: H1(M, R) → R be Mather's α-function. Problem: Relate the regularity of the Mather set Mc and the arithmetic properties of the frequency set ω = Dα (c) ⊂ H1(M, R). \n1", "clean_statement": "3. (Walter Craig) Consider a positive definite Lagrangian L: T M × T → R. Let\n\nα: H1(M, R) → R be Mather's α-function. Problem: Relate the regularity of the Mather set Mc and the arithmetic properties of the frequency set ω = Dα (c) ⊂ H1(M, R).\n1", "statement_status": "exact", "statement_verification": "The canonical record is an OCR extraction from Walter Craig's Problem 13 in the AIM workshop list *New connections between dynamical systems and PDE's* (version dated 15 August 2003). The PDF reads, modulo lost mathematical typography:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[86]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (Walter Craig) Consider a positive definite Lagrangian L: T M × T → R. Let \\n\\nα: H1(M, R) → R be Mather's α-function. Problem: Relate the regularity of the Mather set Mc and the arithmetic properties of the frequency set ω = Dα (c) ⊂ H1(M, R). \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0087", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every nonempty closed set F in the circle, a smooth Tonelli Lagrangian L_F(x,v,t)=1/2(v-b_F(x))^2 can be chosen so that D alpha_F(0)=0 while the lifted zero-class Mather set is exactly F times {0} times the time circle. Thus one fixed rational frequency and differentiability datum permits smooth full, finite, or Cantor-singular Mather geometry. Complementarily, for the integrable torus model L=ell(v), the Mather set is a smooth full graph for every frequency, while total irrationality versus resonance controls unique ergodicity versus multiplicity of minimizing orbit-closure measures.\n\nCandidate contribution (counterexample_family; novelty confidence low): Every nonempty closed subset of S^1 is realizable as the spatial footprint of a smooth Tonelli lifted Mather set at the same differentiable frequency D alpha(0)=0." }, { "id": 20002476, "problem_number": "AIM-PDES-0088", "title": "Two-scale fate of minimax periodic orbits near an irrational cantorus", "statement": "4. (Walter Craig) This question concerns the minimax Birkhoff orbits in an area pre-serving twist map of an annulus. It is well known that\n\n• There exist at least two periodic orbits of a twist map with given rational rotation number α = p/q, namely a minimal orbit and a minimax orbit.\n\n• Taking a limit αj → ω ∈ R \\ Q of rational rotation numbers, the minimal Birkhoff orbits converge to the Mather set Π ω.\n\n• If Π ω is an invariant circle, then the minimax orbits also converge to Π ω.Question: What is the fate of the minimax orbits for the case when the Mather set is a Cantor set?\n1", "original_statement": "4. (Walter Craig) This question concerns the minimax Birkhoff orbits in an area pre-serving twist map of an annulus. It is well known that \n\n• There exist at least two periodic orbits of a twist map with given rational rotation number α = p/q, namely a minimal orbit and a minimax orbit. \n\n• Taking a limit αj → ω ∈ R \\ Q of rational rotation numbers, the minimal Birkhoff orbits converge to the Mather set Π ω.\n\n• If Π ω is an invariant circle, then the minimax orbits also converge to Π ω.Question: What is the fate of the minimax orbits for the case when the Mather set is a Cantor set? \n1", "clean_statement": "4. (Walter Craig) This question concerns the minimax Birkhoff orbits in an area pre-serving twist map of an annulus. It is well known that\n\n• There exist at least two periodic orbits of a twist map with given rational rotation number α = p/q, namely a minimal orbit and a minimax orbit.\n\n• Taking a limit αj → ω ∈ R \\ Q of rational rotation numbers, the minimal Birkhoff orbits converge to the Mather set Π ω.\n\n• If Π ω is an invariant circle, then the minimax orbits also converge to Π ω.Question: What is the fate of the minimax orbits for the case when the Mather set is a Cantor set?\n1", "statement_status": "exact", "statement_verification": "The canonical JSON record is a damaged extraction of Problem 14 in the AIM workshop notes *New connections between dynamical systems and PDE's*. The source page and PDF give the following statement (with only typographical normalization):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[87]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"4. (Walter Craig) This question concerns the minimax Birkhoff orbits in an area pre-serving twist map of an annulus. It is well known that \\n\\n• There exist at least two periodic orbits of a twist map with given rational rotation number α = p/q, namely a minimal orbit and a minimax orbit. \\n\\n• Taking a limit αj → ω ∈ R \\\\ Q of rational rotation numbers, the minimal Birkhoff orbits converge to the Mather set Π ω.\\n\\n• If Π ω is an invariant circle, then the minimax orbits also converge to Π ω.Question: What is the fate of the minimax orbits for the case when the Mather set is a Cantor set? \\n1\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0088", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Mather's classical irrational minimax construction answers the broad question: the Cantor-set analogue is a gap orbit homoclinic to the Mather cantorus, not a second Cantor set. A proved two-scale refinement shows that, when the total minimax/minimal action excess is sublinear in the period (and in particular when Mather's finite action barrier bounds it), the uniform measures on rational minimax orbits converge to the unique irrational Mather measure, while defect-centered lifted configurations can have non-minimizing stationary Birkhoff limits. In the generic Morse index-one case, every finite-window Hessian of such a limit has at most one negative eigenvalue, and a uniformly localized negative mode certifies that the pointed limit lies outside the cantorus.\n\nCandidate contribution (theorem; novelty confidence low): Candidate two-scale compactness theorem: sublinear total minimax action excess forces empirical-measure convergence to the irrational Mather measure, while phase-normalized configurations have stationary Birkhoff subsequential limits; for Morse index-one rational minimax points, all finite-window limiting Hessians have negative index at most one, and a persistent localized negative direction proves that the pointed limit is a non-minimizing defect." }, { "id": 20002477, "problem_number": "AIM-PDES-0089", "title": "Exact-energy reachability at the universal Mañé critical value", "statement": "5. (Gonzalo Contreras) Let M be a compact manifold, L: T M → R a convex superlin-ear Lagrangian, and ˜L: T ˜M → R - lift of L to the universal cover ˜M of M. Let cu\n\nbe the Mane critical value for ˜L.Question: Is it true that\n\ncu = inf {h ∈ R | ∃ x ∈ ˜M ∀y ∈ M ∃trajectory with energy h joining x to y}.", "original_statement": "5. (Gonzalo Contreras) Let M be a compact manifold, L: T M → R a convex superlin-ear Lagrangian, and ˜L: T ˜M → R - lift of L to the universal cover ˜M of M. Let cu\n\nbe the Mane critical value for ˜L.Question: Is it true that \n\ncu = inf {h ∈ R | ∃ x ∈ ˜M ∀y ∈ M ∃trajectory with energy h joining x to y}.", "clean_statement": "5. (Gonzalo Contreras) Let M be a compact manifold, L: T M → R a convex superlin-ear Lagrangian, and ˜L: T ˜M → R - lift of L to the universal cover ˜M of M. Let cu\n\nbe the Mane critical value for ˜L.Question: Is it true that\n\ncu = inf {h ∈ R | ∃ x ∈ ˜M ∀y ∈ M ∃trajectory with energy h joining x to y}.", "statement_status": "exact", "statement_verification": "The record has two recoverable OCR defects: the original item number 15 became 5, and the symbols \\(\\widetilde L\\), \\(\\widetilde M\\), and \\(c_u\\) were partly flattened. More importantly, the mixed quantifiers \\(x\\in\\widetilde M\\) and \\(y\\in M\\) occur in both the official PDF and HTML. They are therefore not an OCR error, but (1) is not literally well typed: a curve in \\(\\widetilde M\\) cannot have an endpoint \\(y\\in M\\) without specifying a lift or asking only that its projected endpoint be \\(y\\).", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[88]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"5. (Gonzalo Contreras) Let M be a compact manifold, L: T M → R a convex superlin-ear Lagrangian, and ˜L: T ˜M → R - lift of L to the universal cover ˜M of M. Let cu\\n\\nbe the Mane critical value for ˜L.Question: Is it true that \\n\\ncu = inf {h ∈ R | ∃ x ∈ ˜M ∀y ∈ M ∃trajectory with energy h joining x to y}.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0089", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After correcting the ill-typed mixed-space source formula and excluding positive-time diagonal self-connections, the exact-energy reachability spectrum is proved to be (c_u, infinity) for every Tonelli Lagrangian whose zero-velocity momentum one-form partial_v L(x,0) is closed. In this class c_u equals e_0=max_x E(x,0); supercritical reachability follows from a complete Finsler metric on the universal cover, while the equilibrium above a maximizer of E(x,0) obstructs reachability at and below c_u. A free-particle circle example proves that the literal all-pairs formulation including x=y is false.\n\nCandidate contribution (special_case; novelty confidence low): For the gauge-closed Tonelli class, the off-diagonal directed exact-energy reachability spectrum on the universal cover is precisely (c_u, infinity), and the critical endpoint is not attained because a maximizer of E(x,0) supports an isolated zero-velocity equilibrium." }, { "id": 20002478, "problem_number": "AIM-PDES-0090", "title": "Five Mane-critical questions and stability of a full Aubry graph", "statement": "16. Let M be a compact manifold, L: T M → R a convex superlinear Lagrangian. The only example known of an energy level without periodic orbits or singularities is\n\n{H = cu} with cu a Mane critical value for the lift of the Lagrangian to ˜M.Question: Is there an example with other energy levels without periodic orbits or singularities? 17. Let M be a compact manifold, H: T M → R a convex superlinear Hamiltonian. The energy level Σ h = {H = h} is of contact type if there exists a 1-form λ on Σ h\n\nsuch that dλ is the symplectic 2-form on T ∗M and λ(XH ) > 0, where XH is the Hamiltonian vector field. Let cu be the critical value on ˜M and let\n\ne0 = min {h ∈ R | π(Σ h) = M }.\n\nQuestion: Is it true that if M 6 = T2, then for all e0 < h < c u the energy level Σ h is not of contact type? 18. Let M be a non-compact manifold and L: T M → R a convex superlinear Lagrangian satisfying appropriate completeness conditions at infinity. There is a compactification 6\n\nof M by adding an \"extended Aubry set\" whose points correspond to \"Busemann viscosity solutions\" of the Hamilton-Jacobi equation (Calc. Var. 13 (2001), 427-458). Problem: Understand the geometry of this compactification. 19. Let L: Tn × Rn → R be an autonomous convex superlinear Lagrangian and H the corresponding Hamiltonian. Let c0 = α(0) be the Mane critical value. Question: Is it true that for any h < c 0 the set\n\n{(x, p ) ∈ R2n | H(x, p ) < h }\n\nhas finite symplectic capacity? 20. Let H: Tn × Rn → R be an autonomous convex superlinear Hamiltonian. Suppose that the Aubry set A0 satisfies π(A0) = Tn and there exists a unique (up to a constant) viscosity solution of the Hamilton-Jacobi equation H(x, Du (x)) = α(0). For c ∈ Rn, let Ac be the Aubry set corresponding to the Hamiltonian Hc(x, p ) =\n\nH(x, p − c). Let vc be a viscosity solution of Hc(x, Dv c(x)) = α(c). Question: is it true that lim\n\n> |c|→ 0\n\n‖Dv c − Du ‖Lip ( π(Ac)) = 0.", "original_statement": "16. Let M be a compact manifold, L: T M → R a convex superlinear Lagrangian. The only example known of an energy level without periodic orbits or singularities is \n\n{H = cu} with cu a Mane critical value for the lift of the Lagrangian to ˜M.Question: Is there an example with other energy levels without periodic orbits or singularities? 17. Let M be a compact manifold, H: T M → R a convex superlinear Hamiltonian. The energy level Σ h = {H = h} is of contact type if there exists a 1-form λ on Σ h\n\nsuch that dλ is the symplectic 2-form on T ∗M and λ(XH ) > 0, where XH is the Hamiltonian vector field. Let cu be the critical value on ˜M and let \n\ne0 = min {h ∈ R | π(Σ h) = M }.\n\nQuestion: Is it true that if M 6 = T2, then for all e0 < h < c u the energy level Σ h is not of contact type? 18. Let M be a non-compact manifold and L: T M → R a convex superlinear Lagrangian satisfying appropriate completeness conditions at infinity. There is a compactification 6\n\nof M by adding an \"extended Aubry set\" whose points correspond to \"Busemann viscosity solutions\" of the Hamilton-Jacobi equation (Calc. Var. 13 (2001), 427-458). Problem: Understand the geometry of this compactification. 19. Let L: Tn × Rn → R be an autonomous convex superlinear Lagrangian and H the corresponding Hamiltonian. Let c0 = α(0) be the Mane critical value. Question: Is it true that for any h < c 0 the set \n\n{(x, p ) ∈ R2n | H(x, p ) < h }\n\nhas finite symplectic capacity? 20. Let H: Tn × Rn → R be an autonomous convex superlinear Hamiltonian. Suppose that the Aubry set A0 satisfies π(A0) = Tn and there exists a unique (up to a constant) viscosity solution of the Hamilton-Jacobi equation H(x, Du (x)) = α(0). For c ∈ Rn, let Ac be the Aubry set corresponding to the Hamiltonian Hc(x, p ) = \n\nH(x, p − c). Let vc be a viscosity solution of Hc(x, Dv c(x)) = α(c). Question: is it true that lim \n\n> |c|→ 0\n\n‖Dv c − Du ‖Lip ( π(Ac)) = 0.", "clean_statement": "16. Let M be a compact manifold, L: T M → R a convex superlinear Lagrangian. The only example known of an energy level without periodic orbits or singularities is\n\n{H = cu} with cu a Mane critical value for the lift of the Lagrangian to ˜M.Question: Is there an example with other energy levels without periodic orbits or singularities? 17. Let M be a compact manifold, H: T M → R a convex superlinear Hamiltonian. The energy level Σ h = {H = h} is of contact type if there exists a 1-form λ on Σ h\n\nsuch that dλ is the symplectic 2-form on T ∗M and λ(XH ) > 0, where XH is the Hamiltonian vector field. Let cu be the critical value on ˜M and let\n\ne0 = min {h ∈ R | π(Σ h) = M }.\n\nQuestion: Is it true that if M 6 = T2, then for all e0 < h < c u the energy level Σ h is not of contact type? 18. Let M be a non-compact manifold and L: T M → R a convex superlinear Lagrangian satisfying appropriate completeness conditions at infinity. There is a compactification 6\n\nof M by adding an \"extended Aubry set\" whose points correspond to \"Busemann viscosity solutions\" of the Hamilton-Jacobi equation (Calc. Var. 13 (2001), 427-458). Problem: Understand the geometry of this compactification. 19. Let L: Tn × Rn → R be an autonomous convex superlinear Lagrangian and H the corresponding Hamiltonian. Let c0 = α(0) be the Mane critical value. Question: Is it true that for any h < c 0 the set\n\n{(x, p ) ∈ R2n | H(x, p ) < h }\n\nhas finite symplectic capacity? 20. Let H: Tn × Rn → R be an autonomous convex superlinear Hamiltonian. Suppose that the Aubry set A0 satisfies π(A0) = Tn and there exists a unique (up to a constant) viscosity solution of the Hamilton-Jacobi equation H(x, Du (x)) = α(0). For c ∈ Rn, let Ac be the Aubry set corresponding to the Hamiltonian Hc(x, p ) =\n\nH(x, p − c). Let vc be a viscosity solution of Hc(x, Dv c(x)) = α(c). Question: is it true that lim\n\n> |c|→ 0\n\n‖Dv c − Du ‖Lip ( π(Ac)) = 0.", "statement_status": "exact", "statement_verification": "The canonical JSON record merges five consecutive questions, Problems 16--20 in the 2003 AIM workshop list *New connections between dynamical systems and PDE's*. The complete record is preserved as one job, but its components are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 16\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[89]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"16. Let M be a compact manifold, L: T M → R a convex superlinear Lagrangian. The only example known of an energy level without periodic orbits or singularities is \\n\\n{H = cu} with cu a Mane critical value for the lift of the Lagrangian to ˜M.Question: Is there an example with other energy levels without periodic orbits or singularities? 17. Let M be a compact manifold, H: T M → R a convex superlinear Hamiltonian. The energy level Σ h = {H = h} is of contact type if there exists a 1-form λ on Σ h\\n\\nsuch that dλ is the symplectic 2-form on T ∗M and λ(XH ) > 0, where XH is the Hamiltonian vector field. Let cu be the critical value on ˜M and let \\n\\ne0 = min {h ∈ R | π(Σ h) = M }.\\n\\nQuestion: Is it true that if M 6 = T2, then for all e0 < h < c u the energy level Σ h is not of contact type? 18. Let M be a non-compact manifold and L: T M → R a convex superlinear Lagrangian satisfying appropriate completeness conditions at infinity. There is a compactification 6\\n\\nof M by adding an \\\"extended Aubry set\\\" whose points correspond to \\\"Busemann viscosity solutions\\\" of the Hamilton-Jacobi equation (Calc. Var. 13 (2001), 427-458). Problem: Understand the geometry of this compactification. 19. Let L: Tn × Rn → R be an autonomous convex superlinear Lagrangian and H the corresponding Hamiltonian. Let c0 = α(0) be the Mane critical value. Question: Is it true that for any h < c 0 the set \\n\\n{(x, p ) ∈ R2n | H(x, p ) < h }\\n\\nhas finite symplectic capacity? 20. Let H: Tn × Rn → R be an autonomous convex superlinear Hamiltonian. Suppose that the Aubry set A0 satisfies π(A0) = Tn and there exists a unique (up to a constant) viscosity solution of the Hamilton-Jacobi equation H(x, Du (x)) = α(0). For c ∈ Rn, let Ac be the Aubry set corresponding to the Hamiltonian Hc(x, p ) = \\n\\nH(x, p − c). Let vc be a viscosity solution of Hc(x, Dv c(x)) = α(c). Question: is it true that lim \\n\\n> |c|→ 0\\n\\n‖Dv c − Du ‖Lip ( π(Ac)) = 0.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0090", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The merged record contains AIM Problems 16-20. For Problem 20, the exact-gauge Tonelli class H(x,p)=a+K(p-Du(x)), with K a superlinear momentum Hamiltonian having positive-definite Hessian, has critical value alpha_bar(c)=a+K(-c), the same full Aubry graph graph(Du) for every shift H_c(x,p)=H(x,p-c), and every critical viscosity solution v_c equals u up to a constant; hence the full Lipschitz gradient difference is identically zero. More generally, a uniformly convex equilibrium critical graph satisfies sup over pi(A_c) of |Dv_c-Du| at most (1+sqrt(M/m))|c|.\n\nCandidate contribution (quantitative_special_case; novelty confidence low): Exact momentum-gauge Hamiltonians solve the full Lipschitz reading of AIM Problem 20 with zero error for every cohomology shift, while uniformly convex equilibrium critical graphs obey the explicit uniform estimate (1+sqrt(M/m))|c| on the projected perturbed Aubry set." }, { "id": 20002479, "problem_number": "AIM-PDES-0091", "title": "Generic Mather measures, periodic support, and rational locking flats", "statement": "21. Mane proved that for a generic positive definite Lagrangian L: T M × T → R and generic c ∈ H1(M, R), the minimizing measure in the Mather set Mc is unique. Problem: For generic L and c is the unique minimizing measure in Mc supported in a periodic orbit? Conjecture (Jeff Xia): For a generic L and all c ∈ H1(M, R), the number of ergodic invariant measures in Mc is finite.\n2", "original_statement": "21. Mane proved that for a generic positive definite Lagrangian L: T M × T → R and generic c ∈ H1(M, R), the minimizing measure in the Mather set Mc is unique. Problem: For generic L and c is the unique minimizing measure in Mc supported in a periodic orbit? Conjecture (Jeff Xia): For a generic L and all c ∈ H1(M, R), the number of ergodic invariant measures in Mc is finite. \n2", "clean_statement": "21. Mane proved that for a generic positive definite Lagrangian L: T M × T → R and generic c ∈ H1(M, R), the minimizing measure in the Mather set Mc is unique. Problem: For generic L and c is the unique minimizing measure in Mc supported in a periodic orbit? Conjecture (Jeff Xia): For a generic L and all c ∈ H1(M, R), the number of ergodic invariant measures in Mc is finite.\n2", "statement_status": "exact", "statement_verification": "The canonical JSON record is an OCR extraction of Problem 21 from the AIM workshop list *New connections between dynamical systems and PDE's*. The original AIM HTML version confirms the following statement (notation modernized only typographically):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 21\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[90]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"21. Mane proved that for a generic positive definite Lagrangian L: T M × T → R and generic c ∈ H1(M, R), the minimizing measure in the Mather set Mc is unique. Problem: For generic L and c is the unique minimizing measure in Mc supported in a periodic orbit? Conjecture (Jeff Xia): For a generic L and all c ∈ H1(M, R), the number of ergodic invariant measures in Mc is finite. \\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0091", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Bernard and Contreras proved Xia's finiteness conjecture in 2008, with the stronger Mañe-generic bound of at most 1+b_1(M) ergodic minimizing measures for every cohomology class. The general periodic-support question remains open in the full time-periodic, arbitrary-dimensional category. This attempt proves that residual occurrence of periodic minimizing measures on an open cohomology region forces dense interiors of rational subgradient fibers of Mather's alpha-function; strict convexity therefore makes periodic minimizing classes meagre. An exact flat-torus model exhibits a residual full-measure set of classes with a unique but nonperiodic minimizing measure.\n\nCandidate contribution (theorem; novelty confidence low): If periodic minimizing measures exist for a residual set of cohomology classes in a nonempty open region U, then the interiors in U of the rational subgradient fibers F_h={c:h in partial alpha(c)} have dense union; consequently, if alpha is strictly convex on a convex U, periodic minimizing classes form a meagre, indeed countable, subset of U." }, { "id": 20002480, "problem_number": "AIM-PDES-0092", "title": "Tensorization of white-noise hyperbolic minimizers", "statement": "2. (Kostia Khanin) Consider a convex superlinear positive definite Lagrangian on Tn ×\n\nRn with white noise perturbation:\n\nL(x, v, t ) = L0(x, v ) +\n\n> N\n\n∑\n\n> i=1\n\nFi(x) ˙ wi(t),\n\nwhere wi(t) are independent Brownian motions. Suppose that the map F = ( F1,..., F n):\n\nTn → RN is an embedding. Then with probability 1 there exists a unique global minimizer γ: R → M.Conjecture: With probability 1, the minimizer γ is a hyperbolic trajectory of the Lagrangian flow. This is proved (E-Khanin-Mazel-Sinai) for n = 1.\n2", "original_statement": "2. (Kostia Khanin) Consider a convex superlinear positive definite Lagrangian on Tn ×\n\nRn with white noise perturbation: \n\nL(x, v, t ) = L0(x, v ) + \n\n> N\n\n∑\n\n> i=1\n\nFi(x) ˙ wi(t),\n\nwhere wi(t) are independent Brownian motions. Suppose that the map F = ( F1,..., F n): \n\nTn → RN is an embedding. Then with probability 1 there exists a unique global minimizer γ: R → M.Conjecture: With probability 1, the minimizer γ is a hyperbolic trajectory of the Lagrangian flow. This is proved (E-Khanin-Mazel-Sinai) for n = 1. \n2", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The official AIM PDF, version dated 15 August 2003, gives this as Problem 22, proposed by Kostia Khanin:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[91]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (Kostia Khanin) Consider a convex superlinear positive definite Lagrangian on Tn ×\\n\\nRn with white noise perturbation: \\n\\nL(x, v, t ) = L0(x, v ) + \\n\\n> N\\n\\n∑\\n\\n> i=1\\n\\nFi(x) ˙ wi(t),\\n\\nwhere wi(t) are independent Brownian motions. Suppose that the map F = ( F1,..., F n): \\n\\nTn → RN is an embedding. Then with probability 1 there exists a unique global minimizer γ: R → M.Conjecture: With probability 1, the minimizer γ is a hyperbolic trajectory of the Lagrangian flow. This is proved (E-Khanin-Mazel-Sinai) for n = 1. \\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0092", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The exact continuous white-noise AIM conjecture is proved for separable higher-dimensional systems assembled from established one-dimensional factors. The pathwise action splits, so the unique global minimizer is the product of the factor minimizers; the derivative cocycle is a block direct sum, so its Lyapunov spectrum is the multiset union of the factor spectra and its gap from zero is the minimum factor gap. Product forcing maps remain embeddings. The source is also repaired conservatively: it is Problem 22, the forcing vector naturally has N components, and the minimizer takes values in the torus rather than an undefined M.\n\nCandidate contribution (special_case; novelty confidence low): For a separable n-dimensional continuous white-noise Tonelli Lagrangian whose one-dimensional factors have unique hyperbolic global minimizers with positive exponents lambda_j, the product has a unique hyperbolic global minimizer, exact Lyapunov spectrum equal to the union of the factor spectra, and spectral gap min_j lambda_j; coordinate-block forcing embeddings tensorize to an embedding of the n-torus." }, { "id": 20002481, "problem_number": "AIM-PDES-0093", "title": "Arnold diffusion, radial genericity, and separatrix-map actions", "statement": "3. (Arnold diffusion) Consider a Cr, 3 ≤ r ≤ ω, Hamiltonian Hε: Tn × Rn × T → R\n\nof the form\n\nHε(x, p, t ) = H0(p) + εH 1(x, p, t ).\n\nSuppose that H0 is positive definite and superlinear. 7\n\nConjecture: Suppose that n ≥ 2. Then for given open sets U0, U 1 ⊂ Rn and typical in the Cr topology H1, there exists ε0 > 0 such that for any ε ∈ (0, ε 0), there exists a trajectory ( x(t), p (t)) of the Hamiltonian system such that p(t0) ∈ U0 and p(t1) ∈ U1.This conjecture seems very hard in the Cω category, when the change of the action is exponentially slow in ε by the Nekhoroshev theorem. The case of finite r should be easier. A precise definition of \"typical\" needs to be established. Mather proved this conjec-ture for n = 2 and for a cusp residual set of perturbations εH 1 in the C∞ topology. This doesn't prove the above conjecture: for given H1 the set of admissible ε does not cover an interval (0, ε 0). Question: Does Mather's theorem holds for an εH 1 in a cusp residual set of trigono-metric polynomials of high order N? Exponentially small perturbations of coefficients are allowed, so the transcendental problem of exponentially small separatrice splitting is avoided. 24. Consider an apriori unstable Hamiltonian of the form\n\nH(θ, I, x, y ) = H0(I) + F (x, y ) + εH 1(θ, I, x, y ),\n\non Tn × Rn × R2, where H0 is convex and superlinear, the Hamiltonian F: R2 → R\n\nhas a separatrix loop, and H1 is a generic perturbation. The large gap problem of Arnold diffusion was overpassed recently for such systems by variational methods of Mather (Xia), by geometrical methods using secondary KAM-tori (de la Llave, Delshams, Seara), and by the method of separatrix map (Treschev). Problem: Understand the relation between the variational method and the method of separatrix map. They seem similar in spirit.\n2", "original_statement": "3. (Arnold diffusion) Consider a Cr, 3 ≤ r ≤ ω, Hamiltonian Hε: Tn × Rn × T → R\n\nof the form \n\nHε(x, p, t ) = H0(p) + εH 1(x, p, t ).\n\nSuppose that H0 is positive definite and superlinear. 7\n\nConjecture: Suppose that n ≥ 2. Then for given open sets U0, U 1 ⊂ Rn and typical in the Cr topology H1, there exists ε0 > 0 such that for any ε ∈ (0, ε 0), there exists a trajectory ( x(t), p (t)) of the Hamiltonian system such that p(t0) ∈ U0 and p(t1) ∈ U1.This conjecture seems very hard in the Cω category, when the change of the action is exponentially slow in ε by the Nekhoroshev theorem. The case of finite r should be easier. A precise definition of \"typical\" needs to be established. Mather proved this conjec-ture for n = 2 and for a cusp residual set of perturbations εH 1 in the C∞ topology. This doesn't prove the above conjecture: for given H1 the set of admissible ε does not cover an interval (0, ε 0). Question: Does Mather's theorem holds for an εH 1 in a cusp residual set of trigono-metric polynomials of high order N? Exponentially small perturbations of coefficients are allowed, so the transcendental problem of exponentially small separatrice splitting is avoided. 24. Consider an apriori unstable Hamiltonian of the form \n\nH(θ, I, x, y ) = H0(I) + F (x, y ) + εH 1(θ, I, x, y ),\n\non Tn × Rn × R2, where H0 is convex and superlinear, the Hamiltonian F: R2 → R\n\nhas a separatrix loop, and H1 is a generic perturbation. The large gap problem of Arnold diffusion was overpassed recently for such systems by variational methods of Mather (Xia), by geometrical methods using secondary KAM-tori (de la Llave, Delshams, Seara), and by the method of separatrix map (Treschev). Problem: Understand the relation between the variational method and the method of separatrix map. They seem similar in spirit. \n2", "clean_statement": "3. (Arnold diffusion) Consider a Cr, 3 ≤ r ≤ ω, Hamiltonian Hε: Tn × Rn × T → R\n\nof the form\n\nHε(x, p, t ) = H0(p) + εH 1(x, p, t ).\n\nSuppose that H0 is positive definite and superlinear. 7\n\nConjecture: Suppose that n ≥ 2. Then for given open sets U0, U 1 ⊂ Rn and typical in the Cr topology H1, there exists ε0 > 0 such that for any ε ∈ (0, ε 0), there exists a trajectory ( x(t), p (t)) of the Hamiltonian system such that p(t0) ∈ U0 and p(t1) ∈ U1.This conjecture seems very hard in the Cω category, when the change of the action is exponentially slow in ε by the Nekhoroshev theorem. The case of finite r should be easier. A precise definition of \"typical\" needs to be established. Mather proved this conjec-ture for n = 2 and for a cusp residual set of perturbations εH 1 in the C∞ topology. This doesn't prove the above conjecture: for given H1 the set of admissible ε does not cover an interval (0, ε 0). Question: Does Mather's theorem holds for an εH 1 in a cusp residual set of trigono-metric polynomials of high order N? Exponentially small perturbations of coefficients are allowed, so the transcendental problem of exponentially small separatrice splitting is avoided. 24. Consider an apriori unstable Hamiltonian of the form\n\nH(θ, I, x, y ) = H0(I) + F (x, y ) + εH 1(θ, I, x, y ),\n\non Tn × Rn × R2, where H0 is convex and superlinear, the Hamiltonian F: R2 → R\n\nhas a separatrix loop, and H1 is a generic perturbation. The large gap problem of Arnold diffusion was overpassed recently for such systems by variational methods of Mather (Xia), by geometrical methods using secondary KAM-tori (de la Llave, Delshams, Seara), and by the method of separatrix map (Treschev). Problem: Understand the relation between the variational method and the method of separatrix map. They seem similar in spirit.\n2", "statement_status": "exact", "statement_verification": "The canonical record is not one problem. It is an OCR merge of two consecutive entries in the AIM workshop list *New connections between dynamical systems and PDE's* (notes by S. Bolotin). The original AIM HTML page presents them as separate list items. In the PDF numbering they are Problems 23 and 24: the extraction lost the leading “2” in “23.” but retained “24.” inside the same JSON string.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[92]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (Arnold diffusion) Consider a Cr, 3 ≤ r ≤ ω, Hamiltonian Hε: Tn × Rn × T → R\\n\\nof the form \\n\\nHε(x, p, t ) = H0(p) + εH 1(x, p, t ).\\n\\nSuppose that H0 is positive definite and superlinear. 7\\n\\nConjecture: Suppose that n ≥ 2. Then for given open sets U0, U 1 ⊂ Rn and typical in the Cr topology H1, there exists ε0 > 0 such that for any ε ∈ (0, ε 0), there exists a trajectory ( x(t), p (t)) of the Hamiltonian system such that p(t0) ∈ U0 and p(t1) ∈ U1.This conjecture seems very hard in the Cω category, when the change of the action is exponentially slow in ε by the Nekhoroshev theorem. The case of finite r should be easier. A precise definition of \\\"typical\\\" needs to be established. Mather proved this conjec-ture for n = 2 and for a cusp residual set of perturbations εH 1 in the C∞ topology. This doesn't prove the above conjecture: for given H1 the set of admissible ε does not cover an interval (0, ε 0). Question: Does Mather's theorem holds for an εH 1 in a cusp residual set of trigono-metric polynomials of high order N? Exponentially small perturbations of coefficients are allowed, so the transcendental problem of exponentially small separatrice splitting is avoided. 24. Consider an apriori unstable Hamiltonian of the form \\n\\nH(θ, I, x, y ) = H0(I) + F (x, y ) + εH 1(θ, I, x, y ),\\n\\non Tn × Rn × R2, where H0 is convex and superlinear, the Hamiltonian F: R2 → R\\n\\nhas a separatrix loop, and H1 is a generic perturbation. The large gap problem of Arnold diffusion was overpassed recently for such systems by variational methods of Mather (Xia), by geometrical methods using secondary KAM-tori (de la Llave, Delshams, Seara), and by the method of separatrix map (Treschev). Problem: Understand the relation between the variational method and the method of separatrix map. They seem similar in spirit. \\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0093", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The canonical OCR record merges AIM Problems 23 and 24. Modern smooth results prove strong Arnold diffusion in two and a half degrees of freedom and weaker O(1) drift in arbitrary time-periodic dimension, while autonomous cusp-residual same-energy connecting theorems are also known; the full fixed-direction, every-small-amplitude, arbitrary-endpoint time-periodic statement and its fixed-order trigonometric-polynomial version were not found as solved. This attempt proves that even open-dense perturbation sets with open-dense amplitude sections on every ray need not contain any terminal amplitude interval, and that ambient C^r genericity need not restrict to a finite Fourier space. It gives a positive leading-jet theorem that upgrades finitely many nonflat analytic coefficient discriminants to all-small-amplitude validity on an open dense set of polynomial directions, and an exact generating-function identity showing that variational critical sequences and twist-kick separatrix-map orbits obey the same local equations.\n\nCandidate contribution (reduction; novelty confidence low): For a finite-dimensional trigonometric coefficient family, if a global diffusion construction can be certified by finitely many discriminants D_j(epsilon,a)=epsilon^{m_j}(d_j(a)+o(1)) locally uniformly, with each leading coefficient d_j real analytic and nonzero, then an open dense set of coefficient directions a satisfies every certification condition for every sufficiently small epsilon; the amplitude cutoff is uniform on compact subsets avoiding the leading zero loci." }, { "id": 20002482, "problem_number": "AIM-PDES-0094", "title": "A conservation-aware calibration of PDE Arnold diffusion", "statement": "5. (Paul Rabinowitz) Is there a reasonable PDE analogue of Arnold's diffusion? It seems that a version of finite dimensional transition chains is not the right mech-anism for PDE. There are results of Kuksin which prove \"diffusion\" for PDE with respect to high order Sobolev norms. The same problem for infinite lattices like Fermi-Pasta-Ulam system.\n2", "original_statement": "5. (Paul Rabinowitz) Is there a reasonable PDE analogue of Arnold's diffusion? It seems that a version of finite dimensional transition chains is not the right mech-anism for PDE. There are results of Kuksin which prove \"diffusion\" for PDE with respect to high order Sobolev norms. The same problem for infinite lattices like Fermi-Pasta-Ulam system. \n2", "clean_statement": "5. (Paul Rabinowitz) Is there a reasonable PDE analogue of Arnold's diffusion? It seems that a version of finite dimensional transition chains is not the right mech-anism for PDE. There are results of Kuksin which prove \"diffusion\" for PDE with respect to high order Sobolev norms. The same problem for infinite lattices like Fermi-Pasta-Ulam system.\n2", "statement_status": "exact", "statement_verification": "The canonical JSON record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[93]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"5. (Paul Rabinowitz) Is there a reasonable PDE analogue of Arnold's diffusion? It seems that a version of finite dimensional transition chains is not the right mech-anism for PDE. There are results of Kuksin which prove \\\"diffusion\\\" for PDE with respect to high order Sobolev norms. The same problem for infinite lattices like Fermi-Pasta-Ulam system. \\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0094", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source is original AIM Problem 25, not 5. The broad question is partially solved: arbitrary finite Sobolev growth and several heteroclinic energy-exchange mechanisms are known for Hamiltonian PDEs, while specially coupled infinite pendulum lattices admit genuine infinite transition chains; unbounded Sobolev growth for the standard cubic defocusing NLS on the square torus and diffusion for classical FPU remain open. The proved contribution shows that, for any mass-conserving Fourier trajectory and s at least 1/2, the characteristic homogeneous Sobolev frequency is comparable by sharp triangle inequalities to the Wasserstein-2s displacement of normalized modal mass. An explicit thin-tail family has divergent high Sobolev/Wasserstein-2s displacement while total variation and Wasserstein-1 displacement vanish, so Sobolev growth does not imply macroscopic mass transport.\n\nCandidate contribution (equivalence_and_counterexample; novelty confidence low): For conserved modal mass, unbounded homogeneous H^s growth is equivalent to unbounded Wasserstein-2s drift of the normalized Fourier-mass distribution, but the explicit family (1-N^{-s}) delta_0 + N^{-s} delta_{N e_1}, for s greater than 1, proves that this can occur while both transported mass and Wasserstein-1 cost tend to zero." }, { "id": 20002483, "problem_number": "AIM-PDES-0095", "title": "Smooth periodic bounded-gradient minimizers need not be Birkhoff", "statement": "6. (Victor Bangert) Let L: Tn × T × Rn → R, L = L(x, u, p ) be a smooth multi-dimensional Lagrangian satisfying the usual convexity assumptions in p ∈ Rn. For example,\n\nL(x, u, p ) = 1\n\n2|p|2 + F (x, u ),\n\nwhere the potential F: Tn+1 → R is periodic in all variables. A function u: Rn → R is called minimal if it minimizes the action functional\n\n∫\n\nL(x, u (x), Du (x)) dx\n\nfor all variations with compact support. The set of minimizers u \"without self-intersections\" is very well understood. Here u: Rn → R is said to be \"without self-intersections\" if the projection to Tn+1 of graph( u) ⊂ Rn+1 is a hypersurface without 8\n\nself-intersections. In particular, for every minimizer u without self-intersections there exists a \"rotation vector\" c ∈ (Rn)∗ such that\n\n|u(x) − u(0) − c · x|\n\nis bounded by a constant that only depends on L. Conversely, for every c ∈ (Rn)∗\n\nthere exists a minimizer u without self-intersections such that |u(x)−c·x| is bounded. Question: Is there an analytical condition implying that the minimizers satisfying this condition have no self-intersections. More concrete question: Suppose u: Rn → R is minimal and |Du (x)| is bounded. Is it true that the graph of u in Tn+1 has no self-intersections. For partial results see V. Bangert, Ann. Inst. Henri Poincar´ e - Analyse non lin´ eaire 6(1989), 95-138, in particular Sect. 8.\n2", "original_statement": "6. (Victor Bangert) Let L: Tn × T × Rn → R, L = L(x, u, p ) be a smooth multi-dimensional Lagrangian satisfying the usual convexity assumptions in p ∈ Rn. For example, \n\nL(x, u, p ) = 1\n\n2|p|2 + F (x, u ),\n\nwhere the potential F: Tn+1 → R is periodic in all variables. A function u: Rn → R is called minimal if it minimizes the action functional \n\n∫\n\nL(x, u (x), Du (x)) dx \n\nfor all variations with compact support. The set of minimizers u \"without self-intersections\" is very well understood. Here u: Rn → R is said to be \"without self-intersections\" if the projection to Tn+1 of graph( u) ⊂ Rn+1 is a hypersurface without 8\n\nself-intersections. In particular, for every minimizer u without self-intersections there exists a \"rotation vector\" c ∈ (Rn)∗ such that \n\n|u(x) − u(0) − c · x|\n\nis bounded by a constant that only depends on L. Conversely, for every c ∈ (Rn)∗\n\nthere exists a minimizer u without self-intersections such that |u(x)−c·x| is bounded. Question: Is there an analytical condition implying that the minimizers satisfying this condition have no self-intersections. More concrete question: Suppose u: Rn → R is minimal and |Du (x)| is bounded. Is it true that the graph of u in Tn+1 has no self-intersections. For partial results see V. Bangert, Ann. Inst. Henri Poincar´ e - Analyse non lin´ eaire 6(1989), 95-138, in particular Sect. 8. \n2", "clean_statement": "6. (Victor Bangert) Let L: Tn × T × Rn → R, L = L(x, u, p ) be a smooth multi-dimensional Lagrangian satisfying the usual convexity assumptions in p ∈ Rn. For example,\n\nL(x, u, p ) = 1\n\n2|p|2 + F (x, u ),\n\nwhere the potential F: Tn+1 → R is periodic in all variables. A function u: Rn → R is called minimal if it minimizes the action functional\n\n∫\n\nL(x, u (x), Du (x)) dx\n\nfor all variations with compact support. The set of minimizers u \"without self-intersections\" is very well understood. Here u: Rn → R is said to be \"without self-intersections\" if the projection to Tn+1 of graph( u) ⊂ Rn+1 is a hypersurface without 8\n\nself-intersections. In particular, for every minimizer u without self-intersections there exists a \"rotation vector\" c ∈ (Rn)∗ such that\n\n|u(x) − u(0) − c · x|\n\nis bounded by a constant that only depends on L. Conversely, for every c ∈ (Rn)∗\n\nthere exists a minimizer u without self-intersections such that |u(x)−c·x| is bounded. Question: Is there an analytical condition implying that the minimizers satisfying this condition have no self-intersections. More concrete question: Suppose u: Rn → R is minimal and |Du (x)| is bounded. Is it true that the graph of u in Tn+1 has no self-intersections. For partial results see V. Bangert, Ann. Inst. Henri Poincar´ e - Analyse non lin´ eaire 6(1989), 95-138, in particular Sect. 8.\n2", "statement_status": "exact", "statement_verification": "The canonical JSON is an OCR extraction of Problem 26 (Victor Bangert) in the AIM list *New connections between dynamical systems and PDE's*. The JSON number “6” has lost its leading “2”; the isolated “8” and final “2” in the prose are page-number artifacts. The official AIM PDF gives the following problem.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[94]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"6. (Victor Bangert) Let L: Tn × T × Rn → R, L = L(x, u, p ) be a smooth multi-dimensional Lagrangian satisfying the usual convexity assumptions in p ∈ Rn. For example, \\n\\nL(x, u, p ) = 1\\n\\n2|p|2 + F (x, u ),\\n\\nwhere the potential F: Tn+1 → R is periodic in all variables. A function u: Rn → R is called minimal if it minimizes the action functional \\n\\n∫\\n\\nL(x, u (x), Du (x)) dx \\n\\nfor all variations with compact support. The set of minimizers u \\\"without self-intersections\\\" is very well understood. Here u: Rn → R is said to be \\\"without self-intersections\\\" if the projection to Tn+1 of graph( u) ⊂ Rn+1 is a hypersurface without 8\\n\\nself-intersections. In particular, for every minimizer u without self-intersections there exists a \\\"rotation vector\\\" c ∈ (Rn)∗ such that \\n\\n|u(x) − u(0) − c · x|\\n\\nis bounded by a constant that only depends on L. Conversely, for every c ∈ (Rn)∗\\n\\nthere exists a minimizer u without self-intersections such that |u(x)−c·x| is bounded. Question: Is there an analytical condition implying that the minimizers satisfying this condition have no self-intersections. More concrete question: Suppose u: Rn → R is minimal and |Du (x)| is bounded. Is it true that the graph of u in Tn+1 has no self-intersections. For partial results see V. Bangert, Ann. Inst. Henri Poincar´ e - Analyse non lin´ eaire 6(1989), 95-138, in particular Sect. 8. \\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0095", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The bounded-gradient implication in AIM Problem 26 is false. For every n at least 8, a non-flat Liu-Wang-Wei global Allen-Cahn minimizer can be spatially rescaled to the exact Junginger-Gestrich-Valdinoci normalization, compressed into the vertical interval (1/4,3/4), and made into a class-A minimizer for a nonnegative smooth one-periodic potential. A clipping argument preserves minimality against competitors leaving that interval, elliptic interior estimates give bounded gradient, and Birkhoff ordering would force hyperplane level sets, contradicting the Lawson-cone asymptotics. Passive-variable lifting gives all higher dimensions.\n\nCandidate contribution (lemma; novelty confidence low): Candidate smooth periodicization-and-lifting lemma: every (0,1)-valued class-A minimizer of the normalized double-well functional integral(|Dh|^2+W(h)) can be compressed to v=1/4+h/2 and transferred to a nonnegative C-infinity one-periodic potential using P(r)=W(2r-1/2)/8; clipping to [1/4,3/4] proves class-A minimality for arbitrary compact competitors, and adding passive variables preserves minimality." }, { "id": 20002484, "problem_number": "AIM-PDES-0096", "title": "Intermediate-dimensional minimizing currents and a flat-torus obstruction", "statement": "7. (Victor Bangert) Let M n be a compact Riemannian manifold and let CqM be the set of closed q-currents on M:\n\nCqM = {T ∈ (Ω q(M )) ∗: T (dα ) = 0 for all α ∈ Ωq−1(M )}.\n\nEvery q-current T defines a homology class [ T ] ∈ Hq(M, R) and a mass\n\nM (T ) = sup {T (ω): ω ∈ Ωq(M ), ‖ω‖∞ = 1 }.\n\nEvery homology class h ∈ Hq(M, R) has a representative with minimal mass. For q = 1 the supports of minimal currents consist of minimal geodesics. For q = n−1any minimizer T is given by a measured lamination by minimizing hypersurfaces (possibly with singularities). Problem: What can one say about minimal currents for 1 < q < n − 1.\n2", "original_statement": "7. (Victor Bangert) Let M n be a compact Riemannian manifold and let CqM be the set of closed q-currents on M:\n\nCqM = {T ∈ (Ω q(M )) ∗: T (dα ) = 0 for all α ∈ Ωq−1(M )}.\n\nEvery q-current T defines a homology class [ T ] ∈ Hq(M, R) and a mass \n\nM (T ) = sup {T (ω): ω ∈ Ωq(M ), ‖ω‖∞ = 1 }.\n\nEvery homology class h ∈ Hq(M, R) has a representative with minimal mass. For q = 1 the supports of minimal currents consist of minimal geodesics. For q = n−1any minimizer T is given by a measured lamination by minimizing hypersurfaces (possibly with singularities). Problem: What can one say about minimal currents for 1 < q < n − 1. \n2", "clean_statement": "7. (Victor Bangert) Let M n be a compact Riemannian manifold and let CqM be the set of closed q-currents on M:\n\nCqM = {T ∈ (Ω q(M )) ∗: T (dα ) = 0 for all α ∈ Ωq−1(M )}.\n\nEvery q-current T defines a homology class [ T ] ∈ Hq(M, R) and a mass\n\nM (T ) = sup {T (ω): ω ∈ Ωq(M ), ‖ω‖∞ = 1 }.\n\nEvery homology class h ∈ Hq(M, R) has a representative with minimal mass. For q = 1 the supports of minimal currents consist of minimal geodesics. For q = n−1any minimizer T is given by a measured lamination by minimizing hypersurfaces (possibly with singularities). Problem: What can one say about minimal currents for 1 < q < n − 1.\n2", "statement_status": "exact", "statement_verification": "The canonical record is OCR from Problem 27 of the AIM workshop list *New connections between dynamical systems and PDE's* (notes by S. Bolotin). The extraction lost the leading “2” in “27.” The original AIM HTML confirms the following notation and question:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[95]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"7. (Victor Bangert) Let M n be a compact Riemannian manifold and let CqM be the set of closed q-currents on M:\\n\\nCqM = {T ∈ (Ω q(M )) ∗: T (dα ) = 0 for all α ∈ Ωq−1(M )}.\\n\\nEvery q-current T defines a homology class [ T ] ∈ Hq(M, R) and a mass \\n\\nM (T ) = sup {T (ω): ω ∈ Ωq(M ), ‖ω‖∞ = 1 }.\\n\\nEvery homology class h ∈ Hq(M, R) has a representative with minimal mass. For q = 1 the supports of minimal currents consist of minimal geodesics. For q = n−1any minimizer T is given by a measured lamination by minimizing hypersurfaces (possibly with singularities). Problem: What can one say about minimal currents for 1 < q < n − 1. \\n2\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0096", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source's mass norm must use pointwise comass, and the structural answer depends on whether one minimizes among real normal currents or integral rectifiable currents. This attempt proves that on a unit flat torus every constant q-vector defines a calibrated real-normal current whose mass equals the algebraic mass norm and which realizes the stable norm. For every intermediate degree 2<=q<=n-2, an explicit nondecomposable constant q-vector yields a diffuse, nonrectifiable minimizing current that cannot be a single measured lamination; nevertheless the same integral homology class also has a calibrated integral minimizer given by the sum of two coordinate q-tori. This supplies a sharp obstruction to extending the endpoint lamination picture to all real minimizers.\n\nCandidate contribution (theorem; novelty confidence low): For each 2<=q<=n-2 on a flat n-torus, the integral class represented by a wedge of a common (q-2)-vector with e_{q-1} wedge e_q plus e_{q+1} wedge e_{q+2} has both a mass-two calibrated integral minimizer and a mass-two calibrated diffuse real-normal minimizer with nondecomposable polar, so the latter is neither rectifiable nor the current of a single oriented measured q-lamination." }, { "id": 20002485, "problem_number": "AIM-PDES-0097", "title": "Aggregate singular sets of minimizing laminations and the codimension-one stable norm", "statement": "8. (Franz Auer) Conjecture: If T ∈ Cn−1M is a minimizing closed current, then the Hausdorff dimension of the corresponding singularity set is at most n − 7. 29. Define a stable norm on Hq(M, R) by\n\n‖h‖ = inf {M (T ): [ T ] = h}.\n\nThis norm is an analog of Mather's β-function). Problem: Study convexity and differentiability properties of the unit ball\n\nB = {h ∈ Hq(M, R): ‖h‖ ≤ 1}.\n\n3", "original_statement": "8. (Franz Auer) Conjecture: If T ∈ Cn−1M is a minimizing closed current, then the Hausdorff dimension of the corresponding singularity set is at most n − 7. 29. Define a stable norm on Hq(M, R) by \n\n‖h‖ = inf {M (T ): [ T ] = h}.\n\nThis norm is an analog of Mather's β-function). Problem: Study convexity and differentiability properties of the unit ball \n\nB = {h ∈ Hq(M, R): ‖h‖ ≤ 1}.\n\n3", "clean_statement": "8. (Franz Auer) Conjecture: If T ∈ Cn−1M is a minimizing closed current, then the Hausdorff dimension of the corresponding singularity set is at most n − 7. 29. Define a stable norm on Hq(M, R) by\n\n‖h‖ = inf {M (T ): [ T ] = h}.\n\nThis norm is an analog of Mather's β-function). Problem: Study convexity and differentiability properties of the unit ball\n\nB = {h ∈ Hq(M, R): ‖h‖ ≤ 1}.\n\n3", "statement_status": "exact", "statement_verification": "The canonical JSON record is visibly concatenated. It begins with", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[96]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"8. (Franz Auer) Conjecture: If T ∈ Cn−1M is a minimizing closed current, then the Hausdorff dimension of the corresponding singularity set is at most n − 7. 29. Define a stable norm on Hq(M, R) by \\n\\n‖h‖ = inf {M (T ): [ T ] = h}.\\n\\nThis norm is an analog of Mather's β-function). Problem: Study convexity and differentiability properties of the unit ball \\n\\nB = {h ∈ Hq(M, R): ‖h‖ ≤ 1}.\\n\\n3\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0097", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The canonical record contains both original Problems 28 and 29. For Problem 28, the report derives a Riemannian disjoint-family singular-set theorem from the Chodosh-Mantoulidis-Schulze Hausdorff-content argument and their explicit variable-metric clustering lemma; combined with the Auer-Bangert measured-leaf decomposition and the strict maximum principle, it gives dim_H sing_AB(T) <= N-8 for a minimizing real codimension-one current on an N-manifold, stronger than Auer's proposed N-7 bound. For Problem 29, it proves the elementary convexity, symmetry, and compactness of the stable unit ball and records the known Auer-Bangert codimension-one differentiability theorem, while leaving the broad general-q program open.\n\nCandidate contribution (theorem_and_application; novelty confidence low): The CMS disjoint-family singular-stratum method, with its C^2-variable-metric clustering extension, applies to the Auer-Bangert leaf family and yields the aggregate bound dim_H sing_AB(T) <= N-8 on every smooth Riemannian N-manifold." }, { "id": 20002486, "problem_number": "AIM-PDES-0098", "title": "Chern and slope obstructions for pseudoholomorphic plane foliations on the four-torus", "statement": "0. (Victor Bangert) Let M 2n be a manifold with an almost complex structure J. Apseudo-holomorphic line is a map f: R2 → M satisfying the PDE: fy = Jf x. Moser proved that for an almost complex structure J on T2n which is close to a standard complex structure J0, and any sufficiently irrational vector v ∈ R2n, there exists a foliation of T2n by pseudo-holomorphic curves which is conjugate to a linear foliation by real 2-planes containing the vector v.Problem: Develop a global (non-perturbative) theory of such foliations or lamina-tions. The theory is expected to be particularly rich for the case T4 due to the positivity of the intersection number of pseudoholomorphic curves. This is an analog of the 9\n\nintersection number for geodesics on T2 - the basis for Hedlund's results on minimal geodesics.\n3", "original_statement": "0. (Victor Bangert) Let M 2n be a manifold with an almost complex structure J. Apseudo-holomorphic line is a map f: R2 → M satisfying the PDE: fy = Jf x. Moser proved that for an almost complex structure J on T2n which is close to a standard complex structure J0, and any sufficiently irrational vector v ∈ R2n, there exists a foliation of T2n by pseudo-holomorphic curves which is conjugate to a linear foliation by real 2-planes containing the vector v.Problem: Develop a global (non-perturbative) theory of such foliations or lamina-tions. The theory is expected to be particularly rich for the case T4 due to the positivity of the intersection number of pseudoholomorphic curves. This is an analog of the 9\n\nintersection number for geodesics on T2 - the basis for Hedlund's results on minimal geodesics. \n3", "clean_statement": "0. (Victor Bangert) Let M 2n be a manifold with an almost complex structure J. Apseudo-holomorphic line is a map f: R2 → M satisfying the PDE: fy = Jf x. Moser proved that for an almost complex structure J on T2n which is close to a standard complex structure J0, and any sufficiently irrational vector v ∈ R2n, there exists a foliation of T2n by pseudo-holomorphic curves which is conjugate to a linear foliation by real 2-planes containing the vector v.Problem: Develop a global (non-perturbative) theory of such foliations or lamina-tions. The theory is expected to be particularly rich for the case T4 due to the positivity of the intersection number of pseudoholomorphic curves. This is an analog of the 9\n\nintersection number for geodesics on T2 - the basis for Hedlund's results on minimal geodesics.\n3", "statement_status": "exact", "statement_verification": "The canonical JSON record is an OCR extraction from Problem 30 in the AIM workshop list *New connections between dynamical systems and PDE's*. It reads “0.” because the leading digit 3 was lost. It also contains the page artifacts “9” and “3,” splits “laminations,” and suppresses superscripts and subscripts.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 0\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[97]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"0. (Victor Bangert) Let M 2n be a manifold with an almost complex structure J. Apseudo-holomorphic line is a map f: R2 → M satisfying the PDE: fy = Jf x. Moser proved that for an almost complex structure J on T2n which is close to a standard complex structure J0, and any sufficiently irrational vector v ∈ R2n, there exists a foliation of T2n by pseudo-holomorphic curves which is conjugate to a linear foliation by real 2-planes containing the vector v.Problem: Develop a global (non-perturbative) theory of such foliations or lamina-tions. The theory is expected to be particularly rich for the case T4 due to the positivity of the intersection number of pseudoholomorphic curves. This is an analog of the 9\\n\\nintersection number for geodesics on T2 - the basis for Hedlund's results on minimal geodesics. \\n3\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0098", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "If a C1 diffeomorphism conjugates a linear plane foliation of T^4 to a foliation by J-holomorphic curves, then c1(TT^4,J)=0. Pulling the standard twistor two-sphere of orthogonal complex structures back along a nonzero-degree map T^2 to S^2 produces explicit almost complex structures with nonzero first Chern class and therefore no such foliation. In the tame case, the transported Ruelle-Sullivan slope class must pair positively with the taming class; in a transverse graph chart, J-invariance and integrability are exactly an algebraic Riccati equation and a commuting-frame first-order PDE.\n\nCandidate contribution (obstruction theorem; novelty confidence low): A C1-conjugate linear pseudoholomorphic plane foliation on T^4 forces c1(TT^4,J)=0, and the pulled-back twistor family gives an explicit infinite class of nonzero-c1 almost complex structures excluded by this condition." }, { "id": 20002487, "problem_number": "AIM-PDES-0099", "title": "Quantizing Mather measures by equilibrium packets and smooth-graph WKB states", "statement": "1. (Craig Evans) Consider the Hamiltonian H(x, p ) = 1\n\n> 2\n\n|p|2 + V (x), where the potential\n\nV is Tn periodic. The corresponding Hamiltonian operator in quantum mechanics is then − h2\n\n> 2\n\n∆ + V (x). Much exciting research in \"semiclassical analysis\" concerns studying the limit as\n\nh → 0 of solutions u(h) of the eigenvalue problem\n\nh2\n\n2 ∆u(h) + V (x)u(h) = E(h)u(h) (∗)and finding connections with classical Hamiltonian dynamics. Problem: Do Mather sets play any role here? Or, conversely, can we somehow \"quantize\" Mather sets? This would presumably mean to build quasimodes (ie ap-proximate solutions of (*)) corresponding to Mather's sets and to prove good error bounds. Would some sort of Diophantine condition be useful here?\n3", "original_statement": "1. (Craig Evans) Consider the Hamiltonian H(x, p ) = 1 \n\n> 2\n\n|p|2 + V (x), where the potential \n\nV is Tn periodic. The corresponding Hamiltonian operator in quantum mechanics is then − h2 \n\n> 2\n\n∆ + V (x). Much exciting research in \"semiclassical analysis\" concerns studying the limit as \n\nh → 0 of solutions u(h) of the eigenvalue problem \n\nh2\n\n2 ∆u(h) + V (x)u(h) = E(h)u(h) (∗)and finding connections with classical Hamiltonian dynamics. Problem: Do Mather sets play any role here? Or, conversely, can we somehow \"quantize\" Mather sets? This would presumably mean to build quasimodes (ie ap-proximate solutions of (*)) corresponding to Mather's sets and to prove good error bounds. Would some sort of Diophantine condition be useful here? \n3", "clean_statement": "1. (Craig Evans) Consider the Hamiltonian H(x, p ) = 1\n\n> 2\n\n|p|2 + V (x), where the potential\n\nV is Tn periodic. The corresponding Hamiltonian operator in quantum mechanics is then − h2\n\n> 2\n\n∆ + V (x). Much exciting research in \"semiclassical analysis\" concerns studying the limit as\n\nh → 0 of solutions u(h) of the eigenvalue problem\n\nh2\n\n2 ∆u(h) + V (x)u(h) = E(h)u(h) (∗)and finding connections with classical Hamiltonian dynamics. Problem: Do Mather sets play any role here? Or, conversely, can we somehow \"quantize\" Mather sets? This would presumably mean to build quasimodes (ie ap-proximate solutions of (*)) corresponding to Mather's sets and to prove good error bounds. Would some sort of Diophantine condition be useful here?\n3", "statement_status": "exact", "statement_verification": "The canonical record is source index 98 of `aim-pdes-notes.json`. It is Craig Evans's problem from the AIM workshop *New connections between dynamical systems and PDE's*. The source PDF/HTML labels it as Problem 31, although the extracted record has lost the leading digit and says “1.”", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[98]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (Craig Evans) Consider the Hamiltonian H(x, p ) = 1 \\n\\n> 2\\n\\n|p|2 + V (x), where the potential \\n\\nV is Tn periodic. The corresponding Hamiltonian operator in quantum mechanics is then − h2 \\n\\n> 2\\n\\n∆ + V (x). Much exciting research in \\\"semiclassical analysis\\\" concerns studying the limit as \\n\\nh → 0 of solutions u(h) of the eigenvalue problem \\n\\nh2\\n\\n2 ∆u(h) + V (x)u(h) = E(h)u(h) (∗)and finding connections with classical Hamiltonian dynamics. Problem: Do Mather sets play any role here? Or, conversely, can we somehow \\\"quantize\\\" Mather sets? This would presumably mean to build quasimodes (ie ap-proximate solutions of (*)) corresponding to Mather's sets and to prove good error bounds. Would some sort of Diophantine condition be useful here? \\n3\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0099", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the Hamiltonian-consistent operator P_h=-h^2 Delta/2+V, all zero-cohomology Mather measures are exactly mu(dx) delta_0 supported over the global maximum set K of V, and every such measure is the Wigner limit of normalized periodic quasimodes satisfying ||(P_h-V_max)u_h||<=C h. In addition, a smooth Hamilton-Jacobi graph carrying a positive smooth invariant density has periodic WKB quasimodes with the exact O(h^2) residual -(h^2/2)e^{iS/h} Delta a whenever the global Bohr-Sommerfeld holonomy condition holds.\n\nCandidate contribution (theorem; novelty confidence low): Every probability measure on the zero-class Mather set K times {0}, including arbitrary singular or nonatomic measures on a degenerate maximum set K, is realized by a diagonal sequence of periodic square-root-h packet superpositions with a uniform O(h) Schrödinger residual independent of the growing packet count." }, { "id": 20002488, "problem_number": "AIM-PDES-0100", "title": "Robust failure in nonconvex stochastic Hamilton-Jacobi homogenization", "statement": "2. (Takis Souganidis) Homogenization problem for random Hamilton-Jacobi equation\n\nF (D2u, Du, u, x/ε, x, ω ) = 0 with non-convex stationary ergodic in x/ε Hamiltonian F.\n3", "original_statement": "2. (Takis Souganidis) Homogenization problem for random Hamilton-Jacobi equation \n\nF (D2u, Du, u, x/ε, x, ω ) = 0 with non-convex stationary ergodic in x/ε Hamiltonian F.\n3", "clean_statement": "2. (Takis Souganidis) Homogenization problem for random Hamilton-Jacobi equation\n\nF (D2u, Du, u, x/ε, x, ω ) = 0 with non-convex stationary ergodic in x/ε Hamiltonian F.\n3", "statement_status": "exact", "statement_verification": "The canonical record is an OCR extraction from the AIM workshop list *New connections between dynamical systems and PDE's*. The official AIM HTML and PDF agree on the following text:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[99]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (Takis Souganidis) Homogenization problem for random Hamilton-Jacobi equation \\n\\nF (D2u, Du, u, x/ε, x, ω ) = 0 with non-convex stationary ergodic in x/ε Hamiltonian F.\\n3\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0100", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted stationary-ergodic nonconvex homogenization question is false in dimension two in both first-order and viscous subclasses. A proved comparison-stability theorem shows more: if two parabolic operators differ uniformly by delta and the reference operator is L-Lipschitz in the scalar unknown, their solutions differ by at most delta times t when L=0, or delta times (exp(Lt)-1)/L in general. Hence a positive subsequential nonhomogenization gap persists under a quantitatively controlled operator perturbation. Applied to Ziliotto's exact almost-sure gap from 1 to 2 at (0,1), every admissible stationary-ergodic Hamiltonian within uniform distance delta less than 1/2 also fails to homogenize, with gap at least 1-2 delta. A proper elliptic analogue gives the stability bound delta/kappa.\n\nCandidate contribution (robustness theorem; novelty confidence low): Ziliotto's first-order counterexample belongs to an explicit admissible uniform open ball of nonhomogenizing stationary-ergodic Hamiltonians: every comparison-admitting perturbation K on the same ergodic environment with sup norm distance less than 1/2 has subsequential gap at least 1 minus twice that distance at (0,1)." }, { "id": 20002489, "problem_number": "AIM-PDES-0101", "title": "Quenched Brownian large deviations in stationary ergodic potentials without mixing", "statement": "3. (Elena Kosygina) Let Ω be a probability space with probability measure P ergodic under a shift transformation group τx: Ω → Ω, x ∈ Rn. Consider the stochastic Hamilton-Jacobi equation\n\nut + H(Dxu, τ xω) = 0, (x, t ) ∈ Rn × (0, ∞),\n\nwhere H(p, ω ) is convex in p and satisfies some regularity assumptions, for example\n\nH(p, ω ) = 1\n\n2|p|2 − V (ω).\n\nA homogenization result for this problem was obtained by Lions and Souganidis. For the Hamiltonian above homogenization is equivalent to a large deviation result for Brownian motion among random obstacles (Snitzman). Problem: Derive a large deviation result for general stationary ergodic setting without any independence or mixing assumptions on P under the shifts.\n3", "original_statement": "3. (Elena Kosygina) Let Ω be a probability space with probability measure P ergodic under a shift transformation group τx: Ω → Ω, x ∈ Rn. Consider the stochastic Hamilton-Jacobi equation \n\nut + H(Dxu, τ xω) = 0, (x, t ) ∈ Rn × (0, ∞),\n\nwhere H(p, ω ) is convex in p and satisfies some regularity assumptions, for example \n\nH(p, ω ) = 1\n\n2|p|2 − V (ω).\n\nA homogenization result for this problem was obtained by Lions and Souganidis. For the Hamiltonian above homogenization is equivalent to a large deviation result for Brownian motion among random obstacles (Snitzman). Problem: Derive a large deviation result for general stationary ergodic setting without any independence or mixing assumptions on P under the shifts. \n3", "clean_statement": "3. (Elena Kosygina) Let Ω be a probability space with probability measure P ergodic under a shift transformation group τx: Ω → Ω, x ∈ Rn. Consider the stochastic Hamilton-Jacobi equation\n\nut + H(Dxu, τ xω) = 0, (x, t ) ∈ Rn × (0, ∞),\n\nwhere H(p, ω ) is convex in p and satisfies some regularity assumptions, for example\n\nH(p, ω ) = 1\n\n2|p|2 − V (ω).\n\nA homogenization result for this problem was obtained by Lions and Souganidis. For the Hamiltonian above homogenization is equivalent to a large deviation result for Brownian motion among random obstacles (Snitzman). Problem: Derive a large deviation result for general stationary ergodic setting without any independence or mixing assumptions on P under the shifts.\n3", "statement_status": "exact", "statement_verification": "The canonical JSON has lost the tens digit in the problem number and contains a terminal OCR artifact. The official AIM HTML and the workshop PDF show that this record is **Problem 33**, not Problem 3. The mathematical text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[100]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (Elena Kosygina) Let Ω be a probability space with probability measure P ergodic under a shift transformation group τx: Ω → Ω, x ∈ Rn. Consider the stochastic Hamilton-Jacobi equation \\n\\nut + H(Dxu, τ xω) = 0, (x, t ) ∈ Rn × (0, ∞),\\n\\nwhere H(p, ω ) is convex in p and satisfies some regularity assumptions, for example \\n\\nH(p, ω ) = 1\\n\\n2|p|2 − V (ω).\\n\\nA homogenization result for this problem was obtained by Lions and Souganidis. For the Hamiltonian above homogenization is equivalent to a large deviation result for Brownian motion among random obstacles (Snitzman). Problem: Derive a large deviation result for general stationary ergodic setting without any independence or mixing assumptions on P under the shifts. \\n3\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0101", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The AIM request, correctly recovered as Problem 33, has been answered for broad stationary-ergodic classes: Armstrong--Tran (2014) prove a quenched LDP for diffusions with absorbing regular random potentials without mixing, and Boivin--Le (2020) remove spatial regularity under explicit moment, spectral, and low-dimensional nondegeneracy hypotheses. As an independently proved special case, for every bounded nonnegative one-dimensional potential stationary under an ergodic real-translation action, the point-hitting Feynman--Kac costs and killed resolvent have deterministic directional Lyapunov exponents without independence, mixing, or ergodicity of the unit shift; an explicit exponential upper bound for ballistic hitting follows.\n\nCandidate contribution (lemma; novelty confidence low): Exact one-dimensional point-hitting additivity plus a bounded boundary-shift comparison forces the unit-cell Birkhoff conditional limit to be invariant under every real translation; therefore ergodicity of the full real action makes it deterministic even when the unit shift is not ergodic. This yields exact cell formulas for both directional Lyapunov exponents, the same exponents for the killed resolvent, and a ballistic hitting upper bound." }, { "id": 20002490, "problem_number": "AIM-PDES-0102", "title": "A quantitative primal-dual certificate for stochastic Mather measures", "statement": "4. (Diogo Gomes) Investigate possible extensions of the techniques used in the Aubry- Mather theory and Monge-Kantorovich problems to study linear programming prob-lems in infinite dimensions. An example of such extensions are the stochastic Mather measures which can be used to analyze second order Hamilton-Jacobi equations.\n3", "original_statement": "4. (Diogo Gomes) Investigate possible extensions of the techniques used in the Aubry- Mather theory and Monge-Kantorovich problems to study linear programming prob-lems in infinite dimensions. An example of such extensions are the stochastic Mather measures which can be used to analyze second order Hamilton-Jacobi equations. \n3", "clean_statement": "4. (Diogo Gomes) Investigate possible extensions of the techniques used in the Aubry- Mather theory and Monge-Kantorovich problems to study linear programming prob-lems in infinite dimensions. An example of such extensions are the stochastic Mather measures which can be used to analyze second order Hamilton-Jacobi equations.\n3", "statement_status": "exact", "statement_verification": "The canonical record is source index 101 of aim-pdes-notes.json. The original AIM PDF, *New connections between dynamical systems and PDE's* (version of August 15, 2003), verifies that this is **Problem 34**, proposed by Diogo Gomes:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[101]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"4. (Diogo Gomes) Investigate possible extensions of the techniques used in the Aubry- Mather theory and Monge-Kantorovich problems to study linear programming prob-lems in infinite dimensions. An example of such extensions are the stochastic Mather measures which can be used to analyze second order Hamilton-Jacobi equations. \\n3\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0102", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For positive diffusion on the flat torus and the shifted quadratic cost L(x,v)=|v-b(x)|^2/2+U(x), the stochastic-holonomic infinite-dimensional linear program and its Hamilton-Jacobi dual are both attained with no duality gap. A principal eigenfunction produces the smooth corrector u, the unique primal optimizer is the invariant graph measure m(x)dx delta_{b-Du}, and every feasible measure has objective gap exactly equal to one half of its squared L2 distance from this feedback graph. The same gap controls the negative-Holder distance of its spatial marginal from m dx at square-root rate.\n\nCandidate contribution (theorem; novelty confidence low): For every feasible occupation measure in the drifted-quadratic stochastic Mather program, the exact graph-distance identity J(mu)-beta=(1/2) integral |v-(b-Du)|^2 dmu combines with an elliptic Poisson estimate to give ||pi_x#mu-m dx||_{(C^alpha)^*} <= C [J(mu)-beta]^{1/2}." }, { "id": 20002491, "problem_number": "AIM-PDES-0103", "title": "From formal graph and density expansions to Mather-measure error bounds", "statement": "5. (Diogo Gomes) For a small perturbation of a completely integrable Hamiltonian system a viscosity solution corresponding to a non-resonant unperturbed torus can be uniformly approximated by formal expansions. Similar expansions can be constructed for the density of the Mather measures. However, it is not known how well Mather measures themselves are approximated by these formal expansions.\n3", "original_statement": "5. (Diogo Gomes) For a small perturbation of a completely integrable Hamiltonian system a viscosity solution corresponding to a non-resonant unperturbed torus can be uniformly approximated by formal expansions. Similar expansions can be constructed for the density of the Mather measures. However, it is not known how well Mather measures themselves are approximated by these formal expansions. \n3", "clean_statement": "**Problem 35 (Diogo Gomes).** For a small perturbation of a completely integrable Hamiltonian system a viscosity solution corresponding to a non-resonant unperturbed torus can be uniformly approximated by formal expansions. Similar expansions can be constructed for the density of the Mather measures. However, it is not known how well Mather measures themselves are approximated by these formal expansions.", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON record is affected by a small OCR/page-boundary error: its displayed number is `5` and it ends with an isolated `3`. The official AIM workshop page identifies it as Problem 35 and gives the following statement:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[102]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"5. (Diogo Gomes) For a small perturbation of a completely integrable Hamiltonian system a viscosity solution corresponding to a non-resonant unperturbed torus can be uniformly approximated by formal expansions. Similar expansions can be constructed for the density of the Mather measures. However, it is not known how well Mather measures themselves are approximated by these formal expansions. \\n3\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0103", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For graph-supported probability measures, an explicit coupling bounds the 1-Wasserstein error by the common-density-weighted graph displacement plus one half of the ambient diameter times the L1 density error. Consequently, order-N formal remainder bounds for both the invariant graph gradient and the projected density give an order-(N+1) Wasserstein bound for the formal Mather measure. A smooth oscillatory counterexample proves that uniform approximation of the viscosity solution and exact density agreement alone cannot control the phase-space measures, so a derivative or invariant-graph estimate is genuinely necessary.\n\nCandidate contribution (lemma; novelty confidence low): The graph-density transfer inequality W1(mu,mu_tilde) <= integral min(m,m_tilde)|G-G_tilde| + (diameter/2)||m-m_tilde||_1, applied to formal Mather-measure truncations, gives a direct quantitative reduction to density and invariant-graph remainder bounds; an explicit oscillatory example shows the graph bound cannot be replaced by uniform control of the Hamilton-Jacobi potential." }, { "id": 20002492, "problem_number": "AIM-PDES-0104", "title": "Flat-germ obstruction to large minimal periods and genuine quasi-periodicity", "statement": "6. (Massimiliano Berti) For a nonlinear wave equation\n\nutt − uxx = f (u), f (0) = f ′(0) = 0 10\n\nwith Dirichlet boundary conditions and general nonlinearity there exist a large num-ber of small amplitude periodic orbit with fixed period. Could one find small ampli-tude periodic orbits of large period and quasi-periodic solutions for the wave equa-tion?", "original_statement": "6. (Massimiliano Berti) For a nonlinear wave equation \n\nutt − uxx = f (u), f (0) = f ′(0) = 0 10 \n\nwith Dirichlet boundary conditions and general nonlinearity there exist a large num-ber of small amplitude periodic orbit with fixed period. Could one find small ampli-tude periodic orbits of large period and quasi-periodic solutions for the wave equa-tion?", "clean_statement": "6. (Massimiliano Berti) For a nonlinear wave equation\n\nutt − uxx = f (u), f (0) = f ′(0) = 0 10\n\nwith Dirichlet boundary conditions and general nonlinearity there exist a large num-ber of small amplitude periodic orbit with fixed period. Could one find small ampli-tude periodic orbits of large period and quasi-periodic solutions for the wave equa-tion?", "statement_status": "exact", "statement_verification": "The canonical record is affected by OCR. The official AIM web rendering and the linked workshop PDF identify this as **Problem 36**, not Problem 6. The isolated `10` after the displayed formula is a page number, and the broken words “num-ber” and “ampli-tude” are line-end hyphenation. The recovered statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: PDEs\nWorkshop: New connections between dynamical systems and PDE's\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/dynpde/dynpde.pdf\nCanonical location: aim-pdes-notes.json notes[103]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"6. (Massimiliano Berti) For a nonlinear wave equation \\n\\nutt − uxx = f (u), f (0) = f ′(0) = 0 10 \\n\\nwith Dirichlet boundary conditions and general nonlinearity there exist a large num-ber of small amplitude periodic orbit with fixed period. Could one find small ampli-tude periodic orbits of large period and quasi-periodic solutions for the wave equa-tion?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 9, "set_id": 14, "source_url": "https://aimath.org/WWN/dynpde/dynpde.pdf", "tags": [ "aim", "AIM-PDES-0104", "aim-domain:pdes", "aim-workshop:dynpde", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 9, "name": "pde", "display_name": "Partial Differential Equations", "description": "PDEs and their applications in physics and geometry.", "slug": "pde", "order_index": 9, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source is AIM Problem 36 and its OCR introduced a wrong problem number, a page-number artifact, and damaged notation. For the Dirichlet wave equation on (0,L), if a smooth nonlinearity vanishes on a neighborhood of zero, then every uniformly small finite-energy solution is linear: it is 2L-periodic, a nonzero solution with active sine modes S has exact minimal period 2L/gcd(S), and its temporal frequency module has rank one. Therefore the literal assumptions f(0)=f'(0)=0 cannot guarantee arbitrarily large minimal periods or genuinely quasi-periodic small solutions for all general nonlinearities; a nonflatness/twist hypothesis is necessary. Known primary results settle substantial fixed-period regimes, while the exact autonomous massless Dirichlet extensions were not verified in the literature searched.\n\nCandidate contribution (obstruction; novelty confidence low): For every nonzero uniformly small finite-energy solution when the nonlinear germ vanishes, the exact minimal period is 2L/gcd(S) and the minimal Bohr frequency module has rank one; this gives a sharp counter-obstruction to interpreting AIM Problem 36 as uniform over all f satisfying only f(0)=f'(0)=0." }, { "id": 20002493, "problem_number": "AIM-PHYSICS-0001", "title": "When semiclassical spectra follow curves—and when they fill area", "statement": "Eigenvalues and Curves\n\nDo the eigenvalues of $a^w(x,hD)$ lie on or cluster around curves in the complex plane $\\mathbb{C}$?", "original_statement": "Eigenvalues and Curves\n\nDo the eigenvalues of $a^w(x,hD)$ lie on or cluster around curves in the complex plane $\\mathbb{C}$?", "clean_statement": "Eigenvalues and Curves\n\nDo the eigenvalues of $a^w(x,hD)$ lie on or cluster around curves in the complex plane $\\mathbb{C}$?", "statement_status": "exact", "statement_verification": "The canonical AIM record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Non-Hermitian quantum mechanics and symplectic geometry\nSection: Eigenvalues of Operators with Analytic Coefficients\nSource item: 1.1\nSource URL: http://aimpl.org/nhquantumsymp/1/\nCanonical location: aim-physics-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Eigenvalues and Curves\\n\\nDo the eigenvalues of $a^w(x,hD)$ lie on or cluster around curves in the complex plane $\\\\mathbb{C}$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/nhquantumsymp/1/", "tags": [ "aim", "AIM-PHYSICS-0001", "aim-domain:physics", "aim-workshop:nhquantumsymp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A bounded perturbation of a normal operator P_h+i epsilon F(P_h) has spectrum in a tube of radius equal to the perturbation norm around the curve t+i epsilon F(t). For the analytic one-dimensional Weyl symbol a=p+i epsilon p^2 with p=x^2+xi^2, the exact identity Op_h^w(p^2)=Q_h^2+h^2 gives eigenvalues E_n+i epsilon(E_n^2+h^2), hence an explicit epsilon h^2 clustering bound around the parabola t+i epsilon t^2. Conversely, the analytic elliptic two-dimensional operator Q_1+iQ_2 has an h-spaced planar eigenvalue grid with an h^{-2} counting law, so no fixed finite union of rectifiable curves can capture it in an o(1) tube. This proves both a sufficient curve criterion and the necessity of the workshop's one-degree-of-freedom restriction.\n\nCandidate contribution (theorem; novelty confidence low): The normal-reference tube criterion converts an operator-norm averaging remainder B_h-F(P_h)=O(delta_h) directly into an epsilon delta_h spectral curve width; for the explicit analytic Weyl symbol p+i epsilon p^2, the exact Weyl correction yields the testable bound dist(Spec, {t+i epsilon t^2}) <= epsilon h^2." }, { "id": 20002494, "problem_number": "AIM-PHYSICS-0002", "title": "Complex Bohr-Sommerfeld action and a stability obstruction", "statement": "Bohr-Sommerfeld\n\nIs there a Bohr-Sommerfeld quantization rule for $a^w(x,hD)$? Is there a link to the usual Bohr-Sommerfeld rule for Hermitian operators?", "original_statement": "Bohr-Sommerfeld\n\nIs there a Bohr-Sommerfeld quantization rule for $a^w(x,hD)$? Is there a link to the usual Bohr-Sommerfeld rule for Hermitian operators?", "clean_statement": "Bohr-Sommerfeld\n\nIs there a Bohr-Sommerfeld quantization rule for $a^w(x,hD)$? Is there a link to the usual Bohr-Sommerfeld rule for Hermitian operators?", "statement_status": "exact", "statement_verification": "The exact corpus record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Non-Hermitian quantum mechanics and symplectic geometry\nSection: Eigenvalues of Operators with Analytic Coefficients\nSource item: 1.2\nSource URL: http://aimpl.org/nhquantumsymp/1/\nCanonical location: aim-physics-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Bohr-Sommerfeld\\n\\nIs there a Bohr-Sommerfeld quantization rule for $a^w(x,hD)$? Is there a link to the usual Bohr-Sommerfeld rule for Hermitian operators?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/nhquantumsymp/1/", "tags": [ "aim", "AIM-PHYSICS-0002", "aim-domain:physics", "aim-workshop:nhquantumsymp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "There is no assumption-free Bohr-Sommerfeld eigenvalue rule for analytic non-selfadjoint Weyl operators, since even the analytic symbol xi+i quantizes to an operator with continuous spectrum and no closed action cycle. For the exact non-selfadjoint shifted oscillator H_{h,theta}=-h^2 d_x^2+(x-i theta)^2, however, the complex canonical translation of the Hermitian cycle has action I(E)=E/2, Maslov index two, and zero subprincipal term, so the usual rule gives the exact spectrum E_n=(2n+1)h. Despite identical action data and eigenvalues, its ground-state Riesz projection has norm exp(theta^2/h), proving that action quantization alone does not control nonnormal spectral stability.\n\nCandidate contribution (obstruction; novelty confidence low): The Hermitian oscillator H_{h,0} and its imaginary translate H_{h,theta} have the same full Bohr-Sommerfeld action function, Maslov correction, Weyl subprincipal term, and exact eigenvalue lattice, but the ground-level resolvent residue changes from norm 1 to exp(theta^2/h); hence complex action data alone cannot determine semiclassical spectral conditioning." }, { "id": 20002495, "problem_number": "AIM-PHYSICS-0003", "title": "Exact zero-Kerr spectrum and high-energy Kerr localization", "statement": "A non-Hermitian Operator from Physics\n\nWhat is the distribution of eigenvalues of the non-Hermitian Hamiltonian\n\\[\nH = \\omega a^\\dagger a + \\chi (a^\\dagger a)^2 - i \\gamma a a^\\dagger + \\beta (a^\\dagger + a)?\n\\]\nHere $a$ and $a^\\dagger$ are the raising and lowering operators respectively and $\\omega$, $\\chi$, $\\gamma$, and $\\beta$ are real parameters.", "original_statement": "A non-Hermitian Operator from Physics\n\nWhat is the distribution of eigenvalues of the non-Hermitian Hamiltonian\n\\[\nH = \\omega a^\\dagger a + \\chi (a^\\dagger a)^2 - i \\gamma a a^\\dagger + \\beta (a^\\dagger + a)?\n\\]\nHere $a$ and $a^\\dagger$ are the raising and lowering operators respectively and $\\omega$, $\\chi$, $\\gamma$, and $\\beta$ are real parameters.", "clean_statement": "A non-Hermitian Operator from Physics\n\nWhat is the distribution of eigenvalues of the non-Hermitian Hamiltonian\n\\[\nH = \\omega a^\\dagger a + \\chi (a^\\dagger a)^2 - i \\gamma a a^\\dagger + \\beta (a^\\dagger + a)?\n\\]\nHere $a$ and $a^\\dagger$ are the raising and lowering operators respectively and $\\omega$, $\\chi$, $\\gamma$, and $\\beta$ are real parameters.", "statement_status": "exact", "statement_verification": "The canonical AIMPL record asks for the eigenvalue distribution of \\[ H=\\omega a^\\dagger a+\\chi(a^\\dagger a)^2-i\\gamma aa^\\dagger +\\beta(a^\\dagger+a), \\qquad \\omega,\\chi,\\gamma,\\beta\\in\\mathbb R. \\] With the standard bosonic commutation relation \\([a,a^\\dagger]=I\\) and number operator \\(N=a^\\dagger a\\), one has \\[ aa^\\dagger=N+1. \\] Therefore the exact canonical operator is \\[ \\boxed{H=\\chi N^2+(\\omega-i\\gamma)N-i\\gamma I+\\beta(a+a^\\dagger).} \\]", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Non-Hermitian quantum mechanics and symplectic geometry\nSection: Eigenvalues of Operators with Analytic Coefficients\nSource item: 1.3\nSource URL: http://aimpl.org/nhquantumsymp/1/\nCanonical location: aim-physics-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A non-Hermitian Operator from Physics\\n\\nWhat is the distribution of eigenvalues of the non-Hermitian Hamiltonian\\n\\\\[\\nH = \\\\omega a^\\\\dagger a + \\\\chi (a^\\\\dagger a)^2 - i \\\\gamma a a^\\\\dagger + \\\\beta (a^\\\\dagger + a)?\\n\\\\]\\nHere $a$ and $a^\\\\dagger$ are the raising and lowering operators respectively and $\\\\omega$, $\\\\chi$, $\\\\gamma$, and $\\\\beta$ are real parameters.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/nhquantumsymp/1/", "tags": [ "aim", "AIM-PHYSICS-0003", "aim-domain:physics", "aim-workshop:nhquantumsymp", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the canonical Hamiltonian with nonzero Kerr coefficient, the operator has compact resolvent and its spectrum lies in explicit disks of radius C sqrt(n+1), for every C>2|beta|, about the diagonal values chi n^2+(omega-i gamma)n-i gamma; all sufficiently high disks are disjoint and contain exactly one algebraically simple eigenvalue, yielding an O(sqrt(n)) asymptotic and an exterior resolvent bound. When chi=0 and omega-i gamma is nonzero, the complete spectrum and eigenfunctions are obtained exactly. A global imaginary-part identity and sign-dependent parabolic enclosure are also proved.\n\nCandidate contribution (theorem; novelty confidence low): For every real parameter choice with chi nonzero and every C>2|beta|, the AIM Jacobi operator has exactly one algebraically simple eigenvalue in each sufficiently high disk |z-[chi n^2+(omega-i gamma)n-i gamma]| 0 consider Aα:= −d2/dx2 + i |x|α in R (or R+ with Dirichlet boundary condition at 0). If α > 2/3, then it is known that the eigenfunctions form a complete system.\n\nOpen problem: Is the same true for 0 < α ≤ 2/3? 1.2. Magnetic Schr¨ odinger operator. Consider\n\nA:= − ∂2\n\n∂x 2 −\n\n( ∂∂y − ix2\n\n2\n\n)2\n\n+ i cy, D(A):= H10 (R2+) ∩ { u: Au ∈ L2(R2+)}\n\nwhere R2+ = {(x, y ) ∈ R2: y > 0}.\n\nOpen problem: Is σ(A) 6 = ∅?It is known that σ(A) 6 = ∅ if |c| << 1 or |c| >> 1. 2. Lyonell Boulton\n\n2.1. Schauder bases of periodic functions and multipliers. Let en(x):= √2 sin( nπx ). Then {en}\n\nis a Schauder basis of Lp(0, 1) for all p > 1. Let f ∈ C(R, C) satisfy f (x + 2) = f (x), f (−x) = −f (x),\n\nf (1 /2 + x) = f (1 /2 − x) and define fn(x):= f (nx ). Let A: Lp(0, 1) → Lp(0, 1) be the linear extension of the map Ae n = fn. Then {fn} is a Schauder basis of Lp(0, 1) if and only if A: Lp(0, 1) −→ Lp(0, 1) is a bounded operator with a bounded inverse. Let {ck} be the Fourier coefficients of f. Then A can be written as A = ∑\n\n> k\n\nckMk where Mk are the linear extensions of the map Mken = ekn.\n\nOpen problem: Find necessary and sufficient conditions on {ck} for 0 /∈ σ(A) whenever p 6 = 2. 3. Amin Boumenir\n\n3.1. Non-self-adjoint inverse problems. We are interested in identifying a non-self-adjoint operator associated with an evolution equation (parabolic or hyperbolic) through \"observations\" of the solution as time evolves. Thus for example in a certain Hilbert space we have\n\nu′(t) = Au (t) and u(0) = f (1) where, for simplicity, we assume that\n\nA = L + B\n\nwith L is a given (known) self-adjoint operator with \"nice properties\" while B is an unknown non-self-adjoint perturbation. For example Ay (x) = y′′ (x) − q(x)y(x) or Au = ∆ u − q(x)u with Im q(x) 6 = 0. We assume that we can observe the solution through a functional 〈·, g 〉 say\n\nω(t) = 〈u(t), g 〉.\n\nFor example if u(x, t ) is the solution of a heat equation, where x ∈ Ω ⊂ Rn, and p ∈ ∂Ω, then ω(t) = u(p, t )(temperature) or ω(t) = ∂nu (p, t ) (heat transfer) are usual observations/readings of the solution on the boundary. Thus we want to recover A or at least its spectrum σA = {λn} ⊂ C from the observation mapping\n\nu(0) → ω(t).\n\n> Date: June 8 - 12, 2015, American Institute of Mathematics, San Jose, California.\n> 1\n\nTo do so, although we do NOT know A, we assume that it has a discrete spectrum {λn} ⊂ C, and in general Im λn → 0 as n → ∞, while Re λn → −∞. If we denote its eigenfunctions by ϕn, 0 and its associated eigenfunctions (roots) by ϕn,ν for ν = 1,..., m n − 1, where mn is the multiplicity of the eigenvalue λn, then we can write a formal solution to the evolution equation\n\nu (t) = ∑\n\n> n≥1\n\neλntmn−1∑\n\n> ν=0\n\ncnν (f ) pnν (t)ϕnν (2) where the Fourier coefficients are cnν (f ) = 〈f, ψ nν 〉 and {ψnν } is the biorthogonal system to {ϕn,ν }. Here\n\npnν are polynomials generated by the multiplicity of the eigenvalue λn. The observation then is given by\n\nω(t) = ∑\n\n> n≥1\n\neλntmn−1∑\n\n> ν=0\n\ncnν (f ) pnν (t)〈ϕnν, g 〉. (3) In the best case, when all cnν (f ) 6 = 0 and 〈ϕnν, g 〉 6 = 0 then it is possible to evaluate/extract all the λn from the observation (2).\n\nOpen problems: i) How do you choose the initial condition f, so we can observe all eλnt, that is all cnν (f ) 6 = 0? We need to know something about the biorthogonal system {ψnν }.ii) How do you choose the observation g so all 〈ϕnν, g 〉 6 = 0? We need to know something about the root functions {ϕn,ν }.iii) How smooth is the sum (2), so we can choose g? We need some information on the type of convergence in (2) so (3) holds. iv) How do we extract the λn and their multiplicity from a given signal given by (3) in finite time? When\n\nλn are complex values and the sum contains polynomials in t, it is much harder than the real case. v) Find the best f and g that allow the identification of A by using the smallest number of observations. Evolution equations are often found in control theory, and for that purpose, we need finite number of observations done in finite time. 4. Marina Chugunova\n\n4.1. Computations of the instability index for a non-self-adjoint operators. The stability of steady states is a basic question about the dynamics of any partial differential equation that models the evolution of a physical system. In order to numerically evaluate the instability index of a given differential operator A, its computation should be reduced to a problem of linear algebra. Particularly for problems with periodic boundary condi-tions, it seems natural to restrict the operator A to a finite-dimensional space of trigonometric polynomials.\n\nOpen problem: Under what conditions the instability index (the total number of unstable eigenvalues) can be computed from the resulting finite dimensional matrix? One difficulty is that the entries of the infinite matrix corresponding to the differential operator A grow with the row and column index, so that any truncation is not a small perturbation. If A is a self-adjoint semi-bounded differential operator of even order, then the instability index can be estimated by variational methods, or computed directly from the zeros of the corresponding Evans function. Understanding the spectrum of a non-self-adjoint operator is a much harder problem. It is not at all obvious how to restrict the computation of its instability index to a finite-dimensional subspace, or how to even estimate its dimension. Furthermore, the numerical calculation of eigenvalues can be extremely ill-conditioned even in finite dimensions. 5. Michael Demuth\n\n5.1. Spectral radius and operator norm. Let A be a bounded linear operator on a Banach space X. Its spectral radius is defined by spr( A):= max {| z|: z ∈ σ(A)}.\n\nGelfand proved the classical formula spr( A) = lim\n\n> n→∞\n\n‖An‖ 1\n\n> n.\n\n> 2\n\nObviously, 0 ≤ spr( A) ≤ ‖ A‖. The question arises: What is the gap between ‖A‖ and spr( A)? Introduce the denotation gap( A):= ‖A‖ − spr( A).\n\nOpen problems: i) For which class of operators holds gap( A) > 0 or gap( A) = 0,\n\nrespectively. ii) What is the smallest m ∈ (0, 1], such that spr( A) ≤ m‖A‖\n\nor gap( A) ≥ (1 − m)‖A‖?\n\nExample 1. Let X = `1(N) and A be the weighted shift-operator defined according to the canonical standard basis by the infinite matrix\n\n\n\n0\n\nb1 0\n\nb2 0\n\nb1 0\n\nb2 0......\n\n\n\nwhere b1, b 2 > 0 and b1b2 = 1. In this case ‖A‖ = max {b1, b 2} and σ(A) = {z ∈ C: |z| ≤ 1} and therefore spr( A) = 1. Thus\n\n• gap( A) = 0: If b1 = b2 = 1 then ‖A‖ = spr( A).\n\n• gap( A) > 0: If b1 6 = b2 then ‖A‖ > spr( A). This kind of estimates are useful in the following situation. Let K be a compact perturbation of A. Study the discrete spectrum of B:= A + K. We are able to analyze the moments and the number of eigenvalues of B outside a ball of radius ‖A‖. It is more interesting and also natural to enlarge this region up to the complement of a ball with radius spr( A). 6. Mark Embree\n\n6.1. Davies' conjecture about approximate diagonalization. Consider a non-normal matrix A ∈\n\nCn×n. Define\n\ns(A, ε ):= inf\n\n> ∆,V V−1(A+∆) Vdiagonal\n\n‖V ‖ ‖ V −1‖ε + ‖∆‖.\n\nOpen problem: Prove Davies' conjecture (2007): There exists a constant Cn > 0, independent of A ∈ Cn×n,such that s(A, ε ) ≤ Cn\n\n√ε.It is known that the conjecture holds for Jordan blocks (then Cn = 2 suffices) and for 3 × 3 matrices with\n\n‖A‖ ≤ 1 (then Cn = 4 suffices). 6.2. Crouzeix' conjecture about the norm of matrix functions. Let A be a bounded linear operator. It is known that ‖Ak‖ ≤ 2 max z∈W (A) |zk|; W (A) denotes the numerical range of A.\n\nOpen problem: Prove Crouzeix' conjecture: There exists a constant C ≥ 2 such that for all analytic functions\n\nf: W (A) → C holds ‖f (A)‖ ≤ C max z∈W (A) |f (z)|.Crouzeix conjectured further that C ≤ 11.08.\n\n> 3", "original_statement": "1. Yaniv Almog \n\n1.1. Completeness of eigenfunctions for Schr¨ odinger operators with complex potentials. For \n\nα > 0 consider Aα:= −d2/dx2 + i |x|α in R (or R+ with Dirichlet boundary condition at 0). If α > 2/3, then it is known that the eigenfunctions form a complete system. \n\nOpen problem: Is the same true for 0 < α ≤ 2/3? 1.2. Magnetic Schr¨ odinger operator. Consider \n\nA:= − ∂2\n\n∂x 2 −\n\n( ∂∂y − ix2\n\n2\n\n)2\n\n+ i cy, D(A):= H10 (R2+) ∩ { u: Au ∈ L2(R2+)}\n\nwhere R2+ = {(x, y ) ∈ R2: y > 0}.\n\nOpen problem: Is σ(A) 6 = ∅?It is known that σ(A) 6 = ∅ if |c| << 1 or |c| >> 1. 2. Lyonell Boulton \n\n2.1. Schauder bases of periodic functions and multipliers. Let en(x):= √2 sin( nπx ). Then {en}\n\nis a Schauder basis of Lp(0, 1) for all p > 1. Let f ∈ C(R, C) satisfy f (x + 2) = f (x), f (−x) = −f (x), \n\nf (1 /2 + x) = f (1 /2 − x) and define fn(x):= f (nx ). Let A: Lp(0, 1) → Lp(0, 1) be the linear extension of the map Ae n = fn. Then {fn} is a Schauder basis of Lp(0, 1) if and only if A: Lp(0, 1) −→ Lp(0, 1) is a bounded operator with a bounded inverse. Let {ck} be the Fourier coefficients of f. Then A can be written as A = ∑ \n\n> k\n\nckMk where Mk are the linear extensions of the map Mken = ekn.\n\nOpen problem: Find necessary and sufficient conditions on {ck} for 0 /∈ σ(A) whenever p 6 = 2. 3. Amin Boumenir \n\n3.1. Non-self-adjoint inverse problems. We are interested in identifying a non-self-adjoint operator associated with an evolution equation (parabolic or hyperbolic) through \"observations\" of the solution as time evolves. Thus for example in a certain Hilbert space we have \n\nu′(t) = Au (t) and u(0) = f (1) where, for simplicity, we assume that \n\nA = L + B\n\nwith L is a given (known) self-adjoint operator with \"nice properties\" while B is an unknown non-self-adjoint perturbation. For example Ay (x) = y′′ (x) − q(x)y(x) or Au = ∆ u − q(x)u with Im q(x) 6 = 0. We assume that we can observe the solution through a functional 〈·, g 〉 say \n\nω(t) = 〈u(t), g 〉.\n\nFor example if u(x, t ) is the solution of a heat equation, where x ∈ Ω ⊂ Rn, and p ∈ ∂Ω, then ω(t) = u(p, t )(temperature) or ω(t) = ∂nu (p, t ) (heat transfer) are usual observations/readings of the solution on the boundary. Thus we want to recover A or at least its spectrum σA = {λn} ⊂ C from the observation mapping \n\nu(0) → ω(t).\n\n> Date: June 8 - 12, 2015, American Institute of Mathematics, San Jose, California.\n> 1\n\nTo do so, although we do NOT know A, we assume that it has a discrete spectrum {λn} ⊂ C, and in general Im λn → 0 as n → ∞, while Re λn → −∞. If we denote its eigenfunctions by ϕn, 0 and its associated eigenfunctions (roots) by ϕn,ν for ν = 1,..., m n − 1, where mn is the multiplicity of the eigenvalue λn, then we can write a formal solution to the evolution equation \n\nu (t) = ∑\n\n> n≥1\n\neλntmn−1∑\n\n> ν=0\n\ncnν (f ) pnν (t)ϕnν (2) where the Fourier coefficients are cnν (f ) = 〈f, ψ nν 〉 and {ψnν } is the biorthogonal system to {ϕn,ν }. Here \n\npnν are polynomials generated by the multiplicity of the eigenvalue λn. The observation then is given by \n\nω(t) = ∑\n\n> n≥1\n\neλntmn−1∑\n\n> ν=0\n\ncnν (f ) pnν (t)〈ϕnν, g 〉. (3) In the best case, when all cnν (f ) 6 = 0 and 〈ϕnν, g 〉 6 = 0 then it is possible to evaluate/extract all the λn from the observation (2). \n\nOpen problems: i) How do you choose the initial condition f, so we can observe all eλnt, that is all cnν (f ) 6 = 0? We need to know something about the biorthogonal system {ψnν }.ii) How do you choose the observation g so all 〈ϕnν, g 〉 6 = 0? We need to know something about the root functions {ϕn,ν }.iii) How smooth is the sum (2), so we can choose g? We need some information on the type of convergence in (2) so (3) holds. iv) How do we extract the λn and their multiplicity from a given signal given by (3) in finite time? When \n\nλn are complex values and the sum contains polynomials in t, it is much harder than the real case. v) Find the best f and g that allow the identification of A by using the smallest number of observations. Evolution equations are often found in control theory, and for that purpose, we need finite number of observations done in finite time. 4. Marina Chugunova \n\n4.1. Computations of the instability index for a non-self-adjoint operators. The stability of steady states is a basic question about the dynamics of any partial differential equation that models the evolution of a physical system. In order to numerically evaluate the instability index of a given differential operator A, its computation should be reduced to a problem of linear algebra. Particularly for problems with periodic boundary condi-tions, it seems natural to restrict the operator A to a finite-dimensional space of trigonometric polynomials. \n\nOpen problem: Under what conditions the instability index (the total number of unstable eigenvalues) can be computed from the resulting finite dimensional matrix? One difficulty is that the entries of the infinite matrix corresponding to the differential operator A grow with the row and column index, so that any truncation is not a small perturbation. If A is a self-adjoint semi-bounded differential operator of even order, then the instability index can be estimated by variational methods, or computed directly from the zeros of the corresponding Evans function. Understanding the spectrum of a non-self-adjoint operator is a much harder problem. It is not at all obvious how to restrict the computation of its instability index to a finite-dimensional subspace, or how to even estimate its dimension. Furthermore, the numerical calculation of eigenvalues can be extremely ill-conditioned even in finite dimensions. 5. Michael Demuth \n\n5.1. Spectral radius and operator norm. Let A be a bounded linear operator on a Banach space X. Its spectral radius is defined by spr( A):= max {| z|: z ∈ σ(A)}.\n\nGelfand proved the classical formula spr( A) = lim \n\n> n→∞\n\n‖An‖ 1 \n\n> n.\n\n> 2\n\nObviously, 0 ≤ spr( A) ≤ ‖ A‖. The question arises: What is the gap between ‖A‖ and spr( A)? Introduce the denotation gap( A):= ‖A‖ − spr( A).\n\nOpen problems: i) For which class of operators holds gap( A) > 0 or gap( A) = 0,\n\nrespectively. ii) What is the smallest m ∈ (0, 1], such that spr( A) ≤ m‖A‖\n\nor gap( A) ≥ (1 − m)‖A‖?\n\nExample 1. Let X = `1(N) and A be the weighted shift-operator defined according to the canonical standard basis by the infinite matrix \n\n\n\n0\n\nb1 0\n\nb2 0\n\nb1 0\n\nb2 0......\n\n\n\nwhere b1, b 2 > 0 and b1b2 = 1. In this case ‖A‖ = max {b1, b 2} and σ(A) = {z ∈ C: |z| ≤ 1} and therefore spr( A) = 1. Thus \n\n• gap( A) = 0: If b1 = b2 = 1 then ‖A‖ = spr( A). \n\n• gap( A) > 0: If b1 6 = b2 then ‖A‖ > spr( A). This kind of estimates are useful in the following situation. Let K be a compact perturbation of A. Study the discrete spectrum of B:= A + K. We are able to analyze the moments and the number of eigenvalues of B outside a ball of radius ‖A‖. It is more interesting and also natural to enlarge this region up to the complement of a ball with radius spr( A). 6. Mark Embree \n\n6.1. Davies' conjecture about approximate diagonalization. Consider a non-normal matrix A ∈\n\nCn×n. Define \n\ns(A, ε ):= inf \n\n> ∆,V V−1(A+∆) Vdiagonal\n\n‖V ‖ ‖ V −1‖ε + ‖∆‖.\n\nOpen problem: Prove Davies' conjecture (2007): There exists a constant Cn > 0, independent of A ∈ Cn×n,such that s(A, ε ) ≤ Cn\n\n√ε.It is known that the conjecture holds for Jordan blocks (then Cn = 2 suffices) and for 3 × 3 matrices with \n\n‖A‖ ≤ 1 (then Cn = 4 suffices). 6.2. Crouzeix' conjecture about the norm of matrix functions. Let A be a bounded linear operator. It is known that ‖Ak‖ ≤ 2 max z∈W (A) |zk|; W (A) denotes the numerical range of A.\n\nOpen problem: Prove Crouzeix' conjecture: There exists a constant C ≥ 2 such that for all analytic functions \n\nf: W (A) → C holds ‖f (A)‖ ≤ C max z∈W (A) |f (z)|.Crouzeix conjectured further that C ≤ 11.08. \n\n> 3", "clean_statement": null, "statement_status": "unrecoverable", "statement_verification": "The canonical JSON record is not one mathematical problem. It is an OCR extraction of the first three pages of the 2015 AIM workshop list *Mathematical aspects of physics with non-self-adjoint operators*. It starts with Yaniv Almog's item 1, but then runs through items contributed by five other participants. The exact OCR text is preserved in `input.json`; it is not silently rewritten here.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Mathematical aspects of physics with non-self-adjoint operators\nSection: \nSource item: 1\nSource URL: https://aimath.org/pastworkshops/nonselfadjointproblems.pdf\nCanonical location: aim-physics-notes.json notes[5]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Yaniv Almog \\n\\n1.1. Completeness of eigenfunctions for Schr¨ odinger operators with complex potentials. For \\n\\nα > 0 consider Aα:= −d2/dx2 + i |x|α in R (or R+ with Dirichlet boundary condition at 0). If α > 2/3, then it is known that the eigenfunctions form a complete system. \\n\\nOpen problem: Is the same true for 0 < α ≤ 2/3? 1.2. Magnetic Schr¨ odinger operator. Consider \\n\\nA:= − ∂2\\n\\n∂x 2 −\\n\\n( ∂∂y − ix2\\n\\n2\\n\\n)2\\n\\n+ i cy, D(A):= H10 (R2+) ∩ { u: Au ∈ L2(R2+)}\\n\\nwhere R2+ = {(x, y ) ∈ R2: y > 0}.\\n\\nOpen problem: Is σ(A) 6 = ∅?It is known that σ(A) 6 = ∅ if |c| << 1 or |c| >> 1. 2. Lyonell Boulton \\n\\n2.1. Schauder bases of periodic functions and multipliers. Let en(x):= √2 sin( nπx ). Then {en}\\n\\nis a Schauder basis of Lp(0, 1) for all p > 1. Let f ∈ C(R, C) satisfy f (x + 2) = f (x), f (−x) = −f (x), \\n\\nf (1 /2 + x) = f (1 /2 − x) and define fn(x):= f (nx ). Let A: Lp(0, 1) → Lp(0, 1) be the linear extension of the map Ae n = fn. Then {fn} is a Schauder basis of Lp(0, 1) if and only if A: Lp(0, 1) −→ Lp(0, 1) is a bounded operator with a bounded inverse. Let {ck} be the Fourier coefficients of f. Then A can be written as A = ∑ \\n\\n> k\\n\\nckMk where Mk are the linear extensions of the map Mken = ekn.\\n\\nOpen problem: Find necessary and sufficient conditions on {ck} for 0 /∈ σ(A) whenever p 6 = 2. 3. Amin Boumenir \\n\\n3.1. Non-self-adjoint inverse problems. We are interested in identifying a non-self-adjoint operator associated with an evolution equation (parabolic or hyperbolic) through \\\"observations\\\" of the solution as time evolves. Thus for example in a certain Hilbert space we have \\n\\nu′(t) = Au (t) and u(0) = f (1) where, for simplicity, we assume that \\n\\nA = L + B\\n\\nwith L is a given (known) self-adjoint operator with \\\"nice properties\\\" while B is an unknown non-self-adjoint perturbation. For example Ay (x) = y′′ (x) − q(x)y(x) or Au = ∆ u − q(x)u with Im q(x) 6 = 0. We assume that we can observe the solution through a functional 〈·, g 〉 say \\n\\nω(t) = 〈u(t), g 〉.\\n\\nFor example if u(x, t ) is the solution of a heat equation, where x ∈ Ω ⊂ Rn, and p ∈ ∂Ω, then ω(t) = u(p, t )(temperature) or ω(t) = ∂nu (p, t ) (heat transfer) are usual observations/readings of the solution on the boundary. Thus we want to recover A or at least its spectrum σA = {λn} ⊂ C from the observation mapping \\n\\nu(0) → ω(t).\\n\\n> Date: June 8 - 12, 2015, American Institute of Mathematics, San Jose, California.\\n> 1\\n\\nTo do so, although we do NOT know A, we assume that it has a discrete spectrum {λn} ⊂ C, and in general Im λn → 0 as n → ∞, while Re λn → −∞. If we denote its eigenfunctions by ϕn, 0 and its associated eigenfunctions (roots) by ϕn,ν for ν = 1,..., m n − 1, where mn is the multiplicity of the eigenvalue λn, then we can write a formal solution to the evolution equation \\n\\nu (t) = ∑\\n\\n> n≥1\\n\\neλntmn−1∑\\n\\n> ν=0\\n\\ncnν (f ) pnν (t)ϕnν (2) where the Fourier coefficients are cnν (f ) = 〈f, ψ nν 〉 and {ψnν } is the biorthogonal system to {ϕn,ν }. Here \\n\\npnν are polynomials generated by the multiplicity of the eigenvalue λn. The observation then is given by \\n\\nω(t) = ∑\\n\\n> n≥1\\n\\neλntmn−1∑\\n\\n> ν=0\\n\\ncnν (f ) pnν (t)〈ϕnν, g 〉. (3) In the best case, when all cnν (f ) 6 = 0 and 〈ϕnν, g 〉 6 = 0 then it is possible to evaluate/extract all the λn from the observation (2). \\n\\nOpen problems: i) How do you choose the initial condition f, so we can observe all eλnt, that is all cnν (f ) 6 = 0? We need to know something about the biorthogonal system {ψnν }.ii) How do you choose the observation g so all 〈ϕnν, g 〉 6 = 0? We need to know something about the root functions {ϕn,ν }.iii) How smooth is the sum (2), so we can choose g? We need some information on the type of convergence in (2) so (3) holds. iv) How do we extract the λn and their multiplicity from a given signal given by (3) in finite time? When \\n\\nλn are complex values and the sum contains polynomials in t, it is much harder than the real case. v) Find the best f and g that allow the identification of A by using the smallest number of observations. Evolution equations are often found in control theory, and for that purpose, we need finite number of observations done in finite time. 4. Marina Chugunova \\n\\n4.1. Computations of the instability index for a non-self-adjoint operators. The stability of steady states is a basic question about the dynamics of any partial differential equation that models the evolution of a physical system. In order to numerically evaluate the instability index of a given differential operator A, its computation should be reduced to a problem of linear algebra. Particularly for problems with periodic boundary condi-tions, it seems natural to restrict the operator A to a finite-dimensional space of trigonometric polynomials. \\n\\nOpen problem: Under what conditions the instability index (the total number of unstable eigenvalues) can be computed from the resulting finite dimensional matrix? One difficulty is that the entries of the infinite matrix corresponding to the differential operator A grow with the row and column index, so that any truncation is not a small perturbation. If A is a self-adjoint semi-bounded differential operator of even order, then the instability index can be estimated by variational methods, or computed directly from the zeros of the corresponding Evans function. Understanding the spectrum of a non-self-adjoint operator is a much harder problem. It is not at all obvious how to restrict the computation of its instability index to a finite-dimensional subspace, or how to even estimate its dimension. Furthermore, the numerical calculation of eigenvalues can be extremely ill-conditioned even in finite dimensions. 5. Michael Demuth \\n\\n5.1. Spectral radius and operator norm. Let A be a bounded linear operator on a Banach space X. Its spectral radius is defined by spr( A):= max {| z|: z ∈ σ(A)}.\\n\\nGelfand proved the classical formula spr( A) = lim \\n\\n> n→∞\\n\\n‖An‖ 1 \\n\\n> n.\\n\\n> 2\\n\\nObviously, 0 ≤ spr( A) ≤ ‖ A‖. The question arises: What is the gap between ‖A‖ and spr( A)? Introduce the denotation gap( A):= ‖A‖ − spr( A).\\n\\nOpen problems: i) For which class of operators holds gap( A) > 0 or gap( A) = 0,\\n\\nrespectively. ii) What is the smallest m ∈ (0, 1], such that spr( A) ≤ m‖A‖\\n\\nor gap( A) ≥ (1 − m)‖A‖?\\n\\nExample 1. Let X = `1(N) and A be the weighted shift-operator defined according to the canonical standard basis by the infinite matrix \\n\\n\\n\\n0\\n\\nb1 0\\n\\nb2 0\\n\\nb1 0\\n\\nb2 0......\\n\\n\\n\\nwhere b1, b 2 > 0 and b1b2 = 1. In this case ‖A‖ = max {b1, b 2} and σ(A) = {z ∈ C: |z| ≤ 1} and therefore spr( A) = 1. Thus \\n\\n• gap( A) = 0: If b1 = b2 = 1 then ‖A‖ = spr( A). \\n\\n• gap( A) > 0: If b1 6 = b2 then ‖A‖ > spr( A). This kind of estimates are useful in the following situation. Let K be a compact perturbation of A. Study the discrete spectrum of B:= A + K. We are able to analyze the moments and the number of eigenvalues of B outside a ball of radius ‖A‖. It is more interesting and also natural to enlarge this region up to the complement of a ball with radius spr( A). 6. Mark Embree \\n\\n6.1. Davies' conjecture about approximate diagonalization. Consider a non-normal matrix A ∈\\n\\nCn×n. Define \\n\\ns(A, ε ):= inf \\n\\n> ∆,V V−1(A+∆) Vdiagonal\\n\\n‖V ‖ ‖ V −1‖ε + ‖∆‖.\\n\\nOpen problem: Prove Davies' conjecture (2007): There exists a constant Cn > 0, independent of A ∈ Cn×n,such that s(A, ε ) ≤ Cn\\n\\n√ε.It is known that the conjecture holds for Jordan blocks (then Cn = 2 suffices) and for 3 × 3 matrices with \\n\\n‖A‖ ≤ 1 (then Cn = 4 suffices). 6.2. Crouzeix' conjecture about the norm of matrix functions. Let A be a bounded linear operator. It is known that ‖Ak‖ ≤ 2 max z∈W (A) |zk|; W (A) denotes the numerical range of A.\\n\\nOpen problem: Prove Crouzeix' conjecture: There exists a constant C ≥ 2 such that for all analytic functions \\n\\nf: W (A) → C holds ‖f (A)‖ ≤ C max z∈W (A) |f (z)|.Crouzeix conjectured further that C ≤ 11.08. \\n\\n> 3\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/nonselfadjointproblems.pdf", "tags": [ "aim", "AIM-PHYSICS-0006", "aim-domain:physics", "aim-workshop:nonselfadjointproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The OCR-damaged composite record is routed to Almog's subproblem 1.1. Primary literature proves Dirichlet half-line completeness at alpha=2/3 and on an interval below it, but the checked results do not cover every alpha>0; Tumanov's stated angle bound cannot reach c=i for alpha<=1/2. The full-line operator is proved to split exactly into Neumann and Dirichlet half-line realizations, so the Dirichlet theorem alone does not settle the full-line wording. An explicit family of normalized packets at scale exp(1/sqrt(alpha)) is constructed which, as alpha tends to zero, approximates every point k^2+i and escapes to infinity, giving a rigorous limiting-pseudospectrum obstruction while preserving compact resolvent for each fixed alpha>0.\n\nCandidate contribution (obstruction; novelty confidence low): For every k>=0 and for each of the Dirichlet half-line, Neumann half-line, and full-line realizations of -d^2/dx^2+i|x|^alpha, there exist normalized compactly supported u_alpha with support in [exp(1/sqrt(alpha)),2 exp(1/sqrt(alpha))] such that ||(A_alpha-(k^2+i))u_alpha|| tends to zero as alpha tends to zero; hence the minimum modulus tends to zero along the entire limiting ray i+[0,infinity)." }, { "id": 20002499, "problem_number": "AIM-PHYSICS-0007", "title": "The sharp Laptev-Safronov threshold and a Lorentz-stable tube obstruction", "statement": "7. Rupert L. Frank\n\n7.1. Laptev-Safronov conjecture. Consider a Schr¨ odinger operator −∆ + V in Rd with a complex po-tential V.\n\nOpen problems: i) What is the largest p such that all non-real eigenvalues lie in a disk around 0 of radius\n\nD(∫\n\n> Rd\n\n|V |p dx)(p−d/ 2) −1\n\n(the constant D > 0 shall not depend on V )? ii) What happens to embedded eigenvalues of self-adjoint Schr¨ odinger operators under non-self-adjoint perturbations? Both problems are related to the Laptev-Safronov conjecture which states that for every d ∈ N and 0 < γ ≤ d/ 2 there exists a constant Dγ,d > 0 such that for every potential V every non-real eigenvalue λ\n\nsatisfies\n\n|λ|γ ≤ Dγ,d\n\n∫\n\n> Rd\n\n|V |γ+ d\n\n> 2\n\ndx;here γ = p − d/ 2 with p from problem i). The Laptev-Safronov conjecture is known to be true in dimension\n\nd = 1 if γ = 1 /2 and in dimension d ≥ 2 if 0 < γ ≤ 1/2. 8. Marcel Hansmann\n\n8.1. Tensor trick for perturbed operators. Let A be a bounded self-adjoint operator in a Hilbert space and let K be a perturbation which is of trace class. Assume that for any ε > 0 there exists a constant\n\nC(ε) > 0 such that ∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) 1+ ε ≤ C(ε)‖K‖1+ ε\n\n> 1+ ε.\n\nOpen problem: Does it follow that there exists C > 0 such that\n\n∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) ≤ C‖K‖1?9. Michael Hitrik\n\n9.1. Upper bounds on the norm of the resolvent. It is well known that the spectrum of a non-self-adjoint operator does not control its resolvent and that the latter may become very large far from the spectrum. Some general upper bounds on resolvents are provided by the abstract operator theory, and restricting the attention to the setting of semiclassical operators on Rn, let us give a rough statement of such bounds. Assume that P = pw(x, hD x) is the semiclassical Weyl quantization on Rn of a nice symbol p\n\nwith Re p ≥ 0, say. Then the norm of the resolvent of P is bounded from above by a quantity of the form\n\nO(1) exp ( O(1) h−n), provided that z ∈ neigh(0, C) is not too close to the spectrum of P. On the other hand, the available lower bounds on the resolvent of P, in the interior of the range of the symbol, coming from the pseudospectral considerations, are typically of the form C−1\n\n> N\n\nh−N, N ∈ N, or (1 /C )e1/(Ch ), provided that p enjoys some analyticity properties, [DSZ04]. There appears to be therefore a substantial gap between the available upper and lower bounds on the resolvent, especially when n ≥ 2, which, to the best of my knowledge, has so far only been bridged in the very special case of elliptic quadratic differential operators, see [HSV13].\n\nOpen problem: Is the upper bound sharp (especially for dimension n ≥ 2)? 10. David Krejˇ ciˇ r´ ık\n\n10.1. Semiclassical pseudomodes of Schr¨ odinger operators with discontinuous potentials. For smooth potentials there exists a quite general theory on the construction of semiclassical pseudomodes, see [DSZ04].\n\nOpen problem: Can the technique be adapted to discontinuous potentials? In my joint paper with Henry [HK15] we have a non-trivial pseudospectrum in a toy model (complex Heaviside-type potential). However, our technique is restricted to the particular situation and the non-trivial pseudospectrum is rather generated by the behavior of the potential at infinity.\n\n> 4", "original_statement": "7. Rupert L. Frank \n\n7.1. Laptev-Safronov conjecture. Consider a Schr¨ odinger operator −∆ + V in Rd with a complex po-tential V.\n\nOpen problems: i) What is the largest p such that all non-real eigenvalues lie in a disk around 0 of radius \n\nD(∫ \n\n> Rd\n\n|V |p dx)(p−d/ 2) −1\n\n(the constant D > 0 shall not depend on V )? ii) What happens to embedded eigenvalues of self-adjoint Schr¨ odinger operators under non-self-adjoint perturbations? Both problems are related to the Laptev-Safronov conjecture which states that for every d ∈ N and 0 < γ ≤ d/ 2 there exists a constant Dγ,d > 0 such that for every potential V every non-real eigenvalue λ\n\nsatisfies \n\n|λ|γ ≤ Dγ,d \n\n∫\n\n> Rd\n\n|V |γ+ d \n\n> 2\n\ndx;here γ = p − d/ 2 with p from problem i). The Laptev-Safronov conjecture is known to be true in dimension \n\nd = 1 if γ = 1 /2 and in dimension d ≥ 2 if 0 < γ ≤ 1/2. 8. Marcel Hansmann \n\n8.1. Tensor trick for perturbed operators. Let A be a bounded self-adjoint operator in a Hilbert space and let K be a perturbation which is of trace class. Assume that for any ε > 0 there exists a constant \n\nC(ε) > 0 such that ∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) 1+ ε ≤ C(ε)‖K‖1+ ε\n\n> 1+ ε.\n\nOpen problem: Does it follow that there exists C > 0 such that \n\n∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) ≤ C‖K‖1?9. Michael Hitrik \n\n9.1. Upper bounds on the norm of the resolvent. It is well known that the spectrum of a non-self-adjoint operator does not control its resolvent and that the latter may become very large far from the spectrum. Some general upper bounds on resolvents are provided by the abstract operator theory, and restricting the attention to the setting of semiclassical operators on Rn, let us give a rough statement of such bounds. Assume that P = pw(x, hD x) is the semiclassical Weyl quantization on Rn of a nice symbol p\n\nwith Re p ≥ 0, say. Then the norm of the resolvent of P is bounded from above by a quantity of the form \n\nO(1) exp ( O(1) h−n), provided that z ∈ neigh(0, C) is not too close to the spectrum of P. On the other hand, the available lower bounds on the resolvent of P, in the interior of the range of the symbol, coming from the pseudospectral considerations, are typically of the form C−1 \n\n> N\n\nh−N, N ∈ N, or (1 /C )e1/(Ch ), provided that p enjoys some analyticity properties, [DSZ04]. There appears to be therefore a substantial gap between the available upper and lower bounds on the resolvent, especially when n ≥ 2, which, to the best of my knowledge, has so far only been bridged in the very special case of elliptic quadratic differential operators, see [HSV13]. \n\nOpen problem: Is the upper bound sharp (especially for dimension n ≥ 2)? 10. David Krejˇ ciˇ r´ ık \n\n10.1. Semiclassical pseudomodes of Schr¨ odinger operators with discontinuous potentials. For smooth potentials there exists a quite general theory on the construction of semiclassical pseudomodes, see [DSZ04]. \n\nOpen problem: Can the technique be adapted to discontinuous potentials? In my joint paper with Henry [HK15] we have a non-trivial pseudospectrum in a toy model (complex Heaviside-type potential). However, our technique is restricted to the particular situation and the non-trivial pseudospectrum is rather generated by the behavior of the potential at infinity. \n\n> 4", "clean_statement": "7. Rupert L. Frank\n\n7.1. Laptev-Safronov conjecture. Consider a Schr¨ odinger operator −∆ + V in Rd with a complex po-tential V.\n\nOpen problems: i) What is the largest p such that all non-real eigenvalues lie in a disk around 0 of radius\n\nD(∫\n\n> Rd\n\n|V |p dx)(p−d/ 2) −1\n\n(the constant D > 0 shall not depend on V )? ii) What happens to embedded eigenvalues of self-adjoint Schr¨ odinger operators under non-self-adjoint perturbations? Both problems are related to the Laptev-Safronov conjecture which states that for every d ∈ N and 0 < γ ≤ d/ 2 there exists a constant Dγ,d > 0 such that for every potential V every non-real eigenvalue λ\n\nsatisfies\n\n|λ|γ ≤ Dγ,d\n\n∫\n\n> Rd\n\n|V |γ+ d\n\n> 2\n\ndx;here γ = p − d/ 2 with p from problem i). The Laptev-Safronov conjecture is known to be true in dimension\n\nd = 1 if γ = 1 /2 and in dimension d ≥ 2 if 0 < γ ≤ 1/2. 8. Marcel Hansmann\n\n8.1. Tensor trick for perturbed operators. Let A be a bounded self-adjoint operator in a Hilbert space and let K be a perturbation which is of trace class. Assume that for any ε > 0 there exists a constant\n\nC(ε) > 0 such that ∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) 1+ ε ≤ C(ε)‖K‖1+ ε\n\n> 1+ ε.\n\nOpen problem: Does it follow that there exists C > 0 such that\n\n∑\n\n> λ∈σd(A+K)\n\ndist( λ, σ (A)) ≤ C‖K‖1?9. Michael Hitrik\n\n9.1. Upper bounds on the norm of the resolvent. It is well known that the spectrum of a non-self-adjoint operator does not control its resolvent and that the latter may become very large far from the spectrum. Some general upper bounds on resolvents are provided by the abstract operator theory, and restricting the attention to the setting of semiclassical operators on Rn, let us give a rough statement of such bounds. Assume that P = pw(x, hD x) is the semiclassical Weyl quantization on Rn of a nice symbol p\n\nwith Re p ≥ 0, say. Then the norm of the resolvent of P is bounded from above by a quantity of the form\n\nO(1) exp ( O(1) h−n), provided that z ∈ neigh(0, C) is not too close to the spectrum of P. On the other hand, the available lower bounds on the resolvent of P, in the interior of the range of the symbol, coming from the pseudospectral considerations, are typically of the form C−1\n\n> N\n\nh−N, N ∈ N, or (1 /C )e1/(Ch ), provided that p enjoys some analyticity properties, [DSZ04]. There appears to be therefore a substantial gap between the available upper and lower bounds on the resolvent, especially when n ≥ 2, which, to the best of my knowledge, has so far only been bridged in the very special case of elliptic quadratic differential operators, see [HSV13].\n\nOpen problem: Is the upper bound sharp (especially for dimension n ≥ 2)? 10. David Krejˇ ciˇ r´ ık\n\n10.1. Semiclassical pseudomodes of Schr¨ odinger operators with discontinuous potentials. For smooth potentials there exists a quite general theory on the construction of semiclassical pseudomodes, see [DSZ04].\n\nOpen problem: Can the technique be adapted to discontinuous potentials? In my joint paper with Henry [HK15] we have a non-trivial pseudospectrum in a toy model (complex Heaviside-type potential). However, our technique is restricted to the particular situation and the non-trivial pseudospectrum is rather generated by the behavior of the potential at infinity.\n\n> 4", "statement_status": "exact", "statement_verification": "The canonical record is visibly corrupted by page-level OCR: it begins with Rupert L. Frank's item 7.1 and then appends the complete items 8.1 (Marcel Hansmann), 9.1 (Michael Hitrik), and 10.1 (David Krejčiřík). The record key is `number: 7`, so this attempt treats only item 7.1. The later items are extraction spillover, not additional assignments.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Mathematical aspects of physics with non-self-adjoint operators\nSection: \nSource item: 7\nSource URL: https://aimath.org/pastworkshops/nonselfadjointproblems.pdf\nCanonical location: aim-physics-notes.json notes[6]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"7. Rupert L. Frank \\n\\n7.1. Laptev-Safronov conjecture. Consider a Schr¨ odinger operator −∆ + V in Rd with a complex po-tential V.\\n\\nOpen problems: i) What is the largest p such that all non-real eigenvalues lie in a disk around 0 of radius \\n\\nD(∫ \\n\\n> Rd\\n\\n|V |p dx)(p−d/ 2) −1\\n\\n(the constant D > 0 shall not depend on V )? ii) What happens to embedded eigenvalues of self-adjoint Schr¨ odinger operators under non-self-adjoint perturbations? Both problems are related to the Laptev-Safronov conjecture which states that for every d ∈ N and 0 < γ ≤ d/ 2 there exists a constant Dγ,d > 0 such that for every potential V every non-real eigenvalue λ\\n\\nsatisfies \\n\\n|λ|γ ≤ Dγ,d \\n\\n∫\\n\\n> Rd\\n\\n|V |γ+ d \\n\\n> 2\\n\\ndx;here γ = p − d/ 2 with p from problem i). The Laptev-Safronov conjecture is known to be true in dimension \\n\\nd = 1 if γ = 1 /2 and in dimension d ≥ 2 if 0 < γ ≤ 1/2. 8. Marcel Hansmann \\n\\n8.1. Tensor trick for perturbed operators. Let A be a bounded self-adjoint operator in a Hilbert space and let K be a perturbation which is of trace class. Assume that for any ε > 0 there exists a constant \\n\\nC(ε) > 0 such that ∑\\n\\n> λ∈σd(A+K)\\n\\ndist( λ, σ (A)) 1+ ε ≤ C(ε)‖K‖1+ ε\\n\\n> 1+ ε.\\n\\nOpen problem: Does it follow that there exists C > 0 such that \\n\\n∑\\n\\n> λ∈σd(A+K)\\n\\ndist( λ, σ (A)) ≤ C‖K‖1?9. Michael Hitrik \\n\\n9.1. Upper bounds on the norm of the resolvent. It is well known that the spectrum of a non-self-adjoint operator does not control its resolvent and that the latter may become very large far from the spectrum. Some general upper bounds on resolvents are provided by the abstract operator theory, and restricting the attention to the setting of semiclassical operators on Rn, let us give a rough statement of such bounds. Assume that P = pw(x, hD x) is the semiclassical Weyl quantization on Rn of a nice symbol p\\n\\nwith Re p ≥ 0, say. Then the norm of the resolvent of P is bounded from above by a quantity of the form \\n\\nO(1) exp ( O(1) h−n), provided that z ∈ neigh(0, C) is not too close to the spectrum of P. On the other hand, the available lower bounds on the resolvent of P, in the interior of the range of the symbol, coming from the pseudospectral considerations, are typically of the form C−1 \\n\\n> N\\n\\nh−N, N ∈ N, or (1 /C )e1/(Ch ), provided that p enjoys some analyticity properties, [DSZ04]. There appears to be therefore a substantial gap between the available upper and lower bounds on the resolvent, especially when n ≥ 2, which, to the best of my knowledge, has so far only been bridged in the very special case of elliptic quadratic differential operators, see [HSV13]. \\n\\nOpen problem: Is the upper bound sharp (especially for dimension n ≥ 2)? 10. David Krejˇ ciˇ r´ ık \\n\\n10.1. Semiclassical pseudomodes of Schr¨ odinger operators with discontinuous potentials. For smooth potentials there exists a quite general theory on the construction of semiclassical pseudomodes, see [DSZ04]. \\n\\nOpen problem: Can the technique be adapted to discontinuous potentials? In my joint paper with Henry [HK15] we have a non-trivial pseudospectrum in a toy model (complex Heaviside-type potential). However, our technique is restricted to the particular situation and the non-trivial pseudospectrum is rather generated by the behavior of the potential at infinity. \\n\\n> 4\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/nonselfadjointproblems.pdf", "tags": [ "aim", "AIM-PHYSICS-0007", "aim-domain:physics", "aim-workshop:nonselfadjointproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The merged OCR record was reconstructed and routed to Rupert Frank's item 7.1. For general complex potentials the uniform eigenvalue-disk threshold is now sharp: p_max=(d+1)/2 for d>=2, with Frank's endpoint bound and Bögli-Cuenin counterexamples for every larger p; in d=1 the largest exponent is p=1. A proved corollary of the Bögli-Cuenin parabolic-tube construction shows that failure persists with any full-space Lorentz L^{q,r} norm, including L^{q,1}, and that every homogeneous angular enclosure based on such a norm must lose at least q-(d+1)/2 powers of normalized distance to [0,infinity). The separate question about perturbing a fixed embedded eigenvalue remains open at this level of generality.\n\nCandidate contribution (lorentz_space_obstruction; novelty confidence low): For every d>=2, q>(d+1)/2, and 1<=r<=infinity, the bounded compactly supported Bögli-Cuenin tube potentials rule out a scale-invariant disk bound using the full-space Lorentz norm L^{q,r}; more generally, a bound |z|^{q-d/2}(dist(z,[0,infinity))/|z|)^beta <= C||V||_{L^{q,r}}^q forces beta>=q-(d+1)/2." }, { "id": 20002500, "problem_number": "AIM-PHYSICS-0008", "title": "Axis and low-dimensional cases of Levitin's disk-intersection conjecture", "statement": "11. Michael Levitin\n\n11.1. Complex eigenvalues of indefinite pencil. For 0 < c < 2 define the 2 n × 2n matrices\n\nA:=\n\n\n\nc 11 c 11 c......... 11 c\n\n, B:=\n\n\n\n1...1\n\n−1...\n\n−1\n\n\n\nwhere the numbers of 1 and −1 in B coincide (equal to n). The eigenvalues of the pencil λ 7 → A − λB are the eigenvalues of B−1A = BA. The spectrum is symmetric with respect to both R and i R. The non-real eigenvalues are contained in the union of the two closed disks {λ ∈ C: |λ ± c| ≤ 2} whereas numerical examples suggest that they lie in their intersection.\n\nOpen problem: Prove that |λ − c| ≤ 2 and |λ + c| ≤ 2 holds for all λ ∈ σ(BA )\\R.11.2. Indefinite Sturm-Liouville. Recently, we have proved some conjectures related to the generalized eigenvalue problem ( − d2\n\n> dx 2\n\n+ c\n\n> 1+ |x|\n\n)ψ = λsgn( x)ψ.\n\nOpen problem: Replace the potential term by a more general function. 12. Marco Marletta\n\n12.1. Zeros and poles of Nevanlinna functions. Let m1, m 2,... be an infinite sequence of meromorphic functions with Im mj (λ) > 0 if Im λ > 0 and Im mj (λ) < 0 if Im λ < 0 (Nevanlinna functions). Suppose that there exists a non-empty interval I ⊆ R such that for every non-empty subinterval J ⊆ I holds lim\n\n> j→∞\n\n#{pole of mj in J} = ∞.\n\nLet g be a function which is analytic in a complex neighborhood of I.\n\nOpen problems: i) Show that for every open complex neighborhood U of J,lim\n\n> j→∞\n\n#{zero of ( mj − g) in U } = ∞.\n\nii) Show that there exists a constant C > 0, independent of j, such that for every open complex neighborhood\n\nU of J, ∣∣#{zero of ( mj − g) in U } − #{pole of mj in U }∣∣ ≤ C.\n\nThe result is known to hold if μ(J):= lim j→∞ j−1 #{pole of mj in J} exists. 13. Boris Mityagin\n\n13.1. Schauder basis for Schr¨ odinger operators with periodic boundary conditions. Consider the Schr¨ odinger operator −d2/dx2 + V on (0, π ) with periodic boundary conditions, where the potential V is a trigonometric polynomial, i.e.,\n\nV (x) =\n\n> m\n\n∑\n\n> k=−m\n\nvke2ikx; vk ∈ C, |k| ≤ m.\n\nOpen Problem: For which sets of coefficients {vk}mk=−m do the eigenfunctions form a Schauder basis for\n\nL2(0, π )? Known Case: Let V (x):= e−2ix + be 2ix. Then the answer is \"yes\" if and only if |b| = 1.\n\n> 5", "original_statement": "11. Michael Levitin \n\n11.1. Complex eigenvalues of indefinite pencil. For 0 < c < 2 define the 2 n × 2n matrices \n\nA:= \n\n\n\nc 11 c 11 c......... 11 c\n\n, B:= \n\n\n\n1...1\n\n−1...\n\n−1\n\n\n\nwhere the numbers of 1 and −1 in B coincide (equal to n). The eigenvalues of the pencil λ 7 → A − λB are the eigenvalues of B−1A = BA. The spectrum is symmetric with respect to both R and i R. The non-real eigenvalues are contained in the union of the two closed disks {λ ∈ C: |λ ± c| ≤ 2} whereas numerical examples suggest that they lie in their intersection. \n\nOpen problem: Prove that |λ − c| ≤ 2 and |λ + c| ≤ 2 holds for all λ ∈ σ(BA )\\R.11.2. Indefinite Sturm-Liouville. Recently, we have proved some conjectures related to the generalized eigenvalue problem ( − d2 \n\n> dx 2\n\n+ c \n\n> 1+ |x|\n\n)ψ = λsgn( x)ψ.\n\nOpen problem: Replace the potential term by a more general function. 12. Marco Marletta \n\n12.1. Zeros and poles of Nevanlinna functions. Let m1, m 2,... be an infinite sequence of meromorphic functions with Im mj (λ) > 0 if Im λ > 0 and Im mj (λ) < 0 if Im λ < 0 (Nevanlinna functions). Suppose that there exists a non-empty interval I ⊆ R such that for every non-empty subinterval J ⊆ I holds lim \n\n> j→∞\n\n#{pole of mj in J} = ∞.\n\nLet g be a function which is analytic in a complex neighborhood of I.\n\nOpen problems: i) Show that for every open complex neighborhood U of J,lim \n\n> j→∞\n\n#{zero of ( mj − g) in U } = ∞.\n\nii) Show that there exists a constant C > 0, independent of j, such that for every open complex neighborhood \n\nU of J, ∣∣#{zero of ( mj − g) in U } − #{pole of mj in U }∣∣ ≤ C. \n\nThe result is known to hold if μ(J):= lim j→∞ j−1 #{pole of mj in J} exists. 13. Boris Mityagin \n\n13.1. Schauder basis for Schr¨ odinger operators with periodic boundary conditions. Consider the Schr¨ odinger operator −d2/dx2 + V on (0, π ) with periodic boundary conditions, where the potential V is a trigonometric polynomial, i.e., \n\nV (x) = \n\n> m\n\n∑\n\n> k=−m\n\nvke2ikx; vk ∈ C, |k| ≤ m. \n\nOpen Problem: For which sets of coefficients {vk}mk=−m do the eigenfunctions form a Schauder basis for \n\nL2(0, π )? Known Case: Let V (x):= e−2ix + be 2ix. Then the answer is \"yes\" if and only if |b| = 1. \n\n> 5", "clean_statement": "11. Michael Levitin\n\n11.1. Complex eigenvalues of indefinite pencil. For 0 < c < 2 define the 2 n × 2n matrices\n\nA:=\n\n\n\nc 11 c 11 c......... 11 c\n\n, B:=\n\n\n\n1...1\n\n−1...\n\n−1\n\n\n\nwhere the numbers of 1 and −1 in B coincide (equal to n). The eigenvalues of the pencil λ 7 → A − λB are the eigenvalues of B−1A = BA. The spectrum is symmetric with respect to both R and i R. The non-real eigenvalues are contained in the union of the two closed disks {λ ∈ C: |λ ± c| ≤ 2} whereas numerical examples suggest that they lie in their intersection.\n\nOpen problem: Prove that |λ − c| ≤ 2 and |λ + c| ≤ 2 holds for all λ ∈ σ(BA )\\R.11.2. Indefinite Sturm-Liouville. Recently, we have proved some conjectures related to the generalized eigenvalue problem ( − d2\n\n> dx 2\n\n+ c\n\n> 1+ |x|\n\n)ψ = λsgn( x)ψ.\n\nOpen problem: Replace the potential term by a more general function. 12. Marco Marletta\n\n12.1. Zeros and poles of Nevanlinna functions. Let m1, m 2,... be an infinite sequence of meromorphic functions with Im mj (λ) > 0 if Im λ > 0 and Im mj (λ) < 0 if Im λ < 0 (Nevanlinna functions). Suppose that there exists a non-empty interval I ⊆ R such that for every non-empty subinterval J ⊆ I holds lim\n\n> j→∞\n\n#{pole of mj in J} = ∞.\n\nLet g be a function which is analytic in a complex neighborhood of I.\n\nOpen problems: i) Show that for every open complex neighborhood U of J,lim\n\n> j→∞\n\n#{zero of ( mj − g) in U } = ∞.\n\nii) Show that there exists a constant C > 0, independent of j, such that for every open complex neighborhood\n\nU of J, ∣∣#{zero of ( mj − g) in U } − #{pole of mj in U }∣∣ ≤ C.\n\nThe result is known to hold if μ(J):= lim j→∞ j−1 #{pole of mj in J} exists. 13. Boris Mityagin\n\n13.1. Schauder basis for Schr¨ odinger operators with periodic boundary conditions. Consider the Schr¨ odinger operator −d2/dx2 + V on (0, π ) with periodic boundary conditions, where the potential V is a trigonometric polynomial, i.e.,\n\nV (x) =\n\n> m\n\n∑\n\n> k=−m\n\nvke2ikx; vk ∈ C, |k| ≤ m.\n\nOpen Problem: For which sets of coefficients {vk}mk=−m do the eigenfunctions form a Schauder basis for\n\nL2(0, π )? Known Case: Let V (x):= e−2ix + be 2ix. Then the answer is \"yes\" if and only if |b| = 1.\n\n> 5", "statement_status": "exact", "statement_verification": "The canonical record is zero-based source index 7 of `aim-physics-notes.json`. It is a composite OCR extraction from page 5 of the AIM PDF *List of Open Problems: Mathematical Aspects of Physics with Non-Self-Adjoint Operators*. The raw record begins with Michael Levitin's item 11 but continues through items 12 and 13 because several page entries were merged into one JSON object.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Mathematical aspects of physics with non-self-adjoint operators\nSection: \nSource item: 11\nSource URL: https://aimath.org/pastworkshops/nonselfadjointproblems.pdf\nCanonical location: aim-physics-notes.json notes[7]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"11. Michael Levitin \\n\\n11.1. Complex eigenvalues of indefinite pencil. For 0 < c < 2 define the 2 n × 2n matrices \\n\\nA:= \\n\\n\\n\\nc 11 c 11 c......... 11 c\\n\\n, B:= \\n\\n\\n\\n1...1\\n\\n−1...\\n\\n−1\\n\\n\\n\\nwhere the numbers of 1 and −1 in B coincide (equal to n). The eigenvalues of the pencil λ 7 → A − λB are the eigenvalues of B−1A = BA. The spectrum is symmetric with respect to both R and i R. The non-real eigenvalues are contained in the union of the two closed disks {λ ∈ C: |λ ± c| ≤ 2} whereas numerical examples suggest that they lie in their intersection. \\n\\nOpen problem: Prove that |λ − c| ≤ 2 and |λ + c| ≤ 2 holds for all λ ∈ σ(BA )\\\\R.11.2. Indefinite Sturm-Liouville. Recently, we have proved some conjectures related to the generalized eigenvalue problem ( − d2 \\n\\n> dx 2\\n\\n+ c \\n\\n> 1+ |x|\\n\\n)ψ = λsgn( x)ψ.\\n\\nOpen problem: Replace the potential term by a more general function. 12. Marco Marletta \\n\\n12.1. Zeros and poles of Nevanlinna functions. Let m1, m 2,... be an infinite sequence of meromorphic functions with Im mj (λ) > 0 if Im λ > 0 and Im mj (λ) < 0 if Im λ < 0 (Nevanlinna functions). Suppose that there exists a non-empty interval I ⊆ R such that for every non-empty subinterval J ⊆ I holds lim \\n\\n> j→∞\\n\\n#{pole of mj in J} = ∞.\\n\\nLet g be a function which is analytic in a complex neighborhood of I.\\n\\nOpen problems: i) Show that for every open complex neighborhood U of J,lim \\n\\n> j→∞\\n\\n#{zero of ( mj − g) in U } = ∞.\\n\\nii) Show that there exists a constant C > 0, independent of j, such that for every open complex neighborhood \\n\\nU of J, ∣∣#{zero of ( mj − g) in U } − #{pole of mj in U }∣∣ ≤ C. \\n\\nThe result is known to hold if μ(J):= lim j→∞ j−1 #{pole of mj in J} exists. 13. Boris Mityagin \\n\\n13.1. Schauder basis for Schr¨ odinger operators with periodic boundary conditions. Consider the Schr¨ odinger operator −d2/dx2 + V on (0, π ) with periodic boundary conditions, where the potential V is a trigonometric polynomial, i.e., \\n\\nV (x) = \\n\\n> m\\n\\n∑\\n\\n> k=−m\\n\\nvke2ikx; vk ∈ C, |k| ≤ m. \\n\\nOpen Problem: For which sets of coefficients {vk}mk=−m do the eigenfunctions form a Schauder basis for \\n\\nL2(0, π )? Known Case: Let V (x):= e−2ix + be 2ix. Then the answer is \\\"yes\\\" if and only if |b| = 1. \\n\\n> 5\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/nonselfadjointproblems.pdf", "tags": [ "aim", "AIM-PHYSICS-0008", "aim-domain:physics", "aim-workshop:nonselfadjointproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the balanced 2n-dimensional Davies-Levitin tridiagonal pencil, the characteristic equation is an exact product identity for consecutive path polynomials. This identity and a strict dominance lemma prove the conjectured two-disk bound for every nonreal eigenvalue on the imaginary axis in every dimension, and a separate quadratic analysis proves the full conjecture for n=1 and n=2. Any counterexample must therefore have n at least 3, be genuinely off-axis, and satisfy an explicit Chebyshev-ratio straddling condition.\n\nCandidate contribution (special_case_and_reduction; novelty confidence low): Every nonreal imaginary-axis eigenvalue obeys the strict disk-intersection bound for all n; the complete 4-by-4 family obeys the conjecture for every 0 6\n\n*Other Open Problems", "original_statement": "14. Kwang Shin \n\n14.1. Non-polynomial complex potentials. Consider \n\nH = − d2\n\ndx 2 + xm + a1xm−1 + · · · + am\n\nin L2(R+) with y(0) cos θ + y′(0) sin θ = 0, where aj ∈ C and θ ∈ C. It is known that H has infinitely many eigenvalues. Moreover, all eigenvalues are real if and only if aj ∈ R for all j and θ ∈ R.\n\nOpen Problem: Is there any non-self-adjoint non-polynomial potential case H, either in L2(R) or in L2(R+),\n\nthat generates infinitely many real eigenvalues and at most finitely many non-real eigenvalues? 15. Petr Siegl \n\n15.1. Riesz basis for Schr¨ odinger operator with complex potential. Consider the self-adjoint har-monic oscillator A0:= −d2/dx2 + x2 in L2(R). Define A:= A0 + V with a complex-valued potential \n\nV ∈ L∞(R). \n\nOpen problem: Is the eigensystem of A a Riesz basis? It is known that the answer is 'yes' if V ∈ Lp(R) for some 1 ≤ p < ∞.16. David A. Smith \n\n16.1. Spectral representation of two-point differential operators. Augmented eigenfunctions are a class of spectral functionals which have been shown to be useful in expressing solutions of initial-boundary value problems [FS15, PSss, Smi14]. This is particularly important in the case where the spatial differential operator is degenerate irregular in the sense of Locker [Loc08], as no other effective solution representation is known. They also provide a spectral theorem where the inverse of the operator is diagonalized. \n\nOpen problems: i) Are there other applications for augmented eigenfunctions? ii) Can a spectral theory be developed using augmented eigenfunctions? \n\n> 6\n\n*Other Open Problems", "clean_statement": "14. Kwang Shin\n\n14.1. Non-polynomial complex potentials. Consider\n\nH = − d2\n\ndx 2 + xm + a1xm−1 + · · · + am\n\nin L2(R+) with y(0) cos θ + y′(0) sin θ = 0, where aj ∈ C and θ ∈ C. It is known that H has infinitely many eigenvalues. Moreover, all eigenvalues are real if and only if aj ∈ R for all j and θ ∈ R.\n\nOpen Problem: Is there any non-self-adjoint non-polynomial potential case H, either in L2(R) or in L2(R+),\n\nthat generates infinitely many real eigenvalues and at most finitely many non-real eigenvalues? 15. Petr Siegl\n\n15.1. Riesz basis for Schr¨ odinger operator with complex potential. Consider the self-adjoint har-monic oscillator A0:= −d2/dx2 + x2 in L2(R). Define A:= A0 + V with a complex-valued potential\n\nV ∈ L∞(R).\n\nOpen problem: Is the eigensystem of A a Riesz basis? It is known that the answer is 'yes' if V ∈ Lp(R) for some 1 ≤ p < ∞.16. David A. Smith\n\n16.1. Spectral representation of two-point differential operators. Augmented eigenfunctions are a class of spectral functionals which have been shown to be useful in expressing solutions of initial-boundary value problems [FS15, PSss, Smi14]. This is particularly important in the case where the spatial differential operator is degenerate irregular in the sense of Locker [Loc08], as no other effective solution representation is known. They also provide a spectral theorem where the inverse of the operator is diagonalized.\n\nOpen problems: i) Are there other applications for augmented eigenfunctions? ii) Can a spectral theory be developed using augmented eigenfunctions?\n\n> 6\n\n*Other Open Problems", "statement_status": "exact", "statement_verification": "The canonical record is item 14.1, attributed to Kwang Shin, in the AIM workshop list *Mathematical aspects of physics with non-self-adjoint operators*. The PDF asks first about the polynomial half-line operator", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Mathematical aspects of physics with non-self-adjoint operators\nSection: \nSource item: 14\nSource URL: https://aimath.org/pastworkshops/nonselfadjointproblems.pdf\nCanonical location: aim-physics-notes.json notes[8]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"14. Kwang Shin \\n\\n14.1. Non-polynomial complex potentials. Consider \\n\\nH = − d2\\n\\ndx 2 + xm + a1xm−1 + · · · + am\\n\\nin L2(R+) with y(0) cos θ + y′(0) sin θ = 0, where aj ∈ C and θ ∈ C. It is known that H has infinitely many eigenvalues. Moreover, all eigenvalues are real if and only if aj ∈ R for all j and θ ∈ R.\\n\\nOpen Problem: Is there any non-self-adjoint non-polynomial potential case H, either in L2(R) or in L2(R+),\\n\\nthat generates infinitely many real eigenvalues and at most finitely many non-real eigenvalues? 15. Petr Siegl \\n\\n15.1. Riesz basis for Schr¨ odinger operator with complex potential. Consider the self-adjoint har-monic oscillator A0:= −d2/dx2 + x2 in L2(R). Define A:= A0 + V with a complex-valued potential \\n\\nV ∈ L∞(R). \\n\\nOpen problem: Is the eigensystem of A a Riesz basis? It is known that the answer is 'yes' if V ∈ Lp(R) for some 1 ≤ p < ∞.16. David A. Smith \\n\\n16.1. Spectral representation of two-point differential operators. Augmented eigenfunctions are a class of spectral functionals which have been shown to be useful in expressing solutions of initial-boundary value problems [FS15, PSss, Smi14]. This is particularly important in the case where the spatial differential operator is degenerate irregular in the sense of Locker [Loc08], as no other effective solution representation is known. They also provide a spectral theorem where the inverse of the operator is diagonalized. \\n\\nOpen problems: i) Are there other applications for augmented eigenfunctions? ii) Can a spectral theory be developed using augmented eigenfunctions? \\n\\n> 6\\n\\n*Other Open Problems\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/nonselfadjointproblems.pdf", "tags": [ "aim", "AIM-PHYSICS-0009", "aim-domain:physics", "aim-workshop:nonselfadjointproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For every real kappa not equal to zero, the zero-free Darboux seed u_kappa(x)=exp(x^2/2)(1+i kappa integral_0^x exp(-t^2) dt) produces a smooth PT-symmetric complex non-polynomial potential V_kappa=x^2-2(log u_kappa)'' on the harmonic-oscillator domain. The resulting closed operator has compact resolvent and exact spectrum {-1,1,3,5,...}; hence it has infinitely many real eigenvalues and no non-real eigenvalues, affirmatively resolving the AIM existential question on L^2(R).\n\nCandidate contribution (explicit_family_and_streamlined_domain_argument; novelty confidence low): The one-parameter zero-free seed above yields a compact specialization in which factorization, operator domains, non-polynomiality, non-self-adjointness, compact resolvent, and the converse exclusion of every additional spectral value are all verified directly." }, { "id": 20002502, "problem_number": "AIM-PHYSICS-0010", "title": "A one-coupling J-Schur certificate for projected spectral enclosures", "statement": "17. Lyonell Boulton\n\n17.1. Numerical approximation of rigorous enclosures for the spectrum of J-self-adjoint opera-tors. The aim of this theme is to device strategies for computing rigorous (hopefully sharp) bounds/enclosures for the spectrum of J-self-adjoint operators by means of projected space methods. The theory of computa-tion for spectra of self-adjoint operators is classical and well developed. J-self-adjoint operators share many properties with their self-adjoint counterpart. I am interested in discussing to what extent general strategies for numerically estimating spectra of J-self-adjoint operators can be devised.\n\nOpen problems: i) Computation of rigorous rough enclosures for J-self-adjoint operators, taking into account the structure of the conjugation J into the projection scheme. ii) Computation of sharp (numerically relevant) enclosures in specific or generic cases. iii) Impact in the study of evolution problems for J-self-adjoint operators. 18. Tanya J. Christiansen\n\n18.1. Isoresonant potentials. Consider the Schr¨ odinger operator −∆ + V on Rd, where the potential\n\nV ∈ L∞\n\n> 0\n\n(Rd). If V ∈ C∞\n\n> c\n\n(Rd; R), then if V is non-trivial the Schr¨ odinger operator has infinitely many resonances. However, if d ≥ 2 there are non-trivial complex-valued potentials V ∈ C∞\n\n> c\n\n(Rd) for which the corresponding Schr¨ odinger operator has no resonances. More generally, one can explicitly construct families of isoresonant, compactly supported complex-valued potentials in dimensions at least 2.\n\nOpen problems: Is there some other data related in some way to spectral or pseudo spectral properties of the operators that distinguish elements (potentials) in these sets? There are related families of isospectral Schr¨ odinger operators in other settings- on the unit circle, for example. One can ask the same question there. 19. Michael Demuth\n\n19.1. Estimates for the resolvent near the spectrum. Let A be a linear operator on a Banach space. Let K be a compact perturbation of A. The approximation numbers of K are defined by\n\nαN (K):= inf {‖ K − F ‖, rank( F ) < N }.\n\nWe consider only compact operators K with lim N →∞ αN (K) = 0. The objective is to estimate the numbers of eigenvalues of the perturbed operator B:= A + K in cer-tain regions of the complex plane. Let Ω t = {λ ∈ C, |λ| > t }. Denote spr( A):= max {| λ|, λ ∈ σ(A)} and assume spr( A) < t < s. De-note by nB (s) the number of eigenvalues of B in Ω s. In [DHHK15] we obtained\n\nnB (s) ≤ (2 e) p\n\n> 2\n\nlog st\n\nsup λ∈Ωt ‖(λ − A)−1‖p\n\n(1 − αN +1 (K) sup λ∈Ωt ‖(λ − A)−1‖)pN∑\n\n> j=1\n\n(αN +1 (K) + αj (K)) p. (4) Here N has to be so large that\n\nαN +1 (K) sup\n\n> λ∈Ωt\n\n‖(λ − A)−1‖ < 1.\n\nThe optimal result depends on the behavior of ‖(λ−A)−1‖ near the spectrum of A, i.e. on Ω s. This is typical for many spectral considerations. It is also related to the pseudospectrum of A. For instance if |λ| > ‖A‖\n\nthen\n\n‖(λ − A)−1‖ ≤ 1\n\n|λ| − ‖ A‖\n\n> 7\n\nand therefore (4) becomes\n\nnB (s) ≤ (2 e) p\n\n> 2\n\nlog st (t − (‖A‖ + αN +1 (K))) pN∑\n\n> j=1\n\n(αN +1 (K) + αj (K)) p.\n\nOpen problems: i) Classify the operators for which the resolvent is polynomially bounded if λ → σ(A)? ii) Classify the operators for which one can find an M ≥ 1 such that\n\n‖(λ − A)−1‖ ≤ M\n\ndist ( λ, σ (A)) for all λ ∈ res( A) or\n\n‖(λ − A)−1‖ ≤ M\n\n|λ| − spr( A)for |λ| > spr( A).", "original_statement": "17. Lyonell Boulton \n\n17.1. Numerical approximation of rigorous enclosures for the spectrum of J-self-adjoint opera-tors. The aim of this theme is to device strategies for computing rigorous (hopefully sharp) bounds/enclosures for the spectrum of J-self-adjoint operators by means of projected space methods. The theory of computa-tion for spectra of self-adjoint operators is classical and well developed. J-self-adjoint operators share many properties with their self-adjoint counterpart. I am interested in discussing to what extent general strategies for numerically estimating spectra of J-self-adjoint operators can be devised. \n\nOpen problems: i) Computation of rigorous rough enclosures for J-self-adjoint operators, taking into account the structure of the conjugation J into the projection scheme. ii) Computation of sharp (numerically relevant) enclosures in specific or generic cases. iii) Impact in the study of evolution problems for J-self-adjoint operators. 18. Tanya J. Christiansen \n\n18.1. Isoresonant potentials. Consider the Schr¨ odinger operator −∆ + V on Rd, where the potential \n\nV ∈ L∞ \n\n> 0\n\n(Rd). If V ∈ C∞ \n\n> c\n\n(Rd; R), then if V is non-trivial the Schr¨ odinger operator has infinitely many resonances. However, if d ≥ 2 there are non-trivial complex-valued potentials V ∈ C∞ \n\n> c\n\n(Rd) for which the corresponding Schr¨ odinger operator has no resonances. More generally, one can explicitly construct families of isoresonant, compactly supported complex-valued potentials in dimensions at least 2. \n\nOpen problems: Is there some other data related in some way to spectral or pseudo spectral properties of the operators that distinguish elements (potentials) in these sets? There are related families of isospectral Schr¨ odinger operators in other settings- on the unit circle, for example. One can ask the same question there. 19. Michael Demuth \n\n19.1. Estimates for the resolvent near the spectrum. Let A be a linear operator on a Banach space. Let K be a compact perturbation of A. The approximation numbers of K are defined by \n\nαN (K):= inf {‖ K − F ‖, rank( F ) < N }.\n\nWe consider only compact operators K with lim N →∞ αN (K) = 0. The objective is to estimate the numbers of eigenvalues of the perturbed operator B:= A + K in cer-tain regions of the complex plane. Let Ω t = {λ ∈ C, |λ| > t }. Denote spr( A):= max {| λ|, λ ∈ σ(A)} and assume spr( A) < t < s. De-note by nB (s) the number of eigenvalues of B in Ω s. In [DHHK15] we obtained \n\nnB (s) ≤ (2 e) p\n\n> 2\n\nlog st\n\nsup λ∈Ωt ‖(λ − A)−1‖p\n\n(1 − αN +1 (K) sup λ∈Ωt ‖(λ − A)−1‖)pN∑\n\n> j=1\n\n(αN +1 (K) + αj (K)) p. (4) Here N has to be so large that \n\nαN +1 (K) sup \n\n> λ∈Ωt\n\n‖(λ − A)−1‖ < 1.\n\nThe optimal result depends on the behavior of ‖(λ−A)−1‖ near the spectrum of A, i.e. on Ω s. This is typical for many spectral considerations. It is also related to the pseudospectrum of A. For instance if |λ| > ‖A‖\n\nthen \n\n‖(λ − A)−1‖ ≤ 1\n\n|λ| − ‖ A‖\n\n> 7\n\nand therefore (4) becomes \n\nnB (s) ≤ (2 e) p\n\n> 2\n\nlog st (t − (‖A‖ + αN +1 (K))) pN∑\n\n> j=1\n\n(αN +1 (K) + αj (K)) p.\n\nOpen problems: i) Classify the operators for which the resolvent is polynomially bounded if λ → σ(A)? ii) Classify the operators for which one can find an M ≥ 1 such that \n\n‖(λ − A)−1‖ ≤ M\n\ndist ( λ, σ (A)) for all λ ∈ res( A) or \n\n‖(λ − A)−1‖ ≤ M\n\n|λ| − spr( A)for |λ| > spr( A).", "clean_statement": "17. Lyonell Boulton\n\n17.1. Numerical approximation of rigorous enclosures for the spectrum of J-self-adjoint opera-tors. The aim of this theme is to devise strategies for computing rigorous (hopefully sharp) bounds/enclosures for the spectrum of J-self-adjoint operators by means of projected space methods. The theory of computa-tion for spectra of self-adjoint operators is classical and well developed. J-self-adjoint operators share many properties with their self-adjoint counterpart. I am interested in discussing to what extent general strategies for numerically estimating spectra of J-self-adjoint operators can be devised.\n\nOpen problems: i) Computation of rigorous rough enclosures for J-self-adjoint operators, taking into account the structure of the conjugation J into the projection scheme. ii) Computation of sharp (numerically relevant) enclosures in specific or generic cases. iii) Impact in the study of evolution problems for J-self-adjoint operators. 18. Tanya J. Christiansen\n\n18.1. Isoresonant potentials. Consider the Schr¨ odinger operator −∆ + V on Rd, where the potential\n\nV ∈ L∞\n0\n\n(Rd). If V ∈ C∞\nc\n\n(Rd; R), then if V is non-trivial the Schr¨ odinger operator has infinitely many resonances. However, if d ≥ 2 there are non-trivial complex-valued potentials V ∈ C∞\nc\n\n(Rd) for which the corresponding Schr¨ odinger operator has no resonances. More generally, one can explicitly construct families of isoresonant, compactly supported complex-valued potentials in dimensions at least 2.\n\nOpen problems: Is there some other data related in some way to spectral or pseudo spectral properties of the operators that distinguish elements (potentials) in these sets? There are related families of isospectral Schr¨ odinger operators in other settings- on the unit circle, for example. One can ask the same question there. 19. Michael Demuth\n\n19.1. Estimates for the resolvent near the spectrum. Let A be a linear operator on a Banach space. Let K be a compact perturbation of A. The approximation numbers of K are defined by\n\nαN (K):= inf {‖ K − F ‖, rank( F ) < N }.\n\nWe consider only compact operators K with lim N →∞ αN (K) = 0. The objective is to estimate the numbers of eigenvalues of the perturbed operator B:= A + K in cer-tain regions of the complex plane. Let Ω t = {λ ∈ C, |λ| > t }. Denote spr( A):= max {| λ|, λ ∈ σ(A)} and assume spr( A) < t < s. De-note by nB (s) the number of eigenvalues of B in Ω s. In [DHHK15] we obtained\n\nnB (s) ≤ (2 e) p\n2\n\nlog st\n\nsup λ∈Ωt ‖(λ − A)−1‖p\n\n(1 − αN +1 (K) sup λ∈Ωt ‖(λ − A)−1‖)pN∑\nj=1\n\n(αN +1 (K) + αj (K)) p. (4) Here N has to be so large that\n\nαN +1 (K) sup\nλ∈Ωt\n\n‖(λ − A)−1‖ < 1.\n\nThe optimal result depends on the behavior of ‖(λ−A)−1‖ near the spectrum of A, i.e. on Ω s. This is typical for many spectral considerations. It is also related to the pseudospectrum of A. For instance if |λ| > ‖A‖\n\nthen\n\n‖(λ − A)−1‖ ≤ 1\n\n|λ| − ‖ A‖\n7\n\nand therefore (4) becomes\n\nnB (s) ≤ (2 e) p\n2\n\nlog st (t − (‖A‖ + αN +1 (K))) pN∑\nj=1\n\n(αN +1 (K) + αj (K)) p.\n\nOpen problems: i) Classify the operators for which the resolvent is polynomially bounded if λ → σ(A)? ii) Classify the operators for which one can find an M ≥ 1 such that\n\n‖(λ − A)−1‖ ≤ M\n\ndist ( λ, σ (A)) for all λ ∈ res( A) or\n\n‖(λ − A)−1‖ ≤ M\n\n|λ| − spr( A)for |λ| > spr( A).", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is an OCR concatenation. Its primary item is Lyonell Boulton's item 17.1 from the 2015 AIM workshop *Mathematical aspects of physics with non-self-adjoint operators*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Mathematical aspects of physics with non-self-adjoint operators\nSection: \nSource item: 17\nSource URL: https://aimath.org/pastworkshops/nonselfadjointproblems.pdf\nCanonical location: aim-physics-notes.json notes[9]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"17. Lyonell Boulton \\n\\n17.1. Numerical approximation of rigorous enclosures for the spectrum of J-self-adjoint opera-tors. The aim of this theme is to device strategies for computing rigorous (hopefully sharp) bounds/enclosures for the spectrum of J-self-adjoint operators by means of projected space methods. The theory of computa-tion for spectra of self-adjoint operators is classical and well developed. J-self-adjoint operators share many properties with their self-adjoint counterpart. I am interested in discussing to what extent general strategies for numerically estimating spectra of J-self-adjoint operators can be devised. \\n\\nOpen problems: i) Computation of rigorous rough enclosures for J-self-adjoint operators, taking into account the structure of the conjugation J into the projection scheme. ii) Computation of sharp (numerically relevant) enclosures in specific or generic cases. iii) Impact in the study of evolution problems for J-self-adjoint operators. 18. Tanya J. Christiansen \\n\\n18.1. Isoresonant potentials. Consider the Schr¨ odinger operator −∆ + V on Rd, where the potential \\n\\nV ∈ L∞ \\n\\n> 0\\n\\n(Rd). If V ∈ C∞ \\n\\n> c\\n\\n(Rd; R), then if V is non-trivial the Schr¨ odinger operator has infinitely many resonances. However, if d ≥ 2 there are non-trivial complex-valued potentials V ∈ C∞ \\n\\n> c\\n\\n(Rd) for which the corresponding Schr¨ odinger operator has no resonances. More generally, one can explicitly construct families of isoresonant, compactly supported complex-valued potentials in dimensions at least 2. \\n\\nOpen problems: Is there some other data related in some way to spectral or pseudo spectral properties of the operators that distinguish elements (potentials) in these sets? There are related families of isospectral Schr¨ odinger operators in other settings- on the unit circle, for example. One can ask the same question there. 19. Michael Demuth \\n\\n19.1. Estimates for the resolvent near the spectrum. Let A be a linear operator on a Banach space. Let K be a compact perturbation of A. The approximation numbers of K are defined by \\n\\nαN (K):= inf {‖ K − F ‖, rank( F ) < N }.\\n\\nWe consider only compact operators K with lim N →∞ αN (K) = 0. The objective is to estimate the numbers of eigenvalues of the perturbed operator B:= A + K in cer-tain regions of the complex plane. Let Ω t = {λ ∈ C, |λ| > t }. Denote spr( A):= max {| λ|, λ ∈ σ(A)} and assume spr( A) < t < s. De-note by nB (s) the number of eigenvalues of B in Ω s. In [DHHK15] we obtained \\n\\nnB (s) ≤ (2 e) p\\n\\n> 2\\n\\nlog st\\n\\nsup λ∈Ωt ‖(λ − A)−1‖p\\n\\n(1 − αN +1 (K) sup λ∈Ωt ‖(λ − A)−1‖)pN∑\\n\\n> j=1\\n\\n(αN +1 (K) + αj (K)) p. (4) Here N has to be so large that \\n\\nαN +1 (K) sup \\n\\n> λ∈Ωt\\n\\n‖(λ − A)−1‖ < 1.\\n\\nThe optimal result depends on the behavior of ‖(λ−A)−1‖ near the spectrum of A, i.e. on Ω s. This is typical for many spectral considerations. It is also related to the pseudospectrum of A. For instance if |λ| > ‖A‖\\n\\nthen \\n\\n‖(λ − A)−1‖ ≤ 1\\n\\n|λ| − ‖ A‖\\n\\n> 7\\n\\nand therefore (4) becomes \\n\\nnB (s) ≤ (2 e) p\\n\\n> 2\\n\\nlog st (t − (‖A‖ + αN +1 (K))) pN∑\\n\\n> j=1\\n\\n(αN +1 (K) + αj (K)) p.\\n\\nOpen problems: i) Classify the operators for which the resolvent is polynomially bounded if λ → σ(A)? ii) Classify the operators for which one can find an M ≥ 1 such that \\n\\n‖(λ − A)−1‖ ≤ M\\n\\ndist ( λ, σ (A)) for all λ ∈ res( A) or \\n\\n‖(λ − A)−1‖ ≤ M\\n\\n|λ| − spr( A)for |λ| > spr( A).\"\nOriginal remarks: [\"Remark: For instance in Hilbert spaces, the bound in ii) is true for normal operators with \\n\\nM = 1. 20. Michael Hitrik \\n\\n20.1. Inverse spectral problems for non-self-adjoint operators, especially in the semiclassical limit. Given a suitable h-pseudodifferential operator P = pw(x, hD x) on Rn or a compact manifold, we would like to understand what information about the classical symbol p can be determined from the spectrum of P, in the semiclassical limit h → 0. We are especially interested in cases when P is non-self-adjoint, with the inverse problems for resonances and for damped wave equations being important sources of motivation. See [DH12], [Hal13], [Pha] for some of the recent works on semiclassical inverse spectral problems in the non-self-adjoint setting. 20.2. Spectra for non-self-adjoint operators in the presence of symmetries. The proof of the reality of the exponentially small eigenvalues of the Kramers-Fokker-Planck type operators in [HHS11] depends on a reflection symmetry for such operators, and there are many natural non-self-adjoint situations where symmetries play a role, including PT-symmetric operators and operators with supersymmetric structures. See also [Shi02], [KS02]. 21. David Krejˇ ciˇ r´ ık \\n\\n21.1. Large-time behavior of the heat equation: subcriticality versus criticality. This open prob-lem is a repetition of the open problem raised during previous meetings in Prague (2010) and Barcelona (2012) \\n\\nhttp://www.ujf.cas.cz/ESFxNSA/ http://gemma.ujf.cas.cz/~david/OTAMP2012/OTAMP2012.html \\n\\nbut little progress has been made so far. Please visit the links above for more details and references. Our conjecture is that the solutions of the heat equation \\\"decay faster\\\" for large times provided that the generator is \\\"more positive\\\" in the sense of the validity of a Hardy-type inequality. There exist both semigroup (with Zuazua [KZ10]) and heat-kernel (with Fraas and Pinchover [FKP10]) versions of the con-jecture and the latter involves non-self-adjoint operators too. The conjecture has been supported by several particular situations, but there exists no general result yet. In the self-adjoint case, the conjectures can be stated as follows. Let Ω be an open connected subset of Rd. Let H0 and H+ be two self-adjoint operators in L2(Ω) such that inf σ(H0) = inf σ(H+) = 0. Assume that H+ is subcritical, in the sense that there is a smooth positive function ρ: Ω → R such that H+ ≥ ρ\\n\\n(Hardy inequality). On the other hand, H0 is assumed to be critical, in the sense that inf σ(H0 − V ) < 0 for any non-negative non-trivial V ∈ C∞ \\n\\n> 0\\n\\n(Ω).\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/pastworkshops/nonselfadjointproblems.pdf", "tags": [ "aim", "AIM-PHYSICS-0010", "aim-domain:physics", "aim-workshop:nonselfadjointproblems", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a bounded operator satisfying A = J A* J with J an antilinear conjugation, every finite-dimensional J-invariant projection preserves complex symmetry and identifies the two off-diagonal coupling norms. A Schur-complement test using the smallest singular value of the finite block, one coupling norm, and a certified tail-resolvent bound rigorously excludes points from the spectrum. For compact operators and strongly convergent J-invariant projections, this gives explicit outer enclosures that eventually exclude every compact subset of the resolvent away from zero. Paired right-left residuals, a persistent pollution example, and a defective 2-by-2 evolution example delimit what J-structure alone can guarantee.\n\nCandidate contribution (enclosure theorem; novelty confidence low): The explicit one-coupling J-Schur exclusion test, its convergent compact-operator outer enclosure, and the paired-residual diagnostic form a testable J-structured certification package for conjugation-self-adjoint finite sections." }, { "id": 20002503, "problem_number": "AIM-PHYSICS-0011", "title": "When holomorphic anomaly equations survive fiber integration", "statement": "Does the holomorphic anomaly equation \"integrate\" over elliptic fibrations?", "original_statement": "Does the holomorphic anomaly equation \"integrate\" over elliptic fibrations?", "clean_statement": "Does the holomorphic anomaly equation \"integrate\" over elliptic fibrations?", "statement_status": "exact", "statement_verification": "The original AimPL URL is currently unavailable through the web interface. The local canonical extraction is internally consistent and contains no visible OCR corruption. The next records are essential context:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Holomorphic anomaly equation\nSource item: 1.1\nSource URL: http://aimpl.org/gromwitnumthry/1/\nCanonical location: aim-physics-notes.json notes[10]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does the holomorphic anomaly equation \\\"integrate\\\" over elliptic fibrations?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/1/", "tags": [ "aim", "AIM-PHYSICS-0011", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The 2012 question is under-specified, but later work proves affirmative numerical and relative regimes for rational elliptic surfaces and the Schoen Calabi-Yau threefold while leaving the general cycle-valued elliptic-fibration HAE conjectural. For any precisely defined class-valued HAE, a proved pushforward-defect identity shows that the pushed-forward equation acquires anomaly-operator, genus-reduction, and boundary-gluing defects. In the multiplicative case the last defect is one half of a fiber covariance, giving both a sufficient fiber-constant product case and an explicit obstruction to naive integration.\n\nCandidate contribution (reduction_and_obstruction; novelty confidence low): The exact defect identity separates failure of HAE pushforward into an anomaly-operator commutator, a linear genus-reduction commutator, and a nonlinear gluing defect; for multiplicative splitting the gluing defect equals one half of the fiber covariance and can be nonzero even when differentiation commutes with integration.", "research_difficulty_suggested": "L4" }, { "id": 20002504, "problem_number": "AIM-PHYSICS-0012", "title": "Differential fingerprints of eta-products", "statement": "Eta-products and root systems and holomorphic anomaly equation\n\nAs it turns out mock modular forms (such as eta-products) do not satisfy the holomorphic anaomaly equation.\n\nDo they satisfy other differential equations?", "original_statement": "Eta-products and root systems and holomorphic anomaly equation\n\nAs it turns out mock modular forms (such as eta-products) do not satisfy the holomorphic anaomaly equation.\n\nDo they satisfy other differential equations?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "This is a plausible reconstruction, not a claim that the inaccessible AimPL page explicitly named Saito. Macdonald's affine-root-system eta identities and Saito's elliptic-root-system eta-products show that the title has a standard mathematical referent.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Holomorphic anomaly equation\nSource item: 1.2\nSource URL: http://aimpl.org/gromwitnumthry/1/\nCanonical location: aim-physics-notes.json notes[11]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Eta-products and root systems and holomorphic anomaly equation\\n\\nAs it turns out mock modular forms (such as eta-products) do not satisfy the holomorphic anaomaly equation.\\n\\nDo they satisfy other differential equations?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Also see problem \\\\ref{hae-cy}.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/1/", "tags": [ "aim", "AIM-PHYSICS-0012", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literal AIM record is conceptually conflated: finite eta-products are modular objects with multiplier systems, not ordinarily genuine mock modular forms with nonzero shadows, and the assigned workshop metadata also disagrees with the source slug. Under the mathematically coherent eta-product reading, every finite eta-product satisfies an exact first-order E2 equation, a Ramanujan-Serre covariant equation with a holomorphic weight-2 modular coefficient, and a finite quasimodular differential system. The q-series of the first-order coefficient determines every eta exponent recursively; for Saito root-system eta-products, the weight and covariant coefficient therefore recover the Coxeter characteristic polynomial.\n\nCandidate contribution (reconstruction_theorem; novelty confidence low): For a finite eta-product, the q-series coefficient of its first-order differential equation is an injective fingerprint of the exponent vector. Equivalently, its weight together with its modular-covariant coefficient determines all eta exponents. Applied to Saito's elliptic eta-products, this reconstructs the Coxeter cyclotomic characteristic polynomial; moreover, the covariant coefficient vanishes exactly for pure powers of eta(tau)." }, { "id": 20002505, "problem_number": "AIM-PHYSICS-0013", "title": "Holomorphic anomaly, fibrations, and the BPS multiple-cover transform", "statement": "Investigate the Holomorphic anomaly equation (HAE) in fibered Calabi-Yau 3-folds and BPS counting. \\label{hae-cy}", "original_statement": "Investigate the Holomorphic anomaly equation (HAE) in fibered Calabi-Yau 3-folds and BPS counting. \\label{hae-cy}", "clean_statement": "Investigate the Holomorphic anomaly equation (HAE) in fibered Calabi-Yau 3-folds and BPS counting. \\label{hae-cy}", "statement_status": "exact", "statement_verification": "The exact canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Holomorphic anomaly equation\nSource item: 1.3\nSource URL: http://aimpl.org/gromwitnumthry/1/\nCanonical location: aim-physics-notes.json notes[12]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate the Holomorphic anomaly equation (HAE) in fibered Calabi-Yau 3-folds and BPS counting. \\\\label{hae-cy}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"For example, does the HAE ``integrate'' over elliptic fibrations or K3 fibers?\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/1/", "tags": [ "aim", "AIM-PHYSICS-0013", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the connected unmarked nonzero-class Calabi--Yau threefold free energy, the Gopakumar--Vafa transform gives the exact coefficient relation N_{g,beta}=sum_{k|beta} k^{2g-3} sum_{h=0}^g c_{h,g-h} n_{h,beta/k}. This has a genus-and-divisibility triangular inverse. An anomaly derivation passes through the multiple-cover operator only with the explicit Adams commutator defect [D,psi_k], and nonlinear HAE boundary products do not pass through unchanged. Coefficientwise finite aggregation along a curve-class monoid homomorphism commutes with multiple covers, while collapsing infinitely many vertical classes to degree zero is not formally defined.\n\nCandidate contribution (formal transport theorem; novelty confidence low): The combined genus/divisibility formula, anomaly--Adams commutator identity, triangular BPS anomaly recursion, and finite-fiber aggregation criterion form an explicit test for whether a fiberwise GW holomorphic anomaly equation survives BPS multiple covers and curve-class integration." }, { "id": 20002506, "problem_number": "AIM-PHYSICS-0014", "title": "Compact-fixed-locus definitions and pole obstructions for noncompact elliptic genera", "statement": "When can the elliptic genus be defined for noncompact Calabi-Yau manifolds?", "original_statement": "When can the elliptic genus be defined for noncompact Calabi-Yau manifolds?", "clean_statement": "When can the elliptic genus be defined for noncompact Calabi-Yau manifolds?", "statement_status": "exact", "statement_verification": "The exact canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Calabi-Yau manifolds\nSource item: 3.1\nSource URL: http://aimpl.org/gromwitnumthry/3/\nCanonical location: aim-physics-notes.json notes[13]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"When can the elliptic genus be defined for noncompact Calabi-Yau manifolds?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/3/", "tags": [ "aim", "AIM-PHYSICS-0014", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the total space of a rank-r circle-equivariant holomorphic vector bundle over a compact n-fold, with no zero fiber weights, the localized equivariant elliptic genus is well-defined and has pole order at most n+r at the nonequivariant point. The possible top Laurent coefficient is an explicit weighted complete-homogeneous characteristic number. This proves exact pole order n+1 for Tot(K_{P^n}) under fiber scaling and a double pole for the volume-preserving weight (1,-1) action on C^2, showing that even equivariant Calabi-Yau triviality is insufficient for a finite ordinary genus.\n\nCandidate contribution (localization_lemma; novelty confidence low): The localized elliptic genus of Tot(E), for a rank-r equivariant bundle E over a compact n-fold with nonzero fiber weights, has pole order at most n+r, with its t^{-(n+r)} coefficient given explicitly by the weighted complete homogeneous class in formula (5.2); in particular Tot(K_{P^n}) has exact pole order n+1." }, { "id": 20002507, "problem_number": "AIM-PHYSICS-0015", "title": "A normal-symbol criterion for lower-dimensional modular reduction", "statement": "When can modular forms on Calabi Yau 3-folds moduli spaces be reduced to lower dimensional forms?", "original_statement": "When can modular forms on Calabi Yau 3-folds moduli spaces be reduced to lower dimensional forms?", "clean_statement": "When can modular forms on Calabi Yau 3-folds moduli spaces be reduced to lower dimensional forms?", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Calabi-Yau manifolds\nSource item: 3.3\nSource URL: http://aimpl.org/gromwitnumthry/3/\nCanonical location: aim-physics-notes.json notes[14]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"When can modular forms on Calabi Yau 3-folds moduli spaces be reduced to lower dimensional forms?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/3/", "tags": [ "aim", "AIM-PHYSICS-0015", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a scalar automorphic object on a domain D/Gamma and a monodromy-equivariant regular interior embedding of a lower-dimensional family, ordinary restriction is automorphic; if it vanishes to exact order nu, the first nonzero transverse coefficient is canonically a section of the pulled-back automorphic line tensored with Sym^nu of the conormal bundle. Hence scalar reduction requires a compatible equivariant normal line or contraction, and its factor is shifted by the inverse normal character. On the diagonal H x H in the genus-two Siegel domain this gives weight (k+nu,k+nu), and for the Igusa cusp form chi_10 it rigorously recovers a nonzero multiple of z^2 Delta(tau_1)Delta(tau_2).\n\nCandidate contribution (reduction criterion; novelty confidence low): The first nonzero transverse jet supplies a testable obstruction-and-weight criterion for the AIM question: along an interior monodromy-equivariant Calabi-Yau subfamily it lies in i^*L^k_chi tensor Sym^nu N^*, so a canonical scalar lower-dimensional form exists only after a compatible equivariant line component is identified, with inverse normal-character weight shift.", "research_difficulty_suggested": "L4" }, { "id": 20002508, "problem_number": "AIM-PHYSICS-0016", "title": "Coefficientwise indices and a level-two Calabi--Yau/CFT test", "statement": "Is there an index theoretic interpretation of elliptic genus? When it can, how do we compute it and what is the geometric interpretation of the coefficients? Check that this matches CFT calculations.", "original_statement": "Is there an index theoretic interpretation of elliptic genus? When it can, how do we compute it and what is the geometric interpretation of the coefficients? Check that this matches CFT calculations.", "clean_statement": "Is there an index theoretic interpretation of elliptic genus? When it can, how do we compute it and what is the geometric interpretation of the coefficients? Check that this matches CFT calculations.", "statement_status": "exact", "statement_verification": "The exact canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Calabi-Yau manifolds\nSource item: 3.2\nSource URL: http://aimpl.org/gromwitnumthry/3/\nCanonical location: aim-physics-notes.json notes[15]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is there an index theoretic interpretation of elliptic genus? When it can, how do we compute it and what is the geometric interpretation of the coefficients? Check that this matches CFT calculations.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/3/", "tags": [ "aim", "AIM-PHYSICS-0016", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a compact complex manifold, every Fourier--Jacobi coefficient of the two-variable elliptic genus is the Fredholm index and holomorphic Euler characteristic of a finite virtual bundle extracted from the formal elliptic bundle. This attempt computes the exact oscillator-level-two bundle. For every compact Calabi--Yau threefold it proves that the six nonzero normalized q^2 charge indices at charges -7/2, -5/2, -1/2, 1/2, 5/2, and 7/2 are respectively -e/2, -e/2, e, e, -e/2, and -e/2. For the quintic e=-200, this agrees with the expansion of the independently localized GLSM/CFT theta quotient.\n\nCandidate contribution (coefficientwise index formula; novelty confidence low): The exact q^2 virtual-bundle formula E_2(y)=Lambda_{-y}Omega^1 tensor V_2(y), with V_2 expressed in symmetric, exterior, and tensor powers of Omega^1 and T, together with its six-charge universal Calabi--Yau threefold Euler-characteristic identity, provides an explicit oscillator-level-two diagnostic for geometric and CFT elliptic genera." }, { "id": 20002509, "problem_number": "AIM-PHYSICS-0017", "title": "Paramodular consistency tests for Calabi-Yau counting series", "statement": "Investigate the role of paramodular groups in counting problems in Calabi Yau 3-folds", "original_statement": "Investigate the role of paramodular groups in counting problems in Calabi Yau 3-folds", "clean_statement": "Investigate the role of paramodular groups in counting problems in Calabi Yau 3-folds", "statement_status": "exact", "statement_verification": "The canonical record is problem 3.4 in the section “Calabi--Yau manifolds”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Calabi-Yau manifolds\nSource item: 3.4\nSource URL: http://aimpl.org/gromwitnumthry/3/\nCanonical location: aim-physics-notes.json notes[16]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Investigate the role of paramodular groups in counting problems in Calabi Yau 3-folds\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/3/", "tags": [ "aim", "AIM-PHYSICS-0017", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a scalar holomorphic paramodular Fricke eigenform in the stated K(N) convention, the Fourier-Jacobi coefficient defect D_{m,n,r}=c_m(n,r)-(-1)^k epsilon c_n(m,r) vanishes identically. Hence the coefficient array is determined by one triangular half, a nonzero diagonal coefficient fixes epsilon=(-1)^k, and any finite nonzero defect refutes the proposed packaging. Combined with the 2024 Aoki-Ibukiyama-Poor theorem, Jacobi membership of all slices plus vanishing defects is also sufficient for convergence to a unique paramodular form. The criterion is audited explicitly on the Igusa cusp form chi_10.\n\nCandidate contribution (obstruction; novelty confidence low): The signed Fourier-Jacobi defect array is a finite-refutation certificate for proposed holomorphic paramodular Calabi-Yau counting series; its diagonal and triangular consequences give a convention-aware preprocessing test before fitting an additive or Borcherds lift." }, { "id": 20002510, "problem_number": "AIM-PHYSICS-0018", "title": "Normalization-aware leading quintic amplitudes at the Gepner point", "statement": "Can we calculate the leading terms of $F_g^B(q)$ at the orbifold point $z=0$ of the quintic\n\\[\n(\\sum x_i^5 + z \\prod x_i =0 )/ \\mathbb{Z}_5^3?\n\\]", "original_statement": "Can we calculate the leading terms of $F_g^B(q)$ at the orbifold point $z=0$ of the quintic\n\\[\n(\\sum x_i^5 + z \\prod x_i =0 )/ \\mathbb{Z}_5^3?\n\\]", "clean_statement": "Can we calculate the leading terms of $F_g^B(q)$ at the orbifold point $z=0$ of the quintic\n\\[\n(\\sum x_i^5 + z \\prod x_i =0 )/ \\mathbb{Z}_5^3?\n\\]", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Specific functions\nSource item: 4.1\nSource URL: http://aimpl.org/gromwitnumthry/4/\nCanonical location: aim-physics-notes.json notes[17]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we calculate the leading terms of $F_g^B(q)$ at the orbifold point $z=0$ of the quintic\\n\\\\[\\n(\\\\sum x_i^5 + z \\\\prod x_i =0 )/ \\\\mathbb{Z}_5^3?\\n\\\\]\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/4/", "tags": [ "aim", "AIM-PHYSICS-0018", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "In the Huang--Klemm--Quackenbush orbifold frame, regularity and residual Z_5 monodromy force F_g(s) to lie in s^{r_g} C[[s^5]], where r_g is the least residue of 2(g-1) modulo 5. In the normalized flat coordinate, F_0(s)=5s^3/6+5s^8/1008+O(s^13) and F_1(s)=C_1-s^5/9+O(s^10). After recovering the AIM pencil coefficient a=-5 phi, these become F_0=-a^3/150+2a^8/(21*5^8)+O(a^13) and F_1=C_1+a^5/28125+O(a^10).\n\nCandidate contribution (lemma; novelty confidence low): For any Z_5-equivariant local reparameterization s_tilde=c s(1+O(s^5)), the monodromy residue class and vanishing order of a regular orbifold amplitude are invariant, while a leading coefficient in degree r rescales by c^{-r}; combining this with the mirror map yields an explicit normalization audit for the ambiguous variables in AIM problem 4.1." }, { "id": 20002511, "problem_number": "AIM-PHYSICS-0019", "title": "Hurwitz class numbers, CM enumeration, and local conductor layers", "statement": "Here $h_m$ is the $m$-th Hurwitz class number, i.e. the number of equivalence classes of positive definite binary quadratic forms of discriminant $-m$ with the class containing $x^2+y^2$ weighted by 1/2 and the class containing $x^2 + xy + y^2$ weighted by 1/3. Moreover by convention $h_0=-1/12.$\n\nFor more information in regards to the connection between physics and number theory, see the following work of\n\n Kathrin Bringmann and Ben Kane\nhttp://arxiv.org/pdf/1305.0112v1.pdf\n\nKatrin Bringmann and Sameer Murthy\nhttp://arxiv.org/pdf/1208.3476v2.pdf\n\nKatrin Bringmann and Jan Manschot\nhttp://arxiv.org/pdf/1304.7208v1.pdf\n\nDo the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \\geq 9$?", "original_statement": "Here $h_m$ is the $m$-th Hurwitz class number, i.e. the number of equivalence classes of positive definite binary quadratic forms of discriminant $-m$ with the class containing $x^2+y^2$ weighted by 1/2 and the class containing $x^2 + xy + y^2$ weighted by 1/3. Moreover by convention $h_0=-1/12.$\n\nFor more information in regards to the connection between physics and number theory, see the following work of\n\n Kathrin Bringmann and Ben Kane\nhttp://arxiv.org/pdf/1305.0112v1.pdf\n\nKatrin Bringmann and Sameer Murthy\nhttp://arxiv.org/pdf/1208.3476v2.pdf\n\nKatrin Bringmann and Jan Manschot\nhttp://arxiv.org/pdf/1304.7208v1.pdf\n\nDo the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \\geq 9$?", "clean_statement": "Here $h_m$ is the $m$-th Hurwitz class number, i.e. the number of equivalence classes of positive definite binary quadratic forms of discriminant $-m$ with the class containing $x^2+y^2$ weighted by 1/2 and the class containing $x^2 + xy + y^2$ weighted by 1/3. Moreover by convention $h_0=-1/12.$\n\nFor more information in regards to the connection between physics and number theory, see the following work of\n\n Kathrin Bringmann and Ben Kane\nhttp://arxiv.org/pdf/1305.0112v1.pdf\n\nKatrin Bringmann and Sameer Murthy\nhttp://arxiv.org/pdf/1208.3476v2.pdf\n\nKatrin Bringmann and Jan Manschot\nhttp://arxiv.org/pdf/1304.7208v1.pdf\n\nDo the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \\geq 9$?", "statement_status": "exact", "statement_verification": "The exact canonical record (item 4.2, “Specific functions”) defines \\(h_m\\) as the weighted number of positive-definite integral binary quadratic forms of discriminant \\(-m\\), gives the exceptional weights \\(1/2\\) for \\(x^2+y^2\\) and \\(1/3\\) for \\(x^2+xy+y^2\\), sets \\(h_0=-1/12\\), cites three papers, and asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Specific functions\nSource item: 4.2\nSource URL: http://aimpl.org/gromwitnumthry/4/\nCanonical location: aim-physics-notes.json notes[18]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Here $h_m$ is the $m$-th Hurwitz class number, i.e. the number of equivalence classes of positive definite binary quadratic forms of discriminant $-m$ with the class containing $x^2+y^2$ weighted by 1/2 and the class containing $x^2 + xy + y^2$ weighted by 1/3. Moreover by convention $h_0=-1/12.$\\n\\nFor more information in regards to the connection between physics and number theory, see the following work of\\n\\n Kathrin Bringmann and Ben Kane\\nhttp://arxiv.org/pdf/1305.0112v1.pdf\\n\\nKatrin Bringmann and Sameer Murthy\\nhttp://arxiv.org/pdf/1208.3476v2.pdf\\n\\nKatrin Bringmann and Jan Manschot\\nhttp://arxiv.org/pdf/1304.7208v1.pdf\\n\\nDo the coefficients of $h_m$ have enumerative significance? Can we find $h_m$ for $m \\\\geq 9$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/4/", "tags": [ "aim", "AIM-PHYSICS-0019", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After reconstructing the malformed phrase \"coefficients of h_m\" as the coefficients H(m) of the standard Hurwitz class-number series, every H(m) is computed by a square-divisor sum of weighted primitive class numbers and by an explicit ring-class formula. For m>0, H(m) is the groupoid cardinality of scale-labeled CM elliptic curves rigidified by {+1,-1}. Exact values are supplied for 9 <= m <= 64. For fixed fundamental discriminant D, the normalized values A_D(f)=H(|D|f^2)/H(|D|) are multiplicative and satisfy A_D(p^a)=1+(p-chi_D(p))(p^a-1)/(p-1).\n\nCandidate contribution (local conductor-layer theorem; novelty confidence low): For fixed negative fundamental discriminant D, the normalized square-family Hurwitz values A_D(f)=H(|D|f^2)/H(|D|) are multiplicative, and the exact contribution added at prime conductor layer p^a is (p-chi_D(p))p^(a-1)H(|D|); hence the first-layer multiplier is p, p+1, or p+2 according as p splits, ramifies, or is inert." }, { "id": 20002512, "problem_number": "AIM-PHYSICS-0020", "title": "Formal, Jacobi, and wave-function transformations of the string coupling", "statement": "Does $F= \\sum f_g(\\tau)\\lambda^{2g-2}$ have transformation properties with respect to $\\lambda$?", "original_statement": "Does $F= \\sum f_g(\\tau)\\lambda^{2g-2}$ have transformation properties with respect to $\\lambda$?", "clean_statement": "Does $F= \\sum f_g(\\tau)\\lambda^{2g-2}$ have transformation properties with respect to $\\lambda$?", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Specific functions\nSource item: 4.3\nSource URL: http://aimpl.org/gromwitnumthry/4/\nCanonical location: aim-physics-notes.json notes[19]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Does $F= \\\\sum f_g(\\\\tau)\\\\lambda^{2g-2}$ have transformation properties with respect to $\\\\lambda$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/4/", "tags": [ "aim", "AIM-PHYSICS-0020", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a formal free energy F(tau,lambda)=sum_g f_g(tau)lambda^(2g-2) whose nonzero coefficients are modular of weights w_g, a scalar law F(gamma tau,j^ell lambda)=j^K F exists exactly when w_g+ell(2g-2)=K for every active genus; two genera determine ell uniquely, and weights w_g=2g-2 force lambda to have weight -1. Quasimodular E_2 terms obstruct this naive law by a calculable quadratic anomaly. Separately, a meromorphic-Jacobi partition-function sector obeys an exact Gaussian modular law and elliptic quasi-periodicity, generic BCOV monodromy acts metaplectically on the partition function rather than by rescaling lambda alone, and the nonconstant Gopakumar--Vafa resummation is even and 2pi-periodic in lambda.\n\nCandidate contribution (criterion; novelty confidence low): A coefficientwise audit combines an if-and-only-if affine-genus-weight criterion for scalar coupling covariance with an explicit recurrence showing how Jacobi index mixes Laurent coefficients; it distinguishes a residual c lambda^2/(c tau+d) index anomaly and a genus-one logarithmic multiplier from a genuine failure of modularity." }, { "id": 20002513, "problem_number": "AIM-PHYSICS-0021", "title": "Mirror-quintic periods, modular-type structures, and a conifold obstruction", "statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?", "original_statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?", "clean_statement": "Is $\\sum \\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\nDo other solutions to the Picard-Fuchs equation have modular properties?", "statement_status": "exact", "statement_verification": "The exact canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Specific functions\nSource item: 4.4\nSource URL: http://aimpl.org/gromwitnumthry/4/\nCanonical location: aim-physics-notes.json notes[20]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is $\\\\sum \\\\frac{(5n)!}{(n!)^5}z^n$ related to (non-holomorphic) forms? Is it related to automorphic objects?\\nDo other solutions to the Picard-Fuchs equation have modular properties?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/4/", "tags": [ "aim", "AIM-PHYSICS-0021", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After the exact normalization x=5^5 z_A, the AIM series is the mirror-quintic hypergeometric period _4F_3(1/5,2/5,3/5,4/5;1,1,1;x). Its four solutions form an Sp(4,Z)-monodromy-equivariant period vector, and the polarized Hodge norm is a canonical real-analytic non-holomorphic invariant; Movasati's published seven-function differential algebra supplies a generalized modular-type answer. A proved additional result is that the conifold Jordan type J_2(1) plus two trivial blocks persists under finite base change and is incompatible with every rank-one-twisted symmetric cube of a rank-two local system, ruling out the standard scalar weight-three symmetric-cube modular construction for the complete period system.\n\nCandidate contribution (obstruction; novelty confidence low): The mirror-quintic conifold monodromy has Jordan partition (2,1,1), preserved by every positive local power, while a rank-one twist of the symmetric cube of any two-dimensional local monodromy can have unipotent partition only (1,1,1,1) or (4); therefore no finite base change at the conifold can produce a twisted symmetric-cube realization of the full rank-four Picard-Fuchs local system." }, { "id": 20002514, "problem_number": "AIM-PHYSICS-0022", "title": "Functional equations and an Euler-product obstruction for mixed mock L-series", "statement": "Do the L-series of mixed mock modular forms have interesting properties?", "original_statement": "Do the L-series of mixed mock modular forms have interesting properties?", "clean_statement": "Do the L-series of mixed mock modular forms have interesting properties?", "statement_status": "exact", "statement_verification": "The sentence has no apparent OCR corruption, but it is mathematically underspecified. It does not define “mixed mock modular form,” choose a cusp or multiplier, or say which of several inequivalent objects is the “L-series.” The nearby records ask whether mixed mock modular forms satisfy differential equations and how mock modular forms relate to geometric invariants. They add motivation but no definitions.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Other problems\nSource item: 5.1\nSource URL: http://aimpl.org/gromwitnumthry/5/\nCanonical location: aim-physics-notes.json notes[21]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Do the L-series of mixed mock modular forms have interesting properties?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/5/", "tags": [ "aim", "AIM-PHYSICS-0022", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The phrase L-series has three distinct readings: the raw coefficient Dirichlet series, the Mellin or test-function transform of the real-analytic modular completion, and shifted Rankin-Selberg series whose special values may generate mixed mock forms. For a rapidly decaying scalar integral-weight completion with a Fricke eigenlaw, the completed Mellin transform satisfies a homogeneous functional equation, while the raw coefficient transform satisfies an exact inhomogeneous equation whose defect is the Mellin transform of the nonholomorphic shadow correction. For the canonical product theta times the Hurwitz class-number mock form, the exact coefficients c(1)=-1/6, c(3)=1/3, c(5)=1, and c(15)=4 violate normalized coprime multiplicativity, proving that its raw coefficient Dirichlet series has no ordinary Euler product.\n\nCandidate contribution (explicit counterexample; novelty confidence low): For the canonical mixed mock product theta(tau) H(tau), the finite identity c(1)c(15)=-2/3 != 1/3=c(3)c(5) proves that the normalized coefficient sequence is not multiplicative and hence that the raw Dirichlet series admits no ordinary absolutely convergent Euler product with local constant terms one." }, { "id": 20002515, "problem_number": "AIM-PHYSICS-0023", "title": "Hauptmodul differential equations for an even-weight mixed-mock class", "statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?", "original_statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?", "clean_statement": "Do mixed mock modular forms satisfy differential equations with respect to a modular function?", "statement_status": "exact", "statement_verification": "The exact canonical record (`aim-physics-notes.json`, zero-based index 22) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Other problems\nSource item: 5.2\nSource URL: http://aimpl.org/gromwitnumthry/5/\nCanonical location: aim-physics-notes.json notes[22]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Do mixed mock modular forms satisfy differential equations with respect to a modular function?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/5/", "tags": [ "aim", "AIM-PHYSICS-0023", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Let Gamma_0(N) have genus zero with Hauptmodul t and v=Dt. For a finite sum F=sum_i A_i f_i^+ of meromorphic modular factors times holomorphic parts of trivial-character harmonic weak Maass forms of even weights w_i<=0, with common total weight W, the normalized function U=v^{-W/2}F is D-finite over C(t). More precisely, each normalized mock component satisfies a projective-Bol inhomogeneous ODE with rational coefficients, and U satisfies a homogeneous rational-coefficient ODE of order at most 1+sum_i(1-w_i). The verified Maass-Poincare series Q(-1,4,9) and the modular factor eta(3 tau)^8 give a symbolic mixed illustration of order at most four.\n\nCandidate contribution (theorem; novelty confidence low): After canonical Dt-weight normalization, every finite mixed sum in the stated genus-zero, trivial-character, even-integral, nonpositive mock-weight class is D-finite over the Hauptmodul field, with order at most 1+sum_i(1-w_i); the construction gives a verified symbolic equation for Q^+(-1,4,9) times eta(3 tau)^8 without asserting an unverified explicit rational right-hand side." }, { "id": 20002516, "problem_number": "AIM-PHYSICS-0024", "title": "Mock modularity, geometric invariants, and finite pole tomography", "statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.", "original_statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.", "clean_statement": "Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.", "statement_status": "exact", "statement_verification": "The exact canonical record is item 5.3 in the section “Other problems”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Physics\nWorkshop: Integrable systems in Gromov-Witten and symplectic field theory\nSection: Other problems\nSource item: 5.3\nSource URL: http://aimpl.org/gromwitnumthry/5/\nCanonical location: aim-physics-notes.json notes[23]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Determine the connection between Mock modular forms and geometric invariants like Gromov-Witten, Donaldson, and OSV conjecture.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "http://aimpl.org/gromwitnumthry/5/", "tags": [ "aim", "AIM-PHYSICS-0024", "aim-domain:physics", "aim-workshop:gromwitnumthry", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The requested connection is family-dependent: proved instances link Donaldson, Vafa-Witten/sheaf, logarithmic Gromov-Witten, and BPS/Donaldson-Thomas series to harmonic Maass, Appell-Lerch, indefinite-theta, or higher-depth mock objects, while OSV remains conjectural. For any 1-periodic meromorphic charge-generating function, a Fourier-contour crossing changes charge-r coefficients by minus 2πi times the twisted polar residues. If an isolated pole has order at most J, phase-normalized jumps form a polynomial of degree at most J-1, so J consecutive charge-sector jumps uniquely reconstruct the full Laurent principal part. Therefore a nonremovable pole obstructs a chamber-independent interpretation of all raw coefficients unless a chamber is fixed or the polar contribution is removed or corrected.\n\nCandidate contribution (reconstruction theorem; novelty confidence low): For an isolated pole of known location and order at most J in a 1-periodic meromorphic charge-generating function, the J contour jumps in charge sectors r=0,...,J-1 uniquely reconstruct every Laurent principal-part coefficient by polynomial interpolation; vanishing of these J jumps already proves the pole removable." }, { "id": 20002517, "problem_number": "AIM-PHYSICS-0025", "title": "A two-vertex counterexample to color-number monotonicity", "statement": "(5) Let G be a finite graph with maximum degree ∆. This problem is about choosing a random coloring from the uniform distribution. Work with Glauber dynamics: pick a site at random, choose a color at random, and try to change the color at the chosen site. If this results in a proper coloring, make the change. If not, keep the current coloring. For q ≥ ∆ + 2 it is known that this Markov chain is ergodic, i.e.,\n\n> 12\n\nthat it connects the set of proper colorings. Let t = t(q) be the mixing time: the smallest time such that the total variation distance from the uniform distribution is at most 1 /e from any starting configuration. The question is: does t(q) decrease as\n\nq increases? Even on the square lattice with n sites there are curious gaps in our knowledge: we know that t(3) is polynomial in n (Luby, Randall and Sinclair: Proc. FOCS 1995) and we know that t(q) = O(n log n) for q ≥ 6. But, for q = 4 and q = 5 we do not even have a polynomial upper bound, even though there is overwhelming numerical evidence that there is a finite correlation length, and therefore that the mixing time is O(n log n). (6) Is there a variant of k-SAT that describes the scaling properties of jamming? In other words, is k-SAT a good \"Boolean idealization\" of the jamming process? Should this involve k-SAT on a lattice? (See Schwartz and Middleton, Phys. Rev. E 70 (2004), 035103(R).) (7) Find a model with constraints and a phase transition, which we can perturb by \"soft-ening\" the constraints. Study the sensitivity of this model's behavior with respect to this perturbation. (8) In constraint satisfaction problems such as k-SAT, what are the useful response/correlation functions? Do they show singular, or power-law, behavior at the transition, analo-gous to spatial correlations and response functions in lattice systems such as the Ising model? (9) The second moment method in k-SAT gives a highly accurate lower bound for the critical density, when the approach of (Achlioptas and Moore: Proc. FOCS 2002) is refined by (Achlioptas and Peres: Proc. STOC 2003). This refinement can be thought of as adding a finite (not infinitesimal) external field which discourages lit-erals from being true. Is there any physical meaning to this value of the external field? For instance, does it minimize fluctuations in, say, the free energy at zero temperature? (10) We believe (and, in some cases, have proved) that below the satisfiability/unsatisfiability transition in k-SAT there is a \"clustering\" transition, in which the solutions clump together in isolated groups. It has been conjectured that this clustering phenomenon makes it hard for search algorithms to find solutions. Can we find algorithms which, in simulation, find solutions in, say, linear time, for values of k and densities at which clustering has been rigorously established? (11) (Following up on the previous question) Can we prove that clustering occurs at some density below the satisfiability transition for k = 3, 4, 5? So far we only have proofs of clustering for larger values of k.(12) Prove an order/disorder phase transition for the equilibrium statistical mechanics model of hard spheres (or hard disks). Simulation evidence of such a transition is 3\n\ngenerally believed to be convincing in both cases, though the nature of the transition is controversial for disks. (13) How should we add friction to jamming? (14) Consider the Ising model on the d-dimensional lattice. Fix β (and perhaps the ex-ternal field strength h) and consider Glauber dynamics. The cutoff phenomenon in Markov chains refers to the existence of a time at which the total variation distance from the stationary (Gibbs) distribution jumps from 1 to 0 in the limit where the size of the lattice goes to infinity. We ask whether this sharp cutoff exists if and only if there is a unique Gibbs state in the infinite system. (15) Give a good definition of the thermodynamic limit of k-SAT. That is, clearly define the mathematical object that corresponds to random formulas Fk(n, m = rn ) with constant density r in the limit n → ∞.(16) Consider the Ising model on a finite box in Zd. Set β > β c and no external field. With fixed boundary conditions where all the boundary sites are set to +1, it is known that Glauber dynamics can take a long time to mix if the initial state consists of a large droplet of −1's. But with a \"warm start,\" where the initial condition consists of all +1's or is random, is the mixing time optimal, i.e., O(n log n) where n is the number of sites in the box?", "original_statement": "(5) Let G be a finite graph with maximum degree ∆. This problem is about choosing a random coloring from the uniform distribution. Work with Glauber dynamics: pick a site at random, choose a color at random, and try to change the color at the chosen site. If this results in a proper coloring, make the change. If not, keep the current coloring. For q ≥ ∆ + 2 it is known that this Markov chain is ergodic, i.e., \n\n> 12\n\nthat it connects the set of proper colorings. Let t = t(q) be the mixing time: the smallest time such that the total variation distance from the uniform distribution is at most 1 /e from any starting configuration. The question is: does t(q) decrease as \n\nq increases? Even on the square lattice with n sites there are curious gaps in our knowledge: we know that t(3) is polynomial in n (Luby, Randall and Sinclair: Proc. FOCS 1995) and we know that t(q) = O(n log n) for q ≥ 6. But, for q = 4 and q = 5 we do not even have a polynomial upper bound, even though there is overwhelming numerical evidence that there is a finite correlation length, and therefore that the mixing time is O(n log n). (6) Is there a variant of k-SAT that describes the scaling properties of jamming? In other words, is k-SAT a good \"Boolean idealization\" of the jamming process? Should this involve k-SAT on a lattice? (See Schwartz and Middleton, Phys. Rev. E 70 (2004), 035103(R).) (7) Find a model with constraints and a phase transition, which we can perturb by \"soft-ening\" the constraints. Study the sensitivity of this model's behavior with respect to this perturbation. (8) In constraint satisfaction problems such as k-SAT, what are the useful response/correlation functions? Do they show singular, or power-law, behavior at the transition, analo-gous to spatial correlations and response functions in lattice systems such as the Ising model? (9) The second moment method in k-SAT gives a highly accurate lower bound for the critical density, when the approach of (Achlioptas and Moore: Proc. FOCS 2002) is refined by (Achlioptas and Peres: Proc. STOC 2003). This refinement can be thought of as adding a finite (not infinitesimal) external field which discourages lit-erals from being true. Is there any physical meaning to this value of the external field? For instance, does it minimize fluctuations in, say, the free energy at zero temperature? (10) We believe (and, in some cases, have proved) that below the satisfiability/unsatisfiability transition in k-SAT there is a \"clustering\" transition, in which the solutions clump together in isolated groups. It has been conjectured that this clustering phenomenon makes it hard for search algorithms to find solutions. Can we find algorithms which, in simulation, find solutions in, say, linear time, for values of k and densities at which clustering has been rigorously established? (11) (Following up on the previous question) Can we prove that clustering occurs at some density below the satisfiability transition for k = 3, 4, 5? So far we only have proofs of clustering for larger values of k.(12) Prove an order/disorder phase transition for the equilibrium statistical mechanics model of hard spheres (or hard disks). Simulation evidence of such a transition is 3\n\ngenerally believed to be convincing in both cases, though the nature of the transition is controversial for disks. (13) How should we add friction to jamming? (14) Consider the Ising model on the d-dimensional lattice. Fix β (and perhaps the ex-ternal field strength h) and consider Glauber dynamics. The cutoff phenomenon in Markov chains refers to the existence of a time at which the total variation distance from the stationary (Gibbs) distribution jumps from 1 to 0 in the limit where the size of the lattice goes to infinity. We ask whether this sharp cutoff exists if and only if there is a unique Gibbs state in the infinite system. (15) Give a good definition of the thermodynamic limit of k-SAT. That is, clearly define the mathematical object that corresponds to random formulas Fk(n, m = rn ) with constant density r in the limit n → ∞.(16) Consider the Ising model on a finite box in Zd. Set β > β c and no external field. With fixed boundary conditions where all the boundary sites are set to +1, it is known that Glauber dynamics can take a long time to mix if the initial state consists of a large droplet of −1's. But with a \"warm start,\" where the initial condition consists of all +1's or is random, is the mixing time optimal, i.e., O(n log n) where n is the number of sites in the box?", "clean_statement": "(5) Let G be a finite graph with maximum degree ∆. This problem is about choosing a random coloring from the uniform distribution. Work with Glauber dynamics: pick a site at random, choose a color at random, and try to change the color at the chosen site. If this results in a proper coloring, make the change. If not, keep the current coloring. For q ≥ ∆ + 2 it is known that this Markov chain is ergodic, i.e.,\n\n> 12\n\nthat it connects the set of proper colorings. Let t = t(q) be the mixing time: the smallest time such that the total variation distance from the uniform distribution is at most 1 /e from any starting configuration. The question is: does t(q) decrease as\n\nq increases? Even on the square lattice with n sites there are curious gaps in our knowledge: we know that t(3) is polynomial in n (Luby, Randall and Sinclair: Proc. FOCS 1995) and we know that t(q) = O(n log n) for q ≥ 6. But, for q = 4 and q = 5 we do not even have a polynomial upper bound, even though there is overwhelming numerical evidence that there is a finite correlation length, and therefore that the mixing time is O(n log n). (6) Is there a variant of k-SAT that describes the scaling properties of jamming? In other words, is k-SAT a good \"Boolean idealization\" of the jamming process? Should this involve k-SAT on a lattice? (See Schwartz and Middleton, Phys. Rev. E 70 (2004), 035103(R).) (7) Find a model with constraints and a phase transition, which we can perturb by \"soft-ening\" the constraints. Study the sensitivity of this model's behavior with respect to this perturbation. (8) In constraint satisfaction problems such as k-SAT, what are the useful response/correlation functions? Do they show singular, or power-law, behavior at the transition, analo-gous to spatial correlations and response functions in lattice systems such as the Ising model? (9) The second moment method in k-SAT gives a highly accurate lower bound for the critical density, when the approach of (Achlioptas and Moore: Proc. FOCS 2002) is refined by (Achlioptas and Peres: Proc. STOC 2003). This refinement can be thought of as adding a finite (not infinitesimal) external field which discourages lit-erals from being true. Is there any physical meaning to this value of the external field? For instance, does it minimize fluctuations in, say, the free energy at zero temperature? (10) We believe (and, in some cases, have proved) that below the satisfiability/unsatisfiability transition in k-SAT there is a \"clustering\" transition, in which the solutions clump together in isolated groups. It has been conjectured that this clustering phenomenon makes it hard for search algorithms to find solutions. Can we find algorithms which, in simulation, find solutions in, say, linear time, for values of k and densities at which clustering has been rigorously established? (11) (Following up on the previous question) Can we prove that clustering occurs at some density below the satisfiability transition for k = 3, 4, 5? So far we only have proofs of clustering for larger values of k.(12) Prove an order/disorder phase transition for the equilibrium statistical mechanics model of hard spheres (or hard disks). Simulation evidence of such a transition is 3\n\ngenerally believed to be convincing in both cases, though the nature of the transition is controversial for disks. (13) How should we add friction to jamming? (14) Consider the Ising model on the d-dimensional lattice. Fix β (and perhaps the ex-ternal field strength h) and consider Glauber dynamics. The cutoff phenomenon in Markov chains refers to the existence of a time at which the total variation distance from the stationary (Gibbs) distribution jumps from 1 to 0 in the limit where the size of the lattice goes to infinity. We ask whether this sharp cutoff exists if and only if there is a unique Gibbs state in the infinite system. (15) Give a good definition of the thermodynamic limit of k-SAT. That is, clearly define the mathematical object that corresponds to random formulas Fk(n, m = rn ) with constant density r in the limit n → ∞.(16) Consider the Ising model on a finite box in Zd. Set β > β c and no external field. With fixed boundary conditions where all the boundary sites are set to +1, it is known that Glauber dynamics can take a long time to mix if the initial state consists of a large droplet of −1's. But with a \"warm start,\" where the initial condition consists of all +1's or is random, is the mixing time optimal, i.e., O(n log n) where n is the number of sites in the box?", "statement_status": "exact", "statement_verification": "The canonical JSON record is a contaminated extraction from the three-page AIM workshop PDF *Phase transitions*. Two page numbers were fused into the text as the block `> 12`, and printed problems (6)--(16) were appended to problem (5). Inspection of the original PDF shows that problem (5) ends with the sentence ending “the mixing time is \\(O(n\\log n)\\).” The next paragraph, beginning “(6) Is there a variant of \\(k\\)-SAT…”, is a separate problem and is not analyzed here.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Phase transitions\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/phasetransition/phasetransition.pdf\nCanonical location: aim-physics-notes.json notes[24]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(5) Let G be a finite graph with maximum degree ∆. This problem is about choosing a random coloring from the uniform distribution. Work with Glauber dynamics: pick a site at random, choose a color at random, and try to change the color at the chosen site. If this results in a proper coloring, make the change. If not, keep the current coloring. For q ≥ ∆ + 2 it is known that this Markov chain is ergodic, i.e., \\n\\n> 12\\n\\nthat it connects the set of proper colorings. Let t = t(q) be the mixing time: the smallest time such that the total variation distance from the uniform distribution is at most 1 /e from any starting configuration. The question is: does t(q) decrease as \\n\\nq increases? Even on the square lattice with n sites there are curious gaps in our knowledge: we know that t(3) is polynomial in n (Luby, Randall and Sinclair: Proc. FOCS 1995) and we know that t(q) = O(n log n) for q ≥ 6. But, for q = 4 and q = 5 we do not even have a polynomial upper bound, even though there is overwhelming numerical evidence that there is a finite correlation length, and therefore that the mixing time is O(n log n). (6) Is there a variant of k-SAT that describes the scaling properties of jamming? In other words, is k-SAT a good \\\"Boolean idealization\\\" of the jamming process? Should this involve k-SAT on a lattice? (See Schwartz and Middleton, Phys. Rev. E 70 (2004), 035103(R).) (7) Find a model with constraints and a phase transition, which we can perturb by \\\"soft-ening\\\" the constraints. Study the sensitivity of this model's behavior with respect to this perturbation. (8) In constraint satisfaction problems such as k-SAT, what are the useful response/correlation functions? Do they show singular, or power-law, behavior at the transition, analo-gous to spatial correlations and response functions in lattice systems such as the Ising model? (9) The second moment method in k-SAT gives a highly accurate lower bound for the critical density, when the approach of (Achlioptas and Moore: Proc. FOCS 2002) is refined by (Achlioptas and Peres: Proc. STOC 2003). This refinement can be thought of as adding a finite (not infinitesimal) external field which discourages lit-erals from being true. Is there any physical meaning to this value of the external field? For instance, does it minimize fluctuations in, say, the free energy at zero temperature? (10) We believe (and, in some cases, have proved) that below the satisfiability/unsatisfiability transition in k-SAT there is a \\\"clustering\\\" transition, in which the solutions clump together in isolated groups. It has been conjectured that this clustering phenomenon makes it hard for search algorithms to find solutions. Can we find algorithms which, in simulation, find solutions in, say, linear time, for values of k and densities at which clustering has been rigorously established? (11) (Following up on the previous question) Can we prove that clustering occurs at some density below the satisfiability transition for k = 3, 4, 5? So far we only have proofs of clustering for larger values of k.(12) Prove an order/disorder phase transition for the equilibrium statistical mechanics model of hard spheres (or hard disks). Simulation evidence of such a transition is 3\\n\\ngenerally believed to be convincing in both cases, though the nature of the transition is controversial for disks. (13) How should we add friction to jamming? (14) Consider the Ising model on the d-dimensional lattice. Fix β (and perhaps the ex-ternal field strength h) and consider Glauber dynamics. The cutoff phenomenon in Markov chains refers to the existence of a time at which the total variation distance from the stationary (Gibbs) distribution jumps from 1 to 0 in the limit where the size of the lattice goes to infinity. We ask whether this sharp cutoff exists if and only if there is a unique Gibbs state in the infinite system. (15) Give a good definition of the thermodynamic limit of k-SAT. That is, clearly define the mathematical object that corresponds to random formulas Fk(n, m = rn ) with constant density r in the limit n → ∞.(16) Consider the Ising model on a finite box in Zd. Set β > β c and no external field. With fixed boundary conditions where all the boundary sites are set to +1, it is known that Glauber dynamics can take a long time to mix if the initial state consists of a large droplet of −1's. But with a \\\"warm start,\\\" where the initial condition consists of all +1's or is random, is the mixing time optimal, i.e., O(n log n) where n is the number of sites in the box?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/phasetransition/phasetransition.pdf", "tags": [ "aim", "AIM-PHYSICS-0025", "aim-domain:physics", "aim-workshop:phasetransition", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For the literal discrete-time coloring Glauber chain in the AIM problem, the edgeless graph on two vertices has d_q(t) = 2^(1-t)(1-1/q)^2 for every t >= 1 and consequently t_mix(2) = 1 while t_mix(3) = 2 at threshold 1/e. Thus mixing time need not decrease as the number of colors increases. More generally, an explicit stochastic coarsening kernel proves d_q(t) <= d_r(t) for every t whenever 2 <= q < r on every edgeless graph.\n\nCandidate contribution (counterexample; novelty confidence low): The exact E_2 formula gives a two-color-to-three-color increase in mixing time, and the displayed color-coarsening kernel proves fixed-time total-variation monotonicity in the opposite direction throughout the edgeless-graph family." }, { "id": 20002518, "problem_number": "AIM-PHYSICS-0026", "title": "Equatorial symmetry and a Jacobian-jet test for local Kerr caustic cusps", "statement": "(1) No analytic description of relativistic caustics (except for the intersections on the equatorial plane. (2) No analytic description of gravitational lensing out of the equatorial plane. (3) No idea (analytical or numerical) for what happens to higher order caustics. (4) Is strong field gravitational lensing observable? (5) Analytical shape of the caustics in Strong Deflection Limit? (6) Gravitational lensing far from the equatorial plane in the Strong Deflection Limit? (7) Description of additional images in the Strong Deflection Limit? (8) What coordinates are best suited for Lensing in Kerr spacetime? (9) Projects for MAXIM? (10) Non-disk lensing events for MAXIM? (11) Gas trajectories in the accretion disk - fully GR code or MHD/Kerr? (12) Are the time delays the same for Kerr and \"Shifted Schwarzschild\"? To what order? (13) Are caustics/critical curves in first order strong field limit equivalent to \"shifted Schwarzschild\"?", "original_statement": "(1) No analytic description of relativistic caustics (except for the intersections on the equatorial plane. (2) No analytic description of gravitational lensing out of the equatorial plane. (3) No idea (analytical or numerical) for what happens to higher order caustics. (4) Is strong field gravitational lensing observable? (5) Analytical shape of the caustics in Strong Deflection Limit? (6) Gravitational lensing far from the equatorial plane in the Strong Deflection Limit? (7) Description of additional images in the Strong Deflection Limit? (8) What coordinates are best suited for Lensing in Kerr spacetime? (9) Projects for MAXIM? (10) Non-disk lensing events for MAXIM? (11) Gas trajectories in the accretion disk - fully GR code or MHD/Kerr? (12) Are the time delays the same for Kerr and \"Shifted Schwarzschild\"? To what order? (13) Are caustics/critical curves in first order strong field limit equivalent to \"shifted Schwarzschild\"?", "clean_statement": "(1) No analytic description of relativistic caustics (except for the intersections on the equatorial plane. (2) No analytic description of gravitational lensing out of the equatorial plane. (3) No idea (analytical or numerical) for what happens to higher order caustics. (4) Is strong field gravitational lensing observable? (5) Analytical shape of the caustics in Strong Deflection Limit? (6) Gravitational lensing far from the equatorial plane in the Strong Deflection Limit? (7) Description of additional images in the Strong Deflection Limit? (8) What coordinates are best suited for Lensing in Kerr spacetime? (9) Projects for MAXIM? (10) Non-disk lensing events for MAXIM? (11) Gas trajectories in the accretion disk - fully GR code or MHD/Kerr? (12) Are the time delays the same for Kerr and \"Shifted Schwarzschild\"? To what order? (13) Are caustics/critical curves in first order strong field limit equivalent to \"shifted Schwarzschild\"?", "statement_status": "exact", "statement_verification": "The canonical problem field concatenates thirteen numbered entries from a one-page AIM list. The original PDF was inspected directly. Its heading is “Gravitational Lensing in the Kerr Spacetime Geometry — Short list of problems/issues,” and its first four lines are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Gravitational lensing in the Kerr spacetime geometry\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/lensing/lensing.pdf\nCanonical location: aim-physics-notes.json notes[25]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(1) No analytic description of relativistic caustics (except for the intersections on the equatorial plane. (2) No analytic description of gravitational lensing out of the equatorial plane. (3) No idea (analytical or numerical) for what happens to higher order caustics. (4) Is strong field gravitational lensing observable? (5) Analytical shape of the caustics in Strong Deflection Limit? (6) Gravitational lensing far from the equatorial plane in the Strong Deflection Limit? (7) Description of additional images in the Strong Deflection Limit? (8) What coordinates are best suited for Lensing in Kerr spacetime? (9) Projects for MAXIM? (10) Non-disk lensing events for MAXIM? (11) Gas trajectories in the accretion disk - fully GR code or MHD/Kerr? (12) Are the time delays the same for Kerr and \\\"Shifted Schwarzschild\\\"? To what order? (13) Are caustics/critical curves in first order strong field limit equivalent to \\\"shifted Schwarzschild\\\"?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/lensing/lensing.pdf", "tags": [ "aim", "AIM-PHYSICS-0026", "aim-domain:physics", "aim-workshop:lensing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The canonical field merges thirteen separate PDF questions; genuine item (1) ends before item (2), and the source PDF itself is missing the closing parenthesis, so the recovered statement is: No analytic description of relativistic caustics (except for the intersections on the equatorial plane). The literal July-2005 status is obsolete because controlled analytic Kerr-caustic descriptions now exist in several regimes, although a globally uniform arbitrary-spin/all-branch description remains harder. For a C^5 equatorially reflection-equivariant rank-one lens-map slice, parity forces the fold derivative to vanish. In adapted coordinates, nonzero lambda_u and lambda_vv certify an ordinary cusp and give U=-(lambda_vv/(2 lambda_u))v^2+O(v^4), V=-(lambda_vv/3)v^3+O(v^5), hence V^2=-(8 lambda_u^3/(9 lambda_vv))U^3+O(U^4).\n\nCandidate contribution (local classification lemma; novelty confidence low): For a reflection-equivariant Kerr lens-map slice at an equatorial rank-one critical point, equatorial parity excludes an ordinary fold with reflection-odd kernel; after the explicit normalization U=X-X(p), v=y, the two Jacobian tests lambda_u not equal to zero and lambda_vv not equal to zero certify a cusp and determine its leading semicubical coefficient -8 lambda_u^3/(9 lambda_vv)." }, { "id": 20002519, "problem_number": "AIM-PHYSICS-0027", "title": "An invariant certificate for caustic-forming Kerr rays", "statement": "(14) Plotting the caustics from conjugate points: Which photons (light rays) form the caustics? (15) What does the lightcone look like? (16) Plot higher order caustic sheets numerically. (17) Plot primary caustic sheet for observer close to Black hole. (18) How do caustics look for observers outside the equatorial plane? (19) How are the astroid caustics formed? (20) Is the birth process of the astroid caustic sheet for Kerr stable? (21) Clarify GR definitions of twist/shear/convergence in terms of lensing definition. (22) Stability of caustics with respect to metric perturbations? (23) Are all stable caustic sheets lightlike?", "original_statement": "(14) Plotting the caustics from conjugate points: Which photons (light rays) form the caustics? (15) What does the lightcone look like? (16) Plot higher order caustic sheets numerically. (17) Plot primary caustic sheet for observer close to Black hole. (18) How do caustics look for observers outside the equatorial plane? (19) How are the astroid caustics formed? (20) Is the birth process of the astroid caustic sheet for Kerr stable? (21) Clarify GR definitions of twist/shear/convergence in terms of lensing definition. (22) Stability of caustics with respect to metric perturbations? (23) Are all stable caustic sheets lightlike?", "clean_statement": "(14) Plotting the caustics from conjugate points: Which photons (light rays) form the caustics? (15) What does the lightcone look like? (16) Plot higher order caustic sheets numerically. (17) Plot primary caustic sheet for observer close to Black hole. (18) How do caustics look for observers outside the equatorial plane? (19) How are the astroid caustics formed? (20) Is the birth process of the astroid caustic sheet for Kerr stable? (21) Clarify GR definitions of twist/shear/convergence in terms of lensing definition. (22) Stability of caustics with respect to metric perturbations? (23) Are all stable caustic sheets lightlike?", "statement_status": "exact", "statement_verification": "The canonical JSON record concatenates ten separately numbered questions from the one-page AIM problem list *Gravitational Lensing in the Kerr Spacetime Geometry*. Inspection of the original PDF shows the exact source boundary:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Gravitational lensing in the Kerr spacetime geometry\nSection: \nSource item: 14\nSource URL: https://aimath.org/WWN/lensing/lensing.pdf\nCanonical location: aim-physics-notes.json notes[26]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"(14) Plotting the caustics from conjugate points: Which photons (light rays) form the caustics? (15) What does the lightcone look like? (16) Plot higher order caustic sheets numerically. (17) Plot primary caustic sheet for observer close to Black hole. (18) How do caustics look for observers outside the equatorial plane? (19) How are the astroid caustics formed? (20) Is the birth process of the astroid caustic sheet for Kerr stable? (21) Clarify GR definitions of twist/shear/convergence in terms of lensing definition. (22) Stability of caustics with respect to metric perturbations? (23) Are all stable caustic sheets lightlike?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/lensing/lensing.pdf", "tags": [ "aim", "AIM-PHYSICS-0027", "aim-domain:physics", "aim-workshop:lensing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For an observer-normalized past null direction in Kerr spacetime, the ray reaches the conjugate caustic exactly at positive affine roots of the determinant of its 2-by-2 screen Jacobi map. Dividing the determinant by lambda squared removes the cone-vertex zero. At a root where tr(adj(D) D') is nonzero, the root has multiplicity one and continues as a unique smooth local function of observer-sky direction. A signed determinant plus smallest-singular-value diagnostic distinguishes ordinary root brackets from possible even or multiple roots, while an explicit Minkowski example proves that a projected coordinate-Jacobian zero can be a false caustic.\n\nCandidate contribution (algorithmic_reduction; novelty confidence low): The observer-normalized, patchwise root/continuation certificate combines the regularized signed Jacobi determinant, smallest singular value, adjugate transversality derivative, and an explicit flat-space projection false-positive test into a reproducible criterion for selecting caustic-forming rays before plotting." }, { "id": 20002520, "problem_number": "AIM-PHYSICS-0028", "title": "Rellich compactness and spectrum on open manifolds", "statement": "1. (Carron) One needs geometrical assumptions on ( M, g ) to make this work. It turns out that bounded Ricci-curvature is not a sufficient condition. Donnelly [3] had shown that if a complete non-compact Riemannian manifold of dimension n has Ric ≥ − (n − 1) g then the essential spectrum of the Laplace operator (on functions) has a non-empty intersection with [0, (n − 1) 2/4]. Moreover, Donnelly and Li [4] had shown that if a Cartan-Hadamard manifold (i.e. a complete simply-connected Riemannian manifold with non-positive curvature) has its curvature going to −∞ at infinity then the essential spectrum of the Laplace operator is empty. 2. It can also be generalized introducing weighted Sobolev spaces - with special conditions on the weights depending on the geometry at infinity (see Joyce's book [7] for the case of ALE manifolds).", "original_statement": "1. (Carron) One needs geometrical assumptions on ( M, g ) to make this work. It turns out that bounded Ricci-curvature is not a sufficient condition. Donnelly [3] had shown that if a complete non-compact Riemannian manifold of dimension n has Ric ≥ − (n − 1) g then the essential spectrum of the Laplace operator (on functions) has a non-empty intersection with [0, (n − 1) 2/4]. Moreover, Donnelly and Li [4] had shown that if a Cartan-Hadamard manifold (i.e. a complete simply-connected Riemannian manifold with non-positive curvature) has its curvature going to −∞ at infinity then the essential spectrum of the Laplace operator is empty. 2. It can also be generalized introducing weighted Sobolev spaces - with special conditions on the weights depending on the geometry at infinity (see Joyce's book [7] for the case of ALE manifolds).", "clean_statement": "1. (Carron) One needs geometrical assumptions on ( M, g ) to make this work. It turns out that bounded Ricci-curvature is not a sufficient condition. Donnelly [3] had shown that if a complete non-compact Riemannian manifold of dimension n has Ric ≥ − (n − 1) g then the essential spectrum of the Laplace operator (on functions) has a non-empty intersection with [0, (n − 1) 2/4]. Moreover, Donnelly and Li [4] had shown that if a Cartan-Hadamard manifold (i.e. a complete simply-connected Riemannian manifold with non-positive curvature) has its curvature going to −∞ at infinity then the essential spectrum of the Laplace operator is empty. 2. It can also be generalized introducing weighted Sobolev spaces - with special conditions on the weights depending on the geometry at infinity (see Joyce's book [7] for the case of ALE manifolds).", "statement_status": "exact", "statement_verification": "The canonical record is not a self-contained open problem. It is the first two answers to a question on page 1 of the AIM workshop notes *Questions arising in open problem sessions in AIM workshop on $L^2$-harmonic forms in geometry and string theory* (notes by Anda Degeratu and Mark Haskins, 17 March 2004). The missing antecedent is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: $L^2$ harmonic forms in geometry and string theory\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/l2harmonic/l2harmonic.pdf\nCanonical location: aim-physics-notes.json notes[27]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. (Carron) One needs geometrical assumptions on ( M, g ) to make this work. It turns out that bounded Ricci-curvature is not a sufficient condition. Donnelly [3] had shown that if a complete non-compact Riemannian manifold of dimension n has Ric ≥ − (n − 1) g then the essential spectrum of the Laplace operator (on functions) has a non-empty intersection with [0, (n − 1) 2/4]. Moreover, Donnelly and Li [4] had shown that if a Cartan-Hadamard manifold (i.e. a complete simply-connected Riemannian manifold with non-positive curvature) has its curvature going to −∞ at infinity then the essential spectrum of the Laplace operator is empty. 2. It can also be generalized introducing weighted Sobolev spaces - with special conditions on the weights depending on the geometry at infinity (see Joyce's book [7] for the case of ALE manifolds).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/l2harmonic/l2harmonic.pdf", "tags": [ "aim", "AIM-PHYSICS-0028", "aim-domain:physics", "aim-workshop:l2harmonic", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The record is recovered as Answers 1 and 2, not an open problem, to the workshop question asking how compact Sobolev embeddings generalize to complete open manifolds; the PDF also corrects the extracted threshold to (n-1)^2/4. As a developed contribution, for a manifold with end dr^2+exp(2r^alpha)g_{S^{n-1}}, the scalar essential spectrum is [0,infinity) for 01. Hence H^1 to L^2 is compact exactly for alpha>1, matching the transition from bounded curvature at infinity to sectional curvature tending to minus infinity.\n\nCandidate contribution (theorem; novelty confidence low): For the explicit exponential-power warped ends f_alpha(r)=exp(r^alpha), the alpha=1 transition simultaneously determines curvature limits, the effective radial Schrödinger potential, the full scalar essential spectrum, and compactness of H^1 into L^2.", "research_difficulty_suggested": "L2" }, { "id": 20002521, "problem_number": "AIM-PHYSICS-0029", "title": "Melrose's answer, compact Sobolev embedding, and escape of mass", "statement": "3. (Melrose) There is no good general answer.\n\nQuestion [ L2-cohomology]:\n\nIs there a proper way to define L2-homology as opposed to L2-cohomology?\n\nMotivation: There is a way to define it for triangulated manifolds ( M, T ), by going up to the universal cover ( ˜M, ˜T ) and taking the L2-chains to be\n\nCi\n\n> (2)\n\n( ˜M ) = {∑ aiσi| ∑ |ai|2 < ∞}.\n\n(and possibly some boundary terms condition coming in also?)\n\nQuestion to ask: How does this depend on the metric? And when do you get something dual to the\n\nL2-cohomology.\n\nQuestion [ L2-cohomology]:\n\nIs there any relation between L2-cohomology and group-cohomology.\n\nAnswer: see L¨ uck in his book [8].\n\nQuestions [Non-parabolicity and exactness of the excision sequence for reduced L2-cohomology - G. Carron's lecture]:\n\n1Let ( M, g ) Riemannian manifold so that d+d∗ is non-parabolic at infinity with respect to K. For a compact\n\n˜K so that K ⊂ ˜K we define the norm\n\nN ˜K (α):= || α|| L2 ( ˜K) + || (d + d∗)α|| L2 (M \\ ˜K).\n\nThen on C∞\n\n> 0\n\n(Λ( M )) all these norms are equivalent. Let W be the completion of C∞\n\n> 0\n\n(Λ( M )) with respect one of them. One of the main points was that\n\nd + d∗: W → L2\n\nis Fredholm. Several related questions arose: 1. Is there any other description for W?2. Is the reduced L2-cohomology ˙Hk\n\n> (2)\n\n(M, g ) the cohomology of a complex/ of many complexes? 3. Can the space W be put into a complex?\n\nQuestion [reduced L2-cohomology - G. Carron's lecture]:\n\nWhen is there a Mayer-Vietoris sequence for ˙H(2) (M, g )?\n\nAnswer: (Carron) The condition of non-parabolicity at infinity is not sufficient for this. One needs a better control over noncompact overlaps. Perhaps if also Range( d) is closed on U ∩ V then it is sufficient??\n\nQuestions [Self-dual Gravitational Instantons - S. Cherkis's lecture]:", "original_statement": "3. (Melrose) There is no good general answer. \n\nQuestion [ L2-cohomology]: \n\nIs there a proper way to define L2-homology as opposed to L2-cohomology? \n\nMotivation: There is a way to define it for triangulated manifolds ( M, T ), by going up to the universal cover ( ˜M, ˜T ) and taking the L2-chains to be \n\nCi\n\n> (2)\n\n( ˜M ) = {∑ aiσi| ∑ |ai|2 < ∞}.\n\n(and possibly some boundary terms condition coming in also?) \n\nQuestion to ask: How does this depend on the metric? And when do you get something dual to the \n\nL2-cohomology. \n\nQuestion [ L2-cohomology]: \n\nIs there any relation between L2-cohomology and group-cohomology. \n\nAnswer: see L¨ uck in his book [8]. \n\nQuestions [Non-parabolicity and exactness of the excision sequence for reduced L2-cohomology - G. Carron's lecture]: \n\n1Let ( M, g ) Riemannian manifold so that d+d∗ is non-parabolic at infinity with respect to K. For a compact \n\n˜K so that K ⊂ ˜K we define the norm \n\nN ˜K (α):= || α|| L2 ( ˜K) + || (d + d∗)α|| L2 (M \\ ˜K).\n\nThen on C∞ \n\n> 0\n\n(Λ( M )) all these norms are equivalent. Let W be the completion of C∞ \n\n> 0\n\n(Λ( M )) with respect one of them. One of the main points was that \n\nd + d∗: W → L2\n\nis Fredholm. Several related questions arose: 1. Is there any other description for W?2. Is the reduced L2-cohomology ˙Hk\n\n> (2)\n\n(M, g ) the cohomology of a complex/ of many complexes? 3. Can the space W be put into a complex? \n\nQuestion [reduced L2-cohomology - G. Carron's lecture]: \n\nWhen is there a Mayer-Vietoris sequence for ˙H(2) (M, g )? \n\nAnswer: (Carron) The condition of non-parabolicity at infinity is not sufficient for this. One needs a better control over noncompact overlaps. Perhaps if also Range( d) is closed on U ∩ V then it is sufficient?? \n\nQuestions [Self-dual Gravitational Instantons - S. Cherkis's lecture]:", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical JSON field is corrupted by a record-boundary error. It begins", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: $L^2$ harmonic forms in geometry and string theory\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/l2harmonic/l2harmonic.pdf\nCanonical location: aim-physics-notes.json notes[28]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (Melrose) There is no good general answer. \\n\\nQuestion [ L2-cohomology]: \\n\\nIs there a proper way to define L2-homology as opposed to L2-cohomology? \\n\\nMotivation: There is a way to define it for triangulated manifolds ( M, T ), by going up to the universal cover ( ˜M, ˜T ) and taking the L2-chains to be \\n\\nCi\\n\\n> (2)\\n\\n( ˜M ) = {∑ aiσi| ∑ |ai|2 < ∞}.\\n\\n(and possibly some boundary terms condition coming in also?) \\n\\nQuestion to ask: How does this depend on the metric? And when do you get something dual to the \\n\\nL2-cohomology. \\n\\nQuestion [ L2-cohomology]: \\n\\nIs there any relation between L2-cohomology and group-cohomology. \\n\\nAnswer: see L¨ uck in his book [8]. \\n\\nQuestions [Non-parabolicity and exactness of the excision sequence for reduced L2-cohomology - G. Carron's lecture]: \\n\\n1Let ( M, g ) Riemannian manifold so that d+d∗ is non-parabolic at infinity with respect to K. For a compact \\n\\n˜K so that K ⊂ ˜K we define the norm \\n\\nN ˜K (α):= || α|| L2 ( ˜K) + || (d + d∗)α|| L2 (M \\\\ ˜K).\\n\\nThen on C∞ \\n\\n> 0\\n\\n(Λ( M )) all these norms are equivalent. Let W be the completion of C∞ \\n\\n> 0\\n\\n(Λ( M )) with respect one of them. One of the main points was that \\n\\nd + d∗: W → L2\\n\\nis Fredholm. Several related questions arose: 1. Is there any other description for W?2. Is the reduced L2-cohomology ˙Hk\\n\\n> (2)\\n\\n(M, g ) the cohomology of a complex/ of many complexes? 3. Can the space W be put into a complex? \\n\\nQuestion [reduced L2-cohomology - G. Carron's lecture]: \\n\\nWhen is there a Mayer-Vietoris sequence for ˙H(2) (M, g )? \\n\\nAnswer: (Carron) The condition of non-parabolicity at infinity is not sufficient for this. One needs a better control over noncompact overlaps. Perhaps if also Range( d) is closed on U ∩ V then it is sufficient?? \\n\\nQuestions [Self-dual Gravitational Instantons - S. Cherkis's lecture]:\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/l2harmonic/l2harmonic.pdf", "tags": [ "aim", "AIM-PHYSICS-0029", "aim-domain:physics", "aim-workshop:l2harmonic", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The original AIM PDF proves that this canonical record is not the later L2-cohomology block: it is exactly Melrose's third answer, 'There is no good general answer,' to the preceding question about extending compact Sobolev--Kondrachov embeddings from compact to complete open manifolds. For the fundamental scalar endpoint on a complete boundaryless manifold, this attempt proves that compactness of H^1(M) into L^2(M), compactness of (Delta+1)^(-1), emptiness of the essential spectrum, and a uniform H^1 tail inequality are equivalent. It further proves that this yes/no property is invariant under uniformly equivalent metrics and exhibits translated Euclidean bumps as the basic obstruction.\n\nCandidate contribution (spectral and escape-of-mass criterion; novelty confidence low): The four-way endpoint diagnostic packages compact H^1-to-L^2 embedding, compact scalar resolvent, empty essential spectrum, and uniform H^1 tightness into one self-contained equivalence, then proves that the package is stable under every uniformly equivalent change of Riemannian metric.", "research_difficulty_suggested": "L2" }, { "id": 20002522, "problem_number": "AIM-PHYSICS-0030", "title": "Why gravitational instantons arise from monopoles", "statement": "1. Why do you expect them to arise as monopole moduli spaces?\n\nAnswer: (Cherkis) string theory argument. 2. Are all 8-dimensional hyperk¨ ahler manifolds (non-compact, with some sort of well-behaved asymptotic behaviour) moduli spaces?\n\nAnswer: (Cherkis) yes, from string theory intuition. 3. What about dimension > 8? 4. One way to get gravitational instantons is by solving the vacuum Einstein equation in Lorentz space, and then performing a wick rotation (meaning t → it ). This worked in the case of Lorentzian Taub-NUT to get the Riemannian Taub-NUT. Which other instantons arrise this way?\n\nAnswer: In order to get a Riemannian metric, one needs t → t + c (time translation) be an isometry. Therefore it does work for Ak-cases but not for the others ( Dk, E6, E 7, E 8). 5. Can they arrive as \"non-trivial\" hyperk¨ ahler reductions of finite dimensional linear spaces?\n\nAnswer: Yes, for ALE spaces (Kronheimer's construction). Yes, for ALF if certain (Dancer) spaces are included with the linear spaces. 6. Is it true that every gravitational instanton has an asymptotic local triholomorphic isometry?\n\nReason: If you have a local R-action at infinity, then the metric is given in terms of a harmonic function, which arises from the hyperk¨ ahler moment map associated to this action (see the work of Gibbons). 7. When is it true that {Solutions of the Einstein equation } > {Self-Dual metrics }?\n\nReason: In general ≥.\n\nAnswers:\n\n(1) In the case of ALF the answer is >, since the Euclidean Schwarzschild metric is not self-dual. (2) Nakajima conjectured that in dimension 4, ALE and Ricci-flat implies self-dual. (3) In dimension > 4 and even, the conjecture is that ALE and Ricci-flat implies K¨ ahler. (4) (Carron) For odd dimensional manifolds an ALE Ricci-flat metric has to be flat since (a) by the Cheeger-Gromoll theorem this manifold has only one end (if not it splits isometrically as\n\nR × N with N compact); (b) The topology at infinity must be of the type ( R, ∞) × S2n/G where G is a finite subgroup of\n\nO(2 n + 1) acting freely on S2n. If g ∈ SO (2 n + 1) it must have 1 as an eigenvalue hence the only such G is\n\n{I, −I}; but the quotient is the real projective space, and there is no odd dimensional compact manifold whose boundary is the real projective space. Therefore G is trivial and the Bishop-Gromov inequality 2implies that M is the Euclidean space. Recall that the Bishop-Gromov inequality states that in a manifold\n\nM n with non-negative Ricci curvature r 7 → vol B(x, r )/w nrn is decreasing with equality everywhere if and only if M n is the Euclidean space. This ratio goes to 1 when r → 0, and in the ALE case we have that this ratio goes to 1 /card G when r → ∞.\n\nQuestions [of S. Cherkis]:", "original_statement": "1. Why do you expect them to arise as monopole moduli spaces? \n\nAnswer: (Cherkis) string theory argument. 2. Are all 8-dimensional hyperk¨ ahler manifolds (non-compact, with some sort of well-behaved asymptotic behaviour) moduli spaces? \n\nAnswer: (Cherkis) yes, from string theory intuition. 3. What about dimension > 8? 4. One way to get gravitational instantons is by solving the vacuum Einstein equation in Lorentz space, and then performing a wick rotation (meaning t → it ). This worked in the case of Lorentzian Taub-NUT to get the Riemannian Taub-NUT. Which other instantons arrise this way? \n\nAnswer: In order to get a Riemannian metric, one needs t → t + c (time translation) be an isometry. Therefore it does work for Ak-cases but not for the others ( Dk, E6, E 7, E 8). 5. Can they arrive as \"non-trivial\" hyperk¨ ahler reductions of finite dimensional linear spaces? \n\nAnswer: Yes, for ALE spaces (Kronheimer's construction). Yes, for ALF if certain (Dancer) spaces are included with the linear spaces. 6. Is it true that every gravitational instanton has an asymptotic local triholomorphic isometry? \n\nReason: If you have a local R-action at infinity, then the metric is given in terms of a harmonic function, which arises from the hyperk¨ ahler moment map associated to this action (see the work of Gibbons). 7. When is it true that {Solutions of the Einstein equation } > {Self-Dual metrics }?\n\nReason: In general ≥.\n\nAnswers: \n\n(1) In the case of ALF the answer is >, since the Euclidean Schwarzschild metric is not self-dual. (2) Nakajima conjectured that in dimension 4, ALE and Ricci-flat implies self-dual. (3) In dimension > 4 and even, the conjecture is that ALE and Ricci-flat implies K¨ ahler. (4) (Carron) For odd dimensional manifolds an ALE Ricci-flat metric has to be flat since (a) by the Cheeger-Gromoll theorem this manifold has only one end (if not it splits isometrically as \n\nR × N with N compact); (b) The topology at infinity must be of the type ( R, ∞) × S2n/G where G is a finite subgroup of \n\nO(2 n + 1) acting freely on S2n. If g ∈ SO (2 n + 1) it must have 1 as an eigenvalue hence the only such G is \n\n{I, −I}; but the quotient is the real projective space, and there is no odd dimensional compact manifold whose boundary is the real projective space. Therefore G is trivial and the Bishop-Gromov inequality 2implies that M is the Euclidean space. Recall that the Bishop-Gromov inequality states that in a manifold \n\nM n with non-negative Ricci curvature r 7 → vol B(x, r )/w nrn is decreasing with equality everywhere if and only if M n is the Euclidean space. This ratio goes to 1 when r → 0, and in the ALE case we have that this ratio goes to 1 /card G when r → ∞.\n\nQuestions [of S. Cherkis]:", "clean_statement": "**Question 1.** Why do you expect self-dual gravitational instantons to arise as monopole moduli spaces?\n\n**Answer (Cherkis).** String theory argument.", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical extraction begins:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: $L^2$ harmonic forms in geometry and string theory\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/l2harmonic/l2harmonic.pdf\nCanonical location: aim-physics-notes.json notes[29]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Why do you expect them to arise as monopole moduli spaces? \\n\\nAnswer: (Cherkis) string theory argument. 2. Are all 8-dimensional hyperk¨ ahler manifolds (non-compact, with some sort of well-behaved asymptotic behaviour) moduli spaces? \\n\\nAnswer: (Cherkis) yes, from string theory intuition. 3. What about dimension > 8? 4. One way to get gravitational instantons is by solving the vacuum Einstein equation in Lorentz space, and then performing a wick rotation (meaning t → it ). This worked in the case of Lorentzian Taub-NUT to get the Riemannian Taub-NUT. Which other instantons arrise this way? \\n\\nAnswer: In order to get a Riemannian metric, one needs t → t + c (time translation) be an isometry. Therefore it does work for Ak-cases but not for the others ( Dk, E6, E 7, E 8). 5. Can they arrive as \\\"non-trivial\\\" hyperk¨ ahler reductions of finite dimensional linear spaces? \\n\\nAnswer: Yes, for ALE spaces (Kronheimer's construction). Yes, for ALF if certain (Dancer) spaces are included with the linear spaces. 6. Is it true that every gravitational instanton has an asymptotic local triholomorphic isometry? \\n\\nReason: If you have a local R-action at infinity, then the metric is given in terms of a harmonic function, which arises from the hyperk¨ ahler moment map associated to this action (see the work of Gibbons). 7. When is it true that {Solutions of the Einstein equation } > {Self-Dual metrics }?\\n\\nReason: In general ≥.\\n\\nAnswers: \\n\\n(1) In the case of ALF the answer is >, since the Euclidean Schwarzschild metric is not self-dual. (2) Nakajima conjectured that in dimension 4, ALE and Ricci-flat implies self-dual. (3) In dimension > 4 and even, the conjecture is that ALE and Ricci-flat implies K¨ ahler. (4) (Carron) For odd dimensional manifolds an ALE Ricci-flat metric has to be flat since (a) by the Cheeger-Gromoll theorem this manifold has only one end (if not it splits isometrically as \\n\\nR × N with N compact); (b) The topology at infinity must be of the type ( R, ∞) × S2n/G where G is a finite subgroup of \\n\\nO(2 n + 1) acting freely on S2n. If g ∈ SO (2 n + 1) it must have 1 as an eigenvalue hence the only such G is \\n\\n{I, −I}; but the quotient is the real projective space, and there is no odd dimensional compact manifold whose boundary is the real projective space. Therefore G is trivial and the Bishop-Gromov inequality 2implies that M is the Euclidean space. Recall that the Bishop-Gromov inequality states that in a manifold \\n\\nM n with non-negative Ricci curvature r 7 → vol B(x, r )/w nrn is decreasing with equality everywhere if and only if M n is the Euclidean space. This ratio goes to 1 when r → 0, and in the ALE case we have that this ratio goes to 1 /card G when r → ∞.\\n\\nQuestions [of S. Cherkis]:\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: context_only; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/l2harmonic/l2harmonic.pdf", "tags": [ "aim", "AIM-PHYSICS-0030", "aim-domain:physics", "aim-workshop:l2harmonic", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "The PDF verifies that 'them' means self-dual gravitational instantons and that the genuine item consists only of Question 1 plus Cherkis's string-theory answer; items 2-7 are extraction spillover. The mathematical core is a proved dimension-and-resultant gateway for ordinary framed charge-k SU(2) monopoles on R^3: the framed, translation-centered, and strongly centered dimensions are 4k, 4k-3, and 4k-4; the only nonflat four-dimensional whole strongly centered case is selected at k=2; in Donaldson's rational-map complex structure its strongly centered slice is the smooth surface b^2+a^2 c=1; and any regular four-dimensional hyperkahler quotient of the strongly centered space by a free proper group H must have dim H=k-2.\n\nCandidate contribution (theorem; novelty confidence low): The dimension-and-resultant gateway combines the four universal monopole zero modes, the explicit charge-two resultant surface b^2+a^2 c=1, the residual finite Z_k cover, and the necessary quotient budget dim H=k-2 into one testable filter for four-dimensional gravitational-instanton realizations.", "research_difficulty_suggested": "L2" }, { "id": 20002523, "problem_number": "AIM-PHYSICS-0031", "title": "A synchronized heat-kernel construction on eight-dimensional hyperkähler products", "statement": "1. There exists an explicit construction of Atiyah [1] and Page [10] for the Green's function on certain ALE\n\nand ALF spaces. Page's construction comes from physics and constructs by hand the Green's functions for both AkALE and AkALF. Atiyah has a formalism which in theory allows one to construct the Green's function on any self-dual gravitational instanton - this formalism involves Serre classes on the corresponding twistor spaces. In the paper he constructs only the Green's function in the AkALE case. (According to Etesi the use of Atiyah's Serre-class formalism is significantly more difficult for the ALF case and has not yet been carried out. The main problem he says is that one cannot find a nice compacification for the twistor space of an ALF self-dual space. Carrying out this construction would allow us to construct L2\n\nharmonic forms over the new DkALF gravitational instantons constructed by Cherkis and Hitchin.)\n\nQuestion: Is there a way to generalize this formalism to 8 dimensional hyperk¨ ahler manifolds?\n\nReason: In the non-compact 4-dimensional case or K3 case, the coefficient of the leading order term in the Green's function gave information about the values of the parameters (e.g. sizes of the cycles, metric). 2. The same story for quaternionic manifolds. How to modify Atiyah's formalism?\n\nReason: There is a reasonable twistor theory for these manifolds. 3. Is it true that any 4-dimensional hyperk¨ ahler manifold (non-compact, with reasonable decay at infinity) can be deformed to a quaternionic one?\n\nFor example: HK has Ric = 0, while quaternionic manifolds satisfy Ric = λg with λ a constant (non-zero cosmological constant). One would like to obtain any HK as a limit of quaternionic spaces with λ → 0.\n\nExample: R4 is the limit of HP 1 (positive λ's) and HH 1 (negative λ's).\n\n2 March 18, 2004 - S. Cherkis as moderator.\n\nQuestion (R. Mazzeo):\n\nWhat are the asymptotics at infinity of the multi-monopole moduli spaces?\n\nAnswer: (Cherkis) They should behave like a N-body problem.\n\nQuestions (K. Lee)\n\nFor these questions, R4\n\n> B\n\ndenotes the non-commutative R4. Here B = Bμν dx μdx ν is a 2-form which gives the non-commutative structure. Then, [ xμ, x ν ] = Bμν. We define non-commutative instantons using the Nahm transform of instanton data\n\nμ1 = [ T1, T 2] − [T3, T 4] − j∗j = B12,\n\nplus two other similar equations (see the work Nekrasov-Schwarz on non-commutative instantons [9]). Let\n\nMk,n B be the moduli space of centered k-instantons on R4\n\n> B\n\nwith gauge group G = U (n). (The fact that the instantons are centered means we consider only instantons with center of mass at the origin) 1. What is the instanton number in the non-commutative set-up?\n\nMotivation: The space Mk,n B is singular for B = 0 but non-singular for B 6 = 0. One would like to understand the limit as B → 0. 2. Does there exists a unique harmonic form in the middle dimension of Mk,n B?\n\nNote: The answer to this question is Yes for the following cases: (i) 1-instantons and any G;(ii) 2-instantons and G = U (1) (see the work of Lee et al). 33. Conjecture: The moduli space of centered 1-instantons on R3 × S1\n\n> B\n\nwith \"given holonomy around S1\n\n> ∞\n\n\"(holonomy is not in the center of G = U (n)) has exactly n L 2-harmonic forms in middle dimension.", "original_statement": "1. There exists an explicit construction of Atiyah [1] and Page [10] for the Green's function on certain ALE \n\nand ALF spaces. Page's construction comes from physics and constructs by hand the Green's functions for both AkALE and AkALF. Atiyah has a formalism which in theory allows one to construct the Green's function on any self-dual gravitational instanton - this formalism involves Serre classes on the corresponding twistor spaces. In the paper he constructs only the Green's function in the AkALE case. (According to Etesi the use of Atiyah's Serre-class formalism is significantly more difficult for the ALF case and has not yet been carried out. The main problem he says is that one cannot find a nice compacification for the twistor space of an ALF self-dual space. Carrying out this construction would allow us to construct L2\n\nharmonic forms over the new DkALF gravitational instantons constructed by Cherkis and Hitchin.) \n\nQuestion: Is there a way to generalize this formalism to 8 dimensional hyperk¨ ahler manifolds? \n\nReason: In the non-compact 4-dimensional case or K3 case, the coefficient of the leading order term in the Green's function gave information about the values of the parameters (e.g. sizes of the cycles, metric). 2. The same story for quaternionic manifolds. How to modify Atiyah's formalism? \n\nReason: There is a reasonable twistor theory for these manifolds. 3. Is it true that any 4-dimensional hyperk¨ ahler manifold (non-compact, with reasonable decay at infinity) can be deformed to a quaternionic one? \n\nFor example: HK has Ric = 0, while quaternionic manifolds satisfy Ric = λg with λ a constant (non-zero cosmological constant). One would like to obtain any HK as a limit of quaternionic spaces with λ → 0. \n\nExample: R4 is the limit of HP 1 (positive λ's) and HH 1 (negative λ's). \n\n2 March 18, 2004 - S. Cherkis as moderator. \n\nQuestion (R. Mazzeo): \n\nWhat are the asymptotics at infinity of the multi-monopole moduli spaces? \n\nAnswer: (Cherkis) They should behave like a N-body problem. \n\nQuestions (K. Lee) \n\nFor these questions, R4 \n\n> B\n\ndenotes the non-commutative R4. Here B = Bμν dx μdx ν is a 2-form which gives the non-commutative structure. Then, [ xμ, x ν ] = Bμν. We define non-commutative instantons using the Nahm transform of instanton data \n\nμ1 = [ T1, T 2] − [T3, T 4] − j∗j = B12,\n\nplus two other similar equations (see the work Nekrasov-Schwarz on non-commutative instantons [9]). Let \n\nMk,n B be the moduli space of centered k-instantons on R4 \n\n> B\n\nwith gauge group G = U (n). (The fact that the instantons are centered means we consider only instantons with center of mass at the origin) 1. What is the instanton number in the non-commutative set-up? \n\nMotivation: The space Mk,n B is singular for B = 0 but non-singular for B 6 = 0. One would like to understand the limit as B → 0. 2. Does there exists a unique harmonic form in the middle dimension of Mk,n B?\n\nNote: The answer to this question is Yes for the following cases: (i) 1-instantons and any G;(ii) 2-instantons and G = U (1) (see the work of Lee et al). 33. Conjecture: The moduli space of centered 1-instantons on R3 × S1 \n\n> B\n\nwith \"given holonomy around S1\n\n> ∞\n\n\"(holonomy is not in the center of G = U (n)) has exactly n L 2-harmonic forms in middle dimension.", "clean_statement": "1. There exists an explicit construction of Atiyah [1] and Page [10] for the Green's function on certain ALE\n\nand ALF spaces. Page's construction comes from physics and constructs by hand the Green's functions for both AkALE and AkALF. Atiyah has a formalism which in theory allows one to construct the Green's function on any self-dual gravitational instanton - this formalism involves Serre classes on the corresponding twistor spaces. In the paper he constructs only the Green's function in the AkALE case. (According to Etesi the use of Atiyah's Serre-class formalism is significantly more difficult for the ALF case and has not yet been carried out. The main problem he says is that one cannot find a nice compacification for the twistor space of an ALF self-dual space. Carrying out this construction would allow us to construct L2\n\nharmonic forms over the new DkALF gravitational instantons constructed by Cherkis and Hitchin.)\n\nQuestion: Is there a way to generalize this formalism to 8 dimensional hyperk¨ ahler manifolds?\n\nReason: In the non-compact 4-dimensional case or K3 case, the coefficient of the leading order term in the Green's function gave information about the values of the parameters (e.g. sizes of the cycles, metric). 2. The same story for quaternionic manifolds. How to modify Atiyah's formalism?\n\nReason: There is a reasonable twistor theory for these manifolds. 3. Is it true that any 4-dimensional hyperk¨ ahler manifold (non-compact, with reasonable decay at infinity) can be deformed to a quaternionic one?\n\nFor example: HK has Ric = 0, while quaternionic manifolds satisfy Ric = λg with λ a constant (non-zero cosmological constant). One would like to obtain any HK as a limit of quaternionic spaces with λ → 0.\n\nExample: R4 is the limit of HP 1 (positive λ's) and HH 1 (negative λ's).\n\n2 March 18, 2004 - S. Cherkis as moderator.\n\nQuestion (R. Mazzeo):\n\nWhat are the asymptotics at infinity of the multi-monopole moduli spaces?\n\nAnswer: (Cherkis) They should behave like a N-body problem.\n\nQuestions (K. Lee)\n\nFor these questions, R4\n\n> B\n\ndenotes the non-commutative R4. Here B = Bμν dx μdx ν is a 2-form which gives the non-commutative structure. Then, [ xμ, x ν ] = Bμν. We define non-commutative instantons using the Nahm transform of instanton data\n\nμ1 = [ T1, T 2] − [T3, T 4] − j∗j = B12,\n\nplus two other similar equations (see the work Nekrasov-Schwarz on non-commutative instantons [9]). Let\n\nMk,n B be the moduli space of centered k-instantons on R4\n\n> B\n\nwith gauge group G = U (n). (The fact that the instantons are centered means we consider only instantons with center of mass at the origin) 1. What is the instanton number in the non-commutative set-up?\n\nMotivation: The space Mk,n B is singular for B = 0 but non-singular for B 6 = 0. One would like to understand the limit as B → 0. 2. Does there exists a unique harmonic form in the middle dimension of Mk,n B?\n\nNote: The answer to this question is Yes for the following cases: (i) 1-instantons and any G;(ii) 2-instantons and G = U (1) (see the work of Lee et al). 33. Conjecture: The moduli space of centered 1-instantons on R3 × S1\n\n> B\n\nwith \"given holonomy around S1\n\n> ∞\n\n\"(holonomy is not in the center of G = U (n)) has exactly n L 2-harmonic forms in middle dimension.", "statement_status": "exact", "statement_verification": "The canonical JSON record is contaminated: after the genuine first item in S. Cherkis's list it contains separately numbered items 2 and 3, material from a March 18 session, and questions attributed to K. Lee and M. Singer. The original six-page AIM workshop PDF was checked directly. On page 2, under “Questions [of S. Cherkis],” item 1 discusses Page's explicit Green functions on certain ALE/ALF gravitational instantons and Atiyah's twistor/Serre-class construction on self-dual four-manifolds. The question and its immediately attached reason are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: $L^2$ harmonic forms in geometry and string theory\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/l2harmonic/l2harmonic.pdf\nCanonical location: aim-physics-notes.json notes[30]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. There exists an explicit construction of Atiyah [1] and Page [10] for the Green's function on certain ALE \\n\\nand ALF spaces. Page's construction comes from physics and constructs by hand the Green's functions for both AkALE and AkALF. Atiyah has a formalism which in theory allows one to construct the Green's function on any self-dual gravitational instanton - this formalism involves Serre classes on the corresponding twistor spaces. In the paper he constructs only the Green's function in the AkALE case. (According to Etesi the use of Atiyah's Serre-class formalism is significantly more difficult for the ALF case and has not yet been carried out. The main problem he says is that one cannot find a nice compacification for the twistor space of an ALF self-dual space. Carrying out this construction would allow us to construct L2\\n\\nharmonic forms over the new DkALF gravitational instantons constructed by Cherkis and Hitchin.) \\n\\nQuestion: Is there a way to generalize this formalism to 8 dimensional hyperk¨ ahler manifolds? \\n\\nReason: In the non-compact 4-dimensional case or K3 case, the coefficient of the leading order term in the Green's function gave information about the values of the parameters (e.g. sizes of the cycles, metric). 2. The same story for quaternionic manifolds. How to modify Atiyah's formalism? \\n\\nReason: There is a reasonable twistor theory for these manifolds. 3. Is it true that any 4-dimensional hyperk¨ ahler manifold (non-compact, with reasonable decay at infinity) can be deformed to a quaternionic one? \\n\\nFor example: HK has Ric = 0, while quaternionic manifolds satisfy Ric = λg with λ a constant (non-zero cosmological constant). One would like to obtain any HK as a limit of quaternionic spaces with λ → 0. \\n\\nExample: R4 is the limit of HP 1 (positive λ's) and HH 1 (negative λ's). \\n\\n2 March 18, 2004 - S. Cherkis as moderator. \\n\\nQuestion (R. Mazzeo): \\n\\nWhat are the asymptotics at infinity of the multi-monopole moduli spaces? \\n\\nAnswer: (Cherkis) They should behave like a N-body problem. \\n\\nQuestions (K. Lee) \\n\\nFor these questions, R4 \\n\\n> B\\n\\ndenotes the non-commutative R4. Here B = Bμν dx μdx ν is a 2-form which gives the non-commutative structure. Then, [ xμ, x ν ] = Bμν. We define non-commutative instantons using the Nahm transform of instanton data \\n\\nμ1 = [ T1, T 2] − [T3, T 4] − j∗j = B12,\\n\\nplus two other similar equations (see the work Nekrasov-Schwarz on non-commutative instantons [9]). Let \\n\\nMk,n B be the moduli space of centered k-instantons on R4 \\n\\n> B\\n\\nwith gauge group G = U (n). (The fact that the instantons are centered means we consider only instantons with center of mass at the origin) 1. What is the instanton number in the non-commutative set-up? \\n\\nMotivation: The space Mk,n B is singular for B = 0 but non-singular for B 6 = 0. One would like to understand the limit as B → 0. 2. Does there exists a unique harmonic form in the middle dimension of Mk,n B?\\n\\nNote: The answer to this question is Yes for the following cases: (i) 1-instantons and any G;(ii) 2-instantons and G = U (1) (see the work of Lee et al). 33. Conjecture: The moduli space of centered 1-instantons on R3 × S1 \\n\\n> B\\n\\nwith \\\"given holonomy around S1\\n\\n> ∞\\n\\n\\\"(holonomy is not in the center of G = U (n)) has exactly n L 2-harmonic forms in middle dimension.\"\nOriginal remarks: [\"Remarks: \\n\\n(i) There exists a physics proof. (ii) For n = 2, this is a special case and the conjectured has been verified. 4. The \\\"massless monopoles\\\" are defined as solutions to the Bogomolny equation: \\n\\nF = ∗∇ φ, \\n\\nplus an auxiliary condition. We think of Φ as a section of the bundle of endomorphisms of the Lie algebra of G. Let ( λ1,..., λ n) be the limit of the eigenvalues of φ as r → ∞. For the massless monopole condition we need λi = λj for some i 6 = j.\\n\\nQuestion: When does the moduli space of massless monopoles have harmonic forms in middle dimension? \\n\\nAnswer: Sergey Cherkis seems to have an answer using a physics argument. \\n\\nConjecture: When all the λi's are equal the moduli space is empty. \\n\\nQuestion (M. Singer): \\n\\nIs the space of all L2-harmonic forms on a manifold M with non-negative sectional curvature finite dimen-sional? \\n\\nHints: \\n\\n(i) By Cheeger-Gromoll, these spaces have finite topology. (ii) Moreover, by the solution of the Soul Conjecture such manifolds are diffeomorphic to normal bundles to totally geodesic compact submanifolds in M.(iii) If Ric ≥ 0 then there exist no L2-harmonic 1-forms (Bochner type argument). \\n\\nQuestion: \\n\\nIntroduction: When (Σ, g ) is a Riemannian surface (possibly with boundary) then a result of Duistermaat-Guillemin [5] gives a relation between the length spectrum {li} of Σ (that is, the lengths of all closed geodesics of Σ) and the spectrum of the Laplacian {λi} on C∞(Σ). Define the (trace of the) wave kernel by \\n\\nZ(t) = \\n\\n> ∞\\n\\n∑\\n\\n> i=0\\n\\ne±t√λi.\\n\\nGenerically, \\n\\nZ(t) ∼ ∑ di\\n\\nt − li.\\n\\nThe wave kernel associated to the spectrum of the eigenvalues of ∆ is a distribution whose singular support must be contained within some set related to the length spectrum?? \\n\\nQuestion: Give a more precise statement of the relationship? \\n\\nQuestions: (S. Cherkis)\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/l2harmonic/l2harmonic.pdf", "tags": [ "aim", "AIM-PHYSICS-0031", "aim-domain:physics", "aim-workshop:l2harmonic", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a product M=X^4 x Y^4 of complete hyperkähler four-manifolds satisfying explicit integrable heat-tail and no-zero-mode hypotheses, the synchronized heat-time integral G_M((x,y),(x',y'))=integral_0^infinity h_X(t,x,x')h_Y(t,y,y') dt is a positive scalar Green kernel. It has the normalized diagonal pole (2 pi^4)^(-1) rho^(-6)+O(rho^(-4)), and is exactly (2 pi^4)^(-1)(|x-x'|^2+|y-y'|^2)^(-3) on R^4 x R^4. By contrast, multiplying factor Green functions gives delta_{x'}G_Y+G_X delta_{y'}, with extended slice support and the wrong homogeneity, so the synchronized heat parameter is essential. This proves a reducible eight-dimensional special case but not a general twistor/Serre-class construction.\n\nCandidate contribution (special_case_and_obstruction; novelty confidence low): Under stated heat-tail hypotheses, synchronized factor heat kernels give a normalized Green kernel on reducible eight-dimensional hyperkähler products, while an exact distributional identity and scaling check rule out the naive product of four-dimensional Green kernels." }, { "id": 20002524, "problem_number": "AIM-PHYSICS-0032", "title": "Audible primitive two-torus areas on a flat three-torus", "statement": "1. When M is a 3-manifold are there some 2-dimensional objects that are related to Spec(∆) in a similar way? 2. Does there exist some analogue for the \"area spectrum\" of minimal surfaces in M 3 (with some suitable geometric assumptions imposed on M 3)? 3. Compute the number of minimal surfaces N (n1, n 2, g ) in T 4 = T 2 × T 2 endowed with the flat metric which represent the homology class n1[T1] + n2[T2], where Ti are the two 2-tori factors of T 4.", "original_statement": "1. When M is a 3-manifold are there some 2-dimensional objects that are related to Spec(∆) in a similar way? 2. Does there exist some analogue for the \"area spectrum\" of minimal surfaces in M 3 (with some suitable geometric assumptions imposed on M 3)? 3. Compute the number of minimal surfaces N (n1, n 2, g ) in T 4 = T 2 × T 2 endowed with the flat metric which represent the homology class n1[T1] + n2[T2], where Ti are the two 2-tori factors of T 4.", "clean_statement": "1. When M is a 3-manifold are there some 2-dimensional objects that are related to Spec(∆) in a similar way? 2. Does there exist some analogue for the \"area spectrum\" of minimal surfaces in M 3 (with some suitable geometric assumptions imposed on M 3)? 3. Compute the number of minimal surfaces N (n1, n 2, g ) in T 4 = T 2 × T 2 endowed with the flat metric which represent the homology class n1[T1] + n2[T2], where Ti are the two 2-tori factors of T 4.", "statement_status": "exact", "statement_verification": "The exact canonical `problem` field contains three numbered questions. This attempt owns only the first:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: $L^2$ harmonic forms in geometry and string theory\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/l2harmonic/l2harmonic.pdf\nCanonical location: aim-physics-notes.json notes[31]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. When M is a 3-manifold are there some 2-dimensional objects that are related to Spec(∆) in a similar way? 2. Does there exist some analogue for the \\\"area spectrum\\\" of minimal surfaces in M 3 (with some suitable geometric assumptions imposed on M 3)? 3. Compute the number of minimal surfaces N (n1, n 2, g ) in T 4 = T 2 × T 2 endowed with the flat metric which represent the homology class n1[T1] + n2[T2], where Ti are the two 2-tori factors of T 4.\"\nOriginal remarks: [\"Remarks: \\n\\n(i) (Cherkis) There is some physical intuition for these kind of relationships. (ii)(Mazzeo) Closed totally geodesic surfaces are not the correct 2-dimensional objects to consider for #1, since there are too few of them. 4(iii) (Mazzeo) There is some version of the Selberg trace formula which applies here. (iv) (Mazzeo) Would also be interesting to look at the eta-invariant. \\n\\nQuestions (M. Jardim): \\n\\nLet M denote the moduli space of k SU( n) (commutative) instantons on R4. Take the Dirac operator DA\\n\\ncoupled with the instanton. Then we have ker DA = (0). Let λ = min Spec DA. λ is a function on M.The questions arrising are: 1. What can we say about the function λ? Is is bounded? \\n\\nMotivation: this is the lightest quark we can have. 2. Same question as above but for instantons on T 4 instead of R4.\\n\\n3 Wrap-up Session, March 21 2004. R. Mazzeo as moderator. \\n\\nQuestion (T. Hausel) \\n\\nMotto: \\\"toric\\\" hyperK¨ ahler quotients are perhaps the simplest examples of complete hyperK¨ ahler metrics. \\n\\nProblem: Understand the Hodge cohomology of these spaces from a systematic point of view. \\n\\nPossible first steps:\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/l2harmonic/l2harmonic.pdf", "tags": [ "aim", "AIM-PHYSICS-0032", "aim-domain:physics", "aim-workshop:l2harmonic", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source question asks whether a three-manifold has two-dimensional objects related to the scalar Laplace spectrum analogously to the wave-trace/closed-geodesic relation. In general the Duistermaat--Guillemin relation remains dimension-independent and still singles out closed geodesics, not surfaces. For a flat three-torus T=R^3/Lambda, this attempt proves a positive special case: primitive dual covectors define connected totally geodesic two-torus fibrations, their fiber area is Vol(T)|xi|, and finite Mobius inversion of scalar eigenvalue multiplicities recovers their primitive norm counts even when primitive and imprimitive vectors have accidental spectral collisions. Thus the scalar spectrum determines the area multiset of unoriented parallel primitive fibration classes.\n\nCandidate contribution (explicit spectral multiplicity formula; novelty confidence low): If m_T(lambda) is the scalar multiplicity extended by zero off the spectrum, then one half of sum_{d>=1} mu(d)m_T(lambda/d^2) is exactly the number of unoriented primitive parallel two-torus fibration classes whose fibers have area Vol(T)sqrt(lambda)/(2pi)." }, { "id": 20002525, "problem_number": "AIM-PHYSICS-0033", "title": "Centered asymptotics for a Gibbons--Hawking subfamily", "statement": "1. Understand the asymptotics of these spaces (for a start see work of Bielawksi-Dancer [2] and Gibbons-Rychnenkova [6]).", "original_statement": "1. Understand the asymptotics of these spaces (for a start see work of Bielawksi-Dancer [2] and Gibbons-Rychnenkova [6]).", "clean_statement": "develop systematic asymptotic descriptions of toric hyperkähler (hypertoric) quotients, in service of their \\(L^2\\) Hodge theory. The problem ranges over dimensions and quotient data. This report proves a quantitative result only for the classical four-dimensional Gibbons--Hawking finite-centre subfamily.", "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: $L^2$ harmonic forms in geometry and string theory\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/l2harmonic/l2harmonic.pdf\nCanonical location: aim-physics-notes.json notes[32]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Understand the asymptotics of these spaces (for a start see work of Bielawksi-Dancer [2] and Gibbons-Rychnenkova [6]).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/l2harmonic/l2harmonic.pdf", "tags": [ "aim", "AIM-PHYSICS-0033", "aim-domain:physics", "aim-workshop:l2harmonic", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For positive integral finite-centre Gibbons--Hawking data, translation to the unique weighted centre removes the dipole. After identifying the nontrivial exterior circle bundles and choosing a decaying gauge, the connection correction is O(r^-3) with all derivatives. Consequently the full metric differs from its centred one-pole model by O(r^-3) in the ALF regime and by O(rho^-4) in the ALE cone regime, improving the respective uncentred orders O(r^-2) and O(rho^-2); the leading surviving scalar term is an explicit quadrupole. This is a proved four-dimensional special case, not a solution for general hypertoric quotients.\n\nCandidate contribution (lemma; novelty confidence low): The unique weighted-centre normalization, propagated through a topology-respecting exterior connection gauge, improves the full multi-centre Gibbons--Hawking metric decay from ALF order r^-2 to r^-3 and from ALE cone order rho^-2 to rho^-4 at every derivative order, with an explicit trace-free quadrupole as the first surviving scalar coefficient." }, { "id": 20002526, "problem_number": "AIM-PHYSICS-0034", "title": "A quantified Gibbons-Hawking N-body cluster model", "statement": "2. (Mazzeo) Combine asymptotics with N -body techniques.", "original_statement": "2. (Mazzeo) Combine asymptotics with N -body techniques.", "clean_statement": "2. (Mazzeo) Combine asymptotics with N -body techniques.", "statement_status": "exact", "statement_verification": "The raw canonical record is preserved in `input.json` and reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: $L^2$ harmonic forms in geometry and string theory\nSection: \nSource item: 2\nSource URL: https://aimath.org/WWN/l2harmonic/l2harmonic.pdf\nCanonical location: aim-physics-notes.json notes[33]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"2. (Mazzeo) Combine asymptotics with N -body techniques.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/l2harmonic/l2harmonic.pdf", "tags": [ "aim", "AIM-PHYSICS-0034", "aim-domain:physics", "aim-workshop:l2harmonic", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a positive multi-center Gibbons-Hawking potential, a selected cluster admits two complementary, uniform expansions. In a ball whose radius is smaller than the separation L from external centers, the external potential is its constant, linear-field, and trace-free tidal terms plus a remainder satisfying |nabla^ell R| <= C_ell M_D |z|^(3-ell)/(L-|z|)^4 for ell=0,1,2. At distances r larger than the internal cluster radius s, the weighted barycenter cancels the cluster dipole exactly, leaving one effective monopole, an explicit quadrupole, and |nabla^ell E| <= C'_ell M_C s^3/(r-s)^(4+ell). A radial gauge also gives explicit connection and coefficientwise metric interaction bounds. This is a rigorous rank-one N-body building block, not a Hodge-cohomology theorem or a full many-body compactification.\n\nCandidate contribution (quantitative_cluster_lemma; novelty confidence low): The near-zone C^2 third-order remainder, far-zone C^2 barycentric multipole remainder, and radial-gauge connection/metric bounds form a single uniform cluster chart for positive Gibbons-Hawking metrics, with constants independent of the number and placement of centers except through the displayed masses, radii, and separations." }, { "id": 20002527, "problem_number": "AIM-PHYSICS-0035", "title": "The L2-signature of toric hyperkahler quotients", "statement": "3. (Hausel) We know that all the L2 Hodge cohomology of these spaces must be in the middle dimension and that any L2-harmonic form in middle dimension is ± self-dual. So it is enough to understand the\n\nL2-signature. This will have consequences for the sign of σL2 (X) and is related to combinatorics and matroids.\n\nMore general problem: Prove a suitable index theorem on such manifolds. Will there be contributions from subsytems??\n\nQuestions (G. Etesi):", "original_statement": "3. (Hausel) We know that all the L2 Hodge cohomology of these spaces must be in the middle dimension and that any L2-harmonic form in middle dimension is ± self-dual. So it is enough to understand the \n\nL2-signature. This will have consequences for the sign of σL2 (X) and is related to combinatorics and matroids. \n\nMore general problem: Prove a suitable index theorem on such manifolds. Will there be contributions from subsytems?? \n\nQuestions (G. Etesi):", "clean_statement": "3. (Hausel) We know that all the L2 Hodge cohomology of these spaces must be in the middle dimension and that any L2-harmonic form in middle dimension is ± self-dual. So it is enough to understand the\n\nL2-signature. This will have consequences for the sign of σL2 (X) and is related to combinatorics and matroids.\n\nMore general problem: Prove a suitable index theorem on such manifolds. Will there be contributions from subsytems??\n\nQuestions (G. Etesi):", "statement_status": "exact", "statement_verification": "The canonical record is item 3 in T. Hausel's question in the wrap-up session of the 2004 AIM workshop *\\(L^2\\) harmonic forms in geometry and string theory*. The preceding lines are necessary to resolve the phrase “these spaces.” They say that “toric” hyperkähler quotients are among the simplest complete hyperkähler metrics and ask for a systematic understanding of their Hodge cohomology. The recovered item is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: $L^2$ harmonic forms in geometry and string theory\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/l2harmonic/l2harmonic.pdf\nCanonical location: aim-physics-notes.json notes[34]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"3. (Hausel) We know that all the L2 Hodge cohomology of these spaces must be in the middle dimension and that any L2-harmonic form in middle dimension is ± self-dual. So it is enough to understand the \\n\\nL2-signature. This will have consequences for the sign of σL2 (X) and is related to combinatorics and matroids. \\n\\nMore general problem: Prove a suitable index theorem on such manifolds. Will there be contributions from subsytems?? \\n\\nQuestions (G. Etesi):\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/l2harmonic/l2harmonic.pdf", "tags": [ "aim", "AIM-PHYSICS-0035", "aim-domain:physics", "aim-workshop:l2harmonic", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The broad AIM item has strong but hypothesis-dependent answers: Hitchin fixes the common middle-degree Hodge-star sign under a linear-growth-primitive hypothesis, Hausel--Swartz prove matroidal definiteness of the compactly supported middle intersection form, and Hausel--Hunsicker--Mazzeo give L2/intersection-cohomology signature formulas for specified fibred ends. A new diagnostic is proved: if every middle L2-harmonic form is literally self-dual or anti-self-dual, linearity forces one common sign; an orientation-reversing isometry exchanges the two sign spaces, so under the common-sign hypothesis the entire middle L2-harmonic space must vanish. Thus any combinatorially certified nonzero L2 class obstructs orientation reversal when the cited analytic hypotheses apply.\n\nCandidate contribution (symmetry_obstruction; novelty confidence low): On a complete oriented 4m-manifold with a finite-dimensional middle L2-harmonic space, literal elementwise plus-or-minus self-duality forces a common Hodge-star sign; if an orientation-reversing isometry preserves the closed L2 de Rham domain, that harmonic space is zero. Consequently a nonzero Hausel--Swartz class with a verified L2 realization obstructs orientation-reversing isometries in the Hitchin class." }, { "id": 20002528, "problem_number": "AIM-PHYSICS-0036", "title": "A smoothness criterion for projective conic compactifications of ALF twistor models", "statement": "1. It is known that there exists a good compactification of ALF spaces for L2-cohomology. What about its twistor space Q?(a) Does it compactify? (b) Does the complex structure extend over the compactification? 2. Study the Yang-Mills functional on open manifolds. For example, for manifolds which are conformally equivalent to a manifold with one cylindrical end, the energy of Yang-Mills instantons (if finite) is con-gruent mod Z to one of the Chern-Simons invariants of the boundary. What about different asymptotic geometries (ALF spaces, etc)?", "original_statement": "1. It is known that there exists a good compactification of ALF spaces for L2-cohomology. What about its twistor space Q?(a) Does it compactify? (b) Does the complex structure extend over the compactification? 2. Study the Yang-Mills functional on open manifolds. For example, for manifolds which are conformally equivalent to a manifold with one cylindrical end, the energy of Yang-Mills instantons (if finite) is con-gruent mod Z to one of the Chern-Simons invariants of the boundary. What about different asymptotic geometries (ALF spaces, etc)?", "clean_statement": "1. It is known that there exists a good compactification of ALF spaces for L2-cohomology. What about its twistor space Q?(a) Does it compactify? (b) Does the complex structure extend over the compactification? 2. Study the Yang-Mills functional on open manifolds. For example, for manifolds which are conformally equivalent to a manifold with one cylindrical end, the energy of Yang-Mills instantons (if finite) is con-gruent mod Z to one of the Chern-Simons invariants of the boundary. What about different asymptotic geometries (ALF spaces, etc)?", "statement_status": "exact", "statement_verification": "The canonical JSON record is contaminated by the next numbered item. Inspection of page 4 of the original AIM workshop PDF gives the complete relevant text as follows (line wrapping and the typography of \\(L^2\\) are normalized, but the symbol \\(Q\\) is preserved):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: $L^2$ harmonic forms in geometry and string theory\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/l2harmonic/l2harmonic.pdf\nCanonical location: aim-physics-notes.json notes[35]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. It is known that there exists a good compactification of ALF spaces for L2-cohomology. What about its twistor space Q?(a) Does it compactify? (b) Does the complex structure extend over the compactification? 2. Study the Yang-Mills functional on open manifolds. For example, for manifolds which are conformally equivalent to a manifold with one cylindrical end, the energy of Yang-Mills instantons (if finite) is con-gruent mod Z to one of the Chern-Simons invariants of the boundary. What about different asymptotic geometries (ALF spaces, etc)?\"\nOriginal remarks: [\"Remarks: \\n\\n(i) Without smoothness and irreducibility assumptions for the YM instantons, the energy cannot be quan-tized by the Chern-Simons invariants. (ii) It is conjectured that the Chern-Simons invariants of a compact 3-manifold are rational, and therefore the same is conjectured for finite energy YM instantons on manifolds with one cylindrical end. If one considers finite energy Yang-Mills instantons on manifolds with different asymptotic geometeries is the YM action still quantized? \\n\\nQuestions (E. Hunsicker):\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/l2harmonic/l2harmonic.pdf", "tags": [ "aim", "AIM-PHYSICS-0036", "aim-domain:physics", "aim-workshop:l2harmonic", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The printed AIM question has an affirmative literature answer for every multi-Taub-NUT space: Etesi-Szabo prove that its twistor space admits a smooth complex compactification. For the explicit conic model Z(q)={xy=q}, the homogeneous closure {xy=qw^2} is a projective conic fibration whose boundary is two disjoint copies of the base and whose exact total-space singular locus is {(b,[0:0:1]):q(b)=0 and dq_b=0}. Applied to one-centre Taub-NUT, the tautological discriminant is transverse, so the fibrewise compactification is smooth and the complex structure extends. Because the minitwistor base is noncompact, this relative construction is not by itself a compact total space; no universal theorem for all ALF types was verified.\n\nCandidate contribution (compactification_criterion; novelty confidence low): For any holomorphic line-bundle conic xy=q, the homogeneous closure has exact singular locus {q=dq=0} at the unique node [0:0:1] of each degenerate fibre; its two added boundary sections are always smooth. This gives a sharp extension/obstruction test and cleanly separates fibrewise projectivity from total compactness in the Taub-NUT twistor model." }, { "id": 20002529, "problem_number": "AIM-PHYSICS-0037", "title": "Topology and character sectors in monopole L2-cohomology", "statement": "1. Is there a topological interpretation for the L2-cohomology of higher dimensional monopole modulii spaces? (i) What are the right tools to use here? (ii) Can we use Saper's L-module techniques? (see [11, Section 12]) 2. These questions involve complete manifolds with special holonomy. (i) Understand asymptotic geometry of these manifolds. 5(ii) Understand their L2-cohomology. (iii) Find general vanishing theorems (possibly even if we do not understand part 1 completely).", "original_statement": "1. Is there a topological interpretation for the L2-cohomology of higher dimensional monopole modulii spaces? (i) What are the right tools to use here? (ii) Can we use Saper's L-module techniques? (see [11, Section 12]) 2. These questions involve complete manifolds with special holonomy. (i) Understand asymptotic geometry of these manifolds. 5(ii) Understand their L2-cohomology. (iii) Find general vanishing theorems (possibly even if we do not understand part 1 completely).", "clean_statement": "1. Is there a topological interpretation for the L2-cohomology of higher dimensional monopole modulii spaces? (i) What are the right tools to use here? (ii) Can we use Saper's L-module techniques? (see [11, Section 12]) 2. These questions involve complete manifolds with special holonomy. (i) Understand asymptotic geometry of these manifolds. 5(ii) Understand their L2-cohomology. (iii) Find general vanishing theorems (possibly even if we do not understand part 1 completely).", "statement_status": "exact", "statement_verification": "The original AIM PDF places this record under the heading **“Questions (E. Hunsicker)”**. The genuine item 1 is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: $L^2$ harmonic forms in geometry and string theory\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/l2harmonic/l2harmonic.pdf\nCanonical location: aim-physics-notes.json notes[36]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"1. Is there a topological interpretation for the L2-cohomology of higher dimensional monopole modulii spaces? (i) What are the right tools to use here? (ii) Can we use Saper's L-module techniques? (see [11, Section 12]) 2. These questions involve complete manifolds with special holonomy. (i) Understand asymptotic geometry of these manifolds. 5(ii) Understand their L2-cohomology. (iii) Find general vanishing theorems (possibly even if we do not understand part 1 completely).\"\nOriginal remarks: [\"Remarks: \\n\\n(a) Some G 2 metrics fiber over S3 with hyperK¨ ahler fibres (although some fibres can degenerate). This may be a good place to test whether L-module techniques are appropriate. (b) In the case of SU(2)-monopole moduli space over R3, the Segal-Selby paper [12] proves (by pure topology) that there is a harmonic form (coming from the topology of the moduli space) as predicted by Sen's conjecture. What remains is to prove that these are the only L2-harmonic forms (surjectivity of the map in the Selby-Segal construction). \\n\\nQuestion: Study extended L2-harmonic forms on noncompact spaces. \\n\\nDefinition: a form ω is extended L2-harmonic if (d + d∗)ω = 0, ω = lim ωj locally, where ωj ∈ C∞ \\n\\n> 0,\\n\\nand (d + d∗)ωj → 0 in L2.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/l2harmonic/l2harmonic.pdf", "tags": [ "aim", "AIM-PHYSICS-0037", "aim-domain:physics", "aim-workshop:l2harmonic", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The recovered Hunsicker problem concerns high-dimensional moduli spaces of ordinary SU(2) monopoles on R3, not the merged special-holonomy or extended-harmonic-form questions. Later work gives a topological interpretation in all coprime electric sectors of the strongly centred charge-k cover via the image of compactly supported in ordinary cohomology, subject to the Fritzsch--Kottke--Singer compactification theorem whose complete proof was deferred; Kottke--Rochon prove the full charge-three Sen statement provided the announced QFB metric hypothesis. The successful verified tools are many-body/fibered-corner compactification and QFB microlocal Hodge theory, not Saper L-modules. A proved finite-cover theorem identifies twisted harmonic forms downstairs with representation multiplicities upstairs, supplies the exact character projector, and reduces each non-coprime cyclic sector to a faithful character on an intermediate quotient.\n\nCandidate contribution (finite_cover_character_diagnostic; novelty confidence low): For a finite regular complete Riemannian cover with right deck group G and irreducible unitary representation rho, H^q_(2)(X;E_{rho*}) is naturally Hom_G(V_rho,H^q_(2)(X_tilde)); evaluation reconstructs the rho-isotypic summand and the central character average is its orthogonal projector. For a cyclic character chi, the upstairs chi-sector is the conjugate-line twisted problem downstairs and, after quotienting by ker chi, becomes a faithful residual-character sector; for monopole electric charge ell the kernel has order gcd(k,ell)." }, { "id": 20002530, "problem_number": "AIM-PHYSICS-0038", "title": "The signed Hill threshold for binary exchange", "statement": "Problem 1. Consider the three-body problem. Let E be the energy, C be the angular momentum, and L = C2E for circular Euler motion. An exchange orbit is an unbounded solution with the property that the distance between one mass m3 and the binary {m1, m 2}\n\ngoes to infinity as t → −∞, m3 exchange with m2 and the distance between m2 and the the binary {m1, m 3} goes to infinity as t → +∞. A solution has Hill type stability if the three bodies cannot undergo an exchange of binary. For exchange orbits, how close to 1 can C2E\n\n> L\n\nbe? (Christian Marchal)", "original_statement": "Problem 1. Consider the three-body problem. Let E be the energy, C be the angular momentum, and L = C2E for circular Euler motion. An exchange orbit is an unbounded solution with the property that the distance between one mass m3 and the binary {m1, m 2}\n\ngoes to infinity as t → −∞, m3 exchange with m2 and the distance between m2 and the the binary {m1, m 3} goes to infinity as t → +∞. A solution has Hill type stability if the three bodies cannot undergo an exchange of binary. For exchange orbits, how close to 1 can C2E\n\n> L\n\nbe? (Christian Marchal)", "clean_statement": "Problem 1. Consider the three-body problem. Let E be the energy, C be the angular momentum, and L = C2E for circular Euler motion. An exchange orbit is an unbounded solution with the property that the distance between one mass m3 and the binary {m1, m 2}\n\ngoes to infinity as t → −∞, m3 exchange with m2 and the distance between m2 and the the binary {m1, m 3} goes to infinity as t → +∞. A solution has Hill type stability if the three bodies cannot undergo an exchange of binary. For exchange orbits, how close to 1 can C2E\n\n> L\n\nbe? (Christian Marchal)", "statement_status": "exact", "statement_verification": "The canonical record is visibly flattened. I checked the compressed text stream of the official AIM PDF itself, not only the extracted JSON. The mathematical content on page 3 is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 1\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[37]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 1. Consider the three-body problem. Let E be the energy, C be the angular momentum, and L = C2E for circular Euler motion. An exchange orbit is an unbounded solution with the property that the distance between one mass m3 and the binary {m1, m 2}\\n\\ngoes to infinity as t → −∞, m3 exchange with m2 and the distance between m2 and the the binary {m1, m 3} goes to infinity as t → +∞. A solution has Hill type stability if the three bodies cannot undergo an exchange of binary. For exchange orbits, how close to 1 can C2E\\n\\n> L\\n\\nbe? (Christian Marchal)\"\nOriginal remarks: [\"Remarks. \\n\\n• Known: E < 0, C large, then C2E < L < 0. \\n\\n• The orbit comes from unstable manifold of E3 (Euler configuration) to the stable manifold of E2. There is a normally hyperbolic invariant 3-sphere S3Euler with W s,\\n\\nW u in dimension 5, and at infinity ( P12 and P13 ) of the Hill's region there are two other invariant 3-spheres S3 \\n\\n> P12, S3 \\n\\n> P13\\n\\nwith 4-dimensional W s, W u. Can we create chain between P12, P13 via S3Euler? (Rick Moeckel) \\n\\n• Suggestion: Consider first the intersection of W u, W s of the different objects. This could be tested numerically.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0038", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the planar Newtonian three-body problem, the sharp Sundman decomposition reduces configuration accessibility at fixed negative energy E and angular momentum C to Phi(s) >= J, where Phi=I U^2 and J=-2EC^2. At an Euler central shape e, the circular value is L_e=-Phi_e/2, so the signed workshop ratio is C^2E/L_e=J/Phi_e. Hence the source's inequality C^2E invariant set\\n\\nFigure 1. Transit orbit\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0039", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The universal all-barbell question remains unverified, but Conley's small-neck theorem and Moeckel's 2005 variational theorem establish substantial special cases. Building on that method, a collision-free rectangular corridor admits an energy-h transit whenever its left, right, and bottom walls satisfy explicit two-sided flow-convexity inequalities (in particular V_y<-2 rho on the bottom), the magnetic Maupertuis integrand can be made positive by an exact gauge change, and one comparison path satisfies the scalar gap Jhat(gamma_0) < m|P-Q^T|-2 area(R). The proof uses the exact reflected distance to the unprotected wall and the flux identity d(x dy-y dx)=2 dx wedge dy. Separately, every regular zero-velocity boundary arc collapses at rate O(sqrt(epsilon)) in Jacobi length, rigorously explaining why connectedness of the barbell alone does not yield a transit by direct minimization.\n\nCandidate contribution (quantitative_variational_certificate; novelty confidence low): For a PCR3BP rectangular corridor satisfying explicit two-sided left/right/bottom flow-convexity, including V_y<-2 rho on the bottom for both signs of tangential velocity, and the stated positivity condition, the gauge-invariant inequality Jhat(gamma_0) < m|P-Q^T|-2 area(R) forces a left-to-right transit; paired with this, regular Hill-boundary arcs have O(sqrt(epsilon)) collapsing Jacobi distance." }, { "id": 20002532, "problem_number": "AIM-PHYSICS-0040", "title": "The four-body super-eight and the rotating-square obstruction", "statement": "Problem 3. Prove the existence of super-eight solution for the 4-body problem with equal masses. (Joseph Gerver)", "original_statement": "Problem 3. Prove the existence of super-eight solution for the 4-body problem with equal masses. (Joseph Gerver)", "clean_statement": "Problem 3. Prove the existence of super-eight solution for the 4-body problem with equal masses. (Joseph Gerver)", "statement_status": "exact", "statement_verification": "The source-verified AIM item is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 3\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[39]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 3. Prove the existence of super-eight solution for the 4-body problem with equal masses. (Joseph Gerver)\"\nOriginal remarks: [\"Remarks. \\n\\n• One difficulty: the square relative equilibrium is absolute minimum in that symmetry class. 4\\n\\n• There is a computer assisted proof by interval arithmetic. Poincar´ e characterize homology class in the planar three-body problem by three inte-gers kij = Deg (xi − xj, 0); that is, the number of oriented turns of the side xi − xj.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0040", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM item is solved: Kapela and Zgliczynski gave a rigorous computer-assisted interval proof in 2003, and Shibayama gave a collisionless analytic variational proof in 2014. Source recovery also shows that the stray numeral 4 is a page number and the Poincare winding-number sentence belongs to Problem 4. As a proved structural contribution, every antipodal simple four-body choreography has an exact one-generator action with four adjacent-pair terms and one combined opposite-pair term; antiperiodicity makes this functional H1-coercive, its dilation law has a unique optimal scale, and the circular generator yields the normalized rotating-square competitor. This pinpoints why naive coercive minimization does not produce the super-eight and why a separate sign component plus collision exclusion is essential.\n\nCandidate contribution (reduced_action_obstruction; novelty confidence low): For an antiperiodic generator x(t+pi)=-x(t) of a four-body simple choreography, the full Newtonian action is exactly the integral of 2|x'|^2 + 4/|x(t)-x(t+pi/2)| + 1/|x(t)|; it controls the full H1 norm, has the exact scale law F(lambda x)=lambda^2 K+lambda^{-1}U with lambda^3=U/(2K), and gives r^3=(2 sqrt(2)+1)/4 for the optimally scaled circular rotating-square generator. The two denominators also exhaust the adjacent- and opposite-pair collision channels." }, { "id": 20002533, "problem_number": "AIM-PHYSICS-0041", "title": "Fixed winding numbers: exact pair bounds, graph coercivity, and collision loss", "statement": "Problem\n4. (Due to Poincar´ e, 1896) Can we obtain periodic solutions by minimizing while fixing the kij 's? (Alain Chenciner)", "original_statement": "Problem \n4. (Due to Poincar´ e, 1896) Can we obtain periodic solutions by minimizing while fixing the kij 's? (Alain Chenciner)", "clean_statement": "Problem\n4. (Due to Poincar´ e, 1896) Can we obtain periodic solutions by minimizing while fixing the kij 's? (Alain Chenciner)", "statement_status": "exact", "statement_verification": "The canonical JSON record is visibly missing the sentence that defines its notation. I checked the immediately preceding paragraph on page 4 of the official AIM workshop PDF. The printed text reads “Poincare characterize homology class ...”; in normalized English, its mathematical content is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 4\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[40]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem \\n4. (Due to Poincar´ e, 1896) Can we obtain periodic solutions by minimizing while fixing the kij 's? (Alain Chenciner)\"\nOriginal remarks: [\"Remarks. \\n\\n• Answers known when kij are all nonzero. When ( k12, k 23, k 31 ) = (1, 1, 1) or ( −1, −1, −1), minimizers are Lagrange's equilateral solutions; otherwise, minimizers have collisions. \\n\\n• In choreographies, kij are all the same. \\n\\n• Question: Is the answer the same in the restricted case as well? (Richard Mont-gomery)\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0041", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For fixed-period, center-of-mass-zero planar Newtonian three-body loops, the action decomposes into three mass-weighted Kepler actions and obeys the sharp Gordon pair lower bound on every nonzero winding edge. The action sublevels in a fixed homology class are H1-bounded if and only if the graph of nonzero pair windings is connected. Connectedness gives compactness of minimizing sequences only up to collision: a local k-turn winding bubble at radius epsilon has action O(sqrt(epsilon)), so weak closure can erase homology at a binary collision. In the massless restricted limit, all full-action control involving the test body degenerates.\n\nCandidate contribution (theorem; novelty confidence low): For the planar three-body action at fixed period and center of mass, fixed-homology action sublevels are H1-bounded exactly when the graph with edge ij for k_ij nonzero is connected; disconnected classes have an explicit cluster-escape family, while connected classes are compact up to collision." }, { "id": 20002534, "problem_number": "AIM-PHYSICS-0042", "title": "Homology-only escape and a strict zero-winding choreography gap", "statement": "Problem 5. Is the figure-8 orbit a minimizer with homology class (0, 0, 0)? (Alain Chenciner)", "original_statement": "Problem 5. Is the figure-8 orbit a minimizer with homology class (0, 0, 0)? (Alain Chenciner)", "clean_statement": "Problem 5. Is the figure-8 orbit a minimizer with homology class (0, 0, 0)? (Alain Chenciner)", "statement_status": "exact", "statement_verification": "The source-verified AIM item is exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 5\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[41]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 5. Is the figure-8 orbit a minimizer with homology class (0, 0, 0)? (Alain Chenciner)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0042", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The abbreviated homology-only formulation has infimum zero and no minimizer: centered constant configurations dilated by R have zero winding and action TU(a)/R tending to zero. For Chenciner's source-supported intended formulation, fixed-period centered three-body choreographies, the action is coercive and has the sharp Lagrange lower bound A_L(T)=(9/2)(2pi/sqrt(3))^(2/3)T^(1/3), attained only by the optimally scaled circular Lagrange choreographies of winding plus or minus one. The common-winding-zero infimum is therefore strictly greater than A_L(T), and a minimizing sequence can lose admissibility only through collision, not escape to infinity or convergence to Lagrange.\n\nCandidate contribution (formulation_counterexample_and_gap; novelty confidence low): The literal zero-homology loop problem admits the explicit action-to-zero escape family, while the centered choreography interpretation has a strict action gap between its zero-winding sector and the sharp Lagrange floor; consequently collision is the only loss-of-compactness mechanism for zero-winding minimizing sequences." }, { "id": 20002535, "problem_number": "AIM-PHYSICS-0043", "title": "Coercivity and collision boundaries for the Broucke-Henon class (1,0,1)", "statement": "Problem 6. Prove the existence of Broucke-H´ enon orbit for the planar 3-body problem. Do Broucke-H´ enon orbits exist for small + big masses? Is the Broucke-H´ enon solution a minimizer in the class (1, 0, 1)? (Andrea Venturelli)", "original_statement": "Problem 6. Prove the existence of Broucke-H´ enon orbit for the planar 3-body problem. Do Broucke-H´ enon orbits exist for small + big masses? Is the Broucke-H´ enon solution a minimizer in the class (1, 0, 1)? (Andrea Venturelli)", "clean_statement": "Problem 6. Prove the existence of Broucke-H´ enon orbit for the planar 3-body problem. Do Broucke-H´ enon orbits exist for small + big masses? Is the Broucke-H´ enon solution a minimizer in the class (1, 0, 1)? (Andrea Venturelli)", "statement_status": "exact", "statement_verification": "The official AIM workshop PDF, *Variational Methods in Celestial Mechanics*, gives the following problem (Problem 6):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 6\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[42]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 6. Prove the existence of Broucke-H´ enon orbit for the planar 3-body problem. Do Broucke-H´ enon orbits exist for small + big masses? Is the Broucke-H´ enon solution a minimizer in the class (1, 0, 1)? (Andrea Venturelli)\"\nOriginal remarks: [\"Remarks. \\n\\n• Numerically, Broucke-H´ enon solution exists for equal masses, at least. \\n\\n• The solution seems to be stable numerically for equal masses. \\n\\n• Solution has symmetry group D2. Just impose D2-symmetry, the action is not coer-cive, so need to impose topological constraints. \\n\\n• Numerically it is a local minimum of A.\\n\\n• Shubart orbit with collision is on the closure of the homology class (1, 0, 1), and it has larger action. \\n\\nFigure 2. Broucke-H´ enon orbit\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0043", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For arbitrary fixed positive masses and fixed period, the centered collision-free three-body loop class with pairwise winding vector (1,0,1) is coercive in H1 and obeys the explicit action lower bound (3/2)(2 pi)^(2/3) T^(1/3) M^(-1/3)(m1 m2 + m3 m1). Any weak-limit loss of either nonzero winding must pass through the corresponding binary-collision stratum. If I_101 is the infimum over the original class, every minimizing sequence in that class has a subsequence converging to q* in its weak closure with A(q*) at most I_101. If a trial loop has action below the collisional-boundary infimum B_coll, then q* cannot be collisional, so q* belongs to the original class, has A(q*)=I_101, and is a collision-free classical minimizer. Branch identification remains a separate gap.\n\nCandidate contribution (lemma; novelty confidence low): The exact AIM homology class (1,0,1), for every positive mass triple and fixed period, has a mass-explicit Gordon action floor, is H1-coercive after center-of-mass reduction, and has a weak-closure defect localized to specified binary-collision strata, yielding a strict collision-barrier criterion." }, { "id": 20002536, "problem_number": "AIM-PHYSICS-0044", "title": "Coercivity, collision support, and the identity braid for the class (1,0,-1)", "statement": "Problem 7. Is (1, 0, −1) realized by a collision-free periodic orbit (see figure 3)? If exists, is it minimizing or not? Is trivial braid realized by a collision-free periodic solution? (Andrea Venturelli)", "original_statement": "Problem 7. Is (1, 0, −1) realized by a collision-free periodic orbit (see figure 3)? If exists, is it minimizing or not? Is trivial braid realized by a collision-free periodic solution? (Andrea Venturelli)", "clean_statement": "Problem 7. Is (1, 0, −1) realized by a collision-free periodic orbit (see figure 3)? If exists, is it minimizing or not? Is trivial braid realized by a collision-free periodic solution? (Andrea Venturelli)", "statement_status": "exact", "statement_verification": "The official AIM workshop PDF gives the needed definition immediately before Problems 4--7:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 7\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[43]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 7. Is (1, 0, −1) realized by a collision-free periodic orbit (see figure 3)? If exists, is it minimizing or not? Is trivial braid realized by a collision-free periodic solution? (Andrea Venturelli)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0044", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For arbitrary fixed positive masses and fixed period, the centered Newtonian loop class with pair windings (k12,k23,k31)=(1,0,-1) is H1-coercive because its nonzero-winding graph is connected, and every minimizing sequence is compact up to collision. Under uniform convergence a winding can fail only if its matching pair collides, so no single binary-collision type can absorb both nonzero entries; a trial action below the explicitly defined collisional-boundary infimum therefore forces a collision-free absolute minimizer. Separately, pair windings are only the abelianization of P3, while every exact collision-free periodic brake orbit is the identity full pure braid because its second half retraces its first. Modern free-fall catalogs thus give strong numerical identity-braid candidates, but not a computer-assisted existence proof.\n\nCandidate contribution (lemma; novelty confidence low): In the exact AIM class (1,0,-1), fixed-action families are H1-bounded and winding loss in a uniform weak limit is edge-local: the entries k12=1 and k31=-1 can fail only on their respective collision strata, so losing both requires both collision types or a triple collision; consequently a strict action gap below the full collisional boundary is sufficient for a collision-free absolute minimizer." }, { "id": 20002537, "problem_number": "AIM-PHYSICS-0045", "title": "Syzygy-word parity and the angular-momentum holonomy obstruction", "statement": "Problem 8. Can any symbol sequence be realized in the planar three-body problem? Are they minimizing or not? Do they have zero angular momentum? (Richard Montgomery)", "original_statement": "Problem 8. Can any symbol sequence be realized in the planar three-body problem? Are they minimizing or not? Do they have zero angular momentum? (Richard Montgomery)", "clean_statement": "Problem 8. Can any symbol sequence be realized in the planar three-body problem? Are they minimizing or not? Do they have zero angular momentum? (Richard Montgomery)", "statement_status": "exact", "statement_verification": "The source-verified AIM item reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 8\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[44]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 8. Can any symbol sequence be realized in the planar three-body problem? Are they minimizing or not? Do they have zero angular momentum? (Richard Montgomery)\"\nOriginal remarks: [\"Remarks. \\n\\n• \\\"Symbol sequence\\\" is given by the crossing of the equator in the shape space, count each time bodies as collinear with which body is in the middle. 5\\n\\nFigure 3. A possible solution with homology class (1, 0, −1) \\n\\n• Could proceed numerically by systematic search and see what sequences are excluded (if any).\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0045", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source question is only partially solved: Moeckel-Montgomery realize every reduced syzygy class at fixed negative energy for equal or near-equal masses and sufficiently small nonzero angular momentum, but zero momentum remains open and the solutions are only periodic modulo rotation. This attempt proves two exact necessary filters. A transverse closed shape loop has an even raw syzygy count with alternating crossing signs, and cyclic stutter reduction preserves parity. For any fixed closed reduced shape-size path and inertial closure sector n, minimizing over rotational lifts has vertical action c_n^2/(2H) and constant angular momentum J=c_n/H, where H is the integral of 1/I and c_n is the gauge-invariant mechanical-holonomy mismatch. Thus a reduced-periodic lift minimizes at J=0, while an inertially closed zero-J lift exists only when the holonomy is trivial.\n\nCandidate contribution (parity_holonomy_reduction; novelty confidence low): A periodic syzygy proposal must be expanded to an even true-period word, and a proposed inertially closed zero-angular-momentum lift must satisfy trivial mechanical holonomy; when holonomy fails, the exact forced angular momentum and action penalty are J=c_n/H and c_n^2/(2H)." }, { "id": 20002538, "problem_number": "AIM-PHYSICS-0046", "title": "When constrained minimizers gain symmetry", "statement": "Problem 9. Often the minimizers in a symmetry class has more symmetry that the class asks for. Is that a coincidence? (Alain Chenciner)", "original_statement": "Problem 9. Often the minimizers in a symmetry class has more symmetry that the class asks for. Is that a coincidence? (Alain Chenciner)", "clean_statement": "Problem 9. Often the minimizers in a symmetry class has more symmetry that the class asks for. Is that a coincidence? (Alain Chenciner)", "statement_status": "exact", "statement_verification": "The canonical record is Problem 9 in the AIM workshop notes *Variational Methods in Celestial Mechanics*, available at . The printed source reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 9\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[45]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 9. Often the minimizers in a symmetry class has more symmetry that the class asks for. Is that a coincidence? (Alain Chenciner)\"\nOriginal remarks: [\"Remarks. Some examples: \\n\\n• Kepler problem with Italian symmetry. Minimizers are circular; that is, with SO (2) symmetry. \\n\\n• Maybe the figure-8 orbit is a minimizer with Z6 and D3 symmetry (v.s. the D6\\n\\nsymmetry the figure-8 has) \\n\\n• In spatial case, can consider D6 symmetry in which the planar figure-8 would pre-sumably by a minimizer, then get Z2 symmetry for free. \\n\\n• Central configurations with four bodies has a line of symmetry. (Wu-Yi Hsiang) It is a phenomenon of optimality, not coincidence.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0046", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Symmetry enhancement is not automatic: the normalizer of the imposed symmetry preserves the minimizing set but may merely permute several minimizers. A rigorous normalizer-orbit theorem shows that enhancement is forced by uniqueness, discreteness under a connected normalizer, or strict isolation modulo phase; convex averaging supplies a second sufficient criterion. For the Italian-symmetry planar Kepler problem, sharp anti-periodic Wirtinger and Jensen inequalities completely determine all minimizers as uniformly parametrized circles of radius (mu T^2/(4 pi^2))^(1/3), with minimum action (3/2)(2 pi)^(2/3) mu^(2/3) T^(1/3).\n\nCandidate contribution (criterion_and_counterexample; novelty confidence low): Candidate novelty: the combined phase-quotient normalizer certificate, explicit finite counterexample, and sharp Italian-Kepler equality analysis give a testable diagnostic for this AIM question: a symmetry-gain claim must identify a rigidity mechanism that collapses the normalizer orbit, while in its absence normalizer elements can simply exchange distinct minima; in the Italian class the certificate yields exactly two oriented phase-orbits and diagonal space-time SO(2) isotropy." }, { "id": 20002539, "problem_number": "AIM-PHYSICS-0047", "title": "Quantitative simplex and dimension bounds for minimum-potential equal-mass configurations", "statement": "Problem 10. Is the central configuration for equal masses with minimal potential on the ellipsoid of moment of inertia 1 necessarily symmetric? (Rick Moeckel)", "original_statement": "Problem 10. Is the central configuration for equal masses with minimal potential on the ellipsoid of moment of inertia 1 necessarily symmetric? (Rick Moeckel)", "clean_statement": "Problem 10. Is the central configuration for equal masses with minimal potential on the ellipsoid of moment of inertia 1 necessarily symmetric? (Rick Moeckel)", "statement_status": "exact", "statement_verification": "Thus the number \\(47\\) is present in the source and is not an OCR error. The PDF is dated 2003 and attributes the notes to Kuo-Chang Chen. It gives no coordinates, proof, citation, ambient dimension, or definition of “symmetric,” “orbit,” or “normalized potential.” In particular, the \\(n=47\\) sentence is evidence of an informal computation or example known at the workshop, not a verifiable counterexample by itself.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 10\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[46]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 10. Is the central configuration for equal masses with minimal potential on the ellipsoid of moment of inertia 1 necessarily symmetric? (Rick Moeckel)\"\nOriginal remarks: [\"Remarks. Appears to be false for n = 47. Should require that the orbit is a minimizer of the normalized potential.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0047", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For n centered unit masses in R^d with moment I and P=n(n-1)/2, the Newtonian force function U is at least P^(3/2)/sqrt(nI) plus 3/[8(nI)^(5/2)] times the squared variance sum of the squared pair distances. The centered Gram matrix further gives a positive explicit dimension-dependent gap when d=n-1, equality uniquely yields the regular simplex and therefore proves symmetry. In fixed dimension the inequalities do not force symmetry; a generic relative rotation of a regular triangle and square attains the optimal planar Gram-variance conditions while having trivial orthogonal stabilizer. The AIM source's n=47 remark is exact source text but remains unverified because no coordinates, proof, or citation were supplied or located.\n\nCandidate contribution (theorem; novelty confidence low): The explicit strong-convexity deficit U-P^(3/2)/sqrt(nI) >= 3/[8(nI)^(5/2)] sum_{i= 2I^2(1/d-1/(n-1)) for d= 0, and the explicit family Q(t) = D R(sqrt(3)t) E is a collision-free three-body full-tensor stress test showing that vanishing internal defect need not imply Euclidean rigidity for a non-Newtonian harmonic force." }, { "id": 20002544, "problem_number": "AIM-PHYSICS-0052", "title": "An affine-defect test for Newtonian motions", "statement": "Problem 15. Are there solutions to the n-body problem that fall in a certain affine class that are not relative equilibrium or some symmetric solutions? (c.f. Gerver's super-8) (Alain Chenciner) 6", "original_statement": "Problem 15. Are there solutions to the n-body problem that fall in a certain affine class that are not relative equilibrium or some symmetric solutions? (c.f. Gerver's super-8) (Alain Chenciner) 6", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "This reconstruction is consistent with the super-eight comparison: the equal-mass four-body super-eight belongs to the centrally symmetric/parallelogram class. It is not verified as Chenciner's uniquely intended definition, so all conclusions below are conditional on this reading.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 15\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[51]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 15. Are there solutions to the n-body problem that fall in a certain affine class that are not relative equilibrium or some symmetric solutions? (c.f. Gerver's super-8) (Alain Chenciner) 6\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0052", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the explicitly stated reconstruction that a fixed affine class consists of labelled configurations Q=AX after center-of-mass reduction, Newtonian tangency is equivalent to the proved basis-independent condition F(AX)(I-P_X)=0. For four planar bodies with one nontrivial affine dependence lambda, this becomes an explicit pairwise vector obstruction and a differentiated velocity obstruction. The analysis also proves that simplex affine classes are locally vacuous and that the equal-mass parallelogram class is invariant and contains local nonhomographic but still symmetric motions. No nonsymmetric periodic orbit is claimed.\n\nCandidate contribution (reduction; novelty confidence low): The affine-defect tensor F(AX)(I-P_X), together with the explicit position and velocity compatibility equations for a planar four-body affine dependence, provides a finite rejection test for proposed nonsymmetric affine-class initial data and cleanly separates vacuous simplex classes from the invariant equal-mass parallelogram class." }, { "id": 20002545, "problem_number": "AIM-PHYSICS-0053", "title": "The syzygy-free three-body solution and quantitative crossing certificates", "statement": "Problem 16. Is the only solution satisfying (i), (ii) and having no syzygies (i.e. eclipses) the Lagrange homothety solutions? (i) Energy < 0. (ii) Angular momentum =\n0. (Richard Montgomery)", "original_statement": "Problem 16. Is the only solution satisfying (i), (ii) and having no syzygies (i.e. eclipses) the Lagrange homothety solutions? (i) Energy < 0. (ii) Angular momentum = \n0. (Richard Montgomery)", "clean_statement": "Problem 16. Is the only solution satisfying (i), (ii) and having no syzygies (i.e. eclipses) the Lagrange homothety solutions? (i) Energy < 0. (ii) Angular momentum =\n0. (Richard Montgomery)", "statement_status": "exact", "statement_verification": "The canonical record is Problem 16 from the AIM workshop list *Variational Methods in Celestial Mechanics*. It reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 16\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[52]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 16. Is the only solution satisfying (i), (ii) and having no syzygies (i.e. eclipses) the Lagrange homothety solutions? (i) Energy < 0. (ii) Angular momentum = \\n0. (Richard Montgomery)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0053", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The OCR-damaged condition is verified as angular momentum equal to zero. Montgomery's 2007 theorem fully answers the intended problem affirmatively for arbitrary positive masses: every maximal negative-energy, zero-angular-momentum Newtonian three-body solution has a syzygy except the two-oriented Lagrange homothetic family, with binary collisions regularized and counted as syzygies. In addition, this attempt proves two local finite-window criteria from Montgomery's shape-height equation: an exact accumulated reciprocal-f threshold for directed crossing and a Riccati/Sturm frequency-window criterion.\n\nCandidate contribution (quantitative_criterion; novelty confidence low): If z>0 and f(t0) z-dot(t0)=-a<0, then a syzygy must occur once the accumulated shape time integral from t0 of ds/f(s) reaches z(t0)/a; independently, if fq is at least omega_0 squared on a shape-time window longer than pi/omega_0, that window contains a syzygy. Equality in the momentum estimate forces the Lagrange tangent degeneracy." }, { "id": 20002546, "problem_number": "AIM-PHYSICS-0054", "title": "Closure-area limit sets and geometric phase on the shape sphere", "statement": "Problem 17. Problems on the shape space for the three-body problem: (Wu-Yi Hsiang) - Are there solutions whose α-(or ω-)limit set is a limit cycle? - The area of the unit shape sphere is π. Conjecture: for any 0 < A 0 ≤ π, there is a solution whose shape curve has its closure with area A0.- Can two closed shape curves with the same homotopy class be deformed to one another by closed shape curves? Conjecture: No. But then determine connected components.", "original_statement": "Problem 17. Problems on the shape space for the three-body problem: (Wu-Yi Hsiang) - Are there solutions whose α-(or ω-)limit set is a limit cycle? - The area of the unit shape sphere is π. Conjecture: for any 0 < A 0 ≤ π, there is a solution whose shape curve has its closure with area A0.- Can two closed shape curves with the same homotopy class be deformed to one another by closed shape curves? Conjecture: No. But then determine connected components.", "clean_statement": "Problem 17. Problems on the shape space for the three-body problem: (Wu-Yi Hsiang) - Are there solutions whose α-(or ω-)limit set is a limit cycle? - The area of the unit shape sphere is π. Conjecture: for any 0 < A 0 ≤ π, there is a solution whose shape curve has its closure with area A0.- Can two closed shape curves with the same homotopy class be deformed to one another by closed shape curves? Conjecture: No. But then determine connected components.", "statement_status": "exact", "statement_verification": "The canonical record is a flattened extraction of Problem 17 in the AIM workshop list *Variational Methods in Celestial Mechanics*. The official PDF, including its page layout, was inspected on PDF page 6. It contains three separate en-dash bullets:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 17\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[53]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 17. Problems on the shape space for the three-body problem: (Wu-Yi Hsiang) - Are there solutions whose α-(or ω-)limit set is a limit cycle? - The area of the unit shape sphere is π. Conjecture: for any 0 < A 0 ≤ π, there is a solution whose shape curve has its closure with area A0.- Can two closed shape curves with the same homotopy class be deformed to one another by closed shape curves? Conjecture: No. But then determine connected components.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0054", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The official source's prescribed area is the two-dimensional area of the set-theoretic closure of a shape trajectory, not area enclosed by a loop. For every complete locally C1 shape trajectory s on the AIM sphere S^2(1/2), the trajectory image has area zero and Area(cl(s(R))) = Area(alpha_s(s) union omega_s(s)); hence positive closure area is entirely a projected-limit-set phenomenon and cannot be realized by a periodic shape curve or finite unions of limit cycles. Separately, signed loop area with multiplicity controls rotation: in an explicit Hopf convention, Delta chi = integral(J/I)dt - 2 A_signed modulo 2 pi, so at zero angular momentum a repeating reduced state closes inertially after finitely many repeats exactly when A_signed/pi is rational. The ordinary same-homotopy deformation question is tautological unless restricted to dynamically realizable reduced-periodic loops, for which a precise realization-space formulation is given.\n\nCandidate contribution (limit_set_area_reduction; novelty confidence low): For locally C1 complete shape trajectories on S^2(1/2), Area(cl(s(R))) equals Area(alpha_s(s) union omega_s(s)); combined with the normalization-correct zero-angular-momentum holonomy Delta chi = -2 A_signed modulo 2 pi, this proves that Hsiang's closure chaoticity and periodic-loop enclosed area are distinct invariants and reduces every positive prescribed closure area to a positive-area projected-limit-set problem." }, { "id": 20002547, "problem_number": "AIM-PHYSICS-0055", "title": "Constant normalized area in the three-body problem", "statement": "Problem 18. Let ∆ be the area of the triangle, I be the moment of inertia. Find periodic solutions to the (planar or spatial) three-body problem such that ∆\n\n> I\n\nis a constant. (Wu-Yi Hsiang)", "original_statement": "Problem 18. Let ∆ be the area of the triangle, I be the moment of inertia. Find periodic solutions to the (planar or spatial) three-body problem such that ∆ \n\n> I\n\nis a constant. (Wu-Yi Hsiang)", "clean_statement": "Problem 18. Let ∆ be the area of the triangle, I be the moment of inertia. Find periodic solutions to the (planar or spatial) three-body problem such that ∆\n\n> I\n\nis a constant. (Wu-Yi Hsiang)", "statement_status": "exact", "statement_verification": "The exact canonical extraction is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 18\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[54]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 18. Let ∆ be the area of the triangle, I be the moment of inertia. Find periodic solutions to the (planar or spatial) three-body problem such that ∆ \\n\\n> I\\n\\nis a constant. (Wu-Yi Hsiang)\"\nOriginal remarks: [\"Remarks. \\n\\n• If the angular momentum is zero, then there is none. \\n\\n• What motions have shapes on the meridian passing the Euler configurations?\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0055", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The OCR-corrupted condition is verified as Delta/I constant. In mass-Jacobi coordinates the normalized quantity a=4 sqrt(m1 m2 m3/M) Delta/I is exactly a shape-sphere latitude and satisfies |a|<=1. The extremal level |a|=1 is dynamically empty for unequal masses; for equal masses it consists of Lagrange homographic motions, whose periodic members are the elliptic Lagrange family. All homographic periodic baselines and their constants are computed, zero angular momentum is rigorously excluded, and every non-extremal planar candidate is reduced to an explicit differential-algebraic system with period-balance and rotational-holonomy closure tests. No nonhomographic periodic solution is claimed.\n\nCandidate contribution (reduction; novelty confidence low): The combined endpoint rigidity theorem and interior shooting obstruction show that the maximum normalized-area latitude admits Newtonian motion only for equal masses and only in the Lagrange homographic family, while any planar interior constant-Delta/I candidate must satisfy the displayed normal-force constraint, two integrated period balances, and the two Hopf-phase closure congruences." }, { "id": 20002548, "problem_number": "AIM-PHYSICS-0056", "title": "A logarithmic-volume identity and zero-angular-momentum obstruction", "statement": "Problem 19. What can be said about the volume of the tetrahedron in the four-body problem? Can it be nonzero forever and stay bounded away from 0 and ∞? (Joseph Gerver)", "original_statement": "Problem 19. What can be said about the volume of the tetrahedron in the four-body problem? Can it be nonzero forever and stay bounded away from 0 and ∞? (Joseph Gerver)", "clean_statement": "Problem 19. What can be said about the volume of the tetrahedron in the four-body problem? Can it be nonzero forever and stay bounded away from 0 and ∞? (Joseph Gerver)", "statement_status": "exact", "statement_verification": "The canonical record is Problem 19 from the AIM workshop *Variational Methods in Celestial Mechanics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 19\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[55]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 19. What can be said about the volume of the tetrahedron in the four-body problem? Can it be nonzero forever and stay bounded away from 0 and ∞? (Joseph Gerver)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0056", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a noncoplanar Newtonian four-body solution, the second derivative of logarithmic tetrahedral volume equals a strictly negative gravitational trace term minus the squared symmetric Jacobi deformation plus an explicit positive angular-momentum term. At zero total angular momentum this makes log-volume strictly concave. Consequently, no complete two-sided zero-angular-momentum solution can have tetrahedral volume uniformly bounded away from zero, even without a positional bound; with all mutual distances at most R, an explicit finite upper bound is obtained for the duration of any prescribed positive volume floor. This complements Montgomery's stronger repeated-coplanarity theorem for bounded zero-angular-momentum solutions but does not settle the unrestricted nonzero-angular-momentum problem.\n\nCandidate contribution (identity_and_obstruction; novelty confidence low): The candidate contribution is the exact decomposition g'' = -G sum_{i0 with |grad U|/K=v^2 constant along the component, equivalently v^2=|grad U|^2/D^2U[T,T]. If the AIM sentence is instead read as requiring every tangent speed, only constant potentials work. In addition, on any regular annulus with compact connected levels, the transnormal condition |grad U|^2=a(U) plus the Levi property forces U to be radial with concentric level circles.\n\nCandidate contribution (theorem; novelty confidence low): Candidate synthesis: a Levi potential on a regular annulus foliated by compact connected levels and satisfying |grad U|^2=a(U)>0 must have concentric circular levels and hence be radial there; separately, the universal-tangent-speed reading forces U to be constant." }, { "id": 20002551, "problem_number": "AIM-PHYSICS-0059", "title": "A Jacobi-Maupertuis and McGehee scale-shape dictionary", "statement": "Problem 22. Assume E = 0 and angular momentum = 0. Can Jacobi's metric give new insights beyond what McGehee's coordinates give? (Richard Montgomery)", "original_statement": "Problem 22. Assume E = 0 and angular momentum = 0. Can Jacobi's metric give new insights beyond what McGehee's coordinates give? (Richard Montgomery)", "clean_statement": "Problem 22. Assume E = 0 and angular momentum = 0. Can Jacobi's metric give new insights beyond what McGehee's coordinates give? (Richard Montgomery)", "statement_status": "exact", "statement_verification": "The canonical record agrees with the PDF; no OCR correction is needed.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 22\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[58]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 22. Assume E = 0 and angular momentum = 0. Can Jacobi's metric give new insights beyond what McGehee's coordinates give? (Richard Montgomery)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0059", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the zero-energy, zero-angular-momentum planar Newtonian three-body problem, the rotation-reduced Jacobi-Maupertuis metric has the homogeneous form 2U(s)(d rho^2 + rho^2 h_shape/4), where rho=2 sqrt(r) and r=sqrt(I). McGehee's Lyapunov variable is exactly the weighted radial geodesic slope v=2U d rho/d ell, with dv/d ell=(U rho/2)||ds/d ell||^2. Radial geodesics are precisely central-configuration homotheties. In each shape Hessian mode, the sharp condition mu<-U/8 is simultaneously the condition for negative Jacobi-Maupertuis second variation and for complex McGehee eigenvalues; it also obstructs a real C2 homogeneous Hamilton-Jacobi branch calibrated by that homothety. Later literature confirms that Jacobi geometry yields complementary curvature, length-space, weak-KAM, minimization, and geodesic-ray insights, but it does not retain the full McGehee collision manifold.\n\nCandidate contribution (equivalence_and_reduction; novelty confidence low): The candidate contribution is the explicit three-way package connecting the radial-flux identities v=2U rho_ell and v_ell=(U rho/2)||s_ell||^2, the sharp Hardy index threshold mu<-U/8, the identical McGehee spiraling discriminant, and the homogeneous shape-eikonal Riccati equation 4A^2+Phi A=Hess U." }, { "id": 20002552, "problem_number": "AIM-PHYSICS-0060", "title": "Limits of action minimization: collisions, constraints, stability, and fixed energy", "statement": "Problem 23. Among the many interesting open questions is the understanding of the nat-ural limits of the minimization method: (Alain Chenciner) - To what extent is it connected to symmetry constraints? - May topological constraints be imposed without forcing the minimizers to have collisions? - To what extent interesting results may be obtained for arbitrary masses? - What can be said of the hyperbolic and elliptic dimensions of a minimizer? - Is it interesting to look at minimizers with fixed energy?", "original_statement": "Problem 23. Among the many interesting open questions is the understanding of the nat-ural limits of the minimization method: (Alain Chenciner) - To what extent is it connected to symmetry constraints? - May topological constraints be imposed without forcing the minimizers to have collisions? - To what extent interesting results may be obtained for arbitrary masses? - What can be said of the hyperbolic and elliptic dimensions of a minimizer? - Is it interesting to look at minimizers with fixed energy?", "clean_statement": "Problem 23. Among the many interesting open questions is the understanding of the nat-ural limits of the minimization method: (Alain Chenciner) - To what extent is it connected to symmetry constraints? - May topological constraints be imposed without forcing the minimizers to have collisions? - To what extent interesting results may be obtained for arbitrary masses? - What can be said of the hyperbolic and elliptic dimensions of a minimizer? - Is it interesting to look at minimizers with fixed energy?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 23 in the 2003 AIM workshop list *Variational Methods in Celestial Mechanics*. The exact record is preserved in `input.json`. Inspection of page 6 of the official PDF shows that the corpus string `nat-ural` is only line-break hyphenation. With that typographical repair, the source reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 23\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[59]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 23. Among the many interesting open questions is the understanding of the nat-ural limits of the minimization method: (Alain Chenciner) - To what extent is it connected to symmetry constraints? - May topological constraints be imposed without forcing the minimizers to have collisions? - To what extent interesting results may be obtained for arbitrary masses? - What can be said of the hyperbolic and elliptic dimensions of a minimizer? - Is it interesting to look at minimizers with fixed energy?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0060", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The five-part 2003 program is partially solved rather than reducible to one yes/no answer: finite-group symmetry now has sharp coercivity and rotating-circle collision tests; Marchal's theorem removes fixed-end interior collisions but not automatically symmetry-boundary collisions; Newtonian topological and arbitrary-mass periodic constructions remain conditional or may be regularizably collisional; and Morse minimality does not determine Floquet hyperbolic/elliptic dimensions. The new proved contribution is a fixed-energy obstruction: for any attractive potential homogeneous of degree -alpha with 0 0 (or find a counterexample). (Rick Moeckel, Marshall Hampton)", "original_statement": "Problem 24. For the n-body problem, n ≥ 4, show that the number of central configurations is finite for all choices of masses mi > 0 (or find a counterexample). (Rick Moeckel, Marshall Hampton)", "clean_statement": "Problem 24. For the n-body problem, n ≥ 4, show that the number of central configurations is finite for all choices of masses mi > 0 (or find a counterexample). (Rick Moeckel, Marshall Hampton)", "statement_status": "exact", "statement_verification": "The source adds that for \\(n>4\\) even generic finiteness was then open, and that the question can be posed algebraically. Comparison with the original PDF found no OCR corruption or missing symbol.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 24\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[60]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 24. For the n-body problem, n ≥ 4, show that the number of central configurations is finite for all choices of masses mi > 0 (or find a counterexample). (Rick Moeckel, Marshall Hampton)\"\nOriginal remarks: [\"Remarks. \\n\\n• For n > 4, even generic finiteness is an open question. \\n\\n• Can be posed as a purely algebraic question about the solutions of a system of polynomial equations.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0061", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The general positive-mass planar finiteness problem remains open, but two sectors are proved finite. For each labeled collinear ordering, the normalized functional U+I/2 is 1-strongly convex in the mass metric and has a unique critical point; an explicit mass-and-order-dependent comparison value bounds its inertia and every mutual separation. In any Euclidean ambient space, an n-body central configuration of affine dimension n-1 is necessarily a regular simplex, and conversely that simplex is central for arbitrary positive masses.\n\nCandidate contribution (quantitative_bound; novelty confidence low): For the unique multiplier-one collinear central configuration in ordering sigma, the explicit comparison value B_sigma defined from centered consecutive integer positions satisfies I <= 2 B_sigma and |q_i-q_j| >= m_i m_j / B_sigma for every pair, while the Hessian of U+I/2 dominates the mass metric." }, { "id": 20002554, "problem_number": "AIM-PHYSICS-0062", "title": "Normal Hessian blocks, an exact four-body index jump, and stability obstructions", "statement": "Problem 25. Give a sharp upper bound for the Morse index of a nonplanar central config-uration (as a critical point of the potential on the normalized configuration space). Give a sharp lower bound for the Morse index of a planar central configuration when it is viewed 7\n\nas part of the nonplanar configuration space. Is the Morse index related to the stability of the rigidly rotating periodic orbits? For example, does a linearly stable relative equilibrium necessarily arise from a minimum of the Newtonian potential? (Rick Moeckel)", "original_statement": "Problem 25. Give a sharp upper bound for the Morse index of a nonplanar central config-uration (as a critical point of the potential on the normalized configuration space). Give a sharp lower bound for the Morse index of a planar central configuration when it is viewed 7\n\nas part of the nonplanar configuration space. Is the Morse index related to the stability of the rigidly rotating periodic orbits? For example, does a linearly stable relative equilibrium necessarily arise from a minimum of the Newtonian potential? (Rick Moeckel)", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is Problem 25 from the AIM workshop list *Variational Methods in Celestial Mechanics*. The official PDF, pages 5--6, gives the following statement after repairing only extraction artifacts:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 25\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[61]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 25. Give a sharp upper bound for the Morse index of a nonplanar central config-uration (as a critical point of the potential on the normalized configuration space). Give a sharp lower bound for the Morse index of a planar central configuration when it is viewed 7\\n\\nas part of the nonplanar configuration space. Is the Morse index related to the stability of the rigidly rotating periodic orbits? For example, does a linearly stable relative equilibrium necessarily arise from a minimum of the Newtonian potential? (Rick Moeckel)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0062", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any collision-free noncollinear planar central configuration q of n positive masses embedded in R^d, the constrained Hessian splits into the planar Hessian and d-2 identical scalar normal blocks V_q, so ind_d(q)=ind_2(q)+(d-2)ind(V_q). The normal block has the two rotation modes in its kernel and between 1 and n-3 negative eigenvalues. Hence for every planar four-body central configuration ind(V_q)=1 exactly and ind_d(q)=ind_2(q)+d-2, with no additional vertical nullity. A separate trace argument gives the non-sharp bound ind_3(q)<=3n-8 for full-dimensional spatial configurations. Current parity theorems relate odd index/nullity to instability, but the conjecture that linear stability forces a nondegenerate minimum remains open.\n\nCandidate contribution (theorem; novelty confidence low): Embedding any genuinely planar four-body Newtonian central configuration of arbitrary positive masses from R^2 into R^d adds exactly d-2 to its Morse index and creates no non-rotational vertical nullity." }, { "id": 20002555, "problem_number": "AIM-PHYSICS-0063", "title": "Compatibility and reversible shooting for connections between relative equilibria", "statement": "Problem 26. Existence proof for homoclinic and heteroclinic orbits between unstable rela-tive equilibrium solutions. (Rick Moeckel)", "original_statement": "Problem 26. Existence proof for homoclinic and heteroclinic orbits between unstable rela-tive equilibrium solutions. (Rick Moeckel)", "clean_statement": "Problem 26. Existence proof for homoclinic and heteroclinic orbits between unstable rela-tive equilibrium solutions. (Rick Moeckel)", "statement_status": "exact", "statement_verification": "The canonical record is Problem 26 from the AIM workshop list *Variational Methods in Celestial Mechanics*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 26\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[62]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 26. Existence proof for homoclinic and heteroclinic orbits between unstable rela-tive equilibrium solutions. (Rick Moeckel)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0063", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For planar Newtonian relative equilibria generated by central configurations a_- and a_+, any collision-free connection at nonzero angular momentum must satisfy the scale-free endpoint identity I(a_-) U(a_-)^2 = I(a_+) U(a_+)^2; after I = 1 normalization this is equality of the two potential values. The equal-mass Euler and Lagrange three-body shapes violate this identity (25/2 versus 9), so no independent rescaling can connect them. In the reduced phase space, a transverse connection also requires dim W^u(x_-) + dim W^s(x_+) at least dim P_mu. Finally, for reflection-time-reversal rho, a nontrivial point in W^u(x) intersect Fix(rho) proves a heteroclinic from x to rho(x), or a homoclinic when rho(x) = x.\n\nCandidate contribution (obstruction_and_reversible_criterion; novelty confidence low): A three-gate certificate combines the exact scale-free energy-angular-momentum endpoint test, the reduced invariant-manifold dimension test, and a reversible one-sided shooting criterion; it includes the explicit scale-independent exclusion of an equal-mass Euler-to-Lagrange connection." }, { "id": 20002556, "problem_number": "AIM-PHYSICS-0064", "title": "Exact reversible shooting criterion for halo-orbit bifurcation", "statement": "Problem 27. Existence proof for the \"halo orbits\" of the three-dimensional restricted three-body problem. (Rick Moeckel)", "original_statement": "Problem 27. Existence proof for the \"halo orbits\" of the three-dimensional restricted three-body problem. (Rick Moeckel)", "clean_statement": "Problem 27. Existence proof for the \"halo orbits\" of the three-dimensional restricted three-body problem. (Rick Moeckel)", "statement_status": "exact", "statement_verification": "The canonical record is Problem 27 from the AIM workshop *Variational Methods in Celestial Mechanics* (PDF version dated June 22, 2003):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 27\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[63]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 27. Existence proof for the \\\"halo orbits\\\" of the three-dimensional restricted three-body problem. (Rick Moeckel)\"\nOriginal remarks: [\"Remarks. Apparently these are born in a bifurcation from elliptical Lagrange orbit but as far as I know, they have only been studied numerically. (Rick Moeckel)\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0064", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The 2003 AIM premise that halo orbits had only been studied numerically is obsolete: Walawska and Wilczak (2019) rigorously validated halo bifurcations from the planar Lyapunov families at all three collinear points for every mass ratio 9.5e-4 <= mu <= 1/2. This attempt additionally proves an exact local reduction in the original CR3BP: after factoring the odd vertical half-return residual as F2=cK, a root of (F1,K) with invertible D_(a,b)(F1,K) generates paired nonplanar reversible periodic branches. Twice differentiating the reduced equations gives an explicit coefficient kappa that determines the Jacobi-constant side and square-root amplitude onset.\n\nCandidate contribution (reduction; novelty confidence low): For an R-symmetric planar CR3BP periodic orbit, the exact certificate F1=K=0 with det D_(a,b)(F1,K) nonzero, where F2=cK is the reflection-forced factorization of the vertical half-return residual, proves a local halo pitchfork; moreover kappa=Omega_x a''(0)-b_* b''(0)+Omega_zz with (a'',b'')^T=-J^{-1} partial_cc(F1,K) determines the Jacobi side and the leading square-root amplitude law." }, { "id": 20002557, "problem_number": "AIM-PHYSICS-0065", "title": "A feasibility sieve for negative-energy binary-single direct scattering", "statement": "Problem 28. Consider negative energy three-body orbits which are unbound in both direc-tions of time. The two Jacobi vectors of such an orbit then asymptote to Keplerian orbits in both the distant past and the distant future, and so associated to such an orbit we have pair of Kepler elements in the past (one elliptic the other hyperbolic) and another pair in the future. The \"direct scattering\" problem is: which pair of Kepler elements can be connected to each other in this way? (Richard Montgomery)", "original_statement": "Problem 28. Consider negative energy three-body orbits which are unbound in both direc-tions of time. The two Jacobi vectors of such an orbit then asymptote to Keplerian orbits in both the distant past and the distant future, and so associated to such an orbit we have pair of Kepler elements in the past (one elliptic the other hyperbolic) and another pair in the future. The \"direct scattering\" problem is: which pair of Kepler elements can be connected to each other in this way? (Richard Montgomery)", "clean_statement": "Problem 28. Consider negative energy three-body orbits which are unbound in both direc-tions of time. The two Jacobi vectors of such an orbit then asymptote to Keplerian orbits in both the distant past and the distant future, and so associated to such an orbit we have pair of Kepler elements in the past (one elliptic the other hyperbolic) and another pair in the future. The \"direct scattering\" problem is: which pair of Kepler elements can be connected to each other in this way? (Richard Montgomery)", "statement_status": "exact", "statement_verification": "The canonical record is Problem 28 in the AIM workshop list *Variational Methods in Celestial Mechanics*. Inspection of page 7 of the official PDF confirms that the only textual defect is line-break hyphenation: “direc-tions” means “directions.” The exact unmodified record remains in input.json. With only that repair, the problem reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 28\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[64]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 28. Consider negative energy three-body orbits which are unbound in both direc-tions of time. The two Jacobi vectors of such an orbit then asymptote to Keplerian orbits in both the distant past and the distant future, and so associated to such an orbit we have pair of Kepler elements in the past (one elliptic the other hyperbolic) and another pair in the future. The \\\"direct scattering\\\" problem is: which pair of Kepler elements can be connected to each other in this way? (Richard Montgomery)\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0065", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every collision-free hyperbolic-elliptic three-body connection, the incoming and outgoing Kepler energy sums agree and the inner-plus-outer angular-momentum vectors equal one conserved total vector. After orbital-plane orientations are forgotten, scalar shape data admit compatible spatial orientations exactly when their two channel-dependent angular-momentum intervals intersect. An explicit equal-mass pair has matching total energy but disjoint intervals, proving it cannot be connected. Separately, under explicit C1 wave-map hypotheses, the unresolved direct-scattering relation is a symplectic range-intersection problem on a six-dimensional spatial or four-dimensional planar reduced space.\n\nCandidate contribution (obstruction; novelty confidence low): The channel-aware interval criterion I^- intersection I^+ nonempty is an exact orientation-feasibility test for scalar incoming and outgoing Kepler shape elements, and the explicit equal-mass data in Corollary 5.2 give an energy-matched but rigorously impossible direct-scattering pair." }, { "id": 20002558, "problem_number": "AIM-PHYSICS-0066", "title": "Full-space mass-splitting criterion for prograde relative periodic orbits", "statement": "Problem 29. Given three different masses that are comparable in size, prove the existence of prograde orbits without assuming one ratio of mutual distances is nearly zero. (Kuo-Chang Chen)", "original_statement": "Problem 29. Given three different masses that are comparable in size, prove the existence of prograde orbits without assuming one ratio of mutual distances is nearly zero. (Kuo-Chang Chen)", "clean_statement": "Problem 29. Given three different masses that are comparable in size, prove the existence of prograde orbits without assuming one ratio of mutual distances is nearly zero. (Kuo-Chang Chen)", "statement_status": "exact", "statement_verification": "The canonical record is Problem 29 in the AIM workshop list *Variational Methods in Celestial Mechanics* (version dated June 22, 2003):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: PDF\nAIM domain: Physics\nWorkshop: Variational Methods in Celestial Mechanics\nSection: \nSource item: 29\nSource URL: https://aimath.org/WWN/varcelest/varcelest.pdf\nCanonical location: aim-physics-notes.json notes[65]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Problem 29. Given three different masses that are comparable in size, prove the existence of prograde orbits without assuming one ratio of mutual distances is nearly zero. (Kuo-Chang Chen)\"\nOriginal remarks: [\"Remarks. \\n\\n• Poincar´ e continuation method can be applied to two cases: one mass is nearly zero, or one ratio of mutual distances is nearly zero. \\n\\n• Known if two masses are equal.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: open; rigor: conditional; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 16, "set_id": 14, "source_url": "https://aimath.org/WWN/varcelest/varcelest.pdf", "tags": [ "aim", "AIM-PHYSICS-0066", "aim-domain:physics", "aim-workshop:varcelest", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 16, "name": "physics", "display_name": "Mathematical Physics", "description": "Problems at the intersection of mathematics and physics.", "slug": "physics", "order_index": 16, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The intended objects are planar relative-periodic direct/prograde orbits in the Chen-Lin braid class, not restricted-problem distant prograde orbits. Chen and Lin's 2009 rigorous theorem still assumes an equal pair. This attempt proves a conditional continuation theorem: if one collision-free equal-pair prograde critical loop has full fixed-period, fixed-twist Hessian kernel consisting only of time-shift and planar-rotation directions, then it persists for every nearby mass triple, including all-distinct triples. Collision clearance, the prograde braid, and an explicit nonhierarchical mutual-distance ratio persist; a chosen mass-comparability bound also persists when the seed satisfies it strictly.\n\nCandidate contribution (conditional continuation criterion; novelty confidence low): For the mass-independent fixed-(T,phi) Jacobi loop action, the full-space kernel condition ker D^2A=span{time shift, planar rotation}, equivalently absence of extra fixed directions of the relative monodromy, is a one-orbit certificate for continuing a Chen-Lin equal-pair prograde orbit to all nearby distinct mass triples while preserving its braid, collision clearance, and a quantitative nonhierarchy functional.", "research_difficulty_suggested": "L4" }, { "id": 20002559, "problem_number": "AIM-PROBABILITY-0001", "title": "Local surjectivity and the boundary-at-infinity gap for RBM(4,3)", "statement": "Is $RBM_{4,3}$ a universal approximator?\n\nDoes the closure of $RBM_{4,3}$ fill the simplex $\\Delta_{15},$ making it a \\textit{universal approximator}?", "original_statement": "Is $RBM_{4,3}$ a universal approximator?\n\nDoes the closure of $RBM_{4,3}$ fill the simplex $\\Delta_{15},$ making it a \\textit{universal approximator}?", "clean_statement": "Is $RBM_{4,3}$ a universal approximator?\n\nDoes the closure of $RBM_{4,3}$ fill the simplex $\\Delta_{15},$ making it a \\textit{universal approximator}?", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Representational Capacity\nSource item: 1.1\nSource URL: http://aimpl.org/boltzmann/1/\nCanonical location: aim-probability-notes.json notes[0]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is $RBM_{4,3}$ a universal approximator?\\n\\nDoes the closure of $RBM_{4,3}$ fill the simplex $\\\\Delta_{15},$ making it a \\\\textit{universal approximator}?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Here, $RBM_{4,3}$ indicates the RBM statistical model with 4 visible and 3 hidden units, belonging to the probability simplex $\\\\Delta_{15}.$ It is known that for universal approximation, it is necessary that the number of hidden units be $m \\\\geq 3,$ and sufficient if $m \\\\geq 7.$ Simulations have suggested that $RBM_{4,3}$ fills the simplex.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/1/", "tags": [ "aim", "AIM-PROBABILITY-0001", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The universality question remains open, but three rigorous reductions are established. First, finite RBM(4,3) laws are normalized Hadamard products of three sums of two positive rank-one tensors, while the associated supermodular-rank-three condition is vacuous because every function on the four-cube has supermodular rank at most three. Second, at b = c = 0 and exp(W) with rows (1,2,3,4), (2,3,4,5), and (3,4,5,6), a specified 15-by-15 log-odds Jacobian minor has exact nonzero determinant -4292546820857203/11884241136598188669271898437500, giving a concrete real local chart. Third, if a strictly positive distribution is in the Euclidean closure but not the finite-parameter image, every approximating parameter sequence has norm tending to infinity, and every normalized convergent subsequence satisfies fifteen explicit ReLU cancellation equations. These imply vanishing of all higher-order Möbius coefficients; conversely, those cancellations recover the ReLU system when combined with four stated first-order identities.\n\nCandidate contribution (explicit_jacobian_and_escape_reduction; novelty confidence low): A concrete exact rational full-rank Jacobian witness, combined with a proof that supermodular rank is an exactly vacuous relaxation at (4,3), reduces every nonexact strictly positive closure point to a finite union of polyhedral tropical escape regimes defined by explicit ReLU and Möbius-cancellation equations." }, { "id": 20002560, "problem_number": "AIM-PROBABILITY-0002", "title": "Exact parity projection and a certified bracket for RBM(3,1)", "statement": "Determine the maximum divergence of $RBM_{3,1}.$\n\nDetermine the maximum divergence of $RBM_{3,1}.$", "original_statement": "Determine the maximum divergence of $RBM_{3,1}.$\n\nDetermine the maximum divergence of $RBM_{3,1}.$", "clean_statement": "Determine the maximum divergence of $RBM_{3,1}.$\n\nDetermine the maximum divergence of $RBM_{3,1}.$", "statement_status": "exact", "statement_verification": "The canonical record contains the sentence", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Representational Capacity\nSource item: 1.2\nSource URL: http://aimpl.org/boltzmann/1/\nCanonical location: aim-probability-notes.json notes[1]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Determine the maximum divergence of $RBM_{3,1}.$\\n\\nDetermine the maximum divergence of $RBM_{3,1}.$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Currently, the maximum divergence of $RBM_{3,1}$ is unknown. The maximum divergence is defined as ${\\\\cal D}_{RBM_{3,1}} := \\\\text{sup}_{p \\\\in \\\\Delta_7} \\\\text{inf}_{q \\\\in RBM_{3,1}} D(p || q),$ where $D$ can be any measure of divergence between probability distributions, such as the KL-divergence.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/1/", "tags": [ "aim", "AIM-PROBABILITY-0002", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For forward KL divergence with natural logarithms, the maximum divergence from RBM_{3,1}, equivalently the mixture of two binary product distributions, lies between (3/4) log(1+2/sqrt(3)) = 0.5757388144... and log 2 = 0.6931471806... nats. The lower endpoint is proved to be the exact distance of each parity distribution. An explicit reverse-I projection is q* = ((3-sqrt(3))/6) delta_000 + ((3+sqrt(3))/6) r_t^{tensor 3}, with t=(3-sqrt(3))/2; the complete boundary-MLE classification proves globality of this inner projection after lower-valued algebraic candidates are rejected by nonnegative-rank arguments. The remaining claim that parity is globally worst is not proved.\n\nCandidate contribution (projection_certificate; novelty confidence low): Instantiating the complete boundary-MLE decomposition at a parity target yields an explicit finite certificate that its four atom-plus-product distributions are global reverse-I projections, including direct support and nonnegative-rank obstructions for every lower-valued algebraic candidate." }, { "id": 20002561, "problem_number": "AIM-PROBABILITY-0003", "title": "A finite RBM separation from equal-parameter naive Bayes, with an SBN caveat", "statement": "What kind of distributions can be represented by an RBM as opposed to directed models?\n\nWhat kind of distributions can be represented by an RBM as opposed to directed models?", "original_statement": "What kind of distributions can be represented by an RBM as opposed to directed models?\n\nWhat kind of distributions can be represented by an RBM as opposed to directed models?", "clean_statement": "What kind of distributions can be represented by an RBM as opposed to directed models?\n\nWhat kind of distributions can be represented by an RBM as opposed to directed models?", "statement_status": "exact", "statement_verification": "The canonical record is item 1.3, in the section “Representational Capacity” of the AIM workshop list *Boltzmann Machines*. Its `problem` field is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Representational Capacity\nSource item: 1.3\nSource URL: http://aimpl.org/boltzmann/1/\nCanonical location: aim-probability-notes.json notes[2]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What kind of distributions can be represented by an RBM as opposed to directed models?\\n\\nWhat kind of distributions can be represented by an RBM as opposed to directed models?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/1/", "tags": [ "aim", "AIM-PROBABILITY-0003", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every m >= 2, an explicit paired-bit RBM with 2m visible and m hidden units and finite weight A > 2 log(2^(m+1)m) defines a strictly positive visible distribution with exactly 2^m strong modes and real nonnegative tensor rank exactly 2^m. It is therefore outside the (m+1)-component mixture-of-products model, although that naive Bayes model and the RBM both have 2m^2+3m raw parameters. The same witness is exactly represented by a same-size sigmoid belief network with independent fair hidden roots, so the separation is deliberately restricted to naive Bayes rather than directed models in general.\n\nCandidate contribution (explicit construction and synthesis; novelty confidence low): The paired-bit finite-weight family has exact nonnegative rank 2^m above the explicit sufficient threshold A > 2 log(2^(m+1)m), giving an equal-raw-parameter naive-Bayes separation while simultaneously admitting an exact same-size SBN representation." }, { "id": 20002562, "problem_number": "AIM-PROBABILITY-0004", "title": "Topology-dependent continuum limits of square binary RBMs", "statement": "Characterize the limiting set of distributions for $\\lim_{n\\to\\infty} RBM_{n,n}.$\n\nConsider the limit as $n \\to \\infty$ for $RBM_{n,n}.$ It was suggested that this limiting set of distributions can be seen as a set of measures on the unit interval. Do all possible measures belong to this set? If not, which ones do?", "original_statement": "Characterize the limiting set of distributions for $\\lim_{n\\to\\infty} RBM_{n,n}.$\n\nConsider the limit as $n \\to \\infty$ for $RBM_{n,n}.$ It was suggested that this limiting set of distributions can be seen as a set of measures on the unit interval. Do all possible measures belong to this set? If not, which ones do?", "clean_statement": "Characterize the limiting set of distributions for $\\lim_{n\\to\\infty} RBM_{n,n}.$\n\nConsider the limit as $n \\to \\infty$ for $RBM_{n,n}.$ It was suggested that this limiting set of distributions can be seen as a set of measures on the unit interval. Do all possible measures belong to this set? If not, which ones do?", "statement_status": "exact", "statement_verification": "The exact canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Representational Capacity\nSource item: 1.4\nSource URL: http://aimpl.org/boltzmann/1/\nCanonical location: aim-probability-notes.json notes[3]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Characterize the limiting set of distributions for $\\\\lim_{n\\\\to\\\\infty} RBM_{n,n}.$\\n\\nConsider the limit as $n \\\\to \\\\infty$ for $RBM_{n,n}.$ It was suggested that this limiting set of distributions can be seen as a set of measures on the unit interval. Do all possible measures belong to this set? If not, which ones do?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: conditional_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/1/", "tags": [ "aim", "AIM-PROBABILITY-0004", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The source limit is undefined without an embedding and topology. Under the standard prefix binary embedding pi_n(x)=sum_i x_i 2^{-i}, the limiting set of finite-parameter RBM(n,n) laws in weak topology (equivalently W_1 on [0,1]) is every Borel probability measure on [0,1]. For every target and every n>=2, an explicit diagonal construction gives W_1 error less than 2/(n+1). Under total-variation convergence, the limiting set is instead exactly the probability measures supported on the dyadic rationals in [0,1). Both characterizations are proved in both directions, and replacing finite models by their closures does not change either limiting set.\n\nCandidate contribution (topology_dependent_limit_characterization; novelty confidence low): For the prefix binary embedding, finite-parameter square binary RBMs are W_1-dense in all probability measures on [0,1] with constructive error below 2/(n+1), whereas their total-variation limiting set is exactly the dyadic-supported measures.", "research_difficulty_suggested": "L4" }, { "id": 20002563, "problem_number": "AIM-PROBABILITY-0005", "title": "Exact determinant-sign characterization of nonnegative rank three for 2x2x2 tensors", "statement": "Can we characterize tensors of non-negative rank at most 3 using the description of $RBM_{3,2}?$\n\nIs it possible to use the implicit description of $RBM_{3,2}$ to determine the characterizing properties of non-negative rank $\\leq 3$ tensors?", "original_statement": "Can we characterize tensors of non-negative rank at most 3 using the description of $RBM_{3,2}?$\n\nIs it possible to use the implicit description of $RBM_{3,2}$ to determine the characterizing properties of non-negative rank $\\leq 3$ tensors?", "clean_statement": "Can we characterize tensors of non-negative rank at most 3 using the description of $RBM_{3,2}?$\n\nIs it possible to use the implicit description of $RBM_{3,2}$ to determine the characterizing properties of non-negative rank $\\leq 3$ tensors?", "statement_status": "exact", "statement_verification": "The canonical AIM record contains two equivalent questions:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Algebraic Statistics and Tensor Characterizations\nSource item: 2.1\nSource URL: http://aimpl.org/boltzmann/2/\nCanonical location: aim-probability-notes.json notes[4]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we characterize tensors of non-negative rank at most 3 using the description of $RBM_{3,2}?$\\n\\nIs it possible to use the implicit description of $RBM_{3,2}$ to determine the characterizing properties of non-negative rank $\\\\leq 3$ tensors?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Recently it has been shown that $RBM_{3,2} = {\\\\cal M}_{3,3},$ where up to scaling, ${\\\\cal M}_{3,3}$ is the set of tensors of non-negative rank at most 3, i.e. ${\\\\cal M_{3,3}} = \\\\{p \\\\in \\\\mathbb{R}^8 : p = \\\\sum_{i=1}^3 a_i \\\\otimes b_i \\\\otimes c_i, \\\\text{ where } a_i,b_i,c_i \\\\in \\\\mathbb{R}^2_{\\\\geq 0} \\\\text{ for } i=1,2,3 \\\\}.$\\n\\nCould this recent finding be used to determine a non-negative rank 3 generalization of the supermodularity and flattening rank constraints known for non-negative rank $\\\\leq 2$ tensors? $$ $$\\nSee $$ $$\\nSeigal, Anna, and Guido Montufar. \\\"Mixtures and products in two graphical models.\\\" arXiv preprint arXiv:1709.05276 (2017).$$ $$\\nand $$ $$\\nAllman, Elizabeth S., et al. \\\"Tensors of nonnegative rank two.\\\" Linear algebra and its applications 473 (2015): 37-53.\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/2/", "tags": [ "aim", "AIM-PROBABILITY-0005", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM question is solved by Seigal and Montufar (2018): a nonnegative 2x2x2 tensor p has nonnegative rank at most three exactly when at least one pair of opposite slice determinants has weakly matching sign, equivalently d_10 d_11 >= 0 or d_20 d_21 >= 0 or d_30 d_31 >= 0. Since every such tensor has nonnegative rank at most four, exact rank four is certified by all three products being strictly negative. Globally, closure(RBM_{3,2}) = M_{3,3}; exact equality holds on the interior, while the RBM boundary image is not closed. As a developed consequence, every nontrivial positive parity tilt has ordinary and complex rank two but nonnegative rank and nonnegative border rank four, including examples arbitrarily close to independence.\n\nCandidate contribution (rank_separation_counterexample; novelty confidence low): For every a,b>0 with a not equal to b, the parity-tilt tensor taking value a on even-parity cells and b on odd-parity cells has real and complex tensor rank exactly two but nonnegative rank and nonnegative border rank exactly four; after normalization these maximal-nonnegative-rank tensors accumulate at the uniform rank-one distribution. A complementary tensor with p_000=0, p_111=2, and all other entries one is a one-zero boundary witness of nonnegative rank four." }, { "id": 20002564, "problem_number": "AIM-PROBABILITY-0006", "title": "Algebraic score equations and an independence saddle for RBM learning", "statement": "What are the algebraic structures relevant for studying learning algorithms for RBMs?\n\nWhat are the algebraic structures relevant for studying learning algorithms for RBMs?", "original_statement": "What are the algebraic structures relevant for studying learning algorithms for RBMs?\n\nWhat are the algebraic structures relevant for studying learning algorithms for RBMs?", "clean_statement": "What are the algebraic structures relevant for studying learning algorithms for RBMs?\n\nWhat are the algebraic structures relevant for studying learning algorithms for RBMs?", "statement_status": "exact", "statement_verification": "The canonical record is item 2.2 in the AIM *Boltzmann Machines* section “Algebraic Statistics and Tensor Characterizations.” Its problem field is", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Algebraic Statistics and Tensor Characterizations\nSource item: 2.2\nSource URL: http://aimpl.org/boltzmann/2/\nCanonical location: aim-probability-notes.json notes[5]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What are the algebraic structures relevant for studying learning algorithms for RBMs?\\n\\nWhat are the algebraic structures relevant for studying learning algorithms for RBMs?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/2/", "tags": [ "aim", "AIM-PROBABILITY-0006", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite binary restricted Boltzmann machine, exponentiated parameters turn the exact observed-data likelihood score equations into a polynomial system after clearing an explicit common denominator, and saturation removes the introduced coordinate and normalization components on the positive parameter torus. In addition, matching the empirical one-bit means produces an explicit stationary independence manifold: its observed Fisher information has rank exactly the number of visible units, and any nonzero empirical pair covariance makes every displayed point an indefinite likelihood saddle along a centered one-hidden-unit perturbation.\n\nCandidate contribution (stationary-manifold theorem; novelty confidence low): Candidate novelty: for any empirical binary distribution with all one-bit means strictly between zero and one, the exact visible-likelihood independence manifold with zero weights and arbitrary hidden biases has an explicitly described Fisher kernel of dimension nm+m; moreover, if any empirical pair covariance is nonzero, a centered one-hidden-unit perturbation has second variation s(1-s) r^T D r, so every point on the displayed manifold is an indefinite saddle." }, { "id": 20002565, "problem_number": "AIM-PROBABILITY-0007", "title": "Exact training oracles from sparse RBM topology", "statement": "Efficient methods to train RBMs with fixed, sparse connectivity\n\nWhat are the most efficient learning algorithms for training sparsely connected RBMs?", "original_statement": "Efficient methods to train RBMs with fixed, sparse connectivity\n\nWhat are the most efficient learning algorithms for training sparsely connected RBMs?", "clean_statement": "Efficient methods to train RBMs with fixed, sparse connectivity\n\nWhat are the most efficient learning algorithms for training sparsely connected RBMs?", "statement_status": "exact", "statement_verification": "The canonical record is AIM Problem Lists, *Boltzmann Machines*, section 3.1, “Effects of Network Connectivity”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Effects of Network Connectivity\nSource item: 3.1\nSource URL: http://aimpl.org/boltzmann/3/\nCanonical location: aim-probability-notes.json notes[6]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Efficient methods to train RBMs with fixed, sparse connectivity\\n\\nWhat are the most efficient learning algorithms for training sparsely connected RBMs?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Consider an RBM with a sparsely connected network topology. The sparse connectivity can be seen as \\\"pre-breaking\\\" weight space symmetries that would otherwise exist for the fully-connected topology. In general it is unknown if weight space symmetries play a central role in the efficiency of machine learning optimization methods.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/3/", "tags": [ "aim", "AIM-PROBABILITY-0007", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a binary restricted Boltzmann machine whose joint visible-hidden incidence graph is a forest, the exact maximum-likelihood gradient and any exact likelihood Hessian-vector product can be computed in O(S(N+e)+N+e) real-arithmetic operations for S samples. A supplied width-tau tree decomposition gives the corresponding fixed-parameter exact junction-tree oracle. A separate symmetry audit proves that sparse topology may remove hidden permutations but always retains hidden-state complement equivalences, and examples show that symmetry count does not control exact message-passing tractability.\n\nCandidate contribution (algorithmic synthesis and symmetry proposition; novelty confidence low): The candidate contribution is the explicit package of an end-to-end linear-time exact likelihood gradient and Hessian-vector oracle for forest-incidence sparse RBMs, its joint-graph bounded-treewidth extension, and a two-sided separation between topology-induced parameter symmetries and exact-inference complexity." }, { "id": 20002566, "problem_number": "AIM-PROBABILITY-0008", "title": "Exact fixed-graph stability bounds for a sparse semi-quantum RBM", "statement": "How do RBMs with quantum effects differ from classical RBMs?\n\nHow are the following models different from classical RBMs:\n\n1) RBMs with sparse connectivity.\n\n2) RBMs with sparse connectivity, and with quantum effects.\n\n3) RBMs with sparse connectivity, and with quantum effects and quantum training algorithm.", "original_statement": "How do RBMs with quantum effects differ from classical RBMs?\n\nHow are the following models different from classical RBMs:\n\n1) RBMs with sparse connectivity.\n\n2) RBMs with sparse connectivity, and with quantum effects.\n\n3) RBMs with sparse connectivity, and with quantum effects and quantum training algorithm.", "clean_statement": "How do RBMs with quantum effects differ from classical RBMs?\n\nHow are the following models different from classical RBMs:\n\n1) RBMs with sparse connectivity.\n\n2) RBMs with sparse connectivity, and with quantum effects.\n\n3) RBMs with sparse connectivity, and with quantum effects and quantum training algorithm.", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Effects of Network Connectivity\nSource item: 3.2\nSource URL: http://aimpl.org/boltzmann/3/\nCanonical location: aim-probability-notes.json notes[7]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How do RBMs with quantum effects differ from classical RBMs?\\n\\nHow are the following models different from classical RBMs:\\n\\n1) RBMs with sparse connectivity.\\n\\n2) RBMs with sparse connectivity, and with quantum effects.\\n\\n3) RBMs with sparse connectivity, and with quantum effects and quantum training algorithm.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/3/", "tags": [ "aim", "AIM-PROBABILITY-0008", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a sparse RBM with classical visible spins and independent hidden qubits carrying longitudinal visible-hidden couplings and transverse hidden fields, the hidden trace is exact and preserves every hidden neighborhood. At fixed classical parameters, C equal to the sum over hidden units of log cosh of the absolute transverse field bounds both directions of KL divergence by min(C, C squared over 8), total variation by tanh(C over 4), and bounded-statistic bias by the statistic range times that total-variation bound. Hidden responses contract, and hidden-induced Fourier terms have degree at most the maximum hidden degree; the full log weight, including visible biases, has degree at most max(1,d).\n\nCandidate contribution (stability_bound; novelty confidence low): Candidate novelty: the assembled fixed-graph perturbation theorem combines the exact independent-hidden trace with two-sided KL bounds, the sharp bounded-likelihood-ratio total-variation bound tanh(C/4), response contraction, and an explicit training-statistic bias certificate; a degree-three example also disproves the inference that sparse connectivity necessarily gives only pairwise visible interactions." }, { "id": 20002567, "problem_number": "AIM-PROBABILITY-0009", "title": "Exact fixed-connectivity inference counts for binary RBMs", "statement": "Given a fixed connectivity structure, how many inference functions can an RBM model compute?\n\nHow many inference functions can an RBM model compute, constrained by a fixed connectivity structure?", "original_statement": "Given a fixed connectivity structure, how many inference functions can an RBM model compute?\n\nHow many inference functions can an RBM model compute, constrained by a fixed connectivity structure?", "clean_statement": "Given a fixed connectivity structure, how many inference functions can an RBM model compute?\n\nHow many inference functions can an RBM model compute, constrained by a fixed connectivity structure?", "statement_status": "exact", "statement_verification": "The canonical AIM record (Boltzmann Machines workshop, section “Effects of Network Connectivity,” problem 3.3) contains:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Effects of Network Connectivity\nSource item: 3.3\nSource URL: http://aimpl.org/boltzmann/3/\nCanonical location: aim-probability-notes.json notes[8]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given a fixed connectivity structure, how many inference functions can an RBM model compute?\\n\\nHow many inference functions can an RBM model compute, constrained by a fixed connectivity structure?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/3/", "tags": [ "aim", "AIM-PROBABILITY-0009", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a binary RBM on a fixed labeled bipartite graph, the standard generic hidden conditional-MAP inference maps are counted exactly by the product over hidden vertices of T(d_j), where T(d) is the number of Boolean linear threshold functions on d labeled inputs; the reverse one-way count is the analogous product over visible degrees. If both directions must share one weight matrix, the count is instead the chamber number of an explicit coupled central hyperplane arrangement. For every graph of maximum degree at most one with r edges, this bidirectional count is exactly 14^r times 2^(n+m-2r).\n\nCandidate contribution (exact counting theorem and reduction; novelty confidence low): Candidate novelty: one-way fixed-connectivity inference capacity factors exactly as the product of threshold-function counts over hidden degrees, while compatible bidirectional pairs are chambers of an explicit shared-weight arrangement; for a matching with r edges their exact number is 14^r 2^(n+m-2r), not the naive product of the two one-way capacities." }, { "id": 20002568, "problem_number": "AIM-PROBABILITY-0010", "title": "Topology overlap and pruning stability for RBMs", "statement": "Given two RBMs with the same number of units but different connectivities, how much do these statistical models overlap?\n\nHow to quantify function approximation given a network topology? In other words, how to measure the proximity between two functions when each are computed by separate networks? Then, can one use this measure to quantify the effects of altering a network's connectivity structure?", "original_statement": "Given two RBMs with the same number of units but different connectivities, how much do these statistical models overlap?\n\nHow to quantify function approximation given a network topology? In other words, how to measure the proximity between two functions when each are computed by separate networks? Then, can one use this measure to quantify the effects of altering a network's connectivity structure?", "clean_statement": "Given two RBMs with the same number of units but different connectivities, how much do these statistical models overlap?\n\nHow to quantify function approximation given a network topology? In other words, how to measure the proximity between two functions when each are computed by separate networks? Then, can one use this measure to quantify the effects of altering a network's connectivity structure?", "statement_status": "exact", "statement_verification": "The accompanying note says that the question was motivated by biological brains having fine-scale topological differences but broadly similar computational properties. The source record is internally coherent and contains no apparent OCR corruption. Nearby records concern the number of inference functions available under fixed connectivity and related representational questions, which supports reading “function” here as the visible probability function or its log-weight/free-energy representative.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Effects of Network Connectivity\nSource item: 3.4\nSource URL: http://aimpl.org/boltzmann/3/\nCanonical location: aim-probability-notes.json notes[9]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Given two RBMs with the same number of units but different connectivities, how much do these statistical models overlap?\\n\\nHow to quantify function approximation given a network topology? In other words, how to measure the proximity between two functions when each are computed by separate networks? Then, can one use this measure to quantify the effects of altering a network's connectivity structure?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This question was motivated by the observation that biological brains often have fine-scale differences in network topology, but overall have highly similar computational properties.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/3/", "tags": [ "aim", "AIM-PROBABILITY-0010", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For 0/1 binary RBMs, zeroing a set of edges whose weights have total deleted l1 norm L changes the visible log-weight function by oscillation at most L, a sharp improvement over the naive 2L Lipschitz certificate. The resulting bounded exponential tilt gives two-sided KL at most min(L,L^2/8), TV at most tanh(L/4), and corresponding Hellinger and normalized-Hamming Wasserstein bounds. These estimates yield directed and symmetric Hausdorff-type bounds for edgewise norm-bounded topology classes. Separately, the common-subgraph model is contained in the intersection of two topology models, but hidden-relabeling and paired-pruning examples prove that neither full overlap nor parameter-paired proximity is determined by topology alone.\n\nCandidate contribution (sharp stability theorem and model-class reduction; novelty confidence low): The candidate contribution is the edge-by-edge proof of the encoding-aware bound d_osc(F_full,F_pruned) no greater than the deleted-edge l1 norm for 0/1 RBMs without sign restrictions, together with explicit two-sided KL, TV, Hellinger, Wasserstein, and norm-bounded directed/Hausdorff consequences and examples separating common-subgraph overlap from hidden-relabeling overlap and paired pruning." }, { "id": 20002569, "problem_number": "AIM-PROBABILITY-0011", "title": "A full-rank tropical RBM certificate for four visible units", "statement": "Is it true that tropical RBMs always have expected dimension?\n\nIs it true that tropical RBMs always have expected dimension, meaning equal to the number of parameters within the model or the dimension of the ambient probability simplex?", "original_statement": "Is it true that tropical RBMs always have expected dimension?\n\nIs it true that tropical RBMs always have expected dimension, meaning equal to the number of parameters within the model or the dimension of the ambient probability simplex?", "clean_statement": "Is it true that tropical RBMs always have expected dimension?\n\nIs it true that tropical RBMs always have expected dimension, meaning equal to the number of parameters within the model or the dimension of the ambient probability simplex?", "statement_status": "exact", "statement_verification": "The two sentences are a short question followed by its clarification; there is no apparent OCR corruption. The original URL, , returned a 502 gateway error when checked on 11 August 2026. The terminology and parameter count agree exactly with the conjecture of Cueto, Morton, and Sturmfels [CMS10].", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Tropical RBMs\nSource item: 4.1\nSource URL: http://aimpl.org/boltzmann/4/\nCanonical location: aim-probability-notes.json notes[10]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is it true that tropical RBMs always have expected dimension?\\n\\nIs it true that tropical RBMs always have expected dimension, meaning equal to the number of parameters within the model or the dimension of the ambient probability simplex?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/4/", "tags": [ "aim", "AIM-PROBABILITY-0011", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The global tropical expected-dimension conjecture remains open in the literature checked, despite the ordinary binary RBM dimension theorem. For the tropical parameterization image, an explicit strict-slicing certificate proves that TM_4^3 has full dimension 15: the cuts L={v:|v|<=1}, U={v:|v|>=3}, and F={v:v_4=1} give an unnormalized local matrix of rank 16. Together with published coding-bound cases and hidden-unit monotonicity, this proves the expected-dimension formula for every hidden-unit count when there are at most four visible units.\n\nCandidate contribution (explicit_rank_certificate; novelty confidence low): Candidate novelty: the strict slicings L={|v|<=1}, U={|v|>=3}, and F={v_4=1} yield a symbolic 16-function basis for the unnormalized local tropical map at (n,m)=(4,3), proving projective dimension 15 and closing the sole gap between the published packing and covering endpoints for four visible variables." }, { "id": 20002570, "problem_number": "AIM-PROBABILITY-0012", "title": "Zonotope support functions characterize tropical RBM parameter images", "statement": "What tropical objects characterize all tropical RBMs?\n\nWhat tropical objects (varieties) characterize all tropical RBMs?", "original_statement": "What tropical objects characterize all tropical RBMs?\n\nWhat tropical objects (varieties) characterize all tropical RBMs?", "clean_statement": "What tropical objects characterize all tropical RBMs?\n\nWhat tropical objects (varieties) characterize all tropical RBMs?", "statement_status": "exact", "statement_verification": "The canonical AIM record is from the 2018 Boltzmann Machines workshop, section “Tropical RBMs,” problem 4.2. It reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Tropical RBMs\nSource item: 4.2\nSource URL: http://aimpl.org/boltzmann/4/\nCanonical location: aim-probability-notes.json notes[11]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What tropical objects characterize all tropical RBMs?\\n\\nWhat tropical objects (varieties) characterize all tropical RBMs?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/4/", "tags": [ "aim", "AIM-PROBABILITY-0012", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For fixed n visible bits and k hidden bits, the max-plus RBM parameter image TM_n^k consists exactly, modulo constants, of the vectors obtained by evaluating the support function of a translated zonotope in R^(n+1) with at most k generators on the Boolean-lift directions (1,v). This image must be distinguished from the generally larger balanced tropical variety TV_n^k. For k=1, every fixed coordinate pair has mixed Boolean face differences of one weak sign across all parallel faces; the three-bit parity score has differences -2 and +2, so it lies outside TM_3^1 while the published equality TV_3^1=TP^7 places it inside TV_3^1.\n\nCandidate contribution (inequality certificate and explicit separator; novelty confidence low): Candidate novelty: a one-hidden-unit tropical RBM score has sign-coherent mixed second differences on every family of parallel Boolean square faces, and the three-bit parity score gives an explicit four-term separator between TM_3^1 and TV_3^1 by attaining -2 and +2 on two parallel faces." }, { "id": 20002571, "problem_number": "AIM-PROBABILITY-0013", "title": "An exact Morse-Bott landscape for leaf-hidden RBMs", "statement": "Characterizing the optimization landscape for RBMs\n\nHow do we characterize local optima on the optimization landscape for RBMs as a function of the architecture and number of empirical samples?", "original_statement": "Characterizing the optimization landscape for RBMs\n\nHow do we characterize local optima on the optimization landscape for RBMs as a function of the architecture and number of empirical samples?", "clean_statement": "Characterizing the optimization landscape for RBMs\n\nHow do we characterize local optima on the optimization landscape for RBMs as a function of the architecture and number of empirical samples?", "statement_status": "exact", "statement_verification": "The canonical AIM record, from the 2018 workshop *Boltzmann Machines*, section “RBM Optimization,” problem 5.2, asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: RBM Optimization\nSource item: 5.2\nSource URL: http://aimpl.org/boltzmann/5/\nCanonical location: aim-probability-notes.json notes[12]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Characterizing the optimization landscape for RBMs\\n\\nHow do we characterize local optima on the optimization landscape for RBMs as a function of the architecture and number of empirical samples?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/5/", "tags": [ "aim", "AIM-PROBABILITY-0013", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For arbitrary binary RBM graphs, exact empirical likelihood factors over visible-containing connected components and depends only on the corresponding empirical block marginals. For the complete architecture class in which every hidden unit has degree at most one, the visible model is product Bernoulli and the full finite-parameter landscape is classified: if every empirical coordinate mean is strictly between zero and one, the critical set is a smooth codimension-n global-maximum fiber of dimension m+e with negative-semidefinite Hessian of rank n and kernel equal to the fiber tangent space; there are no other critical points or local optima. If any coordinate is constant in the weighted sample, there is no finite local extremum and the optimum exists only in the model closure. Thus the exact finite-attainment criterion is coordinatewise empirical support, not sample count alone.\n\nCandidate contribution (theorem; novelty confidence low): The componentwise graph reduction together with a complete Morse-Bott, boundary, and sample-support classification for all binary RBM topologies whose hidden degrees are at most one, including the exact m+e dimension of the maximizer fiber." }, { "id": 20002572, "problem_number": "AIM-PROBABILITY-0014", "title": "Singular EM fixed manifolds and ML-degree-one determinant strata of RBM(3,2)", "statement": "What are the critical points of the EM algorithm for $RBM_{3,2}$?\n\nWhat are the critical points of the EM algorithm for $RBM_{3,2}$? Can the MLE degree for the strata of $RBM_{3,2}$ be determined?", "original_statement": "What are the critical points of the EM algorithm for $RBM_{3,2}$?\n\nWhat are the critical points of the EM algorithm for $RBM_{3,2}$? Can the MLE degree for the strata of $RBM_{3,2}$ be determined?", "clean_statement": "What are the critical points of the EM algorithm for $RBM_{3,2}$?\n\nWhat are the critical points of the EM algorithm for $RBM_{3,2}$? Can the MLE degree for the strata of $RBM_{3,2}$ be determined?", "statement_status": "exact", "statement_verification": "The canonical AIM record is from the 2018 Boltzmann Machines workshop, section “RBM Optimization,” problem 5.1. It asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: RBM Optimization\nSource item: 5.1\nSource URL: http://aimpl.org/boltzmann/5/\nCanonical location: aim-probability-notes.json notes[13]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"What are the critical points of the EM algorithm for $RBM_{3,2}$?\\n\\nWhat are the critical points of the EM algorithm for $RBM_{3,2}$? Can the MLE degree for the strata of $RBM_{3,2}$ be determined?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The EM fixed points may be known for ${\\\\cal M}_{3,3},$ but since these points depend on the specific parameterization of the model, the fixed points of $RBM_{3,2}$ remain unknown.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/5/", "tags": [ "aim", "AIM-PROBABILITY-0014", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For exact EM in the natural eleven-parameter binary RBM(3,2), the fixed-point equations are the eleven posterior moment-matching equations. At a parameter where the visible marginal map has rank seven, exact EM is fixed exactly when the model distribution equals the empirical table, so data outside the visible model can have finite fixed points only on the marginal rank-deficient locus. Every positive empirical table nevertheless has an explicit two-parameter singular fixed manifold: set W=0, set each visible bias to the logit of the corresponding empirical marginal, and leave both hidden biases arbitrary; the visible marginal rank is exactly three there. In addition, each of the six fixed-slice determinant hypersurfaces and each of the three parallel double intersections has ML degree one, with an explicit rational likelihood critical point.\n\nCandidate contribution (exact fixed-point family and lifting lemma; novelty confidence low): Candidate novelty: any finite exact-EM fixed point of a binary RBM(n,m) lifts to a one-parameter exact-EM fixed family of RBM(n,m+1) by appending a zero-weight hidden unit with arbitrary bias; applied twice to the independence fit, this gives an explicit R^2 family of RBM(3,2) fixed points for every positive 2x2x2 data table, and the visible marginal map has rank exactly three along that family." }, { "id": 20002573, "problem_number": "AIM-PROBABILITY-0015", "title": "Exact dropout decomposition for RBM likelihoods and an ordinary-RBM no-go example", "statement": "Is using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?\n\nIs using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?", "original_statement": "Is using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?\n\nIs using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?", "clean_statement": "Is using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?\n\nIs using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?", "statement_status": "exact", "statement_verification": "The canonical AIM record (Boltzmann Machines workshop, section “RBM Optimization,” Problem 5.3) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: RBM Optimization\nSource item: 5.3\nSource URL: http://aimpl.org/boltzmann/5/\nCanonical location: aim-probability-notes.json notes[14]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?\\n\\nIs using dropout during RBM training equivalent to adding a regularizing penalty term on to the training objective function?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This question may have been partially motivated by the observation that alternative regularization techniques for training feedforward neural networks have been shown equivalent to adding a regularizer, or penalty, term to the objective function. The instance that readily comes to mind is that the technique of early stopping is equivalent to adding an $L_2$ penalty term on the size of the network weights.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/5/", "tags": [ "aim", "AIM-PROBABILITY-0015", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For finite binary RBMs under the original independent hidden-unit dropout convention, the expected exact conditional negative log-likelihood is exactly the cross-entropy of a deterministic mean-score (normalized geometric) model plus the nonnegative, data-independent Jensen gap Omega = E[A(r)] - A(E[r]). This baseline is generally not an ordinary RBM. Relative to the ordinary unmasked RBM, the correction is data-dependent and sign-changing, as shown by an explicit one-visible/one-hidden counterexample. The arithmetic mask-mixture likelihood is a third objective, and the CD-1 training dynamics used in the original experiments do not generally admit any scalar objective interpretation.\n\nCandidate contribution (theorem_and_counterexample; novelty confidence low): The expected masked RBM log loss has an exact normalized-geometric/Jensen-gap decomposition whose penalty vanishes exactly when every genuinely random masked hidden unit is disconnected from the visible layer; nevertheless, its correction relative to the ordinary unmasked RBM takes both signs already in a one-visible/one-hidden example." }, { "id": 20002574, "problem_number": "AIM-PROBABILITY-0016", "title": "Exact graph heat flow on a product-RBM manifold", "statement": "How do we use the Wasserstein geometry on RMBs to approximation PDE solutions?\n\nHow do we use the Wasserstein geometry on RMBs to approximation PDE solutions?", "original_statement": "How do we use the Wasserstein geometry on RMBs to approximation PDE solutions?\n\nHow do we use the Wasserstein geometry on RMBs to approximation PDE solutions?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "This is visibly corrupt: “RMBs” is almost certainly “RBMs,” and “to approximation” is almost certainly “to approximate.” The original AIM problem-list page was unavailable during this run, so those corrections could not be verified against the original wording. The official 2018 AIM workshop report does verify the surrounding context: a working group on Wasserstein distance and optimal transport studied Wasserstein natural gradients for Boltzmann machines [AIM18]. Accordingly, the **plausible but not source-verified reconstruction** used here is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: Wasserstein Topics\nSource item: 6.1\nSource URL: http://aimpl.org/boltzmann/6/\nCanonical location: aim-probability-notes.json notes[15]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How do we use the Wasserstein geometry on RMBs to approximation PDE solutions?\\n\\nHow do we use the Wasserstein geometry on RMBs to approximation PDE solutions?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/6/", "tags": [ "aim", "AIM-PROBABILITY-0016", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After explicitly reconstructing the corrupted source as a question about using Wasserstein geometry on RBMs to approximate PDE solutions, this attempt fixes a precise discrete setting. For the coordinate-flip heat equation on the binary hypercube with the logarithmic-mean Maas transport metric, the product-Bernoulli visible family of a hidden-free or leaf-hidden RBM is exactly invariant. Its pullback metric is diagonal with G_ii=1/[kappa_i Lambda(r_i,1-r_i)], its entropy natural-gradient equation is rdot_i=kappa_i(1-2r_i), and the induced density is the full graph heat solution with zero projection residual and an exact dissipation identity. A general pullback projection/Pythagorean residual proposition is proved, and a two-state calculation shows why ordinary static discrete W2 cannot replace this dynamical graph metric.\n\nCandidate contribution (theorem; novelty confidence low): For the weighted coordinate-flip hypercube with the logarithmic-mean Maas metric, the product-Bernoulli RBM manifold has an explicit diagonal pullback tensor, is exactly invariant under entropy gradient flow with zero projection residual, and admits an explicit lift to redundant leaf-hidden RBM parameters." }, { "id": 20002575, "problem_number": "AIM-PROBABILITY-0017", "title": "Certifiably interpretable sparse ferromagnetic RBMs", "statement": "Can we design Boltzmann machines that are interpretable?\n\nCan we design Boltzmann machines that are interpretable? In other words, can we construct useful Boltzmann machines that admit a simple explanation in human terms, and behaves in a predictable manner with certainty?", "original_statement": "Can we design Boltzmann machines that are interpretable?\n\nCan we design Boltzmann machines that are interpretable? In other words, can we construct useful Boltzmann machines that admit a simple explanation in human terms, and behaves in a predictable manner with certainty?", "clean_statement": "Can we design Boltzmann machines that are interpretable?\n\nCan we design Boltzmann machines that are interpretable? In other words, can we construct useful Boltzmann machines that admit a simple explanation in human terms, and behaves in a predictable manner with certainty?", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 7.1 in the “More philosophical questions” section of the Boltzmann Machines workshop list. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: More philosophical questions\nSource item: 7.1\nSource URL: http://aimpl.org/boltzmann/7/\nCanonical location: aim-probability-notes.json notes[16]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Can we design Boltzmann machines that are interpretable?\\n\\nCan we design Boltzmann machines that are interpretable? In other words, can we construct useful Boltzmann machines that admit a simple explanation in human terms, and behaves in a predictable manner with certainty?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/7/", "tags": [ "aim", "AIM-PROBABILITY-0017", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A sparse binary RBM with a fixed labeled support graph and nonnegative visible-hidden weights has monotone, locally bounded hidden responses and an MTP2 visible marginal. More quantitatively, the conditional influence between visible variables i and l is at most tanh((1/16) times the sum of W_ji W_jl over their common hidden neighbors); no common hidden neighbor gives exact conditional independence. A row-sum bound below one then supplies an explicit Dobrushin contraction and mixing-time certificate for random-scan single-visible-site Gibbs inference, while MTP2 exposes negative conditional association and XOR as genuine expressive obstructions.\n\nCandidate contribution (quantitative influence theorem; novelty confidence low): For a sparse ferromagnetic binary RBM, c_il is at most tanh((1/16) sum_j W_ji W_jl), where the sum is only over hidden neighbors shared by visible variables i and l; the corresponding row-sum condition gives a directly auditable Dobrushin mixing certificate." }, { "id": 20002576, "problem_number": "AIM-PROBABILITY-0018", "title": "A gated local neural-sampling extension of Boltzmann learning", "statement": "Are there biologically plausible extensions of Boltzmann Machines for neural networks?\n\nAre there biologically plausible extensions of Boltzmann Machines for neural networks?", "original_statement": "Are there biologically plausible extensions of Boltzmann Machines for neural networks?\n\nAre there biologically plausible extensions of Boltzmann Machines for neural networks?", "clean_statement": "Are there biologically plausible extensions of Boltzmann Machines for neural networks?\n\nAre there biologically plausible extensions of Boltzmann Machines for neural networks?", "statement_status": "exact", "statement_verification": "The question is duplicated verbatim in the extracted `problem` field. This is harmless source duplication; there is no visible OCR corruption. The wording is deliberately broad. “Biologically plausible” is not a mathematical predicate until one specifies which biological constraints are required, and “extension” could refer to neuron dynamics, learning, architecture, or all three. This report therefore gives a conditional existence answer under explicit criteria, not a claim that a particular mechanism is used by an actual brain.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: More philosophical questions\nSource item: 7.2\nSource URL: http://aimpl.org/boltzmann/7/\nCanonical location: aim-probability-notes.json notes[17]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Are there biologically plausible extensions of Boltzmann Machines for neural networks?\\n\\nAre there biologically plausible extensions of Boltzmann Machines for neural networks?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/7/", "tags": [ "aim", "AIM-PROBABILITY-0018", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under explicit idealizations, exact Boltzmann likelihood learning has a biologically motivated three-factor implementation: a synapse-local coactivity trace multiplied by one signed phase signal equals the positive-minus-negative likelihood gradient at stationary equilibrium. Away from equilibrium, its expected bias is bounded by the phase-average total-variation errors and admits an explicit exponential-mixing certificate. Conversely, no memoryless synaptic rule that sees only its own weight and pre/post states under an unlabeled equal mixture of phases can recover all likelihood gradients; a full-support two-visible-unit construction gives identical observations but opposite gradients. These results establish a restricted existence construction, not evidence that actual brains implement Boltzmann learning.\n\nCandidate contribution (no_go_theorem_and_error_bound; novelty confidence low): For exact pairwise Boltzmann likelihood learning, a full-support two-unit indistinguishability example proves that completely unlabeled local coactivity is insufficient, while one signed phase bit is sufficient at equilibrium; the systematic finite-phase bias is at most the sum of the two phase-average total-variation errors, with an explicit burn-in and exponential-mixing bound." }, { "id": 20002577, "problem_number": "AIM-PROBABILITY-0019", "title": "RBMs as a certified bridge from Ising physics to interaction networks", "statement": "How to use RBMs to do statistical inference for a physical system, or properties of a network?\n\nHow to use RBMs to do statistical inference for a physical system, or properties of a network?", "original_statement": "How to use RBMs to do statistical inference for a physical system, or properties of a network?\n\nHow to use RBMs to do statistical inference for a physical system, or properties of a network?", "clean_statement": "How to use RBMs to do statistical inference for a physical system, or properties of a network?\n\nHow to use RBMs to do statistical inference for a physical system, or properties of a network?", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 7.3 in the “More philosophical questions” section of the Boltzmann Machines workshop list. Its exact text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: More philosophical questions\nSource item: 7.3\nSource URL: http://aimpl.org/boltzmann/7/\nCanonical location: aim-probability-notes.json notes[18]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"How to use RBMs to do statistical inference for a physical system, or properties of a network?\\n\\nHow to use RBMs to do statistical inference for a physical system, or properties of a network?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/7/", "tags": [ "aim", "AIM-PROBABILITY-0019", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A spin RBM whose hidden units have degree at most two has an exactly pairwise Ising visible law with explicit additive field and coupling readouts; sign-coherent hidden contributions give a no-cancellation topology certificate. Conversely, every finite pairwise Ising law has an exact representation with one hidden unit per nonzero edge. If a target pairwise Ising law p and a fitted degree-two-RBM law q differ uniformly in log density, up to an additive constant, by eta, then every fitted coupling differs from its target by at most eta and thresholding recovers the exact interaction graph when the minimum nonzero coupling exceeds 2 eta. Total-variation closeness alone cannot certify interactions: an explicit normalized two-spin family has TV at most exp(-2M) while its coupling error equals M.\n\nCandidate contribution (stability theorem and obstruction; novelty confidence low): For strictly positive pairwise Ising laws on the same spin cube, uniform log-density error modulo a constant bounds every coupling error with sharp factor one, yielding exact support recovery above a 2 eta signal gap; a normalized rare-cell tilt shows that no coupling bound can follow from total variation alone without further assumptions. The result is packaged with a sign-coherent degree-two-RBM topology readout." }, { "id": 20002578, "problem_number": "AIM-PROBABILITY-0020", "title": "A definition-aware comparison of classical, complex-amplitude, and quantum Gibbs RBMs", "statement": "The differences between RBM variants\n\nWhat are the differences between classical RBMs, Quantum RBMs, and Complex valued RBMs with quantum states?", "original_statement": "The differences between RBM variants\n\nWhat are the differences between classical RBMs, Quantum RBMs, and Complex valued RBMs with quantum states?", "clean_statement": "The differences between RBM variants\n\nWhat are the differences between classical RBMs, Quantum RBMs, and Complex valued RBMs with quantum states?", "statement_status": "exact", "statement_verification": "The canonical record is item 7.4, \"The differences between RBM variants,\" from the AIM workshop *Boltzmann Machines*, under \"More philosophical questions.\" Its exact question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Boltzmann Machines\nSection: More philosophical questions\nSource item: 7.4\nSource URL: http://aimpl.org/boltzmann/7/\nCanonical location: aim-probability-notes.json notes[19]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The differences between RBM variants\\n\\nWhat are the differences between classical RBMs, Quantum RBMs, and Complex valued RBMs with quantum states?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/boltzmann/7/", "tags": [ "aim", "AIM-PROBABILITY-0020", "aim-domain:probability", "aim-workshop:boltzmann", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After separating probability-law, pure-state, and Gibbs-state meanings, the report proves: diagonal commuting Gibbs RBMs are exactly classical RBMs at the visible-law level; real-parameter m-hidden complex-amplitude RBM Born laws are exactly the tied-pair subfamily of classical 2m-hidden RBMs; finite complex parameters can create destructive-interference support zeros unavailable to finite real classical or finite-temperature Gibbs models; and finite-temperature reduced Gibbs states are positive definite, so they cannot equal nontrivial pure CRBM states. Minimal one-qubit examples also show that identical computational-basis laws need not imply identical quantum states.\n\nCandidate contribution (comparison theorem; novelty confidence low): Candidate novelty: a single definition-aware comparison square combines the exact tied-pair m-to-2m real-amplitude correspondence with finite one-qubit support, rank, coherence, and phase witnesses separating the three RBM meanings." }, { "id": 20002579, "problem_number": "AIM-PROBABILITY-0021", "title": "Counterexamples and a corrected Cauchy theorem for harmonic means", "statement": "Let $\\mu$ be a probability density on $\\mathbb{R}$, with positive density near 0. Let $X_1, X_2, \\ldots, \\overset{i.i.d}\\sim \\mu$, and define the harmonic mean\n\\[H_n = \\frac{n}{\\frac{1}{X_1} + \\cdots + \\frac{1}{X_n}}.\\]\nProve, using Stein's method, that $H_n$ converges to a Cauchy distribution as $n \\rightarrow \\infty$, and find its rate.", "original_statement": "Let $\\mu$ be a probability density on $\\mathbb{R}$, with positive density near 0. Let $X_1, X_2, \\ldots, \\overset{i.i.d}\\sim \\mu$, and define the harmonic mean\n\\[H_n = \\frac{n}{\\frac{1}{X_1} + \\cdots + \\frac{1}{X_n}}.\\]\nProve, using Stein's method, that $H_n$ converges to a Cauchy distribution as $n \\rightarrow \\infty$, and find its rate.", "clean_statement": "Let $\\mu$ be a probability density on $\\mathbb{R}$, with positive density near 0. Let $X_1, X_2, \\ldots, \\overset{i.i.d}\\sim \\mu$, and define the harmonic mean\n\\[H_n = \\frac{n}{\\frac{1}{X_1} + \\cdots + \\frac{1}{X_n}}.\\]\nProve, using Stein's method, that $H_n$ converges to a Cauchy distribution as $n \\rightarrow \\infty$, and find its rate.", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 1.05 from the 2018 workshop *Stein's method and applications in high-dimensional statistics*. It asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Stein's method and applications in high-dimensional statistics\nSection: Problems\nSource item: 1.05\nSource URL: http://aimpl.org/steinhd/1/\nCanonical location: aim-probability-notes.json notes[20]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $\\\\mu$ be a probability density on $\\\\mathbb{R}$, with positive density near 0. Let $X_1, X_2, \\\\ldots, \\\\overset{i.i.d}\\\\sim \\\\mu$, and define the harmonic mean\\n\\\\[H_n = \\\\frac{n}{\\\\frac{1}{X_1} + \\\\cdots + \\\\frac{1}{X_n}}.\\\\]\\nProve, using Stein's method, that $H_n$ converges to a Cauchy distribution as $n \\\\rightarrow \\\\infty$, and find its rate.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/steinhd/1/", "tags": [ "aim", "AIM-PROBABILITY-0021", "aim-domain:probability", "aim-workshop:steinhd", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The literal positivity-only statement is false. For the density f=1/3 on (-1,0) and f=2/3 on (0,1), the reciprocal sum satisfies S_n/(n log n) -> 1/3 in probability, so H_n log n -> 3 and H_n degenerates at zero. Even continuity with equal one-sided limits is insufficient: an explicit continuous-at-zero density has S_n/(n log log n) -> C. A valid qualitative theorem additionally requires balanced reciprocal tails and a finite principal-value center; in the symmetric uniform case the published Stein bounds give Kolmogorov error O(n^{-1/2} log n).\n\nCandidate contribution (counterexample; novelty confidence low): The explicit asymmetric density gives the testable scaling H_n log n -> 3 with an elementary quantitative truncation bound, while a paired density continuous at zero gives H_n -> 0 through an n log log n reciprocal-center drift; together they isolate tail balance and finite principal-value centering as distinct missing hypotheses." }, { "id": 20002580, "problem_number": "AIM-PROBABILITY-0022", "title": "A quantitative arcsine bound for permutation-walk occupation time", "statement": "Let $\\pi$ be a random permutation in $S_n$. For each $1 \\le i \\le n - 1$, define the random variables\n\\begin{align*}\nX_i &=\n\\begin{cases}\n1 &\\mbox{if } \\pi(i + 1) 0\\}}{n} \\rightarrow \\text{Beta}(1/2, 1/2),\\]\nand compute the rate of convergence.", "original_statement": "Let $\\pi$ be a random permutation in $S_n$. For each $1 \\le i \\le n - 1$, define the random variables\n\\begin{align*}\nX_i &=\n\\begin{cases}\n1 &\\mbox{if } \\pi(i + 1) 0\\}}{n} \\rightarrow \\text{Beta}(1/2, 1/2),\\]\nand compute the rate of convergence.", "clean_statement": "Let $\\pi$ be a random permutation in $S_n$. For each $1 \\le i \\le n - 1$, define the random variables\n\\begin{align*}\nX_i &=\n\\begin{cases}\n1 &\\mbox{if } \\pi(i + 1) 0\\}}{n} \\rightarrow \\text{Beta}(1/2, 1/2),\\]\nand compute the rate of convergence.", "statement_status": "exact", "statement_verification": "The canonical record is source index 21 of `aim-probability-notes.json`, from the AIM workshop *Stein's method and applications in high-dimensional statistics*. Its `problem` field is preserved verbatim below. It is visibly truncated and is not a syntactically complete mathematical statement.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Stein's method and applications in high-dimensional statistics\nSection: Problems\nSource item: 1.1\nSource URL: http://aimpl.org/steinhd/1/\nCanonical location: aim-probability-notes.json notes[21]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $\\\\pi$ be a random permutation in $S_n$. For each $1 \\\\le i \\\\le n - 1$, define the random variables\\n\\\\begin{align*}\\nX_i &=\\n\\\\begin{cases}\\n1 &\\\\mbox{if } \\\\pi(i + 1) 0\\\\}}{n} \\\\rightarrow \\\\text{Beta}(1/2, 1/2),\\\\]\\nand compute the rate of convergence.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/steinhd/1/", "tags": [ "aim", "AIM-PROBABILITY-0022", "aim-domain:probability", "aim-workshop:steinhd", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After preserving the corrupt canonical OCR and reconstructing the intended uniform ascent/descent occupation statistic from Fang et al. (2021), this attempt proves that its Wasserstein distance to Beta(1/2,1/2) is at most C n^{-1/5} sqrt(log n) for all sufficiently large n and at least 1/(2 pi n). More generally, the same bounds hold, with a q-dependent upper constant, for the strict-positive occupation fraction of the centered Mallows(q) ascent/descent walk for every fixed q in (0,1]. The upper bound follows from the published strong Brownian embedding, Gaussian small-ball estimates, and an explicit integer-to-continuous Brownian occupation comparison; the lower bound is a universal lattice obstruction.\n\nCandidate contribution (quantitative_bound; novelty confidence low): For fixed q in (0,1], if A_n^(q) is the strict-positive occupation fraction of the centered Mallows(q) ascent/descent walk and Z has the Beta(1/2,1/2) law, then 1/(2 pi n) <= d_W(A_n^(q), Z) <= C_q n^{-1/5} sqrt(log n) for all sufficiently large n." }, { "id": 20002581, "problem_number": "AIM-PROBABILITY-0023", "title": "Time-uniform self-normalization without Cauchy moments", "statement": "Let $X_1, X_2, \\ldots$ be i.i.d. symmetric Cauchy random variables, and let\n\\begin{align*}\nS_t &= \\sum_{i = 1}^t X_i \\\\\nV_t &= \\sum_{i = 1}^t X_i^2\n\\end{align*}\nUsing Stein's method, prove that for all $m, x \\ge 0$,\n\\[\\mathbb{P}\\left( \\exists t \\in \\mathbb{N}: \\frac{S_t}{V_t + m} > x \\right) \\le e^{-2mx^2}.\\]", "original_statement": "Let $X_1, X_2, \\ldots$ be i.i.d. symmetric Cauchy random variables, and let\n\\begin{align*}\nS_t &= \\sum_{i = 1}^t X_i \\\\\nV_t &= \\sum_{i = 1}^t X_i^2\n\\end{align*}\nUsing Stein's method, prove that for all $m, x \\ge 0$,\n\\[\\mathbb{P}\\left( \\exists t \\in \\mathbb{N}: \\frac{S_t}{V_t + m} > x \\right) \\le e^{-2mx^2}.\\]", "clean_statement": "Let $X_1, X_2, \\ldots$ be i.i.d. symmetric Cauchy random variables, and let\n\\begin{align*}\nS_t &= \\sum_{i = 1}^t X_i \\\\\nV_t &= \\sum_{i = 1}^t X_i^2\n\\end{align*}\nUsing Stein's method, prove that for all $m, x \\ge 0$,\n\\[\\mathbb{P}\\left( \\exists t \\in \\mathbb{N}: \\frac{S_t}{V_t + m} > x \\right) \\le e^{-2mx^2}.\\]", "statement_status": "exact", "statement_verification": "The canonical record is Problem 1.15 from the 2018 AIM workshop *Stein's method and applications in high-dimensional statistics*. It asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Stein's method and applications in high-dimensional statistics\nSection: Problems\nSource item: 1.15\nSource URL: http://aimpl.org/steinhd/1/\nCanonical location: aim-probability-notes.json notes[22]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X_1, X_2, \\\\ldots$ be i.i.d. symmetric Cauchy random variables, and let\\n\\\\begin{align*}\\nS_t &= \\\\sum_{i = 1}^t X_i \\\\\\\\\\nV_t &= \\\\sum_{i = 1}^t X_i^2\\n\\\\end{align*}\\nUsing Stein's method, prove that for all $m, x \\\\ge 0$,\\n\\\\[\\\\mathbb{P}\\\\left( \\\\exists t \\\\in \\\\mathbb{N}: \\\\frac{S_t}{V_t + m} > x \\\\right) \\\\le e^{-2mx^2}.\\\\]\"\nOriginal remarks: [\"This inequality can be found as Example 4 in https://arxiv.org/pdf/1808.03204.pdf. There is nothing special about Cauchy, it applies to any symmetric increments. The Cauchy instance just drives home the point that under symmetry, no moments are needed (and hence S_t is not a martingale) to derive a self-normalized concentration inequality as stated.\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/steinhd/1/", "tags": [ "aim", "AIM-PROBABILITY-0023", "aim-domain:probability", "aim-workshop:steinhd", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The requested inequality is fully proved by the verified exponential-supermartingale method: conditional symmetry makes L_t(lambda)=exp(lambda S_t-lambda^2 V_t/2) a nonnegative supermartingale even though the Cauchy sums are not integrable, because each one-step exponential factor is bounded and its conditional mean is at most one. Ville's inequality with lambda=2x gives the bound exp(-2mx^2). The proof extends to arbitrary conditionally symmetric adapted increments with finite predictable weights, and a countable allocation of Ville bounds yields a simultaneous post-hoc tuning boundary.\n\nCandidate contribution (boundary_generalization; novelty confidence low): For conditionally symmetric increments xi_t and arbitrary finite predictable weights A_t, the weighted sums T_t and observed squares Q_t satisfy, for every nonempty countable grid lambda_j>0 with positive weights w_j summing to at most one, the simultaneous anytime boundary T_t <= inf_j{lambda_j Q_t/2 + log(1/(alpha w_j))/lambda_j} with probability at least 1-alpha, together with the stated two-sided analogue." }, { "id": 20002582, "problem_number": "AIM-PROBABILITY-0024", "title": "Boundary corrections and nonexistence after SURE selection", "statement": "Let $X \\sim MVN(\\theta, I_d)$, where $I_d$ denotes the $d$-dimensional identity matrix. Consider the problem of estimating $\\theta$ using estimators of the form $X + h(X)$, where $h$ is in some class $\\mathcal{H}$ of functions. By SURE,\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2\\]\nis an unbiased estimator of the risk associated with $h$. Let\n\\begin{align*}\nh^* &= \\text{arg min} (h \\in \\mathcal{H}: \\hat{r}_h) \\\\\n\\hat{\\theta}^* &= X + h^*(X).\n\\end{align*}\nFind an unbiased estimator $r^{**}$ of the risk $\\mathbb{E}(\\|\\hat{\\theta}^* - \\theta\\|^2)$ of $\\hat{\\theta}^*$.", "original_statement": "Let $X \\sim MVN(\\theta, I_d)$, where $I_d$ denotes the $d$-dimensional identity matrix. Consider the problem of estimating $\\theta$ using estimators of the form $X + h(X)$, where $h$ is in some class $\\mathcal{H}$ of functions. By SURE,\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2\\]\nis an unbiased estimator of the risk associated with $h$. Let\n\\begin{align*}\nh^* &= \\text{arg min} (h \\in \\mathcal{H}: \\hat{r}_h) \\\\\n\\hat{\\theta}^* &= X + h^*(X).\n\\end{align*}\nFind an unbiased estimator $r^{**}$ of the risk $\\mathbb{E}(\\|\\hat{\\theta}^* - \\theta\\|^2)$ of $\\hat{\\theta}^*$.", "clean_statement": "Let $X \\sim MVN(\\theta, I_d)$, where $I_d$ denotes the $d$-dimensional identity matrix. Consider the problem of estimating $\\theta$ using estimators of the form $X + h(X)$, where $h$ is in some class $\\mathcal{H}$ of functions. By SURE,\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2\\]\nis an unbiased estimator of the risk associated with $h$. Let\n\\begin{align*}\nh^* &= \\text{arg min} (h \\in \\mathcal{H}: \\hat{r}_h) \\\\\n\\hat{\\theta}^* &= X + h^*(X).\n\\end{align*}\nFind an unbiased estimator $r^{**}$ of the risk $\\mathbb{E}(\\|\\hat{\\theta}^* - \\theta\\|^2)$ of $\\hat{\\theta}^*$.", "statement_status": "exact", "statement_verification": "The canonical source is Problem 1.2 from the AIM workshop *Stein's method and applications in high-dimensional statistics*. The exact mathematical request is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Stein's method and applications in high-dimensional statistics\nSection: Problems\nSource item: 1.2\nSource URL: http://aimpl.org/steinhd/1/\nCanonical location: aim-probability-notes.json notes[23]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X \\\\sim MVN(\\\\theta, I_d)$, where $I_d$ denotes the $d$-dimensional identity matrix. Consider the problem of estimating $\\\\theta$ using estimators of the form $X + h(X)$, where $h$ is in some class $\\\\mathcal{H}$ of functions. By SURE,\\n\\\\[\\\\hat{r}_h = d + 2\\\\nabla \\\\cdot h(X) + \\\\|h(X)\\\\|^2\\\\]\\nis an unbiased estimator of the risk associated with $h$. Let\\n\\\\begin{align*}\\nh^* &= \\\\text{arg min} (h \\\\in \\\\mathcal{H}: \\\\hat{r}_h) \\\\\\\\\\n\\\\hat{\\\\theta}^* &= X + h^*(X).\\n\\\\end{align*}\\nFind an unbiased estimator $r^{**}$ of the risk $\\\\mathbb{E}(\\\\|\\\\hat{\\\\theta}^* - \\\\theta\\\\|^2)$ of $\\\\hat{\\\\theta}^*$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/steinhd/1/", "tags": [ "aim", "AIM-PROBABILITY-0024", "aim-domain:probability", "aim-workshop:steinhd", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The universal request for an ordinary unbiased risk statistic after pointwise SURE selection is false as stated. For a finite smooth candidate class whose selection regions form a regular partition, the selected risk equals expected regionwise SURE plus the Gaussian pairing with a total signed hypersurface jump-flux measure. Regionwise selected SURE is unbiased for all means exactly when that total measure vanishes. If the measure is nonzero and singular, bilateral Laplace-transform uniqueness proves that no ordinary Borel statistic integrable under every N(theta,I_d) can be unbiased for the risk for all theta. The two-candidate class h_0=0 and h_1=-identity yields an explicit radial hard-threshold counterexample in every dimension.\n\nCandidate contribution (nonexistence criterion; novelty confidence low): Candidate novelty: for finite smooth SURE-selection partitions with compact interfaces, a nonzero singular total normal-flux measure rules out every ordinary single-observation unbiased risk statistic, and the class {0,-identity} generates this obstruction directly by minimizing two individually unbiased SURE criteria." }, { "id": 20002583, "problem_number": "AIM-PROBABILITY-0025", "title": "Sharp total-variation asymptotics for Student t versus the standard normal", "statement": "Let $t_r$ denote the $t$ distribution with $r$ degrees of freedom. As $r \\rightarrow \\infty$, we know that $t_r \\overset{d}\\rightarrow Z$, where $Z$ is distributed as $N(0, 1)$.\n\\begin{itemize}\n\\item[(a)]\nShow that there exists a constant $c > 0$ for which\n\\[d_{TV}(t_r, Z) \\le c/r.\\]\n\\item[(b)]\nFind the best such constant.\n\\item[(c)]\nDoes there exist a constant $c'$ for which\n\\[d_{TV}(t_r, Z) \\ge c'/r?\\]\n\\end{itemize}", "original_statement": "Let $t_r$ denote the $t$ distribution with $r$ degrees of freedom. As $r \\rightarrow \\infty$, we know that $t_r \\overset{d}\\rightarrow Z$, where $Z$ is distributed as $N(0, 1)$.\n\\begin{itemize}\n\\item[(a)]\nShow that there exists a constant $c > 0$ for which\n\\[d_{TV}(t_r, Z) \\le c/r.\\]\n\\item[(b)]\nFind the best such constant.\n\\item[(c)]\nDoes there exist a constant $c'$ for which\n\\[d_{TV}(t_r, Z) \\ge c'/r?\\]\n\\end{itemize}", "clean_statement": "Let $t_r$ denote the $t$ distribution with $r$ degrees of freedom. As $r \\rightarrow \\infty$, we know that $t_r \\overset{d}\\rightarrow Z$, where $Z$ is distributed as $N(0, 1)$.\n\\begin{itemize}\n\\item[(a)]\nShow that there exists a constant $c > 0$ for which\n\\[d_{TV}(t_r, Z) \\le c/r.\\]\n\\item[(b)]\nFind the best such constant.\n\\item[(c)]\nDoes there exist a constant $c'$ for which\n\\[d_{TV}(t_r, Z) \\ge c'/r?\\]\n\\end{itemize}", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 1.25 from the 2018 workshop *Stein's method and applications in high-dimensional statistics*. It asks, for Student \\(t_r\\) with \\(r\\) degrees of freedom and \\(Z\\sim N(0,1)\\):", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Stein's method and applications in high-dimensional statistics\nSection: Problems\nSource item: 1.25\nSource URL: http://aimpl.org/steinhd/1/\nCanonical location: aim-probability-notes.json notes[24]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $t_r$ denote the $t$ distribution with $r$ degrees of freedom. As $r \\\\rightarrow \\\\infty$, we know that $t_r \\\\overset{d}\\\\rightarrow Z$, where $Z$ is distributed as $N(0, 1)$.\\n\\\\begin{itemize}\\n\\\\item[(a)]\\nShow that there exists a constant $c > 0$ for which\\n\\\\[d_{TV}(t_r, Z) \\\\le c/r.\\\\]\\n\\\\item[(b)]\\nFind the best such constant.\\n\\\\item[(c)]\\nDoes there exist a constant $c'$ for which\\n\\\\[d_{TV}(t_r, Z) \\\\ge c'/r?\\\\]\\n\\\\end{itemize}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/steinhd/1/", "tags": [ "aim", "AIM-PROBABILITY-0025", "aim-domain:probability", "aim-workshop:steinhd", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Under the standard convention d_TV=(1/2) integral |f_r-phi|, Pinelis's theorem and the symmetric single-crossing identity d_TV=2d_K give d_TV(t_r,Z)=4 and lim_{r to infinity} r d_TV(t_r,Z)=kappa, where kappa=(1/2)sqrt((7+5sqrt(2))/(pi exp(1+sqrt(2)))). Thus kappa is the exact best coefficient for the source problem's natural asymptotic/r>=4 reading, and every c' in (0,kappa) gives a matching eventual lower bound. For the alternative all-positive-integer reading, a universal upper bound and positive lower bound are proved, while the exact upper constant is rigorously reduced to three low-degree comparisons.\n\nCandidate contribution (asymptotic_refinement; novelty confidence low): If x_r is the positive density crossing and x_0=sqrt(1+sqrt(2)), then x_r=x_0+[(5+4sqrt(2))/(12sqrt(2)x_0)]r^(-1)+O(r^(-2)) and d_TV(t_r,Z)=kappa r^(-1)-(kappa/6)r^(-2)+O(r^(-3))." }, { "id": 20002584, "problem_number": "AIM-PROBABILITY-0026", "title": "Power concavity and mode curvature of symmetric stable densities", "statement": "A random variable $X$ with distribution $f$ is called $s$-concave if, for every $\\lambda \\in [0, 1]$ and all $x$ and $y$ such that $f(x), f(y) > 0$,\n\\[f((1 - \\lambda)x + \\lambda y) > ((1 - \\lambda) f(x)^s + \\lambda f(y)^s)^{1/s}.\\]\nSuppose $Y$ is a random variable with characteristic function\n\\[\\mathbb{E}(e^{itY}) = e^{-|t|^{\\alpha}}.\\]\nShow that $Y$ is $s$-concave for some $s = s(\\alpha)$. Further, if $Y$ has density $f$, show that the curvature of $f$ at its mode is negative.", "original_statement": "A random variable $X$ with distribution $f$ is called $s$-concave if, for every $\\lambda \\in [0, 1]$ and all $x$ and $y$ such that $f(x), f(y) > 0$,\n\\[f((1 - \\lambda)x + \\lambda y) > ((1 - \\lambda) f(x)^s + \\lambda f(y)^s)^{1/s}.\\]\nSuppose $Y$ is a random variable with characteristic function\n\\[\\mathbb{E}(e^{itY}) = e^{-|t|^{\\alpha}}.\\]\nShow that $Y$ is $s$-concave for some $s = s(\\alpha)$. Further, if $Y$ has density $f$, show that the curvature of $f$ at its mode is negative.", "clean_statement": "A random variable $X$ with distribution $f$ is called $s$-concave if, for every $\\lambda \\in [0, 1]$ and all $x$ and $y$ such that $f(x), f(y) > 0$,\n\\[f((1 - \\lambda)x + \\lambda y) > ((1 - \\lambda) f(x)^s + \\lambda f(y)^s)^{1/s}.\\]\nSuppose $Y$ is a random variable with characteristic function\n\\[\\mathbb{E}(e^{itY}) = e^{-|t|^{\\alpha}}.\\]\nShow that $Y$ is $s$-concave for some $s = s(\\alpha)$. Further, if $Y$ has density $f$, show that the curvature of $f$ at its mode is negative.", "statement_status": "exact", "statement_verification": "The canonical record is source index 25 of `aim-probability-notes.json`. Its problem field is preserved exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Stein's method and applications in high-dimensional statistics\nSection: Problems\nSource item: 1.3\nSource URL: http://aimpl.org/steinhd/1/\nCanonical location: aim-probability-notes.json notes[25]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"A random variable $X$ with distribution $f$ is called $s$-concave if, for every $\\\\lambda \\\\in [0, 1]$ and all $x$ and $y$ such that $f(x), f(y) > 0$,\\n\\\\[f((1 - \\\\lambda)x + \\\\lambda y) > ((1 - \\\\lambda) f(x)^s + \\\\lambda f(y)^s)^{1/s}.\\\\]\\nSuppose $Y$ is a random variable with characteristic function\\n\\\\[\\\\mathbb{E}(e^{itY}) = e^{-|t|^{\\\\alpha}}.\\\\]\\nShow that $Y$ is $s$-concave for some $s = s(\\\\alpha)$. Further, if $Y$ has density $f$, show that the curvature of $f$ at its mode is negative.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/steinhd/1/", "tags": [ "aim", "AIM-PROBABILITY-0026", "aim-domain:probability", "aim-workshop:steinhd", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literal AIM formulation is invalid because its strict inequality fails at lambda = 0 and at x = y. After the standard correction from > to >= and restriction to the natural stable range 0 < alpha <= 2, the problem is completely proved: the symmetric stable density is positive, smooth, even, and strictly decreasing away from its unique mode 0; its mode curvature is f_alpha''(0) = -Gamma(3/alpha)/(pi alpha) < 0; and, writing R_alpha = f_alpha f_alpha''/(f_alpha')^2 and M_alpha = sup_{x>0} R_alpha(x), one has M_alpha < infinity and every s <= 1 - M_alpha with s < 0 gives s-concavity. The sharp negative index is s_alpha^* = 1 - M_alpha.\n\nCandidate contribution (variational characterization; novelty confidence low): For the symmetric alpha-stable density, the sharp negative power-concavity index is 1 - sup_{x>0}(f_alpha f_alpha''/(f_alpha')^2); the quotient tends to -infinity at zero and to (alpha+2)/(alpha+1) at infinity for 0 < alpha < 2, reducing sharpness of the tail-forced index -1/(1+alpha) to a global one-variable quotient inequality." }, { "id": 20002585, "problem_number": "AIM-PROBABILITY-0027", "title": "Finite-N boundary and routing-miss Stein comparisons for JSQ approximations", "statement": "Consider a Markov queueing system where there are $N$ servers who serve at rate 1, and new customers arrive into the system at a constant rate of $\\lambda$. When a new customer arrives, they choose to join the queue behind the server with the shortest existing length. If there is more than one queue with the shortest length, simply choose one uniformly at random. Let us assume that $\\lambda < N$ so we have a stationary distribution for this chain. Denote $Q = (q_1, q_2, \\ldots)$ as the stationary distribution of this process, where $q_i$ is the number of queues with \\emph{at least} $i$ people in the queue. This model is known as the Join the shortest queue (JSQ) model.\n\nAs a simplification to this process, suppose that when a new customer arrives, and \\emph{all} of the server queues have 2 people, then this customer leaves straight away. Let $Q_2$ denote the corresponding stationary distribution for this modified process. This is a commonly used approximation for the full JSQ model. Can we use Stein's method to explicitly bound the difference $|\\mathbb{E} f(Q) - \\mathbb{E} f(Q_2)|$ for suitable $f$?\n\nConsider another simplification of the full JSQ process, where an new customer joins the shortest of $k$ randomly chosen queues. Let $Q_3$ denote the stationary distribution of this process. What can we say about $|\\mathbb{E} f(Q) - \\mathbb{E}f(Q_3)|$?", "original_statement": "Consider a Markov queueing system where there are $N$ servers who serve at rate 1, and new customers arrive into the system at a constant rate of $\\lambda$. When a new customer arrives, they choose to join the queue behind the server with the shortest existing length. If there is more than one queue with the shortest length, simply choose one uniformly at random. Let us assume that $\\lambda < N$ so we have a stationary distribution for this chain. Denote $Q = (q_1, q_2, \\ldots)$ as the stationary distribution of this process, where $q_i$ is the number of queues with \\emph{at least} $i$ people in the queue. This model is known as the Join the shortest queue (JSQ) model.\n\nAs a simplification to this process, suppose that when a new customer arrives, and \\emph{all} of the server queues have 2 people, then this customer leaves straight away. Let $Q_2$ denote the corresponding stationary distribution for this modified process. This is a commonly used approximation for the full JSQ model. Can we use Stein's method to explicitly bound the difference $|\\mathbb{E} f(Q) - \\mathbb{E} f(Q_2)|$ for suitable $f$?\n\nConsider another simplification of the full JSQ process, where an new customer joins the shortest of $k$ randomly chosen queues. Let $Q_3$ denote the stationary distribution of this process. What can we say about $|\\mathbb{E} f(Q) - \\mathbb{E}f(Q_3)|$?", "clean_statement": "Consider a Markov queueing system where there are $N$ servers who serve at rate 1, and new customers arrive into the system at a constant rate of $\\lambda$. When a new customer arrives, they choose to join the queue behind the server with the shortest existing length. If there is more than one queue with the shortest length, simply choose one uniformly at random. Let us assume that $\\lambda < N$ so we have a stationary distribution for this chain. Denote $Q = (q_1, q_2, \\ldots)$ as the stationary distribution of this process, where $q_i$ is the number of queues with \\emph{at least} $i$ people in the queue. This model is known as the Join the shortest queue (JSQ) model.\n\nAs a simplification to this process, suppose that when a new customer arrives, and \\emph{all} of the server queues have 2 people, then this customer leaves straight away. Let $Q_2$ denote the corresponding stationary distribution for this modified process. This is a commonly used approximation for the full JSQ model. Can we use Stein's method to explicitly bound the difference $|\\mathbb{E} f(Q) - \\mathbb{E} f(Q_2)|$ for suitable $f$?\n\nConsider another simplification of the full JSQ process, where an new customer joins the shortest of $k$ randomly chosen queues. Let $Q_3$ denote the stationary distribution of this process. What can we say about $|\\mathbb{E} f(Q) - \\mathbb{E}f(Q_3)|$?", "statement_status": "exact", "statement_verification": "The source is Problem 1.35 from the AIM workshop list *Stein's method and applications in high-dimensional statistics*. It asks about \\(N\\) unit-rate exponential servers, a Poisson arrival stream, and the occupancy vector", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Stein's method and applications in high-dimensional statistics\nSection: Problems\nSource item: 1.35\nSource URL: http://aimpl.org/steinhd/1/\nCanonical location: aim-probability-notes.json notes[26]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Consider a Markov queueing system where there are $N$ servers who serve at rate 1, and new customers arrive into the system at a constant rate of $\\\\lambda$. When a new customer arrives, they choose to join the queue behind the server with the shortest existing length. If there is more than one queue with the shortest length, simply choose one uniformly at random. Let us assume that $\\\\lambda < N$ so we have a stationary distribution for this chain. Denote $Q = (q_1, q_2, \\\\ldots)$ as the stationary distribution of this process, where $q_i$ is the number of queues with \\\\emph{at least} $i$ people in the queue. This model is known as the Join the shortest queue (JSQ) model.\\n\\nAs a simplification to this process, suppose that when a new customer arrives, and \\\\emph{all} of the server queues have 2 people, then this customer leaves straight away. Let $Q_2$ denote the corresponding stationary distribution for this modified process. This is a commonly used approximation for the full JSQ model. Can we use Stein's method to explicitly bound the difference $|\\\\mathbb{E} f(Q) - \\\\mathbb{E} f(Q_2)|$ for suitable $f$?\\n\\nConsider another simplification of the full JSQ process, where an new customer joins the shortest of $k$ randomly chosen queues. Let $Q_3$ denote the stationary distribution of this process. What can we say about $|\\\\mathbb{E} f(Q) - \\\\mathbb{E}f(Q_3)|$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/steinhd/1/", "tags": [ "aim", "AIM-PROBABILITY-0027", "aim-domain:probability", "aim-workshop:steinhd", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the capacity-two approximation, the stationary Stein discrepancy localizes exactly to the blocked arrival: if the full-JSQ Poisson equation is A g_f = f - pi f, then pi_2 f - pi f = lambda E_{pi_2}[1{q_2=N} Delta_3^+ g_f], and the blocking probability is at most min{rho, exp[-2N(log(1/rho)-1+rho)]}. For power-of-k routing, the exact generator discrepancy localizes to the event that all sampled queues miss every globally shortest queue; this yields bounds lambda Omega(f)((N-1)/N)^k with replacement and lambda Omega(f)(N-k)/N without replacement. The bounds are completely explicit on the noncircular generator-image class f=A u with normalized increments.\n\nCandidate contribution (finite-N generator comparison theorem; novelty confidence low): The paired boundary/miss identities, combined with a top-state balance, monotone Poisson domination for the capacity-two population, and an arrival-gradient oscillation factor for power-of-k, give explicit finite-N coefficients for both approximations." }, { "id": 20002586, "problem_number": "AIM-PROBABILITY-0028", "title": "Exact non-Gaussian SURE bias and transformed shrinkage criteria", "statement": "Two approaches to generalize Stein's method to non-Gaussian distributions are the following:\n\\begin{itemize}\n\\item[(i)]\nStein coefficients: for a random variable $Y$, the \\emph{Stein coefficient} $T(Y)$ satisfies\n\\[\\mathbb{E}(Y f(Y)) = \\mathbb{E}(T f'(Y)).\\]\n\\item[(ii)]\nZero-biasing: for $Y$ of mean 0 and variance $\\sigma^2$, compute the $Y^*$ that satisfies\n\\[\\mathbb{E}(Yf(Y)) = \\sigma^2 \\mathbb{E} f'(Y^*).\\]\n\nLet $Y_1, \\ldots, Y_d \\overset{i.i.d.}\\sim Y$, and let $\\overline{Y} = (Y_1, \\ldots, Y_d)$. Suppose that $\\mathbb{E} Y_i = 0$, and let $X = Y + \\theta$. Here we will assume $Y$ is a known distribution while $\\theta$ is unknown.\n\nConsider estimators $\\hat{\\theta}$ of $\\Theta$ of the form $\\hat{\\theta} = X + h(X)$. Recall that SURE for Gaussian random variables gives the unbiased estimate of risk\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2.\\]\nCalculate the bias\n\\[e_d := \\mathbb{E}(\\hat{r}_h - \\mathbb{E} \\|\\hat{\\theta} - \\theta\\|^2)\\]\nusing Stein coefficients or zero-biasing. Does $\\frac{e_d}{d}$ converge to 0 as $d \\rightarrow \\infty$? Can we apply this to wavelet shrinkage?\n\\end{itemize}", "original_statement": "Two approaches to generalize Stein's method to non-Gaussian distributions are the following:\n\\begin{itemize}\n\\item[(i)]\nStein coefficients: for a random variable $Y$, the \\emph{Stein coefficient} $T(Y)$ satisfies\n\\[\\mathbb{E}(Y f(Y)) = \\mathbb{E}(T f'(Y)).\\]\n\\item[(ii)]\nZero-biasing: for $Y$ of mean 0 and variance $\\sigma^2$, compute the $Y^*$ that satisfies\n\\[\\mathbb{E}(Yf(Y)) = \\sigma^2 \\mathbb{E} f'(Y^*).\\]\n\nLet $Y_1, \\ldots, Y_d \\overset{i.i.d.}\\sim Y$, and let $\\overline{Y} = (Y_1, \\ldots, Y_d)$. Suppose that $\\mathbb{E} Y_i = 0$, and let $X = Y + \\theta$. Here we will assume $Y$ is a known distribution while $\\theta$ is unknown.\n\nConsider estimators $\\hat{\\theta}$ of $\\Theta$ of the form $\\hat{\\theta} = X + h(X)$. Recall that SURE for Gaussian random variables gives the unbiased estimate of risk\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2.\\]\nCalculate the bias\n\\[e_d := \\mathbb{E}(\\hat{r}_h - \\mathbb{E} \\|\\hat{\\theta} - \\theta\\|^2)\\]\nusing Stein coefficients or zero-biasing. Does $\\frac{e_d}{d}$ converge to 0 as $d \\rightarrow \\infty$? Can we apply this to wavelet shrinkage?\n\\end{itemize}", "clean_statement": "Two approaches to generalize Stein's method to non-Gaussian distributions are the following:\n\\begin{itemize}\n\\item[(i)]\nStein coefficients: for a random variable $Y$, the \\emph{Stein coefficient} $T(Y)$ satisfies\n\\[\\mathbb{E}(Y f(Y)) = \\mathbb{E}(T f'(Y)).\\]\n\\item[(ii)]\nZero-biasing: for $Y$ of mean 0 and variance $\\sigma^2$, compute the $Y^*$ that satisfies\n\\[\\mathbb{E}(Yf(Y)) = \\sigma^2 \\mathbb{E} f'(Y^*).\\]\n\nLet $Y_1, \\ldots, Y_d \\overset{i.i.d.}\\sim Y$, and let $\\overline{Y} = (Y_1, \\ldots, Y_d)$. Suppose that $\\mathbb{E} Y_i = 0$, and let $X = Y + \\theta$. Here we will assume $Y$ is a known distribution while $\\theta$ is unknown.\n\nConsider estimators $\\hat{\\theta}$ of $\\Theta$ of the form $\\hat{\\theta} = X + h(X)$. Recall that SURE for Gaussian random variables gives the unbiased estimate of risk\n\\[\\hat{r}_h = d + 2\\nabla \\cdot h(X) + \\|h(X)\\|^2.\\]\nCalculate the bias\n\\[e_d := \\mathbb{E}(\\hat{r}_h - \\mathbb{E} \\|\\hat{\\theta} - \\theta\\|^2)\\]\nusing Stein coefficients or zero-biasing. Does $\\frac{e_d}{d}$ converge to 0 as $d \\rightarrow \\infty$? Can we apply this to wavelet shrinkage?\n\\end{itemize}", "statement_status": "exact", "statement_verification": "The canonical AIM record is Problem 1.4 from the workshop *Stein's method and applications in high-dimensional statistics*. It proposes two non-Gaussian integration-by-parts devices,", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Stein's method and applications in high-dimensional statistics\nSection: Problems\nSource item: 1.4\nSource URL: http://aimpl.org/steinhd/1/\nCanonical location: aim-probability-notes.json notes[27]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Two approaches to generalize Stein's method to non-Gaussian distributions are the following:\\n\\\\begin{itemize}\\n\\\\item[(i)]\\nStein coefficients: for a random variable $Y$, the \\\\emph{Stein coefficient} $T(Y)$ satisfies\\n\\\\[\\\\mathbb{E}(Y f(Y)) = \\\\mathbb{E}(T f'(Y)).\\\\]\\n\\\\item[(ii)]\\nZero-biasing: for $Y$ of mean 0 and variance $\\\\sigma^2$, compute the $Y^*$ that satisfies\\n\\\\[\\\\mathbb{E}(Yf(Y)) = \\\\sigma^2 \\\\mathbb{E} f'(Y^*).\\\\]\\n\\nLet $Y_1, \\\\ldots, Y_d \\\\overset{i.i.d.}\\\\sim Y$, and let $\\\\overline{Y} = (Y_1, \\\\ldots, Y_d)$. Suppose that $\\\\mathbb{E} Y_i = 0$, and let $X = Y + \\\\theta$. Here we will assume $Y$ is a known distribution while $\\\\theta$ is unknown.\\n\\nConsider estimators $\\\\hat{\\\\theta}$ of $\\\\Theta$ of the form $\\\\hat{\\\\theta} = X + h(X)$. Recall that SURE for Gaussian random variables gives the unbiased estimate of risk\\n\\\\[\\\\hat{r}_h = d + 2\\\\nabla \\\\cdot h(X) + \\\\|h(X)\\\\|^2.\\\\]\\nCalculate the bias\\n\\\\[e_d := \\\\mathbb{E}(\\\\hat{r}_h - \\\\mathbb{E} \\\\|\\\\hat{\\\\theta} - \\\\theta\\\\|^2)\\\\]\\nusing Stein coefficients or zero-biasing. Does $\\\\frac{e_d}{d}$ converge to 0 as $d \\\\rightarrow \\\\infty$? Can we apply this to wavelet shrinkage?\\n\\\\end{itemize}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/steinhd/1/", "tags": [ "aim", "AIM-PROBABILITY-0028", "aim-domain:probability", "aim-workshop:steinhd", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "After correcting the source to X=theta+(Y_1,...,Y_d) and making the variance scaling explicit, the Gaussian SURE bias is exactly e_d=2 sum_i E[(sigma^2-tau(Y_i)) partial_i h_i(X)] and, by coordinate zero biasing, e_d=2 sigma^2 sum_i {E partial_i h_i(X)-E partial_i h_i(X^(i,*))}. There is no unconditional normalized-bias limit: unit Rademacher noise with theta=0 and h_i(x)=x_i^3 has e_d/d=4 for every d. Explicit sufficient conditions for e_d/d to vanish are proved. For an orthogonal transform W, the correct kernel is W diag(tau_i) W^T; row-l4 and cubic-coherence bounds give rigorous transformed-coordinate sufficient conditions without assuming that non-Gaussian wavelet coefficients remain independent.\n\nCandidate contribution (specialization_and_bound; novelty confidence low): For orthogonally transformed iid noise and separable transformed-coordinate shrinkage with |q_j'|<=B_j, the normalized Gaussian-SURE bias is at most (2 sqrt(Var(tau))/d) sum_j B_j (sum_i w_ji^4)^(1/2); for twice-smooth shrinkage it is also at most (2 m_tau/d) sum_{j,i} L_j |w_ji|^3, hence at most 2 L m_tau max_{j,i}|w_ji| under uniform curvature, where m_tau=E|{tau(Y)-sigma^2}Y|. An exact transformed zero-bias replacement formula is also derived." }, { "id": 20002587, "problem_number": "AIM-PROBABILITY-0029", "title": "A dependence-robust quantitative Anscombe-Renyi bound", "statement": "Let $X_1, X_2, \\ldots$ be i.i.d. random variables with mean 0 and variance 1, and let $S_n = \\sum_{i = 1}^n X_i$.\n\nFor $b > 0$, let $T_b$ be a sequence of random times. The following theorem is due to Renyi (1961):\n\\begin{theorem}\nIf $\\frac{T_b}{n_b} \\overset{p}\\rightarrow 1$ as $b \\rightarrow \\infty$, where $n_b \\rightarrow \\infty$ is deterministic, then\n\\[\\frac{S_{T_b}}{\\sqrt{n_b}} \\rightarrow N(0, 1).\\]\nCan we get a bound on $d\\left( \\frac{S_{T_b}}{n_b}, Z\\right)$?\n\\end{theorem}\n\nSome remarks: if $T_b$ is independent of $X_1, X_2, \\ldots$, then Doebler has some results. Also, see ``Randomly Stopped Sums'' by Allan Gut", "original_statement": "Let $X_1, X_2, \\ldots$ be i.i.d. random variables with mean 0 and variance 1, and let $S_n = \\sum_{i = 1}^n X_i$.\n\nFor $b > 0$, let $T_b$ be a sequence of random times. The following theorem is due to Renyi (1961):\n\\begin{theorem}\nIf $\\frac{T_b}{n_b} \\overset{p}\\rightarrow 1$ as $b \\rightarrow \\infty$, where $n_b \\rightarrow \\infty$ is deterministic, then\n\\[\\frac{S_{T_b}}{\\sqrt{n_b}} \\rightarrow N(0, 1).\\]\nCan we get a bound on $d\\left( \\frac{S_{T_b}}{n_b}, Z\\right)$?\n\\end{theorem}\n\nSome remarks: if $T_b$ is independent of $X_1, X_2, \\ldots$, then Doebler has some results. Also, see ``Randomly Stopped Sums'' by Allan Gut", "clean_statement": "Let $X_1, X_2, \\ldots$ be i.i.d. random variables with mean 0 and variance 1, and let $S_n = \\sum_{i = 1}^n X_i$.\n\nFor $b > 0$, let $T_b$ be a sequence of random times. The following theorem is due to Renyi (1961):\n\\begin{theorem}\nIf $\\frac{T_b}{n_b} \\overset{p}\\rightarrow 1$ as $b \\rightarrow \\infty$, where $n_b \\rightarrow \\infty$ is deterministic, then\n\\[\\frac{S_{T_b}}{\\sqrt{n_b}} \\rightarrow N(0, 1).\\]\nCan we get a bound on $d\\left( \\frac{S_{T_b}}{n_b}, Z\\right)$?\n\\end{theorem}\n\nSome remarks: if $T_b$ is independent of $X_1, X_2, \\ldots$, then Doebler has some results. Also, see ``Randomly Stopped Sums'' by Allan Gut", "statement_status": "exact", "statement_verification": "The canonical record is source index 28 of aim-probability-notes.json. Its problem field says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Stein's method and applications in high-dimensional statistics\nSection: Problems\nSource item: 1.45\nSource URL: http://aimpl.org/steinhd/1/\nCanonical location: aim-probability-notes.json notes[28]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $X_1, X_2, \\\\ldots$ be i.i.d. random variables with mean 0 and variance 1, and let $S_n = \\\\sum_{i = 1}^n X_i$.\\n\\nFor $b > 0$, let $T_b$ be a sequence of random times. The following theorem is due to Renyi (1961):\\n\\\\begin{theorem}\\nIf $\\\\frac{T_b}{n_b} \\\\overset{p}\\\\rightarrow 1$ as $b \\\\rightarrow \\\\infty$, where $n_b \\\\rightarrow \\\\infty$ is deterministic, then\\n\\\\[\\\\frac{S_{T_b}}{\\\\sqrt{n_b}} \\\\rightarrow N(0, 1).\\\\]\\nCan we get a bound on $d\\\\left( \\\\frac{S_{T_b}}{n_b}, Z\\\\right)$?\\n\\\\end{theorem}\\n\\nSome remarks: if $T_b$ is independent of $X_1, X_2, \\\\ldots$, then Doebler has some results. Also, see ``Randomly Stopped Sums'' by Allan Gut\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/steinhd/1/", "tags": [ "aim", "AIM-PROBABILITY-0029", "aim-domain:probability", "aim-workshop:steinhd", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The live AIM source really asks about S_{T_b}/n_b, but that normalization converges to zero and has Kolmogorov distance tending to 1/2 from a standard normal; the intended denominator is sqrt(n_b). For the repaired question, a proved finite-sample Kolmogorov bound valid under arbitrary dependence between T and all increments is d_K(S_T/sqrt(n),Z) <= d_K(S_n/sqrt(n),Z) + q_{T,n}(delta) + 3 delta^(1/3)/(4 pi)^(1/3), where q_{T,n}(delta)=P(|T-n|>floor(delta n)). With E|X_1|^3=rho this gives an explicit Berry-Esseen term C_BE rho/sqrt(n). A Gaussian rare-off-scale construction proves that T_n/n -> 1 in probability alone cannot imply any universal rate.\n\nCandidate contribution (quantitative transfer inequality; novelty confidence low): For every nonnegative integer random index T, even one depending on the entire iid increment sequence, the fixed-sum Kolmogorov error transfers to the random sum with the explicit additional term inf_{0floor(delta n)) + 3 delta^(1/3)/(4 pi)^(1/3)}; an rth-moment bound on |T/n-1| then yields exponent 1/(3r+1)." }, { "id": 20002588, "problem_number": "AIM-PROBABILITY-0030", "title": "Counterexample to a malformed CCA limit and a corrected scalar Wasserstein bound", "statement": "Let $(X_i, Y_i)_{i = 1}^n$ be i.i.d. pairs of random variables, where $X_i \\in \\mathbb{R}^p$ and $Y_i \\in \\mathbb{R}^q$.\n\\begin{itemize}\n\\item[1.]\nFind $a \\in \\mathbb{R}^p$ and $b \\in \\mathbb{R}^1$ such that\n\\[\\text{corr}\\left( a^TX, b^T Y \\right)\\]\nis maximized.\n\\item[2.]\nPick $\\hat{a}, \\hat{b}$ such that $\\sum \\hat{a}_i = \\sum \\hat{b}_i = 1$ so that\n\\[\\hat{c} := \\text{corr}\\left( \\{\\hat{a}^TX_i, \\hat{b}^TY_i\\}_{i = 1}^n \\right)\\]\nis maximized.\n\nIt is shown in Bartlett(1939) and Hotelling(1936) that if $X_i$ is independent of $Y_i$, and both have finite fourth moments, then\n\\[A_n := -\\log(1 - \\hat{c})\\left( n - \\frac{1}{2}(p + q + 3) \\right) \\overset{d}\\rightarrow \\chi^2_{p + q - 2}\\]\nas $n \\rightarrow \\infty$. Can we get bounds on $d(A_n, \\chi^2_{p + q - 2})$?\n\\end{itemize}", "original_statement": "Let $(X_i, Y_i)_{i = 1}^n$ be i.i.d. pairs of random variables, where $X_i \\in \\mathbb{R}^p$ and $Y_i \\in \\mathbb{R}^q$.\n\\begin{itemize}\n\\item[1.]\nFind $a \\in \\mathbb{R}^p$ and $b \\in \\mathbb{R}^1$ such that\n\\[\\text{corr}\\left( a^TX, b^T Y \\right)\\]\nis maximized.\n\\item[2.]\nPick $\\hat{a}, \\hat{b}$ such that $\\sum \\hat{a}_i = \\sum \\hat{b}_i = 1$ so that\n\\[\\hat{c} := \\text{corr}\\left( \\{\\hat{a}^TX_i, \\hat{b}^TY_i\\}_{i = 1}^n \\right)\\]\nis maximized.\n\nIt is shown in Bartlett(1939) and Hotelling(1936) that if $X_i$ is independent of $Y_i$, and both have finite fourth moments, then\n\\[A_n := -\\log(1 - \\hat{c})\\left( n - \\frac{1}{2}(p + q + 3) \\right) \\overset{d}\\rightarrow \\chi^2_{p + q - 2}\\]\nas $n \\rightarrow \\infty$. Can we get bounds on $d(A_n, \\chi^2_{p + q - 2})$?\n\\end{itemize}", "clean_statement": "Let $(X_i, Y_i)_{i = 1}^n$ be i.i.d. pairs of random variables, where $X_i \\in \\mathbb{R}^p$ and $Y_i \\in \\mathbb{R}^q$.\n\\begin{itemize}\n\\item[1.]\nFind $a \\in \\mathbb{R}^p$ and $b \\in \\mathbb{R}^1$ such that\n\\[\\text{corr}\\left( a^TX, b^T Y \\right)\\]\nis maximized.\n\\item[2.]\nPick $\\hat{a}, \\hat{b}$ such that $\\sum \\hat{a}_i = \\sum \\hat{b}_i = 1$ so that\n\\[\\hat{c} := \\text{corr}\\left( \\{\\hat{a}^TX_i, \\hat{b}^TY_i\\}_{i = 1}^n \\right)\\]\nis maximized.\n\nIt is shown in Bartlett(1939) and Hotelling(1936) that if $X_i$ is independent of $Y_i$, and both have finite fourth moments, then\n\\[A_n := -\\log(1 - \\hat{c})\\left( n - \\frac{1}{2}(p + q + 3) \\right) \\overset{d}\\rightarrow \\chi^2_{p + q - 2}\\]\nas $n \\rightarrow \\infty$. Can we get bounds on $d(A_n, \\chi^2_{p + q - 2})$?\n\\end{itemize}", "statement_status": "exact", "statement_verification": "The canonical record asks the following (notation preserved, including apparent errors). For i.i.d. pairs $(X_i,Y_i)$ with $X_i\\in\\mathbb R^p$ and $Y_i\\in\\mathbb R^q$:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Stein's method and applications in high-dimensional statistics\nSection: Problems\nSource item: 1.5\nSource URL: http://aimpl.org/steinhd/1/\nCanonical location: aim-probability-notes.json notes[29]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Let $(X_i, Y_i)_{i = 1}^n$ be i.i.d. pairs of random variables, where $X_i \\\\in \\\\mathbb{R}^p$ and $Y_i \\\\in \\\\mathbb{R}^q$.\\n\\\\begin{itemize}\\n\\\\item[1.]\\nFind $a \\\\in \\\\mathbb{R}^p$ and $b \\\\in \\\\mathbb{R}^1$ such that\\n\\\\[\\\\text{corr}\\\\left( a^TX, b^T Y \\\\right)\\\\]\\nis maximized.\\n\\\\item[2.]\\nPick $\\\\hat{a}, \\\\hat{b}$ such that $\\\\sum \\\\hat{a}_i = \\\\sum \\\\hat{b}_i = 1$ so that\\n\\\\[\\\\hat{c} := \\\\text{corr}\\\\left( \\\\{\\\\hat{a}^TX_i, \\\\hat{b}^TY_i\\\\}_{i = 1}^n \\\\right)\\\\]\\nis maximized.\\n\\nIt is shown in Bartlett(1939) and Hotelling(1936) that if $X_i$ is independent of $Y_i$, and both have finite fourth moments, then\\n\\\\[A_n := -\\\\log(1 - \\\\hat{c})\\\\left( n - \\\\frac{1}{2}(p + q + 3) \\\\right) \\\\overset{d}\\\\rightarrow \\\\chi^2_{p + q - 2}\\\\]\\nas $n \\\\rightarrow \\\\infty$. Can we get bounds on $d(A_n, \\\\chi^2_{p + q - 2})$?\\n\\\\end{itemize}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/steinhd/1/", "tags": [ "aim", "AIM-PROBABILITY-0030", "aim-domain:probability", "aim-workshop:steinhd", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The displayed limit is false as written: when p=q=1 under independent Gaussian sampling, the sum-one constraints force the ordinary signed Pearson correlation R_n, and the printed statistic divided by sqrt(n) converges to N(0,1), so it is not tight and cannot converge to chi-square with zero degrees of freedom; its Kolmogorov distance from the point mass chi-square_0 is at least 1/2. The classical repair is the Wilks product over 1-r_j^2 with pq degrees of freedom. For the corrected scalar statistic T_n=-(n-5/2)log(1-R_n^2), an explicit Wasserstein-1 bound of order n^{-1/2} is proved.\n\nCandidate contribution (explicit_bound; novelty confidence low): For independent standard bivariate normal samples and every integer n>6, d_W1(-(n-5/2)log(1-R_n^2), chi-square_1) is at most one half times sqrt((8n+1)/((n-4)(n-6))) plus 3(2n-5)/(4(n-4)(n-6)), hence sqrt(2/n)+O(1/n)." }, { "id": 20002589, "problem_number": "AIM-PROBABILITY-0031", "title": "A uniform-in-q product-chain cutoff and the Potts lower-bound gap", "statement": "Universal lower bound for Potts\n\nConsider the $q$-Potts model or uniform $q$ proper colorings on a general graph $G$. Is there a universal $\\asymp n\\log n$ lower bound on the mixing time uniform in the temperature? Is there always an $\\frac {n\\log n}2$ lower bound?", "original_statement": "Universal lower bound for Potts\n\nConsider the $q$-Potts model or uniform $q$ proper colorings on a general graph $G$. Is there a universal $\\asymp n\\log n$ lower bound on the mixing time uniform in the temperature? Is there always an $\\frac {n\\log n}2$ lower bound?", "clean_statement": "Universal lower bound for Potts\n\nConsider the $q$-Potts model or uniform $q$ proper colorings on a general graph $G$. Is there a universal $\\asymp n\\log n$ lower bound on the mixing time uniform in the temperature? Is there always an $\\frac {n\\log n}2$ lower bound?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Markov chain mixing times*, section *Spin systems*, problem 1.1) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Spin systems\nSource item: 1.1\nSource URL: http://aimpl.org/markovmixing/1/\nCanonical location: aim-probability-notes.json notes[30]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Universal lower bound for Potts\\n\\nConsider the $q$-Potts model or uniform $q$ proper colorings on a general graph $G$. Is there a universal $\\\\asymp n\\\\log n$ lower bound on the mixing time uniform in the temperature? Is there always an $\\\\frac {n\\\\log n}2$ lower bound?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"[Hayes, Sinclair] proved an $\\\\frac {n\\\\log n}d$ lower bound on graphs of maximum degree $d$.\\n\\n[Ding, Peres] proved an $\\\\frac {n\\\\log n}2$ lower bound for the Ising model on general graphs.\"\nResearch attempt: 2; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/1/", "tags": [ "aim", "AIM-PROBABILITY-0031", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For random-scan single-site heat-bath dynamics on the edgeless n-vertex graph with any q=q_n>=2, the total-variation mixing time is n min{log n, (1/2) log(n(q-1))}+O_epsilon(n), uniformly in q. The proof gives explicit moment, chi-square, and cover-time bounds, recovering (n/2) log n for fixed q and showing saturation at n log n when q has order at least n. Separately, disjoint unions of K_q show that uniform proper-coloring heat bath can be frozen and nonirreducible even for fixed q and arbitrarily large n, so the general question requires an ergodicity convention. The arbitrary-graph degree-free Potts/coloring lower bound remains open in the literature checked.\n\nCandidate contribution (sharp worked family; novelty confidence low): Uniformly for every sequence q_n>=2, the edgeless q_n-state random-scan heat-bath chain has total-variation cutoff within an O(n) window at n min{log n, (1/2) log(n(q_n-1))}, with explicit lower bound 1-4/(np_t)-8/[n(q_n-1)p_t^2] and complementary chi-square and strong-stationary-time upper bounds." }, { "id": 20002590, "problem_number": "AIM-PROBABILITY-0032", "title": "Exact tree-uniqueness translation and finite-size optimal mixing transfer", "statement": "Fast mixing for anti-ferromagnetic Ising at high temperature\n\nTake a (random) $d$-regular graph, and consider the anti-ferromagnetic Ising model, \\textit{i.e.} $J_{ij}=-1$ for all $i,j$. Can we show show fast mixing $O(n\\log n)$ up to\n$$(d-1)\\tanh(\\beta)<1 \\, ?$$", "original_statement": "Fast mixing for anti-ferromagnetic Ising at high temperature\n\nTake a (random) $d$-regular graph, and consider the anti-ferromagnetic Ising model, \\textit{i.e.} $J_{ij}=-1$ for all $i,j$. Can we show show fast mixing $O(n\\log n)$ up to\n$$(d-1)\\tanh(\\beta)<1 \\, ?$$", "clean_statement": "Fast mixing for anti-ferromagnetic Ising at high temperature\n\nTake a (random) $d$-regular graph, and consider the anti-ferromagnetic Ising model, \\textit{i.e.} $J_{ij}=-1$ for all $i,j$. Can we show show fast mixing $O(n\\log n)$ up to\n$$(d-1)\\tanh(\\beta)<1 \\, ?$$", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Spin systems\nSource item: 1.3\nSource URL: http://aimpl.org/markovmixing/1/\nCanonical location: aim-probability-notes.json notes[31]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Fast mixing for anti-ferromagnetic Ising at high temperature\\n\\nTake a (random) $d$-regular graph, and consider the anti-ferromagnetic Ising model, \\\\textit{i.e.} $J_{ij}=-1$ for all $i,j$. Can we show show fast mixing $O(n\\\\log n)$ up to\\n$$(d-1)\\\\tanh(\\\\beta)<1 \\\\, ?$$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is known for the ferromagnetic Ising model, \\\\textit{i.e.} when $J_{ij}=1$ [Mossel, Sly 2013].\\nAlso known: explicit sampling, decay of correlations.\"\nResearch attempt: 2; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/1/", "tags": [ "aim", "AIM-PROBABILITY-0032", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Post-workshop optimal-mixing theorems answer the AIM question affirmatively for the standard fixed-parameter interpretation, and in fact for every d-regular graph rather than only a random one. Under the physical zero-field convention exp(-beta sum sigma_u sigma_v), the two-spin edge activity is a=exp(-2 beta), the (d-1)-tree recursion has derivative magnitude (d-1)(1-a)/(1+a)=(d-1)tanh(beta), and hence its uniqueness gap is exactly delta=1-(d-1)tanh(beta). Chen--Feng--Yin--Zhang's modified log-Sobolev theorem then gives rho_GD at least 1/(C(delta)n), with C(delta)=exp(O(1/delta)), and the proved atom bound pi_min at least exp(-n(log 2+d beta)) yields an explicit O(n log n) lazy-chain mixing estimate. Chen--Liu--Vigoda directly gives O(n log n) for the standard nonlazy chain on all bounded-degree graphs in this regime.\n\nCandidate contribution (parameter-translation lemma; novelty confidence low): For zero-field antiferromagnetic Ising in the AIM physical convention, a=exp(-2 beta) converts the published two-spin uniqueness gap exactly to delta=1-(d-1)tanh(beta), and combining the corresponding MLSI with pi_min >= 2^{-n}exp(-2 beta|E|) proves t_mix^lazy(1/4) <= 8C(delta)n[log(n(log 2+d beta))+200] on every d-regular graph." }, { "id": 20002591, "problem_number": "AIM-PROBABILITY-0033", "title": "A one-dimensional counterexample and a bottleneck reduction for spin-glass mixing", "statement": "Spin glass with i.i.d. couplings\n\nConsider the spin glass model on $\\mathbb{Z}_n^d$ with i.i.d. couplings, \\textit{i.e.} with Hamiltonian given by\n$$\nH_n(\\sigma)= \\sum_{(i,j)\\in E_n} J_{ij}\\sigma_i\\sigma_j\\, ,\n$$\nwhere $J=(J_{ij})$ is a random symmetric matrix with i.i.d. $\\pm 1$ entries. The Gibbs distribution is given by\n$$\n\\mu_n(\\sigma)= \\frac{1}{Z_n}\\exp(\\beta H_n)\\, .\n$$\nFix a realization of $J_{ij}$ and consider the Glauber dynamics. Does there exist $\\beta_0$ and $\\varepsilon>0$ such that, for all $\\beta\\geq\\beta_0$, the mixing time of this chain is at least $\\exp(n^\\varepsilon)$?", "original_statement": "Spin glass with i.i.d. couplings\n\nConsider the spin glass model on $\\mathbb{Z}_n^d$ with i.i.d. couplings, \\textit{i.e.} with Hamiltonian given by\n$$\nH_n(\\sigma)= \\sum_{(i,j)\\in E_n} J_{ij}\\sigma_i\\sigma_j\\, ,\n$$\nwhere $J=(J_{ij})$ is a random symmetric matrix with i.i.d. $\\pm 1$ entries. The Gibbs distribution is given by\n$$\n\\mu_n(\\sigma)= \\frac{1}{Z_n}\\exp(\\beta H_n)\\, .\n$$\nFix a realization of $J_{ij}$ and consider the Glauber dynamics. Does there exist $\\beta_0$ and $\\varepsilon>0$ such that, for all $\\beta\\geq\\beta_0$, the mixing time of this chain is at least $\\exp(n^\\varepsilon)$?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record, titled **“Spin glass with i.i.d. couplings,”** considers nearest-neighbor spins on $\\mathbb Z_n^d$ with", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Spin systems\nSource item: 1.2\nSource URL: http://aimpl.org/markovmixing/1/\nCanonical location: aim-probability-notes.json notes[32]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Spin glass with i.i.d. couplings\\n\\nConsider the spin glass model on $\\\\mathbb{Z}_n^d$ with i.i.d. couplings, \\\\textit{i.e.} with Hamiltonian given by\\n$$\\nH_n(\\\\sigma)= \\\\sum_{(i,j)\\\\in E_n} J_{ij}\\\\sigma_i\\\\sigma_j\\\\, ,\\n$$\\nwhere $J=(J_{ij})$ is a random symmetric matrix with i.i.d. $\\\\pm 1$ entries. The Gibbs distribution is given by\\n$$\\n\\\\mu_n(\\\\sigma)= \\\\frac{1}{Z_n}\\\\exp(\\\\beta H_n)\\\\, .\\n$$\\nFix a realization of $J_{ij}$ and consider the Glauber dynamics. Does there exist $\\\\beta_0$ and $\\\\varepsilon>0$ such that, for all $\\\\beta\\\\geq\\\\beta_0$, the mixing time of this chain is at least $\\\\exp(n^\\\\varepsilon)$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: counterexample; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/1/", "tags": [ "aim", "AIM-PROBABILITY-0033", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "The d-unrestricted statement is false. On the one-dimensional cycle, for every fixed finite inverse temperature beta and every realization of the plus-or-minus-one couplings, random-site heat-bath dynamics satisfies tau_mix(epsilon) at most N times (log N plus log(1/epsilon)) divided by 1-tanh(2 beta). A gauge transformation reduces the disorder to all positive bonds with at most one antiperiodic bond, and a canonical-path argument gives polynomial mixing for a box and lazy Metropolis as well. For the intended d>=2 problem, a deterministic boundary-mass lemma shows that a stretched-exponential lower bound would follow from a quenched set A whose internal transition boundary has Gibbs-mass ratio at most exp(-n^epsilon); no such typical-disorder theorem was located, so that intended case remains open.\n\nCandidate contribution (special_case_and_reduction; novelty confidence low): For every N>=3, every coupling realization on the one-dimensional cycle, every fixed finite beta, and every epsilon in (0,1), random-site heat bath has mixing time at most N(log N+log(1/epsilon))/(1-tanh(2 beta)); moreover, for any reversible single-site chain and A of stationary mass at most one half, tau_mix(1/4) is at least pi(A)/(4 pi(B))-1 where B is the internal one-step boundary of A." }, { "id": 20002592, "problem_number": "AIM-PROBABILITY-0034", "title": "Critical Glauber mixing on random regular graphs: corrected bounds and a one-mode reduction", "statement": "Mixing time for ferromagnetic Ising at critical temperature\n\nWhat is the mixing time for the ferromagnetic Ising model at critical temperature $\\beta=\\beta_c$ on a random $d$-regular graph? Is it $n^c$?", "original_statement": "Mixing time for ferromagnetic Ising at critical temperature\n\nWhat is the mixing time for the ferromagnetic Ising model at critical temperature $\\beta=\\beta_c$ on a random $d$-regular graph? Is it $n^c$?", "clean_statement": "Mixing time for ferromagnetic Ising at critical temperature\n\nWhat is the mixing time for the ferromagnetic Ising model at critical temperature $\\beta=\\beta_c$ on a random $d$-regular graph? Is it $n^c$?", "statement_status": "exact", "statement_verification": "The repository text is internally coherent; no OCR correction is needed. The original AIM page (`http://aimpl.org/markovmixing/1/`) returned HTTP 502 when checked on 2026-08-11, so the wording above is verified against the canonical repository record rather than the live page.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Spin systems\nSource item: 1.4\nSource URL: http://aimpl.org/markovmixing/1/\nCanonical location: aim-probability-notes.json notes[33]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Mixing time for ferromagnetic Ising at critical temperature\\n\\nWhat is the mixing time for the ferromagnetic Ising model at critical temperature $\\\\beta=\\\\beta_c$ on a random $d$-regular graph? Is it $n^c$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/1/", "tags": [ "aim", "AIM-PROBABILITY-0034", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For each fixed d >= 3, the zero-field critical heat-bath Glauber dynamics on a uniform random simple d-regular graph mixes in polynomial time. In discrete random-scan updates, current rigorous results give, with high probability, a lower bound of order n^(3/2) and the corrected current upper bound O_d(n^(3+4/(d-2)) log n); the exact exponent remains open. In addition, for d >= 6 the interaction decomposes into the magnetization rank-one direction plus a transverse matrix whose critical operator norm is strictly below one with high probability.\n\nCandidate contribution (reduction; novelty confidence low): For every fixed d >= 6, if A is the adjacency matrix of a uniform random simple d-regular graph and B = A - (d/n)11^T, then at beta_c = atanh(1/(d-1)) one has ||beta_c B||_op < 1 with high probability; conditional on total magnetization, the rank-one energy is constant, leaving only this strictly transverse interaction." }, { "id": 20002593, "problem_number": "AIM-PROBABILITY-0035", "title": "Exact Potts censoring on one edge and a monotonicity obstruction", "statement": "Censoring for the Potts model\n\nConsider the $q$-state Potts model (say with $q=3$ for concreteness) and start from all green. Is it the case that deterministically censoring a sequence of spin flips can only increase the total variation distance to stationarity?", "original_statement": "Censoring for the Potts model\n\nConsider the $q$-state Potts model (say with $q=3$ for concreteness) and start from all green. Is it the case that deterministically censoring a sequence of spin flips can only increase the total variation distance to stationarity?", "clean_statement": "Censoring for the Potts model\n\nConsider the $q$-state Potts model (say with $q=3$ for concreteness) and start from all green. Is it the case that deterministically censoring a sequence of spin flips can only increase the total variation distance to stationarity?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *Markov chain mixing times*, section *Spin systems*, problem 1.5) asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Spin systems\nSource item: 1.5\nSource URL: http://aimpl.org/markovmixing/1/\nCanonical location: aim-probability-notes.json notes[34]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Censoring for the Potts model\\n\\nConsider the $q$-state Potts model (say with $q=3$ for concreteness) and start from all green. Is it the case that deterministically censoring a sequence of spin flips can only increase the total variation distance to stationarity?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This is known for monotone chains by [Peres, Winkler].\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/1/", "tags": [ "aim", "AIM-PROBABILITY-0035", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the q-state Potts heat-bath chain on one edge, started monochromatically, the total-variation distance after any nonempty deterministic update word w is ((q-1)/q) times |(lambda-1)/(lambda+q-1)|^(r(w)-1), where r(w) is the word's number of runs after consecutive repetitions are collapsed. Since a subsequence cannot increase r(w), deterministic censoring cannot improve mixing on this family for any q>=2 and any finite activity lambda>0. At lambda=1 an exact all-graph product formula also proves censoring. An exact two-neighbor calculation shows that for q>=3 and lambda!=1 no total ordering of the colors makes all Potts conditionals stochastically monotone, explaining why the Peres-Winkler theorem does not directly apply. The intended arbitrary-graph ferromagnetic constant-start question remains open; an unrestricted all-temperature/all-q reading is refuted by Holroyd's antiferromagnetic q=4 construction.\n\nCandidate contribution (special-case theorem; novelty confidence low): On K_2, for every q>=2, lambda>0, monochromatic start, deterministic update word w, and deterministic subsequence w', the uncensored law is no farther from the Potts stationary law than the censored law; moreover the exact nonempty-word distance is ((q-1)/q)|(lambda-1)/(lambda+q-1)|^(r(w)-1)." }, { "id": 20002594, "problem_number": "AIM-PROBABILITY-0036", "title": "A two-site Ising counterexample to extremal-pair TV maximization", "statement": "Diagnostics\n\nWe seek diagnostic tests for knowing that a general spin system is mixed on a general graph $G$.\n\n1. Consider the ferromagnetic Ising model on a general graph $G$. Which two $x,y\\in\\Omega$ maximize $\\|P^t(x,\\cdot)-P^t(y,\\cdot)\\|_{TV}$? More generally, is it true that in any monotone reversible chain, the maximal and minimal initial configurations maximize the total variation distance between two chains?\n\n2. Can one use a diagnostic to estimate $t_{\\mbox{mix}}$ up to $O(1)$? Of course, answering question (1) would answer this as well.\n\n3. Consider the $3$-Potts model for a general graph $G$. Can one devise a diagnostic to differentiate between $t_{\\mbox{mix}}=O(n\\log n)$ and $\\exp(\\Omega(n))$?", "original_statement": "Diagnostics\n\nWe seek diagnostic tests for knowing that a general spin system is mixed on a general graph $G$.\n\n1. Consider the ferromagnetic Ising model on a general graph $G$. Which two $x,y\\in\\Omega$ maximize $\\|P^t(x,\\cdot)-P^t(y,\\cdot)\\|_{TV}$? More generally, is it true that in any monotone reversible chain, the maximal and minimal initial configurations maximize the total variation distance between two chains?\n\n2. Can one use a diagnostic to estimate $t_{\\mbox{mix}}$ up to $O(1)$? Of course, answering question (1) would answer this as well.\n\n3. Consider the $3$-Potts model for a general graph $G$. Can one devise a diagnostic to differentiate between $t_{\\mbox{mix}}=O(n\\log n)$ and $\\exp(\\Omega(n))$?", "clean_statement": "Diagnostics\n\nWe seek diagnostic tests for knowing that a general spin system is mixed on a general graph $G$.\n\n1. Consider the ferromagnetic Ising model on a general graph $G$. Which two $x,y\\in\\Omega$ maximize $\\|P^t(x,\\cdot)-P^t(y,\\cdot)\\|_{TV}$? More generally, is it true that in any monotone reversible chain, the maximal and minimal initial configurations maximize the total variation distance between two chains?\n\n2. Can one use a diagnostic to estimate $t_{\\mbox{mix}}$ up to $O(1)$? Of course, answering question (1) would answer this as well.\n\n3. Consider the $3$-Potts model for a general graph $G$. Can one devise a diagnostic to differentiate between $t_{\\mbox{mix}}=O(n\\log n)$ and $\\exp(\\Omega(n))$?", "statement_status": "exact", "statement_verification": "The canonical record is item 1.6, “Diagnostics,” in the “Spin systems” section of the AIM workshop *Markov chain mixing times*. Its three questions are:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Spin systems\nSource item: 1.6\nSource URL: http://aimpl.org/markovmixing/1/\nCanonical location: aim-probability-notes.json notes[35]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Diagnostics\\n\\nWe seek diagnostic tests for knowing that a general spin system is mixed on a general graph $G$.\\n\\n1. Consider the ferromagnetic Ising model on a general graph $G$. Which two $x,y\\\\in\\\\Omega$ maximize $\\\\|P^t(x,\\\\cdot)-P^t(y,\\\\cdot)\\\\|_{TV}$? More generally, is it true that in any monotone reversible chain, the maximal and minimal initial configurations maximize the total variation distance between two chains?\\n\\n2. Can one use a diagnostic to estimate $t_{\\\\mbox{mix}}$ up to $O(1)$? Of course, answering question (1) would answer this as well.\\n\\n3. Consider the $3$-Potts model for a general graph $G$. Can one devise a diagnostic to differentiate between $t_{\\\\mbox{mix}}=O(n\\\\log n)$ and $\\\\exp(\\\\Omega(n))$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"For monotone spin systems, the answer to (1) is known up to a multiplicative factor that is on the order of the volume of the system. This implies that (2) is known up to $O(\\\\log n)$ using the distance between the all plus and all minus initial configurations.\"\nResearch attempt: 1; result status: counterexample; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/1/", "tags": [ "aim", "AIM-PROBABILITY-0036", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-BY-YOU", "research_summary": "For random-scan heat-bath dynamics on two Ising spins with Gibbs weights (w00,w10,w01,w11)=(2,1,3,2), the coupling is strictly ferromagnetic, the chain is monotone and reversible, yet at time one the incomparable pair (10,01) has total-variation distance 3/5 while the bottom/top pair (00,11) has distance 8/15. Exact row-difference formulas show that (10,01) is uniquely maximizing at t=1 and (00,11) is uniquely maximizing for every integer t>=2. This refutes the broad monotone-reversible conjecture and the ferromagnetic-Ising reading that permits site-dependent external fields, but not the zero-field version. The report also proves the standard e(t)<=dbar(t)<=n e(t) extremal diagnostic and gives rigorous sufficient Dobrushin-fast and conductance-slow certificates for 3-Potts.\n\nCandidate contribution (counterexample; novelty confidence low): The explicit two-spin ferromagnetic Ising heat-bath chain with weights (2,1,3,2) has a time-dependent maximizing pair: the incomparable states uniquely maximize pairwise row TV at t=1, whereas the extremal states uniquely maximize it for all t>=2." }, { "id": 20002595, "problem_number": "AIM-PROBABILITY-0037", "title": "A nonreversible stationary-valley reduction for noisy majority", "statement": "Noisy majority model\n\nThe noisy majority model with parameter $\\epsilon\\in (0,1)$ is a spin system on a graph $G$ that is the stationary distribution of the following Markov chain on $\\Omega=\\{\\pm 1\\}^{V(G)}$: assign each vertex a rate-$1$ Poisson clock. When the clock at a site rings, the spin at that vertex randomizes according to a $\\mbox{Ber}(\\frac 12)$ with probability $\\epsilon$ and chooses the majority of the spins of its neighbors and itself with probability $1-\\epsilon$ (flipping a coin in the event of a tie).\n\nConsider the noisy majority model on $\\mathbb Z_n^d$ for $d>1$. Show there exists a fixed $\\epsilon>0$ such that $t_{\\mbox{mix}}\\gtrsim \\exp(n^{\\epsilon})$ (should be true for $\\exp(cn)$).", "original_statement": "Noisy majority model\n\nThe noisy majority model with parameter $\\epsilon\\in (0,1)$ is a spin system on a graph $G$ that is the stationary distribution of the following Markov chain on $\\Omega=\\{\\pm 1\\}^{V(G)}$: assign each vertex a rate-$1$ Poisson clock. When the clock at a site rings, the spin at that vertex randomizes according to a $\\mbox{Ber}(\\frac 12)$ with probability $\\epsilon$ and chooses the majority of the spins of its neighbors and itself with probability $1-\\epsilon$ (flipping a coin in the event of a tie).\n\nConsider the noisy majority model on $\\mathbb Z_n^d$ for $d>1$. Show there exists a fixed $\\epsilon>0$ such that $t_{\\mbox{mix}}\\gtrsim \\exp(n^{\\epsilon})$ (should be true for $\\exp(cn)$).", "clean_statement": "Noisy majority model\n\nThe noisy majority model with parameter $\\epsilon\\in (0,1)$ is a spin system on a graph $G$ that is the stationary distribution of the following Markov chain on $\\Omega=\\{\\pm 1\\}^{V(G)}$: assign each vertex a rate-$1$ Poisson clock. When the clock at a site rings, the spin at that vertex randomizes according to a $\\mbox{Ber}(\\frac 12)$ with probability $\\epsilon$ and chooses the majority of the spins of its neighbors and itself with probability $1-\\epsilon$ (flipping a coin in the event of a tie).\n\nConsider the noisy majority model on $\\mathbb Z_n^d$ for $d>1$. Show there exists a fixed $\\epsilon>0$ such that $t_{\\mbox{mix}}\\gtrsim \\exp(n^{\\epsilon})$ (should be true for $\\exp(cn)$).", "statement_status": "exact", "statement_verification": "The canonical source record reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Spin systems\nSource item: 1.7\nSource URL: http://aimpl.org/markovmixing/1/\nCanonical location: aim-probability-notes.json notes[36]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Noisy majority model\\n\\nThe noisy majority model with parameter $\\\\epsilon\\\\in (0,1)$ is a spin system on a graph $G$ that is the stationary distribution of the following Markov chain on $\\\\Omega=\\\\{\\\\pm 1\\\\}^{V(G)}$: assign each vertex a rate-$1$ Poisson clock. When the clock at a site rings, the spin at that vertex randomizes according to a $\\\\mbox{Ber}(\\\\frac 12)$ with probability $\\\\epsilon$ and chooses the majority of the spins of its neighbors and itself with probability $1-\\\\epsilon$ (flipping a coin in the event of a tie).\\n\\nConsider the noisy majority model on $\\\\mathbb Z_n^d$ for $d>1$. Show there exists a fixed $\\\\epsilon>0$ such that $t_{\\\\mbox{mix}}\\\\gtrsim \\\\exp(n^{\\\\epsilon})$ (should be true for $\\\\exp(cn)$).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/1/", "tags": [ "aim", "AIM-PROBABILITY-0037", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the rate-one asynchronous noisy-majority chain on the side-length-L torus with N=L^d sites, if its stationary law satisfies delta_L = pi(|M| <= 2) <= 1/2, then t_mix(1/4) >= 1/(16 N delta_L), without any reversibility assumption. Consequently, a stationary valley bound delta_L <= exp(-kappa L^beta) implies stretched-exponential mixing. The report also proves nonreversibility by an explicit four-state cycle on the 3-by-3 torus and proves logarithmic mixing in the high-noise regime q > 2d/(2d+1). The missing stationary valley estimate remains open.\n\nCandidate contribution (reduction; novelty confidence low): A parity-sensitive directed-flow reduction for the exact asynchronous AIM model: pi(|M| <= 2) <= exp(-kappa L^beta) implies t_mix(1/4) >= exp(kappa L^beta)/(16 L^d), together with a concrete 3-by-3 Kolmogorov-cycle witness showing that reversibility cannot be assumed.", "research_difficulty_suggested": "L4" }, { "id": 20002596, "problem_number": "AIM-PROBABILITY-0038", "title": "Dyadic critical mixing solved and an exact general flip-graph reduction", "statement": "Edge-flip chain on dyadic tilings and rectangular dissections (Sarah Cannon)\n\nConsider rectangular dissections of an $n\\times n$ lattice into $n$ rectangles of area $n$, where $n=2^k$. The edge-flip chain proceeds by picking an edge bordering two rectangles and replacing it by its bisector. In the \\emph{dyadic} case, the edge is flipped provided the resulting tiling remains dyadic. In the weighted setting, a parameter $\\lambda>0$ is fixed, and the weight of a dissection is given by\n$$\n\\pi(\\sigma)=\\frac{\\lambda^{|\\sigma |}}{Z} ,\n$$\nwhere $|\\sigma|$ is the total edge-length. The chains (in the dyadic and general cases) then correspond to the Glauber dynamics.\n\nFor $\\lambda=1$, can we establish a polynomial upper-bound (fast-mixing) for the mixing time of the edge-flip chain on dyadic tilings or on rectangular dissections?", "original_statement": "Edge-flip chain on dyadic tilings and rectangular dissections (Sarah Cannon)\n\nConsider rectangular dissections of an $n\\times n$ lattice into $n$ rectangles of area $n$, where $n=2^k$. The edge-flip chain proceeds by picking an edge bordering two rectangles and replacing it by its bisector. In the \\emph{dyadic} case, the edge is flipped provided the resulting tiling remains dyadic. In the weighted setting, a parameter $\\lambda>0$ is fixed, and the weight of a dissection is given by\n$$\n\\pi(\\sigma)=\\frac{\\lambda^{|\\sigma |}}{Z} ,\n$$\nwhere $|\\sigma|$ is the total edge-length. The chains (in the dyadic and general cases) then correspond to the Glauber dynamics.\n\nFor $\\lambda=1$, can we establish a polynomial upper-bound (fast-mixing) for the mixing time of the edge-flip chain on dyadic tilings or on rectangular dissections?", "clean_statement": "Edge-flip chain on dyadic tilings and rectangular dissections (Sarah Cannon)\n\nConsider rectangular dissections of an $n\\times n$ lattice into $n$ rectangles of area $n$, where $n=2^k$. The edge-flip chain proceeds by picking an edge bordering two rectangles and replacing it by its bisector. In the \\emph{dyadic} case, the edge is flipped provided the resulting tiling remains dyadic. In the weighted setting, a parameter $\\lambda>0$ is fixed, and the weight of a dissection is given by\n$$\n\\pi(\\sigma)=\\frac{\\lambda^{|\\sigma |}}{Z} ,\n$$\nwhere $|\\sigma|$ is the total edge-length. The chains (in the dyadic and general cases) then correspond to the Glauber dynamics.\n\nFor $\\lambda=1$, can we establish a polynomial upper-bound (fast-mixing) for the mixing time of the edge-flip chain on dyadic tilings or on rectangular dissections?", "statement_status": "exact", "statement_verification": "The preserved input asks about equitable rectangular dissections of an \\(n\\times n\\) lattice square into \\(n\\) rectangles of area \\(n\\), with \\(n=2^k\\), and the edge-flip Glauber chain having stationary weight \\(\\pi(\\sigma)\\propto\\lambda^{|\\sigma|}\\). At \\(\\lambda=1\\), it asks for a polynomial mixing-time upper bound in either of two state spaces:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Edge-flip chains\nSource item: 2.1\nSource URL: http://aimpl.org/markovmixing/2/\nCanonical location: aim-probability-notes.json notes[37]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Edge-flip chain on dyadic tilings and rectangular dissections (Sarah Cannon)\\n\\nConsider rectangular dissections of an $n\\\\times n$ lattice into $n$ rectangles of area $n$, where $n=2^k$. The edge-flip chain proceeds by picking an edge bordering two rectangles and replacing it by its bisector. In the \\\\emph{dyadic} case, the edge is flipped provided the resulting tiling remains dyadic. In the weighted setting, a parameter $\\\\lambda>0$ is fixed, and the weight of a dissection is given by\\n$$\\n\\\\pi(\\\\sigma)=\\\\frac{\\\\lambda^{|\\\\sigma |}}{Z} ,\\n$$\\nwhere $|\\\\sigma|$ is the total edge-length. The chains (in the dyadic and general cases) then correspond to the Glauber dynamics.\\n\\nFor $\\\\lambda=1$, can we establish a polynomial upper-bound (fast-mixing) for the mixing time of the edge-flip chain on dyadic tilings or on rectangular dissections?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Known lower-bound $O(n\\\\log n)$ for dyadic tilings with $\\\\lambda=1$.\\nIn the dyadic case, the edge-flip chain is known to be fast-mixing $O(n^2\\\\log n)$ for $\\\\lambda1$. In the general case, the chain is slowly mixing both for $\\\\lambda1$.\\n\\nReferences:\\n\\n- Sarah Cannon, Sarah Miracle, and Dana Randall, \\\"Phase Transitions in Random Dyadic Tilings and Rectangular Dissections\\\".\\n\\n- Svante Janson, Dana Randall, and Joel Spencer, \\\"Random dyadic tilings of the unit square\\\".\\n\\n- Mike Korm, PhD thesis, Chapter 7.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/2/", "tags": [ "aim", "AIM-PROBABILITY-0038", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The dyadic lambda=1 question was solved by Cannon, Levin, and Stauffer: after accounting exactly for the source chain's extra factor-two lazification, its relaxation time is O(n^{log_2 17}) and mixing time is O(n^{1+log_2 17}). The general rectangular-dissection critical case appears open in the literature checked. For that case, the source-normalized chain is exactly I-L(G_n)/(4n), so polynomial mixing is equivalent up to polynomial exponents to inverse-polynomial edge expansion of the unweighted flip graph. As a sharp worked base case, the general n=4 graph is classified completely: it has nine states, ten edges, edge expansion 1/2, and conductance 1/32; deleting its two non-dyadic leaves gives the seven-state dyadic graph with expansion 2/3.\n\nCandidate contribution (sharp worked finite family; novelty confidence low): For integral equitable dissections of the 4 by 4 square into four area-four rectangles, the general edge-flip graph has exactly nine explicitly classified vertices and ten explicitly listed edges, its sharp unnormalized edge expansion is 1/2, and its CMR-normalized conductance is 1/32; exactly two vertices are non-dyadic leaves, and deleting them gives dyadic edge expansion 2/3." }, { "id": 20002597, "problem_number": "AIM-PROBABILITY-0039", "title": "Dyck random transpositions and a sharp edgewise comparison obstruction", "statement": "Random walks on Dyck's paths\n\nConsider the Markov chain on the set of Dyck's paths, which moves by choosing uniformly at random two coordinates and exchanging their value ($+$ or $-1$) provided the resulting path remains a Dyck's path. What is the mixing time?", "original_statement": "Random walks on Dyck's paths\n\nConsider the Markov chain on the set of Dyck's paths, which moves by choosing uniformly at random two coordinates and exchanging their value ($+$ or $-1$) provided the resulting path remains a Dyck's path. What is the mixing time?", "clean_statement": "Random walks on Dyck's paths\n\nConsider the Markov chain on the set of Dyck's paths, which moves by choosing uniformly at random two coordinates and exchanging their value ($+$ or $-1$) provided the resulting path remains a Dyck's path. What is the mixing time?", "statement_status": "exact", "statement_verification": "The canonical record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Edge-flip chains\nSource item: 2.4\nSource URL: http://aimpl.org/markovmixing/2/\nCanonical location: aim-probability-notes.json notes[38]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Random walks on Dyck's paths\\n\\nConsider the Markov chain on the set of Dyck's paths, which moves by choosing uniformly at random two coordinates and exchanging their value ($+$ or $-1$) provided the resulting path remains a Dyck's path. What is the mixing time?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Cohen, Tetali, Yeliussizov (2015) showed a lower bound of $\\\\Omega(n)$ and an upper bound of $O(n^2\\\\log n)$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/2/", "tags": [ "aim", "AIM-PROBABILITY-0039", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the ordered-with-replacement arbitrary-coordinate Dyck transposition chain K on paths of semilength n, a swap moving an up-step right is legal exactly when all intervening heights are at least 2, while a swap moving an up-step left is always legal. Comparing K with the heat-bath down--up kernel H, adjacent bases sharing an (n-1)-set T satisfy K/H = c(T)/(2n), where c(T) is the number of legal completions. At the alternating path, removing its kth up-step gives exactly c(T)=k. Thus the minimum coefficientwise off-diagonal comparison factor 1/n is attained on a genuine edge, explaining why modern O(n log n) heat-bath results transfer by the standard pointwise method only to the known O(n^2 log n) scale. The exact AIM mixing order remains open between published bounds Omega(n) and O(n^2 log n), with Theta(n log n) conjectured.\n\nCandidate contribution (obstruction; novelty confidence low): For the alternating Catalan basis B_min={1,3,...,2n-1}, the completion degree after deleting position 2k-1 is exactly k; consequently the coefficientwise off-diagonal transition comparison between the AIM proposal kernel and heat-bath down--up has sharp minimum factor 1/n on the edge arising at k=2." }, { "id": 20002598, "problem_number": "AIM-PROBABILITY-0040", "title": "Mixing of the convex-polygon triangulation flip walk", "statement": "Edge-flip on triangulations of a convex polygon\n\nConsider the random walk on triangulations of a convex polygon, driven by flips of a randomly chosen diagonal. What is the mixing time?", "original_statement": "Edge-flip on triangulations of a convex polygon\n\nConsider the random walk on triangulations of a convex polygon, driven by flips of a randomly chosen diagonal. What is the mixing time?", "clean_statement": "Edge-flip on triangulations of a convex polygon\n\nConsider the random walk on triangulations of a convex polygon, driven by flips of a randomly chosen diagonal. What is the mixing time?", "statement_status": "exact", "statement_verification": "The exact source record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Edge-flip chains\nSource item: 2.3\nSource URL: http://aimpl.org/markovmixing/2/\nCanonical location: aim-probability-notes.json notes[39]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Edge-flip on triangulations of a convex polygon\\n\\nConsider the random walk on triangulations of a convex polygon, driven by flips of a randomly chosen diagonal. What is the mixing time?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Known lower bound $\\\\Omega(n^{3/2})$, by Molloy, Reed, Steiger (1998).\\nKnown upper bound $O(n^5\\\\log n)$, by McShine and Tetali.\\nConjecture: the lower bound gives the right order.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/2/", "tags": [ "aim", "AIM-PROBABILITY-0040", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The sharp current bracket for the standard lazy flip walk on triangulations of a convex (n+2)-gon is Omega(n^{3/2}) versus tilde-O(n^2), so the conjectured exponent 3/2 remains open. In addition, for the non-lazy always-flip walk Q, a pentagonal-face argument proves 1+lambda_min(Q) >= ((3-sqrt(5))/8)(n-2)/(n-1) for every n >= 3. Thus negative spectrum is uniformly separated from -1, the non-lazy walk is aperiodic outside the necessary quadrilateral exception, and its absolute spectral gap is tilde-Omega(n^{-2}) using the 2026 lazy-walk gap bound.\n\nCandidate contribution (spectral lemma; novelty confidence low): For the simple non-lazy random walk Q on the associahedron graph of convex (n+2)-gon triangulations, 1+lambda_min(Q) is at least ((3-sqrt(5))/8)(n-2)/(n-1) for every n >= 3." }, { "id": 20002599, "problem_number": "AIM-PROBABILITY-0041", "title": "Corrected bounds and an exact critical-strip reduction for lattice triangulations", "statement": "Edge-flip chain on triangulations\n\nConsider the chain over triangulations of $[0,n]^2$, in which, at each step, an edge is randomly chosen and, if it is the diagonal of convex quadrilateral, flipped to the opposite diagonal. Does this chain have polynomial mixing time?", "original_statement": "Edge-flip chain on triangulations\n\nConsider the chain over triangulations of $[0,n]^2$, in which, at each step, an edge is randomly chosen and, if it is the diagonal of convex quadrilateral, flipped to the opposite diagonal. Does this chain have polynomial mixing time?", "clean_statement": "Edge-flip chain on triangulations\n\nConsider the chain over triangulations of $[0,n]^2$, in which, at each step, an edge is randomly chosen and, if it is the diagonal of convex quadrilateral, flipped to the opposite diagonal. Does this chain have polynomial mixing time?", "statement_status": "exact", "statement_verification": "The exact canonical problem is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Edge-flip chains\nSource item: 2.2\nSource URL: http://aimpl.org/markovmixing/2/\nCanonical location: aim-probability-notes.json notes[40]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Edge-flip chain on triangulations\\n\\nConsider the chain over triangulations of $[0,n]^2$, in which, at each step, an edge is randomly chosen and, if it is the diagonal of convex quadrilateral, flipped to the opposite diagonal. Does this chain have polynomial mixing time?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Known lower bound $O(n^3)$, given by the diameter. Upper-bound $\\\\exp(\\\\Omega(n^2))$.\\n\\nReferences:\\n\\n- Pietro Caputo, Fabio Martinelli, Alistair Sinclair, Alexandre Stauffer, \\\"Random lattice triangulations\\\".\\n\\n- Pietro Caputo, Fabio Martinelli, Alistair Sinclair, Alexandre Stauffer, \\\"Dynamics of Lattice Triangulations on Thin Rectangles\\\".\\n\\n- Alexandre Stauffer, \\\"A Lyapunov function for Glauber dynamics on lattice triangulations\\\".\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/2/", "tags": [ "aim", "AIM-PROBABILITY-0041", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the lazy uniform heat-bath flip chain on full triangulations of the n-by-n lattice square, the source's malformed bounds correct to Omega(n^3) <= t_mix <= exp(O(n^2)); no polynomial upper bound for the critical square was located, so the main problem remains open. A proved special case identifies the critical n-by-1 triangulation chain exactly with half-filled symmetric simple exclusion on a segment of length 2n. It follows that its mixing time is Theta(n^3 log n), and its heat-bath spectral gap is [1-cos(pi/(2n))]/(2n-1), asymptotic to pi^2/(16n^3). The report carefully separates the uniform square from weighted subcritical, supercritical, thin-rectangle, and convex-polygon results.\n\nCandidate contribution (special_case; novelty confidence low): Under the explicit heat-bath normalization, the uniform edge-flip chain on full n-by-1 lattice triangulations has exact spectral gap [1-cos(pi/(2n))]/(2n-1) and total-variation mixing time Theta(n^3 log n), via a move-preserving bijection to half-filled exclusion on 2n sites." }, { "id": 20002600, "problem_number": "AIM-PROBABILITY-0042", "title": "Hitting a moving trajectory on an Eulerian digraph", "statement": "Hitting time of trajectory\n\nConsider a nonreversible (say $\\frac 23$--$\\frac 13$) random walk on an Eulerian digraph $(V,E)$. Prove that the first time, $\\tau_{(X^{(y)}_t)}$ the random walk hits a moving target $X^{(y)}_t$ with $X_0=y$ satisfies the following: $\\max_{x,y} \\mathbb E_x [\\tau_{(X_t^{(y)})}] \\leq |E|\\,|V|$.", "original_statement": "Hitting time of trajectory\n\nConsider a nonreversible (say $\\frac 23$--$\\frac 13$) random walk on an Eulerian digraph $(V,E)$. Prove that the first time, $\\tau_{(X^{(y)}_t)}$ the random walk hits a moving target $X^{(y)}_t$ with $X_0=y$ satisfies the following: $\\max_{x,y} \\mathbb E_x [\\tau_{(X_t^{(y)})}] \\leq |E|\\,|V|$.", "clean_statement": "Hitting time of trajectory\n\nConsider a nonreversible (say $\\frac 23$--$\\frac 13$) random walk on an Eulerian digraph $(V,E)$. Prove that the first time, $\\tau_{(X^{(y)}_t)}$ the random walk hits a moving target $X^{(y)}_t$ with $X_0=y$ satisfies the following: $\\max_{x,y} \\mathbb E_x [\\tau_{(X_t^{(y)})}] \\leq |E|\\,|V|$.", "statement_status": "exact", "statement_verification": "The source record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Non-reversible chains\nSource item: 3.1\nSource URL: http://aimpl.org/markovmixing/3/\nCanonical location: aim-probability-notes.json notes[41]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Hitting time of trajectory\\n\\nConsider a nonreversible (say $\\\\frac 23$--$\\\\frac 13$) random walk on an Eulerian digraph $(V,E)$. Prove that the first time, $\\\\tau_{(X^{(y)}_t)}$ the random walk hits a moving target $X^{(y)}_t$ with $X_0=y$ satisfies the following: $\\\\max_{x,y} \\\\mathbb E_x [\\\\tau_{(X_t^{(y)})}] \\\\leq |E|\\\\,|V|$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Such an upper bound is known for the mixing time.\\nThe current best bound on the hitting time of a trajectory is $|E||V|(1+\\\\log \\\\frac {|E|}{|V|})$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/3/", "tags": [ "aim", "AIM-PROBABILITY-0042", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the corrected lazy deterministic-target formulation, the best verified general theorem remains C|E||V|(1+log(|E|/|V|)); the logarithm-removal question for irregular Eulerian digraphs remains open, while the regular case is known in O(|V|^2). A new special-case theorem proves the literal constant-one |E||V| bound for the meeting time of two independent lazy walks on every finite abelian directed Cayley multigraph. In the advertised 2/3--1/3 biased cycle the proof sharpens this to at most |V|^2/2. The literal non-lazy formulation is false: on an even biased cycle, independent walkers started at opposite parities never meet.\n\nCandidate contribution (special-case theorem; novelty confidence low): If a connected directed Cayley multigraph on a finite abelian group has n vertices, d outgoing arcs per vertex, and m=dn arcs, then two independent 1/2-lazy simple random walks satisfy max_{x,y} E_{x,y}[tau_meet] <= mn; for the lazy 2/3--1/3 cycle the sharper bound E[tau_meet] <= 2k(n-k) <= n^2/2 holds." }, { "id": 20002601, "problem_number": "AIM-PROBABILITY-0043", "title": "A portal-gap reduction for a biased small-world walk", "statement": "Cycle + Erdos-Renyi\n\nConsider the graph $G=\\mathbb Z_n \\cup ER(n,p)$ and make the simple random walk on $G$ nonreversible by adding a counterclockwise drift to the cycle part.\n\nProve that for $p=\\frac \\epsilon n$, $\\epsilon>0$ fixed, the nonreversible chain has $t_{\\mbox{mix}}\\asymp \\log n$.\n\nProve that when $p=\\epsilon n^{-\\frac 32}$ the nonreversible chain has $t_{\\mbox{mix}}=\\tilde O(\\sqrt n)$.", "original_statement": "Cycle + Erdos-Renyi\n\nConsider the graph $G=\\mathbb Z_n \\cup ER(n,p)$ and make the simple random walk on $G$ nonreversible by adding a counterclockwise drift to the cycle part.\n\nProve that for $p=\\frac \\epsilon n$, $\\epsilon>0$ fixed, the nonreversible chain has $t_{\\mbox{mix}}\\asymp \\log n$.\n\nProve that when $p=\\epsilon n^{-\\frac 32}$ the nonreversible chain has $t_{\\mbox{mix}}=\\tilde O(\\sqrt n)$.", "clean_statement": "**Cycle + Erdos-Renyi.** Consider the graph \\(G=\\mathbb Z_n\\cup ER(n,p)\\) and make the simple random walk on \\(G\\) nonreversible by adding a counterclockwise drift to the cycle part.\n\nProve that for \\(p=\\epsilon/n\\), \\(\\epsilon>0\\) fixed, the nonreversible chain has \\(t_{\\mathrm{mix}}\\asymp\\log n\\).\n\nProve that when \\(p=\\epsilon n^{-3/2}\\) the nonreversible chain has \\(t_{\\mathrm{mix}}=\\widetilde O(\\sqrt n)\\).", "statement_status": "reconstructed_unverified", "statement_verification": "The mathematical formulas are readable; the important defect is not OCR but under-specification. “Adding a counterclockwise drift” does not determine transition probabilities, a stationary measure, whether time is discrete or continuous, or whether the claim is quenched or annealed. The original AIM page was unavailable (HTTP 502) when checked on 2026-08-11. The workshop report mentions a working group on “cutoff on the small world,” but supplies no missing kernel definition. Thus the results below use a precise, explicitly labeled reconstruction.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Non-reversible chains\nSource item: 3.2\nSource URL: http://aimpl.org/markovmixing/3/\nCanonical location: aim-probability-notes.json notes[42]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Cycle + Erdos-Renyi\\n\\nConsider the graph $G=\\\\mathbb Z_n \\\\cup ER(n,p)$ and make the simple random walk on $G$ nonreversible by adding a counterclockwise drift to the cycle part.\\n\\nProve that for $p=\\\\frac \\\\epsilon n$, $\\\\epsilon>0$ fixed, the nonreversible chain has $t_{\\\\mbox{mix}}\\\\asymp \\\\log n$.\\n\\nProve that when $p=\\\\epsilon n^{-\\\\frac 32}$ the nonreversible chain has $t_{\\\\mbox{mix}}=\\\\tilde O(\\\\sqrt n)$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"In the reversible case of simple random walk, the former has mixing time $\\\\asymp \\\\log^2 n$ and the latter has mixing time $\\\\tilde O(n)$.\"\nResearch attempt: 1; result status: reduction; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/3/", "tags": [ "aim", "AIM-PROBABILITY-0043", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a precise continuous-time Eulerian reconstruction with fixed counterclockwise excess rate delta>0, a deterministic stopping-time argument gives t_mix(1/4) <= t_S(1/8) + 6(L_max+1)/delta, where S is the set of shortcut endpoints and L_max is the longest portal-free cycle interval. An explicit union bound shows with high probability that L_max=O_epsilon(log n) for p=epsilon/n and L_max=O_epsilon(sqrt(n) log n) for p=epsilon n^{-3/2}. Thus the worst one-dimensional starting trap has the requested order, and the remaining unproved step is global mixing from shortcut endpoints.\n\nCandidate contribution (reduction; novelty confidence low): For the fixed-rate Eulerian interpretation, all non-portal initial states can be removed from the quenched mixing analysis at additive cost O(log n) for p=epsilon/n and O(sqrt(n) log n) for p=epsilon n^{-3/2}, with an explicit geometric hitting-time tail and failure probability.", "research_difficulty_suggested": "L4" }, { "id": 20002602, "problem_number": "AIM-PROBABILITY-0044", "title": "Additive reversibilization, nonnormality, and a sharp normal-chain comparison", "statement": "Nonreversible vs. reversible chains\n\nLet $P$ be the transition matrix of a general nonreversible chain with, say, bounded degree vertices and transition probabilities uniformly bounded from below. Denote its mixing time by $t_{\\mbox{mix}}^P$ and let $\\tilde t_{\\mbox{mix}}$ be the mixing time of its symmetrization.\n\nIs it always true that $t_{\\mbox{mix}}^P\\lesssim \\tilde t_{\\mbox{mix}}$? Is the same also true of the cover time?", "original_statement": "Nonreversible vs. reversible chains\n\nLet $P$ be the transition matrix of a general nonreversible chain with, say, bounded degree vertices and transition probabilities uniformly bounded from below. Denote its mixing time by $t_{\\mbox{mix}}^P$ and let $\\tilde t_{\\mbox{mix}}$ be the mixing time of its symmetrization.\n\nIs it always true that $t_{\\mbox{mix}}^P\\lesssim \\tilde t_{\\mbox{mix}}$? Is the same also true of the cover time?", "clean_statement": "Nonreversible vs. reversible chains\n\nLet $P$ be the transition matrix of a general nonreversible chain with, say, bounded degree vertices and transition probabilities uniformly bounded from below. Denote its mixing time by $t_{\\mbox{mix}}^P$ and let $\\tilde t_{\\mbox{mix}}$ be the mixing time of its symmetrization.\n\nIs it always true that $t_{\\mbox{mix}}^P\\lesssim \\tilde t_{\\mbox{mix}}$? Is the same also true of the cover time?", "statement_status": "exact", "statement_verification": "The canonical record is AIM-PROBABILITY-0044, item 3.3 in the AIM workshop list *Markov chain mixing times*, section “Non-reversible chains.” Its problem field reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Non-reversible chains\nSource item: 3.3\nSource URL: http://aimpl.org/markovmixing/3/\nCanonical location: aim-probability-notes.json notes[43]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Nonreversible vs. reversible chains\\n\\nLet $P$ be the transition matrix of a general nonreversible chain with, say, bounded degree vertices and transition probabilities uniformly bounded from below. Denote its mixing time by $t_{\\\\mbox{mix}}^P$ and let $\\\\tilde t_{\\\\mbox{mix}}$ be the mixing time of its symmetrization.\\n\\nIs it always true that $t_{\\\\mbox{mix}}^P\\\\lesssim \\\\tilde t_{\\\\mbox{mix}}$? Is the same also true of the cover time?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Counterexample by Jonathan Hermon and Hubert Lacoin : consider a $d$-regular tree $T$ of depth $n$. For each non-leaf vertex $x$, if $y$ denotes the right-most child of $x$, then the edge $\\\\{x,y\\\\}$ is replaced by a \\\"diamond\\\", i.e. a lozenge formed by two consecutive up-going edges from $x$ to $y$ and two consecutive down-going edges from $y$ to $x$. Each directed edge has weight $2$, so that the symmetrization is simply obtained by replacing directed edges by non-directed edges. This device allows to separate the harmonic measures of the two walks at the boundary of $T$: for $d$ large enough, the directed walk has about twice more chance to escape through right-most edges. Now, on each leaf of $T$ corresponding to a right-most child, add a path of length $n$. To ensure that the worst starting point of the non-directed walk is still at the root, repeat this graph on $n$ levels. At the leaves of the resulting graph, attach an expander graph, so that the mixing time is given by the hitting time of the expander, which is larger for the directed walk.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/3/", "tags": [ "aim", "AIM-PROBABILITY-0044", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The universal constant-factor mixing comparison with additive reversibilization is refuted by the AIM workshop counterexample, while the corresponding cover-time verdict was not verified and recent work gives a symmetrization bound with an extra nonreversibility-dependent logarithm. As a proved restricted result, every finite irreducible one-half-lazy L2(pi)-normal kernel P satisfies d_2,P(2t) <= d_2,(P+P*)/2(t) for every integer t. The factor two is sharp for the directed lazy cycle P=(I+S)/2; in that family the exact cover times are 2(n-1) for P and n(n-1) for its additive reversibilization.\n\nCandidate contribution (comparison_theorem; novelty confidence low): For every finite irreducible one-half-lazy normal Markov kernel, worst-start chi-square distance after 2t steps is at most that of its additive reversibilization after t steps; the directed lazy cycle attains equality for all t and has the exact cover-time pair 2(n-1) versus n(n-1)." }, { "id": 20002603, "problem_number": "AIM-PROBABILITY-0045", "title": "Non-backtracking versus simple random-walk mixing", "statement": "Non-backtracking vs. simple random walk\n\nIs it true that the lazy non-backtracking random walk (NBRW) mixes faster than the lazy simple random walk (SRW)?", "original_statement": "Non-backtracking vs. simple random walk\n\nIs it true that the lazy non-backtracking random walk (NBRW) mixes faster than the lazy simple random walk (SRW)?", "clean_statement": "Non-backtracking vs. simple random walk\n\nIs it true that the lazy non-backtracking random walk (NBRW) mixes faster than the lazy simple random walk (SRW)?", "statement_status": "exact", "statement_verification": "The exact source question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Non-reversible chains\nSource item: 3.4\nSource URL: http://aimpl.org/markovmixing/3/\nCanonical location: aim-probability-notes.json notes[44]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Non-backtracking vs. simple random walk\\n\\nIs it true that the lazy non-backtracking random walk (NBRW) mixes faster than the lazy simple random walk (SRW)?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Hubert Lacoin and Jonathan Hermon counterexample: let $T$ be a GW tree of depth $n$. The harmonic measures of SRW and NBRW at the boundary of $T$ are singular. On each point of the boundary reached by NBRW, add a path of length $n^{1+\\\\varepsilon}$, with $\\\\varepsilon<1/2$. Repeat this graph on $n^2$ levels. Join the $C^{n^3}$ leaves of this graph by an expander, with mixing time of order $n^3$. With this construction, the worst starting point of both walks is the original root, and mixing comes down to reaching the expander. The NBRW is slowed down by paths, and the ratio of mixing times tends to infinity.\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/3/", "tags": [ "aim", "AIM-PROBABILITY-0045", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The unrestricted claim is false, and its meaning depends on whether NBRW mixing is measured on directed edges or through the current-vertex marginal. On every cycle C_n, the standard lazy directed-edge NBRW has two closed orientation classes, so it never mixes globally to uniform directed-edge measure. Nevertheless each orientation's head marginal does mix, and its exact total-variation profile satisfies d_N,head(2t)=d_SRW(t); hence its epsilon-mixing time is between 2t_mix^SRW(epsilon)-1 and 2t_mix^SRW(epsilon). Thus even the convergent vertex marginal is almost exactly twice slower than lazy SRW under worst directed-edge initialization. The stronger minimum-degree-three Hermon-Lacoin construction described by the AIM synopsis was not located in a primary paper and is not treated as independently verified.\n\nCandidate contribution (exact worked family; novelty confidence low): For standard edge-chain lazification on C_n, if P is lazy SRW and Q_+, Q_- are the two oriented NBRW head kernels, then Q_+^2=RP and Q_-^2=R^{-1}P; consequently d_N,head(2t)=d_SRW(t) and 2t_mix^SRW(epsilon)-1 <= t_mix^N,head(epsilon) <= 2t_mix^SRW(epsilon)." }, { "id": 20002604, "problem_number": "AIM-PROBABILITY-0046", "title": "Reducing interchange comparisons to two label-mixing bounds", "statement": "Comparison of interchange to SRW and Exclusion\n\nConsider the interchange process with $n$ particles on $n$ vertices. Can you say the following?\n\n\\begin{align*}\nt_{\\mbox{mix}}^{IP}\\lesssim & t_{\\mbox{mix}}^{SRW} \\log n\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\lesssim & t_{\\mbox{mix}}^{SRW,n}\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\asymp & t_{\\mbox{mix}}^{SSEP,n/2}\\,.\n\\end{align*}", "original_statement": "Comparison of interchange to SRW and Exclusion\n\nConsider the interchange process with $n$ particles on $n$ vertices. Can you say the following?\n\n\\begin{align*}\nt_{\\mbox{mix}}^{IP}\\lesssim & t_{\\mbox{mix}}^{SRW} \\log n\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\lesssim & t_{\\mbox{mix}}^{SRW,n}\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\asymp & t_{\\mbox{mix}}^{SSEP,n/2}\\,.\n\\end{align*}", "clean_statement": "Comparison of interchange to SRW and Exclusion\n\nConsider the interchange process with $n$ particles on $n$ vertices. Can you say the following?\n\n\\begin{align*}\nt_{\\mbox{mix}}^{IP}\\lesssim & t_{\\mbox{mix}}^{SRW} \\log n\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\lesssim & t_{\\mbox{mix}}^{SRW,n}\\,, \\\\\nt_{\\mbox{mix}}^{IP} \\asymp & t_{\\mbox{mix}}^{SSEP,n/2}\\,.\n\\end{align*}", "statement_status": "exact", "statement_verification": "There is no visible OCR corruption, but several mathematical conventions are omitted:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Exclusion and Interchange processes\nSource item: 4.1\nSource URL: http://aimpl.org/markovmixing/4/\nCanonical location: aim-probability-notes.json notes[45]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Comparison of interchange to SRW and Exclusion\\n\\nConsider the interchange process with $n$ particles on $n$ vertices. Can you say the following?\\n\\n\\\\begin{align*}\\nt_{\\\\mbox{mix}}^{IP}\\\\lesssim & t_{\\\\mbox{mix}}^{SRW} \\\\log n\\\\,, \\\\\\\\\\nt_{\\\\mbox{mix}}^{IP} \\\\lesssim & t_{\\\\mbox{mix}}^{SRW,n}\\\\,, \\\\\\\\\\nt_{\\\\mbox{mix}}^{IP} \\\\asymp & t_{\\\\mbox{mix}}^{SSEP,n/2}\\\\,.\\n\\\\end{align*}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The first is known for $t_{\\\\mbox{rel}}$ [Caputo \\\\emph{et. al}]\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/4/", "tags": [ "aim", "AIM-PROBABILITY-0046", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a connected weighted graph with common symmetric continuous-time edge clocks, single-label random walk and every k-particle SSEP are projections of interchange, while the worst-start product of n independent corresponding walks has mixing time between (1/4)t_rel log n and t_rel(log n+2) for n at least 121. Hence the second AIM comparison implies the first and is equivalent up to constants to t_mix(IP)=O(t_rel(RW) log n); the reverse direction needed for the half-filled SSEP comparison is automatic. The three source questions therefore reduce to two still-open one-sided label-randomization upper bounds.\n\nCandidate contribution (reduction; novelty confidence low): Under a common edge-clock normalization, the three AIM comparisons reduce rigorously to two one-sided inequalities: IP is at most a constant times the n-walk product, equivalently O(t_rel log n), and IP is at most a constant times half-filled SSEP; all other requested directions follow from product estimates or projection.", "research_difficulty_suggested": "L4" }, { "id": 20002605, "problem_number": "AIM-PROBABILITY-0047", "title": "Interchange mixing on the Boolean hypercube", "statement": "Interchange process on the hypercube\n\nObtain good bounds on $t_{\\mbox{mix}}^{IP}$ on the hypercube $\\{0,1\\}^n$.", "original_statement": "Interchange process on the hypercube\n\nObtain good bounds on $t_{\\mbox{mix}}^{IP}$ on the hypercube $\\{0,1\\}^n$.", "clean_statement": "Interchange process on the hypercube\n\nObtain good bounds on $t_{\\mbox{mix}}^{IP}$ on the hypercube $\\{0,1\\}^n$.", "statement_status": "exact", "statement_verification": "The canonical record is AIM-PROBABILITY-0047, item 4.2 in the AIM workshop list *Markov chain mixing times*, section “Exclusion and Interchange processes.” Its complete mathematical prompt is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Exclusion and Interchange processes\nSource item: 4.2\nSource URL: http://aimpl.org/markovmixing/4/\nCanonical location: aim-probability-notes.json notes[46]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Interchange process on the hypercube\\n\\nObtain good bounds on $t_{\\\\mbox{mix}}^{IP}$ on the hypercube $\\\\{0,1\\\\}^n$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/4/", "tags": [ "aim", "AIM-PROBABILITY-0047", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Hermon and Salez proved that the interchange process on the n-dimensional Boolean hypercube mixes in Theta(n)=Theta(log N) time when every edge rings at rate one, resolving the AIM order-of-magnitude problem; this is Theta(N n^2) steps for the standard lazy random-edge chain, where N=2^n. The only precise cutoff claim located, (1/4)log N in unit-edge time, is in a withdrawn 2020 preprint by Chen and Marinho; the official arXiv record says that the paper was withdrawn by Rodrigo Marinho and acknowledges a proof gap in Section 5, so the cutoff claim is unverified. The new proved contribution is an all-coordinate label-position correlation statistic yielding the sharper lower bound t_mix(1/4) >= (1/4)log(nN)-O(1), with exact stationary variance N^2 n/(N-1) and transient variance at most 2nN.\n\nCandidate contribution (lower_bound; novelty confidence low): For unit-edge-rate interchange on Q_n started from the identity, the sum of all n first-level Walsh label-position correlations proves d_TV(t) >= 1-8e^(4t)/(nN)-4e^(4t)/(n(N-1)), and hence t_mix(1/4) >= (1/4)log(nN)-O(1)." }, { "id": 20002606, "problem_number": "AIM-PROBABILITY-0048", "title": "Target-sensitive hitting-time comparison for interchange", "statement": "Hitting time comparison\n\nCan you compare the hitting times of the interchange process to hitting times for the SRW on a graph $G$.", "original_statement": "Hitting time comparison\n\nCan you compare the hitting times of the interchange process to hitting times for the SRW on a graph $G$.", "clean_statement": "Hitting time comparison\n\nCan you compare the hitting times of the interchange process to hitting times for the SRW on a graph $G$.", "statement_status": "exact", "statement_verification": "The canonical record is item 4.3 in the section “Exclusion and Interchange processes” of the AIM workshop *Markov chain mixing times*:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Exclusion and Interchange processes\nSource item: 4.3\nSource URL: http://aimpl.org/markovmixing/4/\nCanonical location: aim-probability-notes.json notes[47]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Hitting time comparison\\n\\nCan you compare the hitting times of the interchange process to hitting times for the SRW on a graph $G$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/4/", "tags": [ "aim", "AIM-PROBABILITY-0048", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "A named label in the weighted continuous-time interchange process has exactly the full path law of the underlying variable-speed random walk, so hitting times of every one-label cylinder target agree in distribution after matching clocks. This cannot extend to arbitrary configuration targets: for half-lazy random-edge interchange on every connected n-vertex graph, the mean time to hit a fixed labeling, averaged over its one-edge neighboring labelings, is exactly 2(n!-1), while every point-to-point hitting mean of one label is at most n^2(n-1)^2. Thus unrestricted comparison fails even up to every fixed polynomial factor, and the well-posed next problem is comparison for targets depending on a specified number k of labels.\n\nCandidate contribution (obstruction; novelty confidence low): For the half-lazy random-edge interchange process on any connected simple n-vertex graph and any target labeling xi, the exact identity (1/|E|) sum_{e in E} E_{tau_e xi}[T_xi] = 2(n!-1) holds; paired with the same-clock marginal bound max_{x,y} E_x[T_y] <= n^2(n-1)^2, it gives a factorial target-resolution obstruction." }, { "id": 20002607, "problem_number": "AIM-PROBABILITY-0049", "title": "Solved cutoff and an exact rank-correlation observable", "statement": "Cutoff for random-to-random shuffle\n\nThe random-to-random card shuffle on $S_n$ has cutoff at $\\frac {3n}4 \\log n$.", "original_statement": "Cutoff for random-to-random shuffle\n\nThe random-to-random card shuffle on $S_n$ has cutoff at $\\frac {3n}4 \\log n$.", "clean_statement": "Cutoff for random-to-random shuffle\n\nThe random-to-random card shuffle on $S_n$ has cutoff at $\\frac {3n}4 \\log n$.", "statement_status": "exact", "statement_verification": "The canonical record is item 5.1 in the AIM workshop section “Chains on \\(S_n\\)”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Chains on $S_n$\nSource item: 5.1\nSource URL: http://aimpl.org/markovmixing/5/\nCanonical location: aim-probability-notes.json notes[48]\nCanonical tag: conjecture\nOriginal extracted problem text (JSON string): \"Cutoff for random-to-random shuffle\\n\\nThe random-to-random card shuffle on $S_n$ has cutoff at $\\\\frac {3n}4 \\\\log n$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The known lower bound is, for any $\\\\epsilon>0$, $(\\\\frac 34-\\\\epsilon)n\\\\log n$ due to Subak and the known upper bounds are $2n\\\\log n$ [Saloff-Coste, Zaniga] improved to $\\\\frac 32 n\\\\log n$ [Morris].\\n\\nIn separation distance there is an $n\\\\log n$ upper bound [Nesterodi, White].\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/5/", "tags": [ "aim", "AIM-PROBABILITY-0049", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:conjecture" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Bernstein and Nestoridi's 2019 theorem, together with Subag's lower bound and the Dieker-Saliola spectrum, resolves the AIM conjecture and refines the cutoff center to (3/4)n log n - (1/4)n log log n with an order-n window. This attempt also proves an elementary finite-n package: a Spearman-type rank-correlation statistic is an eigenfunction with eigenvalue 1-1/n-2/n^2, has exact stationary variance S_n^2/(n-1), has exact maximum one-insertion change n(n-1)/2, and yields an explicit total-variation lower bound, including liminf d_n(floor((1/2)n log n-cn)) >= 1-76 exp(-2c).\n\nCandidate contribution (explicit finite-n eigenfunction bound; novelty confidence low): For standard random-to-random insertion, the rank-correlation statistic F_n has exact uniform variance S_n^2/(n-1), exact one-step oscillation n(n-1)/2, and satisfies d_n(t) >= 1 - 4 lambda_n^(-2t) [36/((n+1)^2(1-lambda_n^2)) + 1/(n-1)], hence the stated asymptotic lower bound with constant 76." }, { "id": 20002608, "problem_number": "AIM-PROBABILITY-0050", "title": "Random biased transpositions: two-class cutoff, parity repair, and heterogeneous product reductions", "statement": "Random biased transpositions\n\nThe random uniform transposition walk on $S_n$ is the discrete time Markov chain that at each time step applies a transposition $(i,j)$ uniformly at random.\n\nThe random nearly-uniform (biased) transposition walk is the corresponding chain that at each time step applies a transposition $(i,j)$ with probability $p_{ij}$ for $p_{ij}$ comparable to the uniform rate up to uniformly bounded constants.\n\nDoes the random nearly-uniform transposition walk on $S_n$ exhibit cutoff?\n\nWhat about in the simpler case where $p_{ij}=p_{i}p_{j}$ for each $1\\leq i,j\\leq n$?", "original_statement": "Random biased transpositions\n\nThe random uniform transposition walk on $S_n$ is the discrete time Markov chain that at each time step applies a transposition $(i,j)$ uniformly at random.\n\nThe random nearly-uniform (biased) transposition walk is the corresponding chain that at each time step applies a transposition $(i,j)$ with probability $p_{ij}$ for $p_{ij}$ comparable to the uniform rate up to uniformly bounded constants.\n\nDoes the random nearly-uniform transposition walk on $S_n$ exhibit cutoff?\n\nWhat about in the simpler case where $p_{ij}=p_{i}p_{j}$ for each $1\\leq i,j\\leq n$?", "clean_statement": "Random biased transpositions\n\nThe random uniform transposition walk on $S_n$ is the discrete time Markov chain that at each time step applies a transposition $(i,j)$ uniformly at random.\n\nThe random nearly-uniform (biased) transposition walk is the corresponding chain that at each time step applies a transposition $(i,j)$ with probability $p_{ij}$ for $p_{ij}$ comparable to the uniform rate up to uniformly bounded constants.\n\nDoes the random nearly-uniform transposition walk on $S_n$ exhibit cutoff?\n\nWhat about in the simpler case where $p_{ij}=p_{i}p_{j}$ for each $1\\leq i,j\\leq n$?", "statement_status": "exact", "statement_verification": "The local JSON record is internally legible and shows no OCR corruption. The linked AIM page returned an HTTP 502 during this run, so I could not compare its current rendering with the preserved record.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Chains on $S_n$\nSource item: 5.2\nSource URL: http://aimpl.org/markovmixing/5/\nCanonical location: aim-probability-notes.json notes[49]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Random biased transpositions\\n\\nThe random uniform transposition walk on $S_n$ is the discrete time Markov chain that at each time step applies a transposition $(i,j)$ uniformly at random.\\n\\nThe random nearly-uniform (biased) transposition walk is the corresponding chain that at each time step applies a transposition $(i,j)$ with probability $p_{ij}$ for $p_{ij}$ comparable to the uniform rate up to uniformly bounded constants.\\n\\nDoes the random nearly-uniform transposition walk on $S_n$ exhibit cutoff?\\n\\nWhat about in the simpler case where $p_{ij}=p_{i}p_{j}$ for each $1\\\\leq i,j\\\\leq n$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/5/", "tags": [ "aim", "AIM-PROBABILITY-0050", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The valid current literature proves total-variation cutoff for the equal two-class product-weight shuffle but does not appear to settle arbitrary bounded product weights or arbitrary comparable edge weights. For the intended with-replacement product model, this run proves the exact tagged-card reduction K = I - 2D_p + 2pp^T, derives its secular equation and the full tagged spectrum for two weight classes, and proves the nonasymptotic lower bound d_TV(t) >= 1 - 6/M_p(t), where M_p(t) = sum_i (1-p_i)^(2t). It also proves that the literal nonlazy formulation cannot mix to uniform on S_n because parity forces d_TV(t) >= 1/2.\n\nCandidate contribution (reduction; novelty confidence low): For every heterogeneous product vector p, the pair of exact formulas K = I - 2D_p + 2pp^T and d_TV(t) >= 1 - 6/[sum_i (1-p_i)^(2t)] gives a finite-rank tagged spectral reduction and an explicit nonasymptotic coupon obstruction; this package specializes to the correct slow-class scale and exact natural-representation spectrum in the two-weight family." }, { "id": 20002609, "problem_number": "AIM-PROBABILITY-0051", "title": "Exact total-variation cutoff profile for random transpositions", "statement": "Cutoff profile of random transpositions\n\nCan we obtain the cutoff profile for the uniform random transposition walk on $S_n$?", "original_statement": "Cutoff profile of random transpositions\n\nCan we obtain the cutoff profile for the uniform random transposition walk on $S_n$?", "clean_statement": "Cutoff profile of random transpositions\n\nCan we obtain the cutoff profile for the uniform random transposition walk on $S_n$?", "statement_status": "exact", "statement_verification": "The canonical record is Problem 5.3 in the AIM workshop list “Markov chain mixing times,” section “Chains on \\(S_n\\)”:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Chains on $S_n$\nSource item: 5.3\nSource URL: http://aimpl.org/markovmixing/5/\nCanonical location: aim-probability-notes.json notes[50]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Cutoff profile of random transpositions\\n\\nCan we obtain the cutoff profile for the uniform random transposition walk on $S_n$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: full_solution; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/5/", "tags": [ "aim", "AIM-PROBABILITY-0051", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Teyssier proved in a peer-reviewed 2020 paper that, for the ordered-pair-with-replacement random-transposition chain P_n(Id)=1/n and P_n((ij))=2/n^2, the total-variation distance at floor((n log n)/2+cn) converges for every fixed c to d_TV(Pois(1+e^(-2c)),Pois(1)). This solves the AIM problem, with center (n log n)/2 and window n. The report separately audits the periodic proper-transposition convention, continuous-time clock scalings, parity, and a May 2026 preprint claim for the distinct separation profile. A new proved finite-n calculation gives exact first and second factorial moments of the fixed-point count and recovers the first two moments of the limiting Poisson law.\n\nCandidate contribution (finite_n_identity; novelty confidence low): For n>=4 and every k>=0, if F_k is the number of fixed points after k ordered-pair random-transposition steps, then E[F_k]=1+(n-1)(1-2/n)^k and E[(F_k)_2]=1+2(n-1)(1-2/n)^k+[n(n-3)/2](1-2/n)^(2k)+[(n-1)(n-2)/2](1-4/n)^k. At k=floor((n log n)/2+cn), the two last eigenmodes each contribute e^(-4c)/2, yielding the first two factorial moments of Pois(1+e^(-2c))." }, { "id": 20002610, "problem_number": "AIM-PROBABILITY-0052", "title": "Coordinate-start sphere cutoff and a support-activation limit law", "statement": "Kac's random walk\n\nConsider the Kac's random walk on the $n$-sphere defined as follows: for a vector $v\\in \\mathcal S^n$, pick two coordinates uniformly at random, and an angle uniformly at random on $[0,2\\pi)$. Then rotate in the circle generated by the two coordinates by that angle.\n\nDoes Kac's random walk on $\\mathcal S^n$ exhibit cutoff?\n\nDoes the analogously defined random walk on $SO(n)$ exhibit cutoff?", "original_statement": "Kac's random walk\n\nConsider the Kac's random walk on the $n$-sphere defined as follows: for a vector $v\\in \\mathcal S^n$, pick two coordinates uniformly at random, and an angle uniformly at random on $[0,2\\pi)$. Then rotate in the circle generated by the two coordinates by that angle.\n\nDoes Kac's random walk on $\\mathcal S^n$ exhibit cutoff?\n\nDoes the analogously defined random walk on $SO(n)$ exhibit cutoff?", "clean_statement": "Kac's random walk\n\nConsider the Kac's random walk on the $n$-sphere defined as follows: for a vector $v\\in \\mathcal S^n$, pick two coordinates uniformly at random, and an angle uniformly at random on $[0,2\\pi)$. Then rotate in the circle generated by the two coordinates by that angle.\n\nDoes Kac's random walk on $\\mathcal S^n$ exhibit cutoff?\n\nDoes the analogously defined random walk on $SO(n)$ exhibit cutoff?", "statement_status": "exact", "statement_verification": "The canonical AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Various\nSource item: 6.1\nSource URL: http://aimpl.org/markovmixing/6/\nCanonical location: aim-probability-notes.json notes[51]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Kac's random walk\\n\\nConsider the Kac's random walk on the $n$-sphere defined as follows: for a vector $v\\\\in \\\\mathcal S^n$, pick two coordinates uniformly at random, and an angle uniformly at random on $[0,2\\\\pi)$. Then rotate in the circle generated by the two coordinates by that angle.\\n\\nDoes Kac's random walk on $\\\\mathcal S^n$ exhibit cutoff?\\n\\nDoes the analogously defined random walk on $SO(n)$ exhibit cutoff?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"For Kac's random walk, it is known that $\\\\frac 12 n\\\\log n \\\\leq t_{\\\\mbox{mix}} \\\\leq 4n\\\\log n$.\\n\\nFor the two coordinate random walk on $SO(n)$, $t_{\\\\mbox{mix}}\\\\leq O(n^4)$ but $\\\\tilde O(n^2)$ is conjectured.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/6/", "tags": [ "aim", "AIM-PROBABILITY-0052", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Jain and Mizgerd's July 2026 preprint proves total-variation cutoff for the sphere Kac walk from a coordinate-vector start at C_BRW n log n with C_BRW approximately 3.8916, but arbitrary deterministic and worst-case starts remain open. For the coordinate-plane walk on SO(n), Pillai and Smith's April 2026 preprint proves an O(n^2 log n) upper bound while cutoff remains open; Hough-Jiang's cutoff theorem concerns the distinct Grassmannian-uniform plane walk. This attempt additionally proves an exact pure-birth representation and gamma-function transform for the time at which all sphere coordinates become nonzero, its fixed-support limit tau/n-log n converging to -(1/2)log(Z_s E), and the resulting total-variation lower curve.\n\nCandidate contribution (exact limit theorem; novelty confidence low): For the standard discrete Kac walk on S^{n-1} from any deterministic start with fixed support size s, the full-coordinate activation time satisfies tau_{n,s}/n-log n converges in distribution to -(1/2)log(Z_s E), where Z_s is Gamma(s,1) and E is independent Exp(1); its tail lower-bounds total variation at floor(n log n+cn), and for s=1 the tail is 1-2 exp(-c) K_1(2 exp(-c))." }, { "id": 20002611, "problem_number": "AIM-PROBABILITY-0053", "title": "Sharp Gibbs status and exact proposal-sensitive Metropolis benchmarks", "statement": "Metropolis on $[0,1]^2$\n\nConsider the stationary distribution on $[0,1]^2$ given by\n\\[\\pi_A(p_1,p_2)=\\mathcal Z^{-1} \\exp(-A|p_1-p_2|^2)\n\\]\nparametrized by $A\\in [10,1000]$.\n\nObtain a good bound on the mixing time of the Metropolis algorithm for this stationary distribution in terms of $A$. Generalize this to $[0,1]^d$.", "original_statement": "Metropolis on $[0,1]^2$\n\nConsider the stationary distribution on $[0,1]^2$ given by\n\\[\\pi_A(p_1,p_2)=\\mathcal Z^{-1} \\exp(-A|p_1-p_2|^2)\n\\]\nparametrized by $A\\in [10,1000]$.\n\nObtain a good bound on the mixing time of the Metropolis algorithm for this stationary distribution in terms of $A$. Generalize this to $[0,1]^d$.", "clean_statement": "Metropolis on $[0,1]^2$\n\nConsider the stationary distribution on $[0,1]^2$ given by\n\\[\\pi_A(p_1,p_2)=\\mathcal Z^{-1} \\exp(-A|p_1-p_2|^2)\n\\]\nparametrized by $A\\in [10,1000]$.\n\nObtain a good bound on the mixing time of the Metropolis algorithm for this stationary distribution in terms of $A$. Generalize this to $[0,1]^d$.", "statement_status": "exact", "statement_verification": "The canonical record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Various\nSource item: 6.2\nSource URL: http://aimpl.org/markovmixing/6/\nCanonical location: aim-probability-notes.json notes[52]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Metropolis on $[0,1]^2$\\n\\nConsider the stationary distribution on $[0,1]^2$ given by\\n\\\\[\\\\pi_A(p_1,p_2)=\\\\mathcal Z^{-1} \\\\exp(-A|p_1-p_2|^2)\\n\\\\]\\nparametrized by $A\\\\in [10,1000]$.\\n\\nObtain a good bound on the mixing time of the Metropolis algorithm for this stationary distribution in terms of $A$. Generalize this to $[0,1]^d$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"The current best known upper bound is $\\\\exp(A^2)$.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/6/", "tags": [ "aim", "AIM-PROBABILITY-0053", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The official AIM workshop-to-paper link identifies the intended chain as random-scan coordinate Gibbs for a target written in the paper as exp(-B^2(u-v)^2), despite the canonical record's generic Metropolis wording and exponent coefficient A. The published two-dimensional total-variation mixing order is Theta(B^2), hence Theta(A) when A is interpreted literally as the exponent coefficient; fixed-dimensional weighted extensions have the same order in a suitable Wasserstein metric, while a general total-variation upper bound remains open. For a fully specified uniform-independence Metropolis proposal, this attempt proves the exact worst-start distance (1-Z)^t, computes Z on the square, derives the diagonal-tube asymptotic in d dimensions, and shows by an irreducible lazy proposal family that no proposal-free Metropolis bound exists.\n\nCandidate contribution (exact_formula_and_obstruction; novelty confidence low): For uniform-independence Metropolis on the literal thin-diagonal square target, the worst-start total-variation distance is exactly (1-Z_A)^t with an explicit error-function formula for Z_A; for the natural extension V_d=2 sum_i (x_i-xbar)^2, Z_{A,d} is asymptotic to sqrt(d)(pi/(2A))^((d-1)/2), and an irreducible lazy proposal gives exact distance (1-rho(A)Z_A)^t, permitting arbitrary target-preserving slowdown." }, { "id": 20002612, "problem_number": "AIM-PROBABILITY-0054", "title": "A four-state sign-valued obstruction to fixed-tolerance function mixing bounds", "statement": "Function-specific mixing time and concentration\n\nLet $(X_n)$ be a stationary Markov chain on a finite state space $\\Omega$ and $f\\colon\\Omega\\to [-1,1]$. Consider\n$$\nS_n=\\frac{1}{n} \\sum_{i=1}^n f(X_i) .\n$$\n\nDoes there exist an absolute constant $c>0$ such that, for all $\\varepsilon>0$ and $n\\geq 1$,\n$$\n\\mathbb{P}\\left(\\Big| S_n-\\mathbb{E} S_n\\Big| \\geq\\varepsilon\\right)\\leq 2\\exp\\left(-\\frac{cn\\varepsilon^2}{t_f(\\delta)}\\right)\\, ,\n$$\nwhere $t_f(\\delta)=\\sup_{x\\in\\Omega}\\left\\{n\\geq 0,\\, \\Big|\\mathbb{E}_x f(X_n) -\\mathbb{E}_\\pi f\\Big|\\leq\\delta\\right\\}$ ? Does it hold for $\\delta=1/4$?", "original_statement": "Function-specific mixing time and concentration\n\nLet $(X_n)$ be a stationary Markov chain on a finite state space $\\Omega$ and $f\\colon\\Omega\\to [-1,1]$. Consider\n$$\nS_n=\\frac{1}{n} \\sum_{i=1}^n f(X_i) .\n$$\n\nDoes there exist an absolute constant $c>0$ such that, for all $\\varepsilon>0$ and $n\\geq 1$,\n$$\n\\mathbb{P}\\left(\\Big| S_n-\\mathbb{E} S_n\\Big| \\geq\\varepsilon\\right)\\leq 2\\exp\\left(-\\frac{cn\\varepsilon^2}{t_f(\\delta)}\\right)\\, ,\n$$\nwhere $t_f(\\delta)=\\sup_{x\\in\\Omega}\\left\\{n\\geq 0,\\, \\Big|\\mathbb{E}_x f(X_n) -\\mathbb{E}_\\pi f\\Big|\\leq\\delta\\right\\}$ ? Does it hold for $\\delta=1/4$?", "clean_statement": "Function-specific mixing time and concentration\n\nLet $(X_n)$ be a stationary Markov chain on a finite state space $\\Omega$ and $f\\colon\\Omega\\to [-1,1]$. Consider\n$$\nS_n=\\frac{1}{n} \\sum_{i=1}^n f(X_i) .\n$$\n\nDoes there exist an absolute constant $c>0$ such that, for all $\\varepsilon>0$ and $n\\geq 1$,\n$$\n\\mathbb{P}\\left(\\Big| S_n-\\mathbb{E} S_n\\Big| \\geq\\varepsilon\\right)\\leq 2\\exp\\left(-\\frac{cn\\varepsilon^2}{t_f(\\delta)}\\right)\\, ,\n$$\nwhere $t_f(\\delta)=\\sup_{x\\in\\Omega}\\left\\{n\\geq 0,\\, \\Big|\\mathbb{E}_x f(X_n) -\\mathbb{E}_\\pi f\\Big|\\leq\\delta\\right\\}$ ? Does it hold for $\\delta=1/4$?", "statement_status": "exact", "statement_verification": "The canonical record asks the following. Let \\((X_i)\\) be a stationary Markov chain on a finite state space \\(\\Omega\\), let \\(f:\\Omega\\to[-1,1]\\), and put \\[ S_N=\\frac1N\\sum_{i=1}^N f(X_i). \\] Is there an absolute constant \\(c>0\\) such that \\[ \\Pr\\bigl(|S_N-\\mathbb E S_N|\\geq\\varepsilon\\bigr) \\leq 2\\exp\\!\\left(-\\frac{cN\\varepsilon^2}{t_f(\\delta)}\\right) \\tag{1} \\] for every \\(\\varepsilon>0\\) and \\(N\\geq1\\)? In particular, can one take the fixed tolerance \\(\\delta=1/4\\)?", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Markov chain mixing times\nSection: Various\nSource item: 6.4\nSource URL: http://aimpl.org/markovmixing/6/\nCanonical location: aim-probability-notes.json notes[53]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Function-specific mixing time and concentration\\n\\nLet $(X_n)$ be a stationary Markov chain on a finite state space $\\\\Omega$ and $f\\\\colon\\\\Omega\\\\to [-1,1]$. Consider\\n$$\\nS_n=\\\\frac{1}{n} \\\\sum_{i=1}^n f(X_i) .\\n$$\\n\\nDoes there exist an absolute constant $c>0$ such that, for all $\\\\varepsilon>0$ and $n\\\\geq 1$,\\n$$\\n\\\\mathbb{P}\\\\left(\\\\Big| S_n-\\\\mathbb{E} S_n\\\\Big| \\\\geq\\\\varepsilon\\\\right)\\\\leq 2\\\\exp\\\\left(-\\\\frac{cn\\\\varepsilon^2}{t_f(\\\\delta)}\\\\right)\\\\, ,\\n$$\\nwhere $t_f(\\\\delta)=\\\\sup_{x\\\\in\\\\Omega}\\\\left\\\\{n\\\\geq 0,\\\\, \\\\Big|\\\\mathbb{E}_x f(X_n) -\\\\mathbb{E}_\\\\pi f\\\\Big|\\\\leq\\\\delta\\\\right\\\\}$ ? Does it hold for $\\\\delta=1/4$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"- This is known for $\\\\delta=\\\\varepsilon^2$, Maxim Rabinovich, Aaditya Ramdas, Michael I. Jordan, and Martin J. Wainwright (2016).\\n\\n - In the reversible case, this holds with $t_f$ replaced by $t_{\\\\text{REL}}$ (Gilman, 1998).\\n\\n - Allan Sly's counterexample for $\\\\delta=1/4$: let $\\\\Omega$ be a segment of length $N$. For all $1\\\\leq i\\\\leq N$, $f(i)=\\\\pm 1$. In each local neighborhood, the number of $1$'s is approximately equal to the number of $-1$'s, but there is a fraction $1/100$ more $1$'s on the right side and a fraction $1/100$ more $-1$'s on the left side.\\n\\n - Roberto Oliveira's progress: it holds with $\\\\delta=\\\\varepsilon/2$. Let $t=t_f(\\\\varepsilon/2)$ and assume $n=kt$. Then\\n\\\\begin{eqnarray*}\\nS_n &=&\\\\frac{1}{t}\\\\sum_{\\\\ell=1}^t\\\\frac{1}{k} \\\\sum_{i=0}^{k-1} f(X_{it+\\\\ell})\\\\, .\\n\\\\end{eqnarray*}\\nLet $\\\\theta\\\\geq 0$. By convexity of $x\\\\mapsto \\\\exp(\\\\theta x)$,\\n\\\\begin{eqnarray*}\\n\\\\mathbb{E}\\\\left[\\\\mathrm{e}^{\\\\theta (S_n-\\\\mathbb{E}_\\\\pi f)}\\\\right]&\\\\leq & \\\\frac{1}{t}\\\\sum_{\\\\ell=1}^t \\\\mathbb{E}\\\\left[\\\\mathrm{e}^{\\\\theta \\\\frac{1}{k} \\\\sum_{i=0}^{k-1} (f(X_{it+\\\\ell})-\\\\mathbb{E}_\\\\pi f)}\\\\right]\\\\, .\\n\\\\end{eqnarray*}\\nNow, by definition of $t$, $\\\\mathbb{E}_\\\\pi f\\\\geq \\\\mathbb{E}\\\\left[f(X_{it+\\\\ell})\\\\big|X_\\\\ell,X_{t+\\\\ell},\\\\dots,X_{(i-1)t+\\\\ell}\\\\right]-\\\\varepsilon/2$. And by Hoeffding's inequality,\\n\\\\begin{eqnarray*}\\n\\\\mathbb{E}\\\\left[\\\\mathrm{e}^{\\\\theta (S_n-\\\\mathbb{E}_\\\\pi f)}\\\\right]&\\\\leq & \\\\frac{1}{t}\\\\sum_{\\\\ell=1}^t \\\\mathrm{e}^{\\\\frac{\\\\theta\\\\varepsilon}{2}}\\\\mathbb{E}\\\\left[\\\\mathrm{e}^{\\\\theta \\\\frac{1}{k} \\\\sum_{i=0}^{k-1} \\\\left(f(X_{it+\\\\ell})-\\\\mathbb{E}\\\\left[f(X_{it+\\\\ell})\\\\big|X_\\\\ell,\\\\dots,X_{(i-1)t+\\\\ell}\\\\right]\\\\right)}\\\\right]\\\\\\\\\\n&\\\\leq & \\\\mathrm{e}^{\\\\frac{\\\\theta^2}{2k}+\\\\frac{\\\\theta\\\\varepsilon}{2}}\\\\, .\\n\\\\end{eqnarray*}\\nApplying the same arguments to $\\\\theta<0$, one obtains\\n\\\\begin{eqnarray*}\\n\\\\mathbb{P}\\\\left(\\\\Big| S_n-\\\\mathbb{E} S_n\\\\Big| \\\\geq\\\\varepsilon\\\\right)&\\\\leq& 2\\\\exp\\\\left(-\\\\frac{k\\\\varepsilon^2}{8}\\\\right)\\\\, ,\\n\\\\end{eqnarray*}\"\nResearch attempt: 1; result status: counterexample; problem status at run: refuted; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/markovmixing/6/", "tags": [ "aim", "AIM-PROBABILITY-0054", "aim-domain:probability", "aim-workshop:markovmixing", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "For every fixed delta in (0,1) and every proposed universal constant c>0, an explicit stationary, reversible, irreducible, aperiodic four-state chain and a sign-valued observable f can have T_f(delta)=1 while its empirical mean has a constant-order deviation probability at a suitably chosen horizon N. Specifically, the construction proves P_pi(|S_N| >= a/2) >= (1-q)^(N-1)(1-2 exp(-N a^2/8)); choosing q=1/(8N) and N large contradicts 2 exp(-c N (a/2)^2). Thus the requested bound fails for delta=1/4, even under stronger structural assumptions on the chain.\n\nCandidate contribution (counterexample; novelty confidence low): An explicit four-state reversible, irreducible, aperiodic stationary hidden-sign chain with f taking exactly the values -1 and +1 has T_f(delta)=1 for either the exact-time or persistence definition, yet satisfies the finite-horizon lower bound P_pi(|S_N| >= a/2) >= (1-q)^(N-1)(1-2 exp(-N a^2/8)) for every fixed delta in (0,1) and any a= alpha remains open; the current general bound is O_alpha(n^2), while O(n log n) is known for almost every monotone set.\n\nCandidate contribution (obstruction; novelty confidence low): For the Ding-Mossel two-subcube family with two disjoint forced blocks of size m, the lazy censored walk has the exact cut conductance Phi(S)=m/[2n(2^m-1)] for S=U_1 minus U_2, yielding the explicit finite-n worst-deterministic-start lower bound t_mix(1/4) >= ceil(n(2^m-1)/(2m)); the literal nonlazy censored flip chain is periodic if and only if the monotone set is the full cube." }, { "id": 20002614, "problem_number": "AIM-PROBABILITY-0056", "title": "Gaussian-side decoupling and free-convolution density", "statement": "Two matrix model with non symmetric potentials\n\nStudy eigenvalue properties (in particular, the limiting density of eigenvalues) of $M_1$ in a two matrix model given by the probability density\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}[V(M_1)+W(M_2)+tM_1M_2]}dM_1 dM_2.\n$$\nIn this context, $V(x)=x^2/2$ and $W(y)$ is a general polynomial of even degree (no symmetry). More in general, consider this model when $V(x)$ is an even quartic polynomial and $W(y)$ is a general polynomial with no symmetries. \\label{two-matrix-x2}", "original_statement": "Two matrix model with non symmetric potentials\n\nStudy eigenvalue properties (in particular, the limiting density of eigenvalues) of $M_1$ in a two matrix model given by the probability density\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}[V(M_1)+W(M_2)+tM_1M_2]}dM_1 dM_2.\n$$\nIn this context, $V(x)=x^2/2$ and $W(y)$ is a general polynomial of even degree (no symmetry). More in general, consider this model when $V(x)$ is an even quartic polynomial and $W(y)$ is a general polynomial with no symmetries. \\label{two-matrix-x2}", "clean_statement": "Two matrix model with non symmetric potentials\n\nStudy eigenvalue properties (in particular, the limiting density of eigenvalues) of $M_1$ in a two matrix model given by the probability density\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}[V(M_1)+W(M_2)+tM_1M_2]}dM_1 dM_2.\n$$\nIn this context, $V(x)=x^2/2$ and $W(y)$ is a general polynomial of even degree (no symmetry). More in general, consider this model when $V(x)$ is an even quartic polynomial and $W(y)$ is a general polynomial with no symmetries. \\label{two-matrix-x2}", "statement_status": "exact", "statement_verification": "The AIM record asks for eigenvalue information, especially the limiting eigenvalue density of \\(M_1\\), in the Hermitian two-matrix model", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Vector equilibrium problems and their applications to random matrix models\nSection: Two-matrix models\nSource item: 1.1\nSource URL: http://aimpl.org/vectorequilib/1/\nCanonical location: aim-probability-notes.json notes[55]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Two matrix model with non symmetric potentials\\n\\nStudy eigenvalue properties (in particular, the limiting density of eigenvalues) of $M_1$ in a two matrix model given by the probability density\\n$$\\n\\\\frac{1}{Z_N}e^{-N\\\\textrm{tr}[V(M_1)+W(M_2)+tM_1M_2]}dM_1 dM_2.\\n$$\\nIn this context, $V(x)=x^2/2$ and $W(y)$ is a general polynomial of even degree (no symmetry). More in general, consider this model when $V(x)$ is an even quartic polynomial and $W(y)$ is a general polynomial with no symmetries. \\\\label{two-matrix-x2}\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"A more general question would be the following: if it is possible to obtain information about the eigenvalues of $M_1$ and $M_2$ in a two-matrix model, and the analysis is significantly simpler for one of the two matrices, could this be used for studying the eigenvalues of the other one?\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/vectorequilib/1/", "tags": [ "aim", "AIM-PROBABILITY-0056", "aim-domain:probability", "aim-workshop:vectorequilib", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For V(x)=x^2/2, square completion gives an exact finite-N representation M1=G-tB and M2=B, where G is standard GUE and B is an independent unitary one-matrix ensemble with effective potential U(y)=W(y)-t^2 y^2/2. Under confinement, the limiting M1 law is SC_1 free-additively convolved with the pushforward of the equilibrium measure mu_U by y -> -ty, equivalently its Cauchy transform solves G(z)=integral [z-G(z)+ty]^{-1} dmu_U(y). The nonsymmetric quadratic family is solved exactly as a shifted GUE. The general even-quartic-V, nonsymmetric-W case remains open in this attempt.\n\nCandidate contribution (reduction; novelty confidence low): The explicit Gaussian-side transfer package combines the exact Jacobian-one decoupling, the effective-potential confinement threshold, the scalar Pastur/Biane density representation, and an exact nonsymmetric quadratic shifted-GUE test family for the AIM formulation." }, { "id": 20002615, "problem_number": "AIM-PROBABILITY-0057", "title": "Negative-quartic two-matrix model is not a real probability measure", "statement": "Coupled random matrix model with negative potentials\n\nAnalysis of the coupled matrix model similar to problem \\ref{two-matrix-x2}, but with potentials\n$$\nV(x)=-(x^4-ax^2), \\qquad W(y)=V(y).\n$$\nIs there any new critical behavior for some value of $a$?", "original_statement": "Coupled random matrix model with negative potentials\n\nAnalysis of the coupled matrix model similar to problem \\ref{two-matrix-x2}, but with potentials\n$$\nV(x)=-(x^4-ax^2), \\qquad W(y)=V(y).\n$$\nIs there any new critical behavior for some value of $a$?", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The canonical record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Vector equilibrium problems and their applications to random matrix models\nSection: Two-matrix models\nSource item: 1.2\nSource URL: http://aimpl.org/vectorequilib/1/\nCanonical location: aim-probability-notes.json notes[56]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Coupled random matrix model with negative potentials\\n\\nAnalysis of the coupled matrix model similar to problem \\\\ref{two-matrix-x2}, but with potentials\\n$$\\nV(x)=-(x^4-ax^2), \\\\qquad W(y)=V(y).\\n$$\\nIs there any new critical behavior for some value of $a$?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: invalid_statement; problem status at run: not_a_problem; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 2, "status": "open", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/vectorequilib/1/", "tags": [ "aim", "AIM-PROBABILITY-0057", "aim-domain:probability", "aim-workshop:vectorequilib", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 2, "level": 2, "name": "L2: Intermediate", "description": "Challenging problems requiring solid mathematical background. Reserved for future additions.", "color_class": "text-blue-600 bg-blue-50 border-blue-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "OPEN-TRIAGE", "research_summary": "Under the real-Hermitian probability interpretation inherited from the preceding AIM record, the displayed potentials V(x)=-x^4+a x^2 give an infinite partition function for every matrix size n, every real a, every real bilinear coupling, and either coupling sign. More generally, for two nonzero quartic leading coefficients, the finite-n real-Hermitian partition function is finite exactly when both coefficients are positive; divergence is proved on disjoint fixed-volume matrix balls. A formal or complex-contour model is possible only after specifying inequivalent extra data such as integration cycles, Stokes sectors, contour coefficients, and filling fractions, so its critical behavior is underdetermined by the record.\n\nCandidate contribution (sharp_non_normalizability_criterion; novelty confidence low): For Z_n equal to the real-Hermitian integral of exp{-n Tr(c1 M1^4+b1 M1^2+c2 M2^4+b2 M2^2+t M1M2)}, with c1 and c2 nonzero, Z_n is finite if and only if c1>0 and c2>0; when a coefficient is negative, divergence occurs on a countable union of pairwise disjoint fixed-volume Frobenius product balls, uniformly in b1,b2,t and for every n>=1.", "research_difficulty_suggested": "L2" }, { "id": 20002616, "problem_number": "AIM-PROBABILITY-0058", "title": "Gaussian source-merger reduction for a two-matrix external-source model", "statement": "$2+1/2$ random matrix models\n\nA general question about Riemann-Hilbert techniques to study the two-matrix plus external source model.", "original_statement": "$2+1/2$ random matrix models\n\nA general question about Riemann-Hilbert techniques to study the two-matrix plus external source model.", "clean_statement": "$2+1/2$ random matrix models\n\nA general question about Riemann-Hilbert techniques to study the two-matrix plus external source model.", "statement_status": "exact", "statement_verification": "The canonical record, in section “Two-matrix models” of the AIM workshop *Vector equilibrium problems and their applications to random matrix models*, reads in full:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Vector equilibrium problems and their applications to random matrix models\nSection: Two-matrix models\nSource item: 1.3\nSource URL: http://aimpl.org/vectorequilib/1/\nCanonical location: aim-probability-notes.json notes[57]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"$2+1/2$ random matrix models\\n\\nA general question about Riemann-Hilbert techniques to study the two-matrix plus external source model.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/vectorequilib/1/", "tags": [ "aim", "AIM-PROBABILITY-0058", "aim-domain:probability", "aim-workshop:vectorequilib", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the precisely defined two-matrix ensemble with density proportional to exp{-n Tr[V(M1)+M2^2/2-tau M1 M2-A M1-B M2]}, integrating the Gaussian matrix M2 exactly gives the one-matrix external-source ensemble with effective potential U(x)=V(x)-tau^2 x^2/2 and effective source C=A+tau B. This requires no commutativity of A and B. If C has r distinct eigenvalues, the M1 average characteristic polynomial therefore satisfies the standard type-II multiple orthogonality relations and the associated (r+1)-by-(r+1) Riemann-Hilbert problem.\n\nCandidate contribution (reduction; novelty confidence low): The explicit source-merger map (V,A,B,tau) -> (V-tau^2 x^2/2, A+tau B), including noncommuting fixed sources, rigorously marks the Gaussian-second-matrix branch of the AIM '2+1/2' program as an existing external-source MOP/RH problem rather than a new RH class.", "research_difficulty_suggested": "L4" }, { "id": 20002617, "problem_number": "AIM-PROBABILITY-0059", "title": "External-source reduction and finite-N response bounds", "statement": "Random matrices with external source. More general cases\n\nStudy the random matrix model with probability measure\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}(V(M)-AM)},\n$$\nwhere $A$ is an $N\\times N$ diagonal matrix with eigenvalues $\\pm a$ (each with equal multiplicity $N/2$, assuming $N$ is even), and $V(M)$ is a general polynomial.", "original_statement": "Random matrices with external source. More general cases\n\nStudy the random matrix model with probability measure\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}(V(M)-AM)},\n$$\nwhere $A$ is an $N\\times N$ diagonal matrix with eigenvalues $\\pm a$ (each with equal multiplicity $N/2$, assuming $N$ is even), and $V(M)$ is a general polynomial.", "clean_statement": "Random matrices with external source. More general cases\n\nStudy the random matrix model with probability measure\n$$\n\\frac{1}{Z_N}e^{-N\\textrm{tr}(V(M)-AM)},\n$$\nwhere $A$ is an $N\\times N$ diagonal matrix with eigenvalues $\\pm a$ (each with equal multiplicity $N/2$, assuming $N$ is even), and $V(M)$ is a general polynomial.", "statement_status": "exact", "statement_verification": "The exact AIM record asks to study the Hermitian external-source ensemble", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Vector equilibrium problems and their applications to random matrix models\nSection: Random matrix models with external source\nSource item: 2.1\nSource URL: http://aimpl.org/vectorequilib/2/\nCanonical location: aim-probability-notes.json notes[58]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Random matrices with external source. More general cases\\n\\nStudy the random matrix model with probability measure\\n$$\\n\\\\frac{1}{Z_N}e^{-N\\\\textrm{tr}(V(M)-AM)},\\n$$\\nwhere $A$ is an $N\\\\times N$ diagonal matrix with eigenvalues $\\\\pm a$ (each with equal multiplicity $N/2$, assuming $N$ is even), and $V(M)$ is a general polynomial.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"A more general situation involves an external source term $A$ with several different eigenvalues $\\\\{a_i\\\\}_{i=1}^{j}$, and multiplicities $\\\\{n_i\\\\}_{i=1}^j$, where $n=n_1+n_2+\\\\ldots+n_j$. It is assumed that all limits $c_j=\\\\lim_{n\\\\to\\\\infty} \\\\frac{n_j}{n}$ exist.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/vectorequilib/2/", "tags": [ "aim", "AIM-PROBABILITY-0059", "aim-domain:probability", "aim-workshop:vectorequilib", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every confining real polynomial V and a balanced external source A=aJ, matrix integration by parts gives the exact finite-N Ward identity E[V'(M)]=aJ, while the normalized free energy is even and convex with alignment derivative equal to Var(Tr(JM)). If V'' is bounded below by kappa>0, Brascamp-Lieb yields the sharp dimension-free bound 0<=q_N'(a)<=1/kappa and 0<=q_N(a)<=a/kappa. The report also proves the exact confluent-HCIZ reduction to multiple orthogonal polynomials, records the necessary confinement condition, and recovers the Gaussian free-convolution transition. It does not claim the full nonsymmetric local-universality problem is solved.\n\nCandidate contribution (theorem; novelty confidence low): The finite-N nonsymmetric response package combines the exact matrix Ward identity, even convex balanced-source free energy, a sharp 1/kappa Lipschitz alignment bound under V''>=kappa, and the resulting necessary Ward moment condition for every candidate limiting density." }, { "id": 20002618, "problem_number": "AIM-PROBABILITY-0060", "title": "Soft d-bar characterization, defect certificate, and exact radial benchmarks", "statement": "Normal matrix model and d-bar problems\n\nIs it possible to use the theory of $\\overline{\\partial}$-problems to analyze the normal matrix models?", "original_statement": "Normal matrix model and d-bar problems\n\nIs it possible to use the theory of $\\overline{\\partial}$-problems to analyze the normal matrix models?", "clean_statement": "Normal matrix model and d-bar problems\n\nIs it possible to use the theory of $\\overline{\\partial}$-problems to analyze the normal matrix models?", "statement_status": "exact", "statement_verification": "The canonical AIM record is problem 3.1 in the section “Normal matrix model” of the workshop *Vector equilibrium problems and their applications to random matrix models*. Its question is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Vector equilibrium problems and their applications to random matrix models\nSection: Normal matrix model\nSource item: 3.1\nSource URL: http://aimpl.org/vectorequilib/3/\nCanonical location: aim-probability-notes.json notes[59]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Normal matrix model and d-bar problems\\n\\nIs it possible to use the theory of $\\\\overline{\\\\partial}$-problems to analyze the normal matrix models?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"Progress on this problem for the case with potential $Q(z)=|z|^2-2c\\\\log|z-a|$, with $a$ and $c$ constant, was reported by Ken McLaughlin, as part of a joint and ongoing work with M. Bertola and S.-Y. Lee.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/vectorequilib/3/", "tags": [ "aim", "AIM-PROBABILITY-0060", "aim-domain:probability", "aim-workshop:vectorequilib", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The broad AIM question has an affirmative answer in regular regimes but not a universal one: modern soft Riemann-Hilbert work rigorously uses the planar d-bar problem for strong asymptotics, while the cited logarithmic-charge model was solved through a contour Riemann-Hilbert reduction. This attempt proves an exact one-row soft d-bar characterization, a compact-support exterior Laurent remainder bound whose forbidden coefficients are precisely the planar orthogonality defects, and a closed-form soft-problem solution for every locally bounded radial weight, including an incomplete-gamma formula for the Gaussian model.\n\nCandidate contribution (quantitative reduction and benchmark theorem; novelty confidence low): The explicit finite-Laurent-jet defect bound, paired with the arbitrary-radial closed-form soft d-bar solution, provides a directly testable normalization, sign, and moment-defect certificate for proposed soft-Riemann-Hilbert parametrices." }, { "id": 20002619, "problem_number": "AIM-PROBABILITY-0061", "title": "Sign normalization and a degree-dependent cut bound for polynomial S-curves", "statement": "$S$-curves for complex potentials in the scalar case\n\nAnalysis of existence and properties of $S$-curves in the scalar case when the potential is $\\phi(z)=\\textrm{Re}\\, z^p$, for $p\\geq 3$, and we take $\\tau$ as the set of continua that join neighboring valleys of $\\phi(z)$ (where $\\phi(z)\\to-\\infty$) in the complex plane. In particular, prove that the support of the equilibrium measure on a curve with the $S$-property is one-cut case, or give bounds on the number of cuts depending on $p$.", "original_statement": "$S$-curves for complex potentials in the scalar case\n\nAnalysis of existence and properties of $S$-curves in the scalar case when the potential is $\\phi(z)=\\textrm{Re}\\, z^p$, for $p\\geq 3$, and we take $\\tau$ as the set of continua that join neighboring valleys of $\\phi(z)$ (where $\\phi(z)\\to-\\infty$) in the complex plane. In particular, prove that the support of the equilibrium measure on a curve with the $S$-property is one-cut case, or give bounds on the number of cuts depending on $p$.", "clean_statement": "$S$-curves for complex potentials in the scalar case\n\nAnalysis of existence and properties of $S$-curves in the scalar case when the potential is $\\phi(z)=\\textrm{Re}\\, z^p$, for $p\\geq 3$, and we take $\\tau$ as the set of continua that join neighboring valleys of $\\phi(z)$ (where $\\phi(z)\\to-\\infty$) in the complex plane. In particular, prove that the support of the equilibrium measure on a curve with the $S$-property is one-cut case, or give bounds on the number of cuts depending on $p$.", "statement_status": "exact", "statement_verification": "The canonical AIM record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Vector equilibrium problems and their applications to random matrix models\nSection: $S$-curves\nSource item: 4.1\nSource URL: http://aimpl.org/vectorequilib/4/\nCanonical location: aim-probability-notes.json notes[60]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"$S$-curves for complex potentials in the scalar case\\n\\nAnalysis of existence and properties of $S$-curves in the scalar case when the potential is $\\\\phi(z)=\\\\textrm{Re}\\\\, z^p$, for $p\\\\geq 3$, and we take $\\\\tau$ as the set of continua that join neighboring valleys of $\\\\phi(z)$ (where $\\\\phi(z)\\\\to-\\\\infty$) in the complex plane. In particular, prove that the support of the equilibrium measure on a curve with the $S$-property is one-cut case, or give bounds on the number of cuts depending on $p$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/vectorequilib/4/", "tags": [ "aim", "AIM-PROBABILITY-0061", "aim-domain:probability", "aim-workshop:vectorequilib", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the standard weighted-energy convention, the literal field Re(z^p) on radial ends in sectors where it tends to minus infinity has energy infimum minus infinity, so the well-posed reading uses Q=-Re(z^p), equivalently a rotation to the usual polynomial growth sectors. Kuijlaars-Silva then supplies a max-min S-curve. Their critical-trajectory polynomial R has degree 2p-2 and every compact support trajectory joins two distinct zeros; consequently the equilibrium support has at most p-1 connected components. In the two-sector case the connected complement makes the support graph a forest, giving at most 2p-3 maximal trajectory arcs.\n\nCandidate contribution (bound; novelty confidence low): For any degree-p polynomial max-min equilibrium measure in the Kuijlaars-Silva class, the number of connected support components is at most p-1; for a two-sector maximizer, the number of maximal support-trajectory edges is at most 2p-3. The bound is paired with an explicit translated-segment obstruction showing why the AIM negative valleys require the sign-normalized field." }, { "id": 20002620, "problem_number": "AIM-PROBABILITY-0062", "title": "Certified natural-coordinate continuation for critical trajectories", "statement": "Numerical computation of $S$-curves\n\nA general question about numerical methods for vector equilibrium problems and computation of corresponding $S$-curves. More in particular, how to produce critical trajectories in a reliable way, given the spectral curve of the problem?", "original_statement": "Numerical computation of $S$-curves\n\nA general question about numerical methods for vector equilibrium problems and computation of corresponding $S$-curves. More in particular, how to produce critical trajectories in a reliable way, given the spectral curve of the problem?", "clean_statement": "Numerical computation of $S$-curves\n\nA general question about numerical methods for vector equilibrium problems and computation of corresponding $S$-curves. More in particular, how to produce critical trajectories in a reliable way, given the spectral curve of the problem?", "statement_status": "exact", "statement_verification": "The canonical AIM record, problem 4.2 in the section “\\(S\\)-curves,” asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Vector equilibrium problems and their applications to random matrix models\nSection: $S$-curves\nSource item: 4.2\nSource URL: http://aimpl.org/vectorequilib/4/\nCanonical location: aim-probability-notes.json notes[61]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Numerical computation of $S$-curves\\n\\nA general question about numerical methods for vector equilibrium problems and computation of corresponding $S$-curves. More in particular, how to produce critical trajectories in a reliable way, given the spectral curve of the problem?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/vectorequilib/4/", "tags": [ "aim", "AIM-PROBABILITY-0062", "aim-domain:probability", "aim-workshop:vectorequilib", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Reliable S-curve computation separates algebraic sheet tracking, local trajectory enclosure, and global graph certification. This attempt proves a branch-sign-invariant natural-coordinate continuation theorem: explicit disk and contraction inequalities certify existence and uniqueness of a horizontal trajectory step, bound its Euler predictor error, and turn a validated integral residual into a position bound. It also derives singularity-aware step scaling and Puiseux launch directions, gives a worked finite critical arc for q(z)=1-z^2, and reduces global certification to finite interval-arithmetic tasks under stated nonrecurrence and separation assumptions.\n\nCandidate contribution (certified continuation lemma; novelty confidence low): The explicit sign-invariant contraction test, Euler-predictor bound, residual-to-position estimate, and singularity-aware rejection rule form a testable local certificate distinguishing a rigorously enclosed critical-trajectory step from an ordinary numerical plot." }, { "id": 20002621, "problem_number": "AIM-PROBABILITY-0063", "title": "A crease-balance law for a hat-shaped external field", "statement": "$S$-curves with piecewise harmonic external field\n\nConsider the external potential\n$$\n\\phi(z)=\\begin{cases} -k_1\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z>0,\\\\ k_2\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z0$. Is there a continuum with the $S$-property connecting the points $z_1=-\\varepsilon_1+i$ and $z_2=\\varepsilon_2-i$ in the complex plane, for small positive $\\varepsilon_1$ and $\\varepsilon_2$? Also, consider the max-min problem in this setting. Does the curve obtained have a portion along the imaginary axis?", "original_statement": "$S$-curves with piecewise harmonic external field\n\nConsider the external potential\n$$\n\\phi(z)=\\begin{cases} -k_1\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z>0,\\\\ k_2\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z0$. Is there a continuum with the $S$-property connecting the points $z_1=-\\varepsilon_1+i$ and $z_2=\\varepsilon_2-i$ in the complex plane, for small positive $\\varepsilon_1$ and $\\varepsilon_2$? Also, consider the max-min problem in this setting. Does the curve obtained have a portion along the imaginary axis?", "clean_statement": "$S$-curves with piecewise harmonic external field\n\nConsider the external potential\n$$\n\\phi(z)=\\begin{cases} -k_1\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z>0,\\\\ k_2\\, \\textrm{Re}\\, z, & \\textrm{Re}\\, z0$. Is there a continuum with the $S$-property connecting the points $z_1=-\\varepsilon_1+i$ and $z_2=\\varepsilon_2-i$ in the complex plane, for small positive $\\varepsilon_1$ and $\\varepsilon_2$? Also, consider the max-min problem in this setting. Does the curve obtained have a portion along the imaginary axis?", "statement_status": "exact", "statement_verification": "The canonical record is visibly truncated. Its formula ends with the literal text", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Vector equilibrium problems and their applications to random matrix models\nSection: $S$-curves\nSource item: 4.3\nSource URL: http://aimpl.org/vectorequilib/4/\nCanonical location: aim-probability-notes.json notes[62]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"$S$-curves with piecewise harmonic external field\\n\\nConsider the external potential\\n$$\\n\\\\phi(z)=\\\\begin{cases} -k_1\\\\, \\\\textrm{Re}\\\\, z, & \\\\textrm{Re}\\\\, z>0,\\\\\\\\ k_2\\\\, \\\\textrm{Re}\\\\, z, & \\\\textrm{Re}\\\\, z0$. Is there a continuum with the $S$-property connecting the points $z_1=-\\\\varepsilon_1+i$ and $z_2=\\\\varepsilon_2-i$ in the complex plane, for small positive $\\\\varepsilon_1$ and $\\\\varepsilon_2$? Also, consider the max-min problem in this setting. Does the curve obtained have a portion along the imaginary axis?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: unknown; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/vectorequilib/4/", "tags": [ "aim", "AIM-PROBABILITY-0063", "aim-domain:probability", "aim-workshop:vectorequilib", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Under the explicitly stated reconstruction of the corrupted AIM formula, any regular active segment on the imaginary-axis crease must satisfy the necessary balance law partial_x U^nu(iy)=(k_1-k_2)/2 for the potential of the off-axis mass. In the collapsed-endpoint case epsilon_1=epsilon_2=0, the segment [-i,i] is a max--min optimizer with value log 2 for every positive k_1,k_2, but it has the S-property if and only if k_1=k_2. The original small-positive-offset existence question remains unresolved.\n\nCandidate contribution (obstruction; novelty confidence low): For the reconstructed piecewise-linear hat field, an active imaginary-axis segment forces the off-axis equilibrium mass to generate the constant horizontal field (k_1-k_2)/2; moreover, the exact collapsed-endpoint optimizer demonstrates that max--min optimality at a crease does not imply the S-property when k_1 differs from k_2." }, { "id": 20002622, "problem_number": "AIM-PROBABILITY-0064", "title": "Weighted image-charge reduction for Green S-curves", "statement": "Green's potential\n\nExistence theory for $S$-curves in a general sense, if the logarithmic potential is replaced by Green's potential.", "original_statement": "Green's potential\n\nExistence theory for $S$-curves in a general sense, if the logarithmic potential is replaced by Green's potential.", "clean_statement": "Green's potential\n\nExistence theory for $S$-curves in a general sense, if the logarithmic potential is replaced by Green's potential.", "statement_status": "exact", "statement_verification": "There is no apparent OCR error, but the statement omits essential data: the Greenian domain \\(D\\), the normalization of its kernel, the external field, whether admissible continua may meet \\(\\partial D\\), the topology of the admissible class, and whether the request concerns equilibrium on a fixed continuum or a max--min free boundary.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Vector equilibrium problems and their applications to random matrix models\nSection: $S$-curves\nSource item: 4.4\nSource URL: http://aimpl.org/vectorequilib/4/\nCanonical location: aim-probability-notes.json notes[63]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Green's potential\\n\\nExistence theory for $S$-curves in a general sense, if the logarithmic potential is replaced by Green's potential.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/vectorequilib/4/", "tags": [ "aim", "AIM-PROBABILITY-0064", "aim-domain:probability", "aim-workshop:vectorequilib", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a nonpolar compact support strictly inside the upper half-plane, weighted Green equilibrium is exactly equivalent to a constrained zero-mass signed logarithmic equilibrium obtained by odd reflection: if nu=mu-mu* then U_H^mu=U^nu, I(nu)=2 I_H(mu), and the reflected weighted logarithmic functional is twice the weighted Green functional. The equivalence preserves the local S-property and transports conformally to simply connected Green domains. In addition, the boundary-approaching segments K_epsilon=[-1,1]+i epsilon satisfy 0 <= V_H(K_epsilon) <= pi epsilon, giving a quantitative obstruction to a general existence theorem based only on Hausdorff compactness in the domain closure.\n\nCandidate contribution (reduction; novelty confidence low): The exact weighted image-charge identity, its local S-property equivalence and conformal transport, together with the explicit boundary-degeneration bound V_H([-1,1]+i epsilon) <= pi epsilon, form a testable reduction-and-obstruction package for formulating Green S-curve existence theorems." }, { "id": 20002623, "problem_number": "AIM-PROBABILITY-0065", "title": "A reverse compatibility certificate from spectral curves to equilibrium measures", "statement": "In a general sense, is it possible to find some explicit connection between the notions of string equations, spectral curve, Lax pair, WKB approximation, etc, and the vector equilibrium problems that appear in the two-matrix or external source models? For example, would it be possible to use string equations and asymptotic information together with the spectral curve, to find heuristically the equilibrium measure?", "original_statement": "In a general sense, is it possible to find some explicit connection between the notions of string equations, spectral curve, Lax pair, WKB approximation, etc, and the vector equilibrium problems that appear in the two-matrix or external source models? For example, would it be possible to use string equations and asymptotic information together with the spectral curve, to find heuristically the equilibrium measure?", "clean_statement": "In a general sense, is it possible to find some explicit connection between the notions of string equations, spectral curve, Lax pair, WKB approximation, etc, and the vector equilibrium problems that appear in the two-matrix or external source models? For example, would it be possible to use string equations and asymptotic information together with the spectral curve, to find heuristically the equilibrium measure?", "statement_status": "exact", "statement_verification": "The AIM record asks whether string equations, spectral curves, Lax pairs, and WKB asymptotics can be connected explicitly to vector equilibrium problems in two-matrix or external-source models, and whether string equations plus asymptotic spectral-curve data can heuristically recover the equilibrium measure.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Vector equilibrium problems and their applications to random matrix models\nSection: Other\nSource item: 5.1\nSource URL: http://aimpl.org/vectorequilib/5/\nCanonical location: aim-probability-notes.json notes[64]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In a general sense, is it possible to find some explicit connection between the notions of string equations, spectral curve, Lax pair, WKB approximation, etc, and the vector equilibrium problems that appear in the two-matrix or external source models? For example, would it be possible to use string equations and asymptotic information together with the spectral curve, to find heuristically the equilibrium measure?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/vectorequilib/5/", "tags": [ "aim", "AIM-PROBABILITY-0065", "aim-domain:probability", "aim-workshop:vectorequilib", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Known two-matrix and external-source frameworks rigorously connect finite-band string/recurrence equations, compatible Lax systems, large-N spectral curves, and model-specific vector equilibrium problems, but not by a universal reversible implication. This attempt proves a scalar six-gate reconstruction theorem: infinity normalization, sheet gluing, positive jumps, and trajectory geometry recover a positive critical measure with componentwise Euler-Lagrange equality, while filling-fraction compatibility and the off-support variational inequality are additional independent gates required to select the physical equilibrium. A Gaussian example audits the normalization y=V'-2C and the jump factor 1/(4 pi i).\n\nCandidate contribution (compatibility theorem and obstruction; novelty confidence low): The four-local-gate plus two-global-gate certificate gives a testable reverse interface from a marked semiclassical spectral curve to a critical or equilibrium measure and proves precisely why an unmarked spectral curve alone is insufficient for physical selection." }, { "id": 20002624, "problem_number": "AIM-PROBABILITY-0066", "title": "When convolution does and does not reduce multiple orthogonality", "statement": "Is it possible to relate multiple orthogonal polynomials to standard orthogonal polynomials, via the so-called ``convolution product\"?", "original_statement": "Is it possible to relate multiple orthogonal polynomials to standard orthogonal polynomials, via the so-called ``convolution product\"?", "clean_statement": "Is it possible to relate multiple orthogonal polynomials to standard orthogonal polynomials, via the so-called ``convolution product\"?", "statement_status": "exact", "statement_verification": "The canonical AIM record asks exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: Vector equilibrium problems and their applications to random matrix models\nSection: Other\nSource item: 5.2\nSource URL: http://aimpl.org/vectorequilib/5/\nCanonical location: aim-probability-notes.json notes[65]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Is it possible to relate multiple orthogonal polynomials to standard orthogonal polynomials, via the so-called ``convolution product\\\"?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/vectorequilib/5/", "tags": [ "aim", "AIM-PROBABILITY-0066", "aim-domain:probability", "aim-workshop:vectorequilib", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Primary-source context identifies Mellin convolution as the likely, though not explicitly verified, meaning of the AIM phrase: Smet and Van Assche had already represented beta=0 Jacobi--Pineiro MOPs as Mellin convolutions of ordinary Jacobi polynomials. This attempt proves an exact constraint-span criterion under which a type II MOP is literally a scalar OP, constructs a positive monomial-block family realizing it, proves that any universal coefficient-diagonal Mellin intertwiner preserving two consecutive moment directions must have a point-mass kernel, and records recurrence-band collapse as a necessary condition for scalarizing an entire step-line sequence. The broad relation now also has a general finite-dimensional, non-convolution answer in arXiv:2606.27594v1.\n\nCandidate contribution (obstruction; novelty confidence low): A universal coefficient-diagonal polynomial transport intertwining positive Mellin convolution with both the zeroth and first moment functionals exists only when the convolution kernel is a point mass; equivalently, the three kernel moments must satisfy c_0 c_2-c_1^2=0, so the only positive case is trivial dilation." }, { "id": 20002625, "problem_number": "AIM-PROBABILITY-0067", "title": "Flat and stationary KPZ statistics with an exact drift audit", "statement": "Statistics of KPZ equation in $1+1$ dimension:\n\nCompute statistics for different initial data including $\\mathcal{Z}_0(x)=1$ (flat) and $\\mathcal{Z}_0(x)=e^{B(x)}$ for $B(x)$ a two-sided Brownian motion (equilibrium).", "original_statement": "Statistics of KPZ equation in $1+1$ dimension:\n\nCompute statistics for different initial data including $\\mathcal{Z}_0(x)=1$ (flat) and $\\mathcal{Z}_0(x)=e^{B(x)}$ for $B(x)$ a two-sided Brownian motion (equilibrium).", "clean_statement": "Statistics of KPZ equation in $1+1$ dimension:\n\nCompute statistics for different initial data including $\\mathcal{Z}_0(x)=1$ (flat) and $\\mathcal{Z}_0(x)=e^{B(x)}$ for $B(x)$ a two-sided Brownian motion (equilibrium).", "statement_status": "exact", "statement_verification": "This is Problem 11.05 in the AIM workshop list *The Kardar--Parisi--Zhang equation and universality class* (canonical source record `aim-probability-notes.json`, index 66). The exact recovered mathematical request is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.05\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[66]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Statistics of KPZ equation in $1+1$ dimension:\\n\\nCompute statistics for different initial data including $\\\\mathcal{Z}_0(x)=1$ (flat) and $\\\\mathcal{Z}_0(x)=e^{B(x)}$ for $B(x)$ a two-sided Brownian motion (equilibrium).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0067", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The literature audit separates formal replica formulas from rigorous results: flat finite-time one-point Fredholm-Pfaffian formulas are replica-derived, stationary finite-time one-point and Baik--Rains asymptotics have a rigorous integrable-probability treatment, and rigorous multipoint formulas apply to the long-time KPZ fixed point rather than the general finite-time continuum equation. In the normalized Ito SHE, the stationary drift family rigorously satisfies Z_u(t,x) =_d exp(ux+u^2t/2) Z_0(t,x+ut), jointly at all finite collections of points, yielding a concrete normalization test for proposed formulas.\n\nCandidate contribution (consistency identity; novelty confidence low): For every finite collection (t_i,x_i), subtracting ux_i+u^2t_i/2 from the drift-u stationary KPZ heights must reproduce the zero-drift joint law evaluated at the sheared points x_i+ut_i; this three-term finite-dimensional audit is proposed as a falsifiable normalization certificate." }, { "id": 20002626, "problem_number": "AIM-PROBABILITY-0068", "title": "Finite-time tilt symmetry and the corrected Airy2 scaling", "statement": "Compute multi-point (spatial) distribution for KPZ with various initial data. For example, with $\\mathcal{Z}_0(x)=\\delta_{x=0}$, let\n$$\\mathcal{Z}(t,x)=e^{-\\frac{x^2}{2t}+t^{1/3}A_t(t^{-2/3}x)-\\frac{t}{24}}.$$\nWhat is the distribution $F(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_t(x_1)\\leq\\xi_1, A_t(x_2)\\leq\\xi_2\\right)=F(\\xi_1,\\xi_2)?$$\nPerhaps easier, show that as $t$ goes to infinity and space is scaled like $t^{2/3}x$, the process $A_{t}(t^{2/3}x)$ converges to the Airy$_2$ process in $x$.", "original_statement": "Compute multi-point (spatial) distribution for KPZ with various initial data. For example, with $\\mathcal{Z}_0(x)=\\delta_{x=0}$, let\n$$\\mathcal{Z}(t,x)=e^{-\\frac{x^2}{2t}+t^{1/3}A_t(t^{-2/3}x)-\\frac{t}{24}}.$$\nWhat is the distribution $F(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_t(x_1)\\leq\\xi_1, A_t(x_2)\\leq\\xi_2\\right)=F(\\xi_1,\\xi_2)?$$\nPerhaps easier, show that as $t$ goes to infinity and space is scaled like $t^{2/3}x$, the process $A_{t}(t^{2/3}x)$ converges to the Airy$_2$ process in $x$.", "clean_statement": "Compute multi-point (spatial) distribution for KPZ with various initial data. For example, with $\\mathcal{Z}_0(x)=\\delta_{x=0}$, let\n$$\\mathcal{Z}(t,x)=e^{-\\frac{x^2}{2t}+t^{1/3}A_t(t^{-2/3}x)-\\frac{t}{24}}.$$\nWhat is the distribution $F(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_t(x_1)\\leq\\xi_1, A_t(x_2)\\leq\\xi_2\\right)=F(\\xi_1,\\xi_2)?$$\nPerhaps easier, show that as $t$ goes to infinity and space is scaled like $t^{2/3}x$, the process $A_{t}(t^{2/3}x)$ converges to the Airy$_2$ process in $x$.", "statement_status": "exact", "statement_verification": "The exact AIM record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.1\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[67]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Compute multi-point (spatial) distribution for KPZ with various initial data. For example, with $\\\\mathcal{Z}_0(x)=\\\\delta_{x=0}$, let\\n$$\\\\mathcal{Z}(t,x)=e^{-\\\\frac{x^2}{2t}+t^{1/3}A_t(t^{-2/3}x)-\\\\frac{t}{24}}.$$\\nWhat is the distribution $F(\\\\xi_1, \\\\xi_2)$ such that\\n$$\\\\mathbb{P}\\\\left(A_t(x_1)\\\\leq\\\\xi_1, A_t(x_2)\\\\leq\\\\xi_2\\\\right)=F(\\\\xi_1,\\\\xi_2)?$$\\nPerhaps easier, show that as $t$ goes to infinity and space is scaled like $t^{2/3}x$, the process $A_{t}(t^{2/3}x)$ converges to the Airy$_2$ process in $x$.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0068", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the standard full-line narrow-wedge stochastic heat equation, the AIM field A_t is already expressed in the t^{2/3}-scaled coordinate. Its finite-time spatial law is stationary and reflection invariant, so the two-point CDF depends only on one separation r and satisfies G_t(r;a,b)=G_t(-r;a,b)=G_t(r;b,a), with the zero-separation law reducing to the known one-point CDF at min(a,b). The standard Airy normalization B_t(r)=2^{1/3}A_t(2^{1/3}r) converges locally uniformly to the conventional Airy2 process by Wu's 2026 theorem, yielding an explicit extended-Airy-kernel formula for the limiting two-point CDF. The source's final A_t(t^{2/3}x) applies the spatial scaling twice under its own displayed definition.\n\nCandidate contribution (reduction; novelty confidence low): Any proposed finite-time narrow-wedge KPZ two-point formula in the AIM normalization must reduce to a function G_t of one separation and two thresholds, obey G_t(r;a,b)=G_t(-r;a,b)=G_t(r;b,a), reduce at r=0 to the one-point law at min(a,b), and transform under B_t(r)=2^{1/3}A_t(2^{1/3}r) to the conventional extended-Airy-kernel limit." }, { "id": 20002627, "problem_number": "AIM-PROBABILITY-0069", "title": "Two-time KPZ laws: fixed-point formulas and a finite-time full-profile obstruction", "statement": "Compute multi-time distribution for KPZ. What is the distribution $F'_{t_1, t_2}(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_{t_1}(x)\\leq\\xi_1, A_{t_2}(x)\\leq\\xi_2\\right)=F'_{t_1, t_2}(\\xi_1,\\xi_2)?$$", "original_statement": "Compute multi-time distribution for KPZ. What is the distribution $F'_{t_1, t_2}(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_{t_1}(x)\\leq\\xi_1, A_{t_2}(x)\\leq\\xi_2\\right)=F'_{t_1, t_2}(\\xi_1,\\xi_2)?$$", "clean_statement": "Compute multi-time distribution for KPZ. What is the distribution $F'_{t_1, t_2}(\\xi_1, \\xi_2)$ such that\n$$\\mathbb{P}\\left(A_{t_1}(x)\\leq\\xi_1, A_{t_2}(x)\\leq\\xi_2\\right)=F'_{t_1, t_2}(\\xi_1,\\xi_2)?$$", "statement_status": "exact", "statement_verification": "The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.15\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[68]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Compute multi-time distribution for KPZ. What is the distribution $F'_{t_1, t_2}(\\\\xi_1, \\\\xi_2)$ such that\\n$$\\\\mathbb{P}\\\\left(A_{t_1}(x)\\\\leq\\\\xi_1, A_{t_2}(x)\\\\leq\\\\xi_2\\\\right)=F'_{t_1, t_2}(\\\\xi_1,\\\\xi_2)?$$\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"This should be much harder since the analogous distribution is not known in the setting of TASEP or LPP. It is not at all clear that there should be a formula.\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0069", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The old literature premise is obsolete for zero-temperature models and the KPZ fixed point: explicit two- and multi-time scaling-limit CDF formulas now exist, including narrow-wedge formulas and recent compact-support extensions. For the literal finite-time narrow-wedge KPZ equation, no rigorous closed two-time CDF was located. This attempt proves an exact normalization-aware identity expressing the requested CDF through the stochastic heat equation's full time-t1 profile and an independent future stochastic heat kernel, and proves that evaluation at one point is not a Markov-sufficient statistic uniformly over deterministic positive initial profiles.\n\nCandidate contribution (obstruction; novelty confidence low): In the AIM normalization, the finite-time two-height CDF equals a precisely thresholded full-profile stochastic-heat-semigroup functional, with distinct formulas for a common scaled coordinate and a common physical coordinate; moreover, two positive profiles with the same value at the observation point but different heat-semigroup averages prove that no universal one-height transition kernel can replace this full-profile conditioning." }, { "id": 20002628, "problem_number": "AIM-PROBABILITY-0070", "title": "A sharp Galerkin GFF cancellation and an ultraviolet obstruction for 2D AKPZ", "statement": "Higher dimensions\n\nMake rigorous sense of the anisotropic KPZ equation\n$$\\partial_t h=(\\partial_x h)^2-(\\partial_y h)^2+\\Delta h+\\xi$$\nand show that the Gaussian free field is invariant.\nIs there a way of getting this equation out of the 2d Schur process dynamics of Borodin-Ferrari? Or perhaps out of the 2d q-Whittaker process dynamics of Borodin-Corwin? Is this related to $2d$ quantum Toda chain?", "original_statement": "Higher dimensions\n\nMake rigorous sense of the anisotropic KPZ equation\n$$\\partial_t h=(\\partial_x h)^2-(\\partial_y h)^2+\\Delta h+\\xi$$\nand show that the Gaussian free field is invariant.\nIs there a way of getting this equation out of the 2d Schur process dynamics of Borodin-Ferrari? Or perhaps out of the 2d q-Whittaker process dynamics of Borodin-Corwin? Is this related to $2d$ quantum Toda chain?", "clean_statement": "Higher dimensions\n\nMake rigorous sense of the anisotropic KPZ equation\n$$\\partial_t h=(\\partial_x h)^2-(\\partial_y h)^2+\\Delta h+\\xi$$\nand show that the Gaussian free field is invariant.\nIs there a way of getting this equation out of the 2d Schur process dynamics of Borodin-Ferrari? Or perhaps out of the 2d q-Whittaker process dynamics of Borodin-Corwin? Is this related to $2d$ quantum Toda chain?", "statement_status": "exact", "statement_verification": "This is Problem 11.2 in the AIM workshop list *The Kardar--Parisi--Zhang equation and universality class*. The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.2\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[69]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Higher dimensions\\n\\nMake rigorous sense of the anisotropic KPZ equation\\n$$\\\\partial_t h=(\\\\partial_x h)^2-(\\\\partial_y h)^2+\\\\Delta h+\\\\xi$$\\nand show that the Gaussian free field is invariant.\\nIs there a way of getting this equation out of the 2d Schur process dynamics of Borodin-Ferrari? Or perhaps out of the 2d q-Whittaker process dynamics of Borodin-Corwin? Is this related to $2d$ quantum Toda chain?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0070", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "On the mean-zero two-torus, the spectrally truncated equation with quadratic form grad(h)^T A grad(h) preserves its cutoff Gaussian free field exactly if and only if the symmetric matrix A is trace-free; this includes the AIM anisotropy diag(1,-1). The proof gives a divergence identity, an exact Dirichlet-energy identity, global finite-dimensional well-posedness, and a witness proving necessity. Nevertheless, for every nonzero symmetric trace-free A and every fixed nonzero Fourier mode, the invariant quadratic drift has variance of order N^2 and is not tight as the cutoff N tends to infinity. Thus cutoff invariance and cancellation of a scalar Wick counterterm do not construct the fixed-coupling ultraviolet equation. The checked literature instead establishes regularized stationary dynamics and a weak-coupling Edwards-Wilkinson limit with enhanced diffusivity; Schur and q-Whittaker results yield GFF or additive limits, not the requested nonlinear continuum construction.\n\nCandidate contribution (Galerkin cancellation and obstruction; novelty confidence low): Candidate trace-and-ultraviolet certificate: trace-free anisotropy is exactly the finite-Galerkin GFF-invariance condition, while every fixed nonzero Fourier mode of the same invariant quadratic drift has variance asymptotic in order to N^2 and is non-tight; therefore GFF preservation plus scalar counterterm cancellation is insufficient for an ultraviolet construction." }, { "id": 20002629, "problem_number": "AIM-PROBABILITY-0071", "title": "A precise high-dimensional Edwards-Wilkinson reduction", "statement": "Prove that above 2 spatial dimensions, the KPZ equation is trivial (i.e., discretizations limit to the linearized equation, perhaps with a larger variance in the noise).", "original_statement": "Prove that above 2 spatial dimensions, the KPZ equation is trivial (i.e., discretizations limit to the linearized equation, perhaps with a larger variance in the noise).", "clean_statement": "Prove that above 2 spatial dimensions, the KPZ equation is trivial (i.e., discretizations limit to the linearized equation, perhaps with a larger variance in the noise).", "statement_status": "exact", "statement_verification": "The AIM record, from the workshop *The Kardar--Parisi--Zhang equation and universality class* (problem 11.25), asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.25\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[70]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Prove that above 2 spatial dimensions, the KPZ equation is trivial (i.e., discretizations limit to the linearized equation, perhaps with a larger variance in the noise).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0071", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The unrestricted statement is not a valid all-coupling theorem: rigorous Edwards-Wilkinson limits in dimensions d >= 3 require a specified regularization and a weak-disorder hypothesis, and are proved throughout the L2 region. For a Gaussian lattice polymer, the report proves a three-part diagnostic: the first-chaos Green sum is finite exactly for d >= 3; the exact second-moment window is pi_d exp(beta^2) < 1; and diffusively rescaled first-chaos spatial averages converge to the additive stochastic heat equation. A separate, explicit nonlinear comparison is still required for the full KPZ height, while strong disorder obstructs the blanket perturbative interpretation.\n\nCandidate contribution (reduction_and_synthesis; novelty confidence low): A normalization-consistent EW admissibility certificate combines the Green-kernel dimension threshold, the exact Gaussian replica L2 threshold, the unique nondegenerate diffusive first-chaos scaling, and an explicit Slutsky reduction for the nonlinear height; failure of the replica inequality gives a concrete check against interpreting d >= 3 alone as linearization." }, { "id": 20002630, "problem_number": "AIM-PROBABILITY-0072", "title": "Critical two-dimensional stochastic heat flow versus dynamic-GFF chaos", "statement": "In 2 spatial dimensions is the measure valued solution of the stochastic heat equation related to the ``exponential'' of the dynamic GFF and hence quantum Louiville gravity?", "original_statement": "In 2 spatial dimensions is the measure valued solution of the stochastic heat equation related to the ``exponential'' of the dynamic GFF and hence quantum Louiville gravity?", "clean_statement": "In 2 spatial dimensions is the measure valued solution of the stochastic heat equation related to the ``exponential'' of the dynamic GFF and hence quantum Louiville gravity?", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *The Kardar--Parisi--Zhang equation and universality class*, “Big picture questions,” item 11.3) asks exactly:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.3\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[71]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"In 2 spatial dimensions is the measure valued solution of the stochastic heat equation related to the ``exponential'' of the dynamic GFF and hence quantum Louiville gravity?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0072", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The natural direct identification is false: a fixed-time exponential of a stationary dynamic GFF is planar GFF Gaussian multiplicative chaos (and gives the LQG area measure), whose normalized two-point correlation has a power singularity, whereas the critical two-dimensional stochastic heat flow has only a logarithmic two-point singularity and is rigorously not any Gaussian multiplicative chaos by the strict moment comparison of Caravenna, Sun, and Zygouras. Precise weaker relations do survive: subcritical SHE/KPZ fluctuations converge to an additive SHE/dynamic-GFF field, and a 2025 preprint proves a conditional GMC relation between continuum polymer path measures at different critical-window parameters; neither is equality of the SHF spatial marginal with LQG.\n\nCandidate contribution (obstruction; novelty confidence low): A two-stage, deterministic-transformation-invariant diagnostic separates the AIM objects: the GFF-GMC power-law two-point singularity cannot match the SHF logarithmic singularity, and any Gaussian field altered to match the latter must be log-log-correlated and then fails the exact Gaussian three-point factorization against the published strict SHF third-moment comparison." }, { "id": 20002631, "problem_number": "AIM-PROBABILITY-0073", "title": "Directional limit shapes and exponent changes for a higher-dimensional directed polymer", "statement": "Study polymer free energy and growth model fluctuation exponents in higher dimension? Compute limit shapes and fluctuation exponents.", "original_statement": "Study polymer free energy and growth model fluctuation exponents in higher dimension? Compute limit shapes and fluctuation exponents.", "clean_statement": "Study polymer free energy and growth model fluctuation exponents in higher dimension? Compute limit shapes and fluctuation exponents.", "statement_status": "exact", "statement_verification": "This is Problem 11.35 from the AIM workshop list *The Kardar--Parisi--Zhang equation and universality class*. The canonical record asks:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.35\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[72]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Study polymer free energy and growth model fluctuation exponents in higher dimension? Compute limit shapes and fluctuation exponents.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0073", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the standard i.i.d. directed polymer with transverse dimension d at least 3, an exact tilted two-replica criterion, exp(lambda(2 beta)-2 lambda(beta)) rho(theta)<1, gives L2 weak disorder at velocity v=grad log M(theta). In that directional region the quenched mesoscopic-tube shape equals the annealed shape lambda(beta)-I(v), with Hessian minus the inverse tilted one-step covariance, and the tilted point-to-line power exponent is zero. At zero tilt, within the verified simple-symmetric-walk scope of Vargas's local limit theorem, the origin point-to-point free energy is n lambda-(d/2)log n+O_P(1), with point-to-point exponent zero and transversal exponent 1/2. At a velocity vertex there is one path, so the quenched shape is instead beta E[omega]-log(2d), strictly below the annealed shape, with Gaussian n^(1/2) free-energy fluctuations and zero transversal width. This proves that neither one exponent nor a globally annealed shape can describe all higher-dimensional regimes and directions.\n\nCandidate contribution (directional phase-and-boundary certificate; novelty confidence low): Candidate combined certificate: the velocity-dependent collision condition exp(gamma(beta)) rho(theta)<1 simultaneously certifies the annealed quenched tube shape and its explicit Hessian in an interior L2 region, necessarily degenerates on approach to each velocity vertex, and matches an exact unique-path boundary formula with a strict quenched/annealed gap and exponent pair (chi,xi)=(1/2,0); at zero velocity the verified local-limit exponent pair is (0,1/2)." }, { "id": 20002632, "problem_number": "AIM-PROBABILITY-0074", "title": "Positive-temperature structure and a finite-rank gap criterion", "statement": "Structure at positive temperature / asymmetry:\n\nPositive temperature polymers have triangular arrays associated with them (which are limits of the Macdonald processes). The marginal measure on a given level is a tropical analog of random matrix eigenvalue ensembles (such as GUE or LUE) which are determinantal point processes. These tropical point processes are no longer determinantal yet, there are still Fredholm determinants for Laplace type transforms of the analog of the largest or smallest eigenvalues. Information about all of the tropical eigenvalues should be accessible via Fredholm determinants (tropical analogs of gap probabilities for example).\n\nWhat is the structure which replaces determinantal point processes and correlation functions and which allows for such computations?", "original_statement": "Structure at positive temperature / asymmetry:\n\nPositive temperature polymers have triangular arrays associated with them (which are limits of the Macdonald processes). The marginal measure on a given level is a tropical analog of random matrix eigenvalue ensembles (such as GUE or LUE) which are determinantal point processes. These tropical point processes are no longer determinantal yet, there are still Fredholm determinants for Laplace type transforms of the analog of the largest or smallest eigenvalues. Information about all of the tropical eigenvalues should be accessible via Fredholm determinants (tropical analogs of gap probabilities for example).\n\nWhat is the structure which replaces determinantal point processes and correlation functions and which allows for such computations?", "clean_statement": "Structure at positive temperature / asymmetry:\n\nPositive temperature polymers have triangular arrays associated with them (which are limits of the Macdonald processes). The marginal measure on a given level is a tropical analog of random matrix eigenvalue ensembles (such as GUE or LUE) which are determinantal point processes. These tropical point processes are no longer determinantal yet, there are still Fredholm determinants for Laplace type transforms of the analog of the largest or smallest eigenvalues. Information about all of the tropical eigenvalues should be accessible via Fredholm determinants (tropical analogs of gap probabilities for example).\n\nWhat is the structure which replaces determinantal point processes and correlation functions and which allows for such computations?", "statement_status": "exact", "statement_verification": "The AIM record asks what replaces determinantal point processes and correlation functions for triangular arrays associated with positive-temperature polymers. These arrays arise as limits of Macdonald processes. Their fixed-level laws resemble GUE/LUE eigenvalue ensembles, but are not determinantal; nevertheless, Laplace transforms of an extremal coordinate can have Fredholm determinant formulas. The question asks for a structure making all analogous eigenvalues, especially joint gap probabilities, accessible.", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.4\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[73]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Structure at positive temperature / asymmetry:\\n\\nPositive temperature polymers have triangular arrays associated with them (which are limits of the Macdonald processes). The marginal measure on a given level is a tropical analog of random matrix eigenvalue ensembles (such as GUE or LUE) which are determinantal point processes. These tropical point processes are no longer determinantal yet, there are still Fredholm determinants for Laplace type transforms of the analog of the largest or smallest eigenvalues. Information about all of the tropical eigenvalues should be accessible via Fredholm determinants (tropical analogs of gap probabilities for example).\\n\\nWhat is the structure which replaces determinantal point processes and correlation functions and which allows for such computations?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0074", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The structural replacement for a determinantal point process is not one kernel but three complementary layers: Macdonald/Whittaker spectral calculus from commuting difference or Toda operators and Cauchy/Bump-Stade identities; geometric-RSK branching and Markov intertwinings for triangular arrays; and exponential-interaction Gibbs resampling for line ensembles. These structures explain selected contour and scalar Fredholm formulas but do not currently provide general all-level gap determinants. The report proves that a full positive-orthant multivariate Laplace transform of nonnegative exponentiated Whittaker coordinates determines all rectangular gaps, and gives a rank-two counterexample showing that every one-coordinate Laplace transform can agree while a joint gap differs.\n\nCandidate contribution (separation_theorem_and_obstruction; novelty confidence low): For finite-rank nonnegative Whittaker or polymer coordinates, the interior multivariate Laplace transform and its finite damped mixed derivatives form a separating transform hierarchy, while coordinate-axis transforms provably do not; the explicit rank-two coupling test is a necessary diagnostic for any proposed all-level Fredholm formalism." }, { "id": 20002633, "problem_number": "AIM-PROBABILITY-0075", "title": "An exact cumulative-color ASEP array and a forced block-move obstruction", "statement": "Associated to q-TASEP there is a triangular array (the q-Whittaker process). This whole array is helpful in computing things about q-TASEP.\n\nIs there such an array for ASEP or for some sort of transformed version of ASEP?", "original_statement": "Associated to q-TASEP there is a triangular array (the q-Whittaker process). This whole array is helpful in computing things about q-TASEP.\n\nIs there such an array for ASEP or for some sort of transformed version of ASEP?", "clean_statement": "Associated to q-TASEP there is a triangular array (the q-Whittaker process). This whole array is helpful in computing things about q-TASEP.\n\nIs there such an array for ASEP or for some sort of transformed version of ASEP?", "statement_status": "exact", "statement_verification": "The record is Problem 11.45 in the AIM workshop list *The Kardar-Parisi-Zhang equation and universality class*. Its exact problem text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.45\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[74]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Associated to q-TASEP there is a triangular array (the q-Whittaker process). This whole array is helpful in computing things about q-TASEP.\\n\\nIs there such an array for ASEP or for some sort of transformed version of ASEP?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0075", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "On a finite ring, the cumulative color thresholds of colored ASEP form an injective nested Markov array. Every threshold row is ordinary two-sided ASEP by an exact generator intertwining. A swap of colors a and b changes exactly |a-b| consecutive rows, so this canonical basic-coupling array necessarily has vertical block moves and cannot be realized by elementary one-row-at-a-time updates when nonadjacent colors swap at positive rate. This answers the broad array question affirmatively while not supplying, or claiming, a q-Whittaker/Gelfand-Tsetlin lift of two-sided ASEP.\n\nCandidate contribution (intertwining_and_obstruction; novelty confidence low): The simultaneous cumulative-threshold projection packages colored ASEP as an injective triangular Markov array, and its exact vertical support rule implies a scoped obstruction: a positive-rate swap of colors a and b requires a simultaneous move in exactly |a-b| rows, excluding a one-row-at-a-time realization of this canonical coupling for nonadjacent colors." }, { "id": 20002634, "problem_number": "AIM-PROBABILITY-0076", "title": "Soft Karlin-McGregor kernels and a Toda determinant obstruction", "statement": "The limit of the triangular array is the diffusion of O'Connell based on the quantum Toda lattice, or (after another limit) the KPZ$_T$ line ensemble. Both have Brownian Gibbs properties allowing paths to cross but at exponential cost (a soft analog of the non-intersecting Brownian Gibbs property for the Airy line ensemble.)The Karlin-McGregor formula underlies the solvability of zero temperature system since it writes non-intersecting line ensembles in terms of determinants.\n\nIs there an analog of the Karlin-McGregor formula when the non-intersecting conditioning is replaced by a softer form of conditioning? Does this explain the solvability and occurrence of Fredholm determinants?", "original_statement": "The limit of the triangular array is the diffusion of O'Connell based on the quantum Toda lattice, or (after another limit) the KPZ$_T$ line ensemble. Both have Brownian Gibbs properties allowing paths to cross but at exponential cost (a soft analog of the non-intersecting Brownian Gibbs property for the Airy line ensemble.)The Karlin-McGregor formula underlies the solvability of zero temperature system since it writes non-intersecting line ensembles in terms of determinants.\n\nIs there an analog of the Karlin-McGregor formula when the non-intersecting conditioning is replaced by a softer form of conditioning? Does this explain the solvability and occurrence of Fredholm determinants?", "clean_statement": "The limit of the triangular array is the diffusion of O'Connell based on the quantum Toda lattice, or (after another limit) the KPZ$_T$ line ensemble. Both have Brownian Gibbs properties allowing paths to cross but at exponential cost (a soft analog of the non-intersecting Brownian Gibbs property for the Airy line ensemble.)The Karlin-McGregor formula underlies the solvability of zero temperature system since it writes non-intersecting line ensembles in terms of determinants.\n\nIs there an analog of the Karlin-McGregor formula when the non-intersecting conditioning is replaced by a softer form of conditioning? Does this explain the solvability and occurrence of Fredholm determinants?", "statement_status": "exact", "statement_verification": "This record is problem 11.5 in the AIM workshop list *The Kardar–Parisi–Zhang equation and universality class*, section “Big picture questions.” The exact extracted text is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.5\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[75]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"The limit of the triangular array is the diffusion of O'Connell based on the quantum Toda lattice, or (after another limit) the KPZ$_T$ line ensemble. Both have Brownian Gibbs properties allowing paths to cross but at exponential cost (a soft analog of the non-intersecting Brownian Gibbs property for the Airy line ensemble.)The Karlin-McGregor formula underlies the solvability of zero temperature system since it writes non-intersecting line ensembles in terms of determinants.\\n\\nIs there an analog of the Karlin-McGregor formula when the non-intersecting conditioning is replaced by a softer form of conditioning? Does this explain the solvability and occurrence of Fredholm determinants?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0076", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The exact soft analogue of the killed/nonintersecting Brownian kernel is the many-body Toda Feynman-Kac bridge kernel: independent Brownian bridges are tilted by the exponential of minus the time-integrated Toda interaction, and a positive Whittaker eigenfunction Doob-transforms this kernel into O'Connell's diffusion. A proved short-time separability theorem shows that if such a kernel were a literal Karlin-McGregor determinant of one common scalar one-particle Schrodinger kernel, its potential would have to be a constant plus a sum of one-body potentials. The finite Toda interaction violates this necessary condition through its nonzero mixed derivatives. This does not rule out Whittaker spectral transforms or Fredholm determinants for selected integrable observables.\n\nCandidate contribution (obstruction; novelty confidence low): For a smooth finite-potential Feynman-Kac kernel, equality at all sufficiently short times with an exponentially shifted determinant of a common scalar one-particle Schrodinger kernel forces the killing potential to be additive in the particle coordinates; hence a nonzero mixed derivative is a local certificate excluding that literal soft Karlin-McGregor representation, and it excludes the Toda exponential pair potential." }, { "id": 20002635, "problem_number": "AIM-PROBABILITY-0077", "title": "A causal max-plus certificate for nearest-neighbor ballistic deposition", "statement": "Expand the universality of the KPZ equation to:\n\nGrowth model such as ballistic deposition.", "original_statement": "Expand the universality of the KPZ equation to:\n\nGrowth model such as ballistic deposition.", "clean_statement": "Expand the universality of the KPZ equation to:\n\nGrowth model such as ballistic deposition.", "statement_status": "exact", "statement_verification": "The exact AIM record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.55\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[76]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Expand the universality of the KPZ equation to:\\n\\nGrowth model such as ballistic deposition.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0077", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the continuous-time unit-block nearest-neighbor ballistic-deposition process on Z^d, the height at a fixed space-time point has an exact direction-sensitive max-plus representation over a finite causal ancestry. Its causal source radius satisfies P(R_t(x) >= r) <= sum_{m>=r} ((2d+1)t)^m/m!, and the common-clock evolution is a pathwise sup-norm contraction in its initial profile. Two exact obstructions delimit direct universality approaches: no scalar vertex weight independent of incoming direction implements the standard update, and deleting a single deposition mark can change a neighboring height by an arbitrarily large amount. These results do not prove a KPZ exponent or scaling limit.\n\nCandidate contribution (structural_obstruction; novelty confidence low): Candidate contribution: for the precise standard continuous-time symmetric nearest-neighbor model, combine the explicit Poisson-series causal-radius bound with its edge-direction-weighted max-plus kernel, and prove by finite witnesses both failure of an incoming-direction-blind scalar vertex-weight update and unbounded single-arrival add-one sensitivity." }, { "id": 20002636, "problem_number": "AIM-PROBABILITY-0078", "title": "Exact block-update heights for finite-range and speed-change exclusion", "statement": "Height functions associated to exclusion processes with longer range jumps, or environment dependent speed-changes.", "original_statement": "Height functions associated to exclusion processes with longer range jumps, or environment dependent speed-changes.", "clean_statement": "Height functions associated to exclusion processes with longer range jumps, or environment dependent speed-changes.", "statement_status": "exact", "statement_verification": "The exact AIM record is the one-line prompt:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.6\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[77]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Height functions associated to exclusion processes with longer range jumps, or environment dependent speed-changes.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0078", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For every bounded finite-range conservative exclusion process on the integer line, including local configuration-dependent speed changes and fixed spatial environments, integrated cut currents define a height whose discrete gradient exactly recovers occupation. A jump from x to y changes precisely the |x-y| height coordinates associated with the crossed bonds; the transported height generator, discrete conservation law, and predictable martingale cross-bracket are explicit. The bracket has bandwidth at most R-1 for jump range R, and any positive-rate jump longer than one rules out an exact single-coordinate corner-flip generator on the same height state space.\n\nCandidate contribution (height_generator_and_covariance_lemma; novelty confidence low): The integrated-current construction yields a support-and-covariance dictionary: displacement z produces an interval update of exactly |z| height coordinates and a height-noise cross-bracket of bandwidth at most R-1, which gives a direct obstruction to single-coordinate corner flips for genuine long jumps and remains valid for bounded local speed changes and fixed spatial environments." }, { "id": 20002637, "problem_number": "AIM-PROBABILITY-0079", "title": "A Gaussian-mode audit for weakly nonlinear stochastic Hamilton-Jacobi equations", "statement": "Stochastic Hamilton-Jacobi equations $\\partial_th=F(\\nabla h)+\\Delta h+\\xi_\\epsilon$ with $F$ scaled appropriately.", "original_statement": "Stochastic Hamilton-Jacobi equations $\\partial_th=F(\\nabla h)+\\Delta h+\\xi_\\epsilon$ with $F$ scaled appropriately.", "clean_statement": "Stochastic Hamilton-Jacobi equations $\\partial_th=F(\\nabla h)+\\Delta h+\\xi_\\epsilon$ with $F$ scaled appropriately.", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.65\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[78]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Stochastic Hamilton-Jacobi equations $\\\\partial_th=F(\\\\nabla h)+\\\\Delta h+\\\\xi_\\\\epsilon$ with $F$ scaled appropriately.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0079", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the explicit one-dimensional weakly asymmetric reconstruction with spatially mollified white noise, the stationary linear gradient has exact variance sigma_rho^2/epsilon. Averaging the scaled nonlinearity over this order-one Gaussian background shows rigorously that the zeroth, first, and second Gaussian Hermite modes select the leading height speed, diverging transport velocity, and KPZ quadratic coupling. An exact affine covariance theorem proves that adding a constant or linear function to F changes only the height counterterm or comoving frame. The effective local quadratic response is one half of E[F''(G)], not F''(0)/2; explicit quartic examples show that the naive Taylor-at-zero test fails in both directions.\n\nCandidate contribution (coefficient criterion; novelty confidence low): For spatial mollification rho in the reconstructed one-dimensional equation, the combined affine-covariant Gaussian-mode test assigns Hermite modes H_0, H_1, and H_2 respectively to height centering, transport, and KPZ coupling, with exact cancellation criterion E[F''(G)]=0 for G of variance ||rho||_2^2/2; the tuned family F_theta(z)=z^4-theta z^2 has predicted coupling 6 sigma_rho^2-theta and cancels at theta=6 sigma_rho^2." }, { "id": 20002638, "problem_number": "AIM-PROBABILITY-0080", "title": "Eden conventions, exponential FPP clocks, and an exact finite-cluster comparison", "statement": "Eden model.", "original_statement": "Eden model.", "clean_statement": "Eden model.", "statement_status": "exact", "statement_verification": "The exact canonical record says only:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.7\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[79]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Eden model.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0080", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite seed on the nearest-neighbor lattice, continuous uniform-boundary-edge Eden growth is exactly exponential bond first-passage percolation: at cluster A the holding time is exponential with rate lambda times the edge-boundary size, and the next site v has probability m_A(v)/B(A). Uniform-boundary-site Eden is instead exponential site FPP, with holding rate lambda times the site-boundary size and uniform next-site law. The two embedded chains agree at A exactly when all boundary sites have equal contact multiplicity. Their one-step total-variation distance and full finite-history likelihood ratio are given explicitly; an L-triomino in Z^2 has one-step distance 3/28. A compensator martingale records the state-dependent conversion between cell count and FPP time.\n\nCandidate contribution (equivalence_and_counterexample; novelty confidence low): At every finite cluster A, the edge-Eden/site-Eden next-site total-variation distance is one half the sum over boundary sites of |m_A(v)/B(A)-1/S(A)|, equality holds exactly for constant boundary multiplicity, and for any ordered growth history their likelihood ratio is the product over steps of m_A(v_next)S(A)/B(A); the reachable L-triomino gives the explicit value 3/28." }, { "id": 20002639, "problem_number": "AIM-PROBABILITY-0081", "title": "KPZ fixed-point status and a scaling-centering obstruction certificate", "statement": "Universality of the KPZ universality class:\n\nProve the existence and uniqueness of the KPZ universality class fixed point introduced by Corwin-Quastel in 1 spatial dimension. That is to say, consider any growth process $h(t,x)$ and show that for an appropriate centering function $\\overline{h}_{\\epsilon}$,\n$$\\lim_{\\epsilon\\rightarrow\\infty}\\epsilon^{1/2}h(\\epsilon^{-3/2}t, \\epsilon^{-1}x)-\\overline{h}_\\epsilon$$\nhas a limit as $\\epsilon\\to 0$ as a space-time process? Then show that the properties of this process identify it uniquely (see the conjectured properties in Corwin-Quastel).", "original_statement": "Universality of the KPZ universality class:\n\nProve the existence and uniqueness of the KPZ universality class fixed point introduced by Corwin-Quastel in 1 spatial dimension. That is to say, consider any growth process $h(t,x)$ and show that for an appropriate centering function $\\overline{h}_{\\epsilon}$,\n$$\\lim_{\\epsilon\\rightarrow\\infty}\\epsilon^{1/2}h(\\epsilon^{-3/2}t, \\epsilon^{-1}x)-\\overline{h}_\\epsilon$$\nhas a limit as $\\epsilon\\to 0$ as a space-time process? Then show that the properties of this process identify it uniquely (see the conjectured properties in Corwin-Quastel).", "clean_statement": "Universality of the KPZ universality class:\n\nProve the existence and uniqueness of the KPZ universality class fixed point introduced by Corwin-Quastel in 1 spatial dimension. That is to say, consider any growth process $h(t,x)$ and show that for an appropriate centering function $\\overline{h}_{\\epsilon}$,\n$$\\lim_{\\epsilon\\rightarrow\\infty}\\epsilon^{1/2}h(\\epsilon^{-3/2}t, \\epsilon^{-1}x)-\\overline{h}_\\epsilon$$\nhas a limit as $\\epsilon\\to 0$ as a space-time process? Then show that the properties of this process identify it uniquely (see the conjectured properties in Corwin-Quastel).", "statement_status": "exact", "statement_verification": "The canonical record (AIM workshop *The Kardar--Parisi--Zhang equation and universality class*, problem 11.75) says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.75\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[80]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Universality of the KPZ universality class:\\n\\nProve the existence and uniqueness of the KPZ universality class fixed point introduced by Corwin-Quastel in 1 spatial dimension. That is to say, consider any growth process $h(t,x)$ and show that for an appropriate centering function $\\\\overline{h}_{\\\\epsilon}$,\\n$$\\\\lim_{\\\\epsilon\\\\rightarrow\\\\infty}\\\\epsilon^{1/2}h(\\\\epsilon^{-3/2}t, \\\\epsilon^{-1}x)-\\\\overline{h}_\\\\epsilon$$\\nhas a limit as $\\\\epsilon\\\\to 0$ as a space-time process? Then show that the properties of this process identify it uniquely (see the conjectured properties in Corwin-Quastel).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0081", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The recorded epsilon limits are inconsistent; setting L=epsilon^(-3/2) recovers the intended long-time 1:2:3 scaling. The KPZ fixed-point law and Feller Markov semigroup are rigorously constructed, and the directed landscape gives the variational all-times/all-initial-data coupling, with a recent preprint characterization of that coupling. However, the literal claim for any growth process is false: spatially constant Poisson deposition is non-tight at one point under KPZ height scaling for every deterministic centering. In addition, for a max-plus kernel semigroup, a time-only deterministic counterterm preserves composition if and only if it is additive, hence linear under measurability or local boundedness, modulo a spatial coboundary.\n\nCandidate contribution (obstruction_and_cocycle_lemma; novelty confidence low): A two-stage screening certificate for KPZ-attraction claims: spatially constant rate-one Poisson deposition cannot be made tight under L^(-1/3) height scaling by any deterministic centering, while a deterministic centering of the form c(t-s)+phi(y)-phi(x) preserves a finite max-plus kernel semigroup exactly when c is additive (and therefore c(r)=vr under measurability or local boundedness)." }, { "id": 20002640, "problem_number": "AIM-PROBABILITY-0082", "title": "A localization-and-stability route to KPZ fixed-point attraction", "statement": "Prove that the fixed point is attractive (i.e., universal in some class of models).", "original_statement": "Prove that the fixed point is attractive (i.e., universal in some class of models).", "clean_statement": "Prove that the fixed point is attractive (i.e., universal in some class of models).", "statement_status": "exact", "statement_verification": "The canonical AIM record states verbatim:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.8\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[81]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Prove that the fixed point is attractive (i.e., universal in some class of models).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0082", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For max-plus variational evolutions on an arbitrary compact target parameter set, a uniform positive quadratic gap between kernel decay and initial-profile growth forces all maximizing starting points into one compact interval. On that interval, the output error is bounded by the sum of the local sup-norm profile error and kernel error. A truncation and continuous-mapping argument gives the corresponding random process-level transfer theorem. Hence KPZ fixed-point attraction follows once a model supplies joint point-to-point kernel convergence, convergent initial data, and uniform localization in probability.\n\nCandidate contribution (lemma; novelty confidence low): A uniform quadratic decay/growth gap yields one common compact maximizer set on any compact target parameter set and the quantitative bound ||T_{K_n}f_n-T_Kf|| <= ||f_n-f||+||K_n-K|| on that set; localization in probability upgrades this to process convergence in C(X)." }, { "id": 20002641, "problem_number": "AIM-PROBABILITY-0083", "title": "Weight universality for LPP and polymers: a finite-volume sensitivity reduction", "statement": "For example, prove universality of LPP and polymer (with respect to weight distributions).", "original_statement": "For example, prove universality of LPP and polymer (with respect to weight distributions).", "clean_statement": "For example, prove universality of LPP and polymer (with respect to weight distributions).", "statement_status": "exact", "statement_verification": "The exact canonical statement is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.85\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[82]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"For example, prove universality of LPP and polymer (with respect to weight distributions).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0083", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For any finite directed path model, the LPP directional derivative and subdifferential are exactly the maximizing-path occupation functional and the convex hull of maximizing occupation vectors, including ties. For the positive-temperature free energy F=beta^{-1} log Z, the gradient is the vector of Gibbs site occupations and the Hessian is beta times their covariance matrix. If all paths have length ell, the squared gradient is the two-replica overlap R and the Hessian trace is beta(ell-R); under independent Brownian perturbations of the environment, R is the martingale quadratic-variation density. An exact third-order Lindeberg replacement ledger expresses the comparison error through occupation cumulants, while deterministic stability and soft-max bounds show why crude comparison fails at full-aspect-ratio KPZ scale. A 2-by-2 example proves that matching mean and variance does not imply finite-size equality.\n\nCandidate contribution (sensitivity_reduction; novelty confidence low): The candidate contribution is a unified, testable finite-volume sensitivity ledger: LPP subgradients with tie handling; polymer occupation/covariance derivatives; the Brownian perturbation identities d_t=R_beta(t)dt and drift beta(ell-R_beta(t))/2; and an exact moment-matched Lindeberg remainder written in the same occupation variables. It reduces a replacement-based universality proof to controlling hybrid Gibbs occupation cumulants/replica intersections, with geodesic-switch control additionally required at zero temperature." }, { "id": 20002642, "problem_number": "AIM-PROBABILITY-0084", "title": "A curvature-localization bridge to the GUE Tracy--Widom law", "statement": "Provide a variational explanation for the actual form of the GUE Tracy-Widom distribution in terms of the properties of the fixed point. Reduce the exact solvability to purely probabilistic terms.", "original_statement": "Provide a variational explanation for the actual form of the GUE Tracy-Widom distribution in terms of the properties of the fixed point. Reduce the exact solvability to purely probabilistic terms.", "clean_statement": "Provide a variational explanation for the actual form of the GUE Tracy-Widom distribution in terms of the properties of the fixed point. Reduce the exact solvability to purely probabilistic terms.", "statement_status": "exact", "statement_verification": "The canonical AIM record (workshop *The Kardar--Parisi--Zhang equation and universality class*, problem 11.9) is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Big picture questions\nSource item: 11.9\nSource URL: http://aimpl.org/kpzuniversality/1/\nCanonical location: aim-probability-notes.json notes[83]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Provide a variational explanation for the actual form of the GUE Tracy-Widom distribution in terms of the properties of the fixed point. Reduce the exact solvability to purely probabilistic terms.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/1/", "tags": [ "aim", "AIM-PROBABILITY-0084", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For parabolic initial data f_kappa(x)=-kappa x^2, the rescaled KPZ fixed-point height at the origin has the exact variational law sup_z{A_2(z)-(1+kappa t)z^2}. Coupled through one Airy_2 path, these variables decrease almost surely from the flat/GOE variational law to A_2(0), every maximizing starting point localizes at zero, and the CDFs increase to the standard GUE Tracy--Widom CDF F_2. A separate unit-conjugacy theorem proves that generic max-plus, Feller, semigroup, vertical-shift, and 1:2:3 scaling properties do not by themselves determine even the scale of the wedge marginal, so the Airy-kernel/Painleve-II formula requires additional normalized probabilistic or integrable input.\n\nCandidate contribution (variational_reduction; novelty confidence low): The family f_kappa(x)=-kappa x^2 yields a monotone one-parameter max-plus bridge sup_z{A_2(z)-(1+kappa t)z^2} down to A_2(0), with localization of every maximizer and monotone convergence of its CDF to F_2." }, { "id": 20002643, "problem_number": "AIM-PROBABILITY-0085", "title": "Spiked polymer formulas and a confluence-safe BBP reduction", "statement": "Exact solvability\n\nUse the Macdonald processes formulas to prove the Baik-Ben Arous-P\\'{e}ch\\'{e} transition for the semi-discrete polymer with a finite number of non-zero drifts for the underlying Brownian motions (tuned critically). In the intermediate disorder scaling, derive the formula for the statistics of the stochastic heat equation started with $\\mathcal{Z}_0(X) = {\\bf 1}_{X\\geq 0} Z^r(X)$ where $Z^N(X)$ is the partition function for the semi-discrete polymer with $r$ levels at time $X$ (the $r$ here represents the number of spikes in BBP and the drifts here of the $r$ Brownian motions can be generally chosen).", "original_statement": "Exact solvability\n\nUse the Macdonald processes formulas to prove the Baik-Ben Arous-P\\'{e}ch\\'{e} transition for the semi-discrete polymer with a finite number of non-zero drifts for the underlying Brownian motions (tuned critically). In the intermediate disorder scaling, derive the formula for the statistics of the stochastic heat equation started with $\\mathcal{Z}_0(X) = {\\bf 1}_{X\\geq 0} Z^r(X)$ where $Z^N(X)$ is the partition function for the semi-discrete polymer with $r$ levels at time $X$ (the $r$ here represents the number of spikes in BBP and the drifts here of the $r$ Brownian motions can be generally chosen).", "clean_statement": "Exact solvability\n\nUse the Macdonald processes formulas to prove the Baik-Ben Arous-P\\'{e}ch\\'{e} transition for the semi-discrete polymer with a finite number of non-zero drifts for the underlying Brownian motions (tuned critically). In the intermediate disorder scaling, derive the formula for the statistics of the stochastic heat equation started with $\\mathcal{Z}_0(X) = {\\bf 1}_{X\\geq 0} Z^r(X)$ where $Z^N(X)$ is the partition function for the semi-discrete polymer with $r$ levels at time $X$ (the $r$ here represents the number of spikes in BBP and the drifts here of the $r$ Brownian motions can be generally chosen).", "statement_status": "exact", "statement_verification": "The canonical AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Open problems\nSource item: 22.02\nSource URL: http://aimpl.org/kpzuniversality/2/\nCanonical location: aim-probability-notes.json notes[84]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Exact solvability\\n\\nUse the Macdonald processes formulas to prove the Baik-Ben Arous-P\\\\'{e}ch\\\\'{e} transition for the semi-discrete polymer with a finite number of non-zero drifts for the underlying Brownian motions (tuned critically). In the intermediate disorder scaling, derive the formula for the statistics of the stochastic heat equation started with $\\\\mathcal{Z}_0(X) = {\\\\bf 1}_{X\\\\geq 0} Z^r(X)$ where $Z^N(X)$ is the partition function for the semi-discrete polymer with $r$ levels at time $X$ (the $r$ here represents the number of spikes in BBP and the drifts here of the $r$ Brownian motions can be generally chosen).\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/2/", "tags": [ "aim", "AIM-PROBABILITY-0085", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "Borodin, Corwin, and Ferrari solved the two requested tasks in 2014: their Theorem 1.3 proves the BBP transition for a fixed finite number of critically tuned O'Connell-Yor Brownian drifts, and their Theorem 1.10 gives the exact Fredholm Laplace transform for the SHE/CDRP with half-line r-spiked initial data. The source's Z^N is a notation error for the fixed-level Z^r. As a supplemental proved result, the BBP rational spike factor is decomposed in a polynomial basis into a rank-at-most-r perturbation of the Airy kernel, yielding an r-by-r determinant formula that is trace-norm continuous when spike parameters coalesce.\n\nCandidate contribution (lemma; novelty confidence low): Writing R_b(z)=product_j(z-b_j), the identity R_b(z)/(R_b(w)(z-w))=1/(z-w)+sum_{p=0}^{r-1} z^p q_p(w)/R_b(w) gives a rank-at-most-r BBP deformation and an r-by-r Fredholm determinant reduction without any divisions by b_i-b_j; under common contour separation and domination, the representation is trace-norm continuous through arbitrary spike confluence." }, { "id": 20002644, "problem_number": "AIM-PROBABILITY-0086", "title": "A direct full-Wick spatial-mollifier construction", "statement": "Stochastic analysis\n\nConsider the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. It is defined via chaos series -- show that it can also be defined by spatially smoothing white-noise and then making sense of the Wick exponential in its formulation as a partition function for non-intersecting Brownian bridges in a space time white-noise environment.", "original_statement": "Stochastic analysis\n\nConsider the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. It is defined via chaos series -- show that it can also be defined by spatially smoothing white-noise and then making sense of the Wick exponential in its formulation as a partition function for non-intersecting Brownian bridges in a space time white-noise environment.", "clean_statement": "Stochastic analysis\n\nConsider the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. It is defined via chaos series -- show that it can also be defined by spatially smoothing white-noise and then making sense of the Wick exponential in its formulation as a partition function for non-intersecting Brownian bridges in a space time white-noise environment.", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Open problems\nSource item: 22.04\nSource URL: http://aimpl.org/kpzuniversality/2/\nCanonical location: aim-probability-notes.json notes[85]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Stochastic analysis\\n\\nConsider the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. It is defined via chaos series -- show that it can also be defined by spatially smoothing white-noise and then making sense of the Wick exponential in its formulation as a partition function for non-intersecting Brownian bridges in a space time white-noise environment.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/2/", "tags": [ "aim", "AIM-PROBABILITY-0086", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For fixed n, t, x, and y, the path expectation over the coalesced Dyson-bridge watermelon of the fully Wick-ordered exponential of spatially mollified white noise converges in L2 of the noise to the O'Connell-Warren chaos partition function. The proof identifies the kth mollified chaos kernel as the spatial convolution of the kth bridge correlation kernel and uses approximate-identity convergence, convolution contraction, and O'Connell-Warren's summability of all chaos norms. The exact finite-scale Wick counterterm is also computed, and its intra-ensemble pair-overlap part is shown to have expectation O(epsilon).\n\nCandidate contribution (theorem_and_quantitative_bound; novelty confidence low): The fully Wick-ordered, spatially mollified coalesced-watermelon partition function converges in L2 to the O'Connell-Warren chaos series; moreover, the expected intra-ensemble part of the Wick counterterm is O(epsilon)." }, { "id": 20002645, "problem_number": "AIM-PROBABILITY-0087", "title": "A finite-window Burke identity and quenched obstruction for a marked Brownian--Poisson polymer", "statement": "Exact solvability\n\nMotivated by success in solvable positive temperature polymers, try to study the polynuclear growth model with an underlying Brownian path measure -- is there a Burkes theorem in this setting? Specifically, study the Poissonian last passage model but at positive temperature and with the Poisson points having random weights.", "original_statement": "Exact solvability\n\nMotivated by success in solvable positive temperature polymers, try to study the polynuclear growth model with an underlying Brownian path measure -- is there a Burkes theorem in this setting? Specifically, study the Poissonian last passage model but at positive temperature and with the Poisson points having random weights.", "clean_statement": "Exact solvability\n\nMotivated by success in solvable positive temperature polymers, try to study the polynuclear growth model with an underlying Brownian path measure -- is there a Burkes theorem in this setting? Specifically, study the Poissonian last passage model but at positive temperature and with the Poisson points having random weights.", "statement_status": "exact", "statement_verification": "The canonical AIM record, from the workshop *The Kardar--Parisi--Zhang equation and universality class*, problem 22.06, is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Open problems\nSource item: 22.06\nSource URL: http://aimpl.org/kpzuniversality/2/\nCanonical location: aim-probability-notes.json notes[86]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Exact solvability\\n\\nMotivated by success in solvable positive temperature polymers, try to study the polynuclear growth model with an underlying Brownian path measure -- is there a Burkes theorem in this setting? Specifically, study the Poissonian last passage model but at positive temperature and with the Poisson points having random weights.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/2/", "tags": [ "aim", "AIM-PROBABILITY-0087", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the finite-radius Brownian polymer in a marked space-time Poisson environment, annealed partition-function tilting leaves the path at its Brownian reference law and makes the encountered points in path-centered coordinates an independent Poisson process with Esscher-tilted marks. The quenched joint law differs by the exact nonconstant factor 1/W_T, whose second moment is an explicit exponential of two-path tube overlap. In a two-corridor soft polymer, Gibbs selection violates the Poisson second-factorial relation at every nonzero inverse temperature and all sufficiently small intensities, proving that a genuine Burke theorem needs additional stationary boundary and reversibility structure.\n\nCandidate contribution (identity_and_obstruction; novelty confidence low): The marked finite-window annealed Poisson output identity, exact 1/W_T quenched correction, closed two-replica overlap formula, and two-corridor factorial obstruction form a concrete diagnostic package for any proposed Burke theorem in the Brownian--Poisson polymer." }, { "id": 20002646, "problem_number": "AIM-PROBABILITY-0088", "title": "A response formula for the full two-replica overlap law", "statement": "Exact solvability\n\nStudy the overlap for replicated paths of a polymer with respect to the quenched Gibbs measure. What is the distribution of the intersection local time for the continuum polymer? Is there replica symmetry or replica breaking in the strong disorder regime or intermediate disorder regime? Try to use the Macdonald processes contour integral formulas to compute more than just the expected overlap.", "original_statement": "Exact solvability\n\nStudy the overlap for replicated paths of a polymer with respect to the quenched Gibbs measure. What is the distribution of the intersection local time for the continuum polymer? Is there replica symmetry or replica breaking in the strong disorder regime or intermediate disorder regime? Try to use the Macdonald processes contour integral formulas to compute more than just the expected overlap.", "clean_statement": "Exact solvability\n\nStudy the overlap for replicated paths of a polymer with respect to the quenched Gibbs measure. What is the distribution of the intersection local time for the continuum polymer? Is there replica symmetry or replica breaking in the strong disorder regime or intermediate disorder regime? Try to use the Macdonald processes contour integral formulas to compute more than just the expected overlap.", "statement_status": "exact", "statement_verification": "The canonical record (AIM workshop *The Kardar--Parisi--Zhang equation and universality class*, Open Problems 22.08) reads:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Open problems\nSource item: 22.08\nSource URL: http://aimpl.org/kpzuniversality/2/\nCanonical location: aim-probability-notes.json notes[87]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Exact solvability\\n\\nStudy the overlap for replicated paths of a polymer with respect to the quenched Gibbs measure. What is the distribution of the intersection local time for the continuum polymer? Is there replica symmetry or replica breaking in the strong disorder regime or intermediate disorder regime? Try to use the Macdonald processes contour integral formulas to compute more than just the expected overlap.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: open; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/2/", "tags": [ "aim", "AIM-PROBABILITY-0088", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For a finite-time Brownian polymer in a spatially mollified Gaussian environment, the entire quenched Laplace transform of the two-replica overlap is exactly the ratio of a source-coupled two-replica partition function to the square of the one-replica partition function; logarithmic source derivatives give every quenched cumulant. Disorder averaging the unnormalized numerator shifts the contact coupling by exactly beta squared, while it does not yield the annealed quenched law. As an exact continuum normalization check, at zero noise two equal-endpoint Brownian bridges have Rayleigh-distributed intersection local time with density (2l/T) exp(-l^2/T).\n\nCandidate contribution (response_identity_and_exact_boundary_case; novelty confidence low): The combined source-response prescription, with quenched/annealed order and the shared-noise contact coefficient audited, together with the explicit zero-noise Rayleigh normalization, is a testable full-law target for any Macdonald or nested-contour computation of this AIM overlap." }, { "id": 20002647, "problem_number": "AIM-PROBABILITY-0089", "title": "The q-exponential martingale, its fixed-q bracket obstruction, and the two-stage SHE limit", "statement": "Stochastic analysis\n\nFind a microscopic G\\\"{a}rtner transform (discrete Hopf-Cole transform) which turns the dynamics of $q$-TASEP into a discrete stochastic heat equation or better yet a polymer. Then prove this converges to the continuum stochastic heat equation.", "original_statement": "Stochastic analysis\n\nFind a microscopic G\\\"{a}rtner transform (discrete Hopf-Cole transform) which turns the dynamics of $q$-TASEP into a discrete stochastic heat equation or better yet a polymer. Then prove this converges to the continuum stochastic heat equation.", "clean_statement": "Stochastic analysis\n\nFind a microscopic G\\\"{a}rtner transform (discrete Hopf-Cole transform) which turns the dynamics of $q$-TASEP into a discrete stochastic heat equation or better yet a polymer. Then prove this converges to the continuum stochastic heat equation.", "statement_status": "exact", "statement_verification": "This is Problem 22.1, “Stochastic analysis,” from the AIM workshop *The Kardar–Parisi–Zhang equation and universality class*. The source record says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Open problems\nSource item: 22.1\nSource URL: http://aimpl.org/kpzuniversality/2/\nCanonical location: aim-probability-notes.json notes[88]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Stochastic analysis\\n\\nFind a microscopic G\\\\\\\"{a}rtner transform (discrete Hopf-Cole transform) which turns the dynamics of $q$-TASEP into a discrete stochastic heat equation or better yet a polymer. Then prove this converges to the continuum stochastic heat equation.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/2/", "tags": [ "aim", "AIM-PROBABILITY-0089", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For continuous-time q-TASEP, Z_n=q^{x_n+n} satisfies an exact pathwise linear nearest-label drift equation dZ_n=(1-q)a_n(Z_{n-1}-Z_n)dt+dM_n with predictable bracket d=(1-q)^2a_n Z_n(Z_n-Z_{n-1})dt. Among local one-coordinate exponentials c_n r^{x_n}, linear two-term drift closure for every gap forces r=q, so the surviving gap factor in the bracket obstructs standard independent multiplicative noise at fixed q within this class. The literature supplies a rigorous iterated route for step/droplet data: centered q-TASEP converges to the O'Connell-Yor semi-discrete SHE, and its intermediate-disorder limit converges to the continuum SHE.\n\nCandidate contribution (obstruction; novelty confidence low): For every particle label n>=2, any nonconstant local exponential F_n=c_n r^{x_n} whose q-TASEP generator closes as A_nF_n+B_nF_{n-1} for all admissible gaps must have r=q; after normalization its predictable bracket necessarily contains 1-F_{n-1}/F_n and hence cannot be a configuration-independent multiple of F_n^2 at fixed q." }, { "id": 20002648, "problem_number": "AIM-PROBABILITY-0090", "title": "Finite-torus stationarity and the infinite-bulk gap", "statement": "Interacting particle systems\n\nBulk of $q$-Whittaker $2d$ dynamics. Invariant measure.", "original_statement": "Interacting particle systems\n\nBulk of $q$-Whittaker $2d$ dynamics. Invariant measure.", "clean_statement": null, "statement_status": "reconstructed_unverified", "statement_verification": "The AIM page was unavailable (HTTP 502 on 2026-08-12), and nearby records only confirm that this item belongs to the q-TASEP/q-Whittaker interacting-particle-system cluster of the workshop. The fragment omits the state space, update rule, boundary conditions, and meaning of “bulk.” There are at least three plausible readings:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Open problems\nSource item: 22.12\nSource URL: http://aimpl.org/kpzuniversality/2/\nCanonical location: aim-probability-notes.json notes[89]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Interacting particle systems\\n\\nBulk of $q$-Whittaker $2d$ dynamics. Invariant measure.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: reduction; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 4, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/2/", "tags": [ "aim", "AIM-PROBABILITY-0090", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 4, "level": 4, "name": "L4: Expert", "description": "Very challenging problems at the frontier of mathematical research.", "color_class": "text-red-600 bg-red-50 border-red-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "The best-supported finite-volume interpretation of the terse problem was solved by Corwin and Toninelli: their periodized two-dimensional q-Whittaker forced-push dynamics preserves an explicit correlated q-Gibbs measure in each nondegenerate fixed-winding torus sector. For the stronger infinite-plane bulk reading, this attempt proves a finite-to-infinite generator lemma and reduces invariance to local Gibbs tightness, control of long forced-push strings, and well-posed conservative infinite dynamics. It also proves that the monotonically growing absolute-height lift cannot have an invariant probability; stationarity concerns particles, dimers, height gradients, or a comoving height.\n\nCandidate contribution (infinite_volume_reduction_and_gauge_obstruction; novelty confidence low): Finite-torus q-Gibbs stationarity transfers to an infinite bulk under an explicit uniform tail bound on remote push strings intersecting a local observable, local rate convergence, and a well-posed generator/core condition; meanwhile any monotone absolute-height lift with positive jump rate admits no invariant probability.", "research_difficulty_suggested": "L4" }, { "id": 20002649, "problem_number": "AIM-PROBABILITY-0091", "title": "Stationary flux and a tangent second-class identity for q-TASEP gaps", "statement": "Interacting particle systems\n\nStudy $q$-TASEP hydrodynamics and other interacting particle systems properties such as second class particles, invariant distributions, and Burkes type theorem.", "original_statement": "Interacting particle systems\n\nStudy $q$-TASEP hydrodynamics and other interacting particle systems properties such as second class particles, invariant distributions, and Burkes type theorem.", "clean_statement": "Interacting particle systems\n\nStudy $q$-TASEP hydrodynamics and other interacting particle systems properties such as second class particles, invariant distributions, and Burkes type theorem.", "statement_status": "exact", "statement_verification": "The exact canonical record (AIM Problem Lists, workshop *The Kardar--Parisi--Zhang equation and universality class*, Open Problem 22.14) says:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Open problems\nSource item: 22.14\nSource URL: http://aimpl.org/kpzuniversality/2/\nCanonical location: aim-probability-notes.json notes[90]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Interacting particle systems\\n\\nStudy $q$-TASEP hydrodynamics and other interacting particle systems properties such as second class particles, invariant distributions, and Burkes type theorem.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/2/", "tags": [ "aim", "AIM-PROBABILITY-0091", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For homogeneous continuous-time q-TASEP with 0=0} alpha q^m/(1-alpha q^m), its label current is j(rho(alpha))=alpha, and its characteristic speed is 1/rho'(alpha). An explicit derivative-tail Palm law for one infinitesimal extra gap unit makes the basic-coupling discrepancy's mean instantaneous excess-clock drift exactly 1/rho'(alpha). Sampling the discrepancy environment from the ordinary stationary marginal instead gives the strictly larger rate (1-q)(1-alpha), so ordinary one-site stationarity is not the correct discrepancy Palm law.\n\nCandidate contribution (lemma; novelty confidence low): The derivative-tail law hat_pi_alpha(k)=partial_alpha P_alpha(G>k)/rho'(alpha) is a probability distribution and satisfies (1-q) E_hat[q^K]=j'(rho(alpha)); moreover j'(rho(alpha))<(1-q)(1-alpha) for alpha>0, explicitly obstructing replacement of the Palm environment by the ordinary q-geometric marginal." }, { "id": 20002650, "problem_number": "AIM-PROBABILITY-0092", "title": "Stationary gaps and a second-class gap discrepancy for two-sided q-ASEP", "statement": "Interacting particle systems\n\nWhat about $q$-ASEP, where particle can jump to right and left and feel a repulsion from their right or left neighbors: Is there Burkes Theorem? Do second class particle methods work?", "original_statement": "Interacting particle systems\n\nWhat about $q$-ASEP, where particle can jump to right and left and feel a repulsion from their right or left neighbors: Is there Burkes Theorem? Do second class particle methods work?", "clean_statement": "Interacting particle systems\n\nWhat about $q$-ASEP, where particle can jump to right and left and feel a repulsion from their right or left neighbors: Is there Burkes Theorem? Do second class particle methods work?", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Open problems\nSource item: 22.16\nSource URL: http://aimpl.org/kpzuniversality/2/\nCanonical location: aim-probability-notes.json notes[91]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Interacting particle systems\\n\\nWhat about $q$-ASEP, where particle can jump to right and left and feel a repulsion from their right or left neighbors: Is there Burkes Theorem? Do second class particle methods work?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/2/", "tags": [ "aim", "AIM-PROBABILITY-0092", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For the intended asymmetric q-simple exclusion process, whose right and left rates are R(1-Q^{right gap}) and L(1-Q^{left gap}), the finite-ring gap chain has an explicit canonical product-form invariant law proportional to the product of 1/(Q;Q)_{g_i}; its q-geometric grand-canonical form yields exact mean velocity and current. The gap dynamics is attractive, and one discrepancy under the basic coupling remains a single second-class gap unit with exact environment-dependent rates R(1-Q)Q^{g_j} and L(1-Q)Q^{g_j}. A published classification implies that the stronger product random-walking-shock closure for this nonlinear rate is unavailable in the genuinely two-sided case, while the literature checked does not establish a two-sided Burke output-independence theorem or turn the gap discrepancy into an ordinary local exclusion second-class particle.\n\nCandidate contribution (lemma_and_synthesis; novelty confidence low): The explicit finite-ring stationarity/current calculation and one-discrepancy generator give a sharp, testable boundary for the AIM question: second-class coupling works at the gap level for all directional parameters, but when both directions are present the nonlinear q-rate lies outside the classified autonomous product random-walking-shock family, and this marker is not a local exclusion second-class particle under the lay-down map." }, { "id": 20002651, "problem_number": "AIM-PROBABILITY-0093", "title": "Fixed-area excursions select Ferrari-Spohn scales, not the literal KPZ scales", "statement": "Gibbs line ensembles\n\nConsider a Brownian excursion on time interval $[-N,N]$ condition on fixed area $N$. Take $1/3, 2/3$ scalings around the edge. Do we get an ``Airy-like'' limiting fluctuation?", "original_statement": "Gibbs line ensembles\n\nConsider a Brownian excursion on time interval $[-N,N]$ condition on fixed area $N$. Take $1/3, 2/3$ scalings around the edge. Do we get an ``Airy-like'' limiting fluctuation?", "clean_statement": "Gibbs line ensembles\n\nConsider a Brownian excursion on time interval $[-N,N]$ condition on fixed area $N$. Take $1/3, 2/3$ scalings around the edge. Do we get an ``Airy-like'' limiting fluctuation?", "statement_status": "exact", "statement_verification": "This is Problem 22.18, “Gibbs line ensembles,” from the AIM workshop *The Kardar–Parisi–Zhang equation and universality class*. The exact source record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Open problems\nSource item: 22.18\nSource URL: http://aimpl.org/kpzuniversality/2/\nCanonical location: aim-probability-notes.json notes[92]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Gibbs line ensembles\\n\\nConsider a Brownian excursion on time interval $[-N,N]$ condition on fixed area $N$. Take $1/3, 2/3$ scalings around the edge. Do we get an ``Airy-like'' limiting fluctuation?\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/2/", "tags": [ "aim", "AIM-PROBABILITY-0093", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "Exact Brownian scaling turns the stated duration-2N, area-N excursion into a unit excursion conditioned to have area 1/(2^{3/2} sqrt(N)), hence a small-area problem. The Airy Hamiltonian shows that area density rho selects height and correlation scales rho and rho^2: more generally, duration N and area N^{1+gamma} select N^gamma and N^{2gamma}. Thus the written area N has constant-order local scales, while the advertised N^{1/3}, N^{2/3} axes require area N^{4/3}. A proved conditional equivalence-of-ensembles theorem reduces the exact microcanonical Ferrari-Spohn process limit to an explicit local density-ratio estimate.\n\nCandidate contribution (scaling obstruction and reduction; novelty confidence low): For a Brownian excursion of duration comparable to N and conditioned area comparable to N^{1+gamma}, 0 <= gamma < 1/2, the canonically matched Ferrari-Spohn height and correlation scales are N^gamma and N^{2gamma}; consequently the AIM statement's area N is incompatible with its proposed 1/3, 2/3 rescaling, which instead selects area N^{4/3}." }, { "id": 20002652, "problem_number": "AIM-PROBABILITY-0094", "title": "Exact discrete Cole–Hopf compatibility and the solved Sasamoto–Spohn convergence problem", "statement": "Stochastic analysis\n\nUse the well-posedness theory for the KPZ equation to redo Bertini-Giacomin or to prove convergence of simpler discretizations of the KPZ equation such as the one previously studied by Sasamoto and Spohn.", "original_statement": "Stochastic analysis\n\nUse the well-posedness theory for the KPZ equation to redo Bertini-Giacomin or to prove convergence of simpler discretizations of the KPZ equation such as the one previously studied by Sasamoto and Spohn.", "clean_statement": "Stochastic analysis\n\nUse the well-posedness theory for the KPZ equation to redo Bertini-Giacomin or to prove convergence of simpler discretizations of the KPZ equation such as the one previously studied by Sasamoto and Spohn.", "statement_status": "exact", "statement_verification": "The exact canonical AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Open problems\nSource item: 22.2\nSource URL: http://aimpl.org/kpzuniversality/2/\nCanonical location: aim-probability-notes.json notes[93]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Stochastic analysis\\n\\nUse the well-posedness theory for the KPZ equation to redo Bertini-Giacomin or to prove convergence of simpler discretizations of the KPZ equation such as the one previously studied by Sasamoto and Spohn.\"\nOriginal remarks: []\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/2/", "tags": [ "aim", "AIM-PROBABILITY-0094", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "SOLVED-IN-LITERATURE", "research_summary": "The AIM alternative asking for convergence of a Sasamoto–Spohn-type discretization has been solved in substantial periodic non-equilibrium and whole-line equilibrium forms by Gubinelli–Perkowski and Jara–Moreno Flores. As an additional proved finite-grid contribution, the report derives the unique nearest-neighbor exponential height drift and the exact Itô counterterm λD/(4νε) that linearize under the discrete Cole–Hopf transform, proves that no quadratic nearest-neighbor drift has exact closure, and shows that the quadratic Cole–Hopf and Sasamoto–Spohn stencils differ exactly by λε²(Δεh)²/12.\n\nCandidate contribution (lemma; novelty confidence low): On a finite periodic nearest-neighbor grid with independent cell noises and Z=exp(λh/(2ν)), exact discrete Cole–Hopf closure forces both a non-polynomial exponential drift and the counterterm λD/(4νε); no degree-at-most-two increment polynomial can close exactly, while the quadratic Cole–Hopf truncation differs from the coefficient-normalized Sasamoto–Spohn stencil by the exact curvature square λε²(Δεh)²/12, and its pointwise Taylor remainder is not perturbatively small under KPZ roughness power counting." }, { "id": 20002653, "problem_number": "AIM-PROBABILITY-0095", "title": "A scaling dictionary and the remaining arbitrary-window polymer problem", "statement": "Polymer universality\n\nConsider intermediate scaling for polymer with $\\beta=n^{-1/4}c(n)$ where $c(n)$ is any function which tends to infinity with $n$. Show universality of the $F_{GUE}$ fluctuations with a fluctuation exponent which may depend on $c(n)$.", "original_statement": "Polymer universality\n\nConsider intermediate scaling for polymer with $\\beta=n^{-1/4}c(n)$ where $c(n)$ is any function which tends to infinity with $n$. Show universality of the $F_{GUE}$ fluctuations with a fluctuation exponent which may depend on $c(n)$.", "clean_statement": "Polymer universality\n\nConsider intermediate scaling for polymer with $\\beta=n^{-1/4}c(n)$ where $c(n)$ is any function which tends to infinity with $n$. Show universality of the $F_{GUE}$ fluctuations with a fluctuation exponent which may depend on $c(n)$.", "statement_status": "exact", "statement_verification": "The exact AIM record is:", "background": "Collected from the canonical American Institute of Mathematics problem corpus and paired with a validated per-problem research attempt.\nSource collection: AIM Problem Lists\nSource type: AimPL\nAIM domain: Probability\nWorkshop: The Kardar-Parisi-Zhang equation and universality class\nSection: Open problems\nSource item: 22.22\nSource URL: http://aimpl.org/kpzuniversality/2/\nCanonical location: aim-probability-notes.json notes[94]\nCanonical tag: problem\nOriginal extracted problem text (JSON string): \"Polymer universality\\n\\nConsider intermediate scaling for polymer with $\\\\beta=n^{-1/4}c(n)$ where $c(n)$ is any function which tends to infinity with $n$. Show universality of the $F_{GUE}$ fluctuations with a fluctuation exponent which may depend on $c(n)$.\"\nOriginal remarks: [\"This requires careful estimation of terms in the discrete chaos series which are further and further out as $n$ increases (reminiscent of Soshnikov's approach to edge universality for Wigner matrices).\"]\nOriginal literature field (JSON string): \"\"\nResearch attempt: 1; result status: partial_result; problem status at run: partially_solved; rigor: proved; novelty confidence: low.\nResearch caveat: machine-generated research aid, not peer reviewed; independently verify before citation.", "difficulty_level_id": 3, "status": "partially_solved", "proposed_by": null, "proposed_year": null, "category_id": 19, "set_id": 14, "source_url": "http://aimpl.org/kpzuniversality/2/", "tags": [ "aim", "AIM-PROBABILITY-0095", "aim-domain:probability", "aim-workshop:kpzuniversality", "aim-source-tag:problem" ], "view_count": 0, "favorite_count": 0, "created_at": "2026-08-14T00:00:00Z", "updated_at": "2026-08-14T00:00:00Z", "published": true, "category": { "id": 19, "name": "probability", "display_name": "Probability", "description": "Problems involving probability theory, stochastic processes, and random structures.", "slug": "probability", "order_index": 19, "created_at": "2026-07-31T15:26:25.671Z" }, "difficulty": { "id": 3, "level": 3, "name": "L3: Advanced", "description": "Difficult problems requiring specialized knowledge and sophisticated techniques.", "color_class": "text-yellow-600 bg-yellow-50 border-yellow-200" }, "set": { "id": 14, "name": "aim_workshop_problem_lists", "display_name": "AIM Workshop Problem Lists", "description": "Canonical open-problem records from American Institute of Mathematics workshop lists, with one validated research report and complete attempt artifacts per record.", "slug": "aim-workshop-problem-lists", "order_index": 14, "created_at": "2026-08-14T00:00:00Z" }, "research_classification": "PARTIAL-PROGRESS", "research_summary": "For beta_n=n^{-1/4}c_n, the intended intermediate-disorder window is 1<